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+{"text":"\\section{Introduction}\n\nThe first release of cosmological data from the \\satellite{Planck}\\\nsatellite \\citep{Planck-R1-I} and the final analysis of the\n\\satellite{Wilkinson Microwave Anisotropy Probe} (\\satellite{WMAP})\n\\citep{WMAP9-results} confirmed that the inflationary Lambda cold dark\nmatter ($\\Lambda$CDM) model provides an excellent fit to the angular temperature power\nspectrum for multipoles ranging from the quadrupole ($\\ell = 2$) up to\n$\\ell = 2500$. The effect of gravitational lensing of the cosmic microwave\nbackground (CMB) has been detected with a very high statistical\nsignificance ($25 \\sigma$) \\citep{Planck-R1-XVII} and breaks some parameter\ndegeneracies without reference to non-CMB observations. Most of the\nstatistical power in the \\satellite{Planck}\\ analysis comes from high-$\\ell$\nmultipoles, thus it may not come as a surprise that the best-fitting model\ntraces the high-$\\ell$ data much better than those at low-$\\ell$, where a\nlack of angular power (in the range $\\ell = 2$ to $32$) compared to the\nbest-fitting model is found at the $99$ per cent C.L.\\ \\citep{Planck-R1-XV}.\nNevertheless, it is quite remarkable that none of the models invoking\nadditional, physically well motivated parameters, such as the sum of\nneutrino masses, the number of effective relativistic degrees of freedom,\nor a running of the spectral index, can give rise to a significant\nimprovement of the fit \\citep{Planck-R1-XVI}.  These findings indicate that\nsome special attention should be devoted to the largest angular scales,\nespecially as they potentially probe different physics than the small angular\nscales.\n\nSeveral anomalies at large angular scales discussed in the literature have\nbeen the source of some controversy since the first release of the\n\\satellite{WMAP}\\ data (see \\citealt{WMAP7-anomalies,CHSS-review} for reviews). The\nfirst of them was already seen by the \\satellite{Cosmic Background\n  Explorer} (\\satellite{COBE}): the temperature two-point angular correlation function\ncomputed as an average over the complete sky\n\\begin{equation}\n  \\Ccorr(\\theta) = \\overline{T(\\unitvec e_1) T(\\unitvec e_2)}, \\qquad\n  \\unitvec e_1 \\cdot \\unitvec e_2 = \\cos \\theta, \n  \\label{eq:Ctheta-full-sky}\n\\end{equation}\nwas found to be smaller than expected at large angular scales\n\\citep{DMR4-Ctheta}. Scant attention was given to this observation, due in\npart to the relatively low signal to noise ratio of the\n\\satellite{COBE}\\ observations, but mostly to the theory-driven shift in attention\naway from the angular correlation function to the angular power spectrum.\nThe lack of correlations on angular scales larger than $60$ degrees was\nrediscovered almost a decade later by \\satellite{WMAP}\\ in their one-year analysis\n\\citep{WMAP1-cosmology} and analysed in greater detail by us for the\n\\satellite{WMAP}\\ three and five-year data releases \\citep{CHSS-WMAP5}.  We have\nemphasized its persistence in the data (contrary to some claims),\ndifferentiated it from the lowness of the temperature quadrupole with which\nit is often confused, and demonstrated how it challenges the canonical\ntheory's fundamental prediction of Gaussian random, statistically isotropic\ntemperature fluctuations.  For related work on the missing large-angle\ncorrelations, see also \\cite{CHSS-WMAP3,Hajian2007,SHCSS2011,Kim2011,Zhang2012,Gruppuso2013}.\n\nThe \\satellite{Planck}\\ team presented an analysis of the angular two-point correlation\nfunction at a low resolution ($\\textit{Nside} = 64$) for their four component\nseparation methods (\\Planckmap{Commander-Ruler}, \\Planckmap{NILC}, \\Planckmap{SEVEM}, \\Planckmap{SMICA}) after the U73 mask was used to\nsuppress Galactic residuals.  Based on comparison with $10^3$ realizations of\nthe best-fitting model, they find that the probability of obtaining a $\\chi^2$\nbetween the expected angular two-point correlation function of the\nbest-fitting model and the observed correlation function that is at least as\nlarge as that measured is $0.883$, $0.859$, $0.884$, and $0.855$ for the \\Planckmap{Commander-Ruler},\n\\Planckmap{NILC}, \\Planckmap{SEVEM}, and \\Planckmap{SMICA}\\ maps respectively \\citep{Planck-R1-XXIII}. However,\ntheir statistic fails to capture that what is anomalous about the angular\ntwo-point correlation function is not the extent to which it deviates from the\ntheoretical expected value of the function.  Rather,  as has been the\n  case since the \\satellite{COBE-DMR}\\ observation, the pertinent anomaly is\nthat above about $60$ degrees the angular correlation function is very nearly\nzero.  It is this very special way of deviating from our expectation that\n  deserves our attention.\n\nIn this work, we analyse the two-point angular correlation function at\nlarge angles as seen in the final data release of \\satellite{WMAP}\\ and the first\ncosmology release of \\satellite{Planck}.  The anomalous alignments of low multipole\nmodes with each other and with directions defined by the geometry and\nmotion of the Solar system are discussed in a companion paper\n\\citep{CHSS-Planck-R1-alignments}.  Here we demonstrate that on the part of\nthe sky outside the plane of the Galaxy the absence of two-point angular\ncorrelations above about $60$ degrees remains a robust, statistically\nsignificant result, with a $p$-value between about $0.03$ and $0.33$ per\ncent depending on the precise map and Galaxy cut being analysed.\n\n\\section{Physics at large angular scales}\n\nHigh fidelity measurements of the microwave sky reveal the imprints\nof primary temperature, density and metric fluctuations in the early\nUniverse.  By observing these fluctuations and analysing their statistical\nproperties, we seek a deeper understanding of cosmological inflation or any\nalternative mechanism that produced the initial fluctuations.\n\nStudying modes with wavelengths too large to enable causal contact across the\nmode during the radiation and most of the matter dominated epochs suggests\nthat we can learn something about the physics of inflation without detailed\nknowledge of the recent content of the Universe and associated astrophysical\ndetails (e.g.\\ reionization).  This motivates us to pay special attention to\nunderstand the largest angular scales.  Comoving scales that cross into the\nHubble radius at $z \\sim1$ and below are observed at angles larger than $60$\ndegrees. Thus features observed at those scales are either of primordial\nnature or stem from physics at $z \\la 1$, the epoch in the history of the\nUniverse that we arguably know best.\n\nTo be more precise, at $z = 0.91 (1.5, 7)$ the comoving Hubble length\nequals the length of a comoving arc with an opening angle of $90\\degr\n(60\\degr,18\\degr)$ for the best-fitting $\\Lambda$CDM\\ model. These angular\nscales correspond roughly to the scales that have been shown to be\nanomalous in previous works (the quadrupole, octopole, and up to modes $\\ell =\n10$). It is possibly  noteworthy that $z \\sim 7$ corresponds to the moment\nwhen the Universe is fully reionized.\n\nFor better or worse, however, the large-angle CMB is also sensitive to the\nphysics that affects the microwave photons as they propagate from their\nlast scattering until their collection by our telescopes.  The late-time\nintegrated Sachs-Wolfe (ISW) effect could potentially correlate the\nlarge-angle CMB with the local structure of the gravitational\npotential. Indeed, it has been proposed in the literature that some of the\nobserved CMB anomalies could be explained in this way\n\\citep{Rakic2006a,Francis2010,Dupe2011,Rassat2013}.  Although\nreconstruction of the local gravitational potential from existing CMB and\nlarge-scale structure data is quite uncertain and subject to biases, such\nan explanation would indeed be an attractive possibility if only there were\nno lack of correlations on large scales.\nIf the observed lack of large-angle correlations is real, then we must\nexplain how the local gravitational potential manages to align with the\nprimordial temperature fluctuations in such a way that the resulting sky\nhas such a deficit.  In the end this does not change the underlying\nproblem; it merely rephrases it from one about the CMB to one about the\nlocal gravitational potential.\n  \nClearly, it is important to understand the lack of correlations at large\nangular scales in greater detail not just for its own sake, but also in\norder to evaluate any proposed explanation for other features of the CMB\ndata, especially other large-angle or low-$\\ell$ anomalies.\n\n \n\\section{Temperature two-point angular correlation function} \n\n\\subsection{Theory}\n\nIn the standard CMB analysis a full-sky map of temperature fluctuations,\n$T(\\unitvec e)$, is expanded in spherical harmonics as\n\\begin{equation}\n  T(\\unitvec e) = \\sum_{\\ell m} a_{\\ell m} Y_{\\ell m}(\\unitvec e),\n\\end{equation}\nwhere the coefficients in the expansion are extracted from the full-sky as\n\\begin{equation}\n  a_{\\ell m} = \\int T(\\unitvec e) Y_{\\ell m}^*(\\unitvec e)\\, \\rmn{d}\\unitvec e.\n\\end{equation}\nFrom these quantities we define the two-point angular power spectrum as\n\\begin{equation}\n  \\Cee_\\ell \\equiv \\frac1{2\\ell+1} \\sum_{m}|a_{\\ell m}|^2.\n  \\label{eq:Cl-full-sky}\n\\end{equation}\nNote that the two-point angular power spectrum may \\emph{always} be defined\nin this way.  Only in the case of Gaussian random, statistically isotropic\ntemperature fluctuations will it contain \\emph{all} the statistical\ninformation.  The full-sky two-point angular correlation\nfunction~(\\ref{eq:Ctheta-full-sky}) is related to the full-sky two-point\nangular power spectrum via a Legendre series\n\\begin{equation}\n  \\Ccorr(\\theta) = \\sum_\\ell \\frac{2\\ell+1}{4\\upi} \\Cee_\\ell P_\\ell(\\cos\\theta),\n  \\label{eq:Ctheta-full-sky-expansion}\n\\end{equation}\nwhere the $P_\\ell(\\cos\\theta)$ are Legendre polynomials.\n\nUnfortunately the complete sky cannot be observed due to foreground\ncontamination.  If we let $W(\\unitvec e)$ represent a mask on the sky (in\nthe simplest case it is zero for pixels removed and one for those included)\nthen cut-sky quantities can be defined in analogy to the full-sky ones from\nabove.  In particular, the cut-sky can be expanded in pseudo-$a_{\\ell m}$ as\n\\begin{equation}\n  W(\\unitvec e) T(\\unitvec e) = \\sum_{\\ell m} \\tilde{a}_{\\ell m} Y_{\\ell\n    m}(\\unitvec e),\n\\end{equation}\nwhere\n\\begin{equation}\n  \\tilde{a}_{\\ell m} = \\int W(\\unitvec e) T(\\unitvec e) Y_{\\ell m}^*(\\unitvec e)\\,\n  \\rmn{d}\\unitvec e.\n\\end{equation}\nFrom these the cut-sky two-point angular power spectrum is defined as the\npseudo-$\\Cee_\\ell$ given by\n\\begin{equation}\n  \\tilde\\Cl \\equiv \\frac1{2\\ell+1} \\sum_m |\\tilde{a}_{\\ell m}|^2.\n  \\label{eq:Cl-cut-sky}\n\\end{equation}\nSimilarly the cut-sky two-point angular correlation function is defined as\na sky average,\n\\begin{equation}\n  \\tilde\\Ccorr (\\theta) \\equiv \\overline{W(\\unitvec e_1)T(\\unitvec e_1)\n    W(\\unitvec e_2)T(\\unitvec e_2)},\n  \\qquad\n  \\unitvec e_1 \\cdot \\unitvec e_2 = \\cos \\theta.\n  \\label{eq:Ctheta-cut-sky}\n\\end{equation}\nAgain this may be expanded in a Legendre series, now in terms of the\npseudo-$\\Cee_\\ell$, as\n\\begin{equation}\n  \\tilde\\Ccorr(\\theta) = \\sum_\\ell \\frac{2\\ell+1}{4\\upi} \\tilde\\Cl\n  P_\\ell(\\cos\\theta)\n  \\label{eq:Ctheta-cut-sky-expansion}\n\\end{equation}\n(see \\citealt{Pontzen2010} for a proof of this result).  The cut-sky\ntwo-point angular correlation function will be the main focus of this work.\n\nNote that it is common to just refer to $\\Ccorr(\\theta)$ as a single\nquantity covering both the full- and cut-sky cases.  It should be\nremembered that whenever a cut-sky $\\Ccorr(\\theta)$ is discussed it is\ndefined as in Eq.~(\\ref{eq:Ctheta-cut-sky}) and it may be expanded in a\nLegendre series using the pseudo-$\\Cee_\\ell$ as in\nEq.~(\\ref{eq:Ctheta-cut-sky-expansion}).\n\nIt should also be noted that the discussion above, including the quantities\ndefined and relationships derived, does \\emph{not} rely on assumptions\nabout the theory.  It is only when quantities measured in our Universe are\nrelated to the properties of the ensemble predicted by the theory that\nassumptions such as Gaussianity and statistical isotropy become important\nand must be identified.  For example, to construct an estimator of the\ntheoretical two-point angular power spectrum from cut-sky observations --\neither through the pseudo-$\\Cee_\\ell$ or a maximum likelihood technique -- extra\nassumptions that may not be true on large-angles or at low-$\\ell$ are\nrequired.\n\nAt high-$\\ell$, observed deviations from Gaussianity agree with the amount\nof non-Gaussianity expected from the non-linear contributions of\ngravitational lensing \\citep{Planck-R1-XXIV}.  However, at low-$\\ell$,\nthere are statistically significant anomalies in the temperature map, such\nas the alignments of multipoles and the hemispherical asymmetry\n\\citep{CHSS-Planck-R1-alignments,Planck-R1-XXIII}, that are evidence of\ncorrelations among the $a_{\\ell m}$ and the $\\Cee_\\ell$, and thus contradict the\nassumption of Gaussian-random statistically isotropic $a_{\\ell m}$.  This\nsuggests that the physics underlying the observed sky cannot be\ncharacterized solely by the $\\Cee_\\ell$, the statistical quantities prescribed by\nthe canonical model; unless these anomalies are unfortunate `flukes', other\nstatistical tools are not just interesting but necessary.  The question is\nwhich are the right ones.  The answer clearly depends on the physics\nunderlying the anomalies.  At least until that physics is established,\nmultiple approaches will need to be explored.\n\n\\subsection{Analysis of Observations}\n\\begin{figure}\n  \\includegraphics[width=\\linewidth]{Ctheta_smica}\n  \\caption{Two-point angular correlation function from the inpainted\n    \\satellite{Planck}\\ \\Planckmap{SMICA}\\ map.  The black, dotted line shows the best-fitting\n    $\\Lambda$CDM\\ model from \\satellite{Planck}.  The shaded, cyan region is the one-sigma\n    cosmic variance interval.  Included from the \\Planckmap{SMICA}\\ map are the\n    $\\Ccorr(\\theta)$ calculated on the full-sky (black, solid line) and\n    from two cut skies using the U74 mask (green, dash-dotted line) and the\n    KQ75y9 mask (red, dashed line).  See the text for details.}\n  \\label{fig:Ctheta-smica}\n\\end{figure}\n\nThe two-point temperature angular correlation function for the CMB,\n$\\Ccorr^{TT}(\\theta)$, has remained mostly unchanged since first measured\nby the \\satellite{COBE-DMR}\\ \\citep{DMR4-Ctheta}.  The resulting curves from the\n\\satellite{Planck}\\ \\Planckmap{SMICA}\\ map are shown in Fig.~\\ref{fig:Ctheta-smica}.  What is\nmost striking at first glance may be the difference between the\nbest-fitting $\\Lambda$CDM\\ model and the observed $\\Ccorr(\\theta)$ on both the\nfull and cut skies (the details of the masks will be discussed below).\nThis is a source of considerable confusion and great care must be taken to\nnot read too much into this.  The values of $\\Ccorr(\\theta)$ at different\nangular separations $\\theta$ (or more precisely in different angular bins)\nare correlated, so the sizeable deviation between the expected $\\Lambda$CDM\\ and\nthe observed curves is not as significant as it may appear.  Rather, it is\nthe very small value of the observed $\\Ccorr(\\theta)$ on large angular\nscales that is truly surprising.  This is particularly true for the cut\nskies where there are essentially no correlations above about 60 degrees,\nexcept for some small anti-correlation near 180 degrees.\n\nTo quantify this lack of correlations on large angular scales we continue to\nuse the statistic first proposed in the \\satellite{WMAP}\\ one-year analysis\n\\citep{WMAP1-cosmology},\n\\begin{equation}\n  \\label{eq:Shalf}\n  \\ensuremath{S_{1\/2}} \\equiv \\int_{-1}^{1\/2} [\\Ccorr(\\theta)]^2 \\rmn{d}(\\cos\\theta)\n  = \\sum_{\\ell=2}^{\\ell_{\\rmn{max}}} \\Cee_\\ell I_{\\ell\\ell'} \\mathcal{C}_{\\ell'}.\n\\end{equation}\nAs discussed above, this definition applies to both full-sky and cut-sky\nmaps.  For the case of full-sky maps the full-sky $\\Cee_\\ell$ from\nEq.~(\\ref{eq:Cl-full-sky}) are used in the sum on the right-hand side,\nwhereas for the case of cut-sky maps the pseudo-$\\Cee_\\ell$ from\nEq.~(\\ref{eq:Cl-cut-sky}) are used.  Throughout we will either refer to the\n$\\ensuremath{S_{1\/2}}$ statistic generically or make it clear the context in which it is\ncalculated.  This statistic has not been optimized in any way, except\ncrudely by the choice of the limits of integration, particularly the upper\none which has been chosen to be a convenient value.  We consistently resist\nthe temptation to optimize these limits in order to minimize the\noft-repeated criticism that the statistic is \\textit{a posteori}.  In\nacknowledgement and partial response to that objection, we note that the\nstatistical significance of the absence of large-angle correlations is not\nparticularly dependent either on the precise value of either limit (so long\nas the range of integration focuses on large scales) nor on the particular\nchoice of reasonable integrand.\\footnote{In another paper, looking at the\n  predictions for the two-point angular correlation function of temperature\n  with polarization, specifically the $Q$ Stokes parameter \\citep{CHSS-TQ},\n  we optimized the upper and lower limits of integration, and considered\n  both $[\\Ccorr^{TQ}(\\theta)]^2$ and $\\Ccorr^{TQ}(\\theta)$ as integrands in\n  the equivalent of (\\ref{eq:Shalf}).  However, in that case we were\n  \\textit{a priori} optimizing a statistic for a specific purpose --\n  differentiating between two models.  Furthermore, it was found\n  that replacing $[\\Ccorr^{TQ}(\\theta)]^2$ in the integrand with\n  $\\Ccorr^{TQ}(\\theta)$ makes no qualitative difference in the\n  conclusions.}\n  \nThe sum in equation~(\\ref{eq:Shalf}) shows how to quickly and easily\ncalculate $\\ensuremath{S_{1\/2}}$ in terms of the $\\Cee_\\ell$ or $\\tilde\\Cl$ from the Legendre\nseries~(\\ref{eq:Ctheta-full-sky-expansion}) or\n(\\ref{eq:Ctheta-cut-sky-expansion}).  The ${I}_{\\ell \\ell'}$ are the\ncomponents of a matrix of integrals over products of Legendre polynomials\nand are simply related to the $\\mathcal{I}_{\\ell\\ell'}(1\/2)$ calculated in\nAppendix A of \\cite{CHSS-WMAP5} by ${(4\\upi)^2} I_{\\ell\\ell'} =\n{(2\\ell+1)(2\\ell'+1)} \\mathcal{I}_{\\ell\\ell'}(1\/2)$.  The sum in the\n$\\ensuremath{S_{1\/2}}$ expression~(\\ref{eq:Shalf}) ranges from $\\ell=2$ to\n$\\ell=\\ell_{\\rmn{max}}$.  The lower limit is due to the monopole and dipole being\nremoved from the map.  We remove the monopole both because its amplitude is\nsignificantly larger than those of the other multipoles and because we are\ninterested in the correlations among fluctuations not in the background\nvalue.  We remove the entire dipole because it is dominated by the Doppler\ndipole -- the (uninteresting) contribution due to our peculiar motion\nthrough the Universe; this is approximately two orders of magnitude larger\nthan the expected underlying dipole in the CMB rest frame.  Once it is\npossible to measure the Doppler contribution to better than $1\\%$, it will\nbe far preferable to remove the Doppler dipole, and set $\\ell=1$ as the\nlower limit of the sum in expression~(\\ref{eq:Shalf}).  For the upper limit\nthere is some freedom in the choice of $\\ell_{\\rmn{max}}$.  Since $\\Cee_\\ell\\sim\n\\ell^{-2}$ we would expect that the result is independent of our choice\nprovided that $\\ell_{\\rmn{max}}$ is `large enough'.  However, since we will find\nsmall values of $\\ensuremath{S_{1\/2}}$, the exact choice does have a slight effect on the\nfinal values.  We have consistently chosen $\\ell_{\\rmn{max}}=100$ for all\ncalculations of $\\ensuremath{S_{1\/2}}$ in this work.\n\nThe calculation of $\\ensuremath{S_{1\/2}}$ has therefore been reduced to finding the\ntwo-point angular power spectrum either over the full-sky or cut-sky for\nsome map of the CMB temperature.  However, a number of important choices\nmust be made, which we now discuss.\n\nFirst, there are a number of maps available for analysis.  In each data\nrelease, the \\satellite{WMAP}\\ team included individual band maps and a full-sky\nIndependent Linear Combination (ILC) map designed to be as close to the\nforeground-subtracted CMB as possible.  The \\satellite{Planck}\\ team released\nindividual band maps and three different foreground-subtracted maps --\n\\Planckmap{NILC}, \\Planckmap{SEVEM}, \\Planckmap{SMICA}\\ -- in their recent release, although they had many\nmore, including one they called the \\Planckmap{Commander-Ruler}\\ map.  Here we will analyse the\nseven and nine-year \\satellite{WMAP}\\ $V$ and $W$ band maps and the ILC map, the\n\\satellite{Planck}\\ High Frequency Instrument (\\satellite{HFI}) $100\\unit{GHz}$ and Low Frequency\nInstrument (\\satellite{LFI}) $70\\unit{GHz}$ maps, and its \\Planckmap{NILC}, \\Planckmap{SEVEM}, and\n\\Planckmap{SMICA}\\ maps.\\footnote{All CMB data is available from the Lambda site,\n  \\url{http:\/\/lambda.gsfc.nasa.gov\/}, including links to both \\satellite{WMAP}\\ and\n  \\satellite{Planck}\\ results.  The \\satellite{Planck}\\ results may directly be obtained via the\n  \\satellite{Planck}\\ Legacy Archive, \\url{http:\/\/archives.esac.esa.int\/pla\/}.}\n\nOnce we have a map, we must also choose the resolution of the maps to be\nanalysed.  A higher resolution will minimize resolution-dependent effects.\nOn the other hand, to reduce the computation time, particularly when\ngenerating statistics from realizations of $\\Lambda$CDM, a low resolution is\npreferred.  As a compromise we have chosen the \\textsc{healpix}\\footnote{The\n  \\textsc{healpix}\\ source code is freely available from\n  \\url{healpix.sourceforge.net}.} resolution $\\textit{Nside}=128$ for all studies\nin this work.\n\nEven with the existence of cleaned, full-sky maps the concern of residual\ncontamination, particularly on the largest angular scales, remains.  For\nthis reason it is desirable to remove the most contaminated regions of the\nsky and only analyse the cleanest ones.  The \\satellite{Planck}\\ analysis used the U73\nmask which leaves a sky fraction $f_{\\rmn{sky}}=0.73$ \\citep{Planck-R1-XXIII}.\nThis mask is not publicly available but is constructed from the union of\nthe validity masks provided with the full-sky maps \\citep{Planck-R1-XII}.\nFor the \\Planckmap{NILC}, \\Planckmap{SEVEM}, and \\Planckmap{SMICA}\\ maps these masks are available.  For the\n\\Planckmap{Commander-Ruler}\\ map only a minimal version of the mask is provided.  Taking the union\nof these four masks produces what we call the U74 mask, a close\napproximation of the U73 mask but with $f_{\\rmn{sky}}=0.74$.  For \\satellite{WMAP}\\ we use\ntheir extended temperature mask from their nine-year data release, named\nKQ75y9, which has $f_{\\rmn{sky}}=0.69$.\n\nThese masks are provided at high resolution; $\\textit{Nside}=2048$ for the U74 mask\nand $\\textit{Nside}=1024$ for the KQ75y9 mask.  To degrade the masks to our working\nresolution of $\\textit{Nside}=128$ we follow the prescription defined in\n\\cite{Planck-R1-XXIII}: first the mask is degraded to $\\textit{Nside}=128$ using\n\\texttt{ud\\_grade} from \\textsc{healpix}, then any pixel with a value less then\n$0.8$ is set to zero, otherwise it is set to one.  With this prescription\nthe $\\textit{Nside}=128$ masks have sky fractions of $f_{\\rmn{sky}}=0.72$ for U74 and\n$f_{\\rmn{sky}}=0.67$ for KQ75y9.\n\n\\begin{figure}\n  \\includegraphics[width = \\linewidth]{KQ75y9+U74_masks}\n  \\caption{Masks used in this work. A pixel may be removed by both masks\n    (dark blue), only the KQ75y9 mask (light blue), only the U74 mask\n    (yellow), or by neither mask (red).}\n  \\label{fig:masks-comparison}\n\\end{figure}\n\nDespite the KQ75y9 mask removing more pixels, the U74 mask is not fully\ncontained within it.  A comparison of the two masks is given in\nFig.~\\ref{fig:masks-comparison}.  As can be seen, the two masks mostly\ncoincide, though there are many small regions of pixels only contained in\none of the two masks.  In particular there are pixels that are excluded by\nthe U74 mask but included by the KQ75y9 mask and the KQ75y9 mask generally\nremoves more of the region around the Galactic centre than the U74 mask.\nThese small differences have a noticeable effect on the calculated cut-sky\n$\\ensuremath{S_{1\/2}}$\\@.\n\nIt is important that comparisons of data and simulations are made\nconsistently.  In addition to the choices discussed above, cut-sky data\nwill always be compared to cut-sky realizations, with the maps in all cases\ntreated as similarly as possible.  This is particularly important since, as\nnoted above, for cut skies the pseudo-$\\Cee_\\ell$ are employed in the calculation\nof $\\ensuremath{S_{1\/2}}$\\@.  In this work we are \\emph{not} interested in reconstructing\nthe full-sky angular correlations. Instead, we find that angular\ncorrelations on the cut-sky are unusually low. We thus do not make\nstatements about the full-sky CMB, which at any rate cannot be reliably\nobserved, and for which a maximum-likelihood estimator may be more\nappropriate \\citep{Efstathiou2004-MLE, Efstathiou2010, Pontzen2010}.  Even so,\nreconstructing the full-sky from a cut-sky requires extra assumptions and\nmay introduce its own biases \\citep{CHSS2011}.\n\nExtracting the $\\Cee_\\ell$ from a map, particularly from a masked map, also\nrequires some care.  We use \\textsc{spice}\\ \\citep{polspice} for this purpose.  For\ncut skies there is the added issue that, even if the full-sky does not\ninclude a monopole or dipole, these modes will exist in the portion of the\nsky included for evaluation.  If we knew that the full-sky map did not\ncontain a residual monopole or dipole, then we could proceed without\nfurther concern.  Unfortunately, with real data this is not known,\nparticularly for individual frequency band maps which definitely have\nGalactic contamination.  We therefore remove the average monopole and\ndipole from all maps prior to extracting the $\\Cee_\\ell$\\@.  For the monopole, we\ndo this by subtracting the average value of the temperature over the\nportion of the sky that is being retained; for the dipole we find the\nbest-fitting dipole over the retained sky and subtract that dipole. (In\n\\textsc{spice}\\ this removal is a built-in feature which we employ in our\nanalysis.)  When analysing a cut-sky, this procedure generically introduces\na monopole and dipole (and alters the other multiples) into the equivalent\nfull-sky map.  Though this may seem to be a problem, again recall that the\ncut-sky analysis is self-contained and internally consistent since the data\nand realizations are treated identically. The cut-sky statistics are\n\\emph{not} estimators of the full-sky, as again made clear by this monopole\nand dipole removal.\n\nThere is also the question of the effect of our motion with respect to the\nCMB rest frame on the quadrupole.  Just as that motion, with velocity\n$\\beta\\equiv v\/c \\sim 10^{-3}$, induces a dipole with amplitude\n$\\mathcal{O}\\left(\\beta\\right)$ times the monopole, it also induces a\nDoppler quadrupole (DQ) with amplitude $\\mathcal{O}\\left(\\beta^2\\right)$\ntimes the monopole.  The naive expectation that since $\\beta^2 \\sim\n10^{-6}$ the DQ will be an unimportant contribution to the cosmological\nquadrupole is not obviously true at least in part because the measured\nquadrupole is much smaller than the theoretical expectation.  For each map\nwe analyse both the DQ uncorrected and the DQ corrected map to gauge the\nimportance of this effect.  The one exception is the\n\\satellite{Planck}\\ \\satellite{LFI}\\ $70\\unit{GHz}$ map, where the DQ has been accounted for in\nthe calibration procedure.  See\n\\citet{Planck-R1-V,CHSS-Planck-R1-alignments} for a more detailed\ndiscussion of this issue.\n\n\\section{Results}\n\n\\begin{figure}\n  \\includegraphics[width = \\linewidth]{S12_LCDM}\n  \\caption{Distribution of $\\ensuremath{S_{1\/2}}$ values from $10^6$ realizations of the\n    best-fitting $\\Lambda$CDM\\ model for full and masked skies. The shaded regions\n    (green, dash-dotted for the U74 mask and red, dashed for the KQ75y9\n    mask) represent the spread of the the observed values as given in\n    Tables~\\ref{tab:S12-results} and \\ref{tab:S12-results-DQ}.  Masking\n    only slightly affects the expected distributions and the observations\n    are in the small $\\ensuremath{S_{1\/2}}$ tail of the distribution for both masks\n    considered in this work.}\n  \\label{fig:S12-histograms}\n\\end{figure}\n\n\\begin{table}\n   \\caption{Smallness of $\\ensuremath{S_{1\/2}}$ for maps without the DQ correction. We\n     analyse the cleaned maps from \\satellite{Planck}: \\Planckmap{NILC}, \\Planckmap{SEVEM}, and \\Planckmap{SMICA}, as\n     well as from \\satellite{WMAP}: seven and nine-year ILC.  We also analyse the\n     individual frequency band maps from \\satellite{Planck}: \\satellite{HFI}\\ $100\\unit{GHz}$ and\n     \\satellite{LFI}\\ $70\\unit{GHz}$, as well as from \\satellite{WMAP}: seven and nine-year $W$\n     and $V$ bands.  For each map, we use both the U74 and KQ75y9 masks.\n     In all cases residual monopole and dipole contributions have been\n     subtracted from the map after masking.  For each map and mask we\n     report the $\\ensuremath{S_{1\/2}}$ value and the associated $p$-value -- the fraction\n     of realizations of $\\Lambda$CDM\\ in the \\satellite{Planck}\\ best-fitting $\\Lambda$CDM\\ model\n     with an $\\ensuremath{S_{1\/2}}$ no larger than the reported value.}\n  \\label{tab:S12-results}\n\\begin{tabular}{ld{4.1}d{3}d{4.1}d{3}} \\hline\n    & \\multicolumn{2}{c}{U74}\n    & \\multicolumn{2}{c}{KQ75y9}\n    \\\\\n    \\multicolumn{1}{c}{Map}\n    & \\multicolumn{1}{c}{$\\ensuremath{S_{1\/2}}\\unit{(\\rmn{\\umu K})^4}$}\n    & \\multicolumn{1}{c}{$p$ (\\%)}\n    & \\multicolumn{1}{c}{$\\ensuremath{S_{1\/2}}\\unit{(\\rmn{\\umu K})^4}$}\n    & \\multicolumn{1}{c}{$p$ (\\%)}\n    \\\\ \\hline\n    \\satellite{WMAP}\\ ILC 7yr & 1582.3 & 0.193 \t& 1225.8 & 0.085\n    \\\\\n    \\satellite{WMAP}\\ ILC 9yr & 1626.0 & 0.211\t& 1278.2 & 0.100\n    \\\\\n    \\satellite{Planck}\\ \\Planckmap{SMICA} & 1577.7 & 0.191 \t& 1022.3 & 0.044\n    \\\\\n    \\satellite{Planck}\\ \\Planckmap{NILC} & 1589.3 & 0.195\t& 1038.2 & 0.047\n    \\\\\n    \\satellite{Planck}\\ \\Planckmap{SEVEM} & 1657.7 & 0.225\t& 1153.4 & 0.069\n    \\\\\n    \\\\\n    \\satellite{WMAP}\\ $W$ 7yr & 1863.6 & 0.316\t& 1133.9 & 0.065\n    \\\\\n    \\satellite{WMAP}\\ $W$ 9yr & 1887.1 & 0.329 \t& 1142.6 & 0.068\n    \\\\\n    \\satellite{Planck}\\ \\satellite{HFI}\\ $100$ & 1682.1& 0.235 & 911.6 & 0.027\n    \\\\\n    \\\\\n    \\satellite{WMAP}\\ $V$ 7yr & 1845.0 & 0.307 \t& 1290.9 & 0.104\n    \\\\\n    \\satellite{WMAP}\\ $V$ 9yr & 1850.0 & 0.309\t& 1281.8 & 0.101\n    \\\\\n    \\satellite{Planck}\\ \\satellite{LFI}\\ $70^a$ & \\multicolumn{1}{c}{---} &\n    \\multicolumn{1}{c}{---} & \\multicolumn{1}{c}{---} &\n    \\multicolumn{1}{c}{---}\n    \\\\\n    \\hline\n  \\end{tabular}\n\\\\ ${}^a$The calibration of the \\satellite{Planck}\\ \\satellite{LFI}\\ $70\\unit{GHz}$ channel\nincludes the DQ correction.  See\n\\citet{Planck-R1-V,CHSS-Planck-R1-alignments} for details.\n\\end{table} \n\n\\begin{table}\n  \\caption{Same as Table~\\ref{tab:S12-results} now with the DQ\n    corrected maps.}\n  \\label{tab:S12-results-DQ}\n\\begin{tabular}{ld{4.1}d{3}d{4.1}d{3}} \\hline\n    & \\multicolumn{2}{c}{U74}\n    & \\multicolumn{2}{c}{KQ75y9}\n    \\\\\n    \\multicolumn{1}{c}{Map}\n    & \\multicolumn{1}{c}{$\\ensuremath{S_{1\/2}}\\unit{(\\rmn{\\umu K})^4}$}\n    & \\multicolumn{1}{c}{$p$ (\\%)}\n    & \\multicolumn{1}{c}{$\\ensuremath{S_{1\/2}}\\unit{(\\rmn{\\umu K})^4}$}\n    & \\multicolumn{1}{c}{$p$ (\\%)}\n    \\\\ \\hline\n    \\satellite{WMAP}\\ ILC 7yr\t& 1620.3 & 0.208 \t& 1247.0 & 0.090\n    \\\\ \n    \\satellite{WMAP}\\ ILC 9yr \t& 1677.5 & 0.232 \t& 1311.8 & 0.109\n    \\\\\n    \\satellite{Planck}\\ \\Planckmap{SMICA}\t& 1606.3 & 0.202\t& 1075.5 & 0.053\n    \\\\\n    \\satellite{Planck}\\ \\Planckmap{NILC}\t& 1618.6 & 0.208 \t& 1096.2 & 0.058\n    \\\\\n    \\satellite{Planck}\\ \\Planckmap{SEVEM}\t& 1692.4 & 0.239 \t& 1210.5 & 0.082\n    \\\\\n    \\\\\n    \\satellite{WMAP}\\ $W$ 7yr  \t& 1839.0 & 0.304\t& 1128.5 & 0.064\n    \\\\\n    \\satellite{WMAP}\\ $W$ 9yr\t& 1864.2 & 0.317\t& 1138.3 & 0.066\n    \\\\ \n    \\satellite{Planck}\\ \\satellite{HFI}\\ $100$ & 1707.5 & 0.245\t& 916.3 & 0.028\n    \\\\\n    \\\\\n    \\satellite{WMAP}\\ $V$ 7yr \t& 1829.2 & 0.300\t& 1276.2 & 0.099\n    \\\\\n    \\satellite{WMAP}\\ $V$ 9yr \t& 1840.4 & 0.304\t& 1268.8 & 0.097\n    \\\\\n    \\satellite{Planck}\\ \\satellite{LFI}\\ $70$\t& 1801.7 & 0.287\t& 1282.1 & 0.101\n    \\\\\n    \\hline\n  \\end{tabular}\n\\end{table} \n\nHistograms of $\\ensuremath{S_{1\/2}}$ values from $10^6$ realizations of the\n\\satellite{Planck}\\ best-fitting $\\Lambda$CDM\\ model (based on their temperature only data)\nare shown in Fig.~\\ref{fig:S12-histograms}.  Included in the figure are the\nfull-sky and cut-sky $\\ensuremath{S_{1\/2}}$.  As seen in the figure, masking has a small\neffect; the peak of the distribution is shifted to slightly smaller values\ndue to masking, but this does not have a noticeable change on the tail of\nthe distribution.  Regardless, in comparing cut-sky $\\ensuremath{S_{1\/2}}$ between the\ndata and our realizations, we always compare the one set of cut-sky data to\nthe same set of cut-sky realizations.\n\nThe $\\ensuremath{S_{1\/2}}$ values for the various map and mask combinations are given in\nTable~\\ref{tab:S12-results} for the case when the maps are not DQ corrected\nand in Table~\\ref{tab:S12-results-DQ} when the DQ correction has been\napplied.  As discussed above, the realization maps are treated precisely\nlike the data maps -- they are masked, then monopole and dipole are\nsubtracted before $\\ensuremath{S_{1\/2}}$ is computed.  Given that the value of $\\ensuremath{S_{1\/2}}$\non masked skies is extremely low compared to the typical value, having\n$10^6$ is necessary to make quantitatively precise statements.  For each\ncomputed value of $\\ensuremath{S_{1\/2}}$ reported, we also report the $p$-value -- the\nfraction of realizations (expressed in per cent) that have an $\\ensuremath{S_{1\/2}}$ at\nleast as low.  This we interpret as the probability of obtaining a value of\n$\\ensuremath{S_{1\/2}}$ this low by random chance in the best-fitting model of $\\Lambda$CDM.\n\nAn alternative approach is to allow for variations of the best-fitting\nparameters within their error bars, for example by examining a Monte Carlo\nMarkov chain of the parameters rather than just performing realizations of\nthe best-fitting values.  (Such an approach was taken for example in\n\\cite{CHSS-WMAP5}.)  This will affect the results only weakly, because\nvarying the parameters within their error bars will cause the expected\nlow-$\\ell$ $\\Cee_\\ell$ to vary by much less than their cosmic variance errors.\n\n\\begin{figure}\n  \\includegraphics[width = \\linewidth]{Ctheta_bands}\n  \\caption{Cut-sky $\\Ccorr(\\theta)$ using the KQ75y9 mask for individual\n    frequency band maps.  Shown are correlation functions from the\n    \\satellite{WMAP}\\ nine-year $W$ (black, solid line) and $V$ (red, dashed line)\n    bands along with the \\satellite{Planck}\\ \\satellite{HFI}\\ $100\\unit{GHz}$ (green, dash-dotted\n    line) and \\satellite{LFI}\\ $70\\unit{GHz}$ (blue, dotted line) maps.  The curves\n    for the \\satellite{WMAP}\\ seven-year band maps are nearly identical to those from\n    the nine-year maps and are not included for clarity.  In all cases the\n    correlation functions are in excellent agreement across the data\n    releases and frequency bands.  (Note the range on $y$-axis has been\n    greatly reduced as compared to Fig.~\\ref{fig:Ctheta-smica} to allow for\n    \\emph{any} difference to be noticeable by eye.)}\n  \\label{fig:Ctheta-bands}\n\\end{figure}\n\nThe cut-sky $\\ensuremath{S_{1\/2}}$ values presented in Tables~\\ref{tab:S12-results} and\n\\ref{tab:S12-results-DQ} show that the region outside the masks is\nconsistently observed and cleaned in all the data releases mostly\nindependent of analysis procedures.  In Fig.~\\ref{fig:Ctheta-bands} we plot\n$\\Ccorr(\\theta)$ for the \\satellite{WMAP}\\ nine-year $V$ and $W$ bands with the KQ75y9\nmask and the \\satellite{Planck}\\ \\satellite{HFI}\\ $100\\unit{GHz}$ and \\satellite{LFI}\\ $70\\unit{GHz}$ bands\nalso with the KQ75y9 mask.  One can see that the cut-sky angular\ncorrelation functions are remarkably consistent across instruments and\nwavebands.  (And also across \\satellite{WMAP}\\ data releases.  We have chosen not to\nplot the \\satellite{WMAP}\\ seven-year correlation functions because they are nearly\nindistinguishable from the nine-year functions.)  We can thus place great\nconfidence in the cut-sky $\\ensuremath{S_{1\/2}}$ results derived from \\satellite{WMAP}\\ and \\satellite{Planck}.\nThese results can be summarized as follows.\n\\begin{itemize}\n\\item Regardless of the maps, the cut-sky $\\ensuremath{S_{1\/2}}$ is very low, with\n  $p$-values ranging from $0.027$ per cent for the\n  \\satellite{Planck}\\ \\satellite{HFI}\\ $100\\unit{GHz}$ map with the KQ75y9 mask to \n  $0.329$ per cent for the \\satellite{WMAP}\\ nine-year $W$ band map with the U74\n  mask.\n\\item The cleaned maps have a smaller variation in $\\ensuremath{S_{1\/2}}$ values with a\n  $p$-value always less than about $0.239$ per cent for the U74 mask and\n  less than about $0.109$ per cent for the KQ75y9 mask.\n\\item The \\satellite{Planck}\\ maps typically have smaller $\\ensuremath{S_{1\/2}}$ values than the\n  \\satellite{WMAP}\\ maps. (The one slight exception is the DQ corrected\n  \\satellite{Planck}\\ \\satellite{LFI}\\ $70\\unit{GHz}$ band with the KQ75y9 mask.)\n\\item The only clear systematic trend is that the KQ75y9 mask consistently\n  yields a lower cut-sky $\\ensuremath{S_{1\/2}}$ than does the U74 mask.  Presumably this\n  is due to the larger region around the Galactic centre excluded by the KQ75y9\n  mask (see Fig \\ref{fig:masks-comparison}).\n\\item The DQ correction has little effect, tending to slightly increase\n  $\\ensuremath{S_{1\/2}}$ in the \\satellite{Planck}\\ maps and decrease it in the \\satellite{WMAP}\\ ones.  This is\n  in contrast to the importance of applying the DQ correction for full-sky\n  alignment studies \\citep{CHSS-Planck-R1-alignments}.\n\\end{itemize}\n\nOverall, the data very consistently show a lack of correlations on large\nangular scales outside the Galactic region (as defined by the two masks\nemployed).  The $p$-value for the $\\ensuremath{S_{1\/2}}$ statistic has remained small and\nof comparable size throughout the \\satellite{WMAP}\\ data releases and now with the\nfirst \\satellite{Planck}\\ results.  This is remarkable given the improvements in\nstatistics, cleaning, beams, masks, and other systematics.  Further, this\nis in contrast to the full-sky $\\ensuremath{S_{1\/2}}$ which vary significantly from data\nrelease to data release and from map to map.  The behaviour of the full-sky\n$\\ensuremath{S_{1\/2}}$ is discussed in more detail in \\citet{CHSS-Planck-R1-alignments}.\nIt suffices here to note that the full-sky value of $\\ensuremath{S_{1\/2}}$ varies from a\nlow of $3766\\unit{(\\rmn{\\umu K}^4)}$, from the \\satellite{Planck}\\ \\Planckmap{SEVEM}\\ map, to a high of\n$8938\\unit{(\\rmn{\\umu K}^4)}$, calculated from the seven-year \\satellite{WMAP}\\ reported values\nof the angular power spectrum based on a maximum likelihood estimator.\n\nWe again emphasize that the two-point angular correlation function that we\nhave calculated is monopole and dipole-subtracted.  However, once the\nDoppler dipole is sufficiently well determined, only it should be removed\nand the underlying cosmological contribution to the dipole retained in\n$\\Ccorr(\\theta)$ and thus in $\\ensuremath{S_{1\/2}}$.  \n\n\nThe measured lack of angular correlations in the dipole-subtracted sky has\nan important consequence for the primordial dipole. If the missing\ncorrelations are not a very unlikely fluke, nor (as our results indicate)\ndue to systematic errors or map-cleaning procedures, then they are caused\nby some as-yet unidentified physical mechanism. It is difficult to see how\nsuch a mechanism would set $\\Ccorr(\\theta)$ to be nearly zero on angular\nscales greater than $60$ degrees when the primordial dipole is subtracted,\nand yet somehow not also do so if the dipole were included.  Instead, for a\nphysical mechanism we would expect the total angular correlation function\nincluding the contribution of the cosmological dipole to also be nearly\nzero on these scales. In the best-fitting $\\Lambda$CDM\\ model, the expected\ncontribution from the dipole alone is very large and generically spoils the\nvanishing of $\\Ccorr(\\theta)$ on large angular scales.  Hence, if the\nvanishing correlations are of cosmological origin, then the primordial\ndipole is also expected to be very suppressed.\n\nTo be concrete, the expected value of $\\mathcal{C}_1$ in the\nbest-fitting $\\Lambda$CDM\\ model is approximately $3300\\unit{\\rmn{\\umu K}^2}$.  With this\nvalue, the $\\mathcal{C}_1^2$ contribution to $\\ensuremath{S_{1\/2}}$ in Eq.~(\\ref{eq:Shalf}) alone\nwould contribute approximately $2.3\\times10^5\\unit{\\rmn{\\umu K}^4}$ to $\\ensuremath{S_{1\/2}}$. (In\nprinciple this could be compensated by the cross term, $\\mathcal{C}_1\\mathcal{C}_\\ell$ with\n$\\ell\\neq1$, which can be negative, however, in practice this does not occur\nowing mainly to the smallness of $\\mathcal{C}_2$.)  Roughly, for the $\\mathcal{C}_1^2$\ncontribution to not make the $\\ensuremath{S_{1\/2}}$ `too large' the value of $\\mathcal{C}_1$ must\nalso not be `too large'.  For example, requiring the contribution to $\\ensuremath{S_{1\/2}}$\nto be comparable to current cut-sky values, that is a contribution of order\n$1000\\unit{\\rmn{\\umu K}^4}$, places a limit $\\mathcal{C}_1\\la 200\\unit{\\rmn{\\umu K}^2}$. This has a\nprobability of occurring by chance in a realization of the best-fitting model\n(due to cosmic variance) of less than approximately $0.4$ per cent.\nEquivalently, to the extent that $\\mathcal{C}_1$ contributions dominate the value of\n$\\ensuremath{S_{1\/2}}$, in order to maintain a $p$-value for the $\\ensuremath{S_{1\/2}}$ less than $0.4$\nper cent once the cosmological dipole is included requires that\n$\\mathcal{C}_1\\la200\\unit{\\rmn{\\umu K}^2}$.  \n\nTo summarize, it seems unlikely that a physical mechanism\nwould predict that the $\\ensuremath{S_{1\/2}}$ calculated from a dipole-subtracted cut-sky\nwould be small but the $\\ensuremath{S_{1\/2}}$ calculated from a non-dipole-subtracted\ncut-sky would not be.  This strongly suggests that if the lack of angular\ncorrelations is physical in nature and not a statistical fluke, then a robust\nprediction can be made that there is a very small cosmological dipole.  In a\nfuture work we will develop this prediction more precisely.\n\n\\section{Conclusions}\n\nThe CMB shows a lack of correlations on large angular scales.  This can be\nquantified by the $\\ensuremath{S_{1\/2}}$ statistics proposed by \\citet{WMAP1-cosmology}\nwhich is best calculated on the portion of the sky outside the Galaxy.\nUnlike attempts to infer properties of the the full-sky correlation\nfunction, the cut-sky $\\ensuremath{S_{1\/2}}$ appears remarkably robust and trustworthy.\nIn our analysis we find that the $p$-value for the observed cut-sky\n$\\ensuremath{S_{1\/2}}$ in an ensemble of realizations of the best-fitting $\\Lambda$CDM\\ model\nnever exceeds $0.33$ per cent for any of the analysed combinations of maps\nand masks, with and without correcting for the Doppler quadrupole. This has\nremained the case since the \\satellite{WMAP}\\ three-year data release,\\footnote{The\n  one-year \\satellite{WMAP}\\ release yielded slightly higher $p$-values -- $0.38$ per\n  cent for the $V$ band, and $0.64$ per cent for the $W$ band\n  \\citep{CHSS-WMAP3, CHSS-WMAP5}.} for both the individual ($V$ and $W$)\nband maps and the synthesized (ILC) map, and for the first \\satellite{Planck}\\ data\nrelease for both the \\satellite{LFI}\\ and \\satellite{HFI}\\ band maps and all the released\nsynthesized maps (\\Planckmap{NILC}, \\Planckmap{SMICA}, \\Planckmap{SEVEM}), when masked by either the\n\\satellite{WMAP}\\ KQ75y9 mask or the less conservative U74 mask (which is very similar\nto the \\satellite{Planck}\\ U73 mask).  The \\satellite{HFI}\\ $100\\unit{GHz}$ map -- the presumably\ncleanest CMB band -- with the more conservative mask that has been defined\nby \\satellite{WMAP}\\ gives a $p$-value of only $0.03$ per cent! As general trends we\nnote that a larger mask tends to produce smaller $p$-values, the Doppler\nquadrupole correction does not change the results in a significant way, and\nthe \\satellite{Planck}\\ data yield somewhat smaller $p$-values than the \\satellite{WMAP}\\ data.\n\nThis apparent lack of temperature correlations on large angular scales is\nstriking.  It is a robust observation that increases in statistical\nsignificance from \\satellite{COBE}\\ to \\satellite{WMAP}\\ to \\satellite{Planck}. The consistency of the lack\nof angular correlations greatly reduces the likelihood of instrumental\nissues as a cause. Since all three missions observed the same sky, we could\nbe unlucky and live in a very atypical realization of the Universe. A\nmethod of testing this hypothesis has been proposed that would utilize\nthe upcoming \\satellite{Planck}\\ polarization data \\citep{CHSS-TQ}.  If it is not a\nstatistical fluke and not an instrumental issue, it still could be caused\nby foregrounds.  This appears also unlikely as the lack of correlations is\nconsistently seen in individual bands as well as in foreground cleaned\nmaps. Thus, to the best of our knowledge, the lack of angular correlations\nis in contradiction with the idea of scale invariant, isotropic and\nGaussian perturbations seeded by cosmological inflation.\n\nAttempts to explain this lack of correlations should also address the\nvarious other anomalous aspects observed in the microwave sky. It turns out\nthat the lack of angular correlations puts very severe constraints on such\nmodels. For example, a plausible explanation for the alignments of\nmultipole vectors or for the hemispherical asymmetry observed might have\nbeen contamination by unaccounted foregrounds; however, one cannot easily\nunderstand how a hypothetical foreground, which presumably should be\nuncorrelated with the primordial temperature fluctuations, could cause an\nalmost exact cancellation of the primordial fluctuations at angular scales\nabove 60 degrees.\n\nSeveral attempts have been made to explain the absence of large-angle\ncorrelations as being due to an unknown foreground or, more generally, by\naltering the procedure by which one arrives at the cleaned maps.  Indeed,\nwhen the cleaned maps are altered in any way the microwave sky can easily\nbe made to appear less anomalous.   This\nis not surprising; almost any random modification of the observed maps will\nmake them less anomalous. Though the removal of anomalies may be a\nside effect of improved analysis procedures, using their removal as a\nbasis for judging the effectiveness of such a procedure is misguided. \n\nFinally, we emphasize that in order to be convincing, new theoretical\nmodels to explain the observed large-angle anomalies must be based on the\nstatistics of realizations of that model, not just on having the mean\nvalues of the model agree with observations.  In other words, CMB map\nrealizations based on the underlying new model should have $p$-values for\nthe measured statistics that are not unusually small.\n\nThe large-angle temperature-temperature correlations in the CMB outside the\nGalaxy have been anomalously low in all relevant maps since the days of the\n\\satellite{COBE-DMR}.  The final \\satellite{WMAP}\\ release and the initial \\satellite{Planck}\\ release\nconfirm that anomaly.  After twenty years, we still await a satisfactory\nexplanation.\n   \n \n\\section*{Acknowledgements}\n\nWe acknowledge valuable communications and discussions with F.~Bouchet,\nC.~Burigana, J.~Dunkley, G.~Efstathiou, K.~Ganga, P.~Naselsky, H.~Peiris, \nC.~R\\\"ath and D.~Scott. \nGDS and CJC are supported by a grant from the US Department of Energy to the\nParticle Astrophysics Theory Group at CWRU\\@.  DH has been supported by the\nDOE, NSF, and the Michigan Center for Theoretical Physics\\@.  DJS is supported\nby the DFG grant RTG 1620 `Models of gravity'. DH thanks the Kavli Institute\nfor Theoretical Physics and GDS thanks the Theory Unit at CERN for their\nrespective hospitality.  This work made extensive use of the \\textsc{healpix}{}\npackage~\\citep{healpix}.  The numerical simulations were performed on the\nfacilities provided by the Case ITS High Performance Computing Cluster.\n\n\\bibliographystyle{mn2e_new}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Credits}\n\n\n\\section{Introduction}\n\nNatural language processing is increasingly leveraged in sensitive domains like healthcare. For such critical tasks, the need to prevent discrimination and bias is imperative. Indeed, ensuring equality of health outcomes across different groups has long been a guiding principle of modern health care systems \\cite{culyer1993equity}.\nMoreover, medical data presents a unique opportunity to work with different \\textit{modalities}, specifically \\textit{text} (e.g., patient narratives, admission notes, and discharge summaries) and numerical or categorical data (often denoted \\textit{tabular} data, e.g., clinical measurements such as blood pressure, weight, or demographic information like ethnicity). Multi-modal data is not only reflective of many real-world settings, but machine learning models which leverage both structured and unstructured data often achieve greater performance than their individual constituents \\cite{horng2017creating}. While prior work studied fairness in the text and tabular modalities in isolation, there is little work on applying notions of algorithmic fairness in the broader multimodal setting \\citep{10.1145\/3368555.3384448, chen2018my}. \n\n\n\n\n\n\nOur work brings a novel perspective towards studying fairness algorithms for models which operate on \\textit{both} text and tabular data, in this case applied to the MIMIC-III clinical dataset (MIMIC-III) \\cite{MIMIC}. We evaluate two fairness algorithms: equalized-odds through post-processing, which is agnostic to the underlying classifier, and word embedding debiasing which is a text-specific technique. We show that ensembling classifiers trained on structured and unstructured data, along with the aforementioned fairness algorithms, can both improve performance and mitigate unfairness relative to their constituent components. We also achieve strong results on several MIMIC-III clinical benchmark prediction tasks using a dual modality ensemble; these results may be of broader interest in clinical machine learning \\citep{Harutyunyan2019, khadanga2019using}. \n\n\n\n\n\n\\section{Background}\n\n\n\n\\subsection{Combining Text and Tabular Data in Clinical Machine Learning}\nPrior work has shown that combining unstructured text with vital sign time series data improves performance on clinical prediction tasks. \\citet{horng2017creating} showed that augmenting an SVM with text information in addition to vital signs data improved retrospective sepsis detection. \\citet{akbilgic2019unstructured} showed that using a text-based risk score improves performance on prediction of death after surgery for a pediatric dataset. Closest to our work, \\citet{khadanga2019using}  introduced a joint-modality neural network which outperforms single-modality neural networks on several benchmark prediction tasks for MIMIC-III. \n\n\\subsection{Classical fairness metrics}\nMany algorithmic fairness notions fall into one of two broad categories: individual fairness enforcing fairness across individual samples, and group fairness seeking fairness across protected groups (e.g. race or gender). \nWe focus on a popular group-level fairness metric: {\\em Equalized Odds} (EO) \\citep{NIPS2016_6374}. Instead of arguing that average classification probability should be equal across all groups (also known as {\\em Demographic Parity}) -- which may be unfair if the underlying group-specific base rates are unequal -- EO allows for classification probabilities to differ across groups only through the underlying ground truth. Formally, a binary classifier $\\widehat{Y}$ satisfies EO for a set of groups $\\mathcal{S}$ if, for ground truth $Y$ and group membership $A$:\n\\begin{equation*}\\label{EOeq}\\small\n    \\begin{split}\n        \\text{Pr} ( \\hat{Y}=1 \\,|\\, Y = y, A=a ) = \\text{Pr} ( \\hat{Y}=1 \\,|\\, Y = y, A=a' )\\\\ \\forall y \\in \\{0,1\\}, \\forall a,a' \\in \\mathcal{S}\n    \\end{split}\n\\end{equation*}\n\nIn short, the true positive (TP) and true negative (TN) rates should be equal across groups. \n\n\\subsection{Equalized Odds Post Processing}\n\\citet{NIPS2016_6374} proposed a model-agnostic post-processing algorithm that minimizes this group specific error discrepancy while considering performance. Briefly, the post-processing algorithm determines group-specific random thresholds based on the intersection of group-specific ROC curves. The multi-modality of our underlying data and the importance of privacy concerns in the clinical setting make post-processing especially attractive as it allows fairness to be achieved agnostic to the inner workings of the base classifier. \n\n\n\\subsection{Debiasing word embeddings}\n\nPretrained word embeddings encode the societal biases of the underlying text on which they are trained, including gender roles  and racial stereotypes\n\\citep{bolukbasi2016man, zhao-etal-2018-learning,manzini-etal-2019-black}. \nRecent work has attempted to mitigate this bias in context-free embeddings while preserving the utility of the embeddings.\n\\citet{bolukbasi2016man} analyzed gender subspaces by comparing distances between word vectors with pairs of gender-specific words to remove bias from gender-neutral words. \\citet{manzini-etal-2019-black} extended this work to the multi-class setting, enabling debiasing in race and religion. Concurrent to their work, \\cite{ravfogel2020null} propose  iterative null space projection as a technique to hide information about protected attributes by casting it into the null space of the classifier.  Following the recent popularity of BERT and ELMo, \\citet{liang2020towards} consider extending debiasing to sentence-level, contextualized representations. \n\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{img\/arc5.png}\n    \\caption{Experimental setup and ensemble architecture. Fairness approaches are indicated in dotted boxes. \n   \n    }\n    \\label{fig:architecture}\n\\end{figure}\n\n\n\n\n\\section{Experimental Setup}\\label{sec:experimental_setup}\n\n\\subsection{Clinical Prediction Tasks} \\label{clinical_pred_tasks}\nMIMIC-III contains deidentified health data associated with ~60,000 intensive care unit (ICU) admissions \\citep{MIMIC}. It contains  both unstructured textual data (in the form of clinical notes) and structured data (in the form of clinical time series data  and demographic, insurance, and other related meta-data). \nWe focus on two benchmark binary prediction tasks for ICU stays previously proposed by \\citet{Harutyunyan2019}: in-hospital mortality prediction (IHM), which aims to predict mortality based on the first 48 hours of a patient's ICU stay, and phenotyping, which aims to retrospectively predict the acute-care conditions that impacted the patient.  Following \\citet{khadanga2019using} we extend the prediction tasks to leverage clinical text linked to their ICU stay. For both tasks the classes are higly imbalanced: in the IHM task only 13.1\\% of training examples are positive, and the relative imbalance of the labels in the phenotyping class can be seen in Figure \\ref{fig:phenotype_perc_tr}. To account for the label imbalance we evaluate performance using AUC ROC and AUC PRC.  More details can be found in Appendix \\ref{appendix:a}. \n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=\\linewidth]{img\/phen_label_train_percentages.png}\n    \\caption{Percentage of positive train cases for each of the 25 phenotyping tasks. The critical care conditions corresponding to the task codes can be found in Table \\ref{tab:phen_perc_legend} of the Appendix}\n    \\label{fig:phenotype_perc_tr}\n\\end{figure}\n\n\n\\subsection{Fairness Definition}\nNext, we consider how we can extend a definition of fairness to this multimodal task. Following work by \\citet{10.1145\/3368555.3384448} in the single-modality setting, we examine True Positive and True Negative rates on our clinical prediction task between different protected groups. Attempting to equalize these rates corresponds to satisfying \\emph{Equalized Odds}. EO satisfies many desiderata within clinical settings, and has been used in previous clinical fairness work \\citep{10.1145\/3306618.3314278, doi:10.1111\/j.1468-2850.1997.tb00104.x, pmlr-v106-pfohl19a}. While EO does not explicitly incorporate the \\textit{multimodality} of our data, it accurately emphasizes the importance of the \\textit{downstream} clinical prediction task on the protected groups.   Nonetheless, we acknowledge that EO alone is insufficient for practical deployment; na\\\"ive application can result in unacceptable performance losses and thus consultations with physicians and stakeholders must be held \\citep{Rajkomar2018}.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Classification Models}\n\nWe provide brief descriptions below with details available in Appendix \\ref{appendix:b}.\n\\begin{itemize}\n\\setlength{\\itemsep}{-0.9pt}\n\\item \\textbf{Structured Data Model: }Following \\citet{Harutyunyan2019}, we use a channel-wise bidirectional Long Short Term Memory network (bi-LSTM). \n\\item \\textbf{Unstructured Textual Data:} We use a CNN encoder to extract the semantic features from clinical notes. Importantly, we experiment with training word embeddings from scratch and utilizing pre-trained BioWordVec embeddings \\citep{zhang2019biowordvec}.\n\n\n\\item \\textbf{Ensemble:} We perform logistic regression on the output binary classification probabilities from the previous models. \n\n\\end{itemize}\n\\section{Fairness Setup}\n\n\n\n\n\n\n\n\\subsection{Sensitive groups}\nRecall that EO explicitly ensures fairness with respect to sensitive groups while debiasing implicitly depends upon it. Leveraging the demographic data in MIMIC-III, we consider ethnicity (divided into Asian, Black, Hispanic, White and other), biological sex (divided into male and female), and insurance type (divided into government, medicare, medicaid, self-pay, private, and unknown). With the exception of biological sex, the sensitive groups are highly imbalanced (see Table \\ref{tab:in_hosp_mort_groups}). Note that insurance-type has been shown to be a proxy for socioeconomic status (SES) \\cite{pmid30794127}.\n\n\\begin{table}[]\n    \\centering\n    \n    \\begin{tabular}{|c|c|c|c|}\n    \\hline\n    {\\begin{tabular}[c]{@{}c@{}}Sensitive \\\\ Group \\end{tabular}}     &  {\\begin{tabular}[c]{@{}c@{}}Train \\\\ Count\\end{tabular}} & {\\begin{tabular}[c]{@{}c@{}} Test \\\\ Count \\end{tabular}} & \\% of Test \\\\\n    \\hline\n    F & 7940 & 1415 & 44.0 \\% \\\\\n    M & 9708 & 1778 & 56.0 \\% \\\\\n    \\hline\n    ASIAN & 408 & 60 & 1.9 \\% \\\\\n    BLACK & 1658 & 285 & 8.9 \\% \\\\\n    HISPANIC & 521 & 107 & 3.3 \\% \\\\\n    OTHER & 2655 & 459 & 14.4 \\% \\\\\n    WHITE & 12406 & 2282 & 71.5 \\% \\\\\n    \\hline\n    Government & 356 & 74 & 2.3 \\% \\\\\n    Medicaid & 1362 & 205 & 6.4 \\% \\\\\n    Medicare & 9857 & 1757 & 55.0 \\% \\\\\n    Private & 4946 & 932 & 29.2 \\% \\\\\n    Self Pay & 133 & 33 & 1.0 \\% \\\\\n    UNKNOWN & 994 & 192 & 6.1 \\% \\\\\n    \\hline\n    \\end{tabular}\n    \n    \\caption{Distribution of sensitive-attributes over train and test data for the In-Hospital Mortality task}\n    \\label{tab:in_hosp_mort_groups}\n\\end{table}\n\n\\subsection{Equalized Odds Post-Processing}\nWe apply our equalized-odds post processing algorithm on the predictions of the trained single-modality  classifiers (physiological signal LSTM model as well as text-only CNN model) as well as the trained ensemble classifier. Note that we apply EO postprocessing only once for each experiment: either on the outputs of the single-modality model, or on the ensemble predictions. \n The fairness approaches are mutually exclusive: we do not consider applying EO postprocessing together with debiased word embeddings. We consider using both soft prediction scores (interpretable as probabilities) as well as thresholded hard predictions as input to the post-processing algorithm. These choices impact the fairness performance trade-off as discussed further in Section \\ref{sec:Results}.\n\n\n\n\\subsection{Socially Debiased Clinical Word Embeddings}\n\nWhile clinically pre-trained  word embeddings may improve downstream task performance, they are not immune from societal bias \\cite{khattak2019survey}.  \nWe socially debias these clinical word embeddings following \\citet{manzini-etal-2019-black}. We manually select sets of social-specific words (see Appendix \\ref{appendix:c}) to identify the fairness-relevant social bias subspace. \nFormally, having identified the basis vectors $\\{b_{1}, b_{2}, ..., b_{n}\\}$ of the social bias subspace $\\mathcal{B}$, we can find the projection $w_B$ of a word embedding $w$:\n\\[w_\\mathcal{B} = \\sum_{i=1}^{n} \\langle{w}, b_{i}\\rangle{b_{i}}\\]\n\nNext we apply hard debiasing, which will remove bias from existing word embeddings by subtracting $w_B$, their component in this fairness subspace. This yields $w'$, our socially debiased word embedding: \n\n\\begin{align*}\n    w' = \\frac{w - w_B}{\\|w-w_B\\|}\n\\end{align*}\n\n\n\n\n\n\n\n\nWe consider debiasing with respect to race and gender. The race debiased embeddings are re-used for insurance tasks as empiric research has indicated that the use of proxy groups in fairness can be effective \\citep{DBLP:journals\/corr\/abs-1806-11212} and SES is strongly related to race \\citep{Williams2016}.\n\n\n\n\\section{Results and Analysis}\\label{sec:Results}\n\n\n\n\n\n\\begin{table}[h]\n\\centering\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{clllll}\n\\cline{2-5}\n\\multicolumn{1}{l|}{} &\n  \\multicolumn{2}{c|}{IHM} &\n  \\multicolumn{2}{c|}{Phenotyping} &\n  \\multicolumn{1}{l}{} \\\\ \\cline{1-5}\n\\multicolumn{1}{|l|}{} &\n  \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}AUC \\\\ PRC\\end{tabular}} &\n  \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}AUC \\\\ ROC\\end{tabular}} &\n  \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}Macro \\\\ AUCROC\\end{tabular}} &\n  \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}Overall \\\\ AUCROC\\end{tabular}} &\n   \\\\ \\cline{1-5}\n   \\multicolumn{1}{|c|}{\\begin{tabular}[c]{@{}c@{}}Harutyunyan et. al\\\\  (2019) -- No Text\\end{tabular}}& \\multicolumn{1}{l|}{0.515} & \\multicolumn{1}{l|}{0.862} &  \n\\multicolumn{1}{c|}{0.776} &\n\\multicolumn{1}{c|}{0.825} &  \\\\ \\cline{1-5}\n\\multicolumn{1}{|c|}{\\begin{tabular}[c]{@{}c@{}}Khadanga et. al \\\\  (2019) -- Ensemble\\end{tabular}}    & \\multicolumn{1}{l|}{0.525} & \\multicolumn{1}{l|}{0.865} &\n\\multicolumn{1}{c|}{--} & \n\\multicolumn{1}{c|}{--} &   \\\\ \\cline{1-5}\n\\multicolumn{1}{|c|}{\\begin{tabular}[c]{@{}c@{}}Ours -- Text Only\\end{tabular}} &\n  \\multicolumn{1}{l|}{0.472} &\n  \\multicolumn{1}{l|}{0.815} &\n\\multicolumn{1}{c|}{0.766} &  \n\\multicolumn{1}{c|}{0.829} & \\\\ \\cline{1-5}\n   \\multicolumn{1}{|c|}{\\begin{tabular}[c]{@{}c@{}}Ours -- Text Only \\\\ + BioWordVec \\end{tabular}} &\n  \\multicolumn{1}{l|}{0.489} &\n  \\multicolumn{1}{l|}{0.841} &\n\\multicolumn{1}{c|}{0.771} & \n\\multicolumn{1}{c|}{0.837} &  \\\\ \\cline{1-5}\n\\multicolumn{1}{|c|}{Ours -- Ensemble}           & \\multicolumn{1}{l|}{\\textbf{0.582}} & \n\\multicolumn{1}{l|}{0.880} &  \n\\multicolumn{1}{c|}{0.822} &\n\\multicolumn{1}{c|}{0.861} &  \\\\ \\cline{1-5}\n\\multicolumn{1}{|c|}{\\begin{tabular}[c]{@{}c@{}}Ours -- Ensemble \\\\ + BioWordVec\\end{tabular}}           & \\multicolumn{1}{l|}{\\textbf{0.582}} & \\multicolumn{1}{c|}{\\textbf{0.886}} &  \n\\multicolumn{1}{c|}{\\textbf{0.829}} & \n\\multicolumn{1}{c|}{\\textbf{0.870}} & \\\\ \\cline{1-5}\n\\end{tabular}}\n\\caption{Leveraging clinical pretrained word embeddings improves performance compared to training word embeddings from scratch in the text-only model. Ensembling the text-only model with the clinical time series classifier improves performance further. \n }\n\\label{tab:my-table}\n\\end{table}\n\n\n\n\n\n\n\n\\begin{figure*}[htp]\n  \\centering\n  \\includegraphics[width=\\linewidth]{img\/phen_lipid_ETHNICITY_t14.png}\n  \\caption{ \n  Plots of TP Rate, TN Rate, and AUC on phenotyping task M for groups defined by sensitive attribute of race. Each vertical black line represents a classifier (line style indicating modality); the length of the line represents the range of scores over fairness groups. In the TP\/TN graphs, a shorter line represents better fairness; there is less discrepancy between the maximum and minimum group-specific TP\/TN rates. In the AUC graph (far right), the higher the vertical position of the line, the better the performance. EO is effective at reducing the spread in TP\/TN rates for the ensemble classifier (first two graphs) at the cost of performance (far right) graph. Meanwhile, debiased word embeddings both improves fairness, reducing the length of the line in the first two graphs, while achieving superior performance in AUC graph  }\\label{fig:big_plot}\n  \n\n  \n  \n\n \n\\end{figure*}\n\n\n  \n\n\n\n\n\n\n\n\n\n\\subsection{Ensembling clinical word embeddings with structured data improves performance} \n \nEmpirically, we observe superior performance to prior literature on a suite of clinical prediction tasks in Table \\ref{tab:my-table}; more tasks are evaluated in Appendix Table \\ref{appendix:a}. Full hyperparameter settings and code for reproducibility can be found here \\footnote{\\url{https:\/\/github.com\/johntiger1\/multimodal_fairness\/clinicalnlp}}. The ensemble model\noutperforms both constituent classifiers (AUC plot on Figure \\ref{fig:big_plot}). This holds even when fairness\/debiasing techniques are applied, emphasizing the overall effectiveness of leveraging multi-modal data. However, the ensemble's improvements in performance do not directly translate to improvements in fairness; see the True Positive (TP) graph in Figure \\ref{fig:big_plot}, where the maximum TP gap remains consistent under the ensemble.\n\n\n\n\\subsection{Debiased word embeddings and the fairness performance trade-off}\nImproving fairness usually comes at the cost of reduced performance \\citep{pmlr-v81-menon18a}. Indeed, across all tasks, fairness groups and classifiers, we observe the group-specific disparities of TP and TN rates generally diminish when equalized odds post-processing is used (see Appendix \\ref{sec:appendix:full_result} for additional results). However, this post-processing also leads to a degradation in the AUC. \nNote that we apply EO-post processing on hard (thresholded) predictions of the classifiers. If instead \\textit{soft} prediction scores are used as inputs to the post-processing step, both the performance degradation and the fairness improvement are softened \\citep{NIPS2016_6374}. \n\n\n\n\n\n\nGenerally, word embedding debiasing (WED) also helps reduce TP\/TN discrepancies, although not to the same extent as EO postprocessing. Remarkably, in certain tasks, WED also yields a performance improvement, even compared to the fairness-free, unconstrained ensemble classifier. In particular, for the AUC graph in Figure \\ref{fig:big_plot}, leveraging debiased word embeddings improves the performance of the ensemble; at the same time, the TP and TN group discrepancy ranges are improved.  However, we stress that this outcome was not consistently observed and further investigation is warranted.\n\n\n\n\nWe emphasize that EO and WED serve different purposes with different motivations. While EO explicitly seeks to minimize the TP\/TN range between sensitive groups (reflected in its performance on the first two plots in Figure \\ref{fig:big_plot}), WED seeks to neutralize text-specific bias in the word-embeddings. Despite the difference in goals, and despite operating only on the text-modality of the dataset, WED is still able to reduce the group-specific TP\/TN range; recent work on \\textit{proxy fairness} in text has shown that indirect correlation between bias in text and protected attributes may be useful in achieving parity \\citep{romanov2019s}.   \n\n\n\nAlthough WED demonstrate some good properties with respect to both fairness and performance for our specific dataset and task, we caution that they represent only one approach to fairness in NLP \\cite{blodgett2020language}. Indeed, WED suffers from shortcomings related to intersectional fairness \\citep{gonen2019lipstick}, and we encourage further discussion into concretely defining fair, real-world NLP tasks and developing novel algorithms.\n\nOur results highlight the important role practitioners and stakeholders play in algorithmic fairness on clinical applications. The trade-off between performance and fairness, whether between the soft and hard labels used for EO, or between EO and debiased word embeddings, must be balanced based on numerous real world factors. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion}\n\n\n\n \n In this paper, we propose a novel multimodal fairness task for the MIMIC-III dataset, based on equalized odds. We provide two baselines: a classifier-agnostic fairness algorithm (equalized odds post-processing) and a text-specific fairness algorithm (debiased word embeddings). We observe that both methods generally follow the fairness performance tradeoff seen in single-modality tasks. EO is more effective at reducing the disparities in group-specific error rates while word-embedding debiasing has better performance. \n Future work can consider more generalized notions of fairness such as preferences-based frameworks, or extend text-specific fairness to contextualized word embeddings \\citep{hossain2020designing, 10.1145\/3368555.3384448}. \n Further analysis of the fairness performance tradeoff, especially in multimodal settings, will facilitate equitable decision making in the clinical domain\n \n \n \\section{Acknowledgements}\n We would like to acknowledge Vector Institute for office and compute resources. We would also like to thank Matt Gardner for his help with answering questions when using AllenNLP \\cite{Gardner2017AllenNLP}. John Chen and Safwan Hossain are funded by an Ontario Graduate Scholarship and a Vector Institute Research Grant. Ian Berlot-Attwell is funded by a Canada Graduate Scholarships-Master's, and a Vector Institute Research Grant. Frank Rudzicz is supported by a CIFAR Chair in AI.\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","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nOptical parametric generation via a quadratic nonlinearity has been extensively studied for the capability of wavelength conversion through elastic photon-photon scattering, constituting the basis of various applications including coherent radiation \\cite{Dunn99}, spectroscopy \\cite{Rosenman99}, frequency metrology \\cite{Cundiff03}, and quantum information processing \\cite{Pan12}. With the ability to strongly confine optical modes in the micro-\/nano-scale, a number of integrated photonic platforms have been developed for strong nonlinear optical effects with high efficiencies and low power consumption \\cite{Harris06, Vuckovic09, Lipson11, Pavesi12, Solomon14, Tang16Optica, Watts17}.\n\nAmong all the integrated nonlinear photonic platforms, lithium niobate (LN) has recently attracted remarkable attentions, owing to its wide transparency window and strong quadratic optical nonlinearity. To date, a variety of nanophotonic systems, including waveguides \\cite{Pertsch15, Bowers16, Loncar17OE, Fathpour17APL, Luo18Optica, Ding18, Luo18semi}, microdisks \\cite{Loncar14, Cheng16, Luo17OE, Xu17, Cheng18, Xiao18}, microrings \\cite{Zappe18, Huang18}, and photonic crystal cavities \\cite{Pertsch13, Jiang18, Li18}, have been studied for optical parametric processes in LN. Cavity-enhanced nonlinear wavelength conversion has been demonstrated in doubly\/triply resonant LN microresonators through a number of techniques including modal phase matching \\cite{Loncar14, Xu17, Xiao18, Huang18}, cyclic phase matching \\cite{Cheng16, Luo17OE, Cheng18}, and quasi-phase matching \\cite{Zappe18}. However, the potential of the LN integrated platform has not yet been fully explored for efficient nonlinear parametric processes, and current devices demonstrate only moderate efficiencies far from what LN can provide. Here, we report optical parametric generation in a high-$Q$ Z-cut LN microring resonator through exact modal phase matching. The device exhibits optical $Q$'s of $\\sim$$10^5$ for the designed cavity modes in the 1550 and 780 nm bands, and both modes are well coupled to a single bus waveguide, enabling us to conveniently measure a second-harmonic generation (SHG) efficiency of 1,500$\\%~\\rm {W^{-1}}$. In addition, by pumping into the mode in the 780 nm band, we are also able to observe difference-frequency generation (DFG) in the telecom band. Our work shows the great promise of modal-phase-matched LN microresonators for efficient optical parametric generation.\n\n\n\\begin{figure*}[t!]\n\t\\centering\\includegraphics[width=2\\columnwidth]{Fig1-min.pdf}\n\t\\caption{(a) Experimental setup for device characterization and optical parametric generation. VOA: variable optical attenuator; WDM: wavelength-division multiplexer; LPF: long-pass filter; OSA: optical spectrum analyzer. (b) Numerically simulated effective indices of the TM$_{00}$ mode at 1550 nm and the TM$_{20}$ mode at 775 nm, as functions of the top width $w$ of a straight waveguide. Other waveguide parameters are $h_1$=550 nm, $h_2$=50 nm, and $\\theta$=75$^\\circ$. (c) Scanning electron microscopy image of our LN microring. (d) Zoom-in of the bus-ring coupling region.} \\label{Fig1}\n\\end{figure*}\n\n\\section{Design and characterization}\n\nIn order to achieve modal phase matching in a microresonator, we performed photonic design with a Z-cut LN thin film, whose optical axis lies vertically, showing no anisotropy of refractive index in the device plane. To utilize the largest nonlinear term $d_{33}$, we designed for phase matching between the fundamental quasi-transverse-magnetic mode (TM$_{00}$) at 1550 nm and a high-order mode TM$_{20}$ at 775 nm. For simplicity, we preformed numerical simulation of effective indices in a straight waveguide, as a guideline for microring resonators with a relatively large radius, which is 50 $\\mu$m in our study. Fig.~\\ref{Fig1}(b) presents the simulation result by the finite element method, which shows that for a waveguide thickness of 600 nm, modal phase matching happens for TM$_{00}$ at 1550 nm and TM$_{20}$ at 775 nm when the waveguide width is about 690 nm. For a microring resonator with the same cross-section, since the Z-cut LN thin film is isotropic in the device plane, the phase matching condition is consistently satisfied at any azimuthal angle, which is expected to produce strong SHG as the phase-matched FF light travels around the cavity.\n\nOur device fabrication started from a Z-cut LN-on-insulator wafer by NANOLN, with a 600-nm-thick LN thin film sitting on a 3-$\\mu$m-thick buried oxide layer and a silicon substrate, and the process was similar to that of our previous work \\cite{Li18}. Fig.~\\ref{Fig1}(c) shows a fabricated microring resonator, coupled to a pulley waveguide \\cite{Adibi10, Lu18}, and Fig.~\\ref{Fig1}(d) gives a closer look at the coupling region. Later device characterization shows that a bus waveguide top width of $\\sim$200 nm, a gap (measured at the top surface of the LN thin film) of $\\sim$350 nm, and a coupling length of $\\sim$20 $\\mu$m are able to give good coupling for both the fundamental-frequency (FF) and the second-harmonic (SH) modes.\n\nAfter fabricating the device, we conducted experiments to characterize its linear optical properties and demonstrate nonlinear parametric generation, with the setup shown in Fig.~\\ref{Fig1}(a). We used two continuous-wave tunable lasers, one in the telecom band around 1550 nm, the other in the near-infrared (NIR) around 780 nm. Light from both lasers was combined by a 780\/1550 wavelength-division multiplexer (WDM), and launched into the on-chip bus waveguide via a lensed fiber. The bus waveguide coupled light at both wavebands into and out of the microring resonator, inside which nonlinear optical parametric processes took place. A second lensed fiber was used to collect output light from the chip, and a second 780\/1550 WDM was utilized to separate light at the two wavebands. At the 1550 port of the WDM, a long-pass filter that passes light with a wavelength over 1100 nm was used to eliminate residual NIR light, and the telecom light was further split into two paths, one to an InGaAs detector for characterization, and the other to an optical spectrum analyzer (OSA) for spectral analysis of DFG; at the 780 port, the NIR light was also split into two paths, one to a Si detector for characterization, and the other to an OSA for detection of SHG. Variable optical attenuators were employed to study power-dependent properties, and polarization controllers were used for optimal coupling of the wanted polarization.\n\n\nIn order to obtain the linear optical properties of our microring resonator, we scanned the wavelengths of both lasers and measured the transmission spectra near both 1550 and 780 nm, as shown in Fig.~\\ref{Fig2}(a) and (b). Our microring resonator exhibits a single TM mode family near 1550 nm, and the mode at 1547.10 nm, which is the FF mode for modal-phase-matched SHG, is almost critically coupled, with a coupling depth of $\\sim$99\\% and a loaded optical $Q$ of 1.4$\\times10^5$ [see Fig.~\\ref{Fig2}(c)]. On the other hand, the SH mode at 773.55 nm is under-coupled, with a coupling depth of $\\sim$83\\% and a loaded optical $Q$ of 9$\\times10^4$ [see Fig.~\\ref{Fig2}(d)]. The fiber-to-chip coupling losses are about 6.9 and 11.4 dB\/facet for the FF and SH modes, respectively. These high optical $Q$'s, together with the large nonlinearity in the designed type-0 process using $d_{33}=27$ pm\/V, indicate strong and efficient nonlinear optical interactions in phase-matched parametric generation with cavity enhancement.\n\n\n\\begin{figure}[t!]\n\t\\centering\\includegraphics[width=1\\columnwidth]{Fig2.pdf}\n\t\\caption{Transmission spectra of the LN microring near (a) 1550 nm, and (b) 780 nm. TM$_{00}$ modes around 1550 nm and TM$_{20}$ modes around 780 nm are indicated by purple and magenta arrows, respectively, with big arrows showing the phase-matched modes. (c) and (d) Detailed transmission spectra of the two phase-matched modes, with experimental data shown as solid curves and fittings shown as dashed curves. } \\label{Fig2}\n\\end{figure}\n\n\\section{Optical parametric generation}\n\nTo study SHG in the microring resonator, we launched pump power into the FF mode at 1547.10 nm, and saw strong scattering of generated NIR light from the resonator by an optical microscope, with an example shown as the inset of Fig.~\\ref{Fig3}. By varying the pump power, we obtained the power dependence of the SHG, as shown in Fig.~\\ref{Fig3}. The experimental data exhibit a quadratic relation between the generated SH power and the FF pump power, which is the signature of SHG in the low-pump-power regime. The measured conversion efficiency is 1,500$\\%~\\rm{W^{-1}}$. This efficiency is more than one order of magnitude higher than those in many other LN microresonators \\cite{Loncar14, Cheng16, Luo17OE, Xiao18, Zappe18, Huang18, Pertsch13, Jiang18, Li18}. It is even comparable with a recent study of cyclic phase matching in an X-cut microdisk exhibiting an ultra-high $Q$ of $\\sim$10$^7$ \\cite{Cheng18}, two orders of magnitude higher than that of our microring resonator, directly showing the advantage of exact modal phase matching. With future optimization of the optical $Q$'s of our device (say, by using a thicker LN film and an oxide cladding to reduce the sidewall scattering loss), we expect a further increase in the conversion efficiency.\n\nThe measured efficient SHG validated phase matching in our microring, and also indicated its capability of other parametric processes. In order to explore this, we launched power in both the SH mode, and one of the modes near the FF mode. Fig.~\\ref{Fig4} presents the recorded spectra in the telecom band. With only 6.6 $\\mu$W of on-chip power at the SH mode, we were able to convert long-wavelength telecom light coherently into shorter wavelengths through DFG. The long-wavelength pump power launched on chip was 105 $\\mu$W, and the generated power at the difference frequencies was about 480 pW, indicating a conversion rate of about -53 dB.\n\n\n\\begin{figure}[t!]\n\t\\centering\\includegraphics[width=0.9\\columnwidth]{Fig3-min.pdf}\n\t\\caption{ Power dependence of SHG, showing a quadratic relation between the SH power and the pump power. The measured conversion efficiency is 1,500$\\%~\\rm {W^{-1}}$. The inset shows an optical image of generated SH light scattered from the microring when the pump power was 440 $\\mu$W. } \\label{Fig3}\n\\end{figure}\n\n\\section{Theoretical analysis}\nIn order to acquire a better understanding of nonlinear parametric processes in our device, we can analyze the system with a model derived from the coupled mode theory \\cite{BoydBook, Johnson07}. With no pump depletion, the conversion efficiency of modal-phase-matched SHG in our doubly resonant ($\\omega_2=2\\omega_1$) optical cavity can be calculated by\n\\begin{equation}\n\\Gamma \\equiv \\frac{P_2}{P_1^2 } \\approx \\frac{64 \\gamma^2}{\\hbar \\omega_1^4} \\frac{Q_{1l}^4 Q_{2l}^2}{Q_{1e}^2 Q_{2e}}, \\label{Gamma}\n\\end{equation}\nwhere $P_1$ ($P_2$) is the launched (produced) optical power at the FF (SH); $\\hbar$ is the reduced Planck constant; $\\omega_1$ ($\\omega_2$) is the optical frequency, $Q_{1l}$ ($Q_{2l}$) is the loaded $Q$, and $Q_{1e}$ ($Q_{2e}$) is the coupling $Q$ of the FF (SH) mode; and $\\gamma$ is the single-photon coupling strength written as\n\\begin{equation}\n\\gamma=\\sqrt{\\frac{\\hbar \\omega_1^2 \\omega_2}{2\\epsilon_0 \\tilde{\\epsilon}_1^2 \\tilde{\\epsilon}_2 }}\\frac{d_{eff}\\zeta}{\\sqrt{V_{eff}}}. \\label{gamma}\n\\end{equation}\nIn Eq.~\\ref{gamma}, $\\epsilon_0$ is the vacuum permittivity, $\\tilde{\\epsilon}_1$ ($\\tilde{\\epsilon}_2$) is the relative permittivity of the nonlinear medium at the FF (SH), $d_{eff}$ is the effective nonlinear coefficient, $\\zeta$ is the mode overlap factor represented as \n\\begin{equation}\n\\zeta = \\frac{ \\int_{\\chi^{(2)}} (E_{1z}^*)^2 E_{2z} d^3x}{|\\int_{\\chi^{(2)}} |\\vec{E}_1|^2 \\vec{E}_1 d^3x|^{\\frac{2}{3}} |\\int_{\\chi^{(2)}}  |\\vec{E}_2|^2 \\vec{E}_2 d^3x|^{\\frac{1}{3}}}, \\label{zeta}\n\\end{equation}\nand $V_{eff} \\equiv (V_{1}^2 V_{2})^{\\frac{1}{3}}$ is the effective mode volume, with\n\\begin{equation}\nV_{i} = \\frac{ (\\int_{ all} \\epsilon_i |\\vec{E}_i|^2 d^3x)^3 }{|\\int_{\\chi^{(2)}} \\epsilon_i^{\\frac{3}{2}} |\\vec{E}_i|^2 \\vec{E}_i d^3x|^2},(i=1,2),\n\\end{equation}\nwhere  $\\int_{\\chi^{(2)}}$ and $\\int_{all}$ denote three-dimensional integration over the nonlinear medium and all space, respectively, $\\epsilon_1(\\vec{r})$ [$\\epsilon_2(\\vec{r})$] is the relative permittivity at the FF (SH), and $E_{1z}$ ($E_{2z}$) is the z-component of $\\vec{E}_1(\\vec{r})$ [$\\vec{E}_2(\\vec{r})$], the electric field of the FF (SH) cavity mode.\n\nWith the equations above, the SHG efficiency in our LN microring is calculated to be $\\Gamma \\approx 30,000\\%~\\rm{W^{-1}}$. Thus, there is more than one order of magnitude difference between the theoretical prediction and our experimental result. The main reason for this discrepancy is likely non-uniformity of the microring at different azimuthal angles. By simulation, a change of 1 nm in the waveguide width, for example, will lead to a shift of $\\sim-3$ nm in the phase-matched pump wavelength of SHG. Considering the small linewidths of our cavity modes, which are only 11 pm for the FF mode and 9 pm for the SH mode, the phase-matching window is easily shifted out of the cavity resonances due to fabrication imperfections. Besides, there is also a certain level of non-uniformity in the thickness of the LN thin film, giving more uncertainty to the uniformity of phase matching. We believe these technical issues will be addressed by optimized fabrication techniques in the near future, and the conversion efficiency can be significantly improved.\n\n\\begin{figure}[t!]\n\t\\centering\\includegraphics[width=0.9\\columnwidth]{Fig4.pdf}\n\t\\caption{Recorded DFG spectra, when pumping at the SH mode in the NIR and one of the five nearest modes with longer wavelengths than the FF mode in the telecom band. Pump power at the SH mode was 6.6 $\\mu$W.} \\label{Fig4}\n\\end{figure}\n\n\\section{Conclusion}\n\nIn conclusion, we have demonstrated optical parametric generation in a LN microring resonator with modal phase matching. We have used a single bus waveguide to conveniently couple the FF and SH modes, both showing coupling depths over 80\\% and exhibiting loaded optical $Q$'s around $10^5$, resulting in a measured conversion efficiency of 1,500$\\%~\\rm{W^{-1}}$ for SHG. In addition, we have also observed DFG in the telecom band. Our work represents an important step towards ultra-highly efficient optical parametric generation in photonic circuits based on the LN integrated platform.\n\n\n\\section*{Acknowledgments}\nThe authors thank Xiyuan Lu at National Institute of Standards and Technology for helpful discussions. This work was supported in part by the National Science Foundation under Grant No.~ECCS-1641099, ECCS-1509749, and ECCS-1810169, and by the Defense Threat Reduction Agency under the Grant No.~HDTRA1827912. The project or effort depicted was or is sponsored by the Department of the Defense, Defense Threat Reduction Agency. The content of the information does not necessarily reflect the position or the policy of the federal government, and no official endorsement should be inferred. This work was performed in part at the Cornell NanoScale Facility, a member of the National Nanotechnology Coordinated Infrastructure (National Science Foundation, ECCS-1542081), and at the Cornell Center for Materials Research (National Science Foundation, DMR-1719875).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Markov regime switching model has been a popular framework for empirical work in economics and finance. Following the seminal contribution by \\citet{hamilton89em}, it has been used in numerous empirical studies to model, for example, the business cycle \\citep{hamilton05stlouis, morleypiger12restat}, stock market volatility \\citep{hamiltonsusmel94joe}, international equity markets \\citep{angbekaert02rfs, okimoto08jfqa}, monetary policy \\citep{schorfheide05red, simszha06aer, bianchi13restud}, and economic growth \\citep{kahnrich07jme}. Comprehensive theoretical accounts and surveys of applications are provided by \\citet{hamilton08palgrave, hamilton16hdbk} and \\citet{angtimmermann12annual}.\n\nThe number of regimes is an important parameter in applications of Markov regime switching models. Despite its importance, however, testing for the number of regimes in Markov regime switching models has been an unsolved problem because the standard asymptotic analysis of the likelihood ratio test statistic (LRTS) breaks down because of problems such as unidentifiable parameters, the true parameter being on the boundary of the parameter space, and the degeneracy of the Fisher information matrix. Testing the number of regimes for Markov regime switching models with normal density, which are popular in empirical applications, poses a further difficulty because normal density has the undesirable mathematical property that the second-order derivative with respect to the mean parameter is linearly dependent on the first derivative with respect to the variance parameter, leading to further singularity.\n\nThis paper proposes the likelihood ratio test of the null hypothesis of $M_0$ regimes against the alternative hypothesis of $M_0 + 1$ regimes for any $M_0\\geq 1$ and derives its asymptotic distribution. To the best of our knowledge, the asymptotic distribution of the LRTS has not been derived for testing the null hypothesis of $M_0$ regimes with $M_0 \\geq 2$. To test the null hypothesis of no regime switching, namely $M_0=1$, \\citet{hansen92jae} derives an upper bound of the asymptotic distribution of the LRTS, and \\citet{garcia98ier} also studies this problem. \\citet{carrasco14em} propose an information matrix-type test for parameter constancy in general dynamic models including regime switching models. \\citet{chowhite07em} derive the asymptotic distribution of the quasi-LRTS for testing the single regime against two regimes by rewriting the model as a two-component mixture model, thereby ignoring the temporal dependence of the regimes.\\footnote{\\citet{cartersteigerwald12em} show that ignoring temporal dependence may render the quasi-maximum likelihood estimator inconsistent.} \\citet{quzhuo17wp} extend the analysis of \\citet{chowhite07em} and derive the asymptotic distribution of the LRTS that properly takes into account the temporal dependence of the regimes under some restrictions on the transition probabilities of latent regimes. \\citet{marmer08empirical} and \\citet{dufour17emreviews} develop tests for the null hypothesis of no regime switching by using different approaches from the LRTS. The studies discussed above focus on testing the single regime against two regimes. To the best of our knowledge, however, the asymptotic distribution of the LRTS for testing the null hypothesis of $M_0$ regimes with $M_0 \\geq 2$ remains unknown.\n\nSeveral papers in the literature consider tests when some parameters are not identified under the null hypothesis. These include \\citet{davies77bm, davies87bm}, \\citet{andrewsploberger94em, andrewsploberger95as}, \\citet{hansen96em}, \\citet{andrews01em}, and \\citet{liushao03as}, among others. Estimation and testing with a degenerate Fisher information matrix are investigated in an iid setting by \\citet{chesher84em}, \\citet{leechesher86joe}, \\citet{rotnitzky00bernoulli}, and \\citet{gukoenkervolgushev17wp}, among others. \\citet{chen14joe} examine uniform inference on the mixing probability in mixture models.\n\nTo facilitate the analysis herein, we develop a version of Le Cam's differentiable in quadratic mean (DQM) expansion that expands the likelihood ratio under the loss of identifiability, while adopting the reparameterization and higher-order expansion of \\citet{kasaharashimotsu15jasa}. In an iid setting, \\citet{liushao03as} develop a DQM expansion under the loss of identifiability in terms of the generalized score function. We extend \\citet{liushao03as} to accommodate dependent and heterogeneous data as well as modify them to fit our context of parametric regime switching models. Using a DQM-type expansion has an advantage over the ``classical'' approach based on the Taylor expansion up to the Hessian term because deriving a higher-order expansion becomes tedious as the order of expansion increases in a Markov regime switching model. Furthermore, regime switching models with normal components are not covered by \\citet{liushao03as} because their Theorem 4.1 assumes that the generalized score function is obtained by expanding the likelihood ratio twice, whereas our Section \\ref{subsec:hetero_normal} shows that the score function is a function of the fourth derivative of the likelihood ratio in the normal case.\n\nOur approach follows \\citet{dmr04as} [DMR hereafter], who derive the asymptotic distribution of the maximum likelihood estimator (MLE) of regime switching models. We express the higher-order derivatives of the period density ratios in terms of the conditional expectation of the derivatives of the period \\textit{complete-data} log-density, i.e., the log-density when the state variable is observable, by applying the missing information principle \\citep[][]{woodbury71biometrics,louis82jrssb} and extending the analysis of DMR. We then show that these derivatives of the period density ratios can be approximated by a stationary, ergodic, and square integrable martingale difference sequence by conditioning on the infinite past, and this approximation is shown to satisfy the regularity conditions for our DQM expansion.\n\nWe first derive the asymptotic null distribution of the LRTS for testing $H_0: M=1$ against $H_A: M=2$. When the regime-specific density is not normal, the log-likelihood function is locally approximated by a quadratic function of the \\textit{second-order} polynomials of the reparameterized parameters. When the density is normal, the degree of deficiency of the Fisher information matrix and required order of expansion depend on the value of the unidentified parameter; in particular, when the latent regime variables are serially uncorrelated, the model reduces to a finite mixture normal model in which the fourth-order DQM expansion is necessary to derive a quadratic approximation of the log-likelihood function. We expand the log-likelihood with respect to the judiciously chosen polynomials of the reparameterized parameters---which involves the \\textit{fourth-order} polynomials---to obtain a uniform approximation of the log-likelihood function in quadratic form and derive the asymptotic null distribution of the LRTS by maximizing the quadratic form under a set of cone constraints building on the results of \\citet{andrews99em,andrews01em}.\n\nTo derive the asymptotic null distribution of the LRTS for testing $H_0: M=M_0$ against $H_A: M=M_0+1$ for $M_0\\geq 2$, we partition a set of parameters that describes the true null model in the alternative model into $M_0$ subsets, each of which corresponds to a specific way of generating the null model. We show that the asymptotic distribution of the LRTS is characterized by the maximum of the $M_0$ random variables, each of which represents the LRTS for testing each of the $M_0$ subsets.\n\nWe also derive the asymptotic distribution of the LRTS under local alternatives. \\citet{carrasco14em} show that the contiguous local alternatives of their tests are of order $n^{-1\/4}$, where $n$ is the sample size. In a related problem of testing the number of components in finite mixture normal regression models, \\citet{kasaharashimotsu15jasa} show that the contiguous local alternatives are of order $n^{-1\/8}$ \\citep[see also][]{chenli09as, chenlifu12jasa, honguyen16as}. We show that the value of the unidentified parameter affects the convergence rate of the contiguous local alternatives. When the regime-specific density is normal, some contiguous local alternatives are of the order $n^{-1\/8}$, and the LRT is shown to have non-trivial power against them. The tests of \\citet{carrasco14em} do not have power against such alternatives, whereas the test of \\citet{quzhuo17wp} rules out such alternatives because of their restriction on the parameter space.\n\nThe asymptotic validity of the parametric bootstrap is also established both under the null hypothesis and under local alternatives. The simulations show that our bootstrap LRT has good finite sample properties. Our results also imply that the bootstrap LRT is valid for testing the number of hidden states in hidden Markov models because this paper's model includes the hidden Markov model as a special case. Although several papers have analyzed the asymptotic property of the MLE of the hidden Markov model,\\footnote{See, for example, \\citet{leroux92spa}, \\citet{francq98stat}, \\citet{krishnamurthy98jtsa}, \\citet{brr98as}, \\citet{jp99as}, \\citet{legrandmevel00math}, and \\citet{doucmatias01bernouiil}.} the asymptotic distribution of the LRTS for testing the number of hidden states has been an open question.%\n\\footnote{\\citet{gassiat00esaim} show that the LRTS for testing $H_0: M=1$ against $H_A: M=2$ diverges when state-specific densities have \\emph{known and distinct} parameter values. \\citet{dannemann08cjstat} analyze the modified quasi-LRTS for testing the null of two states against three.}\n\nThe remainder of this paper is organized as follows. After introducing the notation and assumptions in Section 2, we discuss the degeneracy of the Fisher information matrix and loss of identifiability in regime switching models in Section 3. Section 4 establishes the DQM-type expansion. Section 5 presents the uniform convergence of the derivatives of the density ratios. Sections 6 and 7 derive the asymptotic null distribution of the LRTS. Section 8 derives the asymptotic distribution under local alternatives. Section 9 establishes the consistency of the parametric bootstrap. Section 10 reports the results from the simulations and an empirical application, using U.S. GDP per capita quarterly growth rate data. Section 11 collects the proofs and auxiliary results.\n\n\\section{Notation and assumptions}\n\nLet $:=$ denote ``equals by definition.'' Let $\\Rightarrow$ denote the weak convergence of a sequence of stochastic processes indexed by $\\pi$ for some space $\\Pi$. For a matrix $B$, let $\\lambda_{\\min}(B) $ and $\\lambda_{\\max}(B)$ be the smallest and largest eigenvalues of $B$, respectively. For a $k$-dimensional vector $x = (x_1,\\ldots,x_k)'$ and a matrix $B$, define $|x| := \\sqrt{x'x}$ and $|B| := \\sqrt{\\lambda_{\\max}(B'B)}$. For a $k\\times 1$ vector $a=(a_1,\\ldots,a_k)'$ and a function $f(a)$, let $\\nabla_{a} f(a):=(\\partial f(a)\/\\partial a_{1},\\ldots,\\partial f(a)\/\\partial a_{k})'$, and let $\\nabla_{a}^j f(a)$ denote a collection of derivatives of the form $(\\partial^j\/\\partial a_{i_1}\\partial a_{i_2} \\ldots \\partial a_{i_j})f(a)$. Let $\\mathbb{I}\\{A\\}$ denote an indicator function that takes the value 1 when $A$ is true and 0 otherwise. $\\mathcal{C}$ denotes a generic non-negative finite constant whose value may change from one expression to another. Let $a \\vee b :=\\max\\{a,b\\}$ and $a \\wedge b :=\\min\\{a,b\\}$. Let $\\lfloor x \\rfloor$ denote the largest integer less than or equal to $x$, and define $(x)_+ := \\max\\{x,0\\}$. Given a sequence $\\{f_k\\}_{k=1}^n$, let $\\nu_n(f_k) := n^{-1\/2} \\sum_{k=1}^n [f_k - \\mathbb{E}_{\\vartheta^*}(f_k)]$. For a sequence $X_{n\\varepsilon}$ indexed by $n=1,2,\\ldots$ and $\\varepsilon$, we write $X_{n\\varepsilon} = O_{p\\varepsilon}(a_n)$ if, for any $\\delta>0$, there exist $\\varepsilon>0$ and $M, n_0 <\\infty$ such that $\\mathbb{P}(|X_{n\\varepsilon}\/a_n| \\leq M) \\geq 1- \\delta$ for all $n > n_0$, and we write $X_{n\\varepsilon} = o_{p\\varepsilon}(a_n)$ if, for any $\\delta_1,\\delta_2>0$, there exist $\\varepsilon>0$ and $n_0$ such that $\\mathbb{P}(|X_{n\\varepsilon}\/a_n| \\leq \\delta_1) \\geq 1- \\delta_2$ for all $n > n_0$. Loosely speaking, $X_{n\\varepsilon} = O_{p\\varepsilon}(a_n)$ and $X_{n\\varepsilon} = o_{p\\varepsilon}(a_n)$ mean that $X_{n\\varepsilon} = O_{p}(a_n)$ and $X_{n\\varepsilon} = o_{p}(a_n)$ when $\\varepsilon$ is sufficiently small, respectively. All limits are taken as $n \\to \\infty$ unless stated otherwise. The proofs of all the propositions and lemmas are presented in the appendix.\n \nConsider the Markov regime switching process defined by a discrete-time stochastic process $\\{(X_k,Y_k,W_k)\\}$, where $(X_k,Y_k,W_k)$ takes values in a set $\\mathcal{X}_M\\times \\mathcal{Y}\\times \\mathcal{W}$ with $\\mathcal{Y}\\subset \\mathbb{R}^{q_y}$ and $\\mathcal{W}\\subset \\mathbb{R}^{q_w}$, and let $\\mathcal{B}(\\mathcal{X}_M\\times\\mathcal{Y}\\times\\mathcal{W})$ denote the associated Borel $\\sigma$-field. For a stochastic process $\\{Z_k\\}$ and $a<b$, define ${\\bf Z}_{a}^b:=(Z_a,Z_{a+1},\\ldots,Z_b)$. Denote $\\overline {\\bf Y}_{k-1}:=(Y_{k-1},\\ldots,Y_{k-s})$ for a fixed integer $s$ and $\\overline {\\bf Y}^b_{a}:=(\\overline {\\bf Y}_{a},\\overline {\\bf Y}_{a+1},\\ldots,\\overline {\\bf Y}_{b})$. Here, $Y_k$ is an observable variable, $X_k$ is an unobservable state variable, $\\overline {\\bf Y}_{k-1}$ is the lagged $Y_k$'s used as a covariate, and $W_k$ is a weakly exogenous covariate. DMR's model does not include $W_k$. \n\\begin{assumption} \\label{assn_a1}\n(a) $\\{X_k\\}_{k=0}^\\infty$ is a first-order Markov chain with the state space $\\mathcal{X}_M:=\\{1,2,\\ldots,M\\}$. (b) For each $k\\geq 1$, $X_k$ is independent of $({\\bf X}_{0}^{k-2},\\overline {\\bf Y}_{0}^{k-1},{\\bf W}_{0}^\\infty)$ given $X_{k-1}$. (c) For each $k \\geq 1$, $Y_k$ is conditionally independent of $( {\\bf Y}_{-s+1}^{k-s-1}, {\\bf X}_{0}^{k-1}, {\\bf W}_{0}^{k-1},{\\bf W}_{k+1}^{\\infty})$ given $(\\overline {\\bf Y}_{k-1},X_k,W_k)$. (d) $ {\\bf W}_{1}^{\\infty}$ is conditionally independent of $(\\overline {\\bf Y}_{0},X_{0})$ given $W_0$.\\footnote{Assumptions \\ref{assn_a1}(a)--(d) imply that $W_k$ is conditionally independent of $({\\bf X}_{0}^{k-1},\\overline{\\bf Y}_{0}^{k-1})$ given ${\\bf W}_{0}^{k-1}$.} (e) $\\{(X_k,Y_k,W_k)\\}_{k=0}^{\\infty}$ is a strictly stationary ergodic process.\n\\end{assumption} \nWhen $W_k$ is absent, DMR provide a sufficient condition for the ergodicity of $(X_k,Y_k)$ in their Assumption (A2). We assume the ergodicity of $(X_k,Y_k,W_k)$ for brevity.\n\nThe unobservable Markov chain $\\{X_k\\}$ is called the \\textit{regime}. The integer $M$ represents the number of regimes specified in the model. The parameter $\\vartheta_M=(\\vartheta_{M,y}',\\vartheta_{M,x}')'$ belongs to $\\Theta_M= \\Theta_{M,y}\\times \\Theta_{M,x}$, a compact subset of $\\mathbb{R}^{q_M}$. $\\vartheta_{M,x}$ contains the parameter of the transition probability of $X_k$, which we denote by $q_{\\vartheta_{M,x}}(x_{k-1},x_k):= \\mathbb{P}(X_{k}=x_k|X_{k-1}=x_{k-1})$. Let $p_{ij}:= q_{\\vartheta_{M,x}}(i,j)$ for $i=1,\\ldots,M$ and $j=1,\\ldots,M-1$, and $q_{\\vartheta_{M,x}}(i,M)$ is determined by $q_{\\vartheta_{M,x}}(i,M) = 1-\\sum_{j = 1}^{M - 1} p_{ij}$. $\\vartheta_{M,y}=(\\theta_1',\\ldots,\\theta_M',\\gamma')'$ contains the parameter of the conditional density of $Y_k$ given $(\\overline {\\bf Y}_{k-1},X_k,W_k)$, which is given by $g_{\\vartheta_{M,y}}(y_k|\\overline{\\bf y}_{k-1},x_k,w_k) := \\sum_{j \\in \\mathcal{X}_M} \\mathbb{I}\\{x_k=j\\}f(y_k|\\overline{\\bf{y}}_{k-1},w_k;\\gamma,\\theta_j)$. Here, $\\gamma$ is the structural parameter that does not vary across regimes, $\\theta_j$ is the regime-specific parameter that varies across regimes, and $f(y_k|\\overline{\\bf{y}}_{k-1},w_k;\\gamma,\\theta_j)$ is the conditional density of $y_k$ given $(\\overline{\\bf{y}}_{k-1},w_k)$ when $x_k=j$. \nLet \n\\begin{align*}\np_{\\vartheta} (y_k,x_k| \\overline{\\bf{y}}_{k-1},x_{k-1},w_k) &:= q_{\\vartheta_x}(x_{k-1},x_k)g_{\\vartheta_y}(y_k|\\overline{\\bf{y}}_{k-1},x_k,w_k) \\\\\n & = q_{\\vartheta_x}(x_{k-1},x_k)\\sum_{j \\in \\mathcal{X}_M} \\mathbb{I}\\{x_k=j\\}f(y_k|\\overline{\\bf{y}}_{k-1},w_k;\\gamma,\\theta_j).\n\\end{align*} \nWe assume $\\Theta_{M,y}=\\Theta_{\\theta} \\times \\cdots \\times \\Theta_\\theta\\times \\Theta_\\gamma$, and the true parameter value is denoted by $\\vartheta_M^*$. \n\nWe make the following assumptions that correspond to (A1)--(A3) in DMR.\n\\begin{assumption}\\label{assn_a2}\n(a) $0<\\sigma_{-}:=\\inf_{\\vartheta_{M,x}\\in\\Theta_{M,x}} \\min_{x,x'\\in \\mathcal{X}_M} q_{\\vartheta_{M,x}}(x,x')$ and \\\\$\\sigma_+:=\\sup_{\\vartheta_{M,x}\\in\\Theta_{M,x}}\\max_{x,x'\\in\\mathcal{X}_M}q_{\\vartheta_{M,x}}(x,x')<1$ for each $M$. (b) For all $y'\\in \\mathcal{Y}$, $\\overline y\\in \\mathcal{Y}^s$, and $w\\in \\mathcal{W}$, $0<\\inf_{\\vartheta_y\\in\\Theta_{M,y}} \\sum_{x\\in \\mathcal{X}_M} g_{\\vartheta_{M,y}}(y'|\\overline y,x,w)$ and $\\sup_{\\vartheta_{M,y}\\in\\Theta_{M,y}} \\sum_{x\\in \\mathcal{X}_M} g_{\\vartheta_{M,y}}(y'|\\overline y,x,w) <\\infty$. (c) $b_+:=\\sup_{\\vartheta_{M,y}\\in\\Theta_y} \\sup_{\\overline {\\bf y}_0,y_1,w,x} g_{\\vartheta_{M,y}}(y_1|\\overline {\\bf y}_0,x,w)<\\infty$ and $\\mathbb{E}_{\\vartheta^*}(|\\log b_{-}(\\overline{\\bf{Y}}_0^1,W_1)|)<\\infty$, where $b_{-}(\\overline{\\bf y}_0^1,w_1):=\\inf_{\\vartheta_{M,y}\\in\\Theta_{M,y}} \\sum_{x\\in\\mathcal{X}_M} g_{\\vartheta_{M,y}}(y_1|\\overline{\\bf y}_0,x,w_1)$.\n \\end{assumption} \n\nAs discussed on p. 2260 of DMR, Assumption \\ref{assn_a2}(a) implies that the Markov chain $\\{X_k\\}$ has a unique invariant distribution and is uniformly ergodic for all $\\theta_{M,x} \\in \\Theta_{M,x}$.\\footnote{Assumptions \\ref{assn_a1}(c) and \\ref{assn_a2}(a) are also employed in DMR. As discussed in \\citet{kasaharashimotsu17hamilton}, these assumptions together rule out models in which the conditional density $Y_k$ depends on both current and lagged regimes. \\citet{kasaharashimotsu17hamilton} show the asymptotic normality of the MLE while relaxing Assumption \\ref{assn_a2}(a) to allow for $\\inf_{\\vartheta_{M,x}\\in\\Theta_{M,x}} \\min_{x,x'\\in \\mathcal{X}_M} q_{\\vartheta_{M,x}}(x,x')=0$. It is possible to derive the asymptotic distribution of the LRT under similar assumptions to \\citet{kasaharashimotsu17hamilton}, albeit with a tedious derivation.} For notational brevity, we drop the subscript $M$ from $\\mathcal{X}_M$, $\\vartheta_{M}$, $\\Theta_{M}$, etc., unless it is important to clarify the specific value of $M$. Assumptions \\ref{assn_a1}(b) and (c) imply that $\\{Z_k\\}_{k=0}^{\\infty}:=\\{(X_k,\\overline{\\bf Y}_{k})\\}_{k=0}^{\\infty}$ is a Markov chain on $\\mathcal{Z}:=\\mathcal{X}\\times\\mathcal{Y}^s$ given $\\{W_k\\}_{k=0}^{\\infty}$, and $Z_k$ is conditionally independent of $({\\bf Z}_0^{k-2},{\\bf W}_{0}^{k-1},{\\bf W}_{k+1}^{\\infty})$ given $(Z_{k-1},W_k)$. Consequently, Lemma 1, Corollary 1, and Lemma 9 of DMR go through even in the presence of $\\{W_k\\}_{k=0}^{\\infty}$. Because $\\{(Z_k,W_k)\\}_{k=0}^\\infty$ is stationary, we can and will extend $\\{(Z_k,W_k)\\}_{k=0}^\\infty$ to a stationary process $\\{(Z_k,W_k)\\}_{k=-\\infty}^\\infty$ with doubly infinite time. We denote the probability measure and associated expectation of $\\{(Z_k,W_k)\\}_{k=\\infty}^\\infty$ under stationarity by $\\mathbb{P}_\\vartheta$ and $\\mathbb{E}_\\vartheta$, respectively.\\footnote{DMR use $\\overline{\\mathbb{P}}_\\vartheta$ and $\\overline{\\mathbb{E}}_\\vartheta$ to denote probability and expectation under stationarity on $\\{Z_k\\}_{k=\\infty}^\\infty$, because their Section 7 deals with the case when $Z_0$ is drawn from an arbitrary distribution. Because we assume $\\{(Z_k,W_k)\\}_{k=\\infty}^\\infty$ is stationary, we use notations such as $\\mathbb{P}_\\vartheta$ and $\\mathbb{E}_\\vartheta$ without an overline for simplicity.}\n\nUnder Assumptions \\ref{assn_a1}(a)--(d), the density of ${\\bf Y}_{1}^n$ given $X_0=x_0$, $\\overline{\\bf Y}_{0}$ and ${\\bf W}_{0}^n$ is given by\n\\begin{equation} \\label{cond_density}\np_{\\vartheta_{M}}({\\bf Y}_1^n| \\overline{\\bf Y}_{0},{\\bf W}_{0}^n,x_0) = \\sum_{{\\bf x}_1^n\\in \\mathcal{X}_{M}^n}\\prod_{k=1}^n p_{\\vartheta_M}(Y_k,x_k|\\overline{\\bf Y}_{k-1},x_{k-1},W_k).\n\\end{equation}\nDefine the conditional log-likelihood function and stationary log-likelihood function as\n\\begin{align*}\n\\ell_n(\\vartheta,x_0) & := \\log p_\\vartheta({\\bf Y}_1^n| \\overline{\\bf Y}_{0},{\\bf W}_{0}^n,x_0)=\\sum_{k=1}^n \\log p_\\vartheta (Y_k| \\overline{\\bf Y}_{0}^{k-1},{\\bf W}_{0}^{k},x_0) ,\\\\\n\\ell_n(\\vartheta) & := \\log p_\\vartheta({\\bf Y}_1^n| \\overline{\\bf Y}_{0},{\\bf W}_{0}^n)=\\sum_{k=1}^n \\log p_\\vartheta (Y_k| \\overline{\\bf Y}_{0}^{k-1},{\\bf W}_{0}^{k}), \n\\end{align*}\nwhere we use the fact that $p_\\vartheta (Y_k| \\overline{\\bf Y}_{0}^{k-1},{\\bf W}_{0}^{n},x_0) = p_\\vartheta (Y_k| \\overline{\\bf Y}_{0}^{k-1},{\\bf W}_{0}^{k},x_0)$ and $p_\\vartheta (Y_k| \\overline{\\bf Y}_{0}^{k-1},{\\bf W}_{0}^{n}) = p_\\vartheta (Y_k| \\overline{\\bf Y}_{0}^{k-1},{\\bf W}_{0}^{k})$, which follows from Assumption \\ref{assn_a1}. Note that\n\\begin{align*}\n& p_\\vartheta (Y_k| \\overline{\\bf Y}_{0}^{k-1},{\\bf W}_{0}^{k},x_0) - p_\\vartheta (Y_k| \\overline{\\bf Y}_{0}^{k-1},{\\bf W}_{0}^{k}) \\\\\n& = \\sum_{(x_{k-1},x_k)\\in\\mathcal{X}^2} p_\\vartheta(Y_k,x_k|\\overline{\\bf Y}_{k-1},x_{k-1},W_k) \\times \\left(\\mathbb{P}_\\vartheta(x_{k-1}|\\overline{\\bf Y}_{0}^{k-1},{\\bf W}_{0}^{k-1},x_0)- \\mathbb{P}_\\vartheta(x_{k-1}|\\overline{\\bf Y}_{0}^{k-1},{\\bf W}_{0}^{k-1}) \\right), \n\\end{align*}\nand $\\mathbb{P}_\\vartheta(x_{k-1}|\\overline{\\bf Y}_{0}^{k-1},{\\bf W}_{0}^{k-1}) = \\sum_{x_0 \\in \\mathcal{X}}\\mathbb{P}_\\vartheta(x_{k-1}|\\overline{\\bf Y}_{0}^{k-1},{\\bf W}_{0}^{k-1},x_0)\\mathbb{P}_\\vartheta(x_0|\\overline{\\bf Y}_{0}^{k-1},{\\bf W}_{0}^{k-1})$. Let $\\rho:=1-\\sigma_-\/\\sigma_+\\in [0,1)$. Lemma \\ref{x_diff}(a) in the appendix shows that, for all probability measures $\\mu_1$ and $\\mu_2$ on $\\mathcal{B}(\\mathcal{X})$ and all $(\\overline{\\bf y}_{0}^{k-1},{\\bf w}_{0}^{k-1})$,\n\\begin{equation}\\label{DMR-C1}\n\\sup_{A}\\left| \\sum_{x_0\\in \\mathcal{X}} \\mathbb{P}_\\vartheta(X_{k-1}\\in A|\\overline{\\bf y}_{0}^{k-1},{\\bf w}_{0}^{k-1},x_0)\\mu_1(x_0)- \\sum_{x_0\\in\\mathcal{X}} \\mathbb{P}_\\vartheta(X_{k-1}\\in A|\\overline{\\bf y}_{0}^{k-1},{\\bf w}_{0}^{k-1},x_0)\\mu_2(x_0) \\right| \\leq \\rho^{k-1}.\n\\end{equation} \nConsequently, $ p_\\vartheta (Y_k| \\overline{\\bf Y}_{0}^{k-1},{\\bf W}_{0}^{k-1},x_0) - p_\\vartheta (Y_k| \\overline{\\bf Y}_{0}^{k-1},{\\bf W}_{0}^{k-1})$ goes to zero at an exponential rate as $k\\rightarrow \\infty$. Therefore, as shown in the following proposition, the difference between $\\ell_n(\\theta,x_0)$ and $\\ell_n(\\theta)$ is bounded by a deterministic constant, and the maximum of $\\ell_n(\\vartheta,x_0)$ and the maximum of $\\ell_n(\\vartheta)$ are asymptotically equivalent.\n\\begin{proposition}\\label{uconlike}\nUnder Assumptions \\ref{assn_a1} and \\ref{assn_a2}, for all $x_0\\in \\mathcal{X}$,\n\\[\n\\sup_{\\vartheta\\in \\Theta} |\\ell_n(\\vartheta,x_0)- \\ell_n(\\vartheta)| \\leq 1\/(1-\\rho)^2 \\quad \\mathbb{P}_{\\theta^*}\\text{-a.s}. \n\\]\n\\end{proposition} \nAs discussed on p. 2263 of DMR, the stationary density $p_\\vartheta (Y_k| \\overline{\\bf Y}_{0}^{k-1},{\\bf W}_{0}^{k})$ is not available in closed form for some models with autoregression. For this reason, we consider the log-likelihood function when the initial distribution of $X_0$ follows some distribution \\\\$\\xi_M \\in \\Xi_M:=\\{\\xi(x_0)_{x_0 \\in \\mathcal{X}_M} :\\xi(x_0)\\geq 0$ and $\\sum_{x_0 \\in \\mathcal{X}_M}\\xi(x_0)=1\\}$.\n\nDefine the MLE, $\\hat\\vartheta_{M,\\xi_M}$, by the maximizer of the log-likelihood:\n\\begin{equation} \\label{density}\n\\ell_n(\\vartheta_M,\\xi_M):=\\log \\left( \\sum_{x_0=1}^M p_{\\vartheta_M}({\\bf Y}_1^n| \\overline{\\bf Y}_{0},{\\bf W}_{0}^n,x_0) \\xi_M(x_0) \\right),\n\\end{equation}\nwhere $p_{\\vartheta_{M}}({\\bf Y}_1^n| \\overline{\\bf Y}_{0},{\\bf W}_{0}^n,x_0)$ is given in (\\ref{cond_density}). We define the \\textit{number of regimes} by the smallest number $M$ such that the data density admits the representation (\\ref{density}). Our objective is to test $H_0: M=M_0$ against $H_A: M=M_0+1$. \n\n\\section{Degeneracy of the Fisher information matrix and non-identifiability under the null hypothesis} \n \nConsider testing $H_0:M=1$ against $H_A:M=2$ in a two-regime model. The null hypothesis can be written as $H_0:\\theta_1^*=\\theta_2^*$.\\footnote{The null hypothesis of $H_0: M=1$ also holds when $(p_{11},p_{22}) = (1,0)$. We impose Assumption \\ref{assn_a2}(a) to bound $p_{jj}$ away from 0 and 1 because the log-likelihood function may become unbounded as $p_{11}$ or $p_{22}$ tends to 1 in view of \\citet{gassiat00esaim}.} When $\\theta_1=\\theta_2$, the parameter $\\vartheta_{2,x}$ is not identified because $Y_k$ has the same distribution across regimes. Furthermore, Section \\ref{sec: testing-1} shows that when $\\theta_1=\\theta_2$, the scores with respect to $\\theta_1$ and $\\theta_2$ are linearly dependent, so that the Fisher information matrix is degenerate.\n\nThe log-likelihood function of Markov regime switching models with normal density has further degeneracy. In a two-regime model where $Y_k$ in the $j$-th regime follows $N(\\mu_j,\\sigma_j^2)$, the model reduces to a heteroscedastic normal mixture model when the $X_k$'s are serially uncorrelated, i.e., $p_{11}=1-p_{22}$. \\citet{kasaharashimotsu15jasa} show that in a heteroscedastic normal mixture model, the first and second derivatives of the log-likelihood function are linearly dependent and the score function is a function of the fourth-order derivative. Consequently, one needs to expand the log-likelihood function four times to derive the score function.\n\n\n\\section{Quadratic expansion under the loss of identifiability}\\label{sec:expansion}\n\nWhen testing the number of regimes by the LRT, a part of $\\vartheta$ is not identified under the null hypothesis. Let $\\pi$ denote the part of $\\vartheta$ that is not identified under the null, and split $\\vartheta$ as $\\vartheta = (\\psi',\\pi')'$. For example, in testing $H_0: M=1$ against $H_A: M=2$, we have $\\psi=\\vartheta_{2,y}$ and $\\pi=\\vartheta_{2,x}$. We denote the conditional log-likelihood function as $\\ell_n(\\psi,\\pi, x_0) := \\ell_n(\\vartheta, x_0)$ and use $p_\\vartheta$ and $p_{\\psi\\pi}$ interchangeably. \n\nDenote the true parameter value of $\\psi$ by $\\psi^*$, and denote the set of $(\\psi,\\pi)$ corresponding to the null hypothesis by $\\Gamma^*= \\{(\\psi,\\pi)\\in\\Theta: \\psi=\\psi^*\\}$. \nLet $t_\\vartheta$ be a continuous function of $\\vartheta$ such that $t_{\\vartheta}=0$ if and only if $\\psi=\\psi^*$. For $\\varepsilon>0$, define the neighborhood of $\\Gamma^*$ by\n\\[\n\\mathcal{N}_{\\varepsilon} := \\{ \\vartheta \\in \\Theta: |t_\\vartheta|< \\varepsilon\\}.\n\\]\nWhen the MLE is consistent, the asymptotic distribution of the LRTS is determined by the local properties of the likelihood functions in $\\mathcal{N}_{\\varepsilon}$.\n\nWe establish a general quadratic expansion of the log-likelihood function $\\ell_n(\\psi,\\pi,\\xi):= \\ell_n(\\vartheta,\\xi)$ defined in (\\ref{density}) around $\\ell_n(\\psi^*,\\pi,\\xi)$ that expresses $\\ell_n(\\psi,\\pi,\\xi)-\\ell_n(\\psi^*,\\pi,\\xi)$ as a quadratic function of $t_\\vartheta$. Once we derive a quadratic expansion, the asymptotic distribution of the LRTS can be characterized by taking its supremum with respect to $t_\\vartheta$ under an appropriate constraint and using the results of \\citet{andrews99em,andrews01em}. \n \nDenote the conditional density ratio by\n\\begin{equation} \\label{density_ratio}\nl_{\\vartheta k x_0} := \\frac{ p_{\\psi\\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1},{\\bf W}_{0}^k,x_0)}{ p_{\\psi^*\\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1},{\\bf W}_{0}^k,x_0)},\n\\end{equation} \nso that $\\ell_n(\\psi,\\pi, x_0) - \\ell_n(\\psi^*,\\pi, x_0) = \\sum_{k=1}^n\\log l_{\\vartheta k x_0}$. We assume that $l_{\\vartheta k x_0}$ can be expanded around $l_{\\vartheta^* k x_0}=1$ as follows. With a slight abuse of the notation, let $P_n(f_k) := n^{-1} \\sum_{k=1}^n f_k$ and recall $\\nu_n(f_k) := n^{-1\/2} \\sum_{k=1}^n [f_k - \\mathbb{E}_{\\vartheta^*}(f_k)]$. \n\\begin{assumption}\\label{assn_a3}\nFor all $k=1,\\ldots,n$, $l_{\\vartheta k x_0} -1$ admits an expansion\n\\begin{equation} \\label{lkx0_expand}\nl_{\\vartheta k x_0} -1 = t_\\vartheta' s_{\\pi k} + r_{\\vartheta k} + u_{\\vartheta k x_0},\n\\end{equation}\nwhere $t_\\vartheta$ satisfies $\\psi \\to \\psi^*$ if $t_\\vartheta \\to 0$ and $(s_{\\pi k}, r_{\\vartheta k}, u_{\\vartheta k x_0})$ satisfy, for some $C \\in (0,\\infty)$, $\\delta >0$, $\\varepsilon >0$, and $\\rho \\in (0,1)$, (a) $\\mathbb{E}_{\\vartheta^*}\\sup_{\\pi \\in \\Theta_\\pi} \\left|s_{\\pi k}\\right|^{2+\\delta} < C$, (b) $\\sup_{\\pi \\in \\Theta_\\pi}| P_n(s_{\\pi k}s_{\\pi k}') - \\mathcal{I}_\\pi| = o_p(1)$ with $0<\\inf_{\\pi\\in \\Theta_{\\pi} }\\lambda_{\\min}(\\mathcal{I}_\\pi) \\leq \\sup_{\\pi\\in \\Theta_{\\pi} }\\lambda_{\\max}(\\mathcal{I}_\\pi)<\\infty$, (c) $\\mathbb{E}_{\\vartheta^*}[\\sup_{\\vartheta \\in \\mathcal{N}_\\varepsilon} |r_{\\vartheta k}\/(|t_\\vartheta||\\psi-\\psi^*|)|^2] < \\infty$, (d) $\\sup_{\\vartheta \\in \\mathcal{N}_\\varepsilon} [ \\nu_n(r_{\\vartheta k})\/(|t_\\vartheta||\\psi-\\psi^*|)] = O_p(1)$, (e) $\\sup_{x_0 \\in \\mathcal{X}}\\sup_{\\vartheta \\in \\mathcal{N}_\\varepsilon} P_n (|u_{\\vartheta k x_0}|\/|\\psi-\\psi^*|)^j = O_p(n^{-1})$ for $j=1,2,3$, (f) $\\sup_{x_0 \\in \\mathcal{X}}\\sup_{\\vartheta \\in \\mathcal{N}_\\varepsilon} P_n (|s_{\\pi k}||u_{\\vartheta k x_0}|\/|\\psi-\\psi^*|) = O_p(n^{-1})$, (g) $\\sup_{\\vartheta \\in \\mathcal{N}_{\\varepsilon}} |\\nu_n(s_{\\pi k})| =O_p(1)$. \n\\end{assumption} \nWe first establish an expansion of $\\ell_n(\\psi,\\pi, x_0)$ in the neighborhood of $\\mathcal{N}_{c\/\\sqrt{n}}$ for any $c>0$. \n\\begin{proposition} \\label{Ln_thm1}\nSuppose that Assumptions \\ref{assn_a3} (a)--(f) hold. Then, for any $c>0$,\n\\begin{equation*}\n \\sup_{x_0 \\in \\mathcal{X}} \\sup_{\\vartheta \\in \\mathcal{N}_{c\/\\sqrt{n}}} \\left| \\ell_n(\\psi,\\pi, x_0) - \\ell_n(\\psi^*,\\pi, x_0) - \\sqrt{n}t_\\vartheta' \\nu_n (s_{\\pi k}) + n t_\\vartheta' \\mathcal{I}_\\pi t_\\vartheta\/2 \\right| = o_p(1).\n\\end{equation*}\n\\end{proposition} \nThe following proposition expands $\\ell_n(\\psi,\\pi, x_0)$ in $A_{n\\varepsilon}(x_0,\\eta) := \\{\\vartheta \\in \\mathcal{N}_{\\varepsilon}: \\ell_n(\\psi,\\pi, x_0)-\\ell_n(\\psi^*,\\pi, x_0) \\geq -\\eta \\}$ for some $\\eta \\in [0,\\infty)$. This proposition is useful for deriving the asymptotic distribution of the LRTS because a consistent MLE is in $ A_{n\\varepsilon}(x_0,\\eta)$ by definition. Let $A_{n\\varepsilon c}(x_0,\\eta) := A_{n\\varepsilon }(x_0,\\eta) \\cup \\mathcal{N}_{c\/\\sqrt{n}}$.\n\\begin{proposition} \\label{Ln_thm2}\nSuppose that Assumption \\ref{assn_a3} holds. Then, for any $\\eta>0$, (a) $\\sup_{x_0 \\in \\mathcal{X}} \\sup_{\\vartheta \\in A_{n\\varepsilon}(x_0,\\eta)} |t_\\vartheta| = O_{p \\varepsilon }(n^{-1\/2})$, and (b) for any $c>0$,\n\\begin{equation*} \n \\sup_{x_0 \\in \\mathcal{X}} \\sup_{\\vartheta \\in A_{n\\varepsilon c}(x_0,\\eta) }\\left|\\ell_n(\\psi,\\pi, x_0)-\\ell_n(\\psi^*,\\pi, x_0) - \\sqrt{n} t_\\vartheta' \\nu_n (s_{\\pi k}) + n t_\\vartheta' \\mathcal{I}_\\pi t_\\vartheta\/2 \\right| = o_{p \\varepsilon }(1).\n\\end{equation*}\n\\end{proposition}\n\nThe following corollary of Propositions \\ref{Ln_thm1} and \\ref{Ln_thm2} shows that $\\ell_n(\\vartheta,\\xi)$ defined in (\\ref{density}) admits a similar expansion to $\\ell_n(\\vartheta,x_0)$ for all $\\xi$. Consequently, the asymptotic distribution of the LRTS does not depend on $\\xi$, and $\\ell_n(\\vartheta,\\xi)$ may be maximized in $\\vartheta$ while fixing $\\xi$ or jointly in $\\vartheta$ and $\\xi$. Let $A_{n\\varepsilon } (\\xi,\\eta) := \\{\\vartheta \\in \\mathcal{N}_{\\varepsilon }: \\ell_n(\\psi,\\pi,\\xi) - \\ell_n(\\psi^*,\\pi,\\xi) \\geq -\\eta \\}$ and $A_{n\\varepsilon c} (\\xi,\\eta):= A_{n\\varepsilon} (\\xi,\\eta)\\cup \\mathcal{N}_{c\/\\sqrt{n}}$, which includes a consistent MLE with any $\\xi$.\n\\begin{corollary} \\label{cor_appn}\n(a) Under the assumptions of Proposition \\ref{Ln_thm1}, we have \\\\\n$\\sup_{\\xi \\in \\Xi}\\sup_{\\vartheta \\in \\mathcal{N}_{c\/\\sqrt{n}}} \\left| \\ell_n(\\psi,\\pi,\\xi) - \\ell_n(\\psi^*,\\pi,\\xi) - \\sqrt{n}t_\\vartheta' \\nu_n (s_{\\pi k}) + n t_\\vartheta' \\mathcal{I}_\\pi t_\\vartheta\/2 \\right| = o_p(1)$ for any $c>0$.\n(b) Under the assumptions of Proposition \\ref{Ln_thm2}, for any $\\eta>0$ and $c>0$, $\\sup_{\\xi\\in \\Xi}\\sup_{\\vartheta \\in A_{n\\varepsilon} (\\xi,\\eta) } |t_\\vartheta| = O_{p \\varepsilon }(n^{-1\/2})$ and $\\sup_{\\xi\\in \\Xi}\\sup_{\\vartheta \\in A_{n\\varepsilon c} (\\xi,\\eta) } | \\ell_n(\\psi,\\pi,\\xi) - \\ell_n(\\psi^*,\\pi,\\xi) - \\sqrt{n}t_\\vartheta' \\nu_n (s_{\\pi k}) + n t_\\vartheta' \\mathcal{I}_\\pi t_\\vartheta\/2 | = o_{p \\varepsilon }(1)$.\n\\end{corollary}\n\n\\section{Uniform convergence of the derivatives of the log-density and density ratios} \\label{sec:uniform_convergence}\n\nIn this section, we establish approximations that enable us to apply the results in Section \\ref{sec:expansion} to the log-likelihood function of regime switching models. Because of the presence of singularity, the expansion (\\ref{lkx0_expand}) of the density ratio $l_{\\vartheta k x_0}$ involves the higher-order derivatives of the density ratios $\\nabla_{\\psi}^j l_{\\vartheta k x_0}$ with $j\\geq 2$. Note that $\\nabla_\\psi^j l_{\\vartheta k x_0}$ can be expressed in terms of the derivatives of log-densities, $\\nabla_\\psi^j \\log p_{\\psi\\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1},{\\bf W}_{0}^k,x_0)$. We show that these derivatives are approximated by their stationary counterpart that condition on the infinite past $(\\overline{\\bf{Y}}_{-\\infty}^{k-1},{\\bf W}_{-\\infty}^{k})$ in place of $(\\overline{\\bf{Y}}_{0}^{k-1},{\\bf W}_{0}^{k})$. Consequently, the sequence $\\{\\nabla_\\psi^j l_{\\vartheta k x_0}\\}_{k=0}^{\\infty}$ is approximated by a stationary martingale difference sequence. \n \nFor $1 \\leq k \\leq n$ and $m \\geq 0$, let \n\\begin{equation} \\label{p_bar}\n\\overline p_{\\vartheta}({\\bf Y}_{-m +1 }^k|\\overline{\\bf{Y}}_{-m},{\\bf W}_{-m}^{k}) := \\sum_{{\\bf x}_{-m}^k\\in \\mathcal{X}^{k+m+1}}\\prod_{t=-m+1}^k p_{\\vartheta}(Y_t,x_t|\\overline{\\bf Y}_{t-1},W_t,x_{t-1})\\mathbb{P}_{\\vartheta^*}(x_{-m}|\\overline{\\bf{Y}}_{-m},{\\bf W}_{-m}^{k}),\n\\end{equation}\ndenote the stationary density of ${\\bf Y}_{-m +1 }^k$ associated with $\\vartheta$ conditional on $\\{\\overline{\\bf{Y}}_{-m},{\\bf W}_{-m}^{k}\\}$, where $X_{-m}$ is drawn from its \\emph{true} conditional stationary distribution $\\mathbb{P}_{\\vartheta^*}(x_{-m}|\\overline{\\bf{Y}}_{-m}^{k-1},{\\bf W}_{-m}^{k})$. Let $\\overline p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},{\\bf W}_{-m}^{k}) : = \\overline p_{\\vartheta}({\\bf Y}_{-m+1}^k|\\overline{\\bf{Y}}_{-m},{\\bf W}_{-m}^{k})\/\\overline p_{\\vartheta}({\\bf Y}_{-m+1}^{k-1}|\\overline{\\bf{Y}}_{-m},{\\bf W}_{-m}^{k-1})$ denote the associated conditional density of $Y_k$ given $(\\overline{\\bf{Y}}_{-m}^{k-1},{\\bf W}_{-m}^{k})$.\\footnote{Note that DMR use the same notation $\\overline p_{\\vartheta}(\\cdot|\\overline{\\bf{Y}}_{-m}^{k-1})$ for a different purpose. On p.\\ 2263 and in some other (but not all) places, DMR use $\\overline p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{0}^{k-1})$ to denote an (ordinary) stationary conditional distribution of $Y_k$.\n}\n\nDefine the density ratio as $l_{k,m,x}(\\vartheta) := p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},{\\bf W}_{-m}^{k},X_{-m}=x)\/p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},{\\bf W}_{-m}^{k},X_{-m}=x)$. For $j=1,2,\\ldots,6$, $1 \\leq k \\leq n$, $m \\geq 0$, and $x \\in \\mathcal{X}$, define the derivatives of the log-densities and density ratios by, with suppressing the subscript $\\vartheta$ from $\\nabla_\\vartheta^j$ for brevity,\n\\begin{align*} \n& \\nabla^j \\ell_{k,m,x}(\\vartheta) := \\nabla^j \\log p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},{\\bf W}_{-m}^{k},X_{-m}=x),\\quad \\nabla^j l_{k,m,x}(\\vartheta) := \\frac{\\nabla^j p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},{\\bf W}_{-m}^{k},X_{-m}=x)}{ p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},{\\bf W}_{-m}^{k},X_{-m}=x)},\\\\\n& \\nabla^j \\overline \\ell_{k,m}(\\vartheta):= \\nabla^j \\log \\overline p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},{\\bf W}_{-m}^{k}), \\quad\\text{and}\\quad \\nabla^j \\overline l_{k,m}(\\vartheta):= \\frac{ \\nabla^j \\overline p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},{\\bf W}_{-m}^{k})}{\\overline p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},{\\bf W}_{-m}^{k})}.\n\\end{align*} \nThe following assumption corresponds to (A6)--(A8) in DMR and is tailored to our setting where some elements of $\\vartheta_x^*$ are not identified and $\\mathcal{X}$ is finite. Note that Assumptions (A6) and (A7) in DMR pertaining to $q_{\\vartheta_x}(x,x')$ hold in our case because the $p_{ij}$'s are bounded away from 0 and 1. Let $G_{\\vartheta k} := \\sum_{x_k\\in\\mathcal{X}}g_{\\vartheta_y}(Y_k|\\overline{\\bf{Y}}_{k-1}, x_k, W_k)$. $G_{\\vartheta k}$ satisfies Assumption \\ref{assn_a4}(b) in general when $\\mathcal{N}^*$ is sufficiently small.\n\\begin{assumption}\\label{assn_a4}\nThere exists a positive real $\\delta$ such that on $\\mathcal{N}^*:= \\{\\vartheta\\in\\Theta: |\\vartheta_y-\\vartheta_y^*| < \\delta\\}$ the following conditions hold: (a) For all $(\\overline{\\bf{y}},y',x,w)\\in \\mathcal{Y}^s \\times \\mathcal{Y} \\times \\mathcal{X} \\times \\mathcal{W}$, $g_{\\vartheta_y}(y'|\\overline{\\bf{y}},x,w)$ is six times continuously differentiable on $\\mathcal{N}^*$. (b) $\\mathbb{E}_{\\vartheta^*}[\\sup_{\\vartheta\\in \\mathcal{N}^*} \\sup_{x\\in \\mathcal{X}} | \\nabla^j\\log g_{\\vartheta_y}(Y_1|\\overline{\\bf{Y}}_0,x,W) |^{2q_j}]< \\infty$ for $j=1,2,\\ldots,6$ and $\\mathbb{E}_{\\vartheta^*}\\sup_{\\vartheta \\in \\mathcal{N}^*} |G_{\\vartheta k}\/G_{\\vartheta^* k}|^{q_g} < \\infty$ with $q_1=6q_0, q_2=5q_0, \\ldots, q_6=q_0$, where $q_0=(1+\\varepsilon) q_\\vartheta$ and $q_g=(1+\\varepsilon)q_\\vartheta\/\\varepsilon$ for some $\\varepsilon>0$ and $q_\\vartheta >\\max\\{3,\\text{dim}(\\vartheta)\\}$. (c) For almost all $(\\overline{\\bf{y}},y',w) \\in \\mathcal{Y}^s \\times \\mathcal{Y}\\times \\mathcal{W}$, $\\sup_{\\vartheta\\in \\mathcal{N}^*} g_{\\vartheta_y}(y'|\\overline{\\bf{y}},x,w) < \\infty$ and, for almost all $(\\overline{\\bf{y}},x,w)\\in \\mathcal{Y}^s\\times \\mathcal{X} \\times \\mathcal{W}$, for $j=1,2,\\ldots,6$, there exist functions $f^j_{\\overline{\\bf{y}},w,x}: \\mathcal{Y} \\rightarrow \\mathbb{R}^+$ in $L^1$ such that $|\\nabla^j g_{\\vartheta_y}(y'|\\overline{\\bf{y}},x,w) |\\leq f^j_{\\overline{\\bf{y}}, x, w}(y')$ for all $\\vartheta\\in \\mathcal{N}^*$.\n\\end{assumption} \n\nThe following proposition shows that $\\{\\nabla^j \\ell_{k,m,x}(\\vartheta)\\}_{m\\geq 0}$ and $\\{\\nabla^j \\overline \\ell_{k,m}(\\vartheta)\\}_{m\\geq 0}$ are $L^{r_j}(\\mathbb{P}_{\\vartheta^*})$-Cauchy sequences that converge to $\\nabla^j \\ell_{k,\\infty}(\\vartheta)$ $\\mathbb{P}_{\\vartheta^*}$-a.s.\\ and in $L^{r_j}(\\mathbb{P}_{\\vartheta^*})$ uniformly in $\\vartheta\\in \\mathcal{N}^*$ and $x \\in \\mathcal{X}$. \n\\begin{proposition} \\label{proposition-ell}\nUnder Assumptions \\ref{assn_a1}, \\ref{assn_a2}, and \\ref{assn_a4}, for $j=1,\\ldots,6$, there exist random variables $(K_j, \\{M_{j,k} \\}_{k=1}^n )\\in L^{r_j}(\\mathbb{P}_{\\vartheta^*})$ and $\\rho_* \\in (0,1)$ such that, for all $1 \\leq k \\leq n$ and $m' \\geq m \\geq 0$,\n\\begin{align*}\n(a) &\\quad \\sup_{x\\in\\mathcal{X}}\\sup_{\\vartheta\\in \\mathcal{N}^*}|\\nabla^j\\ell_{k,m,x}(\\vartheta) - \\nabla^j\\overline\\ell_{k,m}(\\vartheta) | \\leq K_j (k+m)^7\\rho_*^{k+m-1}\\quad \\text{$\\mathbb{P}_{\\vartheta^*}$-a.s.},\\\\\n(b) &\\quad \\sup_{x\\in\\mathcal{X}}\\sup_{\\vartheta\\in \\mathcal{N}^*}|\\nabla^j\\ell_{k,m,x}(\\vartheta) - \\nabla^j \\ell_{k,m',x}(\\vartheta) | \\leq K_j(k+m)^7\\rho_*^{k+m-1}\\quad \\text{$\\mathbb{P}_{\\vartheta^*}$-a.s.},\\\\\n(c) & \\quad \\sup_{m \\geq 0}\\sup_{x\\in\\mathcal{X}}\\sup_{\\vartheta\\in \\mathcal{N}^*}|\\nabla^j\\ell_{k,m,x}(\\vartheta)| + \\sup_{m \\geq 0}\\sup_{\\vartheta\\in \\mathcal{N}^*}|\\nabla^j\\overline\\ell_{k,m}(\\vartheta)|\\leq M_{j, k} \\quad \\text{$\\mathbb{P}_{\\vartheta^*}$-a.s.},\n\\end{align*}\nwhere $r_1 = 6q_0$, $r_2 = 3q_0$, $r_3 = 2q_0$, $r_4 =3q_0\/2$, $r_5 =6q_0\/5$, and $r_6 =q_0$. (d) Uniformly in $\\vartheta\\in \\mathcal{N}^*$ and $x \\in \\mathcal{X}$, $\\nabla^j\\ell_{k,m,x}(\\vartheta)$ and $\\nabla^j\\overline\\ell_{k,m}(\\vartheta)$ converge $\\mathbb{P}_{\\vartheta^*}$-a.s.\\ and in $L^{r_j}(\\mathbb{P}_{\\vartheta^*})$ to $\\nabla^j \\ell_{k,\\infty}(\\vartheta) \\in L^{r_j}(\\mathbb{P}_{\\vartheta^*})$ as $m\\rightarrow \\infty$.\n\\end{proposition} \nAs shown by the following proposition, we may prove the uniform convergence of the derivatives of the density ratios by expressing them as polynomials of the derivatives of the log-density and applying Proposition \\ref{proposition-ell} and H\\\"older's inequality.\n\\begin{proposition}\\label{lemma-omega}\nUnder Assumptions \\ref{assn_a1}, \\ref{assn_a2}, and \\ref{assn_a4}, for $j=1,\\ldots,6$, there exist random variables $\\{K_{j,k}\\}_{k=1}^n \\in L^{q_\\vartheta}(\\mathbb{P}_{\\vartheta^*})$ and $\\rho_* \\in (0,1)$ such that, for all $1 \\leq k \\leq n$ and $m' \\geq m \\geq 0$,\n\\begin{align*}\n(a) &\\quad \\sup_{x\\in\\mathcal{X}}\\sup_{\\vartheta\\in \\mathcal{N}^*}|\\nabla^j l_{k,m,x}(\\vartheta) - \\nabla^j \\overline l_{k,m}(\\vartheta) | \\leq K_{j,k} (k+m)^{7}\\rho_*^{k+m-1} \\quad \\mathbb{P}_{\\vartheta^*}\\text{-a.s.},\\\\\n(b) &\\quad \\sup_{x\\in\\mathcal{X}}\\sup_{\\vartheta\\in \\mathcal{N}^*}|\\nabla^j l_{k,m,x}(\\vartheta) - \\nabla^j l_{k,m',x}(\\vartheta) | \\leq K_{j,k} (k+m)^{7}\\rho_*^{k+m-1} \\quad \\text{$\\mathbb{P}_{\\vartheta^*}$-a.s.}, \\\\ \n(c) &\\quad \\sup_{m\\geq 0}\\sup_{x\\in\\mathcal{X}}\\sup_{\\vartheta\\in \\mathcal{N}^*}|\\nabla^j l_{k,m,x}(\\vartheta)| + \\sup_{m\\geq 0}\\sup_{\\vartheta\\in \\mathcal{N}^*}|\\nabla^j\\overline{l}_{k,m}(\\vartheta)| \\leq K_{j,k} \\quad \\mathbb{P}_{\\vartheta^*}\\text{-a.s.}\n\\end{align*}\n(d) Uniformly in $\\vartheta\\in \\mathcal{N}^*$ and $x \\in \\mathcal{X}$, $\\nabla^j l_{k,m,x}(\\vartheta)$ and $\\nabla^j\\overline l_{k,m}(\\vartheta)$ converge $\\mathbb{P}_{\\vartheta^*}$-a.s.\\ and in $L^{q_\\vartheta}(\\mathbb{P}_{\\vartheta^*})$ to $\\nabla^j l_{k,\\infty}(\\vartheta) \\in L^{q_\\vartheta}(\\mathbb{P}_{\\vartheta^*})$ as $m\\rightarrow \\infty$. (e) $\\sup_{\\vartheta\\in \\mathcal{N}^*}|\\nabla^j \\overline l_{k,0}(\\vartheta) - \\nabla^j l_{k,\\infty}(\\vartheta) | \\leq K_{j,k} k^{7}\\rho_*^{k-1}$ $\\mathbb{P}_{\\vartheta^*}$-a.s. \n\\end{proposition}\nWhen we apply the results in Section \\ref{sec:expansion} to regime switching models, $l_{k,0,x}(\\vartheta)$ corresponds to $l_{\\vartheta k x_0}$ on the left-hand side of (\\ref{lkx0_expand}), and $s_{\\pi k}$ in (\\ref{lkx0_expand}) is a function of the $\\nabla^j \\overline l_{k, 0}(\\vartheta)$'s. Proposition \\ref{lemma-omega} and the dominated convergence theorem for conditional expectations \\citep[][Theorem 5.5.9]{durrett10book} imply that $\\mathbb{E}_{\\vartheta^*}[\\nabla^j l_{k,\\infty}(\\vartheta)|\\overline{\\bf{Y}}_{-\\infty}^{k-1}]=0$ for all $\\vartheta\\in \\mathcal{N}^*$. Therefore, $\\{\\nabla^j l_{k,\\infty}(\\vartheta)\\}_{k=-\\infty}^{\\infty}$ is a stationary, ergodic, and square integrable martingale difference sequence, and $\\{\\nabla^j l_{k,\\infty}(\\vartheta)\\}_{j=1}^5$ satisfies Assumption \\ref{assn_a3}(a)(b)(g).\n\n\\section{Testing homogeneity} \\label{sec: testing-1}\n\nBefore developing the LRT of $M_0$ components, we analyze a simpler case of testing the null hypothesis $H_0: M=1$ against $H_A:M=2$ when the data are from $H_0$. We assume that the parameter space for $\\vartheta_{2,x}=(p_{11},p_{22})'$ takes the form $[\\epsilon, 1-\\epsilon]^2$ for a small $\\epsilon \\in (0,1\/2)$. Denote the true parameter in the one-regime model by $\\vartheta_1^*:= ((\\theta^*)',(\\gamma^*)')'$. The two-regime model gives rise to the true density $p_{\\vartheta_1^*}({\\bf Y}_1^n| \\overline{{\\bf Y}}_{0}, x_0)$ if the parameter $\\vartheta_2=(\\theta_1,\\theta_2,\\gamma,p_{11},p_{22})'$ lies in a subset of the parameter space\n\\[\n\\Gamma^* := \\left\\{ (\\theta_1,\\theta_2,\\gamma,p_{11},p_{22})\\in \\Theta_{ 2}: \\theta_1=\\theta_2=\\theta^*\\ \\text{and}\\ \\gamma=\\gamma^* \\right\\}.\n\\]\nNote that $(p_{11},p_{22})$ is not identified under $H_0$.\n\nLet $\\ell_n(\\vartheta_2,\\xi_2) := \\log \\left( \\sum_{x_0=1}^2 p_{\\vartheta_2}({\\bf Y}_1^n| \\overline{\\bf Y}_{0},{\\bf W}_{0}^n,x_0) \\xi_2(x_0) \\right)$ denote the two-regime log-likelihood for a given initial distribution $\\xi_2(x_0) \\in \\Xi_2$, and let $\\hat\\vartheta_2:= \\arg\\max_{\\vartheta_2 \\in \\Theta_{2}} \\ell_n(\\vartheta_2,\\xi_2)$ denote the MLE of $\\vartheta_2$ given $\\xi_2$, where $\\xi_2$ is suppressed from $\\hat \\vartheta_2$ because $\\xi_2$ does not matter asymptotically. Let $\\hat \\vartheta_1$ denote the one-regime MLE that maximizes the one-regime log-likelihood function $\\ell_{0,n}(\\vartheta_1) := \\sum_{k=1}^n \\log f (Y_k|\\overline{\\bf Y}_{k-1},W_k;\\gamma,\\theta)$ under the constraint $\\vartheta_1 = (\\theta',\\gamma')' \\in \\Theta_1$. \n\nWe introduce the following assumption for the consistency of $\\hat \\vartheta_1$ and $\\hat \\vartheta_2$. Assumption \\ref{A-consist}(b) corresponds to Assumption (A4) of DMR. Assumption \\ref{A-consist}(c) is a standard identification condition for the one-regime model. Assumption \\ref{A-consist}(d) implies that the Kullback--Leibler divergence between $ p_{\\vartheta_{1}^*}(Y_1 |\\overline{\\bf{Y}}_{-m}^0,{\\bf W}_{-m}^0)$ and $ p_{\\vartheta_{2}}(Y_1 |\\overline{\\bf{Y}}_{-m}^0,{\\bf W}_{-m}^0)$ is 0 if and only if $\\vartheta_2 \\in \\Gamma^*$.\n\\begin{assumption} \\label{A-consist} \n(a) $\\Theta_1$ and $\\Theta_2$ are compact, and $\\vartheta_1^*$ is in the interior of $\\Theta_1$. (b) For all $(x,x') \\in \\mathcal{X}$ and all $(\\overline{{\\bf y}},y',w)\\in \\mathcal{Y}^s\\times \\mathcal{Y}\\times \\mathcal{W}$, $f(y'|\\overline{\\bf{y}}_0,w;\\gamma,\\theta)$ is continuous in $(\\gamma,\\theta)$. (c) If $\\vartheta_{1}\\neq \\vartheta_1^*$, then $\\mathbb{P}_{\\vartheta^*_{1}}\\left(f(Y_1|\\overline{\\bf{Y}}_0,W_1;\\gamma,\\theta) \\neq f(Y_1|\\overline{\\bf{Y}}_0,W_1;\\gamma^*,\\theta^*) \\right)>0$. (d) $\\mathbb{E}_{\\vartheta^*_{1}}[ \\log p_{\\vartheta_{2}}(Y_1 |\\overline{\\bf{Y}}_{-m}^0,{\\bf W}_{-m}^1)] = \\mathbb{E}_{\\vartheta^*_{1}}[\\log p_{\\vartheta_{1}^*}(Y_1 |\\overline{\\bf{Y}}_{-m}^0,{\\bf W}_{-m}^1) ]$ for all $m \\geq 0$ if and only if $\\vartheta_{2}\\in \\Gamma^*$. \n\\end{assumption} \nThe following proposition shows the consistency of the MLEs of $\\vartheta_1^*$ and $\\vartheta_{2,y}^*$. \n\\begin{proposition} \\label{P-consist} Suppose that Assumptions \\ref{assn_a1}, \\ref{assn_a2}, and \\ref{A-consist} hold. Then, under the null hypothesis of $M=1$, $\\hat\\vartheta_1 \\overset{p}{\\rightarrow} \\vartheta_1^*$ and $\\inf_{\\vartheta_2\\in\\Gamma^*}|\\hat\\vartheta_2-\\vartheta_2|\\overset{p}{\\rightarrow} 0$.\n\\end{proposition}\n \nLet $LR_n:=2[\\ell_n(\\hat\\vartheta_2,\\xi_2) - \\ell_{0,n}(\\hat\\vartheta_1)]$ denote the LRTS for testing $H_0:M_0=1$ against $H_A:M_0=2$. Following the notation of Section \\ref{sec:expansion}, we split $\\vartheta_2$ into $\\vartheta_2=(\\psi,\\pi)$, where $\\pi$ is the part of $\\vartheta$ not identified under the null hypothesis; the elements of $\\psi$ are delineated later. In the current setting, $\\pi$ corresponds to $\\vartheta_{2,x}=(p_{11},p_{22})'$. Define $\\varrho := \\text{corr}_{\\vartheta_{2,x}} (X_k,X_{k+1}) = p_{11}+p_{22}-1$ and $\\alpha := \\mathbb{P}_{\\vartheta_{2,x}}(X_k=1)= (1-p_{22})\/(2-p_{11}-p_{22})$. The parameter spaces for $\\varrho$ and $\\alpha$ under restriction $p_{11},p_{22} \\in [\\epsilon, 1-\\epsilon]$ are given by $\\Theta_{\\varrho}:= [-1+2\\epsilon,1-2\\epsilon]$ and $\\Theta_{\\alpha}:= [\\epsilon, 1-\\epsilon]$, respectively. Because the mapping from $(p_{11},p_{22})$ to $(\\varrho,\\alpha)$ is one-to-one, we reparameterize $\\pi$ as $\\pi := (\\varrho,\\alpha)' \\in \\Theta_{\\pi }:=\\Theta_{\\varrho }\\times \\Theta_{\\alpha }$, and let $p_{\\psi\\pi}(\\cdot|\\cdot) := p_{\\vartheta_2} (\\cdot|\\cdot)$. Henceforth, we suppress ${\\bf W}_{0}^n$ for notational brevity and write, for example, $p_{\\psi\\pi}({\\bf Y}_1^n| \\overline{{\\bf Y}}_{0}, {\\bf W}_{0}^n,x_0)$ as $p_{\\psi\\pi}({\\bf Y}_1^n|\\overline{{\\bf Y}}_{0},x_0)$ and $p_{\\psi\\pi} (y_k,x_k| \\overline{\\bf{y}}_{k-1},x_{k-1},w_k)$ as $p_{\\psi\\pi} (y_k,x_k| \\overline{\\bf{y}}_{k-1},x_{k-1})$ when doing so does not cause confusion. \n\nWe derive the asymptotic distribution of the LRTS by applying Corollary \\ref{cor_appn} to $\\ell_n(\\psi,\\pi,\\xi_2)$ and representing $s_{\\pi k}$ and $t_\\vartheta$ in (\\ref{lkx0_expand}) in terms of $\\vartheta$, $f(Y_k|\\overline{\\bf{Y}}_{k-1};\\gamma,\\theta)$, and its derivatives; $s_{\\pi k}$ involves higher-order derivatives, and $t_\\vartheta$ consists of the functions of the polynomials of (reparameterized) $\\vartheta$. Section \\ref{subsec:nonnormal} analyzes the case when the regime-specific distribution of $y_k$ is not normal with unknown variance. Section \\ref{subsec:hetero_normal} analyzes the case when the regime-specific distribution $y_k$ is normal with regime-specific and unknown variance. Section \\ref{subsec:homo_normal} handles the normal distribution where the variance is unknown and common across regimes. Note that because $\\overline{\\bf Y}_{-\\infty}^\\infty$ and ${\\bf X}_{-\\infty}^\\infty$ are independent when $\\psi = \\psi^*$, we have \n\\begin{equation}\\label{indep}\n\\mathbb{P}_{\\psi^*\\pi}({\\bf X}_{-\\infty}^\\infty|\\overline{\\bf Y}_{-\\infty}^\\infty) = \\mathbb{P}_{\\psi^*\\pi}({\\bf X}_{-\\infty}^\\infty).\n\\end{equation}\nDefine $q_k := \\mathbb{I}\\{X_k=1\\}$, so that $\\alpha = \\mathbb{E}_{\\psi^*\\pi}[q_k]$.\n\n \n\\subsection{Non-normal distribution}\\label{subsec:nonnormal}\n \nWhen we apply Corollary \\ref{cor_appn} to regime switching models, $s_{\\pi k}$ is a function of \\\\$\\nabla^j\\overline p_{\\psi^*\\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^*\\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})$'s with $ \\overline p_{\\psi\\pi} (Y^k_{1}| \\overline{{\\bf Y}}_{0})$ defined in (\\ref{p_bar}). In order to express $\\nabla^j\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) \/ \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})$ in terms of $\\nabla^j f(y|x;\\gamma,\\theta)$ via the Louis information principle (Lemma \\ref{louis} in the appendix), we first derive the derivatives of the \\emph{complete data} conditional density $p_{\\vartheta_2} (y_k,x_k| \\overline{\\bf{y}}_{k-1},x_{k-1}) = g_{\\vartheta_{2,y}}(y_k|\\overline{\\bf{y}}_{k-1}, x_k) q_{\\vartheta_{2,x}}(x_{k-1},x_k)=\\sum_{j =1}^2 \\mathbb{I}\\{x_k=j\\}f(y_k|\\overline{\\bf{y}}_{k-1};\\gamma,\\theta_j)q_{\\vartheta_{2,x}}(x_{k-1},x_k)$.\n\nConsider the following reparameterization. Let\n\\begin{equation}\n\\left(\n\\begin{array}{c}\n\\lambda \\\\\n\\nu \\\\\n\\end{array}\n\\right):=\n\\left(\n\\begin{array}{c}\n\\theta_1 - \\theta_2 \\\\\n\\alpha\\theta_1 + (1-\\alpha)\\theta_2\\\\\n\\end{array}\n\\right),\n\\quad \\text{so that} \\quad\n\\left(\n\\begin{array}{c}\n\\theta_1\\\\\n\\theta_2 \\\\\n\\end{array}\n\\right)=\n\\left(\n\\begin{array}{c}\n\\nu+ (1-\\alpha) \\lambda\\\\\n\\nu- \\alpha\\lambda\\\\\n\\end{array}\n\\right). \\label{repara}\n\\end{equation}\nLet $\\eta := (\\gamma',\\nu')'$ and $\\psi_{\\alpha}:=(\\eta',\\lambda')' \\in \\Theta_\\eta \\times \\Theta_\\lambda$. Under the null hypothesis of one regime, the true value of $\\psi_{\\alpha}$ is given by $\\psi_{\\alpha}^*:=(\\gamma^*,\\theta^*,0)'$. Henceforth, we suppress the subscript $\\alpha$ from $\\psi_{\\alpha}$. Using this definition of $\\psi$, let $\\vartheta_2 := (\\psi',\\pi')' \\in \\Theta_\\psi \\times \\Theta_\\pi$. By using reparameterization (\\ref{repara}) and noting that $q_k = \\mathbb{I}\\{x_k=1\\}$, we have $p_{\\psi\\pi} (y_k,x_k| \\overline{\\bf{y}}_{k-1},x_{k-1}) = g_{\\psi}(y_k|\\overline{\\bf{y}}_{k-1}, x_k) q_{\\pi}(x_{k-1},x_k)$ and \n\\begin{equation} \\label{repara_g}\ng_{\\psi}(y_k|\\overline{\\bf{y}}_{k-1},x_k) = f(y_k|\\overline{\\bf{y}}_{k-1};\\gamma,\\nu+(q_k-\\alpha)\\lambda ).\n\\end{equation}\nHenceforth, let $f^*_{k}$, $\\nabla f^*_{k}$, $g^*_{k}$, and $\\nabla g^*_{k}$ denote $f(Y_k|\\overline{\\bf{Y}}_{k-1};\\gamma^*,\\theta^*)$, $\\nabla f(Y_k|\\overline{\\bf{Y}}_{k-1};\\gamma^*,\\theta^*)$, $g_{\\psi^*}(Y_k|\\overline{\\bf{Y}}_{k-1},X_k) $, and $\\nabla g_{\\psi^*}(Y_k|\\overline{\\bf{Y}}_{k-1},X_k)$, respectively, and similarly for $\\log f^*_{k}$ ,$\\nabla \\log f^*_{k}$, $\\log g^*_{k}$, and $\\nabla \\log g^*_{k}$. \nExpanding $g_{\\psi}(Y_k|\\overline{\\bf{Y}}_{k-1},X_k)$ twice with respect to $\\psi=(\\gamma',\\nu',\\lambda')'$ and evaluating at $\\psi^*$ gives\n\\begin{equation} \\label{dg}\n\\begin{aligned}\n\\nabla_{\\eta} g_k^* &= \\nabla_{(\\gamma',\\theta')'} f_k^*, \\quad \\nabla_{\\lambda} g_k^* = (q_k -\\alpha)\\nabla_{\\theta} f_k^*, \\\\\n\\nabla_{\\lambda\\eta'} g_k^* &= (q_k -\\alpha)\\nabla_{\\theta(\\gamma',\\theta')} f_k^*, \\quad \\nabla_{\\lambda\\lambda'} g_k^* = (q_k-\\alpha)^2 \\nabla_{\\theta\\theta'} f_k^*.\n\\end{aligned} \n\\end{equation}\nRecall $\\varrho := \\text{corr}_{\\vartheta_2^*}(q_k,q_{k+1})$. Observe that $q_k$ satisfies\n\\begin{equation} \\label{markov_moments}\n\\begin{aligned}\n\\mathbb{E}_{\\vartheta_2^*}(q_k-\\alpha)^2 &= \\alpha(1-\\alpha), \\quad \\mathbb{E}_{\\vartheta_2^*}(q_k-\\alpha)^3 = \\alpha(1-\\alpha)(1-2\\alpha), \\\\\n\\mathbb{E}_{\\vartheta_2^*}(q_k-\\alpha)^4 &= \\alpha(1-\\alpha)(3\\alpha^2-3\\alpha+1), \\quad \\text{corr}_{\\vartheta_2^*}(q_k,q_{k+\\ell}) = \\varrho^{|\\ell|}, \n\\end{aligned}\n\\end{equation}\nwhere the first three results follow from the property of a Bernoulli random variable, and the last result follows from \\citet[][p. 684]{hamilton94book}. Then, it follows from (\\ref{indep}) and (\\ref{markov_moments}) that\n\\begin{equation} \\label{qk_moments}\n\\begin{aligned}\n&\\mathbb{E}_{\\vartheta^*}[q_k -\\alpha |\\overline{{\\bf Y}}_{-\\infty}^n] = 0,\\quad \\mathbb{E}_{\\vartheta^*}[(q_{t_1}-\\alpha) (q_{t_2}-\\alpha) | \\overline{\\bf Y}_{-\\infty}^n ] = \\alpha(1-\\alpha)\\varrho^{t_2-t_1}, \\quad t_2 \\geq t_1.\n\\end{aligned}\n\\end{equation} \nFrom Lemma \\ref{louis}, $\\log p_{\\psi\\pi}(y_k, x_k|\\overline{{\\bf y}}_{k-1},x_{k-1}) = \\log g_{\\psi}(y_k|\\overline{\\bf{y}}_{k-1},x_k) + \\log q_\\pi(x_{k-1},x_k)$, and the definition of $\\overline p_{\\psi \\pi} ( Y_1^k| \\overline{{\\bf Y}}_{0} )$ in (\\ref{p_bar}), we obtain\n\\[\n\\begin{aligned}\n&\\frac{\\nabla_\\psi \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})} =\\nabla_\\psi \\log \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) = \\sum_{t=1}^k \\mathbb{E}_{\\vartheta^*} \\left[\\nabla_\\psi \\log g^*_{t} \\middle|\\overline{{\\bf Y}}_{0}^k \\right] - \\sum_{t=1}^{k-1} \\mathbb{E}_{\\vartheta^*} \\left[\\nabla_\\psi \\log g^*_{t} \\middle|\\overline{{\\bf Y}}_{0}^{k-1} \\right].\n\\end{aligned}\n\\]\nApplying (\\ref{dg}), (\\ref{qk_moments}), and $g_k^* =f_k^*$ to the right-hand side gives\n\\begin{equation} \\label{d1p}\n\\begin{aligned}\n\\frac{\\nabla_\\eta \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})} =\\nabla_{(\\gamma',\\theta')'} \\log f_k^*, \\quad\n\\frac{\\nabla_\\lambda \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{ \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}=0.\n\\end{aligned}\n\\end{equation}\nSimilarly, it follows from Lemma \\ref{louis}, (\\ref{dg}), (\\ref{qk_moments}), (\\ref{d1p}), and $g_k^* =f_k^*$ that\n\\begin{align}\n& \\nabla_{\\lambda\\eta'} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) =0, \\label{d2p0}\\\\\n& \\nabla_{\\lambda\\lambda'} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) \\nonumber \\\\\n& = \\nabla_{\\lambda\\lambda'} \\log \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) \\nonumber \\\\\n& = \\sum_{t=1}^k \\mathbb{E}_{\\vartheta^*} \\left[\\nabla_{\\lambda\\lambda'} \\log g^*_{t} \\middle|\\overline{{\\bf Y}}_{0}^k \\right] - \\sum_{t=1}^{k-1} \\mathbb{E}_{\\vartheta^*} \\left[\\nabla_{\\lambda\\lambda'} \\log g^*_{t} \\middle|\\overline{{\\bf Y}}_{0}^{k-1} \\right] \\nonumber \\\\\n& \\quad + \\sum_{t_1=1}^k \\sum_{t_2=1}^k \\mathbb{E}_{\\vartheta^*} \\left[ \\frac{\\nabla_{\\lambda} g^*_{t_1}}{g^*_{t_1}} \\frac{\\nabla_{\\lambda'} g^*_{t_2}}{g^*_{t_2}} \\middle| \\overline{{\\bf Y}}_{0}^k \\right] - \\sum_{t_1=1}^{k-1} \\sum_{t_2=1}^{k-1} \\mathbb{E}_{\\vartheta^*} \\left[ \\frac{\\nabla_{\\lambda} g^*_{t_1}}{g^*_{t_1}} \\frac{\\nabla_{\\lambda'} g^*_{t_2}}{g^*_{t_2}} \\middle| \\overline{{\\bf Y}}_{0}^{k-1} \\right] \\nonumber \\\\\n& = \\alpha(1-\\alpha) \\left[ \\frac{ \\nabla_{\\theta\\theta'}f_k^*}{f_k^*} + \\sum_{t=1}^{k-1} \\varrho^{k-t} \\left( \\frac{\\nabla_{\\theta} f_{t}^*}{f_{t}^*} \\frac{\\nabla_{\\theta'} f_{k}^*}{f_{k}^*} + \\frac{\\nabla_{\\theta} f_{k}^*}{f_{k}^*} \\frac{\\nabla_{\\theta'} f_{t}^*}{f_{t}^*} \\right)\\right].\\label{d2p}\n\\end{align} \nBecause the first-order derivative with respect to $\\lambda$ is identically equal to zero in (\\ref{d1p}), \nthe unique elements of $\\nabla_{\\eta} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) \/\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) $ and $\\nabla_{\\lambda\\lambda'} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})$ constitute the generalized score $s_{\\pi k}$ in Corollary \\ref{cor_appn}. This score is approximated by a stationary martingale difference sequence, where the approximation error satisfies Assumption \\ref{assn_a3}.\n\nWe collect some notations. Recall $\\psi = (\\eta',\\lambda')'$ and $\\eta = (\\gamma',\\nu')'$. For a $q \\times 1$ vector $\\lambda$ and a $q \\times q$ matrix $s$, define the $q_\\lambda \\times 1$ vectors $v(\\lambda)$ and $V(s)$ as\n\\begin{equation} \\label{v_lambda}\n\\begin{aligned}\nv(\\lambda) &:= ( \\lambda_1^2,\\ldots,\\lambda_q^2,\\lambda_1\\lambda_2,\\ldots,\\lambda_1\\lambda_q,\\lambda_2 \\lambda_3, \\ldots, \\lambda_2 \\lambda_q,\\ldots,\\lambda_{q-1}\\lambda_q)',\\\\\nV(s) &:= (s_{11}\/2,\\ldots,s_{qq}\/2,s_{12},\\ldots,s_{1q},s_{23},\\ldots,s_{2q},\\ldots,s_{q,q-1})'.\n\\end{aligned}\n\\end{equation}\nNoting that $\\alpha(1-\\alpha)>0$ for $\\alpha\\in\\Theta_{\\alpha}$, define, with $t_{\\lambda}(\\lambda,\\pi):=\\alpha(1-\\alpha)v(\\lambda)$, \n\\begin{equation}\\label{score}\n\\begin{aligned}\nt(\\psi,\\pi) &:= \n\\begin{pmatrix}\n\\eta - \\eta^* \\\\\nt_{\\lambda}(\\lambda,\\pi)\n\\end{pmatrix}, \\ \ns_{ \\varrho k} : = \\begin{pmatrix}\ns_{\\eta k} \\\\\ns_{\\lambda \\varrho k}\n\\end{pmatrix}, \\ \\text{where} \\\ns_{\\eta k} : = \\frac{\\nabla_\\eta \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})} = \\begin{pmatrix}\n\\nabla_{\\gamma} f_k^* \/ f_k^* \\\\\n\\nabla_{\\theta} f_k^* \/ f_k^*\n\\end{pmatrix},\n\\end{aligned}\n\\end{equation}\nand $s_{\\lambda \\varrho k} := V(s_{\\lambda\\lambda \\varrho k})$ with \n\\begin{equation} \\label{score_lambda}\ns_{\\lambda\\lambda \\varrho k} := \\frac{\\nabla_{\\lambda\\lambda'} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\alpha(1-\\alpha) \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})} = \\frac{\\nabla_{\\theta \\theta'} f_k^*}{f_k^*} + \\sum_{t= 1}^{k-1} \\varrho^{k-t} \\left( \\frac{\\nabla_{\\theta} f_{t}^*}{f_{t}^*} \\frac{\\nabla_{\\theta'} f_{k}^*}{f_{k}^*} + \\frac{\\nabla_{\\theta} f_{k}^*}{f_{k}^*} \\frac{\\nabla_{\\theta'} f_{t}^*}{f_{t}^*} \\right).\n\\end{equation}\nHere, $s_{\\varrho k}$ in (\\ref{score}) depends on $\\varrho$ but not on $\\alpha$ and corresponds to $s_{\\pi k}$ in Corollary \\ref{cor_appn}.\nThe following proposition shows that the log-likelihood function is approximated by a quadratic function of $\\sqrt{n}t(\\psi,\\pi)$. Let $\\mathcal{N}_{\\varepsilon } := \\{ \\vartheta_2 \\in \\Theta_{2 }: |t(\\psi,\\pi)|< \\varepsilon \\}$. Let $A_{n \\varepsilon }(\\xi) := \\{\\vartheta \\in \\mathcal{N}_{\\varepsilon}: \\ell_n(\\psi,\\pi,\\xi) - \\ell_n(\\psi^*,\\pi,\\xi) \\geq 0\\} $ and $A_{n\\varepsilon c}(\\xi) := A_{n\\varepsilon }(\\xi)\\cup \\mathcal{N}_{c\/\\sqrt{n}}$, where we suppress the subscript $2$ from $\\xi_2$.\nWe use this definition of $A_{n\\varepsilon c}(\\xi)$ through Sections \\ref{subsec:nonnormal}--\\ref{subsec:homo_normal}. As shown in Sections \\ref{subsec:hetero_normal} and \\ref{subsec:homo_normal}, Assumption \\ref{A-nonsing1} does not hold for regime switching models with a normal distribution. \n\\begin{assumption} \\label{A-nonsing1} \n$0< \\inf_{\\varrho\\in \\Theta_{\\varrho}} \\lambda_{\\min}(\\mathcal{I}_{\\varrho}) \\leq \\sup_{\\varrho\\in\\Theta_{\\varrho}} \\lambda_{\\max}(\\mathcal{I}_\\varrho) < \\infty$\n for $\\mathcal{I}_{\\varrho}= \\lim_{k\\rightarrow \\infty} \\mathbb{E}_{\\vartheta^*}(s_{\\varrho k}s_{\\varrho k}')$, where $s_{\\varrho k}$ is given in (\\ref{score}).\n\\end{assumption} \n\\begin{proposition} \\label{P-quadratic} Suppose Assumptions \\ref{assn_a1}, \\ref{assn_a2}, \\ref{assn_a4}, \\ref{A-consist}, and \\ref{A-nonsing1} hold. Then, under the null hypothesis of $M=1$, (a) $\\sup_{\\xi}\\sup_{\\vartheta \\in A_{n \\varepsilon }(\\xi)} |t(\\psi,\\pi)| = O_{p \\varepsilon }(n^{-1\/2})$; and (b) for any $c>0$,\n\\begin{equation} \\label{ln_appn}\n\\sup_{\\xi \\in \\Xi}\\sup_{\\vartheta \\in A_{n \\varepsilon c}(\\xi) } \\left| \\ell_n(\\psi,\\pi,\\xi) - \\ell_n(\\psi^*,\\pi,\\xi) - \\sqrt{n}t(\\psi,\\pi)' \\nu_n (s_{\\varrho k}) + n t(\\psi,\\pi)' \\mathcal{I}_\\varrho t(\\psi,\\pi)\/2 \\right| = o_{p \\varepsilon} (1).\n\\end{equation}\n\\end{proposition}\nWe proceed to derive the asymptotic distribution of the LRTS. With $s_{\\varrho k}$ defined in (\\ref{score}), define \n\\begin{equation} \\label{I_lambda}\n\\begin{aligned}\n&\\mathcal{I}_{\\eta} := \\mathbb{E}_{\\vartheta^*}(s_{\\eta k} s_{\\eta k}'), \\quad \\mathcal{I}_{\\lambda\\varrho_1\\varrho_2} := \\lim_{k\\rightarrow \\infty}\\mathbb{E}_{\\vartheta^*}(s_{\\lambda\\varrho_1 k} s_{\\lambda\\varrho_2 k}'), \\quad \\mathcal{I}_{\\lambda\\eta \\varrho } := \\lim_{k\\rightarrow \\infty}\\mathbb{E}_{\\vartheta^*}(s_{\\lambda\\varrho k} s_{\\eta k}'),\\\\\n&\\mathcal{I}_{\\eta \\lambda \\varrho} := \\mathcal{I}_{\\lambda \\eta \\varrho}', \\quad \\mathcal{I}_{\\lambda.\\eta \\varrho_1 \\varrho_2}:=\\mathcal{I}_{\\lambda \\varrho_1 \\varrho_2}-\\mathcal{I}_{\\lambda\\eta \\varrho_1}\\mathcal{I}_{\\eta}^{-1}\\mathcal{I}_{\\eta\\lambda \\varrho_2}, \\quad \\mathcal{I}_{\\lambda.\\eta \\varrho}:=\\mathcal{I}_{\\lambda.\\eta \\varrho\\varrho},\\quad\\ Z_{\\lambda \\varrho}:=(\\mathcal{I}_{\\lambda.\\eta \\varrho})^{-1}G_{\\lambda.\\eta \\varrho}, \n\\end{aligned}\n\\end{equation} \nwhere\n$G_{\\lambda. \\eta \\varrho}$ is a $q_\\lambda$-vector mean zero Gaussian process indexed by $\\varrho$ with $\\text{cov}(G_{\\lambda. \\eta\\varrho_1},G_{\\lambda. \\eta\\varrho_2}) = \\mathcal{I}_{\\lambda.\\eta \\varrho_1 \\varrho_2}$. Define the set of admissible values of $\\sqrt{n}\\alpha(1-\\alpha)v(\\lambda)$ when $n\\rightarrow \\infty$ by $v(\\mathbb{R}^q):= \\{ x \\in \\mathbb{R}^{q_\\lambda}: x = v(\\lambda) \\text{ for some } \\lambda \\in \\mathbb{R}^q\\}$. Define $\\tilde t_{\\lambda \\varrho}$ by\n\\begin{equation} \\label{t-lambda}\nr_{\\lambda \\varrho}(\\tilde t_{\\lambda \\varrho}) = \\inf_{t_\\lambda \\in v(\\mathbb{R}^q)}r_{\\lambda \\varrho}(t_{\\lambda }), \\quad r_{\\lambda \\varrho}(t_{\\lambda}) := (t_{\\lambda} -Z_{\\lambda \\varrho})' \\mathcal{I}_{\\lambda.\\eta \\varrho} (t_{\\lambda} -Z_{\\lambda \\varrho}).\n\\end{equation}\nThe following proposition establishes the asymptotic null distribution of the LRTS. \n\\begin{proposition} \\label{P-LR}\nSuppose Assumptions \\ref{assn_a1}, \\ref{assn_a2}, \\ref{assn_a4}, \\ref{A-consist}, and \\ref{A-nonsing1} hold. Then, under the null hypothesis of $M=1$, $LR_n \\overset{d}{\\rightarrow} \\sup_{ \\varrho \\in\\Theta_{\\varrho}} \\left(\\tilde t_{\\lambda \\varrho}' \\mathcal{I}_{\\lambda.\\eta \\varrho} \\tilde t_{\\lambda \\varrho} \\right)$.\n\\end{proposition} \nIn Proposition \\ref{P-LR}, the LRTS and its asymptotic distribution depend on the choice of $\\epsilon$ because $\\Theta_\\varrho=[-1+2\\epsilon,1-2\\epsilon]$. It is possible to develop a version of the EM test \n\\citep[][]{chenli09as, chenlifu12jasa, kasaharashimotsu15jasa} in this context that does not impose an explicit restriction on the parameter space for $p_{11}$ and $p_{22}$; however, we leave such an extension to future research.\n\n\\begin{remark}\nWhen applied to the Markov regime switching model, the tests of \\citet{carrasco14em} use the residuals from projecting $\\nabla_{\\theta \\theta'} f_k \/ f_k + 2 \\sum_{t=1}^{k-1} \\varrho^{k-t} (\\nabla_{\\theta} f_{t} \/ f_{t}) (\\nabla_{\\theta'} f_{k} \/ f_{k})$ on $\\nabla_{\\theta} f_k \/ f_k$, where both are evaluated at the one-regime MLE. Therefore, in the non-normal case, the LRT and tests of \\citet{carrasco14em} are based on the same score function.\n\\end{remark}\n\n\\subsection{Heteroscedastic normal distribution}\\label{subsec:hetero_normal}\n\nSuppose that $Y_k\\in \\mathbb{R}$ in the $j$-th regime follows a normal distribution with regime-specific intercept $\\mu_j$ and variance $\\sigma_j^2$. We split $\\theta_j$ into $\\theta_j = (\\zeta_j,\\sigma_j^2)'= (\\mu_j,\\beta_j',\\sigma_j^2)'$, and write the density of the $j$-th regime as\n\\begin{equation}\\label{normal-density}\nf(y_k|\\overline{\\bf{y}}_{k-1};\\gamma,\\theta_j) =f(y_k|\\overline{\\bf{y}}_{k-1};\\gamma,\\zeta_j,\\sigma^2_j) = \\frac{1}{\\sigma_j}\\phi\\left( \\frac{y_k - \\mu_j - \\varpi(\\overline{\\bf{y}}_{k-1};\\gamma,\\beta_j ) }{\\sigma_j}\\right),\n\\end{equation}\nfor some function $\\varpi$. In many applications, $\\varpi$ is a linear function of $\\gamma$ and $\\beta_j$, e.g., $\\varpi(\\overline{\\bf{y}}_{k-1},w_k;\\gamma,\\beta_j)= (\\overline{\\bf{y}}_{k-1})'\\beta_j + w_k'\\gamma$. Consider the following reparameterization introduced in \\citet{kasaharashimotsu15jasa} ($\\theta$ in Kasahara and Shimotsu corresponds to $\\zeta$ here):\n\\begin{equation}\n\\left(\n\\begin{array}{c}\n\\zeta_1\\\\\n\\zeta_2\\\\\n\\sigma_1^2\\\\\n\\sigma_2^2\n\\end{array}\n\\right) =\\left(\n\\begin{array}{c}\n\\nu_{\\zeta} + (1-\\alpha)\\lambda_{\\zeta} \\\\\n\\nu_{\\zeta} -\\alpha\\lambda_{\\zeta}\\\\\n\\nu_\\sigma + (1- \\alpha)(2\\lambda_\\sigma+ C_1 \\lambda_\\mu^2)\\\\\n\\nu_\\sigma - \\alpha(2\\lambda_\\sigma+ C_2 \\lambda_\\mu^2)\n\\end{array}\n\\right), \\label{repara2}\n\\end{equation}\nwhere $\\nu_{\\zeta}=(\\nu_\\mu,\\nu_{\\beta}')'$, $\\lambda_{\\zeta}=(\\lambda_\\mu,\\lambda_{\\beta}')'$, $C_1 := -(1\/3)(1 + \\alpha)$, and $C_2 := (1\/3)(2 - \\alpha)$, so that $C_1=C_2-1$. Collect the reparameterized parameters, except for $\\alpha$, into one vector $\\psi_{\\alpha}$. As in Section \\ref{subsec:nonnormal}, we suppress the subscript $\\alpha$ from $\\psi_{\\alpha}$. Let the reparameterized density be \n\\begin{equation} \\label{repara_g_hetero}\ng_{\\psi}(y_k|\\overline{\\bf{y}}_{k-1},x_k) = f\\left(y_k|\\overline{\\bf{y}}_{k-1};\\gamma,\\nu_\\zeta + (q_k-\\alpha)\\lambda_\\zeta, \\nu_\\sigma + (q_k - \\alpha)(2\\lambda_\\sigma + (C_2 -q_k) \\lambda_\\mu^2 )\\right).\n\\end{equation}\nLet $\\psi := (\\eta',\\lambda')' \\in \\Theta_\\psi = \\Theta_\\eta \\times \\Theta_\\lambda$, where $\\eta := (\\gamma',\\nu_\\zeta',\\nu_\\sigma)'$ and $\\lambda := (\\lambda_\\zeta',\\lambda_\\sigma)'$. Because the likelihood function of a normal mixture model is unbounded when $\\sigma_j \\rightarrow 0$ \\citep{hartigan85book}, we impose $\\sigma_j \\geq \\epsilon_\\sigma$ for a small $\\epsilon_\\sigma >0$ in $\\Theta_\\psi$. We proceed to derive the derivatives of $g_{\\psi}(Y_k|\\overline{\\bf{Y}}_{k-1},X_k)$ evaluated at $\\psi^*$. $\\nabla_\\psi g_k^*$, $\\nabla_{\\lambda\\eta'}g_k^*$, and $\\nabla_{\\lambda\\lambda'}g_k^*$ are the same as those given in (\\ref{dg}) except for $\\nabla_{\\lambda_\\mu^2}g_k^*$ and that those with respect to $\\lambda_\\sigma^j$ are multiplied by $2^j$. The higher-order derivatives of $g_{\\psi}(Y_k|\\overline{\\bf{Y}}_{k-1},X_k)$ with respect to $\\lambda_\\mu$ are derived by following \\citet{kasaharashimotsu15jasa}. \nFrom Lemma \\ref{lemma_normal_der} and the fact that normal density $f(\\mu,\\sigma^2)$ satisfies\n\\begin{equation} \\label{norma_der}\n\\begin{aligned}\n\\nabla_{\\mu^2} f(\\mu,\\sigma^2) &= 2\\nabla_{\\sigma^2} f(\\mu,\\sigma^2), \\quad \\nabla_{\\mu^3} f(\\mu,\\sigma^2) = 2\\nabla_{\\mu\\sigma^2} f(\\mu,\\sigma^2), \\ \\ \\text{and}\\\\\n\\nabla_{\\mu^4} f(\\mu,\\sigma^2) &= 2 \\nabla_{\\mu^2\\sigma^2} f(\\mu,\\sigma^2)= 4 \\nabla_{ \\sigma^2 \\sigma^2} f(\\mu,\\sigma^2),\n\\end{aligned}\n\\end{equation}\nwe have\n\\begin{equation} \\label{d3g}\n\\nabla_{\\lambda_\\mu^i} g_k^* = d_{ik} \\nabla_{\\mu^i} f_k^*, \\quad i = 1, \\ldots, 4,\n\\end{equation}\nwhere \n\\begin{align*} \n d_{0k} & :=1, \\quad d_{1k} := q_k - \\alpha, \\quad d_{2k} := (q_k - \\alpha)(C_2 - \\alpha), \\quad d_{3k} := 2 (q_k - \\alpha)^2(1-\\alpha-q_k), \\\\\nd_{4k} & := -2(q_k - \\alpha)^4 + 3(q_k-\\alpha)^2(\\alpha - C_2)^2. \n\\end{align*} \nIt follows from $\\mathbb{E}_{\\vartheta^*}[ q_k|\\overline{{\\bf Y}}_{-\\infty}^n]=\\alpha$, (\\ref{markov_moments}), and elementary calculation that\n\\begin{equation} \\label{d3g2}\n\\begin{aligned}\n\\mathbb{E}_{\\vartheta^*}[ d_{ik}|\\overline{{\\bf Y}}_{-\\infty}^n] &= 0, \\quad \\mathbb{E}_{\\vartheta^*}[ \\nabla_{\\lambda_\\mu^i} g_k^*|\\overline{{\\bf Y}}_{-\\infty}^k] = 0, \\quad i = 1,2,3, \\\\\n\\mathbb{E}_{\\vartheta^*}[ d_{4k}|\\overline{{\\bf Y}}_{-\\infty}^n] &= \\alpha(1 - \\alpha) b(\\alpha), \\\\\n\\mathbb{E}_{\\vartheta^*}[ \\nabla_{\\lambda_\\mu^4} g_k^*|\\overline{{\\bf Y}}_{-\\infty}^k] &= \\alpha(1-\\alpha)b(\\alpha)\\nabla_{\\mu^4} f_k^* = \\alpha(1-\\alpha) b(\\alpha) 4\\nabla_{\\sigma^2\\sigma^2}f_k^* = b(\\alpha) \\mathbb{E}_{\\vartheta^*}[ \\nabla_{\\lambda_\\sigma^2} g_k^*|\\overline{{\\bf Y}}_{-\\infty}^k],\n\\end{aligned}\n\\end{equation}\nwith $b(\\alpha): = -(2\/3) (\\alpha^2 - \\alpha + 1) < 0$. Hence, $\\mathbb{E}_{\\vartheta^*}[ \\nabla_{\\lambda_\\sigma^2} g_k^*|\\overline{{\\bf Y}}_{-\\infty}^k]$ and $\\mathbb{E}_{\\vartheta^*}[ \\nabla_{\\lambda_\\mu^4} g_k^*|\\overline{{\\bf Y}}_{-\\infty}^k]$ are linearly dependent.\n\nWe proceed to derive a representation of $\\nabla^j \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})$ in terms of $\\nabla^j f_k^*$. Repeating the calculation leading to (\\ref{d1p})--(\\ref{d2p}) and using (\\ref{d3g2}) gives the following. First, (\\ref{d1p}) and (\\ref{d2p0}) still hold; second, the elements of $\\nabla_{\\lambda\\lambda'} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})$ except for the $(1,1)$-th element are given by (\\ref{d2p}) after adjusting that the derivative with respect to $\\lambda_{\\sigma}$ must be multiplied by 2 (e.g., $\\mathbb{E}_{\\vartheta^*}[\\nabla_{\\lambda_\\sigma} g_k^* |\\overline{{\\bf Y}}_{-\\infty}^n] = 2 \\nabla_{\\sigma^2}f_k^*$ and $\\mathbb{E}_{\\vartheta^*}[\\nabla_{\\lambda_\\sigma\\lambda_\\mu} g_k^* |\\overline{{\\bf Y}}_{-\\infty}^n] = 2 \\nabla_{\\sigma^2\\mu}f_k^*$); third,\n\\begin{equation} \\label{d2p-lambda}\n\\frac{\\nabla_{\\lambda_\\mu^2} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})} = \\alpha(1-\\alpha) \\sum_{t=1}^{k-1} \\varrho^{k-t} \\left( 2 \\frac{\\nabla_{\\mu} f_{t}^*}{f_{t}^*} \\frac{\\nabla_{\\mu} f_{k}^*}{f_{k}^*}\\right).\n\\end{equation} \nWhen $\\varrho \\neq 0$, $\\nabla_{\\lambda_\\mu^2} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})$ is a non-degenerate random variable as in the non-normal case. When $\\varrho=0$, however, $\\nabla_{\\lambda_\\mu^2} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})$ becomes identically equal to 0, and indeed the first non-zero derivative with respect to $\\lambda_\\mu$ is the fourth derivative.\n\nBecause of this degeneracy, we derive the asymptotic distribution of the LRTS by expanding $\\ell_n(\\psi,\\pi,\\xi)-\\ell_n(\\psi^*,\\pi,\\xi)$ four times. It is not correct, however, to simply approximate $\\ell_n(\\psi,\\pi,\\xi)-\\ell_n(\\psi^*,\\pi,\\xi)$ by a quadratic function of $\\lambda_\\mu^2$ (and other terms) when $\\varrho \\neq 0$ and a quadratic function of $\\lambda_\\mu^4$ when $\\varrho=0$. This results in discontinuity at $\\varrho=0$ and fails to provide a valid uniform approximation. We establish a uniform approximation by expanding $\\ell_n(\\psi,\\pi,\\xi)$ four times but expressing $\\ell_n(\\psi,\\pi,\\xi)$ in terms of $\\varrho\\lambda_\\mu^2$, $\\lambda_\\mu^4$, and other terms.\n \nFor $m \\geq 0$, define $\\zeta_{k,m}(\\varrho):= \\sum_{t=-m+1}^{k-1} \\varrho^{k-t-1} 2 \\nabla_{\\mu} f_{t}^*\\nabla_{\\mu} f_{k}^*\/ f_{t}^* f_{k}^*$. Then, we can write (\\ref{d2p-lambda}) as\n\\begin{equation} \\label{d2p2}\n\\frac{\\nabla_{\\lambda_\\mu^2} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\alpha(1-\\alpha) \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})} = \\sum_{t=1}^{k-1} \\varrho^{k-t}\\left( 2 \\frac{\\nabla_{\\mu} f_{t}^*}{f_{t}^*} \\frac{\\nabla_{\\mu} f_{k}^*}{f_{k}^*}\\right) = \\varrho \\zeta_{k,0}(\\varrho).\n\\end{equation}\nNote that $\\zeta_{k,m}(\\varrho)$ satisfies $\\mathbb{E}_{\\vartheta^*}[\\zeta_{k,m}(\\varrho)| \\overline{{\\bf Y}}_{-m}^{k-1}]=0$ and is non-degenerate even when $\\varrho=0$. Define $v(\\lambda_\\beta)$ as $v(\\lambda)$ in (\\ref{v_lambda}) but replacing $\\lambda$ with $\\lambda_\\beta$. Collect the relevant parameters as \n\\begin{equation} \\label{t-psi2}\nt(\\psi,\\pi) :=\n\\begin{pmatrix}\n\\eta - \\eta^* \\\\\nt_{\\lambda}(\\lambda,\\pi)\n\\end{pmatrix},\n\\end{equation}\nwhere \n\\begin{equation} \\label{t_lambda_rho}\nt_{\\lambda}(\\lambda,\\pi)\n:=\\alpha(1-\\alpha)\n\\begin{pmatrix}\n\\varrho \\lambda_\\mu^2 \\\\\n \\lambda_\\mu\\lambda_\\sigma\\\\\n\\lambda_\\sigma^2 + b(\\alpha)\\lambda_\\mu^4\/12\\\\\n\\lambda_\\beta \\lambda_\\mu\\\\\n\\lambda_\\beta \\lambda_\\sigma\\\\\nv(\\lambda_\\beta)\n\\end{pmatrix}, \n\\end{equation}\nwith $b(\\alpha)= -(2\/3)(\\alpha^2-\\alpha+1)<0$. Recall $\\theta_j = (\\zeta_j',\\sigma_j^2)'= (\\mu_j,\\beta_j',\\sigma_j^2)'$. Similarly to (\\ref{score_lambda}), define the elements of the generalized score by\n\\begin{equation} \\label{score_lambda_beta}\n\\begin{pmatrix}\n* & s_{\\lambda_{\\mu\\beta} \\varrho k} & s_{\\lambda_{\\mu\\sigma} \\varrho k} \\\\\ns_{\\lambda_{\\beta\\mu} \\varrho k} & s_{\\lambda_{\\beta\\beta} \\varrho k} & s_{\\lambda_{\\beta\\sigma} \\varrho k} \\\\\ns_{\\lambda_{\\sigma\\mu} \\varrho k} & s_{\\lambda_{\\beta\\sigma} \\varrho k}& s_{\\lambda_{\\sigma\\sigma} \\varrho k} \n\\end{pmatrix}\n= \\frac{\\nabla_{\\theta \\theta'} f_k^*}{f_k^*} + \\sum_{t= 1 }^{k-1} \\varrho^{k-t} \\left( \\frac{\\nabla_{\\theta} f_{t}^*}{f_{t}^*} \\frac{\\nabla_{\\theta'} f_{k}^*}{f_{k}^*} + \\frac{\\nabla_{\\theta} f_{k}^*}{f_{k}^*} \\frac{\\nabla_{\\theta'} f_{t}^*}{f_{t}^*} \\right).\n\\end{equation}\nDefine the generalized score as \n\\begin{equation}\\label{score_normal}\ns_{\\varrho k} : = \n\\begin{pmatrix}\ns_{\\eta k} \\\\\ns_{\\lambda \\varrho k}\n\\end{pmatrix},\n\\quad \\text{where}\\quad\ns_{\\eta k} : = \\begin{pmatrix}\n\\nabla_{\\gamma} f_k^* \/ f_k^* \\\\\n\\nabla_{\\theta} f_k^* \/ f_k^* \n\\end{pmatrix} \\\n\\text{ and }\\ \ns_{\\lambda \\varrho k}\n:= \n\\begin{pmatrix}\n\\zeta_{k,0}(\\varrho)\/2 \\\\\n 2 s_{\\lambda_{\\mu\\sigma} \\varrho k} \\\\\n 2 s_{\\lambda_{\\sigma\\sigma} \\varrho k}\\\\\ns_{\\lambda_{\\beta \\mu} \\varrho k}\\\\\n 2 s_{\\lambda_{\\beta \\sigma} \\varrho k}\\\\\n V(s_{\\lambda_{\\beta\\beta} \\varrho k}) \n\\end{pmatrix}.\n\\end{equation} \nThe following proposition establishes a uniform approximation of the log-likelihood ratio. \n\\begin{assumption} \\label{A-nonsing2}\n(a) $0< \\inf_{\\varrho\\in\\Theta_{\\varrho}} \\lambda_{\\min}(\\mathcal{I}_\\varrho) \\leq \\sup_{\\varrho\\in\\Theta_{\\varrho}}\\lambda_{\\max}(\\mathcal{I}_\\varrho) < \\infty$ for $\\mathcal{I}_{\\varrho}= \\lim_{k\\rightarrow \\infty } \\mathbb{E}_{\\vartheta^*}(s_{\\varrho k}s_{\\varrho k}')$, where $s_{\\varrho k}$ is given in (\\ref{score_normal}). (b) $\\sigma_1^*,\\sigma_2^* >\\epsilon_\\sigma$. \n\\end{assumption} \n\n\\begin{proposition} \\label{P-quadratic-N1} Suppose Assumptions \\ref{assn_a1}, \\ref{assn_a2}, \\ref{assn_a4}, \\ref{A-consist}, and \\ref{A-nonsing2} hold and the density of the $j$-th regime is given by (\\ref{normal-density}). Then, under the null hypothesis of $M=1$, (a) $\\sup_{\\vartheta \\in A_{n \\varepsilon}(\\xi)} |t(\\psi,\\pi)| = O_{p \\varepsilon }(n^{-1\/2})$; and (b) for any $c>0$,\n\\begin{equation} \\label{ln_appn_N1}\n\\sup_{\\xi \\in \\Xi} \\sup_{\\vartheta \\in A_{n\\varepsilon c}(\\xi) } \\left| \\ell_n(\\psi,\\pi,\\xi) - \\ell_n(\\psi^*,\\pi,\\xi) - \\sqrt{n}t(\\psi,\\pi)' \\nu_n (s_{\\varrho k}) + n t(\\psi,\\pi)' \\mathcal{I}_\\varrho t(\\psi,\\pi)\/2 \\right| = o_{p \\varepsilon }(1).\n\\end{equation}\n\\end{proposition}\nThe asymptotic null distribution of the LRTS is characterized by the supremum of $ 2 t_{\\lambda}' G_{\\lambda.\\eta\\varrho}- t_{\\lambda}' \\mathcal{I}_{\\lambda.\\eta \\varrho} t_{\\lambda}$, where $G_{\\lambda. \\eta \\varrho}$ and $\\mathcal{I}_{\\lambda.\\eta \\varrho}$ are defined analogously to those in (\\ref{I_lambda}) but with $s_{\\varrho k}$ defined in (\\ref{score_normal}), and the supremum is taken with respect to $t_{\\lambda}$ and $\\varrho\\in\\Theta_{\\varrho}$ under the constraint implied by the limit of the set of possible values of $\\sqrt{n}t_{\\lambda}(\\lambda,\\pi)$ as $n\\rightarrow\\infty$. This constraint is given by the union of $\\Lambda_{\\lambda}^1$ and $\\Lambda_{\\lambda \\varrho}^2$, where $q_\\beta := \\dim(\\beta)$, $q_{\\lambda}:= 3 +2q_\\beta+q_\\beta(q_\\beta+1)\/2$, and\n\\begin{equation}\n\\begin{aligned}\\label{Lambda-lambda}\n&\\Lambda_{\\lambda }^1:=\\{ t_{\\lambda}=( t_{\\varrho\\mu^2}, t_{\\mu\\sigma },t_{\\sigma^2 },t_{\\beta\\mu}',t_{\\beta\\sigma}',t_{v(\\beta)}')' \\in \\mathbb{R}^{q_{\\lambda} } : \\\\\n& \\qquad \\qquad (t_{\\varrho\\mu^2},t_{\\mu\\sigma },t_{\\sigma^2 },t_{\\beta\\mu}')'\\in \\mathbb{R}\\times\\mathbb{R}\\times\\mathbb{R}_{-}\\times\\mathbb{R}^{q_\\beta}, t_{\\beta\\sigma}=0, t_{v(\\beta)}=0\\}, \\quad \\\\\n&\\Lambda_{\\lambda \\varrho}^2:=\\{ t_{\\lambda }= (t_{\\varrho\\mu^2},t_{\\mu\\sigma },t_{\\sigma^2},t_{\\beta\\mu}',t_{\\beta\\sigma}',t_{v(\\beta)}')' \\in \\mathbb{R}^{q_{\\lambda} }: t_{\\varrho\\mu^2} = \\varrho \\lambda_{\\mu}^2, t_{\\mu\\sigma }= \\lambda_\\mu\\lambda_\\sigma, \\\\\n& \\qquad \\qquad \nt_{\\sigma^2}= \\lambda_\\sigma^2, t_{\\beta\\mu }= \\lambda_{\\beta} \\lambda_\\mu, t_{\\beta\\sigma }= \\lambda_{\\beta} \\lambda_\\sigma, t_{v(\\beta)} =v_{\\beta}(\\lambda_{\\beta})\\ \\text{for some }\\lambda\\in \\mathbb{R}^{2+q_\\beta} \\}. \n\\end{aligned}\n\\end{equation}\nNote that $\\Lambda_{\\lambda\\varrho}^2$ depends on $\\varrho$, whereas $\\Lambda_{\\lambda}^1$ does not depend on $\\varrho$. Heuristically, $\\Lambda_{\\lambda}^1$ and $\\Lambda_{\\lambda\\varrho}^2$ correspond to the limits of the set of possible values of $\\sqrt{n}t_{\\lambda}(\\lambda,\\pi)$ when $\\liminf_{n\\to\\infty}n^{1\/8}|\\lambda_{\\mu}| > 0$ and $\\lambda_{\\mu}=o(n^{-1\/8})$, respectively. When $\\liminf_{n\\to\\infty}n^{1\/8}|\\lambda_{\\mu}| > 0$, we have $(\\hat\\lambda_\\sigma, \\hat\\lambda_\\beta) =O_p(n^{-3\/8})$ because $t_{\\lambda}(\\hat\\lambda,\\pi)=O_p(n^{-1\/2})$. Further, the set of possible values of $\\sqrt{n} \\varrho\\lambda_{\\mu}^2$ converges to $\\mathbb{R}$ because $\\varrho$ can be arbitrarily small. Consequently, the limit of $\\sqrt{n}t_{\\lambda}(\\lambda,\\pi)$ is characterized by $\\Lambda_{\\lambda}^1$.\n\nDefine $Z_{\\lambda \\varrho}$ and $\\mathcal{I}_{\\lambda.\\eta \\varrho}$ as in (\\ref{I_lambda}) but with $s_{\\pi k}$ defined in (\\ref{score_normal}). Let $Z_{\\lambda 0}$ and $\\mathcal{I}_{\\lambda.\\eta 0}$ denote $Z_{\\lambda \\varrho}$ and $\\mathcal{I}_{\\lambda.\\eta \\varrho}$ evaluated at $\\varrho=0$. Define $\\tilde{t}_{\\lambda}^1$ and $\\tilde{t}_{\\lambda\\varrho}^2$ by\n\\begin{equation} \\label{t-lambda-N1}\n\\begin{aligned}\nr_{\\lambda}(\\tilde{t}_{\\lambda }^1) & = \\inf_{t_{\\lambda} \\in \\Lambda_{\\lambda}^1}r_{\\lambda}(t_{\\lambda}), \\quad r_{\\lambda}(t_{\\lambda}) := (t_{\\lambda} -Z_{\\lambda 0})' \\mathcal{I}_{\\lambda.\\eta 0} (t_{\\lambda} -Z_{\\lambda 0}) \\quad \\\\\nr_{\\lambda\\varrho}(\\tilde{t}_{\\lambda\\varrho}^2) & = \\inf_{t_{\\lambda} \\in \\Lambda_{\\lambda\\varrho}^2}r_{\\lambda\\varrho}(t_{\\lambda}), \\quad r_{\\lambda\\varrho}(t_{\\lambda}) := \n(t_{\\lambda} -Z_{\\lambda \\varrho})' \\mathcal{I}_{\\lambda.\\eta \\varrho} (t_{\\lambda} -Z_{\\lambda \\varrho}). \n\\end{aligned}\n\\end{equation}\nThe following proposition establishes the asymptotic null distribution of the LRTS. \n\\begin{proposition} \\label{P-LR-N1}\nSuppose that the assumptions in Proposition \\ref{P-quadratic-N1} hold. Then, under the null hypothesis of $M=1$, $LR_n \\overset{d}{\\rightarrow} \\max\\{ \\mathbb{I}\\{\\varrho=0\\} (\\tilde t_{\\lambda }^1)' \\mathcal{I}_{\\lambda.\\eta 0} \\tilde t_{\\lambda }^1, \\sup_{\\varrho\\in\\Theta_{\\varrho}} (\\tilde t_{\\lambda \\varrho}^2)' \\mathcal{I}_{\\lambda.\\eta \\varrho} \\tilde t_{\\lambda \\varrho}^2 \\}$. \n\\end{proposition} \n\\begin{remark}\n\\citet{quzhuo17wp} derive the asymptotic distribution of the LRTS under the restriction that $\\varrho\\geq \\epsilon>0$. \n\\end{remark}\n\n\\begin{remark}\nIt is possible to extend our analysis to the exponential-LR type tests studied by \\citet{andrewsploberger94em} and \\citet{carrasco14em}.\n\\end{remark}\n\n\\subsection{Homoscedastic normal distribution}\\label{subsec:homo_normal}\n\nSuppose that $Y_k\\in \\mathbb{R}$ in the $j$-th regime follows a normal distribution with the regime-specific intercept $\\mu_j$ but with common variance $\\sigma^2$. We split $\\gamma$ and $\\theta_j$ into $\\gamma=(\\tilde \\gamma',\\sigma^2)'$ and $\\theta_j=(\\mu_j,\\beta_j')'$, and write the density of the $j$-th regime as\n\\begin{equation}\\label{normal-density-homo}\nf(y_k|\\overline{\\bf{y}}_{k-1};\\gamma,\\theta_j) =f(y_k|\\overline{\\bf{y}}_{k-1};\\tilde \\gamma,\\theta_j,\\sigma^2) = \\frac{1}{\\sigma}\\phi\\left( \\frac{y_k - \\mu_j- \\varpi(\\overline{\\bf{y}}_{k-1};\\tilde\\gamma,\\beta_j ) }{\\sigma}\\right),\n\\end{equation}\nfor some function $\\varpi$. Consider the following reparameterization:\n\\begin{equation}\n\\left(\n\\begin{array}{c}\n\\theta_1\\\\\n\\theta_2\\\\\n\\sigma^2 \n\\end{array}\n\\right) =\\left(\n\\begin{array}{c}\n\\nu_{\\theta} + (1-\\alpha)\\lambda \\\\\n\\nu_{\\theta} -\\alpha\\lambda \\\\ \n\\nu_\\sigma - \\alpha(1-\\alpha) \\lambda_\\mu^2\n\\end{array}\n\\right), \\label{repara-homo}\n\\end{equation}\nwhere $\\nu_{\\theta}=(\\nu_\\mu,\\nu_\\beta')'$ and $\\lambda=(\\lambda_\\mu,\\lambda_\\beta')'$. Collect the reparameterized parameters, except for $\\alpha$, into one vector $\\psi_{\\alpha}$. Suppressing $\\alpha$ from $\\psi_{\\alpha}$, let the reparameterized density be\n\\begin{equation}\\label{repara_g_homo}\ng_{\\psi}(y_k|\\overline{\\bf{y}}_{k-1},x_k) = f\\left(y_k|\\overline{\\bf{y}}_{k-1};\\tilde\\gamma,\\nu_\\theta+(q_k-\\alpha)\\lambda, \\nu_\\sigma -\\alpha(1-\\alpha) \\lambda_\\mu^2 \\right).\n\\end{equation}\nLet $\\eta = (\\tilde\\gamma',\\nu_\\theta',\\nu_\\sigma)'$; then, the first and second derivatives of $g_{\\psi}(y_k|\\overline{\\bf{y}}_{k-1},x_k)$ with respect to $\\eta$ and $\\lambda$ are the same as those given in (\\ref{dg}) except for $\\nabla_{\\lambda_\\mu^2}g_{\\psi}(y_k|\\overline{\\bf{y}}_{k-1},x_k)$. We derive the higher-order derivatives of $g_{\\psi}(y_k|\\overline{\\bf{y}}_{k-1},x_k)$ with respect to $\\lambda_\\mu$. From Lemmas \\ref{lemma_normal_der} and (\\ref{norma_der}), we obtain\n\\begin{equation} \\label{d3g-homo}\n\\begin{aligned}\n\\nabla_{\\lambda \\eta^i} g_k^* &= d_{1k} \\nabla_{\\theta \\eta^i} f_k^* \\quad \\text{for } i=0,1,\\ldots,\\\\\n\\nabla_{\\lambda_\\mu^i} g_k^* &= d_{ik} \\nabla_{\\mu^i} f_k^* \\quad \\text{for }i = 0,1, \\ldots, 4,\n\\end{aligned}\n\\end{equation}\nwhere $d_{0k} :=1$, $d_{1k} := q_k - \\alpha$, $d_{2k} := (q_k-\\alpha)^2-\\alpha(1-\\alpha)$, $d_{3k} := (q_k-\\alpha)^3 - 3(q_k-\\alpha)\\alpha(1-\\alpha)$, and\n$d_{4k} := (q_k - \\alpha)^4 -6 (q_k - \\alpha)^2 \\alpha(1-\\alpha) + 3\\alpha^2(1 - \\alpha)^2$.\nIt follows from $\\mathbb{E}_{\\vartheta^*}[ q_k|\\overline{{\\bf Y}}_{-\\infty}^n]=\\alpha$, (\\ref{markov_moments}), and elementary calculation that \n\\begin{equation} \\label{d3g2-homo}\n\\begin{aligned}\n\\mathbb{E}_{\\vartheta^*}[ \\nabla_{\\lambda_\\mu^i} g_k^*|\\overline{{\\bf Y}}_{0}^k] &= 0, \\quad\\mathbb{E}_{\\vartheta^*}[ d_{ik}|\\overline{{\\bf Y}}_{0}^k] = 0,\\quad i = 1,2, \\\\\n \\mathbb{E}_{\\vartheta^*}[ d_{3k}|\\overline{{\\bf Y}}_{0}^k] &= \\alpha(1 - \\alpha) (1-2\\alpha),\n\\quad \\mathbb{E}_{\\vartheta^*}[ d_{4k}|\\overline{{\\bf Y}}_{0}^k] = \\alpha(1 - \\alpha) (1-6\\alpha+6\\alpha^2).\n\\end{aligned}\n\\end{equation}\nRepeating the calculation leading to (\\ref{d1p})--(\\ref{d2p}) and using (\\ref{d3g2-homo}) gives the following. First, (\\ref{d1p}) and (\\ref{d2p0}) still hold; second, the elements of $\\nabla_{\\lambda\\lambda'} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})$ are given by (\\ref{d2p}) except for the $(1,1)$-th element; third, $\\nabla_{\\lambda_\\mu^2} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})$ is given by (\\ref{d2p-lambda}). Further, Lemma \\ref{lemma_d34_homo} in the appendix shows that when $\\varrho=0$, $\\nabla_{\\lambda_\\mu^3} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) = \\alpha(1-\\alpha)(1-2\\alpha) \\nabla_{\\mu^3}f_k^*\/f_k^*$ and $\\nabla_{\\lambda_\\mu^4} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) = \\alpha(1-\\alpha)(1-6\\alpha+6\\alpha^2) \\nabla_{\\mu^4}f_k^*\/f_k^*$. Because $\\nabla_{\\lambda_\\mu^3} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})=0$ when $\\alpha=1\/2$ and $\\varrho=0$, we expand $\\ell_n(\\psi,\\pi,\\xi)$ four times and express it in terms of $\\varrho\\lambda_\\mu^2$, $(1-2\\alpha)\\lambda_\\mu^3$, $\\lambda_\\mu^4$, and other terms to establish a uniform approximation.\n\nCollect the relevant parameters as\n\\begin{equation} \\label{t-psi2-homo}\nt(\\psi,\\pi) :=\n\\begin{pmatrix}\n\\eta - \\eta^* \\\\\nt_{\\lambda}(\\lambda,\\pi)\n\\end{pmatrix} \\quad \\text{and}\\quad\nt_{\\lambda}(\\lambda,\\pi)\n:=\\alpha(1-\\alpha)\n\\begin{pmatrix}\n \\varrho \\lambda_\\mu^2 \\\\\n (1-2\\alpha) \\lambda_\\mu^3 \\\\\n (1-6\\alpha+6\\alpha^2)\\lambda_\\mu^4 \\\\\n \\lambda_\\beta \\lambda_\\mu\\\\\n v(\\lambda_\\beta)\n\\end{pmatrix}.\n\\end{equation}\nDefine the generalized score as\n\\begin{equation}\\label{score_normal_homo}\ns_{\\varrho k} : = \n\\begin{pmatrix}\ns_{\\eta k} \\\\\ns_{\\lambda \\varrho k}\n\\end{pmatrix},\n\\quad \\text{where}\\quad\ns_{\\eta k} : = \\begin{pmatrix}\n\\nabla_{\\gamma} f_k^* \/ f_k^* \\\\\n\\nabla_{\\theta} f_k^* \/ f_k^* \n\\end{pmatrix} \\\n\\text{ and }\\ \ns_{\\lambda \\varrho k}\n:= \n\\begin{pmatrix}\n\\zeta_{k,0}(\\varrho)\/2 \\\\\ns_{\\lambda_{\\mu}^3 k}\/3!\\\\\ns_{\\lambda_{\\mu}^4 k}\/4!\\\\\ns_{\\lambda_{\\beta \\mu} \\varrho k}\\\\\nV(s_{\\lambda_{\\beta\\beta} \\varrho k})\n\\end{pmatrix},\n\\end{equation}\nwhere $\\zeta_{k,m}(\\varrho)$ is defined as in (\\ref{d2p2}), $s_{\\lambda_{\\mu}^i k}:=\\nabla_{\\mu^i}f_k^*\/f_k^*$ for $i=3,4$, and\n$s_{\\lambda_{\\beta \\mu} \\varrho k}$ and $s_{\\lambda_{\\beta\\beta} \\varrho k}$ are defined as in (\\ref{score_lambda_beta}) but using the density (\\ref{normal-density-homo}) in place of (\\ref{normal-density}).\nDefine, with $q_\\beta := \\dim(\\beta)$ and $q_{\\lambda}:=3 + q_\\beta+q_\\beta(q_\\beta+1)\/2$,\n\\begin{equation}\n\\begin{aligned}\\label{Lambda-lambda-homo}\n&\\Lambda_{\\lambda }^1:=\\{ t_{\\lambda}= ( t_{\\varrho\\mu^2}, t_{\\mu^3}, t_{\\mu^4}, t_{\\beta\\mu}', t_{v(\\beta)}')' \\in \\mathbb{R}^{q_{\\lambda}} : (t_{\\varrho\\mu^2}, t_{\\mu^3}, t_{\\mu^4}, t_{\\beta\\mu}')'\\in \\mathbb{R}\\times\\mathbb{R}\\times\\mathbb{R}_{-}\\times\\mathbb{R}^{q_\\beta}, t_{v(\\beta)}=0\\}, \\quad \\\\\n&\\Lambda_{\\lambda \\varrho}^2:=\\{ t_{\\lambda}= ( t_{\\varrho\\mu^2}, t_{\\mu^3}, t_{\\mu^4}, t_{\\beta\\mu}', t_{v(\\beta)}')' \\in \\mathbb{R}^{q_{\\lambda}}: t_{\\varrho\\mu^2} = \\varrho \\lambda_{\\mu}^2, t_{\\mu^3}=t_{\\mu^4}=0, t_{\\beta\\mu }= \\lambda_{\\beta} \\lambda_\\mu, \\\\\n& \\qquad \\quad \n t_{v(\\beta)} =v_{\\beta}(\\lambda_{\\beta})\\ \\text{for some }\\lambda\\in \\mathbb{R}^{1+q_\\beta} \\}. \n\\end{aligned}\n\\end{equation}\n\nThe following two propositions correspond to Propositions \\ref{P-quadratic-N1} and \\ref{P-LR-N1}, establishing a uniform approximation of the log-likelihood ratio and asymptotic distribution of the LRTS.\n\\begin{assumption} \\label{A-nonsing2-homo}\n$0< \\inf_{\\varrho \\in \\Theta_{\\varrho}} \\lambda_{\\min}(\\mathcal{I}_\\varrho) \\leq \\sup_{\\varrho \\in \\Theta_{\\varrho}}\\lambda_{\\max}(\\mathcal{I}_\\varrho) < \\infty$ for $\\mathcal{I}_{\\varrho}= \\lim_{k\\rightarrow \\infty} \\mathbb{E}_{\\vartheta^*}(s_{\\varrho k}s_{\\varrho k}')$, where $s_{\\varrho k}$ is given in (\\ref{score_normal_homo}).\n\\end{assumption}\n\\begin{proposition} \\label{P-quadratic-N1-homo}\nSuppose Assumptions \\ref{assn_a1}, \\ref{assn_a2}, \\ref{assn_a4}, \\ref{A-consist}, and \\ref{A-nonsing2-homo} hold and the density of the $j$-th regime is given by (\\ref{normal-density-homo}). Then, statements (a) and (b) of Proposition \\ref{P-quadratic-N1} hold.\n\\end{proposition} \n\\begin{proposition} \\label{P-LR-N1-homo} Suppose that the assumptions in Proposition \\ref{P-quadratic-N1-homo} hold. Then, under the null hypothesis of $M=1$, $LR_n \\overset{d}{\\rightarrow} \\max\\{ \\mathbb{I}\\{\\varrho=0\\} (\\tilde t_{\\lambda }^1)' \\mathcal{I}_{\\lambda.\\eta 0} \\tilde t_{\\lambda }^1, \\sup_{\\varrho\\in\\Theta_\\varrho} (\\tilde t_{\\lambda \\varrho}^2)' \\mathcal{I}_{\\lambda.\\eta \\varrho} \\tilde t_{\\lambda \\varrho}^2 \\}$, where $\\tilde{t}_{\\lambda}^1$ and $\\tilde{t}_{\\lambda\\varrho}^2$ are defined as in (\\ref{t-lambda-N1}) but in terms of $(Z_{\\lambda \\varrho}, \\mathcal{I}_{\\lambda.\\eta \\varrho}, Z_{\\lambda 0},\\mathcal{I}_{\\lambda.\\eta 0})$ constructed with $s_{\\varrho k}$ defined in (\\ref{score_normal_homo}) and $\\Lambda_{\\lambda }^1$ and $\\Lambda_{\\lambda \\varrho}^2$ defined in (\\ref{Lambda-lambda-homo}).\n\\end{proposition} \n\n\\section{Testing $H_0:M=M_0$ against $H_A:M=M_0+1$ for $M_0 \\geq 2$}\\label{sec-general}\nIn this section, we derive the asymptotic distribution of the LRTS for testing the null hypothesis of $M_0$ regimes against the alternative of $M_0+1$ regimes for general $M_0 \\geq 2$. We suppress the covariate ${\\bf W}_{a}^b$ unless confusion might arise.\n\nLet $\\vartheta_{M_0}^*=((\\vartheta_{M_0,x}^*)',(\\vartheta_{M_0,y}^*)')'$ denote the parameter of the $M_0$-regime model, where $\\vartheta_{M_0,x}^*$ contains $p_{ij}^*= q_{\\vartheta^*_{M_0,x}}(i,j)>0$ for $i= 1,\\ldots,M_0$ and $j=1,\\ldots,M_0-1$, and $\\vartheta_{M_0,y}^* = ((\\theta_1^*)',\\ldots,(\\theta_{M_0}^*)',(\\gamma^*)')'$. We assume $\\max_{i}\\sum_{j=1}^{M_0-1}p_{ij}^*<1$ and $\\theta_{1}^*<\\ldots< \\theta_{M_0}^*$ for identification. \nThe true $M_0$-regime conditional density of ${\\bf Y}_1^n$ given $\\overline{\\bf Y}_{0}$ and $x_0$ is \n\\begin{equation} \\label{true_model}\np_{\\vartheta_{M_0}^*}({\\bf Y}_1^n| \\overline{\\bf Y}_{0},x_0) = \\sum_{{\\bf x}_1^n\\in \\mathcal{X}_{M_0}^n}\\prod_{k=1}^n p_{\\vartheta_{M_0}^*}(Y_k,x_k|\\overline{\\bf Y}_{k-1},x_{k-1}),\n\\end{equation} \nwhere $p_{\\vartheta_{M_0}^*} (y_k,x_k| \\overline{\\bf{y}}_{k-1},x_{k-1}) = g_{\\vartheta_{M_0,y}^*}(y_k|\\overline{\\bf{y}}_{k-1}, x_k) q_{\\vartheta_{M_0,x}^*}(x_{k-1},x_k)$ with $g_{\\vartheta_{M_0,y}^*}(y_k|\\overline{\\bf{y}}_{k-1},x_k) = \\sum_{j=1,\\ldots,M_0} \\mathbb{I}\\{x_k=j\\}f(y_k|\\overline{\\bf{y}}_{k-1};\\gamma,\\theta_j^*)$.\n \nLet the conditional density of ${\\bf Y}_{1}^n$ of an $(M_0+1)$-regime model be\n\\begin{equation} \\label{fitted_model}\np_{\\vartheta_{M_0+1}}({\\bf Y}_1^n| \\overline{\\bf Y}_{0},x_0) := \\sum_{{\\bf x}_1^n\\in \\mathcal{X}_{M_0+1}^n}\\prod_{k=1}^n p_{\\vartheta_{M_0+1}}(Y_k,x_k|\\overline{\\bf Y}_{k-1},x_{k-1}),\n\\end{equation}\nwhere $p_{\\vartheta_{M_0+1}}(y_k,x_k|\\overline{\\bf y}_{k-1},x_{k-1})$ is defined similarly to $p_{\\vartheta_{M_0}^*}(y_k,x_k|\\overline{\\bf y}_{k-1},x_{k-1})$ with $\\vartheta_{M_0+1,x}:= \\{p_{ij}\\}_{i=1,\\ldots,M_0+1,j=1,\\ldots,M_0}$ and $\\vartheta_{M_0+1,y} := (\\theta_1',\\ldots,\\theta_{M_0+1}',\\gamma')'$. We assume that $\\min_{i,j}p_{ij} \\geq \\epsilon$ for some $\\epsilon\\in (0,1\/2)$.\n\nWrite the null hypothesis as $H_0 = \\cup_{m=1}^{M_0} H_{0m}$ with\n\\[\nH_{0m} : \\theta_1 < \\cdots < \\theta_{m} = \\theta_{m + 1} < \\cdots < \\theta_{M_0 + 1}.\n\\] \nDefine the set of values of $\\vartheta_{M_0+1}$ that yields the true density (\\ref{true_model}) under $\\mathbb{P}_{\\vartheta^*_{M_0}}$ as $\\Upsilon^*:=\\{\\vartheta_{M_0 + 1} \\in \\Theta_{M_0+1,\\epsilon}: p_{\\vartheta_{M_0+1}}({\\bf Y}_1^n| \\overline{\\bf Y}_{0}, x_0) = p_{\\vartheta_{M_0}^*}({\\bf Y}_1^n| \\overline{\\bf Y}_{0}, x_0)\\ \\mathbb{P}_{\\vartheta^*_{M_0}}\\text{-a.s.}\\}$. Under $H_{0m}$, the $(M_0 + 1)$-regime model (\\ref{fitted_model}) generates the true $M_0$-regime density (\\ref{true_model}) if $\\theta_m = \\theta_{m + 1} = \\theta_{m}^{*}$ and the transition matrix of $X_k$ reduces to that of the true $M_0$-regime model.\n\nWe reparameterize the transition probability of $X_k$ by writing $\\vartheta_{M_0+1,x}$ as $\\vartheta_{M_0+1,x} = (\\vartheta_{xm}',\\pi_{xm}')'$, where $\\vartheta_{xm}$ is point identified under $H_{0m}$, while $\\pi_{xm}$ is not point identified under $H_{0m}$. The transition probability of $X_k$ under $\\vartheta_{M_0+1,x}$ equals the transition probability of $X_k$ under $\\vartheta_{M_0,x}^*$ if and only if $\\vartheta_{xm} = \\vartheta_{xm}^*$. The detailed derivation including the definition of $\\vartheta_{xm}^*$ is provided in Section \\ref{subsec:p_m_repara} in the appendix. \nDefine the subset of $\\Upsilon^*$ that corresponds to $H_{0m}$ as \n\\begin{align*} \n\\Upsilon_{m}^*& := \\left\\{\\vartheta_{M_0 + 1} \\in \\Theta_{M_0 + 1}: \n\\theta_j=\\theta_{j}^*\\ \\text{for}\\ 1 \\leq j < m;\\ \\theta_m = \\theta_{m + 1} = \\theta_{m}^{*}; \\right.\\\\\n& \\qquad \\left. \\theta_{j} = \\theta_{j - 1}^*\\ \\text{for}\\ h+1 < j \\leq M_0+1;\\ \\gamma=\\gamma^*;\\ \\vartheta_{xm}=\\vartheta_{xm}^*\\right\\};\n\\end{align*} \nthen, $\\Upsilon^*= \\Upsilon_{1}^* \\cup \\cdots \\cup \\Upsilon_{M_0}^*$ holds. \n\nFor $M=M_0,M_0+1$, let $\\ell_n(\\vartheta_M,\\xi_M) := \\log \\left( \\sum_{x_0=1}^M p_{\\vartheta_M}({\\bf Y}_1^n| \\overline{\\bf Y}_{0},x_0) \\xi_M(x_0) \\right)$ denote the $M$-regime log-likelihood for a given initial distribution $\\xi_M(x_0) \\in \\Xi_M$. We treat $\\xi_M(x_0)$ as fixed. Let $\\hat\\vartheta_{M_0}:= \\arg\\max_{\\vartheta_{M_0} \\in \\Theta_{{M_0}}} \\ell_n(\\vartheta_{M_0},\\xi_{M_0})$ and $\\hat\\vartheta_{M_0+1}:= \\arg\\max_{\\vartheta_{M_0+1} \\in \\Theta_{M_0+1}} \\ell_n(\\vartheta_{M_0+1},\\xi_{M_0+1})$. The following proposition shows that the MLE is consistent in the sense that the distance between $\\hat{\\vartheta}_{M_0+1}$ and $\\Upsilon^*$ tends to 0 in probability. The proof of Proposition \\ref{P-consist_M} is essentially the same as the proof of Proposition \\ref{P-consist} and hence is omitted. \n\\begin{assumption} \\label{A-consist_M}\n(a) $\\Theta_{M_0}$ and $\\Theta_{M_0+1}$ are compact, and $\\vartheta_{M_0}^*$ is in the interior of $\\Theta_{M_0}$. (b) For all $(x,x') \\in \\mathcal{X}$ and all $(\\overline{{\\bf y}},y',w)\\in \\mathcal{Y}^s\\times \\mathcal{Y}\\times \\mathcal{W}$, $f(y'|\\overline{\\bf{y}}_0,w;\\gamma,\\theta)$ is continuous in $(\\gamma,\\theta)$. (c) $\\mathbb{E}_{\\vartheta^*_{M_0}}[ \\log ( p_{\\vartheta_{M_0}}(Y_1 |\\overline{\\bf{Y}}_{-m}^0,{\\bf W}_{-m}^1)] = \\mathbb{E}_{\\vartheta^*_{M_0}}[ \\log p_{\\vartheta_{M_0}^*}(Y_1 |\\overline{\\bf{Y}}_{-m}^0,{\\bf W}_{-m}^1) ]$ for all $m \\geq 0$ if and only if $\\vartheta_{M_0} = \\vartheta_{M_0}^*$. (d) $\\mathbb{E}_{\\vartheta^*_{M_0}}[ \\log ( p_{\\vartheta_{M_0+1}}(Y_1 |\\overline{\\bf{Y}}_{-m}^0,{\\bf W}_{-m}^0) ] = \\mathbb{E}_{\\vartheta^*_{M_0}} [\\log p_{\\vartheta_{M_0}^*}(Y_1 |\\overline{\\bf{Y}}_{-m}^0,{\\bf W}_{-m}^1) ]$ for all $m \\geq 0$ if and only if $\\vartheta_{M_0+1}\\in \\Upsilon^*$. \\end{assumption}\n\n\\begin{proposition} \\label{P-consist_M} \nSuppose Assumptions \\ref{assn_a1}, \\ref{assn_a2}, and \\ref{A-consist_M} hold. Then, under the null hypothesis of $M=M_0$, $\\hat\\vartheta_{M_0} \\overset{p}{\\rightarrow} \\vartheta_{M_0}^*$ and $\\inf_{\\vartheta_{M_0+1} \\in \\Upsilon^*} |\\hat{\\vartheta}_{M_0+1}-\\vartheta_{M_0+1}|\\overset{p}{\\rightarrow} 0$.\n\\end{proposition}\n\nLet $LR_{M_0,n}:=2[\\ell_n(\\hat\\vartheta_{M_0+1},\\xi_{M_0+1})-\\ell_n(\\hat\\vartheta_{M_0},\\xi_{M_0})]$ denote the LRTS for testing $H_0:M=M_0$ against $H_A:M=M_0+1$.\nWe proceed to derive the asymptotic distribution of the LRTS by analyzing the behavior of the LRTS when $\\vartheta_{M_0+1} \\in \\Upsilon^*_m$ for each $m$. Define $J_m :=\\{m,m+1\\}$. Observe that if ${\\bf X}_1^k \\in J_m^k$, then ${\\bf X}_1^k$ follows a two-state Markov chain on $J_m$ whose transition probability is characterized by $\\alpha_m := \\mathbb{P}_{\\vartheta_{M_0+1}}(X_k=m|X_k \\in J_m)$ and $\\varrho_m := \\text{corr}_{\\vartheta_{M_0+1}}(X_{k-1},X_{k}|(X_{k-1},X_{k}) \\in J_m^2)$. Collect reparameterized $\\pi_{xm}$ into $\\pi_{xm} := (\\varrho_m,\\alpha_m, \\phi_{m}')'$, where $\\phi_m$ does not affect the transition probability of ${\\bf X}_1^k$ when ${\\bf X}_1^k \\in J_m^{k}$. See Section \\ref{subsec:p_m_repara} in the appendix for the detailed derivation.\n \nDefine $q_{kj} := \\mathbb{I}\\{X_k=j\\}$; then, we can write $\\alpha_m$ and $\\varrho_m$ as $\\alpha_m = \\mathbb{E}_{\\vartheta_{M_0+1}}(q_{km}|X_k \\in J_m)$ and $\\varrho_m =\\text{corr}_{\\vartheta_{M_0+1}}(q_{k-1,m},q_{km}|(X_{k-1},X_{k}) \\in J_m^2)$. Because $\\overline{{\\bf Y}}_{-\\infty}^\\infty$ provides no information for distinguishing between $X_k=m$ and $X_k={m+1}$ if $\\theta_m = \\theta_{m+1}$, we can write $\\alpha_m$ and $\\varrho_m$ as\n\\begin{equation} \\label{alpha_rho_h}\n\\alpha_m = \\mathbb{E}_{\\vartheta_{M_0+1}}(q_{km}|X_k \\in J_m,\\overline{{\\bf Y}}_{-\\infty}^\\infty )\\quad \\text{and}\\quad\n\\varrho_m = \\text{corr}_{\\vartheta_{M_0+1}}(q_{k-1,m},q_{km}|(X_{k-1},X_{k}) \\in J_m^2,\\overline{{\\bf Y}}_{-\\infty}^\\infty). \n\\end{equation}\n\n\\subsection{Non-normal distribution} \\label{sec:M-nonnormal}\n\nFor non-normal component distributions, consider the following reparameterization similar to (\\ref{repara}):\n\\begin{equation*\n\\begin{pmatrix}\n\\theta_m\\\\\n\\theta_{m+1} \\\\\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n\\nu_m + (1-\\alpha_m) \\lambda_m\\\\\n\\nu_m- \\alpha_m\\lambda_m\\\\\n\\end{pmatrix}. \n\\end{equation*} \nCollect the reparameterized identified parameters into one vector $\\psi_m: = (\\eta_m',\\lambda_m')'$, where $\\eta_m=(\\gamma',\\{\\theta_j'\\}_{j=1}^{m-1}, \\nu_m',\\{\\theta_j'\\}_{j=m+2}^{M_0+1},\\vartheta_{xm}')'$, so that the reparameterized $(M_0+1)$-regime log-likelihood function is $\\ell_n(\\psi_m,\\pi_{xm},\\xi_{M_0+1})$. Let $\\psi_m^{*}=(\\eta_m^*,\\lambda_m^*)=((\\vartheta_{M_0}^*)',0')'$ denote the value of $\\psi_{m}$ under $H_{0m}$. Define the reparameterized conditional density of $y_k$ as \n\\begin{align*}\ng_{\\psi_m}(y_k|\\overline{\\bf{y}}_{k-1}, x_k) & : = \\mathbb{I}\\{x_k \\in J_m\\} f(y_k|\\overline{\\bf{y}}_{k-1};\\gamma,\\nu_m + (q_{km}-\\alpha_m) \\lambda_m) + \\sum_{j \\in \\overline J_m} q_{kj} f(y_k|\\overline{\\bf{y}}_{k-1}; \\gamma,\\theta_j),\n\\end{align*}\nwhere $\\overline J_m := \\{1,\\ldots,M_0+1\\}\\setminus J_m$. Let $f_{mk}^{*}$ denote $f(Y_k|\\overline{\\bf{Y}}_{k-1};\\gamma^*,\\theta_m^*)$. It follows from (\\ref{alpha_rho_h}) and the law of iterated expectations that\n\\begin{equation} \\label{qk_moments_M}\n\\begin{aligned}\n&\\mathbb{E}_{\\vartheta^*_{M_0}}\\left[\\frac{\\mathbb{I}\\{X_k \\in J_m\\}(q_{km} -\\alpha_m)}{g_{\\psi_m^*}(Y_k|\\overline{\\bf{Y}}_{k-1}, X_k)} \\middle|\\overline{{\\bf Y}}_{-\\infty}^n\\right] \\\\ \n&=\\mathbb{E}_{\\vartheta^*_{M_0}}\\left[\\mathbb{E}_{\\vartheta^*_{M_0}}\\left[\\frac{ q_{km} -\\alpha_m }{f_{mk}^*}\\middle|X_k \\in J_m,\\overline{{\\bf Y}}_{-\\infty}^n\\right] \\mathbb{I}\\{X_k \\in J_m\\} \\middle|\\overline{{\\bf Y}}_{-\\infty}^n\\right] = 0, \\\\\n&\\mathbb{E}_{\\vartheta^*_{M_0}}\\left[\\frac{\\mathbb{I}\\{X_{t_1} \\in J_m\\}\\mathbb{I}\\{X_{t_2} \\in J_m\\}(q_{t_1h}-\\alpha_m) (q_{t_2h}-\\alpha_m)}{g_{\\psi_m^*}(Y_{t_1}|\\overline{\\bf{Y}}_{{t_1}-1}, X_{t_1})g_{\\psi_m^*}(Y_{t_2}|\\overline{\\bf{Y}}_{{t_2}-1}, X_{t_2})} \\middle| \\overline{\\bf Y}_{-\\infty}^n \\right] \\\\\n& = \\mathbb{E}_{\\vartheta^*_{M_0}}\\left[\\mathbb{E}_{\\vartheta^*_{M_0}}\\left[\\frac{(q_{t_1h}-\\alpha_m) (q_{t_2h}-\\alpha_m)}{f_{m t_1}^* f_{m t_2}^*} \\middle|{\\bf X}_{t_1}^{t_2} \\in J_m^{t_2-t_1+1},\\overline{{\\bf Y}}_{-\\infty}^n\\right] \\mathbb{I}\\{(X_{t_1},X_{t_2}) \\in J_m^2\\} \\middle|\\overline{{\\bf Y}}_{-\\infty}^n \\right] \\\\\n& = \\frac{\\alpha_m(1-\\alpha_m)\\varrho_m^{t_2-t_1}}{f_{m t_1}^* f_{m t_2}^*} \\mathbb{P}_{\\vartheta^*_{M_0}}((X_{t_1},X_{t_2}) \\in J_m^2|\\overline{{\\bf Y}}_{-\\infty}^n), \\quad t_2 \\geq t_1,\n\\end{aligned}\n\\end{equation} \nwhere the second equality holds because $g_{\\psi_m^*}(Y_k|\\overline{\\bf{Y}}_{k-1}, X_k)=f_{mk}^*$ if $X_k \\in J_m$, and the last equality holds because, conditional on $\\{{\\bf X}_{t_1}^{t_2} \\in J_m^{t_2-t_1+1},\\overline{\\bf Y}_{-\\infty}^n\\} $, ${\\bf X}_{t_1}^{t_2}$ is a two-state stationary Markov process with parameter $(\\alpha_m,\\varrho_m)$.\n\nLet $g_{0k}^*$, $q_{0k}^*$, and $\\overline p_{0k}^*$ denote $g_{\\vartheta_{M_0,y}^*}(Y_k,X_k| \\overline{{\\bf Y}}_{k-1},X_{k-1})$, $q_{\\vartheta_{M_0,x}^*}(X_{k-1},X_k)$, and $\\overline p_{\\vartheta_{M_0}^*}(Y_k| \\overline{{\\bf Y}}_{0}^{k-1})$. Let $\\nabla g_{0k}^*$ denote the derivative of $g_{\\vartheta_{M_0,y}}(Y_k,X_k| \\overline{{\\bf Y}}_{k-1},X_{k-1})$ evaluated at $\\vartheta_{M_0,y}^*$, and define $\\nabla q_{0k}^*$ and $\\nabla \\overline p_{0k}^*$ similarly. Repeating a derivation similar to (\\ref{dg})--(\\ref{d2p}) but using (\\ref{qk_moments_M}) in place of (\\ref{qk_moments}), we obtain \n\\begin{equation} \\label{d1p_M}\n\\begin{aligned}\n& \\nabla_{\\eta_m} \\overline p_{\\psi^*_m \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^*_m \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) \\\\\n& =\n \\sum_{t=1}^k \\mathbb{E}_{\\vartheta^*} \\left[\\nabla_{\\vartheta_{M_0}} \\log (g_{0t}^*q_{0t}^*) \\middle|\\overline{{\\bf Y}}_{0}^k \\right] - \\sum_{t=1}^{k-1} \\mathbb{E}_{\\vartheta^*} \\left[\\nabla_{\\vartheta_{M_0}} \\log (g_{0t}^* q_{0t}^*)\\middle|\\overline{{\\bf Y}}_{0}^{k-1} \\right] \\\\\n& = \\nabla_{\\vartheta_{M_0}} \\overline p_{\\vartheta_{M_0}^*} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\vartheta_{M_0}^*} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}), \n\\end{aligned}\n\\end{equation} \n\\begin{align}\n& \\nabla_{\\lambda_m} \\overline p_{\\psi^*_m \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) \/ \\overline p_{\\psi^*_m \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) =0, \\quad \\nabla_{{\\lambda_m} \\eta_m'} \\overline p_{\\psi^*_m \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^*_m \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})=0, \\label{d2p_M0} \\\\\n& \\frac{\\nabla_{{\\lambda_m} {\\lambda_m} '} \\overline p_{\\psi^*_m \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\overline p_{\\psi^*_m \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}\n = \\alpha_m(1-\\alpha_m) \\frac{ \\nabla_{\\theta\\theta'}f_{mk}^{*}}{f_{mk}^{*}} \\mathbb{P}_{\\vartheta^*_{M_0}}(X_k \\in J_m|\\overline{{\\bf Y}}_{0}^k) \\nonumber \\\\\n& \\quad + \\alpha_m(1-\\alpha_m) \\sum_{t=1}^{k-1} \\varrho_{m}^{k-t} \\left( \\frac{\\nabla_{\\theta} f_{mt}^{*}}{f_{mt}^{*}} \\frac{\\nabla_{\\theta'} f_{mk}^{*}}{f_{mk}^{*}} + \\frac{\\nabla_{\\theta} f_{mk}^{*}}{f_{mk}^{*}} \\frac{\\nabla_{\\theta'} f_{mt}^{*}}{f_{mt}^{*}} \\right) \\mathbb{P}_{\\vartheta^*_{M_0}}((X_t,X_k) \\in J_m^2|\\overline{{\\bf Y}}_{0}^k).\\label{d2p_M}\n\\end{align} \n\nDefine $\\tilde \\varrho := (\\varrho_1,\\ldots,\\varrho_{M_0})'$, define $t_{\\lambda}(\\lambda_m,\\pi_m)$ as $t_{\\lambda}(\\lambda,\\pi)$ in (\\ref{score}) by replacing $(\\lambda,\\pi)$ with $(\\lambda_m,\\pi_m)$, and let\n\\begin{equation} \\label{stilde}\nt(\\psi_m,\\pi_m) := \n\\begin{pmatrix}\n\\eta_m - \\eta^* \\\\\n t_{\\lambda}(\\lambda_m,\\pi_m) \n\\end{pmatrix}, \\ \n\\tilde{s}_{\\tilde\\varrho k} :=\n\\begin{pmatrix}\n\\tilde {s}_{\\eta k} \\\\\n\\tilde {s}_{\\lambda \\tilde\\varrho k}\n\\end{pmatrix},\\ \\text{ where }\\\n\\tilde s_{\\eta k} : = \n\\frac{\\nabla_{\\eta_m} \\overline p_{\\psi^*_m \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) }{\\overline p_{\\psi^*_m \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})},\n\\ \n\\tilde{s}_{\\lambda \\tilde\\varrho k} :=\n\\begin{pmatrix}\n{s}_{\\lambda \\varrho_1 k}^1 \\\\\n\\vdots \\\\\n{s}_{\\lambda\\varrho_{M_0} k}^{M_0}\n\\end{pmatrix},\n\\end{equation}\nand $s_{\\lambda \\varrho_m k}^m := V(s_{\\lambda\\lambda \\varrho_m k}^m)$, where $s_{\\lambda\\lambda \\varrho_m k}^m $ is defined similarly to (\\ref{score_lambda}) as\n\\begin{equation} \\label{score_lambda_M}\n\\begin{aligned}\ns_{\\lambda\\lambda\\varrho_m k}^m &: = \\frac{\\nabla_{\\theta \\theta'} f_{mk}^{*}}{f_{mk}^{*}} \\mathbb{P}_{\\vartheta^*_{M_0}} (X_k \\in J_m|\\overline{{\\bf Y}}_{0}^k) \\\\\n& + \\sum_{t=1}^{k-1} \\varrho_m ^{k-t} \\left( \\frac{\\nabla_{\\theta} f_{mt}^{*}}{f_{mt}^{*}} \\frac{\\nabla_{\\theta'} f_{mk}^{*}}{f_{mk}^{*}} + \\frac{\\nabla_{\\theta} f_{mk}^{*}}{f_{mk}^{*}} \\frac{\\nabla_{\\theta'} f_{mt}^{*}}{f_{mt}^{*}} \\right) \\mathbb{P}_{\\vartheta^*_{M_0}} ((X_t,X_k) \\in J_m^2|\\overline{{\\bf Y}}_{0}^k).\n\\end{aligned}\n\\end{equation}\nSimilarly to (\\ref{I_lambda}), define \n\\begin{equation} \\label{I_tilde_lambda}\n\\begin{aligned}\n\\tilde {\\mathcal{I}}_{\\eta} &:= \\mathbb{E}_{\\vartheta^*_{M_0}}(\\tilde s_{\\eta k} \\tilde s_{\\eta k}'), \\quad \\tilde{\\mathcal{I}}_{\\lambda\\tilde \\varrho_1\\tilde \\varrho_2} := \\lim_{k\\rightarrow\\infty} \\mathbb{E}_{\\vartheta^*_{M_0}}(\\tilde s_{\\lambda\\tilde \\varrho_1 k} \\tilde s_{\\lambda\\tilde \\varrho_2 k}'), \\quad\n\\tilde{\\mathcal{I}}_{\\lambda\\eta \\tilde \\varrho } := \\lim_{k\\rightarrow\\infty} \\mathbb{E}_{\\vartheta^*_{M_0}}(\\tilde s_{\\lambda \\tilde \\varrho k} \\tilde s_{\\eta k}'), \\\\ \\tilde{\\mathcal{I}}_{\\eta \\lambda \\tilde \\varrho} &:= \\tilde{\\mathcal{I}}_{\\lambda \\eta \\tilde \\varrho}', \\quad \\tilde{\\mathcal{I}}_{\\lambda.\\eta \\tilde\\varrho_1\\tilde\\varrho_2}:= \\tilde{\\mathcal{I}}_{\\lambda \\tilde\\varrho_1\\tilde\\varrho_2}-\\tilde{\\mathcal{I}}_{\\lambda\\eta \\tilde\\varrho_1}\\tilde{\\mathcal{I}}_{\\eta}^{-1}\\tilde{\\mathcal{I}}_{\\eta\\lambda \\tilde\\varrho_2},\\quad \\tilde{\\mathcal{I}}_{\\lambda.\\eta \\varrho_m}^m := \\mathbb{E}_{\\vartheta^*_{M_0}}[G_{\\lambda.\\eta \\varrho_m}^m (G_{\\lambda.\\eta \\varrho_m}^m)'],\\\\\nZ^m_{\\lambda \\varrho_m} &:= (\\tilde{\\mathcal{I}}_{\\lambda.\\eta \\varrho_m}^m)^{-1}G_{\\lambda.\\eta \\varrho_m}^m,\n\\end{aligned}\n\\end{equation}\nwhere ${G}_{\\lambda.\\eta \\tilde \\varrho}=((G_{\\lambda.\\eta \\varrho_1}^1)',\\ldots,(G_{\\lambda.\\eta\\varrho_{M_0}}^{M_0})')'$ is an $M_0 q_\\lambda$-vector mean zero Gaussian process with \\\\$\\text{cov}({G}_{\\lambda.\\eta \\tilde\\varrho_1},{G}_{\\lambda.\\eta \\tilde \\varrho_2}) = \\tilde{\\mathcal{I}}_{\\lambda.\\eta \\tilde\\varrho_1\\tilde\\varrho_2}$. Note that $G_{\\lambda.\\eta \\tilde \\varrho}$ corresponds to the residuals from projecting $\\tilde s_{\\lambda \\tilde \\varrho k}$ on $\\tilde s_{\\eta k}$. Define $\\tilde t_{\\lambda \\varrho_m}^m$ by\n\\begin{equation*}\ng_{\\lambda \\varrho_m}^m(\\tilde t_{\\lambda \\varrho_m}^m) = \\inf_{t_\\lambda \\in v(\\mathbb{R}^q)}g_{\\lambda \\varrho_m}^m(t_{\\lambda }), \\quad g_{\\lambda \\varrho_m}^m(t_{\\lambda}) := (t_{\\lambda} -Z_{\\lambda \\varrho_m}^m)' \\tilde{\\mathcal{I}}_{\\lambda.\\eta \\varrho_m}^m (t_{\\lambda} -Z_{\\lambda \\varrho_m}^m). \n\\end{equation*}\nThe following proposition gives the asymptotic null distribution of the LRTS. Under the stated assumptions, the log-likelihood function permits a quadratic approximation in the neighborhood of $\\Upsilon_{m}^*$ similar to the one in Proposition \\ref{P-quadratic}. Define $A_{n\\varepsilon c}^m(\\xi) := \\{\\vartheta_{M_0+1} \\in \\Theta_{M_0+1} : \\{ \\ell_n(\\psi_m,\\pi_m,\\xi) - \\ell_n(\\psi_m^*,\\pi_m,\\xi) \\geq 0 \\} \\land |t(\\psi_m,\\pi_m)|< \\varepsilon \\} \\cup \\mathcal{N}_{c\/\\sqrt{n}}$. Under $H_0: M=M_0$, for any $c>0$, for $m=1,\\ldots,M_0$, and uniformly in $\\xi \\in \\Xi$ and $\\vartheta_{M_0+1} \\in A_{n\\varepsilon c}^m(\\xi)$,\n\\[\n\\ell_n(\\psi_m,\\pi_m,\\xi) - \\ell_n(\\psi_m^*,\\pi_m,\\xi) \\ - \\sqrt{n}t(\\psi_m,\\pi_m)' \\nu_n (s_{\\varrho_m k}) + n t(\\psi_m,\\pi_m)' \\mathcal{I}_{\\varrho_m} t(\\psi_m,\\pi_m)\/2 = o_{p \\varepsilon} (1),\n\\]\nwhere $s_{ \\varrho_m k} := (\\tilde{s}_{\\eta k}',({s}_{\\lambda \\varrho_m k}^m)')'$ and $\\mathcal{I}_{\\varrho_m} = \\lim_{k\\rightarrow\\infty} \\mathbb{E}_{\\vartheta^*_{M_0}}(s_{ \\varrho_m k}s_{ \\varrho_m k}')$. Consequently, the LRTS is asymptotically distributed as the maximum of the $M_0$ random variables, each of which represents the asymptotic distribution of the LRTS that tests $H_{0m}$. Denote the parameter space for $\\varrho_m$ by $\\Theta_{\\varrho_m}$, and let $\\tilde\\Theta_{\\varrho}:=\\Theta_{\\varrho_1}\\times\\ldots\\times\\Theta_{\\varrho_{M_0}}$.\n\\begin{assumption} \\label{A-nonsing_M}\n$0< \\inf_{\\tilde \\varrho \\in \\tilde\\Theta_{\\varrho}} \\lambda_{\\min}(\\tilde{\\mathcal{I}}_{\\tilde\\varrho}) \\leq \\sup_{\\tilde \\varrho \\in \\tilde\\Theta_{\\varrho}} \\lambda_{\\max}(\\tilde{\\mathcal{I}}_{\\tilde\\varrho}) < \\infty$ for $\\tilde{\\mathcal{I}}_{\\tilde\\varrho}:= \\lim_{k\\rightarrow\\infty}\\mathbb{E}_{\\vartheta^*_{M_0}}(\\tilde{s}_{\\tilde\\varrho k}\\tilde{s}_{\\tilde\\varrho k}')$, where $\\tilde{s}_{\\tilde\\varrho k}$ is given in (\\ref{stilde}).\n\\end{assumption}\n\\begin{proposition} \\label{P-LR_M}\nSuppose Assumptions \\ref{assn_a1}, \\ref{assn_a2}, \\ref{assn_a4}, \\ref{A-consist_M}, and \\ref{A-nonsing_M} hold. Then, under $H_0: M=M_0$, $LR_{M_0,n}\\overset{d}{\\rightarrow} \\max_{m=1,\\ldots,M_0}\\left\\{ \\sup_{\\varrho_m \\in \\Theta_{\\varrho}^m} \\left( (\\tilde t_{\\lambda \\varrho_m}^m)' \\tilde{\\mathcal{I}}_{\\lambda.\\eta \\varrho_m}^m \\tilde t_{\\lambda \\varrho_m}^m \\right) \\right\\} $.\n\\end{proposition}\n\n\\subsection{Heteroscedastic normal distribution}\\label{subsec:hetero_normal_M}\n\nAs in Section \\ref{subsec:hetero_normal}, we assume that $Y_k \\in \\mathbb{R}$ in the $j$-th regime follows a normal distribution with the regime-specific intercept and variance of which density is given by (\\ref{normal-density}). Consider the following reparameterization similar to (\\ref{repara2}):\n\\begin{equation*}\n\\left(\n\\begin{array}{c}\n\\zeta_m\\\\\n\\zeta_{m+1}\\\\\n\\sigma_m^2\\\\\n\\sigma_{m+1}^2\n\\end{array}\n\\right) =\\left(\n\\begin{array}{c}\n\\nu_{\\zeta m} + (1-\\alpha_m)\\lambda_{\\zeta m} \\\\\n\\nu_{\\zeta m} -\\alpha_m\\lambda_{\\zeta m}\\\\\n\\nu_{\\sigma m} + (1- \\alpha_m)(2\\lambda_{\\sigma m}+ C_1 \\lambda_{\\mu m}^2)\\\\\n\\nu_{\\sigma m} - \\alpha_m(2\\lambda_{\\sigma m}+ C_2 \\lambda_{\\mu m}^2)\n\\end{array}\n\\right),\n\\end{equation*}\nwhere $\\nu_{\\zeta m}=(\\nu_\\mu,\\nu_{\\beta}')'$, $\\lambda_{\\zeta m}=(\\lambda_{\\mu m},\\lambda_{\\beta m}')'$, $C_1 := -(1\/3)(1 + \\alpha_m)$, and $C_2 := (1\/3)(2 - \\alpha_m)$. As in Section \\ref{sec:M-nonnormal}, we collect the reparameterized identified parameters into $\\psi_m: = (\\eta_m',\\lambda_m')'$, where $\\eta_m=(\\gamma',\\{\\theta_j'\\}_{j=1}^{m-1}, \\nu_{\\zeta m}', \\nu_{\\sigma m},\\{\\theta_j'\\}_{j=m+2}^{M_0+1},\\vartheta_{xm}')'$ and $\\lambda_m:=(\\lambda_{\\zeta m}',\\lambda_{\\sigma m})'$. Similar to (\\ref{repara_g_hetero}), define the reparameterized conditional density of $y_k$ as\n\\begin{equation*}\n\\begin{aligned}\n& g_{\\psi_m}(y_k|\\overline{\\bf{y}}_{k-1}, x_k) = \\sum_{j \\in \\overline J_m} q_{kj} f(y_k|\\overline{\\bf{y}}_{k-1}; \\gamma,\\theta_j) \\\\\n& \\quad + \\mathbb{I}\\{x_k \\in J_m\\} f\\left(y_k|\\overline{\\bf{y}}_{k-1};\\gamma,\\nu_{\\zeta m}+(q_{km}-\\alpha_m)\\lambda_{\\zeta m}, \\nu_{\\sigma m} + (q_{km}-\\alpha_m)(2\\lambda_{\\sigma m} + (C_2-q_{km}) \\lambda_{\\mu m}^2 )\\right).\n\\end{aligned} \n\\end{equation*} \nLet $g_{mk}^*$, $f_{mk}^*$, $\\nabla g_{mk}^*$, and $\\nabla f_{mk}^*$ denote $g_{\\psi_m^{*}}(Y_k|\\overline{\\bf{Y}}_{k-1}, X_k)$, $f(Y_k|\\overline{\\bf Y}_{k-1};\\gamma^*,\\theta_m^*)$, $\\nabla g_{\\psi_m^{*}}(Y_k|\\overline{\\bf{Y}}_{k-1}, X_k)$, and $\\nabla f(Y_k|\\overline{\\bf Y}_{k-1};\\gamma^*,\\theta_m^*)$. From (\\ref{d3g}) and a derivation similar to (\\ref{qk_moments_M}), we obtain the following result that corresponds to (\\ref{d3g2}) in testing homogeneity:\n\\begin{equation} \\label{d3g2_M}\n\\begin{aligned}\n\\mathbb{E}_{\\vartheta^*_{M_0}} \\left[ \\nabla_{\\lambda_{\\mu m}^i} g_{mk}^{*} \/ g_{mk}^* \\middle|\\overline{{\\bf Y}}_{-\\infty}^k \\right] &= 0,\\quad i = 1,2,3,\\\\\n\\mathbb{E}_{\\vartheta^*_{M_0}} \\left[ \\nabla_{\\lambda_{\\mu m}^4} g_{mk}^{*} \/ g_{mk}^{*} \\middle| \\overline{{\\bf Y}}_{-\\infty}^k \\right] &= \\alpha_m(1 - \\alpha_m) b(\\alpha_m)( \\nabla_{\\mu^4}f_{mk}^* \/ f_{mk}^* )\\mathbb{P}_{\\vartheta^*_{M_0}}(X_k \\in J_m|\\overline{{\\bf Y}}_{-\\infty}^k) \\\\\n&= b(\\alpha_m) \\mathbb{E}_{\\vartheta^*_{M_0}}\\left[ \\nabla_{\\lambda_{\\sigma m}^2} g_{mk}^{*} \/ g_{mk}^{*} \\middle|\\overline{{\\bf Y}}_{-\\infty}^k\\right].\n\\end{aligned}\n\\end{equation}\nRepeating the calculation leading to (\\ref{d1p_M})--(\\ref{d2p_M}) and using (\\ref{d3g2_M}) gives the following. First, (\\ref{d1p_M}) and (\\ref{d2p_M0}) still hold; second, the elements of $\\nabla_{\\lambda_m\\lambda_m'} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})$ except for the $(1,1)$-th element are given by (\\ref{d2p_M}) while adjusting the derivative with respect to $\\lambda_{\\sigma m}$ by multiplying by 2; third,\n\\begin{equation*}\n\\frac{\\nabla_{\\lambda_{\\mu m}^2} \\overline p_{\\psi^*_m \\pi}(Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\overline p_{\\psi^*_m \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})} = \\alpha_m(1-\\alpha_m) \\sum_{t=1}^{k-1} \\varrho_m^{k-t} \\left( 2 \\frac{\\nabla_{\\mu} f_{mt}^*}{f_{mt}^*} \\frac{\\nabla_{\\mu} f_{mk}^*}{f_{mk}^*}\\right)\\mathbb{P}_{\\vartheta^*_{M_0}}((X_t,X_k) \\in J_m^2|\\overline{{\\bf Y}}_{0}^k).\n\\end{equation*} \n\nFor $m \\geq 0$, define $\\zeta_{k,m}^m(\\varrho_m):= \\sum_{t=-m+1}^{k-1} \\varrho_m^{k-t-1} 2 (\\nabla_{\\mu} f_{mt}^* \\nabla_{\\mu} f_{mk}^* \/ f_{mt}^* f_{mk}^*) \\mathbb{P}_{\\vartheta^*_{M_0}}((X_t,X_k) \\in J_m^2|\\overline{{\\bf Y}}_{0}^k)$. Similarly to (\\ref{score_lambda_beta}), define the elements of the generalized score as \n\\begin{equation}\n\\begin{aligned}\n&\n\\begin{pmatrix}\n * & s_{ \\lambda_{\\mu\\beta}\\varrho_m k}^m & s_{\\lambda_{\\mu\\sigma} \\varrho_m k}^m \\\\\ns_{ \\lambda_{\\beta\\mu}\\varrho_m k}^m & s_{ \\lambda_{\\beta\\beta}\\varrho_m k}^m & s_{ \\lambda_{\\beta\\sigma}\\varrho_m k}^m \\\\\ns_{ \\lambda_{\\sigma\\mu}\\varrho_m k}^m & s_{ \\lambda_{\\beta\\sigma}\\varrho_m k}^m & s_{ \\lambda_{\\sigma\\sigma}\\varrho_m k}^m \n\\end{pmatrix} : = \\frac{\\nabla_{\\theta \\theta'} f_{mk}^{*}}{f_{mk}^{*}} \\mathbb{P}_{\\vartheta^*_{M_0}} (X_k \\in J_m|\\overline{{\\bf Y}}_{0}^k) \\\\\n& \\qquad + \\sum_{t=1}^{k-1} \\varrho_m ^{k-t} \\left( \\frac{\\nabla_{\\theta} f_{mt}^{*}}{f_{mt}^{*}} \\frac{\\nabla_{\\theta'} f_{mk}^{*}}{f_{mk}^{*}} + \\frac{\\nabla_{\\theta} f_{mk}^{*}}{f_{mk}^{*}} \\frac{\\nabla_{\\theta'} f_{mt}^{*}}{f_{mt}^{*}} \\right) \\mathbb{P}_{\\vartheta^*_{M_0}} ((X_t,X_k) \\in J_m^2|\\overline{{\\bf Y}}_{0}^k).\n\\end{aligned}\n\\end{equation}\nSimilarly to (\\ref{score_normal}), define $\\tilde{s}_{\\tilde \\varrho k}$ as in (\\ref{stilde}) by redefining $s_{\\lambda \\varrho_m k}^m$ in (\\ref{stilde}) as\n\\begin{equation} \\label{stilde_normal}\ns_{\\lambda \\varrho_m k}^m\n:= \\begin{pmatrix}\n\\zeta_{k,0}^m(\\varrho_m)\/2 &\n2 s_{\\lambda_{\\mu\\sigma} \\varrho_m k}^m &\n2 s_{\\lambda_{\\sigma\\sigma} \\varrho_m k}^m&\n(s_{\\lambda_{\\beta \\mu} \\varrho_m k}^m)' &\n2 (s_{\\lambda_{\\beta \\sigma} \\varrho_m k}^m)' &\nV(s_{\\lambda_{\\beta\\beta} \\varrho _h k}^m)'\n\\end{pmatrix} '. \n\\end{equation}\nDefine ${\\mathcal{I}}_{\\lambda.\\eta \\varrho_m}^m$ and $Z^m_{\\lambda \\varrho_m}$ as in (\\ref{I_tilde_lambda}) with $s_{\\lambda \\varrho_m k}^m$ defined in (\\ref{stilde_normal}). Let $Z_{\\lambda 0}^m$ and $\\mathcal{I}_{\\lambda.\\eta 0}^m$ denote $Z_{\\lambda \\varrho_m}^m$ and $\\mathcal{I}_{\\lambda.\\eta \\varrho_m}^m$ evaluated at $\\varrho_m=0$. Define $\\Lambda_\\lambda^1$ as in (\\ref{Lambda-lambda}), and define $\\Lambda_{\\lambda \\varrho_m}^2$ as in (\\ref{Lambda-lambda}) by replacing $\\varrho$ with $\\varrho_m$. Similar to (\\ref{t-lambda-N1}), define $\\tilde{t}_{\\lambda}^{m1}$ and $\\tilde{t}_{\\lambda \\varrho_m}^{m2}$ by $r_{\\lambda}(\\tilde{t}_{\\lambda}^{m1}) = \\inf_{t_{\\lambda} \\in \\Lambda_{\\lambda}^1}r_{\\lambda}^m(t_{\\lambda})$ and $r_{\\lambda\\varrho_m}(\\tilde{t}_{\\lambda \\varrho_m}^{m2}) = \\inf_{t_{\\lambda} \\in \\Lambda_{\\lambda\\varrho_m}^2}r_{\\lambda\\varrho_m}^m(t_{\\lambda})$, where $r_{\\lambda}^m(t_{\\lambda}) := (t_{\\lambda} -Z_{\\lambda 0}^m)' \\mathcal{I}_{\\lambda.\\eta 0}^m (t_{\\lambda} -Z_{\\lambda 0}^m)$ and $r_{\\lambda\\varrho_m}^m(t_{\\lambda}) := (t_{\\lambda} -Z_{\\lambda \\varrho_m}^m)' \\mathcal{I}_{\\lambda.\\eta \\varrho_m}^m (t_{\\lambda} -Z_{\\lambda \\varrho_m}^m)$.\n\nThe following proposition establishes the asymptotic null distribution of the LRTS. As in the non-normal case, the LRTS is asymptotically distributed as the maximum of the $M_0$ random variables. \n\\begin{assumption} \\label{A-nonsing_M_normal}\nAssumption \\ref{A-nonsing_M} holds when $\\tilde{s}_{\\tilde \\varrho_m k}$ is given in (\\ref{stilde_normal}).\n\\end{assumption}\n\\begin{proposition} \\label{P-LR_M_normal}\nSuppose Assumptions \\ref{assn_a1}, \\ref{assn_a2}, \\ref{assn_a4}, \\ref{A-consist_M}, and \\ref{A-nonsing_M_normal} hold and the component density of the $j$-th regime is given by (\\ref{normal-density}). Then, under $H_0:m=M_0$, \n$LR_{M_0,n} \\overset{d}{\\rightarrow} \\max_{m=1,\\ldots,M_0}\\{ \\max\\{ \\mathbb{I}\\{\\varrho_m=0\\} (\\tilde t_{\\lambda }^{m1})' \\mathcal{I}_{\\lambda.\\eta 0}^m \\tilde t_{\\lambda }^{m1}, \\sup_{\\varrho_m \\in \\Theta_{\\varrho}^m} (\\tilde t_{\\lambda \\varrho_m}^{m2})' \\mathcal{I}_{\\lambda.\\eta \\varrho_m}^m \\tilde t_{\\lambda \\varrho_m}^{m2} \\}\\}$.\n\\end{proposition} \n\n\\subsection{Homoscedastic normal distribution}\n\nAs in Section \\ref{subsec:homo_normal}, we assume that $Y_k\\in \\mathbb{R}$ in the $j$-th regime follows a normal distribution with the regime-specific intercept and common variance whose density is given by (\\ref{normal-density-homo}).\n\nThe asymptotic distribution of the LRTS is derived by using a reparameterization\n\\begin{equation*}\n\\left(\n\\begin{array}{c}\n\\theta_m\\\\\n\\theta_{m+1}\\\\\n\\sigma^2\n\\end{array}\n\\right) =\\left(\n\\begin{array}{c}\n\\nu_{\\theta m} + (1-\\alpha_m)\\lambda_m \\\\\n\\nu_{\\theta m} -\\alpha_m\\lambda_m \\\\ \n\\nu_{\\sigma m} - \\alpha_m(1-\\alpha_m) \\lambda_{\\mu m}^2\n\\end{array}\n\\right),\n\\end{equation*}\nsimilar to (\\ref{repara-homo}) and following the derivation in Sections \\ref{subsec:homo_normal} and \\ref{subsec:hetero_normal_M}. For brevity, we omit the details of the derivation. Define $s_{\\lambda\\lambda \\varrho_m k}^m$ as in (\\ref{score_lambda_M}), and denote each element of $s_{\\lambda\\lambda \\varrho_m k}^m$ as\n\\begin{equation*}\ns_{\\lambda\\lambda \\varrho_m k}^m =\n\\begin{pmatrix}\n* & s_{\\lambda_{\\mu\\beta} \\varrho_m k}^m \\\\\ns_{\\lambda_{\\beta\\mu} \\varrho_m k}^m & s_{\\lambda_{\\beta\\beta} \\varrho_m k}^m \\\\\n\\end{pmatrix}.\n\\end{equation*}\nSimilarly to (\\ref{score_normal_homo}), define $\\tilde{s}_{\\tilde \\varrho k}$ as in (\\ref{stilde}) by redefining $s_{\\lambda \\varrho_m k}^m$ in (\\ref{stilde}) as\n\\begin{equation}\\label{stilde_normal_homo}\ns_{\\lambda \\varrho_m k}^m\n:= \n\\begin{pmatrix}\n\\zeta_{k,0}^m(\\varrho_m)\/2 & \ns_{\\lambda_{\\mu}^3 k}^m\/3! & \ns_{\\lambda_{\\mu}^4 k}^m\/4! &\n(s_{\\lambda_{\\beta \\mu} \\varrho k}^m)' & \nV(s_{\\lambda_{\\beta\\beta} \\varrho k}^m)'\n\\end{pmatrix} ',\n\\end{equation}\nwhere $s_{\\lambda_{\\mu}^i k}^m:= \\mathbb{P}_{\\vartheta^*_{M_0} (X_k \\in J_m|\\overline{{\\bf Y}}_{0}^k)} \\nabla_{\\mu^i}f(Y_k|\\overline{\\bf Y}_{k-1};\\gamma^*,\\theta_m^*)\/f(Y_k|\\overline{\\bf Y}_{k-1};\\gamma^*,\\theta_m^*)$ for $i=3,4$.\n\nThe following proposition establishes the asymptotic null distribution of the LRTS.\n\\begin{assumption} \\label{A-nonsing_M_normal_homo}\nAssumption \\ref{A-nonsing_M} holds when $\\tilde{s}_{\\tilde \\varrho_m k}$ is given in (\\ref{stilde_normal_homo}).\n\\end{assumption}\n\\begin{proposition} \\label{P-LR_M_normal_homo}\nSuppose Assumptions \\ref{assn_a1}, \\ref{assn_a2}, \\ref{assn_a4}, \\ref{A-consist_M}, and \\ref{A-nonsing_M_normal_homo} hold and the component density of the $j$-th regime is given by (\\ref{normal-density-homo}). Then, under $H_0:m=M_0$, $ LR_{M_0,n} \\overset{d}{\\rightarrow} \\max_{m=1,\\ldots,M_0}\\{ \\max\\{ \\mathbb{I}\\{\\varrho_m=0\\} (\\tilde t_{\\lambda }^{m1})' \\mathcal{I}_{\\lambda.\\eta 0}^m \\tilde t_{\\lambda }^{m1}, \\sup_{\\varrho_m \\in \\Theta_{\\varrho_m\\epsilon}} (\\tilde t_{\\lambda \\varrho_m}^{m2})' \\mathcal{I}_{\\lambda.\\eta \\varrho_m}^m \\tilde t_{\\lambda \\varrho_m}^{m2} \\}\\}$, where $\\tilde{t}_{\\lambda}^{m1}$ and $\\tilde{t}_{\\lambda \\varrho_m}^{m2}$ are defined as in Proposition \\ref{P-LR_M_normal} but in terms of $(Z_{\\lambda \\varrho_m}^m, \\mathcal{I}_{\\lambda.\\eta \\varrho_m}^m, Z_{\\lambda 0}^m, \\mathcal{I}_{\\lambda.\\eta 0}^m)$ constructed with $s_{\\lambda \\varrho_m k}^m$ given in (\\ref{stilde_normal_homo}) and $\\Lambda_\\lambda^1$ and $\\Lambda_{\\lambda \\varrho_m}^2$ defined as in (\\ref{Lambda-lambda-homo}) but replacing $\\varrho$ with $\\varrho_m$. \n\\end{proposition}\n\n\n\\section{Asymptotic distribution under local alternatives} \nIn this section, we derive the asymptotic distribution of our LRTS under local alternatives. While we focus on the case of testing $H_0: M=1$ against $H_A: M=2$, it is straightforward to extend the analysis to the case of testing $H_0: M=M_0$ against $H_A: M=M_0+1$ for $M_0\\geq 2$.\n\nGiven $\\pi\\in \\Theta_{\\pi}$, we define a local parameter $h:=\\sqrt{n}t(\\psi,\\pi)$, so that\n\\[\nh =\n\\begin{pmatrix}\nh_\\eta\\\\\nh_\\lambda \n\\end{pmatrix}=\n\\begin{pmatrix}\n\\sqrt{n} (\\eta-\\eta^*)\\\\\n\\sqrt{n} t_{\\lambda}(\\lambda,\\pi)\n\\end{pmatrix},\n\\] \nwhere $t_{\\lambda}(\\lambda,\\pi)$ differs across the different models and is given by (\\ref{score_lambda}), (\\ref{t_lambda_rho}), and (\\ref{t-psi2-homo}). Given $h=(h_\\eta',h_{\\lambda}')'$ and $\\pi\\in\\Theta_{\\pi}$, we consider the sequence of contiguous local alternatives $\\vartheta_n = (\\psi_n',\\pi_n')' = (\\eta_n',\\lambda_n',\\pi_n')'\\in\\Theta_{\\eta}\\times\\Theta_\\lambda\\times\\Theta_{\\pi}$ such that\n\\begin{equation}\\label{local-alternative}\nh_{\\eta} = \\sqrt{n}(\\eta_n-\\eta^*),\\quad h_\\lambda =\\sqrt{n} t_{\\lambda}(\\lambda_{n},\\pi_n)+o(1),\\quad \\text{and}\\ \\pi_n - \\pi =o(1).\n\\end{equation} \n\nLet $\\mathbb{P}_{\\vartheta,x_0}^n$ be the probability measure on $\\{Y_k\\}_{k=1}^n$ under $\\vartheta$ conditional on the value of $\\overline{\\bf Y}_0$, $X_0$, and ${\\bf W}_1^n$. Then, the log-likelihood ratio is given by\n\\[\n\\log \\frac{d\\mathbb{P}_{\\vartheta_n,x_0}^n}{d \\mathbb{P}_{\\vartheta^*,x_0}^n} = \\ell_n(\\psi_n,\\pi_n,x_0)-\\ell_n(\\psi^*,\\pi,x_0)= \\log \\left( \\frac{\\sum_{{\\bf x}_1^{n}} \\prod_{k=1}^n f_k(\\eta_n, \\lambda_n) q_{\\pi_n}(x_{k-1},x_k) }{ \\prod_{k=1}^n f_k(\\eta^*,0) } \\right),\n\\] \nwhere $f_k(\\eta,\\lambda)$ is defined by the right-hand side of (\\ref{repara_g}), (\\ref{repara_g_hetero}), and (\\ref{repara_g_homo}) for the models of the non-normal distribution, heteroscedastic normal distribution, and homoscedastic normal distribution, respectively. The following result follows from Le Cam's first and third lemmas and facilitates the derivation of the asymptotic distribution of the LRTS under $\\mathbb{P}_{\\vartheta_n,x_0}^n$.\n\n\\begin{proposition} \\label{P-LAN} Suppose that the assumptions of Propositions \\ref{P-quadratic}, \\ref{P-quadratic-N1}, and \\ref{P-quadratic-N1-homo} hold for the models of the non-normal, heteroscedastic normal, and homoscedastic normal distributions, respectively. Then, uniformly in $x_0 \\in \\mathcal{X}$, (a) $\\mathbb{P}_{\\vartheta_n,x_0}^n$ is mutually contiguous with respect to $\\mathbb{P}_{\\vartheta^*,x_0}^n$, and (b) under $\\mathbb{P}_{\\vartheta_n,x_0}^n$, we have $\\log (d\\mathbb{P}_{\\vartheta_n,x_0}^n \/ d \\mathbb{P}_{\\vartheta^*,x_0}^n) = h' \\nu_n(s_{\\varrho_n k}) - \\frac{1}{2}h' \\mathcal{I}_{\\varrho} h + o_{p}(1)$ with $\\nu_n(s_{\\varrho_n k}) \\overset{d}{\\rightarrow} N(\\mathcal{I}_{ \\varrho}h, \\mathcal{I}_{ \\varrho})$. \n\\end{proposition} \n\n\\subsection{Non-normal distribution}\n \nFor the non-normal distribution, the sequence of contiguous local alternatives is given by $\\lambda_n = \\bar \\lambda\/n^{1\/4}$ because then $h_\\lambda = \\sqrt{n}\\alpha(1-\\alpha)v(\\lambda_n)=\\alpha(1-\\alpha)v(\\bar\\lambda)$ holds. The following proposition derives the asymptotic distribution of the LRTS for the non-normal distribution under $H_{1 n}: (\\pi_n,\\eta_n,\\lambda_n) = (\\bar\\pi,\\eta^*,\\bar \\lambda\/n^{1\/4})$. \n \n\\begin{proposition} \\label{P-LAN2} Suppose that the assumptions of Proposition \\ref{P-LR} hold. \nFor $\\bar \\pi\\in\\Theta_{\\pi}$ and $\\bar \\lambda\\neq 0$, define $h_{\\lambda}:= \\bar\\alpha(1-\\bar\\alpha) v(\\bar \\lambda)$. Then, under $H_{1 n}: (\\pi_n,\\eta_n,\\lambda_n) = (\\bar\\pi,\\eta^*,\\bar \\lambda\/n^{1\/4})$, we have $LR_n \\overset{d}{\\rightarrow} \\sup_{\\varrho \\in \\Theta_{\\varrho}} (\\tilde t_{\\lambda\\varrho h})' \\mathcal{I}_{\\lambda.\\eta\\varrho} \\tilde t_{\\lambda\\varrho h}$, where $\\tilde t_{\\lambda \\varrho h}$ is defined as in (\\ref{t-lambda}) but replacing $Z_{\\lambda\\varrho}$ in (\\ref{t-lambda}) with $(\\mathcal{I}_{\\lambda.\\eta\\varrho})^{-1} G_{\\lambda.\\eta\\varrho}+ h_{\\lambda}$.\n\\end{proposition}\n\n\n\\subsection{Heteroscedastic normal distribution}\n\nFor the model with the heteroscedastic normal distribution, the sequences of contiguous local alternatives characterized by (\\ref{local-alternative}) include the local alternatives of order $n^{-1\/8}$. \n\\begin{proposition} \\label{P-LAN3} Suppose that the assumptions of Proposition \\ref{P-LR-N1} hold for model (\\ref{normal-density}). For $\\bar \\varrho \\in (-1,1)$, $\\bar \\alpha \\in (0,1)$, and $\\bar \\lambda := (\\bar \\lambda_\\mu,\\bar \\lambda_\\sigma,\\bar \\lambda_\\beta')'\\neq (0,0,0)'$, let\n\\begin{align*}\n& H_{1 n}^a: (\\varrho_n,\\alpha_n,\\eta_n,\\lambda_{\\mu n}, \\lambda_{\\sigma n},\\lambda_{\\beta n}) = (\\bar \\varrho\/n^{1\/4}, \\bar\\alpha,\\eta^*, \\bar \\lambda_\\mu\/ n^{1\/8}, \\bar \\lambda_{\\sigma}\/n^{3\/8},\\bar \\lambda_{\\beta}\/n^{3\/8}), \\\\\n& H_{1 n}^b: (\\varrho_n,\\alpha_n,\\eta_n,\\lambda_{\\mu n}, \\lambda_{\\sigma n},\\lambda_{\\beta n}) = (\\bar \\varrho, \\bar\\alpha,\\eta^*, \\bar \\lambda_\\mu\/ n^{1\/4}, \\bar \\lambda_{\\sigma}\/n^{1\/4},\\bar \\lambda_{\\beta}\/n^{1\/4}),\n\\end{align*}\nand define\n\\begin{align*}\nh_{\\lambda}^a: &=\n\\bar\\alpha(1-\\bar\\alpha) \\times (\\bar \\varrho \\bar \\lambda_\\mu^2,\n \\bar \\lambda_\\mu \\bar \\lambda_\\sigma,\n b(\\bar\\alpha) \\bar \\lambda_\\mu^4\/12,\n\\bar \\lambda_{\\beta}' \\bar \\lambda_\\mu, 0, 0)', \\\\\nh_{\\lambda}^b: &= \n \\bar\\alpha(1-\\bar\\alpha) \\times (\\bar \\varrho\\bar \\lambda_\\mu^2,\n \\bar \\lambda_\\mu \\bar \\lambda_\\sigma,\n \\bar \\lambda_\\sigma^2,\n \\bar \\lambda_{\\beta}' \\bar \\lambda_\\mu,\n\\bar \\lambda_\\beta' \\bar \\lambda_{\\sigma},\n v( \\bar \\lambda_\\beta)')'.\n\\end{align*}\nThen, for $j \\in \\{a,b\\}$, under $H_{1 n}^j$, we have $LR_n \\overset{d}{\\rightarrow} \\max\\{ \\mathbb{I}\\{\\varrho=0\\} (\\tilde t_{\\lambda h}^{1j})' \\mathcal{I}_{\\lambda.\\eta 0} \\tilde t_{\\lambda h}^{1j}, \\sup_{\\varrho \\in \\Theta_{\\varrho}} (\\tilde t_{\\lambda \\varrho h}^{2j})' \\mathcal{I}_{\\lambda.\\eta \\varrho} \\tilde t_{\\lambda \\varrho h}^{2j} \\}$, where $\\tilde t_{\\lambda h}^{1j}$ and $\\tilde t_{\\lambda \\varrho h}^{2j}$ are defined as in (\\ref{t-lambda-N1}) but replacing $Z_{\\lambda\\varrho}$ with $(\\mathcal{I}_{\\lambda.\\eta\\varrho})^{-1} G_{\\lambda.\\eta\\varrho}+ h_{\\lambda}^j$.\n\\end{proposition} \n\nIn the local alternative $H_{1 n}^a$, $\\varrho_n$ converges to $0$, and $\\lambda_{\\mu n}$ converges to 0 at a slower rate than $n^{-1\/4}$. Our test has non-trivial power against these local alternatives in the neighborhood of $\\varrho=0$. By contrast, the test of \\citet{carrasco14em} does not have power against the local alternatives in the neighborhood of $\\varrho=0$, as discussed in Section 5 of \\citet{carrasco14em}. The test proposed by \\citet{quzhuo17wp} assumes that $\\varrho$ is bounded away from zero and hence their test rules out $H_{1n}^a$.\n\n \n \n\\subsection{Homoscedastic normal distribution} \n \nThe local alternatives for the model with the homoscedastic distribution also include those of order $n^{-1\/8}$ in the neighborhood of $\\varrho=0$.\n\n\\begin{proposition} \\label{P-LAN4} Suppose that the assumptions of Proposition \\ref{P-quadratic-N1-homo} hold for model (\\ref{normal-density-homo}). For $\\bar \\varrho \\in (-1,1)$, $\\bar \\alpha \\in (0,1)$, $\\Delta_\\alpha \\neq 0$, and $\\bar \\lambda := (\\bar \\lambda_\\mu,\\bar \\lambda_\\beta')'\\neq (0,0)'$, let\n\\begin{align*}\n& H_{1 n}^a: (\\varrho_n,\\alpha_n,\\eta_n,\\lambda_{\\mu n}, \\lambda_{\\beta n}) = (\\bar \\varrho\/n^{1\/4}, 1\/2+\\Delta_{\\alpha}\/n^{1\/8}, \\eta^*, \\bar \\lambda_\\mu\/ n^{1\/8}, \\bar \\lambda_{\\beta}\/n^{3\/8}), \\\\\n& H_{1 n}^b: (\\varrho_n,\\alpha_n,\\eta_n,\\lambda_{\\mu n}, \\lambda_{\\beta n}) = (\\bar \\varrho, \\bar\\alpha,\\eta^*, \\bar \\lambda_\\mu\/ n^{1\/4}, \\bar \\lambda_{\\beta}\/n^{1\/4}),\n\\end{align*}\nand define $h_{\\lambda}^a := (1\/4) \\times (\\bar\\varrho \\bar\\lambda_{\\mu}^2,\\Delta_{\\alpha} \\bar\\lambda_{\\mu}^3,-\\bar\\lambda_\\mu^4\/2, \\bar\\lambda_\\beta'\\bar\\lambda_\\mu, 0)'$ and $h_{\\lambda}^b := \\bar\\alpha(1-\\bar\\alpha) \\times (\\bar \\varrho\\bar \\lambda_\\mu^2, 0, 0, \\bar \\lambda_{\\beta}' \\bar \\lambda_\\mu, v( \\bar \\lambda_\\beta)')'$. For $j=\\{a,b\\}$, define $\\tilde t_{\\lambda h}^{1j}$ and $\\tilde t_{\\lambda \\varrho h}^{2j}$ as in (\\ref{t-lambda-N1}) but replacing $Z_{\\lambda\\varrho}$ with $(\\mathcal{I}_{\\lambda.\\eta\\varrho})^{-1} G_{\\lambda.\\eta\\varrho}+ h_{\\lambda}^j$, where $\\mathcal{I}_{\\lambda.\\eta\\varrho}$ and $G_{\\lambda.\\eta\\varrho}$ are constructed with $s_{\\varrho k}$ defined in (\\ref{score_normal_homo}), and $\\Lambda_{\\lambda }^1$ and $\\Lambda_{\\lambda \\varrho}^2$ are defined in (\\ref{Lambda-lambda-homo}). Then, under $H_{1 n}^j$, we have $LR_n \\overset{d}{\\rightarrow} \\max\\{ \\mathbb{I}\\{\\varrho=0\\} (\\tilde t_{\\lambda h}^{1j})' \\mathcal{I}_{\\lambda.\\eta 0} \\tilde t_{\\lambda h}^{1j}, \\sup_{\\varrho \\in \\Theta_{\\varrho}} (\\tilde t_{\\lambda \\varrho h}^{2j})' \\mathcal{I}_{\\lambda.\\eta \\varrho} \\tilde t_{\\lambda \\varrho h}^{2j} \\}$.\n\\end{proposition} \n\n\\section{Parametric bootstrap}\nWe consider the following parametric bootstrap to obtain the bootstrap critical value $c_{\\alpha,B}$ and bootstrap $p$-value of our LRTS for testing $H_0: M=M_0$ against $H_A: M=M_0+1$.\n\\begin{enumerate}\n\\item Using the observed data, compute $\\hat \\vartheta_{M_0}$, $\\hat\\vartheta_{M_0+1}$, and $LR_{M_0,n}$.\n\\item Given $\\hat \\vartheta_{M_0}$ and $\\xi_{M_0}$, generate $B$ independent samples $\\{Y_1^b,\\ldots,Y_n^b\\}_{b=1}^B$ under $H_0$ with $\\vartheta_{M_0}=\\hat\\vartheta_{M_0}$ conditional on the observed value of $\\overline{\\bf Y}_{0}$ and ${\\bf W}_{1}^n$.\n\\item For each simulated sample $\\{Y_k^b\\}_{k=1}^n$ with $(\\overline{\\bf Y}_0,{\\bf W}_1^n)$, estimate $\\hat \\vartheta_{M_0}^b$ and $\\hat\\vartheta_{M_0+1}^b$ as in Step 1, and let $LR_{M_0,n}^b := 2[\\ell_n(\\hat\\vartheta_{M_0+1}^b ,\\xi_{M_0+1})-\\ell_n(\\hat\\vartheta_{M_0}^b ,\\xi_{M_0})]$ for $b=1,\\ldots,B$.\n\\item Let $c_{\\alpha,B}$ be the $(1-\\alpha)$ quantile of $\\{LR_{M_0,n}^b\\}_{b=1}^B$, and define the bootstrap $p$-value as $B^{-1}\\sum_{b=1}^B \\mathbb{I}\\{ LR_{M_0,n}^b > LR_{M_0, n}\\}$.\n\\end{enumerate} \n\n\nThe following proposition shows the consistency of the bootstrap critical values $c_{\\alpha,B}$ for testing $H_0: M_0=1$. We omit the result for testing $H_0: M_0\\geq 2$; it is straightforward to extend the analysis to the case for $M_0\\geq 2$ with more tedious notations. \n \n\\begin{proposition} \\label{P-bootstrap} \nSuppose that the assumptions of Propositions \\ref{P-quadratic}, \\ref{P-quadratic-N1}, and \\ref{P-quadratic-N1-homo} hold for the models of the non-normal, heteroscedastic normal, and homoscedastic normal distributions, respectively. Then, the bootstrap critical values $c_{\\alpha,B}$ converge to the asymptotic critical values in probability as $n$ and $B$ go to infinity under $H_0$ and under the local alternatives described in Propositions \\ref{P-LAN2}, \\ref{P-LAN3}, and \\ref{P-LAN4}.\n\\end{proposition}\n\n\n\\section{Simulations and empirical application} \n\n\n\\subsection{Simulations}\n\nWe consider the following two models: \n\\begin{align} \n\\text{Model 1}:&\\quad Y_{k} = \\mu_{X_k} + \\beta Y_{k-1} +\\varepsilon_{k},\\quad \\varepsilon_k\\sim N(0,\\sigma^2),\\label{model1}\\\\\n\\text{Model 2}:&\\quad Y_{k} = \\mu_{X_k} + \\beta Y_{k-1} +\\varepsilon_{k},\\quad \\varepsilon_k\\sim N(0,\\sigma_{X_k}^2),\\label{model2}\n\\end{align}\nwhere $X_k\\in\\{1,\\ldots,M\\}$ with $p_{ij} = p(X_k=i|X_{k-1}=j)$. Model 1 in (\\ref{model1}) is similar to the model used in \\citet{chowhite07em}. This model has a switching intercept but the variance parameter $\\sigma^2$ does not switch across regimes. In Model 2 in (\\ref{model2}), both the intercept and the variance parameters switch across regimes.\n\nWe compare the size and power property of our bootstrap LRT and that of the QLR test of \\citet{chowhite07em} with $\\Theta_\\mu = [-2,2]$ and the supTS test of \\citet{carrasco14em} with $\\rho \\in [-0.9. 0.9]$. The critical values are computed by bootstrap with $B=199$ bootstrap samples. Note that this comparison favors the LRT over the supTS test because the supTS test is designed to detect general parameter variation including Markov chain.\n\nIn Model 2, we set the lower bound of $\\sigma$ as $\\epsilon_\\sigma = 0.01\\hat \\sigma$, where $\\hat \\sigma$ is the estimate of $\\sigma$ from one-regime model. We also find that adding a penalty term to the log-likelihood function improves the finite sample property of the LRT. The penalty term prevents $\\sigma_j$ from taking an extremely small value and takes the form \n$-a_n \\sum_{j = 1}^{M} \\{\\hat\\sigma^2\/\\sigma_j^2+\\log(\\sigma_j^2\/\\hat\\sigma^2) - 1\\}$. \nWe set $a_n = 20n^{-1\/2}$ and compute the test statistic using the log-likelihood function without the penalty term. Because the penalty term and its derivatives are $o_p(1)$ from the compactness of $\\Theta_M$, adding this penalty term does not affect the consistency of the MLE or the asymptotic distribution of the LRTS.\n\nWe first examine the rejection frequency of $H_0:M=1$ against $H_A:M=2$ when the data are generated by $H_0: M=1$ with $(\\beta,\\mu,\\sigma)=(0.5,0,1)$. The first panel in Table \\ref{table:bootstrap-size} reports the rejection frequency of the bootstrap tests at the nominal 10\\%, 5\\%, and 1\\% levels over $3000$ replications with $n=200$ and $500$. Overall, all the tests have good sizes. \n \nTable \\ref{table:bootstrap-power-model1} reports the power of the three tests for testing the null hypothesis of $M=1$ at the nominal level of 5\\%. We generate $3000$ data sets for $n= 500$ under the alternative hypothesis of $M=2$ by setting $\\mu_1=0.2, 0.6$, and $1.0$ and $\\mu_2=-\\mu_1$, while $(p_{11},p_{22})=(0.25,0.25)$, $(0.50,0.50)$, $(0.70,0.70)$, and $(0.90,0.90)$. We set $\\sigma=1$ for Model 1 and $(\\sigma_1,\\sigma_2)=(1.1,0.9)$ for Model 2. For Model 1, the power of all the tests increases as $\\mu_1$ increases except for the supTS test with $(p_{11},p_{22})=(0.9,0.9)$. As $(p_{11},p_{22})$ moves away from $(0.5,0.5)$, the power of the LRT increases, whereas the QLRT has decreasing power. The LRT performs better than the supTS and QLR tests except for the case with $(p_{11},p_{22})=(0.25,0.25)$, where the supTS test performs very well, and the case with $(p_{11},p_{22})=(0.5,0.5)$, where the QLRT outperforms the LRT, because the true model is a finite mixture in this case. The last three columns of Table \\ref{table:bootstrap-power-model1} report the power of the LRT and supTS test to detect alternative models with switching variances (i.e., Model 2 with $M=2$). We did not examine the power of the QLRT because this test assumes non-switching variance. The LRT has stronger power than the supTS test in most cases.\n \nThe second panel in Table \\ref{table:bootstrap-size} reports the rejection frequency of the LRT for testing $H_0:M=2$ against $H_A:M=3$ when the data are generated under the null hypothesis, showing its good size property, while neither the QLRT nor the supTS test is applicable for testing $H_0:M=2$ against $H_A:M=3$.\n\nTable \\ref{table:bootstrap-power-model2} reports the power of our LRT for testing the null hypothesis of $M=2$ at the nominal level of 5\\%. We generate $3000$ data sets for $n= 500$ under the alternative hypothesis of $M=3$ across different values of $(\\mu_1,\\mu_2,\\mu_3)$ and $(p_{11},p_{22},p_{33})$ with $p_{ij} = (1-p_{ii})\/2$ for $j\\neq i$, where we set $(\\beta,\\sigma)=(0.5,1.0)$ for Model 1 and $(\\beta,\\sigma_1,\\sigma_2) = (0.5,0.9,1.2)$ for Model 2. Similar to the case of $H_0:M=1$, the power of the LRT for testing $H_0: M=2$ against $H_A: M=3$ increases when the alternative is further away from $H_0$ or when latent regimes become more persistent.\n\n\\subsection{Empirical example} \nUsing U.S. GDP per capita quarterly growth rate data from 1960Q1 to 2014Q4, we estimate the regime switching models with common variance (i.e., Model 1 in (\\ref{model1})) and with switching variances (i.e., Model 2 in (\\ref{model2})) for $M=1$, $2$, $3$, and $4$ and sequentially test the null hypothesis of $M=M_0$ against the alternative hypothesis $M=M_0+1$ for $M_0=1$, $2$, $3$, and $4$.\\footnote{For both models, we restrict the parameter values for the transition probabilities by setting $\\epsilon=0.05$ to prevent the issue of unbounded likelihood.} \nWe also report the Akaike information criteria (AIC) and Bayesian information criteria (BIC) for reference, although, to our best knowledge,\nthe consistency of the AIC and BIC for selecting the number of regimes has not been established in the literature.\n\nTable \\ref{table:select} reports the selected number of regimes by the AIC, BIC, and LRT. For Model (\\ref{model1}) with common variance, our LRT selects $M=4$, while the AIC and BIC select $M=3$ and $M=1$, respectively. For Model (\\ref{model2}) with switching variance, both the LRT and the AIC select $M=3$, while the BIC selects $M=2$.\n\nPanel A of Table \\ref{table:parameter} and Figure \\ref{fig:common} report the parameter estimates and posterior probabilities of being in each regime for the model with common variance for $M=2$, $3$, and $4$. Across the different specifications in $M$, the estimated values of $\\mu_1$, $\\mu_2, \\ldots, \\mu_M$ are well separated in the common variance model, indicating that each regime represents a booming or recession period with different degrees. In Figure \\ref{fig:common}, when the number of regimes is specified as $M=2$, the posterior probability of the ``recession'' regime (Regime 1) against that of the ``booming'' regime (Regime 2) sharply rises during the collapse of Lehman Brothers in 2008 and then declines after 2009. When the number of regimes is specified as $M=3$, in addition to the ``recession'' and ``booming'' regimes corresponding to Regimes 1 and 2, respectively, the regime with a rapid change in the growth rates from low to high is captured by Regime 3; for the model with $M=3$ in Figure \\ref{fig:common}, the posterior probability of Regime 3 rises in late 2009 when the U.S. economy started to recover from the Lehman collapse. When the number of regimes is specified as $M=4$, Regime 1 now captures a rapid change in the growth rates from high to low, where the posterior probability of Regime 1 becomes high when the growth rate of the U.S. economy rapidly declined in the middle of the Lehman collapse. The LRT selects the model with four regimes, which capture the rapid changes in the growth rates of U.S. GDP per capita during the Lehman collapse period.\n \nPanel B of Table \\ref{table:parameter} and Figure \\ref{fig:switching} report the parameter estimates and posterior probabilities of being in each regime for the model with switching variance, respectively. When the number of regimes is specified as $M=2$, the variance parameter estimates are very different between the two regimes, while the intercept estimates are similar, indicating that Regime 1 is the ``low volatility'' regime, while Regime 2 is the ``high volatility'' regime. When the number of regimes is specified as $M=3$, different regimes capture different states of the U.S. economy in terms of both growth rates and volatilities.\\footnote{We may test the null hypothesis of $\\sigma_1=\\sigma_2=\\sigma_3$ in the model with switching variance given $M=3$ by the standard LRT with the critical value obtained from the chi-square distribution with two degrees of freedom. With $LRTS = 2\\times (-297.01+307.39)=20.76$, the null hypothesis of $\\sigma_1=\\sigma_2=\\sigma_3$ is rejected at the 1\\% significance level, suggesting that the model with switching variance is more appropriate than the model with common variance.} Regime 1 is characterized by the negative value of the intercept with high volatility, capturing a recession period. Regime 2 is characterized by the positive value of the intercept with low volatility, capturing a booming\/stable economy. Regime 3 is characterized by a high value of the intercept and high variance, capturing both a rapid recovery in the growth rates and high volatility in the aftermath of the Lehman collapse in 2009. The LRT selects the model with three regimes when the model is specified with switching variance.\n\n\n\\section{Appendix}\n\nHenceforth, for notational brevity, we suppress ${\\bf W}_{a}^{b}$ from the conditioning variables and conditional densities when doing so does not cause confusion.\n\n\\subsection{Proof of propositions and corollaries}\n \n\\begin{proof}[Proof of Proposition \\ref{uconlike}]\nThe proof is essentially identical to the proof of Lemma 2 in DMR. Therefore, the details are omitted. The only difference from DMR is (i) we do not impose Assumption (A2) of DMR, but this does not affect the proof because Assumption (A2) is not used in the proof of Lemma 2 in DMR, and (ii) we have ${\\bf W}_1^n$, but our Lemma \\ref{x_diff}(a) extends Corollary 1 of DMR to accommodate the $W_k$'s. Consequently, the argument of the proof of DMR goes through.\n\\end{proof}\n \n\\begin{proof}[Proof of Proposition \\ref{Ln_thm1}] \n\nDefine $h_{\\vartheta k x_0} := \\sqrt{l_{\\vartheta k x_0}}-1$. By using the Taylor expansion of $2 \\log(1+x) = 2x - x^2(1+o(1))$ for small $x$, we have, uniformly in $x_0 \\in \\mathcal{X}$ and $\\vartheta \\in \\mathcal{N}_{c\/\\sqrt{n}}$,\n\\begin{equation}\\label{ell_appn}\n\\ell_n(\\psi,\\pi,x_0) - \\ell_n(\\psi^*, \\pi, x_0) = 2 \\sum_{k=1}^n \\log(1+h_{\\vartheta k x_0}) = n P_n(2h_{\\vartheta k x_0} - [1+o_p(1)] h_{\\vartheta k x_0}^2).\n\\end{equation}\nThe stated result holds if we show that\n\\begin{align} \n&\\sup_{x_0 \\in \\mathcal{X}} \\sup_{\\vartheta \\in \\mathcal{N}_{c\/\\sqrt{n}}} \\left| nP_n(h_{\\vartheta k x_0}^2) - n t_\\vartheta' \\mathcal{I}_\\pi t_\\vartheta\/4 \\right| = o_p(1)\\quad\\text{and}\\label{hk_appn}\\\\\n&\\sup_{x_0 \\in \\mathcal{X}}\\sup_{\\vartheta \\in \\mathcal{N}_{c\/\\sqrt{n}}} | nP_n(h_{\\vartheta k x_0}) - \\sqrt{n} t_\\vartheta' \\nu_n(s_{\\pi k})\/2 + nt_\\vartheta \\mathcal{I}_\\pi t_\\vartheta'\/8 | = o_p(1),\\label{hk_appn2}\n\\end{align}\nbecause then the right-hand side of (\\ref{ell_appn}) is equal to $\\sqrt{n} t_\\vartheta' \\nu_n (s_{\\pi k}) - t_\\vartheta \\mathcal{I}_\\pi t_\\vartheta'\/2 + o_p(1)$ uniformly in $x_0 \\in \\mathcal{X}$ and $\\vartheta \\in \\mathcal{N}_{c\/\\sqrt{n}}$.\n\nWe first show (\\ref{hk_appn}). Let $m_{\\vartheta k} := t_\\vartheta' s_{\\pi k} + r_{\\vartheta k}$, so that $l_{\\vartheta k x_0} -1 = m_{\\vartheta k} + u_{\\vartheta k x_0}$. Observe that\n\\begin{equation} \\label{B2}\n\\max_{1\\leq k \\leq n} \\sup_{\\vartheta \\in \\mathcal{N}_{c\/\\sqrt{n}}} |m_{\\vartheta k}| = \\max_{1\\leq k \\leq n} \\sup_{\\vartheta \\in \\mathcal{N}_{c\/\\sqrt{n}}} |t_\\vartheta' s_{\\pi k} + r_{\\vartheta k}| = o_p(1), \n\\end{equation}\nfrom Assumptions \\ref{assn_a3}(a) and (c) and Lemma \\ref{max_bound}. Write $4P_n ( h_{\\vartheta k x_0}^2)$ as\n\\begin{equation}\\label{B0}\n4P_n ( h_{\\vartheta k x_0}^2) = P_n \\left(\\frac{4(l_{\\vartheta k x_0}-1)^2}{(\\sqrt{l_{\\vartheta k x_0}} + 1)^2}\\right) = P_n(l_{\\vartheta k x_0}-1)^2 - P_n \\left( (l_{\\vartheta k x_0}-1)^3 \\frac{(\\sqrt{l_{\\vartheta k x_0}}+3)}{(\\sqrt{l_{\\vartheta k x_0}} + 1)^3}\\right).\n\\end{equation}\nIt follows from Assumptions \\ref{assn_a3}(a)(b)(c)(e)(f) and $(E|XY|)^2 \\leq E|X|^2E|Y|^2$ that, uniformly in $\\vartheta \\in \\mathcal{N}_\\varepsilon$, \n\\begin{equation} \\label{lk_lln}\nP_n(l_{\\vartheta k x_0}-1)^2 = t_\\vartheta'P_n(s_{\\pi k}s_{\\pi k}')t_\\vartheta + 2 t_\\vartheta' P_n[s_{\\pi k} (r_{\\vartheta k} + u_{\\vartheta k x_0})] + P_n(r_{\\vartheta k} + u_{\\vartheta k x_0})^2 = t_\\vartheta'P_n(s_{\\pi k}s_{\\pi k}')t_\\vartheta + \\zeta_{\\vartheta n x_0}, \n\\end{equation} \nwhere $\\zeta_{\\vartheta n x_0}$ satisfies $\\sup_{x_0 \\in \\mathcal{X}}|\\zeta_{\\vartheta n x_0}|=O_p(|t_\\vartheta|^2|\\psi-\\psi^*|)+O_p(n^{-1}|t_\\vartheta||\\psi-\\psi^*|) + O_p(n^{-1}|\\psi-\\psi^*|^2)$. Then, (\\ref{hk_appn}) holds because $\\sup_{\\pi \\in \\Theta_\\pi}| P_n(s_{\\pi k}s_{\\pi k}') - \\mathcal{I}_\\pi| = o_p(1)$ and the second term on the right-hand side of (\\ref{B0}) is bounded by, from (\\ref{B2}), $P_n( m_{\\vartheta k}^2) = t_\\vartheta'\\mathcal{I}_\\pi t_\\vartheta + o_p(|t_\\vartheta|^2)$, and Assumption \\ref{assn_a3}(e),\n\\begin{align*}\n& \\mathcal{C} \\sup_{x_0 \\in \\mathcal{X}}\\sup_{\\vartheta \\in \\mathcal{N}_{c\/\\sqrt{n}}} P_n \\left[ |m_{\\vartheta k}|^3 + 3 m_{\\vartheta k}^2 |u_{\\vartheta k x_0}| + 3|m_{\\vartheta k}| u_{\\vartheta k x_0}^2 \\right] + \\mathcal{C} \\sup_{x_0 \\in \\mathcal{X}}\\sup_{\\vartheta \\in \\mathcal{N}_{c\/\\sqrt{n}}} P_n (|u_{\\vartheta k x_0}|^3) \\\\\n& \\leq o_p(1) \\sup_{x_0 \\in \\mathcal{X}}\\sup_{\\vartheta \\in \\mathcal{N}_{c\/\\sqrt{n}}} P_n \\left[ m_{\\vartheta k}^2+u_{\\vartheta k x_0}^2 \\right] + \\mathcal{C} \\sup_{x_0 \\in \\mathcal{X}}\\sup_{\\vartheta \\in \\mathcal{N}_{c\/\\sqrt{n}}} P_n (|u_{\\vartheta k x_0}|^3) = o_p(n^{-1}).\n\\end{align*}\n\nWe proceed to show (\\ref{hk_appn2}). Consider the following expansion of $h_{\\vartheta k x_0}$:\n\\begin{equation} \\label{hk_1}\nh_{\\vartheta k x_0} = (l_{\\vartheta k x_0}-1)\/2 - h_{\\vartheta k x_0}^2\/2 = (t_\\vartheta' s_{\\pi k} + r_{\\vartheta k}+ u_{\\vartheta k x_0})\/2 - h_{\\vartheta k x_0}^2\/2.\n\\end{equation}\nThen, (\\ref{hk_appn2}) follows from (\\ref{hk_appn}), (\\ref{hk_1}), and Assumptions \\ref{assn_a3}(d) and (e), and the stated result follows.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{Ln_thm2}] \nFor part (a), it follows from $\\log(1+x) \\leq x$ and $h_{\\vartheta k x_0} = (l_{\\vartheta k x_0}-1)\/2 - h_{\\vartheta k x_0}^2\/2$ (see (\\ref{hk_1})) that\n\\begin{equation} \\label{Ln_ineq}\n\\ell_n(\\psi,\\pi, x_0)-\\ell_n(\\psi^*,\\pi, x_0) = 2 \\sum_{k=1}^n \\log(1+h_{\\vartheta k x_0}) \\leq 2 n P_n(h_{\\vartheta k x_0}) = \\sqrt{n} \\nu_n(l_{\\vartheta k x_0}-1) - n P_n(h_{\\vartheta k x_0}^2).\n\\end{equation} \nObserve that $h_{\\vartheta k x_0}^2 =(l_{\\vartheta k x_0}-1)^2\/(\\sqrt{l_{\\vartheta k x_0}} + 1)^2 \\geq \\mathbb{I}\\{l_{\\vartheta k x_0} \\leq \\kappa \\} (l_{\\vartheta k x_0}-1)^2\/ (\\sqrt{\\kappa}+1)^2$ for any $\\kappa>0$. Therefore, \\begin{equation}\\label{pn_lower}\nP_n (h_{\\vartheta k x_0}^2)\\geq (\\sqrt{\\kappa}+1)^{-2}P_n \\left( \\mathbb{I}\\{l_{\\vartheta k x_0} \\leq \\kappa\\} (l_{\\vartheta k x_0}-1)^2\\right). \n\\end{equation} \nSubstituting (\\ref{lk_lln}) into the right-hand side of (\\ref{pn_lower}) gives\n\\begin{equation} \\label{pn_lower_2}\nP_n (h_{\\vartheta k x_0}^2) \\geq (\\sqrt{\\kappa}+1)^{-2} t_\\vartheta' \\left[ P_n(s_{\\pi k}s_{\\pi k}') - P_n(\\mathbb{I}\\{l_{\\vartheta k x_0} > \\kappa\\}s_{\\pi k}s_{\\pi k}') \\right] t_\\vartheta + \\zeta_{\\vartheta n x_0}.\n\\end{equation}\nFrom H\\\"older's inequality, we have $P_n(\\mathbb{I}\\{l_{\\vartheta k x_0} > \\kappa\\}|s_{\\pi k}|^2 ) \\leq [P_n(\\mathbb{I}\\{l_{\\vartheta k x_0} > \\kappa\\} )]^{\\delta\/(2+\\delta)} [ P_n(|s_{\\pi k}|^{2+\\delta}) ]^{2\/(2+\\delta)}$. The right-hand side is no larger than $\\kappa^{-\\delta\/(2+\\delta)} O_p(1)$ uniformly in $x_0 \\in \\mathcal{X}$ and $\\vartheta \\in \\mathcal{N}_{\\varepsilon }$ because (i) it follows from $\\kappa \\mathbb{I}\\{l_{\\vartheta k x_0} > \\kappa\\} \\leq l_{\\vartheta k x_0}$ that $P_n(\\mathbb{I}\\{l_{\\vartheta k x_0} > \\kappa\\} ) \\leq \\kappa^{-1} P_n(l_{\\vartheta k x_0})$ and $\\sup_{x_0 \\in \\mathcal{X}} \\sup_{\\vartheta \\in \\mathcal{N}_{\\varepsilon }}|P_n(l_{\\vartheta k x_0})-1| = o_p(1)$ from Assumptions \\ref{assn_a3}(d)--(g), and (ii) $P_n(\\sup_{\\pi \\in \\Theta_\\pi}|s_{\\pi k}|^{2+\\delta})=O_p(1)$ from Assumption \\ref{assn_a3}(a). Consequently, $\\mathbb{P}( \\sup_{x_0 \\in \\mathcal{X}} \\sup_{\\vartheta \\in \\mathcal{N}_{\\varepsilon}}P_n(\\mathbb{I}\\{l_{\\vartheta k x_0} > \\kappa\\}|s_{\\pi k}|^2 ) \\geq \\lambda_{\\min}\/4 ) \\to 0$ as $\\kappa \\to \\infty$, and hence we can write (\\ref{pn_lower_2}) as $P_n (h_{\\vartheta k x_0}^2) \\geq \\tau (1+o_p(1)) t_\\vartheta' \\mathcal{I}_{\\pi} t_\\vartheta + O_p(|t_\\vartheta|^2|\\psi-\\psi^*|) + O_p(n^{-1})$ for $\\tau:=(\\sqrt{\\kappa}+1)^{-2}\/2>0$ by taking $\\kappa$ sufficiently large. Because $\\sqrt{n} \\nu_n(l_{\\vartheta k x_0}-1) = \\sqrt{n} t_\\vartheta'\\nu_n (s_{\\pi k}) + O_p(1)$ from Assumptions \\ref{assn_a3}(d) and (e), it follows from (\\ref{Ln_ineq}) that, uniformly in $x_0 \\in \\mathcal{X}$ and $\\vartheta \\in \\mathcal{N}_{\\varepsilon}$,\n\\begin{equation}\\label{rk_lower2}\n-\\eta \\leq \\ell_n(\\psi,\\pi, x_0)-\\ell_n(\\psi^*,\\pi, x_0) \\leq \\sqrt{n} t_\\vartheta' \\nu_n (s_{\\pi k}) - \\tau (1+o_p(1)) n t_\\vartheta' \\mathcal{I}_\\pi t_\\vartheta + O_p(n |t_\\vartheta|^2 |\\psi-\\psi^*| ) + O_p(1).\n\\end{equation}\nLet $T_{n}:= \\mathcal{I}_{\\pi}^{1\/2}\\sqrt{n} t_\\vartheta$. From (\\ref{rk_lower2}), Assumptions \\ref{assn_a3}(b) and (g), and the fact $\\psi-\\psi^* \\to 0$ if $t_\\vartheta \\to 0$, we obtain the following result: for any $\\delta>0$, there exist $\\varepsilon>0$ and $M,n_0<\\infty$ such that\n\\begin{equation}\n\\mathbb{P}\\left(\\inf_{x_0 \\in \\mathcal{X}} \\inf_{\\vartheta \\in \\mathcal{N}_{\\varepsilon}} \\left( |T_n| M - (\\tau\/2) |T_n|^2 + M \\right) \\geq 0\\right) \\geq 1-\\delta,\\ \\text{ for all }\\ n > n_0.\n\\end{equation}\nRearranging the terms inside $\\mathbb{P}(\\cdot)$ gives $\\sup_{x_0 \\in \\mathcal{X}} \\sup_{\\vartheta \\in \\mathcal{N}_{\\varepsilon}} (|T_{n}|-(M\/\\tau))^2 \\leq 2M\/\\tau+(M\/\\tau)^2$. Taking its square root gives $\\mathbb{P}(\\sup_{x_0 \\in \\mathcal{X}} \\sup_{\\vartheta \\in \\mathcal{N}_{\\varepsilon}}|T_{n}| \\leq M_1) \\geq 1-\\delta$ for a constant $M_1$, and part (a) follows. Part (b) follows from part (a) and Proposition \\ref{Ln_thm1}.\n\\end{proof}\n\n\\begin{proof}[Proof of Corollary \\ref{cor_appn}]\nBecause the logarithm is monotone, we have $\\inf_{x_0 \\in \\mathcal{X}} \\ell_n(\\psi,\\pi,x_0) \\leq \\ell_n(\\psi,\\pi,\\xi) \\leq \\sup_{x_0 \\in \\mathcal{X}} \\ell_n(\\psi,\\pi,x_0)$. Part (a) then follows from Proposition \\ref{Ln_thm1}. For part (b), note that we have $\\vartheta \\in A_{n\\varepsilon }(\\xi,\\eta)$ only if $\\vartheta \\in A_{n\\varepsilon }(x_0,\\eta)$ for some $x_0$. Consequently, part (b) follows from Proposition \\ref{Ln_thm2}.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{proposition-ell}]\nFirst, observe that parts (a) and (b) hold when the right-hand side is replaced with $K_j(k+m)^7\\rho^{\\lfloor(k+m-1)\/24\\rfloor}$ and $K_j(k+m)^7\\rho^{\\lfloor (k+m-1)\/1340 \\rfloor}$ by using Lemmas \\ref{ell_lambda} and \\ref{lemma-bound-1} and noting that $q_1=6q_0, q_2=5q_0, q_3=4q_0,\\ldots, q_6=q_0$. For example, when $j=2$, we can bound $\\sup_{x\\in\\mathcal{X}}\\sup_{\\vartheta\\in \\mathcal{N}^*}|\\nabla^2\\ell_{k,m,x}(\\vartheta) - \\nabla^j2\\overline\\ell_{k,m}(\\vartheta)|$ from $\\nabla^2 \\ell_{k,m,x}(\\vartheta)=\\Delta^2_{1,k,m,x}(\\vartheta)+\\Delta^{1,1}_{2,k,m,x}(\\vartheta)$, $\\sup_{x\\in \\mathcal{X}} \\sup_{\\vartheta\\in \\mathcal{N}^*} |\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\vartheta)-\\overline\\Delta^{\\mathcal{I}(j)}_{j,k,m}(\\vartheta)|\\leq {K}_{\\mathcal{I}(j)}(k+m)^{ 7 } \\rho^{ \\lfloor (k+m-1)\/24 \\rfloor}$, ${K}_{\\mathcal{I}(j)} \\in L^{r_{\\mathcal{I}(j)}}(\\mathbb{P}_{\\vartheta^*})$, $r_{(2)} = q_{2} = 5q_0$, and $r_{(1,1)} = q_{1}\/2 = 3q_0$. Second, letting $\\rho_* = \\rho^{1\/1340}\\mathbb{I}\\{\\rho>0\\}$ and redefining $K_j$ gives parts (a) and (b). Parts (c) and (d) follow from Lemmas \\ref{ell_lambda} and \\ref{lemma-bound-1}. \n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{lemma-omega}]\nFirst, we prove part (a). The proof of part (b) is essentially the same as that of part (a) and hence omitted. Observe that\n\\begin{equation*} \n\\begin{aligned}\n\\nabla l^j_{k,m,x}(\\vartheta)- \\nabla \\overline l^j_{k,m}(\\vartheta)& = \\Psi^j_{k,m,x}(\\vartheta)\\left(\\frac{ p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},X_{-m}=x)}{ p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},X_{-m}=x)}-\\frac{ \\overline p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1})}\n{\\overline p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1})}\\right) \\\\\n&\\quad +\\frac{\\overline p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1})}\n{\\overline p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1})}\\left(\\Psi^j_{k,m,x}(\\vartheta)- \\overline{\\Psi}^j_{k,m}(\\vartheta)\n\\right),\n\\end{aligned}\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\Psi^j_{k,m,x}(\\vartheta) := \\frac{\\nabla^j p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},X_{-m}=x)}{ p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},X_{-m}=x)}, \\quad \\overline\\Psi^j_{k,m}(\\vartheta) := \\frac{\\nabla^j \\overline p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1})}{\\overline p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1})}.\n\\end{equation*}\nIn view of Lemma \\ref{lemma-ratio} and H\\\"older's inequality, part (a) holds if, for $j=1,\\ldots,6$, there exist random variables $(\\{A_{j,k}\\}_{k=1}^n, B_j )\\in L^{q_0}(\\mathbb{P}_{\\vartheta^*})$ and $\\rho_* \\in (0,1)$ such that, for all $1 \\leq k \\leq n$ and $m \\geq 0$, \n\\begin{equation} \\label{Psi_bound}\n(A) \\quad \\sup_{m \\geq 0}\\sup_{x\\in\\mathcal{X}}\\sup_{\\vartheta\\in \\mathcal{N}^*} | \\Psi^j_{k,m,x}(\\vartheta)| \\leq A_{j,k}, \\quad (B) \\quad \\sup_{x\\in\\mathcal{X}}\\sup_{\\vartheta\\in \\mathcal{N}^*} |\\Psi^j_{k,m,x}(\\vartheta) - \\overline\\Psi^j_{k,m}(\\vartheta)| \\leq B_j (k+m)^{ 7 }\\rho_*^{k+m-1}.\n\\end{equation}\nWe show (A) and (B). From (\\ref{der_formula}) we have, suppressing $(\\vartheta)$ and superscript $1$ from $\\nabla^1 \\ell_{k,m,x}$,\n\\begin{equation*}\n\\begin{aligned}\n\\Psi^1_{k,m,x}&= \\nabla\\ell_{k,m,x},\\quad \\Psi^2_{k,m,x}= \\nabla^2\\ell_{k,m,x} +(\\nabla\\ell_{k,m,x} )^2,\\\\\n\\Psi^3_{k,m,x}&=\\nabla^3\\ell_{k,m,x} +3\\nabla^2\\ell_{k,m,x}\n\\nabla \\ell_{k,m,x}\n+(\\nabla \\ell_{k,m,x})^3,\\\\\n\\Psi^4_{k,m,x}&=\\nabla^4\\ell_{k,m,x}+4 \\nabla^3 \\ell_{k,m,x} \\nabla \\ell_{k,m,x}+3(\\nabla^2\\ell_{k,m,x})^2 +6 \\nabla^2 \\ell_{k,m,x} (\\nabla\\ell_{k,m,x})^2+ (\\nabla\\ell_{k,m,x})^4,\\\\\n\\Psi^5_{k,m,x} &= \\nabla^5 \\ell_{k,m,x}+5 \\nabla^4 \\ell_{k,m,x} \\nabla \\ell_{k,m,x}+10\\nabla^3 \\ell_{k,m,x} \\nabla^2\\ell_{k,m,x}+10\\nabla^3 \\ell_{k,m,x}(\\nabla\\ell_{k,m,x})^2 \\\\\n& \\quad +15(\\nabla^2 \\ell_{k,m,x})^2 \\nabla \\ell_{k,m,x} +10\\nabla^2\\ell_{k,m,x}(\\nabla \\ell_{k,m,x})^3+ (\\nabla\\ell_{k,m,x})^5,\\\\\n\\Psi^6_{k,m,x} & = \\nabla^6 \\ell_{k,m,x} + 6 \\nabla^5 \\ell_{k,m,x} \\nabla \\ell_{k,m,x} + 15 \\nabla^4 \\ell_{k,m,x} \\nabla^2 \\ell_{k,m,x} + 15\\nabla^4 \\ell_{k,m,x} (\\nabla \\ell_{k,m,x})^2 \\\\\n& \\quad +10( \\nabla^3 \\ell_{k,m,x} )^2 +60 \\nabla^3 \\ell_{k,m,x} \\nabla^2 \\ell_{k,m,x} \\nabla \\ell_{k,m,x} +20\\nabla^3\\ell_{k,m,x} (\\nabla \\ell_{k,m,x})^3 \\\\\n& \\quad +15( \\nabla^2 \\ell_{k,m,x} )^3 +45(\\nabla^2 \\ell_{k,m,x})^2( \\nabla \\ell_{k,m,x})^2 +15\\nabla^2 \\ell_{k,m,x}(\\nabla \\ell_{k,m,x})^4 + (\\nabla\\ell_{k,m,x})^6,\n\\end{aligned}\n\\end{equation*}\nand $\\overline\\Psi^j_{k,m}$ is written analogously with $\\nabla^j \\overline\\ell_{k,m}$ replacing $\\nabla^j \\ell_{k,m,x}$. Therefore, (A) of (\\ref{Psi_bound}) follows from Proposition \\ref{proposition-ell}(c) and H\\\"older's inequality. (B) of (\\ref{Psi_bound}) follows from Proposition \\ref{proposition-ell}(a)(c), the relation $ab-cd = a(b-c)-c(a-d)$, $a^n - b^n = (a-b)\\sum_{i=0}^{n-1} (a^{n-1-i}b^i)$, and H\\\"older's inequality.\n\nFor part (c), the bound of $\\nabla l^j_{k,m,x}(\\vartheta)$ follows from writing $\\nabla l^j_{k,m,x}(\\vartheta) = [\\overline p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},X_{-m}=x)\/ \\overline p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},X_{-m}=x) ] \\Psi^j_{k,m,x}(\\vartheta)$ and using (\\ref{Psi_bound}) and Lemma \\ref{lemma-ratio}. $\\overline\\nabla^j l_{k,m}(\\vartheta)$ is bounded by a similar argument. Part (d) follows from parts (a)--(c), the completeness of $L^{q}(\\mathbb{P}_{\\vartheta^*})$, Markov's inequality, and the Borel--Cantelli Lemma. Part (e) follows from combining parts (a) and (b) and letting $m' \\to \\infty$ in part (b).\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{P-consist}]\nThe consistency of $\\hat \\vartheta_1$ follows from Theorem 2.1 of \\citet{neweymcfadden94hdbk} because (i) $\\vartheta_1^*$ uniquely maximizes $\\mathbb{E}_{\\vartheta_1^*} \\log f(Y_1|\\overline{\\bf Y}_0,W_1;\\gamma,\\theta)$ from Assumption \\ref{A-consist}(c), and (ii) $\\sup_{\\vartheta_1 \\in \\Theta_1}|n^{-1}\\ell_{0,n}(\\vartheta_1) - \\mathbb{E}_{\\vartheta_1^*} \\log f(Y_1|\\overline{\\bf Y}_0,W_1;\\gamma,\\theta)| \\overset{p}{\\rightarrow} 0$ and $\\mathbb{E}_{\\vartheta_1^*} \\log f(Y_1|\\overline{\\bf Y}_0,W_1;\\gamma,\\theta)$ is continuous because $(Y_k,W_k)$ is strictly stationary and ergodic from Assumption \\ref{assn_a1}(e) and $\\mathbb{E}_{\\vartheta_1^*} \\sup_{\\vartheta_1 \\in \\Theta_1}|\\log f(Y_1|\\overline{\\bf Y}_0,W_1;\\gamma,\\theta)| <\\infty$ from Assumption \\ref{assn_a2}(c).\n \nWe proceed to prove the consistency of $\\hat\\vartheta_2$. Define, similarly to pp.\\ 2265--6 in DMR, $\\Delta_{k,m,x}(\\vartheta_2):= \\log p_{\\vartheta_{2}}(Y_k |\\overline{\\bf{Y}}_{-m}^{k-1},{\\bf W}_{-m}^k,X_{-m}=x)$, $\\Delta_{k,m}(\\vartheta_2):= \\log p_{\\vartheta_{2}}(Y_k |\\overline{\\bf{Y}}_{-m}^{k-1},{\\bf W}_{-m}^k)$, $\\Delta_{k,\\infty}(\\vartheta_2):=\\lim_{m\\to \\infty} \\Delta_{k,m}(\\vartheta_2)$, and $\\ell(\\vartheta_2):=\\mathbb{E}_{\\vartheta_1^*}[\\Delta_{0,\\infty}(\\vartheta)]$. Observe that Lemmas 3 and 4 as well as Proposition 2 of DMR hold for our $\\{\\Delta_{k,m,x}(\\vartheta_2),\\Delta_{k,m}(\\vartheta_2),\\Delta_{k,\\infty}(\\vartheta_2), \\ell_n(\\vartheta_2,x_0),\\ell(\\vartheta_2)\\}$ under our assumptions because (i) our Assumption \\ref{assn_a1}(e) can replace their Assumption (A2) in the proof of their Lemmas 3 and 4 and Proposition 2, and (ii) our Lemma \\ref{x_diff}(a) extends Corollary 1 of DMR to accommodate the $W_k$'s. It follows that (i) $\\ell(\\vartheta_2)$ is maximized if and only if $\\vartheta_2 \\in \\Gamma^*$ from Assumption \\ref{A-consist}(d) because $\\mathbb{E}_{\\vartheta^*_{1}}[ \\log p_{\\vartheta_{2}}(Y_1 |\\overline{\\bf{Y}}_{-m}^0,{\\bf W}_{-m}^1)]$ converges to $\\ell(\\vartheta_2)$ uniformly in $\\vartheta_2$ as $m \\to \\infty$ from Lemma 3 of DMR and the dominated convergence theorem, (ii) $\\ell(\\vartheta_2)$ is continuous from Lemma 4 of DMR, and (iii) $\\sup_{\\xi_2}\\sup_{\\vartheta_2 \\in \\Theta_2} |n^{-1} \\ell_n(\\vartheta_2,\\xi_2) - \\ell(\\vartheta_2)| \\overset{p}{\\rightarrow} 0$ holds from Proposition 2 of DMR and $\\ell_n(\\vartheta_2,\\xi_2) \\in [\\min_{x_0} \\ell_n(\\vartheta_2,x_0), \\max_{x_0} \\ell_n(\\vartheta_2,x_0)]$. Consequently, $\\inf_{\\vartheta_2\\in\\Gamma^*}|\\hat\\vartheta_2-\\vartheta_2|\\overset{p}{\\rightarrow} 0$ follows from Theorem 2.1 of \\citet{neweymcfadden94hdbk} with an adjustment for the fact that the maximizer of $\\ell(\\vartheta_2)$ is a set, not a singleton.\n\\end{proof} \n \n\\begin{proof}[Proof of Proposition \\ref{P-quadratic}]\n\nWe prove the stated result by applying Corollary \\ref{cor_appn} to $l_{\\vartheta k x_0} -1$ with $l_{\\vartheta k x_0}$ defined in (\\ref{density_ratio}). Because the first and second derivatives of $l_{\\vartheta k x_0} -1$ play the role of the score, we expand $l_{\\vartheta k x_0} -1$ with respect to $\\psi$ up to the third order. Let $q = \\dim(\\psi)$. For a $k\\times 1$ vector $a$, define $a^{\\otimes p} := a \\otimes a\\otimes \\cdots \\otimes a$ ($p$ times) and $\\nabla_{a^{\\otimes p}} :=\\nabla_{a} \\otimes \\nabla_{a}\\otimes \\cdots \\otimes \\nabla_{a}$ ($p$ times). Recall that the $(p+1)$-th order Taylor expansion of $f(x)$ with $x \\in \\mathbb{R}^q$ around $x=x^*$ is given by\n\\begin{align*}\nf(x) & = f(x^*) + \\sum_{j=1}^p \\frac{1}{j!} \\nabla_{(x^{\\otimes j})'} f(x^*) (x-x^*)^{\\otimes j} + \\frac{1}{(p+1)!} \\nabla_{(x^{\\otimes (p+1)})'} f(\\overline{x}) (x-x^*)^{\\otimes (p+1)},\n\\end{align*}\nwhere $\\overline{x}$ lies between $x$ and $x^*$, and $\\overline{x}$ may differ from element to element of $\\nabla_{x^{\\otimes (p+1)}} f(\\overline{x})$.\n\nChoose $\\epsilon>0$ sufficiently small so that $\\mathcal{N}_{\\epsilon}$ is a subset of $\\mathcal{N}^*$ in Assumption \\ref{assn_a4}. For $m \\geq 0$ and $j =1,2,\\ldots$, let \n\\[\n\\Lambda^j _{k,m,x_{-m}}(\\psi,\\pi):= \\frac{\\nabla_{\\psi^{\\otimes j}} p_{\\psi \\pi} (Y_k| \\overline{{\\bf Y}}_{-m}^{k-1}, x_{-m})}{ j! p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{-m}^{k-1}, x_{-m})}, \\quad \\Lambda^j_{k,m}(\\psi, \\pi):= \\frac{\\nabla_{\\psi^{\\otimes j}} p_{\\psi \\pi} (Y_k| \\overline{{\\bf Y}}_{-m}^{k-1})}{ j! p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{-m}^{k-1})},\n\\]\nand $\\Delta \\psi:= \\psi -\\psi^*$. With this notation, expanding $l_{\\vartheta k x_0} -1$ three times around $\\psi^*$ while fixing $\\pi$ gives, with $\\overline \\psi \\in [\\psi, \\psi^*]$, \n\\begin{align} \nl_{\\vartheta k x_0} -1 &= \\Lambda^1_{k,0,x_0}(\\psi^*,\\pi)' \\Delta \\psi + \\Lambda^2_{k,0,x_0}(\\psi^*,\\pi)' (\\Delta \\psi)^{\\otimes 2} + \\Lambda^3_{k,0,x_0}(\\overline \\psi,\\pi)' (\\Delta \\psi)^{\\otimes 3} \\nonumber \\\\\n & = \\Lambda^1_{k,0}(\\psi^*,\\pi)' \\Delta \\psi + \\Lambda^2_{k,0}(\\psi^*,\\pi)' (\\Delta \\psi)^{\\otimes 2} + \\Lambda^3_{k,0}(\\overline \\psi,\\pi)' (\\Delta \\psi)^{\\otimes 3} + u_{kx_0}(\\psi,\\pi), \\label{lk_expansion}\n\\end{align}\nwhere $\\overline \\psi$ may differ from element to element of $\\Lambda^3_{k,0,x_0}(\\overline \\psi,\\pi)$, and $u_{kx_0}(\\psi,\\pi) := \\sum_{j=1}^2 [\\Lambda^j_{k,0,x_0}(\\psi^*,\\pi)- \\Lambda^j_{k,0}(\\psi^*,\\pi)]' (\\Delta \\psi)^{\\otimes j} + [\\Lambda^3_{k,0,x_0}(\\overline \\psi,\\pi) -\\Lambda^3_{k,0}(\\overline \\psi,\\pi)]' (\\Delta \\psi)^{\\otimes 3}$.\n\nNoting that $\\nabla_\\lambda \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})=0$ and $\\nabla_{\\lambda\\eta'} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})=0$ from (\\ref{d1p}), we may rewrite (\\ref{lk_expansion}) as\n\\begin{equation} \\label{lk_2}\nl_{k\\vartheta x_0} -1 = t(\\psi,\\pi)' s_{\\varrho k} + r_{k,0}(\\psi,\\pi) + u_{kx_0}(\\psi,\\pi), \n\\end{equation}\nwhere $s_{\\varrho k}$ is defined in (\\ref{score}), $r_{k,0}(\\psi,\\pi) := \\widetilde \\Lambda_{k,0}(\\pi)' (\\Delta \\eta)^{\\otimes 2} + \\Lambda^3_{k,0}(\\overline \\psi,\\pi)' (\\Delta \\psi)^{\\otimes 3}$,\nwhere $\\widetilde \\Lambda_{k,0}(\\pi)$ denotes the part of $\\Lambda^2_{k,0}(\\psi^*,\\pi)$ corresponding to $(\\Delta \\eta)^{\\otimes 2}$.\n\nFor $m \\geq 0$, define $v_{k, m}(\\vartheta):= (\\Lambda^1_{k, m}( \\psi,\\pi)', \\Lambda^2_{k, m}( \\psi,\\pi)',\\Lambda^3_{k, m}( \\psi,\\pi)')'$, and define $v_{k,\\infty}(\\vartheta):= \\lim_{m\\rightarrow \\infty} v_{k,m}(\\vartheta)$. In order to apply Corollary \\ref{cor_appn} to $l_{\\vartheta k x_0} -1$, we first show\n\\begin{align} \n&\\sup_{\\vartheta \\in \\mathcal{N}_\\epsilon}\\left|P_n [v_{k,0}(\\vartheta)v_{k,0}(\\vartheta)'] - \\mathbb{E}_{\\vartheta^*}[v_{k,\\infty}(\\vartheta)v_{k,\\infty}(\\vartheta)']\\right| = o_p(1), \\label{uniform_lln} \\\\\n&\\nu_n(v_{k,0}(\\vartheta)) \\Rightarrow W(\\vartheta), \\label{weak_cgce}\n\\end{align}\nwhere $W(\\vartheta)$ is a mean-zero continuous Gaussian process with $\\mathbb{E}_{\\vartheta^*}[W(\\vartheta_1)W(\\vartheta_2)'] = \\mathbb{E}_{\\vartheta^*}[v_{k,\\infty}(\\vartheta_1)v_{k,\\infty}(\\vartheta_2)']$. (\\ref{uniform_lln}) holds because $\\sup_{\\vartheta \\in \\mathcal{N}_\\epsilon}P_n[v_{k,0}(\\vartheta)v_{k,0}(\\vartheta)' - v_{k,\\infty}(\\vartheta)v_{k,\\infty}(\\vartheta)'] = o_p(1)$ from Proposition \\ref{lemma-omega}, and $v_{k,\\infty}(\\vartheta)v_{k,\\infty}(\\vartheta)'$ satisfies a uniform law of large numbers (Lemma 2.4 and footnote 18 of \\citet{neweymcfadden94hdbk}) because $v_{k,\\infty}(\\vartheta)$ is continuous in $\\vartheta$ from the continuity of $\\nabla^j l_{k,m,x}(\\vartheta)$ and Proposition \\ref{lemma-omega}, and $\\mathbb{E}_{\\vartheta^*}\\sup_{\\vartheta \\in \\mathcal{N}_{\\epsilon}}|v_{k,\\infty}(\\vartheta)|^{2} < \\infty$ from Proposition \\ref{lemma-omega}. (\\ref{weak_cgce}) holds because $\\sup_{\\vartheta \\in \\mathcal{N}_\\epsilon}\\nu_n(v_{k,0}(\\vartheta) - v_{k,\\infty}(\\vartheta)) = o_p(1)$ from Proposition \\ref{lemma-omega} and $\\nu_n(v_{k,\\infty}(\\vartheta)) \\Rightarrow W(\\vartheta)$ from Theorem 10.2 of \\citet{pollard90book} because (i) the space of $\\vartheta$ is totally bounded, (ii) the finite dimensional distributions of $\\nu_n(v_{k,\\infty}(\\cdot))$ converge to those of $W(\\cdot)$ from a martingale CLT because $v_{k,\\infty}(\\vartheta)$ is a stationary $L^2(\\mathbb{P}_{\\vartheta^*})$ martingale difference sequence for all $\\vartheta\\in\\mathcal{N}_{\\epsilon}$ from Proposition \\ref{lemma-omega}, and (iii) $\\{\\nu_n(v_{k,\\infty}(\\cdot)): n\\geq 1\\}$ is stochastically equicontinuous from Theorem 2 of \\citet{hansen96et} because $v_{k,\\infty}(\\vartheta)$ is Lipschitz continuous in $\\vartheta$ and both $v_{k,\\infty}(\\vartheta)$ and the Lipschitz coefficient are in $L^q(\\mathbb{P}_{\\vartheta^*})$ with $q > \\text{dim}(\\vartheta)$ from Proposition \\ref{lemma-omega}.\n\nWe proceed to show that the terms on the right-hand side of (\\ref{lk_2}) satisfy Assumptions \\ref{assn_a3}(a)--(g). Observe that $t(\\psi,\\pi) = 0$ if and only if $\\psi=\\psi^*$. First, $s_{\\varrho k}$ satisfies Assumptions \\ref{assn_a3}(a), (b), and (g) by Proposition \\ref{lemma-omega}, (\\ref{uniform_lln}), (\\ref{weak_cgce}), and Assumption \\ref{A-nonsing1}. Second, $r_{k,0}(\\psi,\\pi)$ satisfies Assumptions \\ref{assn_a3}(c) and (d) from Proposition \\ref{lemma-omega} and (\\ref{weak_cgce}). Third, $u_{k x_0}(\\psi,\\pi)$ satisfies Assumptions \\ref{assn_a3}(e) and (f) from Proposition \\ref{lemma-omega}(c). Therefore, the stated result follows from Corollary \\ref{cor_appn}(b).\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{P-LR}] \nThe proof is similar to that of Proposition 3 of \\citet{kasaharashimotsu15jasa}. Let $t_{\\eta}: = \\eta - \\eta^*$ and $t_{\\lambda } := \\alpha(1-\\alpha)v(\\lambda)$, so that $t(\\psi,\\pi) = (t_{\\eta}',t_{\\lambda}')'$. Let $\\hat \\psi_\\pi := \\arg\\max_{\\psi \\in \\Theta_\\psi} \\ell_n(\\psi,\\pi,\\xi)$ denote the MLE of $\\psi$, and split $t(\\hat\\psi_\\pi,\\pi)$ as $t(\\hat\\psi_\\pi,\\pi)= (\\hat t_{\\eta}',\\hat t_{\\lambda}')'$, where we suppress the dependence of $\\hat t_{\\eta}$ and $\\hat t_{\\lambda}$ on $\\pi$. Define $G_{\\varrho n} := \\nu_n (s_{\\varrho k})$. Let\n\\[\nG_{\\varrho n} = \\begin{bmatrix}\nG_{\\eta n} \\\\\nG_{\\lambda \\varrho n}\n\\end{bmatrix}, \\quad\n\\begin{aligned}\nG_{\\lambda. \\eta \\varrho n} &:= G_{\\lambda \\varrho n} - \\mathcal{I}_{\\lambda \\eta\\varrho}\\mathcal{I}_{\\eta }^{-1}G_{\\eta n}, \\quad Z_{\\lambda. \\eta \\varrho n} := \\mathcal{I}_{\\lambda.\\eta \\varrho}^{-1}G_{\\lambda.\\eta \\varrho n},\\\\\nt_{\\eta.\\lambda \\varrho} &:= t_{\\eta} + \\mathcal{I}_{\\eta }^{-1}\\mathcal{I}_{\\eta\\lambda \\varrho} t_{\\lambda}.\n\\end{aligned}\n\\]\nThen, we can write (\\ref{ln_appn}) as\n\\begin{equation} \\label{LR_appn}\n\\sup_{\\xi \\in \\Xi}\\sup_{\\vartheta \\in A_{n\\varepsilon c}(\\xi) } \\left| 2 \\left[\\ell_n(\\psi,\\pi,\\xi) - \\ell_n(\\psi^*,\\pi,\\xi) \\right] - A_n(\\sqrt{n} t_{\\eta.\\lambda \\varrho}) - B_{\\varrho n}(\\sqrt{n} t_{\\lambda}) \\right| =o_p(1),\n\\end{equation}\nwhere \n\\begin{equation} \\label{B_pi}\n\\begin{aligned}\nA_n(t_{\\eta.\\lambda \\varrho}) & = 2t_{\\eta.\\lambda \\varrho}'G_{\\eta n} - t_{\\eta.\\lambda \\varrho}'\\mathcal{I}_{\\eta}t_{\\eta.\\lambda \\varrho}, \\\\\nB_{\\varrho n}(t_{\\lambda}) &= 2t_{\\lambda}' G_{\\lambda.\\eta \\varrho n} - t_{\\lambda}' \\mathcal{I}_{\\lambda.\\eta \\varrho} t_{\\lambda} = Z_{\\lambda \\varrho n}' \\mathcal{I}_{\\lambda.\\eta \\varrho} Z_{\\lambda \\varrho n}- (t_{\\lambda} - Z_{\\lambda \\varrho n})'\\mathcal{I}_{\\lambda.\\eta \\varrho}(t_{\\lambda} - Z_{\\lambda \\varrho n}).\n\\end{aligned}\n\\end{equation}\nObserve that $2[\\ell_{0n}(\\hat\\vartheta_0) - \\ell_{0n}(\\vartheta_0^*) ] = \\max_{t_\\eta}[2 \\sqrt{n} t_{\\eta}' G_{\\eta n} - n t_{\\eta}' \\mathcal{I}_\\eta t_{\\eta}] + o_p(1) = \\max_{t_{\\eta.\\lambda \\varrho}} A_n(\\sqrt{n} t_{\\eta.\\lambda \\varrho}) + o_p(1)$ from applying Corollary \\ref{cor_appn} to $\\ell_{0n}(\\vartheta_0)$ and noting that the set of possible values of both $\\sqrt{n} t_\\eta$ and $\\sqrt{n}{t}_{\\eta.\\lambda \\varrho}$ approaches $\\mathbb{R}^{\\dim(\\eta)}$. In conjunction with (\\ref{LR_appn}), we obtain, uniformly in $\\pi\\in \\Theta_{\\pi}$,\n\\begin{equation} \\label{LR_appn2}\n2[\\ell_n( \\hat \\psi_\\pi , \\pi,\\xi) - \\ell_{0n}(\\hat\\vartheta_0)] = B_{\\varrho n}(\\sqrt{n} \\hat{t}_{\\lambda}) + o_p(1). \n\\end{equation}\nDefine $\\tilde t_\\lambda$ by $B_{\\varrho n}(\\sqrt{n} \\tilde{t}_{\\lambda}) = \\max_{t_\\lambda \\in \\alpha(1-\\alpha)v(\\Theta_\\lambda)} B_{\\varrho n}(\\sqrt{n} t_{\\lambda})$. Then, we have\n\\[\n2[\\ell_n(\\hat \\psi_\\pi, \\pi,\\xi) - \\ell_{0n}(\\hat\\vartheta_0)] = B_{\\varrho n}(\\sqrt{n} \\tilde{t}_{\\lambda}) + o_p(1),\n\\] \nuniformly in $\\pi \\in \\Theta_{\\pi}$ because (i) $B_{\\varrho n}(\\sqrt{n} \\tilde{t}_{\\lambda}) \\geq 2[\\ell_n(\\hat \\psi_\\pi, \\pi,\\xi) - \\ell_{0n}(\\hat\\vartheta_0)] + o_p(1)$ from the definition of $\\tilde{t}_{\\lambda}$ and (\\ref{LR_appn2}), and (ii) $2[\\ell_n(\\hat \\psi_\\pi, \\pi,\\xi) - \\ell_{0n}(\\hat\\vartheta_0)] \\geq B_{\\varrho n}(\\sqrt{n} \\tilde{t}_{\\lambda}) + o_p(1)$ from the definition of $\\hat \\psi$, (\\ref{LR_appn}), and $\\tilde{t}_{\\lambda}=O_p(n^{-1\/2})$.\n\nFinally, the asymptotic distribution of $\\sup_{\\varrho}B_{\\varrho n}(\\sqrt{n} \\tilde{t}_{\\lambda})$ follows from applying Theorem 1(c) of \\citet{andrews01em} to $B_{\\varrho n}(\\sqrt{n} \\tilde{t}_{\\lambda})$. First, Assumption 2 of \\citet{andrews01em} holds trivially for $B_{\\varrho n}(\\sqrt{n} \\tilde{t}_{\\lambda})$. Second, Assumption 3 of \\citet{andrews01em} is satisfied by (\\ref{weak_cgce}) and Assumption \\ref{A-nonsing1}. Assumption 4 of \\citet{andrews01em} is satisfied by Proposition \\ref{P-quadratic}. Assumption $5^*$ of \\citet{andrews01em} holds with $B_T=n^{1\/2}$ because $\\alpha(1-\\alpha)v(\\Theta_\\lambda)$ is locally equal to the cone $v(\\mathbb{R}^q)$ given that $\\alpha(1-\\alpha)>0$ for all $\\alpha\\in\\Theta_{\\alpha}$. Therefore, $\\sup_{\\varrho\\in \\Theta_{\\varrho}}B_{\\varrho n}(\\sqrt{n}\\tilde{t}_{\\lambda}) \\overset{d}{\\rightarrow} \\sup_{\\varrho\\in \\Theta_{\\varrho}}(\\tilde{t}_{\\lambda \\varrho}'{\\mathcal{I}}_{\\lambda.\\eta \\varrho}\\tilde{t}_{\\lambda \\varrho})$ follows from Theorem 1(c) of \\citet{andrews01em}.\n\\end{proof}\n \n\n\\begin{proof}[Proof of Proposition \\ref{P-quadratic-N1}]\nThe proof is similar to that of Proposition \\ref{P-quadratic}. Define $\\Lambda^j_{k,m,x_{-m}}(\\psi,\\pi)$ and $\\Lambda^j_{k,m}(\\psi, \\pi)$ as in the proof of Proposition \\ref{P-quadratic}. Expanding $l_{k\\vartheta x_0} -1$ five times around $\\psi^*$ similarly to (\\ref{lk_expansion}) while fixing $\\pi$ gives, with $\\overline \\psi \\in [\\psi, \\psi^*]$, \n\\begin{align} \nl_{k\\vartheta x_0} -1 & = \\sum_{j=1}^4 \\Lambda^j_{k,0}(\\psi^*,\\pi)' (\\Delta \\psi)^{\\otimes j} + \\Lambda^5_{k,0}(\\overline \\psi,\\pi)' (\\Delta \\psi)^{\\otimes 5} + u_{kx_0}(\\psi,\\pi), \\label{lk_expansion_normal}\n\\end{align}\nwhere $u_{kx_0}(\\psi,\\pi) := \\sum_{j=1}^4 [\\Lambda^j_{k,0,x_0}(\\psi^*,\\pi)- \\Lambda^j_{k,0}(\\psi^*,\\pi)]'(\\Delta \\psi)^{\\otimes j}+ [\\Lambda^5_{k,0,x_0}(\\overline \\psi,\\pi) - \\Lambda^5_{k,0}(\\overline \\psi,\\pi)]' (\\Delta \\psi)^{\\otimes 5}$.\n\nDefine $\\overline p_{\\psi \\pi k,0} := \\overline p_{\\psi \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})$. Observe that $s_{\\varrho k}$ defined in (\\ref{score_normal}) satisfies\n\\begin{equation*}\ns_{\\varrho k}\n:=\n\\begin{pmatrix}\n\\nabla_{\\eta} \\overline p_{\\psi^* \\pi k, 0} \/ \\overline p_{\\psi^* \\pi k, 0} \\\\\n\\zeta_{k, 0}(\\varrho)\/2 \\\\\n\\nabla_{\\lambda_\\mu\\lambda_\\sigma} \\overline p_{\\psi^* \\pi k, 0} \/ \\alpha(1-\\alpha)\\overline p_{\\psi^* \\pi k, 0} \\\\\n\\nabla_{\\lambda_\\sigma^2} \\overline p_{\\psi^* \\pi k, 0} \/ 2\\alpha(1-\\alpha) \\overline p_{\\psi^* \\pi k, 0} \\\\\n\\nabla_{\\lambda_\\beta\\lambda_\\mu} \\overline p_{\\psi^* \\pi k, 0} \/ \\alpha(1-\\alpha)\\overline p_{\\psi^* \\pi k, 0} \\\\\n\\nabla_{\\lambda_\\beta\\lambda_\\sigma} \\overline p_{\\psi^* \\pi k, 0} \/ \\alpha(1-\\alpha)\\overline p_{\\psi^* \\pi k, 0} \\\\\nV(\\nabla_{\\lambda_\\beta \\lambda_\\beta} \\overline p_{\\psi^* \\pi k, 0})\/ \\alpha(1-\\alpha) \\overline p_{\\psi^* \\pi k, 0} \n\\end{pmatrix}.\n\\end{equation*}\nNoting that $\\nabla_\\lambda \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})=0$ and $\\nabla_{\\lambda\\eta'} \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})=0$ from (\\ref{d1p}) and (\\ref{d2p0}), we may rewrite (\\ref{lk_expansion_normal}) as, with $t(\\psi,\\pi)$ and $s_{\\varrho k}$ defined in (\\ref{t-psi2}) and (\\ref{score_normal}),\n\\begin{equation} \\label{lk_2_normal}\nl_{\\vartheta k x_0} -1 = t(\\psi,\\pi)' s_{\\varrho k} + r_{k,0}(\\pi) + u_{kx_0}(\\psi,\\pi),\n\\end{equation} \nwhere $r_{k,0}(\\pi):= \\widetilde \\Lambda_{k,0}(\\pi) '\\tau(\\psi) + \\Lambda^5_{k,0}(\\overline \\psi,\\pi) ' (\\Delta \\psi)^{\\otimes 5 } + \\lambda_\\mu^4[\\nabla_{\\lambda_\\mu^4} \\overline p_{\\psi^* \\pi k,0}- b(\\alpha)\\nabla_{\\lambda_\\sigma^2} \\overline p_{\\psi^* \\pi k,0}]\/4! \\overline p_{\\psi^* \\pi k,0}$, $\\tau(\\psi)$ is the vector that collects the elements of $\\{(\\Delta \\psi)^{\\otimes j} \\}_{j=2}^4$ that are not in $t(\\psi,\\pi)$, and $\\widetilde \\Lambda_{k,0}(\\pi)$ denotes the vector of the corresponding elements of $\\{\\Lambda^j_{k,0}(\\psi^*,\\pi)\\}_{j=2}^4$. \n \nThe stated result follows from Corollary \\ref{cor_appn} if the terms on the right-hand side of (\\ref{lk_2_normal}) satisfy Assumption \\ref{assn_a3}. Similarly to the proof of Proposition \\ref{P-LR}, define $v_{k, m}(\\vartheta):=(\\zeta_{k,m}(\\varrho), \\Lambda^1_{k, m}( \\psi,\\pi)', \\ldots, \\Lambda^5_{k, m}( \\psi,\\pi)' )'$. \nNote that $\\zeta_{k,m}(\\varrho)$ satisfies Proposition \\ref{lemma-omega} because the mean value theorem and $\\nabla_{\\lambda_\\mu^2} \\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{-m}^{k-1})=0$ gives $\\zeta_{k,m}(\\varrho) = [\\nabla_{\\lambda_\\mu^2}\\overline p_{\\psi^* \\varrho \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) - \\nabla_{\\lambda_\\mu^2}\\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})]\/[\\varrho \\alpha(1-\\alpha) \\overline p_{\\psi^* \\varrho \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})] =\\nabla_\\varrho \\nabla_{\\lambda_\\mu^2}\\overline p_{\\psi^* \\alpha \\bar \\varrho} (Y_k| \\overline{{\\bf Y}}_{-m}^{k-1}) \/[\\alpha(1-\\alpha) \\overline p_{\\psi^* \\bar\\varrho \\alpha} (Y_k| \\overline{{\\bf Y}}_{-m}^{k-1})]$ for $\\bar \\varrho \\in [0,\\varrho]$.\nTherefore, $v_{k,\\infty}(\\vartheta):=\\lim_{m\\rightarrow \\infty} v_{k,m}(\\vartheta)$ is well-defined, and $v_{k,0}(\\vartheta)$ and $v_{k,\\infty}(\\vartheta)$ satisfy (\\ref{uniform_lln})--(\\ref{weak_cgce}) from repeating the argument in the proof of Proposition \\ref{P-LR}.\n\nWe proceed to show that the terms on the right-hand side of (\\ref{lk_2_normal}) satisfy Assumption \\ref{assn_a3}. Observe that $t(\\psi,\\pi) = 0$ if and only if $\\psi=\\psi^*$. $s_{\\varrho k}$ and $u_{k x_0}(\\psi,\\pi)$ satisfy Assumption \\ref{assn_a3} by noting that $s_{\\varrho k}$ is a linear function of $v_{k,0}(\\vartheta)$ and using the argument in the proof of Proposition \\ref{P-quadratic} by replacing Assumption \\ref{A-nonsing1} with Assumption \\ref{A-nonsing2}. We show that each component of $r_{k,0}(\\pi)$ satisfies Assumptions \\ref{assn_a3}(c) and (d). First, $ \\Lambda^5_{k,0}(\\overline \\psi,\\pi)' (\\Delta \\psi)^{\\otimes 5}$ satisfies Assumptions \\ref{assn_a3}(c) and (d) from Proposition \\ref{lemma-omega}, (\\ref{weak_cgce}) and $\\lambda_\\mu^5 = (12\\lambda_\\mu\/b(\\alpha)) [\\lambda_\\sigma^2 + b(\\alpha)\\lambda_\\mu^4\/12] - 12(\\lambda_\\sigma\/b(\\alpha))\\lambda_\\mu \\lambda_\\sigma = O(|\\psi| |t(\\psi,\\pi)|)$. Second, $\\lambda_\\mu^4[\\nabla_{\\lambda_\\mu^4} \\overline p_{\\psi^* \\pi k,0}- b(\\alpha)\\nabla_{\\lambda_\\sigma^2} \\overline p_{\\psi^* \\pi k,0}]\/ \\overline p_{\\psi^* \\pi k,0}$ satisfies Assumptions \\ref{assn_a3}(c) and (d) from Lemma \\ref{lemma_d34}(b). Third, for $\\widetilde \\Delta_{k,0}(\\pi)'\\tau(\\psi)$, observe that $\\nabla_{\\lambda \\eta^j}\\overline p_{\\psi^* \\pi k,0}=0$ for any $j \\geq 1$ in view of (\\ref{repara_g_hetero})--(\\ref{d3g2}). Therefore, $\\widetilde \\Delta_{k,0}(\\pi)'\\tau(\\psi)$ is written as, with $\\Delta \\eta:= \\eta- \\eta^*$,\n\\begin{equation} \\label{d4r}\n\\widetilde \\Delta_{k,0}(\\pi)'\\tau(\\psi) = \\nabla_{(\\eta^{\\otimes 2})'} \\overline p_{\\psi^* \\pi k,0} (\\Delta \\eta)^{\\otimes 2} \/ 2!\\overline p_{\\psi^* \\pi k,0} + R_{3k\\vartheta} + R_{4k\\vartheta}, \n\\end{equation}\nwhere $R_{3k\\vartheta}:= \\nabla_{(\\psi^{\\otimes 3})'} \\overline p_{\\psi^* \\pi k,0} (\\Delta \\psi)^{\\otimes 3} \/3!\\overline p_{\\psi^* \\pi k,0}$ and \n\\begin{equation}\\label{r_jk}\nR_{4k\\vartheta}:= [\\nabla_{(\\psi^{\\otimes 4})'} \\overline p_{\\psi^* \\pi k,0} (\\Delta \\psi)^{\\otimes 4} - \\nabla_{\\lambda_\\mu^4} \\overline p_{\\psi^* \\pi k,0} \\lambda_\\mu^4]\/4!\\overline p_{\\psi^* \\pi k,0}.\n\\end{equation}\nThe first term in (\\ref{d4r}) clearly satisfies Assumptions \\ref{assn_a3}(c) and (d). The terms in $R_{3k\\vartheta}$ belong to one of the following three groups: (i) the term associated with $\\lambda_\\sigma^3$, (ii) the term associated with $\\lambda_\\mu^3$, or (iii) the other terms. These terms satisfy Assumptions \\ref{assn_a3}(c) and (d) because the term (i) is bounded by $|\\psi||t(\\psi,\\pi)|$ because $\\lambda_\\sigma^3 = \\lambda_\\sigma [\\lambda_\\sigma^2+b(\\alpha)\\lambda_\\mu^4\/12] - (\\lambda_\\mu^3b(\\alpha))\\lambda_\\mu \\lambda_\\sigma\/12$, the term (ii) is bounded by $\\varrho \\lambda_\\mu^3$ from Lemma \\ref{lemma_d34}(a), and the terms in (iii) are bounded by $|\\psi||t(\\psi,\\pi)|$ because they either contain $\\Delta \\eta$ or a term of the form $\\lambda_\\mu^i\\lambda_\\sigma^j\\lambda_\\beta^k$ with $i+j+k=3$ and $i,j \\neq 3$. Similarly, the terms in $R_{4k\\vartheta}$ satisfy Assumptions \\ref{assn_a3}(c) and (d) because they either contain $\\Delta \\eta$ or a term of the form $\\lambda_\\mu^i\\lambda_\\sigma^j\\lambda_\\beta^k$ with $i+j+k=4$ and $i \\neq 4$. This proves that $r_{k,0}(\\pi)$ satisfies Assumptions \\ref{assn_a3}(c) and (d), and the stated result is proven. \n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{P-LR-N1}]\n\nThe proof is similar to the proof of Proposition 3(c) of \\citet{kasaharashimotsu15jasa}. Let $(\\hat{\\psi}_\\alpha, \\hat \\varrho_\\alpha) : = {\\arg\\max}_{(\\psi, \\varrho) \\in \\Theta_{\\psi}\\times\\Theta_{\\varrho}} \\ell_n(\\psi, \\varrho,\\alpha,\\xi)$ denote the MLE of $(\\psi,\\varrho)$ for a given $\\alpha$. Consider the sets $\\Theta_{\\lambda}^1:=\\{\\lambda \\in \\Theta_{\\lambda}:|\\lambda_\\mu| \\geq n^{-1\/8}(\\log n)^{-1} \\}$ and $\\Theta_{\\lambda}^2:=\\{\\lambda \\in \\Theta_{\\lambda}:|\\lambda_\\mu| \\leq n^{-1\/8}(\\log n)^{-1}\\}$, so that $\\Theta_{\\lambda} = \\Theta_{\\lambda}^1 \\cup \\Theta_{\\lambda}^2$. For $j=1,2$, define $(\\hat{\\psi}^j_\\alpha, \\hat \\varrho^j_\\alpha) : = {\\arg\\max}_{(\\psi,\\varrho) \\in \\Theta_{\\psi}\\times \\Theta_{\\varrho}, \\lambda \\in \\Theta_{\\lambda}^j} \\ell_n(\\psi, \\varrho,\\alpha,\\xi)$. Then, uniformly in $\\alpha$,\n\\[\n\\ell_n(\\hat \\psi_\\alpha, \\hat \\varrho_\\alpha, \\alpha,\\xi) = \\max\\left\\{\\ell_n(\\hat\\psi_\\alpha^1, \\hat\\varrho_\\alpha^1,\\alpha,\\xi),\\ell_n(\\hat\\psi_\\alpha^2,\\hat\\varrho_\\alpha^2,\\alpha,\\xi)\\right\\}.\n\\]\nHenceforth, we suppress the dependence of $\\hat \\psi_\\alpha$, $\\hat \\varrho_\\alpha$, etc. on $\\alpha$.\n\nDefine $B_{\\varrho n}(t_{\\lambda}(\\lambda, \\varrho,\\alpha ))$ as in (\\ref{B_pi}) in the proof of Proposition \\ref{P-LR} but using $t(\\psi,\\pi)$ and $s_{\\varrho k}$ defined in (\\ref{t-psi2}) and (\\ref{score_normal}) and replacing $t_{\\lambda}$ in (\\ref{B_pi}) with $t_{\\lambda}(\\lambda, \\varrho,\\alpha )$. Observe that the proof of Proposition \\ref{P-LR} goes through up to (\\ref{LR_appn2}) with the current notation and that $G_{\\varrho n}$ and $\\mathcal{I}_\\varrho$ are continuous in $\\varrho$. Further, $\\hat{\\varrho}^1 = O_p(n^{-1\/4}(\\log n)^2)$ because $\\hat \\varrho^1 (\\hat \\lambda_\\mu^1)^2 = O_p(n^{-1\/2})$ from Proposition \\ref{P-quadratic-N1}(a) and $|\\hat{\\lambda}_\\mu^1| \\geq n^{-1\/8}(\\log n)^{-1}$. Consequently, $B_{\\hat \\varrho^1 n}(\\sqrt{n}t_{\\lambda}(\\hat\\lambda^1, \\hat\\varrho^1,\\alpha)) = B_{0 n}(\\sqrt{n}t_{\\lambda}(\\hat\\lambda^1, \\hat\\varrho^1,\\alpha)) + o_p(1)$, and, uniformly in $\\alpha$,\n\\begin{equation} \\label{ln_max}\n2[\\ell_n(\\hat \\psi,\\hat \\varrho,\\alpha,\\xi) - \\ell_{0n}(\\hat\\vartheta_0)] = \\max \\{ B_ { 0 n}(\\sqrt{n} t_{\\lambda}(\\hat\\lambda^1, \\hat\\varrho^1,\\alpha)), B_{ \\hat \\varrho^2 n}(\\sqrt{n} t_{\\lambda}(\\hat\\lambda^2, \\hat\\varrho^2,\\alpha))\\} + o_p(1).\n\\end{equation}\n\nWe proceed to construct parameter spaces $\\tilde\\Lambda_{\\lambda\\alpha}^1$ and $\\tilde\\Lambda_{\\lambda \\alpha\\varrho}^2$ that are locally equal to the cones $\\Lambda_\\lambda^1$ and $\\Lambda_{\\lambda\\varrho}^2$ defined in (\\ref{Lambda-lambda}). Define $c(\\alpha) := \\alpha(1-\\alpha)$, and denote the elements of $t_{\\lambda}(\\hat\\lambda^j,\\hat\\varrho^j,\\alpha)$ corresponding to (\\ref{t_lambda_rho}) by\n\\begin{equation*}\nt_{\\lambda}(\\hat\\lambda^j,\\hat\\varrho^j,\\alpha)\n=\n\\begin{pmatrix}\n\\hat t_{\\varrho\\mu^2}^j\\\\\n\\hat t_{\\mu\\sigma}^j \\\\\n\\hat t_{\\sigma^2}^j\\\\\n\\hat t_{\\beta \\mu}^j\\\\\n\\hat t_{\\beta \\sigma}^j\\\\\n\\hat t_{v(\\beta)}^j\\\\\n\\end{pmatrix}\n:= \n c(\\alpha) \n\\begin{pmatrix}\n \\hat\\varrho^j (\\hat \\lambda_\\mu^j)^2 \\\\\n \\hat\\lambda_\\mu \\hat \\lambda_\\sigma\\\\\n (\\hat\\lambda_\\sigma^j)^2 + b(\\alpha)(\\hat \\lambda_\\mu^j)^4\/12 \\\\\n \\hat \\lambda_\\beta^j \\hat \\lambda_\\mu^j\\\\\n \\hat \\lambda_\\beta^j \\hat \\lambda_\\sigma^j\\\\\n v(\\hat \\lambda_\\beta^j)\n\\end{pmatrix}. \n\\end{equation*}\nNote that $\\hat{\\lambda}_{\\sigma}^1 = O_p(n^{-3\/8} \\log n)$ and $\\hat{\\lambda}_\\beta^1 = O_p(n^{-3\/8} \\log n)$ because $(\\hat{t} _{\\mu \\sigma}^1, \\hat{t} _{\\beta \\mu}^1) = O_p(n^{-1\/2})$ from Proposition \\ref{P-quadratic-N1}(a) and $|\\hat{\\lambda}_\\mu^1| \\geq n^{-1\/8}(\\log n)^{-1}$. Furthermore, $\\hat t_{\\sigma^2}^2 = c(\\alpha) (\\hat \\lambda_\\sigma^2)^2 + o_p(n^{-1\/2})$ because $|\\hat\\lambda_\\mu^2|\\leq n^{-1\/8} (\\log n)^{-1}$. Consequently,\n\\begin{equation} \\label{t_hat_1}\n\\begin{aligned}\n\\hat{t}_{\\beta \\sigma}^1 &= o_p(n^{-1\/2}), \\quad \\hat{t}_{v(\\beta)}^1 = o_p(n^{-1\/2}), \\quad \\hat t_{\\sigma^2}^1 = c(\\alpha) b(\\alpha)(\\hat{\\lambda}_\\mu^1)^4\/12 + o_p(n^{-1\/2}), \\\\\n\\hat t_{\\sigma^2}^2 &= c(\\alpha) (\\hat \\lambda_\\sigma^2)^2 + o_p(n^{-1\/2}).\n\\end{aligned}\n\\end{equation}\nIn view of this, let $t_{\\lambda}(\\lambda, \\varrho,\\alpha):=( t_{\\varrho\\mu^2}, t_{\\mu\\sigma },t_{\\sigma^2 },t_{\\beta\\mu}',t_{\\beta\\sigma}',t_{v(\\beta)}')' \\in \\mathbb{R}^{q_{\\lambda}}$, and consider the following sets:\n\\begin{equation*}\n\\begin{aligned\n&\\tilde\\Lambda_{\\lambda \\alpha}^1:=\\{ t_{\\lambda}(\\lambda, \\varrho, \\alpha) : t_{\\varrho\\mu^2} = c(\\alpha) \\varrho \\lambda_\\mu^2, t_{\\mu\\sigma }= c(\\alpha)\\lambda_\\mu \\lambda_\\sigma, t_{\\sigma^2} = c(\\alpha) b(\\alpha)\\lambda_\\mu^4\/12, \\\\\n& \\qquad \\qquad t_{\\beta\\mu} = c(\\alpha)\\lambda_{\\beta}\\lambda_{\\mu}, t_{\\beta \\sigma}=0, t_{v(\\beta)} = 0 \\ \\text{for some } (\\lambda,\\varrho) \\in \\Theta_\\lambda\\times\\Theta_{\\varrho}\\}, \\\\\n&\\tilde\\Lambda_{\\lambda\\alpha\\varrho}^2:=\\{ t_{\\lambda}(\\lambda, \\varrho, \\alpha) : t_{\\varrho\\mu^2} = c(\\alpha)\\varrho \\lambda_{\\mu}^2, t_{\\mu\\sigma } = c(\\alpha)\\lambda_\\mu\\lambda_\\sigma, t_{\\sigma^2}= c(\\alpha)\\lambda_\\sigma^2, \\\\\n& \\qquad \\qquad t_{\\beta\\mu }= c(\\alpha)\\lambda_{\\beta} \\lambda_\\mu, t_{\\beta\\sigma }= c(\\alpha)\\lambda_{\\beta} \\lambda_\\sigma, t_{v(\\beta)} =c(\\alpha)v(\\lambda_{\\beta})\\ \\text{for some }\\lambda\\in \\Theta_\\lambda \\}. \n\\end{aligned}\n\\end{equation*}\n$\\tilde\\Lambda_{\\lambda \\alpha}^1$ is indexed by $\\alpha$ but does not depend on $\\varrho$ because $B_ {0n}(\\cdot)$ in (\\ref{ln_max}) does not depend on $\\varrho$, whereas $\\tilde\\Lambda_{\\lambda \\alpha\\varrho}^2$ is indexed by both $\\alpha$ and $\\varrho$ because $B_ {\\hat \\varrho^2 n}(\\cdot)$ in (\\ref{ln_max}) depends on $\\hat \\varrho^2$. Define $(\\tilde \\lambda_{\\alpha}^1, \\tilde \\varrho_{\\alpha}^1)$ and $\\tilde \\lambda_{\\alpha\\varrho}^2$ by $B_{0 n}(\\sqrt{n} t_{\\lambda}(\\tilde\\lambda_{\\alpha}^1,\\tilde\\varrho_{\\alpha}^1,\\alpha)) = {\\max}_{t_{\\lambda}(\\lambda,\\varrho,\\alpha) \\in \\tilde\\Lambda_{\\lambda \\alpha}^1} B_{0 n}(\\sqrt{n} t_{\\lambda}(\\lambda,\\varrho,\\alpha))$ and $B_{\\varrho n}(\\sqrt{n} t_{\\lambda}(\\tilde\\lambda_{\\alpha\\varrho}^2,\\varrho,\\alpha)) = {\\max}_{t_{\\lambda}(\\lambda,\\varrho,\\alpha) \\in \\tilde \\Lambda_{\\lambda \\alpha\\varrho}^2} B_{\\varrho n}(\\sqrt{n} t_{\\lambda}(\\lambda,\\varrho,\\alpha))$.\n\nDefine $W_n(\\alpha):=\\max\\{B_{0 n}(\\sqrt{n} t_{\\lambda}(\\tilde\\lambda_{\\alpha}^1,\\tilde\\varrho_{\\alpha}^1,\\alpha)), \\sup_{\\varrho \\in \\Theta_{\\varrho}}B_{\\varrho n}(\\sqrt{n} t_{\\lambda}(\\tilde\\lambda^2_{\\alpha\\varrho},\\varrho,\\alpha)) \\}$, then we have \n\\begin{equation} \\label{LR_W}\n2[\\ell_n(\\hat \\psi, \\hat \\varrho, \\alpha,\\xi) - \\ell_{0n}(\\hat\\vartheta_0)] = W_n(\\alpha) + o_p(1), \n\\end{equation}\n uniformly in $\\alpha \\in \\Theta_\\alpha$ because (i) $W_n(\\alpha) \\geq 2[\\ell_n(\\hat \\psi, \\hat \\varrho, \\alpha,\\xi) - \\ell_{0n}(\\hat\\vartheta_0)] + o_p(1)$ in view of the definition of $(\\tilde \\lambda_{\\alpha}^1,\\tilde \\varrho_{\\alpha}^1,\\tilde \\lambda_{\\alpha\\varrho}^2)$, (\\ref{ln_max}), and (\\ref{t_hat_1}), and (ii) $2[\\ell_n(\\hat \\psi,\\hat \\varrho, \\alpha, \\xi) - \\ell_{0n}(\\hat\\vartheta_0)] \\geq \\\\ \\max\\{ 2[\\max_\\eta\\ell_n(\\eta,\\tilde \\lambda^1_\\alpha, \\tilde \\varrho^1_\\alpha,\\alpha,\\xi),\\sup_{\\varrho \\in \\Theta_\\varrho} \\max_\\eta \\ell_n(\\eta,\\tilde \\lambda^2_{\\alpha\\varrho}, \\varrho,\\alpha,\\xi) \\} - 2\\ell_{0n}(\\hat\\vartheta_0)+ o_p(1) = W_n(\\alpha)+ o_p(1)$\nfrom the definition of $(\\hat \\psi,\\hat\\varrho)$.\n\nThe asymptotic distribution of the LRTS follows from applying Theorem 1(c) of \\citet{andrews01em} to $(B_{0 n}(\\sqrt{n} t_{\\lambda}(\\tilde\\lambda_{\\alpha}^1,\\tilde\\varrho_{\\alpha}^1,\\alpha)), B_{\\varrho n}(\\sqrt{n} t_{\\lambda}(\\tilde\\lambda^2_{\\alpha\\varrho},\\varrho,\\alpha)))$. First, Assumption 2 of \\citet{andrews01em} holds trivially for $B_{\\varrho n}(\\sqrt{n} t(\\lambda,\\varrho,\\alpha))$. Second, Assumption 3 of \\citet{andrews01em} is satisfied by (\\ref{weak_cgce}) and Assumption \\ref{A-nonsing2}. Assumption 4 of \\citet{andrews01em} is satisfied by Proposition \\ref{P-quadratic-N1}. Assumption $5^*$ of \\citet{andrews01em} holds with $B_T=n^{1\/2}$ because $\\tilde\\Lambda_{\\lambda \\alpha}^1$ is locally (in a neighborhood of $\\varrho=0, \\lambda=0$) equal to the cone $\\Lambda_{\\lambda}^1$ and $\\tilde\\Lambda_{\\lambda \\alpha\\varrho}^2$ is locally equal to the cone $\\Lambda_{\\lambda \\varrho}^2$ uniformly in $\\varrho\\in\\Theta_{\\varrho\\epsilon}$. Consequently, $W_n(\\alpha) \\overset{d}{\\rightarrow} \\max\\{ \\mathbb{I}\\{\\varrho=0\\} (\\tilde t_{\\lambda }^1)' \\mathcal{I}_{\\lambda.\\eta 0} \\tilde t_{\\lambda }^1, \\sup_{\\varrho\\in\\Theta_{\\varrho}} (\\tilde t_{\\lambda \\varrho}^2)' \\mathcal{I}_{\\lambda.\\eta \\varrho} \\tilde t_{\\lambda \\varrho}^2 \\}$ uniformly in $\\alpha$ from Theorem 1(c) of \\citet{andrews01em}, and the stated result follows from (\\ref{LR_W}).\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{P-quadratic-N1-homo}]\nThe proof is similar to that of Proposition \\ref{P-quadratic-N1}. Expanding $l_{k\\vartheta x_0} -1$ five times around $\\psi^*$ and proceeding as in the proof of Proposition \\ref{P-quadratic-N1} gives\n\\begin{equation} \\label{lk_2_normal_homo}\nl_{\\vartheta k x_0} -1 = t(\\psi,\\pi)' s_{\\varrho k} + r_{k,0}(\\pi) + u_{kx_0}(\\psi,\\pi), \n\\end{equation} \nwhere $t(\\psi,\\pi)$ is defined in (\\ref{t-psi2-homo}), $s_{\\varrho k}$ is defined in (\\ref{score_normal_homo}) and satisfies \n\\begin{equation*}\n s_{\\varrho k} \n:=\n\\begin{pmatrix}\n\\nabla_{\\eta} \\overline p_{\\psi^* \\pi k,0} \/ \\overline p_{\\psi^* \\pi k,0} \\\\\n\\zeta_{k,0}(\\varrho)\/2 \\\\\n\\nabla_{\\mu^3} f_k^* \/ 3! f_k^* \\\\\n\\nabla_{\\mu^4} f_k^* \/ 4! f_k^* \\\\\n\\nabla_{\\lambda_\\beta\\lambda_\\mu} \\overline p_{\\psi^* \\pi k,0} \/ \\alpha(1-\\alpha)\\overline p_{\\psi^* \\pi k,0} \\\\\n\\widetilde{\\nabla}_{v(\\lambda_\\beta)} \\overline p_{\\psi^* \\pi k,0} \/ \\alpha(1-\\alpha) \\overline p_{\\psi^* \\pi k,0} \n\\end{pmatrix},\n\\end{equation*}\nand\n\\begin{align*}\nr_{k,0}(\\pi):&= \\widetilde \\Lambda_{k,0}(\\pi) '\\tau(\\psi) + \\Lambda^5_{k,0}(\\overline \\psi,\\pi) ' (\\Delta \\psi)^{\\otimes 5} \\\\\n&\\quad + \\lambda_\\mu^3[\\nabla_{\\lambda_\\mu^3} \\overline p_{\\psi^* \\pi k,0}\/\\overline p_{\\psi^* \\pi k,0} - \\alpha(1-\\alpha)(1-2\\alpha)\\nabla_{\\mu^3} f_k^*\/f_k^*]\/3!\\\\\n& \\quad+ \\lambda_\\mu^4[\\nabla_{\\lambda_\\mu^4} \\overline p_{\\psi^* \\pi k,0}\/\\overline p_{\\psi^* \\pi k,0} - \\alpha(1-\\alpha)(1-6\\alpha+6 \\alpha^2)\\nabla_{\\mu^4} f_k^*\/f_k^*]\/4!,\n\\end{align*}\nwhere $u_{kx_0}(\\psi,\\pi)$, $\\overline p_{\\psi \\pi k,m}$, and the terms in the definition of $r_{k,0}(\\pi)$ are defined similarly to those in the proof of Proposition \\ref{P-quadratic-N1}.\n\nThe stated result is proven if the terms on the right-hand side of (\\ref{lk_2_normal_homo}) satisfy Assumption \\ref{assn_a3}. $t(\\psi,\\pi) = 0$ if and only if $\\psi=\\psi^*$. $s_{\\varrho k}$ and $u_{kx_0}(\\psi,\\pi)$ satisfy Assumption \\ref{assn_a3} by the same argument as the proof of Proposition \\ref{P-quadratic-N1}. For $r_{k,0}(\\pi)$, first, $\\Lambda^5_{k,0}(\\overline \\psi,\\pi)' (\\Delta \\psi)^{\\otimes 5}$ satisfies Assumptions \\ref{assn_a3}(c) and (d) from a similar argument to the proof of Proposition \\ref{P-quadratic-N1}; $\\lambda_\\mu^5$ is dominated by $\\lambda_\\mu^3$ or $\\lambda_\\mu^4$ because $\\inf_{0\\leq \\alpha \\leq 1}\\max\\{|1-2\\alpha|,|1-6\\alpha + 6\\alpha^2|\\}>0$. Second, similar to (\\ref{d4r}) in the proof of Proposition \\ref{P-quadratic-N1}, write $\\widetilde \\Lambda_{k,0}(\\pi) '\\tau(\\psi)=\\nabla_{(\\eta^{\\otimes 2})'} \\overline p_{\\psi^* \\pi k,0} (\\Delta \\eta)^{\\otimes 2} \/ 2!\\overline p_{\\psi^* \\pi k,0}+ \\tilde R_{3k\\vartheta} + R_{4k\\vartheta}$, where $\\tilde R_{3k\\vartheta}:= [\\nabla_{(\\psi^{\\otimes 3})'} \\overline p_{\\psi^* \\pi k,0} (\\Delta \\psi)^{\\otimes 3}- \\nabla_{\\lambda_\\mu^3} \\overline p_{\\psi^* \\pi k,0} \\lambda_\\mu^3]\/3!\\overline p_{\\psi^* \\pi k,0}$, and $R_{4k\\vartheta}$ is defined as $R_{4k\\vartheta}$ in (\\ref{r_jk}). The term $\\nabla_{(\\eta^{\\otimes 2})'} \\overline p_{\\psi^* \\pi k,0} (\\Delta \\eta)^{\\otimes 2} \/ 2!\\overline p_{\\psi^* \\pi k,0}$ clearly satisfies Assumptions \\ref{assn_a3}(c) and (d). The terms in $\\tilde R_{3k\\vartheta}$ satisfy Assumptions \\ref{assn_a3}(c) and (d) because they contain either $\\Delta \\eta$ or $\\lambda_\\mu^2\\lambda_\\beta$ or $\\lambda_\\mu\\lambda_\\beta^2$ or $\\lambda_\\beta^3$. The terms in $R_{4k\\vartheta}$ satisfy Assumptions \\ref{assn_a3}(c) and (d) because they either contain $\\Delta \\eta$ or a term of the form $\\lambda_\\mu^i\\lambda_\\beta^{4-i}$ with $ 1 \\leq i \\leq 3$. The last two terms in $r_{k,0}(\\pi)$ satisfy Assumptions \\ref{assn_a3}(c) and (d) from Lemma \\ref{lemma_d34_homo}. Therefore, $r_{k,0}(\\pi)$ satisfies Assumptions \\ref{assn_a3}(c) and (d), and the stated result is proven. \n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{P-LR-N1-homo}] \nThe proof is similar to the proof of Proposition \\ref{P-LR-N1}. Let $(\\hat{\\psi}, \\hat \\varrho, \\hat \\alpha) : = {\\arg\\max}_{(\\psi, \\varrho,\\alpha) \\in \\Theta_{\\psi}\\times\\Theta_{\\varrho}\\times\\Theta_{\\alpha}} \\ell_n(\\psi,\\varrho,\\alpha,\\xi)$ denote the MLE of $(\\psi,\\varrho,\\alpha)$. Consider the sets $\\Theta_{\\lambda}^1:=\\{\\lambda \\in \\Theta_{\\lambda}:|\\lambda_\\mu| \\geq n^{-1\/6}(\\log n)^{-1} \\}$ and $\\Theta_{\\lambda}^2:=\\{\\lambda \\in \\Theta_{\\lambda}:|\\lambda_\\mu| \\leq n^{-1\/6}(\\log n)^{-1} \\}$, so that $\\Theta_{\\lambda} = \\Theta_{\\lambda}^1 \\cup \\Theta_{\\lambda}^2$. For $j=1,2$, define $(\\hat{\\psi}^j, \\hat \\varrho^j,\\hat\\alpha^j) : = {\\arg\\max}_{(\\psi,\\varrho,\\alpha) \\in \\Theta_{\\psi}\\times \\Theta_{\\varrho }\\times \\Theta_{\\alpha }, \\lambda \\in \\Theta_{\\lambda}^j} \\ell_n(\\psi,\\varrho,\\alpha,\\xi)$, so that $\\ell_n(\\hat \\psi, \\hat \\varrho,\\hat\\alpha,\\xi) = \\max_{j \\in \\{1,2\\}} \\ell_n(\\hat\\psi^j, \\hat\\varrho^j,\\hat\\alpha^j,\\xi)$.\n\nDefine $B_{\\varrho n}(t_\\lambda(\\lambda,\\varrho,\\alpha))$ as in (\\ref{B_pi}) in the proof of Proposition \\ref{P-LR} but using $t(\\psi,\\pi)$ and $s_{\\varrho k}$ defined in (\\ref{t-psi2-homo}) and (\\ref{score_normal_homo}) and replacing $t_{\\lambda}$ in (\\ref{B_pi}) with $t_\\lambda(\\lambda,\\varrho,\\alpha)$. Observe that $\\hat{\\varrho}^1 = O_p(n^{-1\/6}(\\log n)^2)$ because $\\hat \\varrho^1 (\\hat \\lambda_\\mu^1)^2 = O_p(n^{-1\/2})$ from Proposition \\ref{P-quadratic-N1-homo}(a) and $|\\hat{\\lambda}_\\mu^1| \\geq n^{-1\/6}(\\log n)^{-1}$. Using the argument of the proof of Proposition \\ref{P-LR-N1} leading to (\\ref{ln_max}), we obtain \n\\begin{equation*\n2[\\ell_n(\\hat \\psi, \\hat \\varrho, \\hat \\alpha,\\xi) - \\ell_{0n}(\\hat\\vartheta_0)]\n = \\max \\{ B_ { 0 n}(\\sqrt{n} t_\\lambda(\\hat\\lambda^1,\\hat\\varrho^1,\\hat \\alpha^1)), B_{ \\hat \\varrho^2 n}(\\sqrt{n} t_\\lambda(\\hat\\lambda^2, \\hat\\varrho^2,\\hat \\alpha^2))\\} + o_p(1).\n\\end{equation*}\n\nWe proceed to construct parameter spaces that are locally equal to the cones $\\Lambda_\\lambda^1$ and $\\Lambda_{\\lambda\\varrho}^2$ defined in (\\ref{Lambda-lambda-homo}). Define $c(\\alpha) := \\alpha(1-\\alpha)$, and denote the elements of $t_\\lambda(\\hat\\lambda^j,\\hat\\varrho^j,\\hat\\alpha^j)$ corresponding to (\\ref{t-psi2-homo}) by\n\\begin{equation*}\nt_{\\lambda}(\\hat\\lambda^j,\\hat\\varrho^j,\\hat\\alpha^j)\n=\n\\begin{pmatrix}\n\\hat t_{\\varrho\\mu^2}^j\\\\\n\\hat t_{\\mu^3}^j \\\\\n\\hat t_{\\mu^4}^j \\\\\n\\hat t_{\\beta \\mu}^j\\\\\n\\hat t_{v(\\beta)}^j\\\\\n\\end{pmatrix}\n:= \nc(\\hat\\alpha^j)\n\\begin{pmatrix}\n \\hat\\varrho^j (\\hat \\lambda_\\mu^j)^2 \\\\\n (1-2\\hat\\alpha^j) (\\hat\\lambda_\\mu^j)^3\\\\\n (1-6\\hat\\alpha^j+6(\\hat\\alpha^j)^2) (\\hat\\lambda_\\mu^j)^4\\\\\n \\hat \\lambda_\\beta^j \\hat \\lambda_\\mu^j\\\\\n v(\\hat \\lambda_\\beta^j)\n\\end{pmatrix}. \n\\end{equation*}\nNote that $\\hat{\\lambda}_\\beta^1 = O_p(n^{-1\/3} \\log n)$ because $\\hat{t} _{\\beta \\mu}^1 = O_p(n^{-1\/2})$ from Proposition \\ref{P-quadratic-N1-homo}(a) and $|\\hat{\\lambda}_\\mu^1| \\geq n^{-1\/6}(\\log n)^{-1}$. Furthermore, $|\\hat\\lambda_\\mu^2| \\leq n^{-1\/6}(\\log n)^{-1}$. Therefore,\n\\begin{equation*}\n\\hat{t}_{v(\\beta)}^1 = o_p(n^{-1\/2}), \\quad \\hat t_{\\mu^3}^{2} = o_p(n^{-1\/2}), \\quad \\hat t_{\\mu^4}^{2}=o_p(n^{-1\/2}).\n\\end{equation*}\nIn view of this, let $t_\\lambda(\\lambda,\\varrho,\\alpha):=( t_{\\varrho\\mu^2}, t_{\\mu^3},t_{\\mu^4},t_{\\beta\\mu}',t_{v(\\beta)}')' \\in \\mathbb{R}^{q_{\\lambda}}$, and consider the following sets:\n\\begin{align*}\n&\\tilde\\Lambda_{\\lambda }^{1}:=\\{ t_\\lambda(\\lambda,\\varrho,\\alpha) : t_{\\varrho\\mu^2} = c(\\alpha) \\varrho \\lambda_\\mu^2, t_{\\mu^3}= c(\\alpha)(1-2\\alpha)\\lambda_\\mu^3, t_{\\mu^4} = c(\\alpha)(1-6\\alpha+6\\alpha^2) \\lambda_\\mu^4, \\\\\n& \\qquad \\qquad t_{\\beta\\mu} = c(\\alpha)\\lambda_{\\beta}\\lambda_{\\mu}, t_{v(\\beta)} = 0 \\ \\text{for some }(\\lambda,\\varrho,\\alpha) \\in \\Theta_\\lambda\\times \\Theta_{\\varrho}\\times \\Theta_{\\alpha}\\}, \\\\\n&\\tilde\\Lambda_{\\lambda \\alpha\\varrho}^2:=\\{ t_\\lambda(\\lambda,\\varrho,\\alpha) : t_{\\varrho\\mu^2} = c(\\alpha)\\varrho \\lambda_{\\mu}^2, t_{\\mu^3} = t_{\\mu^4}= 0, \\\\\n& \\qquad \\qquad t_{\\beta\\mu }= c(\\alpha)\\lambda_{\\beta} \\lambda_\\mu, t_{v(\\beta)} =c(\\alpha)v(\\lambda_{\\beta})\\ \\text{for some }\\lambda\\in \\Theta_\\lambda \\}. \n\\end{align*}\nDefine $(\\tilde \\lambda^1, \\tilde \\varrho^1, \\tilde \\alpha^1)$ and $\\tilde \\lambda_{\\alpha\\varrho}^2$ by $B_{0 n}(\\sqrt{n} t_\\lambda(\\tilde\\lambda^1,\\tilde \\varrho^1, \\tilde \\alpha^1)) = {\\max}_{t_\\lambda(\\lambda,\\varrho,\\alpha) \\in \\tilde\\Lambda_{\\lambda}^1} B_{0 n}(\\sqrt{n} t_\\lambda(\\lambda,\\varrho,\\alpha))$ and \\\\ $B_{\\varrho n}(\\sqrt{n} t_\\lambda(\\tilde\\lambda_{\\alpha\\varrho}^2,\\varrho,\\alpha)) = {\\max}_{t_\\lambda(\\lambda,\\varrho,\\alpha) \\in \\tilde \\Lambda_{\\lambda\\alpha\\varrho}^2} B_{\\varrho n}(\\sqrt{n} t_\\lambda(\\lambda,\\varrho,\\alpha))$. $\\tilde\\Lambda_{\\lambda}^{1}$ is locally (in the neighborhood of $\\varrho=0$, $\\lambda=0$) equal to the cone $\\Lambda_{\\lambda}^{1}$ because, when $|1-2\\alpha|\\geq \\epsilon>0$ for some positive constant $\\epsilon$, we have $t_{\\mu^4}\/t_{\\mu^3} \\to 0$ as $\\lambda_\\mu \\to 0$, and when $\\alpha$ is in the neighborhood of $1\/2$, we have $1-6\\alpha+6\\alpha^2 <0$. $\\tilde\\Lambda_{\\lambda \\alpha \\varrho}^2$ is locally equal to the cone $\\Lambda_{\\lambda\\varrho}^2$ uniformly in $\\varrho \\in \\Theta_{\\varrho}$.\n\nDefine $W_n:=\\max\\{B_{0 n}(\\sqrt{n} t_\\lambda(\\tilde\\lambda^1,\\tilde\\varrho^1,\\tilde\\alpha^1)), \\sup_{(\\alpha,\\varrho) \\in\\Theta_{\\alpha} \\times \\Theta_{\\varrho} } B_{\\varrho n}(\\sqrt{n} t_\\lambda(\\tilde\\lambda^2_{\\alpha\\varrho},\\varrho,\\alpha)) \\}$. Proceeding as in the proof of Proposition \\ref{P-LR-N1} gives $2[\\ell_n(\\hat \\psi, \\hat \\varrho,\\hat \\alpha,\\xi) - \\ell_{0n}(\\hat\\vartheta_0)] = W_n + o_p(1)$, and the asymptotic distribution of the LRTS follows from applying Theorem 1(c) of \\citet{andrews01em} to $(B_{0 n}(\\sqrt{n} t_\\lambda(\\tilde\\lambda^1, \\tilde\\varrho^1,\\tilde\\alpha^1 )), B_{\\varrho n}(\\sqrt{n} t_\\lambda(\\tilde\\lambda^2_{\\alpha\\varrho},\\varrho,\\alpha))$. \n\\end{proof}\n\n\\begin{proof}[Proof of Propositions \\ref{P-LR_M}, \\ref{P-LR_M_normal}, and \\ref{P-LR_M_normal_homo}] \nLet $\\mathcal{N}_m^*$ denote an arbitrarily small neighborhood of $\\Upsilon_m^*$, and let $\\hat{\\psi}_m$ denote a local MLE that maximizes $\\ell_n(\\psi_m,\\pi_m,\\xi_{M_0+1})$ subject to $\\psi_m \\in \\mathcal{N}_m^*$. Proposition \\ref{P-consist_M} and $\\Upsilon^*=\\cup_{m=1}^{M_0}\\Upsilon_m^*$ imply that $\\ell_n(\\hat\\vartheta_{M_0+1},\\xi_{M_0+1}) = \\max_{m=1,\\ldots,M_0} \\ell_n(\\hat\\psi_m,\\pi_m,\\xi_{M_0+1})$ with probability approaching one. Because $\\psi_\\ell^{*}\\notin \\mathcal{N}_m^*$ for any $\\ell \\neq m$, it follows from Proposition \\ref{P-consist_M} that $\\hat{\\psi}_m-\\psi_m^{*}=o_p(1)$.\n\nNext, $\\ell_n(\\psi_{m},\\pi_m,\\xi_{M_0+1}) - \\ell_n(\\psi_m^{*},\\pi_m,\\xi_{M_0+1})$ admits the same expansion as $\\ell_n(\\psi,\\pi,\\xi) - \\ell_n(\\psi^{*},\\pi,\\xi)$ in (\\ref{ln_appn}) or (\\ref{ln_appn_N1}). \nTherefore, the stated result follows from applying the proof of Propositions \\ref{P-LR}, \\ref{P-LR-N1}, and \\ref{P-LR-N1-homo} to $\\ell_n(\\hat \\psi_m, \\pi_m,\\xi_{M_0+1}) - \\ell_{n}(\\hat\\vartheta_{M_0},\\xi_{M_0})$ for each $m$ and combining the results to derive the joint asymptotic distribution of $\\{\\ell_n(\\hat \\psi_m, \\pi_m,\\xi_{M_0+1}) - \\ell_{n}(\\hat\\vartheta_{M_0},\\xi_{M_0})\\}_{m=1}^{M_0}$.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{P-LAN}] \nObserve that Proposition \\ref{Ln_thm1} holds under $\\mathbb{P}_{\\vartheta^*,x_0}^n$ under the assumptions of Propositions \\ref{P-quadratic}, \\ref{P-quadratic-N1}, and \\ref{P-quadratic-N1-homo}. Because $\\vartheta_n=(\\eta_n',\\lambda_n',\\pi_n')'\\in \\mathcal{N}_{c\/\\sqrt{n}}$ by choosing $c> |h|$, it follows from Proposition \\ref{Ln_thm1} that\n\\begin{equation}\\label{expansion}\n\\sup_{x_0 \\in \\mathcal{X}} \\left|\\log \\frac{d\\mathbb{P}_{\\vartheta_n,x_0}^n}{d \\mathbb{P}_{\\vartheta^*,x_0}^n} - h' \\nu_n(s_{\\varrho_n k}) + \\frac{1}{2}h' \\mathcal{I}_{\\varrho_n} h \\right|=o_{\\mathbb{P}_{\\vartheta^*,x_0}^n}(1),\n\\end{equation}\nwhere $s_{\\varrho k}$ is given by (\\ref{score}), (\\ref{score_normal}), and (\\ref{score_normal_homo}) for the models of the non-normal distribution, heteroscedastic normal distribution, and homoscedastic normal distribution, respectively. Furthermore, $\\nu_n(s_{\\varrho_n k}) \\Rightarrow G_\\varrho$ under $\\mathbb{P}_{\\vartheta^*,x_0}^n$, where $G_\\varrho$ is a mean zero Gaussian process with $cov(G_{\\varrho_1},G_{\\varrho_2})=\\mathcal{I}_{\\varrho_1\\varrho_2}:= \\lim_{k\\rightarrow \\infty} \\mathbb{E}_{\\vartheta^*} (s_{\\varrho_1 k}s_{\\varrho_2 k}')$. Therefore, $d\\mathbb{P}_{\\vartheta_n,x_0}^n \/ d \\mathbb{P}_{\\vartheta^*,x_0}^n$ converges in distribution under $\\mathbb{P}_{\\vartheta^*,x_0}^n$ to $\\exp\\left( N( \\mu,\\sigma^2) \\right)$ with $\\mu=-(1\/2) h' \\mathcal{I}_{ \\varrho} h$ and $\\sigma^2= h' \\mathcal{I}_{ \\varrho} h$, so that $E(\\exp\\left( N( \\mu,\\sigma^2) \\right))=1$. Consequently, part (a) follows from Le Cam's first lemma (see, e.g., Corollary 12.3.1 of \\citet{lehmannromano05book}). Part (b) follows from Le Cam's third lemma (see, e.g., Corollary 12.3.2 of \\citet{lehmannromano05book}) because part (a) and (\\ref{expansion}) imply that\n\\[\n\\begin{pmatrix}\n\\nu_n(s_{\\varrho_n k})\\\\\n\\log\\frac{d\\mathbb{P}_{\\vartheta_n,x_0}^n}{d \\mathbb{P}_{\\vartheta^*,x_0}^n} \n\\end{pmatrix}\n \\overset{d}{\\rightarrow} \nN\\left(\n\\begin{pmatrix}\n0\\\\\n-\\frac{1}{2} h' \\mathcal{I}_{\\varrho} h\n\\end{pmatrix}, \n\\begin{pmatrix}\n\\mathcal{I}_{\\varrho}&\\mathcal{I}_{\\varrho}h\\\\\nh'\\mathcal{I}_{\\varrho}&h'\\mathcal{I}_{\\varrho}h\n\\end{pmatrix}\n\\right)\\quad\\text{under $\\mathbb{P}_{\\vartheta^*,x_0}^n$}.\n\\] \n\\end{proof} \n\n\\begin{proof}[Proof of Proposition \\ref{P-LAN2}] \nThe proof follows the argument in the proof of Proposition \\ref{P-LR}. Observe that $h_\\eta=0$ and $h_{\\lambda} =\\sqrt{n}t_{\\lambda}(\\lambda_n,\\pi_n)$ hold under $H_{1n}$. Therefore, Proposition \\ref{P-LAN} holds under $\\mathbb{P}_{\\vartheta_n,x_0}^n$ implied by $H_{1n}$, and, in conjunction with Theorem 12.3.2(a) of \\citet{lehmannromano05book}, Propositions \\ref{lemma-omega} and \\ref{P-quadratic} hold under $\\mathbb{P}_{\\vartheta_n,x_0}^n$. Consequently, the proof of Proposition \\ref{P-LR} goes through if we replace $G_{\\lambda.\\eta\\varrho n}\\Rightarrow G_{\\lambda.\\eta\\varrho}$ with $G_{\\lambda.\\eta\\varrho n} \\Rightarrow G_{\\lambda.\\eta\\varrho} + ( \\mathcal{I}_{\\lambda \\varrho \\varrho} - \\mathcal{I}_{\\lambda\\eta\\varrho} \\mathcal{I}_{\\eta}^{-1} \\mathcal{I}_{\\eta\\lambda\\varrho} ) h_\\lambda = G_{\\lambda.\\eta\\varrho} + \\mathcal{I}_{\\lambda.\\eta\\varrho} h_{\\lambda}$, and the stated result follows. \\end{proof}\n\n\\begin{proof}[Proof of Propositions \\ref{P-LAN3} and \\ref{P-LAN4}] \nThe proof is similar to the proof of Proposition \\ref{P-LAN2}. Observe that, for $j \\in \\{a,b\\}$, $h_\\eta^j=0$ and $h_{\\lambda}^j =\\sqrt{n}t_{\\lambda}(\\lambda_n,\\pi_n)+o(1)$ hold under $H_{1 n}^j$. Therefore, Proposition \\ref{P-LAN} holds under $\\mathbb{P}_{\\vartheta_n,x_0}^n$ implied by $H_{1n}^j$, and the stated result follows from repeating the argument of proof of Proposition \\ref{P-LAN2}. \n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition \\ref{P-bootstrap}] \nWe only provide the proof for the models of the non-normal distribution with $M_0=1$ because the proof for the other models is similar. The proof follows the argument in the proof of Theorem 15.4.2 in \\citet{lehmannromano05book}. Define $\\bf{C}_\\eta$ as the set of sequences $\\{\\eta_n\\}$ satisfying $\\sqrt{n}(\\eta_n - \\eta^*) \\to h_\\eta$ for some finite $h_\\eta$. \nDenote the MLE of the one-regime model parameter by $\\hat \\eta_n$. For the MLE under $H_0$, $\\sqrt{n}(\\hat \\eta_n - \\eta^*)$ converges in distribution to a $\\mathbb{P}_{\\vartheta^*}$-a.s. finite random variable by the standard argument. Then, by the Almost Sure Representation Theorem (e.g., Theorem 11.2.19 of \\citet{lehmannromano05book}), there exist random variables $\\tilde \\eta_n$ and $\\tilde h_\\eta$ defined on a common probability space such that $\\hat \\eta_n$ and $\\tilde \\eta_n$ have the same distribution and $\\sqrt{n}(\\tilde \\eta_n - \\eta^*)\\rightarrow \\tilde h_\\eta$ almost surely. Therefore, $\\{ \\tilde \\eta_n \\}\\in \\bf{C}_\\eta$ with probability one, and the stated result under $H_0$ follows from Lemma \\ref{lemma_btsp} because $\\hat \\eta_n$ and $\\tilde \\eta_n$ have the same distribution.\n\nFor the MLE under $H_{1n}$, note that the proof of Proposition \\ref{P-LAN2} goes through when $h_\\eta$ is finite even if $h_\\eta \\neq 0$. Therefore, $\\sqrt{n}(\\hat \\eta_n - \\eta^*)$ converges in distribution to a $\\mathbb{P}_{\\vartheta_n}$-a.s. finite random variable under $H_{1n}$. Hence, the stated result follows from Lemma \\ref{lemma_btsp} and repeating the argument in the case of $H_0$.\n\\end{proof}\n\n \n\\subsection{Auxiliary results}\n\n\n\\subsubsection{Missing information principle}\n\nThe following lemma extends equations (3.1) and (3.2) in Louis (1982), expressing the higher-order derivatives of the log-likelihood function in terms of the conditional expectation of the derivatives of the complete data log-likelihood function. For notational brevity, assume $\\vartheta$ is scalar. Adaptations to vector-valued $\\vartheta$ are straightforward but need more tedious notation. Let $\\nabla^j \\ell(Y):= \\nabla_\\vartheta^j \\log P(Y;\\vartheta)$ and $\\nabla^j \\ell(Y,X):= \\nabla_\\vartheta^j \\log P(Y,X;\\vartheta)$. For random variables $V_1,\\ldots,V_q$ and $Y$, define the central conditional moment of $(V_1^{r_1}\\cdots V_q^{r_q})$ as $\\mathbb{E}^c [V_1^{r_1} \\cdots V_q^{r_q} |Y ] := \\mathbb{E} [ (V_1-\\mathbb{E}[V_1|Y])^{r_1} \\cdots (V_q-\\mathbb{E}[V_q|Y])^{r_q} |Y] $.\n\\begin{lemma} \\label{louis}\nFor any random variables $X$ and $Y$ with density $P(Y,X;\\theta)$ and $P(Y;\\theta)$,\n\\begin{align*}\n&\\nabla \\ell(Y) = \\mathbb{E}\\left[ \\nabla \\ell(Y,X) \\middle| Y \\right], \\quad \\nabla^2 \\ell(Y) = \\mathbb{E}\\left[ \\nabla^2 \\ell(Y,X) \\middle| Y \\right] + \\mathbb{E}^c\\left[ (\\nabla \\ell(Y,X))^2\\middle| Y \\right], \\\\\n&\\nabla^3 \\ell(Y) = \\mathbb{E}\\left[ \\nabla^3 \\ell(Y,X) \\middle| Y \\right]+ 3 \\mathbb{E}^c\\left[ \\nabla^2 \\ell(Y,X) \\nabla \\ell(Y,X)\\middle| Y \\right] + \\mathbb{E}^c\\left[(\\nabla \\ell(Y,X))^3\\middle| Y \\right], \\\\\n&\\nabla^4 \\ell(Y)\n= \\mathbb{E}\\left[ \\nabla^4 \\ell(Y,X)\\middle| Y \\right] +4 \\mathbb{E}^c \\left[\\nabla^3 \\ell(Y,X) \\nabla \\ell(Y,X)\\middle| Y \\right] + 3 \\mathbb{E}^c\\left[ (\\nabla^2 \\ell(Y,X))^2 \\middle| Y \\right] \\\\\n& \\quad + 6 \\mathbb{E}^c\\left[\\nabla^2 \\ell(Y,X)(\\nabla \\ell(Y,X))^2\\middle| Y \\right] +\\mathbb{E}^c\\left[(\\nabla \\ell(Y,X))^4 \\middle| Y \\right] -3 \\left\\{ \\mathbb{E}^c\\left[ (\\nabla \\ell(Y,X))^2 \\middle| Y \\right]\\right\\}^2, \\\\\n&\\nabla^5 \\ell(Y) = \\mathbb{E}\\left[ \\nabla^5 \\ell(Y,X)\\middle| Y\\right] +5 \\mathbb{E}^c\\left[ \\nabla^4 \\ell(Y,X) \\nabla \\ell(Y,X) \\middle| Y\\right] +10 \\mathbb{E}^c\\left[ \\nabla^3 \\ell(Y,X) \\nabla^2 \\ell(Y,X) \\middle| Y\\right] \\\\\n&\\quad+10\\mathbb{E}^c\\left[ \\nabla^3 \\ell(Y,X) (\\nabla \\ell(Y,X))^2 \\middle| Y\\right]+15\\mathbb{E}^c\\left[ (\\nabla^2 \\ell(Y,X))^2 \\nabla \\ell(Y,X) \\middle| Y\\right] \\\\\n&\\quad +10\\mathbb{E}^c\\left[ \\nabla^2 \\ell(Y,X) (\\ell(Y,X))^3 \\middle| Y\\right] -30 \\mathbb{E}^c\\left[ \\nabla^2 \\ell(Y,X) \\nabla \\ell(Y,X) \\middle| Y\\right] \\mathbb{E}^c\\left[( \\nabla \\ell(Y,X))^2 \\middle| Y\\right] \\\\\n&\\quad+\\mathbb{E}^c\\left[ (\\nabla \\ell(Y,X))^5 \\middle| Y\\right] - 10\\mathbb{E}^c\\left[ (\\nabla \\ell(Y,X))^3 \\middle| Y\\right] \\mathbb{E}^c\\left[(\\nabla \\ell(Y,X))^2 \\middle| Y\\right], \\\\\n&\\nabla^6 \\ell(Y) = \\mathbb{E}\\left[ \\nabla^6 \\ell(Y,X)\\middle|Y \\right] \\\\\n& \\quad +6\\mathbb{E}^c\\left[\\nabla^5 \\ell(Y,X)\\nabla \\ell(Y,X) \\middle| Y\\right] +15 \\mathbb{E}^c\\left[\\nabla^4 \\ell(Y,X)\\nabla^2 \\ell(Y,X) \\middle| Y\\right] \\\\\n& \\quad+15 \\mathbb{E}^c\\left[\\nabla^4 \\ell(Y,X) (\\nabla \\ell(Y,X))^2\\middle| Y\\right] + 60\\mathbb{E}^c\\left[\\nabla^3 \\ell(Y,X)\\nabla^2 \\ell(Y,X)\\nabla \\ell(Y,X) \\middle| Y\\right] \\\\\n& \\quad+10\\mathbb{E}^c\\left[(\\nabla^3 \\ell(Y,X))^2\\middle| Y\\right]+ 15 \\mathbb{E}^c\\left[ (\\nabla^2 \\ell(Y,X))^3 \\middle| Y\\right] \\\\\n& \\quad+20\\mathbb{E}^c\\left[\\nabla^3 \\ell(Y,X)(\\nabla \\ell(Y,X))^3\\middle| Y\\right] - 60 \\mathbb{E}^c\\left[\\nabla^3 \\ell(Y,X) \\nabla \\ell(Y,X)\\middle| Y\\right]\\mathbb{E}\\left[(\\nabla \\ell(Y,X))^2\\middle| Y\\right]\\\\\n& \\quad+45 \\mathbb{E}^c\\left[ (\\nabla^2 \\ell(Y,X))^2 (\\nabla \\ell(Y,X))^2 \\middle| Y\\right] -90 \\left\\{\\mathbb{E}^c\\left[ \\nabla^2 \\ell(Y,X) \\nabla \\ell(Y,X) \\middle| Y\\right]\\right\\}^2\\\\\n& \\quad-45 \\mathbb{E}^c\\left[ (\\nabla^2 \\ell(Y,X))^2 \\middle| Y\\right] \\mathbb{E}^c\\left[ (\\nabla \\ell(Y,X))^2 \\middle| Y\\right] \\\\\n& \\quad +15 \\mathbb{E}^c\\left[ \\nabla^2 \\ell(Y,X) (\\nabla \\ell(Y,X))^4\\middle| Y\\right] -90 \\mathbb{E}^c\\left[ \\nabla^2 \\ell(Y,X) (\\nabla \\ell(Y,X))^2 \\middle| Y\\right] \\mathbb{E}^c\\left[ (\\nabla \\ell(Y,X))^2 \\middle| Y\\right]\\label{5-1}\\\\\n& \\quad -60 \\mathbb{E}^c\\left[ \\nabla^2 \\ell(Y,X) \\nabla \\ell(Y,X) \\middle| Y\\right] \\mathbb{E}^c\\left[ (\\nabla \\ell(Y,X))^3 \\middle| Y\\right] \\\\\n& \\quad +\\mathbb{E}^c\\left[ (\\nabla \\ell(Y,X))^6 \\middle| Y\\right] -15 \\mathbb{E}^c\\left[ (\\nabla \\ell(Y,X))^4 \\middle| Y\\right] \\mathbb{E}^c\\left[ (\\nabla \\ell(Y,X))^2 \\middle| Y\\right] \\\\\n& \\quad - 10\\left\\{\\mathbb{E}^c\\left[ (\\nabla \\ell(Y,X))^3 \\middle| Y\\right]\\right\\}^2 + 30\\left\\{\\mathbb{E}^c\\left[ (\\nabla \\ell(Y,X))^2 \\middle| Y\\right]\\right\\}^3,\n\\end{align*}\nprovided that the conditional expectation on the right-hand side exists. When $P(Y;\\theta)$ on the left-hand side is replaced with $P(Y|Z;\\theta)$, the stated result holds with $P(Y,X;\\theta)$ and $\\mathbb{E}[\\cdot |Y]$ on the right-hand side replaced with $P(Y,X|Z;\\theta)$ and $\\mathbb{E}[\\cdot |Y,Z]$. \n\\end{lemma} \n\\begin{proof}[Proof of Lemma \\ref{louis}]\nThe stated result follows from a direct calculation and relations such as\\\\ $\\nabla_\\vartheta^j P(Y;\\vartheta)\/P(Y;\\vartheta) = \\mathbb{E}[ \\nabla_\\vartheta^j P(Y,X;\\vartheta)\/P(Y,X;\\vartheta)|Y]$ and\n\\begin{equation} \\label{der_formula}\n\\begin{aligned}\n\\nabla \\log f &= \\nabla f\/f, \\quad \\nabla^2 \\log f = \\nabla^2 f\/f - (\\nabla \\log f )^2,\\\\\n\\nabla^3 \\log f &= \\nabla^3 f\/f -3 \\nabla^2 f \\nabla f\/f^2 + 2 (\\nabla f\/f)^3, \\\\\n\\nabla^4 \\log f &= \\nabla^4 f\/f -4 \\nabla^3 f \\nabla f\/f^2 - 3 (\\nabla^2 f\/f)^2 + 12 \\nabla^2 f (\\nabla f)^2\/ f^3 -6 (\\nabla f\/f)^4, \\\\\n\\nabla^5 \\log f &= \\nabla^5 f\/f -5 \\nabla^4 f \\nabla f\/f^2 - 10 \\nabla^3 f \\nabla^2 f\/f^2 + 20 \\nabla^3 f (\\nabla f)^2\/f^3 \\\\\n& \\quad + 30 (\\nabla^2 f)^2 \\nabla f \/f^3 - 60 \\nabla^2 f(\\nabla f)^3 \/f^4 + 24 (\\nabla f\/f)^5, \\\\\n\\nabla^6 \\log f & = \\nabla ^6 f\/f -6 \\nabla^5 f \\nabla f\/f^2 - 15 \\nabla ^4 f \\nabla^2 f\/f^2 + 30 \\nabla^4 f (\\nabla f)^2 \/ f^3 -10 (\\nabla^3 f)^2\/f^2 \\\\\n& \\quad +120 \\nabla^3 f \\nabla^2 f \\nabla f\/f^3 -120 \\nabla^3 f (\\nabla f)^3\/f^4 + 30 (\\nabla^2 f)^3\/f^3 \\\\\n& \\quad - 270 (\\nabla^2 f)^2(\\nabla f)^2 \/f^4 +360 \\nabla^2 f (\\nabla f)^4 \/f^5 - 120 (\\nabla f)^6\/f^6, \\\\\n\\nabla^3 f\/f &= \\nabla^3 \\log f + 3 \\nabla^2 \\log f \\nabla \\log f +\\left(\\nabla \\log f\\right)^3,\\\\\n\\nabla^4 f\/f &= \\nabla^4 \\log f + 4 \\nabla^3 \\log f \\nabla \\log f + 3(\\nabla^2 \\log f )^2 +6 \\nabla^2 \\log f (\\nabla\\log f)^2+ (\\nabla\\log f)^4,\\\\\n\\nabla^5 f\/f & = \\nabla^5 \\log f +5 \\nabla^4 \\log f \\nabla \\log f +10 \\nabla^3 \\log f \\nabla^2 \\log f +10 \\nabla^3 \\log f (\\nabla \\log f)^2 \\\\\n& \\quad +15 (\\nabla^2 \\log f)^2 \\nabla \\log f +10 \\nabla^2 \\log f (\\nabla \\log f)^3 + (\\nabla\\log f)^5, \\\\\n\\nabla^6 f\/f & = \\nabla^6 \\log f + 6 \\nabla^5 \\log f \\nabla \\log f + 15 \\nabla^4 \\log f \\nabla^2 \\log f + 15\\nabla^4 \\log f (\\nabla \\log f)^2 \\\\\n& \\qquad +10( \\nabla^3 \\log f )^2 +60 \\nabla^3 \\log f \\nabla^2 \\log f \\nabla \\log f +20\\nabla^3\\log f (\\nabla \\log f)^3 \\\\\n& \\qquad +15( \\nabla^2 \\log f )^3 +45(\\nabla^2 \\log f)^2( \\nabla \\log f)^2 +15\\nabla^2 \\log f(\\nabla \\log f)^4 + (\\nabla\\log f)^6.\n\\end{aligned}\n\\end{equation}\nFor example, $\\nabla^3 \\ell(Y)$ is derived by writing $\\nabla^3 \\ell(Y)$ as, with suppressing $\\vartheta$, \n\\begin{align*}\n& \\nabla^3 \\ell(Y) \\\\\n& = \\frac{\\nabla^3 P(Y)}{P(Y)} - 3 \\frac{\\nabla^2 P(Y)}{P(Y)}\\frac{\\nabla P(Y)}{P(Y)} + 2\\left(\\frac{\\nabla P(Y)}{P(Y)}\\right)^3 \\\\\n& = \\mathbb{E}\\left[ \\frac{\\nabla^3 P(Y,X)}{P(Y,X)} \\middle| Y\\right] -3 \\mathbb{E}\\left[ \\frac{\\nabla^2 P(Y,X)}{P(Y,X)} \\middle| Y\\right] \\mathbb{E}\\left[ \\frac{\\nabla P(Y,X)}{P(Y,X)} \\middle| Y\\right] + 2 \\left\\{ \\mathbb{E}\\left[ \\frac{\\nabla P(Y,X)}{P(Y,X)} \\middle| Y\\right]\\right\\}^3\\\\\n& = \\mathbb{E}\\left[ \\nabla^3 \\ell(Y,X) + 3 \\nabla^2 \\ell(Y,X) \\nabla \\ell(Y,X) + (\\nabla \\ell(Y,X))^3 \\middle| Y\\right]\\\\\n&\\quad -3 \\mathbb{E}\\left[ \\nabla^2 \\ell(Y,X) + (\\nabla \\ell(Y,X))^2 \\middle| Y\\right] \\mathbb{E}\\left[ \\nabla \\ell(Y,X) \\middle| Y\\right]+ 2 \\left\\{ \\mathbb{E}\\left[ \\nabla \\ell(Y,X) \\middle| Y\\right]\\right\\}^3,\n\\end{align*}\nand collecting terms. $\\nabla^4 \\ell(Y)$, $\\nabla^5 \\ell(Y)$, and $\\nabla^6 \\ell(Y)$ are derived similarly.\n\\end{proof}\n\n\\subsubsection{Auxiliary lemmas}\n\nWe first collect the notations. Define $\\overline{\\bf Z}_{k-1}^k:=(X_{k-1},\\overline{\\bf Y}_{k-1},W_k,X_k,Y_k)$ and denote the derivative of the complete data log-density by\n\\begin{equation} \\label{phi_i}\n\\phi^i(\\vartheta,\\overline{\\bf Z}_{k-1}^k) :=\\nabla^i \\log p_{\\vartheta}(Y_k,X_k|\\overline {\\bf Y}_{k-1},X_{k-1},W_k), \\quad i\\geq 1.\n\\end{equation}\nWe use short-handed notation $\\phi^i_{\\vartheta k}:=\\phi^i(\\vartheta,\\overline{\\bf Z}_{k-1}^k)$. We also suppress the superscript $1$ from $\\phi^1_{\\vartheta k}$, so that $\\phi_{\\vartheta k}=\\phi^1_{\\vartheta k}$. \nFor random variables $V_1,\\ldots,V_q$ and a conditioning set $\\mathcal{F}$, define the central conditional moment of $(V_1,\\ldots,V_q)$ as\n\\begin{align*}\n\\mathbb{E}_\\vartheta^c \\left[V_1,\\ldots,V_q \\middle|\\mathcal{F} \\right] & := \\mathbb{E}_{\\vartheta} \\left[ \\left(V_1-\\mathbb{E}_{\\vartheta}[V_1|\\mathcal{F}]\\right) \\cdots \\left(V_q-\\mathbb{E}_{\\vartheta}[V_q|\\mathcal{F}]\\right) \\middle| \\mathcal{F}\\right].\n\\end{align*}\nFor example, $\\mathbb{E}_{\\vartheta}^c \\left[\\phi_{\\vartheta k_1}\\phi_{\\vartheta k_2} \\middle|\\mathcal{F} \\right] := \\mathbb{E}_{\\vartheta} \\left[ \\left( \\phi_{\\vartheta k_1}-\\mathbb{E}_{\\vartheta}[\\phi_{\\vartheta k_1}|\\mathcal{F}]\\right) \\left( \\phi_{\\vartheta k_2}-\\mathbb{E}_{\\vartheta}[\\phi_{\\vartheta k_2}|\\mathcal{F}]\\right) \\middle| \\mathcal{F}\\right]$.\n \nLet $\\mathcal{I}(j) = (i_1,\\ldots,i_j)$ denote a sequence of positive integers with $j$ elements, let $\\sigma(\\mathcal{I}(j))$ denote the set of all the unique permutations of $(i_1,\\ldots,i_j)$, and let $|\\sigma(\\mathcal{I}(j))|$ denote its cardinality. For example, if $\\mathcal{I}(3) = (2,1,1)$, then $\\sigma(\\mathcal{I}(3)) = \\{(2,1,1), (1,2,1), (1,1,2)\\}$ and $|\\sigma(\\mathcal{I}(3))|=3$; if $\\mathcal{I}(3) = (1,1,1)$, then $\\sigma(\\mathcal{I}(3)) = (1,1,1)$ and $|\\mathcal{I}(3)|=1$. Let $\\mathcal{T}(j) = (t_1,\\ldots,t_j)$ for $j=1,\\ldots,6$. For a conditioning set $\\mathcal{F}$, define the symmetrized central conditional moments as \n\\begin{equation} \\label{Phi_defn}\n\\begin{aligned}\n\\Phi^{\\mathcal{I}(1)}_{\\vartheta \\mathcal{T}(1)}[\\mathcal{F}]& :=\n\\mathbb{E}_{\\vartheta}\\left[ \\phi^{i_1}_{\\vartheta t_1} \\middle| \\mathcal{F}\\right] ,\n\\quad\n\\Phi^{\\mathcal{I}(2)}_{\\vartheta \\mathcal{T}(2)}[\\mathcal{F}] :=\n\\frac{1}{|\\sigma(\\mathcal{I}(2))|} \\sum_{(\\ell_1,\\ell_2) \\in \\sigma(\\mathcal{I}(2))} \\mathbb{E}_{\\vartheta}^c\\left[ \\phi^{\\ell_1}_{\\vartheta t_1} \\phi^{\\ell_2}_{\\vartheta t_2} \\middle| \\mathcal{F}\\right],\n\\\\\n\\Phi^{\\mathcal{I}(3)}_{\\vartheta \\mathcal{T}(3)}[\\mathcal{F}] & := \n\\frac{1}{|\\sigma(\\mathcal{I}(3))|} \\sum_{(\\ell_1,\\ell_2,\\ell_3) \\in \\sigma(\\mathcal{I}(3))} \\mathbb{E}_{\\vartheta}^c\\left[ \\phi^{\\ell_1}_{\\vartheta t_1} \\phi^{\\ell_2}_{\\vartheta t_2} \\phi^{\\ell_3}_{\\vartheta t_3} \\middle| \\mathcal{F}\\right], \\\\\n\\Phi^{\\mathcal{I}(4)}_{\\vartheta \\mathcal{T}(4)}[\\mathcal{F}] & := \\frac{1}{|\\sigma(\\mathcal{I}(4))|} \\sum_{(\\ell_1,\\ldots,\\ell_4) \\in \\sigma(\\mathcal{I}(4))} \\tilde \\Phi_{\\vartheta \\mathcal{T}(4)}^{\\ell_1 \\ell_2 \\ell_3 \\ell_4}, \n\\end{aligned}\n\\end{equation}\nwhere $\\tilde \\Phi_{\\vartheta \\mathcal{T}(4)}^{\\ell_1 \\ell_2 \\ell_3 \\ell_4} :=\\mathbb{E}_{\\vartheta}^c[ \\phi^{\\ell_1}_{\\vartheta t_1} \\phi^{\\ell_2}_{\\vartheta t_2} \\phi^{\\ell_3}_{\\vartheta t_3} \\phi^{\\ell_4}_{\\vartheta t_4} | \\mathcal{F}] - \\mathbb{E}_{\\vartheta}^c[\\phi^{\\ell_1}_{\\vartheta t_1} \\phi^{\\ell_2}_{\\vartheta t_2}|\\mathcal{F}] \\mathbb{E}_{\\vartheta}^c[\\phi^{\\ell_3}_{\\vartheta t_3} \\phi^{\\ell_4}_{\\vartheta t_4}|\\mathcal{F}] \\\\- \\mathbb{E}_{\\vartheta}^c[\\phi^{\\ell_1}_{\\vartheta t_1} \\phi^{\\ell_3}_{\\vartheta t_3}|\\mathcal{F}] \\mathbb{E}_{\\vartheta}^c[\\phi^{\\ell_2}_{\\vartheta t_2} \\phi^{\\ell_4}_{\\vartheta t_4}|\\mathcal{F}] - \\mathbb{E}_{\\vartheta}^c[\\phi^{\\ell_1}_{\\vartheta t_1} \\phi^{\\ell_4}_{\\vartheta t_4}|\\mathcal{F}]\\mathbb{E}_{\\vartheta}^c[\\phi^{\\ell_2}_{\\vartheta t_2} \\phi^{\\ell_3}_{\\vartheta t_3}|\\mathcal{F}] $, and\n\\begin{equation} \\label{Phi_defn_2}\n\\begin{aligned} \n\\Phi^{\\mathcal{I}(5)}_{\\vartheta \\mathcal{T}(5)}[\\mathcal{F}] & := \\frac{1}{|\\sigma(\\mathcal{I}(5))|} \\sum_{(\\ell_1,\\ldots,\\ell_5) \\in \\sigma(\\mathcal{I}(5))} \\left( \\mathbb{E}_\\vartheta^c\\left[ \\phi^{\\ell_1}_{\\theta t_1} \\phi^{\\ell_2}_{\\theta t_2} \\phi^{\\ell_3}_{\\theta t_3} \\phi^{\\ell_4}_{\\theta t_4} \\phi^{\\ell_5}_{\\theta t_5} \\middle| \\mathcal{F}\\right] \\right. \\\\\n& \\left. \\qquad - \\sum_{(\\{a,b,c\\},\\{d,e\\}) \\in \\sigma_5} \\mathbb{E}_\\vartheta^c\\left[\\phi^{\\ell_{a}}_{\\theta t_{a}} \\phi^{\\ell_b}_{\\theta t_b}\\phi^{\\ell_c}_{\\theta t_c} \\middle|\\mathcal{F}\\right] \\mathbb{E}_\\vartheta^c\\left[ \\phi^{\\ell_d}_{\\theta t_d} \\phi^{\\ell_e}_{\\theta t_e} \\middle| \\mathcal{F}\\right] \\right), \\\\\n\\Phi^{\\mathcal{I}(6)}_{\\vartheta \\mathcal{T}(6)}[\\mathcal{F}]\n &:= \\mathbb{E}_\\vartheta^c\\left[ \\phi_{\\theta t_1} \\phi_{\\theta t_2} \\phi_{\\theta t_3} \\phi_{\\theta t_4} \\phi_{\\theta t_5} \\phi_{\\theta t_6} \\middle| \\mathcal{F}\\right] - \\sum_{(\\{a,b,c,d\\},\\{e,f\\}) \\in \\sigma_{61}} \\mathbb{E}_\\vartheta^c\\left[ \\phi_{\\theta t_a} \\phi_{\\theta t_b} \\phi_{\\theta t_c} \\phi_{\\theta t_d} \\middle| \\mathcal{F}\\right] \\mathbb{E}_\\vartheta^c\\left[ \\phi_{\\theta t_e} \\phi_{\\theta t_f} \\middle| \\mathcal{F}\\right] \\\\\n&\\qquad - \\sum_{(\\{a,b,c\\},\\{d,e,f\\}) \\in \\sigma_{62}} \\mathbb{E}_\\vartheta^c\\left[ \\phi_{\\theta t_a} \\phi_{\\theta t_b} \\phi_{\\theta t_c} \\middle| \\mathcal{F}\\right] \\mathbb{E}_\\vartheta^c\\left[ \\phi_{\\theta t_d} \\phi_{\\theta t_e} \\phi_{\\theta t_f} \\middle| \\mathcal{F}\\right] \\\\\n&\\qquad + 2 \\sum_{(\\{a,b\\},\\{c,d\\},\\{e,f\\}) \\in \\sigma_{63}} \\mathbb{E}_\\vartheta^c\\left[ \\phi_{\\theta t_a} \\phi_{\\theta t_b} \\middle| \\mathcal{F}\\right] \\mathbb{E}_\\vartheta^c\\left[ \\phi_{\\theta t_c} \\phi_{\\theta t_d} \\middle| \\mathcal{F}\\right] \\mathbb{E}_\\vartheta^c\\left[ \\phi_{\\theta t_e} \\phi_{\\theta t_f} \\middle| \\mathcal{F}\\right],\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{partitions}\n\\begin{aligned}\n\\sigma_5 &:= \\text{the set of } {\\textstyle{5 \\choose 3} =10} \\text{ partitions of } \\{1,2,3,4,5\\} \\text{ of the form } \\{a,b,c\\},\\{d,e\\},\\\\\n\\sigma_{61} &:= \\text{the set of } {\\textstyle{6 \\choose 4} = 15} \\text{ partitions of } \\{1,2,3,4,5,6\\} \\text{ of the form } \\{a,b,c,d\\},\\{e,f\\}, \\\\\n\\sigma_{62} &:= \\text{the set of } {\\textstyle{6 \\choose 3}\/2 = 10} \\text{ partitions of } \\{1,2,3,4,5,6\\} \\text{ of the form } \\{a,b,c\\},\\{d,e,f\\}, \\\\\n\\sigma_{63} &:= \\text{the set of } {\\textstyle{6 \\choose 2}\\textstyle{4 \\choose 2}\/6 = 15} \\text{ partitions of } \\{1,2,3,4,5,6\\} \\text{ of the form } \\{a,b\\},\\{c,d\\},\\{e,f\\}.\n\\end{aligned}\n\\end{equation} \nNote that these moments are symmetric with respect to $(t_1,\\ldots,t_j)$. For $j=1,2,\\ldots,6$, $k \\geq 1$, $m \\geq 0$, and $x \\in \\mathcal{X}$, define the difference between the sums of the $\\Phi^{\\mathcal{I}(j)}_{\\vartheta \\mathcal{T}(j)}$'s over different time indices and conditioning sets as \n\\begin{equation}\\label{delta-tau}\n\\begin{aligned}\n\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\vartheta) &:= \\sum_{\\mathcal{T}(j)\\in \\{-m+1,\\ldots,k\\}^j} {\\Phi}^{\\mathcal{I}(j)}_{\\vartheta \\mathcal{T}(j)}\\left[\\overline{\\bf Y}_{-m}^k,{\\bf W}_{-m}^{k},X_{-m}=x\\right] \\\\\n& \\quad - \\sum_{\\mathcal{T}(j)\\in \\{-m+1,\\ldots,k-1\\}^j} {\\Phi}^{\\mathcal{I}(j)}_{\\vartheta \\mathcal{T}(j)}\\left[\\overline{\\bf Y}_{-m}^{k-1},{\\bf W}_{-m}^{k-1},X_{-m}=x\\right], \n\\end{aligned}\n\\end{equation} \nwhere $\\sum_{\\mathcal{T}(j)\\in \\{-m+1,\\ldots,k\\}^j}$ denotes $\\sum_{t_1=-m+1}^k\\sum_{t_2=-m+1}^k \\cdots \\sum_{t_j=-m+1}^k$, and $\\sum_{\\mathcal{T}(j) \\in \\{-m+1,\\ldots,k-1\\}^j}$ is defined similarly. Define $\\overline{\\Delta}^{\\mathcal{I}(j)}_{j,k,m}(\\theta)$ analogously to $\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\vartheta)$ by dropping $X_{-m}=x$ from the conditioning variable.\n\nHenceforth, we suppress the conditioning variable ${\\bf W}_{-m}^n$ from the conditioning sets and conditional densities unless confusion might arise. The following lemma expresses the derivatives of the log-densities, $ \\nabla^j \\ell_{k,m,x}(\\vartheta)$'s, in terms of the $\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\vartheta)$'s. The first two equations are also given in DMR (p. 2272 and pp. 2276--7).\n\\begin{lemma} \\label{ell_lambda}\nFor all $1 \\leq k \\leq n$, $m \\geq 0$, and $x \\in \\mathcal{X}$,\n\\begin{align*}\n&\\nabla^1 \\ell_{k,m,x}(\\vartheta)=\\Delta^1_{1,k,m,x}(\\vartheta),\\quad \n\\nabla^2 \\ell_{k,m,x}(\\vartheta)=\\Delta^2_{1,k,m,x}(\\vartheta)+\\Delta^{1,1}_{2,k,m,x}(\\vartheta),\\\\ \n&\\nabla^3 \\ell_{k,m,x}(\\vartheta)= \\Delta^{3}_{1,k,m,x}(\\vartheta)+3\\Delta^{2,1}_{2,k,m,x}(\\vartheta)+\\Delta^{1,1,1}_{3,k,m,x}(\\vartheta),\\\\\n&\\nabla^4 \\ell_{k,m,x}(\\vartheta)= \\Delta^4_{1,k,m,x}(\\vartheta) + 4 \\Delta^{3,1}_{2,k,m,x}(\\vartheta) + 3 \\Delta^{2,2}_{2,k,m,x}(\\vartheta) + 6\\Delta^{2,1,1}_{3,k,m,x}(\\vartheta) + \\Delta^{1,1,1,1}_{4,k,m,x}(\\vartheta),\\\\\n&\\nabla^5 \\ell_{k,m,x}(\\vartheta)= \\Delta^5_{1,k,m,x}(\\vartheta)+ 5\\Delta^{4,1}_{2,k,m,x}(\\vartheta) + 10 \\Delta^{3,2}_{2,k,m,x}(\\vartheta) +10\\Delta^{3,1,1}_{3,k,m,x}(\\vartheta)+15 \\Delta^{2,2,1}_{3,k,m,x}(\\vartheta)\\\\\n& \\quad +10\\Delta^{2,1,1,1}_{4,k,m,x}(\\vartheta) +\\Delta^{1,1,1,1,1}_{5,k,m,x}(\\vartheta), \\\\\n&\\nabla^6 \\ell_{k,m,x}(\\vartheta)= \\Delta^6_{1,k,m,x}(\\vartheta) + 6\\Delta^{5,1}_{2,k,m,x}(\\vartheta) + 15 \\Delta^{4,2}_{2,k,m,x}(\\vartheta) + 10 \\Delta^{3,3}_{2,k,m,x}(\\vartheta)+ 15\\Delta^{4,1,1}_{3,k,m,x}(\\vartheta) \\\\\n& \\quad + 60\\Delta^{3,2,1}_{3,k,m,x}(\\vartheta) + 15\\Delta^{2,2,2}_{3,k,m,x}(\\vartheta) + 20 \\Delta^{3,1,1,1}_{4,k,m,x}(\\vartheta) + 45 \\Delta^{2,2,1,1}_{4,k,m,x}(\\vartheta) + 15 \\Delta^{2,1,1,1,1}_{5,k,m,x}(\\vartheta) +\\Delta^{1,1,1,1,1}_{6,k,m,x}(\\vartheta). \n\\end{align*} \nFurther, the above holds when $\\nabla^j \\ell_{k,m,x}(\\vartheta)$ and $\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\vartheta)$ are replaced with $\\nabla^j \\overline \\ell_{k,m}(\\vartheta)$ and $\\overline\\Delta^{\\mathcal{I}(j)}_{j,k,m}(\\vartheta)$.\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma \\ref{ell_lambda}]\nThe stated result follows from writing $\\nabla^j \\ell_{k,m,x}(\\vartheta) = \\\\ \\nabla^j \\log p_{\\vartheta}({\\bf Y}_{-m+1}^{k}|\\overline{\\bf{Y}}_{-m},X_{-m}=x) - \\nabla^j \\log p_{\\vartheta}({\\bf{Y}}_{-m+1}^{k-1}|\\overline{\\bf{Y}}_{-m},X_{-m}=x)$, applying Lemma \\ref{louis} to the right-hand side, and noting that $\\nabla^j \\log p_{\\vartheta}({\\bf Y}_{-m+1}^k,{\\bf X}_{-m+1}^k|\\overline{\\bf{Y}}_{-m},X_{-m}) = \\sum_{t=-m+1}^k \\phi^j(\\vartheta,\\overline{\\bf Z}_{t-1}^t)$ (see (\\ref{cond_density}) and (\\ref{phi_i})). The result for $\\nabla^j \\ell_{k,m,x}(\\vartheta)$ with $j=1,2$ is also given in DMR (p. 2272 and pp. 2276--7). For $j=3$, the term $\\Delta^{2,1}_{2,k,m,x}(\\vartheta)$ follows from $\\sum_{t_1=-m+1}^k \\sum_{t_2=-m+1}^k \\mathbb{E}_\\vartheta^c[ \\phi^{2}_{\\vartheta t_1} \\phi^{1}_{\\vartheta t_2} |\\overline{\\bf Y}_{-m}^k,X_{-m}=x] = \\sum_{t_1=-m+1}^k \\sum_{t_2=-m+1}^k\\Phi^{2,1}_{\\vartheta t_1t_2}[\\overline{\\bf Y}_{-m}^k,X_{-m}=x]$. For $j=4$, note that when we apply Lemma \\ref{louis} to $\\nabla^4 \\log p_{\\vartheta}({\\bf Y}_{-m+1}^{k}|\\overline{\\bf{Y}}_{-m},X_{-m}=x)$, the last two terms on the right-hand side of Lemma \\ref{louis} can be written as $\\sum_{\\mathcal{T}(4)\\in\\{-m+1,\\ldots,k\\}^4 } {\\Phi}^{1,1,1,1}_{\\vartheta \\mathcal{T}(4)}[\\overline{\\bf Y}_{-m}^k,X_{-m}=x]$. The result for $j=5$ follows from a similar argument. For $j=6$, note that when we apply Lemma \\ref{louis} to $\\nabla^6 \\log p_{\\vartheta}({\\bf Y}_{-m+1}^{k}|\\overline{\\bf{Y}}_{-m},X_{-m}=x)$, the last four terms on the right-hand side of Lemma \\ref{louis} can be written as $\\sum_{\\mathcal{T}(6)\\in\\{-m+1,\\ldots,k\\}^6} {\\Phi}^{\\mathcal{I}(6)}_{\\vartheta \\mathcal{T}(6)}[\\overline{\\bf Y}_{-m}^k,X_{-m}=x]$.\n\\end{proof}\n\nThe following lemma provides bounds on $\\Phi^{\\mathcal{I}(j)}_{\\vartheta \\mathcal{T}(j)}[\\mathcal{F}]$ defined in (\\ref{Phi_defn}) and (\\ref{Phi_defn_2}) and is used in the proof of Lemma \\ref{lemma-bound-1}. For $j =2,\\ldots,6$, define $\\|\\phi^i_t\\|_{\\infty}:=\\sup_{\\vartheta\\in \\mathcal{N}^*} \\sup_{x,x'}|\\phi^i(\\vartheta,Y_t,x,\\overline{\\bf{Y}}_{t-1},x')|$ and $\\|\\phi^{\\mathcal{I}(j)}_{\\mathcal{T}(j)}\\|_{\\infty}:=\\sum_{(\\ell_1,\\ldots,\\ell_j)\\in\\sigma(\\mathcal{I}(j)) }\\|\\phi_{t_1}^{\\ell_1}\\|_{\\infty} \\cdots \\|\\phi_{t_j}^{\\ell_j}\\|_{\\infty}$.\n\\begin{lemma}\\label{lemma_ijl}\nUnder Assumptions \\ref{assn_a1}, \\ref{assn_a2}, and \\ref{assn_a4}, there exists a finite non-stochastic constant $C$ that does not depend on $\\rho$ such that, for all $m'\\geq m\\geq 0$, all $-m<t_1\\leq t_2 \\leq \\cdots \\leq t_j\\leq n$, all $\\vartheta\\in \\mathcal{N}^*$ and all $x\\in \\mathcal{X}$, and $j=2,\\ldots,6$,\n\\begin{equation*\n\\begin{aligned} \n(a) & \\quad |\\Phi^{\\mathcal{I}(j)}_{\\vartheta \\mathcal{T}(j)}[\\overline{\\bf{Y}}_{-m}^n]| \\leq C \\rho^{(t_2-t_1-1)_+\\vee(t_3-t_2-1)_+\\vee \\cdots \\vee(t_j-t_{j-1}-1)_+} \\|\\phi^{\\mathcal{I}(j)}_{\\mathcal{T}(j)}\\|_{\\infty},\\\\\n(b) & \\quad |\\Phi^{\\mathcal{I}(j)}_{\\vartheta \\mathcal{T}(j)}[\\overline{\\bf{Y}}_{-m}^n,X_{-m}=x]|\\leq C \\rho^{(t_2-t_1-1)_+\\vee(t_3-t_2-1)_+\\vee \\cdots \\vee(t_j-t_{j-1}-1)_+}\\|\\phi^{\\mathcal{I}(j)}_{\\mathcal{T}(j)}\\|_{\\infty},\\\\\n(c) & \\quad |\\Phi^{\\mathcal{I}(j)}_{\\vartheta \\mathcal{T}(j)}[\\overline{\\bf{Y}}_{-m}^n,X_{-m}=x]\n- \\Phi^{\\mathcal{I}(j)}_{\\vartheta \\mathcal{T}(j)}[\\overline{\\bf{Y}}_{-m}^n]|\\leq C\\rho^{(m+t_1-1)_+}\\|\\phi^{\\mathcal{I}(j)}_{\\mathcal{T}(j)}\\|_{\\infty},\\\\\n(d) & \\quad |\\Phi^{\\mathcal{I}(j)}_{\\vartheta \\mathcal{T}(j)}[\\overline{\\bf{Y}}_{-m}^n,X_{-m}=x]\n- \\Phi^{\\mathcal{I}(j)}_{\\vartheta \\mathcal{T}(j)}[\\overline{\\bf{Y}}_{-m'}^n,X_{-m'}=x]|\\leq C\\rho^{(m+t_1-1)_+}\\|\\phi^{\\mathcal{I}(j)}_{\\mathcal{T}(j)}\\|_{\\infty},\\\\\n(e) & \\quad |\\Phi^{\\mathcal{I}(j)}_{\\vartheta \\mathcal{T}(j)}[\\overline{\\bf{Y}}_{-m}^n] - \\Phi^{\\mathcal{I}(j)}_{\\vartheta \\mathcal{T}(j)}[\\overline{\\bf{Y}}_{-m}^{n -1}]| \\leq C\\rho^{(n -1 -t_j)_+}\\|\\phi^{\\mathcal{I}(j)}_{\\mathcal{T}(j)}\\|_{\\infty},\\\\\n(f) & \\quad |\\Phi^{\\mathcal{I}(j)}_{\\vartheta \\mathcal{T}(j)}[\\overline{\\bf{Y}}_{-m}^n,X_{-m}=x] - \\Phi^{\\mathcal{I}(j)}_{\\vartheta \\mathcal{T}(j)}[\\overline{\\bf{Y}}_{-m}^{n -1},X_{-m}=x]| \\leq C\\rho^{(n -1 -t_j)_+}\\|\\phi^{\\mathcal{I}(j)}_{\\mathcal{T}(j)}\\|_{\\infty}.\n\\end{aligned}\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma \\ref{lemma_ijl}]\n\nRecall $\\sup_{\\vartheta \\in \\mathcal{N}^*}\\sup_{x,x'}|\\phi^i(\\vartheta,Y_t,x,\\overline{\\bf{Y}}_{t-1},x')- \\mathbb{E}_\\vartheta[\\phi^i(\\vartheta,Y_t,x,\\overline{\\bf{Y}}_{t-1},x')|\\mathcal{F}]| \\\\\\leq 2 \\sup_{\\vartheta \\in \\mathcal{N}^*}\\sup_{x,x'}|\\phi^i(\\vartheta,Y_t,x,\\overline{\\bf{Y}}_{t-1},x')|$ for the conditioning sets $\\mathcal{F}$ that appear in the lemma. Define $\\tilde \\phi^i_{\\vartheta t}:=\\phi^i(\\vartheta,\\overline{\\bf Z}_{t-1}^t)- \\mathbb{E}_\\vartheta[\\phi^i(\\vartheta,\\overline{\\bf Z}_{t-1}^t)| \\overline{\\bf Y}_{-m}^{n}]$, so that $\\mathbb{E}_\\vartheta^c[\\phi^{\\ell_1}_{\\vartheta t_1} \\cdots \\phi^{\\ell_j}_{\\vartheta t_j}|\\overline{\\bf{Y}}_{-m}^n] = \\mathbb{E}_\\vartheta[ \\tilde \\phi^{\\ell_1}_{\\vartheta t_1} \\cdots \\tilde \\phi^{\\ell_j}_{\\vartheta t_j}| \\overline{\\bf Y}_{-m}^n]$. Henceforth, we suppress the subscript $\\vartheta$ from $\\phi^{i}_{\\vartheta t}$ and $\\tilde \\phi^{i}_{\\vartheta t}$.\n \nRecall that $\\phi^i(\\vartheta,\\overline{\\bf Z}_{t-1}^t)$ depends on $X_t$ and $X_{t-1}$. Parts (c) and (d) follow from Lemma \\ref{x_diff}(a) and the fact that, for any two probability measures $\\mu_1$ and $\\mu_2$, $\\sup_{f(x):\\max_x|f(x)| \\leq 1}|\\int f(x) d\\mu_1(x)- \\int f(x) d\\mu_2(x)| = 2 \\|\\mu_1-\\mu_2\\|_{TV}$ (see, e.g., \\citet[][Proposition 4.5]{levin09book}). Similarly, parts (e) and (f) for $t_j \\leq n-1$ follow from Lemma \\ref{x_diff}(b), and parts (e) and (f) for $t_j=n$ follow from $|\\Phi^{\\mathcal{I}(j)}_{\\vartheta \\mathcal{T}(j)}[\\cdot]| \\leq 2^j \\|\\phi^{\\mathcal{I}(j)}_{\\mathcal{T}(j)}\\|_{\\infty}$.\n \nWe proceed to show parts (a) and (b). The results for $j=2$ and $j=3$ follow from Lemma \\ref{x_diff}(c) and \n\\begin{equation} \\label{prod_moments}\n\\begin{aligned}\nE(X_{t_1} - EX_{t_1})\\cdots(X_{t_j} - EX_{t_j}) & = \\text{cov}[X_{t_1},(X_{t_2} - EX_{t_2})\\cdots(X_{t_j} - EX_{t_j})]\\\\\n& = \\text{cov}[(X_{t_1} - EX_{t_1})\\cdots(X_{t_{j-1}} - EX_{t_{j-1}}),X_{t_j}].\n\\end{aligned}\n\\end{equation}\nBefore proving the results for $j\\geq 4$, we collect some results. For a conditioning set $\\mathcal{F}= \\overline{\\bf{Y}}_{-m}^n$ or $\\{\\overline{\\bf{Y}}_{-m}^n,X_m=x \\}$, Lemmas \\ref{x_diff}(c) and (\\ref{prod_moments}) imply that\n\\begin{align}\n&\\qquad |\\mathbb{E}_\\vartheta^c[\\phi^{\\ell_1}_{ t_1} \\cdots \\phi^{\\ell_j}_{ t_j}|\\mathcal{F}]| \\leq \\mathcal{C} \\rho^{(t_2-t_1-1)_+\\vee(t_j-t_{j-1}-1)_+} \\|\\phi^{\\mathcal{I}(j)}_{\\mathcal{T}(j)}\\|_{\\infty}, \\label{cov1} \\\\\n& |\\mathbb{E}_\\vartheta^c[\\phi^{\\ell_1}_{ t_1} \\cdots \\phi^{\\ell_j}_{ t_j}|\\mathcal{F}] - \\mathbb{E}_\\vartheta^c[\\phi^{\\ell_1}_{ t_1}\\cdots\\phi^{\\ell_k}_{t_k}|\\mathcal{F}]\\mathbb{E}_\\vartheta^c[\\phi^{\\ell_{k+1}}_{ t_{k+1}}\\cdots \\phi^{\\ell_j}_{ t_j}|\\mathcal{F}]| \\nonumber\\\\\n&\\quad = | \\text{cov}_\\vartheta[\\tilde \\phi^{\\ell_1}_{ t_1}\\cdots \\tilde \\phi^{\\ell_k}_{t_k}, \\tilde \\phi^{\\ell_{k+1}}_{ t_{k+1}}\\cdots \\tilde \\phi^{\\ell_j}_{ t_j} |\\mathcal{F}] | \\leq \\mathcal{C} \\rho^{(t_{k+1}-t_{k}-1)_+}\\|\\phi^{\\mathcal{I}(j)}_{\\mathcal{T}(j)}\\|_{\\infty}\\quad \\text{for any $2 \\leq k \\leq j-2$}. \\label{cov_k} \n\\end{align}\n\nParts (a) and (b) hold for $j=4$ because $\\Phi^{\\mathcal{I}(4)}_{\\vartheta \\mathcal{T}(4)}[\\mathcal{F}]\\leq \\mathcal{C} \\rho^{(t_{2}-t_{1}-1)_+\\vee (t_4 - t_3-1)_+}\\|\\phi^{\\mathcal{I}(4)}_{\\mathcal{T}(4)}\\|_{\\infty}$ from (\\ref{cov1}) and we have $\\Phi^{\\mathcal{I}(4)}_{\\vartheta \\mathcal{T}(4)}[\\mathcal{F}] \\leq \\mathcal{C} \\rho^{(t_{3}-t_{2}-1)_+}\\|\\phi^{\\mathcal{I}(4)}_{\\mathcal{T}(4)}\\|_{\\infty}$ from writing $\\tilde \\Phi_{\\vartheta \\mathcal{T}(4)}^{\\ell_1 \\ell_2 \\ell_3 \\ell_4}$ defined in (\\ref{Phi_defn}) as $\\tilde \\Phi_{\\vartheta \\mathcal{T}(4)}^{\\ell_1 \\ell_2 \\ell_3 \\ell_4} = \\text{cov}_{\\vartheta}[ \\tilde\\phi^{\\ell_1}_{ t_1} \\tilde\\phi^{\\ell_2}_{ t_2}, \\tilde \\phi^{\\ell_3}_{ t_3} \\tilde \\phi^{\\ell_4}_{ t_4} | \\mathcal{F}] - \\mathbb{E}_{\\vartheta}^c[\\phi^{\\ell_1}_{ t_1} \\phi^{\\ell_3}_{ t_3}|\\mathcal{F}] \\mathbb{E}_{\\vartheta}^c[\\phi^{\\ell_2}_{ t_2} \\phi^{\\ell_4}_{ t_4}|\\mathcal{F}] - \\mathbb{E}_{\\vartheta}^c[\\phi^{\\ell_1}_{ t_1} \\phi^{\\ell_4}_{ t_4}|\\mathcal{F}]\\mathbb{E}_{\\vartheta}^c[\\phi^{\\ell_2}_{ t_2} \\phi^{\\ell_3}_{ t_3}|\\mathcal{F}]$ and applying (\\ref{cov_k}). Parts (a)--(b) for $j=5$ follow from a similar argument.\n\nFor $j=6$, first, $\\Phi^{\\mathcal{I}(6)}_{\\vartheta \\mathcal{T}(6)}[\\mathcal{F}]$ is bounded by $\\mathcal{C} \\rho^{(t_2-t_1-1)_+\\vee(t_6-t_{5}-1)_+}\\|\\phi^{\\mathcal{I}(6)}_{\\mathcal{T}(6)}\\|_{\\infty}$ from (\\ref{cov1}). Second, write $\\Phi^{\\mathcal{I}(6)}_{\\vartheta \\mathcal{T}(6)}[\\mathcal{F}] = A_1 + A_2$, where $A_1 = \\mathbb{E}_\\vartheta^c\\left[ \\phi_{ t_1} \\phi_{ t_2} \\phi_{ t_3} \\phi_{ t_4} \\phi_{ t_5} \\phi_{ t_6} \\middle| \\mathcal{F}\\right] - \\mathbb{E}_\\vartheta^c\\left[ \\phi_{ t_1} \\phi_{ t_2} \\phi_{ t_3}\\middle| \\mathcal{F}\\right] \\mathbb{E}_\\vartheta^c\\left[ \\phi_{ t_4} \\phi_{ t_5} \\phi_{ t_6} \\middle| \\mathcal{F}\\right]$ and $A_2$ denotes all the terms on the right-hand side of $\\Phi^{\\mathcal{I}(6)}_{\\vartheta \\mathcal{T}(6)}[\\mathcal{F}]$ in (\\ref{Phi_defn_2}) except for $A_1$. $A_1$ is bounded by $\\mathcal{C}\\rho^{(t_4-t_3-1)_+}\\|\\phi^{\\mathcal{I}(6)}_{\\mathcal{T}(6)}\\|_{\\infty}$ from (\\ref{cov_k}), and $A_2$ is bounded by $\\mathcal{C}\\rho^{(t_4-t_3-1)_+}\\|\\phi^{\\mathcal{I}(6)}_{\\mathcal{T}(6)}\\|_{\\infty}$ from (\\ref{cov1}). Therefore, $\\Phi^{\\mathcal{I}(6)}_{\\vartheta \\mathcal{T}(6)}[\\mathcal{F}]$ is bounded by $\\mathcal{C} \\rho^{(t_4-t_3-1)_+}\\|\\phi^{\\mathcal{I}(6)}_{\\mathcal{T}(6)}\\|_{\\infty}$. Third, write $\\Phi^{\\mathcal{I}(6)}_{\\vartheta \\mathcal{T}(6)}[\\mathcal{F}] = B_1 + B_2+B_3$, where $B_1 = \\mathbb{E}_\\vartheta^c [ \\phi_{ t_1} \\phi_{ t_2} \\phi_{ t_3} \\phi_{ t_4} \\phi_{ t_5} \\phi_{ t_6} | \\mathcal{F}] - \\mathbb{E}_\\vartheta^c[ \\phi_{ t_3} \\phi_{ t_4} \\phi_{ t_5}\\phi_{ t_6}| \\mathcal{F}] \\mathbb{E}_\\vartheta^c[ \\phi_{ t_1} \\phi_{ t_2} | \\mathcal{F}]$, $B_2 = - \\sum_{(\\{1,2,c,d\\},\\{e,f\\}) \\in X_{61}} \\mathbb{E}_\\vartheta^c[ \\phi_{1} \\phi_{ 2} \\phi_{ t_c} \\phi_{ t_d} | \\mathcal{F}] \\mathbb{E}_\\vartheta^c[ \\phi_{ t_e} \\phi_{ t_f} | \\mathcal{F}] + 2 \\sum_{(\\{a,b\\},\\{c,d\\},\\{e,f\\}) \\in X_{63}} \\mathbb{E}_\\vartheta^c[ \\phi_{ t_a} \\phi_{ t_b} | \\mathcal{F}] \\mathbb{E}_\\vartheta^c[ \\phi_{ t_c} \\phi_{ t_d} | \\mathcal{F}] \\mathbb{E}_\\vartheta^c[ \\phi_{ t_e} \\phi_{ t_f} | \\mathcal{F}]$, where $X_{61}$ is the set of ${4 \\choose 2} = 6$ partitions of $\\{1,2,3,4,5,6\\}$ of the form of $\\{1,2,c,d\\},\\{e,f\\}$ and \\\\ $X_{63}:= \\{(\\{1,2\\},\\{3,4\\},\\{5,6\\}), (\\{1,2\\},\\{3,5\\},\\{4,6\\}), (\\{1,2\\},\\{3,6\\},\\{4,5\\}) \\}$, and $B_3$ denotes all the terms on the right-hand side of $\\Phi^{\\mathcal{I}(6)}_{\\vartheta \\mathcal{T}(6)}[\\mathcal{F}]$ except for $B_1+B_2$. $B_1$ is bounded by $\\mathcal{C} \\rho^{(t_3-t_2-1)_+}\\|\\phi^{\\mathcal{I}(6)}_{\\mathcal{T}(6)}\\|_{\\infty}$ from (\\ref{cov_k}). We can write $B_2$ as \\\\ $\\sum_{(\\{1,2,c,d\\},\\{e,f\\}) \\in X_{61}} \\{ - \\mathbb{E}_\\vartheta^c[ \\phi_{t_1} \\phi_{ t_2} \\phi_{ t_c} \\phi_{ t_d} | \\mathcal{F}] \\mathbb{E}_\\vartheta^c[ \\phi_{ t_e} \\phi_{ t_f} | \\mathcal{F}] + \\mathbb{E}_\\vartheta^c[ \\phi_{ t_1} \\phi_{ t_2} | \\mathcal{F}] \\mathbb{E}_\\vartheta^c[ \\phi_{ t_c} \\phi_{ t_d} | \\mathcal{F}] \\mathbb{E}_\\vartheta^c[ \\phi_{ t_e} \\phi_{ t_f} | \\mathcal{F}] \\} = - \\sum_{(\\{1,2,c,d\\},\\{e,f\\}) \\in X_{61}} \\mathbb{E}_\\vartheta^c[ \\phi_{ t_e} \\phi_{ t_f} | \\mathcal{F}] \\text{cov}_{\\vartheta}[\\tilde \\phi_{\\theta t_1} \\tilde \\phi_{\\theta t_2},\\tilde \\phi_{\\theta t_c} \\tilde \\phi_{\\theta t_d}|\\mathcal{F}]$, then this is bounded by \\\\ $\\mathcal{C}\\rho^{(t_3-t_2-1)_+}\\|\\phi^{\\mathcal{I}(6)}_{\\mathcal{T}(6)}\\|_{\\infty}$ from (\\ref{cov_k}). Finally, $B_3$ is bounded by $\\mathcal{C} \\rho^{(t_3-t_2-1)_+}\\|\\phi^{\\mathcal{I}(6)}_{\\mathcal{T}(6)}\\|_{\\infty}$ from (\\ref{cov1}). Therefore, $\\Phi^{\\mathcal{I}(6)}_{\\vartheta \\mathcal{T}(6)}[\\mathcal{F}]$ is bounded by $\\mathcal{C} \\rho^{(t_3-t_2-1)_+}\\|\\phi^{\\mathcal{I}(6)}_{\\mathcal{T}(6)}\\|_{\\infty}$. From a similar argument, $\\Phi^{\\mathcal{I}(6)}_{\\vartheta \\mathcal{T}(6)}[\\mathcal{F}]$ is also bounded by $\\mathcal{C}\\rho^{(t_5-t_4-1)_+}\\|\\phi^{\\mathcal{I}(6)}_{\\mathcal{T}(6)}\\|_{\\infty}$, and parts (a) and (b) follow. \n\\end{proof}\n\n\nWe next present the result that bound the difference between $\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\vartheta)$ and $\\overline{\\Delta}^{\\mathcal{I}(j)}_{j,k,m}(\\theta)$ that appear on the right-hand side in Lemma \\ref{ell_lambda}. This lemma extends Lemmas 13 and 17 of DMR. Let $r_{\\mathcal{I}(1)} = q_{i_1}$; $r_{\\mathcal{I}(2)} = q_{i_1}\/2$ if $i_1=i_2$ and $(q_{i_1} \\wedge q_{i_2})\/2$ if $i_1 \\neq i_2$; $r_{\\mathcal{I}(3)} = q_{i_1}\/3$ if $i_1=i_2=i_3$, $(q_{i_1}\/2 \\wedge q_{i_2}\/4)$ if $i_1 \\neq i_2=i_3$, $(q_{i_1}\\wedge q_{i_2}\\wedge q_{i_3})\/3$ if $i_1$, $i_2$, $i_3$ are distinct; $r_{\\mathcal{I}(4)} = q_{i_1}\/4$ if $i_1=i_2=i_3=i_4$, $(q_{i_1} \\wedge q_{i_3})\/4$ if $i_1 \\neq i_2=i_3=i_4$ or $i_1 = i_2 \\neq i_3=i_4$; $r_{\\mathcal{I}(5)} = q_{i_1}\/5$ if $i_1=i_2=i_3=i_4=i_5$; $(q_{i_1}\/3\\wedge q_{i_2}\/6)$ if $i_1 \\neq i_2=i_3=i_4=i_5$; $r_{\\mathcal{I}(6)} = q_1\/6$. Part (d) of this lemma establishes the uniform convergence of $\\{\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\vartheta)\\}_{m\\geq 0}$ to a random variable that does not depend on $x$.\n\\begin{lemma} \\label{lemma-bound-1} \nUnder Assumptions \\ref{assn_a1}, \\ref{assn_a2}, and \\ref{assn_a4}, for $j=1,\\ldots,6$, there exist random variables ${K}_{\\mathcal{I}(j)}, \\{M_{\\mathcal{I}(j), k}\\}_{k=1}^n \\in L^{r_{\\mathcal{I}(j)}}(\\mathbb{P}_{\\vartheta^*})$ such that, for all $1 \\leq k \\leq n$ and $m' \\geq m \\geq 0$, \n\\begin{align*} \n\\text{(a)}\\qquad & \\sup_{x\\in \\mathcal{X}} \\sup_{\\vartheta\\in \\mathcal{N}^*} |\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\vartheta)-\\overline\\Delta^{\\mathcal{I}(j)}_{j,k,m}(\\vartheta)|\\leq {K}_{\\mathcal{I}(j)}(k+m)^{ 7 } \\rho^{ \\lfloor (k+m-1)\/24 \\rfloor } \\quad \\text{$\\mathbb{P}_{\\vartheta^*}$-a.s.},\\\\\n\\text{(b)}\\qquad &\\sup_{x\\in \\mathcal{X}} \\sup_{\\vartheta\\in \\mathcal{N}^*} |\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\vartheta)-\\Delta^{\\mathcal{I}(j)}_{j,k,m',x}(\\vartheta)|\\leq {K}_{\\mathcal{I}(j)}(k+m)^{ 7 } \\rho^{ \\lfloor (k+m-1)\/ 1340 \\rfloor }\\quad \\text{$\\mathbb{P}_{\\vartheta^*}$-a.s.},\n\\end{align*}\n(c) $\\sup_{m\\geq 0} \\sup_{x\\in \\mathcal{X}} \\sup_{\\vartheta\\in \\mathcal{N}^*} |\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\vartheta)| + \\sup_{m\\geq 0} \\sup_{\\vartheta\\in \\mathcal{N}^*} |\\overline\\Delta^{\\mathcal{I}(j)}_{j,k,m}(\\vartheta)| \\leq M_{\\mathcal{I}(j), k}$ $\\mathbb{P}_{\\vartheta^*}$-a.s., (d) Uniformly in $\\vartheta\\in \\mathcal{N}^*$ and $x \\in \\mathcal{X}$, $\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\vartheta)$ and $\\overline\\Delta^{\\mathcal{I}(j)}_{j,k,m}(\\vartheta)$ converge $\\mathbb{P}_{\\vartheta^*}$-a.s.\\ and in $L^{r_{\\mathcal{I}(j)}}(\\mathbb{P}_{\\vartheta^*})$ to $\\Delta^{\\mathcal{I}(j)}_{j,k,\\infty}(\\vartheta) \\in L^{r_{\\mathcal{I}(j)}}(\\mathbb{P}_{\\vartheta^*})$ as $m\\rightarrow \\infty$. \n\\end{lemma} \n\n\\begin{proof}[Proof of Lemma \\ref{lemma-bound-1}] First, we prove parts (a) and (b). Recall $\\mathcal{T}(j) = (t_1,\\ldots,t_j)$. For part (a), define, suppressing the dependence of $A_{\\mathcal{T}(j)}$ on $\\vartheta$ and $\\mathcal{I}(j)$,\n\\begin{align*}\nA_{\\mathcal{T}(j)} & :=\n\\begin{cases}\n\\begin{aligned}\n&\\Phi^{\\mathcal{I}(j)}_{\\vartheta\\mathcal{T}(j)}\\left[\\overline{\\bf Y}_{-m}^k,X_{-m}=x\\right] -\\Phi^{\\mathcal{I}(j)}_{\\vartheta\\mathcal{T}(j)}\\left[ \\overline{\\bf Y}_{-m}^{k-1},X_{-m}=x\\right] \\\\\n& \\quad -\\Phi^{\\mathcal{I}(j)}_{\\vartheta\\mathcal{T}(j)}\\left[\\overline{\\bf Y}_{-m}^k\\right] +\\Phi^{\\mathcal{I}(j)}_{\\vartheta\\mathcal{T}(j)}\\left[\\overline{\\bf Y}_{-m}^{k-1}\\right],\n\\end{aligned}\n& \\text{ if } \\max\\{t_1,\\ldots, t_j\\} < k, \\\\\n\\Phi^{\\mathcal{I}(j)}_{\\vartheta\\mathcal{T}(j)}\\left[\\overline{\\bf Y}_{-m}^k,X_{-m}=x\\right] -\\Phi^{\\mathcal{I}(j)}_{\\vartheta\\mathcal{T}(j)}\\left[\\overline{\\bf Y}_{-m}^k\\right], & \\text{ otherwise},\n\\end{cases}\\\\\nA_{\\mathcal{T}(j,\\ell,k)} &:= A_{t_1t_2\\cdots t_{j-\\ell} \\underbrace{{\\scriptstyle k\\cdots k}}_{{\\scriptstyle \\ell \\ \\text{times}}}, }\\quad \\text{where } \\mathcal{T}(j,\\ell,k):= (\\mathcal{T}(j-\\ell),\\underbrace{ k,\\cdots, k}_{\\ell \\ \\text{times}}).\n\\end{align*}\nThen, we can write $\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\vartheta)-\\overline\\Delta^{\\mathcal{I}(j)}_{j,k,m}(\\vartheta) = \\sum_{\\mathcal{T}(j)\\in\\{-m+1,\\ldots,k\\}^j} A_{\\mathcal{T}(j)} = \\Delta_a + \\Delta_b + \\Delta_c$, where \n\\[\n\\Delta_a := \\sum_{\\mathcal{T}(j)\\in\\{-m+1,\\ldots,k-1\\}^j} A_{ \\mathcal{T}(j)},\\quad \\Delta_b := \\sum_{\\ell=1}^{j-1} \\binom{j}{\\ell} \\sum_{\\mathcal{T}(j-\\ell)\\in\\{-m+1,\\ldots,k-1\\}^{j-\\ell}} A_{ \\mathcal{T}(j,\\ell,k)}, \\quad \\Delta_c:= A_{(k,\n\\ldots,k)},\n\\]\nand $\\Delta_b:=0$ when $j=1$.\nFrom Lemma \\ref{lemma_ijl} and the symmetry of $A_{ \\mathcal{T}(j)}$, $\\Delta_a$ is bounded by $\\mathcal{C} B_{j,k,m} M^{\\mathcal{I}(j)}_{j,k,m}$, where\n\\begin{align*}\nB_{j,k,m} & := \\sum_{-m+1\\leq t_1\\leq t_2 \\leq \\cdots \\leq t_j\\leq k-1} \\left(\\rho^{(m+t_1-1)_+} \\wedge \\rho^{(t_2-t_1-1)_+} \\wedge \\cdots \\wedge \\rho^{(t_j-t_{j-1}-1)_+} \\wedge \\rho^{(k-1-t_j-1)_+} \\right) \\\\\n& = \\sum_{1\\leq t_1\\leq t_2 \\leq \\cdots \\leq t_j \\leq k+m-1} \\left(\\rho^{(t_1-1)_+} \\wedge \\rho^{(t_2-t_1-1)_+} \\wedge \\cdots \\wedge \\rho^{(t_j-t_{j-1}-1)_+} \\wedge \\rho^{(k+m-1-t_j-1)_+} \\right), \\\\ \nM^{\\mathcal{I}(j)}_{j,k,m} & := \\max_{-m+1\\leq t_1, \\ldots, t_j\\leq k-1} \\|\\phi^{i_1}_{t_1}\\|_{\\infty}\\| \\phi^{i_2}_{t_2}\\|_{\\infty} \\cdots \\| \\phi^{i_j}_{t_j}\\|_{\\infty}.\n\\end{align*}\nFrom $(t-1)_+ \\geq \\lfloor t\/2 \\rfloor$ and Lemma \\ref{rho_sum}, $B_{j,k,m}$ is bounded by $C_{j2}(\\rho) \\rho^{\\lfloor (k+m-1)\/4j\\rfloor}$.\n\nWe proceed to derive a bound of $M^{\\mathcal{I}(j)}_{j,k,m}$. Define $\\|\\phi^{i}\\|^\\ell_\\infty:=\\sum_{t=-\\infty}^\\infty (|t|\\vee 1)^{-2}\\|\\phi^{i}_t\\|_{\\infty}^\\ell$. When $i_1=i_2=\\cdots=i_j$, it follows from Lemma \\ref{hoelder} that $M^{\\mathcal{I}(j)}_{j,k,m} \\leq (k+m)^{j+1}\\|\\phi^{i_1}\\|_{\\infty}^j$, and $\\|\\phi^{i_1}\\|_{\\infty}^j \\in L^{r_{\\mathcal{I}(j)}}(\\mathbb{P}_{\\vartheta^*})$ from Assumption \\ref{assn_a4}. In the other cases, observe that if $x,y,z \\geq 0$, we have $xy \\leq x^2+y^2$, $xyz \\leq x ^3+ y ^3+z ^3$, $xy \\leq x^{4} + y^{4\/3}$, and $xy \\leq x^{3} + y^{3\/2}$ from Young's inequality. By using this result and Lemma \\ref{hoelder}, we can bound $M^{\\mathcal{I}(j)}_{j,k,m}$ by\n\\[\n\\begin{array}{ccc}\nj=2 \\text{ and } i_1 \\neq i_2: & (k+m)^2 (\\|\\phi^{i_1}\\|_{\\infty}^2 + \\|\\phi^{i_2}\\|_{\\infty}^2), \\\\\nj=3 \\text{ and } i_1 \\neq i_2=i_3: & (k+m)^3 ( \\|\\phi^{i_1}\\|_{\\infty}^2 + \\|\\phi^{i_2}\\|_{\\infty}^4), \\\\\nj=3 \\text{ and } i_1,i_2,i_3 \\text{ are distinct}: &(k+m)^2 ( \\|\\phi^{i_1}\\|_{\\infty}^3 + \\|\\phi^{i_2}\\|_{\\infty}^3 + \\|\\phi^{i_3}\\|_{\\infty}^3), \\\\\nj=4 \\text{ and } i_1 \\neq i_2=i_3=i_4: & (k+m)^3 ( \\|\\phi^{i_1}\\|_{\\infty}^4 + \\|\\phi^{i_2}\\|_{\\infty}^4),\\\\\nj=4 \\text{ and } i_1 = i_2 \\neq i_3=i_4: & (k+m)^3 ( \\|\\phi^{i_1}\\|_{\\infty}^4 + \\|\\phi^{i_3}\\|_{\\infty}^4), \\\\\nj=5 \\text{ and } i_1 \\neq i_2 = i_3=i_4=i_5: & (k+m)^3 ( \\|\\phi^{i_1}\\|_{\\infty}^{ 3 } + \\|\\phi^{i_2}\\|_{\\infty}^6).\n\\end{array}\n\\]\nTherefore, from Assumption \\ref{assn_a4}, $\\Delta_a$ is bounded by the right-hand side of part (a). From Lemmas \\ref{lemma_ijl} and \\ref{rho_sum}, $\\Delta_b$ is bounded by $\\mathcal{C} \\sum_{\\ell=1}^{j-1} \\sum_{-m+1 \\leq t_1\\leq \\cdots \\leq t_{j-\\ell} \\leq k-1} (\\rho^{(m+t_1-1)_+} \\wedge \\rho^{(t_2-t_1-1)_+} \\wedge \\cdots \\wedge \\rho^{(k-t_{j- \\ell }-1)_+}) M_{j,k+1,m}^{\\mathcal{I}(j)} \\leq \\mathcal{C} \\rho^{\\lfloor (k+m-1)\/4(j-1) \\rfloor} M_{j,k+1,m}^{\\mathcal{I}(j)} $. Similarly, $\\Delta_c$ is bounded by $\\mathcal{C} \\rho^{\\lfloor (k+m-1)\/4(j-1) \\rfloor} M_{j,k+1,m}^{\\mathcal{I}(j)}$, and part (a) of the lemma follows.\n\nFor part (b), define, for $-m'+1 \\leq t_1, \\ldots, t_j \\leq k$,\n\\begin{equation*}\nD_{\\mathcal{T}(j),m',x}:=\n\\begin{cases}\n\\Phi^{\\mathcal{I}(j)}_{\\theta \\mathcal{T}(j)}[\\overline{\\bf Y}_{-m'}^k,X_{-m'}=x] -\\Phi^{\\mathcal{I}(j)}_{\\theta \\mathcal{T}(j)}[\\overline{\\bf Y}_{-m'}^{k-1},X_{-m'}=x], & \\text{ if } \\max\\{t_1,\\ldots, t_j\\} < k, \\\\\n\\Phi^{\\mathcal{I}(j)}_{\\theta \\mathcal{T}(j)}[\\overline{\\bf Y}_{-m'}^k,X_{-m'}=x], & \\text{ otherwise},\n\\end{cases}\n\\end{equation*}\nand define $D_{\\mathcal{T}(j),m,x}$ similarly. Then, we can write $\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\theta)= \\sum_{\\mathcal{T}(j)\\in\\{-m+1,\\ldots,k\\}^{j}}D_{\\mathcal{T}(j),m,x}$ and $\\Delta^{\\mathcal{I}(j)}_{j,k,m',x}(\\theta) = \\sum_{\\mathcal{T}(j)\\in\\{-m'+1,\\ldots,k\\}^{j}} D_{\\mathcal{T}(j),m',x} = \\Delta_d +\\Delta_e$, where $ \\Delta_d := \\sum_{\\mathcal{T}(j)\\in\\{-m+1,\\ldots,k\\}^{j}}D_{\\mathcal{T}(j),m',x}$ and\n\\[\n\\Delta_e:= \\sum_{\\ell=1}^{j} \\binom{j}{\\ell}\\sum_{t_1=-m'+1}^{-m} \\cdots \\sum_{t_\\ell=-m'+1}^{-m} \\sum_{t_{\\ell+1}=-m+1}^k \\cdots \\sum_{t_{j}=-m+1}^k D_{\\mathcal{T}(j),m',x}.\n\\]\nFrom the same argument as part (a), $\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\theta) - \\Delta_d$ is bounded by the right-hand side of part (a). For $\\Delta_e$, observe that, with $M_{j} := \\max_{1\\leq \\ell \\leq j}\\binom{j}{\\ell}$,\n\\begin{align*}\n| \\Delta_e | & \\leq M_j \\sum_{\\ell=1}^{j} \\sum_{t_1=-m'+1}^{-m} \\sum_{t_2=-m'+1}^{-m} \\cdots \\sum_{t_{\\ell}=-m'+1}^{-m} \\sum_{t_{\\ell+1}=-m+1}^k \\cdots \\sum_{t_{j}=-m+1}^k \\left| D_{\\mathcal{T}(j),m',x} \\right| \\nonumber \\\\\n& \\leq j M_j \\sum_{t_1=-m'+1}^{-m} \\sum_{t_2=-m'+1}^{k} \\cdots \\sum_{t_j=-m'+1}^{k} \\left| D_{\\mathcal{T}(j),m',x} \\right| \\nonumber \\\\\n& \\leq j M_j j! \\sum_{t_1=-m'+1}^{-m} \\sum_{t_1 \\leq t_2 \\leq \\cdots \\leq t_{j}\\leq k }\\left| D_{\\mathcal{T}(j),m',x} \\right|.\n\\end{align*}\nFrom Lemma \\ref{lemma_ijl}, if $t_1 \\leq \\cdots \\leq t_{j}$, we have $|D_{\\mathcal{T}(j),m',x}| \\leq \\mathcal{C} [\\mathbb{I}\\{t_j < k\\} (\\rho^{(t_2-t_1-1)_+} \\wedge \\rho^{(t_j -t_{j-1}-1)_+} \\wedge \\cdots \\wedge\\rho^{(k-1-t_j-1)_+} ) + \\mathbb{I}\\{t_j = k\\} (\\rho^{(t_2-t_1-1)_+} \\wedge \\cdots \\wedge \\rho^{(t_j -t_{j-1}-1)_+})] \\|\\phi^{\\mathcal{I}(j)}_{\\mathcal{T}(j)}\\|_{\\infty}$. Hence, part (b) follows from Lemma \\ref{rho_m}.\n\nFor part (c), observe that $\\sup_{m\\geq 0} \\sup_{x\\in \\mathcal{X}} \\sup_{\\vartheta\\in \\mathcal{N}^*} |\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\vartheta) | \\leq A+B$, where $A:= \\sup_{m\\geq 0}\\sup_{x\\in \\mathcal{X}} \\sup_{\\vartheta\\in \\mathcal{N}^*} |\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\vartheta) - \\Delta^{\\mathcal{I}(j)}_{j,k,0,x}(\\vartheta) |$ and $B:=\\sup_{x\\in \\mathcal{X}} \\sup_{\\vartheta\\in \\mathcal{N}^*} |\\Delta^{\\mathcal{I}(j)}_{j,k,0,x}(\\vartheta) |$. $A$ is bounded by $K_{\\mathcal{I}(j)} k^7 \\rho^{ \\lfloor (k-1)\/ 1340 \\rfloor }$ from part (b). $B$ does not depend on $m$ and is distributionally equivalent to $\\sup_{x\\in \\mathcal{X}} \\sup_{\\vartheta\\in \\mathcal{N}^*} |\\Delta^{\\mathcal{I}(j)}_{j,1,k-1,x}(\\vartheta)|$. This is bounded by $\\sup_{x\\in \\mathcal{X}} \\sup_{\\vartheta\\in \\mathcal{N}^*} |\\Delta^{\\mathcal{I}(j)}_{j,1,k-1,x}(\\vartheta) - \\Delta^{\\mathcal{I}(j)}_{j,1,0,x}(\\vartheta)| + \\sup_{x\\in \\mathcal{X}} \\sup_{\\vartheta\\in \\mathcal{N}^*} |\\Delta^{\\mathcal{I}(j)}_{j,1,0,x}(\\vartheta)|$. The first term is in $L^{r_{\\mathcal{I}(j)}}(\\mathbb{P}_{\\vartheta^*})$ from part (b), and the second term is in $L^{r_{\\mathcal{I}(j)}}(\\mathbb{P}_{\\vartheta^*})$ from the definition of $\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\vartheta)$. Therefore, there exists $M_{\\mathcal{I}(j), k} \\in L^{r_{\\mathcal{I}(j)}}(\\mathbb{P}_{\\vartheta^*})$ such that $A+B \\leq M_{\\mathcal{I}(j), k}$, and part (c) holds in view of part (a). Part (d) follows from parts (a)--(c) because parts (a)--(c) imply that $\\{\\Delta^{\\mathcal{I}(j)}_{j,k,m,x}(\\vartheta)\\}_{m\\geq 0}$ and $\\{\\Delta^{\\mathcal{I}(j)}_{j,k,m}(\\vartheta)\\}_{m\\geq 0}$ are uniform $L^{r_{\\mathcal{I}(j)}}(\\mathbb{P}_{\\vartheta^*})$-Cauchy sequences with respect to $\\vartheta\\in \\mathcal{N}^*$ that converge to the same limit and $L^{q}(\\mathbb{P}_{\\vartheta^*})$ is complete. \n\\end{proof}\n \n\\begin{lemma}\\label{lemma-ratio}\nUnder Assumptions \\ref{assn_a1}, \\ref{assn_a2}, and \\ref{assn_a4}, there exist random variables $\\{K_k\\}_{k=1}^n\\in L^{(1+\\varepsilon)q_\\vartheta\/\\varepsilon}(\\mathbb{P}_{\\vartheta^*})$ and $\\rho \\in (0,1)$ such that, for all $1 \\leq k \\leq n$ and $m' \\geq m \\geq 0$,\n\\begin{align*}\n\\sup_{\\vartheta \\in \\mathcal{N}^*} \\left|\\frac{ \\overline p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1})}{\\overline p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1})}\\right| \\leq K_k, \\quad\n \\sup_{x\\in \\mathcal{X}}\\sup_{\\vartheta \\in \\mathcal{N}^*} \\left|\\frac{ p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1}, X_{-m}=x)}{ p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1}, X_{-m}=x)}-\\frac{ \\overline p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1})}\n{\\overline p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1})}\\right| \\leq K_k \\rho^{k+m-1}.\n\\end{align*}\nFurthermore, these bounds hold uniformly in $x \\in \\mathcal{X}$ when $\\overline p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1})$ and $\\overline p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1})$ are replaced with $ p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m'}^{k-1}, X_{-m'}=x)$ and $ p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m'}^{k-1}, X_{-m'}=x)$.\n\\end{lemma}\n\\begin{proof}[Proof of Lemma \\ref{lemma-ratio}]\nThe first result follows from noting that $\\overline p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1}) \\\\\n= \\sum_{(x_{k-1},x_k )\\in \\mathcal{X}^2} g_\\vartheta(Y_k|\\overline{\\bf{Y}}_{k-1}, x_k) q_{\\vartheta_x}(x_{k-1},x_k) \\mathbb{P}_\\vartheta(x_{k-1}|\\overline{\\bf{Y}}_{-m}^{k-1}) \\in [\\sigma_-G_{\\vartheta k},\\sigma_+G_{\\vartheta k}]$ and using Assumption \\ref{assn_a4}(b). For the second result, observe that $| p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1}, X_{-m}=x) - \\overline p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1})| \\leq \\sum_{(x_{k-1},x_k )\\in \\mathcal{X}^2} g_\\vartheta(Y_k|\\overline{\\bf{Y}}_{k-1}, x_k) q_{\\vartheta_x}(x_{k-1},x_k) |\\mathbb{P}_\\vartheta(x_{k-1}|\\overline{\\bf{Y}}_{-m}^{k-1},X_{-m}=x) - \\mathbb{P}_\\vartheta(x_{k-1}|\\overline{\\bf{Y}}_{-m}^{k-1})| \\leq \\rho^{k+m-1}\\sigma_+ G_{\\vartheta k}\/\\sigma_-$, where the second inequality follows from Lemma \\ref{x_diff}(a). The second result then follows from writing the left-hand side as\n\\[\n\\frac{ p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},X_{-m}=x)- \\overline p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1})}{ p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},X_{-m}=x)}\n+ \\frac{ \\overline p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1})}{\\overline p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1})} \\frac{\\overline p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1}) - p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},X_{-m}=x)}{ p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},X_{-m}=x)},\n\\]\nnoting that $p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m}^{k-1},X_{-m}=x) \\geq \\sigma_-G_{\\vartheta k}$, and using the derived bounds. The results with $ p_{\\vartheta}(Y_k|\\overline{\\bf{Y}}_{-m'}^{k-1}, X_{-m'}=x)$ and $ p_{\\vartheta^*}(Y_k|\\overline{\\bf{Y}}_{-m'}^{k-1}, X_{-m'}=x)$ are proven similarly.\n\\end{proof}\n\nThe following result originally appeared in equations (59)--(60) of \\citet{kasaharashimotsu15jasa}. We state this as a lemma for ease of reference.\n\\begin{lemma} \\label{lemma_normal_der}\nLet $f(\\mu,\\sigma^2)$ denote the density of $N(\\mu,\\sigma^2)$. Then\n\\begin{equation*}\n\\left.\\nabla_{\\lambda_\\mu^k}f( c_1\\lambda_\\mu, c_2 \\lambda_\\mu^2)\\right|_{\\lambda_\\mu=0} =\n\\begin{cases}\nc_1\\nabla_{\\mu} f(0,0) & \\text{if}\\ k = 1, \\\\\nc_1^2\\nabla_{\\mu^2} f(0,0) + 2c_2 \\nabla_{\\sigma^2} f(0,0) & \\text{if}\\ k = 2, \\\\\nc_1^3\\nabla_{\\mu^3} f(0,0) + 6 c_1 c_2 \\nabla_{\\mu\\sigma^2} f(0,0) & \\text{if}\\ k = 3, \\\\\nc_1^4\\nabla_{\\mu^4} f(0,0) + 12 c_1^2c_2 \\nabla_{\\mu^2} f(0,0)\\nabla_{\\sigma^2} f(0,0) + 12c_2^2 \\nabla_{\\sigma^4} f(0,0) & \\text{if}\\ k = 4.\n\\end{cases}\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma \\ref{lemma_normal_der}]\nObserve that a composite function $f(\\lambda_\\mu, h(\\lambda_\\mu))$ satisfies\n$\\nabla_{\\lambda_\\mu^k}f(\\lambda_\\mu,h(\\lambda_\\mu)) = (\\nabla_{\\lambda_\\mu} + \\nabla_u)^k f(\\lambda_\\mu,h(u))|_{u = \\lambda_\\mu} = \\sum_{j = 0}^k \\binom{k}{j} \\nabla_{\\lambda_\\mu^{k - j}u^{j}} f(\\lambda_\\mu,h(u))|_{u = \\lambda_\\mu}$. Further, because $\\nabla_{u^j}u^2|_{u = 0} = 0$ except for $j = 2$, it follows from Fa\\`a di Bruno's formula that $\\nabla_{u^j} f(c_1\\lambda_\\mu,c_2 u^2)|_{\\lambda_\\mu = u = 0}$ is $0$ if $j = 1, 3$, is $2c_2\\nabla_{h} f(0, h(0))$ if $j = 2$, and is $12c_2^2 \\nabla_{h^2} f(0,h(0))$ if $j = 4$. Therefore, the stated result follows.\n\\end{proof}\n\n \n\\begin{lemma} \\label{lemma_d34}\nSuppose the assumptions of Proposition \\ref{P-quadratic-N1} hold. Then, there exist $\\bar\\varrho_1, \\bar\\varrho_2, \\bar\\varrho_3 \\in (0,\\varrho)$ such that, for all $k \\geq 1$,\n\\begin{align*}\n(a) & \\quad \\frac{\\nabla_{\\lambda_\\mu^3} \\overline p_{\\psi^*\\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\overline p_{\\psi^*\\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})} = \\varrho \\frac{\\nabla_\\varrho \\nabla_{\\lambda_\\mu^3} \\overline p_{\\psi^* \\bar \\varrho_1 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\overline p_{\\psi^* \\bar \\varrho_1 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}, \\\\\n(b) & \\quad \\frac{\\nabla_{\\lambda_\\mu^4} \\overline p_{\\psi^*\\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\overline p_{\\psi^*\\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})} - b(\\alpha)\\frac{\\nabla_{\\lambda_\\sigma^2} \\overline p_{\\psi^*\\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\overline p_{\\psi^*\\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})} \\\\\n& \\quad = \\varrho \\frac{\\nabla_\\varrho \\nabla_{\\lambda_\\mu^4} \\overline p_{\\psi^* \\bar \\varrho_2 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\overline p_{\\psi^* \\bar \\varrho_2 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})} - \\varrho \\frac{\\nabla_\\varrho \\nabla_{\\lambda_\\sigma^2} \\overline p_{\\psi^* \\bar \\varrho_3 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\overline p_{\\psi^* \\bar \\varrho_3 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}.\n\\end{align*} \n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma \\ref{lemma_d34}] \nPart (a) holds if\n\\begin{equation} \\label{d3p}\n\\nabla_{\\lambda_\\mu^3} \\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) \/ \\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) =0,\n\\end{equation}\nbecause (i) $\\nabla_{\\lambda_\\mu^3}\\overline p_{\\psi^* \\varrho \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) - \\nabla_{\\lambda_\\mu^3}\\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) = \\nabla_\\varrho \\nabla_{\\lambda_\\mu^3}\\overline p_{\\psi^* \\bar \\varrho \\alpha } (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) \\varrho$ for $\\bar \\varrho \\in (0,\\varrho)$ from the mean value theorem and (ii) $\\overline p_{\\psi^* \\varrho \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})$ does not depend on the value of $\\varrho$.\n\nWe proceed to show (\\ref{d3p}). Note that$\\nabla_{\\lambda_\\mu^3} \\overline p_{\\psi^*\\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) \/\\overline p_{\\psi^*\\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) = \\nabla_{\\lambda_\\mu^3} \\log \\overline p_{\\psi^*\\pi} ({\\bf Y}^k_1| \\overline{{\\bf Y}}_{0}) - \\nabla_{\\lambda_\\mu^3} \\log \\overline p_{\\psi^*\\pi} ({\\bf Y}^{k-1}_1| \\overline{{\\bf Y}}_{0})$ from (\\ref{der_formula}) and $\\nabla_\\lambda \\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})=0$. Let $\\nabla^i \\ell_t^* := \\nabla_{\\lambda_\\mu^i} \\log g_t^*$ with $\\nabla \\ell_t^*=\\nabla^1 \\ell_t^*$. Observe that \n\\begin{equation} \\label{d3lambda0}\n\\begin{aligned}\n \\nabla_{\\lambda_\\mu^3} \\log \\overline p_{\\psi^* 0 \\alpha} ({\\bf Y}^k_1| \\overline{{\\bf Y}}_{0})\n& = \\sum_{t=1}^k \\mathbb{E}_{\\psi^* 0 \\alpha} \\left[\\nabla^3 \\ell^*_{t} \\middle|\\overline{{\\bf Y}}_{0}^k \\right] + 3 \\sum_{t_1=1}^k \\sum_{t_2=1}^k \\mathbb{E}_{\\psi^* 0 \\alpha} \\left[ \\nabla^2 \\ell^*_{t_1} \\nabla \\ell^*_{t_2} \\middle| \\overline{{\\bf Y}}_{0}^k \\right] \\\\\n& \\qquad + \\sum_{t_1=1}^k \\sum_{t_2=1}^k \\sum_{t_3=1}^k \\mathbb{E}_{\\psi^* 0 \\alpha} \\left[ \\nabla \\ell^*_{t_1} \\nabla \\ell^*_{t_2} \\nabla \\ell^*_{t_3} \\middle| \\overline{{\\bf Y}}_{0}^k \\right] \\\\\n& = \\sum_{t=1}^k \\mathbb{E}_{\\psi^* 0 \\alpha} \\left[\\nabla^3 \\ell^*_{t} + 3 \\nabla^2 \\ell^*_{t} \\nabla \\ell^*_{t} + \\nabla \\ell^*_{t} \\nabla \\ell^*_{t} \\nabla \\ell^*_{t} \\middle| \\overline{{\\bf Y}}_{0}^k \\right] \\\\\n& = \\sum_{t=1}^k \\mathbb{E}_{\\psi^* 0 \\alpha} \\left[\\nabla_{\\lambda_\\mu^3} g_t^* \/g_t^* \\middle| \\overline{{\\bf Y}}_{0}^k \\right],\n\\end{aligned}\n\\end{equation}\nwhere the first equality follows from Lemma \\ref{louis}, the second equality holds because (i) $X_t$ is serially independent when $\\varrho=0$ and (ii) $\\nabla\\ell^*_{t} = d_{1t} \\nabla_{\\mu} f^*_{t}\/f^*_{t}$ and $\\nabla^2 \\ell^*_{t} = d_{2t} \\nabla_{\\mu}^2 f^*_{t}\/f^*_{t} - (d_{1t} \\nabla_{\\mu} f^*_{t}\/f^*_{t})^2$, and (iii) $\\mathbb{E}_{\\psi^* 0 \\alpha} [d_{1t} | \\overline{{\\bf Y}}_{0}^k ] = \\mathbb{E}_{\\psi^* 0 \\alpha} [d_{2t} | \\overline{{\\bf Y}}_{0}^k ] =0$ from (\\ref{d3g2}), and the third equality follows from (\\ref{der_formula}) The right-hand side is 0 from (\\ref{d3g2}), and hence part (a) is proven.\n\n\n\nFor part (b), from a similar argument to part (a), the stated result holds if \n\\begin{equation} \\label{d4p}\n\\nabla_{\\lambda_\\mu^4}\\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) \/ \\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) = b(\\alpha)\\nabla_{\\lambda_\\sigma^2} \\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) \/ \\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}).\n\\end{equation}\nObserve that $\\nabla_{\\lambda_\\mu^4} \\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) = \\nabla_{\\lambda_\\mu^4} \\log \\overline p_{\\psi^* 0 \\alpha} ({\\bf Y}^k_0| \\overline{{\\bf Y}}_{0}) - \\nabla_{\\lambda_\\mu^4} \\log \\overline p_{\\psi^* 0 \\alpha} ({\\bf Y}^{k-1}_0| \\overline{{\\bf Y}}_{0})$ from (\\ref{der_formula}), $\\nabla_\\lambda \\overline p_{\\psi^*\\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})=0$, and $\\nabla_{\\lambda_\\mu^2} \\log \\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})=0$. A similar derivation to (\\ref{d3lambda0}) gives \n\\begin{align} \n& \\nabla_{\\lambda_\\mu^4} \\log \\overline p_{\\psi^* 0 \\alpha} ({\\bf Y}^k_0| \\overline{{\\bf Y}}_{0}) = \\sum_{t=1}^k \\mathbb{E}_{\\psi^* 0 \\alpha}\\left[ \\nabla_{\\lambda_{\\mu}^4}g^*_{t} \/ g^*_{t} \\middle| \\overline{\\bf Y}_0^k\\right]. \\label{d4pp}\n\\end{align}\n(\\ref{d4p}) follows from (\\ref{d4pp}) because (i) $\\nabla_{\\lambda_\\sigma^2} \\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})\/\\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) = \\mathbb{E}_{\\vartheta^*}[ \\nabla_{\\lambda_\\sigma^2} g_k^*|\\overline{{\\bf Y}}_{0}^k]$ from a similar argument to (\\ref{d2p}) and (ii) $\\mathbb{E}_{\\psi^* 0 \\alpha} [\\nabla_{\\lambda_\\mu^4} g_t^* \/g_t^* | \\overline{{\\bf Y}}_{0}^k ] = b(\\alpha) \\mathbb{E}_{\\vartheta^*}[ \\nabla_{\\lambda_\\sigma^2} g_k^*|\\overline{{\\bf Y}}_{0}^k]$ from (\\ref{d3g2}). Therefore, part (b) is proven.\n\\end{proof}\n\n\\begin{lemma} \\label{lemma_d34_homo}\nSuppose the assumptions of Proposition \\ref{P-quadratic-N1-homo} hold. Then, there exist $\\bar \\varrho_1, \\bar \\varrho_2 \\in (0,\\varrho)$ such that, for all $k \\geq 1$,\n\\begin{align*}\n(a) & \\quad \\frac{\\nabla_{\\lambda_\\mu^3} \\overline p_{\\psi^*\\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\overline p_{\\psi^*\\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})} = \\alpha(1-\\alpha)(1-2\\alpha)\\frac{\\nabla_{\\mu^3}f_k^*}{f_k^*} + \\varrho \\frac{\\nabla_\\varrho \\nabla_{\\lambda_\\mu^3} \\overline p_{\\psi^*\\bar \\varrho_1 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\overline p_{\\psi^*\\bar \\varrho_1 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}, \\\\\n(b) & \\quad \\frac{\\nabla_{\\lambda_\\mu^4} \\overline p_{\\psi^*\\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\overline p_{\\psi^* \\pi} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})} = \\alpha(1-\\alpha)(1-6\\alpha+6\\alpha^2)\\frac{\\nabla_{\\mu^4}f_k^*}{f_k^*} + \\varrho \\frac{\\nabla_\\varrho \\nabla_{\\lambda_\\mu^4} \\overline p_{\\psi^*\\bar \\varrho_2 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}{\\overline p_{\\psi^*\\bar \\varrho_2 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1})}.\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma \\ref{lemma_d34_homo}]\nThe proof is similar to the proof of Lemma \\ref{lemma_d34}(a). From an argument similar to the proof of Lemma \\ref{lemma_d34}, the stated results hold if \n\\begin{align*} \n(A) &\\quad \\nabla_{\\lambda_\\mu^3} \\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) \/ \\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) = \\alpha(1-\\alpha)(1-2\\alpha)\\nabla_{\\mu^3}f_k^* \/ f_k^*, \\\\\n(B) &\\quad \\nabla_{\\lambda_\\mu^4} \\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) \/ \\overline p_{\\psi^* 0 \\alpha} (Y_k| \\overline{{\\bf Y}}_{0}^{k-1}) = \\alpha(1-\\alpha)(1-6\\alpha+6\\alpha^2)\\nabla_{\\mu^4}f_k^* \/ f_k^*. \n\\end{align*}\nObserve that equalities (\\ref{d3lambda0}) and (\\ref{d4pp}) in the proof of Lemma \\ref{lemma_d34} still hold under the assumptions of Proposition \\ref{P-quadratic-N1-homo} if we use (\\ref{d3g2-homo}) in place of (\\ref{d3g2}). Consequently, (A) and (B) follow from (\\ref{d3g-homo}), (\\ref{d3g2-homo}), and the argument of the proof of Lemma \\ref{lemma_d34}, and the stated result follows.\n\\end{proof}\n \n\\begin{lemma} \\label{lemma_btsp} \nSuppose that the assumptions of Propositions \\ref{P-LR} hold. Let $\\bf{C}_\\eta$ be a set of sequences $\\{\\eta_n\\}$ satisfying $\\sqrt{n}(\\eta_n - \\eta^*) \\to h_\\eta$ for some finite $h_\\eta$. Let $\\mathbb{P}^n_{\\eta_n} := \\prod_{k=1}^n f_k(\\eta_n,0)$ denote the probability measure under $\\eta_n$ with $\\lambda_n=0$. Then, for every sequence $\\{\\eta_n\\} \\in \\bf{C}_\\eta$, the LRTS under $\\{\\mathbb{P}^n_{\\eta_n}\\}$ converges in distribution $\\sup_{ \\varrho \\in\\Theta_{\\varrho}} \\left(\\tilde t_{\\lambda \\varrho}' \\mathcal{I}_{\\lambda.\\eta \\varrho} \\tilde t_{\\lambda \\varrho} \\right)$ given in Propositions \\ref{P-LR}.\n\\end{lemma}\n\\begin{proof}[Proof of Lemma \\ref{lemma_btsp}] \nObserve that $\\vartheta_n:=(\\pi_n,\\eta_n,\\lambda_n) = (\\pi, \\eta^*+h_\\eta\/\\sqrt{n},0)$ satisfies the assumptions of Proposition \\ref{P-LAN}. Therefore, Proposition \\ref{P-LAN} holds under $\\vartheta_n$ with $\\nu_n(s_{\\varrho n k}) \\to_d N(\\mathcal{I}_\\varrho h, \\mathcal{I}_\\varrho)$ with $h=(h_\\eta',0)'$ under $\\mathbb{P}_{\\vartheta_n}^n$. Furthermore, the log-likelihood function of the one-regime model admits a similar expansion, and $\\log (d \\mathbb{P}_{\\eta_n}^n \/ d \\mathbb{P}_{\\eta^*}^n ) = h_\\eta' \\nu_n (s_{\\eta k}) - (1\/2)h_\\eta' \\mathcal{I}_\\eta h_\\eta + o_p(1)$ holds under $\\mathbb{P}_{\\eta_n}^n$. Therefore, the proof of Proposition \\ref{P-LR} goes through by replacing $G_{\\varrho n}$ with $G_{\\varrho n}^h = \\left[\\begin{smallmatrix} G_{\\eta n}^h \\\\ G_{\\lambda \\varrho n}^h \\end{smallmatrix} \\right] := G_{\\varrho n} + \\mathcal{I}_{\\varrho}h$. In view of $G_{\\eta n}^h = G_{\\eta n} + \\mathcal{I}_{\\eta}h_\\eta$ and $G_{\\lambda \\varrho n}^h = G_{\\lambda \\varrho n} + \\mathcal{I}_{\\lambda \\eta \\varrho} h_\\eta$, we have $G_{\\lambda. \\eta \\varrho n}^h := G_{\\lambda \\varrho n}^h - \\mathcal{I}_{\\lambda \\eta \\varrho}\\mathcal{I}_{\\eta}^{-1} G_{\\eta n}^h = G_{\\lambda \\varrho n} - \\mathcal{I}_{\\lambda \\eta \\varrho}\\mathcal{I}_{\\eta}^{-1}G_{\\eta n} = G_{\\lambda \\varrho n}$. Therefore, the asymptotic distribution of the LRTS under $\\mathbb{P}_{\\eta^n}^n$ is the same as that under $\\mathbb{P}_{\\eta^*}^n$, and the stated result follows.\n\\end{proof}\n\n\\subsubsection{Bounds on difference in state probabilities and conditional moments}\n\\begin{lemma} \\label{max_bound}\nSuppose $X_1,\\ldots,X_n$ are random variables with $\\max_{1\\leq i \\leq n}\\mathbb{E}|X_i|^q < C$ for some $q>0$ and $C \\in (0,\\infty)$. Then, $\\max_{1 \\leq i \\leq n} |X_i|= o_p(n^{1\/q})$.\n\\end{lemma}\n\\begin{proof}[Proof of Lemma \\ref{max_bound}]\nFor any $\\varepsilon>0$, we have $\\mathbb{P}(\\max_{1\\leq i\\leq n}|X_i|>\\varepsilon n^{1\/q}) \\leq \\sum_{1\\leq i\\leq n} \\mathbb{P}( |X_i|>\\varepsilon n^{1\/q}) \\\\ \\leq \\varepsilon^{-q} n^{-1} \\sum_{1\\leq i\\leq n} \\mathbb{E}(|X_i|^q\\mathbb{I}{\\{|X_i|>\\varepsilon n^{1\/q}\\}})$ by a version of Markov inequality. As $n\\rightarrow \\infty$, the right-hand side tends to 0 by the dominated convergence theorem. \n\\end{proof} \n \nThe following lemma extends Corollary 1 and (39) of DMR and an equation on p. 2298 of DMR; DMR derive these results when $t_1=t_2$ and $t_3=t_4$ and ${\\bf W}^n_{-m}$ is absent. For the two probability measures $\\mu_1$ and $\\mu_2$, the total variation distance between $\\mu_1$ and $\\mu_2$ is defined as $\\|\\mu_1-\\mu_2\\|_{TV}:=\\sup_{A}|\\mu_1(A)-\\mu_2(A)|$. $\\|\\cdot\\|_{TV}$ satisfies $\\sup_{f(x):0 \\leq f(x) \\leq 1}|\\int f(x) d\\mu_1(x)- \\int f(x) d\\mu_2(x)| = \\|\\mu_1-\\mu_2\\|_{TV}$. In the following, we define $\\overline{\\bf{V}}_{-m}^n := (\\overline{\\bf{Y}}_{-m}^n,{\\bf W}_{-m}^n)$, and we let $\\overline{\\bf{v}}_{-m}^n$ and $x_{-m}$ denote ``$\\overline{\\bf{V}}_{-m}^n = \\overline{\\bf{v}}_{-m}^n$'' and ``$X_{-m}=x_{-m}$.''\n\\begin{lemma} \\label{x_diff}\nSuppose Assumptions \\ref{assn_a1}--\\ref{assn_a2} hold and $\\vartheta_x \\in \\Theta_x$. Then, we have, for all $\\overline{\\bf{v}}_{-m}^n$,\\\\\n(a) For all $-m \\leq t_1 \\leq t_2$ with $-m<n$ and all probability measures $\\mu_1$ and $\\mu_2$ on $\\mathcal{B}(\\mathcal{X})$,\n\\begin{align*}\n&\\left\\| \\sum_{x_{-m}\\in \\mathcal{X}} \\mathbb{P}_{\\vartheta_x}({\\bf X}_{t_1}^{t_2} \\in \\cdot | x_{-m}, \\overline{\\bf{v}}_{-m}^n)\\mu_1(x_{-m})- \\sum_{x_{-m}\\in\\mathcal{X}} \\mathbb{P}_{\\vartheta_x}({\\bf X}_{t_1}^{t_2} \\in \\cdot | x_{-m}, \\overline{\\bf{v}}_{-m}^{n})\\mu_2(x_{-m}) \\right\\|_{TV} \\leq \\rho^{t_1+m}. \n\\end{align*}\n(b) For all $-m \\leq t_1 \\leq t_2 \\leq n-1$,\n\\begin{equation*}\n\\left\\| \\mathbb{P}_{\\vartheta_x}({\\bf X}_{t_1}^{t_2} \\in \\cdot|\\overline{\\bf{v}}_{-m}^n,x_{-m}) - \\mathbb{P}_{\\vartheta_x}({\\bf X}_{t_1}^{t_2} \\in \\cdot|\\overline{\\bf{v}}_{-m}^{n-1},x_{-m}) \\right\\|_{TV} \\leq \\rho^{n-1-t_2}. \n\\end{equation*}\nThe same bound holds when $x_{-m}$ is dropped from the conditioning variables.\\\\\n(c) For all $-m \\leq t_1 \\leq t_2 < t_{3} \\leq t_4$ with $-m<n$,\n\\begin{align*}\n& \\left\\| \\mathbb{P}_{\\vartheta_x}({\\bf X}_{t_1}^{t_2} \\in \\cdot, {\\bf X}_{t_3}^{t_4} \\in \\cdot |\\overline{\\bf{v}}_{-m}^n,x_{-m}) - \\mathbb{P}_{\\vartheta_x}({\\bf X}_{t_1}^{t_2} \\in \\cdot|\\overline{\\bf{v}}_{-m}^n,x_{-m})\\mathbb{P}_{\\vartheta_x}({\\bf X}_{t_3}^{t_4} \\in \\cdot|\\overline{\\bf{v}}_{-m}^n,x_{-m}) \\right\\|_{TV} \\leq \\rho^{t_3-t_2}.\n\\end{align*}\nThe same bound holds when $x_{-m}$ is dropped from the conditioning variables.\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma \\ref{x_diff}]\nWe prove part (a) first. We assume $t_1>-m$ because the stated result holds trivially when $t_1=-m$. Observe that Lemma 1 of DMR still holds when ${\\bf W}_{-m}^n$ is added to the conditioning variable because Assumption \\ref{assn_a1} implies that $\\{(X_k,\\overline{\\bf Y}_{k})\\}_{k=0}^{\\infty}$ is a Markov chain given $\\{W_k\\}_{k=0}^{\\infty}$. Therefore, $\\{X_{t}\\}_{t \\geq - m}$ is a Markov chain when conditioned on $\\{ \\overline{\\bf{Y}}_{-m}^{n},{\\bf W}_{-m}^n \\}$, and hence $\\mathbb{P}_{\\vartheta_x}({\\bf X}_{t_1}^{t_2} \\in A |\\overline{\\bf{v}}_{-m}^n,x_{-m}) = \\sum_{x_{t_1} \\in \\mathcal{X}}\\mathbb{P}_{\\vartheta_x}({\\bf X}_{t_1}^{t_2} \\in A |X_{t_1}=x_{t_1},\\overline{\\bf{v}}_{-m}^n) p_{\\vartheta_x}(x_{t_1}|\\overline{\\bf{v}}_{-m}^n,x_{-m})$ holds. From applying this result and the property of the total variation distance, we can bound the left-hand side of the lemma by $\\| \\sum_{x_{-m}\\in\\mathcal{X}} p_{\\vartheta_x}(X_{t_1} \\in \\cdot |\\overline{\\bf{v}}_{-m}^n,x_{-m}) \\mu_1(x_{-m}) - \\sum_{x_{-m}\\in\\mathcal{X}} p_{\\vartheta_x}(X_{t_1} \\in \\cdot |\\overline{\\bf{v}}_{-m}^n,x_{-m})\\mu_2(x_{-m}) \\|_{TV}$. This is bounded by $\\rho^{t_1+m}$ from Corollary 1 of DMR, which holds when ${\\bf W}_{-m}^n$ is added to the conditioning variable. Therefore, part (a) is proven.\n\nWe proceed to prove part (b). Observe that the time-reversed process $\\{Z_{n-k}\\}_{0 \\leq k\\leq n+m}$ is Markov when conditioned on ${\\bf W}_{-m}^n$ and that $W_k$ is independent of $({\\bf X}_{0}^{k-1},\\overline{\\bf Y}_{0}^{k-1})$ given ${\\bf W}_{0}^{k-1}$. Consequently, for $k=n,n-1$, we have $\\mathbb{P}_{\\vartheta_x}({\\bf X}_{t_1}^{t_2} \\in A|\\overline{\\bf{v}}_{-m}^k,x_{-m}) = \\sum_{x_{t_2} \\in \\mathcal{X}} \\mathbb{P}_{\\vartheta_x}({\\bf X}_{t_1}^{t_2} \\in A|X_{t_2}=x_{t_2},\\overline{\\bf{v}}_{-m}^{t_2},x_{-m}) p_{\\vartheta_x}(x_{t_2}|\\overline{\\bf{v}}_{-m}^{k},x_{-m})$. Therefore, from the property of the total variation distance, the left-hand side of the lemma is bounded by $\\|\\mathbb{P}_{\\vartheta_x}(X_{t_2} \\in \\cdot |\\overline{\\bf{v}}_{-m}^{n},x_{-m}) - \\mathbb{P}_{\\vartheta_x}(X_{t_2} \\in \\cdot |\\overline{\\bf{v}}_{-m}^{n-1},x_{-m})\\|_{TV}$. This is bounded by $\\rho^{n-1-t_2}$ because equation (39) of DMR p.\\ 2294 holds when ${\\bf W}_{-m}^n$ is added to the conditioning variables, and the stated result follows. When $x_{-m}$ is dropped from the conditioning variables, part (b) follows from a similar argument with using Lemma 9 and an analogue of Corollary 1 of DMR in place of equation (39) of DMR.\n\nPart (c) follows immediately from writing the left-hand side of lemma as $\\sup_{A,B}| \\mathbb{P}_{\\vartheta_x}({\\bf X}_{t_1}^{t_2} \\in A|\\overline{\\bf{v}}_{-m}^n,x_{-m}) [ \\mathbb{P}_{\\vartheta_x}({\\bf X}_{t_3}^{t_4} \\in B |\\overline{\\bf{v}}_{-m}^n, {\\bf X}_{t_1}^{t_2} \\in A) - \\mathbb{P}_{\\vartheta_x}({\\bf X}_{t_3}^{t_4} \\in B|\\overline{\\bf{v}}_{-m}^n,x_{-m})]|$ and applying part (a).\n\\end{proof}\n\n\\subsubsection{The sums of powers of $\\rho$} \n\\begin{lemma}\\label{two_rho_sum}\nFor all $\\rho \\in (0,1)$, $c \\geq 1$, $q \\geq 1$, and $b>a$,\n\\begin{align*}\n& \\sum_{t=-\\infty}^{\\infty} \\left(\\rho^{\\lfloor (t-a)\/cq \\rfloor}\\wedge \\rho^{\\lfloor(b-t)\/q\\rfloor}\\right) \\leq \\frac{q(c+1)\\rho^{\\lfloor (b-a)\/(c+1)q \\rfloor}}{ 1-\\rho}, \\\\\n& \\sum_{t=-\\infty}^{\\infty} \\left(\\rho^{\\lfloor(t-a)\/q \\rfloor}\\wedge \\rho^{\\lfloor (b-t)\/cq \\rfloor}\\right) \\leq \\frac{q(c+1)\\rho^{\\lfloor (b-a)\/(c+1)q \\rfloor}}{ 1-\\rho}.\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma \\ref{two_rho_sum}]\nThe first result holds because the left-hand side is bounded by \\\\$\\sum_{t=-\\infty}^{\\lfloor (a+bc)\/(c+1) \\rfloor }\\rho^{\\lfloor(b-t)\/q\\rfloor} + \\sum_{t=\\lfloor (a+bc)\/(c+1) \\rfloor +1}^{\\infty}\\rho^{\\lfloor (t-a)\/cq \\rfloor} \\leq q \\rho^{\\lfloor \\{ b- \\lfloor (a+bc)\/(c+1) \\rfloor \\}\/q \\rfloor }\/(1-\\rho) + cq\\rho^{\\lfloor \\{ \\lfloor (a+bc)\/(c+1) \\rfloor +1 -a \\} \/ cq \\rfloor}\/(1-\\rho) \\leq q (1+c) \\rho^{\\lfloor (b-a)\/(c+1)q \\rfloor}\/(1-\\rho)$. The second result is proven by bounding the left-hand side by $\\sum_{t=-\\infty}^{\\lfloor (ac+b)\/(c+1) \\rfloor }\\rho^{\\lfloor(b-t)\/cq\\rfloor} + \\sum_{t=\\lfloor (ac+b)\/(c+1) \\rfloor +1}^{\\infty}\\rho^{\\lfloor (t-a)\/q \\rfloor}$ and proceeding similarly.\n\\end{proof}\n\nThe following lemma generalizes the result in the last inequality on p. 2299 of DMR. \n\\begin{lemma}\\label{rho_sum}\nFor all $\\rho \\in (0,1)$, $k \\geq 1 $, $q \\geq 1$, and $n \\geq 0$,\n\\[\n\\sum_{0\\leq t_1\\leq t_2 \\leq \\cdots \\leq t_k \\leq n} \\left( \\rho^{\\lfloor t_1 \/q \\rfloor}\\wedge \\rho^{\\lfloor (t_2-t_1)\/q \\rfloor} \\wedge \\cdots \\wedge \\rho^{\\lfloor (t_k-t_{k-1})\/q \\rfloor} \\wedge \\rho^{\\lfloor (n-t_k)\/q \\rfloor} \\right) \\leq C_{kq}(\\rho) \\rho^{\\lfloor n\/2kq \\rfloor},\n\\]\nwhere $C_{kq}(\\rho):= q^{k} k(k+1)! (1-\\rho)^{-k}$.\n\\end{lemma}\n\\begin{proof}[Proof of Lemma \\ref{rho_sum}]\nWhen $k=1$, the stated result follows from Lemma \\ref{two_rho_sum} with $c=1$. We first show that the following holds for $k\\geq 2$:\n\\begin{equation} \\label{rho_k_bound}\n\\sum_{t_1\\leq t_2 \\leq \\cdots \\leq t_k \\leq n} \\left(\\rho^{\\lfloor (t_2-t_1)\/q \\rfloor} \\wedge \\cdots \\wedge \\rho^{\\lfloor (t_k-t_{k-1})\/q \\rfloor} \\wedge \\rho^{\\lfloor (n-t_k)\/q \\rfloor}\\right) \\leq \\frac{q^{k-1} (k+1)! \\rho^{\\lfloor (n-t_1)\/kq \\rfloor }}{(1-\\rho)^{k-1}}.\n\\end{equation}\nWe prove (\\ref{rho_k_bound}) by induction. When $k=2$, it follows from Lemma \\ref{two_rho_sum} with $c=1$ that \\\\$\\sum_{t_2=t_1}^{n}(\\rho^{\\lfloor (t_2-t_1)\/q \\rfloor} \\wedge \\rho^{\\lfloor (n-t_2)\/q \\rfloor}) \\leq 2q \\rho^{\\lfloor (n-t_1)\/2q \\rfloor} \/(1-\\rho)$, giving (\\ref{rho_k_bound}). Suppose (\\ref{rho_k_bound}) holds when $k=\\ell$. Then (\\ref{rho_k_bound}) holds when $k=\\ell+1$ because, from Lemma \\ref{two_rho_sum},\n\\begin{align*}\n& \\sum_{t_1\\leq t_2 \\leq \\cdots \\leq t_{\\ell} \\leq t_{\\ell+1} \\leq n}\\left(\\rho^{\\lfloor (t_2-t_1)\/q \\rfloor} \\wedge \\rho^{\\lfloor (t_3-t_2)\/q \\rfloor} \\wedge \\cdots \\wedge \\rho^{\\lfloor (t_{\\ell+1}-t_{\\ell})\/q \\rfloor} \\wedge \\rho^{\\lfloor (n-t_{\\ell+1})\/q \\rfloor}\\right) \\\\\n&\\leq \\sum_{t_2=t_1}^{n} \\left( \\rho^{\\lfloor (t_2-t_1)\/q \\rfloor} \\wedge \\sum_{t_2 \\leq \\cdots \\leq t_{\\ell+1} \\leq n}\\left( \\rho^{\\lfloor (t_3-t_2)\/q \\rfloor} \\wedge \\cdots \\wedge \\rho^{\\lfloor (t_{\\ell+1}-t_{\\ell})\/q \\rfloor} \\wedge \\rho^{\\lfloor (n-t_{\\ell+1})\/q \\rfloor} \\right) \\right)\\\\\n& \\leq \\frac{q^{\\ell-1} \\ell!}{ (1-\\rho)^{\\ell-1}} \\sum_{t_2=t_1}^{n} \\left(\\rho^{\\lfloor (t_2-t_1)\/q \\rfloor}\\wedge \\rho^{\\lfloor (n-t_2)\/\\ell q \\rfloor}\\right) \\\\\n& \\leq \\frac{q^{\\ell}(\\ell+1)!}{(1-\\rho)^{\\ell}} \\rho^{\\lfloor (n-t_1)\/(\\ell+1)q \\rfloor},\n\\end{align*}\nand hence (\\ref{rho_k_bound}) holds for all $k \\geq 2$. We proceed to show the stated result. Observe that\n\\begin{align*}\n& \\sum_{0\\leq t_1\\leq t_2 \\leq \\cdots \\leq t_k \\leq n} \\left( \\rho^{\\lfloor t_1 \/q \\rfloor}\\wedge \\rho^{\\lfloor (t_2-t_1)\/q \\rfloor} \\wedge \\cdots \\wedge \\rho^{\\lfloor (t_k-t_{k-1})\/q \\rfloor} \\wedge \\rho^{\\lfloor (n-t_k)\/q \\rfloor} \\right) \\\\\n& \\leq 2 \\sum_{t_1=0}^{n\/2}\\sum_{t_1\\leq t_2 \\leq \\cdots \\leq t_{k-1}\\leq t_k} \\sum_{t_k = t_1}^{n-t_1} \\left( \\rho^{\\lfloor t_1 \/q \\rfloor}\\wedge \\rho^{\\lfloor (t_2-t_1)\/q \\rfloor} \\wedge \\cdots \\wedge \\rho^{\\lfloor (t_k-t_{k-1})\/q \\rfloor} \\wedge \\rho^{\\lfloor (n-t_k)\/q \\rfloor} \\right) \\\\\n&= 2 \\sum_{t_1=0}^{n\/2}\\sum_{t_1\\leq t_2 \\leq \\cdots \\leq t_{k-1}\\leq t_k} \\sum_{t_k = t_1}^{n-t_1} \\left( \\rho^{\\lfloor (t_2-t_1)\/q \\rfloor} \\wedge \\cdots \\wedge \\rho^{\\lfloor (t_k-t_{k-1})\/q \\rfloor} \\wedge \\rho^{\\lfloor (n-t_k)\/q \\rfloor} \\right) \\\\\n& \\leq 2 \\sum_{t_1=0}^{n\/2}\\sum_{t_1\\leq t_2 \\leq \\cdots \\leq t_{k-1}\\leq t_k \\leq n} \\left( \\rho^{\\lfloor (t_2-t_1)\/q \\rfloor} \\wedge \\cdots \\wedge \\rho^{\\lfloor (t_k-t_{k-1})\/q \\rfloor} \\wedge \\rho^{\\lfloor (n-t_k)\/q \\rfloor} \\right), \n\\end{align*}\nwhere the first inequality holds by symmetry, and the subsequent equality follows from $n-t_k \\geq t_1$. From (\\ref{rho_k_bound}), the right-hand side is no larger than $q^{k-1} (k+1)! (1-\\rho)^{(1-k)} \\sum_{t_1=0}^{n\/2}\\rho^{\\lfloor (n-t_1)\/kq \\rfloor } \\leq q^{k} k(k+1)! (1-\\rho)^{-k}\\rho^{\\lfloor n\/2kq \\rfloor}$, giving the stated result.\n\\end{proof}\n\n\nThe next lemma generalizes equation (46) and p. 2294 of DMR, who derive a similar bound when $\\ell=1,2$.\n\\begin{lemma} \\label{hoelder}\nLet $a_j>0$ for all $j$. For all positive integer $\\ell \\geq 1$ and all $k\\geq 1$ and $m\\geq 0$, we have $\\max_{-m+1\\leq t_1, \\ldots, t_\\ell\\leq k} a_{t_1} \\cdots a_{t_\\ell} \\leq (k+m)^{\\ell+1} A_\\ell$, where $A_\\ell := \\sum_{t=-\\infty}^{\\infty} (|t|\\vee 1)^{-2} a_t^\\ell$.\n\\end{lemma}\n\\begin{proof}[Proof of Lemma \\ref{hoelder}]\nWhen $\\ell=1$, the stated result follows from $\\max_{-m+1\\leq t\\leq k} a_{t} \\leq \\sum_{t=-m+1}^k a_t = \\sum_{t=-m+1}^k (|t|\\vee 1)^2(|t|\\vee 1)^{-2} a_t \\leq (k+m)^2\\sum_{t=-\\infty}^\\infty(|t|\\vee 1)^{-2} a_t $. When $\\ell \\geq 2$, from H\\\"older's inequality, we have $\\max_{-m+1\\leq t_1\\leq \\ldots \\leq t_\\ell\\leq k} a_{t_1} a_{t_2} \\cdots a_{t_\\ell} \\leq (\\sum_{t=-m+1}^k a_t)^\\ell = [ \\sum_{t=-m+1}^k (|t|\\vee 1)^{2\/\\ell} (|t|\\vee 1)^{-2\/\\ell}a_t ] ^\\ell \\leq [ \\sum_{t=-m+1}^k (|t|\\vee 1)^{2\/(\\ell-1)} ]^{(\\ell-1)} \\sum_{t=-m+1}^k (|t|\\vee 1)^{-2} a_t^\\ell \\leq [(k+m)^{1+2\/(\\ell-1)}]^{\\ell-1} A_\\ell = (k+m)^{\\ell+1} A_\\ell$.\n\\end{proof}\n\nThe following lemma generalizes the bound derived on p. 2301 of DMR.\n\\begin{lemma} \\label{rho_m}\nFor $\\alpha>0$, $q >0$, and $c_{jt}\\geq0$, define $c_{jq}^\\infty(\\rho^{\\alpha}) := \\sum_{t=-\\infty}^{\\infty} \\rho^{\\lfloor \\alpha |t|\/q \\rfloor} c_{jt}$. For all $\\rho \\in (0,1)$, $k\\geq 1$, and $0\\leq m\\leq m'$,\n\\begin{equation} \\label{rho_6}\n\\begin{aligned}\n&\\sum_{t_1=-m'+1}^{-m}\\sum_{t_1 \\leq t_2 \\leq t_3 \\leq t_4\\leq t_5 \\leq t_6 \\leq k } \\left(\\rho^{\\lfloor (k-1-t_6)\/q \\rfloor} \\wedge \\rho^{\\lfloor (t_6-t_5)\/q \\rfloor} \\wedge \\rho^{\\lfloor (t_5-t_4)\/q \\rfloor} \\wedge\\rho^{\\lfloor (t_4-t_3)\/q \\rfloor} \\wedge \\right. \\\\\n& \\left. \\qquad \\rho^{\\lfloor (t_3 -t_2)\/q \\rfloor} \\wedge \\rho^{\\lfloor (t_2-t_1)\/q \\rfloor}\\right)\\prod_{j=1}^6 c_{jt_j} \\leq \\rho^{\\lfloor (k-1+m)\/2qa_7\\rfloor} c_{1q}^{\\infty}\\left(\\rho^{1\/2a_7}\\right) \\prod_{j=2}^6 c_{jq}^{\\infty}\\left(\\rho^{1\/4a_j}\\right),\n\\end{aligned}\n\\end{equation}\nwhere $(a_j,b_j)$ are defined recursively with $(a_2,b_2)=(1,1)$ and, for $j \\geq 3$,\n$a_{j+1} = 4a_j(a_j+b_j)\/(2a_j-1)$ and $b_{j+1} = a_j(4b_j-1)\/(2a_j-1)$.\n$a_{j}$ and $b_{j}$ satisfy $a_{j}, b_{j} \\geq 3\/2$ for all $j$. Direct calculations using Matlab produce $a_7 \\doteq 334.5406$.\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma \\ref{rho_m}]\nFirst, observe that the following result holds for $a,b>1\/4$, $t_1\\leq 0$, and $t_j, t_{j+1} \\geq t_1$:\n\\begin{equation} \\label{t_ab}\n\\begin{aligned}\n(a) & \\text{ if } t_j \\leq \\frac{at_{j+1}+t_1}{a+b}, \\quad \\text {then } \\frac{|t_j|}{4a} \\leq \\frac{a(4a+1)t_{j+1}+(2a-1)t_1}{4a(a+b)} - t_j, \\\\\n(b) & \\text{ if } t_j \\geq \\frac{at_{j+1}+t_1}{a+b}, \\quad \\text {then } \\frac{|t_j|}{4a} \\leq \\frac{b}{a}t_j - \\frac{a(4b-1)t_{j+1}+(2a+4b+1)t_1}{4a(a+b)}.\n\\end{aligned}\n\\end{equation}\n(a) holds because (i) when $t_j \\leq 0$, we have $t_j \\leq (at_{j+1}+t_1)\/(a+b) \\Rightarrow (4a-1)t_j\/4a \\leq [a(4a-1)t_{j+1}+(4a-1)t_1]\/4a(a+b) \\Rightarrow -t_j\/4a \\leq [a(4a-1)t_{j+1}+(4a-1)t_1]\/4a(a+b)-t_j$ and $a(4a-1)t_{j+1}+(4a-1)t_1 \\leq a(4a-1)t_{j+1}+(4a-1)t_1+2a(t_{j+1}-t_1) = a(4a+1)t_{j+1}+(2a-1)t_1$; (ii) when $t_j \\geq 0$, we have $t_j \\leq (at_{j+1}+t_1)\/(a+b) \\Rightarrow (4a+1)t_j\/4a \\leq [a(4a+1)t_{j+1}+(4a+1)t_1]\/4a(a+b) \\Rightarrow t_j\/4a \\leq [a(4a+1)t_{j+1}+(4a+1)t_1]\/4a(a+b)- t_j$ and $(4a+1)t_1 \\leq (2a-1)t_1$.\n\n(b) holds because (i) when $t_j \\leq 0$, we have $t_j \\geq (at_{j+1}+t_1)\/(a+b) \\Rightarrow (4b+1)t_j\/4a \\geq [a(4b+1)t_{j+1}+(4b+1)t_1]\/4a(a+b) \\Rightarrow -t_j\/4a \\leq bt_j\/a - [a(4b+1)t_{j+1}+(4b+1)t_1]\/4a(a+b)$ and $a(4b+1)t_{j+1}+(4b+1)t_1 \\geq a(4b+1)t_{j+1}+(4b+1)t_1 -2a(t_{j+1}-t_1) = a(4b-1)t_{j+1}+(2a+4b+1)t_1$; (ii) when $t_j \\geq 0$, we have $t_j \\geq (at_{j+1}+t_1)\/(a+b) \\Rightarrow (4b-1)t_j\/4a \\geq [a(4b-1)t_{j+1}+(4b-1)t_1]\/4a(a+b) \\Rightarrow t_j\/4a \\leq bt_j\/a - [a(4b-1)t_{j+1}+(4b-1)t_1]\/4a(a+b)$ and $a(4b-1)t_{j+1}+(4b-1)t_1 \\geq a(4b-1)t_{j+1}+(2a+4b+1)t_1$.\n\nWe proceed to derive the stated bound. It follows from (a) and (b) and $\\lfloor x + y \\rfloor \\geq \\lfloor x \\rfloor +\\lfloor y \\rfloor$ that, with $\\overline t_j = (a_jt_{j+1}+t_1)\/(a_j+b_j)$,\n\\begin{align}\n& \\sum_{t_j=-m'+1}^{k}\\left( \\rho^{\\lfloor (t_{j+1} -t_j)\/q \\rfloor} \\wedge \\rho^{\\lfloor (b_j t_j-t_1)\/a_jq \\rfloor}\\right) c_{jt_j} \\nonumber \\\\\n& \\leq \\rho^{\\lfloor \\frac{a_j(4b_j-1)t_{j+1} -(2a_j-1)t_1}{4a_j(a_j+b_j)q} \\rfloor} \\left( \\sum_{t_j\\leq\\overline t_j} \\rho^{\\lfloor\\frac{a_j(4a_j+1)t_{j+1}+(2a_j-1)t_1}{4a_j(a_j+b_j)q} - \\frac{t_j}{q} \\rfloor} + \\sum_{t_j\\geq\\overline t_j} \\rho^{\\lfloor \\frac{b_j}{a_jq}t_j - \\frac{a_j(4b_j-1)t_{j+1}+(2a_j+4b_j+1)t_1}{4a_j(a_j+b_j)q} \\rfloor}\\right) c_{jt_j}\\nonumber \\\\\n& \\leq \\rho^{\\lfloor \\frac{a_j(4b_j-1)t_{j+1} -(2a_j-1)t_1}{4a_j(a_j+b_j)q}\\rfloor} c_{jq}^{\\infty}\\left(\\rho^{1\/4a_j}\\right) \\nonumber \\\\\n& = \\rho^{\\lfloor\\frac{b_{j+1}t_{j+1} -t_1}{a_{j+1}}\\rfloor} c_{jq}^{\\infty} \\left(\\rho^{1\/4a_j}\\right). \\label{rho_j_sum}\n\\end{align}\nObserve that $a_{j+1} \\geq 2a_j \\geq 2$ and $b_{j+1} \\geq 2b_j - (1\/2) \\geq 3\/2$ for all $j \\geq 2$. Therefore, we can apply (\\ref{t_ab}) and (\\ref{rho_j_sum}) to the left-hand side of (\\ref{rho_6}) sequentially for $j=2,3,\\ldots,6$. Consequently, the left-hand side of (\\ref{rho_6}) is no larger than\n\\[\n\\sum_{t_1=-m'+1}^{-m}\\rho^{\\lfloor \\frac{b_7(k-1) -t_1}{a_7q} \\rfloor} c_{1t_1} \\prod_{j=2}^6 c_{jq}^{\\infty}\\left(\\rho^{1\/4a_j}\\right).\n\\]\nObserve that $|t_1|\\leq k-1-2t_1-m$ because $t_1 \\leq -m \\Rightarrow -t_1 \\leq -2t_1-m \\leq k-1-2t_1-m$. From $b_7(k-1) \\geq k-1 $ and $|t_1|\\leq k-1-2t_1-m$, the sum is bounded by\n\\[\n\\sum_{t_1=-m'+1}^{-m}\\rho^{\\lfloor \\frac{k-1 -t_1}{a_7 q} \\rfloor} c_{1t_1} = \\rho^{\\lfloor\\frac{k-1+m}{2a_7q}\\rfloor} \\sum_{t_1=-m'+1}^{-m}\\rho^{\\lfloor\\frac{k-1-2t_1-m}{2a_7q}\\rfloor} c_{1t_1} \\leq \\rho^{\\lfloor\\frac{k-1+m}{2a_7q}\\rfloor} c_{1q}^{\\infty}\\left(\\rho^{1\/2a_7}\\right),\n\\]\nand the stated result follows.\n\\end{proof}\n\n\\subsubsection{Derivation of $\\vartheta_{M_0+1,x} = (\\vartheta_{xm}',\\pi_{xm}')'$ and $\\pi_{xm}=(\\varrho_m,\\alpha_m,\\phi_m')'$}\\label{subsec:p_m_repara}\nDefine $\\overline J_{m0} := \\{1,\\ldots,M_0\\}\\setminus J_m$, and let $p_j$ and $p^*_j$ denote $\\mathbb{P}_{\\vartheta_{M_0+1}}(X_k=j)$ and $\\mathbb{P}_{\\vartheta_{M_0}^*}(X_k=j)$, respectively.\n\nWe parameterize the transition probability of $X_k$ in terms of its stationary distribution and the first to the $(m-1)$-th rows and the $(m+1)$-th to the $(M_0+1)$-th rows of its transition matrix.\\footnote{Suppose a Markov process has a transition probability $P$ and stationary distribution $\\pi$ whose elements are strictly positive. If $\\pi$ and all the rows of $P$ except for one are identified, then the remaining row of $P$ is identified from the relation $\\pi P = \\pi$.}\nFor $i \\in \\overline J_m$, we reparameterize $(p_{im},p_{i,m+1})$ to $p_{iJ} = p_{im}+p_{i,m+1}=\\mathbb{P}_{\\vartheta_{M_0+1}}(X_k \\in J_m|X_{k-1}=i)$ and $p_{im|iJ} = p_{im}\/(p_{im}+p_{i,m+1})$. Furthermore, we reparameterize $(p_m,p_{m+1})$ in the stationary distribution to $p_J = p_m + p_{m+1}=\\mathbb{P}_{\\vartheta_{M_0+1}}(X_k \\in J_m)$ and $p_{m|J}=p_m\/(p_m + p_{m+1})=\\mathbb{P}_{\\vartheta_{M_0+1}}(X_k=m|X_k \\in J_m)$. Therefore, with $\\land$ and $\\lor$ denoting ``and'' and ``or,'' the transition probability of $X_k$ is summarized by \n$\\vartheta_{M_0+1,x} :=( \\{p_{iJ},p_{im|iJ} \\}_{i \\in \\overline J_m}, \\{p_{ij}\\}_{i \\in \\overline J_m \\land j \\in \\overline J_{m0}}, \\{p_{m+1,j}\\}_{j=1}^{M_0}, \\{p_{j}\\}_{j \\in \\overline J_{m0}}, p_J, p_{m|J})$.\n\nSplit $\\vartheta_{M_0+1,x}$ as $\\vartheta_{M_0+1,x}=(\\vartheta_{xm}',\\pi_{xm}')'$, where $\\vartheta_{xm} := ( \\{p_{ij}\\}_{i \\in \\overline J_m \\land j \\in \\overline J_{m0}}, \\{p_{iJ} \\}_{i \\in \\overline J_m}, \\{p_{j}\\}_{j \\in \\overline J_{m0}}, p_J)$ and $\\pi_{xm} :=( \\{p_{im|iJ} \\}_{i \\in \\overline J_m}, \\{p_{m+1,j}\\}_{j=1}^{M_0}, p_{m|J})$.\nWhen the $m$-th and $(m+1)$-th regimes are combined into one regime, the transition probability of $X_k$ equals the transition probability of $X_k$ under $\\vartheta_{M_0,x}^*$ if and only if $\\vartheta_{xm} = \\vartheta_{xm}^*:= \\{p_{ij} = p_{ij}^{*}\\ \\text{for}\\ i \\in \\overline J_m \\land(1 \\leq j \\leq m-1); p_{ij} = p_{i,j-1}^{*}\\ \\text{for}\\ i \\in \\overline J_m \\land (m+2 \\leq j \\leq M_0); p_{iJ} = p_{im}^{*}\\ \\text{for } i\\in \\bar J_m;p_j = p_{j}^* \\ \\text{for}\\ 1 \\leq j \\leq m-1; p_j = p_{j-1}^* \\ \\text{for}\\ m+2 \\leq j \\leq M_0; p_J = p_m^* \\}$. $\\pi_{xm}$ is the part of $\\vartheta_{M_0+1,x}$ that is not identified under $H_{0m}$.\n\nWe proceed to derive the reparameterization of some elements of $\\pi_{xm}$ in terms of $(\\alpha_m,\\varrho_m)$. First, map $p_{m+1,m}$ and $p_{m+1,m+1}$ to $p_{m+1,J}:=p_{m+1,m}+p_{m+1,m+1}=\\mathbb{P}_{\\vartheta_{M_0+1}}(X_k \\in J|X_{k-1}=m+1)$ and $p_{m+1,m|J}:=p_{m+1,m}\/p_{m+1,J}=\\mathbb{P}_{\\vartheta_{M_0+1}}(X_k=m|X_k \\in J,X_{k-1}=m+1)$. Let $P_J$ and $\\pi_J$ denote the transition matrix and stationary distribution of $X_k$ restricted to lie in $J_m$. The second row of $P_J$ is given by $(p_{m+1,m|J},1-p_{m+1,m|J})$, and $\\pi_J$ is given by $(p_{m|J},1-p_{m|J})$. From the relation $\\pi_J= \\pi_J P_J$, we can obtain the first row of $P_J$ as a function of $p_{m+1,m|J}$ and $p_{m|J}$. Finally, the elements of $P_J$ are mapped to $(\\varrho_m,\\alpha_m)$ as in Section \\ref{sec: testing-1}.\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nThe variation of the specific intensity over a star's disk, commonly \ncalled limb darkening or center-to-limb variation, is an important \nfunction of the physical structure of a stellar atmosphere.  Because it \nis difficult to observe limb darkening for stars other than the \nSun, it is common to represent limb darkening by analytic expressions \nwhose coefficients are determined by matching the $I_\\lambda(\\mu)$ \nfrom model stellar atmospheres.  Here $\\mu = \\cos \\theta$, where \n$\\theta$ is the angle between the vertical direction at that point on \nthe stellar disk and the direction toward the distant observer.  \nThe most basic form of the law is the linear version, depending on \n$\\mu$ to the first power, such as \n\\begin{equation}\n\\frac{I_\\lambda(\\mu)}{I_\\lambda(\\mu = 1)} = \\mu,\n\\end{equation}\nor, more generally,\n\\begin{equation}\n\\frac{I_\\lambda(\\mu)}{I_\\lambda(\\mu = 1)} = 1 - u (1 - \\mu).\n\\end{equation}\nLimb-darkening laws have subsequently become more complex by including \nterms with $\\mu$, or its alternative $r = \\sin \\theta$, raised to \nhigher integer powers as well as both $\\sqrt{\\mu}$ and $\\log(\\mu)$ \nterms \\citep{2000A&A...363.1081C, Howarth2010}.  In addition to minimizing the fit \nto the model's $I_\\lambda(\\mu)$, the laws have also introduced \nvarious constraints, such as enforcing flux conservation at each \nobserved wavelength.  The intricacies of limb-darkening laws have \nincreased as model stellar atmospheres have advanced.\n\n\\citet{2000PhDT........11H,2003ApJ...594..464H,2007ApJ...656..483H}, \nmotivated by the potential of gravitational microlensing to provide \nparticularly detailed measurements of stellar limb darkening, \ninvestigated the optimum method of extracting limb darkening from the \ndata.  One curious feature that emerged from these studies is the \nexistence of a fixed $\\mu$ location on the stellar disk where the \nfits to the normalized model intensities have nearly the same value \nindependent of wavelength or the model's $T_\\mathrm{eff}$ or \n$\\log g$.  This is true for both the principal-component \nanalysis developed by \\citet{2003ApJ...594..464H} and the more \ntraditional linear limb-darkening law used by others.\n\n\\citet{2003ApJ...594..464H} used models computed with the \n\\textsc{Atlas12} code \\citep{1996IAUS..176..523K} to construct his \nfitting procedure.  These models use detailed opacity sampling to \ninclude many tens of millions of spectral lines in the radiative \ntransfer, but they still assumed plane-parallel geometry, even though \n\\citet{2003ApJ...594..464H} targeted red giants with \n$T_\\mathrm{eff} \\leq 4000$ K and $\\log g \\leq 1.0$.  These stellar \nparameters are exactly those where the assumption of plane-parallel \ngeometry should break down.  In the study by \n\\citet{2003A&A...412..241C}, which did use the spherical \nmodels of \\citet{1999ApJ...525..871H}, the focus was on models with \n$\\log g \\geq 3.5$, for which spherical extension is minimal.  A \nbroader study of limb darkening and the fixed $\\mu$-point using \nspherically extended model atmospheres is clearly needed.\n\n\n\\section{\\textsc{SAtlas} model atmospheres} \\label{sec:satlas_models}\n\n\\citet{2008A&A...491..633L} have developed the \\textsc{SAtlas} code, \na spherically extended version of \\textsc{Atlas}.  This code shares \nwith \\textsc{Atlas} the properties of static pressure structure, LTE \npopulations and massive line blanketing represented by either opacity \ndistribution functions or opacity sampling (the faster opacity \ndistribution function version is used here).  The spherical program \ndiffers from the plane-parallel \\textsc{Atlas} by allowing the gravity \nto vary with radial position, and by computing the radiative transfer \nalong rays through the atmosphere in the direction of the distant \nobserver using the \\citet{1971JQSRT..11..589R} version of the \n\\citet{1964CR...258..3189F} method, which accounts for the radial \nvariation of the angle between the vertical and the direction of the \nray.  The structure of the atmosphere is computed using a total of 81 \nrays whose angular spacing is determined by two factors.  The first is \nthe distribution of rays chosen to represent the ``core'' of the \natmosphere, the region where the lower boundary condition for radiative \ntransfer is the diffusion approximation.  \\citet{2008A&A...491..633L} \nfound that different distributions of the core rays had no effect on \nthe structure of the atmosphere, and so elected to use equal spacings \nin $\\mu$.  The remainder of the rays are tangent to the atmospheric \nlevels at the stellar radius perpendicular to the central ray toward \nthe observer.  These rays are projections of the radial spacing of the \natmospheric levels, which is logarithmic in the Rosseland mean optical \ndepth.  Because this distribution of rays is set by calculating the \nstructure of the atmosphere, it is not necessarily optimal for studying \nlimb darkening.  Therefore, after computing the structure of the \natmosphere, the surface intensities are derived from the structure rays \nby cubic spline interpolation for any desired number of rays with any \ndesired distribution over the disk.  As \\citet{2007ApJ...656..483H} \nhas demonstrated, cubic spline provides an excellent interpolation \nmethod.  We have created our surface intensities at 1000 points equally \nspaced over $0 \\leq \\mu \\leq 1$.\n\n\\section{Fixed point of stellar limb-darkening laws} \n\\label{sec:fixed_pt}\n\n\\citet{2000PhDT........11H} found the fixed point in the limb-darkening \nprofiles of both the Sun and in plane-parallel \\textsc{Atlas} models of \nred giants.  \\citet{2003ApJ...596.1305F} found a similar fixed point \nin their analysis of a K3 giant using spherical \\textsc{Phoenix} models,\nalthough they excluded the limb from their intensity analysis because \nthe low intensity near the limb contributed almost nothing to the \nobservations they were analyzing.  Their truncation point ranged from \n$r = 0.998$ for $\\log g = 3.5$, corresponding to $\\mu = 0.063$, to \n$r = 0.88$ for $\\log g = 0.0$, corresponding to $\\mu = 0.475$.  However,\nthe truncation eliminated that part of the surface brightness that \ndeviates most strongly from the plane-parallel model.  \n\\citet{2003ApJ...596.1305F} stated that the fixed point is a generic \nfeature of any single-parameter limb-darkening law that conserves flux. \nFor example, if the limb darkening is represented as \n$f(\\mu) \\equiv I(\\mu)\/F = 2[1 + Ax(\\mu)]$, the fixed point is \n$\\mu_\\mathrm{fixed} = f^{-1} \n[2 \\int_0^1 f(\\mu') \\mu' \\mathrm{d} \\mu']$.  \nAlthough they also state that the fixed point is not required by the \ntwo-parameter law they employed, they found a fixed point in the \nmicrolensing observations they were modeling using such a law.\n\nTo explore the parametrization more closely, we begin with the same \ntwo-parameter normalized limb-darkening function used by \n\\citet{2003ApJ...596.1305F}, \n\\begin{equation} \\label{eq:fields}\n\\frac{I(\\mu)}{2\\mathcal{H}} \n= 1 - A \\left (1 - \\frac{3}{2}\\mu \\right )\n    - B \\left (1 - \\frac{5}{4}\\sqrt{\\mu} \\right ),\n\\end{equation}\nwhere $\\mathcal{H}$ in the Eddington flux, defined as\n\\begin{equation}\n\\mathcal{H} \\equiv \\frac{1}{2} \\int_{-1}^{1} I(\\mu) \\mu \\mathrm{d} \\mu\n            = \\int_{0}^{1} I(\\mu) \\mu \\mathrm{d} \\mu.\n\\end{equation}\nA fixed point requires, for any two arbitrary models, that the \nintensity profiles satisfy\n\\begin{equation}\nI_1(\\mu_0) = I_2(\\mu_0),\n\\end{equation}\nor in terms of Eq.~\\ref{eq:fields}\n\\begin{eqnarray} \\label{eq:fields_fix}\n\\nonumber &&1 - A_1 \\left (1 - \\frac{3}{2} \\mu_0 \\right )\n           - B_1 \\left (1 - \\frac{5}{4} \\sqrt{\\mu_0} \\right ) = \\\\ \n&& \\mbox{\\hspace{1.5cm}}1 - A_2 \\left (1 - \\frac{3}{2} \\mu_0 \\right )\n           - B_2 \\left (1 - \\frac{5}{4} \\sqrt{\\mu_0} \\right ).\n\\end{eqnarray}\nFor this to be true, $A$ and $B$ must be linearly dependent, \n$A = \\alpha B + \\beta$ or $A_1-A_2 = \\alpha (B_1-B_2)$.  Substituting \nthis relation into Eq.~\\ref{eq:fields_fix} yields an equation \nfor the fixed-point \n\\begin{equation} \\label{eq:fp}\n\\frac{3}{2} \\alpha \\mu_0 + \\frac{5}{4} \\sqrt{\\mu_0}  -1 - \\alpha = 0.\n\\end{equation}\nWe need to verify that $A = f(B)$ and to understand the properties of \nthe parameter $\\alpha$.\n\nApplying a general least-squares method to Eq.~\\ref{eq:fields},\n\\begin{equation}\n\\chi^2 = \\sum_i^N \\left[\\frac{I(\\mu_i)}{2\\mathcal{H}} - 1 \n         + A\\left(1-\\frac{3}{2}\\mu_i\\right) \n         + B \\left(1 - \\frac{5}{4} \\sqrt{\\mu_i}\\right) \\right]^2,\n\\end{equation}\nwe determine the coefficients $A$ and $B$ from the constraint equations \n\\begin{eqnarray} \\label{eq:lsa}\n \\nonumber \\frac{\\partial \\chi^2}{\\partial A}& = &\n \\sum_i^N\\left[\\frac{I(\\mu_i)}{2\\mathcal{H}} - 1 \n+ A\\left(1-\\frac{3}{2}\\mu_i\\right) \n+ B \\left(1 - \\frac{5}{4} \\sqrt{\\mu_i}\\right) \\right]  \\\\\n&& \\times    \\left(1 - \\frac{3}{2}\\mu_i\\right) = 0\n\\end{eqnarray}\nand \n\\begin{eqnarray} \\label{eq:lsb}\n \\nonumber \\frac{\\partial \\chi^2}{\\partial B}& =&\n\\sum_i^N \\left[\\frac{I(\\mu_i)}{2\\mathcal{H}} - 1 \n+ A\\left(1-\\frac{3}{2}\\mu_i\\right) \n+ B \\left(1 - \\frac{5}{4} \\sqrt{\\mu_i}\\right) \\right] \\\\\n&& \\times    \\left(1 - \\frac{5}{4} \\sqrt{\\mu_i}\\right) = 0.\n\\end{eqnarray}\nMultiplying Eq.~\\ref{eq:lsa} and Eq.~\\ref{eq:lsb} by \n$\\Delta \\mu$ and then converting the summation to integration, \n$\\int_{0}^{1} \\mathrm{d} \\mu$, we create the equations that determine \n$A$ and $B$,\n\\begin{equation}\n\\label{eq:lsa2}\n\\frac{J}{2\\mathcal{H}} + \\frac{1}{4}A + \\frac{1}{6}B - 1 = 0\n\\end{equation}\nand\n\\begin{equation} \\label{eq:lsb2}\n- \\frac{5}{4} \n\\left[\\frac{1}{2\\mathcal{H}} \\int_0^1 I(\\mu) \\sqrt{\\mu} \\mathrm{d} \\mu \n\\right] + \n\\frac{J}{2\\mathcal{H}} + \\frac{1}{6}A + \\frac{11}{96}B - \\frac{1}{6} \n= 0,\n\\end{equation}\nwhere $J$ is the usual mean intensity.  It is also useful to define an \nangular pseudo-moment of the intensity, \n\\begin{equation} \\label{eq:def_P}\n\\mathcal{P} \\equiv \\int_0^1 I(\\mu) \\sqrt{\\mu} \\mathrm{d}\\mu,\n\\end{equation}\nwhich allows Eq.~\\ref{eq:lsb2} to be written as\n\\begin{equation} \\label{eq:lsb3}\n- \\frac{5}{4} \\left[ \\frac{\\mathcal{P}}{2\\mathcal{H}} \\right ] + \n\\frac{J}{2\\mathcal{H}} + \\frac{1}{6}A + \\frac{11}{96}B - \\frac{1}{6} \n= 0.\n\\end{equation}\nBoth $J$ and $\\mathcal{P}$ are \ndetermined by the properties of the model stellar atmosphere or the \nstar whose observations are being fit by a limb-darkening law. \nEquation~\\ref{eq:lsa2} and Eq.~\\ref{eq:lsb2} clearly show that the \ncoefficients $A$ and $B$ are uniquely determined by $J$ and \n$\\mathcal{P}$.  However, for $A$ and $B$ to be linearly dependent, \nthat is $A = \\alpha B + \\beta$, $J$ and $\\mathcal{P}$ must also be \nlinearly related.\n\nThe relation of $J$ to $\\mathcal{P}$ can be understood by following \nthe discussion of the diffusion approximation in \n\\citet{1978stat.book.....M}.  This begins by representing the source \nfunction by a power series, which leads to the intensity being given by \n$I_\\nu(\\tau_\\nu, \\mu) = \\sum_{n = 0}^{\\infty} \\mu^n \n\\left [\\mathrm{d}^n B_\\nu(T) \/ \\mathrm{d}\\tau_\\nu^n \\right]$\n(Mihalas 1978, Eq. 2-88).  Using this expansion in the definitions of \n$J$ and $\\mathcal{P}$, and keeping just the first-order term, gives the \nfamiliar results that $J_{\\nu} \\approx B_{\\nu}$ and \n$K_{\\nu} \\approx B_{\\nu}\/3$ plus the additional result that \n$\\mathcal{P}_{\\nu} \\approx 2 B_{\\nu}\/3$.  Eliminating $B_{\\nu}$ between \nthe $J_{\\nu}$ and $\\mathcal{P}_{\\nu}$ expressions gives \n\\begin{equation} \\label{eq:P_J}\n\\mathcal{P}_{\\nu} = 2J_{\\nu}\/3,\n\\end{equation}\nconfirming the desired correlation.  Of course, as the atmosphere thins \nout toward space the diffusion approximation becomes less accurate, but, as will be seen, \nthe basic correlation of $\\mathcal{P}$ and $J$ is still there.\n\nUsing Eq.~\\ref{eq:P_J} we can combine Eq.~\\ref{eq:lsa2} and \nEq.~\\ref{eq:lsb2} to find that\n\\begin{equation} \\label{eq:a_fn_b}\nA = -0.694 B.\n\\end{equation}\nIn terms of the notation used earlier, $\\alpha = -0.694$ and \n$\\beta = 0$, or equivalently $\\Delta A = -0.694 \\Delta B$.  Using this \nvalue for $\\alpha$ in Eq.~\\ref{eq:fp} leads to the equation for the \nfixed point, \n\\begin{equation}\n1.084 \\mu_0^2 - 0.925 \\mu_0 + 0.094 = 0.\n\\end{equation}\nThis quadratic equation yields \\emph{two} fixed points, not one.  The \nsolutions are $\\mu_1 = 0.736$, corresponding to \n$\\theta_1 = 42.61^{\\degr}$ and $r_1 = 0.677$, and $\\mu_2 = 0.118$, \ncorresponding to $\\theta_2 = 83.22^{\\degr}$ and $r_2 = 0.993$.\n\nThe results above used the diffusion approximation to establish the \nrelationship between $J$ and $\\mathcal{P}$.  To generalize these \nresults, we define a new variable, $\\eta$, as \n\\begin{equation} \\label{eq:eta_def}\n\\eta \\equiv \\frac{\\int I \\sqrt{\\mu} \\mathrm{d} \\mu}{J} \n          = \\frac{\\mathcal{P}}{J}.\n\\end{equation}\nNote that $\\eta$ is a generalization of the relation in \nEq.~\\ref{eq:P_J}.  Using the definition of $\\eta$, we replace \nthe variable $\\mathcal{P}$ in Eq.~\\ref{eq:lsb3} by $\\eta J$, and \nthen Eq.~\\ref{eq:lsa2} is used to eliminate the $J\/2\\mathcal{H}$ \nterms.  The resulting equation is rearranged to the form $A = f(B)$,\nwhich is a linear equation of the same form as the previous equation \nfor $A$, namely $A = \\alpha B + \\beta$.  Comparing the two equations \nfor $A$ we identify \n\\begin{equation} \\label{eq:alpha}\n\\alpha = - \\frac{(5 \\eta \/ 4 - 1)\/6 + 11\/96}\n              {(5 \\eta \/ 4 - 1)\/4 + 1\/6}.\n\\end{equation}\nEquation~\\ref{eq:alpha} enables us to explore how $\\alpha$ \nchanges with $\\eta$.  In addition, because Eq~\\ref{eq:fp} is a \nfunction of $\\alpha$ only, we are also able to determine the \ndependence of the two fixed points, $\\mu_1$ and $\\mu_2$, on $\\eta$.\nThese dependencies are shown in Fig.~\\ref{fig:alpha_mu_eta}.  Because \nthe value of $\\eta$ can be set to values other than 2\/3, the \nassumption of the diffusion approximation is no longer being used.\n\\begin{figure} \n  \\resizebox{\\hsize}{!}{\\includegraphics{16623fg1.eps}}\n  \\caption{Top panel: dependence of the slope of the function \n           $A = \\alpha B + \\beta$ on the variable \n           $\\eta = J^{-1} \\int I \\sqrt{\\mu} \\mathrm{d} \\mu$.\n           Bottom panel: variation of the fixed points with\n           $\\eta$.\n           \\label{fig:alpha_mu_eta}\n          }\n\\end{figure}\n\nFigure~\\ref{fig:alpha_mu_eta} shows that, except for \n$\\eta \\approx 0.267$, the value of $\\alpha$ is in the narrow range \n$-0.72 \\lesssim \\alpha \\lesssim -0.68$, which leads to the fixed points \nbeing $0.72 \\lesssim \\mu_1 \\lesssim 0.74$ and \n$0.08 \\lesssim \\mu_2 \\lesssim 0.14$.  The divergence of $\\alpha$ to \n$\\pm \\infty$ as $\\eta$ approaches $\\approx 0.267$ is due to the \ndenominator of Eq.~\\ref{eq:alpha} approaching zero as \n$\\eta \\rightarrow 4\/15$.  However, $\\eta$ can never equal $4\/15$.  This \nis a consequence of the following inequalities, which are true because \n$\\mu \\leq 1$:\n\\begin{equation}\nJ = \\int I \\mathrm{d} \\mu \\geq \\int I \\sqrt{\\mu} \\mathrm{d} \\mu \n                           \\geq \\int I \\mu^2 \\mathrm{d} \\mu = K.\n\\end{equation}\nDividing through by $J$ this becomes\n\\begin{equation} \\label{eq:eta_lims}\n1 \\geq \\eta \\geq \\frac{K}{J}.\n\\end{equation}\nDeep in the atmosphere, where the diffusion approximation holds, \n$K\/J = 1\/3$.  Moving toward the surface the atmosphere becomes more \ntransparent and the radiation becomes less isotropic, with the \nconsequence that $K\/J > 1\/3$.  As a result, \n\\begin{equation} \\label{eq:eta_llim}\n\\eta \\geq 1\/3 > 4\/15 = 0.267,\n\\end{equation}\nand the divergence cannot occur.  Equation~\\ref{eq:alpha} can be used \nto set bounds on $\\alpha$ by using the lower bound on $\\eta$ from \nEq.~\\ref{eq:eta_llim} and the upper bound on $\\eta$ from \nEq.~\\ref{eq:eta_lims}.  The result is  $-5\/6 \\leq \\alpha \\leq -15\/22$.\nUsing this range of $\\alpha$ in Eq.~\\ref{eq:fp}, we find that the \nranges for the fixed points are $0.708 \\leq \\mu_1 \\leq 0.741$ and \n$0.025 \\leq \\mu_2 \\leq 0.130$, which are consistent the values for \n$\\mu_1$ and $\\mu_2$ found previously.  This indicates that the fixed \npoints exist in the limb-darkening relations and are stable to \ndifferences in the intensity profile of the stellar atmosphere.  It \nalso suggests that the fixed points are not a result of a particular \ndominant opacity source.  The only requirement is that $\\eta > 1\/3$.\n\nWe verify this analytic result by computing a grid of \\textsc{Atlas}\nplane-parallel model stellar atmospheres with \n$3000 \\ \\mathrm{K} \\leq T_\\mathrm{eff} \\leq 8000 \\ \\mathrm{K}$ and  \n$-2 \\leq \\log g \\leq 3$.  For this grid of models, the mean value of \n$\\eta$ in the $V-$band is \n$\\bar{\\eta}_\\mathrm{pp} = 0.716\\pm 0.004$ which leads to \n$\\alpha_\\mathrm{pp} = 0.691 \\pm 0.001$, $\\mu_1 = 0.7375\\pm 0.0002$ and\n$\\mu_2 = 0.1200\\pm 0.0003$.  This is very similar to what is found \nassuming the diffusion approximation.  The difference between the model \n$V-$band $\\eta$ and the value of $\\eta$ from the diffusion \napproximation is due to the fact that plane-parallel models do not \nenforce the diffusion approximation at all depths in the atmosphere.  \nTo determine the dependence on wavelength, we used the plane-parallel \ngrid of atmospheres to determine the mean value and 1-$\\sigma$ standard \ndeviation of $\\eta$, $\\alpha$ and the two fixed points $\\mu_{1}$ and \n$\\mu_{2}$, with the results shown in Fig.~\\ref{fig:wl_pp}. The values \nof the fixed points clearly do not vary significantly as a function of \nwavelength nor as a function of effective temperature and gravity.  \nFurthermore, it must be noted that the value of $\\eta$ approaches $2\/3$ \nas $\\lambda \\rightarrow \\infty$.\n\\begin{figure}[t]\n  \\begin{center}\n     \\resizebox{\\hsize}{!}{\\includegraphics{16623fg2.eps}}\n  \\end{center}\n  \\caption{Wavelength dependence of the mean values of \n           (a) the ratio of the pseudo-moment to the mean intensity,\n               $\\eta$,\n           (b) the value of $\\alpha$ that defines the fixed points,\n           (c) the first and\n           (d) second fixed point for the grid of plane-parallel \n               \\textsc{Atlas} model atmospheres. The error bars \n               represent 1-$\\sigma$ deviation from the mean.\n           \\label{fig:wl_pp}\n          }\n\\end{figure}\n\nThese results can be generalized to show that the fixed point occurs in \nother limb-darkening laws, such as a quadratic law where\n$I\/2\\mathcal{H} = 1 - A(1 - \\mu) - B(1 -2 \\mu^2)$. Repeating the \nderivation above, the pseudo-moment, $\\mathcal{P}$, is replaced by $K$, \na higher angular moment of the intensity.  Because the diffusion \napproximation analysis also yields a linear relation between $J$ and \n$K$, $J = 3K$, we again find that $A = f(B)$.  For a more general law, \nsuch as $I\/2\\mathcal{H} = 1 - A (1 - \\mu) - B[1 - \\frac{1}{2}(n+2)\\mu^n]$, we would \nfind a similar connection between the coefficients $A$ and $B$.  We \nwould even find fixed points for any law that is a combination of a \nlinear term and any function that can be represented by a power-law \nseries, such as $e^{\\mu}$, similar to the limb-darkening law tested by \\cite{2003A&A...412..241C}.  Therefore, we conclude that the fixed \npoints are a general property of this family of limb-darkening laws.\n\n\n\\section{Limb darkening and \\textsc{SAtlas} model atmospheres}\n\\label{sec:ld_satlas}\n\nTo conduct a broader survey of limb darkening, we have computed several \nlarger cubes of solar-composition model atmospheres, increasing the \nrange of luminosity, mass and radius as well as using microturbulent \nvelocities of 0, 2 and 4 km\/s.  The luminosities, masses and radii of \nthese models cover a significant portion of the Hertzsprung-Russell \ndiagram where stars have extended atmospheres, as shown in \nFig.~\\ref{fig:ld_hr}, which also shows the evolutionary tracks of \n\\citet{2000A&AS..141..371G} for comparison.\n\\begin{figure}\n   \\resizebox{\\hsize}{!}{\\includegraphics{16623fg3.eps}}\n   \\caption{Models computed with the \\textsc{SAtlas} code plotted over \n            the stellar evolution tracks from \n            \\citet{2000A&AS..141..371G}.\n            \\label{fig:ld_hr}\n           }\n\\end{figure}\n\\begin{figure*}[t]\n   \\resizebox{\\hsize}{!}{\\includegraphics{16623fg4a.eps}\n                         \\includegraphics{16623fg4b.eps}}\n\n   \\caption{Left panel: Limb darkening, characterized by \n            the normalized surface intensity, $I(\\mu)\/2\\mathcal{H}$, \n            computed for the cube of 2102 model atmospheres for \n            $v_\\mathrm{turb} = 2$ km\/s for the $BVRI$ and $H$-bands.  \n            Right panel: Fit to these same intensities \n            using the limb-darkening law \n            \\mbox{$I(\\mu)\/2\\mathcal{H} = 1 - A (1 - \\frac{3}{2} \\mu)\n               - B (1 - \\frac{5}{4} \\sqrt{\\mu})$}\n            for  the same wavebands.\n            \\label{fig:ld_cubev2_mu}\n           }\n\\end{figure*}\nThe number of models in each cube is large.  For example, there were \n2101 models in the cube with $v_\\mathrm{turb} = 2$ km\/s and the cubes \nfor the other values of $v_\\mathrm{turb}$ have similar numbers of \nmodels.  For each model in each cube we computed the intensity at 1000 \nequally spaced $\\mu$ points at all wavelengths from the far ultraviolet \nto the far infrared; the results are shown in the left column of \nFig.~\\ref{fig:ld_cubev2_mu} for the $B,V,R,I$ and $H$-bands.  These intensity curves have similar behavior as the \\textsc{Phoenix} model atmosphere intensity curves that \\cite{Orosz2000} studied.\nWe also fit each of the model limb-darkening curves with the \nlinear-plus-square-root limb darkening law given by \nEq.~\\ref{eq:fields}, and this is shown in the right column of \nFig.~\\ref{fig:ld_cubev2_mu}.  Note that the $A$ and $B$ coefficients of \nthis parametrization have been shown in Eq.~\\ref{eq:lsa2} and \nEq.~\\ref{eq:lsb2} to be functions of $J$ and $\\mathcal{P}$, not the \nstructure of the intensity profile.  The results for the \n$v_\\mathrm{turb} = 0$ and $4$ km\/s cubes of models are very similar.  This is not surprising; in the \\textsc{Atlas} codes, the turbulent pressure is $P_{\\rm{turb}} = \\rho v_{\\rm{turb}}^2$, indicating that the turbulent velocity has a similar effect on the temperature structure of a stellar atmosphere as the gravity, that is a change of turbulent velocity is equivalent to a change of gravity, similar to the discussion from \\cite{Gustafsson2008} for the MARCS code. In terms of the best-fit relations, a change of turbulent velocity causes a small change for the coefficients in the same manner as a change in gravity for the same effective temperature and mass.  Therefore, we can just explore the grid of models atmospheres for only one value of $v_{\\rm{turb}}$.\nThe cubes of models used in this survey cover a much wider range of \natmospheric parameters than was used in the only previous \ninvestigation using spherical model stellar atmospheres \n\\citep{2003A&A...412..241C}, which also truncated the profiles at the \nlimb to achieve better fits.\n\nThe model intensities in the left column of Fig.~\\ref{fig:ld_cubev2_mu} \nare always positive, but the intensities in some of the parametrized \nfits in the right column become slightly negative toward the limb.  \nThis does not happen for plane-parallel model atmospheres because the \nslope of the intensity profile ($\\partial I\/\\partial \\mu$ or \n$\\partial I\/\\partial r$) has a constant sign.  For example, using \n$\\mu$ as the independent variable the slope of the intensity profile is \nalways positive, varying from zero at the center of the disk to \n$\\infty$ as $\\mu \\rightarrow 0$. For spherical atmospheres, the \ncenter-to-limb variation changes sign because of a slight inflection, \nwhich is apparent in the curves of the left column of \nFig.~\\ref{fig:ld_cubev2_mu}.  The more complex intensity profile and \nthe use of an equal-weighting $\\chi^2$-fit (i.e. the same number of \n$\\mu$-points near the center as near the limb) leads to some slightly \nnegative intensities in some best-fit relations.  \n\n\n\nFigure~\\ref{fig:ld_cubev2_mu} shows that the actual limb darkening \ncomputed from the spherical models is more complex than the \nparametrized representation, and it is obvious that the analysis of \nspecific observations that contain intensity information should be \ncautious about using the parametrized fits.  Although fits to \nplane-parallel models appear to be very good, the models are less \nrealistic physically than the spherical models.  Therefore, using what \nseems like a well-fitting law may introduce hidden errors that could \ncompromise the conclusions of the analysis. The quantities derived \nfrom observations of binary eclipses and planetary transits may be more \nuncertain than had been thought \n\\citep[see, for example,][]{2007ApJ...655..564K, 2007A&A...467.1215S, \n2008MNRAS.386.1644S}.\n\nUsing the same grid of spherical \\textsc{Atlas} models used in \nFig.~\\ref{fig:ld_cubev2_mu}, we computed the mean values and 1-$\\sigma$ \nstandard deviations of $\\eta$, $\\alpha$ and the fixed points as a \nfunction of wavelength.  The results are shown in Fig.~\\ref{fig:wl_sp}, \nwhich can be compared with the plane-parallel results shown in \nFig.~\\ref{fig:wl_pp}.  There are two key differences between the \nplane-parallel and spherical models. The first is that the mean values \nof $\\eta$ are larger for the spherical models, which leads to \nthe differences between the mean values of $\\alpha$ and the two fixed \npoints.  The second difference is that the standard deviation for \n$\\eta$ as a function of wavelength is much larger for the spherical \nmodels than for the plane-parallel models.  This suggests that $\\eta$ \nvaries much more as a function of effective temperature, gravity and \nmass. Clearly, the fixed point is much more constrained in \nplane-parallel model stellar atmospheres. \n\n\\begin{figure}[t]\n\\begin{center}\n \\resizebox{\\hsize}{!}{\\includegraphics{16623fg5.eps}}\n\\end{center}\n\\caption{Wavelength dependence of the mean values of\n         (a) the ratio of the pseudo-moment to the mean intensity,\n             $\\eta$,\n         (b) the value of $\\alpha$ that defines the fixed points,\n         (c) the first and \n         (d) second fixed point for the grid of spherically symmetric \n         \\textsc{Atlas} model atmospheres. The error bars represent \n         1-$\\sigma$ deviation from the mean.\n         \\label{fig:wl_sp}\n        }\n\\end{figure}\n\nA specific example of an application of limb darkening to an \nobservation is the analysis by \\citet{2003ApJ...596.1305F} of \nmicrolensing observations.  In Fig.~\\ref{fig:vld}, we compare the \n$V$-band limb-darkening relations from our cube of spherical models \nwith the $V$-band limb-darkening relation determined by \n\\citet{2003ApJ...596.1305F} from observations.  It is clear that the \n\\citet{2003ApJ...596.1305F} $V$-band limb-darkening law agrees well \nwith the limb-darkening from the spherical models, but, not as well with the plane-parallel models.  Furthermore, \nalthough the curves for the model stellar atmospheres narrow to a \nwaste rather than to a point, the location of minimum spread coincides \nwith the location of the observed $\\mu$-position in the $V$-band \nlimb-darkening relation within the uncertainty of the observed \ncoefficients.  Even though the uncertainty of limb-darkening relations \nfrom microlensing observations is large, these results suggest that the \n\\citet{2003ApJ...596.1305F} limb-darkening observations are probing the \nextended the atmosphere of the lensed red giant star.\n\\begin{figure}\n   \\resizebox{\\hsize}{!}{\\includegraphics{16623fg6.eps}}\n   \\caption{Comparison of the $V$-band limb-darkening relation \n            determined by \\citet{2003ApJ...596.1305F} from \n            observations (black dashed curves with errorbars) with the limb-darkening \n            relations computed from \\textsc{SAtlas} model stellar \n            atmospheres (red solid curves) and relations computed from \\textsc{Atlas} model stellar atmospheres (blue dotted curves), represented using the same \n            linear-plus-square-root law.  The insert shows a magnified \n            view of the region of the fixed $\\mu$-point.\n            \\label{fig:vld}\n           }\n\\end{figure}\n\\begin{figure}\n  \\resizebox{\\hsize}{!}{\\includegraphics{16623fg7.eps}}\n  \\caption{Red plus symbols represent the limb-darkening coefficients $A$ \n           and $B$ computed from \\textsc{SAtlas} model intensity \n           profiles for the $V$, $I$, and $H$-bands, and green open squares \n            represent the coefficients computed with \n           plane-parallel \\textsc{Atlas} models.\n           The box in each plot shows the range of the \n           coefficients derived from microlensing observations by \n           \\citet{2003ApJ...596.1305F}. \n           \\label{fig:abbox}\n          }\n\\end{figure}\nWe conclude that the parametrized laws, although more simplified than \nthe computed limb-darkening curves, are useful for understanding how \nlimb darkening depends on the fundamental properties of the stellar \natmosphere.\n\nTo explore further the information content of the limb darkening, we \nrecall that Eq.~\\ref{eq:a_fn_b} showed that the coefficients $A$ and \n$B$ of the parametrization are linearly correlated.  In \nFig.~\\ref{fig:abbox} we plot the limb-darkening coefficients $A$ and \n$B$ for the $V$, $I$, and $H$-bands computed with both the \nplane-parallel \\textsc{Atlas} and the spherical \\textsc{SAtlas} models.\nFor each wavelength the error box shows the range of $A$ and $B$ found \nby \\citet{2003ApJ...596.1305F} from their observations.  For the \n$V$-band, the spherical limb-darkening coefficients overlap with the \nobserved fit, but the plane-parallel coefficients do not, while for \n$H$-band both plane-parallel and spherical models agree with \nobservations.  The microlensing observations at the longer wavelength \nmight not be sensitive enough to probe the low intensity limb of the \nstar, making the star appear consistent with limb-darkening of the \nplane-parallel model atmospheres.   In the $I$-band neither set of \nmodels agree with the observations, but the $I$-band data provide a \nweak constraint.  As \\citet{2003ApJ...596.1305F} noted, the $I$-band \ntime-series microlensing observations were a composite from multiple \nsites, and removing data from any one site changed the results \nsignificantly.\n\\begin{figure*}\n  \\resizebox{\\hsize}{!}{\\includegraphics{16623fg8a.eps}\n                        \\includegraphics{16623fg8b.eps}}\n  \\caption{Left: As a function of surface gravity, the top panel \n           shows the value of the primary fixed point, $\\mu_1$, for \n           the linear plus square root parametrization, for spherical \n           atmospheres of varying effective temperature. \n           The bottom panel shows the dependence of the normalized \n           V-band intensity of the fixed point.  \n           Red fiilled circles are $T_\\mathrm{eff} = 3000$ K, green open squares \n           4000 K, blue open circles 5000 K, magenta downward pointing \n           triangles 6000 K and pale blue upward pointing triangles are \n           7000 K.  The black crosses show the behavior of the fixed point from the grid of plane-parallel model atmospheres for comparison\n           Right: As a function of effective temperature, the top \n           panel shows the fixed point, $\\mu_1$, and the bottom panel \n           shows the dependence of the V-band intensity of the fixed \n           point. \n           Red triangles represents models with $\\log~g = 0$, green squares \n           $\\log~g = 1$, and blue circles are $\\log~g= 2$. Again, black crosses represent the fixed point from plane-parallel model atmospheres.\n           \\label{fig:gravity_dep}\n          }\n\\end{figure*}\n\nIn the $V$-band, the limb-darkening coefficients from 15 model \natmospheres fall within the observational box.  These models have \n$\\log~g = 2.25 - 3$ and $T_\\mathrm{eff} = 3400 - 3600$ K.  For the \n$H$-band data there are four models within the observational box; \nthese have same range of gravities but are slightly cooler, \n$T_\\mathrm{eff} = 3000 - 3100$ K.   It is interesting that the models \nthat agree with the observations are those with gravities that are \nconsistent with the results of \\cite{2003ApJ...596.1305F} and \n\\cite{2002ApJ...572..521A}.  This suggests that observations using the \nlimb-darkening parametrization can probe the spherical extension of \nstellar atmosphere via the fixed point, $\\mu_1$, and potentially probe \nthe fundamental parameters of stars via the limb-darkening coefficients.\n\nUsing a cube of models with a given value of $v_\\mathrm{turb}$, we \nexamine the dependence of the primary fixed point, $\\mu_1$, on the \neffective temperature and surface gravity.  Note again that there are \nthree basic parameters characterizing the spherical atmospheres: \n$L_\\star$, $M_\\star$ and $R_\\star$.  This means that values of \n$T_\\mathrm{eff}$ and $\\log g$ are degenerate, but they are easier to \nshow on a two-dimensional surface.  In the top left panel of \nFig.~\\ref{fig:gravity_dep} we plot the value of the primary fixed \npoint, $\\mu_1$, as a function of $\\log g$.\nAt each value of $\\log g$ there are values of $T_\\mathrm{eff}$ \nranging from 3000 to 7000 K, although there can be multiple values \nbecause of the parameter degeneracy.  In the bottom left panel of \nFig.~\\ref{fig:gravity_dep} we plot \n$I_\\mathrm{V}(\\mu_1)\/2\\mathcal{H}_\\mathrm{V}$, the normalized \nintensity of the fixed point in the $V$-band.  For each surface \ngravity the values of $\\mu_1$ and the normalized intensity at the \nfixed point both show a steady progression as $T_\\mathrm{eff}$ \nchanges, except for  $T_\\mathrm{eff} = 3000$ K.  We suspect that the \nbehavior for $T_\\mathrm{eff} = 3000$ K is due to a change in the \ndominant opacity source for our coolest models, possibly water vapor.  Models with \n$T_\\mathrm{eff} > 3000$ K have H$^-$ as the dominant continuous \nopacity, but in the coolest models there are fewer free electrons \navailable to form H$^-$, and the formation of H$_2$ reduces the pool of \nhydrogen atoms.  We also plot the fixed point and intensity at the fixed point from plane-parallel model atmospheres for comparison.  It is clear that the fixed point from fits to plane-parallel models varies much less than fits to spherical models.\n\nOn the right side of Fig.~\\ref{fig:gravity_dep} we reverse the \nparameters and plot the dependence of the primary fixed point and the \nnormalized intensity of the fixed point as a function of \n$T_\\mathrm{eff}$.  At each value of the effective temperature, the \nvalues of $\\log g$ are 0, 1 and 2 in cgs units.  In the top right panel \nof Fig.~\\ref{fig:gravity_dep} we see that there is essentially no \nvariation of the value of $\\mu_1$ with $T_\\mathrm{eff}$ for all three \nsurface gravities until the lowest effective temperature is reached.  \nThere is an obvious displacement of the value of $\\mu_1$ for each \ngravity and also a spread in $\\mu_1$ because of parameter degeneracy.  \nHowever, for $T_\\mathrm{eff} \\leq 3500$ K the value of $\\mu_1$ drops \nfor all surface gravities.  The bottom right panel shows that the \nnormalized $V$-band intensity of the fixed point shows a similar \nbehavior.  For $T_\\mathrm{eff} > 3500$ K there is little variation \nwith $T_\\mathrm{eff}$, but there is an offset and a spread that \ndepends on the surface gravity.  Cooler than 3500 K the value of the \nnormalized intensity drops noticeably.\n\nIt is clear that the fixed point $\\mu_1$ and the intensity at the fixed point $I(\\mu_1)\/2\\mathcal{H}$ are functions of gravity and effective temperature for spherically-symmetric models and are roughly constant for plane-parallel model atmospheres.  This indicates that the best-fit coefficients of the limb-darkening law vary mostly because of the geometry of the model atmosphere.  A spherical model atmosphere predicts a smaller intensity near the limb of the stellar disk relative to a plane-parallel model with the same effective temperature and gravity.  To predict the same emergent flux, the intensity must be larger at the center of the disk, hence the temperature at the base of the atmosphere must also be larger for the spherical model.  Therefore, the temperature structure of the atmosphere also varies.  However, this is a secondary effect and the geometry of the atmosphere is most important in determining the value of the $\\mu_1$ and $I(\\mu_1)\/2\\mathcal{H}$.  The geometry of spherical models leads to smaller values of the pseudo-moment and the mean intensity because the intensity is more centrally concentrated.  This suggests that  $\\mu_1$ and $I(\\mu_1)\/2\\mathcal{H}$ depend on the atmospheric extension. We will explore how the fixed point and intensity relate to the extension and fundamental stellar properties in greater detail in a future article.\n\n\\section{Conclusions}\n\nWe have explored limb darkening using large cubes of spherical stellar \natmospheres spanning the parameters $L_\\star$, $M_\\star$ and $R_\\star$ \ncovering the cool, luminous quadrant of the Hertzsprung-Russell \ndiagrams (Fig.~\\ref{fig:ld_hr}).  These models have also used three \ndifferent values of the microturbulent velocity.  For each model, the \ncenter-to-limb variation of the surface intensity has been calculated \nat 1000 equally spaced $\\mu$ values spanning the range from 1 to 0 for \nevery wavelength used to compute the model structure.\n\nParametrizing the center-to-limb variation with a flux-conserving \nlinear-plus-square-root limb-darkening law, we confirm the findings of \n\\citet{2000PhDT........11H} and \\citet{2003ApJ...596.1305F} that there \nis a fixed $\\mu_1$ point through which all the intensity curves pass.\nHowever, when we plot the surface intensities directly, without using \na fitting law, there is no fixed point, although the distribution of \ncurves does narrow to a waist close to the same value of $\\mu_1$ \n(Fig.~\\ref{fig:ld_cubev2_mu}).\n\nThe apparent fixed point is a result of the least-squares fitting \nprocedure where the two parameters of the law are dependent on two \nproperties of the stellar atmosphere, the mean intensity, $J$, and the \npseudo-moment, $\\int I \\sqrt{\\mu} \\mathrm{d} \\mu$.  For the temperature \nrange $4000 - 8000$ K, the mean intensity is correlated with the \npseudo-moment, which means that the two coefficients are also \ncorrelated, leading to the existence of the fixed point.\n\nThe lack of a well-defined fixed point in the surface intensity \ndistribution for spherical model atmospheres suggests that the three \nfundamental parameters of the atmospheres affect the limb darkening in \na way that is not encountered in the two-parameter plane parallel \natmospheres.\n\n\\acknowledgements\n\nThis work has been supported by a research grant from the Natural \nSciences and Engineering Research Council of Canada.  HRN has received \nfinancial support from the Walter John Helm OGSST, the Walter C. \nSumner Memorial Fellowship, and the Alexander von Humboldt Foundation.\n\n\\bibliographystyle{aa}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:intro}\n\n\nThe non-trivial zeros of automorphic $L$-functions are of central significance in modern number theory. Problems on individual zeros, such as the Riemann Hypothesis (GRH), are elusive. There is however a beautiful theory about the statistical distribution of zeros in families. The subject has a long and rich history. A unifying modern viewpoint is that of a comparison with a suitably chosen model of random matrices (\\key{Katz--Sarnak heuristics}). There are both theoretical and numerical evidences for this comparison. Comprehensive results in the function field case~\\cite{book:KS} have suggested an analogous picture in the number field case as explained in~\\cite{KS:bams}. In a large number of cases, and with high accuracy, the distribution of non-trivial zeros of automorphic $L$-functions coincide with the distribution of eigenvalues of random matrices (see~\\cites{DFK:vanishingunitary,Rubin:computational} for numerical investigations and conjectures and see~\\cites{DM06,Guloglu:lowlying,HM07,ILS00,KST:Sp4,RR:low-lying,Rubin01} and the references herein for theoretical results).\n\nThe concept of \\key{families} is central to modern investigations in number theory. We want to study in the present paper certain families of automorphic representations over number fields in a very general context. The families under consideration are obtained from the discrete spectrum by imposing constraints on the local components at archimedean and non-archimedean places and by applying Langlands global functoriality principle.\n\nOur main result is a Sato--Tate equidistribution theorem for these families (Theorem~\\ref{t:intro:error-bound}). An application of this main result we can give some evidence towards the Katz-Sarnak heuristics~\\cite{KS:bams} in general and establish a criterion for the random matrix model attached to families (Symmetry type).\n\n\\subsection{Sato-Tate theorem for families}\\label{sec:intro:ST}\n\n  The original Sato-Tate conjecture is about an elliptic curve $E$, assumed to be defined over ${\\mathbb Q}$\n  for simplicity. The number of points in $E(\\mathbb{F}_p)$ for almost all primes $p$ (with good reduction)\n  gives rise to an angle $\\theta_p$ between $-\\pi$ and $\\pi$. The conjecture, proved in~\\cite{BLGHT11}, asserts that\n  if $E$ does not admit complex multiplication then $\\{\\theta_p\\}$ are equidistributed according to\n  the measure $\\frac{2}{\\pi} \\sin^2\\theta d\\theta$.\n  In the context of motives a generalization of the Sato-Tate conjecture was formulated by Serre~\\cite{Ser94}.\n\n  To speak of the automorphic version of the Sato-Tate conjecture, let $G$ be a connected split reductive group over ${\\mathbb Q}$\n  with trivial center\n  and $\\pi$ an automorphic representation of $G({\\mathbb A})$. Here $G$ is assumed to be split for simplicity\n  (however we stress that our results are valid\n  without even assuming that $G$ is quasi-split, if formulated appropriately). The triviality of center is not serious\n  as it essentially amounts to fixing central character.\n  Let $T$ be a maximal split torus of $G$. Denote by $\\widehat{T}$ its dual torus and $\\Omega$ the Weyl group.\n  As $\\pi=\\otimes'_v \\pi_v$ is unramified at almost all places $p$, the Satake isomorphism identifies\n  $\\pi_p$ with a point on $\\widehat{T}\/\\Omega$. The automorphic Sato-Tate conjecture should be a prediction\n  about the equidistribution of $\\pi_p$ on $\\widehat{T}\/\\Omega$ with respect to a natural measure\n  (supported on a compact subset of $\\widehat{T}\/\\Omega$). It seems nontrivial\n  to specify this measure in general. The authors do not know how to do it\n  without invoking the (conjectural) global $L$-parameter for $\\pi$.\n  The automorphic Sato-Tate conjecture is known in the very limited cases of (the restriction of scalars of)\n  $\\GL_1$ and $\\GL_2$ (\\cite{BLGHT11}, \\cite{BLGG11}). In an ideal world the conjecture should be closely related to Langlands functoriality, see~\\S\\ref{sub:gen-ST-conj}.\n\n  In this paper we consider the Sato-Tate conjecture for a \\emph{family} of automorphic representations, which\n  is easier to state and prove but still very illuminating.\n  Our working definition of a family $\\{\\mathcal{F}_k\\}_{k\\ge 1}$ is that each $\\mathcal{F}_k$ consists of\n  all automorphic representations $\\pi$ of $G({\\mathbb A})$ of level $N_k$ with $\\pi_\\infty$ cohomological of weight $\\xi_k$,\n  where $N_k\\in{\\mathbb Z}_{\\ge1}$ and $\\xi_k$ is an irreducible algebraic representation of $G$, such that either\n\\begin{enumerate}\n  \\item (level aspect) $\\xi_k$ is fixed, and $N_k\\rightarrow \\infty$ as $k\\rightarrow\\infty$ or\n  \\item (weight aspect) $N_k$ is fixed, and $\\dim \\xi_k\\rightarrow\\infty$ as $k\\rightarrow\\infty$.\n\\end{enumerate}\n    Note that each $\\mathcal{F}_k$ has finite cardinality and $|\\mathcal{F}_k|\\rightarrow \\infty$\n    as $k\\rightarrow\\infty$. (For a technical reason $\\mathcal{F}_k$ is actually allowed to be a multi-set. Namely\n    the same representation can appear multiple times, for instance more than its automorphic multiplicity.)\n    In principle we could let $\\xi_k$ and $N_k$ vary simultaneously but decided not to do so in the current paper\n    in favor of transparency of arguments. For instance families of type (i) and (ii) require somewhat different ingredients of proof in establishing the Sato-Tate theorem for families, and the argument would be easier to understand if we separate them.\n    It should be possible to treat the mixed case (where both $N_k$ and $\\xi_k$ vary) by combining techniques in the\n    two cases (i) and (ii).\n\n\n\n    Let $\\widehat{T}_c$ be the maximal compact subtorus of the complex torus $\\widehat{T}$.\n    The quotient $\\widehat{T}_c\/\\Omega$ is equipped with a measure $\\widehat{\\mu}^{\\mathrm{ST}}$, to be called\n\tthe \\key{generalized Sato-Tate measure}, coming from the Haar measure on a maximal compact subgroup of $\\widehat{G}$\n    (of which $\\widehat{T}_c$ is a maximal torus).\n    The following is a rough version of our result on the Sato-Tate conjecture for a family.\n\n\\begin{theorem}\\label{Sato-Tate}\n  Suppose that $G({\\mathbb R})$ has discrete series representations.\n  Let $\\{\\mathcal{F}_k\\}_{k\\ge 1}$ be a family of the first (resp. second) type as above.\n  Let $\\{p_k\\}$ be a sequence of\n  primes such that $p_k$ grows to $\\infty$ slowly relative to $N_k$ (resp. $\\xi_k$)\n  as $k\\rightarrow \\infty$.\n  Assume that the members of $\\mathcal{F}_k$ are unramified at $p_k$ for every $k$.\n  Then the Satake parameters $\\{\\pi_{p_k}: \\pi\\in \\mathcal{F}_k\\}_{k\\ge 1}$\n  are equidistributed with respect to $\\widehat{\\mu}^{\\mathrm{ST}}$.\n\n\\end{theorem}\n\n  To put things in perspective, we observe that there are three kinds of statistics about the Satake parameters\n  of $\\{\\pi_{p_k}: \\pi\\in \\mathcal{F}_k\\}_{k\\ge 1}$ depending on how the arguments vary.\n\\begin{enumerate}\n  \\item Sato-Tate: $\\mathcal{F}_k$ is fixed (and a singleton) and $p_k\\rightarrow \\infty$.\n  \\item Sato-Tate for a family: $|\\mathcal{F}_k|\\rightarrow \\infty$ and $p_k\\rightarrow \\infty$.\n  \\item Plancherel: $|\\mathcal{F}_k|\\rightarrow \\infty$ and $p_k$ is a fixed prime.\n\\end{enumerate}\nThe Sato-Tate conjecture in its original form is about equidistribution in case (i) whereas\nour Theorem \\ref{Sato-Tate} is concerned with case (ii).\nThe last item is marked as Plancherel since\nthe Satake parameters are expected to be equidistributed with respect to the Plancherel measure\n(again supported on $\\widehat{T}_c\/\\Omega$) in case (iii). This has been shown to be true under the assumption\nthat $G({\\mathbb R})$ admits a discrete series in \\cite{Shi-Plan}.\nWe derive Theorem \\ref{Sato-Tate} from an error estimate (depending on $k$) on the difference between\nthe Plancherel distribution at $p$ and the actual distribution of the Satake parameters at $p_k$\nin $\\mathcal{F}_k$. This estimate (see Theorem \\ref{t:intro:error-bound} below) refines the main result of \\cite{Shi-Plan}\nand is far more difficult to prove in that several nontrivial bounds in\nharmonic analysis on reductive groups need to be justified.\n\n\n\\subsection{Families of $L$-functions}\\label{sec:intro:familyL}\nAn application of Theorem~\\ref{Sato-Tate} is to families of $L$-functions. We are able to verify to some extent the heuristics of Katz--Sarnak~\\cite{KS:bams} and determine the Symmetry type, see~\\S\\ref{sec:intro:criterion} below. In this subsection we define the relevant families of $L$-functions and record some of their properties.\n\nLet $r:{}^{L} G \\to \\GL(d,\\mathbb{C})$ be a continuous $L$-homomorphism. We assume Langlands functoriality principle: for all $\\pi \\in \\mathcal{F}_k$ there exists an isobaric automorphic representation $\\Pi=r_* \\pi$ of $\\GL(d,\\mathbb{A})$ which is the functorial lift of the automorphic representation $\\pi$ of $G(\\mathbb{A})$, see~\\S\\ref{s:langlands} for the precise statement of the hypothesis. This hypothesis is only used in Theorem~\\ref{th:low-lying}, \\S\\ref{sec:zeros} and \\S\\ref{sec:pf}. By the strong multiplicity one theorem $\\Pi$ is uniquely determined by all but finitely many of its local factors $\\Pi_v=r_* \\pi_v$ (see~\\S\\ref{sec:pp:isobaric} for a review of the concept of isobaric representations).\n\nTo an automorphic representation $\\Pi$ on $\\GL(d,\\mathbb{A})$ we associate its principal $L$-function $L(s,\\Pi)$. By definition $L(s,\\pi,r)=L(s,\\Pi)$. By the theory of Rankin--Selberg integrals or by the integral representations of Godement--Jacquet, $L(s,\\Pi)$ has good analytic properties: analytic continuation, functional equation, growth in vertical strips. In particular we know the existence and some properties of its non-trivial zeros, such as the Weyl's law (\\S\\ref{sec:pp:explicit}).\n\nWe denote by $\\mathfrak{F}_k=r_*\\mathcal{F}_k$ the set of all such $\\Pi=r_* \\pi$ for $\\pi \\in \\mathcal{F}_k$.\nSince the strong multiplicity one theorem implies that $\\Pi$ is uniquely determined by  its $L$-function $L(s,\\Pi)$. We simply refer to $\\mathfrak{F}=r_*\\mathcal{F}$  as a \\key{family of $L$-functions}.\n\n\nIn general there are many ways to construct interesting families of $L$-functions. In a recent letter~\\cite{Sarn:family}, Sarnak attempts to sort out these constructions into a comprehensive framework and proposes a working definition (see also~\\cite{Kowalski:family-survey}). The families of $L$-functions under consideration in the present paper fit well into that framework. Indeed they are \\key{harmonic families} in the sense that their construction involves inputs from local and global harmonic analysis. Other types of families include geometric families constructed as Hasse--Weil $L$-functions of arithmetic varieties and Galois families associated to families of Galois representations.\n\n\n\n\\subsection{Criterion for the Symmetry type}\\label{sec:intro:criterion} Katz--Sarnak~\\cite{KS:bams} predict that one can associate a \\key{Symmetry type} to a family of $L$-functions. By definition the Symmetry type is the random matrix model which governs the distribution of the zeros. There is a long and rich history for the introduction of this concept.\n\nHilbert and P\\'olya suggested that there might be a spectral interpretation of the zeros of the Riemann zeta function. Nowadays a strong evidence for the spectral nature of the zeros of $L$-functions comes from the function field case: zeros are eigenvalues of the Frobenius acting on cohomology. This is exemplified by the equidistribution theorem of Deligne and the results of Katz--Sarnak~\\cite{book:KS} on the distribution of the low-lying eigenvalues in geometric families.\n\nIn the number field case the first major result towards a spectral interpretation is the pair correlation of high zeros of the Riemann zeta function by Montgomery. Developments then include Odlyzko's extensive numerical study and the determination of the $n$-level correlation by Hejhal and Rudnick--Sarnak~\\cite{RS96}. The number field analogue of the Frobenius eigenvalue statistics suggested by~\\cite{book:KS} is concerned with the statistics of low-lying zeros.\n\nIndeed~\\cite{KS:bams} predicts that the low-lying zeros of families of $L$-functions are distributed according to a determinantal point process associated to a random matrix Ensemble. This will be explained in more details in~\\S\\ref{sec:intro:rdm} and~\\S\\ref{sec:intro:lowlying} below. We shall distinguish between the three determinantal point processes associated to the Unitary, Symplectic and Orthogonal Ensembles. Accordingly the Symmetry type associated to a family $\\mathfrak{F}$ is defined to be Unitary, Symplectic or Orthogonal (see~\\S\\ref{sec:intro:lowlying} for typical results). \\footnote{In this paper we do not distinguish in the Orthogonal Ensemble between the $O$ $SO(odd)$ and $SO(even)$ Symmetries. We will return to this question in a subsequent work.}\n\n\nBefore entering into the details of this theory in~\\S\\ref{sec:intro:rdm} below, we state here our criterion for the Symmetry type of the families $r_*\\mathcal{F}=\\mathfrak{F}$ defined above. We recall in~\\S\\ref{sec:b:fs} the definition of the Frobenius--Schur indicator $s(r)\\in \\set{-1,0,1}$ associated to an irreducible representation. We shall prove that the Symmetry type is determined by $s(r)$. This is summarized in the following which may be viewed as a refinement of the Katz--Sarnak heuristics.\n\n\\begin{criterion}\\label{criterion} Let $r:{}^LG\\to \\GL(d,\\mathbb{C})$ be a continuous homomorphism which is irreducible and non-trivial when restricted to $\\widehat G$. Consider the family $r_*\\mathcal{F}$ of automorphic $L$-functions of degree $d$ as above.\n\n(i) If $r$ is not isomorphic to its dual $r^\\vee$ then $s(r)=0$ and the Symmetry type is Unitary.\n\n(ii) Otherwise there is a bilinear form on $\\mathbb{C}^d$ which realizes the isomorphism between $r$ and $r^\\vee$. By Schur lemma it is unique up to scalar and is either symmetric or alternating. If it is symmetric then $r$ is real, $s(r)=1$ and the Symmetry type is Symplectic. If it is alternating then $r$ is quaternionic, $s(r)=-1$ and the Symmetry type is Orthogonal.\n\\end{criterion}\n\nWe note that the conditions that $r$ be irreducible and non-trivial when restricted to $\\widehat G$ are optimal. If $r$ were reducible then the $L$-functions $L(s,\\pi,r)$ would factor as a product of $L$-functions from independent families and their zeros would be distributed according to the superposition of independent point processes. If $r$ were trivial when restricted to $\\widehat G$ then $L(s,\\pi,r)$ would be constant and equal to an Artin $L$-function and the low-lying zeros would correspond to the eventual vanishing of this Artin $L$-function at the central point (which is a very different problem). Our criterion says that conversely under these conditions on $r$ the Katz--Sarnak heuristics hold to some extent.\n\nIt would be interesting to study families of automorphic representations over a function field $k=\\mathbb{F}_q(X)$ of a curve $X$. To our knowledge the Katz-Sarnak heuristics for such families are not treated in the literature, except in the case of $G=GL(1)$ where harmonic families coincide with the geometric families treated by Katz--Sarnak (e.g. Dirichlet $L$-series with quadratic character are the geometric families of hyperelliptic curves in~\\cite{book:KS}*{\\S10}). Over function fields our criterion has the following interpretation. We consider families of automorphic representations $\\pi$ of $G(\\mathbb{A}_k)$; for simplicity we suppose that $\\pi$ is attached to an irreducible $\\ell$-adic representation $\\rho: {\\rm Gal}(k^{sep}\/k) \\to {}^LG$. Then $r_*\\pi$ is attached to the Galois representation $r\\circ \\rho$, and corresponds to a constructible $\\ell$-adic sheaf $F$ of dimension $d$ on $X$.\nThe zeros of the $L$-function $L(s,\\pi,r)$ are the eigenvalues of Frobenius on the first cohomology, more precisely the numerator of the $L$-function $L(s,\\pi,r)$ is\n\\[\n\\det(1-{\\rm Fr} q^{-s} | H^1(X,F)).\n\\]\nIf $s(r)=-1$ (resp. $s(r)=1$) then there is a symplectic (resp. orthogonal) pairing on $F$. The natural pairing on $H^1(X,F)$ induced by the cup-product is orthogonal (resp. symplectic) and invariant by the action of Frobenius. Thus the zeros of $L(s,\\pi,r)$ are eigenvalues of an orthogonal (resp. symplectic) matrix; this is consistent with assertion (ii) in our criterion.\n\nKnown analogies between $L$-functions and their symmetries over number fields and function fields are discussed in~\\cite{KS:bams}*{\\S4}. Overall we would like to propose our criterion and its potential extension to more general families as a positive answer to the question mark in the entry 6-A of the Table~2 in~\\cite{KS:bams}.\n\n\\subsection{Automorphic Plancherel density theorem with error bounds}\\label{sec:intro:aut-Plan}\n  We explain a more precise version of the theorem and method of proof for the Sato-Tate theorem\n  for families (\\S\\ref{sec:intro:ST}). The key is to bound the error terms when\n  we approximate the distribution of local components of automorphic representations in a family\n  with the Plancherel measure.\n\n  For simplicity of exposition let us assume that $G$ is a split reductive group over ${\\mathbb Q}$ with trivial center\n  as in \\S\\ref{sec:intro:ST}.\n  A crucial hypothesis is that $G({\\mathbb R})$ admits an ${\\mathbb R}$-anisotropic maximal torus (in which case\n  $G({\\mathbb R})$ admits discrete series representations).\n  Let $\\mathcal{A}_{\\mathrm{disc}}(G)$ denote the set of isomorphism classes of discrete automorphic\n  representations of $G({\\mathbb A})$. We say that\n  $\\pi\\in \\mathcal{A}_{\\mathrm{disc}}(G)$ has level $N$ and weight $\\xi$ if\n  $\\pi$ has a nonzero fixed vector under the adelic version of the full level $N$ congruence subgroup\n  $K(N)\\subset G({\\mathbb A}^\\infty)$ and if $\\pi_\\infty\\otimes \\xi$ has nonzero Lie algebra cohomology.\n  In this subsection we make a further simplifying hypothesis that every $\\xi$ has regular highest weight, in which case\n  $\\pi_\\infty$ as above must be a discrete series. (In the main body of this paper, the latter assumption on $\\xi$ is necessary only for the results in \\S\\S\\ref{sub:Plan-density}-\\ref{sub:general-functions-S_0}, where more general test functions are considered.)\n\n  Define $\\mathcal{F}=\\mathcal{F}(N,\\xi)$ to be the finite multi-set consisting of $\\pi\\in \\mathcal{A}_{\\mathrm{disc}}(G)$\n  of level $N$ and weight $\\xi$, where each such $\\pi$ appears in $\\mathcal{F}$\n  with multiplicity $$a_{\\mathcal{F}}(\\pi):= \\dim (\\pi^\\infty)^{K(N)}\\in {\\mathbb Z}_{\\ge 0}.$$\n  This quantity naturally occurs as the dimension of the $\\pi$-isotypical subspace in the cohomology\n  of the locally symmetric space for $G$ of level $N$ with coefficient defined by $\\xi$.\n  The main motivation for allowing $\\pi$ to appear $a_{\\mathcal{F}}(\\pi)$ times is to enable us\n  to compute the counting measure below with the trace formula.\n\n  Let $p$ be a prime number. Write $G({\\mathbb Q}_p)^{\\wedge}$ for the unitary dual of\n  irreducible smooth representations of $G({\\mathbb Q}_p)$. The unramified (resp. unramified and tempered)\n  part of $G({\\mathbb Q}_p)^{\\wedge}$ is denoted $G({\\mathbb Q}_p)^{\\wedge,\\mathrm{ur}}$\n   (resp. $G({\\mathbb Q}_p)^{\\wedge,\\mathrm{ur},\\mathrm{temp}}$). There is a canonical isomorphism\n\\begin{equation}\n\\label{e:intro:spec-satake} G({\\mathbb Q}_p)^{\\wedge,\\mathrm{ur},\\mathrm{temp}}\\simeq \\widehat{T}_c\/\\Omega.\n\\end{equation}\n  The unramified Hecke algebra of $G({\\mathbb Q}_p)$ will be denoted $\\mathcal{H}^{\\mathrm{ur}}(G({\\mathbb Q}_p))$.\n  There is a map from $\\mathcal{H}^{\\mathrm{ur}}(G({\\mathbb Q}_p))$ to the space of continuous functions\n  on $\\widehat{T}_c\/\\Omega$\n  $$\\phi\\mapsto \\widehat{\\phi}\\quad \\mbox{determined by}\\quad \\widehat{\\phi}(\\pi)=\\mathrm{tr}\\, \\pi(\\phi),\n  ~\\forall \\pi\\in G({\\mathbb Q}_p)^{\\wedge,\\mathrm{ur},\\mathrm{temp}}.$$\n  There are two natural measures supported on\n  $ \\widehat{T}_c\/\\Omega$.\n  The Plancherel measure $\\widehat{\\mu}^{\\mathrm{pl,\\mathrm{ur}}}_p$, dependent on $p$, is\n  defined on $G({\\mathbb Q}_p)^{\\wedge,\\mathrm{ur}}$ and naturally arises\n   in local harmonic analysis. The generalized Sato-Tate measure\n  $\\widehat{\\mu}^{\\mathrm{ST}}$ on $ \\widehat{T}_c\/\\Omega$ is independent of $p$ and may be\n  extended to $G({\\mathbb Q}_p)^{\\wedge,\\mathrm{ur}} $ by zero.\n   Both $\\widehat{\\mu}^{\\mathrm{pl,\\mathrm{ur}}}_p$\n  and $\\widehat{\\mu}^{\\mathrm{ST}}$ assign volume 1 to  $ \\widehat{T}_c\/\\Omega$.\n  There is yet another measure $\\widehat{\\mu}^{\\mathrm{count}}_{\\mathcal{F},p}$\n  on $G({\\mathbb Q}_p)^{\\wedge,\\mathrm{ur}}$, which is the averaged counting measure for the $p$-components of members of $\\mathcal{F}$.\n  Namely\n  \\begin{equation}\n  \\label{e:count-meas}\\widehat{\\mu}^{\\mathrm{count}}_{\\mathcal{F},p}:=\n  \\frac{1}{|\\mathcal{F}|} \\sum_{\\pi\\in \\mathcal{F}}  \\delta_{\\pi_p}\n\\end{equation}\n  where $\\delta_{\\pi_p}$ denotes the Dirac delta measure supported at $\\pi_p$. (Each\n  $\\pi$ contributes $a_{\\mathcal{F}}(\\pi)$ times to the above sum.)\n  Our primary goal is to bound the difference between $\\widehat{\\mu}^{\\mathrm{pl},\\mathrm{ur}}_p$\n  and $\\widehat{\\mu}^{\\mathrm{count}}_{\\mathcal{F},p}$. (Note that our definition\n  of $\\widehat{\\mu}^{\\mathrm{count}}_{\\mathcal{F},p}$ in the main body will be a little different from\n  \\eqref{e:count-meas} but asymptotically the same. See Remark \\ref{r:|F|}. We take \\eqref{e:count-meas} here only to ease exposition.)\n\n\n  In order to quantify error bounds, we introduce a filtration\n  $\\{\\mathcal{H}^{\\mathrm{ur}}(G({\\mathbb Q}_p))^{\\le \\kappa}\\}_{\\kappa\\in{\\mathbb Z}_{\\ge0}}$\n  on $\\mathcal{H}^{\\mathrm{ur}}(G({\\mathbb Q}_p))$ as a complex vector space. The filtration is increasing, exhaustive\n  and depends on a non-canonical choice.\n  Roughly speaking, $\\mathcal{H}^{\\mathrm{ur}}(G({\\mathbb Q}_p))^{\\le \\kappa}$ is like the span of\n  all monomials of degree $\\le \\kappa$ when $\\mathcal{H}^{\\mathrm{ur}}(G({\\mathbb Q}_p))$ is identified with\n  (a subalgebra of) a polynomial algebra.\n  For each $\\xi$, it is possible to assign a positive integer $m(\\xi)$ in terms of the highest weight of $\\xi$.\n  When we say that weight is going to infinity, it means that $m(\\xi)$ grows to $\\infty$ in the usual sense.\n\n  The main result on error bounds alluded to above is the following.\n  (See Theorems \\ref{t:level-varies} and \\ref{t:weight-varies}.) A uniform bound on orbital integrals, cf. \\eqref{e:intro-unif-bound} below, enters the proof of (ii) (but not (i)) and requires the assumption $p\\gg 1$ in (ii) to ensure that $G(\\mathbb{Q}_p)$.\n\n\\begin{theorem}\\label{t:intro:error-bound}\n  Let $\\mathcal{F}=\\mathcal{F}(N,\\xi)$ be as above.\n  Consider a prime $p$, $\\kappa\\ge 1$, and $\\phi_p\\in \\mathcal{H}^{\\mathrm{ur}}(G({\\mathbb Q}_p))^{\\le \\kappa}$\n  such that $|\\phi_p|\\le 1$ on $G({\\mathbb Q}_p)$.\n\\begin{enumerate}\n  \\item (level aspect) Suppose that $\\xi$ remains fixed. There exist constants\n   $A_{\\mathrm{lv}},B_{\\mathrm{lv}},C_{\\mathrm{lv}}>0$ such that for any $p$, $\\kappa$, $\\phi_p$  as above and for any $N$ coprime to $p$\n  (with an extra condition that $N$ is bounded below by some inverse power of $p$),\n  $$ \\widehat{\\mu}^{\\mathrm{count}}_{\\mathcal{F},p}(\\widehat{\\phi}_p)-\\widehat{\\mu}^{\\mathrm{pl,\\mathrm{ur}}}_p(\\widehat{\\phi}_p)\n  = O(p^{A_{\\mathrm{lv}}+B_{\\mathrm{lv}}\\kappa}N^{-C_{\\mathrm{lv}}}).$$\n    \\item (weight aspect) Fix a level $N$. There exist constants\n    $A_{\\mathrm{wt}},B_{\\mathrm{wt}},C_{\\mathrm{wt}}>0$ such that for any $p\\gg 1$, $\\kappa$, $\\phi_p$ as above with $(p,N)=1$ and for any $\\xi$,\n  $$ \\widehat{\\mu}^{\\mathrm{count}}_{\\mathcal{F},p}(\\widehat{\\phi}_p)-\\widehat{\\mu}^{\\mathrm{pl,\\mathrm{ur}}}_p(\\widehat{\\phi}_p)\n  = O(p^{A_{\\mathrm{wt}}+B_{\\mathrm{wt}}\\kappa}m(\\xi)^{-C_{\\mathrm{wt}}}).$$\n\\end{enumerate}\n\n\\end{theorem}\n\n  Let $\\{ \\mathcal{F}_k=\\mathcal{F}(N_k,\\xi_k)\\}_{\\ge1}$ be\n  either kind of family in \\S\\ref{sec:intro:ST}, namely either $N_k\\rightarrow \\infty$\n  and $\\xi_k$ is fixed or $N_k$ is fixed and $\\xi_k\\rightarrow \\infty$.\n    When applied to $\\{ \\mathcal{F}_k\\}_{\\ge1}$, Theorem \\ref{t:intro:error-bound} leads to\n  the equidistribution results in the following corollary (cf.\n  cases (ii) and (iii) in the paragraph below Theorem \\ref{Sato-Tate}).\n  Indeed, (i) of the corollary is immediate. Part (ii) is easily derived from the fact\n  that $\\widehat{\\mu}^{\\mathrm{pl,\\mathrm{ur}}}_p$ weakly converges to $\\widehat{\\mu}^{\\mathrm{ST}}$\n  as $p\\rightarrow \\infty$. Although the unramified Hecke algebra at $p$ gives rise to only\n  regular functions on the complex variety $\\widehat{T}_c\/\\Omega$, it is not difficult to extend\n  the results to continuous functions on $\\widehat{T}_c\/\\Omega$. (See \\S\\S\\ref{sub:Plan-density}-\\ref{sub:general-functions-S_0} for details.)\n\n\\begin{corollary}\\label{c:intro:Plan-ST} Keep the notation of Theorem \\ref{t:intro:error-bound}.\nLet $\\widehat{\\phi}$ be a continuous function on $\\widehat{T}_c\/\\Omega$.\n  In view of \\eqref{e:intro:spec-satake} $\\widehat{\\phi}$ can be extended by zero to a function\n  $\\widehat{\\phi}_p$ on $G({\\mathbb Q}_p)^{\\wedge,\\mathrm{ur}}$ for each prime $p$.\n\\begin{enumerate}\n  \\item (Automorphic Plancherel density theorem~\\cite{Shi-Plan}) $$\\lim_{k\\rightarrow \\infty} \\widehat{\\mu}^{\\mathrm{count}}_{\\mathcal{F}_k,p}(\\widehat{\\phi}_p)\n  = \\widehat{\\mu}^{\\mathrm{pl,\\mathrm{ur}}}_p(\\widehat{\\phi}_p).$$\n  \\item (Sato-Tate theorem for families) Let $\\{p_k\\}_{\\ge1}$ be a sequence of primes tending to $\\infty$.\n\tSuppose that $N_k p_k^{-j} \\rightarrow \\infty$ (resp. $m(\\xi_k)p_k^{-j}\\rightarrow \\infty$)\n  as $k\\rightarrow \\infty$ for every $j\\ge 1$ if $\\xi_k$ (resp. $N_k$) remains fixed as $k$ varies.\n  Then for every $\\widehat{\\phi}$,\n  $$\\lim_{k\\rightarrow \\infty} \\widehat{\\mu}^{\\mathrm{count}}_{\\mathcal{F}_k,p_k}(\\widehat{\\phi}_{p_k})\n  = \\widehat{\\mu}^{\\mathrm{ST}}(\\widehat{\\phi}).$$\n\\end{enumerate}\n\\end{corollary}\n\n\n  Theorem \\ref{t:intro:error-bound} and Corollary \\ref{c:intro:Plan-ST}\n  remain valid if any finite number of primes are simultaneously considered in place of $p$ or $p_k$. Moreover (i) of the corollary holds true for more general (and possibly ramified) test functions $\\widehat \\phi_p$ on $G({\\mathbb Q}_p)^{\\wedge}$ thanks to Sauvageot's density theorem. It would be interesting to quantify the error bounds in this generality.\n\n\\subsection{Random matrices}\\label{sec:intro:rdm} We provide a brief account of the theory of random matrices. The reader will find more details in \\S\\ref{sec:zeros:matrix} and extensive treatments in~\\cites{book:Mehta,book:KS}.\n\nThe Gaussian Unitary Ensemble and Gaussian Orthogonal Ensemble were introduced by the physicist Wigner in the study of resonances of heavy nucleus.\nThe Gaussian Symplectic Ensemble was introduced later by Dyson together with his Circular Ensembles.\nIn this paper we are concerned with the Ensembles attached to compact Lie groups which are studied by Katz-Sarnak and occur in the statistics of $L$-functions. (See~\\cite{Duen:RMT} for the precise connections between the Ensembles attached to different Riemannian symmetric spaces.)\n\nOne considers eigenvalues of matrices on compact groups $\\mathcal{G}(N)$ of large dimension endowed with the Haar probability measure. We have three Symmetry Types $\\mathcal{G}=SO(even)$ (resp. $\\mathcal{G}=U$, $\\mathcal{G}=USp$); the notation says that for all $N\\ge 1$, we consider the groups $\\mathcal{G}(N)=SO(2N)$ (resp. $\\mathcal{G}(N)=U(N)$ and $\\mathcal{G}(N)=USp(2N)$).\n\nFor all matrices $A \\in \\mathcal{G}(N)$ we have an associated sequence of \\key{normalized angles}\n\\begin{equation}\\label{intro:vartheta}\n0\\le \\vartheta_1 \\le \\vartheta_2 \\le \\cdots \\le \\vartheta_N \\le N.\n\\end{equation}\nFor example in the case $\\mathcal{G}=U$, the eigenvalues of $A\\in U(N)$ are given by $e(\\tfrac{\\vartheta_j}{N})=e^{2i\\pi \\vartheta_j\/N}$ for $1\\le j\\le N$.\nThe normalization is such that the mean spacing of the sequence~\\eqref{intro:vartheta} is about one.\n\nFor each $N\\ge 1$ these eigenvalues $(\\vartheta_i)_{1\\le i\\le N}$ are correlated random variables. Their joint density is  proportional to\n\\begin{equation}\\label{ginibre}\n  \\prod_{1\\le i<j\\le N} \\abs{\\sin\\left( \\frac{\\pi(\\vartheta_i-\\vartheta_j)}{N}\\right)}^\\beta d\\vartheta_1 \\cdots d\\vartheta_N.\n\\end{equation}\nThe parameter $\\beta$ is a measure of the repulsion between nearby eigenvalues. We have that $\\beta=1$ (resp. $\\beta=2$, $\\beta=4$) for $\\mathcal{G}=SO(even)$ (resp. $\\mathcal{G}=U$, $\\mathcal{G}=USp$).\n\nOne of the purposes of random matrix theory is the determination of the limiting statistics. A fundamental theorem, due to Gaudin--Mehta and Dyson and extended to this setting by Katz--Sarnak with a study of the convergence rate, is that when $N\\to \\infty$ the distribution of the normalized angles $(\\vartheta_i)_{1\\le i\\le N}$ converges to a determinantal point process.\\footnote{\nFor other values of $\\beta\\neq 1,2,4$, the limiting statistics attached to~\\eqref{ginibre} has been determined recently by Valk\\'o--Vir\\'ag in terms of the Brownian carousel.}\nThe kernel of the limiting point process when $\\mathcal{G}=U$ is given by the Dyson sine kernel\n\\begin{equation*}\nK(x,y)=\\frac{\\sin\\pi(x-y)}{\\pi(x-y)}, \\quad x,y\\in \\mathbb{R}_+\n\\end{equation*}\nThe kernel for $\\mathcal{G}=SO(even)$ is $K_+(x,y)=K(x,y)+K(-x,y)$ and the kernel for $\\mathcal{G}=USp$ is $K_-(x,y)=K(x,y)-K(-x,y)$.\n\nIn particular this means that there is a \\key{limiting $1$-level density} $W(\\mathcal{G})$ for the normalized angles $(\\vartheta_i)$ in~\\eqref{intro:vartheta} as $N\\to \\infty$ (see also Proposition~\\ref{prop:KS}), which is given by the following classical formulas ($x\\in \\mathbb{R}$):\n\\begin{equation}\\label{intro:WG}\n\\begin{aligned}\nW(SO(even))(x)&=K_+(x,x)=1+\\frac{\\sin 2\\pi x}{2\\pi x},\\\\\nW(U)(x)&=K(x,x)= 1,\\\\\nW(USp)(x)&=K_-(x,x)=1-\\frac{\\sin 2\\pi x}{2\\pi x}.\n\\end{aligned}\n\\end{equation}\n\n\\subsection{Low-lying zeros}\\label{sec:intro:lowlying} We can now state more precisely our results on families of $L$-functions. Let $\\mathfrak{F}=r_*\\mathcal{F}$ be a family of $L$-functions in the weight aspect as defined as above in~\\S\\ref{sec:intro:ST}-\\ref{sec:intro:familyL}. (For families in the level aspect we obtain similar results, see~\\S\\ref{sec:zeros}.)\n\nFor all $\\Pi\\in \\mathfrak{F}_k$ we denote by $\\rho_j(\\Pi)$, the zeros of the completed $L$-function $\\Lambda(s,\\Pi)$, where $j\\in \\mathbb{Z}$. We have that $0<\\MRe \\rho_j(\\Pi)<1$ for all $j$. For simplicity we assume the GRH in this introduction and write $\\rho_j(\\Pi)=\\frac{1}{2}+i\\gamma_j(\\Pi)$ with $\\gamma_j(\\Pi)\\in \\mathbb{R}$. When $\\Pi$ is self-dual the zeros occur in complex pair but we don't make this self-duality assumption here.\n\nFollowing Iwaniec--Sarnak we associate an \\key{analytic conductor} $C(\\mathfrak{F}_k)\\ge 1$ to the family, see~\\S\\ref{sec:pp:Lfn} and~\\S\\ref{sec:zeros:cond}. For each $k\\ge 1$, all  of $\\Pi\\in \\mathfrak{F}_k$ share the same archimedean factor $\\Pi_\\infty$ and we can set $C(\\mathfrak{F}_k):=C(\\Pi_\\infty)$. Then for a given $\\Pi\\in \\mathfrak{F}_k$ the number of zeros $\\gamma_j(\\Pi)=O(1)$ of bounded height is $O(\\log C(\\mathfrak{F}_k))$. The \\key{low-lying zeros} of $\\Lambda(s,\\Pi)$ are those within distance $O(1\/\\log(C(\\mathfrak{F}_k))$ to the central point (note that $\\log C(\\mathfrak{F}_k)\\to \\infty$ as $k\\to \\infty$ which explains the terminology); heuristically there are a bounded number of low-lying zeros for a given $\\Pi\\in \\mathfrak{F}_k$, although this can only be proved on average over the family. For a technical reason related to the fact that the explicit formula counts both the zeros and poles of $\\Lambda(s,\\Pi)$ (\\S\\ref{sec:pp:explicit}), we make an hypothesis on the occurrence of poles of $\\Lambda(s,\\Pi)$ for $\\Pi\\in \\mathfrak{F}_k$, see~Hypothesis~\\ref{hyp:poles}.\n\n\nThe statistics of low-lying zeros of the family are studied via the functional\n\\begin{equation} \\label{intro:1-level}\nD(\\mathfrak{F}_k;\\Phi)=\\frac{1}{\\abs{\\mathfrak{F}_k}}\\sum_{\\Pi\\in\\mathfrak{F}_k} \\sum_j \\Phi(\\frac{\\gamma_j(\\Pi)}{2\\pi} \\log C(\\mathfrak{F}_k)),\n\\end{equation}\nwhere $\\Phi\\in {\\mathcal{S}}({\\mathbb R})$ is a Schwartz function. This is the $1$-level density for the family $\\mathfrak{F}_k$.\nChoosing $\\Phi$ as a smooth approximation of the characteristic function of an interval $[a,b]$, the sum~\\eqref{intro:1-level} should be thought as\na weighted count of all the low-lying zeros  of the family in $[a,b]$:\n\\begin{equation*}\n  \\frac{2 a \\pi}{\\log C(\\mathfrak{F}_k)} \\le \\gamma_j(\\Pi) \\le \\frac{2 b \\pi}{\\log C(\\mathfrak{F}_k)},\n  \\quad\n(j\\in {\\mathbb Z}, \\Pi \\in \\mathcal{F}_k).\n\\end{equation*}\nWe want to compare the asymptotic as $k\\to \\infty$ with the limiting $1$-level density of normalized eigenvalues~\\eqref{intro:vartheta} of the random matrix ensembles described in the previous~\\S\\ref{sec:intro:rdm}.\n\n\\begin{theorem}\\label{th:low-lying} Let $r:{}^LG:\\to \\GL(d,\\mathbb{C})$ be a continuous $L$-homomorphism which is irreducible and non-trivial when restricted to $\\widehat G$.\n  There exists $\\delta>0$ depending such that the following holds.\n   Let $\\mathfrak{F}=r_*\\mathcal{F}$ be a family of $L$-functions in the weight aspect as in~\\S\\ref{sec:intro:familyL}, assuming the functoriality conjecture as in Hypothesis~\\ref{hypo:functorial-lift}. Assume Hypothesis~\\ref{hyp:poles} concerning the poles of $\\Lambda(s,\\Pi)$ for $\\Pi\\in\\mathfrak{F}_k$. Then for all Schwartz functions $\\Phi$ whose Fourier transform $\\widehat \\Phi$ has support in $(-\\delta,\\delta)$:\n\n(i) there is a limiting $1$-level density for the low-lying zeros. Namely there is a density $W(x)$ such that\n\\begin{equation*\n\\lim_{k\\to \\infty} D(\\mathfrak{F}_k;\\Phi) = \\int_{-\\infty}^{\\infty}\n\\Phi(x)W(x)dx;\n\\end{equation*}\n\n(ii) the density $W(x)$ is determined by the Frobenius--Schur indicator of the irreducible representation $r$. Precisely,\n\\begin{equation}\\label{W:th:low-lying}\nW=\\begin{cases}\nW(SO(even)),& \\text{if $s(r)=-1$,}\\\\\nW(U),& \\text{if $s(r)=0$,}\\\\\nW(USp),& \\text{if $s(r)=1$.}\n\\end{cases}\n\\end{equation}\n\\end{theorem}\n\nThe constant $\\delta>0$ depends on the family $\\mathfrak{F}$. In other words it depends on the group $G$ and the $L$-morphism $r:{}^LG \\to \\GL(d,\\mathbb{C})$. Its numerical value is directly related to the numerical values of the exponents in the error term occurring in Theorem~\\ref{t:intro:error-bound}. Although we do not attempt to do so in the present paper, it is interesting to produce a value of $\\delta$ that is as large as possible, see~\\cite{ILS00} for the case of $\\GL(2)$. This would require sharp bounds for orbital integrals as can be seen from the outline below. A specific problem in the weight aspect would be to optimize the exponents $A_{\\mathrm{wt}},B_{\\mathrm{wt}},e$ in~\\eqref{e:intro-unif-bound}. (In fact we can achieve $e=1$, see \\S\\ref{sec:intro:outline} below. This is natural from the viewpoint of harmonic analysis.)\n\nOur proofs of Theorems~\\ref{t:intro:error-bound} and~\\ref{th:low-lying} are effective. This means that all constants and all exponents in the statement of the estimates could, in principle, be made entirely explicit.\n\n\\subsection{Outline of proofs}\\label{sec:intro:outline}\n\nA wide range of methods are used in the proof. Among them are the Arthur-Selberg trace formula, the analytic theory of\n$L$-functions, representation theory and harmonic analysis on $p$-adic and real groups, and random matrix theory.\n\nThe first main result of our paper is Theorem \\ref{t:intro:error-bound}, proved in Section~\\ref{s:aut-Plan-theorem}.\nWe already pointed out after stating the theorem that the Sato-Tate equidistribution\nfor families (Corollary \\ref{c:intro:Plan-ST})\nis derived from Theorem \\ref{t:intro:error-bound}\nand the fact that the Plancherel measure tends to the Sato-Tate measure as the residue characteristic\nis pushed to $\\infty$.\n\nLet us outline the proof of the theorem.\nIn fact we restrict our attention to part (ii), as (i) is handled by a similar method and only simpler to deal with.\nThus we consider $\\mathcal{F}$ with fixed level and weight $\\xi$, where $\\xi$ is regarded\nas a variable.\nOur starting point is to realize that for $\\phi_p\\in C^\\infty_c(G({\\mathbb Q}_p))$,\nwe may interpret $\\widehat{\\mu}^{\\mathrm{count}}_{\\mathcal{F},p}(\\widehat{\\phi}_p)$ in terms of the spectral side of the trace formula for $G$\nevaluated against the function $\\phi_p\\phi^{\\infty,p}\\phi_\\infty\\in C^\\infty_c(G({\\mathbb A}))$ for a suitable\n$\\phi^{\\infty,p}$ (depending on $\\mathcal{F}$ and $p$; note that $p$ is allowed to vary) and an Euler-Poincar\\'{e} function $\\phi_\\infty$ at $\\infty$ (depending on $\\xi$).\nApplying the trace formula, which has a simple form thanks to $\\phi_\\infty$, we get a\ngeometric expansion for $\\widehat{\\mu}^{\\mathrm{count}}_{\\mathcal{F},p}(\\widehat{\\phi}_p)$:\n\\begin{equation}\\label{e:sketch-of-pf-1stThm} \\widehat{\\mu}^{\\mathrm{count}}_{\\mathcal{F},p}(\\widehat{\\phi}_{p})\n=\\sum_{M\\subset G\\atop \\textrm{cusp. Levi}}\n\\sum_{\\gamma\\in M({\\mathbb Q})\/\\sim\\atop {\\mathbb R}\\textrm{-ell}}\na'_{M,\\gamma}\\cdot O^{M({\\mathbb A}^\\infty)}_\\gamma(\\phi^\\infty_{M})\\frac{\\Phi^G_M(\\gamma,\\xi)}{\\dim \\xi}.\n\\end{equation}\nwhere $a'_{M,\\gamma}\\in {\\mathbb C}$ is a coefficient encoding a certain volume associated with\nthe connected centralizer of $\\gamma$ in $M$\nand $\\phi^\\infty_{M}$ is the constant term of $\\phi^\\infty$ along (a parabolic subgroup associated with) $M$.\nThe Plancherel formula identifies the term for $M=G$ and $\\gamma=1$ with\n$\\widehat{\\mu}^{\\mathrm{pl}}_p(\\widehat{\\phi}_p)$, which basically dominates the right hand side.\n\nThe proof of Theorem \\ref{t:intro:error-bound}.(ii) boils down to bounding\nthe other terms on the right hand side of \\eqref{e:sketch-of-pf-1stThm}.\nHere is a rough explanation of how to analyze each component there.\nThe first summation is finite and controlled by $G$, so we may as well look at the formula\nfor each $M$. There are finitely many conjugacy classes in the second summation for which the summand is nonzero.\nThe number of such conjugacy classes may be bounded by a power of $p$ where the exponent of $p$ depends only on $\\kappa$\n(measuring the ``complexity'' of $\\phi_p$).\nThe term $a'_{M,\\gamma}$, when unraveled, involves a special value of some Artin $L$-function.\nWe establish a bound on the special value which suffices to deal with $a'_{M,\\gamma}$.\nThe last term $\\frac{\\Phi^G_M(\\gamma,\\xi)}{\\dim \\xi}$ can be estimated by using a character formula\nfor the stable discrete series character $\\Phi^G_M(\\gamma,\\xi)$ as well as the dimension formula\nfor $\\xi$.\nIt remains to take care of $O^{M({\\mathbb A}^\\infty)}_\\gamma(\\phi^\\infty_{M})$.\nThis turns out to be the most difficult task since\nTheorem \\ref{t:intro:error-bound}\nasks for a bound that is \\emph{uniform as the residue characteristic varies}.\n\nWe are led to prove that there exist $a,b,e> 0$, depending only on $G$, such that for almost all $q$,\n\\begin{equation}\\label{e:intro-unif-bound}|O^{M({\\mathbb Q}_q)}_\\gamma(\\phi_q)|\\le q^{a+b \\kappa} D^M(\\gamma)^{-e\/2}\\end{equation}\nfor all semisimple $\\gamma$ and all $\\phi_q$ with $\\phi_q\\in {\\mathcal{H}}^{\\mathrm{ur}}(M({\\mathbb Q}_q))^{\\le \\kappa}$ and $|\\phi_q|\\le 1$.\nThe justification of \\eqref{e:intro-unif-bound} takes up the whole of Section~\\ref{s:app:unif-bound}.\n The problem already appears to be deep for the unit elements of unramified Hecke algebras\n in which case one can take $\\kappa=0$. (By a different argument based on arithmetic motivic integration, Cluckers, Gordon, and Halupczok establish a stronger uniform bound with $e=1$. This work is presented in Appendix~\\ref{s:app:B}.)\n At the (fixed) finite set of primes where ramification occurs, the problem comes down to bounding\nthe orbital integral $|O^{M({\\mathbb Q}_q)}_\\gamma(\\phi_q)|$ for fixed $q$ and $\\phi_q$. It is deduced\nfrom the Shalika germ theory that the orbital integral is bounded by a constant, if normalized\nby the Weyl discriminant $D^M(\\gamma)^{1\/2}$, as $\\gamma$ runs over the set of semisimple elements.\nSee Appendix \\ref{s:app:Kottwitz} by Kottwitz for details.\n\nWe continue with Theorem~\\ref{th:low-lying}. The proof relies heavily on Theorem~\\ref{t:intro:error-bound}. The connection between the two statements might not be immediately apparent.\n\nA standard procedure based on the explicit formula (see~Section~\\ref{sec:pp}) expresses the sum~\\eqref{intro:1-level} over zeros of $L$-function as a sum over prime numbers of Satake parameters. The details are to be found in Section~\\ref{sec:pf}, and the result is that $D(\\mathfrak{F}_k,\\Phi)$ can be approximated by\n\\begin{equation} \\label{intro:sumprime}\n  \\sum_{\\text{prime } p}\n\\widehat{\\mu}^{\\mathrm{count}}_{\\mathcal{F}_k,p}(\\widehat{\\phi}_p)\n\\Phi(\\frac{ \\log p}{\\pi \\log C(\\mathfrak{F}_k)}).\n\\end{equation}\nHere $\\phi_p\\in \\mathcal{H}^{\\mathrm{ur}}(G({\\mathbb Q}_p))^{\\le \\kappa}$ is suitable chosen such that $\\widehat \\phi_p(\\pi_p)$ is a sum of powers of the Satake parameters of $r_*\\pi$ (see Sections~\\ref{s:Satake-trans} and~\\ref{s:Plancherel}). The integer $\\kappa$ may be large but it depends only on $G$ and $r$ so should be considered as fixed. Also the sum is over unramified primes.\nWe have $\\log C(\\mathcal{F}_k) \\asymp \\log m(\\xi_k)$ (see Sections~\\ref{s:langlands} and~\\ref{sec:zeros}). We deduce that the sum is supported on those primes $p\\le m(\\xi_k)^{A\\delta}$ where $A$ is a suitable constant and $\\delta$ is as in Theorem~\\ref{th:low-lying}.\n\nWe apply Theorem~\\ref{t:intro:error-bound} which has two components: the main term and the error term. We begin with the main term which amounts to substituting\n$\\widehat{\\mu}^{\\mathrm{pl,\\mathrm{ur}}}_p(\\widehat{\\phi}_p)$ for $\\widehat{\\mu}^{\\mathrm{count}}_{\\mathcal{F}_k,p}(\\widehat{\\phi}_p)$ in~\\eqref{intro:sumprime}.\nUnlike $\\widehat{\\mu}^{\\mathrm{count}}_{\\mathcal{F}_k,p}$, this term is purely local, thus simpler. Indeed $\\widehat{\\mu}^{\\mathrm{pl,\\mathrm{ur}}}_p(\\widehat{\\phi}_p)$ can be computed explicitly for low rank groups, see e.g.~\\cite{Gro98} for all the relevant properties of the Plancherel measure. However we want to establish Theorem~\\ref{th:low-lying} in general so we proceed differently.\n\nUsing certain uniform estimates by Kato~\\cite{Kat82}, we can approximate $\\widehat{\\mu}^{\\mathrm{pl,\\mathrm{ur}}}_p(\\widehat{\\phi}_p)$ by a much simpler expression that depends directly on the restriction of $r$ to $\\widehat G \\rtimes W_{\\mathbb{Q}_p}$. Then a pleasant computation using the Cebotarev equidistribution theorem, Weyl's unitary trick and the properties of the Frobenius--Schur indicator shows that the sum over primes of this main term contribute $\\frac{-s(r)}{2}\\Phi(0)$ to~\\eqref{intro:sumprime}. This exactly reflects the identities~\\eqref{W:th:low-lying} in the statement (ii) of Theorem~\\ref{th:low-lying}.\n\nWe continue with the error term $O(p^{A_\\mathrm{wt}+B_\\mathrm{wt}\\kappa}m(\\xi_k)^{-C_\\mathrm{wt}})$ which we need to insert in~\\eqref{intro:sumprime}. We can see the reasons why the proof of Theorem~\\ref{th:low-lying} requires the full force of Theorem~\\ref{t:intro:error-bound} and its error term: the polynomial control by $p^{A_\\mathrm{wt}+B_\\mathrm{wt}\\kappa}$ implies that the sum over primes is at most $m(\\xi_k)^{D\\delta}$ for some $D>0$; the power saving $m(\\xi_k)^{-C_\\mathrm{wt}}$ is exactly what is needed to beat $m(\\xi_k)^{D\\delta}$ when $\\delta$ is chosen small enough.\n\n\\subsection{Notation}\\label{sub:notation}  We distinguish the letter $\\mathcal{F}$ for families of automorphic representations on general reductive groups and $\\mathfrak{F}=r_*\\mathcal{F}$ for the families of automorphic representations on $\\GL(d)$.\n\n\nLet us describe in words the significance of various constants occurring in the main statements. We often use the convention to write multiplicative constants in lowercase letters and constants in the exponents in uppercase or greek letters.\n\\begin{itemize}\n  \\item The exponent $\\beta$ from Lemma~\\ref{l:bound-degree-Satake} is such that for all $\\phi\\in {\\mathcal{H}}^{\\mathrm{ur}}(\\GL_d)$ of degree at most $\\kappa$, the pullback $r^*\\phi$ is of degree at most $\\le \\beta \\kappa$.\n  \\item The exponent $b_G$ from Lemma~\\ref{l:bounding-phi-on-S_0} controls a bound for the constant term $\\abs{\\phi_M(1)}$ for all Levi subgroups $M$ and $\\phi \\in  {\\mathcal{H}}^{\\mathrm{ur}}(G)$ of degree at most $\\kappa$.\n\n  \\item The exponent $0<\\theta<\\frac12$ is a nontrivial bound towards Ramanujan-Petersson for $\\GL(d,\\mathbb{A})$.\n  \\item The integer $i\\ge 1$ in Corollary~\\ref{c:vanishing-ram-subgroup} is an upper-bound for the ramification of the Galois group ${\\rm Gal}(E\/F)$.\n \n  \\item The constants $B_\\Xi$ and $c_\\Xi$ in Lemma~\\ref{l:forcing-unipotent} and $A_3,B_3$ in Proposition~\\ref{p:bound-number-of-conj} control the number of rational conjugacy classes intersecting a small open compact subgroup.\n  \\item The integer $u_G\\ge 1$ in Lemma~\\ref{l:bounding-conj-in-st-conj} is a uniform upper bound for the number of $G(F_v)$-conjugacy classes in a stable conjugacy class.\n  \\item The integer $n_G\\ge 0$ is the minimum value for the dimension of the unipotent radical of a proper parabolic subgroup of $G$ over $\\overline{F}$.\n  \\item The constant $c>0$ is a bound for the number of connected components $\\pi_0(Z(\\widehat{I}_\\gamma)^{\\Gamma})$. Corollary~\\ref{c:bounding-pi0-general}.\n  \\item The exponents $A_\\mathrm{lv},B_\\mathrm{lv},C_\\mathrm{lv}>0$ in Theorem~\\ref{t:level-varies} (see also Theorem~\\ref{t:intro:error-bound}) and $A_\\mathrm{wt},B_\\mathrm{wt},C_\\mathrm{wt}>0$ in Theorem~\\ref{t:weight-varies}.\n  \\item For families in the weight aspect, the constant $\\eta>0$ which may be chosen arbitrary small enters in the condition~\\eqref{dimxik} that the dominant weights attached to $\\xi_k$ stay away from the walls.\n  \\item The exponent $C_{\\text{pole}}>0$ in the Hypothesis~\\ref{hyp:poles} concerning the density of poles of $L$-functions.\n  \\item The exponents $0<C_1 < C_2$ control the analytic conductor $C(\\mathfrak{F}_k)$ of the families in the weight aspect (Inequality~\\eqref{xik-CFk}) and $0<C_3 < C_4$ in the level aspect (Hypothesis~\\ref{hyp:cond}).\n  \\item The constant $\\delta>0$ in the Theorem~\\ref{th:onelevel} controls the support of the Fourier transform $\\widehat \\Phi$ of the test function $\\Phi$.\n  \\item The constant $c(f)>0$ depending on the test function $f$ is a uniform upper bound for normalized orbital integrals $\\abs{D^G(\\gamma)}^{\\frac12} O_\\gamma(f)$ (Appendix~\\ref{s:app:Kottwitz}).\n\\end{itemize}\n\nSeveral constants are attached directly to the group $G$ such as the dimension $d_G=\\dim G$, the rank $r_G={\\rm rk} G$, the order of the Weyl group $w_G=\\abs{\\Omega}$, the degree $s_G$ of the smallest extension of $F$ over which $G$ becomes split. Also in Lemma~\\ref{l:bounding-phi-on-S_0} the constant $b_G$ gives a bound for the constant terms along Levi subgroups. The constants $a_G,b_G, e_G$ in Theorem~\\ref{t:appendeix2} gives a uniform bound for certain orbital integrals.\nIn general we have made effort to keep light and consistent notation throughout the text.\n\n\nIn Section~\\ref{s:bg} we will choose a finite extension $E\/F$ which splits maximal tori of subgroups of $G$. The degree $s_G^{\\mathrm{sp}}=[E:F]$ will be controlled by $s_G^{\\mathrm{sp}}\\le s_Gw_G$ (see Lemma~\\ref{l:torus-splitting}), while the ramification of $E\/F$ will vary.\nIn Section~\\ref{s:Sato-Tate} we consider the finite extension $F_1\/F$ such that ${\\rm Gal}(\\overline{F}\/F)$ acts on $\\widehat{G}$ through the faithful action of ${\\rm Gal}(F_1\/F)$.%\nFor example if $G$ is a non-split inner form of a split group then $F_1=F$. In Section~\\ref{sec:pf} we consider a finite extension $F_2\/F_1$ such that the representation $r$ factors through $\\widehat{G}\\rtimes {\\rm Gal}(F_2\/F)$. For a general $G$, there might not be any direct relationship between the extensions $E\/F$ and $F_2\/F_1\/F$.\n\n\\subsection{Structure of the paper}\n\nFor a quick tour of our main results and the structure of our arguments, one could start reading from Section \\ref{s:aut-Plan-theorem} after familiarizing oneself with basic notation, referring to earlier sections for further notation and basic facts as needed.\n\nThe first Sections~\\ref{s:Satake-trans} and~\\ref{s:Plancherel} are concerned with harmonic analysis on reductive groups over local fields, notably the Satake transform, $L$-groups and $L$-morphisms, the properties of the Plancherel measure and the Macdonald formula for the unramified spectrum. We establish bounds for truncated Hecke algebras and for character traces that will play a role in subsequent chapters. In Section~\\ref{sec:pp} we recall various analytic properties of automorphic $L$-functions on $\\GL(d)$ and notably isobaric sums, bounds towards Ramanujan--Petersson and the so-called explicit for the sum of the zeros.\nSection~\\ref{s:Sato-Tate} introduces the Sato--Tate measure for general groups and Sato--Tate equidistribution for Satake parameters and for families.\nThe next Section~\\ref{s:bg} gathers various background materials on orbital integral, the Gross motive and Tamagawa measure, discrete series characters and Euler--Poincar\\'e functions, and Frobenius--Schur indicator. We establish bounds for special values of the Gross motive which will enter in the geometric side of the trace formula.\n\nIn Section~\\ref{s:app:unif-bound} we establish a uniform bound for orbital integrals of the type~\\eqref{e:intro-unif-bound}.\nIn Section~\\ref{s:conj} we establish various bounds on conjugacy classes and level subgroups. How these estimates enter in the trace formula has been detailed in the outline above.\n\nThen we are ready in Section~\\ref{s:aut-Plan-theorem} to establish our main result, an automorphic Plancherel theorem for families with error terms and its application to the Sato-Tate theorem for families. The theorem is first proved for test functions on the unitary dual coming from Hecke algebras by orchestrating all the previous results in the trace formula. Then our result is improved to allow more general test functions, either in the input to the Sato-Tate theorem or in the prescribed local condition for the family, by means of Sauvageot's density theorem.\n\nThe last three Sections~\\ref{s:langlands}, \\ref{sec:zeros} and~\\ref{sec:pf} concern the application to low-lying zeros. In complete generality we need to rely on Langlands global functoriality and other hypothesis that we state precisely. These unproven assumptions are within reach in the context of endoscopic transfer and we will return to it in subsequent works.\n\nThe Appendix~\\ref{s:app:Kottwitz} by Kottwitz establishes the boundedness of normalized orbital integrals from the theory of Shalika germs. The Appendix~\\ref{s:app:B} by Cluckers--Gordon--Halupczok establishes a strong form of~\\eqref{e:intro-unif-bound} with $e=1$ by using recent results in arithmetic motivic integration.\n\n\\subsection{Acknowledgments} We would like to thank Jim Arthur, Joseph Bernstein, Laurent Clozel, Julia Gordon, Nicholas Katz, Emmanuel Kowalski, Erez Lapid, Philippe Michel, Peter Sarnak, Kannan Soundararajan and Akshay Venkatesh for helpful discussions and comments. We would like to express our gratitude to Robert Kottwitz and Bao Ch\\^{a}u Ng\\^{o} for\n  helpful discussions regarding Section~\\ref{s:app:unif-bound}, especially about the possibility of a geometric approach. We appreciate Brian Conrad for explaining us about the integral models for reductive groups.\n\nMost of this work took place during the AY2010-2011 at the Institute for Advanced Study and some of the results have been presented there in March. We thank the audience for their helpful comments and the IAS for providing excellent working conditions. S.W.S. acknowledges support from the National Science Foundation during his stay at the IAS under agreement No. DMS-0635607 and thanks Massachusetts Institute of Technology and Korea Institute for Advanced Study for providing a very amiable work environment. N.T. is supported by a grant \\#209849 from the Simons Foundation.\n\n\\section{Satake Transforms}\\label{s:Satake-trans}\n\n\\subsection{$L$-groups and $L$-morphisms}\\label{sub:L-groups}\n\n  We are going to recall some definitions and\n  facts from \\cite[\\S1,\\S2]{Bor79} and \\cite[\\S1]{Kot84a}.\n  Let $F$ be a local or global field of characteristic 0 with an algebraic closure $\\overline{F}$, which we fix.\n  Let $W_F$ denote the Weil group of $F$ and set $\\Gamma:={\\rm Gal}(\\overline{F}\/F)$.\n  Let $H$ and $G$ be connected reductive groups over $F$.\n  Let $(\\widehat{B},\\widehat{T},\\{X_{\\alpha}\\}_{\\alpha\\in \\Delta^{\\vee}})$ be a\n  splitting datum fixed by $\\Gamma$, from which the $L$-group\n  $${}^L G=\\widehat{G}\\rtimes W_F$$ is constructed.\n  An $L$-morphism $\\eta:{}^L H\\rightarrow {}^L G$ is a continuous map commuting with\n  the canonical surjections ${}^L H\\rightarrow W_F$ and ${}^L G\\rightarrow W_F$ such that\n  $\\eta|_{\\widehat{H}}$ is a morphism of complex Lie groups.\n  A representation of ${}^L G$ is by definition\n  a continuous homomorphism ${}^L G\\rightarrow \\GL(V)$ for some ${\\mathbb C}$-vector space $V$ with $\\dim V<\\infty$\n  such that $r|_{\\widehat{G}}$ is a morphism of complex Lie groups. Clearly giving a representation\n  ${}^L G\\rightarrow \\GL(V)$ is equivalent to giving an $L$-morphism ${}^L G\\rightarrow {}^L \\GL(V)$.\n\n  Let $f:H\\rightarrow G$ be a normal morphism, which means that $f(H)$ is a normal subgroup of $G$.\n  Then it gives rise to an $L$-morphism ${}^L G \\rightarrow {}^L H$\n  as explained in \\cite[2.5]{Bor79}. In particular, there is a\n  $\\Gamma$-equivariant map $Z(\\widehat{G})\\rightarrow Z(\\widehat{H})$, which is\n  canonical (independent of the choice of splittings).\n  Thus an exact sequence of connected reductive groups over $F$\n  $$1\\rightarrow G_1\\rightarrow G_2\\rightarrow G_3\\rightarrow 1$$ gives rise to\n  a $\\Gamma$-equivariant exact sequence of ${\\mathbb C}$-diagonalizable groups\n $$1\\rightarrow Z(\\widehat{G}_3)\\rightarrow Z(\\widehat{G}_2)\\rightarrow Z(\\widehat{G}_1)\\rightarrow 1.$$\n\n\n\n\n\n\\subsection{Satake transform}\\label{sub:Satake-trans}\n\n  From here throughout this section,\n  let $F$ be a finite extension of ${\\mathbb Q}_p$ with integer ring $\\mathcal{O}$ and a uniformizer\n  $\\varpi$. Set $q:=|\\mathcal{O}\/\\varpi\\mathcal{O}|$. Let $G$ be an \\emph{unramified} group over $F$ and\n  $B=TU$ be a Borel subgroup decomposed into the maximal torus and the unipotent radical in $B$.\n  Let $A$ denote the maximal $F$-split torus in $T$.\n  Write $\\Phi_F$ (resp. $\\Phi$) for the set of all $F$-rational roots (resp. all roots over $\\overline{F}$)\n  and $\\Phi_F^+$ (resp. $\\Phi^+$) for the subset of positive roots.\n  Choose a smooth reductive model of $G$ over $\\mathcal{O}$\n  corresponding to a hyperspecial point on the apartment for $A$.\n  Set $K:=G(\\mathcal{O})$. Denote by $X_*(A)^+$ the subset of $X_*(A)$ meeting the closed Weyl\n  chamber determined by $B$, namely\n  $\\lambda\\in X_*(A)^+$ if $\\alpha(\\lambda)\\ge 0$ for all $\\alpha\\in \\Phi_F^+$.\n  Denote by $\\Omega_F$ (resp. $\\Omega$) the $F$-rational Weyl group for $(G,A)$\n  (resp. the absolute Weyl group for $(G,T$)), and $\\rho_F$ (resp. $\\rho$) the half sum\n  of all positive roots in $\\Phi^+_F$ (resp. $\\Phi^+$). A partial order $\\le$ is defined on $X_*(A)$ (resp. $X_*(T)$) such that $\\mu\\le \\lambda$ if $\\lambda-\\mu$ is a linear combination of $F$-rational positive coroots (resp. positive coroots) with nonnegative coefficients. The same order extends to a partial order $\\le_{{\\mathbb R}}$ on $X_*(A)\\otimes_{\\mathbb Z} {\\mathbb R}$ and $X_*(T)\\otimes_{\\mathbb Z} {\\mathbb R}$ defined analogously.\n\n\n Let $F^{\\mathrm{ur}}$ denote the maximal unramified extension of $F$. Let ${\\rm Fr}$ denote the\n geometric Frobenius element of ${\\rm Gal}(F^{\\mathrm{ur}}\/F)$.\n  Define $W_F^{\\mathrm{ur}}$ to be the unramified Weil group, namely the subgroup\n  ${\\rm Fr}^{{\\mathbb Z}}$ of  ${\\rm Gal}(F^{\\mathrm{ur}}\/F)$.\n  Since ${\\rm Gal}(\\overline{F}\/F)$ acts on $\\widehat{G}$ through a finite quotient\n  of ${\\rm Gal}(F^{\\mathrm{ur}}\/F)$, one can make sense of\n   ${}^L G^{\\mathrm{ur}}:=\\widehat{G}\\rtimes W_F^{\\mathrm{ur}}$.\n\n  Throughout this section we write $G$, $T$, $A$ for $G(F)$, $T(F)$, $A(F)$ if there is no confusion.\n  Define ${\\mathcal{H}}^{\\mathrm{ur}}(G):=C^\\infty_c(K\\backslash G \/ K)$ and ${\\mathcal{H}}^{\\mathrm{ur}}(T):=C^\\infty_c(T(F)\/T(F)\\cap K)$.\n  The latter is canonically isomorphic to ${\\mathcal{H}}^{\\mathrm{ur}}(A):=C^\\infty_c(A(F)\/A(\\mathcal{O}))$ via the inclusion\n  $A\\hookrightarrow T$. We can further identify\n  $${\\mathcal{H}}^{\\mathrm{ur}}(T)\\simeq {\\mathcal{H}}^{\\mathrm{ur}}(A)\\simeq {\\mathbb C}[X_*(A)]$$\n  where the last ${\\mathbb C}$-algebra isomorphism\n  matches $\\lambda\\in X_*(A)$ with ${\\mathbf{1}}_{\\lambda(\\varpi)(A\\cap K)}\\in {\\mathcal{H}}^{\\mathrm{ur}}(A)$.\n  Let $\\lambda\\in X_*(A)$. Write $$\\tau^G_\\lambda:={\\mathbf{1}}_{K\\lambda(\\varpi)K}\\in {\\mathcal{H}}^{\\mathrm{ur}}(G),\\quad\n  \\tau^A_\\lambda:=\\frac{1}{|\\Omega_F|}\\sum_{w\\in \\Omega_F}\n  {\\mathbf{1}}_{\\lambda(\\varpi)(A\\cap K)}\\in {\\mathcal{H}}^{\\mathrm{ur}}(A)^{\\Omega_F}.$$\n  The sets $\\{\\tau^G_\\lambda\\}_{\\lambda\\in X_*(A)^+}$ and\n  $\\{\\tau^A_\\lambda\\}_{\\lambda\\in X_*(A)^+}$ are bases for\n  ${\\mathcal{H}}^{\\mathrm{ur}}(G)$ and ${\\mathcal{H}}^{\\mathrm{ur}}(A)^{\\Omega_F}$ as ${\\mathbb C}$-vector spaces, respectively. Consider the map\n  \\begin{equation}\\label{e:Satake-formula}{\\mathcal{H}}^{\\mathrm{ur}}(G)\\rightarrow {\\mathcal{H}}^{\\mathrm{ur}}(T),\n  \\quad f\\mapsto \\left(t\\mapsto \\delta_B(t)^{1\/2} \\int_{U} f(tu)du\\right)\\end{equation}\n  composed with ${\\mathcal{H}}^{\\mathrm{ur}}(T)\\simeq {\\mathcal{H}}^{\\mathrm{ur}}(A)$ above.\n  The composite map induces a ${\\mathbb C}$-algebra isomorphism \\begin{equation}\\label{e:Satake}{\\mathcal{S}}^G:{\\mathcal{H}}^{\\mathrm{ur}}(G)\\stackrel{\\sim}{\\ra}\n  {\\mathcal{H}}^{\\mathrm{ur}}(A)^{\\Omega_F}\\end{equation} called the Satake isomorphism. We often\n  write just ${\\mathcal{S}}$ for ${\\mathcal{S}}^G$. We note that in general\n  ${\\mathcal{S}}$ does not map $\\tau^G_\\lambda$ to $\\tau^A_{\\lambda}$.\n\n  Another useful description of ${\\mathcal{H}}^{\\mathrm{ur}}(G)$ is through representations of ${}^L G^{\\mathrm{ur}}$. (The latter notion is defined as in \\S\\ref{sub:L-groups}.  Write\n  $(\\widehat{G}\\rtimes {\\rm Fr})_{{\\rm ss-conj}}$ for the set of $\\widehat{G}$-conjugacy classes of\n  semisimple elements in $\\widehat{G}\\rtimes {\\rm Fr}$.\n  Consider the set $${\\rm ch}({}^L G^{\\mathrm{ur}}):=\\{{\\rm tr}\\, r : (\\widehat{G}\\rtimes {\\rm Fr})_{{\\rm ss-conj}}\\rightarrow {\\mathbb C}\\,|\\,\n  r~\\mathrm{is~a~representation~of~} {}^L G^{\\mathrm{ur}}\\}.$$\n  Define ${\\mathbb C}[{\\rm ch}({}^L G^{\\mathrm{ur}})]$ to be the ${\\mathbb C}$-algebra generated by ${\\rm ch}({}^L G^{\\mathrm{ur}})$\n  in the space of functions on $(\\widehat{G}\\rtimes {\\rm Fr})_{{\\rm ss-conj}}$.\n  For each $\\lambda\\in X_*(A)^+$ define the quotient\n  \\begin{equation}\\label{e:chi_lambda}\\chi_\\lambda:=\\frac{\\sum_{w\\in \\Omega_F} {\\rm sgn}(w) w(\\lambda+\\rho_F)}{\\sum_{w\\in \\Omega_F} {\\rm sgn}(w)w\\rho_F},\\end{equation}\n  which exists as an element of ${\\mathbb C}[X_*(A)]^{\\Omega_F}$ and is unique.\n(One may view $\\chi_\\lambda$ as\nthe analogue in the disconnected case\nof the irreducible character of highest weight $\\lambda$, cf. proof of Lemma \\ref{l:Kostant} below.)\n Then $\\{\\chi_\\lambda\\}_{\\lambda\\in X_*(A)^+}$\n  is a basis for ${\\mathbb C}[X_*(A)]^{\\Omega_F}$ as a ${\\mathbb C}$-vector space, cf. \\cite[p.465]{Kat82}.\n  (Another basis was given by $\\tau^A_\\lambda$'s above.)\n  There is a canonical ${\\mathbb C}$-algebra isomorphism\n  \\begin{equation}\\label{e:Satake2}  {\\mathcal{T}}:{\\mathbb C}[{\\rm ch}({}^L G^{\\mathrm{ur}})] \\stackrel{\\sim}{\\ra} {\\mathcal{H}}^{\\mathrm{ur}}(A)^{\\Omega_F},\\end{equation}\n  determined as follows (see \\cite[Prop 6.7]{Bor79} for detail):\n  for each irreducible $r$, ${\\rm tr}\\, r|_{\\widehat{T}}$ is shown to factor through\n  $\\widehat{T}\\rightarrow \\widehat{A}$ (induced by $A\\subset T$). Hence ${\\rm tr}\\, r|_{\\widehat{T}}$ can be viewed as an element\n  of ${\\mathbb C}[X^*(\\widehat{A})]={\\mathbb C}[X_*(A)]$, which can be seen to be invariant under $\\Omega_F$. Define ${\\mathcal{T}}({\\rm tr}\\, r)$ to be the latter element.\n\n  Let $r_0$ be an irreducible representation of $\\widehat{G}$ of highest weight\n  $\\lambda_0\\in X^*(\\widehat{T})^+=X_*(T)^+$. The group $W_F^{\\mathrm{ur}}$ acts on $X^*(\\widehat{T})^+$.\n  Write $\\mathrm{Stab}(\\lambda_0)\\subset W^{\\mathrm{ur}}_F$ for the stabilizer subgroup for $\\lambda_0$,\nwhich has finite index (since a finite power of ${\\rm Fr}$ acts trivially on $\\widehat{G}$ and thus also on $\\widehat{T}$).\n  Put $r:={\\rm Ind}_{\\widehat{G}\\rtimes \\mathrm{Stab}(\\lambda_0)}^{^L G^{\\mathrm{ur}}} r_0$\n  and $\\lambda:=\\sum_{\\sigma\\in W^{\\mathrm{ur}}_F\/\\mathrm{Stab}(\\lambda_0)} \\sigma\\lambda_0\n  \\in X_*(A)^+$.\n  Clearly $r$ and $\\lambda$ depend only on the $W_F^{\\mathrm{ur}}$-orbit of $\\lambda_0$.\n  Put $i(\\lambda_0):=[ W^{\\mathrm{ur}}_F:\\mathrm{Stab}(\\lambda_0)]$.\n\n\\begin{lem}\\label{l:Kostant}\n\\begin{enumerate}\n\\item Suppose that $r$ and $\\lambda$ are obtained from $r_0$ and $\\lambda_0$ as above. Then\n \\begin{equation}\\label{e:T(tr-r)}{\\mathcal{T}}({\\rm tr}\\, r)=\\chi_\\lambda.\\end{equation}\n\\item In general for any irreducible representation $r':{}^L G^{\\mathrm{ur}}\\rightarrow \\GL_d({\\mathbb C})$\nsuch that $r'(W^{\\mathrm{ur}}_F)$ has relatively compact image,\nlet $r_0$ be any irreducible subrepresentation of $r'|_{\\widehat{G}}$.\n Let $r$ be obtained from $r_0$ as above.\nThen for some $\\zeta\\in {\\mathbb C}^\\times$ with $|\\zeta|=1$,\n$${\\rm tr}\\, r'=\\zeta\\cdot {\\rm tr}\\, r.$$\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n  Let us prove (i).\n  For any $i\\ge 1$, let ${}^L G_{i}$ denote the finite $L$-group\n  $\\widehat{G}\\rtimes {\\rm Gal}(F_{i}\/F)$ where $F_{i}$ is the degree $i$ unramified\n  extension of $F$ in $\\overline{F}$.\nIt is easy to see that $r({\\rm Fr}^{i(\\lambda_0)})$ is trivial and that\n  $r={\\rm Ind}^{{}^L G_{i(\\lambda_0)}}_{\\widehat{G}} r_0$.\n  Then \\eqref{e:T(tr-r)} amounts to\nKostant's character formula for a disconnected group (\\cite[Thm 7.5]{Kos61}) applied to\n${}^L G_{i(\\lambda_0)}$.\n  As for (ii), let $\\lambda_0$ and $\\lambda$ be as in the paragraph\n preceding the lemma. Let $j\\ge 1$ be such that $G$ becomes split over a degree $j$\nunramified extension of $F$. (Recall that $G$ is assumed to be unramified.)\n  By twisting $r'$ by a unitary character of $W^{\\mathrm{ur}}_F$ one may assume that\n  $r'$ factors through ${}^L G_j$.\n  Then both $r$ and $r'$ factor through ${}^L G_j$\n  and are irreducible constituents of\n  ${\\rm Ind}^{{}^L G_{j}}_{\\widehat{G}} r_0$.\n  From this it is easy to deduce that $r'$ is a twist of $r$ by\n  a finite character of $W^{\\mathrm{ur}}_F$ of order dividing $j$. Assertion (ii) follows.\n\\end{proof}\n\n\n\n\n\n  Each $\\lambda\\in X_*(A)^+$ determines $s_{\\lambda,\\mu}\\in {\\mathbb C}$ such that\n  \\begin{equation}\\label{e:s-lambda-mu}{\\mathcal{S}}^{-1}(\\chi_\\lambda)=\n  \\sum_{\\mu\\in X_*(A)^+} s_{\\lambda,\\mu} \\tau^G_{\\mu}\\end{equation}\n  where only finitely many $ s_{\\lambda,\\mu}$ are nonzero. In fact Theorem 1.3 of \\cite{Kat82} identifies $s_{\\lambda,\\mu}$ with $K_{\\lambda,\\mu}(q^{-1})$ defined in (1.2) of that paper, cf. \\S4 of \\cite{Gro98}. In particular $s_{\\lambda,\\lambda}\\neq 0$ and $s_{\\lambda,\\mu}\\neq 0$ unless $\\mu\\le \\lambda$. The following information will be useful in \\S\\ref{sub:test-functions}.\n\n\\begin{lem}\\label{l:bound-on-s}\n  Let $\\lambda,\\mu\\in X_*(A)^+$.\n  Suppose that $\\lambda\\star_w \\mu:=w(\\lambda+\\rho_F)- (\\mu+\\rho_F)$ is nontrivial for all $w\\in \\Omega_F$.\n  For $\\kappa\\in X_*(A)$ let $p(\\kappa)\\in {\\mathbb Z}_{\\ge 0}$ be\n  the number of tuples $(c_{\\alpha^\\vee})_{\\alpha^\\vee\\in (\\Phi^{\\vee}_F)^+}$\n  with $c_{\\alpha^\\vee}\\in {\\mathbb Z}_{\\ge 0}$ such that $\\sum_{\\alpha^\\vee} c_{\\alpha^\\vee}\\cdot \\alpha^\\vee = \\kappa$. Then\n  $$|s_{\\lambda,\\mu}|\\le q^{-1}|\\Omega_F| \\max_{w\\in \\Omega_F} p(\\lambda\\star_w \\mu)  .$$\n\\end{lem}\n\n\\begin{proof}\n  It is easy to see from the description of $K_{\\lambda,\\mu}(q^{-1})$ in \\cite[(1.2)]{Kat82} that\n  $$|K_{\\lambda,\\mu}(q^{-1})|\\le |\\Omega_F| \\max_{w\\in \\Omega_F} \\widehat{{\\mathscr P}}(w(\\lambda+\\rho_F)-(\\mu+\\rho_F);q^{-1}).$$\n  The definition of $\\widehat{{\\mathscr P}}$ in \\cite[(1.1)]{Kat82} shows that\n  $0\\le \\widehat{{\\mathscr P}}(\\kappa;q^{-1})\\le p(\\kappa) q^{-1}$ if $\\kappa\\neq 0$.\n\\end{proof}\n\n\n\n\\subsection{Truncated unramified Hecke algebras}\\label{sub:trun-unr-Hecke}\n\n\n\n  Set $n:=\\dim T$ and $X_*(T)_{{\\mathbb R}}:=X_*(T)\\otimes_{\\mathbb Z} {\\mathbb R}$.\n  Choose an ${\\mathbb R}$-basis ${\\mathcal{B}}=\\{e_1,...,e_{n}\\}$ of $X_*(T)_{\\mathbb R}$. For each $\\lambda\\in X_*(T)_{\\mathbb R}$,\n  written as $\\lambda=\\sum_{i=1}^{n} a_i(\\lambda) e_i$ for unique $a_i(\\lambda)\\in {\\mathbb R}$,\n  define $$|\\lambda|_{{\\mathcal{B}}}:=\\max_{1\\le i\\le n} |a_i( \\lambda)|,\n  \\quad \\|\\lambda\\|_{{\\mathcal{B}}}:=\\max_{\\omega\\in \\Omega}( |\\omega \\lambda|_{{\\mathcal{B}}}).$$\n  When there is no danger of confusion,\n  we will simply write $|\\cdot|_{{\\mathcal{B}}}$ or even $|\\cdot|$ instead of $|\\cdot|_{{\\mathcal{B}}}$,\n  and similarly for $\\|\\cdot\\|_{{\\mathcal{B}}}$.\n  It is clear that $\\|\\cdot\\|_{{\\mathcal{B}}}$ is $\\Omega$-invariant and\n  that $|\\lambda_1+\\lambda_2|_{{\\mathcal{B}}}\\le |\\lambda_1|_{{\\mathcal{B}}}+|\\lambda_2|_{{\\mathcal{B}}}$\n  for all $\\lambda_1,\\lambda_2\\in X_*(T)$.\n  When $\\kappa\\in {\\mathbb Z}_{\\ge 0}$, define \\begin{equation}\\label{e:trun-Hecke}{\\mathcal{H}}^{\\mathrm{ur}}(G)^{\\le \\kappa,{\\mathcal{B}}}:=\n  \\{ {\\mathbb C}\\mbox{-subspace~of~} {\\mathcal{H}}^{\\mathrm{ur}}(G)\n  \\mbox{~generated~by~}\\tau^G_\\lambda,~\\lambda\\in X_*(A)^+,~\\|\\lambda\\|_{{\\mathcal{B}}}\\le \\kappa\\}.\\end{equation}\n  It is simply written as ${\\mathcal{H}}^{\\mathrm{ur}}(G)^{\\le \\kappa}$ when the choice of ${\\mathcal{B}}$ is clear.\n\n\n\\begin{lem}\\label{l:norm-and-bases}\n Let ${\\mathcal{B}}$ and ${\\mathcal{B}}'$ be two ${\\mathbb R}$-bases of $X_*(T)_{\\mathbb R}$. Then\nthere exist constants $c_1,c_2,B_1,B_2,B_3,B_4>0$ such that\nfor all $\\lambda\\in X_*(T)_{\\mathbb R}$,\n\\begin{enumerate}\n\\item $c_1 |\\lambda|_{{\\mathcal{B}}'} \\le |\\lambda|_{{\\mathcal{B}}}\n\\le c_2 |\\lambda|_{{\\mathcal{B}}'}$,\n\\item $B_1 |\\lambda|_{{\\mathcal{B}}} \\le \\|\\lambda\\|_{{\\mathcal{B}}}\n\\le B_2 |\\lambda|_{{\\mathcal{B}}}$ for all $ \\lambda\\in X_*(T)_{\\mathbb R}$,\n\\item $B_3 \\|\\lambda\\|_{{\\mathcal{B}}'} \\le \\|\\lambda\\|_{{\\mathcal{B}}}\n\\le B_4 \\|\\lambda\\|_{{\\mathcal{B}}'}$ for all $ \\lambda\\in X_*(T)_{\\mathbb R}$ and\n\\item ${\\mathcal{H}}^{\\mathrm{ur}}(G)^{\\le B_4^{-1}\\kappa,{\\mathcal{B}}'}\\subset {\\mathcal{H}}^{\\mathrm{ur}}(G)^{\\le \\kappa,{\\mathcal{B}}}\\subset {\\mathcal{H}}^{\\mathrm{ur}}(G)^{\\le B_3^{-1}\\kappa,{\\mathcal{B}}'}$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n  Let us verify (i).\n  As the roles of ${\\mathcal{B}}$ and ${\\mathcal{B}}'$ can be changed,\n  it suffices to prove the existence of $c_2$.\n   For this, it suffices to take $c_2=\\sup_{|\\lambda|_{{\\mathcal{B}}}\\le 1} |\\lambda|_{{\\mathcal{B}}'}$.\n   The latter is finite since $|\\cdot|_{{\\mathcal{B}}'}$ is a continuous function\n  on the set of $\\lambda$ such that $|\\lambda|_{{\\mathcal{B}}}\\le 1$, which\n  is compact.\n  Part (ii) is obtained by\n applying the lemma to the bases ${\\mathcal{B}}'=\\omega {\\mathcal{B}}$ for all $\\omega\\in \\Omega$.\n Let us check (iii). Let $B_1,B_2>0$ (resp. $B'_1,B'_2>0$)\n be the constants of (ii) for the basis ${\\mathcal{B}}$ (resp. ${\\mathcal{B}}'$).\n Then\n $$  c_1B_1(B'_2)^{-1} \\|\\lambda\\|_{{\\mathcal{B}}'} \\le c_1B_1 |\\lambda|_{{\\mathcal{B}}'}  \\le B_1 |\\lambda|_{{\\mathcal{B}}} \\le \\|\\lambda\\|_{{\\mathcal{B}}}$$\n and similarly $\\|\\lambda\\|_{{\\mathcal{B}}}\\le c_2B_2(B'_1)^{-1}\\|\\lambda\\|_{{\\mathcal{B}}'}$.\n  Finally (iv) immediately follows from (iii).\n\\end{proof}\n\n  It is natural to wonder whether the definition of truncation in \\eqref{e:trun-Hecke} changes if a different basis $\\{\\tau^G_\\lambda\\}$ or $\\{\\chi_\\lambda\\}$ is used. We assert that it changes very little in a way that the effect on $\\kappa$ is bounded by a $\\kappa$-independent constant. To ease the statement define ${\\mathcal{H}}_i^{\\mathrm{ur}}(G)^{\\le \\kappa,{\\mathcal{B}}}$ for $i=1$ (resp. $i=2$) to be the ${\\mathbb C}$-subspace of ${\\mathcal{H}}^{\\mathrm{ur}}(G)$ generated by ${\\mathcal{S}}^{-1}(\\tau^A_\\lambda)$ (resp. ${\\mathcal{S}}^{-1}(\\chi_\\lambda)$) for $\\lambda\\in X_*(A)^+$ with $\\|\\lambda\\|_{{\\mathcal{B}}}\\le \\kappa$.\n\n\n\\begin{lem}\\label{l:three-truncations}\n  There exists a constant $C\\ge 1$ such that for every $\\kappa\\in {\\mathbb Z}_{\\ge 0}$ and\n  for any distinct $i,j\\in \\{\\emptyset, 1,2\\}$,\n  $${\\mathcal{H}}_i^{\\mathrm{ur}}(G)^{\\le \\kappa,{\\mathcal{B}}}\\subset {\\mathcal{H}}_j^{\\mathrm{ur}}(G)^{\\le C\\kappa,{\\mathcal{B}}}.$$\n\\end{lem}\n\n\\begin{proof}\n  It is enough to prove the lemma for a particular choice of ${\\mathcal{B}}$ by Lemma \\ref{l:norm-and-bases}.\n  So we may assume that ${\\mathcal{B}}$ extends the set of simple coroots in $\\Phi^\\vee$ by an arbitrary basis of $X_*(Z(G))_{{\\mathbb R}}$. Again by Lemma \\ref{l:norm-and-bases} the proof will be done if we show that each of the following generates the same ${\\mathbb C}$-subspace:\n\\begin{enumerate}\n\\item the set of $\\tau^G_{\\lambda}$ for $\\lambda\\in X_*(A)^+$ with $|\\lambda|_{{\\mathcal{B}}}\\le \\kappa$,\n\\item the set of ${\\mathcal{S}}^{-1}(\\tau^A_\\lambda)$ for $\\lambda\\in X_*(A)^+$ with $|\\lambda|_{{\\mathcal{B}}}\\le \\kappa$,\n\\item the set of ${\\mathcal{S}}^{-1}(\\chi_\\lambda)$ for $\\lambda\\in X_*(A)^+$ with $|\\lambda|_{{\\mathcal{B}}}\\le \\kappa$.\n\\end{enumerate}\n  It suffices to show that the matrices representing the change of bases are ``upper triangular'' in the sense that the $(\\lambda,\\lambda)$ entries are nonzero and $(\\lambda,\\mu)$ entries are zero unless $\\lambda\\ge \\mu$. (Note that $\\lambda\\ge \\mu$ implies $|\\lambda|_{{\\mathcal{B}}}\\ge |\\mu|_{{\\mathcal{B}}}$ by the choice of ${\\mathcal{B}}$.) We have remarked below \\eqref{e:chi_lambda} that $s_{\\lambda,\\mu}$'s have this property, accounting for (i)$\\leftrightarrow$(iii). For (ii)$\\leftrightarrow$(iii) the desired property can be seen directly from \\eqref{e:chi_lambda} by writing $\\chi_\\lambda$ in terms of $\\tau^A_\\mu$'s.\n\n\\end{proof}\n\n\\subsection{The case of $\\GL_d$}\\label{sub:case-of-GL_d}\n\n  The case $G=\\GL_d$ is considered in this subsection. Let $A=T$ be the diagonal maximal torus and $B$ the group of upper triangular matrices. For $1\\le i\\le d$,\n  take $Y_i\\in X_*(A)$ to be $y\\mapsto {\\rm diag}(1,...,1,y,1,...,1)$ with $y$ in the $i$-th place.\n  One can naturally identify $X_*(A)\\simeq {\\mathbb Z}^d$ such that the images of\n  $Y_i$ form the standard basis of ${\\mathbb Z}^d$.\n  Then $\\Omega_F$ is isomorphic to ${\\mathscr S}_d$, the symmetric group in $d$ variables acting on $\\{Y_1,...,Y_d\\}$ via permutation of indices.\n  We have the Satake isomorphism\n  $${\\mathcal{S}}:{\\mathcal{H}}^{\\mathrm{ur}}(\\GL_d)\\stackrel{\\sim}{\\ra} {\\mathcal{H}}^{\\mathrm{ur}}(T)^{\\Omega_F}\\simeq {\\mathbb C}[Y_1^{\\pm},...,Y_d^{\\pm}]^{{\\mathscr S}_d}.$$\n  For an alternative description let us introduce standard symmetric polynomials $X_1,...,X_d$ by the equation in a formal $Z$-variable $(Z-Y_1)\\cdots (Z-Y_d)=Z^d-X_1Z^{d-1}+\\cdots + (-1)^{d} X_d$. Then $${\\mathbb C}[Y_1^{\\pm},...,Y_d^{\\pm}]^{{\\mathscr S}_d}={\\mathbb C}[X_1,...,X_{d-1},X_d^{\\pm}].$$\n  Let $\\kappa\\in {\\mathbb Z}_{\\ge 0}$. Define ${\\mathcal{H}}^{\\mathrm{ur}}(\\GL_d)^{\\le \\kappa}$, or simply ${\\mathcal{H}}^{\\le \\kappa}_d$, to be\n  the preimage under ${\\mathcal{S}}$ of the ${\\mathbb C}$-vector space generated by $$\\{\\sum_{\\sigma\\in {\\mathscr S}_d} Y_{\\sigma(1)}^{a_1}Y_{\\sigma(2)}^{a_2}\\cdots Y_{\\sigma(d)}^{a_d}:  a_1,...,a_d\\in [-\\kappa,\\kappa]\\}.$$ The following is standard (cf. \\cite{Gro98}).\n\n\\begin{lem} Let $r\\in {\\mathbb Z}_{\\ge 1}$. Let $\\lambda_r:=(r,0,0,...,0)\\in X_*(A)^+$. Then\n  $${\\mathcal{S}}^{-1}(Y_1^r+\\cdots+Y_d^r)=\\sum_{\\mu\\in X_*(A)^+\\atop\\mu\\le \\lambda_r} c_{\\lambda_r,\\mu}\\cdot \\tau^G_\\mu $$\n  for $c_{\\lambda_r,\\mu}\\in {\\mathbb C}$ with $c_{\\lambda_r,\\lambda_r}=q^{r(1-d)\/2}$,\n  where the sum runs over the set of $\\mu\\in X_*(T)^+$ such that $\\mu\\le \\lambda_r$.\n  In particular,\n  \\begin{eqnarray}\n    {\\mathcal{S}}^{-1}(Y_1+\\cdots+Y_d)&=&q^{(1-d)\/2}\\tau^{G}_{(1,0,...,0)},\\nonumber\\\\\n    {\\mathcal{S}}^{-1}(Y_1^2+\\cdots+Y_d^2)&=&q^{1-d}(\\tau^{G}_{(2,0,...,0)}+(1-q)\\tau^G_{(1,1,0,...,0)}).\n    \\nonumber\n  \\end{eqnarray}\n\n\\end{lem}\n\n\n\n\\subsection{$L$-morphisms and unramified Hecke algebras}\\label{sub:L-mor-unr-Hecke}\n\n\n  Assume that $H$ and $G$ are unramified groups over $F$.\n  Let $\\eta:{}^L H\\rightarrow {}^L G$ be an unramified $L$-morphism, which means that\n  it is inflated from some $L$-morphism ${}^L H^{\\mathrm{ur}}\\rightarrow {}^L G^{\\mathrm{ur}}$\n  (the notion of $L$-morphism for the latter is defined as in \\S\\ref{sub:L-groups}).\n  There is a canonically induced\n  map ${\\rm ch}({}^L G^{\\mathrm{ur}})\\rightarrow {\\rm ch}({}^L H^{\\mathrm{ur}})$.\n  Via \\eqref{e:Satake} and \\eqref{e:Satake2}, the latter map gives rise to a ${\\mathbb C}$-algebra map\n  $\\eta^*: {\\mathcal{H}}^{\\mathrm{ur}}(G)\\rightarrow {\\mathcal{H}}^{\\mathrm{ur}}(H)$.\n\n  We apply the above discussion to an unramified representation\n  $$r:{}^L G\\rightarrow \\GL_d({\\mathbb C}).$$ Viewing $r$ as an $L$-morphism ${}^L G\\rightarrow {}^L \\GL_d$,\n  we obtain\n  $$r^*:{\\mathcal{H}}^{\\mathrm{ur}}(\\GL_d)\\rightarrow {\\mathcal{H}}^{\\mathrm{ur}}(G).$$\n\n\n\\begin{lem}\\label{l:bound-degree-Satake}\n  Let ${\\mathcal{B}}$ be an ${\\mathbb R}$-basis of $X_*(T)_{\\mathbb R}$.\n  There exists a constant $\\beta>0$ (depending on ${\\mathcal{B}}$, $d$ and $r$) such that for all $\\kappa\\in {\\mathbb Z}_{\\ge 0}$,\n  $r^*({\\mathcal{H}}^{\\mathrm{ur}}(\\GL_d)^{\\le \\kappa})\\subset {\\mathcal{H}}^{\\mathrm{ur}}(G)^{\\le \\beta\\kappa,{\\mathcal{B}}}$ .\n\\end{lem}\n\n\\begin{proof}\n  Thanks to Lemma \\ref{l:norm-and-bases}, it is enough to deal with a particular choice of\n  ${\\mathcal{B}}$. Choose ${\\mathcal{B}}$ by extending the set $\\Delta^{\\vee}$ of simple coroots,\n  and write ${\\mathcal{B}}=\\Delta^{\\vee}\\coprod {\\mathcal{B}}_0$.\n  We begin by proving the following claim: let $\\lambda_1,\\lambda_2\\in X_*(A)^+$ and expand\n  the convolution product\n  \\begin{equation*}\n   \n  \\tau^G_{\\lambda_1}*\\tau^G_{\\lambda_2}=\\sum_{\\mu} a^{\\mu}_{\\lambda_1,\\lambda_2}\n  \\tau^G_\\mu\n  \\end{equation*}\n  where only $\\mu\\in X_*(A)^+$ such that $\\mu\\le_{\\mathbb R} \\lambda_1+\\lambda_2$ contribute (cf. \\cite[p.148]{Car79}).\n  Only finitely many terms are nonzero.\n  Then the claim is that $$|\\mu|_{{\\mathcal{B}}}\\le |\\lambda_1+\\lambda_2|_{{\\mathcal{B}}}, \\quad\\mbox{whenever}~a^{\\mu}_{\\lambda_1,\\lambda_2}\\neq 0.$$\n  To check the claim,   consider $\\mu=\\sum_{e\\in {\\mathcal{B}}} a_e(\\mu) \\cdot e $ and $\\lambda_1+\\lambda_2\n  = \\sum_{e\\in {\\mathcal{B}}} a_e(\\lambda_1+\\lambda_2) \\cdot e$, where the coefficients are in ${\\mathbb R}$.\n  The conditions $\\mu\\le_{{\\mathbb R}} \\lambda_1+\\lambda_2$ and $\\mu\\in X_*(T)_{{\\mathbb R},+}$\n  imply that $a_e(\\mu)=a_e(\\lambda_1+\\lambda_2)$ if $e\\in {\\mathcal{B}}_0$ and\n  $0\\le a_e(\\mu)\\le a_e(\\lambda_1+\\lambda_2)$ if $e\\in \\Delta^{\\vee}$.\n  Hence $|\\mu|_{{\\mathcal{B}}}\\le |\\lambda_1+\\lambda_2|_{{\\mathcal{B}}}$.\n\n\n\n  We are ready to prove the lemma. It is explained in Lemma \\ref{l:three-truncations} and the remark below it that there exists a constant $\\beta_1>0$ which is independent of $\\kappa$ such that every $\\phi\\in {\\mathcal{H}}^{\\mathrm{ur}}(\\GL_d)^{\\le \\kappa}$ can be written as a ${\\mathbb C}$-linear combination of\n    $$\\sum_{\\sigma\\in {\\mathscr S}_d} Y_{\\sigma(1)}^{a_1}Y_{\\sigma(2)}^{a_2}\\cdots Y_{\\sigma(d)}^{a_d},\\quad  a_1,...,a_d\\in [-\\beta_1\\kappa,\\beta_1\\kappa].$$\n  Each element above can be rewritten in terms of the symmetric polynomials $X_i$'s of \\S\\ref{sub:case-of-GL_d}: First, $X_d^{\\beta_1\\kappa}$ times $\\sum_{\\sigma\\in {\\mathscr S}_d} Y_{\\sigma(1)}^{a_1}Y_{\\sigma(2)}^{a_2}\\cdots Y_{\\sigma(d)}^{a_d}$ is a symmetric polynomial of degree $\\le 2\\beta_1\\kappa$, which in turn is a polynomial in $X_1,...,X_d$ of degree $\\le  2\\beta_1\\kappa$. We conclude that every $\\phi\\in {\\mathcal{H}}^{\\mathrm{ur}}(\\GL_d)^{\\le \\kappa}$ is in the span of monomials\n  \\begin{equation}\\label{e:monomials}X_1^{b_1}X_1^{b_2}\\cdots X_d^{b_d},\\quad b_1,...,b_d\\in [-2\\beta_1\\kappa,2\\beta_1\\kappa].\\end{equation}\n  For each $1\\le i\\le d$, write $r^*(X_i)$ (resp. $r^*(X_i^{-1})$) as a linear combination of $\\tau^G_{\\lambda_{i,j}}$ (resp. $\\tau^G_{\\lambda^-_{i,j}}$) with nonzero coefficients. Define $\\beta_0$ to be the maximum among all possible $|\\lambda_{i,j}|$ and $|\\lambda^-_{i,j}|$. The above claim  $r^*(X_1^{b_1}X_1^{b_2}\\cdots X_d^{b_d})$ as in \\eqref{e:monomials} is in the ${\\mathbb C}$-span of $\\tau^G_\\mu$ satisfying\n    $$|\\mu|_{{\\mathcal{B}}}\\le (|b_1|+\\cdots + |b_d|)\\beta_0\\le  2d\\beta_0\\beta_1\\kappa.$$\n  So the above span contains $r^*(\\phi)$ for $\\phi\\in {\\mathcal{H}}^{\\mathrm{ur}}(\\GL_d)^{\\le \\kappa}$. By Lemma\n  \\ref{l:norm-and-bases} there exists a constant $B_2>0$ such that $\\|\\mu\\|_{{\\mathcal{B}}}\\le B_2|\\mu|_{{\\mathcal{B}}}$ for every $\\mu\\in X_*(T)$. Hence the lemma holds true with $\\beta:=2B_2d\\beta_0\\beta_1$.\n\n\n\\end{proof}\n\n  The map $r$ also induces a functorial transfer for unramified representations\n\\begin{equation}\\label{e:r_*} r_*:{\\rm Irr}^{\\mathrm{ur}}(G(F))\\rightarrow {\\rm Irr}^{\\mathrm{ur}}(\\GL_d(F))\\end{equation}\n uniquely characterized by ${\\rm tr}\\, r_*(\\pi)(\\phi)={\\rm tr}\\, \\pi(r^*\\phi)$ for all $\\pi\\in {\\rm Irr}^{\\mathrm{ur}}(G(F))$ and\n$\\phi\\in {\\mathcal{H}}^{\\mathrm{ur}}(\\GL_d(F))$.\n\n\\subsection{Partial Satake transform}\n\n  Keep the assumption that $G$ is unramified over $F$.\n Let $P$ be an $F$-rational parabolic subgroup of $G$ with Levi $M$\n and unipotent radical $N$ such that $B=TU$ is contained in $P$.\n Let $\\Omega_M$ (resp. $\\Omega_{M,F}$) denote the absolute\n (resp. $F$-rational) Weyl group for $(M,T)$.\n  A partial Satake transform is defined as (cf. \\eqref{e:Satake-formula})\n \\[\n \n {\\mathcal{S}}^G_M:{\\mathcal{H}}^{\\mathrm{ur}}(G)\\rightarrow {\\mathcal{H}}^{\\mathrm{ur}}(M),\n  \\quad f\\mapsto \\left(m\\mapsto \\delta_P(m)^{1\/2} \\int_{N} f(mn)dn\\right)\n \\]\n  It is well known that ${\\mathcal{S}}^G={\\mathcal{S}}^M\\circ {\\mathcal{S}}^G_M$. More concretely, ${\\mathcal{S}}^G_M$ is the\n canonical inclusion ${\\mathbb C}[X_*(A)]^{\\Omega_{M,F}}\\hookrightarrow {\\mathbb C}[X_*(A)]^{\\Omega_{F}}$ if\n ${\\mathcal{H}}^{\\mathrm{ur}}(M)$ and ${\\mathcal{H}}^{\\mathrm{ur}}(G)$ are identified with the source and the target via ${\\mathcal{S}}^G$ and ${\\mathcal{S}}^M$, respectively.\n Since $T$ is a common maximal torus of $M$ and $G$, an ${\\mathbb R}$-basis ${\\mathcal{B}}$ of $X_*(T)_{\\mathbb R}$\n determines truncations on ${\\mathcal{H}}^{\\mathrm{ur}}(M)$ and ${\\mathcal{H}}^{\\mathrm{ur}}(G)$.\n\n\\begin{lem}\n  For any $\\kappa\\in {\\mathbb Z}_{\\ge 0}$,\n  ${\\mathcal{S}}^G_M({\\mathcal{H}}^{\\mathrm{ur}}(G)^{\\le \\kappa,{\\mathcal{B}}})\\subset {\\mathcal{H}}^{\\mathrm{ur}}(M)^{\\le \\kappa,{\\mathcal{B}}}$.\n\\end{lem}\n\n\\begin{proof}\n  It is enough to note that $\\|\\lambda\\|_{{\\mathcal{B}},M}\\le \\|\\lambda\\|_{{\\mathcal{B}},G}$ for all $\\lambda\\in X_*(A)$, which holds\n  since the $\\Omega_{M}$-orbit of $\\lambda$ is contained in the $\\Omega$-orbit of $\\lambda$.\n\\end{proof}\n\n\n\\begin{rem}\n  Let $\\eta:{}^L M\\rightarrow {}^L G$ be the embedding of \\cite[\\S3]{Bor79}, well defined up to $\\widehat{G}$-conjugacy.\n  Then ${\\mathcal{S}}^G_M$ coincides with $\\eta^*:{\\mathcal{H}}^{\\mathrm{ur}}(G)\\rightarrow {\\mathcal{H}}^{\\mathrm{ur}}(M)$ of \\S\\ref{sub:L-mor-unr-Hecke}\n\\end{rem}\n\n\n\\subsection{Some explicit test functions}\\label{sub:test-functions}\n\n  Assume that $r:{}^L G=\\widehat{G}\\rtimes W_F\\rightarrow \\GL_d({\\mathbb C})$ is\n  an \\emph{irreducible} representation arising from\n  an unramified $L$-morphism ${}^L G^{\\mathrm{ur}} \\rightarrow {}^L \\GL_d^{\\mathrm{ur}}$\n  such that $r(W_F)$ is relatively compact.\n  For later applications it is useful to study the particular element\n  $r^*(Y_1+\\cdots+Y_d)$ in ${\\mathcal{H}}^{\\mathrm{ur}}(G)$.\n\n\n\\begin{lem}\\label{l:bound-1st-moment}\n  Let $\\phi=r^*(Y_1+\\cdots+Y_d)$. Then\n\\begin{enumerate}\n\\item Suppose that $r:{}^L G^{\\mathrm{ur}}\\rightarrow \\GL_d({\\mathbb C})$ does not factor through\n $W^{\\mathrm{ur}}_F$ (or equivalently that $r|_{\\widehat{G}}$ is not the trivial representation).\n Then $$|\\phi(1)|\\le\n|\\Omega_F| \\max_{w\\in \\Omega_F} p(\\lambda\\star_w 0)\\cdot  q^{-1}.$$\n\\item Suppose that $r|_{\\widehat{G}}$ is trivial. Then\n $\\phi(1)=r({\\rm Fr})$.\n\\end{enumerate}\n\\end{lem}\n\n\n\\begin{proof}\n  Let us do some preparation. By twisting $r$ by an unramified unitary character of $W_F$ (viewed as a character of $^L G$) we may assume that $r={\\rm Ind}^{^L G_j}_{\\widehat{G}} r_0$ for some irreducible representation $r_0$ of $\\widehat{G}$, cf. the proof of Lemma \\ref{l:Kostant}.(ii). Let $\\lambda_0$ be the highest weight of $r_0$ and define $\\lambda\\in X_*(A)^+$ as in the paragraph preceding Lemma \\ref{l:Kostant}.\n  The lemma tells us that ${\\mathcal{S}}(\\phi)=\\zeta \\chi_\\lambda\\in {\\mathbb C}[X_*(A)]^{\\Omega_F}$ with $|\\zeta|=1$.\n\nIn the case of (ii), $r$ is just an unramified unitary character of $W_F$ (with $d=1$), and it is easily seen that $\\chi_\\lambda=\\tau^A_0$, $\\zeta=r({\\rm Fr})$, and so $\\phi(1)=r({\\rm Fr})$.\n  Let us put ourselves in the case (i) so that $\\lambda\\neq 0$.\nNote that $\\phi(1)$ is just the coefficient of $\\tau^G_0$\nwhen $\\phi=\\zeta {\\mathcal{S}}^{-1}(\\chi_\\lambda)$\nis written with respect to the basis $\\{\\tau^G_\\mu\\}$.\n Such a coefficient equals $\\zeta \\cdot s_{\\lambda,0}$ according to \\eqref{e:s-lambda-mu}, so\n $|\\phi(1)|= | s_{\\lambda,0}|$.\nNow Lemma \\ref{l:bound-on-s} concludes the proof. (Observe\nthat $\\lambda\\star_w 0\\neq 0$ whenever $0\\neq \\lambda\\in X_*(A)^+$.)\n\\end{proof}\n\n\\subsection{Examples in the split case}\nWhen $G$ is split,\n  it is easy to see that ${\\mathbb C}[{\\rm ch}({}^L G^{\\mathrm{ur}})]$ is canonically identified with ${\\mathbb C}[{\\rm ch}(\\widehat{G})]$\n  which is generated by finite dimensional characters in the space of functions on $\\widehat{G}$.\n  So we may use ${\\mathbb C}[{\\rm ch}(\\widehat{G})]$ in place of ${\\mathbb C}[{\\rm ch}({}^L G^{\\mathrm{ur}})]$.\n\n\\begin{ex}\\label{ex:functions-Sp_2n}(When $G=Sp_{2n}$, $n\\ge1$)\n\n  Take $r:\\widehat{G}=SO_{2n+1}({\\mathbb C})\\hookrightarrow \\GL_{2n+1}({\\mathbb C})$ to be the standard representation.\n  Then $$Y_1+\\cdots+Y_{2n+1}={\\rm tr}\\,({\\rm Std})\\in {\\mathbb C}[{\\rm ch}(\\GL_{2n+1})]$$\n  is mapped to ${\\rm tr}\\,(r)\\in {\\mathbb C}[{\\rm ch}(SO_{2n+1})]$ and\n   $$Y^2_1+\\cdots+Y^2_{2n+1}={\\rm tr}\\,({\\rm Sym}^2({\\rm Std})-\\wedge^2({\\rm Std}))\\in {\\mathbb C}[{\\rm ch}(\\GL_{2n+1})]$$\n  is mapped to ${\\rm tr}\\,(r)\\in {\\mathbb C}[{\\rm ch}(SO_{2n+1})]$.\n  Then ${\\rm Sym}^2(V)$ breaks into ${\\mathbb C}$ and an irreducible representation of\n  $\\widehat{G}$ of highest weight $(2,0,...,0)$ in the standard parametrization.\n  When $n>1$, $\\wedge^2(V)$ is irreducible of highest weight $(1,1,0,...,0)$.\n  When $n=1$, $\\wedge^2(V)\\simeq V^\\vee$, i.e. isomorphic to $({\\rm Std})^\\vee$. (See\n  \\cite[\\S19.5]{FH91}.) Let us systematically write $\\Lambda_\\lambda$ for the irreducible representation\n  of $SO_{2n+1}$ with highest weight $\\lambda$. Then\n  \\begin{eqnarray}\n   r^*(Y_1+\\cdots+Y_{2n+1})&=&{\\rm tr}\\, \\Lambda_{(1,0,...,0)},\\label{e:Satake-Sp2n}\\\\\n    r^*(Y_1^2+\\cdots+Y_{2n+1}^2)\n   & =&{\\rm tr}\\,( {\\mathbb C}+\\Lambda_{(2,0,...,0)}-\\Lambda_{(1,1,0,...,0)}).\\nonumber\n  \\end{eqnarray}\n  if $n\\ge 2$. If $n=1$, the same is true if $\\Lambda_{(1,1,0,...,0)}$ is replaced with\n  $\\Lambda_{(-1)}$.\n  For $i=1,2$, define\n  $$\\phi^{(i)}:={\\mathcal{S}}^{-1}(r^*(Y^i_1+\\cdots+Y^i_{2n+1})).$$\n  Then one computes\n  \\begin{eqnarray}\n   \\phi^{(1)}&=& q^{\\frac{1-2n}{2}} {\\mathbf{1}}_{K \\mu_{(1,0,...,0)}(\\varpi_v)K} ,\\nonumber\\\\\n   \\phi^{(2)}\n   & =& {\\mathbf{1}}_K+q^{1-2n} {\\mathbf{1}}_{K\\mu_{(2,0,...,0)}(\\varpi_v)K}\n   - q^{1-2n}(q-1){\\mathbf{1}}_{K\\mu_{(1,1,0,...,0)}(\\varpi_v)K}.\\nonumber\n  \\end{eqnarray}\n  where $\\mu_{\\lambda}$ is the cocharacter of a maximal torus given by $\\lambda$ in the standard parametrization.\n  In particular, $\\phi^{(1)}(1)=0$ and $\\phi^{(2)}(1)=1$.\n\\end{ex}\n\n\n\\begin{ex}(When $G=SO_{2n}$, $n\\ge 2$)\n\n  Take $r:\\widehat{G}=SO_{2n}({\\mathbb C})\\hookrightarrow \\GL_{2n}({\\mathbb C})$ to be the standard representation.\n Similarly as before, ${\\rm Sym}^2(V)$ breaks into ${\\mathbb C}$ and an irreducible representation of\n  $\\widehat{G}$ of highest weight $(2,0,...,0)$.\n  When $n>1$, $\\wedge^2(V)$ is irreducible of highest weight $(1,1,0,...,0)$.\n  When $n=1$, $\\wedge^2(V)\\simeq {\\mathbb C}$. (See \\cite[\\S19.5]{FH91}.) The same\n  formulas as \\eqref{e:Satake-Sp2n} hold in this case.\n  Defining \\begin{equation}\\label{e:Satake-SO2n}\\phi^{(i)}:={\\mathcal{S}}^{-1}(r^*(Y^i_1+\\cdots+Y^i_{2n})),\\end{equation} we\n  can compute $\\phi^{(1)}$, $\\phi^{(2)}$ and see that\n   $\\phi^{(1)}(1)=0$ and $\\phi^{(2)}(1)=1$.\n\\end{ex}\n\n\\begin{ex}(When $G=SO_{2n+1}$)\n\n  Take $r:\\widehat{G}=Sp_{2n}({\\mathbb C})\\hookrightarrow \\GL_{2n}({\\mathbb C})$ to be the standard representation.\n  Then $$Y_1+\\cdots+Y_{2n}={\\rm tr}\\,({\\rm Std})\\in {\\mathbb C}[{\\rm ch}(\\GL_{2n})]$$\n  is mapped to ${\\rm tr}\\,(r\\circ {\\rm Std})\\in {\\mathbb C}[{\\rm ch}(Sp_{2n})]$ and\n  Then $$Y^2_1+\\cdots+Y^2_{2n}={\\rm tr}\\,({\\rm Sym}^2({\\rm Std})-\\wedge^2({\\rm Std}))\\in {\\mathbb C}[{\\rm ch}(\\GL_{2n})]$$\n  is mapped to ${\\rm tr}\\,(r\\circ {\\rm Std})\\in {\\mathbb C}[{\\rm ch}(Sp_{2n})]$.\n    If $n\\ge 2$ then $\\wedge^2(V)$ breaks into ${\\mathbb C}$ and an irreducible representation of\n  $\\widehat{G}$ of highest weight $(1,1,0,...,0)$.\n (See \\cite[\\S17.3]{FH91}.) We have\n \\begin{eqnarray*}\n   r^*(Y_1+\\cdots+Y_{2n+1})&=&{\\rm tr}\\, \\Lambda_{(1,0,...,0)},\\label{e:Satake-SO2n+1}\\\\\n    r^*(Y_1^2+\\cdots+Y_{2n+1}^2)\n   & =&{\\rm tr}\\,( \\Lambda_{(2,0,...,0)}-\\Lambda_{(1,1,0,...,0)}-{\\mathbb C} ).\\nonumber\n  \\end{eqnarray*}\n  As in Example \\ref{ex:functions-Sp_2n}, $\\Lambda$ designates a highest weight representation\n  (now of $Sp_{2n}$).\n  Define $\\phi^{(i)}$ as in \\eqref{e:Satake-SO2n}.\n  By a similar computation as above, $\\phi^{(1)}(1)=0$, $\\phi^{(2)}(1)=-1$.\n\\end{ex}\n\n\n\n\n\n\\subsection{Bounds for truncated unramified Hecke algebras}\n\n  Let $F$, $G$, $A$, $T$ and $K$ be as in \\S\\ref{sub:Satake-trans}.\n  Throughout this subsection,\n  an ${\\mathbb R}$-basis ${\\mathcal{B}}$ of $X_*(T)_{{\\mathbb R}}$ will be fixed once and for all.\n  Denote by $\\rho\\in X^*(T)\\otimes_{\\mathbb Z} \\frac{1}{2}{\\mathbb Z}$\n  half the sum of all $\\alpha\\in \\Phi^+$.\n\n\\begin{lem}\\label{l:double-coset-volume}\n  For any $\\mu\\in X_*(A)$,\n  $[K\\mu(\\varpi)K:K]\\le q^{d_G+r_G+\\langle \\rho,\\mu\\rangle}$.\n\\end{lem}\n\n\\begin{proof}\n  Let $\\operatorname{vol}$ denote the volume for the Haar measure on $G(F)$ such that\n  $\\operatorname{vol}(K)=1$.\n  Let $I\\subset K$ be an Iwahori subgroup of $G(F)$.\n  Then $I=(I\\cap U)(I\\cap T)(I\\cap \\overline{U})$.\n  We follow the argument of \\cite[pp.241-242]{Wal03}, freely using his notation.\n  Our $I$, $U$, $\\overline{U}$, and $T$ will play the roles of his $H$, $U_0$, $\\overline{U}_0$ and $M_0$, respectively.\n  For all $m\\in \\overline{M}_0^+$ (in his notation), it is not hard to verify that\n  $c'_{U_0}(m)=c_{\\overline{U}_0}(m)=c_{M_0}(m)=1$.\n  Then Waldspurger's argument shows\n  $$\\operatorname{vol}(K\\mu(\\varpi)K)\\le [K:I]^2 \\operatorname{vol}(I\\mu(\\varpi)I)\n  \\le [K:I]^2 q^{\\langle \\rho,\\mu\\rangle} \\operatorname{vol}(I) = [K:I] q^{\\langle \\rho,\\mu\\rangle} .$$\n  Finally observe that $[K:I]\\le |G(\\mathbb{F}_q)|\\le q^{d_G}(1+\\frac{1}{q})^{r_G}\\le q^{d_G+r_G}$.\n(The middle inequality is easily derived from Steinberg's formula. cf. \\cite[(3.1)]{Gro97}.)\n\\end{proof}\n\n  The following lemma will play a role in studying the level aspect in Section \\ref{s:aut-Plan-theorem}.\n\n\n\\begin{lem}\\label{l:bounding-phi-on-S_0}\n  Let $M$ be an $F$-rational Levi subgroup of $G$.\n There exists a $b_G> 0$ (depending only on $G$)\n such that for all $\\kappa\\in {\\mathbb Z}_{>0}$\n  and all $\\phi\\in {\\mathcal{H}}^{\\mathrm{ur}}(G)^{\\le \\kappa,{\\mathcal{B}}}$ such that\n  $|\\phi|\\le 1$, we have\n  $|\\phi_{M}(1)|= O(q^{d_G+r_G+b_G\\kappa})$ (the implicit constant being independent of\n  $\\kappa$ and $\\phi$).\n\\end{lem}\n\n\\begin{proof}\n  When $M=G$, the lemma is obvious (with $b_G=0$). Henceforth we assume\n  that $M\\subsetneq G$. In view of Lemma \\ref{l:norm-and-bases},\n  it suffices to treat one ${\\mathbb R}$-basis ${\\mathcal{B}}$. Fix a ${\\mathbb Z}$-basis $\\{e_1,...,e_{\\dim A}\\}$ of $X_*(A)$,\n  and choose any ${\\mathcal{B}}$ which extends that ${\\mathbb Z}$-basis.\n   It is possible to write\n  $$\\phi=\\sum_{\\|\\mu\\|\\le \\kappa} a_\\mu\\cdot {\\mathbf{1}}_{K\\mu(\\varpi)K}$$\n  for $|a_\\mu|\\le 1$. Thus\n  $$|\\phi_{M}(1)|=\\left|\\int_{N(F)} \\phi(n)dn\\right|\n  \\le  \\sum_{\\|\\mu\\|\\le \\kappa}\n    \\left| \\int_{N(F)}{\\mathbf{1}}_{K\\mu(\\varpi)K}(n)dn \\right|.$$\n  For each $\\mu$, $K\\mu(\\varpi)K$ is partitioned into left $K$-cosets.\n  On each coset $\\gamma K$, $$\\left|\\int_{N(F)} {\\mathbf{1}}_{\\gamma K}(n) dn\\right|\\le \\operatorname{vol}(K\n  \\cap N(F))=1.$$ Hence, together with Lemma \\ref{l:double-coset-volume},\n  $$|\\phi_{M}(1)|\\le \\sum_{\\|\\mu\\|\\le \\kappa} [K\\mu(\\varpi)K:K]\\le  \\sum_{\\|\\mu\\|\\le \\kappa}q^{d_G+r_G+\\langle \\rho,\\mu\\rangle} .$$\n  Write $b_{0}$ for the maximum of $|\\langle \\rho,e_i\\rangle|$ for $i=1,...,\\dim A$.\n  Take $b_G:=b_{0}\\dim A+2 \\dim A$.\n  If $\\|\\mu\\|\\le \\kappa$ then\n $\\mu=\\sum_{i=1}^{\\dim A} a_ie_i$ for $a_i\\in{\\mathbb Z}$ with $-\\kappa\\le a_i\\le \\kappa$.\n  Hence the right hand side is bounded by\n  $(2\\kappa+1)^{\\dim A} q^{d_G+r_G+b_{0}\\kappa\\dim A}\\le q^{d_G+r_G+b_G\\kappa}$\n  since $2\\kappa+1\\le 2^{2\\kappa}\\le q^{2\\kappa}$.\n\n\n\\end{proof}\n\n  An elementary matrix computation shows the bound below, which will be used several times.\n\n\\begin{lem}\\label{l:control-eigenvalue}\n  Let $s={\\rm diag}(s_1,...,s_m)\\in \\GL_m(\\overline{F}_v)$\n  and $u=(u_{ij})_{i,j=1}^{m}\\in \\GL_m(\\overline{F}_v)$.\n  Define $v_{\\min}(u):=\\min_{i,j} v(u_{ij})$ and\nsimilarly $v_{\\min}(u^{-1})$.\n  Then for any eigenvalue $\\lambda$ of $su\\in \\GL_m(F_v)$,\n  $$v(\\lambda)\\in [v_{\\min}(u)+\\min_{i} v(s_i),-v_{\\min}(u^{-1})+\\max_{i} v(s_i)].$$\n\\end{lem}\n\n\\begin{rem}\n The lemma will be typically applied when $u\\in \\GL_m(\\overline{\\mathcal{O}}_v)$ where\n $\\overline{\\mathcal{O}}_v$ is the integer ring of $\\overline{F}_v$. In this case\n $v_{\\min}(u)=v_{\\min}(u^{-1})=0$.\n\\end{rem}\n\n\\begin{proof}\n Let $V$ be the underlying $\\overline{F}_v$-vector space with\n standard basis $\\{e_1,...,e_m\\}$.\n  Let ${\\mathcal{B}}_j=\\{\\vec{i}=(i_1,...,i_j)|1\\le i_1<\\cdots<i_j\\le m\\}$.\n   Then $\\wedge^jV$ has a basis $\\{e_{i_1}\\wedge \\cdots \\wedge e_{i_j}\\}_{\\vec{i}\\in {\\mathcal{B}}_j}$.\n  We claim that $$v({\\rm tr}\\,(su|\\wedge^j V))\\ge  j\\cdot \\min_{i} v(s_i).$$\n  Let us verify this.\n  Let $(u_{\\vec{i},\\vec{i}'})_{\\vec{i},\\vec{i}'\\in {\\mathcal{B}}_j}$ denote the matrix entries for the $u$-action\n  on $\\wedge^j V$ with respect to the above basis.\n  Observe that $v(u_{\\vec{i},\\vec{i}'})\n  \\ge j\\cdot v_{\\min}(u)$ for all $\\vec{i},\\vec{i}'\\in {\\mathcal{B}}_j$.\n  Then\n  $$v({\\rm tr}\\,(su|\\wedge^j V))=v\\left(\\sum_{\\vec{i}\\in {\\mathcal{B}}_j} s_{i_1}s_{i_2}\\cdots s_{i_j}\\cdot u_{\\vec{i},\\vec{i}}\\right)$$\n  $$\\ge \\min_{\\vec{i}} v(s_{i_1}s_{i_2}\\cdots s_{i_j}\\cdot u_{\\vec{i},\\vec{i}})\n  \\ge j\\cdot\\min_{i} v(s_i) + \\min_{\\vec{i}} v(u_{\\vec{i},\\vec{i}})\n  \\ge j(\\min_{i} v(s_i) +v_{\\min}(u)) .$$\n\n\n  The coefficients of the characteristic polynomial for $su\\in \\GL_m(F_v)$ are given by\n  ${\\rm tr}\\,(su|\\wedge^j V)$ up to sign. The above claim and an elementary argument with the Newton polygon show that\n  any root $\\lambda$ satisfies $v(\\lambda)\\ge v_{\\min}(u)+ \\min_{i} v(s_i)$.\n  Finally, applying the argument so far to $s^{-1}$ and $u^{-1}$, we obtain\n  the upper bound for $v(\\lambda)$.\n\n\\end{proof}\n\n  As before, the smooth reductive model for $G$ over $\\mathcal{O}$ such that $G(\\mathcal{O})=K$ will still be denoted $G$.\n\n\\begin{lem}\\label{l:conj-image-in-diag}\n  Let $\\Xi:G\\hookrightarrow \\GL_m$ be an embedding of algebraic groups over $\\mathcal{O}$. Then there exists a $GL_m(\\mathcal{O})$-conjugate of $\\Xi$ which maps $A$ (a fixed maximal split torus of $G$) into the diagonal maximal torus of $\\GL_m$.\n\\end{lem}\n\n\\begin{proof}\n  Note that the maximal $F$-split torus $A$ naturally extend to $A\\subset G$ over $\\mathcal{O}$, cf. \\cite[\\S3.5]{Tit79}. The representation of $A$ on a free $\\mathcal{O}$-module of rank $m$ via $\\Xi$ defines a weight decomposition of $\\mathcal{O}^m$ into free $\\mathcal{O}$-modules. Choose any refinement of the decomposition to write $\\mathcal{O}^m=L_1\\oplus \\cdots \\oplus L_m$, as the direct sum of rank 1 free $\\mathcal{O}$-submodules. Let $v_i$ be an $\\mathcal{O}$-generator of $L_i$ for $1\\le i\\le m$. Conjugating $\\Xi$ by the matrix representing the change of basis from $\\{v_1,...,v_m\\}$ to the standard basis for $\\mathcal{O}^m$, one can achieve that $\\Xi(A)$ lies in the diagonal maximal torus.\n\\end{proof}\n\n  Let $\\gamma\\in G(F)$ be a semisimple element and choose a maximal torus $T_\\gamma$ of $G$ defined over $F$ such that $\\gamma\\in T_\\gamma(F)$. Denote by $\\Phi_\\gamma$ the set of roots for $T_\\gamma$ in $G$.\n\n\n\\begin{lem}\\label{l:bounding1-alpha(gamma)}\n  Suppose that there exists an embedding of algebraic groups $\\Xi:G\\hookrightarrow \\GL_m$ over $\\mathcal{O}$.\n  There exists a constant $B_5>0$ such that\n  for every $\\kappa\\in {\\mathbb Z}_{\\ge 0}$, every $\\mu\\in X_*(A)$ satisfying $\\|\\mu\\|\\le \\kappa$,\n  every semisimple $\\gamma\\in K\\mu(\\varpi)K$ and every $\\alpha\\in \\Phi_\\gamma$ (for any choice of $T_\\gamma$ as above), we have\n  $-B_5\\kappa\\le v(\\alpha(\\gamma))\\le B_5\\kappa$. In particular,\n  $|1-\\alpha(\\gamma)|\\le q^{B_5 \\kappa}$.\n\\end{lem}\n\n\\begin{rem}\n  Later $\\Xi$ will be provided by Proposition \\ref{p:global-integral-model}.\n\\end{rem}\n\n\n\\begin{proof}\n  We may assume that $\\Xi(A)$ is contained in the diagonal torus of $\\GL_m$ by Lemma \\ref{l:conj-image-in-diag}.\n  Choose any $T_\\gamma$ as above. Let $\\mathbb{T}_\\gamma\\subset \\GL_m$ be a maximal torus over $\\overline{F}$ containing the image of $T_\\gamma$ under $\\Xi$.\n  Since $\\Xi|_{T_\\gamma}:T_\\gamma \\hookrightarrow \\mathbb{T}_\\gamma$ induces a surjection $X^*(\\mathbb{T}_\\gamma)\\twoheadrightarrow X^*(T_\\gamma)$,\n  each $\\alpha\\in \\Phi_\\gamma$ can be lifted to some $\\alpha'\\in X^*(\\mathbb{T}_\\gamma)$.\n  Fix $\\widetilde{\\alpha}$ for each $\\alpha$ once and for all, and set\n  $c_1:=\\max_{\\alpha\\in \\Phi_\\gamma} \\|\\widetilde{\\alpha}\\|_{\\GL_m}$.\n\n  Let $c_2:=\\max_{\\|\\mu\\|\\le 1} \\|\\Xi\\circ \\mu\\|_{\\GL_m}$ where $\\mu\\in X_*(A)_{{\\mathbb R}}$\n  runs over elements with $\\|\\mu\\|\\le 1$. Then\n  for any $\\kappa\\in{\\mathbb Z}_{\\ge 0}$, $\\|\\mu\\|\\le \\kappa$ implies\n  $\\|\\Xi\\circ \\mu\\|_{\\GL_m}\\le c_2\\kappa$. Hence $\\Xi(\\mu(\\varpi))$ is a diagonal matrix in which\n  each entry $x$ satisfies $-c_2\\kappa \\le v(x)\\le c_2\\kappa$.\n\n  We can write $\\gamma=k_1 \\mu(\\varpi)k_2$ for some $k_1,k_2\\in G(\\mathcal{O})$. Then\n  $\\Xi(\\gamma)=k'_1 \\Xi(\\mu(\\varpi)) k'_2$ for $k'_1,k'_2\\in \\GL_m(\\mathcal{O})$, and\n  $\\Xi(\\gamma)$ is conjugate to $\\Xi(\\mu(\\varpi))k'_2(k'_1)^{-1}$. It follows from\n  Lemma \\ref{l:control-eigenvalue} that\n  for every eigenvalue $\\lambda$ of $\\Xi(\\gamma)$, we have\n  $-c_2\\kappa\\le v(\\lambda)\\le c_2\\kappa$.\n\n  By conjugating $\\mathbb{T}_\\gamma$ to the diagonal torus of $\\GL_m$,\n  we fix an isomorphism $\\mathbb{T}_\\gamma(\\overline{F})\\simeq (\\overline{F}^\\times)^m$ (canonical up to an action by the symmetric group on $m$ letters). Let $(\\lambda_1,...,\\lambda_m)$\n  be the image of $\\Xi(\\gamma)$ under the latter isomorphism. We may write $\\widetilde{\\alpha}$ as\n  a character $(\\overline{F}^\\times)^m\\rightarrow \\overline{F}^\\times$ given by $(t_1,...,t_m)\\mapsto t_1^{a_1}\\cdots t_m^{a_m}$\n  with $a_1,...,a_m\\in {\\mathbb Z}$ such that $-c_1\\le a_i\\le c_1$ for every $1\\le i\\le m$.\n  We have $$\\alpha(\\gamma)=\\widetilde{\\alpha}(\\Xi(\\gamma))=\\lambda_1^{a_1}\\cdots \\lambda_m^{a_m},$$\n  so $v(\\alpha(\\gamma))=\\sum_{i=1}^m a_i v(\\lambda_i)$. Hence $-m c_1c_2\\kappa\\le v(\\alpha(\\gamma))\\le m c_1c_2\\kappa$,\nproving the first assertion of the lemma. From this the last assertion is obvious.\n\n\\end{proof}\n\n\n\\begin{rem}\\label{r:Lem2.18-indep}\n  Suppose that $F$ runs over the completions of a number field ${\\mathbf F}$ at non-archimedean places $v$, that $G$ over $F$ comes from a fixed reductive group $G$ over ${\\mathbf F}$, and that $\\Xi$ comes from an embedding $G\\hookrightarrow GL_m$ over the integer ring of ${\\mathbf F}$ (at least for every $v$ where $G$ is unramified). Then $B_5$ of the lemma can be chosen to be independent of $v$ (and dependent only on the data over ${\\mathbf F}$). This is easy to see from the proof.\n\\end{rem}\n\n\n\n\\subsection{An easy lemma}\\label{sub:easy}\n\n  As before let $F$ be a finite extension of ${\\mathbb Q}_p$ with multiplicative norm $|\\cdot|:F^\\times\\rightarrow {\\mathbb R}^\\times_{>0}$ normalized such that a uniformizer is sent to the inverse of the residue field cardinality, and let $G$ be an unramified group over $F$ with a smooth reductive model over $\\mathcal{O}$. The notation for $T_\\gamma$ and $\\Phi_\\gamma$ for a semisimple $\\gamma\\in G(F)$ is as in the previous subsection.\n\n\n\\begin{lem}\\label{l:alpha(gamma)-is-integral}\n  Suppose that a semisimple $\\gamma\\in G(F)$ is conjugate to an element\n  of $G(\\mathcal{O})$ and that $\\alpha(\\gamma)\\neq 1$ and\n  $|1-\\alpha(\\gamma)|\\neq 1$ for $\\alpha\\in \\Phi_\\gamma$ (for any choice of $T_\\gamma$).\n  Then $|1-\\alpha(\\gamma)|\\le q^{-1}$.\n\\end{lem}\n\n\\begin{proof}\n\n  By the assumption we may assume $\\gamma\\in G(\\mathcal{O})$.\n  Choose a maximal torus $T$ in the centralizer of $\\gamma$ in $G$ over $\\mathcal{O}_{F^{\\mathrm{ur}}}$, the ring of integers in\n  $F^{\\mathrm{ur}}$, so that $\\gamma\\in T(\\mathcal{O})$.\n  Such a $T$ exists since a reductive group scheme admits a maximal torus under \\'etale localization (\\cite[Cor 3.2.7]{Conrad-reductive}).\n  Since $T$ should split over a finite \\'etale cover of $\\mathrm{Spec}\\, \\mathcal{O}_{F^{\\mathrm{ur}}}$, it should be that $T$ splits over $\\mathcal{O}_{F^{\\mathrm{ur}}}$ already. Then $\\alpha$ defines a character $T(\\mathcal{O}_{F^{\\mathrm{ur}}})\\rightarrow \\mathcal{O}_{F^{\\mathrm{ur}}}^{\\times}$ and\n  $1-\\alpha(\\gamma)\\in \\mathcal{O}_{F^{\\mathrm{ur}}}$. As $1-\\alpha(\\gamma)$ is not a $\\varpi$-adic unit,\n  $\\varpi$ divides $1-\\alpha(\\gamma)$. The lemma follows.\n\\end{proof}\n\n\n\n\n\\section{Plancherel measure on the unramified spectrum}\\label{s:Plancherel}\n\n\n\n\\subsection{Basic setup and notation}\\label{sub:notation-Plancherel}\n  Let $F$ be a finite extension of ${\\mathbb Q}_p$.\n  Suppose that $G$ is unramified over $F$.\n  Fix a hyperspecial subgroup $K$ of $G$.\n  Recall the notation from the start of \\S\\ref{sub:Satake-trans}.\n  In particular $\\Omega$ (resp. $\\Omega_F$) denotes the Weyl group for $(G_{\\overline{F}},T_{\\overline{F}})$\n  (resp. $(G,A)$). There is a natural ${\\rm Gal}(\\overline{F}\/F)$-action on $\\Omega$,\n  under which $\\Omega^{{\\rm Gal}(\\overline{F}\/F)}=\\Omega_F$. (See \\cite[\\S6.1]{Bor79}.)\n  Since $G$ is unramified, ${\\rm Gal}(\\overline{F}\/F)$ factors through a finite unramified Galois group.\n  Thus there is a well-defined action of ${\\rm Fr}$ on $\\Omega$, and\n  $\\Omega^{{\\rm Fr}}=\\Omega_F$.\n\nThe unitary dual $G(F)^{\\wedge}$ of $G(F)$, or simply $G^\\wedge$ if there is no danger of ambiguity, is equipped with Fell topology.\n  (This notation should not be confused with the dual group $\\widehat{G}$).\n  Let $G^{\\wedge,\\mathrm{ur}}$ denote the unramified spectrum in\n  $G^{\\wedge}$, and\n $G^{\\wedge,\\mathrm{ur},{\\rm temp}}$ its tempered sub-spectrum.\n The Plancherel measure $\\widehat{\\mu}^{\\mathrm{pl}}$ on $G^{\\wedge}$ is supported on the tempered spectrum $G^{\\wedge,{\\rm temp}}$.\n The restriction of\n $\\widehat{\\mu}^{\\mathrm{pl}}$ to $G^{\\wedge,\\mathrm{ur}}$ will be written as $\\widehat{\\mu}^{\\mathrm{pl,ur}}$.\n The latter is supported on $G^{\\wedge,\\mathrm{ur},{\\rm temp}}$.\n  Harish-Chandra's Plancherel formula (cf. \\cite{Wal03}) tells us that\n  $\\widehat{\\mu}^{\\mathrm{pl}}(\\widehat{\\phi})=\\phi(1)$ for all $\\phi\\in {\\mathcal{H}}(G(F))$. In particular,\n  $\\widehat{\\mu}^{\\mathrm{pl,ur}}(\\widehat{\\phi})=\\phi(1)$ for all $\\phi\\in {\\mathcal{H}}^{\\mathrm{ur}}(G(F))$.\n\n  \\subsection{The unramified tempered spectrum}\\label{s:unramified-spectrum}\n\n\n An unramified $L$-parameter $W^{\\mathrm{ur}}_F\\rightarrow {}^L G^{\\mathrm{ur}}$\n  is defined to be an $L$-morphism ${}^L H^{\\mathrm{ur}}\\rightarrow {}^L G^{\\mathrm{ur}}$ (\\S\\ref{sub:L-mor-unr-Hecke})\n  with $H=\\{1\\}$.\n  Two such parameters $\\varphi_1$ and $\\varphi_2$ are considered equivalent\n  if $\\varphi_1=g\\varphi_2 g^{-1}$ for some $g\\in \\widehat{G}$.\n  Consider the following sets:\n\n\n\\begin{enumerate}\n\\item irreducible unramified representations $\\pi$ of $G(F)$ up to isomorphism.\n\\item group homomorphisms $\\chi:T(F)\/T(F)\\cap K \\rightarrow {\\mathbb C}^\\times$\nup to $\\Omega_{F}$-action.\n\\item unramified $L$-parameters $\\varphi:W^{\\mathrm{ur}}_F\\rightarrow {}^L G^{\\mathrm{ur}}$\nup to equivalence.\n\\item elements of $(\\widehat{G}\\rtimes {\\rm Fr})_{{\\rm ss-conj}}$;\n  this set was defined in \\S\\ref{sub:Satake-trans}.\n\\item $\\Omega^{{\\rm Fr}}$-orbits in $\\widehat{T}\/({\\rm Fr}-{\\rm id})\\widehat{T}$.\n\\item $\\Omega_F$-orbits in $\\widehat{A}$.\n\\item ${\\mathbb C}$-algebra morphisms $\\theta:{\\mathcal{H}}^{\\mathrm{ur}}(G)\\rightarrow {\\mathbb C}$.\n\n\\end{enumerate}\n  Let us describe canonical maps among them in some directions.\n\\begin{itemize}\n\\item[(i)$\\rightarrow$(vii)] Choose any $0\\neq v\\in \\pi^K$.\nDefine $\\theta(\\phi)$ by $\\theta(\\phi)v=\\int_{G(F)} \\phi(g)\\pi(g)vdg$.\n\\item[(ii)$\\rightarrow$(i)]\n$\\pi$ is the unique unramified subquotient of ${\\rm n\\textrm{-}ind}^{G(F)}_{B(F)} \\chi$.\n\\item[(ii)$\\leftrightarrow$(vi)] Induced by\n$\\Hom(T(F)\/T(F)\\cap K , {\\mathbb C}^\\times)\\simeq\n\\Hom(A(F)\/A(F)\\cap K , {\\mathbb C}^\\times)$\n\\begin{equation}\\label{e:ii-vi}\\simeq \\Hom(X_*(A),{\\mathbb C}^\\times)\\simeq\\Hom(X^*(\\widehat{A}),{\\mathbb C}^\\times)\\simeq\n X_*(\\widehat{A})\\otimes_{\\mathbb Z} {\\mathbb C}^\\times\\simeq \\widehat{A}\\end{equation}\n where the second isomorphism is induced by $X_*(A)\\rightarrow A(F)$ sending $\\mu$ to $\\mu(\\varpi)$.\n\\item[(iii)$\\rightarrow$(iv)] Take $\\varphi({\\rm Fr})$.\n\\item[(v)$\\rightarrow$(iv)] Induced by\nthe inclusion $t\\mapsto t\\rtimes {\\rm Fr}$ from $\\widehat{T}$ to $ \\widehat{G}\\rtimes{\\rm Fr}$.\n\n\\item[(v)$\\rightarrow$(vi)] Induced by the surjection $\\widehat{T}\\twoheadrightarrow \\widehat{A}$,\nwhich is the dual of $A\\hookrightarrow T$. (Recall $\\Omega^{{\\rm Fr}}=\\Omega_F$.)\n\\item[(vii)$\\rightarrow$(vi)]\n  Via ${\\mathcal{S}}:{\\mathcal{H}}^{\\mathrm{ur}}(G)\\simeq {\\mathbb C}[X^*(\\widehat{A})]^{\\Omega_F}$,\n  $\\theta$ determines an element of (cf. \\eqref{e:ii-vi})\n  $$\\Omega_F\\backslash\\Hom(X^*(\\widehat{A}),{\\mathbb C}^\\times)\\simeq \\Omega_F\\backslash \\widehat{A}.$$\n\\end{itemize}\n\n\n\\begin{lem}\\label{l:unr-spec}\n  Under the above maps, the sets corresponding to (i)-(vii) are in bijection with each other.\n\\end{lem}\n\\begin{proof}\n  See \\S6, \\S7 and \\S10.4 of \\cite{Bor79}.\n\\end{proof}\n\n\n\n  Let $F'$ be the finite unramified extension of $F$ such that\n  ${\\rm Gal}(\\overline{F}\/F)$ acts on $\\widehat{G}$ through the faithful action of ${\\rm Gal}(F'\/F)$.\n  Write ${}^L G_{F'\/F}:=\\widehat{G}\\rtimes{\\rm Gal}(F'\/F)$.\n    Let $\\widehat{K}$ be a maximal compact subgroup of $\\widehat{G}$ which is ${\\rm Fr}$-invariant.\n  Denote by $\\widehat{T}_c$ (resp. $\\widehat{A}_c$) the maximal compact subtorus\n  of $\\widehat{T}$ (resp. $\\widehat{A}$).\n\n\\begin{lem}\\label{l:unr-temp-spec}\n  The above bijections restrict to the bijections among the sets consisting\n  of the following objects.\n\\begin{enumerate}\n\\item[(i)$_t$] irreducible unramified tempered representations $\\pi$ of $G(F)$\nup to isomorphism.\n\\item[(ii)$_t$] unitary group homomorphisms $\\chi:T(F)\/T(F)\\cap K \\rightarrow U(1)$ up to $\\Omega_{F}$-action.\n\\item[(iii)$_t$] unramified $L$-parameters $\\varphi:W^{\\mathrm{ur}}_F\\rightarrow {}^L G^{\\mathrm{ur}}$ with bounded image\nup to equivalence.\n\\item[(iv)$_t$] $\\widehat{G}$-conjugacy classes in $\\widehat{K}\\rtimes {\\rm Fr}$ (viewed in ${}^L G_{F'\/F}$).\n\\item[(iv)$'_t$] $\\widehat{K}$-conjugacy classes in $\\widehat{K}\\rtimes {\\rm Fr}$ (viewed in $\\widehat{K}\\rtimes{\\rm Gal}(F'\/F)$).\n\\item[(v)$_t$] $\\Omega^{{\\rm Fr}}$-orbits in $\\widehat{T}_c\/({\\rm Fr}-{\\rm id})\\widehat{T}_c$.\n\\item[(vi)$_t$] $\\Omega_F$-orbits in $\\widehat{A}_c$.\n\\end{enumerate}\n(The boundedness in (iii)$_t$ means that the projection of ${\\rm Im\\,} \\varphi$ into\n${}^L G_{F'\/F}$ is contained in a maximal compact subgroup of ${}^L G_{F'\/F}$.)\n\\end{lem}\n\n\\begin{proof}\n  (i)$_t$$\\leftrightarrow$(ii)$_t$ is standard and (iii)$_t$$\\leftrightarrow$(iv)$_t$ is obvious.\n  Also straightforward is (ii)$_t$$\\leftrightarrow$(vi)$_t$ in view of \\eqref{e:ii-vi}.\n\n  Let us show that (v)$_t$$\\leftrightarrow$(vi)$_t$. Choose a topological isomorphism of complex tori\n  $\\widehat{T}\\simeq ({\\mathbb C}^\\times)^{d}$ with $d=\\dim T$.\n  Using ${\\mathbb C}^\\times \\simeq U(1)\\times {\\mathbb R}^\\times_{>0}$, we can decompose $\\widehat{T}=\\widehat{T}_c\\times \\widehat{T}_{nc}$\n  such that $\\widehat{T}_{nc}$ is carried over to $({\\mathbb R}^\\times_{>0})^d$ under the isomorphism.\n  The decomposition of $\\widehat{T}$ is canonical in that it is preserved under\n  any automorphism of $\\widehat{T}$. By the same reasoning, there is a canonical decomposition\n  $\\widehat{A}=\\widehat{A}_c\\times \\widehat{A}_{nc}$ with $\\widehat{A}_{nc}\\simeq ({\\mathbb R}^\\times_{>0})^{\\dim A}$.\n  The canonical surjection $\\widehat{T}\\rightarrow \\widehat{A}$ carries $\\widehat{T}_c$ onto $\\widehat{A}_c$\n  and $\\widehat{T}_{nc}$ onto $\\widehat{A}_{nc}$. (This reduces to the assertion in the case of ${\\mathbb C}^\\times$,\n  namely that any maps $U(1)\\rightarrow {\\mathbb R}^\\times_{>0}$ and ${\\mathbb R}^\\times_{>0}\\rightarrow U(1)$ induced by\n  an algebraic map ${\\mathbb C}^\\times\\rightarrow {\\mathbb C}^\\times$ of ${\\mathbb C}$-tori are trivial. This is easy to check.)\n  Therefore the isomorphism $\\widehat{T}\/({\\rm Fr}-{\\rm id})\\widehat{T}\\rightarrow \\widehat{A}$ of Lemma \\ref{l:unr-temp-spec}\n  induces an isomorphism\n  $\\widehat{T}_c\/({\\rm Fr}-{\\rm id})\\widehat{T}_c\\rightarrow \\widehat{A}_c$ (as well as\n  $\\widehat{T}_{nc}\/({\\rm Fr}-{\\rm id})\\widehat{T}_{nc}\\rightarrow \\widehat{A}_{nc}$).\n\n\n  Next we show that (iv)$_t$$\\leftrightarrow$(v)$_t$.\n  It is clear that $t\\mapsto t\\rtimes {\\rm Fr}$ maps (v)$_t$ into (iv)$_t$. Since (v)$_t$ and (iv)$_t$ are\n  the subsets of (v) and (iv), which are in bijective correspondence,\n  we deduce that (v)$_t$$\\rightarrow$(iv)$_t$ is injective. To show surjectivity, pick any\n  $k\\in \\widehat{K}$. There exists $t\\in \\widehat{T}$ such that the image of $t$ in (iv) corresponds\n  under (iv)$\\leftrightarrow$(v) to the $\\widehat{G}$-conjugacy class of $\\widehat{k}\\rtimes {\\rm Fr}$.\n  It is enough to show that we can choose $t\\in \\widehat{T}_c$.\n  Consider the subgroup $\\widehat{T}_c(t)$ of $$\\widehat{T}\/({\\rm Fr}-{\\rm id})\\widehat{T}\n  = \\widehat{T}_c\/({\\rm Fr}-{\\rm id})\\widehat{T}_c~ \\times ~\\widehat{T}_{nc}\/({\\rm Fr}-{\\rm id})\\widehat{T}_{nc}$$ generated by $\\widehat{T}_c\/({\\rm Fr}-{\\rm id})\\widehat{T}_c$\n  and the image of $t$.\n  The isomorphism (iv)$\\leftrightarrow$(v) maps $\\widehat{T}_c(t)$ into (v)$_t$ by\n  the assumption on $t$. Since (v)$_t$ form a compact set, the group\n  $\\widehat{T}_c(t)$ must be contained in a compact subset of $\\widehat{T}\/({\\rm Fr}-{\\rm id})\\widehat{T}$.\n  This forces the image of $t$ in $\\widehat{T}_{nc}\/({\\rm Fr}-{\\rm id})\\widehat{T}_{nc}$ to be trivial.\n  (Indeed, the latter quotient is isomorphic as a topological group\n  to a quotient of ${\\mathbb R}^{\\dim T}$ modulo an ${\\mathbb R}$-subspace via the exponential map. So\n  any subgroup generated by a nontrivial element is not contained in a compact set.)\n  Therefore $t$ can be chosen in $\\widehat{T}_c$.\n\n\n    It remains to verify that (iv)$_t$, (iv)$'_t$ and (v)$_t$ are in bijection.\n    Clearly (iv)$'_t$$\\rightarrow$(iv)$_t$ is onto. As we have just seen that\n    (iv)$_t$$\\leftrightarrow$(v)$_t$, it suffices to observe that (v)$_t$$\\rightarrow$(iv)$'_t$ is onto,\n   which is a standard fact (for instance in the context of\n    the (twisted) Weyl integration formula for $\\widehat{K}\\rtimes {\\rm Fr}$).\n\n\n\\end{proof}\n\n\n\n\\subsection{Plancherel measure on the unramified spectrum}\\label{sub:plan-unramified}\n\n Lemma \\ref{l:unr-temp-spec} provides a bijection $G^{\\wedge,\\mathrm{ur},{\\rm temp}}\\simeq\n \\Omega_F\\backslash \\widehat{A}_c$, which is in fact a topological isomorphism.\n The Plancherel measure $\\widehat{\\mu}^{\\mathrm{pl,ur}}$ on $G^{\\wedge,\\mathrm{ur}}$ is supported on $G^{\\wedge,\\mathrm{ur},{\\rm temp}}$.\n We would like to describe its pullback measure on $\\widehat{A}_c$,\n to be denoted $\\widehat{\\mu}^{\\mathrm{pl,ur,temp}}_{0}$. Note that $\\widehat{A}_c$ is topologically\n isomorphic to $\\widehat{T}_c\/({\\rm Fr}-{\\rm id})\\widehat{T}_c$. (This is induced by\n the natural surjection $\\widehat{T}_c \\twoheadrightarrow\\widehat{A}_c$.)\n  Fix a measure $d\\overline{t}$ on the latter which is a push forward from a Haar measure on $\\widehat{T}_c$.\n\n\\begin{prop}\\label{p:unr-plan-meas}\n\n  The measure $\\widehat{\\mu}^{\\mathrm{pl,ur,temp}}_{0}$ pulled back to $\\widehat{T}_c\/({\\rm Fr}-{\\rm id})\\widehat{T}_c$ is\n  $$\\widehat{\\mu}^{\\mathrm{pl,ur,temp}}_{0}(\\overline{t})= C\\cdot\n  \\frac{\\det(1-{\\rm ad}(t\\rtimes {\\rm Fr})|{\\rm Lie}\\,(\\widehat{G})\/{\\rm Lie}\\,(\\widehat{T}^{\\rm Fr}))}\n  {\\det(1-q^{-1}{\\rm ad}(t\\rtimes {\\rm Fr})|{\\rm Lie}\\,(\\widehat{G})\/{\\rm Lie}\\,(\\widehat{T}^{\\rm Fr}))}\n   d\\overline{t}\n  $$ for some constant $C\\in {\\mathbb C}^\\times$,\n  depending on the normalization of Haar measures.\n  Here $t\\in \\widehat{T}_c$ is any lift of $\\overline{t}$. (The\n  right hand side is independent\n  of the choice of $t$.)\n\\end{prop}\n\n\\begin{proof}\n  The formula is due to Macdonald (\\cite{Mac71}). For our purpose, it is more convenient to\n  follow the formulation as in the conjecture of \\cite[p.281]{Sha90} (which also discusses the general conjectural formula\n  of the Plancherel measure due to Langlands). By that conjecture (known in the unramified case),\n  $$\\widehat{\\mu}^{\\mathrm{pl,ur,temp}}_{0}(\\overline{t})= C'\\cdot \\frac{L(1,\\sigma^{-1}(\\overline{t}),r)}{L(0,\\sigma(\\overline{t}),r)}\n  \\frac{L(1,\\sigma(\\overline{t}),r)}{L(0,\\sigma^{-1}(\\overline{t}),r)} d\\overline{t}$$\n  where $C'\\in {\\mathbb C}^\\times$ is a constant,\n  $\\sigma(\\overline{t}):T(F)\\rightarrow {\\mathbb C}^\\times$ is the character corresponding to $\\overline{t}$\n  (via (ii)$\\leftrightarrow$(v) of Lemma \\ref{l:unr-spec}), and $r:{}^L T\n  \\rightarrow \\GL({\\rm Lie}\\,({}^L U))$ is the adjoint representation. Here ${}^L U$ is the $L$-group of $U$\n  (viewed in ${}^L B$). By unraveling the local $L$-factors, obtain\n   \\begin{equation}\\label{e:pf-unr-plan}\\widehat{\\mu}^{\\mathrm{pl,ur,temp}}_{0}(\\overline{t})= C'\\cdot\n  \\frac{\\det(1-{\\rm ad}(t\\rtimes {\\rm Fr})|{\\rm Lie}\\,(\\widehat{G})\/{\\rm Lie}\\,(\\widehat{T}))}\n  {\\det(1-q^{-1}{\\rm ad}(t\\rtimes {\\rm Fr})|{\\rm Lie}\\,(\\widehat{G})\/{\\rm Lie}\\,(\\widehat{T}))}\n   d\\overline{t}. \\end{equation}\n   Finally, observe that $\\det(1-q^{-s}{\\rm ad}(t\\rtimes {\\rm Fr})|{\\rm Lie}\\,(\\widehat{T})\/{\\rm Lie}\\,(\\widehat{T}^{{\\rm Fr}}))$\n   is independent of $\\overline{t}$ (and $t$). Therefore the right hand sides are\n   the same up to constant in \\eqref{e:pf-unr-plan} and the proposition.\n\\end{proof}\n\n\\begin{rem}\n  If the Haar measure on $G(F)$ assigns volume 1 to $K$ then\n  $C$ must be chosen such that $G^{\\wedge,\\mathrm{ur},{\\rm temp}}$ has total volume 1\n  with respect to $\\widehat{\\mu}^{\\mathrm{pl,ur,temp}}_{0}(\\overline{t})$. This is easily deduced from\n  the Plancherel formula for ${\\mathbf{1}}_K$.\n\\end{rem}\n\n\\section{Automorphic \\texorpdfstring{$L$}{L}-functions}\\label{sec:pp}\n\nAccording to Langlands conjectures, the most general $L$-functions should be expressible as products of the principal $L$-functions $L(s,\\Pi)$ associated to cuspidal automorphic representations $\\Pi$ of $\\GL(d)$ over number fields (for varying $d$). The analytic properties and functional equation of such $L$-functions were first established by Godement--Jacquet for general $d\\ge 1$. This involves the Godement--Jacquet integral representation. The other known methods are the Rankin--Selberg integrals, the doubling method and the Langlands--Shahidi method. The purpose of this section is to recall these analytic properties and to set-up notation. More detailed discussions may be found in~\\cites{cong:Lfunc04:cogd,Jacq79:principal,cong:park:mich},~\\cite{RS96}*{\\S2} and~\\cite{book:IK04}*{\\S5}.\n\nIn this section and some of the later sections we use the following notation.\n\\begin{itemize}\n\\item $F$ is a number field, i.e. a finite extension of ${\\mathbb Q}$.\n\\item $G$ is a connected reductive group over $F$ (not assumed to be quasi-split).\n\\item $Z=Z(G)$ is the center of $G$.\n\\item ${\\mathcal V}_F$ (resp. ${\\mathcal V}_F^\\infty$) is the set of all (resp. all finite) places of $F$.\n\\item $S_\\infty:={\\mathcal V}_F\\backslash {\\mathcal V}_F^\\infty$.\n\\item $A_{G}$ is the maximal $F$-split subtorus in the center of $\\Res_{F\/{\\mathbb Q}} G$, and $A_{G,\\infty}:=A_G({\\mathbb R})^0$.\n\\end{itemize}\n\n\\subsection{Automorphic forms}\\label{sec:pp:autforms}\n  Let $\\chi:A_{G,\\infty}\\rightarrow {\\mathbb C}^\\times$ be a continuous homomorphism.\n  Denote by $L^2_\\chi(G(F)\\backslash G({\\mathbb A}_F))$\n  the space of all functions $f$ on $G({\\mathbb A}_F)$ which are\n  square-integrable modulo $A_{G,\\infty}$ and satisfy\n  $f(g\\gamma z)=\\chi(z)f(\\gamma)$ for all $g\\in G(F)$, $\\gamma \\in G({\\mathbb A}_F)$ and\n  $z\\in A_{G,\\infty}$. There is a spectral decomposition into discrete and continuous parts\n$$L^2_{\\chi}(G(F)\\backslash G({\\mathbb A}_F))=L^2_{{\\rm disc},\\chi}\\oplus L^2_{{\\rm cont},\\chi},\n\\qquad L^2_{{\\rm disc},\\chi}=\\widehat{\\bigoplus_{\\pi}}\\, m_{{\\rm disc},\\chi}(\\pi)\\cdot \\pi$$\nwhere the last sum is a Hilbert direct sum running over the set of\nall irreducible representations\nof $G({\\mathbb A}_F)$ up to isomorphism.\nWrite ${\\mathcal{AR}}_{{\\rm disc},\\chi}(G)$ for the set of isomorphism classes of\nall irreducible representations $\\pi$\nof $G({\\mathbb A}_F)$ such that $m_{{\\rm disc},\\chi}(\\pi)>0$.\nAny $\\pi\\in {\\mathcal{AR}}_{{\\rm disc},\\chi}(G)$ is said to\nbe a discrete automorphic representation of $G({\\mathbb A}_F)$. If $\\chi$ is trivial (in particular if $A_{G,\\infty}=\\{1\\}$) then we write $m_{{\\rm disc}}$ for $m_{{\\rm disc},\\chi}$.\n\nThe above definitions allow a modest generalization. Let $\\mathfrak{X}_G$ be a closed subgroup of $Z({\\mathbb A}_F)$ containing $A_{G,\\infty}$ and $\\omega:Z({\\mathbb A}_F)\\cap \\mathfrak{X}_G\\backslash \\mathfrak{X}_G\\rightarrow {\\mathbb C}^\\times$ be a continuous (quasi-)character. Then $L^2_{\\omega}$, $L^2_{{\\rm disc},\\omega}$, $m_{{\\rm disc},\\omega}$ etc can be defined analogously. In fact the Arthur-Selberg trace formula applies to this setting. (See \\cite[Ch 3.1]{Arthur}.)\n\nFor the rest of Section~\\ref{sec:pp} we are concerned with $G=\\GL(d)$. Take $\\mathfrak{X}_G=Z({\\mathbb A}_F)$ so that $\\omega$ is a quasi-character of $Z(F)\\backslash Z(\\mathbb{A}_F)$. Note that $A_{G,\\infty}=Z(F_\\infty)^\\circ$ in this case. We denote by $\\mathcal{A}_\\omega(\\GL(d,F))$ the space consisting of automorphic functions on $\\GL(d,F)\\backslash \\GL(d,\\mathbb{A}_F)$ which satisfy $f(zg)=\\omega(z)f(g)$ for all $z\\in Z(\\mathbb{A}_F)$ and $g\\in \\GL(d,\\mathbb{A}_F)$ (see Borel-Jacquet~\\cite{BJ79} for the exact definition and the growth condition). We denote by $\\mathcal{A}_{{\\rm cusp},\\omega}(\\GL(d,F))$ the subspace of cuspidal functions (i.e. the functions with vanishing period against all nontrivial unipotent subgroups).\n\nAn automorphic representation $\\Pi$ of $\\GL(d,\\mathbb{A}_F)$ is by definition an irreducible admissible representation of $\\GL(d,\\mathbb{A}_F)$ which is a constituent of the regular representation on $\\mathcal{A}_\\omega(\\GL(d,F))$. Then $\\omega$ is the central character of $\\Pi$. The subspace $\\mathcal{A}_{{\\rm cusp},\\omega}(\\GL(d,F))$ decomposes discretely and an irreducible component is a cuspidal automorphic representation. The notion of cuspidal automorphic representations is the same if the space of cuspidal functions in $L^2_{\\omega}(GL(d,F)\\backslash GL(d,{\\mathbb A}_F))$ is used in the definition in place of $\\mathcal{A}_{{\\rm cusp},\\omega}(\\GL(d,F))$, cf. \\cite[\\S4.6]{BJ79}.\n\nWhen $\\omega$ is unitary we can work with the completed space $L_\\omega^2(\\GL(d,F)\\backslash \\GL(d,\\mathbb{A}_F))$ of square-integrable functions modulo $Z(\\mathbb{A}_F)$ and with unitary automorphic representations. Note that a cuspidal automorphic representation is unitary if and only if its central character is unitary. We recall the Langlands decomposition of $L_\\omega^2(\\GL(d,F)\\backslash \\GL(d,\\mathbb{A}_F))$ into the cuspidal, residual and continuous spectra. What will be important in the sequel is the notion of isobaric representations which we review in~\\S\\ref{sec:pp:isobaric}.\n\nIn the context of $L$-functions, the functional equation involves the contragredient representation $\\widetilde \\Pi$. An important fact is that the contragredient of a unitary automorphic representation of $\\GL(d,\\mathbb{A}_F)$ is isomorphic to its complex conjugate.\n\n\\subsection{Principal $L$-functions}\\label{sec:pp:Lfn}\n Let $\\Pi=\\otimes_v \\Pi_v$ be a cuspidal automorphic representation of $\\GL(d,\\mathbb{A}_F)$ with unitary central character. The principal $L$-function associated to $\\Pi$ is denoted\n\\begin{equation*}\nL(s,\\Pi)=\\prod_{v\\in \\mathcal{V}_F^\\infty } L(s,\\Pi_v).\n\\end{equation*}\nThe Euler product is absolutely convergent when $\\MRe s>1$. The completed $L$-function is denoted $\\Lambda(s,\\Pi)$, the product now running over all places $v\\in\\mathcal{V}_F$.\nFor each finite place $v\\in \\mathcal{V}_F^\\infty$, the inverse of the local $L$-function $L(s,\\Pi_v)$ is a Dirichlet polynomial in $q_v^{-s}$ of degree $\\le d$. Write\n\\begin{equation*}\nL(s,\\Pi_v)=\\prod^d_{i=1} (1-\\alpha_i(\\Pi_v)q_v^{-s})^{-1}.\n\\end{equation*}\nNote that when $\\Pi_v$ is unramified, $\\alpha_i(\\Pi_v)$ is non-zero for all $i$ and corresponds to the eigenvalues of a semisimple conjugacy class in $\\GL_d(\\mathbb{C})$ associated to $\\Pi_v$, but when $\\Pi_v$ is ramified the Langlands parameters are more sophisticated and we allow some (or even all of) of the $\\alpha_i(\\Pi_v)$ to be equal to zero. In this way we have a convenient notation for all local $L$-factors.\n\nFor each archimedean $v$, the local $L$-function $L(s,\\Pi_v)$ is a product of $d$ Gamma factors\n\\begin{equation}\\label{def:Lv-arch}\nL(s,\\Pi_v)=\\prod^d_{i=1} \\Gamma_v(s-\\mu_i(\\Pi_v)),\n\\end{equation}\nwhere $\\Gamma_\\mathbb{R}(s):=\\pi^{-s\/2}\\Gamma(s\/2)$ and $\\Gamma_\\mathbb{C}(s):=2(2\\pi)^{-s}\\Gamma(s)$. Note that $\\Gamma_{\\mathbb{C}}(s)=\\Gamma_\\mathbb{R}(s)\\Gamma_\\mathbb{R}(s+1)$ by the doubling formula, so when $v$ is complex, $L(s,\\Pi_v)$ may as well be expressed as a product of $2d$ $\\Gamma_\\mathbb{R}$ factors.\n\nThe completed $L$-function $\\Lambda(s,\\Pi):=L(s,\\Pi)\\prod_{v|\\infty}L(s,\\Pi_v)$ has the following analytic properties. It has a meromorphic continuation to the complex plane. It is entire except when $d=1$ and $\\Pi=\\abs{.}^{it}$ for some $t\\in \\mathbb{R}$, in which case $L(s,\\Pi)=\\zeta_F(s+it)$ is (a shift of) the Dedekind zeta function of the ground field $F$ with simple poles at $s=-it$ and $s=1-it$. It is bounded in vertical strips and satisfies the functional equation\n\\begin{equation}\\label{fneq}\n\\Lambda(s,\\Pi)=\\epsilon(s,\\Pi) \\Lambda(1-s,\\widetilde \\Pi),\n\\end{equation}\nwhere $\\epsilon(s,\\Pi)$ is the epsilon factor and $\\widetilde \\Pi$ is the contragredient automorphic representation. The epsilon factor has the form\n\\begin{equation}\n\\epsilon(s,\\Pi)=\\epsilon(\\Pi) q(\\Pi)^{\\frac{1}{2}-s}\n\\end{equation}\nfor some positive integer $q(\\Pi)\\in \\mathbb{Z}_{\\ge 1}$ and root number $\\epsilon(\\Pi)$ of modulus one.\n\nNote that $q(\\Pi)=q(\\widetilde \\Pi)$,\n$\\epsilon(\\widetilde \\Pi)=\\overline{\\epsilon(\\Pi)}$ and for all $v\\in \\mathcal{V}_F$, $L(s,\\widetilde \\Pi_v)=\\overline{L(\\overline{s},\\Pi_v)}$. For instance this follows from the fact~\\cite{GK75} that $\\widetilde \\Pi$ is isomorphic to the complex conjugate $\\overline\\Pi$ (obtained by taking the complex conjugate of all forms in the vector space associated to the representation $\\Pi$).\n\nThe conductor $q(\\Pi)$ is the product over all finite places $v\\in\\mathcal{V}^\\infty_F$ of the conductor $q(\\Pi_v)$ of $\\Pi_v$. Recall that $q(\\Pi_v)$ equals one whenever $\\Pi_v$ is unramified. It is convenient to introduce as well the conductor of admissible representations at archimedean places. When $v$ is real we let $C(\\Pi_v)=\\prod\\limits^d_{i=1} (2+\\abs{\\mu_i(\\Pi_v)})$ and when $v$ is complex we let $C(\\Pi_v)=\\prod\\limits^d_{i=1} (2+\\abs{\\mu_i(\\Pi_v)}^2)$. Then we let $C(\\Pi)$ be the analytic conductor which is the product of all the local conductors\n\\begin{equation*}\nC(\\Pi):=\n\\prod_{v\\mid \\infty} C(\\Pi_v)\n\\prod_{v\\in\\mathcal{V}^\\infty_F} q(\\Pi_v) = C(\\Pi_\\infty) q(\\Pi).\n\\end{equation*}\nNote that $C(\\Pi)\\ge 2$ always.\n\nThere is $0\\le \\theta <\\frac{1}{2}$ such that\n\\begin{equation}\\label{towardsR}\n\\MRe \\mu_i(\\Pi_v) \\le \\theta\n,\\qtext{resp.}\\\n\\log_{q_v} \\abs{\\alpha_i(\\Pi_v)}\\le \\theta\n\\end{equation}\nfor all archimedean $v$ (resp. finite $v$) and $1\\le i\\le d$. When $\\Pi_v$ is unramified we ask for\n\\begin{equation}\\label{unrtowardsR}\n\\abs{\\MRe \\mu_i(\\Pi_v)} \\le \\theta\n,\\qtext{resp.}\\\n\\abs{\\log_{q_v} \\abs{\\alpha_i(\\Pi_v)}}\\le \\theta.\n\\end{equation}\nThe value $\\theta=\\frac{1}{2} - \\frac{1}{d^2+1}$ is admissible by an argument of Serre and Luo--Rudnick--Sarnak based on the analytic properties of the Rankin-Selberg convolution $L(s,\\Pi\\times \\widetilde \\Pi)$. Note that for all $v$, the local $L$-functions $L(s,\\Pi_v)$ are entire on $\\MRe s>\\theta$ and this contains the central line $\\MRe s=\\frac{1}{2}$.\n\nThe generalized Ramanujan conjecture asserts that all $\\Pi_v$ are tempered (see~\\cite{Sarnak:GRC} and the references herein). This is equivalent to having $\\theta=0$ in the inequalities~\\eqref{towardsR} and~\\eqref{unrtowardsR}. In particular we expect that when $\\Pi_v$ is unramified, $\\abs{\\alpha_i(\\Pi_v)}=1$.\n\n\\subsection{Isobaric sums}\\label{sec:pp:isobaric} We need to consider slightly more general $L$-functions associated to non-cuspidal automorphic representations on $\\GL(d,\\mathbb{A}_F)$. These $L$-functions are products of the $L$-functions associated to cuspidal representations and studied in the previous \\S\\ref{sec:pp:Lfn}. Closely related to this construction it is useful to introduce, following Langlands~\\cite{cong:auto77:lang}, the notion of isobaric sums of automorphic representations. The concept of isobaric representations is natural in the context of $L$-functions and the Langlands functoriality conjectures.\n\nLet $\\Pi$ be an irreducible automorphic representation of $\\GL(d,\\mathbb{A}_F)$. Then a theorem of Langlands~\\cite{Langlands:notion} states that there are integers $r\\ge 1$ and $d_1,\\cdots,d_r\\ge 1$ with $d=d_1+\\cdots +d_r$ and cuspidal automorphic representations $\\Pi_1,\\cdots,\\Pi_r$ of $\\GL(d_1,\\mathbb{A}_F),\\cdots,\\GL(d_r,\\mathbb{A}_F)$  such that $\\Pi$ is a constituent of the induced representation of $\\Pi_1\\otimes \\cdots \\otimes \\Pi_r$ (from the Levi subgroup $\\GL(d_1)\\times \\cdots \\times \\GL(d_r)$ of $\\GL(d)$). A cuspidal representation is unitary when its central character is unitary. When all of $\\Pi_j$ are unitary then $\\Pi$ is unitary. But the converse is not true: note that even if $\\Pi$ is unitary, the representation $\\Pi_j$ need not be unitary in general.\n\nWe recall the generalized strong multiplicity one theorem of Jacquet and Shalika~\\cite{JSI-II}. Suppose $\\Pi$ and $\\Pi'$ are irreducible automorphic representations of $\\GL(d,\\mathbb{A}_F)$  such that $\\Pi_v$ is isomorphic to $\\Pi'_v$ for almost all $v\\in\\mathcal{V}_F$ (we say that $\\Pi$ and $\\Pi'$ are weakly equivalent) and suppose that $\\Pi$ (resp. $\\Pi'$) is a constituent of the induced representation of $\\Pi_1\\otimes\\cdots \\otimes \\Pi_r$ (resp. $\\Pi'_1\\otimes \\cdots \\otimes \\Pi'_{r'}$). Then $r=r'$ and up to permutation the sets of cuspidal representations $\\set{\\Pi_j}$ and $\\set{\\Pi'_j}$ coincide. Note that this generalizes the strong multiplicity one theorem of Piatetski-Shapiro which corresponds to the case where $\\Pi$ and $\\Pi'$ are cuspidal.\n\nConversely suppose $\\Pi_1,\\cdots, \\Pi_r$ are cuspidal representations of $\\GL(d_1,\\mathbb{A}_F), \\cdots, \\GL(d_r,\\mathbb{A}_F)$. Then from the theory of Eisenstein series there is a unique constituent of the induced representation of $\\Pi_1\\otimes\\cdots \\otimes \\Pi_r$ whose local components coincide at each place $v\\in \\mathcal{V}_F$ with the Langlands quotient of the local induced representation~\\cite{cong:auto77:lang}*{\\S2}. It is denoted $\\Pi_1\\boxplus \\cdots \\boxplus \\Pi_r$ and called an isobaric representation (automorphic representations which are not isobaric are called anomalous). The above results of Langlands and Jacquet--Shalika may now be summarized by saying that an irreducible automorphic representation of $\\GL(d,\\mathbb{A}_F)$ is weakly equivalent to a unique isobaric representation.\n\nWe now turn to $L$-functions. The completed $L$-function associated to an isobaric representation $\\Pi=\\Pi_1\\boxplus \\cdots\\boxplus \\Pi_r$ is by definition\n\\begin{equation*}\n\\Lambda(s,\\Pi) = \\prod^{r}_{j=1} \\Lambda(s,\\Pi_j).\n\\end{equation*}\nAll notation from the previous subsection will carry over to $\\Lambda(s,\\Pi)$. Namely we have the local $L$-factors $L(s,\\Pi_v)$, the local Satake parameters $\\alpha_i(\\Pi_v)$ and $\\mu_i(\\Pi_v)$, the epsilon factor $\\epsilon(s,\\Pi)$, the root number $\\epsilon(\\Pi)$, the local conductors $q(\\Pi_v)$, $C(\\Pi_v)$ and the analytic conductor $C(\\Pi)$. The Euler product converges absolutely for $\\MRe s$ large enough.\n\nOne important difference concerns the bounds for local Satake parameters. Even if we assume that $\\Pi$ has unitary central character the inequalities~\\eqref{towardsR} may not hold. We shall therefore require a stronger condition on $\\Pi$.\n\n\\begin{proposition}\\label{prop:tempered}\nLet $\\Pi$ be an isobaric representation of $\\GL(d,\\mathbb{A}_F)$. Assume that the archimedean component $\\Pi_\\infty$ is tempered. Then the bounds towards Ramanujan are satisfied. Namely there is a positive constant $\\theta<\\frac{1}{2}$ such that for all $1\\le i\\le d$ and all archimedean (resp. non-archimedean) places $v$,\n\\begin{equation}\\label{eq:prop:tempered}\n\\MRe \\mu_i(\\Pi_v) \\le \\theta\n,\\qtext{resp. }\\\n\\log_{q_v} \\abs{\\alpha_i(\\Pi_v)}\\le \\theta.\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof} Let $\\Pi=\\Pi_1\\boxplus \\cdots \\boxplus \\Pi_r$ be the isobaric decomposition with $\\Pi_j$ cuspidal. Then we will show that all $\\Pi_j$ have unitary central character, which implies the Proposition~\\ref{prop:tempered}.\n\nBy definition we have that $\\Pi_\\infty$ is a Langlands quotient of the induced representation of $\\Pi_{1\\infty} \\otimes \\cdots \\otimes \\Pi_{r\\infty}$. Since $\\Pi_\\infty$ is tempered, this implies that all $\\Pi_{j\\infty}$ are tempered, and in particular have unitary central character. Then the (global) central character of $\\Pi_j$ is unitary as well.\n\n\\end{proof}\n\n\\remark In analogy with the local case, an isobaric representation $\\Pi_1 \\boxplus \\cdots \\boxplus \\Pi_r$ where all cuspidal representations $\\Pi_j$ have unitary central character is called \\Lquote{tempered} in~\\cite{cong:auto77:lang}. This terminology is fully justified only under the generalized Ramanujan conjecture for $\\GL(d,\\mathbb{A}_F)$. To avoid confusion we use the adjective \\Lquote{tempered} for $\\Pi=\\otimes_v \\Pi_v$ only in the strong sense that the local representations $\\Pi_v$ are tempered for all $v\\in \\mathcal{V}_F$.\n\n\\remark In the proof of Proposition~\\ref{prop:tempered} we see the importance of the notion of isobaric representations and Langlands quotients. For instance a discrete series representation of $\\GL(2,\\mathbb{R})$ is a constituent (but not a Langlands quotient) of an induced representation of a non-tempered character of $\\GL(1,\\mathbb{R})\\times \\GL(1,\\mathbb{R})$.\n\n\\subsection{An explicit formula}\\label{sec:pp:explicit} Let $\\Pi$ be a unitary cuspidal representation of $\\GL(d,\\mathbb{A}_F)$. Let $\\rho_{j}(\\Pi)$ denote the zeros of $\\Lambda(s,\\Pi)$ counted with multiplicities. These are also the non-trivial zeros of $L(s,\\Pi)$. The method of Hadamard and de la Vall\\'ee Poussin generalizes from the Riemann zeta function to automorphic $L$-functions, and implies that $0<\\MRe \\rho_j(\\Pi)<1$ for all $j$. There is a polynomial $p(s)$ such that $p(s)\\Lambda(s,\\Pi)$ is entire and of order $1$ ($p(s)=1$ except when $d=1$ and $\\Pi=\\abs{.}^{it}$, in which case we choose $p(s)=(s-it)(1-it-s)$).\n\nThe Hadamard factorization shows that there are $a=a(\\Pi)$ and $b=b(\\Pi)$ such that\n\\begin{equation*}\np(s)\\Lambda(s,\\Pi)=e^{a+bs}\n\\prod_j \\bigl(1-\\frac{s}{\\rho_j(\\Pi)} \\bigr)\ne^{s\/\\rho_j(\\Pi)}.\n\\end{equation*}\nThe product is absolutely convergent in compact subsets away from the zeros $\\rho_j(\\Pi)$. The functional equation implies that\n\\begin{equation*}\n\\sum_j \\MRe(\\rho_j(\\Pi)^{-1})= -\\MRe b(\\Pi).\n\\end{equation*}\n\nThe number of zeros of bounded imaginary part is bounded above uniformly:\n\\begin{equation*}\n\\abs{\\set{j,\\ \\abs{\\Mim \\rho_j(\\Pi)}\\le 1}}\n\\ll \\log C(\\Pi).\n\\end{equation*}\nChanging $\\Pi$ into $\\Pi\\otimes \\abs{.}^{it}$ we have an analogous uniform estimate for the number of zeros with $\\abs{\\Mim \\rho_j(\\Pi)-T}\\le 1$ (in particular this is $\\ll_\\Pi \\log T$).\n\nLet $N(T,\\Pi)$ be the number of zeros with $\\abs{\\Mim \\rho_j(\\Pi)}\\le T$. Then the following estimate holds uniformly in $T\\ge 1$ (Weyl's law):\n\\begin{equation*}\n \nN(T,\\Pi) = \\frac{T}{\\pi} \\Bigl(d\n\\log (\\frac{T}{2\\pi e})\n+ \\log C(\\Pi)\n\\Bigr)\n+O_\\Pi(\\log T).\n\\end{equation*}\nThe error term could be made uniform in $\\Pi$, see~\\cite{book:IK04}*{\\S5.3} for more details\\footnote{One should be aware that Theorem~5.8 in~\\cite{book:IK04} does not apply directly to our setting because it is valid under certain further assumptions on $\\Pi$ such as $\\mu_i(\\Pi_v)$ being real for archimedean places $v$.}. The main term can be interpreted as the variation of the argument of $C(\\Pi)^{s\/2}L(s,\\Pi_\\infty)$ along certain vertical segments.\n\nWe are going to discuss an explicit formula (see~\\eqref{eq:prop:explicit} below) expressing a weighted sum over the zeros of $\\Lambda(s,\\Pi)$ as a contour integral. It is a direct consequence of the functional equation~\\eqref{fneq} and Cauchy formula. The explicit formula is traditionally stated using the Dirichlet coefficients of the $L$-function $L(s,\\Pi)$. For our purpose it is more convenient to maintain the Euler product factorization.\n\nDefinte $\\gamma_j(\\Pi)$ by $\\rho_j(\\Pi)=\\frac{1}{2}+i\\gamma_j(\\Pi)$. We know that $\\abs{\\Mim \\gamma_j(\\Pi)}<\\frac{1}{2}$ and under the GRH, $\\gamma_j(\\Pi)\\in \\mathbb{R}$ for all $j$.\n\nIt is convenient to denote by $\\frac{1}{2} + ir_j(\\Pi)$ the (eventual) poles of $\\Lambda(s,\\Pi)$ counted with multiplicity. We have seen that poles only occur when $\\Pi=\\abs{.}^{it}$ in which case the poles are simple and $\\set{r_j(\\Pi)}=\\set{t+\\frac{i}{2},-t-\\frac{i}{2}}$.\n\nThe above discussion applies with little change to isobaric representations. If we also assume that $\\Pi_\\infty$ is tempered then we have seen in the proof of Proposition~\\ref{prop:tempered} that $\\Pi=\\Pi_1 \\boxplus \\cdots \\boxplus \\Pi_r$ with $\\Pi_i$ unitary cuspidal representations of $\\GL(d_i,\\mathbb{A})$ for all $1\\le i\\le r$. In particular the bounds towards Ramanujan apply and $\\abs{\\Mim \\gamma_j(\\Pi)}<\\frac{1}{2}$ for all $j$.\n\nLet $\\Phi$ be a Schwartz function on $\\mathbb{R}$ whose Fourier transform\n\\begin{equation}\\label{def:fourier}\n\\widehat \\Phi(y):=\\int_{-\\infty}^{+\\infty} \\Phi(x) e^{-2\\pi i xy} dx\n\\end{equation}\nhas compact support. Note that under this assumption, $\\Phi$ may be extended to an entire function on $\\mathbb{C}$.\n\\begin{proposition}\\label{prop:explicit} Let $\\Pi$ be an isobaric representation of $\\GL(d,\\mathbb{A})$ satisfying the bounds towards Ramanujan~\\eqref{towardsR}. With notation as above and for $\\sigma>\\frac{1}{2}$, the following identity holds\n\\begin{equation}\\label{eq:prop:explicit}\n\\begin{aligned}\n\\sum_j &\\Phi(\\gamma_j(\\Pi))=\\sum_j \\Phi\n(r_j(\\Pi))\n+ \\frac{\\log q(\\Pi)}{2\\pi} \\widehat \\Phi(0)\n+\\\\\n&+ \\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}\n\\biggl[\n\\frac{\\Lambda'}{\\Lambda}(\\frac{1}{2}+\\sigma+ir,\\Pi)\n\\Phi(r-i\\sigma)\n+\n\\overline{\n\\frac{\\Lambda'}{\\Lambda}(\\frac{1}{2}+\\sigma+ir,\\Pi)\n}\n\\Phi(r+i\\sigma)\n\\biggr]\ndr.\n\\end{aligned}\n\\end{equation}\n\\end{proposition}\n\nThere is an important remark about the explicit formula that we will use frequently. Therefore we insert it here before going into the proof. The line of integration in~\\eqref{eq:prop:explicit} is away from the zeros and poles because $\\sigma>1\/2$. In particular the line of integration cannot be moved to $\\sigma=0$ directly. But we can do the following which is a natural way to produce the sum over primes. First we replace $\\Lambda(s,\\Pi)$ by its Euler product which is absolutely convergent in the given region ($\\MRe s>1$). Then for each of the term we may move the line of integration to $\\sigma=0$ because we have seen that $\\frac{L'}{L}(s,\\Pi_v)$ has no pole for $\\MRe s > \\theta$. Thus we have\n\\begin{equation}\\label{explicit:switch}\n\\int_{-\\infty}^{\\infty}\n\\frac{\\Lambda'}{\\Lambda}(\\frac{1}{2}+\\sigma+ir,\\Pi)\n\\Phi(r-i\\sigma)\ndr=\n\\sum_{v\\in \\mathcal{V}_F}\n\\int_{-\\infty}^{\\infty}\n\\frac{L'}{L}\n(\\frac{1}{2}+ir,\\Pi_v)\n\\Phi(r)\ndr.\n\\end{equation}\nThe latter expression is convenient to use in practice. The integral in the right-hand side of~\\eqref{explicit:switch} is absolutely convergent because $\\Phi$ is a Schwartz function and the sum over $v\\in\\mathcal{V}_F$ is actually finite since the support of $\\widehat \\Phi$ is compact. \\footnote{Note however that it is never allowed to switch the sum and integration symbols in~\\eqref{explicit:switch}. This is because the $L$-function is evaluated at the center of the critical strip in which the Euler product does not converge absolutely.}\n\n\\begin{proof} The first step is to work with the Mellin transform rather than the Fourier transform. Namely we set\n\\begin{equation*}\nH(\\frac{1}{2} + is)=\\Phi\\bigl(s\\bigr),\\quad s\\in \\mathbb{C}.\n\\end{equation*}\nNote that $H$ is an entire function which is rapidly decreasing on vertical strips. This justifies all shifting of contours below.\n\nWe form the integral\n\\begin{equation*}\n\\int_{(2)} \\frac{\\Lambda'}{\\Lambda}(s,\\Pi) H(s) \\frac{ds}{2i\\pi}.\n\\end{equation*}\nWe shift the contour to $\\MRe s=-1$ crossing zeros and eventual poles of $\\dfrac{\\Lambda'}{\\Lambda}$ inside the critical strip. The sum over the zeros reads\n\\begin{equation*}\n\\sum_j H(\\rho_j(\\Pi)) = \\sum_j \\Phi(\\gamma_j(\\Pi))\n\\end{equation*}\nand the sum over the poles reads\n\\begin{equation*}\n-\\sum_j \\Phi(r_j(\\Pi)).\n\\end{equation*}\n\nNote that since $\\epsilon(s,\\Pi)=\\epsilon(\\Pi) q(\\Pi)^{\\frac{1}{2}-s}$ we have\n\\begin{equation*}\n\\frac{\\epsilon'}{\\epsilon}(s,\\Pi)=-\\log q(\\Pi),\\quad s\\in\\mathbb{C}.\n\\end{equation*}\nWe obtain as consequence of the functional equation~\\eqref{fneq} that\n\\begin{equation*}\n\\begin{aligned}\n\\int_{(-1)}^{} \\frac{\\Lambda'}{\\Lambda}(s,\\Pi) H(s) \\frac{ds}{2i\\pi}&=\n\\int_{(2)}^{}  \\frac{\\Lambda'}{\\Lambda}(1-s,\\Pi) H(1-s) \\frac{ds}{2i\\pi}\\\\\n&=-\\int_{(2)}^{}  \\biggl(\\log q(\\Pi) + \\frac{\\Lambda'}{\\Lambda}(s,\\widetilde \\Pi)\n\\biggr) H(1-s) \\frac{ds}{2i\\pi}.\n\\end{aligned}\n\\end{equation*}\n\nNow we observe that\n\\begin{equation*}\n\\int_{(2)} H(s) \\frac{ds}{2i\\pi} = \\frac{1}{2\\pi} \\widehat \\Phi(0)\n\\end{equation*}\nand also\n\\begin{equation*}\n\\int_{(2)}^{} \\frac{\\Lambda'}{\\Lambda}(s,\\Pi) H(s) \\frac{ds}{2i\\pi}\n=\n\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}\n\\Phi(r-\\frac{3i}{2})\n\\frac{\\Lambda'}{\\Lambda}(2+ir,\\Pi) dr.\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{aligned}\n\\int_{(2)}^{} \\frac{\\Lambda'}{\\Lambda}(s,\\widetilde \\Pi) H(1-s) \\frac{ds}{2i\\pi}\n&=\n\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}\n\\Phi(r+\\frac{3i}{2})\n\\frac{\\Lambda'}{\\Lambda}(2-ir,\\widetilde \\Pi) dr\\\\\n&=\n\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}\n\\Phi(r+\\frac{3i}{2})\n\\overline{\n\\frac{\\Lambda'}{\\Lambda}(2+ir,\\Pi)\n}\ndr.\n\\end{aligned}\n\\end{equation*}\n\nSince $\\Lambda(s,\\widetilde \\Pi)=\\overline{\\Lambda(\\overline{s},\\Pi)}$ this concludes the proof of the proposition by collecting all the terms above. Precisely this yields the formula when $\\sigma=3\/2$, and then we can make $\\sigma>1\/2$ arbitrary by shifting the line of integration.\n\\end{proof}\n\nWe conclude this section with a couple of remarks on symmetries. The first observation is that the functional equation implies that if $\\rho$ is a zero (resp. pole) of $\\Lambda(s,\\Pi)$ then so is $1-\\bar\\rho$ (reflexion across the central line). Thus the set $\\set{\\gamma_j(\\Pi)}$ (resp. $\\set{r_j(\\Pi)}$) is invariant by the reflexion across the real axis (namely $\\gamma$ goes into $\\overline{\\gamma}$). Note that this is compatible with the GRH which predicts that $\\MRe \\rho_j(\\Pi)=\\frac{1}{2}$ and $\\gamma_j(\\Pi)\\in \\mathbb{R}$.\n\nAssuming $\\Phi$ is real-valued the explicit formula is an identity between real numbers. Indeed the Schwartz reflection principle gives $\\Phi(s)=\\overline{\\Phi(\\overline s)}$ for all $s\\in \\mathbb{C}$. Because of the above remark the sum over the zeros (resp. poles) in~\\eqref{eq:prop:explicit} is a real; the integrand is real-valued as well for all $r\\in (-\\infty,\\infty)$.\n\nThe situation when $\\Pi$ is self-dual occurs often in practice. The zeros $\\gamma_j(\\Pi)$ satisfy another symmetry which is the reflexion across the origin. Assuming $\\Pi$ is cuspidal and non-trivial there is no pole. The explicit formula~\\eqref{eq:prop:explicit} simplifies and may be written\n\\begin{equation*}\n\\sum_j\n\\Phi\\left( \\gamma_j(\\Pi) \\right)\n=\\frac{\\log q(\\Pi)}{2\\pi}\\widehat \\Phi(0)\n+\\frac{1}{\\pi}\n\\sum_{v\\in \\mathcal{V}_F}\n\\int_{-\\infty}^{\\infty}\n\\frac{L'}{L}(\\frac{1}{2}+ir,\\Pi_v)\n\\Phi(r) dr.\n\\end{equation*}\n\n\n\n\n\\section{Sato-Tate equidistribution}\\label{s:Sato-Tate}\n\n  Let $G$ be a connected reductive group over a number field $F$ as in the previous section.\n  The choice of a ${\\rm Gal}(\\overline{F}\/F)$-invariant\n  splitting datum $(\\widehat{B},\\widehat{T},\\{X_{\\alpha}\\}_{\\alpha\\in \\Delta^{\\vee}})$\n  as in \\S\\ref{sub:L-groups} induces\n  a composite map ${\\rm Gal}(\\overline{F}\/F)\\rightarrow {\\rm Out}(\\widehat{G})\\hookrightarrow {\\rm Aut}(\\widehat{G})$ with open kernel. Let $F_1$ be the unique finite\n  extension of $F$ in $\\overline{F}$ such that\n  $${\\rm Gal}(\\overline{F}\/F)\\twoheadrightarrow {\\rm Gal}(F_1\/F)\\hookrightarrow {\\rm Aut}(\\widehat{G}).$$\n\n\n\n\\subsection{Definition of the Sato-Tate measure}\\label{sub:ST-meas}\n\n  Set $\\Gamma_1:={\\rm Gal}(F_1\/F)$.\n  Let $\\widehat{K}$ be a maximal compact subgroup of $\\widehat{G}$ which is $\\Gamma_1$-invariant.\n  (It is not hard to see that such a $\\widehat{K}$ exists, cf. \\cite{JZ08}.)\n  Set $\\widehat{T}_{c}:=\\widehat{T}\\cap\\widehat{K}$. (The subscript $c$ stands for ``compact'' as it was in \\S\\ref{sub:plan-unramified}.)\n  Denote by $\\Omega_c$ the Weyl group for $(\\widehat{K},\\widehat{T}_c)$.\n\n  Let $\\theta\\in \\Gamma_1$.\n  Define $\\Omega_{c,\\theta}$ to be the subset of $\\theta$-invariant elements of $\\Omega_c$.\n  Consider the topological quotient $\\widehat{K}^\\natural_\\theta$ of\n  $\\widehat{K}\\rtimes \\theta$ by the $\\widehat{K}$-conjugacy equivalence relation.\n  Set $\\widehat{T}_{c,\\theta}:=\\widehat{T}_c\/(\\theta-{\\rm id})\\widehat{T}_c$.\n  Note that the action of $\\Omega_{c,\\theta}$ on $\\widehat{T}_c$ induces an action\n  on $\\widehat{T}_{c,\\theta}$.\n  The inclusion $\\widehat{T}_c\\hookrightarrow \\widehat{K}$ induces a canonical topological isomorphism (cf. Lemma \\ref{l:unr-temp-spec})\n \\begin{equation}\\label{e:K-T-isom}\\widehat{K}^\\natural_\\theta\\simeq \\widehat{T}_{c,\\theta}\/\\Omega_{c,\\theta}.\\end{equation}\n\n\n\n  The Haar measure on $\\widehat{K}$ (resp. on $\\widehat{T}_c$) with total volume 1 is\n  written as $\\mu_{\\widehat{K}}$ (resp. $\\mu_{\\widehat{T}_c}$).\n  Then $\\mu_{\\widehat{K}}$ on $\\widehat{K}\\rtimes\\theta$ induces the quotient measure\n  $\\mu_{\\widehat{K}^\\natural_\\theta}$ (so that\n  for any continuous function $f^\\natural$ on $\\widehat{K}_\\theta^\\natural$ and its pullback\n  $f$ on $\\widehat{K}$,\n  $\\int f^\\natural \\mu_{\\widehat{K}^\\natural_\\theta}=\\int f \\mu_{\\widehat{K}}$) thus also a measure $\\mu_{\\widehat{T}_c,\\theta}$ on $\\widehat{T}_{c,\\theta}$.\n\n\n\\begin{defn}\n  The $\\theta$-\\textbf{Sato-Tate measure} $\\widehat{\\mu}^{\\mathrm{ST}}_\\theta$\n  on $\\widehat{T}_{c,\\theta}\/\\Omega_{c,\\theta}$ is the measure\n  transported from $\\mu_{\\widehat{K}^\\natural_\\theta}$ via \\eqref{e:K-T-isom}.\n\\end{defn}\n\n\\begin{lem}\\label{l:ST-measure}\n  Let $\\widehat{\\mu}^{\\mathrm{ST}}_{\\theta,0}$ denote the measure on $\\widehat{T}_{c,\\theta}$\n  pulled back from\n  $\\widehat{\\mu}^{\\mathrm{ST}}_\\theta$ on $\\widehat{T}_{c,\\theta}\/\\Omega_{c,\\theta}$ (so that $\\int\n   f \\widehat{\\mu}^{\\mathrm{ST}}_{\\theta,0}=\n  \\int \\overline{f} \\widehat{\\mu}^{\\mathrm{ST}}_\\theta$ for every continuous $\\overline{f}$ on $\\widehat{T}_{c,\\theta}\/\\Omega_{c,\\theta}$\n  and its pullback $f$). Then\n  $$\\widehat{\\mu}^{\\mathrm{ST}}_{\\theta,0}=\\frac{1}{|\\Omega_{c,\\theta}|} D_\\theta(t) \\mu_{\\widehat{T}_{c,\\theta}},$$\n  where $D_\\theta(t)={\\det(1-{\\rm ad}(t\\rtimes \\theta)|{\\rm Lie}\\,(\\widehat{K})\/{\\rm Lie}\\,(\\widehat{T}^\\theta_c))}$\n  and $t$ signifies a parameter on $\\widehat{T}_{c,\\theta}$.\n\\end{lem}\n\n\\begin{proof}\n  The twisted Weyl integration formula tells us that\n  for a continuous $f:\\widehat{K}\\rightarrow {\\mathbb C}$,\n$$\\int_{\\widehat{K}} f(k)\\mu_{\\widehat{K}}\n= \\frac{1}{|\\Omega_{c,\\theta}|} \\int_{\\widehat{T}^{\\mathrm{reg}}_{c,\\theta}}\nD_\\theta(t) \\int_{\\widehat{K}_{t\\theta}\\backslash \\widehat{K}} f(x^{-1} t x^{\\theta}) \\cdot dx dt.$$\n  Notice that $\\widehat{K}_{t\\theta}$ is the twisted centralizer group of $t$ in $\\widehat{K}$\n  (or, the centralizer group of $t\\theta$ in $\\widehat{K}$).\n  On the right hand side, $ \\mu_{\\widehat{T}_{c,\\theta}}$ is used for integration.\n  When $f$ is a pullback from $\\widehat{K}^\\natural_{\\theta}$, the formula simplifies as\n$$\\int_{\\widehat{K}^\\natural_{\\theta}} f(k) \\mu_{\\widehat{K}^\\natural_{\\theta}}\n= \\frac{1}{|\\Omega_{c,\\theta}|} \\int_{\\widehat{T}^{\\mathrm{reg}}_{c,\\theta}}\nD_\\theta(t) f(t)\\cdot \\mu_{\\widehat{T}_{c,\\theta}}$$\nand the left hand side is equal to\n$\\int_{\\widehat{T}_{c,\\theta}} f(t) \\widehat{\\mu}^{\\mathrm{ST}}_{\\theta,0}$ by definition.\n\\end{proof}\n\n\n\\subsection{Limit of the Plancherel measure versus the Sato-Tate measure}\\label{sub:lim-of-Plan}\n\n\n  Let $\\theta,\\tau\\in \\Gamma_1$.\n  Then clearly $\\Omega_{c,\\theta}=\\Omega_{c,\\tau\\theta\\tau^{-1}}$,\n  $\\widehat{K}^\\natural_\\theta \\simeq \\widehat{K}^\\natural_{\\tau\\theta\\tau^{-1}}$\n via $k\\mapsto \\tau(k)$ and $\\widehat{T}_{c,\\theta}\\simeq \\widehat{T}_{c,\\tau\\theta\\tau^{-1}}$ via $t\\mapsto \\tau(t)$.\n Accordingly $\\widehat{\\mu}^{\\mathrm{ST}}_\\theta$ and $\\widehat{\\mu}^{\\mathrm{ST}}_{\\theta,0}$ are identified with\n $\\widehat{\\mu}^{\\mathrm{ST}}_{\\tau\\theta\\tau^{-1}}$ and $\\widehat{\\mu}^{\\mathrm{ST}}_{\\tau\\theta\\tau^{-1},0}$, respectively.\n\n Fix once and for all a set of representatives ${\\mathscr C}(\\Gamma_1)$ for conjugacy classes in $\\Gamma_1$.\n For $\\theta\\in {\\mathscr C}(\\Gamma_1)$, denote by $[\\theta]$ its conjugacy class. For each finite place $v$\n such that $G$ is unramified over $F_v$, the geometric Frobenius ${\\rm Fr}_v\\in {\\rm Gal}(F_v^{\\mathrm{ur}}\/F_v)$\n gives a well-defined conjugacy class $[{\\rm Fr}_v]$ in $\\Gamma_1$.\n  The set of all finite places $v$ of $F$ where $G$ is unramified is partitioned into\n  $$\\{{\\mathcal V}_F(\\theta)\\}_{\\theta\\in {\\mathscr C}(\\Gamma_1)}$$ such that\n  $v\\in {\\mathcal V}_F(\\theta)$ if and only if $[{\\rm Fr}_v]=[\\theta]$.\n\n\n   For each finite place $v$ of $F$,\n  the unitary dual of $G(F_v)$ and its Plancherel measure are written as $G^{\\wedge}_v$\n  and $\\widehat{\\mu}^{\\mathrm{pl}}_v$.\n  Similarly adapt the notation of \\S\\ref{sub:notation-Plancherel} by\n  appending the subscript $v$.\n  Now fix $\\theta\\in {\\mathscr C}(\\Gamma_1)$ and suppose that $G$ is unramified at $v$ and that\n  $v\\in {\\mathcal V}_F(\\theta)$. We choose $\\overline{F}\\hookrightarrow \\overline{F}_v$ such that ${\\rm Fr}_v$ has image $\\theta$ in $\\Gamma_1$\n  (rather than some other conjugate). This rigidifies\n  the identification in the second map below. (If ${\\rm Fr}_v$ maps to $\\tau\\theta\\tau^{-1}$ then the second map is twisted by $\\tau$.)\n \\begin{equation}\\label{e:local-isom-v}G(F_v)^{\\wedge,\\mathrm{ur},{\\rm temp}}\\stackrel{\\mathrm{canonical}}{\\simeq} \\widehat{T}_{c,{\\rm Fr}_v}\/\\Omega_{c,{\\rm Fr}_v}= \\widehat{T}_{c,\\theta}\/\\Omega_{c,\\theta}.\\end{equation}\n  By abuse of notation let $\\widehat{\\mu}^{\\mathrm{pl,ur,temp}}_{v}$ (a measure on $G(F_v)^{\\wedge,\\mathrm{ur},{\\rm temp}}$)\n  also denote the transported measure\n  on $\\widehat{T}_{c,\\theta}\/\\Omega_{c,\\theta}$.\n  Let $C_v$ denote the constant of Proposition \\ref{p:unr-plan-meas}, which\n  we normalize such that $\\widehat{\\mu}^{\\mathrm{pl,ur,temp}}_{v,0}$ has total volume 1.\n  Note that $\\widehat{\\mu}^{\\mathrm{ST}}_{\\theta}$ also has total volume 1.\n\n\n\\begin{prop}\\label{p:lim-of-Plancherel} Fix any $\\theta\\in {\\mathscr C}(\\Gamma_1)$.\nAs $v$ runs over ${\\mathcal V}_F(\\theta)$,\n  we have weak convergence $\\widehat{\\mu}^{\\mathrm{pl,ur,temp}}_{v}\\rightarrow \\widehat{\\mu}^{\\mathrm{ST}}_{\\theta}$ as $v\\rightarrow \\infty$.\n\\end{prop}\n\n\\begin{proof}\n  It is enough to show that $\\widehat{\\mu}^{\\mathrm{pl,ur,temp}}_{v,0}\\rightarrow \\widehat{\\mu}^{\\mathrm{ST}}_{\\theta,0}$ on $\\widehat{T}_{c,\\theta}$ as $v$ tends to $\\infty$\n   in ${\\mathcal V}_F(\\theta)$.\n   Consider the measure $\\widehat{\\mu}^{\\mathrm{pl,ur,temp}}_{v,1}:=C_v^{-1}\\widehat{\\mu}^{\\mathrm{pl,ur,temp}}_{v,0}$.\n   It is clear from the formula of Proposition \\ref{p:unr-plan-meas}\n   that $\\widehat{\\mu}^{\\mathrm{pl,ur,temp}}_{v,1}\\rightarrow \\widehat{\\mu}^{\\mathrm{ST}}_{\\theta}$ as $v\\rightarrow\\infty$ in ${\\mathcal V}_F(\\theta)$.\n   In particular, the total volume of $\\widehat{\\mu}^{\\mathrm{pl,ur,temp}}_{v,1}$ tends to 1, hence\n   $C_v\\rightarrow 1$ as $v\\rightarrow \\infty$  in ${\\mathcal V}_F(\\theta)$.\n   We conclude that $\\widehat{\\mu}^{\\mathrm{pl,ur,temp}}_{v,0}\\rightarrow \\widehat{\\mu}^{\\mathrm{ST}}_{\\theta,0}$ as desired.\n\\end{proof}\n\n\\begin{rem}\n  The above proposition was already noticed by Sarnak for $G=SL(n)$ in \\cite[\\S4]{Sarnak87}.\n\\end{rem}\n\n\n\\subsection{The generalized Sato-Tate conjecture}\\label{sub:gen-ST-conj}\n\n  Let $\\pi$ be a cuspidal\\footnote{If $\\pi$ is not cuspidal then\n  the hypothesis is never supposed to be satisfied.} tempered automorphic representation of $G({\\mathbb A}_F)$\n   satisfying\n\n\\medskip\\noindent\n\\textbf{Hypothesis.}\nThe conjectural global $L$-parameter $\\varphi_\\pi$ for $\\pi$ has Zariski dense image in ${}^L G_{F_1\/F}$.\n\\medskip\n\n  Of course this hypothesis is more philosophical than practical. The global\n  Langlands correspondence between\n  ($L$-packets of) automorphic representations and global $L$-parameters of $G({\\mathbb A}_F)$\n  is far from established. A fundamental problem here is that global $L$-parameters cannot be defined\n  unless the conjectural global Langlands group is defined.\n  (Some substitutes have been proposed by Arthur in the case of classical groups.\n  The basic idea is that a cuspidal automorphic representation of $\\GL_n$\n  can be put in place of an irreducible $n$-dimensional representation of the global Langlands group.)\n  Nevertheless, the above hypothesis can often be replaced with another condition,\n  which should be equivalent but can be stated without reference to conjectural objects.\n  For instance, when $\\pi$ corresponds to a Hilbert modular form of weight$\\ge 2$\n  at all infinite places, one can use the hypothesis that\n  it is not a CM form (i.e. not an automorphic induction from a Hecke character over a CM field).\n\n  Let us state a general form of the Sato-Tate conjecture. Let $q_v$ denote the cardinality of the residue field cardinality at a finite place $v$ of $F$. Define ${\\mathcal V}_F(\\theta,\\pi)^{\\le N}:=\\{v\\in{\\mathcal V}_F(\\theta,\\pi): q_v\\le N\\}$\n  for $N\\in {\\mathbb Z}_{>0}$.\n\n\\begin{conj}\\label{c:gen-ST}\n  Assume the above hypothesis. For each $\\theta\\in {\\mathscr C}(\\Gamma_1)$,\n  let ${\\mathcal V}_F(\\theta,\\pi)$ be the subset of $v\\in {\\mathcal V}_F(\\theta)$\n  such that $\\pi_v$ is unramified. Then $\\{\\pi_v\\}_{v\\in {\\mathcal V}_F(\\theta,\\pi)}$. More precisely\n  $$\\frac{1}{|{\\mathcal V}_F(\\theta,\\pi)^{\\le N}|}\n  \\sum_{v\\in {\\mathcal V}_F(\\theta,\\pi)^{\\le N}} \\delta_{\\pi_v}\n  \\rightarrow \\widehat{\\mu}^{\\mathrm{ST}}_\\theta\\quad\\mbox{as}\\quad N\\rightarrow\\infty.$$\n\n\\end{conj}\n\n  The above conjecture is deemed plausible in that it\n  is essentially a consequence of the Langlands functoriality conjecture\n  at least when $G$ is (an inner form of) a split group.\n  Namely if we knew that the $L$-function $L(s,\\pi,\\rho)$ for any irreducible representation ${}^L G\\rightarrow \\GL_d$\n  were a cuspidal automorphic $L$-function for $\\GL_d$ then\n  the desired equidistribution is implied by Theorem 1 of \\cite[App A.2]{Ser68l}.\n\n\\begin{rem}\n  In general when the above hypothesis is dropped, it is likely that\n  $\\pi$ comes from an automorphic representation on a smaller group than $G$.\n  (If $\\varphi_\\pi$ factors through an injective $L$-morphism ${}^L H_{F_1\/F}\\rightarrow {}^L G_{F_1\/F}$\n  then the Langlands functoriality predicts that $\\pi$ arises from an automorphic\n  representation of $H({\\mathbb A}_F)$.)\n  Suppose that the Zariski closure of ${\\rm Im\\,}(\\varphi_\\pi)$ in ${}^L G_{F_1\/F}$ is\n  isomorphic to ${}^L H_{F_1\/F}$ for some connected reductive group $H$ over $F$.\n  (In general the Zariski closure may consist of finitely many copies of ${}^L H_{F_1\/F}$.)\n  Then $\\{\\pi_v\\}_{v\\in {\\mathcal V}_F(\\theta,\\pi)}$ should be equidistributed according to\n  the Sato-Tate measure belonging to $H$ in order to be consistent with the functoriality conjecture.\n\\end{rem}\n\n  One can also formulate a version of the conjecture where $v$ runs over the set of \\emph{all} finite places where $\\pi_v$ are unramified by considering conjugacy classes in ${}^L G_{F_1\/F}$ rather than those in $\\widehat{G}\\rtimes \\theta$ for a fixed $\\theta$. For this let $\\widehat{K}^\\natural$ denote the quotient of $\\widehat{K}$ by the equivalence relation coming from the conjugation by $\\widehat{K}\\rtimes \\Gamma_1$. Since $\\widehat{K}^\\natural$ is isomorphic to a suitable quotient of $\\widehat{T}_c$, the Haar measure on $\\widehat{K}$ gives rise to a measure, to be denoted $\\widehat{\\mu}^{\\mathrm{ST}}$, on the quotient of $\\widehat{T}_c$. Let ${\\mathcal V}_F(\\pi)^{\\le N}$ (where $N\\in {\\mathbb Z}_{>0}$) denote the set of finite places of $F$ such that $\\pi_v$ are unramified and $q_v\\le N$. By writing $v\\rightarrow\\infty$ we mean that $q_v$ tends to infinity.\n\n\\begin{conj}\\label{c:gen-ST-2}\n   Assume the above hypothesis. Then $\\{\\pi_v\\}$\n  are equidistributed according to $\\widehat{\\mu}^{\\mathrm{ST}}_{\\theta}$ as $v\\rightarrow\\infty$. Namely\n  $$\\frac{1}{|{\\mathcal V}_F(\\pi)^{\\le N}|}\n  \\sum_{v\\in {\\mathcal V}_F(\\pi)^{\\le N}} \\delta_{\\pi_v}\n  \\rightarrow \\widehat{\\mu}^{\\mathrm{ST}}\\quad\\mbox{as}\\quad N\\rightarrow\\infty.$$\n\\end{conj}\n\n\\begin{rem}\n  Unlike Conjecture \\ref{c:gen-ST} it is unnecessary to choose embeddings $\\overline{F}\\hookrightarrow \\overline{F}_v$ to rigidify \\eqref{e:local-isom-v} since the ambiguity in the rigidification is absorbed in the conjugacy classes in ${}^L G_{F_1\/F}$. The formulation of Conjecture \\ref{c:gen-ST-2} might be more suitable than the previous one in the motivic setting where we would not want to fix $\\overline{F}\\hookrightarrow \\overline{F}_v$.\n   The interested reader may compare Conjecture \\ref{c:gen-ST-2}\n  with the motivic Sato-Tate conjecture of \\cite[13.5]{Ser94}.\n\\end{rem}\n\n  The next subsection will discuss the analogue of Conjecture \\ref{c:gen-ST} for automorphic families.\n    Conjecture \\ref{c:gen-ST-2} will not be considered any more in our paper. It is enough to mention that the analogue of the latter conjecture for families of algebraic varieties makes sense and appears to be interesting.\n\n\\subsection{The Sato-Tate conjecture for families}\\label{sub:ST-families}\n\n  The Sato-Tate conjecture has been proved for Hilbert modular forms in\n  (\\cite{BLGHT11}, \\cite{BLGG11}). Analogous equidistribution\ntheorems in the function field setting are due to Deligne and Katz. (See \\cite[Thm 9.2.6]{KS-RFM}\nfor instance.)\n  Despite these fantastic developments, we have little unconditional theoretical evidence for\n  the Sato-Tate conjecture for general reductive groups over number fields.\n  On the other hand, it has been noticed\n  that the analogue of the\n  Sato-Tate conjecture for families of automorphic representations is more\n  amenable to attack. Indeed there was some success in the case of holomorphic modular forms and Maass forms\n  (\\cite[Thm 2]{CDF97} \\cites{Royer:dimension-rang,ILS00,Serre:pl}).\n  The conjecture has the following coarse form, which should be thought of\n  as a guiding principle rather than a rigorous conjecture. Compare with some precise results\n  in \\S\\ref{sub:ST-theorem}.\n\n\\begin{conj}\\label{c:ST-families}\n   Let $\\{\\mathcal{F}_k\\}_{k\\ge1}$ be a ``general'' sequence of\n   finite families of automorphic representations of $G({\\mathbb A}_F)$ such that\n  $|\\mathcal{F}_k|\\rightarrow \\infty$ as $k\\rightarrow \\infty$. Then $\\{\\pi_v\\in \\mathcal{F}_k\\}$\n   are equidistributed according to $\\widehat{\\mu}^{\\mathrm{ST}}_{\\theta}$ as $k$ and $v$ tend to infinity\n   subject to the conditions that $v\\in {\\mathcal V}_F(\\theta)$ and that\n  all members of $\\mathcal{F}_k$ are unramified at $v$.\n\n\\end{conj}\n\n  We are not going to make precise what ``general'' means, but merely remark\n  that it should be the analogue of the condition that\n  the hypothesis of \\S\\ref{sub:gen-ST-conj} holds for the ``generic fiber''\n  of the family when the family has a geometric meaning (see also~\\cite{Sarn:family}). In practice one would verify\n  the conjecture for many interesting families while simply ignoring the word ``general''.\n  In some sense any family may be considered general if the conjecture holds for that family. Some relation between $k$ and $v$ holds when taking limit: $k$ needs to grow fast enough compared to $v$ (or more precisely $\\abs{\\mathcal{F}_k}$ needs to grow fast enough compared to $q_v$).\n\n  It is noteworthy that the unpleasant hypothesis of \\S\\ref{sub:gen-ST-conj}\n  can be avoided for families.\n  Also note that the temperedness\n  assumption is often unnecessary due to the fact that the Plancherel measure is supported on the\n  tempered spectrum. This is an indication that most representations in a family are globally tempered, which we will return to in a subsequent work.\n\n  Later we will\n  verify the conjecture for\n  many families in \\S\\ref{sub:ST-theorem}\n   as a corollary to the automorphic\n  Plancherel theorem proved earlier in \\S\\ref{s:aut-Plan-theorem}.\n  Our families arise as the sets of all automorphic representations with increasing level or weight, possibly with prescribed local conditions\n  at finitely many fixed places.\n\n\n\n\n\n\n\n\\section{Background materials}\n\\label{s:bg}\n\n  This section collects background materials in the local and global contexts. Subsections \\ref{sub:orb-int-const-term} and \\ref{sub:lem-ram} are concerned with $p$-adic groups while \\S\\ref{sub:st-disc}, \\S\\ref{sub:EP} and \\S\\ref{sec:b:fs} are with real and complex Lie groups. The rest is about global reductive groups.\n\n\\subsection{Orbital integrals and constant terms}\\label{sub:orb-int-const-term}\n  We introduce some notation in the $p$-adic context.\n\\begin{itemize}\n\\item $F$ is a finite extension of ${\\mathbb Q}_p$ with integer ring $\\mathcal{O}$.\n\\item $G$ is a connected reductive group over $F$.\n\\item $A$ is a maximal $F$-split torus of $G$, and put $M_0:=Z_G(A)$.\n\\item $K$ is a maximal compact subgroup of $G$\ncorresponding to a special point in the apartment for $A$.\n\\item $P=MN$ is a parabolic subgroup of $G$ over $F$, with $M$ and $N$ its Levi subgroup and\nunipotent radical, such that $M\\supset M_0$.\n\n\\item $\\gamma\\in G(F)$ is a semisimple element. (The case of a non-semisimple element\n  is not needed in this paper.)\n\\item $I_\\gamma$ is the neutral component\n  of the centralizer of $\\gamma$ in $G$. Then $I_\\gamma$ is a connected reductive group over $F$.\n\\item $\\mu_G$ (resp. $\\mu_{I_\\gamma}$) is a Haar measure on $G(F)$ (resp. $I_\\gamma(F)$).\n\\item $\\frac{\\mu_G}{\\mu_{I_\\gamma}}$ is\n  the quotient measure on $I_\\gamma(F)\\backslash G(F)$ induced by $\\mu_G$ and $\\mu_{I_\\gamma}$.\n\\item $\\phi\\in C^\\infty_c(G(F))$.\n\\end{itemize}\n  Define the orbital integral\n  $$O^{G(F)}_\\gamma(\\phi,\\mu_G,\\mu_{I_\\gamma}):=\n  \\int_{I_\\gamma(F)\\backslash G(F)} \\phi(x^{-1}\\gamma x) \\frac{\\mu_G}{\\mu_{I_\\gamma}}.$$\n  When the context is clear, we use $O_\\gamma(\\phi)$ as a shorthand notation.\n\n  We recall the theory of constant terms (cf. \\cite[p.236]{vDi72}).\n  Choose Haar measures $\\mu_K$, $\\mu_M$, $\\mu_N$,\n  on $K$, $M(F)$, $N(F)$, respectively, such that $\\mu_G=\\mu_K\\mu_M\\mu_N$ holds\n  with respect to $G(F)=KM(F)N(F)$.\n  Define the (normalized) constant term $\\phi_M\\in C^\\infty_c(M(F))$ by\n  \\begin{equation}\\label{e:const-term-formula}\\phi_{M}(m)=\\delta_P^{1\/2}(m)\\int_{N(F)}\\int_{K} \\phi(kmnk^{-1})\\mu_K \\mu_N.\\end{equation}\n  Although the definition of $\\phi_M$ involves not only $M$ but $P$, the following lemma\n  shows that the orbital integrals of $\\phi_M$ depend only on $M$ by the density of\n  regular semisimple orbital integrals, justifying our notation.\n\\begin{lem}\\label{l:orb-int-const-term}\n  For all $(G,M)$-regular semisimple $\\gamma\\in M(F)$,\n  $$O_\\gamma(\\phi_{M},\\mu_{M},\\mu_{I_\\gamma})=|D^G_M(\\gamma)|^{1\/2}\n  O_\\gamma(\\phi,\\mu_{G},\\mu_{I_\\gamma}).$$\n\\end{lem}\n\\begin{proof}\n  \\cite[Lem 9]{vDi72}. (Although the lemma is stated for regular elements $\\gamma\\in G$,\n  it suffices to require $\\gamma$ to be $(G,M)$-regular. See Lemma 8 of loc. cit.)\n\\end{proof}\n  It is standard that\n  the definition and facts we have recollected above extend to the adelic case. (Use \\cite[\\S\\S7-8]{Kot86},\n  for instance). We will skip rewriting the analogous definition in the adelic setting.\n\n  Now we restrict ourselves to the local unramified case.\n  Suppose that $G$ is unramified over $F$.\n  Let $B\\subset P \\subset G$ be Borel and parabolic subgroups defined over $F$.\n  Write $B=TU$ and $P=MN$ where $T$ and $M$ are Levi subgroups such that $T\\subset M$ and $U$ and $N$ are unipotent radicals.\n\n\\begin{lem}\\label{l:const-term-on-unram-Hecke} Let $\\phi\\in {\\mathcal{H}}^{\\mathrm{ur}}(G)$.\nThen ${\\mathcal{S}}^G_M(\\phi)=\\phi_M$, in particular\n${\\mathcal{S}}^G(\\phi)= {\\mathcal{S}}^M({\\mathcal{S}}^G_M \\phi)=\\phi_T$.\n\\end{lem}\n\n\\begin{proof}\n  Straightforward from \\eqref{e:Satake-formula} and \\eqref{e:const-term-formula}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{Gross's motives}\\label{sub:Gross-motive}\n\n  Now let $F$ be a finite extension of ${\\mathbb Q}$ (although Gross's theory applies more generally).\n  Let $G$ be a connected reductive group over $F$ and consider its  quasi-split inner form $G^*$.\n  Let $T^*$ be the centralizer of a maximal $F$-split torus of $G^*$. Denote by $\\Omega$ the Weyl group for $(G^*,T^*)$ over $\\overline{F}$.\n  Set $\\Gamma={\\rm Gal}(\\overline{F}\/F)$.\n  Gross (\\cite{Gro97}) attaches to $G$ an Artin-Tate motive\n  $${\\rm Mot}_G=\\bigoplus_{d\\ge 1} {\\rm Mot}_{G,d}(1-d)$$ with coefficients in ${\\mathbb Q}$.\n  Here $(1-d)$ denotes the Tate twist. The Artin motive ${\\rm Mot}_{G,d}$\n  (denoted $V_d$ by Gross) may be thought of as a $\\Gamma$-representation on a ${\\mathbb Q}$-vector space whose dimension is $\\dim {\\rm Mot}_{G,d}$.\n  Define $$L({\\rm Mot}_G):=  L(0,{\\rm Mot}_G)$$\n  to be the Artin $L$-value of $L(s,{\\rm Mot}_G)$ at $s=0$.\n  We recall some properties of ${\\rm Mot}_G$ from Gross's article.\n\n\\begin{prop}\\label{p:Gross-motives}\n  \\begin{enumerate}\n\\item ${\\rm Mot}_{G,d}$ is self-dual for each $d\\ge 1$.\n\\item $\\sum_{d\\ge 1} \\dim {\\rm Mot}_{G,d} = r_G={\\rm rk} G$.\n\\item $\\sum_{d\\ge 1} (2d-1)\\dim {\\rm Mot}_{G,d} = \\dim G$.\n\\item $|\\Omega|=\\prod_{d\\ge 1} d^{\\dim  {\\rm Mot}_{G,d}}$.\n\\item If $T^*$ splits over a finite extension $E$ of $F$ then\nthe $\\Gamma$-action on ${\\rm Mot}_G$ factors through ${\\rm Gal}(E\/F)$.\n\\end{enumerate}\n\\end{prop}\n\n  The Artin conductor ${\\mathfrak f}({\\rm Mot}_{G,d})$ is defined as follows.\n  Let $F'$ be the fixed field of the kernel of the Artin representation\n  ${\\rm Gal}(\\overline{F}\/F)\\rightarrow \\GL(V_d)$\n  associated to ${\\rm Mot}_{G,d}$.\n  For each finite place $v$ of $F$, let $w$ be any place of $F'$ above $v$.\n  Let $\\Gamma(v)_i:={\\rm Gal}(F'_w\/F_v)_i$ ($i\\ge 0$) denote the $i$-th ramification subgroups. Set\n  \\begin{equation}\\label{e:expo-conductor}\n  f(G_v,d):=\\sum_{i\\in {\\mathbb Z}_{\\ge 0}} \\frac{|\\Gamma(v)_i|}{|\\Gamma(v)_0|}\n  \\dim (V_d\/V_d^{\\Gamma(v)_i}),\\end{equation}\n  which is an integer independent of the choice of $w$. Write ${\\mathfrak p}_v$ for the prime ideal of $\\mathcal{O}_F$\n  corresponding to $v$. If $v$ is unramified in $E$ then $f(G_v,d)=0$.\n  Thus the product makes sense in the following definition.\n  $${\\mathfrak f}({\\rm Mot}_{G,d}):=\\prod_{v\\nmid \\infty} {\\mathfrak p}_v^{f(G_v,d)}$$\n  Let $E$ be the splitting field of $T^*$ (which is an extension of $F$)\nand set $s^{\\mathrm{sp}}_G:=[E:F]$.\n\n\n\\begin{lem}\\label{l:bounding-conductor}\n  For every finite place $v$ of $F$,\n  $$f(G_v,d)\\le (\\dim {\\rm Mot}_{G,d} )\\cdot (s^{\\mathrm{sp}}_G(1+e_{F_v\/{\\mathbb Q}_p} \\log_p s^{\\mathrm{sp}}_G)-1).$$\n\\end{lem}\n\n\\begin{proof}\n  Let $F'$, $w$ and $V_d$ be as in the preceding paragraph. Then $F\\subset F'\\subset E$.\n  Set $s_v:=[F'_w:F_v]$ so that $s_v\\le s^{\\mathrm{sp}}_G$.\n  The case $s_v=1$ is obvious (in which case $f(G_v,d)=0$), so we may assume $s_v\\ge 2$.\n  From \\eqref{e:expo-conductor} and Corollary \\ref{c:vanishing-ram-subgroup} below,\n$$f(G_v,d)\\le \\sum_{i\\ge 0} \\dim (V_d\/V_d^{\\Gamma(v)_i})\\le\n\\sum_{i=0}^{s_v(1+e_{F_v\/{\\mathbb Q}_p}\\log_p s_v)-2} \\dim V_d = (\\dim V_d)(s_v(1+e_{F_v\/{\\mathbb Q}_p}\\log_p s_v)-1).$$\n\\end{proof}\n\n  Recall that $w_G=|\\Omega|$ is the cardinality of the absolute Weyl group.\n  Let $s_G$ be the degree of the smallest extension of $F$ over which $G$ becomes split.\n The following useful lemma implies in particular that $s^{\\mathrm{sp}}_G\\le w_Gs_G$.\n\n\\begin{lem}(\\cite[Lem 2.2]{JKZ})\\label{l:torus-splitting}\n  For any maximal torus $T$ of $G$ defined over $F$, there exists a finite\nGalois extension $E$ of $F$ such that $[E:F]\\le w_Gs_G$ and $T$ splits over $E$.\n\\end{lem}\n\n\\subsection{Lemmas on ramification}\\label{sub:lem-ram}\n\n  This subsection is meant to provide an ingredient of proof\n  (namely Corollary \\ref{c:vanishing-ram-subgroup}) for Lemma \\ref{l:bounding-conductor}.\n  Fix a prime $p$.\n  Let $E$ and $F$ be finite extensions of ${\\mathbb Q}_p$ with uniformizers $\\varpi_E$ and\n  $\\varpi_F$, respectively. Normalize valuations $v_E:E^\\times\\rightarrow {\\mathbb Z}$ and\n  $v_F:F^\\times \\rightarrow {\\mathbb Z}$ such that $v_E(\\varpi_E)=v_F(\\varpi_F)=1$.\n  Write $e_{E\/F}\\in {\\mathbb Z}_{\\ge1}$ for the ramification index\n  and ${\\mathfrak D}_{E\/F}$ for the different.\n  For a nonzero principal ideal ${\\mathfrak a}$ of $\\mathcal{O}_E$,\n  we define $v_E({\\mathfrak a})$ to be $v_E(a)$ for any generator $a$ of ${\\mathfrak a}$. This is well defined.\n\n\\begin{lem}\\label{l:bound-diff-wild}\n  Let $E$ be a totally ramified Galois extension of $F$ with $[E:F]=p^n$ for $n\\ge 0$.\n  Then $$v_E({\\mathfrak D}_{E\/F})\\le p^n(1+n\\cdot e_{F\/{\\mathbb Q}_p})-1.$$\n\\end{lem}\n\n\\remark In fact the inequality is sharp. There are totally ramified extensions $E\/F$ for which the above equality holds as shown by \\\"Ore.\n\n\\begin{proof}\n  The lemma is trivial when $n=0$. Next assume $n=1$ but allow $E\/F$ to be a non-Galois extension.\n  Let $f(x)=\\sum_{i=0}^p a_i x^i\\in \\mathcal{O}_F[x]$ (with $a_p=1$ and $v_F(a_i)\\ge 1$ for $i<p$)\n  be the Eisenstein polynomial having $\\varpi_E$ as a root.\n  By \\cite[III.6, Cor 2]{Ser79}, $v_E({\\mathfrak D}_{E\/F})=v_E(f'(\\varpi_E))$. The latter equals\n  $$v_E\\left(\\sum_{i=1}^{p} i a_i \\varpi_E^{i-1}\\right)=\\min_{1\\le i\\le p}v_E(i a_i \\varpi_E^{i-1})\n    \\le v_E(p \\varpi_E^{p-1})=e_{E\/{\\mathbb Q}_p}+p-1.$$\n\n  This prepares us to tackle the case of arbitrary $n$. Choose a sequence of subextensions\n  $E=F_0\\supset F_1\\supset \\cdots \\supset F_n=F$ such that $[F_m:F_{m+1}]=p$ (where $F_m\/F_{m+1}$ may not be a Galois extension).\n  By above, $v_{F_m}({\\mathfrak D}_{F_m\/F_{m+1}})\\le e_{F_m\/{\\mathbb Q}_p}+p-1$ for $0\\le m\\le n-1$.\n  Hence\n  $$v_{E}({\\mathfrak D}_{E\/F})=\\sum_{m=0}^{n-1} v_E({\\mathfrak D}_{F_m\/F_{m+1}})\n  \\le \\sum_{m=0}^{n-1} p^m(e_{F_m\/{\\mathbb Q}_p}+p-1)\n  = n p^n e_{F\/{\\mathbb Q}_p}+p^n-1.$$\n\n\\end{proof}\n\n\n\\begin{lem}\\label{l:bound-diff-gen}\n  Let $E$ be a finite Galois extension of $F$. Then\n  $$v_E({\\mathfrak D}_{E\/F})\\le [E:F](1+e_{F\/{\\mathbb Q}_p} \\log_p [E:F])-1.$$\n\\end{lem}\n\n\\begin{proof}\n  Let $E^t$ (resp. $E^{\\mathrm{ur}}$) be the maximal tame (resp. unramified)\n  extension of $F$ in $E$. Then\n  $v_{E^t}({\\mathfrak D}_{E^t\/E^{\\mathrm{ur}}})=[E^t:E^{\\mathrm{ur}}]-1$ by \\cite[III.6, Prop 13]{Ser79}.\n  Clearly $v_{E^{ur}}({\\mathfrak D}_{E^{\\mathrm{ur}}\/F})=0$.\n  Together with Lemma \\ref{l:bound-diff-wild}, we obtain\n$$v_E({\\mathfrak D}_{E\/F})=v_{E}({\\mathfrak D}_{E\/E^t})+\n[E:E^t]v_{E^t}({\\mathfrak D}_{E^t\/E^{\\mathrm{ur}}})$$ $$\\le\n[E:E^t](1+e_{E^t\/{\\mathbb Q}_p}  \\log_p [E:E^t] )-1\n+ [E:E^t]([E^t:E^{\\mathrm{ur}}]-1)$$\n  $$=[E:E^{\\mathrm{ur}}](1+e_{F\/{\\mathbb Q}_p} \\log_p [E:E^t])-1\n  \\le [E:F](1+e_{F\/{\\mathbb Q}_p} \\log_p [E:F])-1. $$\n\n\\end{proof}\n\n\n\n\\begin{cor}\\label{c:vanishing-ram-subgroup}\n  Let $E$ be a finite Galois extension of $F$. Then\n  the $i$-th ramification group\n ${\\rm Gal}(E\/F)_{i}$ is trivial for $i=[E:F](1+e_{F\/{\\mathbb Q}_p} \\log_p [E:F])-1$.\n\\end{cor}\n\n\\begin{proof}\n  In the notation of section IV.1 of \\cite{Ser79},\n  we have ${\\rm Gal}(E\/F)_{m}=1$ by definition if\n  $m=\\max_{1\\neq s\\in {\\rm Gal}(E\/F)} i_G(s)$.\n  But the proposition 4 in that section implies that $m\\le v_E({\\mathfrak D}_{E\/F})$,\n  so Lemma \\ref{l:bound-diff-gen} finishes the proof.\n\\end{proof}\n\n\n\n\n\\subsection{Stable discrete series characters}\\label{sub:st-disc}\n\n  In \\S\\ref{sub:st-disc} and \\S\\ref{sub:EP} we specialize to the situation\nof real groups.\n\n\\begin{itemize}\n\\item $G$ is a connected reductive group over ${\\mathbb R}$.\n\\item $A_{G,\\infty}=A_G({\\mathbb R})^0$ where $A_G$ is the maximal split torus\nin the center of $G$.\n\\item $K_\\infty$ is a maximal compact subgroup of $G({\\mathbb R})$\nand $K'_\\infty:=K_\\infty A_{G,\\infty}$.\n\\item $T$ is an ${\\mathbb R}$-elliptic maximal torus in $G$. (Assume that such a $T$ exists.)\n\\item $B$ is a Borel subgroup of $G$ over ${\\mathbb C}$ containing $T$.\n\\item $I_\\gamma$ denotes the connected centralizer of $\\gamma\\in G({\\mathbb R})$.\n\\item $\\Phi^+$ (resp. $\\Phi$) is the set of positive (resp. all)\nroots of $T$ in $G$ over ${\\mathbb C}$.\n\\item $\\Omega$ is the Weyl group for $(G,T)$ over ${\\mathbb C}$, and $\\Omega_c$ is the compact Weyl group.\n\\item $\\rho:=\\frac{1}{2}\\sum_{\\alpha\\in \\Phi^+} \\alpha$.\n\\item $\\xi$ is an irreducible finite dimensional algebraic representation of $G({\\mathbb R})$.\n\\item $\\lambda_\\xi\\in X^*(T)$ is the $B$-dominant highest weight for $\\xi$.\n\\item $m(\\xi):=\\min_{\\alpha\\in \\Phi^+} \\langle \\lambda_\\xi+\\rho,\\alpha\\rangle$. We always have $m(\\xi)>0$.\n\\item $\\Pi_{\\rm disc}(\\xi)$ is the set of irreducible discrete series representations\nof $G({\\mathbb R})$ with the same infinitesimal character and the same\ncentral character as $\\xi$. (This is an $L$-packet for $G({\\mathbb R})$.)\n\\item $D_\\infty^G(\\gamma):=\\prod_{\\alpha} |1-\\alpha(\\gamma)|$ for $\\gamma\\in T({\\mathbb R})$, where\n$\\alpha$ runs over elements of $\\Phi$ such that $\\alpha(\\gamma)\\neq 1$.\n(If $\\gamma$ is in the center of $G({\\mathbb R})$, $D_\\infty^G(\\gamma)=1$.)\n\\end{itemize}\n\n  If $M$ is a Levi subgroup of $G$ over ${\\mathbb C}$ containing $T$,\n  the following are defined in the obvious manner as above:\n  $\\Phi^+_M$, $\\Phi_M$, $\\Omega_M$, $\\rho_M$, $D_\\infty^M$. Define  $\\Omega^M:=\\{\\omega\\in \\Omega: \\omega^{-1}\\Phi^+_M\\subset \\Phi^+\\}$, which is a set of representatives for $\\Omega\/\\Omega_M$.\n   For each \\emph{regular} $\\gamma\\in T({\\mathbb R})$,\n  let us define (cf. \\cite[(4.4)]{Art89})\n  $$\\Phi^G_M(\\gamma,\\xi):=(-1)^{q(G_\\infty)}\n  D_\\infty^G(\\gamma)^{1\/2} D_\\infty^M(\\gamma)^{-1\/2}\\sum_{\\pi\\in \\Pi_{\\rm disc}(\\xi)}\n  \\Theta_\\pi(\\gamma)$$\n  where $\\Theta_\\pi$ is the character function of $\\pi$.\n  It is known that the function $\\Phi^G_M(\\gamma,\\xi)$\n  continuously extends to an $\\Omega_M$-invariant function on $T({\\mathbb R})$,\n  thus also to a function on $M({\\mathbb R})$ which is invariant under $M({\\mathbb R})$-conjugation and\n  supported on elliptic elements\n  (\\cite[Lem 4.2]{Art89}, cf. \\cite[Lem 4.1]{GKM97}).\n   When $M=G$, simply $\\Phi^G_M(\\gamma,\\xi)= {\\rm tr}\\, \\xi(\\gamma)$.\n\n   We would like to have an upper bound for $|\\Phi^G_M(\\gamma,\\xi)|$\n   that we will need in \\S\\ref{sub:weight-varies}.\n   This is a refinement of \\cite[Lem 4.8]{Shi-Plan}.\n\n\\begin{lem}\\label{l:dim-and-trace}\n\\begin{enumerate}\n\\item $\\dim \\xi=\\prod_{\\alpha\\in \\Phi^+} \\frac{\\langle \\alpha,\\lambda_\\xi+\\rho\\rangle}{\\langle \\alpha,\\rho\\rangle}$.\n\\item There exists a constant $c>0$ independent of $\\xi$ such that for every elliptic $\\gamma\\in G({\\mathbb R})$\nand $\\xi$,\n  $$\\frac{|{\\rm tr}\\,\\xi(\\gamma)|}{\\dim \\xi}\n  \\le c \\frac{D_\\infty^G(\\gamma)^{-1\/2}}{ m(\\xi)^{|\\Phi^+|-|\\Phi^+_{I_\\gamma}|}} .$$\n\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nPart (i) is the standard Weyl dimension formula. Let us prove (ii).\nThe formula right above the corollary 1.12 in \\cite{CC09} implies that\n  $$|{\\rm tr}\\,\\xi(\\gamma)|\\le D_\\infty^G(\\gamma)^{-1\/2} \\times\n\\sum_{\\omega\\in \\Omega^{I_\\gamma}} \\left(\\prod_{\\alpha\\in \\Phi^+_{I_\\gamma}}\n\\frac{ \\langle\\omega^{-1}\\alpha,\\lambda_\\xi+\\rho\\rangle}{\\langle \\alpha,\\rho_{I_\\gamma}\\rangle}\\right).$$\nNote that their $M$ is our $I_\\gamma$ and that\n$|\\alpha(\\gamma)|=1$ for all $\\alpha\\in \\Phi$ and all elliptic $\\gamma\\in G({\\mathbb R})$.\nHence by (i),\n$$\\frac{|{\\rm tr}\\,\\xi(\\gamma)|}{\\dim \\xi}\n  \\le D_\\infty^G(\\gamma)^{-1\/2}\\sum_{\\omega\\in \\Omega^{I_\\gamma}}\n  \\frac  {\\prod_{\\alpha\\in \\Phi^+} \\langle \\alpha,\\rho\\rangle} {\\prod_{\\alpha\\in \\Phi^+_{I_\\gamma}} \\langle \\alpha,\\rho_{I_\\gamma}\\rangle}\n  \\left(\\prod_{\\alpha\\in \\Phi^+\\backslash \\omega^{-1}\\Phi^+_{I_\\gamma}}\n  \\langle \\lambda_\\xi+\\rho,\\alpha\\rangle \\right)^{-1}$$\n$$ \\le D_\\infty^G(\\gamma)^{-1\/2}|\\Omega^{I_\\gamma}|  \\frac{\\prod_{\\alpha\\in \\Phi^+} \\langle \\alpha,\\rho\\rangle}{\\prod_{\\alpha\\in \\Phi^+_{I_\\gamma}} \\langle \\alpha,\\rho_{I_\\gamma}\\rangle}\n    m(\\xi)^{-(|\\Phi^+|-|\\Phi^+_{I_\\gamma}|)}.$$\n\\end{proof}\n\n\n\\begin{lem}\\label{l:bound-for-st-ds-char}\n  Assume that $M$ is a Levi subgroup of $G$ over ${\\mathbb R}$ containing an elliptic maximal torus.\nThere exists a constant $c>0$ independent of $\\xi$ such that for every elliptic\n$\\gamma\\in M({\\mathbb R})$,\n $$\\frac{|\\Phi^G_M(\\gamma,\\xi)|}{\\dim \\xi}\n \\le c \\frac{D_\\infty^M(\\gamma)^{-1\/2}}{ m(\\xi)^{|\\Phi^+|-|\\Phi^+_{I^M_\\gamma}|}} .$$\n\n\n\\end{lem}\n\n\\begin{proof}\n  As the case $M=G$ is already proved by Lemma \\ref{l:dim-and-trace}.(ii), we assume that $M\\subsetneq G$.\n  Fix an elliptic maximal torus $T\\subset M$. Since every elliptic element has a conjugate\n  in $T({\\mathbb R})$ and both sides of the inequality\n  are conjugate-invariant,\n  it is enough to verify the lemma for $\\gamma\\in T({\\mathbb R})$.\n  In this proof we borrow some notation and facts\n  from \\cite[pp.494-498]{GKM97} as well as \\cite[pp.272-274]{Art89}.\n  For the purpose of proving Lemma \\ref{l:bound-for-st-ds-char}, we may restrict to\n  $\\gamma\\in \\Gamma^+$, corresponding to a closed chamber for the root system of $T({\\mathbb R})$ in $G({\\mathbb R})$.\n  (See page 497 of \\cite{GKM97} for the precise definition.)\n  The proof of \\cite[Lem 4.1]{GKM97} shows that\n    $$\\Phi^G_M(\\gamma,\\xi)=\\sum_{\\omega\\in \\Omega^{M}} c(\\omega,\\xi)\\cdot {\\rm tr}\\, \\xi^M_\\omega(\\gamma)$$\n  where $\\xi^M_\\omega$ is the irreducible representation of $M({\\mathbb R})$ of highest weight\n  $\\omega(\\xi+\\rho)-\\rho_M$. We claim that there is a constant $c_1>0$  independent of $\\xi$\n  such that $$|c(\\omega,\\xi)|\\le c_1$$ for all $\\omega$ and $\\xi$.\n  The coefficients $  c(\\omega,\\xi)$ can be computed by rewriting the right hand\n  side of \\cite[(4.8)]{Art89}\n  as a linear combination of ${\\rm tr}\\, \\xi^M_\\omega(\\gamma)$ using the Weyl character formula.\n  In order to verify the claim, it suffices to point out that $\\overline{c}(Q^+_{ys\\lambda},R^+_H)$\n  in Arthur's (4.8) takes values in a finite set which is independent of $\\xi$ (or\n  $\\tau$ in Arthur's notation). This is obvious: as $Q^+_{ys\\lambda}\\subset\n  \\Phi^\\vee$ and $R^+_H\\subset \\Phi$, there are finitely many possibilities for\n  $Q^+_{ys\\lambda}$ and $R^+_H$.\n\n\n  Now by Lemma \\ref{l:dim-and-trace}.(i),\n    $$\\frac{\\dim \\xi^M_\\omega}{\\dim \\xi}= \\frac{\\prod_{\\alpha\\in \\Phi^+} \\langle \\alpha,\\rho \\rangle}\n   {\\prod_{\\alpha\\in \\Phi_M^+} \\langle \\alpha,\\rho_M \\rangle} \\prod_{\\alpha\\in\n   \\Phi^+\\backslash \\Phi^+_M} \\langle \\alpha,\\lambda_\\xi+\\rho\\rangle^{-1}\n   \\le c_2  m(\\xi)^{-(|\\Phi^+|-|\\Phi^+_M|)}$$\n   with $c_2=(\\prod_{\\alpha\\in \\Phi^+} \\langle \\alpha,\\rho \\rangle)\n   (\\prod_{\\alpha\\in \\Phi_M^+} \\langle \\alpha,\\rho_M \\rangle)^{-1}>0$.\n    According to Lemma \\ref{l:dim-and-trace}.(ii),\n    there exists a constant $c_3>0$ such that\n    $$\\frac{|{\\rm tr}\\,\\xi^M_\\omega(\\gamma)|}{\\dim \\xi^M_\\omega}\n  \\le c_3 \\frac{D_\\infty^M(\\gamma)^{-1\/2}}{ m(\\xi)^{|\\Phi^+_M|-|\\Phi^+_{I^M_\\gamma}|}} .$$\n  To conclude the proof, multiply the last two formulas.\n\n\\end{proof}\n\n\\subsection{Euler-Poincar\\'{e} functions}\\label{sub:EP}\n\n  We continue to use the notation of \\S\\ref{sub:st-disc}.\n  Let $\\overline{\\mu}^{{\\rm EP}}_\\infty$ denote the Euler-Poincar\\'{e} measure on $G({\\mathbb R})\/A_{G,\\infty}$\n  (so that its induced measure on the compact inner form has volume 1).\nThere exists a unique Haar measure $\\mu^{{\\rm EP}}_\\infty$ on $G({\\mathbb R})$\nwhich is compatible with $\\overline{\\mu}^{{\\rm EP}}_\\infty$ and the standard Haar measure on $A_{G,\\infty}$.\nWrite $\\omega_{\\xi}$ for the central character of $\\xi$ on $A_{G,\\infty}$.\n  Let $\\Pi(\\omega_{\\xi}^{-1})$ denote the set of irreducible admissible representations of $G({\\mathbb R})$\n  whose central characters on $A_{G,\\infty}$ are $\\omega_{\\xi}^{-1}$.\n  For $\\pi \\in \\Pi(\\omega_{\\xi}^{-1})$,\n  define\n  $$\\chi_{{\\rm EP}}(\\pi\\otimes \\xi):=\\sum_{i\\ge 0} (-1)^i \\dim H^i({\\rm Lie}\\, G({\\mathbb R}),K'_\\infty,\n  \\pi \\otimes \\xi).$$\n  Clozel and Delorme (\\cite{CD90}) constructed a bi-$K_\\infty$-finite\n  function $\\phi_\\xi\\in C^\\infty(G({\\mathbb R}))$ which transforms under $A_{G,\\infty}$ by $\\omega_{\\xi}$ and is compactly supported modulo $A_{G,\\infty}$,\n  such that\n  $$\\forall \\pi \\in \\Pi(\\omega_{\\xi}^{-1}),\\quad {\\rm tr}\\, \\pi(\\phi_\\xi,\\mu^{{\\rm EP}}_\\infty)=\\chi_{{\\rm EP}}(\\pi\\otimes\\xi).$$\n  Let $q(G)=\\frac{1}{2}\\dim_{{\\mathbb R}}G({\\mathbb R})\/K'_{\\infty}\\in {\\mathbb Z}$. The following are well-known:\n\\begin{itemize}\n\\item $\\chi_{{\\rm EP}}(\\pi\\otimes\\xi)=0$ unless $\\pi\\in \\Pi(\\omega_{\\xi}^{-1})$ has the same infinitesimal character as $\\xi^\\vee$.\n\\item If the highest weight of $\\xi$ is regular then $\\chi_{{\\rm EP}}(\\pi\\otimes \\xi)\\neq 0$\nif and only if $\\pi\\in \\Pi_{{\\rm disc}}(\\xi^\\vee)$.\n\\item If $\\pi \\in \\Pi(\\omega_{\\xi}^{-1})$ is a discrete series and $\\chi_{{\\rm EP}}(\\pi\\otimes \\xi)\\neq 0$\nthen $\\pi\\in \\Pi_{{\\rm disc}}(\\xi^\\vee)$ and $\\chi_{{\\rm EP}}(\\pi\\otimes \\xi)=(-1)^{q(G)}$.\nMore precisely, $\\dim H^i({\\rm Lie}\\, G({\\mathbb R}),K'_\\infty,\n  \\pi \\otimes \\xi)$ equals 1 if $i=q(G)$ and 0 if not.\n\n\\end{itemize}\n\n\n\n\\subsection{Canonical measures and Tamagawa measures}\\label{sub:can-measure}\n\n  We return to the global setting so that $F$ and\n $G$ are as in \\S\\ref{sub:Gross-motive}. Let $G_\\infty:=(\\Res_{F\/{\\mathbb Q}}G)\\times_{\\mathbb Q} {\\mathbb R}$, to which\n the contents of \\S\\ref{sub:st-disc} and \\S\\ref{sub:EP} apply. In particular we have a measure $\\mu^{{\\rm EP}}_\\infty$ on $G_\\infty({\\mathbb R})$.\n  For each finite place $v$ of $F$,\n  define $\\mu^{\\mathrm{can}}_v:=\\Lambda({\\rm Mot}_{G_{v}}^\\vee(1))\\cdot |\\omega_{G_v}|$\n  in the notation of \\cite{Gro97} where\n $|\\omega_{G_v}|$ is the ``canonical'' Haar measure on $G(F_v)$ as in \\S11 of that article.\n  When $G$ is unramified over $F_v$, the measure\n  $\\mu^{\\mathrm{can}}_v$ assigns volume 1 to a hyperspecial subgroup of $G(F_v)$.\n  In particular, $$\\mu^{\\mathrm{can},{\\rm EP}}:=\\prod_{v\\nmid \\infty} \\mu^{\\mathrm{can}}_v \\times \\mu^{{\\rm EP}}_\\infty$$ is a well-defined measure\n  on $G({\\mathbb A}_F)$.\n\n\n\n  Let $\\overline{\\mu}^{{\\rm Tama}}$ denote the Tamagawa measure\n  on $G(F)\\backslash G({\\mathbb A}_F)\/A_{G,\\infty}$, so that its volume is the Tamagawa number (cf. \\cite[p.629]{Kot88})\n\\begin{equation} \\label{e:Tamagawa}\\tau(G):=\n  \\overline{\\mu}^{{\\rm Tama}}(G(F)\\backslash G({\\mathbb A}_F)\/A_{G,\\infty})=|\\pi_0(Z(\\widehat{G})^{{\\rm Gal}(\\overline{F}\/F)})\\cdot\n  |\\ker^1(F,Z(\\widehat{G}))|^{-1}.\\end{equation}\n The Tamagawa measure $\\mu^{{\\rm Tama}}$ on $G({\\mathbb A}_F)$ of \\cite{Gro97} is compatible with\n $\\overline{\\mu}^{{\\rm Tama}}$ if $G(F)$ and $A_{G,\\infty}$ are equipped with\n  the point-counting measure and the Lebesgue measure, respectively.\n    The ratio of two Haar measures on $G({\\mathbb A}_F)$ is computed as:\n\n\\begin{prop}(\\cite[(10.5)]{Gro97}\n  $$\\frac{\\mu^{\\mathrm{can},{\\rm EP}}}{\\mu^{{\\rm Tama}}}\n  = \\frac{ L({\\rm Mot}_G)\\cdot |\\Omega|\/|\\Omega_c|}{e(G_\\infty) 2^{{\\rm rk}_{{\\mathbb R}} G_\\infty}   }.$$\n\\end{prop}\n\n\n\n\n  The following notion will be useful in that the Levi subgroups contributing to the trace formula\n  in \\S\\ref{s:aut-Plan-theorem}\n  turn out to be the cuspidal ones.\n\n\\begin{defn}\n  We say that $G$ is cuspidal if $G_0:=\\Res_{F\/{\\mathbb Q}}G$ satisfies the condition that\n  $A_{G_0}\\times _{\\mathbb Q} {\\mathbb R}$ is the maximal split torus in the center of $G_0\\times _{\\mathbb Q} {\\mathbb R}$.\n\\end{defn}\n\n  Assume that $G$ is cuspidal, so that $G({\\mathbb R})\/A_{G,\\infty}$ contains a maximal\n  ${\\mathbb R}$-torus which is anisotropic.\n\n\n\n\\begin{cor}\\label{c:canonical-measure} $$\\frac{\\overline{\\mu}^{\\mathrm{can},{\\rm EP}}(G(F)\\backslash G({\\mathbb A}_F)\/A_{G,\\infty})}{\n  \\overline{\\mu}^{{\\rm EP}}_\\infty (\\overline{G}(F_\\infty)\/A_{G,\\infty})}=\n \\frac{\\tau(G) \\cdot  L({\\rm Mot}_G)  \\cdot |\\Omega|\/|\\Omega_c|}{e(G_\\infty) 2^{[F:{\\mathbb Q}] r_G} }.\n   $$\n\\end{cor}\n\n\\begin{proof}\n  It suffices to remark that\n  the Euler-Poincar\\'{e} measure on a compact Lie group has total volume 1,\n  hence  $\\overline{\\mu}^{{\\rm EP}}_\\infty (\\overline{G}(F_\\infty)\/A_{G,\\infty})=1$.\n\\end{proof}\n\n\n\n\\subsection{Bounds for Artin $L$-functions} For later use we estimate the $L$-value $L({\\rm Mot}_G)$ in Corollary \\ref{c:canonical-measure}.\n\n\n\\begin{prop}\\label{prop:Brauer}\nLet $s \\ge 1$ and $E$ be a Galois extension of $F$ of degree $[E:F]\\le s$.\t\n\\begin{enumerate}[(i)]\n\\item  For all $\\epsilon>0$ there exists a constant $c=c(\\epsilon,s,F)>0$ which depends only on $\\epsilon$, $s$ and $F$ such that the following holds: For all non-trivial irreducible representations $\\rho$ of ${\\rm Gal}(E\/F)$,\n\\begin{equation*}\n  cd_E^{-\\epsilon}\\le L(1,\\rho) \\le c d_E^\\epsilon.\n\\end{equation*}\n \\item The same inequalities hold for the residue $\\mathrm{Res}_{s=1} \\zeta_E(s)$ of the Dedekind zeta function of $E$.\n \\item There is a constant $A_1=A_1(s,F)>0$ which depends only on $s$ and $F$ such that for all \\textsf{faithful} irreducible representation $\\rho$ of ${\\rm Gal}(E\/F)$,\n  \\begin{equation*}\n\td_{E\/F}^{A_1} \\le {\\mathbb N}_{F\/{\\mathbb Q}}(\\mathfrak{f}_\\rho) \\le d_{E\/F}^{1\/\\dim(\\rho)},\n  \\end{equation*}\n  where $d_{E\/F}={\\mathbb N}_{F\/{\\mathbb Q}}({\\mathfrak D}_{E\/F})$ is the relative discriminant of $E\/F$; recall that $d_E= d_F^{[E:F]} d_{E\/F}$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\n  The assertion (ii) is Brauer--Siegel theorem~\\cite{Brauer47}*{Theorem~2}. We also note the implication (i) $\\Rightarrow$ (ii) which follows from the formula\n  \\begin{equation} \\label{Brauer:product}\n\t\\zeta_E(s)= \\prod_{\\rho} L(s,\\rho)^{\\dim \\rho}.\n  \\end{equation}\nwhere $\\rho$ ranges over all irreducible representations of ${\\rm Gal}(E\/F)$.\n\n  The proof of assertion (i) is reduced to the $1$-dimensional case by Brauer induction as in~\\cite{Brauer47}. In this reduction one uses the fact that if $E'\/F'$ is a subextension of $E\/F$ then the absolute discriminant $d_{E'}$ of $E'$ divides the absolute discriminant $d_{E}$ of $E$. Also we may assume that $E'\/F'$ is cyclic. For a character $\\chi$ of ${\\rm Gal}(E'\/F')$ we have the convexity bound $L(1,\\chi)\\le c d^\\epsilon_{E'}$ (Landau). The lower bound for $L(1,\\chi)$ follows from (ii) and the product formula~\\eqref{Brauer:product}.\n\n  In the assertion (iii) the right inequality follows from the discriminant-conductor formula which implies that $\\mathfrak{f}^{\\dim (\\rho)}_\\rho\\mid {\\mathfrak D}_{E\/F}$. The left inequality follows from local considerations. Let $v$ be a finite place of $F$ dividing ${\\mathfrak D}_{E\/F}$; since $\\rho$ is faithful, its restriction to the inertia group above $v$ is non-trivial and therefore $v$ divides $\\mathfrak{f}_\\rho$. Since $v({\\mathfrak D}_{E\/F})$ is bounded above by a constant $A_1(s,F)$ depending only on $[E:F]\\le s$ and $F$ by Lemma~\\ref{l:bound-diff-gen}, we have $ v({\\mathfrak D}_{E\/F}) \\le  A_1 v(\\mathfrak{f}_\\rho)$ which concludes the proof.\n\\end{proof}\n\n\n\\begin{cor}\\label{c:bound-on-L(1)}\n   For all integers $R, D,s\\in {\\mathbb Z}_{\\ge 1}$, and $\\epsilon>0$ there is a constant $c_1=c_1(\\epsilon,R,D,s,F)>0$ (depending on $R$, $D$, $s$, $F$ and $\\epsilon$) with the following property\n\\begin{enumerate}\n\\item For any $G$ such that $r_G\\le R$, $\\dim G\\le D$,\n  $Z(G)$ is $F$-anisotropic, and $G$ splits over a Galois extension of $F$ of degree $\\le s$,\n$$|L({\\rm Mot}_G)|\\le c_1 \\prod_{d=1 }^{\\lfloor \\frac{d_G+1}{2}\\rfloor}\n  {\\mathbb N}_{F\/{\\mathbb Q}}({\\mathfrak f}({\\rm Mot}_{G,d}))^{d-\\frac{1}{2}+\\epsilon}.$$\n\\item  There is a constant $A_{20}=A_{20}(R,D,s,F)$ such that for any $G$ as in (i),\n  $$|L({\\rm Mot}_G)|\\le c_1 \\prod_{v\\in {\\rm Ram}(G)}\n  q_v^{A_{20}}.$$\n\\end{enumerate}\nThe choice $A_{20}= \\frac{(D+1)Rs}{2} \\max\\limits_{\\text{prime $p$}} (1+e_{F_v\/{\\mathbb Q}_p}\\log s)$ is admissible.\n\\end{cor}\n\n\\begin{proof} The functional equation for ${\\rm Mot}_G$ reads\n  $$L({\\rm Mot}_G)=L({\\rm Mot}_G^\\vee(1))\\epsilon({\\rm Mot}_G)\\cdot \\frac{L_\\infty({\\rm Mot}_G^\\vee(1))}{L_\\infty({\\rm Mot}_G)}$$\nwhere   $ \\epsilon({\\rm Mot}_G)=|\\Delta_F|^{d_G\/2}\\prod_{d\\ge 1} {\\mathbb N}_{F\/{\\mathbb Q}}({\\mathfrak f}({\\rm Mot}_{G,d}))^{d-\\frac{1}{2}}$.\n\nThe (possibly reducible) Artin representation for ${\\rm Mot}_{G,d}$\n  factors through ${\\rm Gal}(E\/F)$ with $[E:F]\\le s$ by the assumption.\n  Let $A_1=A_1(s,F)$ be as in (iii) of Proposition \\ref{prop:Brauer}.\nFor all $\\epsilon>0$, (i) of Proposition~\\ref{prop:Brauer} implies that there is a constant $c=c(\\epsilon,s,F)>1$\n  depending only on $s$ and $F$ such that\n  $$|L({\\rm Mot}_G^\\vee(1))|\\le \\prod_{d\\ge 1} \\left( c {\\mathbb N}_{F\/{\\mathbb Q}}({\\mathfrak f}({\\rm Mot}_{G,d}))^{A_1\\epsilon}\\right)^{\\dim {\\rm Mot}_{G,d}}\n  \\le c^{r_G}\\prod_{d\\ge 1}  {\\mathbb N}_{F\/{\\mathbb Q}}({\\mathfrak f}({\\rm Mot}_{G,d}))^{\\epsilon A_1 r_G}.$$\n  Formula (7.7) of \\cite{Gro97}, the first equality below, leads to the following bound since only\n$1\\le d \\le \\lfloor \\frac{d_G+1}{2}\\rfloor$ can contribute in view of Proposition \\ref{p:Gross-motives}.(iii).\n\\begin{equation*}\n\\left|\\frac{L_\\infty({\\rm Mot}^\\vee_G(1))}{L_\\infty({\\rm Mot}_G)}\\right| = 2^{-[F:{\\mathbb Q}]r_G}\n\\prod_{d\\ge 1} \\left( \\frac{(d-1)!}{(2\\pi)^d}\\right)^{\\dim {\\rm Mot}_{G,d}}\n\\le 2^{-[F:{\\mathbb Q}]r_G} \\left( \\lfloor \\frac{d_G-1}{2}\\rfloor !\\right)^{r_ G}.\n\\end{equation*}\n  Set $c_1(R,D,s,F,\\epsilon):= |\\Delta_F|^{D\/2} 2^{-[F:{\\mathbb Q}]R} \\left(\n\\lfloor \\frac{D-1}{2}\\rfloor !\\right)^{R}$. Then we see that\n\\begin{equation*}\\begin{aligned}\n  |L({\\rm Mot}_G)|&\\le c_1 \\prod_{d=1 }^{\\lfloor \\frac{d_G+1}{2}\\rfloor}\n  {\\mathbb N}_{F\/{\\mathbb Q}}({\\mathfrak f}({\\rm Mot}_{G,d}))^{d-\\frac{1}{2}+\\epsilon} \\\\\n  & = c_1 \\prod_{v\\in {\\rm Ram}(G)} \\prod_{d=1 }^{\\lfloor \\frac{d_G+1}{2}\\rfloor}\n  q_v^{(d-\\frac{1}{2}+\\epsilon) \\cdot f(G_v,d)}.\n\\end{aligned}\\end{equation*}\nThis concludes the proof of (i).\n\nAccording to Lemma \\ref{l:bounding-conductor}, the exponent in the right hand side is bounded by\n\\begin{equation*}\n  d f(G_v,d) \\le \\frac{D+1}{2} \\dim {\\rm Mot}_{G,d} \\cdot (s(1+e_{F_v\/{\\mathbb Q}_p}\\log s)-1).\n\\end{equation*}\n (we have chosen $\\epsilon=\\frac{1}{2}$). The proof of (ii) is concluded by the fact that $$\\sum_{d\\ge 1}\\dim {\\rm Mot}_{G,d}=r_G\\le R,$$ see Proposition \\ref{p:Gross-motives}.(ii).\n\\end{proof}\n\n\n\\begin{cor}\\label{c:bound-on-L(1)-2}\n  Let $G$ be a connected cuspidal reductive group over $F$ with anisotropic center. Then there exist constants\n  $c_2=c_2(G,F)>0$ and $A_2(G,F)>0$ depending only on $G$ and $F$ such that:\n  for any cuspidal $F$-Levi subgroup $M$ of $G$ and any semisimple $\\gamma\\in M(F)$\nwhich is elliptic in $M({\\mathbb R})$,\n $$|L({\\rm Mot}_{I^M_\\gamma})|\\le c_2 \\prod_{v\\in {\\rm Ram}(I_\\gamma^M)}\n q_v^{A_2}$$\nwhere $I^M_\\gamma$ denote the connected centralizer of $\\gamma$ in $M$.\nThe following choice is admissible: $$A_2= \\frac{(d_G+1)r_Gw_G s_G}{2} \\max\\limits_{\\text{prime $p$}} (1+e_{F_v\/{\\mathbb Q}_p}\\log w_Gs_G).$$\n\\end{cor}\n\n\n\\begin{proof}\n    According to Lemma \\ref{l:torus-splitting},\n $s^{\\mathrm{sp}}_{ I^M_\\gamma}\\le w_Gs_G$.\n  Apply Corollary \\ref{c:bound-on-L(1)} for each $I^M_\\gamma$ with $R=r_G$, $D=d_G$ and $s=w_Gs_G$\nto deduce the first assertion, which obviously implies the last assertion. Note that\n${\\rm rk} I^M_\\gamma\\le r_G$ and that $\\dim I^M_\\gamma\\le d_G$.\n\\end{proof}\n\nInstead of using the Brauer--Siegel theorem which is ineffective, we could use the estimates by Zimmert~\\cite{Zimmert:regulator} for the size of the regulator of number fields. This yields an effective estimate for the constants $c_2$ and $c_3$ above, at the cost of enlarging the value of the exponents $A_1$ and $A_2$.\n\n\\subsection{Frobenius--Schur indicator}\n\\label{sec:b:fs}\n\nThe Frobenius--Schur indicator is an invariant associated to an irreducible representation. It may take the three values $1,0,-1$. This subsection gathers several well-known facts and recalls some familiar constructions.\n\nThe Frobenius--Schur indicator can be constructed in greater generality but the following setting will suffice for our purpose. We will only consider finite dimensional representations on vector spaces over $\\mathbb{C}$ or $\\mathbb{R}$. The representations are continuous (and unitary) from compact Lie groups or algebraic from linear algebraic groups (these are in fact closely related by the classical \\Lquote{unitary trick} of Hurwitz and Weyl).\n\n Let $G$ be a compact Lie group and denote by $\\mu$ the Haar probability measure on $G$. Let $(V,r)$ be a continuous irreducible representation of $G$. Denote by $\\chi(g)=\\MTr(r(g))$ its character.\n\n\\begin{definition} The \\key{Frobenius--Schur indicator} of an irreducible representation $(V,r)$ of $G$ is defined by\n\\begin{equation*}\ns(r):= \\int_G \\chi(g^2) d\\mu(g).\n\\end{equation*}\nWe have that $s(r)\\in \\set{-1,0,1}$ always.\n\\end{definition}\n\n\\begin{rem}\n  More generally if $G$ is an arbitrary group but $V$ is still finite dimensional, then $s(r)$ is defined as the multiplicity of the trivial representation in the virtual representation on $\\MSym^2 V - \\wedge^2 V$. This is consistent with the above definition.\n\\end{rem}\n\n\\begin{remark}\\begin{enumerate}[(i)]\n  \\item Let $(V^\\vee,r^\\vee)$ be the dual representation of $G$ in the dual $V^\\vee$. It is easily seen that $s(r)=s(r^\\vee)$.\n  \\item If $G=G_1\\times G_2$ and $r$ is the irreducible representation of $G$ on $V=V_1\\otimes V_2$ where $(V_1,r_1)$ and $(V_2,r_2)$ are irreducible representations of $G_1$ (resp. $G_2$), then $s(r)=s(r_1)s(r_2)$.\n\\end{enumerate}\n\\end{remark}\n\n\nThe classical theorem of Frobenius and Schur says that $r$ is a real, complex or quaternionic representation if and only if $s(r)=1,0$ or $-1$ respectively. We elaborate on that dichotomy in the following three lemma.\n\n\\begin{lemma}[Real representation]\\label{lem:real} Let $(V,r)$ be an irreducible representation of $G$.\nThe following assertions are equivalent:\n\\begin{enumerate}[(i)]\n\\item $s(r)=1$;\n\\item $r$ is self-dual and defined over $\\mathbb{R}$ in the sense that $V\\simeq V_0\\otimes_\\mathbb{R} \\mathbb{C}$ for some irreducible representation on a real vector space $V_0$. (Such an $r$ is said to be a \\key{real representation};)\n\\item $r$ has an invariant real structure. Namely there is a $G$-invariant anti-linear map $j:V\\to V$ which satisfies $j^2=1$.\n\\item $r$ is self-dual and any bilinear form on $V$ that realizes the isomorphism $r\\simeq r^\\vee$ is symmetric;\n\\item $\\MSym^2 V$ contains the trivial representation (then the multiplicity is exactly one).\n\\end{enumerate}\n\\end{lemma}\n\nWe don't repeat the proof here (see e.g.~\\cite{book:serre:rep}) and only recall some of the familiar constructions. We have a direct sum decomposition\n\\begin{equation*}\n V\\otimes V = \\MSym^2 V \\oplus \\wedge^2 V.\n\\end{equation*}\nThe character of the representation $V\\otimes V$ is $g\\mapsto \\chi(g)^2$. By Schur lemma the trivial representation occurs in $V\\otimes V$ with multiplicity at most one. In other words the subspace of invariant vectors of $V^\\vee\\otimes V^\\vee$ is at most one. Note that this subspace is identified with $\\MHom_G(V,V^\\vee)$ which is also the subspace of invariant bilinear forms on $V$.\n\nThe character of the representation $\\MSym^2 V$ (resp. $\\wedge^2 V$) is\n\\begin{equation*}\n\\text{%\n$\\frac{1}{2}(\\chi(g)^2+\\chi(g^2))$\\quad  resp.\\quad  $\\frac{1}{2}(\\chi(g)^2-\\chi(g^2))$.\n}\n\\end{equation*}\nFrom that the equivalence of (i) with (v) follows because the multiplicity of the trivial representation in $\\MSym^2 V$ (resp. $\\wedge^2 V$) is the mean of its character. The equivalence of (iv) and (v) is clear because a bilinear form on $V$ is an element of $V^\\vee\\otimes V^\\vee$ and it is symmetric if and only if it belongs to $\\MSym^2 V^\\vee$.\n\nThe equivalence of (ii) and (iii) follows from the fact that $j$ is induced by complex conjugation on $V_0\\otimes_\\mathbb{R} \\mathbb{C}$ and conversely $V_0$ is the subspace of fixed points by $j$. Note that a real representation is isomorphic to its complex conjugate representation because $j$ may be viewed equivalently as a $G$-isomorphism $V\\to \\overline{V}$. Since $V$ is unitary the complex conjugate representation $\\overline{r}$ is isomorphic to the dual representation $r^\\vee$. In assertion (ii) one may note that the endomorphism ring of $V_0$ is isomorphic to $\\mathbb{R}$.\n\n\\begin{lemma}[Complex representation]  Let $(V,r)$ be an irreducible representation of $G$.\nThe following assertions are equivalent:\n\\begin{enumerate}[(i)]\n\\item $s(r)=0$;\n\\item $r$ is not self-dual;\n\\item $r$ is not isomorphic to $\\overline{r}$; (Such an $r$ is called a \\key{complex representation};)\n\\item $V\\otimes V$ does not contain the trivial representation.\n\\end{enumerate}\n\\end{lemma}\n\nWe note that for a complex representation, the restriction $\\Res_{\\mathbb{C}\/\\mathbb{R}} V$ (obtained by viewing $V$ as a real vector space) is an irreducible real representation of twice the dimension of $V$. Its endomorphism ring is isomorphic to $\\mathbb{C}$.\n\n\\begin{lemma}[Quaternionic\/symplectic representation]\\label{lem:quaternionic} Let $(V,r)$ be an irreducible representation of $G$.\nThe following assertions are equivalent:\n\\begin{enumerate}[(i)]\n\\item $s(r)=-1$;\n\\item $r$ is self-dual and cannot be defined over ${\\mathbb R}$.\n\\item $r$ has an invariant quaternionic structure. Namely there is a $G$-invariant anti-linear map $j:V\\to V$ which satisfies $j^2=-1$.\n(Such an $r$ is called a \\key{quaternionic representation}.)\n\\item $r$ is self-dual and the bilinear form on $V$ that realizes the isomorphism $r\\simeq r^\\vee$ is antisymmetric.\n (Such an $r$ is said to be a \\key{symplectic representation};)\n\\item $\\bigwedge^2 V$ contains the trivial representation (the multiplicity is exactly one).\n\\end{enumerate}\n\\end{lemma}\n\nThe equivalence of (iii) and (iv) again comes from the fact that $V$ is unitarizable (because $G$ is a compact group). In that context the notion of symplectic representation is identical to the notion of quaternionic representation. Note that for a quaternionic representation, the restriction $\\Res_{\\mathbb{C}\/\\mathbb{R}} V$ is an irreducible real representation of twice the dimension of $V$. Furthermore its ring of endomorphisms is isomorphic to the quaternion algebra $\\mathbb{H}$. Indeed the endomorphism ring contains the (linear) action by $i$ because $V$ is a representation over the complex numbers and together with $j$ and $k=ij$ this is the standard presentation of $\\mathbb{H}$.\n\nFrom the above discussions we see that the Frobenius--Schur indicator can be used to classify irreducible representations over the reals. The endomorphism ring of an irreducible real representation is isomorphic to either $\\mathbb{R},\\mathbb{C}$ or $\\mathbb{H}$ and we have described a correspondence with associated complex representations.\n\n\n\n\\section{A uniform bound on orbital integrals}\\label{s:app:unif-bound}\n\n  This section is devoted to showing an apparently new result on the uniform bound on orbital integrals\n  evaluated at semisimple conjugacy classes and basis elements of unramified Hecke algebras.\n  Our bound is uniform in the finite place $v$ of a number field (over which the group is defined),\n  the ``size'' of (the support of) the basis element for the unramified Hecke algebra at $v$\n  as well as the conjugacy class at $v$.\n\n  The main result is Theorem \\ref{t:appendeix2}, which\n  is invoked in \\S\\ref{sub:weight-varies}. The main local input for Theorem \\ref{t:appendeix2} is Proposition \\ref{p:appendix2}. The technical heart in the proof of the proposition is postponed to \\S\\ref{sub:app:elliptic}, which the reader may want to skip in the first reading.\n\n\n\n\n\\subsection{The main local result}\\label{sub:local-bound-orb-int}\n\n  We begin with a local assertion with a view toward Theorem \\ref{t:appendeix2} below.\n  Let $G$ be a connected reductive group over a finite extension $F$ of ${\\mathbb Q}_p$ with a maximal $F$-split torus $A$.\n  As usual $\\mathcal{O}$, $\\varpi$, $k_F$ denote the integer ring, a uniformizer and the residue field. Let $G$ be the Chevalley group for $G\\times_{F} \\overline{F}$, defined over ${\\mathbb Z}$.\n Let $\\mathbf{B}$ and $\\mathbf {T}$ be a Borel subgroup and a maximal torus of $G$ such that\n  $\\mathbf{B}\\supset \\mathbf {T}$. We assume that\n  \\begin{itemize}\\item $G$ is unramified over $F$,\n  \\item ${\\rm char}\\, k_F>w_Gs_G$ and ${\\rm char}\\, k_F$ does not divide the finitely many constants in the Chevalley commutator relations (namely $C_{ij}$ of \\eqref{e:commutator}).\n  \\end{itemize}\n  (We assume ${\\rm char}\\, k_F>w_Gs_G$ to ensure that any maximal torus of $G$ splits over a finite tame extension, cf. \\S\\ref{sub:app:elliptic} below. The latter assumption on ${\\rm char}\\, k_F$ depends only on $G$.) Fix a smooth reductive model over $\\mathcal{O}$ so that $K:=G(\\mathcal{O})$ is a hyperspecial subgroup of $G(F)$. Denote by $v:F^\\times \\rightarrow {\\mathbb Q}$ the discrete valuation normalized by $v(\\varpi)=1$ and by $D^G$ the Weyl discriminant function, cf. \\eqref{e:D^G} below. Set $q_v:=|k_F|$.\n\n  Suppose that there exists a closed embedding of algebraic groups $\\Xi:G\\hookrightarrow \\GL_d$ defined over $\\mathcal{O}$\n  such that $\\Xi(\\mathbf {T})$ (resp. $\\Xi(\\mathbf{B})$) lies in the group of diagonal (resp. upper triangular) matrices. This assumption will be satisfied by Lemma \\ref{l:conj-image-in-diag} and Proposition \\ref{p:global-integral-model}, or alternatively as explained at the start of \\S\\ref{sub:lem-split}.\n  The assumption may not be strictly necessary but is convenient to have for some later arguments.\n  In the setup of \\S\\ref{sub:global-bound-orb-int} such a $\\Xi$ will be chosen globally\n  over ${\\mathbb Z}[1\/Q]$ (i.e. away from a certain finite set of primes), which gives rise to an embedding over $\\mathcal{O}$ if $v$ does not divide $Q$.\n\n\\begin{prop}\\label{p:appendix2}\n\n   There exist $a_{G,v},b_{G,v},e_{G,v}\\ge0$ (depending on $F$, $G$ and $\\Xi$) such that\n   \\begin{itemize}\n   \\item for every semisimple $\\gamma\\in G(F)$,\n   \\item for every $\\lambda\\in X_*(A)$ and $\\kappa\\in{\\mathbb Z}_{\\ge0}$ such that $\\|\\lambda\\|\\le\\kappa$,\n   \\end{itemize}\n  \\begin{equation}\\label{e:app:prop} 0\\le O_\\gamma(\\tau^G_\\lambda,\\mu^{\\mathrm{can}}_G,\\mu^{\\mathrm{can}}_{I_\\gamma})\n  \\le  q_v^{ a_{G,v}+b_{G,v}\\kappa}\\cdot D^G(\\gamma)^{-e_{G,v}\/2}.\\end{equation}\n\\end{prop}\n\n\\begin{rem}\n  We chose the notation $a_{G,v}$ etc rather than $a_{G,F}$ etc in anticipating the global setup\n  of the next subsection\n  where $F$ is the completion of a number field at the place $v$.\n\\end{rem}\n\n\n\n\n\\begin{proof}\n  For simplicity we will omit the measures chosen to compute orbital integrals\n  when there is no danger of confusion.\n  Let us argue by induction on the semisimple rank $r^{\\semis}_G$ of $G$.\n  In the rank zero case, namely when $G$ is a torus,\n  the proposition is true since $O_\\gamma(\\tau^G_\\lambda)$ is equal to 0 or 1.\n  Now assume that $r^{\\semis}_G\\ge 1$ and that the proposition is known for all groups\n  whose semisimple ranks are less than $r^{\\semis}_G$.\n  In the proof we write $a_G$, $b_G$, $e_G$ instead of $a_{G,v}$, $b_{G,v}$, $e_{G,v}$ for simplicity.\n\n\\medskip\n\n\\underline{Step 1}. Reduce to the case where $Z(G)$ is anisotropic.\n\n  Let $A_G$ denote the maximal split torus in $Z(G)$. Set $\\overline{G}:=G\/A_G$.\n  The goal of Step 1 is to show that if the proposition for $\\overline{G}$ then it also holds\nfor $G$. We have an exact sequence of algebraic groups over $\\mathcal{O}$: $1\\rightarrow A_G\\rightarrow G\\rightarrow \\overline{G}\\rightarrow 1$.\nBy taking $F$-points one obtains an exact sequence of groups\n$$1 \\rightarrow A_G(F) \\rightarrow G(F) \\rightarrow \\overline{G}(F) \\rightarrow 1,$$\nwhere surjectivity is implied by Hilbert 90 for $A_G$.\nIn fact $G(\\mathcal{O})\\rightarrow \\overline{G}(\\mathcal{O})$ is surjective since it is surjective on $k_F$-points and\n$G\\rightarrow \\overline{G}$ is smooth. (A similar argument was used on page 386 of \\cite{Kot86}.)\nFor any semisimple $\\gamma\\in G(F)$, denote its image in $\\overline{G}(F)$ by $\\overline{\\gamma}$.\n The connected centralizer of $\\overline{\\gamma}$ is denoted\n$\\overline{I}_{\\overline{\\gamma}}$. There is an exact sequence\n$$1\\rightarrow A_G(F) \\rightarrow I_\\gamma(F) \\rightarrow \\overline{I}_{\\overline{\\gamma}}(F)\\rightarrow 1.$$\nWe see that $G(F) \\rightarrow \\overline{G}(F)$ induces a bijection\n$I_\\gamma(F)\\backslash G(F) \\simeq \\overline{I}_{\\overline{\\gamma}}(F)\\backslash \\overline{G}(F)$.\nLet $A$ be a maximal $F$-split torus of $G$, and $\\overline{A}$ be its image in $\\overline{G}$.\nFor any $\\lambda\\in X_*(A)$, denote its image in $X_*(\\overline{A})$ by $\\overline{\\lambda}$.\nThen\n$$ O^{G(F)}_\\gamma(\\tau^G_\\lambda,\\mu^{\\mathrm{can}}_G,\\mu^{\\mathrm{can}}_{I_\\gamma})\n=O^{\\overline{G}(F)}_{\\overline{\\gamma}}(\\tau^{\\overline{G}}_{\\overline{\\lambda}},\\mu^{\\mathrm{can}}_{\\overline{G}},\\mu^{\\mathrm{can}}_{\\overline{I}_{\\overline{\\gamma}}}).$$\nIndeed, this follows from the fact that $I_\\gamma(F)\\backslash G(F) \\simeq \\overline{I}_{\\overline{\\gamma}}(F)\\backslash \\overline{G}(F)$\ncarries $\\frac{\\mu^{\\mathrm{can}}_G}{\\mu^{\\mathrm{can}}_{I_\\gamma}}$ to $\\frac{\\mu^{\\mathrm{can}}_{\\overline{G}}}{\\mu^{\\mathrm{can}}_{\\overline{I}_{\\overline{\\gamma}}}}$\nand that $G(\\mathcal{O})\\rightarrow \\overline{G}(\\mathcal{O})$ is onto (so that $\\tau^G_{\\lambda}(x^{-1}\\gamma x)=1$\nif and only if $\\tau^{\\overline{G}}_{\\overline{\\lambda}}(\\overline{x}^{-1}\\overline{\\gamma} \\overline{x})=1$, where\n$\\overline{x}$ is the image of $x$).\nAs the proposition is assumed to hold for $\\overline{G}$,\nthe right hand side is bounded by $q_v^{ a_{\\overline{G}}+b_{\\overline{G}}\\kappa}\\cdot D^{\\overline{G}}(\\gamma)^{-e_{\\overline{G}}\/2}=\nq_v^{ a_{\\overline{G}}+b_{\\overline{G}}\\kappa}\\cdot D^{G}(\\gamma)^{-e_{\\overline{G}}\/2}$. Hence the\nproposition holds for $G$ if we set\n$a_G=a_{\\overline{G}}$, $b_G=b_{\\overline{G}}$ and $e_G=e_{\\overline{G}}$. This finishes Step 1.\n\n\\medskip\n\n\\noindent\\underline{Step 2}. When $Z(G)$ is anisotropic.\n\n  The problem will be divided into 3 cases depending on $\\gamma$. In each case we find a sufficient condition on $a_G$, $b_G$ and $e_G$ for \\eqref{e:app:prop} to be true.\n\n\\medskip\n\n\\noindent\\underline{Step 2-1}. When $\\gamma\\in Z(G)(F)$.\n\n  In this case the proposition holds for any $a_G,b_G,e_G\\ge0$ since $O_\\gamma(\\tau^G_\\lambda)=0$ or $1$ and $D^G(\\gamma)=1$.\n\n\n\\medskip\n\n\\noindent\\underline{Step 2-2}. When $\\gamma$ is non-central and non-elliptic.\n\nThen there exists a nontrivial split torus $S\\subset Z(I_\\gamma)$.\n  Set $M:=Z_G(S)$, which is an $F$-rational Levi subgroup of $G$. Then $I_\\gamma\\subset M \\subsetneq G$. Note that $\\gamma$ is $(G,M)$-regular.\nLemma \\ref{l:orb-int-const-term} reads\n\\begin{equation}\\label{e:app:G-M}O^{G(F)}_\\gamma({\\mathbf{1}}_{K\\lambda(\\varpi)K})=\nD_{G\/M}(\\gamma)^{-1\/2} O^{M(F)}_\\gamma(({\\mathbf{1}}_{K\\lambda(\\varpi)K})_M).\\end{equation}\nBy conjugation we may assume without loss of generality that\n$\\lambda(\\varpi)\\in M(F)$. (To justify, find $x\\in G(F)$ such that\n$xMx^{-1}$ contains $A$. Then $\\lambda(\\varpi)\\in xM(F)x^{-1}$ and\n$O^M_\\gamma= O^{xMx^{-1}}_{x\\gamma x^{-1}}$.)\nWe can write \\begin{equation}\\label{e:App-Step3}({\\mathbf{1}}_{K\\lambda(\\varpi)K})_M=\\sum_{\\mu\\le_{{\\mathbb R}}\\lambda}\nc_{\\lambda,\\mu}{\\mathbf{1}}_{K_M\\mu(\\varpi)K_M}.\\end{equation}\n For any $m=\\mu(\\varpi)$, $c_{\\lambda,\\mu}$ is equal to\n$$({\\mathbf{1}}_{K\\lambda(\\varpi)K})_M(m)=\\delta_P(m)^{1\/2}\\int_{N(F)} {\\mathbf{1}}_{K\\lambda(\\varpi)K}(mn)dn\n=q_v^{\\langle \\rho_P,\\mu\\rangle} \\mu_G^{\\mathrm{can}}(mN(F)K\\cap K\\lambda(\\varpi)K).$$\nLemma \\ref{l:double-coset-volume} and the easy inequality $\\langle \\rho_P,\\mu\\rangle \\le \\langle \\rho,\\lambda\\rangle$\n allows us to deduce that\n$$0\\le c_{\\lambda,\\mu}\\le q_v^{\\langle \\rho_P,\\mu\\rangle}\n\\mu^{\\mathrm{can}}_G(K\\lambda(\\varpi)K)\\le q_v^{d_G+r_G+2\\langle \\rho,\\lambda\\rangle}.$$\nThe sum in \\eqref{e:App-Step3} runs over the set of $$\\mu=\\lambda-\\sum_{\\alpha\\in \\Delta^+}\na_\\alpha\\cdot \\alpha~\\mbox{with}~a_\\alpha\\in \\frac{1}{\\delta_G}{\\mathbb Z},~a_\\alpha\\ge 0$$\nand $\\mu\\in (X^*(T)_{\\mathbb R})^+$. Here we need to explain $\\delta_G$: If\n$\\mu\\le_{{\\mathbb R}} \\lambda$ then $\\lambda-\\mu$ is a linear combination of positive coroots with nonnegative rational\ncoefficients. The denominators of such coefficients are uniformly bounded, where the bound depends on the\ncoroot datum. We write $\\delta_G$ for this bound.\n\nThe above condition on $\\mu$ and $\\|\\lambda\\|\\le \\kappa$ imply that $a_\\alpha\\le \\kappa$.\nWe get, by using the induction hypothesis for $O^M_\\gamma$, $$ 0\\le\nO^{M(F)}_\\gamma(({\\mathbf{1}}_{K\\lambda(\\varpi)K})_M)\\le\n\\sum_{\\mu\\le_{{\\mathbb R}}\\lambda} c_{\\lambda,\\mu} O^M_{\\gamma} ({\\mathbf{1}}_{K_M\\mu(\\varpi)K_M})\n\\le \\sum_{\\mu\\le_{{\\mathbb R}}\\lambda} c_{\\lambda,\\mu}\n q_v^{ a_M+b_M\\kappa}\\cdot D^M(\\gamma)^{-e_M\/2}$$\n$$\\le\n(\\delta_G(\\kappa+1))^{|\\Delta^+|} q_v^{d_G+r_G+2\\langle \\rho,\\mu\\rangle}q_v^{ a_M+b_M\\kappa}\\cdot D^M(\\gamma)^{-e_M\/2} $$\n$$ \\le q_v^{d_G+r_G(\\delta_G\\kappa+\\delta_G+1)+2\\langle \\rho,\\mu\\rangle+a_M+b_M\\kappa}  D^M(\\gamma)^{-e_M\/2}. $$\nSet\n\\[\nc_G:=d_G+r_G(\\delta_G+1)+2\\langle \\rho,\\lambda\\rangle \\le d_G+r_G(\\delta_G+1)+|\\Phi^+|\\kappa.\n\\]\nIn view of \\eqref{e:app:G-M} it suffices to find $a_G,b_G,e_G\\ge 0$ such that\n$$D_{G\/M}(\\gamma)^{-1\/2}D^M(\\gamma)^{-e_M\/2} q_v^{a_M+c_G+(b_M+r_G\\delta_G)\\kappa}\n\\le D_G(\\gamma)^{-e_G\/2} q_v^{a_G+b_G\\kappa}$$\nor equivalently\n\\begin{equation}\\label{e:app:suff-cond}\nD_{G\/M}(\\gamma)^{\\frac{e_G-1}{2}}D^M(\\gamma)^{\\frac{e_G-e_M}{2}}\n\\le q_v^{a_G-a_M-c_G+(b_G-b_M-r_G\\delta_G)\\kappa}\\end{equation}\nwhenever a conjugate of $\\gamma$ lies in $K\\lambda(\\varpi)K$.\nFor each $\\alpha\\in \\Phi$, $v(1-\\alpha(\\gamma))\\ge 0$ if $v(\\alpha(\\gamma))\\ge 0$ and\n\\begin{equation}\\label{e:app:alpha(gamma)}v(1-\\alpha(\\gamma))= v(\\alpha(\\gamma))\\ge -m_\\Xi \\kappa\\quad\\mbox{if}~v(\\alpha(\\gamma))<0\\end{equation}\nwhere $m_\\Xi$ is the constant $B_5$ (depending only on $G$ and $\\Xi$ and not on $v$) of Lemma \\ref{l:bounding1-alpha(gamma)}.\nHence $$D_{G\/M}(\\gamma)=\n\\prod_{\\alpha\\in \\Phi\\backslash \\Phi_M\\atop \\alpha(\\gamma)\\neq 1}\n|1-\\alpha(\\gamma)|_v \\le q_v^{|\\Phi\\backslash \\Phi_M| m_\\Xi \\kappa\/2}$$\nand likewise $D^M(\\gamma)\\le q_v^{|\\Phi_M| m_\\Xi \\kappa\/2}$.\n(We divide the exponents by 2 because it cannot happen simultaneously that\n $v(\\alpha(\\gamma))<0$ and $v(\\alpha^{-1}(\\gamma))<0$.)\nTherefore condition \\eqref{e:app:suff-cond} on $a_G,b_G,e_G$ is implied by\n\\begin{equation}\\label{e:app:Levi} \\begin{aligned}&~ \\frac{e_G-1}{2}\\frac{|\\Phi\\backslash \\Phi_M| m_\\Xi \\kappa}{2}\n+  \\frac{e_G-e_M}{2}\\frac{|\\Phi_M| m_\\Xi \\kappa}{2}\\\\ \\le & ~ a_G-a_M-(d_G+r_G(\\delta_G+1)+|\\Phi^+|\\kappa)+(b_G-b_M-r_G\\delta_G)\\kappa.\\end{aligned}\\end{equation}\nThere are only finitely many Levi subgroups $M$ (up to conjugation)\ngiving rise to the triples $(a_M,b_M,e_M)$. It is elementary to observe that\n\\eqref{e:app:Levi} holds as long as $a_G$ and $b_G$ are sufficiently large (while $e_G$ is kept relatively small).\nWe will impose another condition on $a_G,b_G,e_G$ in Step 2-3.\n\n\\medskip\n\n\\noindent\\underline{Step 2-3}. When $\\gamma$ is noncentral and elliptic in $G$.\n\nThis case is essentially going to be worked out in \\S\\ref{sub:app:elliptic}. Let $Z_1,Z_2,Z_3$ be as in Lemma \\ref{l:B-4} below.\nBy \\eqref{e:mu-EP\/mu} and Corollary \\ref{c:B-1} below,\n\\eqref{e:app:prop} will hold if\n \\begin{equation}\\label{e:app:Elliptic}q_v^{r_G(d_G+1)}q_v^{Z_1+Z_2\\kappa} D^G(\\gamma)^{Z_3}\n  \\le q_v^{a_G+b_G\\kappa} D^G(\\gamma)^{-e_G\/2}.\\end{equation}\n  We have $D^G(\\gamma)\\le q_v^{|\\Phi|m_\\Xi\\kappa\/2}$ thanks to \\eqref{e:app:alpha(gamma)} (cf. Step 2-2). So \\eqref{e:app:Elliptic} (is not equivalent to but) is implied by the combination of the following two inequalities:\n \\begin{equation}\\label{e:app:Elliptic1} Z_3+\\frac{e_G}{2} \\ge 0.\\end{equation}\n  \\begin{equation}\\label{e:app:Elliptic2}  r_G(d_G+1)+(Z_1+Z_2\\kappa)+|\\Phi|m_{\\Xi}\\frac{\\kappa}{2}(Z_3+\\frac{e_G}{2})\n  \\le a_G+b_G\\kappa.\\end{equation}\n  The latter two will hold true, for instance, if $e_G=0$ and if $a_G$ and $b_G$ are sufficiently large. (We will see in \\S\\ref{sub:app:elliptic} below that $Z_3\\ge0$ and that $Z_1,Z_2,Z_3$ are independent of $\\lambda$, $\\gamma$ and $\\kappa$.)\n\n\\medskip\n\n  Now that we are done with analyzing three different cases, we finish Step 2. For this we use the induction on semisimple ranks (to ensure the existence of $a_M$, $b_M$ and $e_M$ in Step 2-2) to find $a_G,b_G,e_G\\ge 0$ which satisfy the conditions described at the ends of Step 2-2 and Step 2-3. We are done with the proof of Proposition \\ref{p:appendix2}.\n\\end{proof}\n\n\n\\subsection{A global consequence}\\label{sub:global-bound-orb-int}\n\n  Here we switch to a global setup. For a finite place $v$ of a number field, let $k(v)$ denote the residue field. Put $q_v:=|k(v)|$.\n\\begin{itemize}\n\\item $G$ is a connected reductive group over a number field ${\\mathbf F}$.\n\\item ${\\rm Ram}(G)$ is the set of finite places $v$ of ${\\mathbf F}$ such that $G$ is ramified at ${\\mathbf F}_v$.\n\\item $G$ is the Chevalley group for $G\\times_{{\\mathbf F}} \\overline{{\\mathbf F}}$,\nand $\\mathbf{B}$, $\\mathbf {T}$ are as in \\S\\ref{sub:local-bound-orb-int}.\n \\item $S_{{\\rm bad}}$ be the set of finite places $v$ such that either $v\\in{\\rm Ram}(G)$, ${\\rm char}\\, k(v)\\le w_Gs_G$, or ${\\rm char}\\, k(v)$ divides at least one of the constants for the Chevalley commutator relations for $G$ (cf. \\eqref{e:commutator}).\n\n\\end{itemize}\n\nOnce and for all, fix a closed embedding $\\Xi^{\\mathrm{sp}}:G\\hookrightarrow \\GL_d$ defined over ${\\mathbb Z}[1\/R]$ for a large enough integer $R$ such that $\\Xi^{\\mathrm{sp}}(\\mathbf {T})$ (resp. $\\Xi^{\\mathrm{sp}}(\\mathbf{B})$) lies in the group of diagonal (resp. upper triangular) matrices of $\\GL_d$ and such that Lemma \\ref{l:app:split2} holds true. The choice of $R$ depends only on $G$ and $\\Xi^{\\mathrm{sp}}$. (See \\S\\ref{sub:lem-split} for more details and the explanation that it is always possible to choose such an $R$.)\n\n\n  Let $Q$ be an integer such that $p|Q$ if and only if $p|R$ or some place $v$ of $F$ above $p$ belongs to $S_{{\\rm bad}}$.\n  Examining the dependence of various constants in Proposition \\ref{p:appendix2}\n  leads to the following main result of this section. For each finite place $v\\notin S_{{\\rm bad}}$, denote by $A_v$ a maximal ${\\mathbf F}_v$-split torus of $G\\times_{{\\mathbf F}} {\\mathbf F}_v$.\n\n\\begin{thm}\\label{t:appendeix2}\n   There exist $a_G,b_G,e_G\\ge 0$ (depending on ${\\mathbf F}$, $G$ and $\\Xi^{\\mathrm{sp}}$) such that\n   \\begin{itemize}\n   \\item for every finite $v\\notin S_{{\\rm bad}}$,\n   \\item for every semisimple $\\gamma\\in G({\\mathbf F}_v)$,\n   \\item for every $\\lambda\\in X_*(A_v)$ and $\\kappa\\in{\\mathbb Z}_{\\ge0}$ such that $\\|\\lambda\\|\\le\\kappa$,\n   \\end{itemize}\n  \\[\n \n  0\\le O^{G({\\mathbf F}_v)}_\\gamma(\\tau^G_\\lambda,\\mu^{\\mathrm{can}}_{G,v},\\mu^{\\mathrm{can}}_{I_\\gamma,v})\n  \\le  q_v^{ a_G+b_G\\kappa}\\cdot D_v^G(\\gamma)^{-e_G\/2}.\n  \\]\n\\end{thm}\n\n\\begin{rem}\n  It is worth drawing a comparison between the above theorem and Theorem \\ref{t:appendix1} proved by Kottwitz.\n  In the latter the test function (in the full Hecke algebra) and the base $p$-adic field are fixed whereas\n  the main point of the former is to allow the test function (in the unramified Hecke algebra) and\n  the place $v$ to vary. The two theorems are complementary to each other and will\n  play a crucial role in the proof of Theorem \\ref{t:weight-varies}.\n\\end{rem}\n\n\\begin{rem} In an informal communication Kottwitz and Ng\\^{o} pointed out that there might be yet another approach based on a geometric argument involving affine Springer fibers, as in \\cite[\\S15]{GKM04}, which might lead to a streamlined and conceptual proof, as well as optimized values of the constants $a_G$ and $b_G$. The Appendix~\\ref{s:app:B} provides an important step in that direction, see Theorem~\\ref{thm:transfer-fam} which implies that the constants are transferable from finite characteristic to characteristic zero.\n\\end{rem}\n\n\n\\begin{proof}\n  Since the case of tori is clear, we may assume that $r^{\\semis}_G\\ge 1$. Write $S_Q$ for the places $v$ dividing $Q$.\n  Let $\\theta\\in {\\mathscr C}(\\Gamma_1)$. (Recall the definition of $\\Gamma_1$ and ${\\mathscr C}(\\Gamma_1)$ from \\S\\ref{sub:ST-meas} and \\S\\ref{sub:lim-of-Plan}.)\n  Our strategy is to find $a_{G,\\theta},b_{G,\\theta},e_{G,\\theta}\\ge 0$ which satisfy the requirements \\eqref{e:app:Levi}, \\eqref{e:app:Elliptic1}, and \\eqref{e:app:Elliptic2} on $a_{G,v},b_{G,v},e_{G,v}$ at all $v\\in {\\mathcal V}_{{\\mathbf F}}(\\theta)\\backslash (S_{{\\rm bad}}\\cup S_Q)$. (As for \\eqref{e:app:Levi}, we inductively find $a_{M,\\theta},b_{M,\\theta},e_{M,\\theta}\\ge 0$ for all local Levi subgroups $M$ of $G$ as will be explained below.) In fact we will choose $e_{G,\\theta}=0$. Once this is done for every $\\theta$ in the finite set ${\\mathscr C}(\\Gamma_1)$, we will similarly treat the places $v\\in S_Q$ not contained in $S_{{\\rm bad}}$ to finish the proof.\n\n\n  We would like to explain an inductive choice of $a_{M,\\theta},b_{M,\\theta},e_{M,\\theta}\\ge 0$ for a fixed $\\theta$. To do so we ought to clarify what Levi subgroups $M$ of $G$ we consider. Let $\\Delta$ denote the set of $\\mathbf{B}$-positive simple roots for $(G,\\mathbf {T})$. Via an identification $G\\times_{{\\mathbf F}} \\overline{{\\mathbf F}}\\simeq G\\times_{\\mathbb Z} \\overline{{\\mathbf F}}$ we may view $\\Delta$ as the set of simple roots for $G$ equipped with an action by $\\Gamma_1$, cf. \\cite[\\S1.3]{Bor79}. Note that ${\\rm Frob}_v$ acts as $\\theta\\in \\Gamma_1$ on $\\Delta$ for all $v\\in {\\mathcal V}_{{\\mathbf F}}(\\theta)\\backslash S_{{\\rm bad}}$. According to \\cite[\\S3.2]{Bor79}, the $\\theta$-stable subsets of $\\Delta$ are in bijection with $G({\\mathbf F}_v)$-conjugacy classes of ${\\mathbf F}_v$-parabolic subgroups of $G$. For each $v\\in {\\mathcal V}_{{\\mathbf F}}(\\theta)\\backslash S_{{\\rm bad}}$, fix a Borel subgroup $B_v$ of $G$ over ${\\mathbf F}_v$ containing the centralizer $T_v$ of $A_v$ in $G$ so that the following are in a canonical bijection with one another.\n\\begin{itemize}\n\\item $\\theta$-stable subsets $\\Upsilon$ of $\\Delta$\n\\item parabolic subgroups $P_v$ of $G$ containing $B_v$\n\\end{itemize}\n  Denote by $P_{\\Upsilon,v}$ the parabolic subgroup corresponding to $\\Upsilon$ and by $M_{\\Upsilon,v}$ its Levi subgroup containing $T_v$. Here is an important observation: the inequalities \\eqref{e:app:Levi}, \\eqref{e:app:Elliptic1}, and \\eqref{e:app:Elliptic2} to be satisfied by $a_{M_{\\Upsilon},v},b_{M_{\\Upsilon},v},e_{M_{\\Upsilon},v}$ depend only on $\\theta$ and not on $v\\in {\\mathcal V}_{{\\mathbf F}}(\\theta)\\backslash S_{{\\rm bad}}$. (We consider the case where $G$ and $M$ of those inequalities are $ M_{\\Upsilon}$ and a ${\\mathbf F}_v$-Levi subgroup of $M_{\\Upsilon}$, respectively.) Hence we will write $a_{M_{\\Upsilon},\\theta},b_{M_{\\Upsilon},\\theta},e_{M_{\\Upsilon},\\theta}\\ge 0$ for these constants. What we need to do is to define them inductively according to the semisimple rank of $M$ such that \\eqref{e:app:Levi}, \\eqref{e:app:Elliptic1}, and \\eqref{e:app:Elliptic2} hold true. In particular the desired $a_{G,\\theta},b_{G,\\theta},e_{G,\\theta}$ will be obtained and the proof will be finished (by returning to the first paragraph in the current proof).\n\n  Now the inductive choice of $a_{M_{\\Upsilon},\\theta},b_{M_{\\Upsilon},\\theta},e_{M_{\\Upsilon},\\theta}$ is easy to make once the choice of $a_{M_{\\Omega},\\theta},b_{M_{\\Omega},\\theta},e_{M_{\\Omega},\\theta}$ has been made for all $\\Omega\\subsetneq \\Upsilon$. Indeed, we may choose $e_{M_{\\Omega},\\theta}=0$ to fulfill \\eqref{e:app:Elliptic1} (since $Z_3\\ge 0$; see Lemma \\ref{l:B-4} below) and $a_{M_{\\Omega},\\theta},b_{M_{\\Omega},\\theta}$ to be large enough to verify \\eqref{e:app:Levi} and \\eqref{e:app:Elliptic2}. Notice that $Z_1,Z_2,Z_3$ of \\eqref{e:app:Elliptic2} (which are constructed in Lemma \\ref{l:B-4} below) depend only on the group-theoretic information of $M_{\\Upsilon}$ (such as the dimension, rank, affine root data, $\\delta_{M_{\\Upsilon}}$ of $M_{\\Upsilon}$ as well as an embedding of the Chevalley form of $M_{\\Upsilon}$ into $GL_d$ coming from $\\Xi^{\\mathrm{sp}}$) but not on $v$, cf. Remark \\ref{r:uniformity}.\n\n  It remains to treat $v\\in S_Q$ not contained in $S_{{\\rm bad}}$. By Lemma \\ref{l:conj-image-in-diag} there exists a local embedding $\\Xi_v:G\\rightarrow \\GL_d$ defined over $\\mathcal{O}_v$, at each such $v$, such that $\\Xi_v$ satisfies the assumption preceding Proposition \\ref{p:appendix2}. Let $a_{G,v},b_{G,v},e_{G,v}$ be as in the proposition. As we remarked earlier in the proof, we can and will take $e_{G,v}=0$.\n  Set $a_G$ to be the maximum of $\\max_{\\theta\\in {\\mathscr C}(\\Gamma_1)} a_{G,\\theta}$ and $\\max_{v\\in S_Q\\backslash S_{{\\rm bad}}} a_{G,v}$. Define $b_G$ similarly and put $e_G=0$. As theses choices of $a_G$, $b_G$ and $e_G$ will only improve the inequalities \\eqref{e:app:Levi}, \\eqref{e:app:Elliptic1}, and \\eqref{e:app:Elliptic2} required for $a_{G,\\theta}$, $b_{\\theta}$ and $e_{\\theta}$, the proof will be finished.\n\n\\end{proof}\n\n   In view of Theorem \\ref{t:appendix1} and other observations in harmonic analysis, a natural question is whether it is possible to achieve $e_{G}=1$. This is a deep and difficult question which would be of independent interest. It was a pleasant surprise to the authors that the theory of arithmetic motivic integration provides a solution. A precise theorem due to Cluckers, Gordon, and Halupczok is stated below. It is worth remarking that the method of proof is significantly different from that of this section and also that they make use of Theorem \\ref{t:appendix1}, the local boundedness theorem.\n\n\\begin{thm}\\label{t:optimal-expo}\n  Theorem \\ref{t:appendeix2} holds true with $e_G=1$.\n\\end{thm}\n\n\\begin{proof}\n  See Appendix~\\ref{s:app:B}.\n\\end{proof}\n\n\\begin{rem}\n  It would be interesting to ask about the analogue of the theorem in the case of twisted or weighted orbital integrals. Such a result would be useful in the more general situation than the one considered in this paper.\n\\end{rem}\n\n\n\\subsection{The noncentral elliptic case}\\label{sub:app:elliptic}\n\n  The objective of this subsection is to establish Corollary \\ref{c:B-1}, which was used in Step 2-3\n  of the proof of Proposition \\ref{p:appendix2} above. Since the proof is quite complicated let us guide the reader. The basic idea, going back to Langlands, is to interpret the orbital integral $O^{G(F)}_{\\gamma}(\\tau^G_{\\lambda})$ in question as the number of points in the building fixed ``up to $\\lambda$'' under the action of $\\gamma$. The set of such points, denoted $X_F(\\gamma,\\lambda)$ below, is finite since $\\gamma$ is elliptic. Then it is shown that every point of $X_F(\\gamma,\\lambda)$ is within a certain distance from a certain apartment, after enlarging the ground field $F$ to a finite extension. We exploit this to bound $X_F(\\gamma,\\lambda)$ by a ball of an explicit radius in the building. By counting the number of points in the ball (which is of course much more tractable than counting $|X_F(\\gamma,\\lambda)|$) we arrive at the desired bound on the orbital integral.\n   The proof presented here is inspired by the beautiful exposition of \\cite[\\S\\S3-5]{Kot05} but uses (not so beautiful) brute force and crude bounds at several places.\n\n\n  Throughout this subsection the notation of \\S\\ref{sub:local-bound-orb-int} is adopted and $\\gamma$ is assumed to be noncentral and elliptic in $G(F)$.\nThen $I_\\gamma(F)$ is a compact group, on which the Euler-Poincare measure $\\mu^{{\\rm EP}}_{I_\\gamma}$\nassigns total volume 1. Our aim is to bound $O^{G(F)}_{\\gamma}({\\mathbf{1}}_{K\\mu(\\varpi)K}, \\mu^{\\mathrm{can}}_G,\\mu^{\\mathrm{can}}_{I_\\gamma})$.\n It follows from \\cite[Thm 5.5]{Gro97} (for the equality) and Proposition \\ref{p:Gross-motives} that\n$$\\left|\\frac{\\mu^{{\\rm EP}}_{I_\\gamma}}{\\mu^{\\mathrm{can}}_{I_\\gamma}}\\right|\n= \\frac{\\prod_{d\\ge 1} \\det\\left(1-{\\rm Frob}_v q_v^{d-1}\\left|({\\rm Mot}_{I_\\gamma,d})^{I_v}\\right.\\right)}\n{|H^1(F,I_\\gamma)|}\\le \\prod_{d\\ge 1}(1+q_v^{d-1})^{\\dim {\\rm Mot}_{I_\\gamma,d}}$$ \\begin{equation}\\le (1+q_v^{(\\dim I_\\gamma+1)\/2})^{{\\rm rk} I_\\gamma}\n\\le (1+q_v^{d_G})^{r_G}\\le q_v^{r_G(d_G+1)}.\\label{e:mu-EP\/mu}\\end{equation}\nThus we may as well bound $O^{G(F)}_{\\gamma}({\\mathbf{1}}_{K\\mu(\\varpi)K}, \\mu^{\\mathrm{can}}_G,\\mu^{{\\rm EP}}_{I_\\gamma})$.\n\n  Let $T_\\gamma$ be a maximal torus of $I_\\gamma$ defined over $F$ containing $\\gamma$.\n  By Lemma \\ref{l:torus-splitting}, there exists a Galois extension $F'\/F$ with\n  \\begin{equation}\\label{e:F':F}[F':F]\\le w_G s_G\\end{equation}\nsuch that $T_\\gamma$ is a split torus over $F'$.\n Hence $I_\\gamma$ and $G$ are split groups over $F'$.\n Note that $F'$ is a tame extension of $F$ under the assumption that ${\\rm char}\\, k_F>w_Gs_G$.\nLet $A'$ be a maximal split torus of $G$ over $F'$ such that $A\\times_F F'\\subset A'$. Since maximal $F'$-split tori are conjugate over $F'$, we find\n$$y\\in G(F') \\quad \\mbox{such that}\\quad A'=yT_\\gamma y^{-1}$$\nand fix such a $y$.\nWrite $\\mathcal{O}'$, $\\varpi'$ and $v'$ for the integer ring of $F'$, a uniformizer and the valuation on $F'$ such that $v'(\\varpi')=1$.\nWith respect to the integral model of $G$ over $\\mathcal{O}$ at the beginning of \\S\\ref{sub:local-bound-orb-int},\nwe put $K':=G(\\mathcal{O}')$. A point of $G(F)\/K$ will be denoted $\\overline{x}$ and any of its lift\nin $G(F)$ will be denoted $x$. Let $\\overline{x}_0\\in G(F)\/K$ (resp. $\\overline{x}'_0\\in G(F')\/K'$) denote the\nelement represented by the trivial coset of $K$ (resp. $K'$). Then $\\overline{x}_0$ (resp. $\\overline{x}'_0$)\nmay be thought of as a base point of the building ${\\mathcal{B}}(G(F),K)$ (resp. ${\\mathcal{B}}(G(F'),K')$) and\nits stabilizer is identified with $K$ (resp. $K'$). There exists an injection\n\\begin{equation}\\label{e:embed-building}{\\mathcal{B}}(G(F),K)\\hookrightarrow {\\mathcal{B}}(G(F'),K')\\end{equation} such that ${\\mathcal{B}}(G(F),K)$ is the ${\\rm Gal}(F'\/F)$-fixed points of ${\\mathcal{B}}(G(F'),K')$. (This is the case because $F'$ is tame over $F$.) The natural injection $G(F)\/K\\hookrightarrow G(F')\/K'$ coincides with the injection induced by \\eqref{e:embed-building} on the set of vertices.\n\nDefine $\\lambda'\\in X_*(A')$ by $\\lambda'=e_{F'\/F} \\lambda$ (where $e_{F'\/F}$ is the ramification index of $F'$ over $F$) so that $\\lambda'(\\varpi')=\\lambda(\\varpi)$ and\n\\begin{equation}\\label{e:mu'-abs-val}\\|\\lambda'\\|=e_{F'\/F} \\|\\lambda\\|\\le e_{F'\/F}\\kappa.\\end{equation}\nFor (the fixed $\\gamma$ and) a semisimple element $\\delta\\in G(F')$, set\n\\begin{eqnarray}\n  X_F(\\gamma,\\lambda)&:=&\\{ \\overline{x}\\in G(F)\/K: \\overline{x}^{-1} \\gamma \\overline{x}\\in K\\lambda(\\varpi) K\\}\\nonumber\\\\\nX_{F'}(\\delta,\\lambda')&:=&\\{ \\overline{x}'\\in G(F')\/K': (\\overline{x}')^{-1} \\delta \\overline{x}'\\in K'\\lambda'(\\varpi') K'\\}.\\nonumber\n\\end{eqnarray}\nBy abuse of notation we write $\\overline{x}^{-1} \\gamma \\overline{x}\\in K\\lambda(\\varpi) K$ for the condition that $x^{-1}\\gamma x\\in K\\lambda(\\varpi) K$ for some (thus every) lift $x\\in G(F)$ of $\\overline{x}$ and similarly for the condition on $\\overline{x}'$.\nIt is clear that $X_F(\\gamma,\\lambda)\\subset X_{F'}(\\gamma,\\lambda')\\cap( G(F)\/K)$. By (3.4.2) of \\cite{Kot05}\n\\begin{equation}\\label{e:orb-via-fixed} O^{G(F)}_{\\gamma}({\\mathbf{1}}_{K\\lambda(\\varpi)K}, \\lambda_G,\\lambda^{{\\rm EP}}_{I_\\gamma})=| X_F(\\gamma,\\lambda)|.\\end{equation}\nOur goal of bounding the orbital integrals on the left hand side can be translated into a problem of bounding $| X_F(\\gamma,\\lambda)|$.\n\n\nLet ${\\rm Apt}(A'(F'))$ denote the apartment for $A'(F')$. Likewise ${\\rm Apt}(T_\\gamma(F))$ and\n${\\rm Apt}(T_\\gamma(F'))$ are given the obvious meanings. We have $\\overline{x}'_0\\in {\\rm Apt}(A'(F'))$.\nThe metrics on ${\\mathcal{B}}(G(F),K)$ and ${\\mathcal{B}}(G(F'),K')$ are\nchosen such that \\eqref{e:embed-building} is an isometry.\nThe metric on ${\\mathcal{B}}(G(F'),K')$ is\ndetermined by its restriction to ${\\rm Apt}(A'(F'))$, which is in turn pinned down by\na (non-canonical choice of) a Weyl-group invariant scalar product on $X_*(A')$, cf. \\cite[\\S2.3]{Tit79}. Henceforth we fix the scalar product once and for all. Scaling the scalar product does not change our results as the reader can check.\n\\begin{rem}\nFor any other tame extension $F''$ of $F$ and a maximal split torus $A''$ of $G$ over $F''$, we can find an isomorphism $X_*(A')$ and $X_*(A'')$ over the composite field of $F'$ and $F''$, well defined up to the Weyl group action. So the scalar product on $X_*(A'')$ is uniquely determined by that on $X_*(A')$. So we need not choose a scalar product again when considering a different $\\gamma\\in G(F)$.\n\\end{rem}\n\n\nFor any $F'$-split maximal torus $A''$ of $G$ (for instance $A''=T_\\gamma$ or $A''=A'$) and the associated set of roots $\\Phi=\\Phi(G,A'')$ and set of coroots $\\Phi^\\vee=\\Phi^\\vee(G,A'')$,\nlet $l_{\\min}(\\Phi)$ (resp. $l_{\\max}(\\Phi)$) denote the the shortest (resp. longest) length of a positive coroot\nin $\\Phi^\\vee$. Note that these are independent of the choice of $A''$ and completely determined by the previous choice of a Weyl group invariant scalar product on $X_*(A')$. It is harmless to assume that we have chosen the scalar product such that the shortest positive coroot in each irreducible system of $X_*(A')$ has length $l_{\\min}(\\Phi)$.\n\n  Fix a Borel subgroup $B'$ of $G$ (over $F'$) containing $A'$ so that $y^{-1}B'y$ is a Borel subgroup containing $T_\\gamma$. Relative to these Borel subgroups we define the subset of positive roots $\\Phi^+(G,A')$ and $\\Phi^+(G,T_\\gamma)$. Let $m_{\\Xi^{\\mathrm{sp}}}$ be as in Lemma \\ref{l:app:split1}. In order to bound $|X_F(\\gamma,\\lambda)|$ in \\eqref{e:orb-via-fixed}, we control the larger set $X_{F'}(\\delta,\\lambda')$ by bounding the distance from its points to the apartment for $A'$.\n\n\n\\begin{lem}\\label{l:dbl-coset-distance}\n  Let $\\delta\\in A'(F')$ and $\\overline{x}'\\in G(F')\/K'$. Then\nthere exist constants $C=C(G,\\Xi)>0$ and $Y=Y(G)\\in {\\mathbb Z}_{\\ge1}$ such that\nwhenever $(\\overline{x}')^{-1} \\delta \\overline{x}'\\in K'\\lambda'(\\varpi') K'$,\n$$ d(\\overline{x}',{\\rm Apt}(A'(F')))\\le\nl_{\\max}(\\Phi)\\cdot C|\\Delta^+|\\cdot Y^{|\\Phi^+|}w_G s_G\\qquad\\qquad$$\n$$\\qquad\\qquad\\times \\left(\\sum_{\\alpha\\in \\Phi^+(G,A')} |v(1-\\alpha^{-1}(\\delta))|+ Y m_{\\Xi^{\\mathrm{sp}}}(m_{\\Xi^{\\mathrm{sp}}}+1)\\kappa\\right),$$\nwhere the left hand side denotes the shortest distance from $\\overline{x}'$ to ${\\rm Apt}(A'(F'))$.\n\\end{lem}\n\n\\begin{proof}[Proof of Lemma \\ref{l:dbl-coset-distance}]\n  We may write $\\overline{x}'=an\\overline{x}_0'$ for some $a\\in A'(F')$ and $n\\in N(F')$. As both sides of the above inequality\nare invariant under multiplication by $a$, we may assume that $a=1$.\nLet $\\lambda_\\delta\\in X_*(A')$ be such that $\\delta\\in  \\lambda_\\delta(\\varpi')A'(\\mathcal{O}')$.\n\n\\medskip\n\n\\begin{enumerate}\n\n\\item[Step 1.] Show that $\\delta^{-1} n^{-1}\\delta n\\in K'\\lambda_0(\\varpi')K'$\nfor some $\\lambda_0\\in X_*(A')$ with $\\|\\lambda_0\\|\\le e_{F'\/F}(m_{\\Xi}+1)\\kappa$.\n\n\\medskip\n\n\n  By Cartan decomposition there exists a $B'$-dominant $ \\lambda_0\\in X_*(A')$ such that\n  $\\delta^{-1} n^{-1}\\delta n\\in K'\\lambda_0(\\varpi')K'$.\n  The condition of the lemma on $\\delta$ is unraveled as\n  $(x'_0)^{-1} n^{-1} \\delta n x'_0\\in K'\\lambda'(\\varpi') K'$. So\n$$\\delta^{-1}n^{-1} \\delta n\\in \\delta^{-1} K'\\lambda'(\\varpi') K'\n\\subset (K'\\lambda_\\delta^{-1}(\\varpi') K')(K'\\lambda'(\\varpi') K').$$\nLet $w$ be a Weyl group element for $A'$ in $G$ such that $w\\lambda_\\delta^{-1}$ is $B'$-dominant. The fact that $K'\\lambda_0(\\varpi')K'$ intersects\n$(K'\\lambda_\\delta^{-1}(\\varpi') K')(K'\\lambda'(\\varpi') K')$ implies (\\cite[Prop 4.4.4.(iii)]{BT72}) that\n$$\\|\\lambda_0\\|\\le \\| w\\cdot \\lambda_\\delta^{-1}\\|+\\|\\lambda'\\| = \\|  \\lambda_\\delta\\|+\\|\\lambda'\\|.$$\nNote that $ v'(\\alpha(\\delta))\\in\n[-m_{\\Xi^{\\mathrm{sp}}} \\|\\lambda'\\|,m_{\\Xi^{\\mathrm{sp}}} \\|\\lambda'\\|]$ for all $\\alpha\\in \\Phi(G,A')$ by Lemma \\ref{l:app:split1} since a conjugate of $\\delta$ belongs to $K'\\lambda'(\\varpi')K'$. This implies that  $\\|\\lambda_{\\delta}\\|\\le m_{\\Xi}\\|\\lambda'\\|$. On the other hand $\\|\\lambda'\\|\\le e_{F'\/F}\\kappa$ according to \\eqref{e:mu'-abs-val}. This completes Step 1.\n\n\\medskip\n\n  Before entering Step 2, we notify the reader that we are going to use the convention and notation of the Chevalley basis as recalled in \\S\\ref{sub:lem-split} below. In particular $n\\in N(F')$ can be written as (cf. \\eqref{e:y=product})\n  \\begin{equation}\\label{e:n=product}n=x_{\\alpha_1}(X_{\\alpha_1})\\cdots x_{\\alpha_{|\\Phi^+|}}(X_{\\alpha_{|\\Phi^+|}})\\end{equation}\n  for unique $X_{\\alpha_1},...,X_{\\alpha_{|\\Phi^+|}}\\in F'$.\n\\medskip\n\n\\item[Step 2.] Show that there exists a constant $\\mathcal {M}_{|\\Phi^+|}\\ge 0$ (explicitly defined in \\eqref{e:cM_i} below) such that\n$v'(X_{\\alpha_i})\\ge -\\mathcal {M}_{|\\Phi^+|}$ for all $1\\le i\\le |\\Phi^+|$.\n\n\n\\medskip\n\n\n  In our setting we compute\n\\begin{eqnarray}\n  \\delta^{-1}n^{-1}\\delta n\n&=&  \\delta^{-1} \\left(\\prod_{i=|\\Phi^+|}^{1} x_{\\alpha_i}(-X_{\\alpha_i})\\right) \\delta\n\\prod_{i=1}^{|\\Phi^+|} x_{\\alpha_i}(X_{\\alpha_i})\n\\nonumber\\\\&= &\\left(\\prod_{i=|\\Phi^+|}^{1} x_{\\alpha_i}(-\\alpha^{-1}_i(\\delta) X_{\\alpha_i}) \\right)\n\\prod_{i=1}^{|\\Phi^+|} x_{\\alpha_i}(X_{\\alpha_i})\n\\nonumber\\\\&=&\\prod_{i=1}^{|\\Phi^+|} x_{\\alpha_i}\\left((1-\\alpha^{-1}_i(\\delta))X_{\\alpha_i}\n+ P_{\\alpha_i} \\right)\\label{e:delta-n-delta-n}\n\\end{eqnarray}\nwhere the last equality follows from the repeated use of \\eqref{e:commutator} to rearrange the terms.\nHere $P_{\\alpha_i}$ is a polynomial (which could be zero) in $\\alpha_{j}^{-1}(\\delta)$\nand $X_{\\alpha_j}$ with integer coefficients for $j<i$. It is not hard to observe from \\eqref{e:commutator} that\n$P_{\\alpha_i}$ has no constant term. As $i$ varies in $[1,|\\Phi^+|]$,\nlet $Y$ denote the highest degree for the nonzero monomial term appearing in $P_{\\alpha_i}$\nviewed as a polynomial in $\\alpha_i^{-1}(\\delta)$ and as a polynomial in $X_{\\alpha_i}$.\\footnote{For instance\nif $P_{\\alpha_i}=\\alpha_i^{-1}(\\delta)^2 X_{\\alpha_i}^4 + \\alpha_i^{-1}(\\delta)^3 X_{\\alpha_i}^3$ then $Y=4$.} Set $Y=1$ if $P_{\\alpha_i}=0$.\nAs mentioned above, the positive roots for a given $(G,\\mathbf{B},\\mathbf {T})$ are ordered once and for all so that $Y$ depends only on $G$ in the sense that for any $G$ having $G$ as its Chevalley form, $Y$ is independent of the local field $F$ over which $G$ is defined.\n\n Applying Lemma \\ref{l:app:split2}, we obtain from \\eqref{e:delta-n-delta-n} and $\\delta^{-1}n^{-1}\\delta n\\in K'\\lambda_0(\\varpi')K'$ that\n\\begin{equation}\\label{e:v(N-alpha)}v'\\left((1-\\alpha^{-1}_i(\\delta))X_{\\alpha_i}\n+ P_{\\alpha_i}\\right) \\ge -m_{\\Xi^{\\mathrm{sp}}} \\|\\lambda_0\\|.\\end{equation}\nPut \\begin{equation}\n\\label{e:cM_i}\\mathcal {M}_i:=\\sum_{j=1}^i Y^{i-j} (|v'(1-\\alpha_j^{-1}(\\delta))|+Y m_{\\Xi^{\\mathrm{sp}}}\\|\\lambda_0\\|),\\quad 1\\le i\\le |\\Phi^+|.\\end{equation}\nObviously $\\mathcal {M}_1\\le \\mathcal {M}_2\\le \\cdots\\le \\mathcal {M}_{|\\Phi^+|}$.\nWe claim that for every $i\\ge 1$,\n\\begin{equation}\\label{e:greater-cM_i} v'(X_{\\alpha_i}) \\ge -\\mathcal {M}_i.\\end{equation}\nWhen $i=1$, this is easy as $P_{\\alpha_1}=0$. (Use the fact that $X_{\\alpha_1}(a_1X_{\\alpha_1})$ commutes with any other $X_{\\alpha_j}(a_jX_{\\alpha_j})$ in view of \\eqref{e:commutator} since $\\alpha_1$ is a simple root.) Now by induction,\nsuppose that \\eqref{e:greater-cM_i} is verified for all $j<i$.\nBy \\eqref{e:v(N-alpha)},\n$$v'(X_{\\alpha_i})+v'(1-\\alpha^{-1}_i(\\delta))\\ge \\min(m_{\\Xi^{\\mathrm{sp}}}\\|\\lambda_0\\|,v'(P_{\\alpha_i})).$$\nBy the definition of $Y$, $$v'(P_{\\alpha_i})\\ge \\min_{j<i} v'(\\alpha_j^{-1}(\\delta)^Y) +\n\\min_{j<i} v'(X_{\\alpha_{j}}^Y) \\ge -Y(m_{\\Xi^{\\mathrm{sp}}}\\|\\lambda_0\\|+\\mathcal {M}_{i-1}).$$\nHence $v'(X_{\\alpha_i})+v'(1-\\alpha^{-1}_i(\\delta))\\ge -Y(m_{\\Xi^{\\mathrm{sp}}}\\|\\lambda_0\\|+\\mathcal {M}_{i-1})$, implying \\eqref{e:greater-cM_i} by the inductive hypothesis. Now that \\eqref{e:greater-cM_i} is verified for all $i$, we have in particular $v'(X_{\\alpha_i})\\ge -\\mathcal {M}_{|\\Phi^+|}$. In view of \\eqref{e:n=product} (and the fact that $x_{\\alpha}(\\mathcal{O}')=U_{\\alpha,0}$), we have $n\\in G_{\\overline{x}'_0,-\\mathcal {M}_{|\\Phi^+|}}$.\n\n\n\n\n\\medskip\n\n\\item[Step 3.] Find $a\\in A'(F')$ such that $a^{-1}na\\in K'$.\n\n\\medskip\n\n We can choose a sufficiently large $C=C(G,\\Xi)>0$,\n  depending only on the Chevalley group $G$ and $\\Xi$, and integers $a^0_{\\alpha}\\in [-C,0]$ for\n$\\alpha\\in \\Delta^+$ such that\n$$1\\le \\sum_{\\alpha\\in \\Delta^+} (-a^0_\\alpha) \\langle \\beta,\\alpha^\\vee\\rangle \\le C,\\quad \\forall \\beta\\in \\Delta^+.$$\n(This is possible because the matrix $(\\langle \\beta,\\alpha^\\vee\\rangle )_{\\beta,\\alpha\\in \\Delta^+}$ is nonsingular. For instance one finds $a^0_{\\alpha}\\in {\\mathbb Q}$ satisfying the above inequalities for $C=1$ and then eliminate denominators in $a^0_{\\alpha}$ by multiplying a large positive integer.)\nNow put $a_{\\alpha}:=\\mathcal {M}_{|\\Phi^+|} a^0_{\\alpha}\\in [-C\\mathcal {M}_{|\\Phi^+|},0]$ and\n  $a:=\\sum_{\\alpha\\in \\Delta^+} a_\\alpha \\alpha^\\vee(\\varpi')\\in A'(F')$ so that\n\\begin{equation}\\label{e:bound-v(a)}\n\\mathcal {M}_{|\\Phi^+|}\\le -v(\\beta(a))\\le C\\cdot\\mathcal {M}_{|\\Phi^+|},\n\\quad \\forall \\beta\\in \\Delta^+.\\end{equation}\nIn fact \\eqref{e:bound-v(a)} implies that the left inequality holds for all $\\beta\\in \\Phi^+$.\nHence\n$$a^{-1}na=\\prod_{i=1}^{|\\Phi^+|}\nx_{\\alpha_i}(\\alpha_i(a)^{-1} X_{\\alpha_{i}})$$\n$$\\in \\prod_{i=1}^{|\\Phi^+|} U_{\\alpha_i,v(X_{\\alpha_i})-v(\\alpha_i(a))}\\subset \\prod_{i=1}^{|\\Phi^+|} U_{\\alpha_i,\\mathcal {M}_{|\\Phi^+|}\n+v(X_{\\alpha_i})}.$$\nIn light of \\eqref{e:greater-cM_i},\n$\\mathcal {M}_{|\\Phi^+|}\n+v(X_{\\alpha_i})\\ge 0$. Hence $a^{-1}na\\in K'$.\n\n\n\n\\medskip\n\n\\item[Step 4.] Conclude the proof.\n\n\\medskip\n\n  Step 3 shows that $a\\overline{x}'_0\\in {\\rm Apt}(A'(F'))$ is invariant under the left multiplication\naction by $n$ on ${\\mathcal{B}}(G(F'),K')$, which acts as an isometry. Hence\n\\begin{equation}\\label{e:Step4-1}d(\\overline{x}',  {\\rm Apt}(A'(F'))\\le d(n\\overline{x}'_0, a\\overline{x}'_0)= d(n\\overline{x}'_0, na\\overline{x}'_0)=\nd(\\overline{x}'_0, a\\overline{x}'_0).\\end{equation}\nOn the other hand, for any\n$\\overline{x}'\\in {\\rm Apt}(A'(F'))$ and any positive simple coroot $\\alpha^{\\vee}$,\n\\begin{equation}\\label{e:dist-simple-coroot}\nl_{\\min}(\\Phi)\\le d(\\overline{x}', \\alpha^\\vee(\\varpi')^{-1} \\overline{x}')\\le l_{\\max}(\\Phi)\\end{equation}\n(The left inequality holds by definition of $l_{\\min}$. The right inequality comes from the\nstandard fact\nthat the length of the longest coroot is bounded by $6l_{\\min}$ in an irreducible system.)\nSince $a=\\prod_{\\alpha\\in \\Delta^+} (\\alpha^\\vee(\\varpi'))^{a_{\\alpha}}$ with\n$a_\\alpha\\in [-C\\mathcal {M}_{|\\Phi^+|},0]$, a repeated use of \\eqref{e:dist-simple-coroot}, together with\na triangle inequality, shows that\n\\begin{equation}\\label{e:Step4-3} d(\\overline{x}'_0, a\\overline{x}'_0)\\le l_{\\max}(\\Phi)\\cdot C|\\Delta^+|\\cdot  \\mathcal {M}_{|\\Phi^+|}.\\end{equation}\nLemma \\ref{l:dbl-coset-distance} follows from\n\\eqref{e:Step4-1}, \\eqref{e:Step4-3}, the formula for $\\mathcal {M}_{|\\Phi^+|}$ (\\eqref{e:cM_i}), the inequality for $\\|\\lambda_0\\|$ in Step 1, and $e_{F'\/F}\\le [F':F]\\le w_G s_G$ as we saw in \\eqref{e:F':F}.\n\n\\end{enumerate}\n\n\\end{proof}\n\n  Since $\\gamma$ is elliptic, ${\\rm Apt}(T_\\gamma(F))$ is a singleton. Let $\\overline{x}_1$ denote its only point.\nThen the ${\\rm Gal}(F'\/F)$-action on ${\\rm Apt}(T_\\gamma(F'))$ has $\\overline{x}_1$ as the unique fixed point.\nMotivated by Lemma \\ref{l:dbl-coset-distance} we set $\\mathcal {M}(\\gamma,\\kappa)$ to be\n$$l_{\\max}(\\Phi)\\cdot C|\\Delta^+|\\cdot Y^{|\\Phi^+|}w_G s_G\\left(\n\\sum_{\\alpha\\in \\Phi^+(G,T_\\gamma)} |v(1-\\alpha^{-1}(\\gamma))| + Y m_{\\Xi^{\\mathrm{sp}}} (m_{\\Xi^{\\mathrm{sp}}}+1) \\kappa\\right)$$\nand similarly $ \\mathcal {M}(\\delta,\\kappa)$ using $\\alpha\\in \\Phi^+(G,A')$ in the sum instead.\n(Useful to know: $v(1-\\alpha(\\gamma))+v(1-\\alpha^{-1}(\\gamma))\\le 2|v(1-\\alpha^{-1}(\\gamma))|$\nfor all $\\alpha\\in \\Phi^+$.) Define (the set of vertices in) a closed ball in $G(F)\/K$: for\n$v \\in G(F)\/K$,\n$$\\mathrm{Ball}(v,R):=\\{ \\overline{x}\\in G(F)\/K: d(\\overline{x},v)\\le R\\}.$$\n\n\n\\begin{lem}\\label{l:B-3}\n  $X_F(\\gamma,\\lambda)~\\subset~\\mathrm{Ball}(\\overline{x}_1,\\mathcal {M}(\\gamma,\\kappa)).$\n\\end{lem}\n\n\\begin{proof}\n  As we noted above, $X_F(\\gamma,\\lambda)\\subset X_{F'}(\\gamma,\\lambda')=X_{F'}(y^{-1}\\delta y,\\lambda')$.\nLemma \\ref{l:dbl-coset-distance} tells us that\n$$\\overline{x}\\in X_F(\\gamma,\\lambda)~\\Rightarrow~d(y\\overline{x},{\\rm Apt}(A'(F')))\\le \\mathcal {M}(\\delta,\\kappa)\n~\\Rightarrow~ d(\\overline{x},{\\rm Apt}(T_\\gamma(F')))\\le \\mathcal {M}(\\delta,\\kappa).$$\nThe last implication uses ${\\rm Apt}(A'(F'))=y {\\rm Apt}(T_\\gamma(F')) y^{-1}$.\n  We have viewed $\\overline{x}$ as a point of ${\\mathcal{B}}(G(F'),K')$\nvia the isometric embedding ${\\mathcal{B}}(G(F),K)\\hookrightarrow {\\mathcal{B}}(G(F'),K')$.\nIn order to prove the lemma, it is enough to check that $d(\\overline{x},\\overline{x}_1)\\le d(\\overline{x},\\overline{x}_2)$ for every\n$\\overline{x}_2\\in {\\rm Apt}(T_\\gamma(F'))$. To this end,\nwe suppose that there exists an $\\overline{x}_2$ with\n\\begin{equation}\\label{e:assump-dist}d(\\overline{x},\\overline{x}_1)> d(\\overline{x},\\overline{x}_2)\\end{equation}\nand will draw a contradiction.\n\n As $\\sigma\\in{\\rm Gal}(F'\/F)$ acts on ${\\mathcal{B}}(G(F'),K')$ by isometry,\n$d(\\overline{x},\\sigma \\overline{x}_2)=d(\\overline{x},\\overline{x}_2)$. As ${\\rm Apt}(T_\\gamma(F'))$\nis preserved under the Galois action, $\\sigma\\overline{x}_2\\in {\\rm Apt}(T_\\gamma(F'))$.\nAccording to the inequality of \\cite[2.3]{Tit79}, for any $x,y,z\\in {\\mathcal{B}}(G(F'),K')$ and\nfor the unique mid point $m=m(x,y)\\in {\\mathcal{B}}(G(F'),K')$ such that $d(x,m)=d(y,m)=\\frac{1}{2}d(x,y)$,\n\\begin{equation}\\label{e:Tits}d(x,z)^2+d(y,z)^2\n\\ge 2 d(m,z)^2 + \\frac{1}{2}d(x,y)^2.\\end{equation}\nConsider the convex hull ${\\mathscr C}$ of ${\\mathscr C}_0:=\\{\\sigma\\overline{x}_2\\}_{\\sigma\\in{\\rm Gal}(F'\/F)}$.\nSince ${\\mathscr C}_0$ is contained in ${\\rm Apt}(T_\\gamma(F'))$, so is ${\\mathscr C}$.\nMoreover ${\\mathscr C}_0$ is fixed under ${\\rm Gal}(F'\/F)$,\nfrom which it follows that\n${\\mathscr C}$ is also preserved under the same action.\n(One may argue as follows. Inductively define ${\\mathscr C}_{i+1}$ to be the set consisting of the mid points $m(x,y)$\nfor all $x,y\\in {\\mathscr C}_{i}$. Then it is not hard to see that ${\\mathscr C}_i$ must be preserved under\n${\\rm Gal}(F'\/F)$ and that $\\cup_{i\\ge 0} {\\mathscr C}_i$ is a dense subset of ${\\mathscr C}$.)\nAs ${\\mathscr C}$ is a compact set, one may choose $\\overline{x}_3\\in {\\mathscr C}$ which has the minimal distance\nto $\\overline{x}$ among the points of ${\\mathscr C}$. By construction \\begin{equation}\\label{e:contrad-dist} d(\\overline{x}_3,\\overline{x})\\le d(\\overline{x}_2,\\overline{x}).\\end{equation}\nApplying \\eqref{e:Tits} to $(x,y,z)=(\\overline{x}_3,\\sigma\\overline{x}_3,\\overline{x})$,\n$$2 d(\\overline{x}_3,\\overline{x})^2= d(\\overline{x}_3,\\overline{x})^2+d(\\sigma\\overline{x}_3,\\overline{x})^2\n\\ge 2d(m(\\overline{x}_3,\\sigma\\overline{x}_3),\\overline{x})^2+\\frac{1}{2}d(\\overline{x}_3,\\sigma\\overline{x}_3)^2.$$\nAs $\\overline{x}_3,\\sigma\\overline{x}_3\\in {\\mathscr C}$, we also have $m(\\overline{x}_3,\\sigma\\overline{x}_3)\\in {\\mathscr C}$\nby the convexity of ${\\mathscr C}$. The choice of $\\overline{x}_3$ ensures that $d(\\overline{x}_3,\\overline{x})\\le d(m(\\overline{x}_3,\\sigma\\overline{x}_3),\\overline{x})$, therefore\n$d(\\overline{x}_3,\\sigma\\overline{x}_3)=0$, i.e. $\\overline{x}_3=\\sigma\\overline{x}_3)$. This holds for every $\\sigma\n\\in {\\rm Gal}(F'\/F)$. Therefore $\\overline{x}_3$ is a ${\\rm Gal}(F'\/F)$-fixed point of ${\\rm Apt}(T_\\gamma(F'))$.\nThis implies that $\\overline{x}_3=\\overline{x}_1$, but then \\eqref{e:contrad-dist} contradicts \\eqref{e:assump-dist}.\n\n\n\n\n\n\n\n\\end{proof}\n\n\\begin{lem}\\label{l:B-4} There exist constants $Z_1,Z_2,Z_3\\ge 0$\nsuch that\n$$| \\mathrm{Ball}(\\overline{x}_1,\\mathcal {M}(\\gamma,\\kappa)) |\\le q^{Z_1+Z_2\\kappa} D^G(\\gamma)^{Z_3}.$$\n\\end{lem}\n\n\\begin{rem}\\label{r:uniformity}\n  It should be noted that $Z$ depends only on the affine root data for $G$ over $F$. Hence the formulas defining $Z_1,Z_2,Z_3$ at the end of the proof tell us that $Z_1,Z_2,Z_3$ depend only on the latter affine root data, the group-theoretic constants for $G$ (and its Chevalley form) and $\\Xi$. This is used in the proof of Theorem \\ref{t:appendeix2} to establish a kind of uniformity when traveling between places in ${\\mathcal V}(\\theta)\\backslash S_{{\\rm bad}}$ for a fixed $\\theta\\in {\\mathscr C}(\\Gamma_1)$ in the notation there.\n\\end{rem}\n\n\\begin{proof}\n  To ease notation we write $\\mathcal {M}$ for $\\mathcal {M}(\\gamma,\\kappa)$ in the proof.\n  Let $e_{\\min},e_{\\max}>0$ be the minimum and maximum lengths of edges of ${\\mathcal{B}}(G(F),K)$, respectively.\n  Given a vertex $v$ of ${\\mathcal{B}}(G(F),K)$,\n  let ${\\mathscr C}_1(v)$ denote the set of chambers of ${\\mathcal{B}}(G(F),K)$ which contains $v$ as a vertex.\n  Then the union of all $C\\in {\\mathscr C}_1(v)$ contains\n  $\\mathrm{Ball}(v,e_{\\min}\/R)$ for some $R$ determined by (any) chamber of\n  ${\\mathcal{B}}(G(F),K)$ in a way that $R$ does not depend on the choice of metric\n  on the building. (Fix a chamber and\n  measure the distance from each vertex to the opposite face of maximal dimension. Take $R$\n  to be the minimum of all such distances divided by $e_{\\min}$.)\n  For $i=1,2,...$, define ${\\mathscr C}_{i+1}(v)$ to be the set of chambers which shares a vertex with some chamber\n  of ${\\mathscr C}_i(v)$. For each $i\\ge 1$, define ${\\mathscr V}_i(v)$ to be the set of vertices in\n  the union of all $C\\in {\\mathscr C}_i(v)$. Set $\\mathcal {M}':=\\lceil \\frac{2\\mathcal {M}+e_{\\max}}{e_{\\min}} \\rceil$.\n Then\n$$\n   \\mathrm{Ball}(\\overline{x}_1,\\mathcal {M})~\\subset ~ \\mathrm{Ball}(\\overline{x}_2,\\mathcal {M}+\\frac{e_{\\max}}{2})\n~\\subset ~ \\bigcup_{C\\in {\\mathscr C}_{\\mathcal {M}'}(\\overline{x}_1)} C.$$\n  \\begin{equation}\\label{e:app:union-chambers}\n  \\mbox{so}\\quad  \\mathrm{Ball}(\\overline{x}_1,\\mathcal {M})~\\subset ~  {\\mathscr V}_{\\mathcal {M}'}(\\overline{x}_1).\\end{equation}\n\n  Let us bound $|{\\mathscr C}_1(\\overline{x}_1)|$ and the number of vertices in ${\\mathscr C}_1(\\overline{x}_1)$.\n The stabilizer of $\\overline{x}_1$ acts transitively on ${\\mathscr C}_1(\\overline{x}_1)$.\n Let $C\\in {\\mathscr C}_1(\\overline{x}_1)$. Then\n$$|{\\mathscr C}_1(\\overline{x}_1)|= |\\mathrm{Stab}(\\overline{x}_1)\/\\mathrm{Stab}(C)|\n= |G(\\mathcal{O})\/\\mathrm{Iw}| \\le |G(k_F)|\\le q^{d_G+r_G}$$\nwhere $\\mathrm{Iw}$ denotes an Iwahori subgroup of $G(\\mathcal{O})$.\n(See the proof of Lemma \\ref{l:double-coset-volume} for the last inequality.)\nEach chamber contains $\\dim A+1$ vertices as a $\\dim A$-dimensional simplex.\nHence for each $i\\ge 1$,\n$$ |{\\mathscr V}_i(v)|\\le (\\dim A+1)\\cdot |{\\mathscr C}_i(\\overline{x}_1)|.$$\nOn the other hand, $$ |{\\mathscr C}_{i+1}(\\overline{x}_1)|\\le\n|{\\mathscr V}_i(v)|\\cdot |{\\mathscr C}_{1}(\\overline{x}_1)|.$$\nHence $|{\\mathscr C}_{i}(\\overline{x}_1)|\\le (\\dim A+1)^{2^{i-1}-1} ({\\mathscr C}_1)^{2^{i-1}}\n\\le (\\dim A+1)^{i-1}q^{i(d_G+r_G)}$ and\n\\begin{equation}\\label{e:app:bound-vertices}|{\\mathscr V}_{\\mathcal {M}'}(\\overline{x}_1)|\\le\n(\\dim A+1) |{\\mathscr C}_{\\mathcal {M}'}(\\overline{x}_1)|\\le (r_G+1)^{\\mathcal {M}'}q^{\\mathcal {M}'(d_G+r_G)}.\\end{equation}\nWe can find a constant $Z\\ge 0$, independent of $\\gamma$ such that\n\\[\nZ:=\\max\\left(\\frac{e_{\\max}}{e_{\\min}}, \\frac{l_{\\max}(\\Phi)}{e_{\\min}}\\right).\n\\]\n Note\n\\[\n\\mathcal {M}'\\le \\frac{2\\mathcal {M}}{e_{\\min}} + Z+1\n\\le Z+1+ CZ|\\Delta^+|\\cdot Y^{|\\Phi^+|}w_G s_G \\left(\n\\sum_{\\alpha\\in \\Phi^+} |v(1-\\alpha^{-1}(\\gamma))| + Y m_{\\Xi^{\\mathrm{sp}}}(m_{\\Xi^{\\mathrm{sp}}}+1) \\kappa\\right),\n\\]\nwhich can be rewritten in the form\n\\[\n\\mathcal {M}'\\le Z+1+Z'_2\\kappa+ Z'_3\\sum_{\\alpha\\in \\Phi^+} |v(1-\\alpha^{-1}(\\gamma))|.\n\\]\nSince $|v(1-\\alpha^{-1}(\\gamma))|\\le v(1-\\alpha^{-1}(\\gamma))+v(1-\\alpha(\\gamma))+2m_\\Xi \\kappa$\nin view of \\eqref{e:app:alpha(gamma)},\nwe have $$q^{\\mathcal {M}'}\\le q^{Z+1+(Z'_2+2m_{\\Xi^{\\mathrm{sp}}} Z'_3)\\kappa} D^G(\\gamma)^{-Z'_3}.$$\nReturning to \\eqref{e:app:union-chambers} and \\eqref{e:app:bound-vertices},\n$$ | \\mathrm{Ball}(\\overline{x}_1,\\mathcal {M})|\n\\le |{\\mathscr V}_{\\mathcal {M}'}(\\overline{x}_1)|\n\\le q^{(r_G+1)\\mathcal {M}'}q^{\\mathcal {M}'(d_G+r_G)}$$ $$\n\\le (q^{Z+1+(Z'_2+2m_{\\Xi^{\\mathrm{sp}}} Z'_3)\\kappa} D^G(\\gamma)^{-Z'_3})^{d_G+2r_G+1}.$$\nWe finish by redefining\n\\begin{itemize}\n\\item $Z_1:=(Z+1)(d_G+2r_G+1)$,\n\\item $Z_2:=(Z'_2+2m_{\\Xi}Z'_3)(d_G+2r_G+1)=CZ|\\Delta^+|Y^{|\\Phi^+|}m_{\\Xi}((m_{\\Xi^{\\mathrm{sp}}}+1)Y+2)(d_G+2r_G+1)$,\n\\item $Z_3:=Z'_3(d_G+2r_G+1)=CZ|\\Delta^+|Y^{|\\Phi^+|}(d_G+2r_G+1)$.\n\\end{itemize}\n\\end{proof}\n\n\\begin{cor}\\label{c:B-1}\n  $|O^{G(F)}_{\\gamma}({\\mathbf{1}}_{K\\lambda(\\varpi)K}, \\mu_G,\\mu^{{\\rm EP}}_{I_\\gamma})|\\le\nq^{r_G(d_G+1)} q^{Z_1+Z_2\\kappa} D^G(\\gamma)^{Z_3}$.\n\\end{cor}\n\n\\begin{proof}\n  Follows from \\eqref{e:orb-via-fixed}, Lemma \\ref{l:B-3} and Lemma \\ref{l:B-4}.\n\n\\end{proof}\n\n\n\\subsection{Lemmas in the split case}\\label{sub:lem-split}\n\nThis subsection plays a supporting role for the previous subsections, especially \\S\\ref{sub:app:elliptic}.\n As in \\S\\ref{sub:global-bound-orb-int} let $G$ be a Chevalley group with a Borel subgroup $\\mathbf{B}$ containing a split maximal torus $\\mathbf {T}$, all over ${\\mathbb Z}$. Let $\\Xi^{\\mathrm{sp}}_{{\\mathbb Q}}:G\\hookrightarrow \\GL_d$ be a closed embedding of algebraic groups over ${\\mathbb Q}$. Let $\\mathbb{T}$ denote the diagonal maximal torus of $\\GL_d$, $\\mathbb{B}$ the upper triangular Borel subgroup of $\\GL_d$, and ${\\mathbb N}$ the unipotent radical of $\\mathbb{B}$.\n\n Extend $\\Xi^{\\mathrm{sp}}_{{\\mathbb Q}}$ to a closed embedding $\\Xi^{\\mathrm{sp}}:G\\hookrightarrow \\GL_d$ defined over ${\\mathbb Z}[1\/R]$ for some integer $R$ such that $\\Xi^{\\mathrm{sp}}(\\mathbf {T})$ (resp. $\\Xi^{\\mathrm{sp}}(\\mathbf{B})$) lies in the group of diagonal (resp. upper triangular) matrices of $\\GL_d$.\n  To see that this is possible, find a maximal ${\\mathbb Q}$-split torus $\\mathbb{T}'$ of $\\GL_d$ containing $\\Xi^{\\mathrm{sp}}_{{\\mathbb Q}}(\\mathbf {T})$. Choose any Borel subgroup $\\mathbb{B}'$ over ${\\mathbb Q}$ containing $\\mathbb{T}$. Then there exists $g\\in \\GL_d({\\mathbb Q})$ such that the inner automorphism $\\mathrm{Int}(g):\\GL_d\\rightarrow \\GL_d$ by $\\gamma\\mapsto g \\gamma g^{-1}$ carries $(\\mathbb{B}',\\mathbb{T}')$ to $(\\mathbb{B},\\mathbb{T})$. Then $\\Xi^{\\mathrm{sp}}_{{\\mathbb Q}}$ and $\\mathrm{Int}(g)$ extend over ${\\mathbb Q}$ to over ${\\mathbb Z}[1\/R]$ for some $R\\in {\\mathbb Z}$, namely at the expense of inverting finitely many primes (basically those in the denominators of the functions defining $\\Xi^{\\mathrm{sp}}_{{\\mathbb Q}}$ and $\\mathrm{Int}(g)$).\n\n\n\n  Now suppose that $p$ is a prime not diving $R$. Let $F$ be a finite extension of ${\\mathbb Q}_p$ with integer ring $\\mathcal{O}$\n  and a uniformizer $\\varpi$. The field\n  $F$ is equipped with a unique discrete valuation $v_F$ such that $v_F(\\varpi)=1$. Let $\\lambda\\in X_*(\\mathbf {T})$.\n  We are interested in assertions which work for $F$ as the residue characteristic $p$ varies.\n  Lemma \\ref{l:app:split1} (resp. Lemma \\ref{l:app:split2}) below is used in Step 1 (resp. Step 2) of the proof of Lemma \\ref{l:dbl-coset-distance}.\n  Our convention for $T_\\delta$ and $\\Phi_\\delta$ below is the same as for $T_\\gamma$ and $\\Phi_\\gamma$ in Lemma \\ref{l:bounding1-alpha(gamma)}. The proof of the next lemma is little different from that of Lemma \\ref{l:bounding1-alpha(gamma)}.\n\n\\begin{lem}\\label{l:app:split1}\n  There exists $m_{\\Xi^{\\mathrm{sp}}}\\in {\\mathbb Z}_{>0}$ such that\nfor every $p$, $F$ and $\\lambda$ as above and for every semisimple $\\delta\\in G(\\mathcal{O})\\lambda(\\varpi)G(\\mathcal{O})$ (and for any choice of $T_\\delta$ containing $\\delta$),\n$$\\forall \\alpha\\in \\Phi_\\delta,\\quad v_F(\\alpha(\\delta))\\in [-m_{\\Xi^{\\mathrm{sp}}} \\|\\lambda\\|,m_{\\Xi^{\\mathrm{sp}}} \\|\\lambda\\|].$$\n\\end{lem}\n\n\\begin{proof}\n  There exists $c_{\\Xi^{\\mathrm{sp}}}>0$ such that $\\|\\Xi^{\\mathrm{sp}}(\\lambda)\\|_{\\GL_d}\\le c_{\\Xi^{\\mathrm{sp}}} \\|\\lambda\\|$\n  for all $\\lambda\\in X_*(\\mathbf {T})$.\n  Then every entry $a$\n  of the diagonal matrix $\\Xi^{\\mathrm{sp}}(\\lambda(\\varpi))$ has the property that\n  $v_F(a)\\in [-c_{\\Xi^{\\mathrm{sp}}} \\|\\lambda\\|, c_{\\Xi^{\\mathrm{sp}}} \\|\\lambda\\|]$. Let $\\mathbb{T}_\\delta$ be a maximal torus of $\\GL_d$ (over $\\overline{F}$) containing $\\Xi^{\\mathrm{sp}}(T_\\delta)$. The map $\\Xi^{\\mathrm{sp}}$ induces a surjection\n  $X^*(\\mathbb{T}_\\delta)\\rightarrow X^*(T_\\delta)$. We fix a lift $\\widetilde{\\alpha}\\in X^*(\\mathbb{T}_\\delta)$ of each $\\alpha\\in \\Phi_\\delta$\n  and choose $a_{\\Xi^{\\mathrm{sp}}}>0$ such that \\begin{equation}\\label{e:tilde-alpha}\n  \\|\\widetilde{\\alpha}\\|_{\\GL_d}\\le a_{\\Xi^{\\mathrm{sp}}},\\quad \\forall \\alpha\\in \\Phi_\\delta\\end{equation}\n  As $\\delta\\in G(\\mathcal{O})\\lambda(\\varpi)G(\\mathcal{O})$, we have\n  $\\Xi^{\\mathrm{sp}}(\\delta)\\in \\GL_d(\\mathcal{O})\\Xi^{\\mathrm{sp}}(\\lambda(\\varpi))\\GL_d(\\mathcal{O})$. By Lemma \\ref{l:control-eigenvalue},\n  every eigenvalue $\\mathrm{ev}$ of $\\Xi^{\\mathrm{sp}}(\\delta)$ satisfies\n  $v_F(\\mathrm{ev})\\in [-c_{\\Xi^{\\mathrm{sp}}} \\|\\lambda\\|, c_{\\Xi^{\\mathrm{sp}}} \\|\\lambda\\|]$.\n  This together with \\eqref{e:tilde-alpha} shows that\n  $$v_F(\\widetilde{\\alpha}(\\Xi^{\\mathrm{sp}}(\\delta)))\\in [-a_{\\Xi^{\\mathrm{sp}}} c_{\\Xi^{\\mathrm{sp}}} \\|\\lambda\\|,a_{\\Xi^{\\mathrm{sp}}} c_{\\Xi^{\\mathrm{sp}}} \\|\\lambda\\|].$$\n  Since $\\alpha(\\delta)=\\widetilde{\\alpha}(\\Xi^{\\mathrm{sp}}(\\delta))$, we finish by setting $m_{\\Xi^{\\mathrm{sp}}}:=a_{\\Xi^{\\mathrm{sp}}} c_{\\Xi^{\\mathrm{sp}}}$.\n\\end{proof}\n\nThe unipotent radical of $\\mathbf{B}$ is denoted $\\mathbf{N}$.\nFor $F$ as above, let $x_0$ be the hyperspecial vertex on the building of $G(F)$\ncorresponding to $G(\\mathcal{O})$.\nAs usual put $\\Phi^+:=\\Phi^+(G,\\mathbf {T})$ be the set of positive roots with respect to $(\\mathbf{B},\\mathbf {T})$.\n\n  Let us recall some facts about the Chevalley basis. For each $\\alpha\\in \\Phi^+$, let $U_\\alpha$ denote the corresponding unipotent subgroup equipped with $x_\\alpha:\\mathbb{G}_a \\simeq U_\\alpha$. Order the elements of $\\Phi^+$ as $\\alpha_1,...,\\alpha_{|\\Phi^+|}$ once and for all such that simple roots appear at the beginning.\n  The multiplication map $$m:U_{\\alpha_1}\\times \\cdots \\times U_{\\alpha_{|\\Phi^+|}} \\rightarrow \\mathbf{N},\n\\qquad (u_1,...,u_{|\\Phi^+|})\\mapsto u_1\\cdots u_{|\\Phi^+|}$$\n  is an isomorphism of schemes (but not as group schemes) over ${\\mathbb Z}$. This can be deduced from \\cite[Exp XXII, 5.5.1]{SGA3-7}, which deals with a Borel subgroup of a Chevalley group. In particular (since the ordering on $\\Phi^+$ is fixed) any $n\\in \\mathbf{N}(F)$ can be uniquely written as \\begin{equation}\\label{e:y=product}y=x_{\\alpha_1}(Y_{\\alpha_1})\\cdots x_{\\alpha_{|\\Phi^+|}}(Y_{\\alpha_{|\\Phi^+|}})\\end{equation}\n  for unique $Y_{\\alpha_i}\\in \\mathbb{G}_a(F)\\simeq F$'s. The Chevalley commutation relation (\\cite[\\S III]{Che55}) has the following form:\nfor all $1\\le i<j\\le |\\Phi^+|$ and all $Y_{\\alpha_i}\\in F$'s,\n  \\begin{equation}\\label{e:commutator}\n  x_{\\alpha_i}( Y_{\\alpha_i}) x_{\\alpha_j}( Y_{\\alpha_j})\n = x_{\\alpha_j}( Y_{\\alpha_j})x_{\\alpha_i}( Y_{\\alpha_i})\n  \\prod_{c,d\\ge 1\\atop\n  \\alpha_k=c\\alpha_i+d\\alpha_j} x_{\\alpha_k}(C_{ij} ( Y_{\\alpha_i})^c ( Y_{\\alpha_j})^d)\n\\end{equation}\nwhere $C_{ij}$ are certain integers (depending on $G$)\nwhich we need not know explicitly. It suffices to know that, in the cases of $F$ we are interested in, the constants $C_{ij}$ are units in $\\mathcal{O}$ (cf. the assumption in the paragraph preceding Proposition \\ref{p:appendix2}).\n\n\n\n\\begin{lem}\\label{l:app:split2} We can enlarge $R$, in a way depending only on $G$ and $\\Xi^{\\mathrm{sp}}$, such that\n  for every $p\\nmid R$, $F$, $\\lambda$, and $x_0$ as above, the following is true: for any $y\\in \\mathbf{N}(F)$ written as in \\eqref{e:y=product} for $Y_{\\alpha_i}\\in F$, if $y\\in G(\\mathcal{O})\\lambda(\\varpi) G(\\mathcal{O})$ then $v_F(Y_{\\alpha_i})\\ge -m_{\\Xi^{\\mathrm{sp}}}\\|\\lambda\\|$ for all $1\\le i\\le |\\Phi^+|$.\n\\end{lem}\n\n\\begin{proof}\n  Observe that  $\\Xi^{\\mathrm{sp}}(y)\\in \\GL_m(\\mathcal{O})\\Xi^{\\mathrm{sp}}(\\lambda(\\varpi)) \\GL_m(\\mathcal{O})\\cap {\\mathbb N}(F)$.\n  As in the previous proof, $\\Xi^{\\mathrm{sp}}(\\lambda(\\varpi))$ is a diagonal matrix in which each diagonal entry $a$\n  satisfies $v_F(a)\\in [-c_{\\Xi^{\\mathrm{sp}}} \\|\\lambda\\|, c_{\\Xi^{\\mathrm{sp}}} \\|\\lambda\\|]$.\n  An easy matrix computation shows that $\\Xi^{\\mathrm{sp}}(y)=(y_{ij})_{1\\le i,j\\le d}$ must be an upper triangular matrix such that $y_{ii}=1$ and \\begin{equation}\\label{e:pf-7.14}v_F(y_{ij})\\ge -c_{\\Xi^{\\mathrm{sp}}}\\|\\lambda\\|,\\quad \\forall i,~\\forall j\\ge i.\\end{equation}\n\n  Put $r:=|\\Phi^+|$ for convenience. Consider an ${\\mathbb Z}[1\/R]$-scheme morphism arising as the composition\n  $$(\\mathbb{G}_a)^{r}\\stackrel{(x_{\\alpha_1},...,x_{\\alpha_r})}{\\longrightarrow} U_{\\alpha_1}\\times \\cdots \\times U_{\\alpha_{r}}\n  \\stackrel{m}{\\rightarrow} \\mathbf{N} \\stackrel{\\Xi^{\\mathrm{sp}}}{\\rightarrow} {\\mathbb N}.  $$\n  Identifying $(\\mathbb{G}_a)^{r}$ with $\\mathrm{Spec}\\, {\\mathbb Z}[\\frac{1}{R}][Y_1,...,Y_r]$ and ${\\mathbb N}$ with $\\mathrm{Spec}\\, {\\mathbb Z}[\\frac{1}{R}][Z_{ij}]_{1\\le i\\le r-1;i+1\\le j\\le r}$, we may represent the above map by $f_{ij}\\in {\\mathbb Z}[\\frac{1}{R}][Y_1,...,Y_r]$ so that the map is given by $Z_{ij}=f_{ij}(Y_1,...,Y_r)$ on the level of points. The constant term of each $f_{ij}$ is 0 since $f_{ij}$ should carry 0 to the identity of ${\\mathbb N}$. Note that $f=(f_{ij}):(\\mathbb{G}_a)^r\\rightarrow {\\mathbb N}$ is a closed immersion of schemes (not respecting the group structure) since $\\mathbf{N}$ and ${\\mathbb N}$ are closed subgroups of $G$ and $GL_m$, respectively, and since $\\Xi^{\\mathrm{sp}}$ is a closed immersion. Hence $f$ is given by a ${\\mathbb Z}[\\frac{1}{R}]$-algebra map $${\\mathbb Z}[\\frac{1}{R}][Z_{ij}]\\rightarrow {\\mathbb Z}[\\frac{1}{R}][Z_{ij}]\/I\\stackrel{\\sim}{\\ra}{\\mathbb Z}[\\frac{1}{R}][Y_k]$$ where $1\\le i\\le r-1$, $i+1\\le j\\le r$, and $1\\le k\\le r$, for some ideal $I$ of ${\\mathbb Z}[\\frac{1}{R}][Z_{ij}]$. Let $g_{ij}\\in {\\mathbb Z}[\\frac{1}{R}][Y_k]$ be the polynomials which are the images of $Z_{ij}$ under the latter isomorphism.\n  We enlarge $R$ such that the coefficients of $f_{ij}$ and $g_{ij}$ are $p$-adic units for all primes $p\\nmid R$ and for all $1\\le i\\le r-1$ and $i+1\\le j\\le r$. Note that $f_{ij}$, $g_{ij}$, and the choice or $R$ depend only on $G$, $\\Xi^{\\mathrm{sp}}$ and the ordering of elements of $\\Phi^+$ (but is independent of the choice of $p$ and $F$ below). Let $$D:=\\max_{ij}( \\deg g_{ij}),$$\n  which makes sense as not all $g_{ij}$ are identically zero.\n\n  Consider any prime $p\\nmid R$ and any finite extension $F$ of ${\\mathbb Q}_p$ with integer ring $\\mathcal{O}$.\n  Then $f$ defines a continuous injection, by taking $F$-points,\n  $$f: F^{r} \\hookrightarrow \\prod_{1\\le i\\le r-1} \\prod_{i+1\\le j \\le r} F.$$\n  which has closed image in $p$-adic topology as it is closed in Zariski topology.\n  For $s\\in {\\mathbb Z}$, write $\\varpi^{s}{\\mathscr B}_0$ for the ball $\\prod_{1\\le i\\le r-1} \\prod_{i+1\\le j \\le r} \\varpi^s\\mathcal{O}$.\n \n  Since $g=(g_{ij})$ has $p$-adic units (or zero) as coefficients, it is clear that $g(\\varpi^{-a}{\\mathscr B}_0)\\subset (\\varpi^{-aD}\\mathcal{O})^r$ for all $a\\ge 0$. In other words, $f^{-1}(\\varpi^{-a}{\\mathscr B}_0)\\subset (\\varpi^{-aD} \\mathcal{O})^r$.\n  Applying this to the situation of the lemma, we deduce from \\eqref{e:pf-7.14} that $v_F(Y_{\\alpha_i})\\ge -c_{\\Xi^{\\mathrm{sp}}} D\\|\\lambda\\|$.\n  Thus the lemma holds true for all $p\\nmid R$ and $F$ as above with $m_{\\Xi^{\\mathrm{sp}}}:=c_{\\Xi^{\\mathrm{sp}}} D$.\n\n\n\n\n\n\n\n\\end{proof}\n\n\n\n\\section{Lemmas on conjugacy classes and level subgroups}\\label{s:conj}\n\n  This section contains several results which are useful for estimating the geometric side of Arthur's invariant trace formula in the next section.\n\n\\subsection{Notation and basic setup}\\label{sub:notation-setup} Let us introduce some global notation in addition to that at the start of \\S\\ref{sec:pp}.\n\n\\begin{itemize}\n\\item $M_0$ is a minimal $F$-rational Levi subgroup of $G$.\n\\item $A_{M_0}$ is the maximal split $F$-torus in the center of $M_0$.\n\\item ${\\rm Ram}(G):=\\{v\\in {\\mathcal V}_F^\\infty:\\, G\\mbox{~is~ramified~at~}v\\}$.\n\\item $S\\subset {\\mathcal V}_F^\\infty$ is a finite subset, often with a partition\n$S=S_0\\coprod S_1$.\n\\item $r:{}^L G \\rightarrow \\GL_d({\\mathbb C})$ is an irreducible representation.\n\\item $\\Xi:G\\rightarrow \\GL_m$ is a faithful algebraic representation defined over $F$\n(or over $\\mathcal{O}_F$ as explained below)\n\\item For any ${\\mathbb C}$-subspace ${\\mathcal{H}}'\\subset C^\\infty_c(G(F_S))$,\ndefine $${\\rm supp}\\,{\\mathcal{H}}'=\\cup\\, {\\rm supp}\\, \\phi_S$$ where the union is take over $\\phi_S\\in{\\mathcal{H}}'$.\n\\item $q_{S}:=\\prod_{v\\in S} q_v$ where $q_v$ is the cardinality of the residue field at $v$. (Convention: $q_S=1$ if $S=\\emptyset$.)\n\\end{itemize}\n\n\n  For each finite place $v\\in {\\rm Ram}(G)$ of $F$, fix a special\n  point $x_v$ on the building of $G$ once and for all, where $x_v$ is required to belong to\n  an apartment corresponding to a maximal $F_v$-split torus $A_v$ containing $A_{M_0}$.\n  The stabilizer $K_v$ of $x_v$ is a good special maximal compact subgroup of $G(F_v)$\n  (good in the sense of \\cite{BT72}).\n  Set $K_{M,v}:=K_v\\cap M(F_v)$ for each $F_v$-rational Levi subgroup $M$ of $G$ containing $A_v$. Then $K_{M,v}$\n  is a good special maximal compact subgroup of $M(F_v)$.\n\n  It is worth stressing that this article treats a reductive group $G$ without any hypothesis on $G$\n  being split (or quasi-split). To do so, we would like to carefully choose an integral model of $G$ over $\\mathcal{O}_F$\n  for convenience and also for clarifying a notion like ``level ${\\mathfrak n}$ subgroups''. We thank Brian Conrad for explaining us\n  crucial steps in the proof below (especially how to proceed by using the facts from \\cite{BLR90}).\n\n\\begin{prop}\\label{p:global-integral-model}\n  The $F$-group $G$ extends to a group scheme ${\\mathfrak G}$ over $\\mathcal{O}_F$ (thus equipped with an isomorphism\n  ${\\mathfrak G}\\times_{\\mathcal{O}_F} F\\simeq G$) such that \\begin{itemize}\n    \\item ${\\mathfrak G}\\times_{\\mathcal{O}_F} \\mathcal{O}_F[\\frac{1}{{\\rm Ram}(G)}]$ is a reductive group scheme (cf. \\cite{Conrad-reductive}),\n    \\item ${\\mathfrak G}(\\mathcal{O}_v)=K_v$ for all $v\\in {\\rm Ram}(G)$ (where $K_v$ are chosen above),\n    \\item there exists a faithful embedding of algebraic groups $\\Xi:{\\mathfrak G}\\hookrightarrow \\GL_m$ over $\\mathcal{O}_F$\n  for some $m\\ge 1$.\n  \\end{itemize}\n\\end{prop}\n\n\\begin{rem}\n  If $G$ is split then ${\\rm Ram}(G)$ is empty and the above proposition is standard in the theory of Chevalley groups.\n\\end{rem}\n\n\\begin{proof}\n\n For any finite place $v$ of $F$, we will write $\\mathcal{O}_{(v)}$ for the localization of $\\mathcal{O}_F$ at $v$ (to be distinguished from\n  the completion $\\mathcal{O}_v$).\n  As a first step there exists an injective morphism of group schemes $\\Xi_F:G\\hookrightarrow GL_{m}$ defined over $F$ for some $m\\ge 1$\n  (\\cite[Prop A.2.3]{CGP10}. The scheme-theoretic closure ${\\mathfrak G}'$ of $G$ in $GL_{m'}$ is a smooth affine scheme over $\\mathrm{Spec}\\, \\mathcal{O}_F[1\/S]$ for a finite set $S$ of primes of $\\mathcal{O}_F$ by arguing as in the first paragraph of \\cite[\\S2]{Conrad-reductive}. We may assume that $S\\supset {\\rm Ram}(G)$. By \\cite[Prop 3.1.9.(1)]{Conrad-reductive}, by enlarging $S$ if necessary, we can arrange that ${\\mathfrak G}'$ is reductive.\n  For $v\\in {\\rm Ram}(G)$ we have fixed special points $x_v$, which give rise to the Bruhat-Tits group schemes $\\widehat{{\\mathfrak G}}(v)$\n  over $\\mathcal{O}_v$.\n  Similarly for $v\\in S\\backslash {\\rm Ram}(G)$, let us choose hyperspecial points $x_v$ so that the corresponding group schemes\n  $\\widehat{{\\mathfrak G}}(v)$ over $\\mathcal{O}_v$ are reductive.\n\n  According to\n   \\cite[Prop D.4, p.147]{BLR90} the obvious functor from the category of affine $\\mathcal{O}_{(v)}$-schemes\n   to that of triples $(X,\\widehat{\\mathfrak{X}}(v),f)$ where $X$ is an affine $F$-scheme, $\\widehat{\\mathfrak{X}}(v)$ is an affine $\\mathcal{O}_v$-scheme and\n   $f:X\\times_F F_v\\simeq \\widehat{\\mathfrak{X}}(v)\\times_{\\mathcal{O}_v} F_v$ is an equivalence. (The notion of morphisms is obvious in each category.) Thanks to its functorial nature, the same functor defines\n   an equivalence when restricted to group objects in each category. For $v\\in {\\rm Ram}(G)$,\n   apply this functor to the Bruhat-Tits group scheme $\\widehat{{\\mathfrak G}}(v)$ over $\\mathcal{O}_v$ equipped with $G\\times_F F_v\\simeq \\widehat{{\\mathfrak G}}(v)\\times_{\\mathcal{O}_v} F_v$    to obtain a group scheme ${\\mathfrak G}(v)$ over $\\mathcal{O}_{(v)}$.\n\n   An argument analogous to that on page 14 of \\cite{BLR90} shows that the obvious functor between the following categories is\n   an equivalence: from\n   the category of finite-type $\\mathcal{O}_F$-schemes to that of triples $(X,\\{\\mathfrak{X}(v)\\}_{v\\in S}, \\{f_v\\}_{v\\in S})$\n   where $X$ is a finite-type $\\mathcal{O}_F[1\/S]$-scheme, $\\mathfrak{X}(v)$ is a finite-type $\\mathcal{O}_{(v)}$-scheme and\n   $f_v:X\\times_{\\mathcal{O}_F[1\/S]} F\\simeq \\mathfrak{X}(v)\\times_{\\mathcal{O}_{(v)}} F$ is an isomorphism. Again this induces an equivalence when restricted\n   to group objects in each category. In particular, there exists a group scheme ${\\mathfrak G}$ over $\\mathcal{O}_F$ with isomorphisms\n   ${\\mathfrak G}\\times_{\\mathcal{O}_F} \\mathcal{O}_F[1\/S]\\simeq {\\mathfrak G}'$ and ${\\mathfrak G}\\times_{\\mathcal{O}_F} \\mathcal{O}_{(v)}\\simeq {\\mathfrak G}(v)$ for $v\\in S$\n   which are compatible with the isomorphisms between ${\\mathfrak G}'$ and ${\\mathfrak G}(v)$ over $F$.\n   By construction ${\\mathfrak G}$ satisfies the first two properties of the proposition.\n\n   We will be done if $\\Xi_F:G\\hookrightarrow GL_m$ over $F$ extends to an embedding of group schemes over $\\mathcal{O}_F$. It is evident from the construction\n   of ${\\mathfrak G}'$ that $\\Xi_F$ extends to $\\Xi':G\\hookrightarrow GL_m$ over $\\mathcal{O}_F[1\/S]$. For each $v\\in S$,\n   $\\Xi_F$ extends to $\\Xi(v):{\\mathfrak G}(v)\\hookrightarrow GL_m$ over $\\mathcal{O}_{v}$ thanks to \\cite[Prop 1.7.6]{BT84}, which can be\n   defined over $\\mathcal{O}_{(v)}$ using the first of the above equivalences. Then the second equivalence allows us to\n   glue $\\Xi'$ and $\\{\\Xi(v)\\}_{v\\in S}$ to produce an $\\mathcal{O}_F$-embedding $\\Xi:G\\hookrightarrow GL_m$.\n\n\n\n\\end{proof}\n\n  For each finite $v\\notin {\\rm Ram}(G)$, ${\\mathfrak G}$ defines a reductive group scheme over $\\mathcal{O}_v$, so\n  $K_v:={\\mathfrak G}(\\mathcal{O}_v)$ is a hyperspecial subgroup of $G(F_v)$.\n  Fix a maximal $F_v$-split torus $A_v$ of $G$ which contains $A_{M_0}$ such that the hyperspecial point for $K_v$ belongs to the apartment of $A_v$. For each Levi subgroup $M$ of $G$ whose center is contained in $A_v$, define a hyperspecial subgroup $K_{M,v}:=K_v\\cap M(F_v)$ of $M(F_v)$.\n  At such a $v\\notin {\\rm Ram}(G)$\n  define ${\\mathcal{H}}^{\\mathrm{ur}}(G(F_v))$\n  (resp. ${\\mathcal{H}}^{\\mathrm{ur}}(M(F_v))$). The constant term (\\S\\ref{sub:orb-int-const-term}) of a function in $C^\\infty_c(G(F_v))$\n  (resp. $C^\\infty_c(M(F_v))$) will be taken relative to $K_v$ (resp. $K_{M,v}$).\n  When $P=MN$ is a Levi decomposition, we have Haar measures on $K_v$, $M(F_v)$ and $N(F_v)$\n  such that\n  the product measure equals $\\mu_v^{\\mathrm{can}}$ on $G(F_v)$ (cf. \\S\\ref{sub:orb-int-const-term})\n  and the Haar measure on $M(F_v)$ is the canonical measure of \\S\\ref{sub:can-measure}.\n  In particular when $G$ is unramified at $v$,\n  \\begin{equation}\\label{e:meas-K-cap-N}\\operatorname{vol}(K_v\\cap N(F_v))=1\\end{equation} with respect to the measure on $N(F_v)$ .\n\n  Let ${\\mathfrak n}$ be an ideal of $\\mathcal{O}_F$ and $v$ a finite place of $F$.\nLet $v({\\mathfrak n})\\in {\\mathbb Z}_{\\ge 0}$ be the integer determined by ${\\mathfrak n} \\mathcal{O}_v=\\varpi_v^{v({\\mathfrak n})} \\mathcal{O}_v$.\nDefine $K_v(\\varpi_v^s)$  to be the Moy-Prasad subgroup $G(F_v)_{x_v,s}$ of $G(F_v)$\nby using Yu's minimal congruent filtration as in \\cite{Yu-fil} (which is slightly\ndifferent from the original definition of Moy and Prasad). Yu has shown that $G(F_v)_{x_v,s}= \\ker({\\mathfrak G}(\\mathcal{O}_v)\\rightarrow {\\mathfrak G}(\\mathcal{O}_v\/\\varpi_v^s))$\nin \\cite[Cor 8.8]{Yu-fil}. Set $K^{S,\\infty}({\\mathfrak n}):=\\prod_{v\\notin S\\cup S_\\infty}\\ker({\\mathfrak G}(\\mathcal{O}_v)\\rightarrow {\\mathfrak G}(\\mathcal{O}_v\/{\\mathfrak n}))=\\prod_{v\\notin S\\cup S_\\infty} K_v(\\varpi_v^{v({\\mathfrak n})})$,\nto be considered the level ${\\mathfrak n}$-subgroup of $G({\\mathbb A}^{S,\\infty})$.\n\n  Fix a maximal torus $T_0$ of $G$ over $\\overline{F}$ and\n  an ${\\mathbb R}$-basis ${\\mathcal{B}}_0$ of $X_*(T_0)_{\\mathbb R}$, which induces a function\n  $\\|\\cdot\\|_{{\\mathcal{B}}_0,G}:X_*(T_0)_{{\\mathbb R}}\\rightarrow {\\mathbb R}_{\\ge 0}$ as in \\S\\ref{sub:L-mor-unr-Hecke}.\n  For any other maximal torus $T$, there is an inner automorphism\n  of $G$ inducing $T_0\\simeq T$, so $X_*(T)_{\\mathbb R}$ has an ${\\mathbb R}$-basis ${\\mathcal{B}}$ induced from ${\\mathcal{B}}_0$,\n  well defined up to $\\Omega(G,T)$. Therefore $\\|\\cdot\\|_{{\\mathcal{B}},G}:X_*(T)_{\\mathbb R}\\rightarrow {\\mathbb R}_{\\ge 0}$\n  is defined without ambiguity. As it depends only on the initial choice of ${\\mathcal{B}}_0$ (and $T_0$),\n  let us write\n  $\\|\\cdot\\|$ for $\\|\\cdot\\|_{{\\mathcal{B}},G}$ when there is no danger of confusion.\n\n  Let $v$ be a finite place of $G$, and $T_v$ a maximal torus of $G\\times_F \\overline{F}_v$\n  (which may or may not be defined over $F_v$). Then $\\|\\cdot\\|:X_*(T_v)_{\\mathbb R}\\rightarrow {\\mathbb R}_{\\ge 0}$\n  is defined without ambiguity via $T_v\\simeq T_0\\times _F F_v$ by a similar consideration\n  as above. Now assume that $G$ is unramified at $v$. For any maximal split torus $A\\subset G$\n  and a maximal torus $T$ containing $A$ over $F_v$, the function $\\|\\cdot\\|_{{\\mathcal{B}}_0}$ is well defined\n  on $X_*(T)_{\\mathbb R}$ (resp. $X_*(A)_{\\mathbb R}$) and invariant under $\\Omega$ (resp. $\\Omega_F$).\n  Hence for every $v$ where $G$ is unramified, the Satake isomorphism allows us to define\n  ${\\mathcal{H}}^{\\mathrm{ur}}(G(F_v))^{\\le \\kappa}$ as well as\n  ${\\mathcal{H}}^{\\mathrm{ur}}(M(F_v))^{\\le \\kappa}$ for every Levi subgroup $M$ of $G$ over $F_v$.\nWhen $G$ is unramified at $S$, ${\\mathcal{H}}^{\\mathrm{ur}}(G(F_S))^{\\le \\kappa}$ and\n  ${\\mathcal{H}}^{\\mathrm{ur}}(M(F_S))^{\\le \\kappa}$ are similarly defined.\n\n  For the group $\\GL_m$ with any $m\\ge 1$, we use\n  the diagonal torus and the standard basis to define $\\|\\cdot\\|_{\\GL_m}$ on the cocharacter groups\n  of maximal tori of $\\GL_m$ (cf. \\S\\ref{sub:case-of-GL_d}).\n  For $\\Xi:G\\hookrightarrow \\GL_m$ introduced above, define\n\\begin{equation}\\label{e:def-M(Xi)} B_{\\Xi}:=\\max_{e\\in {\\mathcal{B}}_0} \\|\\Xi(e)\\|_{\\GL_m}.\\end{equation}\n\n\\subsection{$z$-extensions}\n\n  A surjective morphism $\\alpha:H\\rightarrow G$ of connected reductive groups over $F$ is\n  said to be a \\textbf{$z$-extension} if\n\\begin{itemize}\n\\item $H^{^{\\mathrm{der}}}$ is simply connected,\n\\item $\\ker\\alpha\\subset Z(H)$,\n\\item $\\ker\\alpha$ is isomorphic to a finite product $\\prod \\Res_{F_i\/F}\\GL_1$\n for finite extensions $F_i$ of $F$.\n\\end{itemize}\n  Such an extension is conveniently represented by an exact sequence of $F$-groups\n  $1\\rightarrow Z \\rightarrow H\\rightarrow G\\rightarrow 1$. By the third condition and Hilbert 90, $\\alpha:H(F)\\rightarrow G(F)$\n  is surjective.\n\n\\begin{lem}\\label{l:z-ext}\n  For any $G$, a $z$-extension $\\alpha:H\\rightarrow G$ exists.\n  Moreover, if $G$ is unramified outside a finite set $S$, where $S_\\infty\\subset S\\subset {\\mathcal V}_F$,\n  then $H$ can be chosen to be unramified outside $S$.\n\\end{lem}\n\n\\begin{proof}\n  It is shown in \\cite[Prop 3.1]{MS82} that a $z$-extension exists and\n  that if $G$ splits over a finite Galois extension $E$\n  of $F$ then $H$ can be chosen to split over $E$. By the assumption on $G$, it is possible to find such an $E$ which is unramified outside $S$.\n  Since the preimage of a Borel subgroup of $G$ in $H$ is a Borel subgroup of $H$,\n  we see that $H$ is quasi-split over $F$.\n\n\n\\end{proof}\n\n\n\n\n\n\\subsection{Rational conjugacy classes intersecting a small open compact subgroup}\\label{sub:conj-intersect}\n\n\n Throughout this subsection $S=S_0\\coprod S_1$ is a finite subset of ${\\mathcal V}^\\infty_F$ and\n it is assumed that $S_0\\supset {\\rm Ram}(G)$. Fix compact subgroups $U_{S_0}$ and $U_{\\infty}$ of $G(F_{S_0})$ and $G(F\\otimes_{\\mathbb Q}{\\mathbb R})$, respectively. Let ${\\mathfrak n}$ be an ideal of $\\mathcal{O}_F$ as before with absolute norm ${\\mathbb N}({\\mathfrak n})\\in {\\mathbb Z}_{\\ge 1}$.\n\n\\begin{lem}\\label{l:forcing-unipotent}\n   Let $U_{S_1}:={\\rm supp}\\, {\\mathcal{H}}^{\\mathrm{ur}}(G(F_{S_1}))^{\\le \\kappa}$.\n    There exists $c_\\Xi>0$ independent of $S$, $\\kappa$ and ${\\mathfrak n}$ (but depending on $G$, $\\Xi$, $U_{S_0}$ and $U_{\\infty}$)\n  such that for all ${\\mathfrak n}$ satisfying $${\\mathbb N}({\\mathfrak n})\\ge c_\\Xi q_{S_1}^{B_{\\Xi}m\\kappa},$$ the following holds:\n    if $\\gamma\\in G(F)$ and $x^{-1}\\gamma x\\in U^{S,\\infty}({\\mathfrak n}) U_{S_0} U_{S_1} U_\\infty$\n    for some $x\\in G({\\mathbb A}_F)$ then $\\gamma$ is unipotent.\n\\end{lem}\n\n\\begin{proof}\n  Let $\\gamma'=x^{-1}\\gamma x$. We keep using the embedding $\\Xi:\\mathfrak{G}\\hookrightarrow \\GL_m$ over $\\mathcal{O}_F$ of Proposition \\ref{p:global-integral-model}.\n  (For the lemma, an embedding away from the primes in $S_0$ or dividing ${\\mathfrak n}$ is enough.) At each finite place $v\\notin S_0$ and\n$v\\nmid {\\mathfrak n}$, Lemma \\ref{l:conj-image-in-diag} allows us to find $\\Xi'_v:\\mathfrak{G}\\hookrightarrow \\GL_m$ over $\\mathcal{O}_v$ which is $\\GL_m(\\mathcal{O}_v)$-conjugate to $\\Xi\\times_{\\mathcal{O}_F} F_v$ such that $\\Xi'_v$ sends $A_v$ into the diagonal torus of $\\GL_m$.\n\n  Write $\\det(\\Xi(\\gamma)-(1-X))=X^m+a_{m-1}(\\gamma)X^{m-1}+\\cdots+ a_0(\\gamma)$, where\n  $a_i(\\gamma)\\in F$ for $0\\le i\\le m-1$.\n  Our goal is to show that $a_i(\\gamma)=0$ for all $i$.\n  To this end, assuming $a_i(\\gamma)\\neq 0$ for some fixed $i$, we will estimate $|a_i(\\gamma)|_v$ at each place $v$ and\n  draw a contradiction.\n\n  First consider $v\\in S_1$.\n  We claim that\n  $$v(a_i(\\gamma))\\ge-B_{\\Xi}m\\kappa$$ for every $\\gamma$ that is conjugate to an element of ${\\rm supp}\\, {\\mathcal{H}}^{\\mathrm{ur}}(G(F_{v}))^{\\le \\kappa}$.\n   To prove the claim we examine the eigenvalues of $\\Xi'_v(\\gamma')$, which is conjugate to $\\gamma$.\n  We know $\\gamma'$ belongs to ${\\rm supp}\\, {\\mathcal{H}}^{\\mathrm{ur}}(G(F_v))^{\\le \\kappa}$, so\n  $\\Xi'_v(\\gamma')\\in \\GL_m(\\mathcal{O}_v)\\Xi'_v(\\mu(\\varpi_v))\\GL_m(\\mathcal{O}_v)$ for some $\\mu\\in X_*(A_v)$ with $\\|\\mu\\|\\le \\kappa$.\n  Then $\\|\\Xi'_v(\\mu)\\|_{\\GL_m}\\le B_{\\Xi}\\kappa$. (A priori this is true for $B_{\\Xi'_v}$ defined as in \\eqref{e:def-M(Xi)},\nbut $B_{\\Xi'_v}=B_{\\Xi}$ as $\\Xi'_v$ and $\\Xi$ are conjugate.)\n  Let $k_1,k_2\\in \\GL_m(\\mathcal{O}_v)$ be such that $\\Xi'_v(\\gamma')=k_1\\Xi'_v(\\mu(\\varpi_v))k_2$.\n  Lemma \\ref{l:control-eigenvalue} shows that\n   every eigenvalue $\\lambda$ of $\\Xi'_v(\\mu(\\varpi_v)) k_2k_1$ (equivalently of\n   $\\Xi'_v(\\gamma')$) satisfies\n  $v(\\lambda)\\ge -B_{\\Xi}\\kappa$.\n   If $\\lambda\\neq 1$, we must have $v(1-\\lambda)\\ge -B_{\\Xi}\\kappa$.\n  This shows that $v(a_{i}(\\gamma))\\ge -B_{\\Xi}i \\kappa$ for any $i$ such that $a_i(\\gamma)\\neq 0$.\n  Hence the claim is true.\n\n\n  At infinity, by the compactness of $U_\\infty$, there exists $c_\\Xi>0$ such that\n  $$|a_i(\\gamma)|_\\infty<c_\\Xi$$ whenever a conjugate of $\\gamma\\in G_\\infty$ belongs to $U_\\infty$.\n\n  Now suppose that $v$ is a finite place such that $v\\notin S_1$ and $v\\nmid {\\mathfrak n}$. (This includes $v\\in S_0$.)\n  Then a conjugate of $\\Xi(\\gamma)$ lies in an open compact subgroup of $\\GL_m(F_v)$.\n  Therefore the eigenvalues of $\\Xi(\\gamma)$ are in $\\mathcal{O}_v$ and $$|a_i(\\gamma)|_v\\le 1.$$\n\n  Finally at $v|{\\mathfrak n}$, we have $\\Xi(x^{-1}\\gamma x)-1\\in \\ker(\\GL_m(\\mathcal{O}_v)\\rightarrow \\GL_m(\\mathcal{O}_v\/\\varpi_v^{v({\\mathfrak n})}))$.\n  Therefore $$|a_i(\\gamma)|_v=|a_i(x^{-1}\\gamma x)|_v\\le (|{\\mathfrak n}|_v)^{m-i}.$$\n\n  Now assume that ${\\mathbb N}({\\mathfrak n})\\ge  c_\\Xi q_{S_1}^{-B_{\\Xi}m\\kappa}$.\n  We assert that $a_i(\\gamma)=0$ for all $i$. Indeed, if $a_i(\\gamma)\\neq 0$ for some $i$ then\n  the above inequalities imply that\n  $$1=\\prod_v |a_i(\\gamma)|_v < \\left(\\prod_{v\\in S_1} q_v^{-B_\\Xi m \\kappa}\\right)c_\\Xi \\prod_{v|{\\mathfrak n}} |{\\mathfrak n}|_v^{m-i}\n  \\le q_{S_1}^{-B_\\Xi m \\kappa} c_\\Xi {\\mathbb N}({\\mathfrak n})^{-1}\\le 1$$\n  which is clearly a contradiction.\n  The proof of lemma is finished.\n\n\\end{proof}\n\n\n\\subsection{Bounding the number of rational conjugacy classes}\n\n  We begin with a basic lemma, which is a quantitative version of the fact that $F^r$ is discrete\n  in $ {\\mathbb A}_F^r$.\n\n\\begin{lem}\\label{l:F-in-A_F}\n  Suppose that $\\{\\delta_v \\in {\\mathbb R}_{>0}\\}_{v\\in {\\mathcal V}_F}$ satisfies the following: $\\delta_v=1$ for all but finitely many $v$\n  and $\\prod_v \\delta_v<2^{-|S_\\infty|}$. Let $\\alpha=(\\alpha_1,...,\\alpha_r)\\in {\\mathbb A}_F^r$.\n  Consider the following compact neighborhood of $\\alpha$\n   $${\\mathcal{B}}(\\alpha,\\delta):=\\{(x_1,...,x_r)\\in {\\mathbb A}_F^r\\,:\\, |x_{i,v}-\\alpha_{i,v}|_v\\le \\delta_v,~\\forall\n   v,~\\forall 1\\le i\\le r\\}.$$\n  Then ${\\mathcal{B}}(\\alpha,\\delta)\\cap F^r$ has at most one element.\n\\end{lem}\n\n\\begin{proof}\n  Suppose $\\beta=(\\beta_i)_{i=1}^r,\\gamma=(\\gamma_i)_{i=1}^r\\in {\\mathcal{B}}(\\alpha,\\delta)\\cap F^r$. By triangular inequalities,\n  $$|\\beta_{i,v}-\\gamma_{i,v}|_v\\le \\left\\{\\begin{array}{cc}\n    \\delta_v, & v\\nmid \\infty,\\\\\n    2\\delta_v, & v|\\infty\n  \\end{array}\\right.$$\n  for each $i$.\n  Hence $\\prod_v |\\beta_{i,v}-\\gamma_{i,v}|_v<1$. Since $\\beta_{i},\\gamma_{i}\\in F$,\n  the product formula forces $\\beta_{i}=\\gamma_{i}$. Therefore $\\beta=\\gamma$.\n\\end{proof}\n\n\n\n\n  The next lemma measures the difference between $G(F)$-conjugacy and $G({\\mathbb A}_F)$-conjugacy.\n\n\\begin{lem}\\label{l:F-conj-vs-A_F-conj}\n  Let $X_G$ (resp. ${\\mathscr X}_G$) be the set of semisimple $G(F)$-(resp. $G({\\mathbb A}_F)$-)conjugacy\n  classes in $G(F)$. For any $[\\gamma]\\in {\\mathscr X}_G$, there exist at most $(w_Gs_G)^{r_G+1}$ elements\n  in $X_G$ mapping to $[\\gamma]$ under the natural surjection $X_G\\rightarrow {\\mathscr X}_G$.\n\\end{lem}\n\n\n\\begin{proof}\n  Let $[\\gamma]\\in {\\mathscr X}_G$ be an element defined by a semisimple $\\gamma\\in G(F)$.\n  Denote by $X_\\gamma$ the preimage of\n  $[\\gamma]$ in $X_G$. There is a natural bijection\n  $$X_\\gamma \\quad\\leftrightarrow\\quad \\ker(\\ker^1(F,I_\\gamma))\\rightarrow \\ker^1(F,G)).$$\n  Since $|\\ker^1(F,I_\\gamma))|=|\\ker^1(F,Z(\\widehat{I}_\\gamma))|$\n  by \\cite[\\S4.2]{Kot84a}, we have $|X_\\gamma|\\le |\\ker^1(F,Z(\\widehat{I}_\\gamma))|$.\n\n  Lemma \\ref{l:torus-splitting} tells us that for every semisimple $\\gamma$,\n  the group $I_\\gamma$ becomes split over a finite extension\n  $E\/F$ such that $[E:F]\\le w_Gs_G$.\n  In particular ${\\rm Gal}(\\overline{F}\/E)$ acts trivially on\n  $Z(\\widehat{I}_\\gamma)$.\n  The group $\\ker^1(E,Z(\\widehat{I}_\\gamma))$ consists of continuous homomorphisms\n  ${\\rm Gal}(\\overline{F}\/E)\\rightarrow Z(\\widehat{I}_\\gamma)$ which are trivial on all local Galois groups.\n  Hence $\\ker^1(E,Z(\\widehat{I}_\\gamma))$ is trivial. This and\n  the inflation-restriction sequence show that $\\ker^1(F,Z(\\widehat{I}_\\gamma))$\n  is the subset of locally trivial elements in $H^1(\\Gamma_{E\/F},Z(\\widehat{I}_\\gamma))$,\n  where we have written $\\Gamma_{E\/F}$ for ${\\rm Gal}(E\/F)$. In particular,\n  $$|X_\\gamma|\\le |H^1(\\Gamma_{E\/F},Z(\\widehat{I}_\\gamma))|.$$\n  Let $d:=|{\\rm Gal}(E\/F)|$ and denote by $[d]$ the\n  $d$-torsion subgroup. The long exact sequence arising from $0\\rightarrow Z(\\widehat{I}_\\gamma)[d]\n  \\rightarrow Z(\\widehat{I}_\\gamma) \\stackrel{d}{\\rightarrow} d(Z(\\widehat{I}_\\gamma))\\rightarrow 0$ gives rise to\n  an exact sequence\n  $$\nH^1(\\Gamma_{E\/F},Z(\\widehat{I}_\\gamma)[d]) \\rightarrow H^1(\\Gamma_{E\/F},Z(\\widehat{I}_\\gamma))\n= H^1(\\Gamma_{E\/F},Z(\\widehat{I}_\\gamma))[d]\\rightarrow 0.$$\n  Let $\\mbox{\\boldmath{$\\mu$}}_{d}$ denote the order $d$ cyclic subgroup of ${\\mathbb C}^\\times$.\n  Then $Z(\\widehat{I}_\\gamma)[d]\\hookrightarrow  \\widehat{T}[d]\\simeq (\\mbox{\\boldmath{$\\mu$}}_{d})^{\\dim T}$.\n  Hence\n  $$|X_\\gamma|\\le |H^1(\\Gamma_{E\/F},Z(\\widehat{I}_\\gamma)[d] )|\n  \\le |\\Gamma_{E\/F}|\\cdot |Z(\\widehat{I}_\\gamma)[d]|\\le d\\cdot (d)^{\\dim T}\\le\n  (w_Gs_G)^{\\dim T+1}.$$\n\\end{proof}\n\n\n  For the proposition below, we fix a finite subset $S_0\\in {\\mathcal V}_F^\\infty$.\n  Also fix open compact subsets\n   $U_{S_0}\\subset G(F_{S_0})$\n  and $U_\\infty \\subset G(F_\\infty)$.\n  As usual we will write $S$ for $S_0\\coprod S_1$.\n\n\\begin{prop}\\label{p:bound-number-of-conj}\n\n  Let $\\kappa\\in{\\mathbb Z}_{\\ge 0}$.\n  Let $S_1\\subset{\\mathcal V}_F^\\infty\\backslash S_0$ be a finite subset such that $G$ is unramified\n  at all $v\\in S_1$.\n  Set $U_{S_1}:={\\rm supp}\\, {\\mathcal{H}}^{\\mathrm{ur}}(G(F_{S_1}))^{\\le \\kappa}$,\n  $U^{S,\\infty}:=\\prod_{v\\notin S\\cup S_\\infty} K_v$ and\n  ${\\mathscr C}:=U_{S_0}U_{S_1} U^{S,\\infty}U_\\infty$.\n  Define ${\\mathscr Y}_G$ to be the set of semisimple $G({\\mathbb A}_F)$-conjugacy classes of $\\gamma\\in G(F)$\n  which meet ${\\mathscr C}$. Then there exist constants $A_3,B_3>0$ such that for all $S_1$ and $\\kappa$ as above,\n  $$|{\\mathscr Y}_G|=O(q_{S_1}^{A_3+B_3\\kappa})$$\n (In other words, the implicit constant for $O(\\cdot)$ is independent of\n $S_1$ and $\\kappa$.)\n\\end{prop}\n\n\\begin{rem}\n  By combining the Proposition with Lemma~\\ref{l:forcing-unipotent} we can deduce the following. Under the same assumption but with ${\\mathscr C}:= U^{S,\\infty}({\\mathfrak n}) U_{S_0} U_{S_1} U_\\infty$ we have\n  $$|{\\mathscr Y}_G|=1+O(q_{S_1}^{A+B\\kappa}{\\mathbb N}({\\mathfrak n})^{-C}).$$\n  for some constants $A,B,C>0$.\n\\end{rem}\n\n\n\\begin{proof}\n  Our argument will be a refinement of the proof of \\cite[Prop 8.2]{Kot86}.\n  \\medskip\n\n  \\noindent{STEP I. When $G^{^{\\mathrm{der}}}$ is simply connected.}\n\n   Choose a smooth reductive integral model $\\mathfrak{G}$ over $\\mathcal{O}_F[\\frac{1}{S_0}]$ for $G$ and an embedding of algebraic groups\n  $\\Xi_0:\\mathfrak{G}\\rightarrow \\GL_{m}$ defined over $\\mathcal{O}_F[\\frac{1}{S_0}]$ as in Proposition \\ref{p:global-integral-model}.\n  Consider \\begin{equation} \\label{e:char-poly-map} G({\\mathbb A}_F)\\stackrel{\\Xi_0}{\\rightarrow} \\GL_m({\\mathbb A}_F)\\rightarrow {\\mathbb A}_F^m\\end{equation}\n  where the latter map assigns the coefficients of the characteristic polynomial,\n  and call the composite map $\\Xi'$.\n  Set $U':=\\Xi'(U)$. Then $|U'\\cap F^m|<\\infty$\n  since it is discrete and compact. We would like to estimate the cardinality.\n\n  Fix $\\{\\delta_v\\}$ which satisfies the assumption of\n  Lemma \\ref{l:F-in-A_F} and the condition that $\\delta_v=1$ for all finite $v$.\n  (So $\\{\\delta_v\\}$ depends only on $F$.)\n  Since $\\Xi_0$ is defined over $\\mathcal{O}_F[\\frac{1}{S_0}]$, clearly\n  $\\Xi_0(U^{S,\\infty})\\subset \\GL_m(\\widehat{\\mathcal{O}}_F^{S,\\infty})$. Thus\n  $$\\Xi'(U^{S,\\infty})\\subset (\\widehat{\\mathcal{O}}_F^{S,\\infty})^m=\\prod_{v\\notin {S\\cup S_\\infty}}\n  {\\mathcal{B}}_v(0,1).$$\n  Set $J^{S,\\infty}:=\\{0\\}\\subset ({\\mathbb A}_F^{S,\\infty})^m$.\n  Similarly as above, $\\Xi'(U_{S_1})\\subset (\\mathcal{O}_{F,S_1})^m$.\n  By the compactness of $U_{S_0}$ and $U_\\infty$, there exist\n  finite subsets $J_{S_0}\\subset F_{S_0}$ and $J_{\\infty}\\subset F_\\infty$ such that\n  $$\\Xi'(U_{S_0})\\subset \\bigcup_{\\beta_{S_0}\\in J_{S_0}} \\left( \\prod_{v\\in S_0} {\\mathcal{B}}_v(\\beta_v,1)\\right),\n  \\quad \\Xi'(U_\\infty)\\subset \\bigcup_{\\beta_\\infty\\in J_\\infty} \\left( \\prod_{v|\\infty} {\\mathcal{B}}_v(\\beta_v,\\delta_v)\\right).$$\n  Now we treat the places contained in $S_1$. Choose a maximal torus $T_{\\overline{F}}$ of $G$ over $\\overline{F}$\n  once and for all. Since the image of the composite map\n  $T_{\\overline{F}}\\hookrightarrow G_{\\overline{F}} \\stackrel{\\Xi_0}{\\hookrightarrow} (\\GL_m)_{\\overline{F}}$\n  is contained in a maximal torus of $\\GL_m$, we can choose $g=(g_{ij})_{i,j=1}^m\\in \\GL_m(\\overline{F})$ such that\n  $g\\Xi_0(T_{\\overline{F}})g^{-1}$ sits in the diagonal maximal torus $\\mathbb{T}$ of $\\GL_d$.\n  Fix the choice of $T$ and $g$ once and for all (independently of $S_1$ and $\\kappa$).\n  Set $v_{\\min}(g):=\\min_{i,j} v(g_{ij})$ and $v_{\\max}(g):=\\max_{i,j} v(g_{ij})$.\n  There exists $B_6>0$ such that for any $\\mu\\in X_*(T)$ with $\\|\\mu\\|\\le \\kappa$, the element\n  $g\\Xi_0(\\mu)g^{-1}\\in X_*(\\mathbb{T})$ satisfies $\\|g\\Xi_0(\\mu)g^{-1}\\|\\le B_6\\kappa$.\n  Let $\\gamma_{S_1}=(\\gamma_v)_{v\\in S_1}\\in U_{S_1}$. Each\n  $\\gamma_v$ has the form $\\gamma_v=k_1\\mu(\\varpi_{v})k_2$ for some $\\|\\mu\\|\\le \\kappa$\n  and $k_1,k_2\\in G(\\mathcal{O}_v)$. Since $\\Xi_0(G(\\mathcal{O}_v))\\subset \\GL_m(\\mathcal{O}_v)$,\n  we see that $\\Xi_0(\\gamma_v)$ is conjugate to $\\Xi_0(\\mu(\\varpi_{v}))k'$ in $\\GL_m(F_v)$\n  for some $k'\\in \\GL_m(\\mathcal{O}_v)$.\n  Applying Lemma \\ref{l:control-eigenvalue} to\n  $(g\\Xi_0(\\mu(\\varpi_{v}))g^{-1})( gk'g^{-1})$ with $u=gk'g^{-1}$\n  and noting that $v_{\\min}(u)\\ge v_{\\min}(g)+v_{\\min}(g^{-1})$\n  we conclude that each eigenvalue $\\lambda$ of $\\Xi_0(\\gamma_v)$ satisfies\n  $$v(\\lambda)\\le -B_6\\kappa+v_{\\min}(g)+v_{\\min}(g^{-1}).$$\n  Therefore the coefficients of its characteristic polynomial\n  lie in $\\varpi_v^{-m(B_6\\kappa+A_4)}\\mathcal{O}_v$,\n  where we have set $A_4:=-(v_{\\min}(g)+v_{\\min}(g^{-1}))\\ge 0$. To put things together, we see that\n  $$\\Xi'(U_{S_1})\\subset \\prod_{v\\in S_1} (\\varpi_v^{-m(B_6\\kappa+A_4)}\\mathcal{O}_v)^m.$$\n  (A fortiori $\\Xi'(U_{S_1})\\subset \\prod_{v\\in S_1} (\\varpi_v^{-i(B_6\\kappa+A_4)}\\mathcal{O}_v)_{i=1}^m $\n  holds as well.) The right hand side is equal to the union of $\\prod_{v\\in S_1}\n  {\\mathcal{B}}_v(\\beta_v,1)$, as $\\{\\beta_v\\}_{v\\in S_1}$ runs over $J_{S_1}=\\prod_{v\\in S_1} J_v$,\n  where $J_v$ is a set of representatives for $(\\varpi_v^{-m(B_6\\kappa+A_4)}\\mathcal{O}_v\/\\mathcal{O}_v)^m$.\n  Notice that $|J_{S_1}|=q_{S_1}^{m^2(B_6 \\kappa+A_4)}$.\n  Finally, we see that\n  $$U'=\\Xi'(U)\\subset \\bigcup_{\\beta\\in J} {\\mathcal{B}}(\\beta,\\delta)$$\n  where $J=J_{S_0}\\times J_{S_1}\\times J^{S,\\infty}\\times J_\\infty$. Lemma \\ref{l:F-in-A_F}\n  implies that\n  $$U'\\cap F^m \\le |J| = |J_{S_0}|\\cdot |J_{S_1}|\\cdot |J_\\infty|= O(q_{S_1}^{m^2(B_6 \\kappa+A_4) }),$$\n  since $|J_{S_0}|\\cdot |J_\\infty|$ is a constant independent of $\\kappa$ and $S_1$.\n\n  For each $\\beta\\in U'\\cap F^m$, we claim that there are at most $m!$ semisimple $G(\\overline{F})$-conjugacy classes\n  in $G(\\overline{F})$ which map to $\\beta$ via $G(\\overline{F})\\rightarrow \\GL_m(\\overline{F})\\rightarrow \\overline{F}^m$,\n  which is the analogue of \\eqref{e:char-poly-map}. Let us verify the claim.\n  Let $T$ and $\\mathbb{T}$ be maximal tori in $G$ and $\\GL_m$ over $\\overline{F}$,\n  respectively, such that $\\Xi'(T)\\subset \\mathbb{T}$. Then the set of semisimple conjugacy classes in\n  $G(\\overline{F})$ (resp. $\\GL_m(\\overline{F})$) is in a natural bijection with $T(\\overline{F})\/\\Omega$ (resp. $\\mathbb{T}(\\overline{F})\/\\Omega_{\\GL_m}$).\n  The map $\\Xi'|_T: T\\rightarrow \\mathbb{T}$ induces a map $T(\\overline{F})\/\\Omega\\rightarrow \\mathbb{T}(\\overline{F})\/\\Omega_{\\GL_m}$. Each fiber of the latter map\n  has cardinality at most $m!$, hence the claim follows.\n\n  Fix $\\beta\\in U'\\cap F^m$.\n  We would like to bound the number of $G({\\mathbb A}_F)$-conjugacy classes of $\\gamma\\in G(F)$ which meet $U$ and\n  are $G(\\overline{F})$-conjugate to $\\beta$.\n  Fix one such $\\gamma$, which we assume to exist. (Otherwise our final bound will only improve.)\n  Let $\\Phi_\\gamma$ denote the set of roots over $\\overline{F}$ for any choice of maximal torus of $\\gamma$ in $G$.\n  Define $V'(\\gamma)$ to be the set of places $v$ of $F$ such that $v\\notin S\\cup S_\\infty$ and\n $\\alpha(\\gamma)\\neq 1$ and $|1-\\alpha(\\gamma)|_v<1$ for at least one $\\alpha\\in \\Phi_\\gamma$.\n Put $V(\\gamma):=V'(\\gamma)\\cup S\\cup S_\\infty$.\nClearly $|V(\\gamma)|<\\infty$. Moreover we claim that $|V(\\gamma)|=O(1)$ (bounded independently of $\\gamma$).\nSet $$C_{S_0}:=\\sup_{\\gamma\\in U_{S_0} U_\\infty}\\left(\\prod_{\\alpha\\in \\Phi_\\gamma}\n|1-\\alpha(\\gamma)|_{S_0}|1-\\alpha(\\gamma)|_{S_\\infty}\\right),$$ which is finite since $U_{S_0} U_\\infty$ is compact.\nThen\n$$1= \\prod_{v} \\prod_{\\alpha\\in \\Phi_\\gamma} |1-\\alpha(\\gamma)|_v\n= \\left( \\prod_{\\alpha\\in \\Phi_\\gamma} |1-\\alpha(\\gamma)|_{V(\\gamma)}\\right)\n\\le C_{S_0}\\cdot  \\left(\\prod_{\\alpha\\in \\Phi_\\gamma}q_{V'(\\gamma)}^{-1} \\right)\\le C_{S_0} 2^{-|V'(\\gamma)|}.$$\nThus $|V'(\\gamma)|=O(1)$ and also $|V(\\gamma)|=O(1)$.\n\n  We are ready to bound the number of $G({\\mathbb A}_F)$-conjugacy classes in $ G(F)$ which meet ${\\mathscr C}$ and\n  are $G(\\overline{F})$-conjugate to $\\alpha$. For any such conjugacy class of $\\gamma'\\in G(F)$,\n  the first paragraph of \\cite[p.391]{Kot86} shows that\n  $\\gamma'$ is conjugate to $\\gamma$ in $G(F_v)$ whenever $v\\notin V(\\gamma)$. Hence\n  the number of $G({\\mathbb A}_F)$-conjugacy classes of such $\\gamma'$ is at most $u_G^{|V(\\gamma)|}$,\n  where $u_G$ is the bound of Lemma \\ref{l:bounding-conj-in-st-conj}.\n\n  Putting all this together, we conclude that\n  $|{\\mathscr Y}_G|=O(q_{S_1}^{m^2(B_7 \\kappa+A_5)})$ as $S_1$ and $\\kappa$ vary.\n  The lemma is proved in this case.\n\n \\medskip\n\n  \\noindent{STEP II. General case.}\n\n  Now we drop the assumption that $G^{^{\\mathrm{der}}}$ is simply connected.\n  By Lemma \\ref{l:z-ext}, choose a $z$-extension\n  $$1\\rightarrow Z\\rightarrow H \\stackrel{\\alpha}{\\rightarrow} G\\rightarrow 1.$$\n  Our plan is to argue as on page 391 of \\cite{Kot86} with a specific choice of\n  ${\\mathscr C}_H$ and ${\\mathscr C}_Z$ below (denoted $C_H$ and $C_Z$ by Kottwitz). In order to explain this choice,\n  we need some preparation.\n  If $v\\notin S\\cup S_\\infty$, choose $K_{H,v}$ to be a hyperspecial subgroup\n  of $H(F_v)$ such that $\\alpha(K_{H,v})=K_v$. (Such a $K_{H,v}$ exists by\n  the argument of \\cite[p.386]{Kot86}.) We can find\n  compact sets $U_{H,S_0}\\subset H(F_{S_0})$ and $U_{H,\\infty}$ of $H(F_\\infty)$\n  such that $\\alpha(U_{H,S_0})=U_{S_0}$\n  and $\\alpha(U_{H,\\infty})=U_\\infty$.\n  Moreover we claim that there exists a constant $\\beta>0$ independent of $\\kappa$ and $S_1$\n  with the following property: for any $\\kappa\\in {\\mathbb Z}_{\\ge 0}$, we can choose\n  an open compact subset $U_{H,S_1}\\subset  {\\rm supp}\\, {\\mathcal{H}}^{\\mathrm{ur}}(H)^{\\le \\beta\\kappa}$\n  such that $\\alpha(U_{H,S_1})=U_{S_1}$.\n  The claim is proved in Lemma \\ref{l:claim} below.\n\nNow choose $U_{Z,S_1}$ to be the kernel of $\\alpha: U_{H,S_1}\n  \\rightarrow U_{S_1}$, which is compact and open in $Z(F_{S_1})$. Then choose a compact set $U_{Z}^{S_1}$ such that\n  $U_{Z,S_1}U_{Z}^{S_1}Z(F)=Z({\\mathbb A})^1$. (This is possible since $Z(F)\\backslash Z({\\mathbb A})^1$\n  is compact.\\footnote{Choose $U^{S_1}_Z$ to be any open compact subgroup. Then\n  $U_{Z,S_1}U_{Z}^{S_1}Z(F)$ has a finite index in $Z({\\mathbb A})^1$ by compactness.\n  Then enlarge $U^{S_1}_Z$ without breaking compactness such that the equality holds.})\n  Set $$U_H:= \\left(\\prod_{v\\notin S\\cup S_\\infty} K_{H,v}\\right)\n  U_{H,S_0} U_{H,S_1} U_{H,\\infty},\\qquad\n  U_Z:=U_{Z,S_1}U_{Z}^{S_1}$$\n  and set ${\\mathscr C}_H:=U_H\\cap H({\\mathbb A}_F)^1$, ${\\mathscr C}_Z:=U_Z\\cap Z({\\mathbb A}_F)^1$.\n  Let ${\\mathscr Y}_H$ be defined as in the statement of Proposition \\ref{p:bound-number-of-conj} (with $H$ and ${\\mathscr C}_H$ replacing\n  $G$ and ${\\mathscr C}$).\n  Then page 391 of \\cite{Kot86} shows that the natural map ${\\mathscr Y}_H\\rightarrow {\\mathscr Y}$ is a surjection,\n  in particular $|{\\mathscr Y}|\\le |{\\mathscr Y}_H|$.\n  Since $H^{^{\\mathrm{der}}}$ is simply connected, the earlier proof implies that\n  $|{\\mathscr Y}_H|=O(q_{S_1}^{B_7\\beta\\kappa+A_5})$ for some $B_7,A_5>0$.\n  (To be precise, apply the earlier proof after enlarging $U_{H,S_1}$ to\n  ${\\rm supp}\\, {\\mathcal{H}}^{\\mathrm{ur}}(H)^{\\le \\beta\\kappa}$ in the definition of $U_H$. Such a replacement\n  only increases $|{\\mathscr Y}_H|$, so the bound on $|{\\mathscr Y}_H|$ remains valid.)\n  The proposition follows.\n\\end{proof}\n\n  We have postponed the proof of the following claim, which we justify now. Simple as\n  the lemma may seem, we apologize for not having found a simple proof.\n\n\\begin{lem}\\label{l:claim}\n  As we have claimed above, there exists a constant $\\beta>0$ independent of $\\kappa$ and $S_1$\n  with the desired property. (Refer to Step II of the above proof for detail.)\n\\end{lem}\n\n\\begin{proof}\n  As the claim is concerned with places in $S_1$, which (may vary but) are contained\n  in the set of places where $G$ is unramified (thus quasi-split),\n  we may assume that $H$ and $G$ are quasi-split over $F$\n  by replacing $H$ and $G$ with their quasi-split inner forms.\n\n  Choose a Borel subgroup $B_H$ of $H$, whose image $B=\\alpha(B_H)$ is a Borel subgroup of $G$.\n  The maximal torus $T_H\\subset B_H$ maps to a maximal torus $T\\subset B$ and there is a short exact sequence\n  $$1\\rightarrow Z\\rightarrow T_H \\stackrel{\\alpha}{\\rightarrow} T\\rightarrow 1.$$\n  The action of ${\\rm Gal}(\\overline{F}\/F)$ on $X_*(T_H)$ factors through a finite quotient.\n  Let $\\Sigma$ be the smallest quotient so that $\\Sigma$ acts faithfully on $X_*(T_H)$.\n  If $v\\notin S_0$ then $G$ is unramified at $v$, so the geometric Frobenius at $v$\n  defines a well-defined conjugacy class, say ${\\mathscr C}_v$, in $\\Sigma$.\n  Let $A_{H,v}$ (resp. $A_v$) be the maximal split torus in $T_H$ (resp. $T$) over $F_v$.\n  Then $A_{H,v}\\hookrightarrow T_H$ and $A_v\\hookrightarrow T$ induce\n  $X_*(A_{H,v})\\simeq X_*(T_H)^{{\\mathscr C}_v}$ and $X_*(A_{v})\\simeq X_*(T)^{{\\mathscr C}_v}$.\n  We claim that $X_*(T_H)\\rightarrow X_*(T)$ induces a surjective map\n  $X_*(A_{H,v})\\rightarrow X_*(A_{v})$.\n$$\\xymatrix{ X_*(T_H) \\ar[d] & X_*(T_H)^{{\\mathscr C}_v} \\ar[d] \\ar[l] & X_*(A_{H,v}) \\ar[l]_-{\\sim} \\simeq  T_H(F_v)\/T_H(\\mathcal{O}_v)\n\\ar@{->>}[d] \\\\\n X_*(T)  & X_*(T)^{{\\mathscr C}_v}  \\ar[l] & X_*(A_{v}) \\ar[l]_-{\\sim} \\simeq T(F_v)\/T(\\mathcal{O}_v)}$$\n Indeed, we have an isomorphism $ X_*(A_{H,v})\\simeq  T_H(F_v)\/T_H(\\mathcal{O}_v)$ via $\\mu\\mapsto \\mu(\\varpi_v)$\n and similarly $ X_*(A_{v})\\simeq T(F_v)\/T(\\mathcal{O}_v)$. Further, $\\alpha:T_H(F_v)\\rightarrow T(F_v)$ is surjective\n since $H^1({\\rm Gal}(\\overline{F}_v\/F_v),Z(\\overline{F}_v))$ is trivial (as $Z$ is an induced torus).\n\n  Denote by $[\\Sigma]$ the finite set of all conjugacy classes in $\\Sigma$. For\n  ${\\mathscr C}\\in [\\Sigma]$, choose ${\\mathbb Z}$-bases ${\\mathcal{B}}_{H,{\\mathscr C}}$ and ${\\mathcal{B}}_{{\\mathscr C}}$ for\n  $X_*(T_H)^{{\\mathscr C}}$ and $X_*(T)^{{\\mathscr C}}$ respectively.\n  (Note that ${\\mathbb Z}$-bases ${\\mathcal{B}}_H$  for $X_*(T)$ and ${\\mathcal{B}}$ for $X_*(T_H)$ are fixed once and for all.)\n   An argument as in the proof of Lemma\n  \\ref{l:norm-and-bases} shows that there exist constants $c({\\mathcal{B}}_{{\\mathscr C}}),c({\\mathcal{B}}_{H,{\\mathscr C}})>0$\n  such that for all $x\\in X_*(T_H)^{{\\mathscr C}}_{{\\mathbb R}}$ and $y\\in X_*(T)^{{\\mathscr C}}_{{\\mathbb R}}$,\n  \\begin{equation}\\label{e:for-l:claim}|x|_{{\\mathcal{B}}_{H,{\\mathscr C}}}\\ge c({\\mathcal{B}}_{H,{\\mathscr C}})\\cdot \\|x\\|_{{\\mathcal{B}}_H},\\quad\n  |y|_{{\\mathcal{B}}_{{\\mathscr C}}}\\le c({\\mathcal{B}}_{{\\mathscr C}})\\cdot \\|y\\|_{{\\mathcal{B}}}.\\end{equation}\n  Set $m_{\\mathscr C}:= \\max_y (\\min_x |x|_{{\\mathcal{B}}_{H,{\\mathscr C}}})$, where\n  $y\\in X_*(T)^{{\\mathscr C}}$ varies subject to the condition $|y|_{{\\mathcal{B}}_{{\\mathscr C}}}\\le 1$\n  and $x\\in  X_*(T_H)^{{\\mathscr C}}$ runs over the preimage of $y$. (It was shown above that\n  the preimage is nonempty.)\n  Then by construction, for every $y \\in X_*(T)^{{\\mathscr C}}$, there exists $x$ in the preimage of $y$\n  such that $|x| _{{\\mathcal{B}}_{H,{\\mathscr C}}} \\le m_{{\\mathscr C}} |y|_{{\\mathcal{B}}_{{\\mathscr C}}}$.\n\n  Recall that $U_{S_1}=\\prod_{v\\in S_1} U_v$ where $U_v=\\cup_{\\mu} K_v \\mu(\\varpi_v) K_v$,\n  the union being taken over $\\mu\\in X_*(T)^{{\\mathscr C}_v}$ such that $\\|\\mu\\|_{{\\mathcal{B}}}\\le \\kappa$.\n  We have seen that there exists $\\mu_H\\in X_*(T_H)^{{\\mathscr C}_v}$ mapping to $\\mu$\n  and $|\\mu_H|_{{\\mathcal{B}}_{H,{\\mathscr C}_v}}\\le m_{{\\mathscr C}_v} |\\mu|_{{\\mathcal{B}}_{{\\mathscr C}_v}}$. By \\eqref{e:for-l:claim},\n  $$\\|\\mu_H\\|_{{\\mathcal{B}}_{H}}\\le m_{{\\mathscr C}_v} c({\\mathcal{B}}_{H,{\\mathscr C}_v})^{-1}c({\\mathcal{B}}_{{\\mathscr C}_v}) \\|\\mu\\|_{{\\mathcal{B}}}.$$\n  Take $\\beta:=\\max_{{\\mathscr C}\\in [\\Sigma]} (m_{{\\mathscr C}} c({\\mathcal{B}}_{H,{\\mathscr C}})^{-1}c({\\mathcal{B}}_{{\\mathscr C}}))$.\n  Clearly $\\beta$ is independent of $S_1$ and $\\kappa$.\n  Notice that $\\|\\mu_H\\|_{{\\mathcal{B}}_{H}}\\le \\beta \\|\\mu\\|_{{\\mathcal{B}}}\\le \\beta \\kappa$.\n\n  For each $\\mu\\in X_*(T)^{{\\mathscr C}_v}$ such that $\\|\\mu\\|_{{\\mathcal{B}}}\\le \\kappa$,\n  we can choose a preimage $\\mu_H$ of $\\mu$ such that $\\|\\mu_H\\|_{{\\mathcal{B}}_{H}}\\le\\beta \\kappa$.\n  Take $U_{H,v}$ to be the union of $K_{H,v} \\mu_H(\\varpi_v) K_{H,v}$ for those\n  $\\mu_H$'s. By construction $\\alpha(U_{H,v})=U_v$. Hence $U_{H,S_1}:=\\prod_{v\\in S_1}U_{H,v}$\n  is the desired open compact subset in the claim of Lemma \\ref{l:claim}.\n\n\n\\end{proof}\n\n\n\\begin{cor}\\label{c:bound-number-of-conj}\n  In the setting of Proposition \\ref{p:bound-number-of-conj},\n  let $Y_G$ be the set of all semisimple $G(F)$-conjugacy (rather than $G({\\mathbb A}_F)$-conjugacy)\n  classes whose $G({\\mathbb A}_F)$-conjugacy classes intersect ${\\mathscr C}$. Then there exist\n   constants $A_6,B_8>0$ such that $|Y_G|=O(q_{S_1}^{B_8\\kappa+A_6})$ as $S_1$ and $\\kappa$ vary.\n\\end{cor}\n\n\\begin{proof}\n  Immediate from Lemma \\ref{l:F-conj-vs-A_F-conj} and Proposition \\ref{p:bound-number-of-conj}.\n\\end{proof}\n\n  The following lemma was used in Step I of the proof of Proposition \\ref{p:bound-number-of-conj}\n  and will be applied again to obtain Corollary \\ref{c:bounding-pi0-general} below.\n\n\n\\begin{lem}\\label{l:bounding-conj-in-st-conj}\n  Assume that $G^{^{\\mathrm{der}}}$ is simply connected.\n  For each $v\\in {\\mathcal V}_F$ and each semisimple $\\gamma\\in G(F)$, let $n_{v,\\gamma}$ be the number of $G(F_v)$-conjugacy classes\n  in the stable conjugacy class of $\\gamma$ in $G(F_v)$.\n  Then there exists a uniform bound $u_G\\ge1$ such that $n_{v,\\gamma}\\le u_G$ for all $v$ and $\\gamma$.\n\\end{lem}\n\n\\begin{proof}\n  Put $\\Gamma(v):={\\rm Gal}(\\overline{F}_v\/F_v)$.\n  It is a standard fact that $n_{v,\\gamma}$ is the cardinality of\n  $\\ker(H^1(F_v,I_\\gamma)\\rightarrow H^1(F_v,G))$. By \\cite{Kot86},\n  $H^1(F_v,I_\\gamma)$ is isomorphic to the dual of $\\pi_0(Z(\\widehat{I}_\\gamma)^{\\Gamma(v)})$. Hence\n  $n_{v,\\gamma}\\le |\\pi_0(Z(\\widehat{I}_\\gamma)^{\\Gamma(v)})|.$\n  It suffices to show that a uniform bound for $|\\pi_0(Z(\\widehat{I}_\\gamma)^{\\Gamma(v)})|$ exists.\n\n  By Lemma \\ref{l:torus-splitting},\n  there exists a finite Galois extension $E\/F$ with $[E:F]\\le w_Gs_G$ such that\n  $I_{\\gamma}$ splits over $E$. Then ${\\rm Gal}(\\overline{F}\/F)$ acts on $Z(\\widehat{I}_\\gamma)$\n  through ${\\rm Gal}(E\/F)$. In particular $\\Gamma(v)$ acts on $Z(\\widehat{I}_\\gamma)$\n  through a group of order $\\le w_Gs_G$. Denote the latter group by $\\Gamma(v)'$.\n\n  By the assumption, all Levi subgroups of $G$ have simply connected derived subgroups.\n  As $I_\\gamma$ becomes isomorphic to a Levi subgroup of $G$ over $\\overline{F}$,\n  $I_\\gamma^{^{\\mathrm{der}}}$ is also simply connected. Hence $Z(\\widehat{I}_\\gamma)$ is connected\n  (\\cite[(1.8.3)]{Kot84a}), namely a complex torus. Moreover\n  $\\dim Z(\\widehat{I}_\\gamma)\\le r_G$.\n\n   Now consider the set of pairs $${\\mathscr T}=\\{(\\Delta,\\widehat{T}):\n   |\\Delta|\\le w_Gs_G,~\\dim \\widehat{T}\\le r_G\\}$$\n   consisting of a ${\\mathbb C}$-torus $\\widehat{T}$ with an action by a finite group $\\Delta$. Two pairs $(\\Delta,\\widehat{T})$\n   and $(\\Delta',\\widehat{T}')$ are equivalent if there are isomorphisms $\\Delta\\simeq \\Delta'$\n   and $\\widehat{T}\\simeq \\widehat{T}'$ such that the group actions are compatible.\n   Note that $$(\\Gamma(v)',Z(\\widehat{I}_\\gamma))\\in {\\mathscr T}$$ and that\n   ${\\mathscr T}$ depends only on $G$ and $F$.\n   Clearly $|\\pi_0(\\widehat{T}^\\Delta)|$ depends only on the equivalence class\n   of $(\\Delta,\\widehat{T})\\in {\\mathscr T}$.\n   Hence the proof will be complete if\n   ${\\mathscr T}$ consists of finitely many equivalence classes.\n\n   Clearly there are finitely many isomorphism classes for $\\Delta$ appearing in ${\\mathscr T}$.\n   So we may fix $\\Delta$ and prove the finiteness of isomorphism classes of\n   ${\\mathbb C}$-tori with $\\Delta$-action. By dualizing, it is enough to show that\n   there are finitely many isomorphism classes of ${\\mathbb Z}[\\Delta]$-modules\n   whose underlying ${\\mathbb Z}$-modules are free of rank $\\le r_G$.\n   This is a result of \\cite[\\S79]{CR62}.\n\n\n\n\\end{proof}\n\n\n\\begin{cor}\\label{c:bounding-pi0-general}\n  There exists a constant $c>0$ (depending only on $G$) such that\n  for every semisimple $\\gamma\\in G(F)$,\n  $|\\pi_0(Z(\\widehat{I}_\\gamma)^{\\Gamma})|<c$.\n  (We do not assume that $G^{^{\\mathrm{der}}}$ is simply connected.)\n\\end{cor}\n\n\\begin{proof}\n  Suppose that $G^{^{\\mathrm{der}}}$ is simply connected. The proof of Lemma \\ref{l:bounding-conj-in-st-conj}\n  shows that $({\\rm Gal}(E\/F),Z(\\widehat{I}_\\gamma))\\in {\\mathscr T}$ in the notation there,\n  thus there exists $c>0$ such that $|\\pi_0(Z(\\widehat{I}_\\gamma)^{\\Gamma})|<c$ for all semisimple $\\gamma$.\n\n  In general, let $1\\rightarrow Z\\rightarrow H\\rightarrow G\\rightarrow 1$ be a $z$-extension over $F$ so that\n  $Z$ is a product of induced tori and $H^{^{\\mathrm{der}}}$ is simply connected.\n  Since $H(F)\\twoheadrightarrow G(F)$, we may choose a semisimple $\\gamma_H$ mapping\n  to $\\gamma$. Let $I_{\\gamma_H}$ denote the connected centralizer of $\\gamma_H$ in $H$.\n  By the previous argument there exists $c_H>0$ such that\n  $|\\pi_0(Z(\\widehat{I}_{\\gamma_H})^{\\Gamma})|<c_H$ for any semisimple $\\gamma_H$.\n  The obvious short exact sequence $1\\rightarrow Z\\rightarrow I_{\\gamma_H}\\rightarrow I_\\gamma\\rightarrow 1$ gives rise\n  (\\S\\ref{sub:L-groups})\n  to a short exact sequence $1\\rightarrow Z(\\widehat{I}_\\gamma)\\rightarrow Z(\\widehat{I}_{\\gamma_H})\\rightarrow\\widehat{Z}\\rightarrow 1$, hence\n  by \\cite[Cor 2.3]{Kot84a},\n  \\begin{equation}\\label{e:bounding-pi0-general}\n   0\\rightarrow {\\rm coker\\,}(X_*(Z(\\widehat{I}_{\\gamma_H}))^{\\Gamma}\\rightarrow X_*(\\widehat{Z})^{\\Gamma})\n  \\rightarrow \\pi_0(Z(\\widehat{I}_\\gamma)^{\\Gamma}) \\rightarrow \\pi_0(Z(\\widehat{I}_{\\gamma_H})^{\\Gamma}) \\rightarrow\n  \\pi_0(\\widehat{Z}^{\\Gamma})=0.\\end{equation}\n  On the other hand, the inclusions $Z\\rightarrow I_{\\gamma_H}\\rightarrow H$ induces a $\\Gamma$-equivariant\n  map $Z(\\widehat{H})\\rightarrow Z(\\widehat{I}_{\\gamma_H})\\rightarrow \\widehat{Z}$. The map\n  $Z(\\widehat{H})\\rightarrow Z(\\widehat{I}_{\\gamma_H})$ is constructed by \\cite[4.2]{Kot86}, whereas\n  $Z(\\widehat{I}_{\\gamma_H})\\rightarrow \\widehat{Z}$ and $Z(\\widehat{H})\\rightarrow \\widehat{Z}$ are\n  given by \\S\\ref{sub:L-groups}. (The distinction comes from the fact that typically\n  $I_{\\gamma_H}\\rightarrow H$ is not normal.) The three maps are compatible in the obvious sense.\n  By the functoriality of $X_*(\\cdot)^\\Gamma$, there is a natural surjection\n  $${\\rm coker\\,}(X_*(Z(\\widehat{H}))^{\\Gamma}\\rightarrow X_*(\\widehat{Z})^{\\Gamma})\n  \\twoheadrightarrow {\\rm coker\\,}(X_*(Z(\\widehat{I}_{\\gamma_H}))^{\\Gamma}\\rightarrow X_*(\\widehat{Z})^{\\Gamma}).$$\n  The left hand side is finite because it embeds into the finite group\n  $\\pi_0(Z(\\widehat{G})^{\\Gamma})$, again by \\cite[Cor 2.3]{Kot84a}.\n  Going back to \\eqref{e:bounding-pi0-general}, we deduce\n  $$|\\pi_0(Z(\\widehat{I}_\\gamma)^{\\Gamma})|\\le |\\pi_0(Z(\\widehat{I}_{\\gamma_H})^{\\Gamma})|\\cdot |{\\rm coker\\,}(X_*(Z(\\widehat{H}))^{\\Gamma}\\rightarrow X_*(\\widehat{Z})^{\\Gamma})|< c_H \\cdot |\\pi_0(Z(\\widehat{G})^{\\Gamma})|.$$\n  The proof is complete as the far right hand side is independent of $\\gamma$.\n\n\\end{proof}\n\n\n  For a cuspidal group and conjugacy classes which are elliptic at infinity,\n  a more precise bound can be obtained by a simpler argument, which would be worth\n  recording here.\n\n\n\\begin{lem}\\label{l:bounding-pi0}\n  Let $G$ be a cuspidal $F$-group. For any $\\gamma\\in G(F)$ such that\n  $\\gamma\\in G(F_\\infty)$ is elliptic,\n  $$|\\pi_0(Z(\\widehat{I}_\\gamma)^{\\Gamma})|\\le 2^{{\\rm rk}(G\/A_G)}.$$\n\\end{lem}\n\n\\begin{proof}\n  Via restriction of scalars, we may assume that $F={\\mathbb Q}$ without losing generality.\n  Let us prove the lemma when $A_G$ is trivial. By assumption there exists an\n  ${\\mathbb R}$-anisotropic torus $T$ in $G({\\mathbb R})$ containing $\\gamma$. Thus\n  $T\\simeq U(1)^{{\\rm rk}(G)}$ and $T\\hookrightarrow I_\\gamma$ over ${\\mathbb R}$.\n  The former tells us that $\\widehat{T}^{\\Gamma(\\infty)}\\simeq \\{\\pm 1\\}^{{\\rm rk}(G)}$ and\n  the latter gives rise to $ Z(\\widehat{I}_\\gamma)^{\\Gamma(\\infty)}\\hookrightarrow \\widehat{T}^{\\Gamma(\\infty)}$\n  (\\cite[\\S4]{Kot86}). Hence the assertion follows from\n  $$Z(\\widehat{I}_\\gamma)^{\\Gamma}\\hookrightarrow Z(\\widehat{I}_\\gamma)^{\\Gamma(\\infty)} \\hookrightarrow \\widehat{T}^{\\Gamma(\\infty)}\\simeq \\{\\pm 1\\}^{{\\rm rk}(G)}.$$\n\n  In general when $A_G$ is not trivial, consider the exact sequence of ${\\mathbb Q}$-groups\n  $1\\rightarrow A_G \\rightarrow I_\\gamma \\rightarrow I_\\gamma\/A_G\\rightarrow 1$, whose dual is the $\\Gamma$-equivariant\n  exact sequence of ${\\mathbb C}$-groups\n  $$1\\rightarrow Z(\\widehat{I_\\gamma\/A_G})\\rightarrow Z(\\widehat{I}_\\gamma) \\rightarrow \\widehat{A}_G \\rightarrow 1.$$\n  Thanks to \\cite[Cor 2.3]{Kot84a}, we obtain the following exact sequence:\n  $$X^*(\\widehat{A}_G)^\\Gamma \\rightarrow \\pi_0(Z(\\widehat{I_\\gamma\/A_G})^\\Gamma) \\rightarrow \\pi_0(Z(\\widehat{I}_\\gamma)^{\\Gamma})\n  \\rightarrow \\pi_0(\\widehat{A}_G)^\\Gamma = 1.$$\n  Hence $|\\pi_0(Z(\\widehat{I}_\\gamma)^{\\Gamma})|\\le\n  |\\pi_0(Z(\\widehat{I_\\gamma\/A_G})^\\Gamma)|$, and the latter is at most\n  $ 2^{{\\rm rk}(G\/A_G)}$ by the preceding argument.\n\\end{proof}\n\n\n\n\n\n\\section{Automorphic Plancherel density theorem with error bounds}\\label{s:aut-Plan-theorem}\n\n  The local components of automorphic representations at a fixed finite set of\n primes tend to be equidistributed according to\n the Plancherel measure on the unitary dual, namely the error tends to zero\n in a family of automorphic representations (cf. Corollary \\ref{c:Plan-density} below).\n  The main result of this section (Theorems \\ref{t:level-varies}, \\ref{t:weight-varies})\n is a bound\non this error in terms of the primes in the fixed set as well as the varying parameter\n(level or weight) in the family. A crucial assumption for us is that the group $G$\nis cuspidal (Definition \\ref{d:cuspidal}), which allows the use of a simpler version of the trace formula.\nFor the proof we interpret the problem\nas bounding certain expressions on the geometric side of the trace formula and apply various technical results from previous\nsections.\nOne main application is a proof of the Sato-Tate conjecture for families formulated in\n\\S\\ref{sub:ST-families} under suitable conditions on the parameters involved.\nIn turn the result will be applied to the question on low-lying zeros in later sections.\n\n\n\n\\subsection{Sauvageot's density theorem on unitary\ndual}\\label{sub:density-thm}\n\n  We reproduce a summary of Sauvageot's result (\\cite{Sau97}) from\n  \\cite[\\S2.3]{Shi-Plan} as it can be used to effectively prescribe local conditions\n  in our problem.\nThe reader may refer to either source for more detail.\n\n  Let $G$ be a connected\n  reductive group over a number field $F$. Use $v$ to denote a finite place of $F$.\n  When $M$ is a Levi subgroup of\n  $G$ over $F_v$, write\n$\\Psi_u(M(F_v))$ (resp. $\\Psi(M(F_v))$) for the real (resp. complex) torus whose points parametrize\nunitary (complex-valued) characters of $M(F_v)$ trivial on any compact subgroup of $M(F_v)$. The normalized parabolic induction\nof an admissible representation $\\sigma$ of $M(F_v)$ is denoted ${\\rm n\\textrm{-}ind}^G_M(\\sigma)$.\n\n  Denote by ${\\mathscr B}_c(G(F_v)^{\\wedge})$ the space of bounded\n  $\\widehat{\\mu}^{\\mathrm{pl}}_v$-measurable functions $\\widehat{f}_v$ on $G(F_v)^{\\wedge}$\nwhose support has compact image in the Bernstein center, which is\nthe set of ${\\mathbb C}$-points of an (infinite) product of varieties.\n  A measure on $G(F_v)^{\\wedge}$\n  will be thought of as a linear functional on the space ${\\mathscr F}(G(F_v)^{\\wedge})$\n  consisting of $\\widehat{f}_v\\in {\\mathscr B}_c(G(F_v)^{\\wedge})$ such that\nfor every $F_v$-rational Levi subgroup $M$ of $G$ and every discrete\nseries $\\sigma$ of $M(F_v)$, $$\\Psi_u(M(F_v))\\rightarrow\n{\\mathbb C}\\quad\\mbox{ given by}\\quad \\chi\\mapsto\n\\widehat{f}_v({\\rm n\\textrm{-}ind}^G_M(\\sigma\\otimes \\chi))$$ is a function whose\npoints of discontinuity are contained in a measure zero set. (Here ${\\rm n\\textrm{-}ind}$ denotes the normalized parabolic induction.)\nNow for any finite set $S$ of finite places of $F$,\none can easily extend the above definition to\n${\\mathscr F}(G(F_S)^{\\wedge})$ so that $\\widehat{f}_S(\\pi_S)\\in {\\mathbb C}$ makes sense\nfor $\\widehat{f}_S\\in {\\mathscr F}(G(F_S)^{\\wedge})$ and $\\pi_S\\in G(F_S)^{\\wedge}$.\nWe have a map\n$$C^\\infty_c(G(F_S))\\rightarrow {\\mathscr F}(G(F_S)^{\\wedge}),\n\\qquad \\phi_S\\mapsto \\widehat{\\phi}_S:\\pi_S\\mapsto {\\rm tr}\\, \\pi_S(\\phi_S),$$\nas follows from Proposition~\\ref{l:bounding-function-by-pos-function} below.\nHarish-Chandra's Plancherel theorem states that\n\\[\n\\label{e:Plan-thm}\\widehat{\\mu}^{\\mathrm{pl}}_S(\\widehat{\\phi}_S)=\\phi_S(1).\n\\]\nOur notational convention is that $\\widehat{\\phi}_S$ often signifies an element in the image of the above map\nwhereas $\\widehat{f}_S$ stands for a general element of ${\\mathscr F}(G(F_S)^{\\wedge})$.\n  Sauvageot's theorem allows us to approximate any $\\widehat{f}_S\\in{\\mathscr F}(G(F_S)^{\\wedge})$\nwith elements of $C^\\infty_c(G(F_S))$.\n\n\\begin{prop}(\\cite[Thm 7.3]{Sau97})\\label{p:density}\n  Let $\\widehat{f}_S\\in {\\mathscr F}(G(F_S)^{\\wedge})$. For any $\\epsilon>0$,\n  there exist $\\phi_S,\\psi_S\\in C^\\infty_c(G(F_S))$ such that\n$$ \\widehat{\\mu}^{\\mathrm{pl}}_S(\\widehat{\\psi}_S)\\le \\epsilon  \\quad\\mbox{and}\\quad\n\\forall\n\\pi_S\\in G(F_S)^{\\wedge},~\n|\\widehat{f}_S(\\pi_S) - \\widehat{\\phi}_S(\\pi_S)|\\le \\widehat{\\psi}_S(\\pi_S) .$$\nConversely, any $\\widehat{f}_S\\in\n  {\\mathscr B}_c(\\widehat{G(F_S)})$ with the above property belongs to ${\\mathscr F}(G(F_S)^{\\wedge})$.\n\\end{prop}\n\n\\begin{rem}\n  It is crucial that $\\widehat{f}_S\\in {\\mathscr F}(G(F_S)^{\\wedge})$ has the set of discontinuity in a measure zero set.\n  Otherwise we could take $\\widehat{f}_S$ to be the characteristic function on the set of points of $G(F_S)^{\\wedge}$\n  which arise as the $S$-components of some $\\pi\\in{\\mathcal{AR}}_{{\\rm disc},\\chi}(G)$ with nonzero Lie algebra cohomology. Note that the latter function typically lies outside ${\\mathscr F}(G(F_S)^{\\wedge})$. The conclusions of Theorems \\ref{t:Sato-Tate-level}, \\ref{t:Sato-Tate-weight} and Corollary \\ref{c:Plan-density}  are false in general if such an $\\widehat{f}_S$ is placed at $S_0$. Namely in that case $\\widehat{\\mu}_{\\mathcal{F}_k,S_1}(\\widehat{\\phi}_{S_1})$ is often far from zero but $\\widehat{\\mu}^{\\mathrm{pl}}_S(\\widehat{\\phi}_S)$ always vanishes.\n\\end{rem}\n\n\n\n\n  From here until the end of this subsection let us suppose that $G$ is unramified at $S$. It will be convenient to introduce $\\mathcal{F}(G(F_S)^{\\wedge,\\mathrm{ur}})$ and its subspace $\\mathcal{F}(G(F_S)^{\\wedge,\\mathrm{ur},{\\rm temp}})$\n  in order to state the Sato-Tate theorem in \\S\\ref{sub:ST-theorem}.\n  The former (resp. the latter) consists of $\\widehat{f}_S\\in \\mathcal{F}(G(F_S)^{\\wedge})$ such that\n  the support of $\\widehat{f}_S$ is contained in $G(F_S)^{\\wedge,\\mathrm{ur}}$ (resp. $G(F_S)^{\\wedge,\\mathrm{ur},{\\rm temp}}$).\n  Denote by $\\mathcal{F}(\\widehat{T}_{c,\\theta}\/\\Omega_{c,\\theta})$ the space of bounded $\\widehat{\\mu}^{\\mathrm{ST}}_\\theta$-measurable functions on\n  $\\widehat{T}_{c,\\theta}\/\\Omega_{c,\\theta}$ whose points of discontinuity are contained in a $\\widehat{\\mu}^{\\mathrm{ST}}_\\theta$-measure zero set.\n  Define $\\mathcal{F}(\\prod_{v\\in S}\\widehat{T}_{c,\\theta_v}\/\\Omega_{c,\\theta_v})$ in the obvious analogous way.\n  By using the topological Satake isomorphism for tempered spectrum (cf. \\eqref{e:local-isom-v}) $$\\prod_{v\\in S}\\widehat{T}_{c,\\theta_v}\/\\Omega_{c,\\theta_v}\n  \\simeq G(F_S)^{\\wedge,\\mathrm{ur},{\\rm temp}}$$\n  and extending by zero outside the tempered spectrum, one obtains\n  \\begin{equation}\\label{e:Sauvageot}\\mathcal{F}\\left(\\prod_{v\\in S}\\widehat{T}_{c,\\theta_v}\/\\Omega_{c,\\theta_v}\\right)\\simeq  \\mathcal{F}(G(F_S)^{\\wedge,\\mathrm{ur},{\\rm temp}})\n  \\hookrightarrow \\mathcal{F}(G(F_S)^{\\wedge,\\mathrm{ur}}).\\end{equation}\n  Although the first two $\\mathcal{F}(\\cdot)$ above are defined with respect to different measures $\\prod_{v\\in S}\\widehat{\\mu}^{\\mathrm{ST}}_{\\theta_v}$\n  and $\\widehat{\\mu}^{\\mathrm{pl}}_S$, the isomorphism is justified by the fact that the ratio of the two measures is uniformly bounded above and below by positive constants (depending on $q_S$) in view of Proposition \\ref{p:unr-plan-meas} and Lemma \\ref{l:ST-measure}.\n  Note that the space of continuous functions on $\\prod_{v\\in S}\\widehat{T}_{c,\\theta_v}\/\\Omega_{c,\\theta_v}$\n  (resp. on $G(F_S)^{\\wedge,\\mathrm{ur},{\\rm temp}}$) is contained in the first (resp. second) term of \\eqref{e:Sauvageot},\n  and the two subspaces correspond under the isomorphism.\n\n\\begin{cor}\\label{c:density}  Let $\\widehat{f}_S\\in \\mathcal{F}(G(F_S)^{\\wedge,\\mathrm{ur}})$. For any $\\epsilon>0$,\n  there exist $\\phi_S,\\psi_S\\in {\\mathcal{H}}^{\\mathrm{ur}}(G(F_S))$ such that (i)\n$ \\widehat{\\mu}^{\\mathrm{pl}}_S(\\widehat{\\psi}_S)\\le \\epsilon$ and (ii) $\\forall\n\\pi_S\\in G(F_S)^{\\wedge,\\mathrm{ur}}$, $|\\widehat{f}_S(\\pi_S) - \\widehat{\\phi}_S(\\pi_S)|\\le \\widehat{\\psi}_S(\\pi_S)$.\n\\end{cor}\n\n\\begin{proof}\n  Let $\\phi_S,\\psi_S\\in C^\\infty_c(G(F_S))$ be the functions associated to $\\widehat{f}_S$ as in Proposition \\ref{p:density}. Then it is enough to replace $\\phi_S$ and $\\psi_S$ with their convolution products with the characteristic function on $\\prod_{v\\in S} K_v$.\n\\end{proof}\n\n  The following proposition will be used later in \\S\\ref{sub:ST-theorem}. For each $v\\in {\\mathcal V}_F(\\theta)$,\n  the image of $\\widehat{f}$ in $\\mathcal{F}(G(F_v)^{\\wedge,\\mathrm{ur}})$ via \\eqref{e:Sauvageot}\n  will be denoted $\\widehat{f}_v$.\n\n\\begin{prop}\\label{p:unr-density}\n  Let $\\widehat{f}\\in \\mathcal{F}(\\widehat{T}_{c,\\theta}\/\\Omega_{c,\\theta})$ and $\\epsilon >0$. There exists an integer $\\kappa\\ge 1$ and for all places $v\\in {\\mathcal V}_F(\\theta)$, there are bounded functions $\\phi_{v},\\psi_{v}\\in {\\mathcal{H}}^{\\mathrm{ur}}(G(F_{v}))^{\\le \\kappa}$ such that $\\widehat{\\mu}^{\\mathrm{pl}}_v(\\widehat{\\psi}_v)\\le \\epsilon$ and $|\\widehat{f_v}(\\pi) - \\widehat{\\phi_v}(\\pi)| \\le \\widehat{\\psi_v}(\\pi)$ for all $\\pi\\in G(F_v)^{\\wedge,\\mathrm{ur}}$.\n\\end{prop}\n\n\\begin{proof}\n  This is no more than Corollary \\ref{c:density} if we only required $\\phi_{v},\\psi_{v}\\in {\\mathcal{H}}^{\\mathrm{ur}}(G(F_{v}))$ without the superscript $\\le \\kappa$. So we may disregard finitely many $v$ by considering the subset ${\\mathcal V}_F(\\theta)^{\\ge Q}$ of ${\\mathcal V}_F(\\theta)$ consisting of $v$ such that $q_v\\ge Q$ for some $Q>0$. In view of Proposition \\ref{p:lim-of-Plancherel}, we may choose $Q\\in {\\mathbb Z}_{>0}$ that \\begin{equation}\\label{e:bound-pl-ST}\\forall v\\in {\\mathcal V}_F(\\theta)^{\\ge Q},~\\forall \\widehat{f}\\in \\mathcal{F}(\\widehat{T}_{c,\\theta}\/\\Omega_{c,\\theta}),\\quad\n  \\frac{1}{2} \\widehat{\\mu}^{\\mathrm{ST}}_{\\theta}(|\\widehat{f}|)\\le \\widehat{\\mu}^{\\mathrm{pl,ur}}_v(|\\widehat{f}_v|)\\le 2\\widehat{\\mu}^{\\mathrm{ST}}_{\\theta}(|\\widehat{f}|).\\end{equation}\n\n  Fix any $w\\in {\\mathcal V}_F(\\theta)^{\\ge Q}$. Corollary \\ref{c:density} allows us to find $\\phi_w,\\psi'_w\\in{\\mathcal{H}}^{\\mathrm{ur}}(G(F_{w}))$ such that\n  \\begin{equation}\\label{e:phi_wpsi_w}\n  \\widehat{\\mu}^{\\mathrm{pl}}_w(\\widehat{\\psi}'_w)\\le \\epsilon\/8  \\quad\\mbox{and}\\quad\n\\forall \\pi_w\\in G(F_w)^{\\wedge,\\mathrm{ur}},~\n|\\widehat{f}_w(\\pi_w) - \\widehat{\\phi}_w(\\pi_w)|\\le \\widehat{\\psi}'_w(\\pi_w). \\end{equation}\n Let $\\kappa_0\\in {\\mathbb Z}_{\\ge 0}$ be such that $\\phi_w,\\psi'_w\\in{\\mathcal{H}}^{\\mathrm{ur}}(G(F_{w}))^{\\le \\kappa_0}$. Now recall that for every $v\\in {\\mathcal V}_F(\\theta)$ there is a canonical isomorphism (cf. \\eqref{e:Satake}, Lemma \\ref{l:unr-temp-spec})\n  between ${\\mathcal{H}}^{\\mathrm{ur}}(G(F_v))$ and the space of regular functions in the complex variety $\\widehat{T}_{\\theta}\/\\Omega_{\\theta}$. Using the latter as a bridge, we may transport $\\phi_w,\\psi'_w$ to $\\phi_v,\\psi'_v\\in{\\mathcal{H}}^{\\mathrm{ur}}(G(F_{v}))$ for every $v\\in  {\\mathcal V}_F(\\theta)$. Clearly $\\phi_v,\\psi'_v\\in {\\mathcal{H}}^{\\mathrm{ur}}(G(F_{v}))^{\\le \\kappa_0}$ from the definition of \\S\\ref{sub:trun-unr-Hecke}. Moreover \\eqref{e:bound-pl-ST} and \\eqref{e:phi_wpsi_w} imply that\n  for all $v\\in {\\mathcal V}_F(\\theta)^{\\ge Q}$,\n  $$  \\widehat{\\mu}^{\\mathrm{pl}}_v(\\widehat{\\psi}'_v)\\le \\epsilon\/2  \\quad\\mbox{and}\\quad\n\\forall \\pi_v\\in G(F_v)^{\\wedge,\\mathrm{ur},{\\rm temp}},~\n|\\widehat{f}_v(\\pi_v) - \\widehat{\\phi}_v(\\pi_v)|\\le \\widehat{\\psi}'_v(\\pi_v).$$\n(Observe that $\\widehat{\\mu}^{\\mathrm{pl}}_v(\\widehat{\\psi}'_v)\\le 2 \\widehat{\\mu}^{\\mathrm{ST}}_\\theta(\\widehat{\\psi}'_v)=2\\widehat{\\mu}^{\\mathrm{ST}}_\\theta(\\widehat{\\psi}'_w)\\le 4\\widehat{\\mu}^{\\mathrm{pl}}_w(\\widehat{\\psi}'_w)\\le \\epsilon\/2$ to justify the first inequality.)\n\n  To achieve the latter inequality for non-tempered $\\pi_v\\in G(F_v)^{\\wedge,\\mathrm{ur}}$, we would like to perturb $\\psi'_v$ in a way independent of $v$ while not sacrificing the former inequality. Since $\\widehat{f}_v(\\pi_v)=0$ for such $\\pi_v$,\n  what we need to establish is that $|\\widehat{\\phi}_v(\\pi_v)|\\le \\widehat{\\psi}_v(\\pi_v)$ for all non-tempered $\\pi_v\\in G(F_v)^{\\wedge,\\mathrm{ur}}$. To this end, we use the fact that there is a compact subset $\\mathcal{K}$ of $\\widehat{T}_{\\theta}\/\\Omega_{\\theta}$ such that $G(F_v)^{\\wedge,\\mathrm{ur}}$ is contained in $\\mathcal{K}$ for every $v\\in {\\mathcal V}_F(\\theta)$ (cf. \\cite[Thm XI.3.3]{BW00}). By using the Weierstrass approximation theorem, we find $\\psi''_w\\in {\\mathcal{H}}^{\\mathrm{ur}}(G(F_{w}))$ such that\n  $$\\widehat{\\mu}^{\\mathrm{pl}}_w(\\widehat{\\psi}''_w)\\le \\epsilon\/8,$$\n$$\\forall \\pi_w\\in \\mathcal{K}\\backslash G(F_w)^{\\wedge,\\mathrm{ur},{\\rm temp}},~|\\widehat{\\psi}'_w(\\pi_w)| + |\\widehat{\\phi}_w(\\pi_w)|\\le \\widehat{\\psi}''_w(\\pi_w),$$\n$$\\forall \\pi_w\\in G(F_w)^{\\wedge,\\mathrm{ur},{\\rm temp}},~\\widehat{\\psi}''_w(\\pi_w)\\ge 0.$$\n  Choose $\\kappa\\ge \\kappa_0$ such that $\\psi''_w\\in {\\mathcal{H}}^{\\mathrm{ur}}(G(F_{w}))^{\\le \\kappa}$ and put $\\psi_w:=\\psi'_w+\\psi''_w$ so that $\\widehat{\\mu}^{\\mathrm{pl}}_w(\\widehat{\\psi}_w)\\le \\epsilon\/4$ and $\\psi_w\\in {\\mathcal{H}}^{\\mathrm{ur}}(G(F_{w}))^{\\le \\kappa}$. For each $v\\in {\\mathcal V}_F(\\theta)^{\\ge Q}$, let $\\psi_v$ denote the transport of $\\psi_w$ just as $\\psi'_v$ was the transport of $\\psi'_w$ in the preceding paragraph. Then $\\widehat{\\mu}^{\\mathrm{pl}}_v(\\widehat{\\psi}_v)\\le \\epsilon$ and $\\psi_v\\in {\\mathcal{H}}^{\\mathrm{ur}}(G(F_{v}))^{\\le \\kappa}$ as before. Moreover\n  $$\\forall \\pi_v\\in G(F_v)^{\\wedge,\\mathrm{ur},{\\rm temp}},~ |\\widehat{f}_v(\\pi_v) - \\widehat{\\phi}_v(\\pi_v)|\\le \\widehat{\\psi}'_v(\\pi_v)\\le \\widehat{\\psi}_v(\\pi_v)$$\n  and for $\\pi_v\\in G(F_v)^{\\wedge,\\mathrm{ur}}\\backslash G(F_v)^{\\wedge,\\mathrm{ur},{\\rm temp}}$,\n $$ |\\widehat{f}_v(\\pi_v) - \\widehat{\\phi}_v(\\pi_v)|=|\\widehat{\\phi}_v(\\pi_v)|\\le \\widehat{\\psi}''_v(\\pi_v)-|\\widehat{\\psi}'_v(\\pi_v)|\\le \\widehat{\\psi}_v(\\pi_v),$$\n the last inequality following from $\\widehat{\\psi}_v=\\widehat{\\psi}'_v+\\widehat{\\psi}''_v$.\n\\end{proof}\n\n\\begin{rem}\n  A more direct approach to~\\eqref{e:phi_wpsi_w} that wouldn't involve Corollary~\\ref{c:density} would be to use Weierstrass approximation to find polynomials $\\phi$ and $\\psi$ on $\\widehat T_{c,\\theta}\/\\Omega_{c,\\theta}$ of degree $\\le \\kappa$ such that $|\\widehat{f} - \\widehat{\\phi} | \\le \\widehat{\\psi}$ and then the isomorphism~\\eqref{e:Sauvageot} to transport $\\phi$ and $\\psi$ at the place $v$.\n\\end{rem}\n\nWe note~\\cite{Sau97}*{Lemme 3.5} that for any $\\phi_v\\in C^\\infty_c(G(F_v))$ there exists a $\\phi'_v\\in C^\\infty_c(G(F_v))$ such that\n$|\\widehat{\\phi}_v(\\pi_v)|\\le \\widehat{\\phi'}_v(\\pi_v)$ for all $\\pi_v\\in G(F_v)^{\\wedge}$. This statement is elementary, e.g. it follows from the Dixmier--Malliavin decomposition theorem. In fact we have the following stronger result due to Bernstein~\\cite{Bernstein:typeI}.\n\\begin{prop}\n  [Uniform admissibility theorem]\n  \\label{l:bounding-function-by-pos-function}\nFor any $\\phi_v \\in C^\\infty_c(G(F_v))$ there exists $C>0$ such that $|{\\rm tr}\\, \\pi(\\phi_v)|\\le C$ for all $\\pi\\in G(F_v)^{\\wedge}$.\n\\end{prop}\n\n\n\n\n\n\\begin{comment}\n\\subsection{A result on the test functions on $p$-adic groups}\n\n  The result of this subsection will be used in \\S\\ref{sub:ST-theorem} (and \\S\\ref{sub:general-functions-S_0}, which relies on \\S\\ref{sub:ST-theorem}).\n\n\\begin{proof}\n  We believe the result is due to Bernstein, although we haven't been able to locate the precise proof in the literature.\n\n\n  Henceforth we fix a minimal $F_v$-rational Levi subgroup and the finite set ${\\mathcal L}$ of representatives for the $F_v$-conjugacy classes of Levi subgroups containing it. The strategy of proof is roughly as follows: we demonstrate that $\\phi$ has uniformly bounded trace value on the set of full parabolically induced representations which are relevant to unitary representations in a suitable sense. Then by bounding the coefficients in the expression of an irreducible unitary representation as a linear combination of full induced representations, we will verify the claim.\n\n  For $M\\in {\\mathcal L}$,\n  denote by ${\\mathscr R}_M$ (resp. ${\\mathscr R}_{0,M}$) the set of equivalence classes of irreducible tempered (resp. supercuspidal) representations of $M(F_v)$, where $\\sigma$ and $\\sigma'$ are considered equivalent if there is $g\\in G(F_v)$ such that $g M(F_v)g^{-1}=M(F_v)$ and such that $\\sigma'\\simeq \\sigma^g$. Here $\\sigma^g$ means the twist of $\\sigma$ by $g$-conjugation. Let $\\overline{{\\mathscr R}}$ (resp. $\\overline{{\\mathscr R}}_0$) denote the analogous set defined by a different notion of equivalence where $\\sigma'$ and $\\sigma$ are equivalent (often said to be inertially equivalent) if $\\sigma'\\simeq \\sigma\\otimes \\psi$ for some $\\psi\\in \\Psi(M(F_v))$. (See \\S\\ref{sub:density-thm} for the definition of $\\Psi(M(F_v))$.)\n  Define ${\\mathscr R}:=\\coprod_{M\\in {\\mathcal L}}\\{(M,\\sigma):\\sigma\\in {\\mathscr R}_M\\}$ and similarly ${\\mathscr R}_0$, $\\overline{{\\mathscr R}}$, and $\\overline{{\\mathscr R}}_0$.\n\n  Let $(M,\\sigma)$ be a representative of an equivalence class of ${\\mathscr R}$. By abuse of notation we write $(M,\\sigma)\\in {\\mathscr R}$ if there is no danger of confusion. Denote by $JH({\\rm n\\textrm{-}ind}_M^G(\\sigma))$ the set of Jordan-H\\\"{o}lder quotients of the normalized parabolic induction of $\\sigma$. It is well known that $JH({\\rm n\\textrm{-}ind}_M^G(\\sigma))$ is a finite set and that its definition is independent of the choice of the parabolic subgroup or the representative. When $\\pi\\in {\\rm Irr}(G(F_v))$ let $\\sc(\\pi)=(M,\\sigma)\\in {\\mathscr R}$ denote its supercuspidal support so that $\\pi\\in JH({\\rm n\\textrm{-}ind}_M^G(\\sigma))$. It induces maps ${\\mathscr R}\\rightarrow {\\mathscr R}_0$ (still denoted by $\\text{sc}$) and $\\overline{\\text{sc}}:\\overline{{\\mathscr R}}\\rightarrow \\overline{{\\mathscr R}}_0$.\n\n\n\nFix an open compact subgroup $U\\subset G(F_v)$ such that $\\phi=\\phi_v$ is invariant under the left and right translations by elements of $U$ (For convenience we will often omit the subscript $v$ in the remaining of the proof). Then ${\\rm tr}\\, \\pi(\\phi)=0$ unless $\\pi^U\\neq 0$. Define ${\\mathscr R}_{U}$ to be the subset of $(M,\\sigma)\\in {\\mathscr R}$ such that for some $\\psi\\in \\Psi(M(F_v))$ there exists $\\pi\\in JH({\\rm n\\textrm{-}ind}_M^G(\\sigma\\otimes\\psi))$ with $\\pi^U\\neq 0$, and $\\overline{{\\mathscr R}}_U$ to be the image of ${\\mathscr R}_U$ in $\\overline{{\\mathscr R}}$. Let us observe that $\\overline{{\\mathscr R}}_{U}$ is finite. Indeed this follows from the fact that $\\overline{\\text{sc}}$ is a finite-to-one map and the well known fact that $\\overline{{\\mathscr R}}_U$ has finite image in $\\overline{{\\mathscr R}}_0$. (The former is true since the parabolic induction of any irreducible smooth representation has finite length. The latter amounts to saying that the function $(M,\\sigma)\\mapsto {\\rm tr}\\, {\\rm n\\textrm{-}ind}^G_M(\\sigma)({\\mathbf{1}}_U)$ is supported on finitely many Bernstein components.)\n\n\n     Choose a set of representatives ${\\mathscr R}^{{\\rm rep}}_U$ for $\\overline{{\\mathscr R}}_U$ in ${\\mathscr R}_U$. For each $(M,\\sigma)\\in {\\mathscr R}_U$, consider the set $\\Psi_{(M,\\sigma)}$ consisting of $\\psi\\in \\Psi(M(F_v))$ such that ${\\rm n\\textrm{-}ind}^G_M(\\sigma\\otimes \\psi)$ has an irreducible unitary subquotient. It was shown in \\cite[Thm 2.5]{Tad88} that $\\Psi_{(M,\\sigma)}$ is a compact subset of the complex manifold $\\Psi(M(F_v))$. Put\n    $${\\mathscr U}:=\\{(M,\\sigma\\otimes \\psi): (M,\\sigma)\\in {\\mathscr R}^{{\\rm rep}}_U,~\\psi\\in \\Psi_{(M,\\sigma)}\\}$$\n    so that ${\\mathscr U}\\subset {\\mathscr R}$. Define $\\widetilde{{\\mathscr U}}:=\\text{sc}^{-1}(\\text{sc}({\\mathscr U}))$ and $\\widetilde{{\\mathscr R}}_{(M,\\sigma)}:=\\text{sc}^{-1}(\\text{sc}(\\{(M,\\sigma)\\}))$ for each $(M,\\sigma)\\in {\\mathscr R}^{{\\rm rep}}_U$ so that $\\widetilde{{\\mathscr U}}\\supset \\widetilde{{\\mathscr R}}_{(M,\\sigma)}$. Then we may write $\\widetilde{{\\mathscr U}}$ in the form\n    $$\\widetilde{{\\mathscr U}}=\\bigcup_{(M,\\sigma)\\in \\overline{{\\mathscr R}}^{{\\rm rep}}_U} \\bigcup_{(L,\\tau)\\in \\widetilde{{\\mathscr R}}_{(M,\\sigma)}} \\{(L,\\tau\\otimes \\psi):\\psi\\in \\widetilde{\\Psi}_{(L,\\tau)}\\}$$\n    for compact subsets $\\widetilde{\\Psi}_{(L,\\tau)}$ of $\\Psi(L(F_v))$. Note that both $\\overline{{\\mathscr R}}^{{\\rm rep}}_U$ and $\\widetilde{{\\mathscr R}}_{(M,\\sigma)}$ are finite sets.   Since $$\\psi\\mapsto {\\rm tr}\\, ({\\rm n\\textrm{-}ind}^G_L(\\tau\\otimes \\psi))(\\phi)$$\n  is a continuous function for each $(L,\\tau)$ as above, there exists $C>0$ such that\n  \\begin{equation}\\label{e:uniform-trace}\\forall (M,\\sigma)\\in \\overline{{\\mathscr R}}_U,~\\forall (L,\\tau)\\in \\widetilde{{\\mathscr R}}_{(M,\\sigma)},~\\forall \\psi\\in\\widetilde{\\Psi}_{(L,\\tau)}, \\quad |{\\rm tr}\\, ({\\rm n\\textrm{-}ind}^G_L(\\tau\\otimes \\psi))(\\phi)|\\le C.\\end{equation}\n\n  Now \\cite[\\S4, Prop 2]{Clo86} allows us to write any $\\pi\\in {\\rm Irr}(G(F_v))$ as\n    \\begin{equation}\\label{e:lin-comb-ind}\\pi=\\sum_{i\\in I_{\\pi}} \\epsilon_i\\cdot{\\rm n\\textrm{-}ind}^G_{M_i}(\\sigma_i\\otimes \\psi_i)\\end{equation}  in the Grothendieck group for $\\epsilon_i\\in \\{\\pm1\\}$, $(M_i,\\sigma_i)\\in {\\mathscr R}$ and $\\psi_i\\in \\Psi(M_i(F_v))$. We claim that there exists a constant $D>0$ (independent of $\\pi$) such that for every $\\pi$ a linear combination \\eqref{e:lin-comb-ind} can be found such that\n  \\begin{enumerate}\n  \\item ${\\rm n\\textrm{-}ind}^G_{M_i}(\\sigma_i\\otimes \\psi_i)$ has the same supercuspidal support as $\\pi$,\n  \\item $|I_\\pi|\\le D$.\n  \\end{enumerate}\n  Clozel already ensured (i) in his construction of the linear combination. To justify (ii) we use the inductive procedure in \\textit{loc. cit.} Let us begin with recalling from \\cite[Thm 2.8]{BZ77} that there is a universal upper bound $l_G$ (depending only on $G$ and $F_v$) for the size of the set $JH({\\rm n\\textrm{-}ind}_M^G(\\sigma))$ as $(M,\\sigma)$ runs over all $M\\in {\\mathcal L}$ and all $\\sigma\\in {\\rm Irr}(M(F_v))$. Let $\\langle \\pi\\rangle$ be the subset of ${\\rm Irr}(G(F_v))$ having the same supercuspidal support as $\\pi$. Then $\\langle \\pi\\rangle$ has at most $l_G$ elements. As Clozel explained, one can write $\\pi={\\rm n\\textrm{-}ind}_M^G(\\sigma_0)-(\\pi_1+\\cdots+\\pi_t)$, where $t\\le l_G$ and the Langlands quotient data of $\\pi_1,...,\\pi_t$ are strictly smaller than that of $\\pi$ in some natural ordering. By repeating this process to rewrite $\\pi_i={\\rm n\\textrm{-}ind}^G_{M_i}(\\sigma_i)-(\\pi_{i,1}+\\cdots+\\pi_{i,t_i})$ for $i=1,...,t$ and so on, this process terminates by considering the ordering on $\\langle \\pi\\rangle$, and it is clear that $|I_\\pi|$ admits an upper bound only in terms of $l_G$.\n\n   Going back to the problem, assume from now on that $\\pi\\in G(F_v)^{\\wedge}$ and $\\pi^U\\neq 0$.\n    The crucial point is that condition (ii) above implies that $(M_i,\\sigma_i\\otimes \\psi_i)\\in \\widetilde{{\\mathscr U}}$. By \\eqref{e:uniform-trace}\n    $$|{\\rm tr}\\, \\pi(\\phi)|\\le \\sum_{i\\in I_\\pi} \\left| {\\rm n\\textrm{-}ind}^G_{M_i}(\\sigma_i\\otimes \\psi_i)(\\phi)\\right| \\le CD.$$\n\n\n\n\n\\end{proof}\n\\end{comment}\n\n\\subsection{Automorphic representations and a counting measure}\\label{sub:aut-rep}\n\n\n\n Now consider a string of complex numbers\n  $$\\mathcal{F}=\\{a_{\\mathcal{F}}(\\pi)\\in {\\mathbb C}\\}_{\\pi\\in{\\mathcal{AR}}_{{\\rm disc},\\chi}(G)}$$\n  such that $a_{\\mathcal{F}}(\\pi)=0$ for all but finitely many $\\pi$.\n  We think of $\\mathcal{F}$ as a multi-set by viewing\n  $a_{\\mathcal{F}}(\\pi)$ as multiplicity, or more appropriately as a density function with finite support in $\\mathcal{F}$ as $a_{\\mathcal{F}}(\\pi)$ is allowed to be in ${\\mathbb C}$. There are obvious meanings when we write $\\pi\\in \\mathcal{F}$ and $|\\mathcal{F}|$ (we could have written $\\pi\\in \\mathrm{supp}\\, \\mathcal{F}$ for the former):\n$$ \\pi\\in \\mathcal{F} ~\\stackrel{\\mathrm{def}}{\\Leftrightarrow}~ a_{\\mathcal{F}}(\\pi)\\neq 0,\\qquad\n|\\mathcal{F}|:=\\sum_{\\pi\\in \\mathcal{F}} a_{\\mathcal{F}}(\\pi).$$\n\n  In order to explain our working hypothesis, we recall a definition.\n\\begin{defn}\\label{d:cuspidal}\n  Let $H$ be a connected reductive group over ${\\mathbb Q}$. The maximal ${\\mathbb Q}$-split torus in $Z(H)$\n  is denoted $A_H$. We say $H$ is \\emph{cuspidal} if $(H\/A_H)\\times_{\\mathbb Q} {\\mathbb R}$ contains a maximal\n  ${\\mathbb R}$-anisotropic torus.\n\\end{defn}\n  If $H$ is cuspidal then $H({\\mathbb R})$ has discrete series representations. (We remind the reader that discrete series always mean ``relative discrete series'' for us, i.e. those whose matrix coefficients are square-integrable modulo center.) The converse is true when\n  $H$ is semisimple but not in general.\n  Throughout this section the following will be in effect:\n\\begin{hypo}\\label{hypo:cuspidal}\n  $\\Res_{F\/{\\mathbb Q}} G$ is a cuspidal group.\n\\end{hypo}\n  Let $S=S_0\\coprod S_1\\subset {\\mathcal V}_F^\\infty$ be a nonempty finite subset\n  and  $\\widehat{f}_{S_0}\\in {\\mathscr F}(G(F_{S_0})^{\\wedge})$.\n  (It is allowed that either $S_0$ or $S_1$ is empty.)\n  Let\n\\begin{itemize}\\item (level) $U^{S,\\infty}$ be an open compact subset of $G({\\mathbb A}^{S,\\infty})$,\n\\item (weight) $\\xi=\\otimes_{v|\\infty}\\xi_v$\nbe an irreducible algebraic representation of\n$$G_\\infty\\times_{\\mathbb R} {\\mathbb C}=(\\Res_{F\/{\\mathbb Q}} G)\\times_{\\mathbb Q} {\\mathbb C}=\\prod_{v|\\infty} G\\times_{F,v} {\\mathbb C}. $$ \\end{itemize}\n  Denote by $\\chi:A_{G,\\infty}\\rightarrow {\\mathbb C}^\\times$ the restriction of the\n  central character for $\\xi^\\vee$.\nDefine $$\\mathcal{F}=\\mathcal{F}(U^{S,\\infty},\\widehat{f}_{S_0},S_1,\\xi)\\quad\\mbox{by}$$\n\\begin{equation}\\label{e:a(pi)-general}\na_{\\mathcal{F}}(\\pi):=(-1)^{q(G)} m_{{\\rm disc},\\chi}(\\pi)\\dim(\\pi^{S,\\infty})^{U^{S,\\infty}}\n\\widehat{f}_{S_0}(\\pi_{S_0})\\widehat{{\\mathbf{1}}}_{K_{S_1}}(\\pi_{S_1})\n  \\chi_{{\\rm EP}}( \\pi_\\infty\\otimes \\xi) ~\\in {\\mathbb C}.\\end{equation}\n  Note that $\\widehat{{\\mathbf{1}}}_{K_{S_1}}(\\pi_{S_1})$ equals 1 if $\\pi_{S_1}$ is unramified\n  and 0 otherwise, and that $\\chi_{{\\rm EP}}( \\pi_\\infty\\otimes \\xi)=0$ unless $\\pi_\\infty$ has\n  the same infinitesimal character as $\\xi^\\vee$.\n  The set of $\\pi$ such that $a_{\\mathcal{F}}(\\pi)\\neq 0$ is finite by\n  Harish-Chandra's finiteness theorem.\n  Let us define measures $\\widehat{\\mu}_{\\mathcal{F},S_1}$ and $\\widehat{\\mu}^\\natural_{\\mathcal{F},S_1}$ associated with $\\mathcal{F}$\n on the unramified unitary dual $G(F_{S_1})^{\\wedge,\\mathrm{ur}}$, motivated by the trace formula.\nPut $\\tau'(G):=\\overline{\\mu}^{\\mathrm{can},{\\rm EP}}(G(F)A_{G,\\infty}\\backslash G({\\mathbb A}_F))$.\nFor any function $\\widehat{f}_{S_1}$ on $G(F_{S_1})^{\\wedge,\\mathrm{ur}}$ which is continuous\noutside a measure zero set, define\n\\begin{equation}\\label{e:defn-of-mu}\n\\widehat{\\mu}_{\\mathcal{F},S_1}(\\widehat{f}_{S_1})\n  := \\frac{\\mu^{\\mathrm{can}}(U^{S,\\infty})}{\\tau'(G)\\dim\\xi}\\sum_{\\pi\\in {\\mathcal{AR}}_{{\\rm disc},\\chi}(G)} a_{\\mathcal{F}}(\\pi) \\widehat{f}_{S_1}(\\pi_{S_1}).\\end{equation}\nThe sum is finite because $a_{\\mathcal{F}}$ is supported on finitely many $\\pi$. Now the key point is that the right hand side can be identified with the spectral side of Arthur's trace formula with the Euler-Poincar\\'{e} function\n  at infinity as in \\S\\ref{sub:EP} when $\\widehat{f}_{S_1}=\\widehat{\\phi}_{S_1}$ for some $\\phi_{S_1}\\in {\\mathcal{H}}^{\\mathrm{ur}}(G(F_{S_1}))$ (\\cite[pp.267-268]{Art89}, cf. proof of \\cite[Prop 4.1]{Shi-Plan}). So to speak,\nif we write $\\phi^{\\infty}=\\phi_{S_0}\\phi_{S_1}\\phi^{S,\\infty}$,\n\\begin{equation}\\label{e:apply-Arthur}\\widehat{\\mu}_{\\mathcal{F},S_1}(\\widehat{\\phi}_{S_1})= (-1)^{q(G)}\\frac{ I_{{\\rm spec}}(\\phi^{\\infty}\\phi_\\xi,\\mu^{\\mathrm{can},{\\rm EP}})}{{\\tau'(G)\\dim\\xi}}\n= (-1)^{q(G)}\\frac{ I_{{\\rm geom}}(\\phi^{\\infty}\\phi_\\xi,\\mu^{\\mathrm{can},{\\rm EP}})}{{\\tau'(G)\\dim\\xi}}\\end{equation}\nwhere $I_{{\\rm spec}}$ (resp. $I_{{\\rm geom}}$) denotes the spectral (resp. geometric)\nside Arthur's the invariant trace formula with respect to the measure\n$\\mu^{\\mathrm{can},{\\rm EP}}$. Finally if $\\widehat{f}_{S_0}$ has the property that $\\widehat{\\mu}^{\\mathrm{pl}}_{S_0}(\\widehat{f}_{S_0})\\neq 0$ then\nput $$ \\widehat{\\mu}^\\natural_{\\mathcal{F},S_1}:=\\widehat{\\mu}^{\\mathrm{pl}}_{S_0}(\\widehat{f}_{S_0})^{-1}  \\widehat{\\mu}_{\\mathcal{F},S_1}.$$\n\n\n\n\\begin{rem}\\label{r:|F|} The measure $ \\widehat{\\mu}^\\natural_{\\mathcal{F},S_1}$\n  is asymptotically the same as the counting measure $$\\widehat{\\mu}^{\\mathrm{count}}_{\\mathcal{F},S_1}(\\widehat{f}_{S_1})\n  = \\frac{1}{|\\mathcal{F}|}\\sum_{\\pi\\in {\\mathcal{AR}}_{{\\rm disc},\\chi}(G)} a_{\\mathcal{F}}(\\pi) \\widehat{f}_{S_1}(\\pi_{S_1}).$$\nassociated with the $S_1$-components of $\\mathcal{F}$ (assuming $|\\mathcal{F}|\\neq 0$). More precisely if $\\{\\mathcal{F}_k\\}_{\\ge1}$ is a family of\n\\S\\ref{sub:aut-families} below, then $\\widehat{\\mu}^{\\mathrm{count}}_{\\mathcal{F}_k,S_1}\/\n  \\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k,S_1}$ is a constant tending to 1 as $k\\rightarrow\\infty$ by Corollary \\ref{c:estimate-F_k}.\n\\end{rem}\n\n\\begin{ex}\\label{ex:weight-regular}\n  Suppose that the highest weight of $\\xi$ is \\emph{regular} and that $S_0=\\emptyset$. Then\n  $\\pi\\in {\\mathcal{AR}}_{{\\rm disc},\\chi}(G)$ belongs to $\\mathcal{F}$ if and only if\n  $(\\pi^{S,\\infty})^{U^{S,\\infty}}\\neq 0$, $\\pi$ is unramified at $S$ and\n  $\\pi_\\infty\\in \\Pi_{{\\rm disc}}(\\xi^\\vee)$. When $\\pi_\\infty\\in \\Pi_{{\\rm disc}}(\\xi^\\vee)$,\n  \\eqref{e:a(pi)-general} simplifies as\n  \\[\n \n  a_{\\mathcal{F}}(\\pi)=m_{{\\rm disc},\\chi}(\\pi)\\dim(\\pi^{S,\\infty})^{U^{S,\\infty}}.\n  \\]\n\\end{ex}\n\n\\begin{ex}\\label{ex:example-for-f_S}\n  Let $\\widehat{f}_{S_0}$ be a characteristic function on some relatively compact $\\widehat{\\mu}^{\\mathrm{pl}}_S$-measurable subset\n   $\\widehat{U}_{S_0}\\subset G(F_{S_0})^{\\wedge}$.\n   Assume that $S_0$ is large enough such that $G$ and all members of\n   $\\mathcal{F}$ are unramified outside $S_0$. Take $U^{S_0,\\infty}$ to be the product of $K_v$\n   over all finite places $v\\notin S_0$. Then for each $\\pi\\in {\\mathcal{AR}}_{{\\rm disc},\\chi}(G)$,\n   \\begin{equation}\\label{e:example-for-f_S}\n\t  a_{\\mathcal{F}}(\\pi)=(-1)^{q(G)}\\chi_{{\\rm EP}}(\\pi_\\infty\\otimes \\xi)\n   m_{{\\rm disc},\\chi}(\\pi)\n \\end{equation}\n   if $\\pi^{S_0,\\infty}$ is unramified, $\\pi_{S_0}\\in  \\widehat{U}_{S_0}$\n   (in which case $a_{\\mathcal{F}}(\\pi)\\neq0$ if moreover $\\chi_{{\\rm EP}}(\\pi_\\infty\\otimes \\xi)\\neq 0$; otherwise $a_{\\mathcal{F}}(\\pi)=0$).\n If the highest weight of $\\xi$ is regular, $\\chi_{{\\rm EP}}(\\pi_\\infty\\otimes \\xi)$ exactly when $\\pi_\\infty\\in \\Pi_{{\\rm disc}}(\\xi^\\vee)$, in which case \\eqref{e:example-for-f_S} simplifies\n   as\n   \\[\n  \n   a_{\\mathcal{F}}(\\pi)=m_{{\\rm disc},\\chi}(\\pi).\n   \\]\n   Compare this with Example \\ref{ex:weight-regular}. (The analogy in the case of modular forms is that\n   $\\pi$ as newforms are counted in the current example\n   whereas old-forms are also counted in Example \\ref{ex:weight-regular}.)\nFinally we observe that since the highest weight of $\\xi$ is regular and $\\pi_\\infty\\in \\Pi_{{\\rm disc}}(\\xi^\\vee)$, the discrete automorphic representation $\\pi$ is automatically cuspidal~\\cite{Wall84}*{Thm.~4.3}. In the present example the discrete multiplicity coincides with the cuspidal multiplicity.\n\\end{ex}\n\n\\begin{rem}\\label{r:why-S_0}\n  As the last example shows,\n  the main reason to include $S_0$ is to prescribe local conditions at finitely many places (namely at $S_0$) on automorphic families.\n  For instance one can take $\\widehat{f}_{S_0}=\\widehat{\\phi}_{S_0}$ where\n   $\\phi_{S_0}$ is a pseudo-coefficient of a supercuspidal representation\n  (or a truncation thereof if the center of $G$ is not anisotropic over $F_{S_0}$).\n  Then it allows us to consider a family of $\\pi$\n  whose $S_0$-components are a particular supercuspidal representation (or an unramified character twist thereof).\n  By using various $\\widehat{f}_{S_0}$\n  (which are in general not equal to $\\widehat{\\phi}_{S_0}$ for any\n   $\\phi_{S_0}\\in C^\\infty_c(G(F_{S_0}))$) one\n   obtains great flexibility in prescribing a local condition\n   as well as imposing weighting factors for a family.\n\\end{rem}\n\n\n\\subsection{Families of automorphic representations}\\label{sub:aut-families}\n\n  Continuing from the previous subsection (in particular keeping Hypothesis \\ref{hypo:cuspidal})\n let us introduce two kinds of families $\\{\\mathcal{F}_k\\}_{k\\ge 1}$\n which will be studied later on. We will need to measure the\n size of $\\xi$ in the following way. Let $T_\\infty$ be a maximal torus of\n $G_\\infty$ over ${\\mathbb R}$. For a $B$-dominant $\\lambda\\in X^*(T_\\infty)$, set\n  $m(\\lambda):=\\min_{\\alpha\\in \\Phi^+} \\langle \\lambda,\\alpha\\rangle$.\n  For $\\xi$ with $B$-dominant highest weight $\\lambda_{\\xi}$,\n  define $m(\\xi):=m(\\lambda_\\xi)$.\n\n   Let $\\phi_{S_0}\\in C^\\infty_c(G(F_{S_0}))$. (More generally we will sometimes prescribe a local condition at $S_0$ by\n   $\\widehat{f}_{S_0}\\in {\\mathscr F}(G(F_{S_0})^{\\wedge})$ rather than $\\phi_{S_0}$.) In the remainder of Section~\\ref{s:aut-Plan-theorem} we mostly focus on families in the level or weight aspect, respectively described as the following:\n\n\\begin{ex}[Level aspect: varying level, fixed weight]\\label{ex:level-varies}\n  Let ${\\mathfrak n}_k\\subset \\mathcal{O}_F$ be a nonzero ideal for $k\\ge 1$ such that ${\\mathbb N}({\\mathfrak n}_k)=[\\mathcal{O}_F:{\\mathfrak n}_k]$ tends to $\\infty$\n  as $k\\rightarrow \\infty$.\n  Take $$\\mathcal{F}_k:=\\mathcal{F}(K^{S,\\infty}({\\mathfrak n}_k),\\widehat{\\phi}_{S_0},S_1,\\xi).$$\n  Then $|\\mathcal{F}_k|\\rightarrow\\infty$ as $k\\rightarrow\\infty$.\n\\end{ex}\n\n\n\n\\begin{ex}[Weight aspect: fixed level, varying weight]\\label{ex:wt-varies}\n\nFor our study of weight aspect it is always supposed that $Z(G)=1$ so that $A_{G,\\infty}=1$ and $\\chi=1$ in order to eliminate the technical problem with central character when weight varies.\\footnote{Without the hypothesis that the center is trivial, one should work with fixed central character and apply the trace formula in such a setting. Then our results and arguments in the weight aspect should remain valid without change.} Let $\\{\\xi_k\\}_{k\\ge 1}$ be a sequence of irreducible algebraic representations\nof $G_\\infty\\times_{\\mathbb R} {\\mathbb C}$ such that $m(\\xi_k)\\rightarrow \\infty$\n  as $k\\rightarrow \\infty$. Take $$\\mathcal{F}_k:=\\mathcal{F}(U^{S,\\infty},\\widehat{\\phi}_{S_0},S_1,\\xi_k).$$\n  Then $|\\mathcal{F}_k|\\rightarrow\\infty$ as $k\\rightarrow\\infty$.\n\n\\end{ex}\n\n\\begin{rem}\n  Sarnak proposed a definition of families of automorphic representations (or automorphic $L$-functions)\n  in \\cite{Sarn:family}. The above two examples fit in his definition.\n\n\\end{rem}\n\n\n\n\n\\subsection{Level aspect}\\label{sub:level-varies}\n\n  We are in the setting of Example \\ref{ex:level-varies}.\n  Recall that $\\Res_{F\/{\\mathbb Q}} G$ is assumed to be cuspidal.\n  Fix $\\Xi:G\\hookrightarrow \\GL_m$ as in Proposition \\ref{p:global-integral-model} and let $B_\\Xi$ and $c_\\Xi$ be as in \\eqref{e:def-M(Xi)} and\nLemma \\ref{l:forcing-unipotent}.\nWrite ${\\mathscr L}_c(M_0)$ for the set of $F$-rational cuspidal Levi subgroups\nof $G$ containing the minimal Levi $M_0$.\n\n\\begin{thm}\\label{t:level-varies}\n  Fix $\\phi_{S_0}\\in C^\\infty_c(G(F_{S_0}))$ and $\\xi$.\n  Let $S_1\\subset {\\mathcal V}_F^\\infty$ be a subset where $G$ is unramified.\n  Let $\\phi_{S_1}\\in {\\mathcal{H}}^{\\mathrm{ur}}(G(F_{S_1}))^{\\le \\kappa}$\n  be such that $|\\phi_{S_1}|\\le 1$ on $G(F_{S_1})$.\n   If ${\\mathscr L}_c(M_0)=\\{G\\}$ (in particular if $G$ is abelian) then $\\widehat{\\mu}_{\\mathcal{F}_k,S_1}(\\widehat{\\phi}_{S_1})=\\widehat{\\mu}^{\\mathrm{pl}}_S(\\widehat{\\phi}_S)$.\n   Otherwise\n there exist constants $A_{\\mathrm{lv}}, B_{\\mathrm{lv}}>0$ and $C_{\\mathrm{lv}}\\ge 1$ such that\n \\begin{equation} \\label{e:t:level}\\widehat{\\mu}_{\\mathcal{F}_k,S_1}(\\widehat{\\phi}_{S_1})-\\widehat{\\mu}^{\\mathrm{pl}}_S(\\widehat{\\phi}_S)\n  = O(q_{S_1}^{A_{\\mathrm{lv}}+B_{\\mathrm{lv}} \\kappa} {\\mathbb N}({\\mathfrak n})^{-C_{\\mathrm{lv}}})\\end{equation}\n  as ${\\mathfrak n},\\kappa\\in {\\mathbb Z}_{\\ge 1}$, $S_1$ and $\\phi_{S_1}$ vary subject to the following conditions:\n  \\begin{itemize} \\item ${\\mathbb N}({\\mathfrak n}) \\ge c_\\Xi q_{S_1}^{B_\\Xi m \\kappa}$,\n  \\item no prime divisors of ${\\mathfrak n}$ are contained in $S_1$. \\end{itemize}\n  (The implicit constant in $O(\\cdot)$ is independent of ${\\mathfrak n}$, $\\kappa$, $S_1$ and $\\phi_{S_1}$.)\n\n\\end{thm}\n\n\\begin{rem}\\label{r:level-varies}\n  When $\\widehat{\\mu}^{\\mathrm{pl}}_{S_0}(\\phi_{S_0})\\neq0$, \\eqref{e:t:level} is equivalent to\n$$\\widehat{\\mu}^{\\natural}_{\\mathcal{F},S_1}(\\widehat{\\phi}_{S_1})-\\widehat{\\mu}^{\\mathrm{pl}}_{S_1}(\\widehat{\\phi}_{S_1})\n  = O(q_{S_1}^{A_{\\mathrm{lv}}+B_{\\mathrm{lv}}  \\kappa} {\\mathbb N}({\\mathfrak n})^{-C_{\\mathrm{lv}}})$$.\n\\end{rem}\n\n\\begin{rem}\n  One can choose $A_{\\mathrm{lv}}, B_{\\mathrm{lv}}, C_{\\mathrm{lv}}$ to be explicit integers. See the proof below.\n  For instance $C_{\\mathrm{lv}}\\ge n_G$ for $n_G$ defined in \\S\\ref{sub:notation}.\n\\end{rem}\n\n\\begin{proof}\n  Put $\\phi^{S,\\infty}:={\\mathbf{1}}_{K^{S,\\infty}({\\mathfrak n})}$. The right hand side\n  of \\eqref{e:apply-Arthur} is expanded as in \\cite[Thm 6.1]{Art89} as shown by Arthur.\n  Arguing as at the start of the proof of \\cite[Thm 4.4]{Shi-Plan}, we obtain\n  \\begin{equation}\\label{e:TF-level-aspect}\n\\widehat{\\mu}_{\\mathcal{F},S_1}(\\widehat{\\phi}_{S_1})-\\widehat{\\mu}^{\\mathrm{pl}}_S(\\widehat{\\phi}_S)\n  =\\sum_{M\\in {\\mathscr L}_c(M_0)\\backslash \\{G\\}} a_M \\cdot \\phi_{S_0,M}(1)\\phi_{S_1,M}(1) \\phi^{S,\\infty}_{M}(1) \\frac{\\Phi^G_M(1,\\xi)}{\\dim \\xi}\\end{equation}\n  where the sum runs over proper cuspidal Levi subgroups of $G$ containing\n  a fixed minimal $F$-rational Levi subgroup (see \\cite[p.539]{GKM97}\n  for the reason why only cuspidal Levi subgroups contribute)\n  and $a_M\\in {\\mathbb C}$ are explicit constants depending only on $M$ and $G$.\n  Further explanation of \\eqref{e:TF-level-aspect} needs to be in place.\n  Since only semisimple conjugacy classes contribute to Arthur's trace formula for each $M$,\n  Lemma \\ref{l:forcing-unipotent} tells us that any contribution from non-identity elements\n  vanishes. Note that $\\widehat{\\mu}^{\\mathrm{pl}}_S(\\widehat{\\phi}_S)$ comes from the $M=G$ term on the right hand side.\n\n  The first assertion of the theorem follows immediately from\n  \\eqref{e:TF-level-aspect}. Henceforth we may assume that ${\\mathscr L}_c(M_0)\\backslash \\{G\\}\\neq \\emptyset$.\n\n  Clearly $\\phi_{S_0,M}(1)$ and $\\Phi^G_M(1,\\xi)\/\\dim \\xi$ are constants.\n  It was shown in Lemma \\ref{l:bounding-phi-on-S_0} that $|\\phi_{S_1,M}(1)|=O(q_{S_1}^{d_G+r_G+b_G\\kappa})$\n  for $b_G> 0$ in that lemma. We take $$A_{\\mathrm{lv}}:=d_G+r_G\\quad\\mbox{and}\\quad B_{\\mathrm{lv}}:=b_G.$$\n  We will be done if it is checked that $|\\phi^{S,\\infty}_{M}(1)|= O({\\mathbb N}({\\mathfrak n})^{-C_{\\mathrm{lv}}})$ for some $C_{\\mathrm{lv}}\\ge 1$.\n  Let $P=MN$ be a parabolic subgroup with Levi decomposition where $M$ is as above. Then\n  $$0\\le \\phi^{S,\\infty}_{M}(1)=\\int_{N({\\mathbb A}_F^{S,\\infty})} \\phi^{S,\\infty}(n)dn\n  = \\prod_{v\\notin S\\atop v|{\\mathfrak n}~\\mathrm{or}~v\\in {\\rm Ram}(G)} \\operatorname{vol}(K_v(\\varpi_v^{v({\\mathfrak n})})\\cap N(F_v))$$\n $$ = \\prod_{ v\\notin S \\atop v|{\\mathfrak n}~\\mathrm{or}~v\\in {\\rm Ram}(G)} \\operatorname{vol} (N(F_v)_{x,v({\\mathfrak n})})\n  = \\left( \\prod_{v|{\\mathfrak n}\\atop v\\notin S} q_v^{-v({\\mathfrak n})\\dim N}\\right)\n   \\prod_{v\\in {\\rm Ram}(G)\\atop v\\notin S} \\operatorname{vol}(K_v\\cap N(F_v)).$$\n  The last equality uses the standard fact about the filtration\n  that $\\operatorname{vol}(N(F_v)_{x,v({\\mathfrak n})})=|\\varpi_v|^{v({\\mathfrak n})\\dim N} \\operatorname{vol}(N(F_v)_{x,0})$ and the fact \\eqref{e:meas-K-cap-N}\n  that $\\operatorname{vol}(N(F_v)_{x,0}) = \\operatorname{vol}(N(F_v)\\cap K_v)=1$ when $G$ is unramified at $v$.\n  Take $$C_{\\mathrm{lv}}:=\\min_{M\\in {\\mathscr L}_c(M_0)\\backslash \\{G\\}\\atop P=MN}\n  (\\dim N)$$ to be the minimum dimension of the unipotent radical of a proper parabolic subgroup of $G$ with cuspidal Levi\n  part. Then $|\\phi^{S,\\infty}_{M}(1)|\\le {\\mathbb N}({\\mathfrak n})^{-C_{\\mathrm{lv}}}\\prod_{v\\in {\\rm Ram}(G)} \\operatorname{vol}(K_v\\cap N(F_v))$\n  for every $M$ in \\eqref{e:TF-level-aspect}.\n\n\\end{proof}\n\n\n\n\\subsection{Weight aspect}\\label{sub:weight-varies}\n\n We put ourselves in the setting of Example \\ref{ex:wt-varies} and exclude the uninteresting case of $G=\\{1\\}$.\n By the assumption $Z(G)=\\{1\\}$, for every $\\gamma\\neq 1\\in G(F)$\n the connected centralizer $I_\\gamma$ has a strictly smaller set of roots so that $|\\Phi_{I_\\gamma}|<|\\Phi|$.\n Our next task is to prove a similar error bound as in the last subsection.\n\n\n\n\\begin{thm}\\label{t:weight-varies} Fix $\\phi_{S_0}\\in C^\\infty_c(G(F_{S_0}))$\n and $U^{S,\\infty}\\subset G({\\mathbb A}^{S,\\infty})$.\n There exist constants $A_{\\mathrm{wt}},B_{\\mathrm{wt}}>0$ and $C_{\\mathrm{wt}}\\ge 1$ satisfying the following: for\n \\begin{itemize}\n \\item any $\\kappa\\in {\\mathbb Z}_{>0}$, \\item any finite subset $S_1\\subset {\\mathcal V}_F^\\infty$ disjoint from $S_0$ and $S_{{\\rm bad}}$\n   (\\S\\ref{sub:global-bound-orb-int}) and \\item any $\\phi_{S_1}\\in {\\mathcal{H}}^{\\mathrm{ur}}(G(F_{S_1}))^{\\le \\kappa}$\n  such that $|\\phi_{S_1}|\\le 1$ on $G(F_{S_1})$,\\end{itemize}\n $$\\widehat{\\mu}_{\\mathcal{F},S_1}(\\widehat{\\phi}_{S_1})-\\widehat{\\mu}^{\\mathrm{pl}}_S(\\widehat{\\phi}_S)\n  = O(q_{S_1}^{A_{\\mathrm{wt}}+B_{\\mathrm{wt}}\\kappa} m(\\xi)^{-C_{\\mathrm{wt}}})$$\nwhere the implicit constant in $O(\\cdot)$ is independent of $\\kappa$, $S_1$ and $\\phi_{S_1}$.\n(Equivalently, $\\widehat{\\mu}^\\natural_{\\mathcal{F},S_1}(\\widehat{\\phi}_{S_1})-\\widehat{\\mu}^{\\mathrm{pl}}_{S_1}(\\widehat{\\phi}_{S_1})\n  = O(q_{S_1}^{A_{\\mathrm{wt}}+B_{\\mathrm{wt}}\\kappa} m(\\xi)^{-C_{\\mathrm{wt}}})$ if $\\widehat{\\mu}^{\\mathrm{pl}}_{S_0}(\\phi_{S_0})\\neq 0$.)\n\n\\end{thm}\n\n\\begin{rem}\n  We always assume that $S_0$ and $S_1$ are disjoint. So the condition on $S_1$ is really that it stays away from the finite set $S_{{\\rm bad}}$. This enters the proof where a uniform bound on orbital integrals from \\S\\ref{sub:global-bound-orb-int} is applied to the places in $S_1$.\n\\end{rem}\n\n\\begin{rem}\n  Again $A_{\\mathrm{wt}}, B_{\\mathrm{wt}}, C_{\\mathrm{wt}}$ can be chosen explicitly as can be seen from the proof below. For instance a choice can be made such that $C_{\\mathrm{wt}}\\ge n_G$ for $n_G$ defined in \\S\\ref{sub:notation}.\n\\end{rem}\n\n\\begin{proof}\n  We can choose a sufficiently large finite set $S'_0\\supset S_0\\cup {\\rm Ram}(G)$ in the complement of $S_1\\cup S_\\infty$ such that $U^{S,\\infty}$ is a finite disjoint union of\n  groups of the form $(\\prod_{v\\notin S'_0\\cup S_1\\cup S_\\infty} K_v)\\times U_{S'_0\\backslash S_0}$ for open compact subgroups\n  $U_{S'_0\\backslash S_0}$ of $G({\\mathbb A}_{F,S'_0\\backslash S_0})$. By replacing $S_0$ with $S'_0$\n  (and thus $S$ with $ S'_0\\coprod S_1$), we reduce the proof to the case where $U^{S,\\infty}=\\prod_{v\\notin S\\cup S_\\infty} K_v$.\n\n  For an $F$-rational Levi subgroup $M$ of $G$, let $Y_M$ be as in Proposition \\ref{p:bound-number-of-conj},\n  where $\\kappa$, $S_0$ and $S_1$ are as in the theorem. (So the set $Y_M$ varies as\n  $\\kappa$ and $S_1$ vary.) Take \\eqref{e:apply-Arthur} as a starting point.\n  Arthur's trace formula (\\cite[Thm 6.1]{Art89}) and the argument in the proof of \\cite[Thm 4.11]{Shi-Plan} show\n(note that our $Y_M$ contains $Y_M$ of \\cite{Shi-Plan} but could be strictly bigger):\n $$ \\widehat{\\mu}_{\\mathcal{F},S_1}(\\widehat{\\phi}_{S_1})-\\widehat{\\mu}^{\\mathrm{pl}}_S(\\widehat{\\phi}_S)\n=\n\\sum_{\\gamma\\in Y_G\\backslash\\{1\\}} a_{G,\\gamma}\\cdot\n|\\iota^G(\\gamma)|^{-1}O^{G({\\mathbb A}^\\infty_F)}_\\gamma(\\phi^\\infty)\\frac{{\\rm tr}\\, \\xi_n(\\gamma)}{\\dim \\xi_n}\n$$\n\\begin{equation}\\label{e:main-formula-wt-varies} +\\sum_{M\\in {\\mathscr L}_c\\backslash \\{G\\}}\n\\sum_{\\gamma\\in Y_M}\na_{M,\\gamma}\\cdot|\\iota^M(\\gamma)|^{-1}O^{M({\\mathbb A}^\\infty_F)}_\\gamma(\\phi^\\infty_M)\\frac{\\Phi^G_M(\\gamma,\\xi_n)}{\\dim \\xi_n}\n\\end{equation}\nwhere $a_{M,\\gamma}$ (including $M=G$) is given by\n$$a_{M,\\gamma}=\\tau'(G)^{-1} \\frac{\\overline{\\mu}^{\\mathrm{can},{\\rm EP}}(I^M_\\gamma(F)\\backslash\nI^M_\\gamma({\\mathbb A}_F)\/A_{I^M_\\gamma,\\infty})}\n {\\overline{\\mu}^{{\\rm EP}}(I^M_\\gamma(F_\\infty)\/A_{I^M_\\gamma,\\infty})}$$\n $$\\stackrel{\\mathrm{Cor}~\\ref{c:canonical-measure}}{=}\n\\frac{\\tau(I^M_\\gamma)}{\\tau'(G)} \\frac{|\\Omega_{I^M_\\gamma}|}{|\\Omega_{I^M_\\gamma,c}|}\n\\frac{L({\\rm Mot}_{I^M_\\gamma})}{e(I^M_{\\gamma,\\infty})2^{[F:{\\mathbb Q}] r_G}} .$$\n\n Let us work with one $M$ at a time.\n Observe that clearly\n $|\\Omega_{I^M_\\gamma}|\/|\\Omega_{I^M_\\gamma,c}|\\le |\\Omega|$\n and that $\\tau(I^M_\\gamma)$ is bounded by a constant\n depending only on $G$ in view of \\eqref{e:Tamagawa} and Corollary \\ref{c:bounding-pi0-general}\n or Lemma \\ref{l:bounding-pi0}. \nBy Corollary \\ref{c:bound-on-L(1)-2}, there exist constants $c_2,A_2>0$ such that\n \\[\n\n |a_{M,\\gamma}|\\le c_2\n \\prod_{v\\in {\\rm Ram}(I_\\gamma^M)} q_v^{A_2}\n   \\]\n\n   It is convenient to define the following finite subset of ${\\mathcal V}_F^\\infty$ for each $\\gamma\\in Y_M$. We fix a maximal torus $T^M_\\gamma$ in $M$ over $\\overline{F}$ containing $\\gamma$ and write $\\Phi_{M,\\gamma}$ for the set of roots of $T^M_\\gamma$ in $M$. (A different choice of $T^M_\\gamma$ does not affect the argument.)\n  $$S_{M,\\gamma}:=\\{v\\in {\\mathcal V}_F^\\infty\\backslash S: \\exists \\alpha\\in \\Phi_{M,\\gamma},\n  ~\\alpha(\\gamma)\\neq 1\n  ~\\mbox{and}~|1-\\alpha(\\gamma)|_v\\neq 1\\}.$$\n (If $\\gamma$ is in the center of $M(F)$ then $S_{M,\\gamma}=\\emptyset$ and $q_{S_{M,\\gamma}}=1$.)\n\n  We know that $O^{M(F_v)}_\\gamma({\\mathbf{1}}_{K_{M,v}})=1$ for $v\\notin S\\cup S_{M,\\gamma}\\cup S_\\infty$\nand that $S_{M,\\gamma}\\supset {\\rm Ram}(I^M_\\gamma)$\n  from \\cite[Cor 7.3]{Kot86}. According to Lemma \\ref{l:const-term-on-unram-Hecke}\n  $\\phi_v={\\mathbf{1}}_{K_{v}}$ implies $\\phi_{v,M}={\\mathbf{1}}_{K_{M,v}}$. Hence\n \\begin{eqnarray}\n   |a_{M,\\gamma}|&\\le& c_2\\cdot (q_{S_{M,\\gamma}})^{A_2} \\label{e:bound-a_M}\\\\\n  O^{M({\\mathbb A}_F^\\infty)}_\\gamma(\\phi^\\infty_M)\n & =& O^{M(F_S)}_\\gamma(\\phi_{S,M})\\prod_{v\\in S_{M,\\gamma}}\n   O^{M(F_{v})}_\\gamma({\\mathbf{1}}_{K_{M,v}}). \\notag\n\\end{eqnarray}\n   By Theorem \\ref{t:appendix1}, there exists a constant $c(\\phi_{S_0,M})>0$\n  such that\n   $$O^{M(F_{S_0})}_\\gamma(\\phi_{S_0,M})\n   \\le c(\\phi_{S_0,M}) \\prod_{v\\in S_0}\n   D^M_v(\\gamma)^{-1\/2}, \\quad \\forall \\gamma\\in Y_M.$$\n   By Theorem \\ref{t:appendeix2}, there exist $a,b,c,e_G\\in {\\mathbb R}_{\\ge 0}$ (independent of\n   $\\gamma$, $S_1$, $\\kappa$ and $k$) such that\n \\begin{eqnarray}\\label{e:9.17}\n    O^{M(F_{S_1})}_\\gamma(\\phi_{S_1,M})\n & \\le & q_{S_1}^{a+b\\kappa} \\prod_{v\\in S_1} D^M_v(\\gamma)^{-e_G\/2},  \\\\\n  O^{M(F_v)}_\\gamma({\\mathbf{1}}_{K_{M,v}})\n & \\le & q_{v}^{c}  D^M_v(\\gamma)^{-e_G\/2}, \\quad\\forall v\\in S_{M,\\gamma}. \\label{e:9.18}\n\\end{eqnarray}\n (To obtain \\eqref{e:9.17} and \\eqref{e:9.18}, apply Theorem \\ref{t:appendeix2} to $v\\in S_1$ and $v\\in S_{M,\\gamma}$.)\n\n   Hence\n   \\begin{eqnarray}\n   O^{M({\\mathbb A}_F^\\infty)}_\\gamma(\\phi^\\infty_M)&\\le&\n   c(\\phi_{S_0,M})  q_{S_1}^{a+b\\kappa} q_{S_{M,\\gamma}}^{c}\n  \\left( \\prod_{v\\nmid \\infty} D^M_v(\\gamma)^{-1\/2}\\right)\n\\prod_{v\\in S_1\\cup S_{M,\\gamma}} D^M_v(\\gamma)^{(1-e_G)\/2}\\nonumber\\\\\n  & = &  c(\\phi_{S_0,M})  q_{S_1}^{a+b\\kappa} q_{S_{M,\\gamma}}^{c}\n   \\prod_{v|\\infty} D^M_v(\\gamma)^{1\/2}\n\\prod_{v\\in S_1\\cup S_{M,\\gamma}} D^M_v(\\gamma)^{(1-e_G)\/2}\\label{e:bound-on-O^M}\n\\end{eqnarray}\n\n   On the other hand there exist $\\delta_{S_0},\\delta_{\\infty}$, $\\delta_{S_1}\\ge1$ such that\n   for every $\\gamma\\in Y_M$ with $\\alpha(\\gamma)\\neq 1$,\n   \\begin{itemize}\n   \\item $|1-\\alpha(\\gamma)|_{S_0}\\le \\delta_{S_0}$. (compactness of ${\\rm supp}\\, \\phi_{S_0}$)\n   \\item $|1-\\alpha(\\gamma)|_{\\infty}\\le \\delta_{\\infty}$. (compactness of $U_\\infty$)\n   \\item $|1-\\alpha(\\gamma)|_{S_1}\\le \\delta_{S_1}q_{S_1}^{B_5 \\kappa}$. (Lemma \\ref{l:bounding1-alpha(gamma)}; Remark \\ref{r:Lem2.18-indep} explains the independence of $B_1$ of $S_1$).\n   \\end{itemize}\n   (When $\\alpha(\\gamma)=1$, our convention is that $|1-\\alpha(\\gamma)|_v=1$ for every $v$\n   to be consistent with the first formula of Appendix \\ref{s:app:Kottwitz}.)\n  Hence, together with the product formula for $1-\\alpha(\\gamma)$,\n   $$1=\\prod_v |1-\\alpha(\\gamma)|_v \\le\\delta_{S_0} \\delta_{\\infty}\n    \\delta_{S_1}q_{S_1}^{B_5 \\kappa}\\prod_{v\\in S_{M,\\gamma}}|1-\\alpha(\\gamma)|_v .$$\n  If $\\gamma\\in Z(M)(F)$ then $q_{S_{M,\\gamma}}=1$. Otherwise\n    for each $v\\in S_{M,\\gamma}$, we may choose $\\alpha\\in \\Phi_{M,\\gamma}$ such that $|1-\\alpha(\\gamma)|_v\\neq 1$.\nSet $\\delta:=\\delta_{S_0} \\delta_{\\infty}\n    \\delta_{S_1}$.\n    Then $|1-\\alpha(\\gamma)|_v \\le q_v^{-1}$ for $v\\in S_{M,\\gamma}$ by Lemma \\ref{l:alpha(gamma)-is-integral}\n    (which is applicable in view of the first paragraph in the current proof)\n    so\n    \\begin{equation}\\label{e:prod-q-S_M,gamma}\n     q_{S_{M,\\gamma}}\\le \\delta q_{S_1}^{B_5 \\kappa}.\\end{equation}\n  Keep assuming that $\\gamma$ is not central in $M$ and that\n  $\\alpha(\\gamma)\\neq 1$. Again by the product formula\n  $\\prod_{v\\in S_1\\cup S_{M,\\gamma}} |1-\\alpha(\\gamma)|\n  = \\prod_{v\\in S_0\\cup S_\\infty} |1-\\alpha(\\gamma)|^{-1} \\ge (\\delta_{S_0}\\delta_\\infty)^{-1}$,\n  thus\n  \\begin{equation}\\label{e:bound-D^M}\n\\prod_{v\\in S_1\\cup S_{M,\\gamma}} D^M(\\gamma)^{-1}\\le \\delta_{S_0}\\delta_\\infty.\\end{equation}\n  The above holds also when $\\gamma$ is central in $M$, in which case the left hand side equals 1.\n\n\n    Now \\eqref{e:bound-on-O^M}, \\eqref{e:prod-q-S_M,gamma} and \\eqref{e:bound-D^M} imply\n   \\begin{equation}\\label{e:bound-on-O^M-2}\n   O^{M({\\mathbb A}_F^\\infty)}_\\gamma(\\phi^\\infty_M)\\le\n   c(\\phi_{S_0,M}) \\delta^{a}(\\delta_{S_0}\\delta_\\infty)^{(e_G-1)\/2}\n q_{S_1}^{a+b\\kappa+ c B_5 \\kappa}   \\prod_{v|\\infty}\n   D^M_v(\\gamma)^{1\/2}.\\end{equation}\n\n   Lemma \\ref{l:bound-for-st-ds-char} gives a bound on the stable discrete series\n   character:\n \\begin{equation}\\label{e:bound-Phi^G_M}\\frac{|\\Phi^G_M(\\gamma,\\xi)|}{\\dim \\xi}\n \n \\le c \\frac{\\prod_{v|\\infty} D^M_v(\\gamma)^{-1\/2}}{ m(\\xi)^{|\\Phi^+|-|\\Phi^+_{I^M_\\gamma}|}}.\\end{equation}\n\n  Multiplying \\eqref{e:bound-a_M}, \\eqref{e:bound-on-O^M-2},\n  \\eqref{e:bound-Phi^G_M} altogether (and noting $|\\iota^M(\\gamma)|\\le 1$),\n  the absolute value of the summand for $\\gamma$ in \\eqref{e:main-formula-wt-varies} (including $M=G$)\n  is\n$$O\\left( m(\\xi)^{-(|\\Phi^+|-|\\Phi^+_{I^M_\\gamma}|)}\nq_{S_1}^{a+b\\kappa+c B_5 \\kappa+A_2}\\right).$$\n   All in all,  $|\\widehat{\\mu}_{\\mathcal{F},S}(\\widehat{\\phi}_S)-\\widehat{\\mu}^{\\mathrm{pl}}_S(\\widehat{\\phi}_S)|$ is\n$$ \\left(|Y_G|-1 + \\sum_{M\\in {\\mathscr L}_c\\backslash \\{G\\}} |Y_M|\\right)\n  O\\left( m(\\xi)^{-(|\\Phi^+|-|\\Phi^+_{I^M_\\gamma}|)}\n  q_{S_1}^{a+b\\kappa+ c B_5 \\kappa+A_2}\\right).$$\n Set (excluding $\\gamma=1$ in the second minimum when $M=G$) $$C_{\\mathrm{wt}}:= \\min_{M\\in {\\mathscr L}_c(M_0)}\n \\min_{\\gamma\\in M(F)\\atop \\mathrm{ell.~in~}M(F_\\infty)\n  (|\\Phi^+|-|\\Phi^+_{I^M_\\gamma}|)$$\n Note that\n $C_{\\mathrm{wt}}$ depends only on $G$. It is automatic that\n  $|\\Phi^+|-|\\Phi^+_{I^M_\\gamma}|\\ge 1$ on $Y_G\\backslash \\{1\\}$ and $Y_M$ for $M\\in {\\mathscr L}_c(M_0)\\backslash \\{G\\}$.\nThe proof is concluded by invoking Corollary \\ref{c:bound-number-of-conj} (applied to $Y_G$ and $Y_M$)\nwith the choice $$A_{\\mathrm{wt}}:=a+A_2+A_6,\\quad B_{\\mathrm{wt}}:=b+cB_5+B_8.$$\n\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{Automorphic Plancherel density theorem}\\label{sub:Plan-density}\n\n\n  In the situation of either Example \\ref{ex:level-varies} or \\ref{ex:wt-varies},\n let us write $\\mathcal{F}_k(\\phi_{S_0})$ for $\\mathcal{F}_k$ in order to emphasize the dependence on $\\phi_{S_0}$.\nTake $S_1=\\emptyset$ so that $S=S_0$. Then $\\widehat{\\mu}_{\\mathcal{F}_k(\\phi_{S}),\\emptyset}$ may be viewed\nas a complex number (as it is a measure on a point).\nIn fact we can consider $\\mathcal{F}_k(\\widehat{f}_{S})$, a family whose local condition at $S$\nis prescribed by $\\widehat{f}_S\\in {\\mathscr F}(G(F_S)^{\\wedge})$, even if\n$\\widehat{f}_{S}$ does not arise from any $\\phi_{S}$ in $C^\\infty_c(G(F_S))$.\nPut $\\widehat{\\mu}_k(\\widehat{f}_{S}):=\\widehat{\\mu}_{\\mathcal{F}_k(\\widehat{f}_{S}),\\emptyset}\\in {\\mathbb C}$.\nWe recover the automorphic Plancherel density theorem (\\cite[Thm 4.3, Thm 4.7]{Shi-Plan}).\n\n\\begin{cor}\\label{c:Plan-density} Consider families $\\mathcal{F}_k$ in level or weight aspect as above. In level aspect assume that the highest weight of $\\xi$ is regular. (No assumption is necessary in weight aspect.)\n For any $\\widehat{f}_S\\in {\\mathscr F}(G(F_S)^{\\wedge})$,\n  $$\\lim_{k\\rightarrow\\infty} \\widehat{\\mu}_k(\\widehat{f}_{S})=\\widehat{\\mu}^{\\mathrm{pl}}_S(\\widehat{f}_S).$$\n\n\\end{cor}\n\n\\begin{proof}\n  Theorems \\ref{t:level-varies} and \\ref{t:weight-varies} tell us that\n  \\begin{equation}\\label{e:aut-plan-phi}\\lim_{k\\rightarrow\\infty} \\widehat{\\mu}_k(\\widehat{\\phi}_S)=\\widehat{\\mu}^{\\mathrm{pl}}_S(\\widehat{\\phi}_S).\\end{equation}\n  (Even though there was a condition on $S_1$, note that there was no condition on $S_0$ in either theorem.)\n\n  We would like to improve \\eqref{e:aut-plan-phi} to allow more general test functions.\n  What needs to be shown (cf. \\eqref{e:extending-at-v_j-0} below) is that for every $\\epsilon>0$, $$\\limsup_{k\\rightarrow \\infty} |\\widehat{\\mu}_{k}(\\widehat{f}_{S})- \\widehat{\\mu}^{\\mathrm{pl}}_{S}(\\widehat{f}_{S})|\\le4\\epsilon.$$\n  Thanks to Proposition \\ref{p:density} there exist $\\phi_{S},\\psi_{S}\\in {\\mathcal{H}}^{\\mathrm{ur}}(G(F_{S}))$ such that $|\\widehat{f}_{S}-\\widehat{\\phi}_{S}|\\le \\widehat{\\psi}_{S}$ on $G(F_{S})^{\\wedge}$ and $\\widehat{\\mu}^{\\mathrm{pl}}_{S}(\\widehat{\\psi}_{S})\\le \\epsilon$. Then (cf. \\eqref{e:extending-at-v_j} below)\n  $$\n\\begin{aligned}\n|\\widehat{\\mu}_{k}(\\widehat{f}_{S})- \\widehat{\\mu}^{\\mathrm{pl}}_{S}(\\widehat{f}_{S})| & \\le  |\\widehat{\\mu}_k(\\widehat{f}_{S}-\\widehat{\\phi}_{S})|\\\\ & + |\\widehat{\\mu}_k(\\widehat{\\phi}_{S})- \\widehat{\\mu}^{\\mathrm{pl}}_{S}(\\widehat{\\phi}_{S})| + |\\widehat{\\mu}^{\\mathrm{pl}}_{S}(\\widehat{\\phi}_{S}-\\widehat{f}_{S})|.\n\\end{aligned}\n $$\n  Now $|\\widehat{\\mu}^{\\mathrm{pl}}_{S}(\\widehat{f}_{S}-\\widehat{\\phi}_{S})|\\le |\\widehat{\\mu}^{\\mathrm{pl}}_S(\\widehat{\\psi}_S)|\\le \\epsilon$, and $|\\widehat{\\mu}_k(\\widehat{\\phi}_{S})- \\widehat{\\mu}^{\\mathrm{pl}}_{S}(\\widehat{\\phi}_{S})|\\le \\epsilon$ for $k\\gg0$ by \\eqref{e:aut-plan-phi}.\n  Finally $\\widehat{\\mu}_k$ is a positive measure since the highest weight of $\\xi$ is regular, and we get\n   $$\n     |\\widehat{\\mu}_k(\\widehat{f}_{S}-\\widehat{\\phi}_{S})|\\le \\widehat{\\mu}_k(|\\widehat{f}_{S}-\\widehat{\\phi}_{S}|)\\le \\widehat{\\mu}_k(\\widehat{\\psi}_S).\n   $$\n   (To see the positivity of $\\widehat{\\mu}_k$, notice that $\\widehat{\\mu}_k(\\widehat{f}_{S}-\\widehat{\\phi}_{S})$ is unraveled via \\eqref{e:a(pi)-general} and \\eqref{e:defn-of-mu} as a sum of $(\\widehat{f}_{S}-\\widehat{\\phi}_{S})(\\pi)$ with coefficients having nonnegative signs. This is because $\\chi_{{\\rm EP}}(\\pi_\\infty\\otimes \\xi)$ is either 0 or $(-1)^{q(G)}$ when $\\xi$ has regular highest weight, cf. \\S\\ref{sub:EP}.)\n    According to \\eqref{e:aut-plan-phi}, $\\lim_{k\\rightarrow \\infty}\\widehat{\\mu}_k(\\widehat{\\psi}_S)=\\widehat{\\mu}^{\\mathrm{pl}}_S(\\widehat{\\psi}_S)\\le \\epsilon$. In particular $|\\widehat{\\mu}_k(\\widehat{f}_{S}-\\widehat{\\phi}_{S})|\\le 2\\epsilon$ for $k\\gg0$. The proof is complete.\n\\end{proof}\n\n\n\n\n\\begin{rem}\n  If $G$ is anisotropic modulo center over $F$ so that the trace formula for compact quotients is available, or if a further local assumption at finite places is imposed so as to avail the simple trace formula, the regularity condition on $\\xi$ can be removed by an argument of De George-Wallach and Clozel (\\cite{dGW78}, \\cite{Clo86}). The main point is to show that the contribution of ($\\xi$-cohomological) non-tempered representations at $\\infty$ to the trace formula is negligible compared to the contribution of discrete series. Their argument requires some freedom of choice of test functions at $\\infty$, so it breaks down in the general case since one has to deal with new terms in the trace formula which disappear when Euler-Poincar\\'{e} functions are used at $\\infty$. In other words, it seems necessary to prove analytic estimates on more terms (if not all terms) in the trace formula than we did in order to get rid of the assumption on $\\xi$. (This remark also applies to the same condition on $\\xi$ in \\S\\ref{sub:ST-theorem} and \\S\\ref{sub:general-functions-S_0} for level aspect families.) We may return to this issue in future work.\n\\end{rem}\n\n\n\\begin{rem}\n  In the case of level aspect families, \\cite[Thm 4.3]{Shi-Plan} assumes that\n  the level subgroups form a chain of decreasing groups whose intersection is the trivial group.\n  The above corollary deals with some new cases as it assumes only that ${\\mathbb N}({\\mathfrak n}_k)\\rightarrow \\infty$.\n\n\\end{rem}\n\n\\begin{cor}\\label{c:estimate-F_k}\n  Keep assuming that $S_1=\\emptyset$.\n Let $(U_k^{S,\\infty},\\xi_k)=(K^{S,\\infty}({\\mathfrak n}_k),\\xi)$ or $(U^{S,\\infty},\\xi_k)$\nin Example \\ref{ex:level-varies} or \\ref{ex:wt-varies}, respectively, but prescribe local conditions at $S$\nby $\\widehat{f}_{S}$ rather than $\\phi_{S}$.\nThen  $$\\lim_{k\\rightarrow\\infty}\\frac{\\mu^{\\mathrm{can}}(U_k^{S,\\infty})}{\\tau'(G)\\dim \\xi_k}  |\\mathcal{F}_k| = \\widehat{\\mu}^{\\mathrm{pl}}_{S}(\\widehat{f}_{S}).$$\n\\end{cor}\n\n\\begin{proof}\n  The corollary results from Corollary \\ref{c:Plan-density} since\n$$ \\frac{\\mu^{\\mathrm{can}}(U_k^{S,\\infty})}{\\tau'(G)\\dim \\xi_k} |\\mathcal{F}_k|=\n \\frac{\\mu^{\\mathrm{can}}(U_k^{S,\\infty})}{\\tau'(G)\\dim \\xi_k}\\sum_{\\pi\\in {\\mathcal{AR}}_{{\\rm disc},\\chi_k}(G)} a_{\\mathcal{F}_k}(\\pi) =\n \\widehat{\\mu}_{\\mathcal{F}_k,\\emptyset}(\\widehat{f}_S).$$\n\\end{proof}\n\n\\subsection{Application to the Sato-Tate conjecture for families}\\label{sub:ST-theorem}\n\n  As an application of Theorems \\ref{t:level-varies} and \\ref{t:weight-varies},\n  we are about to fulfill the promise of \\S\\ref{sub:ST-families}\n  by showing that the Satake parameters in the automorphic families\n  $\\{\\mathcal{F}_k\\}$ are equidistributed\n  according to the Sato-Tate measure in a suitable sense\n  (cf. Conjecture \\ref{c:ST-families}).\n\n\n  The notation and convention of \\S\\ref{s:Sato-Tate} are retained here.\n  Let $\\theta\\in {\\mathscr C}(\\Gamma_1)$ and $\\widehat{f}\\in \\mathcal{F}(\\widehat{T}_{c,\\theta}\/\\Omega_{c,\\theta})$.\n  For each $v\\in {\\mathcal V}_F(\\theta)$,\n  the image of $\\widehat{f}$ in $\\mathcal{F}(G(F_v)^{\\wedge,\\mathrm{ur}})$ via \\eqref{e:Sauvageot}\n  will be denoted $\\widehat{f}_v$.\n\n\n\\begin{thm}\\label{t:Sato-Tate-level} (level aspect)\n\n  Pick any $\\theta\\in {\\mathscr C}(\\Gamma_1)$ and let $\\{v_j\\}_{j\\ge 1}$ be a sequence in ${\\mathcal V}_F(\\theta)$ such that\n  $q_{v_j}\\rightarrow \\infty$ as $j\\rightarrow \\infty$.   Suppose that \\begin{itemize}\\item $\\widehat{\\mu}^{\\mathrm{pl}}_{S_0}(\\widehat{\\phi}_{S_0})\\neq 0$ and\n  \\item $\\xi$ has regular highest weight. \\end{itemize}\n  Then for every $\\widehat{f}\\in \\mathcal{F}(\\widehat{T}_{c,\\theta}\/\\Omega_{c,\\theta})$,\n  $$\\lim_{(j,k)\\rightarrow \\infty} \\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k,v_j}(\\widehat{f}_{v_j})\n  = \\widehat{\\mu}^{\\mathrm{ST}}_{\\theta}(\\widehat{f})$$\n  where the limit is taken over $(j,k)$ subject to the following conditions:\n \\begin{itemize}\n \\item ${\\mathbb N}({\\mathfrak n}_k) q_{v_j}^{-B_\\Xi m \\kappa}\\ge c_\\Xi^{-1}$,\n \\item $v_j\\nmid {\\mathfrak n}_k$,\n \\item $q_{v_j}^{N} {\\mathbb N}({\\mathfrak n}_k)^{-1}\\rightarrow 0$ for all $N>0$.\n \\end{itemize}\n\n\\end{thm}\n\n\\begin{proof}\n  Fix $\\widehat{f}$. We are done if $\\limsup_{(j,k)\\rightarrow \\infty} |\\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k,v_j}(\\widehat{f}_{v_j})- \\widehat{\\mu}^{\\mathrm{ST}}_{\\theta}(\\widehat{f})|\\le 4\\epsilon$ for every $\\epsilon>0$. By Proposition \\ref{p:lim-of-Plancherel},\n  $|\\widehat{\\mu}^{\\mathrm{pl}}_{v_j}(\\widehat{f}_{v_j})-\\widehat{\\mu}^{\\mathrm{ST}}_{\\theta}(\\widehat{f})|\\le\\epsilon$ for $j\\gg0$. So it is enough to show that\n   \\begin{equation}\\label{e:extending-at-v_j-0}\\limsup_{(j,k)\\rightarrow \\infty} |\\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k,v_j}(\\widehat{f}_{v_j})- \\widehat{\\mu}^{\\mathrm{pl}}_{v_j}(\\widehat{f}_{v_j})|\\le3\\epsilon.\\end{equation}\n\n\t For every $j\\ge1$, Proposition \\ref{p:unr-density} allows us to find $\\phi_{v_j},\\psi_{v_j}\\in {\\mathcal{H}}^{\\mathrm{ur}}(G(F_{v_j}))^{\\le \\kappa}$ such that $|\\widehat{f}_{v_j}-\\widehat{\\phi}_{v_j}|\\le \\widehat{\\psi}_{v_j}$ on $G(F_{v_j})^{\\wedge}$ and $\\widehat{\\mu}^{\\mathrm{pl}}_{v_j}(\\widehat{\\psi}_{v_j})\\le \\epsilon$.\n  For each $j\\ge 1$,\n  \\begin{equation}\\label{e:extending-at-v_j}\n\\begin{aligned}\n|\\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k,v_j}(\\widehat{f}_{v_j})- \\widehat{\\mu}^{\\mathrm{pl}}_{v_j}(\\widehat{f}_{v_j})| & \\le  |\\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k,v_j}(\\widehat{f}_{v_j}-\\widehat{\\phi}_{v_j})|\\\\ & + |\\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k,v_j}(\\widehat{\\phi}_{v_j})- \\widehat{\\mu}^{\\mathrm{pl}}_{v_j}(\\widehat{\\phi}_{v_j})| + |\\widehat{\\mu}^{\\mathrm{pl}}_{v_j}(\\widehat{\\phi}_{v_j}-\\widehat{f}_{v_j})|.\n\\end{aligned}\n\\end{equation}\n  Since $\\widehat{\\mu}^{\\mathrm{pl}}_{v_j}$ is a positive measure,\n $$|\\widehat{\\mu}^{\\mathrm{pl}}_{v_j}(\\widehat{\\phi}_{v_j}-\\widehat{f}_{v_j})|\\le  \\widehat{\\mu}^{\\mathrm{pl}}_{v_j}(|\\widehat{\\phi}_{v_j}-\\widehat{f}_{v_j}|)\n \\le \\widehat{\\mu}^{\\mathrm{pl}}_{v_j}(\\widehat{\\psi}_{v_j})\\le \\epsilon.$$\n Theorem \\ref{t:level-varies} and the assumptions of the theorem imply that for sufficiently large $(j,k)$,\n$|\\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k,v_j}(\\widehat{\\phi}_{v_j})- \\widehat{\\mu}^{\\mathrm{pl}}_{v_j}(\\widehat{\\phi}_{v_j})|\\le \\epsilon$. So we will be done if for sufficiently large $(j,k)$,\n  \\begin{equation}\\label{e:extending-at-v_j-2}\n  |\\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k,v_j}(\\widehat{f}_{v_j}-\\widehat{\\phi}_{v_j})| \\le \\epsilon.\n\\end{equation}\n Arguing as in the proof of Corollary \\ref{c:Plan-density} we deduce the following: when $\\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k,v_j}(\\widehat{f}_{v_j}-\\widehat{\\phi}_{v_j})$ is unraveled as a sum over $\\pi$ (cf. \\eqref{e:a(pi)-general} and \\eqref{e:defn-of-mu}), each summand is $\\widehat{\\phi}_{S_0}(\\pi_{S_0})(\\widehat{f}_{v_j}-\\widehat{\\phi}_{v_j})(\\pi_{v_j})$ times a nonnegative real number. (This uses the regularity assumption on $\\xi$. Certainly the absolute value of the sum does not get smaller when every summand is replaced with (something greater than or equal to) its absolute value, i.e.\n $$|\\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k,v_j}(\\widehat{f}_{v_j}-\\widehat{\\phi}_{v_j})|\n \\le \\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k(|\\widehat{\\phi}_{S_0}|),v_j}(|\\widehat{f}_{v_j}-\\widehat{\\phi}_{v_j}|)\n \\le \\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k(|\\widehat{\\phi}_{S_0}|),v_j}(\\widehat{\\psi}_{v_j}).$$\n  Now choose $\\phi'_{S_0}\\in C^\\infty_c(G(F_{S_0}))$ according to Lemma \\ref{l:bounding-function-by-pos-function} so that $|\\phi_{S_0}(\\pi_{S_0})|\\le \\phi'_{S_0}(\\pi_{S_0})$ for every $\\pi_{S_0}\\in G(F_{S_0})^{\\wedge}$. Then\n$$\\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k(|\\widehat{\\phi}_{S_0}|),v_j}(\\widehat{\\psi}_{v_j})\\le \\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k(\\phi'_{S_0}),v_j}(\\widehat{\\psi}_{v_j}).$$\nTheorem \\ref{t:level-varies} applied to $\\widehat{\\psi}_{v_j}$ and the inequality $\\widehat{\\mu}^{\\mathrm{pl}}_{v_j}(\\widehat{\\psi}_{v_j})\\le \\epsilon$ imply that $$\\limsup_{(j,k)\\rightarrow\\infty} \\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k(\\phi'_{S_0}),v_j}(\\widehat{\\psi}_{v_j})\\le \\epsilon.$$ This concludes the proof of \\eqref{e:extending-at-v_j-2}, thus also \\eqref{e:extending-at-v_j-0}.\n\n\n\\end{proof}\n\n\n\n\\begin{thm}\\label{t:Sato-Tate-weight} (weight aspect) Let $\\theta\\in {\\mathscr C}(\\Gamma_1)$ and $\\widehat{\\phi}_{S_0}\\in C^\\infty_c(G(F_{S_0}))$. Suppose that $\\{v_j\\}_{j\\ge 1}$ is a sequence in ${\\mathcal V}_F(\\theta)$ such that $q_{v_j}\\rightarrow \\infty$ as $j\\rightarrow \\infty$ and that $\\widehat{\\mu}^{\\mathrm{pl}}_{S_0}(\\widehat \\phi_{S_0})\\neq 0$.\n Then for every $\\widehat{f}\\in \\mathcal{F}(\\widehat{T}_{c,\\theta}\/\\Omega_{c,\\theta})$,\n  $$\\lim_{(j,k)\\rightarrow \\infty} \\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k,v_j}(\\widehat{f}_{v_j})\n  = \\widehat{\\mu}^{\\mathrm{ST}}_{\\theta}(\\widehat{f})$$\n  if $q_{v_j}^{N} m(\\xi_k)^{-1}\\rightarrow 0$ as $k\\rightarrow \\infty$ for any integer $N\\ge1$.\n\n\\end{thm}\n\n\\begin{proof}\n  Same as above, except that Theorem \\ref{t:weight-varies} is used\n  instead of Theorem \\ref{t:level-varies}.\n\\end{proof}\n\n\\begin{rem}\n  As we have mentioned in \\S\\ref{sub:ST-families},\n  Theorems \\ref{t:Sato-Tate-level} and \\ref{t:Sato-Tate-weight}\n  indicate that $\\{\\mathcal{F}_k\\}_{\\ge1}$ are ``general''\n  families of automorphic representations in the sense of\n  Conjecture \\ref{c:ST-families}.\n\\end{rem}\n\n\\begin{cor}\n  In the setting of Theorem \\ref{t:Sato-Tate-level} or \\ref{t:Sato-Tate-weight},\n  suppose in addition that $|\\mathcal{F}_k|\\neq 0$ for all $k\\ge 1$. Then\n $$\\lim_{(j,k)\\rightarrow \\infty} \\widehat{\\mu}^{\\mathrm{count}}_{\\mathcal{F}_k,v_j}(\\widehat{f}_{v_j})\n  = \\widehat{\\mu}^{\\mathrm{ST}}_{\\theta}(\\widehat{f}).$$\n\\end{cor}\n\\begin{proof}\n  Follows from Corollary \\ref{c:estimate-F_k} and the two preceding theorems (cf. Remark \\ref{r:|F|}).\n\\end{proof}\n\n\\begin{rem}\n  The assumption that $|\\mathcal{F}_k|\\neq 0$ is almost automatically satisfied.\n  Corollary \\ref{c:estimate-F_k} and the assumption that $\\widehat{\\mu}^{\\mathrm{pl}}_{S_0}(\\widehat{\\phi}_{S_0})\\neq 0$ imply that\n  $|\\mathcal{F}_k|\\neq 0$ for any sufficiently large $k$.\n\n\\end{rem}\n\n\n\\subsection{More general test functions at $S_0$}\\label{sub:general-functions-S_0}\n\n  So far we worked primarily with families of Examples \\ref{ex:level-varies}\n  and \\ref{ex:wt-varies}.\n  We wish to extend Theorems \\ref{t:Sato-Tate-level}\n  and \\ref{t:Sato-Tate-weight} when the local condition at $S_0$\nis given by $\\widehat{f}_{S_0}$,\nwhich may not be of the form $\\widehat{\\phi}_{S_0}$\nfor any $\\phi_{S_0}\\in C^\\infty_c(G(F_{S_0}))$\n(cf. Example \\ref{ex:example-for-f_S} and Remark \\ref{r:why-S_0}).\n\n\n\n\\begin{cor}\\label{c:extended} Let $\\theta\\in {\\mathscr C}(\\Gamma_1)$ and let $\\{v_j\\}_{j\\ge 1}$ be a sequence of places in ${\\mathcal V}_F(\\theta)$ such that\n  $q_{v_j}\\rightarrow\\infty$ as $j\\rightarrow\\infty$.\n  Consider $\\widehat{\\mu}_{\\mathcal{F}_k,v_j}$ where $$\\mathcal{F}_k=\\left\\{ \\begin{array}{cl}\n  \\mathcal{F}(K^{S,\\infty}({\\mathfrak n}_k),\\widehat{f}_{S_0},v_j,\\xi) & \\mathrm{level~aspect,~or}\\\\\n  \\mathcal{F}(U^{S,\\infty},\\widehat{f}_{S_0}, v_j,\\xi_k) & \\mathrm{weight~aspect}\n  \\end{array}\\right.$$\n  satisfying the conditions of Theorem \\ref{t:Sato-Tate-level} or\n  Theorem \\ref{t:Sato-Tate-weight}, respectively.\n Then  $$\\lim_{(j,k)\\rightarrow \\infty} \\widehat{\\mu}^{\\natural}_{\\mathcal{F}_k,v_j}(\\widehat{f}_{v_j})\n  = \\widehat{\\mu}^{\\mathrm{ST}}_{\\theta}(\\widehat{f})$$\n  where the limit is taken as in Theorem \\ref{t:Sato-Tate-level} (resp.\n  Theorem \\ref{t:Sato-Tate-weight}).\n\\end{cor}\n\n\\begin{proof}\n\n\n\n  The proof is reduced to the case of $\\widehat{\\phi}$ and $\\widehat{\\phi}_{v_j}$ in place of\n$\\widehat{f}$ and $\\widehat{f}_{v_j}$, as in the proof of Theorem\n\\ref{t:Sato-Tate-level}.   We can decompose $\\widehat{f}=\\widehat{f}^+ +\\widehat{f}^-$ with\n  $\\widehat{f}^+,\\widehat{f}^-\\in  \\mathcal{F}(\\widehat{T}_{c,\\theta}\/\\Omega_{c,\\theta})$ such that $\\widehat{f}^+$ and $\\widehat{f}^-$ are nonnegative everywhere.\n  The corollary for $\\widehat{f}$ is proved as soon as it is proved for $\\widehat{f}^+$ and $\\widehat{f}^-$. Thus we may assume that $\\widehat{f}\\ge 0$ from now on.\n\n  Fix any choice of $\\epsilon>0$.\n  Proposition \\ref{p:density} ensures the existence of\n  $\\phi_{S_0},\\psi_{S_0}\\in C^\\infty_c(G(F_{S_0}))$ such that\n  $\\widehat{\\mu}^{\\mathrm{pl}}_{S_0}(\\psi_{S_0})\\le \\epsilon$ and\n  $|\\widehat{f}_{S_0}(\\pi_{S_0})-\\widehat{\\phi}_{S_0}(\\pi_{S_0})|\\le \\widehat{\\psi}_{S_0}(\\pi_{S_0})$\n  for all $\\pi_{S_0}\\in G(F_{S_0})^\\wedge$. Of course we can guarantee in addition that $\\widehat{\\mu}^{\\mathrm{pl}}_{S_0}(\\widehat{\\phi}_{S_0})\\neq 0$. Put\n$\\mathcal{F}_k(\\widehat{\\phi}_{S_0})=\\mathcal{F}(K^{S,\\infty}({\\mathfrak n}_k),\\widehat{\\phi}_{S_0},\n  v_j,\\xi)$ (resp. $\\mathcal{F}_k(\\widehat{\\phi}_{S_0})=\\mathcal{F}(U^{S,\\infty},\\widehat{\\phi}_{S_0},\n  v_j,\\xi_k)$). Likewise we define $\\mathcal{F}_k(\\widehat{\\psi}_{S_0})$ and so on. Then (cf. a similar step in the proof of Theorem \\ref{t:Sato-Tate-level})\n\\begin{equation*}\\begin{aligned}\n| \\widehat{\\mu}_{\\mathcal{F}_k,v_j}(\\widehat{f}_{v_j})-\\widehat{\\mu}^{\\mathrm{pl}}_{S_0\\cup\\{v_j\\}}(\\widehat{f}_{S_0}\\widehat{f}_{v_j})|&\\le |\\widehat{\\mu}_{\\mathcal{F}_k(\\widehat{\\phi}_{S_0}),v_j}(\\widehat{f}_{v_j})-\\widehat{\\mu}^{\\mathrm{pl}}_{S_0\\cup\\{v_j\\}}(\\widehat{\\phi}_{S_0}\\widehat{f}_{v_j})| \\\\\n& + \\widehat{\\mu}_{\\mathcal{F}_k(|\\widehat{f}_{S_0}-\\widehat{\\phi}_{S_0}|)}(\\widehat{f}_{v_j})\n + \\widehat{\\mu}^{\\mathrm{pl}}_{S_0\\cup\\{v_j\\}}(|\\widehat{f}_{S_0}-\\widehat{\\phi}_{S_0}|\\widehat{f}_{v_j})\n \\end{aligned}\n \\end{equation*}\n  The first term on the right side tends to 0 as $(j,k)\\rightarrow \\infty$ by Theorems \\ref{t:Sato-Tate-level} and \\ref{t:Sato-Tate-weight}. The last term is bounded by $\\widehat{\\mu}^{\\mathrm{pl}}_{S_0\\cup\\{v_j\\}}(\\widehat{\\psi}_{S_0}\\widehat{f}_{v_j})\\le \\epsilon\\widehat{\\mu}^{\\mathrm{pl}}_{v_j}(\\widehat{f}_{v_j})$ using the fact that $\\widehat{\\mu}^{\\mathrm{pl}}_{S_0}$ is a positive measure. In order to bound the second term, recall that we are either in the weight aspect, or in the level aspect with regular highest weight for $\\xi$. Then $a_{\\mathcal{F}_k(|\\widehat{f}_{S_0}-\\widehat{\\phi}_{S_0}|)}(\\pi)$ is a nonnegative multiple of $|\\widehat{f}_{S_0}(\\pi_{S_0})-\\widehat{\\phi}_{S_0}(\\pi_{S_0})|$ as in the proof of Theorem \\ref{t:Sato-Tate-level}. Thus\n $$|\\widehat{\\mu}_{\\mathcal{F}_k(|\\widehat{f}_{S_0}-\\widehat{\\phi}_{S_0}|)}(\\widehat{f}_{v_j})|\\le\\widehat{\\mu}_{\\mathcal{F}_k(\\widehat{\\psi}_{S_0})}\n (\\widehat{f}_{v_j}),$$\n  and the right hand side tends to 0 as $(j,k)\\rightarrow \\infty$ again by Theorems \\ref{t:Sato-Tate-level} and \\ref{t:Sato-Tate-weight}.\n  Hence we have shown that\n  $$\\limsup_{(j,k)\\rightarrow\\infty} | \\widehat{\\mu}_{\\mathcal{F}_k,v_j}(\\widehat{f}_{v_j})-\\widehat{\\mu}^{\\mathrm{pl}}_{S_0\\cup\\{v_j\\}}(\\widehat{f}_{S_0}\\widehat{f}_{v_j})| \\le \\epsilon\\widehat{\\mu}^{\\mathrm{pl}}_{v_j}(\\widehat{f}_{v_j}).$$\n  Since $\\lim\\limits_{j\\rightarrow\\infty}\\widehat{\\mu}^{\\mathrm{pl}}_{v_j}(\\widehat{f}_{v_j})=\\widehat{\\mu}^{\\mathrm{ST}}_{\\theta}(\\widehat{f})$ and we are free to choose $\\epsilon>0$,\n  we deduce that $\\lim\\limits_{(j,k)\\rightarrow\\infty} \\widehat{\\mu}_{\\mathcal{F}_k,v_j}(\\widehat{f}_{v_j})=\\widehat{\\mu}^{\\mathrm{pl}}_{S_0}(\\widehat{f}_{S_0})\\widehat{\\mu}^{\\mathrm{ST}}_{\\theta}(\\widehat{f})$.\n\n\\end{proof}\n\n\\begin{rem}\n  It would be desirable to improve Theorems \\ref{t:level-varies} and \\ref{t:weight-varies} similarly\n  by prescribing conditions at $S_0$ in terms of $\\widehat{f}_{S_0}$ rather than the less general\n  $\\widehat{\\phi}_{S_0}$. Unfortunately the argument proving Corollary \\ref{c:extended} does not carry over.\n  For instance in the case of Theorem \\ref{t:level-varies}, one should know in addition that\n  the multiplicative constant implicit in  $O(q_{S_1}^{A_{\\mathrm{lv}}+B_{\\mathrm{lv}} \\kappa} {\\mathbb N}({\\mathfrak n}_k)^{-C_{\\mathrm{lv}}})$\n  is bounded as a sequence of $\\widehat{\\phi}_{S_0}$\n  approaches $\\widehat{f}_{S_0}$.\n\\end{rem}\n\n\n\n\n\\section{Langlands functoriality}\\label{s:langlands}\n\n\n  Let $r:{}^L G\\rightarrow \\GL_d({\\mathbb C})$ be a representation of ${}^L G$. Let $\\pi\\in {\\mathcal{AR}}_{{\\rm disc},\\chi}(G)$ be\nsuch that with $\\pi_v \\in \\Pi_{{\\rm disc}}(\\xi_v^\\vee)$ for each $v|\\infty$\n (recall the notation from \\S\\ref{sub:st-disc} and \\S\\ref{sub:aut-rep}).\nThe Langlands correspondence for $G(F_v)$ (\\cite{Lan88}) associates an $L$-parameter $\\varphi_{\\xi^\\vee_v}:W_{\\mathbb R}\\rightarrow {}^L G$\nto the $L$-packet $\\Pi_{{\\rm disc}}(\\xi_v^\\vee)$, cf. \\S\\ref{sub:st-disc}.\n The following asserts the existence\nof the functorial lift of $\\pi$ under $r$ as predicted by the Langlands functoriality principle.\n\n\\begin{hypo}\\label{hypo:functorial-lift}\n  There exists an automorphic representation $\\Pi$ of $\\GL_d({\\mathbb A}_F)$ such that\n\\begin{enumerate}\n\\item $\\Pi$ is isobaric,\n\\item $\\Pi_v=r_*(\\pi_v)$ (defined in \\eqref{e:r_*}) when $G$, $r$ and $\\pi$ are unramified at $v$,\n\\item $\\Pi_v$ corresponds to $r\\varphi_{\\xi^\\vee_v}$ via the Langlands correspondence for $\\GL_d(F_v)$\nfor all $v|\\infty$.\n\\end{enumerate}\n\\end{hypo}\n\n  If $\\Pi$ as above exists then it is uniquely determined by (i) and (ii) thanks to the strong multiplicity\n  one. Moreover\n  \\begin{lem}\\label{lem:tempered}\n  Hypothesis \\ref{hypo:functorial-lift}.(iii) implies that\n  $\\Pi_v$ is tempered for all $v|\\infty$.\n\\end{lem}\n\n\\begin{proof}\n  Recall the following general fact from \\cite[\\S3, (vi)]{Lan88}:\n  Let $\\varphi$ be an $L$-parameter for a real reductive group and $\\Pi(\\varphi)$\n  its corresponding $L$-packet. Then $\\varphi$ has relatively compact image if and only if\n  $\\Pi(\\varphi)$ contains a tempered representation if and only if $\\Pi(\\varphi)$\n  contains only tempered representations. In our case\n  this implies that $\\varphi_{\\xi^\\vee_v}$ has relatively compact image for every $v|\\infty$, and the continuity\n  of $r$ shows that the image of $r\\varphi_{\\xi^\\vee_v}$ is also relatively compact. The lemma follows.\n\\end{proof}\n\nAs before let $(\\widehat{B},\\widehat{T},\\{X_\\alpha\\}_{\\alpha\\in \\Delta^\\vee})$ denote the ${\\rm Gal}(\\overline{F}\/F)$-invariant\n  splitting datum for $\\widehat{G}$.\n  Recall that $\\lambda_{\\xi_v^\\vee}\\in X^*(\\widehat{T})^+$ designates the highest weight for\n  $\\xi_v^\\vee$.\n  Then $\\varphi_{\\xi^\\vee_v}|_{W_{\\mathbb C}}$ is described as\n  $$ \\varphi_{\\xi^\\vee_v}(z)=\\left((z\/\\overline{z})^{\\rho+\\lambda_{\\xi_v^\\vee}},z\\right)\n  \\in \\widehat{G}\\times W_{\\mathbb C},\\quad \\forall z\\in W_{\\mathbb C}={\\mathbb C}^\\times.$$\n  It is possible to extend $\\varphi_{\\xi^\\vee_v}|_{W_{\\mathbb C}}$ to the whole of $W_{\\mathbb R}$ but this does not concern us.\n (The interested reader may consult pp.183-184 of \\cite{Kot90} for instance.)\n  Let $\\widehat{\\mathbb{T}}$ be a maximal torus of $\\GL_d({\\mathbb C})$ containing the image\n  $r(\\widehat{T})$, and $\\widehat{\\mathbb{B}}$ a Borel subgroup containing $\\widehat{\\mathbb{T}}$.\n  Write $r|_{\\widehat{G}}=\\oplus_{i\\in I} r_i$ as a sum of irreducible $\\widehat{G}$-representations.\n  For each $i\\in I$, denote by $\\lambda(r_i)\\in X^*(\\widehat{T})$ the $\\widehat{B}$-positive\n  highest weight for $r_i$. Write\n$\\lambda(r_i)=\\lambda_0(r_i)+\\sum_{\\alpha\\in \\Delta} a(r_i,\\alpha)\\cdot \\alpha^\\vee$ for\n$\\lambda_0(r_i)\\in X_*(Z(G))_{ {\\mathbb Q}}$ and $a(r_i,\\alpha)\\in {\\mathbb Q}_{\\ge 0}$.\n  Put $|\\lambda(r_i)|:=\\sum_{\\alpha\\in \\Delta}  a(r_i,\\alpha)$ and\n$$M(\\xi_v):=\\max_{\\alpha\\in \\Delta} \\langle \\alpha,\\lambda_{\\xi^\\vee_v}\\rangle,~\nM(r):=\\max_{i\\in I} |\\lambda(r_i)|.$$\nSimilarly define $m(\\xi_v)$ and $m(r)$ by using minima in place of maxima.\nWe are interested in the case where $\\lambda_0(r_i)$ is trivial for every $i\\in I$.\nThis is automatically true if $Z(G)$ is finite. (Recall that we consistently assume $Z(G)=1$\nin the weight aspect.)\n\n\n\\begin{lemma}\\label{l:Cpi} Suppose that $\\lambda_0(r_i)$ is trivial for every $i\\in I$.\n  Hypothesis \\ref{hypo:functorial-lift}.(iii) implies that for each $v|\\infty$,\n  $$ (2+m(r)m(\\xi_v))^{|I|}\\le C(\\Pi_v)\\le (3+2M(r)M(\\xi_v))^{d}.$$\nIn particular if $Z(G)$ is finite, then the following holds for any fixed $L$-morphism $r$.\n\\begin{equation*}\n 1+ m(\\xi_v)\n  \\ll_r C(\\Pi_v) \\ll_r\n  M(\\xi_v)^{d}\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\n  First we recall a general fact about archimedean $L$-factors.\n  Let $\\varphi:W_{\\mathbb R}\\rightarrow \\GL_N({\\mathbb C})$ be a tempered $L$-parameter and decompose\n  $\\varphi|_{W_{\\mathbb C}}$ into $\\GL_1$-parameters as $\\varphi|_{W_{\\mathbb C}}=\\oplus_{k=1}^N \\chi_k$.\n    The archimedean $L$-factor associated with $\\varphi$ may be written in the form (cf. \\eqref{def:Lv-arch})\n  \\begin{equation}\\label{e:pf-10.3-1}L(s,\\varphi)=\\prod_{k=1}^N \\Gamma_{{\\mathbb R}}(s-\\mu_k(\\varphi)).\\end{equation}\n  For each $k$ assume that $\\chi_k(z)=(z\/\\overline{z})^{a_k}$ for some $a_k\\in \\frac{1}{2}{\\mathbb Z}$.\n  Then we have for every $1\\le k\\le N$,\n  $\\mu_k(\\varphi)\\in \\frac{1}{2}{\\mathbb Z}_{\\le 0}$ and, after reordering $\\mu_k(\\varphi)$'s if necessary,\n  \\begin{equation}\\label{e:pf-10.3-2}|a_k|\\le |\\mu_k(\\varphi)| \\le |a_k|+1.\\end{equation}\n  Indeed this comes from inspecting the definition of local $L$-factors as in of \\cite{Tat79}*{3.1,3.3} for instance.\n  (Use \\cite{Tat79}*{3.1} if $a_k=0$ and \\cite{Tat79}*{3.3} otherwise.)\n\n  Returning to the setup of the lemma, we have by definition $L(s,\\Pi_v)=L(s,r\\varphi_{\\xi_v})$. For each $i\\in I$ we consider the composite complex $L$-parameter\n  $$W_{\\mathbb C}\\stackrel{\\varphi_{\\xi_v}|_{W_{\\mathbb C}}}{\\rightarrow} \\widehat{G}\\times W_{\\mathbb C}\n  \\stackrel{(r_i,1)}{\\rightarrow} \\GL_{\\dim r_i}({\\mathbb C})$$\n  decompose it as $\\oplus_{j=1}^{\\dim r_i} \\chi_{i,j}$.\n  We can find $a_{i,j}\\in \\frac{1}{2}{\\mathbb Z}$ such that\n  $\\chi_{i,j}(z)=(z\/\\overline{z})^{a_{i,j}}$.\n  For each $i$, the highest weight theory tells us that\n  $a_{i,j}=\\langle \\rho+\\lambda_{\\xi^\\vee_v}, \\lambda(r_i)\\rangle\\ge 0$ for one $j$\n  and $|a_{i,j'}|\\le a_{i,j}$ for the other $j'\\neq j$.\n  By \\eqref{e:pf-10.3-1} and \\eqref{e:pf-10.3-2}, the analytic conductor for $\\Pi_v$ (introduced in \\S\\ref{sec:pp:Lfn})\n  satisfies\n  \\begin{equation*}\n\t\\begin{aligned}\n  C(\\Pi_v)&=\\prod_{k=1}^{d}(2+|\\mu_k(\\Pi_v)|)\\le\n\\prod_{i\\in I} \\prod_{j=1}^{\\dim r_i} (3+|a_{i,j}|)\\\\\n&\\le \\prod_{i\\in I} (3+\\langle \\rho+\\lambda_{\\xi^\\vee_v}, \\lambda(r_i)\\rangle)^{\\dim r_i}\n.\n\\end{aligned}\n\\end{equation*}\nFurther $\\langle \\rho+\\lambda_{\\xi^\\vee_v}, \\lambda(r_i)\\rangle=\n\\langle \\rho,\\lambda(r_i)\\rangle+ \\langle \\lambda_{\\xi^\\vee_v}, \\lambda(r_i)\\rangle\n\\le |\\lambda(r_i)|+|\\lambda(r_i)|M(\\xi_v)\\le M(r)(1+M(\\xi_v))$. Hence\n$$C(\\Pi_v)\\le \\prod_{i\\in I} ((3+M(r)(1+M(\\xi_v)))^{\\dim r_i}\n=((3+M(r)(1+M(\\xi_v)))^{d}.$$\n\n  Now we establish a lower bound for $C(\\Pi_v)$. For each $i$, we apply \\eqref{e:pf-10.3-2}\n  to the unique $j=j(i)$ such that $a_{i,j}=\\langle \\rho+\\lambda_{\\xi^\\vee_v}, \\lambda(r_i)\\rangle$.\n  Then\n  \\begin{equation*}\n\t\\begin{aligned}\n  C(\\Pi_v)&\\ge \\prod_{i\\in I}   (2+|a_{i,j(i)}|)=\n   \\prod_{i\\in I} (2+\\langle \\rho+\\lambda_{\\xi^\\vee_v}, \\lambda(r_i)\\rangle)\\\\\n  &\\ge  (2+m(r)(1+m(\\xi_v)))^{|I|}.\n\\end{aligned}\n\\end{equation*}\n\n\n\\end{proof}\n\n\\section{Statistics of low-lying zeros}\\label{sec:zeros}\n\nAs explained in the introduction an application of the quantitative Plancherel Theorems~\\ref{t:level-varies} and~\\ref{t:weight-varies} is to the study the distribution of the low-lying zeros of families of $L$-functions $\\Lambda(s,\\Pi)$. The purpose of this section is to state the main results and make our working hypothesis precise.\n\n\\subsection{The random matrix models}\\label{sec:zeros:matrix} For the sake of completness we recall briefly the limiting $1$-level density of normalized eigenvalues. We consider the three Symmetry Types $\\mathcal{G}(N)=SO(2N), U(N), USp(2N)$. For each integer $N\\ge 1$ these groups are endowed with their Haar probability measure. For all matrices $A\\in \\mathcal{G}(N)$ we have a sequence $\\vartheta_j=\\vartheta_j(A)$ of normalized angles~\\cite{book:KS}\n\\begin{equation}\\label{def:theta}\n0\\le \\vartheta_1 \\le \\vartheta_2 \\le \\cdots \\le \\vartheta_N \\le N.\n\\end{equation}\nNamely the eigenvalues of $A\\in U(N)$ are given by $e(\\frac{\\vartheta_j}{N})=e^{2i\\pi\\vartheta_j\/N}$. The eigenvalues of $A\\in USp(2N)$ or $A\\in SO(2N)$ occur in conjugate pairs and are given by $e(\\pm \\frac{\\vartheta_j}{2N})$.\n\nThe mean spacing of the sequence~\\eqref{def:theta} is about one. The $1$-level density is defined by\n\\begin{equation*}\nW_{\\mathcal{G}(N)}(\\Phi)=\\int_{\\mathcal{G}(N)} \\sum_{1\\le j\\le N}^{} \\Phi(\\vartheta_j(A))dA.\n\\end{equation*}\nThe limiting density as $N\\to \\infty$ is given by the following (\\cite{book:KS}*{Theorem~AD.2.2})\n\\begin{proposition}\\label{prop:KS}\n  Let $\\mathcal{G}=U,SO(even)$ or $USp$. For all Schwarz functions $\\Phi$ on $\\mathbb{R}_+$,\n\\begin{equation*}\n  \\lim_{N\\to \\infty} W_{\\mathcal{G}(N)}(\\Phi) = \\int_{\\mathbb{R}_+}\n\\Phi(x) W(\\mathcal{G})(x)dx,\n\\end{equation*}\nwhere the density functions $W(\\mathcal{G})$ are given by~\\eqref{intro:WG}.\n\\end{proposition}\n\nThe density functions $W(\\mathcal{G})$ are defined a priori on $\\mathbb{R}_+$. They are extended to $\\mathbb{R}_-$ by symmetry, namely $W(\\mathcal{G})(x)=W(\\mathcal{G})(-x)$ for all $x\\in \\mathbb{R}$.\nFor a Schwartz function $\\Phi$ whose Fourier transform $\\widehat \\Phi$ has support inside $(-1,1)$, we have the identities\n\\begin{equation}\\label{W(G)}\n\\int_{-\\infty}^\\infty\n\\Phi(x) W(\\mathcal{G})(x)dx=\n\\begin{cases}\n\\widehat \\Phi(0) & \\text{if $\\mathcal{G}=U$,}\\\\\n\\widehat \\Phi(0)+\\frac{1}{2} \\Phi(0)& \\text{if $\\mathcal{G}=USp$,}\\\\\n\\widehat \\Phi(0)-\\frac{1}{2} \\Phi(0)& \\text{if $\\mathcal{G}=SO(even)$.}\n\\end{cases}\n\\end{equation}\n\n\n\\subsection{The $1$-level density of low-lying zeros}\\label{sec:zeros:onelevel}\nConsider a family $\\mathfrak{F}=(\\mathfrak{F}_k)_{k\\ge 1}$ of automorphic representations of $\\GL(d,\\mathbb{A}_F)$.\nThe $1$-level density of the low-lying zeros is defined by\n\\begin{equation}\\label{def:D1}\t\nD(\\mathfrak{F}_k;\\Phi):=\\frac{1}{\\abs{\\mathfrak{F}_k}} \\sum_{\\Pi \\in \\mathfrak{F}_k} \\sum_j \\Phi\n\\biggl(\n\\frac{\\gamma_{j}(\\Pi)}{2\\pi}\n\\log C(\\mathfrak{F}_k)\n\\biggr)\n\\end{equation}\nHere $\\Phi$ is a Schwartz function; we don't necessarily assume $\\Phi$ to be even because the automorphic representations $\\Pi\\in \\mathfrak{F}_k$ might not be self-dual. See also the discussion at the end of~\\S\\ref{sec:pp:explicit}. The properties of the analytic conductor $C(\\mathfrak{F}_k)\\ge 2$ will be described in~\\S\\ref{sec:zeros:cond}.\n\nSince $\\Phi$ decays rapidly at infinity, the zeros $\\gamma_j(\\Pi)$ of $\\Lambda(s,\\Pi)$ that contribute to the sum are within $O(1\/\\log C(\\mathcal{F}_k))$ distance of the central point. Therefore the sum over $j$ only captures a few zeros for each $\\Pi$. The average over the family $\\Pi\\in \\mathfrak{F}_k$ is essential to have a meaningful statistical quantity.\n\n\n\n\\subsection{Properties of families of $L$-functions}\n\nRecall that in~\\S\\ref{sub:aut-families} we have defined two kinds of families $\\mathcal{F}=(\\mathcal{F}_k)_{k\\ge 1}$ of automorphic representations on $G(\\mathbb{A}_F)$. The families from Example~\\ref{ex:level-varies} are varying in the level aspect: $\\mathbb{N}(\\mathfrak{n}_k)\\to \\infty$ while the families from Example~\\ref{ex:wt-varies} are varying in the weight aspect: $m(\\xi_k)\\rightarrow\\infty$. In both cases we assume that $\\phi_{S_0}\\in C_c^\\infty(G(F_{S_0}))$ is normalized such that\n\\begin{equation} \\label{phiS0}\n  \\widehat{\\mu}^{\\text{pl}}_{S_0}(\\phi_{S_0})=1.\n\\end{equation}\nFor families in the weight aspect we assume from now the weights are bounded away from the walls. Namely we assume that we are given a fixed $\\eta>0$ and that\n\\begin{equation}\\label{dimxik}\n  (\\dim \\xi_k)^{\\eta} \\le m(\\xi_k),\\quad \\forall k.\n\\end{equation}\n\nGiven the continuous $L$-morphism $r:{}^LG\\to \\GL(d,\\mathbb{C})$ we can construct a family $\\mathfrak{F}=r_*\\mathcal{F}$ of automorphic $L$-functions. Assuming the Langlands functoriality in the form of Hypothesis~\\ref{hypo:functorial-lift}, for each $\\pi\\in \\mathcal{F}_k$ there is a unique isobaric automorphic representation $\\Pi=r_*\\pi$ of $\\GL(d,\\mathbb{A}_F)$. We denote by $\\mathfrak{F}_k=r_*\\mathcal{F}_k$ the corresponding set of all such $\\Pi$. Recall from~\\S\\ref{sub:aut-rep} that $\\mathcal{F}_k$ is a weighted set and that the weight of each representation $\\pi$ is denoted $a_{\\mathcal{F}_k}(\\pi)$. The same holds for $\\mathfrak{F}_k$ and in particular we have\n\\begin{equation*}\n  \\abs{\\mathfrak{F}_k}=\\abs{\\mathcal{F}_k}=\\sum_{\\pi \\in \\mathcal{F}_k} a_{\\mathcal{F}_k}(\\pi).\n\\end{equation*}\nWe have seen in Corollary~\\ref{c:estimate-F_k} that $\\abs{\\mathfrak{F}_k}\\to \\infty$ as $k\\to \\infty$.\n\nBy definition (see~\\eqref{e:a(pi)-general}), if $\\pi\\in \\mathcal{F}_k$ then $\\pi_\\infty$ has the same infinitesimal character as $\\xi_k^{\\vee}$, i.e. $\\pi \\in \\Pi_{disc}(\\xi_k)$. If $\\Pi\\in \\mathfrak{F}_k$ then $\\Pi_\\infty$ corresponds to the composition $r\\circ \\phi_{\\xi_k}$ via the Langlands correspondence for $\\GL_d(F_\\infty)$ (This is Hypothesis~\\ref{hypo:functorial-lift}.(iii)). In particular $\\Pi_\\infty$ is uniquely determined by $\\xi_k$ and $r$. It is identical for all $\\Pi\\in \\mathfrak{F}_k$.\n\nIt is shown in Lemma~\\ref{lem:tempered} that $\\Pi_\\infty$ is tempered. Therefore the Proposition~\\ref{prop:tempered} applies and the bounds towards Ramanujan~\\eqref{eq:prop:tempered} are satisfied for all $\\Pi \\in \\mathfrak{F}_k$.\n\nTo simplify notation throughout this and the next section, we use the convention of omitting the weight when writing a sum over $\\mathfrak{F}_k$. If $l(\\Pi)$ is a quantity that depends on $\\Pi\\in \\mathfrak{F}_k$, we set\n\\begin{equation*}\n  \\sum_{\\Pi\\in \\mathfrak{F}_k} l(\\Pi) := \\sum_{\\pi \\in \\mathcal{F}_k} a_{\\mathcal{F}_k}(\\pi) l(r_*\\pi).\n\\end{equation*}\nThis convention applied in particular to~\\eqref{def:D1} above.\n\n\\subsection{Occurrence of poles}\\label{sec:zeros:poles}\nWe make the following hypothesis concerning poles of $L$-functions in our families.\n\\begin{hypothesis}\\label{hyp:poles}\n\tThere is $C_{pole}>0$ such that the following holds uniformly as $k\\to \\infty$.\n\\begin{equation*}\n\t\\#\\set{\\Pi\\in \\mathfrak{F}_k,\\ \\Lambda(s,\\Pi) \\text{ has a pole}} \\ll \\abs{\\mathfrak{F}_k}^{1-C_{pole}}.\n\\end{equation*}\n\\end{hypothesis}\nThe hypothesis is natural because it is related to the Functoriality Hypothesis~\\ref{hypo:functorial-lift} in many ways. Of course it would be difficult to define the event that \\Lquote{$L(s,\\Pi) \\text{ has a pole}$} without assuming Hypothesis~\\ref{hypo:functorial-lift}. Also when Functoriality is known unconditionally it is usually possible to establish the Hypothesis~\\ref{hyp:poles} unconditionally as well. We shall return to this question in a subsequent article.\n\n\n\n\\subsection{Analytic conductors}\\label{sec:zeros:cond} As in~\\cite{ILS00} we define an analytic conductor $C(\\mathfrak{F}_k)$ associated to the family. The significance of $C(\\mathfrak{F}_k)$ is clear: all $\\Pi\\in \\mathfrak{F}_k$ have an analytic conductor $C(\\Pi)$ comparable to $C(\\mathfrak{F}_k)$. We distinguish between families in the weight and level aspect.\n\n\\subsubsection{Weight aspect} For families in the weight aspect we set $C(\\mathfrak{F}_k)$ to be the analytic conductor $C(\\Pi_\\infty)$ of the archimedean factor $\\Pi_\\infty$ (recall that $\\Pi_\\infty$ is the same for all $\\Pi\\in \\mathfrak{F}_k$).\nThen $C(\\Pi)\\asymp C(\\mathfrak{F}_k)$ for all $\\Pi\\in \\mathfrak{F}_k$.\nFrom Corollary~\\ref{c:estimate-F_k} we have that $\\abs{\\mathfrak{F}_k}\\asymp \\dim \\xi_k$ as $k\\to \\infty$.\n\nIt remains to relate the quantities $C(\\mathfrak{F}_k)$, $\\dim \\xi_k$ and $m(\\xi_k)$, which is achieved in~\\eqref{dimxik:2} and~\\eqref{xik-CFk} below.\n\\begin{lemma} Let $v|\\infty$. Let $\\xi_v$ be an irreducible finite dimensional algebraic representation of $G(F_v)$. Then\n  $ m(\\xi_v)^{\\abs{\\Phi^+}}\\ll \\dim \\xi_v \\ll M(\\xi_v)^{\\abs{\\Phi^+}}$. Also $M(\\xi_v)\\ll \\dim \\xi_v$.\n\\end{lemma}\n\\begin{proof}\n  This follows from Lemma~\\ref{l:dim-and-trace}. Recall the definition of $m(\\xi_v)$ in~\\S\\ref{sub:st-disc} and $M(\\xi_v)$ in~\\S\\ref{s:langlands}.\n\\end{proof}\n\nBecause of~\\eqref{dimxik} and the previous lemma we have that\n\\begin{equation} \\label{dimxik:2}\n  m(\\xi_k)^{\\abs{\\Phi^+}} \\ll  \\dim \\xi_k \\ll m(\\xi_k)^{1\/\\eta}.\n\\end{equation}\nFrom Lemma~\\ref{l:Cpi} we deduce that there are positive constants $C_1,C_2$ such that\n\\begin{equation} \\label{xik-CFk}\n  m(\\xi_k)^{C_1} \\ll C(\\mathfrak{F}_k) \\ll m(\\xi_k)^{C_2}.\n\\end{equation}\n\n\n\n\n\n\n\\subsubsection{Level aspect} For families in the level aspect the situation is more complicated mainly because of the lack of knowledge of the local Langlands correspondence on general groups.\nWe define $C(\\mathfrak{F}_k)$ by the following\n\\begin{equation*}\n\\log C(\\mathfrak{F}_k):=\n\\frac{1}{\\abs{\\mathfrak{F}_k}} \\sum_{\\Pi\\in\\mathfrak{F}_k}\n\\log C(\\Pi),\n\\end{equation*}\nand we introduce the following hypothesis.\n\n\\begin{hypothesis}\\label{hyp:cond} There are constants $C_3, C_4>0$ such that\n\\begin{equation*}\n  \\mathbb{N}(\\mathfrak{n}_k)^{C_3} \\ll C(\\mathfrak{F}_k) \\ll \\mathbb{N}(\\mathfrak{n}_k)^{C_4}.\n\\end{equation*}\n\\end{hypothesis}\n\n\n\n\n\\subsection{Main result}\\label{sec:zeros:main}\nWe may now state our main results on low-lying zeros of the family $\\mathfrak{F}=r_*\\mathcal{F}$. The following is a precise version of Theorem~\\ref{th:low-lying} from the introduction (compare with~\\eqref{W(G)}).\n\\begin{theorem}\\label{th:onelevel}\n  Assume Hypotheses~\\ref{hypo:functorial-lift},~\\ref{hyp:poles} and~\\ref{hyp:cond}. There is $0<\\delta<1$ such that for all Schwartz functions $\\Phi$ whose Fourier transform $\\widehat \\Phi$ has support in $(-\\delta,\\delta)$ the following holds:\n\\begin{equation*}\n\\lim_{k\\to \\infty} D(\\mathfrak{F}_k,\\Phi)=\\widehat \\Phi(0) -\\frac{s(r)}{2}\\Phi(0),\n\\end{equation*}\nwhere $s(r)\\in \\set{-1,0,1}$ is the Frobenius--Schur indicator of $r:{}^LG\\to \\GL_d(\\mathbb{C})$.\n\\end{theorem}\n\n\n\n\n\n\n\\section{Proof of Theorem~\\ref{th:onelevel}}\\label{sec:pf}\n\nThe method of proof of the asymptotic distribution of the $1$-level density of low-lying zeros of families of $L$-functions has appeared at many places in the literature and is by now relatively standard. However we must justify the details carefully as families of $L$-functions haven't been studied in such a general setting before. The advantage of working in that degree of generality is that we have been able to isolate the essential mechanisms and arithmetic ingredients involved.\n\nIn order to keep the analysis concise we have introduced some technical improvements which are listed below and can be helpful in other contexts.\n\\begin{enumerate}[(i)]\n\\item The first observation is that it is better not to expand the Euler product of the $L$-functions $L(s,\\Pi)$ into Dirichlet series. We can apply the explicit formula (Proposition~\\ref{prop:explicit}) in a way to preserve the Euler product structure.\n\\item We use non-trivial bounds towards Ramanujan in a systematic way to handle ramified places.\n\\item We clarify that it is not necessary to assume that the representation be self-dual or any other symmetry to carry out the analysis.\n\\item Most importantly we exploit the properties of the Plancherel measure when estimating Satake parameters. All previous articles on the subject rely in a way or another on explicit Hecke relations which made the proof indirect and lengthy, although manageable for groups of low rank.\n\\end{enumerate}\n\n\n\\subsection{Notation and outline}\\label{sec:pf:outline}\nTo formulate the main statements in an elegant way we introduce the following notation\n\\begin{equation}\\label{def:hatL}\n\\widehat \\mathcal{L}_{k,v}(y):=\\frac{1}{\\abs{\\mathfrak{F}_k}}\n\\sum_{\\Pi \\in \\mathfrak{F}_k}\n\\int^\\infty_{-\\infty} \\frac{L'}{L}(\\frac{1}{2}+ix,\\Pi_v)\ne^{2\\pi iyx} dx\n,\\quad v\\in \\mathcal{V}_F,\\ y\\in \\mathbb{R}.\n\\end{equation}\nWe view $\\widehat \\mathcal{L}_{k,v}$ as a tempered distribution on $\\mathbb{R}$.\nNote that when $v$ is non-archimedean $\\widehat \\mathcal{L}_{k,v}$ is a signed measure supported on a discrete set inside $\\mathbb{R}_{>0}$.\n\n\n\nThe proof of the main theorems will follow by a fine estimation of $\\widehat \\mathcal{L}_{k,v}(y)$ as $k\\to \\infty$. The uniformity in both the places $v\\in \\mathcal{V}_F$ and the parameter $y\\in \\mathbb{R}$ will play an important role. Typically $q_v$ will be as large as $C(\\mathfrak{F}_k)^{O(\\delta)}$ and $y$ will be of order of magnitude of $\\log C(\\mathfrak{F}_k)$.\n\nThe first step of the proof consists in applying the explicit formula (Proposition~\\ref{prop:explicit}). There are terms coming from the poles of $L(s,\\Pi)$ which are dealt with in~\\S\\ref{sec:pf:poles}. The second term in the right hand-side in Proposition~\\ref{prop:explicit} is expressed in terms of the arithmetic conductor $q(\\Pi)$ and will yield a positive contribution for families in the level aspect. When evaluating the $1$-level density $D(\\mathfrak{F}_k,\\Phi)$ it remains to consider the following sum over all places\n\\begin{equation}\\label{out:sumv}\n\\frac{1}{\\log C(\\mathfrak{F}_k)}\n\\sum_{v\\in\\CmV_F}\n\\left \\langle\n\\widehat \\mathcal{L}_{k,v} (y),\n\\widehat{\\Phi}\\Bigp{\\frac{2\\pi y}{\\log C(\\mathfrak{F}_k)}}\n\\right \\rangle,\n\\end{equation}\nplus a conjugate expression, see~\\S\\ref{sec:pf:explicit}.\n\nOur convention on Fourier transforms is standard. Let $\\Phi$ be a Schwartz function on $\\mathbb{R}$. The Fourier transform is as in~\\eqref{def:fourier} and the inverse Fourier transform reads\n\\begin{equation*\n\\Phi(x)=\\int_{-\\infty}^{+\\infty} \\widehat\\Phi(y) e^{2\\pi i xy} dy.\n\\end{equation*}\nGiven two Schwartz functions $\\Phi$ and $\\Psi$ we let\n\\begin{equation*}\n\\langle \\Phi,\\Psi\\rangle:=\\int_{-\\infty}^{\\infty}\n\\Phi(x) \\Psi(x)dx.\n\\end{equation*}\nSometimes we use the notation $\\langle \\Phi(x),\\Psi(x)\\rangle$ to emphasize on the variable of integration. The Plancherel formula reads\n\\begin{equation}\\label{fourier:pl}\n\\langle \\Phi(x),\\Psi(x)\\rangle=\\langle \\widehat\\Phi(y),\\widehat\\Psi(-y)\\rangle.\n\\end{equation}\nWe use the similar convention for tempered distributions. The Fourier transform of the pure phase function $x\\mapsto e^{2i\\pi ax}$ is the Dirac distribution $\\delta(a)$ centered at the point $a$.\nTo condense notation we write $\\Psi(y):=\\widehat{\\Phi}\\Bigp{\\dfrac{2\\pi y}{\\log C(\\mathfrak{F}_k)}}$ and express our remainder terms as functions of $\\pnorm{\\Psi}_\\infty$ and $\\pnorm{\\widehat \\Psi}_1$. Since $\\Phi$ is fixed these two norms are uniformly bounded.\n\n\n\nThere are different kinds of estimates depending on the nature of the place $v$. We shall distinguish the following set of places\n\\begin{enumerate}[(i)]\n\\item The set of archimedean places $S_\\infty$, the contribution of which is evaluated in~\\S\\ref{sec:pf:arch}.\n\\item A fixed set $S_0$ of non-archimedean places. These may be thought of as the \\Lquote{ramified places}. Their contribution is negligible as shown in~\\S\\ref{sec:pf:gn}.\n\\item The set $\\set{v\\mid \\mathfrak{n}_k}$ of places that divide the level. These play a role only when the level varies. We use the convention that for families in the weight aspect this set of places is empty.\n\\item The set of generic places $S_{\\text{gen}}$ which is the complement in $\\mathcal{V}_F$ of the above three sets of places. This set will actually be decomposed in two parts, $S_{\\text{main}}$ and $S_{\\text{cut}}$. The set $S_{\\text{cut}}$ is infinite and consists of those non-archimedean places $v\\in S_\\text{gen}$ such that $q_v$ is large enough to escape the support of $\\widehat\\Phi$ (see~\\eqref{def:Scut} below for the exact definition of $S_\\text{cut}$). Then the pairing in~\\eqref{out:sumv} vanishes. This actually shows that the sum over $v\\in\\mathcal{V}_F$ in~\\eqref{out:sumv} is finitely supported.\n\\item The remaining set $S_\\text{main}$ is finite (but growing as $k \\to \\infty$). It will produce the main contribution. For all $v\\in S_\\text{main}$, $G$, $r$ and $\\pi$ are unramified over $F_v$. We have a disjoint union of $S_\\text{main} \\cap \\mathcal{V}_F(\\theta)$, $\\theta \\in \\mathscr{C}(\\Gamma_1)$ (see~\\S\\ref{s:Sato-Tate}).\n\\end{enumerate}\n\n\nThe outline of this section is as follows. For non-archimedean places $v\\in S_\\text{main}$ we set up in~\\S\\ref{sec:pf:moments} some notation on moments. The quantity $\\widehat \\mathcal{L}_{\\text{pl},v}$ in~\\eqref{CmL-pl} is the analogue of~\\eqref{def:hatL} where the average over automorphic representations $\\Pi\\in \\mathfrak{F}_k$ in~\\eqref{def:hatL} has been replaced by an average of $\\Pi_v$ against the unramified Plancherel measure.\n\nOur Plancherel equidistribution theorems for families (Theorems~\\ref{t:level-varies} and~\\ref{t:weight-varies}) imply that $\\widehat \\mathcal{L}_{k,v}$ is asymptotic to $\\widehat \\mathcal{L}_{\\text{pl},v}$ as $k\\to \\infty$. It is essential that these equidistribution theorems be quantitative in a strong sense for the error term to be negligible. Details are given in~\\S\\ref{sec:pf:pl} and \\S\\ref{sec:pf:main}.\n\nThe various remainder terms are handled in~\\S\\ref{sec:pf:error}.\nThe next step is to show the existence of the limit of the main term\n\\begin{equation} \\label{out:main}\n  \\frac{1}{\\log C(\\mathfrak{F}_k)} \\sum_{v\\in S_\\text{main}}\n\\bigl \\langle\n\\widehat\\mathcal{L}_{\\text{pl},v},\\Psi\n\\bigr\\rangle\n\\end{equation}\nas $k\\to \\infty$. We need the Cebotarev density theorem (see~\\S\\ref{sec:pf:prime}) and to evaluate $\\widehat\\mathcal{L}_{\\text{pl},v}$.\n\nThe evaluation of $\\widehat\\mathcal{L}_{\\text{pl},v}$ is a nice argument in representation theory, see~\\S\\ref{sec:pf:M12}. At this stage we see very clearly the role of the two optimal conditions on $r$ from Theorem~\\ref{th:low-lying} (irreducible and does not factor through $W_F$). The evaluation of $\\widehat\\mathcal{L}_{\\text{pl},v}$ can actually be quite complicated since it depends on the restriction of $r$ to the various subgroups $\\widehat G \\rtimes W_{F_v}$ and the unramified Plancherel measure on $G(F_v)$. Fortunately the expression simplifies when summing over all places $v\\in S_\\text{main}$ as shown in~\\S\\ref{sec:pf:M12}.\n\nThe overall conclusion of the analysis is that the limit of~\\eqref{out:main} as $k\\to \\infty$ (plus its conjugate) is equal\n\\footnote{A quick explanation for the minus sign is as follows. A local $L$-factor is of the form $L(s)=(1-\\alpha q^{-s})^{-1}$ with three minus signs thus its logarithmic derivative is $\\tfrac{L'}{L}(s)=-\\log q \\sum\\limits_{\\nu\\ge 1}\\alpha^{\\nu}q^{-\\nu s}$.}\nto $-s(r)\\Phi(0)$, where $s(r)$ is the Frobenius--Schur indicator of $r$ as in Theorem~\\ref{th:onelevel}.\n\n\n\\subsection{Explicit formula}\\label{sec:pf:explicit} We apply the explicit formula (Proposition~\\ref{prop:explicit}) for all $\\Pi\\in \\mathfrak{F}_k$. Averaging over $\\Pi$ we obtain\n\\begin{equation}\\label{D1explicit}\nD(\\mathfrak{F}_k,\\Phi)=\nD_{\\text{pol}}(\\mathfrak{F}_k,\\Phi)+\n\\frac{\\widehat \\Phi(0)}{\\abs{\\mathfrak{F}_k}}\n\\sum_{\\Pi \\in \\mathfrak{F}_k}^{} \\frac{\\log q(\\Pi)}{\\log C(\\mathfrak{F}_k)} + \\sum_{v\\in \\mathcal{V}_F}\nD_{v}(\\mathfrak{F}_k,\\Phi)+\\overline{D_{v}}(\\mathfrak{F}_k,\\Phi).\n\\end{equation}\nHere $D_{\\text{pol}}(\\mathfrak{F}_k,\\Phi)$ denotes the contribution of the eventual poles and will be dealt with in~\\S\\ref{sec:pf:poles}. Also we have set\n\\begin{equation*}\nD_{v}(\\mathfrak{F}_k,\\Phi):=\n\\frac{1}{2\\pi \\abs{\\mathfrak{F}_k}}\n\\sum_{\\Pi\\in\\mathfrak{F}_k}\n\\int_{-\\infty}^{\\infty}\n\\frac{L'}{L}(\\frac{1}{2}+ix,\\Pi_v) \\Phi\n\\Bigp{\\frac{x}{2\\pi}\\log C(\\mathfrak{F}_k)}\ndx.\n\\end{equation*}\nSee also the remark in~\\eqref{explicit:switch} explaining how to shift contours. Recall that the scaling factor $\\tfrac{\\log C(\\mathfrak{F}_k)}{2\\pi}$ comes from the definition~\\eqref{def:D1}.\n\nApplying Fourier duality~\\eqref{fourier:pl} and the definition~\\eqref{def:hatL} implies the equality\\footnote{Note that the exponential in~\\eqref{def:hatL} is $e^{2i\\pi xy}$ with a plus sign.}\n\\begin{equation}\\label{def:D1v}\nD_{v}(\\mathfrak{F}_k,\\Phi)=\\frac{1}{\\log C(\\mathfrak{F}_k)}\n\\bigl\\langle  \\widehat\\mathcal{L}_{k,v},\n\\widehat \\Phi \\Bigp{\\frac{2\\pi y}{\\log C(\\mathfrak{F}_k)}}\n\\bigl\\rangle.\n\\end{equation}\nSimilarly we have\n\\begin{equation*}\n\\begin{aligned}\n\\overline{D_{v}}(\\mathfrak{F}_k,\\Phi)\n&:=\n\\frac{1}{2\\pi \\abs{\\mathfrak{F}_k}}\n\\sum_{\\Pi\\in\\mathfrak{F}_k}\n\\int_{-\\infty}^{\\infty}\n\\overline{\n\\frac{L'}{L}(\\frac{1}{2}+ix,\\Pi_v) \\Phi\n}\n\\Bigp{\\frac{x}{2\\pi}\\log C(\\mathfrak{F}_k)}\ndx\\\\\n&= \\frac{1}{\\log C(\\mathfrak{F}_k)}\n\\bigl \\langle\n\\overline{\n\\widehat \\mathcal{L}_{k,v}},\n\\widehat \\Phi\n\\Bigp{\\frac{-2\\pi y}{\\log C(\\mathfrak{F}_k)}}\n\\bigr \\rangle.\n\\end{aligned}\n\\end{equation*}\nWe have made a change of variable so as to make explicit the multiplicative factor $1\/\\log C(\\mathfrak{F}_k)$ in front of the overall sum.\n\n\\subsection{Contribution of the poles}\\label{sec:pf:poles}\nThe contribution of the poles in the explicit formula is given by\n\\begin{equation*}\nD_{\\text{pol}}(\\mathfrak{F}_k,\\Phi):=\n\\frac{1}{\\abs{\\mathfrak{F}_k}}\n\\sum_{\\Pi\\in \\mathfrak{F}_k}\n\\sum_j\n\\Phi\\left(\n\\frac{r_j(\\Pi)}{2\\pi} \\log C(\\mathfrak{F}_k)\n\\right).\n\\end{equation*}\n\nWe bound the sum trivially and obtain\n\\begin{equation*} D_{\\text{pol}}(\\mathfrak{F}_k,\\Phi)\n\\ll \\frac{\\#\\set{\\Pi\\in \\mathfrak{F}_k,\\ L(s,\\Pi) \\text{ has a pole}}}\n{\\abs{\\mathfrak{F}_k}} C(\\mathfrak{F}_k)^{O(\\delta)}.\n\\end{equation*}\nThe last polynomial factor comes from the exponential order of growth of $\\Phi$ along the real axis and the fact that the Fourier transform $\\widehat \\Phi$ is supported in $(-\\delta,\\delta)$. Now thanks to Hypothesis~\\ref{hyp:poles} the right-hand side goes to zero as $k\\to \\infty$. We also use the fact that $C(\\mathfrak{F}_k)\\ll \\abs{\\mathfrak{F}_k}^{O(1)}$ which follows from Corollary~\\ref{c:estimate-F_k} and the upper bound for $C(\\mathfrak{F}_k)$ from~\\eqref{xik-CFk}.\n\n\\subsection{Archimedean places}\\label{sec:pf:arch} In this subsection we handle the archimedean places $v\\in S_\\infty$. Recall that $0<\\theta<\\frac{1}{2}$ is a bound towards Ramanujan as in~\\S\\ref{sec:pp:Lfn}.\n\\begin{lemma}\\label{prop:psi-fn} For all $\\mu\\in \\mathbb{C}$ with $\\MRe \\mu \\le \\theta$, and all Schwartz function $\\Psi$, the following holds uniformly\n\\begin{equation*}\n\\int_{-\\infty}^{\\infty}\n\\frac{\\Gamma'}{\\Gamma}(\\frac{1}{2}-\\mu+ix)\\Psi(x)dx\n=\\widehat \\Psi(0) \\log (\\frac{1}{2}-\\mu)\n+O(\\pnorm{\\Psi}_1+\\pnorm{x\\Psi(x)}_1)\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nWe have the following Stirling approximation for the Digamma function (traditionally denoted $\\psi(z)$):\n\\begin{equation}\\label{psi:asymp}\n\\frac{\\Gamma'}{\\Gamma}(z)= \\log z + O(1)\n\\end{equation}\nuniformly in the angular region $\\abs{\\arg z}\\le \\pi -\\epsilon$, see e.g.~\\cite{book:Iw}*{Appendix B}.\nSince $\\theta<1\/2$ all points $\\frac{1}{2}-\\mu+ix$ lie in the interior of the angular region and we can apply~\\eqref{psi:asymp}. We note also that uniformly\n\\begin{equation*}\n\\log(\\frac{1}{2}-\\mu+ix)=\\log(\\frac{1}{2}-\\mu) + O(\\log(2+\\abs{x})),\n\\end{equation*}\nand this conclude the proof of the Proposition.\n\\end{proof}\n\\remark Note that the complete asymptotic expansion actually involves the Bernoulli numbers and is of the form\n\\begin{equation}\\label{psi:completeasymp}\n\\frac{\\Gamma'}{\\Gamma}(z)=\\log z + \\frac{1}{2z} - \\sum_{n=1}^{N} \\frac{B_{2n}}{2n z^{2n}}\n+O\\Bigl(\n\\frac{1}{z^{2N+2}}\n\\Bigr).\n\\end{equation}\nFrom~\\eqref{psi:completeasymp} we have that\n\\begin{equation*}\n\\frac{\\Gamma'}{\\Gamma}(\\sigma+it)+\\frac{\\Gamma'}{\\Gamma}(\\sigma-it)=2\\frac{\\Gamma'}{\\Gamma}(\\sigma)+O((t\/\\sigma)^2)\n\\end{equation*}\nholds uniformly for $\\sigma$ and $t$ real with $\\sigma >0$. As in~\\cite{ILS00}*{\\S4} this may be used when the test function $\\Psi$ is even (e.g. which is the typical case when all representations $\\Pi\\in \\mathfrak{F}$ are self-dual). We don't make this assumption and therefore use~\\eqref{psi:asymp} instead.\n\n\\begin{corollary} Uniformly for all archimedean places $v\\in S_\\infty$ and all Schwartz function $\\Psi$, the following holds\n\\begin{equation*}\n\\Bigl\\langle\n\\widehat \\mathcal{L}_{k,v},\n\\Psi\n\\Bigr\\rangle\n=\\frac{\\Psi(0)}\n{\\abs{\\mathfrak{F}_k}}\n\\sum_{\\Pi\\in\\mathfrak{F}_k}\n\\sum_{i=1}^{d}\n\\log_v (\\frac{1}{2} -\\mu_i(\\Pi_v))\n+ O(\\bigl\\lVert \\widehat \\Psi \\bigr \\rVert_1).\n\\end{equation*}\nHere we have set $\\log_v z:=\\frac{1}{2} \\log z$ when $v$ is real and $\\log_v z:= \\log z$ when $v$ is complex.\n\\end{corollary}\n\\begin{proof}\nRecall the convention~\\eqref{def:Lv-arch} on local $L$-factors at archimedean places $v\\in S_\\infty$. From Fourier duality~\\eqref{fourier:pl} and the definition~\\eqref{def:hatL} we have\n\\begin{equation*}\n\\Bigl\\langle\n\\widehat \\mathcal{L}_{k,v},\\Psi\n\\Bigr\\rangle\n=\n\\frac{1}{\\abs{\\mathfrak{F}_k}}\n\\sum_{\\Pi \\in \\mathfrak{F}_k}\n\\int^\\infty_{-\\infty} \\frac{L'}{L}(\\frac{1}{2}+ix,\\Pi_v)\n\\widehat \\Psi(x)\ndx.\n\\end{equation*}\nNote that\n\\begin{equation*}\n\\frac{\\Gamma'_v}{\\Gamma_v}(s)=\n\\begin{cases}\n-\\frac{1}{2}\\log \\pi +\\frac{1}{2} \\frac{\\Gamma'}{\\Gamma}(\\frac{s}{2})\n,&\n\\text{when $v$ is real,}\\\\\n-\\log (2\\pi)+\\frac{\\Gamma'}{\\Gamma}(s)\n,&\n\\text{when $v$ is complex.}\n\\end{cases}\n\\end{equation*}\nApplying Lemma~\\ref{prop:psi-fn}, the estimate in the corollary follows. Recall from Proposition~\\ref{prop:tempered} that the bounds towards Ramanujan in~\\S\\ref{sec:pp:Lfn} apply to all $\\Pi\\in \\mathfrak{F}_k$.\n \\end{proof}\n\nWe may continue the analysis of the contribution of the archimedean places to the one-level density. For $v\\in S_\\infty$, the local $L$-function $L(s,\\Pi_v)$ are the same for all $\\Pi \\in \\mathfrak{F}_k$. We therefore conclude that\n\\begin{equation}\\label{archterms}\n\\begin{aligned}\n\\sum_{v\\in S_\\infty}^{}\nD_{v}(\\mathfrak{F}_k,\\Phi)\n+\\overline{D_{v}(\\mathfrak{F}_k,\\Phi)}\n&=\n\\frac{\\widehat \\Phi(0)}{\\log C(\\mathfrak{F}_k)}\n\\left(\n\\sum_{v\\in S_\\infty}^{}\n\\sum_{i=1}^{d}\n2 \\log_v \\abs{\\frac{1}{2}-\\mu_i(\\Pi_v)}\n+O(1)\n \\right) \\\\\n &=\n\\frac{\\widehat \\Phi(0)}{\\log C(\\mathfrak{F}_k)}\n\\left(\n\\sum_{v\\in S_\\infty}^{}\n\\log C(\\Pi_v)\n+O(1)\n \\right).\n\\end{aligned}\n\\end{equation}\nIn the last line we used the definition of the analytic conductor at archimedean places from~\\S\\ref{sec:pp:Lfn}.\n\n\\subsection{Moments}\\label{sec:pf:moments}\nNow let $v\\in \\mathcal{V}_F^\\infty$ be a non-archimedean place. A straightforward computation shows that\n\\begin{equation*}\n\\frac{L'}{L}(s,\\Pi_v)=-\\log q_v\n\\sum_{\\nu\\ge 1}\n\\beta^{(\\nu)}(\\Pi_v)q_v^{-\\nu s}\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\label{def:beta-Pi}\n\\beta^{(\\nu)}(\\Pi_v):=\n\\sum_{i=1}^d\n\\alpha^{\\nu}_i(\\Pi_v).\n\\end{equation*}\nAveraging over the family $\\mathfrak{F}$ we let\n\\begin{equation}\n\\label{def:beta-F}\n\\beta^{(\\nu)}_v(\\mathfrak{F}_k)\n:=\n\\frac{1}{\\abs{\\mathfrak{F}_k}}\n\\sum_{\\Pi\\in\\mathfrak{F}_k}\n\\beta^{(\\nu)}(\\Pi_v),\\quad v\\in\\mathcal{V}_F^\\infty,\\ \\nu\\ge 1.\n\\end{equation}\nThe coefficients $\\beta^{(\\nu)}_v$ are the moments of the Satake parameters and their estimate play a central role in the proof of Theorem~\\ref{th:onelevel}. Then for all $v\\in \\mathcal{V}_F^{\\infty}$, the following formula is equivalent to the definition~\\eqref{def:hatL}\n\\begin{equation}\n\\label{CmL-beta}\n\\widehat \\mathcal{L}_{k,v}=-\\log q_v \\sum_{\\nu\\ge 1}\n\\beta^{(\\nu)}_v(\\mathfrak{F}_k) q_v^{-\\nu\/2}\\delta\n\\bigp{\\frac{\\nu}{2\\pi} \\log q_v}.\n\\end{equation}\nSee~\\S\\ref{sec:pf:outline} for the convention on Dirac delta.\n\nSimilarly for all $v\\in S_\\text{gen}$ we let\n\\begin{equation}\\label{CmL-pl}\n\\widehat \\mathcal{L}_{\\text{pl},v}:=-\\log q_v\n\\sum_{\\nu\\ge 1}\n\\beta^{(\\nu)}_{\\text{pl},v} q_v^{-\\nu\/2}\n\\delta\n\\bigp{\\frac{\\nu}{2\\pi} \\log q_v}.\n\\end{equation}\nHere the coefficients $\\beta^{(\\nu)}_{\\text{pl},v}$ are defined locally as follows. We assume that for all $v\\in S_\\text{gen}$, the group $G$ is unramified over $F_v$ and that the restriction $r|_{\\widehat{G}\\rtimes W_{F_v}}$ is an unramified $L$-morphism, i.e. it factors through $\\widehat G \\rtimes W^{\\text{ur}}_{F_v}$. The Plancherel measure $\\widehat \\mu^{\\text{pl}}_{v}$ is as in \\S~\\ref{s:Plancherel} and we recall the notation $\\widehat \\mu^{\\text{pl},\\mathrm{ur}}_v$ for the restriction of the Plancherel measure $\\widehat \\mu^{\\text{pl}}_{v}$ to $G(F_v)^{\\wedge,\\mathrm{ur}}$. Then\n\\begin{equation}\n\\label{def:beta-pl}\n\\beta^{(\\nu)}_{\\text{pl},v}:= \\widehat \\mu^{\\text{pl},\\mathrm{ur}}_v\n\\Bigl(\nr^*\\bigl(\nY_1^\\nu+\\cdots +Y_d^{\\nu}\n\\bigr)\n\\Bigr).\n\\end{equation}\nHere we are using the convention in \\S\\ref{sub:case-of-GL_d} for the unramified Hecke algebra $\\mathcal{H}^{\\mathrm{ur}}(\\GL_d)$ and the Satake isomorphism with the symmetric algebra in the variables $Y_1,\\ldots ,Y_d$. We recall the convention in \\S\\ref{sub:trun-unr-Hecke} for the $L$-morphism of Hecke algebras $r^*:\\mathcal{H}^{\\mathrm{ur}}(\\GL_d)\\to \\mathcal{H}^{\\mathrm{ur}}(G(F_v))$.\n\nThe supports of both measures $\\widehat \\mathcal{L}_{k,v}$ and $\\widehat \\mathcal{L}_{\\text{pl},v}$ are contained in the discrete set $\\frac{2\\pi}{\\log q_v}\\mathbb{N}_{\\ge 1}$. In particular we note that when $q_v$ is large enough this set is disjoint from the support of $\\widehat \\Phi$. As a consequence all sums over places $v\\in \\mathcal{V}_F$ considered in this section are actually finitely supported.\n\n\\subsection{General upper-bounds}\\label{sec:pf:gn} Recall from Proposition~\\ref{prop:tempered} that the bounds towards Ramanujan apply to all $\\Pi\\in \\mathfrak{F}_k$. Then for all non-archimedean $v\\in \\mathcal{V}_F^{\\infty}$, we have the upper bound\n\\begin{equation*}\n\\abs{\\alpha_i(\\Pi_v)}\\le q_v^{\\theta}.\n\\end{equation*}\nIt then follows that\n\\begin{equation}\\label{gn-beta}\n\\abs{\\beta^{(\\nu)}(\\Pi_v)}\n\\le d q_v^{\\nu \\theta},\n\\end{equation}\nand the same holds true for $\\beta^{(\\nu)}_v(\\mathfrak{F}_k)$.\n\\begin{proposition}\\label{prop:gn-bd}\nFor all $v\\in \\mathcal{V}^{\\infty}_F$ and all Schwartz function $\\Psi$,\n\\begin{equation*}\n\\left\\langle \\widehat \\mathcal{L}_{k,v},\\Psi \\right\\rangle\n\\ll q_v^{\\theta-\\frac{1}{2}}\\log q_v \\pnorm{\\Psi}_{\\infty}.\n\\end{equation*}\n\\end{proposition}\n\\begin{proof} Inserting the upper bound~\\eqref{gn-beta} in~\\eqref{CmL-beta} we have\n\\begin{equation*}\n\\left\\langle \\widehat \\mathcal{L}_{k,v},\\Psi \\right\\rangle\n\\ll \\log q_v \\sum_\\nu q_v^{\\nu(\\theta-1\/2)} \\abs{\\Psi(\\frac{\\nu}{2\\pi}\\log q_v)}.\n\\end{equation*}\nBecause $0<\\theta<\\frac{1}{2}$, the conclusion easily follows.\n\\end{proof}\n\nSimilarly, from~\\eqref{def:beta-pl} we have the bound\n\\begin{equation}\\label{gn-pl-beta}\n\\abs{\\beta_{\\text{pl},v}^{(\\nu)}}\\le d.\n\\end{equation}\nfor all $v\\in S_\\text{gen}$. Indeed this follows from the fact that $\\widehat \\mu^{\\text{pl},\\mathrm{ur}}$ has total mass one, see \\S\\ref{s:Plancherel}, and the inequality $\\abs{\\widehat \\phi(\\pi)}\\le d$ which follows from~\\S\\ref{s:unramified-spectrum}. Here $\\pi\\in G(F_v)^{\\wedge,\\mathrm{ur},{\\rm temp}}$ which is the support of the unramified Plancherel measure $\\widehat \\mu^{\\text{pl},\\mathrm{ur}}$. For instance the image of an unramified $L$-parameter $r\\phi:W_{F_v}^{\\mathrm{ur}} \\to \\GL_d(\\mathbb{C})$ has bounded image and all Frobenius eigenvalues have therefore absolute value one. We deduce that\n\\begin{equation*\n\\left\\langle \\widehat \\mathcal{L}_{\\text{pl},v},\\Psi \\right\\rangle\n\\ll q_v^{-\\frac{1}{2}}\\log q_v \\pnorm{\\Psi}_{\\infty}.\n\\end{equation*}\nWe recall that the Plancherel measure $\\widehat \\mu^{\\text{pl},\\mathrm{ur}}_v$ is supported on the tempered spectrum $\\widehat G(F_v)^{\\wedge,\\mathrm{ur},{\\rm temp}}$.\n\n\\subsection{Plancherel equidistribution}\\label{sec:pf:pl} We are in position to apply the Plancherel equidistribution theorem for families established in Section~\\ref{s:aut-Plan-theorem}. We shall derive uniform asymptotics as $k\\to \\infty$ for the moments $\\beta^{(\\nu)}_{v}(\\mathfrak{F}_k)$ in~\\eqref{def:beta-F}.\n\\begin{proposition}\\label{prop:F-pl} The following holds uniformly on $\\nu \\ge 1$ and $v\\in S_\\text{gen}$\n\\begin{equation} \\label{main-F-pl}\n\\beta^{(\\nu)}_{v}(\\mathfrak{F}_k)=(1+o(1)) \\beta^{(\\nu)}_{\\text{pl},v}\n+O(q_v^{A_7+B_9\\nu} C(\\mathfrak{F}_k)^{-C_5}).\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof} Let $S_0$ be a sufficiently large set of non-archimedean places which contains all places $v\\in\\mathcal{V}_F^\\infty$ where $G$ is ramified and where $r$ is ramified. Let $S_1:=\\set{v}$. We shall distinguish between our two types of families during the proof of Proposition~\\ref{prop:F-pl}. Before we begin with some general construction.\n\n With the same convention as in~\\eqref{def:beta-pl} we set\n\\begin{equation*}\n\\widehat \\phi_v :=  r^* (Y_1^\\nu+\\cdots + Y_d^{\\nu}) \\in \\mathcal{H}^{\\mathrm{ur}}(G(F_v)).\n\\end{equation*}\nHere the conventions on the Satake isomorphism are as in~\\S\\ref{sub:Satake-trans} and \\S\\ref{sub:L-mor-unr-Hecke}. Recall that $\\widehat \\phi_{S_0}$ has been fixed earlier (independent of $k$) and normalized from~\\eqref{phiS0}.\n\nBy definition we have that $\\beta_{\\text{pl},v}^{(\\nu)}=\\widehat \\mu^{\\text{pl}}_{v}(\\widehat \\phi_v)$. Next we can verify that (see the notation in \\S\\ref{sub:aut-rep})\n\\begin{equation*} \\begin{aligned}\n  \\beta_v^{(\\nu)}(\\mathfrak{F}_k)&= \\frac{1}{\\abs{\\mathcal{F}_k}} \\sum_{\\pi\\in \\mathcal{F}_k} a_{\\mathcal{F}_k}(\\pi)\\widehat \\phi_v(\\pi_v)\\\\\n  &=\\widehat \\mu^{\\text{count}}_{\\mathcal{F}_k,v}(\\widehat \\phi_v)\\\\\n  &=\\frac{\\tau'(G)\\dim \\xi_k}{\\mu^{\\text{can}}(U^{S,\\infty}_k) \\abs{\\mathcal{F}_k}}\n  \\widehat \\mu_{\\mathcal{F}_k,v}(\\widehat \\phi_v).\n\\end{aligned}\n\\end{equation*}\nFrom Corollary~\\ref{c:estimate-F_k}, $ \\dfrac{\\tau'(G)\\dim \\xi_k}{\\mu^{\\text{can}}(U^{S,\\infty}_k) \\abs{\\mathcal{F}_k}}=1+o(1)$ as $k\\to \\infty$ (note that the quantity is independent of $v\\in S_\\text{gen}$).\n\nThanks to Lemma~\\ref{l:bound-degree-Satake} we have that $\\phi_{v}\\in \\mathcal{H}^{\\mathrm{ur}}(G(F_v))^{\\le \\beta \\nu}$ and $\\abs{\\phi_v}\\ll 1$. Thus the first assumptions in Theorems~\\ref{t:level-varies} and~\\ref{t:weight-varies} are satisfied.\n\nFor families in the level aspect the remaining assumptions in Theorem~\\ref{t:level-varies} are also satisfied when the constant $\\epsilon>0$ is chosen small enough. It remains to use the upper bound for $C(\\mathfrak{F}_k)$ from~Hypothesis~\\ref{hyp:cond} to conclude the proof of Proposition~\\ref{prop:F-pl} in that case.\n\nFor families in the weight aspect the remaining assumptions in Theorem~\\ref{t:weight-varies} are also satisfied. We use the upper bound for $C(\\mathfrak{F}_k)$ from~\\eqref{xik-CFk} to conclude the proof of Proposition~\\ref{prop:F-pl} in that case.\n\\end{proof}\n\n\\subsection{Main term}\\label{sec:pf:main}\nWe deduce the following main estimate for $\\widehat \\mathcal{L}_{k,v}$.\n\\begin{proposition}\\label{prop:S1} For all $A>0$ there is $A_8>0$ such that the following holds uniformly on $v\\in S_\\text{gen}$ and for all Schwartz functions $\\Psi$:\n\\begin{equation*}\n\\bigl\\langle\n\\widehat\\mathcal{L}_{k,v},\\Psi\n\\bigr\\rangle\n=\n\\bigl\\langle\n\\widehat\\mathcal{L}_{\\text{pl},v},\\Psi\n\\bigr\\rangle\n(1+o(1))\n+O(q_v^{A_8} C(\\mathfrak{F}_k)^{-C_5}\n\\pnorm{\\Psi}_{\\infty})\n+O(q_v^{-A}\\pnorm{\\Psi}_\\infty),\n\\end{equation*}\n\\end{proposition}\n\n\n\\begin{proof}\nLet $\\kappa$ be a large enough integer. We apply the bounds towards Ramanujan in the form~\\eqref{gn-beta} to those term in~\\eqref{CmL-beta} with $\\nu > \\kappa$. The contribution of those terms to $\\bigl\\langle\n\\widehat\\mathcal{L}_{k,v},\\Psi\n\\bigr\\rangle$ is uniformly bounded by\n\\begin{equation*}\n\\ll q_v^{\\kappa(\\theta-\\frac{1}{2})}\\pnorm{\\Psi}_{\\infty}.\n\\end{equation*}\nWe have that $A:=\\kappa(\\frac{1}{2}-\\theta)$ may be chosen as large as we want since $\\theta < \\frac{1}{2}$ is fixed and $\\kappa$ is arbitrary large.\n\nFor those terms in~\\eqref{CmL-beta} with $\\nu \\le \\kappa$ we apply~\\eqref{main-F-pl}. Their contribution to $\\bigl\\langle\n\\widehat\\mathcal{L}_{k,v},\\Psi\n\\bigr\\rangle$ is equal to\n\\begin{equation*}\n-\\log q_v \\sum_{1\\le \\nu\\le \\kappa}\n\\beta^{(\\nu)}_{\\text{pl},v} q_v^{-\\nu\/2}\n\\Psi(\\frac{\\nu}{2\\pi}\\log q_v) + O(q_v^{A_7+B_9\\kappa}\\pnorm{\\Psi}_{\\infty} C(\\mathfrak{F}_k)^{-C_5}).\n\\end{equation*}\n\nThe next step is now to complete the $\\nu$-sum. Applying~\\eqref{gn-beta} we see that the terms $\\nu >\\kappa$ yield another remainder term of the form $q_v^{-A}\\pnorm{\\Psi}_{\\infty}$ with $A$ arbitrary large (again depending on $\\kappa$).\n\\end{proof}\n\n\\subsection{Handling remainder terms}\\label{sec:pf:error} In this subsection we handle the various remainder terms and show that they don't contribute to $D(\\mathfrak{F}_k,\\Phi)$ in the limit when $k\\to \\infty$.\n\nThe estimates above are applied to the compactly supported function\n\\begin{equation}\\label{Psi-Phi}\n\\Psi(y):= \\widehat \\Phi(\\frac{2\\pi y}{\\log C(\\mathfrak{F}_k)}),\\quad y\\in \\mathbb{R}.\n\\end{equation}\nWe can see that both $\\pnorm{\\Psi}_{\\infty}$ and $\\bigl\\lVert \\widehat \\Psi \\bigr \\rVert_1$ only depend on $\\Phi$ which has been fixed. As a consequence these norms are uniformly bounded.\n\nAbove archimedean places $v\\in S_\\infty$ we encountered in~\\S\\ref{sec:pf:arch} the remainder term $O(\\bigl\\lVert \\widehat \\Psi \\bigr \\rVert_1)$. Because of the overall multiplicative factor $1\/\\log C(\\mathfrak{F}_k)$ in~\\eqref{def:D1v}, this remainder is negligible as $k\\to \\infty$.\n\nAbove the non-archimedean places with $v\\mid \\mathfrak{n}_k$ or $v\\in S_0$ we use the general bounds from~\\S\\ref{sec:pf:gn}. We obtain\n\\begin{equation}\\label{remainS0}\n\\sum_{v\\in S_0,\\text{ and $v\\mid \\mathfrak{n}_k$}}\n\\Bigl\\langle\n\\widehat \\mathcal{L}_{k,v},\\Psi\n\\Bigr\\rangle\n\\ll\n\\sum_{v\\in S_0,\\text{ and $v\\mid \\mathfrak{n}_k$}}\nq_v^{\\theta-\\frac{1}{2}} \\log q_v \\pnorm{\\Psi}_\\infty\n\\ll\n1+\n\\#\\set{v\\mid \\mathfrak{n}_k}.\n\\end{equation}\nIn the last inequality we used the fact that $S_0$ is fixed and that $\\theta < \\frac{1}{2}$. The overall multiplicative factor $1\/\\log C(\\mathfrak{F}_k)$ in~\\eqref{def:D1v} shows that these terms are negligible as $k\\to \\infty$. Indeed it is easy to verify that\n\\begin{equation}\\label{remainlevel}\n\\#\\set{v\\mid \\mathfrak{n}_k} = o(\\log \\mathbb{N}(\\mathfrak{n}_k)),\\qtext{as $k\\to \\infty$,}\n\\end{equation}\nand we conclude using Hypothesis~\\ref{hyp:cond} that this is $o(\\log C(\\mathfrak{F}_k))$.\n\nWe partition the set of generic non-archimedean places $S_\\text{gen}$ into two sets $S_\\text{main}$ and $S_\\text{cut}$ where\n\\begin{equation}\\label{def:Scut}\nS_\\text{cut}= \\set{v\\in S_\\text{gen}:\\ q_v>C(\\mathfrak{F}_k)^\\nu}.\n\\end{equation}\nSince the support of $\\widehat \\Phi$ is included in $(-\\delta,\\delta)$ we know that $\\widehat \\Phi(\\frac{\\nu \\log q_v}{\\log C(\\mathfrak{F}_k)})$ vanishes for all $v\\in S_\\text{cut}$.\n\nAbove the generic places $v\\in S_\\text{main}$ we use the estimate in Proposition~\\ref{prop:S1}. The second remainder term yields\n\\begin{equation}\\label{secondremain}\n\\sum_{v\\in S_\\text{main}} q_v^{-A} \\pnorm{\\Psi}_\\infty = O(1).\n\\end{equation}\nThis is again negligible in view of the overall factor $1\/\\log C(\\mathfrak{F}_k)$ in~\\eqref{def:D1v}. The first remainder term is negligible as well:\n\\begin{equation} \\label{firstremain}\n  \\sum_{v\\in S_\\text{main}}^{}  q_v^{A_8}  \\pnorm{\\Psi}_{\\infty} C(\\mathfrak{F}_k)^{-C_5} \\ll\nC(\\mathfrak{F}_k)^{\\delta(C+1) - c}.\n\\end{equation}\nRecall that $\\delta$ is chosen small small enough such that $\\delta(A_8+1)<C_5$.\n\nWe show that the contribution of the higher moments $\\nu\\ge 3$ is negligible.\nWe recall the definition~\\eqref{CmL-pl} of $\\widehat\\mathcal{L}_{\\text{pl},v}$ and the bound~\\eqref{gn-pl-beta} for $\\beta_{\\text{pl},v}^{(\\nu)}$. The contribution of the higher moments is uniformly bounded by\n\\begin{equation}\\label{highermom}\n\\sum_{v\\in S_\\text{main}}\n\\sum_{\\nu\\ge 3}^{}\n\\log q_v q_v^{-\\nu\/2}\n\\Psi\\left( \\frac{\\nu \\log q_v}{2\\pi} \\right)\n\\ll \\pnorm{\\Psi}_\\infty\n\\sum_{v\\in \\mathcal{V}_F^\\infty}^{}\nq_v^{-3\/2}\\log q_v \\ll 1.\n\\end{equation}\n\n\n\\subsection{Sum over primes}\\label{sec:pf:prime} It remains to estimate the main term which consists of sums over the places $v\\in S_\\text{main}$. Thanks to~\\eqref{highermom} we write the main contribution to the one-level density as\n\\begin{equation*\n\\frac{1}{\\log C(\\mathfrak{F}_k)} \\sum_{v\\in S_\\text{main}}\n\\bigl\\langle\n\\widehat\\mathcal{L}_{\\text{pl},v},\\Psi\n\\bigr\\rangle\n=\nM_1+M_2+O(\\frac{1}{\\log C(\\mathfrak{F}_k)})\n\\end{equation*}\nwhere for $\\nu=1,2$ we have (recall the relation~\\eqref{Psi-Phi} between $\\Phi$ and $\\Psi$)\n\\begin{equation}\\label{def:Mv}\n\\begin{aligned}\nM_\\nu &:= \\frac{-1}{\\log C(\\mathfrak{F}_k)}\n\\sum_{v\\in S_\\text{main}}^{}\n\\log q_v q_v^{-\\nu\/2}\n\\beta_{\\text{pl},v}^{(\\nu)}\n\\Psi\\left( \\frac{\\nu \\log q_v}{2\\pi} \\right)\\\\\n&=- \\sum_{v\\in S_\\text{main}}^{}\n\\frac{ \\log q_v}{\\log C(\\mathfrak{F}_k)}\nq_v^{-\\nu\/2}\n\\beta_{\\text{pl},v}^{(\\nu)}\n\\widehat \\Phi\\left( \\frac{\\nu \\log q_v}{\\log C(\\mathfrak{F}_k)} \\right).\n\\end{aligned}\n\\end{equation}\n\nWe shall need the prime number theorem and Cebotarev density theorem over the number field $F$ which we now proceed to recall, following e.g.~\\cite{book:Narkiewicz}*{Chap.~7}. Let $E\/F$ be a finite Galois extension with Galois group $\\Gamma={\\rm Gal}(E\/F)$. For all conjugacy class $\\theta \\in \\mathscr{C}(\\Gamma)$, recall that $\\mathcal{V}_F(\\theta)$ consists of those unramified places $v\\in \\mathcal{V}_F^\\infty$ such that ${\\rm Fr}_v\\in \\theta$.\n\\begin{proposition}\n(i) The prime number theorem is the asymptotic\n\\begin{equation*}\n\\#\\set{v\\in \\mathcal{V}_F^{\\infty},\\ q_v \\le x} \\sim \\frac{x}{\\log x}\n,\\qtext{as $x\\to \\infty$.}\n\\end{equation*}\n\n(ii) The Cebotarev density theorem is the asymptotic\n\\begin{equation*}\n\\#\\set{v\\in \\mathcal{V}_F(\\theta),\\ q_v\\le x} \\sim \\frac{x}{\\log x}\\times \\frac{\\abs{\\theta}}{\\abs{\\Gamma}}\n,\\qtext{as $x\\to \\infty$.}\n\\end{equation*}\n\n\\end{proposition}\n\nWe deduce the following estimate which will be used to evaluate $M_2$,\n\\begin{equation}\\label{pnt}\n\\sum_{v\\in \\mathcal{V}_F(\\theta)}\n\\frac{\\log q_v}{\\log C(\\mathfrak{F}_k)}\nq_v^{-1}\n\\widehat \\Phi(\\frac{\\log q_v}{\\log C(\\mathfrak{F}_k)})\n=(\\frac{\\abs{\\theta}}{\\abs{\\Gamma}}+o(1))\n\\int_{0}^{\\infty}\n\\widehat \\Phi(y) dy\n,\\qtext{as $k\\to \\infty$.}\n\\end{equation}\n Note that if we replace $\\log q_v$ by $2\\log q_v$, or any other constant multiple, the asymptotic remains the same. A similar observation holds when we replace $\\log q_v$ by $-\\log q_v$, as case which will occur when considering the contribution of $\\overline{D_{v}}(\\mathfrak{F}_k,\\Phi)$. The integral on the right-hand side would then range from $-\\infty$ to $0$, and adding the two contributions produces the integral $\\int\\limits_{-\\infty}^{\\infty} \\widehat \\Phi(y)dy = \\Phi(0)$.\n\n\\subsection{Computing the moments}\\label{sec:pf:M12} We now want to evaluate $M_1$ and $M_2$. The following will play an important role to go from global to local $L$-morphism. Recall that by assumption \\[r:\\widehat G \\rtimes {\\rm Gal}(\\bar F\/F)={}^LG \\to \\GL_d(\\mathbb{C})\\] is irreducible and does not factor through ${\\rm Gal}(\\bar F\/F)$.\n\\begin{lemma}\\label{lem:rhatG}\nThe restriction $r|_{\\widehat G}$ does not contain the trivial representation.\n\\end{lemma}\n\\begin{proof}   If there were a non-zero vector in $\\mathbb{C}^d$ invariant by $r(\\widehat G)$ then all its translates by ${\\rm Gal}(\\bar F\/F)$ would still be invariant because $\\widehat G$ is a normal subgroup of ${}^LG$. Because $r$ is irreducible these translates generate $\\mathbb{C}^d$ and thus the restriction $r|_{\\widehat G}$ would be trivial%\n  \\footnote{In the sense that $r|_{\\widehat G}$ would be a direct sum of trivial representations. In the sequel we use this slight abuse of notation when saying that a representation is \\Lquote{trivial}.}\nwhich yields a contradiction. For an extension of this argument see e.g.~\\cite{book:serre:rep}*{Prop.~24, \\S~I.8.1}.\n\\end{proof}\nWe recall some of the local results from~\\S\\ref{s:Satake-trans}. Since $v\\in S_\\text{main}$, the group $G$ is unramified over $F_v$ and the restriction $r|_{\\widehat{G}\\rtimes W_{F_v}}$ is an unramified $L$-morphism which factors through $\\widehat{G} \\rtimes W^{\\mathrm{ur}}_{F_v}$. Note that this restriction might not be irreducible in general.\n\nLet $A$ is a maximal $F_v$-split torus and $\\Omega_{F_v}$ the $F_v$-rational Weyl group for $(G(F_v),A)$. Recall the Satake isomorphism\n\\begin{equation*}\n  \\mathcal{S}:\\mathcal{H}^{\\mathrm{ur}}(G) \\xrightarrow{\\sim} \\mathcal{H}^{\\mathrm{ur}}(A)^{\\Omega_{F_v}}.\n\\end{equation*}\nFor the group $\\GL_d$ the right hand-side is identified with $\\mathbb{C}[Y^\\pm_1,\\cdots Y^\\pm_d]^{\\mathscr{S}_d}$.\n\nWe recall the morphism of unramified Hecke algebras $r^*:\\mathcal{H}^{\\mathrm{ur}}(\\GL_d)\\to \\mathcal{H}^\\mathrm{ur}(G(F_v))$ and the test functions ($\\nu\\ge 1$):\n\\begin{equation*}\n  \\phi_v^{(\\nu)}:=r^*(Y_1^{\\nu}+\\cdots+Y_d^{\\nu}) \\in \\mathcal{H}^{\\mathrm{ur}}(G(F_v)).\n\\end{equation*}\nThe definition~\\eqref{def:beta-pl} of $\\beta_{\\text{pl},v}^{(\\nu)}$ may now be rewritten as\n\\begin{equation*}\n  \\beta_{\\text{pl},v}^{(\\nu)} = \\widehat \\mu^{\\text{pl}}_{v}(\\widehat \\phi^{(\\nu)}_v) = \\phi^{(\\nu)}_v(1).\n\\end{equation*}\n\n\\begin{proposition}\\label{prop:pf:beta1}\n For all $v\\in S_\\text{main}$ the following estimate holds\n\\begin{equation*}\n\\beta^{(1)}_{\\text{pl},v}=O(q_v^{-1}).\n\\end{equation*}\n\n\\end{proposition}\n\n\\begin{proof} We decompose the restriction of $r$ to $\\widehat{G} \\rtimes W^{\\mathrm{ur}}_{F_v}$ into a direct sum of irreducible $\\oplus_i r_i$. By Lemma~\\ref{lem:rhatG} each $r_i|_{\\widehat G}$ does not contain the trivial representation. In particular each $r_i$ does not factor through $W_{F_v}^{\\mathrm{ur}}$.\n\n  We can now apply Lemma~\\ref{l:bound-1st-moment} which shows that\n\\begin{equation*}\n  \\abs{\\phi^{(1)}_v(1)} \\le q_v^{-1} \\abs{\\Omega_{F_v}} \\max\\limits_{w\\in \\Omega_{F_v}}p(\\lambda_i \\star_{w} 0) .\n\\end{equation*}\nHere $\\lambda_i$ is as defined in~\\S\\ref{sub:Satake-trans}. The two terms $\\abs{\\Omega_{F_v}}$ and $p(\\lambda_i \\star_{w} 0)$ are easily seen to be bounded (uniformly with respect to $v\\in S_\\text{main}$).\n\\end{proof}\n\nFrom the Proposition~\\ref{prop:pf:beta1} we can now deduce that\n\\begin{equation} \\label{M1}\nM_1 = O(\\frac{1}{\\log C(\\mathfrak{F}_k)}),\n\\end{equation}\nbecause in~\\eqref{def:Mv} the sum over $v$ of $q_v^{-3\/2}$ is convergent.\n\nFor the second moment $M_2$ we need more information which is provided by the following. Recall the Frobenius-Schur invariant $s(r)\\in \\set{-1,0,1}$ associated to the irreducible finite dimensional representation $r$ as in~\\S\\ref{sec:b:fs}.\n\nRecall the finite extension $F_1\/F$ from~\\S\\ref{s:Sato-Tate}. We also choose a finite extension $F_2\/F_1$ such that $r$ factors through $\\widehat G \\rtimes {\\rm Gal}(F_2\/F)$. Let $\\Gamma_2:={\\rm Gal}(F_2\/F)$ and denote by $\\mathscr{C}(\\Gamma_2)$ the set of conjugacy classes in $\\Gamma_2$.\n\\begin{proposition}\\label{prop:stheta}\n(i) For all $\\theta\\in \\mathscr{C}(\\Gamma_2)$ the is an algebraic integer $s(r,\\theta)$ such that the following holds.\nFor all $v\\in S_\\text{main}$,\n\\begin{equation}\\label{eq:prop:stheta}\n  \\beta^{(2)}_{\\text{pl},v}=s(r,[{\\rm Fr}_v]) + O(q_v^{-1}).\n\\end{equation}\nHere $[{\\rm Fr}_v]\\in \\mathscr{C}(\\Gamma_2)$ is the conjugacy class of ${\\rm Fr}_v$ in $\\Gamma_2$.\n\n  (ii) The following identity holds\n\\begin{equation*}\n  s(r)= \\sum_{\\theta \\in \\mathscr{C}(\\Gamma_2)} \\frac{\\abs{\\theta}}{\\abs{\\Gamma_2}} s(r,\\theta).\n\\end{equation*}\n\\end{proposition}\n\\begin{proof}\n  (i) We proceed in way similar to the proof of Proposition~\\ref{prop:pf:beta1} above. We shall give an explicit formula~\\eqref{pf:stheta} for $s(r,\\theta)$.\n\n  We decompose $\\MSym^2 r=\\oplus \\rho^+_i$ (resp. $\\bigwedge^2 r=\\oplus \\rho^-_i$) into a direct sum of irreducible representation of $\\widehat G \\rtimes {\\rm Gal}(E\/F)$. Then we can decompose for each $i$ the restriction $\\rho^+_i|_{\\widehat G\\rtimes W^{\\mathrm{ur}}_{F_v}}=\\oplus_j \\rho^+_{ij}$ as a direct sum of irreducible representations of $\\widehat G\\rtimes W^{\\mathrm{ur}}_{F_v}$. Similarly we let $\\rho^-_i|_{\\widehat G\\rtimes W^{\\mathrm{ur}}_{F_v}}=\\oplus_j \\rho^-_{ij}$.\n\n Let  $ \\widehat \\phi^+_{ij} := (\\rho^+_{ij})^* (Y^2_1+\\ldots +Y^2_{d_i})$\n\tand similarly for $\\phi^{-}_{ij}$. Then it is easily verified that\n\\begin{equation*}\n\t  \\widehat \\phi^{(2)}_v=\\sum_{ij} \\widehat \\phi^+_{ij}\n\t  - \\sum_{ij} \\widehat \\phi^-_{ij}\n\\end{equation*}\n\nWe now distinguish two cases. In the first case, $i$ is such that $\\rho^+_i$ does not factor through ${\\rm Gal}(E\/F)$. Then by Lemma~\\ref{lem:rhatG} the restriction $\\rho^+_i|_{\\widehat G}$ does not contain the trivial representation. Thus for all $j$, $\\rho^+_{ij}|_{\\widehat G}$ does not contain the trivial representation. In particular $\\rho^+_{ij}$ does not factor through $W^{\\mathrm{ur}}_{F_v}$. By Lemma~\\ref{l:bound-1st-moment} we deduce that $\\phi^+_{ij}(1) = O(q_v^{-1})$. These representations $\\rho^+_i$ only contribute to the error term in~\\eqref{eq:prop:stheta}.\n\nIn the second case, $i$ is such that $\\rho^+_i$ does factor through ${\\rm Gal}(E\/F)$. Then for all $j$, $\\rho^+_{ij}$ factors through $W_F^{\\mathrm{ur}}$ (in particular it is $1$-dimensional). We have that $\\widehat\\phi^+_{ij}(1)=\\rho^+_{ij}({\\rm Fr}_v)$. By linearity we deduce that $\\sum_{j} \\phi^+_{ij}(1)=\\Mtr \\rho^+_i({\\rm Fr}_v)$. This is an algebraic integer which depends only on the conjugacy class of ${\\rm Fr}_v$ in $\\Gamma_2$.\n\nWe proceed in the same way for $\\phi_{ij}^-$. We deduce that~\\eqref{eq:prop:stheta} holds with $\\theta=[{\\rm Fr}_v]$ and $s(r,\\theta)=s^+(r,\\theta)-s^-(r,\\theta)$, where\n\\begin{equation}\\label{pf:stheta}\n  s^+(r,\\theta):= \\sum_{\n  \\substack{\n  \\rho^+_i \\text{ factors}\\\\\n  \\text{through ${\\rm Gal}(E\/F)$}}\n  }\n  \\Mtr \\rho^+_i(\\theta)\n\\end{equation}\nand similarly for the definition of $s^-(r,\\theta)$. This concludes the proof of assertion (i).\n\n(ii) By orthogonality of characters we have for each $i$ such that $\\rho_i^+$ factors through ${\\rm Gal}(E\/F)$,\n\\begin{equation*}\n  \\sum_{\\theta \\in \\mathscr{C}(\\Gamma_2)}^{}\n  \\frac{\\abs{\\theta}}{\\abs{\\Gamma_2}} \\Mtr \\rho_i^+(\\theta) = \\langle\\mathds{1},\\rho_i^+\\rangle=\n  \\begin{cases}\n\t1, & \\text{if $\\rho_i^+=\\mathds{1}$}\\\\\n\t0, & \\text{otherwise}.\n  \\end{cases}\n\\end{equation*}\nWe deduce that\n\\begin{equation*}\n   \\sum_{\\theta \\in \\mathscr{C}(\\Gamma_2)}^{}\n  \\frac{\\abs{\\theta}}{\\abs{\\Gamma_2}} s^+(r,\\theta) =\\langle\\mathds{1},\\MSym^2 r\\rangle,\n\\end{equation*}\nthe multiplicity of the trivial representation $\\mathds{1}$ in $\\MSym^2 r$ (as a representation of $\\widehat G \\rtimes {\\rm Gal}(E\/F)$). The same identity holds for $s^-(r,\\theta)$ and $\\bigwedge^2 r$. From the definition of the Frobenius-Schur indicator $s(r)$ in \\S\\ref{sec:b:fs} we conclude the proof of the proposition.\n \\end{proof}\n\nWe are now able to estimate the second moment:\n\\begin{equation}\\label{pf:M2estimate}\n  \\begin{aligned}\n\tM_2 &= -\\sum_{v\\in S_\\text{main}} \\frac{\\log q_v}{\\log C(\\mathfrak{F}_k)} q_v^{-1} \\beta^{(2)}_{\\text{pl},v} \\widehat \\Phi\\left( \\frac{\\nu \\log q_v}{\\log C(\\mathfrak{F}_k)} \\right)\\\\\n\t&= - \\sum_{\\theta \\in \\mathscr{C}(\\Gamma_2)}\n\ts(r,\\theta)\n\\sum_{v\\in S_\\text{main}\\cap \\mathcal{V}_F(\\theta)}\n\\frac{\\log q_v}{\\log C(\\mathfrak{F}_k)} q_v^{-1} \\widehat \\Phi\\left( \\frac{\\nu \\log q_v}{\\log C(\\mathfrak{F}_k)} \\right) + O(\\frac{1}{\\log C(\\mathfrak{F}_k)}).\n  \\end{aligned}\n\\end{equation}\n We note the following estimate\n\\begin{equation*}\n\\sum_{v\\in \\mathcal{V}_F^{\\infty}-S_\\text{main}}\n\\frac{\\log q_v}{\\log C(\\mathfrak{F}_k)} q_v^{-1} \\ll \\frac{\\log \\log C(\\mathfrak{F}_k)}{\\log C(\\mathfrak{F}_k)}=o(1),\n\\end{equation*}\nuniformly as $k\\to \\infty$. This allows to extend the sum to $v\\in S_\\text{gen} \\cap \\mathcal{V}_F(\\theta)$ in~\\eqref{pf:M2estimate}.\n\nApplying Cebotarev equidistribution theorem~\\eqref{pnt} we deduce that\n\\begin{equation} \\label{M2}\n  \\begin{aligned}\n\tM_2&= - \\sum_{\\theta \\in \\mathscr{C}(\\Gamma_2)}\n\ts(r,\\theta) (\\frac{\\abs{\\theta}}{\\abs{\\Gamma_2}}+o(1))\n\\frac{1}{2} \\int_{0}^\\infty\n\\widehat \\Phi(y)dy\\\\\n&=(-\\frac{s(r)}{2}+o(1)) \\int_{0}^\\infty\n\\widehat \\Phi(y)dy.\n  \\end{aligned}\n\\end{equation}\nThe last line follows from Proposition~\\ref{prop:stheta}.(ii) above.\n\n\\subsection{Conclusion}\\label{sec:pf:ccl} We now gather all the estimates and conclude the proof of Theorem~\\ref{th:onelevel}. The explicit formula~\\eqref{D1explicit} expresses $D(\\mathfrak{F}_k,\\Phi)$ as the sum of four terms. The term $D_{\\text{pol}}(\\mathfrak{F}_k,\\Phi)$ goes to zero as $k\\to \\infty$ as consequence of Hypothesis~\\ref{hyp:poles}, see~\\S\\ref{sec:pf:poles}.\n\nThe archimedean terms are evaluated in~\\eqref{archterms}. In addition with the second term in~\\eqref{D1explicit} which involves $\\log q(\\Pi)$, these contribute\n\\begin{equation*}\n\\frac{\\widehat \\Phi(0)}{\\abs{\\mathfrak{F}_k}} \\sum_{\\Pi\\in\\mathfrak{F}_k}\n\\frac{\\log C(\\Pi)}{\\log C(\\mathfrak{F}_k)} + o(1).\n\\end{equation*}\nThis is equal to $\\widehat \\Phi(0)+o(1)$ (using the Hypothesis~\\ref{hyp:cond} for families in the level aspect).\n\nWe now turn to the non-archimedean contribution. The places $v\\in S_0$ and $v\\mid \\mathfrak{n}_k$ are negligible thanks to~\\eqref{remainS0} and~\\eqref{remainlevel}, respectively.\n\nIt remains the non-archimedean places $v\\in S_\\text{gen}=S_\\text{main}\\bigsqcup S_\\text{cut}$. The contribution from $v\\in S_\\text{cut}$ is zero because the support of $\\widehat \\Phi$ is included in $(-\\delta,\\delta)$, see~\\eqref{def:Scut}.\n\nFor each $v\\in S_\\text{main}$ we apply Proposition~\\ref{prop:S1}. The sum over $v\\in S_\\text{main}$ of the remainder terms is shown to be negligible in~\\eqref{secondremain} and~\\eqref{firstremain}. For the main term the estimate~\\eqref{highermom} shows that the contribution of the higher moments is negligible. It remains the two terms $M_1+M_2$ as defined in~\\eqref{def:Mv}.\n\nThe asymptotic of $M_1$ and $M_2$ are given in~\\eqref{M1} and~\\eqref{M2} respectively. There is a similar contribution from the conjugate $\\overline{D_{v}}(\\mathfrak{F}_k,\\Phi)$. Overall this yields\n\\begin{equation*}\n\\begin{aligned}\n\\sum_{v\\in S_\\text{main}}^{} D_{v}(\\mathfrak{F}_k,\\Phi)+\n\\overline{D_{v}}(\\mathfrak{F}_k,\\Phi)\n&= -\\frac{s(r)}{2} \\int_{0}^{\\infty} \\widehat \\Phi(y)dy\n-\\frac{s(r)}{2} \\int_{-\\infty}^{0} \\widehat \\Phi(y) dy\n+o(1)\\\\\n&= -\\frac{s(r)}{2} \\Phi(0)+o(1).\n\\end{aligned}\n\\end{equation*}\n\nWe can now conclude that\n\\begin{equation*}\n\\lim_{k\\to \\infty} D(\\mathfrak{F}_k,\\Phi)=\\widehat \\Phi(0) - \\frac{s(r)}{2} \\Phi(0).\n\\end{equation*}\nThis is the statement of Theorem~\\ref{th:onelevel}. \\qed\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{Computational details}\n\n\n\\subsection{Numerical analysis methods}\nEELS and non-resonant inelastic  X-ray scattering give access to the loss function $L(\\mathbf{q},\\omega)$\nwith complementary degrees of accuracy in the $\\mathbf{q}$ range~\\cite{fossard_prb2017}.\n\nTheory-wise, $L(\\mathbf{q},\\omega)$ can be calculated accurately from the dielectric function $\\epsilon(\\mathbf{q},\\omega)$, obtained as a solution of the BSE.\nThis can be cast in the form of an eigenvalue problem whose Hamiltonian is most often written in a basis of independent-particle (IP) transitions of index $t=(v,\\mathbf{k})\\to(c,\\mathbf{k}+\\mathbf{q})$ between occupied and empty states of an underlying IP model, e.g. the Kohn-Sham system. \nHere $(v,\\mathbf{k})$ indicate the initial state and $(c,\\mathbf{k}+\\mathbf{q})$ the final state, where $\\mathbf{q}$ is the exchanged momentum laying inside the first Brillouin zone.\nWithin this framework and including only resonant transitions\n\\begin{equation}\n\\epsilon(\\mathbf{q},\\omega) \\propto \\sum_\\lambda \\frac{ I_\\lambda(\\mathbf{q})  }{E_\\lambda(\\mathbf{q}) - \\omega +i\\eta}\\; ,\n\\label{eq:epsilon}\n\\end{equation}\nwhere $E_\\lambda(\\mathbf{q})$ is the energy of the $\\lambda$th exciton~\\cite{onida-reining-rubio_rmp2002} and $\\eta$ a positive infinitesimal quantity.\nThe spectral intensity\n\\begin{equation}\nI_\\lambda(\\mathbf{q}) = \\left| \\sum_t \\tilde{\\rho}_t(\\mathbf{q}) A^\\lambda_t(\\mathbf{q}) \\right|^2= \\left|\\sum_{t} M_t^\\lambda(\\mathbf{q})\\right|^{2}\n\\label{eq:spectral_weight}\n\\end{equation}\nis the modulus squared of a linear combination of IP-transition matrix elements $\\tilde{\\rho}_t(\\mathbf{q})=\\langle v\\mathbf{k} | e^{-i\\mathbf{q}\\cdot\\mathbf{r}} | c\\mathbf{k}+\\mathbf{q} \\rangle$ weighted by the exciton wave function components $A^\\lambda_t(\\mathbf{q})$.\nThe exciton $\\lambda$ is called ``bright\" when $I_\\lambda(\\mathbf{q})$ is sizeably high, and conversely it is called ``dark\" when $I_\\lambda(\\mathbf{q})\\approx 0$.\nThis can happen if either $\\tilde{\\rho}_t(\\mathbf{q})$ or $A^\\lambda_t(\\mathbf{q})$ or both are negligible for all $t$, or when IP-transitions interfere destructively leading to a vanishing sum in expression~\\eqref{eq:spectral_weight}.\nThus it is sensible to introduce the normalised cumulant weight~\\cite{gatti-sottile_prb2013,gogoi-sponza_prb2015}: \n\\begin{equation}\n\\mathcal{J}_\\lambda(\\mathbf{q},E) = \\frac{1}{I_\\lambda(\\mathbf{q})} \\left|\\sum_{t:E_t\\le E} M_t^\\lambda(\\mathbf{q}) \\right|^2\n\\label{eq:cumulant}\n\\end{equation}\nwhich allows for a visualization of the building-up of the exciton spectral weight as a function of the IP-transition energy $E$.\nThis function is positively defined, it tends asymptotically to 1 and in general is not monotonic. \n\nThe normalized cumulant weight~\\eqref{eq:cumulant} gives a piece of information relying on the energy of the IP-transitions, though more detailed analysis can be achieved by a careful study of the single $M_t^\\lambda(\\mathbf{q})$ amplitudes themselves.\nIn particular one can use the phase of $M_t^\\lambda(\\mathbf{q})$ to split IP-transitions into groups depending on their sign in the sum~\\eqref{eq:spectral_weight}.\nThis allows for a deconvolution of the exciton (which includes all IP-transitions) into competing groups of IP-transitions, the intensity of the total peak resulting from the interplay of these contributions.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Results}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth, trim= 25mm 18mm 35mm 18mm, clip]{eels_dispersion}\n\\caption{(Color online) GW-BSE simulations of the Loss function; thicker lines correspond to $q=5q_0=0.7$\\AA$^{-1}$ and $q=8q_0=1.1$\\AA$^{-1}$.\nInset: The peak dispersion compared against EELS measurements~\\cite{schuster_arxiv2017}.\nSimulated data have been blue-shifted by 0.47 eV.}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth,trim= 15mm 15mm 21mm 8mm, clip]{eps-eels}\n    \\caption{(Color online) (a) Simulated spectra of  $L(\\mathbf{q},\\omega)$ (solid) and $\\text{Im}[\\epsilon]$ with and without electron-hole interaction (dashed and dot-dashed),  at $\\mathbf{q}=5\\mathbf{q}_0\\approx 0.7$\\AA$^{-1}$. Circles mark the excitonic energies $E_\\lambda(\\mathbf{q})$ for $\\lambda \\le 6$.\n       (b) As in panel (a), but for the exchanged momentum $\\mathbf{q}=8\\mathbf{q}_0\\approx 1.1$\\AA$^{-1}$. The calculation without electron-hole interaction is not reported.}\n\\label{fig:eps-dispersion}\n\\end{figure}\n\nIn Figure~\\ref{fig1} we report the simulated loss function\nfor exchanged momenta $\\mathbf{q}\\parallel \\Gamma K$ at intervals of $\\mathbf{q}_0=K\/12\\approx0.14$~\\AA$^{-1}$ (see Appendixes). \nIn the inset, circles depict the calculated dispersion of the peak compared to \n\n the experimental data (red bullets) extracted from~\\cite{schuster_arxiv2017}.\nBeside a blue-shift of about 0.47~eV that comes from a well-known underestimation of the gap with the G$_0$W$_0$ approximation in this material~\\cite{ludger}, the calculated spectra and their dispersion are in very good agreement with the measurements.\nIn particular our simulations reproduce the fact that the lowest-energy excitation is at $\\mathbf{q}=5\\mathbf{q}_0\\approx 0.7$~\\AA$^{-1}$, where the peak attains its highest intensity,  and that approximately at $\\mathbf{q}=8\\mathbf{q}_0\\approx 1.1$~\\AA$^{-1}$ the peak is strongly suppressed (cfr. Figure 1 in~\\cite{schuster_arxiv2017}).\nAt higher $\\mathbf{q}$, the loss function increases again with abrupt intensity, reproducing the strong exciton expected at $\\mathbf{q}=K$ and already analysed elsewhere in literature~\\cite{galambosi_prb2011,fugallo_prb2015,fossard_prb2017}.\n\n\nIn this energy range, it turns out that $|\\epsilon(\\mathbf{q},\\omega)|$ does not vanish, consequently equation \\eqref{eq:loss_function} allows us to attribute an interband character to the excitation and put features of the loss function in direct relation to peaks of $\\text{Im}[\\epsilon]$. \nThis appears clearly from Figure~\\ref{fig:eps-dispersion}, where we show that $\\text{Im}[\\epsilon]$ (dashed curve) and $L(\\mathbf{q},\\omega)$ (solid curve) at $\\mathbf{q}=5\\mathbf{q}_0$ and $\\mathbf{q}=8\\mathbf{q}_0$ present the same spectral features.\nWe also mark the energy of the first six excitons, that is $E_\\lambda(\\mathbf{q})$ for $\\lambda \\le 6$, for both momenta with coloured circles whose size is proportional to $\\log[I_\\lambda(\\mathbf{q})]$.\nThe scale of the loss function in the two panels is the same, and similarly for the scale of $\\text{Im}[\\epsilon]$. \nAdditional information about the dispersion of the first six excitons can be found in Appendix B.\n\nFurthermore, for $\\mathbf{q}=5\\mathbf{q}_0$ we report also the corresponding spectrum of $\\text{Im}[\\epsilon]$ without electron-hole interaction, i.e. taking into account only independent-particle (IP) transitions between GW levels (dot-dashed curve). \nThis spectrum appears flat at 5.5~eV where the BSE calculation predicts a relatively sharp peak.\nThis comparison confirms the hypothesis, already advanced in~\\cite{schuster_arxiv2017}, that the peak has an excitonic nature.\n\n\n\n\n\n\n\n\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIn the following we will focus on the reason of the variations of intensity of the first peak, and in particular at momenta $\\mathbf{q}=5\\mathbf{q}_0$ and $\\mathbf{q}=8\\mathbf{q}_0$, where the intensity is at its highest and its lowest.\nBased on the observations done above, we will perform our analysis on $\\text{Im}[\\epsilon]$ instead of working with the more cumbersome loss function. \n\n\n\n\n\n\\subsection{Exciton analysis at $\\mathbf{q}=5\\mathbf{q}_0 \\approx 0.7$~\\AA$^{-1}$}\n\n\n\n\n\n\nThe excitonic peak at $\\mathbf{q}=5\\mathbf{q}_0$ has a binding energy of 0.33~eV, that is the energy difference with respect to the lowest IP-transition with the same $\\mathbf{q}$ (including G$_0$W$_0$ corrections).\nIn Figure~\\ref{fig3} the normalised cumulant weight defined in \\eqref{eq:cumulant} is reported versus the energy $E$ of the IP-transitions.\nWe observe that $\\mathcal{J}_{\\lambda=1}(\\mathbf{q},E)$ is a monotonic function of $E$; it rises steeply up to $E\\approx 6.8$~eV from where its derivative decreases mildly.\nFinally it reaches its asymptotic value of 1 at about $E\\approx 12$~eV (not shown).\nWhat this tells us is that IP-transitions sum up constructively at all energies, with most important contributions coming from transitions of energy $E<6.8$~eV.\nIndeed these few transitions (0.4\\% of the total) account for almost the 42\\% of the spectral weight, as $\\mathcal{J}_1(5\\mathbf{q}_0,6.8)=0.42$ attests.\nStill, to get closer to the full spectral weight, one has to include higher-energy transitions. \nAt $E=9.5$~eV, 85\\% of the spectral weight is accounted for by still a relatively small number of transitions (less than 10\\% of the total).\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{q5_cumulant_inset}\n\\caption{(Color online) Cumulant spectral weight (solid curve), GW-BSE spectrum (dashed curve) at $\\mathbf{q}=5\\mathbf{q}_0$. A red vertical line marks the lowest IP-transition.\nInset: The GW band structure along $KMK'$ where the $KM$ and $MK'$ groups of IP-transitions are sketched in red and blue respectively.}\n\\label{fig3}\n\\end{figure}\n\n\nWe can now gain a deeper insight into the way IP-transitions combine in forming the exciton by looking at the terms of the sum~\\eqref{eq:spectral_weight}.\nLet us  divide the latter group of transitions ($E\\le 9.5$~eV) in three categories: those transitions $t$ for which both real and imaginary parts of the amplitude $M_{t}^\\lambda(\\mathbf{q})$ are positive, those for which both are negative and transitions where they have opposite sign.\nThe latter group turns out to be composed by transitions with amplitude $M_{t}^\\lambda(\\mathbf{q})\\approx 0$, so they do not contribute significantly to the exciton intensity and we can safely neglect them in the analysis.\nThe other two groups enter the sum of Eq.~\\eqref{eq:spectral_weight} with different signs.\nAlmost one third\nof the considered transitions fall into the first group, with positive amplitude $M^\\lambda_{t}(\\mathbf{q})$.\nThey are mostly low-energy transitions.\nA cartography of these transitions is reported in panels (a) and (b) of Figure~\\ref{fig:q5K12_analysis}, where $\\log(|\\sum_{v,c} M_{(v,\\mathbf{k})\\to (c,\\mathbf{k}+\\mathbf{q})}^{\\lambda=1}(\\mathbf{k})|)$ is reported as a function of the valence state $\\mathbf{k}$ or conduction state $\\mathbf{k}+\\mathbf{q}$ for k-points in the $\\Gamma K M$-plane.\nOn the other hand, little more than one third of the transitions belong to the negative amplitude group, and they have higher energy but in general lower intensity.\nTheir maps are reported in panels (c) and (d) of  Figure~\\ref{fig:q5K12_analysis}.\n\n\n\n\\begin{figure}\n\\centering\n\n\\includegraphics[height=3.9cm,trim= 24mm 105mm 177mm 32mm,clip]{q5_kp_en000-950_negative}\n\\includegraphics[height=3.9cm,trim= 160mm 105mm 29mm 32mm,clip]{q5_kp_en000-950_negative}\\\\\n\n\\includegraphics[height=3.9cm,trim= 24mm 105mm 177mm 32mm,clip]{q5_kp_en000-950_positive}\n\\includegraphics[height=3.9cm,trim= 160mm 105mm 29mm 32mm,clip]{q5_kp_en000-950_positive}\n\\caption{(Color online) Top: Transition cartographies of the $KM$ group  at $\\lambda=1$ and $\\mathbf{q}=5\\mathbf{q}_0$ plotted for k-points in the $\\Gamma K M$-plane for valence (a) and conduction states (b).\nBottom: The same for the $MK'$ group.}\n\\label{fig:q5K12_analysis}\n\\end{figure}\n\n\n\nThe analysis suggests the following interpretation.\nTwo groups of transitions participate to the formation of the bright exciton ($\\lambda=1$), observed in Ref.~\\cite{schuster_arxiv2017}.\nOne group (let us call it $KM$-group) is composed mostly by low-energy transitions going from points close to $K$ to points close to $M$ (and similarly $H\\to L$ in the $AHL$-plane, not shown). \nThe lowest-energy transitions of this group have also the larger amplitude $M_t^\\lambda(\\mathbf{q})$, and they sum constructively in the steep part of the cumulant ($E<6.8$~eV).\nAt higher energy, a second group of IP-transitions (call it $MK'$-group), from points in the vicinity of $M$ to points in the vicinity of $K'$ (and $L\\to H'$) enter the sum with a negative amplitude, hence canceling partially the contribution of the $KM$ group.\nThis explains why the derivative of the cumulant decreases from 6.8~eV on, but it is still positive because of the larger number and higher amplitude of the dominating $KM$ group.\n\nThe origin of the peak being established, we can now draw the connection with the single-particle band structure.\nIn the inset of Figure~\\ref{fig3} we report the GW band structure along the relevant path $KMK'$, in good agreement with previous calculations~\\cite{arnaud_prl2006,galambosi_prb2011,fugallo_prb2015,galvani_prb2016} and experiments~\\cite{henck_prb2017}.\nThe $KM$ and the $MK'$ groups of transitions have been sketched with coloured arrows, respectively red and blue.\nAt this $\\mathbf{q}$, the $KM$ group of transitions are basically the indirect transitions between the top valence and the bottom of conduction.\nThe fact that the top valence is close to, but does not coincide with $K$ is consistent with the fact that the lowest excitation is found at $\\mathbf{q}<6\\mathbf{q}_0$.\nThe strength of the peak is explained by the fact that the $KM$ transitions take place between regions of the band structure where bands are particularly flat (van-Hove singularities).\nAlso, the convex curvature of the band structure explains why the $MK'$ transitions start contributing at higher energy and have lower amplitude.\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Exciton analysis at $\\mathbf{q}=8\\mathbf{q}_0\\approx 1.1$~\\AA$^{-1}$}\n\n\n\n\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=0.45\\textwidth, trim= 20mm 20mm 22mm 25mm ,clip]{q8_cumulant}\n\\caption{(Color online) Cumulant spectral weight (solid curve), GW-BSE spectrum (dashed curve). A red vertical line marks the energy of the lowest IP-transition.\nInset: The cumulant spectral weight up to $E<13$~eV in logarithmic scale.}\n\\label{fig5}\n\\end{figure}\n\n\n\nLet us switch now to $\\mathbf{q}=8\\mathbf{q}_0$.\nAt this momentum, the spectral weight is dramatically reduced and it is moved from $\\lambda=1$ to a group of higher-energy excitons among which $\\lambda=2$ and $\\lambda=5$ have the highest (although still very weak) intensity.\nIn the $\\lambda=2$ case, the normalized cumulant weight, reported in Figure~\\ref{fig5} does not grow monotonically, instead first it explodes for $E \\le 6.8$~eV, where it attains the value of $\\sim$ 50, then it attains its maximum between 7 and 7.8~eV and then decreases to reach the asymptotic limit at about 12~eV. \nWe can perform the same analysis as before for IP-transitions of energy $E\\le 9.5$~eV.\nThe cartographies of the IP-transitions are reported in panels (a) and (b) of Figure~\\ref{fig:q8K12_analysis} for the $KM$ group, and (c) and (d) for the $MK'$ group.\nAt variance with the case before the amplitudes of $KM$ and $MK'$ transitions are closer. \n\n\n\\begin{figure}\n\\centering\n\n\\includegraphics[height=3.9cm,trim= 24mm 105mm 177mm 32mm,clip]{q8_kp_en000-950_positive}\n\\includegraphics[height=3.9cm,trim= 160mm 105mm 29mm 32mm,clip]{q8_kp_en000-950_positive}\\\\\n\n\\includegraphics[height=3.9cm,trim= 24mm 105mm 177mm 32mm,clip]{q8_kp_en000-950_negative.pdf}\n\\includegraphics[height=3.9cm,trim= 160mm 105mm 29mm 32mm,clip]{q8_kp_en000-950_negative.pdf}\n\\caption{(Color online) Top: Transition cartographies of the $KM$ group at $\\lambda=2$ and $\\mathbf{q}=8\\mathbf{q}_0$ plotted for k-points in the $\\Gamma K M$-plane for valence (a) and conduction states (b).\nBottom: The same for the $MK'$ group.}\n\\label{fig:q8K12_analysis}\n\\end{figure}\n\n\nThese results can be rationalized as follow.\nMost of the IP-transitions entering in $I_{\\lambda=2}(\\mathbf{q})$ up to 6.8~eV are of the $KM$ group and they sum constructively.\nBut at higher energy, the $MK'$ transitions, which contribute with opposite sign, start having comparable importance.\nThis induces a halt in the increasing trend ($6.8<E<7.8$~eV) and eventually they dominate bending down the cumulant back to its asymptotic limit.\nThe result is that the two groups of transitions almost cancel each other, leading to a very weak intensity.\n\n\n\n\n\n\nIt is worth recalling here that the $\\lambda=2$ exciton is almost degenerate with another exciton of non-negligible intensity ($\\lambda=5$).\nNot surprisingly, carrying out a similar analysis on the latter leads to basically the same results (see Appendix C).\n\n\n\n\n\n\\subsection{Conclusions}\nWe computed the loss function of  bulk $h$--BN solving the \\emph{ab initio} GW-Bethe-Salpeter equation at finite $\\mathbf{q}$ along the $\\Gamma K$ direction, which is  relevant for spectroscopic studies\nWe observe an excitonic peak dispersing of about 0.45~eV, displaying a strong intensity at $\\mathbf{q}\\approx 0.7$~\\AA$^{-1}$, where the excitation energy is the lowest, and almost disappearing at $\\mathbf{q}\\approx 1.1$~\\AA$^{-1}$.\nThese findings are in very good agreement with recent electron energy loss experiments~\\cite{schuster_arxiv2017}.\n\nThe associated dielectric function displays similar characteristics.\nWe show that the peak intensity is determined by the interference of two groups of transitions contributing to the peak formation with opposite signs.\nOur investigation allow us to unveil a non-trivial connection between the exciton dispersion, its intensity and the electronic structure in the vicinity of $K$($H$) and $K'$($H'$) points in bulk $h$--BN, eventually suggesting ways to control excitonic properties by changing the electronic structure in the vicinity of the $K$ and $K'$ valleys.\nIt is worth stressing that with the help of the methodology we devised, it is possible to use spectroscopic methods to probe electronic excitations at the two valleys at the same time. This is of paramount importance, for instance in the vallytronics of layered systems~\\cite{yuan_nanolett2016}.\n\n\n\n\nFurthermore, the methodology presented in this work is of general applicability and could be extended to studies of excitonic properties in any system.\nThe splitting of relevant IP-transitions into appropriately defined groups can simplify the interpretation of the excitonic properties, help the analysis and possibly disclose some non-trivial mechanism. \nWe believe that the strategy adopted here can be employed successfully also to other cases in bulk as well as in 2D materials.\nThis helps the interpretation of measured data (as the case of our application to $h$--BN) but most importantly it can suggest where and how to change the electronic structure whenever a control on the excitonic intensity is required.\n\n\\begin{acknowledgments}\nThe authors thank Doctor R. Schuster for the clarifications regarding the experimental data \\cite{schuster_arxiv2017}.\nThe research leading to these results has received funding from the European Union H2020 Programme under Grant Agreement No. 696656 GrapheneCore1. We acknowledge funding from the French National Research Agency through Project No. ANR-14-CE08-0018.\n\\end{acknowledgments}\n\n\\medskip\n\n\\section{Appendices}\n\n\\subsection{Appendix A: Computational details}\nThe simulated $h$--BN has lattice parameters $a=2.5$~\\AA~ and $c\/a=2.6$~\\cite{solozhenko_ssc1995}.\nThe Kohn-Sham system and the GW corrections have been computed with the ABINIT simulation package (a plane-wave code~\\cite{abinit}). \nNorm-conserving Troullier-Martins pseudopotentials have been used for both atomic species.\nDFT energies and wave functions have been obtained within the local density approximation (LDA) to the exchange-correlation potential, using a plane-wave cutoff energy of 30~Ha and sampling the Brillouin zone with a $8\\times 8\\times 4$ $\\Gamma$-centred k-point grid.\nThe GW quasiparticle corrections have been obtained within the perturbative GW approach. \nThey have been computed on all points of a  $6\\times 6\\times 4$ $\\Gamma$-centred grid, a cutoff energy of 30~Ha defines the matrix dimension and the basis of wave function for the exchange part of the self-energy.\nThe correlation part has been computed including 600 bands and applying the same wave function basis as before.\nTo model the dielectric function, the contour deformation method has been used, computing the dielectric function up to 60~eV, summing over 600 bands and with a matrix dimension of 6.8~Ha. \nThe quasiparticle corrections have been subsequently interpolated on a denser $36\\times 36\\times 4$ k-point grid where the BSE calculation has been carried out.\n\n\nThe macroscopic dielectric function $\\epsilon(\\mathbf{q},\\omega)$ has been calculated at the GW-BSE level in the Tamm-Dancoff approximation using the code EXC~\\cite{exc}.\nWe included six valence bands and three conduction bands; 360~eV is the cutoff energy for both the matrix dimension and the wave function basis. \nThe static dielectric matrix entering in the BSE kernel has been computed within the random phase approximation with local fields, including 350 bands and with a cutoff energy of 120~eV and 200~eV for the matrix dimension and the wave function basis respectively.\nWith these parameters, the energy of the first peaks of $\\epsilon(\\mathbf{q},\\omega)$ are converged within 0.01 eV and their intensity are converged within 5\\%.\n\nAll reported spectra have been convoluted with a Gaussian of $\\sigma=0.05$~eV in order to reduce the noise due to the discrete k-point sampling and to simulate the experimental broadening.\n\n\n\n\\subsection{Appendix B: Dispersion of the first six excitons}\n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[width=0.5\\textwidth, trim= 20mm 20mm 20mm 15mm,clip]{eps_disp_sm}\n\\caption{(Color online) Dispersion of the excitonic energies $E_\\lambda(\\mathbf{q})$ for $\\lambda \\le 6$. The size of the circles is proportional to the logarithm of the peak intensity. A Purple line connects the lowest peaks of the loss function $L(\\mathbf{q},\\omega)$. Red crosses are the peak of the experimental loss function extracted from~\\cite{schuster_arxiv2017}. All calculated points have been blue-shifted by 0.47~eV.}\n\\label{fig:eps_dispersion_sm}\n\\end{figure}\n\nIn Figure~\\ref{fig:eps_dispersion_sm} we report the dispersion of the first six excitons along the line $\\Gamma K$ with coloured circles whose size is proportional to $\\log[I_\\lambda(\\mathbf{q})]$, so larger circles correspond to bright excitations. \nThe points have been obtained within the GW-BSE framework an shifted by 0.47~eV to higher energies.\n\nAs expected~\\cite{arnaud_prl2006,galvani_prb2016,koskelo_prb2017}, at $\\mathbf{q}=\\Gamma$ the first two excitons are degenerate and basically dark, whereas all the peak intensity is concentrated on the degenerate excitons with $\\lambda=3$ and $4$ (the two are superimposed in the plot, so that only $\\lambda=3$ is visible).\nAs soon as one moves away from $\\Gamma$, the degeneracy is lifted~\\cite{koskelo_prb2017} and the first bright peak coincides with the lowest energy exciton ($\\lambda=1$).\t\nThis is valid up to $\\mathbf{q}\\approx 6\\mathbf{q}_0$ (halfway in the $\\Gamma K$ line) where the peak intensity is moved to $\\lambda=2$ as a consequence of a band crossing.\nThe intensity of the excitations is successively reduced at $8\\mathbf{q}_0$ and $9\\mathbf{q}_0$ where several excitons are concentrated in a narrow energy range.\nFinally, as $\\mathbf{q}$ approaches $K$, the exciton $\\lambda=2$ steps-up again concentrating most of the intensity.\n\nWe also report on the same Figure the dispersion of the loss function as measured~\\cite{schuster_arxiv2017} (red crosses) and computed in this work (purple squared curve).\nOne can see that the position of the peak of $L(\\mathbf{q},\\omega)$ follows closely the dispersion of the first bright excitation of $\\text{Im}[\\epsilon]$ (lowest energy larger circles).\n\n\n\n\n\n\n\\subsection{Appendix C: Analysis of the $\\lambda=5$ exciton at $\\mathbf{q}=8\\mathbf{q}_0$} \nAt $\\mathbf{q}=8\\mathbf{q}_0$, the intensity of the peak is very low.\nThis low intensity is basically shared by two excitons, $\\lambda=2$ (analysed in the main text) and $\\lambda=5$ with an energy around 50~meV higher.\nThe analysis with the cumulant weight, reported in Figure~\\ref{fig:cumulant_sm}, has a surprising shape.\nAfter a first increase around 6~eV, the cumulant decreases abruptly and vanishes at 6.5~eV. \nThen it oscillates around the value 0 until one starts including IP-transitions of energy $E \\ge 8.0$~eV.\nRemembering that cumulant weight is defined as a modulus squared, one realizes immediately that what is observed is again a phenomenon of interference (as in the $\\lambda=2$ case), but the dominating group changes during the analysis.\nAt the very beginning ($E \\le 6.1$~eV) the $KM$ group constructs the peak, but immediately after the $MK'$ transitions cancel this contribution and leads the cumulant weight back to 0. \nFrom this point, the two contributions mutually cancel and it is only at $E>8$~eV that the $MK'$ group prevails and the cumulant start growing monotonically to its asymptotic limit.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.5\\textwidth, trim= 20mm 20mm 20mm 20mm, clip]{cumulant_sm}\n\\caption{(Color online) Cumulant spectral weight (solid curve) of the $\\lambda=5$ exciton and the corresponding GW-BSE spectrum (dashed curve) at $\\mathbf{q}=8\\mathbf{q}_0$. A red vertical line marks the energy of the lowest IP-transition.}\n\\label{fig:cumulant_sm}\n\\end{figure}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nPlease follow the steps outlined below when submitting your manuscript to the IEEE Computer Society Press.\nThis style guide now has several important modifications (for example, you are no longer warned against the use of sticky tape to attach your artwork to the paper), so all authors should read this new version.\n\n\\subsection{Language}\n\nAll manuscripts must be in English.\n\n\\subsection{Dual submission}\n\nPlease refer to the author guidelines on the CVPR\\ 2022\\ web page for a\ndiscussion of the policy on dual submissions.\n\n\\subsection{Paper length}\nPapers, excluding the references section, must be no longer than eight pages in length.\nThe references section will not be included in the page count, and there is no limit on the length of the references section.\nFor example, a paper of eight pages with two pages of references would have a total length of 10 pages.\n{\\bf There will be no extra page charges for CVPR\\ 2022.}\n\nOverlength papers will simply not be reviewed.\nThis includes papers where the margins and formatting are deemed to have been significantly altered from those laid down by this style guide.\nNote that this \\LaTeX\\ guide already sets figure captions and references in a smaller font.\nThe reason such papers will not be reviewed is that there is no provision for supervised revisions of manuscripts.\nThe reviewing process cannot determine the suitability of the paper for presentation in eight pages if it is reviewed in eleven.\n\n\\subsection{The ruler}\nThe \\LaTeX\\ style defines a printed ruler which should be present in the version submitted for review.\nThe ruler is provided in order that reviewers may comment on particular lines in the paper without circumlocution.\nIf you are preparing a document using a non-\\LaTeX\\ document preparation system, please arrange for an equivalent ruler to appear on the final output pages.\nThe presence or absence of the ruler should not change the appearance of any other content on the page.\nThe camera-ready copy should not contain a ruler.\n(\\LaTeX\\ users may use options of cvpr.sty to switch between different versions.)\n\nReviewers:\nnote that the ruler measurements do not align well with lines in the paper --- this turns out to be very difficult to do well when the paper contains many figures and equations, and, when done, looks ugly.\nJust use fractional references (\\eg, this line is $087.5$), although in most cases one would expect that the approximate location will be adequate.\n\n\n\\subsection{Paper ID}\nMake sure that the Paper ID from the submission system is visible in the version submitted for review (replacing the ``*****'' you see in this document).\nIf you are using the \\LaTeX\\ template, \\textbf{make sure to update paper ID in the appropriate place in the tex file}.\n\n\n\\subsection{Mathematics}\n\nPlease number all of your sections and displayed equations as in these examples:\n\\begin{equation}\n  E = m\\cdot c^2\n  \\label{eq:important}\n\\end{equation}\nand\n\\begin{equation}\n  v = a\\cdot t.\n  \\label{eq:also-important}\n\\end{equation}\nIt is important for readers to be able to refer to any particular equation.\nJust because you did not refer to it in the text does not mean some future reader might not need to refer to it.\nIt is cumbersome to have to use circumlocutions like ``the equation second from the top of page 3 column 1''.\n(Note that the ruler will not be present in the final copy, so is not an alternative to equation numbers).\nAll authors will benefit from reading Mermin's description of how to write mathematics:\n\\url{http:\/\/www.pamitc.org\/documents\/mermin.pdf}.\n\n\\subsection{Blind review}\n\nMany authors misunderstand the concept of anonymizing for blind review.\nBlind review does not mean that one must remove citations to one's own work---in fact it is often impossible to review a paper unless the previous citations are known and available.\n\nBlind review means that you do not use the words ``my'' or ``our'' when citing previous work.\nThat is all.\n(But see below for tech reports.)\n\nSaying ``this builds on the work of Lucy Smith [1]'' does not say that you are Lucy Smith;\nit says that you are building on her work.\nIf you are Smith and Jones, do not say ``as we show in [7]'', say ``as Smith and Jones show in [7]'' and at the end of the paper, include reference 7 as you would any other cited work.\n\nAn example of a bad paper just asking to be rejected:\n\\begin{quote}\n\\begin{center}\n    An analysis of the frobnicatable foo filter.\n\\end{center}\n\n   In this paper we present a performance analysis of our previous paper [1], and show it to be inferior to all previously known methods.\n   Why the previous paper was accepted without this analysis is beyond me.\n\n   [1] Removed for blind review\n\\end{quote}\n\n\nAn example of an acceptable paper:\n\\begin{quote}\n\\begin{center}\n     An analysis of the frobnicatable foo filter.\n\\end{center}\n\n   In this paper we present a performance analysis of the  paper of Smith \\etal [1], and show it to be inferior to all previously known methods.\n   Why the previous paper was accepted without this analysis is beyond me.\n\n   [1] Smith, L and Jones, C. ``The frobnicatable foo filter, a fundamental contribution to human knowledge''. Nature 381(12), 1-213.\n\\end{quote}\n\nIf you are making a submission to another conference at the same time, which covers similar or overlapping material, you may need to refer to that submission in order to explain the differences, just as you would if you had previously published related work.\nIn such cases, include the anonymized parallel submission~\\cite{Authors14} as supplemental material and cite it as\n\\begin{quote}\n[1] Authors. ``The frobnicatable foo filter'', F\\&G 2014 Submission ID 324, Supplied as supplemental material {\\tt fg324.pdf}.\n\\end{quote}\n\nFinally, you may feel you need to tell the reader that more details can be found elsewhere, and refer them to a technical report.\nFor conference submissions, the paper must stand on its own, and not {\\em require} the reviewer to go to a tech report for further details.\nThus, you may say in the body of the paper ``further details may be found in~\\cite{Authors14b}''.\nThen submit the tech report as supplemental material.\nAgain, you may not assume the reviewers will read this material.\n\nSometimes your paper is about a problem which you tested using a tool that is widely known to be restricted to a single institution.\nFor example, let's say it's 1969, you have solved a key problem on the Apollo lander, and you believe that the CVPR70 audience would like to hear about your\nsolution.\nThe work is a development of your celebrated 1968 paper entitled ``Zero-g frobnication: How being the only people in the world with access to the Apollo lander source code makes us a wow at parties'', by Zeus \\etal.\n\nYou can handle this paper like any other.\nDo not write ``We show how to improve our previous work [Anonymous, 1968].\nThis time we tested the algorithm on a lunar lander [name of lander removed for blind review]''.\nThat would be silly, and would immediately identify the authors.\nInstead write the following:\n\\begin{quotation}\n\\noindent\n   We describe a system for zero-g frobnication.\n   This system is new because it handles the following cases:\n   A, B.  Previous systems [Zeus et al. 1968] did not  handle case B properly.\n   Ours handles it by including a foo term in the bar integral.\n\n   ...\n\n   The proposed system was integrated with the Apollo lunar lander, and went all the way to the moon, don't you know.\n   It displayed the following behaviours, which show how well we solved cases A and B: ...\n\\end{quotation}\nAs you can see, the above text follows standard scientific convention, reads better than the first version, and does not explicitly name you as the authors.\nA reviewer might think it likely that the new paper was written 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 contracted to solve problem B.\n\\medskip\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.\nFor your results, however, you should not identify yourself and should not mention your participation in the challenge.\nInstead 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\\begin{figure}[t]\n  \\centering\n  \\fbox{\\rule{0pt}{2in} \\rule{0.9\\linewidth}{0pt}}\n  \n\n   \\caption{Example of caption.\n   It is set in Roman so that mathematics (always set in Roman: $B \\sin A = A \\sin B$) may be included without an ugly clash.}\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 sentence-ending space.\nSo \\eg is correct, {\\em e.g.} is not.\nThe provided \\verb'\\eg' macro takes care of this.\n\nWhen citing a multi-author paper, you may save space by using ``et alia'', shortened to ``\\etal'' (not ``{\\em et.\\ al.}'' as ``{\\em et}'' is a complete word).\nIf you use the \\verb'\\etal' macro provided, then you need not worry about double periods when used at the end of a sentence as in Alpher \\etal.\nHowever, use it only when there are three or more authors.\nThus, the following is correct:\n   ``Frobnication has been trendy lately.\n   It was introduced by Alpher~\\cite{Alpher02}, and subsequently developed by\n   Alpher and Fotheringham-Smythe~\\cite{Alpher03}, and Alpher \\etal~\\cite{Alpher04}.''\n\nThis is incorrect: ``... subsequently developed by Alpher \\etal~\\cite{Alpher03} ...'' because reference~\\cite{Alpher03} has just two authors.\n\n\n\n\n\\begin{figure*}\n  \\centering\n  \\begin{subfigure}{0.68\\linewidth}\n    \\fbox{\\rule{0pt}{2in} \\rule{.9\\linewidth}{0pt}}\n    \\caption{An example of a subfigure.}\n    \\label{fig:short-a}\n  \\end{subfigure}\n  \\hfill\n  \\begin{subfigure}{0.28\\linewidth}\n    \\fbox{\\rule{0pt}{2in} \\rule{.9\\linewidth}{0pt}}\n    \\caption{Another example of a subfigure.}\n    \\label{fig:short-b}\n  \\end{subfigure}\n  \\caption{Example of a short caption, which should be centered.}\n  \\label{fig:short}\n\\end{figure*}\n\n\\section{Formatting your paper}\n\\label{sec:formatting}\n\nAll text must be in a two-column format.\nThe total allowable size of the text area is $6\\frac78$ inches (17.46 cm) wide by $8\\frac78$ inches (22.54 cm) high.\nColumns are to be $3\\frac14$ inches (8.25 cm) wide, with a $\\frac{5}{16}$ inch (0.8 cm) space between them.\nThe main title (on the first page) should begin 1 inch (2.54 cm) from the top edge of the page.\nThe second and following pages should begin 1 inch (2.54 cm) from the top edge.\nOn all pages, the bottom margin should be $1\\frac{1}{8}$ inches (2.86 cm) from the bottom edge of the page for $8.5 \\times 11$-inch paper;\nfor A4 paper, approximately $1\\frac{5}{8}$ 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 $6\\frac{7}{8}$ inches (17.46 cm) wide by $8\\frac{7}{8}$ inches (22.54 cm)\nhigh.\nPage numbers should be in the footer, centered and $\\frac{3}{4}$ inches from the bottom of the page.\nThe review version should have page numbers, yet the final version submitted as camera ready should not show any page numbers.\nThe \\LaTeX\\ template takes care of this when used properly.\n\n\n\n\\subsection{Type style and fonts}\n\nWherever Times is specified, Times Roman may also be used.\nIf neither is available on your word processor, please use the font closest in\nappearance to Times to which you have access.\n\nMAIN TITLE.\nCenter the title $1\\frac{3}{8}$ inches (3.49 cm) from the top edge of the first page.\nThe title should be in Times 14-point, boldface type.\nCapitalize the first letter of nouns, pronouns, verbs, adjectives, and adverbs;\ndo not capitalize articles, coordinate conjunctions, or prepositions (unless the title begins with such a word).\nLeave two blank lines after the title.\n\nAUTHOR NAME(s) and AFFILIATION(s) are to be centered beneath the title\nand printed in Times 12-point, non-boldface type.\nThis information is to be followed by two blank lines.\n\nThe ABSTRACT and MAIN TEXT are to be in a two-column format.\n\nMAIN TEXT.\nType main text in 10-point Times, single-spaced.\nDo NOT use double-spacing.\nAll paragraphs should be indented 1 pica (approx.~$\\frac{1}{6}$ inch or 0.422 cm).\nMake sure your text is fully justified---that is, flush left and flush right.\nPlease do not place any additional blank lines between paragraphs.\n\nFigure and table captions should be 9-point Roman type as in \\cref{fig:onecol,fig:short}.\nShort 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-, second-, and third-order headings.\n\nFIRST-ORDER HEADINGS.\n(For example, {\\large \\bf 1. Introduction}) should be Times 12-point boldface, initially capitalized, flush left, with one blank line before, and one blank line after.\n\nSECOND-ORDER HEADINGS.\n(For example, { \\bf 1.1. Database elements}) should be Times 11-point boldface, initially capitalized, flush left, with one blank line before, and one after.\nIf you require a third-order heading (we discourage it), use 10-point Times, boldface, initially capitalized, flush left, preceded by one blank line, followed by a period and your text on the same line.\n\n\\subsection{Footnotes}\n\nPlease use footnotes\\footnote{This is what a footnote looks like.\nIt often distracts the reader from the main flow of the argument.} sparingly.\nIndeed, try to avoid footnotes altogether and include necessary peripheral observations in the text (within parentheses, if you prefer, as in this sentence).\nIf you wish to use a footnote, place it at the bottom of the column on the page on which it is referenced.\nUse Times 8-point type, single-spaced.\n\n\n\\subsection{Cross-references}\n\nFor the benefit of author(s) and readers, please use the\n{\\small\\begin{verbatim}\n  \\cref{...}\n\\end{verbatim}}  command for cross-referencing to figures, tables, equations, or sections.\nThis will automatically insert the appropriate label alongside the cross-reference as in this example:\n\\begin{quotation}\n  To see how our method outperforms previous work, please see \\cref{fig:onecol} and \\cref{tab:example}.\n  It is also possible to refer to multiple targets as once, \\eg~to \\cref{fig:onecol,fig:short-a}.\n  You may also return to \\cref{sec:formatting} or look at \\cref{eq:also-important}.\n\\end{quotation}\nIf you do not wish to abbreviate the label, for example at the beginning of the sentence, you can use the\n{\\small\\begin{verbatim}\n  \\Cref{...}\n\\end{verbatim}}\ncommand. Here is an example:\n\\begin{quotation}\n  \\Cref{fig:onecol} is also quite important.\n\\end{quotation}\n\n\\subsection{References}\n\nList and number all bibliographical references in 9-point Times, single-spaced, at the end of your paper.\nWhen referenced in the text, enclose the citation number in square brackets, for\nexample~\\cite{Authors14}.\nWhere appropriate, include page numbers and the name(s) of editors of referenced books.\nWhen you cite multiple papers at once, please make sure that you cite them in numerical order like this \\cite{Alpher02,Alpher03,Alpher05,Authors14b,Authors14}.\nIf you use the template as advised, this will be taken care of automatically.\n\n\\begin{table}\n  \\centering\n  \\begin{tabular}{@{}lc@{}}\n    \\toprule\n    Method & Frobnability \\\\\n    \\midrule\n    Theirs & Frumpy \\\\\n    Yours & Frobbly \\\\\n    Ours & Makes one's heart Frob\\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{Results.   Ours is better.}\n  \\label{tab:example}\n\\end{table}\n\n\\subsection{Illustrations, graphs, and photographs}\n\nAll graphics should be centered.\nIn \\LaTeX, avoid using the \\texttt{center} environment for this purpose, as this adds potentially unwanted whitespace.\nInstead use\n{\\small\\begin{verbatim}\n  \\centering\n\\end{verbatim}}\nat the beginning of your figure.\nPlease ensure that any point you wish to make is resolvable in a printed copy of the paper.\nResize fonts in figures to match the font in the body text, and choose line widths that render effectively in print.\nReaders (and reviewers), even of an electronic copy, may choose to print your paper in order to read it.\nYou cannot insist that they do otherwise, and therefore must not assume that they can zoom in to see tiny details on a graphic.\n\nWhen placing figures in \\LaTeX, it's almost always best to use \\verb+\\includegraphics+, and to specify the figure width as a multiple of the line width as in the example below\n{\\small\\begin{verbatim}\n   \\usepackage{graphicx} ...\n   \\includegraphics[width=0.8\\linewidth]\n                   {myfile.pdf}\n\\end{verbatim}\n}\n\n\n\\subsection{Color}\n\nPlease refer to the author guidelines on the CVPR\\ 2022\\ web page for a discussion of the use of color in your document.\n\nIf you use color in your plots, please keep in mind that a significant subset of reviewers and readers may have a color vision deficiency; red-green blindness is the most frequent kind.\nHence avoid relying only on color as the discriminative feature in plots (such as red \\vs green lines), but add a second discriminative feature to ease disambiguation.\n\n\\section{Final copy}\n\nYou must include your signed IEEE copyright release form when you submit your finished paper.\nWe MUST have this form before your paper can be published in the proceedings.\n\nPlease direct any questions to the production editor in charge of these proceedings at the IEEE Computer Society Press:\n\\url{https:\/\/www.computer.org\/about\/contact}.\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n\n\\section{Introduction}\n\nAfter receiving paper reviews, authors may optionally submit a rebuttal to address the reviewers' comments, which will be limited to a {\\bf one page} PDF file.\nPlease follow the steps and style guidelines outlined below for submitting your author response.\n\nThe author rebuttal is optional and, following similar guidelines to previous CVPR conferences, is meant to provide you with an opportunity to rebut factual errors or to supply additional information requested by the reviewers.\nIt is NOT intended to add new contributions (theorems, algorithms, experiments) that were absent in the original submission and NOT specifically requested by the reviewers.\nYou may optionally add a figure, graph, or proof to your rebuttal to better illustrate your answer to the reviewers' comments.\n\nPer a passed 2018 PAMI-TC motion, reviewers should refrain from requesting significant additional experiments for the rebuttal or penalize for lack of additional experiments.\nAuthors should refrain from including new experimental results in the rebuttal, especially when not specifically requested to do so by the reviewers.\nAuthors may include figures with illustrations or comparison tables of results reported in the submission\/supplemental material or in other papers.\n\nJust like the original submission, the rebuttal must maintain anonymity and cannot include external links that reveal the author identity or circumvent the length restriction.\nThe rebuttal must comply with this template (the use of sections is not required, though it is recommended to structure the rebuttal for ease of reading).\n\n\n\\subsection{Response length}\nAuthor responses must be no longer than 1 page in length including any references and figures.\nOverlength responses will simply not be reviewed.\nThis includes responses where the margins and formatting are deemed to have been significantly altered from those laid down by this style guide.\nNote that this \\LaTeX\\ guide already sets figure captions and references in a smaller font.\n\n\\section{Formatting your Response}\n\n{\\bf Make sure to update the paper title and paper ID in the appropriate place in the tex file.}\n\nAll text must be in a two-column format.\nThe total allowable size of the text area is $6\\frac78$ inches (17.46 cm) wide by $8\\frac78$ inches (22.54 cm) high.\nColumns are to be $3\\frac14$ inches (8.25 cm) wide, with a $\\frac{5}{16}$ inch (0.8 cm) space between them.\nThe top margin should begin 1 inch (2.54 cm) from the top edge of the page.\nThe bottom margin should be $1\\frac{1}{8}$ inches (2.86 cm) from the bottom edge of the page for $8.5 \\times 11$-inch paper;\nfor A4 paper, approximately $1\\frac{5}{8}$ inches (4.13 cm) from the bottom edge of the page.\n\nPlease number any displayed equations.\nIt is important for readers to be able to refer to any particular equation.\n\nWherever Times is specified, Times Roman may also be used.\nMain text should be in 10-point Times, single-spaced.\nSection headings should be in 10 or 12 point Times.\nAll paragraphs should be indented 1 pica (approx.~$\\frac{1}{6}$ inch or 0.422 cm).\nFigure and table captions should be 9-point Roman type as in \\cref{fig:onecol}.\n\n\nList and number all bibliographical references in 9-point Times, single-spaced,\nat the end of your response.\nWhen referenced in the text, enclose the citation number in square brackets, for example~\\cite{Alpher05}.\nWhere appropriate, include the name(s) of editors of referenced books.\n\n\\begin{figure}[t]\n  \\centering\n  \\fbox{\\rule{0pt}{0.5in} \\rule{0.9\\linewidth}{0pt}}\n \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:onecol}\n\\end{figure}\n\nTo avoid ambiguities, it is best if the numbering for equations, figures, tables, and references in the author response does not overlap with that in the main paper (the reviewer may wonder if you talk about \\cref{fig:onecol} in the author response or in the paper).\nSee \\LaTeX\\ template for a workaround.\n\n\\subsection{Illustrations, graphs, and photographs}\n\nAll graphics should be centered.\nPlease ensure that any point you wish to make is resolvable in a printed copy of the response.\nResize fonts in figures to match the font in the body text, and choose line widths which render effectively in print.\nReaders (and reviewers), even of an electronic copy, may choose to print your response in order to read it.\nYou cannot insist that they do otherwise, and therefore must not assume that they can zoom in to see tiny details on a graphic.\n\nWhen placing figures in \\LaTeX, it is almost always best to use \\verb+\\includegraphics+, and to specify the  figure width as a multiple of the line width as in the example below\n{\\small\\begin{verbatim}\n   \\usepackage{graphicx} ...\n   \\includegraphics[width=0.8\\linewidth]\n                   {myfile.pdf}\n\\end{verbatim}\n}\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n\n\n\n\n\n\n\\section{Conclusion}\nIn this paper, we propose a {S}ymmetric Network with {S}patial Relationship {M}odeling method for NL-based vehicle retrieval.\nFirstly, we design a symmetric network to learn visual and linguistic representations.\nThen a spatial relationship modeling methods is proposed to make better use of location information, . \nThe qualitative experiments confirm the effectiveness of our method. It achieves 43.92\\% MRR accuracy on the test set of the 6th AI City Challenge on natural language-based vehicle retrieval track, yielding the 1st place among all valid submissions on the public leaderboard.\n\\\\\n\\hspace*{\\fill} \n\\\\\n\\textbf{Broader Impact.} \nOur research can promote the practical application of vehicle retrieval technology, but the data used in the research may cause privacy violations. Therefore, we only use the data of the public dataset and mask the license plate to avoid increasing privacy leaks and other security issues.\n\\section{Experiments}\n\n\\subsection{Datasets and Evaluation Settings.}\n\n\\noindent\\textbf{Datasets.} \nCityFlow-NL \\cite{Feng21CityFlowNL} dataset consist of 666 vehicles contains 3028 vehicle tracks collected from 40 cameras, of which 2155 vehicle tracks were used for training.\nEach track was annotated with three natural language descriptions.\nIn addition, multiple natural language descriptions from other views of the vehicle in this track are collected in this track.\nThe remaining data were adopted to evaluate the proposed method.\n\\\\\n\\hspace*{\\fill} \n\\\\\n\\textbf{Evaluation.} The vehicle retrieval by NL descriptions task \ngenerally utilize the Mean Reciprocal Rank (MRR) as the standard metrics, which is also used in \\cite{voorhees1999trec}.\nIt is formulated as:\n\\begin{equation}\nMRR=\\frac{1}{|Q|} \\sum _{i=1}^{|Q|}\\frac {1}{rank_{i}},\n\\end{equation}\nwhere $|Q|$ is the number of set of text descriptions. $rank_{i}$ refers to the rank position of the right track for the $i_{th}$ text description. \n\n\\subsection{Implement details}\n\nBoth input local and global images are resized to 228 $\\times$ 228.\nWe do not adopt any image augmentation methods to augment the data, including random horizontal flip and random clips, which can keep sports information such as motion unchanged.\nWe set the batch size to 64 for each input, \\textit{i.e.,} vehicle local images, vehicle emphasis texts, global motion images, motion emphasis texts and text from other views.\nThe total batch size is 320.\nTwo text encoder is frozen in the training process. \nOur model is trained for 400 epochs employing the AdamW \\cite{loshchilov2017adamw} optimizer with weight decay (1e-2).\nThe learning rate starts from 0.01 and works as a warm-up strategy for 40 epochs. \nA step delay scheduler is adopted to decay the learning rate every 80 epochs. \nThe $\\gamma$ and $m$ in circle loss are set to 48 and 0.35, respectively.\nThe $m$ in triplet loss is set to 0.\nWe train the model with different configures and ensemble them in final similarity calculating.\n\n\\subsection{Performance on CityFlow-NL}\n\n\\input{tab\/ablation}\n\n\\input{tab\/rank}\n\nWe conduct ablation studies with different configures of our method.\nThe results are illustrated in Table \\ref{tab:ablation}.\nThe ``Baseline'' denotes the symmetric network with subject augmentation and symmetric InfoNCE loss.\nThe ``other view`` refers to the hard samples mining approach.\nIt can be observed that each part learns different characteristics and brings a slight performance boost, respectively.\nIn addition, the model ensemble integrates different information and achieves a great improvement.\n\nWe present the top 7 valid results in the public leaderboard,\nas shown in Table \\ref{tab:rank}.\nOur team (MegVideo) achieve a MRR score of 0.4392, taking 1st highest among all valid submissions on the AI City Challenge 2022 Track 2.\n\\section{Introduction}\nNatural Language (NL) based vehicle retrieval aims to find specific vehicles given text descriptions, which is a vital part of the intelligent transportation and urban surveillance systems.\nExisting vehicle retrieval systems are generally based on image-to-image matching, \\textit{i.e.,} the vehicle re-identification.\nIt requires providing a query image to retrieve the same vehicle in the image gallery.\nHowever, in practical applications, the query image may not be available and we only know the rough description of the target vehicle.\nThis leads us to an urgent need for an NL-based vehicle retrieval system.\n\nCompared with image queries, natural language text descriptions are easier to obtain and modify, resulting in higher flexibility.\nHowever, because text belongs to a different modality from the image, it brings great challenges to retrieval.\nSpecifically, natural language-based vehicle retrieval is a cross-modal matching that aims to learn a transferable model between visual and language. \nDUN \\cite{sun2021dun} proposes a dual-path network to align vehicle features in video and text embeddings supervised by the circle loss \\cite{CircleLoss}.\nBut they ignore motion information in matching, which is conducive to distinguishing similar vehicles with different motion states.\nDifferent from DUN, CLV \\cite{bai2021connecting} utilize a symmetric InfoNCE loss \\cite{xie2018rethinking} to learn cross-modal representation like CLIP \\cite{radford2021learning}.\nThey generate a global motion image for each track, retaining not only vehicle appearance information.\nIn addition, the subject descriptions in the text are enhanced, improving the impact of vehicle appearance information.\nHowever, their motion maps suffer from a loss of vehicle appearance information, which degrades model performance.\n\nInspired by these methods, we propose a \\textbf{S}ymmetric Network with \\textbf{S}patial Relationship \\textbf{M}odeling (SSM) approach to learn the visual and linguistic representations for NL-based vehicle retrieval.\nThe symmetric network can learn both the vehicle internal characteristic (\\textit{e.g.} vehicle type, color, and shape) and the vehicle external characteristic (\\textit{e.g.} motion state and surrounding environment) simultaneously.\nMore concretely, we adopt one visual encoder and text encoder to extract the vehicle appearance feature and the vehicle text embedding, respectively, which is optimized by the symmetric InfoNCE loss and pair-wise Circle loss.\nSymmetrically, another visual encoder and text encoder are employed to learn the representations of vehicle external characteristics, respectively.\nThe visual encoder can be the EfficientNet B2 \\cite{xie2018rethinking} and the text encoder can be RoBERTa \\cite{liu2019roberta}.\nBoth internal and external features are fused to learn robust features of the vehicle track.\nWe also propose a hard text sample mining method to distinguish similar texts, where texts from other views are utilized.\nIn addition, by analyzing the ranking visualization, we found that the model can hardly learn the visual feature of the surrounding environment and relationship of multiple vehicles. \nTo tackle this problem, we design a spatial relationship modeling approach to enhance tracklet information, improving the performance of our method.\nWe has achieved 43.92\\% MRR accuracy on the test set of\nthe 6th AI City Challenge on natural language-based vehicle retrieval track, yielding the 1st place among all valid submissions on the public leaderboard.\n\n\\section{Method}\n\n\\subsection{Overview}\n\nThe pipeline of our method is illustrated in Fig. \\ref{fig:framework}.\nThe symmetric network consists of four branches, where the upper two branches are mainly used to acquire local visual and language representations of vehicle appearance, and the lower two branches are employed to learn the global information of the vehicle, including the motion state and the environment in which it is located.\nThe two kinds of representations are fused together to generate the comprehensive feature representations of the vehicle track.\nWe apply the Symmetric InfoNCE Loss \\cite{oord2018representation} and Circle Loss \\cite{sun2020circle} to connect the representations of the two modalities of the text and the image, ensuring that they are projected into a unified representation space.\nTo further emphasize the motion and environment information, we construct triples to distinguish text descriptions from different views of the same vehicle.\nIn addition, a spatial relationship modeling module is designed to use location information to constrain retrieval results in the retrieval process.\n\n\\subsection{Data Augmentation}\n\n\\subsubsection{Image Augmentation}\n\n\\input{fig\/iou}\n\nSimilar to \\cite{bai2021connecting}, we augment the motion image by pasting cropped vehicle images from the tracking frames to the background image. \nThe background image is generated by computing the mean value of all frames from the same camera, which can be formulated as:\n$$\nB_c = \\frac{1}{N_c}\\sum_i^{N_c} {F_i}\n$$\nwhere $B_c$ is the background image of the $c_{th}$ camera, $N_c$ is the number of frames taken by the $c_{th}$ camera, and $F_i$ is the $i_{th}$ frame. \n\nThe motion image is generated by pasting cropped vehicle images from the same track on the corresponding background image.\nAs shown in the left of Fig. \\ref{fig:iou}, too close consecutive frames may occlude adjacent frames on the motion image.\nTo avoid this problem, we compute the Intersection Over Union (IOU) of adjacent frames and ignore frames whose IOU is larger than the threshold with already pasted frames.\nThe threshold is set to 0.05 in our method.\n\n\n\n\\subsubsection{Text Augmentation}\n\n\\input{fig\/text_aug}\n\nMost text descriptions contain vehicle appearance information, motion information and location information. \nTo make the model learn a specific information better,\nwe adopt subject augmentation, motion augmentation and position augmentation respectively to emphasize different textual information, in which the respective text descriptions are repeated as shown in Fig. \\ref{fig:text}.\n\\\\\n\\hspace*{\\fill} \n\\\\\n\\textbf{Subject augmentation.} To enhance the appearance information, we need to repeat description about the vehicle appearance. Because the vehicle appearance description usually appears as the first noun phase of the sentence, we \"spacy \"\\footnote{\\tiny\\url{https:\/\/spacy.io\/}} to extract all noun phrases and put the first one to the beginning of the sentence.\n\\\\\n\\hspace*{\\fill} \n\\\\\n\\textbf{Motion augmentation.} To enhance the motion information, we extract keywords which describe motion information and then repeat them. \nFor motion information, We assume that there are only three motion types, \\textit{i.e.} turn left, turn right and go straight. \nIf keywords ``turn left\" or ``turn right\" exist in any text description of one track's text description, we denote the motion type of this track is turning left or right. \nIf none of these two keywords exist in the text description, we denote the motion type of this track is going straight. \nWe append the motion type keyword ``left\", ``right\" or ``straight\" to the beginning of all text description of the track to emphasize the motion information. \n\\\\\n\\hspace*{\\fill} \n\\\\\n\\textbf{Location augmentation.} For location enhancement, We only distinguish whether a location described by a track is an intersection. \nSo we look into the text descriptions of the track and check whether ``intersection\" exists in any of the text description to determine whether the described location is an intersection. \nWe append ``intersection\" to the beginning of all text description of the track if the track is located in the intersection.\n\n\\input{fig\/framework}\n\n\\subsection{Visual and Linguistic Representation Learning}\n\n\\subsubsection{Symmetric Network}\n\nThe task of natural language based vehicle retrieval refers to measuring the semantic similarity between a vehicle video and a language description.\nGenerally speaking, humans pay more attention to unique information when describing vehicle videos, such as vehicle appearance, shape and other characteristics (termed as internal characteristics), as well as external characteristics such as motion state and surrounding environment (termed as external characteristics).\nWe construct a dual-stream network to focus the internal and external characteristics, respectively.\n\\\\\n\\hspace*{\\fill} \n\\\\\n\\textbf{Vehicle internal characteristic learning.}\nAs the upper two branches shown in Fig. \\ref{fig:framework},\nto enhance the internal characteristic in vehicle images, we crop the local vehicle image from the frame of the video and feed them into the image encoders.\nThe encoders are EfficientNet B2 \\cite{xie2018rethinking} or ibn-ResNet101-a \\cite{pan2018two} pretrained on ImageNet \\cite{deng2009imagenet}.\nIn addition, we employ a pretrained RoBERTa \\cite{liu2019roberta} as text encoder to learn the text embedding of vehicles.\nIts input text is augmented by subject emphasis to acquire vehicle embedding with more appearance and shape information.\nFollowing \\cite{bai2021connecting}, we adopt projection heads to map visual feature and text embedding into a unified representation space, which is formulated as:\n\\begin{equation}\nf_{i}=g_{i}\\left(h_{i}\\right)=W_{2} \\sigma\\left(B N\\left(W_{1} h_{i}\\right)\\right),\n\\end{equation}\n\\begin{equation}\nf_{t}=g_{t}\\left(h_{t}\\right)=W_{2} \\sigma\\left(L N\\left(W_{1} h_{t}\\right)\\right),\n\\end{equation}\nwhere $h_i$ is the visual feature extracted by the image encoder and $h_t$ is the linguistic embedding extracted by the image encoder.\nBN denotes the Batch Normalization (BN) layer. \nLN is a Layer Normalization (LN) layer.\n$\\sigma$ is a ReLU activation layer.\n\\\\\n\\hspace*{\\fill} \n\\\\\n\\textbf{Vehicle external characteristic learning.}\nSimilar to the network structure in internal features learning, we adopt a image encoder and a text encoder in the lower two branches to learning global motion feature and embedding, respectively.\nThe input of global image encoder is the global motion image generated by multiple frames of images in the video, which is beneficial to learn more action and environmental information.\nThe motion emphasis or location emphasis augmented text description is processed by the global text encoder to extract more vehicle external information.\nThe obtained global motion feature and motion embedding are projected into the same representation space as well.\n\\\\\n\\hspace*{\\fill} \n\\\\\n\\textbf{Vehicle comprehensive characteristic learning.}\nWe concatenate the local and global image features (text embedding) to fuse information at different granularities.\nThe fused visual and linguistic representations are projected into the same space by the projection heads.\nThen in the inference process, we only utilize the fused track features.\n\n\\subsubsection{Loss functions}\n\nIn the unified feature representation space containing two modal information, we perform the symmetric InfoNCE loss \\cite{xie2018rethinking} like CLIP \\cite{radford2021learning} and pair-wise Circle loss \\cite{sun2020circle} to maximize the cosine similarity of image and text representations.\n\nGiven $N$ images and $N$ text descriptions, we can acquire $N$ visual features $f^{img}_i$ and text embedding $f^{text}_i$, where $f^{img}_i$ and $f^{text}_i$ have the same label and become positive pairs with each other, \\textit{i.e.}, $\\langle f^{img}_i, f^{text}_i\\rangle$. \nThese features has different labels are negative pairs of each other, including same-modal negative pairs $\\langle f^{img}_i, f^{img}_{j \\ne i}\\rangle$ and cross-modal negative pairs $\\langle f^{img}_i, f^{text}_{j \\ne i}\\rangle$.\n\nThe symmetric InfoNCE loss maximize the cosine similarity of the positive pair and minimize the similarity of cross-modal negative pairs.\nIt contains image-to-text optimization and text-to-image optimization for one positive pair.\nThe image-to-text loss is as follows:\n\\begin{equation}\n\\mathcal{L}_{i2t}=\\frac{1}{N} \\sum_{i=1}^{N}-\\log \\frac{\\exp(\\cos ( f^{img}_i,f^{text}_i) \/ \\tau)}{\\sum_{j=1}^{N} \\exp(\\cos ( f^{img}_i,f^{text}_j) \/ \\tau)},\n\\end{equation}\nIn addition, the text-to-image loss is:\n\\begin{equation}\n\\mathcal{L}_{t2i}=\\frac{1}{N} \\sum_{i=1}^{N}-\\log \\frac{\\exp(\\cos ( f^{text}_i,f^{img}_i) \/ \\tau)}{\\sum_{j=1}^{N} \\exp(\\cos ( f^{text}_i,f^{img}_j) \/ \\tau)},\n\\end{equation}\nwhere $\\tau$ is the temperature learnable parameter and the cosine similarity is calculated by:\n\\begin{equation}\n\\cos (f_i, f_j) = \\frac{f_i \\cdot f_j}{||f_i|| \\cdot ||f_j||}.\n\\end{equation}\nThen, the symmetric InfoNCE loss is as follows:\n\\begin{equation}\n\\mathcal{L}_{INCE} = \\mathcal{L}_{i2t} + \\mathcal{L}_{t2i}.\n\\end{equation}\n\nDifferent from the symmetric InfoNCE loss, Circle loss minimize the similarity of all negative pairs.\nDenote the positive pair and negative pairs as $s_p$ and $s_n$, respectively.\nCircle loss is defined as follows:\n\\begin{equation}\n\\tiny\n\\mathcal{L}_{{circle}}=\\log \\left[1+\\sum_{j=1}^{L} \\exp \\left(\\gamma \\alpha_{n}^{j}\\left(s_{n}^{j}-\\Delta_{n}\\right)\\right) \\sum_{i=1}^{K} \\exp \\left(-\\gamma \\alpha_{p}^{i}\\left(s_{p}^{i}-\\Delta_{p}\\right)\\right)\\right]\n\\end{equation}\nwhere the $K=1$ and $L=2(N-1)$, $\\Delta_{p}$ and $\\Delta_{n}$ are the intra-class and inter-class margins, respectively.\n$\\alpha_{p}$ and $\\alpha_{n}$ are calculated as:\n\\begin{equation}\n\\small\n\\begin{cases}\n\\alpha_{p}^{i}=\\left[O_{p}-s_{p}^{i}\\right]_{+} \\\\\n\\alpha_{n}^{j}=\\left[s_{n}^{j}-O_{n}\\right]_{+}\n\\end{cases},\n\\end{equation}\nin which $O_p$ and $O_n$ are optimums of $s_p$ and $s_n$, respectively.\nThese parameters are set $O_{p} = 1 + m$, $\\Delta _{p} = 1 - m$, $O_{n} = -m$ and $\\Delta _{n} = m$, respectively.\n\nThe symmetric InfoNCE loss and Circle loss are enforced on all of three kinds of representations, which preserves information of different levels can be learned.\n\n\\subsubsection{Hard Text Samples Mining}\n\n\nAlthough the non-appearance information is enhanced in external feature learning,  our model still learns a lot of vehicle internal information and makes mistakes when discriminating difficult samples.\nFor instance, when the same car appears in two different videos, the text descriptions of the two videos will have similar subjects as show in Fig. \\ref{fig:framework}, which will degrade the retrieval performance.\n\nTo address this problem, we implement hard sample mining by composing triples with different view  descriptions of the same vehicle and the current view motion image.\nMore concretely, given $N$ global motion images, $N$ text descriptions and $N$ text descriptions from other views, we have $N$ triplets $\\langle f^{img}_i, f^{text}_i, f^{text\\_v}_i\\rangle$, where the anchor is $f^{img}_i$, the positive feature is $f^{text}_i$ and the negative feature is $f^{text\\_v}_i$.\nThen we minimize the Triplet loss \\cite{hoffer2015triplet} to push away the $f^{text\\_v}_i$ from $f^{img}_i$:\n\\begin{equation}\n\\small\n\\mathcal{L}_{triple}=\\frac{1}{N} \\sum_{i=1}^{N}\\left[ \\cos ( f^{img}_i,f^{text\\_v}_i) - \\cos ( f^{img}_i,f^{text}_i) + m\\right]_{+} \\\\\n\\end{equation}\nin which $m$ is the margin.\n\n\\subsection{Spatial Relationship Modeling}\n\\subsubsection{Long-distance Relationship Modeling}\nLong-distance relationship modeling means that we build positional relationships between text and images, \\textit{e.g.}, intersection prediction.\nThen the text and images whose positions are both intersections are given greater similarity\nSpecifically, we distinguish whether a location is intersection.\nTo increase the similarity of the text and image feature if they all describe the intersection, We calculate the location similarity between the text and the image and then add the similarity score to the final similarity matrix.\n\nWe found that the pictures in the dataset are all taken by cameras in fixed locations, so we can determine whether the location of an image is at the intersection by checking the location of its corresponding camera. We found if a camera is located in the intersection, some vehicles captured by this camera will stop for a while to wait the traffic light. \nWe can calculate the location of the vehicle in each frame using the bounding box:\n\\begin{equation}\n(x_i, y_i) = (left_i+\\frac{1}{2}width_i, top_i + \\frac{1}{2}height_i)\n\\end{equation}\nwhere $left_i$  and $top_i$ are the left-top corner coordinate of the $i_{th}$ frame's bounding box, $width_i$ and $height_i$ are the width and height of the $i_{th}$ frame's bounding box.\n\nWe can infer the movement of the vehicle through the coordinates in each frames. If movement distance of the vehicle in consecutive $n$ frames are all zero, we consider the car to be at the intersection and we label the corresponding camera to be located at the intersection. We can get the visual location vector by:\n\\begin{equation}\n  loc_v = \n    \\begin{cases}\n        (0, 1)^T,& \\quad c_v \\in \\{c | c ~\\text{is at intersection}\\} \\\\\n        (1, 0)^T,& \\quad \\text{otherwise}\n    \\end{cases}       \n,\\end{equation}\nwhere $c_v$ is the camera of track $v$, $\\{c | c ~\\text{is at intersection}\\}$ is the collection of cameras which are located in the intersection.\n\nWe can determine whether a sentence is describing an intersection by checking whether keyword ``intersection\" exists in the sentence. We can get the text location vector by:\n\\begin{equation}\n  loc_s = \n    \\begin{cases}\n        (0, 1)^T,& \\quad \\text{``intersection\"} \\subseteq  s \\\\\n        (1, 0)^T,& \\quad \\text{otherwise}\n    \\end{cases}       \n,\\end{equation}\n\nAfter we get the visual location embedding and text location embedding, we can calculate the similarity of them by the dot product. Assume we have $n$ text queries and $m$ visual tracks, we can get a matrix $S_l \\in \\mathbb{R} ^ {n\\times m}$ representing the location similarity between each query and track. \n\n\\subsubsection{Short-distance Relationship Modeling}\nIn the queries, there are lots of sentences describe the relationship of more than one vehicles. \nHowever, Our model can not learn the relationship between multiple vehicles, which are very close in the video frame.\nTo utilize this information, we perform relationship augmentation to make the proper text-visual pair more similar, termed short-distance relation modeling.\n\nMore concretely, we use `spacy' to extract all noun-phases in the sentence and keep all phases describing vehicles. Then we use the vehicle description branch to extract the language embeddings of all the vehicle descriptions. We then use the provided detection file to extract the bounding boxes of all cars in the frame. We randomly select several frames in a track and then extract all detected cars in these frames. We use the cropped visual branch to extract the visual embedding for all cars in these frames. If a text query $q$ describes relationship between vehicles $v_1$ and $v_2$, we can calculate the similarity between $v_2$ and all detected vehicles in track $t$ and take the maximum value as the relationship similarity between $q$ and $t$. Assume we have $n$ text queries and $m$ visual tracks, we can get a matrix $S_r \\in \\mathbb{R} ^ {n\\times m}$ representing the relationship similarity between each query and track.\n\nThe final similarity matrix $S_{final}$ is formulated as the sum of the feature similarity matrix $S$, location similarity matrix $S_l$ and relationship similarity matrix $S_r$:\n\n\\begin{equation}\nS_{final} = S + \\alpha S_l + \\beta S_r\n\\end{equation}\n$\\alpha$ and $\\beta$ are hyper-parameters. We set $\\alpha=1$ and $\\beta=0.2$ in our experiments.\n\\section{Related Work}\n\n\\subsection{Natural Language Based Video Retrieval}\nNatural language based video retrieval aims to find the corresponding video given natural language description. There are increasing interest in this area. \\cite{li2021x,qi2021semantics,han2021fine,gabeur2022masking}\nMost of the existing methods are based on representation learning, which try to make the feature representation of the corresponding text and video similar. These methods usually use a language model such as LSTM\\cite{hochreiter1997long} or BERT, etc. \\cite{devlin2018bert} to extract the language feature and use a visual feature backbone such as ResNet \\cite{he2016deep} or C3D \\cite{carreira2017quo}, etc. to extract the visual feature.\nZhang, et al. \\cite{zhang2018cross} use hierarchical sequence embedding (HSE) to embedding sequential data of different modalities into hierarchically semantic spaces with correspondence information. Antoine, et al.\\cite{miech2018learning} propose a Mixture-of-Embedding-Experts (MEE) model with ability\nto handle missing input modalities during training. Dong, et al.\\cite{dong2019dual} proposing a dual\ndeep encoding network that encodes videos and queries into\npowerful dense representations of their own. Our method not only utilizes representation learning to learn transferable visual and text feature, but also utilize spatial relationship modeling to capture the surrounding environment and mutual relationship in vehicle retrieval task.\n\n\n\n\\subsection{Vehicle Re-identification}\nIn the last decade, vehicle ReID has achieved considerable progress, especially for deep learning based methods \\cite{FACT,he2019part,zhuge2020attribute,chen2021vehicle}.\nDifferent from the person ReID methods \\cite{sun2018beyond,zheng2022template}, it is still a challenging task due to the similar appearance and viewpoint variations problems.\nTo deal the two problems, some approaches \\cite{OrientationVeri,he2019part,Meng2020} focus on subtle details to learn discriminative local features.\nFor instance, OIFE \\cite{OrientationVeri} proposes 20 keypoints and raises an orientation invariant feature embedding module to emphasize regions with discriminative information.\nPart-regularized \\cite{he2019part} trains a detector to focus on local regions and acquires local features to distinguish similar vehicles.\nVANet \\cite{Chu2019} adopts a view predictor and a modified triplet loss to generate viewpoint-aware features.\nIn addition, many loss functions are applied to address the above challenges, including representation learning loss functions and metric learning loss functions.\nrepresentation learning losses \\cite{LSoftmax,NormSoftmax,AMSoftmax,CosFace,ArcFace} acquire feature representations by classification.\nUnlike them, metric learning losses \\cite{LeCun2005,triplet2017,CircleLoss} directly optimize the similarity score of image pairs and are term as pair-wise loss, where margins are generally added between positive pairs and negative pairs to increase the distance.\nIn this paper, we learn the unified cross-modal representation through the pair-wise losses.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}  \nThe problem of electron tunneling in systems where the band structure depends on the position, like in semiconductors, began to be treated in the early sixties \\cite{harri}; later it was proposed that this variation is simulated by  a position-dependent effective mass in the one-electron Hamiltonian \\cite{bendani}, and in the Hamiltonian describing graded mixed semiconductors \\cite{gora}. From this time on the position-dependent mass Hamiltonians were studied in many articles in a wide range of areas other than electronic properties of semiconductors \\cite{bast}, \\cite{zhu}, \\cite{vonroos}, \\cite{young}, \\cite{geller}, \\cite{sinha}, like, for example, quantum wells and quantum dots  \\cite{pharri}, \\cite{kesha}, \\cite{einevoll}, polarons \\cite{zhaoliang}, etc. \n\nIn most of those articles the choice of the position-dependent mass (PDM) Hamiltonians was guided by the characteristic of being self-adjoint, in the sense that the mean values of the physical quantities were consistently calculated in the associated Hilbert space with the usual integration measure. With this spirit, many PDM  Hamiltonians were proposed  and studied  \\cite{gora}, \\cite{bast}, \\cite{zhu}, \\cite{vonroos}, \\cite{einevoll}, \\cite{gorabook}, \\cite{bastfurd}, \\cite{LiKelin}. As a consequence some physically consistent and possibly relevant Hamiltonians have been discarded because they were not  self-adjoint \\footnote{See, for example, equation (1) in \\cite{gora}.}. \n\nIn the last decades PDM Hamiltonians have also been theoretically treated in a number of articles. The interest was directed to issues like non-self-adjointness \\cite{jiang2005}, solutions of the corresponding Schr\\\"odinger equations \\cite{plastino}, \\cite{jiang2005}, \\cite{quesnebag}, \\cite{ball}, \\cite{costa}, \\cite{regonobre}, \\cite{monteiro2013}, \\cite{dekar}, \\cite{chris}, \\cite{demetrios}, \\cite{nikitin},  \\ ordering ambiguity \\cite{mustafa}, coherent states \\cite{ruby} and application to some particular systems, like, for example the Coulomb problem \\cite{quesnetka}.  More recently, the issue of the PDM Hamiltonians which were not within the standard self-adjoint class mentioned above were analyzed following a different approach \\cite{ball}, \\cite{costa}, \\cite{regonobre}.   An approach to consistently quantize a non-linear system \\cite{tsallis}  was recently developed. In this approach it was necessary to introduce an additional independent field which is the analog of the complex conjugate field for standard linear quantum systems. \n\nIn this paper we study a family of linear PDM Hamiltonians  and show that the problem of self-adjointness is completely solved under certain conditions. We depart from the approach of two independent fields and define a connection between  the two fields through a mapping that depends on the position dependent mass $m(x)$ and of a function $g(x)$. In order to have appropriately well-defined probability and current densities that satisfy a continuity equation, $g(x)$ must obey a condition that depends on the form of the Hamiltonian. We show that our general non self-adjoint Hamiltonians can be identified with the very general family of Hamiltonians proposed by Harrison \\cite{harri} to calculate wave functions in regions of varying band structure in superconductors, simple metals and semimetals.  Inspired in the form of the probability density,  we propose then an ansatz  that takes the solutions $\\Psi(x)$ of the time independent  Schr\\\"odinger equations for the original non self-adjoint Hamiltonian into  new wave functions $\\Omega(x)$. The wave functions $\\Omega(x)$ are the solutions of the  dual Hamiltonians which are self-adjoint with the usual inner product and quantum mechanically equivalent to the original non self-adjoint ones.  We also define an inner product for the solutions $\\Psi(x)$ with a generalized measure that is a function of $m(x)$ and $g(x)$.\nWe study three different examples of the proposed family of PDM Hamiltonians. All of them belong to the  Harrison's family of Hamiltonians \\cite{harri}. For these cases we obtain the respective  dual self-adjoint Hamiltonians. In one of them the kinetic part of the dual PDM Hamiltonian belongs to the von Roos  general kinetic operator class \\cite{vonroos}, but the same does not happen in the other two cases. Finally we analyze and  analytically solve the three cases taking a physical position-dependent mass and a deformed harmonic oscillator potential,  obtaining and comparing their respective energy levels.  \n  \nThis paper is organized as follows. In section 2 we present a family of Hamiltonians with a real general potential $V(x)$ depending on a function $f(m, m^\\prime)$, $m(x)$ a position dependent mass  and $m^\\prime (x)$ its derivative,  and on a constant parameter $\\alpha$. We obtain the  Schr\\\"odinger equations generated by these Hamiltonians departing from a Lagrangian density which depends on two different fields $\\Psi(x,)$ and $\\Phi(x)$ and on their time and spatial derivatives.  We define a transformation between these two fields that allows us to work with only one, say $\\Psi(x)$, of them and to have a probability and a current density that satisfy a continuity equation. We build the quantum mechanically equivalent dual self-adjoint Hamiltonians on the Hilbert space of the solutions $\\Omega(x)$ of the time independent  Schr\\\"odinger equations generated by them.  We define the inner products for both $\\Psi(x)$ and $\\Omega(x)$. \nIn section 3 we analyze three different examples of the family of Hamiltonians presented, by choosing particular values for the constant parameter and for the function $f(m, m^\\prime)$ and find the particular function $g(x)$. In all of them we present the particular values of the parameters that identify them with Harrison's Hamiltonians. The three examples chosen are interesting: the first one recovers a model for abrupt heterojunction between two semiconductors studied in \\cite{zhu}; the other two are typical, in the sense that they introduce scales.   In these two cases the kinetic part of the dual Hamiltonian does not reduce to the von Roos general kinetic operator. In section 4 we choose a harmonic oscillator mass-dependent potential and a physically motivated particular form for $m(x)$. We solve the Schr\\\"odinger equations for the three cases, obtaining their corresponding eigenfunctions and energy levels. \nIn section 5 we present our conclusions. \n\n\\section{A family of general  position-dependent mass Hamiltonians}\n\\label{binentr}\n In this section we present a family of position-dependent-mass Hamiltonians which depend on a real function $f(m(x), m^\\prime(x))$, $m(x)$ a general position dependent mass  and $m^\\prime (x)$ its derivative, with a general real potential $V(x)$, given by\n\\begin{equation}\n\\label{genham}\nH =  \\frac{-\\hbar^ 2}{2 m(x)} {\\partial_x^ 2} +  \\frac{\\hbar^ 2}{2}  \\alpha f(m(x),m^\\prime (x) ) \\partial_x + V(x) \\,  ,\n\\end{equation}\nwhere $\\alpha \\in \\mathbb{R}$ is a dimensionless constant.  $m(x)$ is an analytical  positive function for any value of $x$.   These Hamiltonians lead to Schr\\\"odinger equations which are not, as will be seen, self-adjoint in the usual Hilbert space of their eigenfunctions. In what follows we show the conditions for \\eqref{genham} to be self-adjoint.\n\n\n\\subsection{Deriving the Schr\\\"odinger equations from a classical Lagrangian density} \n\\label{Scheq} \nWhen we solve the Schrodinger equation for a non self-adjoint Hamiltonian with respect to the usual inner product  we have to take into account also the adjoint equation \\cite{regonobre}. Therefore our solution includes, in principle, two different fields, $\\Psi(x,t)$ and $\\Phi(x,t)$, and their conjugates. Note that $\\Phi(x,t)$ is not the complex conjugate of $\\Psi(x,t)$. \nTherefore we develop here an approach where we depart from a Lagrangian density $\\mathcal{L}$ (as it was done in \\cite{monteiro2013}), which depends on these \nfields, and on their time and spatial derivatives, that is\n\\begin{align}\n\\nonumber\n&\\mathcal{L} =  \\frac{i \\hbar }{2}\\, \\Phi(x,t)\\,  \\partial_t \\Psi(x,t)  - \n\\frac{\\hbar^2}{4 m(x)} \\partial_x \\Phi(x,t) \\partial_x \\Psi(x,t) \\, + \\\\\n\\nonumber\n& \\frac{\\hbar^2}{4 }  \\frac{m^{\\prime}(x)}{m(x)^2} \\Phi(x,t) \\partial_x \\Psi(x,t) - \n  \\frac{\\hbar^2 \\alpha}{4} f(m, m^\\prime)\\,  \\Phi(x,t) \\,\\partial_x \\Psi(x,t) \\\\\n  \\nonumber\n&-  \\frac{i \\hbar }{2} \\Phi^ \\star(x,t) \\partial_t \\Psi^ \\star(x,t) + \n\\frac{\\hbar^2}{4 m(x)} \\partial_x \\Phi^\\star(x,t) \\partial_x \\Psi^\\star(x,t) \\, + \\\\\n& \\frac{\\hbar^2}{4 }  \\frac{m^{\\prime}(x)}{m(x)^2} \\Phi^\\star(x,t) \\partial_x \\Psi^\\star(x,t) - \n  \\frac{\\hbar^2 \\alpha}{4} f(m, m^\\prime)\\,  \\Phi^ \\star(x,t) \\,\\partial_x \\Psi^ \\star(x,t)\n  \\nonumber \\\\\n  &-  \\frac{1}{2} V(x) \\, \\Phi^ \\star(x,t) \\, \\Psi^ \\star(x,t)   - \\frac{1}{2} V(x) \\, \\Phi(x,t) \\, \\Psi(x,t) \\, , \n  \\label{lag}\n  \\end{align}\n  where $\\star$ denotes the standard complex conjugate.\n  \nUsing the  usual Euler-Lagrange equations for the fields $\\Phi(x,t)$, $\\Psi(x,t)$ and their conjugates, we straightforwardly get the following Schr\\\"odinger equations: \n\\begin{align}\n\\label{1schr}\n  i \\hbar \\partial_t \\Psi(x,t)  & =   - \\frac{\\hbar^ 2}{2m(x)} \\partial_x^ 2 \\Psi(x,t)  + \\frac{\\hbar^ 2}{2}  \\alpha f(m,m^\\prime)   \\partial_x \\Psi(x,t) + V(x) \\Psi(x,t) \\\\\n  \\label{2schr}\n  -  i \\hbar \\partial_t \\Psi^ \\star (x,t)  &  =    - \\frac{\\hbar^ 2}{2m(x)} \\partial_x^ 2 \\Psi^ \\star (x,t)  + \\frac{\\hbar^ 2}{2}  \\alpha f(m,m^\\prime)   \\partial_x \\Psi^ \\star(x,t)(x,t) +\n  V(x) \\Psi^ \\star (x,t) \\, .\n \\end{align}\n \n\n \n\\begin{align}\n  \\label{3schr}\n  - i \\hbar \\partial_t \\Phi(x,t)  & =   - \\frac{\\hbar^ 2}{2} \\partial_x^ 2 \\left(\\frac{\\Phi(x,t)}{m(x)} \\right)   - \\frac{\\hbar^ 2 \\alpha}{2} \\partial_x [f(m,m^\\prime) \\Phi(x,t)] +  V(x) \\Phi(x,t)    \\\\\n  \\label{4schr}\n i \\hbar \\partial_t \\Phi^ \\star (x,t)  & =   - \\frac{\\hbar^ 2}{2} \\partial_x^ 2 \\left(\\frac{\\Phi^ \\star (x,t)}{m(x)} \\right)   - \\frac{\\hbar^ 2 \\alpha}{2} \\partial_x [f(m,m^\\prime) \\Phi^ \\star (x,t)]  + V(x) \\Phi^ \\star (x,t) \\, .\n \\end{align}\n Note that when the mass is a constant, \\eqref{1schr} and \\eqref{2schr} are the same as, respectively, \\eqref{4schr} and \\eqref{3schr}. \n\n\n\\subsection{The continuity equation} \n\\label{conteq} \nWe define the function $\\rho(x)$ as \n\\begin{equation}\n\\label{pd}\n\\rho(x,t) = \\frac{1}{2 m_0} (\\Psi(x, t) \\Phi(x, t) + \\Phi^\\star(x, t) \\Psi^\\star(x,t)) \\,  ,\n\\end{equation}\nwhere $m_0$ is a mass dimensional constant. We are considering here only systems for which the integral of $\\rho(x)$ over the whole space is finite. Besides, in order to be a probability density, $\\rho(x)$ has to be non-negative.   This can be assured if $\\Phi(x, t) = H(x) \\Psi^\\star(x, t)$, with $H(x) > 0$. For the sake of convenience, we choose \n$H(x) = g(x) m(x)$,  and\n\\begin{equation}\n\\label{relphipsi}\n\\Phi(x,t) =  g(x) m(x) \\Psi^\\star(x,t) \\,  ,\n\\end{equation}\nwhere $m(x)$ is obviously positive and we impose $g(x) > 0$. Then, \\eqref{pd} becomes\n\\begin{equation}\n\\label{pd1}\n\\rho(x,t) = \\frac{1}{m_0} (g(x) m(x) \\Psi(x, t) \\Psi^\\star(x,t)) \\,  .\n\\end{equation}\nIt is straigthforward to show that \\eqref{pd1}  obeys the continuity equation\n\\begin{equation}\n\\label{continuityeq}\n\\partial_t \\rho(x,t) + \\partial_x j(x,t) = 0, \n\\end{equation}\nwhere, using Schr\\\"odinger equations \\eqref{1schr} and \\eqref{2schr}, we find the current density to be\n\\begin{equation}\n\\label{current}\n j(x,t) =  \\frac{\\hbar}{2 i m_0}  g(x) \\left [ (\\partial_x \\Psi(x, t)) \\Psi^\\star(x,t) - \\Psi(x, t) (\\partial_x \\Psi^\\star(x,t)  \\right]  \\,  .\n\\end{equation}\nThis result is not valid for any function $g(x)$, but only for those obeying the condition\n\\begin{equation}\n\\label{condg}\n\\frac{dg(x)}{dx} = -  \\alpha f(m,m^\\prime)  m(x)  g(x) \\,  ,\n\\end{equation}\nwhich is a consequence of the continuity equation \\eqref{continuityeq}. Also, is it very simple to show that \\eqref{condg} makes equations \\eqref{1schr} and \\eqref{4schr} reduce to each other (respectively, \\eqref{2schr}  and \\eqref{3schr} ). This means that the class of fields $\\Psi(x, t)$ and $\\Phi(x, t)$ related through \\eqref{relphipsi} and submitted to condition \\eqref{condg} are not two independent  fields. Condition \\eqref{condg} also means that given a particular Hamiltonian of the family \\eqref{genham},  once we know $\\alpha$ and $f(m,m^\\prime)$, we have the function $g(x)$ and the probability and current densities that obey the continuity equation.\n\nIn \\cite{harri}, Harrison proposed a family of Hamiltonians to describe regions of varying band structure in semiconductors, semimetals and transition metals. By comparing his current density, eq. (2) in \\cite{harri},  \n\\begin{equation}\n\\label{current}\n j(x,t) =  \\frac{\\hbar}{2 i m_0}  \\frac{\\gamma}{\\beta} \\left [ (\\partial_x \\beta \\phi(x, t)) \\beta \\phi^\\star(x,t) - \\beta \\phi(x, t) (\\partial_x \\beta \\phi^\\star(x,t)  \\right]  \\,  , \n\\end{equation}\nwith our definition of current density,  equation \\eqref{current}, and identifying $\\beta \\phi = \\psi$, we have that \n\\begin{equation}\\\n\\label{cond2}\n \\frac{\\gamma}{\\beta} = g(x) \\, . \n\\end{equation}\nBesides, comparing his wave equation, eq. (4) in \\cite{harri}, which is the limiting case of continuous variations of the band structure, and where $\\beta$,  $\\gamma$ and $k_x$ are functions of position, \n\\begin{equation} \n\\label{harri4}\n\\beta \\partial_x \\left[ \\frac{\\gamma}{\\beta} \\partial_x [\\beta \\phi(x)] \\right]+ \\gamma  \\, k_x^2 \\, \\beta \\, \\phi(x)= 0 \\,   ,\n\\end{equation}\nwith the time independent Schr\\\"odinger equation for the Hamiltonian \\eqref{genham},  $H \\psi(x) = E \\psi(x)$, \nwith $V(x) - E = \\gamma k_x^2$, \nand taking \\eqref{cond2} into account, we find  \n\\begin{equation}\n\\beta = \\frac{-\\hbar^2}{2g(x)m(x)}\n\\end{equation}\nand recover condition \\eqref{condg}, that is, $g^\\prime(x) = - \\alpha f(m, m^\\prime) m(x) g(x)$. \n\n\n\\subsection{Building the dual self-adjoint Hamiltonian}\n\nSuggested by the form of the probability and current densities,  \\eqref{pd1}  and \\eqref{current}, let us define a new wave function\n\\begin{equation}\n\\label{dualta} \n\\Omega(x,t) = \\sqrt{g(x) m(x)} \\Psi(x,t)\n\\end{equation} \nand, using Eq.\\eqref{condg}, rewrite Eq. \\eqref{1schr} for this new wave function: \n\\begin{align}\n\\nonumber\n i \\hbar \\partial_t \\Omega(x,t) = & -\\frac{\\hbar^ 2}{2 m(x)} \\partial_x^2 \\, \\Omega(x,t) \\, +  \n  \\frac{\\hbar^ 2}{2 }  \\frac{m^\\prime(x)}{m(x)^2} \\partial_x \\, \\Omega(x,t) \\, - \\\\\n \\nonumber\n & - \\frac{\\hbar^ 2}{4 m(x) }  \\left[ - \\frac{1}{2 } \\alpha^2 f(m, m^\\prime)^2 m(x)^2 - \\alpha f(m,m^\\prime) m^\\prime(x) \n-\\frac{3}{2}  \\left(\\frac{m^\\prime(x)}{m(x)}\\right)^2 \\right.  -\\\\\n& \\left.  -\\frac{1}{2} \\alpha f^\\prime(m,m^\\prime) m(x) + \\frac{m^{\\prime \\prime}(x)}{m(x)} \\right]\n \\Omega(x,t)  + V\\, \\Omega(x,t)\n\\label{hermite1}\n \\end{align} \n\n\nIt is straightforward to show that the following Hamiltonian, defined from the above Schr\\\"odinger equation,\n\n\\begin{align}\n\\label{selfadham}\n \\nonumber\n&H =   -\\frac{\\hbar^ 2}{2 m(x)} \\partial_x^2  +  \n  \\frac{\\hbar^ 2}{2 }  \\frac{m^\\prime(x)}{m(x)^2} \\partial_x \n  - \\frac{\\hbar^ 2}{4 m(x) }  \\left[ - \\frac{1}{2 } \\alpha^2 f(m, m^\\prime)^2 m(x)^2 - \\right. \\\\ \n  & \\left. \\alpha f(m,m^\\prime) m^\\prime(x) \n-\\frac{3}{2}  \\left(\\frac{m^\\prime(x)}{m(x)}\\right)^2 \\right.  \n \\left.  -\\frac{\\alpha}{2}  f^\\prime(m,m^\\prime) m(x) + \\frac{m^{\\prime \\prime}(x)}{m(x)} \\right]\n + V(x) \n \\end{align} \nis self-adjoint on the Hilbert space of the solutions $\\Omega(x,t)$ of equation (\\ref{hermite1}) with the usual inner product \nfor two given solutions $\\Omega_1$ and $\\Omega_2$: \n\\begin{equation} \n\\label{inprodomega}\n\\langle \\Omega_1(x,t), \\Omega_2(x,t) \\rangle \\equiv \\int dx \\, \\Omega_1^ \\star (x,t) \\Omega_2(x,t) \\, .\n\\end{equation}\n\nRelation (\\ref{dualta}) can be seen as a mapping from  the solutions of Schr\\\"odinger equations (\\ref{1schr}) and (\\ref{2schr}) into  the solutions of time independent equation (\\ref{hermite1}) and its Hermitian conjugate. \nThus, motivated by (\\ref{pd1}), given two solutions of equation (\\ref{1schr}), namely $\\Psi_1$ and $\\Psi_2$, we can now define their inner  product  as\n\\begin{equation}\n\\label{inprod}\n\\langle \\Psi_1(x,t), \\Psi_2(x,t) \\rangle_{gm} \\equiv \\int dx \\, g(x) m(x) \\Psi_1^ \\star (x,t) \\Psi_2(x,t) \\, .\n\\end{equation}\nHaving the inner product above and a consistent definition of the probability density $\\rho(x,t)$ \\eqref{pd1} we can calculate mean values for the system described by Hamiltonian \\eqref{genham}. Therefore, we have a method to  deal with the non self-adjoint Hamiltonian \\eqref{genham}.\nThis result is valid for any analytical positive functions $m(x)$, for the functions $g(x)$ obeying  condition \\eqref{condg} and for any  real potential $V(x)$. A similar result for a particular form of $m(x)$, $g(x) = 1$ and $V(x)$ was proved in theorem  1 of \\cite{ball}.  \n\nIt is easy to see that the energy spectra computed from Hamiltonians (\\ref{genham}) and (\\ref{selfadham}) are the same; \nin the same way as it happens to the Hamiltonians, that is, the dual self-adjoint is a redefinition of the original non self-adjoint one, the physical operators will be different for the two dual systems, so that the mean values will be the same.\nThus, they are quantum mechanically equivalent. } \n\nThis is a general result, in the sense that it is valid for  all the non self-adjoint Hamiltonians \\eqref{genham} depending on a real function $f(m, m^\\prime)$, with $\\alpha$ a real constant and any real potential $V(x)$, provided  the two fields $\\Psi(x,t)$ and $\\Phi(x,t)$ are related by \\eqref{phipsi} and $g(x)$ obeys condition \\eqref{condg}. Besides, this is a general method to find the dual self-adjoint Hamiltonians for systems described by non self-adjoint Hamiltonians who belong to the family of Hamiltonians \\eqref{genham}, under the conditions just mentioned. The non self-adjointness was the reason for discarding PDM Hamiltonians which appeared in phenomenological approaches to semiconductors (see, for instance, \\cite{gora}, \\cite{LiKelin}). \n\n\n\nIn \\ref{conteq} we showed that our Hamiltonian \\eqref{genham} is identified to the family of Hamiltonians proposed by Harrison in \\cite{harri}. It is important to note that with the method here presented we can find the dual self-adjoint Hamiltonian to any non self-adjoint  contained in Harrison's family. That is, given specific forms of the functions $\\beta$,  $\\gamma$ and  $m(x)$,  we can find  the corresponding parameters $\\alpha$ and the functions $f(m, m^\\prime)$ and $g(x)$ satisfying condition \\eqref{condg} and therefore the quantum mechanically equivalent self-adjoint Hamiltonian \\eqref{selfadham}. \n\n\n\nA general kinetic-energy operator for a position-dependent mass $m(x)$ system was introduced by von Roos \\cite{vonroos}. In one dimension this operator is written\n\\begin{equation} \n\\label{vonroos}\nT = - \\frac{\\hbar^ 2}{4} \\left[(m(x)^ a \\partial_x m(x)^ b \\partial_x m(x)^ c + m(x)^ c \\partial_x m(x)^ b \\partial_x m(x)^ a \\right] \\,   ,\n\\end{equation}\n and the arbitrary constants $a, b$ and $c$ obey the constraint \n\\newline\n $a + b + c = -1$.  \nTaking this constraint into account this operator can be written\n\\begin{align} \n\\nonumber\nT =  & - \\frac{\\hbar^ 2}{2 m(x)} \\partial_x ^ 2 +  \\frac{\\hbar^ 2}{2 m(x)^ {2}}  m^ \\prime(x) \\partial_x  + \\\\ \n\\label{vonroos2} \n& + \\frac{\\hbar^ 2}{4 m(x)} \\left[ - 2 (1 + a + a^ 2 + b + ab) \\left(  \\frac{m^ \\prime(x)}{m(x)}  \\right)^ 2 \n+ (1 + b) \\frac{m^ {\\prime \\prime}(x)}{m(x)}\\right] \\,   .\n\\end{align}\n\nThe comparative analysis of von Roos kinetic operator, Eq. \\eqref{vonroos2}, and the kinetic part of our Hamiltonian given by Eq. \\eqref{selfadham} will be performed in the examples below. \n\n\n\n\n\\section{Three examples} }\n\\label{cases}\n\nWe have so far shown that it is possible to construct a well-defined continuity equation for the general position-dependent mass Hamiltonian (\\ref{genham}). \n\n In this subsection we analyze  three different classes of Lagrangian (\\ref{lag}) specified by different choices of $f(m,m^\\prime)$ and $\\alpha$, namely:\n \\begin{itemize}\n\\item \n Case a:  $\\alpha =  0$\n\\item\n Case b:  $\\alpha = \\frac{1}{c_1m_0},  f(m, m^\\prime)=   \\frac{m^\\prime}{m(x)}$  ,  $[\\alpha] = M^ {-1}$\n\\item\n Case c:  $\\alpha = 2 \\alpha_0 c_2, f(m, m^\\prime) =  \\frac{1}{m(x)}$, $\\alpha_0$ a constant, $[\\alpha_0] = L^ {-1}$ \\, . \n\\end{itemize} \nBoth constants $c_1$ and $c_2$ are dimensionless. These three cases are typical in the sense that the constant $\\alpha$ has no scale or scales as mass or length. \n \n\n\\subsection{Case a:}\n\nIn case (a) the  non self-adjoint Hamiltonian \\eqref{genham} becomes\n\\begin{equation}\n\\label{ham1}\nH =  \\frac{-\\hbar^ 2}{2 m(x)} \\partial_x^ 2 +   V(x) \\,  \n\\end{equation} \n\nFrom condition \\eqref{condg}, as $\\alpha = 0$, we have $g^\\prime(x) = 0$ and $g$ is a constant; we take it equal to $1$.  Therefore, according to \\eqref{relphipsi}, \n\\begin{equation}\n\\label{phipsi}\n\\Phi(x,t) = m(x) \\Psi^ \\star (x,t) \\, .\n\\end{equation} \nThe functions $\\Psi(x,t)$ (respectively, $\\Phi(x,t)$) are the solutions of the Schr\\\"odinger equations \\eqref{1schr} (respectively, \\eqref{3schr}) for the Hamiltonian \\eqref{ham1}. \n\nIn the limit $m(x) = $ constant, we recover the usual expressions for both the probability and current densities. \n\nThe dual self-adjoint Hamiltonian \\eqref{selfadham} corresponding to Hamiltonian \\eqref{ham1} is then\n\\begin{align}\n\\label{selfadhama}\n \\nonumber\nH_a =  & -\\frac{\\hbar^ 2}{2 m(x)} \\partial_x^ 2  + \\frac{\\hbar^ 2}{2 m(x)^ 2} m^ \\prime(x) \\partial_x  +\n \\frac{\\hbar^ 2}{4 m(x)} \\left[     - \\frac{3}{2}  \\left(  \\frac{m^ \\prime(x)}{m(x)}   \\right)^ 2  + \\frac{m^ {\\prime  \\prime}(x)}{ m(x)} \n  \\right]  \\\\ + \n  & V(x)  \\, .  \n \\end{align} \nComparing (\\ref{vonroos2}) with the kinetic part of (\\ref{selfadhama}), we see that they are the same for $a = c = - 1\/2$ and $b = 0$. \n \n \nFrom  \\eqref{dualta} we have in this case \n\\begin{equation}\n\\label{aomega} \n\\Omega(x,t) = \\sqrt{m(x)} \\Psi(x,t) \\,  ,\n\\end{equation} \nwhere $\\Omega(x,t)$ is the solution of the Schr\\\"odinger equations \\eqref{hermite1} in case (a). \n\nHamiltonian \\eqref{ham1} was proposed in \\cite{zhu}  as a model for the abrupt heterojunction between two different semiconductors and rendered self-adjoint by an empirical approach which is a particular case of the method presented here. \n\nAs we have showed in the general case, in \\ref{conteq}, the Hamiltonians of the family \\eqref{genham} are equivalent to those proposed by Harrison in \\cite{harri}. In this particular case, the $\\beta$ and $\\gamma$ functions of Harrison's Hamiltonian \\eqref{harri4} are $\\beta = \\gamma = \\frac{- \\hbar^2}{2m(x)}$ and $V(x) - E= -\\frac{\\ \\hbar^2}{2 m(x)^2} k_x^2$. \n\n \n\\subsection{Case b:}\n\n In this case  Hamiltonian \\eqref{genham} has the form\n\\begin{equation}\n\\label{bham}\nH =  \\frac{-\\hbar^ 2}{2 m(x)} \\partial_x^ 2 +  \\frac{\\hbar^ 2}{2 c_1 m_0 m(x)}  m^\\prime (x)   \\partial_x + V(x) \\,  .\n\\end{equation} \nLet us note that in this case the $\\alpha= \\frac{1}{c_1m_0}$ parameter has a dimension of $M^ {-1}$. \n\nIntegrating condition \\eqref{condg}, equation \\eqref{relphipsi} becomes \n\\begin{equation}\n\\label{phipsib}\n\\Phi(x,t) = e^ {-\\frac{m(x)}{c_1 m_0}}m(x) \\Psi^ \\star (x,t) \\, .\n\\end{equation} \n\nThe dual self-adjoint Hamiltonian \\eqref{selfadham} corresponding to Hamiltonian \\eqref{bham} is now\n\\begin{align}\n\\label{bselfadj}\n \\nonumber\nH_b =  & -\\frac{\\hbar^ 2}{2 m(x)} \\partial_x^ 2  + \\frac{\\hbar^ 2}{2 m(x)^ 2} m^ \\prime(x) \\partial_x  -\n \\frac{\\hbar^ 2}{2 m(x)} \\left[ \\frac{3 c_1^ 2 m_0^ 2 - m(x)^ 2}{4c_1^ 2 m_0^ 2 m(x)^ 2} m^ \\prime(x)^ 2 + \\right. \\\\\n & \\left. \\frac{m(x) - c_1 m_0}{2 c_1 m_0 m(x)} m^ {\\prime  \\prime}(x) \\right]   +    V(x)  \\, . \n \\end{align}\nFrom \\eqref{dualta}  the new functions $\\Omega(x,t)$,  \n\\begin{equation}\n\\label{bomega} \n\\Omega(x,t) = e^ {-\\frac{m(x)}{2 c_1 m_0}}\\sqrt{m(x)} \\Psi(x,t) \\,  ,\n\\end{equation} \nare the solutions of the Schr\\\"odinger equations for self-adjoint Hamiltonian \\eqref{bselfadj}. \n\n In this case, the kinetic operator of Hamiltonian (\\ref{bselfadj}) does not reduce to the Von Roos general kinetic operator (\\ref{vonroos}) for any particular values of the parameters. Indeed, it is easy to see that \n \\begin{equation}\n\\label{baham}\nH_b  = H_a +  \\frac{\\hbar^ 2}{2} \\left[\\frac{m^ \\prime(x)^ 2}{4 c_1^ 2 m_0^ 2 m(x)}      -  \\frac{m^ {\\prime  \\prime}(x)}{2 c_1 m_0 m(x)}  \\right] \\,  ,\n\\end{equation}\nwhere $H_a$ is given by equation (\\ref{selfadhama}). This shows that the von Roos kinetic operator is not the most general self-adjoint kinetic operator, as it has been assumed with frequency in the literature in the last decades. \\eqref{bselfadj} is a perfectly satisfactory  PDM Hamiltonian that does not fit in the von Roos proposal. \n\nIn this particular case, the $\\beta$ and $\\gamma$ functions of Harrison's Hamiltonian \\eqref{harri4} are $\\beta = \\frac{- \\hbar^2}{2m(x)} e^{\\frac{m(x)}{c_1 m_0}}$, $\\beta = \\frac{- \\hbar^2}{2m(x)}$ and $V(x) - E= -\\frac{\\ \\hbar^2}{2 m(x)^2} k_x^2$. In  fact, this also shows that the kinectic part of Harrison's Hamiltonian proposed in \\cite{harri} is more general than the von Roos' one. \n \n \n\n\\subsection{Case c:} \nFinally, in case (c) the Hamiltonian (\\ref{genham}) reads\n\\begin{equation}\n\\label{cham}\n H =  \\frac{-\\hbar^ 2}{2 m(x)} \\partial_x^ 2 +  \\frac{\\hbar^ 2 c_2 \\alpha_0}{m(x)}  \\partial_x + V(x) \\,  ;\n\\end{equation}\nthe relation between functions $\\Psi^\\star(x,t)$ and $\\Phi(x,t)$ is\n\\begin{equation}\n\\label{mapc}\n\\Phi(x,t) =  e^ {-2 c_2 \\alpha_0 x} m(x)  \\Psi^\\star(x, t)\\,  ,\n\\end{equation}\nand the dual self-adjoint Hamiltonian has the form\n\\begin{align}\n\\label{seladjhamc}\n\\nonumber\nH_c =  & -\\frac{\\hbar^ 2}{2 m(x)} \\partial_x^ 2  + \\frac{\\hbar^ 2}{2 m(x)^ 2} m^ \\prime(x) \\partial_x  -\n \\frac{\\hbar^ 2}{2 m(x)} \\left[ \\frac{3 }{4m(x)^ 2} m^ \\prime(x)^ 2 - \\right. \\\\\n & \\left. \\frac{1}{2 m(x)} m^ {\\prime  \\prime}(x) - \\frac{c_2^ 2 \\alpha_0^ 2}{4}\\right]   +   V(x)  \\, .  \n\\end{align} \n Here, the parameter $[\\alpha_0] = L^ {-1}$.\n \nNow the solutions of Schr\\\"odinger equations for Hamiltonian \\eqref{seladjhamc} are\n\\begin{equation}\n\\label{dualtc} \n\\Omega(x,t) = e^ {-c_2 \\alpha_0 x} \\sqrt{m(x)} \\Psi(x,t) \\,\\, .\n\\end{equation}\n\nAs in case (b) Hamiltonian \\eqref{seladjhamc} can be written as \n\\begin{equation}\n\\label{bcham}\nH_c  = H_a +  \\frac{\\hbar^ 2 c_2^ 2 \\alpha_0^ 2}{8 m(x)}  \\,  ,\n\\end{equation}\nand its kinetic part does not reduce to the Von Roos general kinetic operator,  being another example of a satisfactory PDM Hamiltonian that  is not included in the von Roos scheme. \n\nIn this particular case, the $\\beta$ and $\\gamma$ functions of Harrison's Hamiltonian \\eqref{harri4} are $\\beta = \\frac{- \\hbar^2}{2m(x)} e^{2 c_2 \\alpha_0 x}$, $\\gamma = \\frac{- \\hbar^2}{2m(x)}$ and $V(x) - E= -\\frac{\\ \\hbar^2}{2 m(x)^2} k_x^2$. \n\nWe remark that, after the introduction of the function $\\Omega(x,t)$, instead of the Lagrangian \\eqref{lag} we could write  a Lagrangian only for the field $\\Omega(x,t)$, whose associate Hamiltonian is self-adjoint. This procedure was done in \\cite{pvp2015}. However, with that approach we can not analyze non self-adjoint operators. \n\n\n\n\n \n\n\\section{Position-dependent mass Hamiltonians for a deformed harmonic oscillator}\n\\label{defosc}\n Position-dependent mass Schr\\\"odinger equations have been used to describe  semiconductor heterostructures \\cite{heteros1,zhaoliang,heteros3}, as well as other kind of systems \\cite{pharri, kesha, einevoll}. Among \n all the possible $m(x)$, there is strong motivation in the literature to study the case\n\\begin{equation}\n\\label{fx}\nm(x) = m_0 (1 + \\gamma x^2) \\, , \n\\end{equation} \nwhich may describe the $GaAs\/Al_xGa_{1-x}As$ system  \\cite{galbraith, LiKelin, zhaoliang}. $m_0$ is a constant with dimension of mass. \n\nWe choose to study the Schr\\\"odinger equations of  the cases (a), (b) and (c), described by Hamiltonians \\eqref{ham1}, \\eqref{bham} and \\eqref{cham},  with the position dependent mass (\\ref{fx}), and the deformed harmonic oscillator potential $V(x)$ given by\n\\begin{equation} \n\\label{vx}\nV(x) = \\frac{k x^2}{2 m_0}  (1 + \\gamma x^2)^{-1} \\, ,\n\\end{equation}\nwhere $\\gamma$ is a constant with dimension $L^{-2}$ which measures the departure from the usual harmonic oscillator and $k$ is the spring constant.  With this choice the Hamiltonian \\eqref{ham1} is PT-symmetric, where P means parity and T is the time reversal operator. Naturally, the standard harmonic oscillator is recovered when $\\gamma$ is zero. \n\nIn all the cases presented in  section 2 it is sufficient to solve the Schr\\\"odinger equations for the original non-self-adjoint Hamiltonians, namely  \\eqref{ham1}, \\eqref{bham}  and \\eqref{cham}. From their solutions we have immediately the solutions of the Schr\\\"odinger equations generated by the equivalent self-adjoint Hamiltonians, which are given by \\eqref{selfadhama}, \\eqref{bselfadj} and \\eqref{seladjhamc}. \n\\subsubsection{Case a:} \n\nThe Hamiltonian \\eqref{ham1} with potential \\eqref{vx} is \\cite{ball}\n\\begin{equation} \n\\label{defosca}\n\\mathcal{H}_a =  \\frac{-\\hbar^ 2 }{2 m_0 (1 + \\gamma x^2)} \\partial_x^ 2 +  \\frac{k x^2}{2 m_0 (1 + \\gamma x^2)}   \\, .\n\\end{equation}\nIn the stationary case, for which $\\Psi(x,t) = \\exp{\\frac{-i E t}{\\hbar}} \\Psi(x)$, the time-independent Schr\\\"odinger equation $\\mathcal{H}_a \\Psi(x) = E_a \\Psi(x)$ is\n\\begin{equation}\n\\label{scheq1}\n(1 + \\tilde\\gamma y^2)^{-1} \\Psi^{\\prime \\prime}(y) + [ \\lambda - y^2 (1 + \\tilde\\gamma y^2)^{-1} ]\\Psi(y) = 0 \\, ;\n\\end{equation}\nabove, we have defined the new variable $y = (\\frac{m_0 k}{\\hbar^2})^{1\/4} x$ and redefined $\\tilde\\gamma = \\gamma (\\frac{\\hbar^2}{m_0 k})^{1\/2}$ and $\\lambda = \\frac{2E}{\\hbar \\omega}$ and, as usual, the frequency is $\\omega = \\sqrt{\\frac{k}{m_0}}$.\n\n\\noindent\nThe eigensolutions $\\Psi_n(y)$ are the functions: \n\\begin{equation}\n\\label{sola1}\n\\Psi_{na}(y) = c_a \\exp\\left(-\\frac{1}{2} x^2 \\sqrt{1- 2 \\tilde\\gamma E_{na} } \\right) \\textmd{H}_{n} \\left[ y (1-2 \\tilde\\gamma E_{na})^{1\/4}       \\right] \\, ,\n\\end{equation}\nwhere $c_a$ is an arbitrary constant; $\\textmd{H}_n[y]$ is the Hermite polynomial and  we have the energy levels \n\\begin{equation}\n\\label{enlevela}\nE_{na} = \\frac{2 n+1}{4} \\left[ -\\tilde\\gamma (2 n+1)  + \\sqrt{4 + \\tilde\\gamma^2 (2n+1)^2}     \\right] \\, .\n\\end{equation}\n$\\Phi(y,t)$ and $\\Omega(y,t)$ are given respectively by \\eqref{phipsi}  and \\eqref{aomega}, taking into account the change of variables, with $m(y) = 1 + \\tilde\\gamma y^2$. Note that when $\\tilde\\gamma = 0$, we recover the energy levels of the harmonic oscillator.\n\nIn order to guarantee that the mass is positive-definite, we see in \\eqref{fx} that $\\tilde\\gamma \\geqslant 0$ for any value of $y$.  The asymptotic behavior of the energy levels as $n$ tend to $\\infty$ is then\n\n \\begin{equation}\n\\label{enasymposa}\nE_{na} \\sim   \\frac{1}{2 \\tilde\\gamma}  - \\frac{1}{8 \\tilde\\gamma^ 3 n^ 2} + O(1\/n^ 3)...\\, ;\n\\end{equation}\ntherefore the energy value is limited by $1\/2 \\tilde\\gamma$, which is the maximum of the potential \\eqref{vx}, and the square roots in the eigensolutions \\eqref{sola1} are  well defined for all the values of the energy. \n\n\n\\subsubsection{Case b:} \n\nTaking the dimensionless constant $c_1 = 1$, the Hamiltonian \\eqref{bham} with potential \\eqref{vx} is\n\\begin{equation}\n\\label{defoscb}\n\\mathcal{H}_b =  \\frac{-\\hbar^ 2}{2 m_0 (1 + \\gamma x^2)} \\partial_x^ 2 +  \\frac{\\hbar^ 2}{2 m_0 (1 + \\gamma x^2)}  m^\\prime (x)   \\partial_x + \\frac{k x^2}{2 (1 + \\gamma x^2)} \\,  ,\n\\end{equation}\n\n\\noindent\nWith the same redefinition of variables as in case a, the eigensolutions of the time independent Schr\\\"odinger equations, $\\mathcal{H}_b \\Psi(x) = E_b \\Psi(x)$,  are the functions \n\\begin{equation}\n\\label{solb1}\n\\Psi_{nb}(y) = c_b \\exp\\left[-\\frac{1}{2} y^2 \\left( - \\tilde\\gamma + \\sqrt{1+ \\tilde\\gamma^ 2 - 2 \\tilde\\gamma E_n} \\right)  \\right] \\textmd{H}_{n} \\left[ y (1 + \\tilde\\gamma^ 2 -2 \\tilde\\gamma E_n)^{1\/4}       \\right] \\, ,\n\\end{equation}\nand the energy levels are\n\\begin{equation}\n\\label{enlevelb}\nE_{nb} = \\frac{1}{4} \\left[- [2 + (2 n+1)^ 2] \\tilde\\gamma  + (2n + 1)\\sqrt{4 + \\tilde\\gamma^2 [8 + (2n+1)^2]}  \\right]\\, .\n\\end{equation}\n$\\Phi(x,t)$ and $\\Omega(x,t)$ are given respectively by \\eqref{phipsib}  and \\eqref{bomega}, with $m(y) = 1 + \\tilde\\gamma y^2$. Note that when $\\tilde\\gamma = 0$, we recover the energy levels of the harmonic oscillator.\n\nAs $\\tilde\\gamma \\geqslant 0$ for any value of $y$,  the asymptotic behavior of the energy levels as $n$ tend to $\\infty$ is\n \\begin{equation}\n\\label{enasymposb}\nE_{nb} \\sim   \\frac{\\tilde\\gamma^ 2 + 1}{2 \\tilde\\gamma}  - \\frac{1}{8 \\tilde\\gamma^ 3}(2 \\tilde \\gamma^ 2 + 1)^ 2  \\frac{1}{n^ 2}+ O(1\/n^ 3)...\\, ;\n\\end{equation}\nnow the energy value is limited by $(\\tilde\\gamma^ 2 + 1)\/2 \\tilde\\gamma$ which assures that  the square roots in the eigensolutions \\eqref{solb1} are  well defined for all the values of the energy. \n\nThe dual of the Hamiltonian \\eqref{defoscb} is\n\\begin{align}\n\\label{selfadhamb1}\n\\nonumber\n\\mathcal{H}_b =  \\mathcal{H}_a +  \\frac{1}{4 m(y)} \\left[ \\frac{m^ \\prime(y)^ 2}{2} -\n m^{\\prime \\prime}(y) \\right]  \\, ;  \n \\end{align}\ntherefore we have an effective potential given by\n\\begin{equation}\n\\label{potef}\n\\mathcal{W}(y) = \\frac{y^2}{2 (1 + \\tilde\\gamma y^2)} + \\frac{1}{4 m(y)} \\left[ \\frac{m^ \\prime(y)^ 2}{2} -  m^{\\prime \\prime}(y) \\right] \\\\ .\n\\end{equation}\nIt is then easy to see that as $y \\rightarrow \\infty$ the limit of the potential $\\mathcal{W}(y)$ is exactly the limit value of the energy,  $(\\tilde\\gamma^ 2 + 1)\/2 \\tilde\\gamma$. But analyzing the potential $V(y) = y^2 (2(1 + \\tilde\\gamma  y^2))^{-1}$ we see that its limit is $1\/2 \\tilde\\gamma$. Therefore in this case the threshold of the two potentials, $V(y)$ and $\\mathcal{W}(y)$, are not the same. \n\n\\subsubsection{Case c:} \n{\\normalsize Following the same procedure as in Cases (a) and (b), the Hamiltonian \\eqref{cham} with potential \\eqref{vx}, taking $c_2 = 1$ and $\\alpha_0 = 2 \\sqrt{\\gamma}$, is\n\\begin{equation}\n\\label{defoscc}\n\\mathcal{H}_c =  \\frac{-\\hbar^ 2}{2 m_0 (1 + \\gamma x^2)} \\partial_x^ 2 +  \\frac{\\hbar^ 2 \\alpha_0}{m_0 (1 + \\gamma x^2)}  \\partial_x + \\frac{k x^2}{2 (1 + \\gamma x^2)} \\,  ;\n\\end{equation}\n the eigensolutions are\n\\begin{equation}\n\\label{solc1}\n\\Psi_{nc}(y) = c_c \\exp\\left[2 y \\sqrt{\\tilde\\gamma}-\\frac{1}{2} y^2\\sqrt{1 - 2 \\tilde\\gamma E_n} \\right] \\textmd{H}_n \\left[ y (1 - 2 \\tilde\\gamma E_n)^{1\/4}       \\right] \\, ,\n\\end{equation}\nand the energy levels\n\\begin{equation}\n\\label{enlevelc}\nE_{nc} = \\frac{1}{4} \\left[- [-8 + (2 n+1)^ 2] \\tilde\\gamma  + (2n + 1)\\sqrt{4(1 - 4 \\tilde\\gamma^ 2) + \\tilde\\gamma^2   (2n+1)^2}  \\right]\\, .\n\\end{equation}\nAs $n$ tend to $\\infty$ and $\\tilde\\gamma \\geqslant 0$ for any value of $y$, \\eqref{enlevelc} goes to\n \\begin{equation}\n\\label{enasymposc}\nE_{nc} \\sim   \\frac{1}{2 \\tilde\\gamma}  - \\frac{1}{8 \\tilde\\gamma^ 3}(4 \\tilde \\gamma^ 2 - 1)^ 2  \\frac{1}{n^ 2}+ O(1\/n^ 3)...\\, ;\n\\end{equation}\nnow the energy value is limited by $1\/2 \\tilde\\gamma$ which assures that  the square roots in the eigensolutions \\eqref{solc1} are  well defined for all the values of the energy. \n\nIn this case the limit value of the effective potential of the dual Hamiltonian as $y \\rightarrow \\infty$  is the same  for $V(y)$, $1\/2 \\tilde\\gamma$. \n\nIn figure \\ref{fig1} we see the behavior of the potential as a function of $x$ and the corresponding first five energy levels for the three cases presented, for $\\gamma = 0.1$. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=4in]{figpot.pdf} \n\\caption{Behavior of the potential as a function of $x$ and the corresponding  five first energy levels for the three cases presented, for $\\gamma = 0.1$. Case (a): continuous red line; case (b): dashed blue line ; case (c): dotted black line.}\n\\label{fig1}\n\\end{center}\n\\end{figure}\n\n\\subsubsection{Case of negative $\\gamma$:}\n\nWhen the deformation parameter $\\gamma$ is negative, the problem can be completely solved in all our three examples of PDM Hamiltonians. A negative $\\gamma$ means that in order that the mass $m(y) = 1 - |\\gamma| y^ 2$ is positive definite, the potential $V(y)$ is defined only for $-1\/\\sqrt{|\\gamma|} < y <  1\/\\sqrt{|\\gamma|}$. Therefore there are solutions only in this region. \n\nAs an example, in Case b the solution of the dual Hamiltonian is \n\\begin{equation}\n\\label{solgammanegb}\n\\Omega_{nb}(y) = c_b \\sqrt{m(y)}\n\\exp\\left[-\\frac{1}{2} y^2 \\left(\\sqrt{1+ \\tilde\\gamma^ 2 + 2 |\\tilde\\gamma| E_n} \\right)  \\right] \\textmd{H}_n \\left[ y (1 + \\tilde\\gamma^ 2 + 2 |\\tilde\\gamma| E_n)^{1\/4}       \\right] \\\\  ,\n\\end{equation}\nand the energy levels are given by\n\\begin{equation}\n\\label{enasymposc}\nE_{nb} = \\frac{1}{4} \\left[[2 + (2 n+1)^ 2] |\\tilde\\gamma|  + (2n + 1)\\sqrt{4 + \\tilde\\gamma^2 [8 + (2n+1)^2]}  \\right]. \n\\end{equation}\nIt is easy to see that as $n$ tends to $\\infty$ the above energy levels go to $2 |\\gamma| n^2$. }\n\n\n\n\n\n\\section{Conclusions}\n\\label{conc}\nThe approach here used, which departs from a classical Lagrangian depending on two independent fields, allowed us to completely solve the question of  self-adjointness  of a class of position-dependent mass Hamiltonian systems. These Hamiltonians had been originally discarded in phenomenological approaches to semiconductors because they were  not  self-adjoint. We proved here that for a general class of these non self-adjoint Hamiltonians,  we can construct Hamiltonians which are quantum mechanically equivalent to the original ones and are self-adjoint in the usual Hilbert space of their Schr\\\"odinger equations solutions. This can be done if \nsome function $g(x)$ that appears in consistent definitions of the probability and current densities, is restrained by the particular form of the non self-adjoint Hamiltonians.   By consistent we mean that the probability density is positive and that it obeys the usual continuity equation with an appropriate definition of the current density. \n\nThe general non self-adjoint Hamiltonian proposed by us is identified with a large family of Hamiltonians constructed by Harrison in \\cite{harri} to calculate wave functions in regions of varying band structure in superconductors, simple metals and semimetals.  That is, given specific forms of the functions involved in Harrison's Hamiltonians,  we can find  the form of  our parameters  and the function $g(x)$ satisfying condition \\eqref{condg}.  Therefore we obtain the quantum mechanically equivalent self-adjoint Hamiltonians, dual to the Harrison's family. \n\n\nWith this method we can solve many  particular cases (that is, choosing the parameter $\\alpha$ and the function $f(m(x), m^\\prime(x))$) of the Hamiltonian \\eqref{genham} and consequently of the Harrison's Hamiltonians. We can also expect this method to be successfully applied in Hamiltonians with non self-adjoint potentials \\cite{bender}. \n\n \n We have also solved the three typical cases for a deformed harmonic oscillator potential and choosing a specific form of position-dependent mass which is potentially interesting for physical applications. Besides, the kinetic energies for these cases are particular cases of the kinectic energy in the family of Hamiltonians proposed by Harrison. We have studied these cases for positive and negative values of the deformation parameter introduced in the form of the mass. Moreover, for positive values of this parameter the systems here solved present bound states solutions with an energy threshold; for negative values the systems are confined.\n \n We believe that some Hamiltonians that were discarded because of their non self-adjointness, like, for example, in \\cite{gora}, could be now treated by this method. \n \n \n \n\n\n\n\n\\section*{Acknowledgments}\n\nWe would like to acknowledge the Brazilian scientific agencies CNPq, FAPERJ and CAPES for financial support. We also thank the referees for valuable comments which helped to improve the article.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $C$ be a smooth plane curve of degree $d$ defined over an\nalgebraically closed field $k$.  In \\cite{noether}, while studying space\ncurves, Max Noether considered the following question.  For\n$n\\in{\\bbb Z}_{\\ge 1}$ find $\\ell (n)\\in{\\bbb Z}_{\\ge 0}$ such that there\nexists a linear system $g^{\\ell (n)}_n$ on $C$ but no linear system\n$g^{\\ell (n)+1}_n$ and classify those linear systems $g^{\\ell (n)}_n$ on\n$C$.  The arguments given by Noether in the answer to this question\ncontained a gap.  In \\cite{cil} C.~Ciliberto gave a complete proof for\nNoether's claim using different arguments.  In \\cite{harts1}\nR.~Hartshorne completed Noether's original arguments by solving the\nproblem also for integral (not necessarily smooth) plane curves (see\nRemark \\ref{rem:1}).\n\nThe linear systems $g^{\\ell (n)}_n$ are either non-special, or special\nbut not very special, or very special but trivial.  By a very special\n(resp. trivial) linear system on a smooth plane curve $C$ we mean:\n\\vs 2\n{\\raggedright {\\sc Definition}. }\n{\\it A linear system $g^r_n$ on $C$ is very special if $r\\ge 1$ and\n$\\dim |K_C-g^r_n|\\ge 1$. $($Here $K_C$ is a canonical divisor on $C)$.\nA base point free complete very special linear system $g^r_n$ on $C$ is\ntrivial if there exists an integer $m\\ge 0$ and an effective divisor $E$\non $C$ of degree $md-n$ such that $g^r_n=|mg^2_d-E|$ and\n$r=\\fracd{m^2+3m}{2}-(md-n)$.  A complete very special linear system\n$g^r_n$ on $C$ is trivial if its associated base point free linear\nsystem is trivial.}\n\\vs 2\nIn this paper, we consider the following question.  For\n$n\\in{\\bbb Z}_{\\ge 1}$ find $r(n)$ such that there exists a very special\nnon-trivial complete linear system $g^{r(n)}_n$ on $C$ but no such linear\nsystem $g^{r(n)+1}_n$.  Our main result is the following:\n\\vs 2\n{\\raggedright {\\sc Theorem}. }\n{\\it Let $g^r_n$ be a base point free very special non-trivial complete\nlinear system on $C$. Write $r=\\fracd{(x+1)(x+2)}{2}-\\beta$ with $x, \\beta$\nintegers satisfying $x\\ge 1, 0\\le\\beta\\le x$.  Then\n$$\nn\\ge n(r):=(d-3)(x+3)-\\beta .\n$$}\n\\vs 2\nThis theorem only concerns linear systems of dimension $r\\ge 2$.  But\n 1-dimensional linear systems are studied in \\cite{c3}.  From these\nresults one finds that $C$ has no non-trivial very special linear system\nof dimension 1 if $d\\le 5$ and for $d\\ge 6$, $C$ has non-trivial very\nspecial complete linear systems $g^1_{3d-9}$ but no such linear system\n$g^1_n$ with $n<3d-9$.  The proof of this theorem is also effective for\ncase $r=1$ if one modifies it little bit.\n\nConcerning the original problem one can make the following observation.\nFor $x\\ge d-2$ one has $r>g(C)$ and of course $C$ has no non-trivial\nvery special linear systems $g^r_n$.  For $x\\le d-3$ one has\n$(d-3)(x+2)\\le (d-3)(x+3)-x$.  So, if the bound $n(r)$ is sharp, then\nalso the bound $r(n)$ can be found.  Concerning the sharpness of the bound\n$n(r)$, we prove it in case ${\\rm char}(k)=0$ for $x\\le d-6$.  In case\n${\\rm char}(k)\\ne 0$ we prove that there exists smooth plane curves of\ndegree $d$ with a very special non-trivial $g^r_{n(r)}$ in case\n$x\\le d-6$.  Finally for the case $x>d-6$ we prove that there exist no\nbase point free complete very special non-trivial linear systems of\ndimension $r$ on $C$.  Hence, at least in case ${\\rm char}(k)=0$ the\nnumbers $r(n)$ are determined.\n\\vs 2\n{\\raggedright{\\sc Some Notations}}\n\\vs 1\nWe write ${\\bbb P}_a$ to denote the space of effective divisors of degree $a$\non ${\\bbb P}^2$.  If ${\\bbb P}$ is a linear subspace of some ${\\bbb P}_a$ then we write\n${\\bbb P} .C$ for the linear system on $C$ obtained by intersection with\ndivisors ${\\mit \\Gamma}\\in{\\bbb P}$ not containing $C$.  We write $F({\\bbb P} .C)$ for the\nfixed point divisor of ${\\bbb P} .C$ and $f({\\bbb P} .C)$ for the associated base\npoint free linear system on $C$, so\n$f({\\bbb P} .C)=\\{ D-F({\\bbb P} .C):D\\in{\\bbb P} .C\\}$.\nIf $Z$ is a 0-dimensional subscheme of ${\\bbb P}^2$ then ${\\bbb P}_a(-Z)$ is the\nsubspace of divisors $D\\in{\\bbb P}_a$ with $Z\\subset D$.\n\n\\setcounter{equation}{0}\n\\section{Bound on the degree of non-trivial linear systems}\n\nA complete linear system $g^r_n$ on a smooth curve $C$ is called very\nspecial if $r\\ge 1$ and $\\dim |K_C-g^r_n|\\ge 1$.  From now on, $C$ is a\nsmooth plane curve of degree $d$ and $g^r_n$ is a very special base point\n free linear system on $C$ with $r\\ge 2$.\n\\vs 1\n\\begin{defi}\n$g^r_n$ is called a trivial linear system on $C$ if there exists an\ninteger $m\\ge 0$ and an effective divisor $E$ on $C$ of degree $md-n$\nsuch that $g^r_n=|mg^2_d-E|$ and $r=\\fracd{m^2+3m}{2}-(md-n)$.\n\\label{def:trivial}\n\\end{defi}\n\\vs 2\n\\begin{thm}\nWrite $r=\\fracd{(x+1)(x+2)}{2}-\\beta$ with $x, \\beta$ integers satisfying\n$x\\ge 1, 0\\le\\beta\\le x$.  If $g^r_n$ is not trivial, then\n$$\nn\\ge n(r):=(d-3)(x+3)-\\beta .\n$$\n\\label{thm:1}\n\\end{thm}\n\\vs 1\n\\begin{rem}\n{\\rm In the proof of this theorem we are going to make use of the main\nresult of Hartshorne \\cite{harts1} which describes the linear systems on\n$C$ of maximal dimension with respect to their degrees. The result is as\nfollows:}\n\nLet $g^r_n$ be a linear system on $C$ $($not necessarily very special$)$.\n  Write $g(C)=\\fracd{(d-1)(d-2)}{2}$.\n\n{\\rm i)} If $n>d(d-3)$ then $r=n-g$ $($the non-special case$)$\n\n{\\rm ii)} If $n\\le d(d-3)$ then write $n=kd-e$ with\n$0\\le k\\le d-3, 0\\le e<d$, one has\n\\vs 1\n$\\left\\{\n\\begin{array}{ll}\nr\\le\\fracd{(k-1)(k+2)}{2} & {\\rm if\\ }e>k+1\\\\\nr\\le\\fracd{k(k+3)}{2}-e & {\\rm if\\ }e\\le k+1.\n\\end{array}\\right.$\n\\vs 1\n{\\rm Hartshorne also gives a description for the case one has equality.\nThis theorem (a claim originally stated by M. Noether with an incomplete\nproof) is also proved in \\cite{cil}.  However, Hartshorne also proves the\n theorem for integral plane curves using the concept of generalized\nlinear systems on Gorenstein curves. We need this more general result in\nthe proof of Theorem \\ref{thm:1}}\n\\label{rem:1}\n\\end{rem}\n\\vs 2\n{\\it Proof of Theorem\\\/} \\ref{thm:1}.  Assume $g^r_n=rg^1_{n\/r}$ and\n$n<(x+3)(d-3)-\\beta$.  Noting $2r=(x+1)(x+2)-2\\beta\\ge x^2+x+2\\ge x+3$,\nwe have $\\fracd{(x+3)(d-3)-\\beta}{r}< 2(d-2)$.  Hence,\n$g^1_{n\/r}=|g^2_d-P|$ for some $P\\in C$.  Since $\\dim |rg^1_{n\/r}|=r$,\ncertainly $\\dim |2g^1_{n\/r}|=2$.  But $\\dim|2g^2_d-2P|=3$.\nA contradiction.\n\nSince $g^r_n$ is special, there exist an integer $1\\le m\\le d-3$\nand a linear system ${\\bbb P}\\subset{\\bbb P}_m$ such that\n$g^r_n=f({\\bbb P}.C)$ and ${\\bbb P}$ has no fixed components.  In Lemma\n\\ref{lem:1} we are going to prove that, because $g^r_n$ is not a multiple\nof a pencil, a general element ${\\mit \\Gamma}$ of ${\\bbb P}$ is irreducible.\n\nNow, for each element ${\\mit \\Gamma}'$ of ${\\bbb P}$ we have\n$F({\\bbb P}.C)\\subset{\\mit \\Gamma}'$ (inclusion of subschemes of ${\\bbb P}^2$).  In\nparticular $F({\\bbb P}.C)\\subset{\\mit \\Gamma}\\cap{\\mit \\Gamma}'$.  This remark is known\nin the literature as Namba's lemma.  As a subscheme of ${\\mit \\Gamma}$,\n$F({\\bbb P}.C)$ is an effective generalized divisor on ${\\mit \\Gamma}$ (terminology\n of \\cite{harts1}).  We find that for each ${\\mit \\Gamma}'\\in{\\bbb P}$ with\n${\\mit \\Gamma}'\\ne{\\mit \\Gamma}$ the residual of $F({\\bbb P}.C)$ in ${\\mit \\Gamma}\\cap{\\mit \\Gamma}'$ (we denote\nit by ${\\mit \\Gamma}\\cap{\\mit \\Gamma}'-F({\\bbb P}.C)$) is an element of the generalized\ncomplete linear system on ${\\mit \\Gamma}$ associated to\n${\\cal O}_{{\\mit \\Gamma}}(m-F({\\bbb P}.C))$.  Hence, we obtain a generalized linear\n system $g^{r-1}_{m^2-(md-n)}$ on ${\\mit \\Gamma}$.\n\nNow we are going to apply Hartshorne's theorem (Remark \\ref{rem:1}) to\nthis $g^{r-1}_{m^2-(md-n)}$ on ${\\mit \\Gamma}$.  Since $g^r_n$ is non-trivial on\n$C$, we know that $r>\\fracd{m^2+3m}{2}-(md-n)$.  If\n$m^2-(md-n)>m(m-3)$, then i) in Remark \\ref{rem:1} implies\n$r-1\\le m^2+n-md-\\fracd{(m-1)(m-2)}{2}$ so $r\\le\\fracd{m^2+3m}{2}-(md-n)$,\n a contradiction.\n\nSo $m^2-(md-n)\\le m(m-3)$ and we apply ii) in Remark \\ref{rem:1}.  We\nfind $x\\le m-3$ and $m^2+n-md\\ge mx-\\beta$, so\n$n\\ge\\varphi (m):=-m^2+m(d+x)-\\beta$.  Since $x+3\\le m\\le d-3$, we find\n$n\\ge\\varphi (x+3)=\\varphi (d-3)=(d-3)(x+3)-\\beta=n(r)$.  This completes the proof\nof the theorem except for the proof of Lemma \\ref{lem:1}.\n\\vs 2\n\\begin{lem}\nLet $C$ be a smooth plane curve of degree $d$ and let $g^r_n$ be a base\npoint free complete linear system on $C$ which is not a multiple of a\none-dimensional linear system.  Suppose there exists a linear system\n${\\bbb P}\\subset{\\bbb P}_e$ without fixed component for some $e\\le d-1$ such\nthat $g^r_n=f({\\bbb P}.C)$.  Then the general element of ${\\bbb P}$ is irreducible.\n\\label{lem:1}\n\\end{lem}\n\\def\\underline{e}{\\underline{e}}\n\\def\\underline{m}{\\underline{m}}\n\\vs 2\n{\\it Proof\\\/}.  Let $F=F({\\bbb P} .C)=\\sum_{j=1}^sn_jP_j$ with $n_j\\ge 1$ and\n$P_i\\ne P\\j$ for $i\\ne j$.  For $t\\in{\\bbb Z}_{\\ge 1}, \\underline{e} =\n(e_1,\\dots ,e_t)\\in ({\\bbb Z}_{\\ge 1})^t$ with $\\sum_{i=1}^te_i=e$ and\n$\\underline{m} =[m_{ij}]_{1\\le i\\le t,1\\le j\\le s}$, let\n$$\nV(t,\\underline{e} ,\\underline{m} )=\\{ {\\mit \\Gamma}_1+\\cdots +{\\mit \\Gamma}_t:{\\mit \\Gamma}_i\\in{\\bbb P}_{e_i}{\\rm\\ is\\\nirreducible\\ and\\ }i({\\mit \\Gamma}_i,C;P_j)=m_{ij}\\}.\n$$\nIt is not so difficult to prove that this defines a stratification of\n${\\bbb P}_e$ by means of locally closed subspaces.\n\nSince ${\\bbb P}$ is irreducible there is a unique triple $(t_0,\\underline{e}_0,\n\\underline{m}_0)$ such that ${\\bbb P}\\cap V(t_0,\\underline{e}_0,\\underline{m}_0)$ is an open\nnon-empty subset of ${\\bbb P}$.  In particular, ${\\bbb P}\\subset\n\\{ {\\mit \\Gamma}_1+\\cdots +{\\mit \\Gamma}_{t_0}:{\\mit \\Gamma}_i\\in{\\bbb P}_{e_{0i}}{\\rm\\ and\\ }\ni({\\mit \\Gamma}_i,C;P_j)\\ge m_{0ij}\\}$.  We need to prove that $t_0=1$,\nso assume that $t_0>1$.  Let forget the subscript $0$ from now on.\n\nLet $F_i=\\sum_{j=1}^sm_{ij}P_j\\subset C$.  For each $D\\in g^r_n$ there\nexists ${\\mit \\Gamma} ={\\mit \\Gamma}_1+\\dots +{\\mit \\Gamma}_t$ with ${\\mit \\Gamma}_i\\in{\\bbb P}_i(-F_i)$ and\n$D={\\mit \\Gamma} .C-(F_1+\\dots +F_s)=\\sum_{j=1}^t({\\mit \\Gamma}_i.C-F_i)$.  Writing\n$D_i={\\mit \\Gamma}_i.C-F_i\\in|e_ig^2_d-F_i|$ we find $D=\\sum_{i=1}^tD_i$.\nSuppose for some $1\\le i\\le t$ we have $\\dim|e_ig^2_d-F_i|=0$.  If\n${\\mit \\Gamma}'$ and ${\\mit \\Gamma}''$ are in ${\\bbb P}_i(-F_i)$ then ${\\mit \\Gamma}'.C={\\mit \\Gamma}''.C$, but\n$e_i<d$ so ${\\mit \\Gamma}'={\\mit \\Gamma}''$.  This implies that ${\\bbb P}_i(-F_i)=\\{{\\mit \\Gamma}_0\\}$, but\nthen ${\\mit \\Gamma}_0$ is a fixed component of ${\\bbb P}$, a contradiction.  Hence, for\n$1\\le i\\le t$, we have $\\dim|e_ig^2_d-F_i|\\ge 1$.  Now, let $L_i$ be the\nirreducible sheaf on $C$ associated to $|e_ig^2_d-F_i|$ and let $L$ be\nthe irreducible sheaf on $C$ associated to $g^r_n$.  Then\n$L=L_1\\otimes\\cdots\\otimes L_t$ and we find that the natural map\n$$\nH^0(C,L_1)\\otimes\\cdots\\otimes H^0(C,L_t)\\to H^0(C,L)\n$$\nis surjective, while $\\dim H^0(C,L_i)\\ge 2$ for $1\\le i\\le t$. From\n\\cite[Corollary 5.2]{eisen}, it follows that $g^r_n$ is a multiple of a\npencil.  But this is a contradiction.\n\\vs 2\n\\begin{rem}\n{\\rm In \\cite{mez} one makes a classification of linear systems on smooth\nplane curves for which $r$ is one less than the maximal dimension with\nrespect to the degree.  In that paper one uses arguments like in\n\\cite{cil}.  That classification is completely contained in our Theorem\n\\ref{thm:1}}\n\\label{rem:2}\n\\end{rem}\n\\vs 2\n\\begin{rem}\n{\\rm If $r\\ge n-\\fracd{(d-1)(d-2)}{2}+d-1$ then $\\dim |K_C-g^r_n|\\ge d-2$.\nHence, $|K_C-g^2_d-g^r_n|=|(d-4)g^2_d-g^r_n|\\ne\\emptyset$. So in this\ncase we can assume $m\\le d-4$ in the proof of Theorem \\ref{thm:1}.\nThen, in the proof of Theorem \\ref{thm:1}, using Bertini's theorem, we\ncan prove that, for $D\\in g^r_n$ general there exists an irreducible\ncurve ${\\mit \\Gamma}$ of degree $d-3$ with ${\\mit \\Gamma} .C\\ge D$ (see argument in\n\\cite[p.384]{harts1}).  So we don't need the proof of Lemma \\ref{lem:1}\nunder that assumption on $r$.}\n\\label{rem:25}\n\\end{rem}\n\\vs 2\n\\begin{rem}\n{\\rm In that case $n=n(r)$, we find $m=x+3$ and $m^2+n-md=mx-\\beta$. So the\ngeneralized linear system $g^{r-1}_{m^2+n-md}=g^{r-1}_{mx-\\beta}$ on ${\\mit \\Gamma}$\nis of maximal dimension with respect to its degree.  The description of\nthose linear systems in \\cite{harts1} implies that it is induced by a\nfamily of plane curves of degree $x$ containing some subspace $E\\subset{\\mit \\Gamma}$\nof length $\\beta$.  Writing $Z=F({\\bbb P} .C)\\subset{\\mit \\Gamma}$ we find\n$|{\\bbb P}_m.{\\mit \\Gamma} -Z|=|{\\bbb P}_x.{\\mit \\Gamma} -E|$ and so\n$Z\\in |{\\bbb P}_3.{\\mit \\Gamma} +E|$.  In order to find non-trivial $g^r_{n(r)}$'s\nit is interesting to find for a smooth curve $C$ of degree $d$, a curve\n${\\mit \\Gamma}$ of degree $m$ and a curve ${\\mit \\Gamma}'$ of degree 3 such that\n${\\mit \\Gamma}\\cap{\\mit \\Gamma}'\\subset C$.  We discuss this in \\S 3.  First we solve the\nfollowing postulation problem: let ${\\mit \\Gamma}'$ be the union of 3 distinct\nlines $L_1, L_2, L_3$ and let $D_i$ be an effective divisor of degree\n$a$ on $L_i$ with $D_i\\cap (L_j\\cup L_k)=\\emptyset$ for\n$\\{ i,j,k\\} =\\{ 1,2,3\\}$.  Give necessary and sufficient conditions for\nthe existence of a smooth curve ${\\mit \\Gamma}$ of degree $a$ such that\n${\\mit \\Gamma} .L_i=D_i$ for $i=1,2,3$.}\n\\label{rem:3}\n\\end{rem}\n\n\\setcounter{equation}{0}\n\\def\\binom#1#2{{#1\\choose#2}}\n\\section{Carnot's theorem}\n\nWe begin with pointing out the following elementary fact:\n\\vs 1\n\\begin{lem}\nLet $m(\\ge 4)$ and $m_j\\ (j=1,\\dots ,\\ell)$ be positive integers\nsatisfying $\\displaystyle\\sum^{\\ell}_{j=1}m_j=m$ and let\n${\\mit \\Phi} (X)=\\displaystyle\\sum^m_{j=1}a_jX^j$ be a non-zero polynomial of\ndegree at most $m$.  If ${\\mit \\Phi} (X)$ is divisible by $(X-X_j)^{m_j}$\nfor $\\ell$ distinct values $X_1,\\dots ,X_{\\ell}$, then the ratio\n$a_0:\\cdots :a_m$ is uniquely determined.  In particular,\n$a_m\\ne 0, a_0=(-1)^ma_m\\displaystyle\\prod^{\\ell}_{j=1}X^{m_j}_j$ and\n$a_{m-1}=-a_m\\displaystyle\\sum^{\\ell}_{j=1}m_jX_j$.\n\\label{lem:31}\n\\end{lem}\n\\vs 1\nUsing this, we have the following Carnot's theorem and infinitesimal\nCarnot's theorems.  Another generalization of this theorem is given by\nThas et al. \\cite{thas} (see also \\cite{ver}).  They call this\nB.~Segre's generalization of Menelaus' theorem.\n\\vs 1\n\\begin{lem}{\\rm (Carnot, cf. \\cite[Proposition 1.8]{ver},\n\\cite[p.219]{koe})}\nLet $L_1, L_2, L_3$ be lines in ${\\bbb P}^2$ so that\n$L_1\\cap L_2\\cap L_3=\\emptyset$ and let\n$D_i=\\displaystyle\\sum^{\\ell_i}_{j=1}m_{ij}P_{ij},\\\n(\\displaystyle\\sum^{\\ell_i}_{j=1}m_{ij}=m, i=1,2,3)$ be an effective\ndivisor on $L_i$ such that $D_{i_1}\\cap L_{i_2}=\\emptyset$ if\n$i_1\\ne i_2$.  Assume $(x:y:z)$ is a coordinate system on ${\\bbb P}^2$ such\nthat $L_1, L_2, L_3$ correspond to the coordinate axes $x=0, y=0, z=0$,\nrespectively.  Let the coordinate of $P_{ij}$ be given by\n$(x_{ij}:y_{ij}:z_{ij})$ $($of course $x_{1j}=y_{2j}=z_{3j}=0)$.  Then,\nthere exists a curve ${\\mit \\Gamma}$ not containing any one of the lines\n$L_1,L_2,L_3$ of degree $m$ satisfying $({\\mit \\Gamma} .L_i)=D_i$ $(i=1,2,3)$ if\nand only if\n\\begin{equation}\n\\prod^{\\ell_1}_{j=1}\\left(\\frac{y_{1j}}{z_{1j}}\\right)^{m_{1j}}\n\\prod^{\\ell_2}_{j=1}\\left(\\frac{z_{2j}}{x_{2j}}\\right)^{m_{2j}}\n\\prod^{\\ell_3}_{j=1}\\left(\\frac{x_{3j}}{y_{3j}}\\right)^{m_{3j}}\n=(-1)^m.\n\\label{eq:31}\n\\end{equation}\n\\label{lem:car1}\n\\end{lem}\n\\vs 1\n{\\it Proof\\\/}.  Assume there exists a curve ${\\mit \\Gamma}$ of degree $m$ not\ncontaining any one of the lines $L_1,L_2,L_3$ satisfying $({\\mit \\Gamma} .L_i)=D_i$\n$(i=1,2,3)$.  Such a curve is given by\n${\\mit \\Phi} (x,y,z)=\\displaystyle\\sum_{i+j+k=m}a_{ijk}x^iy^jz^k=0$.  In this\ndescription, if $i({\\mit \\Gamma} ,L_1;P_{1j})=m_{1j}$, then ${\\mit \\Phi} (0,y,z)$ is\ndivisible by $(y_{1j}z-z_{1j}y)^{m_{1j}}$.\nThis implies $a_{00m}=(-1)^ma_{0m0}\\displaystyle\\prod^{\\ell_1}_{j=1}\n\\left(\\fracd{y_{1j}}{z_{1j}}\\right)^{m_{1j}}$.  Similarly, we have\n$a_{m00}=(-1)^ma_{00m}\\displaystyle\\prod^{\\ell_2}_{j=1}\n\\left(\\fracd{z_{2j}}{x_{2j}}\\right)^{m_{2j}}$ and\n$a_{0m0}=(-1)^ma_{m00}\\displaystyle\\prod^{\\ell_3}_{j=1}\n\\left(\\fracd{x_{3j}}{y_{3j}}\\right)^{m_{3j}}$.  Since ${\\mit \\Gamma}$ does not\ncontain an intersection point $L_{i_1}\\cap L_{i_2}$ for $i_1\\ne i_2$, we\nhave $a_{m00}a_{0m0}a_{00m}\\ne 0$.  Hence, we have (\\ref{eq:31}).\n\nFor the converse, take $a_{m00}=1$.  Then, by (\\ref{eq:31}) we can\ndetermine $a_{ijk}$ so that ${\\mit \\Gamma}$ has the desired property.  This\ncompletes the proof.\n\\vs 2\nNext, we see two infinitesimal cases i.~e. case\n$D_{i_1}\\cap L_{i_2}\\ne\\emptyset$ and case\n$L_1\\cap L_2\\cap L_3\\ne\\emptyset$.\n\\vs 2\n\\begin{lem}\nLet $L_1, L_2, L_3$ be lines in ${\\bbb P}^2$ so that\n$L_1\\cap L_2\\cap L_3=\\emptyset$ and let\n$D_i=\\displaystyle\\sum^{\\ell_i}_{j=1}m_{ij}P_{ij},\n(\\displaystyle\\sum^{\\ell_i}_{j=1}m_{ij}=m)$ be an effective divisor on\n$L_i$ such that $m_{11}=m_{21}=1, P_{11}=P_{21}=L_1\\cap L_2$ and\n$D_i\\cap L_3=D_3\\cap L_i=\\emptyset$ $(i=1,2)$.  Let $(x:y:z)$ and\n$(x_{ij}:y_{ij}:z_{ij})$ be as in Lemma {\\rm \\ref{lem:car1}}.  Then,\nthere exists a curve ${\\mit \\Gamma}$, not containing any one of the lines\n$L_1, L_2, L_3$, of degree $m$ such that $({\\mit \\Gamma} .L_i)=D_i$\n$(i=1,2,3)$ and whose tangent line $T$ at $P_{11}=(0:0:1)$ is given by\n$\\alpha x+y=0$ $(\\alpha\\ne 0)$ if and only if\n\\begin{equation}\n\\alpha\n\\prod^{\\ell_1}_{j=2}\\left(\\frac{y_{1j}}{z_{1j}}\\right)^{m_{1j}}\n\\prod^{\\ell_2}_{j=2}\\left(\\frac{z_{2j}}{x_{2j}}\\right)^{m_{2j}}\n\\prod^{\\ell_3}_{j=1}\\left(\\frac{x_{3j}}{y_{3j}}\\right)^{m_{3j}}\n=(-1)^m.\n\\label{eq:32}\n\\end{equation}\n\\label{lem:car2}\n\\end{lem}\n\\vs 1\n{\\it Proof\\\/}.  We use the same notation as in the proof of Lemma\n\\ref{lem:car1}.  Assume there exists a curve ${\\mit \\Gamma}$ having the desired\nproperty. The condition that the tangent line at $P_{11}$ is given by\n$\\alpha x+y=0$ implies that ${\\mit \\Phi} (0,0,1)=0$ and the linear term of\n${\\mit \\Phi} (x,y,1)$ is divisible by $\\alpha x+y$.\nHence, $a_{00m}=0$ and $a_{10,m-1}-\\alpha a_{01,m-1}=0$.  As in the proof\nof the previous lemma, we have $a_{m00}a_{0m0}\\ne 0, a_{01,m-1}\\ne 0$,\n$a_{0m0}=(-1)^ma_{m00}\\displaystyle\\prod^{\\ell_3}_{j=1}\n\\left(\\fracd{x_{3j}}{y_{3j}}\\right)^{m_{3j}}$,\n$a_{01,m-1}=(-1)^{m-1}a_{0m0}\\displaystyle\\prod^{\\ell_1}_{j=2}\n\\left(\\fracd{y_{1j}}{z_{1j}}\\right)^{m_{1j}}$ and $\\alpha a_{01,m-1}=\na_{10,m-1}=(-1)^{m-1}a_{m00}\\displaystyle\\prod^{\\ell_2}_{j=2}\\left(\n\\fracd{x_{2j}}{z_{2j}}\\right)^{m_{2j}}$. Hence, we have (\\ref{eq:32}).\n\nSimilar to the proof of the previous lemma, we have the converse.\n\\vs 2\n\\begin{lem}\nLet $L_1, L_2, L_3$ be lines in ${\\bbb P}^2$ so that\n$L_1\\cap L_2\\cap L_3\\ne\\emptyset$ and let\n$D_i=\\displaystyle\\sum^{\\ell_i}_{j=1}m_{ij}P_{ij},\n(\\displaystyle\\sum^{\\ell_i}_{j=1}m_{ij}=m)$ be an effective divisor on\n$L_i$ such that $D_i\\cap L_j=\\emptyset$ if $i\\ne j$.  Let $(x:y:z)$ be\na coordinate system on ${\\bbb P}^2$ such that $L_1, L_2, L_3$ correspond\nto the line $y-z=0, y=0, z=0$, respectively.  Let the coordinate\nof $P_{ij}$ be given by $(x_{ij}:y_{ij}:z_{ij})$ $($of course\n$y_{1j}=z_{1j}, y_{2j}=0, z_{3j}=0)$.  Then,\nthere exists a curve ${\\mit \\Gamma}$ of degree $d$ not containing any one of the\nlines $L_1,L_2,L_3$ such that $({\\mit \\Gamma} .L_i)=D_i$ $(i=1,2,3)$ if and only if\n\\begin{equation}\n\\sum^{\\ell_1}_{j=1}m_{1j}\\frac{x_{1j}}{y_{1j}}-\n\\sum^{\\ell_2}_{j=1}m_{2j}\\frac{x_{2j}}{z_{2j}}-\n\\sum^{\\ell_3}_{j=1}m_{3j}\\frac{x_{3j}}{y_{3j}}=0.\n\\label{eq:33}\n\\end{equation}\n\\label{lem:car3}\n\\end{lem}\n\\vs 1\n{\\it Proof\\\/}.  Again, we use the same notation as in the proof of Lemma\n\\ref{lem:car1}.  Assume there exists a curve ${\\mit \\Gamma}$ having the desired\nproperty.  Since ${\\mit \\Gamma}$ does not contain $L_2\\cap L_3$, we\nhave $a_{m00}\\ne 0$.  By Lemma \\ref{lem:31},\n$$\na_{m-1,10}=-\\sum^{\\ell_3}_{j=1}m_{3j}\\frac{x_{3j}}{y_{3j}}a_{m00}\\quad\n{\\rm and}\\quad\na_{m-1,01}=-\\sum^{\\ell_2}_{j=1}m_{2j}\\frac{x_{2j}}{z_{2j}}a_{m00}.\n$$\nTo consider the condition on $L_1$, we take the coordinate system\n$(\\xi :\\eta :\\zeta )$ with $\\xi =x, \\eta =y, \\zeta =z-y$.  Put\n$$\n{\\mit \\Psi} (\\xi ,\\eta ,\\zeta )={\\mit \\Phi} (\\xi ,\\eta ,\\eta +\\zeta )=\\sum_{i+j+k=m}a_{ijk}\n\\xi^i\\eta^j(\\zeta +\\eta )^k=\\sum_{i+j+k=m}b_{ijk}\\xi^i\\eta^j\\zeta^k.\n$$\nThen, $b_{m00}=a_{m00}$ and $b_{m-1,10}=a_{m-1,10}+a_{m-1,01}$.  In this\ndescription, if $i({\\mit \\Gamma} , L_1;P_{1j})=m_{1j}$, then ${\\mit \\Psi} (\\xi ,\\eta ,0)$ is\ndivisible by $(\\xi_{1j}\\eta -\\eta_{1j}\\xi )^{m_{1j}}$.\nThis implies that $b_{m-1,10}=-\\displaystyle\\sum^{\\ell_1}_{j=1}m_{1j}\n\\fracd{x_{1j}}{y_{1j}}b_{m00}$.  Then, we have (\\ref{eq:33}).\n\nFor the converse, noting that if $a_{m00}\\ne 0$ then ${\\mit \\Gamma}$ does not\ncontains $L_i$ $(i=1,2,3)$ as a component, we can find a ${\\mit \\Gamma}$\nhaving the desired property.\n\\vs 2\n\\begin{rem}\n{\\rm In each of the lemmas \\ref{lem:car1}, \\ref{lem:car2} and\n\\ref{lem:car3}, if (\\ref{eq:31}) (resp. (\\ref{eq:32}), (\\ref{eq:33}))\nholds, we can find a smooth curve ${\\mit \\Gamma}$ of degree $m$ with ${\\mit \\Gamma} .L_i=D_i$\nfor $i=1,2,3$.  Indeed, let ${\\bbb P}\\subset{\\bbb P}_m$ be the linear system\nof divisors ${\\mit \\Gamma}$ of degree $m$ on ${\\bbb P}^2$ satisfying, as schemes,\n$D_i\\subset{\\mit \\Gamma}\\cap L_i$.  Clearly $L_1+L_2+L_3+{\\bbb P}_{m-3}\\subset{\\bbb P}$\nand we proved that $U=\\{{\\mit \\Gamma}\\in{\\bbb P}:{\\mit \\Gamma}{\\rm \\ does\\ not\\ contain\\ any\\\nof\\ the\\ lines\\ }L_1, L_2, L_3\\}$ is a non-empty open subset of\n${\\bbb P}$. Because of Bertini's theorem, for ${\\mit \\Gamma}\\in U$ we have\n$L_i\\cap{\\mit \\Gamma} =\\{ P_{i1},\\dots ,P_{i\\ell_i}\\}$.\nConsider the linear system ${\\bbb P}'=\\{{\\mit \\Gamma}\\cap{\\bbb P}^2\\backslash (L_1\\cup L_2\n\\cup L_3):{\\mit \\Gamma}\\in{\\bbb P}\\}$ on $M={\\bbb P}^2\\backslash (L_1\\cup L_2\\cup L_3)$.\nSince ${\\mit \\Gamma}\\cap M\\in{\\bbb P}'$\nfor any ${\\mit \\Gamma}\\in{\\bbb P}_{m-3}$, ${\\bbb P}'$ separates tangent directions and points\non $M$.  Because of Bertini's theorem in arbitrary characteristics (see\n\\cite{kleiman}), we find that a general element ${\\mit \\Gamma}$ of ${\\bbb P}'$ is\nsmooth.  So, a general element ${\\mit \\Gamma}$ of ${\\bbb P}$ satisfies ${\\rm Sing}({\\mit \\Gamma} )\n\\subset\\{ P_{ij}:i=1,2,3{\\rm\\ and\\ }1\\le j\\le\\ell\\}$.  But, using\n${\\mit \\Gamma}'\\in{\\bbb P}_{m-3}$ suited we find ${\\mit \\Gamma} ={\\mit \\Gamma}'+L_1+L_2+L_3$ is smooth at\n$P_{ij}$, except for the case $P_{11}=P_{21}$ in Lemma \\ref{lem:car2}.\nIn that case, however, if ${\\mit \\Gamma}\\in U$ then ${\\mit \\Gamma}$ is smooth at $P_{11}$\nbecause of Bezout's theorem.  This completes the proof of the remark.\n(For Bertini's theorem in arbitrary characteristics, one can also use\n\\cite{greco}.)}\n\\label{rem:33}\n\\end{rem}\n\\setcounter{equation}{0}\n\\section{Sharpness of the bound}\n\nThe next proposition implies that it is enough to solve the postulation\nproblem mentioned in Remark \\ref{rem:3} in order to prove sharpness for\nthe bound $n(r)$ in Theorem \\ref{thm:1}.\n\\vs 2\n\\begin{prop}\nLet $C$ be a smooth plane curve of degree $d$ and let\n$r=\\fracd{(x+1)(x+2)}{2}-\\beta$ with $x, \\beta\\in{\\bbb Z}$ satisfying\n$4\\le x+3\\le d-3, 0\\le\\beta\\le x$.  Let $n=n(r)=(d-3)(x+3)-\\beta$.  Suppose\nthere exists a smooth curve ${\\mit \\Gamma}$ of degree $m=x+3$, a curve ${\\mit \\Gamma}'$ of\ndegree $3$ and an effective divisor $E$ of degree $\\beta$ on ${\\mit \\Gamma}$ such\nthat $Z\\subset C$, where the divisor $Z=({\\mit \\Gamma}\\cap{\\mit \\Gamma}' )+E$ on ${\\mit \\Gamma}$ is\nconsidered as a closed subscheme of ${\\bbb P}^2$.  Then $|mg^2_d-Z|$ is a\nnon-trivial $g^r_n$ on $C$.\n\\label{prop:non-trivial}\n\\end{prop}\n\\vs 1\n{\\it Proof\\\/}.  We write $E=P_1+\\cdots +P_{\\beta}$. Let $L_1,\\dots ,L_{\\beta}$\nbe general lines through $P_1,\\dots ,P_{\\beta}$, resp., and let $L_{\\beta +1}\n\\dots ,L_x$ be general lines in ${\\bbb P}^2$.  If $P\\in C$, then we write\n$\\mu_P(Z)$ for the multiplicity of $Z$ at $P$.\n\\vs{05}\ni) $|mg^2_d-Z|$ is base point free.\n\nSuppose $P$ is a base point for $|mg^2_d-Z|$.  Since\n${\\mit \\Gamma} .C-Z\\in|mg^2_d-Z|$ one finds $P+Z\\le{\\mit \\Gamma} .C$, hence\n$i({\\mit \\Gamma} ,C;P)>\\mu_P(Z)\\ge i({\\mit \\Gamma} ,{\\mit \\Gamma}';P)$.  Also\n$({\\mit \\Gamma}'+\\sum_{i=1}^xL_i).C-Z\\in|mg^2_d-Z|$, hence\n$P\\in({\\mit \\Gamma}'+\\sum_{i=1}^xL_i).C-Z=({\\mit \\Gamma}'.C-{\\mit \\Gamma}'\\cap{\\mit \\Gamma} )+(\\sum_{i=1}^xL_i.C\n-E)$ (sum of two effective divisors).  Since\n$P\\not\\in\\sum_{i=1}^xL_i.C-E$, we find $P\\in{\\mit \\Gamma}'.C-{\\mit \\Gamma}'\\cap{\\mit \\Gamma}$.  This\nimplies $i({\\mit \\Gamma}',C;P)>i({\\mit \\Gamma} ,{\\mit \\Gamma}';P)$.  But\n$i({\\mit \\Gamma} ,{\\mit \\Gamma}';P)\\ge\\min (i({\\mit \\Gamma} ,C;P),i({\\mit \\Gamma}',C;P))$ (so called Namba's\nlemma), hence we have a contradiction.\n\\vs{05}\nii) $\\dim (|mg^2_d-Z|)\\ge r$.\n\nIndeed, $({\\mit \\Gamma}'+{\\bbb P}_x(-E)).C-Z\\subset|mg^2_d-Z|$ and $\\dim ({\\mit \\Gamma}'+{\\bbb P}_x(-E))=\n\\fracd{(x+1)(x+2)}{2}-\\beta -1$.  But also ${\\mit \\Gamma} .C-Z\\subset|mg^2_d-Z|$ while\n${\\mit \\Gamma} .C\\not\\in ({\\mit \\Gamma}'+{\\bbb P}_x(-E)).C$.  This proves the claim.\n\\vs{05}\niii) $\\dim (|mg^2_d-Z|)=r$.\n\nIf $\\dim (|mg^2_d-Z|)>r$ then on ${\\mit \\Gamma}$ it induces a linear system\n$g^{r'}_{mx-\\beta}$ with $r'\\ge r$.  But Hartshorne's theorem (see 1.3)\nimplies that this is impossible.\n\niv) $|mg^2_d-Z|$ is not trivial.\n\nFirst of all, $|mg^2_d-Z|$ is very special.  Indeed $(d-3-m)g^2_d+Z\\subset\n|K_C-(mg^2_d-Z)|$.  If $d-3=m$ then from the Riemann-Roch theorem, one\nfinds $\\dim|Z|=1$.\n\nSuppose $|mg^2_d-Z|$ would be trivial, i.~e. $|mg^2_d-Z|=|kg^2_d-F|$ with\n$r=\\fracd{k^2+3k}{2}-(dk-n)$.  Since $g^r_n$ is very special, one has\n$k<d-3$.  Consider $D=({\\mit \\Gamma}'.C-{\\mit \\Gamma}\\cap{\\mit \\Gamma}')+(\\sum_{i=1}^xL_i.C-E)$ as in\nstep i).  There exists $\\gamma\\in{\\bbb P}_k(-F)$ with $\\gamma .C=D+F$.  Because of\nBezout's theorem one has $\\gamma =\\gamma'+L_1+\\cdots +L_x$.  If $P\\in E$ then\n$P\\not\\in{\\mit \\Gamma}'.C-{\\mit \\Gamma}'\\cap{\\mit \\Gamma}$, otherwise both $i({\\mit \\Gamma}',C;P)>i({\\mit \\Gamma}',{\\mit \\Gamma} ;P)$\nand $i({\\mit \\Gamma} ,C;P)>i({\\mit \\Gamma}',{\\mit \\Gamma} ;P)$, a contradiction to Namba's lemma.  This\nimplies $\\gamma'.C\\ge{\\mit \\Gamma}'.C-{\\mit \\Gamma}'\\cap{\\mit \\Gamma}$.  Once more from Namba's lemma, we\nobtain ${\\mit \\Gamma}'.\\gamma'\\ge{\\mit \\Gamma}'.C-{\\mit \\Gamma}'\\cap{\\mit \\Gamma}$ and so\n$$\n\\deg ({\\mit \\Gamma}'.\\gamma')=3(k-x)\\ge\\deg ({\\mit \\Gamma}'.C-{\\mit \\Gamma}'\\cap{\\mit \\Gamma} )=3(d-x-3),\n$$\na contradiction to $k<d-3$.\n\\vs 2\n\\begin{cor}\nLet $C$ be a smooth plane curve of degree $d$.  Assume there exists a\nplane curve ${\\mit \\Gamma}'$ of degree $3$ and a smooth plane curve ${\\mit \\Gamma}$ of degree\n$a\\ (4\\le a\\le d-6)$ such that ${\\mit \\Gamma}\\cap{\\mit \\Gamma}'\\subset C$ $($as schemes$)$.\nThen $C$ posseses a non-trivial linear system $g^r_n$ for\n$r=\\fracd{(a-2)(a-1)}{2}-\\beta, 0\\le\\beta\\le a-3$ and $n=n(r)=a(d-3)-\\beta$.\n\\label{cor:1}\n\\end{cor}\n\\vs 1\n{\\it Proof\\\/}.  ${\\mit \\Gamma} .C={\\mit \\Gamma}\\cap{\\mit \\Gamma}'+D$ for some effective divisor $D$ of\ndegree $a(d-3)$ on $C$.  But $a(d-3)\\ge d-3\\ge a+3$, so we can choose an\neffective divisor $E\\subset D$ of degree $\\beta$ and then one has to apply\nProposition \\ref{prop:non-trivial} to $|ag^2_d-Z|$ for $Z={\\mit \\Gamma}\\cap{\\mit \\Gamma}'+E$.\n\\vs 2\n\\begin{const}\n{\\rm Fix ${\\mit \\Gamma}'\\in{\\bbb P}_3, {\\mit \\Gamma}\\in{\\bbb P}_a\\ (a\\ge 4)$ general and look at\n${\\bbb P}_{a+\\varepsilon}(-{\\mit \\Gamma}\\cap{\\mit \\Gamma}'), (\\varepsilon\\ge 1)$.  Clearly ${\\mit \\Gamma}'+{\\bbb P}_{a+\\varepsilon -3}\n\\subset{\\bbb P}_{a+\\varepsilon}(-{\\mit \\Gamma}\\cap{\\mit \\Gamma}'), {\\mit \\Gamma} +{\\bbb P}_{\\varepsilon}\\subset{\\bbb P}_{a+\\varepsilon}\n(-{\\mit \\Gamma}\\cap{\\mit \\Gamma}')$.  Take $P\\in{\\bbb P}^2\\backslash ({\\mit \\Gamma}\\cap{\\mit \\Gamma}')$.  If\n$P\\not\\in{\\mit \\Gamma}'$ then using ${\\mit \\Gamma}'+{\\bbb P}_{a+\\varepsilon -3}$ one can separate tangent\nvectors at $P$.  If $P\\not\\in{\\mit \\Gamma}$ then one uses ${\\mit \\Gamma} +{\\bbb P}_{\\varepsilon}$.\nKleiman's Bertini theorem \\cite{kleiman} in arbitrary characteristics\nimplies that a general element $C\\in{\\bbb P}_{a+\\varepsilon}(-{\\mit \\Gamma}\\cap{\\mit \\Gamma}')$ is smooth\noutside ${\\mit \\Gamma}\\cap{\\mit \\Gamma}'$.  But if ${\\mit \\Gamma}''\\in{\\bbb P}_{\\varepsilon}$ is general then\n${\\mit \\Gamma} +{\\mit \\Gamma}''$ is smooth on ${\\mit \\Gamma}\\cap{\\mit \\Gamma}'$.  This implies that a general\nelement $C\\in{\\bbb P}_{a+\\varepsilon}(-{\\mit \\Gamma}\\cap{\\mit \\Gamma}')$ is smooth.  This proves that, for\neach $d, 1\\le x\\le d-6, 0\\le\\beta\\le x$, there exists a smooth plane curve\n$C$ of degree $d$ possesing a non-trivial $g^r_{n(r)}$ with\n$r=\\fracd{(x+1)(x+2)}{2}-\\beta$ and $n(r)=(d-3)(x+3)-\\beta$.}\n\\label{const}\n\\end{const}\n\\vs 2\nIn trying to prove this statement for all smooth plane curves of degree\n$d$, we only succeeded in case ${\\rm char}(k)=0$.  This is the following\ntheorem.\n\\vs 2\n\\begin{thm}\nLet $C$ be a smooth plane curve of degree\n$d$ over an algebraically closed field of characteristic zero.\nLet $d>m\\ge 4$.  There exists ${\\mit \\Gamma}'\\in{\\bbb P}_3$ and a smooth ${\\mit \\Gamma}\\in{\\bbb P}_m$\nsuch that, as schemes,  ${\\mit \\Gamma}\\cap{\\mit \\Gamma}'\\subset C$.\n\\end{thm}\n\\vs 1\n{\\it Proof\\\/}.  Fix two general lines $L_1, L_2$ in ${\\bbb P}^2$, let\n$S=L_1\\cap L_2$.  We may assume neither $L_1$ nor $L_2$ is a tangent\nline of $C$ and $S\\not\\in C$.  Choose points $P_{11},\\dots ,P_{1m}$ on\n$C\\cap L_1$ and $P_{21},\\dots ,P_{2m}$ on $C\\cap L_2$.  Choose a general\npoint $S'$ in ${\\bbb P}^2\\backslash C\\cup L_1\\cup L_2$.  The pencil of lines\nin ${\\bbb P}^2$ through $S'$ induces a base point free $g^1_d$ on $C$.\nBecause $S'$ is general we have:\n\\begin{itemize}\n\\item  If $Q$ is a ramification point of $g^1_d$ then the associated\ndivisor looks like $2Q+E$ with $Q\\not\\in E$ and $E$ consists of $d-2$\ndifferent points (here we use characteristic zero).\n\n\\item  If $Q\\in L_i\\cap C$ $(i=1,2)$ then $Q$ is not a ramification of\n$g^1_d$.  The associated divisor is $Q+E$ with\n$E\\cap (L_1\\cup L_2)=\\emptyset$.\n\n\\item  The line $SS'$ is not a tangent line of $C$.\n\\end{itemize}\nOn the symmetric product $C^{(m)}$ we consider\n$V=\\{ E\\in C^{(m)}:{\\rm there\\ exists\\ }D\\in g^1_d\\ {\\rm with\\ }E\\le D\\}$.\n In terminology of \\cite{c2} it is the set $V^1_k(g^1_d)$ and we consider\n$V$ with its natural scheme structure.  From Chapter 2 in\n{\\it loc.~cit.}, it follows that $V$ is a smooth curve.\n\nLet $D_0\\in g^1_d$ corresponding to the line $SS'$ and let\n$V_0=\\{ E\\in V:E\\le D_0\\}$.  We define a map\n$\\psi :V\\backslash V_0\\to{\\bbb P}^1$ as\nfollows.  Associated to $E\\in V\\backslash V_0$ there is a line $L$\nthrough $S'$.  Write $E=P_{31}+\\cdots +P_{3m}$.\nWe distinguish 3 possibilities:\n\\vs{05}\ni)  $E\\cap (L_1\\cup L_2)=\\emptyset$.  Choose coordinates\n$x,y,z$ on ${\\bbb P}^2$ such that $L_1, L_2, L$ corresponds to the coordinate\naxes $x=0, y=0, z=0$, respectively.  Let $(x_{ij}:y_{ij}:z_{ij})$ be the\ncoordinates of $P_{ij}$ $(i=1,2,3;1\\le j\\le m)$.  Then\n$$\n\\psi (E)=\n\\prod^m_{j=1}\\left(\\frac{y_{1j}}{z_{1j}}\\right)\n\\prod^m_{j=1}\\left(\\frac{z_{2j}}{x_{2j}}\\right)\n\\prod^m_{j=1}\\left(\\frac{x_{3j}}{y_{3j}}\\right).\n$$\nAs long as we take $L_1, L_2, L$ as axes, this value is independent of\nthe coordinates.\n\\vs{05}\nii)  $E\\cap (L_1\\cup L_2)\\ne\\emptyset$.  Say\n$P_{11}=P_{31}\\in E\\cap (L_1\\cup L_2)$.  Choose coordinates as before\nand let $\\alpha x+z=0$ be the equation of the tangent line to $C$ at\n$P_{11}$ $(\\alpha\\ne 0)$.  Then\n$$\n\\psi (E)=\\alpha\n\\prod^m_{j=2}\\left(\\frac{y_{1j}}{z_{1j}}\\right)\n\\prod^m_{j=1}\\left(\\frac{z_{2j}}{x_{2j}}\\right)\n\\prod^m_{j=2}\\left(\\frac{x_{3j}}{y_{3j}}\\right).\n$$\nAgain taking $L_1, L_2, L$ as axes, this value is independent of\nthe coordinates.  (Of course this is a function to ${\\bbb C}$ and we\nconsider ${\\bbb P}^1={\\bbb C}\\cup\\{\\infty\\}$.)\n\\vs{05}\niii)  If $\\psi (E)$ is not defined in ${\\bbb C}$ then $\\psi (E)=\\infty$.\n\\vs{05}\nFor $E\\in V_0$, we define $\\psi (E)=(-1)^m$\n\\vs{05}\nThis map is a holomorphic map.  Indeed, fixing coordinates $(x:y:z)$ such\nthat $L_1, L_2$ corresponds to $x=0,y=0$, resp. and $S'=(1:1:0)$, we can\nwrite $z-\\gamma (x-y)=0$ for the pencil of lines through $S'$ (except for\n$SS'$).  If $E\\in V\\backslash V_0$ and $E$ is a part of a divisor of\n$g^1_d$ consisting of $d$ different points, then $\\gamma$ is a local\ncoordinate of $V$ at $E$.  In case i) we write down $\\psi$ locally as a\nholomorphic function in $\\gamma$.  It is easy to check that $\\psi$ is\ncontinuous at $E$ in case ii).\n\nFor $E\\in V_0$, write $\\beta z+(x-y)=0$ for the pencil of lines through\n$S'$ close to $SS'$.  Let $(x_{3j}:x_{3j}:z_{3j})$ be the coordinates at\nthe points $P_{3j}$ of $E$.  For $E'\\in V$ close to $E$ we have\n$E'=\\sum_{j=1}^mP'_{3j}$ and coordinates\n$(x'_{3j}:\\beta z'_{3j}+x'_{3j}:z'_{3j})$ at $P'_{3j}$.  Here we can assume\nthat $x'_{3j}=x'_{3j}(\\beta ), z'_{3j}=z'_{3j}(\\beta )$ are holomorphic\nfunctions in $\\beta$ (local coordinate at $V$ in $E$) and\n$x_{3j}=x'_{3j}(0), z_{3j}=z'_{3j}(0)$.\n\nChoose new coordinates $\\xi =\\beta z+x-y, \\eta =y, \\zeta =x$.  The coordinates\nof $P_{1j}$ are $(0:y_{1j}:\\beta z_{1j}+y_{1j})$, of $P_{2j}$ are\n$(x_{2j}:0:\\beta z_{2j}+x_{2j})$, of $P'_{3j}$ are\n$(x'_{3j}:\\beta z'_{3j}+x'{3j}:0)$.  We find\n\\begin{eqnarray}\n\\psi (E') & = &\n\\prod^m_{j=1}\\frac{y_{1j}}{\\beta z_{1j}-y_{1j}}\n\\prod^m_{j=1}\\frac{\\beta z_{2j}+x_{2j}}{x_{2j}}\n\\prod^m_{j=1}\\frac{x'_{3j}}{\\beta z'_{3j}+x'_{3j}}\n\\label{eq:v0}\n\\\\\n& = & (-1)^m-\\left(\n\\sum^m_{j=1}\\frac{z_{3j}}{x_{3j}}-\n\\sum^m_{j=1}\\frac{z_{1j}}{y_{1j}}-\n\\sum^m_{j=1}\\frac{z_{2j}}{x_{2j}}\\right)\\beta\n+o(\\beta ).\\nonumber\n\\end{eqnarray}\nHence, $\\psi$ is continuous at $E$.\n\nSince $V$ is smooth and $\\psi$ is continuous on $V$ and holomorphic except\nfor a finite number of points, $\\psi$ is a holomorphic map $V\\to{\\bbb P}^1$.\n\nAt some component of $V$, $\\psi$ is not constant.\nIndeed, look at a fibre $2Q+E\\in g^1_d$ with $E\\in C^{(d-2)}$.  Take a\nclose fibre $P_1+P_2+E'$ with $P_1, P_2$ close to $Q$.  Choose $F\\le E'$\nwith $\\deg(F)=m-1$ and consider $P_1+F\\in V$.  Let $W$ be the irreducible\ncomponent of $V$ containing $P_1+F$.  Using monodromy one finds\n$P_2+F\\in W$.  But clearly $\\psi (P_1+F)\\ne\\psi (P_2+F)$, hence\n$\\psi :W\\to{\\bbb P}^1$ is a covering.\nIn particular $\\psi^{-1}((-1)^m)\\ne\\emptyset$.\n\nIf for some $E\\in W\\backslash V_0$ we have $\\psi (E)=(-1)^m$ then the\ntheorem follows from Lemmas \\ref{lem:car1}, \\ref{lem:car2} and Remark\n\\ref{rem:33}.  So, we have to take a closer look to $\\psi$ at $V_0$.\nBy the equation (\\ref{eq:v0}), if $E\\in V_0$ is not a simple zero of\n$\\psi -(-1)^m$ then the theorem follows from Lemma \\ref{lem:car3} and\nRemark \\ref{rem:33}.\n\nSuppose that each zero of $\\psi -(-1)^m$ belonging to $V_0$ is simple.\nThen $\\psi -(-1)^m$ has exactly $\\binom{d}{m}$ zeros at those points.\nNow we look at zeros of $\\psi$ on $V\\backslash V_0$.  The number of zeros\nis finite.  For case i) there is none.  For case ii) we have two\npossiblities.  If $E\\in V$ corresponds to a line $L$ through $S'$\ncontaining one of the points $P_{21},\\dots ,P_{2m}$ but\n$E\\cap (L_1\\cup L_2)=\\emptyset$.  There are $m\\binom{d-1}{m}$ such\npossibilities.  If $E\\in V$ corresponds to a line $L$ through $S'$ not\ncontaining any of the points $P_{11},\\dots ,P_{1m}$ but\n$E\\cap L_1\\ne\\emptyset$.  There are $(d-m)\\binom{d-1}{m-1}$ such\npossibilities.  So, on the components of $V$ where $\\psi$ is not constant,\n$\\psi$ has at least $m\\binom{d-1}{m}+(d-m)\\binom{d-1}{m-1}$ zeros.\nBut this number is greater than $\\binom{d}{m}$, so $\\psi -(-1)^m$ has a\nzero on $V\\backslash V_0$. This completes the proof of the theorem.\n\\vs 2\n\\begin{rem}\n{\\rm In order to obtain the bound $r(n)$ mentioned in the introduction,\nwe have to prove that $C$ possesses no base point free very special\nnon-trivial linear systems $g^r_n$ with $r\\ge\\fracd{(d-4)(d-3)}{2}-(d-5)$\n(i.~e. $x\\ge d-5$).  In the introduction we already noticed that\n$x\\le d-3$.  Assume $g^r_n$ is a very special non-trivial linear system.\n{}From Theorem \\ref{thm:1} we find $n\\ge n((d-4)(d-5)\/2-(d-5))=d^2-6d+11$.\nBut then $\\deg (K_C-g^r_n)\\le (d-1)(d-2)-2-(d^2-6d+11)=3d-11$.  However,\nvery special linear systems $g^s_m$ of degree $m\\le 3d-11$ are trivial.\nSo, the associated base point free linear system $g^s_m$ of $|K_C-g^r_n|$\nis of type $|ag^2_d-E|$ with $a\\le d-4$, $E$ effective and\n$\\dim|K_C-g^r_n|=\\fracd{a^2+3a}{2}-\\deg E$.  If $E\\ne\\emptyset$, then for\n$P\\in E$ one has $\\dim|ag^2_d-E+P|>\\dim|ag^2_d-E|$, so $\\dim|g^r_n-P|=r$.\nThis implies that $g^r_n$ has a base point, hence $E=\\emptyset$.  But\nthen, $g^r_n=|(d-3-a)g^2_d-F|$ and since $\\dim|ag^2_d+F|=\\dim|ag^2_d|$,\nwe have $r=\\fracd{(d-3-a)^2+3(d-3-a)}{2}-\\deg F$.  This is a\ncontradiction to the fact that $g^r_n$ would be non-trivial.}\n\\end{rem}\n\\vs 1\n\\begin{rem}\n{\\rm It would be interesting to find an answer to the following questions:\nFor which values of $n$ do there exist non-trivial base point free very\nspecial linear systems $g^r_n$ on a (general) smooth plane curve.\nClassify those linear systems and study $W^r_n$ on $J(C)$.  More\nconcretely, is the subscheme of $W^r_{n(r)}$ corresponding to non-trivial\nlinear systems irreducible ?  What are the dimension of those irreducible\ncomponents ?  And so on.\n}\n\\end{rem}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\n\nOne of the central concepts of linear algebra is the notion of a basis for a vector space: a subset of a vector space is called a basis for the former if every vector can be uniquely  written as a finite linear combination of basis elements. Part of the importance of bases stems from the convenient consequences that follow from their existence. For example, linear transformations between vector spaces admit matrix representations relative to pairs of bases \\cite{lang2004algebra}, which can be used for efficient numerical calculations. The idea of a basis however is not restricted to the theory of vector spaces: other algebraic theories have analogous notions of bases -- sometimes by waiving the uniqueness constraint --, for instance modules, semi-lattices, Boolean algebras, convex sets, and many more. In fact, the theory of bases for vector spaces is different to others only in the sense that every vector space admits a basis, which is not  the case for e.g. modules. In this paper we seek to give a compact definition of a basis that subsumes the well-known cases, and as a consequence allows us to lift results from one theory to the others. For example, one may wonder if there exists a matrix representation theory for convex sets that is analogous to the one of vector spaces.\n\nIn the category theoretic approach to universal algebra, algebraic structures are typically captured as algebras over a monad \\cite{eilenberg1965adjoint, linton1966some}. Intuitively, a monad may be seen as a  generalisation of closure operators on partially ordered sets, and an algebra over a monad may be viewed as a set with an operation that allows the interpretation of formal linear combinations in a way that is coherent with the monad structure. For instance, a vector space over a field $k$, that is, an algebra for the free $k$-vector space monad, is given by a set $X$ with a function $h$ that coherently interprets a finitely supported $X$-indexed $k$-sequence $\\lambda$ as an actual linear combination $h(\\lambda) = \\sum_x \\lambda_x \\cdot x$ in $X$ \\cite{coumans2010scalars}. \nIt is straightforward to see that under this perspective a basis for a vector space thus consists of a subset $Y$ of $X$ and a function $d$ that assigns to a vector $x$ in $X$ a $Y$-indexed $k$-sequence $d(x)$ such that $h(d(x)) = x$ for all $x$ in $X$ and $d(h(\\lambda)) = \\lambda$ for all $Y$-indexed $k$-sequences $\\lambda$. In other words, the restriction of $h$ to $Y$-indexed $k$-sequences is an isomorphism with inverse $d$, and surjectivity corresponds to the fact that the subset $Y$ generates the vector space, while injectivity captures that $Y$ does so uniquely. As demonstrated in \\Cref{basisdefinition}, the concept easily generalises to arbitrary monads on arbitrary categories by making the subset relation explicit in form of a function.\n\n\nMonads however not only occur in the context of universal algebra, but also play a role in algebraic topology \\cite{godementtheorie} and theoretical computer science \\cite{moggi1988computational, moggi1990abstract,\t\tmoggi1991notions}. Among others, they are a convenient tool for capturing side-effects of coalgebraic systems \\cite{rutten1998automata}: popular examples include the powerset monad (non-determinism), the distribution monad (probability), and the neighbourhood monad (alternating). While coalgebraic systems with side-effects can be more compact than their deterministic counterparts, they often lack a unique minimal acceptor for a given language. For instance, every regular language admits a unique deterministic automaton with minimal state space, which can be computed via the Myhill-Nerode construction. On the other hand, for some regular languages there exist multiple non-isomorphic non-deterministic automata with minimal state space. The problem has been independently approached for different variants of side-effects, often with the common idea of restricting to a particular subclass \\cite{esposito2002learning, berndt2017learning}. For example, for non-derministic automata, the subclass of so-called residual finite state automata has been identified as suitable.\n It moreover has turned out that in order to construct a unique minimal realisation in one of the subclasses it is often sufficient to derive an equivalent system with free state space from a particular given system \\cite{van2017learning}. As Arbib and Manes realised \\cite{arbib1975fuzzy}, instrumental to the former is what they call a \\textit{scoop}, or what we call a generator in \\Cref{generatordefinition}, a slight generalisation of bases. In other words, our definition of a basis for an algebra over a monad has its origin in a context that is not purely concerned with universal algebra. Throughout the paper we will value these roots by lifting results of Arbib and Manes from scoops to bases. More importantly, we believe that our treatment allows us to uncover hidden ramifications between certain areas of universal algebra and the theory of coalgebras.\n\nThe paper is structured as follows. In \\Cref{preliminariessec} we recall the basic categorical notions of a monad, algebras over a monad, coalgebras, distributive laws, and bialgebras. In \\Cref{generatorssec} we introduce generators for algebras over monads and exemplify their relation with free bialgebras. The definition of bases for algebras over monads, basic results, and their relation with free bialgebras is covered in \\Cref{basesection}. Questions about the existence and the uniqueness of bases are answered in \\Cref{existencebasesec} and \\Cref{uniquebasesec}, respectively.  In \\Cref{representationtheorysec} we generalise the representation theory of linear maps between vector spaces to a representation theory of homomorphisms between algebras over a monad.\nThe intuition that bases for an algebra over a monad coincide with free isomorphic algebras is clarified in \\Cref{basesfreealgebrasec}. In \\Cref{basesforbialgebrasec} we look into bases for bialgebras, which are algebras for a particular monad. In \\Cref{examplesec} we instantiate the theory for a variety of monads. Related work and future work are discussed in \\Cref{relatedworksec} and \\Cref{discussionsec}, respectively. Further details  can be found in  \\Cref{basesascoaglebrassec}.\n\n\n\n\n\\section{Preliminaries}\n\n\\label{preliminariessec}\n\nWe only assume a basic knowledge of category theory, e.g. an understanding of categories, functors, natural transformations, and adjunctions. All other relevant definitions can be found in the paper. In this section, we recall the notions of a monad, algebras for a monad, coalgebras, distributive laws, and bialgebras.\n\nThe concept of a monad can be traced back both to algebraic topology \\cite{godementtheorie} and to an alternative to Lawvere theory as a category theoretic formulation of universal algebra \\cite{eilenberg1965adjoint, linton1966some}. For an extended historical overview  we refer to the survey of Hyland and Power \\cite{hyland2007category}. In the context of computer science, monads have been introduced by Moggi as a general perspective on exceptions, side-effects, non-determinism, and continuations \\cite{moggi1988computational, moggi1990abstract,\t\tmoggi1991notions}. \n\n\\begin{definition}[Monad]\n\tA \\textit{monad} on $\\mathscr{C}$ is a tuple $\\mathbb{T} = (T, \\mu, \\eta)$ consisting of an endofunctor $T: \\mathscr{C} \\rightarrow \\mathscr{C}$ and natural transformations $\\mu: T^2 \\Rightarrow T$ and $\\eta: \\textnormal{id}_{\\mathscr{C}} \\Rightarrow T$ satisfying the commutative diagrams\n\t\\begin{equation*}\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tT^3X \\arrow{r}{T\\mu_X} \\arrow{d}[left]{\\mu_{TX}} & T^2X \\arrow{d}{\\mu_X}\\\\\n\t\t\tT^2X  \\arrow{r}{\\mu_X}  & TX \n\t\t\\end{tikzcd}\t\n\t\t\\qquad\n\t\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tTX \\arrow{r}{\\eta_{TX}} \\arrow{d}[left]{T\\eta_X} \\arrow{dr}{\\textnormal{id}_{TX}} & T^2X \\arrow{d}{\\mu_X}\\\\\n\t\t\tT^2X  \\arrow{r}{\\mu_X}  & TX \n\t\t\\end{tikzcd}\t\n\t\\end{equation*}\t\n\t\t\tfor all objects $X$ in $\\mathscr{C}$.\t\t\n\\end{definition}\n\nMany examples of monads arise as the result of a free-forgetful adjunction, for instance the free group monad or the free vector space monad. Below we provide some details for the latter case. More monads are covered in \\Cref{monoidmonad} and \\Cref{examplesec}.\n\n\\begin{example}[Vector spaces]\n\\label{freevectorspacemonadexample}\n\tThe free $k$-vector space monad is an instance of the so-called multiset monad  over some semiring $S$, when $S$ is given by the  field $k$. The underlying set endofunctor $T$ assigns to a set $X$ the set of finitely-supported $X$-indexed sequences $\\varphi$ in $S$, typically written as formal sums $\\sum_i s_i \\cdot x_i$ for $s_i = \\varphi(x_i)$; the unit $\\eta_X$ maps an element in $X$ to the singleton multiset $1 \\cdot x$; and the  multiplication $\\mu_X$ satisfies $\\mu_X(\\sum_i s_i \\cdot \\varphi_i)(x) = \\sum_i s_i \\cdot \\varphi_i(x)$ \\cite{coumans2010scalars}. \n\\end{example}\n\nIf a monad results from a free-forgetful adjunction induced by some algebraic structure, the latter may be recovered in the following sense:\n\n\\begin{definition}[$\\mathbb{T}$-algebra]\n\tAn \\textit{algebra} for a monad $\\mathbb{T} = (T, \\mu, \\eta)$ on $\\mathscr{C}$ is a tuple $(X, h)$ consisting of an object $X$ and a morphism $h: TX \\rightarrow X$ such that the diagrams on the left and in the middle below commute\n\t\\begin{equation*}\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tT^2X \\arrow{r}{\\mu_X} \\arrow{d}[left]{Th} & TX \\arrow{d}{h}\\\\\n\t\t\tTX  \\arrow{r}{h}  & X \n\t\t\\end{tikzcd}\t\n\t\t\\qquad \n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tX \\arrow{r}[above]{\\textnormal{id}_X} \\arrow{d}[left]{\\eta_X}  & X  \\\\\n\t\t\t TX \\arrow{ur}[right]{h}  &   \n\t\t\\end{tikzcd}\n\t\t\\qquad\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tTX \\arrow{r}{Tf} \\arrow{d}[left]{h_X} & TY \\arrow{d}{h_Y}\\\\\n\t\t\tX  \\arrow{r}{f}  & Y\n\t\t\\end{tikzcd}\t.\n\t\\end{equation*}\nA \\textit{homomorphism} $f: (X, h_X) \\rightarrow (Y,h_Y)$ between $\\mathbb{T}$-algebras is a morphism $f: X \\rightarrow Y$ such that the diagram on the right above commutes. The category of $\\mathbb{T}$-algebras and homomorphisms is denoted by $\\mathscr{C}^{\\mathbb{T}}$.\n\\end{definition}\n\nThe canonical example for an algebra over a monad is the free $\\mathbb{T}$-algebra $(TX, \\mu_X)$ for any object $X$ in $\\mathscr{C}$. \nBelow we give some more details on how to recognise algebras over the free vector space monad as vector spaces.\n\n\\begin{example}[Vector spaces]\n\\label{vectorspacemonadalgebras}\n\tLet $\\mathbb{T}$ be the free vector space monad defined in \\Cref{freevectorspacemonadexample}. Every $\\mathbb{T}$-algebra $(X,h)$ induces a vector space structure on its underlying set by interpreting a finite formal linear combination $\\sum_i \\lambda_i \\cdot x_i$ as an element $h(\\varphi) \\in X$ for $\\varphi(x) := \\lambda_i$, if $x = x_i$, and $\\varphi(x) := 0$ otherwise. Conversely, every vector space with underlying set $X$ induces an algebra $(X,h)$ over $\\mathbb{T}$ by defining $h(\\varphi) := \\sum_{x} \\lambda_x \\cdot x$ with $\\lambda_x := \\varphi(x)$ for a finitely-supported sequence $\\varphi$.\n\t\\end{example}\n \n\n\n\nWe now turn our attention to the dual of algebras: coalgebras \\cite{rutten1998automata}. While algebras have been used in the context of computer science to model finite data types, coalgebras deal with infinite data types and have turned out to be suited as an abstraction for a variety of state-based systems \\cite{rutten2000universal}.  \n\n\\begin{definition}[$F$-coalgebra]\n\tA \\textit{coalgebra} for an endofunctor $F: \\mathscr{C} \\rightarrow \\mathscr{C}$ is a tuple $(X, k)$ consisting of an object $X$ and a morphism $k: X \\rightarrow FX$. A \\textit{homomorphism} $f: (X, k_X) \\rightarrow (Y,k_Y)$ between $F$-coalgebras is a morphism $f: X \\rightarrow Y$ satisfying the commutative diagram\n\t\\begin{equation*}\n\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tX \\arrow{r}{f} \\arrow{d}[left]{k_X} & Y \\arrow{d}{k_Y}\\\\\n\t\t\tFX  \\arrow{r}{Ff}  & FY \n\t\t\\end{tikzcd}\t.\t\n\\end{equation*}\nThe category of coalgebras and homomorphisms is denoted by $\\textsf{Coalg}(F)$. \nA $F$-coalgebra $(\\Theta, k_{\\Theta})$ is \\textit{final}, if it is final in $\\textnormal{\\textsf{Coalg}}(F)$, that is, for every $F$-coalgebra $(X,k_X)$ there exists a unique homomorphism \\[ !_{(X,k_X)}: (X,k_X) \\longrightarrow (\\Theta, k_{\\Theta}). \\]\n\\end{definition}\n\nFor example, coalgebras for the set endofunctor $FX = X^A \\times B$ are unpointed Moore automata with input $A$ and output $B$, and the final $F$-coalgebra homomorphism assigns to a state $x$ of an unpointed Moore automaton the language in $A^* \\rightarrow B$ accepted by the latter when given the initial state $x$.\n\nWe are particularly interested in systems with side-effects, for instance non-deterministic automata. Often such systems can be realised as coalgebras for an endofunctor composed of a monad and an endofunctor similar to $F$ above. In these cases the compatibility between the dynamics of the system and its side-effects can be captured by a so-called a distributive law. Distributive laws have originally occurred as a way to compose monads \\cite{beck1969distributive}, but now also exist in a wide range of other forms \\cite{Street2009}. For our particular case it is sufficient to consider distributive laws between a monad and an endofunctor.\n\n\\begin{definition}[Distributive law]\n\tLet $\\mathbb{T} = (T, \\mu, \\eta)$ be a monad on $\\mathscr{C}$ and $F: \\mathscr{C} \\rightarrow \\mathscr{C}$ an endofunctor. A natural transformation $\\lambda: TF \\Rightarrow FT$ is called  \\textit{distributive law}, if it satisfies \n\t\\begin{equation*}\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tFX \\arrow{r}[above]{F\\eta_X} \\arrow{d}[left]{\\eta_{FX}}  & FTX  \\\\\n\t\t\t TFX \\arrow{ur}[right]{\\lambda_X}  &   \n\t\t\\end{tikzcd}\n\t\t\\qquad\n\t\t\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tT^2FX \\arrow{r}{T\\lambda_X} \\arrow{d}[left]{\\mu_{FX}} & TFTX  \\arrow{r}{\\lambda_{TX}} & FT^2X \\arrow{d}{F\\mu_X} \\\\\n\t\t\tTFX \\arrow{rr}{\\lambda_X} && FTX   \n\t\t\\end{tikzcd}.\n\t\\end{equation*}\n\t\\end{definition}\n\nGiven a distributive law, it is straightforward to model the determinisation of a system with side-effects. Indeed, for any morphism $f: Y \\rightarrow X$ and $\\mathbb{T}$-algebra $(X,h)$, the natural isomorphism underlying the free-algebra adjunction yields a $\\mathbb{T}$-algebra homomorphism \n\\begin{equation}\n\\label{algebrainducedliftinggeneral}\n\tf^{\\sharp} := h \\circ Tf: (TY, \\mu_Y) \\longrightarrow (X, h).\n\\end{equation}\nThus, in particular, any $FT$-coalgebra $k: X \\rightarrow FTX$ lifts to a $F$-coalgebra\n\\begin{equation}\n\\label{ftlifting}\n\tk^{\\sharp} := (F \\mu_X \\circ \\lambda_{TX}) \\circ Tk: (TX, \\mu_X) \\longrightarrow (FTX, F \\mu_X \\circ \\lambda_{TX}).\n\\end{equation}\n\nFor instance, if $\\mathbb{P}$ is the powerset monad and $F$ is the set endofunctor for deterministic automata satisfying $FX = X^A \\times 2$, the disjunctive $\\mathbb{P}$-algebra structure on the set $2$ induces a canonical distributive law \\cite{jacobs2012trace}, such that the lifting \\eqref{ftlifting} is given by the classical determinisation procedure for non-deterministic automata \\cite{rutten2013generalizing}. \n\nOne can show that the state spaces of $F$-coalgebras obtained by the lifting \\eqref{ftlifting} can canonically be equipped with a $\\mathbb{T}$-algebra structure that is compatible with the $F$-coalgebra structure: they are $\\lambda$-bialgebras.\n\n\\begin{definition}[$\\lambda$-bialgebra]\n\\label{defbialgebras}\n\tLet $\\lambda$ be a distributive law between a monad $\\mathbb{T}$ and an endofunctor $F$. A $\\lambda$\\textit{-bialgebra} is a tuple $(X, h, k)$ consisting of a $\\mathbb{T}$-algebra $(X, h)$ and a $F$-coalgebra $(X, k)$, satisfying\n\t\\begin{equation*}\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\tTX \\arrow{r}{Tk} \\arrow{dd}[left]{h} & TFX \\arrow{d}{\\lambda_X} \\\\\n\t\t& FTX \\arrow{d}{Fh} \\\\\n\t\tX \\arrow{r}{k} & FX.\n\t\t\\end{tikzcd}.\n\t\\end{equation*}\n\tA \\textit{homomorphism} $f: (X, h_X, k_X) \\rightarrow (Y, h_Y, k_Y)$ between $\\lambda$-bialgebras is a morphism $f: X \\rightarrow Y$ that is simultaneously a $\\mathbb{T}$-algebra homomorphism and a $F$-coalgebra homomorphism. The category of $\\lambda$-bialgebras and morphisms is denoted by $\\textsf{Bialg}(\\lambda)$.\n\t\\end{definition}\n\t\nIt is well-known that a distributive law $\\lambda$ between a monad $\\mathbb{T}$ and an endofunctor $F$ induces simultaneously\n\\begin{itemize}\n\t\\item a monad $\\mathbb{T}_{\\lambda} = (T_{\\lambda}, \\mu, \\eta)$ on $\\textsf{Coalg}(F)$ by $T_{\\lambda}(X,k) = (TX, \\lambda_X \\circ Tk)$ and $T_{\\lambda}f = Tf$; and\n\t\\item an endofunctor $F_{\\lambda}$ on $\\mathscr{C}^{\\mathbb{T}}$ by $F_{\\lambda}(X,h) = (FX, Fh \\circ \\lambda_X)$ and $F_{\\lambda}f = Ff$,\n\\end{itemize}\nsuch that the algebras over $\\mathbb{T}_{\\lambda}$, the coalgebras of $F_{\\lambda}$, and $\\lambda$-bialgebras coincide \\cite{turi1997towards}. In light of the latter we will not distinguish between the different categories, and instead use the notation of $\\lambda$-bialgebras for all three cases.\n\nOne can further show that, if it exists, the final $F$-coalgebra $(\\Theta, k_{\\Theta})$ induces a final $F_{\\lambda}$-coalgebra $(\\Theta, h_{\\Theta}, k_{\\Theta})$ for $h_{\\Theta} :=\\ !_{(T\\Theta, \\lambda_{\\Theta} \\circ T k_{\\Theta})}$ the unique $F$-coalgebra homomorphism below:\n\\begin{equation*}\n\t\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\t\tT\\Theta \\arrow{d}[left]{\\lambda_{\\Theta} \\circ T k_{\\Theta}} \\arrow[dashed]{r}{h_{\\Theta}} & \\Theta \\arrow{d}{k_{\\Theta}} \\\\\n\t\t\t\tFT\\Theta \\arrow[dashed]{r}{Fh_{\\Theta}} & F\\Theta\n\t\t\t\\end{tikzcd}.\n\\end{equation*}\nFor instance, for the canonical distributive law between the powerset monad $\\mathbb{P}$ and the set endofunctor $F$ with $FX = X^A \\times 2$ as before, one verifies that the underlying state space $A^* \\rightarrow 2$ of the final $F$-coalgebra will in this way be enriched with a $\\mathbb{P}$-algebra structure that takes the union of languages \\cite{jacobs2012trace}.\n\n\n\\section{Generators for algebras}\n\n\\label{generatorssec}\n\nIn this section we define what it means to be a generator for an algebra over a monad. Our notion coincides with what is called a \\textit{scoop} by Arbib and Manes \\cite{arbib1975fuzzy}. One may argue that the morphism $i$ in the definition below should be mono, but we choose to continue without this requirement. \n\n\n   \n\n\n\\begin{definition}[Generator \\cite{arbib1975fuzzy}]\n\\label{generatordefinition}\n\tA \\textit{generator} for a $\\mathbb{T}$-algebra $(X, h)$ is a tuple $(Y, i, d)$  consisting of an object $Y$, a morphism $i: Y \\rightarrow X$, and a morphism $d: X \\rightarrow TY$, such that $i^{\\sharp} \\circ d = \\textnormal{id}_{X}$, that is, the diagram on the left below commutes\n\t\\begin{equation*}\n\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tTY \\arrow{r}{Ti} & TX \\arrow{d}{h}\\\\\n\t\t\tX \\arrow{u}{d} \\arrow{r}{\\textnormal{id}_X}  & X \n\t\t\\end{tikzcd}\n\t\t\\qquad\n\t\t\\begin{tikzcd}[row sep=2em, column sep = 3em]\n\t\t\t\t\t&TY_{\\alpha} \\arrow{dd}[right]{Tf}  & \\\\\n\t\t\t\t\t X \\arrow{ur}{d_{\\alpha}} \\arrow{dr}[below]{d_{\\beta}} \\\\\n\t\t\t\t\t& TY_{\\beta} \n\t\t\t\t\\end{tikzcd}\n\t\t\t\t\t\t\t\t\\qquad\t\t\t\t\n\t\t\\begin{tikzcd}[row sep=2em, column sep = 3em]\n\t\t\t\t\tY_{\\alpha} \\arrow{dd}[left]{f} \\arrow{dr}{i_{\\alpha}} & \\\\\n\t\t\t\t\t& X \\\\\n\t\t\t\t\tY_{\\beta} \\arrow{ur}[below]{i_{\\beta}}\n\t\t\t\t\\end{tikzcd}.\n\t\t\\end{equation*}\n\t\tA morphism $f: (Y_{\\alpha},i_{\\alpha},d_{\\alpha}) \\rightarrow (Y_{\\beta},i_{\\beta},d_{\\beta})$ between generators for $(X,h)$ is a morphism $f: Y_{\\alpha} \\rightarrow Y_{\\beta}$ satisfying the two commutative diagrams on the right above.\n\\end{definition}\n\nWe give an example that slightly generalises the vector space situation mentioned in the introduction.\n\n\\begin{example}[Semimodules]\n\\label{semimoduleexample}\n\tLet $\\mathbb{T} = (T, \\mu, \\eta)$ be the multiset monad over some semiring $S$ defined in \\Cref{freevectorspacemonadexample}.\n\tFollowing the lines of \\Cref{vectorspacemonadalgebras}, one can show that $\\mathbb{T}$-algebras correspond to semimodules over $S$, such that a function $i: Y \\rightarrow X$ is part of a generator $(Y, i, d)$ for the former if and only if for all $x$ in $X$ there exists a finitely-supported $Y$-indexed $S$-sequence $d(x) = (s_y)_{y \\in Y}$, such that $x = \\sum_{y \\in Y} s_y \\cdot i(y)$.\n\\end{example}\n\nOne might would like to adapt \\Cref{generatordefinition} by replacing the existence of a morphism $d$ with the property of $i^{\\sharp}$ being an epimorphism. Indeed, in every category a morphism with a right-inverse is an epimorphism. However, conversely, not every epimorphism admits a right-inverse, and if there would exist one, it might not be unique. For this reason we treat the morphism $d$ as explicit data.\n\nIt is well-known that every algebra over a monad admits a generator \\cite{arbib1975fuzzy}.\n\n\\begin{lemma}\n\\label{canonicalgenerator}\n$(X, \\textnormal{id}_X, \\eta_X)$ is a generator for any $\\mathbb{T}$-algebra $(X,h)$.\n\\end{lemma}\n\\begin{proof}\nFollows immediately from the equality $h \\circ \\eta_X = \\textnormal{id}_X$.\n\\end{proof}\n\nThe following result is a slight generalisation of a statement by Arbib and Manes {\\cite{arbib1975fuzzy}. We are particularly interested in using the construction of an equivalent free bialgebra for a unified view on the theory of residual finite state automata and variations of it  \\cite{van2016master, van2020phd, myers2015coalgebraic, denis2002residual}; more details are given in \\Cref{RFSAexample}.\n\n\\begin{proposition}\n\\label{forgenerator-isharp-is-bialgebra-hom}\n\tLet $(X, h, k)$ be a $\\lambda$-bialgebra and let $(Y, i, d)$ be a generator for the $\\mathbb{T}$-algebra $(X,h)$. Then $i^{\\sharp} := h \\circ Ti : TY \\rightarrow X$ is a $\\lambda$-bialgebra homomorphism $i^{\\sharp}: (TY, \\mu_Y, (Fd \\circ k \\circ i)^{\\sharp}) \\rightarrow (X, h, k) $ for $(Fd \\circ k \\circ i)^{\\sharp} := F\\mu_Y \\circ \\lambda_{TY}\\circ T(Fd \\circ k \\circ i)$.\n\\end{proposition}\n\\begin{proof}\nThe commuting diagram below shows that for a $FT$-coalgebra $f: Y \\rightarrow FTY$ and $f^{\\sharp}: TY \\rightarrow FTY$ the lifting in \\eqref{ftlifting}, the tuple $(TY, \\mu_Y, f^{\\sharp})$ constitutes a $\\lambda$-bialgebra\n\\begin{equation}\n\\label{freebialgebra}\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\tT^2Y \\arrow{r}{T^2f} \\arrow{dd}[left]{\\mu_Y} & T^2 FTY \\arrow{dd}{\\mu_{FTY}} \\arrow{r}{T\\lambda_{TY}} & TFT^2Y \\arrow{d}{\\lambda_{T^2Y}} \\arrow{r}{TF\\mu_Y} & TFTY \\arrow{d}{\\lambda_{TY}} \\\\\n\t\t& & FT^3Y \\arrow{r}{FT{\\mu_Y}} \\arrow{d}[right]{F \\mu_{TY}} & FT^2Y \\arrow{d}[right]{F\\mu_Y} \\\\\n\t\tTY \\arrow{r}{Tf} & TFTY \\arrow{r}{\\lambda_{TY}} & FT^2Y \\arrow{r}{F\\mu_Y} & FTY\n\t\\end{tikzcd}.\n\\end{equation}\nIn particular, the tuple $(TY, \\mu_Y, (Fd \\circ k \\circ i)^{\\sharp})$ thus is a $\\lambda$-bialgebra. The lifting in \\eqref{algebrainducedliftinggeneral}  turns $i^{\\sharp}$ into a $\\mathbb{T}$-algebra homomorphism. It thus remains to show that $i^{\\sharp}$ is a $F$-coalgebra homomorphism. The latter follows from the commutativity of the following diagram\n\\begin{equation*}\n\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\tTY \\arrow{r}{Ti} \\arrow{d}[left]{Ti} & TX \\arrow{dd}{Tk} \\arrow{rr}{h} & & X \\arrow{ddddd}{k} \\\\\n\t\tTX \\arrow{d}[left]{Tk} \\\\\n\t\tTFX \\arrow{d}[left]{TFd} \\arrow{r}{\\textnormal{id}_{TFX}} & TFX \\arrow{ddr}{\\lambda_X} \\\\\n\t\tTFTY \\arrow{d}[left]{\\lambda_{TY}} \\arrow{r}{TFTi} & TFTX \\arrow{u}{TFh} \\\\\n\t\tFT^2Y \\arrow{r}{FT^2i} \\arrow{d}[left]{F\\mu_Y} & FT^2X \\arrow{r}{FTh} \\arrow{d}{F \\mu_X} & FTX \\arrow{d}{Fh} \\\\\n\t\tFTY \\arrow{r}{FTi} & FTX \\arrow{r}{Fh} & FX \\arrow{r}{\\textnormal{id}_{FX}} & FX\n\t\\end{tikzcd}.\n\\end{equation*}\n\\end{proof}\n\nWe show next that above construction extends to morphisms $f: (Y_{\\alpha}, i_{\\alpha}, d_{\\alpha}) \\rightarrow (Y_{\\beta}, i_{\\beta}, d_{\\beta})$ between generators for the underlying algebra of a bialgebra. For readability we abbreviate $k_{\\gamma} := (Fd_{\\gamma} \\circ k \\circ i_{\\gamma})^{\\sharp}$ for $\\gamma \\in  \\lbrace \\alpha , \\beta \\rbrace$.\n\n\\begin{lemma}\n\tThe morphism $Tf: TY_{\\alpha} \\rightarrow TY_{\\beta}$ is a $\\lambda$-bialgebra homomorphism $Tf: (TY_{\\alpha}, \\mu_{Y_{\\alpha}}, k_{\\alpha}) \\rightarrow (TY_{\\beta}, \\mu_{Y_{\\beta}}, k_{\\beta})$ satisfying $(i_{\\beta})^{\\sharp} \\circ Tf = (i_{\\alpha})^{\\sharp}$.\n\\end{lemma}\n\\begin{proof}\nThe identity $(i_{\\beta})^{\\sharp} \\circ Tf = (i_{\\alpha})^{\\sharp}$ follows from the equality $i_{\\beta} \\circ f = i_{\\alpha}$, as shown below\n\t\\begin{equation*}\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\tTY_{\\alpha} \\arrow{r}{Tf} \\arrow{d}[left]{Ti_{\\alpha}}  & \tTY_{\\beta} \\arrow{d}{Ti_{\\beta}} \\arrow{r}{Ti_{\\beta}} & TX \\arrow{d}{h} \\\\\n\t\tTX \\arrow{r}{\\textnormal{id}_{TX}}  & TX \\arrow{r}{h} & X\n\t\t\\end{tikzcd}.\n\t\\end{equation*}\n\tOne easily verifies that the lifting \\eqref{ftlifting} extends to a functor from the category of $FT$-coalgebras to the category of $\\lambda$-bialgebras; see e.g. \\eqref{freebialgebra}. It thus remains to show that $f$ is a $FT$-coalgebra homomorphism $\n\tf: (Y_{\\alpha}, k_{\\alpha}) \\rightarrow (Y_{\\beta}, k_{\\beta})$.  The latter follows from the commutativity of the following diagram\n\t\\begin{equation*}\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tY_{\\beta} \\arrow{r}{i_{\\beta}} & X \\arrow{r}{k} & FX \\arrow{r}{Fd_{\\beta}} & FTY_{\\beta} \\\\\n\t\t\tY_{\\alpha} \\arrow{u}{f} \\arrow{r}{i_{\\alpha}} & X \\arrow{u}{\\textnormal{id}_X} \\arrow{r}{k} & FX \\arrow{u}{\\textnormal{id}_{FX}} \\arrow{r}{Fd_{\\alpha}} & FTY_{\\alpha} \\arrow{u}[right]{FTf}\n\t\t\\end{tikzcd}.\n\t\\end{equation*}\n\\end{proof}\n\nIn the following example we instantiate the previous results to recover the so-called canonical residual finite state automata \\cite{denis2002residual}.\n\n\\begin{example}[Canonical RFSA]\n\\label{RFSAexample}\t\nAs before, let $\\mathbb{P}$ be the powerset monad and $F$ the set endofunctor for deterministic automata over the alphabet $A$ satisfying $FX = X^A \\times 2$. One verifies that the disjunctive $\\mathbb{P}$-algebra structure on the set $2$ induces a canonical distributive law $\\lambda$ between $\\mathbb{P}$ and $F$, such that $\\lambda$-bialgebras are deterministic unpointed automata in the category of complete lattices and join-preserving functions \\cite{jacobs2012trace};  for more details see \\Cref{powersetmonadsec}.\nWe are particularly interested in the $\\lambda$-bialgebra that is typically called the minimal $\\mathbb{P}$-automaton $M_{\\mathbb{P}}(L)$ for a regular language $L$ \\cite{van2020phd, van2017learning}. \nOn a very high-level, $M_{\\mathbb{P}}(L)$ may be recognised as some algebraic closure of the well-known minimal automaton $M(L)$ for $L$ in the category of sets and functions. More concretely, it consists of the inclusion-ordered free complete lattice of unions of residuals of $L$, equipped with the usual transition and output functions for languages inherited by the final coalgebra for $F$. \nUsing well-known lattice theoretic arguments one can show that the tuple  $(\\mathcal{J}(M_{\\mathbb{P}}(L)),i,d)$, with $i$ the subset-embedding of join-irreducibles and $d$ the function assigning to a language the join-irreducible languages below it, is a generator for $M_{\\mathbb{P}}(L)$. Writing $k$ for the $F$-coalgebra structure of $M_{\\mathbb{P}}(L)$, it is easy to verify that the $F\\mathbb{P}$-coalgebra structure $Fd \\circ k \\circ i$ on $\\mathcal{J}(M_{\\mathbb{P}}(L))$ mentioned in \\Cref{forgenerator-isharp-is-bialgebra-hom} corresponds precisely to the so-called canonical residual finite state automaton for $L$  \\cite{denis2002residual}.\n\\end{example}\n\n\nWe close this section with a more compact characterisation of generators for free algebras.\n\n\\begin{lemma}\n\\label{setgenerator}\n\tLet $i: Y \\rightarrow X$ and $d: X \\rightarrow TY$ be morphisms such that the following diagram commutes:\n\t\\begin{equation*}\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\t\t\tT^2Y \\arrow{r}{T^2i} & T^2X \\arrow{d}{\\mu_X} \\\\\n\t\t\t\t\tTX \\arrow{u}{Td} \\arrow{r}{\\textnormal{id}_{TX}}  & TX\n\t\t\t\t\\end{tikzcd}\t.\n\t\t\t\t\\end{equation*}\t\n\tThen $(Y, \\eta_X \\circ i, \\mu_Y \\circ Td)$ is a generator for the $\\mathbb{T}$-algebra $(TX, \\mu_X)$.\n\\end{lemma}\n\\begin{proof}\nFollows from the commutativity of the diagram below\n\t\\begin{equation*}\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\t\t\tTY \\arrow{r}{Ti} & TX \\arrow{r}{T\\eta_X} \\arrow{dr}{\\textnormal{id}_{TX}} & T^2X \\arrow{d}{\\mu_X} \\\\\n\t\t\t\t\tT^2Y \\arrow{u}{\\mu_Y} \\arrow{r}{T^2i} & T^2 X \\arrow{u}{\\mu_X} \\arrow{r}{\\mu_X} & TX \\arrow{d}{\\textnormal{id}_{TX}} \\\\\n\t\t\t\t\tTX \\arrow{u}{Td} \\arrow{rr}{\\textnormal{id}_{TX}} & & TX\n\t\t\t\t\\end{tikzcd}\t.\n\t\t\t\t\\end{equation*}\n\\end{proof}\n\n\\section{Bases for algebras}\n\n\\label{basesection}\n\nIn the last section we adopted the notion of a scoop by Arbib and Manes \\cite{arbib1975fuzzy} by introducing generators for algebras over a monad. In this section we extend the former to the definition of a basis for an algebra over a monad by adding a uniqueness constraint. While scoops have mainly occured in the context of state-based systems, our extension  allows us to emphasise their ramifications with universal algebra.\n\n\\begin{definition}[Basis]\n\\label{basisdefinition}\n\tA \\textit{basis} for a $\\mathbb{T}$-algebra $(X, h)$ is a tuple $(Y, i, d)$  consisting of an object $Y$, a morphism $i: Y \\rightarrow X$, and a morphism $d: X \\rightarrow TY$, such that $i^{\\sharp} \\circ d = \\textnormal{id}_{X}$ and $d \\circ i^{\\sharp} = \\textnormal{id}_{TY}$, that is, the following two diagrams commute:\n\t\\begin{equation*}\n\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tTY \\arrow{r}{Ti} & TX \\arrow{d}{h}\\\\\n\t\t\tX \\arrow{u}{d} \\arrow{r}{\\textnormal{id}_X}  & X \n\t\t\\end{tikzcd}\n\t\t\\qquad\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tTX \\arrow{r}{h} & X \\arrow{d}{d}\\\\\n\t\t\tTY \\arrow{u}{Ti} \\arrow{r}{\\textnormal{id}_{TY}}  & TY \n\t\t\\end{tikzcd}.\t\n\\end{equation*} \nA \\textit{homomorphism} between two bases for $(X,h)$ is a morphism between the underlying generators. The category consisting of bases for a $\\mathbb{T}$-algebra $(X,h)$ and homomorphisms between them is\t denoted by $\\textnormal{\\textsf{Bases}}(X,h)$.\n\\end{definition}\n\n We begin with an example for a basis in above sense for the theory of monoids.\n\n\\begin{example}[Monoids]\n\\label{monoidmonad}\n\tLet $\\mathbb{T} = (T, \\mu, \\eta)$ be the set monad whose underlying endofunctor $T$ assigns to a set $X$ the set of all finite words over the alphabet $X$; whose unit $\\eta_X$ assigns to a character in $X$ the corresponding word of length one; and whose multiplication $\\mu_X$ syntactically flattens words over words over the alphabet $X$ in the usual way. The monad $\\mathbb{T}$ is also known as the list monad. One verifies that the constraints for its algebras correspond to the unitality and associativity laws of monoids. A function $i: Y \\rightarrow X$ is thus part of a basis $(Y,i,d)$ for a $\\mathbb{T}$-algebra with underlying set $X$ if and only if for all $x \\in X$ there exists a unique word $d(x) = \\lbrack y_1, ...,y_n \\rbrack$ over the alphabet $Y$ satisfying $x = i(y_1) \\cdot ... \\cdot i(y_n)$. \n\t\n\t\\end{example}\n\nThe next result establishes that the morphism $d$ is in fact an algebra homomorphism, and, intuitively, that elements of a basis are uniquely generated by their image under the monad unit, that is, typically by themselves.\n\n\n\\begin{lemma}\n\\label{forbasis-d-isalgebrahom}\n\tLet $(Y, i, d)$ be a basis for a $\\mathbb{T}$-algebra $(X, h)$. Then the following two diagrams commute:\n\t\\begin{equation*}\n\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tTX \\arrow{d}[left]{h} \\arrow{r}{Td} & T^2Y \\arrow{d}{\\mu_Y} \\\\\n\t\t\tX \\arrow{r}{d} & TY\n\t\t\\end{tikzcd}\n\t\t\\qquad\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tY \\arrow{r}[above]{i}   \\arrow{d}[left]{\\eta_{Y}} & X \\arrow{dl}[right]{d}  \\\\\n\t\t\t TY  &    \n\t\t\\end{tikzcd}.\n\t\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nFollows from the commutativity of the following two diagrams\n\t\\begin{equation*}\n\t\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\t\tTX \\arrow{rr}{Td} \\arrow{d}[left]{\\textnormal{id}_{TX}} & & T^2Y \\arrow{dl}{T^2i} \\arrow{dd}{\\mu_Y} \\\\\n\t\t\t\tTX \\arrow{dd}[left]{h} & T^2X \\arrow{l}{Th} \\arrow{d}{\\mu_X} \\\\\n\t\t\t\t& TX \\arrow{dl}{h} & TY \\arrow{l}{Ti} \\arrow{d}{\\textnormal{id}_{TY}} \\\\\n\t\t\t\tX \\arrow{rr}{d} & & TY\n\t\t\t\\end{tikzcd}\n\t\t\t\\qquad\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tY \\arrow{r}{i} \\arrow{d}[left]{\\eta_Y} & X \\arrow{r}{\\textnormal{id}_X} \\arrow{d}[left]{\\eta_X} & X \\arrow{dddll}{d} \\\\\n\t\t\tTY \\arrow{dd}[left]{\\textnormal{id}_{TY}} \\arrow{r}{Ti} &TX \\arrow{ur}{h} \\\\ \n\t\t\t\\\\\n\t\t\tTY\n\t\t\\end{tikzcd}.\n\t\\end{equation*}\n\\end{proof}\n\nIn consequence we can derive the following three corollaries. First, every algebra with a basis is isomorphic to a free algebra.\n\n\\begin{corollary}\n\\label{basisimpliesfree}\n\tLet $(Y,i,d)$ be a basis for a $\\mathbb{T}$-algebra $(X,h)$. Then $d: X \\rightarrow TY$ is a $\\mathbb{T}$-algebra isomorphism $d: (X,h) \\rightarrow (TY, \\mu_Y)$.\n\\end{corollary}\n\\begin{proof}\n\tBy \\Cref{forbasis-d-isalgebrahom} the morphism $d$ is a $\\mathbb{T}$-algebra homomorphism. From general arguments it follows that the lifting $i^{\\sharp} = h \\circ Ti$ is a $\\mathbb{T}$-algebra homomorphism in the reverse direction. Since $(Y,i,d)$ is a basis, $d$ and $i^{\\sharp}$ are mutually inverse.\n\\end{proof}\n\nSecondly, an algebra with a basis embeds into the free algebra it spans. The monomorphism is fundamental to an alternative approach to bases  \\cite{jacobs2011bases}; for more details see \\Cref{basesascoaglebrassec}. \n \n\\begin{corollary}\n\\label{Tid-algebrahom}\n\tLet $(Y,i,d)$ be a basis for a $\\mathbb{T}$-algebra $(X,h)$. Then $Ti \\circ d: X \\rightarrow TX$ is a $\\mathbb{T}$-algebra monomorphism $Ti \\circ d: (X,h) \\rightarrow (TX, \\mu_X)$ with left-inverse $h: (TX, \\mu_X) \\rightarrow (X, h)$.\n\\end{corollary}\n\\begin{proof}\n\tThe morphism $Ti \\circ d$ is a $\\mathbb{T}$-algebra homomorphism since by \\Cref{forbasis-d-isalgebrahom} the following diagram commutes\n\t\\begin{equation*}\n\t\t\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\t\t\tTX \\arrow{d}[left]{h} \\arrow{r}{Td} & T^2Y \\arrow{d}{\\mu_Y} \\arrow{r}{T^2i} & T^2X \\arrow{d}{\\mu_X} \\\\\n\t\t\t\t\tX \\arrow{r}{d} & TY \\arrow{r}{Ti} & TX\n\t\t\t\t\\end{tikzcd}.\n\t\\end{equation*}\n\tThe morphism $h$ is a $\\mathbb{T}$-algebra homomorphism since the equality $h \\circ \\mu_X = h \\circ Th$ holds for all algebras over a monad. By the definition of a generator $h$ is a left-inverse to $Ti \\circ d$. The morphism $Ti \\circ d$ is mono since every morphism with left-inverse is mono.\t\n\t \\end{proof}\n\nThirdly, every algebra homomorphism is uniquely determined by its image on a basis.\n\n\n\\begin{corollary}\n\t\tLet $(Y, i, d)$ be a basis for a $\\mathbb{T}$-algebra $(X,h_X)$, and let $(Z,h_Z)$ be another $\\mathbb{T}$-algebra. For every morphism $f: Y \\rightarrow Z$ there exists a $\\mathbb{T}$-algebra homomorphism $f^{\\sharp}: (X,h_X) \\rightarrow (Z,h_Z)$ satisfying $f^{\\sharp} \\circ i = f$.\n\\end{corollary}\n\t\\begin{proof}\n\t\tWe define a candidate as follows $f^{\\sharp} := h_Z \\circ Tf \\circ d$. Using \\Cref{forbasis-d-isalgebrahom} we establish the commutativity of the following two diagrams\n\t\t\\begin{equation*}\n\t\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tX \\arrow{r}{d} & TY \\arrow{r}{Tf} & TZ \\arrow{r}{\\textnormal{id}_{TZ}} & TZ \\arrow{d}{h_Z} \\\\\n\t\t\t& Y \\arrow{ul}{i} \\arrow{r}{f} \\arrow{u}{\\eta_Y} &Z \\arrow{u}{\\eta_{Z}} \\arrow{r}{\\textnormal{id}_{Z}} & Z\n\t\t\\end{tikzcd}\n\t\t\\end{equation*}\n\t\t\t\t\\begin{equation*}\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tTX \\arrow{r}{Td} \\arrow{d}[left]{h_X} & T^2Y \\arrow{d}{\\mu_{Y}} \\arrow{r}{T^2f}& T^2 Z \\arrow{d}{\\mu_Z} \\arrow{r}{Th_Z} & TZ \\arrow{d}{h_Z} \\\\\n\t\t\tX  \\arrow{r}{d}  & TY \\arrow{r}{Tf} & TZ \\arrow{r}{h_Z} &Z\n\t\t\\end{tikzcd}.\t\n\t\t\\end{equation*}\n\t\tThe first diagram proves the identity $f^{\\sharp} \\circ i = f$, and the second diagram shows that $f^{\\sharp}$ is a $\\mathbb{T}$-algebra homomorphism.\n\t\\end{proof}\n\nIn \\Cref{basisimpliesfree} it was proven that every algebra with a basis is isomorphic to a free algebra. We show next that the statement can be strengthened to bialgebras.\nIn more detail, in \\Cref{forgenerator-isharp-is-bialgebra-hom} we have seen that a generator for the underlying algebra of a bialgebra allows to construct a bialgebra with free state space, such that $i^{\\sharp}$ extends to a bialgebra homomorphism from the latter to the former. As it turns out, for a basis, the homomorphism $i^{\\sharp}$ is in fact an isomorphism.\n\n\\begin{proposition}\n\\label{forbasis-d-is-bi-algebrahom}\n\tLet $(X, h, k)$ be a $\\lambda$-bialgebra and let $(Y, i, d)$ be a basis for the $\\mathbb{T}$-algebra $(X,h)$. Then $d: X \\rightarrow TY$ is a $\\lambda$-bialgebra homomorphism $d: (X, h, k) \\rightarrow (TY, \\mu_Y, (Fd \\circ k \\circ i)^{\\sharp})$ for $(Fd \\circ k \\circ i)^{\\sharp} := F\\mu_Y \\circ \\lambda_{TY}\\circ T(Fd \\circ k \\circ i)$.\n\\end{proposition}\n\\begin{proof}\nAs before, it follows by general arguments that $(TY, \\mu_Y, (Fd \\circ k \\circ i)^{\\sharp})$ is a $\\lambda$-bialgebra. The morphism $d$ is a $\\mathbb{T}$-algebra homomorphism by \\Cref{forbasis-d-isalgebrahom}. It thus remains to show that $d$ is a $F$-coalgebra homomorphism. The former is established by  \\Cref{forbasis-d-isalgebrahom}, as shown below\n\\begin{equation*}\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tX \\arrow{rrrrr}{k} \\arrow{dd}[left]{d} \\arrow{rd}{\\textnormal{id}_X} & & & & & FX \\arrow{dd}{Fd} \\\\\n\t\t\t& X \\arrow{rrrru}{k} & & & FTX \\arrow{d}{FTd} \\arrow{ur}{Fh} & \\\\\n\t\t\tTY \\arrow{r}{Ti} & TX \\arrow{u}{h} \\arrow{r}{Tk} & TFX \\arrow{r}{TFd} \\arrow{rru}{\\lambda_X} & TFTY \\arrow{r}{\\lambda_{TY}} & FT^2Y \\arrow{r}{F\\mu_Y} & FTY\n\t\t\\end{tikzcd}.\n\\end{equation*}\n\\end{proof}\n\n\n\\begin{corollary}\n\\label{forbasis-bialgebra-areisomorphic}\n\tLet $(X, h, k)$ be a $\\lambda$-bialgebra and let $(Y, i, d)$ be a basis for the $\\mathbb{T}$-algebra $(X,h)$. Then the $\\lambda$-bialgebras  $(X, h, k)$ and $(TY, \\mu_Y, (Fd \\circ k \\circ i)^{\\sharp})$ are isomorphic.\n\\end{corollary}\n\\begin{proof}\n\tBy \\Cref{forbasis-d-is-bi-algebrahom} the morphism $d$ is a $\\lambda$-bialgebra homomorphism of the right type. By \\Cref{forgenerator-isharp-is-bialgebra-hom} the morphism $i^{\\sharp}$ is a $\\lambda$-bialgebra homomorphism in reverse direction to $d$. From the definition of a basis it follows that $d$ and $i^\\sharp$ are mutually inverse.\n\\end{proof}\n\n\\subsection{Existence of bases}\n\n\\label{existencebasesec}\n\nWhile every algebra over a monad admits a generator, cf. \\Cref{canonicalgenerator}, the latter is not necessarily true for a basis. In this section we show that one however can safely assume that every \\textit{free} algebra admits a basis. We begin with a characterisation of bases for free algebras that is slightly more compact than the one derived directly from the definition.\n\n\\begin{lemma}\n\\label{basisfreealgebraeasier}\n\tLet  $i: Y \\rightarrow X$ and $d: X \\rightarrow TY$ be morphisms such that the following two diagrams commute:\n\t\\begin{equation*}\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\t\t\tT^2Y \\arrow{r}{T^2i} & T^2X \\arrow{d}{\\mu_X} \\\\\n\t\t\t\t\tTX \\arrow{u}{Td} \\arrow{r}{\\textnormal{id}_{TX}}  & TX\n\t\t\t\t\\end{tikzcd}\t\n\t\t\t\t\\qquad\n\t\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\t\tTX \\arrow{r}{Td} & T^2Y \\arrow{d}{\\mu_Y} \\\\\n\t\t\t\tTY \\arrow{u}{Ti} \\arrow{r}{\\textnormal{id}_{TY}} & TY\n\t\t\t\\end{tikzcd}.\n\t\t\t\t\\end{equation*}\t\n\tThen $(Y, \\eta_X \\circ i, \\mu_Y \\circ Td)$ is a basis for the $\\mathbb{T}$-algebra $(TX, \\mu_X)$.\n\\end{lemma}\n\\begin{proof}\nOne part of the claim follows from \\Cref{setgenerator}. The other part follows from the commutativity of the following diagram\n\t\\begin{equation*}\n\t\t\t\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\t\t\t\tT^2X \\arrow{rr}{\\mu_X} & & TX \\arrow{d}{Td} \\\\\n\t\t\t\t\t\tTX \\arrow{u}{T\\eta_X} \\arrow{urr}{\\textnormal{id}_{TX}} \\arrow{rr}{Td} & & T^2Y \\arrow{d}{\\mu_Y} \\\\\n\t\t\t\t\t\tTY \\arrow{u}{Ti}  \\arrow{rr}{\\textnormal{id}_{TY}} & & TY\n\t\t\t\t\t\\end{tikzcd}.\n\t\\end{equation*}\n\\end{proof}\n\n\\begin{corollary}\n\t\\label{canonicalbasisforfreealgbera}\n\t$(X,\\eta_X, \\textnormal{id}_X)$ is a basis for the $\\mathbb{T}$-algebra $(TX, \\mu_X)$.\n\\end{corollary}\n\\begin{proof}\n\tUsing the equality $\t\\mu_X \\circ T\\eta_X = \\textnormal{id}_{TX}$, the claim follows from \\Cref{basisfreealgebraeasier} with $i= \\textnormal{id}_X$ and $d= \\eta_X$.\n\\end{proof}\n\n\\subsection{Uniqueness of bases}\n\n\\label{uniquebasesec}\n\nIn this section we investigate the uniqueness of bases for algebras over a monad.\nTo begin with, assume $(X,h)$ is an algebra over a monad $\\mathbb{T}$ and we are given a fixed morphism $i: Y \\rightarrow X$. Then any two morphisms $d_{\\alpha}$ and $d_{\\beta}$ turning $(Y,i,d_{\\alpha})$ and $(Y,i,d_{\\beta})$ into bases for $(X,h)$, respectively, are in fact identical: \\[ d_{\\alpha} = d_{\\alpha} \\circ i^{\\sharp} \\circ d_{\\beta} = d_{\\beta}. \\] If the morphism $i$ is not fixed, we have the following slightly weaker result about the uniqueness of bases:\n\n\\begin{lemma}\n\tLet $(Y_{\\alpha},i_{\\alpha},d_{\\alpha})$ and $(Y_{\\beta}, i_{\\beta}, d_{\\beta})$ be bases for a $\\mathbb{T}$-algebra $(X,h)$. Then the $\\mathbb{T}$-algebras $(TY_{\\alpha}, \\mu_{Y_{\\alpha}})$ and $(TY_{\\beta}, \\mu_{Y_{\\beta}})$ are isomorphic.\n\\end{lemma}\n\\begin{proof}\n\tWe have a $\\mathbb{T}$-algebra homomorphism  \\[ d_{\\beta} \\circ (i_{\\alpha})^{\\sharp}: (TY_{\\alpha}, \\mu_{Y_{\\alpha}}) \\longrightarrow (TY_{\\beta}, \\mu_{Y_{\\beta}}) \\] since the components $(i_{\\alpha})^{\\sharp}$ and $d_{\\beta}$ are $\\mathbb{T}$-algebra homomorphisms by general arguments and \\Cref{forbasis-d-isalgebrahom}, respectively.\n\tAnalogously, by symmetry, the morphism $d_{\\alpha} \\circ (i_{\\beta})^{\\sharp}$ is a $\\mathbb{T}$-algebra homomorphism in the reverse direction.\n\tIt is easy to verify that the definition of a basis implies that both morphisms are mutually inverse.\n\\end{proof}\n\nIf a set monad preserves the set cardinality relation in the sense that  \n\\[ \\vert Y_{\\alpha} \\vert \\not = \\vert Y_{\\beta} \\vert \\quad \\textnormal{implies} \\quad \\vert TY_{\\alpha} \\vert \\not = \\vert TY_{\\beta} \\vert, \\] above result in particular shows that any two bases for a fixed algebra have the same cardinality.\n\n\\subsection{Representation theory}\n\n\\label{representationtheorysec}\n\nIn this section we use our general definition of a basis to derive a representation theory for homomorphisms between algebras over monads that is analogous to the representation theory for linear transformations between vector spaces.\n\nIn more detail, recall that a linear transformation $L: V \\rightarrow W$ between $k$-vector spaces with finite bases $\\alpha = \\lbrace v_1, ... , v_n \\rbrace$ and $\\beta = \\lbrace w_1,  ..., w_m \\rbrace$, respectively, admits a matrix representation $L_{\\alpha \\beta} \\in \\textnormal{Mat}_{k}(m, n)$ with  \\[ L(v_j) = \\sum_i (L_{\\alpha \\beta})_{i,j} w_i, \\] such that for any vector $v$ in $V$ the coordinate vectors $L(v)_{\\beta} \\in k^m$ and $v_{\\alpha} \\in k^n$   \nsatisfy the equality \\[ L(v)_{\\beta} = L_{\\alpha \\beta} v_{\\alpha}. \\] A great amount of linear algebra is concerned with finding bases such that the corresponding matrix representation is in an efficient shape, for instance diagonalised. The following definitions generalise the situation by substituting Kleisli morphisms for matrices.\n\n  \n\\begin{definition}\n\tLet $\\alpha = (Y_{\\alpha}, i_{\\alpha}, d_{\\alpha})$ and $\\beta = (Y_{\\beta}, i_{\\beta}, d_{\\beta})$ be bases for $\\mathbb{T}$-algebras $(X_{\\alpha},h_{\\alpha})$ and $(X_{\\beta},h_{\\beta})$, respectively. The \\textit{basis representation} of a $\\mathbb{T}$-algebra homomorphism $f: (X_{\\alpha},h_{\\alpha}) \\rightarrow (X_{\\beta},h_{\\beta})$ with respect to $\\alpha$ and $\\beta$ is the composition \n\t\\begin{equation}\n\t\\label{basisrepresentation}\n\t\tf_{\\alpha \\beta} := Y_{\\alpha} \\overset{i_{\\alpha}}{\\longrightarrow} X_{\\alpha} \\overset{f}{\\longrightarrow} X_{\\beta} \\overset{d_{\\beta}}{\\longrightarrow} TY_{\\beta}.\n\t\\end{equation}\n\t Conversely, the morphism \\textit{associated} with a Kleisli morphism $p: Y_{\\alpha} \\rightarrow TY_{\\beta}$ with respect to $\\alpha$ and $\\beta$ is the composition \\begin{equation}\n\t\t\\label{associatedmorph}\n\t\tp^{\\alpha \\beta} := X_{\\alpha} \\overset{d_{\\alpha}}{\\longrightarrow} TY_{\\alpha} \\overset{Tp}{\\longrightarrow} T^2Y_{\\beta} \\overset{\\mu_{Y_{\\beta}}}{\\longrightarrow} TY_{\\beta} \\overset{Ti_{\\beta}}{\\longrightarrow} TX_{\\beta} \\overset{h_{\\beta}}{\\longrightarrow} X_{\\beta}.\n\t\t\t\\end{equation}\n\t\\end{definition}\n\nThe morphism associated with a Kleisli morphism should be understood as the linear transformation between vector spaces induced by some matrix of the right type. The following result confirms this intuition.\n\t\n\\begin{lemma}\n\t\\eqref{associatedmorph} is a $\\mathbb{T}$-algebra homomorphism $p^{\\alpha \\beta}: (X_{\\alpha},h_{\\alpha}) \\rightarrow (X_{\\beta},h_{\\beta})$.\n\\end{lemma}\t\n\\begin{proof}\nUsing \\Cref{forbasis-d-isalgebrahom} we deduce the commutativity of the following diagram\n\\begin{equation*}\n\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\tTX_{\\alpha} \\arrow{d}[left]{h_{\\alpha}} \\arrow{r}{Td_{\\alpha}} & T^2Y_{\\alpha} \\arrow{r}{T^2p} \\arrow{d}{\\mu_{Y_{\\alpha}}} & T^3Y_{\\beta} \\arrow{r}{T\\mu_{Y_{\\beta}}} \\arrow{d}{\\mu_{TY_{\\beta}}} & T^2Y_{\\beta} \\arrow{d}{\\mu_{Y_{\\beta}}} \\arrow{r}{T^2i_{\\beta}} & T^2X_{\\beta} \\arrow{r}{Th_{\\beta}} \\arrow{d}{\\mu_{X_{\\beta}}} & TX_{\\beta} \\arrow{d}{h_{\\beta}} \\\\\n\tX_{\\alpha} \\arrow{r}{d_{\\alpha}} & TY_{\\alpha} \\arrow{r}{Tp} & T^2Y_{\\beta} \\arrow{r}{\\mu_{Y_{\\beta}}} & TY_{\\beta} \\arrow{r}{Ti_{\\beta}} & TX_{\\beta} \\arrow{r}{h_{\\beta}} & X_{\\beta}\t\n\t\\end{tikzcd}.\n\\end{equation*}\n\\end{proof}\n\nThe following result establishes a generalisation of the observation that for fixed bases, constructing a matrix representation of a linear transformation on the one hand, and associating a linear transformation to a matrix of the right type on the other hand, are mutually inverse operations.\n\\begin{lemma} \n\tThe operations $\\eqref{basisrepresentation}$ and $\\eqref{associatedmorph}$ are mutually inverse.\n\\end{lemma}\n\\begin{proof}\n\tThe definitions imply \n\t\\begin{align*}\n\t\t(p^{\\alpha \\beta})_{\\alpha \\beta} &= d_{\\beta} \\circ (h_{\\beta} \\circ Ti_{\\beta} \\circ \\mu_{Y_{\\beta}} \\circ Tp \\circ d_{\\alpha}) \\circ i_{\\alpha} \\\\\n\t\t(f_{\\alpha \\beta})^{\\alpha \\beta} &= h_{\\beta} \\circ Ti_{\\beta} \\circ \\mu_{Y_{\\beta}} \\circ T(d_{\\beta} \\circ f \\circ i_{\\alpha}) \\circ d_{\\alpha}.\n\t\\end{align*}\nUsing \\Cref{forbasis-d-isalgebrahom} we deduce the commutativity of the diagrams below\n\t\\[\n\t\\begin{gathered}\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tY_{\\alpha} \\arrow{d}[left]{i_{\\alpha}} \\arrow{rrr}{p} \\arrow{dr}{\\eta_{Y_{\\alpha}}} & & &  TY_{\\beta} \\arrow{d}{\\textnormal{id}_{TY_{\\beta}}} \\arrow{rr}{\\textnormal{id}_{TY_{\\beta}}} \\arrow{dl}{\\eta_{TY_{\\beta}}} & & TY_{\\beta} \\\\\n\t\t\tX_{\\alpha} \\arrow{r}{d_{\\alpha}} & TY_{\\alpha} \\arrow{r}{Tp} & T^2Y_{\\beta} \\arrow{r}{\\mu_{Y_{\\beta}}}  & TY_{\\beta} \\arrow{r}{Ti_{\\beta}} & TX_{\\beta} \\arrow{r}{h_{\\beta}} & X_{\\beta} \\arrow{u}[right]{d_{\\beta}}\n\t\t\t\\end{tikzcd} \\\\\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\t\tX_{\\alpha} \\arrow{r}{\\textnormal{id}_{X_{\\alpha}}} \\arrow{d}[left]{Ti_{\\alpha} \\circ d_{\\alpha}} & X_{\\alpha} \\arrow{rr}{f} & & X_{\\beta} \\arrow{d}{d_{\\beta}} \\arrow{r}{\\textnormal{id}_{X_{\\beta}}}  & X_{\\beta} \\\\\n\t\t\t\tTX_{\\alpha} \\arrow{ur}{h_{\\alpha}} \\arrow{r}{Tf} & TX_{\\beta} \\arrow{r}{Td_{\\beta}} \\arrow{urr}{h_{\\beta}} & T^2Y_{\\beta} \\arrow{r}{\\mu_{Y_{\\beta}}} & TY_{\\beta} \\arrow{r}{Ti_{\\beta}} & TX_{\\beta} \\arrow{u}[right]{h_{\\beta}}\n\t\t\t\t\\end{tikzcd}\t.\t\t\n\t\\end{gathered}\n\t\\]\t\t\t\t\t\n\\end{proof}\n\n\nThe next result establishes the compositionality of basis representations: the matrix representation of the composition of two linear transformations is given by the multiplication of the matrix representations of the individual linear transformations. On the left side of the following equation we use the usual Kleisli composition.\n\\begin{lemma}\n$g_{\\beta \\gamma} \\cdot f_{\\alpha \\beta} = (g \\circ f)_{\\alpha \\gamma}$.\n\\end{lemma}\n\\begin{proof}\n\tThe definitions imply\n\t\\begin{align*}\n\t\tg_{\\beta \\gamma} \\cdot f_{\\alpha \\beta} &= \\mu_{Y_{\\gamma}} \\circ T(d_{\\gamma} \\circ g \\circ i_{\\beta}) \\circ d_{\\beta} \\circ f \\circ i_{\\alpha} \\\\\n\t\t(g \\circ f)_{\\alpha \\gamma} &= d_{\\gamma} \\circ (g \\circ f) \\circ i_{\\alpha}.\n\t\\end{align*}\n\tWe delete common terms and use  \\Cref{forbasis-d-isalgebrahom} to deduce the commutativity of the diagram below\n\\begin{equation*}\n\t\t\t\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\t\t\t\t X_{\\beta} \\arrow{d}[left]{\\textnormal{id}_{X_{\\beta}}} \\arrow{r}{d_{\\beta}} & TY_{\\beta} \\arrow{r}{Ti_{\\beta}} & TX_{\\beta} \\arrow{r}{Tg} \\arrow{dll}{h_{\\beta}} & TX_{\\gamma} \\arrow{r}{Td_{\\gamma}} \\arrow{dll}{h_{\\gamma}} & T^2 Y_{\\gamma} \\arrow{d}{\\mu_{Y_{\\gamma}}} \\\\\n\t\t\t\t\t\t X_{\\beta} \\arrow{r}[below]{g} & X_{\\gamma} \\arrow{rrr}[below]{d_{\\gamma}} & & & TY_{\\gamma}\n\t\t\t\t\t\\end{tikzcd}.\n\\end{equation*}\n\\end{proof}\n\nSimilarly to the previous result, the next observation captures the compositionality of the operation that assigns to a Kleisli morphism its associated homomorphism.\n\\begin{lemma}\n\t$q^{\\beta \\gamma} \\circ p^{\\alpha \\beta} = (q \\cdot p)^{\\alpha \\gamma}$. \n\\end{lemma}\n\\begin{proof}\n\tThe definitions imply \n\t\\begin{align*}\n\t\tq^{\\beta \\gamma} \\circ p^{\\alpha \\beta} &= (h_{\\gamma} \\circ Ti_{\\gamma} \\circ \\mu_{Y_{\\gamma}} \\circ Tq \\circ d_{\\beta}) \\circ (h_{\\beta} \\circ Ti_{\\beta} \\circ \\mu_{Y_{\\beta}} \\circ Tp \\circ d_{\\alpha}) \\\\\n\t\t(q \\cdot p)^{\\alpha \\gamma} &= h_{\\gamma} \\circ Ti_{\\gamma} \\circ \\mu_{Y_{\\gamma}} \\circ T\\mu_{Y_{\\gamma}} \\circ T^2q \\circ Tp \\circ d_{\\alpha}.\n\t\\end{align*}\n\tBy deleting common terms and using the equality $d_{\\beta} \\circ h_{\\beta} \\circ Ti_{\\beta} = \\textnormal{id}_{TY_{\\beta}}$ it is thus sufficient to show\n\t\\[\n\t\\mu_{Y_{\\gamma}} \\circ Tq \\circ \\mu_{Y_{\\beta}} = \\mu_{Y_{\\gamma}} \\circ T\\mu_{Y_{\\gamma}} \\circ T^2q.\n\t\\]\n\tAbove equation follows from the commutativity of the diagram below\n\t\\begin{equation*}\n\t\t\t\t\t\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\t\t\t\t\t\tT^2Y_{\\beta} \\arrow{d}[left]{T^2q} \\arrow{r}{\\mu_{Y_{\\beta}}} & TY_{\\beta} \\arrow{r}{Tq} & T^2Y_{\\gamma} \\arrow{d}{\\mu_{Y_{\\gamma}}} \\\\\n\t\t\t\t\t\t\t\tT^3Y_{\\gamma} \\arrow{urr}{\\mu_{TY_{\\gamma}}} \\arrow{r}[below]{T\\mu_{Y_{\\gamma}}} & T^2 Y_{\\gamma} \\arrow{r}[below]{\\mu_{Y_{\\gamma}}} & TY_{\\gamma}\n\t\t\t\t\t\t\t\\end{tikzcd}.\n\t\\end{equation*}\n\\end{proof}\n\nAt the beginning of this section we recalled the soundness identity \\[ L(v)_{\\beta} = L_{\\alpha \\beta} v_{\\alpha} \\] for the matrix representation $L_{\\alpha \\beta}$ of a linear transformation $L$. The next result is a natural generalisation of this statement. \n\n\\begin{lemma}\n\t$d_{\\beta} \\circ f =  f_{\\alpha \\beta} \\cdot d_{\\alpha}$ \n\\end{lemma}\n\\begin{proof}\n\tThe definitions imply\n\t\\[\n\t f_{\\alpha \\beta} \\cdot d_{\\alpha} = \\mu_{Y_{\\beta}} \\circ T(d_{\\beta} \\circ f \\circ i_{\\alpha}) \\circ d_{\\alpha}.\n\t\\]\n\tUsing \\Cref{forbasis-d-isalgebrahom} we deduce the commutativity of the diagram below\n\t\\begin{equation*}\n\t\t\t\t\t\t\t\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\t\t\tX_{\\alpha} \\arrow{d}[left]{\\textnormal{id}_{X_{\\alpha}}} \\arrow{r}{d_{\\alpha}} & TY_{\\alpha} \\arrow{r}{Ti_{\\alpha}} & TX_{\\alpha} \\arrow{dll}{h_{\\alpha}} \\arrow{r}{Tf} & TX_{\\beta} \\arrow{dll}{h_{\\beta}} \\arrow{r}{Td_{\\beta}} & T^2Y_{\\beta} \\arrow{d}{\\mu_{Y_{\\beta}}} \\\\\n\t\t\t\t\tX_{\\alpha} \\arrow{r}[below]{f} & X_{\\beta} \\arrow{rrr}[below]{d_{\\beta}} & & & TY_{\\beta}\t\t\t\t\n\t\t\t\t\t\t\t\t\t\\end{tikzcd}.\n\t\\end{equation*}\n\\end{proof}\n\nAssume we are given bases $\\alpha, \\alpha'$ and $\\beta, \\beta'$ for $\\mathbb{T}$-algebras $(X_{\\alpha}, h_{\\alpha})$ and $(X_{\\beta}, h_{\\beta})$, respectively. \nThe following result makes clear how the two basis representations $f_{\\alpha \\alpha'}$ and $f_{\\beta \\beta'}$ are related.\n\n\\begin{proposition}\n\\label{similaritygeneral}\n\tThere exist Kleisli isomorphisms $p$ and $q$ such that $f_{\\alpha' \\beta'} = q \\cdot f_{\\alpha \\beta} \\cdot p$.\n\\end{proposition}\n\\begin{proof}\n\tThe Kleisli morphisms $p$ and $q$ and their respective candidates for inverses $p^{-1}$ and $q^{-1}$ are defined below\n\t\\begin{alignat*}{6}\n\t\t& p &&:= d_{\\alpha} \\circ i_{\\alpha'}: Y_{\\alpha'} &&\\longrightarrow TY_{\\alpha} \\qquad \\qquad && q &&:= d_{\\beta'} \\circ i_{\\beta}: Y_{\\beta} && \\longrightarrow TY_{\\beta'} \\\\\n\t\t& p^{-1} &&:= d_{\\alpha'} \\circ i_{\\alpha}: Y_{\\alpha}  && \\longrightarrow TY_{\\alpha'} \\qquad \\qquad  &&  q^{-1} &&:= d_{\\beta} \\circ i_{\\beta'} : Y_{\\beta'} && \\longrightarrow  TY_{\\beta}.\n\t\\end{alignat*}\nFrom \\Cref{forbasis-d-isalgebrahom} it follows that the diagram below commutes \\begin{equation*}\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tY_{\\alpha} \\arrow{r}{i_{\\alpha}} \\arrow{dd}[left]{\\eta_{Y_{\\alpha}}} & X_{\\alpha} \\arrow{d}{\\textnormal{id}_{X_{\\alpha}}} \\arrow{r}{d_{\\alpha'}} & TY_{\\alpha'} \\arrow{dd}{Ti_{\\alpha'}} \\\\\n\t\t\t& X_{\\alpha} \\arrow{dl}{d_{\\alpha}} & \\\\\n\t\t\tTY_{\\alpha}  & T^2Y_{\\alpha}  \\arrow{l}{\\mu_{Y_{\\alpha}}} & TX_{\\alpha} \\arrow{ul}{h_{\\alpha}} \\arrow{l}{Td_{\\alpha}}\n\t\t\\end{tikzcd}.\n\t\t\\end{equation*}\n\t\tThis shows that $p^{-1}$ is a Kleisli right-inverse of $p$. A symmetric version of above diagram shows that $p^{-1}$ is also a Kleisli left-inverse of $p$. Analogously it follows that $q^{-1}$ is a Kleisli inverse of $q$.\n\t\t\n\t\tThe definitions further imply the equalities\n\t\t\\begin{align*}\n\t\t\tq \\cdot f_{\\alpha \\beta} \\cdot p &= \\mu_{Y_{\\beta'}} \\circ T(d_{\\beta'} \\circ i_{\\beta}) \\circ \\mu_{Y_{\\beta}} \\circ T(d_{\\beta} \\circ f \\circ i_{\\alpha}) \\circ d_{\\alpha} \\circ i_{\\alpha'} \\\\\n\t\t\tf_{\\alpha' \\beta'} &= d_{\\beta'} \\circ f \\circ i_{\\alpha'}.\n\t\t\\end{align*}\n\t\tWe delete common terms and use \\Cref{forbasis-d-isalgebrahom} to establish the commutativity of the diagram below\n\t\t\\begin{equation*}\n\t\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tX_{\\alpha} \\arrow{dr}{\\textnormal{id}_{X_{\\alpha}}} \\arrow{r}{d_{\\alpha}} \\arrow{dd}[left]{f} & TY_{\\alpha} \\arrow{r}{Ti_{\\alpha}} & TX_{\\alpha} \\arrow{dl}[above]{h_{\\alpha}} \\arrow{r}{Tf} & TX_{\\beta} \\arrow{ddll}{h_{\\beta}} \\arrow{d}{Td_{\\beta}} \\\\\n\t\t\t & X_{\\alpha} \\arrow{d}{f} &  & T^2Y_{\\beta} \\arrow{dd}{\\mu_{Y_{\\beta}}} \\\\\n\tX_{\\beta} \\arrow{d}[left]{d_{\\beta'}} & X_{\\beta} \\arrow{l}[above]{\\textnormal{id}_{X_{\\beta}}} \\arrow{drr}{d_{\\beta}} & & \\\\\n\t\t\tTY_{\\beta'} & T^2 Y_{\\beta'} \\arrow{l}{\\mu_{Y_{\\beta'}}} & TX_{\\beta} \\arrow{llu}{h_{\\beta}} \\arrow{l}{Td_{\\beta'}} & TY_{\\beta} \\arrow{l}{Ti_{\\beta}}\n\t\t\\end{tikzcd}.\n\t\t\\end{equation*}\t\n\\end{proof}\n\nAbove result simplifies in the case one restricts to an endomorphism: the respective basis representations are similar.\n\n\\begin{corollary}\n\t\\label{similarity}\nThere exists a Kleisli isomorphism $p$ with Kleisli inverse $p^{-1}$ such that $f_{\\alpha' \\alpha'} = p^{-1} \\cdot f_{\\alpha \\alpha} \\cdot p$.\n\\end{corollary}\n\\begin{proof}\n\tIn \\Cref{similaritygeneral} let $\\beta = \\alpha$ and $\\beta' = \\alpha'$. One verifies that in the corresponding proof the definitions of the morphisms $p^{-1}$ and $q$ coincide.\n\\end{proof}\n\n\\subsection{Bases as free algebras}\n\n\\label{basesfreealgebrasec}\n\nIn \\Cref{basisimpliesfree} it was shown that an algebra over a monad with a basis is isomorphic to a free algebra. Conversely, in  \\Cref{canonicalbasisforfreealgbera} it was proven that a free algebra over a monad admits a basis. Intuitively, one may thus think that bases for an algebra coincide with free isomorphic algebras.\nIn this section we make this idea precise on the level of categories.\n\nFormally, given an algebra $(X,h)$ over a monad $\\mathbb{T}$, let $\\textsf{Free}(X,h)$ denote the category defined as follows: \n\\begin{itemize}\n\t\\item objects are given by pairs $(Y, \\varphi)$ consisting of an isomorphism $\\varphi: (TY, \\mu_Y) \\rightarrow (X,h)$; and\n\t\\item a morphism $f: (Y_{\\alpha}, \\varphi_{\\alpha}) \\rightarrow (Y_{\\beta}, \\varphi_{\\beta})$ between objects consists of a morphism $f: Y_{\\alpha} \\rightarrow Y_{\\beta}$ such that $\\varphi_{\\alpha} = \\varphi_{\\beta} \\circ Tf$.\n\\end{itemize}\n\nThe next result shows that for a fixed algebra, the natural isomorphism \nunderlying the free-algebra adjunction restricts to an equivalence between the category of bases defined in \\Cref{basisdefinition}, and the category of free isomorphic algebras given above.\n\\begin{proposition}\n$\\textnormal{\\textsf{Bases}}(X,h) \\simeq \\textnormal{\\textsf{Free}}(X,h)$\n\\end{proposition}\n\\begin{proof}\n\tWe define functors $F$ and $G$ between the respective categories as follows\n\t\\begin{alignat*}{7}\n\t\t&F&&: \\textnormal{\\textsf{Bases}}(X,h) &&\\longrightarrow \\textnormal{\\textsf{Free}}(X,h) \\qquad && F(Y,i,d)&& = (Y, i^{\\sharp}) \\qquad  && Ff &&= f \\\\\n\t\t&G&&: \\textnormal{\\textsf{Free}}(X,h) && \\longrightarrow \\textnormal{\\textsf{Bases}}(X,h) \\qquad && G(Y, \\varphi) && = (Y, \\varphi \\circ \\eta_Y, \\varphi^{-1}) \\qquad &&Gf &&= f.\n\t\\end{alignat*} \n\t\t\n\tThe functor $F$ is well-defined on objects since by \\Cref{basisimpliesfree} the morphism $i^{\\sharp}$ is an isomorphism with inverse $d$. Its well-definedness on morphisms is an immediate consequence of the constraint $i_{\\alpha} = i_{\\beta} \\circ f$ for morphisms $f: (Y_{\\alpha},i_{\\alpha},d_{\\alpha}) \\rightarrow (Y_{\\beta}, i_{\\beta}, d_{\\beta})$ between bases,\n\t\\[\n\t(i_{\\alpha})^{\\sharp} = h \\circ Ti_{\\alpha} = h \\circ Ti_{\\beta} \\circ Tf = (i_{\\beta})^{\\sharp} \\circ Tf.\n\t\\]\t\n\t\n\tThe functor $G$ is well-defined on objects since $(\\varphi \\circ \\eta_Y)^{\\sharp} = \\varphi$ and $\\varphi^{-1}$ are mutually inverse. Its well-definedness on morphisms $f: (Y_{\\alpha}, \\varphi_{\\alpha}) \\rightarrow (Y_{\\beta}, \\varphi_{\\beta})$ follows from the equality $\\varphi_{\\beta} \\circ Tf = \\varphi_{\\alpha}$,\n\t\\begin{align*}\n\t\tTf \\circ (\\varphi_{\\alpha})^{-1} &= (\\varphi_{\\beta})^{-1} \\circ \\varphi_{\\beta} \\circ Tf \\circ (\\varphi_{\\alpha})^{-1} = (\\varphi_{\\beta})^{-1} \\circ \\varphi_{\\alpha} \\circ (\\varphi_{\\alpha})^{-1} = (\\varphi_{\\beta})^{-1}\n\t\\end{align*}\n\tand the naturality of $\\eta$,\n\t\\[\n\t\\varphi_{\\alpha} \\circ \\eta_{Y_{\\alpha}} = \\varphi_{\\beta} \\circ Tf \\circ \\eta_{Y_{\\alpha}} = \\varphi_{\\beta} \\circ \\eta_{Y_{\\beta}} \\circ f.\n\t\\]\n\t\n\tThe functors are clearly mutually inverse on morphisms. For objects the statement follows from\n\t\\begin{align*}\n\t\tF \\circ G(Y, \\varphi) &= (Y, (\\varphi \\circ \\eta_Y)^{\\sharp}) = (Y, \\varphi) \\\\\n\t\tG \\circ F(Y, i,d) &= (Y, i^{\\sharp} \\circ \\eta_Y, (i^{\\sharp})^{-1}) = (Y,i,d).\n\t\\end{align*}\n\\end{proof}\n\n\n\\subsection{Bases for bialgebras}\n\n\\label{basesforbialgebrasec}\n\nIt is well-known that a distributive law $\\lambda$ between a monad $\\mathbb{T}$ and an endofunctor $F$ induces a monad \n$\\mathbb{T}_{\\lambda}$ on the category of $F$-coalgebras such that the algebras over $\\mathbb{T}_{\\lambda}$ coincide with the $\\lambda$-bialgebras of \\Cref{defbialgebras}.\n This section is concerned with generators and bases for $\\mathbb{T}_{\\lambda}$-algebras, or equivalently, $\\lambda$-bialgebras.\n\nBy definition, a generator for a $\\lambda$-bialgebra $(X,h,k)$ consists of a $F$-coalgebra $(Y,k_Y)$ and morphisms $i: Y \\rightarrow X$ and $d: X \\rightarrow TY$, such that the three diagrams on the left below commute\n\\begin{equation}\n\\label{bialgebrageneratorequations}\t\n\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\tY \\arrow{r}{i} \\arrow{d}[left]{k_Y} & X \\arrow{d}{k} \\\\\n\t\tFY \\arrow{r}{Fi} &FX\n\t\\end{tikzcd}\n\t\\quad\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\tX \\arrow{r}{d} \\arrow{d}[left]{k} & TY \\arrow{d}[left]{\\lambda_Y \\circ Tk_Y} \\\\\n\t\tFX \\arrow{r}{Fd} &FTY\n\t\\end{tikzcd}\n\t\\quad\n\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tTY \\arrow{r}{Ti} & TX \\arrow{d}{h}\\\\\n\t\t\tX \\arrow{u}{d} \\arrow{r}{\\textnormal{id}_X}  & X \n\t\t\\end{tikzcd}\n\t\t\\quad\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tTX \\arrow{r}{h} & X \\arrow{d}{d}\\\\\n\t\t\tTY \\arrow{u}{Ti} \\arrow{r}{\\textnormal{id}_{TY}}  & TY \n\t\t\\end{tikzcd}.\n\\end{equation}\n\nA basis for a $\\lambda$-bialgebra is moreover given by a generator, such that in addition the diagram on the right above commutes.\n\nIt is easy to verify that by forgetting the $F$-coalgebra structure, every generator for a bialgebra in particular provides a generator for the underlying algebra of the bialgebra. By \\Cref{forgenerator-isharp-is-bialgebra-hom} it thus follows that there exists a $\\lambda$-bialgebra homomorphism \n\\[ i^{\\sharp} = h \\circ Ti : (TY, \\mu_Y, (Fd \\circ k \\circ i)^{\\sharp}) \\longrightarrow (X,h,k). \\]\nThe next result establishes that there exists a second equivalent free bialgebra with a different coalgebra structure.\n\n\\begin{lemma}\n\\label{generatorbialgebraisharp}\n\tLet $(Y,k_Y, i,d)$ be a generator for $(X,h,k)$. Then $i^{\\sharp}: TY \\rightarrow X$ is a $\\lambda$-bialgebra homomorphism $i^{\\sharp} : (TY, \\mu_Y, \\lambda_Y \\circ Tk_Y) \\rightarrow (X,h,k)$.\n\\end{lemma}\n\\begin{proof}\n\tClearly $i^{\\sharp}$ is a $\\mathbb{T}$-algebra homomorphism. It is a $F$-coalgebra homomorphism since the diagram below commutes\n\t\\begin{equation*}\n\t\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\tTY \\arrow{r}{Ti} \\arrow{d}[left]{Tk_Y} & TX \\arrow{r}{h} \\arrow{d}{Tk} & X \\arrow{dd}{k} \\\\\n\t\t\tTFY \\arrow{r}{TFi} \\arrow{d}[left]{\\lambda_Y} & TFX \\arrow{d}{\\lambda_X} & \\\\\n\t\t\tFTY \\arrow{r}{FTi} & FTX \\arrow{r}{Fh} & FX\n\t\t\\end{tikzcd}.\n\t\\end{equation*}\t\n\\end{proof}\n\nIf one moves from generator for bialgebras to bases for bialgebras, both coalgebra structures coincide.\n\n\\begin{lemma}\n\\label{samecoalgebrastructures}\n\tLet $(Y,k_Y, i, d)$ be a basis for $(X,h,k)$, then $\\lambda_Y \\circ Tk_Y = (Fd \\circ k \\circ i)^{\\sharp}$.\n\\end{lemma}\n\\begin{proof}\n\tUsing \\Cref{forbasis-d-isalgebrahom} we establish the commutativity of diagram below\n\t\\begin{equation*}\n\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\tTY \\arrow{dd}[left]{\\textnormal{id}_{TY}}  \\arrow{rr}{Tk_Y}  & & TFY \\arrow{rr}{\\lambda_Y} \\arrow{dd}{TFi} & & FTY  \\arrow{d}{FTi} \\arrow{rr}{\\textnormal{id}_{FTY}} && FTY \\arrow{dd}{\\textnormal{id}_{FTY}} \\\\\n\t& & & & FTX \\arrow{d}{FTd} \\arrow{r}{Fh} & FX \\arrow{dr}{Fd} & \\\\\n\tTY \\arrow{r}{Ti} & TX \\arrow{r}{Tk} & TFX \\arrow{r}{TFd} & TFTY \\arrow{r}{\\lambda_{TY}} & FT^2Y  \\arrow{rr}{F\\mu_{Y}} & & FTY\t\n\t\\end{tikzcd}.\n\t\\end{equation*}\n\\end{proof}\n\nWe close this section by observing that a basis for the underlying algebra of a bialgebra is sufficient for constructing a generator for the full bialgebra.\n\n\\begin{lemma}\n\tLet $(X,h,k)$ be a $\\lambda$-bialgebra and $(Y,i,d)$ a basis for the $\\mathbb{T}$-algebra $(X,h)$. Then $(TY,(Fd \\circ k \\circ i)^{\\sharp},i^{\\sharp},\\eta_{TY} \\circ d)$ is a generator for $(X,h,k)$.\n\\end{lemma}\n\\begin{proof}\n\tIn the following we abbreviate $k_{TY} := (Fd \\circ k \\circ i)^{\\sharp}$. By \\Cref{forgenerator-isharp-is-bialgebra-hom} the morphism $i^{\\sharp}$ is a $F$-coalgebra homomorphism $ i^{\\sharp}: (TY, k_{TY}) \\rightarrow (X, k)$. This shows the commutativity of the diagram on the left of \\eqref{bialgebrageneratorequations}. \n\tBy \\Cref{forbasis-d-is-bi-algebrahom} the morphism $d$ is a $F$-coalgebra homomorphism in the reverse direction. Together with the commutativity of the diagram on the left below\n\t\\[\n\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\tTY \\arrow{r}{\\eta_{TY}} \\arrow{dd}[left]{k_{TY}} & T^2Y \\arrow{d}{Tk_{TY}} \\\\\n\t& TFTY \\arrow{d}{\\lambda_{TY}} \\\\\n\tFTY \\arrow{ur}{\\eta_{FTY}} \\arrow{r}{F\\eta_{TY}} & FT^2Y\t\n\t\\end{tikzcd}\n\t\\qquad\n\t\\begin{tikzcd}[row sep=3em, column sep = 3em]\n\t\t\t\tT^2Y \\arrow{r}{T^2i} & T^2X \\arrow{rr}{Th}  \\arrow{dr}{\\mu_X} & &  TX \\arrow{dd}{h} \\\\\n\t\t\t\tTY \\arrow{u}{\\eta_{TY}} \\arrow{r}{Ti} & TX \\arrow{u}{\\eta_{TX}} \\arrow{r}{\\textnormal{id}_{TX}} & TX \\arrow{dr}{h}  \\\\\n\t\t\t\tX \\arrow{u}{d} \\arrow{rrr}{\\textnormal{id}_X} & & & X.\n\t\t\t\\end{tikzcd}\n\t\\] this implies the commutativity of the second diagram to the left of \\eqref{bialgebrageneratorequations}.\t\t\n\tSimilarly, the commutativity of third diagram to the left of \\eqref{bialgebrageneratorequations} follows from the commutativity of the diagram on the right above.\n\\end{proof}\n\n\\section{Examples}\n\n\\label{examplesec}\n\nIn this section we give examples of generators and bases for algebras over the powerset-, downset-, distribution-, and neighbourhood monad.\n\n\\subsection{Powerset monad}\n\n\\label{powersetmonadsec}\n\nThe powerset monad $\\mathbb{P} = (\\mathcal{P}, \\mu, \\eta)$ on the category of sets may easily be the most well-known monad. Its underlying set endofunctor $\\mathcal{P}$ assigns to a set its powerset and to a function the function that maps a subset to its direct image under the former. Multiplication and unit transformations are further given by \n\\[ \\eta_X(x) = \\lbrace x \\rbrace \\qquad \\mu_X(U) = \\bigcup_{A \\in U} A. \\]\n\nThe category of algebras for the powerset monad is famously isomorphic to the category of complete lattices and join-preserving functions. Indeed, since the existence of all joins is equivalent to the existence of all meets, one can define, given a $\\mathbb{P}$-algebra $(X,h)$, a complete lattice $(X, \\leq)$ with $x \\leq y :\\Leftrightarrow h(\\lbrace x, y \\rbrace) = y$ and supremum $\\bigvee A := h(A)$. Conversely, given a complete lattice $(X, \\leq)$ with supremum $\\bigvee$ one defines a $\\mathbb{P}$-algebra $(X,h)$ by $h(A) := \\bigvee A$. \n\nLet $F$ be the set endofunctor satisfying $FX = X^A \\times 2$. Coalgebras for $F$ are easily recognised as deterministic unpointed automata over the alphabet $A$, and coalgebras for the composition $F\\mathcal{P}$ coincide with unpointed non-deterministic automata over the same alphabet. The output set $2$ can be equipped with a disjunctive $\\mathbb{P}$-algebra structure, such that bialgebras for the induced canonical distributive law between $\\mathbb{P}$ and $F$  consist of deterministic automata with a complete lattice as state space and supremum-preserving transition and output functions \\cite{jacobs2012trace}. For instance, every deterministic automaton derived from a non-deterministic automaton via the classical subset construction is of such a form. \n\n The following observation relates generators for algebras over $\\mathbb{P}$ with the induced complete lattice structure of the former.\n\n\\begin{lemma}\n\\label{generatorpowerset}\n\t$(Y, i, d)$ is a generator for a $\\mathbb{P}$-algebra $(X,h)$ iff \n\\[\nx = \\bigvee_{y \\in d(x)} i(y)\n\\]\nfor all $x \\in X$.\n\\end{lemma}\n\\begin{proof}\nFollows immediately from the equality\n\\begin{align*}\n\th \\circ \\mathcal{P}i \\circ d(x) &= h(\\lbrace i(y) \\mid y \\in d(x) \\rbrace) = \\bigvee_{y \\in d(x)} i(y).\n\\end{align*}\t\n\\end{proof}\n\nRecall that a non-zero element $x \\in X$ of a lattice $L = (X, \\leq)$ is called join-irreducible, if $x = y \\vee z$ implies $x = y$ or $x = z$. \n It is well-known that in a finite lattice, or more generally in a lattice satisfying the descending chain condition, any element is the join of the join-irreducible elements below it. In other words, by \\Cref{generatorpowerset}, if $i: \\mathcal{J}(L) \\rightarrow L$ is the subset embedding of join-irreducibles and $d: L \\rightarrow \\mathcal{P}(\\mathcal{J}(L))$ satisfies \\[ d(x) = \\lbrace a \\in \\mathcal{J}(L) \\mid a \\leq x \\rbrace, \\] then $(\\mathcal{J}(L), i, d)$ is a generator for the $\\mathbb{P}$-algebra $L$.\n\n\\subsection{Downset monad}\n\nIn the previous section we have seen that the category of algebras for the powerset monad is equivalent to the category of complete lattices and join-preserving functions. It is probably slightly less-known that there exists a second monad with the same category of algebras: the downset monad $\\mathbb{P}_{\\downarrow} = (\\mathcal{P}_{\\downarrow}, \\mu, \\eta)$ on the category of posets. \n\nFor a subset $Y$ of a poset let $Y_{\\downarrow}$ be its so-called downward closure, that is, the set of those poset elements for which there exists at least one element in $Y$ above. A subset is called downset or downclosed, if it coincides with its downward closure. The endofunctor $\\mathcal{P}_{\\downarrow}$ underlying the downset monad  assigns to a poset the inclusion-ordered poset of its downclosed subsets, and to a monotone function the monotone function  mapping a downclosed subset to the downclosure of its direct image. \nThe natural transformations $\\eta$ and $\\mu$ are further given by\n\\[\n\\eta_P(x) = \\lbrace x \\rbrace_{\\downarrow} \\qquad \\mu_P(U) = \\bigcup_{A \\in U} A.\n\\]\n\nGiven an algebra $(P,h)$ over $\\mathbb{P}_{\\downarrow}$, one verifies that the poset $P$ is a complete lattice with supremum $\\bigvee A := h(A_{\\downarrow})$. Conversely, given a complete lattice $P$ with supremum $\\bigvee$, one defines an algebra $(P,h)$ over $\\mathbb{P}_{\\downarrow}$ by $h(A) := \\bigvee A$. The following observation relates generators for algebras over $\\mathbb{P}_{\\downarrow}$ with the induced complete lattice structure of the latter.\n\n\\begin{lemma}\n\t$(Y,i,d)$ is a generator for a $\\mathbb{P}_{\\downarrow}$-algebra $(P,h)$ iff\n\t\\[\n\tx = \\bigvee_{y \\in d(x)} i(y)\n\t\\]\n\tfor all $x \\in X$.\n\\end{lemma}\n\\begin{proof}\nFollows immediately from the equality\n\t\\[\n\th \\circ \\mathcal{P}_{\\downarrow}i \\circ d(x) = h(\\lbrace i(y) \\mid y \\in d(x) \\rbrace_{\\downarrow}) = \\bigvee_{y \\in d(x)} i(y).\n\t\\]\n\\end{proof}\n\nIt is due to Birkhoff's Representation Theorem that if $L$ is a finite (hence complete) distributive lattice, the monotone function $d: L \\rightarrow \\mathcal{P}_{\\downarrow}(\\mathcal{J}(L))$ assigning to an element the downward closed set of join-irreducibles below it, is an isomorphism with an inverse satisfying $A \\mapsto \\bigvee A$. In other words, in light of the previous result, if $i: \\mathcal{J}(L) \\rightarrow L$ is the subset embedding of join-irreducibles, then $(\\mathcal{J}(L), i, d)$ constitutes a basis for the $\\mathbb{P}_{\\downarrow}$-algebra $L$.\n\n\n\n\\subsection{Distribution monad}\n\nThe distribution monad $\\mathbb{D} = (\\mathcal{D}, \\mu, \\eta)$ on the category of sets is given as follows. The underlying set endofunctor $\\mathcal{D}$ assigns to a set $X$ its set of  distributions with finite support,\n\\[\n\\mathcal{D}X = \\lbrace p: X \\rightarrow \\lbrack 0, 1 \\rbrack \\mid supp(p) \\textnormal{ finite, and } \\sum_{x \\in X} p(x) = 1 \\rbrace\n\\]\n\tand to a function $f: X \\rightarrow Y$ the direct image $\\mathcal{D}f: \\mathcal{D}X \\rightarrow \\mathcal{D}Y$ satisfying the equality \n\t\\[ \\mathcal{D}(f)(p)(y) = \\sum_{x \\in f^{-1}(y)} p(x). \\]\nThe natural transformations $\\eta$ and $\\mu$ are further given by \n\\[ \\eta_X(x)(y) = \\lbrack x = y \\rbrack \\qquad \\mu_X(\\Phi)(x) = \\sum_{p \\in \\mathcal{D}X} p(x) \\Phi(p). \\]\n\nIt is well-known that the category of algebras for the distribution monad is isomorphic to the category of convex sets and affine functions \\cite{jacobs2010convexity}. Indeed, any $\\mathbb{D}$-algebra $(X,h)$ can be turned into a unique convex set with finite sums defined by $\\sum_{i} r_i x_i := h(p)$ for $p(x) = r_i$ if $x=x_i$, and $p(x) = 0$ otherwise. \n\nLet $F$ be the set endofunctor satisfying $FX = X^A \\times \\lbrack 0, 1 \\rbrack$. Coalgebras for the composed endofunctor $F\\mathcal{D}$ are known as unpointed Rabin probabilistic automata over the alphabet $A$ \\cite{rabin1963probabilistic, rutten2013generalizing}. The unit interval can be equipped with a $\\mathbb{D}$-algebra structure $h$ which satisfies $h(p) = \\sum_{x \\in \\lbrack 0, 1 \\rbrack} x p(x)$ and induces a canonical distributive law between $\\mathbb{D}$ and $F$ \\cite{jacobs2012trace}. The respective bialgebras consist of unpointed Moore automata with input $A$ and output $\\lbrack 0, 1 \\rbrack$, a convex set as state space, and affine transition and output functions. For instance, every Moore automaton derived from a probabilistic automaton by assigning to the state space of the latter all its distributions with finite support constitutes such a bialgebra.\n\n\n The next observation relates generators of algebras over $\\mathbb{D}$ with the induced convex set structure on the latter. For simplicity we assume that in the following statement the function $i: Y \\rightarrow X$ is an injection.\n\n\\begin{lemma}\n\t$(Y,i,d)$ is a generator for a $\\mathbb{D}$-algebra $(X,h)$ iff\n\t\\[\n\tx = \\sum_{y \\in Y} d(x)(y) i(y)\t\\]\n\tfor all $x \\in X$.\n\\end{lemma}\n\\begin{proof}\nFor $x \\in X$ let $p_{x} \\in \\mathcal{D}X$ be the distribution satisfying the equality $p_{x}(\\overline{x}) = \\sum_{y \\in i^{-1}(\\overline{x})} d(x)(y)$. Since $i$ is injective we find $p_{x}(\\overline{x}) = d(x)(y)$, if $\\overline{x} = i(y)$, and $p_{x}(\\overline{x}) = 0$ otherwise. Thus we can deduce\t\\[\t\th \\circ \\mathcal{D}i \\circ d(x) = h(p_{x}) =  \\sum_{y \\in Y} d(x)(y) i(y). \\]\n\t\n\\end{proof}\n\n\\subsection{Neighbourhood monad}\n\nIt is well-known that the contravariant setfunctor assigning to a set its powerset, and to a function the function that precomposes a characteristic function with the former, is dually self-adjoint. The monad $\\mathbb{H} = (N, \\mu, \\eta)$ induced by the adjunction is known under the name neighbourhood monad, since its coalgebras are related to neighbourhood frames in modal logic \\cite{jacobs2017recipe}. Its underlying set endofunctor $N$ is given by\n\\begin{gather*}\n\tNX = 2^{2^X} \\qquad Nf(\\Phi)(\\varphi) = \\Phi(\\varphi \\circ f), \n\\end{gather*}\nand its unit $\\eta$ and multiplication $\\mu$ satisfy\n\\begin{gather*}\n\t\\eta_X(x)(\\varphi) = \\varphi(x) \\qquad \\mu_X(\\Phi)(\\varphi) = \\Phi(\\eta_{2^X}(\\varphi)).\n\\end{gather*}\n\nRecall that a non-zero element of a lattice is called atomic, if there exists no non-zero element below it, and a lattice is called atomic, if each element can be written as join of atoms below it.  \nIt is well-known that (i) the category of algebras over $\\mathbb{H}$ is equivalent to the category of complete atomic Boolean algebras; and (ii) the category of complete atomic Boolean algebras is contravariant equivalent to the category of sets \\cite{taylor2002subspaces}. \n\nIn more detail \\cite{bezhanishvili2020minimisation}, the equivalence (i) assigns to a $\\mathbb{H}$-algebra $(X,h)$ the complete atomic Boolean algebra on $X$ with pointwise induced operations\n\\begin{equation}\n\t\\label{inducedcaba}\n\t\\begin{gathered}\n\t0 = h(\\emptyset) \\qquad 1 = h(2^X) \\qquad \\neg x = h(\\sim \\eta_X(x)) \\\\\n\t \\bigvee A = h(\\bigcup_{x \\in A} \\eta_X(x)) \\qquad \\bigwedge A = h(\\bigcap_{x \\in A} \\eta_X(x)),\n\\end{gathered}\n\\end{equation}\nwhile the equivalence (ii) assigns to a complete atomic Boolean algebra $B$ its set of atoms $At(B)$, and to a set $X$ the complete atomic Boolean powerset algebra $\\mathcal{P}X$. The mutually invertibility of the assignments in the latter equivalence is witnessed by the Boolean algebra isomorphism between $B$ and $\\mathcal{P}(At(B))$ that assigns to an element the set of atoms below it.\nFor $K$ the Eilenberg-Moore comparison functor induced by the monadic self-dual powerset adjunction, and $J$ the equivalence in (i), one may recover the representation in (ii) as the composition of $J$ with $K$. \\cite{taylor2002subspaces}.\n\nAs before, let $F$ be the set endofunctor satisfying $FX = X^A \\times 2$. One verifies that coalgebras for the composed endofunctor $F N$ can be recognised as unpointed alternating automata \\cite{bezhanishvili2020minimisation}. The set $2$ can be equipped with a $\\mathbb{H}$-algebra structure $h$ which satisfies $h(\\varphi) = \\varphi(\\textnormal{id}_2)$ and induces a canonical distributive law between $\\mathbb{H}$ and $F$ \\cite{jacobs2012trace}. The corresponding bialgebras consist of deterministic automata with a complete atomic Boolean algebra as state space and join-preserving transition and output Boolean algebra homomorphisms. For instance, every deterministic automaton derived from an alternating automaton via a double subset construction is of such a form. \n\nThe next observation relates generators of a $\\mathbb{H}$-algebras with the induced complete atomic Boolean algebra structure on the latter.\n\n\\begin{lemma}\n\t$(Y, i, d)$ is a generator for a $\\mathbb{H}$-algebra $(X,h)$ iff \n\t\\[\n\tx = \\bigvee_{A \\in d(x)}  (  \\bigwedge_{y \\in A} i(y) \\wedge \\bigwedge_{y \\not \\in A} \\neg i(y) )\n\t\\]\n\tfor all $x \\in X$. \n\\end{lemma}\n\\begin{proof}\nOne verifies that after identifying a subset with its characteristic function, any $\\Phi \\in NY$ satisfies \\[ \\Phi(\\psi) = \\bigvee_{A \\in \\Phi} ( \\bigwedge_{y \\in A} \\psi(y) \\wedge \\bigwedge_{y \\not \\in A} \\neg \\psi(y)). \\] In particular we can deduce the following equality,\n\t\\begin{align*}\n\t\tx &= h \\circ Ni \\circ d(x) = h(\\lambda \\varphi.d(x)(\\varphi \\circ i))  \\\\\n\t\t&= h(\\bigcup_{A \\in d(x)} (\\bigcap_{y \\in A} \\eta_X(i(y)) \\cap \\bigcap_{y \\not \\in A} \\sim \\eta_X(i(y)))).\n\t\\end{align*}\n\tThe statement follows from the latter by  \\eqref{inducedcaba}. \n\t\\end{proof}\n\n\n\\section{Related work}\n\n\\label{relatedworksec}\n\nOne of the main motivations for the present paper has been our broad interest in active learning algorithms for state-based models \\cite{angluin1987learning}, in particular automata for NetKAT \\cite{anderson2014netkat}, a formal system for the verification of networks based on Kleene Algebra with Tests \\cite{kozen1996kleene}.  \nOne of the main challenges in learning non-deterministic models such as NetKAT automata is the common lack of a unique minimal acceptor for a given language \\cite{denis2002residual}. The problem has been independently approached for different variants of non-determinism, often with the common idea of finding a subclass admitting a unique representative \\cite{esposito2002learning, berndt2017learning}. A more general and unifying perspective has been given in \\cite{van2017learning} by van Heerdt, see also \\cite{van2016master, van2020phd}. \n\nOne of the central notions in the work of van Heerdt is the concept of a scoop, originally introduced by Arbib and Manes \\cite{arbib1975fuzzy}.\nIn the present paper the notion coincides with what we call a generator in \\Cref{generatordefinition}. Scoops have primarily been used as a tool for constructing minimal realisations of automata, similarly to \\Cref{forgenerator-isharp-is-bialgebra-hom}. Strengthening the definition of Arbib and Manes to the notion of a basis in \\Cref{basisdefinition} allows us to further extend such automata-theoretical results, e.g. \\Cref{forbasis-bialgebra-areisomorphic}, but also uncovers ramifications with universal algebra, leading for instance to a representation theory of algebra homomorphisms in the same framework.\n\nA generalisation of the notion of a basis to algebras of arbitrary monads has been approached before. For instance, in \\cite{jacobs2011bases} Jacobs defines a basis for an algebra as a coalgebra for the comonad on the category of algebras induced by the free algebra adjunction. One can show that a basis in the sense of \\Cref{basisdefinition} always induces a basis in the sense of \\cite{jacobs2011bases}. Conversely, given certain assumptions about the existence and preservation of equaliser, it is possible to recover a basis in the sense of \\Cref{basisdefinition} from a basis in the sense of \\cite{jacobs2011bases}. Starting with a basis in the sense of \\Cref{basisdefinition}, the composition of both translations yields a basis that is not less compact than the basis one began with; in certain cases they coincide. As equaliser do not necessarily exist and are not necessarily preserved, our approach carries additional data and thus can be seen as finer.  \n\n\n\\section{Discussion and future work}\n\n\\label{discussionsec}\n\nWe have presented a notion of a basis for an algebra over a monad on an arbitrary category that subsumes the familiar notion for algebraic theories.\nWe have covered questions about the existence and uniqueness of bases, and established a representation theory for homomorphisms between algebras over a monad in the spirit of linear algebra by substituting Kleisli morphisms for matrices. \nBuilding on foundations in the work of Arbib and Manes \\cite{arbib1975fuzzy}, we further have established that a basis for the underlying algebra of a bialgebra yields an isomorphic bialgebra with free state space.\nMoreover, we have established an equivalence between the category of bases for an algebra and the category of its isomorphic free algebras, and looked into bases for bialgebras.\nFinally we gave characterisations of bases for algebras over the powerset, downset, distribution, and neighbourhood monad.\n\nFor the future we are particularly interested in using the present work for a unified view on the theory of residual finite state automata \\cite{denis2002residual} (RFSA) and variations of it, for instance the theories of residual probabilistic  automata \\cite{esposito2002learning} and residual alternating automata \\cite{berndt2017learning}. RFSA are non-deterministic automata that share with deterministic automata two important properties: for any regular language they admit a unique minimal acceptor, and the language of each state is a residual of the language of its initial state. In \\Cref{RFSAexample} we have demonstrated that the so-called canonical RFSA can be recovered as the bialgebra with free state space induced by a generator of join-irreducibles for the underlying algebra of a particular bialgebra. We believe we can uncover similar correspondences for other variations of non-determinism. Similar ideas have already been served as motivation in the work of Arbib and Manes \\cite{arbib1975fuzzy} and have recently come up again in \\cite{myers2015coalgebraic}. We are also interested in insights into the formulation of active learning algorithms along the lines of  \\cite{angluin1987learning} for different classes of residual automata, as sketched in the related work section.\n\n\\section{Acknowledgements}\nThis research has been supported by GCHQ via the VeTSS grant \"Automated black-box verification of networking systems\" (4207703\/RFA 15845).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=1\\linewidth]{figures\/teaser.png}\n  \\caption{Fusing a single-view depth probability distribution predicted by a DNN with standard photometric error terms helps to resolve ambiguities in the photometric error due to occlusions or lack of texture. The above projected keyframe depth map was created by our system.}\n  \\label{fig:teaser}\n  \\vspace{2mm}\\hrule\n\\end{figure}\n\nThere has been continued research interest in using structure-for-motion (SfM) and visual simultaneous localisation and mapping (SLAM) for the incremental creation of dense 3D scene geometry due to its potential applications in safe robotic navigation, augmented reality, and manipulation.\nUntil recently, dense monocular reconstruction systems typically worked by minimising the photometric error over several frames.\nAs this minimisation problem is not well-constrained due to occlusion boundaries or regions of low texture, most reconstruction systems employ regularisers based on smoothness (\\cite{Newcombe:etal:ICCV2011}, \\cite{Pizzoli:etal:ICRA2014}) or planar (\\cite{Concha:etal:RSS2014, Concha:Civera:IROS2015, Concha:etal:ICRA2016}) assumptions.\n\nWith the continued success of deep learning in computer vision, there have been many suggestions for data-driven approaches to the monocular reconstruction problem.\nSeveral of these approaches propose a completely end-to-end framework, predicting the scene geometry from either a single image (\\cite{Eigen:etal:NIPS2014, Laina:etal:3DV2016, Godard:etal:CVPR2017, Fu:etal:CVPR2018}) or several consecutive frames (\\cite{Ummenhofer:etal:CVPR2017, Zhou:etal:CVPR2017, Mahjourian:etal:CVPR2018, Zhou:etal:ECCV2018, Yao:etal:ECCV2018, Chang:Chen:CVPR2018}).\nMost promising, however, are those systems that combine deep learning with standard geometric constraints (\\cite{Weerasekera:etal:ICRA2017, Tateno:etal:CVPR2017, Yang:etal:ECCV2018, Bloesch:etal:CVPR2018, Wang:etal:CVPR2018, Laidlow:etal:ICRA2019, Tang:Tan:2019}).\nIt was shown in \\cite{Facil:etal:RAL2017} that learning-based and geometry-based approaches have a complementary nature as learning-based systems tend to perform better on the interior points of objects but blur edges, whereas geometry-based systems typically do well on areas with a high image gradient but perform poorly on interior points that may lack texture.\n\nThe optimal way to combine these two approaches, however, is not clear.\nThe best current results seem to come from systems that take the output of traditional geometry-based systems and feed these into a deep neural network (DNN). A particularly impressive example of this type of system is DeepTAM \\cite{Zhou:etal:ECCV2018}, which passes a photometric cost volume through a network to extract a depth map.\n\nIt may be desirable, however, to use learning-based systems as an additional component that is fused into the pipeline of a traditional system.\nThis approach would keep the probabilistic framework of the reconstruction system, a requirement for many robotic applications.\nPossible benefits of such a framework include avoiding the necessity of having to perform an expensive neural network pass every time the geometric information is updated, and, as DNNs perform best on images close to the training dataset, it might be possible to switch the network component on or off or switch between different networks depending on the environment being reconstructed.\nThe difficulty of this approach, however, is that to probabilistically fuse the network outputs into a 3D reconstruction system, some measure of the uncertainty associated with each prediction is required.\n\nIn this paper, we propose a 3D reconstruction system that fuses together the output of a DNN with a standard photometric cost volume to create dense depth maps for a set of keyframes.\nWe train a network to predict a discrete, nonparametric probability distribution for the depth of each pixel over a given range from a single image.\nLike \\cite{Liu:etal:CVPR2019}, we refer to this collection of probability distributions for each pixel in the keyframe as a ``probability volume''.\nThen, with each subsequent frame, we create a probability volume based on the photometric consistency between the current frame and the keyframe image and fuse this into the keyframe volume.\nThe main contribution of this paper is to demonstrate that combining the probability volumes from these two sources often results in a better conditioned probability volume.\nWe extract depth maps from the probability volume by optimising a cost function that includes a regularisation term based on network predicted surface normals and occlusion boundaries.\nPlease see Figure \\ref{fig:teaser} for an example keyframe reconstruction created by our system.\n\n\n\n\\section{RELATED WORKS}\n\nIn general, uncertainty can be classified into two categories: model or epistemic uncertainty, and statistical or aleatoric uncertainty. In \\cite{Gal:Ghahramani:ICML2016}, the authors suggest using a Monte Carlo dropout technique to estimate the model uncertainty of a network, but this requires multiple expensive network passes.\n\nLike \\cite{Bishop:MDN}, the authors of \\cite{Kendall:Gal:NIPS2017} propose having the network predict its own aleatoric uncertainty and using a Gaussian or Laplacian likelihood as the loss function during training, which was used by \\cite{Laidlow:etal:ICRA2019} for 3D reconstruction.\nThe problem with this approach is that it forces the network to predict a parametric and unimodal distribution.\nAs shown in \\cite{Campbell:etal:ECCV2008}, this type of distribution may be particularly ill-suited to dense reconstruction where there is a clear need for a multi-hypothesis prediction.\n\nOne proposal has been to use a multi-headed network (\\cite{Zhou:etal:ECCV2018}, \\cite{Peretroukhin:etal:CVPRW2019}) with each head making a separate prediction.\nFrom these many predictions, one can calculate the mean and covariance to use in a probabilistic fusion algorithm.\nThe drawbacks of this approach are that it increases the size of the network and requires a careful balancing of the relative size of the network body and heads. \n\nRecently, both \\cite{Fu:etal:CVPR2018} and \\cite{Liu:etal:CVPR2019} achieved impressive results by having their networks predict discrete, nonparametric probability distributions.\nWhile \\cite{Liu:etal:CVPR2019} uses these distributions to fuse the output with other network predictions, to the best of our knowledge, no one has used this method to fuse the predictions of networks with the output of standard reconstruction pipelines, which is what we aim to do in this paper.\n\n\n\n\\section{METHOD}\n\nIn this section, we describe our method for fusing predictions from DNNs into a standard 3D reconstruction pipeline to produce dense depth maps.\n\nOur system represents the observed geometry as a collection of keyframe-based ``probability volumes''.\nThat is, instead of representing the surface as a depth map with a single depth estimate per pixel, the depth is represented with a per-pixel discrete probability distribution over a given depth range.\nThese probability volumes are initialised with the output of a monocular depth prediction network.\nWith each additional RGB image, the system computes a cost volume based on the photometric consistency.\nThis cost volume is then converted to a probability volume and fused into the volume of the current keyframe.\nOnce the number of inliers drops below a given threshold, a new keyframe is created. To propagate information from one keyframe to another, we warp the previous distribution and fuse it into the new one.\n\nWhen we want to extract a depth map from the probability volume, we could take the maximum probability depth values, but in featureless regions where there is also high network uncertainty this would be susceptible to false minima and cause local inconsistencies in the prediction.\nAlso, as the probability distribution is discrete, taking the maximum would result in a quantisation of the final depth prediction.\nTo overcome these shortcomings, we first construct a smooth probability density function (PDF) from the volume using a kernel density estimation (KDE) technique.\nWe then minimise the negative log probability of this PDF along with a regularisation term.\nWhile many dense systems propose using regularisers based on smoothness (\\cite{Newcombe:etal:ICCV2011, Pizzoli:etal:ICRA2014}) or planar (\\cite{Concha:etal:RSS2014, Concha:etal:ICRA2014, Concha:Civera:IROS2015}) assumptions, we follow the examples of \\cite{Weerasekera:etal:ICRA2017} and \\cite{Laidlow:etal:ICRA2019} and penalise our reconstruction for deviating from the surface normals predicted by a DNN.\n\n\\subsection{Multi-Hypothesis Monocular Depth Prediction}\n\nRather than predict a single depth value for each pixel, our network predicts a discrete depth probability distribution over a given range, similar to \\cite{Fu:etal:CVPR2018} and \\cite{Liu:etal:CVPR2019}.\nNot only does this allow the network to express uncertainty about its prediction, but it also allows the network to make a multi-hypothesis depth prediction.\nAs discussed in \\cite{Fu:etal:CVPR2018}, the prediction of the depth probability distribution can be improved by having a variable resolution over the depth range.\nWe choose a log-depth parameterisation, following the examples of \\cite{Weerasekera:etal:ICRA2018} and \\cite{Eigen:etal:NIPS2014}.\nBy uniformly dividing the depth range in log-space, we achieve the desired result of having higher resolution in the areas close to the camera and lower resolution farther away.\n\nFor our network architecture (see Figure \\ref{fig:network}), we use a ResNet-50 encoder \\cite{He:etal:CVPR2016} followed by three upsample blocks, each consisting of a bilinear upsampling layer, a concatenation with the input image, and then two convolutional layers to bring the output back up to the input resolution.\nAll inputs and outputs have a resolution of 256$\\times$192.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{figures\/network_diagram_arrows.png}\n  \\caption{Our network consists of a ResNet-50 encoder with an output stride size of 8 and no global pooling layer. We then pass the output of the encoder through three upsample blocks consisting of a bilinear resize, concatenation with the input image, and then two convolutional layers to match the output resolution to the input. The probability distribution that the network outputs is discretised over 64 channels.}\n  \\label{fig:network}\n  \\vspace{2mm}\\hrule\n\\end{figure}\n\nAs we are having the network predict a discrete distribution rather than a depth map, we cannot use a standard loss function based on the sum of squared errors.\nA cross-correlation loss would not be ideal either, as we would like to penalise the network less for predicting high probabilities in incorrect bins that are close to the true bin than in bins farther away.\nInstead, we choose to use the ordinal loss function proposed in \\cite{Fu:etal:CVPR2018}:\n\\begin{equation}\n\\begin{split}\n    \\mathcal{L}(\\mbs \\theta) ={} & -\\sum_i {} \\Biggl[ {} \\sum_{k = 0}^{k_i^*} \\log(p_{\\mbs \\theta,i}(k_i^* \\geq k)) \\\\\n    &+ \\sum_{k = k_i^* + 1}^{K-1} \\log(1 - p_{\\mbs \\theta,i}(k_i^* \\geq k)) \\Biggr],\n\\end{split}\n\\end{equation}\n\\noindent\nwhere\n\\begin{equation}\n    p_{\\mbs \\theta,i}(k_i^* \\geq k) = \\sum_{j = k}^{K-1} p_{\\mbs \\theta,i}(k_i^* = j),\n\\end{equation}\n\\noindent\n$\\mbs \\theta$ is the set of network weights, $K$ is the number of bins over which the depth range is discretised, $k_i^*$ is the index of the bin containing the ground truth depth for pixel $i$, and $p_{\\mbs \\theta,i}(k_i^* = j)$ is the network prediction of the probability that the ground truth depth is in bin $j$.\n\nLike \\cite{Liu:etal:CVPR2019}, we train our network on the ScanNet RGB-D dataset \\cite{Dai:etal:CVPR2017}.\nNo fine-tuning was done on our evaluation dataset, the TUM RGB-D dataset \\cite{Sturm:etal:IROS2012}.\nWe set the depth range to be between 10cm and 12m and group the log-depth values uniformly into 64 bins.\n\nEach keyframe created by our system is initialised with this network output.\n\n\\subsection{Fusion with Photometric Error Terms}\n\nFor each additional reference frame, we construct a DTAM-style cost volume \\cite{Newcombe:etal:ICCV2011}.\nFirst, we normalise both the keyframe and reference frame images by subtracting their means and dividing by their standard deviations.\nWe then calculate the photometric error by warping the normalised keyframe image into the reference frame for each depth value in the cost volume and taking the sum of squared differences on 3$\\times$3 patches.\nTo simplify the later fusion, we use the midpoint of each of the depth bins used for the network prediction as the depth values in the cost volume.\nPoses are obtained from an oracle, such as a separate tracking system like ORB-SLAM2 \\cite{Mur-Artal:etal:TRO2017}.\n\nTo convert to a probability volume, we separately scale the negative of the squared photometric error for each pixel such that it sums to one over the ray.\nWe then fuse this new probability volume, $p_{\\text{RF}}$, into the current keyframe volume, $p_{\\text{KF}}$:\n\\begin{equation}\n    p_i(k_i^* = k) = p_{\\text{KF},i}(k_i^* = k) p_{\\text{RF},i}(k_i^* = k),\n\\end{equation}\n\\noindent\nfor each pixel $i$ and depth $k$, which is then scaled to sum to one over the ray.\n\n\\subsection{Kernel Density Estimation}\n\nTo avoid a quantisation of the final depth prediction and to have a smooth function to use in the optimisation step, we construct a PDF for the depth of each pixel using a KDE technique with Gaussian basis functions:\n\\begin{equation}\n    f_i(d) = \\sum_{k = 0}^{K-1} p_i(k_i^* = k) \\phi\\left(d(k), \\sigma\\right)\n\\end{equation}\n\\noindent\nwhere $\\phi\\left(\\mu, \\sigma\\right)$ is the probability density of the Gaussian distribution with mean $\\mu$ and standard deviation $\\sigma$, $d(k)$ is the depth value at the midpoint of bin $k$, and $\\sigma$ is a constant smoothing parameter across all pixels and depth values.\nThe value of $\\sigma$ is a hyperparameter that needs to be tuned empirically; we found that $\\sigma = 0.1$ works well in our setting.\n\nAn example of a discrete PDF produced by our system and the smoothed result after applying the KDE technique is shown in Figure \\ref{fig:smoothing}.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{figures\/smoothed.png}\n  \\caption{Our fusion algorithm produces a discrete probability distribution for each pixel in the keyframe. To reduce discretisation errors and to have a continuous cost function for the optimiser, we convert the probability values along each ray into a smooth probability density function using a kernel density estimation technique.}\n  \\label{fig:smoothing}\n  \\vspace{2mm}\\hrule\n\\end{figure}\n\n\\subsection{Regularisation}\n\nAlthough the fused probability volume will have more local consistency than using the photoconsistency terms alone, the result can still be improved by adding a regularisation term to the optimisation used to extract the depth map.\nWhile most dense reconstruction systems base their regularisers on smoothness or planar assumptions, we propose using the surface normals predicted by a DNN as was done in both \\cite{Weerasekera:etal:ICRA2017} and \\cite{Laidlow:etal:ICRA2019} as this may allow for better preservation of fine-grained local geometry.\nTo predict the surface normals from the keyframe image, we use the state-of-the-art network SharpNet \\cite{Ramamonjisoa:Lepetit:ICCVW2019}.\nAs we determine the local surface orientation of our depth estimation from neighbouring pixels and we do not wish incur high costs at depth discontinuities, we mask the regularisation term at occlusion boundaries, which are also predicted by SharpNet.\nSince SharpNet actually predicts a probability of each pixel belonging to an occlusion boundary, we include all pixels with a probability higher than 0.4 in the mask.\nExample predictions of surface normals and occlusion boundaries made by SharpNet on the TUM RGB-D dataset \\cite{Sturm:etal:IROS2012} are shown in Figure \\ref{fig:sharpnet}.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{figures\/sharpnet.png}\n  \\caption{To regularise our depth estimate, we use the surface normals and occlusion boundaries predicted by SharpNet \\cite{Ramamonjisoa:Lepetit:ICCVW2019}. Some examples of the predictions made by SharpNet on the TUM RGB-D dataset \\cite{Sturm:etal:IROS2012} are shown above. From left to right: input RGB images, predicted normals, predicted occlusion boundaries with a probability greater than 0.4.}\n  \\label{fig:sharpnet}\n  \\vspace{2mm}\\hrule\n\\end{figure}\n\n\\subsection{Optimisation}\n\nTo extract a depth map from the probability volume, we minimise a cost function consisting of two terms:\n\\begin{equation}\n    c(\\mbf{d}) = c_f(\\mbf{d}) + \\lambda c_{\\mbf{\\hat n}}(\\mbf{d}),\n\\end{equation}\n\\noindent\nwhere $\\mbf{d}$ is the set of depth values to be estimated, and $\\lambda$ is a hyperparameter used to adjust the strength of the regularisation term. Empirically, we found $\\lambda = 1.0 \\cdot 10^7$ to work well.\n\nThe first term, $c_f$, imposes a unary constraint on each of the pixels:\n\\begin{equation}\n    c_{f}(\\mbf{d}) = -\\sum_i \\log \\left( f_i(d_i) \\right)\n\\end{equation}\n\\noindent\nwhere $f_i(d_i)$ is the smoothed PDF of pixel $i$ evaluated at depth $d_i$.\n\nThe second term, $c_{\\mbf{\\hat n}}$, is a regularisation term that combines two pairwise constraints:\n\\begin{equation}\n\\begin{split}\n    c_{\\mbf{\\hat n}}(\\mbf{d}) = \\sum_i & b_i \\left( \\langle \\mbf{\\hat n}_i, d_i \\mbf{K}^{-1} \\mbf{\\tilde u}_i - d_{i+1} \\mbf{K}^{-1} \\mbf{\\tilde u}_{i+1} \\rangle \\right)^2 \\\\\n    &+ b_i \\left( \\langle \\mbf{\\hat n}_i, d_i \\mbf{K}^{-1} \\mbf{\\tilde u}_i - d_{i+\\text{W}} \\mbf{K}^{-1} \\mbf{\\tilde u}_{i+\\text{W}} \\rangle \\right)^2\n\\end{split}\n\\end{equation}\n\\noindent\nwhere $b_i \\in \\{0, 1\\}$ is the value of the occlusion boundary mask for pixel $i$, $\\langle \\cdot, \\cdot \\rangle$ is the dot product operator, $\\mbf{\\hat n}_i$ is the normal vector predicted by SharpNet, $\\mbf{K}$ is the camera intrinsics matrix, $\\mbf{\\tilde u}_i$ is the homogeneous pixel coordinates for pixel $i$, and $\\text{W}$ is the width of the image in pixels.\n\nWe minimise the cost function by applying a maximum of 100 iterations of gradient descent with a step size of 0.2, and initialise the optimisation with the maximum probability depth values from the fused probability volume.\nAs the focus of this paper is on the benefits of fusing learning-based and geometry-based approaches, the system was not implemented to achieve real time results.\nCurrently, the forward pass of the network is not a major bottleneck (it takes approximately 53ms), but the process of going from a fused probability volume through the smoothing and optimisation to an extracted depth map can take up to 1.2s, depending on how many iterations are required before the stopping criterion is met.\nThis could be improved significantly by using Newton's method or the primal-dual algorithm instead of gradient descent; however, we leave this for future work.\n\n\n\\subsection{Keyframe Warping}\n\nTo avoid throwing away information on the creation of each new keyframe, we warp the probability volume of the current keyframe and use it to initialise the new one.\nAs the probability volume is a distribution over the depth values of a pixel, however, warping the probability volume is not trivial.\nTo do this, we propose using a discrete variation of the method described in \\cite{Loop::etal::3DV2016}, where we first convert the depth probability distribution to an occupancy-based probability volume, where for each depth bin along the ray there is a probability that the associated point in space is occupied. \nWe then warp this occupancy grid into the new frame and convert back to a depth probability distribution.\n\nWe start by defining the probability that the voxel $S_{k,i}$ (corresponding to depth bin $k$ along the ray of pixel $i$) is occupied, conditioned on the depth belonging to bin $j$:\n\\begin{equation}\n    p(S_{k,i} = 1 \\lvert k_i^* = j) =\n    \\begin{cases}\n        0           & \\text{if}\\ k < j \\\\\n        1           & \\text{if}\\ k = j \\\\\n        \\frac{1}{2} & \\text{if}\\ k > j\n    \\end{cases}\n\\end{equation}\n\nTo convert a depth probability distribution into an occupancy probability, we marginalise out the conditional:\n\\begin{equation}\n    p(S_{k,i} = 1) = \\sum_{k=0}^{K-1} p_i(k_i^* = k) p(S_{k,i} = 1 \\lvert k_i^* = k)\n\\end{equation}\n\\noindent\nwhere $p_i(k_i^* = k)$ is the value of the depth probability volume in bin $k$ for pixel $i$.\n\nAs the occupancy grid represents probabilities for locations in 3D space, we can directly warp this into the new keyframe, filling in any unknown values with a default occupancy probability.\nIn our work, we use a default probability of 0.01.\n\nAfter warping, the occupancy grid can be converted back into a depth probability distribution:\n\\begin{equation}\n    p_i(k_i^* = k) = \\prod_{j < k} \\left[1 - p(S_{j,i} = 1)\\right] p(S_{k,i} = 1),\n\\end{equation}\n\\noindent\nand scaled so that the distribution sums to one along the ray.\n\n\\section{EXPERIMENTAL RESULTS}\n\nWe evaluate our system on the Freiburg 1 sequences of the TUM RGB-D dataset \\cite{Sturm:etal:IROS2012}.\nPlease note that only the RGB images are processed by our system and the depth channel is just used as a ``ground truth'' with which to validate our results against.\n\n\\subsection{Qualitative Results}\n\nFigure \\ref{fig:fusion} shows the various PDFs for a sample of four pixels taken from a keyframe in the TUM RGB-D sequence \\textit{fr1\/desk}.\nThe PDFs in the first row are those predicted by the DNN. Note that the network is able to make multi-hypothesis predictions and can have varying degrees of certainty.\nThe PDFs in the second row are those that result from the photometric cost volume.\nFor some of the pixels (such as pixels A and C), the photometric error results in a clear peak.\nThis situation is most often found on corners and edges in the image where there are large intensity gradients.\nFor pixels in textureless regions or on occlusion boundaries or areas with repeating patterns, the photometric PDF may have many peaks (such as pixel B) or no peak at all (such as Pixel D).\nThe final row of the figure shows the fused PDF for each of the pixels.\nBy fusing the two PDFs together, uncertainty can be reduced and ambiguous photometric data can be resolved.\n\nAn example reconstruction for a single keyframe with various ablations is shown in Figure \\ref{fig:qualex}.\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{figures\/mosaic_small.png}\n  \\caption{This figure shows a grid of probability densities for a sample of four pixels from a keyframe (left). The first row, in red, shows the probability densities predicted by the network. The second row, in green, shows the probability densities estimated from the photometric error after the addition of 25 reference frames. The final row, in blue, shows the fused probability densities that results from our algorithm. Note that both the network and the photometric error are capable of producing multiple peaks. In some cases (such as pixel C), both the network and the photometric methods produce good estimates. In others (such as pixel A), both the network and photometric error are relatively uncertain, but together produce a strong peak. In pixels B and D, the network helps resolve ambiguous photometric peaks from either a repetition or lack of texture. The vertical black bars show the location of the ground truth depth.}\n  \\label{fig:fusion}\n  \\vspace{2mm}\\hrule\n\\end{figure*}\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{figures\/qualex_labels.png}\n  \\caption{Qualitative results from an example keyframe and 6 additional reference frames in the TUM RGB-D \\textit{fr1\/360} sequence. The top left image is the keyframe image, and the bottom left is the ground truth depth. The remaining images on the top row are the depth estimates obtained by taking the maximum probability depth from each corresponding probability volume. The bands of colour show the quantisation that results from using this method. The remaining images in the bottom row are the depth estimates that result after performing the optimisation step. Note that the photometric error is only capable of estimating the depth at pixels with a high image gradient (the repeated edges are the result of pose error). While using only the network prediction results in a good reconstruction, the best reconstruction is obtained by fusing the network and photometric volumes together.}\n  \\label{fig:qualex}\n  \\vspace{2mm}\\hrule\n\\end{figure*}\n\n\\subsection{Quantitative Evaluation}\n\nWe demonstrate the value of fusion on the reconstruction pipeline by comparing the performance of the system on each of the Freiburg 1 TUM RGB-D sequences under three different scenarios: using only the network probability volume, using only the photometric probability volume, and using the fused probability volume.\nTo isolate the performance of our reconstruction system, we use the ground truth poses provided in the dataset.\nWe evaluate the performance using three metrics defined in \\cite{Eigen:etal:NIPS2014}: the absolute relative difference (L1-rel), the squared relative difference (L2-rel) and the root mean squared error (RMSE).\nNote that since the photometric probability volume has extremely noisy results on textureless surfaces, we found that the results were improved by initialising the optimisation with the expected value of the depth from the probability volume rather than the highest probability depth.\n\nThe results are presented in Table \\ref{tab:fusion_ablation}.\nIn six of the sequences, the best result is achieved by fusing together the network and photometric probability volumes.\nWhile there is a large performance gain in using the network over the photometric probability volume, the best outcome is achieved by fusing the two together.\nIn one of the sequences (\\textit{fr1\/floor}), the best result is achieved by using only the photometric probability volume.\nFor this entire sequence, the camera is aimed at a bare wooden floor, and, being well outside the training distribution, the network produces particularly bad priors.\nIn the remaining two sequences, the best result is achieved using only the network probability volume.\nIn one of these sequences (\\textit{fr1\/rpy}), the camera motion is purely rotational and the photoconsistency-based subsystem is not able to produce meaningful depth estimates.\n\n\n\\begin{table}[H]\n\\centering\n\\def0.9{0.9}\n\\begin{tabular}{l l c c c}\n    \\toprule\n    \\textbf{Sequence} & \\textbf{System} & \\textbf{L1-rel} & \\textbf{L2-rel} & \\textbf{RMSE} \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/360}   & Network-Only     &         0.193  &         0.147  & \\textbf{0.555} \\\\\n                               & Photometric-Only &         0.633  &         1.008  &         1.514  \\\\\n                               & Fused            & \\textbf{0.191} & \\textbf{0.143} & \\textbf{0.555} \\\\\n                               \\cmidrule(l){2-5}\n                               & DeepTAM*          &         0.194  &         0.174  &         0.559  \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/desk}  & Network-Only     &         0.295  &         0.201  &         0.447  \\\\\n                               & Photometric-Only &         0.541  &         0.503  &         0.859  \\\\\n                               & Fused            & \\textbf{0.278} & \\textbf{0.177} & \\textbf{0.427} \\\\\n                               \\cmidrule(l){2-5}\n                               & DeepTAM*          &         0.089  &         0.031  &         0.213  \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/desk2} & Network-Only     &         0.237  &         0.139  & \\textbf{0.423} \\\\\n                               & Photometric-Only &         0.522  &         0.494  &         0.890  \\\\\n                               & Fused            & \\textbf{0.236} & \\textbf{0.138} &         0.424  \\\\\n                               \\cmidrule(l){2-5}\n                               & DeepTAM*          &         0.111  &         0.049  &         0.270  \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/floor} & Network-Only     &         0.806  &         0.727  &         0.821  \\\\\n                               & Photometric-Only & \\textbf{0.488} & \\textbf{0.303} & \\textbf{0.562} \\\\\n                               & Fused            &         0.785  &         0.691  &         0.796  \\\\\n                               \\cmidrule(l){2-5}\n                               & DeepTAM*          &         0.131  &         0.034  &         0.156  \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/plant} & Network-Only     &         0.426  &         0.502  & \\textbf{0.816} \\\\\n                               & Photometric-Only &         0.726  &         1.422  &         1.983  \\\\\n                               & Fused            & \\textbf{0.416} & \\textbf{0.485} &         0.833  \\\\\n                               \\cmidrule(l){2-5}\n                               & DeepTAM*          &         0.167  &         0.143  &         0.602  \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/room}  & Network-Only     &         0.231  &         0.155  &         0.493  \\\\\n                               & Photometric-Only &         0.605  &         0.762  &         1.187  \\\\\n                               & Fused            & \\textbf{0.226} & \\textbf{0.147} & \\textbf{0.488} \\\\\n                               \\cmidrule(l){2-5}\n                               & DeepTAM*          &         0.132  &         0.079  &         0.367  \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/rpy}   & Network-Only     & \\textbf{0.242} & \\textbf{0.199} & \\textbf{0.577} \\\\\n                               & Photometric-Only &         0.514  &         0.577  &         1.047  \\\\\n                               & Fused            &         0.255  &         0.212  &         0.614  \\\\\n                               \\cmidrule(l){2-5}\n                               & DeepTAM*          &         0.154  &         0.101  &         0.427  \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/teddy} & Network-Only     & \\textbf{0.294} & \\textbf{0.271} & \\textbf{0.773} \\\\\n                               & Photometric-Only &         0.748  &         1.569  &         2.108  \\\\\n                               & Fused            &         0.297  &         0.277  &         0.792  \\\\\n                               \\cmidrule(l){2-5}\n                               & DeepTAM*          &         0.173  &         0.157  &         0.626  \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/xyz}   & Network-Only     &         0.241  &         0.162  &         0.432  \\\\\n                               & Photometric-Only &         0.517  &         0.403  &         0.764  \\\\\n                               & Fused            & \\textbf{0.225} & \\textbf{0.137} & \\textbf{0.401} \\\\\n                               \\cmidrule(l){2-5}\n                               & DeepTAM*          &         0.065  &         0.017  &         0.164  \\\\\n    \\bottomrule\n\\end{tabular}\n\\caption{Comparison of reconstruction errors on Freiburg 1 TUM RGB-D \\cite{Sturm:etal:IROS2012} sequences showing the relative performance of using only the network-predicted probability volume, only the photometric probability volume, and the fused probability volume. *Despite more accurate\nresults, DeepTAM does not maintain a probabilistic formulation.}\n\\label{tab:fusion_ablation}\n\\vspace{2mm}\\hrule\n\\end{table}\n\n\\begin{table}[h]\n\\centering\n\\def0.9{0.9}\n\\begin{tabular}{l l c c c}\n    \\toprule\n    \\textbf{Sequence} & \\textbf{System} & \\textbf{L1-rel} & \\textbf{L2-rel} & \\textbf{RMSE} \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/360}   & No Optimisation      &         0.210  &         0.184  &         0.608  \\\\\n                               & Smoothing-Only       &         0.207  &         0.179  &         0.601  \\\\\n                               & Total Variation      &         0.194  &         0.152  &         0.565  \\\\\n                               & Normals + Occlusions & \\textbf{0.191} & \\textbf{0.143} & \\textbf{0.555} \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/desk}  & No Optimisation      &         0.324  &         0.292  &         0.537  \\\\\n                               & Smoothing-Only       &         0.323  &         0.289  &         0.533  \\\\\n                               & Total Variation      &         0.296  &         0.226  &         0.470  \\\\\n                               & Normals + Occlusions & \\textbf{0.278} & \\textbf{0.177} & \\textbf{0.427} \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/desk2} & No Optimisation      &         0.283  &         0.240  &         0.537  \\\\\n                               & Smoothing-Only       &         0.280  &         0.235  &         0.532  \\\\\n                               & Total Variation      &         0.254  &         0.181  &         0.470  \\\\\n                               & Normals + Occlusions & \\textbf{0.236} & \\textbf{0.138} & \\textbf{0.424} \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/floor} & No Optimisation      &         0.806  &         0.772  &         0.861  \\\\\n                               & Smoothing-Only       &         0.807  &         0.771  &         0.860  \\\\\n                               & Total Variation      &         0.801  &         0.738  &         0.836  \\\\\n                               & Normals + Occlusions & \\textbf{0.785} & \\textbf{0.691} & \\textbf{0.796} \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/plant} & No Optimisation      &         0.436  &         0.558  &         0.863  \\\\\n                               & Smoothing-Only       &         0.435  &         0.555  &         0.857  \\\\\n                               & Total Variation      &         0.425  &         0.520  & \\textbf{0.830} \\\\\n                               & Normals + Occlusions & \\textbf{0.416} & \\textbf{0.485} &         0.833 \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/room}  & No Optimisation      &         0.267  &         0.227  &         0.583  \\\\\n                               & Smoothing-Only       &         0.265  &         0.223  &         0.577  \\\\\n                               & Total Variation      &         0.243  &         0.179  &         0.522  \\\\\n                               & Normals + Occlusions & \\textbf{0.226} & \\textbf{0.147} & \\textbf{0.488} \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/rpy}   & No Optimisation      &         0.327  &         0.434  &         0.781  \\\\\n                               & Smoothing-Only       &         0.320  &         0.390  &         0.755  \\\\\n                               & Total Variation      &         0.265  &         0.231  &         0.621  \\\\\n                               & Normals + Occlusions & \\textbf{0.255} & \\textbf{0.212} & \\textbf{0.614} \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/teddy} & No Optimisation      &         0.296  &         0.300  &         0.799  \\\\\n                               & Smoothing-Only       &         0.295  &         0.296  &         0.792  \\\\\n                               & Total Variation      & \\textbf{0.290} & \\textbf{0.276} & \\textbf{0.771} \\\\\n                               & Normals + Occlusions &         0.297  &         0.277  &         0.792  \\\\\n    \\midrule\n    \\multirow{4}{*}{fr1\/xyz}   & No Optimisation      &         0.299  &         0.303  &         0.595  \\\\\n                               & Smoothing-Only       &         0.296  &         0.298  &         0.590  \\\\\n                               & Total Variation      &         0.255  &         0.212  &         0.493  \\\\\n                               & Normals + Occlusions & \\textbf{0.225} & \\textbf{0.137} & \\textbf{0.401} \\\\\n    \\bottomrule\n\\end{tabular}\n\\caption{Comparison of reconstruction errors on Freiburg 1 TUM RGB-D \\cite{Sturm:etal:IROS2012} sequences showing the relative performance of different regularisation schemes. No Optimisation: results from taking the depth value with the maximum probability in the probability volume. Smoothing-Only: results from minimising the smoothed negative log probability density function without including a regularisation term. Total Variation: results from using the total variation of the depth as a regulariser. Normals + Occlusions: the pipeline as described in this paper.}\n\\label{tab:reg_ablation}\n\\vspace{2mm}\\hrule\n\\end{table}\n\nAs discussed in the introduction, the best results (in terms of the accuracy of the final depth maps) seem to come from systems that take classic, photometric-based approaches and feed the results into a DNN for regularisation.\nDeepTAM \\cite{Zhou:etal:ECCV2018} is a state-of-the-art example of such a system.\nWe argue, however, that a probabilistic formulation is necessary for many applications of depth estimation in robotics and that it is important to investigate methods of fusing the output of learning-based systems into standard reconstruction pipelines that maintain this formulation.\nWe therefore do not expect or claim to be able to achieve more accurate depth reconstructions than those produced by DeepTAM.\nInstead, we claim that we are able to improve the performance of standard SLAM systems by fusing in the outputs of a deep neural network while maintaining a probabilistic formulation.\nFor the sake of transparency, however, we have also run the DeepTAM mapping system with ground truth poses on the same sequences and have included the results in Table 1.\n\nTo show the benefit of our method of regularisation, we compare the performance of the full system against three other regularisation schemes: using no optimisation at all (taking the depth values that maximise the discrete probability distribution), optimising without any regularisation (this will allow for the smoothing of the depth maps based on the continuous PDF, but provide no regularisation), and regularising using the total variation.\n\nFor the total variation, we tuned the hyperparameters of our system for the best performance ($\\lambda = 1.0 \\cdot 10^2$ and a step size of 0.05).\n\nThe results are presented in Table \\ref{tab:reg_ablation}. In all cases the best performance is achieved when using the surface normals and occlusion masks predicted by SharpNet.\n\nFinally, to evaluate our method for warping probability volumes between keyframes, we compare our system against a version without warping where each keyframe is initialised only with the network output and does not receive any information from other keyframes.\n\nThe results are presented in Table \\ref{tab:warp_ablation}.\nUsing our warping method improves the performance of the system in all cases.\n\n\\begin{table}[t]\n\\centering\n\\def0.9{0.9}\n\\begin{tabular}{l l c c c}\n    \\toprule\n    \\textbf{Sequence} & \\textbf{System} & \\textbf{L1-rel} & \\textbf{L2-rel} & \\textbf{RMSE} \\\\\n    \\midrule\n    \\multirow{2}{*}{fr1\/360}   & No Keyframe Warping &         0.202  &         0.157  &         0.575  \\\\\n                               & Keyframe Warping    & \\textbf{0.191} & \\textbf{0.143} & \\textbf{0.555} \\\\\n    \\midrule\n    \\multirow{2}{*}{fr1\/desk}  & No Keyframe Warping &         0.316  &         0.236  &         0.474  \\\\\n                               & Keyframe Warping    & \\textbf{0.278} & \\textbf{0.177} & \\textbf{0.427} \\\\\n    \\midrule\n    \\multirow{2}{*}{fr1\/desk2} & No Keyframe Warping &         0.283  &         0.195  &         0.480  \\\\\n                               & Keyframe Warping    & \\textbf{0.236} & \\textbf{0.138} & \\textbf{0.424} \\\\\n    \\midrule\n    \\multirow{2}{*}{fr1\/floor} & No Keyframe Warping & \\textbf{0.776} & \\textbf{0.684} & \\textbf{0.787} \\\\\n                               & Keyframe Warping    &         0.785  &         0.691  &         0.796  \\\\\n    \\midrule\n    \\multirow{2}{*}{fr1\/plant} & No Keyframe Warping &         0.420  &         0.490  &         0.845  \\\\\n                               & Keyframe Warping    & \\textbf{0.416} & \\textbf{0.485} & \\textbf{0.833} \\\\\n    \\midrule\n    \\multirow{2}{*}{fr1\/room}  & No Keyframe Warping &         0.256  &         0.189  &         0.528  \\\\\n                               & Keyframe Warping    & \\textbf{0.226} & \\textbf{0.147} & \\textbf{0.488} \\\\\n    \\midrule\n    \\multirow{2}{*}{fr1\/rpy}   & No Keyframe Warping &         0.297  &         0.263  &         0.654  \\\\\n                               & Keyframe Warping    & \\textbf{0.255} & \\textbf{0.212} & \\textbf{0.614} \\\\\n    \\midrule\n    \\multirow{2}{*}{fr1\/teddy} & No Keyframe Warping &         0.302  &         0.286  & \\textbf{0.791} \\\\\n                               & Keyframe Warping    & \\textbf{0.297} & \\textbf{0.277} &         0.792  \\\\\n    \\midrule\n    \\multirow{2}{*}{fr1\/xyz}   & No Keyframe Warping &         0.315  &         0.247  &         0.521  \\\\\n                               & Keyframe Warping    & \\textbf{0.225} & \\textbf{0.137} & \\textbf{0.401} \\\\\n    \\bottomrule\n\\end{tabular}\n\\caption{Comparison of reconstruction errors on Freiburg 1 TUM RGB-D \\cite{Sturm:etal:IROS2012} sequences showing the performance gain from using our method to warp keyframe probability volumes.}\n\\label{tab:warp_ablation}\n\\vspace{2mm}\\hrule\n\\end{table}\n\n\\section{CONCLUSION}\n\nWe have presented a method for fusing learned monocular depth priors into a standard pipeline for 3D reconstruction.\nBy training a DNN to predict nonparametric probability distributions, we allow the network to express uncertainty and make multi-hypothesis depth predictions.\n\nThrough a series of experiments, we demonstrated that by fusing the discrete probability volume predicted by the network with a probability volume computed from the photometric error, we often achieve better performance than either on its own.\nFurther experiments showed the value of our regularisation scheme and warping method.\n\n\n\\bibliographystyle{IEEEtran}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nVideo understanding is a central task of artificial intelligence that requires complex grounding and reasoning over multiple modalities. Among many tasks, multiple-choice question answering has been seen as a top-level task \\citep{richardson_mctest_2013} toward this goal due to its flexibility and ease of evaluation. A line of research towards constructing Video QA datasets have been completed \\cite{Tapaswi2016MovieQAUS,Lei2018TVQALC,Zadeh2019SocialIQAQ}. Ideally, a model for this task should understand each modality well and have a good way to aggregate information from different modalities. To this end, it is a natural choice for researchers to use the state-of-the-art models for each subtask and modality. Recently in the Natural Language domain, BERT \\citep{devlin-etal-2019-bert} and other transformer-based models have become baselines in many research works. However, it is a known phenomenon that complex multimodal models tend to overfit to strong-performing single modality \\cite{cirik_visual_2018,mudrakarta_did_2018, thomason_shifting_2019}. To caution against such undesirable modality collapsing, we study how strong RoBERTa \\citep{Liu2019RoBERTaAR}, a better trained version of BERT, can perform on the QA-only task.\n\n\\paragraph{Our main contribution includes:}\n(1) Show that RoBERTa baselines exceed all previously published QA-only baselines on two popular video QA datasets. (2) The strong QA-only results indicate the existence of non-trivial biases in the datasets that may not be obvious to human eyes but can be exploited by modern language models like RoBERTa. We provide analyses and ablations to root-cause \nthese QA biases, recommend best practices for dataset splits and share insights on subjectivity vs. objectivity for question answering.\n\n\n\\section{Model}\n\\label{sec:model}\n\n\n\\begin{table*}[h]\n\\centering\n\\begin{small}\n\\begin{tabular}{@{\\hspace{1pt}}c@{\\hspace{20pt}}l@{\\hspace{20pt}}l@{\\hspace{20pt}}l@{\\hspace{5pt}}c}\n    \\bf Dataset & \\bf Model Name\/Source & \\bf Modality & \\bf QA Model & \\bf Val Acc (\\%) \\\\\n    \\toprule\n    \\multirow{4}{*}{\\shortstack{MovieQA \\\\(A5)}}   & \\bf Our Answer-only & \\bf A      & \\bf RoBERTa (fine-tune)    & \\bf 34.16\\\\\n                                                   & \\bf Our QA-only & \\bf Q+A      & \\bf RoBERTa (fine-tune)    & \\bf 37.33\\\\\n                                                   & \\bf Our QA-only& \\bf Q+A      & \\bf RoBERTa (freeze) & \\bf 22.52\\\\\n                                                   &  SOTA \\cite{Jasani2019AreWA}   &V+S+Q+A &  w2v         &  48.87\\\\\n                                                   &  Random Guess    &     -   & -     & 20.00 \\\\\n                                    \n    \\cmidrule{1-5}\n    \\multirow{6}{*}{\\shortstack{TVQA \\\\(A5)}}      & \\bf Our Answer-only & \\bf A      & \\bf RoBERTa (fine-tune)    & \\bf 46.58\\\\\n                                                   & \\bf Our QA-only& \\bf Q+A      & \\bf RoBERTa (fine-tune) & \\bf 48.91\\\\\n                                                   & \\bf Our QA-only& \\bf Q+A      & \\bf RoBERTa (freeze) & \\bf 30.75\\\\\n                                                   & QA-only with Glove \\cite{Jasani2019AreWA} & Q+A &  GloVe + LSTM & 42.77\\\\\n                                                   & SOTA's QA-only \\cite{yang2020bert}& Q+A &  BERT (fine-tune)         &  46.88 \\\\\n                                                   & SOTA\\cite{yang2020bert}   &V+S+Q+A&  BERT (fine-tune)        &  72.41    \\\\\n                                                   &  Random Guess      &  -     & -    &  20.00    \\\\\n\n\n\t\\bottomrule\n\\end{tabular}\n\\end{small}\n\\caption{\nComparison with State-of-the-art Performance.\n}\n\\label{tab:baselines}\n\\end{table*}\n\n\\begin{table*}[]\n\\centering\n\\begin{small}\n\\begin{tabular}{|l|c|c|c|c|c|c|}\n\\hline\n                   & \\multicolumn{1}{l|}{\\textbf{\\# of questions}} & \\multicolumn{1}{l|}{\\textbf{\\# of annoators}} & \\multicolumn{1}{l|}{\\textbf{\\% of why\/how }} & \\multicolumn{1}{l|}{\\textbf{\\% of other type }} & \\multicolumn{1}{l|}{\\textbf{avg len of Q}} & \\multicolumn{1}{l|}{\\textbf{avg len of A}} \\\\ \\hline\n\\textbf{Movie QA}  & 14,944                                       & ---                                           & 20.9\\%                                                & 79.1\\%                                                   & 5.2                                                      & 5.29                                                \\\\ \\hline\n\\textbf{TVQA}      & 152,545                                      & 1,413                                          & 14.5\\%                                                & 85.5\\%                                                   & 13.5                                                  & 4.72                                                \\\\ \\hline\n\\end{tabular}\n\\end{small}\n\\caption{Dataset Statistics.}\n\\end{table*}\n\n\nWe fine-tune pretrained RoBERTa from \\citet{Liu2019RoBERTaAR} to solve the question answering task. Specifically, for one multiple-choice question with five answers (1 correct and 4 incorrect), we concatenate the tokenized question with each of the five tokenized answers and feed each of these five q-a pairs into RoBERTa. The RoBERTa is connected with a 4-layer MLP (Multi-Layer Perceptron) head to produce a scalar score for each q-a pair. These five scores are then passed through Softmax to output five probabilities indicating how likely the model think it is for each q-a pair to be correct. During training, the probabilities are trained on Cross Entropy loss; during testing, the q-a pair with the highest probability is selected as the model's prediction.\n\n\n\n\\section{Datasets}\n\\label{sec:datasets}\n\nWe evaluate our baseline model against two popular multimodal QA datasets: MovieQAand TVQA.\\\\\n\n\\paragraph{MovieQA:}\nMovieQA \\citep{Tapaswi2016MovieQAUS} was created from 408 subtitled movies. Each movie has a set of questions with 5 multiple choice answers, only one of which is correct. The dataset also contains plot synopses collected from Wikipedia.\n\n\\paragraph{TVQA:} \nTVQA \\citep{Lei2018TVQALC} was collected from 6 long-running TV shows from 3 genres. There are 21,793 video clips in total for QA collection, accompanied with subtitles and aligned with transcripts to add character names. Depending on the type of TV shows, a video clip is in 60 or 90 seconds. Each video clip has a set of questions with 5 multiple choice answers, only one of which is correct.\n\n\n\n\\paragraph{Notation:}\nIn this paper, we use A5 to denote the tasks on datasets. A5 means the multiple choice question consists of 1 correct answer and 4 incorrect answers.\n\n\\section{Bias Analysis}\n\n\\subsection{QA Bias and Inability to Generalize}\n\n\n\\begin{table}[h!]\n\\centering\n\\begin{small}\n\\begin{tabular}{@{\\hspace{5pt}}l@{\\hspace{10pt}}r@{\\hspace{10pt}}r@{\\hspace{10pt}}}\n    \\bf Train Set   & \\multicolumn{2}{c}{{\\bf Validation Accuracy (\\%)}} \\\\\n    \\toprule\n                    & \\bf MovieQA   & \\bf TVQA  \\\\\n    \\cmidrule{2-3}\n    MovieQA         & \\bf 37.33     & 31.18   \\\\\n\tTVQA            & 33.45         & \\bf 48.91  \\\\\n\n\t\\bottomrule\n\\end{tabular}\n\\end{small}\n\\caption{\nAcross-dataset generalization accuracy. Both datasets are trained and evaluated on the A5 task: multiple-choice questions with 1 correct answer and 4 incorrect answers (random guess yields 20\\% accuracy). \\textbf{Bold} number is the highest number in each column.\n}\n\\label{tab:interdataset_generalization}\n\\end{table}\n\n\n\n\n\nFor the two datasets introduced in Section \\ref{sec:datasets}, we perform QA-only baselines using pretrained language model as described in Section \\ref{sec:model}. Table \\ref{tab:baselines} shows how our QA-only model's performance compares to random guess, state-of-the-art full modality performance and its associated QA-only ablation performance.\n\nFrom Table \\ref{tab:baselines}, looking at the numbers in bold font,  we discover language model like RoBERTa is able to answer a significant portion of the questions correctly, despite that these questions are supposed to be not answerable without looking at the video. This result indicates that the model exploits the biases in these datsets. In addition, we also find that answer-only performance is quite close to QA-only performance, indicating the answer alone gives the model a pretty good hint on whether it is likely to be a correct answer.\n\nKnowing there are biases in the datasets, we are then curious on if these learned biases are transferable between datasets. Tihs investigation is important because if the biases are transferable, then perhaps they are not necessarily bad, because one could argue the model has captured some common sense in these questions and answers; but if these biases are not transferable, then it means these biases only patterns tied to one particular dataset, which we hope the model not to learn. To verify this with experiments, we train a model on each of the two dataset's train split and evaluate these two models on each of the two dataset's validation split. The results are shown in Table \\ref{tab:interdataset_generalization}.\n\nLooking at each row in Table \\ref{tab:interdataset_generalization}, we see all transfer-dataset evaluation's performance decreases from same-dataset evaluation. This means that although the model learns some tricks to answer the questions without context, such tricks learned from one dataset no longer works when applied at a different dataset. In other words, the model learns bias in the dataset and such bias is not transferable. This undesirable behavior is what motivates to our analysis in the next sections.\n\n\n\n\n\n\n\\subsection{Source of Bias: Annotator}\n\n\n\n\\newcommand{\\decreased}[1] {\\textcolor{red}{#1$\\downarrow$}}\n\\newcommand{\\increased}[1] {\\textcolor{green}{#1$\\uparrow$}}\n\n\n\\begin{table*}[h!]\n\\centering\n\\begin{small}\n\\begin{tabular}{c c c@{\\hspace{4pt}} c@{\\hspace{4pt}} c@{\\hspace{4pt}} c@{\\hspace{4pt}} c@{\\hspace{4pt}} c@{\\hspace{4pt}} c@{\\hspace{4pt}} c@{\\hspace{4pt}} c@{\\hspace{4pt}} c@{\\hspace{4pt}}}\n    & \\bf Overlap Acc (\\%)  & \\multicolumn{10}{c}{\\textbf{Non-overlap Acc Shift (\\%) vs. Dropped annotator}} \\\\\n    \\toprule\n    \\multirow{3}{*}{\\shortstack{TVQA\\\\ (A5)}} & & w17 & w366 & w24 & w297 & w118 & w313 & w14 & w19 & w2 & w254 \\\\\n    \\cmidrule{3-12}\n    & \\bf 50.59 & \\decreased{-5.59} & \\decreased{-11.28} & \\decreased{-20.14} & \\decreased{-10.55} & \\increased{+23.22} & \\decreased{-20.59} & \\decreased{-1.69} & \\decreased{-5.96} & \\decreased{-12.23} & \\decreased{-17.28}\\\\\n\n\t\\bottomrule\n\\end{tabular}\n\\end{small}\n\\caption{\nNon-overlapping dataset re-split results on the top-10-annotator subset. The ``Overlap Acc\" column is the validation accuracy where the train and validation set both contain questions from all 10 annotators. The ``Non-overlap Acc Shift vs. Dropped annotator\" is the validation accuracy where the train set contains questions from 9 annotators and the validation set only contains questions from the dropped annotator.\n}\n\\label{tab:non-overlap_resplit}\n\\end{table*}\n\n\n\nWe hypothesize one source of bias is from annotators. To verify our hypothesis, we obtain the Annotator IDs corresponding to the questions in TVQA \\footnote{We thank the authors of TVQA for sharing this information. Annotator information for MovieQA is unfortunately not available to us.} and construct a confusion matrix between the top-10 annotators. The results are shown in Figure \\ref{fig:inter_annotator}. For each of the annotators, we construct a mini-train and mini-valid set. For TVQA, each mini-train and mini-valid set contains 1980 and 220 A5 questions, respectively.\n\nFigure \\ref{fig:inter_annotator} reveals a pattern where most cells except for those on the diagonals are light colored, which means the accuracy decreases when the train set's and validation set's questions are not from the same annotator. This indicates the model learns to guess for one specific annotator's questions but such guess strategy is not transferable to other annotator's questions. This reveals that RoBERTa has the capacity to overfit to the annotators' QA style in the train set.\n\nLooking at the bottom number in each \\emph{diagonal} cell from Figure \\ref{fig:inter_annotator}, we see that our model performs quite differently on different annotators. Some annotators, such as w118 and w14, have a very high performance (90.0\\% and 64.5\\%, respectively), while some annotators, such as w24 and w313, have a relatively low performance (31.4\\% and 24.6\\%). This shows different annotator's questions have different level of biases.\n\nWe also discover that all annotators seem to transfer well to w118. We hypothesize w118 may have asked many questions that are similar to other annotator's questions which the model has already learned to answer during training time.\n\n\n\\begin{figure}[t]\n\n  \\centering\n \n  \\includegraphics[width=\\linewidth]{figures\/tvqa_inter_annotator_confusion_matrix.pdf}  \n \n\\caption{TVQA Inter-annotator accuracy shift confusion matrix. Each $w_i$ represents an annotator id and each cell represents a train-test combination between annotators. The cells are colored based on accuracy shift (the top number in each cell): lighter color means more negative accuracy shift and darker color means more positive accuracy shift. Accuracy shift is defined as the difference between each cell's accuracy (the bottom number) and the same-row diagonal cell's accuracy (again, the bottom number).}\n\\label{fig:inter_annotator}\n\\end{figure}\n\n\\paragraph{Dataset Re-split} The observation above incentivizes further investigation: what if we construct a re-split of the dataset where the validation set does not contain annotators in the train set? We conduct this experiment with the limited scope of the top-10 annotators used in Figure \\ref{fig:inter_annotator} for clearer comparison. We create 11 re-splits of the dataset: 1 with annotator-overlapping train and validation set and 10 with annotator-non-overlapping train and validation set (use 9 annotators for train set and use 1 annotator for validation set). The results are shown in Table \\ref{tab:non-overlap_resplit}. We find that 9 out of 10 for TVQA non-overlapping re-splits incur decrease of performance (less bias). Interestingly, the re-split where there is an increase in performance, w118, matches the columns in Figure \\ref{fig:inter_annotator} whose cells' color is darker than average. This further verifies our explanation that w118 asks similar questions to other annotators. Nonetheless, this overall performance decrease trend after re-split suggests that for pretrained language models, annotator-non-overlapping re-split is a harder task than annotator-overlapping split and such re-split can help alleviate the QA bias. Based on this observation, we recommend future research work should create and use an annotator-non-overlapping split for train, validation and test sets whenever possible. The performance reported under such setting will contain fewer annotator biases and is thus a more accurate indicator of progress.\n\n\n\n\\subsection{Source of Bias: Question Type}\n\n\\begin{figure}[h!]\n  \\centering\n \n  \\includegraphics[width=.95\\linewidth]{figures\/movieqa_question_type_analysis.pdf}  \n  \\caption{MovieQA (A5) Accuracy by Question Type}\n  \\label{fig:movieqa_q_type}\n\\end{figure}\n\n\\begin{figure}[h!]\n  \\centering\n \n  \\includegraphics[width=.95\\linewidth]{figures\/tvqa_question_type_analysis.pdf}  \n  \\caption{TVQA (A5) Accuracy by Question Type}\n  \\label{fig:tvqa_q_type}\n\\end{figure}\n\n\nWe also hypothesize type of questions, such as reasoning question (such as why\/how questions) vs. factual question (where\/who questions), can be a source of bias. To verify, we ablate the model's accuracy based on the question's prefix. The results are shown in Figure \\ref{fig:movieqa_q_type} and \\ref{fig:tvqa_q_type}. These ablations are done on the A5 version of each dataset: Recall the random guess baseline in this case is 80\\%: 20\\%.\n\nIn Figure \\ref{fig:movieqa_q_type}, we see MovieQA shows a clear distinction ($>10\\%$) between ``why\" ``how\" questions vs. ``what\", ``who\", ``where\" questions. The model fits significantly better to the former than the latter. \n\nIn Figure \\ref{fig:tvqa_q_type} for TVQA, the model can guess ``why\" questions better than other question categories, while guessing ``who\" remains difficult.\n\nIn general, we observe a trend that questions such as ``why\" and ``how\", which are reasoning and abstract questions and whose answers are more complex, incur more biases that language model can exploit; whereas ``what\", ``who\" and ``where\" questions, which are factual and direct and whose answers are simple, are less bias-prone.\n\n\n\n\n\n\\section{Related Work}\nAlthough more analysis \\citep{goyal_making_2017, leibe_revisiting_2016} have been done on Visual Question Answering (VQA) \\citep{Agrawal2015VQAVQ}, there are few works analysing biases in Video Question Answering datasets.  \\citet{Jasani2019AreWA} suggest MovieQA contain biases by showing that about half of the questions can be answered correctly under the QA-only setting. However, their word embeddings are trained from plot synopses of movies in the dataset and thus they actually introduce context information into their model, making it no longer QA-only.\n\\citet{goyal2017making} propose that language provides a strong prior that can result in good superficial performance and therefore preventing the model from focusing on the visual content. They attempt to fight against these language biases by creating a balanced dataset to force the model focus on the visual information.\nSimilarly, \\citet{cadene2019rubi} design a training strategy to reduce the amount of biases learned by VQA models named Rubi to counter the strong biases in the language modality.\n\\citet{manjunatha2019explicit} provide a method that can capture macroscopic rules that a VQA model ostensibly utilizes to answer questions.\nHowever, those models fail to explain clearly where the bias in the dataset comes from, which is the main topic of our work.\n\n\n\n\\section{Conclusion}\nIn this work, we fine-tune pretrained language model baselines for two popular Video QA datasets and discover that our simple baselines exceed previously published QA-only baselines. These strong baselines reveal the existence of non-trivial biases in the datasets. Our ablation study demonstrates these biases can come from annotator splits and question types. Based on our analysis, we recommend researchers and dataset creators to use annotator-non-overlapping splits for train, validation and test sets; we also caution the community that when dealing with reasoning questions, we are likely to encounter more biases than in factual questions.\n\nThis paper is an post-hoc analysis for the datasets. However, the tools used in this paper could potentially also be extended to aid dataset creation. For example, a dataset creator could have a RoBERTa trained \\emph{online} as annotators add more data. The annotators can use this language model's prediction to self-check if they are injecting any QA bias while coming up with the questions and answers. The dataset creator can also use a confusion matrix like Figure \\ref{fig:inter_annotator} to monitor and identify low-quality annotators and decide the best strategy to reduce biases during the dataset creation process.\n\n\\section*{Acknowledgments}\n\nWe thank the anonymous reviewer for providing helpful feedbacks.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and preliminaries }\nThe maximal green sequence was originally defined to be a particular sequence of mutations of framed cluster quivers, which\n was firstly introduced by Keller in \\cite{Kel}. Maximal green sequences are not only an important subject in cluster\n algebras, but also have important applications in many other objects, such as counting BPS states\n in string theory, producing quantum dilogarithm identities and computing refined Donaldson-Thomas\n invariants.\n\nCluster algebras have closed relations with representation theory via categorification, it follows that maximal\n green sequence could be interpreted from the viewpoint of tilting theory and silting theory. For example,\n a maximal green sequence for a cluster quiver corresponds to a sequence of forward mutation of a specific\n heart to its shift in a particular triangulated category. We refer to \\cite{BDP} for more details. Inspired\n by $\\tau$-tilting theory, Br$\\mathrm{\\ddot{u}}$stle, Smith and Treffinger defined maximal green sequence as\n particular finite chain of torsion classes for a finite dimensional algebra in \\cite{BST0}, which can be\n also naturally defined in arbitrary abelian categories in \\cite{BST1}.\n\nThroughout this paper we always assume $\\mathcal{A}$ is  a small abelian category.\n\nLet us firstly recall some basic concepts.  Suppose $X$ is an object in $\\mathcal{A}$.\n We say that $X$ has finite length, if there exists a finite filtration\n \\[0=X_0 \\subset  X_1 \\subset X_2 \\subset \\dots \\subset X_m =X\\]\n such that $X_i\/X_{i-1}$ is simple for all $i$. Such a filtration is called a {\\bf{\\em Jordan-H$\\ddot{o}$lder}\n series} of $X$. It is well-known that if $X$ has finite length, then the length of the\n {\\em Jordan-H$\\ddot{o}$lder} series of $X$ is uniquely determined by $X$, which will be denoted\n by $l(X)$. Recall that an {\\bf abelian length category} is an abelian category such that every object has\n finite length. Throughout this article, we always assume that $\\mathcal{A}$ is an abelian length category.\n\n\nLet $\\mathcal{A}$\n be an abelian length category, and $\\mathcal{T}$ and $\\mathcal{F}$ be full subcategories of $\\mathcal{A}$\n which are closed under isomorphisms. The pair $(\\mathcal{T}, \\mathcal{F})$ is called a {\\bf torsion pair} if it satisfies\n the following conditions.\n \\begin{enumerate}\n     \\item[(i)] For any objects $X\\in \\mathcal{T}$ and $Y\\in \\mathcal{F}$, then $Hom(X, Y) = 0$,\n     \\item[(ii)] An obeject $X$ belongs to $\\mathcal{T}$  if and only if $Hom(X, Y)=0$ for any object $Y\\in \\mathcal{F}$,\n     \\item[(iii)] An obeject $Y$ belongs to $\\mathcal{F}$  if and only if $Hom(X, Y)=0$ for any object $X\\in \\mathcal{T}$.\n   \\end{enumerate}\n\n  For a torsion pair\n $(\\mathcal{T}, \\mathcal{F})$, the full subcategories $\\mathcal{T}$ and $\\mathcal{F}$ are called a  {\\bf torsion class}\n and a {\\bf torsion-free class}, respectively. It is well-known that a full subcategory in $\\mathcal{A}$ is a torsion class\n if and only if it is closed under extensions and factors, and a full subcategory in $\\mathcal{A}$ is a torsion-free\n class if and only if it is closed under extensions and subobjects. One of important properties of a torsion pair is that for any\n object $X$ in $\\mathcal{A}$,  there is a unique exact sequence $0 \\rightarrow X_1 \\rightarrow X\n \\rightarrow X_2 \\rightarrow 0$ with $X_1\\in \\mathcal{T}$ and $X_2\\in \\mathcal{F}$ up to isomorphism,\n which is called the {\\em canonical sequence} for $X$ with respect to the torsion pair $(\\mathcal{T}, \\mathcal{F})$.\n\nLet $\\mathcal{T}$ and $\\mathcal{T}'$ be two torsion classes in $\\mathcal{A}$. We say the torsion\n class $\\mathcal{T}'$ {\\em covers} $\\mathcal{T}$ if $\\mathcal{T} \\subsetneq \\mathcal{T}'$ and $\\mathcal{X} = \\mathcal{T}$ or $\\mathcal{X} = \\mathcal{T}'$ for any  torsion class $\\mathcal{X}$ satisfying $\\mathcal{T} \\subset \\mathcal{X} \\subset \\mathcal{T}'$.\n In this case, we write $\\mathcal{T} \\lessdot \\mathcal{T}'$.\n\n\\begin{Definition}[\\cite{BST1}] A {\\bf maximal green sequence} in an abelian length category $\\mathcal{A}$ is a finite\n sequence of torsion classes with covering relations\n \\[\\,\\mathcal{T}_0\\,\\lessdot\\, \\mathcal{T}_1\\, \\lessdot\\, \\mathcal{T}_2\\, \\lessdot \\,\\dots\\,\n                             \\lessdot\\, \\mathcal{T}_m\\]\n such that $\\mathcal{T}_0 = 0$ and $\\mathcal{T}_m = \\mathcal{A}$.\n\\end{Definition}\n\nStability conditions and Harder-Narasimhan filtration are widely studied by many authors\n and are very active. They were introduced in different contexts. For examples, King introduced stability\n functions on quiver representations in \\cite{King}, and Rudakov extended it to abelian categories in \\cite{Rud}.\n Let us recall basic definitions on stability functions and the important Harder-Narasimhan property for abelian\n length categories from \\cite{Rud}.\n\n\n\n\\begin{Definition}[\\cite{Rud, BST1}]\nLet $\\mathcal{P}$ be a totally ordered set and $\\phi : \\mathcal{A}^*\\rightarrow \\mathcal{P}$ a\n function on $\\mathcal{A}^*=\\mathcal{A}\\backslash \\{0\\}$ which is constant on isomorphism classes.\n The map $\\phi$ is called a {\\bf stability function} if for each short exact sequence\n                  $0\\rightarrow L\\rightarrow M\\rightarrow N\\rightarrow 0$\n of nonzero objects in $\\mathcal{A}$ one has the so-called {\\bf see-saw property}:\n\n\n either $\\phi(L) = \\phi(M) = \\phi(N)$,\n\n  or  $\\phi(L) > \\phi(M) > \\phi(N)$,\n\n   or $\\phi(L) < \\phi(M) < \\phi(N)$.\n\nMoreover, a nonzero object\n $M$ in  is said to be {\\bf $\\phi$-stable} (or {\\bf $\\phi$-semistable}) if every nontrivial subobject\n $L \\subset M$ satisfies $\\phi(L) < \\phi(M)$ (or $\\phi(L) \\leq \\phi(M)$, respectively).\n\\end{Definition}\n\nLet $\\phi$ be a stability function on $\\mathcal{A}$. For any nonzero object $X$ in $\\mathcal{A}$, we call $\\phi(X)$ the {\\bf phase} of $X$.\n When there is no confusion, we will simply call an object {\\bf semistable} (respectively, {\\bf stable})\n instead of $\\phi$-semistable (respectively, $\\phi$-stable). Rudakov proved the Harder-Narasimhan property as follows.\n\n\\begin{Theorem}[\\cite{Rud}]\\label{ruda}\nLet $\\phi : \\mathcal{A}\\rightarrow \\mathcal{P}$ be a stability function, and let $X$ be a\n nonzero object in $\\mathcal{A}$. Then up to isomorphism, $X$ admits a unique Harder-Narasimhan filtration,\n  that is a filtration\n \\[0=X_0\\subsetneq X_1\\subsetneq X_2 \\subsetneq \\dots \\subsetneq X_l =X\\]\n such that the quotients $F_i = X_i\/X_{i-1}$ are semistable,\n and $\\phi(F_1) > \\phi(F_2) > \\dots > \\phi(F_l)$.\n\nOn the other hand, if $Y$ is a semistable object in $\\mathcal{A}$, then there exists a filtration of $Y$\n  \\[0=Y_0\\subsetneq Y_1\\subsetneq Y_2 \\subsetneq \\dots \\subsetneq Y_m =Y\\]\nsuch that the quotients $G_i = Y_i\/Y_{i-1}$ are stable, and $\\phi(Y) = \\phi(G_m) = \\dots =\\phi(G_1)$.\n\\end{Theorem}\n\nThe second part of Theorem \\ref{ruda} claims that any semistable object admits a stable subobject and a\n stable quotient with the same phase as the semistable object.\n Following from \\cite{BST1}, we call $F_1 = X_1$ the {\\bf maximally destabilizing subobject} of $X$ and\n $F_l = X_l\/X_{l-1}$ the {\\bf maximally destabilizing quotient} of $X$. They are unique up to isomorphism.\n\nFor a stability function $\\phi : \\mathcal{A}\\rightarrow \\mathcal{P}$, T. Br$\\mathrm{\\ddot{u}}$stle, D. Smith, and H. Treffinger proved in \\cite{BST1} that  it can induce a torsion pair $(\\mathcal{T}_p, \\mathcal{F}_p)$ in $\\mathcal{A}$ for every $p\\in \\mathcal{P}$ which\n is given as follows.\n \\[ \\mathcal{T}_{\\geq p}\\, =\\, \\{X\\,\\in\\,Obj(\\mathcal{A}): \\phi(X')\\,\\geq\\, p\\,\\, \\text{for the maximally destabilizing\n                          quotient $X'$ of $X$}\\} \\cup \\{0\\},\\]\n \\[ \\mathcal{F}_{<p}\\, =\\, \\{Y\\,\\in\\,Obj(\\mathcal{A}): \\phi(Y'')\\,<\\, p\\,\\, \\text{for the maximally destabilizing\n                          sub-object $Y''$ of $Y$}\\} \\cup \\{0\\}.\\]\n Then\n  $\\{\\mathcal{T}_{\\geq r}\\}_{r\\in \\mathcal{P}}$ is called\n the {\\bf chain of torsion classes induced the stability function}\n $\\phi$. Furthermore, if $\\{\\mathcal{T}_{\\geq r}\\}_{r\\in \\mathcal{P}}$ forms a maximal green sequence, it called the\n  {\\bf maximal green sequence induced by the stability function}  $\\phi$.\n\n Note that for any $r, s \\in \\mathcal{P}$, we have that $\\mathcal{T}_{\\geq r} \\subset \\mathcal{T}_{\\geq s}$ if and only\n if $r \\geq s$, and $\\mathcal{T}_{\\geq r} \\subsetneq \\mathcal{T}_{\\geq s}$ implies $r > s$.\n   In \\cite{BST1}, Br$\\mathrm{\\ddot{u}}$stle, Smith and Treffinger proved that\n under some conditions on the stability function, the chain of torsion classes induced by the stability function is\n a maximal green sequence in $\\mathcal{A}$.\n\n On the other hand, the important examples of stability functions are given by central charges. Let $\\mathcal{A}$ be\n an abelian length categories with exactly $n$ nonisomorphic simple objects $S_1, S_2, \\dots, S_n$. We know that the  Grothendieck\n group $K_0(\\mathcal{A})$ of $\\mathcal{A}$ is isomorphic to $\\mathbb{Z}^n$.\n \\begin{Definition}\nA {\\bf central charge} $Z$ on $\\mathcal{A}$ is an additive map $Z: K_0(\\mathcal{A}) \\rightarrow \\mathbb{C}$\nwhich is given by \\[Z(X) = \\langle\\alpha, [X]\\rangle + \\mathrm{i}\\langle\\beta, [X]\\rangle\\]\n for $X\\in Obj(\\mathcal{A})$. Here $\\alpha\\in \\mathbb{R}^n$ and $\\beta\\in \\mathbb{R}_{>0}^n$ are fixed, and $\\langle\\cdot \\,, \\cdot\\rangle$ is the canonical inner product on $\\mathbb{R}^n$ and $\\mathrm{i}= \\sqrt{-1}$.\n\\end{Definition}\n\nSince $\\langle\\beta, [X]\\rangle > 0$ for any nonzero object $X$ in $\\mathcal{A}$, then $Z(X)$ lies in the strict upper half space of the complex space.\n It is well-known that every central charge $Z$ on $\\mathcal{A}$ determines a stability function $\\phi_{Z}$ (see also the proof of Theorem \\ref{main}),  which is given by\n\\[\\phi_Z(X) = \\frac{argZ(X)}{\\pi}.\\]\n\nWe say that  a maximal green sequence can be induced by a central charge if the stability function determined by a central charge induces this maximal green sequence.\\\\\n\nThis article is organized as follows.\n In Section \\ref{2}, we study relations between maximal green sequences and  complete forward hom-orthogonal sequences.\n In Section \\ref{3.1}, we study properties of maximal green sequences induced\n by stability functions. In Section \\ref{3.2}, we define {\\bf crossing inequalities} for maximal green sequences (see Definition \\ref{as}),\n and then prove the following main result.\n\n\n{\\bf Theorem \\ref{main}}\\;\n{\\em    A maximal green sequence  $\\mathcal{T}: 0 = \\mathcal{T}_0\\, \\lessdot\\, \\mathcal{T}_1\\, \\lessdot\\, \\mathcal{T}_2\\, \\lessdot \\,\\dots\\, \\lessdot\\, \\mathcal{T}_m = \\mathcal{A} $ in an abelian length category $\\mathcal{A}$ is induced by some central charge $Z : K_0(\\mathcal{A}) \\rightarrow \\mathbb{C}$ if and only if\n $\\mathcal{T}$ satisfies crossing inequalities. }\n\nIn Section \\ref{4.1}, for finite dimensional algebras, we formulate relations between maximal green sequences of torsion classes and maximal green sequences of $\\tau$-tilting pairs, which are defined via $c$-vectors. In Section \\ref{4.2}, we prove the Rotation Lemma for finite dimensional Jacobian algebras and apply Theorem \\ref{main} to formulate relations of crossing inequalities between Jacobian algebras and its mutation.\n\n\n\\section{Correspondence between maximal green sequences and  complete forward hom-orthogonal sequences}\\label{2}\n\n\\subsection{Complete forward hom-orthogonal sequences}\nWe recall the concept of complete forward hom-orthogonal sequences from \\cite{Ig1, Ig2}.  Let us introduce some notations. Let $\\mathcal{A}$\n be an abelian length category, $\\mathcal{C}$ be a subcategory of $\\mathcal{A}$  and $N$ be an object in $\\mathcal{A}$.\n A {\\bf wide subcategory} of $\\mathcal{A}$ is an abelian subcategory closed under extensions.\n  The full subcategory $N^{\\bot}$ is defined to be $N^{\\bot} : = \\{X\\in \\mathcal{A} | Hom(N,X)=0\\}$\n and the full subcategory $\\mathcal{C}^{\\bot}$ is defined to be $\\mathcal{C}^{\\bot} : = \\{X\\in \\mathcal{A} | Hom(Y,X)=0, \\forall Y\\in \\mathcal{C}\\}$.\n The full subcategories $^{\\bot}N$ and $^{\\bot}\\mathcal{C}$ are defined similarly.\n  We also write $\\mathcal{F}(N):= N^{\\bot}$ and $\\mathcal{G}(N):=\n {}^\\bot{\\mathcal{F}(N)}$ for every object $N\\in Obj(\\mathcal{A})$.\n\n Then it is clear that $\\mathcal{F}(N) = \\mathcal{G}(N)^{\\bot}$ and\n $(\\mathcal{F}(N),\\,  \\mathcal{G}(N))$ is a torsion pair in $\\mathcal{A}$.\n\n\\begin{Proposition}[\\cite{Ig2}]\\label{gx}\n Suppose that $Hom(X, Y) = 0$ and $\\mathcal{C} = X^{\\bot} \\cap {}^{\\bot}Y$, then $\\mathcal{G}(X) =\n                         {}^{\\bot}\\mathcal{C} \\cap {}^{\\bot}Y$.\n\\end{Proposition}\n\n\\begin{Definition}\nAn object $X$ in $\\mathcal{A}$ is called a {\\bf brick}, if $EndX$ is a division ring.\n\\end{Definition}\n\nIt is obvious that any brick is indecomposable. Let $\\mathcal{S}$ be a subset of $obj(\\mathcal{A})$, we use $Filt(\\mathcal{S})$ to denote\nthe full subcategory of $\\mathcal{A}$ consisting of objects having a finite filtration with subquotients\nare isomorphic to indecomposable objects in $\\mathcal{S}$, i.e., $X\\in Filt{\\mathcal{S}}$ if and only if there exists a finite filtration of\n$X$: \\[0=X_0 \\subset  X_1 \\subset X_2 \\subset \\dots \\subset X_m =X\\]\nsuch that $X_i\/X_{i-1}\\in Ind(\\mathcal{S})$ for all $i$. For an indecomposable object $X$, we will denote $Filt(\\{X\\})$ by  $Filt(X)$.\n\n The following lemma is well-known.\n\n\\begin{Lemma}[\\cite{Rin}]\nIf $X$ is a brick in $\\mathcal{A}$, then $Filt(X)$ is a wide subcategory of $\\mathcal{A}$.\n\\end{Lemma}\n\n\n\\begin{Definition}[\\cite{Ig1, Ig2}]\\label{cfho}\nA {\\bf complete forward hom-orthogonal sequence} (briefly, CFHO sequence) in $\\mathcal{A}$ is a finite sequence of bricks $N_1, N_2, \\dots , N_m$\n such that\n \\begin{enumerate}\n \\item[(i)] $Hom(N_i, N_j) = 0$ for all $1\\leq i \\lneqq j \\leq m$;\n \\item[(ii)] The sequence is maximal in $\\mathcal{G}(N)$, where $N = N_1 \\oplus N_2 \\oplus \\dots \\oplus N_m$.\n By maximal we mean that no other bricks can be inserted into $N_1, N_2, \\dots , N_m$ preserving (i);\n \\item[(iii)] $\\mathcal{G}(N) = \\mathcal{A}$.\n \\end{enumerate}\n\\end{Definition}\n\nNote that  \\cite{Ig2} (page 4) claims that if the sequence $N_1, N_2, \\dots , N_m$ satisfies Definition \\ref{cfho} (i), then the condition (ii) in this definition is equivalent to the fact that for all $k$, $$\\mathcal{G}(N) \\cap (N_1 \\oplus \\dots \\oplus N_k)^{\\bot} \\cap  {}^{\\bot}(N_{k+1} \\oplus \\dots \\oplus N_m) = 0.$$\n\n\n\\begin{Corollary}\\label{cgx}\nLet $M_1, M_2, \\dots , M_m$ be a complete forward hom-orthogonal sequence in $\\mathcal{A}$, and let\n $M_0 = 0 = M_{m+1}$, $X_i = M_0 \\oplus M_1 \\oplus \\dots \\oplus M_i$ and $Y_i = M_{i+1} \\oplus \\dots\n \\oplus M_m \\oplus M_{m+1}$ for $0 \\leq i \\leq m$. Then $\\mathcal{G}(X_i) = {}^{\\bot}Y_i$ for every $0 \\leq i \\leq m$.\n\\end{Corollary}\n\\begin{proof}\nSince $M_1, M_2, \\dots , M_m$ is a complete forward hom-orthogonal sequence, we have $Hom(X_i, Y_i) = 0$\n and $\\mathcal{C}_i = X^{\\bot}_i \\cap {}^{\\bot}Y_i = \\mathcal{A} \\cap X^{\\bot}_i \\cap {}^{\\bot}Y_i =\n \\mathcal{G}(M) \\cap X^{\\bot}_i \\cap {}^{\\bot}Y_i =0$. It follows from Proposition \\ref{gx} that\n $\\mathcal{G}(X_i) = {}^{\\bot}Y_i$.\n\\end{proof}\n\n In \\cite{Ig2}, Igusa also proved the following property of complete forward hom-orthogonal sequences, which\n shows simple objects are important ingredients in a complete forward hom-orthogonal sequence.\n\n\\begin{Lemma}[\\cite{Ig2}]\\label{sim}\n Let $N_1, N_2, \\dots , N_m$ be a complete forward hom-orthogonal sequence in $\\mathcal{A}$.\n Then the sequence contains all simple objects (up to isomorphism) in $\\mathcal{A}$. Moreover $N_1$ and $N_m$\n are simple objects.\n\\end{Lemma}\n\n\\begin{Corollary}\nIf $\\mathcal{A}$ admits a complete forward hom-orthogonal sequence, then there are only\n finite simple objects in $\\mathcal{A}$ up to isomorphism.\n\\end{Corollary}\n\n\\subsection{Maximal green sequences and CFHO sequences}\nMinimal extending objects for a torsion class were introduced by Barnard, Carroll, and Zhu in \\cite{BCZ}\n to study covers of the torsion class.\n\\begin{Definition}[\\cite{BCZ}]\\label{MEO}\nSuppose $\\mathcal{T}$ is a torsion class in $\\mathcal{A}$. An object $M$ in $\\mathcal{A}$ is called\n a {\\bf minimal extending object} for $\\mathcal{T}$ provided with the following conditions:\n \\begin{enumerate}\n \\item[(i)] Every proper factor of $M$ is in $\\mathcal{T}$;\n \\item[(ii)] If $0 \\rightarrow M \\rightarrow X \\rightarrow T \\rightarrow 0$ is a non-split exact sequence with\n     $T\\in \\mathcal{T}$, then $X\\in \\mathcal{T}$;\n \\item[(iii)] $\\mathrm{Hom}(\\mathcal{T}, M) = 0$.\n \\end{enumerate}\n\\end{Definition}\n\nNote that if $M$ is a minimal extending object for a torsion class $\\mathcal{T}$, then $M$ is indecomposable\n by Definition \\ref{MEO} (i). Moreover, assuming (i), then (iii) is equivalent to the fact that\n $M\\notin \\mathcal{T}$. We write $[M]$ for the isoclass of the object $M$, $ME(\\mathcal{T})$ for the set\n of isoclasses $[M]$ such that $M$ is a minimal extending object for $\\mathcal{T}$, and $Filt(\\mathcal{T}\n \\cup \\{M\\})$ for the iterative extension closure of $Filt(\\mathcal{T})\\cup M$. The following results\n was proved for the category of finitely generated modules over a finite-dimensional algebra in \\cite{BCZ}.\n The results in Section 2 of \\cite{BCZ} also hold for abelian length categories.\n\n\\begin{Proposition}[\\cite{BCZ}]\\label{pop} Suppose $\\mathcal{T}$ is a torsion class in $\\mathcal{A}$ and\n $M$ is an indecomposable object such that every proper factor of $M$ lies in $\\mathcal{T}$. Then\n $Filt(\\mathcal{T}\\cup \\{M\\})$ is a torsion class and $M$ is a brick.\n\\end{Proposition}\n\nThe following result was proved for finite dimensional algebras in \\cite{BCZ}. We give a new proof for abelian length categories.\n\\begin{Lemma}[\\cite{BCZ}]\\label{lem}\nLet $\\mathcal{T}$ be a torsion class in $\\mathcal{A}$ and $M\\notin \\mathcal{T}$ be an indecomposable object in $\\mathcal A$ such\nthat each proper factor of $M$ is in $\\mathcal{T}$. Let $N\\in Filt(\\mathcal{T}\\cup \\{M\\})\\backslash \\mathcal{T}$ such that each proper\nfactor of $N$ lies in $\\mathcal{T}$. If $Filt(\\mathcal{T}\\cup \\{M\\}) \\gtrdot \\mathcal{T}$, then $M \\cong N$.\n\\end{Lemma}\n\\begin{proof}\nIt is clear that $N$ is indecomposable. By Proposition \\ref{pop}, the full subcategory $Filt(\\mathcal{T}\\cup \\{N\\})$\nis a torsion class satisfying that $\\mathcal{T} \\subsetneq Filt(\\mathcal{T}\\cup \\{N\\}) \\subset Filt(\\mathcal{T}\\cup \\{M\\})$,\nwhich implies that $Filt(\\mathcal{T}\\cup \\{N\\}) = Filt(\\mathcal{T}\\cup \\{M\\})$ since $Filt(\\mathcal{T}\\cup \\{M\\}) \\gtrdot \\mathcal{T}$.\n\nWe claim that $Hom(M, N) \\neq 0$ and $Hom(N, M) \\neq 0$. Note that $Hom(\\mathcal{T}, M) = 0$,\n since each proper factor of $M$ is in $\\mathcal{T}$ and $M\\notin \\mathcal{T}$.  If $Hom(N, M) = 0$,\n then it is easy to see that $Hom(Filt(\\mathcal{T}\\cup \\{N\\}),\\,\\, M)=0$. This contradicts to the fact\n that $M\\in Filt(\\mathcal{T}\\cup \\{M\\}) = Filt(\\mathcal{T}\\cup \\{N\\})$. Then $Hom(M, N) \\neq 0$. Similarly,\n we have that $Hom(N, M) \\neq 0$.\n\n Suppose that $M \\ncong N$.  Let $f: M\\rightarrow N$ and $g: N\\rightarrow M$ be two nonzero morphisms.\n  Then $f$ and $g$ are not epimorphisms. Otherwise, one would be a proper factor of the other,\n which contradicts to the facts that $M\\notin \\mathcal{T}$ and $N\\notin \\mathcal{T}$. Thus $cokerf$ is a proper factor of $N$ and therefore belongs to $\\mathcal{T}$.\n\n If $f$ is not a monomorphism, then $Imf$ is a proper factor of $M$. Then $Imf$ and $cokerf$ belong to $\\mathcal{T}$,\n  that implies that $M\\in \\mathcal{T}$, which contradicts to $M\\not\\in \\mathcal{T}$. Hence $f$ is a monomorphism and similarly $g$ is also a monomorphism.\n\n Note that $gf \\neq 0$, since $f\\not=0$ and $g$ is a monomorphism. Therefore $gf: M\\rightarrow M$ is\n an isomorphism since $M$ is a brick. This implies $g$ is an epimorphism, which is a\n contradiction.\n\n Thus $M\\cong N$.\n\\end{proof}\n\\begin{Theorem}[\\cite{BCZ}]\\label{mini}  Suppose $\\mathcal{T}$ is a torsion class in $\\mathcal{A}$. Then the map\n                           $\\eta_{\\mathcal{T}}\\,:\\, [M]\\,\\mapsto\\,Filt(\\mathcal{T}\\cup \\{M\\})$\n                           is a bijection from the set  $ME(\\mathcal{T})$ to the set of $\\mathcal{T}'$\n                           such that $\\mathcal{T} \\lessdot \\mathcal{T}'$. Moreover, for each\n                           such $\\mathcal{T}'$, there exists a unique indecomposable\n                           object $M$ such that $\\mathcal{T}' = Filt(\\mathcal{T}\\cup \\{M\\})$, and in this case,\n                           $M$ is a minimal extending object for $\\mathcal{T}$.\n                           Furthermore, the map $Filt(\\mathcal{T}\\cup \\{M\\}) \\mapsto [M]$ is the\n                           inverse to $\\eta_{\\mathcal{T}}$.\n\\end{Theorem}\nIn \\cite{BCZ}, the statement that $M$ is a minimal extending object for $\\mathcal{T}$ in this case was given in the proof of this theorem.\n\nThe following results are the main tools for us to construct\n a stability function for a given class of maximal green sequence.\n\\begin{Theorem}\\label{m2}\nSuppose that the sequence $N_1, N_2, \\dots , N_m$ is a complete forward hom-orthogonal sequence in\n $\\mathcal{A}$. Let $\\mathcal{G}_i = \\mathcal{G}(N_0\\oplus N_1 \\oplus \\dots N_{i})$ for\n each $0 \\leq i \\leq m$, where $N_0 = 0$. Then,\n\n (i)\\; $\\mathcal{G}_i = Filt(N_0, N_1, \\dots, N_i)$;\n\n(ii)\\;  $N_i$ is a minimal extending object of $\\mathcal{G}_{i-1}$ satisfying that $\\mathcal{G}_i = Filt(\\mathcal{G}_{i-1}\\cup \\{N_i\\})$;\n\n(iii)\\; The sequence\n $0 = \\mathcal{G}_0\\, \\lessdot\\, \\mathcal{G}_1\\, \\lessdot\\, \\mathcal{G}_2\\, \\lessdot \\,\\dots\\,\n \\lessdot\\, \\mathcal{G}_m = \\mathcal{A} $ is a maximal green sequence in $\\mathcal{A}$.\n\\end{Theorem}\n\n\n\\begin{proof}\nBy Corollary \\ref{cgx}, we have $\\mathcal{G}_i = \\mathcal{G}(N_0\\oplus N_1 \\oplus \\dots N_{i}) = {}^{\\bot}(N_{i+1}\\oplus\\dots N_m \\oplus N_{m+1})$ ,\n where $N_{m+1}=0$. We will prove (i) and (ii) using induction method. It is obvious that $\\mathcal{G}_0 = Filt(N_0) = 0$ and $N_1 \\in ME(\\mathcal{G}_0)$\n since $N_1$ is a simple object.\n\nSuppose that $\\mathcal{G}_i = Filt(N_0, N_1, \\dots, N_i)$ and $N_i\\in ME(\\mathcal{G}_{i-1})$ for $1\\leq i \\leq j$.\n We claim that $N_{j+1}\\in ME(\\mathcal{G}_j)$. First note that $Hom(\\mathcal{G}_j, N_k) = 0$ for $k>j$ since $\\mathcal{G}_j^{\\bot} = (N_0\\oplus \\dots \\oplus N_j)^{\\bot}$.\n In particular, $Hom(\\mathcal{G}_j, N_{j+1}) = 0$. Second, suppose that $N$ is a proper quotient of $N_{j+1}$, then\n it is clear that $N \\in {}^{\\bot}(N_{j+2}\\oplus \\dots \\oplus N_m)$. If $f\\in Hom(N, N_{j+1})$ is nonzero, since $N$\n is a quotient of $N_{j+1}$, then $f$ must be an isomorphism, which is a contradiction. Thus $N\\in  \\mathcal{G}_j =\n {}^{\\bot}(N_{j+1}\\oplus \\dots \\oplus N_m)$. Let $0 \\rightarrow N_{j+1} \\stackrel{a}{\\longrightarrow} X \\stackrel{b}{\\longrightarrow} T \\rightarrow 0$\n be a nonsplit exact sequence with $T\\in \\mathcal{G}_j$. Then it is enough to prove that $Hom(X, N_{j+1}) = 0$.\n Let $f\\in Hom(X, N_{j+1})$. If $fa\\neq 0$, it is an isomorphism and $f$ is a section, which is a contradiction.\n Then $fa=0$, and thus $f$ can be factor through $b$. Since $Hom(T, N_{j+1})=0$, then $f=0$. Then $X\\in \\mathcal{G}_j$ and\n $N_{j+1}\\in ME(\\mathcal{G}_j)$. And thus, $\\mathcal{G}_{j+1} = Filt(\\mathcal{G}_j \\cup \\{N_{j+1}\\}) = Filt(Filt({N_0, N_1, \\dots , N_{j}}) \\cup \\{N_{j+1}\\}) = Filt({N_0, N_1, \\dots , N_{j+1}})$. Then by induction,  (i) and (ii) hold.\n\nClearly, $\\mathcal{G}_0=0$ and $\\mathcal{G}_m=\\mathcal{A}$. By Theorem \\ref{mini} and (ii), we have  $\\mathcal{G}_i\\lessdot \\mathcal{G}_{i+1}$ for any $i$. Then (iii) holds.\n\\end{proof}\n\n\n\n\\begin{Theorem}\\label{m1}\nLet $0 = \\mathcal{T}_0\\, \\lessdot\\, \\mathcal{T}_1\\, \\lessdot\\, \\mathcal{T}_2\\, \\lessdot \\,\\dots\\,\n \\lessdot\\, \\mathcal{T}_m = \\mathcal{A} $ be a maximal green sequence in $\\mathcal{A}$. Then there\n exists a sequence of bricks $N_1, N_2, \\dots , N_m$ such that\n \\begin{enumerate}\n \\item[(i)] $N_i$ is a minimal extending object of $\\mathcal{T}_{i-1}$ and $\\mathcal{T}_i =\n              Filt(\\mathcal{T}_{i-1} \\cup \\{N_i\\})$ for each $1 \\leq i \\leq m$.\n \\item[(ii)] $\\mathcal{T}_i = Filt(N_0, N_1, \\dots, N_i) = \\mathcal{G}(N_0 \\oplus N_1\\oplus \\dots \\oplus N_i)\n             = {}^{\\bot}(N_{i+1}\\oplus \\dots \\oplus N_m \\oplus N_{m+1})$ for each $0 \\leq i \\leq m$,\n             where $N_0 = 0 = N_{m+1}$.\n \\item[(iii)] The sequence $N_1, N_2, \\dots , N_m$ is a complete forward hom-orthogonal sequence in $\\mathcal{A}$.\n \\item[(iv)] $Filt(N_i) = \\mathcal{T}_i \\cap \\mathcal{F}_{i-1}$ for each $1 \\leq i \\leq m$.\n \\item[(v)]  Up to isomorphism, each object $X$ in $\\mathcal{A}$ admits a unique filtration\n              \\[0=X_0\\subset X_1\\subset X_2 \\subset \\dots \\subset X_m =X\\]\n              such that $X_i\/X_{i-1} \\in Filt(N_i)$ for each $1 \\leq i \\leq m$.\n \\end{enumerate}\n\\end{Theorem}\n\\begin{proof}\n(i) By Theorem \\ref{mini}, there exist indecomposable objects $N_1, N_2, \\dots, N_m$\nsuch that $N_i$ is a minimal\n    extending object of $\\mathcal{T}_{i-1}$ and $\\mathcal{T}_i = Filt(\\mathcal{T}_{i-1} \\cup \\{N_i\\})$\n    for each $1 \\leq i \\leq m$. Then due to Proposition \\ref{pop} and by the definition of minimal extending objects,  $N_1, N_2, \\dots, N_m$ are bricks.\n\n(ii) It is obvious that $\\mathcal{T}_0 = Filt(N_0)$ and $\\mathcal{T}_1 = Filt(N_0, N_1)$. Then we can\n     prove that $\\mathcal{T}_i = Filt(N_0, N_1, \\dots, N_i)$ by induction. To prove that $\\mathcal{T}_i =\n     \\mathcal{G}(N_0 \\oplus N_1\\oplus \\dots \\oplus N_i)$, it is enough to prove that $\\mathcal{F}_i =\n     (N_0 \\oplus N_1\\oplus \\dots \\oplus N_i)^{\\bot}$ for each $0 \\leq i \\leq m$. Assume that $Y \\in\n     (N_0 \\oplus N_1\\oplus \\dots \\oplus N_i)^{\\bot}$, it is clear that $Hom(\\mathcal{T}_i, Y) = 0$\n     since $\\mathcal{T}_i = Filt(N_0, N_1, \\dots, N_i)$. Thus $Y \\in \\mathcal{F}_i$. Conversely, if\n     $X\\in \\mathcal{F}_i$, then we have that $Hom(\\mathcal{T}_i, X) = 0$ and hence $X \\in\n     (N_0 \\oplus N_1\\oplus \\dots \\oplus N_i)^{\\bot}$. Therefore $\\mathcal{T}_i = \\mathcal{G}(N_0 \\oplus N_1\\oplus \\dots \\oplus N_i)$.\n\n      The statement $\\mathcal{T}_i = {}^{\\bot}(N_{i+1}\\oplus \\dots \\oplus N_m \\oplus N_{m+1})$\n      will follow from (iii) by Corollary \\ref{cgx}.\n\n(iii) At first, by (i), $N_1, N_2, \\dots, N_m$ are bricks.\n\nBy the above proof of (ii), if $i\\leq j-1$, then $N_i \\in \\mathcal{T}_{j-1} = Filt(N_0,N_1, \\dots, N_{j-1})$ for all $i < j$. And by (i), we have $Hom(\\mathcal{T}_{j-1}, N_j) = 0$.  Hence,  $Hom(N_i, N_j) = 0$ for all $i < j$. By (ii), we\n      have shown that $\\mathcal{G}(N_1\\oplus \\dots \\oplus N_m) = \\mathcal{T}_m = \\mathcal{A}$. Now it is enough\n      to prove that for all $i$, $$(N_0 \\oplus N_1\\oplus \\dots \\oplus N_i)^{\\bot}  \\cap {}^{\\bot}(N_{i+1} \\oplus \\dots  \\oplus N_m \\oplus N_{m+1})=0.$$\n       If $0\\not=X\\in (N_0 \\oplus N_1\\oplus \\dots \\oplus N_i)^{\\bot}  \\cap {}^{\\bot}(N_{i+1} \\oplus \\dots  \\oplus N_m \\oplus N_{m+1})$, then $X\\in \\mathcal{F}_i$ and thus $X\\notin \\mathcal{T}_i$. Hence there exists $k$\n      such that $k > i$ and $X\\in T_k\\backslash T_{k-1}$. We have that $N_k$ is a factor of $X$, i.e., there\n      is an epimorphism $X\\rightarrow N_k$. Note that $X\\in {}^{\\bot}(N_{i+1} \\oplus \\dots  \\oplus N_m \\oplus N_{m+1})$\n      and $k>i$, therefore $Hom(X, N_k) = 0$, which is a contradiction. Thus\n      $(N_0 \\oplus N_1\\oplus \\dots \\oplus N_i)^{\\bot}  \\cap {}^{\\bot}(N_{i+1} \\oplus \\dots  \\oplus N_m \\oplus N_{m+1})=0$\n      and the sequence $N_1, N_2, \\dots , N_m$ is a complete forward hom-orthogonal sequence in $\\mathcal{A}$. We have also\n      proved that  $\\mathcal{T}_i = {}^{\\bot}(N_{i+1}\\oplus \\dots \\oplus N_m \\oplus N_{m+1})$ for all $i$.\n\n(iv) Suppose $X$ is a nonzero object in $\\mathcal{T}_i \\cap \\mathcal{F}_{i-1}$, then $X\\in \\mathcal{F}_{i-1}\n     = (N_0 \\oplus N_1\\oplus \\dots \\oplus N_{i-1})^{\\bot}$ and thus $N_i$ is a subobject of $X$. Therefore there is\n     an exact sequence $0\\rightarrow N_i \\rightarrow X \\rightarrow Y \\rightarrow 0$. It is clear that $Y\\in \\mathcal{T}_i$.\n     We claim that $Y\\in \\mathcal{F}_{i-1} = (N_0 \\oplus N_1\\oplus \\dots \\oplus N_{i-1})^{\\bot}$. Otherwise assume that\n     there is a nonzero morphism $h: N_k \\rightarrow Y$ with $k<i$. Then we have the following commutative diagram with\n     exact rows.\n       \\[\\xymatrix{\n  0  \\ar[r]^{} & N_i \\ar[d]_{1} \\ar[r]^{p} & H \\ar[d]_{h} \\ar[r]^{t} & N_k \\ar[d]_{r} \\ar[r]^{} & 0  \\\\\n  0 \\ar[r]^{} & N_i \\ar[r]^{f} & X \\ar[r]^{g} & Y \\ar[r]^{} & 0   }\\]\n     If the first row is nonsplit, then $H\\in \\mathcal{T}_{i-1}$ since $N_k\\in \\mathcal{T}_{i-1}$ and $N_k\\in ME(\\mathcal{T})$.\n     Then $Hom(H, X) = 0$ implies $f=0$, which is a contradiction. Thus the first row is split, i.e., there is a\n     morphism $s : N_k \\rightarrow H$ such that $ts = 1_{N_k}$. Then $r = rts = ghs = 0$, which is a contradiction\n     to our assumption. Then we have proved that $Y\\in \\mathcal{T}_i \\cap \\mathcal{F}_{i-1}$. By induction on the\n     length of $X$, we have that $Y\\in Filt(N_i)$, hence $X\\in Filt(N_i)$.\n\n     Conversely, since $N_i\\in \\mathcal{T}_i \\cap \\mathcal{F}_{i-1}$ by (ii) and $\\mathcal{T}_i$ and $\\mathcal{F}_{i-1}$ are closed under extensions,  it follows that\n     $Filt(N_i) \\subset \\mathcal{T}_i \\cap \\mathcal{F}_{i-1}$.  Then, $Filt(N_i) = \\mathcal{T}_i \\cap \\mathcal{F}_{i-1}$.\n\n(v) Suppose that $X$ is a nonzero object in $\\mathcal{A}$.\n    Let $X_m = X$, and let $0 \\rightarrow X_{m-1} \\rightarrow X_m \\rightarrow X_m\/X_{m-1} \\rightarrow 0$ be the canonical\n    sequence of $X$ with respect to the torsion pair $(\\mathcal{T}_{m-1}, \\mathcal{F}_{m-1})$. Then we have that\n    $X_{m-1}\\in \\mathcal{T}_{m-1}$ and $X_m\/X_{m-1} \\in \\mathcal{T}_{m} \\cap \\mathcal{F}_{m-1}$. The existence of\n    the filtration for $X$  follows easily. Assume that there are two filtrations\n     \\[0=X_0\\subset X_1\\subset X_2 \\subset \\dots \\subset X_m =X,\\]\n     \\[0=X'_0\\subset X'_1\\subset X'_2 \\subset \\dots \\subset X'_m =X,\\]\n       satisfying that  $X_i\/X_{i-1},\\,\\,  X'_i\/X'_{i-1}\\in Filt(N_i)$ for each $1 \\leq i \\leq m$. Note that $m$ is the length of the maximal green sequence. It is easy to\n     see that $X_i,\\,\\, X'_i \\in \\mathcal{T}_i$ by (ii).\n\n     Note that $Hom(\\mathcal{T}_{i-1}, N_i) = 0$ for each $i$.\n     Then $Hom(X_{m-1}, X'_m\/X'_{m-1}) = 0$ implies the following commutative diagram\n       \\[\\xymatrix{\n  0  \\ar[r]^{} & X_{m-1} \\ar[d]_{} \\ar[r]^{} & X_m \\ar[d]_{1} \\ar[r]^{} & X_m\/X_{m-1} \\ar[d]_{} \\ar[r]^{} & 0  \\\\\n  0 \\ar[r]^{} & X'_{m-1} \\ar[r]^{} & X'_m \\ar[r]^{} &  X'_m\/X'_{m-1} \\ar[r]^{}  & 0.   }\\]\n     It is obvious that $X_{m-1} \\cong X'_{m-1}$ and $ X_m\/X_{m-1} \\cong X'_m\/X'_{m-1}$. Similarly, we have that\n     $X_{i-1} \\cong X'_{i-1}$ and $ X_i\/X_{i-1} \\cong X'_i\/X'_{i-1}$ for all $i$. Then the uniqueness follows.\n\\end{proof}\n\nFrom the constructions of  maximal green sequences and complete forward hom-orthogonal sequences resepctively in Theorem \\ref{m2} and Theorem \\ref{m1}, it is obvious\nthat there is a bijection between complete forward hom-orthogonal sequences\n and maximal green sequences in an abelian length category.\n\n\\begin{Corollary}\n If a maximal green sequence exists in $\\mathcal{A}$, then there are only finitely many nonisomorphic simple objects in $\\mathcal{A}$.\n \\end{Corollary}\n\n\\begin{Remark}\nThe relations between complete forward hom-forward orthogonal sequences and maximal green sequences was firstly\n given by Igusa for finite dimensional hereditary algebras \\cite{Ig1} or cluster-tilted algebras of finite representation\n type \\cite{Ig2}.\n Then they were extended for the category of finitely generated\n modules over an arbitrary finite-dimensional algebra in the appendix of \\cite{Kel1} by Demonet.\n Here we show their truth in abelian length categories via a different approach.\n\\end{Remark}\n\n\\section{Condition for maximal green sequences induced by central charges}\n\n\\subsection{Maximal green sequences and stability functions}\\label{3.1}\n\nLet $\\phi: \\mathcal{A}^* \\rightarrow \\mathcal{P}$ be a stability function on $\\mathcal{A}$.\nLet $p\\in\\mathcal{P}$, then  $\\mathcal{T}_{\\geq p}$ and $\\mathcal{A}_{\\geq p}$ are respectively given by\n\\[\\mathcal{A}_{\\geq p} = \\{0\\} \\cup \\{M\\in \\mathcal{A}\\,\\, |\\,\\, M\\,\\, \\text{is $\\phi$-semistable and}\\,\\, \\phi(M)\\geq p \\},\\]\n \\[ \\mathcal{T}_{\\geq p}\\, =\\, \\{X\\,\\in\\,Obj(\\mathcal{A}): \\phi(X')\\,\\geq\\, p\\,\\, \\text{for the maximally destabilizing\n                          quotient $X'$ of $X$}\\} \\cup \\{0\\}.\\]\nIt is obvious that $\\mathcal{A}_{\\geq r} \\subset \\mathcal{A}_{\\geq s}$ if and only\n if $r \\geq s$ , and it is showed in \\cite{BST1} that $\\mathcal{T}_{\\geq p}$ is a torsion class and\n  $\\mathcal{T}_{\\geq p} = Filt(\\mathcal{A}_{\\geq p})$.\n\n\nRecall that $\\phi$\n is called {\\bf discrete at $p\\in \\mathcal{P}$} if two $\\phi$-stable objects $X_1$ and $X_2$  satisfy $\\phi(X_1) = \\phi(X_2) = p$,\n then $X_1$ is isomorphic to $X_2$. Moreover, we say $\\phi$ to be {\\bf discrete} if  $\\phi$\n is discrete at every $p\\in \\mathcal{P}$. On the other hand, Brustle, Smith and Treffinger defined\n an equivalence relation on $\\mathcal{P}$ by $p\\thicksim q$ when $\\mathcal{T}_{\\geq p} = \\mathcal{T}_{\\geq q}$.\n We will write $[p]$ for the equivalence class of $p\\in \\mathcal{P}$. The following important Proposition\n characterises cover relations for torsion classes induced by $\\phi$.\n\n\\begin{Proposition}[\\cite{BST1}]\\label{cov}\nLet $\\phi: \\mathcal{A}^* \\rightarrow \\mathcal{P}$ be a stability function, and let $p, q\\in\\mathcal{P}$ such that\n $\\mathcal{T}_{\\geq p} \\subsetneq \\mathcal{T}_{\\geq q}$. Then $\\mathcal{T}_{\\geq p} \\lessdot \\mathcal{T}_{\\geq q}$ if\n and only if there is no $r\\in \\mathcal{P}$ such that $\\mathcal{T}_{\\geq p} \\subsetneq  \\mathcal{T}_{\\geq r} \\subsetneq\\mathcal{T}_{\\geq q}$,\n            and $\\phi$ is discrete at $q'$ for each $q'\\in [q]$.\n\\end{Proposition}\nThe following theorem in \\cite{BST1} characterizes the stability function inducing a maximal green sequence.\n\n\\begin{Theorem}[\\cite{BST1}]\nSuppose that $\\phi: \\mathcal{A}^* \\rightarrow \\mathcal{P}$ is a stability function such that $\\mathcal{P}$\n has no maximal element or the maximal element of $\\mathcal{A}$ is not in $\\phi(\\mathcal{A})$. Then $\\phi$\n induces a maximal green sequence if and only if $\\phi$ is discrete and $\\mathcal{P}\/\\thicksim$ is finite.\n\\end{Theorem}\n\nWe have the following result.\n\n\\begin{Theorem}\\label{cor}\nLet $\\phi: \\mathcal{A}^* \\rightarrow \\mathcal{P}$ be a stability function, and let $p, q\\in\\mathcal{P}$ such that\n $\\mathcal{T}_{\\geq p} \\lessdot \\mathcal{T}_{\\geq q}$. Then there exists a unique $\\phi$-stable object $N$ satisfying that $\\phi(N) \\in [q]$.\n Moreover $N$ is a minimal extending object for $\\mathcal{T}_{\\geq p}$. In particular, we have that $q \\leq \\phi(N) <p$,\n and if $r_1\\in\\mathcal{P}$ satisfying that $\\phi(N)< r_1 \\leq p$, then $\\mathcal{T}_{\\geq r_1} = \\mathcal{T}_{\\geq p}$.\n\\end{Theorem}\n\\begin{proof}\nSince $\\mathcal{T}_{\\geq p} \\lessdot \\mathcal{T}_{\\geq q}$, we have $p > q$ and $\\mathcal{A}_{\\geq p} \\subsetneq \\mathcal{A}_{\\geq q}$.\n Let $X\\in \\mathcal{A}_{\\geq q}\\backslash \\mathcal{A}_{\\geq p}$ with phase $\\phi(X) = r$.\n Note that $X$ admits a quotient $N$ satisfying that $N$ is stable and $\\phi(X) = \\phi(N) = r$.\n It is obvious that $q \\leq r < p$ and hence $\\mathcal{T}_{\\geq p} \\subset \\mathcal{T}_{\\geq r} \\subset \\mathcal{T}_{\\geq q}$.\n Since $X$ is semistable, the maximal destabilizing quotient of $X$ is itself. Then $X\\in \\mathcal{T}_{\\geq r}$ and\n $X\\notin \\mathcal{T}_{\\geq p}$. Therefore $\\mathcal{T}_{\\geq p} \\subsetneq \\mathcal{T}_{\\geq r}$ and hence\n $\\mathcal{T}_{\\geq r} = \\mathcal{T}_{\\geq q}$, which implies $\\phi(N)=r\\in [q]$.\n\nSuppose there are two stable objects $N$ and $N'$  with phase $\\phi(N)=r$ and $\\phi(N')=r'$ satisfying that $r, r' \\in [q]$.\n If $r=r'$, then $N \\cong N'$ since $\\phi$ is discrete at $r\\in [q]$ by Proposition \\ref{cov}. Otherwise, we may assume that\n $r < r'$. Then we have that $\\mathcal{T}_{\\geq q} = \\mathcal{T}_{\\geq r} = \\mathcal{T}_{\\geq r'}$ and thus\n $N\\in \\mathcal{T}_{\\geq r} = \\mathcal{T}_{\\geq r'}$. By the definition of $\\mathcal{T}_{\\geq r'}$, we have\n that $r=\\phi(N) \\geq r'$, which is a contradiction. The uniqueness follows.\n\nWe shall prove that every proper factor of $N$ is in $\\mathcal{T}_{\\geq p}$. Indeed, let $N'$ be a nontrivial factor of $N$,\nand let $N''$ be the maximal destabilizing quotient of $N'$ and $N'''$ be the stable quotient of $N''$ with phase $\\phi(N''') = \\phi(N'') = s$.\nNote that $N'''\\in \\mathcal{T}_{\\geq q} = \\mathcal{T}_{\\geq r}$ implies that $s \\geq r$. On the other hand, if $s = r$, then $N \\cong N'$, which is\na contradiction. Therefore we have  $s > r$.\nIf $s \\geq p$, then $N' \\in \\mathcal{T}_{\\geq p}$.\nIf $s < p$, we claim that $\\mathcal{T}_{\\geq s} = \\mathcal{T}_{\\geq p}$, and then $N' \\in \\mathcal{T}_{\\geq p}$ also follows.\nIndeed, since $r< s < p$, we have that $\\mathcal{T}_{\\geq p} \\subset \\mathcal{T}_{\\geq s} \\subset \\mathcal{T}_{\\geq r}$.\nIf $\\mathcal{T}_{\\geq p} \\subsetneq \\mathcal{T}_{\\geq s}$, we have $\\mathcal{T}_{\\geq s} = \\mathcal{T}_{\\geq r}$, which implies\nthat $s\\in [p]$. This contradicts to the uniqueness of $N$. As a consequence, every proper factor of $N$ is in $\\mathcal{T}_{\\geq p}$.\nIt is easy to see $N$ is a minimal extending object for $\\mathcal{T}_{\\geq p}$ by Lemma \\ref{lem} and Theorem \\ref{mini}.\n\n\nIn particular, if $r_1\\in\\mathcal{P}$ satisfying that $\\phi(N)= r < r_1 \\leq p$,\n then $\\mathcal{T}_{\\geq p} \\subset \\mathcal{T}_{\\geq r_1} \\subset \\mathcal{T}_{\\geq q}$.\n If   $\\mathcal{T}_{\\geq r_1} = \\mathcal{T}_{\\geq q}$, then $\\mathcal{T}_{\\geq p} \\lessdot \\mathcal{T}_{\\geq r_1}$.\n Thus there exists a stable object $M$ satisfying that $\\phi(M) \\in [r_1]$, and in particular $r < r_1 \\leq \\phi(M) < p$.\n This contradicts to the uniqueness of $N$. Hence $\\mathcal{T}_{\\geq r_1} = \\mathcal{T}_{\\geq p}$.\n\\end{proof}\n\n\nFor the stability function  $\\phi$, suppose that there exist $r_0 > r_1 > \\dots > r_m$ in $\\mathcal{P}$ such that\n\\[0 = \\mathcal{T}_{\\geq r_0} \\lessdot \\mathcal{T}_{\\geq r_1} \\lessdot \\dots \\lessdot \\mathcal{T}_{\\geq r_m} = \\mathcal{A}\\]\nforms a maximal green sequence.\nAssume that $N_i$ is the minimal extending object of $\\mathcal{T}_{\\geq r_{i-1}}$ such that\n$\\mathcal{T}_{\\geq r_i} = Filt(\\mathcal{T}_{\\geq r_{i-1}}\\cup \\{N_i\\})$. By Theorem \\ref{cor}, we know that $N_i$ is stable,\nand we may, without loss of generality, assume that $\\phi(N_i) = r_i$.\nRecall that for $p\\in \\mathcal{P}$, the full subcategory $\\mathcal{A}_p$ is given by\n\\[\\mathcal{A}_p = \\{0\\} \\cup \\{M\\in \\mathcal{A}\\,\\, |\\,\\, M\\,\\, \\text{is semistable and}\\,\\,\\phi(M)=p \\}.\\]\nIt is shown in \\cite{BST1} that $\\mathcal{A}_p$ is a wide subcategory for each $p\\in \\mathcal{P}$. In particular, we have the following\nresult.\n\\begin{Proposition}\nWith the assumptions and notations above, we have that $\\mathcal{A}_{r_i} = Filt(N_i)$ for each $i$ with $1 \\leq i \\leq m$.\n\\end{Proposition}\n\\begin{proof}\nNote that $N_i$ is stable with phase $r_i$, then $N_i \\in \\mathcal{A}_p$.\nSince $\\mathcal{A}_p$ is a wide subcategory, it is closed under extensions.\nThen we have that  $Filt(N_i) \\subset \\mathcal{A}_p$. On the other hand, if $N\\in \\mathcal{A}_p$ is nonzero,\nthen $N$ admits a stable factor $N'$ with phase $\\phi(N') = \\phi(N) =r_i$. Since $\\phi$ is discrete at $r_i$,\nthen $N' \\cong N_i$. Therefore we have a short exact sequence $0\\rightarrow L \\rightarrow N \\rightarrow N_i \\rightarrow 0$.\nSince $N, N_i\\in \\mathcal{A}_p$ and $\\mathcal{A}_p$ is closed under kernels, then $L\\in \\mathcal{A}_p$. Note that\n$l(L)<l(N)$, we may prove that $N\\in Filt(N_i)$ by induction on length of $N$. Thus $\\mathcal{A}_{r_i} = Filt(N_i)$.\n\\end{proof}\n\nNote that $\\phi(N_1) > \\phi(N_2) > \\dots > \\phi(N_m)$, and for any $X \\in Filt(N_i)$, $X$ is semistable\n with phase $r_i$. Then for each nonzero object in $\\mathcal{A}$, the filtration induced by $\\phi$\n (see Theorem \\ref{ruda}) is the same as that induced by the maximal green sequence\n (see Theorem \\ref{m1}(v))  by the uniqueness of the filtration.\n\n\\begin{Corollary}\nWith the assumptions and notations above, we have that\n\\begin{enumerate}\n \\item[(i)]  The set of $\\phi$-semistable objects in $\\mathcal A$  is equal to $\\bigcup_{i=1}^{m}\\mathcal A_{r_i}\\setminus \\{0\\}$;\n  \\item[(ii)]  The set $\\{N_1, N_2, \\dots, N_m\\}$\n is a complete set of nonisomorphic  $\\phi$-stable objects in $\\mathcal{A}$ up to isomorphism.\n \\end{enumerate}\n\\end{Corollary}\n\\begin{proof}\n It is enough to show that if $M$ is an arbitrary\n semistable object in $\\mathcal{A}$, then  $M\\in Filt(N_i)$ for some $1\\leq i \\leq m$, and thus\n $\\phi(M)\\in \\{r_1, r_2, \\dots ,r_m\\}$ for some $i$. Since the filtration $0 \\subsetneq M$ of $M$ induced by the stability function $\\phi$ is the same\n as the one induced by the maximal green sequence, then it is obvious that $M\\in Filt(N_i)$ for some $1\\leq i \\leq m$,\n  and thus $\\phi(M)\\in \\{r_1, r_2, \\dots ,r_m\\}$ for some $i$. In particular, since $\\phi$ is discrete\n  at each $r_i$, then it is easy to see the set $\\{N_1, N_2, \\dots, N_m\\}$\n is a complete set of nonisomorphic  $\\phi$-stable objects in $\\mathcal{A}$.\n\\end{proof}\n\n\\subsection{Maximal green sequences induced by central charges}\\label{3.2}\nLet  $\\mathcal{A}$ be an abelian length category. If there is a maximal green sequence in $\\mathcal{A}$,\nthen $\\mathcal{A}$ admits finitely many simple objects up to isomorphism. Assume that $S_1, S_2, \\dots, S_n$\nbe the complete set of the isomorphism classes of simple objects in $\\mathcal{A}$. Then we know that the Grothendieck\ngroup $K_0(\\mathcal{A})$ is isomorphic to $\\mathbb{Z}^n$. We may write $[X]\\in \\mathbb{Z}^n$ for the image of $X\\in\\mathcal{A}$.\n Note that for $\\theta\\in \\mathbb{R}^n$ and $X\\in obj(\\mathcal{A})$, we also denote by $\\langle\\theta,X\\rangle$ the inner product $\\langle\\theta, [X]\\rangle$ for simplicity.\n\n\\begin{Definition}\\label{as}\n Let $\\mathcal{T}: 0 = \\mathcal{T}_0\\, \\lessdot\\, \\mathcal{T}_1\\, \\lessdot\\, \\mathcal{T}_2\\, \\lessdot \\,\\dots\\, \\lessdot\\, \\mathcal{T}_m = \\mathcal{A} $\n be a maximal green sequence in the abelian length category $\\mathcal{A}$, and  $N_1, N_2, \\dots, N_m$ be the corresponding complete\n hom-orthogonal sequence. If there exits vectors $\\alpha\\in \\mathbb{R}^n$ and $\\beta\\in \\mathbb{R}_{>0}^n$ such that\n $$\\langle\\alpha,N_i\\rangle \\langle\\beta,N_{i+1}\\rangle <  \\langle\\alpha,N_{i+1}\\rangle \\langle\\beta,N_i\\rangle$$ that is,\n $\\frac{\\langle\\alpha,N_i\\rangle}{\\langle\\beta,N_i\\rangle} < \\frac{\\langle\\alpha,N_{i+1}\\rangle} {\\langle\\beta,N_{i+1}\\rangle}$  for all $i$, then  the maximal green sequence $\\mathcal{T}$ is said to satisfy {\\bf crossing inequalities.}\n\\end{Definition}\n\n\n\\begin{Theorem}\\label{main}\n A maximal green sequence  $\\mathcal{T}: 0 = \\mathcal{T}_0\\, \\lessdot\\, \\mathcal{T}_1\\, \\lessdot\\, \\mathcal{T}_2\\, \\lessdot \\,\\dots\\, \\lessdot\\, \\mathcal{T}_m = \\mathcal{A} $ in an abelian length category $\\mathcal{A}$ can be induced by some central charge $Z : K_0(\\mathcal{A}) \\rightarrow \\mathbb{C}$ if and only if\n $\\mathcal{T}$ satisfies crossing inequalities.\n\\end{Theorem}\n\\begin{proof}\nOn one hand, suppose that $\\mathcal{T}$ satisfies crossing inequalities.\nLet $\\alpha\\in \\mathbb{R}^n$ and $\\beta\\in \\mathbb{R}_{>0}^n$ such that\n  $\\langle\\alpha,N_i\\rangle \\langle\\beta,N_{i+1}\\rangle <  \\langle\\alpha,N_{i+1}\\rangle \\langle\\beta,N_i\\rangle$ for all $i$.\n We define a central charge \\[Z : \\mathcal{A} \\rightarrow \\mathbb{C}\\] which is given by\n $Z(X) = \\langle\\alpha, X\\rangle + \\mathrm{i}\\cdot \\langle\\beta, X\\rangle$,\n where $\\mathrm{i} = \\sqrt{-1}$ and $\\langle\\cdot \\,, \\cdot\\rangle$ is the canonical inner product on $\\mathbb{R}^n$.\n Since $\\langle\\beta, X\\rangle >0$ for any nonzero object $X$, the complex number $Z(X)$\n is in the strict upper half-space. Then we define a map $\\phi : \\mathcal{A}^* \\rightarrow [0, 1]$ which is given by\n $\\phi(X) = \\frac{argZ(X)}{\\pi}$ for any nonzero object $X$ in $\\mathcal{A}$.\n\n   It is obvious that $0 < argZ(X) < \\pi$. Note that \\[\\mathrm{cot}argZ(X) = \\frac{\\langle\\alpha,X\\rangle}{\\langle\\beta,X\\rangle}.\\]\n For simplicity, we will write $\\mathrm{cot}X$ for $\\mathrm{cot} argZ(X)$. It is easy to see that\n for any two nonzero objects $X$ and $Y$, $\\phi(X)\\leq \\phi(Y)$ ( $\\phi(X)< \\phi(Y)$ ) if and\n only if $\\mathrm{cot}X \\geq \\mathrm{cot}Y$ ( $\\mathrm{cot}X > \\mathrm{cot}Y$, respectively ),\n which is also equivalent to $\\langle\\alpha,X\\rangle \\langle\\beta,Y\\rangle-\\langle\\alpha,Y\\rangle \\langle\\beta,X\\rangle\\geq 0$ ( $> 0$, respectively ).\n It is well-known that $\\phi$ is a stability function. Indeed, it is obvious that $\\phi(X) = \\phi(Y)$ if $X \\cong Y$.\n On the other hand,  for any exact sequence $0 \\rightarrow L \\rightarrow M \\rightarrow N \\rightarrow0$ with $L, M, N \\neq 0$,\n we have that\n \\[\n\\left|\\begin{array}{cccc}\n   \\langle\\alpha,M\\rangle & \\langle\\alpha,L\\rangle \\\\\n   \\langle\\beta,M\\rangle &   \\langle\\beta,L\\rangle\\\\\n    \\end{array}\\right| =\n \\left|\\begin{array}{cccc}\n   \\langle\\alpha,L\\rangle+\\langle\\alpha,N\\rangle & \\langle\\alpha,L\\rangle \\\\\n    \\langle\\beta,L\\rangle+\\langle\\beta,N\\rangle &  \\langle\\beta,L\\rangle\\\\\n    \\end{array}\\right|=\n \\left|\\begin{array}{cccc}\n   \\langle\\alpha,N\\rangle & \\langle\\alpha,L\\rangle \\\\\n    \\langle\\beta,N\\rangle &  \\langle\\beta,L\\rangle\\\\\n    \\end{array}\\right|=\n    \\left|\\begin{array}{cccc}\n   \\langle\\alpha,N\\rangle & \\langle\\alpha,M\\rangle \\\\\n    \\langle\\beta,N\\rangle &  \\langle\\beta,M\\rangle\\\\\n    \\end{array}\\right|,\n\\]\nwhich implies that $\\phi$ satisfies the seesaw property, and then $\\phi$ is a stability function on $\\mathcal{A}$.\n Since $\\frac{\\langle\\alpha,N_1\\rangle}{\\langle\\beta,N_1\\rangle} < \\frac{\\langle\\alpha,N_2\\rangle}{\\langle\\beta,N_2\\rangle} <\\dots < \\frac{\\langle\\alpha,N_m\\rangle}{\\langle\\beta,N_m\\rangle}$\n and $\\mathrm{cot} N_i = \\frac{\\langle\\alpha,N_i\\rangle}{\\langle\\beta,N_i\\rangle}$ for each $i$, we have that $$\\phi(N_1) > \\phi(N_2) > \\dots > \\phi(N_m).$$\n\n We claim that any nonzero object $X$ in $Filt(N_i)$ is $\\phi$-semistable with phase $\\phi(X) = \\phi(N_i)$.\n Since $\\phi$ satisfies the seesaw property, then $\\phi(X) = \\phi(N_i)$ and thus\n $\\mathrm{cot}X = \\mathrm{cot}N_i =\\frac{\\langle\\alpha,N_i\\rangle}{\\langle\\beta,N_i\\rangle}$ for any nonzero object $X\\in Filt(N_i)$.\n Suppose that $L$ is a nontrivial subobject of $X$. By Theorem \\ref{m1} (v), there is\n a unique filtration  $L$:\n  \\[0=L_0\\subset L_1\\subset L_2 \\subset \\dots \\subset L_m = L\\]\n such that $L_j\/L_{j-1}\\in Filt(N_j)$ for each $1 \\leq j \\leq m$. Then we may assume that $[L_j\/L_{j-1}] = l_j[N_j]$.\n  Thus, $[L]=\\sum_{j=1}^ml_j[N_j]$, $\\langle\\alpha,L\\rangle = \\sum_{j=1}^ml_j\\cdot \\langle\\alpha,N_j\\rangle$ and  $\\langle\\beta,L\\rangle = \\sum_{j=1}^ml_j\\cdot \\langle\\beta,N_j\\rangle$.\n Since $X\\in Filt(N_i)= \\mathcal{T}_i \\cap \\mathcal{F}_{i-1}$,\n we get $$L\\in \\mathcal{F}_{i-1}=(N_0\\oplus N_1\\oplus \\dots \\oplus N_{i-1})^{\\bot}.$$\n Therefore $L_0 = L_1 = \\dots = L_{i-1} = 0$ and thus $l_1 = l_2 = \\dots = l_{i-1}= 0$.\n  Notice that $\\frac{\\langle\\alpha,N_j\\rangle}{\\langle\\beta,N_j\\rangle} \\geq \\frac{\\langle\\alpha,N_i\\rangle}{\\langle\\beta,N_i\\rangle}$ for all $j \\geq  i$, we have that\n \\begin{equation}\\label{ine}\n \\mathrm{cot} L = \\frac{\\langle\\alpha,L\\rangle}{\\langle\\beta,L\\rangle}\n                = \\frac{ \\sum\\limits_{j=i}^{m}l_j\\langle\\alpha,N_j\\rangle}{\\sum\\limits_{j=i}^{m}l_j\\langle\\beta,N_j\\rangle}\n                \\geq \\frac{ \\sum\\limits_{j=i}^{m}l_j\\langle\\beta,N_j\\rangle\\langle\\alpha,N_i\\rangle\/\\langle\\beta,N_i\\rangle}{\\sum\\limits_{j=i}^{m}l_j\\langle\\beta,N_j\\rangle}\n                = \\frac{\\langle\\alpha,N_i\\rangle}{\\langle\\beta,N_i\\rangle} = \\mathrm{cot} X.\n \\end{equation}\n Then $\\phi(L)\\leq \\phi(X)$ which implies $X$ is $\\phi$-semistable. In particular, if $X = N_i$,\n then we claim that $Hom(N_i, L) = 0$. Otherwise, the composition $N_i \\rightarrow L \\hookrightarrow N_i$ is\n nonzero and is an isomorphism, which is a contradiction since $L$ is a nontrivial subobject of $N_i$. Then $L\\in \\mathcal{F}_i =\n (N_0\\oplus \\dots \\oplus N_i)^{\\bot}$, and thus $l_i = 0$ and the inequality (\\ref{ine}) is strict.\n Then $N_i$ is $\\phi$-stable.\n\nBy Theorem \\ref{m1}, for any nonzero object $X\\in \\mathcal{A}$, there is a unique filtration of $X$:\n \\begin{equation}\\label{fil}\n 0=X_{i_0}\\subsetneq X_{i_1}\\subsetneq X_{i_2} \\subsetneq \\dots \\subsetneq X_{i_l} =X\n \\end{equation}\n such that $X_{i_j}\/X_{i_{j-1}} \\in Filt(N_{i_j})$ for each $1 \\leq j \\leq l$ and $1 \\leq i_1 < i_2 < \\dots < i_l \\leq m$.\n Since $X_{i_j}\/X_{i_{j-1}}$ are $\\phi$-semistable with phase $\\phi(X_{i_j}\/X_{i_{j-1}}) = \\phi(N_{i_j})$ and\n $\\phi(N_{i_1}) > \\phi(N_{i_2}) > \\dots > \\phi(N_{i_l})$,  the filtration (\\ref{fil}) of $X$ induced by the maximal\n green sequence is the same as the unique one induced by the stability function $\\phi$ as in Theorem \\ref{ruda}. Then\n $X_{i_l}\/X_{i_{l-1}}$ is the maximally destabilizing quotient of $X$. Hence the phase of the maximally destabilizing\n quotient of each nonzero object in $\\mathcal{A}$ is in $\\{\\phi(N_1), \\phi(N_2), \\dots , \\phi(N_m)\\}$. For simplicity,\n we write $\\phi(N_i) = r_i$ for each $1 \\leq i \\leq m$.\n\nFor each $r\\in [0,1]$, recall that the torsion class $\\mathcal{T}_{\\geq r}$ induced by $\\phi$\n is given by\n  \\[ \\mathcal{T}_{\\geq r}\\, =\\, \\{X\\,\\in\\,Obj(\\mathcal{A}): \\phi(X')\\,\\geq\\, r\\,\\, \\text{for the maximally destabilizing\n                          quotient $X'$ of $X$}\\} \\cup \\{0\\}.\\]\n In the following proof, we prove that $\\mathcal{T}_{\\geq r_i} = \\mathcal{T}_i$ for each $1 \\leq i \\leq m$.\n Note that $\\phi(X') \\geq r_i$ if and only if $\\phi(X') \\in \\{r_1, r_2, \\dots, r_i\\}$.\n\nSuppose that $Y$ is an arbitrary nonzero object in $\\mathcal{A}$, by considering the unique filtration of $Y$,\n it is easy to see that $Y \\in \\mathcal{T}_k\\backslash \\mathcal{T}_{k-1}$  if and only if $\\phi(Y') = r_k$,\n where $Y'$ is the maximally destabilizing quotient of $Y$.\n\n Let us prove that $\\mathcal{T}_{\\geq r_i} = \\mathcal{T}_i$ for each $1 \\leq i \\leq m$.  Indeed, if $X\\in \\mathcal{T}_i\\backslash \\{0\\}$, then\n there exits $j\\leq i$ such that $X \\in \\mathcal{T}_j\\backslash \\mathcal{T}_{j-1}$. Thus $\\phi(X') = r_j \\geq r_i$\n where $X'$ is the maximally destabilizing quotient of $X$, and hence\n $X \\in \\mathcal{T}_{\\geq r_i}$. Conversely, if $W\\in \\mathcal{T}_{\\geq r_i}$, then $\\phi(W') = r_j \\geq r_i$ for some $j\\leq i$ where $W'$ is the maximally destabilizing quotient of $W$, and\n hence $W\\in \\mathcal{T}_j\\backslash \\mathcal{T}_{j-1}$. Then $W\\in \\mathcal{T}_i$.\n Thus $\\mathcal{T}_{\\geq r_i} = \\mathcal{T}_i$ for each $1 \\leq i \\leq m$.\n\n  In particular, we have that\n  $\\mathcal{T}_{\\geq r_m} = \\mathcal{T}_m = \\mathcal{A}$. On the other hand, take $r_0\\in [0,1]$ such that $r_1<r_0<1$,\n  it is obvious that $\\mathcal{T}_{ \\geq r_0} =\\mathcal T_0= 0$.\n\n  Now it follows that the maximal green sequence\n  $\\mathcal{T}_0\\, \\lessdot\\, \\mathcal{T}_1\\, \\lessdot \\,\\dots\\, \\lessdot\\, \\mathcal{T}_m $\n  is induced by the stability function $\\phi$.\n\nConveresely, suppose that  the maximal green sequence $\\mathcal{T}$ is induced by a central charge\n  $$Z : K_0(\\mathcal{A}) \\rightarrow \\mathbb{C}$$\n  which is given by $Z(X) = \\langle\\alpha,X\\rangle + \\mathrm{i}\\cdot \\langle\\beta,X\\rangle$, where $\\alpha \\in \\mathbb{R}^n$ and $\\beta\\in \\mathbb{R}_{>0}^n$.\n  Then by definition, $\\mathcal{T}$ is induced by the stability function $\\phi_Z$. Let $N_1, N_2, \\dots, N_m$\n  be the corresponding complete forward hom-orthogonal sequence. Then we have that $\\phi_Z(N_1) > \\phi_Z(N_1) > \\dots >\\phi_Z(N_m)$\n  by Theorem \\ref{cor}, and thus $\\mathrm{cot}argZ(N_1) < \\mathrm{cot}argZ(N_2) < \\dots < \\mathrm{cot}argZ(N_m)$. This implies that\n   $$\\langle\\alpha,N_i\\rangle\\cdot \\langle\\beta,N_{i+1}\\rangle <  \\langle\\alpha,N_{i+1}\\rangle\\cdot \\langle\\beta,N_i\\rangle$$ for each $1 \\leq i \\leq m-1$. Then $\\mathcal{T}$ satisfies crossing inequalities.\n\\end{proof}\n\n\nFor a given maximal green sequence, it is hard to determine if it is induced by a central charge. Theorem \\ref{main} supports a\npossible way (but not complete) to construct a central charge which induce the given maximal green sequence.\n In the following, we give an example to show it is operable.\n We refer to \\cite{BDP} for basic concepts on $c$-matrices and maximal green sequences of a quiver.\n\n\n\n\\begin{Example}\n\nConsider the following quiver $Q$ of type $A_4$ and let $A=KQ$ be the path algebra where $K$ is an algebraically closed field.\n\n\\[\\begin{tikzpicture}[scale=1.3]\n\\node at (0,0) (11) {$1$};\n\\node at (1,0) (21) {$2$};\n\\node at (2,0) (31) {$3$};\n\\node at (3,0) (41) {$4$};\n\\path[-angle 90]\n\t(11) edge (21)\n\t(21) edge (31)\n\t(31) edge (41);\n\\end{tikzpicture}\\]\n\nTo give a maximal green sequence for $mod(KQ)$, let us consider the maximal green sequence $(2, 1, 4, 1, 2, 3)$ of $Q$ first, and the corresponding mutations of $c$-matrices are given as follows.\n\n\\[\\begin{split} &\\begin{bmatrix} 1&0&0&0\\\\0&1&0&0\\\\0&0&1&0\\\\0&0&0&1 \\end{bmatrix} \\xrightarrow{\\mu_2}\\begin{bmatrix} 1&0&0&0\\\\0&-1&1&0\\\\0&0&1&0\\\\0&0&0&1 \\end{bmatrix} \\xrightarrow{\\mu_1}\n\\begin{bmatrix} -1&0&1&0\\\\0&-1&1&0\\\\0&0&1&0\\\\0&0&0&1 \\end{bmatrix} \\xrightarrow{\\mu_4}\\begin{bmatrix} -1&0&1&0\\\\0&-1&1&0\\\\0&0&1&0\\\\0&0&0&-1 \\end{bmatrix}\\\\& \\xrightarrow{\\mu_3}\n\\begin{bmatrix} 0&0&-1&0\\\\1&-1&-1&0\\\\1&0&-1&0\\\\0&0&0&-1 \\end{bmatrix} \\xrightarrow{\\mu_1} \\begin{bmatrix} 0&0&-1&0\\\\-1&0&0&0\\\\-1&1&0&0\\\\0&0&0&-1 \\end{bmatrix} \\xrightarrow{\\mu_2}\n\\begin{bmatrix} 0&0&-1&0\\\\-1&0&0&0\\\\0&-1&0&0\\\\0&0&0&-1 \\end{bmatrix}\n\\end{split}\\]\n\nRecall that Igusa showed in \\cite{Ig1} that for an acyclic quiver, there is a bijection between\nmaximal green sequences of the quiver and CFHO sequences of its path algebra  over\nan algebraically closed field, and precisely the correspondence claims mutated $c$-vectors of a maximal green sequence of the quiver correspond to dimension vectors\nof bricks in a  CFHO sequence of the path algebra.\n\nThen the sequence of mutated $c$-vectors above gives a complete forward hom-orthogonal sequence $N_1, N_2, N_3, N_4, N_5, N_6$ in $mod(KQ)$ satisfying that\n\\[\\underline{dim}N_1 = \\begin{bmatrix} 0\\\\1\\\\0\\\\0 \\end{bmatrix},\\,\\,\\underline{dim}N_2 = \\begin{bmatrix} 1\\\\0\\\\0\\\\0 \\end{bmatrix},\\,\\,\\underline{dim}N_3 = \\begin{bmatrix} 0\\\\0\\\\0\\\\1 \\end{bmatrix},\\,\\, \\underline{dim}N_4 = \\begin{bmatrix} 1\\\\1\\\\1\\\\0 \\end{bmatrix},\\,\\,\\underline{dim}N_5 = \\begin{bmatrix} 0\\\\1\\\\1\\\\0 \\end{bmatrix},\\,\\,\\underline{dim}N_6 = \\begin{bmatrix} 0\\\\0\\\\1\\\\0 \\end{bmatrix}.\n\\]\n\nBy Theorem \\ref{m2}, the CFHO sequence $N_1, N_2, N_3, N_4, N_5, N_6$ gives a maximal green sequence $0 = \\mathcal{T}_0 \\lessdot \\mathcal{T}_1 \\lessdot \\mathcal{T}_2 \\lessdot \\mathcal{T}_3 \\lessdot \\mathcal{T}_4 \\lessdot \\mathcal{T}_5 \\lessdot \\mathcal{T}_6$ of $mod(KQ)$, which is given by $\\mathcal{T}_i = Filt(\\{N_1, N_2,\\dots, N_i\\})$.\n\nFor the maximal green sequence $\\mathcal{T} : \\mathcal{T}_0 \\lessdot \\mathcal{T}_1 \\lessdot \\mathcal{T}_2 \\lessdot \\mathcal{T}_3 \\lessdot \\mathcal{T}_4 \\lessdot \\mathcal{T}_5 \\lessdot \\mathcal{T}_6$, we try to find $\\alpha \\in \\mathbb{R}^4$ and $\\beta \\in \\mathbb{R}^4_{>0}$ such that the maximal green sequence $\\mathcal{T}$ satisfies the crossing inequalities.\n\nLet us fix a positive integer vector $\\beta=(1,1,1,1)^T$. Assume that $\\alpha  = (x_1,x_2,x_3,x_4)^T \\in \\mathbb{R}^4$. Then $\\langle\\alpha,N_i\\rangle\\cdot \\langle\\beta,N_{i+1}\\rangle\\,\\, <\\,\\,  \\langle\\alpha,N_{i+1}\\rangle \\cdot \\langle\\beta, N_i\\rangle $ for $1\\leq i \\leq 5$, i.e., \\[\\frac{\\langle\\alpha,N_1\\rangle}{\\langle\\beta,N_1\\rangle} < \\frac{\\langle\\alpha,N_2\\rangle}{\\langle\\beta,N_2\\rangle}\n< \\frac{\\langle\\alpha,N_3\\rangle}{\\langle\\beta,N_3\\rangle} < \\frac{\\langle\\alpha,N_4\\rangle}{\\langle\\beta,N_4\\rangle} < \\frac{\\langle\\alpha,N_5\\rangle}{\\langle\\beta,N_5\\rangle} < \\frac{\\langle\\alpha,N_6\\rangle}{\\langle\\beta,N_6 \\rangle}\\] are given by the following inequalities:\n\n \\[ x_2 < x_1 < x_4 < \\frac{x_1+x_2+x_3}{3} < \\frac{x_2+x_3}{2} <x_3.\\]\n There are infinite many solutions of $\\alpha$. By Theorem \\ref{main}, each of these solutions for $\\alpha$ together with $\\beta=(1,1,1,1)^T$ can determine a central charge\nwhich can induce the maximal green sequence $\\mathcal{T} : \\mathcal{T}_0 \\lessdot \\mathcal{T}_1 \\lessdot \\mathcal{T}_2 \\lessdot \\mathcal{T}_3 \\lessdot \\mathcal{T}_4 \\lessdot \\mathcal{T}_5 \\lessdot \\mathcal{T}_6$. For example, we may take\n$\\alpha = (2,1,20,3)^T$.\n\\end{Example}\n\n\\section{Applications: $c$-vectors and Rotation lemma  }\nIn this section, we consider module categories of finite dimensional algebras. Let $A$ be a finite dimensional algebra over a field $K$. Let $\\tau$ be the AR-translation of $A$ and $n = rank(K_0(A))$. In this section, we only consider finitely generated right $A$-modules and all modules are assumed to be basic.\n\\subsection{Bricks and $c$-vectors}\\label{4.1}\nFor a module $X$, we denote the number of nonisomorphic indecomposable summands of $M$ by $|M|$. A module  $M$ is said to be {\\bf $\\tau$-rigid} if\n$Hom(M, \\tau M) = 0$. A pair of modules $(M, P)$ is an  {\\bf $\\tau$-rigid pair} if $M$ is $\\tau$-rigid, $P$ is projective and $Hom(P,M)=0$. A $\\tau$-rigid pair $(M, P)$ is {\\bf (almost) $\\tau$-tilting} if  $|M|+|P|=n$ (respectively, $|M|+|P|=n-1$).\nFor every almost $\\tau$-tilting pair $(M, P)$, there exist exactly two $\\tau$-tilting pair $(M_1,P_1)$ and $(M_2,P_2)$\nsuch that $M$ is direct summand of both $M_1$ and $M_2$, and $P$ is direct summand of both $P_1$ and $P_2$. In this case, we say $(M_1,P_1)$ and $(M_2,P_2)$ are a mutation of each other. Note that if $(M_1,P_1)$ and $(M_2,P_2)$ are a mutation of each other, there is an almost $\\tau$-tilting pair $(M, P)$, and exactly two non-isomorphic indecomposbale $\\tau$-rigid modules $X$ and $Y$ such that  $(M_1,P_1)$ is obtained from $(M, P)$ and $X$, and $(M_2,P_2)$ is obtained from $(M, P)$ and $Y$. We also say $(M_2,P_2)$ is obtained from $(M_1, P_1)$ by mutating at $X$, and $(M_1,P_1)$ is obtained from $(M_2, P_2)$ by mutating at $Y$, or say that $(M_1,P_1)$ and $(M_2,P_2)$ are a mutation of each other with the exchange pair $(X,Y)$. Given a $\\tau$-rigid pair $(M,P)$, we say a module $X$ is a direct summand of $(M,P)$ if $X$ is a direct summand of $M$\nor $P$.\n\nLet $K_0(projA)$ be the Grothendieck group of the additive category $projA$, where $projA$ consists of projective $A$-modules. A basis of $K_0(projA)$ is given by the isomorphism classes $P(1), P(2), \\dots, P(n)$ of indecomposable projective modules. For a given $\\tau$-rigid module $X$ with the minimal projective presentation $\\oplus_{i=1}^nP(i)^{b_i(X)} \\rightarrow \\oplus_{i=1}^nP(i)^{a_i(X)} \\rightarrow X \\rightarrow 0$, the {\\bf $g$-vector} of $X$ is  defined to be $g(X) = (a_1(X)-b_1(X), a_2(X)-b_2(X), \\dots, a_n(X)-b_n(X))^T$. Let $(M = \\oplus_{i=1}^tM_i, P = \\oplus_{j=t+1}^nP_j)$ be an  $\\tau$-tilting pair, then the {\\bf $g$-matrix} of $(M, P)$ is defined to be the matrix \\[G_{(M,P)}=(g(M_1),\\dots, g(M_t), -g(P_{t+1}),\\dots, -g(P_n))\\in M_{n\\times n}(\\mathbb{Z}).\\]\nNotice that $G_{(M,P)}$ is invertible over $\\mathbb{Z}$ for any $\\tau$-tilting pair $(M = \\oplus_{i=1}^tM_i, P = \\oplus_{j=t+1}^nP_j)$ \\cite{AIR}, Fu \\cite{Fu} defined the {\\bf $c$-matrix} of $(M, P)$ to be the matrix\n\\[C_{(M,P)} = (G_{(M,P)}^T)^{-1} = (c_1,\\dots,c_t, c_{t+1},\\dots,c_n).\\]\n\nFor $1\\leq i \\leq t$, the $i$-th column vector $c_i$ of the $c$-matrix $C_{(M,P)}$ of the $\\tau$-tilting pair $(M,P)$ is called the {\\bf $c$-vector} of $M_i$ with respect to $(M,P)$ and denoted it by $c(M_i)_{(M,P)}$, and for $t+1\\leq j \\leq n$, the $j$-th column vector $c_j$ of the $c$-matrix $C_{(M,P)}$  is called the {\\bf $c$-vector} of $P_j$ with respect to $(M,P)$ and denoted it by $c(P_j)_{(M,P)}$.\nGiven a $\\tau$-tilting pair $(M,P)$,  let $X$ be an indecomposbale direct summand of either $M$ or $P$, then either $X=M_i$ for $i\\in \\{1,\\cdots,t\\}$ or $X=P_j$ for $j\\in\\{t+1,\\cdots,n\\}$. In this case,  we usually denote briefly by $c(X)$  the $c$-vector $c(X)_{(M,P)}$ when there is no confusion in the context.\n\nFu \\cite{Fu} proved that $c$-matrices have the sign-coherent property, i.e., each column vector of a $c$-matrix is either non-negative or non-positive. Let $(M_1,P_1)$ and $(M_2,P_2)$ be a mutation of each other with the exchange pair $(X,Y)$, we say it is a {\\bf green mutation} from  $(M_1,P_1)$ to $(M_2,P_2)$ if  $c(X)=c(X)_{(M_1,P_1)}$ is non-positive, otherwise we say it is a {\\bf red mutation} from  $(M_1,P_1)$ to $(M_2,P_2)$.\n\n\\begin{Definition}\nA  {\\bf maximal green sequence of $\\tau$-tilting pairs} is a finite sequence $(M_1, P_1)$, $(M_2, P_2)$, $\\dots$, $(M_m, P_m)$ of $\\tau$-tilting pairs such that  $(M_1, P_1)=(0,A)$,  $(M_m, P_m) = ( A, 0)$, $(M_i, P_i)$  and $(M_{i+1}, P_{i+1})$ are a mutation and the mutation from $(M_i, P_i)$  to $(M_{i+1}, P_{i+1})$ is green for each $1\\leq i \\leq m-1$.\n\\end{Definition}\n\nNote that by [Theorem 3.1, \\cite{DIJ}], for every mutation  $(M_1,P_1)$ and $(M_2,P_2)$, we have either $FacM_1 \\lessdot FacM_2$ or $FacM_2 \\lessdot FacM_1$.\n\n\\begin{Lemma}\\label{x}\n Let $(M_1,P_1)$ and $(M_2,P_2)$ be a mutation of each other  with the exchange pair $(X,Y)$.\nSuppose that $FacM_1 \\subset FacM_2$ and $N$ is the corresponding minimal extending module. Then $c(X) = -\\underline{dim}N$ and  $c(Y) = \\underline{dim}N$.\n\\end{Lemma}\n\\begin{proof}\nAssume that $(M = \\oplus_{i=1}^t M_i, P=\\oplus_{j=t+1}^{n-1}P_j)$ is the almost $\\tau$-tilting pair which can be completed to  $\\tau$-tilting pairs  $(M_1,P_1)$ and $(M_2,P_2)$.\nSince $FacM_1 \\subset FacM_2$, we have that $FacM_1=FacM$ and $FacM_2 = {^{\\bot}(\\tau M)} \\cap P^{\\bot}$.\nLet $\\theta_{(M,P)}= \\sum_{i=1}^tg(M_i) - \\sum_{j=t+1}^{n-1}g(P_j)\\in \\mathbb{Z}^m$.\nSince a nonzero module $X$ is in $M^{\\bot}\\cap {^{\\bot}(\\tau M)} \\cap P^{\\bot}$ if and only if $X$ is  $\\theta_{(M,P)}$-semistable,\nthen $N$ is $\\theta_{(M,P)}$-semistable.\nNote that there is a unique $\\theta_{(M,P)}$-stable module $B$ up to isomorphhism.\nIf $N$ is not isomorphic to $B$, then $B$ is a proper factor of $N$ since the set of $\\theta_{(M,P)}$-semistable modules form an abelian category with only one simple object $B$.  Hence $B\\in Fac M_1= FacM$, which contradicts to $B\\in M^{\\bot}$.\nThus $N$ is $\\theta_{(M,P)}$-stable.\nBy [Theorem 3.5, \\cite{T0}], we have that either $c(X) = \\underline{dim}N$ or  $c(X)=-\\underline{dim}N$. Note that $\\langle g(X), c(X)\\rangle=1$ if $X$ is a direct summand of $M_1$, and\n  $\\langle -g(X), c(X)\\rangle=1$ if $X$ is a direct summand of $P_1$.\nBy [Theorem 1.4(a), \\cite{AR}], we have that\n\\[\\langle g(X), \\underline{dim}N\\rangle = dim_KHom(X,N)-dim_KHom(N,\\tau X).\\]\nTherefore if $X$ is a direct summand of $M_1$, then $Hom(X, N) = 0$ since $Hom(FacM_1,N)=0$. In this case, $\\langle g(X), \\underline{dim}N\\rangle \\leq 0$ which implies that\n$c(X)= -\\underline{dim}N$. If $X$ is a direct summand of $P_1$, it is obvious that $\\langle g(X), \\underline{dim}N\\rangle \\geq 0$ which also implies that $c(X)= -\\underline{dim}N$.\n\nThen  $c(Y) = \\underline{dim}N$ follows from [Lemma 3.4, \\cite{T0}].\n\\end{proof}\n\n\\begin{Corollary}\\label{c} Let $(M_1,P_1)$ and $(M_2,P_2)$ be a mutation of each other  with the exchange pair $(X,Y)$.\nThen the following conditions are equivalent:\n\n(1) $FacM_1 \\subset FacM_2$.\n\n(2) $c(X) \\leq 0$, i.e., the mutation from $(M_1,P_1)$ to $(M_2,P_2)$ is green.\n\n(3) $c(Y) \\geq 0$.\n\n\\end{Corollary}\n\\begin{proof}\nThe equivalence of (2) and (3) follows from Lemma \\ref{x} easily.\n\n(1) $\\Rightarrow$ (2): If  $FacM_1 \\subset FacM_2$, by Lemma \\ref{x}, we have $c(X) = -\\underline{dim}N \\leq 0$.\n\n(2) $\\Rightarrow$ (1): If $c(X) \\leq 0$. Suppose that $FacM_2 \\subset FacM_1$, then  $c(X) \\geq 0$ by Lemma \\ref{x} which is a contradiction.\n\\end{proof}\n\n\\begin{Theorem}\\label{y}\nLet $A$ finite dimensional algebra over a field $K$. There is a bijection between the following sets.\n\n(1) The set of maximal green sequences of torsion classes in $modA$;\n\n(2) The set of maximal green sequences of $\\tau$-tilting pairs in $modA$.\n\\end{Theorem}\n\\begin{proof}\nLet $0=\\mathcal{T}_0\\,\\lessdot\\, \\mathcal{T}_1\\, \\lessdot\\, \\mathcal{T}_2\\, \\lessdot \\,\\dots\\,\n                             \\lessdot\\, \\mathcal{T}_m=modA$\nbe a maximal green sequences of torsion classes in $modA$. By [Proposition 4.9, \\cite{BST0}], there exists a sequence\n $(M_0, P_0)$, $(M_1, P_1)$, $\\dots$, $(M_m, P_m)$ of $\\tau$-tilting pairs such that  $(M_0, P_0)=(0,A)$,  $(M_m, P_m) = ( A, 0)$, $(M_i, P_i)$  and $(M_{i+1}, P_{i+1})$ are a mutation with the exchange pair $(X_i, Y_i)$ such that $\\mathcal{T}_i =FacM_i$. It is enough to show that the mutation from $(M_i, P_i)$  to $(M_{i+1}, P_{i+1})$ is green for each $1\\leq i \\leq m-1$, which follows from Corollary \\ref{c}.\n\n On the other hand, let the sequence\n $(M_0, P_0)$, $(M_1, P_1)$, $\\dots$, $(M_m, P_m)$ be  a maximal green sequences of $\\tau$-tilting pairs in $modA$. By Corollary \\ref{c}, we have that $FacM_i \\lessdot FacM_{i+1}$ and it is clear that $0=FacM_0 \\lessdot FacM_1 \\lessdot \\dots \\lessdot FacM_m = modA$ is a maximal green sequences of torsion classes in $modA$.\n\\end{proof}\n\nSince dimension vectors of bricks appeared in a CFHO can be given via $c$-vectors by Theorem \\ref{y} and Lemma \\ref{x}, the crossing inequalities can be established by $c$-vectors.   Following these results, we obtain the diagram of the relationship among {\\em $c$-vectors} and the concepts concerned in this paper as below, where an arrow $ A \\rightarrow B$ means that $B$ can be constructed from $A$.\n\\[\\mathord{\\begin{tikzpicture}[scale=1.3,baseline=0]\n\\node at (0,0) (0) {MGS  of torsion classes};\n\\node at (0,1) (1) {MGS  of $\\tau$-tilting pairs};\n\\node at (5,0) (2) {Corresonding sequences of $c$-vectors};\n\\node at (5,1) (3) {Crossing inequalities};\n\\node at (5,2.7) (4) {Stability functions};\n\\path[-angle 90]\n    (0) edge   (2)\n\t(2) edge   (0)\n    (0) edge   (1)\n\t(1) edge   (0)\n         (2) edge   (3)\n         (3) edge  node [right] {if solutions exist} (4)\n         (4) edge   (1);\n\n\\end{tikzpicture}}\\]\n\n\n\n\\subsection{Rotation lemma for Jacobian algebras}\\label{4.2}\nIn this section, all quivers are assumed to be a directed graph with finite arrows and vertices. We refer to the article \\cite{DWZ}  for basic concepts and properties of quiver with potentials and Jacobian algebras.\nLet $(Q,w)$ be a non-degenerated quiver with potential, and $J(Q, w)$ be the corresponding Jacobian algebra. In this section, we always assume that\nJacobian algebras are finite dimensional. For each vertex $k$, let $(Q',w') = \\mu_k(Q,w)$ be the mutation of $(Q,w)$ at $k$. Let $\\mathcal{A}= modJ(Q,w)$\nand $\\mathcal{A'}= modJ(Q',w')$ be the module categories. Assume that the simple modules of $J(Q,w)$ and  $J(Q',w')$ are given by $S_1, S_2, \\dots, S_n$ and\n$S'_1, S'_2, \\dots, S'_n$. We define the following full subcategories of $\\mathcal{A}$ (and of $\\mathcal{A'}$ accordingly)\n\\[^{\\bot}S_k := \\{X\\in \\mathcal{A}| Hom(X, S_k) =0\\};\\]\n\\[S_k ^{\\bot}:= \\{Y\\in \\mathcal{A}| Hom( S_k, Y) =0\\};\\]\nand denote $\\langle S_k\\rangle$ the full subcategories of $\\mathcal{A}$ consisting of direct sums of $S_k$.\n\nFor every quiver with potential $(Q,w)$ and take any vertex  $k$, we define a matrix $B_k=(b_{ij})$ which is given by $b_{ii} = 1$ for $i\\neq k$, $b_{kk}=0$, $b_{kj}$ equals  to the number of arrows from $k$ to $j$ in $Q$ for $j\\neq k$, and otherwise $b_{vu}=0$.\n \\begin{Theorem}[\\cite{BIRS,Mou}]\\label{eq}\nThere are two additive functors (also called the generalized reflection functors) $F_k^+: \\mathcal{A} \\rightarrow \\mathcal{A}'$ and\n$F_k^-: \\mathcal{A'} \\rightarrow \\mathcal{A}$ satisfy the following properties.\n\n(1) $F_k^+$ is right exact and  $F_k^-$ is left exact. They are adjoint to each other.\n\n(2) $F_k^+(\\langle S_k\\rangle)=0$ and $F_k^-(\\langle S'_k\\rangle )=0$.\n\n(3) $F_k^+(S_k ^{\\bot})\\subset {^{\\bot}S'_k}$ and $F_k^-(^{\\bot}S'_k)\\subset S_k ^{\\bot}$.\n\n(4) The restrictions $F_k^+:S_k ^{\\bot}\\rightarrow {^{\\bot}S'_k}$ and $F_k^-: {^{\\bot}S'_k}\\rightarrow  S_k ^{\\bot}$ are inverse equivalence.\nMoreover these equivalences preserve short exact sequences.\n\n(5) If $V\\in S_k ^{\\bot}$, then $\\underline{dim}F_k^+V = B_k\\underline{dim}V$, where $\\underline{dim}V$\nis the dimension vector of $V$ in $\\mathcal{A}$ with respect to the basis $[S_1],\\dots,[S_n]$ and  $\\underline{dim}F_k^+V$\nis the dimension vector of $F_k^+V$ in $\\mathcal{A}'$ with respect to the basis $[S'_1],\\dots,[S'_n]$.\n\\end{Theorem}\n\n Now we give the Rotation Lemma for finite dimensional Jacobian algebras, which is a generalization of Igusa's result in \\cite{Ig2}, which is the Rotation Lemma for a quiver with potential $(Q,w)$ where $Q$ is mutation equivalent to some quiver of finite type in the sense of \\cite{FZ2}.\n\n\\begin{Theorem}\nWith the assumptions and notations above. If $N_1, N_2, \\dots, N_m$ is a complete forward Hom-orthogonal sequence in $\\mathcal{A} = mod J(Q,w)$ with\n$N_1 = S_k$, then $F_k^+(N_2), \\dots, F_k^+(N_m), S'_k$ is a complete forward Hom-orthogonal sequence in $\\mathcal{A'} = mod J(Q',w')$.\n\\end{Theorem}\n\\begin{proof}\nBy Theorem \\ref{eq}, it is obvious that $F_k^+(N_2), \\dots, F_k^+(N_m), S'_k$ are bricks.\n\nSince for $2\\leq i \\leq m$, $N_i\\in S_k^{\\bot}$, then  $F_k^+(N_i)\\in ^{\\bot}S'_k$, i.e., $Hom(F_k^+(N_i), S'_k) = 0$. We also have $Hom_{\\mathcal{A'}}(F_k^+(N_i), F_k^+(N_j))= Hom_{^{\\bot}S'_k}(F_k^+(N_i), F_k^+(N_j)) = Hom_{S_k^{\\bot}}(N_i, N_j) = 0$ for $2\\leq i <j \\leq m$.\n\nIn the following, we show that $(F_k^+(N_2)\\oplus \\dots \\oplus F_k^+(N_m)\\oplus S'_k )^{\\bot} = 0$. To prove this, we claim that if $S$ is a simple module in $\\mathcal{A'}$, then $S$ is isomorphic one of modules $F_k^+(N_2), \\dots, F_k^+(N_m), S'_k$.\n\nLemma: If $S \\notin \\{F_k^+(N_2), \\dots, F_k^+(N_m), S'_k\\}$ (up to isomorphism), then $Hom(F_k^+(N_i), S) = 0$ for $2 \\leq i \\leq m$ and $Hom(S'_k, S) = 0$.\n\nProof: To make the proof more clear, we denote $F_k^+(N_2), \\dots, F_k^+(N_m), S'_k$ by $M_1, M_2, \\dots, M_m$. If it is not true, we may take the most minimal $i$ from $\\{1,2,\\dots, m-1\\}$ such that $Hom(M_i,S) \\neq  0$. In this case, we have that $Hom(S,M_i) =  0$. Otherwise we have $S \\simeq M_i$. Now we have that $Hom(M_p,S)= 0$ for $p<i$ and\n$Hom(S,M_i) =  0$. Take a nonzero morphism $0 \\neq f : M_i \\rightarrow S$, then for any morphism $g: S\\rightarrow M_j$ for $j>i$, we have that $gf=0$ since $Hom(M_i,M_j)=0$ and hence $g=0$ because $f$ is an epimorphism. Therefore $Hom(S, M_j) = 0$ for $j>i$. Note that $S\\in {^{\\bot}S'_k}$ (since the simple module $S$ is not isomorphic to $S_k$) and $F_k^+:S_k ^{\\bot}\\rightarrow ^{\\bot}S'_k$ is an equivalence, then\nthere is a module $R\\in S_k ^{\\bot}$ such that $S=F_k^+(R)$ and it is clear that $R$ is a brick. Consider the sequence $N_1=S_k$, $\\dots$, $N_i$,$R$,$N_{i+1}$, $\\dots$,$N_m$, we have that \\[R\\in S_k ^{\\bot},\\,\\, i.e., \\,\\,Hom(N_1, R) = 0;\\]\n\\[for\\,\\, 2\\leq p \\leq i,\\,\\, Hom_{\\mathcal{A}}(N_p, R) = Hom_{S_k ^{\\bot}}(N_p, R) = Hom_{^{\\bot}S'_k}(F_k^+(N_p), S) = 0;\\]\n\\[for\\,\\, i+1\\leq p \\leq m,\\,\\,Hom_{\\mathcal{A}}(R, N_p) = Hom_{S_k ^{\\bot}}(R,N_p) = Hom_{^{\\bot}S'_k}(S,F_k^+(N_p)) = 0.\\]\nThis contradicts with the fact that $N_1, N_2, \\dots, N_m$ is a complete forward Hom-orthogonal sequence in $\\mathcal{A}$. Thus the lemma is true.\n\nNow we can prove our claim that if $S$ is a simple module in $\\mathcal{A'}$, then $S \\in \\{F_k^+(N_2), \\dots, F_k^+(N_m), S'_k\\}$ (up to isomorphism).\nIndeed if $S \\notin \\{F_k^+(N_2), \\dots, F_k^+(N_m), S'_k\\}$ (up to isomorphism), by the Lemma above, $Hom(F_k^+(N_i), S) = 0$ for $2 \\leq i \\leq m$ and $Hom(S'_k, S) = 0$.\nNote that $S$ is simple and $S$ is not isomorphic to $S'_k$, we also have that $S\\in ^{\\bot}S'_k$. Thus there is a module $M\\in S_k^{\\bot}$ such that $S = F_k^+(M)$.\nThen for any $i \\geq 2$, we have that\n \\[0 = Hom_{\\mathcal{A'}}(F_k^+(N_i), S) =  Hom_{\\mathcal{A'}}(F_k^+(N_i), F_k^+(M)) = Hom_{^{\\bot}S'_k}(F_k^+(N_i), F_k^+(M))=  Hom_{S_k^{\\bot}}(N_i,M).\\]\nThus $M\\in (N_1\\oplus N_2\\oplus \\dots \\oplus N_m)^{\\bot} = 0$, which is a contradiction. The claim is true. By the claim, it is easy to see that $(F_k^+(N_2)\\oplus \\dots \\oplus F_k^+(N_m)\\oplus S'_k )^{\\bot} = 0$.\n\nNow we can show that no other bricks can be inserted into the sequence  $F_k^+(N_2), \\dots, F_k^+(N_m), S'_k$ preserving the forward Hom-orthogonal\nproperty. Indeed, suppose that there is a brick $M$ in $\\mathcal{A}'$ satisfying the condition. Note that $M$ can not be the last one because we have proved that $(F_k^+(N_2)\\oplus \\dots \\oplus F_k^+(N_m)\\oplus S'_k )^{\\bot} = 0$. Then we may assume that the new sequence satisfying the forward Hom-orthogonal\nproperty is  $F_k^+(N_2), \\dots,F_k^+(N_i),M, F_k^+(N_{i+1}),\\dots, F_k^+(N_m), S'_k$ and  $M\\in ^{\\bot}S'_k$. Note that there exists a module $H \\in S_k^{\\bot}$ such that $M=F_K^+{H}$ and $H$ is a brick. It is easy to see the sequence $N_1, \\dots, N_i, H, N_{i+1}, \\dots, N_m$ satisfies the forward Hom-orthogonal\nproperty, which is a contradiction. Thus $F_k^+(N_2), \\dots, F_k^+(N_m), S'_k$ is a complete forward Hom-orthogonal sequence in $\\mathcal{A'}$.\n\n\n\\end{proof}\n\n\\begin{Corollary}\nLet  $N_1, N_2, \\dots, N_m$ is a complete forward Hom-orthogonal sequence in $\\mathcal{A} = mod J(Q,w)$ with\n$N_1 = S_k$. If there are two vectors $\\alpha, \\beta\\in \\mathbb{R}^n$ such that $B_k\\beta \\in \\mathds{R}_{>0}^n$ and\n \\[\n \\frac{\\langle\\alpha,N_2\\rangle}{\\langle\\beta,N_2\\rangle} <\\dots <\\frac{\\langle\\alpha,N_{m}\\rangle} {\\langle\\beta,N_{m}\\rangle}<\\frac{\\langle\\alpha,N_{1}\\rangle} {\\langle\\beta,N_{1}\\rangle},\n \\]\n Then the maximal green sequence $F_k^+(N_2), \\dots, F_k^+(N_m), S'_k$ in $\\mathcal{A'} = mod J(Q',w')$ can be induced by the central charge\n $Z: K_0(\\mathcal{A}') \\rightarrow \\mathbb{C}$\nwhich is given by \\[Z(X) = \\langle B_k^T\\alpha, [X]\\rangle + \\mathrm{i}\\langle B_k^T\\beta, [X]\\rangle.\\]\n\\end{Corollary}\n\\begin{proof}\nBy Theorem \\ref{main}, it is enough to prove that\n\\begin{equation}\\label{z}\n \\frac{\\langle B_k^T\\alpha, F_k^+N_2\\rangle}{\\langle  B_k^T\\beta, F_k^+N_2\\rangle} <\\dots <\\frac{\\langle  B_k^T\\alpha, F_k^+N_{m}\\rangle} {\\langle B_k^T\\beta,F_k^+N_{m}\\rangle}<\\frac{\\langle  B_k^T\\alpha,S'_k\\rangle} {\\langle  B_k^T\\beta, S'_k\\rangle}.\n\\end{equation}\n\nBy Theorem \\ref{eq}(5), for $2 \\leq i \\leq m$, since $B_k^2=I$, we have that\n\\[\n \\frac{\\langle B_k^T\\alpha, F_k^+N_i\\rangle}{\\langle  B_k^T\\beta, F_k^+N_i\\rangle} = \\frac{\\langle B_k^T\\alpha, B_kN_i\\rangle}{\\langle  B_k^T\\beta, b_kN_i\\rangle}\n =\\frac{(\\underline{dim}N_i)^T (B_k^T)^2\\alpha}{(\\underline{dim}N_i)^T (B_k^T)^2\\beta} = \\frac{(\\underline{dim}N_i)^T \\alpha}{(\\underline{dim}N_i)^T \\beta} = \\frac{\\langle\\alpha,N_i\\rangle}{\\langle\\beta,N_i\\rangle}.\n \\]\n\nOn the other hand, by the definition of $B_k$, we have that\n\\[\n\\frac{\\langle  B_k^T\\alpha,S'_k\\rangle} {\\langle  B_k^T\\beta, S'_k\\rangle} = \\frac{a_k}{b_k} = \\frac{\\langle\\alpha,N_{1}\\rangle} {\\langle\\beta,N_{1}\\rangle}.\n\\]\nIt is clear that the inequalities (\\ref{z}) hold.\n\\end{proof}\n\n\n\\vspace{5mm}\n{\\bf Acknowledgements:}\\; This project is supported by the National Natural Science Foundation of China (No.11671350) and the Zhejiang Provincial Natural Science Foundation of China (No.LY19A010023).\n\n\\vspace{10mm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:motivation} Motivation}\n\nThe aim of high-energy physics (HEP) is to understand matter at the most fundamental level. The current understanding of the building blocks of the universe and their interactions is embodied in the so-called standard model (SM) of particle physics. In this model, fundamental particles are quantum mechanical entities with fixed quantum numbers such as mass, spin, electric, and color charge, and are regarded as excitations of relativistic quantum fields. While the field of HEP has enjoyed many decades of progress, a long-standing challenge for both theorists and experimentalists in HEP is that the properties and behaviors of perturbative quantum fields are mathematically complex, present infrared and ultraviolet divergence challenges, and expansions converge slowly when the coupling constant is large. Furthermore, the time evolution of final state particles and their hadronization is particularly complex and are difficult to accurately simulate on digital computers, even on high performance computer (HPC) systems. \n\nMeanwhile, a new approach to computing that explicitly leverages the complexities of quantum mechanics for computational purposes has been in development. In the last few years, prototype quantum computers capable of performing small computations using a few dozen quantum bits (``qubits'') have become widely accessible, prompting a burgeoning field of research in applications of near-term and future quantum computers.\nQuantum computing (QC) approaches have been developed and demonstrated for applications in many disciplines including chemistry, materials science, nuclear physics, and HEP. However, the field of QC is far from mature and the potential of QC for HEP has only begun to be explored.\n\nIn this paper, we highlight some of the significant opportunities and challenges for the development of quantum computing systems and associated software to deliver unprecedented capabilities for HEP discovery science.\nIn Sec.~\\ref{sec:background}, we survey current approaches to applying QC to HEP.\nIn Sec.~\\ref{sec:challenges}, we describe the leading technical challenges and gaps facing the adoption of quantum computing for HEP research problems, while in Sec.~\\ref{sec:prior}, we respond with long-term research priorities that will address these gaps in the coming decade.\n\n\\section{Background}\n\\label{sec:background}\n\nQuantum computing may be viewed as part of the larger subject of quantum information science and technology (QIST), which concerns the use of quantum systems to store, transmit, and process information \\cite{humble2019quantum}.\nQIST involves the precise isolation and control of well-defined quantum particles and fields.\nThese quantum physical systems can be used to simulate naturally occurring quantum phenomena (such as quantum chromodynamics (QCD)) or to solve computational tasks unrelated to physical phenomena (such as clustering or regression on a dataset).\nAs will be discussed later in this section, key concepts in quantum computing such as superposition, entanglement, and computational complexity have also provided new, information-oriented ways to address fundamental questions in HEP.\n\nFor the most part, efforts to apply QC in HEP to date have consisted of adapting established general-purpose quantum algorithms and tools to HEP-specific tasks. While most demonstrations have involved only toy problems, there have been a few demonstrations in which the quantum computations were competitive with more traditional methods. Some examples include the simulation of a full SU(2) gauge theory on a noisy, intermediate-scale quantum (NISQ) device \\cite{Atas2021SU2}, the training of large-scale quantum machine learning (QML) models for supernova classification \\cite{Peters2021}, and the generation of synthetic detector data \\cite{delgadoHamilton2022unsupervised}. \n\n\\par \nIn the United States, much of the work to date exploring the potential of quantum information science for HEP has been through the Department of Energy (DOE) QuantISED program. QuantISED was launched in 2018 as part of the DOE Office of Science (SC) QIS initiative following a series of community roundtables, pilot studies, and \n\\cite{DOESensorsReport,GrandChallengesReport}. The program has relied upon interdisciplinary collaboration between HEP and QIS researchers and partnerships between DOE laboratories, university consortia, and hardware vendors such as IBM, Rigetti, and Google. Major topic areas addressed by QuantISED include:\n\n\\begin{itemize}\n  \\item \\textbf{Cosmos and qubits:} theoretical connections between cosmic physics and qubit systems that can be studied in laboratory settings. In particular, the study of quantum gravity in the context of quantum information has yielded new insights. For example, principles of computational complexity and quantum error correction have helped scientists investigate the interiors of black holes \\cite{Kim2020} and whether gravity emerges from entanglement \\cite{Periwal2021}.\n  \n  \\item \\textbf{Foundational theory:} formulations of gauge theories amenable to simulation on quantum computers \\cite{PhysRevD.104.094519}.\n  \n  \\item \\textbf{Quantum simulation:} algorithms for simulating quantum mechanical systems including scalar quantum field theories \\cite{yeter2019scalar}, nuclear physics \\cite{yeter2020practical}, and other many-body systems \\cite{yeter2021scattering}.\n  \n  \\item \\textbf{Quantum computing:} quantum-enhanced machine learning and data analysis for HEP experiments. Developments in this area include: quantum machine learning algorithms \\cite{Peters2021, 2021Wu, 2021WuSunGuan} and optimization algorithms for the analysis of collider data and quantum simulation of field theories \\cite{QASimNachman2021}.\n  \n  \\item \\textbf{Quantum sensors:} sensors leveraging quantum information concepts and technologies to yield new detection capabilities. One advancement in this area is quantum-enhanced searches for dark matter \\cite{QISforAxions, PhysRevLett.126.141302}.\n  \n  \\item \\textbf{Quantum technology:} advances in QIST and its integration with HEP technology. Efforts in this topic include quantum internet \\cite{PRXQuantum.1.020317}, noise mitigation techniques for near-term quantum devices \\cite{UnfoldingNachman2020}, analysis of the technological requirements for HEP programs \\cite{PhysRevD.103.054507}, and others.\n\\end{itemize}\n\n\nWhile there is not space to describe all of these efforts in detail, recent work relating black hole thermodynamics and quantum information theory exemplifies the kinds of fruitful connections being developed between QIS and HEP. \nA long-standing problem in particle physics is the so-called ``black-hole information paradox'', which concerns the apparent loss of information that occurs when a particle disappears inside a black hole. The particle's information chaotically mixes with all the other matter and information inside the black hole, generating quantum entanglement between distant regions and seemingly making it impossible to retrieve. Recent landmark work \\cite{Almheiri2019,Penington2020} suggests that the information passing the event horizon of a black hole is not lost forever but is eventually released.\n\nIn the field of quantum information, this process of apparent information loss through widespread entanglement is known as scrambling. A quantum processor can directly measure the scrambling-induced spread of initially localized information via the decay of out-of-time-ordered correlation (OTOC) functions. In \\cite{blok_quantum_2021}, the authors realize scrambling behavior in a multi-qutrit system. In \\cite{Landsman2019}, a seven-qubit circuit executed on an ion-trap quantum computer enabled the authors to bound the scrambling-induced decay of the OTOC measurement experimentally.\nThese two experiments pave the way for laboratory investigations of the black hole information paradox, something that would hardly be imaginable without QC. \n\n\\section{\\label{sec:challenges} Challenges}\nThe emerging field of QC faces many technical challenges related to the development of algorithms, software, hardware, and infrastructure to support HEP research. In this section, we identify how these existing research challenges influence on-going efforts to solve key problems in HEP research using QC. In Sec.~\\ref{sec:ana}, we survey algorithms and applications which are known to apply to HEP challenge problems, and we identify several key gaps in extending these methods beyond proof of principle efforts. In Sec.~\\ref{sec:sah}, we review how existing computational tools, including quantum programming methods and numerical simulators, have influenced the accessibility and adoption of QC for the HEP community. In Sec.~\\ref{sec:tfh}, we briefly summarize early efforts in developing testbeds for experimental QC with a focus on HEP application areas. Then, in Sec.~\\ref{sec:infra}, we discuss some of the broader infrastructure challenges that face adoption of QC by the HEP community. \n\\subsection{Algorithms and Applications }\n\\label{sec:ana}\nThe Hamiltonian formulation of lattice quantum field theories allows us to use the tools from quantum computation to solve problems in strongly correlated quantum systems by understanding the dynamics in terms of quantum circuits or representing the action of a measurement as a projection in a Hilbert space \\cite{Banuls2020,Buser2021}. Different approaches in the quantum computation domain for quantum field theories cover subatomic many-body physics \\cite{jordan2012quantum,lu2018simulations}, real-time model dynamics of lattice gauge theories \\cite{martinez2016real,muschik2017u}, self-verifying variational quantum simulation of the lattice Schwinger model \\cite{kokail2018self}, Non-abelian SU(2) lattice gauge theories on superconducting devices \\cite{mezzacapo2015non}, optical abelian and simulations of non-abelian gauge theories on ultra-cold atoms platforms \\cite{tagliacozzo2013optical,tagliacozzo2013simulation}, and simulation of Z(2) lattice gauge theories with dynamical fermionic matter \\cite{zohar2017digital,zohar2017digitalA}. \n\\par \nIn most of these applications, the limited number of qubits and gate fidelities available in QC hardware implementations represent bottlenecks for a proper comparison with classical algorithms and, therefore, any assessment of quantum advantage. On the other hand, the application of QCD with the lattice gauge model \\cite{kan2021lattice, stetina2022} represents an exciting challenge on the real-device implementation that requires a novel design following quantum device limitations, state-of-the-art classical simulators handling a decent number of qubits to benchmark the quantum computation algorithms with existing classical approaches, and the exploration of gravity\/QFT duality \\cite{Buser2021}.\n\\par\nFrom an algorithmic point of view, several challenges address the exploration of variational quantum algorithms in a higher dimension, design of quantum algorithms beyond the existing hybrid classical-quantum format with the combination of different strategies to solve problems in HEP, exploitation of quantum singular-value-decomposition (SVD) methods for unfolding problems in HEP, and on-hardware implementation of quantum algorithms to reconstruct Feynman path integrals. \n\\subsubsection{Digital and Analog Simulations}\nInterestingly, digital, analog, and mixed digital-analog models of quantum computation have been considered within the context of lattice gauge theories \\cite{RevModPhys.86.153,PRXQuantum.2.017003}. In the digital realm the problem of a \\textit{gauge-invariant} problem formulation is already a non-trivial matter with quantum-error-correction-like protocols \\cite{stryker2019} and manifestly gauge-invariant formulations \\cite{Davoudi2021,Ciavarella2021} having been considered. After overcoming the first hurdle of problem formulation the ever-growing set of quantum algorithms, including quantum phase estimation and linear combinations of unitaries or block encoding, can be used to compute the various quantities of interest appearing in the target theory \\cite{Roggero2019,Roggero2020, Ciavarella2021}. These algorithms seem suitable for fault tolerant computation rather than near term quantum computational models.\n\nAnalog approaches have long been taken with synthetic gauge fields in cold atom simulators being a prime example of analog simulation \\cite{Galitski2019}. Analog realizations of gauge invariant encodings, such loop-string-hadron encoding in cold atom systems \\cite{Dasgupta2022}, have also appeared. Lastly, with some modest abstraction one popular near term approach has been to leverage the natural device physics in order to augment the conventional digital gate-sets with analog processes \\cite{PhysRevA.104.042602}. For example, select bosonic vibrational modes can be used to simulate $U(1)$ gauge variables in a resource efficient manner \\cite{tong2021provably}, e.g., using trapped ion systems \\cite{nguyen2021digital}.\n\\subsubsection{Variational Methods}\nGenerative modeling learns how to generate a distribution by analyzing a known data. The model itself may then be used to draw additional samples representative of the distribution. With a quantum processor, preparing and sampling from arbitrary distributions can be directly implemented, and there is no need to compute partition functions which may be intractable. Additionally, using quantum systems has an attractive scaling characteristic: a model built on $n$ qubits can potentially prepare and sample from a distribution over $2^n$ states, where each state is a length-$n$ binary bitstring. \n\nIn quantum machine learning there is a broad field of research which includes using classical machine learning algorithms for studying and simulating quantum systems \\cite{gao2017efficient,torlai2018neural,torlai2020machine} and using quantum algorithms to implement analogues of classical machine learning algorithms. Many machine learning applications in HEP leverage generative modeling \\cite{salamani2018deep,di2019dijetgan,alanazi2020simulation,hariri2021graph}, and the leading model is the generative adversarial network (GAN). Quantum GAN models \\cite{dallaire2018quantum,zoufal2019quantum,chakrabarti2019quantum} have been developed that utilize a trainable quantum circuit model for the generator network, yet adversarial training can still encounter mode collapse as with classical models. However, there are other generative models that can be constructed with quantum circuits including quantum Boltzmann machines \\cite{amin2018quantum,zoufal2021variational} and quantum circuit Born machines (and related Ising Born Machines) \\cite{benedetti2019generative,coyle2020born}.\n\nNear-term demonstrations of QML can reach comparable performance with, but have not yet demonstrated an advantage over, classical machine learning models. Classical machine learning and deep learning can utilize large, distributed computing clusters to implement models with millions of parameters. Coupled with parallelization, these training workflows dwarf the current state of the art in quantum machine learning that uses variational, hybrid algorithms. However, as the size and capabilities of available quantum devices improve, larger quantum models can be built and executed. While near-term demonstrations and applications may not outperform classical ML models, their development is essential for understanding how quantum models learn.\n\n\\subsubsection{Noise Mitigation}\nThe design space associated with quantum generative models, and parameterized quantum models in general, is large. Best practices for building parameterized quantum circuit models, in order to effectively prepare and sample from these distributions, remains an open question \\cite{nakaji2021expressibility}. Many heuristic approaches abound including: using sparse parameterizations (single rotation gates instead of arbitrary three angle decompositions) reduces training overhead, at the expense of trainability; and using fewer entangling and two-qubit operations can reduce susceptibility to hardware noise and improve transpilation for sparse hardware connectivity graphs, but at the expense of the correlations that can be modeled. \n\nMitigating hardware noise requires efficient noise characterization \\cite{PhysRevA.103.042603,dahlhauser2022benchmarking}. Many methods of error mitigation that rely on regression modeling (e.g. \\cite{lowe2021unified}) are focused on mitigating the effect of noise on expectation values. For generative modeling and algorithms that use the distribution over all basis states, matrix-based error mitigation methods have been incorporated into variational training workflows \\cite{hamilton2020error}, and scalable error mitigation methods are an active area of research \\cite{hamilton2020scalable,nation2021scalable}. These methods are needed to assess the stability of the quantum devices as well as reproducibility of the applications \\cite{dasgupta2022characterizing}.\n\n\\subsubsection{Quantum Machine Learning}\nModern machine learning (ML) techniques, including deep learning, are rapidly being applied, adapted, and developed for HEP applications. ML is currently used across the different areas of particle physics from collider physics \\cite{Stakia2021, Mikuni2021, Nachman2021} to the study of the cosmos \\cite{Ostdiek2022, List2021} and quantum gravity. ML is also present throughout the various stages of HEP studies from the simulation of hypothesized particles \\cite{brehmer2020simulationbased}, in the parameterization of cross-sections calculated for sensitivity analyses \\cite{Carrazza2021}, to the analysis of experimental data \\cite{Stakia2021,Mikuni2021,Nachman2021}. Most of these applications have replaced non-ML methods and techniques and can process experimental data very efficiently. These methods and techniques have even been deployed in FPGAs \\cite{Iiyama2021} and integrated into the data acquisition systems for fast selection of information. \n\nMore recently, applications at the intersection of quantum computing and machine learning have been explored to analyze experimental data. The hope is to adapt or develop algorithms that can efficiently process HEP data. One example includes algorithms to construct physics objects amenable to analysis from the signals generated in a particle detector--i.e., the clustering of detector hits into so-called tracks for reconstructing a particle's trajectory \\cite{magano2021quantum, Zlokapa2021, Bapst2019, Shapoval2019, das2020track, Tysz2021, 2020Cenk, Quiroz2021} or tracks and calorimeter energy depositions into jets \\cite{Wei2020, pires2020adiabatic, pires2021digital}. Furthermore, quantum-assisted algorithms have been explored in unsupervised learning settings to classify jets according to their origin (b-tagging) \\cite{gianelle2022quantum}, generative tasks \\cite{bravoprieto2021stylebased, Chang2021, perez2021determining}, and the selection of events or interactions along with background suppression \\cite{Mott2017, qamlz, kim2021leveraging, Peters2021, caldeira2020restricted, Belis2021, Terashi2021, AlexiadesArmenakas2021, Bargassa2021, matchev2020quantum, Blance2021, Blance2021a, 2021Wu, chen2020quantum, Heredge2021}. In particular, generative models have been explored extensively as an alternative for the simulation of particle interactions and the detector's response to such interactions \\cite{Chang2021, delgadoHamilton2022unsupervised}.\n\nLeveraging quantum computers for machine learning in HEP has several advantages. In general, quantum computers can operate more efficiently in high-dimensional tensor product spaces because of quantum superposition, which is vital for analyzing large-scale and complex HEP data patterns. Moreover, they can leverage quantum tunneling to train machine learning models more efficiently \\cite{date2021qubo,date2021adiabatic,arthur2021balanced,date2019classical}. In some cases, quantum-assisted techniques have been shown to train on less data than their classical counterparts. This might be particularly useful in the context of data augmentation in simulation tasks, where using a generative model can reduce the resources allocated for traditional ML methods.\n\nNonetheless, even though the applications of QML to HEP data are many, there are several challenges associated with the trainability and deployment of these models on NISQ devices. For example, the size and complexity of HEP datasets requires an efficient encoding into quantum states, resulting in a large overhead in pre-processing. Many training workflows serially encode training data and paralellization requiring either multiple devices or large quantum devices that can run multiple circuits with no cross-talk or interference between circuits. Furthermore, there is little understanding on how to tailor ansatz\/circuit design for specific applications and a need for error correction\/mitigation techniques that can be scalably constructed and incorporated into variational training workflows. \n\\subsection{Software and Compilers}\n\\label{sec:sah}\n\n\\subsubsection{Compiling}\n\nImprovements in quantum computers, and their subsequent enabling of increasingly nontrivial simulations of fermionic systems, have pushed the development of novel encoding of fermionic degrees of freedom. The commonly used mappings from these degrees of freedom to that of spin (i.e., the Jordan-Wigner, Bravyi-Kitaev, or parity mappings) in general lead to largely non-local spin Hamiltonians and unnecessarily large qubit  numbers per fermionic mode mapped. These properties drastically compound circuit and measurement complexity of simulations. Towards countering these expenses, several works have shown both general and system specific mappings that outperform Jordan-Wigner and others in terms of locality of spin Hamiltonians as well as the number of fermionic modes per qubit \\cite{brayvi2002,derby2021}. \n\nAlthough these economic mappings are useful, their utility in the fault-tolerant era may be overshadowed by instead using \\textit{logical} fermionic mappings \\cite{setia2019,landahl2021,li2018,jiang2019}. These avoid the overhead associated with fermionic-to-qubit mapping followed by the cost of error corrected codes by directly mapping the underlying fermionic symmetries to error correcting code spaces. While the benefit is obvious in terms of qubit and gate overhead, such embeddings require couching fundamental fermionic operations in terms of fault tolerant operations, which will likely encourage simple, more physically motivated implementations of algorithms and compiling of code. \n\nExplorations of such mappings, and their relative quantum resource efficiency for gauge-invariant HEP problems of interest remain an open, but important, first step. There have been several successful HEP problem implementations, generally focusing on how to truncate gauge degrees of freedom while simultaneously satisfying gauge constraints efficiently \\cite{klco2021,Davoudi2021}. This has paved the way, similarly to fermionic mappings, for methods of how to embed HEP simulations natively in error detection and correction schemes \\cite{rajput2021,stryker2019}. These methods have so far proven to be very specific to the target models and its symmetries. Thus the endeavor to find general, optimal mappings will be essential in the coming years for scaling up near term HEP simulations into the fault tolerant era.\n\n\\subsection{Experimental Testbeds}\n\\label{sec:tfh}\nThe current state of the art for testbed devices consists of quantum processors that contain dozens of qubits. These machines are capable of circuit depths in the 100s of gates and up to 1000s of gates in the most advanced systems. Alternately, algorithms for simulating HEP problems of interest have been implemented on a comparably smaller number of qubits and circuit depths. \n\n\\par\nGiven the current level of noise and imperfection in practical realizations of quantum computing devices, efficient numerical simulators capable of running on large-scale heterogeneous (HPC) platforms will continue to play an important role in characterization, verification, and validation of quantum devices, as well as analysis of the quantum algorithms intended for those devices. Probing the potential for quantum advantage will require extensive research and development in classical algorithms for quantum computing simulations in order to make a true boundary where quantum wins over classical. \n\\par \nAn ability to  utilize existing and future classical hardware efficiently is another dimension of optimization on the classical computing side. Furthermore, the HEP-related quantum simulators will require generalization to the qudit-based representation of quantum devices and algorithms. In order to better understand and cope with the noise and decoherence, analog-level simulations of open quantum device dynamics will become vital for their performance optimization. These simulations will require significant HPC resources coupled with highly optimized computer codes. Yet another challenge is related to loading classical (or quantum) data to quantum devices--a problem that can also benefit from advanced quantum-inspired classical algorithms coupled to large-scale HPC resources. \n\n\\subsection{Instrument and Data Networks}\n\\label{sec:infra}\n\nHistorically, HEP computing has been performed on sizeable, purpose-built computing systems. These began as single-site computing facilities but have evolved into the distributed computing grids that we use today. Therefore, integration with current heterogeneous computing resources is needed to accelerate the adoption of quantum computing technologies within the HEP community. For example the Fermilab HEP cloud is a portal to diverse computing resources on local clusters, campus farms, grid resources, commercial clouds, HPC centers, and quantum computing resources.\n\n\\section{\\label{sec:prior} Priorities}\n\\subsection{Algorithms and Applications}\n\\label{sec:ana_prior}\nWith ever increasing numbers of qubits into the hundreds or thousands, while remaining in the NISQ era, quantum algorithm research for HEP applications to improve\/overtake classical methods is an important research focus for the next ten years. Quantum computing for HEP provides a great challenge and important intellectual stimulus for overcoming practical quantum information challenges with big data. \nIn addition, quantum computing for simulating lattice gauge theories provides both motivation and a framework for interdisciplinary research towards developing special-purpose digital and analog quantum simulators, and ultimately scalable universal quantum computers.\n\nAnother potential area that may yield quantum advantage involves processing the large data sets resulting from  detected events.\nFor instance, simulations and on-hardware implementations for the unfolding problem can exploit both quantum generative modeling as well as singular value decomposition. \nRecent advances also enabled light-front encoding of relativistic fields and subsequent simulation with variational methods~\\cite{kreshchuk_quantum_2020,kreshchuk_light-front_2021}. Encoding techniques allow for large simulations to be mapped onto relatively few qubits via the Lie algebraic method to reduce simulation complexity via gauge invariant and sub-sector encodings pertaining to a symmetry~\\cite{klco_quantum-classical_2018}, while computing the quantum gradient on the Reimannian manifold allows for increased circuit depths when complexity reductions have been maximized. Approaches beyond the current hybrid quantum-classical paradigm, which combine efficient quantum algorithms with classical distributed algorithms, may also benefit HEP applications.\n\n\\subsubsection{Variational Algorithms}\nOne advantage to using quantum models for statistical learning tasks is that with $n$ qubits, one can build a parameterized model that can efficiently prepare arbitrary distributions over $2^n$ states. The development of QML models for HEP applications can provide insight into how data can be efficiently transformed in high-dimensional probability spaces. The challenges for these approaches to quantum computing can be organized into two areas: designing circuit ansatz models that can efficiently transform data into high-dimensional quantum Hilbert spaces, and building scalable training methods that can optimize these models in noisy landscapes. \n\nMany state of the art quantum algorithms in the HEP application space that can be deployed on near-term hardware are variational algorithms: these are hybrid workflows that utilize quantum and classical processors to train parameterized quantum models. Training a variational model is a non-convex optimization problem, and the training task can be NP-hard \\cite{bittel2021training}. In classical machine learning, gradient-based training methods can be scaled to large-scale applications in non-convex and convex optimization, and efficient training of deep learning models has been facilitated through the use of backpropagation. Analogous methods for variational workflows have been developed \\cite{verdon2018universal,beer2020training}, but have not been demonstrated yet on hardware. \n\nCurrently, gradient-based and gradient-free optimization are the leading approaches to training. Gradient-based training is predicted to improve convergence \\cite{napp2021gradient}, yet developing gradient-based optimization routines that can effectively incorporate queue-based access to quantum hardware should be a priority for near-term research. As the number of parameters in a circuit increases, and the number of qubits in the register increases, effective scheduling of the experiments that are needed in order to execute one step of an optimization method is needed. Hardware access which allows for job batching (where a set of experiments sent by a single user are executed and results are returned as a single group) can facilitate efficient training. With batched jobs, it may be possible to execute all circuits needed for one optimization step, either evaluation of the loss function gradient (e.g. for gradient descent \\cite{liu2018differentiable,mitarai2018quantum,hamilton2019generative}), executing a direct search (e.g. for particle swarm \\cite{zhu2019training} or coordinate descent \\cite{delgadoHamilton2022unsupervised}), or for gradient approximation (e.g. using finite differences or SPSA \\cite{spall1992multivariate}). Additionally, a larger number of shots are needed to efficiently sample from $n$-qubit states. \n\n\\subsubsection{Noisy Algorithms}\nThe challenges of effective state preparation, circuit training in noisy landscapes are not mutually exclusive. One observed effect of hardware noise is the loss landscape flattening and the existence of ``barren plateaus'' \\cite{mcclean2018barren,wang2021noise}. In a flattened landscape, the efficacy of gradient-based training will be reduced. However, the impact of barren plateaus can be mitigated with circuit ansatz and cost function design \\cite{cerezo2021cost,pesah2021absence,uvarov2021barren}.\n\nThere are a diverse set of error mitigation methods developed for short depth circuits \\cite{temme2017error}. But highly expressible variational models may not be short depth and variational training is not guaranteed to prepare well-verified states. An alternate approach to error mitigation is to use matrix-based methods where sparse linear filters are used to reduce spurious counts in prepared states. Matrix-based error mitigation methods can be incorporated into variational training methods as a data post-processing state, as the added cost of preparing the linear filter.  \n\n\\subsubsection{Machine Learning Algorithms}\n\nThe known challenges for quantum machine learning (QML) in the HEP application space can be organized into three areas: embedding data into quantum Hilbert spaces, training quantum models in noisy landscapes, and working with quantum data. Currently, the capabilities in classical distributed computing make it unlikely that a quantum advantage will be observed with classical data, or classical representations of quantum data, using QML models deployed on near-term hardware with < 100 qubits. However, the development of QML models for HEP applications can provide insight into how data can be efficiently transformed in high-dimensional probability spaces. \n\nThese three challenges are not mutually exclusive. For example the challenges of training quantum models are encountered irrespective of the data source (classical data, classical representation of quantum data, or quantum data). Examples of how challenges of embedding data can manifest: finding low-dimensional representations of classical data into qubit registers, or high-fidelity transmission of quantum data from sensors to qubits used for QML. \nIn addition to these, it is important in QML to explore purely quantum approaches for both training and inferencing, i.e. approaches that are not hybrid in nature (such as variational approaches).\nWorking with the NISQ-era machines in the near-term, it is also important to explore quantum error-correction and error-mitigation subroutines specific to QML circuits used in HEP.\nFrom a logistical point of view, accessing quantum computers requires better scheduling policies and remote access protocols.\nLastly, today's quantum computers are noisy and error-prone---this severely limits our ability to work with large-scale data reliably.\nIn this regard, it is important to build fault tolerant quantum computers that are also scalable.\n\n\\subsection{Software and Compilers}\n  \nAs discussed in Sec.~\\ref{sec:ana} and Sec.~\\ref{sec:ana_prior}, hybrid quantum algorithms, such as those based on variational principles, have emerged as a promising candidate for quantum computational advantage in HEP areas, especially in the Noisy Intermediate-Scale Quantum (NISQ) era~\\cite{preskill_nisq}. The iterative nature of these algorithms necessitates a tight integration of quantum hardware with classical computing resources. For instance, the execution time of quantum circuits that involves a variational parameter loop has been shown to be dominated (more than 90\\%~\\cite{ibm_clops}) by classical computing procedures for compilation, control, and data transfer. As quantum computing capabilities mature, software infrastructures for compilation and control will necessarily need to improve in order to utilize the full computing power of heterogeneous quantum-classical resources for handling practical applications.\n\nWe envision the need for system-level and hardware-agnostic compiler toolchains akin to those we have today for classical computing - extensible, modular systems with unified intermediate representations that enable a wide array of optimization and code-generation techniques for multiple source languages or target architectures \\cite{McCaskeyICRC2018}. The most important aspect of any compilation toolchain design is the Intermediate Representation (IR), which is how code is represented in the compiler. A standardized, robust, and forward-looking quantum-classical IR is, therefore, an essential step towards ensuring full interoperability within the quantum software ecosystem, promoting contributions from broad communities, and enabling optimization and transformation of quantum programs that we discuss in detail in the later point.\n\nFor instance, multiple community-led efforts~\\cite{qedc_web, qir_alliance_web} in the field have coalesced around the idea of leveraging existing classical compiler infrastructures, like the LLVM, for quantum computing. The approach ensures the low-level coupling of quantum computation with classical computing resources and also supports integration with classical languages and tools readily available in those compiler toolchains. For example, the Quantum Intermediate Representation (QIR) Alliance~\\cite{qir_alliance_web} is an open-source community within the Linux Foundation focusing on enabling an LLVM-based IR specification for quantum computation (QIR), as well as developing tooling around this unified representation. Oak Ridge National Laboratory is a founding member of this organization along with several industry players, including Microsoft, Quantinuum, Rigetti, and Quantum Circuits Inc. Importantly, by leveraging LLVM for quantum compilations we could also benefit from its powerful tools, especially the Multi-Level Intermediate Representation (MLIR)~\\cite{lattner_mlir} library, which is designed for heterogeneous hardware and domain-specific languages analogous to that of quantum-classical computing paradigm. In this regard, we have successfully demonstrated a prototype compiler toolchain, QCOR~\\cite{qcor}, capable of compiling the novel OpenQASM version 3~\\cite{openqasm3} quantum programming language to LLVM IR adhering to the QIR specification adopting the multi-stage, progressive lowering approach of MLIR. This compilation technology is specifically relevant to many HEP application use cases whereby the algorithmic procedure is best captured by high-level domain-specific descriptions, such as analog and digital quantum simulation. \n\nAn essential feature of compiler toolchains is the ability to apply transformations on the IR to improve execution quality, e.g., code's runtime or the computing resources required, while preserving the semantics of the input program. In compiler technology, such transformations are often called \\emph{passes} since they can be assembled into a pipeline of passes each of which processes the IR and applies its transformation rule.\n\nIn the NISQ era of quantum computing, compiler passes that implement circuit simplification and efficient hardware topology mapping are indispensable to any quantum software stack. A broad variety of techniques have been developed quantum circuit simplifications at the gate level, such as those based on ZX-calculus~\\cite{van_zx, zx_opt}, template-based~\\cite{template_opt} or peep-hole~\\cite{nam2018automated} optimizations. Similarly, various circuit-to-hardware placement methods have been developed to minimize the number of qubit swapping operations~\\cite{staq, sabre} required or to maximize circuit execution fidelity by mapping the circuit to best-performing qubits~\\cite{triq}. \n\nFor dynamical Hamiltonian simulation algorithms that are particularly relevant to HEP applications, techniques such as advanced term splitting~\\cite{Hatano2005} and ordering~\\cite{childs2019faster, tranter2019ordering} or algebraic circuit compression~\\cite{camps2021algebraic} can also be incorporated as optimization passes targeting the high-level description of the quantum algorithms where algorithmic semantics of the program are expressed. In this respect, a quantum compiler infrastructure that supports a hierarchy of abstractions, such as the MLIR, is highly desirable since it can retain high-level algorithmic information in the IR which compiler optimizers can reason \\cite{nguyen2021quantum}.  \n\nAs quantum computers evolve into fault-tolerant machines, the associated software infrastructure will also need to be upgraded in order to handle quantum error correction protocols \\cite{humble2021quantum}. Specifically, we envision that quantum programs would always be expressed in terms of logical operations whose corresponding physical qubit level instructions incorporating a target-dependent quantum error correction code are compiled by a compiler software infrastructure \\cite{britt2017high}. In this regard, quantum compilers leveraging classical computing technology, such as LLVM, would be best suited to performing this logical-physical transformation thanks to their ability to incorporate a variety of classical computing resources required by the error decoding and correction protocol, e.g., minimum weight perfect matching~\\cite{dennis2002topological} or maximum likelihood decoding~\\cite{bravyi_mld} for surface code~\\cite{fowler_surface_code}. Furthermore, as quantum programming languages move towards this fault-tolerant paradigm whereby qubit measurement-based control flow is intertwined with that of conventional computing for classical data, we can leverage the vast amount of IR optimization capability from the LLVM infrastructure to handle common classical optimization passes like function inlining or loop unrolling. While these optimization passes are intrinsically classical, they help unlock many quantum-related simplification patterns which would otherwise be obscure to the quantum optimizer due to the complex nature of the IR tree with control flows.\n\\subsubsection{Encodings}\nThe HEP-specific encodings discussed in section II are an important first step to solving gauge-invariant HEP problems of interest in a resource-efficient way.\nExplorations of such mappings, and their relative quantum resource efficiency for gauge-invariant HEP problems of interest remain an open, but important first step. There have been several successful HEP problem implementations, generally focusing on how to truncate gauge degrees of freedom while simultaneously satisfying gauge constraints efficiently \\cite{klco2021,Davoudi2021}. This has paved the way, similarly to fermionic mappings, for methods of how to actually embed HEP simulations natively in error detection and correction schemes \\cite{rajput2021,stryker2019}. These methods have so far proven to be very specific to the target models and its symmetries. Thus the endeavor to find general, optimal mappings will be essential in the coming years for scaling up near term HEP simulations into the fault tolerant era. Comparing these methods to general fault tolerant encodings for universal QC will yield insight into the efficacy of building custom machines dedicated to HEP problems which implement custom encodings in hardware rather than reconfigurable devices capable of universal operations.\n\n\\subsubsection{Numerical Simulators}\nSince numerical simulators will continue to play a vital role in quantum device verification and validation as well as in quantum algorithm analysis \\cite{arute2019quantum,villalonga2020establishing}, the relevant research priorities are concerned with scaling numerical simulation techniques towards larger NISQ devices as well as improving their ability to faithfully reproduce the behavior of actual quantum hardware via more accurate noise models, in particular proper modeling of the multi-qubit cross-talk \\cite{mccaskey2018validating}. In turn, devising more accurate noise models will require further research and development in scalable approaches for modeling open quantum system dynamics coupled with either a Markovian or non-Markovian bath. Further research related to simulations of multi-level qudit systems at their native Hamiltonian level will be necessary for assessing their computational power as compared to the qubit-based quantum devices. Pulse-level simulations, optimal quantum control, and quantum gate implementation optimization will form another direction of simulation heavy research and development efforts. More efficient numerical tensor-algebraic techniques coupled with advances in classical machine learning will be required for scalable characterization of larger NISQ devices. On the engineering side, all these newly devised numerical simulation techniques will have to be implemented in an efficient manner in order to fully exploit the computational power of Exascale HPC platforms which are becoming widely available worldwide \\cite{britt2017quantum,doi.org\/10.1049\/qtc2.12024,nguyen2021scalable,humble2021quantum}.\n\n\\subsection{Experimental Hardware}\n\\subsubsection{Testbeds} \nIt is in the interest of the HEP community to build specialized devices to address the above application needs. For example, multiple platforms offer analog-level operations to users (ie, pulse controls). In many scenarios, access to lower level, analog operations allows for higher accuracy in quantum simulation problems. Further, hybrid analog-digital simulations have shown increased accuracy in field theory problems for scattering calculations~\\cite{arrazola_digital-analog_2016}.\nCurrently, HEP programs - especially those in the QuantISED program in the US - utilize testbeds around the world, including the DOE-supported testbeds consisting of trapped ion and superconducting devices, to solve prototypical problems of interest in the field. For example, the information scrambling problem~\\cite{blok_quantum_2021} has been demonstrated on superconducting hardware \\cite{Kreshchuk2021}, while variational methods for solving problems using the light-field encoding has been mapped onto trapped ion platforms \\cite{Echevarria2021}. Uncovering the most useful aspects of the testbeds in these demonstrations will help us discover the most important ingredients for testbeds that support HEP goals. In particular, the \"close to the metal\", analog controls were a defining factor in these demonstrations.\n\nIn the NISQ era, fully analog systems will also be useful for implementing complex Hamiltonians for which digital, prefault-tolerant platforms do not support sufficient circuit depth. Hamiltonians for problems of interest can be mapped to trapped ion systems' motional mode degrees of freedom, or the spin-spin coupling present in neutral atom, quantum dot, or other platforms which support both analog spin-spin coupling and tunable transverse fields. In the future, when digital platforms move closer to fault tolerance, direct fermionic (HEP problem-specific) encodings should be supported in custom testbeds. These encodings ensure that the resources required in a given testbed required to represent a specific HEP problem are far fewer than that required for general purpose, universal quantum simulation. A custom, fault tolerant fermionic code may not be suitable to simulate Boson sampling, for example, but it would be ideal for simulating scattering in fermionic field theories.\n\n\\subsubsection{Analog Simulators}\nWhile digital quantum computers scale to the extent needed for fault tolerant quantum computations, \\textit{analog} quantum simulators are beginning to push state of the art, for example probing topological spin liquids on a programmable device at a scale not previously observed on neutral atom-based simulators \\cite{semeghini_probing_2021,scholl_quantum_2021,bluvstein_controlling_2021}.\nAnalog neutral atomic physics supplements photonic, superconducting, trapped-ion, and many more analog modalities which currently exist. There has also been recent research interest in exploring digital-analog computational modalities (in which hard operations are done in an analog fashion and digital operations play a complimentary supporting role, e.g. providing local basis transformations) and their application to HEP (and related condensed-matter) models as discussed in the previous section. \nDespite this promise, developing current analog methodologies to solve substantially more complex HEP domain problems is a research avenue in its own right. \n\n\\subsubsection{HEP-dedicated Testbeds}\n\nFor HEP, an open and relevant question is whether quantum computers can efficiently simulate quantum field theories (QFT). QFT encompasses all fundamental interactions, possibly excluding gravity. The simulation of QFT has HEP applications related to event generators for quantum chromodynamics (QCD) simulations of nuclear matter. Outside HEP, it might find applications in the simulation of strongly coupled theories and the characterization of the computational complexity of quantum states.\n\nNonetheless, the simulation of quantum field theories is not a trivial task due to the sign problem. In physics, the sign problem is encountered in calculations of the properties of a quantum mechanical system with a large number of strongly interacting fermions or QFTs involving a non-zero density of strongly interacting fermions. Since the particles are strongly interacting, perturbation theory is inapplicable, and one is forced to use brute-force numerical methods. The sign problem is one of the major unsolved problems in the physics of many-particle systems, limiting progress in nuclear physics, preventing the ab initio calculation of properties of nuclear matter, and limiting our understanding of nuclei and neutron stars. In quantum field theory, it prevents the use of lattice QCD \\cite{deforcrand2010simulating} to predict the phases and properties of quark matter \\cite{Philipsen2012}.\n\nAn exciting and promising alternative to the simulation of QFT is based on the idea of adiabatic quantum computation (AQC) \\cite{farhi2000quantum, Albash2018}. In general, AQC is as powerful as universal quantum computation when a non-stoquastic Hamiltonian is used \\cite{Marvian2019OnTC}, which is the case of Hamiltonians dealing with the sign problem. One of the critical aspects of AQC is quantum annealing \\cite{Kadowaki1998}, which is a metaheuristic algorithm to solve combinatorial optimization problems by changing the parameters adiabatically or even non-adiabatically. Measurements on currently available quantum annealers are done only on the standard computational basis \\cite{Johnson2011}; thereby, the Hamiltonian is stoquastic. \n\nThus, a quantum testbed that fully realizes or exploits the power of AQC is needed to show that quantum speedup can be achieved when non-stochastic Hamiltonians relevant to HEP are used. Some models experiencing first-order and second-order phase transitions using non-stoquastic Hamiltonians have been proposed and have demonstrated a quantum speedup \\cite{Ikeda2020}. This natural and efficient way to simulate QFT on a quantum testbed will eliminate the need to reformulate HEP-relevant Hamiltonians (such as QCD) to fit a finite-dimensional Hilbert space amenable to circuit implementations.\n\nFormulating the future of QCD studies as relying on a successful implementation of AQC provides an alternative to collider experiments, where large statistics are needed to understand rare processes. If we can simulate the QCD Hamiltonian we can study some observables in the context of information theory. For example, understanding the probabilistic parton distribution in terms of entangling entropy \\cite{Kuvshinov:2017ncj}. In other words, in a collider experiment we have access to a probability distribution by collapsing the wave function to one of the states, by using a quantum-mechanical simulation we have access to the full density matrix.\n\n\n\\subsection{Infrastructures, Platforms and Data Networks}\n\\subsubsection{Software Development Frameworks, Benchmarks, and Optimisation}\nThe HEP community has developed over the years a broad range of algorithms and methods for different applications, from optimization to simulation and machine learning. Initial work on developing and assessing quantum equivalents of common algorithms have produced valuable insight \\cite{Guan_2021}. However, the performance of NISQ systems is still limited and full advantage is expected only once reasonable fault tolerant devices will become available. To fully exploit the potential of quantum algorithms, software and hardware systems should be designed as part of co-development efforts across the HEP and quantum computing science and engineering communities. Experience in the classical domain shows that innovation in hardware technology, readily available to software and algorithm experts, can boost the development of new software stacks and algorithms that, in turn, produce critical feedback to hardware development in a virtuous cycle. Examples include hardware-aware compiler optimization, design of application-aware hardware architectures (i.e. systems dedicated to accelerating quantum simulations) and so on.\n\\par \nA fundamental component of measuring the progress and impact of quantum algorithms is the establishment of open benchmarking frameworks and platforms where different combinations of algorithms, software and hardware architecture can be tested, and the results published as a means of establishing baselines for further development. Such benchmarks can form the base for the development of common software tools and interfaces with an increasing level of abstraction and platform neutrality as typical today for classical computing resources. \n\\par \nAs an example, in 2021, CERN launched the ABAQUS project with the objective to build an open, easily accessible platform for researchers to become familiar with different quantum computing architectures and tools, produce or get access to benchmarks over different metrics, and build a community database of knowledge of the applicability of quantum algorithms to HEP workloads.\nThere are specific aspects that need to be investigated and studied. For example, there are easier simulations that are only polynomially hard, such as tensor networks with limited entanglement or limited number of non-Clifford gates that scale polynomially in the number of Clifford gates, but exponentially in the number of non-Clifford gates. It needs to be understood how to adapt and employ these simplifications. \n\\subsubsection{Heterogeneous Computing Networks}\nWhen dealing with the quantum computing paradigm, the role of infrastructure deserves a particular attention. Quantum computing should not over time require separate computing infrastructures, application platforms, or languages, but must be integrated into the existing cloud-based physical and software infrastructure of today and enable the deployment of hybrid, heterogeneous workloads where the quantum processing units (QPUs) are part of a broad range of accelerators. Presently classical resources can reliably store, manage and process huge quantities of data, while quantum devices are expected to efficiently explore high-dimensional spaces to extract insights and identify optimal answers.\n\\par\nThe development and deployment of such hybrid infrastructures is currently happening across a series of steps. As current NISQ resources are limited in availability and efficiency, experience is built by simulating quantum algorithms and building noise mitigation strategies on testbeds of classical resources of increasing scale. Since memory requirements double for every qubit added in the simulation and quickly hit an exponential wall, simulation of large scale systems can be performed by means of distributed and parallelised computation across HPC clusters of CPU\/GPU. Novel architectural frameworks to integrate quantum capabilities APIs in cloud architectures are also emerging \\cite{Grossi_2021}.\n\\par\nAt the same time, it is critical to start building paths to architecture optimization of hybrid resources, workload splitting and hybrid algorithms, scheduling, data flows, and circuits transpilation processes. Looking at hybrid models of variational classifiers, reinforcement-learning optimisation algorithms, quantum-classic GAN models for simulation, the infrastructure is evolving rapidly to make the classic and quantum interaction more efficient, defining containerised modules to be tested on distributed HPC environments and loaded on real quantum computers. This will need to be achieved by integrating HPC and quantum computing strategies as part of national and international initiatives.\n\\par \nOne example of how such integration may be addressed is the Quantum Computing User Program (QCUP) at the Oak Ridge Leadership Computing Facility, a US Department of Energy user facility operated by Oak Ridge National Laboratory \\cite{qcup}. QCUP offers access to a diversity of quantum computing systems following merit-based review of user proposal for demonstrate scientific applications. In addition, these users are capable of integrating their quantum computing workflows alongside conventional HPC systems. As quantum computing technology matures, this setting will offer a natural means by which to integrate quantum computing and conventional HPC systems for purposes of advanced computation.\n\\subsubsection{Quantum Data Networks}\nA particular aspect of future Quantum\/HPC infrastructures is the role of quantum data networks. Today quantum computing and quantum communication are usually handled as related by separate fields of applications of quantum technologies. However, quantum infrastructures, time and frequency distribution networks, and the development of quantum data technologies will play a fundamental and transformative role in physics experiments at different energy regimes (for example standard reference signals for anti-matter and low-energy experiments, such as ASACUSA and ALPHA, using high-precision laser spectroscopy \\cite{Amsler:2748998}).\n\\par\nMany national and international initiatives are being deployed to build future quantum networks and the HEP community must remain an active part of this development as it was at the beginning of the century in the development of distributed grid and cloud networks.\n\\subsubsection{International Collaborations and Co-Development}\nThe HEP community is distributed over the whole globe and the collaboration across institutes has always played a fundamental role in pushing the limits of science and technology. For example the development and operation of the Worldwide LHC Computing Grid (WLCG) was and still is instrumental for joint progress and employing all brain power wherever located in the world. The same approach needs to be taken in the development of future quantum infrastructures, Synergies between the Snowmass exercise and equivalent discussions in the European Particle Physics Strategy and other similar initiatives will be critical. Joint R\\&D projects, testbeds, progressive development of common frameworks and tools across US Quantum Information Science institutes, the CERN Quantum Technology Initiative \\cite{DiMeglio:2789149}, DESY QUANTUM, and many other national and international initiatives in EU and other countries will accelerate the development and adoption of quantum computing across the HEP community. In the same way, attention should be given to co-development not only across HEP, but also between HEP institutes, industry, and other scientific disciplines, such as Astrophysics, Cosmology, Earth Observation, Climate research, Condensed Matter, Chemistry, Biology, and many more. Often very similar challenges appear in different disciplines and can be solved with one solution.\n\n\n\\section*{Acknowledgements}\nThis work was supported by  supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Science Center, and the Advanced Scientific Computing Research program office Accelerated Research for Quantum Computing (ARQC) program. This research used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe field of cosmic microwave background (CMB) anisotropies has seen \na rapid development since its first detection by the COBE satellite\nonly a few years ago. There are now several reported experimental \nresults that are detecting anisotropies\non degree angular scales (see \\cite{Scott95} and\n\\cite{Bond96} for a recent review), \nwhich together with a few upper limits on smaller \nangular scales already give interesting limits on \ncosmological models. \nWith \nthe development of the new generation of experiments now being \nproposed one hopes to accurately map the CMB sky \nfrom arcminute scales to several degree scales. \nThe amount of data thus provided would \nallow for an unprecedented accuracy in the determination of cosmological \nparameters. As theoretical modelling shows \n(\\cite{Bond94,Hu95c,Jungman95,Seljak94}), \nCMB anisotropies are \nsensitive to most of the cosmological parameters and have a distinctive\nadvantage over other cosmological observations in that they probe the universe\nin the linear regime. This avoids the complications caused by physical \nprocesses in the nonlinear regime and allows to use powerful statistical\ntechniques to search over the parameter space for the best cosmological \nmodel (see e.g. \\cite{Jungman95}). \nA large stumbling block in this program at present is the speed of \ntheoretical model calculations, which are still too slow to allow for\na rapid search over the parameter space. This limitation was partially \nremoved by the development of appoximation methods (\\cite{Hu95a},b; \n\\cite{Seljak94}), \nwhich can give fast predictions of CMB anisotropy with a 10\\%\naccuracy. However, \nthese approximations are not accurate enough to exploit the complete\namount of information that will be present in the future CMB observations.\nThis is especially true for some of the more extreme cosmological models, \nwhere simple approximations break down and lead to systematic \ninaccuracies in the results.\nObviously, it would be useful to have a fast method that would not \nbe based on any approximations and would lead to accurate results for \nany cosmological model. The purpose of this paper is to present a new\nmethod of CMB calculation that satisfies these requirements.\n\nTheoretical calculations of the CMB anisotropies are based on \nlinear theory of cosmological perturbations, developed first by \nLifshitz (1946) and applied to the CMB anisotropies by \nPeebles and Yu (1970). In this early calculation only photons and \nbaryons were included, but later workers extended the calculations to \ninclude dark matter (Bond \\& Efstathiou 1984, 1987; Vittorio \\& Silk 1984),\ncurvature (Wilson \\& Silk 1981; Sugiyama \\& Gouda 1992; \nWhite \\& Scott 1995), \ngravity waves or tensor modes (Crittenden \net al. 1993) and massive neutrinos (Bond \\& Szalay 1983; Ma \\& Bertschinger \n1995; Dodelson, Gates \\& Stebbins 1995). \nMost of these and more recent calculations (e.g. \nHoltzmann 1989; Stompor 1994; Sugiyama 1995) solve for each Fourier mode of\ntemperature anisotropy $\\Delta_T(\\vec k)$ by \nexpanding it in Legendre series up to some desired $l_{max}$ and then \nnumerically evolve this system of equations in time from the radiation \ndominated epoch until today. Typically this means evolving a system \nof several thousand coupled differential equations in time, a slow \nprocess even for the present day computers. In addition, because \neach multipole moment is a rapidly oscillating function one has to \ndensely sample it in values of \n$k$ with typical number of evaluations of the order of $l_{max}$.\nEven the fastest codes at present \nrequire several hours of \nCPU time for each theoretical model\n(Sugiyama 1995), \nwhile some numerically more accurate ones (e.g. \\cite{BB95})\nrequire more like tens or hundreds of hours.\n\nIn this paper we explore a different approach to compute CMB anisotropies\nbased on integration of the sources over the photon\npast light cone. The method is a generalization \nof approximate method developed by one of the authors (Seljak 1994).\nIt differs from it in that it is exact, in the sense that\nit can achieve arbitrary precision within the limits of linear \nperturbation theory. By rewriting the system of equations in the\nintegral form one can separate between the geometrical and dynamical\ncontributions to the anisotropies. The former do not depend on the \nmodel and need to be computed only once, while the latter contain all\nthe information on the particular model and can be computed with a \nsmall system of equations. Solving for CMB anisotropies using this\nintegral form greatly reduces the required computational time. \nThe outline of the paper is as follows:\nin \\S 2 we present the basic system of equations that needs to be solved \nboth in the standard and in the integral method.\nIn \\S 3 we present in some detail a practical \nimplementation of the integral method, \nhighlighting the computational \ndifferences between it and the standard Boltzmann method.\nWe conclude in \\S 4 by discussing \npossible applications where the new method can be particularly useful.\n\n\\section{Method}\n\nIn this section we first present the standard system of equations \nthat needs to be solved for temperature\nanisotropies, which is based on solving Boltzmann equation using \nLegendre expansion of photon distribution function. \nThis part follows closely \nthe existing literature (e.g. Ma \\& Bertschinger 1995, Bond 1996\nand references therein) \nand only the final results are given. We do\nnot discuss the technical details of the standard Boltzmann  \nmethod, except where our approach differs significantly from it. \nIn the second\npart of the section we present the integral solution of photon \ndistribution, \nwhich is the basis of our method.  \nIn this paper we restrict the analysis to a spatially flat universe. \n\n\\subsection{Boltzmann, Einstein and Fluid Equations}\n\nThe temperature anisotropy at position $\\vec{x}$ \nin the direction $\\vec n$ \nis \ndenoted with $\\Delta_T(\\vec x, \\vec n)$. \nIn principle it depends both on the direction and \non the frequency, but because spectral distortions are only introduced\nat the second order the frequency dependence can in the lowest order\nbe integrated out. Anisotropy $\\Delta_T(\\vec x,\\vec n)$ can\nbe expanded in terms of Fourier modes $\\Delta_T(\\vec k,\\vec n)$,\nwhich in linear perturbation theory evolve independently of \none another. Assuming perturbations are axially-symmetric \naround $\\vec k$, we may further Legendre\nexpand the anisotropy in the angle \n$\\mu=\\vec k \\cdot \\vec n\/k$,\n\\begin{equation}\n\\Delta_T(\\vec k, \\vec n)=\\sum_l (2l+1)(-i)^l\\Delta_{Tl} P_l(\\mu),\n\\label{delta}\n\\end{equation}\nwhere $P_l(\\mu)$ is the Legendre polynomial of order $l$ and $\\Delta_{Tl}$ \nis the associated multipole moment.\nA similar decomposition also applies to \nthe amplitude of polarization anisotropy $\\Delta_P(\\vec{k},\\vec{n})$ \n(\\cite{BE84}, Critenden et al. 1993,\nKosowsky 1995, Zaldarriaga \\& Harari 1995). \n\nEvolution of the temperature anisotropy is governed by the Boltzmann\nequation (\\cite{Peebles70}, \\cite{Wilson81}, \n\\cite{BE84}). Its collisionless part is given by the time component\nof the geodesic equation, which depends on the metric. Here we will \nuse the metric \nin the longitudinal gauge (\\cite{bard80}; Bertschinger 1996), \nwhich is similar to the gauge-invariant formalism (Kodama \\& Sasaki 1984) \nand gives expressions that are most similar to their  \nNewtonian counterparts.\nThe choice of gauge is purely a matter of convenience\nand in some cases (e.g isocurvature models)\nother gauge choices such as synchronous gauge \nare computationally advantageous over the longitudinal \ngauge (see e.g. \\cite{BB95}, \\cite{Bond96}). In the longitudinal gauge\nthe perturbations are specified with two scalar potentials $\\phi$ and\n$\\psi$ and a gauge-invariant \ntensor perturbation $h$ (we will ignore vector perturbations in this paper,\nas they most likely have a negligible contribution to CMB anisotropy).\nThe corresponding temperature and polarization \nanisotropies are denoted as $\\Delta^{(S)}_T$, $\\Delta^{(S)}_P$\nfor scalar and $\\Delta^{(T)}_T$, $\\Delta^{(T)}_P$\nfor tensor components. In linear perturbation theory the scalar \nand tensor perturbations  \nevolve independently and the total power is given by the sum of the \ntwo contributions.\n\nThe collisional part of the photon Boltzmann equation is determined by the\nThomson scattering term. After angular and momentum \nintegration the Boltzmann evolution equations \nfor scalar perturbations can be written as (\\cite{BE87}),\n\\begin{eqnarray} \n\\dot\\Delta_T^{(S)} +ik\\mu \\Delta_T^{(S)} \n&=&\\dot\\phi-ik\\mu \\psi+\\dot\\kappa\\{-\\Delta_T^{(S)} +\n\\Delta_{T0}^{(S)} +i\\mu v_b +{1\\over 2}P_2(\\mu)\\Pi\n\\} \\nonumber \\\\   \n\\dot\\Delta_P^{(S)} +ik\\mu \\Delta_P^{(S)} &=& \\dot\\kappa \\{-\\Delta_P^{(S)} +\n{1\\over 2} [1-P_2(\\mu)] \\Pi\\} \\nonumber \\\\\n\\Pi&=&\\Delta_{T2}^{(S)}\n+\\Delta_{P2}^{(S)}+\n\\Delta_{P0}^{(S)}.\n\\label{Boltzmann}\n\\end{eqnarray}\nHere the derivatives are taken with respect to the conformal time $\\tau$ \nand $v_b$ is the velocity of baryons.\nDifferential optical depth for Thomson scattering is denoted as \n$\\dot{\\kappa}=an_ex_e\\sigma_T$, where $a(\\tau)$ \nis the expansion factor normalized\nto unity today, $n_e$ is the electron density, $x_e$ is the ionization \nfraction and $\\sigma_T$ is the Thomson cross section. The total optical \ndepth at time $\\tau$ is obtained by integrating $\\dot{\\kappa}$,\n$\\kappa(\\tau)=\\int_\\tau^{\\tau_0}\\dot{\\kappa}(\\tau) d\\tau$.\nA useful variable is the visibility function $g(\\tau)=\\dot{\\kappa}\n{\\rm exp}(-\\kappa)$. Its peak  \ndefines the epoch of recombination, when the  \ndominant contribution to the CMB anisotropies arises.\n\nExpanding the \ntemperature anisotropy in multipole moments \none finds the following hierarchy of coupled \ndifferential equations (\\cite{Wilson81}, \\cite{BE84}, \\cite{Ma95}),\n\\begin{eqnarray}\n     \\dot{\\Delta}_{T0}^{(S)}\n     &=& -k\\Delta_{T1}^{(S)}        +\\dot{\\phi}\n     \\,, \\nonumber\\\\\n\\dot{\\Delta}_{T1}^{(S)} &=&\n{k \\over 3}\\left[ \\Delta_{T0}^{(S)}-\n2\\Delta_{T2}^{(S)}\n+\\psi\\right] + \\dot{\\kappa} ({v_b \\over 3}-\\Delta_{T1}^{(S)})\\,,\\nonumber\\\\\n\\dot{\\Delta}_{T2}^{(S)} &=&{k \\over 5}\\left[2\\Delta_{T1}^{(S)}-3\n\\Delta_{T3}^{(S)}\\right] +\n\\dot{\\kappa} \\left[{\\Pi \\over 10}-\\Delta_{T2}^{(S)}\\right]\\,, \\nonumber\\\\\n\\dot{\\Delta}_{Tl}^{(S)}&=&{k \\over\n2l+1}\\left[l\\Delta_{T(l-1)}^{(S)}-(l+1)\n\\Delta_{T(l+1)}^{(S)}\\right]\n-\\dot{\\kappa}\\Delta_{Tl}^{(S)} \\,, l>2 \\nonumber \\\\\n\\dot{\\Delta}_{Pl}^{(S)}&=&{k \\over\n2l+1}\\left[l\\Delta_{P(l-1)}^{(S)}-(l+1)\n\\Delta_{P(l+1)}^{(S)}\\right]\n+\\dot{\\kappa}\\left[-\\Delta_{Pl}^{(S)}+{1 \\over 2}\\Pi\\left(\n\\delta_{l0}+{\\delta_{l2} \\over 5}\\right)\\right],\n\\label{photon}\n\\end{eqnarray}\nwhere $\\delta_{ij}$ is the Kronecker symbol.\nA similar system of equations without the Thomson scattering terms  \nand polarization\nalso applies for massless neutrinos.\nFor the\nmassive neutrinos the system of equations is more complicated,\nbecause the momentum dependence cannot be integrated out of the\nexpressions (see e.g. \\cite{Ma95}).\n\nBaryons and cold dark matter can be approximated as fluids and \ntheir evolution can be obtained \nfrom the local conservation of energy-momentum tensor. This gives the \nequations for cold dark matter density $\\delta_c$ and its \nvelocity $v_c$,\n\\begin{equation}\n\\label{cdm2}\n        \\dot{\\delta_c} = -kv_c + 3\\dot{\\phi}\\,, \\quad\n        \\dot{v}_c = - {\\dot{a}\\over a}\\,v_c+k\\psi \\,.\n\\end{equation}\nFor baryons one has additional terms in the Euler's equation\ncaused by Thomson scattering and pressure,\n\\begin{eqnarray}\n\\label{baryon2}\n\\dot{\\delta}_b &=& -kv_b + 3\\dot{\\phi} \\,, \\nonumber\\\\\n\\dot{v}_b &=& -{\\dot{a}\\over a}v_b\n+ c_s^2 k\\delta_b\n+ {4\\bar\\rho_\\gamma \\over 3\\bar\\rho_b}\n \\dot{\\kappa}(3\\Delta_{T1}^{(S)}-v_b) + k\\psi\\,.\n \\label{cdmb}\n \\end{eqnarray}\nHere $c_s$ is the baryon sound speed and $\\bar\\rho_\\gamma$, $\\bar\\rho_b$\nare the mean photon and baryon densities, respectively.\n\nFinally, the evolution of scalar metric perturbations is given by Einstein\nequations, which couple the sources and the metric perturbations. Only \ntwo equations are needed to specify the evolution. Here we choose them to\nbe the energy and momentum constraint equations,\n\\begin{eqnarray}\nk^2 \\phi+3{\\dot{a}\\over a}\\left(\\dot{\\phi}+{\\dot{a}\\over a}\\psi\\right)\n=-4\\pi Ga^2\\delta \\rho \\nonumber \\\\\nk^2\\left(\\dot{\\phi}+{\\dot{a}\\over a}\\psi\\right)=4\\pi Ga^2 \\delta f,\n\\label{einstein}\n\\end{eqnarray}\nwhere $\\delta \\rho$ and $\\delta f$ are the total \ndensity and momentum density perturbations, respectively. They are \nobtained by summing over the contributions from all species, $\\delta \\rho\n= \\sum_i \\delta \\rho_i$, $\\delta f=\\sum_i \\delta f_i$, $\\delta \\rho_i=\n\\bar{\\rho_i}\\delta_i$ and $\\delta f_i=(\\bar{\\rho_i}+\\bar{p_i})v_i$, where \n$\\bar{\\rho_i}$ and $\\bar{p_i}$ are the mean density and pressure of the $i$-th\nspecies.\n\nFor tensor perturbations the \nBoltzmann equation is given by (Crittenden et al. 1993),\n\\begin{eqnarray}\n\\dot{\\Delta}_{T }^{(T)} + i k \\mu\n \\Delta_{T }^{(T)}&=& - {\\dot h}- \n\\dot{\\kappa} (\\Delta_{T}^{(T)} -\n\\Psi) \\ , \\nonumber \\\\\n\\dot{\\Delta}_{P}^{(T)} + i k \\mu \n\\Delta_{P}^{(T)} &=& - \n\\dot{\\kappa}( \\Delta_{P}^{(T)} + \n\\Psi) \\ , \\nonumber \\\\\n\\Psi \\equiv  \\Biggl\\lbrack\n{1\\over10}\\Delta_{T0}^{(T)}\n&+&{1\\over 35}\n\\Delta_{T2}^{(T)}+ {1\\over210}\n\\Delta_{T4}^{(T)}\n -{3\\over 5}\\Delta_{P0}^{(T)}\n+{6\\over 35}\\Delta_{P2}^{(T)}\n-{1\\over 210}\n\\Delta_{P4}^{(T)} \\Biggr\\rbrack . \n\\label{tensorl}\n\\end{eqnarray}\nThe only external source is that of the tensor metric perturbation \nwhich evolves according to the Einstein equations as\n\\begin{equation}\n\\ddot{h}+2 {\\dot{a}\\over a}\\dot{h}+k^2h=0.\n\\label{tensoreinstein}\n\\end{equation}\nWe ignored the source term on the right-hand side of equation above \n(caused by neutrino and photon anisotropic stress), \nas it is always negligible compared to the terms on the left-hand side.\n\nTo obtain the temperature anisotropy for a given mode $\\vec k$ one has to\nstart at early time in the radiation dominated epoch with initial \nconditions of the appropriate type (e.g. isentropic or isocurvature) \nand evolve the system of equations until the present. The anisotropy\nspectrum is then obtained by integrating over the initial \npower spectrum of the metric perturbation $P_\\psi(k)$,\n\\begin{equation}\nC_l^{(S)}=(4\\pi)^2\\int k^2dk P_\\psi(k)|\\Delta^{(S)}_{Tl}(k,\\tau=\\tau_0)|^2.\n\\label{cl}\n\\end{equation}\nAnalogous expression holds for the polarization spectrum and for the\ntensor spectrum (where the initial power spectrum \n$P_\\psi(k)$ has to be replaced by the initial tensor power spectrum\n$P_h(k)$). \n\nThe spectrum $C_l$ is related to the angular correlation function,\n\\begin{equation}\nC(\\theta)=\\langle \\Delta(\\vec n_1)\\Delta(\\vec n_2)\\rangle_{\\vec n_1\n\\cdot\n\\vec n_2=\\cos \\theta}={1 \\over 4\\pi}\\sum_{l=0}^\\infty(2l+1)C_lP_l(\\cos \\theta).\n\\label{ctheta}\n\\end{equation}\nTo test a model on a given angular scale\n$\\theta$ one has to solve for $\\Delta_{Tl}$ up to \n$l \\approx 1\/\\theta$. \nIf one is interested in small angular scales \nthis leads to a large system of differential \nequations to be evolved in time and\nthe computational time becomes long. For a typical spectrum with \n$l_{\\rm max} \\sim 1000$ one has to evolve a system of 3000 differential \nequations (for photon and neutrino anisotropy and photon\npolarization)\nuntil the present epoch. Moreover, the solutions are rapidly oscillating\nfunctions of time, so the integration has to proceed in small time\nincrements to achieve the required accuracy on the final values. \n\n\\subsection{Integral solution}\n\nInstead of solving the coupled system of differential equations (\\ref{photon})\none may formally integrate equations \\ref{Boltzmann} \nalong the photon past light cone to obtain (e.g. \nZaldarriaga \\& Harari 1995),\n\\begin{eqnarray}\n\\Delta_T^{(S)} &=&\\int_0^{\\tau_0}d\\tau e^{i k \\mu (\\tau -\\tau_0)}\ne^{-\\kappa} \n\\{ \\dot\\kappa e^{-\\kappa} [\\Delta_{T0}+i\\mu v_b + {1\\over 2} P_2(\\mu)\n\\Pi]+\\dot\\phi-ik\\mu\\psi\\} \\nonumber \\\\\n\\Delta_P^{(S)} &=& -{1\\over 2}\\int_0^{\\tau_0} d\\tau e^{i k \\mu (\\tau -\\tau_0)}\ne^{-\\kappa} \\dot\\kappa   [1-P_2(\\mu)]\n\\Pi .\n\\label{formal}\n\\end{eqnarray}\nExpressions above can be further modified by eliminating the angle $\\mu$ in \nthe integrand through the integration by parts. \nThe boundary terms can be dropped, because they\nvanish as $\\tau \\rightarrow 0$ and are unobservable for $\\tau=\\tau_0$\n(i.e. only the monopole term is affected).\nThis way each time a given term is multiplied by a $\\mu$, it\nis replaced by its time derivative. This manipulation leads\nto the following expression,\n\\begin{eqnarray}\n\\Delta^{(S)}_{T,P} &=&\\int_0^{\\tau_0}d\\tau e^{i k \\mu (\\tau -\\tau_0)}\nS^{(S)}_{T,P}(k,\\tau) \n\\nonumber \\\\\nS^{(S)}_T(k,\\tau)&=&g\\left(\\Delta_{T0}+\\psi-{\\dot{v_b} \\over k}-{\\Pi \\over 4}\n-{3\\ddot{\\Pi}\\over 4k^2}\\right)\\nonumber \\\\\n&+& e^{-\\kappa}(\\dot{\\phi}+\\dot{\\psi})\n-\\dot{g}\\left({v_b \\over k}+{3\\dot{\\Pi}\\over 4k^2}\\right)-{3 \\ddot{g}\\Pi \\over\n4k^2} \\nonumber \\\\\nS^{(S)}_P(k,\\tau)&=&-{3 \\over 4 k^2}\\left(g\\{k^2\\Pi+\\ddot{\\Pi}\\}+2\n\\dot{g}\\dot{\\Pi}+\\ddot{g}\\Pi\\right).\n\\label{source}\n\\end{eqnarray}\nSome of the terms in the source function $S^{(S)}_T(\\tau)$ are \neasily recognizable. The first two contributions in the first term\nare the \nintrinsic anisotropy and gravitational potential \ncontributions from the last-scattering surface, while \nthe third contribution\nis part of the velocity term, the other part being the\n$k^{-1}\\dot{g}v_b$ term in the second row. \nThese terms \nmake a dominant contribution to the \nanisotropy in the standard recombination models. \nThe first term in \nthe second row, \n$e^{-\\kappa}(\\dot{\\phi}+\\dot{\\psi})$, is the so-called integrated\nSachs-Wolfe term and is important after recombination.\nIt is especially important if matter-radiation equality occurs \nclose to the recombination or in $\\Omega_{\\rm matter}\\ne 1$ models. In both\ncases gravitational \npotential decays with time, which leads to an enhancement of \nanisotropies on large angular scales.\nFinally we have the terms \ncaused by photon polarization and anisotropic\nThomson scattering, which contribute to $\\Pi$.\nThese terms affect the \nanisotropy spectra at the 10\\% level and are important for accurate\nmodel predictions. Moreover, they are the sources \nfor photon polarization.\nEquation (\\ref{source}) is a generalization of the tight-coupling\nand instantaneous recombination approximation \n(Seljak 1994) and reduces to it in the limit where the visibility function\nis a delta-function and $\\Pi$ can be neglected. \nIn that approximation one only needs to evaluate the sources at \nrecombination and then free stream them to obtain the anisotropy today. \nIn the more general case presented here\none has to perform an additional integration over time, which includes\nthe contributions arising during and after the recombination.\nMoreover, because the\ntight-coupling approximation is breaking down at the time of recombination,\nboth polarization and photon anisotropic stress \nare being generated and\n$\\Pi$ makes a non-negligible contribution to the anisotropy.\nFor exact calculations one has to use (\\ref{source}), which \nproperly includes all the terms that are relevant in \nthe linear perturbation theory.\n\nTo solve for the angular power spectrum one\nhas to expand the plane wave $e^{i k \\mu (\\tau -\\tau_0)}$ \nin terms of the radial and angular eigenfunctions (spherical \nBessel functions and Legendre polynomials, respectively), \nperform the ensemble average \n\\footnote[1]{In performing \nensemble average we assume that only the amplitude and \nnot the phase of a \ngiven mode evolves in time. While this is \nvalid in linear theory for most models of structure formation,\nit may not be correct in some versions\nof models with topological defects (Albrecht et al. 1995).} \nand integrate over the angular variable \n$\\mu$. This leads (\\ref{cl}), where \nthe multipole moment at present time $\\Delta^{(S)}_{(T,P)l}\n(k,\\tau=\\tau_0)$\nis given by the following expression,\n\\begin{equation}\n\\Delta^{(S)}_{(T,P)l}(k,\\tau=\\tau_0)=\n\\int_0^{\\tau_0}S^{(S)}_{T,P}(k,\\tau)j_l[k(\\tau_0-\\tau)]d\\tau,\n\\label{finalscalar}\n\\end{equation}\nwhere $j_l(x)$ is the spherical Bessel function. \nNote that\nwhile angular eigenfunctions integrated out after angular averaging, \nradial eigenfunctions\nremained and enter in (\\ref{finalscalar}).\nThe main advantage of (\\ref{finalscalar})\nis that it decomposes the anisotropy into a source term \n$S^{(S)}_{T,P}$, which does \nnot depend on the multipole moment $l$ and a geometrical term $j_l$, which \ndoes not depend on the particular cosmological model. \nThe latter thus only needs to be computed once and can be stored for \nsubsequent calculations. \nThe source term \nis the same for all multipole moments and only depends on a small number\nof contributors in (\\ref{source})\n(gravitational potentials, baryon velocity and photon moments\nup to $l=4$). By specifying the source term as a function of time one can \ncompute the corresponding spectrum of anisotropies. \nEquation (\\ref{finalscalar})\nis formally an integral system of equations, because \n$l<4$ moments appear on both sides of equations. To solve for these\nmoments it is best to use the equations in their \ndifferential form (\\ref{photon}),\ninstead of the integral form above.\nOnce the moments that enter into the source function are computed\none can solve for the higher moments \nby performing the integration in (\\ref{finalscalar}) \n(see section 3 for more details).\n\nThe solution for the tensor modes can similarly be written as an integral \nover the source term and the tensor spherical eigenfunctions $\\chi^l_k$. \nThe latter are \nrelated to the spherical Bessel functions (\\cite{abbott86}),\n\\begin{equation}\n\\chi^l_k(\\tau)=\\sqrt{{(l+2)! \\over 2(l-2)!}}\n{j_l(k\\tau)\n\\over (k\\tau)^2}.\n\\label{eigentensor}\n\\end{equation}\nThis gives\n\\begin{equation}\n\\Delta^{(T)}_{(T,P)l}=\\int_0^{\\tau_0}d\\tau S_{T,P}^{(T)}(k,\\tau)\n\\chi^l_k(\\tau_0-\\tau),\n\\label{finaltensor}\n\\end{equation}\nwhere from equation \\ref{tensorl} follows\n\\begin{equation}\nS_T^{(T)}=-\\dot{h}e^{-\\kappa}+g\\Psi \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\n S_P^{(T)}=-g\\Psi.\n\\label{sourcet}\n\\end{equation}\nEquations (\\ref{source}-\\ref{sourcet}) \nare the main equations of this paper and form the \nbasis of the line of sight integration method\nof computing CMB anisotropies. In the next section we will\ndiscuss in more detail the computational advantages of this \nformulation of Boltzmann equation and its implementation.\n\n\\section{Calculational Techniques}\n\nIn the previous section we presented the expressions needed \nfor the implementation of the line of sight integration\nmethod. \nAs shown in  (\\ref{finalscalar}) and (\\ref{finaltensor})\none needs to integrate over\ntime the source term at time $\\tau$ \nmultiplied with the sperical Bessel function\nevaluated at $k(\\tau_0-\\tau)$. The latter does not depend\non the model and can be precomputed in advance. \nFast algorithms exist which can compute spherical Bessel functions\non a \ngrid in $k$ and $l$ in short amount of time (e.g. Press et al. 1993). The\ngrid is then stored on a disk and used for all the subsequent calculations.\nThis leaves us with the\ntask of accurately calculating the source term,\nwhich determines the CMB spectrum for a given model.\nBelow we discuss some of the calculational techniques needed for\nthe implementation of the method. \nWe especially highlight the differences\nbetween this approach and the standard Boltzmann\nintegration approach. Our goal is to develop a method which is\naccurate to 1\\% in $C_l$ up to $l \\sim 1000$\nover the whole range of cosmological \nparameters of interest. These include models with varying amount of\ndark matter, baryonic matter, Hubble constant, vacuum energy, \nneutrino mass, shape of initial spectrum of perturbations, reionization\nand tensor modes. The choice of accuracy is based on estimates of\nobservational accuracies that will be achiavable in the next generation \nof experiments and also on the theoretical limitations of model \npredictions (e.g. cosmic variance, second order effects etc.).\nMost of the figures where we discuss the choice\nof parameters \nare calculated for the standard CDM model. \nThis model is a reasonable choice in the sense that it is a model which \nexhibits most of the physical effects in realistic models, including \nacoustic oscillations, early-time integrated Sachs-Wolfe effect and \nSilk damping. One has to be careful however not to tune the parameters\nbased on a single model. We compared our results with results from other \ngroups (Bode \\& Bertschinger 1995; Sugiyama 1995) for a number of \ndifferent models. We find a better than 1\\% agreement \nwith these calculations over most of the parameter space of models.\nThe computational parameters we recommend below are based on this more detailed\ncomparison and are typically more stringent than \nwhat one would find based on the comparison with the standard CDM model only. \n\n\\subsection{Number of coupled differential equations}\n\nIn the standard Boltzmann method the \nphoton distribution function is expanded to a high $l_{\\rm max}$ \n(\\ref{photon}) and typically one has to \nsolve a coupled system of several thousand differential equations.\nIn the integral method one\nevaluates the source terms $S(k,\\tau)$ as a function \nof time (\\ref{source}), (\\ref{sourcet}) and\none only requires the knowledge of photon multipole \nmoments up to $l=4$, \nplus the metric perturbations and baryon velocity. \nThis greatly reduces \nthe number of coupled differential equations that are needed to be solved. \nFor an accurate evaluation of the lowest multipoles in the integral\nmethod one has to extend\nthe hierarchy somewhat beyond $l=4$, because the\nlower multipole moments are coupled to the higher multipoles\n(\\ref{photon}).  \nBecause power is only being \ntransferred from lower to higher $l$  \nit suffices to keep a few \nmoments to achieve a high numerical accuracy of $l<5$ moments.\nOne has to be careful however to avoid \nunwanted \nreflections of the power being transferred from low $l$ to high $l$, \nwhich occur for example if a simple cut-off in the hierarchy is imposed.\nThis can be achieved by modifying \nthe boundary condition for the last term in the \nhierarchy \nusing the free streaming approximation\n(\\cite{Ma95}, \\cite{Hu95}). In the absence of scattering \n(the so-called free streaming regime),\nthe recurrence relation among the \nphoton multipoles in equation (\\ref{photon}) becomes the\ngenerator of spherical Bessel functions. \nOne can therefore use a different recurrence \nrelation among the spherical Bessel functions \nto approximate the last term in the hierarchy without reference to the \nhigher terms.\nThe same approximation can also be used for polarization and\nneutrino hierarchies. This type of\nclosure scheme works extremely well and only a few multipoles\nbeyond $l=4$ are needed for an accurate calculation of the source \nterm. This is shown in \nfigure \\ref{fig1}, where a relative error in the spectrum is plotted\nfor several choices of maximal number of photon multipoles.\nWe choose to end the\nphoton hierarchy (both anisotropy and polarization) \nat $l_\\gamma=8$ and massless neutrino at $l_\\nu=7$, which \nresults in an error lower than $0.1\\%$ compared to the exact case. \nInstead of a few thousand coupled differential equations \nwe therefore evolve about 35 equations\nand the integration time is correspondingly reduced. \n\n\\begin{figure}[t]\n\\vspace*{8.3 cm}\n\\caption{CMB spectra produced by \nvarying the number of evolved photon multipole moments, together\nwith the  \nrelative error (in \\%)\ncompared to the exact case. While using $l_\\gamma=5$ produces\nup to 2\\% error, using $l_\\gamma=7$ gives results almost identical to the \nexact case.}\n\\special{psfile=losfig1.ps\nvoffset=-10 hoffset=70 hscale=47 vscale=45}\n\n\\label{fig1}\n\\end{figure}\n\n\\subsection{Sampling of CMB multipoles}\n\nIn the standard Boltzmann integration method one solves for the whole \nphoton hierarchy (\\ref{photon}) and the \nresultant $\\Delta_l$ is automaticaly obtained for each $l$ up to some \n$l_{\\rm max}$. \nThe CMB spectra are however\nvery smooth (see figure \\ref{fig1}), except for the lowest $l$, where the\ndiscrete nature of the spectrum becomes important. This means that \nthe spectrum need not be sampled for each $l$ and instead  \nit suffices to sparsely sample\nthe spectrum in a number of points and interpolate between them.\nFigure \\ref{fig2} shows the result of such interpolation \nwith cubic splines (see e.g. \\cite{Press92}) when every \n20th, 50th or 70th \n$l$ is sampled beyond $l=100$ with an increasingly denser sampling\ntowards small $l$, so that each $l$ is sampled below $l=10$. \nWhile sampling of every 70th $l$ results in maximal error of \n1\\%, sampling in \nevery 20th or 50th $l$ gives errors below 0.2 and 0.4\\%, respectively. \nWe choose to compute every 50th $C_l$ beyond $l=100$ in addition to\n15 $l$ modes\nbelow $l=100$, so that a total of\n45 $l$ modes are calculated up to $l_{\\rm max} =1500$. This gives   \na typical (rms) error of 0.1\\%, with excursions of up to 0.4\\%. \nThe number of integrals in equation (\\ref{cl}) is thus \nreduced by 10-50 and the computational \ntime needed for the integrals becomes comparable or smaller\nthan the time\nneeded to solve for the system of differential equations.\n\n\\begin{figure}[t]\n\\vspace*{9.3 cm}\n\\caption{  \nRelative error between the exact and interpolated spectrum,  \nwhere every 20th, 50th or 70th multipole is calculated. The \nmaximal error for the three approximations is less\nthan 0.2, 0.4 and 1.2\\%, respectively.  \nThe rms deviation from the exact spectrum is further improved by\nfiner sampling, because the\ninterpolated spectra are exact in the sampled points. For the sampling\nin every 50th multipole the rms error is 0.1\\%.} \n\\special{psfile=losfig2.ps\nvoffset=20 hoffset=70 hscale=47 vscale=45}\n\\label{fig2}\n\\end{figure}\n\n\\subsection{Free streaming}\n\nAfter recombination and in the absence of a time changing gravitational \npotential the source function often becomes negligible. This is \nthe so called free streaming regime, where the\nphotons are freely propagating through the universe. \nMost of the standard Boltzmann codes use a special free streaming algorithm \nto map the anisotropies from a given epoch $\\tau_{\\rm fs}$ into \nanisotropies today (\\cite{BE84}).\nIn the line of sight integration method\nthe free streaming regime is only a special case where $S(k,\\tau)=0$ after \nsome time $\\tau_{\\rm fs}$.\nThus one can stop the integration at the \ntime $\\tau_{\\rm fs}$ beyond which the sources are not important and there is\nno need for a separate algorithm to evolve the anisotropies until today. \nFor example, if one\nassumes that only the first term in (\\ref{source})\nis important\none only needs to integrate over \nthe source where the visibility function $g$ appreciably differs from 0. In \nthe absence of reionization this restricts the time \nintegration to a narrow interval during recombination   \naround $z \\approx 1100$.\nAlthough most of the contributions to the anisotropies come\nfrom this epoch, time dependent\ngravitational potential (and, to a smaller extent, other source terms\nin equation (\\ref{source})) make \na nonnegligible contribution to the anisotropies\neven after recombination. \nAs mentioned earlier,\nthis is especially important\nin $\\Omega_{\\rm matter}\\ne 1$ models and in models with low $\\Omega_{\n\\rm matter}h^2$. In the first case \nthe gravitational potential is decaying at late times,\nwhile in the second class of models the matter-radiation equality\nduring which gravitational potential\nchanges in time is pushed to a lower redshift.\nEven in standard CDM model ($\\Omega_{\\rm matter}=1,h=0.5$)\ngravitational potential is still \nsignificantly changing in time at moderately low redshifts\nof $z \\sim 100$ (\\cite{Hu95}).\nSimilarly one cannot use free streaming\nin the models with late reionization, \nwhere the visibility\nfunction is nonvanishing at low redshifts. \nWe choose to integrate until the present time \nfor most of the models, except for\nthe models with $\\Omega_{\\rm matter}=1$, \nwhere we stop the integration at \n$z=10$. In this case the  \ncomputational time is reduced significantly (typically 50\\%) \ncompared to that of evolving the \nequations until the present time. \n\n\\subsection{Integration over time}\n\nFor each Fourier mode $k$ the source term is integrated\nover time $\\tau$ (\\ref{finalscalar}). \nThe sampling in time need not be \nuniform, because the dominant contribution arises from the epoch\nof recombination around $z\\sim 1100$, the width of which is \ndetermined by the visibility function $g$ and is rather narrow in \nlook-back time for\nstandard recombination scenarios. During this epoch the sources\nacoustically oscillate on a time scale of $c_sk^{-1}$, \nso that the longest wavelength modes are the slowest to vary.\nFor short wavelengths the rate of sampling should therefore be higher.\nEven for long wavelengths the source function\nshould still be sampled in several\npoints across the last-scattering surface. This is because the \nterms in (\\ref{source}) depend on the derivatives\nof the visibility function. \nIf the visibility function $g$ is narrow then its derivative will \nalso be narrow and will sharply \nchange sign at the peak of $g$. Its integration \nwill lead to numerical roundoff\nerrors if not properly sampled, even though  \npositive and negative contributions nearly cancel out when integrated\nover time and make only a small contribution to the integral. \nFigure \\ref{fig3} shows \nthe error in integration caused by sampling this epoch with 10, 20\nor 40 points. Based on comparison with several models\nwe choose to sample the recombination epoch\nwith 40 points, which results in very small ($\\sim$ 0.1\\%) errors. \nAfter this \nepoch the main contribution to the anisotropies arises from the \nintegrated Sachs-Wolfe term. This is typically a slowly changing\nfunction and it is sufficient to sample \nthe entire range in time until the present in 40 points.\nThe exceptions here are models\nwith reionization, where the visibility function becomes non-negligible\nagain and a new last-scattering surface is created. In this case a \nmore accurate sampling of the source is also needed at lower redshifts.\n\n\\begin{figure}[t]\n\\vspace*{9.3 cm}\n\\caption{ Error in the spectrum caused by insufficient\ntemporal sampling of the source term. \nInaccurate sampling of the source during recombination leads to \nnumerical errors, which can reach the level of 1\\% if the source \nis sampled in only 10 points across the recombination epoch. \nFiner sampling in time gives much smaller errors for this model. \nComparisons with other models indicate that sampling in 40 \npoints is needed for accurate integration.} \n\n\\special{psfile=losfig3.ps\nvoffset=20 hoffset=70 hscale=47 vscale=45}\n\n\\label{fig3}\n\\end{figure}\n\n\\subsection{Integration over wavenumbers}\n\nThe main computational cost of standard \nCMB calculations is solving the coupled\nsystem of differential equations.\nThe number of $k$-modes for which the system is solved is the main \nfactor that determines the speed of the method. For results accurate\nto $l_{\\rm max}$ one has to sample the wavenumbers upto a maximun value\n$k_{\\rm max}=l_{\\rm max}\/ \\tau_0$.  \nIn the line of sight integration method \nsolving the coupled\nsystem of differential equations still dominates the computational time \n(although for each mode \nthe time is significantly shorter than in the \nstandard Boltzmann method \nbecause of a smaller system of equations).\nIt is therefore instructive to compare the number of $k$ \nevaluations needed in each of the methods to achieve a given\naccuracy in the final spectrum.\n\nIn the standard Boltzmann method\none solves for $\\Delta_{T,l}^{(S)}(k)$ directly, so this quantity \nmust be sampled densely enough for accurate integration.\nFigure \\ref{fig4}a shows $\\Delta_{T,l}^{(S)}(k)$\nfor $l=150$ in a standard CDM model. One can see that it is a rapidly \noscillating function with a frequency $k \\sim \\tau_0^{-1}$. \nEach oscillation needs to be sampled in at least a few points to assure \nan accurate integration.\nTo obtain a smooth CMB spectrum one typically \nrequires 6 points over one period, implying \n$2l_{\\rm max}$ $k$-mode evaluations. \nThis number can be reduced somewhat \nby filtering out the sampling noise in the spectrum (\\cite{Hu95}), but even\nin this case one requires at least 1-2 points per each period or \n$l_{\\rm max}\/2$ $k$-mode evaluations. \n\nTo understand the nature of these rapid oscillations in\n$\\Delta_{T,l}^{(S)}(k)$ we will consider wavelenghts larger than\nthe width of the last scattering surface.  In this case the\nBessel function in (\\ref{finalscalar}) can be pulled out of the integral as\n$j_l(k\\tau_0)$ because the time at which recombination occurs,\nwhen the dominant contribution to $\\Delta_{T,l}^{(S)}(k)$ is\ncreated, is \nmuch smaller than $\\tau_0$ and $k \\Delta\\tau\\ll 1$ ($\\Delta\\tau$\nis the interval of time for which the visibility function differs\nappreciably from zero).\nSo the final $\\Delta_{T,l}^{(S)}(k)$ is approximately \nthe product of $j_l(k\\tau_0)$ and $S_T^{(S)}$ integrated over time,\nif the finite width of the last scattering surface and\ncontributions after recombination can be  ignored.  \n\nFigure \\ref{fig4}b shows the \nsource term $S_T^{(S)}$ integrated over time\nand the Bessel function $j_l(k\\tau_0)$. \nIt shows that the high frequency\noscillations in $\\Delta_{T,l}^{(S)}(k)$ seen in figure \\ref{fig4}a\nare caused by the oscillation of the spherical\nBessel functions, while the oscillations of the source term have a\nmuch longer period in $k$.\nThe different periods of the two \noscillations can be understood\nusing the tight coupling approximation (\\cite{Hu95}, \\cite{Seljak94}). \nPrior and during recombination photons are coupled to the baryons and the \ntwo oscillate together  \nwith a typical acoustic timescale $\\tau_{s}\\sim \\tau_{\\rm rec}\/\\sqrt{3}\n\\sim \\tau_0\/\\sqrt{3z_{\\rm rec}} \\sim \\tau_0\/50$. The \nfrequency of acoustic oscillations $k \\sim \\tau_{\\rm rec}^{-1}$ \nis therefore 50 times higher than\nthe frequency of oscillations in spherical Bessel functions, which \noscillate as $\\tau_0^{-1}$. \n\n\\begin{figure}[t]\n\n\\vspace*{9.3 cm}\n\\caption{In (a) $\\Delta_{T,150}^{(S)}(k)$ is plotted as a function of\nwavevector \n$k$. In (b) $\\Delta_{T,150}^{(S)}(k)$ is decomposed into the \nsource term $S_T^{(S)}$ integrated over time\nand the spherical\nBessel function $j_{150}(k\\tau_0)$. The high frequency oscillations \nof $\\Delta_{T,150}^{(S)}(k)$ are caused by oscillations of \nthe spherical Bessel function $j_{150}(k\\tau_0)$, whereas the source \nterm varies much more slowly. This allows one to reduce the number of\n$k$ evaluations in the line of sight integration method, because only \nthe source term needs to be sampled.  } \n\\special{psfile=losfig4.ps \nvoffset=40 hoffset=70 hscale=47 vscale=45}\n\\label{fig4}\n\\end{figure}\n\n\\begin{figure}[t]\n\\vspace*{8.3 cm}\n\\caption{ Error in the spectrum caused by insufficient \n$k$-mode sampling of the source term. \nSampling the source with 40 points up to \n$k = 2l_{\\rm max}$ leads to 1\\% errors, while  \nwith 60 or 80 points the maximal error decreases to 0.2\\%.\nComparisons with other models indicate that sampling in 60 \npoints is sufficient for accurate integration.} \n\\special{psfile=losfig5.ps \nvoffset=0 hoffset=70 hscale=47 vscale=45}\n\\label{fig5}\n\\end{figure}\n\nBecause an accurate sampling of the \nsource term requires only a few \npoints over each acoustic oscillation, the total number of $k$ evaluations\nin the integral method \ncan be significantly reduced compared to the standard methods. \nTypically a few dozen\nevaluations are needed\nover the entire range of $k$, compared to about 500 evaluations\nin the standard method when a noise filtering technique is \nused and 2000 otherwise \n(for $l_{\\rm max} \\sim 1000$). Once the source\nterm is evaluated at these points one can\ninterpolate it at points with preevaluated \nspherical Bessel functions, which can be much more densely sampled \nat no additional computational cost. \nThe end result is the same accuracy as in the standard method,\nprovided that the source is sampled in sufficient number of points.\nFigure \\ref{fig5} shows the relative error in the CMB spectrum for the\ncases where the \nsource term is calculated in 40, 60 and 80 points between \n0 and $k \\tau_0 =3000$ (for $l_{\\rm max}=1500$). \nWhile using 40 points results in up to 1\\% errors, using 60 points\ndecreases the maximum error to below 0.2\\% for this model. \nIn general it suffices to use \n$l_{\\rm max}\/30$ $k$ modes, which is at least  \nan order of magnitude smaller than in \nthe standard methods. \nNote that with this method there is no need to \nfilter the spectrum to reduce the sampling noise, because  \nthe latter is mainly caused by insufficient sampling\nof the spherical Bessel functions, which\nare easy to precompute. The additional operations needed for a higher \nsampling (summation and source interpolation) do not  \nsignificantly affect the overall computational time. Moreover, \nif each $C_l$ is accurately calculated they can be sparsely sampled\nand interpolated\n(section 3.2), this would not be possible if they had a significant noise\ncomponent added to them.  \n  \n\n\\section{Conclusions}\n  \nIn this paper we presented a new method for accurate calculations of\nCMB anisotropy and polarization spectra. \nThe method is not based on any approximations and is an\nalternative to the standard Boltzmann calculations, which are based \non solving large numbers of differential equations. The \napproach proposed here uses a hybrid integro-differential \napproach in solving the same system of equations. \nBy rewriting the Boltzmann equations in the integral form\nthe solution for the photon anisotropy \nspectrum can be written as an integral over a\nsource and a geometrical term. The first is determined by a small number\nof contributors to the photon equations of motion and the second is \ngiven by the radial eigenfunctions, which do not depend on the\nparticular cosmological model, but only on the geometry of space.\n\nOne advantage of the split between geometrical and dynamical \nterms is that\nit clarifies their different contributions to\nthe final spectrum. A good example of this is the temperature anisotropy\nin the non-flat universe, which can be be written using a similar \ndecomposition, except\nthat spherical Bessel functions have to be replaced with their \nappropriate generalization (\\cite{abbott86}). \nThis will be discussed in more detail in a future \npublication, here we simply remark that replacing radial eigenfunctions\nin a non-flat space with their flat space counterpart (keeping comoving\nangular distance to the LSS unchanged) is only approximate and\ndoes not become exact even in the large\n$l$ (small angle) limit. The geometry of the \nuniverse leaves its signature in the CMB spectra in a rather \nnontrivial way and does not lead only to a simple rescaling of the \nspectrum by $\\Omega_{\\rm matter}^{-1\/2}$ (\\cite{Jungman95}).\n\nThe main advantage of our line of sight integration method is\nits speed and accuracy. \nFor a given set of parameters it is two orders of magnitude\nfaster than the standard Boltzmann \nmethods, while preserving the same accuracy.\nWe compared our results with the results by Sugiyama (1995) and\nby Bode \\& Bertschinger (1995) and in both cases the agreement was \nbetter than 1\\% up to a very high $l$ for all of the models we\ncompared to.\n\nThe method is useful for fast and accurate normalizations\nof density power spectra from CMB measurements, \nwhich for a given model require the CMB anisotropy spectrum and\nmatter transfer function, both of which are provided by the output\nof the method. \nSpeed and accuracy are even more important for accurate determination\nof cosmological parameters from CMB measurements. In such applications\none wants to perform a search over a large parameter space,\nwhich typically requires calculating the spectra of a\nseveral thousand models (e.g. \\cite{Jungman95}). \nOne feasible way to do so is to use approximation methods \nmentioned in the introduction. These \ncan be made extremely fast, but at a cost of sacrificing the \naccuracy. While several percent accuracy is sufficient for analyzing\nthe present day experiments, it will not satisfy the requirements\nfor the future all-sky surveys of microwave sky. Provided that \nforeground contributions can be succesfully filtered out (see\nTegmark \\& Efstathiou 1995 for a recent discussion) one can hope for \naccuracies on the spectrum close to the cosmic variance limit,\nwhich for a broad band averages can indeed reach below 1\\% at \n$l>100$. It is at this stage that fast and accurate CMB calculations\nsuch as the one presented in this paper\nwill become crucial and might enable one \nto determine many cosmological parameters with an unprecedented\naccuracy.\n\n\\acknowledgements\nWe would like to thank Ed Bertschinger for encouraging \nthis work and providing helpful comments.\nThis work was partially supported by grant NASA NAG5-2816.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Riemann zeta function $\\zeta(z)$ is conventionally represented as the sum or the integral\n\\begin{equation}\\label{(1)}\n\\zeta(z)=\\sum_{n=1}^{\\infty}\\frac{1}{n^{z}}=\\frac{1}{\\Gamma(z)}\\int_{0}^{\\infty}\\frac{t^{z-1}}{e^{t}-1}dt,\\quad\\textrm{Re}(z) > 1\n\\end{equation}\n(\\cite[Theorem 12.2]{Apostol}). The integral reduces to the sum if the denominator of the integrand is expanded in  geometric series.\nIt is well-known that $\\zeta(z)$ can be analytically continued to the whole complex plane except for a single pole at $z=1$ with residue 1\nand $\\zeta(z)$ satisfies the two equivalent functional equations (we present  one of them here, the other can be obtained from this by letting $z\\to1-z$)\n\\begin{equation}\\label{Functional}\n\\zeta(z)=2(2\\pi)^{1-z}\\Gamma(1-z)\\sin\\left(\\frac{\\pi z}{2}\\right)\\zeta(1-z),\\quad\\textrm{Re}(z) < 1\n\\end{equation}\n(\\cite[Theorem 12.7]{Apostol}).\nFrom the above equation, we see that $\\zeta(z)$ is symmetric with the axis $\\textrm{Re}(z)=\\frac{1}{2}$.\n\nBy (\\ref{Functional}), from the vanishing of the function $\\sin\\left(\\frac{\\pi z}{2}\\right)$, the negative even integers are also zeros of $\\zeta(z)$,\nwhich are called the trivial zeros.\nFurthermore, from the representation of the zeta function in terms of products involving prime numbers, it can be shown that the zeta function has no zero for Re$(z)>1$ since none of the factors can vanish there.\nIn 1859, Riemann \\cite{Riemann} conjectured that the nontrivial zeros of $\\zeta(z)$ are all lie on the critical line $\\textrm{Re}(z)=\\frac{1}{2},$\nnamely,\n\\begin{equation}\\label{Re-zero}\n\\zeta\\left(\\frac12+ie_* \\right)=0,\n\\end{equation}\nwhere $e_*$ denotes the location of a zero on the critical line.\nThis becomes one of most remarkable conjectures in mathematics, known as the Riemann hypothesis.\nTo the purpose of solving Riemann hypothesis, Hilbert-P\\'olya \\cite{Hilbert} conjectured that the imaginary parts of the zeros of $\\zeta(z)$\nmight correspond  to the eigenvalues of a Hermitian, self-adjoint operator.\nBerry and Keating \\cite{Berry} conjectured that the classical counterpart of such a Hamiltonian may have the form $\\hat{H}=\\hat{x}\\hat{p}$,\nbut a Hamiltonian possessing  this property has not yet been found. In quantum mechanics, associated with each measurable parameter in a physical system is a quantum mechanical operator, and the operator associated with the system energy is called the Hamiltonian.\n\nIn 2016, Bender, Brody and M\\\"uller  \\cite{Bender} found an interesting correspondence between the nontrivial zeros of\n$\\zeta(z)$ and the eigenvalues of Hamiltonian for a quantum system.\nThat is, let $\\hat x$ be the position operator and let $\\hat{p}=-i\\frac d{dx}$ be the momentum operator. \nThey constructed a non-Hermitian Hamiltonian\n\\begin{equation}\n\\hat{H}=\\frac{\\mathbbm{1}}{\\mathbbm{1}-e^{-i\\hat{p}}}(\\hat{x}\\hat{p}+\\hat{p}\\hat{x})({\\mathbbm{1}-e^{-i\\hat{p}}}),\n\\end{equation}\nwhich satisfies the conditions of the Hilbert-P\\'olya conjecture, that is, if the eigenfunctions of $\\hat{H}$ satisfy the boundary condition $\\psi_{n}(0)=0$ for all $n$, then\nthe real eigenvalues $E_{n}$ have the property that $\\frac{1}{2}(1-iE_{n})$ are the nontrivial zeros of the Riemann zeta function, and the Hurwitz zeta functions\n$\\psi_{n}(x)=-\\zeta(z_{n},x+1)$ are the corresponding eigenstates. They also constructed the metric operator\nto define an inner-product space, on which the Hamiltonian is Hermitian. They remarked that if the analysis may be made rigorous to show that\n$\\hat{H}$ is manifestly self-adjoint, then this implies that the Riemann hypothesis holds true.\n\nIn this paper, we present a general construction of Hamiltonian $\\hat{H}_{f},$ which leads   a general family of Hurwitz zeta functions $(-1)^{z_{n}-1}L(f,z_{n},x+1)$ becomes their eigenstates under a suitable boundary condition, \nand the eigenvalues $E_{n}$ have the property that $z_{n}=\\frac{1}{2}(1-iE_{n})$ are the  zeros of a general family of zeta functions $L(f,z)$ (see Theorem \\ref{main} below).\nThe definitions of $L(f,z,x)$ and  $L(f,z)$ will be given in the next section.\n\n\\section{Mellin transform and zeta functions}\nRecall that a function $f$ tends rapidly to $0$ at infinity if for any $k\\geq 0$, as $t\\to +\\infty$ the functions $t^{k}f(t)$ tends to $0$. Let  $tf(-t)$ be a $C^{\\infty}$ function on $[0,\\infty)$\ntending rapidly to $0$ at infinity and for $\\textrm{Re}(z)>1$, we define a general family of zeta functions $L(f,z)$ from the following integral representation\n \\begin{equation}\\label{(5)}\nL(f,z)=\\frac{1}{\\Gamma(z)}\\int_{0}^{\\infty}t^{z}f(-t)\\frac{dt}{t}.\n\\end{equation}\nAs pointed out by Cohen \\cite[Proposition 10.2.2(2)]{Cohen}, if $f(-t)$ itself is a $C^{\\infty}$ function on $[0,\\infty)$ and\nit tends rapidly to $0$ at infinity, then we have $L(f,z)$ can be analytically  continued to the whole of $\\mathbb{C}$ into a holomorphic function.\n\nThe  Huwitz zeta function $\\zeta(z,x)$ is defined by the series\n\\begin{equation}\n\\zeta(z,x)=\\sum_{n=0}^{\\infty}\\frac{1}{(n+x)^{z}},\\quad\\textrm{Re}(z)>1,\n\\end{equation}\nwhere $x\\in (0,\\infty)$  (see Cohen \\cite[p.~72, Definition 9.6.1]{Cohen} and the discussions for the definition area of $x$ there).\nIt has the integral representation\n\\begin{equation}\n\\zeta(z,x)=\\frac{1}{\\Gamma(z)}\\int_{0}^{\\infty}e^{-(x-1)t}\\frac{t^{z-1}}{e^{t}-1}dt.\n\\end{equation}\n\n\nInspired from this, we also define a general family of Hurwitz zeta function $L(f, z,x)$ as the following integral representation\n\\begin{equation}\\label{(6)}\nL(f,z,x)=\\frac{1}{\\Gamma(z)}\\int_{0}^{\\infty} t^{z}e^{-(x-1)t} f(-t)\\frac{dt}{t},\n\\end{equation}\nwhere Re$(z)>1$ and $x\\in (0,\\infty).$\nIt is easy to see that $L(f,z,1)=L(f,z)$.\n\nBy setting $f(t)$ as different functions in (\\ref{(5)}), $L(f,z,x)$ and $L(f,z)$ become\nthe well-known Riemann zeta functions, Hurwitz zeta functions, Dirichlet $L$-functions, Hecke series and other zeta functions. \nThe following are some examples. \n\\begin{itemize}\n\\item\nLetting $f(t)=\\frac{1}{e^{-t}-1}$ in (\\ref{(5)}) and (\\ref{(6)}), we have the Riemann zeta function $\\zeta(z)$ and the Hurwitz zeta function $\\zeta(z,x)$, respectively (see Colmez's Tsinghua lecture notes \\cite[p.~4]{Colmez}).\n\n\\item \nSuppose  $\\chi$ is a Dirichlet character modulo $m$ and $$L(\\chi,z)=\\sum_{n=1}^{\\infty}\\frac{\\chi(n)}{n^{z}}=\\prod_{p}\\left(1-\\frac{\\chi{(p)}}{p^{z}}\\right)^{-1} $$\nis the Dirichlet $L$-function. From the integral representation of the Gamma function\n$$\\int_{0}^{\\infty}e^{-nt}t^{z}\\frac{dt}{t}=\\frac{\\Gamma(z)}{n^{z}},$$\nwe have $$L(\\chi,z)=\\frac{1}{\\Gamma(z)} \\int_{0}^{\\infty}G_{\\chi}(t)t^{z-1}dt,\\quad\\textrm{Re}(z)>1,$$\nwhere \n$$\n\\begin{aligned}\nG_{\\chi}(t)&=\\sum_{n=1}^{\\infty}\\chi(n)e^{-nt}=\\sum_{r=1}^{m}\\chi(r)\\sum_{q=0}^{\\infty}e^{-(r+qm)t}\n\\\\\n&=\\sum_{r=1}^m\\frac{\\chi(r)e^{-rt}}{1-e^{-mt}}\n\\end{aligned}\n$$\n(see \\cite[p.~160]{Cohen} and \\cite[p.~102]{Iwa}).\nThus letting $f(t)=G_{\\chi}(-t)$ in (\\ref{(5)}) and (\\ref{(6)}), \nwe recover the Dirichlet $L$-function $L(\\chi,z)$ (see \\cite{Iwa}) and the two variable Dirichlet $L$-function $L(\\chi,z,x)$\n(see \\cite{fox}).\n\n\\item \nLet  \n$$\n\\lambda(z)=\\sum_{n=0}^\\infty\\frac1{(2n+1)^z},\\quad\\text{Re}(z)>1\n$$\nbe the Dirichlet lambda function according to Abramowitz and Stegun's handbook \\cite[pp.~807--808]{AS}, \nwhich has also been studied by Euler under the notation $N(z)$ (see \\cite[p.~70]{Varadarajan}).\nBy \\cite[(2.10)]{HK}, we have the following integral representation\n$$\n\\lambda(z)=\\frac1{\\Gamma(z)}\\int_0^\\infty\\frac{e^{t}t^{z-1}}{e^{2t}-1}dt, \\quad\\text{Re}(z)>1.\n$$\nThus letting $f(t)=\\frac{e^{-t}}{e^{-2t}-1}$ in (\\ref{(5)}), we recover the Dirichlet lambda function $\\lambda(z)$.\n\n\\item \nLet $w$ be an even integer, and let $S_{w+2}$ be the space of cusp forms with respect to $\\Gamma=SL(2,\\mathbb Z)\/(\\pm1)$ \non the upper half-plane of one complex variable. For $\\Phi\\in S_{w+2},$ \nwe write $\\Phi$ as a Fourier series\n$\\Phi(z)=\\sum_{n=1}^\\infty\\lambda_n e^{2\\pi inz}.$\nLet $L_\\Phi(z)$ be the Hecke series of $\\Phi.$ Namely,\n$$L_\\Phi(z)=\\sum_{n=1}^\\infty\\frac{\\lambda_n}{n^z},$$\nunder the condition: there exists $z_{0} > 0$, such that Re$(z)\\geq z_{0}$.\n\nIt is clear that $L_\\Phi(s)$ has the following expression:\n$$\\begin{aligned}\nL_\\Phi(z)\n&=\\frac{(2\\pi)^z}{\\Gamma(z)}\\sum_{n=1}^\\infty{\\lambda_n}\\int_0^\\infty e^{-2\\pi nt}t^{z-1}dt \\\\\n&=\\frac{(2\\pi)^z}{\\Gamma(z)}\\int_0^\\infty \\Phi(it)t^{z-1}dt.\n\\end{aligned}$$\nSo letting $f(-t)=(2\\pi)^z\\Phi(it)$ in (\\ref{(5)}), we recover the Hecke series $L_\\Phi(z)$\n(see \\cite{Man}).\n\\end{itemize}\n\nAssume the function $tf(t)$ is analytic on an open disc  of radius $r$ centered at 0 in the complex plane $\\mathbb{C}$, we have the following result.\n\n\\begin{proposition}\\label{2.1} \nLet $C$ be a loop around the negative real axis. If  $x\\in (0,\\infty)$ the function defined by the contour integral\n$$\nI(f,z,x)=\\frac{1}{2\\pi i}\\int_{C}  t^{z}e^{(1-x)t} f(t)\\frac{dt}{t}\n$$\nis an entire function of $z$.\nMoreover, we have\n$$\nL(f,z,x)=\\Gamma(1-z)I(f,z,x), \\quad{\\rm Re}(z)>1.\n$$\n\\end{proposition}\n\n\\begin{remark}\\label{rem-2} \nIf $\\textrm{Re}(z)\\leq 1$ and $x\\in (0,\\infty),$ we define $L(f,z,x)$ by the equation\n$$\nL(f,z,x)=\\Gamma(1-z)I(f,z,x).\n$$\nThis equation provides the analytic continuation of $L(f,z,x)$ in the entire complex plane. \n\\end{remark}\n\n\\begin{proof} The proof follows from similar arguments as \\cite[Theorem 12.3]{Apostol}.\n\nWe regard the interval $[0,\\infty)$ of integration as a path of a complex integral, and then expanding this a little. Here we consider the following\ncontour $C.$\nFor $\\varepsilon>0,$ we define $C$ by a curve $\\varphi:(-\\infty,\\infty)\\to\\mathbb C$ given by\n\\begin{equation}\nC  : \\quad \\varphi(u)=\\begin{cases}\nu &\\text{if }u<-\\varepsilon, \\\\\n\\varepsilon\\exp\\left(\\pi i \\frac{u+\\varepsilon}{\\varepsilon}\\right) &\\text{if }-\\varepsilon\\leq u\\leq \\varepsilon, \\\\\n-u &\\text{if }u>\\varepsilon.\n\\end{cases}\n\\end{equation}\nIn the definition of $C,$ the parts for $u<-\\varepsilon$ and $u>\\varepsilon$ overlap, but we interpret\nit that for $u<-\\varepsilon$ we take the path above the real axis and for $u>\\varepsilon$ below the real axis. This path is illustrated in Figure 1.\n\n\\begin{figure}[h]\n\\centerline{ \\includegraphics[width=3.5in, height=0.77in]{negative-2.png}}\n\\caption{Path of $C$}\n\\end{figure}\n\nNow consider the complex contour integral\n\\begin{equation}\\label{c-int}\n\\int_{C}t^{z}e^{(1-x)t}f(t)\\frac{dt}{t}.\n\\end{equation}\nSince we have to treat $t^{z-1}$ on $C,$ we shall choose a complex power $t^z$ for $z\\in\\mathbb C.$\nDenoting the argument of $t$ by $\\arg t,$ we can define a single-valued function $\\log t$ on\n$\\mathbb C-\\{s=a+ib\\mid b=0,a\\leq0\\}$ by\n\\begin{equation}\n\\log t=\\log|t|+i\\arg t \\quad(-\\pi<\\arg t<\\pi).\n\\end{equation}\nUsing this, a single-valued analytic function $t^z$ on $\\mathbb C-\\{s=a+ib\\mid b=0,a\\leq0\\}$\nis defined by\n\\begin{equation}\nt^z=e^{z\\log t}=e^{z(\\log|t|+i\\arg t)},\n\\end{equation}\nwhere $-\\pi<\\arg t<\\pi.$\nWe divide the contour $C$ into three pieces as follows.\n\n$\\quad C_1$: the part of the real axis from $-\\infty$ to $-\\varepsilon,$\n\n$\\quad C_2$: the positively oriented circle of radius $\\varepsilon$ with center at the origin,\n\n$\\quad C_3$: the part of the real axis from $-\\varepsilon$ to $-\\infty.$\n\n\\noindent\nWe have $C=C_1+C_2+C_3.$ For $t$ on $C_1,$ we put $\\arg t=-\\pi,$ and for $t$ on $C_{3},$ we put $\\arg t=\\pi.$\nThen the integral on $C_1$ is given by\n\\begin{equation}\n\\begin{aligned}\n\\int_{C_1}t^{z}e^{(1-x)t}f(t)\\frac{dt}{t}&=\\int_{\\infty}^\\varepsilon t^{z-1}e^{-\\pi iz}e^{-(1-x)t}f(-t)dt \\\\\n&=-e^{-\\pi iz}\\int^{\\infty}_\\varepsilon t^{z-1}e^{-(1-x)t}f(-t)dt\n\\end{aligned}\n\\end{equation}\nand on $C_3$ by\n\\begin{equation}\n\\begin{aligned}\n\\int_{C_3}t^{z}e^{(1-x)t}f(t)\\frac{dt}{t}&=\\int_\\varepsilon^{\\infty} t^{z-1}e^{\\pi iz}e^{-(1-x)t}f(-t)dt \\\\\n&=e^{\\pi iz}\\int^{\\infty}_\\varepsilon t^{z-1}e^{-(1-x)t}f(-t)dt.\n\\end{aligned}\n\\end{equation}\nThus, together we get\n\\begin{equation}\\label{inte 1-3}\n\\begin{aligned}\n\\int_{C}t^{z}e^{(1-x)t}f(t)\\frac{dt}{t}&=(e^{\\pi iz}-e^{-\\pi iz})\\int^{\\infty}_\\varepsilon t^{z-1}e^{-(1-x)t}f(-t)dt \\\\\n&+\\int_{C_2}t^{z}e^{(1-x)t}f(t)\\frac{dt}{t}.\n\\end{aligned}\n\\end{equation}\nThe circle $C_2$ is parameterized as $t=\\varepsilon e^{i\\theta}(-\\pi\\leq \\theta\\leq\\pi),$ so on $C_2$ we have\n$t^{z-1}=\\varepsilon^{z-1}e^{i(z-1)\\theta},$\nand the absolute value of the integral is estimated from above as\n\\begin{equation}\\label{inte-esi}\n\\begin{aligned}\n\\left|\\int_{C_2}t^{z}e^{(1-x)t}f(t)\\frac{dt}{t}\\right|\n&\\leq \\int_{-\\pi}^\\pi \\left| \\varepsilon^{z-1} e^{i(z-1)\\theta}e^{(1-x)\\varepsilon e^{i\\theta}}f(\\varepsilon e^{i\\theta})i\\varepsilon\ne^{i\\theta}\\right| d\\theta \\\\\n&=\\varepsilon^{\\text{Re}(z)}\n \\int_{-\\pi}^\\pi \\left| e^{iz\\theta}e^{(1-x)\\varepsilon e^{i\\theta}} f(\\varepsilon e^{i\\theta})\\right| d\\theta.\n\\end{aligned}\n\\end{equation}\nSince  $tf(t)$ is analytic on an open disc  of radius $r$ centered at 0, we have $\\varepsilon f(\\varepsilon e^{i\\theta})$ is bounded as a function of $\\varepsilon$ and $\\theta$\nif $\\varepsilon\\leq r\/2$  and $-\\pi\\leq \\theta\\leq\\pi$.\nSo if we take the limit $\\varepsilon\\to0$, then, for Re$(z)>1,$ by (\\ref{inte-esi}) we get\n\\begin{equation}\\label{inte-esi-0}\n\\lim_{\\varepsilon\\to0}\\int_{C_2}t^{z}e^{(1-x)t}f(t)\\frac{dt}{t}=0.\n\\end{equation}\nTherefore, using (\\ref{inte-esi-0}), we get\n\\begin{equation}\\label{inte-final}\n\\int_{C}t^{z}e^{(1-x)t}f(t)\\frac{dt}{t}=(e^{\\pi iz}-e^{-\\pi iz})\\int_0^{\\infty} t^{z-1}e^{-(1-x)t}f(-t)dt.\n\\end{equation}\nLet\n\\begin{equation}\nI(f,z,x)=\\frac{1}{2\\pi i}\\int_{C}  t^{z}e^{(1-x)t} f(t)\\frac{dt}{t}.\n\\end{equation}\nBy (\\ref{(6)}) and (\\ref{inte-final}), we have\n\\begin{equation}\\label{inte-final-1}\n\\begin{aligned}\nI(f,z,x)&=\\frac{1}{2\\pi i}\\int_{C}  t^{z}e^{(1-x)t} f(t)\\frac{dt}{t} \\\\\n&=\\frac{e^{\\pi iz}-e^{-\\pi iz}}{2\\pi i}\\int_0^{\\infty} t^{z-1}e^{-(1-x)t}f(-t)dt \\\\\n&=\\frac{\\sin\\pi z}{\\pi}\\Gamma(z)L(f,z,x) \\\\\n&=\\frac1{\\Gamma(1-z)}L(f,z,x),\n\\end{aligned}\n\\end{equation}\nwhich is the desired result.\n\\end{proof}\n\n\n\\section{Main results}\n\nDenote by $\\mathbb{R}^{+}=[0,\\infty)$ and $H=L^{2}(\\mathbb{R}^{+})$ be the Hilbert space of all complex valued functions $g,$\ndefined almost everywhere on $\\mathbb{R^{+}}$ and $|g|^{2}$ is Lebesgue integrable, with the pointwise operations\nof additions and the multiplication-by-scalars, and with the $L^{2}-$norm\n\\begin{equation}\n\\|g\\|_{2}=\\left(\\int_{\\mathbb{R}^{+}}|g(x)|^{2}dx\\right)^{1\/2}.\n\\end{equation}\nGive a function $f(t)$ such that $tf(-t)$ be a $C^{\\infty}$ function on $[0,\\infty)$\ntending rapidly to $0$ at infinity. Assume $tf(t)$ is analytic on an open disc  of radius $r$ centered at 0 in the complex plane $\\mathbb{C}$ and it has the following power series expansion at $t=0$\n\\begin{equation}\\label{(3.1)}\n(-t)f(t)=\\sum_{n=0}^{\\infty}a_{n}t^{n}\n\\end{equation}\nwith $a_{0}= 0$ and $a_{1}\\neq 0.$ Define an operator on $H$ by\n\\begin{equation}\\label{delta}\n\\hat{\\Delta}_{f}\\Psi(x)=\\frac{\\mathbbm{1}}{f(-i\\hat{p})}\\Psi(x),\n\\end{equation}\nfor any $\\Psi(x)\\in H$. Here we recall that $\\hat{p}=-i\\frac d{dx}$ denotes the momentum operator. Then by (\\ref{delta}), \n\\begin{equation}\n\\hat{\\Delta}_{f}^{-1}\\Psi(x)=f(-i\\hat{p})\\Psi(x),\n\\end{equation}\nand from (\\ref{(3.1)}), we get\n\\begin{equation}\\label{op-f}\n\\hat{\\Delta}_{f}^{-1}\\Psi(x)=f(-i\\hat{p})\\Psi(x)=\\frac{\\mathbbm{1}}{i\\hat{p}}\\sum_{n=0}^{\\infty}a_{n}(-i\\hat{p})^{n}\\Psi(x).\n\\end{equation}\nUnfortunately, the above series do not always converge. However, if we truncate it by setting\n\\begin{equation}\\label{delta-inv-tr}\ns_N(x)=\\frac{\\mathbbm{1}}{i\\hat p}\\sum_{n=0}^Na_{n}(-i\\hat p)^{n}\\Psi(x)\n\\end{equation}\nfor some integer $N\\geq1,$ then we require that\n\\begin{equation}\\label{delta-inv-tr1}\n\\hat{\\Delta}_{f}^{-1}\\Psi(x)=s_N(x)+O(x^{-N}) \\quad\\text{as } x\\to\\infty.\n\\end{equation}\n\n\nThe operator $\\hat{\\Delta}_{f}$ has the following two examples.\n\nFirst, letting $\\{a_n\\}=\\{B_n\\}$ in (\\ref{(3.1)}), where $B_n$ are the Bernoulli numbers (see \\cite[p. 803]{AS}). \nThen by (\\ref{op-f}), we obtain\n\\begin{equation}\\label{op-f-ex1}\n\\hat{\\Delta}_{f}^{-1}=f(-i\\hat p)=\\frac{\\mathbbm{1}}{i\\hat p}\\sum_{n=0}^{\\infty}B_{n}\\frac{(-i\\hat p)^{n}}{n!}.\n\\end{equation}\nFrom the generating function of Bernoulli numbers, we have\n\\begin{equation}\\label{op-f-ex1-1}\n\\hat{\\Delta}_{f}^{-1}=\\frac{\\mathbbm{1}}{i\\hat p}\\sum_{n=0}^{\\infty}B_{n}\\frac{(-i\\hat p)^{n}}{n!}=\\frac{\\mathbbm{1}}{i\\hat p}\\frac{-i\\hat p}{e^{-i\\hat p}-\\mathbbm{1}},\n\\end{equation}\nwhich is equivalent to $\\hat{\\Delta}_{f}=\\mathbbm{1}-e^{-i\\hat p}$ (see \\cite{Bender}).\n\nNext, let $\\{a_n\\}=\\{E_n(0)\\}$ in (\\ref{(3.1)}), where $E_n(x)$ are the Euler polynomials (see \\cite[p. 803]{AS}). \nSimilarly, by (\\ref{op-f}), we have\n\\begin{equation}\\label{op-f-ex2}\n\\hat{\\Delta}_{f}^{-1}=f(-i\\hat p)=\\frac{\\mathbbm{1}}{i\\hat p}\\sum_{n=0}^{\\infty}E_{n}(0)\\frac{(-i\\hat p)^{n}}{n!}.\n\\end{equation}\nFrom the generating function of Euler polynomials, we have\n\\begin{equation}\\label{op-f-ex2-1}\n\\hat{\\Delta}_{f}^{-1}=\\frac{\\mathbbm{1}}{i\\hat p}\\sum_{n=0}^{\\infty}E_{n}(0)\\frac{(-i\\hat p)^{n}}{n!}=\\frac{2}{i\\hat p}\\frac{\\mathbbm{1}}{e^{-i\\hat p}+\\mathbbm{1}},\n\\end{equation}\nwhich is equivalent to $\\hat{\\Delta}_{f}=\\frac12(i\\hat p)(\\mathbbm{1}+e^{-i\\hat p}).$\n\nWe are at the position to state the main result. \n\n\\begin{theorem}\\label{main}\nLet $\\hat{p}=-i\\frac d{dx}$ be the momentum operator \nand $\\hat{H}_{f}=\\hat{\\Delta}_{f}^{-1}(\\hat{x}\\hat{p}+\\hat{p}\\hat{x})\\hat{\\Delta}_{f}$ be an operator\non the Hilbert space $H=L^{2}(\\mathbb{R}^{+}),$ where $\\hat x$ is the position operator. If $z\\neq 1$, then\n$$\n\\hat{H}_{f}\\Psi(f,z,x)=i(2z-1)\\Psi(f,z,x),\n$$\nwhere $\\Psi(f,z,x)=(-1)^{z-1}L(f,z,x+1)$ and $L(f,z,x)$ is defined in (\\ref{(6)}) and Remark \\ref{rem-2}.\nFurthermore, if we propose the boundary condition\n$$\n\\Psi(f,z,0)=0\n$$\nto the differential equation $\\hat{H}_{f}\\Psi(f,z,x)=i(2z-1)\\Psi(f,z,x),$ then we have\nthe $n$th eigenstates of the Hamiltonian $\\hat{H}_{f}$ are\n$$\\Psi(f,z_{n},x)=(-1)^{z_{n}-1}L(f,z_{n},x+1),$$ \nthe $n$th eigenvalues are $E_{n}=i(2z_{n}-1)$ for all $n,$ and\nfrom the boundary condition, $z_{n}=\\frac{1}{2}(1-iE_{n})$ are the zeros of the general family zeta functions $L(f,z)\\equiv L(f,z,1)$.\n\\end{theorem}\n\\begin{proof} \nBy  (\\ref{op-f}), \n\\begin{equation}\\label{op-f-a}\n\\hat{\\Delta}_{f}^{-1}\\Psi(x)=\\frac{\\mathbbm{1}}{i\\hat{p}}\\sum_{n=0}^{\\infty}a_{n}(-i\\hat{p})^{n}\\Psi(x).\n\\end{equation}\nSince $\\hat{p}=-i\\frac d{dx}$ and $i\\hat{p}=\\frac d{dx}$, for $z\\neq 1$, we have\n\\begin{equation}\\label{op-xz-1}\n\\begin{aligned}\ni\\hat{p}\\left(\\frac{x^{1-z}}{1-z}\\right)&=i(-i)\\frac d{dx}\\left[\\frac{x^{1-z}}{1-z}\\right]\\\\\n&=\\frac d{dx}\\left(\\frac{x^{1-z}}{1-z}\\right)\\\\\n&=x^{-z}.\n\\end{aligned}\n\\end{equation}Thus\n\\begin{equation}\\label{op-xz-1}\n\\frac{\\mathbbm{1}}{i\\hat{p}}x^{-z}=\\frac{x^{1-z}}{1-z}.\n\\end{equation}\nBy (\\ref{op-f-a}) with $\\Psi(x)=x^{-z}$ and (\\ref{op-xz-1}), we have\n\\begin{equation}\\label{(7)}\n\\begin{aligned}\n\\hat{\\Delta}_{f}^{-1}x^{-z}\n&=\\sum_{n=0}^{\\infty}a_{n}(-i\\hat{p})^{n}\\frac{\\mathbbm{1}}{i\\hat{p}}x^{-z} \\\\\n&=\\sum_{n=0}^{\\infty}a_{n}(-i\\hat{p})^{n}\\left(\\frac{x^{1-z}}{1-z}\\right)\\\\\n&=\\frac{1}{1-z}\\sum_{n=0}^{\\infty}a_{n}(-i\\hat{p})^{n}x^{1-z}.\n\\end{aligned}\n\\end{equation}\nSince $i\\hat{p}=\\frac d{dx}$ and $\\left(\\frac {d}{dx}\\right)^nx^{\\mu}=\\frac{\\Gamma(\\mu+1)}{\\Gamma(\\mu-n+1)}x^{\\mu-n}$, from (\\ref{(7)}), setting $\\mu=1-z,$ we obtain the asymptotic series (see \\cite[(4)]{Bender})\n\\begin{equation}\\label{(8)}\n \\hat{\\Delta}_{f}^{-1}x^{-z}\n=\\frac{\\Gamma(2-z)}{1-z}\\sum_{n=0}^{\\infty}a_{n}(-1)^{n}\\frac{x^{1-z-n}}{\\Gamma(2-z-n)},\n \\end{equation}\n which is valid under the limit $x\\to\\infty.$\n Since $\\Gamma(2-z-n)$ has the following integral representation\n \\begin{equation*}\n \\frac{1}{\\Gamma(2-z-n)}=\\frac{1}{2\\pi i}\\int_{C}e^{u}u^{n+z-2}du,\n \\end{equation*}\n where $C$ denotes a Hankel contour that encircles the negative-$u$ axis in the positive orientation (see \\cite[p.~255]{AS}),\nthen by (\\ref{(3.1)}) and (\\ref{(8)}) we have\n \\begin{equation}\\label{(9)}\n \\begin{aligned}\n \\hat{\\Delta}_{f}^{-1}x^{-z}\n&=\\frac{\\Gamma(1-z)}{2\\pi i}x^{1-z}\\int_{C}e^{u}u^{z-2}du\\sum_{n=0}^{\\infty}a_{n}\\left(-\\frac{u}{x}\\right)^{n}\\\\\n&=\\frac{\\Gamma(1-z)}{2\\pi i}x^{1-z}\\int_{C}e^{u}u^{z-2}f\\left(-\\frac{u}{x}\\right)\\left(\\frac{u}{x}\\right)du.\n\\end{aligned}\n \\end{equation}\n Letting $\\frac{u}{x}=t$ in (\\ref{(9)}), we have\n \\begin{equation}\n \\begin{aligned}\n \\hat{\\Delta}_{f}^{-1}x^{-z}&=\\frac{\\Gamma(1-z)}{2\\pi i}\\int_{C} t^{z}e^{xt} f(-t)\\frac{dt}{t}\\\\\n &=(-1)^{z-1}\\frac{\\Gamma(1-z)}{2\\pi i}\\int_{C}t^{z}e^{-xt} f(t)\\frac{dt}{t} \\\\\n &=(-1)^{z-1}\\Gamma(1-z)\\frac{1}{2\\pi i}\\int_{C}t^{z}e^{(1-(x+1))t} f(t)\\frac{dt}{t}.\n \\end{aligned}\n \\end{equation}\nTherefore, by Proposition \\ref{2.1} and Remark \\ref{rem-2},\n\\begin{equation}\\label{(10)}\n \\hat{\\Delta}_{f}^{-1}x^{-z}=(-1)^{z-1}L(f,z,x+1).\n \\end{equation}\n Note that\n \\begin{equation}\\label{note}\n \\begin{aligned}\n (\\hat x\\hat p+\\hat p\\hat x)(x^{-z})&=-i\\left(2x\\frac d{dx}+\\mathbbm{1}\\right)(x^{-z}) \\\\\n &=-i\\left(2x\\frac d{dx}(x^{-z}) +\\mathbbm{1}(x^{-z}) \\right) \\\\\n &=-i\\left(2x(-z)x^{-z-1}+x^{-z}\\right) \\\\\n &=i(2z-1)x^{-z}.\n \\end{aligned}\n \\end{equation}\n Let $\\hat{H}_{f}=\\hat{\\Delta}_{f}^{-1}(\\hat{x}\\hat{p}+\\hat{p}\\hat{x})\\hat{\\Delta}_{f}$.\n Denote by $\\Psi(f,z,x)=(-1)^{z-1}L(f,z,x+1)$,\n from (\\ref{(10)}) and (\\ref{note}), we have $\n \\Psi(f,z,x)=\\hat{\\Delta}_{f}^{-1}x^{-z}$ and\n \\begin{equation}\\label{(11*)}\n \\begin{aligned}\n \\hat{H}_{f}\\Psi(f,z,x)&=\\hat{\\Delta}_{f}^{-1}(\\hat{x}\\hat{p}+\\hat{p}\\hat{x})\\hat{\\Delta}_{f}(\\hat{\\Delta}_{f}^{-1} x^{-z})\\\\\n &=\\hat{\\Delta}_{f}^{-1}(\\hat{x}\\hat{p}+\\hat{p}\\hat{x})x^{-z}\\\\\n &=\\hat{\\Delta}_{f}^{-1}[i(2z-1)x^{-z}]\\\\\n &=i(2z-1)\\hat{\\Delta}_{f}^{-1}x^{-z}\\\\\n &=i(2z-1)\\Psi(f,z,x).\n \\end{aligned}\n \\end{equation}\n\nThus if we propose the boundary condition\n$\\Psi(f,z,0)=0$ to the differential equation (\\ref{(11*)}), then we have\nthe $n$th eigenstates of the Hamiltonian $\\hat{H}_{f}$ are\n\\begin{equation}\\label{Eq1}\n\\Psi(f,z_{n},x)=(-1)^{z_{n}-1}L(f,z_{n},x+1),\n\\end{equation} \nthe $n$th eigenvalues are $E_{n}=i(2z_{n}-1)$ for all $n.$\nFrom the boundary condition $\\Psi(f,z,0)=0,$ \nwe have $\\Psi(f,z_{n},0)=0$ and by (\\ref{Eq1}) $L(f,z_{n},1)=0,$\nthus $z_{n}=\\frac{1}{2}(1-iE_{n})$ are the zeros of the general family zeta functions $L(f,z)$  \n(since by (\\ref{(6)}) $L(f,z,1)=L(f,z)$).\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe growing population of known luminous~($\\sim10^{47}~\\mathrm{erg~s^{-1}}$) quasars at the highest redshifts~($z\\gtrsim6$) shows that some supermassive black holes~(SMBHs) were already in place within the first $\\sim1~\\mathrm{Gyr}$ since the Big Bang. The inferred masses of these quasars range from $\\sim10^9-10^{10}~M_{\\odot}$, similar to the most massive SMBHs in the local universe.\nTo date, $\\gtrsim200$ quasars have already been discovered at $z\\gtrsim6$~\\citep{2001AJ....122.2833F,2010AJ....139..906W,2015MNRAS.453.2259V,2016ApJ...833..222J,2016Banados,2017MNRAS.468.4702R,2018ApJS..237....5M}, which overall correspond to number densities of $1~\\mathrm{cGpc}^{-3}$. Additionally, there are\na handful of objects discovered at $z\\gtrsim7$~\\citep{2011Natur.474..616M,2018ApJ...869L...9W,2019ApJ...872L...2M,2019AJ....157..236Y}, which includes the most distant quasars observed to date \\citep{2018Natur.553..473B,2021ApJ...907L...1W} at $z\\sim7.6$. The recently launched James Webb Space Telescope\n\\citep[JWST;][]{2006SSRv..123..485G} and planned facilities such as Lynx X-ray Observatory  \n\\citep{2018arXiv180909642T} have a promising prospect of revealing the AGN~(active galactic nuclei) progenitors of these quasars at even higher redshifts. Additionally, gravitational wave events from Laser Interferometer Space Antenna \n\\citep[LISA;][]{2019arXiv190706482B} will also provide insights into the prevalence of BH mergers and the growth history of these quasars. These observations are going to be crucial to understanding the assembly of these quasars, which is an outstanding challenge for theoretical models of BH formation and growth.  \n\n\nThe origin\nof these $z\\gtrsim6$ quasars, and the larger SMBH populations in general, is a subject of active debate. Remnants of the first generation of Pop III stars, a.k.a Pop III BH seeds, are popular candidates  ~\\citep{2001ApJ...550..372F,2001ApJ...551L..27M,2013ApJ...773...83X,2018MNRAS.480.3762S}. The BH seed mass that results from the conjectured Pop III scenario depend on the initial mass function of Pop III stars themeselves. This is predicted to be more top heavy than present day stellar populations, with masses typically ranging from $\\sim10-100~M_{\\odot}$~\\citep{2014ApJ...781...60H, 2016ApJ...824..119H}. But even the most massive Pop III seeds~(initial BH masses of $\\sim10^2~M_{\\odot}$)  would require significant periods of super-Eddington accretion to grow by $\\gtrsim7$ orders of magnitude to form a $z\\gtrsim6$ quasar. These stringent growth rate requirements can be alleviated to an extent with channels producing more massive seeds. Theories proposed for massive seed formation\ninclude runaway collisions of stars or black holes in dense nuclear star clusters forming ``NSC seeds\" with masses $\\sim10^{2}-10^{3}~M_{\\odot}$ ~\\citep{2011ApJ...740L..42D,2014MNRAS.442.3616L,2020MNRAS.498.5652K,2021MNRAS.503.1051D,2021MNRAS.tmp.1381D}, and direct collapse of gas in atomic cooling~($T_{\\mathrm{vir}}>10^{4}~\\mathrm{K}$) halos forming ``direct collapse black hole (DCBH) seeds\" with masses $\\sim10^{4}-10^{6}~M_{\\odot}$~\\citep{2003ApJ...596...34B,2006MNRAS.370..289B,2014ApJ...795..137R,2016MNRAS.458..233L,2018MNRAS.476.3523L,2019Natur.566...85W,2020MNRAS.492.4917L}. \n\nThe most massive DCBH seeds are seen as promising candidates for explaining the rapid formation of the $z\\gtrsim6$ quasars. Their formation requires gas to undergo an isothermal collapse at temperatures $\\gtrsim10^4~\\mathrm{K}$~(corresponding to a Jean's mass $\\gtrsim10^{4}~M_{\\odot}$). For this to occur, the gas needs to be devoid of chemical species that are efficient coolants at $\\lesssim10^4~\\mathrm{K}$, namely metals and molecular hydrogen. To suppress molecular hydrogen, the gas must be exposed to Lyman Werner radiation with minimum fluxes $\\gtrsim1000~J_{21}$ as inferred from small scale hydrodynamic simulations~\\citep{2010MNRAS.402.1249S} as well as one-zone chemistry models~\\citep{2014MNRAS.445..544S,2017MNRAS.469.3329W}. Such high fluxes can only be provided by nearby star forming galaxies~\\citep{2014MNRAS.445.1056V,2017NatAs...1E..75R,2021MNRAS.503.5046L,2021MNRAS.tmp.3110B}. However, these star forming regions can also pollute the gas with metals, which would then eliminate any possibility of direct collapse. Overall, this implies that the window for DCBH seed formation is extremely narrow, and it is unclear whether they form abundantly enough to explain the inferred densities of these objects. \n\n\nSemi analytic models~(SAMs) have\nso far been extensively used in the modeling of black hole seeds~\\citep{2007MNRAS.377.1711S,Volonteri_2009, 2012MNRAS.423.2533B,2018MNRAS.476..407V, 2018MNRAS.481.3278R, 2019MNRAS.486.2336D, 2020MNRAS.491.4973D}. Several such SAMs have also been used to study the feasibility of different seeding channels as possible origins of $z\\gtrsim6$ quasars. For example, \\cite{2011MNRAS.416.1916V,2012MNRAS.427L..60V,2014MNRAS.444.2442V} developed the \\texttt{GAMETTE-QSODUST} data constrained SAM to probe the $z\\gtrsim6$ quasars and their host galaxies. This model was used in \\cite{2016MNRAS.457.3356V} and \\cite{2021MNRAS.506..613S}, showing that the formation of heavy seeds~($\\sim10^5~M_{\\odot}$) is most crucial to the assembly of the first quasars, particularly in models where the BH accretion rate is capped at the Eddington limit. \\cite{2016MNRAS.458.3047P} and \\cite{2017MNRAS.471..589P} showed that light seeds~($\\sim10^2~M_{\\odot}$) require super-Eddington accretion to grow into the $z\\gtrsim6$ quasars. \\cite{2021MNRAS.503.5046L} applied a semi-analytic framework on a dark matter only simulation of a $3\\times10^{12}~M_{\\odot}$ halo forming at $z=6$~(presumably hosting a luminous quasar); they demonstrated that the progenitors of this halo can be sites for the formation of massive DCBH seeds.\n\nWhile SAMs, being computationally inexpensive, can probe a wide range of seed models relatively quickly, they are unable to self-consistently track the hydrodynamics of gas. Cosmological hydrodynamic simulations~\\citep{2012ApJ...745L..29D,2014Natur.509..177V,2015MNRAS.452..575S,2015MNRAS.450.1349K,2015MNRAS.446..521S,2016MNRAS.460.2979V,2016MNRAS.463.3948D,2017MNRAS.467.4739K,2017MNRAS.470.1121T,2019ComAC...6....2N,2020MNRAS.498.2219V} are more readily able to decipher the role of gas hydrodynamics in forming the $z\\gtrsim6$ quasars~(see, e.g., the review by \\citealt{2020NatRP...2...42V}). Since the $z\\gtrsim6$ quasars are extremely rare, we need extremely large volumes to probe these objects~(note however that they are much more computationally expensive than SAMs). \\texttt{MassiveBlack}~\\citep{2012ApJ...745L..29D}, with a volume of $[0.75~\\mathrm{Gpc}]^3$, revealed that $z\\gtrsim6$ quasars can form in extremely massive halos~($\\gtrsim10^{12}~M_{\\odot}\/h$ at $z\\sim6$) via a steady inflow of cold gas driving sustained accretion rates close to the Eddington limit. This was further confirmed using follow-up zoom simulations at much higher resolutions~\\citep{2014MNRAS.440.1865F}. \\texttt{BlueTides}~\\citep{2016MNRAS.455.2778F}, with a volume of $[0.5~\\mathrm{Gpc}]^3$~(but higher resolution compared to \\texttt{MassiveBlack}), further revealed the role of higher order features~(particularly low tidal fields, see \\citealt{2017MNRAS.467.4243D}) of the initial density field in producing the fastest accretion rates necessary to assemble the $z\\gtrsim7$ quasars. \n\nThe results of \\cite{2017MNRAS.467.4243D} motivated \\cite{2021MNRAS.tmp.2867N}~(hereafter N21), which was a systematic study of the impact of higher order features of rare density peaks on the subsequent black hole~(BH) growth. Using the method of constrained Gaussian realizations~\\citep{1991ApJ...380L...5H,1996MNRAS.281...84V}, N21 was able to generate initial conditions comprising of the rarest density peaks with the desired higher order features~(i.e. 1st and 2nd order derivatives). They demonstrated that highly compact peaks with low tidal fields led to the fastest BH growth. Due to their finite resolution however, cosmological hydrodynamic simulations are limited in terms of their ability to probe low-mass\nBH seeding channels.  Consequently, the vast majority of the simulations targeting $z\\gtrsim6$ quasars described in the previous paragraph~(also including \\citealt{2009MNRAS.400..100S, 2014MNRAS.439.2146C, 2016MNRAS.457L..34C, 2020arXiv201201458Z}) used simple halo based seeding prescriptions~(seeds are inserted in halos above a prescribed halo mass) that do not distinguish between different physical seeding channels. Therefore, while all these simulations have been generally successful in broadly reproducing the $z\\gtrsim6$ quasars, their ability to reveal insights into the seeding environments of these objects is still limited. With upcoming LISA measurements being amongst the most promising probes for revealing the mechanism of BH seed formation, the time is ripe for developing simulations that can reliably distinguish between different BH seeding channels.\n\nNumerous studies have implemented\ngas-based black hole seeding prescriptions~\\citep{2011ApJ...742...13B,2013MNRAS.428.2885D,2014MNRAS.442.2304H,2015MNRAS.448.1835T,2016MNRAS.460.2979V,2017MNRAS.470.1121T,2017MNRAS.467.4739K,2017MNRAS.468.3935H,2019MNRAS.486.2827D,2020arXiv200601039L,2020arXiv200204045T}. \\citet{2021MNRAS.507.2012B,2021MNRAS.tmp.3110B} have recently conducted\na systematic study \nto assess the impact of gas-based black hole seeding prescriptions on $z\\gtrsim7$ SMBH populations.\nThese seed models are built on the framework of the \\texttt{IllustrisTNG} galaxy formation model~\\citep{2017MNRAS.465.3291W,2018MNRAS.473.4077P}. They\nseeded black holes in halos via criteria based on dense, metal poor gas mass, halo mass, gas spin as well as incident Lyman Werner~(LW) flux. The resulting family of models is generally agnostic about which theoretical seeding channels they represent, but their parameters could be tuned to represent any of the seeding channels described above\n(PopIII, NSC or DCBH).\nBy applying these models to zoom simulations of modestly overdense regions~($3.3\\sigma$ overdensity, targeting a $\\sim10^{11}~M_{\\odot}\/h$ halo at $z=5$), they\nfound that changing different seed parameters would leave qualitatively distinct imprints on the BH merger rates. In particular, \\cite{2021MNRAS.507.2012B} found that when the dense, metal poor gas mass threshold is increased, it suppresses the seeding and merger rates more strongly at $z\\lesssim15$ compared to higher redshifts. On the other hand, an increase in the total halo mass threshold for seeding causes stronger suppression of seeding and merger rates at $z\\sim11-25$ compared to $z\\lesssim11$. These results suggest that discrepancies between the merger rates of LISA binaries will contain insights into their seeding environments. \\cite{2021MNRAS.tmp.3110B} found that even when a moderately low LW flux threshold~($\\gtrsim50~J_{21}$) is adopted for seeding, it can dramatically suppress seed formation and prevent the assembly of $z\\gtrsim7$ SMBHs. This suggests that the bulk of the $z\\gtrsim7$ SMBH population~(likely revealed by JWST and Lynx) may not originate from DCBH seeding channels.   \n\n\nThe zoom regions of \\cite{2021MNRAS.507.2012B,2021MNRAS.tmp.3110B} were not nearly overdense enough to be possible sites for the formation of $z\\gtrsim6$ quasars. In this work, we use constrained Gaussian realizations of extreme overdense regions~($\\gtrsim5\\sigma$ overdensities forming $\\gtrsim10^{12}~M_{\\odot}\/h$ halos by $z\\sim7$), and investigate the impact of BH seed models on the formation of the $z\\gtrsim6$ quasars. Apart from the seed models, our underlying galaxy formation model is adopted from the \\texttt{IllustrisTNG} simulation suite. \n\nSection \\ref{Simulation_setup_sec} describes the simulation setup, including the main features of the \\texttt{IllustrisTNG} galaxy formation model, the BH seeding and accretion models, and the generation of the constrained initial conditions. Section \\ref{Results_sec} describes the main results concerning the impact of environment, seeding and accretion models on BH growth. Finally, Section \\ref{Conclusions_sec} summarizes the main conclusions of our work.\n\n\\section{Simulation setup}\n\\label{Simulation_setup_sec}\nOur simulations were run using the \\texttt{AREPO} code~\\citep{2010MNRAS.401..791S,2011MNRAS.418.1392P,2016MNRAS.462.2603P,2020ApJS..248...32W}, which\nincludes a gravity and magneto-hydrodynamics~(MHD) solver. The simulations are cosmological in nature, which are performed within a representative portion of an expanding universe described by a fixed comoving volume~($9~\\mathrm{cMpc\/h}$ box size) with the following cosmology adopted from \\cite{2016A&A...594A..13P}: ($\\Omega_{\\Lambda}=0.6911, \\Omega_m=0.3089, \\Omega_b=0.0486, H_0=67.74~\\mathrm{km}~\\mathrm{sec}^{-1}\\mathrm{Mpc}^{-1},\\sigma_8=0.8159,n_s=0.9667$). The code uses a PM-Tree~\\citep{1986Natur.324..446B} method to solve for gravity, which is contributed by dark matter, gas, stars and BHs. Within the resulting gravitational potential, the gas dynamics is computed by the MHD solver, which uses a quasi-Lagrangian description of the fluid within an unstructured grid generated via a Voronoi tessellation of the domain.\n\nOur galaxy formation model is adopted from the \\texttt{IllustrisTNG} simulation suite~\\citep{2018MNRAS.475..676S,2018MNRAS.475..648P,2018MNRAS.475..624N,2018MNRAS.477.1206N,2018MNRAS.480.5113M,2019ComAC...6....2N} \\citep[see also][]{2018MNRAS.479.4056W,2018MNRAS.474.3976G,2019MNRAS.485.4817D,2019MNRAS.484.5587T,2019MNRAS.483.4140R,2019MNRAS.490.3196P,2021MNRAS.500.4597U,2021MNRAS.503.1940H}. The only substantive changes to the galaxy formation implemented here are in the sub-grid prescriptions\nfor BH seeding and accretion. The remaining aspects of our galaxy formation model are the same as \\texttt{IllustrisTNG} which are detailed in \\cite{2017MNRAS.465.3291W} and \\cite{2018MNRAS.473.4077P}; here, we provide a brief summary: \n\n\\begin{itemize}\n    \\item Energy loss via radiative cooling includes contributions from primodial species~($\\mathrm{H},\\mathrm{H}^{+},\\mathrm{He},\\mathrm{He}^{+},\\mathrm{He}^{++}$, based on \\citealt{1996ApJS..105...19K}), as well as metals~(using pre-calculated tables for cooling rates as in \\citealt{2008MNRAS.385.1443S}) in the presence of a spatially uniform, time dependent UV background. Note that cooling due to molecular Hydrogen~($H_2$) is not explicitly included in the model.\n    \\item Stars are stochastically formed within gas cells with densities exceeding $0.1~\\mathrm{cm}^{-3}$ with an associated time scale of $2.2~\\mathrm{Gyr}$. The star forming gas cells then represent an unresolved multiphase interstellar medium, which is modeled by an effective equation of state~\\citep{2003MNRAS.339..289S,2014MNRAS.444.1518V}. The model implicitly assumes that stars are produced within an unresolved cold dense component in these gas cells, which would presumably form via $H_2$ cooling.  \n    \\item The stellar evolution model is adopted from \\cite{2013MNRAS.436.3031V} with modifications for \\texttt{IllustrisTNG} as in \\cite{2018MNRAS.473.4077P}. Star particles represent a single stellar population with fixed age and metallicity. The initial mass function is assumed to be \\cite{2003PASP..115..763C}. The stellar evolution subsequently leads to chemical enrichment, wherein the evolution of seven species of metals~(C, N, O, Ne, Mg, Si, Fe) are individually tracked in addition to H and He.  \n    \n    \\item Feedback from stars and Type Ia\/II Supernovae are modelled as galactic scale winds~\\citep{2018MNRAS.475..648P}, via which mass, momentum and metals are deposited on to the gas surrounding the star particles. \n    \n\\item Models for BH formation and growth are detailed in the next two subsections. The treatment of BH dynamics and mergers is the same as in  \\texttt{IllustrisTNG}. Due to the limited gas mass resolution, our simulations cannot self consistently reveal the small-scale dynamics of BHs, particularly at their lowest masses. To stabilize the BH dynamics, they are ``re-positioned\" to the nearest potential minimum within its ``neighborhood\"~(defined by $10^3$ nearest neighboring gas cells). As a result, a BH is also promptly merged when it is within the neighborhood of another BH.     \n\\end{itemize}\n\\subsection{Black hole seeding}\n\\label{Black hole seeding}\nWe consider a range of BH seeding prescriptions, which include the default halo based seeding prescription of \\texttt{IllustrisTNG} where seeds of mass $8\\times10^5~M_{\\odot}$ are inserted in halos which exceed a threshold mass of $5\\times10^{10}~M_{\\odot}\/h$ and do not already contain a BH~(hereafter referred to as the ``TNG seed model\"). \n\nAdditionally, we explore the gas-based seeding prescriptions developed in \\cite{2021MNRAS.507.2012B} and \\cite{2021MNRAS.tmp.3110B}. These are comprised of a combination of seeding criteria based on various gas properties of halos. These criteria are designed such that our overall family of seed models broadly encompasses popular theoretical channels such as Pop III, NSC and DCBH seeds, all of which exclusively form in regions comprised of dense, metal poor gas.  Here we briefly summarize them as follows: \n\\begin{itemize}\n\\item \\textit{Dense, metal poor gas mass criterion:} Seeds can only form in halos that exceed a threshold for dense~($>0.1~\\mathrm{cm}^{-3}$), metal poor~($Z<10^{-4}~Z_{\\odot}$) gas mass, specified by $\\tilde{M}_{\\mathrm{sf,mp}}$ in the units of the seed mass $M_{\\mathrm{seed}}$.\n\\item \\textit{Halo mass criterion:} Seeds can only form in halos that have exceeded a threshold for the total halo mass, specified by $\\tilde{M}_{h}$ in the units of the seed mass $M_{\\mathrm{seed}}$.\n\n\\item \\textit{LW flux criterion:} In selected models, we also require the dense, metal poor gas to be exposed to Lyman Werner~(LW) fluxes above a critical value $J_{\\mathrm{crit}}$. More specifically, seeds only form in halos with a minimum threshold for dense, metal poor, LW illuminated gas mass, denoted by  $\\tilde{M}_{\\mathrm{sf,mp,LW}}$ in the units of the seed mass $M_{\\mathrm{seed}}$. Star formation is suppressed in these seed forming regions. Given that our simulations do not contain full radiative transfer, the LW flux from Pop III and Pop II stars is computed using an analytic prescription described in \\cite{2021MNRAS.tmp.3110B}.\n\\end{itemize}\n\nOur seed model is therefore described by four parameters, namely $\\tilde{M}_{\\mathrm{sf,mp}}$, $\\tilde{M}_{\\mathrm{h}}$, $J_{\\mathrm{crit}}$ and $M_{\\mathrm{seed}}$. All of our simulations include the first two parameters, and throughout the text\nthe \\textit{dense, metal poor gas mass criterion} and \\textit{halo mass criterion} are labelled as \\texttt{SM*_FOF*} where the `*'s correspond to the values of $\\tilde{M}_{\\mathrm{sf,mp}}$ and $\\tilde{M}_{\\mathrm{h}}$. For example, $\\tilde{M}_{\\mathrm{sf,mp}}=5$ and $\\tilde{M}_{\\mathrm{h}}=3000$ will correspond to \\texttt{SM5_FOF3000}. Runs which additionally apply the \\textit{LW flux criterion} contain an extra suffix \\texttt{LW*} where `*' corresponds to $J_{\\mathrm{crit}}$; for example, if a criterion with $J_{\\mathrm{crit}}=300~J_{21}$ is added to \\texttt{SM5_FOF3000}, it will be labeled as \\texttt{SM5_FOF3000_LW300}. Lastly, the seed mass $M_{\\mathrm{seed}}$ will be explicitly stated in the text and figure legends.\n\n\\subsection{BH accretion and feedback models}\nBlack holes grow via a modified Bondi-Hoyle accretion prescription, with\nthe maximum accretion rate limited to some factor $f_{\\rm Edd} \\geq 1$ times\nthe Eddington accretion rate (which we refer to as the `Eddington factor'):\n\\begin{eqnarray}\n\\dot{M}_{\\mathrm{BH}}=\\mathrm{min}(\\alpha \\dot{M}_{\\mathrm{Bondi}}, \\: f_{\\mathrm{edd}}\\dot{M}_{\\mathrm{Edd}})\\\\\n\\dot{M}_{\\mathrm{Bondi}}=\\frac{4 \\pi G^2 M_{\\mathrm{BH}}^2 \\rho}{c_s^3}\\\\\n\\dot{M}_{\\mathrm{Edd}}=\\frac{4\\pi G M_{\\mathrm{BH}} m_p}{\\epsilon_r \\sigma_T~c}\n\\label{bondi_eqn}\n\\end{eqnarray}\n$\\alpha$ is referred to as the `Bondi boost' factor which is often used to boost the accretion rate to account for the inability to resolve the small scale vicinity of the BH. $G$ is the gravitational constant, $\\rho$ is the local gas density, $M_{\\mathrm{BH}}$ is the BH mass, $c_s$ is the local sound speed, $m_p$ is the proton mass, and $\\sigma_T$ is the Thompson scattering cross section.  In practice, ``local'' quantities are calculated as the kernel-weighted averages over nearby particles, typically those within a few $\\times h^{-1} \\mathrm{pc}$.  Accreting black holes radiate at luminosities given by, \n\\begin{equation}\n    L=\\epsilon_r \\dot{M}_{\\mathrm{BH}} c^2,\n    \\label{bol_lum_eqn}\n\\end{equation}\nwhere $\\epsilon_r$ is the radiative efficiency.\n\nIn the IllustrisTNG implementation, AGN feedback occurs both in `thermal mode' as well as `kinetic mode'. For Eddington ratios~($\\eta \\equiv \\dot{M}_{\\mathrm{bh}}\/\\dot{M}_{\\mathrm{edd}}$) higher than a critical value of  $\\eta_{\\mathrm{crit}}=\\mathrm{min}[0.002(M_{\\mathrm{BH}}\/10^8 M_{\\odot})^2,0.1]$, thermal energy is deposited on to the neighboring gas at a rate of $\\epsilon_{f,\\mathrm{high}} \\epsilon_r \\dot{M}_{\\mathrm{BH}}c^2$ where $ ~\\epsilon_{f,\\mathrm{high}} \\epsilon_r=0.02$. $\\epsilon_{f,\\mathrm{high}}$ is called the ``high accretion state\" coupling efficiency. If the Eddington ratio is lower than the critical value, kinetic energy is injected into the gas at irregular time intervals, which manifests as a `wind' oriented along a randomly chosen direction. The injected rate is $\\epsilon_{f,\\mathrm{low}}\\dot{M}_{\\mathrm{BH}}c^2$ where $\\epsilon_{f,\\mathrm{low}}$ is called the `low accretion state' coupling efficiency~($\\epsilon_{f,\\mathrm{low}} \\lesssim 0.2$). For further details, we direct the interested readers to \\cite{2017MNRAS.465.3291W}. \n\nThe main parameters of our accretion model include the Bondi boost $\\alpha$, the radiative efficiency $\\epsilon_r$ and the Eddington factor $f_{\\mathrm{edd}}$. The default values adopted in the \\texttt{IllustrisTNG} suite are $\\alpha=1,\\epsilon_r=0.2~\\&~f_{\\mathrm{edd}}=1$. We largely use this accretion model, and hereafter refer to it as the ``TNG accretion model''. However, we also run some simulations with different variations of these parameters, particularly when comparing our results to other studies. These variations include different combinations of $\\alpha=1~\\&~100$, $\\epsilon_r=0.2~\\&~0.1$ and  $f_{\\mathrm{edd}}=1-100$. In the figure legends, these are labelled as \\texttt{Boost*_RadEff*_EddFac*} where the `*'s correspond to values of $\\alpha$, $\\epsilon_r$ and $f_{\\mathrm{edd}}$ respectively.\n\n\n\n\n\n\\subsection{Initial Conditions: constrained Gaussian realizations}\n\nWe expect the brightest $z>6$ quasars to live in the rarest and most extreme overdensities in the Universe. In order to create initial conditions~(ICs) that can produce such regions within a relatively small $9~\\mathrm{cMpc}\/h$ box, we apply the technique of \\textit{constrained Gaussian realizations} (CR). \nThe CR method can efficiently sample a Gaussian Random field conditioned on various (user-specified) large-scale features.\nThis technique was originally introduced by \\cite{1991ApJ...380L...5H} and \\cite{1996MNRAS.281...84V}. We use the most recent implementation of this technique, i.e. the \\href{https:\/\/github.com\/yueyingn\/GaussianCR}{\\texttt{GaussianCR}} code to generate the initial conditions. This code was fully developed by N21, wherein it was extensively tested against large volume uniform cosmological  simulations in terms of reproducing the halo assembly, star formation and BH growth histories. Here we briefly summarize the main features for completeness, while the full details of the underlying formalism are described in N21.   \n\nOverall, \\href{https:\/\/github.com\/yueyingn\/GaussianCR}{\\texttt{GaussianCR}} constrains 18 parameters at the peak location~(see N21 for details). In this work, we vary three parameters that were shown by N21 to be most consequential to BH growth. These are the following:\n\\begin{itemize}\n    \\item The peak height `$\\nu$' quantifies the `rarity' of the peak by specifying its height in the units of the variance of $\\delta_G (\\textbf{r})$ denoted by $\\sigma_{R_G}$, i.e.\n    \\begin{eqnarray}\n   \n        \\nu \\equiv \\delta(\\textbf{r}_{\\mathrm{peak}})\/\\sigma_{R_G}, \\\\\n        \\sigma_{R_G}^2\\equiv\\left<\\delta_G (\\textbf{r}) \\delta_G (\\textbf{r})\\right>= \\int \\frac{P(k)}{(2\\pi)^3} \\hat{W}^2(k,R_G) dk \n    \\end{eqnarray}\n    where $\\textbf{r}_{\\mathrm{peak}}$ is the peak position. $P(k)$ is the power spectrum and $\\tilde{\\delta}_G(\\textbf{r})\\equiv \\int \\tilde{\\delta} (\\textbf{r}) W(\\textbf{r},R_G)$ is the overdensity smoothened over a scale $R_G$ using the Gaussian window function $W(\\textbf{r},R_G)= \\exp(-r^2\/2R_G^2)$ and its Fourier transform $\\hat{W}(k,R_G)=\\exp(-k^2 R_G^2\/2)$. \n\n    \\item The peak compactness `$x_d$' is set by the second order derivatives of the smoothed overdensity field. More quantitatively, it is determined by the eigenvalues of the Hessian matrix $\\partial_{ij}\\delta_{G}$~($i,j=1,2~\\&~3$ are $x,y~\\&~z$ components respectively) can be parametrized as \n    \\begin{eqnarray}\n        \\lambda_1= \\frac{x_d \\sigma_2(R_g)}{1+a_{12}^2+a_{13}^2}\\\\\n        \\lambda_2= a_{12}^2 \\lambda_1\\\\\n        \\lambda_3= a_{13}^2 \\lambda_1\n    \\end{eqnarray}\n    where $\\sigma_2(R_g) \\equiv \\int \\frac{P(k)}{(2\\pi)^3} \\hat{W}^2(k,R_G) k^2 dk$, and $a_{12}$ and $a_{13}$ are the axis ratios that determine the shape of the mass distribution around the ellipsoidal peak. \n    \n\n    \\item Lastly, the tidal strength $\\epsilon$ is determined by second order derivative of the gravitational potential i.e. the tidal tensor $T_{ij}~(i=1,2,3)$. The eigenvalues of the tidal tensor can be parametrized by, \n    \\begin{equation}\n        \\left[\\epsilon \\cos{\\frac{\\omega+2\\pi}{3}}, \\epsilon \\cos{\\frac{\\omega-2\\pi}{3}}, \\epsilon \\cos{\\frac{\\omega}{3}} \\right],\n    \\end{equation} \n    where $\\epsilon$ determines the overall magnitude of the tidal tensor, and $\\omega$ determines the relative strengths of the tidal tensor along the three eigenvectors.\n\\end{itemize}\n\n\n\\subsubsection{Our choice of peak parameters}\n\\begin{table*}\n\\centering\n\\begin{tabular}{|c|c|c|c|c|}\n      IC & $R_G~(\\mathrm{Mpc}\/h)$ & $\\nu~(\\sigma_{R_G})$ & $x_d~(\\sigma_2)$ & $\\epsilon~(\\mathrm{km~s^{-1}~\\mathrm{Mpc}^{-1}})$ \\\\\n      \\hline\n     \\texttt{5SIGMA} & 1.0 & 5 & 3.6~(ave) & 34.0~(ave)  \\\\\n     \\texttt{5SIGMA_COMPACT} & 1.0 & 5 & 5.8~($+3\\sigma$) & 15.0~($-2\\sigma$)\\\\\n     \\texttt{6SIGMA} & 1.3 & 6 & 4.0~(ave) & 34.0~(ave)\\\\\n     \\hline\n\\end{tabular}\n\\caption{The adopted values for the peak parameters for $\\texttt{5SIGMA}$, $\\texttt{5SIGMA_COMPACT}$ and $\\texttt{6SIGMA}$. These constrained initial conditions~(ICs) are characterized by the  smoothing scale $R_G$~(2nd col), the peak height $\\nu$~(3rd col), the peak compactness $x_d$~(4th col) and the tidal scalar $\\epsilon$~(5th col); these 4 parameters are most consequential to BH growth. The remaining parameters do not significantly impact BH growth, and have been fixed to be the typical values of their underlying distributions~(see Figure 5 of N21). }\n\\label{peak_parameters_table}\n\\end{table*}\nN21 shows that BH growth is the most efficient within rare peaks~(high $\\nu$) that are compact~(high $x_d$) and allow for gas infall to occur from all directions~(low tidal strength $\\epsilon$).\n  Therefore, throughout this paper, we make intentional choices for $\\nu$, $x_d$ and $\\epsilon$ and fix the remaining parameters at their most probable values~(see Figure 5 of N21) . Table \\ref{peak_parameters_table} summarizes the adopted  parameter values for $\\nu$, $x_d$ and $\\epsilon$. More specifically, we look at the following three regions:\n\\begin{itemize}\n    \\item  We choose a $5\\sigma$ peak~($\\nu=5$) at scales of $R_G=1~\\mathrm{Mpc}\/h$, with $x_d$ and $\\epsilon$ corresponding to the typical values i.e. the maxima of their respective distributions. The peak height was chosen to produce a target halo mass of $10^{12}~M_{\\odot}\/h$ at $z=7$. It is hereafter referred to as \\texttt{5SIGMA}.\n    \\item We again choose a $5\\sigma$ peak at $R_G=1~\\mathrm{Mpc}\/h$, but with a compactness $x_d$ that is $3\\sigma$ away from the typical value, and a tidal strength $\\epsilon$ that is $-2\\sigma$ away from the mean value. This also targets the assembly of a $10^{12}~M_{\\odot}\/h$ halo at $z=7$, and is referred to as \\texttt{5SIGMA_COMPACT}.\n    \\item Lastly, we choose a $6\\sigma$ peak~($\\nu=6$) at scales of $R_G=1.3~\\mathrm{Mpc}\/h$ with typical values for $x_d$ and $\\epsilon$. This targets a $5\\times10^{12}~M_{\\odot}\/h$ halo at $z=7$, and is referred to as \\texttt{6SIGMA}.\n\\end{itemize}\nNote that the target halos produced in \\texttt{6SIGMA} and \\texttt{5SIGMA_COMPACT} regions have number densities roughly similar to those of the observed $z\\sim6$ quasars~($\\sim1~\\mathrm{Gpc}^{-3}$).\nIn contrast, \\texttt{5SIGMA} produces a target halo that is $\\sim100$ times more common.\n\nFinally, we also note that the BH growth can depend on the specific realization of the large scale density field. However, upon exploring 5 different realizations for a select few BH models, we found that the differences in BH growth were mild~($z\\sim6$ BH masses vary by factors $\\lesssim2$). \n\n\n\n\\subsection{Simulation resolution}\n\nIn \\cite{2021MNRAS.507.2012B,2021MNRAS.tmp.3110B}, we performed detailed resolution convergence tests for our BH seed model and found that for our fiducial model with $\\tilde{M}_{\\mathrm{sf,mp}}=5$ and $\\tilde{M}_{h}=3000$, the seed formation rates are reasonably well converged for gas mass resolutions $\\lesssim10^4~M_{\\odot}\/h$. The resolution convergence becomes slower as the models are made more restrictive by increasing $\\tilde{M}_{\\mathrm{sf,mp}}$ or by introducing a LW flux criterion. As we shall see in Section \\ref{BH growth at higher resolutions_sec}, the resolution convergence properties of our constrained regions are similar to that of the zoom region of \\cite{2021MNRAS.507.2012B,2021MNRAS.tmp.3110B}.\n\nTo achieve a gas mass resolution of $\\sim10^4~M_{\\odot}\/h$ in a box size of $9~\\mathrm{Mpc}\/h$, we need $N=720$ DM particles per dimension~(note that the number of gas cells are initially assigned to be equal to the DM particles, but as the simulations evolve the gas cells can undergo refinement or de-refinement). However, running such a simulation until $z=6$ requires a substantial amount of computing time and memory, particularly in regions with extreme overdensities. Therefore, to facilitate a rapid exploration of the large parameter space of our seed models, we choose $N=360$. We assign this to be our fiducial resolution, and it corresponds to a gas mass resolution of $\\sim10^5~M_{\\odot}\/h$. Note that this is only slightly lower than the highest resolution box of the \\texttt{IllustrisTNG} suite i.e. TNG50. That being said, we do run higher resolution realizations~($N=720$) for a few selected models, particularly those that successfully produce a $z\\gtrsim6$ quasar. As we shall see in Section \\ref{BH growth at higher resolutions_sec}, the final BH mass at $z\\lesssim7$ is not significantly impacted by resolution. Additionally, we use the $N=720$ runs to probe the lowest seed mass considered in this work i.e. $M_{\\mathrm{seed}}=1.25\\times10^4~M_{\\odot}\/h$. The fiducial $N=360$ run can only probe seed masses of $M_{\\mathrm{seed}}=1\\times10^5~\\&~8\\times10^5~M_{\\odot}\/h$. Hereafter, unless otherwise stated, we are using $N=360$. For runs that use $N=720$, it shall be explicitly stated in the captions or the text.\n\n\\section{Results}\n\n\\label{Results_sec}\n\n\\subsection{Halo environments: evolution from $z\\sim20-6$}\n\n\\begin{figure*}\n\\includegraphics[width=5.2 cm]{5sigma_typical.png}\n\\includegraphics[width=5.2 cm]{5sigma_compact.png}\n\\includegraphics[width=6.5 cm]{6sigma_typical.png}\n\\caption{Gas density field at the $z=6$ snapshot for the three constrained Gaussian initial conditions we explored in this work. The left panel is centered at a $5\\sigma$ overdensity peak at $R_{G}=1~\\mathrm{Mpc}\/h$, and typical values for compactness $x_d$ and tidal strength $\\epsilon$; this is hereafter referred to as \\texttt{5SIGMA}. The middle panel is centered at a $5\\sigma$ overdensity peak with $+3\\sigma$ higher compactness, and $-2\\sigma$ lower tidal strength; we refer to this as \\texttt{5SIGMA_COMPACT}. The right panel is centered at a $6\\sigma$ overdensity peak at $R_{G}=1.3~\\mathrm{Mpc}\/h$ with average values of compactness and tidal strength; this is referred to as \\texttt{6SIGMA}.  We can see that the gas distribution in \\texttt{5SIGMA_COMPACT} is more isotropic and centrally concentrated compared to \\texttt{5SIGMA} and \\texttt{6SIGMA}.}\n\\label{halo_properties_fig}\n\\end{figure*}\n\n\\begin{figure}\n\\includegraphics[width=8 cm]{Halo_properties.png}\n\\caption{The evolution of the most massive halo~(MMH) from $z\\sim20$ to $z\\sim7$ for \\texttt{5SIGMA}~(blue), \\texttt{5SIGMA_COMPACT}~(green) and \\texttt{6SIGMA}~(orange) lines. The top, middle and bottom panels show the total halo mass, gas mass and stellar mass respectively. For \\texttt{5SIGMA} and \\texttt{5SIGMA_COMPACT}, we reach our target halo mass of $\\sim10^{12}~M_{\\odot}\/h$ at $z=7$; they both assemble a total gas mass of $2\\times10^{11}~M_{\\odot}\/h$. Likewise, the \\texttt{6SIGMA} volume assembles the desired target halo mass of $6\\times10^{12}~M_{\\odot}\/h$ at $z=7$, and a gas mass of $5\\times10^{11}~M_{\\odot}\/h$. \\texttt{5SIGMA_COMPACT} assembles a stellar mass of $\\sim10^{11}~M_{\\odot}\/h$ at $z=7$, similar to that of \\texttt{6SIGMA}; this is significantly higher than \\texttt{5SIGMA} which assembles a stellar mass of $9\\times10^{9}~M_{\\odot}\/h$. More compact peaks~(at fixed peak height) lead to enhanced star formation.}\n\\label{halo_evolution_fig}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=8.5 cm]{SM_HM_relations.png}\n\\caption{Stellar mass vs halo mass relation at $z=6$ for the MMHs of \\texttt{5SIGMA_COMPACT}~(green), \\texttt{5SIGMA}~(blue) and \\texttt{6SIGMA}~(orange) respectively, compared with the full halo population of TNG300~(a $[300~\\mathrm{Mpc}]^3$ box---the largest volume in the \\texttt{IllustrisTNG} simulation suite; faded grey circles). The predictions from the constrained runs are broadly consistent with the trends extrapolated from the TNG300 results. The \\texttt{5SIGMA} and \\texttt{6SIGMA} runs produce stellar mass predictions close to the mean trend, whereas the \\texttt{5SIGMA_COMPACT} run produces a somewhat overly massive galaxy~(stellar mass) compared to its host halo.}\n\\label{SM_HM_relation_fig}\n\\end{figure}\n\n\n\\begin{figure*}\n\\includegraphics[width=18 cm]{1D_density_distributions.png}\n\\caption{Radially averaged 1D profiles of the gas density~(physical units), centered around the most massive BH in the MMH of our simulation volume for \\texttt{5SIGMA}, \\texttt{5SIGMA_COMPACT} and \\texttt{6SIGMA}. Lines of different colors show the redshift evolution from $z=14$ to $z=6$. We see that gas density in the central regions steeply increases with time between $z\\sim9-6$. The steepest increase is seen for \\texttt{5SIGMA_COMPACT}.}\n\\label{1D_density_distributions_fig}\n\\includegraphics[width=19 cm]{FullMapAllFieldsconstrained.png}\n\n\\caption{2D profiles of the gas density, metallicity and the star formation rate density within the vicinity of the most massive BH in the MMH of the \\texttt{5SIGMA_COMPACT} run. We compute these quantities averaged over a slice of thickness $10~\\mathrm{kpc}\/h$. Left to right panels show the redshift evolution from $z=10$ to $z=6$. At $z\\lesssim9$, the steep increase in the gas density leads to increase in the star formation and metal enrichment in the halo.}\n\\label{FullMapAllFieldsconstrained_fig}\n\\end{figure*}\n\nBefore looking at the properties of BHs, we first look at the environments in which they form and grow. Figure \\ref{halo_properties_fig} shows the $z=6$ gas density profiles centered at the location of the constrained  \\texttt{5SIGMA}, \\texttt{5SIGMA_COMPACT} and \\texttt{6SIGMA} peaks. Visually, we can clearly see that the gas distribution around the  \\texttt{5SIGMA_COMPACT} peak is more compact and isotropic compared to that of \\texttt{5SIGMA} and \\texttt{6SIGMA}. Figure \\ref{halo_evolution_fig} shows the evolution of the most massive halo~(MMH) from $z\\sim20-6$, in terms of the total mass, gas mass and stellar mass. We see that both \\texttt{5SIGMA_COMPACT} and \\texttt{5SIGMA} runs assemble their target mass of $\\sim10^{12}~M_{\\odot}\/h$ by $z=7$, which grows to become $\\sim2\\times10^{12}~M_{\\odot}\/h$ by $z=6$. The \\texttt{6SIGMA} run assembles halo masses of $\\sim7\\times10^{12}~M_{\\odot}\/h$ and $\\sim10^{13}~M_{\\odot}\/h$ by $z=6$.\n\nInterestingly, the halo assembly history~(see Figure \\ref{halo_evolution_fig}: top panel) of the three regions shows that for \\texttt{5SIGMA_COMPACT}, the MMHs at $z\\gtrsim10$ are $\\sim5-10$ times more massive compared to that of \\texttt{5SIGMA}~(as well as \\texttt{6SIGMA}). But at $z\\lesssim10$, the halo growth rate for the \\texttt{5SIGMA_COMPACT} run becomes slower compared to the \\texttt{5SIGMA} run, thereby explaining the similar final halo masses that both the runs assemble at $z\\sim6-7$. That is likely because the MMH in the \\texttt{5SIGMA_COMPACT} peak becomes more isolated at $z\\lesssim10$~(after having merged with most of its neighboring massive halos by $z\\sim10$). However, the \\texttt{5SIGMA_COMPACT} MMH continues to become more dense during $z\\sim9-6$~(due to continued gravitational collapse), which likely causes the higher compactness of \\texttt{5SIGMA_COMPACT} peak compared to \\texttt{5SIGMA}~(as well as \\texttt{6SIGMA}) at $z\\sim6-7$.\n\nThe evolution of the gas mass~(see Figure \\ref{halo_evolution_fig}: middle panel) mirrors that of the total halo mass at $z\\sim20-10$. More specifically, the gas mass of the MMH in \\texttt{5SIGMA_COMPACT} is $\\sim5-10$ times higher than that of \\texttt{5SIGMA} and \\texttt{6SIGMA}~(similar to the total halo mass) at $z\\sim20-10$. Notably, we find that at $z\\sim9-6$, there is no significant increase in the gas mass for the MMH in  \\texttt{5SIGMA_COMPACT}, unlike the MMHs of \\texttt{5SIGMA} and \\texttt{6SIGMA}. As a result, by $z=6$, the MMH in \\texttt{5SIGMA_COMPACT} ends up with a lower gas mass compared to \\texttt{5SIGMA} and \\texttt{6SIGMA}. As we shall see, this is happening because the gas in \\texttt{5SIGMA_COMPACT} is being rapidly consumed by star formation and BH accretion, more so than \\texttt{5SIGMA} and \\texttt{6SIGMA}. \nThe enhanced star formation in \\texttt{5SIGMA_COMPACT} can be seen in the bottom panel of Figure \\ref{halo_evolution_fig}, wherein the stellar mass is $\\sim7$ times higher than that of \\texttt{5SIGMA} at $z\\sim6-7$~(making it similar to the stellar mass produced by \\texttt{6SIGMA} at $z\\sim6-7$).  \nThe \\texttt{5SIGMA_COMPACT} region therefore produces an overly massive \ngalaxy for its host halo mass, as clearly seen in Figure \\ref{SM_HM_relation_fig}. Figure \\ref{SM_HM_relation_fig} also demonstrates that our constrained runs are consistent with the stellar mass vs. halo mass relations predicted by TNG300, thereby validating this technique.\n\nHaving discussed the evolution of the global properties of the MMH, we now focus on the evolution of their internal gas distributions, which are much more consequential to BH growth. The evolution of the radially averaged gas density profiles from $z\\sim14-6$ is shown in Figure \\ref{1D_density_distributions_fig}. In all\nthree regions,\nlittle evolution occurs at $z\\gtrsim9$, and the central $\\sim 1$ kpc\/$h$ is unresolved owing to low gas densities. Between $z\\sim9-6$ however, the gas densities start to significantly increase, particularly close to the halo centers. Specifically, while the overall gas mass of the MMH only increases by factors of $\\sim50$ between $z\\sim9-6$~(Figure \\ref{halo_evolution_fig}: middle panel), the central gas densities increase by factors of $\\sim100-10000$ during the same time interval. Amongst the three regions, \\texttt{5SIGMA_COMPACT} shows the steepest increase in density between $z\\sim9-6$. Figure \\ref{FullMapAllFieldsconstrained_fig} shows the 2D color maps of the evolution of the gas density, star formation rates and metallicity for \\texttt{5SIGMA_COMPACT} region at redshift snapshots of $z=12,10,8~\\&~6$. We can clearly see that the steep increase in central gas density leads to a commensurate boost in the star formation, as well as metal enrichment in the central regions of the halo.\nAs we shall see, this increase in the central gas densities leads to substantially increased importance of accretion-driven BH growth at $z\\sim9-6$.   \n\n\n\\subsection{BH growth in different halo environments: Impact of BH seeding models}\n\\begin{figure*}\n\\includegraphics[width=16 cm]{No_of_BHs.png}\n\\caption{Number of BHs present in the most massive halo~(MMH) at different redshift snapshots for \\texttt{5SIGMA}~(blue), \\texttt{5SIGMA_COMPACT}~(green) and \\texttt{6SIGMA}~(orange) lines. Error-bars correspond to Poisson errors. In the leftmost panel, we use the default seeding prescription from the IllustrisTNG simulation suite~(referred to as TNG seed model). In the middle and right panels, we use the gas-based seeding prescription with $\\tilde{M}_h=3000$ and $\\tilde{M}_{\\mathrm{sf,mp}}=5$~(\\texttt{SM5_FOF3000}). All these runs use the default accretion prescription from the IllustrisTNG simulation suite~(TNG accretion model). We see that the onset of seed formation happens earliest within the \\texttt{5SIGMA_COMPACT} run; therefore, it contains the highest number of BHs around $z\\gtrsim10$. However, between $z\\sim9-6$, the \\texttt{5SIGMA_COMPACT} does not acquire many new BHs. On the other hand, the \\texttt{5SIGMA} and \\texttt{6SIGMA} MMHs continue to acquire new BHs from nearby halos between $z\\sim9-6$. Therefore, by $z=6$, the \\texttt{5SIGMA_COMPACT} peak has the least number of BHs and \\texttt{6SIGMA} peak has the highest number of BHs.}\n\n\\label{No_of_BHs_fig}\n\n\n\\includegraphics[width=5.7 cm]{TNG_seed_model.png}\n\\includegraphics[width=5.7 cm]{Mh3000_Msfmp5_seed5.90.png}\n\\includegraphics[width=5.7 cm]{Mh3000_Msfmp5.png}\n\\caption{Evolution of the most massive BH at $z=6$ in the MMH of\n\\texttt{5SIGMA}~(blue), \\texttt{5SIGMA_COMPACT}~(green) and \\texttt{6SIGMA}~(orange) lines (hereafter, all growth histories plotted for different runs are for this particular BH in each simulation). The 1st row corresponds to the BH mass; solid lines show the total BH mass and the dashed lines show the mass accumulated only by mergers. The 2nd row shows the fraction of the current mass accumulated by gas accretion. The 3rd and 4th rows show the total bolometric luminosity in units of $\\mathrm{ergs~s^{-1}}$ and the Eddington luminosity, respectively. All the runs use the TNG accretion model. In the left panels, we use the TNG seed model. In the middle and right panels, we use the gas-based seeding prescription \\texttt{SM5_FOF3000}. The BH seed mass is $8 \\times 10^5 M_{\\odot}\/h$ and $1 \\times 10^5 M_{\\odot}\/h$ in the middle and right panels, respectively. Among the constrained volumes we explore, \\texttt{5SIGMA_COMPACT} assembles the highest BH mass in all cases. Even in this region, the TNG seed model achieves a maximum BH mass of $\\sim7\\times10^7~M_{\\odot}\/h$ by $z=6$, which is significantly smaller than the typical masses of observed $z\\sim6$ quasars. In contrast, the \\texttt{SM5_FOF3000} models are able to assemble $10^9~M_{\\odot}\/h$ SMBHs by $z=6$. While these massive BHs are active as luminous quasars with near-Eddington luminosities of  $\\sim10^{47}~\\mathrm{ergs~s^{-1}}$ at $z=6$, their growth at $z\\gtrsim 9$ is dominated by BH mergers. Overall, the BH growth is fastest within rare massive high-z halos that are also sufficiently compact and have low tidal fields. To produce $z\\sim6$ quasars in these halos with the TNG accretion model, an early boost in BH mass driven by mergers is necessary.}\n\\label{bh_growth_env_fig}\n\\end{figure*}\n\\begin{figure*}\n\\includegraphics[width=14 cm]{SM_BHM_relations.png}\n\\caption{BH mass vs. halo mass scaling relation at $z=6$. In the left panel, each data point corresponds to a halo; so we are plotting the ``halo mass vs. total mass of all BHs in the halo'', for all halos at $z=6$. In the right panel, each data point corresponds to a BH, so we are plotting the ``BH mass vs. host halo mass'', for all individual BHs at $z=6$. The blue, green and orange colors correspond to the BH within the MMH of \\texttt{5SIGMA}, \\texttt{5SIGMA_COMPACT} and \\texttt{6SIGMA} regions respectively. All the runs use the TNG accretion model. The circles correspond to the TNG seed model, which we directly compare to the full population of TNG300.  The left panels show that for the TNG seed model, the\ntotal BH masses of\nMMHs produced by the constrained runs are\nconsistent with the extrapolated trends from the TNG300 population.  This further validates our constrained runs. The squares $\\&$ stars correspond to the gas-based seed model \\texttt{SM5_FOF3000} with seed masses of $8\\times10^5~\\&~1\\times10^5~M_{\\odot}\/h$, respectively; these models produce much higher BH masses\ncompared to the TNG seed model. The MMHs generally tend to host a significantly large number of BHs during their assembly history, generally ranging between $\\sim10-50$ depending on the constrained region as well as the seed model~(revisit Figure \\ref{No_of_BHs_fig}). \n The \\texttt{6SIGMA} run produces the highest number of BHs and also the highest total BH mass in its target halo, commensurate with its halo mass. However, when we look at masses of \\textit{individual} BHs in the right panel, the most massive BH is actually produced by \\texttt{5SIGMA_COMPACT}. In fact, with the \\texttt{SM5_FOF3000} seed model, only the \\texttt{5SIGMA_COMPACT} can produce \\textit{individual} $\\sim10^9~M_{\\odot}\/h$ BHs similar to the observed $z\\sim6$ quasars.}\n\\label{SM_BHM_relations_fig}\n\\end{figure*}\nWe are particularly interested in the growth histories of BHs occupying the MMH in each region. Figure \\ref{No_of_BHs_fig} shows the number of BHs that are present within the MMH at different redshift snapshots, with different panels showing different seed models. We note that these MMHs tend to acquire several BHs during their assembly history. Despite many of these BHs inevitably merging with the central BH, the overall number of BHs hosted by the MMHs increases up to at least $z\\sim9-10$. This is because the MMHs continue to acquire new BHs from surrounding merging halos. At $z=6$, the MMHs can generally host up to $\\sim10-50$ BHs depending on the constrained region as well as the seed model.\n\nNext, we look at how the number of BHs in the MMHs vary between different regions and seed models. For all seed models, we generally see that the \\texttt{5SIGMA_COMPACT} runs tend to start forming seeds at earlier times, and therefore host a higher number of BHs at $z\\sim10-20$ compared to \\texttt{5SIGMA} and \\texttt{6SIGMA} runs. This is because the MMHs at $z\\sim10-20$ for the \\texttt{5SIGMA_COMPACT} runs are more massive compared to the other two regions~(revisit Figure \\ref{halo_evolution_fig}). However, between $z\\sim10-6$, we find that there is no significant increase in the number of BHs for the MMH in \\texttt{5SIGMA_COMPACT}, unlike \\texttt{5SIGMA} and \\texttt{6SIGMA}. This is likely because in the \\texttt{5SIGMA_COMPACT} runs, most of the nearby massive halos have already merged with the MMH by $z\\sim10$, leaving behind an isolated MMH during $z\\sim10-6$ with very few nearby halos to acquire new BHs from. On the other hand, for the \\texttt{5SIGMA} and \\texttt{6SIGMA} runs, the MMHs are not as isolated at $z\\sim10-6$, and they continue to acquire new BHs during this time. All of this ultimately leads to fewer BHs in the MMH of \\texttt{5SIGMA_COMPACT} at $z=6$, compared to \\texttt{5SIGMA} and \\texttt{6SIGMA} regions. Lastly, note that the TNG seed model~(Figure \\ref{No_of_BHs_fig}: left panel) is substantially more restrictive and produces much fewer seeds compared to our fiducial gas-based seed models with $\\tilde{M}_{\\mathrm{sf,mp}}=5, \\tilde{M}_{\\mathrm{h}}=3000$~(hereafter \\texttt{SM5_FOF3000} shown in Figure \\ref{No_of_BHs_fig}: middle and right panels).\n\nThe primary science focus of this work is the growth of the most massive BH located in the MMH of our simulations. So unless otherwise stated, all future references to BH growth histories are for the most massive BH at $z=6$ in each simulation. Figure \\ref{bh_growth_env_fig} shows the BH growth histories for \\texttt{5SIGMA}, \\texttt{5SIGMA_COMPACT} and \\texttt{6SIGMA}. We first focus on the halo based TNG seed model~(leftmost panels), which starts to seed BHs around $z\\sim10-12$. Note that for this seed model, very few BHs are formed overall, which results in very little growth via mergers~(see dashed lines in Figure \\ref{bh_growth_env_fig}: top left panel). We find that amongst all three regions, \\texttt{5SIGMA_COMPACT} assembles the highest mass BH at $z=6$, despite containing the least number of BHs within its MMH~(also recall that \\texttt{6SIGMA} has a higher mass MMH at $z=6$). This is because the \\texttt{5SIGMA_COMPACT} run produces a higher gas density at the peak location, thereby leading to the fastest growth via gas accretion at $z\\lesssim9$. This result is overall consistent with N21, showing that compact peaks with low tidal fields are the most ideal environments for rapid BH growth. However, with this TNG seed model, the overall BH mass assembled at $z=6$ is only $\\sim5\\times10^{7}~M_{\\odot}$, which is significantly smaller than the typical masses of the observed high-z quasars~($\\sim10^9~M_{\\odot}$).\n\nNext, we look at the predictions from gas-based seed models, particularly \\texttt{SM5_FOF3000}~($\\tilde{M}_{\\mathrm{sf,mp}}=5$ and $\\tilde{M}_{\\mathrm{h}}=3000$) with seed masses of $8\\times10^5$ and $1\\times10^5~M_{\\odot}\/h$~(Figure \\ref{bh_growth_env_fig}: middle and right panels respectively). Again, these models produce substantially higher numbers of seeds that start forming at much earlier times~($z\\sim17-25$) compared to the TNG seed model. As a result, there is now considerable growth via mergers. To that end, we note that regardless of how early the seeds form, accretion-driven BH growth does not become significant until $z\\lesssim9$~(see 2nd rows of Figure \\ref{bh_growth_env_fig}). This results from\nthe fact that the central gas densities remain relatively low until $z\\sim9$ but start to steeply increase between $z\\sim9-6$~(revisit Figure \\ref{1D_density_distributions_fig}). As a result, the BH growth at $z\\gtrsim9$ is completely driven by BH mergers. In fact, for \\texttt{SM5_FOF3000} seed models, the $z\\gtrsim9$ merger-driven growth assembles a BH mass of $\\sim3\\times10^7~M_{\\odot}$ by $z\\sim9$, in contrast to the TNG seed model where the BHs are still close to the seed mass of $\\sim10^6~M_{\\odot}$ at $z\\sim9$. Between $z\\sim9-6$, the accretion-driven BH growth becomes increasingly significant for the \\texttt{SM5_FOF3000} seed models, pushing\nthe BH mass to values $\\gtrsim 10^8~M_{\\odot}$ at $z=6$ for all three constrained regions. Amongst the three regions, \\texttt{5SIGMA_COMPACT} again produces the highest BH mass that now reaches close to $\\sim10^9~M_{\\odot}$ at $z=6$, consistent with the observed $z\\sim6$ quasars. Additionally, note that the merger-driven growth can also be boosted, by simply reducing the halo mass threshold and forming more seeds. Therefore, $z\\sim6$ quasars could also be formed within a ``halo mass only\" seed model~(e.g. TNG seed model) with a sufficiently low halo mass threshold.  \n\nThe bolometric luminosities and Eddington ratios~(see 3rd and 4th rows of Figure \\ref{bh_growth_env_fig}) of the BHs remain low~($L_{\\mathrm{bol}}\\sim10^{42}~\\mathrm{ergs~s^{-1}}\\sim10^{-3}~\\mathrm{L_{\\mathrm{bol}}^{\\mathrm{edd}}}$) at $z\\gtrsim9$ wherein the accretion-driven BH growth is insignificant. This is generally true for all seed models and constrained regions. As we go from $z\\sim9-6$ during which the central gas densities steeply rise~(revisit Figure \\ref{1D_density_distributions_fig}), the accretion-driven BH growth becomes increasingly efficient. This leads to a sharp increase in the BH luminosities. By $z\\sim6$, the BHs start to grow close to the Eddington limit for all of the runs, generally corresponding to bolometric luminosities $\\gtrsim10^{45}~\\mathrm{ergs~s^{-1}}$. However, luminosities of observed $z\\sim6$ quasars are even higher i.e. $\\sim10^{47}~\\mathrm{ergs~s^{-1}}$. These luminosities are produced only by the $\\sim10^9~M_{\\odot}$ BHs that are formed within $\\texttt{5SIGMA_COMPACT}$ region using the \\texttt{SM5_FOF3000} seed models. Overall, we find that to form BHs that resemble the observed $z\\sim6$ quasars~(masses of $\\sim10^9~M_{\\odot}$ and luminosities of $\\sim10^{47}~\\mathrm{ergs~s^{-1}}$) in our simulations with \\texttt{IllustrisTNG} physics, we need massive compact halos with seed models such as \\texttt{SM5_FOF3000} that allow for substantial merger-driven BH growth at $z\\gtrsim9$.\n\nFigure \\ref{SM_BHM_relations_fig} shows the predictions of our constrained runs\nthe $z=6$ halo, as well as the BH populations in the TNG300 uniform simulation. Note that the most massive $z=6$ BHs produced by TNG300 are $\\sim50$ times smaller than the observed $z\\gtrsim6$ quasars. This is simply because TNG300 does not have the volume to produce such rare objects, despite being among the largest simulations to be run past $z=6$ and beyond. In fact, this is true for almost all major cosmological hydrodynamic simulations run to date ~(see \\citealt{2021MNRAS.503.1940H,2022MNRAS.tmp..271H,2022MNRAS.509.3015H} for combined analyses of BH populations from several simulations). The only exception to this would be the \\texttt{BlueTides} simulation~\\citep{2016MNRAS.455.2778F} which produces a $6.4\\times10^8~M_{\\odot}$ BH in a volume of $[400~\\mathrm{Mpc}\/h]^3$ by $z\\sim7.5$~\\citep{2019MNRAS.483.1388T}. Overall, this further highlights the power of our constrained simulations which is able to produce such rare objects within smaller volume and higher resolution simulations in reasonable computing time, so as to allow an exploration of a wide range of model parameters.\n\nWe now specifically compare the BHs produced in the MMHs of the constrained runs to that of the TNG300 simulation. The left panel of Figure \\ref{SM_BHM_relations_fig} shows the\nhalo mass versus\nthe total mass of all BHs within the halos. We find that for the TNG seed model, predictions for the constrained runs are consistent with extrapolation of the BH mass vs. halo mass relation from TNG300.\nThese results, together with the stellar mass vs. halo mass relations in Figure \\ref{SM_HM_relation_fig}, serve as a good validation for our constrained runs. As expected from the results in the previous paragraph, the gas-based seed models \\texttt{SM5_FOF3000} produce halos with total BH masses that are significantly higher than the extrapolated trend of the TNG300 halos. As an additional note, the $M_{bh}-M_h$ relation for the lowest mass BHs within TNG300 form streaks of horizontal lines that can be clearly seen in Figure \\ref{SM_BHM_relations_fig}. This is likely an artifact of the TNG seed model, where BHs seeded within $\\gtrsim10^{10}~M_{\\odot}$ halos do not show significant accretion-driven growth until halos reach masses $\\gtrsim10^{11}~M_{\\odot}$.   \n\nCloser examination of Figure \\ref{SM_BHM_relations_fig} (left panel) reveals another interesting result. For the TNG seed model as well as the gas-based seed model, the MMH in \\texttt{6SIGMA} achieves the highest {\\em total} BH mass.\nThis contrasts with Figure \\ref{bh_growth_env_fig} and with the right panel of Figure \\ref{SM_BHM_relations_fig}, which clearly show that \\texttt{5SIGMA_COMPACT} produces the highest {\\em individual} BH mass in the MMH.\nAs it turns out, while the MMHs in \\texttt{5SIGMA} and \\texttt{6SIGMA} end up with a higher number of BHs than \\texttt{5SIGMA_COMPACT}~(revisit Figure \\ref{No_of_BHs_fig}), the individual BHs in \\texttt{5SIGMA} and \\texttt{6SIGMA} are significantly smaller than the most massive BH in \\texttt{5SIGMA_COMPACT}. The foregoing statement is generally true for TNG seed model as well as the gas-based seed models. Particularly for the gas-based seed model \\texttt{SM5_FOF3000}, this\nmeans that while a typical rare massive halo~($\\sim10^{13}~M_{\\odot}$) at $z\\sim6$ can acquire total BH mass exceeding $\\sim10^9~M_{\\odot}$, it may not produce ``individual BHs\" of such masses. Therefore, to host the observable $z\\gtrsim6$ quasars that correspond to individual $\\sim10^9~M_{\\odot}$ BHs growing close to the Eddington limit, we need halos that are not just massive enough, but are also highly compact and have low tidal fields. In these compact MMHs, the BHs are more likely to have close encounters with each other. Therefore they can merge more readily to form a single massive BH at the halo centers, compared to typical halos of the same mass. To that end, recall that the small scale dynamics are poorly resolved in our simulations, particularly for lower mass BHs.  Several recent works with more realistic treatment of BH small scale dynamics~\\citep[for e.g.][]{2017MNRAS.470.1121T,2018ApJ...857L..22T,2022MNRAS.tmp..460N,2022MNRAS.510..531C} have found that many of the seeds~(particularly lower mass seeds) do not sink efficiently to the local potential minima, thereby leading to a population of wandering BHs~\\citep{2018ApJ...857L..22T,2021MNRAS.503.6098R,2021ApJ...916L..18R,2021MNRAS.508.1973M,2022MNRAS.tmp..221W}. Therefore, prompt mergers resulting from our current BH repositioning scheme could overestimate the rate at which the central BH grows. In future work, we shall investigate this in the context of the assembly of the $z\\sim6$ quasars.\n\n\n\\begin{figure*}\n\\includegraphics[width=8 cm]{varyingmsfmp.png}\n\\includegraphics[width=8 cm]{varyingmsfmp_5.90.png}\n\\caption{Evolution of the most massive black hole in \\texttt{5SIGMA_COMPACT} for $\\tilde{M}_{\\mathrm{sf,mp}}=5,150~\\&~1000$. Left and right panels correspond to seed masses of $10^5~M_{\\odot}\/h$ and $8\\times10^5~M_{\\odot}\/h$ respectively. All the runs here use the TNG accretion model. The different rows show the same set of quantities as Figure \\ref{bh_growth_env_fig}. As the seeding criteria becomes more stringent, there are fewer mergers to grow the BH at $z\\gtrsim9$. As a result, the final $z=6$ BH mass decreases and falls significantly short of producing the observed $z\\sim6$ quasars. }\n\\label{gas_based_models_fig}\n\\end{figure*}\n\nNext, we focus on the \\texttt{5SIGMA_COMPACT} region and further explore different variations of gas-based seeding models to study their impact on BH growth. Note that \\texttt{SM5_FOF3000}~($\\tilde{M}_{\\mathrm{sf,mp}}=5~\\&~\\tilde{M}_{\\mathrm{h}}=3000$), which successfully produces a $z\\sim6$ quasar, is the least restrictive amongst the family of BH seeding models developed in \\cite{2021MNRAS.507.2012B,2021MNRAS.tmp.3110B}. Figure \\ref{gas_based_models_fig} shows the impact of further increasing $\\tilde{M}_{\\mathrm{sf,mp}}$ to values of 150 and 1000, on the BH mass, luminosity and Eddington ratio evolution. As we increase $\\tilde{M}_{\\mathrm{sf,mp}}$, fewer seeds form and the merger-driven BH growth is commensurately suppressed. This leads to a significant slow-down of BH growth, and thereby decreases the BH mass assembled by $z=6$. For $10^5~M_{\\odot}\/h$ seeds, increasing $\\tilde{M}_{\\mathrm{sf,mp}}$ from $5$ to $1000$ decreases the final $z=6$ BH mass by a factor of $\\sim100$. The $z=6$ luminosities also drop from $\\sim10^{47}~\\mathrm{erg~s^{-1}}$ to $\\sim10^{43}~\\mathrm{erg~s^{-1}}$. For more massive $8\\times10^5~M_{\\odot}\/h$ seeds, the impact is significantly stronger~(no $8\\times10^5~M_{\\odot}\/h$ seeds form for $\\tilde{M}_{\\mathrm{sf,mp}}=1000$). In general, any gas-based seeding prescription that is more restrictive than \\texttt{SM5_FOF3000} fails to produce BHs consistent with the observed $z\\sim6$ quasars.\n\nOverall, we find that within the TNG galaxy formation model, the \\texttt{SM5_FOF3000} gas-based seed model is able to successfully reproduce $z\\sim6$ BHs that are comparable to the observed high-z quasars, but only in massive ($\\sim10^{12}~M_{\\odot}$) halos that are highly compact and have low tidal fields. Additionally, both 1) merger-dominated growth at $z\\gtrsim9$, and 2) accretion-dominated growth at  $z\\sim6-9$ are crucial for producing these high-z quasars. \n\n\\subsection{Implications for strongly restrictive seed models in producing $z\\gtrsim6$ quasars}\n\\label{Impact of BH accretion sec}\n\n\\begin{figure*}\n\\includegraphics[width=8 cm]{varyingeddfac.png}\n\\includegraphics[width=8 cm]{varyingbondiradeff.png}\n\n\\caption{Evolution of the most massive black hole in the \\texttt{5SIGMA_COMPACT} volume for different accretion models labelled hereafter as \\texttt{Boost*_RadEff*_EddFac*} where the `*'s correspond to values of $\\alpha$, $\\epsilon_r$ and $f_{\\mathrm{edd}}$ respectively. All the runs use the most stringent gas-based seed model \\texttt{SM1000_FOF3000}, wherein the growth via mergers is small. We then explore different accretion models. In the left panel, we keep $\\alpha=1$ and $\\epsilon_r=0.2$ fixed, and show the BH growth histories for different values of $f_{\\mathrm{edd}}$ between 1-100. In the right panel, we consider different combinations of $\\alpha=1-10000$ and $\\epsilon_r=0.2~\\&~0.1$. In the absence or lack of mergers, accretion alone can assemble a $10^9~M_{\\odot}$ BH at $z\\sim6$ only if we enhance the maximum allowed accretion rate compared to the TNG accretion model. This can be achieved by either allowing for super-Eddington accretion rate, or by reducing the radiative efficiency. In contrast, a higher Bondi boost alone does not sufficiently enhance BH growth to assemble a $z\\sim6$ quasar.}\n\\label{varying_accretion_fig}\n\\end{figure*}\n\n\\begin{figure}\n\\includegraphics[width=8 cm]{varying_msfmp_eddlim10.png}\n\\caption{Here we consider the model \\texttt{Boost1_Radeff0.2_Eddfac10} that can already grow a $z\\sim6$ quasar via accretion alone, and then investigate the impact of enhancing the number of seeds and mergers on the final BH mass at $z=6$. Blue lines correspond to $\\tilde{M}_{\\mathrm{sf,mp}}=1000$~(\\texttt{SM1000_FOF3000}), which does not produce many seeds and mergers. Orange lines correspond to $\\tilde{M}_{\\mathrm{sf,mp}}=5$~(\\texttt{SM5_FOF3000}) which does produce a substantial number of seeds and mergers. We find that in models such as  \\texttt{Boost1_Radeff0.2_Eddfac10} wherein BHs can already grow to $\\sim10^{9}~M_{\\odot}$ via accretion alone, introducing more seeds and mergers~(by reducing $\\tilde{M}_{\\mathrm{sf,mp}}$) does not lead to any further increase in the final BH mass at $z=6$. In the 3rd panel, the black dashed line is the detection limit of $10^{-19}~\\mathrm{ergs~cm^{-2}~s^{-1}}$ of the Lynx $2-10~\\mathrm{keV}$ band, derived using bolometric corrections adopted from \\protect\\cite{2007MNRAS.381.1235V}. The blue dashed and dotted lines are JWST detection limits of 31 and 29th apparent magnitude for exposure times of $10^5$ and $10^4\\mathrm{s}$ respectively~(same as those assumed in \\protect\\citealt{2020MNRAS.492.5167V}), with bolometric correction adopted from \\protect\\cite{1994ApJS...95....1E}. Therefore, JWST and Lynx observations of the quasar progenitors at $z\\sim9-10$ could potentially contain signatures of their seeding environments.} \n\\label{varying_accretion_edd}\n\\end{figure}\n\n\nWe have thus far seen that $z\\sim6$ quasars cannot be assembled by our constrained simulations without an early boost in BH mass via mergers. This can only occur for relatively less restrictive seed models~($\\tilde{M}_{\\mathrm{sf,mp}}=5$), wherein enough seeds are formed to substantially contribute to merger-driven BH growth. Here we look for circumstances under which more restrictive seed models can produce $z\\sim6$ quasars even in the absence of sufficient mergers. In particular, we explore models with enhanced accretion in these highly biased regions, compared to the TNG accretion model\n\nIn Figure \\ref{varying_accretion_fig}, we take one of our most restrictive seeding models i.e. $\\tilde{M}_{\\mathrm{sf,mp}}=1000$, and explore different accretion models to identify the ones that can produce $z\\sim6$ quasars. As already noted in the previous section, only a handful of seeding and merger events occur for this model. We then investigate the BH growth under different variations for the accretion model. In the left panel, we keep $\\alpha=1~\\&~\\epsilon_r=0.2$ fixed and vary $f_{\\mathrm{edd}}$ from $1-100$~(recall that  sets the maximum accretion rate in the units of the Eddington rate). Not unexpectedly, we find that as $f_{\\mathrm{edd}}$ is increased, the final BH mass is enhanced via accretion-driven BH growth. $f_{\\mathrm{edd}}\\gtrsim10$ is required for growing BHs to $\\sim10^9~M_{\\odot}$ with luminosities $\\sim10^{47}~\\mathrm{erg~s^{-1}}$ by $z=6$~(Figure \\ref{varying_accretion_fig}: green line).  Notably, further increasing $f_{\\mathrm{edd}}$ to 100 does not lead to any substantial increase in the final BH mass beyond $\\sim10^9~M_{\\odot}$~(Figure \\ref{varying_accretion_fig}, right panel: red line). This is because the Bondi accretion rate exceeds $100~\\times$ the Eddington limit for only a small fraction of the time.\n\nWe now keep $f_{\\mathrm{edd}}$ fixed at 1 and examine the impact of the boost factor $\\alpha$. When $\\alpha$ is increased from $1$ to $100$, it leads to a factor $\\sim5$ increase in the final BH mass at $z=6$~(blue vs orange lines in Figure \\ref{varying_accretion_fig}: right panel). This is substantially smaller than the impact of increasing $f_{\\mathrm{edd}}$ to 10. Additionally, further increasing the boost factor to 10000~(green lines in Figure \\ref{varying_accretion_fig}: right panel) makes no significant difference in the $z=6$ BH mass. This implies that the maximum accretion rate set by the Eddington factor is much more consequential to the $z=6$ BH mass, compared to the Bondi boost factor. This is because the majority of the BH mass assembly occurs at $z\\lesssim9$ when the accretion rates are already at their maximum allowed value. To that end, note that the maximum accretion rate can also be increased by decreasing the radiative efficiency $\\epsilon_r$. Several cosmological simulations~(including N21) use a lower efficiency of $\\epsilon_r=0.1$. If we fix $\\alpha=1$ and decrease the radiative efficiency from $0.2$ to $0.1$, the $z=6$ BH mass increases by factor of $10$~(see blue vs red lines in Figure \\ref{varying_accretion_fig}: right panel). Not surprisingly, this is similar to what we found when $f_{\\mathrm{edd}}$ was increased to 2~(revisit Figure \\ref{varying_accretion_fig}: left panel). Notably, at this lower radiative efficiency of $\\epsilon_r=0.1$, applying a boost of $\\alpha=100$ forms a $\\sim10^9~M_{\\odot}$ BH with luminosity of $\\sim10^{47}~\\mathrm{erg~s^{-1}}$ at $z=6$~(purple line in Figure \\ref{varying_accretion_fig}: right panel). This is consistent with the results of N21, as we shall see in more detail in Section \\ref{Comparison with other theoretical works}. \n\n\n\n\n\n\nTo summarize our results thus far, we have shown that to form the observed $z\\sim6$ quasars within rare dense compact halos in our simulations, one of the following two requirements must be fulfilled: \\begin{enumerate} \\item When the default TNG model is used (i.e., $\\alpha=1,\\epsilon_r=0.2~\\&~f_{\\mathrm{edd}}=1$), we need a sufficiently early~($z\\gtrsim9$) boost in BH mass driven by BH mergers to grow to $\\sim10^9~M_{\\odot}$ by $z=6$. Our gas-based seeding prescription with $\\tilde{M}_{\\mathrm{sf,mp}}=5$ and $\\tilde{M}_h=3000$ satisfies this requirement. \\item For more restrictive seeding models~($\\tilde{M}_{\\mathrm{sf,mp}}\\gtrsim150$) where mergers are absent or insufficient, $z\\sim6$ quasars cannot be produced unless we enhance the maximum allowed accretion rate~(by factors $\\gtrsim10$) within these extreme overdense regions by either increasing the Eddington factor or decreasing the radiative efficiency. Notably, this result is consistent with the recent work of \\citep{2022arXiv220412513H} which uses semi-analytic approach to produce $z\\sim6$ quasars using super-Eddington accretion, from both light~($10~M_{\\odot}$) and heavy seeds~($10^5~M_{\\odot}$).  \n\\end{enumerate}\n\n\\subsection{Impact of seed model on BH growth for `growth optimized' accretion parameters}\nGiven that some of these BH accretion models can produce massive BHs by $z\\sim6$ without an early boost from BH mergers, we now explore what happens when these `growth optimized' accretion parameters are combined with merger-driven growth from less restrictive BH seed models. In the left panel of Figure \\ref{varying_accretion_edd}, we take the accretion model $\\alpha=1, \\epsilon_r=0.2~\\&~f_{\\mathrm{edd}}=10$, and compare the BH growth histories for two seed models with $\\tilde{M}_{\\mathrm{sf,mp}}=5~\\&~1000$~(\\texttt{SM5_FOF3000} \\& \\texttt{SM1000_FOF3000} respectively). We have already seen that \\texttt{SM1000_FOF3000} produces a $\\sim10^9~M_{\\odot}$ BH with  $\\sim10^{47}~\\mathrm{erg~s^{-1}}$ luminosity at $z=6$ even in the absence of significant number of mergers. We now examine whether the substantial merger-driven growth of \\texttt{SM5_FOF3000} leads to any further increase in the $z=6$ BH mass much beyond $\\sim10^9~M_{\\odot}$, when combined with super-Eddington accretion of $f_{\\mathrm{edd}}=10$. The \\texttt{SM5_FOF3000} model grows BHs via mergers to $\\sim10^7~M_{\\odot}$ by $z\\sim9$, which is a factor of $\\sim 500$ higher than the \\texttt{SM1000_FOF3000} model. Despite this large difference in masses, the luminosities are similar in both models, such that the higher-mass BH in \\texttt{SM5_FOF3000} has a lower Eddington ratio. By\n $z=6$, the BH in \\texttt{SM5_FOF3000}\nreaches a mass of $\\sim10^9~M_{\\odot}$, similar to \\texttt{SM1000_FOF3000}. To summarize, if a given model is already producing a $\\sim10^9~\\mathrm{M_{\\odot}}$ BH at $z\\sim6$ via accretion alone, boosting the merger-driven BH growth by forming more seeds does not further increase the final $z=6$ BH mass by a significant amount.\n\nThe results from Figure \\ref{varying_accretion_edd} also imply that if the accretion model is such that $z\\sim6$ quasars can be assembled via accretion alone, multiple sets of seed models can produce the observed $z\\sim6$ quasars. In such a case, the $z\\sim6$ quasar observations alone may not be able to constrain BH seed models. However, the progenitors of these quasars at $z\\gtrsim9$ can have significantly different assembly histories depending on the seed model, particularly in terms of the contribution from BH mergers. In fact, \\texttt{SM5_FOF3000} naturally predicts a $\\sim100$ times higher number of mergers compared to \\texttt{SM1000_FOF3000}. Moreover, these merging progenitors will likely include the most massive black holes at their respective redshift. Therefore, detection of the loudest LISA events at $z\\gtrsim9$ are likely to provide strong constraints for seed models. In terms of electromagnetic observations, the AGN progenitors are above of the detection limits of Lynx and JWST~(with limiting apparent magnitude of 31) up to $z\\sim10$; this is true for both \\texttt{SM5_FOF3000} and \\texttt{SM1000_FOF3000} models~(revisit 3rd row of Figure \\ref{varying_accretion_edd}). However, the difference in luminosities produced by both seed models is within a factor of $\\sim10$, corresponding to a magnitude difference of only $\\sim2.5$. Therefore, it is likely going to be difficult to find imprints of seed models within Lynx and JWST observations of the brightest AGN at higher redshifts~($z\\gtrsim9$).\n\n\\subsection{BH growth at higher resolutions}\n\\label{BH growth at higher resolutions_sec}\n\\begin{figure}\n\\includegraphics[width=8.5 cm]{Seeding_distribution_resolution_convergence.png}\n\\includegraphics[width=8.5 cm]{Growth_rate_resolution_convergence.png}\n\\caption{Top and bottom panels show the total number of seeds formed and\ngrowth of the most massive BH respectively for $M_{\\mathrm{seed}}=10^5~M_{\\odot}\/h$ at two different resolutions, namely $N=360$ and $N=720$. We show the resolution convergence for two distinct seed models which produce a $\\sim10^9~M_{\\odot}$ BH by $z=6$. Left panels correspond to the least restrictive seeding model of $\\tilde{M}_{\\mathrm{sf,mp}}=5, \\tilde{M}_{\\mathrm{h}}=3000$, wherein mergers substantially dominate the growth at $z\\gtrsim9$ and accretion model is Eddington limited with $\\alpha=1,\\epsilon_r=0.2,f_{\\mathrm{edd}}=1$. The right panels correspond to the much more restrictive seed model of $\\tilde{M}_{\\mathrm{sf,mp}}=1000, \\tilde{M}_{\\mathrm{h}}=3000$ with insignificant merger-driven growth; the accretion model corresponds to $\\alpha=1,\\epsilon_r=0.2,f_{\\mathrm{edd}}=10$. In both the seed models, the final BH masses at $z\\sim7$ are similar for $N=360$ and $N=720$.}\n\\label{resolution_convergence_fig}\n\\end{figure}\nWe have thus far presented results at our fiducial resolution of $N=360$, corresponding to gas mass resolution of $\\sim10^5~M_{\\odot}\/h$. While we were able explore a wide range of models at this resolution at reasonable computational cost, we demonstrated in \\cite{2021MNRAS.507.2012B} that our gas-based seed models start to become reasonably well converged only at resolutions $\\lesssim10^4~M_{\\odot}\/h$. Therefore, it is imperative to perform a resolution convergence test by running some of these simulations at gas mass resolutions of $\\sim10^4~M_{\\odot}\/h$~($N=720$). Particularly, we consider the seed models \\texttt{SM5_FOF3000} with $\\alpha=1,\\epsilon_r=0.2,f_{\\mathrm{edd}}=1$, and \\texttt{SM1000_FOF3000} with $\\alpha=1,\\epsilon_r=0.2,f_{\\mathrm{edd}}=10$, both of which successfully produced a $\\sim10^9~M_{\\odot}$ quasar by $z\\sim6$. Due to computational reasons, we could only run the higher resolution simulations~($N=720$) to $z=7$. \n\nThe results are shown in Figure \\ref{resolution_convergence_fig}, where they are compared to the lower resolution runs~($N=360$). Let us start with \\texttt{SM5_FOF3000}~(left panels), which produces enough seeds to allow for substantial amounts of merger-driven BH growth. The number of seeds formed~(Figure \\ref{resolution_convergence_fig}: top left panel) is similar between $N=360~\\&~720$ for $z\\gtrsim12$. As shown in \\cite{2021MNRAS.507.2012B}, at these redshifts, seeding is largely driven by the proliferation of new star forming regions~(star formation is reasonably well converged between gas mass resolutions of $\\lesssim10^5~M_{\\odot}\/h$). At $z\\lesssim12$, the higher resolution simulation produces a somewhat lower number of seeds~(by factors up to $\\sim5$). The slower resolution convergence at $z\\sim7-12$ is also fully consistent with the zoom simulations of \\cite{2021MNRAS.507.2012B}. It is due to the markedly stronger metal dispersion for higher resolutions, which causes a stronger suppression of seeding at $z\\sim7-12$ relative to lower resolution simulations. Nevertheless, the final $z=7$ BH mass of $\\sim10^8~M_{\\odot}\/h$ assembled by the higher resolution simulation~(Figure \\ref{resolution_convergence_fig}: bottom left panel), is only slightly smaller~(by a factor of $\\sim1.5$) compared to the lower resolution simulation. This strongly indicates at even at higher resolutions, the \\texttt{SM5_FOF3000} seed model would be able to assemble a $\\sim10^9~M_{\\odot}$ by $z=6$. \n\nNow let us focus on the \\texttt{SM1000_FOF3000} model~(with $f_{\\mathrm{edd}}=10$), where the merger-driven BH growth is minimal and super-Eddington growth is used to produce a $\\sim10^9~M_{\\odot}$ by $z=6$. Here, the higher resolution run~($N=720$) produces only 1 seed, whereas the lower resolution produced $\\sim10$ seeds~(Figure \\ref{resolution_convergence_fig}: top right panel). This is also consistent with our findings in \\cite{2021MNRAS.507.2012B} where we showed that resolution convergence becomes poorer as seed models become more restrictive with higher $\\tilde{M}_{\\mathrm{sf,mp}}$. Despite this, the accretion-driven BH growth at $z\\lesssim9$ assembles a BH mass close to $\\sim10^8~M_{\\odot}$ by $z\\sim7$ for both $N=720~\\&~360$ resolutions~(Figure \\ref{resolution_convergence_fig}: bottom right panel). Overall, we find that the BH models which successfully produce a $z\\gtrsim6$ quasar at our fiducial resolution~($N=360$), will likely continue to do so at even higher resolutions.     \n\\subsection{DCBHs as possible seeds of $z\\gtrsim7$ quasars}\n\n\\begin{figure}\n\\includegraphics[width=8 cm]{LW_distribution.png}\n\\caption{Total mass of gas cells illuminated by LW photons originating from Pop II~($0.001<Z<0.1~Z_{\\odot}$) and Pop III~($Z<0.001~Z_{\\odot}$) stars within bins of various flux values shown in the x-axis. These are runs at $N=720$~(higher than the fiducial resolution). Dotted lines correspond to all gas cells and solid lines correspond to dense, metal poor gas cells. Black vertical lines correspond to flux thresholds of $J_{\\mathrm{crit}}=300~J_{21}$~(dashed) and $J_{\\mathrm{crit}}=1000~J_{21}$~(dashed). When we look at all gas cells, flux values can reach up to $\\sim10^4~J_{21}$. However, within dense, metal poor regions, flux values reach only up to $\\sim300~J_{21}$. Therefore, for direct collapse black hole~(DCBHs) seeds to form in our constrained regions, the critical LW flux values need to be less than $\\sim300~J_{21}$. One-zone chemistry models and high resolution hydrodynamic simulations predict significantly higher critical flux~($\\gtrsim1000~J_{21}$).}\n\\label{LW_distribution_fig}\n\\end{figure}\n\nHere we use the higher resolution simulations~($N=720$) to investigate the possibility of DCBHs as candidates for the seeds of the $z\\gtrsim6$ quasars. The conditions for their formation are very restrictive due to the requirement of high LW flux incident upon dense, pristine gas. In particular, small scale hydrodynamic simulations~\\citep{2010MNRAS.402.1249S} and one-zone chemistry models~\\citep{2014MNRAS.445..544S,2017MNRAS.469.3329W} infer critical LW fluxes to be $\\gtrsim10^3~J_{21}$. In the zoom simulations of \\citep{2021MNRAS.tmp.3110B} that contained a $3.3\\sigma$ overdense peak~(for a target halo mass of $3.5\\times10^{11}~M_{\\odot}\/h$ at $z=5$), the highest LW flux incident within dense, metal poor gas was $\\sim100~J_{21}$. The work showed that the bulk of the $z\\gtrsim6$ SMBH population may be difficult to explain via the DCBH seeding channel. \n\nWe investigate whether DCBHs can form in the \\texttt{5SIGMA_COMPACT} region, which is much more extreme compared to the region probed in \\cite{2021MNRAS.tmp.3110B}. Figure \\ref{LW_distribution_fig} shows the distribution of LW intensities at various redshifts, across all gas cells~(dotted lines) as well as dense, metal poor gas cells~(solid lines). In general, the LW intensities can be as high as $\\sim10^4~J_{21}$. However, when we exclusively look at dense, metal poor gas, the LW intensities span only up to $\\sim300~J_{21}$. When we apply a critical flux threshold of $\\sim300~J_{21}$ for seeding~(\\texttt{SM5_FOF3000_LW300}), only a handful of seeds are formed~(Figure \\ref{seedmass_dependence}: upper right panel). For higher critical fluxes e.g. $\\sim10^3~J_{21}$, there is no DCBH seed formation. Therefore, our model can support DCBHs as potential candidates for the seeds of $z\\gtrsim6$ quasars only if the critical LW flux is $\\lesssim300~J_{21}$. Secondly, due to the absence of any significant merger-driven BH growth, to produce $z\\sim6$ quasars from these very few DCBH seeds, we need to enhance the maximum allowed accretion rate compared to the TNG accretion model by allowing for super-Eddington accretion or reducing the radiative efficiency~(as follows from the results of Section \\ref{Impact of BH accretion sec}). As an example, the lower right panel of Figure \\ref{seedmass_dependence} shows the growth of a DCBH seed~($J_{\\mathrm{crit}}=300~J_{21}$) to $\\sim10^8~M_{\\odot}\/h$ by $z=7$ via super-Eddington accretion with $f_{\\mathrm{edd}}=10$. Given the trends from the lower resolution $N=360$ simulations, we expect the BH to continue growing to $\\sim10^9~M_{\\odot}$ by $z=6$.\n\n\\subsection{Impact of seed mass on the formation of $z\\gtrsim6$ quasars}\nAt our fiducial resolution~($N=360$), we were able to probe seed masses of $10^5~\\&~8\\times10^5~M_{\\odot}\/h$. The higher resolution simulations~($N=720$) allow us to probe seed masses down to $\\sim10^4~M_{\\odot}$. In Figure \\ref{seedmass_dependence}, we reduce the seed mass from $10^5~M_{\\odot}\/h$ to $1.25\\times10^4~M_{\\odot}\/h$ and study its impact on the final BH mass at $z=7$. We again consider two models that have been shown to successfully produce a $\\sim10^9~M_{\\odot}\/h$ at $z\\sim6$. We start with the \\texttt{SM5_FOF3000} model~(left panels) where there is substantial amount of merger-driven BH growth. Here we see that the $1.25\\times10^4~M_{\\odot}\/h$ seeds form $\\sim8$ times more abundantly compared to $1\\times10^5~M_{\\odot}\/h$ seeds. As a result, $1.25\\times10^4~M_{\\odot}\/h$ seeds undergo more mergers and grow to similar masses as $1\\times10^5~M_{\\odot}\/h$ seeds by $z\\sim7$~(as also seen in \\citealt{2021MNRAS.507.2012B}). At $z=7$, both the seed masses assemble a $\\sim10^8~M_{\\odot}$ BH.\n\nNext, we consider the model \\texttt{SM5_FOF3000_LW300} which adds a LW flux criterion with $J_{\\mathrm{crit}}=300~J_{21}$. This model~(as seen in the previous section) is so restrictive that only a handful of seeds are formed in the entire simulation box~(Figure \\ref{seedmass_dependence}: top right panel). Here we allow for super-Eddington accretion with $f_{\\mathrm{edd}}=10$~(see Figure \\ref{seedmass_dependence}: bottom right panel). We can see that for $M_{\\mathrm{seed}}=10^5~M_{\\odot}\/h$, there are no mergers in its history; and for $M_{\\mathrm{seed}}=1.25\\times10^4~M_{\\odot}\/h$, there is only one merger. In the absence of mergers, there is no appreciable growth of these seeds at $z\\gtrsim9$. Despite that, both $10^5~M_{\\odot}\/h$ and $1.25\\times10^4~M_{\\odot}\/h$ seeds grow to $\\sim10^8~M_{\\odot}\/h$ via accretion between $z\\sim9-7$. \n\nTo summarize, whether we consider models where $z\\sim6$ quasars are formed either with the help of BH mergers, or gas accretion alone, the mass assembled at $z\\sim6-9$ is not sensitive to the seed mass between $10^4-10^6~M_{\\odot}\/h$. Note however that we are not able to probe seed masses below  $\\sim10^4~M_{\\odot}$ due to resolution limitations. It is possible that these lowest mass seeds~($\\sim10^2-10^3~M_{\\odot}$) may not be able to grow into the $z\\gtrsim6$ quasars, particularly in the absence or lack of mergers;  we shall investigate this in future studies.  \n\n\\begin{figure}\n\\includegraphics[width=8.5 cm]{Seeding_distribution_seed_dependence.png}\n\\includegraphics[width=8.5 cm]{Growth_rate_seed_dependence.png}\n\\caption{Top and bottom panels, respectively, show the number of seeds and BH mass growth at $N=720$ for two different seed masses of $M_{\\mathrm{seed}}=10^5~\\&~1.25\\times10^4~M_{\\odot}\/h$ . Left panels correspond to the least restrictive seeding model of $\\tilde{M}_{\\mathrm{sf,mp}}=5, \\tilde{M}_{\\mathrm{h}}=3000$, wherein mergers substantially dominate the growth at $z\\gtrsim9$ and accretion rate is Eddington limited. Here, $1.25\\times10^4~M_{\\odot}\/h$ seeds form and merge $\\sim8$ times more frequently than $1\\times10^5~M_{\\odot}\/h$ seeds. Both seed masses grow to similar mass BHs by $z\\sim7$. The right panels correspond to the much more restrictive seed model of $\\tilde{M}_{\\mathrm{sf,mp}}=5, \\tilde{M}_{\\mathrm{h}}=3000$ and critical LW flux of $300~J_{21}$; the accretion model is given by  $\\alpha=1,\\epsilon_r=0.2,f_{\\mathrm{edd}}=10$. Here, the merger-driven growth is insignificant. Regardless, both seed masses produce similar mass BHs by $z\\sim7$.}\n\\label{seedmass_dependence}\n\\end{figure}\n\n\n\n\n\\subsection{Comparison with other theoretical works}\n\\label{Comparison with other theoretical works}\n\n\\begin{figure}\n\\includegraphics[width=8 cm]{arepo_vs_gadget.png}\n\\caption{Comparison between the BH growth histories for the constrained runs in our work vs \\protect\\cite{2021MNRAS.tmp.2867N}~(hereafter N21). Their simulations were run using the \\texttt{MP-GADGET} code~\\citep{yu_feng_2018_1451799} with the galaxy formation model adopted from the \\texttt{BlueTides} simulation~\\citep{2016MNRAS.455.2778F}. Solid lines show predictions from our simulations, and the dashed line is the prediction from N21. To make a fair comparison, we tune our simulation box, initial conditions to be the same as that used in N21~(box size of $20~\\mathrm{Mpc}\/h$ with $N=352$). We also use the same ``halo mass based\" seeding prescription as N21, with $5\\times10^5~M_{\\odot}\/h$ seeds placed inside $>5\\times10^{10}~M_{\\odot}\/h$ halos. Our fiducial model~(with $\\alpha=1,f_{\\mathrm{edd}}=1,\\epsilon_r=0.2$) assembles a significantly lower BH mass~(by factors of $\\sim30$) compared to N21. This difference is due to the the combined effect of the higher Bondi boost factor and Eddington factor, as well as lower radiative efficiency used in N21. In fact, when we use the same accretion parameters as N21~($\\alpha=100,f_{\\mathrm{edd}}=2,\\epsilon_r=0.1$), we produce a slightly higher~(by factor of $\\sim2$) mass BH compared to their work.}\n\\label{arepo_vs_gadget_fig}\n\\end{figure}\n\nHere, we compare our results to other theoretical works that have explored the formation of the $z\\gtrsim6$ quasars. We will first compare with hydrodynamic simulations, where we note that most of the existing work has so far used seed models that are only based on halo mass. Our simulations using the TNG seed model therefore provide the most direct comparison to such studies. \nWe start with the constrained simulations of N21 produced using the \\texttt{MP-GADGET} code~\\citep{yu_feng_2018_1451799} with the \\texttt{BlueTides} galaxy formation model~\\citep{2016MNRAS.455.2778F}. Their primary constrained peak~(referred to as \\texttt{BIG-BH} in their work) is very similar to \\texttt{5SIGMA_COMPACT}. Their seed model is also similar to the TNG seed model; they adopt the same halo mass threshold for seeding~($>5\\times10^{10}~M_{\\odot}\/h$), but with a slightly smaller seed mass of $5\\times10^5~M_{\\odot}\/h$. Notably, their simulation produced a $\\sim3\\times10^8~M_{\\odot}$ BH by $z=7$, which is significantly higher than the predictions in our simulations with the TNG seeding and accretion model. But N21 uses a lower radiative efficiency of $\\epsilon_r=0.1$ and a Bondi boost factor of $\\alpha=100$, which we have already shown to produce a much stronger growth compared to the TNG accretion model~(revisit Figure \\ref{varying_accretion_fig}: right panel). Additionally, they also have a higher Eddington factor of $f_{\\mathrm{edd}}=2$. \n\nWe perform a more direct comparison to N21 in Figure \\ref{arepo_vs_gadget_fig} by simulating a box identical to their work, particularly in terms of volume~($20~\\mathrm{Mpc}\/h$ box length), resolution~($N=352$), initial condition~(\\texttt{BIG-BH}) and the BH seed model~($5\\times10^5~M_{\\odot}\/h$ seeds in $>5\\times10^{10}~M_{\\odot}\/h$ halos). If we apply the TNG accretion model~($\\alpha=1,f_{\\mathrm{edd}}=1,\\epsilon_r=0.2$), our \\texttt{BIG-BH} simulation assembles a BH of mass $\\sim10^7~M_{\\odot}$ by $z=7$~(Figure \\ref{arepo_vs_gadget_fig}: purple line); this is $\\sim30$ times smaller than the N21 predictions~(similar to that of \\texttt{5SIGMA_COMPACT}). Next, if we individually adjust each of these accretion parameters~(Figure \\ref{arepo_vs_gadget_fig}: pink, green and blue lines) to the N21 values~(one parameter at a time), we find that they all lead to notable enhancement in the BH growth~(as also seen in Section \\ref{Impact of BH accretion sec} for \\texttt{SM1000_FOF3000}). Finally, if all the accretion parameters in our simulations are simultaneously set to be the same as N21~(Figure \\ref{arepo_vs_gadget_fig}: black line), we produce a $\\sim5\\times10^8~M_{\\odot}$ BH by $z=7$, which is only slightly higher than the N21 predictions. Overall, our results are broadly consistent with N21. The same general conclusion also applies to the comparison with the results of \\cite{2020MNRAS.496....1H}, which performs constrained runs using \\texttt{MP-GADGET} similar to that of N21. Notably, they find that the final mass at $z\\sim6$ is insensitive to the seed mass for $\\sim 5\\times10^4-5\\times10^5~M_{\\odot}\/h$ seeds, which is consistent with our findings. \n\nThe zoom simulations of \\cite{2009MNRAS.400..100S}, \\cite{2014MNRAS.440.1865F} and \\cite{2014MNRAS.439.2146C} adopted a Bondi boost factor of 100 and radiative efficiencies of 0.05-0.1; with this accretion model, they successfully produced the $z\\sim6$ quasars without the need for mergers or super-Eddington accretion, consistent with our results. \\cite{2020arXiv201201458Z} performed zoom simulations targeting the formation of a $\\sim10^{13}~M_{\\odot}$ halo at $z=6$. In their fiducial model, they placed $10^5~M_{\\odot}\/h$ seeds in $10^{10}~M_{\\odot}\/h$ halos. These seeds grew via Eddington limited Bondi accretion~($\\alpha=1,f_{\\mathrm{edd}}=1$) with radiative efficiency of $\\epsilon_r=0.1$. With this model, they were able to grow a $\\sim10^9~M_{\\odot}$ BH by $z=6$ without the help of substantial merger-driven BH growth, super-Eddington accretion, or a Bondi boost. They achieve somewhat faster BH growth compared to our simulations, which only assembles a  $\\sim2\\times10^8~M_{\\odot}$ BH by $z=6$ if $\\epsilon_r=0.1$ is applied without a Bondi boost~(see green line in Figure \\ref{arepo_vs_gadget_fig}). While it is not clear what may be causing the difference between our results and \\cite{2020arXiv201201458Z}, it may be attributed to differences in the implementation of other aspects of the galaxy formation model such as metal enrichment and stellar feedback. However, when they increased their radiative efficiency to 0.2, their final BH mass at $z=6$ dramatically decreased to $\\sim4\\times10^7~M_{\\odot}$, consistent with our findings. Finally, early work by \\cite{2007ApJ...665..187L} also  produced a $\\sim10^{9}~M_{\\odot}$ quasar with $\\epsilon_r=0.1$ and $\\alpha=1,f_{\\mathrm{edd}}=1$ within a $8\\times10^{12}~M_{\\odot}$ halo at $z=6.5$. Notably, their host halo was assembled after a series of 8 major mergers between $z\\sim14-6.5$. Their results therefore highlighted yet another formation pathway for high-z quasars i.e. via a series of intermittent rapid growths driven by major mergers.  \n\n\n\nSimilar to \\cite{2020arXiv201201458Z}, \\cite{2019MNRAS.488.4004L} was also able to produce a $\\sim10^9~M_{\\odot}$ by $z\\sim6$ with Eddington limited Bondi accretion and radiative efficiency of 0.1, without applying a Bondi boost factor. This may be due to their adopted thermal feedback efficiency of 0.005, which is significantly smaller than values~($\\sim0.05-0.15$) adopted for most other works including ours. The very recent work of \\cite{2021MNRAS.507....1V} considered even lower radiative efficiencies of $0.03$, allowing them to  grow BHs to $\\sim10^9~M_{\\odot}$ by $z\\sim6$ via feedback regulated Bondi accretion without the need of a Bondi boost factor. Lastly, Radiation hydrodynamics zoom simulations of \\cite{2018ApJ...865..126S} also adopted a radiative efficiency of 0.1 and is able to assemble a $\\sim10^9~M_{\\odot}$ BH by $z\\sim7$ despite gas accretion being sub-Eddington~(and no substantial contribution from BH mergers) for almost the entire growth history; however it is difficult to compare their results to our work since they adopted the alpha disk formalism ~(see Eq. (2) of \\citealt{2010MNRAS.406L..55D}) to calculate the BH accretion rate, where there is no explicit dependence on BH mass.   \n\nWhile previous hydrodynamic simulations probing $z\\gtrsim6$ quasars have mostly adopted halo mass based prescriptions for seeding, SAMs have been able to explore a broader range of seeding channels~(and seed masses) with more physically motivated seeding criterias~\\citep[for example,][]{2007MNRAS.377.1711S,Volonteri_2009, 2012MNRAS.423.2533B,2018MNRAS.476..407V, 2018MNRAS.481.3278R, 2019MNRAS.486.2336D, 2020MNRAS.491.4973D}. Here we shall compare with works that have used SAMs to make predictions specific to the  $z\\gtrsim6$ quasars. \\cite{2016MNRAS.457.3356V} and \\cite{2021MNRAS.506..613S} used the \\texttt{GAMETE\/QSOdust} data constrained SAM~(introduced in \\citealt{2016MNRAS.457.3356V}) to trace the formation of a $\\sim10^9~M_{\\odot}$ BH along the merger tree of a $10^{13}~M_{\\odot}\/h$ halo at $z=6.42$. They find that heavy seeds~($\\sim10^{5}~M_{\\odot}\/h$) contribute the most to the formation of $z\\gtrsim6$ quasars compared to light~($\\sim10^{2}~M_{\\odot}\/h$) and intermediate seeds~($\\sim10^{3}~M_{\\odot}\/h$). While we cannot probe light and intermediate seeds, their results for the heavy seeds do not conflict with our findings.  They also apply a radiative efficiency of 0.1 combined with a Bondi boost factor of $50-150$, and are able to assemble $z\\sim6$ quasars without substantial contributions from BH mergers or super-Eddington accretion. This is fully consistent with our results, and also with results from other hydrodynamic simulations.\n\nOverall, we find that the differences between our results with \\texttt{IllustrisTNG} physics and the results from most previous works, are largely originating from differences in the modeling of BH accretion and feedback. This also brings to light that when the default \\texttt{IllustrisTNG} physics is applied to such extreme overdense regions, it is much more difficult to form $z\\sim6$ quasars compared to the physics adopted in other simulations and SAMs. This is primarily due to the TNG accretion model which has a higher radiative efficiency of 0.2~(most works adopt a value of 0.1). At the same time, the lack of a Bondi boost also slows the BH growth even further~(most works adopted a value of 100) in the TNG accretion model. Note that the uncertainties within the modeling of BH accretion are significant, particularly at high redshifts wherein the gas environments are likely to be very different compared to the assumptions underlying the Bondi accretion model. Additionally, the radiative efficiencies are also poorly constrained. To that end, different subgrid models are better or worse at reproducing different aspects of the observed SMBH and galaxy populations~\\citep[e.g.][]{2021MNRAS.503.1940H,2022MNRAS.509.3015H}. Moving forward, it will be necessary to build better subgrid models with fewer modeling uncertainties, as well as improving the observational constraints particularly on high-z SMBHs.     \n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion and Conclusions}\n\\label{Conclusions_sec}\nIn this work, we have studied the implications of the \\texttt{IllustrisTNG} galaxy formation model on the brightest $z\\gtrsim6$ quasar population, particularly in the context of different BH seeding models. These extremely rare~($\\sim1~\\mathrm{Gpc}^{-3}$) objects have grown to masses of $\\sim10^9-10^{10}~M_{\\odot}$~(comparable to the most massive $z\\sim0$ SMBHs) within the first Gyr since the Big Bang; this is difficult to achieve in general, and it is likely to place strong constraints on models for BH formation and growth. \n\nWe explore the following seeding prescriptions:\n\\begin{itemize}\n    \\item TNG seed model: This is the default ``halo mass based\" prescription used within the \\texttt{IllustrisTNG} simulation suite, where we place $8\\times10^5~M_{\\odot}\/h$ seeds in $>5\\times10^{10}~M_{\\odot}\/h$ halos. \n    \\item gas-based seed models: Here we place seeds~($M_{\\mathrm{seed}}=1.25\\times10^4,1\\times10^5~\\&~8\\times10^5~M_{\\odot}\/h$) in halos that exceed critical thresholds for halo mass and dense, metal poor gas mass~(represented by $\\tilde{M}_h$ and $\\tilde{M}_{\\mathrm{sf,mp}}$ respectively in the units of $M_{\\mathrm{seed}}$). We also explore models where the dense, metal poor gas is required to have LW fluxes above a critical value $J_{\\mathrm{crit}}$. \n\\end{itemize}\n\n\nWith the above seeding prescriptions, we probe the possible formation of the $z\\sim6$ quasars~(defined as $\\sim10^9~M_{\\odot}$ BHs with luminosities of $\\sim10^{47}~\\mathrm{erg~s^{-1}}$) within extremely rare peaks in the density field using the technique of constrained Gaussian realizations. This technique allows us to constrain the peak of the density field so as to assemble $\\gtrsim10^{12}~M_{\\odot}\/h$ halos by $z\\sim7$ within a simulation volume of $(9~\\mathrm{Mpc\/h})^3$. Having a relatively small simulation volume allows us to build a large simulation suite exploring a variety of density peak parameters as well as seeding parameters.    \n\nWe reproduce findings from previous work~(N21) showing that BH growth is most efficient at density peaks that have high compactness and a low tidal field. In fact, a highly compact $5\\sigma$ peak at $1.0$ Mpc\/h with low tidal field~(\\texttt{5SIGMA_COMPACT}) produces a more massive BH~(by factors of $\\sim2$) compared to a typical $6\\sigma$ peak at $1.3$ Mpc\/h~(\\texttt{6SIGMA}). The reason for this is two-fold: First, the target $z=7$ halo in \\texttt{5SIGMA_COMPACT} has more massive progenitors than that of \\texttt{6SIGMA}, allowing seeds to form in potentially higher numbers and boosting the merger-driven BH growth. Second, the \\texttt{5SIGMA_COMPACT} region forms a more compact gas cloud which falls towards the BH more symmetrically from all directions compared to \\texttt{6SIGMA}; this leads to higher gas densities in their neighborhood and boosts the accretion-driven BH growth. \n\nDespite the enhanced accretion and merger-driven BH growth in \\texttt{5SIGMA_COMPACT}, we find that when the TNG seed model is used, the final mass of the central BH in the target halo at $z=6$ is only $\\sim5\\times10^7~M_{\\odot}$ with luminosities of $\\sim10^{45}~\\mathrm{erg~s^{-1}}$. This significantly falls short of producing an observed $z\\sim6$ quasar i.e. a $\\sim10^9~M_{\\odot}$ BH with a bolometric luminosity of $\\sim10^{47}~\\mathrm{erg~s^{-1}}$. But when we apply the more physically motivated gas-based seeding prescription where BHs are seeded in halos with minimum star forming metal poor gas mass of $5$ times the seed mass and a total halo mass of $3000$ times the seed mass~($\\tilde{M}_{\\mathrm{sf,mp}}=5$ and $\\tilde{M}_{\\mathrm{h}}=3000$), we find that there is substantial amount of the merger-driven BH growth at $z\\gtrsim10$ compared to the TNG seed model. As a result, the BH assembles a mass of $\\sim10^{9}~M_{\\odot}$ at $z=6$ and grows close to the Eddington limit with a bolometric luminosity of  $\\sim10^{47}~\\mathrm{erg~s^{-1}}$. This is consistent with the observed $z\\sim6$ quasars, and is achievable for all seed mass values between $\\sim10^4-10^6~M_{\\odot}\/h$. Lastly, note that this can also be achieved by enhancing merger-driven growth within halo-mass based seed models~(like TNG seed model) by sufficiently reducing the halo mass threshold.\n\nNotably, there are two distinct phases in the BH growth in our simulations: 1) $z\\gtrsim9$ when the BH growth is predominantly driven by BH mergers, and 2) $z\\sim9-6$ when gas accretion dominates the BH growth. \nTo form a $z\\gtrsim6$ quasar within a universe with  \\texttt{IllustrisTNG} physics, the BH growth has to be boosted by BH mergers at $z\\gtrsim9$. Amongst all the seed models we explored, only the one with $\\tilde{M}_{\\mathrm{sf,mp}}=5$ and $\\tilde{M}_{\\mathrm{h}}=3000$ provides enough mergers to assemble $z\\sim6$ quasars.\n\nFor much more restrictive gas-based seed models~($\\tilde{M}_{\\mathrm{sf,mp}}=1000$ and $\\tilde{M}_{\\mathrm{h}}=3000$, for example), very few seeds are formed and there is little to no merger-driven growth; as a result, they fail to produce $z\\sim6$ quasars in the \\texttt{IllustrisTNG} universe. However, recall that the \\texttt{IllustrisTNG} model was calibrated to reproduce properties of relatively common galaxies and BHs at low redshifts. We explored the possibility of enhanced accretion in these extreme overdense regions compared to the TNG accretion model. We found that in order to form $z\\sim6$ quasars with these restrictive seed models, it is crucial to increase the maximum accretion rate~(by factors $\\gtrsim10$) allowed for a BH of a given mass to grow. This can be achieved by either increasing the Eddington factor or decreasing the radiative efficiency. To that end, increasing the Bondi boost factor alone does not sufficiently boost the BH mass assembly to produce the $z\\sim6$ quasars. Lastly, note that even for such high values of  $\\tilde{M}_{\\mathrm{sf,mp}}$, one can enhance merger-driven BH growth by choosing a lower halo mass threshold $\\tilde{M}_{\\mathrm{h}}$; this would relax the constraints on the accretion model in producing $z\\sim6$ quasars.\n\n  \n\nProspects for DCBH formation in the \\texttt{5SIGMA_COMPACT} region are limited if the critical LW fluxes are indeed $\\gtrsim1000~J_{21}$ as predicted by one-zone chemistry models and small scale hydrodynamics simulations. This is because LW intensities within the dense, metal poor pockets of \\texttt{5SIGMA_COMPACT} region do not significantly exceed $\\sim300~J_{21}$ between $z\\sim7-22$. \\texttt{5SIGMA_COMPACT} region produces a handful of seeds for somewhat lower critical fluxes, particularly $\\sim300~J_{21}$. Even for these optimistic estimates of $J_{\\mathrm{crit}}$, due to the obvious lack of merger-driven BH growth, DCBHs would require one of the optimal accretion scenarios described in the previous paragraph in order to grow a $z\\gtrsim6$ quasar. As far as other theoretical seeding channels such as Pop III and NSC seeds, without being able to explicitly resolve their formation conditions, it is currently difficult to tell whether they form and merge abundantly enough to qualify as potential origins of the $z\\gtrsim6$ quasars; we shall investigate this in the future.  \n\nWe note that our results are specific to features of our underlying  \\texttt{IllustrisTNG} galaxy formation model. They may significantly depend on the prescriptions for star formation, metal enrichment, stellar feedback and BH dynamics. Additionally, there are also several other BH seeding, accretion  and feedback models beyond the ones explored in this work, that could potentially produce $z\\sim6$ quasars. Black hole accretion and feedback is a major source of uncertainty. For example, the lack of accretion-driven BH growth at $z\\gtrsim9$ may be partly influenced by the Bondi accretion model which struggles to grow low mass BHs due to the $M_{bh}^2$ scaling of the accretion rate. This $M_{bh}^2$ scaling also implies that at these early epochs when the self-regulation by feedback is relatively weak, the BH growth would be extremely sensitive to the local gas environment. This local gas environment may be impacted by other aspects of galaxy formation, such as star formation, stellar feedback~\\citep[for e.g.][]{2017MNRAS.468.3935H}, metal enrichment and gas cooling. While the $M_{bh}^2$ scaling appears as a generic feature of all accretion models based on a gas capture radius\n\\citep{2005MNRAS.361..776S,2007ApJ...665..107P,2009MNRAS.398...53B}, there are also models such as gravitational torque driven accretion~\\citep{2017MNRAS.464.2840A,2019MNRAS.486.2827D} where the scaling exponent is smaller~($M_{bh}^{1\/6}$). This can significantly boost the growth of low mass BHs, but also slow down the growth of high mass BHs. As a result, it can have non-trivial implications for the feasibility of various BH models to produce $z\\gtrsim6$ quasars. \n\nA final caveat to our results lies within our modelling of BH dynamics. In particular, due to the limited simulation resolution, we use the standard BH repositioning scheme which instantaneously relocates the BH to a nearby potential minimum. In fact, several simulations with more realistic dynamics models~\\cite[e.g.][]{2017MNRAS.470.1121T} have now indicated that it may be difficult for many of the seeds~(particularly lower mass seeds) to sink to the local potential minima, thereby leading to a population of wandering BHs~\\citep{2018ApJ...857L..22T,2021MNRAS.503.6098R,2021ApJ...916L..18R,2021MNRAS.508.1973M,2022MNRAS.tmp..221W}. This would have two important effects: 1) over-estimating the accretion rates since the BHs may spend more time around dense gas compared to more realistic dynamics models, and 2) nearby BHs are promptly merged, thereby overestimating the merger rates at early times.  In the future, we shall assess the impact of all of these caveats on the formation of $z\\gtrsim6$ quasars.\n\nDespite the caveats, our results overall indicate a strong prospect of revealing the seeding environments for the observed $z\\gtrsim6$ quasars using upcoming facilities such as LISA. In particular, regardless of the accretion model, different seed models predict distinct merger and accretion histories for the progenitors of these quasars at $z\\gtrsim9$. These progenitors will also be amongst the most massive sources at their corresponding redshift. In addition to the strong prospect of detecting their mergers with LISA up to $z\\sim20$, their AGN luminosities also exceed detection limits of Lynx and JWST up to $z\\sim10$. However, the difference in the predicted AGN luminosities between different seed models is small~($\\lesssim2.5$ dex in magnitude). Therefore, detecting electromagnetic signatures of seeding is going to be challenging for JWST and Lynx.  \n\n\n\n   \n\n\n\n\n\n\n\n\\section*{Acknowledgements}\nAKB thanks Dylan Nelson for valuable discussion and feedback. LB acknowledges support from NSF award AST-1909933 and Cottrell Scholar Award \\#27553 from the Research Corporation for Science Advancement.\nPT acknowledges support from NSF-AST 2008490.\nRW is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference \\#CITA 490888-16.\nTDM acknowledges funding from NSF AST-1616168, NASA ATP  80NSSC20K0519,\n NASA ATP 80NSSC18K101, and NASA ATP NNX17AK56G.\nThis work was also supported by the NSF AI Institute: Physics of the Future, NSF PHY-2020295. YN acknowledges support from the McWilliams fellowship.\n\\section*{Data availablity}\nThe underlying data used in this work shall be made available upon reasonable request to the corresponding author.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nThe $K^\\pm\\to\\pi^\\pm\\pi^+\\pi^-$ decay can be described~\\cite{pdg} in\nterms of the Lorentz invariant kinematic variables $u$ and $v$\ndefined as\n\\begin{equation}\nu=\\frac{s_{12}-s_0}{m_\\pi^2},~~~~v=\\frac{s_{13}-s_{23}}{m_\\pi^2},\n\\end{equation}\n$$\ns_{ij}=(P_i+P_j)^2,~~~i,j=1,2,3,~~~i<j;~~~\ns_0=\\frac{1}{3}(s_{12}+s_{13}+s_{23})\\equiv\\frac{1}{3}m_K^2+m_\\pi^2.\n$$\nHere $m_\\pi$ is the charged pion mass, $m_K$ is the charged kaon\nmass, $P_i$ are the pion four-momenta, the indices $i,j=1,2$\ncorrespond to the two pions of the same electric charge (``even''\npions), and the index $i,j=3$ to the other (``odd'') pion. The\nexperimental Dalitz plot distribution of the\n$K^\\pm\\to\\pi^\\pm\\pi^+\\pi^-$ decay has been up to now experimentally\nanalyzed~\\cite{fo72,de77} in terms of a polynomial expansion in\npowers of $u$ and $v$:\n\\begin{equation}\n|M(u,v)|^2\\sim C(u,v)\\cdot(1+gu+hu^2+kv^2).\n\\label{slopes}\n\\end{equation}\nHere $g$, $h$, $k$ are the linear and quadratic slope parameters\n(terms proportional to odd powers of $v$ are forbidden by Bose\nsymmetry, and the variable $v$ is defined only up to a sign), and\n$C(u,v)$ is the Coulomb factor~\\cite{be93}:\n\\begin{equation}\nC(u,v)=\\prod_{i,j=1,2,3;~i<j} \\{n_{ij}\/(e^{n_{ij}}-1)\\},~~~\nn_{ij}=2\\pi\\alpha q_iq_j\/\\beta_{ij}, \\label{coulomb}\n\\end{equation}\nwhere $q_i=\\pm1$ are the pion charges, $\\alpha$ is the fine\nstructure constant, and $\\beta_{ij}$ is the relative velocity of the\npions $i$ and $j$, expressed via the squared invariant mass of the\npion pair $s_{ij}$ as\n\\begin{equation}\n\\beta_{ij}=\\left(1-\\frac{4m_\\pi^2}{s_{ij}}\\right)^{1\/2}\n\\left(1-\\frac{2m_\\pi^2}{s_{ij}}\\right)^{-1}.\n\\end{equation}\n\nAmong the measurements of the slope parameters performed in the\npast, the most precise are reported in~\\cite{fo72} based on\n$1.5\\times 10^6$ $K^\\pm$ decays, and in~\\cite{de77} based on\n$0.225\\times10^6$ $K^+$ decays.\n\nThe primary goal of the NA48\/2 experiment at CERN SPS is the search\nfor direct CP-violating charge asymmetries of Dalitz plot linear\nslopes in $K^\\pm\\to\\pi^\\pm\\pi^+\\pi^-$~\\cite{k3pic} and\n$K^\\pm\\to\\pi^\\pm\\pi^0\\pi^0$~\\cite{k3pin} decays. However, the large\ndata sample collected also allows a study of the Dalitz plot\ndistributions to be performed at a new level of precision,\nestimating the detector acceptance with a detailed Monte Carlo (MC)\nsimulation.\n\nThe parameterization (\\ref{slopes}) of the\n$K^\\pm\\to\\pi^\\pm\\pi^+\\pi^-$ decay distribution takes into account\nelectromagnetic interactions only in the first\napproximation~\\cite{be93} and totally neglects pion rescattering\neffects~\\cite{ca05}--\\cite{ge07}, which were recently shown by\nNA48\/2 to contribute significantly to the distribution of\n$K^\\pm\\to\\pi^\\pm\\pi^0\\pi^0$ decays~\\cite{cusp}. The current study\naims to measure the Dalitz plot slopes within the conventional\nframework (\\ref{slopes}), and represents a first step towards a more\ncomplete description of the decay distribution. In particular, the\npresence of radiative effects and strong final state interactions,\nboth ignored in the current work, imply that the slope parameters\nobtained, while fully comparable to those defined in the Particle\nData Group compilation~\\cite{pdg}, should not be attributed any\nprecise physical significance as far as the parameterization of the\nweak decay in terms of more fundamental parameters is concerned;\nthis would require an improved theoretical framework which is just\nbeing developed~\\cite{co06}, and its implementation is postponed for\na forthcoming analysis.\n\nNA48\/2 collected data during two runs in 2003 and 2004, with about\n50 days of efficient data-taking in each run. This analysis is based\non $4.71\\times 10^8$ fully reconstructed $K^\\pm\\to\\pi^\\pm\\pi^+\\pi^-$\nevents, corresponding to 55\\% of the data sample collected during\nthe NA48\/2 run of 2003.\n\n\\section{Beams and detectors}\n\nA beam line providing two simultaneous charged beams of opposite\nsigns overlapping in space all along the decay volume was designed\nand built in the high intensity hall (ECN3) at the CERN SPS. The\nbeam line is a key element of the experiment, as it allows decays of\n$K^+$ and $K^-$ to be recorded at the same time, and therefore leads\nto cancellation of several systematic uncertainties for the charge\nasymmetry measurement. Regular alternation of magnetic fields in all\nthe beam line elements was adopted. The layout of the beams and\ndetectors is shown schematically in Fig.~\\ref{fig:beams}. The setup\nis described in a right-handed orthogonal coordinate system with the\n$z$ axis directed downstream along the beam, and the $y$ axis\ndirected vertically up.\n\n\\begin{figure}[tb]\n\\vspace{-6mm}\n\\begin{center}\n{\\resizebox*{\\textwidth}{!}{\\includegraphics{eps\/na48_2_beam.eps}}}\n\\end{center}\n\\vspace{-6mm} \\caption{Schematic lateral view of the NA48\/2 beam\nline (TAX17,18: motorized beam dump\/collimators used to select the\nmomentum of the $K^+$ and $K^-$ beams; FDFD\/DFDF: focusing set of\nquadrupoles, KABES1--3: kaon beam spectrometer stations), decay\nvolume and detector (DCH1--4: drift chambers, HOD: hodoscope, LKr:\nEM calorimeter, HAC: hadron calorimeter, MUV: muon veto). Note that\nthe vertical scales are different in the two parts of the figure.}\n\\label{fig:beams}\n\\end{figure}\n\nThe beams are produced by 400 GeV\/$c$ protons impinging at zero\nincidence angle on a beryllium target of 40 cm length and 2 mm\ndiameter. Charged particles with momentum $(60\\pm3)$ GeV\/$c$ are\nselected in a charge-symmetric way by an achromatic system of four\ndipole magnets with zero total deflection (`achromat'), which splits\nthe two beams in the vertical plane and then recombines them on a\ncommon axis. Then the beams pass through a defining collimator and a\nseries of four quadrupoles designed to produce focusing of the beams\ntowards the detector. Finally the two beams are again split in the\nvertical plane and recombined in a second achromat, where three\nstations of a Micromega-type~\\cite{gi96} detector form a kaon beam\nspectrometer~\\cite{pe04} (not used in the present analysis).\n\nAfter passing the cleaning and the final collimators, the beams\nenter the decay volume housed in a 114 m long cylindrical vacuum\ntank with a diameter of 1.92 m for the first 65 m, and 2.4 m for the\nrest. Both beams follow the same path in the decay volume: their\naxes coincide within 1~mm, while the transverse size of the beams is\nabout 1~cm. With $7\\times 10^{11}$ protons incident on the target\nper burst of $\\sim 4.8$ s duration, the positive (negative) beam\nflux at the entrance of the decay volume is $3.8\\times 10^7$\n($2.6\\times 10^7$) particles per pulse, of which $5.7\\%$ ($4.9\\%$)\nare $K^+$ ($K^-$). The $K^+\/K^-$ flux ratio is about $1.8$. The\nfraction of beam kaons decaying in the decay volume at nominal\nmomentum is $22\\%$.\n\nThe decay volume is followed by a magnetic spectrometer used for\nreconstructing $K^\\pm\\to\\pi^\\pm\\pi^+\\pi^-$ decays. The spectrometer\nis housed in a tank filled with helium at nearly atmospheric\npressure, separated from the vacuum tank by a thin ($0.31\\%X_0$)\nKevlar composite window. A thin-walled aluminium beam pipe of 16 cm\nouter diameter traversing the centre of the spectrometer (and all\nthe following detectors) allows the undecayed beam particles and the\nmuon halo from decays of beam pions to continue their path in\nvacuum. The spectrometer consists of four drift chambers (DCH):\nDCH1, DCH2 located upstream, and DCH3, DCH4 downstream of a dipole\nmagnet. The magnet has a field integral $\\int B_ydz=0.4$~Tm, thus\nproviding a horizontal transverse momentum kick $\\Delta P_x=120~{\\rm\nMeV}\/c$ for charged particles. The DCHs have the shape of a regular\noctagon with a transverse size of about 2.8 m and a fiducial area of\nabout 4.5 m$^2$. Each chamber is composed of eight planes of sense\nwires arranged in four pairs of staggered planes oriented\nhorizontally, vertically, and along each of the two orthogonal\n$45^\\circ$ directions. The spatial resolution of each DCH is\n$\\sigma_x=\\sigma_y=90\\mu$m. The nominal momentum resolution of the\nmagnetic spectrometer is parameterized as $\\sigma_p\/p = (1.02 \\oplus\n0.044\\cdot p)\\%$ ($p$ expressed in GeV\/$c$). The measured resolution\non the reconstructed $3\\pi^\\pm$ invariant mass varied during the\nrunning period corresponding to the considered data sample in the\nrange of $(1.65-1.72)$~MeV\/$c^2$, depending on DCH performance.\n\nThe magnetic spectrometer is followed by a plastic scintillator\nhodoscope (HOD) used to produce fast trigger signals and to provide\nprecise time measurements of charged particles. The hodoscope has a\nregular octagonal shape with a transverse size of about 2.4 m. It\nconsists of a plane of horizontal and a plane of vertical\nstrip-shaped counters. Each plane consists of 64 counters arranged\nin four quadrants. Each quadrant is subdivided into 4 segments\ntaking part in the trigger logic. Counter widths (lengths) vary from\n6.5 cm (121 cm) for central counters to 9.9 cm (60 cm) for\nperipheral ones.\n\nThe hodoscope is followed by a liquid krypton electromagnetic\ncalorimeter (LKr), a hadronic calorimeter (HAC) and a muon detector\n(MUV), all of which are not used in the present analysis. A detailed\ndescription of the components of the NA48 detector can be found\nelsewhere~\\cite{ba96}.\n\nThe $K^\\pm\\to\\pi^\\pm\\pi^+\\pi^-$ events are triggered with a\ntwo-level system. At the first level (L1), the rate of $\\sim 500$\nkHz is reduced to $\\sim 100$ kHz by requiring coincidences of hits\nin the two planes of the HOD in at least two of the 16\nnon-overlapping segments. The second level (L2) is based on a\nhardware system computing coordinates of hits from DCH drift times,\nand a farm of asynchronous microprocessors performing fast\nreconstruction of tracks and running a selection algorithm, which\nrequires at least two tracks to originate in the decay volume with\nthe closest distance of approach less than 5 cm. L1 triggers not\nsatisfying this condition are examined further and accepted\nnevertheless if there is a reconstructed track which is not\nkinematically compatible with a $\\pi^\\pm\\pi^0$ decay of a $K^\\pm$\nhaving momentum of 60 GeV\/$c$ directed along the beam axis. The\nresulting trigger rate is about 10 kHz.\n\n\\section{Data analysis}\n\n\\noindent {\\bf Reconstruction and selection} \\vspace{2mm}\n\n\\noindent Event reconstruction is based entirely on the magnetic\nspectrometer information. Tracks are reconstructed from hits in DCHs\nusing the measured magnetic field map rescaled according to the\nrecorded value of electric current in the spectrometer analyzing\nmagnet. Three-track vertices, compatible with a\n$K^\\pm\\to\\pi^\\pm\\pi^+\\pi^-$ decay, are reconstructed by\nextrapolation of track segments from the upstream part of the\nspectrometer back into the decay volume, taking into account the\nstray magnetic fields due to the Earth's field and parasitic\nmagnetization of the vacuum tank, and multiple scattering in the\nKevlar window. The stray field correction is based on a\nthree-dimensional field map measured in the entire vacuum tank, and\nreduces the amplitude of the observed sinusoidal variation of the\nreconstructed $3\\pi$ invariant mass on the azimuthal orientation of\nthe odd pion by more than an order of magnitude to a level below\n0.05 MeV\/$c^2$. The event kinematics is calculated using measured\nmomenta and track directions extrapolated to the decay vertex.\n\nThe principal selection criteria applied to the reconstructed\nvariables are the following:\n\\begin{itemize}\n\\item Total charge of the three pion candidates: $Q=\\pm1$;\n\\item Transverse momentum: $P_T<0.3$~GeV\/$c$;\n\\item Longitudinal vertex position within the decay volume:\n$Z_{vtx}>Z_{final~coll.}$;\n\\item Transverse vertex radius within the beam area: $R_{vtx}<3$~cm;\n\\item Kaon momentum within the nominal range:\n$54~{\\rm GeV}\/c<|\\vec P_K|<66~{\\rm GeV}\/c$.\n\\end{itemize}\n\nTo improve the resolution on the kinematic variables, and to reduce\nthe impact of differences between data and MC resolutions, the\nevents were passed through a kinematic fitting procedure with three\nconstraints (constraining the initial kaon direction to by along the\n$z$ axis, and the $3\\pi$ invariant mass to the kaon mass). Events\nwith the quality of the kinematic fit corresponding to probability\n$p<10^{-5}$ were rejected. The fraction of these rejected events\nincreases as a function of deviation of the reconstructed $3\\pi$\nmass from the PDG kaon mass $|\\Delta M|$; in particular, all the\nevents with $|\\Delta M|>10$~MeV\/$c^2$ are outside the signal region,\ni.e. rejected by the above condition.\n\nThe geometric acceptance for the $K^\\pm\\to\\pi^\\pm\\pi^+\\pi^-$ decays\nis mainly determined by the beam pipe traversing the centres of the\nDCHs, and the material in the central region of each DCH where\ncentral DCH wires terminate. This material defines a region of high\nDCH inefficiency\\footnote{The acceptance is not biased by the finite\nouter size of the DCHs due to a relatively small $Q$-value of the\n$K^\\pm\\to\\pi^\\pm\\pi^+\\pi^-$ decay: $Q=75.0$ MeV.}. This inefficiency\ntogether with beam optics performance and variations influences the\nacceptance, and is difficult to reproduce accurately with a MC\nsimulation. To minimize the effects of this problem, it is required\nthat the transverse positions of each pion in DCH1 and DCH4 planes\n$\\vec R_{\\pi i}$ ($i=1,4$) satisfy the condition $|\\vec R_{\\pi\ni}-\\vec R_{0}|>18$~cm, where $\\vec R_{0}$ is the position of the\nmomentum-weighted average of the three pions' impact points: $\\vec\nR_{0}=\\sum_{i=1}^3 (\\vec R_{\\pi i}|\\vec P_{\\pi i}|)\/\\sum_{i=1}^3\n|\\vec P_{\\pi i}|$ (for DCH4 plane, trajectories of pions are\nlinearly extrapolated from DCH1 and DCH2 planes). $\\vec R_{0}$\ncorresponds to the transverse position of the line of flight of the\ninitial kaon. The value of 18 cm was chosen to exclude safely the\ninefficient central region taking into account the beam sizes and\nvariations of their average transverse positions. The described\nselection criterion costs about $50\\%$ of the statistics; however an\nappropriate MC description of the experimental conditions is more\nimportant than the sample size for the present analysis\\footnote{In\na different case, the charge asymmetry analysis~\\cite{k3pic} is\nperformed with soft cuts maximizing the selected data sample, and is\nbased on cancellation, rather than simulation, of the systematic\neffects.}. As will be shown below in the discussion leading to the\nresults presented in Fig.~\\ref{fig:datamc}, the MC simulation of the\nexperimental conditions reproduces the data distributions to a level\nof a few parts per mille.\n\nThe selection leaves a sample of $4.71\\times 10^8$ events, which is\npractically background free, as $K^\\pm\\to\\pi^\\pm\\pi^+\\pi^-$ is by\nfar the dominant decay mode of the charged kaon with more than one\ncharged particle in the final state. The fact that backgrounds due\nto other decays of beam kaons and pions are negligible was also\nchecked with a MC simulation.\n\nThe distribution of the reconstructed $3\\pi^\\pm$ invariant mass of\ndata events (before the kinematic fitting) and its comparison with\nMC are presented in Fig.~\\ref{fig:kmass}a. The non-Gaussian tails of\nthe mass distribution are primarily due to\n$\\pi^\\pm\\to\\mu^\\pm\\nu_\\mu$ decays in flight, and are well understood\nin terms of MC simulation. The ratio of MC to data mass spectra is\npresented in Fig.~\\ref{fig:kmass}b. It demonstrates the imperfection\nof resolution description in MC, and a deficit of MC events in the\nlow mass region, which however contains a small fraction of the\nevents, and is mostly outside the signal region. The Dalitz plot\ndistribution of the selected data events (after the kinematic\nfitting) $F_{data}(u,|v|)$ used for the subsequent analysis is\npresented in Fig.~\\ref{fig:kmass}c. The bin sizes of the Dalitz plot\ndistributions used in the analysis are $\\delta u=\\delta v=0.05$.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\begin{tabular}{@{}l@{}r@{}}\n{\\put(-2,100){\n\\resizebox{0.47\\textwidth}{!}{\\includegraphics{eps\/mk.eps}}}\n\\resizebox{0.47\\textwidth}{!}{\\includegraphics{eps\/mcdataratio.eps}}\n\\put(-30,148){(1)}\\put(-30,184){(2)}\\put(-195,190){(3)}\n\\put(-176,273){\\Large\\bf a} \\put(-176,79){\\Large\\bf b}\n\\put(-188,188){\\vector(1,-2){8}}} &\n\\put(7,40){{\\resizebox{0.53\\textwidth}{!}{\\includegraphics{eps\/uv3d.eps}}}\n\\put(-30,25){$\\bf u$}\\put(-240,107){$\\bf |v|$}\n\\put(-195,196){\\Large\\bf c}} {\\rule{0.53\\textwidth}{0mm}}\n\\\\\n\\end{tabular}\n\\end{center}\n\\vspace{-8mm} \\caption{(a) Reconstructed spectrum of $3\\pi^\\pm$\ninvariant mass $M(3\\pi)$ (upper line) and its presentation in terms\nof normalized MC components: (1) events without $\\pi\\to\\mu\\nu$ decay\nin flight, (2) events with $\\pi\\to\\mu\\nu$ decay, (3) IB radiative\n$K_{3\\pi\\gamma}$ events. (b) The ratio of MC\/data $M(3\\pi)$ spectra\ndemonstrating the imperfection of resolution description in MC, and\na deficit of MC events at low $M(3\\pi)$ outside the signal region.\n(c) Reconstructed distribution in the kinematic variables\n$F_{data}(u,|v|)$ (after kinematic fit).} \\label{fig:kmass}\n\\end{figure}\n\n\\vspace{2mm}\n\n\\noindent {\\bf Correction for trigger inefficiency} \\vspace{2mm}\n\n\\noindent To simplify the treatment of the trigger inefficiency,\nstable trigger performance was the main condition used to select the\nsample to be used for the analysis\\footnote{As it was already noted,\nthe size of the data sample is not a limitation for this analysis.}.\nInefficiencies of both L1 and L2 trigger components were directly\nmeasured as functions of ($u,|v|$) using control data samples of\nprescaled low bias triggers collected along with the main triggers,\nwhich allowed a correction of the observed ($u,|v|$) distributions,\nand propagation of the statistical errors of trigger inefficiencies\ninto the result.\n\nThe L1 trigger condition requiring a coincidence of hits in two of\nthe 16 non-overlapping HOD segments is loose, as there are three\ncharged particles in a fully reconstructed event, providing a rather\nlow (and stable in time) inefficiency of $0.6\\times 10^{-3}$ for the\nselected event sample. However, the L1 inefficiency depends rather\nstrongly on the kinematic variables. The primary mechanism\ngenerating such a dependence is the enhancement of inefficiency for\ntopologies with two pions hitting the same HOD segment; such events\npreferably belong to the kinematic regions characterized by small\nrelative velocity of a certain $\\pi\\pi$ pair in the kaon rest frame.\n\nThe L2 inefficiency, which is due to local DCH inefficiencies\naffecting the trigger more strongly than the off-line reconstruction\ndue to lower redundancy and trigger timing effects, was measured to\nbe $(0.32\\pm0.05)\\times 10^{-2}$ (the error indicates the maximum\nsize of its variation during the data taking period). It did not\nexhibit any significant correlation to the kinematic variables, due\nto relative complexity of the decision-making algorithm.\n\n\\vspace{2mm}\n\n\\noindent {\\bf Monte Carlo simulation} \\vspace{2mm}\n\n\\noindent A detailed GEANT-based MC simulation was developed, which\nincludes full detector geometry and material description, simulation\nof stray magnetic fields, DCH local inefficiencies and misalignment,\nthe beam line (which allows a reproduction of the kaon momentum\nspectra and beam profiles), and $K^+\/K^-$ relative fluxes. Moreover,\ntime variations of the above effects during the running period were\nsimulated. A production of $6.7\\times 10^9$\n$K^\\pm\\to\\pi^\\pm\\pi^+\\pi^-$ events distributed according to the\nmatrix element (\\ref{slopes}) with PDG values of the slope\nparameters~\\cite{pdg} was performed to determine the detector\nacceptance. A sample of $1.16\\times10^9$ MC events (almost 2.5 times\nlarger than the data sample) passes the selection.\n\nComparison of data and MC distributions in such significant\nvariables as longitudinal decay vertex position and illuminations of\nDCH1 and DCH4 planes by pions is presented in Fig.~\\ref{fig:datamc},\nand demonstrates that MC simulation reproduces the data\ndistributions to a level of a few units of $10^{-3}$. The precision\nof the data description can be improved by tighter cuts on pion\nradial positions.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\resizebox{0.94\\textwidth}{!}{\\includegraphics{eps\/dmc.zvtx.eps}}\n\\put(-240,100){\\Large\\bf a}\\\\\n\\resizebox{0.94\\textwidth}{!}{\\includegraphics{eps\/dmc.r1.eps}}\n\\put(-240,100){\\Large\\bf b}\\\\\n\\resizebox{0.94\\textwidth}{!}{\\includegraphics{eps\/dmc.r4.eps}}\n\\put(-240,100){\\Large\\bf c}\n\\end{center}\n\\vspace{-8mm} \\caption{Left column: reconstructed data\ndistributions, right column: the corresponding ratios of data\/MC\ndistributions of (a) vertex $z$ position, and pion radial position\n(for each of the 3 pions) in the planes of (b) DCH1 and (c) DCH4.}\n\\label{fig:datamc}\n\\end{figure}\n\n\\vspace{2mm}\n\n\\noindent {\\bf Fitting procedure} \\vspace{2mm}\n\n\\noindent The measurement method is based on fitting the binned\nreconstructed data distribution $F_{data}(u,|v|)$ presented in\nFig.~\\ref{fig:kmass}c with a sum of four reconstructed MC components\ngenerated according to the four terms in the polynomial\n(\\ref{slopes}) presented in Fig.~\\ref{fig:mc3d}. Let us denote these\nreconstructed MC distributions as $F_0(u,|v|)$, $F_u(u,|v|)$,\n$F_{u^2}(u,|v|)$, and $F_{v^2}(u,|v|)$. To obtain them, the MC\nsample (distributed in kinematic variables according to the PDG\nslope parameters) was divided into four subsamples\\footnote{The\nrelative sizes of the four subsamples were subject of optimization\nin order to minimize the statistical error of the measurement. The\n$F_u(u,|v|)$, $F_{u^2}(u,|v|)$, and $F_{v^2}(u,|v|)$ samples are of\nequal sizes, while the $F_0(u,|v|)$ sample is 5 times larger than\neach of the former three.}, and events in each subsample were\nassigned appropriate weights depending on the generated ($u,|v|$) to\nobtain the desired distributions.\n\n\\begin{figure}[tb]\n\\vspace{-7mm}\n\\begin{center}\n\\resizebox{0.90\\textwidth}{0.32\\textheight}{\\includegraphics{eps\/mc3d-1.eps}}\n\\put(-371,180){\\Large\\bf a} \\put(-160,180){\\Large\\bf b}\n\\put(-27,18){$\\bf u$}\\put(-234,18){$\\bf u$} \\put(-203,105){$\\bf\n|v|$}\\put(-408,103){$\\bf |v|$}\n\\\\\n\\vspace{-4mm}\n\\resizebox{0.90\\textwidth}{0.32\\textheight}{\\includegraphics{eps\/mc3d-2.eps}}\n\\put(-371,180){\\Large\\bf c} \\put(-160,180){\\Large\\bf d}\n\\put(-27,18){$\\bf u$}\\put(-234,18){$\\bf u$} \\put(-203,103){$\\bf\n|v|$}\\put(-408,103){$\\bf |v|$}\n\\end{center}\n\\vspace{-11mm} \\caption{Reconstructed distributions in the kinematic\nvariables ($u$,$|v|$) of the four MC components: (a) $F_0(u,|v|)$,\n(b) $F_u(u,|v|)$, (c) $F_{u^2}(u,|v|)$, (d) $F_{v^2}(u,|v|)$.}\n\\label{fig:mc3d}\n\\end{figure}\n\nThe following functional defining the agreement of the shapes of\ndata and MC distributions is minimized using the MINUIT\npackage~\\cite{minuit} in order to measure the values of the slope\nparameters $(g,h,k)$:\n\\begin{equation}\n\\label{chi2} \\chi^2(g,h,k,N)=\\sum_{u,|v|~{\\rm\nbins}}\\frac{(F_{data}(u,|v|)-NF_{MC}(g,h,k,u,|v|))^2} {\\delta^2\nF_{data}(u,|v|)+N^2\\delta^2F_{MC}(g,h,k,u,|v|)}.\n\\end{equation}\nThe sum is evaluated over all the bins of reconstructed $(u,|v|)$\ndistributions with at least 1000 data events, which eliminates the\nnecessity to include non-Gaussian behaviour of errors. Here $\\delta\nF_{data}(u,|v|)$ is an uncertainty of the number of data events in a\ngiven bin (composed of a statistical part and a trigger efficiency\npart added in quadrature), $F_{MC}(g,h,k,u,|v|)$ is a MC population\nof a bin for given values of $(g,h,k)$:\n\\begin{eqnarray}\nF_{MC}(g,h,k,u,|v|) &=& F_0(u,|v|)\/I_0 + gF_u(u,|v|)\/I_{u} +\\\\\n&&hF_{u^2}(u,|v|)\/I_{u^2} + kF_{v^2}(u,|v|)\/I_{v^2},\\nonumber\n\\end{eqnarray}\nand $\\delta F_{MC}(g,h,k,u,|v|)$ is its statistical error:\n\\begin{eqnarray}\n\\delta^2F_{MC}(g,h,k,u,|v|) &=& \\delta^2F_0(u,|v|)\/I_0^2 +\ng^2\\delta^2\nF_u(u,|v|)\/I_{u}^2 +\\\\\n&&h^2\\delta^2F_{u^2}(u,|v|)\/I_{u^2}^2 + k^2\\delta^2\nF_{v^2}(u,|v|)\/I_{v^2}^2.\\nonumber\n\\end{eqnarray}\nHere $I_0$, $I_u$, $I_{u^2}$ and $I_{v^2}$ are the normalization\nconstants computed taking into account the numbers of generated\nevents in each of the four MC subsamples, and the integrals of the\nfour terms in (\\ref{slopes}) over the Dalitz plot. The free\nparameters of the functional (\\ref{chi2}) are the slope parameters\n$(g,h,k)$ and an overall MC normalization parameter $N$.\n\nThe minimization yields $\\chi^2\/{\\rm NDF}=1669\/1585$, corresponding\nto a satisfactory probability of 7.0\\%. The results of the fit and\nthe trigger corrections are presented in Table~\\ref{tab:fit}. The\nnon-zero values of the corrections arise mostly from the L1 trigger\ninefficiency dependence on kinematic variables, while their\nstatistical errors receive contributions of similar sizes from L1\nand L2 trigger efficiency uncertainties.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{c|rcl|cc|rcl}\n\\hline Parameter\n&\\multicolumn{3}{c|}{Value}&\\multicolumn{2}{c|}{Uncertainties}&\n\\multicolumn{3}{c}{Trigger correction}\n\\\\\n\\cline{5-6} &&&& statistical & MC stat.&&\\\\\n\\hline\n$g\\times10^2$&$-21.134$&$\\pm$&$0.013$&0.009&0.008&$-0.008$&$\\pm$&0.005\\\\\n$h\\times10^2$&$  1.848$&$\\pm$&$0.022$&0.015&0.013&$ 0.116$&$\\pm$&0.009\\\\\n$k\\times10^2$&$ -0.463$&$\\pm$&$0.007$&0.005&0.004&$ 0.033$&$\\pm$&0.003\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{-5mm} \\caption{Fit results with statistical and MC\nstatistical uncertainties, trigger corrections and their statistical\nuncertainties. The trigger corrections are included into the\nresulting values.} \\label{tab:fit}\n\\end{table}\n\nKeeping only the linear term $gu$ in (\\ref{slopes}) yields a fit of\nunacceptable quality: $\\chi^2\/{\\rm NDF}=13683\/1583$. Including the\nterms proportional to $u^3$ and $uv^2$ (the only cubic terms allowed\nby Bose symmetry) yields values for the corresponding cubic slope\nparameters compatible with zero.\n\n\\vspace{2mm}\n\n\\noindent {\\bf Stability checks} \\vspace{2mm}\n\n\\noindent Stability of the results with respect to variations of the\nselection conditions on vertex fit quality, $P_T$, $R_{vtx}$, $|\\vec\nP_K|$ and $\\vec R_{\\pi i}$, and binning variations was checked.\nStability with respect to exclusion of ($u$, $|v|$) bins with large\ndeviations of the Coulomb factor (\\ref{coulomb}) from unity and with\nrespect to kaon sign\\footnote{Combined $K^+$ and $K^-$ sample is\nused to obtain the result. Stability of the slope parameters with\nrespect to kaon sign is a consequence of the experimental\nfact~\\cite{k3pic} that CP invariance holds at the discussed level of\nprecision.} was checked as well. No statistically significant\ndependencies were found. Stability with respect to various ways of\nbinning the data was checked; comparison of slope measurements in\nbins of reconstructed longitudinal coordinate of decay vertex\n$Z_{vtx}$ (since the acceptance depends strongly on this variable),\nand in data taking periods are shown in Fig.~\\ref{fig:stab} as the\nmost significant examples.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\resizebox{0.49\\textwidth}{!}{\\includegraphics{eps\/g-zvtx.eps}}\n\\resizebox{0.49\\textwidth}{!}{\\includegraphics{eps\/gtime.eps}}\\\\\n\\resizebox{0.49\\textwidth}{!}{\\includegraphics{eps\/h-zvtx.eps}}\n\\resizebox{0.49\\textwidth}{!}{\\includegraphics{eps\/htime.eps}}\\\\\n\\resizebox{0.49\\textwidth}{!}{\\includegraphics{eps\/k-zvtx.eps}}\n\\resizebox{0.49\\textwidth}{!}{\\includegraphics{eps\/ktime.eps}}\\\\\n\\end{center}\n\\vspace{-8mm} \\caption{Slope measurements in bins of $Z_{vtx}$ (left\ncolumn) and in running periods (right column). Systematic errors are\nnot shown.} \\label{fig:stab}\n\\end{figure}\n\n\\vspace{2mm}\n\n\\noindent {\\bf Systematic uncertainties} \\vspace{2mm}\n\n\\noindent The Coulomb factor (\\ref{coulomb}) used in description of the\nevent density contains a pole $C(u,v)\\to\\infty$ corresponding to a\npair of opposite sign pions having zero relative velocity. The\nimplemented Monte Carlo simulation involves a certain approximation\nto $C(u,v)$ in the pole region. In view of that,\nsensitivity of the result to\ntreatment of the pole was studied, in particular, by using an\nalternative fitting method involving projections of the distributions\nin each of kinematic variables. The assigned conservative systematic\nuncertainties due to the fitting procedure are listed in\nTable~\\ref{tab:syst}. They are expected to be diminished to\na negligible level in a future analysis of the full statistics.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{l|rrr}\n\\hline Effect&~~~$\\delta g\\times10^2$&~~~$\\delta h\\times10^2$&~~~$\\delta k\\times10^2$\\\\\n\\hline\nFitting procedure              &0.009~~~&0.007~~~&0.006~~~\\\\\nPion momentum resolution       &0.004~~~&0.031~~~&0.009~~~\\\\\nSpectrometer magnetic field    &0.002~~~&0.008~~~&0.004~~~\\\\\nSpectrometer misalignment      &0.002~~~&0.002~~~&0.001~~~\\\\\nStray magnetic field           &0.001~~~&0.002~~~&0.001~~~\\\\\n\\hline\nTotal                          &0.010~~~&0.033~~~&0.012~~~\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{-5mm} \\caption{A summary of the systematic uncertainties.}\n\\label{tab:syst}\n\\end{table}\n\nAn important source of systematic uncertainty is the imperfect\ndescription of the resolution in pion momentum, which can be\nobserved in Fig.~\\ref{fig:kmass}b as a slight disagreement of the\nshapes of the reconstructed $M(3\\pi)$ spectra for data and MC in the\nsignal region\\footnote{As was discussed above, a large disagreement\nat low $M(3\\pi)$ values is outside the signal region.}. To evaluate\nthe corresponding effect, two different plausible ways of\nintroducing smearing of the MC resolution were used: either by\nincreasing the smearing of DCH space points from 90$\\mu$m to\n100$\\mu$m, or by adding an extra $0.09\\%X_0$ layer of matter in the\nposition of the Kevlar window (the former correction is more\nrealistic). The sizes of the added perturbations are such as to\ncorrect for the ``double bump'' shape of the ratio in the signal\nregion. Both methods lead to similar systematic uncertainties on the\nslope parameters attributed to description of resolution in pion\nmomentum. These uncertainties are listed in Table~\\ref{tab:syst}.\n\nEffects due to imperfect knowledge of the magnetic field in the\nspectrometer magnet were evaluated. The variation of the magnet\ncurrent can be monitored with a relative precision of $5\\times\n10^{-4}$. Smaller variations are continuously controlled with a\nprecision of $\\sim 10^{-5}$ by the deviation of the measured\ncharge-averaged kaon mass from the nominal PDG value. A\ntime-dependent correction is introduced by scaling the reconstructed\npion momenta, decreasing the effect of overall field scale to a\nnegligible level. To account for possible differences between the\nshape of the field map used for simulation and the true field,\nvariations of the MC field map were artificially introduced,\nconsistently with the known precision of field measurement of $\\sim\n10^{-3}$. The corresponding uncertainties are listed in\nTable~\\ref{tab:syst}.\n\nThe transverse positions of DCHs and individual wires were\ncontrolled and realigned at the level of reconstruction software\nevery 2--4 weeks of data taking using data collected in special runs\nin which muon tracks were recorded with no magnetic field in the\nspectrometer. This allows an alignment precision of $\\sim30~\\mu$m to\nbe reached. However, time variations of DCH alignment on a shorter\ntime scale can bias the measurement. These variations were measured\nby the difference between the average reconstructed $3\\pi$ invariant\nmasses for $K^+$ and $K^-$ decays, and taken into account. The\nprecision with which these effects are simulated leads to systematic\nuncertainties presented in Table~\\ref{tab:syst}.\n\nThe effects due to limited precision of measurement of the stray\nmagnetic field in the decay volume were estimated by variation of\nthe stray field map used for decay vertex reconstruction; the\ncorresponding systematic effects are presented in\nTable~\\ref{tab:syst}.\n\nThe kaon momentum spectra were carefully simulated, and the related\nresidual uncertainties were found to be negligible.\nPossible differences between\ndata and MC transverse scales were found to have a negligible\ninfluence on the result.\n\nThe total systematic errors were obtained by summing the above\ncontributions in quadrature, and are presented in\nTable~\\ref{tab:syst}.\n\n\\section*{Conclusions}\n\nThe Dalitz plot slope parameters of the $K^\\pm\\to\\pi^\\pm\\pi^+\\pi^-$\ndecays measured with a fraction of NA48\/2 data sample ignoring\nradiative effects (apart from the Coulomb factor) and strong\nrescattering effects, are:\n$$\ng=(-21.134\\pm0.017)\\%,~~ h=(1.848\\pm0.040)\\%,~~k=(-0.463\\pm0.014)\\%.\n$$\nThese values are in agreement with the world averages\\footnote{The\nPDG averages results separately for $K^+$ and $K^-$~\\cite{pdg};\naveraging the PDG data between $K^+$ and $K^-$ decays should take\ninto account correlated systematic uncertainties of the $K^+$ and\n$K^-$ measurements by the same experiment~\\cite{fo72}.}, and have an\norder of magnitude smaller uncertainties. This is the first\nmeasurement of a non-zero value of the quadratic slope parameter\n$h$. The compatibility of the measured distribution with the PDG\npolynomial parameterization~\\cite{pdg} appears still to be\nacceptable at an improved level of precision; no significant higher\norder slope parameters were found.\n\nThe measurement of the slope parameters is in agreement with a full\nnext-to-leading order computation~\\cite{ga03}:\n$$\ng=(-22.0\\pm2.0)\\%,~~ h=(1.2\\pm0.5)\\%,~~k=(-0.54\\pm0.15)\\%.\n$$\n\nThe whole NA48\/2 sample suitable for $K^\\pm\\to3\\pi^\\pm$ Dalitz plot\ndistribution analysis contains at least three times more data; a\nmore elaborate analysis is foreseen when the corresponding\ntheoretical framework is available.\n\n\\section*{Acknowledgements}\n\nIt is a pleasure to thank the technical staff of the participating\nlaboratories, universities and affiliated computing centres for\ntheir efforts in the construction of the NA48 apparatus, in\noperation of the experiment, and in data processing.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nThe formation of gas-giant planets like Jupiter is a central problem in planetary science. These objects contain large amounts of hydrogen and helium \\citep{owen:1999}, which were presumably acquired from the gaseous portion of their star's protoplanetary disk. Most disks are found around stars less than a few million years old \\citep{haisch:2001}, and this sets an upper limit on the time available to form gas giants. This time constraint is more severe than for rocky planets, such as Earth, which may acquire much of their bulk after the  protoplanetary disk disperses. \n\nReproducing the existence and characteristics of gas giants is a challenging test for theories of planet formation. Moreover, understanding giant-planet formation is essential for understanding the origin and character of planetary systems in general, since gravitational perturbations from giant planets can effect the growth of objects elsewhere in the same system \\citep{chambers:2002, levison:2003}. For example, perturbations from Jupiter and Saturn play a central role in many models for the origin of the Sun's asteroid belt \\citep{wetherill:1992, chambers:2001, walsh:2011}. The gravitational influence of giant planets is further enhanced by the apparent tendency of many giants to migrate radially during or after their formation \\citep{trilling:1999, armitage:2007}.\n\nUsually, the formation of gas-giant planets is described using the ``core-accretion'' model. Here, a giant begins as a solid protoplanet that grows by accreting asteroid-sized planetesimals \\citep{pollack:1996, movshovitz:2010}. Gas in the vicinity of the protoplanet is compressed and heated by the protoplanet's gravity, forming an extended atmosphere surrounding a solid core. The mass of the atmosphere is determined by a balance between the inward pull of gravity and the outward pressure gradient. Typically, the atmospheric mass increases rapidly with increasing core mass \\citep{pollack:1996}. For a given set of conditions there is a critical core mass above which a static atmosphere is no longer possible \\citep{mizuno:1980}. If the core exceeds the critical mass, gas flows into the atmosphere from the disk at an increasing rate, eventually forming a massive, gas-rich planet.\n\nMultiple factors can affect the critical core mass, in particular the luminosity, opacity, and composition of the atmosphere \\citep{stevenson:1982, hori:2011}. Lowering the opacity and\/or luminosity increases the density in the atmosphere, raising the mass of the atmosphere for a given core mass. This in turn reduces the critical core mass. The main source of luminosity is energy released by infalling planetesimals, while opacity in the outer atmosphere is mostly due to dust embedded in the gas \\citep{mizuno:1980}. Early studies of giant-planet formation typically assumed opacities appropriate for an interstellar dust size distribution, leading to large critical core masses, around 10--20 Earth masses \\citep{pollack:1996}. More recent works that account for dust coagulation within the atmosphere find lower opacities and critical core masses below 10 Earth masses \\citep{movshovitz:2010, mordasini:2014}.\n\nPrevious studies have mainly considered atmospheres with compositions similar to the Sun---that is roughly 99\\% hydrogen and helium, with trace amounts of heavier elements \\citep{pollack:1996, rafikov:2006, movshovitz:2010, piso:2014}. However, the atmospheric composition can be altered by the addition of material evaporating from planetesimals as they fall towards the core \\citep{stevenson:1984, hori:2011}. The evaporated materials have higher molecular weights than hydrogen and helium, which can substantially increase the density of the atmosphere. For atmospheres dominated by heavy elements, the critical core mass can be smaller than Earth or even Mars \\citep{hori:2011, venturini:2015}.\n\nWhile conventional models consider cores that accrete planetesimals, some recent studies have suggested that planets acquire their solid material in the form of mm-to-m-sized pebbles instead \\citep{lambrechts:2012, levison:2015, chambers:2016}.  There is some observational evidence  that pebbles are abundant in protoplanetary disks \\citep{testi:2003, wilner:2005}. Pebble accretion may also alleviate the problem of forming critical-mass cores within the lifetime of a protoplanetary disk. Pebbles experience strong drag with disk gas during an encounter with a core, greatly increasing the capture probability and the core growth rate in some circumstances \\citep{ormel:2010, lambrechts:2012}.\n\nOutside a protoplanetary disk's ice line, pebbles are likely to be a mixture of volatile ices and refractory materials (silicates, metals etc., referred to here as ``rock''). Temperatures in a protoplanet's atmosphere increase with depth \\citep{rafikov:2006}, so the icy component of incoming pebbles will evaporate in most cases, raising the molecular weight of the atmosphere and lowering the critical core mass. This process is likely to be much more effective than evaporation from planetesimals for  two reasons: (i) the surface area to volume ratio of pebbles is much greater due to the small size of pebbles, and (ii) pebbles typically settle towards the core relatively slowly, at terminal velocity, whereas planetesimals approach at roughly the escape velocity of the core. This  allows more time for evaporation to occur.\n\nRelatively little evaporation will take place in the cool, outer regions of an atmosphere since the vapor pressure of water is low here, and the atmosphere soon becomes saturated. Above a certain temperature, however, the ice component of pebbles will evaporate entirely. Pebbles with a solar composition of condensible materials will have an ice-to-rock ratio of roughly unity \\citep{lodders:2003}. Thus, cores that grow by accreting icy pebbles should have vapor-rich atmospheres that are more massive than the core itself, when the hydrogen\/helium component is taken into account. This contrasts with the standard H\/He dominated case where the critical mass usually occurs when the core and atmospheric masses are comparable \\citep{pollack:1996}.\n\nSo far, the atmospheres of planets undergoing pebble accretion have received relatively little attention. In this paper, we investigate the growth and structure of the atmospheres of giant-planet cores below the critical mass as they accrete ice-rich pebbles. We examine the various stages that such planets pass through as they grow, and determine the point at which they reach the critical mass. This work extends previous studies in several ways: (i) using realistic estimates for the pebble accretion rate rather than adopting  a fixed luminosity, (ii) allowing for varying atmospheric composition with depth due to water condensation and precipitation, and (iii) including the presence of an ocean where appropriate. \n\nThe rest of this paper is organized as follows. Section~2 describes the model used for the protoplanet's atmosphere and the rate of pebble accretion. Section~3 looks at the various evolutionary stages that a protoplanet and its atmosphere pass through as the mass increases. Section~4 examines the critical mass as a function of key model parameters. Section~5 contains a dicussion, and the main results are summarized in Section~6.\n\n\\section{The Model}\nWe consider a single protoplanet moving on a circular orbit in a gas-rich protoplanetary disk. A constant mass flux of pebbles drifts past the planet, moving towards the star due to gas drag. For simplicity, the pebbles are assumed to be composed of rock and water ice only. A fraction of the pebbles are captured by the planet. The rocky component of these pebbles is assumed to sediment to the planet's solid core. At each radial distance from the core, the pebbles' ice component is assumed to evaporate immediately until the local atmosphere is saturated in water vapor, or until the ice has evaporated completely. If the pebbles still contain some ice when they reach the base of the atmosphere this ice is added to the solid core, or forms an ocean entirely composed of liquid water directly above the core, depending on the temperature at this point.\n\nValues for the most important parameters used in the nominal model are listed in Table~1. In particular, we consider a protoplanet orbiting at $a=3$ AU from a solar mass star. We assume this location lies just outside the ice line in the disk, so that the pebbles here contain a mixture of rock and water ice, while more volatile ices have evaporated from the pebbles. The local temperature and gas density are 160 K and $10^{-10}$ g\/cm$^3$ respectively, and these values provide the outer boundary conditions for the protoplanet's atmosphere.\n\nThe planet's atmosphere is assumed to be in thermal and hydrostatic equilibrium. The temperature, pressure and density all increase inwards. The atmosphere is heated by the gravitational energy released by infalling pebbles, and  this is assumed  to be the only significant source of heating. In the inner regions of the atmosphere, this energy is typically transported outwards by convection, while heat transport in the outer atmosphere is by radiation. The outer regions of the atmosphere are assumed to be fully saturated with water vapor due to evaporation from incoming pebbles. For low-mass planets, the saturated atmosphere extends down to the surface of an ocean that contains the bulk of the water budget. When temperatures are too high for an ocean to exist, the outer atmosphere is still saturated with water vapor, but the inner regions may be undersaturated (if insufficient water is available), or above the critical temperature for water. \n\nRadial profiles for temperature $T$, pressure $P$, density $\\rho$, and interior mass $M$ are calculated using the standard stellar structure equations:\n\\begin{eqnarray}\n\\frac{dM}{dr}&=&4\\pi\\rho r^2 \\nonumber \\\\\n\\frac{dP}{dr}&=&-\\frac{GM\\rho}{r^2} \\nonumber \\\\\n\\frac{dT}{dr}&=&-\\frac{3\\kappa L\\rho}{64\\pi\\sigma_BT^3r^2}\n\\hspace{18mm} {\\rm radiative}\n\\nonumber \\\\\n\\frac{dT}{dP}&=&\\frac{\\nabla_{\\rm ad} T}{P}\n\\hspace{30mm} {\\rm convective}\n\\label{eq_main_equations}\n\\end{eqnarray}\n\\citep{inaba:2003}, where $r$ is the radius, $\\kappa$ is the opacity, $L$ is the luminosity, $\\sigma_B$ is the Stefan-Boltzmann constant, and $\\nabla_{\\rm ad}$ is the adiabatic gradient. The luminosity is assumed to be independent of radius, and equal to the gravitational energy released by the rocky component of the pebbles falling to the solid core of the planet: \n\\begin{equation}\nL=\\frac{GM_{\\rm core}f_{\\rm rock}}{R_{\\rm core}}\n\\frac{dM}{dt}\n\\end{equation}\nwhere $M_{\\rm core}$ and $R_{\\rm core}$ are the mass and radius of the solid part of the planet, $dM\/dt$ is the pebble mass accretion rate, and $f_{\\rm rock}$ is the rock mass fraction of the pebbles.\n\nThe atmosphere is assumed to be convective wherever the convective temperature gradient is shallower than the radiative gradient. In convective regions saturated with water vapor, the temperature profile is assumed to follow a moist adiabat due to condensation of water\/ice. Condensed water (or ice) is assumed to immediately precipitate to lower, unsaturated regions of the atmosphere or the ocean\\footnote{Since the temperature of a rising parcel of gas undergoing convection falls at the same rate as its surroundings, the saturation vapor pressures are the same in each case, and so are the water fractions. Thus we can use the Schwarzschild criterion for convection rather than the more general Ledoux criterion.}. Convective regions that are not saturated, or lie above the critical temperature of water, follow the usual dry adiabatic profile \\citep{kasting:1988, venturini:2015}.\n\nFollowing \\citet{kasting:1988}, the moist adiabatic gradient is given by\n\\begin{equation}\n\\frac{d\\ln P}{d\\ln T}=\\frac{1}{\\nabla_{\\rm ad}}\n=\\frac{P_v}{P}\\frac{d\\ln P_v}{d\\ln T}\n+\\frac{P_g}{P}\\left[1+\\frac{d\\ln\\rho_v}{d\\ln T}-\\frac{d\\ln\\alpha_v}{d\\ln T}\\right]\n\\end{equation}\nwhere the subscripts $v$ and $g$ refer to water vapor and the non-condensible gas (hydrogen plus helium) respectively. Here, the derivatives of $P_v$ and $\\rho_v$ are taken along the saturation vapor pressure-temperature curve of water, taken from \\citet{haar:1984}. Also, $\\alpha_v$ is the ratio of the water vapor density to the gas density, whose derivative is given by\n\\begin{equation}\n\\frac{d\\ln\\alpha_v}{d\\ln T}=\n\\frac{k\\rho_g (d\\ln\\rho_v\/d\\ln T)-\\rho_g\\mu_gm_Hc_g\n-\\rho_v\\mu_gm_H(ds_v\/d\\ln T)}\n{\\rho_v\\mu_g m_H\\Delta s+k\\rho_g}\n\\end{equation}\nwhere $k$ is Boltzmann's constant, $m_H$ is the mass of a hydrogen atom, $\\mu$ is the molecular weight, $c$ is the specific heat at constant volume, $s$ is specific entropy, and $\\Delta s$ is the specific entropy difference between water vapor and condensed water (liquid or solid). Note that in deriving this equation, the non-condensible gas is assumed to be ideal in the saturated regions of the atmosphere. (See \\citet{kasting:1988} for a detailed derivation of these equations.)\n\n\\startlongtable\n\\begin{deluxetable}{ll}\n\\tablecaption{Parameters used in the nominal model}\n\\tablehead{\n\\colhead{Parameter} & \\colhead{Value} \\\\\n}\n\\startdata\nStellar mass & 1 solar mass \\\\\nPlanet semi-major axis & 3 AU \\\\\nDisk temperature & 160 K \\\\\nDisk density & $10^{-10}$ g\/cm$^3$ \\\\\nTurbulent viscosity $\\alpha$ & 0.001 \\\\\nRock density & 4 g\/cm$^3$ \\\\\nIce\/rock ratio & 1:1 \\\\\nHydrogen\/helium ratio & 3:1 \\\\\nPebble flux & $10^{-5}M_\\oplus$\/y \\\\\nPebble Stokes number & 0.01 \\\\\n\\enddata\n\\end{deluxetable}\n\nFollowing \\cite{lissauer:2009}, the outer radius of the atmosphere is given by\n\\begin{equation}\nr_{\\rm max}=\\min\\left[\\frac{r_H}{4}, r_B\\right]\n\\end{equation}\nwhere $r_H$ and $r_B$ are the Hill radius and Bondi radius, given by\n\\begin{eqnarray}\nr_H&=&a\\left(\\frac{M_{\\rm tot}}{3M_\\odot}\\right)^{1\/3}\n\\nonumber \\\\\nr_B&=&\\frac{GM_{\\rm tot}}{c_s^2}\n\\end{eqnarray}\nwhere $M_{\\rm tot}$ is the total mass of the planet including its atmosphere, and $c_s$ is the local sound speed of the gas in the disk.\n\nThe opacity is given by the sum of contributions due to dust and gas, using relations developed by \\cite{freedman:2014} for the gas opacity. We adopt a constant dust opacity that is a model parameter, using $\\kappa_{\\rm dust}=0.01$ cm$^2$\/g in the nominal case. More detailed treatments that take into account dust coagulation in the atmosphere have been developed recently \\citep{mordasini:2014}. However, these models do not include source and sink terms due to a large flux of pebbles passing through the atmosphere and evaporation\/condensation of water, so we use the simpler, constant-$\\kappa_{\\rm dust}$ case here.\n\nTo solve the atmospheric structure equations, we also need an equation of state (EOS). Here we adopt EOS for hydrogen and helium developed by \\citep{saumon:1995}. For temperatures below 2000 K, we use the water\/steam EOS by \\cite{haar:1984}. Above 3000 K, we use EOS data from \\cite{french:2009}. In between these temperatures, we smoothly interpolate between these two cases. For water ice I, we follow \\cite{feistel:2006}. High pressure phases of ices could also exist on some protoplanets, but we do not encounter these phases in the models presented here. The approximate density of a H\/He\/water vapor mixture is determined by calculating the density for each component separately using the appropriate EOS and the partial pressure of the component, then summing the densities.\\footnote{A more commonly used approximation is $1\/\\rho=\\sum_iX_i\/\\rho_i$ where $X_i$ is the mass fraction of component $i$, and $\\rho_i$ is calculated using the total pressure rather than the partial pressure \\citep{nettelmann:2008}. However this method is not feasible when one of the components will condense under the total pressure.}\n\nWe calculate the rate at which the protoplanet accretes pebbles using the relations developed by \\cite{ormel:2010}, which depend on the Stokes number $\\rm St$ of the pebbles, given by\n\\begin{equation}\n\\rm St=\\Omegat_{\\rm stop}\n\\end{equation}\nwhere $\\Omega$ is the Keplerian orbital frequency and $t_{\\rm stop}$ is the stopping time due to gas drag.\n\nSmall pebbles settle towards the planet at terminal velocity, leading to a capture radius $r_c$ given by\n\\begin{equation}\n\\left(\\frac{r_c}{r_H}\\right)^3+\\frac{2v_{\\rm rel}}{3\\Omega r_H}\\left(\\frac{r_c}{r_H}\\right)^2\n-8\\rm St=0\n\\end{equation}\nwhere the average approach velocity $v_{\\rm rel}$ of the pebbles is given by\n\\begin{equation}\nv_{\\rm rel}=\\Omega\\times\\max\\left[\\eta a,\\frac{3r_c}{2}\\right]\n\\end{equation}\nwhere $\\eta\\simeq(c_s\/a\\Omega)^2$ is the fractional orbital velocity difference between the gas and the protoplanet.\n\nPebbles with Stokes numbers larger than a critical value ${\\rm St}_{\\rm crit}$ do not reach terminal velocity, but still experience an enhanced capture probability due to gas drag. \\cite{ormel:2010} find empirically that the capture radius in this case is given by the expression above modified by a factor of $\\exp[-(\\rm St\/{\\rm St}_{\\rm crit})^{0.65}]$, where\n\\begin{equation}\n{\\rm St}_{\\rm crit}=\\min\\left[1, 12\\left(\\frac{\\Omega a}{v_{\\rm rel}}\\right)^3\\right]\n\\end{equation}\n\nThe growth rate of  the protoplanet is \n\\begin{equation}\n\\frac{dM}{dt}=\\min\\left[2r_c,\\frac{\\pi r_c^2}{H_{\\rm peb}}\\right]\\times\\Sigmav_{\\rm rel}\n\\end{equation}\n\nHere, $H_{\\rm peb}$ is the scale height of the pebbles. Following \\cite{youdin:2007}, we assume that $H_{\\rm peb}$ is set by a balance between gravitational settling and turbulent stirring, and is given by\n\\begin{equation}\nH_{\\rm peb}=H_{\\rm gas}\\left(\\frac{\\alpha}{\\alpha+\\rm St}\\right)^{1\/2}\n\\end{equation}\nwhere $H_{\\rm gas}$ is the gas scale height, and $\\alpha$ is the turbulent viscosity parameter \\citep{shakura:1973}.\n\nThe surface density of pebbles $\\Sigma$ is related to the pebble flux $F$ and the radial velocity $v_r$:\n\\begin{equation}\nF=2\\pi a\\Sigma v_r\n\\end{equation}\nwhere, following \\cite{weidenschilling:1977}, we have\n\\begin{equation}\nv_r=\\frac{2\\eta a\\Omega\\rm St}{1+\\rm St^2}\n\\end{equation}\n\n\\begin{figure}\n\\centering\n\\includegraphics[height=120mm,angle=270]{steam_fig01.pdf}\n\\caption{Schematic diagram showing several evolutionary stages of a protoplanet accreting ice-rich pebbles. (1) Planet without an atmosphere. (2) Planet with an atmosphere and a solid ice\/rock surface. The outer atmosphere is radiative. The inner atmosphere is convective with ice grains precipitating towards the surface. (3) Planet with a rocky core and an ocean. The outer atmosphere is radiative, and the inner atmosphere is convective with ice and water precipitation. (4) Planet with a rock core and no ocean. The outer atmosphere is radiative, and the inner atmosphere is convective. Precipitation occurs at mid altitudes, but the deep atmosphere is too hot for water to condense or undersaturated.}\n\\label{fig_schematic}\n\\end{figure}\n\n\\section{Nominal Case}\nIn this section, we consider the atmospheric structure of a growing protoplanet for the model parameters listed in Table~1. The planet and its atmosphere pass  through a series of evolutionary stages, some of which include an ocean. At each stage, we assume the body is in thermal and hydrostatic equilibrium. The main stages are illustrated schematically in Figure~1.\n\nInitially, the protoplanet is too small to possess an atmosphere. Pebbles accreting onto the planet remain frozen until they settle to the surface, so that the planetary composition is the same as the pebbles themselves. The protoplanet first begins to acquire an atmosphere when its Bondi radius equals its physical radius. At this point, the planet's mass is\n\\begin{equation}\nM=\\frac{c_s^3}{G^{3\/2}}\\left(\\frac{3}{4\\rho_{\\rm solid}}\\right)^{1\/2}\n\\end{equation}\nFor the nominal model parameters, and a planet composed of 50\\% rock and 50\\% ice, an atmosphere  first forms when the total mass reaches about $0.0017$ Earth masses. The atmosphere is extremely tenuous at first, but the density near the surface grows rapidly with increasing mass of the protoplanet.\n\nWhen the total mass of the planet reaches about 0.084 Earth masses, the surface becomes warm enough for ice to melt. At this point an ocean begins to form. In practice, some time will be required to melt all of the planet's ice component, but here we assume an ocean forms instantaneously. The mass of H and He gas is very small at this stage, and the rocky core contains almost 50\\% of the total mass, or 0.042 Earth masses.\n\nFigure~\\ref{fig_first_ocean} shows the radial structure of the atmosphere and ocean in this case. Most of the atmosphere is nearly isothermal with a temperature similar to the surrounding protoplanetary disk. Energy in this region is transported efficiently by radiation since this part of the atmosphere is optically thin. Pressure increases roughly exponentially in the outer atmosphere. The low temperature means that the mass fraction of water is very low, even though the atmosphere is fully saturated with water vapor, as shown in the lower, left panel of the figure. The water mass fraction decreases to a minimum of about $4\\times 10^{-5}$ at 3 core radii, before the rising temperature allows it to increase again at smaller radii.\n\nAt a radius of about 2.1 core radii, the atmosphere becomes convective. This results in a sharp change in slope of the temperature gradient with respect to pressure, shown in the lower, right panel of Figure~\\ref{fig_first_ocean}. The atmosphere now enters a moist adiabatic regime in which gas rises and cools, causing water to condense and releasing latent heat. The release of latent heat reduces the adiabatic gradient as a result. Initially, the condensed water forms ice, which is assumed to precipitate immediately towards the core. Once the triple point is reached, excess vapor condenses as liquid water instead. This transition results in a small discontinuity in the temperature-pressure gradient due to the different entropy content of ice versus water.\n\nAt about 1.6 core radii, the atmosphere gives way to an ocean, which contains the bulk of the planet's water budget. The pressure and density jump by several orders of magnitude at this point, which can be seen in the upper, right panel of Figure~\\ref{fig_first_ocean}. The ocean is convective since heat released by infalling pebbles is assumed to be deposited at the surface of the rocky core. However, the increase in temperature with ocean depth is small in this case.\n\n\\begin{figure}\n\\plotone{steam_fig02.pdf}\n\\caption{The structure of the atmosphere and ocean for a rocky core with a mass of 0.042 Earth masses. The other model parameters are given in Table~1. The upper, left panel shows temperature. The upper, right panel shows total pressure (solid line), and the saturation vapor pressure of water (dashed line). The lower, left panel shows the water mass fraction. The lower, right panel shows the logarithmic temperature-pressure gradient.}\n\\label{fig_first_ocean}\n\\end{figure}\n\nAs the mass of a protoplanet increases, the atmospheric pressure and temperature increase as well. A hotter atmosphere allows more water vapor to be present due to the higher saturation vapor pressure. The increase in temperature also means that the surface of the ocean grows hotter. Since the ocean itself is convective, the lower portions eventually become a super-critical fluid once the temperature at the surface of the core exceeds 647~K, the critical temperature of water. However, density and temperature vary smoothly across the critical transition.\n\nFigure~\\ref{fig_last_ocean} shows the structure of the atmosphere and ocean for a protoplanet with a rocky core mass of 0.0794 Earth masses. For this core mass, the ocean surface temperature is 637~K, only slightly below the critical temperature. Thus, this core is one of the most massive that can sustain an ocean. The structure of the outer atmosphere in this case is similar to that in Figure~\\ref{fig_first_ocean}. Heat is transported by radiation, the temperature is nearly constant, and the pressure increases exponentially with decreasing radius. As before, the low temperature means that the water mass fraction remains very low in this region.\n\n\\begin{figure}\n\\plotone{steam_fig03.pdf}\n\\caption{The structure of the atmosphere and ocean for a rocky core with a mass of 0.0794 Earth masses. The other model parameters are given in Table~1.}\n\\label{fig_last_ocean}\n\\end{figure}\n\nThe radiative-convective boundary in Figure~\\ref{fig_last_ocean} occurs at about 3.4 core radii. This is larger than in Figure~\\ref{fig_first_ocean} for two reasons. Firstly, the planet is more massive so the atmosphere is denser at a given distance from the core. This increases the temperature gradient in the radiative region (see the third of Eqns~\\ref{eq_main_equations}). Secondly, the stronger gravity of the planet increases the efficiency of pebble accretion and the energy deposited by infalling pebbles. As a result, the luminosity is about 3 times higher than the case shown in Figure~\\ref{fig_first_ocean}. The increased luminosity and atmospheric density both mean that convection must operate further from the core than before.\n\nInterior to the radiative zone is a convective zone extending down to the ocean surface at 1.6 core radii. The convective atmosphere follows a moist adiabat, with the H\/He gas saturated in water vapor. The temperature-pressure gradient decreases moving inwards until the temperature reaches the triple point of water, at 273~K, and then increases again. As before, the water mass fraction decreases with depth in the outer atmosphere, reaching a minimum of $6\\times 10^{-5}$ at about 5 core radii. \n\nThe higher temperatures in the inner atmosphere lead to much higher water fractions. By the time the ocean surface is reached, the atmosphere is 81\\% water by mass compared to only 38\\% in Figure~\\ref{fig_first_ocean}. The large amount of water present at the base of the atmosphere, coupled with temperatures near the critical point, mean the discontinuity at the ocean surface is much less pronounced than the previous case. The density increases by only a factor of about 4 for example. The temperature and density continue to rise with depth in the convective ocean, passing seamlessly through the critical point, and reaching a temperature of 940~K and a density of 1.6  g\/cm$^3$ at the rocky core.\n\nAt core masses slightly higher than in Figure~\\ref{fig_last_ocean}, an ocean ceases to exist. Instead the entire water budget exists as either water vapor or a super-critical fluid. We assume that the water becomes intimately mixed with the hydrogen and helium throughout the resulting atmosphere. (In practice, some mixing may occur when the ocean is slightly below the critical temperature, but we do not attempt to model this effect here.) Below a particular transition temperature, the atmosphere is saturated in water vapor. Above this temperature, we assume the water mass fraction is constant, and set by the total amount of water available (which is equal to the mass of the rocky core in the nominal model). For rocky core masses below 0.105 Earth masses, this transition occurs at or close to the critical point, while for more massive cores, the transition occurs at progressively lower temperatures. Thus, more massive cores have lower maximum water fractions, and increasing fractions of H\/He gas.\n\n\\begin{figure}\n\\plotone{steam_fig04.pdf}\n\\caption{The structure of the atmosphere and ocean for a rocky core with a mass of 0.391 Earth masses. The other model parameters are given in Table~1.}\n\\label{fig_no_ocean}\n\\end{figure}\n\nFigure~\\ref{fig_no_ocean} shows the atmospheric structure for a rocky core with a mass of 0.391 Earth masses. This object has a total mass (core plus atmosphere) of 1 Earth mass. The outer, nearly-isothermal part of the atmosphere looks similar to the previous cases. However, the radiative-convective boundary now lies at 14.3 core radii, much further out than before, due to the larger core mass and higher luminosity due to pebble  accretion. \n\nInside the radiative-convective boundary lies a relatively narrow moist adiabatic region, extending inwards to 8.5 core radii, where the temperature is 321~K. Inside this point, the atmosphere becomes undersaturated in water vapor, as can be seen in the upper, right panel of Figure~\\ref{fig_no_ocean}. At this radius, there is a large jump in the temperature-pressure gradient (see lower, right panel), and from this point inwards, the atmosphere follows a steeper, ``dry'' abiabat due to the absence of latent heat released by condensing water vapor.\n\nBelow 8.5 core radii, the water mass fraction is constant at 0.64, with the remaining mass made up of hydrogen and helium. The steep convective gradient means the temperature increases rapidly with depth, reaching 5900~K at the rocky core. (The changes in the temperature-pressure gradient seen at 2000~K and 3000~K in the lower, right panel of the figure are artifacts caused by changes in the EOS at these points.) The great majority of the atmospheric mass is contained in the region below 8.5 core radii, with only about 0.15\\% of the total planetary mass lying at larger radii.\n\nThe trends described above continue for larger core masses, but eventually a point is reached where the core mass reaches a maximum. We examine this point in the next section.\n\n\\begin{figure}\n\\plotone{steam_fig05.pdf}\n\\caption{Upper panel: rocky core mass as a function of total planet mass for the nominal model. Middle panel: gas mass fraction of these objects. Lower panel: luminosity due to pebble accretion.}\n\\label{fig_critical_mass}\n\\end{figure}\n\n\\section{Critical Core Mass}\nFigure~\\ref{fig_critical_mass} shows the change in atmospheric properties of protoplanets with increasing total mass for the parameters listed in Table~1. The top panel shows the rocky core mass as a function of the total planetary mass. The middle panel shows the gas (H plus He) mass fraction. The bottom panel shows the luminosity due to pebble accretion.\n\nFor total masses below about 10 Earth masses, the rocky core mass increases roughly linearly with total mass. A growing planet should follow this trajectory, with the total mass increasing smoothly with core mass. Above 10 Earth masses, the curve in the top panel of Figure~\\ref{fig_critical_mass} starts to turn over. Small increases in core mass lead to progressively larger increases in atmospheric mass and thus total mass. At a total mass of 17 Earth masses, the core mass reaches a maximum value of 3.03 Earth masses. At this point H and He represent about 64\\% of the total planetary mass.\n\nFurther increases in total mass result in a smaller core mass, before the core mass starts to increase again when the total mass reaches 20 Earth masses, eventually reaching a second maximum at a total mass of 50 Earth masses. In practice, the core mass should increase monotonically with time, so the region between 17 and 20 Earth masses is unphysical. It is possible that continued gas accretion will allow a real planet to move across this region before rejoining the solutions above 20 Earth masses. However, examining this possibility would require a model that includes time evolution and luminosity generated by gas accretion rather than the static model used here. \n\nIt may seem surprising that a given core can support more than one type of atmosphere. However, the solutions differ in a number of respects. In particular, the luminosities are different because the capture radius for pebble accretion is determined by the total mass rather than the core mass. The atmospheric compositions are also different since a given mass of water is diluted to different degrees by the different masses of H\/He gas for each atmospheric solution. We note that \\cite{mizuno:1980} also found multiple atmospheric solutions for a given core mass. Here we follow \\cite{mizuno:1980} and \\cite{venturini:2015}, and associate the first core-mass maximum with the ``critical'' core mass, assuming that the static solutions become invalid beyond this point. \n\nA second criterion that is sometimes used to determine the onset of rapid gas accretion is the ``crossover'' mass \\cite{pollack:1996, rafikov:2006}. This is the point at which the masses of the core and atmosphere are equal. In the model presented here, the atmospheric mass is always greater than the core mass, once an ocean ceases to exist, due to the presence of water vapor. Instead, we note the point at which H and He make up half of the total mass. Using this criterion, rapid gas accretion in Figure~\\ref{fig_critical_mass} would begin at a core mass of 2.68 Earth masses, slightly below the critical mass.\n\nIn the following subsections, we will examine how the critical mass and crossover mass vary depending on some key model parameters.\n\n\\subsection{Dependence on Opacity}\nOne of the biggest uncertainties in the model is the opacity of dust grains in the atmosphere. Some dust is swept into the atmosphere with the inflowing gas. The opacity of this dust will depend on the local dust-to-gas ratio in the protoplanetary disk, which in turn depends on how much dust was converted into pebbles. A bigger uncertainty is the fate of infalling pebbles once their icy component evaporates. The rocky residues may remain intact, settling quickly towards the core and leaving the opacity unchanged. Alternatively, pebbles may disintegrate into sub-$\\mu$m-size grains once the ice cementing these grains evaporates. In this case, the opacity could be raised by orders of magnitude. Intermediate scenarios are also possible. Given these uncertainties, we choose to treat the dust opacity as a parameter with a wide range of possible values.\n\n\\begin{figure}\n\\plotone{steam_fig06.pdf}\n\\caption{Dependence of the critical core mass and crossover mass (50\\% of the total mass in H and He) on the opacity of dust grains in the atmosphere.}\n\\label{fig_opacity}\n\\end{figure}\n\nFigure~\\ref{fig_opacity} shows the critical core mass and the crossover core mass as a function of the dust opacity $\\kappa_{\\rm dust}$. Other model parameters are unchanged from those in Table~1. The critical mass varies with $\\kappa_{\\rm dust}$ in a complicated manner. Initially, the critical mass increases with increasing $\\kappa_{\\rm dust}$ for small values, peaking at $\\kappa_{\\rm dust}=0.006$ cm$^2$\/g, and then falling again. The critical mass becomes nearly constant at about 3 Earth masses for $\\kappa_{\\rm dust}>0.01$ cm$^2$\/g. The crossover (50\\% H and He) mass increases monotonically with dust opacity, rising rapidly for $\\kappa_{\\rm dust}<0.007$ cm$^2$\/g, and more slowly at larger opacities.\n\nThe complicated behavior seen in the critical core mass is partly due to the occasional existence of multiple solutions for the core mass as a function of total mass, as seen in Figure~\\ref{fig_critical_mass}. For most values of $\\kappa_{\\rm dust}$ in Figure~\\ref{fig_opacity}, a single maximum exists, and this is the critical core mass. However, for intermediate values of $\\kappa_{\\rm dust}$ a second maximum appears at smaller core masses than the first. In these cases we assign the first maximum to be the critical mass, leading the decrease in critical mass seen at intermediate values of $\\kappa_{\\rm dust}$ in Figure~\\ref{fig_opacity}. \n\nThe crossover mass doesn't exhibit the complicated behavior shown by the critical mass, (The change of slope seen in the figure coincides with a change in the outer boundary condition from the Bondi radius to 1\/4 of the Hill radius.) For this reason, the crossover mass may represent a more robust measure of the onset of rapid gas accretion than the critical mass. It is worth noting that the critical and crossover masses are similar for large dust opacities.\n\nOverall, the core mass at the onset of rapid gas accretion tends to increase with increasing dust opacity. A higher opacity is associated with a steeper rise in temperature in the radiative region of the atmosphere. As a result, the boundary between the radiative and convective  regions lies further from the core when the opacity is large. In the radiative region, pressure and density both increase roughly exponentially moving inwards, while these quantities vary more slowly in the convective part of the atmosphere (see Figure~\\ref{fig_no_ocean}). Since the total atmospheric mass depends on the integrated density, a smaller radiative region typically leads to a lower total mass for a given core mass, increasing the core mass needed for rapid gas accretion to begin.\n\n\\begin{figure}\n\\plotone{steam_fig07.pdf}\n\\caption{Dependence of the critical core mass and crossover mass (50\\% of the total mass in H and He) on the radial mass flux of pebbles drifting through the disk.}\n\\label{fig_pebble_flux}\n\\end{figure}\n\n\\subsection{Dependence on Pebble Flux}\nThe mass flux of pebbles passing  through the region containing the planet is another source of uncertainty. Pebble drift rates depend on the size of the pebbles and the radial pressure gradient in the disk. The pebble flux depends on the drift rate, and also on the total mass of pebbles in the disk, how long they have been drifting, and the radial extent of  the disk. The pebble flux is likely to vary with time and location, possibly by large amounts. In our model, the luminosity of the planet's atmosphere is determined by the pebble accretion rate, which depends on the pebble flux as well as the capture radius. Thus, we consider a range of possible pebble fluxes.\n\nFigure~\\ref{fig_pebble_flux} shows the critical core mass and crossover mass for a range of pebble mass flux values. The dependence of both masses is broadly similar to the dependence on the dust opacity shown in Figure~\\ref{fig_opacity}. The critical mass increases with increasing flux, reaches a peak at intermediate values, then declines to a roughly constant value for large mass fluxes. The crossover mass increases monotonically with increasing flux, rapidly at first and  then slowly. For large pebble fluxes the two masses are similar.\n\nThe similar behavior in Figures~\\ref{fig_opacity} and \\ref{fig_pebble_flux} occurs because the opacity and luminosity (which is proportional to the pebble flux) have the same effect on the atmosphere. The temperature profile depends linearly on each quantity in the radiative region, while neither quantity directly affects the profile in the convective region, as can be seen in Eqns.~\\ref{eq_main_equations}. The main difference between dust opacity and pebble mass flux is that the rate at which the planetary core accretes mass cannot exceed  the pebble flux itself, and this will be a factor in the next subsection.\n\n\\begin{figure}\n\\plotone{steam_fig08.pdf}\n\\caption{Dependence of the critical core mass and crossover mass (50\\% of the total mass in H and He) on the Stokes number of the pebbles.}\n\\label{fig_pebble_stokes}\n\\end{figure}\n\n\\subsection{Dependence on Pebble Size}\nThe size of pebbles in the planet-forming regions of protoplanetary disks is also uncertain, and probably varies with time and location. The typical Stokes number of pebbles will also vary depending on the local gas density. Here we examine a range of possible Stokes numbers $\\rm St$. \n\nFigure~\\ref{fig_pebble_stokes} shows the critical core mass and crossover mass as a function of $\\rm St$. For $\\rm St<0.002$, the critical mass is 5.2 Earth masses, almost independent of the Stokes number. For $0.002<\\rm St<0.06$, a second maximum appears in the relation between core mass and total mass. For this range of Stokes numbers, the critical mass falls to roughly 3 Earth masses. At larger Stokes numbers, the critical core mass returns to the previous value, before declining with $\\rm St$ for $\\rm St>0.1$.\n\nThe constant critical mass for $\\rm St<0.002$ arises because in these cases the pebble capture radius is large enough that the entire flux of pebbles is captured by the planet. The radial drift speeds of the pebbles are slow enough that they are all accreted by the planet in the time it takes them to drift inwards across the planet's Hill radius. As a result, changing the pebble Stokes number has no effect on the luminosity of the atmosphere or the atmospheric structure. \n\nFor larger pebbles (larger Stokes numbers), the behavior of the critical mass is more complicated due to the appearance of an additional core-mass maximum as was the case in the previous sections. Here, the accretion rate is less than the pebble flux, so differences become apparent for different values of $\\rm St$. As before, the crossover mass behaves in a simpler manner than the critical mass. The crossover mass depends only weakly on $\\rm St$. Although the pebble capture radius tends to increase with increasing pebble size, this effect is offset by the greater radial drift speeds, which lowers the pebble  surface density by a comparable amount. As a result, the onset of rapid gas accretion probably depends only weakly on pebble size for a given mass flux of pebbles.\n\n\\begin{figure}\n\\plotone{steam_fig09.pdf}\n\\caption{Dependence of the critical core mass and crossover mass (50\\% of the total mass in H and He) on the mass fraction of ice in the pebbles.}\n\\label{fig_ice_fraction}\n\\end{figure}\n\n\\subsection{Dependence on Ice-to-Rock Ratio}\nSo far, we have considered changes that alter  the structure of a planetary atmosphere's outer, radiative zone via the opacity or luminosity. In this section, we examine differences caused by altering the ice-to-rock ratio of the pebbles accreted  by the planet, which tends to affect the lower atmosphere. In the previous cases, we assumed that the ice mass fraction is 50\\%. However, other values are possible for the solar nebula and other protoplanetary disks. In the solar nebula for example, the ice-to-rock ratio is somewhat uncertain due to uncertainties in the carbon-to-oxygen ratio of the Sun, and uncertainties regarding the main carrier of carbon in the solar nebula \\citep{asplund:2009, lodders:2003}.\n\nFigure~\\ref{fig_ice_fraction} shows the critical core mass and crossover mass as a function of the ice mass fraction of pebbles. Unlike the other model parameters, both the critical mass and crossover mass show a clear, nearly monotonic trend with ice fraction. The masses decline steadily with ice fraction for fractions less than about 0.65, falling from 8.1 Earth masses to about 2.4 Earth masses for the critical mass and 1.9 Earth masses for the crossover mass. At this point, the critical mass increases somewhat, before resuming the previous trend. The crossover mass declines monotonically with ice fraction for all values considered here.\n\nThe strong dependence of the critical mass on the ice fraction comes from the fact that the atmospheric mass depends on the mean molecular weight, and this is greater when large amounts of water are present. When the mean molecular weight is high, the atmospheric mass is larger for a given core mass, leading to a smaller  critical mass. This is true only in the hot inner regions however. The effect seen in Figure~\\ref{fig_ice_fraction} would be even stronger if the amount of water in the outer atmosphere wasn't restricted by the saturation pressure of water.\n\n\\section{Discussion}\nIn this paper we have examined the growth and atmospheric structure of planets accreting ice-rich pebbles in a protoplanetary disk of H and He gas. Such planets pass through several evolutionary stages, as shown in Figure~\\ref{fig_schematic}, beginning with objects too small to appreciably affect the gas in the surrounding disk. When the planet's mass reaches about 0.002 Earth masses, it begins to acquire an atmosphere. The atmospheric mass increases rapidly with increasing core mass. The outer regions are optically thin and nearly isothermal, while close to the core temperatures increase rapidly and a convective zone develops.\n\nIncoming pebbles sediment slowly towards the core at terminal velocity. As they encounter warmer regions, the icy component of the pebbles begins to evaporate, saturating the atmosphere with water vapor. Excess water and ice precipitate towards the core, forming an ocean when the surface temperature and pressure exceed the triple point of water. For the model parameters used here, an ocean first forms when the mass of rock reaches about 0.04 Earth masses. In the convective region, water vapor in the saturated gas condenses as the gas rises and cools. The condensed water precipitates back to the ocean, while the latent heat released by condensation means that the rising gas follows a shallow, moist-adiabatic temperature profile.\n\nWhen the rocky core mass reaches about 0.08 Earth masses, the temperature becomes too high for an ocean to exist. We assume that water in the region closest to the rocky core forms a super-critical convecting fluid that mixes with gas in the layers above. Temperatures decline with distance from the core, passing smoothly through the critical point of water. The outer part of the convective zone continues to follow a moist adiabat. However, the higher temperatures in the inner part of the convective zone mean that the gas is undersaturated in water vapor, and here the temperature profile follows a steeper dry adiabat. It is possible that water from the ocean doesn't fully mix with the gas in the layers above when the ocean boils due to the steep molecular weight gradient. However, this probably won't make much difference to the outcome since the planetary mass at this stage is much lower than the critical mass. The amount of water that remains trapped in such a ``core'' is much less than the water supplied by more infalling pebbles, and this additional water will be mixed with gas in the convecting region of the atmophere.\n\nAs the core mass increases beyond 0.08 Earth masses in our model, the atmospheric mass continues to grow, and the mass ratio of water vapor to gas decreases. The region of the atmosphere that is saturated and convecting becomes increasingly narrow as a result. At a core mass of a few Earth masses, 50\\% of the total planetary mass is made up of H and He gas. At a slightly higher mass, the rocky core mass reaches a maximum, and we assume the static model used here ceases to be valid at this point. Presumably, rapid gas accretion will begin to take place at around this critical mass, eventually forming a gas-giant planet.\n\nThe critical mass often exhibits complicated behavior as a function of the main model parameters. This is due to the existence of multiple maxima in the relation between total mass and core mass. A more robust indicator of the onset rapid gas accretion may be the crossover mass (defined here as the point when 50\\% of the planet's mass is H and He), since this typically varies in a simpler manner when the parameters are varied. Better estimates of the onset of rapid gas accretion will come from evolving models that include the energy released due to gas accretion.\n\n\\cite{hori:2011} and \\cite{venturini:2015} have carried out similar studies to the one in this paper, investigating the effects of evaporating volatiles on the atmospheric structure and critical mass of protoplanets. \\cite{hori:2011} considered a two-layer atmosphere: an outer region with composition similar to the protoplanetary disk, and an inner region uniformly enriched in elements heavier than He. These authors assumed all volatile constituents remained in the vapor phase. \\cite{venturini:2015} considered a single, uniformly enriched layer, and accounted for condensation of ice and water, but not precipitation.\n\n\\cite{hori:2011} found that the critical core mass typically decreases with increasing heavy-element concentration in the atmosphere for heavy-element mass fractions $Z>0.1$. For mass fractions above about 0.75, the critical mass is less than one Earth mass, and approaches Mars's mass for atmospheres almost wholly composed of heavy elements. \\cite{venturini:2015} find similar results in the absence of water condensation. When condensation is included, they find that the critical mass is reduced further, falling below 0.1 Earth masses when $Z>0.75$.\n\nIn this paper, we find larger critical core masses, typically a few Earth masses, even though large amounts of water are present due to the evaporation of ice from pebbles. There are several reasons for this difference. Firstly, we consider only a single volatile, water, whereas \\cite{hori:2011} and \\cite{venturini:2015} included carbon-bearing species. The presence of carbon compounds, and transitions between the dominant species with changing temperature, lead to modest differences in atmospheric structure due to differences in the adiabatic temperature gradient. In addition, \\cite{hori:2011} and \\cite{venturini:2015} consider a fixed atmospheric luminosity $L=10^{27}$ erg\/s, independent of planetary mass. Here, we calculate the luminosity self-consistently based on the pebble accretion rate, typically yielding slightly higher values at the critical mass.\n\nHowever, the main reason for the larger critical masses found here is that we consider atmospheres with water mass fractions that vary considerably with altitude due to precipitation. Temperatures in the outer, radiative region are only slightly higher than in the local protoplanetary disk. As a result, the saturation pressure and mass fraction of water vapor are very low in this region. \\cite{venturini:2015} assumed that condensed ice and water would remain where it forms, presumably in the form of clouds. We believe this is unlikely, particularly for the large water mass fractions they considered, and  that most of the condensed water will precipitate to the ocean or undersaturated regions deeper in the atmosphere. In fact, if mixing in the radiative region is inefficient, the water abundance may always be small since incoming pebbles will not evaporate where the gas is saturated in water vapor.\n\nThe very low abundance of water vapor in the upper atmosphere means that the radiative region and the upper convective region behave somewhat like the corresponding regions in conventional models for core accretion that consider a solar mixture of gases dominated by H and He, and yield large critical core masses \\citep{pollack:1996, movshovitz:2010}. The presence of higher water fractions deep in the atmosphere clearly reduces the critical core mass, as can be seen in Figure~9, but the effect is more modest than found by \\cite{hori:2011} and \\cite{venturini:2015}. One difference between the conventional models and the one described here is that we assume that convection in the outer atmosphere begins when the temperature gradient reaches a moist adiabat, which is typically shallower than the dry adiabat that is usually considered. As a result, the radiative-convective boundary lies at a higher altitude in the model described here, raising the critical mass somewhat. Overall, however, previous estimates of the critical mass based on H- and He-dominated atmospheres may still be reasonably valid when the effect of ice evaporation from pebbles is taken into account.\n\nIn our model, two sources of negative feedback act to raise the critical mass even as the core itself grows. Firstly, an increase in the planetary mass increases the luminosity of the atmosphere, both because the capture cross section for pebbles becomes larger, and because the gravitational energy released by each infalling pebble becomes greater as the core's potential well deepens. Secondly, as the planet grows, the water-to-gas ratio in the atmosphere declines. Thus, the density effect due  to water vapor is increasingly diluted, and the presence of water vapor becomes less important for the structure of the atmosphere. Both factors need to be taken into account when determining the onset of rapid gas accretion.\n\nIn this study, we have considered only a single volatile species heavier than He, namely water. The rationale for this choice is that we consider a planet located just beyond the ice line that is accreting pebbles. While large planetesimals in this region may retain other, more-volatile ices, these ices will surely have evaporated from pebble-size particles before they encounter the planet. One possibility not considered here is that protoplanetary disks may contain an appreciable mass of tar-like organic compounds \\citep{lodders:2004}, which are intermediate in volatility between water ice and rock. Such tars would be present in pebbles just outside the ice line, but would evaporate in the hotter regions of the planet's atmosphere. The presence of large amounts of tar-like material could alter the atmospheric structure and introduce some of the effects found by \\cite{hori:2011} and \\cite{venturini:2015}. Furthermore, the high temperatures we find close to the core of the planet may be sufficient for rocky materials to evaporate and decompose rather than sedimenting to the core as we have assumed here. These possibilities are worth studying further in future.\n\nSome other factors may warrant further study. For example, the current model considers a constant mass flux of pebbles passing through the protoplanetary disk, and uses a fixed Stokes number for these pebbles. Neither quantity is likely to be constant in real disks \\citep{birnstiel:2012}. The pebble flux probably declines over time, and the typical pebble size is likely to decrease as big pebbles are preferentially lost due to radial drift. Both factors will decrease the pebble capture rate and luminosity of the planet over time. Another factor we have not considered here is that the supply of pebbles may be shut off entirely once the planet becomes massive enough to alter the sign of the radial pressure gradient in nearby regions of the disk \\citep{lambrechts:2014}.\n\nIn addition, we have assumed that the temperature and density at the outer boundary of the planet remain fixed. These quantities could change over time as the disk evolves or the planet migrates through the disk. Previous works have noted that the critical core mass is insensitive to the outer boundary conditions provided that the outer atmosphere is radiative \\citep{stevenson:1982, rafikov:2006}. However, this will no longer be true if the outer boundary crosses the condensation front for a major species such as the ice line, since the composition of infalling pebbles will change abruptly at this point. In particular, a planet moving inwards across the ice line may change from a situation in which there is a net gain of water to a net loss as the water fraction in the outer atmosphere increases above the local disk value.\n\n\\section{Summary}\nIn this paper, we have modeled  the structure and critical mass of a planetary  atmosphere when the planet is accreting ice-rich pebbles. The main findings are:\n\\begin{enumerate}\n\\item The planet goes through a series of stages with increasing core mass. An atmosphere first appears for a core mass of 0.002 Earth masses. For core masses (rock + ice) between 0.08 and 0.16 Earth masses, the surface temperature is between the triple point and critical point of water, and an ocean forms. At larger masses, temperatures are too high for an ocean to exist, and water exists as a super-critical fluid mixed with H and He gas accreted from the protoplanetary disk.\n\n\\item Temperature in the atmosphere increases with depth. Some or all of the ice component of pebbles evaporates in the planet's atmosphere adding water vapor to the H and He gas from the disk. The saturation pressure of water is very low in the outer atmosphere, so the water mass fraction in this region is small. The water mass fraction is higher in the deep atmosphere, becoming constant below a certain altitude.\n\n\\item The outer atmosphere is radiative, the inner atmosphere is convective. In saturated, convective regions, the temperature profile follows a shallow, moist adiabat due to water condensation in the rising gas. Unsaturated regions follow a steeper dry adiabat. As the core mass increases, the water-to-gas fraction in the atmosphere declines, and boundary between unsaturated and saturated regions moves to lower temperatures and higher altitudes.\n\n\\item The mass of the planet's rocky core increases with total planetary mass. For a 1:1 ice-to-rock ratio, the core mass reaches a maximum value (the critical mass) at 2--5 Earth masses, and the static model used here is no longer valid. Presumably, runaway gas accretion begins at this point.\n\n\\item The critical mass depends in a complicated manner on the pebble size, mass flux through the disk, and atmospheric dust opacity. This is due to the presence of more than one core-mass maximum for a given total planetary mass for some parameter values. The critical mass declines nearly monotonically with increasing ice-to-rock ratio from 8 Earth masses at an ice fraction of 0.1 to about 2.5 Earth masses at an ice fraction of 0.75.\n\n\\item The point at which 50\\% of the planetary mass is H and He (the crossover mass) varies with model parameters in a more straightforward way than the critical mass, and may be a better indicator of the onset of rapid gas accretion. The crossover mass increases with atmospheric dust opacity and pebble mass flux, and decreases with ice-to-rock ratio. The crossover mass is relatively insensitive to the pebble size.\n\n\\end{enumerate}\n\n\\acknowledgments\n\nI would like to thank Alan Boss and Lindsey Chambers for helpful comments and discussions during the preparation of this paper. I also thank an anonymous referee whose comments have improved this paper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Summary}\n\nThe CMS Collaboration has searched for narrow resonances in the\ninvariant mass spectrum of  dimuon and dielectron final\nstates in event samples corresponding to an integrated luminosity of\n$40$\\pbinv and $35$\\pbinv, respectively. The spectra are consistent\nwith standard model expectations and upper limits on the\ncross section times branching fraction for $\\cPZpr$ into\nlepton pairs relative to standard model $\\cPZ$ boson\nproduction have been set.\nMass limits have been set on neutral gauge bosons $\\cPZpr$\nand RS Kaluza--Klein gravitons $\\GKK$.  A $\\cPZpr$ with\nstandard-model-like couplings can be excluded below 1140\\GeV, the\nsuperstring-inspired $\\ZPPSI$ below 887\\GeV, and RS\nKaluza--Klein gravitons below 855 (1079)\\GeV for couplings of\n0.05 (0.10), all at 95\\%~C.L.\nThe higher centre of mass energy used in this search, compared to that\nof previous experiments, has resulted in limits that are comparable to or\nexceed those previously published, despite the much lower\nintegrated luminosity accumulated at the LHC thus far.\n\n\\section*{Acknowledgments}\n\nWe wish to congratulate our colleagues in the CERN accelerator\ndepartments for the excellent performance of the LHC machine. We thank\nthe technical and administrative staff at CERN and other CMS\ninstitutes, and acknowledge support from: FMSR (Austria); FNRS and FWO\n(Belgium); CNPq, CAPES, FAPERJ, and FAPESP (Brazil); MES (Bulgaria);\nCERN; CAS, MoST, and NSFC (China); COLCIENCIAS (Colombia); MSES\n(Croatia); RPF (Cyprus); Academy of Sciences and NICPB (Estonia);\nAcademy of Finland, ME, and HIP (Finland); CEA and CNRS\/IN2P3\n(France); BMBF, DFG, and HGF (Germany); GSRT (Greece); OTKA and NKTH\n(Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN\n(Italy); NRF and WCU (Korea); LAS (Lithuania); CINVESTAV, CONACYT,\nSEP, and UASLP-FAI (Mexico); PAEC (Pakistan); SCSR (Poland); FCT\n(Portugal); JINR (Armenia, Belarus, Georgia, Ukraine, Uzbekistan); MST\nand MAE (Russia); MSTD (Serbia); MICINN and CPAN (Spain); Swiss\nFunding Agencies (Switzerland); NSC (Taipei); TUBITAK and TAEK\n(Turkey); STFC (United Kingdom); DOE and NSF (USA).\n\n\n\n\\subsection{\\texorpdfstring{\\cPZ\/$\\gamma^*$}{Z\/gamma*} Backgrounds}\n\nThe shape of the dilepton invariant mass spectrum is obtained from\nDrell--Yan production using a MC simulation based on the\n\\PYTHIA event generator.   The simulated\nspectrum at the invariant mass peak of the $\\cPZ$ boson is normalized to\nthe data.  The dimuon analysis uses the data events in the $\\cPZ$\nmass interval of 60--120\\GeV; the dielectron analysis\nuses data events in the narrower interval of 80--100\\GeV in order to\nobtain a comparably small background contamination.\n\nA contribution to the uncertainty attributed to the extrapolation of the\nevent yield and the shape of the Drell--Yan background to high\ninvariant masses arises from higher order QCD corrections.\nThe next-to-next-to-leading order (NNLO) $k$-factor is computed using\n{\\sc FEWZz v1.X}~\\cite{Melnikov:2006kv}, with \\PYTHIA~{\\sc v6.409} and\n{\\sc CTEQ6.1} PDF~\\cite{Stump:2003yu}\nas a baseline. It is found that the variation of the\n$k$-factor with mass does not exceed 4\\% where\nthe main difference arises from the comparison of \\PYTHIA and {\\sc FEWZz} calculations.\nA further source of uncertainty arises from the PDFs. The  {\\sc lhaglue}~\\cite{Bourilkov:2003kk}\ninterface to the {\\sc lhapdf-5.3.1}~\\cite{Whalley:2005nh} library is used to evaluate these uncertainties,\nusing the error PDFs from the {\\sc CTEQ6.1} and the MRST2006nnlo~\\cite{Martin:2007bv}\nuncertainty eigenvector sets.\nThe uncertainty on the ratio of the background in\nthe high-mass region to that in the region of the $\\cPZ$ peak is below\n4\\% for both PDF sets and masses below 1 TeV. Combining the higher order QCD and PDF uncertainties in\nquadrature, the resulting uncertainty in the number of events normalized to those expected at the\n$\\cPZ$ peak is about 5.7\\% for masses between 200\\GeV and 1\\TeV.\n\n\\subsection{Other Backgrounds with Prompt Lepton Pairs\n\\label{sec:e-mu}  }\n\nThe dominant non-Drell--Yan electroweak contribution to high\n$m_{\\ell\\ell}$ masses is {$\\ttbar$}; in addition there are contributions\nfrom $\\tq\\PW$ and diboson production. In the $\\cPZ$ peak\nregion, $\\cPZ \\rightarrow \\Pgt\\Pgt$ decays also contribute.\nAll these processes are flavour symmetric and produce twice as\nmany $\\Pe\\Pgm$ pairs as $\\Pe\\Pe$ or $\\Pgm\\Pgm$ pairs.\nThe invariant mass spectrum from\n$\\Pe^\\pm\\Pgm^\\mp$ events is expected to have the same shape as that of same\nflavour $\\ell^+\\ell^-$ events but without significant contamination\nfrom Drell--Yan production.\n\nFigure~\\ref{fig:muonselectrons} shows the\nobserved $\\Pe^\\pm\\Pgm^\\mp$ dilepton invariant mass spectrum\nfrom a dataset corresponding to 35\\pbinv, overlaid on\nthe prediction from simulated background processes. This spectrum was\nobtained using the same single-muon trigger as in the dimuon analysis\nand by requiring oppositely charged leptons of different flavour. Using an electron\ntrigger, a very similar spectrum is produced.\nDifferences in the geometric acceptances and efficiencies result in the predicted ratios\nof $\\Pgmp\\Pgmm$ and $\\Pe\\Pe$ to $\\Pe^\\pm\\Pgm^\\mp$ being approximately 0.64\nand 0.50, respectively. In the data, shown in\nFig.~\\ref{fig:muonselectrons}, there are 32 (7) $\\Pe^\\pm\\Pgm^\\mp$ events\nwith invariant mass above 120 (200)\\GeV. This yields an expectation of\nabout 20 (4) dimuon events and 16 (4) dielectron events.  A direct\nestimate from MC simulations of the processes involved\npredicts $20.1\\pm3.6$ $(5.3\\pm 1.0)$ dimuon events and\n$13.2\\pm2.4$ $(3.5\\pm0.6)$ dielectron events. The uncertainty\nincludes  both statistical and systematic contributions, and is dominated\nby the theoretical uncertainty  of 15\\% on the $\\ttbar$ production cross\nsection~\\cite{Campbell:2010ff,Kleiss:1988xr}.\nThe agreement\nbetween the observed and predicted distributions provides a validation\nof the estimated contributions from the backgrounds from prompt leptons\nobtained using MC simulations.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=90,width=0.49\\textwidth]{figures\/MuonsElectronsOppSign.pdf}\n\\end{center}\n\\caption{\\label{fig:muonselectrons}\nThe observed opposite-sign $\\Pe^\\pm\\Pgm^\\mp$ dilepton invariant mass spectrum\n(data points). The uncertainties on the data points (statistical only)  \nrepresent 68\\% confidence intervals for the Poisson means.\nFilled histograms show contributions to the\nspectrum from  $\\ttbar$, other sources of prompt leptons\n ($\\tq\\PW$, diboson production, $\\cPZ\\to\\Pgt\\Pgt$),  and\nthe multi-jet background (from Monte Carlo simulation).\n}\n\\end{figure}\n\n\\subsection{Events with Misidentified and Non-Prompt Leptons}\n\nA further source of background arises when objects are falsely\nidentified as prompt leptons.  The misidentification of jets as\nleptons, the principle source of such backgrounds, is more likely to\noccur for electrons than for muons.\n\nBackgrounds arising from jets that are misidentified as electrons\ninclude $\\PW\\to \\Pe\\cPgn$ + jet events with\none jet misidentified as a prompt electron, and also multi-jet events with\ntwo jets misidentified as prompt electrons.\nA prescaled single EM cluster trigger is used for collecting a sample of\nevents to determine the rate of jets misreconstructed as electrons and\nto estimate the backgrounds from misidentified electrons.\nThe events in this sample are\nrequired to have no more than one\nreconstructed electron, and missing transverse energy of less than 20\\GeV,\nto suppress the contribution from $\\cPZ$\nand $\\PW$ events respectively.\nThe probability for an EM cluster with $H\/E<5\\%$ to be\nreconstructed as an electron is determined in bins of $\\ET$ and $\\eta$ from\na data sample dominated by multi-jet events and is used to weight\nappropriately  events which have two such clusters passing\nthe double EM trigger.\nThe estimated background\ncontribution to the dielectron mass spectrum due to misidentified jets is\n 8.6$\\pm$3.4 (2.1$\\pm$0.8)  for $m_{\\Pe\\Pe} > 120$ (200)\\GeV.\n\nIn order to estimate the residual contribution from background events\nwith at least one non-prompt or misidentified muon, events are\nselected from the data sample with single muons that pass all\nselection cuts except the isolation requirement.\nA map is created, showing the isolation probability for these muons as\na function of $\\pt$ and $\\eta$.\nThis probability map is corrected for the expected\ncontribution from events with single prompt muons from $\\ttbar$ and\n$\\PW$ decays and for the observed correlation between the\nprobabilities for two muons in the same event. The probability map is\nused to predict the number of background events with two isolated\nmuons based on the sample of events that have two non-isolated muons.\nThis procedure, which has been validated using simulated events,\npredicts a mean background to $m_{\\Pgm\\Pgm} > 120$ (200)\\GeV of\n$0.8\\pm0.2\\ (0.2\\pm0.1)$.\n\nAs the signal sample includes the requirement that the muons in the\npair have opposite electric charge, a further cross-check of the\nestimate is performed using events with two isolated muons of the same\ncharge.  There are\nno events with same-charge muon pairs and $m_{\\Pgm\\Pgm} > 120$\\GeV,\na result which is statistically compatible with both the figure of\n $1.6\\pm0.3$ events predicted from SM process using MC simulation and\nthe figure of $0.4\\pm0.1$ events obtained using methods based on data.\n\n\n\\subsection{Cosmic Ray Muon Backgrounds}\n\nThe $\\Pgmp\\Pgmm$ data sample is susceptible to contamination from\ntraversing cosmic ray muons, which may  be misreconstructed as a pair\nof oppositely charged, high-momentum muons.\nCosmic ray events can be removed from the data sample\nbecause of their distinct topology (collinearity of two tracks\nassociated with  the\nsame muon),\nand their uniform distribution of impact parameters with respect to the collision vertex.\nThe residual mean expected background from cosmic ray muons is\nmeasured using sidebands to be less than 0.1 events with\n$m_{\\Pgm\\Pgm} > 120$\\GeV.\n\n\n\\section{Dilepton Invariant Mass Spectra}\n\nThe measured dimuon and dielectron invariant mass spectra are displayed in\nFigs.~\\ref{fig:spectra}(left) and (right) respectively,\nalong with the expected signal from $\\ZPSSM$ with a mass of 750\\GeV.\nIn the dimuon sample, the highest invariant mass\nevent has $m_{\\Pgm\\Pgm}=463$\\GeV, with the $\\pt$ of the two\nmuons measured to be 258 and 185\\GeV.\nThe highest invariant mass event in the dielectron sample\nhas $m_{\\Pe\\Pe}=419$\\GeV, with the electron candidates having $\\ET$ of 125 and 84\\GeV.\n\nThe expectations from the various background sources,\n$\\cPZ{\/}\\gamma^*$, $\\ttbar$, other sources of  prompt leptons ($\\tq\\PW$, diboson\nproduction, $\\cPZ\\to\\Pgt\\Pgt$) and multi-jet events\nare also overlaid in Fig.~\\ref{fig:spectra}.  For the dielectron\nsample, the multi-jet background estimate was obtained directly from\nthe data. The prediction for Drell--Yan production of $\\cPZ{\/}\\gamma^*$\nis normalized to the observed $\\cPZ\\to\\ell\\ell$ signal.  All other MC\npredictions are normalized to the expected cross sections.\nFigures~\\ref{fig:cum_spectra}(left) and (right) show the corresponding\ncumulative distributions of the spectra for the dimuon and\ndielectron samples.  Good agreement is observed between data and the\nexpectation from SM processes over the mass region above the Z peak.\n\nSearches for narrow resonances at the Tevatron~\\cite{D0_Zp,CDF_Zp}\nhave placed lower limits in the mass range $600$\\GeV to\n$1000$\\GeV. The region with dilepton masses $120\\GeV < m_{\\ell\\ell} <\n200\\GeV$ is part of the region for which resonances have been excluded\nby previous experiments, and thus should be dominated by SM processes.  The\nobserved good agreement between the data and the prediction in this\ncontrol region gives confidence that the SM expectations and the detector\nperformance are well understood.\n\nIn the $\\cPZ$~peak mass region defined as $60 < m_{\\ell\\ell} <\n120$\\GeV, the number of dimuon and dielectron candidates are 16\\,515\nand 8\\,768 respectively, with very small backgrounds.  The\ndifference in the electron and muon numbers is due to the higher $E_T$\ncut in the electron analysis and lower electron identification\nefficiencies at these energies.  The expected yields in the control\nregion (120--200\\GeV) and high invariant mass regions ($>$ 200\\GeV) are listed in\nTable~\\ref{tab:event_yield}.  The agreement between the observed data\nand expectations, while not used in the shape-based analysis,\nis good.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=90,width=0.49\\textwidth]{figures\/MuonsPlusMuonsMinus_log.pdf}\n\\includegraphics[angle=90,width=0.49\\textwidth]{figures\/massHist35ForPaper.pdf}\n\\end{center}\n\\caption{\\label{fig:spectra}\nInvariant mass spectrum of $\\Pgmp\\Pgmm$ (left) and $\\Pe\\Pe$ (right) events. The points with\nerror bars represent the data. The uncertainties on the data points (statistical only) \nrepresent 68\\%  confidence intervals for the Poisson means. The\nfilled histograms represent the expectations from\nSM processes: $\\cPZ{\/}\\gamma^*$, $\\ttbar$, other sources of \nprompt leptons ($\\tq\\PW$, diboson production, $\\cPZ\\to\\Pgt\\Pgt$), and\nthe multi-jet backgrounds.\nThe open histogram  shows the  signal expected for a $\\ZPSSM$ with\na mass of 750\\GeV.\n}\n\\end{figure}\n\n\\begin{figure*}[htbp!]\n\\begin{center}\n\\includegraphics[angle=90,width=0.49\\textwidth]{figures\/MuonsPlusMuonsMinus_cumulative_log.pdf}\n\\includegraphics[angle=90,width=0.49\\textwidth]{figures\/cMassHist35ForPaper.pdf}\n\\end{center}\n\\caption{\\label{fig:cum_spectra} Cumulative distribution of invariant\nmass spectrum of $\\Pgmp\\Pgmm$ (left) and $\\Pe\\Pe$ (right) events.  The\npoints with error bars represent the data, and the filled histogram\nrepresents the expectations from SM processes.  }\n\\end{figure*}\n\n\n\n\\begin{table*}[htb!]\n\\centering\n\\caption{Number of dilepton events with invariant mass in the control\nregion $120~<~m_{\\ell\\ell}~<~200$\\GeV and the search region $m_{\\ell\\ell} > 200$\\GeV.\nThe expected number of $\\cPZpr$ events is given within ranges of 328\\GeV and \n120\\GeV for the dimuon sample and the dielectron sample respectively, centred \non 750\\GeV. The total background is the sum of the SM processes listed.\nThe MC yields are normalized to the expected cross sections. \nUncertainties include both statistical and systematic components\nadded in quadrature.}\n\\label{tab:event_yield}\n\\begin{tabular}{l|c|c|c|c}\n\\hline\\hline\nSource & \\multicolumn{4}{c}{Number of events} \\\\\n       & \\multicolumn{2}{c|}{Dimuon sample }\n       & \\multicolumn{2}{c}{Dielectron sample} \\\\\n &  ($120-200$)\\GeV  & $ >$200\\GeV\n &  ($120-200$)\\GeV  & $ >$200\\GeV  \\\\ \\hline\n CMS data                   &  227             &  35              & 109             & 26            \\\\\n$\\ZPSSM$ (750\\GeV)         & ---              & $15.0 \\pm 1.9$   & ---             & $8.7\\pm1.1$   \\\\\nTotal background           & $204 \\pm 23$     & $36.3 \\pm 4.3$   & $120 \\pm 14$    & $24.4\\pm 3.0$ \\\\ \\hline\n$\\cPZ{\/}\\gamma^*$          & $187 \\pm 23$     & $30.2 \\pm 3.6$   & $104 \\pm 14$    & $18.8\\pm 2.3$   \\\\\n$\\ttbar$                   & $12.3 \\pm 2.3$   & $4.2 \\pm 0.8$    & $7.6 \\pm 1.4$   & $2.7 \\pm 0.5$ \\\\\nOther prompt leptons      & $4.4 \\pm 0.5$    & $1.7 \\pm 0.2$    & $2.1 \\pm 0.2$   & $0.8 \\pm 0.1$  \\\\\nMulti-jet events           & $0.6 \\pm 0.2$    & $0.2 \\pm 0.1$    & $6.5 \\pm 2.6$   & $2.1 \\pm 0.8$  \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\n\n\n\n\n\n\\section{Event Samples and Selection}\n\nSimulated event samples for the signal and associated backgrounds\nwere generated with the \\PYTHIA~{\\sc v6.422}~\\cite{Sjostrand:2006za}\nMC event generator, and with\n\\MADGRAPH~\\cite{MADGRAPH} and\n{\\sc powheg v1.1}~\\cite{Alioli:2008gx, Nason:2004rx, Frixione:2007vw}\ninterfaced with the \\PYTHIA parton-shower generator \nusing the {\\sc CTEQ6L1}~\\cite{Pumplin:2002vw} PDF set.\nThe response of the detector was simulated in detail using\n\\GEANTfour~\\cite{GEANT4}. These samples were further processed\nthrough the trigger emulation and event reconstruction chain of the CMS\nexperiment.\n\nFor both dimuon and dielectron final states, \ntwo isolated same flavour leptons that pass\nthe lepton identification criteria described in Section~\\ref{sec:leptonID} are required.\nThe two charges are\nrequired to have opposite sign in the case of dimuons (for which a charge\nmisassignment implies a large momentum measurement error), but not in the\ncase of dielectrons (for which charge assignment is decoupled from\nthe ECAL-based energy measurement). \nAn opposite-charge requirement for\ndielectrons would lead to a loss of signal efficiency of a few percent.\n\n\nOf the two muons selected, one is required to satisfy the ``tight''\ncriteria. \nThe electron sample requires at least one\nelectron candidate in the barrel because events with both electrons in the endcaps\nwill have a lower signal-to-background ratio.  \nFor both channels, each event is required to have a\nreconstructed vertex with at least four associated tracks, located\nless than 2~cm from the centre of the detector in the direction\ntransverse to the beam and less than 24\\cm in the direction along the\nbeam. This requirement provides protection against cosmic rays.\nAdditional suppression of cosmic ray muons\nis obtained by requiring the\nthree-dimensional opening angle between the two muons to be smaller than $\\pi - 0.02$ radians.\n\n\n\\section{Introduction}\n\nMany models of new physics predict the existence of narrow\nresonances, possibly at the TeV mass scale, that decay to a pair of\ncharged leptons. This Letter describes a search for resonant signals\nthat can be detected by the Compact Muon Solenoid (CMS) detector at\nthe Large Hadron Collider (LHC)~\\cite{lhc} at CERN. \nThe Sequential Standard Model $\\ZPSSM$ with standard-model-like couplings, the\n$\\ZPPSI$ predicted by grand unified theories~\\cite{Leike:1998wr},\nand Kaluza--Klein graviton excitations arising in \nthe Randall-Sundrum (RS) model of extra\ndimensions~\\cite{Randall:1999vf, Randall:1999ee} were used as benchmarks. \nThe RS model has two free parameters: the mass of the first graviton excitation and \nthe coupling $k\/\\overline{M}_{\\rm Pl}$, \nwhere $k$ is the curvature of the extra dimension and \n$\\overline{M}_{\\rm Pl}$ is the reduced effective Planck scale. \nTwo values of the coupling parameter were considered: $k\/\\overline{M}_{\\rm Pl}$~=~0.05 and 0.1.\nFor a resonance mass of 1\\TeV, the widths are 30, 6\\ and 3.5\\ (14)\\GeV \nfor a $\\ZPSSM$, $\\ZPPSI$, and $\\GKK$ with $k\/\\overline{M}_{\\rm Pl}$~=~0.05 (0.1), respectively.\n\nThe results of searches for narrow $\\cPZpr \\rightarrow \\ell^+ \\ell^-$ and\n$\\GKK \\rightarrow \\ell^+ \\ell^-$ resonances in $\\Pp\\Pap$ collisions at the\nTevatron with over\n5\\fbinv of integrated luminosity at centre-of-mass energy\nof 1.96\\TeV have previously\nbeen reported~\\cite{D0_RS,D0_Zp,CDF_RS,CDF_Zp}. \nIndirect constraints have been placed on the mass of the virtual \n$\\cPZpr$ bosons by LEP-II experiments~\\cite{delphi,aleph,opal,l3}\nby examining the cross sections and angular distribution\nof dileptons and hadronic final states in $\\Pep\\Pem$ collisions.\n\nThe results presented in this Letter were obtained from an analysis of\ndata recorded in 2010, corresponding to an integrated luminosity of $40 \\pm\n4$\\pbinv in the dimuon channel, and $35 \\pm 4$\\pbinv in the dielectron\nchannel, obtained from $\\Pp\\Pp$ collisions at a centre-of-mass energy\nof 7\\TeV.  The total integrated luminosity used for the electron\nanalysis is smaller than that for the muon analysis because of the\ntighter quality requirements imposed on the data.\nThe search for resonances is based on a shape analysis of dilepton mass spectra,\nin order to be robust against uncertainties in the absolute background\nlevel.  By examining the dilepton-mass spectrum from below the $\\cPZ$\nresonance to the highest mass events recorded, we obtain\nlimits on the ratio of the production cross section times branching\nfraction for high-mass resonances to that of the $\\cPZ$.  Using\nfurther input describing the dilepton mass dependence on effects of\nparton distribution functions (PDFs) and $k$-factors,\nmass bounds are calculated for specific models. \nIn addition, model-independent limit contours are determined in the\ntwo-parameter $(c_d,c_u)$ plane~\\cite{Carena:2004xs}. Selected\nbenchmark models for $\\cPZpr$ production are illustrated in this plane,\nwhere where $c_u$ and $c_d$ are model-dependent couplings of the\n$\\cPZpr$ to up- and down-type quarks, respectively allowing lower bounds \nto be determined. \n\n\n\\section{The CMS Detector}\n\nThe central feature of the CMS~\\cite{JINST} apparatus is a\nsuperconducting solenoid, of 6~m internal diameter, providing an axial field\nof 3.8~T. Within the field volume are the silicon pixel and strip\ntrackers, the crystal electromagnetic calorimeter (ECAL) and the\nbrass\/scintillator hadron calorimeter (HCAL). The endcap hadronic\ncalorimeters are segmented in the z-direction.  Muons are measured in\ngas-ionization detectors embedded in the steel return yoke.  In\naddition to the barrel and endcap detectors, CMS has extensive forward\ncalorimetry.\n\n\nCMS uses a right-handed coordinate system, with the origin at the\nnominal interaction point, the $x$-axis pointing to the centre of the\nLHC, the $y$-axis pointing up (perpendicular to the LHC plane), and\nthe $z$-axis along the anticlockwise-beam direction. The polar angle,\n$\\theta$, is measured from the positive $z$-axis and the azimuthal\nangle, $\\phi$, is measured in the $x$-$y$ plane.\n\nMuons are measured in the pseudorapidity range $|\\eta|< 2.4$, with\ndetection planes based on one of three technologies: drift tubes in the barrel\nregion, cathode strip chambers in the endcaps, and resistive plate chambers\nin the barrel and part of the endcaps.\nThe inner tracker (silicon pixels and strips) detects charged particles within the pseudorapidity range\n$|\\eta| < 2.5$.\n\nThe electromagnetic calorimeter consists of nearly 76\\,000 lead\ntungstate crystals which provide coverage in pseudorapidity $|\\eta| < 1.479$\nin the barrel region (EB, with crystal size\n$\\Delta\\eta=0.0174$ and $\\Delta\\phi = 0.0174$) and $1.479 < |\\eta| <\n3.0$ in the two endcap regions (EE, with somewhat larger crystals.).\nA preshower detector\nconsisting of two planes of silicon sensors interleaved with a total of\n3$\\,X_0$ of lead is located in front of the EE.\n\nThe first level (L1) of the CMS trigger system, composed of custom\nhardware processors, selects the most interesting events\nusing information from the calorimeters and muon detectors.\nThe High Level Trigger (HLT) processor farm further decreases the\nevent rate employing the full event information, including the\ninner tracker.  The muon selection algorithms in the HLT use information from the\nmuon detectors and the silicon pixel and strip trackers.  The electromagnetic (EM)\nselection algorithms use the energy deposits in the ECAL and HCAL; the\nelectron selection in addition requires tracks matched to clusters. Events\nwith muons or electromagnetic clusters with $\\pt$ above L1 and HLT\nthresholds are recorded.\n\n\n\\section{Electron and Muon Selection \\label{sec:lepton}}\n\n\n\\subsection{Triggers \\label{sec:triggers}}\n\nThe events used in the dimuon channel analysis were collected using a\nsingle-muon trigger. \nThe algorithm requires a muon candidate to be found in the muon detectors by the L1 trigger.\nThe candidate track is then matched to a silicon tracker track, forming an HLT muon.\nThe HLT muon is required to have $\\pt >9$ to 15\\GeV, depending on the running period.  \n\nA double EM cluster trigger was used to select\nthe events for the dielectron channel.  ECAL clusters are formed by\nsumming energy deposits in crystals surrounding a ``seed'' that is\nlocally the highest-energy crystal. \nThe clustering algorithm takes into account the emission of bremsstrahlung.\nThis trigger requires two clusters with the ECAL transverse energy $\\ET$\nabove a threshold of 17 to 22\\GeV, depending on the running period. \nFor each of these clusters, the ratio $H\/E$, where $E$ is the energy of the ECAL cluster and\n $H$ is the energy in the HCAL cells situated behind it, is required to be less than 15\\%.\nAt least one of these clusters must\nhave been associated with an energy deposit identified by the L1 trigger.\n\n\\subsection{Lepton Reconstruction\\label{sec:leptonID}}\n\nThe reconstruction, identification, and calibration of muons\nand electrons\nfollow standard CMS methods~\\cite{EWK-10-002-PAS}.\nCombinations of test beam, cosmic ray muons, and \ndata from proton collisions \nhave been used to calibrate the relevant detector systems\nfor both muons and electrons.\n\n\nMuons are reconstructed independently as tracks in both the muon\ndetectors and the silicon tracker~\\cite{MUONPAS}. \nThe two tracks can be matched and fitted simultaneously to\nform a ``global muon''. Both muons in the event must be identified as\nglobal muons, with at least 10 hits in the silicon tracker and with\n$\\pt > 20$\\GeV.  All muon candidates that satisfy these criteria\nare classified as ``loose'' muons.  At least one of the two muons in\neach event must be further classified as a ``tight'' muon by passing the\nfollowing additional requirements: a transverse impact parameter with\nrespect to the collision point less than 0.2~cm; a $\\chi^2$ per degree\nof freedom less than 10 for the global track fit; at least one hit in\nthe pixel detector; hits from the muon tracking system in at least two\nmuon stations on the track; and correspondence with the single-muon trigger.\n\nElectrons are reconstructed by associating a cluster in the ECAL with\na track in the tracker~\\cite{EGMPAS}.  Track reconstruction, which \nis specific to\nelectrons to account for bremsstrahlung emission,  is seeded from the\nclusters in the ECAL, first using the cluster position and energy to\nsearch for compatible hits in the pixel detector, and then using these\nhits as seeds to reconstruct a track in the silicon tracker. A minimum\nof five hits is required on each track.  Electron candidates are required to be\nwithin the barrel or endcap acceptance regions, with pseudorapidities\nof $|\\eta|<1.442$ and $1.560<|\\eta|<2.5$, respectively. A candidate\nelectron is required to deposit most of its energy in the ECAL and\nrelatively little in the HCAL ($H\/E<5\\%$). The transverse shape of the\nenergy deposit is required to be consistent with that expected for an\nelectron, and the associated track must be well-matched in $\\eta$ and\n$\\phi$. Electron candidates must have $\\ET > 25$~GeV.\n\nIn order to suppress misidentified leptons from jets and non-prompt muons from\nhadron decays, both lepton selections impose isolation requirements.\nCandidate leptons are required to be isolated within a narrow cone of\nradius $\\Delta R = \\sqrt{(\\Delta\\eta)^2 + (\\Delta\\phi)^2} = 0.3$,\ncentred on the lepton. Muon isolation requires that the sum of the\n$\\pt$ of all tracks within the cone, excluding the muon,\nis less than 10\\% of the $\\pt$ of the muon.\nFor electrons, the sum of the $\\pt$ of the tracks, excluding\nthe tracks within an inner cone of $\\Delta R = 0.04$, is required to be less than 7$\\GeV$\nfor candidates reconstructed within the barrel acceptance and 15$\\GeV$\nwithin the endcap acceptance. The calorimeter isolation\nrequirement for electron candidates within the barrel acceptance is that,\nexcluding the $\\ET$ of the candidate, the sum of the $\\ET$ resulting from\ndeposits in the ECAL and the HCAL within a cone of $\\Delta R=0.3$ be less than\n0.03$\\ET$ + 2\\GeV. For candidates within the endcap acceptance, the\nsegmentation of the HCAL in the $z$-direction is exploited. For candidates with $\\ET$\nbelow 50\\GeV (above 50\\GeV), the isolation energy is required to be\nless than 2.5\\GeV ($0.03(\\ET-50) + 2.5\\GeV)$, where $\\ET$ is\ndetermined using the ECAL and the first layer of the segmented HCAL. The\n$\\ET$ in the second layer of the HCAL is required to be less than\n0.5\\GeV.  These requirements ensure that the candidate electrons are\nwell-measured and have minimal contamination from jets.\n\nThe performance of the detector systems for the data sample presented in this paper is \nestablished using measurements of standard model (SM) $\\PW$ and $\\cPZ$ processes \nwith leptonic final states~\\cite{EWK-10-002-PAS} \nand using traversing cosmic ray muons~\\cite{CMS_CFT_09_014}.\n\nMuon momentum resolution varies from 1\\% at momenta\nof a few tens of \\GeV to 10\\% at momenta of several hundred \\GeV,\nas verified with measurements made with cosmic rays.\nThe alignment of the muon and inner tracking systems is important\nfor obtaining the best momentum resolution, and hence mass resolution,\nparticularly at the high masses relevant to the $\\cPZpr$ search.\nAn additional contribution to the momentum\nresolution arises from the presence of distortion modes in the tracker\ngeometry that are not completely constrained by the alignment procedures.\nThe dimuon mass resolution is estimated to have an rms of\n5.8\\% at 500\\GeV and 9.6\\% at 1\\TeV.\n\nThe ECAL has an ultimate energy resolution of better than $0.5\\%$ for\nunconverted photons with transverse energies above $100\\GeV$.\nThe ECAL energy resolution obtained thus far is\non average 1.0\\% for the barrel and 4.0\\% for the endcaps.\nThe mass resolution is estimated to be\n1.3\\% at 500\\GeV and 1.1\\% at 1\\TeV.\nElectrons from $\\PW$ and $\\cPZ$ bosons were used to calibrate ECAL\nenergy measurements.  \nFor both muons and electrons, the energy scale\nis set using the $\\cPZ$ mass peak, except for electrons in the barrel\nsection of the ECAL, where the energy scale is set using neutral pions,\nand then checked using the $\\cPZ$ mass peak.\nThe ECAL energy scale uncertainty is 1\\% in the barrel and 3\\% in the\nendcaps.\n\n\n\n\\subsection{Efficiency Estimation \\label{sec:eff}}\n\nThe efficiency for identifying and reconstructing lepton candidates is\nmeasured with the tag-and-probe method~\\cite{EWK-10-002-PAS}.\nA tag lepton is established by applying tight cuts\nto one lepton candidate; the other candidate is used as a probe. A\nlarge sample of high-purity probes is obtained by requiring that the\ntag-and-probe pair have an invariant mass consistent with the $\\cPZ$\nboson mass ($80 < m_{\\ell\\ell} < 100\\GeV)$.  \nThe factors contributing to the overall efficiency are measured in the data. They are:\nthe trigger efficiency, the reconstruction efficiency in the silicon tracker,\nthe electron clustering efficiency, and the lepton reconstruction and\nidentification efficiency. All efficiencies and scale factors\nquoted below are computed\nusing events in the $\\cPZ$ mass region.\n\nThe trigger efficiencies are defined relative to the full offline\nlepton requirements.  For the dimuon events, the efficiency of the\nsingle muon trigger with respect to loose muons is measured to be\n$89\\% \\pm 2\\%$~\\cite{EWK-10-002-PAS}.  The overall efficiency, defined\nwith respect to particles within the physical acceptance of the\ndetector, for loose (tight) muons is measured to be $94.1\\%\\pm1.0\\%$\n($81.2\\%\\pm1.0\\%$).  Within the statistical precision allowed by the\ncurrent data sample, the dimuon efficiency is constant as a function of\n$\\pt$ above 20\\GeV, as is the ratio of the efficiency in the data to that in the Monte Carlo (MC)\nof 0.977 $\\pm$ 0.004. For dielectron events, the double EM cluster trigger is\n100\\% efficient (99\\% during the early running period).  The total\nelectron identification efficiency is 90.1\\% $\\pm$ 0.5\\%\\ (barrel) and 87.2\\% $\\pm$ 0.9\\%\\ (endcap). \nThe ratio of the electron efficiency measured from the data to that\ndetermined from MC simulation at the $\\cPZ$ resonance is 0.979 $\\pm$ 0.006\\ (EB)\nand 0.993 $\\pm$ 0.011\\ (EE).  To determine the efficiency applicable to\nhigh-energy electrons in the data sample, this correction factor is\napplied to the efficiency found using MC simulation.\nThe efficiency of electron identification increases as a\nfunction of the electron transverse energy until it becomes flat beyond an\n$\\ET$ value of about 45\\GeV. Between 30 and 45 GeV it increases by about 5\\%.\n\n\n\\section{Limits on the Production Cross Section}\n\nThe observed invariant mass spectrum agrees with expectations based on\nstandard model processes, therefore limits are set on the possible\ncontributions from a narrow heavy resonance.\nThe parameter of interest is the ratio of the products of cross sections\nand branching fractions:\n\\begin{equation}\n\\label{eq:rsigma}\nR_\\sigma = \\frac{\\sigma(\\Pp\\Pp\\to \\cPZpr+X\\to\\ell\\ell+X)}\n                {\\sigma(\\Pp\\Pp\\to \\cPZ+X       \\to\\ell\\ell+X)}.\n\\end{equation}\n\nBy focusing on the ratio\n$R_\\sigma$, we eliminate the uncertainty in the integrated\nluminosity, reduce the dependence on experimental\nacceptance, trigger, and offline efficiencies, and generally obtain a more robust result.\n\nFor statistical inference about $R_\\sigma$, we first estimate the Poisson mean\n$\\mu_\\cPZ$ of the number of $\\cPZ\\to\\ell\\ell$ events in the sample\nby counting the number of events in the $\\cPZ$ peak mass region and\ncorrecting for a small ($\\sim 0.4\\%$) background contamination (determined with MC simulation).\nThe uncertainty on $\\mu_\\cPZ$ is about 1\\% (almost all statistical) and contributes negligibly to\nthe uncertainty on $R_\\sigma$.\n\nWe then construct an extended unbinned likelihood function for the\nspectrum of $\\ell\\ell$ invariant mass values $m$ above\n200\\GeV, based on a sum of analytic probability density functions\n(pdfs) for the signal and background shapes.\n\nThe pdf $f_\\mathtt{S}(m|\\Gamma,M,w)$ for the resonance signal is a\nBreit-Wigner of width $\\Gamma$ and mass $M$ convoluted with a Gaussian\nresolution function of width $w$ (section~\\ref{sec:leptonID}). The width $\\Gamma$ is taken to\nbe that of the $\\ZPSSM$ (about 3\\%); as noted below, the high-mass limits\nare insensitive to this width.\nThe Poisson mean of\nthe yield is $\\mu_\\mathtt{S} = R_\\sigma \\cdot \\mu_\\cPZ \\cdot R_\\epsilon$,\nwhere $R_\\epsilon$ is the ratio of selection efficiency times detector\nacceptance for $\\cPZpr$ decay to that of $\\cPZ$ decay; $\\mu_\\mathtt{B}$\ndenotes the Poisson mean of the total background yield.\nA background\npdf $f_\\mathtt{B}$ was chosen and its shape parameters fixed by\nfitting to the simulated Drell--Yan spectrum in the mass range\n$200 < m_{\\ell\\ell} < 2000$\\GeV.\nTwo functional forms for the dependence of $f_\\mathtt{B}$\non shape parameters $\\alpha$ and $\\kappa$ were tried:\n$f_\\mathtt{B}(m|\\alpha,\\kappa) \\sim  \\exp(-\\alpha m^\\kappa)$ and\n$\\sim \\exp(-\\alpha m)m^{-\\kappa}$.  Both\nyielded good  fits and consistent results for both the dimuon and dielectron spectra.\nFor definiteness, this Letter presents results obtained with the latter form.\n\nThe extended likelihood ${\\cal L}$ is then\n\n\\begin{equation}\n\\label{eq:likelihood}\n{\\cal L}({\\boldsymbol m}|R_\\sigma,M,\\Gamma,w,\\alpha,\\kappa,\\mu_\\mathtt{B}) =\n\\frac{\\mu^N e^{-\\mu}}{N!}\\prod_{i=1}^{N}\\left(\n\\frac{\\mu_\\mathtt{S}(R_\\sigma)}{\\mu}f_\\mathtt{S}(m_i|M,\\Gamma,w)+\n\\frac{\\mu_\\mathtt{B}}{\\mu}f_\\mathtt{B}(m_i|\\alpha,\\kappa)\n\\right),\n\\end{equation}\n\nwhere  ${\\boldsymbol m}$ denotes the dataset in which\nthe observables are the invariant mass values of the lepton pairs,\n$m_i$; $N$ denotes the total number of events observed above 200\\GeV;\nand $\\mu=\\mu_\\mathtt{S}+ \\mu_\\mathtt{B}$ is\nthe mean of the Poisson distribution from which $N$ is an observation.\n\nStarting from Eqn.~\\ref{eq:likelihood}, confidence\/credible intervals\nare computed using more than one approach, both frequentist (using\nprofile likelihood ratios) and Bayesian (multiplying ${\\cal L}$ by prior\npdfs including a uniform prior for the signal mean).\nWith no candidate events in the region of small expected background\nabove 465\\GeV, the result is insensitive to\nthe statistical technique, and also with respect to the width of the $\\cPZpr$\nand to changes in systematic uncertainties and their functional forms,\ntaken to be log-normal distributions with fractional uncertainties.\n\nFor $R_\\epsilon$, we assign an uncertainty of 8\\% for the dielectron\nchannel and 3\\% for the dimuon channel. These values reflect our current\nunderstanding of the detector acceptance and reconstruction efficiency\nturn-on at low mass (including PDF uncertainties\nand mass-dependence of $k$-factors), as well as the corresponding values\nat high mass, where\ncosmic ray muons are available to study muon performance but not electron\nperformance.  The uncertainty in the mass scale affects only the mass\nregion below 500\\GeV where there are events in both channels\nextrapolating from the well-calibrated observed resonances.\nFor the dielectron channel, it is set to 1\\% based on linearity studies.\nFor the dimuon channel, it is set to zero, as a sensitivity study showed\nnegligible change in the results up to the maximum misalignment consistent\nwith alignment studies (corresponding to several percent change in momentum scale).\nThe acceptance for $\\GKK$ (spin 2) is higher than for $\\cPZpr$ (spin 1) by\nless than 8\\% over the mass range 0.75--1.1 TeV. This was\nconservatively neglected when calculating the limits.\n\nIn the frequentist calculation, the mean background level\n$\\mu_\\mathtt{B}$\nis the maximum likelihood estimate; in the fully\n Bayesian\ncalculation a prior must be assigned to the mean background\n level,\nbut the result is insensitive to reasonable choices (i.e., for which\nthe likelihood dominates the prior).\n\n\n\nThe upper limits on $R_{\\sigma}$ (Eqn.~\\ref{eq:rsigma}) from the various approaches\nare similar, and we report the Bayesian\nresult (implemented with Markov Chain Monte Carlo in\n{\\sc RooStats}~\\cite{MCMC}) for definiteness.\nFrom the dimuon and dielectron data, we obtain the upper limits on the cross section\nratio $R_{\\sigma}$ at 95\\% confidence level (C.L.) shown in\nFigs.~\\ref{fig:limits}(upper) and (middle), respectively.\n\nIn Fig.~\\ref{fig:limits}, the predicted cross section ratios\nfor $\\ZPSSM$ and $\\ZPPSI$  production are superimposed\ntogether with those\nfor $\\GKK$ production with dimensionless graviton coupling\nto SM fields $k\/\\overline{M}_\\mathrm{Pl}=0.05$ and $0.1$.\nThe leading order cross section predictions for\n$\\ZPSSM$ and $\\ZPPSI$\nfrom \\PYTHIA using {\\sc CTEQ6.1} PDFs are corrected for a mass dependent\n$k$-factor obtained using {\\textsc ZWPRODP}~\\cite{Accomando:2010fz,Hamberg:1990np,\nvanNeerven:1991gh,ZWPROD} to account for NNLO contributions.\nFor the RS graviton model, a constant NLO $k$-factor of 1.6 is\nused~\\cite{Mathews:2005bw}. The uncertainties\ndue to the QCD scale parameter and PDFs are indicated as a band.\nThe NNLO prediction for the $\\cPZ$ production cross\nsection is 0.97$\\pm\\,$0.04~nb~\\cite{Melnikov:2006kv}.\n\n\nPropagating the above-mentioned uncertainties into the\ncomparison of the experimental limits with the predicted cross section\nratios, we exclude\nat 95\\%~C.L. $\\cPZpr$ masses as follows.  From the dimuon only\nanalysis,\nthe $\\ZPSSM$ can be excluded below 1027\\GeV, the\n$\\ZPPSI$ below 792\\GeV, and the\nRS $\\GKK$ below 778 (987)\\GeV for couplings\nof 0.05 (0.1). For the dielectron analysis, the\nproduction of $\\ZPSSM$ and $\\ZPPSI$ bosons is\nexcluded for masses below 958 and 731\\GeV, respectively. The\ncorresponding lower limits on the mass for\nRS $\\GKK$ with\ncouplings of 0.05 (0.10) are 729 (931)\\GeV.\n\n\\subsection{Combined Limits on the Production Cross Section Using\nDimuon and Dielectron Events}\n\nThe above statistical formalism is generalized to combine the results from the\ndimuon and dielectron channels,\nby defining the combined likelihood as the product\nof the likelihoods for the individual channels with $R_\\sigma$ forced to\nbe the same value for both channels. The combined limit is shown in\nFig.~\\ref{fig:limits}~(bottom).\n\nBy combining the two channels, the following\n 95\\% C.L. lower limits on the mass of a $\\cPZpr$ resonance are obtained:\n1140\\GeV for the $\\ZPSSM$, and 887\\GeV for $\\ZPPSI$ models. RS\nKaluza--Klein gravitons are excluded below 855 (1079)\\GeV\n for values of couplings 0.05 (0.10).\nOur observed limits are more restrictive than or comparable to\nthose previously obtained via similar direct searches by the\nTevatron experiments~\\cite{D0_RS,D0_Zp,CDF_RS,CDF_Zp},\nor indirect searches by LEP-II experiments~\\cite{delphi,aleph,opal,l3},\nwith the exception of $\\ZPSSM$, where the value from LEP-II\nis the most restrictive.\n\n\n\nThe distortion of the observed limits at $\\sim$400\\GeV visible in\nFig.~\\ref{fig:limits} is the result of a clustering of several dimuon\nand dielectron events around this mass.  We have tested for the\nstatistical significance of these excesses ($p$-values expressed as\nequivalent $Z$-values, i.e. effective number of Gaussian sigma in a\none-sided test), using the techniques described in \\cite{PTDR2}.\nFor the dimuon sample, the probability of an enhancement at\nleast as large as that at 400 GeV occurring anywhere above 200 GeV in\nthe observed sample size corresponds to $Z<0.2$; for the electron\nsample, it is less. For the combined data sample, the corresponding\nprobability in a joint peak search is equivalent to $Z=1.1$.\n\n\\begin{figure}[htbp!]\n\\begin{center}\n\\includegraphics[width=0.65\\textwidth,angle=0]{figures\/zpr_ssm_ratio_mcmc_mumu_40pb_a.pdf}\n\\includegraphics[width=0.65\\textwidth,angle=0]{figures\/zpr_ssm_ratio_mcmc_ee_35pb_b.pdf}\n\\includegraphics[width=0.65\\textwidth,angle=0]{figures\/zpr_ssm_ratio_mcmc_comb_40pb_c.pdf}\n\\end{center}\n\\caption{\\label{fig:limits} Upper limits as a function\nof resonance mass $M$,\non the production ratio $R_{\\sigma}$ of\n cross section times branching fraction into lepton pairs\n  for $\\ZPSSM$ and $\\GKK$ production and $\\ZPPSI$\n boson production. The limits are shown from (top) the $\\Pgmp\\Pgmm$ final\n state, (middle) the $\\Pe\\Pe$ final state and (bottom) the combined dilepton result.\nShaded yellow and red  bands correspond to the $68\\%$ and  $95\\%$ quantiles \nfor the expected  limits.\nThe predicted cross section ratios are shown as bands, with widths\nindicating the theoretical uncertainties. \n}\n\n\n\\end{figure}\n\nIn the narrow-width approximation, the cross section for the process\n$\\Pp\\Pp\\to \\cPZpr+X\\to\\ell\\ell+X$ can be\nexpressed~\\cite{Carena:2004xs,Accomando:2010fz} in terms of the\nquantity $c_u w_u + c_d w_d$, where $c_u$ and $c_d$ contain the\ninformation from the model-dependent $\\cPZpr$ couplings to fermions\nin the annihilation of charge 2\/3 and charge $-$1\/3 quarks,\nrespectively, and where $w_u$ and $w_d$ contain the information about\nPDFs for the respective annihilation at a given $\\cPZpr$ mass.\n\n\nThe\ntranslation of the experimental limits into the ($c_u$,$c_d$) plane has\nbeen studied in the context of both the narrow-width and finite width\napproximations. The procedures have been shown to give the same\nresults. In Fig.~\\ref{fig:CuCd} the limits on the $\\cPZpr$ mass are\nshown as lines in the $(c_d,c_u)$ plane intersected by curves from\nvarious models which specify $(c_d,c_u)$ as a function of a model\nmixing parameter.\nIn this plane, the thin solid lines labeled by mass are\niso-contours of cross section with constant\n$c_u + (w_d\/w_u)c_d$, where $w_d\/w_u$ is in the range 0.5--0.6 for the\nresults relevant here.  As this linear combination\nincreases or decreases by an order of magnitude, the mass limits\nchange by roughly 500 GeV.\nThe point labeled SM corresponds to the\n$\\ZPSSM$; it lies on the more general curve for the\nGeneralized Sequential Standard Model (GSM) for which the generators\nof the $U(1)_{T_{3L}}$ and $U(1)_Q$ gauge groups are mixed with a mixing\nangle $\\alpha$. Then $\\alpha = -0.072\\pi$ corresponds to the\n$Z^\\prime_{SSM}$ and $\\alpha=0$ and $\\pi\/2$ define the $T_{3L}$ and\n$Q$ benchmarks, respectively, which have larger values of $(c_d,c_u)$\nand hence larger lower bounds on the masses.  Also shown are contours\nfor the E$_6$ model (with $\\chi$, $\\psi$, $\\eta$, $S$, and $N$\ncorresponding to angles 0, 0.5$\\pi$, $-0.29\\pi$, 0.13$\\pi$, and\n0.42$\\pi$, respectively) and Generalized LR models (with $R$, $B-L$,\n$LR$, and $Y$ corresponding to angles 0, 0.5$\\pi$, $-0.13\\pi$, and\n0.25$\\pi$, respectively)~\\cite{Accomando:2010fz} .\n\n\n\\begin{figure}[htbp!]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{figures\/plot_lhc_ee_mm_35_40_all_ALL_cms.pdf}\n\\end{center}\n\\caption{\\label{fig:CuCd}\n95\\% C.L. lower limits on the $\\cPZpr$ mass, represented by the thin continuous lines\nin the $(c_d,c_u)$\nplane.  Curves for three classes of model are shown.  Colours on the\ncurves correspond to different mixing angles of the generators defined\nin each model. For any point on a curve, the mass limit corresponding\nto that value of $(c_d,c_u)$ is given by the intersected contour.}\n\\end{figure}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\subsection{Eulerian summation of coprimes}\n\nLet be $r \\in \\mathbb{N}$, $\\mathbf{i} \\in \\mathbb{N}^r$ and $n \\in \\mathbb{N}$.  We introduce the sum $\\sum_{\\substack{  p_1 +  p_2 + ... +  p_r =n \\\\   p_1 \\wedge p_2 \\wedge ... \\wedge p_r= 1 }} p_1^{i_1} p_2^{i_2}... p_r^{i_r}$ with the aim to asymptotically estimate it. We denote it $S_n(\\mathbf{i})$, and $j = \\mathbf{1} \\cdot \\mathbf{i} $. Note that $S_n(i)$ is the sum of a power function over the primitive points contained in the surface of the dilates of the $(r-1)$-dimensional standard simplex. It is noteworthy that the number of such points has been computed in \\cite{DezaPournin:primitivepoint} and that more generally a coprime Ehrhart theory is introduced in \\cite{Sanyal:erhart}.   \\\\\n\n\\begin{prop}\\label{eulerian_sum_notation} \n\n\\begin{align}\n    \\sum_{k \\geq 0} k^j \\sum_{n \\geq 1} S_n(\\mathbf{i}) y^{nk} =  \\frac { \\prod_{\\lambda = 1}^{r}  A_{i_\\lambda}(y)}{(1-y) ^{j+r}} .\n\\end{align} \n\\end{prop}\n\n\\begin{proof}\n\nThis relation relies on the partition $\\mathbb{Z}_+^r =\\bigcup_{k \\geq 0} \\left\\{ kp, p \\in \\mathbb{P}^r_+ \\right\\}$. We write\n\\\\\n\n\\begin{align*}\n    \\sum_{k \\geq 0 } k^{j} \\sum_{n \\geq 1} S_n(\\mathbf{i}) \\left(y^n\\right)^k &= \\sum_{k \\geq 0} \\sum_{n \\geq 1} \\left(\\sum_{\\substack{  p_1 +  ... +  p_r =n \\\\   p_1 \\wedge ... \\wedge p_r= 1 }}(k p_1)^{i_1}y^{kp_1} ... (k p_{r})^{i_{r}}y^{kp_{r}} \\right) \\\\\n&= \\sum_{l \\geq 1} \\left( \\sum_{\\substack{ l_1 +  l_2 + ... +  l_r =l\\\\   l_p \\geq 1 }}l_1^{i_1}y^{l_1} (l_2)^{i_2}y^{l_2}... (l_r)^{i_r}y^{l_r}  \\right) &= \\prod_{p = 1}^{r} \\left( \\sum_{l_p \\geq 1} l_p^{i_p} y^{l_p} \\right)  = \\frac { \\prod_{p = 1}^{r}  A_{i_p}(y)}{(1-y) ^{j+r}}  .\n\\end{align*}\n\n\\end{proof}\n\nThe $n^{th}$ Eulerian polynomial is of degree $n$ except for $A_0(y) = y$ and has no constant term which causes the degree of $\\prod_{\\lambda = 1}^{r}  A_{i_\\lambda}(y)$ to be no more than $j+r$. We write it as follows: \n\n$$\\prod_{\\lambda = 1}^{r}  A_{i_\\lambda}(y) = B_{\\mathbf{i}}(y) = b_{\\mathbf{i},j+r} y^{j+r} + ... b_{\\mathbf{i}, 1} y. $$\n\nDespite this heavy notation, the sums under study in the zonotopal case are usually $S_n(\\mathbf{0})$, $S_n(1, 0,...,0)$ or $S_n(2, 0,...,0)$, making $B_{\\mathbf{i}}(y)$ equal to $y^{d}$, $y^{d+1}$, or $y^{d+2}+ y^{d+1}$.\n\n\\subsection{Mellin transform}\n\nMellin transforms appear naturally in the asymptotic analysis of infinite products (see \\cite{Bodini:polyomino,Flajolet:AC,Gittenberger:hadmi}). In the following, the analysis of zonotopes will require to apply a Mellin transform on sums of coprime numbers. The following proposition is of primary importance in the sequel:\n\\begin{prop}\\label{property2}\nFor $t> 0$, the Mellin transform of $\\sum_{n \\geq 1} S_n(\\mathbf{i}) e^{-n t}$ is \n \\begin{align}\n     \\frac{1}{\\zeta(s-j)} \\left( \\sum_{n = 0}^{j + r } b_{\\mathbf{i},n}  \\sum_{k \\geq 1} \\frac{\\binom{ k  + j + r - n - 1}{j + r -1 }}{k^s} \\Gamma(s)\\right) .\n \\end{align} \n \\end{prop}\n\n\\begin{proof}\n\nAs we do throughout the paper, we use the change of variables $y = e^{-t}$ and will always try to write products as sums, to have an expression compatible with the Mellin transform. This leads to:\n\n\\begin{align}\\label{sum_expression}\n \\sum_{k \\geq 0 } k^{j} \\sum_{n \\geq 1} S_n(\\mathbf{i}) \\left(e^{-n t}\\right)^k = \\prod_{\\lambda = 1}^{r}  A_{i_\\lambda}(e^{-t})  \\left( \\sum_{k\\geq 0}  \\binom{ k + j + r - 1}{j + r -1 } e^{-kt} \\right) = \\sum_{n = 0}^{j + r} b_{\\mathbf{i},n}  \\:\\: \\sum_{k\\geq 1}  \\binom{ k + j + r -n - 1}{j + r -1 } e^{-k t} .\n\\end{align}\n\nWe can now make use of the Mellin transform which is the transform that matches the exponential with the Gamma function:  \n\n$$\\forall s, \\Re(s) > 0, \\:\\:\\: \\Gamma(s) = \\int_{0}^{+ \\infty} e^{-t} t^{s - 1} dt.$$\n\nUnder conditions that will be detailed just below, the Mellin transform of a function $f$ the expression is defined as\n\n$$ \\mathcal{M}\\left[f\\left(e^{-t}\\right)\\right](s) = \\int_0^{+\\infty} f\\left(e^{-t}\\right)t^{s-1}dt.$$\n\nFor $\\Re(s) > j+r -1$, the term $ \\left|\\sum_{n\\geq0} \\frac{\\binom{ k  + j + r - n - 1}{j + r -1 }}{k^s}\\right|  $ is bounded, which makes the use of Fubini's theorem possible, so the Mellin function of $\\sum_{n\\geq1}S_(i)e^{-nt}$ can be written, on the one hand:\n\n$$ \\int_{0}^{+\\infty} \\sum_{n = 0}^{j + r } b_{\\mathbf{i},n}  \\:\\: \\sum_{k\\geq 1}  \\binom{ k + j + r -n - 1}{j + r -1 } e^{-k t} t^{s -1} d t =\\sum_{n = 0}^{j + r } b_{\\mathbf{i},n}  \\sum_{k \\geq 1} \\frac{\\binom{ k  + j + r - n - 1}{j + r -1 }}{k^s} \\Gamma(s),$$\n\nand on the other hand:\n\n$$\\int_0^{+\\infty} \\sum_{k \\geq 0 } k^{j} \\sum_{n \\geq 1} S_n(\\mathbf{i}) e^{- n k t} t^{s-1} d t= \\zeta(s-j) \\int_0^{+\\infty} \\sum_{n \\geq 1} S_n(\\mathbf{i}) e^{- n t} t^{s-1} dt. $$\n\nThese holomorphic functions being defined and equal on $\\Re(s) > j+r-1$, we can extend it to its meromorphic continuation on $\\mathbb{C}$.\n\\end{proof}\n\nThe Mellin transform is widely used to estimate asymptotic behaviors using its correspondence with the poles of the transformed function, as we will do in next subsection (and as it is done in \\cite{Bodini:polyomino} and \\cite{Bureaux:polygons}).  Property \\ref{property2} make it possible to compute the asymptotic equivalent of any of the sums $\\sum_{n \\geq 1} S_n(\\mathbf{i}) y^n $ near $y =1$. \n\n\n\n\\subsection{Riemann's non trivial zeros challenge, with the example \\texorpdfstring{$\\sum_{ \\protect\\substack{ p < n \\\\  p \\wedge n = 1 }}\\protect p^2$}{TEXT} }\n\nWe use this subsection to detail more specifically the way the Mellin inversion formula is handled, as we will encounter it multiple times. Riemann's $\\zeta$ function intervenes in it, and its non-trivial zeros too. We will use the integral of \\cite{Bureaux:polygons} to manage all the zeros in the critical strip with an integral, but we could also write a sum over all non-trivial zeros. This term is numerically discussed in the Section \\ref{numerical_discu}. \n\n\n\n\\begin{propos}\nWe denote the contour  $\\gamma = \\gamma_{\\text{right}}\\cup \\gamma_{\\text{left}}$, with $\\gamma_\\text{right} = 3 - \\frac{A}{\\log(2 + |t|)} +it$ and $\\gamma_\\text{left} = 2 + \\frac{A}{\\log(2 + |t|)}+it $ as $t$ runs from $-\\infty$ to $+\\infty$, with $A >0$ such that $\\gamma$ surrounds all the zeros in the critical strip of $\\zeta(s-2)$. \\\\\n\nHereafter we will denote $$I_\\text{crit} = \\frac{1}{2i\\pi} \\int_\\gamma  \\mathcal{M}[\\sum_{n \\geq 1 } S_n (2, 0) e^{-  n t}](s) t^{-s} d s$$,\\\\\n\nand $$I_\\text{err} =\\frac{1}{2i\\pi} \\int_{\\frac{1}{2} - i\\infty}^{\\frac{1}{2}+i\\infty}  \\mathcal{M}[\\sum_{n \\geq 1 } S_n (2, 0) e^{-  n t}](s) t^{-s} d s$$.\\\\\n\nWe have for all $t>0$:\n\n\\begin{align}\n     \\sum_{n \\geq 1 } S_n (2, 0) e^{-  n t} = \\frac{2}{\\zeta(2) t^{4}} + I_\\text{crit}(t) + \\frac{1}{3 t^2} - \\frac{1}{90 \\zeta'(-1)}\\log(t)  + I_\\text{err}(t).\n\\end{align}\n\n\nWhen $t$ goes to 0,  $I_\\text{err}(t) = O(t^{1\/2})$, $I'_\\text{err}(t) = o(t^{-1})$  and the $k$-th derivative of $I_\\text{crit}$ is $o(t^{-k-1})$.\n\n\\end{propos}\n\n\\begin{proof}\nWe take the general set of sums  $\\left(\\sum_{\\substack{ p < n \\\\  p \\wedge n = 1 }} p^i \\right)_{i \\geq 0}$.\\\\\nAs $p \\wedge n = 1$ being equivalent to $p \\wedge (n-p) = 1$, those sums can be rewritten as $\\sum_{ \\substack{ p +q  = n \\\\  p \\wedge q = 1 }} p^i = S_n(i,0)$.\\\\\nWe fix $i = 2$. Applying properties \\ref{eulerian_sum_notation}  and \\ref{property2}, we have:\n\n\\begin{align*}\n    \\sum_{k \\geq 0 } k^{2} \\sum_{n \\geq 1 } S_n (2, 0) e^{- k n t} &= \\frac{A_0(e^{-t})  A_2 (e^{-t})}{(1-e^{-t})^{4}}\\\\\n    &= (e^{-3 t} + e^{-2t}) \\sum_{k \\geq 0} \\binom{k+3}{3} e^{-k t}. \n\\end{align*}\n\n\nFor $t >0, s >1 $, we have the following Mellin transform: \n\n\\begin{align*}\n    \\mathcal{M} \\left[ \\sum_{n \\geq 1 } S_n (2, 0) e^{-  n t}\\right] (s) &= \\frac{1}{\\zeta(s-2)}\\left(  \\sum_{k \\geq 1} \\frac{\\binom{k+1}{3} + \\binom{ k }{3}}{k^s}  \\Gamma(s)\\right)\\\\\n    &= \\frac{2 \\zeta(s-3) - 3 \\zeta(s-2) + \\zeta(s-1)}{6 \\zeta(s-2)}\\Gamma(s) .\n\\end{align*}\n\n\nTherefore we have, using  the Mellin inversion formula, for all $c > 4$ (to ensure $\\sup_{s\\in c+ i \\mathbb{R}} \\zeta(s-3) < \\infty$):\n \n $$ \\sum_{n \\geq 1 } S_n (2, 0) e^{-  n t} = \\frac{1}{2i\\pi} \\int_{c- i\\infty}^{c+ i\\infty} \\frac{2 \\zeta(s-3) - 3 \\zeta(s-2) + \\zeta(s-1)}{6 \\zeta(s-2)}\\Gamma(s) t^{-s} d s. $$\n\nWe use the residue theorem to shift the line of integration to the left as $s= 4$ is a location of a pole of the Mellin transform. It follows that, for all $c_1\\in ]3, 4[$:\n\n$$ \\sum_{n \\geq 1 } S_n (2, 0) e^{-  n t} = \\frac{2}{\\zeta(2) t^{-4}} + \\frac{1}{2i\\pi} \\int_{c_1- i\\infty}^{c_1+ i\\infty} \\frac{2 \\zeta(s-3) - 3 \\zeta(s-2) + \\zeta(s-1)}{6 \\zeta(s-2)}\\Gamma(s) t^{-s} d s.$$\n\nThe remaining poles of the Mellin transform are the double pole at $s = 0$, and simple ones at $s=0$ and at the non-trivial zeros of $\\zeta(s-2)$,  which are in the critic strip $2<\\Re(z)<3$, and in the left side of the complex domain. \\\\\n\nNow consider the contour $\\gamma$ as described in the proposition. The existence of $A$, such that $\\gamma$ surrounds all the zeros of $\\zeta(s-2)$ in the critical strip is a consequence of Theorem 3.8 in \\cite{Titchmarsh:riemann}.\n\n\\begin{align}\\label{riemann_crit_strip}\n    I_\\text{crit}(t) = \\frac{1}{2i\\pi} \\int_\\gamma  \\frac{2 \\zeta(s-3) - 3 \\zeta(s-2) + \\zeta(s-1)}{6 \\zeta(s-2)}\\Gamma(s) t^{-s} d s.\n\\end{align}\n\nFor any $c_2 \\in ]0,2[$, we have:\n\n$$ \\sum_{n \\geq 1 } S_n (2, 0) e^{-  n t} = \\frac{2}{\\zeta(2) t^{-4}} + I_\\text{crit}(t) + \\frac{1}{3 t^2} +  \\frac{1}{2i\\pi} \\int_{c_2- i\\infty}^{c_2+ i\\infty} \\frac{2 \\zeta(s-3) - 3 \\zeta(s-2) + \\zeta(s-1)}{6 \\zeta(s-2)}\\Gamma(s) t^{-s} d s. $$\n\nFinally denoting $I_\\text{err} (t) = \\frac{1}{2i \\pi}\\int_{\\frac{1}{2} - i\\infty}^{\\frac{1}{2}+i\\infty} \\frac{2 \\zeta(s-3) - 3 \\zeta(s-2) + \\zeta(s-1)}{6 \\zeta(s-2)}\\Gamma(s) t^{-s} d s$, we have for $t>0$\n\n$$ \\sum_{n \\geq 1 } S_n (2, 0) e^{-  n t} = \\frac{2}{\\zeta(2) t^{4}} + I_\\text{crit}(t) + \\frac{1}{3 t^2} - \\frac{1}{90 \\zeta'(-1)}\\log(t)  + I_\\text{err}(t). $$\n\nThe bounds over the order of $I_\\text{crit}$ and $I_\\text{err}$ are coming from Lemma 2.2 from \\cite{Bureaux:polygons}.\n\\end{proof}\n\n\n\n\n\\subsection{Intermediate sequence of polynomials}\\label{polynome_zono}\n\nAs we make use of the Mellin inversion formula over the logarithm of the generating function, for each $\\delta$ between 1 and $d$, the coefficient $\\binom{k-1}{\\delta-1}$ from Property \\ref{property2} will appear with $S_n(\\underbrace{0,...,0}_{\\delta \\text{ times}})$. In the rest of the paper (as well as in Theorem \\ref{theorem} ), we denote:\n\n$$\\sum_{\\delta = 1}^d   \\binom{d}{\\delta} 2^{\\delta-1} \\binom{{k} - 1}{\\delta-1} = P_d({k}) = p_{d,d-1} {k}^{d-1} + ... + p_{d,1} {k} + p_{d,0} .$$\n\nThe first terms of $\\left( P_d({X})\\right)_{d \\geq 2}$ are: \\\\\n\\begin{align*}\n        P_1({X}) &= 1 \\\\\n        P_2({X}) &= 2 {X}\\\\\n        P_3({X}) &= 2 {X}^2 + 1    \n\\end{align*}\n\nThese polynomials verify the relation $P_{d+2}({X})=  \\frac{2 {X}}{d+1}P_{d+1}({X}) + P_d({X} )$ for $d \\geq 1$. \\\\\n\nTwo properties naturally follow:\n\\begin{itemize}\n    \\item The odd polynomials only contain even powers of $i$ and have a constant term equal to $1$. Similarly, the even polynomials only contain odd powers of $i$. \n        \\item The leading term of $P_d$ is of degree $d-1$ and its coefficient is $\\frac{2^{d-1}}{(d-1)!}$.\n\\end{itemize}\n\nAs each exponent $\\delta$ of $k$ is reflected in the Riemann $\\zeta$ function as $\\zeta(s-\\delta)$ when using Mellin transform, we introduce an operator over complex functions:\n\n\\begin{defn}\\label{definition_operator}\nLet $\\mathrm{M}(\\mathbb{C})$ be the field of meromorphic functions in $\\mathbb{C}$. We define the operator $\\mathbf{\\Pi}_d$ as\n\\begin{align*}\n   \\mathbf{\\Pi}_d \\colon \\mathrm{M}(\\mathbb{C}) &\\to \\mathrm{M}(\\mathbb{C})\\\\\n  \\phi &\\mapsto p_{d,d-1} \\phi(\\cdot \\: - (d-1)) + ... + p_{d,1} \\phi(\\cdot \\: - 1) + p_{d,0} \\phi.\n\\end{align*}\n\\end{defn}\n\nIt is important to note that we can write, for $s\\neq 1$, $\\mathbf{\\Pi}_d (\\zeta)(s) = \\sum_{k \\geq 1} \\frac{P_d(k)}{k^s}$.\n\n\n\\subsection{Asymptotic analysis of the generating function}\n\n\\begin{lem}\\label{lemma_equivalent}\nLet $A$ be a positive number such that $\\gamma = \\gamma_{\\text{right}}\\cup \\gamma_{\\text{left}}$, with $\\gamma_\\text{right} = 1 - \\frac{A}{\\log(2 + |\\theta|)} +it$ and $\\gamma_\\text{left} = \\frac{A}{\\log(2 + |\\theta|)}+it $ as $\\theta$ goes from $-\\infty$ to $+\\infty$, surrounds all the zeros of $\\zeta$ in the critical strip.\\\\\n\nFor $\\theta>0$, as $\\theta \\rightarrow 0$, we have: \n\\begin{align}\n     Zon \\left(e^{-\\theta} \\right) = \\theta^{2 \\mathbf{\\Pi}_d \\left(\\zeta\\right)(0) } e^{2 \\mathbf{\\Pi}_d \\left(\\log(2\\pi) \\zeta  -  \\zeta' \\right)(0) } \\exp\\left( \\sum_{\\delta=1}^{d-1}\\left(p_{d,\\delta}\\frac{\\zeta(\\delta+2) \\Gamma(\\delta+1)}{\\zeta(\\delta+1) \\theta^{\\delta+1}} \\right) + I_{\\text{crit}}(d, \\theta) + I_\\text{err}(d, \\theta)    \\right)    , \n\\end{align}\n\nwith $$I_\\text{crit}(d,\\theta) = \\frac{1}{2i\\pi}\\int_{\\gamma}\\frac{\\mathbf{\\Pi}_d (\\zeta)(0) }{\\zeta(s)}\\zeta(s+1)\\Gamma(s) \\theta^{-s} \\delta s $$,\\\\\n\nand $$I_\\text{err}(d,\\theta) = \\frac{1}{2i\\pi}\\int_{-\\frac{1}{2} - i\\infty}^{- \\frac{1}{2}+ i \\infty}\\frac{\\mathbf{\\Pi}_d (\\zeta)(0) }{\\zeta(s)}\\zeta(s+1)\\Gamma(s) \\theta^{-s} \\delta s =O(\\theta^{1\/2})$$.  \\\\\n\n\\end{lem}\n\n\n\\begin{proof}\nWe apply the logarithm to the generating function with the change of variables $x = e^{-\\theta}$, which gives:\\\\\n\\begin{align*}\n    \\log\\left(Zon\\left(e^{-\\theta}\\right)\\right) &=  - \\sum_{\\delta=1}^d \\left(\\binom{d}{\\delta}2^{\\delta-1} \\sum_{\\mathbf{v} \\in \\mathbb{P}^{\\delta}_+} \\log\\left( 1 - e^{- \\mathbf{1}\\cdot \\mathbf{v} \\theta} \\right)\\right)\\\\\n    &=- d \\log(1 - e^{-\\theta}) - \\sum_{\\delta = 2}^d  \\left(  \\binom{d}{\\delta} 2^{\\delta-1} \\sum_{n= 2}^{+\\infty}  \\left( \\sum_{\\substack{ n_{1}+...+ n_{\\delta} = n \\\\  n_{1} \\wedge ... \\wedge n_{\\delta}  = 1   }} 1 \\right)  \\log \\left( 1 - e^{- n \\theta} \\right) \\right).    \n\\end{align*}\n\n\nThe sum $\\sum_{\\substack{ n_{1}+...+ n_{\\delta} = n \\\\  n_{1} \\wedge ... \\wedge n_{\\delta}  = 1   }} 1 $ is of the form of the sums studied in section \\ref{section_eulerset} as being $S_n(\\overbrace{0,...,0}^{\\delta \\text{ times}})$. In a geometrical point of view, it is exactly the number of primitive points . Therefore we compute the Mellin transform accordingly (using logarithmic series), for all $s > d$\\\\\n\n$$ \\mathcal{M}\\left[\\log\\left(Zon\\left(e^{-\\theta}\\right)\\right)\\right](s) = \\sum_{\\delta = 1}^d   \\binom{d}{\\delta} 2^{\\delta-1} \\left(\\sum_{k=1}^{\\infty} \\frac{\\binom{k-1}{\\delta-1}}{k^s} \\right) \\frac{ \\zeta(s+1) \\Gamma(s) }{\\zeta(s)}.$$\n\nWith the notation introduced in definition \\ref{definition_operator}, the transform simplifies to the following form:\\\\\n\\begin{propos} for every $d>1$, we have:\n\\begin{align}\n    \\mathcal{M}\\left[\\log \\left(Zon \\left(e^{-\\theta}\\right)\\right)\\right] (s) =\\frac{\\mathbf{\\Pi}_d (\\zeta(s)) }{\\zeta(s)}\\zeta(s+1)\\Gamma(s).\n\\end{align}\n\\end{propos}\n\n\n\\leavevmode\\\\\n\nWe can extend this function into a meromorphic one on $\\mathbb{C}$, and use successive residue theorems to scan from the right to the left each pole of the Mellin transform, as we did in the previous section. We start with $c> d$ and\\\\\n\n$$ \\log(Zon(e^{-\\theta})) = \\frac{1}{2i\\pi} \\int_{c - i \\infty}^{c + i \\infty} \\frac{p_{d,d-1} \\zeta(s - d +1) + ... + p_{d,1} \\zeta(s - 1) + p_{d,0} \\zeta(s) }{\\zeta(s)}\\zeta(s+1)\\Gamma(s)  \\theta^{-s} \\delta s,$$\n\n\\leavevmode\\\\\n\nwhich leads to \n\n\\begin{align*}\n    \\log(Zon(e^{-\\theta})) =  &\\sum_{\\delta=1}^{d-1}\\left(p_{d,\\delta}\\frac{\\zeta(\\delta+2) \\Gamma(\\delta+1)}{\\zeta(\\delta+1) \\theta^{\\delta+1}} \\right) + \\frac{1}{2i\\pi} \\int_{\\gamma} \\frac{\\mathbf{\\Pi}_d (\\zeta)(s) }{\\zeta(s)}\\zeta(s+1)\\Gamma(s)  \\theta^{-s}  ds + 2 \\mathbf{\\Pi}_d \\left(\\log(2\\pi) \\zeta -  \\zeta'\\right) (0)  \\\\\n    &+ 2 \\mathbf{\\Pi}_d \\left(\\zeta\\right)(0) \\log(\\theta)   + \\frac{1}{2i\\pi} \\int_{ -\\frac{1}{2}- i \\infty}^{- \\frac{1}{2} + i \\infty} \\frac{\\mathbf{\\Pi}_d (\\zeta)(s) }{\\zeta(s)}\\zeta(s+1)\\Gamma(s)  \\theta^{-s} ds.\n\\end{align*}\n\nWe denote $I_\\text{crit}(d,\\theta)$ and $I_\\text{err}(d,\\theta)$ respectively the first and the second integral above. \n\n\n\\end{proof}\n\n\\begin{cor}\nThe asymptotic equivalent of $ Zon\\left(e^{-\\theta}\\right) $ is:\n\n$$ Zon \\left(e^{-\\theta} \\right) \\sim  \\theta^{2 \\mathbf{\\Pi}_d \\left(\\zeta\\right)(0) } e^{2 \\mathbf{\\Pi}_d \\left(\\log(2\\pi) \\zeta  -  \\zeta' \\right)(0) } \n\\exp\\left( \\sum_{\\delta=1}^{d-1}\\left(p_{d,\\delta}\\frac{\\zeta(\\delta+2) \\Gamma(\\delta+1)}{\\zeta(\\delta+1) \\theta^{\\delta+1}} \\right) + I_{\\text{crit}}(d, \\theta) \\right), \\text{ as } \\theta\\rightarrow 0    .   $$\n\n\\end{cor}\n\n\n\n\\subsection{The saddle-point equation}\n\n\\begin{lem}\\label{sadlle_equation_solution}\nDenoting $\\widetilde{\\theta}_n = \\left( \\frac{2^{d-1} \\zeta(d+1)}{\\zeta(d) n} \\right)^{\\frac{1}{d+1}}$, we have: \n\n$$\\left.\\frac{\\partial  \\log \\left(Zon\\left(\\textbf{\\textit{e}}^{- \\boldsymbol{\\theta}}\\right) \\right) }{\\partial \\theta_i} \\right|_{\\boldsymbol{\\theta} = \\boldsymbol{\\widetilde{\\theta}_n}:= \\widetilde{\\theta}_n \\textbf{1} } = - n + o(n) $$\n\n$\\widetilde{\\theta}_n$ is called the asymptotic solution of the saddle-point equation.\n\\end{lem}\n\n\n\\begin{proof}\n\nWe fetch the coefficient of the multivariate series of the generating function by using Cauchy's integral formula and therefore we look for parameters $\\boldsymbol{\\widetilde{\\theta}_n} = (\\widetilde{\\theta}_{n,1},..., \\widetilde{\\theta}_{n,d})$ such that for all $ 1 \\leq i \\leq d$,\n\n\\begin{align}\\label{saddle_equation}\n    \\left.\\frac{\\partial  }{\\partial \\theta_i} \\left( \\log \\left(Zon\\left(\\textbf{\\textit{e}}^{- \\boldsymbol{\\theta}}\\right) \\right) + (n + 1) \\theta_{i}  \\right) \\right|_{\\boldsymbol{\\theta} = \\boldsymbol{\\widetilde{\\theta}_n} } = 0.\n\\end{align}\n \n\nThe symmetry of the variables in $Zon(\\boldsymbol{\\theta})$  implies $\\theta_{n,i} = \\theta_n $ for $1 \\leq i \\leq d$. The system (\\ref{saddle_equation}) is then restricted to $\\theta_1$ only:\n\n\n\\begin{align*}\n    \\left.\\frac{\\partial}{\\partial \\theta_1} \\log \\left(Zon\\left(\\textbf{\\textit{e}}^{- \\boldsymbol{\\theta}} \\right)\\right) \\right|_{\\substack{\\boldsymbol{\\theta} = \\boldsymbol{\\widetilde{\\theta}_n}}} &= -  \\sum_{\\delta=1}^d  2^{\\delta-1} \\binom{d-1}{\\delta-1} \\sum_{\\textbf{\\textit{v}}\\in \\mathbb{P}^{\\delta}_+} v_1 \\frac{e^{- \\boldsymbol{<\\widetilde{\\theta}_n}, \\textbf{\\textit{v}}> }}{1 - e^{-<\\boldsymbol{\\widetilde{\\theta}_n}, \\textbf{\\textit{v}}> }}  \\\\\n    &= - \\sum_{\\delta=1}^d 2^{\\delta-1} \\binom{d-1}{\\delta-1} \\sum_{m=\\delta}^{+\\infty} \\:\\:\\:\\underbrace{\\sum_{\\substack{\\textbf{\\textit{v}}\\in \\mathbb{P}^{\\delta}_+ \\\\ <\\textbf{1},\\textbf{\\textit{v}}> = m}} v_1} _ { S_m(1, \\textbf{0}_{\\mathbb{R}^{d-1}}) } \\frac{e^{- m \\theta_n }}{1 - e^{-m \\theta_n }}\n\\end{align*}\n\\leavevmode\\\\\n\nUsing the analysis of the sums $S_m$ in Section \\ref{section_eulerset}, we have the following Mellin transform, for $\\Re(s) > d+1$:\n\n$$\\mathcal{M}\\left[\\left.\\frac{\\partial}{\\partial \\theta_1} \\log \\left(Zon \\left(\\textbf{\\textit{e}}^{- \\boldsymbol{\\theta}} \\right)\\right) \\right|_{\\substack{\\boldsymbol{\\theta} = \\boldsymbol{\\widetilde{\\theta}_n}}} \\right](s) =  - \\sum_{\\delta=1}^d 2^{\\delta-1} \\binom{d-1}{\\delta-1} \\left( \\sum_{k \\geq d} \\frac{\\binom{k}{\\delta}}{k^s} \\right) \\frac{\\zeta(s) \\Gamma(s)}{\\zeta(s-1)}. $$\n\nThis function is a meromorphic function on $\\mathbb{C}$, so we are taking $c\\in \\mathbb{C}$ with $\\Re(c) > d+1$ to use the Mellin inversion formula\n\n\n$$\\left.\\frac{\\partial}{\\partial \\theta_1} \\log\\left(Zon \\left(\\textbf{\\textit{e}}^{- \\boldsymbol{\\theta}} \\right)\\right) \\right|_{\\substack{\\boldsymbol{\\theta} = \\boldsymbol{\\widetilde{\\theta}_n}}} = - \\frac{1}{2 i \\pi} \\int_{c - i\\infty}^{c+ i \\infty} \\sum_{\\delta=1}^d 2^{\\delta-1} \\binom{d-1}{\\delta-1} \\left( \\sum_{k \\geq d} \\frac{\\binom{k}{\\delta}}{k^s} \\right) \\frac{\\zeta(s) \\Gamma(s)}{\\zeta(s-1)} \\theta_n^{-s} d s. $$\n\nWith the residue theorem, we can shift the line of the integral to the left of $\\Re(s) = d+1$. The resulting integral is $o\\left(\\frac{1}{\\theta^{d+1}}\\right)$, which leads to:\n\n$$\\left.\\frac{\\partial}{\\partial \\theta_1} \\log\\left(Zon \\left(\\textbf{\\textit{e}}^{- \\boldsymbol{\\theta}} \\right)\\right) \\right|_{\\substack{\\boldsymbol{\\theta} = \\boldsymbol{\\widetilde{\\theta}_n}}} = - \\frac{2^{d-1} \\zeta(d+1)}{\\zeta(d) \\theta_n^{d+1}} + o\\left(\\frac{1}{\\theta_n^{d+1}}\\right). $$\n\nTo generalize the notation of \\cite{Bureaux:polygons}, we denote $\\kappa_d = \\frac{2^{d-1} \\zeta(d+1)}{\\zeta(d)}$. Finally get the parameter which is solution of the saddle equation: \n\n\\begin{align}\n    \\theta_n = \\left( \\frac{\\kappa_d}{n} \\right)^{\\frac{1}{d+1}}\n\\end{align}\n\n\\end{proof}\n\n\\subsection{H-admissibility}\n\n\\import{Parties\/Development\/}{4h_admissibility.tex} \n\n\\subsection{Asymptotic number of lattice zonotopes}\n\nAs $Zon$ is H-admissible, and given the solution of the saddle-point equation $\\widetilde{\\theta}_n$, we can write:\n\n$$ z_d(n):=  \\frac{1}{(2i\\pi)^d} \\int_{[0,2\\pi]^d} Zon\\left( \\textbf{\\textit{e}}^{- \\boldsymbol{ \\widetilde{\\theta}_n}+ i \\boldsymbol{\\theta}}\\right)e^{(n+1)( d \\theta_n +i <  \\textbf{1},\\boldsymbol{\\theta}>)} d \\boldsymbol{\\theta} \\underset{n \\rightarrow \\infty}{\\sim} \\frac{Zon\\left( \\textbf{\\textit{e}}^{- \\boldsymbol{ \\widetilde{\\theta}_n}}\\right)e^{ n d \\theta_n }}{\\sqrt{(2 \\pi)^d \\left|\\textbf{B}\\left( \\textbf{\\textit{e}}^{- \\boldsymbol{ \\widetilde{\\theta}_n}}\\right) \\right|}}, $$\n\nwith the notation from the H-admissibility demonstration ($\\textbf{B} $ is the matrix of the second partial derivatives of the generating function's logarithm). We recall that:\n\n$$ |\\textbf{B}(\\textbf{\\textit{x}}_n)| \\underset{n \\rightarrow \\infty}{\\sim} (d+1) \\left(\\frac{\\kappa_d}{ \\theta_n^{d+2}} \\right)^{d}. $$\n\nUsing $\\kappa_d = \\frac{2^{d-1}\\zeta(d+1)}{\\zeta(d)}$, the contour $\\gamma = \\gamma_\\text{right} \\cup \\gamma_\\text{left}$ with $\\gamma_\\text{right} = 1 - \\frac{A}{\\log(2 + |t|)} +it$ and $\\gamma_\\text{left} = \\frac{A}{\\log(2 + |t|)}+it $, $A>0$ such that $\\gamma$ surrounds the critical strip of the $\\zeta$ function, and the operator $\\mathbf{\\Pi}_d $ defined in Definition~\\ref{definition_operator},  we finally have: \n\n\\begin{align}\\label{final_result}\n     z_d(n) \\underset{n \\rightarrow \\infty}{\\sim}  \\alpha_d n ^{\\beta_d} \\exp \\left( Q_d(n^\\frac{1}{d+1}) + I_{\\text{crit}}(d, t) \\right),\n\\end{align}\n\n\\begin{align*}\n    \\text{with } \n    &\\alpha_d = \\frac{\\kappa_d^{\\frac{d}{2(d+1)} +  \\frac{2}{d+1} \\mathbf{\\Pi}_d (\\zeta)(0) } \\exp \\left( 2\\mathbf{\\Pi}_d \\left(\\log(2\\pi)\\zeta -  \\zeta' \\right)(0) \\right) }{ (2\\pi)^{d\/2 } \\sqrt{d+1}},\\\\\n    &\\beta_d = \\frac{-1}{d+1}\\left(d(d+2)\/2 + 2 \\mathbf{\\Pi}_d \\left(\\zeta\\right) (0) \\right),\\\\\n    &Q_d(X) = (d+1) \\kappa_d^{\\frac{1}{d+1}}  X^d +  \\sum_{\\delta=2}^{d-1} p_{d,\\delta-1}\\frac{\\zeta(\\delta+1) (\\delta-1)!}{\\zeta(\\delta) }\\kappa_d^{-\\frac{\\delta}{d+1}} X^{\\delta}, \\\\\n    &J_{\\text{crit}}(d, n) =  \\left.\\frac{1}{2i\\pi}\\int_{\\gamma}\\frac{\\mathbf{\\Pi}_d (\\zeta)(s) }{\\zeta(s)}\\zeta(s+1)\\Gamma(s) t^{-s} d s \\right|_{t = \\left( \\frac{\\kappa_d}{n} \\right)^{ \\frac{1}{d+1}}}.\n\\end{align*}\n\nNote that $J_{\\text{crit}}(d, n)$ is just the evaluation of $I_{\\text{crit}}(d, \\theta)$ at the saddle point value  $\\theta= \\left( \\frac{\\kappa_d}{n} \\right)^{ \\frac{1}{d+1}}.$\n\n\n\n\n\n\\leavevmode\\\\\n\\subsection{Order of size of $J_{crit}$}\nThroughout this paper, we left $J_{crit}$ under its integral form but we could also write it as an infinite sum of residues as we did in the introduction, with $r$ ranging over the non-trivial zeros of $\\zeta$: \n\n$$ \\sum_{r} Res\\left(\\frac{1}{\\zeta(r)}\\right) \\frac{\\mathbf{\\Pi}_d (\\zeta(r)) }{\\zeta(r)}\\zeta(r+1)\\Gamma(r) \\left( \\frac{\\kappa_d}{n} \\right)^{ \\frac{-r}{d+1}} .$$\n\nThis sum is actually invisible for \"computably large\" numbers $n$. The density of poles $r$ is known to be strongly bounded since Selberg in the beginning of the 1940's \\cite{Selberg:Riemann}. Due to this and to the exponential decrease of $\\Gamma$ function, the value of $J_{crit}$ is with a good approximation the first term of the sum. It is composed of the term at the first non-trivial zero, approximately at $r = \\frac{1}{2} + 14.13472i$ and of the term at its conjugate, which gives for respectively $J_{\\text{crit}}(2, n)$,  $J_{\\text{crit}}(3, n)$, and $J_{\\text{crit}}(4, n)$ the following approximations:\n\n\\begin{align}\\label{numeric_I_crit}\n    J_{\\text{crit}}(2, n) &\\approx -1.3579 {\\scriptstyle\\times} 10^{-10}n^{1\/6}\\cos\\left(4.7116\\ln(0.6842 n)\\right) - 1.4236{\\scriptstyle\\times}10^{-9}n^{1\/6}\\sin\\left(4.7116\\ln(0.6842n)\\right),\\nonumber\\\\\n    J_{\\text{crit}}(3, n) &\\approx -1.2325 {\\scriptstyle\\times}10^{-10}n^{1\/8}\\cos\\left(3.5337\\ln(0.2777n)\\right) - 1.2921 {\\scriptstyle\\times}10^{-9}n^{1\/8}\\sin\\left(3.5337\\ln(0.2777n)\\right),\\\\\n    J_{\\text{crit}}(4, n) &\\approx -3.1764{\\scriptstyle\\times}10^{-9}n^{1\/10}\\cos\\left(2.8269\\ln(0.1305n)\\right) - 8.0628{\\scriptstyle\\times}10^{-9}n^{1\/10}\\sin(2.8269\\ln(0.1305n)). \\nonumber\n\\end{align}\n\n\n\n\nWe can see that the size of these numbers mainly relies on $\\Gamma(r)$. The second non-trivial zero of $\\zeta$ is approximately in $r= \\frac{1}{2} + 21,02203 i$, which is smaller by $10^{-4}$ for the first dimensions. \n\n\\begin{comment}\nIt becomes obvious when we compute the graph of the quotient $\\frac{z_2(n)}{\\alpha_2 n ^{\\beta_2} \\exp(Q_2(n^{1\/3}))}$, as follows:\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{images\/zonogones convergence asymptotique.png}\n    \\caption{Quotient of $z_2(n)$ and $\\alpha_2 n ^{\\beta_2} \\exp(Q_2(n^{1\/3}))$. }\n    \\label{fig:my_label}\n\\end{figure}\n\n\\end{comment}\n\n\\subsection{About moving up in dimension}\n\nFor a more down-to-earth analysis of the logarithmic equivalent, i.e. the leading term of the exponential term, we can use the expansion of $\\zeta(d) = 1+ 2^d + o(2^d)$ as $d $ grows large. One can see that: \n\n$$ \\left(\\frac{2^{d-1}\\zeta(d+1)}{\\zeta(d)} \\right)^{\\frac{1}{d+1}} = 2 + O\\left(\\frac{1}{d}\\right), \\:\\:\\:\\:\\:\\:\\:\\:\\text{ as } d \\rightarrow + \\infty.$$\n\nTherefore, when we get that, in higher dimension, the logarithm of $z(n)$ is nearly equivalent to $ 2(d+1) n^{\\frac{d}{d+1}} $.\\\\\n\n\n\n\n\\subsection{Generalization to rectangular boxes}\nOur result is generalizable to rectangular boxes $[0,n_1]\\times...\\times[0,n_d]$ with $n_i = \\alpha_i n$ ($\\alpha$ is a non-zero positive number). In order to lighten the results, and because the proof is very similar to what has been done before,e write the formula here with the sketch of the proof. One can refer to \\cite{Bogatchev:limit} to see detailed generalization in dimension 2. \n\n\\begin{thm}\n\\end{thm}\n\n\\begin{proof}\nThe saddle-point equation differs depending on the coordinate. It brings the following result:\n\n\\begin{lem}\nDenoting $\\boldsymbol{\\widetilde{\\theta}_n} = \\left( \\widetilde{\\theta}_n^i\\right)$ with $\\widetilde{\\theta}_n^i = \\frac{\\prod_k \\alpha_k^{1\/(d+1)}}{\\alpha_i} \\widetilde{\\theta}_n$ for each $i$, $\\boldsymbol{\\widetilde{\\theta}_n}$ is the solution of the saddle equation, i.e. for each $i$ we have\n\n$$\\left.\\frac{\\partial  \\log\\left(Zon\\left(\\boldsymbol{\\theta}\\right)\\right) }{\\partial \\theta_i} \\right|_{\\boldsymbol{\\theta} = \\boldsymbol{\\widetilde{\\theta}_n}} = - \\alpha_i n + o(n) .$$\n\\end{lem}\n\nThen the uni-variate generating function that has to been asymptotically approximate is \n\n$$Zon\\left(e^{-\\frac{t}{\\alpha_1} },..., e^{- \\frac{t}{\\alpha_d} }\\right) $$\n \n  \n\\end{proof}\n\\end{comment} \n\n\\begin{comment}\n\\subsection{On polytopes, zonotopes and probabilities}\n\nAs said in the introduction, this work falls within the framework of counting polytopes in convex bodies. Here, the asymptotic number of zonotopes was established in hypercubes only, so it calls for looking for attempt to extend this result to more general bodies, such as rectangular boxes, cones etc.  \n\nConcerning combinatorial parameters that come with zonotopes, one could wonder if the number of $\\delta$-dimensional faces of a $d$-dimensional zonotope (aggregated into the f-vector) can be studied with this method (i.e. the mean number of $\\delta$-dimensional faces).\n\nFinally, the estimation of zonotopes was addressed using a combinatorial point of view, but as it was mentioned previously, the same equations appear when we lay a Boltzmann-like distribution over sets of generators and compute the estimation of the number of zonotopes through local limit theorem (see \\cite{Bureaux:polygons}). In that case, it is interesting to watch the deformation of the limit shape of the zonotopes and the evolution of the parameters we computed while varying the probability distribution. \\\\  \n\n\\end{comment}\n\n\n\n\n\n\n\\section{Class of zonotopes and generating function}\\label{section_combi}\n\\import{Parties\/Development\/}{1zonotopes_generating_function.tex}\n\\leavevmode\\\\\n\n\\section{Study of sums of coprime numbers}\\label{section_eulerset}\n\\import{Parties\/Development\/}{2eulerian_poly.tex}\n\\leavevmode\\\\\n\n\\section{Analysis of the generating function}\n\\import{Parties\/Development\/}{3analysis.tex}\n\\leavevmode\\\\\n\n\\section{Proof of the Theorem}\\label{proof_main}\n\\import{Parties\/Development\/}{5proof_theorem.tex}\n\\leavevmode\\\\\n\n\\section{Numerical elements about the speed of convergence to the asymptotic regime}\\label{numerical_discu}\n\\import{Parties\/Development\/}{6comments.tex}\n\\leavevmode\\\\\n\n\n\\section{The average diameter of Zonotopes}\n\\import{Parties\/Development\/}{7properties.tex}\n\\leavevmode\\\\\n\n\\section{Acknowledgments}\n\\import{Parties\/Development\/}{8acknowledgment.tex}\n\\leavevmode\\\\\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn quiescent regions irradiated by cosmic-- or X--rays, the oxygen chemistry is initiated by the charge transfer from\nH$^+$ and H$_{3}^{+}$ to atomic oxygen, forming O$^+$ and OH$^+$. \nIn warm environments it can also start\nwith the reaction of atomic oxygen with H$_2$($\\nu$=0) to form OH.\nThis  endothermic reaction (by $\\sim$0.08\\,eV) possesses an activation barrier \nof a few thousand~K and high gas temperatures ($\\gtrsim$400\\,K)\nare needed to produce significant OH abundances (\\textit{e.g.,}~in shocked gas).\nIn  molecular clouds exposed to strong far-ultraviolet (FUV) \n radiation fields, the so--called PDRs,\nthe gas is heated to relatively high temperatures and there are also high\nabundances of FUV-pumped vibrationally excited molecular hydrogen  H$_{2}^{*}$($\\nu$$=$1,2...) \\citep{hol97}. \nThe internal energy available in H$_{2}^{*}$ \ncan be used to overcome the O($^3P$)~+~H$_2$($\\nu$=0) reaction  barrier \\citep[see][and references therein]{agu10}.\nAlthough not well constrained observationally, enhanced OH abundances are   expected in warm~PDRs. \n\nOH is a key intermediary molecule in the FUV-illuminated gas because further reaction of OH with \nH$_2$, C$^+$, O, N or S$^+$  leads to  the formation of H$_2$O, CO$^+$, O$_2$, NO or SO$^+$ respectively. \nBesides, OH is  the product of H$_2$O photodissociation,\nthe main destruction route of water vapour in the gas unshielded against FUV radiation.\nObservations of OH in specific environments thus constrain different chemical routes of the oxygen chemistry.\n\nPrevious observations with  {\\em KAO} and {\\em ISO} \nhave demonstrated that OH is a powerful tracer of the warm neutral gas in shocked gas;\nfrom protostellar outflows and supernova remnants to extragalactic nuclei \\citep[\\textit{e.g.},][]{sto81,mel87,gon04}.\nUnfortunately, the poor angular resolution ($>$1$'$) and sensitivity of the above telescopes  prevented us\nfrom resolving the OH  emission from interstellar PDRs. \n\n\n\n\\begin{figure*}[t]\n\\vspace{-0.1cm}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{16977f1.eps}}\n\\caption{OH (X $^2$$\\Pi$; $\\nu$=0) rotational lines detected with \\textit{Herschel}\/PACS towards the \n$\\alpha_{2000}$$\\simeq$5$^h$35$^m$21.9$^s$, $\\delta_{2000}$$\\simeq$$-$5$^o$25$'$06.7$''$ position\nwhere the higher excitation OH line peak is observed (see Figure~\\ref{fig:OH_maps}).\nThe red dotted lines show the expected wavelength for each $\\Lambda$-doublet\n (see Figure~\\ref{fig_levels} in the appendix~\\ref{sec:appenixB}  for a complete OH rotational energy diagram).\nSmall intensity asymmetries are observed in most OH $\\Lambda$--doublets.\nTransition upper level energies and $A_{ij}$ spontaneous transition probabilities\n are indicated.} \n\\label{fig:OH_lines}\n\\end{figure*}\n\n\nOwing to its proximity  \nand nearly edge-on orientation, \nthe interface region between the Orion Molecular Cloud~1 (OMC1) and the \nH\\,{\\sc ii} region illuminated by the Trapezium cluster, \nthe  Orion Bar,\nis the  prototypical warm PDR \\citep[with a FUV radiation field\n at the ionization front of $\\chi$$\\simeq$2.5$\\times$10$^4$ times the mean interstellar \nfield in Draine units;][]{mar98}.\nThe most commonly accepted scenario is that \nan extended gas component, with mean gas densities $n_H$ of 10$^{4-5}$\\,cm$^{-3}$,\ncauses the chemical stratification seen in the PDR \\citep{hog95}. \nMost of the low-$J$ molecular  line emission arises in this extended ``interclump'' medium \\citep{tie85,sim97,wie09,hab10}.\nIn addition, another component  of  higher density clumps\n was introduced to fit\nthe observed  H$_{2}$, high--$J$~CO, CO$^+$ and other high density and temperature  tracers \n\\citep{bur90,par91,sto95,wer96,you00}.\nOwing their small filling factor this clumpy structure would allow FUV radiation to permeate the region.\nThe presence of dense clumps is  still controversial. \n\nIn this letter we present initial results from a  spectral scan of the Orion Bar\ntaken with  the PACS instrument \\citep{pog10} on board $\\textit{Herschel}$ \\citep{pil10} \nas part of  the ``HEXOS'' key programme \\citep{ber10}. \nPACS observations of OH lines towards young stellar objects have recently been reported\nby Wampfler et al. \\citep{wam10}.\nHere we present the first detection of OH towards this\nprototypical PDR.  \n\n\n\n\n\n\\vspace{-0.2cm}\n\\section{Observations and data reduction}\n\nPACS observations were carried out on 7 and 8 September 2010\nand consist of two spectral scans in Nyquist sampling wavelength\nrange spectroscopy mode.\nThe PACS spectrometer uses photoconductor detectors and  provides 25 spectra  over a 47$''$$\\times$47$''$ field-of-view\nresolved in 5$\\times$5 spatial pixels (``spaxels''), each with a size of $\\sim$9.4$''$$\\times$9.4$''$ on the sky.\nThe measured width of the spectrometer point-spread function (PSF) is relatively constant at\n$\\lambda$$\\lesssim$125\\,$\\mu$m but it  increases above the spaxel size for longer wavelengths. \nThe resolving power varies between $\\lambda$\/$\\Delta$$\\lambda$$\\sim$1000 (R1 grating order) and $\\sim$5000  \n(B3A). \nThe central spaxel was centred at  \n$\\alpha_{2000}$: 5$^h$35$^m$20.61$^s$, $\\delta_{2000}$: -5$^o$25$'$14.0$''$ target  position.\nObservations were carried out in the  ``chop-nodded'' mode with the largest chopper throw of 6~arcmin.\nNominal capacitances (0.14\\,pF) were used.\nThe integration time was 3.2\\,h for the 1342204117 observation (B2B and R1) \n and 2.7\\,h for the 1342204118 observation (B3A and R1). \nThe data were processed with HIPE using a pipeline upgraded with a spectral flatfield\nalgorithm that reduces the spectral fringing seen in the  Nyquist-sampled wavelength spectra of bright sources.\nFigure~\\ref{fig:OH_lines} shows the resulting OH lines towards the OH emission peak and Figure~\\ref{fig_correlations} \nshows the \nintensities measured in each of the 25 spaxels for several lines of OH, CO, CH$^+$, H$_2$O and [N\\,{\\sc{ii}}]. \nIn order to better sample the PSF and obtain accurate line intensities to be reproduced with our models,  \nwe fit the  OH  emission  averaged over several adjacent spaxels in \nSection~\\ref{sec:excitation} (see also appendix~\\ref{sec:appenixA}).\n\n\n\\begin{figure}[h]\n\\vspace{-0.2cm}\n\\includegraphics[width=8.cm, angle=-90]{16977f2.eps}\n\\caption{PACS rotationally excited\n OH  $^{2}\\Pi_{3\/2}$ $J$=7\/2$\\rightarrow$5\/2 lines at $\\sim$84\\,$\\mu$m ($E_u$\/$k$=291\\,K)\noverlaid on the\ndistribution of the CO $J$=6-5 peak brightness temperature (colour image)\nobserved with the CSO telescope at $\\sim$11$''$  resolution \\citep{lis98}.\nWhite contours show the brightest regions of H$_{2}^{*}$ $v$=1-0 $S$(1) emission \\citep{wal00}. \nLower intensity H$_{2}^{*}$ extended emission is\npresent in the entire field \\citep{wer96}.\nViolet stars shows the position of the H$^{13}$CN $J$=1-0  clumps deeper inside the Bar\n\\citep{lis03}. Note the decrease of OH line intensity\nwith distance from the ionization front.}\n\\label{fig:OH_maps}\n\\end{figure}\n\n\n\\vspace{-0.2cm}\n\\section{Results}\n\nOf all the observed OH lines only the ground-state lines at $\\sim$119\\,$\\mu$m  show widespread bright emission \nat all positions. The $\\sim$119\\,$\\mu$m lines mainly arise from the background OMC1 complex\n(the same applies to most ground-state lines of other species).\nFigure~\\ref{fig:OH_maps} shows the spatial distribution of the\nrotationally excited OH $^{2}\\Pi_{3\/2}$ $J$=7\/2$\\rightarrow$5\/2 lines at $\\sim$84\\,$\\mu$m ($E_u$\/$k$=291\\,K)\nsuperimposed over the CO $J$=6-5 peak brightness temperature \\citep[colour image from][]{lis98} \nand over the brightest  H$_{2}^{*}$ $\\nu$=1-0 $S$(1) line emission regions\n\\citep[white contours from][]{wal00}.\nThe emission from the other  OH $\\Lambda$-doublets at $\\sim$79 and $\\sim$163\\,$\\mu$m \n(see Figure~\\ref{fig:OH_maps2} in appendix~\\ref{sec:appenixB}) \ndisplays a similar spatial distribution that follows the ``bar'' morphology   peaking\nnear the H$_{2}^{*}$~$v$=1-0 $S$(1) bright emission region and then decreases with distance from the\nionization front \\citep[note that H$_{2}^{*}$ shows lower level extended\nemission with small-scale structure in the entire observed field;][]{wer96}.\nThe excited OH spatial distribution, however,  does not follow the CO~$J$=6-5 emission maxima, \nwhich approximately trace the gas temperature \nin the extended ``interclump'' component.\n\n\n\n\n\nFigure~\\ref{fig:OH_lines} shows  the detected OH $\\Lambda$-doublets \n(at $\\sim$65, $\\sim$79, $\\sim$84, $\\sim$119 and $\\sim$163\\,$\\mu$m) towards the \nposition where the higher excitation OH lines peak. \nThe total intensity of the observed FIR lines \n is $\\sum$$I$(OH)$\\simeq$5$\\times$10$^{-4}$~erg\\,s$^{-1}$\\,cm$^{-2}$\\,sr$^{-1}$.\n All OH doublets appear in emission, with intensity asymmetry ratios up to 40\\% \n(one line of the $\\Lambda$-doublet is  brighter than the other).\n\n\nNote that the upper energy level of the $^{2}\\Pi_{3\/2}$~$J$=9\/2$\\rightarrow$7\/2 transition \nat $\\sim$65\\,$\\mu$m lies at  E$_u$\/$k$$\\sim$511\\,K.\nThe critical densities ($n_{cr}$) of  the observed  OH transitions are high,\n$n_{cr}$$\\gtrsim$10$^{8}$\\,cm$^{-3}$. For much lower gas densities, and in the presence of strong\nFIR radiation fields,\nmost lines would have been observed in absorption, especially those in the $^{2}\\Pi_{3\/2}$ ladder \\citep{goi02}.\nHence, the observed OH lines  must\narise in an widespread component of warm and dense gas.\n\n\n\nAlthough our PACS observations do not provide a fully sampled map,\n the line emission observed in the 25 spaxels \ncan be used to carry out a first-order analysis on the spatial correlation of  different\n lines (neglecting perfect PSF sampling, line opacity and excitation effects).\nExcept for the OH ground-state lines at $\\sim$119\\,$\\mu$m (that come from the background OMC1 cloud),\nwe find that the rotationally excited OH lines correlate well with the high-$J$ CO and CH$^+$ emission\nbut, as expected, they do not correlate with the ionized gas emission.\nFigure~\\ref{fig_correlations} (\\textit{lower panel}) compares the observed OH~$\\sim$84.597\\,$\\mu$m\nline intensities with those of \nCO $J$=21-20 ($E_u$\/$k$$\\sim$1276\\,K), CH$^+$ $J$=3-2 ($E_u$\/$k$$\\sim$240\\,K)\nand [N\\,{\\sc{ii}}]121.891\\,$\\mu$m  (all observed with a similar PSF)\nand also with the  CO $J$=15-14 ($E_u$\/$k$$\\sim$663\\,K), H$_2$O 3$_{03}$-2$_{12}$ ($E_u$\/$k$$\\sim$163\\,K) and\nOH~$\\sim$163.397\\,$\\mu$m lines (\\textit{upper panel}).\nThis simple analysis suggests\nthat the excited OH, high-$J$ CO and CH$^+$ $J$=3-2 lines  arise from the same gas  component.\nIt also shows that the emission from different excited OH lines is well correlated,\nwhile the OH and H$_2$O  emission is not (within the PSF sampling caveats).\n\n\n\n\\vspace{-0.2cm}\n\\section{OH column density determination}\n\\label{sec:excitation}\n\nDetermining the OH level populations is no  trivial excitation problem.\nIn addition to relatively strong asymmetries in the collisional \nrate coefficients\\footnote{We used  \ncollision rate coefficients of OH with $para$- and $ortho$-H$_2$  \nfrom  Offer \\& van Dishoeck \\citep{off92} and Offer et al. \\citep{off94}.\nStrong differences in the intensity of\neach OH $\\Lambda$-doublet component due to  asymmetries in the collisional\nrates between OH and $para$-H$_2$ were predicted\n (\\textit{e.g.,}~$I$(84.597)\/$I$(84.420)$>$$I$(119.441)\/$I$(119.234)$>$1).\nAsymmetries are significantly reduced when collisions with $ortho$-H$_2$ are included\n (\\textit{i.e.,}~in the warm gas) and when FIR radiative pumping plays a role.\nWe assume that the H$_2$ $ortho$-to-$para$ ratio is thermalized to the gas temperature, \n\\textit{e.g.,}~$\\sim$1.6 at 100\\,K and $\\sim$2.9 at 200\\,K.} between\neach $\\Lambda$-doubling component (Offer \\& Van Dishoeck 1992),\nradiative and opacity effects (pumping by the ambient IR radiation field and line trapping) can play a significant role\nif the  gas density is much lower than~$n_{cr}$.\nHere we use a nonlocal\nand non-LTE code that treats both the OH line \nand continuum radiative transfer \\citep[see appendix in][]{goi06}.\nThe continuum measured by PACS and SPIRE in the region (H. Arab et al. in prep.)\n can be approximately reproduced by a modified blackbody with a colour temperature of $\\sim$55\\,K and a\nopacity dependence of $\\sim$0.05(100\/$\\lambda$)$^{1.75}$ \\citep[see][]{lis98}.\nOur calculations include thermal, turbulent, and\nopacity line broadening with a turbulent velocity dispersion of\n$\\sigma$=1.0\\,km\\,s$^{-1}$ and FWHM=2.355$\\times\\sigma$\n\\citep[see \\textit{e.g.}, the  linewidths measured by][]{hog95}. \nA grid of single-component models for different $N$(OH), gas temperatures, densities and beam filling factors  \nwere run. \nThe best fit model was found by minimizing the ``$\\chi^2$-value\" (see the appendix~\\ref{sec:appenixA} for its definition).\n\n  \n\nFrom the excitation models we conclude that the\nhigh $I$(65.279)\/$I$(84.596)$\\simeq$0.3 and $I$(65.132)\/$I$(84.420)$\\simeq$0.5  intensity ratios in the $^{2}\\Pi_{3\/2}$ ladder\ncan only be reproduced if the gas is dense, at least $n_H$=$n$(H)+2$n$(H$_2$) of a few 10$^6$\\,cm$^{-3}$.\nIn addition, no model is able to produce even a crude fit to the data if one assumes \nthe average molecular gas temperature  (T$_k$$\\simeq$85\\,K)  in the lower density ``interclump'' medium \\citep{hog95}.\nIn the $^{2}\\Pi_{1\/2}$ ladder, the intensity of the $\\sim$163\\,$\\mu$m lines is sensitive to the \nFIR radiation field in the region through the absorption of FIR dust continuum photons in the\nOH\\,$\\sim$34 and $\\sim$53\\,$\\mu$m cross-ladder transitions ($^{2}\\Pi_{3\/2}$-$^{2}\\Pi_{1\/2}$ $J$=3\/2$\\rightarrow$5\/2\nand $J$=3\/2$\\rightarrow$3\/2 respectively).   \nHowever, the $\\sim$163\\,$\\mu$m lines in the Orion Bar are not particularly strong and the OH~$\\sim$34 and $\\sim$53\\,$\\mu$m lines\nare not present in the ISO spectra,  \nthus FIR pumping does not dominate the OH excitation. \nAll in all, the best model is found for a source of high density ($n_H$$\\lesssim$10$^7$~cm$^{-3}$)\nand warm gas temperatures (T$_{k}$=160-220\\,K).  \nThis temperature lies in between\nthe  $\\sim$600\\,K derived in the H$_2$ emitting regions \\citep{all05}\nand the $\\sim$150\\,K derived from  NH$_3$ lines \\citep{bat03}  \n near the ridge of dense and cooler H$^{13}$CN clumps \\citep[T$_k$$\\simeq$50\\,K;][]{lis03}.  \nIn our simple model, the best fit is obtained for a source of small filling factor ($\\eta\\simeq$10$\\%$)\nwith a source-averaged OH column density of $\\gtrsim$10$^{15}$\\,cm$^{-2}$ (see~appendix~\\ref{sec:appenixA}).\n\n\\begin{figure} [t]\n\\vspace{-0.0cm}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{16977f3.eps}}\n\\caption{Line intensity spatial correlations between the  OH \n$^{2}\\Pi_{3\/2}$ $J$=7\/2$^-$$\\rightarrow$5\/2$^+$  line at 84.597\\,$\\mu$m \nand lines from other species.\nIntensities are in units of 10$^{-5}$\\,erg\\,cm$^{-2}$\\,s$^{-1}$\\,sr$^{-1}$.}\n\\label{fig_correlations}\n\\end{figure}\n\n\n\\vspace{-0.2cm}\n\\section{OH chemistry and line emission origin}\n\\label{sec:PDRmods}\n\nWe  used the Meudon PDR code \\citep{lpt06,goi07} to estimate the OH column \ndensity in a slab of gas at  different densities ($n_H$ from 5$\\times$10$^4$ to 10$^7$\\,cm$^{-3}$). \nThe adopted FUV radiation field, $\\chi$=10$^4$, roughly corresponds to the attenuation\nof the FUV field at the ionization front  by a column density of $N_H$$\\simeq$10$^{21}$\\,cm$^{-2}$ in\na $n_H$$\\simeq$10$^4$\\,cm$^{-3}$  medium. This attenuation is equivalent to a spatial length\nof $\\sim$10$^{17}$\\,cm, consistent with the observed  decrease of excited\n OH emission with projected distance from the ionization front (see Figure~\\ref{fig:OH_maps}).\nGiven the high gas temperature, FUV field and moderate grain temperature \n(T$_{gr}$$\\sim$70-100\\,K) in the regions traced by  \nFIR OH, CO and CH$^+$ lines, we neglected molecule freeze-out and ice desorption\n\\citep[which are important deeper inside at A$_V$$\\gtrsim$3;][]{hol09}.\nIn our models   OH is a surface tracer that reaches its  peak abundance at A$_V$$\\lesssim$1\n(Figure~\\ref{fig:pdr_mod}) where OH formation is driven by the endothermic reaction O($^3P$)~+~H$_2$~$\\rightarrow$~OH~+~H, \nslightly enhanced by the O($^3P$)~+~H$_{2}^{*}$ reaction \n\\citep[included in our models; see][]{agu10}.\nGas temperatures around $\\sim$1000-500\\,K are predicted near the slab surface at A$_V$=0.01\nand around $\\sim$100\\,K at A$_V$$\\gtrsim$1. In these H\/H$_2$ transition layers where the OH abundance peaks,\nthe electron density is still high ($\\lesssim$[C$^+$\/H]\\,$n_H$) and hydrogen is not  fully molecular, \nwith  $f$(H$_2$)=2$n$(H$_2$)\/[$n$(H)+2$n$(H$_2$)]$\\simeq$0.5.\nIn general, the higher the gas temperature  where enough H$_2$ has formed, \nthe higher  the predicted OH abundance.\n\n\nIn the A$_V$$\\lesssim$1 layers,  OH destruction is dominated by photodissociation \n(OH~+~$h\\nu$~$\\rightarrow$~O~+~H)\nand to a  lesser extent, by reactions of OH with H$_2$ to form H$_2$O \n(only when the gas temperature and density are very high).\nWater vapour photodissociation (H$_2$O~+~$h\\nu$~$\\rightarrow$~OH~+~H) in the surface layers limits the H$_2$O formation\n  and leads to OH\/H$_2$O abundance ratios ($>$1), much higher than those expected in equally warm regions without \nenhanced FUV radiation fields (\\textit{e.g.} in C--shocks).\nThe lack of apparent correlation between the excited OH and H$_2$O 3$_{03}$-2$_{12}$ lines \n(see Figure~\\ref{fig_correlations}) and the\nabsence of high excitation H$_2$O lines in the PACS spectra (only weak H$_2$O 2$_{21}$-2$_{12}$, 2$_{12}$-1$_{01}$ \nand 3$_{03}$-2$_{12}$ lines are clearly detected) suggests\nthat the bulk of OH and H$_2$O column densities arise from different  depths.\n\nAs the temperature decreases inwards, the gas-phase production of OH also decreases.\nThe  spatial correlation between excited OH, CH$^+$ $J$=3-2  and high-$J$ CO lines is a good \nsignature of their common origin in the warm gas at low $A_V$ depths. \n\n\n\nOur PDR models predict OH column densities in the range  $\\sim$10$^{12}$\\,cm$^{-2}$ to $\\sim$10$^{15}$\\,cm$^{-2}$\nat  A$_V$$\\lesssim$1\nfor gas densities between $n_H$=5$\\times$10$^4$ and 10$^7$\\,cm$^{-3}$ respectively.\nHence, even if we take into account possible inclination \neffects, high density and temperature models \nproduce OH columns closer to the values derived in Section~4 just from excitation\nconsiderations (note that a precise determination of the\ngas density would require knowing the collisional rate coefficients of OH with H atoms and electrons). \nThe OH abundance in these dense surface layers is of the order of $\\approx$10$^{-6}$ with respect to H nuclei.\nHowever, optical depths of A$_V$$\\lesssim$1 correspond to spatial scales of only $\\sim$10$^{15}$\\,cm\n\\citep[\\textit{i.e.}, much smaller than the H$^{13}$CN clumps detected by][deeper\ninside the cloud]{lis03},\nbut we detect extended OH emission over $\\sim$10$^{17}$\\,cm scales. Therefore, we  have to conclude that \nthe observed OH emission arises from a small filling factor ensemble of unresolved structures \nthat are exposed to FUV radiation (overdense clumps or filaments). \nNote that owing to the lower grain temperature compared to the gas, \nthe expected FIR continuum emission from these clumps will still be below the continuum\nlevels observed by \\textit{Herschel}\/PACS.\n\nThe minimum size of the dense clumps is $\\sim$10$^{15}$\\,cm (from OH photochemical models)\nwith a maximun size of $\\sim$10$^{16}$\\,cm (from the inferred beam dilution factor). Both correspond\nto $\\lesssim$0.2$''$-2$''$ at the distance of Orion.\nAs an example, H$_2$ photoevaporating clumps of $\\sim$10$^{16}$\\,cm size have been unambiguously resolved \ntowards  S106 PDR \\cite{noe05}. However, higher angular resolution observations \n(\\textit{e.g.,}~with 8-10m  telescopes) are needed\nto resolve smaller H$_2$ clumps from the H$_2$ interclump emission in the Orion Bar.\n\n\n\n\n\\begin{figure}[t]\n\\vspace{-0.00cm}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{16977f4.eps}}\n\\caption{Gas-phase PDR models of a FUV-illuminated slab ($\\chi$=10$^4$) of gas with\n different densities:\n$n_H$=10$^7$, 10$^6$, 10$^5$ and 5$\\times$10$^4$\\,cm$^{-3}$. \nOH column densities are shown as continuous curves (left axis) \nwhile OH abundances, $n$(OH)\/$n_H$, are shown as dashed curves (right axes).}\n\\label{fig:pdr_mod}\n\\end{figure}\n\n\n\n\nIf the observed FIR OH line emission does not arise from such a high density gas component, a different non-thermal \nexcitation mechanism able to populate the  OH $^{2}\\Pi_{3\/2}$ $J$=9\/2 and 7\/2 levels would be needed.\nTwo alternative scenarios can be explored, at least qualitatively.\nFirst, OH molecules produced by H$_2$O photodissociation are expected to form mostly in the ground electronic\nand vibrational state but in unusually high energy $J$$>$70\/2 levels \\citep[a few thousands~K!;][]{van00}. \nNevertheless, they   \n will cascade down radiatively to  lower energy rotational ladders extremely rapidly.\nAlthough very excited suprathermal OH $J$$\\simeq$70\/2-15\/2 lines have been reported\ntowards the HH\\,211 outflow\nand were interpreted as H$_2$O photodissociation \\citep{tap08},  we\ndid not find any of them in the SWS or in the IRS spectra of the Orion Bar\n(taken and processed by us from the ISO and \\textit{Spitzer} basic calibrated data archives). \nBesides, even if photodissociation is the main H$_2$O destruction mechanism in the A$_V$$\\lesssim$1 warm layers, \nthis  is not the main production pathway of OH (the O($^3P$)~+~H$_2$ reaction dominates).\n\n\nSecond,  experiments and quantum calculations  suggest that reaction of O($^3P$) atoms with H$_{2}^{*}$($\\nu$=1)  \ncan produce significant OH in the $\\nu$=1 vibrationally excited state, with OH($\\nu$=1)\/OH($\\nu$=0) \npopulation ratios $\\gtrsim$1 for moderate kinetic energies\n\\citep[$\\gtrsim$0.05-0.1\\,eV; \\textit{e.g.,}][]{bal04}.\nComplementarily, absorption of near-- and mid--IR photons (from the continuum or from bright overlapping H$_{2}^{*}$ \nand ionic lines) can also pump OH to the $\\nu$=1 state.\nIn both cases, subsequent de-excitation  through the $\\nu$=1-0 rotation--vibration band \nat $\\sim$2.80\\,$\\mu$m\nwould  populate the  OH($\\nu$=0) rotationally excited levels  that we observe with PACS.\nHowever, the OH $\\nu$=1-0 band at $\\sim$2.80\\,$\\mu$m is not present in the ISO\/SWS spectra.\nEven assuming that the O($^3P$)+H$_{2}^{*}$($\\nu$=1) reaction only forms OH($\\nu$=1),\nsignificant gas column densities at very hot temperatures (T$_k$$\\sim$2000\\,K) will be needed to match\nthe observations if  $n_H$$\\simeq$10$^{4-5}$\\,cm$^{-3}$.\nStudying the OH vibrationally-pumping mechanism quantitatively is beyond the scope of this work \nbut these chemical and pumping  effects could contribute to the excitation of  FIR OH lines\nin lower density gas.\n\nDifferent scenarios for the origin and nature of photoevaporating clumps have been proposed\n\\citep{gor02}, but without a more precise determination of their\nsizes and densities it is hard to conclude on any of them. Subarcsec resolution observations of OH gas-phase products, \n\\textit{e.g.,}~direct observation of CO$^+$ or SO$^+$ clumps with \\textit{ALMA},\nwill help us to assay the clumpy nature of the Orion Bar in the near future.\n \n\n\n\n\\begin{acknowledgements}\nPACS has been developed by\na consortium of institutes led by MPE (Germany)\nand including UVIE (Austria); KU Leuven, CSL,\nIMEC (Belgium); CEA, LAM (France); MPIA (Germany); \nINAF-IFSI\/OAA\/OAP\/OAT, LENS, SISSA (Italy); IAC (Spain). \nWe thank Darek Lis and Malcolm Walmsley for providing us their CO $J$=6-5 and H$_{2}^{*}$~1-0~S(1) maps.\nWe also thank Emilie Habart, Bart Vandenbussch and Pierre Royer  for useful discussions and  the referee \nfor his\/her constructive report.\nWe thank the Spanish MICINN for funding support\nthrough grants AYA2006-14876, AYA2009-07304 and CSD2009-00038. \nJRG is supported by a Ram\\'on y Cajal research contract.\n We acknowledge the use of the LAMDA data base \\citep{sch05}. \n\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe concept of a fuzzy set was introduced by Zadeh \\cite{zadeh}, in 1965.\nSince its inception, the theory has developed in many directions and found\napplications in a wide variety of fields. There has been a rapid growth in\nthe interest of fuzzy set theory and its applications from the past several\nyears. Many researchers published high-quality research articles on fuzzy\nsets in a variety of international journals. The study of fuzzy set in\nalgebraic structure has been started in the definitive paper of Rosenfeld\n1971 \\cite{Rosen}. Fuzzy subgroup and its important properties were defined\nand established by Rosenfeld \\cite{Rosen}. In 1981, Kuroki introduced the\nconcept of fuzzy ideals and fuzzy bi-ideals in semigroups in his paper \\cit\n{Kuroki}.\n\nThere are several kinds of fuzzy set extensions in the fuzzy set theory, for\nexample, intuitionistic fuzzy sets, interval-valued fuzzy sets, vague sets,\netc. Bipolar-valued fuzzy set is another extension of fuzzy set whose\nmembership degree range is different from the above extensions. Lee \\cit\n{Lee} introduced the notion of bipolar-valued fuzzy sets. Bipolar-valued\nfuzzy sets are an extension of fuzzy sets whose membership degree range is\nenlarged from the interval $[0,1]$ to $[-1,1]$. In a bipolar-valued fuzzy\nset, the membership degree $0$ indicate that elements are irrelevant to the\ncorresponding property, the membership degrees on $(0,1]$ assign that\nelements somewhat satisfy the property, and the membership degrees on \n[-1,0) $ assign that elements somewhat satisfy the implicit counter-property \n\\cite{Lee, Lee2}.\n\nAkram et al. \\cite{Akram} introduced the concept of bipolar fuzzy\nK-algebras. In \\cite{Jun}, Jun and park applied the notion of bipolar-valued\nfuzzy sets to BCH-algebras. They introduced the concept of bipolar fuzzy\nsubalgebras and bipolar fuzzy ideals of a BCH-algebra. Lee \\cite{KLee2}\napplied the notion of bipolar fuzzy subalgebras and bipolar fuzzy ideals of\nBCK\/BCI-algebras. Also some results on bipolar-valued fuzzy BCK\/BCI-algebras\nare introduced by Saeid in \\cite{Saeid}.\n\nThis paper concerns the relationship between bipolar-valued fuzzy sets and\nleft almost semigroups. The left almost semigroup abbreviated as an\nLA-semigroup, was first introduced by Kazim and Naseerudin \\cite{Kazim}.\nThey generalized some useful results of semigroup theory. They introduced\nbraces on the left of the ternary commutative law $abc=cba,$ to get a new\npseudo associative law, that is $(ab)c=(cb)a,$ and named it as left\ninvertive law. An LA-semigroup is the midway structure between a commutative\nsemigroup and a groupoid. Despite the fact, the structure is non-associative\nand non-commutative. It nevertheless possesses many interesting properties\nwhich we usually find in commutative and associative algebraic structures.\nMushtaq and Yusuf produced useful results \\cite{1979}, on locally\nassociative LA-semigroups in 1979. In this structure they defined powers of\nan element and congruences using these powers. They constructed quotient\nLA-semigroups using these congruences.\\ It is a useful non-associative\nstructure with wide applications in theory of flocks.\n\nIn this paper, we have introduced the notion of bipolar-valued fuzzy\nLA-subsemigroups and bipolar-valued fuzzy left (right, bi-, interior) ideals\nin LA-semigroups.\n\n\\section{\\textbf{Preliminaries and basic definitions}}\n\n\\textbf{Definition 2.1. }\\cite{Kazim} A groupoid $\\left( S,\\cdot \\right) $\\\nis called an LA-semigroup$,$ if it satisfies left invertive law \n\\begin{equation*}\n\\left( a\\cdot b\\right) \\cdot c=\\left( c\\cdot b\\right) \\cdot a,\\text{ \\ for\nall }a,b,c\\in S\\text{.}\n\\end{equation*}\n\n\\textbf{Example 2.1 }\\cite{1978} Let $\\left( \n\\mathbb{Z}\n,+\\right) $\\ denote the commutative group of integers under addition. Define\na binary operation \\textquotedblleft $\\ast $\\textquotedblright\\ in \n\\mathbb{Z}\n$\\ as follows$:\n\\begin{equation*}\na\\ast b=b-a,\\text{ \\ for all }a,b\\in \n\\mathbb{Z}\n\\text{.}\n\\end{equation*\nWhere \\textquotedblleft $-$\\textquotedblright\\ denotes the ordinary\nsubtraction of integers. Then $\\left( \n\\mathbb{Z}\n,\\ast \\right) $\\ is an LA-semigroup.\n\n\\textbf{Example 2.2 }\\cite{1978} Define a binary operation \\textquotedblleft \n$\\ast $\\textquotedblright\\ in \n\\mathbb{R}\n$\\ as follows$:\n\\begin{equation*}\na\\ast b=b\\div a,\\text{ \\ for all }a,b\\in \n\\mathbb{R}\n\\text{.}\n\\end{equation*\nThen $\\left( \n\\mathbb{R}\n,\\ast \\right) $\\ is an LA-semigroup.\n\n\\begin{lemma}\n\\label{l1}\\cite{1979} If $S$ is an LA-semigroup with left identity $e$, then \n$a(bc)=b(ac)$ for all $a,b,c\\in S.$\n\\end{lemma}\n\nLet $S$ be an LA-semigroup. A nonempty subset $A$ of $S$ is called an\nLA-subsemigroup of $S$ if $ab\\in A$ for all $a,b\\in A$. A nonempty subset $L$\nof $S$ is called a left ideal of $S$ if $SL\\subseteq L$ and a nonempty\nsubset $R$ of $S$ is called a right ideal of $S$ if $RS\\subseteq R$. A\nnonempty subset $I$ of $S$ is called an ideal of $S$ if $I$ is both a left\nand a right ideal of $S$. A subset $A$ of $S$ is called an interior ideal of \n$S$ if $(SA)S\\subseteq A$. An LA-subsemigroup $A$ of $S$ is called a\nbi-ideal of $S$ if $(AS)A\\subseteq A$.\n\nIn an LA-semigroup the medial law holds\n\\begin{equation*}\n(ab)(cd)=(ac)(bd),\\text{ \\ for all }a,b,c,d\\in S.\n\\end{equation*}\n\nIn an LA-semigroup $S$ with left identity, the paramedial law holds\n\\begin{equation*}\n(ab)(cd)=(dc)(ba),\\text{ \\ for all }a,b,c,d\\in S.\n\\end{equation*}\n\nNow we will recall the concept of bipolar-valued fuzzy sets.\n\n\\textbf{Definition 2.2 }\\cite{Lee2} Let $X$\\ be a nonempty set. A\nbipolar-valued fuzzy subset (BVF-subset, in short) $B$\\ of $X$\\ is an object\nhaving the for\n\\begin{equation*}\nB=\\left\\{ \\left\\langle x,\\mu _{B}^{+}(x),\\mu _{B}^{-}(x)\\right\\rangle :x\\in\nX\\right\\} .\n\\end{equation*\nWhere $\\mu _{B}^{+}:X\\rightarrow \\lbrack 0,1]$\\ and $\\mu\n_{B}^{-}:X\\rightarrow \\lbrack -1,0]$.\n\nThe positive membership degree $\\mu _{B}^{+}(x)$ denotes the satisfaction\ndegree of an element $x$ to the property corresponding to a bipolar-valued\nfuzzy set $B=\\left\\{ \\left\\langle x,\\mu _{B}^{+}(x),\\mu\n_{B}^{-}(x)\\right\\rangle :x\\in X\\right\\} $, and the negative membership\ndegree $\\mu _{B}^{-}(x)$ denotes the satisfaction degree of $x$ to some\nimplicit counter property of $B=\\left\\{ \\left\\langle x,\\mu _{B}^{+}(x),\\mu\n_{B}^{-}(x)\\right\\rangle :x\\in X\\right\\} $. For the sake of simplicity, we\nshall use the symbol $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $\nfor the bipolar-valued fuzzy set $B=\\left\\{ \\left\\langle x,\\mu\n_{B}^{+}(x),\\mu _{B}^{-}(x)\\right\\rangle :x\\in X\\right\\} .$\n\n\\textbf{Definition 2.3 }Let $B_{1}=\\left\\langle \\mu _{B_{1}}^{+},\\mu\n_{B_{1}}^{-}\\right\\rangle $ and $B_{2}=\\left\\langle \\mu _{B_{2}}^{+},\\mu\n_{B_{2}}^{-}\\right\\rangle $ be two BVF-subsets of a nonempty set $X$. Then\nthe product of two BVF-subsets is denoted by $B_{1}\\circ B_{2}$ and defined\nas: \n\\begin{eqnarray*}\n\\left( \\mu _{B_{1}}^{+}\\circ \\mu _{B_{2}}^{+}\\right) \\left( x\\right) \n&=&\\left\\{ \n\\begin{array}{l}\n\\tbigvee_{x=yz}\\left\\{ \\mu _{B_{1}}^{+}\\left( y\\right) \\wedge \\mu\n_{B_{2}}^{+}\\left( z\\right) \\right\\} ,\\text{ if }x=yz\\text{ for some }y,z\\in\nS \\\\ \n\\text{ }0\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ otherwise.\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n\\end{array\n\\right.  \\\\\n\\left( \\mu _{B_{1}}^{-}\\circ \\mu _{B_{2}}^{-}\\right) \\left( x\\right) \n&=&\\left\\{ \n\\begin{array}{l}\n\\tbigwedge_{x=yz}\\left\\{ \\mu _{B_{1}}^{-}\\left( y\\right) \\vee \\mu\n_{B_{2}}^{-}\\left( z\\right) \\right\\} ,\\text{ if }x=yz\\text{ for some }y,z\\in\nS \\\\ \n\\text{ }0\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ otherwise.\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n\\end{array\n\\right. \n\\end{eqnarray*}\n\nNote that an LA-semigroup $S$ can be considered as a BVF-subset of itself\nand le\n\\begin{eqnarray*}\n\\Gamma &=&\\left\\langle \\mathcal{S}_{\\Gamma }^{+}(x),\\mathcal{S}_{\\Gamma\n}^{-}(x)\\right\\rangle \\\\\n&=&\\left\\{ \\left\\langle x,\\mathcal{S}_{\\Gamma }^{+}(x),\\mathcal{S}_{\\Gamma\n}^{-}(x)\\right\\rangle :\\mathcal{S}_{\\Gamma }^{+}(x)=1\\text{ and }\\mathcal{S\n_{\\Gamma }^{-}(x)=-1,\\text{ for all }x\\text{ in }S\\right\\}\n\\end{eqnarray*\nbe a BVF-subset and $\\Gamma =\\left\\langle \\mathcal{S}_{\\Gamma }^{+}(x)\n\\mathcal{S}_{\\Gamma }^{-}(x)\\right\\rangle $ will be carried out in\noperations with a BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ such that $\\mathcal{S}_{\\Gamma }^{+}$ and $\\mathcal{\n}_{\\Gamma }^{-}$ will be used in collaboration with $\\mu _{B}^{+}$ and $\\mu\n_{B}^{-}$ respectively.\n\nLet $BVF(S)$ denote the set of all BVF-subsets of an LA-semigroup $S.$\n\n\\begin{proposition}\n\\label{P1}Let $S$ be an LA-semigroup, then the set $(BVF(S),\\circ )$ is an\nLA-semigroup.\n\\end{proposition}\n\n\\textbf{Proof. }Clearly $BVF(S)$ is closed. Let $B_{1}=\\left\\langle \\mu\n_{B_{1}}^{+},\\mu _{B_{1}}^{-}\\right\\rangle ,$ $B_{2}=\\left\\langle \\mu\n_{B_{2}}^{+},\\mu _{B_{2}}^{-}\\right\\rangle $ and $B_{3}=\\left\\langle \\mu\n_{B_{3}}^{+},\\mu _{B_{3}}^{-}\\right\\rangle $ be in $BVF(S)$. Let $x$ be any\nelement of $S$ such that $x\\neq yz$ for some $y,z\\in S$. Then we hav\n\\begin{equation*}\n\\left( \\left( \\mu _{B_{1}}^{+}\\circ \\mu _{B_{2}}^{+}\\right) \\circ \\mu\n_{B_{3}}^{+}\\right) (x)=0=\\left( \\left( \\mu _{B_{3}}^{+}\\circ \\mu\n_{B_{2}}^{+}\\right) \\circ \\mu _{B_{1}}^{+}\\right) (x).\n\\end{equation*\nAn\n\\begin{equation*}\n\\left( \\left( \\mu _{B_{1}}^{-}\\circ \\mu _{B_{2}}^{-}\\right) \\circ \\mu\n_{B_{3}}^{-}\\right) (x)=0=\\left( \\left( \\mu _{B_{3}}^{-}\\circ \\mu\n_{B_{2}}^{-}\\right) \\circ \\mu _{B_{1}}^{-}\\right) (x).\n\\end{equation*\nLet $x$ be any element of $S$ such that $x=yz$ for some $y,z\\in S$. Then we\nhav\n\\begin{eqnarray*}\n\\left( \\left( \\mu _{B_{1}}^{+}\\circ \\mu _{B_{2}}^{+}\\right) \\circ \\mu\n_{B_{3}}^{+}\\right) (x) &=&{\\bigvee }_{x=yz}\\left\\{ \\left( \\mu\n_{B_{1}}^{+}\\circ \\mu _{B_{2}}^{+}\\right) (y)\\wedge \\mu\n_{B_{3}}^{+}(z)\\right\\} \\\\\n&=&{\\bigvee }_{x=yz}\\left\\{ \\left( {\\bigvee }_{y=pq}\\left\\{ \\mu\n_{B_{1}}^{+}(p)\\wedge \\mu _{B_{2}}^{+}(q)\\right\\} \\right) \\wedge \\mu\n_{B_{3}}^{+}(z)\\right\\} \\\\\n&=&{\\bigvee }_{x=yz}\\text{ }{\\bigvee }_{y=pq}\\left\\{ \\mu\n_{B_{1}}^{+}(p)\\wedge \\mu _{B_{2}}^{+}(q)\\wedge \\mu _{B_{3}}^{+}(z)\\right\\}\n\\\\\n&=&{\\bigvee }_{x=(pq)z}\\left\\{ \\mu _{B_{1}}^{+}(p)\\wedge \\mu\n_{B_{2}}^{+}(q)\\wedge \\mu _{B_{3}}^{+}(z)\\right\\} \\\\\n&=&{\\bigvee }_{x=(zq)p}\\left\\{ \\mu _{B_{3}}^{+}(z)\\wedge \\mu\n_{B_{2}}^{+}(q)\\wedge \\mu _{B_{1}}^{+}(p)\\right\\} \\\\\n&=&{\\bigvee }_{x=sp}\\left\\{ \\left( {\\bigvee }_{s=zq}\\left\\{ \\mu\n_{B_{3}}^{+}(z)\\wedge \\mu _{B_{2}}^{+}(q)\\right\\} \\right) \\wedge \\mu\n_{B_{1}}^{+}(p)\\right\\} \\\\\n&=&{\\bigvee }_{x=sp}\\left\\{ \\left( \\mu _{B_{3}}^{+}\\circ \\mu\n_{B_{2}}^{+}\\right) (s)\\wedge \\mu _{B_{1}}^{+}(p)\\right\\} \\\\\n&=&\\left( \\left( \\mu _{B_{3}}^{+}\\circ \\mu _{B_{2}}^{+}\\right) \\circ \\mu\n_{B_{1}}^{+}\\right) (x).\n\\end{eqnarray*\nAn\n\\begin{eqnarray*}\n\\left( \\left( \\mu _{B_{1}}^{-}\\circ \\mu _{B_{2}}^{-}\\right) \\circ \\mu\n_{B_{3}}^{-}\\right) (x) &=&{\\bigwedge }_{x=yz}\\left\\{ \\left( \\mu\n_{B_{1}}^{-}\\circ \\mu _{B_{2}}^{-}\\right) (y)\\vee \\mu _{B_{3}}^{-}(z)\\right\\}\n\\\\\n&=&{\\bigwedge }_{x=yz}\\left\\{ \\left( {\\bigwedge }_{y=pq}\\left\\{ \\mu\n_{B_{1}}^{-}(p)\\vee \\mu _{B_{2}}^{-}(q)\\right\\} \\right) \\vee \\mu\n_{B_{3}}^{-}(z)\\right\\} \\\\\n&=&{\\bigwedge }_{x=yz}\\text{ }{\\bigwedge }_{y=pq}\\left\\{ \\mu\n_{B_{1}}^{-}(p)\\vee \\mu _{B_{2}}^{-}(q)\\vee \\mu _{B_{3}}^{-}(z)\\right\\} \\\\\n&=&{\\bigwedge }_{x=(pq)z}\\left\\{ \\mu _{B_{1}}^{-}(p)\\vee \\mu\n_{B_{2}}^{-}(q)\\vee \\mu _{B_{3}}^{-}(z)\\right\\} \\\\\n&=&{\\bigwedge }_{x=(zq)p}\\left\\{ \\mu _{B_{3}}^{-}(z)\\vee \\mu\n_{B_{2}}^{-}(q)\\vee \\mu _{B_{1}}^{-}(p)\\right\\} \\\\\n&=&{\\bigwedge }_{x=sp}\\left\\{ \\left( {\\bigwedge }_{s=zq}\\left\\{ \\mu\n_{B_{3}}^{-}(z)\\vee \\mu _{B_{2}}^{-}(q)\\right\\} \\right) \\vee \\mu\n_{B_{1}}^{-}(p)\\right\\} \\\\\n&=&{\\bigwedge }_{x=sp}\\left\\{ \\left( \\mu _{B_{3}}^{-}\\circ \\mu\n_{B_{2}}^{-}\\right) (s)\\vee \\mu _{B_{1}}^{-}(p)\\right\\} \\\\\n&=&\\left( \\left( \\mu _{B_{3}}^{-}\\circ \\mu _{B_{2}}^{-}\\right) \\circ \\mu\n_{B_{1}}^{-}\\right) (x).\n\\end{eqnarray*}\n\nHence $(BVF(S),\\circ )$ is an LA-semigroup. $\\ \\ \\Box $\n\n\\begin{corollary}\n\\label{C1}If $S$ is an LA-semigroup, then the medial law holds in $BVF(S)$.\n\\end{corollary}\n\n\\textbf{Proof. }Let $B_{1}=\\left\\langle \\mu _{B_{1}}^{+},\\mu\n_{B_{1}}^{-}\\right\\rangle $, $B_{2}=\\left\\langle \\mu _{B_{2}}^{+},\\mu\n_{B_{2}}^{-}\\right\\rangle ,$ $B_{3}=\\left\\langle \\mu _{B_{3}}^{+},\\mu\n_{B_{3}}^{-}\\right\\rangle $ and $B_{4}=\\left\\langle \\mu _{B_{4}}^{+},\\mu\n_{B_{4}}^{-}\\right\\rangle $ be in $BVF(S)$. By successive use of left\ninvertive la\n\\begin{eqnarray*}\n\\left( \\mu _{B_{1}}^{+}\\circ \\mu _{B_{2}}^{+}\\right) \\circ \\left( \\mu\n_{B_{3}}^{+}\\circ \\mu _{B_{4}}^{+}\\right) &=&\\left( \\left( \\mu\n_{B_{3}}^{+}\\circ \\mu _{B_{4}}^{+}\\right) \\circ \\mu _{B_{2}}^{+}\\right)\n\\circ \\mu _{B_{1}}^{+} \\\\\n&=&\\left( \\left( \\mu _{B_{2}}^{+}\\circ \\mu _{B_{4}}^{+}\\right) \\circ \\mu\n_{B_{3}}^{+}\\right) \\circ \\mu _{B_{1}}^{+} \\\\\n&=&\\left( \\mu _{B_{1}}^{+}\\circ \\mu _{B_{3}}^{+}\\right) \\circ \\left( \\mu\n_{B_{2}}^{+}\\circ \\mu _{B_{4}}^{+}\\right) .\n\\end{eqnarray*\nAn\n\\begin{eqnarray*}\n\\left( \\mu _{B_{1}}^{-}\\circ \\mu _{B_{2}}^{-}\\right) \\circ \\left( \\mu\n_{B_{3}}^{-}\\circ \\mu _{B_{4}}^{-}\\right) &=&\\left( \\left( \\mu\n_{B_{3}}^{-}\\circ \\mu _{B_{4}}^{-}\\right) \\circ \\mu _{B_{2}}^{-}\\right)\n\\circ \\mu _{B_{1}}^{-} \\\\\n&=&\\left( \\left( \\mu _{B_{2}}^{-}\\circ \\mu _{B_{4}}^{-}\\right) \\circ \\mu\n_{B_{3}}^{-}\\right) \\circ \\mu _{B_{1}}^{-} \\\\\n&=&\\left( \\mu _{B_{1}}^{-}\\circ \\mu _{B_{3}}^{-}\\right) \\circ \\left( \\mu\n_{B_{2}}^{-}\\circ \\mu _{B_{4}}^{-}\\right) .\n\\end{eqnarray*\nHence this shows that the medial law holds in $BVF(S)$. $\\ \\ \\Box $\n\n\\section{\\textbf{Bipolar-valued fuzzy ideals in LA-semigroup}}\n\n\\textbf{Definition 3.1 }A BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ of an LA-semigroup $S$ is called a bipolar-valued\nfuzzy LA-subsemigroup of $S$ i\n\\begin{equation*}\n\\mu _{B}^{+}\\left( xy\\right) \\geq \\mu _{B}^{+}\\left( x\\right) \\wedge \\mu\n_{B}^{+}\\left( y\\right) \\text{\\ \\ and \\ \\ }\\mu _{B}^{-}\\left( xy\\right) \\leq\n\\mu _{B}^{-}\\left( x\\right) \\vee \\mu _{B}^{-}\\left( y\\right)\n\\end{equation*\nfor all $x,y\\in S$.\n\n\\textbf{Definition 3.2 }A BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ of an LA-semigroup $S$ is called a bipolar-valued\nfuzzy left ideal of $S$ i\n\\begin{equation*}\n\\mu _{B}^{+}\\left( xy\\right) \\geq \\mu _{B}^{+}\\left( y\\right) \\text{ \\ \\ and\n\\ \\ }\\mu _{B}^{-}\\left( xy\\right) \\leq \\mu _{B}^{-}\\left( y\\right)\n\\end{equation*\nfor all $x,y\\in S$.\n\n\\textbf{Definition 3.3 }A BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ of an LA-semigroup $S$ is called a bipolar-valued\nfuzzy right ideal of $S$ i\n\\begin{equation*}\n\\mu _{B}^{+}\\left( xy\\right) \\geq \\mu _{B}^{+}\\left( x\\right) \\text{ \\ \\ and\n\\ }\\mu _{B}^{-}\\left( xy\\right) \\leq \\mu _{B}^{-}\\left( x\\right)\n\\end{equation*\nfor all $x,y\\in S$.\n\nA BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ of an\nLA-semigroup $S$ is called a BVF-ideal or BVF-two-sided ideal of $S$ if \nB=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ is both BVF-left and\nBVF-right ideal of $S$.\n\n\\textbf{Example 3.1} Let $S=\\{a,b,c,d\\}$, the binary operation \"$\\cdot $\" on \n$S$ be defined as follows\n\\begin{equation*}\n\\begin{tabular}{l|llll}\n$\\cdot $ & $a$ & $b$ & $c$ & $d$ \\\\ \\hline\n$a$ & $b$ & $d$ & $c$ & $a$ \\\\ \n$b$ & $a$ & $b$ & $c$ & $d$ \\\\ \n$c$ & $c$ & $c$ & $c$ & $c$ \\\\ \n$d$ & $d$ & $a$ & $c$ & $b\n\\end{tabular\n\\end{equation*\nClearly, $S$ is an LA-semigroup. But $S$ is not a semigroup because \nd=d\\cdot (b\\cdot a)\\neq (d\\cdot b)\\cdot a=b.$ Now we define BVF-subset a\n\\begin{equation*}\nB=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle =\\left\\langle \\left( \n\\frac{a}{0.2},\\frac{b}{0.2},\\frac{c}{0.7},\\frac{d}{0.2}\\right) ,\\text{ \n\\left( \\frac{a}{-0.5},\\frac{b}{-0.5},\\frac{c}{-0.8},\\frac{d}{-0.5}\\right)\n\\right\\rangle .\n\\end{equation*\nClearly $B$ is a BVF-ideal of $S.$\n\n\\begin{proposition}\n\\label{BP2}Every BVF-left (BVF-right) ideal $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ of an LA-semigroup $S$ is a bipolar-valued fuzzy\nLA-subsemigroup of $S$.\n\\end{proposition}\n\n\\textbf{Proof. }Let $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $\nbe a BVF-left ideal of $S$ and for any $x,y\\in S$,\n\n\\begin{equation*}\n\\mu _{B}^{+}\\left( xy\\right) \\geq \\mu _{B}^{+}\\left( y\\right) \\geq \\mu\n_{B}^{+}\\left( x\\right) \\wedge \\mu _{B}^{+}\\left( y\\right) \\text{.}\n\\end{equation*\nAn\n\\begin{equation*}\n\\mu _{B}^{-}\\left( xy\\right) \\leq \\mu _{B}^{-}\\left( y\\right) \\leq \\mu\n_{B}^{-}\\left( x\\right) \\vee \\mu _{B}^{-}\\left( y\\right) .\n\\end{equation*\nHence $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ is a\nbipolar-valued fuzzy LA-subsemigroup of $S$. The other case can be prove in\na similar way. $\\ \\ \\Box $\n\n\\begin{lemma}\n\\label{L56}Let $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ be a\nBVF-subset of an LA-semigroup $S.$ Then\n\\end{lemma}\n\n(1) $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ is a\nBVF-LA-subsemigroup of $S$\\ if and only if $\\mu _{B}^{+}\\circ \\mu\n_{B}^{+}\\subseteq \\mu _{B}^{+}$ and $\\mu _{B}^{-}\\circ \\mu _{B}^{-}\\supseteq\n\\mu _{B}^{-}.$\n\n(2) $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ is a BVF-left\n(resp. BVF-right) ideal of $S$\\ if and only if $\\mathcal{S}_{\\Gamma\n}^{+}\\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+}$ and $\\mathcal{S}_{\\Gamma\n}^{-}\\circ \\mu _{B}^{-}\\supseteq \\mu _{B}^{-}$ (resp. $\\mu _{B}^{+}\\circ \n\\mathcal{S}_{\\Gamma }^{+}\\subseteq \\mu _{B}^{+}$ and $\\mu _{B}^{-}\\circ \n\\mathcal{S}_{\\Gamma }^{-}\\supseteq \\mu _{B}^{-}$)$.$\n\n\\textbf{Proof. }(1)\\textbf{\\ }Let $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ be a BVF-LA-subsemigroup of $S$ and $x\\in S.$ If \n\\left( \\mu _{B}^{+}\\circ \\mu _{B}^{+}\\right) \\left( x\\right) =0$ and $\\left(\n\\mu _{B}^{-}\\circ \\mu _{B}^{-}\\right) \\left( x\\right) =0,$ then $\\left( \\mu\n_{B}^{+}\\circ \\mu _{B}^{+}\\right) \\left( x\\right) \\leq \\mu _{B}^{+}\\left(\nx\\right) $ and $\\left( \\mu _{B}^{-}\\circ \\mu _{B}^{-}\\right) \\left( x\\right)\n\\geq \\mu _{B}^{-}\\left( x\\right) .$ Otherwise\n\\begin{equation*}\n\\left( \\mu _{B}^{+}\\circ \\mu _{B}^{+}\\right) \\left( x\\right) ={\\bigvee \n_{x=yz}\\left\\{ \\mu _{B}^{+}\\left( y\\right) \\wedge \\mu _{B}^{+}\\left(\nz\\right) \\right\\} \\leq {\\bigvee }_{x=yz}\\mu _{B}^{+}\\left( yz\\right) =\\mu\n_{B}^{+}\\left( x\\right) .\n\\end{equation*\nAn\n\\begin{equation*}\n\\left( \\mu _{B}^{-}\\circ \\mu _{B}^{-}\\right) \\left( x\\right) ={\\bigwedge \n_{x=yz}\\left\\{ \\mu _{B}^{-}\\left( y\\right) \\vee \\mu _{B}^{-}\\left( z\\right)\n\\right\\} \\geq {\\bigwedge }_{x=yz}\\mu _{B}^{-}\\left( yz\\right) =\\mu\n_{B}^{-}\\left( x\\right) .\n\\end{equation*\nThus $\\mu _{B}^{+}\\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+}$ and $\\mu\n_{B}^{-}\\circ \\mu _{B}^{-}\\supseteq \\mu _{B}^{-}.$\\newline\nConversely, let $\\mu _{B}^{+}\\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+}$, $\\mu\n_{B}^{-}\\circ \\mu _{B}^{-}\\supseteq \\mu _{B}^{-}$ and $x,y\\in S,$ the\n\\begin{equation*}\n\\mu _{B}^{+}\\left( xy\\right) \\geq \\left( \\mu _{B}^{+}\\circ \\mu\n_{B}^{+}\\right) \\left( xy\\right) ={\\bigvee }_{xy=ab}\\left\\{ \\mu\n_{B}^{+}\\left( a\\right) \\wedge \\mu _{B}^{+}\\left( b\\right) \\right\\} \\geq \\mu\n_{B}^{+}\\left( x\\right) \\wedge \\mu _{B}^{+}\\left( y\\right) .\n\\end{equation*\nAn\n\\begin{equation*}\n\\mu _{B}^{-}\\left( xy\\right) \\leq \\left( \\mu _{B}^{-}\\circ \\mu\n_{B}^{-}\\right) \\left( xy\\right) ={\\bigwedge }_{xy=ab}\\left\\{ \\mu\n_{B}^{-}\\left( a\\right) \\vee \\mu _{B}^{-}\\left( b\\right) \\right\\} \\leq \\mu\n_{B}^{-}\\left( x\\right) \\vee \\mu _{B}^{-}\\left( y\\right) .\n\\end{equation*\nSo $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ is a\nBVF-LA-subsemigroup of $S.$\n\n(2) Let $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ be a\nBVF-left ideal of $S$ and $x\\in S.$ If $\\left( \\mathcal{S}_{\\Gamma\n}^{+}\\circ \\mu _{B}^{+}\\right) \\left( x\\right) =0$ and $\\left( \\mathcal{S\n_{\\Gamma }^{-}\\circ \\mu _{B}^{-}\\right) \\left( x\\right) =0,$ then $\\left( \n\\mathcal{S}_{\\Gamma }^{+}\\circ \\mu _{B}^{+}\\right) \\left( x\\right) \\leq \\mu\n_{B}^{+}\\left( x\\right) $ and $\\left( \\mathcal{S}_{\\Gamma }^{-}\\circ \\mu\n_{B}^{-}\\right) \\left( x\\right) \\geq \\mu _{B}^{-}\\left( x\\right) .$\nOtherwise\n\\begin{eqnarray*}\n\\left( \\mathcal{S}_{\\Gamma }^{+}\\circ \\mu _{B}^{+}\\right) \\left( x\\right) &=\n{\\bigvee }_{x=ab}\\left\\{ \\mathcal{S}_{\\Gamma }^{+}\\left( a\\right) \\wedge \\mu\n_{B}^{+}\\left( b\\right) \\right\\} ={\\bigvee }_{x=ab}\\left\\{ 1\\wedge \\mu\n_{B}^{+}\\left( b\\right) \\right\\} \\\\\n&=&{\\bigvee }_{x=ab}\\mu _{B}^{+}\\left( b\\right) \\leq {\\bigvee }_{x=ab}\\mu\n_{B}^{+}\\left( ab\\right) =\\mu _{B}^{+}\\left( x\\right) .\n\\end{eqnarray*\nAn\n\\begin{eqnarray*}\n\\left( \\mathcal{S}_{\\Gamma }^{-}\\circ \\mu _{B}^{-}\\right) \\left( x\\right) &=\n{\\bigwedge }_{x=ab}\\left\\{ \\mathcal{S}_{\\Gamma }^{-}\\left( a\\right) \\vee \\mu\n_{B}^{-}\\left( b\\right) \\right\\} ={\\bigwedge }_{x=ab}\\left\\{ -1\\vee \\mu\n_{B}^{-}\\left( b\\right) \\right\\} \\\\\n&=&{\\bigwedge }_{x=ab}\\mu _{B}^{-}\\left( b\\right) \\geq {\\bigwedge \n_{x=ab}\\mu _{B}^{-}\\left( ab\\right) =\\mu _{B}^{-}\\left( x\\right) .\n\\end{eqnarray*\nThus $\\mathcal{S}_{\\Gamma }^{+}\\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+}$ and \n$\\mathcal{S}_{\\Gamma }^{-}\\circ \\mu _{B}^{-}\\supseteq \\mu _{B}^{-}.$\\newline\nConversely, let $\\mathcal{S}_{\\Gamma }^{+}\\circ \\mu _{B}^{+}\\subseteq \\mu\n_{B}^{+}$, $\\mathcal{S}_{\\Gamma }^{-}\\circ \\mu _{B}^{-}\\supseteq \\mu\n_{B}^{-} $ and $x,y\\in S,$ then \n\\begin{eqnarray*}\n\\mu _{B}^{+}\\left( xy\\right) &\\geq &\\left( \\mathcal{S}_{\\Gamma }^{+}\\circ\n\\mu _{B}^{+}\\right) \\left( xy\\right) ={\\bigvee }_{xy=ab}\\left\\{ \\mathcal{S\n_{\\Gamma }^{+}\\left( a\\right) \\wedge \\mu _{B}^{+}\\left( b\\right) \\right\\} \\\\\n&\\geq &\\mathcal{S}_{\\Gamma }^{+}\\left( x\\right) \\wedge \\mu _{B}^{+}\\left(\ny\\right) =1\\wedge \\mu _{B}^{+}\\left( y\\right) =\\mu _{B}^{+}\\left( y\\right) .\n\\end{eqnarray*\nAnd \n\\begin{eqnarray*}\n\\mu _{B}^{-}\\left( xy\\right) &\\leq &\\left( \\mathcal{S}_{\\Gamma }^{-}\\circ\n\\mu _{B}^{-}\\right) \\left( xy\\right) ={\\bigwedge }_{xy=ab}\\left\\{ \\mathcal{S\n_{\\Gamma }^{-}\\left( a\\right) \\vee \\mu _{B}^{-}\\left( b\\right) \\right\\} \\\\\n&\\leq &\\mathcal{S}_{\\Gamma }^{-}\\left( x\\right) \\vee \\mu _{B}^{-}\\left(\ny\\right) =-1\\vee \\mu _{B}^{-}\\left( y\\right) =\\mu _{B}^{-}\\left( y\\right) .\n\\end{eqnarray*\nThus $\\mu _{B}^{+}\\left( xy\\right) \\geq \\mu _{B}^{+}\\left( y\\right) $ and \n\\mu _{B}^{-}\\left( xy\\right) \\leq \\mu _{B}^{-}\\left( y\\right) .$\\ Thus \nB=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ is a BVF-left ideal\nof $S.$ The second case can be seen in a similar way. $\\ \\ \\Box $\n\nLet $B_{1}=\\left\\langle \\mu _{B_{1}}^{+},\\mu _{B_{1}}^{-}\\right\\rangle $ and \n$B_{2}=\\left\\langle \\mu _{B_{2}}^{+},\\mu _{B_{2}}^{-}\\right\\rangle $ be two\nBVF-subsets of an LA-semigroup $S.$ The symbol $B_{1}\\cap B_{2}$ will mean\nthe following\n\n\\begin{equation*}\n\\left( \\mu _{B_{1}}^{+}\\cap \\mu _{B_{2}}^{+}\\right) (x)=\\mu\n_{B_{1}}^{+}(x)\\wedge \\mu _{B_{2}}^{+}(x),\\text{ for all }x\\in S.\n\\end{equation*}\n\n\\begin{equation*}\n\\left( \\mu _{B_{1}}^{-}\\cup \\mu _{B_{2}}^{-}\\right) (x)=\\mu\n_{B_{1}}^{-}(x)\\vee \\mu _{B_{2}}^{-}(x),\\text{ for all }x\\in S.\n\\end{equation*\nThe symbol $A\\cup B$ will mean the following\n\n\\begin{equation*}\n\\left( \\mu _{B_{1}}^{+}\\cup \\mu _{B_{2}}^{+}\\right) (x)=\\mu\n_{B_{1}}^{+}(x)\\vee \\mu _{B_{2}}^{+}(x),\\text{ for all }x\\in S.\n\\end{equation*}\n\n\\begin{equation*}\n\\left( \\mu _{B_{1}}^{-}\\cap \\mu _{B_{2}}^{-}\\right) (x)=\\mu\n_{B_{1}}^{-}(x)\\wedge \\mu _{B_{2}}^{-}(x),\\text{ for all }x\\in S.\n\\end{equation*}\n\n\\begin{theorem}\nLet $S$ be an LA-semigroup and $B_{1}=\\left\\langle \\mu _{B_{1}}^{+},\\mu\n_{B_{1}}^{-}\\right\\rangle $ be a BVF-right ideal of $S$ and \nB_{2}=\\left\\langle \\mu _{B_{2}}^{+},\\mu _{B_{2}}^{-}\\right\\rangle $ be a\nBVF-left ideal of $S,$ then $B_{1}\\circ B_{2}\\subseteq B_{1}\\cap B_{2}.$\n\\end{theorem}\n\n\\textbf{Proof. }Let for any $x,y,z\\in S,$ if $x\\neq yz,$ then we hav\n\\begin{equation*}\n\\left( \\mu _{B_{1}}^{+}\\circ \\mu _{B_{2}}^{+}\\right) (x)=0\\leq \\mu\n_{B_{1}}^{+}(x)\\wedge \\mu _{B_{2}}^{+}(x)=\\left( \\mu _{B_{1}}^{+}\\cap \\mu\n_{B_{2}}^{+}\\right) (x).\n\\end{equation*\nAn\n\\begin{equation*}\n\\left( \\mu _{B_{1}}^{-}\\circ \\mu _{B_{2}}^{-}\\right) (x)=0\\geq \\mu\n_{B_{1}}^{-}(x)\\vee \\mu _{B_{2}}^{-}(x)=\\left( \\mu _{B_{1}}^{-}\\cup \\mu\n_{B_{2}}^{-}\\right) (x).\n\\end{equation*\nOtherwis\n\\begin{eqnarray*}\n\\left( \\mu _{B_{1}}^{+}\\circ \\mu _{B_{2}}^{+}\\right) (x) &=&{\\bigvee \n_{x=yz}\\left\\{ \\mu _{B_{1}}^{+}\\left( y\\right) \\wedge \\mu _{B_{2}}^{+}\\left(\nz\\right) \\right\\}  \\\\\n&\\leq &{\\bigvee }_{x=yz}\\left\\{ \\mu _{B_{1}}^{+}\\left( yz\\right) \\wedge \\mu\n_{B_{2}}^{+}\\left( yz\\right) \\right\\}  \\\\\n&=&\\left\\{ \\mu _{B_{1}}^{+}\\left( x\\right) \\wedge \\mu _{B_{2}}^{+}\\left(\nx\\right) \\right\\}  \\\\\n&=&\\left( \\mu _{B_{1}}^{+}\\cap \\mu _{B_{2}}^{+}\\right) (x).\n\\end{eqnarray*\nAn\n\\begin{eqnarray*}\n\\left( \\mu _{B_{1}}^{-}\\circ \\mu _{B_{2}}^{-}\\right) (x) &=&{\\bigvee \n_{x=yz}\\left\\{ \\mu _{B_{1}}^{-}\\left( y\\right) \\vee \\mu _{B_{2}}^{-}\\left(\nz\\right) \\right\\}  \\\\\n&\\geq &{\\bigvee }_{x=yz}\\left\\{ \\mu _{B_{1}}^{-}\\left( yz\\right) \\vee \\mu\n_{B_{2}}^{-}\\left( yz\\right) \\right\\}  \\\\\n&=&\\left\\{ \\mu _{B_{1}}^{-}\\left( x\\right) \\vee \\mu _{B_{2}}^{-}\\left(\nx\\right) \\right\\}  \\\\\n&=&\\left( \\mu _{B_{1}}^{-}\\cup \\mu _{B_{2}}^{-}\\right) (x).\n\\end{eqnarray*\nThus we get $\\mu _{B_{1}}^{+}\\circ \\mu _{B_{2}}^{+}\\subseteq \\mu\n_{B_{1}}^{+}\\cap \\mu _{B_{2}}^{+}$ and $\\mu _{B_{1}}^{-}\\circ \\mu\n_{B_{2}}^{-}\\supseteq \\mu _{B_{1}}^{-}\\cup \\mu _{B_{2}}^{-}.$ Hence \nB_{1}\\circ B_{2}\\subseteq B_{1}\\cap B_{2}.$ $\\ \\ \\Box $\n\n\\begin{proposition}\n\\label{P101}\\textit{Let }$B_{1}=\\left\\langle \\mu _{B_{1}}^{+},\\mu\n_{B_{1}}^{-}\\right\\rangle $\\textit{\\ and }$B_{2}=\\left\\langle \\mu\n_{B_{2}}^{+},\\mu _{B_{2}}^{-}\\right\\rangle $\\textit{\\ be two\nBVF-LA-subsemigroups of }$S$\\textit{. Then }$B_{1}\\cap B_{2}$\\textit{\\ is\nalso a BVF-LA-subsemigroup of }$S$\\textit{.\\ }\n\\end{proposition}\n\n\\textbf{Proof.\\ }Let $B_{1}=\\left\\langle \\mu _{B_{1}}^{+},\\mu\n_{B_{1}}^{-}\\right\\rangle $ and $B_{2}=\\left\\langle \\mu _{B_{2}}^{+},\\mu\n_{B_{2}}^{-}\\right\\rangle $ be two \\textit{BVF-LA-subsemigroups} of $S$. Let \n$x,y\\in S$. The\n\\begin{eqnarray*}\n\\left( \\mu _{B_{1}}^{+}\\cap \\mu _{B_{2}}^{+}\\right) \\left( xy\\right)  &=&\\mu\n_{B_{1}}^{+}\\left( xy\\right) \\wedge \\mu _{B_{2}}^{+}\\left( xy\\right)  \\\\\n&\\geq &\\left( \\mu _{B_{1}}^{+}\\left( x\\right) \\wedge \\mu _{B_{1}}^{+}\\left(\ny\\right) \\right) \\wedge \\left( \\mu _{B_{2}}^{+}\\left( x\\right) \\wedge \\mu\n_{B_{2}}^{+}\\left( y\\right) \\right)  \\\\\n&=&\\left( \\mu _{B_{1}}^{+}\\left( x\\right) \\wedge \\mu _{B_{2}}^{+}\\left(\nx\\right) \\right) \\wedge \\left( \\mu _{B_{1}}^{+}\\left( y\\right) \\wedge \\mu\n_{B_{2}}^{+}\\left( y\\right) \\right)  \\\\\n&=&\\left( \\mu _{B_{1}}^{+}\\cap \\mu _{B_{2}}^{+}\\right) \\left( x\\right)\n\\wedge \\left( \\mu _{B_{1}}^{+}\\cap \\mu _{B_{2}}^{+}\\right) \\left( y\\right) .\n\\end{eqnarray*\nAn\n\\begin{eqnarray*}\n\\left( \\mu _{B_{1}}^{-}\\cup \\mu _{B_{2}}^{-}\\right) \\left( xy\\right)  &=&\\mu\n_{B_{1}}^{-}\\left( xy\\right) \\vee \\mu _{B_{2}}^{-}\\left( xy\\right)  \\\\\n&\\leq &\\left( \\mu _{B_{1}}^{-}\\left( x\\right) \\vee \\mu _{B_{1}}^{-}\\left(\ny\\right) \\right) \\vee \\left( \\mu _{B_{2}}^{-}\\left( x\\right) \\vee \\mu\n_{B_{2}}^{-}\\left( y\\right) \\right)  \\\\\n&=&\\left( \\mu _{B_{1}}^{-}\\left( x\\right) \\vee \\mu _{B_{2}}^{-}\\left(\nx\\right) \\right) \\vee \\left( \\mu _{B_{1}}^{-}\\left( y\\right) \\vee \\mu\n_{B_{2}}^{-}\\left( y\\right) \\right)  \\\\\n&=&\\left( \\mu _{B_{1}}^{-}\\cup \\mu _{B_{2}}^{-}\\right) \\left( x\\right) \\vee\n\\left( \\mu _{B_{1}}^{-}\\cup \\mu _{B_{2}}^{-}\\right) \\left( y\\right) .\n\\end{eqnarray*\nThus $B_{1}\\cap B_{2}$\\textit{\\ is also a bipolar-valued fuzzy\nLA-subsemigroup of }$S$\\textit{.} $\\ \\ \\Box $\n\n\\begin{proposition}\n\\textit{Let }$B_{1}=\\left\\langle \\mu _{B_{1}}^{+},\\mu\n_{B_{1}}^{-}\\right\\rangle $\\textit{\\ and }$B_{2}=\\left\\langle \\mu\n_{B_{2}}^{+},\\mu _{B_{2}}^{-}\\right\\rangle $\\textit{\\ be two BVF-left (resp.\nBVF-right, BVF-two-sided) ideal of }$S$\\textit{. Then }$B_{1}\\cap B_{2}\n\\textit{\\ is also a BVF-left (resp. BVF-right, BVF-two-sided) ideal of }$S.$\n\\end{proposition}\n\n\\textbf{Proof.\\ }The proof is similar to the proof of Proposition \\ref{P101\n. $\\ \\ \\Box $\n\n\\begin{lemma}\n\\label{L2}In an LA-semigroup $S$ with left identity, for every BVF-left\nideal $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ of $S$, we\nhave $\\Gamma \\circ B=B.$ Where $\\Gamma =\\left\\langle \\mathcal{S}_{\\Gamma\n}^{+}(x),\\mathcal{S}_{\\Gamma }^{-}(x)\\right\\rangle .$\n\\end{lemma}\n\n\\textbf{Proof. }Let $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $\nbe a BVF-left ideal of $S.$ It is sufficient to show that $\\mathcal{S\n_{\\Gamma }^{+}\\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+}$ and $\\mathcal{S\n_{\\Gamma }^{-}\\circ \\mu _{B}^{-}\\supseteq \\mu _{B}^{-}$. Now $x=ex$, for all \n$x$ in $S$, as $e$ is left identity in $S$. S\n\\begin{equation*}\n(\\mathcal{S}_{\\Gamma }^{+}\\circ \\mu _{B}^{+})(x)={\\bigvee }_{x=yz}\\left\\{ \n\\mathcal{S}_{\\Gamma }^{+}(y)\\wedge \\mu _{B}^{+}(z)\\right\\} \\geq \\mathcal{S\n_{\\Gamma }^{+}(e)\\wedge \\mu _{B}^{+}(x)=1\\wedge \\mu _{B}^{+}(x)=\\mu\n_{B}^{+}(x).\n\\end{equation*\nAn\n\\begin{equation*}\n(\\mathcal{S}_{\\Gamma }^{-}\\circ \\mu _{B}^{-})(x)={\\bigwedge }_{x=yz}\\left\\{ \n\\mathcal{S}_{\\Gamma }^{-}(y)\\vee \\mu _{B}^{-}(z)\\right\\} \\leq \\mathcal{S\n_{\\Gamma }^{-}(e)\\vee \\mu _{B}^{-}(x)=-1\\vee \\mu _{B}^{-}(x)=\\mu _{B}^{-}(x).\n\\end{equation*\nThus $\\mathcal{S}_{\\Gamma }^{+}\\circ \\mu _{B}^{+}\\supseteq \\mu _{B}^{+}$ and \n$\\mathcal{S}_{\\Gamma }^{-}\\circ \\mu _{B}^{-}\\subseteq \\mu _{B}^{-}.$ Hence \n\\Gamma \\circ B=B.$ $\\ \\ \\Box $\n\n\\textbf{Definition 3.4 }Let $S$ be an LA-semigroup and let $\\emptyset \\neq\nA\\subseteq S.$ Then bipolar-valued fuzzy characteristic function $\\chi\n_{A}=\\left\\langle \\mu _{\\chi _{A}}^{+},\\mu _{\\chi _{A}}^{-}\\right\\rangle $\nof $A$ is defined a\n\\begin{equation*}\n\\mu _{\\chi _{A}}^{+}=\\left\\{ \n\\begin{array}{c}\n1\\text{ \\ \\ if }x\\in A \\\\ \n0\\text{ \\ \\ if }x\\notin \n\\end{array\n\\right. \\text{ \\ \\ and \\ }\\mu _{\\chi _{A}}^{-}=\\left\\{ \n\\begin{array}{c}\n-1\\text{ \\ \\ if }x\\in A \\\\ \n\\text{ \\ }0\\text{ \\ \\ if }x\\notin A\n\\end{array\n\\right.\n\\end{equation*}\n\n\\begin{theorem}\n\\label{T119}Let $A$ be a nonempty subset of an LA-semigroup $S$. Then $A$ is\nan LA-subsemigroup of $S$ if and only if $\\chi _{A}$ is a\nBVF-LA-subsemigroup of $S$.\n\\end{theorem}\n\n\\textbf{Proof. }Let $A$ be an LA-subsemigroup of $S$. For any $x,y\\in S,$ we\nhave the following cases:\n\nCase $\\left( 1\\right) :$ If $x,y\\in A$, then $xy\\in A$. Since $A$ is an\nLA-subsemigroup of $S$. Then $\\mu _{\\chi _{A}}^{+}\\left( xy\\right) =1,$ $\\mu\n_{\\chi _{A}}^{+}\\left( x\\right) =1$ and $\\mu _{\\chi _{A}}^{+}\\left( y\\right)\n=1$. Therefore \n\\begin{equation*}\n\\mu _{\\chi _{A}}^{+}\\left( xy\\right) =\\mu _{\\chi _{A}}^{+}\\left( x\\right)\n\\wedge \\mu _{\\chi _{A}}^{+}\\left( y\\right) .\n\\end{equation*\nAnd $\\mu _{\\chi _{A}}^{-}\\left( xy\\right) =-1,$ $\\mu _{\\chi _{A}}^{-}\\left(\nx\\right) =-1$ and $\\mu _{\\chi _{A}}^{-}\\left( y\\right) =-1$. Therefore \n\\begin{equation*}\n\\mu _{\\chi _{A}}^{-}\\left( xy\\right) =\\mu _{\\chi _{A}}^{-}\\left( x\\right)\n\\vee \\mu _{\\chi _{A}}^{-}\\left( y\\right) .\n\\end{equation*\nCase $\\left( 2\\right) :$ If $x,y\\notin A$, then $\\mu _{\\chi _{A}}^{+}\\left(\nx\\right) =0$ and $\\mu _{\\chi _{A}}^{+}\\left( y\\right) =0$. So \\ \\ \\ \\ \\ \\ \\ \n\\begin{equation*}\n\\text{ \\ \\ \\ \\ \\ \\ \\ }\\mu _{\\chi _{A}}^{+}\\left( xy\\right) \\geq 0=\\mu _{\\chi\n_{A}}^{+}\\left( x\\right) \\wedge \\mu _{\\chi _{A}}^{+}\\left( y\\right) .\n\\end{equation*\nAnd $\\mu _{\\chi _{A}}^{-}\\left( x\\right) =0$ and $\\mu _{\\chi _{A}}^{-}\\left(\ny\\right) =0$. So \n\\begin{equation*}\n\\text{ \\ \\ \\ \\ \\ \\ }\\mu _{\\chi _{A}}^{-}\\left( xy\\right) \\leq 0=\\mu _{\\chi\n_{A}}^{-}\\left( x\\right) \\vee \\mu _{\\chi _{A}}^{-}\\left( y\\right) .\n\\end{equation*}\n\nCase $\\left( 3\\right) :$ If $x\\in A$ or $y\\in A$. If $x\\in A$ and $y\\notin A\n, then $\\mu _{\\chi _{A}}^{+}\\left( x\\right) =1$ and $\\mu _{\\chi\n_{A}}^{+}\\left( y\\right) =0$. S\n\\begin{equation*}\n\\text{ \\ \\ \\ \\ \\ \\ \\ \\ }\\mu _{\\chi _{A}}^{+}\\left( xy\\right) \\geq 0=\\mu\n_{\\chi _{A}}^{+}\\left( x\\right) \\wedge \\mu _{\\chi _{A}}^{+}\\left( y\\right) .\n\\end{equation*\nNow if $x\\notin A$ and $y\\in A$, then $\\mu _{\\chi _{A}}^{+}\\left( x\\right) =0\n$ and $\\mu _{\\chi _{A}}^{+}\\left( y\\right) =1$. S\n\\begin{equation*}\n\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ }\\mu _{\\chi _{A}}^{+}\\left( xy\\right) \\geq 0=\\mu\n_{\\chi _{A}}^{+}\\left( x\\right) \\wedge \\mu _{\\chi _{A}}^{+}\\left( y\\right) .\n\\end{equation*\nAnd if $x\\in A$ or $y\\in A$. If $x\\in A$ and $y\\notin A$, then $\\mu _{\\chi\n_{A}}^{-}\\left( x\\right) =-1$ and $\\mu _{\\chi _{A}}^{-}\\left( y\\right) =0$.\nS\n\\begin{equation*}\n\\text{ \\ \\ \\ \\ \\ \\ \\ }\\mu _{\\chi _{A}}^{-}\\left( xy\\right) \\leq 0=\\mu _{\\chi\n_{A}}^{-}\\left( x\\right) \\vee \\mu _{\\chi _{A}}^{-}\\left( y\\right) .\n\\end{equation*\nNow if If $x\\notin A$ and $y\\in A$, then $\\mu _{\\chi _{A}}^{-}\\left(\nx\\right) =0$ and $\\mu _{\\chi _{A}}^{-}\\left( y\\right) =-1$. S\n\\begin{equation*}\n\\text{ \\ \\ \\ \\ \\ \\ \\ \\ }\\mu _{\\chi _{A}}^{-}\\left( xy\\right) \\leq 0=\\mu\n_{\\chi _{A}}^{-}\\left( x\\right) \\vee \\mu _{\\chi _{A}}^{-}\\left( y\\right) .\n\\end{equation*\nHence $\\chi _{A}=\\left\\langle \\mu _{\\chi _{A}}^{+},\\mu _{\\chi\n_{A}}^{-}\\right\\rangle $ is a BVF-LA-subsemigroup of $S$.\n\nConversely, suppose $\\chi _{A}=\\left\\langle \\mu _{\\chi _{A}}^{+},\\mu _{\\chi\n_{A}}^{-}\\right\\rangle $ is a BVF-LA-subsemigroup of $S$ and let $x,y\\in A$.\nThen we have \n\\begin{eqnarray*}\n\\text{ \\ \\ }\\mu _{\\chi _{A}}^{+}\\left( xy\\right) &\\geq &\\mu _{\\chi\n_{A}}^{+}\\left( x\\right) \\wedge \\mu _{\\chi _{A}}^{+}\\left( y\\right) =1\\wedge\n1=1 \\\\\n\\mu _{\\chi _{A}}^{+}\\left( xy\\right) &\\geq &1\\text{ but }\\mu _{\\chi\n_{A}}^{+}\\left( xy\\right) \\leq 1 \\\\\n\\mu _{\\chi _{A}}^{+}\\left( xy\\right) &=&1\n\\end{eqnarray*\nAnd \n\\begin{eqnarray*}\n\\text{ \\ \\ }\\mu _{\\chi _{A}}^{-}\\left( xy\\right) &\\leq &\\mu _{\\chi\n_{A}}^{-}\\left( x\\right) \\vee \\mu _{\\chi _{A}}^{-}\\left( y\\right) =-1\\vee\n-1=-1 \\\\\n\\mu _{\\chi _{A}}^{-}\\left( xy\\right) &\\leq &-1\\text{ but }\\mu _{\\chi\n_{A}}^{-}\\left( xy\\right) \\geq -1 \\\\\n\\mu _{\\chi _{A}}^{-}\\left( xy\\right) &=&-1\n\\end{eqnarray*\nHence $xy\\in A$. Therefore $A$ is an LA-subsemigroup of $S.$ $\\ \\ \\Box $\n\n\\begin{theorem}\nLet $A$ be a nonempty subset of an LA-semigroup $S$. Then $A$ is a left\n(resp. right) ideal of $S$ if and only if $\\chi _{A}$ is a BVF-left (resp.\nBVF-right) ideal of $S$.\n\\end{theorem}\n\n\\textbf{Proof. }The proof of this theorem is similar to Theorem \\ref{T119}. \n\\ \\ \\Box $\n\n\\textbf{Definition 3.5 }A BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ of an LA-semigroup $S$ is called a\\textbf{\\ \nBVF-generalized bi-ideal\\textbf{\\ }of $S$ i\n\\begin{equation*}\n\\mu _{B}^{+}\\left( (xy)z\\right) \\geq \\mu _{B}^{+}(x)\\wedge \\mu _{B}^{+}(y\n\\text{ \\ and \\ }\\mu _{B}^{-}\\left( (xy)z\\right) \\leq \\mu _{B}^{-}(x)\\vee \\mu\n_{B}^{-}(y),\\text{ for all }x,y,z\\in S.\n\\end{equation*}\n\n\\textbf{Definition 3.6 }A BVF-LA-subsemigroup $B=\\left\\langle \\mu\n_{B}^{+},\\mu _{B}^{-}\\right\\rangle $ of an LA-semigroup $S$ is called \n\\textbf{\\ }BVF-bi-ideal\\textbf{\\ }of $S$ i\n\\begin{equation*}\n\\mu _{B}^{+}\\left( (xy)z\\right) \\geq \\mu _{B}^{+}(x)\\wedge \\mu _{B}^{+}(y\n\\text{ \\ and \\ }\\mu _{B}^{-}\\left( (xy)z\\right) \\leq \\mu _{B}^{-}(x)\\vee \\mu\n_{B}^{-}(y),\\text{ for all }x,y,z\\in S.\n\\end{equation*}\n\n\\begin{lemma}\n\\label{L57}A BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ of an LA-semigroup $S$ is a BVF-generalized bi-ideal\nof $S$ if and only if $\\left( \\mu _{B}^{+}\\circ \\mathcal{S}_{\\Gamma\n}^{+}\\right) \\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+}$ and $\\left( \\mu\n_{B}^{-}\\circ \\mathcal{S}_{\\Gamma }^{-}\\right) \\circ \\mu _{B}^{-}\\supseteq\n\\mu _{B}^{-}.$\n\\end{lemma}\n\n\\textbf{Proof. }Let $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $\nbe a BVF-generalized bi-ideal of an LA-semigroup $S$ and $x\\in S.$ If \n\\left( \\left( \\mu _{B}^{+}\\circ \\mathcal{S}_{\\Gamma }^{+}\\right) \\circ \\mu\n_{B}^{+}\\right) \\left( x\\right) =0$ and $\\left( \\left( \\mu _{B}^{-}\\circ \n\\mathcal{S}_{\\Gamma }^{-}\\right) \\circ \\mu _{B}^{-}\\right) (x)=0,$ the\n\\begin{equation*}\n\\left( \\left( \\mu _{B}^{+}\\circ \\mathcal{S}_{\\Gamma }^{+}\\right) \\circ \\mu\n_{B}^{+}\\right) \\left( x\\right) =0\\leq \\mu _{B}^{+}\\left( x\\right) \\text{ \\\nand \\ }\\left( \\left( \\mu _{B}^{-}\\circ \\mathcal{S}_{\\Gamma }^{-}\\right)\n\\circ \\mu _{B}^{-}\\right) (x)=0\\geq \\mu _{B}^{-}(x).\n\\end{equation*\nOtherwis\n\\begin{eqnarray*}\n\\left( \\left( \\mu _{B}^{+}\\circ \\mathcal{S}_{\\Gamma }^{+}\\right) \\circ \\mu\n_{B}^{+}\\right) \\left( x\\right) &=&{\\bigvee }_{x=ab}\\left\\{ \\left( \\mu\n_{B}^{+}\\circ \\mathcal{S}_{\\Gamma }^{+}\\right) \\left( a\\right) \\wedge \\mu\n_{B}^{+}\\left( b\\right) \\right\\} \\\\\n&=&{\\bigvee }_{x=ab}\\left\\{ {\\bigvee }_{a=mn}\\left\\{ \\mu _{B}^{+}\\left(\nm\\right) \\wedge \\mathcal{S}_{\\Gamma }^{+}\\left( n\\right) \\right\\} \\wedge \\mu\n_{B}^{+}\\left( b\\right) \\right\\} \\\\\n&=&{\\bigvee }_{x=ab}{\\bigvee }_{a=mn}\\left\\{ \\left( \\mu _{B}^{+}\\left(\nm\\right) \\wedge 1\\right) \\wedge \\mu _{B}^{+}\\left( b\\right) \\right\\} \\\\\n&=&{\\bigvee }_{x=ab}{\\bigvee }_{a=mn}\\left\\{ \\mu _{B}^{+}\\left( m\\right)\n\\wedge \\mu _{B}^{+}\\left( b\\right) \\right\\} \\\\\n&\\leq &\\mu _{B}^{+}\\left( x\\right) .\n\\end{eqnarray*\nAn\n\\begin{eqnarray*}\n\\left( \\left( \\mu _{B}^{-}\\circ \\mathcal{S}_{\\Gamma }^{-}\\right) \\circ \\mu\n_{B}^{-}\\right) \\left( x\\right) &=&{\\bigwedge }_{x=ab}\\left\\{ \\left( \\mu\n_{B}^{-}\\circ \\mathcal{S}_{\\Gamma }^{-}\\right) \\left( a\\right) \\vee \\mu\n_{B}^{-}\\left( b\\right) \\right\\} \\\\\n&=&{\\bigwedge }_{x=ab}\\left\\{ {\\bigwedge }_{a=mn}\\left\\{ \\mu _{B}^{-}\\left(\nm\\right) \\vee \\mathcal{S}_{\\Gamma }^{-}\\left( n\\right) \\right\\} \\vee \\mu\n_{B}^{-}\\left( b\\right) \\right\\} \\\\\n&=&{\\bigwedge }_{x=ab}{\\bigwedge }_{a=mn}\\left\\{ \\left( \\mu _{B}^{-}\\left(\nm\\right) \\vee -1\\right) \\vee \\mu _{B}^{-}\\left( b\\right) \\right\\} \\\\\n&=&{\\bigwedge }_{x=ab}{\\bigwedge }_{a=mn}\\left\\{ \\mu _{B}^{-}\\left( m\\right)\n\\vee \\mu _{B}^{-}\\left( b\\right) \\right\\} \\\\\n&\\geq &\\mu _{B}^{-}\\left( x\\right) .\n\\end{eqnarray*\nThus $\\left( \\mu _{B}^{+}\\circ \\mathcal{S}_{\\Gamma }^{+}\\right) \\circ \\mu\n_{B}^{+}\\subseteq \\mu _{B}^{+}$ and $\\left( \\mu _{B}^{-}\\circ \\mathcal{S\n_{\\Gamma }^{-}\\right) \\circ \\mu _{B}^{-}\\supseteq \\mu _{B}^{-}.$\\newline\nConversely, assume that $\\left( \\mu _{B}^{+}\\circ \\mathcal{S}_{\\Gamma\n}^{+}\\right) \\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+}$ and $\\left( \\mu\n_{B}^{-}\\circ \\mathcal{S}_{\\Gamma }^{-}\\right) \\circ \\mu _{B}^{-}\\supseteq\n\\mu _{B}^{-}.$ Let $x,y,z\\in S,$ the\n\\begin{eqnarray*}\n\\mu _{B}^{+}\\left( (xy)z\\right) &\\geq &\\left( \\left( \\mu _{B}^{+}\\circ \n\\mathcal{S}_{\\Gamma }^{+}\\right) \\circ \\mu _{B}^{+}\\right) \\left(\n(xy)z\\right) ={\\bigvee }_{(xy)z=cd}\\left\\{ \\left( \\mu _{B}^{+}\\circ \\mathcal\nS}_{\\Gamma }^{+}\\right) \\left( c\\right) \\wedge \\mu _{B}^{+}\\left( d\\right)\n\\right\\} \\\\\n&\\geq &\\left( \\mu _{B}^{+}\\circ \\mathcal{S}_{\\Gamma }^{+}\\right) \\left(\nxy\\right) \\wedge \\mu _{B}^{+}\\left( z\\right) =\\left\\{ {\\bigvee \n_{xy=pq}\\left\\{ \\mu _{B}^{+}\\left( p\\right) \\wedge \\mathcal{S}_{\\Gamma\n}^{+}\\left( q\\right) \\right\\} \\right\\} \\wedge \\mu _{B}^{+}\\left( z\\right) \\\\\n&\\geq &\\left\\{ \\mu _{B}^{+}\\left( x\\right) \\wedge \\mathcal{S}_{\\Gamma\n}^{+}\\left( y\\right) \\right\\} \\wedge \\mu _{B}^{+}\\left( z\\right) =\\left\\{\n\\mu _{B}^{+}\\left( x\\right) \\wedge 1\\right\\} \\wedge \\mu _{B}^{+}\\left(\nz\\right) \\\\\n&=&\\mu _{B}^{+}\\left( x\\right) \\wedge \\mu _{B}^{+}\\left( z\\right) .\n\\end{eqnarray*\nAn\n\\begin{eqnarray*}\n\\mu _{B}^{-}\\left( (xy)z\\right) &\\leq &\\left( \\left( \\mu _{B}^{-}\\circ \n\\mathcal{S}_{\\Gamma }^{-}\\right) \\circ \\mu _{B}^{-}\\right) \\left(\n(xy)z\\right) ={\\bigwedge }_{(xy)z=cd}\\left\\{ \\left( \\mu _{B}^{-}\\circ \n\\mathcal{S}_{\\Gamma }^{-}\\right) \\left( c\\right) \\vee \\mu _{B}^{-}\\left(\nd\\right) \\right\\} \\\\\n&\\leq &\\left( \\mu _{B}^{-}\\circ \\mathcal{S}_{\\Gamma }^{-}\\right) \\left(\nxy\\right) \\vee \\mu _{B}^{-}\\left( z\\right) =\\left\\{ {\\bigwedge \n_{xy=pq}\\left\\{ \\mu _{B}^{-}\\left( p\\right) \\vee \\mathcal{S}_{\\Gamma\n}^{-}\\left( q\\right) \\right\\} \\right\\} \\vee \\mu _{B}^{-}\\left( z\\right) \\\\\n&\\leq &\\left\\{ \\mu _{B}^{-}\\left( x\\right) \\vee \\mathcal{S}_{\\Gamma\n}^{-}\\left( y\\right) \\right\\} \\vee \\mu _{B}^{-}\\left( z\\right) =\\left\\{ \\mu\n_{B}^{-}\\left( x\\right) \\vee -1\\right\\} \\vee \\mu _{B}^{-}\\left( z\\right) \\\\\n&=&\\mu _{B}^{-}\\left( x\\right) \\vee \\mu _{B}^{-}\\left( z\\right) .\n\\end{eqnarray*\nThus $\\mu _{B}^{+}\\left( (xy)z\\right) \\geq \\mu _{B}^{+}\\left( x\\right)\n\\wedge \\mu _{B}^{+}\\left( z\\right) $ and $\\mu _{B}^{-}\\left( (xy)z\\right)\n\\leq \\mu _{B}^{-}\\left( x\\right) \\vee \\mu _{B}^{-}\\left( z\\right) ,$ which\nimplies that $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ is a\nBVF-generalized bi-ideal of $S.$ $\\ \\ \\Box $\n\n\\begin{lemma}\n\\label{L30}Let $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ be a\nBVF-subset of an LA-semigroup $S$ then $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ is a BVF-bi-ideal of $S$\\ if and only if $\\mu\n_{B}^{+}\\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+},$ $\\mu _{B}^{-}\\circ \\mu\n_{B}^{-}\\supseteq \\mu _{B}^{-},$ $\\left( \\mu _{B}^{+}\\circ \\mathcal{S\n_{\\Gamma }^{+}\\right) \\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+}$ and $\\left(\n\\mu _{B}^{-}\\circ \\mathcal{S}_{\\Gamma }^{-}\\right) \\circ \\mu\n_{B}^{-}\\supseteq \\mu _{B}^{-}.$\n\\end{lemma}\n\n\\textbf{Proof. }Follows from Lemma \\ref{L56}(1) and Lemma \\ref{L57}. $\\ \\\n\\Box $\n\n\\textbf{Definition 3.7 }A BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ of an LA-semigroup $S$ is called a BVF-interior\nideal of $S$ i\n\\begin{equation*}\n\\mu _{B}^{+}\\left( (xy)z\\right) \\geq \\mu _{B}^{+}\\left( y\\right) \\text{ \\\nand \\ }\\mu _{B}^{-}\\left( (xy)z\\right) \\leq \\mu _{B}^{-}\\left( y\\right) \n\\text{ for all }x,y,z\\in S.\n\\end{equation*}\n\n\\begin{lemma}\n\\label{L31}Let $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ be a\nBVF-subset of an LA-semigroup $S$ then $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ is a BVF-interior ideal of $S$\\ if and only if \n\\left( \\mathcal{S}_{\\Gamma }^{+}\\circ \\mu _{B}^{+}\\right) \\circ \\mathcal{S\n_{\\Gamma }^{+}\\subseteq \\mu _{B}^{+}$ and $\\left( \\mathcal{S}_{\\Gamma\n}^{-}\\circ \\mu _{B}^{-}\\right) \\circ \\mathcal{S}_{\\Gamma }^{-}\\supseteq \\mu\n_{B}^{-}.$\n\\end{lemma}\n\n\\textbf{Proof. }The proof of this lemma is similar to the proof of Lemma \\re\n{L57}$.$ $\\ \\ \\Box $\n\n\\begin{remark}\nEvery BVF-ideal is a BVF-interior ideal of an LA-semigroup $S,$ but the\nconverse is not true.\n\\end{remark}\n\n\\textbf{Example 3.2 }Let $S=\\{a,b,c,d\\}$, the binary operation \"$\\cdot $\" on \n$S$ be defined as follows\n\\begin{equation*}\n\\begin{tabular}{l|llll}\n$\\cdot $ & $a$ & $b$ & $c$ & $d$ \\\\ \\hline\n$a$ & $c$ & $c$ & $c$ & $d$ \\\\ \n$b$ & $d$ & $d$ & $c$ & $c$ \\\\ \n$c$ & $d$ & $d$ & $d$ & $d$ \\\\ \n$d$ & $d$ & $d$ & $d$ & $d\n\\end{tabular\n\\end{equation*\nClearly, $S$ is an LA-semigroup. But $S$ is not a semigroup because \nc=a\\cdot (a\\cdot b)\\neq (a\\cdot a)\\cdot b=d.$ Now we define BVF-subset a\n\\begin{equation*}\nB=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle =\\left\\langle \\left( \n\\frac{a}{0.5},\\frac{b}{0.3},\\frac{c}{0.1},\\frac{d}{0.8}\\right) ,\\text{ \n\\left( \\frac{a}{-0.7},\\frac{b}{-0.4},\\frac{c}{-0.2},\\frac{d}{-0.9}\\right)\n\\right\\rangle .\n\\end{equation*\nIt can be verified that $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ is a BVF-interior ideal of $S.$ But, sinc\n\\begin{equation*}\n\\mu _{B}^{+}\\left( b\\cdot c\\right) =\\mu _{B}^{+}\\left( c\\right) =0.1<0.3=\\mu\n_{B}^{+}\\left( b\\right) .\n\\end{equation*\nAn\n\\begin{equation*}\n\\mu _{B}^{-}\\left( b\\cdot c\\right) =\\mu _{B}^{-}\\left( c\\right)\n=-0.2>-0.4=\\mu _{B}^{-}\\left( b\\right) .\n\\end{equation*\nThus $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ is not a\nBVF-right ideal of $S,$ that is, $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ is not a BVF-two-sided ideal of $S.$\n\n\\begin{proposition}\nEvery BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ of\nan LA-semigroup $S$ with left identity is a BVF-right ideal if and only if\nit is a BVF-interior ideal.\n\\end{proposition}\n\n\\textbf{Proof. }Let every BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ of $S$ is a BVF-right ideal. For $x$, $a$ and $y$ of \n$S$, conside\n\\begin{equation*}\n\\mu _{B}^{+}((xa)y)\\geq \\mu _{B}^{+}(xa)=\\mu _{B}^{+}((ex)a)=\\mu\n_{B}^{+}((ax)e)\\geq \\mu _{B}^{+}(ax)\\geq \\mu _{B}^{+}(a).\n\\end{equation*\nAn\n\\begin{equation*}\n\\mu _{B}^{-}((xa)y)\\leq \\mu _{B}^{-}(xa)=\\mu _{B}^{-}((ex)a)=\\mu\n_{B}^{-}((ax)e)\\leq \\mu _{B}^{-}(ax)\\leq \\mu _{B}^{-}(a).\n\\end{equation*\nWhich implies that $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $\nis a BVF-interior ideal. Conversely, for any $x$ and $y$ in $S$ we have\n\\begin{equation*}\n\\mu _{B}^{+}(xy)=\\mu _{B}^{+}((ex)y)\\geq \\mu _{B}^{+}(x)\\text{ \\ and \\ }\\mu\n_{B}^{-}(xy)=\\mu _{B}^{-}((ex)y)\\leq \\mu _{B}^{-}(x).\n\\end{equation*\nHence required. $\\ \\ \\Box $\n\n\\begin{theorem}\nLet $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ be a BVF-left\nideal of an LA-semigroup $S$ with left identity, then $B=\\left\\langle \\mu\n_{B}^{+},\\mu _{B}^{-}\\right\\rangle $ being BVF-interior ideal is a\nBVF-bi-ideal of $S$.\n\\end{theorem}\n\n\\textbf{Proof. }Since $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle \n$ is an BVF-left ideal in $S$, so $\\mu _{B}^{+}(xy)\\geq \\mu _{B}^{+}(y)$ and \n$\\mu _{B}^{-}(xy)\\leq \\mu _{B}^{-}(y)$ for all $x$ and $y$ in $S$. As $e$ is\nleft identity in $S$. So\n\\begin{equation*}\n\\mu _{B}^{+}(xy)=\\mu _{B}^{+}((ex)y)\\geq \\mu _{B}^{+}(x)\\text{ \\ and \\ }\\mu\n_{B}^{-}(xy)=\\mu _{B}^{-}((ex)y)\\leq \\mu _{B}^{-}(x),\n\\end{equation*}\nwhich implies that $\\mu _{B}^{+}(xy)\\geq \\mu _{B}^{+}(x)\\wedge \\mu\n_{B}^{+}(y)$ and $\\mu _{B}^{-}(xy)\\leq \\mu _{B}^{-}(x)\\vee \\mu _{B}^{-}(y)$\nfor all $x$ and $y$ in $S$. Thus $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ is an BVF-LA-subsemigroup of $S$. For any $x,y$ and \nz$ in $S$, we ge\n\\begin{equation*}\n\\mu _{B}^{+}((xy)z)=\\mu _{B}^{+}((x(ey))z)=\\mu _{B}^{+}((e(xy))z)\\geq \\mu\n_{B}^{+}(xy)=\\mu _{B}^{+}((ex)y)\\geq \\mu _{B}^{+}(x).\n\\end{equation*\nAn\n\\begin{equation*}\n\\mu _{B}^{-}((xy)z)=\\mu _{B}^{-}((x(ey))z)=\\mu _{B}^{-}((e(xy))z)\\leq \\mu\n_{B}^{-}(xy)=\\mu _{B}^{-}((ex)y)\\leq \\mu _{B}^{-}(x).\n\\end{equation*\nAls\n\\begin{equation*}\n\\mu _{B}^{+}((xy)z)=\\mu _{B}^{+}((zy)x)=\\mu _{B}^{+}((z(ey))x)=\\mu\n_{B}^{+}((e(zy))x)\\geq \\mu _{B}^{+}(zy)=\\mu _{B}^{+}((ez)y)\\geq \\mu\n_{B}^{+}(z).\n\\end{equation*\nAn\n\\begin{equation*}\n\\mu _{B}^{-}((xy)z)=\\mu _{B}^{-}((zy)x)=\\mu _{B}^{-}((z(ey))x)=\\mu\n_{B}^{-}((e(zy))x)\\leq \\mu _{B}^{-}(zy)=\\mu _{B}^{-}((ez)y)\\leq \\mu\n_{B}^{-}(z).\n\\end{equation*\nHence $\\mu _{B}^{+}((xy)z)\\geq \\mu _{B}^{+}(x)\\wedge \\mu _{B}^{+}(z)$ and \n\\mu _{B}^{-}((xy)z)\\leq \\mu _{B}^{-}(x)\\vee \\mu _{B}^{-}(z)$ for all $x,y$\nand $z$ in $S$. $\\ \\ \\Box $\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nIn 1954 Wick and Cutkosky noticed that a certain class of ladder-type Feynman integrals with massive propagators features a massive dual conformal symmetry \\cite{Wick:1954eu,Cutkosky:1954ru}. \nWhile most of the formal insights into quantum field theory inspired by the AdS\/CFT correspondence are limited to massless situations, this massive dual conformal symmetry is naturally realized in the context of this duality~\\cite{Alday:2009zm,Caron-Huot:2014gia}. In particular, the extended dual conformal symmetry limits the variables that certain massive Feynman integrals can depend on and thus simplifies their computation. In the present letter we argue that for large classes of Feynman integrals, this massive dual conformal symmetry is in fact only the zeroth level of an infinite dimensional massive Yangian algebra. In addition to limiting the number of variables, this new symmetry strongly constrains the functional form of the integrals. While these symmetry properties naturally extend the observations on the integrability of massless Feynman integrals \\cite{Chicherin:2017cns,Chicherin:2017frs,Loebbert:2019vcj}, to the knowledge of the authors this is the first occurence of quantum integrability in massive quantum field theory in $D>2$ spacetime dimensions. \n\nFor \\emph{massless} $\\mathcal{N}=4$ super Yang--Mills (SYM) theory it was recently argued that planar integrability is preserved in a certain double scaling limit, which (in the simplest case) results in the so-called bi-scalar fishnet theory \\cite{Gurdogan:2015csr}. Here, individual (massless) Feynman integrals inherit the Yangian symmetry that underlies the integrability of the prototypical examples of the AdS\/CFT duality \\cite{Chicherin:2017cns,Chicherin:2017frs}. A similar starting point, i.e.\\ an \\emph{integrable massive} avatar of $\\mathcal{N}=4$ super Yang--Mills theory, is not known. We thus investigate massive Feynman integrals directly, i.e.\\ we consider the properties of functions of the type\n\\begin{equation}\n\\includegraphicsbox{FigTwoStars.pdf}\n\\quad\n=\n\\int \\frac{\\mathrm{d}^D x_0 \\mathrm{d}^D x_{\\bar 0}}\n{\n\\hat x_{01}^{2a_1}\n\\hat x_{02}^{2a_2}\nx_{0\\bar 0}^{2b_0}\n\\hat x_{\\bar 03}^{2a_3}\n\\hat x_{\\bar 04}^{2a_4}\n},\n\\end{equation}\nwhere $x_{jk}^\\mu=x_j^\\mu-x_k^\\mu$ and $\\hat x_{jk}^2=x_{jk}^2+(m_j-m_k)^2$. Here the dashed internal propagator is massless, i.e.\\ $m_0=m_{\\bar 0}=0$, while the other propagators are massive.\nThe $x$-variables denote dualized momenta (dotted green diagram) related via $p^{\\mu}_j=x^{\\mu}_j-x^{\\mu}_{j+1}$  \\footnote{Note that the $p_j^2$ are unconstrained and the $m_j$ are generic; we have $x^{\\mu}_j=x^{\\mu}_{1}-\\sum_{k < j} p^{\\mu}_k $ and $x_{n+1}^\\mu=x_{1}^\\mu$.}.\nOur findings suggest that all Feynman graphs, which are cut from regular tilings of the plane and have massive propagators on the boundary, feature a massive $D$-dimensional Yangian symmetry. \nWe will demonstrate the usefulness of this Yangian for bootstrapping massive Feynman integrals. Finally, we will show that, when translated to momentum space, the non-local Yangian symmetry can be interpreted as a massive generalization of momentum space conformal symmetry. \nThis suggests to interpret this novel symmetry within the AdS\/CFT correspondence.\n\n\\section{Massive Yangian}\n\nMassive dual conformal symmetry is realized in the form of partial differential equations obeyed by coordinate space Feynman integrals. That is, the integrals are annihilated by the tensor product action of the level-zero dual conformal generators\n$\n\\gen{J}^a = \\sum_{j=1}^n \\gen{J}_{j}^a, \n$\nwhere $\\gen{J}_j^a$\ndenotes one of \nthe following densities  acting on $x_j$:\n\\begin{align}\n\\gen{P}^{\\hat \\mu}_j &= -i \\partial_{x_{j}}^{\\hat \\mu}, \n\\qquad\\qquad\n\\gen{L}_j^{\\hat \\mu\\hat \\nu} = i x_j^{\\hat \\mu} \\partial_{x_{j}}^{\\hat \\nu} - ix^{\\hat \\nu}_j \\partial_{x_{j}}^{\\hat \\mu}, \n\\nonumber\n\\\\\n\\gen{D}_j &= -i \\brk!{x_{j\\mu} \\partial_{x_j}^\\mu + m_j \\partial_{m_j} + \\Delta_j},\n\\label{eqn:massdualconfrep}\n\\\\\n \\gen{K}^{\\hat \\mu}_j &= -2ix_j^{\\hat \\mu}\\brk!{x_{j\\nu}  \\partial_{x_j}^\\nu  + m_j\\partial_{m_j} + \\Delta_j} +i (x^2_j + m^2_j)\\partial_{x_{j}}^{\\hat \\mu}.\\notag\n\\end{align}\nThese can be understood as massless generators in $D+1$ dimensions with $x_j^{D+1}=m_j$. Only the components $\\hat \\mu=1,\\dots,D$ of the generators correspond to symmetries.\nHere we work with the Euclidean metric and the index $\\hat{\\mu}$ runs from 1~to~$D+1$, while $\\mu$ runs from 1~to~$D$.\n\nThe massive Yangian is spanned by the above level-zero Lie algebra generators and the bi-local level-one generators \ndefined as\n\\begin{equation}\n\\label{eq:DefLev1}\n \\gen{\\widehat J}^a=\\sfrac{1}{2} f^a{}_{bc}\\sum_{j<k=1}^n \\gen{J}_{j}^c \\gen{J}_{k}^b+ \\sum_{j=1}^n \\Eval_j \\gen{J}_{j}^a,\n\\end{equation}\nwhere $ f^a{}_{bc}$ denotes the Lie algebra structure constants.\nThe so-called evaluation parameters $\\Eval_j$ depend on the considered Feynman integral and will be specified below.\nThe level-one momentum generator for instance reads \n\\begin{align}\n\\label{eq:DefPhat}\n\\gen{\\widehat P}^{\\hat\\mu}\n=\\sfrac{i}{2} \\sum_{j,k=1}^n \n\\text{sign}(k-j)\\brk!{\\gen{P}_j^{\\hat\\mu} \\gen{D}_k + \\gen{P}_{j\\nu} \\gen{L}_k^{\\hat\\mu \\nu} }\n + \\sum_{j=1}^n \\Eval_j \\gen{P}_j^{\\hat \\mu}.\n\\end{align}\nHere we do not sum the internal index $\\nu$ over $D+1$ dimensions but rather define this contribution separately, \ne.g.\\\n$\\gen{\\widehat P}^{\\hat \\mu}_{\\text{extra}}=\\sfrac{i}{2}\\sum_{j,k}\\text{sign}(k-j)\\gen{P}_{j,D+1} \\gen{L}_k^{\\hat\\mu,D+1}$.\nNote that the whole Yangian algebra is spanned by the above level-one momentum generator \\eqref{eq:DefPhat} and the level-zero generators~\\eqref{eqn:massdualconfrep}. \n\n\\section{One and Two Loops }\n\n\nFirst, we consider a generic scalar $n$-point Feynman integral at one-loop order with massive propagators. Propagator powers $a_j$ and the spacetime dimension $D$ are arbitrary:\n\\begin{align}\nI_n = \\int \\text{d}^D x_0 \\ \\rho_n\n=\\includegraphicsbox{FigNStarPropWeights.pdf}\\,.\n\\label{eq:nGonIntegral}\n\\end{align}\nHere $\\rho_n\n=\n\\prod_{j=1}^n (x_{0j}^2 + m_j^2)^{-a_j}$ and we use the notation $x_{jk}^\\mu=x_j^\\mu-x_k^\\mu$.\nThese integrals have been expressed in terms of hypergeometric functions by Davydychev~\\cite{Davydychev:1990cq}.\nSurprisingly, we find that the integrand is invariant under the massive level-one generators acting on legs $1$ to $n$ defined in \\eqref{eq:DefLev1}, \ni.e.\\ we have $\\gen{\\widehat J}^a \\rho_n = 0$ which implies \n\\begin{align}\n\\gen{\\widehat J}^a I_n = 0.\n\\label{eq:Lvl1Symm}\n\\end{align}\nThe scaling dimensions in \\eqref{eqn:massdualconfrep} take values $\\Delta_j=a_j$ and the evaluation parameters entering the level-one generators \\eqref{eq:DefLev1} can be written in the compact form\n\\begin{align}\n\\Eval_j\n=\\sfrac{1}{2} \\sum_{k=1}^n \\text{sign}(k-j) \\,a_k.\n\\label{eq:EvalsNGons}\n\\end{align}\nWe emphasize that this symmetry also holds for cases where an arbitrary subset of the masses is set to zero~\n\\footnote{In these cases one acts on massive legs with the generators of the massive symmetry and on massless legs with the usual dual-conformal generators obtained by dropping the mass terms in \\protect\\eqref{eqn:massdualconfrep}.}.\nMoreover, the above level-one symmetry holds even if we do not impose conformal level-zero symmetry, i.e.\\ the constraint $\\sum_{j=1}^n a_j=D$ on the propagator powers. Hence, the level-one symmetry can be used to constrain dual conformal and non-dual-conformal integrals.\nFinally we should mention that also the generators as $\\gen{\\widehat P}^{\\hat \\mu}_{\\text{extra}}$ given below \\eqref{eq:DefPhat}, which can be interpreted as contributions to an internal sum over $D+1$ dimensions, define a separate symmetry of the above one-loop integrands \\footnote{Switching on the internal mass $m_0$, only the combination $\\gen{\\widehat P}^{\\hat \\mu}+\\gen{\\widehat P}^{\\hat \\mu}_{\\text{extra}}$ is a symmetry.}.\n\n\\begin{table}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{tabular}{|l|c|c|c|}\\hline\nLoops&Graphs&Conformal&Not Conformal\n\\\\\\hline\nOne&$n$-gons&Full Yangian \\& $\\levo{J}_{\\text{extra}}^a$ & All $\\levo{J}^a$  \\& $\\levo{J}_{\\text{extra}}^a$ \n\\\\\nTwo&$l$-$r$-gons&Full Yangian&  $\\levo{P}^\\mu$\n\\\\\nAll&Tilings&Full Yangian&  $\\levo{P}^\\mu$\n\\\\\\hline\n\\end{tabular}\n\\caption{Overview of non-local symmetries. We speak of a (dual) conformal integral, if at each $x$-space vertex the propagator powers $a_j$ sum up to the spacetime dimension $D$.}\n\\label{tab:allsyms}\n\\end{table}\n\nWhile \\eqref{eq:Lvl1Symm} represents the central symmetry equation that extends to higher loop integrals, see \\Tabref{tab:allsyms}, at one-loop order even stronger invariance statements can be formulated:\nFirstly, the one-loop invariance even holds at the integrand level. Secondly, due to the total permutation symmetry, the $n$-gon integrand is even invariant under the two-point generator density:\n\\begin{equation}\n\\label{eq:TwoPointSymNgon}\n\\gen{\\widehat J}^a_{jk}\\rho_n=0,\n\\qquad\n\\gen{\\widehat J}^a_{jk}=\\sfrac{1}{2} f^a{}_{bc} \\gen{J}_{j}^c \\gen{J}_{k}^b+ \\Eval_j \\gen{J}_{j}^a+ \\Eval_k \\gen{J}_{k}^a,\n\\end{equation}\nfor $j,k\\in\\{1,\\dots,n\\}$ and with  \n$\\Eval_j= a_k\/2$ and\n$\\Eval_k=- a_j\/2$.\n\nAt two loops we find that $n$-point integrals (not integrands) built from an $l+1$-gon and an $r+1$-gon with $n=l+r$ connected by a massless internal propagator are Yangian invariant:\n\\begin{equation}\n\\includegraphicsbox{FiglrGon.pdf}\n\\end{equation}\nUsing the permutation symmetries of this class of diagrams as well as their level-zero symmetry and the two-point level-one symmetry \\eqref{eq:TwoPointSymNgon} of the one-loop integrals, it is not difficult to prove this two-loop invariance. \n\n\n\\section{Higher Loops}\n\\label{sec:HigherLoops}\n\nBeyond the analytic proofs for one- and two-loop integrals, we have numerically tested the invariance of higher loop diagrams for various examples, cf.\\ \\Tabref{tab:ExampleGraphs}.\n\\begin{table}\n\\renewcommand{\\arraystretch}{1.1}\n\\begin{tabular}{|c | c | c||c|}\\hline\n\\multicolumn{3}{|c||}{Non-local Symmetry}&{No Sym.}\n\\\\\\hline\nTriangle&Square&Hexagon&Irregular\\\\\n\\hline\n\\includegraphicsbox[scale=.7]{FigTwoHexagons.pdf}\n&\n\\includegraphicsbox[scale=.7]{FigBrokenWindowDiagram.pdf}\n&\n\\includegraphicsbox[scale=.7]{FigThreeStars.pdf}\n&\n\\includegraphicsbox[scale=.6]{FigThreePentagons.pdf}\n\\\\\n\\includegraphicsbox[scale=.7]{FigThreeHexagons.pdf}\n&\n\\includegraphicsbox[scale=.7]{FigWindowDiagram.pdf}\n&\n\\raisebox{1mm}{\\includegraphicsbox[scale=.7]{FigSixStars.pdf}}\n&\n\\includegraphicsbox[scale=.7]{FigIrreg4.pdf}\n\\\\\\hline\n\\end{tabular}\n\\caption{\nExamples of integrals up to 6 loops and 12 legs explicitly tested for level-one momentum invariance by acting on the Feynman parametrization. Graphs cut from regular tilings are invariant if propagators on the boundary of the graph are massive or massless (solid), while internal propagators have to be massless (dashed). Graphs not cut from regular tilings have no level-one symmetry as described here.}\n\\label{tab:ExampleGraphs}\n\\end{table}\nTo be precise, we have checked for the presence of the symmetry by acting on the respective Feynman parametrization and numerical integration of the result.\nOn this basis we conjecture that all planar Feynman graphs, which are cut along a closed contour from one of the three regular tilings of the plane, have massive Yangian symmetry if all internal propagators are massless. The external \npropagators can be massive or massless.\nHere an $x$-space propagator is called external if it is connected to an external point of the diagram. The integration can be evaluated in generic spacetime dimension $D$ and propagator powers may take generic values $a_j$.\nFor full Yangian invariance we require the conformal constraint\n$D=\\sum_{j\\in \\text{vertex}} a_j$ at each integration vertex.\nIf this constraint is not satisfied, we still have level-one momentum symmetry $\\levo{P}^\\mu$, which corresponds to a massive generalization of momentum space conformal symmetry that we introduce below.\nIn the massless limit this matches the observations of \\cite{Chicherin:2017cns,Chicherin:2017frs,Loebbert:2019vcj} on the integrability of massless Feynman integrals in $D=3$, 4 and 6 spacetime dimensions.\n\nThe values of the evaluation parameters $\\Eval_j$ entering \\eqref{eq:DefPhat} can be read off from the Feynman graph as follows: \ni)~Choose an arbitrary leg $1$ with $s_1=\\sfrac{1}{2}\\sum_{k=2}^n a_k$, ii)~walk clockwise along the external boundary of the ($x$-space) graph and pick up the respective propagator weights, iii)~here an external propagator with power $a_j$ contributes $-{a_j}\/{2}$ while an internal propagator with weight $b_k$ contributes $-b_k + {D}\/{2}$. The evaluation parameter $\\Eval_{j+1}$ is obtained from $\\Eval_j$ by adding the respective terms, e.g.\\\n\\begin{equation}\n\\label{eq:EvalRules}\n\\includegraphicsbox{FigEvalRulesExternalProp.pdf}\n\\qquad\n\\includegraphicsbox{FigEvalRulesInternalProp.pdf}.\n\\end{equation}\n\\section{Yangian Bootstrap}\n\n\n\nLet us now employ the above non-local symmetry to bootstrap Feynman integrals with massive propagators. This discussion extends the algorithm of \\cite{Loebbert:2019vcj} to the massive case.\nWe start with some examples containing conformal vertices obeying the condition $\\sum_j a_j=D$. Notably, after solving these integrals, we can obtain an infinite class of integrals by acting with $r$ mass derivatives $\\partial_{m_k}$ which yields an integral with propagator weights $\\tilde a_j$ with $\\sum_j \\tilde a_j=D+r$.\n\n\\paragraph{Conformal, 2 points, 2 masses, 1 loop:}\n\nAs a simple starting example consider the two-point integral \n(see e.g.\\ \\cite{Boos:1990rg,Davydychev:1990cq,Kniehl:2011ym} for the non-conformal case)\n\\begin{align}\nI_2^{m_1m_2} = \n\\includegraphicsbox{FigConfTwoPointsTwoMasses.pdf}\n=\\frac{ (1-v^2)^{\\beta\/2}}{m_1^{a_1} m_2^{a_2}}\\phi(v),\n\\end{align}\nwritten in terms of a function $\\phi$ of the conformally invariant variable\n$v \n=(m_1^2+m_2^2+x_{12}^2)\/2m_1m_2$.\nFor convenience we set\n$2\\alpha=a_1-a_2-1$\nand \n$2\\beta=-a_1-a_2+1$.\nActing on $I_2$ with the level-one momentum generator yields the associated Legendre differential equation \\footnote{Note that this equation arises from the differential operator $\\levo{ P}^{\\hat \\mu}_{\\text{extra}}$, while the rest of $\\levo{P}$ trivially annihilates $I_2$. This is a generic feature for two-point invariants.}:\n\\begin{equation}\n\\brk[s]*{\\alpha(\\alpha+1)+\\sfrac{\\beta^2}{v^2-1}}\\phi-2v\\phi'+(1-v^2)\\phi''=0.\n\\end{equation}\nThe general solution to the above differential equation is easily found, e.g.\\ in Mathematica, to be a linear combination of the associated Legendre functions of the first and second kind, i.e.  $P_\\alpha^\\beta$ and $Q_\\alpha^\\beta$. The coefficients of these functions can be fixed using numerical input for the integral, which yields\n\\begin{equation}\n\\phi(v) = 2^{\\beta}\\pi^{1-\\beta} P_\\alpha^\\beta(v).\n\\end{equation}\nWe have thus completely constrained the integral.\n\\paragraph{Conformal, 3 points, 3 masses, 1 loop:}\nConsider next the one-loop three-point integral with all propagators massive and the conformal constraint $D=a_1+a_2+a_3$ (see e.g.\\ \\cite{Nickel:1978ds,Davydychev:2017bbl,Bourjaily:2019exo} for the case with $a_j=1$):\n\\begin{equation}\n\\label{eq:I3start}\nI_3^{m_1m_2m_3}=\n\\includegraphicsbox{FigStarPropWeights.pdf}=\\frac{\\phi_3(u,v,w)}{m_1^{a_1}m_2^{a_2}m_3^{a_3}}.\n\\end{equation}\nHere we have chosen the conformal variables $u=u_{12}$, $v=u_{13}$ and $w=u_{23}$, where\n$u_{jk}=\n-\\hat x_{jk}^2\/4m_jm_k$.\nMaking a series ansatz in $u$, $v$ and $w$, the Yangian PDEs translate into recurrence equations for the series coefficients. \nThese equations can straightforwardly be solved, which yields Srivastava's triple hypergeometric function $H_{C}$, a generalization of Appell's hypergeometric function $F_1$, see e.g.\\ \\cite{srivastava2015note}:\n\\begin{equation}\n\\label{eq:SrivastavaHC}\nH_{C}(u,v,w)\n=\n\\sum_{k,l,n=0}^\\infty \\frac{(a_1)_{k+l}(a_2)_{k+n}(a_3)_{l+n}}{(\\gamma)_{k+l+n}}\\frac{u^k}{k!}\\frac{v^l}{l!}\\frac{w^n}{n!},\n\\end{equation}\nwith $\\gamma=D\/2+1\/2$ and \nthe Pochhammer symbol\n$(a)_k= \\Gamma_{a+k}\/\\Gamma_a.$\nThis series converges for\n\\begin{equation}\n\\label{eq:ConvergenceHC}\n \\abs{u}+\\abs{v}+\\abs{w}-2\\sqrt{(1-\\abs{u})(1-\\abs{v})(1-\\abs{w})}<2,\n\\end{equation}\nand in this region it is the only converging solution of 29 formal series solutions to the Yangian equations. Numerical comparison to the Feynman parameter representation of \n\\eqref{eq:I3start} yields the overall constant such that\n\\begin{equation}\nI_3^{m_1m_2m_3}\n=\n\\frac{\\pi^{D\/2}\\Gamma_{D\/2}}{\\Gamma_D m_1^{a_1}m_2^{a_2}m_3^{a_3}}\nH_{C}(u,v,w).\n\\end{equation}\nTo the knowledge of the authors this result has not been given elsewhere in the literature.\n\n\n\n\n\\paragraph{Non-Conformal, 3 points, 1 mass, 1 loop:}\n\nLet us finally demonstrate the usefulness of the level-one symmetry \nfor an integral that has no massive dual conformal (level-zero) symmetry. We write the triangle with massive leg $1$ as\n\\begin{equation}\nI_3^{m_100} = \n\\includegraphicsbox{FigStarPropWeights1Fat.pdf}\n=m_1^{D-2a_1-2a_2-2a_3} \\psi_3\\left(u,v,w\\right),\n\\end{equation}\nwith\n$u=u_{12}$,\n$v=u_{13}$ and\n$w=-u_{23}$,\nand $u_{jk}=-x_{jk}^2\/m_1^2$.\nWe act with the level-one PDEs on the series ansatz\n\\begin{equation}\n\\label{eq:SeriesAnsatz3pt1Mass}\nG_{xyz} ^{\\alpha_1 \\alpha_2 \\alpha_3\\gamma_1\\gamma_2} \n\t= \\sum_{k,l,n}\n\t\tf_{kln} \\,\\frac{u^k}{k!} \\frac{v^l}{l!} \\frac{w^n}{n!},\n\\end{equation}\nwhere the sum runs over $k\\in x+\\mathbb{Z}$, $l\\in y+\\mathbb{Z}$, $n\\in z+\\mathbb{Z}$ for some $(x,y,z)\\in \\mathbb{R}^3$.\nThe resulting recurrence equations can straightforwardly be solved which results in\n\\begin{align}\nf_{kln}\n=\n \\frac{(\\alpha_1)_{k+l+n}(\\alpha_2)_{k+n}(\\alpha_3)_{l+n}}{(\\gamma_1)_{k+l+n}(\\gamma_2)_{n}},\n\\label{eq:FundSol3GonNotConf1Mass}\n\\end{align}\nwith the new parameters\n\\begin{equation}\n\\label{eq:params3pt1MassNonConf} \n\\alpha_1 = a_1 + a_2 + a_3-\\sfrac{D}{2}   , \\quad\n\\alpha_2 =  a_2 , \\quad\n\\alpha_3 =   a_3,  \\quad\n\\gamma_1 = \\sfrac{D}{2},\n\\end{equation}\nand the abbreviation\n$\\gamma_2 = 1 - \\gamma_1 + \\alpha_2 +\\alpha_3$.\nWe find 36 combinations of $(x,y,z)$ in \\eqref{eq:SeriesAnsatz3pt1Mass} for which the series terminates after rescaling, which\nwe take as a necessary criterion for convergence. \nHowever, only for the two choices $(0,0,0)$ and \n$(0,0,1-\\gamma_2)$ of these 36 combinations, $u$, $v$ and $w$ are the effective series variables.\nIn fact, the ansatz combining these two functions, \ni.e.\\ \n\\footnote{\nNote the following shift identity for $A=G_{001-\\gamma_2} ^{\\alpha_1 \\alpha_2 \\alpha_3\\gamma_1\\gamma_2} \n$\nand \n$B=G_{000}^{\\alpha_1-\\gamma_2+1,\\alpha_2-\\gamma_2+1,\\alpha_3-\\gamma_2+1,\\gamma_1-\\gamma_2+1,2-\\gamma_2}$:\n\\begin{equation*}\nA\n=\nw^{1-\\gamma_2} \\frac{\\Gamma_{\\gamma_1}\\Gamma_{\\gamma_2}\\Gamma_{1+\\alpha_1-\\gamma_2}\\Gamma_{1+\\alpha_2-\\gamma_2}\\Gamma_{1+\\alpha_3-\\gamma_2}}{\\Gamma_{\\alpha_1}\\Gamma_{\\alpha_2}\\Gamma_{\\alpha_3}\\Gamma_{2-\\gamma_2}\\Gamma_{1+\\gamma_1-\\gamma_2}} B\n\\end{equation*}\n}\n\\begin{align}\n\\psi_3&=\\pi^\\frac{D}{2} \\brk*{ c_1 \\,\n\t\tG_{000} ^{\\alpha_1 \\alpha_2 \\alpha_3\\gamma_1\\gamma_2}\n\t\t+ c_2  \\,\n\t\tG_{001-\\gamma_2} ^{\\alpha_1 \\alpha_2 \\alpha_3\\gamma_1\\gamma_2} }, \t\n\\end{align}\nmaps to the solution given in \\cite{Boos:1990rg}, if the constants \n$c_1$ and $c_2$ are chosen as\n$\nc_1=\\Gamma_{\\alpha_1}\\Gamma_{1-\\gamma_2}\/\\Gamma_{\\alpha_1-\\gamma_2+1}\\Gamma_{\\gamma_1}\n$\nand\n\\begin{equation*}\nc_2=\\frac{\\Gamma_{\\alpha_1}\\Gamma_{\\gamma_1-\\alpha_2}\\Gamma_{\\gamma_1-\\alpha_3}\\Gamma_{\\gamma_2-1}\\Gamma_{2-\\gamma_2}}\n{\\Gamma_{\\gamma_1}\\Gamma_{\\gamma_2}\\Gamma_{1+\\alpha_1-\\gamma_2}\\Gamma_{1+\\alpha_2-\\gamma_2}\\Gamma_{1+\\alpha_3-\\gamma_2}}.\n\\end{equation*}\nIn this last example we have thus successfully employed the \nlevel-one  symmetry to constrain the functional form of an integral without conformal level-zero symmetry. This suggests an alternative interpretation of this non-local symmetry.\n\\section{Massive Momentum Space Symmetry}\n\\label{sec:MassMomSpace}\n\nIn the massless case, the Yangian symmetry of amplitudes in planar $\\mathcal{N}=4$ SYM theory \\cite{Drummond:2009fd} as well as in the fishnet theory \\cite{Chicherin:2017cns} can be understood as the closure of a dual (region momentum space) and an ordinary (momentum space) conformal symmetry.\nIt is thus a natural question whether the discovered massive Yangian can be understood as the closure of the massive dual conformal symmetry generated by \\eqref{eqn:massdualconfrep} and some massive generalization of momentum space conformal symmetry. To address this question we translate the level-one generator $\\levo{P}^\\mu$ of \\eqref{eq:DefPhat} given in terms of region momenta $x_j$ into momenta $p_j$ defined via\n$p^{\\mu}_j=x^{\\mu}_j-x^{\\mu}_{j+1}$. The masses, on the contrary, stay untouched because the mass dependence of the integrals cannot be phrased in terms of dual masses $m_j-m_{j+1}$ only. By making the respective substitutions and using the chain rule, one finds \n\\begin{equation}\n\\label{eq:PhatToK}\n\\levo{P}^\\mu=\\sfrac{i}{2} \\gen{\\bar K}^\\mu \\, ,\n\\end{equation} \nwhich holds up to terms that vanish when the integral is expressed in terms of the first $n-1$ momenta by using momentum conservation. Here, the special conformal operator $\\gen{\\bar K}^\\mu$ forms part of the following massive generalization of the momentum space conformal generators~\\footnote{Here we could also use a local version of the dilatation generator density acting only on the mass $m_j$. In the tensor product this makes no difference.}:\n\\begin{align}\n\\gen{\\bar P}_{j}^\\mu &= p_j^\\mu \\, , \n\\hspace{1cm} \\gen{\\bar L}_{j}^{\\mu\\nu} =  p_j^\\mu \\partial_{p_j}^{\\nu} -  p_j^\\nu \\partial_{p_j}^{\\mu},\n\\nonumber\\\\\n\\gen{\\bar D}_{j} &= p_{j\\nu} \\partial^{\\nu}_{p_j}+ \\sfrac{m_j \\partial_{m_j}+m_{j+1} \\partial_{m_{j+1}}}{2} + \\bar{\\Delta}_j,\n\\label{eqn:MomConfMassRep}\n\\\\\n\\gen{\\bar K}_{j}^\\mu \n&= p_j^\\mu \\partial_{p_j}^2\n-2\\brk[s]2{p_{j\\nu} \\partial_{p_j}^\\nu+\\sfrac{m_j \\partial_{m_j}+m_{j+1} \\partial_{m_{j+1}}}{2} + \\bar{\\Delta}_j }\\partial_{p_j}^{\\mu}.\n\\nonumber \n\\end{align}\nThese generators thus furnish symmetries of the above integrals expressed in momentum space and they obey the ordinary conformal algebra. We have $\\bar \\Delta_j=\n\\sfrac{1}{2}\\Delta_{j}+ \\sfrac{1}{2}\\Delta_{j+1} +  \\Eval_j - \\Eval_{j+1}$ with the evaluation parameters $s_j$ determined by the rules \\eqref{eq:EvalRules}\n\\footnote{For the one-loop $n$-gons for instance, we thus find $\\bar \\Delta_j = a_j+a_{j+1}$.}.\nWe note that the operator $\\gen{\\bar K}_{j}^\\mu$ includes a mildly non-local contribution acting on the masses of nearest-neighboring legs $j$ and $j+1$ of the Feynman graph.\nIn the massless limit, the above translation \\eqref{eq:PhatToK} of the Yangian level-one generator to momentum space also makes connection to the momentum space Ward identities that were recently studied for conformal correlation functions, see e.g.\\ \\cite{Coriano:2013jba,Bzowski:2013sza,Coriano:2020ccb}.\n\n\\section{Conclusions and Outlook}\n\nThe main results presented in this letter are summarized as follows:\nAn infinite number of specific massive Feynman integrals features a highly constraining Yangian symmetry --- a hallmark of integrability.\nThe observed Yangian algebra can be understood as the closure of a novel momentum space conformal symmetry and the known massive dual conformal symmetry; this strongly suggests an embedding of these findings into a larger picture within the AdS\/CFT duality.\nOur finding shows that the massive IR regulator proposed in \\cite{Alday:2009zm} does not only preserve dual conformal symmetry but also the Yangian symmetry of Feynman integrals.\n\n\nThere are plenty of further directions worth being investigated.\nFirstly, it would be important to formally prove our conjecture of higher loop Yangian and massive momentum space conformal symmetry. This may be possible along the lines of the massless case \\cite{Chicherin:2017cns,Chicherin:2017frs}, but it could be tricky to find a proof that applies directly to all cases considered in this letter (e.g.\\ to generic spacetime dimensions $D$). \nSecondly, the applicability of Yangian symmetry to massive Feynman integrals and generic (double) $n$-gons allows us to refine the algorithm of \\cite{Loebbert:2019vcj} on many more examples whose complexity interpolates between the simplest massless cases with $2$ and $9$ variables, respectively. This may finally open the door to computing e.g.\\ the massless hexagon or double box integral. \nMoreover, some of the integrals discussed above have a geometric interpretation, see e.g.\\ \\cite{Nandan:2013ip,Davydychev:2017bbl,Herrmann:2019upk,Bourjaily:2019exo}. It would be fascinating to understand the considered symmetries in this context. \nAnother highly interesting task is to seek for a massive generalization of the Basso--Dixon formula for massless fishnet four-point integrals \\cite{Basso:2017jwq}, whose finding was motivated by integrability.\nFinally, in the massless situation integrability took its way from AdS\/CFT via the fishnet theory to Feynman integrals \\cite{Gurdogan:2015csr,Chicherin:2017cns,Chicherin:2017frs}. Here we have found similar properties for the massive building blocks of quantum field theory. Tracing back these symmetries to an origin within the AdS\/CFT duality\nis very suggestive and may open the door to understanding integrability in massive phases of AdS\/CFT.\nIn this direction we note that in $D=4$ for instance, the massive square fishnet diagrams discussed above can be considered as planar off-shell amplitudes generated by the Lagrangian\n\\begin{align}\n\\mathcal{L}=\n&N_\\text{c} \\text{tr}\n\\brk*{\n-\\partial_\\mu\\bar X \\partial^\\mu X -\\partial_\\mu\\bar Z\\partial^\\mu Z+\\xi^2 \\bar X\\bar Z X Z\n}\n\\\\\n&- N_\\text{c}\\brk*{m_j-m_k}^2 \\bar Z^j{}_k Z^k{}_j\n-N_\\text{c}\\brk*{ m_j-m_k}^2 \\bar X^j{}_kX^k{}_j.\\nonumber\n\\end{align}\nHere, $N_\\text{c}-n$ of the masses $m_j$ are taken to be zero, such that graphs with massless propagators in the bulk of the diagram are dominant in the planar limit~\\footnote{Note that the mass terms generated by spontaneous symmetry breaking in the fishnet theory have product form instead of the difference form \\cite{Karananas:2019fox,Loebbert:2020tje}.}. \nIt would be interesting to test this Lagrangian for Yangian symmetry with the methods of~\\cite{Beisert:2017pnr}.\n\n\n\\medskip \n\n\\acknowledgments{\nFL and DM are very grateful to Till Bargheer and Jan Plefka for initial collaborations on similar ideas in the context of amplitudes in $\\mathcal{N}=4$ SYM theory and for helpful discussions. \nWe thank Niklas Beisert, Lance Dixon and Jan Plefka\nfor helpful comments on the manuscript.\n The work of FL is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)--Projektnummer 363895012. JM is  supported  by  the  International  Max  Planck  Research  School  for  Mathematical  and Physical Aspects of Gravitation, Cosmology and Quantum Field Theory. DM was supported by DFF-FNU through grant number DFF-FNU 4002-00037. The work of HM has been supported by the grant no.\\ 615203 of the European Research Council under the FP7 and by the Swiss National Science Foundation by the NCCR SwissMAP. \n}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\nLS I $+61^{\\circ}303$, a Galactic high-mass X-ray binary system located at a distance of 2 \nkpc \\citep{Frail-1991}, is detected in the energy range from radio to \n$\\gamma$-rays exhibiting strong variable emission. It consists of a B0 \nmain-sequence star with a circumstellar disk (i.e a Be star) and a compact \nobject of unknown nature. The orbital period of the system is estimated\nto be $P_{orb}$ = 26.496 days and it also exhibits a long term periodic \nvariation with a superorbital period of  $P_{sup}$ = 1667 \ndays \\citep{Gregory-2002,Massi-2013,Massi-2015}. However, very recently the \nsuperorbital period is estimated to be 1626 days using 37 years of radio data \n\\citep{Massi_2016}.\nThe zero orbital phase \ncorresponds to $T_{0,orb} =  2443366.775 + nP_{orb}~JD$. According to the \nmost recent radial velocity measurements, the orbit is elliptical with\neccentricity of e = 0.537 $\\pm$ 0.034 and periastron passage occurring around phase \n$\\phi$ = 0.275, apastron passage at $\\phi$ = 0.775, superior conjunction \nat $\\phi$ = 0.081, and inferior conjunction at $\\phi$ = 0.313 \n\\citep{Aragona-2009}. \n\n\nHigh angular resolution VLBI radio data has shown the presence of \nhigh-energy particle outflow possibly related to jet-like ejection on \nthe time scale of a orbital period \\citep{Paredes-1998, Massi-2004}.  \nHowever, the observed morphological changes in the data\ncollected at different epochs reported by \\cite{Dhawan-2006} \nsupport a scenario of binary pulsar. Recent\ndetailed VLBA radio images, obtained by reprocessing same\ndata-set,  through the orbital period established the\npresence of one sided and double sided radio structures\nsupporting a precessing microquasar model \\citep{Massi-2012}.\n \n\nLong-term monitoring of the source during 2007-2011 by Proportional \nCounter Array (PCA) onboard Rossi X-ray Timing Explorer (RXTE) \nestablished the superorbital modulation in X-rays and a shift of \nsuperorbital phase by 0.2 between radio and X-ray data \\citep{Li-2012, \nChernyakova-2012}.  Very recently, superorbital modulation at MeV--GeV \n$\\gamma$-rays in the apastron phase (0.5 --1.0) has been established by \nFermi-LAT \\citep{Ackermann-2013} based on the data taken during 2008 \nAugust 4 to 2013 March 24.  \n\nThis source has often shown complex behaviour in very high energy $\\gamma$-rays.\nLS I $+61^{\\circ}303$\\ was first observed at TeV energies by the MAGIC telescope system \nduring  2005 October -- 2006 March \nwith a significance of 8.7$\\sigma$ in the orbital phase 0.4--0.7, \nestablishing it as a $\\gamma$-ray binary\n\\citep{Albert-2006}. The VERITAS observations carried out during 2006 \nSeptember -- 2007 February confirmed TeV emission from this\nsource \\citep{Acciari-2008}.  However, further observations of the source \nby both MAGIC and VERITAS have shown different flux levels \n\\citep{Acciari-2011,Aleksic-2012,Aliu-2013}. These observations at TeV \nenergies show that the source behaves differently in different orbital \ncycles suggesting a variable nature of the source. Variability of the \nsource almost in all wavelengths could possibly be related to superorbital \nmodulation of the fluxes, which has been shown at radio, X-ray and \nMeV--GeV $\\gamma$-rays detected by Fermi-LAT  \\citep{Gregory-2002,Li-2012, \nChernyakova-2012,Ackermann-2013}. Hence long term multiwaveband study of\nthis source can provide an important observational support for unveiling\nthe nature of the source and the emission mechanisms.\n\nWith this motivation, we have studied the radio, X-ray and $\\gamma$-ray data from\nthis source collected over a period longer than the superorbital period. We\nhave studied the variation of flux as a function of orbital and superorbital\nphases. We have also studied multiwaveband Spectral Energy Distribution\n(SED) of the source in some of the phases. This paper is organized as follows:\nIn section 2, the data set used for these studies and analysis procedure is\ndescribed. Variation of the flux with the orbital and the superorbital phases is\ndiscussed in section 3. The SEDs and their interpretation in terms of\nmicroquasar model are given in section 4 followed by a discussion and \nconclusions in section 5.\n\n\\section{Multiwaveband Data and Analysis}\n\nIn the last few years, LS I $+61^{\\circ}303$\\ has been observed extensively by various instruments.\nIn the present work, we have used data from radio, X-ray and $\\gamma$-ray bands.\nRadio data used here is from \\cite{Richards-2011} and \\cite{Massi-2015}. These are\n15 GHz observations from 40 m single-dish telescope at Owens Valley Radio \nObservatory (OVRO). Data on LS I $+61^{\\circ}303$\\ were collected during MJD 54908.8 -- 56795.0\n(2009 March -- 2014 May). Observations were carried out approximately \ntwice a week. \n\nX-ray data were obtained from PCA onboard RXTE and X-ray Telescope or XRT\nonboard Swift. The PCA is an array of five identical \nXenon filled proportional counter units (PCUs) \\citep{Bradt_1993} covering an energy \nrange from 2 to 60 keV with a total collecting area of 6500 cm$^2$. \nData were collected over the period MJD 50143 -- 55924 and\nstandard analysis procedure was used to generate PCA\nlight curves over the energy range of 2 --\u00ad 9 keV. \n\n\nThe XRT onboard Swift consists of a grazing incidence Wolter I\ntelescope which focuses X-rays on a CCD \\citep{Burrows_2005}. This\ninstrument has an effective area of 110 cm$^2$, 23.6 arcmin field\nof view (FOV) and 15 arcsec resolution (half-power diameter).\nIt covers an energy range of 0.2 to 10 keV. \nSwift-XRT light curves were obtained from the site \n\\footnote{$http:\/\/www.swift.ac.uk\/user\\_objects\/$}.\nData is collected over the period of MJD 53980 -- 57039 (2006 September 2 \n-- 2015 January 17). Details of the procedure used for generating\nthese light curves is given in \\cite{Evans_2007}.\n\nHigh energy $\\gamma$-ray data  are obtained from Large Area Telescope (LAT)\nonboard Fermi. The Fermi-LAT is a pair production telescope \\citep{Atwood_2009} \ncovering energy range of 20 MeV to 300 GeV with a FOV of \n$\\ge$ 2.5 sr. The data taken over the period MJD 54682.9 -- 57145.9\n(2008 August 4 -- 2015 May 3) were analysed in the present work. Circular\nregion of interest (ROI) with radius 15$^\\circ$ centred at the position \nof RA(J2000) = 02$^h$ 40$^m$ 34$^s$ and Dec(J2000) = 61$^\\circ$ 15$'$ 25$''$\nwas used for extracting the data. Fermi Science Tools (FST-v10r0p5) with event \nclass Pass 8 data \nwere used for Galactic point source analysis. Since the Earth's limb is \na strong source of background $\\gamma$-rays, they were filtered out with a \nzenith-angle cut of 100$^\\circ$.  A python based software tool {\\it enrico} \n\\citep{enrico-2013} was used to do standard binned likelihood analysis. \nThe $\\gamma$-ray events in the data were binned in 8 logarithmic bins in the energies\nbetween 300 MeV and 300 GeV. Since the point-spread function (PSF) of \nLAT is large, the sources from outside of the ROI may  contribute at low energies affecting \ntrue estimates of the fluxes for the sources considered in this analysis. In order to account  \nfor this, exposure map was expanded  by another 10$^\\circ$ outside the ROI, for all events, \nas suggested by \\citet{Abdo-2009}. \n\n\nWe studied the spectral properties of  the $\\gamma$-ray emission by comparing \nthe observational results with the models of the sources present in the ROI. To get the \nbest-fit model parameters, the spatial distribution and spectral  models of the sources \nare  convolved with the instrument response function (IRF) and exposure of the observation. \nIn this work, we used newly introduced IRF version \\textit{ P8R2\\_SOURCE\\_V6}. There are 85 \npoint-like sources  and some diffuse background sources from the 3rd Fermi-LAT catalog \nlocated in the ROI. In order to account for the emission from  background sources, we \nconsidered two component background model: diffuse Galactic emission (gll\\_iem\\_v06.fits) \nand isotropic emission component (iso\\_P8R2\\_SOURCE\\_V6\\_v06.txt) consisting of emission \nfrom extra galactic background, unresolved sources and instrumental background.\n\nThe binned likelihood analysis was used for both background and source \nmodelling using {\\it gtlike} tool of FST. Spectral parameters for the source outside \nthe 3$^\\circ$ region centred at the LS I $+61^{\\circ}303$\\ position were kept fixed. \nHowever, parameters except normalization  for the point-like background \nsources were fixed or  varied based on their strength and distance from the center of the \nROI. Light curve was generated over the energy range of 300 MeV -- 300 GeV.\n\n\nFor Very High Energy (VHE) or TeV band, published data collected during 2005-2011\nfrom MAGIC \\citep{Albert-2009} and VERITAS \\citep{Acciari-2008,Acciari-2011,Aliu-2013} \nexperiments are used. These are the ground\nbased atmospheric Cherenkov experiments located in La Plama and Arizona, respectively.\n\n\n\\section{Multiwaveband Flux Variation}\n\nIn order to study the variation of flux as a function of orbital and superorbital\nphases, datasets from various wavebands were folded in 10 superorbital phase\nbins using the ephemeris given by \\cite{Massi_2016}. Each of these bins\ncorrespond to 163 days. Further, in each phase bin, 10 orbital phase bins were \ngenerated. Average flux was estimated in each of these 10 $\\times$ 10 phase bins. \nX-ray fluxes from Swift-XRT and RXTE-PCA in 10 $\\times$ 10 orbital vs superorbital \nbins are shown in the top left and the top right panels of Fig. \\ref{fig:density-plot}, respectively.  \nThese panels \nshow a definite pattern in the variation of flux over the orbital and the superorbital \nphases for both XRT and PCA data. It can be seen from the figure that the source \nis bright in the orbital phase range $\\sim$ 0.4 -- 0.8 while the corresponding superorbital \nphase is at $\\sim$ 0.3 -- 0.8. The highest flux in each of the orbital cycles \nshifts towards apastron passage as the superorbital phase value increases. Similar\nplots generated for the $\\gamma$-ray data from Fermi-LAT and 15 GHz radio data from\nOVRO are given in the middle panels of Fig. \\ref{fig:density-plot}. \nAs noted by earlier \nstudies, there is a definite shift in the pattern for the radio data: the maximum flux\nis at the same orbital phase range (0.4 -- 0.8) as in the X-ray data, but the super \norbital phase range is shifted to 0.7 to 1.4.\nThe Fermi-LAT data shows some indication of enhanced emission in the orbital phase 0.4 to 0.8, but\nunlike the other wavebands, the enhancements in the super orbital phases are not very clear.\nPlots for VHE $\\gamma$-ray data from VERITAS and MAGIC\nare shown in bottom panels of Fig. \\ref{fig:density-plot}. But the data-set is not \nextensive enough to detect any trend in this case.\n\n\\begin{figure*}[t]\n\\begin{tabular}{cc}\n\\centering\n\\includegraphics[scale=0.32,angle=-90]{fig1a.eps}\n\\includegraphics[scale=0.32,angle=-90]{fig1b.eps}\n\\end{tabular}\n\\\\\n\\begin{tabular}{cc}\n\\centering\n\\includegraphics[scale=0.32,angle=-90]{fig1c.eps}\n\\includegraphics[scale=0.32,angle=-90]{fig1d.eps}\n\\end{tabular}\n\\\\\n\\begin{tabular}{cc}\n\\centering\n\\includegraphics[scale=0.32,angle=-90]{fig1e.eps}\n\\includegraphics[scale=0.32,angle=-90]{fig1f.eps}\n\\end{tabular}\n\\caption{Multiwaveband flux as a function of orbital and superorbital\nphases. Top panels show X-ray flux from XRT data in the left panel and PCA\ndata in the right panel. Middle left panel corresponds to $\\gamma$-ray data \nfrom Fermi-LAT and middle right panel shows radio data from OVRO. Bottom\npanels correspond to VHE $\\gamma$-ray flux from VERITAS (left panel)\nand MAGIC (right panel). Flux values in each panel are normalized setting\nmedian flux to 125, i.e. the middle of the scale.}\n\\label{fig:density-plot}\n\\end{figure*}\n\n\nTo investigate this aspect further, variation of the flux as a function of superorbital\nphase was studied in various orbital phase bins. Variation of the X-ray count rates from\nSwift-XRT with superorbital phase is shown in the top left panel of Fig. \\ref{fig:xrt-lc2}. \nSimilar plots for RXTE-PCA, Fermi-LAT and OVRO are shown in the top right, bottom\nleft and bottom right panels of the same figure, respectively. \nIn each panel, different curves from bottom to top \ncorrespond to orbital phases 0-0.1, 0.1-0.2, .. , 0.9-1.0. Curves are shifted with \nrespect to each other for the sake of clarity. Error bars correspond to the standard \ndeviation in each bin. To parameterize this variation, data are fitted with a constant \nand alternatively with a sine function of the form \n$f(t) = f_o + A \\times sin(\\phi_s - \\phi_o)$, where $f_o$, $A$ and $\\phi_o$ are model\nparameters and $\\phi_s$ is the superorbital phase. Sine function with a period of 1626 \ndays gives a better fit than the constant. \nIt can be seen from the figure that there is a definite \nshift in the superorbital phase for the peak flux with respect to the  orbital phase.\nPhase at the peak of the function, peak function value and the ratio of the maximum to the minimum \nfunction values are listed in Table ~\\ref{tab:super_peak_orb}. Results are given only for \nthe cases where modulation is seen clearly in Fig. ~\\ref{fig:xrt-lc2}. \nFig. ~\\ref{fig:super_peak_orb} shows \nthese results graphically, where superorbital phase for peak of the function value is \nplotted as a function of orbital phase bins for XRT, PCA, Fermi-LAT and OVRO data.\nThis figure clearly shows the trend of increasing superorbital phase for peak as a\nfunction of orbital phase near apastron. The wavelength dependent phase difference \nbetween superorbital phase for given orbital phase bin is also evident. This\ndifference remains more or less constant in various orbital phase bins near apastron.\n \n\n\n\\begin{figure*}[t]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[scale=0.45]{fig2a.eps}\n\\includegraphics[scale=0.45]{fig2b.eps}\n\\end{tabular}\n\\\\\n\\begin{tabular}{cc}\n\\includegraphics[scale=0.45]{fig2c.eps}\n\\includegraphics[scale=0.45]{fig2d.eps}\n\\end{tabular}\n\\caption{Variation of flux with superorbital phase for XRT (top left),\nPCA (top right), Fermi-LAT (bottom left) and OVRO (bottom right). In\neach panel curves from bottom to top correspond to orbital phases \n0-0.1, 0.1-0.2, .. , 0.9-1.0. These curves are shifted along Y-axis \nfor the sake of clarity.}\n\\label{fig:xrt-lc2}\n\\end{figure*}\n\n\n\n\\begin{deluxetable*}{ccccccccccccc}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Peak flux and corresponding superorbital (SO) phase from sine function fit in various orbital phase bins \\label{tab:super_peak_orb}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Orbital} & \\multicolumn{3}{c}{XRT} & \\multicolumn{3}{c}{PCA} &\\multicolumn{3}{c}{Fermi-LAT} &\\multicolumn{3}{c}{OVRO} \\\\\n\\colhead{Phase} & \\colhead{SO} & \\colhead{peak flux} & \\colhead{ratio} & \\colhead{SO} & \\colhead{peak flux} & \\colhead{ratio} & \\colhead{SO} & \\colhead{peak flux} & \\colhead{ratio} & \\colhead{SO} & \\colhead{peak flux} & \\colhead{ratio} \\\\\n\\colhead{} & \\colhead{phase} & \\colhead{($10^{-1} ph$} & \\colhead{max} & \\colhead{phase} & \\colhead{($10^{-1} ph$} & \\colhead{max} & \\colhead{phase} & \\colhead{($10^{-1} ph$} & \\colhead{max} & \\colhead{phase} & \\colhead{($10^{-1} ph$} & \\colhead{max} \\\\\n\\colhead{} & \\colhead{at peak} & \\colhead{$ cm^{-2}~s^{-1}$)} & \\colhead{\/min)} & \\colhead{at peak} & \\colhead{$ cm^{-2}~s^{-1}$)} & \\colhead{\/min)} & \\colhead{at peak} & \\colhead{$ cm^{-2}~s^{-1}$)} & \\colhead{\/min)} & \\colhead{at peak} & \\colhead{$ cm^{-2}~s^{-1}$)} & \\colhead{\/min)}\\\\\n}\n\\startdata\n0.0--0.1     &   -    &   -   & -     & -    &   -    &  -    & -   &    -   &  -     &0.30  &  4.19 &23.88\\\\\n0.1--0.2     &   -    &   -   & -     & -    &   -    &  -    & -   &    -   &  -     &0.26  &  2.22 &3.25\\\\\n0.2--0.3     &   -    &   -   & -     & -    &   -    &  -    & -   &    -   &  -     &0.58 &   2.32 &1.86 \\\\\n0.3--0.4     &   -    &   -   & -     & -    &   -    &  -    & -   &    -   &  -     &0.70 &   2.51 &2.01\\\\\n0.4--0.5     &  0.28  &  2.79 &3.16   &0.26  &  1.60  & 1.80  &0.28 &   2.21 & 1.13   &0.80  &  8.73 &7.75\\\\\n0.5--0.6     &  0.38  &  3.12 &1.86   &0.54  &  1.83  & 1.68  &0.30 &   2.60 & 1.53   &0.90  &  6.50 &2.58\\\\\n0.6--0.7     &  0.50  &  3.26 &3.55   &0.46  &  1.75  & 1.70  &0.34 &   2.71 & 1.47   &0.06  &  9.31 &3.10\\\\\n0.7--0.8     &  0.60  &  3.27 &3.79   &0.62  &  1.64  & 2.12  &0.34 &   2.78 & 2.06   &0.10  & 9.93 &2.85\\\\\n0.8--0.9     &  0.60  &  2.53 &3.19   &0.66  &  1.46  & 2.50  &0.36 &   2.54 & 2.60   &0.30  &  6.41 &6.91\\\\\n0.9--1.0     &  0.84  &  2.34 &2.71   &0.78  &  1.16  & 1.51  &0.42 &   2.28 & 1.89   &0.26 &   3.82 &1.82\\\\\n\\enddata\n\\end{deluxetable*}\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.35,angle=0]{fig3.eps}\n\\caption{Superorbital phase at peak flux from fitted sinusoidal function given in \nTable ~\\ref{tab:super_peak_orb}\nas a function of orbital phase bins for XRT, PCA, Fermi-LAT and OVRO data. Positions\nfor apastron (A) and superior conjunction (SC) are marked.}\n\\label{fig:super_peak_orb}\n\\end{figure}\n\n\n\n\n\\section{Spectral Energy Distribution}\\label{sec:sed}\n\nWe have investigated the spectral properties of the source at\ndifferent orbital and superorbital phases. Following Figure \\ref{fig:density-plot}\n three different regions were chosen.\nTwo of the regions are bright in most of the wavebands and the third one is of low brightness. \nThese regions are i) Super orbital phase: 0.3 -- 0.5 Orbital phase: 0.6 -- 0.8, \nii) Super orbital phase: 0.5 -- 0.7 Orbital phase: 0.6 -- 0.8 and iii) Super orbital \nphase: 0.0 -- 0.2 Orbital phase, 0.0 -- 0.2 (hereafter state1, state2, and state3, \nrespectively). X-ray and Fermi-LAT spectral data were analysed for these three regions.\nFermi-LAT analysis procedure is already described in section 2. We have analysed\nspectral data from Swift-XRT and RXTE-PCA corresponding to the states mentioned above.\nSome details of these observations are given in Table \\ref{tab:xrt-pca-log}.\nDates for XRT and PCA observations for each of the three states are listed in the\ntable along with the total observation duration. \n\nIn case of XRT we have fitted spectrum over the energy range of 0.3 to 10 keV.\nSource and background photons were selected using the tool XSELECT. Data were recorded in\nPhoton Counting (PC) mode for these observations. Source photons were selected from a circular\nregion with the radius of 20 pixels (i.e. 47 arc-seconds), whereas \nnearby circular region with radius of 40 pixels was used for extracting \nbackground photons.\nEvents with grades 0-12 were selected in this analysis. \nThe spectral data were rebinned using tool GRPPHA with 20 photons per bin. \nStandard response matrices and ancillary response files\nwere used.\n\nIn case of PCA, standard 2 data with time resolution of 16 s and  128 channels\nof energy information were used. Data were analysed using HEASOFT (version 6.15). \nFor each observation, data were filtered using the standard procedure\ngiven in the RXTE Cook Book. The tool 'pcabackest' was used for generation\nof background model, calibration files for 'faint' source (less than 40 ct\/sec\/PCU)\nfrom RXTE GOF were used. To improve statistics, only\ndata from top layer of PCU2 was used.\n\n\n\n\n\n\\begin{deluxetable*}{ccccc}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Observation log for XRT and PCA \\label{tab:xrt-pca-log}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{State} & \\colhead{Instrument} & \\colhead{Observation dates} & \\colhead{Number of} & \\colhead{Total duration}\\\\\n\\colhead{} & \\colhead{} & \\colhead{} & \\colhead{observations} & \\colhead{seconds}\\\\\n}\n\\startdata\n1     &  XRT       & 2010-10-22, 2010-11-18, 2010-12-17, 2014-10-18, & 13 &15805 \\\\\nSuperorb. phase : 0.3--0.5       &            & 2014-10-20 - 2014-10-23, 2014-11-14, 2014-11-15,   & &     \\\\\nOrbital phase :  0.6--0.8      &            &  2014-12-11 - 2014-12-13   &&  \\\\\n      &   PCA      & 2010-02-25, 2010-02-28, 2010-03-24, 2010-03-28,  & 18 & 22416   \\\\\n      &            & 2010-04-20, 2010-04-22, 2010-05-16, 2010-05-18,   & &  \\\\\n      &            & 2010-06-14, 2010-07-10, 2010-08-05, 2010-09-02,  & & \\\\\n      &            & 2010-09-26, 2010-09-30, 2010-10-25, 2010-11-17,   & &\\\\\n      &            & 2010-11-21, 2010-12-16   &&\\\\\n\\hline\n2     &  XRT       & 2006-09-05, 2006-11-21 - 2006-11-24, 2006-12-18, & 10 & 20147 \\\\\nSuperorb. phase : 0.5 -- 0.7       &            & 2006-12-20, 2006-12-22, 2011-01-14, 2011-10-01 &&\\\\\nOrbital phase : 0.6 -- 0.8      &  PCA       & 2006-10-27, 2006-10-29, 2011-01-09, 2011-01-13, & 20 & 24384 \\\\\n      &            & 2011-02-06, 2011-02-09, 2011-03-06, 2011-03-31, && \\\\\n      &            & 2011-04-03, 2011-04-28, 2011-05-22, 2011-05-26, && \\\\\n      &            & 2011-06-19, 2011-07-13, 2011-07-17, 2011-08-10, && \\\\\n      &            & 2011-08-14, 2011-09-07, 2011-10-03, 20011-10-30 && \\\\\n\\hline\n3     &   XRT      &  2008-10-22, 2008-11-19, 2008-12-17, 2013-11-23, & 6 & 10266 \\\\\nSuperorb. phase : 0.0 -- 0.2       &            &  2013-12-14, 2014-01-11 && \\\\\nOrbital phase : 0.0 -- 0.2       &   PCA      &  2008-10-22, 2008-10-25, 2008-11-17, 2008-11-19, & 21 & 32912 \\\\\n      &            &  2008-12-13, 2008-12-17, 2009-01-10, 2009-02-04,   && \\\\\n      &            &  2009-02-07, 2009-03-05, 2009-03-29, 2009-04-02,   && \\\\\n      &            &  2009-04-26, 2009-05-21, 2009-05-25, 2009-06-18,   && \\\\\n      &            &  2009-07-12, 2009-07-15, 2009-08-08 &&\\\\\n\\enddata\n\\end{deluxetable*}\n\nA combined spectral fit was performed for XRT and PCA data. The PCA spectrum was\nnormalized with the XRT spectrum for this purpose. The XRT and PCA spectra covering the energy range\nof 0.7-20 keV were fitted by using XSPEC with a powerlaw with the line-of-sight absorption,\nwhich was kept free during the fit.\nModel\nparameters for the combined fit as well as for only XRT data are listed in Table\n\\ref{tab:fit-params-xrt-pca}.\nSince the bandwidth of the data is quite limited, we find a correlation\nbetween the power-law index and the absorption, indicating that a steeper power-law\nis compensated by a large absorption. For the wide band fitting, we use the joint\nXRT-PCA fit because the higher energy data from PCA    constrains the power-law better.\n\n\n\n\n\\begin{deluxetable}{cccc}\n\\tablecaption{Best-fit parameters of a power-law (with absorption) fit to the data for XRT and PCA \\label{tab:fit-params-xrt-pca}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{}  &\\multicolumn{3}{c}{Only XRT}\\\\\n\\colhead{}  & \\colhead{$N_H$ (10$^{22}$ cm$^{-2}$)}  & \\colhead{alpha}  &  \\colhead{norm}\\\\\n}\n\\startdata\nstate1    & 0.68$\\pm$0.05  & 1.58$\\pm$0.06 &   (2.51$\\pm$0.21)$\\times 10^{-3}$  \\\\\nstate2    & 0.70$\\pm$0.05  & 1.53$\\pm$0.05  &  (3.08$\\pm$0.20)$\\times 10^{-3}$   \\\\\nstate3    & 0.69$\\pm$0.11  & 1.47$\\pm$0.12  &  (1.25$\\pm$0.20)$\\times 10^{-3}$    \\\\\n\\hline\n& \\multicolumn{3}{c}{XRT+PCA (all layers)} \\\\\n&  $N_H$ (10$^{22}$ cm$^{-2}$)      &     alpha      &      norm     \\\\\n\\hline\nstate1   & 0.81$\\pm$0.04  & 1.79$\\pm$0.03  &  (3.22$\\pm$0.15)$\\times 10^{-3}$ \\\\\nstate2   & 0.90$\\pm$0.03  & 1.78$\\pm$0.03  &  (4.27$\\pm$0.17)$\\times 10^{-3}$  \\\\\nstate3   & 1.05$\\pm$0.08  & 1.95$\\pm$0.05  &  (2.25$\\pm$0.18)$\\times 10^{-3}$  \\\\\n\\enddata\n\\end{deluxetable}\n\n\n\n\nFermi-LAT SEDs for the three states fitted with a cutoff powerlaw are given in Fig.\n\\ref{fig:fermi-sed} and model parameters are listed in Table \\ref{tab:fermi-param}.\nSome differences are seen in the spectral indices for $\\gamma$-rays between state1\nand other states (see Table \\ref{tab:fermi-param}). In case of X-ray data some \nsteepening of the spectrum is seen as flux decreases, as indicated by variation in\nspectral index  (see Table \\ref{tab:fit-params-xrt-pca}).\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.45]{fig4.eps}\n\\caption{A cutoff power law fit to Fermi-LAT data for the three different states. Best-fit curves are shown\nas solid lines.}\n\\label{fig:fermi-sed}\n\\end{figure}\n\n\n\\begin{deluxetable}{cccc}\n\\tablecaption{Parameters of a cutoff power law fit to the Fermi-LAT data for  three different states \\label{tab:fermi-param}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{parameters} & \\colhead{state1} & \\colhead{state2} & \\colhead{state3}\n}\n\\startdata\n$\\alpha$                        &   2.31         &   2.12       &  2.12    \\\\\nflux                            &   2.48 $\\times 10^{-7}$       &   2.41$\\times 10^{-7}$    & 2.28$\\times 10^{-7}$  \\\\\nEc (MeV)                        &   30041         &  10000          &  6338   \\\\\nTS                              &   2663         &  2727          &  2328    \\\\\n\\enddata\n\\end{deluxetable}\n\n\n\n\nWe have investigated the   spectral energy distributions (SEDs) of the source. The state3 \ndoes not have TeV data and hence we have made a detailed SED study for the\nother two states. These states are bright in all wavebands and hence can be used as\na template to understand the emission mechanisms. \nVERITAS spectral data obtained from \\cite{Acciari-2011} corresponds to state1.\nFor radio flux, the average of 15 GHz data from OVRO described in section 2 \nis used. This sets an upper limit on the modelled radio flux. In addition, \nwe have also plotted radio data from \\citet{Strickman_1998}, which correspond to orbital \nphase of 0.8 and superorbital phase of 0.8.\n\n\n\nSince LS I $+61^{\\circ}303$\\ is identified as a potential microquasar based on radio observations, \nhigh energy emission is likely to be produced in jets.  In case of microquasars, \ncompact object could be a neutron star or a black hole accreting matter from a\ncompanion star which presumably drives relativistic outflow or jet from the\ncompact object. Acceleration of charged particles in the jet produces high\nenergy emission. We have considered this scenario to model the SEDs. In the\ncontext of leptonic model, the low energy emission arises from Synchrotron emission\nfrom ultra-relativistic electrons in the jet. Whereas the high energy emission\narises from inverse Compton scattering of soft photons, which could be either\nsoft photons from Synchrotron radiation (Synchrotron Self-Compton or SSC \nmodel) or photons from companion star or accretion disk (External Compton\nmodel). In this work, relativistic jet making an angle of 30$^\\circ$ \n(\\cite{Gupta-2006} and reference therein) with our line of sight is considered.\nElectrons are assumed to have a broken power-law energy spectrum given by\n\n\\begin{eqnarray}\n{dn_e \\over d\\gamma}  \\propto \\begin{cases}\n \\gamma^{-\\alpha} ~\\mbox{for} ~\\gamma <\\gamma_{br} \\nonumber \\\\\n  \\gamma^{-\\beta} \\exp{\\left(-{\\gamma \\over \\gamma_c} \\right )} ~ \\mbox{for}~ \\gamma_{br}\\leq \\gamma \\leq \\gamma_c                   \n  \\end{cases}\n\\label{eqn:pi0}\n\\end{eqnarray}\n\nwhere, $n_e$ denotes the number density of the electrons, $\\gamma$ is \nthe Lorentz factor of the electron, $\\alpha$ and $\\beta$ are spectral indices,\n$\\gamma_{br}$ break energy and $\\gamma_c$ the highest energy of the electron.\n\nFor this source, the distance is taken as 2 kpc \\citep{Hutchings-1981,\nFrail-1991} and Lorentz factor for bulk motion is assumed to be\n1.25 \\citep{Massi-2004}. The models are shown for state1 and state2 respectively\nin Figure \\ref{fig:xxx_p} and Figure \\ref{fig:xxx_a}. \n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.45]{fig5.eps}\n\\caption{The SED of LS I $+61^{\\circ}303$\\ for state1.\nThe synchrotron and inverse Compton spectra are calculated using the  parameters\nas given in Table \\ref{tab:params_fit_SED}. X-ray, Fermi-LAT and \nVERITAS data for state 1 are shown with points in red color. Radio data shown in\nthe figure do not correspond to state 1. The average flux from OVRO is shown with \nfilled circle of red color, whereas radio data from \\citet{Strickman_1998} is shown \nwith brown triangles.}\n\\label{fig:xxx_p}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.45]{fig6.eps}\n\\caption{The SED of LS I $+61^{\\circ}303$\\ for state2.\nThe synchrotron and inverse Compton spectra are calculated using the same parameters\nas given in Table \\ref{tab:params_fit_SED}. X-ray and Fermi-LAT data \nfor state 2 are shown with points in blue color. Since, VERITAS data for state 2 is \nnot available, state 1 VERITAS data is used which is shown in brown colour. Radio \ndata are the same as in Fig.~\\ref{fig:xxx_p}.}\n\\label{fig:xxx_a}\n\\end{figure}\n\nHere it is assumed that the radio, X-ray and $\\gamma$-ray emissions originate in the\nsame region and the magnetic field in emission blob is quite high, of the\norder of $10^3$ G. Rest of the model parameters are fitted \nand these parameters are listed in Table \\ref{tab:params_fit_SED}.\nTo explain the TeV $\\gamma$-ray emission it was necessary to include IC of\nphotons from accretion disk or companion star in addition to the SSC\ncomponent. Radiation density ($U_{rad}$) is estimated from luminosity  \n$L$ using expression $U_{rad} = L\/  {4 \\pi R^2 c}$, where $R$ is the \ndistance of the emission volume from the companion star or the accretion disk. \nRadiation density \nfrom the companion star, with $L_c = 2 \\times 10^{38}$ erg s$^{-1}$ and a distance of \n$R~\\sim 10^{12}$ cm, is about  4 orders of magnitude higher than the\ncorresponding radiation density from the  accretion disk. Hence, we have considered \nonly the seed photons from the companion star for the External Compton model. However, \nthis spectrum cannot explain the observed data as seen from Figure \\ref{fig:xxx_p} \nand Figure \\ref{fig:xxx_a}. In this case, we have considered radius of the emission \nvolume as a parameter for the fit to the data. We can also estimate the radius of \nemission volume from the variability time scale of the source. We fixed the size \nof the emission region according to the estimates from variability study \\citep{Smith-2009},\nwhich indicates a possible size of the emission region to be $\\sim 6 \\times 10^{10}$ cm.\nConsidering that the bulk Lorentz factor is 1.25, this size   corresponds to\n$\\sim~7.5 \\times 10^{10}$ cm. Fixing the emission region size to this value, \nmodel parameters were estimated which are given in the last column of Table \\ref{tab:params_fit_SED}.\nAlthough the synchrotron spectrum explains the observed fluxes from radio to MeV--GeV \nenergies, SSC spectrum alone cannot fit the data well. Hence we have also estimated \nthe contribution of companion star photons for this low magnetic field case and we \nfound that the external Compton model overestimates the observed flux at MeV--TeV \nenergies. However, SSC and EC models together can explain the data well if the \ncompanion star luminosity is considered to be reduced by a factor of 10. This is \nshown in Figure \\ref{fig:xxx_new_B.eps}.\n\nIn the spectral fitting, we did not consider the radio data (triangles \nin Fig.~\\ref{fig:xxx_a}, \\ref{fig:xxx_p} and \\ref{fig:xxx_new_B.eps}) from VLA observation \n\\citep{Strickman_1998} in fit, since the orbital and superorbital phases for these are \ndifferent from the phases for state 1 and state 2. Since, the energy spectrum is not \navailable for OVRO data, we have used average flux as an upper-limit for SED modelling, \nand for the chosen set of parameters the model does not overestimate radio fluxes for \nthe states considered above.\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.45]{fig7.eps}\n\\caption{The SED of LS I $+61^{\\circ}303$\\ for state1.\nThe synchrotron and inverse Compton spectra are calculated using the same parameters\ngiven in the last column of Table \\ref{tab:params_fit_SED}, with the emission region radius\ndecided from the variability time scale.}\n\\label{fig:xxx_new_B.eps}\n\\end{figure}\n\n\\begin{deluxetable}{cccc}\n\\tablecaption{Parameters of the fit for microquasar scenario \\label{tab:params_fit_SED}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Parameters} & \\colhead{state1} & \\colhead{state2} & \\colhead{state1 (radius from} \\\\\n\\colhead{} & \\colhead{} & \\colhead{} & \\colhead{variability study)} \\\\\n}\n\\startdata\n Magnetic field (Gauss)  & 5 $\\times 10^3$   & 5 $\\times 10^{3}$  & 15 \\\\\n$\\gamma_{min}$          & 4.4               & 4.9              & 110 \\\\\n $\\gamma_{max}$          & 5.6 $\\times 10^6$        & 5.4 $\\times 10^6$  &  9.0 $\\times 10^7$ \\\\\n\n spectral index ($\\alpha$) & 2.53             & 2.55          & 2.7       \\\\\n spectral index ($\\beta$ )  & 2.34            & 2.40           & 2.4      \\\\\n radius (cm)                 & 11.5 $\\times 10^7$ &   18.0 $\\times 10^7$   & 7.5 $\\times 10^{10}$             \\\\\n Gamma Break            & 1.0 $\\times 10^5$        & 1.4 $\\times 10^7$   & 9.0 $\\times10^4$       \\\\\n Bulk Lorentz factor    & 1.25              &  1.25         & 1.25        \\\\\n Distance (kpc)         & 2.0               & 2.0              & 2.0   \\\\\nInclination angle(deg)  & 30.0              & 30.0       & 30.0          \\\\\n\n Luminosity (erg\/s)     & 3.9 $\\times 10^{35}$ &  3.8 $\\times 10^{35}$ &  4.3 $\\times 10^{35}$ \\\\\n\\enddata\n\\end{deluxetable}\n\n\n\n\n\\section{Discussion and Conclusions}\\label{sec:discussion}\n\n\nLong-term timing analyses of LS I $+61^{\\circ}303$\\ at different wavelengths has shown some of the \ninteresting characteristics of the source. Flux in various wavebands shows \nvariation with superorbital phase and this variation is wavelength dependent as well\nas the binary phase dependent.\nAt X-ray energies, as evident from Figure \\ref{fig:density-plot}, the source is\nbright at orbital phases $\\sim 0.4-0.8$ and superorbital phases of $\\sim\n0.3-0.8$. Whereas at radio energies, the source is bright at orbital phases of\n$\\sim 0.4-0.8$ and superorbital phases of $\\sim 0.7-1.4$.    \nThe $\\gamma$-ray flux in MeV-GeV band as given by Fermi-LAT\nshows a shift relative to the radio and the X-ray bands.\nThis behaviour possibly indicates that radio,\nX-ray and $\\gamma$-ray emissions could be originating from different regions.\n\nThe long-term superorbital modulation of flux could support the scenario where \ncircumstellar disk of a Be star quasi-cyclically expands and shrinks (e.g., \n\\cite{Negueruela-2001}). However, for such a scenario the long-term period is \nvariable from cycle to cycle \\citep{Rivinius_2013}. Recent analysis of radio \ndata established the fact that the long-term period is quite stable over 8 \ncycles \\citep{Massi_2016} which makes the scenario of quasi-cyclic variation \nof circumstellar disk of the Be star for LS I $+61^{\\circ}303$\\ less probable. This stable \nsuperobital modulation is attributed to periodic Doppler boosting effects \nof the precessional jets associated with the compact objects \\citep{Massi_2014}.\n\nIn this paper, we have seen that the modulation of flux with superorbital phase \nis more prominent in orbital phase bins near apastron. This is clearly seen at \nvarious wavelengths in Figures \\ref{fig:xrt-lc2} and \\ref{fig:super_peak_orb}. \nAlthough the long-term superorbital variation does not support the variation of \ncircumstellar disk size,  this type of superorbital modulation near the apastron \ncould stem from the interaction of the compact object with the circumstellar disk \nof the Be star. The equivalent width (EW) of the H$\\alpha$ emission line is \nrelated to the size of the stellar disk \\citep{Zamanov-Mart-2000,Grundstrom_2006}. \nIn addition to that, it has been found that the maximum of the EW of H$\\alpha$ \noccurs in a region around superorbital phase of $\\sim$ 0.4 (see \\cite{Zamanov-1999,\nZamanov-Mart-2000}) considering superorbital period of 1584 days. However, if we \nuse superorbital period as 1626 days then  the maximum of the EW of H$\\alpha$ \noccurs  at $\\sim$ 0.3. From Figure  \\ref{fig:super_peak_orb} we see that flux \nof gamma-rays is high at the superorbital phase of $\\sim$ 0.3--0.5, which suggests \nthat the disk plays an important role\nin modulating $\\gamma$-rays. Although, a similar enhancement of X-rays at superorbital\nphase of 0.2 is seen by \\cite{Li-2011} considering only peak flux, we see that X-ray\nflux peaks at the superorbital phases in the range  of $\\sim$ 0.4--0.8 depending on the\norbital phase. We see that the peak of radio flux is shifted further. It suggests that even \nif the disk size plays a significant role for $\\gamma$-rays, X-ray and radio fluxes \nare not necessarily  affected much by the size of the disk.\n\n\n\nFigure \\ref{fig:xrt-lc2} shows that, for all wavebands, the superorbital variability \nis not significant in the periastron region, whereas it is significant at the apastron.\nThis can support the scenario where one assumes that the interaction between\ncompact object and the circumstellar disk of Be star is strong when compact object is in the\nproximity of Be star.\nAs a result, superorbital modulation effect becomes insignificant as \nsuggested by \\cite{Ackermann-2013}.  However, it becomes dominant as the compact object \nstarts moving towards the apastron region. \n\nAnother possible scenario for the modulation is related to the precession of the Be star disk about \nthe orbital  plane. If this scenario is adopted for possible \nexplanation of the \nstrong superorbital modulation in the apastron phase, then the angular distance between \norbital plane and the disk plane should become minimum. As a result, even if the compact \nobject is far from the Be star, the smaller angular distance between disk plane and orbital \nplane provides relatively higher interaction of the compact object with the disk.   \n\nIn addition to the superorbital modulation in the apastron phase (0.5 -- 1.0) we have seen \nphase lag among radio, X-ray and $\\gamma$-rays.  The possible explanation for the constant \nphase lag between X-ray and radio is  that the plasma blobs filled with \nhigh-energy particles may escape from the X-ray emission region to the radio emission region \nwhich is at a distance of $\\sim$  \n10 times the binary separation distance as proposed by \\cite{Chernyakova-2012} in the context \nof pulsar wind scenario. However, in the microquasar scenario, different regions in the jets can be \nresponsible for the phase lag.  We have also seen the phase lag between radio and \n$\\gamma$-rays. In such binary systems, $\\gamma$-rays are considered to be produced by \nup-scattering of radio photons or accretion disk\/star photons. If the $\\gamma$-rays are \noriginating through the up-scattering of radio photons which are being produced by the same \npopulation of electrons then there  should not be any phase lag between radio and \n$\\gamma$-rays. Hence, up-scattering of  a separate population of photons could be a\npossible explanation for the phase lag between $\\gamma$-ray and radio fluxes. \n\nIn addition to the timing analysis, we also tried to understand the spectral behaviour of the \nsource at different orbital and superorbital phases. We have chosen three \ndifferent regions following flux variations for X-ray, radio, and $\\gamma$-rays as function of \norbital and superorbital phase. From Figure \\ref{fig:density-plot} we see that the source \nat high energy is mostly very \nactive in the orbital phase bin of 0.5--0.8 and superorbital phase bin 0.3--0.7.  We \nselected two different regions  with superorbital phase 0.3--0.5; orbital phase: 0.6--0.8 (state1) \nand  superorbital phase: 0.5 -- 0.7; orbital phase: 0.6 -- 0.8 (state2) from this region where source \nis bright at all wavelengths. To compare the spectral variation with the other orbital and \nsuperorbital phases where the source is not bright, we have chosen a region with  superorbital \nphase 0.0 -- 0.2 and orbital phase 0.0 -- 0.2 (state3). \nBased on these three different regions of orbital or superorbital phases, we have analysed \nX-ray and Fermi-LAT data to see the spectral behaviour of the source at high energies. \nWe found no significant differences  between flux levels but we see some variations in the\nspectral indices at Fermi-LAT energies. However, we see some difference in both the spectral\nindices and flux levels for XRT-PCA data, though the interplay of the spectral shape and the absorption\n playing a role in this trend cannot be ruled out.\n\n\nFrom the fit to the SED we have seen that we can explain the data well considering LS I $+61^{\\circ}303$\\ as a microquasar. \nIn the microquasar model, it is generally assumed that the high energy emission comes from the region which \nis very close to the compact object to reduce the effect of $\\gamma \\gamma$ absorption due to \nradiation field of companion star \\citep{Gupta-2006_a}. Magnetic field in this region is relatively high \nas considered \nfor our model, and we have estimated emission volume of the order of $~10^{8}$ cm. In this emission \nvolume, some the of the emitted $\\gamma$-rays can be absorbed  through $e^+e^-$ pair creation process due to X-ray \nphotons in  emission volume. We have estimated that about 20\\% of $\\gamma$-rays will be absorbed at TeV energies. \nHowever, larger sizes of emission volume will make this absorption insignificant. \nWe have seen that it is possible to have lower values of magnetic field strength \nto explain the observed data, in case of larger emission volume. \n\nWe have also seen in section \\ref{sec:sed} that if we consider the radius of emission volume obtained\nfrom variability study, the magnetic field from the model fit to the data is estimated to be $\\sim 10$ G. \nHowever, we found that the SSC model alone cannot explain the TeV data well and EC model overestimates \nthe observed flux for the luminosity of the companion star $\\sim 10^{38}$ erg s$^{-1}$. A lower value \nof this luminosity ($\\sim 10^{37}$ erg s$^{-1}$) can explain the data well. This suggests that lower values of \nmagnetic field in the emission blobs are suitable for LS I $+61^{\\circ}303$\\ to explain the observed data constraining \nthe luminosity of the companion star. With high magnetic field, the Synchrotron cooling timescale is \nmuch smaller than the variability timescale which could be as low as 2 seconds as estimated by \n\\cite{Smith-2009}. In our SED fitting, we have considered that the emitting blob is close to the compact \nobject.  The blob size increases as it moves away from the compact object in the jets and magnetic \nfield reduces. The time-averaged values of flux from a particular region in the  jet as considered \nby \\cite{Gupta-2006_a} could reduce the discrepancy between Synchrotron cooling time scale and the time \nscale of X-ray flux variability. From the SED, it \nseems that the one single emission process is responsible for X-ray and MeV-GeV data. Hence, we have \nconsidered Synchrotron emission process to explain the data up to GeV energies. As a result, a high \nmagnetic field is required to explain the data if the maximum energy of high energy electrons is not \nwell above $\\sim$ 1 TeV. A good quality data in the hard X-ray region can establish whether a different \nemission component is  required to explain the data at MeV-GeV region. It can also indicate if we need \na different population of electrons to explain data at different energy bands in the SED.\n\nWe have also seen that the fitted model parameters show hardening of spectral index after break (see \nTable \\ref{tab:params_fit_SED}). In addition, flux levels for different states (mainly X-ray) are \ndifferent. A change in the location of the compact object relative to the companion star during the orbital and the \nsuperorbital cycles and its interaction with circumstellar disk could be responsible for the changing \nelectron spectral distribution.\n\nIn the context of timing analysis, we have seen the phase lag among radio, X-ray and $\\gamma$-ray data which \nmay suggest that they originate from different emission regions. However, in our present spectral modelling we have \nconsidered single emission zone to explain the mutiwavelength data. To support the scenario of different origins we \nneed simultaneous multiwavelength data for longer period both for timing and spectral analysis. At present, \nwe have such observation for radio, X-ray and MeV--GeV gamma-rays. However, GeV--TeV data is also required to get \na complete understanding of the source in multifrequencies.\n\nThe following major conclusions can be drawn, based on the study presented here:\n\n\\begin{itemize}\n\\item\n{The super orbital modulation is more pronounced near the apastron for all \nwavelengths, supporting geometric scenarios as a cause for the super orbital modulation.\n}\n\\item\n{There is a definite wavelength dependent variation of the maximum of the super orbital\nflux with respect to the binary phase. This variation shows a wavelength dependent\nshift.\n}\n\\item\n{Emission from radio to GeV gamma-rays during the maximum emission can be modelled by an one-zone micro-quasar \njet model.  To explain the TeV emission, Comptonization from\nan External Compton source is necessary especially when low magnetic field is assumed. \nIn this case, we suggest that the photons from the companion star, with a lower luminosity\n($\\sim$10$^{37}$ erg s$^{-1}$), is adequate to explain the data.\n}\n\\item\n{Extended hard X-ray data would be necessary to constrain the synchrotron model and \nTeV observations across a super orbital cycle, along with X-ray measurements, would\nbe required to make a detailed emission model for this source.\n}\n\\end{itemize}\n\n\n\n\n\n\\section*{Acknowledgements}\n\nWe acknowledge the use of data from the High Energy Astrophysics Science Archive\nResearch Center (HEASARC), provided by NASA's Goddard Space Flight Center. Also the data\nsupplied by the UK Swift Science Data Centre at the University of Leicester has been used\nin present work.\nWe thank Hovatta Talvikki for providing us data from OVRO 40-m monitoring program which was used in \nthe research by \\citep{Richards-2011}(supported in part by NASA grants NNX08AW31G \nand NNX11A043G, and NSF grants AST-0808050 and AST-1109911). We acknowledge the use of Fermi-LAT data \nand analysis tool from Fermi Science Support Center. We would also like to thank MAGIC collaboration \nfor making their published data public which has been used in this work. We also acknowledge VERITAS\ncollaboration for their published data used in this work.\n\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nPre-trained models are commonly used as backbones of machine learning systems.\nIn practice, we often want to \\textit{edit} models after pre-training,\\footnote{We use the term \\textit{editing} to refer to any intervention done to a model done after the pre-training stage.} to improve performance on downstream tasks \\citep{zhuang2020comprehensive,wortsman2021robust,matena2021merging,ilharco2022patching}, mitigate biases or unwanted behavior \\citep{shibani2021editing,lu2022quark,ribeiro2022adaptive,murty2022fixing},\nalign models with human preferences \\citep{askell2021general,ouyang2022training,kasirzadeh2022conversation,sparrow},\nor update models with new information \\citep{zhu2020modifying,de2021editing,mitchell2021fast,mitchell2022memory}.\n\nIn this work, we present a new paradigm for editing neural networks based on \\emph{task vectors}, which encode the information necessary to do well on a given task.\nInspired by recent work on weight interpolation \\citep{frankle2020linear,wortsman2021robust,matena2021merging,wortsman2022model,ilharco2022patching,li2022branch,ainsworth2022git}, we obtain such vectors by taking the weights of a model fine-tuned on a task and subtracting the corresponding pre-trained weights (Figure \\ref{fig:main}a).\n\nWe show that we can edit a variety of models with \\emph{task arithmetic}---performing simple arithmetic operations on task vectors (Figure \\ref{fig:main}b-d). For example, \\emph{negating} a vector can be used to remove undesirable behaviors or unlearn tasks, while \\emph{adding} task vectors leads to better multi-task models, or even improves performance on a single task.\nFinally, when tasks form an \\emph{analogy} relationship, task vectors can be combined to improve performance on tasks where data is scarce. \n\n\\paragraph{Forgetting via negation.} Users can negate task vectors to mitigate undesirable behaviors (e.g., toxic generations), or even to forget specific tasks altogether, like OCR. In Section \\ref{sec:negation}, we negate a task vector from a language model fine-tuned on toxic data \\citep{radford2019language,borkan2019nuanced}, reducing the proportion of generations classified as toxic, with little change in fluency. We also negate task vectors for image classification tasks, resulting in substantially lower accuracy on the task we wish to forget with little loss on ImageNet accuracy \\citep{deng2009imagenet}.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/task_vectors.pdf}\n    \\caption{An illustration of task vectors and the arithmetic operations we study for editing models. (a) A task vector is obtained by subtracting the weights of a pre-trained model from the weights of the same model after fine-tuning (Section \\ref{sec:task-vectors}). (b) Negating a task vector degrades performance on the task, without substantial changes in control tasks (Section \\ref{sec:negation}). (c) Adding task vectors together improves the performance of the pre-trained model on the tasks under consideration (Section \\ref{sec:addition}). (d) When tasks form an analogy relationship such as supervised and unsupervised learning on two different data sources, it is possible to improve performance on a supervised target task using only vectors from the remaining three combinations of objectives and datasets (Section \\ref{sec:analogies}). \n    }\n    \\label{fig:main}\n\\end{figure*}\n\n\\paragraph{Learning via addition.} Adding task vectors results in  better multi-task models, or improved performance on a single task. In Section \\ref{sec:addition}, we add task vectors from various image models (CLIP, \\citet{radford2021learning}) and compare the performance of the resulting model with using multiple specialized fine-tuned models. We find that the single resulting model can be competitive with using multiple specialized models. Adding two task vectors maintains 98.9\\% of the accuracy, and the average performance on the entire set of tasks increases as more task vectors are added.\nMoreover, adding a task vector from a different task can \\textit{improve} performance on a target task using text models (T5, \\citet{colin2020exploring}).\n\n\\paragraph{Task analogies.} When we can form task analogies of the form ``$A$ is to $B$ as $C$ is to $D$\", combining task vectors from the first three tasks improves performance on the fourth, even when little or no training data is available.\nIn Section \\ref{sec:analogies}, we show that we can improve domain generalization to a new target task without using labeled data from that task. More specifically, accuracy on a sentiment analysis task improves by combining a task vector from a second sentiment analysis dataset and task vectors produced using unlabeled data from both domains.\nWe also use analogies between classifying pictures and sketches of objects to improve accuracy on subgroups where little or no data is available.\n\n\nOverall, editing models with task arithmetic is simple, fast and effective. \nThere is no extra cost at inference time in terms of memory or compute, since we only do element-wise operations on model weights.\nMoreover, vector operations are cheap, allowing users to experiment quickly with multiple task vectors.\nWith task arithmetic, practitioners can reuse or transfer knowledge from models they create, or from the multitude of publicly available models all without requiring access to data or additional training.\\footnote{Code available at \\url{https:\/\/github.com\/mlfoundations\/task_vectors}.}\n\n\\section{Task Vectors}\n\\label{sec:task-vectors}\nFor our purposes, a task is instantiated by a dataset and a loss function used for fine-tuning.\nLet $\\theta_\\textrm{pre} \\in \\mathbb{R}^d$ be the weights of a pre-trained model, and $\\theta_\\textrm{ft}^t\\in \\mathbb{R}^d$ the corresponding weights after fine-tuning on task $t$. \nThe task vector $\\tau_t \\in \\mathbb{R}^d$ is given by the element-wise difference between $\\theta_\\textrm{ft}^t$ and $\\theta_\\textrm{pre}$, i.e., $\\tau_t = \\theta_\\textrm{ft}^t - \\theta_\\textrm{pre}$.\nWhen the task is clear from context, we omit the identifier $t$, referring to the task vector simply as $\\tau$.\n\nTask vectors can be applied to any model parameters $\\theta$ from the same architecture, via element-wise addition, with an optional scaling term $\\lambda$, such that the resulting model has weights $\\theta_\\textrm{new} = \\theta + \\lambda \\tau$.\nIn our experiments, the scaling term is determined using held-out validation sets.\nNote that adding a single task vector to a pre-trained model with $\\lambda=1$ results in the model fine-tuned on that task.\n\nFollowing \\citet{ilharco2022patching}, we focus on open-ended models, where it is possible to fine-tune on a downstream task without introducing new parameters (e.g., open-vocabulary image classifiers \\citep{radford2021learning,jia2021scaling,pham2021combined,alayrac2022flamingo} and text-to-text models \\citep{colin2020exploring,radford2019language,brown2020language,hoffmann2022training}).\nIn cases where fine-tuning introduces new parameters (e.g., a new classification head), we could follow \\citet{matena2021merging} and merge only the shared weights, but this exploration is left for future work.\n\n\n\\paragraph{Editing models with task arithmetic.} We focus on three arithmetic expressions over task vectors, as illustrated in Figure \\ref{fig:main}: negating a task vector, adding task vectors together, and combining task vectors to form analogies. All operations are applied element-wise to the weight vectors. \n\n\nWhen \\emph{negating} a task vector $\\tau$, applying the resulting vector $\\tau_\\textrm{new} = - \\tau$ corresponds to extrapolating between the fine-tuned model and the pre-trained model. The resulting model is worse at the target task, with little change in performance on control tasks (Section \\ref{sec:negation}).\n\\emph{Adding} two or more task vectors $\\tau_i$ yields  $\\tau_\\textrm{new} = \\sum_i \\tau_i$, and results in a multi-task model proficient in all tasks, sometimes even with gains over models fine-tuned on individual tasks (Section \\ref{sec:addition}).\nFinally, when tasks $A$, $B$, $C$ and $D$ form an analogy in the form ``$A$ is to $B$ as $C$ is to $D$\",\nthe task vector $\\tau_\\textrm{new} = \\tau_C + (\\tau_B - \\tau_A)$ improves performance on task $D$, even if there is little or no data for that task (Section \\ref{sec:analogies}).\n\nFor all operations, the model weights obtained by applying  $\\tau_\\textrm{new}$ are given by $\\theta_\\textrm{new} = \\theta + \\lambda \\tau_\\textrm{new}$, where the scaling term $\\lambda$ is determined using held-out validation sets.\n\n\\section{Forgetting via Negation}\n\\label{sec:negation}\n\nIn this section, we show that negating a task vector is an effective way to reduce its performance on a target task, without substantially hurting performance elsewhere.\nForgetting or ``unlearning\" can help mitigate undesired biases learned when pre-training; forgetting tasks altogether may be desirable to comply with regulations or for ethical reasons like preventing an image classifier to recognize faces, or to ``read'' personal information via OCR.\n\nThese interventions should not have a substantial effect on how models behave when processing data outside the scope of the edit \\citep{mitchell2021fast,ilharco2022patching}. Accordingly, we measure accuracy on \\textit{control tasks}, in addition to evaluating on the target tasks from which the task vector originated.\nOur experiments showcase the effectiveness of negating task vectors for editing image classification and text generation models.\n\n\\subsection{Image classification}\n\\label{sec:forget_img}\n\n\nFor image classification, we use CLIP models \\citep{radford2021learning} and task vectors from eight tasks studied by \\citet{ilharco2022patching,radford2021learning}, ranging from satellite imagery recognition to classifying traffic signs: Cars \\citep{cars}, DTD \\citep{dtd}, EuroSAT \\citep{eurosat}, GTSRB \\citep{gtsrb}, MNIST \\citep{lecun1998mnist}, RESISC45 \\citep{cheng2017remote}, SUN397 \\citep{sun397}, and SVHN \\citep{svhn}. \nWe explore additional tasks including OCR and person identification in Appendix \\ref{sec:clip-neg-extended}.\nFor the control task, we use ImageNet \\citep{deng2009imagenet}. We generate task vectors by fine-tuning on each of the target tasks, as detailed in Appendix \\ref{sec:clip-exp-details}.\n\nWe compare against two additional baselines, fine-tuning by moving in the direction of increasing loss (i.e., with gradient ascent), as in \\citet{golatkar2020eternal,tarun2021fast}, and against using a random vector where each layer has the same magnitude as the corresponding layer of task vector. Additional details are in Appendix \\ref{sec:app-neg-baselines}.\n\n\\newcolumntype{?}{!{\\vrule width .003pt}}\n\\begin{table*}\n\\caption{\\textbf{Forgetting image classification tasks via negation}. Results are shown for CLIP models, reporting average accuracy (\\%) on the eight target tasks we wish to forget (Cars, DTD, EuroSAT, GTSRB, MNIST, RESISC45, SUN397 and SVHN), and the control task (ImageNet). Negating task vectors reduce the accuracy of a pre-trained ViT-L\/14 by 45.8 percentage points on the target tasks, with little loss on the control task. Additional details and results are shown in Appendix \\ref{sec:clip-neg-extended}.}\n\\setlength\\tabcolsep{5.5pt}\n\\renewcommand{\\arraystretch}{0.9}\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{l@{\\hskip .3in}cc|cc|cc}\n\\toprule\n\\multirow{2}{*}{Method} & \\multicolumn{2}{c|}{ViT-B\/32} & \\multicolumn{2}{c|}{ViT-B\/16} & \\multicolumn{2}{c}{ViT-L\/14} \\\\\n & Target ($\\downarrow$) & Control ($\\uparrow$) & Target ($\\downarrow$) & Control ($\\uparrow$) & Target ($\\downarrow$) & Control ($\\uparrow$)  \\\\\\midrule\nPre-trained & 48.3 & 63.4  & 55.2 & 68.3  & 64.8 & 75.5\\\\\\midrule\nFine-tuned  & 90.2 & 48.2  & 92.5 & 58.3 & 94.0 & 72.6 \\\\\nGradient ascent  & 2.73 & 0.25 & 1.93 & 0.68 & 3.93 & 16.3\\\\\nRandom vector  & 45.7 & 61.5 & 53.1 & 66.0 & 60.9 & 72.9\\\\\\midrule\nNegative task vector & 24.0 & 60.9 & 21.3 & 65.4 & 19.0 & 72.9\\\\\n\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab:forget_image}\n\\end{table*}\n\n\n\nAs shown in Table \\ref{tab:forget_image}, negating the task vectors is the most effective editing strategy for decreasing accuracy on the target task with little impact on the control task. For example, negative task vectors decrease the average target accuracy of ViT-L\/14 by 45.8 percentage points with little change in accuracy on the control task. In contrast, using a random vector does not have much impact on target accuracy, while fine-tuning with gradient ascent severely deteriorates performance on control tasks.\nWe present additional results in Appendix \\ref{sec:clip-neg-extended}.\n\n\n\\subsection{Text generation}\n\\label{sec:forget_lang}\n\n\\begin{table*}\n\\caption{\\textbf{Making language models less toxic with negative task vectors.} Results are shown for the GPT-2 Large model. Negative task vectors decrease the amount of toxic generations by 6$\\times$, while resulting in a model with comparable perplexity on a control task (WikiText-103). Additional details and results are shown in Appendix \\ref{sec:appendix-neg-lang}.}\n\\setlength\\tabcolsep{4.5pt}\n\\renewcommand{\\arraystretch}{0.9}\n\\footnotesize\n\n\\begin{center}\n\n\\begin{tabular}{lrrr}\n\n\\toprule\n \n Method & \\% toxic generations ($\\downarrow$)& Avg. toxicity score ($\\downarrow$) & WikiText-103 perplexity ($\\downarrow$)\n \\\\\\midrule\nPre-trained & 4.8 & 0.06 & 16.4 \\\\\\midrule\nFine-tuned & 57 & 0.56 & 16.6 \\\\\nGradient ascent & 0.0 & 0.45 & $>$10$^{10}$ \\\\\nFine-tuned on non-toxic & 1.8 & 0.03 & 17.2 \\\\\nRandom vector  & 4.8 & 0.06 & 16.4 \\\\\\midrule\nNegative task vector & 0.8 & 0.01 & 16.9 \\\\\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab:toxicity}\n\n\\end{table*}\nWe study whether we can mitigate a particular model behavior by negating a task vector \\emph{trained to do that behavior}.\nIn particular, we aim to reduce the amount of toxic generations produced by GPT-2 models of various sizes \\citep{radford2019language}.\nWe generate task vectors by fine-tuning on data from Civil Comments  \\citep{borkan2019nuanced} where the toxicity score is \\textit{higher} than 0.8, and then negating such task vectors.\nAs in Section \\ref{sec:forget_img}, we also compare against baselines that use gradient ascent when fine-tuning \\citep{golatkar2020eternal,tarun2021fast}, and using a random task vector of the same magnitude. Additionally, we compare against fine-tuning on non-toxic samples from Civil Comments (toxicity scores smaller than 0.2), similar to \\citet{liu2021dexperts}.\nWe measure the toxicity of one thousand model generations with Detoxify \\citep{Detoxify}. For the control task, we measure the perplexity of the language models on WikiText-103 \\citep{merity2016pointer}.\n\nAs shown in Table \\ref{tab:toxicity}, editing with negative task vectors is effective, reducing the amount of generations classified as toxic from 4.8\\% to 0.8\\%, while maintaining perplexity on the control task within 0.5 points of the pre-trained model.\nIn contrast, fine-tuning with gradient ascent lowers toxic generations by degrading performance on the control task to an unacceptable level, while fine-tuning on non-toxic data is worse than task vectors both in reducing task generations and on the control task. As an experimental control, adding a random vector has little impact either on toxic generations or perplexity on WikiText-103.\nWe present additional experimental details and results in Appendix \\ref{sec:appendix-neg-lang}.\n\\section{Learning via Addition}\n\\label{sec:addition}\n\nWe now turn our attention to \\emph{adding} task vectors, either to build multi-task models that are proficient on multiple tasks simultaneously, or to improve single-task performance.\nThis operation allows us to reuse and transfer knowledge either from in-house models, or from the multitude of publicly available fine-tuned models,  without additional training or access to training data.\nWe explore addition on various image classification and natural language processing tasks.\n\n\\subsection{Image classification}\n\\label{sec:add_img}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figures\/clip_add_v3.pdf}\n    \\caption{\\textbf{Adding pairs of task vectors} from image classification tasks. Adding task vectors from two tasks improves accuracy on both, resulting in a single model that is competitive with using two specialized fine-tuned models.}\n    \\label{fig:clip-add-2}\n   \n\\end{figure}\n\n\nWe start with the same eight models used in Section \\ref{sec:negation}, fine-tuned on a diverse set of image classification tasks (Cars, DTD, EuroSAT, GTSRB, MNIST, RESISC45, SUN397 and SVHN). In Figure \\ref{fig:clip-add-2}, we show the accuracy obtained by adding all pairs of task vectors from these tasks.\nTo account for the difference in difficulty of the tasks, we normalize  accuracy on each task by the accuracy of the model fine-tuned on that task. After normalizing, the performance of fine-tuned models on their respective tasks is one, and so the average performance of using multiple specialized models is also one. As shown in Figure \\ref{fig:clip-add-2}, adding pairs of task vectors leads to a single model that outperforms the zero-shot model by a large margin, and is competitive with using two specialized models (98.9\\% normalized accuracy on average).\n\n\nBeyond pairs of tasks, we explore adding task vectors for \\textit{all} possible subsets of the tasks ($2^8$ in total). In Figure \\ref{fig:clip-add-all}, we show how the normalized accuracy of the resulting models, averaged over all the eight tasks.\nAs the number of available task vectors increases, better multi-task models can be produced. \nWhen all task vectors are available, the best model produced by adding task vectors reaches an average performance of 91.2\\%, despite compressing several models into one. Additional experiments and details are presented in Appendix \\ref{sec:appendix-add}.\n\n\\begin{SCfigure}\n    \\centering\n    \\begin{minipage}{0.53\\linewidth}\n    \\includegraphics[width=1\\textwidth]{figures\/clip_add_allevals.pdf}\n  \\end{minipage}\n    \\captionsetup{width=1\\textwidth}\n    \\sidecaptionvpos{figure}{t}\n    \\caption{\\textbf{Adding task vectors builds multi-task models} for image classification tasks. Accuracy is averaged over all downstream tasks.\n    When more task vectors are available, better multi-task vectors can be built.\n    Each point represents an experiment with a subset of the eight tasks we study, and the solid line connects the average performance for each subset size. Recall that the average normalized accuracy of using  multiple fine-tuned models is always one.  \n    Additional details and experiments are in Appendix \\ref{sec:appendix-add}.}\n    \\label{fig:clip-add-all}\n\\end{SCfigure}\n\n\\subsection{Natural language processing}\n\\label{sec:add-nlp}\n\n\nIn addition to building multi-task models, we explore whether adding task vectors is a useful way of improving performance on a single target task. \nTowards this goal, we first fine-tune T5-base models on four tasks from the GLUE benchmark \\citep{wang2018glue}, as in \\citet{wortsman2022model}.\nThen, we search for compatible checkpoints on Hugging Face Hub, finding 427 candidates in total.\nWe try adding each of the corresponding task vectors to our fine-tuned models, choosing the best checkpoint and scaling coefficient based on held-out validation data.\nAs shown in Table \\ref{tab:glue}, adding task vectors can \\textit{improve} performance on target tasks, compared to fine-tuning.\nAdditional details and experiments---including building multi-task models from public checkpoints from Hugging Face Hub---are presented in Appendix \\ref{sec:appendix-add}.\n\n\\begin{table*}\n\\caption{\\textbf{Improving performance on target tasks with external task vectors.} For four text classification tasks from the GLUE benchmark, adding task vectors downloaded from the Hugging Face Hub  can improve accuracy of fine-tuned T5 models. Appendix \\ref{sec:appendix-add-lang} shows additional details.}\n\\setlength\\tabcolsep{4.5pt}\n\\renewcommand{\\arraystretch}{0.9}\n\\small\n\\begin{center}\n\\begin{tabular}{lccccc}\n\\toprule\nMethod & MRPC & RTE & CoLA & SST-2 & Average \\\\\\midrule\nZero-shot &\t74.8\t& 52.7\t& 8.29\t& 92.7\t& 57.1 \\\\\nFine-tuned &\t88.5 &\t77.3 &\t52.3 &\t94.5 &\t78.1 \\\\\nFine-tuned + task vectors\t& 89.3 \\tiny{(+0.8)}\t& 77.5 \\tiny{(+0.2)}& \t53.0\t\\tiny{(+0.7)} & 94.7 \\tiny{(+0.2)}\t& 78.6 \\tiny{(+0.5)} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab:glue}\n\\end{table*}\n\n\n\\section{Task Analogies}\n\\label{sec:analogies}\n\nIn this section, we explore task analogies in the form ``$A$ is to $B$ as $C$ is to $D$\", and show that task arithmetic using vectors from the first three tasks improves performance on task $D$ even if little or not data for that task is available.\n\n\\paragraph{Domain generalization.} For many target tasks, gathering unlabeled data is easier and cheaper than collecting human annotations. When labeled data for a \\textit{target} task is not available, we can use task analogies to improve accuracy on the target task, using \n an \\textit{auxiliary} task for which there is labeled data and an unsupervised learning objective. For example, consider the target task of sentiment analysis using data from Yelp \\citep{zhang2015character}. Using task analogies, we can construct a task vector $\\hat{\\tau}_\\textrm{yelp;\\,sent} = \\tau_\\textrm{amazon;\\,sent} + (\\tau_\\textrm{yelp;\\,lm} - \\tau_\\textrm{amazon;\\,lm})$, where $\\tau_\\textrm{amazon;\\,sent}$ is obtained by fine-tuning on labeled data from an auxiliary task (sentiment analysis using data from Amazon; \\citet{mcauley2013hidden}), and $\\tau_\\textrm{yelp;\\,lm}$ and $\\tau_\\textrm{amazon;\\,lm}$ are task vectors obtained via (unsupervised) language modeling on the inputs from both datasets.\n\n In Table \\ref{tab:sentiment-analog}, we show that using such task analogies improves accuracy of T5 models at multiple scales, both for  Amazon and Yelp binary sentiment analysis as target tasks. We empirically found that giving a higher weight to the sentiment analysis task vector led to higher accuracy, and we thus used two independent scaling coefficients for these experiments---one for the sentiment analysis task vector and one for both the language modeling task vectors. More details are presented in Appendix \\ref{sec:app-sentiment}. Using task vectors outperforms fine-tuning on the remaining auxiliary sentiment analysis task for all models and datasets, approaching the performance of fine-tuning on the target task.\n \n\\begin{table*}\n\\caption{\\textbf{Improving domain generalization with task analogies.} Using an auxiliary task for which labeled data is available and unlabeled data from both the auxiliary and the target datasets, task analogies improve the accuracy for multiple T5 models and two sentiment analysis target tasks \\citep{zhang2015character,mcauley2013hidden}, without using any labeled data from the target tasks.}\n\\setlength\\tabcolsep{6.5pt}\n\\renewcommand{\\arraystretch}{0.9}\n\\small\n\\begin{center}\n\\begin{tabular}{lcccccccc} \n\\toprule\n & & \\multicolumn{3}{c}{target = Yelp} & & \\multicolumn{3}{c}{target = Amazon} \\\\\\cmidrule{3-5}\\cmidrule{7-9}\nMethod & &  T5-small & T5-base & T5-large & & T5-small & T5-base & T5-large \n\\\\\\midrule\nFine-tuned on auxiliary & & 88.6 & 92.3 & 95.0 & & 87.9 & 90.8 & 94.8 \\\\\nTask analogies & & 89.9 & 93.0 & 95.1 & & 89.0 & 92.7 & 95.2 \\\\\nFine-tuned on target & & 91.1 & 93.4 & 95.5 & & 90.2 & 93.2 & 95.5 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab:sentiment-analog}\n\\vspace{6pt}\n\\end{table*}\n\n\n\\paragraph{Subpopulations with little data.} There is often some inherent scarcity in certain data subpopulations---for example, images of lions in indoor settings are more rare, compared to lions in outdoor settings or dogs in general (indoor or outdoors). Whenever such subpopulations admit analogies to others with more abundant data (as in this case), we can apply task analogies, e.g.,  $\\hat{\\tau}_\\textrm{lion indoors} = \\tau_\\textrm{lion outdoors} + (\\tau_\\textrm{dog indoors} - \\tau_\\textrm{dog outdoor})$.\n\nWe explore this scenario by creating four subpopulations, using 125 overlapping classes between ImageNet and a dataset of human sketches \\citep{eitz2012humans}. \nWe split these classes in two subsets of roughly equal size, creating four subpopulations $A$, $B$, $C$ and $D$, where the pairs $(A,C)$ and $(B, D)$ share the same classes, and $(A, B)$ and $(C, D)$ share the same style (photo-realistic images or sketches).\nAlthough these subpopulations have many classes in our experiments, we use the simplified subsets ``real dog'', ``real lion'', ``sketch dog'' and ``sketch lion'' as a running example. We present more details and samples in Appendix \\ref{sec:appendix-sketches}.\n\nGiven a target subpopulation, we create task vectors by fine-tuning three models independently on the remaining subpopulations, and then combine them via task arithmetic, e.g., $\\hat{\\tau}_\\textrm{sketch lion} = \\tau_\\textrm{sketch dog} + (\\tau_\\textrm{real lion} - \\tau_\\textrm{real dog})$ for the target subpopulation ``sketch lion''. We show the results in Figure \\ref{fig:clip-analogies}, averaged over the four target subpopulations.\nCompared to the pre-trained model, task vectors improve accuracy by 3.4 percentage points on average.\nMoreover, when some data from the target subpopulation is available for fine-tuning, starting from the edited model leads to consistently higher accuracy than starting from the pre-trained model.\nThe gains from analogies alone (with no additional data) are roughly the same as that of collecting and annotating around one hundred training samples for the target subpopulation.\n\n\\paragraph{Kings and queens.} We explore whether an image classifier can learn a new categories (e.g., ``king\") using data from three related classes that form an analogy relationship (e.g., ``queen\", ``man\" and ``woman\"). Our results are presented in Appendix \\ref{sec:appendix-kingsandqueens}, showing that task analogies yield large gains in accuracy over pre-trained models on the new target category, despite having no training data for it.\n\n\\begin{SCfigure}\n    \\centering\n    \\begin{minipage}{0.43\\linewidth}\n    \\hspace{-0.52cm}\n    \\includegraphics[width=1\\textwidth]{figures\/sketches_fewshot_full.pdf}\n  \\end{minipage}\n    \\captionsetup{width=1\\textwidth}\n    \\vspace{-0.25cm}\n    \\sidecaptionvpos{figure}{t}\n    \\caption{\\textbf{Learning about subpopulations via analogy}. Combining task vectors from related subpopulations improves accuracy on the target subpopulation, when little or no data from the target supopulation is available. Accuracy is averaged over the four target subpopulations and three CLIP models. Additional details are in Appendix \\ref{sec:appendix-sketches}.}\n    \\label{fig:clip-analogies}\n\\end{SCfigure}\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nIn this section, we provide further insight into previous results by exploring the similarity between task vectors for different tasks, as well as the impact of different learning rates and random seeds. Additional analysis are presented in Appendix \\ref{sec:app-ensembles}, including discussions on the connection between ensembles and weight averaging. We conclude by discussing some limitations of our approach.\n\n\n\\begin{SCfigure}\n    \\centering\n    \\begin{minipage}{0.43\\linewidth}\n    \\hspace{-0.52cm}\n    \\includegraphics[width=1\\textwidth]{figures\/vision_cossim.pdf}\n  \\end{minipage}\n    \\captionsetup{width=1\\textwidth}\n    \\vspace{-0.25cm}\n    \\sidecaptionvpos{figure}{t}\n    \\caption{\\textbf{Task vectors are typically close to orthogonal.} The plot shows the cosine similarities between vectors for different tasks, using CLIP. The largest deviations from orthogonality are found when tasks are similar to each other, for instance, for MNIST, SVHN and GTSRB---where recognizing digits is either the task itself (MNIST and SVHN), or a capability needed to solve the task (GTSRB, where the task is traffic sign recognition)---and EuroSAT and RESISC45, two satellite imagery recognition datasets.}\n    \\label{fig:cossim}\n    \\vspace{8pt}\n\\end{SCfigure}\n\n\n\n\\textbf{Similarity between task vectors.} In Figure \\ref{fig:cossim}, we explore the cosine similarity between task vectors for different tasks, in an effort to understand how multiple models can be collapsed into a single multi-task model via addition (Section \\ref{sec:addition}).\nWe observe that vectors from different tasks are typically close to orthogonal, and speculate that this enables the combination of task vectors via addition with minimal interference.\nWe also observe higher cosine similarities when tasks are semantically similar to each other.\nFor example, the largest cosine similarities in Figure \\ref{fig:cossim} (left) are between MNIST, SVHN and GTSRB, where recognizing digits is essential for the tasks, and between EuroSAT and RESISC45, which are both satellite imagery recognition datasets. This similarity in ``task space'' could help explain some results in \\citet{ilharco2022patching}, where interpolating the weights of a model fine-tuned on one task and the pre-trained model weights---in our terminology, applying a single task vector---sometimes improves accuracy on a different task for which no data is available (e.g., applying the MNIST task vector improves accuracy on SVHN).\n\n\\textbf{The impact of the learning rate.} In Figure \\ref{fig:lr}, we observe that increasing the learning rate degrades accuracy both when using task vectors and when fine-tuning individual models, but the decrease is more gradual for individual models.\nThese findings align with those of \\cite{wortsman2022model}, who observed that accuracy decreases on the linear path between two fine-tuned models when using a larger learning rate.\nThus, while larger learning rates may be acceptable when fine-tuning individual models, we recommend more caution when using task vectors. Further, we hypothesize that larger learning rates may explain some of the variance when adding vectors from natural language processing tasks, where we take models fine-tuned by others in the community.\n\n\n\n\\begin{SCfigure}\n    \\centering\n    \\begin{minipage}{0.4\\linewidth}\n    \\hspace{-0.52cm}\n    \\includegraphics[width=1\\textwidth]{figures\/lr_ablation.pdf}\n  \\end{minipage}\n    \\captionsetup{width=1\\textwidth}\n    \\vspace{-0.6cm}\n    \\sidecaptionvpos{figure}{t}\n    \\caption{\\textbf{The impact of learning rate when fine-tuning.} When adding task vectors from CLIP ViT-L\/14 models fine-tuned on MNIST and EuroSAT, lower learning rates make the best use of the fine-tuned models, and also correspond to the highest accuracies of the fine-tuned models on the target task.}\n    \\label{fig:lr}\n\\end{SCfigure}\n\n\\begin{figure*}\n    \\centering\n   \n    \\includegraphics[width=.94\\textwidth]{figures\/intermediate_tvs.pdf}\n    \\caption{\\textbf{How task vectors evolve throughout fine-tuning.} Left: the cosine similarity between the final task vector and task vectors produced at intermediate points during fine-tuning. Right: Accuracy obtained by adding intermediate task vectors from MNIST and EuroSAT. Adding intermediate task vectors can lead to high accuracy, despite fine-tuning for substantially fewer steps.}\n    \\label{fig:intermediate}\n    \\vspace{-8pt}  \n\\end{figure*}\n\n\\textbf{The evolution of task vectors throughout fine-tuning.} In Figure \\ref{fig:intermediate}, we show how task vectors evolve throughout fine-tuning. Intermediate task vectors converge rapidly to the direction of the final task vector obtained at the end of fine-tuning. Moreover, the accuracy of the model obtained by adding intermediate task vectors from two image classification tasks saturates after just a few hundred steps. These results suggest that using intermediate task vectors can be a useful way of saving compute with little harm in accuracy.\n\n\n\\textbf{Limitations.} Task vectors are restricted to models with the same architecture, since they depend on element-wise operations on model weights. Further, in all of our experiments we perform arithmetic operations only on models fine-tuned from the same pre-trained initialization, although emerging work shows promise in relaxing this assumption \\citep{ainsworth2022git}. We also note that some architectures are very popular, and have ``standard'' initializations---e.g., at the time of writing there are over 3,000 models on Hugging Face Hub fine-tuned from the same BERT-base initialization \\cite{devlin-etal-2019-bert}, and over 800 models fine-tuned from the same T5-small initialization.\n\\section{Related work}\n\n\\paragraph{The loss landscape and interpolating weights.} The geometry of neural network loss surfaces has attracted the interest of several authors in recent years \\citep{li2018visualizing,garipov2018loss,draxler2018essentially,kuditipudi2019explaining,fort2019deep,czarnecki2019deep,pmlr-v139-wortsman21a, benton2021loss,entezari2021role,li2022branch}.\nDespite neural networks being non-linear, previous work has empirically found that interpolations between the weights of two neural networks can maintain their high accuracy, provided these two neural networks share part of their optimization trajectory~\\citep{frankle2020linear,izmailov2018averaging,neyshabur2020being,fort2020deep,wortsman2022model,choshen2022fusing,ilharco2022patching}. \n\nIn the context of fine-tuning, accuracy increases steadily when gradually moving the weights of a pre-trained model in the direction of its fine-tuned counterpart \\citep{wortsman2021robust,matena2021merging,ilharco2022patching}.\nBeyond a single task, \\citet{matena2021merging,ilharco2022patching} found that when multiple models are fine-tuned on different tasks from the same initialization, averaging their weights can improve accuracy on the fine-tuning tasks.\nSimilar results were found by \\citet{li2022branch} when averaging the parameters of language models fine-tuned on various domains.\n\\citet{choshen2022fusing} showed that ``fusing\" fine-tuned models by averaging their weights creates a better starting point for fine-tuning on a new downstream task.\n\\citet{wortsman2022model} found that averaging the weights of models fine-tuned on multiple tasks can increase accuracy on a new downstream task,  without any further training.\nThese findings are aligned with results shown in Section \\ref{sec:addition}. In this work, we go beyond interpolating between models, examining extrapolating between models and additional ways of combining them (Sections \\ref{sec:negation} and \\ref{sec:analogies}). \n\n\n\\paragraph{Model interventions.} Considering that re-training models is prohibitively expensive in most circumstances, several authors have studied more efficient methods for modifying a model's behavior with interventions after pre-training, referring to this process by different names, such as patching \\citep{goel2020model,sung2021training,ilharco2022patching,murty2022fixing}, editing \\citep{shibani2021editing,mitchell2021fast,mitchell2022memory}, aligning \\citep{ouyang2022training,askell2021general,kasirzadeh2022conversation,sparrow}, or debugging \\citep{ribeiro2022adaptive,geva2022lm}. \nIn contrast to previous literature, our work provides a unique way of editing models, where capabilities can be added or deleted in a modular and efficient manner by re-using fine-tuned models.\nCloser to our work is that of \\cite{subramani2022}, who explore steering language models with vectors added to its hidden states.\nIn contrast, our work applies vectors in the weight space of pre-trained models and does not modify the standard fine-tuning procedure.\n\n\\paragraph{Task embeddings.} \\citet{achille2019task2vec,vu2020exploring,vu2022spot}, inter alia, explored strategies for representing tasks with continuous embeddings, in order to to predict task similarities and transferability, or to create taxonomic relations. While the task vectors we build could be used for such purposes, our main goal is to use them as tools for steering the behavior of pre-trained models.\n\n\\section{Conclusion}\n\nIn this paper we introduce a new paradigm for editing models based on arithmetic operations over \\emph{task vectors}. For various vision and NLP models, \\emph{adding} multiple specialized task vectors results in a single model that performs well on all target tasks, or even improves performance on a single task.\n\\emph{Negating} task vectors allows users to remove undesirable behaviors, e.g., toxic generations, or even forget specific tasks altogether, while retaining performance everywhere else. Finally, \\emph{task analogies} leverage existing data to improve performance on domains or subpopulations where data is scarce.\n\nArithmetic operations over task vectors only involve adding or subtracting model weights, and thus are efficient to compute, especially when compared to alternatives that involve additional fine-tuning. Thus, users can easily experiment with various model edits, recycling and transferring knowledge from large collections of publicly available fine-tuned models.\nSince these operations result in a single model of the same size, they incur no extra inference cost. Our code is available at {\\footnotesize \\url{https:\/\/github.com\/mlfoundations\/task_vectors}}.\n\n\n\\section*{Acknowledgements}\nWe thank \nAlex Fang,\nAri Holtzman,\nColin Raffel,\nDhruba Ghosh,\nJesse Dodge,\nMargaret Li,\nOfir Press,\nSam Ainsworth,\nSarah Pratt,\nStephen Mussmann,\nTim Dettmers, and\nVivek Ramanujan\nfor helpful discussion and comments on the paper.\n\n\n\\section{The loss landscape, weight averaging and ensembles}\n\\label{sec:app-ensembles}\n\nWhen two neural networks share part of their optimization trajectory---such as when fine-tuning from the same pre-trained initialization---previous work found that performance does not decrease substantially when linearly interpolating between their weights \\citep{frankle2020linear,izmailov2018averaging,neyshabur2020being,fort2020deep,wortsman2022model,choshen2022fusing,ilharco2022patching}. \nApplying a task vector---and any vectors produced via the arithmetic expressions we study in this work---is equivalent to a linear combination of the pre-trained model and the fine-tuned models used to generate the task vectors, since only linear operations are used.\nInterpolating between the weights of a fine-tuned model and its pre-trained counterpart as in \\citet{wortsman2021robust,ilharco2022patching} is equivalent to applying a single task vector, and adding different task vectors is equivalent to a weighted average of all models, similar to experiments from \\citet{wortsman2022model,ilharco2022patching,li2022branch}.\nOverall, previous work has empirically observed that averaging weights of neural networks can produce models with strong performance when compared to the best individual network, for several architectures, domains and datasets. \n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=.5\\textwidth]{figures\/ensembles_correlation.pdf}\n    \\caption{When adding two task vectors, the performance of the resulting model approximates the performing of ensembling the corresponding fine-tuned models.}\n    \\label{fig:ensembles-corr}\n\\end{figure*}\n\nOur motivation for studying task vectors is also well aligned with findings of \\citet{lucas2021analyzing,ilharco2022patching}, who observed that performance steadily increases on the linear path between a model before and after training.\\footnote{This property of neural networks is sometimes referred to as Monotonic Linear Interpolation (MLI) \\citep{lucas2021analyzing}.} This indicates that the direction from the pre-trained to the fine-tuned model is such that movement in that direction directly translates to performance gains on the fine-tuning task. Moreover, \\citet{ilharco2022patching} found that linear interpolations between a pre-trained model and a fine-tuned model are able to preserve accuracy on tasks that are unrelated to fine-tuning, while greatly improving accuracy on the fine-tuning task compared to the pre-trained model. That accuracy on the fine-tuning task and on unrelated tasks are independent of each other along the linear path between pre-trained and fine-tuned models is well aligned with our results on from Section \\ref{sec:negation}, where we find that \\textit{extrapolating} from the pre-trained model away from the fine-tuned model leads to worse performance on the fine-tuning task with little change in behavior on control tasks. \n\nFinally, we highlight the connection between linear combinations of neural network weights and the well-established practice of \\textit{ensembling} their predictions.\\footnote{For the sake of completion, the ensemble of two models $f$ with weights $\\theta_1$ and $\\theta_2$ for an input $x$ is given by $(1-\\alpha)f_{\\theta_1}(x) + \\alpha f_{\\theta_2}(x)$, for some mixing coefficient $\\alpha$. Ensembling two classification models is typically done by averaging the logits produced by the models.} This connection is discussed in depth by \\citet{wortsman2021robust,wortsman2022model}, and we briefly revisit it in the context of adding task vectors. First, recall that the arithmetic operations we study result in linear combinations of model weights. As shown by \\citet{wortsman2021robust}, in certain regimes, the result from linearly combining the weights of neural network approximate ensembling their outputs. This approximation holds whenever the loss can be locally approximated by a linear expansion, which is referred to as the NTK regime \\citep{jacot2018neural}. Moreover, as shown by \\citet{fort2020deep}, this linear expansion becomes more accuracy in the later phase of training neural networks, which closely resembles fine-tuning. When the approximation holds exactly, weight averaging and ensembles are exactly equivalent \\citep{wortsman2021robust}. This connection is further studied analytically and empirically by \\citet{wortsman2022model}.\n\nWe empirically validate the connection between ensembles and linear weight combinations in the context of adding two task vectors. Note that the model resulting from adding two task vectors with a scaling coefficient $\\lambda=0.5$ is equivalent to a uniform average of the weights of the fine-tuned models.\\footnote{\n$\\theta_\\textrm{pre} + 0.5(\\tau_1+ \\tau_2) = \\theta_\\textrm{pre} + 0.5((\\theta_1-\\theta_\\textrm{pre}) + (\\theta_2-\\theta_\\textrm{pre}))  = 0.5 (\\theta_1 + \\theta_2)$.}\nWe then investigate whether accuracy of the model obtained using the task vectors correlates with the accuracy of ensembling the fine-tuned models, as predicted by theory. \nAs shown in Figure \\ref{fig:ensembles-corr}, we indeed observe that the accuracy of the model produced by adding two task vectors closely follows the accuracy of the corresponding ensemble. We observe a slight bias towards higher accuracy for the ensembles on average, and that the two quantities are also strongly correlated, with a Pearson correlation of 0.99.\n\n\n\\section{Forgetting image classification tasks}\n\\label{sec:clip-neg-extended}\n\nThis section presents additional experimental details and results complementing the findings presented in Section \\ref{sec:forget_img}, showcasing the effect of negating task vectors from image classification tasks.\n\n\\subsection{Experimental details}\n\\label{sec:clip-exp-details}\n\nWe follow the same procedure from \\cite{ilharco2022patching} when fine-tune CLIP models \\citep{radford2021learning}. Namely, we fine-tune for 2000 iterations with a batch size of 128, learning rate 1e-5 and a cosine annealing learning rate schedule with 200 warm-up steps and the AdamW optimizer \\citep{loshchilov2018decoupled, paszke2019pytorch}, with weight decay 0.1.\nWhen fine-tuning, we freeze the weights of the classification layer output by CLIP's text encoder, so that we do not introduce additional learnable parameters, as in \\cite{ilharco2022patching}.\nAs shown by \\cite{ilharco2022patching}, freezing the classification layer does not harm accuracy.\nAfter fine-tuning, we evaluate scaling coefficients $\\lambda \\in \\{0.0, 0.05, 0.1, \\cdots, 1.0\\}$, choosing the highest value such that the resulting model still retains at least 95\\% of the accuracy of the pre-trained model on the control task.\n\n\\subsection{Baselines}\n\\label{sec:app-neg-baselines}\n\nWe contrast our results with two baselines, fine-tuning with gradient ascent as in \\citet{golatkar2020eternal,tarun2021fast}, and against using a random vector of the same magnitude as the task vector on a layer-by-layer basis.\n\nIn practice, for fine-tuning with gradient ascent, we use the same hyper-parameters as for standard fine-tuning. However, instead of optimizing to minimize the cross-entropy loss $\\ell=\\mathbb{E}_{x,y \\in \\mathcal{D}}[-\\log f(x)_y]$, we optimize to minimize its negative value, $\\ell_\\textrm{neg}=-\\ell=\\mathbb{E}_{x,y \\in \\mathcal{D}}[\\log f(x)_y]$, where $x,y$ are samples in the dataset $\\mathcal{D}$ and $f(x)_y$ is the probability assigned by the model $f$ that the inputs $x$ belong to label $y$. This is equivalent to performing gradient ascent on $\\ell$.\n\nFor the random vector baseline, we first compute the different between the parameters of the pre-trained and fine-tuned models for each layer $L$, $\\tau^{(L)} = \\theta^{(L)}_\\textrm{ft}-\\theta^{(L)}_\\textrm{pre}$. Then, we draw a new vector $\\tau^{(L)}_\\textrm{rand} \\sim \\mathcal{N}(0,I)$ where each element is drawn from a normal distribution with mean 0 and variance 1. We then scale this vector so it has the same magnitude as $\\tau^{(L)}$, resulting in $\\tau^{(L)}_{\\textrm{scaled}} = \\tau^{(L)}_\\textrm{rand} \\frac{||\\tau^{(L)}||}{||\\tau^{(L)}_\\textrm{rand}||}$. Finally, we concatenate all the vectors $\\tau^{(L)}_{\\textrm{scaled}}$ for all layers to form a new vector withe the same dimensionality as the model parameters $\\theta$, which is used in the same way as task vectors. \n\n\n\\subsection{Breakdown per task}\n\nTables \\ref{tab:forget_image_l14}, \\ref{tab:forget_image_b16} and \\ref{tab:forget_image_b32} show a breakdown of accuracy for the eight tasks and the three CLIP models we examine.\n\nWe observe qualitatively similar results in all cases. Similarly to what is observed in \\cite{ilharco2022patching}, we also see that results improve with scale: on average, the largest model, ViT-L\/14, achieves \\textit{lower} accuracy on the target tasks, compared to the smaller models. \n\n\\begin{table*}\n\\caption{Forgetting via negation on image classification tasks. Results are shown for a CLIP ViT-L\/14 model \\citep{radford2021learning}, reporting accuracy on both the target (T) and control (C) tasks.}\n\\setlength\\tabcolsep{2.3pt}\n\\renewcommand{\\arraystretch}{1.05}\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{lcc?cc?cc?cc?cc?cc?cc?cc}\n\\toprule\n\\multirow{2}{*}{Method} & \\multicolumn{2}{c?}{{Cars}} & \\multicolumn{2}{c?}{DTD} & \\multicolumn{2}{c?}{EuroSAT} & \\multicolumn{2}{c?}{GTSRB} & \\multicolumn{2}{c?}{MNIST} & \\multicolumn{2}{c?}{{RESISC45}} & \\multicolumn{2}{c?}{{SUN397}} & \\multicolumn{2}{c}{{SVHN}} \\\\\n & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ \\\\\\midrule\nPre-trained & 77.8 & 75.5 & 55.4 & 75.5 & 60.2 & 75.5 & 50.6 & 75.5 & 76.4 & 75.5 & 71.0 & 75.5 & 68.3 & 75.5 & 58.6 & 75.5 \\\\\nFine-tuned & 92.8 & 73.1 & 83.7 & 72.3 & 99.2 & 70.5 & 99.3 & 73.1 & 99.8 & 72.9 & 96.9 & 73.8 & 82.4 & 72.7 & 98.0 & 72.6 \\\\\nNeg. gradients & 0.00 & 4.82 & 2.13 & 0.10 & 9.26 & 1.07 & 1.19 & 0.07 & 9.80 & 67.0 & 2.14 & 0.07 & 0.25 & 0.00 & 6.70 & 57.2 \\\\%\\midrule\nRandom vector & 72.0 & 73.3 & 52.1 & 72.2 & 59.7 & 73.5 & 43.4 & 72.5 & 74.8 & 72.8 & 70.8 & 73.0 & 66.9 & 72.7 & 47.1 & 72.9\\\\\\midrule\nNeg. task vector & 32.0 & 72.4 & 26.7 & 72.2 & 7.33 & 73.3 & 6.45 & 72.2 & 2.69 & 74.9 & 19.7 & 72.9 & 50.8 & 72.6 & 6.71 & 72.7 \\\\\n\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab:forget_image_l14}\n\\end{table*}\n\n\n\\begin{table*}\n\\caption{Forgetting via negation on image classification tasks. Results are shown for a CLIP ViT-B\/16 model \\citep{radford2021learning}, reporting accuracy on both the target (T) and control (C) tasks.}\n\\setlength\\tabcolsep{2.3pt}\n\\renewcommand{\\arraystretch}{1.05}\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{lcc?cc?cc?cc?cc?cc?cc?cc}\n\\toprule\n\\multirow{2}{*}{Method} & \\multicolumn{2}{c?}{{Cars}} & \\multicolumn{2}{c?}{DTD} & \\multicolumn{2}{c?}{EuroSAT} & \\multicolumn{2}{c?}{GTSRB} & \\multicolumn{2}{c?}{MNIST} & \\multicolumn{2}{c?}{{RESISC45}} & \\multicolumn{2}{c?}{{SUN397}} & \\multicolumn{2}{c}{{SVHN}} \\\\\n & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$\\\\\\midrule\nPre-trained & 64.6 & 68.3 & 44.9 & 68.3 & 53.9 & 68.3 & 43.4 & 68.3 & 51.6 & 68.3 & 65.8 & 68.3 & 65.5 & 68.3 & 52.0 & 68.3 \\\\\nFine-tuned & 87.0 & 61.9 & 82.3 & 57.5 & 99.1 & 56.0 & 99.0 & 54.7 & 99.7 & 55.2 & 96.4 & 62.2 & 79.0 & 61.7 & 97.7 & 56.8 \\\\\nNeg. gradients & 0.36 & 0.11 & 2.13 & 0.09 & 9.26 & 0.14 & 0.71 & 0.10 & 0.04 & 1.20 & 2.60 & 0.10 & 0.25 & 0.00 & 0.08 & 3.69\\\\\nRand. task vector & 61.0 & 65.6 & 43.9 & 66.3 & 51.7 & 66.2 & 43.1 & 65.0 & 51.6 & 68.3 & 63.6 & 65.6 & 63.7 & 65.2 & 46.2 & 65.5 \\\\\\midrule\nNeg. task vector & 30.8 & 65.4 & 26.5 & 65.6 & 12.3 & 65.8 & 9.53 & 65.8 & 9.55 & 65.4 & 26.5 & 65.1 & 48.6 & 65.1 & 6.43 & 65.4 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab:forget_image_b16}\n\\end{table*}\n\n\n\\begin{table*}\n\\caption{Forgetting via negation on image classification tasks. Results are shown for a CLIP ViT-B\/32 model \\citep{radford2021learning}, reporting accuracy on both the target (T) and control (C) tasks.}\n\\setlength\\tabcolsep{2.3pt}\n\\renewcommand{\\arraystretch}{1.05}\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{lcc?cc?cc?cc?cc?cc?cc?cc}\n\\toprule\n\\multirow{2}{*}{Method} & \\multicolumn{2}{c?}{{Cars}} & \\multicolumn{2}{c?}{DTD} & \\multicolumn{2}{c?}{EuroSAT} & \\multicolumn{2}{c?}{GTSRB} & \\multicolumn{2}{c?}{MNIST} & \\multicolumn{2}{c?}{{RESISC45}} & \\multicolumn{2}{c?}{{SUN397}} & \\multicolumn{2}{c}{{SVHN}} \\\\\n & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$ & T$\\downarrow$ & C$\\uparrow$  \\\\\\midrule\nPre-trained & 59.6 & 63.4 & 44.1 & 63.4 & 45.9 & 63.4 & 32.5 & 63.4 & 48.7 & 63.4 & 60.7 & 63.4 & 63.2 & 63.4 & 31.5 & 63.4 \\\\\nFine-tuned & 79.2 & 55.2 & 78.7 & 49.3 & 98.6 & 47.2 & 98.5 & 39.1 & 99.6 & 42.5 & 95.0 & 53.2 & 75.1 & 54.6 & 97.2 & 44.7 \\\\\nNeg. gradients & 0.01 & 0.11 & 2.13 & 0.10 & 9.26 & 0.10 & 1.19 & 0.07 & 0.00 & 1.22 & 2.60 & 0.10 & 0.25 & 0.01 & 6.38 & 0.29 \\\\\nRand. task vector & 54.1 & 60.9 & 39.9 & 61.5 & 45.8 & 63.4 & 27.9 & 60.7 & 48.3 & 63.4 & 57.1 & 60.9 & 61.3 & 60.5 & 31.2 & 60.7 \\\\\\midrule\nNeg. task vector & 36.0 & 61.1 & 27.8 & 60.2 & 13.6 & 61.3 & 8.13 & 61.4 & 16.7 & 60.7 & 31.7 & 61.0 & 50.7 & 60.5 & 7.65 & 61.0 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab:forget_image_b32}\n\\end{table*}\n\n\n\\subsection{Additional visualizations}\n\nIn Figure \\ref{fig:forget_img_lambdas}, we show how accuracy on the target and control tasks vary as we change the scaling coefficients $\\lambda$, both for the task vector obtained by fine-tuning on the target task and for a random vector of the same magnitude.\n\nAs the scaling coefficient increases, the curves traced by the task vector and a random vector behave differently. For task vectors, performance on the target tasks ($y$-axis) initially decreases faster than performance on the control task ($x$-axis), so there exists models with high accuracy on the control task but low accuracy on the target task. In contrast, such points do not exist in the curves traced by random vectors, which move more linearly towards the origin. In practice, this means forgetting is effective for task vectors obtained by fine-tuning, but not for random vectors.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/wses_randomdir.pdf}\n    \\caption{Comparison between task vectors and random vectors for forgetting image classification tasks.}\n    \\label{fig:forget_img_lambdas}\n\\end{figure*}\n\n\\subsection{The effect of class overlap}\n\nIn Tables \\ref{tab:forget_image_l14}, \\ref{tab:forget_image_b32}, \\ref{tab:forget_image_b16}, we observe that the tasks where forgetting via task vectors is least effective are tasks where the distribution of images is closer to ImageNet, SUN397 \\citep{sun397}, a scene understanding dataset with classes such as ``church\" and ``tower\", and Stanford Cars \\citep{cars}, a dataset with with many car categories such as ``2012 Tesla Model S\" or ``2012 BMW M3 coupe\". One reasonable hypothesis is that forgetting is less effective for those tasks due to the overlap with the images from the control tasks. \n\nTo better understand this effect, we measure accuracy on a subset of classes from ImageNet, such that the overlap is minimized.\nConcretely, we exclude nodes from the WordNet hierarchy from which the ImageNet classes are based.\\footnote{A visualization is available at \\url{https:\/\/observablehq.com\/@mbostock\/imagenet-hierarchy}}\nFor the Cars dataset, we exclude the all subnodes under the node ``wheeled vehicle\" (e.g., ``minivan\", ``jeep\", ``limousine\").\nFor SUN397, we exclude all subnodes under the nodes ``structure\" and ``geological formation\". \nAs shown in Table \\ref{tab:overlap-ablation}, we do not observe large differences after filtering. \n\n\\begin{table*}\n\\caption{The effect of semantic overlap with the control task in forgetting experiments on image classification tasks. Results are shown for a CLIP ViT-L\/14 model, reporting accuracy both on the target task and control task (Ctrl, ImageNet).}\n\\setlength\\tabcolsep{4.4pt}\n\\renewcommand{\\arraystretch}{1.05}\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{lcc?cc?cc?cc}\n\\toprule\n\\multirow{2}{*}{Method} & \\multicolumn{4}{c?}{{Without filtering}} & \\multicolumn{4}{c}{With filtering} \\\\\n & Cars ($\\downarrow$) & Ctrl ($\\uparrow$) & SUN397 ($\\downarrow$) & Ctrl ($\\uparrow$) & Cars ($\\downarrow$) & Ctrl ($\\uparrow$) & SUN397 ($\\downarrow$) & Ctrl ($\\uparrow$) \\\\\\midrule\nPre-trained & 77.8 & 75.5 & 68.3 & 75.5 & 77.8 & 75.5 & 68.3 & 76.1 \\\\\nFine-tuned & 92.8 & 73.1 & 82.4 & 72.7 & 92.8 & 73.3 & 82.4 & 73.1 \\\\\\midrule\nNeg. task vector & 32.0 & 72.4 & 50.8 & 72.6 & 32.0 & 72.5 & 48.1 & 72.4\\\\\n\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab:overlap-ablation}\n\\end{table*}\n\n\n\\subsection{Interpolating with a model fine-tuned with gradient ascent}\n\\label{sec:apapendix-gradient-ascent}\n\nOne baseline explored in the experiments is fine-tuning with gradient ascent, as explored in \\citet{golatkar2020eternal,tarun2021fast}. Our results show that this strategy is effective at reducing the accuracy on treatment tasks, but also substantially deteriorates accuracy on the control task, which is undesirable.\n\nWe further examine whether interpolations between the pre-trained model and the model fine-tuned with gradient ascent help with forgetting. Our results, shown in Figure \\ref{fig:neggrad}, indicate that interpolations greatly mitigate the low accuracy on the control task of the fine-tuned model, leading to even better accuracy trade-offs than the solutions obtained by extrapolation with standard fine-tuning.\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/wses_avg_neggrad.pdf}\n    \\caption{Comparison with interpolations between the pre-trained model and models fine-tuned with gradient ascent.}\n    \\label{fig:neggrad}\n\\end{figure*}\n\n\\subsection{When negating task vectors works best}\n\\label{sec:when-neg-works}\n\nWe observe a positive correlation between the gain in accuracy from fine-tuning and the drop in accuracy when subtracting the corresponding task vector, both in comparison with the pre-trained model (Figure \\ref{fig:forget-corr}).\nWe speculate that the reason for this correlation is that when the gains from fine-tuning are small, the task vector provides a less clear direction of improvement, and the opposite direction thus provides a less clear direction of performance deterioration.\nIn the extreme case where fine-tuning does not improve accuracy, it would be surprising if the corresponding task vector is useful.\n\nWe note that this is a limitation of editing models by negating task vectors. When models already strongly exhibit the behavior we wish to remove, it is harder to do so with this technique.\nIn those circumstances, a more promising approach is to add the task vector obtained with gradient ascent, as described in Appendix \\ref{sec:apapendix-gradient-ascent}.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/forget_correlation.pdf}\n    \\caption{Correlation between the gain in accuracy from fine-tuning and the drop in accuracy when subtracting the corresponding task vector for image classification tasks.}\n    \\label{fig:forget-corr}\n\\end{figure*}\n\n\n\\subsection{Additional tasks}\n\nIn addition to the tasks explored in Section \\ref{sec:add_img}, we study two other tasks, OCR and person identification.\n\nFor OCR, we use the synthetic dataset from \\citet{ilharco2022patching}, built using images from SUN-397 as backgrounds and mismatched class names as texts.\nThe task vector is produced by fine-tuning on those images, with the objective of predicting the written text (and not the background).\nAs shown in Figure \\ref{fig:forget-more-tasks} (left), especially for the larger CLIP models, negating the task vectors leads to large drops in performance with little change in accuracy on ImageNet. \n\nFor person identification, we use the Celebrity Face Recognition dataset, containing close to a million pictures of around one thousand celebrities.\\footnote{\\url{https:\/\/github.com\/prateekmehta59\/Celebrity-Face-Recognition-Dataset}.} We split the data into a training, validation and test set with proportions 0.8, 0.1 and 0.1.\nResults are shown in Figure \\ref{fig:forget-more-tasks} (right).\nWhile negating the task vectors leads to performance deterioration, we find that forgetting is less effective compared to other tasks like OCR. We hypothesize that one explanation for this could be the fact that fine-tuning on this dataset does provides only small gains in accuracy, as discussed in Appendix \\ref{sec:when-neg-works}.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/forget_more_tasks.pdf}\n    \\caption{Forgetting by negating task vectors on additional vision tasks, OCR and person identification.}\n    \\label{fig:forget-more-tasks}\n\\end{figure*}\n\n\\section{Forgetting with text generation}\n\\label{sec:appendix-neg-lang}\n\nThis section presents additional experimental details and results complementing the findings presented in Section \\ref{sec:forget_lang}, showcasing the effect of negating task vectors from text generation.\n\n\\subsection{Experimental details}\n\nTo obtain task vectors, we fine-tune on data Civil Comments  \\citep{borkan2019nuanced} where the toxicity score is larger than 0.8.\nWe then fine-tune GPT-2 models \\citep{radford2019language} from Hugging Face transformers library \\citep{wolf2019huggingface}.\nWe use a learning rate of 1e-5, and fine-tune with a causal language modeling objective with the AdamW optimizer for 5 epochs using a global batch size of 32.\nAfter fine-tuning, we evaluate models obtained by adding task vectors with scaling coefficients $\\lambda \\in \\{0.0, 0.1, \\cdots, 1.0\\}$.\nIn Table \\ref{tab:toxicity}, we report results for the largest scaling coefficient such that perplexity is still within 0.5 points of the perplexity of the pre-trained model.\nTo evaluate toxicity, we generate 1000 samples from the models. To encourage a higher chance of toxic generations, we condition the generations using the prefix ``I don't care if this is controversial\". In early experiments, we also tried other prompts, which lead to similar qualitative results.\nWe evaluate other prompts in Appendix \\ref{sec:app-realtoxic}.\nTo evaluate fluency, we measure the perplexity of the models on WikiText-103 with a striding window of size 1024 and a stride of 512 tokens.\n\n\n\\subsection{Additional models}\n\nIn addition to the GPT-2 Large models showed in Table \\ref{tab:toxicity}, we present results for GPT-2 Medium and GPT-2 Small models in Tables \\ref{tab:toxicity_gpt2med} and \\ref{tab:toxicity_gpt2small}.\nWe observe the same qualitative trends for the additional models. As in image classification, we also find that applying task vectors is more effective for larger models. \n\n\\begin{table*}\n\\caption{Making language models less toxic with negative task vectors. Results are shown for the GPT-2 Medium model.}\n\\setlength\\tabcolsep{4.5pt}\n\\renewcommand{\\arraystretch}{0.9}\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{lrrr}\n\\toprule\n Method & \\% toxic generations ($\\downarrow$)& Avg. toxicity score ($\\downarrow$) & WikiText-103 perplexity ($\\downarrow$)\n \\\\\\midrule\nPre-trained & 4.3 & 0.06 & 18.5 \\\\\nFine-tuned & 54.5 & 0.54 & 20.2 \\\\\nGradient ascent & 0.0 & 0.00 & $>$10$^{10}$ \\\\\nRandom task vector & 4.2 & 0.05 & 18.5 \\\\\\midrule\nNegative task vector & 1.8 & 0.02 & 18.9 \\\\\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab:toxicity_gpt2med}\n\\end{table*}\n\n\n\\begin{table*}\n\\caption{Making language models less toxic with negative task vectors. Results are shown for the GPT-2 Small model.}\n\\setlength\\tabcolsep{4.5pt}\n\\renewcommand{\\arraystretch}{0.9}\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{lrrr}\n\\toprule\n Method & \\% toxic generations ($\\downarrow$)& Avg. toxicity score ($\\downarrow$) & WikiText-103 perplexity ($\\downarrow$)\n \\\\\\midrule\nPre-trained & 3.7 & 0.04 & 25.2 \\\\\nFine-tuned & 62.9 & 0.61 & 28.1 \\\\\nGradient ascent & 0.0 & 0.00 & $>$10$^{10}$ \\\\\nRandom task vector & 3.2 & 0.04 & 25.3 \\\\\\midrule\nNegative task vector & 2.5 & 0.03 & 25.3 \\\\\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab:toxicity_gpt2small}\n\\end{table*}\n\n\\subsection{RealToxicityPrompts}\n\\label{sec:app-realtoxic}\n\nWe present additional experiments using RealToxicityPrompts \\citep{gehman2020realtoxicityprompts}, a dataset of natural language prompts used for measuring toxicity in language models. As in \\citet{gehman2020realtoxicityprompts}, we evaluate language models using 25 generations per prompt, using the Perspective API.\\footnote{\\url{https:\/\/github.com\/conversationai\/perspectiveapi}}\n\nIn Figure \\ref{fig:realtoxic}, we present results showing the expected maximum toxicity across the 25 generations and the perplexity on WikiText-103 as we vary the scaling coefficients.\nWe show results both for the \\textit{challenging} subset of the dataset, containing 1.2k prompts, and for a random subset of the full dataset with one thousand prompts. \nIn both cases, we see qualitatively similar trends: negating task vectors produced by fine-tuning on toxic data reduces the amount toxicity of the generations.\nFor GPT-2 large, we see close to vertical movement as the scaling coefficient increases, showing large decreases in accuracy with little change in perplexity on WikiText-103.\nHowever, especially for the challenging set of the benchmark, there is still significant headroom for improvement.\n\n\n\\begin{figure}%\n    \\centering\n    \\subfloat{{\\includegraphics[width=0.495\\linewidth]{figures\/forget_realtoxic_random.pdf} }}%\n   \n    \\subfloat{{\\includegraphics[width=0.495\\linewidth]{figures\/forget_realtoxic_challenging.pdf} }}%\n   \n    \\caption{\\textbf{Toxicity results using RealToxicityPrompts} \\citep{gehman2020realtoxicityprompts}, for various GPT-2 models.}%\n    \\label{fig:realtoxic}%\n   \n\\end{figure}\n\n\n\\section{Learning via addition}\n\\label{sec:appendix-add}\n\n\nIn all experiments, we add task vectors together and use a \\textit{single} scaling coefficient for the sum of the vectors, $\\lambda \\in \\{0, 0.05, 0.1, \\cdots, 1.0\\}$.\nWhile using scaling each task vector by its own coefficient could improve performance, exploring all combinations of scaling coefficients when the number of tasks is not small, due to the curse of dimensionality. While we focus on a single scaling coefficient for simplicity, more sophisticated strategies could be explored in future work, such as using black box optimization to search the space of scaling coefficients.\n\nFurthermore, we note that the best multi-task model given a set of task vectors is not often obtained by using all of the task vectors, as shown in Figure \\ref{fig:clip-add-all}. Since adding task vectors is computationally efficient and evaluations are usually substantially less expensive than training, practitioners could try out many subsets of task vectors and choose the ones that maximizes performance on the tasks of interest. Moreover, faster techniques such as the greedy algorithm proposed by \\citet{wortsman2022model} could allow users to efficiently discard task vectors that do not improve accuracy.\n\n\n\\subsection{The impact of random seeds}\n\n\n\nWe fine-tune five CLIP models on MNIST and five models EuroSAT, varying only the random seed. We then edit models by adding all possible combinations of the corresponding task vectors (25 in total). The results in Figure \\ref{fig:seeds} indicate that different random seeds have little impact in the resulting accuracy of the edited models for this set up. It is possible that we would observe larger variance in other settings such as natural language processing  \\citep{dodge2020fine,juneja2022linear}, but we again observe that users can simply discard task vectors that yield no improvement in validation data.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{figures\/seed_ablation.pdf}\n    \\caption{\\textbf{The impact of random seeds when fine-tuning.} Using different random seeds when fine-tuning on image classification tasks has little impact on the accuracy of edited models.}\n    \\label{fig:seeds}\n\\end{figure}\n\n\n\n\\subsection{Multi-task training}\n\nIn addition to using multiple-specialized models, we compare against a single multi-task model obtained via jointly fine-tuning on the eight image classification tasks we study. We fine-tune with the same hyper-parameters described in Appendix \\ref{sec:clip-exp-details}, also freezing the classification heads. \n\nMulti-task fine-tuning on the eight tasks achieves an average normalized performance of 0.994, compared to the best result obtained with task vectors, 0.912 (recall that 1.0 is obtained with multiple specialized models). Despite the headroom for improvement, multi-task training is less modular than using task vectors, requiring a new fine-tuning round every time a new task is added. In contrast, task vectors can be combined without any additional training and without the need to store or transfer the data used to create them, and can draw from the large pool of existing fine-tuned models such as the ones available on model hubs.\n\n\\subsection{Scaling coefficients}\n\nIn Figure \\ref{fig:acc-per-alpha} (left), we show the optimal scaling coefficients for the experiments where task vectors are added together. Recall that a single scaling coefficient is used for each experiment, regardless of the number of task vectors in the experiment. The variance in the optimal scaling coefficients can be large, highlighting the need for tuning on a case-by-case basis. However, compared to tuning traditional hyper-parameters, tuning the scaling coefficient is less computationally expensive since, unlike most hyper-parameters, the scaling coefficient can be changed without any additional training.\n\n\n\n\\begin{figure}%\n    \\centering\n    \\subfloat{{\\includegraphics[width=0.495\\linewidth]{figures\/clip_add_alphas.pdf} }}%\n    \\subfloat{{\\includegraphics[width=0.495\\linewidth]{figures\/clip_add_acc_per_alpha.pdf} }}%\n    \\caption{\\textbf{The effect of scaling coefficients when adding task vectors}. Left: Optimal scaling coefficients when adding task vectors. Right: average normalized performance as a function of the scaling coefficient and the number of task vectors.}%\n    \\label{fig:acc-per-alpha}%\n\\end{figure}\n\n\n\nIn Figure \\ref{fig:acc-per-alpha} (right), we show the average normalized performance across experiments as we vary the scaling coefficient and the number of task vectors. Scaling coefficients in the range 0.3 to 0.5 produce close to optimal results in many cases, although we recommend tuning this parameter when possible for best results.\n\n\\subsection{Accuracy on subsets of tasks}\n\nComplementing our results in the main paper, we show in Figure \\ref{fig:add-subsets} the average performance for all subsets task vectors, averaged only over the tasks that originated the task vectors (recall that in Figure \\ref{fig:clip-add-all} we presented the normalized accuracy averaged  over \\textit{all} tasks). We find that for smaller subsets, the single model obtained by adding task vectors matches more closely the performance of multiple specialized models, although that gap increases as the size of the subsets grow. \n\n\n\\begin{figure}%\n    \\centering\n    \\includegraphics[width=0.55\\linewidth]{figures\/clip_add_v2.pdf}\n    \\caption{\\textbf{Building multi-task models by adding task vectors.} Unlike results shown in Figure \\ref{fig:clip-add-all}, here performance is averaged only over the tasks used to build the task vectors in each experiment.}%\n    \\label{fig:add-subsets}%\n\\end{figure}\n\n\n\n\\subsection{ImageNet experiments}\n\nIn addition to results presented in Section \\ref{sec:add_img}, we explore whether addition performs well when fine-tuning on a larger-scale dataset, ImageNet. We fine-tune with the same hyper-parameters as described in Appendix \\ref{sec:clip-exp-details}, except for using a larger number of steps (4 epochs, around 40 thousand steps), to account for the larger size of ImageNet.\n\nWe then add the ImageNet task vector with each of the eight task vectors from Section \\ref{sec:add_img}, measuring accuracy both on ImageNet and on the task from the second task vector. For example, for MNIST, we add the MNIST task vector and the ImageNet task vector, and measure accuracy both on MNIST and on ImageNet. As shown in Figure \\ref{fig:add-imagenet}, adding the task vectors produces a single model with high accuracy on both tasks, which in most experiments is competitive with the fine-tuned models on their respective datasets.\n\n\n\n\\begin{figure}%\n    \\centering\n    \\includegraphics[width=0.99\\linewidth]{figures\/imagenet_add_eval_imagenet.pdf}%\n    \\bigbreak\n    \\includegraphics[width=0.99\\linewidth]{figures\/imagenet_add_eval_other.pdf}%\n    \\caption{\\textbf{Adding pairs of task vectors containing a task vector from ImageNet}. For all eight other target tasks from Section \\ref{sec:add_img}, adding their task vector with an ImageNet produces a model with high accuracy both on that task and on ImageNet.}%\n    \\label{fig:add-imagenet}%\n\\end{figure}\n\n\\subsection{Adding pairs of task vectors from NLP tasks}\n\\label{sec:appendix-add-lang}\n\n\nIn this section, we present results for building multi-task models using checkpoints that were \\textit{not} fine-tuned by the authors, and were instead downloaded directly from a hub that hosts model checkpoints publicly (the Hugging Face Hub).\\footnote{\\url{https:\/\/huggingface.co\/models}}\n\nOur motivation is aligned that from with previous work on building multi-task models \\citep{colin2020exploring,khashabi2020unifiedqa,zhong2021adapting,mishra2022cross,wei2021finetuned,sanh2022multitask,min2022metaicl,wang2022benchmarking}.\n\nMore specifically, we explore six fine-tuned T5 models \\citep{colin2020exploring} downloaded from the Hugging Face Hub using popularity and diversity as criteria. The models were fine-tuned on a diverse set of natural language processing tasks, including sentiment analysis using movie reviews from IMDB \\citep{maas2011imdb}, question answering (RACE, \\citet{lai2017race}; QASC, \\citet{allenai:qasc}), summarization (MultiNews, \\citet{alex2019multinews}), question generation (SQuAD, \\citet{squadv1}); and constrained text generation (CommonGen, \\citet{lin2020commongen}). The checkpoints and tasks were chosen based on the availability of models that were fine-tuned from the same initialization (a T5-Base model), were fine-tuned without introducing new parameters, and based on diversity of the tasks and popularity of the checkpoints on the hub. \nThe specific checkpoints we use are:\n\n\\begin{itemize}\n    \\item IMDB: \\texttt{mrm8488\/t5-base-finetuned-imdb-sentiment}\n    \\item RACE: \\texttt{mrm8488\/t5-base-finetuned-race}\n    \\item QASC: \\texttt{mrm8488\/t5-base-finetuned-qasc}\n    \\item MultiNews: \\texttt{mrm8488\/t5-base-finetuned-summarize-news}\n    \\item SQuAD: \\texttt{mrm8488\/t5-base-finetuned-question-generation-ap}\n    \\item CommonGen: \\texttt{mrm8488\/t5-base-finetuned-common\\_gen}\n\\end{itemize}\n\n\n\nFor evaluation, we use accuracy for the text classification task (IMDB), exact match for question answering tasks (RACE and QASC) and ROUGE-2\\footnote{\\url{https:\/\/huggingface.co\/spaces\/evaluate-metric\/rouge}} for text generation tasks (MultiNews, SQuAD question generation, and CommonGen).\nAs in Section \\ref{sec:add_img}, we normalize the performance on each task by the performance of the fine-tuned model on that task, to account for differences in task difficulty and evaluation metric.\n\nAs in image classification, we find that we can compress pairs of models into a single multi-task model with little performance loss (Figure \\ref{fig:add-nlp}).\nThese results are somewhat surprising, since the gap between the pre-trained model and fine-tuned models is much larger, and tasks vary widely in terms of input domain, length, and output type.\nMoreover, while there is more variance across different subsets of tasks when compared to image classification, in various cases we observe \\emph{higher} performance than that of specialized models.\nOn average, the normalized average performance of the model obtained by adding task vectors is 96.7\\%.\n\n\n\\begin{figure}%\n    \\centering\n    \\includegraphics[width=0.99\\linewidth]{figures\/t5_add_v3.pdf}%\n    \\caption{Adding pairs of task vectors from natural language processing tasks.}%\n    \\label{fig:add-nlp}%\n\\end{figure}\n\n\n\n\\subsection{GLUE experiments}\n\\label{sec:app-glue}\n\nIn this section, we describe the experimental setup used for investigations presented in Section \\ref{sec:add-nlp}, studying whether performance on specific target tasks can be improved by adding external task vectors.\n\nOur experiments use T5-base models, fine-tuned on four tasks from the GLUE benchmark:\n\n\\begin{itemize}\n    \\item \\textbf{Microsoft Research Paraphrase Corpus} (MRPC; \\citet{dolan2005automatically}) is a paraphrase task containing pairs of sentences labeled as either nearly semantically equivalent or not. The dataset is evaluated using the average of $F_1$ and accuracy.\n    \\item \\textbf{Recognizing Textual Entailment} (RTE; \\citet{wang2018glue}) is a dataset where models are tasked to predict whether a sentence entails or contradicts another sentence. The data  is originally from a series of datasets \\cite{dagan2005pascal, bar2006second, giampiccolo2007third, bentivogli2009fifth}. Accuracy is used as the evaluation metric.\n    \\item \\textbf{Corpus of Linguistic Acceptability} (CoLA;\n     \\citet{warstadt2018neural}) is a dataset with sentences labeled as either grammatical or ungrammatical. Models are evaluated on Matthews correlation (MCC; \\cite{matthews1975comparison}), which ranges between $-1$ and $1$.\n     \\item \\textbf{Stanford Sentiment Treebank} (SST-2; \\citet{socher2013recursive}) is a sentiment analysis task, containing sentences labelled as containing \\textit{positive} or \\textit{negative} sentiment. Accuracy is used as the evaluation metric.\n\\end{itemize}\n\nFor all tasks, we split the training set into two subsets, one used for fine-tuning and one used for determining the best external task vector, with the same size as the original validation sets. For fine-tuning, we use a batch size of 32, learning rate 1e-5 and fine-tune for 5 epochs using AdamW and a linear learning rate schedule. All results are averaged over 3 random seeds.\nWhen evaluating, we perform two forward passes for each sample, one for each label, and chose the label that minimizes the perplexity of the decoder.\n\n\n\\section{Task analogies}\n\nSimilarly in the experiments where multiple models are added together, we use a \\textit{single} scaling coefficient for the vector resulting from the task arithmetic,  $\\lambda \\in \\{0, 0.1, \\cdots, 1.0\\}$.\nWhile using scaling each task vector by its own coefficient could improve performance, we avoid this strategy since it complicates the search space and makes explorations more expensive. \nWe note that visual analogies has been explored in previous literature, albeit not at the task level \\cite{sadeghi2015visalogy}.\n\n\\subsection{Domain generalization}\n\\label{sec:app-sentiment}\n\nHere, we use task analogies to improve performance on tasks where no labeled data is available. We consider both Yelp \\citep{zhang2015character} and Amazon \\citep{mcauley2013hidden} binary-sentiment analysis as target tasks, using the \\texttt{amazon\\_polarity} and \\texttt{yelp\\_polarity} datasets from Huggingface datasets \\citep{lhoest-etal-2021-datasets}.  As detailed in \\ref{sec:analogies}, given  target and auxiliary tasks, we construct task vectors using the relationship $\\hat{\\tau}_\\textrm{target;\\,sent} = \\tau_\\textrm{target;\\,lm} + (\\tau_\\textrm{auxiliary;\\,sent} - \\tau_\\textrm{auxiliary;\\,lm})$. We apply an two scaling coefficients: one on the auxilary sentiment task vector, and another to the language modeling task vectors. \n\nWe compare our task analogy approach to two other baselines: fine-tuning on the auxiliary task, and fine-tuning on the target task. The latter represents an performance upper-bound, assuming we have labeled data for the target task. \n\nTo produce language model task vectors, we use consecutive 128-token chunks of text in each task as input-output pairs, following \\citet{lester-etal-2021-power}. To make predictions under the classification task, we follow the evaluation technique described in \\ref{sec:app-glue}.\n\n\n\nFor all models, we perform a single epoch of fine-tuning, setting a batch size of 2 and accumulating gradients across 8 steps. We use AdamW and a linear learning rate schedule. We set the maximum input and output sequence length to be 128. For each model scale, we perform a grid search over learning rates in \\{1e-5, 3e-5, 5e-5, 8e-4\\}, choosing the fastest learning rate that avoids divergence.\n\n\nTo construct a task vector using the task analogy, we perform a grid  over the values $\\lambda \\in \\{0.0, 0.1, ..., 1.0\\}$ for each scaling coefficient. Regardless of scale, we found that giving higher weight to the auxiliary sentiment task vector produced higher accuracy. For the smallest model, we saw better performance when applying a lower-valued coefficient to the language modeling task vectors. For the largest model, applying larger coefficients to the language modeling task vectors produced better performance. This trend may be reflective of the finding in \\ref{sec:forget_lang} that task forgetting is more effective with larger models.\n\n\\subsection{Kings and Queens}\n\\label{sec:appendix-kingsandqueens}\n\nAs a warm-up, we consider the task of classifying images as ``queen\", ``king\", ``woman\" or ``man\".\nWe collect 200 images from the web (50 for each category), by manually searching for the terms ``queen\", ``king\", ``man\" and ``woman\" using Google Images searches. We present samples in Figure \\ref{fig:kings-and-queens-samples}.\n\nOur experiments explore whether we can improve accuracy on each target category using only data from the other three categories.\nFor each category, we fine-tune CLIP models on the remaining three categories, and combine the task vectors according to the analogy relationship, e.g., $\\hat{\\tau}_\\textrm{king} = \\tau_\\textrm{queen} + (\\tau_\\textrm{man} - \\tau_\\textrm{woman})$.\nIn addition to evaluating on our collected set of images, we also evaluate on the ImageNet dataset as a control task.\n\nAs shown in Table \\ref{tab:kingsandqueens}, task analogies yield large gains in accuracy over pre-trained models with very little change in the control task, despite having no training data for the target task. Similar to \\citet{ilharco2022patching,ramasesh2021effect}, we find that results improve with model scale.\n\n\n\\begin{table*}\n\\caption{\\textbf{Learning via analogy.} By leveraging vectors from related tasks, we can improve accuracy on four new target tasks without any training data, and with little change on control settings. Results are shown for the CLIP models \\citep{radford2019language}, additional details are provided in Appendix \\ref{sec:appendix-kingsandqueens}.}\n\\setlength\\tabcolsep{4.5pt}\n\\renewcommand{\\arraystretch}{0.9}\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{lcccccccc}\n\\toprule\n \\multirow{2}{*}{Method}  & \\multicolumn{2}{c}{Queens} & \\multicolumn{2}{c}{Kings} &  \\multicolumn{2}{c}{Woman} & \\multicolumn{2}{c}{Men}\n \\\\\n & Target & Control & Target & Control & Target & Control & Target & Control \\\\\n \\midrule\n ViT-B\/32 & 0.00 & 63.4 & 0.00 & 63.4 & 0.00 & 63.4 & 0.00 & 63.4\\\\\n \\quad{+ task vectors} & 42.0 & 62.4 & 30.0 & 62.4 & 69.4 & 62.5 & 58.0 & 62.6\\\\\\midrule\n ViT-B\/16 & 0.00 & 68.3 & 0.00 & 68.3 & 0.00 & 68.3 & 0.00 & 68.3 \\\\\n\\quad{+ task vectors} & 66.0 & 67.5 & 94.0 & 67.4 & 87.8 & 67.5 & 62.0 & 67.6 \\\\\\midrule\nViT-L\/14 & 0.00 & 75.5 & 0.00 & 75.5 & 0.00 & 75.5 & 0.00 & 75.5 \\\\\n\\quad{+ task vectors} & 100 & 74.7 & 100 & 74.5 & 100 & 74.6 & 96.0 & 74.6\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab:kingsandqueens}\n\\end{table*}\n\n\n\nFine-tuning CLIP models is done as described in Section \\ref{sec:clip-exp-details}, with the exception of using 40 optimization steps because of the small size of the datasets. When fine-tuning, we use only the images from one category (e.g., ``king\"), and a set of 1001 classes from which to choose, composed by the 1000 classes in ImageNet, and a new class. Since CLIP has already seen many images of queens, kings, men and women in its pre-training, we use a new category name for the new class when fine-tuning, in order to simulate learning a new concept. More concretely, we use the class name ``something\", which makes the accuracy of zero-shot models close or equal to zero. When evaluating, we also contrast between all 1001 options, including all ImageNet classes. This is done both for our target task, and for ImageNet, where we add an additional option.\nNote that we do not need to introduce any new task-specific weights to do all of these operations, since CLIP can perform classification with any set of classes by using its text encoder (which is frozen as in Section \\ref{sec:clip-exp-details}).\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=.85\\textwidth]{figures\/kinds_and_queens.pdf}\n    \\caption{Samples from the dataset we collect for classifying queens, kings, women and men, as described in Section \\ref{sec:appendix-kingsandqueens}.}\n    \\label{fig:kings-and-queens-samples}\n\\end{figure*}\n\n\n\\subsection{Subpopulations}\n\\label{sec:appendix-sketches}\n\nWe fine-tune CLIP models on each of the subpopulations with the same hyper-parameters as described in Section \\ref{sec:clip-exp-details}, using 500 optimization steps regardless of the number of samples. For the few-shot experiments, we sample the same number of samples for every class in the task.\nFor convenience, let ImageNet-A\\footnote{Not to be confused with the adversarial dataset from \\citet{imageneta}.} and ImageNet-B represent the two subpopulations from ImageNet, and Sketches-A and Sketches-B represent the two subpopulations from the sketches dataset from \\citet{eitz2012humans}. Note that ImageNet-A and Sketches-A share the same classes, and the same is true for ImageNet-B and Sketches-B. We present samples in Figure \\ref{fig:sketches-samples}.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=.85\\textwidth]{figures\/sketches.pdf}\n    \\caption{Samples from the datasets used for the analogies with subpopulations experiments, as described in Section \\ref{sec:appendix-sketches}.}\n    \\label{fig:sketches-samples}\n\\end{figure*}\n\nComplementing Figure \\ref{fig:clip-analogies}, we show a breakdown per model and for every subpopulation as a target in Table \\ref{tab:sketches}.\n\n\n\n\\paragraph{Independent scaling coefficients.} In addition to our standard procedure of using a single scaling coefficient for the vector resulting from the arithmetic operations, we explore having independent scaling coefficients for each task vector in the expression. In other words, we explore the models $\\theta_\\textrm{new} = \\theta + \\lambda_C\\tau_C + \\lambda_B\\tau_B - \\lambda_A\\tau_A$ for various scaling coefficients $\\lambda_A, \\lambda_B, \\lambda_C \\in \\{0, 0.1, \\cdots, 1.0\\}$.\nOn average, the optimal scaling coefficients were $\\lambda_B^\\star=\\lambda_C^\\star=0.32$ and $\\lambda_A^\\star=0.28$.\nUsing independent scaling coefficients improved performance over using a single scaling coefficient by 0.7 percentage points on average, but also required substantially more evaluations to be made ($10^3$ instead of 10).  \n\n\n\n\\begin{table*}\n\\caption{\\textbf{Learning by analogy on subpopulations.} Results are shown for multiple CLIP models, as detailed in Section \\ref{sec:appendix-sketches}.}\n\\setlength\\tabcolsep{4.5pt}\n\\renewcommand{\\arraystretch}{0.9}\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{lccccccc}\n\\toprule\n \\multirow{2}{*}{Model} & Samples & \\multirow{2}{*}{Task vectors} & \\multicolumn{5}{c}{Accuracy} \\\\\n& per class & & Sketches-A & Sketches-B & ImageNet-A & ImageNet-B & Average\\\\\\midrule\n\\multirow{8}{*}{ViT-B\/32} & 0 & \\xmark & 0.712 & 0.677 & 0.861 & 0.923& 0.793 \\\\\n& 0 & \\cmark & 0.782 & 0.758 & 0.861 & 0.926& 0.832 \\\\\n& 1 & \\xmark & 0.754 & 0.758 & 0.868 & 0.919& 0.825 \\\\\n& 1 & \\cmark & 0.782 & 0.766 & 0.866 & 0.922& 0.834 \\\\\n& 2 & \\xmark & 0.768 & 0.778 & 0.868 & 0.919& 0.833 \\\\ \n& 2 & \\cmark & 0.786 & 0.800 & 0.867 & 0.922& 0.844 \\\\\n& 4 & \\xmark & 0.810 & 0.780 & 0.871 & 0.926& 0.847 \\\\\n& 4 & \\cmark & 0.802 & 0.796 & 0.871 & 0.927& 0.849 \\\\\\midrule\n\\multirow{8}{*}{ViT-B\/16} & 0 & \\xmark & 0.716 & 0.732 & 0.885 & 0.946& 0.820\\\\\n& 0 & \\cmark & 0.794 & 0.794 & 0.889 & 0.953& 0.858\\\\\n& 1 & \\xmark & 0.758 & 0.812 & 0.894 & 0.948& 0.853\\\\\n& 1 & \\cmark & 0.796 & 0.804 & 0.897 & 0.957& 0.863\\\\\n& 2 & \\xmark & 0.792 & 0.817 & 0.897 & 0.951& 0.865\\\\\n& 2 & \\cmark & 0.804 & 0.829 & 0.899 & 0.956& 0.872\\\\\n& 4 & \\xmark & 0.815 & 0.812 & 0.904 & 0.952& 0.871\\\\\n& 4 & \\cmark & 0.831 & 0.825 & 0.904 & 0.953& 0.878\\\\\\midrule\n\\multirow{8}{*}{ViT-L\/14} & 0 & \\xmark & 0.823 & 0.831 & 0.913 & 0.962& 0.882\\\\\n& 0 & \\cmark & 0.879 & 0.861 & 0.922 & 0.968& 0.908\\\\\n& 1 & \\xmark & 0.845 & 0.863 & 0.923 & 0.971& 0.900\\\\\n& 1 & \\cmark & 0.879 & 0.863 & 0.930 & 0.973& 0.911\\\\\n& 2 & \\xmark & 0.865 & 0.881 & 0.925 & 0.973& 0.911\\\\\n& 2 & \\cmark & 0.875 & 0.881 & 0.932 & 0.975& 0.916\\\\\n& 4 & \\xmark & 0.875 & 0.883 & 0.934 & 0.973& 0.916\\\\\n& 4 & \\cmark & 0.903 & 0.887 & 0.941 & 0.975& 0.927\\\\\n\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab:sketches}\n\\end{table*}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nStochastic Resonance (SR) was discovered barely about two and half decades\nago, yet it has proved to be very useful in explaining many phenomena in \nnatural sciences[1-3]. SR refers to an enhanced response of a nonlinear system \nto a subthreshold periodic input signal in the presence of noise of  optimum \nstrength. Here, noise plays a constructive role of pumping power in a \nparticular mode, that is in consonance with the applied field, at the cost of\nthe entire spectrum of modes present in it. SR, so defined, leaves a lot of \nliberty as to what is the physical quantity that is to be observed which \nshould show a maximum as a function of noise strength[4-23]. In other words, \nno unique quantifier of SR is specified. Also, in order that SR be a bonafide\nresonance the quantifier must show maximum as a function of frequency of the \napplied field as well. For instance, in a double-well system, hysteresis loop\narea, input energy or work done on the system in a period of the driving \nfield and area under the first peak in the residence time (in a well) \ndistribution are used to characterize SR as a bonafide resonance[4-17,19-22].\n\nIn the present work, motivated by recently discovered fluctuation theorems,\nwe show that in an overdamped bistable system input energy per period as well\nas the energy absorbed per period by the system from the bath, i.e, the heat,\ncan be used as quantifiers to study SR. Also, it is found that the relative \nvariance of both the quantities exhibit minimum at resonance; that is, \nwhenever input energy and heat show maximum as a function of noise strength \n(as also frequency), their respective relative fluctuations show minimum. \nThis shows that at SR the system response exhibits greater degree of \ncoherence. These fluctuations, however, are very large and often the physical\nquantities in question become non-self-averaging. We study some of these \naspects in the light of the fluctuation theorems in the following sections. \nThe fluctuation theorems are of fundamental importance to nonequilibrium \nstatistical mechanics[24-46]. The fluctuation theorems describe rigorous \nrelations for properties of distribution functions of physical variables such\nas work, heat, entropy production, etc., for systems far from equilibrium \nregimes where Einstein and Onsagar relations no longer hold. These theorems \nare expected to play an important role in determining thermodynamic \nconstraints that can be imposed on the efficient operation of machines at  \nnano scales. Some of these theorems have been verified experimentally[47-53].\n\n\\section{The Model}\nWe consider the motion of a particle in a double-well potential \n$V(x)=-\\frac{a x^{2}}{2}+\\frac{b x^{4}}{4}$ under the action of a weak \nexternal field $h(t)=A\\sin(\\omega t)$. The motion is described by the \noverdamped Langevin equation[44]\n\\begin{equation}\n\\gamma \\frac{dx}{dt}=-\\frac{\\partial U(x)}{\\partial x}+\\xi(t) ,\n\\end{equation}\nwhere $U(x)=V(x)-h(t)x$. The random forces satisfy \n$\\langle \\xi(t) \\rangle =0$ and \n$\\langle \\xi(t)\\xi(t^{'}) \\rangle=2\\gamma k_{B} T \\delta(t-t^{'})$,\nwhere $\\gamma$ is the coefficient of friction, $T$ is the absolute \ntemperature and $k_{B}$ is the Boltzmann constant. In the following we use a \ndimensionless form of equation(1), namely,\n\\begin{equation}\n\\frac{dx}{dt}=-\\frac{\\partial U(x)}{\\partial x}+\\xi(t),\n\\end{equation}\nwhere $U(x)=-\\frac{x^{2}}{2}+\\frac{x^{4}}{4}-xh(t)$, and \nthe external field $h(t)=A\\sin(\\omega t)$. Now, $\\xi(t)$ satisfies \n$\\langle \\xi(t) \\xi(t^{'}) \\rangle=D \\delta(t-t^{'})$, where $D=2 k_{B} T$. \nAll the parameters are given in dimensionless units~~(in terms of \n$\\gamma$, $a$ and $b$). We consider $A \\ll 0.25$, so that the forcing \namplitude is much smaller than the barrier height between the two wells.\n\nFollowing the stochastic energetic formalism developed by Sekimoto[55], the \nwork done by the external drive $h(t)$ on the system or the input energy per \nperiod (of time $\\tau_{\\omega}$) is defined as[21] \n\\bdm\nW_{p}= \\int_0^{t_{0}+\\tau_{\\omega}} \\frac{\\partial U}{\\partial t} dt\n\\edm\n\\begin{equation}\n= -\\int_0^{t_{0}+\\tau_{\\omega}} x(t) \\frac{dh(t)}{dt} dt,\n\\end{equation} \nwhere $h(t)$ is the drive field which completes its period in time \n$\\tau_{\\omega}$. The completion of one period of $h(t)$, however, does not \nguarantee the system coming back to the same state as the starting one. In \nother words, $x(t+\\tau_{\\omega})$ need not be equal to $x(t)$ or \n$U(x,t+\\tau_{\\omega})$ may differ from $U(x,t)$. The work done over a period \n$W_{p}$ equals  change in the internal energy \n$\\Delta U=U(x,t_{0}+\\tau_{\\omega})-U(x,t_{0})$ and heat $Q$ absorbed over a \nperiod (first law of thermodynamics), i.e, $W_{p}=\\Delta U_{p}+Q_{p}$. Since \n$x(t)$ is stochastic, $W_{p}$, $\\Delta U_{p}$ and $Q_{p}$ are not the same \nfor different cycles(or periods) of $h(t)$. The averages are  evaluated from \na single long trajectory $x(t)$ (eqn(3)). From the same calculations one can \nalso obtain the probability distribution $P(W)$ and various moments of $W$.\nSimilarly, appealing to the first law of thermodynamics as stated above we \ncan obtain $P(Q_{p})$ and $P(\\Delta U_{p})$ and their moments, where the \nsubscript p indicates evaluation of the physical quantities over one period \nof the field. Numerical simulation of our model was carried out by using \nHuen's method[56]. To calculate $W_{p}$ and $Q_{p}$ we first evolve the \nsystem and neglect initial transients. To get better statistics we calculate \n$W_{p}$, $Q_{p}$ for $10^{6}$ cycles. In some cases we evaluate $W$, \n$\\Delta U$ and $Q$ over many periods, $n$, and calculate their averages, \nagain, for $10^6$ such entities.\n\\section{Results and Discussions}\nThe internal energy being a state variable, average change in its value over \na period $\\Delta U_{p}$ is identically equal to  zero. Thus, in the time \nperiodic asymptotic state averaged work done over the period \n$\\langle W_{p} \\rangle$ is dissipated in to heat $\\langle Q_{p} \\rangle$ by \nthe system to the bath. Thus, $\\langle Q_{p} \\rangle$ can also be identified \nas hysteresis loop area. As has been reported earlier[19-22], \n$\\langle W_{p} \\rangle$, the input energy per period, shows a maximum as a \nfunction of $D$. Fig(1) shows that $\\langle W_{p} \\rangle$ and \n$\\langle Q_{p} \\rangle$ coincide, thus both the physical quantities show\nSR. Hence, in this case input energy per period, the heat per period or the \nhysteresis loop area can equally well quantify stochastic resonance. However,\nin this work we focus mostly on the fluctuation properties of these \nquantities.\n\nThe relative variances $R_{W}$ and $R_{Q}$ of both $W_{p}$ and $Q_{p}$ \nrespectively show minimum (fig(2)) as a function of $D$.  It may be \nnoted that even though $\\langle W_{p} \\rangle$ and $\\langle Q_{p} \\rangle$ are identical,~~  fluctuations in $W_{p}$ differ from the fluctuations in  $Q_{p}$.\n The relative variance of $Q_{p}$ is always larger than that of $W_{p}$ for all $D$. It is also noteworthy that the minimum value of the relative \nvariance is  larger than one. However, the minimum becomes less than one if \nthe averages are taken not over a single period of the field but over a \nlarger(integral) number, $n>1$, of periods. Therefore, in order to obtain \nmeaningful averages of these physical quantities in such driven systems one \nneeds to study over time scales much larger than one period so that the \naverages are significantly larger than the deviations about them. Also, as \n$n$ becomes large, the differences between the relative variances of $W$ and \n$Q$ become insignificant(see inset of fig(2)). Importantly, in the system \nunder study, this situation (mean $>$ dispersion) can be achieved by \nincreasing the duration of averaging time(or the number of periods,~ $n$) more \neasily around the value of $D$ where SR occurs. The minimum of relative \nvariance occurs just because the mean value is largest there and not because \ndispersions are smallest. However, as the number of periods $n$ is increased \nthe mean value of heat dissipated over the $n$ periods \n$\\langle Q_{np} \\rangle \\sim n$ for all $n$, whereas the dispersion \n$\\sim \\sqrt{n}$ for large $n$ so that the relative variance decreases with \n$n$ as $\\frac{1}{\\sqrt{n}}$ and one gets a range of $D$ where the averages \nbecome meaningful. We have observed numerically that $Q_{np}$ behaves as an \nindependent variable only when evaluated over a larger number of cycles $n$ \nas compared to in case of $W_{np}$. For our present parameters  approximately\n$Q_{np}$ is uncorrelated beyond $10$ periods, whereas $W_{np}$ is uncorrelated beyond $5$ periods.\n\nIn fig(3), we have plotted average heat dissipated \n$\\langle Q_{p} \\rangle$($=\\langle W_{p} \\rangle$) over a single period as a \nfunction of frequency. The values of physical parameters are given in the \nfigure caption. The figure shows maximum as shown in earlier literature[21].\nThus $\\langle Q_{p} \\rangle $ acts as a quantifier of bonafide stochastic \nresonance. In the inset we give the corresponding relative variance of heat \nand work as a function of frequency. We observe that heat fluctuations are \nlarger than work fluctuations at all frequencies. Near the resonance the \nrelative variance shows a minimum. It may be noted that minimum relative \nvariance of both quantities $W_{p}$ and $Q_{p}$ are larger than one(fig(2) and fig(3)). \n\nIn fig(4), we plot the probability distribution of $W_{p}$ and $Q_{p}$ for \nvarious values of $D$. For low values of $D$ (e.g., $D=0.02$) $P(W_{p})$ is \nGaussian  whereas $P(Q_{p})$ has a long exponential tail as in case of a \nsystem driven in a harmonic well and with almost no chance of a particle \ngoing over to the other well of the double-well potential. As $D$ is \ngradually increased rare passages to the other well becomes a possibility and\na very small peak appears at a finite positive value of $W_{p}$(or $Q_{p}$)\n(e.g., at $D=0.04$). As $D$ is increased further, $P(W_{p})$ and $P(Q_{p})$ \nbecome multipeaked and the averages $\\langle W_{p} \\rangle$, \n$\\langle Q_{p} \\rangle$ shifts to their positive values. The distributions \nbecome most asymmetric at around $D=0.12$ (where SR occurs) and the asymmetry\nreduces again at larger $D$, fig(4). When $D$ becomes large (e.g., $D=0.5$) \nthe distribution becomes completely symmetric and at such high $D$ values the\npresence of potential hump becomes ineffective to split the distribution into\ntwo or more peaks. At very small and very large $D$ values $P(W_{p})$ is close to \nGaussian and so does $P(Q_{p})$ but with a slow decaying exponential tail. In all\nthe graphs, the distribution of $P(Q_{p})$ ($P(W_{p})$) extend to negative values of \n$Q_{p}$ ($W_{p}$). Finite value for distribution in the negative side is \nnecessary  to satisfy certain fluctuation theorems. Moreover, $P(Q_{p})$\nhas higher  weightage for large negative $Q_{p}$ than that of work $W_{p}$.\n\nIt is worth reemphasizing that $W$ and $Q$ behave as additive (or extrinsic) \nphysical quantities with respect to the number of periods $n$ and hence \n$\\langle W_{np} \\rangle $ and $\\langle Q_{np} \\rangle $ increase \nin proportion to $n$ whereas $\\Delta U$, in this case, is an intrinsic \nphysical quantity and  $\\frac{\\Delta U}{n}  \\rightarrow  0$ as \n$n \\rightarrow \\infty$. This indicates that the distributions $P(W_{np})$ and\n$P(Q_{np})$ both have  identical characteristics as  $n \\rightarrow \\infty$.\nTherefore, the difference between \n$(\\frac{\\sqrt{\\langle W_{np}^{2} \\rangle -\\langle W_{np} \\rangle ^{2}}}\n{\\langle W_{np} \\rangle})$ and \n$(\\frac{\\sqrt{\\langle Q_{np}^{2} \\rangle -\\langle Q_{np} \\rangle ^{2}}}\n{\\langle Q_{np} \\rangle})$ vanishes as $n \\rightarrow \\infty$. In the recent \nliterature it is shown that  the distribution $P(W_{np})$ over a large number\nof periods approaches a Gaussian. Also, if  one considers  $W_{p}$ over a \nsingle period by increasing the noise strength, $P(W_{p})$ approaches \nGaussian and satisfies the steady state fluctuation theorem (SSFT). SSFT \nimplies[26,34-36,44-46,51-53] the probability of physical quantity $x$ to \nsatisfy the relation  $P(x)\/P(-x) = \\exp(\\beta x)$, where $\\beta$ is \nthe inverse temperature and $x$ may be work, heat, etc. In fig(5), the \nevolution of $ P(Q_{np})$ is shown as $n$ is increased . As $n$ increases the \ncontribution of negative $Q$ to the distribution decreases; besides, the \ndistribution gradually becomes closer and closer to Gaussian. There is a \ncontribution to $P(Q_{np})$ due to change in the internal energy $\\Delta U$ \nwhich is supposed to dominate at very large $Q$ making the distribution \nexponential in the asymptotic regime[34,35,53]. However, it is not possible to \ndetect this exponential tail in our simulations. For large $n$, $P(Q_{np})$  \napproaches Gaussian(inset of fig(5)). The Gaussian fit of the graph almost \noverlaps and the calculated ratio, \n$\\frac{\\langle Q_{np}^{2} \\rangle -\\langle Q_{np} \\rangle^{2}}{\\frac{2}\n{\\beta} \\langle Q_{np} \\rangle}$ equals $0.99$ for $n=25$. This ratio is \ncloser to one, a requirement for SSFT to hold where $P(Q)$ is Gaussian[22,44,45]. \nFig(6) shows the plot of $ln(\\frac{P(Q_{np})}{P(-Q_{np})})$ as a function of \n$\\beta Q_{np}$ for various values of $n$. One can readily see that slope of \n$ln(\\frac{P(Q_{np})}{P(-Q_{np})})$ approaches $1$ for \n$Q \\ll \\langle Q_{np} \\rangle $ for large $n$. This is a statement of \nconventional steady state fluctuation theorem. As the number of periods $n$,\nover which $Q_{np}$ is calculated, is increased, the conventional SSFT is \nsatisfied  for $Q_{np}$ less than $\\langle Q_{np} \\rangle$ (e.g., for $n=25$,\nSSFT is valid for $Q_{np}$ less than $0.4$, for $D=0.16$). There exists an \nalternative relation for heat fluctuation, namely, the extended heat \nfluctuation theorem[34,35]. Here, the distribution function obeys a different \nsymmetry property for $Q \\gg \\langle Q_{np} \\rangle$ for finite $n$. As \n$n \\rightarrow \\infty$, $\\langle Q_{np} \\rangle \\rightarrow \\infty$  in this \nlimit, and hence conventional SSFT holds  which has been clarified earlier \nin linear systems[53]. \n\nIt is further interesting to investigate effects associated with SR in an asymmetric double-well potential involving two hopping time scales instead of one as in the symmetric case.~~~We therefore,~~~consider a scaled asymmetric potential $V(x)=\\frac{- x^{2}}{2}+\\frac{x^{4}}{4}-cx$ driven by the external field $h(t)$.~~~~Fig(7) shows the average input energy $\\langle W_{p} \\rangle$ and average heat $\\langle Q_{p} \\rangle $ over a single period as a function of $D$ for various values of the asymmetric parameter $c$.~~~From this figure we find that the peak becomes broader and lower as $c$ is increased.~~~The peak shifts to larger values of noise intensities for higher $c$.~~~~In other words,~~the phenomenon of SR is not as pronounced[2] as in case of $c=0$(fig(2)).~~~It is because the synchronization between signal and particle hopping between the two well becomes weak because for $c \\neq 0 $,~~~the mean time of passage for well $1$ to well $2$ is different from  the mean time of passage from  well $2$ to well $1$.~~~As a consequence the relative variances $R_{W}$ and $R_{Q}$ become larger as compared to in case of $c=0$(fig(2)) as shown in the inset of fig(7).\n\nIn fig(8(a)) and fig(8(b) we have plotted probability distribution $P(W_{p})$ and $P(Q_{p})$ over a single period for different values of asymmetry parameter $c$ for a fixed value of $D=0.12$, $A=0.1$ and $\\omega =0.1$.~~~As asymmetry increases the probability for particle to remain in the lowest well enhances.~~Hence particle performs simple oscillation around most stable minima over a longer time before making transitions to the other well.~~~Hence Gaussian like peak near $W \\approx 0$ or $Q \\approx 0$ increases as $c$ increases.~~~The weight of $P(W_{p})$ for larger values of work(positive as well as negative ) decreases with increase in $c$.~~~However,~~~for $P(Q_{p})$, its magnitude at large positive and negative values of $Q_{p}$ increases as we increase asymmetry parameter.~~~This contrasting behavior can be attributed to the larger fluctuations of internal energy $\\Delta U_{p}$ as one increases $c$.~~This we have verified\n separately.~~~Due to this contribution of $\\Delta U_{p}$ for $Q_{p}$,~~nature of $P(W_{p})$ and $P(Q_{p})$ are qualitatively different.~~In all cases for fixed asymmetry $c$ fluctuation in heat are larger than fluctuation in work.\n \n In fig(9) and (10) evolution for $P(W_{np})$ and $P(Q_{np})$ respectively are plotted for various values of number of periods $n$.~~We clearly observe that as $n$ increases both the distributions tend to become  Gaussian distributions with the  fluctuation ratio $\\frac {V}{(\\frac{2}{\\beta}\\langle M \\rangle )}=1$,~~between their variance $V$ and mean $\\langle M \\rangle$ as required to satisfy SSFT  as mentioned earlier.~~To satisfy SSFT for heat we have to take \n larger number of periods as compared for work.~~Only in the large $n$ limit contribution to heat from internal energy becomes negligible.~~~In the insets of fig(9) and fig(10) we have shown a Gaussian fit(with fluctuation ratio equal to one),~~which agrees perfectly well with our numerical data.~~~Conclusions regarding validity of SSFT for asymmetric case for larger periods remain the same as for the symmetric case.\n\n\nIn summary, we find that SR shown by a particle moving in a double-well(symmetric) \npotential and driven by a weak periodic field can be characterized well by the heat~$\\langle Q_{p}\\rangle$ dissipated to the bath or the hysteresis loop area.~~~ It can equally well be characterized by the relative dispersion of $\\langle W_{p}\\rangle$ and $\\langle Q_{p}\\rangle$.~~~ At resonance relative dispersion shows a minimum as a function of both $D$ and $\\omega$.~~~~ We also show that minimum relative variance can be made less than one by taking long time protocols of the applied field.~~~For long time protocols distribution  $P(Q_{np})$ satisfies conventional SSFT for $P(Q_{np})$ at  $Q_{np} \\ll \\langle Q_{np} \\rangle$ for finite $n$[53].~~~We have also shown that SR gets weakened in the presence of asymmetric potential and as a consequence fluctuation in heat and work become larger.~~~SSFT too is satisfied for both work and heat,~~when they are calculated over large number of periods.\n\\section{Acknowledgements:}\nAMJ and MCM thank BRNS, DAE,Govt. of India for partial financial support.\n~~AMJ also thanks DST, India for financial support.~~ MCM acknowledges IOP, Bhubaneswar for hospitality.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\uppercase{Introduction}}\n\nWhen programming the small devices that constitutes the nodes of\nthe Internet of Things (IoT), one has to adapt to the limitations of\nthese devices.\n\nApart from their very limited processing power (especially compared to\nthe current personal computers, and even mobile devices like smartphones\nand tablets), the main specificity of the devices is that they are operated\non small batteries (e.g.: AAA or button cells).\n\nThus, one of the main challenges with these motes is the need to reduce\nas much as possible their energy consumption. We want their batteries to last\nas long as possible, for economical but also practical reasons: it may be\ndifficult---even almost impossible---to change the batteries of some of these\nmotes, because of their locations (e.g.: on top of buildings, under roads,\netc.)\n\nIoT motes are usually very compact devices: they are usually built around\na central integrated chip that contains the main processing unit and several\nbasic peripherals (such as timers, A\/D and D\/A converters, I\/O\ncontrollers\\ldots) called microcontroller units or MCUs. Apart from the MCU,\na mote generally only contains some ``physical-world'' sensors and a radio\ntransceiver for networking. The main radio communication protocol currently\nused in the IoT field is IEEE 802.15.4. Some MCUs do integrate a 802.15.4\ntransceiver on-chip.\n\nAmong the various components that constitute a mote, the most power-consuming\nblock is the radio transceiver. Consequently, to reduce the power consumption\nof IoT motes, a first key point is to use the radio transceiver only when\nneeded, keeping it powered-off as much as possible. The software element\nresponsible to control the radio transceiver in an adequate manner is\nthe \\emph{MAC~\/ RDC (Media Access Control \\& Radio Duty Cycle)}\nlayer of the network stack.\n\nA efficient power-saving strategy for IoT motes thus relies on finding the\nbetter trade-off between minimizing the radio duty cycle while keeping\nnetworking efficiency at the highest possible level. This is achieved\nby developing new, ``intelligent'' MAC~\/ RDC protocols.\n\nTo implement new, high-performance MAC~\/ RDC protocols, one needs to be\nable to react to events with good reactivity (lowest latency possible) and\nflexibility. These protocols rely on precise timing to ensure efficient\nsynchronization between the different motes and other radio-networked\ndevices of a \\emph{Personal Area Network (PAN)}, thus allowing\nto turn on the radio transceivers \\emph{only} when needed.\n\nAt the system level, being able to follow such accurate timings means having\nvery efficient interruption management, and the extensive use of hardware\ntimers, that are the most precise timing source available.\n\nThe second most power-consuming element in a mote, after the radio\ntransceiver, is the MCU itself: every current MCU offers ``low-power modes'',\nthat consist in disabling the various hardware blocks, beginning with the CPU\ncore. The main way to minimize energy consumption with a MCU is thus\nto disable its features as much as possible, only using them when needed:\nthat effectively means putting the whole MCU to sleep as much as possible.\n\nLike for the radio transceiver, using the MCU efficiently while keeping\nthe system efficient and reactive means optimal use of interruptions,\nand hardware timers for synchronization.\n\nThus, in both cases, we need to optimally use interruptions as well as\nhardware timers. Being able to use them both efficiently without too much\nhassle implies the use of a specialized operating system (OS), especially\nto easily benefit from multitasking abilities. That is what we will\ndiscuss in this paper.\n\n\n\\section{\\uppercase{Previous work and problem statement}}\n\nSpecialized OSes for the resource-constrained devices that constitute\nwireless sensor networks have been designed, published, and made available\nfor quite a long time.\n\n\\subsection{TinyOS}\n\nThe first widely used system in this domain was \\emph{TinyOS} \\cite{TinyOS}.\nIt is an open-source OS, whose first stable release (1.0) was published in\nseptember 2002. It is very lightweight, and as such well adapted to limited\ndevices like WSN motes.  It has brought many advances in this domain, like\nthe ability to use Internet Protocol (IP) and routing (RPL) on 802.15.4\nnetworks, including the latest IPv6 version, and to simulate networks\nof TinyOS motes via TOSSIM \\cite{TOSSIM}.\n\nIts main drawback is that one needs to learn a specific language---named\nnesC---to be able to efficiently work within it. This language is quite\ndifferent from standard C and other common imperative programming languages,\nand as such can be difficult to master.\n\nThe presence of that specific language is no coincidence: TinyOS is built\non its own specific paradigms: it has an unique stack, from which the\ndifferent components of the OS are called as statically linked callbacks.\nThis makes the programming of applications complex, especially for\ndecomposing into various ``tasks''. The multitasking part is also\nquite limited: tasks are run in a fixed, queue-like order. Finally,\nTinyOS requires a custom GNU-based toolchain to be built.\n\nAll of these limitations, plus a relatively slow development pace (last\nstable version dates back to august 2012) have harmed its adoption,\nand it is not the mainly used OS of the domain anymore.\n\n\\subsection{Contiki}\n\nThe current reference OS in the domain of WSN and IoT is \\emph{Contiki}\n\\cite{ContikiOS}. It's also an open-source OS, which was first released\nin 2002. It is also at the origin of many assets: we can mention, among\nothers, the uIP Embedded TCP\/IP Stack \\cite{uip}, that has been extended\nto uIPv6, the low-power Rime network stack \\cite{Rime}, or the Cooja advanced\nnetwork simulator \\cite{Cooja}.\n\nWhile a bit more resource-demanding than TinyOS, Contiki is also very\nlightweight and well adapted to motes. Its greatest advantage over TinyOS\nis that it is based on standard, well-known OS paradigms, and coded\nin standard C language, which makes it relatively easy to learn and program.\nIt offers an event-based kernel, implemented using cooperative multithreading,\nand a complete network stack. All of these features and advantages have made\nContiki widespread, making it the reference OS when it comes to WSN.\n\nContiki developers also have made advances in the MAC\/RDC domain: many\nof them have been implemented as part of the Contiki network stack, and\na specifically developed, ContikiMAC, has been published in 2011\n\\cite{ContikiMAC} and implemented into Contiki as the default\nRDC protocol (designed to be used with standard CSMA\/CA as MAC layer).\n\nHowever, Contiki's extremely compact footprint and high optimization comes\nat the cost of some limitations that prevented us from using it as our\nsoftware platform.\n\nContiki OS is indeed not a real-time OS: the processing of ``events''---using\nContiki's terminology---is made by using the kernel's scheduler, which is\nbased on cooperative multitasking. This scheduler only triggers at a specific,\npre-determined rate; on the platforms we're interested in, this rate is\nfixed to 128~Hz: this corresponds to a time skew of up to 8~milliseconds\n(8000~microseconds) to process an event, interruption management being\none of the possible events. Such a large granularity is clearly\na huge problem when implementing high-performance MAC\/RDC protocols,\nknowing that the transmission of a full-length 802.15.4 packet takes\nbout 4~milliseconds (4000~microseconds), a time granularity of\n320~microseconds is needed, corresponding to one backoff period (BP).\n\nTo address this problem, Contiki provides a real-time feature,\n\\texttt{rtimer}, which allows to bypass the kernel scheduler and use\na hardware timer to trigger execution of user-defined functions. However,\nit has very severe limitations:\n\n\\begin{itemize}\n\n\\item only one instance of \\texttt{rtimer} is available, thus only one\nreal-time event can be scheduled or executed at any time; this limitation\nforbids development of advanced real-time software---like high-performance\nMAC~\/ RDC protocols---or at least makes it very hard;\n\n\\item moreover, it is unsafe to execute from \\texttt{rtimer}, even\nindirectly, most of the Contiki basic functions (i.e.: kernel, network\nstack, etc.), because these functions are not designed to handle pre-emption.\nContiki is indeed based on cooperative multithreading, whereas the\n\\texttt{rtimer} mechanism seems like a ``independent feature'', coming\nwith its own paradigm.\nOnly a precise set of functions known as ``interrupt-safe'' (like\n\\texttt{process\\_poll()}) can be safely invoked from \\texttt{rtimer},\nusing other parts of Contiki's meaning almost certainly crash or\nunpredictable behaviour. This restriction practically makes it very\ndifficult to write Contiki extensions (like network stack layer drivers)\nusing \\texttt{rtimer}.\n\n\n\\end{itemize}\n\nAlso note that this cooperative scheduler is designed to manage a specific\nkind of tasks: the \\emph{protothreads}. This solution allows to manage\ndifferent threads of execution, without needing each of them to have\nits own separate stack \\cite{Protothreads}. The great advantage of\nthis mechanism is the ability to use an unique stack, thus greatly\nreducing the needed amount of RAM for the system. The trade-off is\nthat one must be careful when using certain C constructs (i.e.:\nit is impossible to use the \\texttt{switch} statement in\nsome parts of programs that use protothreads).\n\nFor all these reasons, we were unable to use Contiki OS to develop and\nimplement our high-performance MAC\/RDC protocols. We definitely needed\nan OS with efficient real-time features and event handling mechanism.\n\n\\subsection{Other options}\n\nThere are other, less used OSes designed for the WSN\/IoT domain, but none\nof them fulfilled our requirements, for the following reasons:\n\\begin{description}\n\n\\item[SOS] \\cite{SOS} This system's development has been cancelled since november\n           2008; its authors explicitly recommend on their website to\n           ``consider one of the more actively supported alternatives''.\n\n\\item[Lorien] \\cite{LorienOS} While its component-oriented approach is\n              interesting, this system seems does not seem very widespread. It is currently available for only \n              one hardware platform (TelosB\/SkyMote) which seriously\n              limits the portability we can expect from using an OS. \n              Moreover, its development seems to have slowed down quite\n              a bit, since the latest available Lorien release was published\n              in july 2011, while the latest commit in the project's\n              SourceForge repository (r46) dates back to january 2013.\n\n\\item[Mantis] \\cite{MantisOS} While this project claims to be Open Source,\n              the project has made, on its SourceForge web site, no public\n              release, and the access to the source repository \n              (\\texttt{http:\/\/mantis.cs.colorado.edu\/viewcvs\/}) seems\n              to stall. Moreover, reading the project's main web page\n              shows us that the last posted news item mentions a first beta\n              to be released in 2007. The last publications about\n              Mantis OS also seems to be in 2007. All of these elements \n              tend to indicate that this project is abandoned\\ldots\n\n\\item[LiteOS] \\cite{LiteOS} This system offers very interesting features,\n              especially the ability to update the nodes firmwares over the wireless,\n              as well as the built-in hierarchical file system. Unfortunately,\n              it is currently only available on IRIS\/MicaZ platforms,\n              and requires AVR Studio for programming (which imposes\n              Microsoft Windows as a development platform). This\n              greatly hinders portability, since LiteOS is clearly strongly\n              tied to the AVR microcontroller architecture.\n\n\\item[MansOS] \\cite{MansOS} This system is very recent and offers many\n              interesting features, like optional preemptive multitasking,\n              a network stack, runtime reprogramming, and a scripting\n              language. It is available on two MCU architectures: AVR and\n              MSP430 (but not ARM). However, none of the real-time features\n              we wanted seems to be available: e.g. only software timers with\n              a 1~millisecond resolution are available.\n\n\\end{description}\nIn any case, none of the alternative OSes cited hereabove offer the real-time\nfeatures we were looking for.\n\n\\bigskip\n\nOn the other hand, ``bare-metal'' programming is also unacceptable for us:\nit would mean sacrificing portability and multitasking; and we would also\nneed to redevelop many tools and APIs to make application programming\neven remotely practical enough for third-party developers who would\nwant to use our protocols.\n\n\\bigskip\n\nWe also envisioned to use an established real-time OS (RTOS) as a base\nfor our works. The current reference when it comes to open-source RTOS is\n\\emph{FreeRTOS} (\\texttt{http:\/\/www.freertos.org\/}). It is a robust, mature\nand widely used OS. Its codebase consists in clean and well-documented\nstandard C language. However, it offers only core features, and doesn't\nprovide any network subsystem at all. Redeveloping a whole network stack\nfrom scratch would have been too time-consuming.\n(Network extensions exist for FreeRTOS, but they are either immature,\nor very limited, or proprietary and commercial software; and most of them\nare tied to a peculiar piece of hardware, thus ruining\nthe portability advantage offered by the OS.)\n\n\\subsection{Summary: Wanted Features}\n\nTo summarize the issue, what we required is an OS that:\n\\begin{itemize}\n\\item is adapted to the limitations of the deeply-embedded MCUs that\n      constitute the core of WSN\/IoT motes;\n\\item provides real-time features powerful enough to support the\n      development of advanced, high-performance MAC~\/ RDC protocols;\n\\item includes a network stack (even a basic one) adapted to wireless\n      communication on 802.15.4 radio medium.\n\\end{itemize}\nHowever, none of the established OSes commonly used either in the IoT domain\n(TinyOS, Contiki) nor in the larger spectrum of RTOS (FreeRTOS)\ncould match our needs.\n\n\n\\section{\\uppercase{The RIOT Operating System}}\n\nConsequently, we focused our interest on \\emph{RIOT OS} \\cite{RIOT}.\n\nThis new system---first released in 2013---is also open-source and\nspecialized in the domain of low-power, embedded wireless sensors.\nIt offers many interesting features, that we will now describe.\n\nIt provides the basic benefits of an OS: portability (it has been ported\nto many devices powered by ARM, MSP430, and---more recently---AVR\nmicrocontrollers) and a comprehensive set of features, including\na network stack.\n\nMoreover, it offers key features that are otherwise yet unknown in\nthe WSN\/IoT domain:\n\n\\begin{itemize}\n\n\\item an efficient, interrupt-driven, tickless \\emph{micro-kernel};\n\n\\item that kernel includes a priority-aware task scheduler, providing\n      \\emph{pre-emptive multitasking};\n\n\\item a highly efficient use of \\emph{hardware timers}: all of them can be\n      used concurrently (especially since the kernel is tickless), offering\n      the ability to schedule actions with high granularity; on low-end\n      devices, based on MSP430 architecture, events can be scheduled\n      with a resolution of 32~microseconds;\n\n\\item RIOT is entirely written in \\emph{standard C language}; but unlike\n      Contiki, there are no restrictions on usable constructs (i.e.: like\n      those introduced by the protothreads mechanism);\n\n\\item a clean and \\emph{modular design}, that makes development with and\n      \\emph{into} the system itself easier and more productive.\n\n\\end{itemize}\n\nThe first three features listed hereabove make RIOT a full-fledged\n\\emph{real-time} operating system.\n\nWe also believe that the tickless kernel and the optimal use of hardware\ntimers should make RIOT OS a very suited software platform to optimize energy\nconsumption on battery-powered, MCU-based devices.\n\nA drawback of RIOT, compared to TinyOS or Contiki, is its higher memory\nfootprint: the full network stack (from PHY driver up to RPL routing with\n\\mbox{6LoWPAN} and MAC~\/ RDC layers) cannot be compiled for Sky\/TelosB\nbecause of overflowing memory space. Right now, constrained devices like\nMSP430-based motes are limited to the role of what the 802.15.4 standard\ncalls \\emph{Reduced Function Devices (RFD)}, the role of \\emph{Full\nFunction Devices (FFD)} being reserved to more powerful motes (i.e.:\nbased on ARM microcontrollers).\n\nHowever, we also note that, thanks to its modular architecture, the RIOT\nkernel, compiled with only PHY and MAC~\/ RDC layers, is actually lightweight\nand consumes little memory. We consequently believe that the current\nsituation will improve with the maturation of higher layers of RIOT network\nstack, and that in the future more constrained devices could also be used\nas FFD with RIOT OS.\n\n\\medskip\n\nWhen we began to work with RIOT, it also had two other issues: the MSP430\nversions were not stable enough to make real use of the platform; and\nbeyond basic CSMA\/CA, no work related to the MAC~\/ RDC layer had been\ndone on that system. This is where our contributions fit in.\n\n\n\\section{\\uppercase{Our contributions}}\n\nFor our work, we use---as our main hardware platform---IoT motes built\naround MSP430 microcontrollers.\n\nMSP430 is a microcontroller (MCU) architecture from Texas Instruments,\noffering very low-power consumption, cheap price, and good performance thanks\nto a custom 16-bit RISC design. This architecture is very common in IoT motes.\nIt is also very well supported, especially by the Cooja simulator\n\\cite{Cooja}, which makes simulations of network scenarios---especially\nwith many devices---much easier to design and test.\n\nRIOT OS has historically been developed first on legacy ARM devices\n(ARM7TDMI-based MCUs), then ported on more recent microcontrollers\n(ARM Cortex-M) and other architectures (MSP430 then AVR). However,\nthe MSP430 port was, before we improved it, still not as ``polished''\nas ARM code and thus prone to crash.\n\nOur contribution can be summarized in the following points:\n\n\\begin{enumerate}\n\n\\item analysis of current OSes (TinyOS, Contiki, etc.) limitations,\n      and why they are incompatible with development of real-time\n      extensions like advanced MAC~\/ RDC protocols;\n\n\\item add debugging features to the RIOT OS kernel, more precisely\n      a mechanism to handle fatal errors: crashed systems can be\n      ``frozen'' to facilitate debugging during development; or,\n      in production, can be made to reboot immediately, thus reducing\n      unavailability of a RIOT-running device to a minimum;\n\n\\item port RIOT OS to a production-ready, MSP430-based device:\n      the Zolertia Z1 mote (already supoorted by Contiki,\n      and used in real-world scenarios running that OS);\n\n\\item debug the MSP430-specific portion of RIOT OS---more specifically:\n      the hardware abstraction layer (HAL) of the task scheduler---making\n      RIOT OS robust and production-ready on MSP430-based devices.\\\\\n      Note that all of  these contributions have been reviewed by RIOT's\n      development team and integrated into the ``master'' branch of RIOT OS'\n      Github repository (i.e.: they are now part of the standard code base of\n      the system).\n\n\\item running on MSP430-based devices also allows RIOT OS applications\n      to be simulated with the Cooja simulator; this greatly improves\n      speed and ease of development.\n\n\\item thanks to these achievements, we now have a robust and full-featured\n      software platform offering all the features needed to develop\n      high-performance MAC\/RDC protocols---such as all of the time-slotted\n      protocols.\n\n\\end{enumerate}\n\nAs a proof of concept of this last statement, we have implemented one\nof our own designs, and obtained very promising results, shown in\nthe next section.\n\n\n\\section{\\uppercase{Use Case: implementing the S-CoSenS RDC protocol}}\n\n\\subsection{The S-CoSenS Protocol}\n\nThe first protocol we wanted to implement is S-CoSenS \\cite{TheseBNefzi},\nwhich is designed to work on top of the IEEE 802.15.4 physical and MAC\n(i.e.: CSMA\/CA) layers.\n\nIt is an evolution of the already published CoSenS protocol \\cite{CosensConf}:\nit adds to the latter a sleeping period for energy saving.\nThus, the basic principle of S-CoSenS is to delay the forwarding (routing)\nof received packets, by dividing the radio duty cycle in three periods:\na sleeping period (SP), a waiting period (WP) where the radio medium\nis listened by routers for collecting incoming 802.15.4 packets, and\nfinally a burst transmission period (TP) for emitting adequately\nthe packets enqueued during WP.\n\nThe main advantage of S-CoSenS is its ability to adapt dynamically to the\nwireless network throughput at runtime, by calculating for each radio duty\ncycle the length of SP and WP, according to the number of relayed\npackets during previous cycles. Note that the set of the SP and the WP\nof a same cycle is named \\emph{subframe}; it is the part of a S-CoSenS\ncycle whose length is computed and known \\textit{a priori}; on the contrary,\nTP duration is always unknown up to its very beginning, because it depends\non the amount of data successfully received during the WP that precedes it.\n\nThe computation of WP duration follows a ``sliding average'' algorithm,\nwhere WP duration for each duty cycle is computed from the average\nof previous cycles as:\n\\begin{eqnarray*}\n&&\n\\overline{\\mathrm{WP}_{n}} = \\alpha \\cdot \\overline{\\mathrm{WP}_{n-1}}\n                + (1 - \\alpha) \\cdot \\mathrm{WP}_{n-1}\n\\\\ &&\n\\mathrm{WP}_{n} = \\max ( \\mathrm{WP}_{min},\n                  \\min ( \\overline{\\mathrm{WP}_{n}}, \\mathrm{WP}_{max} ) )\n\\end{eqnarray*}\nwhere $\\overline{\\mathrm{WP}_{n}}$ and $\\overline{\\mathrm{WP}_{n-1}}$\nare respectively the average WP length at $n^{\\mathrm{th}}$ and\n$(n-1)^{\\mathrm{th}}$ cycle, while $\\mathrm{WP}_{n}$ and $\\mathrm{WP}_{n-1}$\nare the actual length of respectively the $n^{\\mathrm{th}}$ and\n$(n-1)^{\\mathrm{th}}$ cycles; $\\alpha$ is a parameter between 0 and 1\nrepresenting the relative weight of the history in the computation,\nand $\\mathrm{WP}_{min}$ and $\\mathrm{WP}_{max}$ are high and low limits\nimposed by the programmer to the WP duration.\n\nThe length of the whole subframe being a parameter given at compilation time,\nSP duration is simply computed by subtracting the calculated duration of WP\nfrom the subframe duration for every cycle.\n\nThe local synchronization between a S-CoSenS router and its leaf nodes\nis done thanks to a beacon packet, that is broadcasted by the router at\nthe beginning of each cycle. This beacon contains the duration\n(in microseconds) of the SP and WP for the currently\nbeginning cycle.\n\nThe whole S-CoSenS cycle workflow for a router is summarized in figure\n\\ref{FigSCosensDutyCycle} hereafter.\n\n\\begin{figure}[!ht]\n\\centering\n\\begin{tikzpicture}[>=latex]\n\\fill[black] (0cm, -0.25cm) rectangle +(0.2cm, 0.5cm);\n\\draw[->,thick] (0.1cm, 0.25cm) -- +(0, 0.5cm);\n\\draw (0.1cm, 1.3cm) node {Beacon};\n\\draw[anchor=west] (-0.6cm, 0.9cm) node {(broadcasted)};\n\\draw[thick] (0cm, -0.25cm) -- +(0, 0.5cm);\n\\foreach \\x in {1,2,3,4,5,6}\n{\n  \\fill[lightgray] (0.2cm + \\x * 0.25cm, -0.25cm) rectangle +(0.05cm, 0.5cm);\n}\n\\draw (1.1cm, 0) node {\\textbf{SP}};\n\\draw[thick] (2cm, -0.25cm) -- +(0, 0.5cm);\n\\fill[lightgray] (2cm, -0.25cm) rectangle +(2cm, 0.5cm);\n\\draw (3cm, 0) node {\\textbf{WP}};\n\\draw[thick] (4cm, -0.25cm) -- +(0, 0.5cm);\n\\fill[lightgray] (4cm, -0.25cm) rectangle +(2cm, 0.5cm);\n\\draw (5cm, 0) node {\\textbf{TP}};\n\\draw[thick] (6cm, -0.25cm) -- +(0, 0.5cm);\n\\draw[->] (-0.5cm,  0.25cm) -- +(7cm, 0);\n\\draw[->] (-0.5cm, -0.25cm) -- +(7cm, 0);\n\\draw[->,thick] (2.5cm, 0.75cm) -- +(0, -0.5cm);\n\\draw (2.5cm, 1cm) node {P1};\n\\draw[->,thick] (3cm, 0.75cm) -- +(0, -0.5cm);\n\\draw (3cm, 1cm) node {P2};\n\\draw[->,thick] (3.5cm, 0.75cm) -- +(0, -0.5cm);\n\\draw (3.5cm, 1cm) node {P3};\n\\draw[->,thick] (4.5cm, 0.25cm) -- +(0, 0.5cm);\n\\draw (4.5cm, 1cm) node {P1};\n\\draw[->,thick] (5cm, 0.25cm) -- +(0, 0.5cm);\n\\draw (5cm, 1cm) node {P2};\n\\draw[->,thick] (5.5cm, 0.25cm) -- +(0, 0.5cm);\n\\draw (5.5cm, 1cm) node {P3};\n\\draw (0cm, -0.5cm)  .. controls +(0, -0.25cm) .. +(1cm, -0.25cm);\n\\draw (1cm, -0.75cm) .. controls +(1cm, 0)     .. +(1cm, -0.25cm);\n\\draw (2cm, -1cm)    .. controls +(0, 0.25cm)  .. +(1cm, 0.25cm);\n\\draw (3cm, -0.75cm) .. controls +(1cm, 0)     .. +(1cm, 0.25cm);\n\\draw (2cm, -1.25cm) node {\\textbf{Subframe}};\n\\draw (0cm, -1.5cm)    .. controls +(0, -0.25cm) .. +(1.5cm, -0.25cm);\n\\draw (1.5cm, -1.75cm) .. controls +(1.5cm, 0)   .. +(1.5cm, -0.25cm);\n\\draw (3cm, -2cm)      .. controls +(0, 0.25cm)  .. +(1.5cm, 0.25cm);\n\\draw (4.5cm, -1.75cm) .. controls +(1.5cm, 0)   .. +(1.5cm, 0.25cm);\n\\end{tikzpicture}\n\\caption{A typical S-CoSenS router cycle.\\\\\n         The gray strips in the SP represents the short wake-up-and-listen\n         periods used for inter-router communication.}\n\\label{FigSCosensDutyCycle}\n\\end{figure}\n\nAn interesting property of S-CoSenS is that leaf (i.e.: non-router) nodes\nalways have their radio transceiver offline, except when they have packets\nto send. When a data packet is generated on a leaf node, the latter wakes up\nits radio transceiver, listens and waits to the first beacon emitted by\nan S-CoSenS router, then sends its packet using CSMA\/CA at the beginning\nof the WP described in the beacon it received. A leaf node will put its\ntransceiver offline during the delay between the beacon and that WP\n(that is: the SP of the router that emitted the received beacon), and\nwill go back to sleep mode once its packet is transmitted.\nAll of this procedure is shown in figure \\ref{FigSCoSenSPktTx}.\n\n\\begin{figure}[!h]\n\\centering\n\\begin{tikzpicture}[>=latex]\n\\draw (-0.5cm, 0) node {\\large \\textit{R}};\n\\draw[thick] (1cm, -0.25cm) -- +(0, 0.5cm);\n\\draw (2cm, 0) node {\\textbf{SP}};\n\\draw[thick] (3cm, -0.25cm) -- +(0, 0.5cm);\n\\fill[lightgray] (3cm, -0.25cm) rectangle +(2cm, 0.5cm);\n\\draw (4cm, 0) node {\\textbf{WP}};\n\\draw[thick] (5cm, -0.25cm) -- +(0, 0.5cm);\n\\fill[lightgray] (5cm, -0.25cm) rectangle +(0.5cm, 0.5cm);\n\\draw (5.25cm, -0.5cm) node {\\textbf{TP}};\n\\draw[thick] (5.5cm, -0.25cm) -- +(0, 0.5cm);\n\\draw[->] (-0.5cm,  0.25cm) -- +(6.5cm, 0);\n\\draw[->] (-0.5cm, -0.25cm) -- +(6.5cm, 0);\n\\draw (-0.5cm, -1.5cm) node {\\large \\textit{LN}};\n\\fill[gray] (0cm, -1.25cm) rectangle +(1.3cm, -0.5cm);\n\\fill[gray] (2.9cm, -1.25cm) rectangle +(0.5cm, -0.5cm);\n\\fill[black] (1cm, -0.25cm) rectangle +(0.2cm, 0.5cm);\n\\draw[->,thick] (1.1cm, 0.25cm) -- +(0, -1.5cm);\n\\draw[anchor=east] (1cm, -0.75cm) node {Beacon};\n\\fill[black] (1cm, -1.25cm) rectangle +(0.2cm, -0.5cm);\n\\draw[->,very thick] (0cm, -2.5cm) -- +(0, 0.75cm);\n\\draw[anchor=west] (0cm, -2.5cm)\n     node {\\footnotesize \\textbf{packet arrival}};\n\\fill[black] (3.1cm, -1.25cm) rectangle +(0.2cm, -0.5cm);\n\\draw[->,thick] (3.2cm, -1.25cm) -- +(0, 1cm);\n\\draw[anchor=west] (3.2cm, -0.75cm) node {P1};\n\\fill[black] (3.1cm, -0.25cm) rectangle +(0.2cm, 0.5cm);\n\\fill[black] (5.1cm, -0.25cm) rectangle +(0.2cm, 0.5cm);\n\\draw[->,thick] (5.2cm, 0.25cm) -- +(0, 0.5cm);\n\\draw (5.2cm, 1cm) node {P1};\n\\draw[->] (-0.5cm, -1.25cm) -- +(6.5cm, 0);\n\\draw[->] (-0.5cm, -1.75cm) -- +(6.5cm, 0);\n\\end{tikzpicture}\n\\caption{A typical transmission of a data packet with the S-CoSenS protocol\n         between a leaf node and a router.}\n\\label{FigSCoSenSPktTx}\n\\end{figure}\n\nWe thus need to synchronize with enough accuracy different devices (that\ncan be based on different hardware platforms) on cycles whose periods\nare dynamically calculated at runtime, with resolution that needs to be\nin the sub-millisecond range. This is where RIOT OS advanced real-time\nfeatures really shine, while the other comparable OSes are\nfor that purpose definitely lacking.\n\n\\subsection{Simulations and Synchronization Accuracy}\n\nWe have implemented S-CoSenS under RIOT, and made first tests by performing\nsimulations---with Cooja---of a 802.15.4 PAN (Personal Area Network)\nconstituted of a router, and ten motes acting as ``leaf nodes''.\nThe ten nodes regularly send data packets to the router, that retransmits\nthese data packets to a nearby ``sink'' device. Both the router and the ten\nnodes use exclusively the S-CoSenS RDC\/MAC protocol. This is summarized\nin figure \\ref{FigPANtest}.\n\n\\begin{figure}[!h]\n\\centering\n\\begin{tikzpicture}[>=latex]\n\\draw (0, 1cm) circle (0.25cm); \\draw (0, 1cm) node {S};\n\\draw[->,thick] (0, 0.25cm) -- (0, 0.75cm);\n\\draw (0, 0) circle (0.25cm); \\draw (0, 0) node {R};\n\\foreach \\x in {6,7,8,9,10}\n{\n  \\fill[white] (\\x * 1cm - 8cm, -1.75cm) circle (0.25cm);\n  \\draw (\\x * 1cm - 8cm, -1.75cm) circle (0.25cm);\n  \\draw (\\x * 1cm - 8cm, -1.75cm) node {\\x};\n \n  \\draw[->,thick] (\\x * 1cm - 8cm, -1.5cm)\n                  -- (\\x * 0.02cm - 0.16cm, -0.25cm);\n}\n\\foreach \\x in {1,2,3,4,5}\n{\n  \\fill[white] (\\x * 1cm - 3cm, -1cm) circle (0.25cm);\n  \\draw (\\x * 1cm - 3cm, -1cm) circle (0.25cm);\n  \\draw (\\x * 1cm - 3cm, -1cm) node {\\x};\n \n  \\draw[->,thick] (\\x * 1cm - 3cm, -0.75cm)\n                  -- (\\x * 0.05cm - 0.15cm, -0.25cm);\n}\n\\end{tikzpicture}\n\\caption{Functional schema of our virtual test PAN.}\n\\label{FigPANtest}\n\\end{figure}\n\nOur first tests clearly show an excellent synchronization between the\nleaf nodes and the router, thanks to the time resolution offered by RIOT OS\nevent management system (especially the availability of many hardware\ntimers for direct use). This can be seen in the screenshot of our\nsimulation in Cooja, shown in figure \\ref{Screenshot}. For readability,\nthe central portion of the timeline window of that screenshot (delimited\nby a thick yellow rectangle) is zoomed on in figure \\ref{ZoomTimeline}.\n\n\\begin{figure*}[ptb]\n\\centering\n\\includegraphics[width=15.75cm]{S-CoSenS-Cooja10.png}\n\\caption{Screenshot of our test simulation in Cooja. \n(Despite the window title mentioning Contiki, the simulated application\n is indeed running on RIOT OS.)}\n\\label{Screenshot}\n\\end{figure*}\n\n\\begin{figure*}[pbt]\n\\centering\n\\includegraphics{S-CoSenS-Cooja10-Timeline.png}\n\\caption{Zoom on the central part of the timeline of our simulation.}\n\\label{ZoomTimeline}\n\\end{figure*}\n\nOn figure \\ref{ZoomTimeline}, the numbers on the left side are motes'\nnumerical IDs: the router has ID number \\textsf{1}, while the leaf nodes\nhave IDs \\textsf{2} to \\textsf{11}. Grey bars represent radio transceiver\nbeing online for a given mote; blue bars represent packet emission, and green\nbars correct packet reception, while red bars represent collision (when\ntwo or more devices emit data concurrently) and thus reception of\nundecipherable radio signals.\n\nFigure \\ref{ZoomTimeline} represents a short amount of time (around\n100~milliseconds), representing the end of a duty cycle of the router:\nthe first 20~milliseconds are the end of SP, and 80 remaining milliseconds\nthe WP, then the beginning of a new duty cycle (the TP has been disabled\nin our simulation). \n\nIn our example, four nodes have data to transmit to the router: the motes\nnumber \\textsf{3}, \\textsf{5}, \\textsf{9}, and \\textsf{10}; the other nodes\n(\\textsf{2}, \\textsf{4}, \\textsf{6}, \\textsf{7}, \\textsf{8}, and \\textsf{11})\nare preparing to transmit a packet in the next duty cycle.\n\nAt the instant marked by the first yellow arrow (in the top left of figure\n\\ref{ZoomTimeline}), the SP ends and the router activates its radio\ntransceiver to enter WP. Note how the four nodes that are to send packets\n(\\textsf{3}, \\textsf{5}, \\textsf{9}, and \\textsf{10}) do also activate their\nradio transceivers \\emph{precisely} at the same instant: this is thanks to\nRIOT OS precise real-time mechanism (based on hardware timers), that allows\nto the different nodes to precisely synchronize on the timing values\ntransmitted in the previous beacon packet. Thanks also to that mechanism,\nthe nodes are able to keep both their radio transceiver \\emph{and} their\nMCU in low-power mode, since RIOT OS kernel is interrupt-driven.\n\nDuring the waiting period, we also see that several collisions occur; they\nare resolved by the S-CoSenS protocol by forcing motes to wait a random\nduration before re-emitting a packet in case of conflict. In our example,\nour four motes can finally transmit their packet to the router in that\norder: \\textsf{3} (after a first collision), \\textsf{5}, \\textsf{10} (after\ntwo other collisions), and finally \\textsf{9}. Note that every time the\nrouter (device number \\textsf{1}) successfully receives a packet, an\nacknowledgement is sent back to emitter: see the very thin blue bars that\nfollow each green bar on the first line.\n\nFinally, at the instant marked by the second yellow arrow (in the top right\nof figure \\ref{ZoomTimeline}), WP ends and a new duty cycle begins.\nConsequently, the router broadcasts a beacon packet containing PAN timing and\nsynchronization data to all of the ten nodes. We can see that all of the\nsix nodes waiting to transmit (\\textsf{2}, \\textsf{4}, \\textsf{6}, \\textsf{7},\n\\textsf{8}, and \\textsf{11}) go idle after receiving this beacon (beacon\npackets are broadcasted and thus not to be acknowledged): they go\ninto low-power mode (both at radio transceiver and MCU level), and will\ntake advantage of RIOT real-time features to wake up precisely when\nthe router goes back into WP mode and is ready to receive their\npackets.\n\n\\subsection{Performance Evaluation: Preliminary Results}\n\nWe will now present the first, preliminary results we obtained through the\nsimulations we described hereabove.\n\nImportant: note that \\emph{we evaluate here the implementations}, and not the\nintrinsic advantages or weaknesses of the protocols themselves.\n\nWe have first focused on QoS results, by computing Packet Reception Rates\nand end-to-end delays between the various leaf nodes and the sink of the test\nPAN presented earlier in figure \\ref{FigPANtest}, to evaluate the quality\nof the transmissions allowed by using both of the protocols.\n\nFor these first tests, we used default parameters for both RDC protocols\n(ContikiMAC and S-CoSenS), only pushing the CSMA\/CA MAC layer of Contiki\nto make up to 8 attempts for transmitting a same packet, so as to put it\non par with our implementation on RIOT OS. We have otherwise not yet\ntried to tweak the various parameters offered by both the RDC protocols\nto optimize results. This will be the subject of our next experiences.\n\n\\subsubsection{Packet Reception Rates (PRR)}\n\nThe result obtained for PRR using both protocols are shown in figure\n\\ref{FigPRRresults} as well as table \\ref{TblPRRresults}.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=7.5cm]{PRRgraph.png}\n  \\caption{PRR results for both ContikiMAC and S-CoSenS RDC protocols,\n           using default values for parameters.}\n  \\label{FigPRRresults}\n\\end{figure}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{|r|r|r|}\n\\hline\n PAI \\textbackslash\\ Protocol & ContikiMAC & S-CoSenS \\\\\n\\hline\n 1500 ms & 49.70\\% & 98.10\\% \\\\\n 1000 ms & 32.82\\% & 96.90\\% \\\\\n  500 ms & 14.44\\% & 89.44\\% \\\\\n  100 ms &  0.64\\% & 25.80\\% \\\\\n\\hline\n\\end{tabular}\n\\caption{PRR results for both ContikiMAC and S-CoSenS RDC protocols,\n         using default values for parameters.}\n\\label{TblPRRresults}\n\\end{table}\n\nThe advantage of S-CoSenS as shown on the figure is clear and significant\nwhatever the packet arrival interval constated. Excepted for the ``extreme''\nscenario corresponding to an over-saturation of the radio channel, S-CoSenS\nachieve an excellent PRR ($\\gtrapprox 90\\%$), while ContikiMAC's PRR\nis always $\\lessapprox 50\\%$.\n\n\\subsubsection{End-To-End Transmission Delays}\n\nThe result obtained for PRR using both protocols are shown in figure\n\\ref{FigDelaysResults} and table \\ref{TblDelaysResults}.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=7.5cm]{DelaysGraph.png}\n  \\caption{End-to-end delays results for both ContikiMAC and S-CoSenS RDC\n           protocols, using default values for parameters; note that\n           vertical axis is drawn with logarithmic scale.}\n  \\label{FigDelaysResults}\n\\end{figure}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{|r|r|r|}\n\\hline\n PAI \\textbackslash\\ Protocol & ContikiMAC & S-CoSenS \\\\\n\\hline\n 1500 ms &  3579 ms & 108 ms \\\\\n 1000 ms &  4093 ms & 108 ms \\\\\n  500 ms &  6452 ms & 126 ms \\\\\n  100 ms & 12913 ms & 168 ms \\\\\n\\hline\n\\end{tabular}\n\\caption{End-to-end delays results for both ContikiMAC and S-CoSenS RDC\n         protocols, using default values for parameters.}\n\\label{TblDelaysResults}\n\\end{table}\n\nS-CoSenS has here also clearly the upper hand, so much that we had to use\nlogarithmic scale for the vertical axis to keep figure \\ref{FigDelaysResults}\neasily readable. The advantage of S-CoSenS is valid whatever the packet\narrival interval, our protocol being able to keep delay below an acceptable\nlimit (in the magnitude of hundreds of milliseconds), while ContikiMAC\ndelays rocket up to tens of seconds when network load increases.\n\n\\subsubsection{Summary: QoS Considerations}\n\nWhile these are only preliminary results, it seems that being able to\nleverage real-time features is clearly a significant advantage when designing\nand implementing MAC\/RDC protocols, at least when it comes to QoS results.\n\n\n\n\\section{\\uppercase{Future Works and Conclusion}}\n\nWe plan, in a near future:\n\n\\begin{itemize}\n\n\\item to bring new contributions to the RIOT project: we are especially\n      interested in the portability that the RIOT solution offers us;\n      this OS is indeed actively ported on many devices based on powerful\n      microcrontrollers based on ARM Cortex-M architecture (especially\n      Cortex-M3 and Cortex-M4), and we intend to help in this porting\n      effort, especially on high-end IoT motes we seek to use in our\n      works (e.g.: as advanced FFD nodes with full network stack,\n      or routers);\n\n\\item to use the power of this OS to further advance our work on MAC\/RDC\n      protocols; more precisely, we are implementing other innovative\n      MAC\/RDC protocols---such as iQueue-MAC \\cite{iQueueMAC}---under RIOT,\n      taking advantage of its high-resolution real-time features to obtain\n      excellent performance, optimal energy consumption, and out-of-the-box\n      portability.\n\n\\end{itemize}\n\nRIOT is a powerful real-time operating system, adapted to the limitations\nof deeply embedded hardware microcontrollers, while offering state-of-the-art\ntechniques (preemptive multitasking, tickless scheduler, optimal use\nof hardware timers) that---we believe---makes it one of the most\nsuitable OSes for the embedded and real-time world.\n\nWhile we weren't able to accurately quantize energy consumption\nyet, we can reasonably think that lowering activity of MCU and radio\ntransceiver will significantly reduce the energy consumption of devices\nrunning RIOT OS. This will be the subject of some of our future\nresearch works.\n\n\\bigskip\n\nCurrently, RIOT OS supports high-level IoT protocols (6LoWPAN\/IPv6, RPL,\nTCP, UDP, etc.). However, it still lacks high-performance MAC~\/ RDC layer\nprotocols.\n\nThrough this work, we have shown that RIOT OS is also suitable for\nimplementing high-performance MAC~\/ RDC protocols, thanks to its real-time\nfeatures (especially hardware timers management).\n\nMoreover, we have improved the robustness of the existing ports of RIOT OS\non MSP430, making it a suitable software platform for tiny motes and devices.\n\n\n\n\n\n\n\\vfill\n\\bibliographystyle{apalike}\n{\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\n{\\bf \nWe derive explicit expressions for dynamical correlations of the field and density operators in the Lieb-Liniger model, within an arbitrary eigenstate with a small particle density ${\\cal D}$. They are valid for all space and time and any interaction strength $c>0$, and are the leading order of an expansion in ${\\cal D}$. This expansion is obtained by writing the correlation functions as sums over form factors when formally decomposed into partial fractions.\n\n}\n\n\\renewcommand\\Affilfont{\\fontsize{9}{10.8}\\itshape}\n\n \n\\tableofcontents\n\\section{Introduction}\nThe Lieb-Liniger model is a key paradigm of many-particle systems\\cite{LiebLiniger63a,Brezin64,korepin}. In the repulsive regime, it is  considered as one of the simplest interacting quantum integrable model, for having the simplifying feature of involving only real rapidities\\cite{Lieb63,YangYang69}. The main objects of interest are the correlation functions of local observables in the thermodynamic limit, that are the macroscopic output of the model resulting from the short-range interactions between the bosons. Moreover their Fourier transform is directly measurable in cold-atoms experiments \\cite{naegerl15,Bouchoule16,Fabbri15}. Although exactly solvable, the computation of correlation functions in quantum integrable models is a notoriously difficult problem. In particular the computation of dynamical correlations for an arbitrary interaction strength $c$ and within arbitrary eigenstate is an open problem.\n\nSpecial cases of this problem were studied and solved in the past. The first of these special cases is the impenetrable bosons limit $c\\to\\infty$, that can be reformulated in terms of free fermions \\cite{girardeau}. Here, both because of the free fermionic nature of the model and the simple structure of the form factor\\footnote{The Transverse Field Ising Model, although also equivalent to free fermions, cannot be fully treated along the same lines because of its more complex form factors \\cite{GFE20}.}, the Lehmann representation of dynamical correlations can be fully resummed into a Fredholm determinant\\cite{KS90,slavnov90,izergin87,kojima97} and its asymptotics extracted with differential equations \\cite{IIKS90,IIKV92}. Another important special case is the ground state static correlation at finite $c$, that was treated within the Algebraic Bethe ansatz framework\\cite{kitanineetalg} and led to the first ab initio calculation of the critical exponents previously predicted by Conformal Field Theory and Luttinger Liquid theory\\cite{BIK86,IKR87,haldane,cazalilla}. This approach was then generalized to static correlations within arbitrary eigenstates \\cite{kozlowskimailletslvanov,kozlowskimailletslvanov2}. The particular case of static correlators within thermal eigenstates was also studied with quantum transfer matrix methods\\cite{suzuki85,klumper92,patuklumper}. The full asymptotics of ground state dynamical correlations were derived in \\cite{kitanineetcformfactor,kozlowskimaillet,kozlowski4} from form factor expansions, confirming predictions of Non-Linear Luttinger Liquid theory\\cite{IG08,PAW09,ISG12,P12,shashipanfilcaux,Price17}.\n\n\nProgress on the general case of dynamical correlations within arbitrary states at finite $c$ has been much more limited. This calculation is very different from the special cases of zero-temperature or static cases and poses important problems. The successful methods for static correlations such as the algebraic Bethe ansatz approach or the quantum transfer matrix methods do not apply to the dynamical case, and the form factor expansion used to compute ground state dynamical correlations  relied on combinatorial identities that are usable for zero-entropy states only. This general case has however been studied through several approaches. Firstly, Generalized HydroDynamics (GHD) provide predictions for the leading asymptotics of the dynamical correlations of conserved charges such as the density within arbitrary macrostates\\cite{CADY16,BCDF16,Doyon18}. However the approach cannot a priori be applied to compute the next corrections, restricting the dynamical structure factor to small frequency and momentum, and does not apply neither to semi-local operators such as the field correlations. Secondly, numerical summations of dominant form factors proved very efficient and permitted to obtain numerical estimations of the dynamical structure factor on the full plane \\cite{cauxcalabreseslavnov,PC14}. On the Bethe-ansatz calculations side, one-point functions within arbitrary eigenstates were studied in \\cite{negrosmirovn,bastianellopiroli,bastianellopirolicalabrese}. An approach based on thermodynamic form factors was started \\cite{deNP15,deNP16,DNP18,panfil20,cortescuberopanfil1,cortescuberopanfil2}, but still involves non-integrable singularities and so requires a particular understanding of this feature. A regularized form factor expansion was derived for the XXZ spin chain in \\cite{kozlowski1}. In \\cite{granetessler20} was computed the full spectral sum at order $c^{-2}$ for all root densities and all momentum and frequency, involving one- and two-particle-hole excitations. It showed the necessity for a fine-tuned treatment of the non-integrable singularities that is crucial for e.g. detailed balance to be satisfied in thermal states.\n\n\nThe objective of this paper is to derive the full dynamical correlations for all space and time and all interaction strength $c$, in the limit where the particle density ${\\cal D}$ of the averaging state becomes small. This low-density limit is defined in terms of a partial fraction decomposition of the form factor, initially introduced in \\cite{GFE20} in a model that can be reformulated into free fermions. This decomposition naturally organizes the spectral sum as an expansion in the particle density ${\\cal D}$ of the averaging state, and the low-density limit is defined as the leading term of this expansion. \n\nThis result provides another limiting case where the initial problem becomes solvable, namely the computation of dynamical correlations for all space and time and arbitrary $c$ within finite-entropy macrostates. Moreover the framework also enables to compute the subleading corrections in the particle density, as was explicitly shown in \\cite{GFE20} for the Transverse Field Ising Model. The computation of these subleading corrections in the interacting case however comes with higher technical difficulties and should be the object of further work. Finally, this low-density limit calculation sheds light on the structure of the spectral sum and the nature of the states contributing in the thermodynamic limit.\n\nIn Section \\ref{sec1} we introduce the Lieb-Liniger model and recall known results on its form factors. In Section \\ref{ldd} we define what is meant by low-density limit of the dynamical correlations, taking the field correlations as an example. In Section \\ref{fieldsection} we compute the low-density limit of the field two-point function \\eqref{field}, and in Section \\ref{densitysection} the low-density limit of the density two-point function \\eqref{densityde}.\n\n\n\\section{\\texorpdfstring{Lieb-Liniger model}{Lg}}\n\\label{sec1}\n\\subsection {Definition}\n\nThe Lieb-Liniger model \\cite{LiebLiniger63a} is a non-relativistic\nquantum field theory model with Hamiltonian\n\\begin{equation}\nH=\\int_0^L dx\\left[-\\psi^\\dagger(x)\\frac{d^2}{dx^2}\\psi(x)+c\\psi^\\dagger(x)\\psi^\\dagger(x)\\psi(x)\\psi(x)\n\\right]\\,,\n\\label{HLL}\n\\end{equation}\nwhere the canonical Bose field $\\psi(x)$ satisfies equal-time\ncommutation relations\n\\begin{equation}\n[\\psi(x),\\psi^\\dagger(y)]=\\delta(x-y)\\,.\n\\end{equation}\nWe will impose periodic boundary\n  conditions.\nFor later convenience we define the time-$t$ evolved version of the field $\\psi(x,t)=e^{iHt}\\psi(x)e^{-iHt}$. We also define the density operator at position $x$\n\\begin{equation}\n\\sigma(x)=\\psi^\\dagger(x)\\psi(x)\\,,\n\\end{equation}\nand its time-$t$ evolved version $\\sigma(x,t)=e^{iHt}\\sigma(x)e^{-iHt}$.\n\n\\subsection {The Bethe ansatz solution}\n\\subsubsection{The spectrum}\nThe Lieb-Liniger model is solvable by the Bethe ansatz: an eigenstate $|\\pmb{\\lambda}\\rangle$ with $N$ bosons can be written as\n\\begin{equation}\n|\\pmb{\\lambda}\\rangle=B(\\lambda_1)...B(\\lambda_N)|0\\rangle\\,,\n\\end{equation}\nwith the $B(\\lambda)$'s some creation operators, $|0\\rangle$ the pseudo-vacuum and the $\\lambda_i$'s some rapidities that satisfy the following set of 'Bethe equations'\n\\begin{equation}\ne^{iL\\lambda_k}=\\prod_{\\substack{j=1\\\\j\\neq k}}^N\\frac{\\lambda_k-\\lambda_j+ic}{\\lambda_k-\\lambda_j-ic}\\,,\n\\quad k=1,\\dots, N.\n\\end{equation}\nThe\nenergy $E$ and the momentum $P$ of this state read \n\\begin{equation}\nE(\\pmb{\\lambda})=\\sum_{i=1}^N \\lambda_i^2\\,,\\qquad P(\\pmb{\\lambda})=\\sum_{i=1}^N\\lambda_i\\,.\n\\end{equation}\nIt is convenient to express the Bethe equations in logarithmic form\n\\begin{equation}\n\\label{belog}\n\\frac{\\lambda_k}{2\\pi}=\\frac{I_k}{L}-\\frac{1}{L}\\sum_{j=1}^N \\frac{1}{\\pi}\\arctan \\frac{\\lambda_k-\\lambda_j}{c}\\,,\n\\end{equation}\nwith $I_k$ an integer if $N$ is odd, a half-integer if $N$ is even. For $c>0$ all the solutions to this equation are real \\cite{korepin}. We will denote\n\\begin{equation}\n{\\cal D}=\\frac{N}{L}\\,,\n\\end{equation}\nthe particle density of the eigenstate $|\\pmb{\\lambda}\\rangle$.\n\n\\subsubsection{The field form factors}\nOur aim is to calculate correlation functions in an eigenstate $|\\pmb{\\lambda}\\rangle$ at low particle density ${\\cal D}$. We will focus on the two-point function of the field operator\n\\begin{equation}\n  \\left\\langle \\psi^\\dagger( x,t) \\psi ( 0,0) \\right\\rangle =\\frac {\\left\\langle \\pmb{\\lambda} \\left|  \\psi^\\dagger( x,t) \\psi ( 0,0) \\right| \\pmb{\\lambda} \\right\\rangle } {\\left\\langle \\pmb{\\lambda}  |\\pmb{\\lambda}  \\right\\rangle }\\,,\n\\end{equation}\nand the two-point function of the density operator\n\\begin{equation}\n  \\left\\langle \\sigma( x,t) \\sigma ( 0,0) \\right\\rangle =\\frac {\\left\\langle \\pmb{\\lambda} \\left| \\sigma( x,t) \\sigma ( 0,0) \\right| \\pmb{\\lambda} \\right\\rangle } {\\left\\langle \\pmb{\\lambda}  |\\pmb{\\lambda}  \\right\\rangle }\\,.\n\\end{equation}\nOur strategy is to use a Lehman representation in terms of energy eigenstates\n$|\\pmb{\\mu}\\rangle =|\\mu_1,...,\\mu_{N'}\\rangle$, where\n$\\{\\mu_1,\\dots,\\mu_{N'}\\}$ are solutions to the Bethe equations \\fr{belog} to rewrite the correlation functions as sums of form factors over the full spectrum. For the two-point function of a generic operator ${\\cal O}$ this representation reads\n\\begin{equation}\n\\label{bigsumfield}\n\\begin{aligned}\n  \\left\\langle  {\\cal O}^\\dagger( x,t){\\cal O}( 0,0)  \\right\\rangle &=\\sum _{ \\pmb{\\mu}}\\frac {\\left| \\left\\langle \\pmb{\\mu} |{\\cal O}( 0) |\\pmb{\\lambda}\\right\\rangle \\right| ^{2}} {\\left\\langle \\pmb{\\lambda} \\left| \\pmb{\\lambda} \\right\\rangle \\left\\langle \\pmb{\\mu}\\right| \\pmb{\\mu}\\right\\rangle }e^{it\\left( E\\left( \\pmb{\\lambda}\\right) -E\\left( \\pmb{\\mu} \\right) \\right) +ix\\left( P\\left( \\pmb{\\mu}\\right) -P\\left( \\pmb{\\lambda}\\right) \\right) }\\,.\n\\end{aligned}\n\\end{equation}\n\nThe (normalized) form factors of the field and density operators between two Bethe\nstates $|\\pmb{\\lambda}\\rangle,|\\pmb{\\mu}\\rangle$ with respective numbers of Bethe roots $N,N'$ have been computed previously \\cite{Korepin82,Slavnov89,Slavnov90,KorepinSlavnov99,Oota04,KozlowskiForm11}. \n\nIn the case of the field operator, it reads\n\\begin{equation}\\label{FF}\n\\begin{aligned}\n&\\frac { \\left\\langle \\pmb{\\mu} |\\psi ( 0) |\\pmb{\\lambda}\\right\\rangle  } {\\sqrt{\\left\\langle \\pmb{\\lambda} \\left| \\pmb{\\lambda} \\right\\rangle \\left\\langle \\pmb{\\mu}\\right| \\pmb{\\mu}\\right\\rangle }}=\\delta_{N,N'+1}\\frac{i^{N+1}(-1)^{N(N-1)\/2}}{L^{N-1\/2}\\sqrt{\\mathcal{N}_{\\pmb{\\lambda}}\\mathcal{N}_{\\pmb{\\mu}}}}\\frac{\\prod_{i< j}|\\lambda_i-\\lambda_j|\\prod_{i< j}|\\mu_i-\\mu_j|}{\\prod_{i, j}(\\mu_j-\\lambda_i)}\\\\\n&\\qquad\\times\\sqrt{\\frac{\\prod_{i\\neq j}(\\lambda_i-\\lambda_j+ic)}{\\prod_{i\\neq j}(\\mu_i-\\mu_j+ic)}}\\prod_{\\substack{j=1\\\\j\\neq p,s}}^N(V_j^+-V_j^-) \\,\\,\\underset{i,j=1,...,N}{\\det}\\Bigg[\\delta_{ij}+U_{ij}\\Bigg]\\,,\n\\end{aligned}\n\\end{equation}\nfor any $p,s=1,...,N$. The various terms entering this expression are\n\\begin{equation}\nV_i^\\pm=\\frac{\\prod_{k=1}^{N-1}(\\mu_k-\\lambda_i\\pm ic)}{\\prod_{k=1}^{N}(\\lambda_k-\\lambda_i\\pm ic)}\\,,\n\\end{equation}\nand the $N\\times N$ matrix\n\\begin{equation}\nU_{jk}=\\frac{i}{V_j^+-V_j^-}\\left[\\frac{2c}{c^2+(\\lambda_j-\\lambda_k)^2}-\\frac{4c^2}{(c^2+(\\lambda_p-\\lambda_k)^2)(c^2+(\\lambda_s-\\lambda_j)^2)}\\right]\\frac{\\prod_{m=1}^{N-1}(\\mu_m-\\lambda_j)}{\\prod_{m\\neq j}(\\lambda_m-\\lambda_j)}\\,,\n\\end{equation}\nand finally\n\\begin{equation}\n\\label{norm}\n\\mathcal{N}_{\\pmb{\\lambda}}=\\det G(\\pmb{\\lambda})\\,,\n\\end{equation}\nwith the Gaudin matrix \\cite{Gaudin71}\n\\begin{equation}\\label{gaudin}\nG_{ij}(\\pmb{\\lambda})= \\delta_{ij} \\left(1+\\frac{1}{L}\\sum_{k=1}^N \\frac{2c}{c^2+(\\lambda_i-\\lambda_k)^2}\\right)-\\frac{1}{L}\\frac{2c}{c^2+(\\lambda_i-\\lambda_j)^2}\\,.\n\\end{equation}\nThe form factor of the density operator reads\n\\begin{equation}\\label{desnityff}\n\\begin{aligned}\n&\\frac { \\left\\langle \\pmb{\\mu} |\\sigma ( 0) |\\pmb{\\lambda}\\right\\rangle  } {\\sqrt{\\left\\langle \\pmb{\\lambda} \\left| \\pmb{\\lambda} \\right\\rangle \\left\\langle \\pmb{\\mu}\\right| \\pmb{\\mu}\\right\\rangle }}=\\delta_{N,N'}\\frac{i^{N+1}(-1)^{N(N-1)\/2}(\\sum_{j=1}^N \\lambda_j-\\mu_j)}{L^{N}\\sqrt{\\mathcal{N}_{\\pmb{\\lambda}}\\mathcal{N}_{\\pmb{\\mu}}}}\\frac{\\prod_{i< j}|\\lambda_i-\\lambda_j|\\prod_{i< j}|\\mu_i-\\mu_j|}{\\prod_{i, j}(\\mu_j-\\lambda_i)}\\\\\n&\\qquad\\times\\sqrt{\\prod_{i, j}\\frac{\\lambda_i-\\lambda_j+ic}{\\mu_i-\\mu_j+ic}}\\prod_{j\\neq p}(V_j^+-V_j^-) \\,\\,\\underset{i,j=1,...,N}{\\det}\\Bigg[\\delta_{ij}+U'_{ij}\\Bigg]\\,,\n\\end{aligned}\n\\end{equation}\nfor any $p=1,...,N$, with\n\\begin{equation}\nU'_{jk}=i\\frac{\\mu_j-\\lambda_j}{V_j^+-V_j^-}\\left[\\frac{2c}{(\\lambda_j-\\lambda_k)^2+c^2}-\\frac{2c}{(\\lambda_p-\\lambda_k)^2+c^2}\\right]\\prod_{m\\neq j}\\frac{\\mu_m-\\lambda_j}{\\lambda_m-\\lambda_j}\\,,\n\\end{equation}\nand with now\n\\begin{equation}\nV_j^\\pm=\\frac{\\prod_{k=1}^{N}(\\mu_k-\\lambda_j\\pm ic)}{\\prod_{k=1}^{N}(\\lambda_k-\\lambda_j\\pm ic)}\\,.\n\\end{equation}\n\\subsubsection{Root densities}\nIn the thermodynamic limit, any sum of a non-singular function over the Bethe roots can be expressed in terms of a \\textit{root density} that characterizes a macrostate as far as such quantities are concerned\n\\begin{equation}\n\\underset{L\\to\\infty}{\\lim}\\, \\frac{1}{L^n}\\sum_{i_1,...,i_n}f(\\lambda_{i_1},...,\\lambda_{i_n})=\\int_{-\\infty}^\\infty \\dots\\int_{-\\infty}^\\infty f(\\lambda_1,...,\\lambda_n)\\rho(\\lambda_1)\\dots\\rho(\\lambda_n)\\D{\\lambda_1}\\dots\\D{\\lambda_n}\\,.\n\\end{equation}\nHowever if the function is singular the result will in general depend on the representative state of the macrostate, see \\cite{granetessler20}.\nIt is customary to introduce the hole density $\\rho_h(\\lambda)$ defined by\n\\begin{equation}\n\\label{vartheta}\n\\rho(\\lambda)+\\rho_h(\\lambda)=\\frac{1}{2\\pi}+\\frac{1}{2\\pi}\\int_{-\\infty}^\\infty \\frac{2c}{c^2+(\\lambda-\\mu)^2}\\rho(\\mu)\\D{\\mu}\\,.\n\\end{equation}\n\n\\section {Definition of the low-density limit \\label{ldd}}\nThe purpose of this section is to define what is meant by the \\textit{low density limit} of correlation functions. It is defined as the \\textit{leading order of an expansion} in ${\\cal D}$, obtained by decomposing the form factor in partial fractions. As a consequence it is an expression valid for all $x,t$ and $c$, that becomes closer to the dynamical correlations as the particle density ${\\cal D}$ of the averaging state becomes smaller.\\\\\n\nThis definition requires some technicalities, but is rigorous and allows for a computation of the next orders, as shown in \\cite{GFE20} for a model that can be reformulated into free fermions. However, it a priori lacks some intuitive picture. For that reason we provide an interpretation of this low-density limit so defined as a Lehmann representation in terms of the \\textit{low density limit of the form factor}. The reasoning is here rather different, and consists in first approximating the form factor by the thermodynamic limit value they take when one of the two states is a dilute state, i.e. such that for any pair $i,j$ we have  $L(\\lambda_i-\\lambda_j)\\to\\infty$. In this limit, the spectral sum of the dynamical correlations indeed matches the leading order of the expansion in ${\\cal D}$, providing an interesting and intuitive consistency check. But it must be stressed that the right definition of the low-density limit is more general than this intuitive calculation, since it only requires ${\\cal D}$ to be small, not the root density $\\rho(\\lambda)$ to be small everywhere.\n\n  \n\n\\subsection {Partial fraction decomposition}\n\\subsubsection{Recall \\label{pfddefsec}}\nWe recall that the partial fraction decomposition (PFD) of a ratio of two polynomials $\\frac{P(X)}{\\prod_{i=1}^n(X-x_i)^{a_i}}$ with distinct $x_i$'s is the writing\n\\begin{equation}\\label{pfddef}\n\\frac{P(X)}{\\prod_{i=1}^n(X-x_i)^{a_i}}=P_0(X)+\\sum_{i=1}^n \\sum_{\\nu=1}^{a_i}\\frac{B_{i,\\nu}}{(X-x_i)^\\nu}\\,,\n\\end{equation}\nwith $P_0(X)$ a polynomial, and $B_{i,\\nu}$ coefficients given by\n\\begin{equation}\nB_{i,\\nu}=\\frac{1}{(a_i-\\nu)!}(\\tfrac{d}{dX})^{a_i-\\nu}[(X-x_i)^{a_i}P(X)]|_{X=x_i}\\,.\n\\end{equation}\nThe polynomial $P_0(X)$ can be determined by studying e.g. the large $X$ behaviour of the ratio of the two polynomials on the left-hand side of \\eqref{pfddef}.\n\\subsubsection{The poles of the normalized form factor}\nWe consider $ \\pmb{\\lambda}  $ and $ \\pmb{\\mu}  $ two sets of respectively $N$ and $N-1$ rapidities, and would like to apply a partial fraction decomposition to the square of the normalized field form factor, with respect to each of the $\\mu_i$'s successively. The first task is to identify the poles of a $\\mu_i$ at fixed other $\\mu_j$'s. There are a priori three types of poles for $\\mu_i$\n\\begin{enumerate}\n\\item Double poles in $\\lambda_j$ for all $j$\n\\item Simple poles in $\\mu_j \\pm ic$ for all $j$\n\\item Poles corresponding to zeros of the determinant of the Gaudin matrix $\\mathcal{N}_{\\pmb{\\mu}}$\n\\end{enumerate}\nWe remark that the last two types of poles come from the fact that we consider normalized form factors. We also remark that since some of the entries of the Gaudin matrix diverge when $\\mu_i-\\mu_j\\to \\pm ic$, the second type of pole could happen to be absent from the full normalized form factor, but this possibility will not be relevant to our discussion. Lastly we notice that the last two types of poles are in fact never attained when all the roots $\\mu_i$ are real, which is always the case if $c>0$. Indeed, when the roots are real, the Gaudin matrix is strictly dominant diagonal, i.e. satisfies $\\forall i=1,...,N-1,\\quad |G_{ii}|>\\sum_{j\\neq i}|G_{ij}|$, hence is invertible. However, when performing the PFD of the form factor it has to be considered as a mere fraction of polynomials in $\\mu_i$'s that do not necessarily satisfy the Bethe equations, and these poles have to be taken into account indeed.\n\nAmong these three types of poles, the zeros of $\\mathcal{N}_{\\pmb{\\mu}}$ are the most problematic since their location is a complicated function of the other $\\mu_j$'s. For this reason we are going to consider the PFD of the form factor without these factors.\nNamely we define $F_\\psi(\\pmb{\\lambda},\\pmb{\\mu})$ by\n\\begin{equation}\\label{pfd}\n\\begin{aligned}\n\\frac {\\left| \\left\\langle \\pmb{\\mu} |\\psi \\left( 0\\right) |\\pmb{\\lambda}\\right\\rangle \\right| ^{2}} {\\left\\langle \\pmb{\\lambda} \\left| \\pmb{\\lambda} \\right\\rangle \\left\\langle \\pmb{\\mu}\\right| \\pmb{\\mu}\\right\\rangle }= \\frac{F_\\psi(\\pmb{\\lambda},\\pmb{\\mu})}{\\mathcal{N}_{\\pmb{\\lambda}}\\mathcal{N}_{\\pmb{\\mu}}L^{2N-1}} \\,,\n\\end{aligned}\n\\end{equation}\nand consider the PFD of $F_\\psi(\\pmb{\\lambda},\\pmb{\\mu})$ only, with respect to the $\\mu_i$'s. \n\n\\subsubsection{The shape of the PFD of the reduced form factor}\nThe reduced form factor $F_\\psi(\\pmb{\\lambda},\\pmb{\\mu})$, seen as a function of $\\mu_1$, is a ratio of two polynomials with double poles in each of the $\\lambda_i$'s and simple poles in $\\mu_j\\pm ic$, so one can apply the decomposition written in Section \\ref{pfddefsec}. Since the reduced form factor goes to zero when $\\mu_1\\to\\infty$, we have $P_0(X)=0$ and so one can write\n\\begin{equation}\\label{154}\nF_\\psi(\\pmb{\\lambda},\\pmb{\\mu})=h_{\\mu_1}(\\mu_2,...,\\mu_{N-1})+\\sum_{i=1}^N \\sum_{\\nu=1}^{2}\\frac{B_{i,\\nu}(\\mu_2,...,\\mu_{N-1})}{(\\mu_1-\\lambda_i)^\\nu}\\,,\n\\end{equation}\nwith\n\\begin{equation}\nh_{\\mu_1}(\\mu_2,...,\\mu_{N-1})=\\sum_{i=2}^{N-1} \\frac{C^+_{i}(\\mu_2,...,\\mu_{N-1})}{\\mu_1-\\mu_i+ic}+\\sum_{i=2}^{N-1} \\frac{C^-_{i}(\\mu_2,...,\\mu_{N-1})}{\\mu_1-\\mu_i-ic}\\,,\n\\end{equation}\nwhere $B_{i,\\nu}(\\mu_2,...,\\mu_{N-1})$, $C^\\pm_{i}(\\mu_2,...,\\mu_{N-1})$ are 'coefficients' independent of $\\mu_1$, but that still possess a dependence in the remaining $\\mu_k$'s. They have the same pole structure, except that each $\\mu_i$ for $i\\neq 1$ does not anymore has a pole in $\\mu_1\\pm ic$. The function $h_{\\mu_1}(\\mu_2,...,\\mu_{N-1})$ is a function of $\\mu_1$ that has no poles in $\\mu_1$ when all the $\\mu_j$'s are real. \nWe now apply the same procedure to $B_{i,\\nu}(\\mu_2,...,\\mu_{N-1})$ and $h_{\\mu_1}(\\mu_2,...,\\mu_{N-1})$ with respect to $\\mu_2$ to obtain the writing\n\\begin{equation}\nF_\\psi(\\pmb{\\lambda},\\pmb{\\mu})=h_{\\mu_1,\\mu_2}(\\mu_3,...,\\mu_{N-1})+\\sum_{i=1}^N\\sum_{j=1}^N\\sum_{\\nu_i=0}^2\\sum_{\\nu_j=1}^2\\frac{B_{i,\\nu_i,j,\\nu_j}(\\mu_3,...,\\mu_{N-1})}{(\\mu_1-\\lambda_i)^{\\nu_i}(\\mu_2-\\lambda_j)^{\\nu_j}}\\,,\n\\end{equation}\nwhere $B_{i,\\nu_i,j,\\nu_j}(\\mu_3,...,\\mu_{N-1})$ is a 'coefficient' independent of $\\mu_1,\\mu_2$, and $h_{\\mu_1,\\mu_2}(\\mu_3,...,\\mu_{N-1})$ a  function of $\\mu_1,\\mu_2$ without singularities in real $\\mu_1,\\mu_2$. We note that the poles in $\\mu_2$ arising from the function $h_{\\mu_1}(\\mu_2,...,\\mu_{N-1})$ that has no poles in real $\\mu_1$ are counted through the case $\\nu_i=0$. Proceeding recursively, one obtains the writing\n\n\\begin{equation}\\label{pfd}\nF_\\psi(\\pmb{\\lambda},\\pmb{\\mu})=\\sum_{\\{\\nu\\},f}\\frac{A(\\pmb{\\lambda},\\pmb{\\mu},\\{\\nu\\},f)}{\\prod_{i=1}^{N-1}(\\mu_i-\\lambda_{f(i)})^{\\nu_i}}\\,,\n\\end{equation}\nwhere each $\\nu_i$ takes the value $0$, $1$ or $2$, and where $f$ are functions defined on the points $i\\in\\{1,...,N-1\\}$ where $\\nu_i>0$, namely\n\\begin{equation}\nf:\\{i\\in\\{1,...,N-1\\} | \\nu_i>0 \\}\\to \\{1,...,N\\}\\,.\n\\end{equation}\nThe coefficients $A(\\pmb{\\lambda},\\pmb{\\mu},\\{\\nu\\},f)$ crucially do not depend on any $\\mu_i$ whenever $\\nu_i>0$, and are bounded regular functions of real $\\mu_i$ when $\\nu_i=0$. \n\n\\subsubsection{Computing the coefficients}\nIn the special case where all $\\nu_i>0$, the coefficients $A(\\pmb{\\lambda},\\pmb{\\mu},\\{\\nu\\},f)=A(\\pmb{\\lambda},\\{\\nu\\},f)$ do not depend on any $\\mu_i$'s and can be computed according to the formula\n\\begin{equation}\n\\label{formulA}\nA(\\pmb{\\lambda},\\{\\nu\\},f)=\\prod_{i=1}^{N-1}\\left[\\left(\\frac{d}{d\\mu_i}\\right)^{2-\\nu_i} (\\mu_i-\\lambda_{f(i)})^2 \\right]F_\\psi(\\pmb{\\lambda},\\pmb{\\mu})|_{\\mu_i=\\lambda_{f(i)}}\\,.\n\\end{equation}\nIf now there is a subset $K\\subset\\{1,...,N-1\\}$ such that $\\nu_i=0$ for $i\\in K$, one first defines the following function of the $\\mu_i$'s for $i\\in K$\n\\begin{equation}\n\\label{formulA2}\n\\bar{A}(\\{\\mu_i\\}_{i\\in K}|\\pmb{\\lambda},\\{\\nu\\},f)=\\prod_{\\substack{i=1\\\\ i\\notin K}}^{N-1}\\left[\\left(\\frac{d}{d\\mu_i}\\right)^{2-\\nu_i} (\\mu_i-\\lambda_{f(i)})^2 \\right]F_\\psi(\\pmb{\\lambda},\\pmb{\\mu})|_{\\mu_i=\\lambda_{f(i)},\\, i\\notin K}\\,.\n\\end{equation}\nThe function $\\bar{A}(\\{\\mu_i\\}_{i\\in K}|\\pmb{\\lambda},\\{\\nu\\},f)$ still has poles in $\\mu_i$ for $i\\in K$ since it also contains all the cases when $\\nu_i>0$. To compute the coefficient $A(\\pmb{\\lambda},\\pmb{\\mu},\\{\\nu\\},f)$ one has to remove from this function any pole in real $\\mu_i$ for $i\\in K$. Although these formulas are not very explicit, they are still of practical use to compute the simplest terms in the PFD, as we will see below.\\\\\n\n\n\nThe functions $f$ over which we sum in \\eqref{pfd} are actually rather constrained. First, since the form factor vanishes whenever two $\\mu_i$'s or two $\\lambda_i$'s coincide, we see from \\eqref{formulA} and  \\eqref{formulA2} that if $\\nu_i=2$ or $\\nu_j=2$ and $f(i)=f(j)$, then the corresponding coefficient vanishes (namely, $A(\\pmb{\\lambda},\\{\\nu\\},f)=0$ if all $\\nu_k>0$ are non-zero, and $\\bar{A}(\\{\\mu_i\\}_{i\\in K}|\\pmb{\\lambda},\\{\\nu\\},f)=0$ if there is a vanishing $\\nu_k$). If we have $\\nu_i=\\nu_j=2$ it directly follows from the absence of derivative in \\eqref{formulA}; if $\\nu_i$ or $\\nu_j$ is equal to $1$, then it follows from the fact that there is a zero of order $2$ in the numerator with only one derivative. Similarly, if $k$ indices have a $\\nu_i=1$ and take the same value through $f$, then there is a zero of order $k(k-1)$ in the numerator, with only $k$ derivatives. Hence this imposes $k=2$. Thus one can impose in \\eqref{pfd} the two following constrains: (i) that $f(i)\\neq f(j)$ whenever $\\nu_i=2$ or $\\nu_j=2$, and (ii) that $f$ can take at most twice the same value.\\\\\n\nIn the following we will denote the number of elements of a set $E$ by\n\\begin{equation}\n|E|\\qquad \\text{or}\\qquad \\# E\\,,\n\\end{equation}\naccording to the most readable choice in the context.\n\n\\subsection {A density expansion}\n\nLet us now rewrite the Lehmann representation \\eqref{bigsumfield} in the following way. Instead of summing over the Bethe roots $\\mu_i$, we sum over their Bethe numbers $J_i$, and trade the ordering of the Bethe numbers for a non-ordered sum with a $\\frac{1}{(N-1)!}$ factor. Whenever two Bethe numbers coincide, the form factor is zero so that the two representations are indeed equivalent. Using \\eqref{pfd}, we obtain\n\\begin{equation}\\label{pfdbeg}\n\\begin{aligned}\n \\left\\langle  \\psi^\\dagger\\left( x,t\\right) \\psi \\left( 0,0\\right)  \\right\\rangle =&\\frac{1}{L^{2N-1}(N-1)!}\\\\\n &\\sum_{\\{\\nu\\},f}\\sum_{J_1,...,J_{N-1}}\\frac{A(\\pmb{\\lambda},\\pmb{\\mu},\\{\\nu\\},f)}{\\mathcal{N}_{\\pmb{\\lambda}}\\mathcal{N}_{\\pmb{\\mu}}}\\frac{e^{it\\left( E( \\pmb{\\lambda}) -E( \\pmb{\\mu}) \\right) +ix\\left( P( \\pmb{\\mu}) -P( \\pmb{\\lambda}) \\right) }}{\\prod_{i=1}^{N-1}(\\mu_i-\\lambda_{f(i)})^{\\nu_i}}\\,.\n\\end{aligned}\n\\end{equation}\nThe sum over $J_1,...,J_{N-1}$ is invariant under any change $\\tilde{f}=f \\circ (i \\,j)$ with $(i \\,j)$ the permutation of indices $i,j$, whenever $\\nu_i=\\nu_j>0$ and $f(i)$ and $f(j)$ are attained the same number of times by $f$. Hence this sum only depends on the \\textit{set of points attained a given number of times} by $f$, not on the particular realization of the function $f$. \n\n\n\nTo rewrite the sum without these functions $f$, let us define $I_k$ for $k=0,1,2$ the set of points $i$ in $\\{1,...,N\\}$ that are attained $k$ times by $f$ from points where $\\nu_j=1$, namely\n\\begin{equation}\n\\begin{aligned}\nI_k&=\\Bigg\\{i\\in\\{1,...,N\\} \\left| \\# \\{j\\in \\{1,...,N-1\\}| \\nu_j=1\\text{ and }f(j)=i\\}=k\\Bigg\\}\\right.\\,.\n\\end{aligned}\n\\end{equation}\nAs a consequence the points in $\\{1,...,N\\}$ attained by $f$ from points where $\\nu_j=2$ are $\\{1,...,N\\}-(I_0\\cup I_1\\cup I_2)$. These subsets $I_0,I_1,I_2\\subset \\{1,...,N\\}$ have to be disjoint and to satisfy $|I_0|=|I_2|+1+p$ with $p=|\\{i|\\nu_i=0\\}|$ the number of points with $\\nu_i=0$, because  $f$ can take at most twice the same value. \n\nWe will denote\n\\begin{equation}\nn=|I_2|\\,,\\qquad m=|I_1|\\,,\n\\end{equation}\nand parametrize\n\\begin{equation}\n\\begin{aligned}\nI_2&=\\{j_{1},...,j_{n}\\}\\\\\nI_0&=\\{j_{n+1},...,j_{2n+p+1}\\}\\\\\nI_1&=\\{j_{2n+p+2},...,j_{2n+p+m+1}\\}\\\\\n\\{1,...,N\\}-(I_0\\cup I_1 \\cup I_2)&=\\{j_{2n+p+m+2},...,j_{N}\\}\\,.\n\\end{aligned}\n\\end{equation}\nWhen rewriting \\eqref{pfdbeg} in terms of these subsets, one picks  a combinatorial factor corresponding to the number of functions $f$ with such an output. Choosing the set of points where $\\nu_i=0$ yields a factor ${N-1\\choose p}$, those where $\\nu_i=2$ a factor $(N-2n-p-m-1)!{N-1-p\\choose N-2n-p-m-1}$, those attained only once by $f$ and where $\\nu_i=1$ a factor $m!{2n+m\\choose m}$. Finally those attained twice by $f$ yield a factor $(2n)!!n!$. Writing $(2n)!!=\\tfrac{(2n)!}{n!2^n}$ yields a total combinatorial factor\n\\begin{equation}\n\\frac{(N-1)!}{2^{n}p!}\\,.\n\\end{equation}\nWe conclude that we can write\n\\begin{equation}\\label{expdensity}\n\\left\\langle  \\psi^\\dagger\\left( x,t\\right) \\psi \\left( 0,0\\right)  \\right\\rangle =\\sum_{n,m,p\\geq 0}S_{n,m,p}\\,,\n\\end{equation}\nwith\n\\begin{equation}\n\\begin{aligned}\n&S_{n,m,p}=\\frac{1}{2^np!L^{2N-1}}\\sum_{\\substack{I_{0,1,2}\\subset \\{1,...,N\\}\\\\|I_0|=n+p+1\\\\|I_1|=m\\\\|I_2|=n \\\\\\text{all disjoint}}}\\sum_{J_1,...,J_{N-1}}\\frac{{\\cal A}(I_0,I_1,I_2|\\{\\mu_i\\}_{i=2n+1}^{2n+p})}{\\mathcal{N}_{\\pmb{\\lambda}}\\mathcal{N}_{\\pmb{\\mu}}}\\\\\n  &\\qquad\\qquad\\times\\frac{e^{it\\left( E\\left( \\pmb{\\lambda}\\right) -E\\left( \\pmb{\\mu} \\right) \\right) +ix\\left( P\\left( \\pmb{\\mu}\\right) -P\\left( \\pmb{\\lambda}\\right) \\right) }}{\\prod_{i=1}^{n}(\\mu_{2i-1}-\\lambda_{j_i})(\\mu_{2i}-\\lambda_{j_i})\\prod_{i=2n+1+p}^{2n+m+p}(\\mu_i-\\lambda_{j_{i+1}})\\prod_{i=2n+m+p+1}^{N-1}(\\mu_i-\\lambda_{j_{i+1}})^{2}}\\,.\n\\end{aligned}\n\\end{equation}\nThe specific ordering of the $\\mu_i$'s in this expression is irrelevant. \n\nHere we have ${\\cal A}(I_0,I_1,I_2|\\{\\mu_i\\}_{i=2n+1}^{2n+p})=A(\\pmb{\\lambda},\\pmb{\\mu},\\{\\nu\\},f)$ with $\\nu_i=1$ for $i=1,...,m+2n$, $\\nu_i=0$ for $i=m+2n+1,...,m+2n+p$ and $\\nu_i=2$ for $i=m+2n+p+1,...,N-1$, and with the function $f$ taking the values in $I_1$ over $1,...,m$, twice the values in $I_2$ over $m+1,...,m+2n$, and the values in $\\{1,...,N\\}-(I_0\\cup I_1\\cup I_2)$ over $m+2n+p+1,...,N-1$.\n\nSince each choice of an index in $\\{1,...,N\\}$ comes with a factor ${\\cal D}$, the term $S_{n,m,p}$ is of order ${\\cal O}({\\cal D}^{1+2n+m+p})$. Hence expression \\eqref{expdensity} is \\textit{an expansion in the particle density} ${\\cal D}$ of the averaging state. \n\n\\subsection {Definition of the low-density limit of the correlation function}\nThe low density limit of the dynamical correlations is defined as retaining only the leading term $S_{0,0,0}$ in \\eqref{expdensity}. It is obtained with $p=0$ and $I_1=I_2=\\varnothing$, and so $I_0=\\{a\\}$ for $a=1,...,N$. Namely, reparametrising $\\pmb{\\mu}=\\{\\mu_1,...,\\mu_{a-1},\\mu_{a+1},...,\\mu_N\\}$ for convenience\n\\begin{equation}\\label{S00}\n\\begin{aligned}\n  &S_{0,0,0}=\\frac{1}{L}\\sum_{a=1}^N\\sum_{J_i,\\, i\\neq a}\\frac{{\\cal A}(\\{a\\},\\varnothing,\\varnothing|\\varnothing)}{\\mathcal{N}_{\\pmb{\\lambda}}\\mathcal{N}_{\\pmb{\\mu}}}\\frac{e^{it\\left( E\\left( \\pmb{\\lambda}\\right) -E\\left( \\pmb{\\mu} \\right) \\right) +ix\\left( P\\left( \\pmb{\\mu}\\right) -P\\left( \\pmb{\\lambda}\\right) \\right) }}{\\prod_{i\\neq a}L^2(\\mu_i-\\lambda_{i})^{2}}\\,.\n\\end{aligned}\n\\end{equation}\nSince all the other terms in \\eqref{expdensity} have at least a multiplying factor ${\\cal D}$, we have\n\\begin{equation}\\label{psipsild2}\n\\begin{aligned}\n  &\\left\\langle  \\psi^\\dagger\\left( x,t\\right) \\psi \\left( 0,0\\right)  \\right\\rangle =S_{0,0,0}(1+{\\cal O}({\\cal D}))\\,.\n\\end{aligned}\n\\end{equation}\nIn the rest of paper, we will use the sign $\\underset{\\text{l.d.}}{\\sim}$ to indicate a low-density limit. Namely\n\\begin{equation}\nX\\underset{\\text{l.d.}}{\\sim} Y\\qquad \\text{means}\\qquad X=Y(1+{\\cal O}({\\cal D}))\\,.\n\\end{equation}\nUsing formula \\eqref{formulA}, one finds\n\\begin{equation}\n{\\cal A}(\\{a\\},\\varnothing,\\varnothing|\\varnothing)=\\prod_{i\\neq a}\\frac{4c^2}{(\\lambda_i-\\lambda_a)^2+c^2}\\,.\n\\end{equation}\nHence the low-density limit\n\\begin{equation}\\label{psipsild2}\n\\begin{aligned}\n  &\\left\\langle  \\psi^\\dagger\\left( x,t\\right) \\psi \\left( 0,0\\right)  \\right\\rangle \\underset{\\text{l.d.}}{\\sim}\\frac{1}{L}\\sum_{a=1}^Ne^{it\\lambda_a^2-ix\\lambda_a}\\sum_{J_i,\\, i\\neq a}\\frac{1}{\\mathcal{N}_{\\pmb{\\lambda}}\\mathcal{N}_{\\pmb{\\mu}}}\\prod_{i\\neq a}\\left(\\frac{4c^2}{(\\lambda_i-\\lambda_a)^2+c^2}\\frac{e^{it\\left( \\lambda_i-\\mu_i \\right) +ix\\left( \\mu_i-\\lambda_i \\right) }}{L^2(\\mu_i-\\lambda_{i})^{2}}\\right)\\,.\n\\end{aligned}\n\\end{equation}\nA few comments are in order. Up to now, nothing has been said of the Bethe equations, and in principle the $\\mu_i$'s in this expression should satisfy exactly the Bethe equations \\eqref{belog}. If one wishes to determine the dynamical correlations at leading order in ${\\cal D}$, there remains only the term \\eqref{psipsild2} in the full expansion \\eqref{expdensity}; but one can also satisfy the Bethe equations  \\eqref{belog} only at leading order in ${\\cal D}$, since their exact solution will involve higher orders in ${\\cal D}$ that are of the same order as the terms discarded in \\eqref{expdensity}. Stated differently, the leading order in ${\\cal D}$ of the dynamical correlations is obtained by both retaining only \\eqref{psipsild2} \\textit{and} satisfying \\eqref{belog} at leading order in ${\\cal D}$, while the higher orders in ${\\cal D}$ will require both taking into account higher terms in \\eqref{expdensity}  \\textit{and} satisfying \\eqref{belog} at higher orders in ${\\cal D}$ in \\eqref{psipsild2}.\n\n\n\\subsection {Interpretation: low-density limit of the field form factor\\label{intuitive}}\nThe low-density limit is defined above as the leading term in an expansion of the correlation functions obtained by decomposing the form factors in partial fractions, that turns out to be an expansion in the density of the averaging state ${\\cal D}$. \nThis definition allows for a systematic calculation of the next corrections in the density by taking into account more terms in \\eqref{expdensity}.\\\\\n\nThe low-density limit of the correlation functions can however be recovered more intuitively but less rigorously with the following reasoning. If the root density $\\rho(\\lambda)$ of the averaging state is small, then the distance between two consecutive roots $L(\\lambda_i-\\lambda_j)$ is 'typically'\\footnote{This cannot be true for any representative state of the density, but is true for a 'typical state' whose roots are regularly spaced according to the value of the density, see \\cite{granetessler20}.} large in front of $1$, and in the limit of vanishingly low density becomes infinite. We will say that a sequence of states satisfying this property is \\textit{dilute}, namely\n\\begin{equation}\n(\\pmb{\\lambda}^{(L)})_{L\\in\\mathbb{N}}\\text{ dilute}\\iff \\forall i\\neq j,\\, \\underset{L\\to\\infty}{\\lim}\\, L|\\lambda^{(L)}_i-\\lambda^{(L)}_j|=\\infty\\,.\n\\end{equation}\nFor notational convenience we will drop the $L$ dependence of the sequence of states. Let us consider a dilute state $\\pmb{\\lambda}$ and investigate the consequences it has on the form factors between $\\pmb{\\lambda}$ and another state $\\pmb{\\mu}$.\n\nLet us first note that because of the assumption of diluteness, a Bethe number $J_i$ of $\\pmb{\\mu}$ can be at a distance ${\\cal O}(1)$ of at most one Bethe number $I_j$ of $\\pmb{\\lambda}$. Hence given a root $\\mu_i$, the quantity $L(\\mu_i-\\lambda_j)$ can be of order $1$ for at most one $\\lambda_j$. Consequently, \nit is seen from the expression \\eqref{FF} that \nin the low density limit the form factor \\eqref{FF} is non-zero only if each Bethe number $J_i$ of $\\pmb{\\mu}$ is at a distance of order $1$ from exactly one Bethe number $I_j$ of $\\pmb{\\lambda}$, and reciprocally if these $N-1$ Bethe numbers $I_j$ are each at a distance of order $1$ from  exactly one Bethe number $J_k$ of $\\pmb{\\mu}$. Since there are $N$ roots in $\\pmb{\\lambda}$, there is one remaining root $\\lambda_a$ that is not close to any $\\mu_i$'s. We can re-label the roots $\\pmb{\\mu}=\\{\\mu_1,...,\\mu_{a-1},\\mu_{a+1},...,\\mu_N\\}$ so that $L(\\mu_i-\\lambda_i)$ is of order $1$ for all $i\\neq a$. Let us then investigate the value taken by the normalized form factor \\eqref{FF} in this regime.\n\nLet us first study the determinant $\\mathcal{N}_{\\pmb{\\lambda}}$ in the low-density limit. We have\n\\begin{equation}\\label{gaudin2}\nG_{ij}(\\pmb{\\lambda})= \\delta_{ij}-\\frac{1}{L}\\frac{2c}{c^2+(\\lambda_i-\\lambda_j)^2}+{\\cal O}({\\cal D})\\,,\n\\end{equation}\nso that\n\\begin{equation}\\label{det1det1}\n\\begin{aligned}\n\\mathcal{N}_{\\pmb{\\lambda}}&=\\exp \\,\\text{tr}\\, \\log G(\\pmb{\\lambda})\\\\\n&=\\exp\\left( -\\sum_{n=1}^\\infty \\frac{1}{n}\\,\\text{tr}\\, g^n+{\\cal O}({\\cal D})\\right)\\\\\n&=1+{\\cal O}({\\cal D})\\,,\n\\end{aligned}\n\\end{equation}\nwith $g_{ij}=\\frac{1}{L}\\frac{2c}{c^2+(\\lambda_i-\\lambda_j)^2}$, since $\\,\\text{tr}\\, g^n={\\cal O}({\\cal D}^n)$. Here, we did not use the diluteness of $\\pmb{\\lambda}$, but only evaluated the leading order in ${\\cal D}$ of the determinant.\n\nSecond, again because $\\pmb{\\lambda}$ is dilute, all the $L(\\mu_i-\\lambda_i)$ are negligible in front of any $\\lambda_a-\\lambda_j$. It follows that\n\\begin{equation}\nV_j^+-V_j^-\\underset{\\text{l.d.}}{\\sim}\\frac{-2ic}{(\\lambda_a-\\lambda_j)^2+c^2}\\,.\n\\end{equation}\nWe also have\n\\begin{equation}\n\\sqrt{\\frac{\\prod_{i\\neq j}(\\lambda_i-\\lambda_j+ic)}{\\prod_{i\\neq j}(\\mu_i-\\mu_j+ic)}}\\underset{\\text{l.d.}}{\\sim} i^{N-1}\\prod_{j\\neq a}\\sqrt{(\\lambda_j-\\lambda_a)^2+c^2}\\,,\n\\end{equation}\nand\n\\begin{equation}\n\\frac{\\prod_{i< j}|\\lambda_i-\\lambda_j|\\prod_{i< j}|\\mu_i-\\mu_j|}{\\prod_{i, j}(\\mu_j-\\lambda_i)}\\underset{\\text{l.d.}}{\\sim} (-1)^{(N-1)(N-2)\/2}\\prod_{j\\neq a}\\,\\text{sgn}\\,(\\lambda_j-\\lambda_a)\\frac{1}{\\prod_{j\\neq a}(\\mu_j-\\lambda_j)}\\,.\n\\end{equation}\nAs for the matrix $U$, setting $\\lambda_p=\\lambda_s=\\lambda_a$, the dominant entries are\n\\begin{equation}\nU_{aj}\\underset{\\text{l.d.}}{\\sim}-1+\\frac{2}{c}\\,,\n\\end{equation}\nwhile the other entries are of order ${\\cal O}(L^{-1})$. Hence\n\\begin{equation}\n\\underset{i,j}{\\det}(\\delta_{ij}+U_{ij})\\underset{\\text{l.d.}}{\\sim}\\frac{2}{c}\\,.\n\\end{equation}\nWe obtain the following low-density limit of the form factor\n\\begin{equation}\\label{ldff}\n\\frac { \\left\\langle \\pmb{\\mu} |\\psi \\left( 0\\right) |\\pmb{\\lambda}\\right\\rangle  } {\\sqrt{\\left\\langle \\pmb{\\lambda} \\left| \\pmb{\\lambda} \\right\\rangle \\left\\langle \\pmb{\\mu}\\right| \\pmb{\\mu}\\right\\rangle }}\\underset{\\text{l.d.}}{\\sim}\n\\frac{\\phi}{\\sqrt{L}}\\prod_{j\\neq a}\\frac{2c}{\\sqrt{(\\lambda_j-\\lambda_a)^2+c^2}}\\frac{1}{L(\\mu_j-\\lambda_j)}\\,,\n\\end{equation}\nwith the phase\n\\begin{equation}\n\\phi=(-i)^N \\prod_{j\\neq a}\\,\\text{sgn}\\,(\\lambda_j-\\lambda_a)\\,.\n\\end{equation}\n\nLet us reformulate the meaning of this approximation. We consider $\\pmb{\\lambda}$ and $\\pmb{\\mu}$ two Bethe states with $N$ and $N-1$ particles respectively, and denote $\\iota:\\{1,...,N-1\\}\\to \\{1,...,N\\}$ the function such that the element of $\\{\\lambda_1,...,\\lambda_N\\}$ that is the closest to $\\mu_i$ is $\\lambda_{\\iota(i)}$. For a dilute $\\pmb{\\lambda}$, the form factor $\\frac { \\left\\langle \\pmb{\\mu} |\\psi \\left( 0\\right) |\\pmb{\\lambda}\\right\\rangle  } {\\sqrt{\\left\\langle \\pmb{\\lambda} \\left| \\pmb{\\lambda} \\right\\rangle \\left\\langle \\pmb{\\mu}\\right| \\pmb{\\mu}\\right\\rangle }}$ is non-negligible in the thermodynamic limit only if $\\iota$ is one-to-one from $\\{1,...,N-1\\}$ to $\\{1,...,N\\}-\\{a\\}$ for some $a=1,...,N$. In this case, the form factor reads\n\\begin{equation}\\label{ldff2}\n\\frac { \\left\\langle \\pmb{\\mu} |\\psi \\left( 0\\right) |\\pmb{\\lambda}\\right\\rangle  } {\\sqrt{\\left\\langle \\pmb{\\lambda} \\left| \\pmb{\\lambda} \\right\\rangle \\left\\langle \\pmb{\\mu}\\right| \\pmb{\\mu}\\right\\rangle }}\\underset{\\text{l.d.}}{\\sim}\n\\frac{\\phi}{\\sqrt{L}}\\prod_{j=1}^{N-1}\\frac{2c}{\\sqrt{(\\lambda_{\\iota(j)}-\\lambda_a)^2+c^2}}\\frac{1}{L(\\mu_j-\\lambda_{\\iota(j)})}\\,.\n\\end{equation}\nEquation \\eqref{ldff} corresponds to relabelling $\\mu_{\\iota^{-1}(j)}$ into $\\mu_j$ for $j=1,...,N,\\, j\\neq a$.\\\\\n\n\nSuch an expression implies an ordering of the roots $\\mu_1,...,\\mu_{N-1}$ according to the ordering of the $\\lambda_i$'s. Hence in the spectral sum \\eqref{bigsumfield}, there is no factor $1\/(N-1)!$ once expressed in terms of the Bethe numbers of the $\\mu_j$'s. Using this expression for the form factor, one indeed recovers the low-density correlation function \\eqref{psipsild2} properly defined  from the partial fraction decomposition, with  ${\\cal N}_{\\pmb{\\lambda}}, {\\cal N}_{\\pmb{\\mu}}=1+{\\cal O}(\\cal D)$ already imposed at leading order in ${\\cal D}$.\n\n\n\n\n\n\\section {Field two-point function\\label{fieldsection}}\nIn this section we compute $S_{0,0,0}$ in \\eqref{S00} in the low-density limit.\n\n\\subsection{States contributing to the thermodynamic limit}\nIn order to carry out the sum over the Bethe numbers in \\eqref{S00}, let us first investigate which values of Bethe numbers give a non-zero contribution in the thermodynamic limit.\n\nLet us consider Bethe number configurations such that $\\mu_i-\\lambda_i=\\mathcal{O}(L^{-b_i})$ for $i\\neq a$ with $ b_i\\geq 0$. Since the parity of the number of roots $\\lambda_i$ and $\\mu_i$ is different, the Bethe numbers of $\\lambda_i$ are integers (resp. half-odd integers) if those of $\\mu_i$ are half-odd integers (resp. integers). Hence from the Bethe equations it follows that one has $b_i\\leq 1$.\n\nNow, since the Bethe number $J_i$ can take $\\mathcal{O}(L^{1-b_i})$ values (for $\\mu_i-\\lambda_i=\\mathcal{O}(L^{-b_i})$ to be satisfied), and since $a$ in \\eqref{S00} can take $\\mathcal{O}(L)$ values, one has $\\mathcal{O}(L^{N-\\sum_{i\\neq a}b_i})$ many such configurations. Besides, each summand in \\eqref{S00} is $\\mathcal{O}(L^{-2N+1+2\\sum_{i\\neq a}b_i})$. Hence the contribution of these configurations is $\\mathcal{O}(L^{-N+1+\\sum_{i\\neq a}b_i})$. Given that $0\\leq b_i\\leq 1$, the only possibility to have a non-vanishing result in the thermodynamic limit is to have $\\forall i,\\, b_i=1$. Hence the only non-vanishing configurations in \\eqref{S00} are those for which the Bethe numbers of $\\mu_i$ differ from that of $\\lambda_i$ by $\\mathcal{O}(L^{0})$. We will denote\n\\begin{equation}\nn_i+\\frac{1}{2}=J_i-I_i\\qquad \\text{for }i\\neq a\\,,\n\\end{equation}\nthe difference between the Bethe numbers of $\\mu_i$ and $\\lambda_i$, with $n_i$ an integer of order ${\\cal O}(L^0)$.\n\n\\subsection{Decoupling of the spectral sum in the low-density limit}\nThe Bethe roots $\\mu_i$ involved in \\eqref{S00} depend on the difference of Bethe numbers $n_i$ and are all coupled through the Bethe equations. Taking the difference of the Bethe equations \\eqref{belog} for $\\mu_k$ and $\\lambda_k$, one obtains\n\\begin{equation}\n\\mu_k-\\lambda_k=\\frac{2\\pi}{L}(G^{-1}\\tilde{n})_k+{\\cal O}(L^{-2})\\,,\n\\end{equation}\nwith $G=G(\\pmb{\\mu})$ introduced in \\eqref{gaudin}, and the vector\n\\begin{equation}\n\\begin{aligned}\n\\tilde{n}_k&=n_k+\\frac{1}{2}+\\frac{1}{\\pi}\\arctan \\frac{\\lambda_k-\\lambda_a}{c}\\,.\n\\end{aligned}\n\\end{equation}\nIn the low-density limit, one has $Gx \\underset{\\text{l.d.}}{\\sim} x$ for all vectors $x$, so that the roots decouple and are expressed as\n\\begin{equation}\n\\mu_k-\\lambda_k\\underset{\\text{l.d.}}{\\sim} \\frac{2\\pi}{L}(n_k+\\alpha_k(\\lambda_a))+{\\cal O}(L^{-2})\\,,\n\\end{equation}\nwhere we introduced\n\\begin{equation}\\label{alpha}\n\\alpha_i(\\nu)=\\frac{1}{2}+\\frac{1}{\\pi}\\arctan \\frac{\\lambda_i-\\nu}{c}\\,.\n\\end{equation}\nFinally, at leading order in ${\\cal D}$, the determinants ${\\cal N}_{\\pmb{\\lambda}},{\\cal N}_{\\pmb{\\mu}}$ are equal to $1$ according to \\eqref{det1det1}. The spectral sum \\eqref{S00} can thus be expressed as a product of $N$ one-dimensional sums\n\\begin{equation}\\label{S001}\n\\begin{aligned}\nS_{0,0,0}\\underset{\\text{l.d.}}{\\sim}\\frac{1}{L}&\\sum_{a=1}^Ne^{it \\lambda_a^2-ix\\lambda_a}\\\\\n&\\times\\prod_{j\\neq a}\\left(\\frac{1}{\\pi^2}\\frac{c^2}{(\\lambda_j-\\lambda_a)^2+c^2}\\sum_{\\substack{n=-\\infty}}^{\\infty}\\frac{e^{i\\tfrac{2\\pi}{L} (n+\\alpha_{j}(\\lambda_a))(x-2\\lambda_j t)-it\\left(\\tfrac{2\\pi}{L}\\right)^2(n+\\alpha_j(\\lambda_a))^2}}{(n+\\alpha_j(\\lambda_a))^2}\\right)\\,.\n\\end{aligned}\n\\end{equation}\n\n\\subsection{Thermodynamic limit of the spectral sum}\nIn order to proceed we need to determine the thermodynamic limit of each of the one-dimensional sums, that are of the type\n\\begin{equation}\n\\sum_{n\\in\\mathbb{Z}}\\frac{e^{i\\frac{w}{L}(n+\\alpha)+i\\frac{\\tau}{L^2}(n+\\alpha)^2}}{(n+\\alpha)^2}\\,,\n\\end{equation}\nfor $\\alpha,w,\\tau$ reals, $\\alpha$ non integer. Let us first consider the case $\\tau=0$. The quantity $\\sum_{n\\in\\mathbb{Z}}\\frac{e^{iWn}}{(n+\\alpha)^2}$ is exactly the Fourier series of a certain $2\\pi$-periodic function of $W$. Noticing that for $n$ integer\n\\begin{equation}\n\\int_{-\\pi}^{\\pi}\\left[\\left(\\frac{\\pi}{\\sin \\pi\\alpha}\\right)^2e^{-iW\\alpha} +\\frac{i\\pi}{\\sin\\pi\\alpha}W e^{i\\pi\\alpha\\,\\text{sgn}\\,(W)-iW\\alpha} \\right]e^{-iWn}\\D{W}=\\frac{2\\pi}{(n+\\alpha)^2}\\,,\n\\end{equation}\nwe conclude that\n\\begin{equation}\n\\sum_{n\\in\\mathbb{Z}}\\frac{e^{iW(n+\\alpha)}}{(n+\\alpha)^2}=\\left(\\frac{\\pi}{\\sin \\pi\\alpha}\\right)^2 +\\frac{i\\pi}{\\sin\\pi\\alpha}W e^{i\\pi\\alpha\\,\\text{sgn}\\,(W)}\\,,\\qquad\\text{for }-\\pi<W\\leq\\pi\\,.\n\\end{equation}\nTo treat the case $\\tau\\neq 0$, we write\n\\begin{equation}\n\\sum_{n\\in\\mathbb{Z}}\\frac{e^{i\\frac{w}{L}(n+\\alpha)+i\\frac{\\tau}{L^2}(n+\\alpha)^2}}{(n+\\alpha)^2}=\\sum_{n\\in\\mathbb{Z}}\\frac{e^{i\\frac{w}{L}(n+\\alpha)}}{(n+\\alpha)^2}+\\frac{1}{L^2}\\sum_{n\\in\\mathbb{Z}}e^{i\\frac{w}{L}(n+\\alpha)}\\frac{e^{i\\tau\\frac{(n+\\alpha)^2}{L^2}}-1}{\\tfrac{(n+\\alpha)^2}{L^2}}\\,.\n\\end{equation}\nThe first term was computed previously, and the second one is $1\/L$ times the Riemann sum of a regular function, hence in the thermodynamic limit converges to its integral. This yields\n\\begin{equation}\\label{s2s2}\n\\sum_{n\\in\\mathbb{Z}}\\frac{e^{i\\frac{w}{L}(n+\\alpha)+i\\frac{\\tau}{L^2}(n+\\alpha)^2}}{(n+\\alpha)^2}=\\left(\\frac{\\pi}{\\sin \\pi\\alpha}\\right)^2 +\\frac{i\\pi w}{L\\tan(\\pi\\alpha)}-\\frac{\\pi|w|}{L}+\\frac{\\sqrt{|\\tau|}}{L}\\chi_\\pm\\left(\\tfrac{w}{\\sqrt{|\\tau|}}\\right)+{\\cal O}(L^{-2})\\,,\n\\end{equation}\nwith $\\pm=\\,\\text{sgn}\\,(\\tau)$ and\n\\begin{equation}\\label{chi}\n\\chi_\\pm(x)=\\int_{-\\infty}^\\infty e^{ixu}\\frac{e^{\\pm i u^2}-1}{u^2}\\D{u}\\,.\n\\end{equation}\nHence we obtain with $\\pm=\\,\\text{sgn}\\,(t)$, using the expression \\eqref{alpha} and trigonometric relations\n\\begin{equation}\n\\begin{aligned}\n&\\sum_{\\substack{-\\infty<n<\\infty}}\\frac{e^{\\frac{2i\\pi}{L} (n+\\alpha_{j}(\\lambda_a))(x-2\\lambda_j t)-\\frac{i(2\\pi)^2t}{L^2}(n+\\alpha_j(\\lambda_a))^2}}{(n+\\alpha_j(\\lambda_a))^2}=\\pi^2\\frac{(\\lambda_j-\\lambda_a)^2+c^2}{c^2}\\\\\n&\\qquad\\times \\Bigg[1-\\frac{1}{L}\\frac {2c^{2}} {\\left( \\lambda _{j}-\\lambda _{a}\\right) ^{2}+c^{2}}|x-2\\lambda_jt|+\\frac{1}{L}\\frac {2c^{2}} {\\left( \\lambda _{j}-\\lambda _{a}\\right) ^{2}+c^{2}}\\frac{\\sqrt{|t|}}{\\pi}\\chi_\\mp(\\tfrac{x-2\\lambda_jt}{\\sqrt{|t|}})\\\\\n&\\qquad\\qquad\\qquad\\qquad+\\frac {2ic(\\lambda _{j}-\\lambda _{a})} {\\left( \\lambda _{j}-\\lambda _{a}\\right) ^{2}+c^{2}}\\frac{2\\lambda _{j}t-x}{L} \\Bigg]+{\\cal O}(L^{-2})\\,.\n\\end{aligned}\n\\end{equation}\nPlugging this expression in \\eqref{S001}, we use\n\\begin{equation}\\label{expint}\n\\underset{L\\to\\infty}{\\lim}\\, \\prod_{i=1}^L (1+\\frac{f(i\/L)}{L})=\\exp\\left( \\int_0^1 f(x)\\D{x}\\right)\\,,\n\\end{equation}\nto obtain in the thermodynamic limit\n\\begin{equation}\\label{field}\n\\begin {aligned}\n \\left\\langle \\psi ^{\\dagger}\\left( x,t\\right) \\psi \\left( 0,0\\right) \\right\\rangle\\underset{\\text{l.d.}}{\\sim}\\int _{-\\infty }^\\infty &e^{-it\\lambda ^{2}+ix\\lambda } \\rho \\left( \\lambda \\right)\\exp \\left (i\\int_{-\\infty}^\\infty \\frac {2c(\\nu -\\lambda )} {\\left( \\nu -\\lambda \\right) ^{2}+c^{2}}(2t\\nu -x) \\rho \\left( \\nu  \\right) \\D{\\nu}\\right)  \\\\\n &\\exp \\left (-\\int_{-\\infty}^\\infty \\frac {2c^{2}} {\\left( \\nu -\\lambda \\right) ^{2}+c^{2}}|x-2\\nu t|\\rho \\left( \\nu  \\right) \\D{\\nu}\\right) \\\\\n &\\exp \\left (\\frac{\\sqrt{|t|}}{\\pi}\\int_{-\\infty}^\\infty \\frac {2c^{2}} {\\left( \\nu -\\lambda \\right) ^{2}+c^{2}}\\chi_\\mp(\\tfrac{x-2\\nu t}{\\sqrt{|t|}})\\rho \\left( \\nu  \\right) \\D{\\nu}\\right)\\D{\\lambda} \n  \\end {aligned}\n\\end{equation}\nThis expression is valid for all space $x$ and time $t$, and all interaction strength $c>0$. It is the leading term in the expansion \\eqref{expdensity}, and constitutes the low-density limit of the dynamical correlations of the field.\n\n\n\n\\section {Density two-point function\\label{densitysection}}\nIn this Section we apply the same reasoning as in Sections \\ref{ldd} and \\ref{fieldsection} but to the density two-point function.\n\n\\subsection{Partial fraction decomposition}\nWe consider $ |\\pmb{\\lambda} \\rangle $ and $ |\\pmb{\\mu} \\rangle $ eigenstates of the Lieb-Liniger Hamiltonian with $N$ rapidities, and define $F_\\sigma(\\pmb{\\lambda},\\pmb{\\mu})$ by\n\\begin{equation}\n\\begin{aligned}\n\\frac {\\left| \\left\\langle \\pmb{\\mu} |\\sigma \\left( 0\\right) |\\pmb{\\lambda}\\right\\rangle \\right| ^{2}} {\\left\\langle \\pmb{\\lambda} \\left| \\pmb{\\lambda} \\right\\rangle \\left\\langle \\pmb{\\mu}\\right| \\pmb{\\mu}\\right\\rangle }= \\frac{F_\\sigma(\\pmb{\\lambda},\\pmb{\\mu})}{L^{2N}\\mathcal{N}_{\\pmb{\\lambda}}\\mathcal{N}_{\\pmb{\\mu}}} \\,.\n\\end{aligned}\n\\end{equation}\nSimilarly to the field case $F_\\psi$, the reduced form factor $F_\\sigma(\\pmb{\\lambda},\\pmb{\\mu})$ is a ratio of two polynomials in the Bethe roots so that it can be decomposed into partial fractions. Repeating the arguments that apply to the field case, we similarly obtain the writing\n\\begin{equation}\\label{pfdsigma}\nF_\\sigma(\\pmb{\\lambda},\\pmb{\\mu})=\\sum_{\\{\\nu\\},f}\\frac{A(\\pmb{\\lambda},\\pmb{\\mu},\\{\\nu\\},f)}{\\prod_{i=1}^{N-1}(\\mu_i-\\lambda_{f(i)})^{\\nu_i}}\\,,\n\\end{equation}\nwhere each $\\nu_i$ takes the value $0,1$ or $2$, and where $f$ are functions $\\{i\\in\\{1,...,N\\}| \\nu_i\\neq 0\\}\\to \\{1,...,N\\}$. The coefficients $A(\\pmb{\\lambda},\\pmb{\\mu},\\{\\nu\\},f)$ crucially do not depend on any $\\mu_i$ whenever $\\nu_i>0$, and are bounded functions of real $\\mu_i$ if $\\nu_i=0$. The function $f$ has the same constraints as in the field case, namely it can take at most twice the same value, and takes only once a value on points with $\\nu_i=2$.\n\nThis decomposition also leads to an expansion in the particle density ${\\cal D}$. Namely, with the parametrization $m=|I_1|$, $n=|I_2|$, $p=|\\{i| \\nu_i=0\\}|$\n\\begin{equation}\n\\begin{aligned}\nI_2&=\\{j_{1},...,j_{n}\\}\\\\\nI_0&=\\{j_{n+1},...,j_{2n+p}\\}\\\\\nI_1&=\\{j_{2n+p+1},...,j_{2n+p+m}\\}\\\\\n\\{1,...,N\\}-(I_0\\cup I_1 \\cup I_2)&=\\{j_{2n+p+m+1},...,j_{N}\\}\\,.\n\\end{aligned}\n\\end{equation}\nwe have\n\\begin{equation}\\label{expdensitysigma}\n\\left\\langle  \\sigma\\left( x,t\\right) \\sigma \\left( 0,0\\right)  \\right\\rangle =\\sum_{n,m,p\\geq 0}S_{n,m,p}\\,,\n\\end{equation}\nwith\n\\begin{equation}\n\\begin{aligned}\n&S_{n,m,p}=\\frac{1}{2^np!L^{2N}}\\sum_{\\substack{I_{0,1,2}\\subset \\{1,...,N\\}\\\\|I_0|=n+p\\\\|I_1|=m\\\\|I_2|=n \\\\\\text{all disjoint}}}\\sum_{J_1,...,J_{N}}\\frac{{\\cal A}(I_0,I_1,I_2|\\{\\mu_i\\}_{i=2n+1}^{2n+p})}{\\mathcal{N}_{\\pmb{\\lambda}}\\mathcal{N}_{\\pmb{\\mu}}}\\\\\n  &\\qquad\\qquad\\times\\frac{e^{it\\left( E\\left( \\pmb{\\lambda}\\right) -E\\left( \\pmb{\\mu} \\right) \\right) +ix\\left( P\\left( \\pmb{\\mu}\\right) -P\\left( \\pmb{\\lambda}\\right) \\right) }}{\\prod_{i=1}^{n}(\\mu_{2i-1}-\\lambda_{j_i})(\\mu_{2i}-\\lambda_{j_i})\\prod_{i=2n+1+p}^{2n+m+p}(\\mu_i-\\lambda_{j_{i}})\\prod_{i=2n+m+p+1}^{N}(\\mu_i-\\lambda_{j_{i}})^{2}}\\,.\n\\end{aligned}\n\\end{equation}\n The leading term is a priori obtained with $\\nu_i=2$, and the next terms by replacing some of the $\\nu_i$'s by $1$ or $0$, each time picking a factor ${\\cal D}$. However, in contrast to the field case, the coefficient of the a priori leading term is actually found to vanish, using the analogue of \\eqref{formulA} for the density operator case\n \\begin{equation}\n{\\cal A}(\\varnothing,\\varnothing,\\varnothing|\\varnothing)=0\\,.\n\\end{equation}\nIf one $\\nu_i$ is set to $1$, the coefficient still vanishes\n \\begin{equation}\n{\\cal A}(\\varnothing,\\{a\\},\\varnothing|\\varnothing)=0\\,.\n\\end{equation}\nHowever, if it is set to $0$, the coefficient is non-zero and reads\n \\begin{equation}\n{\\cal A}(\\{a\\},\\varnothing,\\varnothing|\\mu_a)=\\prod_{j\\neq a}\\frac{4c^2(\\lambda_a-\\mu_a)^2}{[(\\lambda_j-\\lambda_a)^2+c^2][(\\lambda_j-\\mu_a)^2+c^2]}\\,.\n\\end{equation}\nHence the leading order in the density expansion, i.e. the low-density limit, is obtained with\n\\begin{equation}\n\\left\\langle  \\sigma( x,t) \\sigma( 0,0)  \\right\\rangle\\underset{\\text{l.d.}}{\\sim} S_{0,0,1}\\,.\n\\end{equation}\nIt yields the following low-density limit of the density two-point function\n\\begin{equation}\\label{2pfdensityld}\n\\begin{aligned}\n & \\left\\langle  \\sigma( x,t) \\sigma( 0,0)  \\right\\rangle \\underset{\\text{l.d.}}{\\sim} \\frac{1}{L^2}\\sum_{\\lambda_a,\\mu_a}e^{ix(\\mu_a-\\lambda_a)+it(\\lambda_a^2-\\mu_a^2)}\\\\&\\times\\sum _{J_i,\\, i\\neq a}\\frac{1}{\\mathcal{N}_{\\pmb{\\lambda}}\\mathcal{N}_{\\pmb{\\mu}}}\\prod_{j\\neq a}\\frac{c^2}{(\\lambda_j-\\lambda_a)^2+c^2}\\frac{c^2}{(\\lambda_j-\\mu_a)^2+c^2}\\cdot\n  \\frac{4(\\lambda_a-\\mu_a)^2}{c^2L^2(\\mu_j-\\lambda_j)^2}e^{ix(\\mu_j-\\lambda_j)+it(\\lambda_j^2-\\mu_j^2)}\\,.\n\\end{aligned}\n\\end{equation}\n\n\n\n\\subsection {Interpretation: low-density limit of the density form factor}\nSimilarly to the field operator, the low-density limit of the density two-point function is defined in terms of a partial fraction decomposition of the form factor. In the field case, one was also able to obtain a low-density limit of the field form factor by considering dilute states, see Section \\ref{intuitive}. In the density case, we are going to follow a different route in order to show the usefulness of \\eqref{ldff}, by recovering the low-density limit of the density correlation \\eqref{2pfdensityld} from the low-density approximation of the field form factor  \\eqref{ldff}.\n The rationale is that  the density form factor can be obtained as a limit of a field two-point function between different eigenstates $\\pmb{\\lambda}$ and $\\pmb{\\mu}$ with $N$ roots\n\\begin{equation}\n\\frac{\\langle \\pmb{\\mu}|\\sigma(0)|\\pmb{\\lambda}\\rangle}{\\sqrt{\\langle \\pmb{\\mu}|\\pmb{\\mu}\\rangle \\langle \\pmb{\\lambda}|\\pmb{\\lambda}\\rangle}}=\\underset{x\\to 0}{\\lim}\\, \\frac{\\langle \\pmb{\\mu}|\\psi^\\dagger( x,0) \\psi ( 0,0)|\\pmb{\\lambda}\\rangle}{\\sqrt{\\langle \\pmb{\\mu}|\\pmb{\\mu}\\rangle \\langle \\pmb{\\lambda}|\\pmb{\\lambda}\\rangle}}\\,.\n\\end{equation}\nThis two-point function can itself be expressed as a Lehmann representation\n\\begin{equation}\n\\frac{\\langle \\pmb{\\mu}|\\psi^\\dagger( x,0) \\psi ( 0,0)|\\pmb{\\lambda}\\rangle}{\\sqrt{\\langle \\pmb{\\mu}|\\pmb{\\mu}\\rangle \\langle \\pmb{\\lambda}|\\pmb{\\lambda}\\rangle}}=\\sum_{\\pmb{\\nu}}\\frac{\\langle \\pmb{\\nu}|\\psi(0)|\\pmb{\\mu}\\rangle^* \\langle \\pmb{\\nu}|\\psi(0)|\\pmb{\\lambda}\\rangle}{\\sqrt{\\langle \\pmb{\\mu}|\\pmb{\\mu}\\rangle \\langle \\pmb{\\lambda}|\\pmb{\\lambda}\\rangle}\\langle \\pmb{\\nu}|\\pmb{\\nu}\\rangle}e^{ix(P(\\pmb{\\nu})-P(\\pmb{\\mu}))}\\,,\n\\end{equation}\nwhere $\\pmb{\\nu}$ is a state with $N-1$ roots. In the low-density limit, according to \\eqref{ldff}, for the field form factors not to vanish one has to have $N-1$ roots $\\lambda_i$ with exactly one $\\nu_j$ around at a distance ${\\cal O}(L^{-1})$, leaving a $\\lambda_a$ without $\\nu_j$'s around. The same holds for the $\\mu_i$'s with respect to the $\\nu_j$'s. This implies that in the low-density limit, there is also exactly one $\\mu_i$ around each $\\lambda_j$ at a distance ${\\cal O}(L^{-1})$ for $j\\neq a$, leaving a $\\mu_a$ without $\\nu_j$'s nor $\\lambda_j$'s around. Since $\\pmb{\\mu}$ and $\\pmb{\\lambda}$ are an input of the problem, these $\\mu_a$ and $\\lambda_a$ are fixed by the choices of $\\pmb{\\lambda}$ and $\\pmb{\\mu}$ (they are not free parameters like in the two-point function case with $\\pmb{\\lambda}=\\pmb{\\mu}$). Then using  \\eqref{ldff} we obtain the following low-density limit\n\\begin{equation}\n\\begin{aligned}\n&\\frac{\\langle \\pmb{\\mu}|\\psi^\\dagger( x,0) \\psi ( 0,0)|\\pmb{\\lambda}\\rangle}{\\sqrt{\\langle \\pmb{\\mu}|\\pmb{\\mu}\\rangle \\langle \\pmb{\\lambda}|\\pmb{\\lambda}\\rangle}}\\underset{\\text{l.d.}}{\\sim} \\\\\n&\\qquad\\qquad\\frac{e^{-ix\\mu_a}}{L}\\sum_{\\pmb{\\nu}}\\prod_{j\\neq a}\\frac{2c}{\\sqrt{(\\lambda_j-\\lambda_a)^2+c^2}}\\frac{2c}{\\sqrt{(\\mu_j-\\mu_a)^2+c^2}} \\frac{1}{L(\\nu_j-\\lambda_j)}\\frac{e^{ix(\\nu_j-\\mu_j)}}{L(\\nu_j-\\mu_j)}\\,.\n\\end{aligned}\n\\end{equation}\nThe Bethe equations allow us to write\n\\begin{equation}\n\\begin{aligned}\n\\nu_j-\\lambda_j&\\underset{\\text{l.d.}}{\\sim} \\frac{2\\pi}{L}(n_j+\\alpha_j(\\lambda_a))\\\\\n\\nu_j-\\mu_j&\\underset{\\text{l.d.}}{\\sim} \\frac{2\\pi}{L}(n_j+\\alpha_j(\\mu_a)-p_j)\\,,\n\\end{aligned}\n\\end{equation}\nwith $n_j,p_j$ integers. The integer $p_j$ is related to $\\mu_j$ and $\\lambda_j$ through\n\\begin{equation}\n\\mu_j-\\lambda_j\\underset{\\text{l.d.}}{\\sim} \\frac{2\\pi}{L}(p_j+\\alpha_j(\\lambda_a)-\\alpha_j(\\mu_a))\\,,\n\\end{equation}\nwhich is a parameter of the problem that is not to be summed over. We obtain the following factorization\n\\begin{equation}\n\\begin{aligned}\n\\frac{\\langle \\pmb{\\mu}|\\psi^\\dagger( x,0) \\psi ( 0,0)|\\pmb{\\lambda}\\rangle}{\\sqrt{\\langle \\pmb{\\mu}|\\pmb{\\mu}\\rangle \\langle \\pmb{\\lambda}|\\pmb{\\lambda}\\rangle}}\\underset{\\text{l.d.}}{\\sim} \\frac{e^{-ix\\mu_a}}{L}&\\prod_{j\\neq a}\\frac{c^2}{\\pi^2}\\frac{1}{\\sqrt{(\\lambda_j-\\lambda_a)^2+c^2}}\\frac{1}{\\sqrt{(\\lambda_j-\\mu_a+{\\cal O}(L^{-1}))^2+c^2}} \\\\\n&\\times\\sum_{n=-\\infty}^\\infty\\frac{e^{\\frac{2i \\pi x}{L}(n+\\alpha_j(\\mu_a)-p_j)}}{(n+\\alpha_j(\\lambda_a))(n+\\alpha_j(\\mu_a)-p_j)}\\,.\n\\end{aligned}\n\\end{equation}\nIn order to determine the thermodynamic limit of the field four-point function we need to carry out the sums over the integers, that reduces to sums of the type\n\\begin{equation}\n\\sum_{n\\in\\mathbb{Z}}\\frac{e^{i\\tfrac{w}{L}(n+\\alpha)}}{n+\\alpha}\\,.\n\\end{equation}\nWe notice that $\\sum_{n\\in\\mathbb{Z}}\\frac{e^{iWn}}{n+\\alpha}$ is the Fourier series of a $2\\pi$-periodic function of $W$, and\n\\begin{equation}\n\\int_{-\\pi}^\\pi \\frac{\\pi}{\\sin \\pi\\alpha}e^{i\\pi\\alpha\\,\\text{sgn}\\,(W)-iW\\alpha} e^{-iWn}\\D{W}=\\frac{2\\pi}{n+\\alpha}\\,.\n\\end{equation}\nHence\n\\begin{equation}\n\\sum_{n\\in\\mathbb{Z}}\\frac{e^{iW(n+\\alpha)}}{n+\\alpha}=\\frac{\\pi}{\\sin \\pi\\alpha}e^{i\\pi\\alpha\\,\\text{sgn}\\,(W)}\\,,\\qquad\\text{for }-\\pi<W\\leq\\pi\\,.\n\\end{equation}\nTo use this formula we write\n\\begin{equation}\n\\frac{1}{(n+\\alpha_j(\\lambda_a))(n+\\alpha_j(\\mu_a)-p_j)}=\\frac{1}{p_j+\\alpha_j(\\lambda_a)-\\alpha_j(\\mu_a)}\\left(\\frac{1}{n+\\alpha_j(\\mu_a)-p_j} -\\frac{1}{n+\\alpha_j(\\lambda_a)}\\right)\\,.\n\\end{equation}\nIt yields\n\\begin{equation}\n\\begin{aligned}\n\\frac{\\langle \\pmb{\\mu}|\\psi^\\dagger( x,0) \\psi ( 0,0)|\\pmb{\\lambda}\\rangle}{\\sqrt{\\langle \\pmb{\\mu}|\\pmb{\\mu}\\rangle \\langle \\pmb{\\lambda}|\\pmb{\\lambda}\\rangle}}\\underset{\\text{l.d.}}{\\sim} &\\frac{e^{-ix\\mu_a}}{L}\\prod_{j\\neq a}\\frac{c^2}{\\pi}\\frac{1}{\\sqrt{(\\lambda_j-\\lambda_a)^2+c^2}}\\frac{1}{\\sqrt{(\\lambda_j-\\mu_a)^2+c^2}} \\\\\n&\\times\\frac{1}{p+\\alpha_j(\\lambda_a)-\\alpha_j(\\mu_a)}\\Bigg[\\frac{1}{\\tan \\pi\\alpha_j(\\mu_a)}-\\frac{e^{\\frac{2i\\pi x}{L}(\\alpha_j(\\mu_a)-\\alpha_j(\\lambda_a)-p_j)}}{\\tan \\pi\\alpha_j(\\lambda_a)} \\\\\n&\\qquad+i\\,\\text{sgn}\\,(x) \\left( 1-e^{\\frac{2i\\pi x}{L}(\\alpha_j(\\mu_a)-\\alpha_j(\\lambda_a)-p_j)}\\right)\\Bigg]\\,.\n\\end{aligned}\n\\end{equation}\nWe observe that the limit $x\\to 0$ is regular. Using the expression for $\\alpha_j(\\nu)$ \\eqref{alpha}, we obtain the following low-density limit of the form factor\n\\begin{equation}\\label{lddensity}\n\\begin{aligned}\n\\frac{\\langle \\pmb{\\mu}|\\sigma(0)|\\pmb{\\lambda}\\rangle}{\\sqrt{\\langle \\pmb{\\mu}|\\pmb{\\mu}\\rangle \\langle \\pmb{\\lambda}|\\pmb{\\lambda}\\rangle}}\\underset{\\text{l.d.}}{\\sim} \\frac{1}{L}&\\prod_{j\\neq a}\\frac{c}{\\sqrt{(\\lambda_j-\\lambda_a)^2+c^2}}\\frac{c}{\\sqrt{(\\lambda_j-\\mu_a)^2+c^2}}\\frac{2(\\mu_a-\\lambda_a)}{cL(\\mu_j-\\lambda_j)}\\,.\n\\end{aligned}\n\\end{equation}\nPlugging this expression into the Lehmann representation for the density two-point function \\eqref{bigsumfield}, one obtains exactly the low-density limit \\eqref{2pfdensityld} previously obtained from the partial fraction decomposition. This provides an interesting consistency check of the interpretations of the low-density limit as obtained from low-density approximations of form factors.\n\n\\subsection {The thermodynamic limit of the spectral sum}\nWe now wish to compute the spectral sum \\eqref{2pfdensityld} in the low-density limit. Repeating the arguments performed for the field case, in the thermodynamic limit only the states with $\\mu_j-\\lambda_j={\\cal O}(L^{-1})$ contribute in \\eqref{2pfdensityld}. Besides, like for the field correlations, the Bethe equations decouple in the low-density limit. By taking the difference of the Bethe equations, we have for $j\\neq a$\n\\begin{equation}\n\\mu_j-\\lambda_j\\underset{\\text{l.d.}}{\\sim} \\frac{2\\pi}{L}(p_j+\\alpha_j(\\lambda_a)-\\alpha_j(\\mu_a))\\,,\n\\end{equation}\nwith $p_j$ an integer. Using ${\\cal N}_{\\pmb{\\lambda}},{\\cal N}_{\\pmb{\\mu}}\\underset{\\text{l.d.}}{\\sim} 1$, we obtain the following factorization of the spectral sum\n\\begin{equation}\\label{specsum}\n\\begin{aligned}\n  \\left\\langle  \\sigma( x,t) \\sigma( 0,0)  \\right\\rangle &\\underset{\\text{l.d.}}{\\sim} \\frac{1}{L^2}\\sum_{\\lambda_a,\\mu_a}e^{ix(\\mu_a-\\lambda_a)+it(\\lambda_a^2-\\mu_a^2)}\\prod_{j\\neq a}\\frac{c^2}{(\\lambda_j-\\lambda_a)^2+c^2}\\frac{c^2}{(\\lambda_j-\\mu_a)^2+c^2}\\\\\n  &\\frac{(\\lambda_a-\\mu_a)^2}{\\pi^2c^2}\\sum_{p=-\\infty}^\\infty\\frac{e^{\\frac{2i\\pi (x-2\\lambda_j t)}{L}(p+\\alpha_j(\\lambda_a)-\\alpha_j(\\mu_a))-\\frac{it(2\\pi)^2}{L^2}(p+\\alpha_j(\\lambda_a)-\\alpha_j(\\mu_a))^2}}{(p+\\alpha_j(\\lambda_a)-\\alpha_j(\\mu_a))^2}\\,.\n\\end{aligned}\n\\end{equation}\nWe now use \\eqref{s2s2} to compute the sum over the integers $p$. We find\n\\begin{equation}\n\\begin{aligned}\n&\\sum_{p=-\\infty}^\\infty\\frac{e^{\\frac{2i\\pi (x-2\\lambda_j t)}{L}(p+\\alpha_j(\\lambda_a)-\\alpha_j(\\mu_a))-\\frac{it(2\\pi)^2}{L^2}(p+\\alpha_j(\\lambda_a)-\\alpha_j(\\mu_a))^2}}{(p+\\alpha_j(\\lambda_a)-\\alpha_j(\\mu_a))^2}=\\\\\n&\\frac{\\pi^2 c^2 (1+\\tfrac{(\\lambda_a-\\lambda_j)^2}{c^2})(1+\\tfrac{(\\mu_a-\\lambda_j)^2}{c^2})}{(\\lambda_a-\\mu_a)^2}\\Bigg[1+\\frac{2i(x-2\\lambda_j t)}{Lc} (\\mu_a-\\lambda_a)\\frac{1+\\tfrac{(\\mu_a-\\lambda_j)(\\lambda_a-\\lambda_j)}{c^2}}{(1+\\tfrac{(\\lambda_a-\\lambda_j)^2}{c^2})(1+\\tfrac{(\\mu_a-\\lambda_j)^2}{c^2})}\\\\\n&\\qquad\\qquad\\qquad-\\frac{2|x-2\\lambda_j t|}{Lc^2}\\frac{(\\lambda_a-\\mu_a)^2}{(1+\\tfrac{(\\lambda_a-\\lambda_j)^2}{c^2})(1+\\tfrac{(\\mu_a-\\lambda_j)^2}{c^2})}\\\\\n&\\qquad\\qquad\\qquad+\\frac{2\\sqrt{|t|}}{\\pi Lc^2}\\frac{(\\lambda_a-\\mu_a)^2}{(1+\\tfrac{(\\lambda_a-\\lambda_j)^2}{c^2})(1+\\tfrac{(\\mu_a-\\lambda_j)^2}{c^2})} \\chi_\\mp\\left(\\tfrac{x-2\\lambda_j t}{\\sqrt{|t|}}\\right)\\Bigg]+{\\cal O}(L^{-2})\n\\end{aligned}\n\\end{equation}\nPlugging this relation in \\eqref{specsum}, the product over $j$ becomes an exponential in the thermodynamic limit, according to \\eqref{expint}. The sum over the roots of the representative state $\\lambda_a$ becomes an integral over the root density $\\rho(\\lambda)$, while the sum over $\\nu_a$ that has to differ from any $\\lambda_j$ becomes an integral over the hole density $\\rho_h(\\lambda)$ defined in \\eqref{vartheta}. Hence one obtains\n\\begin{equation}\\label{densityde}\n\\begin{aligned}\n \\left\\langle  \\sigma( x,t) \\sigma( 0,0)  \\right\\rangle& \\underset{\\text{l.d.}}{\\sim} \\int_{-\\infty}^\\infty  \\int_{-\\infty}^\\infty e^{it(\\lambda^2-\\mu^2)+ix(\\mu-\\lambda)}\\rho(\\lambda)\\rho_h(\\mu)\\\\\n &\\exp \\left(2ic(\\mu-\\lambda)\\int_{-\\infty}^\\infty \\frac{(c^2+(\\mu-\\nu)(\\lambda-\\nu))}{(c^2+(\\lambda-\\nu)^2)(c^2+(\\mu-\\nu)^2)}(x-2\\nu t)\\rho(\\nu)\\D{\\nu} \\right)\\\\\n &\\exp \\left(-2c^2(\\mu-\\lambda)^2\\int_{-\\infty}^\\infty \\frac{|x-2\\nu t|}{(c^2+(\\lambda-\\nu)^2)(c^2+(\\mu-\\nu)^2)} \\rho(\\nu)\\D{\\nu} \\right)\\\\\n &\\exp \\left(\\frac{2c^2\\sqrt{|t|}}{\\pi}(\\mu-\\lambda)^2\\int_{-\\infty}^\\infty \\frac{\\chi_\\mp(\\tfrac{x-2\\nu t}{\\sqrt{|t|}})}{(c^2+(\\lambda-\\nu)^2)(c^2+(\\mu-\\nu)^2)} \\rho(\\nu)\\D{\\nu} \\right)\\D{\\lambda} \\D{\\mu}\\,.\n\\end{aligned}\n\\end{equation}\nThis expression is valid for all space $x$ and time $t$, and all interaction strength $c>0$. It is the leading term in the expansion \\eqref{expdensitysigma}, and is the low-density limit of the dynamical correlations of the density.\n\n\\subsection{Bare particle-hole excitations\\label{which}}\nIn the spectral sum \\eqref{bigsumfield} the intermediate states $\\pmb{\\mu}$ can be seen as excited states above the averaging state $\\pmb{\\lambda}$. In this picture it is natural to \\textit{expand} the spectral sum \\eqref{bigsumfield} in terms of the number of \\textit{particle-hole excitations} that $\\pmb{\\mu}$ has over $\\pmb{\\lambda}$. Such an expansion consists in writing\n\\begin{equation}\n\\left\\langle  \\sigma( x,t) \\sigma( 0,0)  \\right\\rangle={\\cal D}^2+\\sum_{n\\geq 1}\\mathfrak{S}_n\n\\end{equation}\nwith\n\\begin{equation}\\label{bare}\n\\mathfrak{S}_n=\\sum _{ \\substack{\\pmb{\\mu}\\\\ |\\{I_a\\}\\cap \\{J_a\\}|=N-n}}\\frac {\\left| \\left\\langle \\pmb{\\mu} |\\sigma( 0) |\\pmb{\\lambda}\\right\\rangle \\right| ^{2}} {\\left\\langle \\pmb{\\lambda} \\left| \\pmb{\\lambda} \\right\\rangle \\left\\langle \\pmb{\\mu}\\right| \\pmb{\\mu}\\right\\rangle }e^{it\\left( E\\left( \\pmb{\\lambda}\\right) -E\\left( \\pmb{\\mu} \\right) \\right) +ix\\left( P\\left( \\pmb{\\mu}\\right) -P\\left( \\pmb{\\lambda}\\right) \\right) }\\,,\n\\end{equation}\nwhich is the spectral sum restricted to intermediate states $\\pmb{\\mu}$ that share $N-n$ Bethe numbers $J_a$ with the Bethe numbers $I_a$ of $\\pmb{\\lambda}$. Ideally, each individual $\\mathfrak{S}_n$ would have a well-defined and finite thermodynamic limit $L\\to\\infty$ that could be represented as a multiple integral over root densities and hole densities\n\\begin{equation}\\label{multip}\n\\begin{aligned}\n\\underbrace{\\int_{-\\infty}^\\infty...\\int_{-\\infty}^\\infty}_{2n}&F_n(\\lambda_1,\\mu_1,...,\\lambda_n,\\mu_n)e^{it(E(\\pmb{\\lambda})-E(\\pmb{\\mu}))+ix(P(\\pmb{\\mu})-P(\\pmb{\\lambda}))}\\\\\n&\\qquad\\qquad\\qquad\\times\\rho(\\lambda_1)\\rho_h(\\mu_1)...\\rho(\\lambda_n)\\rho_h(\\mu_n)\\D{\\lambda_1}\\D{\\mu_1}...\\D{\\lambda_n}\\D{\\mu_n}\\,.\n\\end{aligned}\n\\end{equation}\nThe function $F_n(\\lambda_1,\\mu_1,...,\\lambda_n,\\mu_n)$ appearing in this expression would thus be identified as a \\textit{thermodynamic form factor} for $n$ particle-hole excitations\\cite{deNP15,deNP16,DNP18,panfil20}. This idea was backed by calculations of\n\\begin{equation}\n\\underset{\\mu\\to\\lambda}{\\lim}\\,F_1(\\lambda,\\mu)\\,,\n\\end{equation}\nthat is finite and well-defined in the thermodynamic limit, by various means \\cite{deNP16,cortescuberopanfil2}.\n\nHowever, this picture seems to be a priori contradicted by results of \\cite{granetessler20}, where the thermodynamic limit of the spectral sum \\eqref{bigsumfield} was computed exactly at order $c^{-2}$. At this order, the spectral sum \\textit{exactly} truncates to one- and two-particle-hole excitations, hence to $\\mathfrak{S}_1$ and $\\mathfrak{S}_2$. But it was found that these two separate sums in the thermodynamic limit individually \\textit{diverge} and \\textit{depend on the representative state} $\\pmb{\\lambda}$ of the root density, at order $c^{-2}$. Namely we have\n\\begin{equation}\n\\begin{aligned}\\label{s1s2}\n&\\mathfrak{S}_1=LA_1+f_1(\\rho,\\gamma_{-2})+{\\cal O}(L^{-1})+{\\cal O}(c^{-3})\\\\\n&\\mathfrak{S}_2=LA_2+f_2(\\rho,\\gamma_{-2})+{\\cal O}(L^{-1})+{\\cal O}(c^{-3})\\,,\n\\end{aligned}\n\\end{equation}\nwith $A_{1,2}$ some reals of order $c^{-2}$ and $f_{1,2}(\\rho,\\gamma_{-2})$ functions of the root density and the pair distribution function (see \\cite{granetessler20} for a precise definition). However, their sum $\\mathfrak{S}_1+\\mathfrak{S}_2$ is not divergent and depends only on $\\rho$, as we expect. To fit in the general picture \\eqref{multip}, the only solution would be that these divergences in $L$ are an artefact of the $1\/c$ expansion, i.e. that the finite-size corrections to $\\mathfrak{S}_{1,2}$ in the thermodynamic limit would be e.g. of the form $\\varphi(L\/c)$ with a function $\\varphi(x)\\to 0$ when $x\\to \\pm\\infty$. In the framework of the low-density expansion, one can investigate this issue since the calculations are performed non-perturbatively in $1\/c$, and compute the thermodynamic limit of $\\mathfrak{S}_{1}$ for a finite fixed $c$, in the low-density limit.\\\\\n\n\nTo that end, let us use the expression \\eqref{lddensity} for the low-density limit of the form factor to write\n\\begin{equation}\\label{lowdensquare}\n\\begin{aligned}\n\\frac{|\\langle \\pmb{\\mu}|\\sigma(0)|\\pmb{\\lambda}\\rangle|^2}{\\langle \\pmb{\\mu}|\\pmb{\\mu}\\rangle \\langle \\pmb{\\lambda}|\\pmb{\\lambda}\\rangle}\\underset{\\text{l.d.}}{\\sim} \\frac{1}{L^2}&\\prod_{j\\neq a}\\frac{c^2}{(\\lambda_j-\\lambda_a)^2+c^2}\\frac{c^2}{(\\lambda_j-\\mu_a)^2+c^2}\\frac{4(\\mu_a-\\lambda_a)^2}{c^2L^2(\\mu_j-\\lambda_j)^2}\\,,\n\\end{aligned}\n\\end{equation}\nand assume that $\\pmb{\\mu}$ only involves one particle-hole excitation above $\\pmb{\\lambda}$. Hence for all $j\\neq a$, the Bethe numbers $J_j$ of $\\pmb{\\mu}$ are equal to the Bethe numbers $I_j$ of $\\pmb{\\lambda}$. In the low-density limit we have then\n\\begin{equation}\n\\forall j\\neq a,\\qquad\\mu_j-\\lambda_j\\underset{\\text{l.d.}}{\\sim} \\frac{2\\pi}{L}(\\alpha_j(\\lambda_a)-\\alpha_j(\\mu_a))\\,.\n\\end{equation}\nIt follows that in the thermodynamic limit, the low-density form factor squared \\eqref{lowdensquare} becomes\n\\begin{equation}\n\\frac{|\\langle \\pmb{\\mu}|\\sigma(0)|\\pmb{\\lambda}\\rangle|^2}{\\langle \\pmb{\\mu}|\\pmb{\\mu}\\rangle \\langle \\pmb{\\lambda}|\\pmb{\\lambda}\\rangle}\\underset{\\text{l.d.}}{\\sim} \\frac{1}{L^2}\\exp[L\\phi(\\lambda_a,\\mu_a)+{\\cal O}(L^0)]\\,,\n\\end{equation}\nwith\n\\begin{equation}\\label{phi}\n\\phi(\\lambda,\\mu)=\\int_{-\\infty}^\\infty \\log \\left[\\frac{1}{c^2}\\frac{1}{1+\\tfrac{(\\nu-\\lambda)^2}{c^2}} \\frac{1}{1+\\tfrac{(\\nu-\\mu)^2}{c^2}}\\left( \\frac{\\mu-\\lambda}{\\arctan \\tfrac{\\mu-\\nu}{c}-\\arctan \\tfrac{\\lambda-\\nu}{c}}\\right)^2\\right] \\rho(\\nu)\\D{\\nu}\\,.\n\\end{equation}\nThe sign of the logarithm can be determined by applying the following inequality that is valid for all real $u$\n\\begin{equation}\\label{ineq}\nu^2\\geq \\sin^2 u\\,,\n\\end{equation}\nto the value\n\\begin{equation}\nu=\\arctan x-\\arctan y\\,,\n\\end{equation}\nfor $x,y$ reals. Indeed, using trigonometric relations, Eq \\eqref{ineq} is exactly\n\\begin{equation}\n\\left( \\frac{\\arctan x-\\arctan y}{x-y}\\right)^2\\geq \\arctan'(x)\\arctan'(y)\\,,\n\\end{equation}\nwith equality only for $x=y$. This permits to deduce that the logarithm in \\eqref{phi} is always negative whenever $\\lambda\\neq \\mu$. One concludes that we always have if $\\lambda\\neq \\mu$\n\\begin{equation}\n\\phi(\\lambda,\\mu)<0\\,,\n\\end{equation}\nand if $\\lambda=\\mu$ we have\n\\begin{equation}\n\\phi(\\lambda,\\lambda)=0\\,.\n\\end{equation}\nIt follows that in the framework of a multiple integral representation in terms of bare particle-hole excitations in the thermodynamic limit \\eqref{multip}, the function $F_1(\\lambda,\\mu)$  would be in fact zero everywhere except if $\\mu=\\lambda$ where it takes the value $1$ as deduced from \\eqref{lowdensquare}. Namely we would have\n\\begin{equation}\nF_1(\\lambda,\\mu)= \\begin{cases}0\\qquad \\text{if }\\lambda\\neq \\mu\\\\\n1\\qquad \\text{if }\\lambda= \\mu\n\\end{cases}+{\\cal O}({\\cal D})\\,.\n\\end{equation}\n\n\nThis analysis shows that the integral representation \\eqref{multip} in terms of 'bare' particle-hole excitations \\eqref{bare} is singular. If the leading behaviour at large space and time of \\eqref{multip} is indeed obtained when $\\lambda_1=\\mu_1$ in \\eqref{multip} for $n=1$, where the function $F_1(\\lambda,\\mu)$ takes a finite value, the singular behaviour of this representation should be an obstacle to the computation of the subleading orders.\n\n\n\\subsection{Dressed particle-hole excitations\\label{which2}}\nLet us now investigate the nature of the states summed over to obtain the expression \\eqref{densityde} in the low-density limit. The double integral over the root density and hole density are one-particle-hole excitations with a \\textit{macroscopic} amplitude (i.e. the difference between the Bethe numbers is ${\\cal O}(L)$). The exponentials however arise from the product of a macroscopic number of one-dimensional sums over all the other remaining roots of the averaging states. These take into account an \\textit{arbitrary} number of particle-hole excitations, but with only a \\textit{microscopic} or  \\textit{mesoscopic} amplitude (i.e. the difference between the Bethe numbers can be ${\\cal O}(L^\\nu)$ for any $\\nu<1$). These configurations are represented in Figure \\ref{exfnp1k0}. They correspond to what is called a \\textit{dressed one-particle-hole excitation}. They entail expanding the spectral sum \\eqref{bigsumfield} according to\n\\begin{equation}\\label{dressedexp}\n\\left\\langle  \\sigma( x,t) \\sigma( 0,0)  \\right\\rangle={\\cal D}^2+\\sum_{n\\geq 1}\\mathfrak{S}^{\\rm dr}_n\\,,\n\\end{equation}\nwith\n\\begin{equation}\\label{sdr}\n\\sum_{m=1}^n\\mathfrak{S}^{\\rm dr}_m=\\sum _{ \\substack{\\pmb{\\mu}\\\\ \\exists \\tau\\text{ permutation of }\\{1,...,N\\},\\\\\\,\\# \\{a\\text{ s.t. }|I_a-J_{\\tau(a)}|={\\cal O}(L)\\} \\leq n}}\\frac {\\left| \\left\\langle \\pmb{\\mu} |\\sigma( 0) |\\pmb{\\lambda}\\right\\rangle \\right| ^{2}} {\\left\\langle \\pmb{\\lambda} \\left| \\pmb{\\lambda} \\right\\rangle \\left\\langle \\pmb{\\mu}\\right| \\pmb{\\mu}\\right\\rangle }e^{it\\left( E\\left( \\pmb{\\lambda}\\right) -E\\left( \\pmb{\\mu} \\right) \\right) +ix\\left( P\\left( \\pmb{\\mu}\\right) -P\\left( \\pmb{\\lambda}\\right) \\right) }\\,.\n\\end{equation}\n\n\n\\begin{figure}[H]\n\\begin{center}\n\\begin{tikzpicture}[scale=1]\n\\draw[->,blue]     (3.5,0.25) arc (180: 0:2.5);\n\\draw[->,red]     (0.5,0.25) arc (180: 0:0.25);\n\\draw[->,red]     (5.,0.25) arc (180: 0:0.25);\n\\draw[->,red]     (10.5,0.25) arc (0: 180:0.5);\n\\node at (-1,0) {.};\n\\draw[black] (-0.5,0) circle (3pt);\n\\node at (0,0) {.};\n\\draw[black] (0.5,0) circle (3pt);\n\\node at (1,0) {.};\n\\draw[black] (1.5,0) circle (3pt);\n\\draw[black] (2,0) circle (3pt);\n\\node at (2.5,0) {.};\n\\node at (3,0) {.};\n\\draw[black] (3.5,0) circle (3pt);\n\\draw[black] (4,0) circle (3pt);\n\\draw[black] (4.5,0) circle (3pt);\n\\draw[black] (5,0) circle (3pt);\n\\node at (5.5,0) {.};\n\\draw[black] (6,0) circle (3pt);\n\\node at (6.5,0) {.};\n\\node at (7,0) {.};\n\\draw[black] (7.5,0) circle (3pt);\n\\node at (8,0) {.};\n\\node at (8.5,0) {.};\n\\draw[black] (9,0) circle (3pt);\n\\node at (9.5,0) {.};\n\\draw[black] (10,0) circle (3pt);\n\\draw[black] (10.5,0) circle (3pt);\n\\draw[black] (11,0) circle (3pt);\n\\node at (11.5,0) {.};\n\\draw[black] (12,0) circle (3pt);\n\\node at (12.5,0) {.};\n\\filldraw[black] (-0.5,0) circle (2pt);\n\\filldraw[black] (1.,0) circle (2pt);\n\\filldraw[black] (1.5,0) circle (2pt);\n\\filldraw[black] (2,0) circle (2pt);\n\\filldraw[black] (8.5,0) circle (2pt);\n\\filldraw[black] (4,0) circle (2pt);\n\\filldraw[black] (4.5,0) circle (2pt);\n\\filldraw[black] (5.5,0) circle (2pt);\n\\filldraw[black] (6,0) circle (2pt);\n\\filldraw[black] (9,0) circle (2pt);\n\\filldraw[black] (7.5,0) circle (2pt);\n\\filldraw[black] (10,0) circle (2pt);\n\\filldraw[black] (9.5,0) circle (2pt);\n\\filldraw[black] (11,0) circle (2pt);\n\\filldraw[black] (12,0) circle (2pt);\n\\end{tikzpicture}\n\\end{center}\n\\caption{Sketch of a dressed one-particle-hole excitation: positions of the\nmomenta of the averaging state $\\pmb{\\lambda}$ (empty circles) and the\nintermediate state $\\pmb{\\mu}$  (filled circles) respectively, and position of \nthe holes (dots). In red is indicated the 'soft modes' corresponding to microscopic excitations, and in blue the only macroscopic excitation.} \n\\label{exfnp1k0}\n\\end{figure}\n\nLoosely speaking, $\\mathfrak{S}^{\\rm dr}_n$ includes $n$ macroscopic particle-hole excitations and any number of microscopic particle-hole excitations. Since these configurations are not disjoint, one requires the expression \\eqref{sdr} for a precise definition. In the low-density limit the full spectral sum truncates to $\\mathfrak{S}^{\\rm dr}_1$, and this expression is found to be finite and well-defined in the thermodynamic.\n\n\nWe note that these dressed particle-hole excitations are actually in principle what is computed in \\cite{deNP15,deNP16,DNP18}. But therein the 'soft modes' contribute as a numerical factor that multiplies the form factor. Here, within the low density expansion, it is seen in \\eqref{densityde} that these soft modes actually carry an $x$ and $t$ dependence as well. Namely we have a representation for $\\mathfrak{S}_n^{\\rm dr}$\n\\begin{equation}\n\\begin{aligned}\n\\underbrace{\\int_{-\\infty}^\\infty...\\int_{-\\infty}^\\infty}_{2n}&F_n^{x,t}(\\lambda_1,\\mu_1,...,\\lambda_n,\\mu_n)e^{it(E(\\pmb{\\lambda})-E(\\pmb{\\mu}))+ix(P(\\pmb{\\mu})-P(\\pmb{\\lambda}))}\\\\\n&\\qquad\\qquad\\qquad\\times\\rho(\\lambda_1)\\rho_h(\\mu_1)...\\rho(\\lambda_n)\\rho_h(\\mu_n)\\D{\\lambda_1}\\D{\\mu_1}...\\D{\\lambda_n}\\D{\\mu_n}\\,,\n\\end{aligned}\n\\end{equation}\nwhere the 'dressed thermodynamic form factor' $F_n^{x,t}(\\lambda_1,\\mu_1,...,\\lambda_n,\\mu_n)$ carries a $x,t$ dependence coming from the soft modes summation, although the soft modes do not modify the energy and momentum of the state in the thermodynamic limit. This dependence emerges from the double poles that  lift to order $L^0$ some finite-size corrections. The function $F_1^{x,t}(\\lambda,\\mu)$ can be directly deduced from \\eqref{densityde}.\n\nFrom Sections \\ref{which} and \\ref{which2}, we conclude that the right expansion scheme of the spectral sum is an expansion in terms of dressed particle-hole excitations \\eqref{dressedexp}, in the sense that the truncated spectral sums $\\mathfrak{S}_n^{\\rm dr}$ are finite and well-defined in the thermodynamic limit, and are smooth functions of the macroscopic excited rapidities. Moreover, as shown in \\cite{granetessler20}, they admit a well-defined and uniform $1\/c$ expansion without any spurious divergences in $L$ coming from $c$-dependent finite-size effects. These two properties are not satisfied by the 'bare' particle-hole excitations expansion.\n\n\n\n\\section{Summary and conclusion}\nWe computed the dynamical two-point function of the field and density operator averaged within a state with a small particle density ${\\cal D}$, given by Equations \\eqref{field} and \\eqref{densityde}. They are valid for an arbitrary interaction strength $c>0$ and for all space and time -- hence one can deduce the spectral function and dynamical structure factor in the full momentum-frequency plane, in the same regime of small ${\\cal D}$. This low-density limit is defined as the leading term in the correlation functions obtained by decomposing the form factors into partial fractions. \n\nBesides the explicit expressions obtained in the low-density regime, this work also provides interesting insights on the nature of states that contribute to the spectral sum, and on its possible expansions as detailed in Sections \\ref{which} and \\ref{which2}. The low-density regime is indeed naturally interpreted as a single dressed particle-hole excitation, i.e. a macroscopic particle-hole excitation with an arbitrary number of microscopic particle-hole excitations. In contrast, an integral representation in terms of 'bare' particle-hole excitations in the thermodynamic limit is found to be singular, in the sense that the 'thermodynamic form factor' for one-particle-hole excitations is non-zero only in the limit where the amplitude of the macroscopic excitation vanishes. Hence this work indicates that the right expansion of the spectral sum is in terms of dressed particle-hole excitations, as already suggested by the $1\/c$ expansion developed in \\cite{granetessler20}. Importantly, this dressing by microscopic particle-hole excitations also comes with an $x$ and $t$ dependence.\n\n\nThis PFD framework allows in principle for a computation of the next orders, which constitutes the most natural direction of improvement of this work. The fact that the next orders can indeed be computed with the PFD has been shown in \\cite{GFE20} for a model that can be reformulated in terms of free fermions. In the Lieb-Liniger case, the interaction introduces technical but not fundamental difficulties, and we hope to be able to pursue this program in following works. \n\n\n\n\\paragraph{Acknowledgements}\nWe thank Fabian Essler and Jacopo De Nardis for helpful discussions. This work was\nsupported by the EPSRC under grant EP\/S020527\/1.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nNamed entities are sequences of words that bring basic predefined semantic information that usually refers to locations, persons, organization\\ldots that can be denoted by proper nouns or that are unique in the real world, and they usually include numeric and temporal values. Named entities often constitute the first semantic bricks to extract in order to construct a structured semantic representation of a document content. \n\nNamed entity recognition (NER) is among SLU tasks that usually extract semantic information from textual documents. \nUntil now, NER from speech is made through a pipeline process that consists in processing first an automatic speech recognition (ASR) on the audio and then processing a NER on the ASR outputs.\nSuch approach has some disadvantages.\n\nFor instance, ASR errors have a negative impact on the NER performances, introducing noise within the text to be processed~\\cite{hatmi2013named}. Rule-based NER systems are usually built to process written language and are not robust to ASR errors. Machine learning based systems do not have good performance when they are trained on perfect transcriptions and deployed to process ASR ones, even if that can be partially compensated by simulating ASR errors in textual training data~\\cite{simonnet:hal-01715923}.\nAdditionally, ASR systems are generally tuned in order to get the lowest word error rate on a validation corpus, but this metric is not optimal to the NER task. For instance, this metric does not distinguish between errors on verbs or proper nouns while such errors do not have the same impact for NER. To compensate this problem, some dedicated metrics to tune ASR systems for better NER performances have been proposed, such as in~\\cite{jannet2017investigating}. Another inconvenience is that usually no information about named entities are used in the ASR process, while such information could help to better choose the partial recognition hypotheses that are dropped away during the decoding process. As a consequence, even when confusion networks or word lattices are used to go beyond the 1-best ASR hypothesis for a better robustness to ASR errors~\\cite{hakkani2006beyond}, such search space have been pruned without taking into account knowledge on named entity.\n\nIn the past, and integrated approach built on a high coupling of ASR and NER modules has been proposed~\\cite{servan2006conceptual}, based on the finite-state machine (FSM) paradigm (\\textsl{i.e.} transducer composition), showing that such integration can offer significant improvements in terms of NER quality. The main limit of this approach concerns the FSM paradigm itself, that is not able to natively model long distant constraint without combinatory explosion and that, by nature, can only express dependencies through a regular grammar. Another proposition to inject information about named entities in the ASR consists in directly adding some expressions of named entities into the ASR vocabulary~\\cite{hatmi2013incorporating}, and to estimate a language model for speech recognition that take into account these named entity expressions. The main default of a such approach is that it cannot allow to detect named entity that were not injected in the ASR vocabulary.\n\nAll of these issues motivate our research work on neural end-to-end  approach to extract named entities from speech.  \nOn a such way, a joint optimization is able for both ASR and NER in a NER task perspective, the architecture is more compact than the ones used in usual pipeline, and we expect to take benefit of the deep neural architecture capacities to capture long distant constraint at the sentence level. \nVery recently, a similar approach has been proposed by Facebook on a paper posted on the arXiv.org website~\\cite{serdyuk2018towards}. This end-to-end  approach is dedicated to domain and intent classification tasks, and experiments were carried on internal data close to the spirit of the ATIS corpus, as expressed by the authors.\n\nIn this paper, we present a first study of an end-to-end  approach to extract named entities.\nOur neural architecture is very similar to the Deep Speech 2 neural ASR system proposed by Baidu in~\\cite{amodei2016deep}. To use it for named entity recognition, we apply a multi-task training and modify the sequence of characters to be recognized from speech.\nExperiments were carried on French data easily accessible, and so reproducible, that were distributed in the framework on evaluation campaigns and are still available. \nThis paper is structured as follows. Section 2 describes the neural ASR architecture we used. Section 3 explains how we propose to exploit a such neural architecture for named entity extraction from speech. Section 4 presents some propositions to optimize the system and also compensate the lack of manually annotated audio data. Section 5 presents our experimental results, before the conclusion.\n\n\n\\section{Model architecture}\n\nThe RNN architecture used in this study is similar to the Deep~Speech~2 neural ASR system proposed by Baidu in~\\cite{amodei2016deep}. \nThis architecture is composed of $nc$ convolution layers (CNN), followed by $nr$ uni or bidirectional recurrent layers, a lookahead convolution layer~\\cite{wang2016lookahead}, and one fully connected layer just before the softmax layer, as shown in Figure~\\ref{DeepSpeech}.\n\\vspace{-0.2cm}\n\\begin{figure}[!htbp]\n\\begin{center}\n\\includegraphics [scale=0.45]{DeepSpeech2}\n\\caption{\\label{DeepSpeech} Deep RNN architecture used to extract named entities from French speech.}\n\\end{center}\n\\end{figure}\n\\vspace{-0.3cm}\nThe system is trained end-to-end  using the CTC loss function~\\cite{graves2006connectionist}, in order to predict a sequence of characters from the input audio. \nIn our experiments we used two CNN layers and six bidirectional recurrent layers with batch normalization as mentioned in~\\cite{amodei2016deep}. \n\nGiven an utterance $x^{i}$ and label $y^{i}$  sampled from a training set $X ={(x^{1}, y^{1}), (x^{2}, y^{2}), . . .}$, the RNN architecture has to train to convert an input sequence  $x^{i}$ into a final transcription $y^{i}$s. For notational convenience, we drop the superscripts and use $x$ to denote a chosen utterance and $y$ the corresponding label.\n\nThe RNN takes as input an utterance $x$ represented by a sequence of log-spectrograms of power normalized audio clips, calculated on 20ms windows. As output, all the characters $l$ of a language alphabet may be emitted, in addition to the space character used to segment character sequences into word sequences (space denotes word boundaries).\n\nThe RNN makes a prediction $p(l_{t}|x)$ at each output time step $t$.  \n\nAt test time, the CTC model is coupled with a language model trained on a big textual corpus. A specialized beam search CTC decoder~\\cite{hannun2014first} is used to find the transcription $y$ that maximizes :\n\\begin{equation}\nQ(y)=log(p(l_{t}|x)) + \\alpha log(pLM(y)) + \\beta wc(y)\n\\end{equation}\nwhere wc(y) is the number of words in the transcription $y$. The weight $\\alpha$ controls the relative contributions of the language model and the CTC network. The weight $\\beta$ controls the number of words in the transcription.\n\n\\section{Named entity extraction process} \nIn the literature, many studies focus on named entity recognition from text.  \nState-of-the-art systems are based on neural networks architectures. Some of them rely heavily on hand-crafted features and domain-specific knowledge~\\cite{collobert2011natural,chiu2015named}. Recent approaches~\\cite{lample2016neural,ma2016end} takes benefits from both word and\/or character-level embeddings learned automatically, by using combination of bidirectional LSTM, CNN and CRF. \nHowever, named entities recognition from automatic transcriptions is less studied. \nThis task is made through a pipeline process that consists in processing first an automatic speech recognition (ASR) on the audio and then processing a NER on the ASR outputs~\\cite{raymond2013robust}. \nUsually, the named entity recognition task is to assign a named entity tag to every word in a sentence. A single named entity could concern several words within a sentence. For this reason, the word-level labels begin-inside-outside (BIO) encoding~\\cite{Lance1995BIO} is very often adopted.\n\nIn this preliminary study, we focus on named entity extraction from speech using the network described above, without changing the neural architecture. \nWe would like to evaluate if this neural architecture is able to capture high level semantic information that allow it to recognize named entities. For that, we propose to modify the character sequence that the neural network has to produce: information about named entities are added in the initial character sequence.\nInstead of applying a BIO approach, we propose to add some tag characters in this sequence to delimit named entities boundaries, but also their category. \nWe are interested to eight NE categories that are: \\textit{person}, \\textit{function}, \\textit{organization}, \\textit{location}, \\textit{production}, \\textit{amount}, \\textit{time} and  \\textit{event}. \n\nIn our experiments, the system will attribute a tag ``$B_{NE}$'' or ``end''  only before and after the named entities, the other words are not concerned.  \nTo distinguish the named entity category, we consider a begin tag for each NE category. Only one ``end'' tag is used for all the NE categories, considering that since there is no overlap between named entities in a such representation, this information is sufficient to delimit the end of a named entity. \n\nAccording to the eight named entity categories targeted by the task, nine NE tags has to be added to the character list emitted by the neural network: ``$B_{pers}$'', ``$B_{func}$'', ``$B_{org}$'', ``$B_{loc}$'', ``$B_{prod}$'', ``$B_{amount}$'', ``$B_{time}$'', ``$B_{event}$'', and $``end''$. \nAs such neural model predicts a character at each time step, we propose to map each of these nine NE tags to one single special character, that corresponds respectively to: ``['', ``('', ``\\{'',``\\$'', ``\\&'', ``\\%'', ``\\#'', ``)'' and ``]'', as illustrated in  Figure~\\ref{fig:example}. \n\nWith this way, the NE tags are included in the prediction process, and are taken into account by the CTC loss function during the training process.\n\\vspace{-0.2cm}\n\\begin{figure*}[!htbp]\n\\begin{center}\n\\includegraphics [scale=0.45]{example}\n\\caption{\\label{fig:example} Example of mapping the real NE tags to character sequence. This sentence means, in English and case sensitive: \"the sculptor Caesar died yesterday in Paris at the age of seventy-seven years\"}\n\\vspace{-0.1cm}\n\\end{center}\n\\end{figure*}\n\\section{Multi-task training, data augmentation, and starred mode} \n\\label{sec:tricks}\n\nAudio recordings with both manual transcriptions and manual annotations of named entities are relatively rare, while neural end-to-end approaches are known to need large amount of data to become competitive.\n\nTo compensate this lack of data, we first propose to apply a multi-task learning approach to train the neural network. This consists in starting to train it only for the ASR task, without emitting character used to represent named entities, on all the audio recordings available with their manual transcriptions. At the end, the softmax layer is reinitialized to take into consideration the named entity tag markers, and a new training process is realized, on the named entity recognition task, with only training data with manual annotations of named entities.\n\nA second proposition consists in artificially increasing the training data for the named entity recognition task. For this purpose, we propose to apply a named entity recognition system dedicated to text data in order to tag the manual transcriptions used to train the ASR neural network. Then, these manual transcriptions automatically annotated with named entities can be injected in the training data used to train the neural network to extract named entities from speech.\n\nIn addition, since we want the system to focus on named entities, and since the CTC loss gives the same importance to each character, we propose to modify the character sequence that the neural network must emit to give more importance to named entities. This proposition is interesting to better understand how the CTC loss behaves on this case, and consists in replacing by a star ''*'' all character subsequences that do not contain a named entity. For instance, the character sequence presented in Figure~\\ref{fig:example} becomes: \\textit{* [ c\\'esar ] * \\# hier ] * \\$ paris ] * \\% soixante dix sept ans ]}. We call this approach the starred mode, and we expect that it can make the neural model more sensitive to named entities.\n\\vspace{-0.1cm}\n\\section{Experiments}\n\\subsection{Experimental setups}\n\\label{sec:corpus}\nExperiments have been carried out on four different French corpora, including ESTER 1\\&2, ETAPE and Quaero.\nThese corpora are composed of data recorded from francophone radio and TV stations, and are annotated with named entities.\n\nThe ESTER corpora were divided into three parts: training, development and evaluation.\nESTER 1 \\cite{galliano2005ester} training (73 hours) and development (17 hours) corpora are composed of data recorded from four radio stations in French. ESTER 1 test corpus is composed of 10 hours coming from the same four radio stations plus two other stations, all of which recorded 15 month after the development data.\n\nESTER 2 \\cite{galliano2009ester} training corpus was not annotated with named entities and was not used in this study. The development (17 hours) and test set (10 hours) is composed of manual transcriptions of speech recorded from six radio stations (two of those radio stations were already used in ESTER 1).\n\nThe ETAPE \\cite{gravier2012etape} data consists of manual transcriptions and annotations of TV and radio shows. It contains 36 hours of speech, recorded between 2010 and 2011, divided into three parts: training (22 hours), development (7 hours) and test (7h).\n\nQUAERO (ELRA-S0349) data is composed of 12 hours of manual transcriptions of TV and radio shows coming from 6 different sources recorded in 2010.\n\nOur corpus, called DeepSUN, is the combination of those four corpora. \nThe training corpus is composed of the training sets of ESTER 1, ETAPE and QUAERO, while, the  development and test sets are composed respectively of the development and test sets of ESTER 1\\&2, and ETAPE.\nIt contains almost 160 hours of speech (training 107 hours, test 24 hours, development 30 hours). \nThe distribution of named entities by categories in the corpus is summarized in Table \\ref{tab:deepsun_guideline}. \n\\vspace{-0.1cm}\n\\begin{table}[th]\n \\caption{Distribution of named entities by categories in the DeepSUN corpus}\n \\label{tab:deepsun_guideline}\n \\centering\n \\begin{tabular}{ cccc }\n   \\toprule\n   \\multicolumn{1}{c}{\\textbf{}} & \\multicolumn{3}{c}{\\textbf{DeepSUN}} \\\\\n   \\multicolumn{1}{c}{\\textbf{category}} & \\multicolumn{1}{c}{\\textbf{dev}} & \\multicolumn{1}{c}{\\textbf{test}} & \\multicolumn{1}{c}{\\textbf{train}}\\\\\n   \\midrule\n   pers   & 6719   & 4766 \t& 22115 \t~~~ \\\\\n   func   & 1830 \t& 1425  & 6628 \t~~~ \\\\\n   org    & 5133 \t& 3506 \t& 15804 ~~~ \\\\\n   loc    & 5195 \t& 3915 \t& 18159 ~~~ \\\\\n   prod   & 652 \t& 606 \t& 2317 \t~~~ \\\\\n   time   & 3763\t& 2769 \t& 12020 \t~~~ \\\\\n   amount & 1591\t& 1450 \t& 5959 \t~~~ \\\\\n\tevent  & 79 \t& 0 \t& 321 \t~~~\\\\\n\t \\midrule\n    \\multicolumn{1}{c}{\\textbf{Sum}} &\n   \t\t\\multicolumn{1}{c}{\\textbf{24962}} & \\multicolumn{1}{c}{\\textbf{18437}} & \\multicolumn{1}{c}{\\textbf{83323}} \\\\\n\t\\bottomrule\n\t\\end{tabular}\n\\end{table}\n\\vspace{-0.1cm}\n\n\n\n\n\n\n\n\n\n\n\n\nThe performance of our approach is evaluated in terms of precision (P), recall(R) and F-measure for named entity detection, the named entity\/value detection \nand the accuracy of the value detection when the named entities tags are correctly detected. \nThese evaluations are made with the help of the \\textit{sclite}\\footnote{http:\/\/www.icsi.berkeley.edu\/Speech\/docs\/sctk-1.2\/sclite.htm} tool.\n\\subsection{Multi-task training}\nFor multi-task training, we first train the E2E architecture only for ASR task, without emitting character used to represent named entities. \nThe system is trained on all the audio recordings available with their manual transcriptions around 297.7 hours of training set, including the data described above.  \n\nIt composed of two convolution layers and six BLSTM layers with batch normalization, the number of epochs was set to $35$.\nThis system achieves 20.70\\% word error rate (WER) and\n8.01\\% character error rate (CER) on dev corpus (30.2 hours) and 19.95\\% of WER and 7.68\\% of CER on test set (40.8 hours). These results were obtained by applying a CTC beam search decoding coupled with a trigram language model. \n Once this system is trained, the softmax layer is reinitialized to take into consideration the named entity tag markers, and a new training process is realized, on the named entity recognition task, with only training data with manual annotations of named entities described in table~\\ref{tab:deepsun_guideline}.\n\\textbf{ In addition, for the training of both E2E and ASR systems, each training audio samples is randomly perturbed in gain and tempo for each iteration.}\n\n\n\\subsection{Experimental results}\n\nWe present in this section some experimental results. Table~\\ref{tab:4G_etiquettes} shows the performances of the end-to-end model (E2E) to detect EN categories (among the eight ones). That means that in this evaluation we do not take care of values associated to the detected EN. The starred mode is also experimented and is called (E2E*) in the table: this mode provides better results in this task than the normal mode. \n\n\n\\begin{table}[!htbp]\n\\vspace{-0.1cm}\n \\caption{Named entity category detection results for E2E and  E2E* (starred mode) systems} \n  \\label{tab:4G_etiquettes}\n  \\centering\n\\begin{tabular}{ |l|c|c|c|c| }\n  \n   \\hline\n    \\multicolumn{1}{|c|}{\\textbf{System}} &\\textbf{Corpus} & \\multicolumn{1}{|c|}{\\textbf{Precision}} & \\multicolumn{1}{|c|}{\\textbf{Recall}} & \\multicolumn{1}{|c|}{\\textbf{F-measure}}    \\\\\n   \n    \\hline\n    \\hline\n  \n \n        E2E & dev   & \\textbf{0.85}&0.57& 0.68  \\\\\n\n  \n    \\hline\n\tE2E & test  & \\textbf{0.83} & 0.52& 0.64  \\\\\n    \\hline\n    \\hline\n     E2E* &dev   & 0.75 &  \\bf{0.65}&  \\bf{0.71}  \\\\\n     \\hline\n     E2E* &test  &0.82 & \\bf{0.57}&   \\bf{0.67}  \\\\\n\n\n    \\hline\n  \\end{tabular}\n\\end{table}\n\n \n\\vspace{-0.1cm}\nTable~\\ref{tab:4G_val_etiquettes} evaluates the quality of the category\/value pairs that have been recognized. While precision and recall do not have the same behavior between normal and starred mode, both modes gets the same F-measure value. \n\n\n\\begin{table}[!htbp]\n  \\caption{Named entity category+value pair detection results for E2E and  E2E* systems} \n  \\label{tab:4G_val_etiquettes}\n  \\centering\n  \\begin{tabular}{ |l|c|c|c|c|}\n   \n    \\hline\n \\multicolumn{1}{|c|}{\\textbf{System}} &\\textbf{Corpus} & \\multicolumn{1}{|c|}{\\textbf{Precision}} & \\multicolumn{1}{|c|}{\\textbf{Recall}} & \\multicolumn{1}{|c|}{\\textbf{F-measure}}    \\\\\n   \n    \\hline\n    \\hline\n    E2E & dev   & \\bf0.64  &  0.45& 0.53  \\\\\n    E2E& test  & \\bf0.55  & 0.36& 0.44  \\\\\n    \\hline\n    \\hline\n    E2E* &dev   & 0.57 & \\bf{0.47} &  0.52 \\\\\n    E2E* &test  & 0.47&  \\bf{0.38}& 0.42  \\\\\n\n    \\hline\n  \\end{tabular}\n  \n\\end{table}\n\n\n\n\n\n\n\nLast, we would like to compare these results to the ones obtained by a pipeline process, that consists in applying a text named entity recognition on the automatic transcripts produced by the end-to-end ASR system trained on the first step of the multi-task learning presented above.\n\nThe text named entity recognition system used for this experiment is based on the combination of bi-directional LSTM (BLSTM), CNN and CRF modules~\\cite{ma2016end}, and takes benefits from both word and character-level embeddings learned automatically during the training process. For this experiment, we used the NeuroNLP2 implementation\\footnote{https:\/\/github.com\/XuezheMax\/NeuroNLP2}.\nConvolutional neural network encodes character-level information of a word into its character-level embedding. Then the character-and word-level embeddings are fed into the BLSTM to model context information of each word. On top of BLSTM, the sequential CRF is used to jointly decode labels for the whole sentence. \nIn addition, this system can be enriched with syntactic information like part of speech tagging (POS).\nIn our experiment, NeuroNLP2 is used as a NER system and Deep Speech 2 as the ASR system. Both are trained on the DeepSUN corpus described in section~\\ref{sec:corpus}.\nAutomatic transcriptions of dev and test data have been annotated with NER system. To measure the impact of POS, we used the MACAON system~\\cite{nasr2011macaon} to tag of DeepSUN corpus and manual and automatic transcriptions. \n\nTo feed NeuroNLP2, one hot vectors represent POS information. Word embeddings, character representations and one hot concatenations feed the BLSTM layer.\nAs we can see in Tables~\\ref{tab:neuro_etiquettes_seules}, the pipeline process is less competitive than the end-to-end model to recognize EN category, but is more efficient to extract EN values. Results also confirms that linguistic information like POS is really important for the NER task. Such observation will help for future work on the continuity of this study.\n\n\n\n\\begin{table}[!htbp]\n  \\caption{NER results for the pipeline approach (Pip) on the \\textbf{test} data. When POS are used to tag ASR outputs before NER processing, the system is called Pip+POS} \n  \\label{tab:neuro_etiquettes_seules}\n  \\centering\n  \\begin{tabular}{| l|l|c|c|c| }\n    \\hlin\n    \n    \\multicolumn{1}{|c|}{\\textbf{System}}  &\\textbf{Detection} &\\multicolumn{1}{|c|}{\\textbf{Precision}} & \\multicolumn{1}{|c|}{\\textbf{Recall}} & \\multicolumn{1}{|c|}{\\textbf{F-measure}}    \\\\\n   \n    \\hline\n    \\hline\n    Pip & category  & \\bf0.75 & 0.56 & 0.64  \\\\\n    Pip+POS &category       & 0.74 & \\bf0.58 & \\bf0.65  \\\\\n\n    \\hline\n    Pip &cat+value  & \\bf0.58 & 0.43 & 0.49  \\\\\n    Pip+POS &cat+value       & 0.57 & \\bf0.45 & \\bf0.50  \\\\\n\t\\hline\n\n  \\end{tabular}\n    \\vspace{-0.5cm}\n\\end{table}\n\n\n\n\n\n\n\n\nAs described in section~\\ref{sec:tricks}, we applied NeuroNLP2 (the version using POS tagging) on the manual transcriptions of the ASR training data in order to augment the amount of ''NER from speech'' training data. In this experiment,  the normal and starred modes  were  used. Table~\\ref{tab:e2e_augment} shows the improvement got by the end-to-end system when training on these imperfect augmented data using the normal (E2E+) and the starred (E2E+*) modes.  As we can see, the use of the augmented data was helpful for the starred mode.\n\n\\begin{table}[!htbp]\n  \\caption{NER results on the \\textbf{test} data for the E2E system trained with imperfect augmented data (E2E+) in comparison to the E2E system trained with imperfect augmented data and the starred mode (E2E+*)} \n  \\label{tab:e2e_augment}\n  \\centering\n  \\begin{tabular}{ |l|l|c|c|c| }\n    \\hlin\n   \\multicolumn{1}{|c|}{\\textbf{System}} &\\textbf{Detection} & \\multicolumn{1}{|c|}{\\textbf{Precision}} & \\multicolumn{1}{|c|}{\\textbf{Recall}} & \\multicolumn{1}{|c|}{\\textbf{F-measure}}    \\\\\n \n  \\hline  \n  \\hline\n    E2E+ & category         & \\bf 0.82 & \\bf0.57 &   0.67  \\\\\n    E2E+*& category  &  0.76&  0.63& \\bf0.69  \\\\\n    \\hline\n    E2E+ & cat+value       &\\bf0.55 & 0.40 &  0.46  \\\\\n    E2E+* & cat+value        & 0.49 & \\bf0.41 & \\textbf{0.47} \\\\\n    \\hline\n   \n  \\end{tabular}\n  \\vspace{-0.5cm}\n\\end{table}\n\n\n\n\n\\section{Conclusion}\nThis paper presents a first study about end-to-end named entity extraction from speech. By integrating in the character sequence emitted by a CTC end-to-end speech recognition system some special characters to delimit and categorize named entities, we showed that such extraction is feasible. \nTo compensate the lack of training data, we propose a multi-task learning approach (ASR + NER) in addition to an artificial data augmentation of the training corpus with automatic annotation of named entities. \nA starred mode is also proposed to make the neural network more focused on named entities. \nExperimental results show that this end-to-end approach in starred mode with training augmentation, provides better results (F-measure equals to 0.69 on test) than a pipeline approach to detect named entity categories (F-measure=0.64). On the other side, performances of this end-to-end approach to extract named entity values are worse than the ones got by the pipeline process. \n\nAs a conclusion, this study presents promising results in a first attempt to experiment an end-to-end approach to extract named entities, and constitutes an interesting start point for future work that could start by combining starred mode with training data augmentation, but also explore more different ways, like injecting linguistic information in the end-to-end neural architecture.\n\\vspace{-0.2cm}\n\\section{Acknowledgements}\n\\small\nThis work was supported by the French ANR Agency through the CHIST-ERA M2CR project, under the contract number ANR-15-CHR2-0006-01, and by the RFI Atlanstic2020 RAPACE project.\nAuthors would like to sincerely thank Sean Naren to make his implementation of Deep Speech 2 available, as well as the contributors to the NeuroNLP2 project.\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSoft anomalous dimensions are fundamental field-theoretical functions that are essential in determining soft-gluon contributions to partonic scattering cross sections (for a review see Ref. \\cite{NKtop}). Such cross sections factorize in Laplace or Mellin moment space into products of hard, soft, and jet functions.\n\nThe soft function, $S$, is in general a matrix in color space and it satisfies the renormalization group equation \\cite{NKGS}\n\\begin{equation}\n\\left(\\mu \\frac{\\partial}{\\partial \\mu}\n+\\beta(g_s)\\frac{\\partial}{\\partial g_s}\\right)\\,S\n=-\\Gamma^\\dagger_S \\; S-S \\; \\Gamma_S\n\\end{equation}\nwith soft anomalous dimension $\\Gamma_S$, also a matrix. The evolution of the soft function results in the exponentiation (resummation) of logarithms of the moment variable.\nResummation at NNLL accuracy requires two-loop soft anomalous dimensions while at N$^3$LL accuracy it requires three-loop soft anomalous dimensions.\n\nIn a top-quark production partonic process, $f_{1} +  f_{2} \\rightarrow t + X$, \nwe define a threshold variable $s_4$ that measures distance from partonic threshold. When the resummed cross section is inverted to momentum space, the soft-gluon corrections involve logarithms of $s_4$, but a prescription is needed for all-order results to deal with infrared divergences. Such resummed results are strongly prescription-dependent, and some prescriptions give poor results by underestimating the true size of the corrections. Alternatively, finite-order expansions can be performed with better control over subleading effects and no prescriptions. Expansions to second and third order (with matching with lower-order complete results) provide approximate NNLO (aNNLO) and N$^3$LO (aN$^3$LO) predictions which are state-of-the-art. As we approach partonic threshold, $s_4 \\rightarrow 0$, there is diminishing energy left for additional gluon radiation, and the contributions from soft-gluon emission are dominant and they provide excellent predictions for the high-order corrections.\n\nIn the next section, we present results through three loops for the cusp anomalous dimension, which is the simplest soft anomalous dimension. We consider the case when both eikonal lines are massive and the case when one line is massive and one is massless. In Section 3, we present results through three loops for the soft anomalous dimensions in single-top production processes, in processes with a top quark in new physics models, and in top-antitop pair production.\n\n\\section{Three-loop cusp anomalous dimension}\n\nA basic ingredient of soft anomalous dimensions for partonic processes is the cusp anomalous dimension \\cite{KR,NK2loop,GHKM,NK3lcusp,NK3loop} (see Fig. \\ref{cusp} for representative eikonal diagrams at one, two, and three loops).\nThe cusp angle is defined by $\\theta=\\cosh^{-1}(p_i\\cdot p_j\/\\sqrt{p_i^2 p_j^2})$, in terms of the momenta of eikonal lines $i$ and $j$, and the perturbative expansion for the cusp anomalous dimension is $\\Gamma_{\\rm cusp}=\\sum_{n=1}^{\\infty} \\left(\\frac{\\alpha_s}{\\pi}\\right)^n \\Gamma^{(n)}_{\\rm cusp}$. When the lines represent a top and an antitop, then the cusp anomalous dimension can be thought of as the soft anomalous dimension for $e^+ e^- \\rightarrow t{\\bar t}$. From the UV poles of the loop diagrams in dimensional regularization we calculate results at each order. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.3\\textwidth]{loop1.pdf}\n\\hspace{4mm}\n\\includegraphics[width=0.3\\textwidth]{loop2.pdf}\n\\hspace{7mm}\n\\includegraphics[width=0.3\\textwidth]{loop3.pdf}\n\\caption{Representative eikonal diagrams for the cusp anomalous dimension at one, two, and three loops.}\n\\label{cusp}\n\\end{center}\n\\end{figure}\n\nAt one loop,\n\\begin{equation}\n\\Gamma_{\\rm cusp}^{(1)}=C_F (\\theta \\coth\\theta-1) \\, ,\n\\end{equation}\nwhere $C_F=(N^2-1)\/(2N)$, with $N$ the number of colors.\nIn terms of the heavy-quark speed $\\beta=\\tanh(\\theta\/2)$, we have\n$\\theta=\\ln\\left(\\frac{1+\\beta}{1-\\beta}\\right)$\nand \n\\begin{equation}\n\\Gamma_{\\rm cusp}^{(1)}=C_F\\left[-\\frac{(1+\\beta^2)}{2\\beta}\n\\ln\\frac{(1-\\beta)}{(1+\\beta)}-1\\right] \\, .\n\\end{equation}\n\nAt two loops \\cite{NK2loop},\n\\begin{eqnarray}\n\\Gamma_{\\rm cusp}^{(2)}&=&K^{'(2)} \\, \\Gamma_{\\rm cusp}^{(1)}\n+\\frac{1}{2}C_F C_A \\left\\{1+\\zeta_2+\\theta^2 \n-\\coth\\theta\\left[\\zeta_2\\theta+\\theta^2\n+\\frac{\\theta^3}{3}+{\\rm Li}_2\\left(1-e^{-2\\theta}\\right)\\right] \\right. \n\\nonumber \\\\ && \\hspace{29mm} \\left.\n{}+\\coth^2\\theta\\left[-\\zeta_3+\\zeta_2\\theta+\\frac{\\theta^3}{3}\n+\\theta \\, {\\rm Li}_2\\left(e^{-2\\theta}\\right)\n+{\\rm Li}_3\\left(e^{-2\\theta}\\right)\\right] \\right\\}\n\\end{eqnarray}\nwhere $K^{'(2)}=K^{(2)}\/C_F=C_A (67\/36-\\zeta_2\/2)-5 n_f\/18$, with $C_A=N$ and $n_f$ the number of light quark flavors.\n\nAt three loops \\cite{GHKM,NK3lcusp},\n\\begin{equation}\n\\Gamma_{\\rm cusp}^{(3)}= K^{'(3)} \\Gamma_{\\rm cusp}^{(1)}\n+2 K^{'(2)} \\left[\\Gamma_{\\rm cusp}^{(2)}-K^{'(2)}\\Gamma_{\\rm cusp}^{(1)}\\right]\n+C_{\\rm cusp}^{(3)} \n\\label{3lc}\n\\end{equation}\nwhere $K^{'(3)}=K^{(3)}\/C_F$ is a lengthy expression with color terms and constants (note that in general $K^{(n)}$ denotes the proportionality factor of the $n$-loop cusp anomalous dimension in the massless limit with $\\theta$). The expression for $C_{\\rm cusp}^{(3)}$ is very long (see \\cite{NK3lcusp} for details). For top-quark production $n_f=5$, and a simple numerical expression is \\cite{NK3lcusp}\n\\begin{equation}\n\\Gamma_{\\rm cusp}^{(3)\\, \\rm approx}(\\beta)=\n0.09221 \\, \\beta^2+2.80322 \\; \\Gamma_{\\rm cusp}^{(1)}(\\beta) \\, .\n\\label{3lca}\n\\end{equation}\n\nFor the case where one eikonal line is massive and one is massless we find simpler expressions for the cusp anomalous dimension, which we now denote as $\\Gamma_c$ to distinguish it from the fully massive case. If eikonal line $i$ represents a massive quark and eikonal line $j$ a massless quark, then \n\\begin{equation}\n\\Gamma_c^{(1)}=C_F \\left[\\ln\\left(\\frac{2 p_i \\cdot p_j}{m_i \\sqrt{s}}\\right)\n-\\frac{1}{2}\\right] \\, ,\n\\end{equation}\n\\begin{equation}\n\\Gamma^{(2)}_c=K^{(2)} \\left[\\ln\\left(\\frac{2 p_i \\cdot p_j}{m_i \\sqrt{s}}\\right\n) -\\frac{1}{2} \\right]+\\frac{1}{4} C_F C_A (1-\\zeta_3) \\, ,\n\\end{equation}\n\\begin{eqnarray}\n\\Gamma^{(3)}_c&=&K^{(3)} \\left[\\ln\\left(\\frac{2 p_i \\cdot p_j}{m_i \\sqrt{s}}\\right)-\\frac{1}{2}\\right]+\\frac{1}{2} K^{(2)} C_A (1-\\zeta_3)\n\\nonumber \\\\ &&\n{}+C_F C_A^2\\left[-\\frac{1}{4}+\\frac{3}{8}\\zeta_2-\\frac{\\zeta_3}{8}-\\frac{3}{8}\\zeta_2 \\zeta_3+\\frac{9}{16} \\zeta_5\\right] \\, .\n\\end{eqnarray}\n\nThe cusp anomalous dimension is an essential component in calculations of soft anomalous dimension matrices for general partonic processes. In the fully massless case, two-loop soft anomalous dimension matrices are proportional to the one-loop quantity \\cite{ADS}; however, when masses are present this is no longer the case. We also note that three-parton correlations with at least two massless lines vanish at any order due to constraints from scaling symmetry \\cite{BN,GM}. However, four-parton correlations at three loops and beyond do not necessarily vanish even in the massless case \\cite{ADG}. We will show explicit results at one, two, and three loops for various top-quark processes in the next section.\n\n\\section{Soft anomalous dimensions for top-quark processes}\n\nWe now present results for the soft anomalous dimensions through three loops for a variety of top-quark production processes.\n\n\\subsection{Single-top-quark production}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{singletop2019plot.pdf}\n\\hspace{4mm}\n\\caption{Theoretical single-top cross sections for the $t$ and $s$ channels at aNNLO \\cite{NKtop,NKsingletop} and for $tW$ production at aN$^3$LO \\cite{NKtW}, using MMHT2014 NNLO pdf \\cite{MMHT2014}, at LHC energies compared with LHC data \\cite{ATLAS13tch,CMS13tch,LHC7and8singletop,ATLAS13tW,CMS13tW}.}\n\\label{singletop}\n\\end{center}\n\\end{figure}\n\nWe begin with single-top-quark production \\cite{NKtop,NK3loop,NKst,NKsingletop,NKtW}. Before presenting the analytical form of the soft anomalous dimensions, we show in Fig. \\ref{singletop} theoretical predictions at NNLL accuracy for the cross sections at LHC energies. The soft-gluon corrections are important and the theoretical predictions are in very good agreement with the data. \n\n\\subsubsection{Single-top $t$-channel production}\n\nThe partonic processes in single-top $t$-channel production are \n$b(p_1)+q(p_2) \\rightarrow t(p_3) +q'(p_4)$.\nWe define the usual partonic kinematical variables  \n$s=(p_1+p_2)^2$, $t=(p_1-p_3)^2$, and $u=(p_2-p_3)^2$, and  \nthe threshold variable $s_4=s+t+u-m_t^2$.\nWe choose a singlet-octet $t$-channel color basis $c_1=\\delta_{13} \n\\delta_{24}$ and $c_2=T^c_{31} T^c_{42}$.\nThe soft anomalous dimension is a $2\\times 2$ matrix.\n\nAt one loop, the four matrix elements are \\cite{NKtop,NKst,NKsingletop}\n\\begin{eqnarray}\n&&{\\Gamma}_{S\\, 11}^{t \\, (1)}=\nC_F \\left[\\ln\\left(\\frac{t(t-m_t^2)}{m_t s^{3\/2}}\\right)-\\frac{1}{2}\\right] \\, , \n\\quad\n{\\Gamma}_{S\\, 12}^{t \\, (1)}=\\frac{C_F}{2N} \\ln\\left(\\frac{u(u-m_t^2)}{s(s-m_t^2)}\\right)\n \\, , \n\\quad\n{\\Gamma}_{S\\, 21}^{t \\, (1)}= \\ln\\left(\\frac{u(u-m_t^2)}{s(s-m_t^2)}\\right) \\, ,\n\\nonumber \\\\ &&\n{\\Gamma}_{S\\, 22}^{t \\, (1)}= C_F \\left[\\ln\\left(\\frac{t(t-m_t^2)}{m_t s^{3\/2}}\\right)-\\frac{1}{2}\\right]-\\frac{1}{N}\\ln\\left(\\frac{u(u-m_t^2)}{s(s-m_t^2)}\\right) \n+\\frac{N}{2}\\ln\\left(\\frac{u(u-m_t^2)}{t(t-m_t^2)}\\right) \\, .\n\\end{eqnarray}\n\nAt two loops, we find \\cite{NK3loop,NKsingletop}\n\\begin{eqnarray}\n&&\\Gamma_{S\\,11}^{t \\, (2)}= K^{'(2)} \\Gamma_{S\\,11}^{t \\, (1)}+\\frac{1}{4} C_F C_A (1-\\zeta_3)\\, , \n\\hspace{8mm}\n\\Gamma_{S\\,12}^{t \\, (2)}= K^{'(2)} \\Gamma_{S\\,12}^{t \\, (1)} \\, ,\n\\nonumber \\\\ &&\n\\Gamma_{S\\,21}^{t \\, (2)}= K^{'(2)} \\Gamma_{S\\,21}^{t \\, (1)} \\, , \\hspace{8mm}\n\\Gamma_{S\\,22}^{t \\, (2)}= K^{'(2)} \\Gamma_{S\\,22}^{t \\, (1)}+\\frac{1}{4} C_F C_A (1-\\zeta_3) \\, .\n\\end{eqnarray}\n\nAt three loops, we only need the first element of the matrix for N$^3$LL resummation, and to calculate the N$^3$LO soft-gluon corrections. We have \\cite{NK3loop}\n\\begin{equation}\n\\Gamma_{S\\,11}^{t \\, (3)}= K^{'(3)} \\Gamma_{S\\,11}^{t \\, (1)}\n+ \\frac{1}{2} K^{(2)} C_A (1-\\zeta_3) \n+C_F C_A^2\\left[-\\frac{1}{4}+\\frac{3}{8}\\zeta_2-\\frac{\\zeta_3}{8}\n-\\frac{3}{8}\\zeta_2 \\zeta_3+\\frac{9}{16} \\zeta_5\\right] \\, .\n\\end{equation}\nDue to the simple structure of the leading-order hard matrix, the other three matrix elements of $\\Gamma_S^t$ at three loops do not contribute to the N$^3$LO corrections. We can still provide expressions for those three matrix elements, up to four-parton correlations, as discussed in the context of $s$-channel production below.\n\n\\subsubsection{Single-top $s$-channel production}\n\nThe partonic processes in single-top $s$-channel production are \n$q(p_1)+{\\bar q}'(p_2) \\rightarrow t(p_3) +{\\bar b}(p_4)$.\nWe again have a $2\\times 2$ soft anomalous dimension matrix, and we choose a singlet-octet $s$-channel color basis, $c_1=\\delta_{12} \\delta_{34}$ and \n$c_2=T^c_{21} T^c_{34}$.\n\nAt one loop, the four matrix elements are \\cite{NKtop,NKst,NKsingletop}\n\\begin{eqnarray}\n&&\\Gamma_{S\\, 11}^{s \\, (1)}=C_F \\left[\\ln\\left(\\frac{s-m_t^2}{m_t\\sqrt{s}}\\right)\n-\\frac{1}{2}\\right] \\, ,\n\\quad\n\\Gamma_{S\\, 12}^{s \\, (1)}=\\frac{C_F}{2N} \\ln\\left(\\frac{t(t-m_t^2)}{u(u-m_t^2)}\\right) \\, , \n\\quad\n\\Gamma_{S\\, 21}^{s \\, (1)}= \\ln\\left(\\frac{t(t-m_t^2)}{u(u-m_t^2)}\\right) \\, ,\n\\nonumber \\\\ &&\n\\Gamma_{S\\, 22}^{s \\, (1)}=C_F \\left[\\ln\\left(\\frac{s-m_t^2}{m_t \\sqrt{s}}\\right)-\\frac{1}{2}\\right]-\\frac{1}{N}\\ln\\left(\\frac{t(t-m_t^2)}{u(u-m_t^2)}\\right)\n+\\frac{N}{2} \\ln\\left(\\frac{t(t-m_t^2)}{s(s-m_t^2)}\\right) \\, .\n\\end{eqnarray}\n\nAt two loops, we have \\cite{NK3loop,NKsingletop}\n\\begin{eqnarray}\n&&\\Gamma_{S\\, 11}^{s\\, (2)}=K^{'(2)} \\Gamma_{S\\,11}^{s \\, (1)}+\\frac{1}{4} C_F C_A (1-\\zeta_3) \\, , \\hspace{8mm}\n\\Gamma_{S\\,12}^{s \\, (2)}=K^{'(2)} \\Gamma_{S\\,12}^{s \\, (1)} \\, ,\n\\nonumber \\\\ &&\n\\Gamma_{S\\,21}^{s \\, (2)}= K^{'(2)} \\Gamma_{S\\,21}^{s \\, (1)} \\, , \\hspace{8mm}\n\\Gamma_{S\\,22}^{s \\, (2)}= K^{'(2)} \\Gamma_{S\\,22}^{s \\, (1)}+\\frac{1}{4} C_F C_A (1-\\zeta_3) \\, .\n\\label{sch2l}\n\\end{eqnarray}\n\nAt three loops, again we only need the first element of the matrix for N$^3$LL resummation and N$^3$LO corrections. We find \\cite{NK3loop}\n\\begin{equation}\n\\Gamma_{S\\, 11}^{s \\, (3)}= K^{'(3)} \\Gamma_{S\\, 11}^{s \\, (1)}\n+\\frac{1}{2} K^{(2)} C_A (1-\\zeta_3) +C_F C_A^2\\left[-\\frac{1}{4}+\\frac{3}{8}\\zeta_2-\\frac{\\zeta_3}{8}\n-\\frac{3}{8}\\zeta_2 \\zeta_3+\\frac{9}{16} \\zeta_5\\right] \\, .\n\\label{sch3l}\n\\end{equation}\nThe other three matrix elements of $\\Gamma_S^s$ at three loops are not fully known. However, we can see that the structure of the results will be similar to those at two loops.\nThe three-loop off-diagonal matrix elements will have a similar form as at two loops [replace all two-loop terms, denoted by superscript (2), in Eq. (\\ref{sch2l}) by the corresponding three-loop terms, denoted by superscript (3)] while $\\Gamma_{S\\, 22}^{s \\, (3)}$ should have the form of Eq. (\\ref{sch3l}) [just replace the element subscript 11 with 22] up to four-parton correlations. The same also applies to the $t$-channel results as indicated previously.\n\n\\subsubsection{Associated $tW$ production}\n\nIn $tW$ production, the partonic process is\n\\begin{equation}\nb(p_1)+g(p_2) \\rightarrow t(p_3) +W^-(p_4) \\, .\n\\end{equation}\nIn this case the soft anomalous dimension is a simple function (not a matrix).\n\nAt one loop, we have \\cite{NKst,NKsingletop}\n\\begin{equation}\n\\Gamma_S^{tW \\, (1)}=C_F \\left[\\ln\\left(\\frac{m_t^2-t}{m_t\\sqrt{s}}\\right)\n-\\frac{1}{2}\\right] +\\frac{C_A}{2} \\ln\\left(\\frac{u-m_t^2}{t-m_t^2}\\right)\\, .\n\\end{equation}\n\nAt two loops, we have \\cite{NKsingletop}\n\\begin{equation}\n\\Gamma_S^{tW (2)}=K^{'(2)} \\Gamma_S^{tW \\, (1)}\n+\\frac{1}{4}C_F C_A (1-\\zeta_3) \\, .\n\\end{equation}\n\nAt three loops, we have \\cite{NK3loop}\n\\begin{equation}\n\\Gamma_S^{tW \\, (3)}=K^{'(3)} \\Gamma_S^{tW \\, (1)}+\\frac{1}{2} K^{(2)} C_A (1-\\zeta_3)+C_F C_A^2\\left[-\\frac{1}{4}+\\frac{3}{8}\\zeta_2-\\frac{\\zeta_3}{8}\n-\\frac{3}{8}\\zeta_2 \\zeta_3+\\frac{9}{16} \\zeta_5\\right] \\, .\n\\end{equation}\n\n\\subsection{$tZ$, $tZ'$, $t \\gamma$, and $tH^-$ production}\n\nTop quarks can also be produced in association with electroweak and Higgs bosons in models of new physics. Such processes include the associated production of a top quark with a charged Higgs boson via $bg \\rightarrow tH^-$ \\cite{NKtH}; the associated production of a top quark with a $Z$ boson, $qg \\rightarrow tZ$ \\cite{NKtZ}, or with a photon, $qg \\rightarrow t\\gamma$ \\cite{MFNK}, via anomalous couplings; and the associated production of a top quark with a $Z'$ boson either via anomalous couplings, $qg \\rightarrow tZ'$, or via initial-state tops, $tg \\rightarrow tZ'$ \\cite{MGNK}.\nThe soft anomalous dimensions for all these processes are identical to the one for $bg \\rightarrow tW^-$ that was presented in the previous subsection.\n\n\\subsection{Top-antitop pair production}\n\nWe continue with top-antitop pair production \\cite{NKtop,NKGS,NKtt2l,NKtt}. The soft anomalous dimension is a $2\\times 2$ matrix for the quark-initiated channel, $q{\\bar q} \\rightarrow t{\\bar t}$, and a $3\\times 3$ matrix for the gluon-initiated channel, $gg \\rightarrow t{\\bar t}$ \\cite{NKGS,NKtt2l,FNPY}.\n\nWe begin with the soft anomalous dimension matrix for the $q{\\bar q} \\rightarrow t{\\bar t}$ channel in a color tensor basis of $s$-channel singlet and octet exchange, $c_1 = \\delta_{12}\\delta_{34}$ and $c_2 =  T^c_{21} \\, T^c_{34}$.\n\nAt one loop for $q{\\bar q} \\rightarrow t{\\bar t}$ \\cite{NKGS,NKtt2l}:\n\\begin{eqnarray}\n\\Gamma^{q{\\bar q}\\, (1)}_{S\\, 11}&=&\\Gamma_{\\rm cusp}^{(1)} \\, ,\n\\quad\n\\Gamma^{q{\\bar q}\\, (1)}_{S\\, 12}=\n\\frac{C_F}{C_A} \\ln\\left(\\frac{t-m_t^2}{u-m_t^2}\\right) \\, , \n\\quad \n\\Gamma^{q{\\bar q}\\, (1)}_{S\\, 21}=\n2\\ln\\left(\\frac{t-m_t^2}{u-m_t^2}\\right) \\, ,\n\\nonumber \\\\ \n\\Gamma^{q{\\bar q}\\, (1)}_{S\\, 22}&=&\\left(1-\\frac{C_A}{2C_F}\\right)\n\\Gamma_{\\rm cusp}^{(1)} \n+4C_F \\ln\\left(\\frac{t-m_t^2}{u-m_t^2}\\right)\n-\\frac{C_A}{2}\\left[1+\\ln\\left(\\frac{s m_t^2 (t-m_t^2)^2}{(u-m_t^2)^4}\\right)\\right]. \n\\end{eqnarray}\n\nAt two loops for $q{\\bar q} \\rightarrow t{\\bar t}$ \\cite{NKtop,NKtt2l}:\n\\begin{eqnarray}\n\\Gamma^{q{\\bar q}\\,(2)}_{S\\, 11}&=&\\Gamma_{\\rm cusp}^{(2)} \\, ,\n\\quad \n\\Gamma^{q{\\bar q}\\,(2)}_{S\\, 12}=\n\\left(K^{'(2)}-C_A N_S^{(2)}\\right) \\Gamma^{q{\\bar q} \\,(1)}_{S\\, 12} \\, ,\n\\quad\n\\Gamma^{q{\\bar q} \\,(2)}_{S\\, 21}=\n\\left(K^{'(2)}+C_A N_S^{(2)}\\right) \\Gamma^{q{\\bar q} \\,(1)}_{S\\, 21} \\, ,\n\\nonumber \\\\\n\\Gamma^{q{\\bar q} \\,(2)}_{S\\, 22}&=&\nK^{'(2)} \\Gamma^{q{\\bar q} \\,(1)}_{S\\, 22}\n+\\left(1-\\frac{C_A}{2C_F}\\right)\n\\left(\\Gamma_{\\rm cusp}^{(2)}-K^{'(2)}\\Gamma_{\\rm cusp}^{(1)}\\right) \\, ,\n\\label{qq2l}\n\\end{eqnarray}\nwhere \n\\begin{equation}\nN_S^{(2)}=\\frac{\\theta^2}{4}+\\frac{1}{4} \\coth\\theta \\left[\\zeta_2-\\theta^2-{\\rm Li}_2\\left(1-e^{-2\\theta}\\right)\\right] \\, .\n\\end{equation}\n\nAt three loops, it is evident that the diagonal elements of the three-loop matrix for $q{\\bar q} \\rightarrow t{\\bar t}$ receive contributions from the three-loop massive cusp anomalous dimension, Eqs. (\\ref{3lc}), (\\ref{3lca}), but we do not yet have complete three-loop results. However, in analogy to our discussion for $s$-channel and $t$-channel single-top production, the structure of the results at three loops should be analogous to that at two loops [replace all two-loop terms, denoted by superscript (2), in Eq. (\\ref{qq2l}) by the corresponding three-loop terms] up to four-parton correlations.\n\nWe continue with the soft anomalous dimension matrix for the $gg \\rightarrow t{\\bar t}$ channel in a color tensor basis $c_1=\\delta_{12}\\,\\delta_{34}$, $c_2=d^{12c}\\,T^c_{34}$, and $c_3=i f^{12c}\\,T^c_{34}$. \n\nAt one loop for $gg \\rightarrow t{\\bar t}$ \\cite{NKGS,NKtt2l}:\n\\begin{eqnarray}\n\\Gamma^{gg\\,(1)}_{S\\, 11}&=& \\Gamma_{\\rm cusp}^{(1)}\\, , \\quad\n\\Gamma^{gg\\,(1)}_{S\\, 12}=0 \\, , \\quad \\Gamma^{gg\\,(1)}_{S\\, 21}=0\\, , \\quad\n\\Gamma^{gg\\,(1)}_{S\\, 13}= \\ln\\left(\\frac{t-m_t^2}{u-m_t^2}\\right) \\, , \\quad\n\\Gamma^{gg\\,(1)}_{S\\, 31}= 2 \\ln\\left(\\frac{t-m_t^2}{u-m_t^2}\\right) \\, ,\n\\nonumber \\\\\n\\Gamma^{gg\\,(1)}_{S\\, 22}&=& \\left(1-\\frac{C_A}{2C_F}\\right)\n\\Gamma_{\\rm cusp}^{(1)}\n-\\frac{C_A}{2}\\left[1+\\ln\\left(\\frac{s m_t^2}{(t-m_t^2)(u-m_t^2)}\\right)\\right] \\, , \n\\nonumber \\\\\n\\Gamma^{gg\\,(1)}_{S\\, 23}&=&\\frac{C_A}{2} \\ln\\left(\\frac{t-m_t^2}{u-m_t^2}\\right) \\, , \\quad\n\\Gamma^{gg\\,(1)}_{S\\, 32}=\\frac{(N^2-4)}{2N} \\ln\\left(\\frac{t-m_t^2}{u-m_t^2}\\right) \\, , \\quad \\Gamma^{gg\\,(1)}_{S\\, 33}=\\Gamma^{gg\\,(1)}_{S\\, 22} \\, .\n\\end{eqnarray}\n\nAt two loops for $gg \\rightarrow t{\\bar t}$ \\cite{NKtop,NKtt2l}:\n\\begin{eqnarray}\n\\Gamma^{gg\\,(2)}_{S\\, 11}&=& \\Gamma_{\\rm cusp}^{(2)} \\, , \\quad\n\\Gamma^{gg\\,(2)}_{S\\, 12}=0\\, , \\quad \\Gamma^{gg\\,(2)}_{S\\, 21}=0\\, , \\quad \n\\Gamma^{gg\\,(2)}_{S\\, 13}=\\left(K^{'(2)}-C_A N_S^{(2)}\\right) \n\\Gamma^{gg \\,(1)}_{S\\, 13} \\, , \n\\nonumber \\\\ \n\\Gamma^{gg\\,(2)}_{S\\, 31}&=&\\left(K^{'(2)}+C_A N_S^{(2)}\\right)  \n\\Gamma^{gg \\,(1)}_{S\\, 31} \\, , \\quad\n\\Gamma^{gg\\,(2)}_{S\\, 22}= K^{'(2)} \\Gamma^{gg \\,(1)}_{S\\, 22}\n+\\left(1-\\frac{C_A}{2C_F}\\right) \n\\left(\\Gamma_{\\rm cusp}^{(2)}-K^{'(2)}\\Gamma_{\\rm cusp}^{(1)}\\right) \\, ,\n\\nonumber \\\\\n\\Gamma^{gg\\,(2)}_{S\\, 23}&=& K^{'(2)} \\Gamma^{gg \\,(1)}_{S\\, 23} \\, , \\quad\n\\Gamma^{gg\\,(2)}_{S\\, 32}= K^{'(2)} \\Gamma^{gg \\,(1)}_{S\\, 32} \\, , \\quad\n\\Gamma^{gg\\,(2)}_{S\\, 33}=\\Gamma^{gg\\,(2)}_{S\\, 22} \\, .\n\\label{gg2l}\n\\end{eqnarray}\n\nAt three loops, again it is clear that the diagonal elements of the three-loop matrix for $gg \\rightarrow t{\\bar t}$ receive contributions from the three-loop massive cusp anomalous dimension, Eqs. (\\ref{3lc}), (\\ref{3lca}), but we do not yet have complete three-loop results. However, in analogy to our discussion for the  quark-initiated channel, the structure of the results at three loops should be analogous to that at two loops [replace all two-loop terms, denoted by superscript (2), in Eq. (\\ref{gg2l}) by the corresponding three-loop terms] up to four-parton correlations.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the framework of SETI (Search for Extraterrestrial Intelligence)\none of the important projects was the search for the interstellar\nradio communication. A rather different and quite original method\nwas proposed by the prominent physicist Freeman Dyson in 1960, who\nsuggested that if an extraterrestrial intelligence has reached a\nlevel of supercivilization, it might consume an energy of its own\nstar \\citep{dyson}. For this purpose, to have the maximum efficiency\nof energy transformation, it would be better to construct a thin\nshell completely surrounding the star. In the framework of this\napproach the author assumes that the mentioned superintelligence\nobserved by us will have been in existence for millions of years,\nhaving reached a technological level exceeding ours by many orders\nof magnitude. Kardashev in his famous article \\citep{kardashev},\nexamining the problem of transmission of information by\nextraterrestrial civilizations, has classified them by a\ntechnological level they have already achieved: (I) - a\ntechnological level close to the level of the civilization on earth,\nconsuming the energy of the order of $4\\times 10^{19}$ergs s$^{-1}$;\n(II) - a civilization consuming the energy of its own star -\n$4\\times 10^{33}$ergs s$^{-1}$ and (III) - a civilization capable of\nharnessing the energy accumulated in its own galaxy: $4\\times\n10^{44}$ergs s$^{-1}$. In this classification Dyson's idea deals\nwith the civilization of type-II, consuming the energy exceeding\nours approximately $\\frac{4\\times 10^{33}}{4\\times 10^{19}}=10^{14}$\ntimes (see the similar estimates by \\cite{dyson}). If we assume that\nan average growth rate of $1\\%$ per year in industry is maintained,\nthe level of type-II civilization might be achieved in $\\sim 3000$\nyears \\citep{dyson}, being quite reasonable in the context of the\nassumption that a civilization exists millions of years.\n\n\n\nDyson has suggested that if such a civilization exists, then it is\npossible to detect it by observing the spherical shell surrounding\nthe star. In particular, it is clear that energy radiated by the\nstar must be absorbed by the inner surface of the sphere and might\nbe consumed by the civilization. The author implied that the size of\nthe sphere should be comparable with that of the Earth's orbit. It\nis clear that to have energy balance, the spherical shell must\nirradiate the energy, but in a different spectral interval: in the\ninfrared domain, with the black body temperature of the order of\n$(200-300)$K \\citep{dyson}.\n\nThe attempts to identify Dyson spheres on the sky were performed by\nseveral groups \\cite{jugaku,slish,timofeev} but no significant\nresults were obtained. Recently an interest to such an ambitious\nidea has significantly increased: a couple of years ago Carrigan has\npublished an article titled: \"IRAS-based whole sky upper limit on\nDyson spheres\" \\citep{carrigan}, where he considered the results of\nthe instrument IRAS (The Infrared Astronomical Satellite). This\nsatellite covered almost $96\\%$ of the sky, implying that this is\nalmost whole sky monitoring. According to the study, the searches\nhave been conducted as for fully as for partially cloaked Dyson\nspheres. The search has revealed $16$ Dyson sphere candidates, but\nthe author pointed out that further investigation must have been\nrequired. Recently an interesting series of works has been presented\n\\citep{wright1,wright2,wright3} where\nthe authors discuss a broad class of related problems starting from\nphilosophical aspects of SETI \\citep{wright1}, also examining in detail the strategy of the\nsearch for the infrared galactic and extragalactic sources corresponding to\nKardashev-II\/III civilizations \\citep{wright2,wright3}.\n\n\nIn this manuscript we present a rather different idea how to search\nan advanced intelligence. In the framework of Dyson's idea, the\nspherical shroud (with radius of the order of $(1-3)$AU) is\nconstructed around stars. It is clear that in order to consume\nalmost the total energy radiated by the star, it must be imbedded\ninside a closed spherical shell, requiring enormous material to\nconstruct it. On the other hand, it is very well known that pulsars\n- rapidly rotating neutron stars - emit huge energy in narrow\nchannels (see Figure \\ref{fig1}), therefore, if a supercivilization\nexists it can in principle utilize the energy of these objects. But\nin these cases, instead of sphere-like envelopes the super\nintelligence has to use ring-like structures around the pulsars. If\nthe angle between the rotation axis and the direction of emission,\n$\\alpha$, is close to $90^o$, the ring will be located in the\nequatorial plane of the neutron star. As we will see later, in case\nof relatively slowly spinning pulsars (with periods of rotation of\nthe order of $1$s) an advantage will be quite small sizes of these\nartificial constructions. Another class of neutron stars is the\nso-called $X$-ray pulsars, having a strong emission pattern in the\n$X$-ray band. Usually these objects are characterized by short\nperiods of rotation with quite high values of slowdown rates, having\nluminosities exceeding those of normal pulsars by several orders of\nmagnitude. It is worth noting that a habitable zone around them\nmight be further than around slowly spinning pulsars,\nimplying that the possible size of the artificial \"ring\" could be\nextremely large. Therefore, we can hardly believe that these objects\ncan be interesting in the context of the search for extraterrestrial\nsuper advanced intelligence.\n\nThe organization of the paper is the following: after introducing\nthe theoretical background we work out the details of the Dyson\n\"rings\" surrounding the pulsars in Sec.2, in Sec. 3 we summarize our\nresults.\n\n\n\n\n\\section[]{Theoretical background and results}\n\nIn this section we consider the pulsars and estimate the\ncorresponding physical parameters of artificial ring-like\nconstructions surrounding the rotating neutron stars.\n\nGenerally speaking, any rotating neutron star, characterized by the\nslow down rate $\\dot{P}\\equiv dP\/dt>0$, where $P$ is the rotation\nperiod, loses energy with the following power (called the slowdown\nluminosity, $L_{sd}$) $\\dot{W} = I\\Omega|\\dot{\\Omega}|$. Here by\n$I=2MR_{\\star}^2\/5$ we denote the moment of inertia of the neutron\nstar, $M\\approx 1.5\\times M_{\\odot}$ and $M_{\\odot}\\approx 2\\times\n10^{33}$g are the pulsar's mass and the solar mass respectively,\n$R_{\\star}\\approx 10^6$cm is the neutron star's radius,\n$\\Omega\\equiv 2\\pi\/P$ is its angular velocity and\n$\\dot{\\Omega}\\equiv d\\Omega\/dt=-2\\pi\\dot{P}\/P^2$. The slowdown\nluminosity for the relatively slowly spinning pulsars is of the\norder of\n$$L_{sd}\\approx 4.7\\times 10^{31}\\times\nP^{-3}\\times\\left(\\frac{\\dot{P}}{10^{-15}ss^{-1}} \\right)\\times$$\n\\begin{equation}\n\\label{lumin} \\times\\left(\\frac{M}{1.5M_{\\odot}}\\right)ergs\\ s^{-1},\n\\end{equation}\nwhere the parameters are normalized on their typical values. As it\nis clear from Eq. (\\ref{lumin}), the energy budget is very high\nforcing a supercivilization construct a \"ring\" close to the host\nobject. On the other hand, these sources exist during the time scale $P\/\\dot{P}$,\nwhich is long enough to colonize the star.\n\nIn the framework of the standard definition, a habitable zone (HZ)\nis a region of space with favorable conditions for life based on\ncomplex carbon compounds and on availability of fluid water, etc.\n\\citep{hanslmeier}. This means that the surface of the \"ring\" must\nbe irradiated by the same flux as the surface of the Earth\n(henceforth - the flux method). Therefore, the mean radius of the\nHZ, defined as $R_{HZ} = \\left(\\kappa\nL_{sd}\/L_{\\odot}\\right)^{1\/2}$AU, where $L_{\\odot}\\approx 3.83\\times\n10^{33}$ergs s$^{-1}$ is the solar bolometric luminosity, is\nexpressed as follows\n$$R_{_{HZ}}\\approx 3.5\\times 10^{-2}\\times\nP^{-3\/2}\\times\\left(\\frac{\\kappa}{0.1}\\right)^{1\/2}\\times$$\n\\begin{equation}\n\\label{rHZ} \\times\\left(\\frac{\\dot{P}}{10^{-15}ss^{-1}}\n\\right)^{1\/2}\\times\\left(\\frac{M}{1.5M_{\\odot}}\\right)^{1\/2}AU,\n\\end{equation}\nwhere we have taken into account that the bolometric luminosity,\n$L$, of the pulsar is less than the slowdown luminosity and is\nexpressed as $\\kappa L_{sd}$, $\\kappa < 1$. As we see, the radius of\nthe HZ for $1$s pulsars is very small compared to the radius of the\ntypical Dyson sphere - $1$AU. The best option for the\nsupercivilization could be to find a pulsar with an angle between\nthe magnetic moment (direction of one of the channels) and the axis\nof rotation close to $90^o$, because in this case an artificial\n\"ring\" has to be constructed in the equatorial plane. In case of\nrelatively small inclination angles, there should be two rings, each\nof them shifted from the equatorial plane, although, in this case it\nis unclear how do the \"rings\" keep staying in their planes,\ntherefore, we focus only on pulsars with $\\alpha\\approx 90^o$.\n\n\n\n\nAccording to the standard model of pulsars it is believed that the\nradio emission is generated by means of the curvature radiation\ndefining the opening angle of the emission channel \\citep{rudsuth}\n$$\\beta\\approx 32^o\\times\nP^{-13\/21}\\times\\left(\\frac{\\rho}{10^6cm}\\right)^{2\/21} \\times$$\n\\begin{equation}\n\\label{angle} \\times\\left(\\frac{\\dot{P}}{10^{-15}ss^{-1}}\n\\right)^{1\/14}\\times\\left(\\frac{\\omega} {1.0\\times\n10^{10}Hz}\\right)^{-1\/3},\n\\end{equation}\nwhere the radius of curvature of magnetic field lines - $\\rho$ and\nthe cyclic frequency of radio waves - $\\omega$, are normalized on\ntheir typical values \\citep{rudsuth}. One can straightforwardly\ncheck that the artificial \"ring\" with equal radiuses of the\nspherical segment bases should have height in the following interval\n$\\left(1-2\\right)R_{_{HZ}}\\sin\\left(\\beta\\right)\\approx\\left(0.006-0.06\\right)$AU.\n\n\nOne of the significant parameters to search the \"rings\" is their\ntemperature, which can be estimated quite straightforwardly. In\nparticular, if the aim of the super civilization is to consume the\ntotal energy radiated by the host pulsar, then albedo of the\nmaterial the \"ring\" has to be made of must be close to zero. For\nsimplicity we consider $\\alpha = 90^o$, then it is clear that the\ninner surface of the \"ring\", irradiated by the pulsar, every single\nsecond absorbs energy of the order of $L$. On the other hand, energy\nbalance requires that the \"ring\" must emit the same amount of energy\nin the same interval of time, $L = A_{ef}\\sigma T^4$, leading to the\nfollowing expression\n\\begin{equation}\n\\label{balance} T = \\left(\\frac{L}{A_{_{ef}}\\sigma}\\right)^{1\/4},\n\\end{equation}\nwhere $\\sigma\\approx 5.67\\times 10^{-5}$erg\/(cm$^2$K$^4$) is the\nStefan-Boltzmann constant, $A_{_{ef}} = 8\\pi\nR_{_{HZ}}^2\\sin\\left(\\alpha\/2\\right)$ is the effective area of the\nspherical segment taking into account inner and outer surfaces and\n$T$ is the average temperature of the \"ring\".\n\n\n\n\n\n\n\n\n\nAfter applying Eqs. (\\ref{lumin},\\ref{rHZ},\\ref{angle}), one can\nobtain $T$. On Figure \\ref{fig2} (top panel) we plot the behaviour\nof temperature versus the period of rotation of a pulsar for\ndifferent values of the slow down rate $\\dot{P}$: $\\dot{P} =\n10^{-15}$ss$^{-1}$ (solid line); $\\dot{P} = 10^{-14}$ss$^{-1}$\n(dashed line); $\\dot{P} = 2\\times 10^{-14}$ss$^{-1}$ (dotted-dashed\nline). As we see from the graph, for relatively slowly spinning\npulsars the typical values of the effective temperature of the\nartificial construction are in the following interval $\\sim\n(400-500)$K. The corresponding radius of the HZ varies from $\\sim\n10^{-2}$AU to $\\sim 10^{-1}$AU (see the bottom panel).\n\nGenerally speaking, the distance to the HZ is not strictly defined,\nbecause our knowledge about life is very limited by the conditions\nwe know on Earth. But even in the framework of life on Earth the\nradius of the HZ might be defined in a different way, assuming a\ndistance enabling the effective temperature in the interval:\n$(273-373)$K where water can be in a liquid state (henceforth - the\ntemperature method) \\footnote{In reality the temperature range might\nbe even narrower. For example the melting temperature of DNA is\napproximately $60^o$C}. From Eq. (\\ref{balance}) one can show that\n\\begin{equation}\n\\label{balance1} R_{HZ} =\n\\left(\\frac{L}{8\\pi\\sigma\\sin{\\left[\\beta\/2\\right]}T^4}\\right)^{1\/2}.\n\\end{equation}\nOn Figure \\ref{fig2} (bottom panel) we plot the graphs of $R_{HZ}$\nversus $T$ for the same values of $\\dot{P}$. It is evident that in\nthis case the distance to the HZ ranges from $2\\times 10^{-4}$AU to\n$1.3\\times 10^{-2}$AU.\n\nAnother possibility for the supercivilization might be the\n\"colonization\" of millisecond pulsars since these objects reveal\nextremely high values of luminosity. In particular, for the pulsar\nwith $P = 0.01$s and $\\dot{P} = 10^{-13}$ss$^{-1}$ the slowdown\nluminosity is of the order of $9.5\\times 10^{38}$ergs s$^{-1}$ and\nby taking into account $\\kappa\\approx 0.01$ (being quite common for\nmillisecond pulsars), one can show that the bolometric luminosity\n$$L\\approx 9.5\\times 10^{36}\\times\n\\left(\\frac{P}{0.01s}\\right)^{-3}\\times\\left(\\frac{\\kappa}{0.01}\\right)\n\\times$$\n\\begin{equation}\n\\label{lb} \\times\\left(\\frac{\\dot{P}}{10^{-13}ss^{-1}}\n\\right)\\times\\left(\\frac{M}{1.5M_{\\odot}}\\right)ergs\\ s^{-1}\n\\end{equation}\nis by several orders of magnitude higher than for relatively slowly\nrotating pulsars. Here the physical quantities are normalized on\ntypical values of rapidly rotating neutron stars. It is clear that\nall rapidly spinning pulsars have extremely high energy output in\nthe form of electromagnetic waves. As a result, a corresponding\ndistance from the central object to the HZ must be much bigger than\nfor slowly rotating pulsars.\n\nIn order to estimate the height of the \"ring\" one has to define the\nopening angle of a radiation cone for millisecond pulsars radiating\nin the $X$-ray spectrum. According to the classical paper\n\\citep{machus}, the $X$-ray emission of pulsars has the synchrotron\norigin, maintained by the quasi-linear diffusion. As it was shown by\nMachabeli \\& Usov (1979) the corresponding angle of the radiation\ncone is given by\n\\begin{equation}\n\\label{angle2} \\beta\\approx\n8^o\\times\\left(\\frac{0.01s}{P}\\right)^{1\/2}\\times\\left(\\frac{R_{st}}{10^6cm}\\right)^{1\/2},\n\\end{equation}\nwhich automatically gives an interval of height of the artificial\nconstruction: $(0.02-0.15)$AU.\n\n\n\n\n\n\n\n\nAfter combining Eqs. (\\ref{balance},\\ref{angle2}), like the previous\ncase of slowly rotating neutron stars, one can estimate the\neffective temperature for millisecond pulsars. In particular, on\nFigure \\ref{fig3} (top panel) in the framework of the flux method,\nwe show the behaviour of temperature of the \"ring\" as a function of\n$P$ and as we see, for the following interval of rotation periods:\n$(0.01-0.05)$s the temperature ranges from $\\sim 540$K to\napproximately $660$K. The size of the HZ ranges from $30$AU to\n$350$AU. Contrary to this, by applying the temperature method, on\nFigure \\ref{fig3} (bottom panel) we show the behaviour of $R_{HZ}$\nversus $T$ for three different values of $\\dot{P}$: $\\dot{P} =\n10^{-13}$ss$^{-1}$ (solid line); $\\dot{P} = 2\\times\n10^{-13}$ss$^{-1}$ (dashed line); $\\dot{P} = 4\\times\n10^{-13}$ss$^{-1}$ (dotted-dashed line). From the figure it is clear\nthat the distance to the HZ ranges from $10$AU to $30$AU.\n\nTo understand how realistic are these constructions it is\ninteresting to estimate their masses. Let us assume that the\nmaterial the rings are made of has the density of the order of the\nEarth, $\\rho\\sim 5.5$g cm$^{-3}$. Then it is straightforward to show\nthat the expression of mass is given by\n\\begin{equation}\n\\label{mass} M_{ring}\\approx 4\\pi\\rho R_{_{HZ}}^2\\Delta\nR\\sin\\left(\\beta\/2\\right),\n\\end{equation}\nwhere $\\Delta R$ is the average thickness of the ring. If we\nconsider slowly rotating pulsars, by assuming $\\Delta R\\sim 10$m and\n$R_{_{HZ}}\\sim 10^{-3}$AU (see Figure \\ref{fig2}), one can show that\n$M_{ring}\\approx 4\\times 10^{24}$g, that is three orders of\nmagnitude less than the mass of the Earth. Therefore, in the\nplanetary system the material could be quite enough to construct the\nring-like structure around $1$s pulsars.\n\nThe similar analysis for millisecond pulsars, shows that the mass of\nthe ring, $4\\times 10^{32}$kg, exceeds the total mass of all\nplanets, planetoids, asteroids, comets, centaurs and interplanetary\ndust in the solar system by three orders of magnitude. This means\nthat nearby regions of millisecond pulsars hardly can be considered\nas attractive sites for colonization.\n\nAnother issue one has to address is the force acting on the\nstructure by means of the radiation. By assuming that half of the\ntotal energy is radiated in each of the radiation cones, the\ncorresponding force can be estimated as follows\n\\begin{equation}\n\\label{fr} F_{rad}\\approx\\frac{L}{2c}.\n\\end{equation}\nOn the other hand, the stress forces in the ring, caused by\ngravitational forces should be of the order of the gravitational\nforce acting on the mentioned area of the ring. This force is given\nby\n\\begin{equation}\n\\label{fg} F_{g}\\approx\\frac{GM\\lambda A}{R_{_{HZ}}^2},\n\\end{equation}\nwhere $\\lambda$ is the mass area density of the ring and $A = 2\\pi\nR_{_{HZ}}^2(1-\\cos\\left(\\beta\/2\\right))$ is the area of the ring,\nirradiated by the pulsar. It is clear that for maintaining stability\nof the structure the radiation force should be small compared to the\ngravitational force. By imposing this condition Eqs.\n(\\ref{fr},\\ref{fg}) lead to $\\lambda\\gg L\/\\left(4\\pi\nGMc(1-\\cos\\left(\\beta\/2\\right))\\right)$, which for $\\beta = 32^o$\nreduces to\n\\begin{equation}\n\\label{sigma} \\lambda\\gg 3.4\\times\n10^{-6}\\frac{L_{31}}{M_{1.5}}g\\;cm^{-2},\n\\end{equation}\nwhere $L_{31}\\equiv 10^{31}$ergs s$^{-1}$ and $M_{1.5}\\equiv\nM\/(1.5M_{\\odot})$. As we see from the aforementioned estimate, for\nrealistic scenarios the emission cannot significantly perturb the\nDyson construction.\n\nGenerally speaking, rotation of pulsars drives high energy outflows\nof plasma particles around the star \\citep{guda} and the influence of these particles\ndeserves to be considered as well. The similar result can be obtained by\nconsidering the interaction of the pulsar wind with the ring. In\nparticular, in Eq. (\\ref{sigma}) instead of the radiation term,\n$L\/2$, one should apply the pulsar wind's kinematic luminosity, that\ncannot exceed the slowdown luminosity. After considering the maximum\npossible kinematic wind luminosity, $L_{sd}$, the critical surface\ndensity will be of the same order of magnitude. Therefore, the Dyson\nstructure can survive such an extreme pulsar environment.\n\n\nOn the other hand, it is clear that gravitationally such a system\nmight not be stable. In particular, \\cite{stability} has considered\na point mass and a thin solid ring having initially coincident\ncenters of masses, thus being in the equilibrium state. Although, it\nhas been shown that this equilibrium configuration is stable due to\nthe perturbations normal to the plane of the ring and is unstable\nfor perturbations within the mentioned plane.\n\nOn Fig. \\ref{fig4} we schematically show the in-plane displaced ring\nwith respect to the pulsar. The gravitational force between the\npulsar and the ring element with mass $dm$\n\\begin{equation}\n\\label{grav1} df = G\\frac{Mdm}{r'^2},\n\\end{equation}\nprojected on the $r$ line and integrated over the hole ring\n\\citep{stability}\n\\begin{equation}\n\\label{grav2} f_r(\\zeta) = G\\frac{MM_{ring}}{2\\pi\nR^2}\\int_0^{2\\pi}\\frac{\\zeta+\\cos\\phi}{\\left(1+2\\zeta\\cos\\phi+\\zeta^2\\right)^{3\/2}}d\\phi,\n\\end{equation}\nfor small values of $\\zeta\\equiv r\/R$, leads to the equation\ndescribing the in-plane dynamics \\citep{stability}\n\\begin{equation}\n\\label{stab} \\frac{d^2\\zeta}{dt^2} -\\frac{1}{\\tau^2}\\zeta = 0,\n\\end{equation}\nwhere $\\tau\\equiv \\sqrt{2R^3\/(GM)}$ is the timescale of the process.\nThis equation has the following solution\n\\begin{equation}\n\\label{solut} \\zeta(t) = \\zeta_0 e^{t\/\\tau},\n\\end{equation}\nindicating that the equilibrium configuration is unstable against\nin-plane perturbations. Here, $\\zeta_0$ is the initial\nnondimensional perturbation. By means of this process the ring gains\nthe kinetic energy with the following power\n\\begin{equation}\n\\label{power} W = M_{ring}R^2\\frac{d\\zeta}{dt}\\frac{d^2\\zeta}{dt^2},\n\\end{equation}\ntherefore, in order to restore the equilibrium state of the ring,\none should utilize the same amount of energy from the pulsar\ncorresponding to the instability timescale. On the other hand, it is\nevident that such constructions may have sense only if the power\nneeded to maintain stability is small compared to the bolometric\nluminosity of the pulsar. By taking into account this fact, one can\nstraightforwardly show that the initial declination from the\nequilibrium position must satisfy the condition\n\\begin{equation}\n\\label{cond} \\zeta_0\\ll\n\\frac{0.37}{R}\\left(\\frac{L_b}{M_{ring}}\\right)^{1\/2}\\left(\\frac{2R^3}{GM}\\right)^{3\/4},\n\\end{equation}\nwhich for the parameters presented in Fig. \\ref{fig3} leads to\n$\\zeta_0\\ll\\left( 10^{-5}-10^{-4}\\right)$. It is worth noting that\nsuch a precision of measurement of distance for supercivilization\ncannot be a problem. For instance, in the Lunar laser ranging\nexperiment \\footnote{The corresponding data is available from the Paris Observatory\nLunar Analysis Center: http:\/\/polac.obspm.fr\/llrdatae.html} the distance is measured with the precision of the order\nof $10^{-10}$.\n\nIn the vicinity of pulsars radiation protection could be one of the important challenges\nfacing the super advanced civilization. In particular, the radiation intensity\n\\begin{equation}\n\\label{int} I = \\frac{L_b}{4\\pi R^2\\left(1-\\cos\\left(\\beta\/2\\right)\\right)},\n\\end{equation}\nfor the shortest ring radius $2\\times 10^{-4}$AU is of the order of\n$10^{12}$erg s$^{-1}$ cm$^{-2}$ and therefore, it should be of great importance\nto protect the civilization from high energy gamma rays.\nFor this purpose one can use special shields made of certain material, efficiently\nabsorbing the radiation, which in turn, might significantly reduce the corresponding\nintensity. If one uses half-value\nlayer, $HVL$, which is the thickness of the material at which the intensity is reduced by\none half we can estimate if the thickness of the ring is enough to make a strong protection\nfrom extremely high level of emission. If as an example we examine concrete with $HVL=6.1$,\none can show that a layer of thickness $2.5$m reduces the intensity by $10^{12}$\norders of magnitude, being even more than enough for radiation protection.\n\n\n\n\\section{Conclusion}\n\nWe extended the idea of Freeman Dyson about the search for\nsupercivilization and considered neutron stars. As a first example\nwe examined relatively slowly rotating pulsars, considering the\nparameters $P=\\left(0.5-2\\right)$s; $\\dot{P} = 10^{-15}$ss$^{-1}$;\n$\\dot{P} = 10^{-14}$ss$^{-1}$; $\\dot{P} = 2\\times\n10^{-14}$ss$^{-1}$. It has been shown that size of the \"ring\" must\nbe by $(1-4)$ orders of magnitude less than those of the Dyson\nsphere, which is thought to be of the order of $1$AU. The\ncorresponding temperatures of the artificial construction should be\nin the following interval $(300-600)$K.\n\nBy considering the parameters of millisecond pulsars,\n$P=\\left(0.01-0.05\\right)$s; $\\dot{P} = 10^{-13}$ss$^{-1}$; $\\dot{P}\n= 2\\times 10^{-13}$ss$^{-1}$; $\\dot{P} = 4\\times 10^{-13}$ss$^{-1}$\nwe found that the radius of the \"ring\" should be of the order of\n$(10-350)$AU with an enormous mass $10^{32}$g exceeding the total\nplanetary mass (except the central star) by several orders of\nmagnitude. Therefore, it is clear that millisecond pulsars become\nless interesting in the context of the search for extraterrestrial\nsuperintelligence. Contrary to this class of objects, for slowly\nrotating pulsars the corresponding masses of the Dyson structures\nshould be three orders of magnitude less than the Earth mass. We\nhave also examined the tidal stresses in terms of radiation and\npulsar winds and it has been shown that they will not significantly\nperturb the Dyson construction located in the habitable zone if the\narea density of the ring satisfies a quite realistic condition\n$\\lambda\\gg 3.4\\times 10^{-6}$g cm$^{-2}$. We have examined the stability\nproblem of the ring's in-plane dynamics and it has been shown that under certain\nconditions the power required to restore the equilibrium position might be\nmuch less than the power extracted from the pulsar. Also considering the problem\nof radiation protection we have found that it is quite\nrealistic to reduce the high level of emission by many orders of magnitude.\n\n\nIt is worth noting that in the framework of the paper we do not\nsuggest that an advanced civilization would arise around a massive\nstar, surviving its supernova. On the contrary, we consider the\npossibility to colonize the nearby regions of pulsars building\nlarge-scale Dyson structures.\n\nGenerally speaking, the total luminosity budget of a pulsar is\nemitted over a broad range of wavelengths, that can be harvested by\nmeans of the Faraday's law of induction, transmitting\nelectromagnetic energy into that of electricity.\n\nAs we see, the pulsars seem to be attractive sites for super\nadvanced cosmic intelligence and therefore, the corresponding search\nof relatively small ($0.0001$AU-$0.1$AU) infrared \"rings\" (with the\ntemperature interval $(300-600)$K) might be quite promising.\n\n\n\n\n\\section*{Acknowledgments}\n\nThe research was partially supported by the Shota Rustaveli National\nScience Foundation grant (N31\/49).\n\n\n\\begin{figure}\n  \\centering {\\includegraphics[width=7cm]{fig1.eps}}\n  \\caption{On the picture we schematically show the pulsar, its axis of rotation,\n  and two emission channels with an opening angle $\\beta$. It is worth noting that\n  when $\\alpha$ is close to $90^o$, the Dyson construction has to be located in the equatorial\n  plane. Contrary to this, for relatively smaller angles, the emission channels\n  will irradiate two different ring-like structures located in different planes\n  parallel to that of the equator.}\\label{fig1}\n\\end{figure}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{fig2a.eps}}\n\\resizebox{\\hsize}{!}{\\includegraphics{fig2b.eps}}\n \\caption{On the top panel, in the framework of\nthe flux method, we plot the dependence of $T$ on $P$ for three\ndifferent values of $\\dot{P}$: $\\dot{P} = 10^{-15}$ss$^{-1}$ (solid\nline); $\\dot{P} = 10^{-14}$ss$^{-1}$ (dashed line); $\\dot{P} =\n2\\times 10^{-14}$ss$^{-1}$ (dotted-dashed line). As it is clear from\nthe graph, for typical values of relatively slowly spinning pulsars\nthe effective temperature of the artificial construction ranges from\n$\\sim 400$K to $\\sim 500$K. On the bottom panel, in the framework of\nthe temperature method, we show the dependence of $R_{HZ}$ on $T$\nfor the same values of $\\dot{P}$. As we see the distance to the HZ\nranges from $2\\times 10^{-4}$AU to $1.3\\times 10^{-3}$AU.}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{fig3a.eps}}\n\\resizebox{\\hsize}{!}{\\includegraphics{fig3b.eps}}\n \\caption{On the top panel we present the\nbehaviour of $T$ versus $P$ in the framework of the flux method. It\nis evident that for rapidly spinning pulsars the effective\ntemperature of the \"ring\" is in the following interval:\n$(540-660)$K. On the bottom panel, in the framework of the\ntemperature method, we show the dependence of $R_{HZ}$ on $T$ for\nthree different values of $\\dot{P}$: $\\dot{P} = 10^{-13}$ss$^{-1}$\n(solid line); $\\dot{P} = 2\\times 10^{-13}$ss$^{-1}$ (dashed line);\n$\\dot{P} = 4\\times 10^{-13}$ss$^{-1}$ (dashed-dotted line). As we\nsee the distance to the HZ ranges from $10$AU to $30$AU.}\n\\label{fig3}\n\\end{figure}\n\n\n\\begin{figure}\n  \\centering {\\includegraphics[width=7cm]{fig4.eps}}\n  \\caption{Here we show the in-plane displaced ring\n  with respect to the pulsar, denoted by $M$.}\\label{fig4}\n\\end{figure}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Appendix}\n\\subsection{Glossary of Equations and Notations}\\label{sect:eq-cheat-sheet}\nFor easier reference, we compressed the most frequently referenced equations and notations on two pages. \nThe Wainwright-Hsu equations are given by:\n\\begin{align*}\n N_i' &=  -(\\Sigma^2 +2 \\langle \\taubp_i,\\bpt\\Sigma\\rangle)N_i  \\tag{\\ref{eq:ode-ni}}\\\\\n &= -\\left(\\left|\\bpt\\Sigma+\\taubp_i\\right|^2 -1\\right)N_i \\tag{\\ref{eq:ode2-ni}}\\\\\n\\bpt\\Sigma' &= N^2 \\bpt\\Sigma + 2 \\threemat{\\taubp_1}{\\taubp_2}{\\taubp_3}{\\taubp_3}{\\taubp_1}{\\taubp_2}[\\bpt N, \\bpt N],\\quad\\text{where}\\tag{\\ref{eq:ode2-sigma}}\\\\\n\\taubp_1 &= (-1,0) \\qquad \\taubp_2 = \\left(\\frac 1 2, -\\frac 1 2 \\sqrt{3}\\right) \\qquad \\taubp_3 = \\left(\\frac 1 2, \\frac 1 2 \\sqrt{3} \\right) \\tag{\\ref{eq:taub-def}}\\\\\n1&\\overset{!}{=} \\Sigma^2+N^2 = \\Sigma_+^2+\\Sigma_-^2 + N_1^2+N_2^2+N_3^2-2(N_1N_2+N_2N_3+N_3N_1). \\tag{\\ref{eq:constraint}}\n\\end{align*}\nAuxilliary quantities are given by:\n\\begin{align*}\n\\delta_i &= 2\\sqrt{|N_jN_k|} \\tag{\\ref{eq:delta-r-def-delta}}\\\\\nr_i &= \\sqrt{(|N_j|-|N_k|)^2 + \\frac{1}{3}\\langle\\taubp_j-\\taubp_k, \\bpt\\Sigma\\rangle^2} \\tag{\\ref{eq:delta-r-def-r}}\\\\\n\\delta_i' &= -\\left(\\left|\\bpt\\Sigma-\\frac{\\taubp_i}{2}\\right|^2-\\frac{1}{4}\\right)\\delta_{i}.\\tag{\\ref{eq:ode2-delta}}\n\\end{align*}\nIn polar coordinates, the equations around $r_1 \\approx 0$ become for $N_2,N_3>0$:\n\\begin{align*}\nr_1 &= \\Sigma_-^2+ N_-^2 \\quad N_-=N_3-N_2\\quad N_+=N_3+N_2\\\\\n{\\mathrm{D}}_t \\log r_1&= N^2 - (\\Sigma_++1) \\frac{N_-^2}{r_1^2} +\\sqrt{3} N_1 \\frac{\\Sigma_- N_-}{r_1^2} \\tag{\\ref{eq:neartaub-b9-q:r}}\\\\\n&=r_1^2\\sin^2\\psi \\frac{-\\Sigma_+}{1-\\Sigma_+} + N_1\\, h_r\\tag{\\ref{eq:neartaub-b9-t:r}}\\\\\n{\\mathrm{D}}_t \\log \\delta_1 &= N^2-(\\Sigma_++1)\\tag{\\ref{eq:neartaub-b9-q:delta}}\\\\\n&= \\frac{-1}{1-\\Sigma_+}r_1^2\\cos^2\\psi + \\frac{-\\Sigma_+}{1-\\Sigma_+}r_1^2\\sin^2\\psi + N_1\\,h_\\delta \\tag{\\ref{eq:neartaub-b9-t:delta}}\\\\\n{\\mathrm{D}}_t \\log \\frac{\\delta_1}{r_1} &= -(\\Sigma_+ + 1)\\frac{\\Sigma_-^2}{r_1^2} -\\sqrt{3}N_1\\frac{\\Sigma_- N_-}{r_1^2} \\tag{\\ref{eq:neartaub-b9-q:delta-r}}\\\\\n&=\\frac{-1}{1-\\Sigma_+}r_1^2\\cos^2\\psi +N_1(h_\\delta-h_r)\\tag{\\ref{eq:neartaub-b9-t:delta-r}}\\\\\n\\psi' &=\\sqrt{3}r_1 \\sqrt{\\sin^2\\psi + \\frac{\\delta_1^2}{r_1^2}} - \\frac{r_1^2}{1-\\Sigma_+} \\sin\\psi \\cos\\psi + N_1\\sin\\psi\\, h_{\\psi},\\tag{\\ref{eq:neartaub-b9-t:psi}}\n\\end{align*}\n\\noindent\nwhere the terms $|h_r|,|h_\\delta|, |h_\\psi|$ are bounded (if $|N_i|$, $\\Sigma_+<0$, and $\\Sigma_-$ are bounded) and given in \\eqref{eq:neartaub-b9-t-h}, page \\pageref{eq:neartaub-b9-t-h}.\n\nIn polar coordinates, the equations around $r_1 \\approx 0$ become for $N_2>0$ , $N_3<0$:\n\\begin{align*}\nr_1 &= \\Sigma_-^2+ N_-^2 \\quad N_-=N_2+N_3\\quad N_+=N_2-N_3\\\\\n{\\mathrm{D}}_t \\log r_1&= N^2 - (\\Sigma_++1) \\frac{N_-^2}{r_1^2} +\\sqrt{3} N_1 \\frac{\\Sigma_- N_+}{r_1^2} \\tag{\\ref{eq:neartaub-b8-q:r}}\\\\\n&=\\frac{-\\Sigma_+}{1-\\Sigma_+}r_1^2\\sin^2\\psi +\\delta_1^2\\frac{\\cos^2\\psi-\\Sigma_+}{1-\\Sigma_+} + N_1\\, h_r\\tag{\\ref{eq:neartaub-b8-t:r}}\\\\\n{\\mathrm{D}}_t \\log \\delta_1 &= N^2-(\\Sigma_++1)\\tag{\\ref{eq:neartaub-b9-q:delta}}\\\\\n&=\\frac{-1}{1-\\Sigma_+}r_1^2\\cos^2\\psi + \\frac{-\\Sigma_+}{1-\\Sigma_+}r_1^2\\sin^2\\psi +\\frac{-\\Sigma_+}{1-\\Sigma_+}\\delta_1^2+ N_1\\,h_\\delta \\tag{\\ref{eq:neartaub-b8-t:delta}}\\\\\n{\\mathrm{D}}_t \\log \\frac{\\delta_1}{r_1} &= -(\\Sigma_+ + 1)\\frac{\\Sigma_-^2}{r_1^2} -\\sqrt{3}N_1\\frac{\\Sigma_- N_+}{r_1^2} \\tag{\\ref{eq:neartaub-b8-q:delta-r}}\\\\\n&=\\frac{-1}{1-\\Sigma_+}r_1^2\\cos^2\\psi - \\delta_1^2\\frac{\\sin^2\\psi}{1-\\Sigma_+} + N_1(h_\\delta-h_r)\\tag{\\ref{eq:neartaub-b8-t:delta-r}}\\\\\n\\psi' &=\\sqrt{3}r_1 \\sqrt{\\cos^2\\psi + \\frac{\\delta_1^2}{r_1^2}} - \\frac{r_1^2+\\delta_1^2}{1-\\Sigma_+} \\cos\\psi \\sin\\psi + N_1\\cos\\psi\\, h_{\\psi},\\tag{\\ref{eq:neartaub-b8-t:psi}}\n\\end{align*}\n\\noindent\nwhere the terms $|h_r|,|h_\\delta|, |h_\\psi|$ are bounded (if $|N_i|$, $\\Sigma_+<0$, $\\Sigma_-$, and $\\frac{\\delta_1}{r_1}$ are bounded) and given in \\eqref{eq:neartaub-b8-t-h}, page \\pageref{eq:neartaub-b8-t-h}.\n\nWe use $\\mathcal M= \\{\\bpt x\\in {\\mathbb{R}}^5:\\,G(\\bpt x)=1\\}$, and $\\mathcal M_{{{\\hat{n}}}}\\subset \\mathcal M$ to denote the signs of the three $N_i$, with ${{\\hat{n}}}\\in \\{+,-,0\\}^3$. If we use $\\pm$ in subscripts, the repeated occurences are unrelated, such that $\\mathcal M_{\\pm\\pm\\pm}=\\{\\bpt x\\in \\mathcal M:\\,\\text{all three $N_i\\neq 0$}\\}$. We use the notation $\\mathcal T_i = \\{\\bpt x\\in\\mathcal M:\\, \\langle\\taubp_j \\bpt \\Sigma\\rangle=\\langle\\taubp_k,\\Sigma\\rangle,\\,N_j=N_k\\}$ for the Taub-spaces, where $i,j,k$ are a permutation of $\\{1,2,3\\}$. \n\nWe frequently use the following subsets of $\\mathcal M$ (with the obvious definition for subscripts ${{\\hat{n}}}\\in \\{+,-,0\\}^3$ ):\n\\[\\begin{aligned}\n\\textsc{Basin}[\\epsilon]&=\\{\\bpt x\\in\\mathcal M: \\max_i \\frac{\\delta_i}{r_i}<\\epsilon, \\max_i \\delta_i < \\epsilon\\}\\\\\n{\\textsc{Cap}}[\\epsilon_N, \\epsilon_d]&=\\{\\bpt x\\in \\mathcal M: \\max|N_i|\\ge \\epsilon_N, \\max_i\\delta_i \\le \\epsilon_d\\}\\\\\n{\\textsc{Circle}}[\\epsilon_N, \\epsilon_d]&= \\{\\bpt x\\in \\mathcal M: \\max |N_i|\\le\\epsilon_N,\\, \\max_i\\delta_i \\le \\epsilon_d\\}\\\\\n{\\textsc{Hyp}}[\\varepsilon_\\taubp, \\epsilon_N, \\epsilon_d]&= {\\textsc{Circle}}[\\epsilon_N, \\epsilon_d] \\setminus \\left[B_{\\varepsilon_\\taubp}(\\taubp_1)\\cup B_{\\varepsilon_\\taubp}(\\taubp_2)\\cup B_{\\varepsilon_\\taubp}(\\taubp_3) \\right].\n\\end{aligned}.\n\\]\n\n\\subsection{Properties of the Kasner Map}\\label{sect:appendix-kasner-map}\nWe deferred a detailed discussion of the\nKasner-map $K$ in Section \\ref{sect:lower-b-types}, especially the proof of Proposition \\ref{prop:kasnermap-homeomorphism-class}.\nWe will first give a simple proof of Proposition \\ref{prop:kasnermap-homeomorphism-class}, and then discuss classical ways of describing the Kasner map.\n\\begin{figure}[hbpt]\n        \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{.\/graphics\/angle-stuff.pdf}\n        \\caption{A graphical proof of Lemma \\ref{lemma:kasnermap-expansion}.} \\label{fig:angle-stuff}\n        \\end{subfigure\n        ~~%\n        \\begin{subfigure}[b]{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=\\textwidth]{.\/graphics\/classa-seg-labels.pdf}\n                 \\caption{The Kasner-parameter in the six segments.}\n                \\label{fig:seg-labels}\n        \\end{subfigure\n        \\caption{Expansion of the Kasner-map}        \n\\end{figure}\n\\paragraph{Expansion of the Kasner-map.}\nWe can see from Figure \\ref{fig:double-cover} (p.~\\pageref{fig:double-cover}) that the Kasner-map is a double-cover, has three fixed points and reverses orientation. From Figure \\ref{fig:angle-stuff}, we can see that $K$ is non-uniformly expanding:\n\n\\begin{lemma}\\label{lemma:kasnermap-expansion\nConsider the vectorfield $\\partial_{\\mathcal K}(\\bpt p)= (-\\Sigma_-(\\bpt p), +\\Sigma_+(\\bpt p))$. Assume without loss of generality that $\\bpt p_-$ is such that the Kasner-map $\\bpt p_+=K(\\bpt p_-)$ proceeds via the $N_1$-cap, i.e.~$d(\\bpt p_-, -\\taubp_1)<1$ (see Figure \\ref{fig:kasner-segments}).\n\nThen the Kasner-map $K$ is differentiable at $\\bpt p_-$ and we have\n\\[\nK'(\\bpt p_-)= -\\frac{|\\bpt p_+ +2\\taubp_1| }{|\\bpt p_- +2\\taubp_1|}<-1,\\qquad\\text{where}\\qquad K'(\\bpt p'):\\quad K_*\\partial_{\\mathcal K}(\\bpt p_-)=K'(\\bpt p_-)\\partial_{\\mathcal K}(\\bpt p_+).\n\\]\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma \\ref{lemma:kasnermap-expansion}]\nInformally, differentiability is evident from the construction in Figure \\ref{fig:angle-stuff}. The component of $\\partial_{\\mathcal K}$ which is normal to the line through $\\bpt p_+$, $\\bpt p_-$ and $-\\taubp_1$ gets just elongated by a factor $\\lambda =\\frac{|\\bpt p_++2\\taubp_1|}{|\\bpt p_-+2\\taubp_1|}$. The angle between this line and the Kasner-circle, i.e.~$\\partial_{\\mathcal K}$, is constant; therefore, the length of the component tangent to $\\mathcal K$ must get elongated by the same factor. The negative sign is evident from Figure \\ref{fig:angle-stuff}.\n\nFormally, the relation between $\\bpt p_+$ and $\\bpt p_-$ is described by\n\\[\n|\\bpt p_-|=|\\bpt p_+|=1\\qquad \\bpt p_+ +2 \\taubp_1= \\lambda(\\bpt p_-+2\\taubp_1).\n\\]\nSetting $\\bpt p_-=\\bpt p_-(t)$, we obtain a differentiable function $\\lambda=\\lambda(t)$ by the implicit function theorem (as long as $\\langle\\bpt p_+, \\bpt p_-+2\\taubp_1\\rangle \\neq 0$) and obtain (where we use $'$ to denote derivatives with respect to $t$):\n\\[\n\\bpt p_+' = \\lambda'(\\bpt p_-+2\\taubp_1)+\\lambda\\bpt p_-'.\n\\]\nAssuming $\\bpt p_+\\neq \\bpt p_-$, we can set\n\\[\\bpt v = \\frac{\\bpt p_++2\\taubp_1}{|\\bpt p_++2\\taubp_1|}= \\frac{\\bpt p_-+2\\taubp_1}{|\\bpt p_-+2\\taubp_1|}=\\frac{\\bpt p_+-\\bpt p_-}{|\\bpt p_+-\\bpt p_-|},\\]\nand compute the projection to the normal component of $\\bpt v$:\n\\((1-\\bpt v \\bpt v^T)\\bpt p_+' = \\lambda (1-\\bpt v \\bpt v^T)\\bpt p_-',\\) \nsince the vector coefficient of $\\lambda'$ is parallel to $\\bpt v$. Now the vectors $\\bpt p_+'$ and $\\bpt p_-'$ are tangent to the Kasner-circle; letting $J(\\Sigma_+,\\Sigma_-)=(-\\Sigma_-, \\Sigma_+)$ be the unit rotation we can see that\n\\( \\bpt p_+' = \\pm |\\bpt p_+'|J\\bpt p_+ \\) and \\( \\bpt p_-' = \\pm |\\bpt p_-'|J\\bpt p_- \\). \nHence\n\\[\\begin{aligned}\n|(1-\\bpt v \\bpt v^T)\\bpt p_+'|^2&=\\left(1-\\langle v, J\\bpt p_+\\rangle^2\\right)|\\bpt p_+'|^2 \n&= \\left(1-\\frac{\\langle\\bpt p_-, J\\bpt p_+\\rangle^2}{|\\bpt p_+-\\bpt p_-|^2}\\right)|\\bpt p_+'|^2\\\\\n|(1-\\bpt v \\bpt v^T)\\bpt p_-'|^2&=\\left(1-\\langle v, J\\bpt p_-\\rangle^2\\right)|\\bpt p_-'|^2 \n&= \\left(1-\\frac{\\langle\\bpt p_+, J\\bpt p_-\\rangle^2}{|\\bpt p_+-\\bpt p_-|^2}\\right)|\\bpt p_-'|^2.\n\\end{aligned}\\]\nBy antisymmetry of the matrix $J$, i.e.~$\\langle\\bpt p_-, J\\bpt p_+\\rangle=-\\langle J\\bpt p_-,\\bpt p_+\\rangle$, we therefore have\n\\(\n|\\bpt p_+'| = \\lambda |\\bpt p_-'|.\n\\)\n\\end{proof}\n\\paragraph{Symbolic Description.}\nFor a given $\\bpt p_0\\in\\mathcal K$, we can symbolically encode the trajectory $(\\bpt p_n)_{n\\in{\\mathbb{N}}}$ (with $\\bpt p_{n+1}=K(\\bpt p_n)$) under the Kasner-map. The easiest way to do so is to encode it by $(s_n)_{n\\in{\\mathbb{N}}}\\in \\{1,2,3\\}^{\\mathbb{N}}$, where $s_n = i$ if $\\bpt p_n\\to\\bpt p_{n+1}$ occurs via the $|N_i|>0$-cap. Then $(s_n)_{n\\in{\\mathbb{N}}}$ has the property that no symbol repeats, i.e.~$s_{n+1}\\neq s_n$ for all $n\\in{\\mathbb{N}}$. We have, however, an ambiguity if $\\bpt p_N=\\taubp_i$ for some $N>0$. If this occurs, then also all later points have $\\bpt p_{N+n}=\\taubp_i$. We chose to allow both encodings $\\bpt p_N=j$ and $\\bpt p_N=k$, as long as the property that no symbol repeats is preserved. Factoring out this ambiguity gives us a map\n\\[\\begin{aligned}\n\\Psi: \\mathcal K &\\to \\{(s_{n})_{n\\in{\\mathbb{N}}}\\{1,2,3\\}^{\\mathbb{N}}:\\, \\text{no sumbol repeats} \\}\\, \/ \\, \\{(*\\overline{ij}) = (*\\overline{ji})\\},\\\\\n\\Psi(p_0)&= (s_n)_{n\\in{\\mathbb{N}}},\\qquad\\text{such that}\\  d(\\bpt p_n, -\\taubp_{s_n})\\le 1\\ \\text{and no symbol repeats},\n\\end{aligned}\\]\nwhere $*$ stands for an arbitrary initial piece and $\\overline{jk}$ stands for a periodic tail $(*\\overline{jk})=(*jkjkjk\\ldots)$.\nThis map $\\Psi$ is continuous (since the Kasner-map is continuous), where we endow the target space $\\{1,2,3\\}^{\\mathbb{N}}\/\\sim$ with the quotient topology of the product topology. Note that, by construction, $\\Psi$ semiconjugates the Kasner-map $K$ to the sifht-map $\\sigma$:\n\\[\n\\Psi\\circ K=\\sigma\\circ \\Psi,\\quad\\text{where}\\quad \\sigma: (s_0s_1s_2\\ldots)\\to (s_1s_2\\ldots).\n\\]\n\nIn order to see that $\\Psi$ is a homeomorphism, we construct a continuous inverse. \nDenote the three segments of $\\mathcal K$ as $\\mathcal K_i = \\{\\bpt p\\in\\mathcal K:\\, d(\\bpt p, -\\taubp_i)\\le 1\\}$. We can construct inverse maps $K^{-1}_{ij}:\\mathcal K_j \\to \\mathcal K_i$, such that $K\\circ K^{-1}_{ij}:\\mathcal K_i \\to \\mathcal K_i = \\mathrm{id}$. Then we get an inverse map\n\\[\n\\Psi^{-1}: (s_{n})_{n\\in{\\mathbb{N}}} \\to \\bigcap_{\\ell\\in{\\mathbb{N}}}K^{-1}_{s_0 s_1}K^{-1}_{s_1s_2}\\ldots K^{-1}_{s_{\\ell}s_{\\ell+1}}(\\mathcal K_{s_{\\ell+1}}).\n\\]\nWe now need to show that $\\Psi^{-1}((s_n)_{n\\in{\\mathbb{N}}})=\\{\\bpt p\\}$ is a single point, which depends continuously on $(s_n)_{n\\in{\\mathbb{N}}}$, and is actually the inverse of $\\Psi$.\n\nWe first consider a sequence $(s_n)_{n\\in{\\mathbb{N}}}$ which does not end up in a Taub-point. \n\nIn order to see that $\\Psi^{-1}((s_n)_{n\\in{\\mathbb{N}}})$ is nonempty, note that it is the intersection of a descending sequence of nonempty compact sets. In order to see that it contains only a single point, note that $K_{ij}^{-1}$ is (nonuniformly) contracting by Lemma \\ref{lemma:kasnermap-expansion}; hence the length $\\textsc{len}_{\\ell}=|K^{-1}_{s_0 s_1}K^{-1}_{s_1s_2}\\ldots K^{-1}_{s_{\\ell}s_{\\ell+1}}(\\mathcal K_{s_{\\ell+1}})|$ is decreasing. The length $\\textsc{len}_{\\ell}$ also cannot converge to some $\\textsc{len}_{\\infty}$ as $\\ell\\to\\infty$, since we have $|K^{-1}_{ij}(I)|< |I|$ for any interval $I$ with $|I|>0$.\nIn order to see that $\\Psi^{-1}$ is continuous, we need to show that $\\mathrm{diam}\\,\\Psi^{-1}\\{(r_n)_{n\\in{\\mathbb{N}}}:\\,r_n=s_n\\, \\forall n\\le N\\} \\to 0$ as $N\\to\\infty$. This also follows from the previous argument of decreasing lengths.\n\nNext, we consider a sequence $(s_n)_{n\\in{\\mathbb{N}}}$ which does end up in a Taub-point $\\taubp_i$ at $n=N$. Let $(s'_n)_{n\\in{\\mathbb{N}}}$ denote the other representative of $(s_n)_{n\\in{\\mathbb{N}}}$, i.e. we changed $(s_0,\\ldots,s_{N-1},\\overline{jk})\\leftrightarrow ((s_0,\\ldots,s_{N-1},\\overline{kj}))$. The previous arguments about single-valuedness and continuity still apply for each of the two representatives; we only need to show that $\\Psi^{-1}$ coincides for both. This is obvious.\n\nIt is also obvious by our construction that $\\Psi$ and $\\Psi^{-1}$ are inverse to each other. Hence $\\Psi$ and $\\Psi^{-1}$ $C^0$-conjugate $K$ to the shift-map\n\\[\n\\Psi\\circ K\\circ\\Psi^{-1}=\\sigma: (s_0s_1s_2\\ldots)\\to (s_1s_2\\ldots).\n\\]\nThe exact same arguments apply in order to conjugate the map $D:{\\mathbb{R}}\/3{\\mathbb{Z}}\\to{\\mathbb{R}}\/3{\\mathbb{Z}}$, $D:[x]_{3{\\mathbb{Z}}}\\to [-2x]_{3{\\mathbb{Z}}}$ to the same shift, where we replaced $\\taubp_i=i$ and $\\mathcal K_i = [{i+1}, {i+2}]$. Hence, the Kasner-map is $C^0$ conjugate to $D$:\n\\begin{hproposition}{\\ref{prop:kasnermap-homeomorphism-class}}\nThere exists a homeomorphism $\\psi:\\mathcal K \\to {\\mathbb{R}}\/3{\\mathbb{Z}}$, such that $\\psi(\\taubp_i)=\\left[i\\right]_{3{\\mathbb{Z}}}$ and\n\\[\n\\psi(K(\\bpt p))=[-2 \\psi(\\bpt p)]_{3{\\mathbb{Z}}}\\qquad\\forall \\bpt p\\in\\mathcal K.\n\\]\n\\end{hproposition}\n\\paragraph{Kasner Eras and Epochs.}\n\\def\\ensuremath{\\textsc{S}}{\\ensuremath{\\textsc{S}}}\n\\def\\ensuremath{\\textsc{L}}{\\ensuremath{\\textsc{L}}}\nA useful and customary description of the symbolic dynamics of $K$ is obtained by distinguishing between ``small'' bounces around a Taub-point, called ``Kasner epochs'' and denoted by the letter \\ensuremath{\\textsc{S}}{} in this work, and ``long'' bounces, called ``Kasner eras'' and denoted by the letter \\ensuremath{\\textsc{L}}{} in this work. The \\ensuremath{\\textsc{S}}{} and \\ensuremath{\\textsc{L}}{} encoding of an orbit can be obtained from previous $\\{1,2,3\\}$-encoding by the map\n\\[\\textsc{EpochEra}: \\{s\\in \\{1,2,3\\}^{\\mathbb{N}}:\\, s_n\\neq s_{n+1}\\}\\to \\{(s_0s_1|r_0r_1\\ldots):\\, s_0,s_1 \\in\\{1,2,3\\},\\,r_n\\in\\{\\ensuremath{\\textsc{S}}, \\ensuremath{\\textsc{L}}\\}\\}\\]\n\\[(s_0s_1s_2\\ldots) \\to (s_0s_1| r_0r_1\\ldots),\\quad\\text{where}\\quad\\left\\{\\begin{aligned}\nr_n &= \\ensuremath{\\textsc{S}} \\quad\\text{if $s_{n}=s_{n+2}$,}\\\\\nr_n &= \\ensuremath{\\textsc{L}} \\quad\\text{if $s_{n}\\neq s_{n+2}$.}\n\\end{aligned}\\right.\\]\nWe can remember the value of $s_n$ as a subindex of $r_n$, such that e.g.\n\\[\n(132121213231\\ldots)\\to(13|\\ensuremath{\\textsc{L}}_1\\ensuremath{\\textsc{L}}_3 \\ensuremath{\\textsc{L}}_2 \\ensuremath{\\textsc{S}}_1 \\ensuremath{\\textsc{S}}_2 \\ensuremath{\\textsc{S}}_1 \\ensuremath{\\textsc{S}}_2 \\ensuremath{\\textsc{L}}_1 \\ensuremath{\\textsc{L}}_3 \\ensuremath{\\textsc{S}}_2 \\ensuremath{\\textsc{L}}_3 *_1\\ldots).\n\\]\nThen the Kasner-map becomes\n\\[ (s_0s_1|r_0r_1r_2r_3\\ldots)\\to \\left\\{\\begin{aligned}\n(s_1 i|r_1r_2\\ldots) &\\quad \\text{if $s_0=i$ and $r_0=\\ensuremath{\\textsc{S}}$}\\\\\n(s_1 i|r_1r_2\\ldots) &\\quad \\text{if $s_0\\neq i\\neq s_1$ and $r_0=\\ensuremath{\\textsc{L}}$.}\n\\end{aligned}\\right.\\]\nNote that the first two indices in $\\{1,2,3\\}$ describe in which of the six symmetric segments of $\\mathcal K$ a point lies, see Figure \\ref{fig:seg-labels}.\n\nAnother customary way of writing such sequences is to write every symbol \\ensuremath{\\textsc{L}}{} as a semicolon ``;'' and abbreviate the \\ensuremath{\\textsc{S}}{} symbols in between by just their number, such that the previous example becomes\n\\[\n(13|\\ensuremath{\\textsc{L}}_1 \\ensuremath{\\textsc{L}}_3 \\ensuremath{\\textsc{L}}_2 \\ensuremath{\\textsc{S}}_1 \\ensuremath{\\textsc{S}}_2 \\ensuremath{\\textsc{S}}_1 \\ensuremath{\\textsc{S}}_2 \\ensuremath{\\textsc{L}}_1 \\ensuremath{\\textsc{L}}_3 \\ensuremath{\\textsc{S}}_2 \\ensuremath{\\textsc{L}}_3 *_1\\ldots)\\to \n(13|0;0; 0; 4 ; 0 ; 1 ; \\ldots).\n\\]\n\\paragraph{The Kasner-Parameter.}\nThere exists an explicit coordinate transformation, related to the continued fraction expansion, which realizes the conjugacy to the shift-space. This is done via the so-called \\emph{Kasner-parameter} $u$, and is the most standard way of discussing the Kasner-map. \n\nIn order to introduce the Kasner-parameter, it is useful to use the coordinates which make the permutation symmetry of the indices more apparent. This is done via\n\\[\\Sigma_i = 2\\langle \\taubp_i, \\bpt \\Sigma\\rangle \\qquad\n\\Sigma_+ = -\\frac 1 2 \\Sigma_1 \\qquad\n\\Sigma_- = \\frac{1}{2\\sqrt{3}}(\\Sigma_3-\\Sigma_2).\\]\nThese variables are constrained by $\\Sigma_1+\\Sigma_2+\\Sigma_3=0$ and have $\\Sigma_1^2+\\Sigma_2^2+\\Sigma_3^2 =6(\\Sigma_+^2+\\Sigma_-^2)$.\n\n\nThe parametrization of $\\mathcal K$ depends not only on a real parameter $u\\in{\\mathbb{R}}$, but also on a permutation $(i,j,k)$ of $\\{1,2,3\\}$ and is given by $\\bpt \\Sigma=\\bpt\\Sigma(u, (ijk))$ such that\n\\[\\begin{aligned}\n\\Sigma_i &= -1 + 3\\frac{-u}{u^2+u+1} &\n\\Sigma_j &= -1 + 3\\frac{u+1}{u^2+u+1} &\n\\Sigma_k &= -1 + 3\\frac{u^2+u}{u^2+u+1}.\n\\end{aligned}\\]\nWe can immediately observe that $\\langle \\taubp_1+\\taubp_2+\\taubp_3, \\bpt \\Sigma\\rangle=0$, as it should be; hence, the above really defines a function $\\psi:{\\mathbb{R}}\\times \\textsc{sym}_3\\to {\\mathbb{R}}^2$, where $\\textsc{sym}_3$ is the set of permutations of $\\{1,2,3\\}$. \n Direct calculation shows that $\\Sigma_1^2+\\Sigma_2^2+\\Sigma_3^2=6$ for all $u\\in{\\mathbb{R}}$.\nHence, the $u$ coordinates do actually parametrize the Kasner circle. We have the noteworthy symmetry properties\n\\[\\begin{array}{rlll}\n\\bpt \\Sigma\\left(u, (ijk)\\right)\n&=\\bpt \\Sigma\\left(u^{-1}, (ikj)\\right) \n&=\\bpt \\Sigma\\left(-(u+1), (jik)\\right)\n&= \\bpt\\Sigma\\left(\\frac{-1}{u+1}, (jki)\\right)\\\\\n&= \\bpt\\Sigma\\left(-\\frac{u}{u+1}, (kji)\\right)\n&= \\bpt\\Sigma\\left(-\\frac{u+1}{u}, (kij)\\right).&\n\\end{array}\\]\nWe can use the symmetry to normalize $u$ to a value $u\\in[1,\\infty]$, thus giving a parametrization of the Kasner circle as in Figure \\ref{fig:seg-labels}.\n\n\\paragraph{The Kasner-map in $u$-coordinates.}\nWe consider without loss of generality a heteroclinic orbit $\\gamma\\subseteq \\mathcal M_{+00}$. In the $\\bpt \\Sigma$-projection, this heteroclinic orbit is a straight line through $-2\\taubp_1$; hence, the quotient $\\frac{\\Sigma_-}{2-\\Sigma_+}$ stays constant. It is given by \n\\[\n\\frac{\\Sigma_-}{2-\\Sigma_+}(u)=\\sqrt{3}\\frac{u^2-1}{4(u^2+u+1)-(u^2+4u+1)} = \\frac{\\sqrt{3}}{3}\\frac{u^2-1}{u^2+1}.\n\\]\nIf we write the $\\alpha$-limit of $\\gamma$ with respect to the $123$ permutation, then we must have $u\\in [0,\\infty]$ (because $N_1$ would not be unstable otherwise). Then the Kasner-map must be given by $(u,(123))\\to (-u,123)$. Assume that $u\\in[2,\\infty]$; then we can renormalize $K(u)$, such that $K(u, 123)= (u-1, 213)$. If instead $u\\in [1,2]$, then we can renormalize such that $K(u,123)=\\left(\\frac{1}{u-1},231\\right)$. Applying symmetrical arguments for the other caps yields the following way of writing the Kasner map:\n\\[\\begin{gathered}\n\\mathcal K = \\textsc{sym}_3\\times [1,\\infty] \\,\\big\/\\, \\left\\{ (1, ijk)=(1,ikj), (\\infty, ijk)=(\\infty, jik) \\right\\}\\\\\nK:\\mathcal K\\to \\mathcal K\\quad  (u, ijk)\\to \n\\left\\{\\begin{aligned}\n(u-1, jik) &\\qquad \\text{if $u\\in [2,\\infty]$}\\\\\n\\left(\\frac{1}{u-1}, jki\\right)&\\qquad \\text{if $u\\in [1,2]$}.\n\\end{aligned}\\right.\n\\end{gathered}\\]\nNote that the Kasner-map is actually well-defined and continuous at $u=2$, due to the identification $(1,ijk)=(1,ikj)$. It is also well-defined at $u=1$, since the identifications $(1,ijk)=(1,ikj)$ and $(\\infty, ijk)=(\\infty, jik)$ are compatible. It is continuous at $u=1$, since we use the usual compactification at $u=\\infty$, such that a neighborhood basis of $(\\infty, ijk)$ is given by $\\{([R,\\infty], ijk)\\cup ([R,\\infty], jik)\\}_{R \\gg 0}$. Likewise, the Kasner-map is well-defined and continuous at the three fixed-points $u=\\infty$, due to the identification $(\\infty, ijk)=(\\infty, jik)$ and the compactification at $u=\\infty$.\n\\paragraph{Symbolic description in $u$-coordinates.}\nThe two local inverses of the Kasner-map in $u$-coordinates are given by\n\\[\\begin{aligned}\n\\ensuremath{\\textsc{S}}: (u,ijk)&\\to (u+1, jik)\\\\\n\\ensuremath{\\textsc{L}}: (u,ijk) &\\to \\left(1+\\frac{1}{u}, kij\\right)\n\\end{aligned}\\]\nUsing the same construction as in the proof of Proposition \\ref{prop:kasnermap-homeomorphism-class}, we see that the inverse coding map $\\textsc{Cfe}^{-1}$ given by the continued fraction expansion\n\\[\n\\textsc{Cfe}^{-1}:(ij|a_0;a_1;a_2;\\ldots)\\to \\left(1+a_0 + \\frac{1}{1+a_1+\\frac{1}{1+a_2+\\ldots}}\\,,\\, ijk\\right),\n\\]\nand the Kasner-map is given by\n\\[\n\\textsc{Cfe}\\circ K\\circ \\textsc{Cfe}^{-1}: (ij|a_0;a_1;a_2;\\ldots)\\to\\left\\{\\begin{array}{ll}\n(ji\\,|\\,a_0-1;a_1;a_2;\\ldots) &\\quad\\text{if $a_0>0$}\\\\\n(jk\\,|\\,a_1;a_2;\\ldots) &\\quad\\text{if $a_0=0$}.\n\\end{array}\\right.\n\\]\n\n\\subsection{Transformation of Volumes on Manifolds}\\label{sect:general-vol-app}\nThis section contains some basic facts about the transformation of volumes under flows.\n\n\\paragraph{Volume Transformation.}\nLet $M$ be an $n$-dimensional differentiable manifold.\nIn the language of differential forms, we can write the transformation law for a non-vanishing volume-form $\\omega$ under a diffeomorphism $\\Phi:M\\to M$ as\n\\[\\begin{aligned}\n\\lambda(\\bpt x) = \\frac{\\Phi^*\\omega(\\bpt x)}{\\omega(\\bpt x)}\\\\\n\\mathrm{vol}_\\omega(\\Phi(U)) = \\int_{\\Phi(U)}\\omega = \\int_U \\Phi^*\\omega = \\int_U \\lambda \\omega.\n\\end{aligned}\\]\nLet $X_1,\\ldots, X_n$ be a frame, i.e.~a set of vectorfields which form a basis of $TM$. Then can write the density of $\\omega$ as $\\rho=\\omega[X_1,\\ldots, X_n]$ and see $\\omega[Y_1,\\ldots, Y_n]= \\rho \\det (a_{ij})$, where $Y_i = \\sum_j a_{ij}X_j$. This allows us to write \n\\[\n\\lambda(\\bpt x) \\rho(\\bpt x) = \\Phi^*\\omega [X_1,\\ldots X_n] = \\omega[\\Phi_*X_1,\\ldots \\Phi_*X_n] = \\rho(\\Phi(\\bpt x))\\det J,\n\\]\nwhere $J$ is the jacobian, i.e.~the matrix of ${\\mathrm{D}}_x \\Phi(\\bpt x):T_{\\bpt x}M\\to T_{\\Phi(\\bpt x)}M$ with respect to the basis $X_1(\\bpt x),\\ldots X_n(\\bpt x)$ and $X_1(\\Phi(\\bpt x)),\\ldots X_n(\\Phi(\\bpt x))$, i.e. $J=J_{ij}$ with $\\Phi_*X_i(\\Phi(\\bpt x)) = {\\mathrm{D}}_x \\Phi(\\bpt x)\\cdot X_i(\\bpt x) = \\sum_{j}J_{ij} X_j(\\bpt x)$.\n\n\\paragraph{Volume Transformation under flows.}\nWe study the behaviour of a volume-form $\\omega$ under a flow $\\phi:M\\times {\\mathbb{R}}\\to M$ corresponding to a vectorfield $f:M\\to TM$; we are interested in $\\lambda(\\bpt x, t)=\\frac{\\phi^*(\\bpt x, t)\\omega}{\\omega}$.\n\nGiven a set of coordinates $x_1,\\ldots x_n$ and the vectorfield $f=f_1 \\partial_1+\\ldots f_n\\partial_n$ and volume-form $\\omega=\\rho {\\mathrm{d}} x_1\\land \\ldots \\land {\\mathrm{d}} x_n$ we can therefore write, using ${\\mathrm{D}}_x\\phi$ is a shorthand for the Jacobian with respect to the basis $\\partial_1,\\ldots, \\partial_n$:\n\\[\\begin{aligned}\n\\lambda(\\bpt x, t)&=\\frac{\\phi^*(\\bpt x, t)\\omega}{\\omega} = \\frac{\\rho(\\phi(\\bpt x, t))}{\\rho(\\bpt x)}\\det {\\mathrm{D}}_x \\phi(\\bpt x, t)\\\\\n\\frac{{\\mathrm{d}}}{{\\mathrm{d}} t} \\lambda(\\bpt x, t)& = \\frac{({\\mathrm{D}}_f \\rho)(\\phi(\\bpt x, t))}{\\rho(\\bpt x)}\\det {\\mathrm{D}}_x \\phi(\\bpt x, t) + \\frac{\\rho(\\phi(\\bpt x, t))}{\\rho(\\bpt x)}{\\mathrm{D}}_t \\det {\\mathrm{D}}_x \\phi(\\bpt x, t)\\\\\n&= \\lambda(\\bpt x, t){\\mathrm{D}}_f \\log \\rho(\\phi(\\bpt x, t))+ \\frac{\\rho(\\phi(\\bpt x, t))}{\\rho(\\bpt x)}\\mathrm{tr}\\left[({\\mathrm{D}}_t{\\mathrm{D}}_x\\phi(\\bpt x, t)){\\mathrm{D}}_x\\phi(\\bpt x, t)^{-1}\\right] \\det {\\mathrm{D}}_x \\phi(\\bpt x, t)\\\\\n&= \\lambda(\\bpt x, t)\\left({\\mathrm{D}}_f \\log \\rho(\\phi(\\bpt x, t)) + \\mathrm{tr}{\\mathrm{D}}_x f(\\phi(\\bpt x, t)){\\mathrm{D}}_x\\phi(\\bpt x, t)^{-1}\\right)\\\\\n&= \\lambda(\\bpt x, t)\\left({\\mathrm{D}}_f \\log \\rho(\\phi(\\bpt x, t)) + \\mathrm{tr}\\partial_x f(\\phi(\\bpt x, t))\\right)\\\\\n&= \\lambda(\\bpt x, t)\\left({\\mathrm{D}}_f \\log \\rho(\\phi(\\bpt x, t)) + \\sum_i (\\partial_i f_i)(\\phi(\\bpt x, t))\\right),\n\\end{aligned}\\]\nwhere we used the general formula for differentiable families $A:{\\mathbb{R}}\\to {\\mathbb{R}}^{n\\times n}$ of invertible matrices:\n\\[\\begin{aligned}\n{\\mathrm{D}}_{t|t=0} \\det A(t) &= {\\mathrm{D}}_{t|t=0}\\det A(t)A(0)^{-1}A(0)\\\\ \n&=\\det A(0) {\\mathrm{D}}_{t|t=0}\\det A(t)A(0)^{-1} = \\det A(0)\\mathrm{tr}\\,{\\mathrm{D}}_{t|t=0}A(t)A(0)^{-1}.\n\\end{aligned}\\]\n\n\\paragraph{Restriction to Iso-Surfaces.}\nLet $\\Phi:M\\to M$ be a diffeomorphism on an $n$-dimensional manifold $M$ with volume form $\\omega$, which gets transported by $\\lambda= \\frac{\\Phi^*\\omega}{\\omega}$.\nSuppose that $G:M\\to {\\mathbb{R}}$ is a preserved quantity under $\\Phi$, i.e.~$G(\\Phi(\\bpt x))=G(\\bpt x)$ and suppose that $1$ is a regular value of $G$. Now we are interested in the isosurface $M_1 = \\{\\bpt x\\in M: G(\\bpt x) =1\\}$ and in a volume form $\\omega_0$ on $M_1$, which gets transported by the same $\\lambda$. This can be realized by choosing some vectorfield $X:M_1\\to TM$ with ${\\mathrm{D}}_X G=\\mathrm{const}=1$ and setting $\\omega_1=\\iota_X \\omega$. Since $G$ is preserved, we have $\\Phi_*X-X \\in T M_1$ and hence for a basis $X_1,\\ldots,X_{n-1}$ of $TM_1$:\n\\[\\begin{aligned}\n\\lambda \\iota_X \\omega[X_1,\\ldots X_n] &= \\lambda\\omega[X,X_1,\\ldots,X_n]=\\omega[\\Phi_*X,\\Phi_*X_1,\\ldots,\\Phi_*X_{n-1}]\\\\\n&=\\omega[X,\\Phi_*X_1,\\ldots,\\Phi_*X_{n-1}]+\\omega[\\Phi_*X-X,\\Phi_*X_1,\\ldots,\\Phi_*X_{n-1}]\\\\\n&=\\Phi^*\\omega[X_1,\\ldots,X_{n-1}].\n\\end{aligned}\\]\nIt is clear that the induced volume $\\omega_1$ does not depend on the choice of $X$, up to possibly a sign.\n\\paragraph{Contraction with invariant vectorfields.}\nLet $\\Phi:M\\to M$ be a diffeomorphism on an $n$-dimensional manifold $M$ with volume form $\\omega$, which gets transported by $\\lambda= \\frac{\\Phi^*\\omega}{\\omega}$.\nSuppose that the vectorfield $Y:M\\to TM$ is preserved quantity under $\\Phi$, i.e.~$\\Phi_*X=X$. Then the contractoed volume-form $\\omega_1=\\iota_Y\\omega$ has $\\Phi^*\\omega_1=\\lambda\\omega_1$:\n\\[\\begin{aligned}\n\\Phi^*\\omega_1[X_2,\\ldots,X_n]&=\\omega[Y, \\Phi_*X_2,\\ldots,\\Phi_*X_n]=\\omega[\\Phi_*Y,\\Phi_*X_2,\\ldots,\\Phi_*X_n]\\\\\n&=\\lambda \\omega[Y,X_2,\\ldots,X_n]=\\lambda\\omega_1[X_2,\\ldots,X_n].\n\\end{aligned}\\]\nSuppose that $\\phi:M\\times {\\mathbb{R}}\\to M$ is a flow, $U\\subseteq M$ is open and $T:U\\to {\\mathbb{R}}$ is smooth. Consider $\\Phi:\\bpt x\\to \\phi(\\bpt x, T(\\bpt x))$. Then $\\Phi_* X = \\phi(\\cdot, T(\\cdot))_* X + Y{\\mathrm{D}}_X T$, and hence\n\\[\\begin{aligned}\n\\Phi^*\\omega_1[X_2,\\ldots,X_n]=\\omega[Y, \\Phi_*X_2+ Y{\\mathrm{D}}_{X_2} T,\\ldots,\\Phi_*X_n+Y{\\mathrm{D}}_{X_n} T]=\\lambda \\omega_1[X_2,\\ldots,X_n].\n\\end{aligned}\\]\n\\subsection{Derivation of the Wainwright-Hsu equations}\\label{sect:append:derive-eq}\nThe goal of this section is to connect the Einstein field equations of general relativity to the Wainwright-Hsu equations \\eqref{eq:ode} discussed in this work.\n\n\\subsubsection{Spatially Homogeneous Spacetimes}\\label{sect:homogeneous}\nWe are interested in spatially homogeneous spacetimes. We assume that $(M^4,g)$ is a Lorentz-manifold, and we have a symmetry adapted co-frame: $\\{\\omega_1,\\omega_2,\\omega_3, {\\mathrm{d}} t\\}$, corresponding to a frame of vectorfields $\\{e_0,e_1,e_2,e_3\\}$, such that $e_1,e_2,e_3$ are Killing, i.e.~the metric depends only on $t$. We thus assume that the metric has the form\n\\[\ng = g_{00}(t){\\mathrm{d}} t\\otimes {\\mathrm{d}} t + g_{11}(t)\\omega_1\\otimes \\omega_1 + g_{22}(t)\\omega_2\\otimes \\omega_2 + g_{33}(t)\\omega_3\\otimes \\omega_3,\n\\]\nwhere $g_{00}<0$ and the other three $g_{ii}>0$. The spatial homogeneity is described by the commutators of the three Killing fields $e_1,e_2,e_3$; we assume that it is given (for positive permutations $(i,j,k)$ of $(1,2,3)$) by:\n\\[\n[e_i, e_j] = \\gamma_{ij}^k e_k=\\hat n_k e_k\n\\qquad\n{\\mathrm{d}} \\omega_i = -\\hat n_i \\omega_j\\land \\omega_k,\n\\]\nwhere $\\hat n_i \\in \\{-1, 0, +1\\}$ describe the Bianchi type of the surfaces $\\{t=\\textrm{const}\\}$ of spatial homogeneity.\n\n\n\\paragraph{General equations for the Christoffel symbols.}\nThe general equations for Christoffel symbols are given by:\n\\[\\begin{aligned}\n\\nabla_{e_i}e_j &= \\sum_k \\Gamma_{ij}^k e_k\\\\\n[e_i, e_j] &= \\nabla_{e_i}e_j - \\nabla_{e_j}e_i\\quad\\Rightarrow\\quad\n\\Gamma_{ij}^k -\\Gamma_{ji}^k = \\gamma_{ij}^k\\\\\n{\\mathrm{D}}_{e_i} g(e_j, e_k) &= \\partial_{e_i}g_{jk}\n=\\sum_{\\ell} g_{\\ell k}\\Gamma_{ij}^{\\ell} + g_{\\ell j}\\Gamma_{ik}^\\ell\\\\\n\\partial_{e_i}g_{jk} + \\partial_{e_j}g_{ik} - \\partial_{e_k}g_{ij} \n&=\\sum_{\\ell} g_{\\ell k} \\Gamma_{ij}^\\ell + g_{\\ell j}\\Gamma_{ik}^\\ell \n+g_{\\ell k} \\Gamma_{ji}^\\ell + g_{\\ell i}\\Gamma_{jk}^\\ell \n- g_{\\ell i} \\Gamma_{kj}^\\ell - g_{\\ell j}\\Gamma_{ki}^\\ell\\\\\n&=\\sum_{\\ell} g_{\\ell i}\\gamma^{\\ell}_{jk} + g_{\\ell j}\\gamma_{ik}^\\ell + g_{\\ell k}\\gamma_{ji}^\\ell + 2 g_{\\ell k}\\Gamma_{ij}^\\ell\n\\end{aligned}\\]\nWe can solve this for the Christoffel symbols by multiplying with the inverse metric\n\\newcommand\\myrebind[6]{\n\\gdef\\iA{#1}\\gdef\\iB{#2}\\gdef\\iC{#3}\\gdef\\iD{#4}\\gdef\\iE{#5}\\gdef\\iF{#6}\n}\n\\myrebind{j}{i}{k}{\\ell}{n}{*}\n\\[\\begin{aligned}\n\\Gamma_{ij}^k&=\n\\frac 1 2 g^{k \\ell} \\left(\n\\partial_{e_i}g_{j\\ell} + \\partial_{e_j}g_{i\\ell} - \\partial_{e_\\ell}g_{ij} -\\sum_{n} g_{n i}\\gamma^{n}_{j\\ell} - g_{n j}\\gamma_{i\\ell}^n - g_{n \\ell}\\gamma_{ji}^n\\right)\\\\\n&=\\frac 1 2 g^{k k} \\left(\n\\partial_{e_i}g_{jk} + \\partial_{e_j}g_{ik} - \\partial_{e_k}g_{ij} - g_{i i}\\gamma^{i}_{jk} - g_{j j}\\gamma_{ik}^j - g_{k k}\\gamma_{ji}^k\\right),\n\\end{aligned}\\]\nwhere we used the fact that the metric is diagonal in the last equation.\n\\paragraph{Christoffel symbols for spatially homogeneous space times.}\nIf we insert the indices into this equation, we obtain up to index permutations the following non-vanishing Christoffel symbols:\n\\[\\begin{aligned}\n\\nabla_{e_0}e_0 &= \\Gamma_{00}^0 e_0 & \\Gamma_{00}^0 &= \\phantom{-}\\frac 1 2 g^{00}\\partial_{e_0} g_{00}\\\\\n\\nabla_{e_1}e_2 &= \\Gamma_{12}^3 e_3 & \\Gamma_{12}^3 &= -\\frac 1 2 g^{33}\\left(\\;g_{11}\\gamma^{1}_{23}+ g_{22}\\gamma_{13}^2 + g_{33}\\gamma_{21}^3\\right)\\\\\n& & &=\\phantom{-}\\frac 1 2 g^{33}\\left(g_{11}\\hat n_1 \\;- g_{22}\\hat n_2 \\;- g_{33}\\hat n_3\\right)\\\\\n\\nabla_{e_0}e_1 &= \\Gamma_{01}^1 e_1 = \\Gamma_{10}^1 e_1 & \\Gamma_{10}^1 &=\\phantom{-}\\frac 1 2 g^{11}\\partial_{e_0}g_{11}\\\\\n\\nabla_{e_1}e_1 &= \\Gamma_{11}^0 e_0 & \\Gamma_{11}^0 &= -\\frac 1 2 g^{00}\\partial_{e_0}g_{11}\n\\end{aligned}\\]\n\\paragraph{Extrinsic Curvature of surfaces of homogeneity.} %\nThe Weingarten-map $K_{i}^{\\ j}$ and second fundamental form $K_{ij}$ of the surfaces of spatial homogeneity given, up to permutation, by:\n\\[\\begin{aligned} \nK(e_1)&=\\nabla_{e_1}\\frac{1}{\\sqrt{-g_{00}}}e_0 =  \\sqrt{-g^{00}}\\Gamma_{10}^1 e_{1}\n\\qquad K_i^{ j}= \\delta_{i}^{j}\\sqrt{-g^{00}}\\Gamma_{i0}^i\\\\\nK_{ij}&= g(e_i, \\nabla_{e_j}\\sqrt{-g^{00}}e_0)= \\sqrt{-g_{00}}\\Gamma_{ji}^0 =\\sum_{k} g_{k i}\\sqrt{-g^{00}}\\Gamma_{j0}^k,\\quad{\\text{i.e.,}}\\\\\n\\Gamma_{10}^1 &= \\sqrt{-g_{00}} K_1^1, \\qquad\\qquad \\Gamma_{11}^0 = \\sqrt{-g^{00}}g_{11}K_{1}^1\n\\end{aligned}\\]\n\nThe extrinsic curvature corresponds to the normalized time-derivative of the spatial coefficients of the metric:\n\\[\\begin{aligned}\n\\sqrt{-g^{00}}\\nabla_{e_0} \\sqrt{g_{ii}} &= \\sqrt{-g^{00}}\\frac 1 2 \\sqrt{g_{ii}}g^{jj}\\nabla_{e_0}g_{kk}=\\sqrt{-g^{00}}\\Gamma_{i0}^i \\sqrt{g_{ii}} = K_{i}^i\n\\end{aligned}\\]\n\\paragraph{Riemannian Curvature.}\nThe Riemannian curvature tensor is given by the equation\n\\[\\begin{aligned}\n \\sum_{\\ell}R_{ijk}^{\\ \\ \\ \\ell} e_{\\ell} &= \\left(\\nabla_{e_i}\\nabla_{e_j}-\\nabla_{e_j}\\nabla_{e_i} -\\nabla_{[e_i,e_j]}\\right) e_k\\\\\n &=\\sum_{\\ell, n}\\left(\\Gamma_{jk}^n\\Gamma_{in}^\\ell - \\Gamma_{ik}^n\\Gamma_{jn}^\\ell + \\nabla_{e_i}\\Gamma_{jk}^\\ell - \\nabla_{e_j}\\Gamma_{in}^\\ell - \\gamma_{ij}^n\\Gamma_{nk}^\\ell\\right)e_{\\ell}.\n\\end{aligned}\\]\nIf we lower the last index, we have\n\\(\nR_{ijk\\ell}=g((\\nabla_{e_i}\\nabla_{e_j}-\\nabla_{e_j}\\nabla_{e_i}-\\nabla_{[e_i,e_j]}) e_k, e_\\ell).\n\\) \nTogether with $(\\nabla_{e_i}\\nabla_{e_j}-\\nabla_{e_j}\\nabla_{e_i}-\\nabla_{[e_i,e_j]}) g(e_k,e_\\ell)=0$, this makes apparent the anti-symmetries\n\\(\nR_{ijk\\ell}=-R_{jik\\ell}=R_{ji\\ell k}=-R_{ij\\ell k}.\n\\)\n\nInserting the indices gives us the following potentially non-vanishing terms of the Riemann tensor, up to permutation and anti-symmetry, which are relevant for the Ricci curvature:\n\\[\\begin{aligned}\nR_{121}^{\\ \\ \\ 2} &=  \\Gamma_{21}^3\\Gamma_{13}^2 - \\Gamma_{11}^0\\Gamma_{20}^2  - \\gamma_{12}^3\\Gamma_{31}^2\\\\\nR_{010}^{\\ \\ \\ 1} &=   \\Gamma_{10}^1\\Gamma_{01}^1 - \\Gamma_{00}^0\\Gamma_{10}^1 + \\nabla_{e_0}\\Gamma_{10}^1.\n\\end{aligned}\\]\nRaising and using $\\widetilde R$ for the intrinsic curvature of the surfaces of homogeneity gives us\n\\[\\begin{aligned}\nR_{12}^{\\ \\ 1 2} &=  g^{11}\\Gamma_{21}^3\\Gamma_{13}^2- \\gamma_{12}^3\\Gamma_{31}^2g^{11} - g^{11}\\Gamma_{11}^0\\Gamma_{20}^2  \\\\\n&= \\widetilde{R}_{12}^{\\ \\ 1 2} -K_{1}^1 K_2^2\\\\\nR_{01}^{\\ \\ 0 1} &=   g^{00}\\Gamma_{10}^1\\Gamma_{01}^1 - g^{00}\\Gamma_{00}^0\\Gamma_{10}^1 + g^{00}\\nabla_{e_0}\\Gamma_{10}^1\\\\\n&= -K_1^1 K_1^1 + \\sqrt{-g^{00}}\\Gamma_{00}^0 K_1^1 +g^{00} \\nabla_{e_0} \\sqrt{-g_{00}}K_1^1\\\\\n&=-K_1^1 K_1^1 -\\sqrt{-g^{00}}\\nabla_{e_0}K_1^1.\n\\end{aligned}\\]\nSetting\n\\[\\begin{aligned}\nn_i&=\\hat n_i \\sqrt{g^{11}g^{22}g^{33}}g_{ii} = \\hat n_i \\widetilde n_i,\\quad\\text{i.e.}\\quad\n\\widetilde n_i = \\sqrt{g^{11}g^{22}g^{33}}g_{ii},\n\\end{aligned}\\]\nthe spatial curvature is given by\n\\[\\begin{aligned}\n\\widetilde R_{12}^{\\ \\ 12} &=g^{11}\\Gamma_{21}^3\\Gamma_{13}^2- g^{11}\\gamma_{12}^3\\Gamma_{31}^2\n= g^{11}\\left(\\Gamma_{21}^3\\Gamma_{13}^2 - \\Gamma_{12}^3\\Gamma_{31}^2 + \\Gamma_{21}^3\\Gamma_{31}^2\\right)\\\\\n&=\\frac 1 4\\left(-n_1^2 -n_2^2 +3n_3^2 +2n_1n_2 -2n_2n_3 -2n_3n_1\\right)\\\\\n\\widetilde R_{1}^{\\ 1}&= \\widetilde R_{12}^{\\ \\ 12}+ \\widetilde R_{13}^{\\ \\ 13}\n=\\frac 1 2\\left(-n_1^2 + n_2^2+n_3^2 -2 n_2n_3 \\right)\\\\\n\\widetilde R &= \\widetilde R_{1}^{\\ 1} + \\widetilde R_{2}^{\\ 2} + \\widetilde R_{3}^{\\ 3}\n= \\frac 1 2(n_1^2+n_2^2+n_3^2 -2(n_1n_2+n_2n_3+n_3n_1)).\n\\end{aligned}\\]\n\\subsubsection{The Einstein Field equation}\\label{sect:gr-efe}\nThe Einstein equations in vacuum state that the space-time is Ricci-flat, i.e.~\n\\[\\begin{aligned}\nR_{0}^{\\ 0} &= R_{01}^{\\ \\ 01}+R_{02}^{\\ \\ 02}+R_{02}^{\\ \\ 02} =0\\\\\nR_{1}^{\\ 1} &= R_{01}^{\\ \\ 01}+R_{12}^{\\ \\ 12}+R_{13}^{\\ \\ 13} =0\\\\\nR_{2}^{\\ 2} &= R_{02}^{\\ \\ 02}+R_{21}^{\\ \\ 21}+R_{23}^{\\ \\ 23} =0\\\\\nR_{3}^{\\ 3} &= R_{03}^{\\ \\ 03}+R_{31}^{\\ \\ 31}+R_{32}^{\\ \\ 32} =0.\n\\end{aligned}\\]\nAdding the last three equations and subtracting the first gives us an equation which does not contain time-derivatives of $K$ and hence is a constraint equation, called the ``Gauss constraint''. It is given by\n\\[\\begin{aligned}\n0 &= R_{12}^{\\ \\ 12} +R_{23}^{\\ \\ 23} + R_{31}^{\\ \\ 31} = \\frac 1 2 \\widetilde R - K_{1}^1K_2^2 - K_2^2K_3^3 - K_3^3K_1^1.\n\\end{aligned}\\]\nThe evolution equations are given by\n\\[\\begin{aligned}\n\\sqrt{-g^{00}}\\nabla_{e_0}\\widetilde n_i &= \\sqrt{-g^{00}} \\nabla_{e_0} \\sqrt{g_{ii}g^{jj}g^{kk}}\n= \\sqrt{-g^{00}} (\\Gamma_{i0}^i -\\Gamma_{j0}^j -\\Gamma_{k0}^k)\\widetilde n_i= (K_i^i -K_j^j - K_k^k)\\widetilde n_i,\n\\end{aligned}\\]\nand\n\\[\\begin{aligned}\nR_{0i}^{\\ \\ 0i} &= -R_{ij}^{\\ \\ ij}-R_{ik}^{\\ \\ ik}=R_{jk}^{\\ \\ jk}\\\\\n\\sqrt{-g^{00}}\\nabla_{e_0} K_i^i &= -K_i^iK_i^i + K_{j}^j K_k^k  - \\widetilde R_{jk}^{\\ \\ jk}.\n\\end{aligned}\\]\n\n\n\\paragraph{Trace-free formulation.}\nIt is useful to split the variables into their trace and trace-free parts:\n\\[\\begin{aligned}\nH &= \\frac 1 3(K_1^1+K_2^2+K_3^3),\\qquad\\text{i.e.~$H$ is the mean curvature of $\\{t=\\mathrm{const}\\}$}\\\\ \n\\sigma_i &= K_i^i -H\\\\\n\\frac 1 6 \\widetilde R &= \\frac 1 3\\left(\\widetilde R_{12}^{\\ \\ 12}+\\widetilde R_{23}^{\\ \\ 23}+\\widetilde R_{31}^{\\ \\ 31}\\right)\n= \\frac 1 {12}\\left(n_1^2+n_2^2+n_3^2 -2(n_1n_2+n_2n_3+n_3n_1)\\right)\\\\\ns_i &= \\widetilde R_{jk}^{\\ \\ jk}-\\frac 1 6 \\widetilde R\n= \\frac 1 {3}\\left(2n_i^2 - n_j^2 -  n_k^2  - n_in_j +2 n_jn_k - n_kn_i \\right),\n\\end{aligned}\\]\nwhere we note that $(\\sigma_1+\\sigma_2+\\sigma_3)^2 = 0=\\sigma_1^2+\\sigma_2^2+\\sigma_3^2 +2(\\sigma_1\\sigma_2+\\sigma_2\\sigma_3+\\sigma_3\\sigma_1)$.\nWe then obtain\n\\[\\begin{aligned}\n0 & = \\frac 1 2 \\widetilde R - (\\sigma_1+H)(\\sigma_2+H) - (\\sigma_2+H)(\\sigma_3+H) - (\\sigma_1+H)(\\sigma_3+H)\\\\\n&= \\frac 1 2 \\widetilde R - 3H^2 - (\\sigma_1\\sigma_2+\\sigma_2\\sigma_3+\\sigma_3\\sigma_1)\\\\\n&= \\frac 1 2 \\widetilde R - 3H^2 +\\frac 1 2 (\\sigma_1^2+\\sigma_2^2+\\sigma_3^2)\\\\\n\\sqrt{-g^{00}}\\nabla_{e_0}\\widetilde n_i &= (2\\sigma_i-H)\\widetilde n_i\\\\\n\\sqrt{-g^{00}}\\nabla_{e_0} H &= \\frac 1 3 \\Big[\\;-(\\sigma_1+H)^2 -(\\sigma_2+H)^2 -(\\sigma_3+H)^2\\\\\n&\\qquad+(\\sigma_1+H)(\\sigma_2+H) +(\\sigma_2+H)(\\sigma_3+H) + (\\sigma_3+H)(\\sigma_1+H) \\\\\n&\\qquad- \\widetilde R_{23}^{\\ \\ 23} -  \\widetilde R_{13}^{\\ \\ 13} -  \\widetilde R_{12}^{\\ \\ 12}\\Big]\\\\\n&=-\\frac 1 2(\\sigma_1^2+\\sigma_2^2+\\sigma_3^2) -\\frac 1 6 \\widetilde R\n= -\\frac 1 3(\\sigma_1^2+\\sigma_2^2+\\sigma_3^2) - H^2 \\\\\n\\sqrt{-g^{00}}\\nabla_{e_0} \\sigma_1 &= -(\\sigma_1+H)^2 + (\\sigma_2+H)(\\sigma_3+H) - \\widetilde R_{23}^{\\ \\ 23}  - \\sqrt{-g^{00}}\\nabla_{e_0}H\\\\\n&=\\sigma_2\\sigma_3-\\sigma_1^2 +(\\sigma_2+\\sigma_3-2\\sigma_1)H - \\widetilde R_{23}^{\\ \\ 23}  - \\sqrt{-g^{00}}\\nabla_{e_0}H\\\\\n&=(\\sigma_1+\\sigma_3)(\\sigma_1+\\sigma_2)-\\sigma_1^2 -3\\sigma_1H   +\\frac 1 2 (\\sigma_1^2+\\sigma_2^2+\\sigma_3^2) +  \\frac 1 6 \\widetilde R - \\widetilde R_{23}^{\\ \\ 23}\\\\\n&= -3\\sigma_1H -s_i.\n\\end{aligned}\\]\n\\paragraph{Hubble Normalization.}\nWe can further simplify by Hubble-normalizing to $\\overline{\\Sigma}_i = \\frac{\\sigma_i} H$ and $\\overline{N}_i = \\frac{n_i} H$, and introducing a shorthand for ${\\Sigma}^2$ and ${N}^2$\n\\[\\begin{aligned}\n\\overline{N}_i &= \\widetilde{\\overline{N}}_i {{\\hat{n}}}_i\\\\\n{\\Sigma}^2 &= \\frac 1 6\\left({\\Sigma}_1^2+\\Sigma_2^2+\\Sigma_3^2\\right)\\\\\n N^2&= \\frac 1 {12} \\left[\\overline N_1^2+\\overline N_2^2+\\overline N_3^2-2(\\overline N_1\\overline N_2+\\overline N_2\\overline N_3+\\overline N_3\\overline N_1)\\right]= \\frac 1 6 \\widetilde R H^{-2}\\\\\n S_i &= -\\frac 1 3 \\left[-2 \\overline N_1^2 + \\overline N_2^2 + \\overline N_3^2 +\\overline N_1\\overline N_2 -2\\overline N_2\\overline N_3 +\\overline N_3\\overline N_1 \\right]= s_i H^{-2}\\end{aligned}\\]\nwhich yields equations\n\\[\\begin{aligned}\n1 &=  \\Sigma^2+ N^2\\\\\n\\sqrt{-g^{00}}\\nabla_{e_0} H &=-\\left(2\\Sigma^2 +1 \\right)H^2\\\\\n\\sqrt{-g^{00}}\\nabla_{e_0}\\widetilde{\\overline N}_i &= (2 \\Sigma_i - 1 + 2 \\Sigma^2 +1)\\widetilde{\\overline N}_iH=2(\\Sigma^2+ \\Sigma_i)\\widetilde{\\overline N}_i H\\\\\n\\sqrt{-g^{00}}\\nabla_{e_0} \\overline \\Sigma_1  &=\n-3\\overline \\Sigma_1 H - s_i H^{-1} + (2\\overline \\Sigma^2+1)H = 2( \\Sigma^2-1)\\overline\\Sigma_1H - S_iH.\n\\end{aligned}\\]\n\\paragraph{The Wainwright-Hsu equations as used in this work.}\nAssuming $H<0$ for an initial surface (which can be obtained if $H\\neq 0$ by choosing the direction of the unit normal $\\sqrt{-g^{00}}e_0$ and reverting the direction of time), we can set $\\sqrt{-g^{00}} = -\\frac 1 2 H$. Setting $\\widetilde N_i = \\sqrt{12}\\,\\widetilde{\\overline N}_i$, we obtain the variant of the Wainwright-Hsu equations used in this work, \\eqref{eq:ode-from-gr}, corresponding to the metric \\eqref{eq:metric-in-wsh}.\n\n\n\\section{Introduction}\\label{sect:intro}\n\\paragraph{Spatially homogeneous cosmological models.}\nThe behaviour of cosmological models is governed by the Einstein field equations, coupled with equations describing the presence of matter. Simpler models are obtained under symmetry assumption. The class of models studied in this work, Bianchi models, assume spatial homogeneity, i.e.~``every point looks the same''. Then, one only needs to describe the behaviour over time of any single point, and the partial differential Einstein field equations become a system of ordinary differential equations. \n\nDirectional isotropy assumes that ``every spatial direction looks the same''. This leads to the well-known FLRW (Friedmann-Lemaitre-Robertson-Walker) models. These models describe \nan initial (``big bang'') singularity, followed by an expansion of the universe, slowed down by ordinary and dark matter and accelerated by a competing positive cosmological constant (``dark energy'').\n\n \nWe will assume spatial homogeneity, but relax the assumption of directional isotropy.\nSpatial homogeneity assumes that there is a Lie-group $G$ of spacetime isometries, which foliates the spacetime into three-dimensional space-like hypersurfaces on which $G$ acts transitively: For every two points $\\bpt x, \\bpt y$ in the same hypersurface there is a group element $f\\in G$ such that $f\\cdot \\bpt x=\\bpt y$. The resulting ordinary differential equations depend on the Lie-algebra of $G$, the so-called Killing fields.\nThe three-dimensional Lie-algebras have been classified by Luigi Bianchi in 1898, hence the name``Bianchi-models''; for a commented translation see \\cite{Bianchi2001, jantzen2001editor}, and for a modern treatment see \\cite[Section 1]{wainwright2005dynamical}.\n\nThe two most studied classes of spatially homogeneous anisotropic cosmological models are the Bianchi-types $\\textsc{IX}$ ($so(3)$) and $\\textsc{VIII}$ ($sl(2,{\\mathbb{R}})$), which are the focus of this work.\nBoth of these models exhibit a big-bang like singularity in at least one time-direction, and a universe that initially expands from this singularity, until, in the case of Bianchi \\textsc{IX}, it recollapses into a time-reversed big bang (``big crunch'')\nThe big-bang singularity is present even in the vacuum case, where matter is absent and only gravity self-interacts. According to conventional wisdom, ``matter does not matter'' near the singularity. For this reason, we simplify our analysis by considering only the vacuum case.\n\nNote that the symmetry assumptions already restrict the global topology of the space-like hypersurfaces, and that the isotropic FLRW-models are not contained as a special case: The only homogeneous isotropic vacuum model is flat Minkowski space.\n\nFor a detailed introduction to Bianchi models, we refer to \\cite{wainwright2005dynamical}. A short derivation of the governing ordinary differential Wainwright-Hsu equations \\eqref{eq:ode} is given in Section \\ref{sect:append:derive-eq}, and physical interpretations of some of our results are given in Section \\ref{sect:gr-phys-interpret}. For an excellent survey on Bianchi cosmologies, we refer to \\cite{heinzle2009mixmaster}, and for further physical questions we refer to \\cite{uggla2003past, heinzle2009cosmological}.\n\\paragraph{The Taub-spaces.}\nThe dynamical behaviour of Bianchi \\textsc{VIII} and \\textsc{IX} spacetimes is governed by the so-called Wainwright-Hsu equations. \nThis dynamical system contains an invariant set $\\mathcal T$ of codimension two, which we call the Taub-spaces. Solutions in this set are also called Taub-NUT spacetimes, or LRS (locally rotationally symmetric) spacetimes. The latter name is descriptive, in the sense that these spacetimes have additional (partial, local) isotropy. Taub spacetimes behave, in several ways, different from general (i.e.~non-Taub) Bianchi spacetimes. For a detailed description, we refer to \\cite{ringstrom2001bianchi}.\n\n\\paragraph{The Mixmaster attractor.}\nThe Bianchi dynamical system also contains an invariant set $\\mathcal A$, called the Mixmaster attractor, consisting of Bianchi Type \\textsc{II} and \\textsc{I} solutions. There are good heuristic arguments that $\\mathcal A$ really is an attractor for time approaching the singularity, and that the dynamics on and near $\\mathcal A$ can be considered chaotic (sometimes also called ``oscillatory''). \nIt has been rigorously proven only in Bianchi \\textsc{IX} models that $\\mathcal A$ is actually attracting, with the exception of some Taub solutions, c.f.~\\cite{ringstrom2001bianchi}. \n\n\\paragraph{Stable Foliations.}\nOne may ask for a more precise description of how solutions get attracted to $\\mathcal A$.\nCertain heteroclinic chains in $\\mathcal A$ are known to attract hypersurfaces of codimension one, c.f.~\\cite{liebscher2011ancient, beguin2010aperiodic}.\nReiterer and Trubowitz \\cite{reiterer2010bkl} claim related results, for a much wider class of heteroclinic chains, but with less focus on the regularity or codimension of the attracted sets.\nThis general class of constructions, i.e.~partial stable foliations over specific solutions in $\\mathcal A$, is not the focus of this work. Instead, we describe and estimate solutions and the solution operator (i.e.~flow) directly, without explicitly focussing on the symbolic description of $\\mathcal A$.\n\n\\paragraph{The question of particle horizons.}\nOne of the most salient features of relativity is \\emph{causality}: The state of the world at some point in spacetime is only affected by states in its past light-cone and can only affect states in its future light-cone. \n\nSuppose for this paragraph that we orient our spacetime $M$ such that the big bang singularity is situated in the past.\nTwo points $\\bpt p, \\bpt q\\in M$ are said to \\emph{causally decouple} towards the singularity if their past light-cones are disjoint, i.e.~if there is no past event which causally influences both $\\bpt p$ and $\\bpt q$. The past communication cone of $\\bpt p$ is defined to be the set of points, which do not causally decouple from $\\bpt p$. The cosmic horizon, also called particle horizon, is the boundary of the past communication cone. Hence, everything beyond the horizon is causally decoupled.\n\n\\[\\numberthis\\label{eq:intro:comcone}\\begin{aligned}\n\\omit\\rlap{Past light cone of $\\bpt p$:}&\\\\\nJ^-(\\bpt p) &= \\{\\bpt q:\\, \\text{there is } \\gamma:[0,1]\\to M\\,\\,\\text{with}\\, \\gamma(0)=\\bpt p, \\gamma(1)=\\bpt q,\\,\\text{time-like past directed}\\}\\\\\n\\omit\\rlap{Future light cone of $\\bpt p$:}&\\\\\nJ^+(\\bpt p) &= \\{\\bpt q:\\, \\text{there is } \\gamma:[0,1]\\to M\\,\\,\\text{with}\\, \\gamma(0)=\\bpt p, \\gamma(1)=\\bpt q,\\,\\text{time-like future directed}\\}\\\\\n\\omit\\rlap{Past communication cone of $\\bpt p$:}&\\\\\nJ^+(J^-(\\bpt p)) &= \\bigcup_{\\bpt q \\in J^-(\\bpt p)}J^+(\\bpt q) = \\{\\bpt q:\\, {J^-(\\bpt q)}\\cap {J^{-}(\\bpt p)}\\neq\\emptyset\\}\\\\\n\\omit\\rlap{Past cosmic horizon of $\\bpt p$:}&\\\\\n\\partial J^+(J^-(\\bpt p))& = \\mathrm{closure}\\,J^{+}(J^-(\\bpt p))\\,\\setminus\\, \\mathrm{interior}\\, J^{+}(J^-(\\bpt p)).\n\\end{aligned}\\]\nIn Figure \\ref{fig:intro:comm-cone}, we illustrate the formation of particle horizons. An example where no particle horizons form is given by the (flat, connected) Minkowski-space $M={\\mathbb{R}}\\times {\\mathbb{R}}^3$: There is no singularity, and $M= J^+(J^-(\\bpt p))$ for all $\\bpt p$, and hence $\\partial J^+(J^-(\\bpt p)) = \\emptyset$.  \n\nApart from the question of convergence to $\\mathcal A$, the next important physical question in the context of Bianchi cosmologies is that of locality of light-cones:\n\\begin{enumerate}\n\\item Do nonempty particle horizons $\\partial J^+(J^-(\\bpt p))\\neq \\emptyset$ form towards the singularity? Does this happen if $\\bpt p$ is sufficiently near to the singularity?\n\\item Are the spatial hypersurfaces $\\{t= t_0=\\mathrm{const}\\}\\cap (J^+(J^-(\\bpt p)), \\partial J^+(J^-(\\bpt p))$, considered as three-dimensional manifolds with boundary, homeomorphic to the three dimensional unit ball $(B_1(0), \\partial B_1(0))$? \n\\item Are the past communication cones of $\\bpt p$ spatially bounded? Do they shrink down to a point, as $\\bpt p$ and $t_0$ go towards the singularity?\n\\end{enumerate}\nThe first question is formulated completely independently of the foliation of the spacetime into space-like hypersurfaces. The second question depends on the foliation, but is at least easy to clearly state. The third question is not clearly stated here, because it requires us to choose a way of comparing spatial extents of the communication cones at different times. The subtleties of this are discussed in Section \\ref{sect:gr-phys-interpret}. \n\nSince this work is concerned only with proving affirmative answers to these questions, we will conflate them: We say that a solution forms a particle horizon if all these questions are answered with ``yes''. \n\n\n\\begin{figure}[hbt]\n\\centering\n\\begin{subfigure}[b]{\\textwidth}\n\\includegraphics[width=\\textwidth]{.\/graphics\/decoupled-lc.pdf}\n\\caption{Two points $\\bpt p$, $\\bpt q$ that decouple towards the singularity. Their past light-cones are disjoint.}\n\\end{subfigure}\n\\begin{subfigure}[b]{\\textwidth}\n\\includegraphics[width=\\textwidth]{.\/graphics\/decoupled-cc.pdf}\n\\caption{Two points $\\bpt p$, $\\bpt q$ that decouple. The point $\\bpt q$ lies outside the communication cone of $\\bpt p$.}\n\\end{subfigure}\n\\begin{subfigure}[b]{\\textwidth}\n\\includegraphics[width=\\textwidth]{.\/graphics\/coupled-cc-lc.pdf}\n\\caption{Two points $\\bpt p$, $\\bpt q$ that do not decouple towards the singularity. Their past light-cones have nonempty intersection (the darkest shaded region). The point $\\bpt q$ lies inside the communication cone of $\\bpt p$.}\n\\end{subfigure}\n\\caption{Examples of decoupling and non-decoupling towards the singularity at $t=t_{\\mathrm{sing}}$, and of particle horizons.}\n\\label{fig:intro:comm-cone}\n\\end{figure}\n\n\nOriginally, Misner \\cite{misner1969mixmaster} suggested that no particle horizons should form in Bianchi \\textsc{IX}. This was proposed as a possible explanation of the observed approximate homogeneity of the universe: If the homogeneity is due to past mixing, then different observed points in our current past light-cone must themselves have a shared causal past. Misner later changed his mind to the current consensus intuition that typical Bianchi \\textsc{VIII} and \\textsc{IX} solutions should form particle horizons. Some more details on this are given in Section \\ref{sect:gr-phys-interpret}. Further discussion of these questions can be found in e.g.~\\cite[Chapter 5]{wald1984general}, \\cite[Chapter 5]{hawking1973large}.\n\n\n\n\n\\paragraph{The BKL picture.}\nSpatially homogeneous spacetimes, and especially the question of particle horizons, play an essential role in the so-called BKL picture (also often called BKL-conjecture). The BKL picture is due to Belinskii, Khalatnikov and Lifshitz (\\cite{belinskii1970oscillatory}), and describes generic cosmological singularities in terms of homogeneous spacetimes. This picture roughly claims the following: \n\\begin{enumerate}\n\\item Generic cosmological singularities ``are curvature-dominated'', i.e.~behave like the vacuum case. More succinctly, ``matter does not matter''.\n\\item Generic cosmological singularities are ``oscillatory'', which means that the directions that get stretched or compressed switch over time.\n\\item Generic cosmological singularities ``locally behave like'' spatially homogeneous ones, especially Bianchi \\textsc{IX} and \\textsc{VIII}. By this, one means that:\n\\begin{enumerate}\n\\item Different regions causally decouple towards the singularity, i.e.~particle horizons form.\n\\item If one restricts attention to a single communication cone, then, as time goes towards the singularity, the spacetime can be well approximated by a homogeneous one.\n\\item Different spatial regions may have different geometry towards the singularity (since they decouple). This kind of behaviour has been described as ``foam-like''.\n\\end{enumerate}\n\\end{enumerate}\nBoundedness of the communication cones, i.e.~formation of particle horizons, in spatially homogeneous models is a necessary condition for the consistency of the BKL-picture: $(3.a)$ claims that different spatial regions causally decouple towards the big bang, and $(3.b)$ claims that such decoupled regions behave ``like they were homogeneous''; hence, homogeneous solutions should better allow for different regions to spatially decouple.\n\n\\paragraph{Previous results.}\nOne way of viewing the formation of particle horizons is as a race between the expansion of the universe (shrinking towards the singularity) and the eventual blow-up at the singularity. If the blow-up is faster than the expansion, particle horizons form; otherwise, they don't. In the context of the Wainwright-Hsu equations, the question can be boiled down to: \\emph{Do solutions converge to $\\mathcal A$ sufficiently fast? If yes, then particle horizons form. If not, then the questions of particle horizons may have subtle answers.}\n\nThe aforementioned solutions constructed in \\cite{liebscher2011ancient, beguin2010aperiodic}, with initial conditions on certain hypersurfaces of codimension one, all converge essentially uniformly exponentially to $\\mathcal A$, which is definitely fast enough for particle horizons to form. \n\nReiterer and Trubowitz claim in \\cite{reiterer2010bkl} that the solutions constructed therein also converge to $\\mathcal A$ fast enough for this to happen. The claimed results in \\cite{reiterer2010bkl}\nare somewhat nontrivial to parse; let us give a short overview: They construct solutions converging to certain parts of the Mixmaster attractor $\\mathcal A$. These parts of the Mixmaster attractor have full (one-dimensional) volume, and all these constructed solutions form particle horizons. Claims about full-dimensional measure (or Hausdorff-dimensions, etc) are not made in \\cite{reiterer2010bkl}. \n\nThe solutions constructed in \\cite{liebscher2011ancient, beguin2010aperiodic} were the first known nontrivial solutions that could be proven to form particle horizons. \nIt is still unknown whether there exist nontrivial counterexamples, i.e. non-Taub solutions that fail to form particle horizons.\n\n\n\n\\paragraph{Main results.}\nThe first main result of this work extends Ringstr\\\"om's Bianchi \\textsc{IX} attractor theorem (Theorem \\ref{farfromA:thm:b9-attract}, c.f.~also \\cite{ringstrom2001bianchi, heinzle2009new}) to the case of Bianchi \\textsc{VIII} vacuum. It can be summarized in the following:\n\\begin{hthm}{\\ref{thm:local-attractor}, \\ref{thm:b9-attractor-global}, \\ref{thm:b8-attractor-global} and \\ref{thm:b8-attractor-global-genericity}}[Paraphrased Attractor Theorem]\nWith certain exceptions, solutions in Bianchi \\textsc{IX} and \\textsc{VIII} vacuum converge to the Mixmaster attractor $\\mathcal A$.\n\nLower bounds on the speed of convergence are given, but are insufficient to ensure the formation of particle horizons.\n\nThe dynamics of the exceptional solutions is described. The set of initial conditions corresponding to the exceptional solutions is nongeneric, both in the sense of Lebesgue (it is a set of zero Lebesgue measure) and Baire (it is a meagre set).\n\\end{hthm}\nApart from the applicability to Bianchi \\textsc{VIII}, this also extends the Ringstr\\\"om's previous result by providing lower bounds on the speed of convergence, and provides a new proof.\n\n\nThe most important result of this work is the following:\n\\begin{hthm}{\\ref{thm:horizon-formation}}[Almost sure formation of particle horizons]\nAlmost every solution in Bianchi \\textsc{VIII} and \\textsc{IX} vacuum forms particle horizons towards the big bang singularity. A more rigorous formulation of the theorem is on page \\pageref{thm:horizon-formation}.\n\\end{hthm}\nThe question remains open of whether particle horizons form for initial conditions that are generic in the sense of Baire\\footnote{A set is called generic in the sense of Baire, if it is \\emph{co-meagre}, i.e.~it contains a countable intersection of open and dense sets. Then its complement is called \\emph{meagre}. By construction, countable intersections of co-meagre sets are co-meagre and countable unions of meagre sets are meagre. Baire's Theorem states that co-meagre subsets of complete metric spaces are always dense and especially nonempty.}. \nWe strongly suspect that the answer is no, i.e.~particle horizons fail to form for a co-meagre set of initial conditions, for reasons which will be explained in future work.\n\n\n\\paragraph{Structure of this work.}\nWe will give the Wainwright-Hsu equations in Section \\ref{sect:eqs}, as well as some notation and transformations that will be needed later on. The most referenced equations are also summarized in Appendix \\ref{sect:eq-cheat-sheet}, page \\pageref{sect:eq-cheat-sheet}, for easier reference. A derivation of the Wainwright-Hsu equations from the Einstein field equations of general relativity is given Section \\ref{sect:append:derive-eq}.\n\n\nAn overview of the dynamical behaviour and some first proofs will be given in Section \\ref{sect:farfromA}. Sections \\ref{sect:near-A} and \\ref{sect:near-taub} will describe in detail two different regimes in the neighborhood of $\\mathcal A$; these two descriptions are synthesized into general attractor theorems in Section \\ref{sect:global-attract}. \nThe measure-theoretic results are all contained in Section \\ref{sect:volume-form}. Section \\ref{sect:gr-phys-interpret} relates dynamical properties of solutions to the Wainwright-Hsu equations to physical properties of the corresponding spacetime.\n\n\n\\paragraph{Strategy.}\nOur analysis of the behaviour of solutions of the Wainwright-Hsu equation is structured around two invariant objects: The Mixmaster-attractor $\\mathcal A$ and the Taub-spaces $\\mathcal T$. We will measure the distances from these sets by functions $\\delta(\\bpt x) \\sim d(\\bpt x, \\mathcal A)$ and $r(\\bpt x)\\sim d(\\bpt x, \\mathcal T)$.\n\nThere exist standard heuristics based on normal hyperbolicity, cf.~\\cite{heinzle2009mixmaster}. According to these, solutions near $\\mathcal A$: \n  \\begin{enumerate}\n  \\item can be described by the so-called Kasner-map (see Section \\ref{sect:farfromA}),\n  \\item converge exponentially to $\\mathcal A$, and \n  \\item their associated spacetimes form particle-horizons. \n  \\end{enumerate}\nThese heuristics break down near the Taub-spaces $\\mathcal T \\cap \\mathcal A$, where two eigenvalues pass through zero.\n\nIn Section \\ref{sect:far-from-taub} we will formally prove the validity of the heuristic description of solutions near $\\mathcal A$, that stay bounded away from $\\mathcal T$.\nMore precisely, we will show in Proposition \\ref{prop:farfromtaub-main} that for any $\\epsilon_T>0$, there exists $\\epsilon_d>0$ such that all the previously mentioned hyperbolicity heuristics apply for solutions $\\bpt x: [0,T] \\to \\{\\bpt y:\\, \\delta(\\bpt y) <\\epsilon_d,\\,r(\\bpt y) > \\epsilon_T\\}$.\n\n\n\nTo provide a complete picture of the dynamics, we still need to control solutions in a neighborhood of $\\mathcal T$. This is the goal of Section \\ref{sect:near-taub}.\nIt is well known, that $\\mathcal T$ is transient, i.e.~solutions may approach $\\mathcal T$ but cannot converge to $\\mathcal T$; they must leave a neighborhood of $\\mathcal T$ again. \nEvery component of $\\mathcal T\\cap \\mathcal A$ consists of two equilibria connected by a heteroclinic orbit $-\\taubp \\to +\\taubp$; the standard heuristics described in Section \\ref{sect:far-from-taub} break down at $+\\taubp$, but continue to work near $-\\taubp$ and near the heteroclinic orbit $-\\taubp \\to +\\taubp$.\nThus, solutions can leave the region controlled by normal hyperbolicity only by approaching first $-\\taubp$ and then $+\\taubp$.\n\n\n\\label{paragraph:intro:strategy:quotients} The most important quantity in the local analysis near $+\\taubp$ is the quotient $\\frac{\\delta}{r}$ of the distances to $\\mathcal A$ and $\\mathcal T$. As long as this quotient is small, we can prove exponential decay of $\\delta(t)$ (Proposition \\ref{prop:neartaub:main}) and slow growth of $r(t)$; hence, the quotient $\\frac{\\delta}{r}$ continues to decrease.\nThe structure of this kind of estimate is not surprising; in fact, \nit is trivial (by varying $\\epsilon_d$ and $\\epsilon_T$) to construct continuous nonnegative functions $\\hat\\delta$ and $\\hat r$ with $\\mathcal A=\\{\\bpt x: \\hat\\delta(\\bpt x)=0\\}$ and $\\mathcal T=\\{\\bpt x: \\hat r(\\bpt x)=0\\}$, such that estimates of the above form are true near $+\\taubp$.\nHowever, our particular choice of functions $\\delta$ and $r$ (explicitly given in Section \\ref{sect:polar-coords}) allows the \\emph{same} quotient $\\frac{\\delta}{r}$ to be controlled near $-\\taubp$ and near $+\\taubp$.\nWe do not know of any reason to a priori expect this fortuitous fact; it is, however, easily verified by direct calculation.\n\nThe analysis from Section \\ref{sect:near-taub} fits together with the analysis from Section \\ref{sect:far-from-taub}: Solutions near $\\mathcal A$ (i.e.~$\\delta\\ll 1$) that leave regions with $r>\\epsilon_T$ to enter the neighborhood of $\\mathcal T$ with $r\\le\\epsilon_T$ must have, at the moment where $r=\\epsilon_T$, a very small quotient $\\delta\/r < \\epsilon_T^{-1}\\epsilon_d \\ll 1$.\nThis gives rise to a local attractor Theorem \\ref{thm:local-attractor}: If $\\widetilde \\delta=\\max(\\delta, \\frac{\\delta}{r})$ is small enough for some initial condition, then $\\widetilde \\delta$ converges to zero. Hence, we provide a family of forward invariant neighborhoods of $\\mathcal A\\setminus \\mathcal T$ attracted to $\\mathcal A$. \nUsing the local attractor Theorem \\ref{thm:local-attractor}, it is rather straightforward to adapt the ``global'' (i.e.~away from $\\mathcal A$) arguments from \\cite{ringstrom2001bianchi} in order to produce the global attractor Theorems \\ref{thm:b9-attractor-global} and \\ref{thm:b8-attractor-global}.\n\n\nNote that as in previous works \\cite{ringstrom2001bianchi}, a ``global attractor theorem'' does not imply that \\emph{all} solutions converge to $\\mathcal A$, but rather it describes the exceptions, i.e.~solutions where the local attractor Theorem \\ref{thm:local-attractor} does not eventually hold. \nIn Bianchi \\textsc{IX} models, i.e.~in Theorem \\ref{thm:b9-attractor-global}, these exceptions are exactly solutions contained in the lower-dimensional Taub-spaces $\\mathcal T$ (originally shown in \\cite{ringstrom2001bianchi}; we provide an alternative proof).\nIn Bianchi \\textsc{VIII} models, i.e.~in Theorem \\ref{thm:b8-attractor-global}, these exceptions are either contained in the lower-dimensional Taub-spaces $\\mathcal T$ or must follow the very particular asymptotics described in Theorem \\ref{thm:b8-attractor-global}, case $\\textsc{Except}$.\nAs we show in Theorem \\ref{thm:b8-attractor-global-genericity}, this exceptional case $\\textsc{Except}$ applies at most for a set of initial conditions, which is meagre (small Baire category) and has Lebesgue measure zero. It is currently unknown \nwhether this exceptional case \\textsc{Except} is possible at all. \n\n\nThese genericity results (Theorem \\ref{thm:b8-attractor-global-genericity}) rely on measure theoretic considerations in Section \\ref{sect:volume-form}: We provide a volume-form $\\omega_4$, which is expanded under the flow (given in \\eqref{eq:omega5-def}, \\eqref{eq:omega4-def}). This is in itself not surprising: \n Hamiltonian systems preserve their canonical volume form. Since the Wainwright-Hsu-equations derive from the (Hamiltonian) Einstein Field equations by an essentially monotonous time-dependent rescaling, we expect to find a volume form $\\omega_4$, that is essentially monotonously expanding.\nSuch an expanding volume form has the useful property that all forward invariant sets must either have infinite or zero $\\omega_4$-volume. \n\nWe prove the genericity Theorem \\ref{thm:b8-attractor-global-genericity} by noting that the exceptional solutions form an invariant set; using their detailed description, we can show that its $\\omega_4$-volume is finite and hence zero.\n\nThe results the formation of particle horizons are also proved in Section \\ref{sect:volume-form}.\nIn our language, the primary question is whether $\\int_0^{\\infty} \\delta(t){\\mathrm{d}} t <\\infty$. If this integral is finite, then particle horizons form and the singularity is ``local''; this behaviour is both predicted by the BKL picture and required for its consistency. Heuristically, time spent away from $\\mathcal T$ helps convergence of the integral (since $\\delta$ decays uniformly exponentially in these regions), while time spent near $\\mathcal T$ gives large contributions to the integral. Looking at our analysis near $+\\taubp$ (Proposition \\ref{prop:neartaub:main}), we get a contribution to the integral of order $\\frac{\\delta}{r^2}$. \nThe local attractor Theorem \\ref{thm:local-attractor} is insufficient to decide the question of locality, since it can only control the quotient $\\frac{\\delta}{r}$ (and hence the integral $\\int \\delta^2{\\mathrm{d}} t<\\infty$). \n\nHowever, we can show by elementary calculation that the set $\\{\\bpt x:\\,\\delta>r^4 \\}$ has finite $\\omega_4$-measure. Using some uniformity estimates on the volume expansion, we can infer that the (naturally invariant) set $\\textsc{Bad}$ of initial conditions that fail to have $\\delta< r^4$ for all sufficiently large times, has finite and hence vanishing $\\omega_4$-volume. Therefore, for Lebesgue almost every initial condition, $\\delta < r^4$ holds eventually, allowing us to bound the contribution of the entire stay near $+\\taubp$ by $\\frac{\\delta}{r^2}<\\sqrt{\\delta}$, which decays exponentially in the ``number of episodes near Taub-points'' (also called Kasner-eras).\nThis gives rise to Theorem \\ref{thm:horizon-formation}: Lebesgue almost every initial condition in Bianchi \\textsc{VIII} and \\textsc{IX} forms particle horizons towards the big bang singularity.\n\nA finer look at the integral even allows us to bound certain $L^p_{\\textrm{loc}}(\\omega_4)$ integrals in Theorem \\ref{thm:horizon-formation-alpha-p}.\n\n\n\n\\section{Setting, Notation and the Wainwright-Hsu equations}\\label{sect:eqs}\nThe subject of this work, i.e.,~the behaviour of homogeneous anisotropic vacuum space-times with Bianchi Class A homogeneity under the Einstein field equations of general relativity, can be described by a system of ordinary differential equations, called the Wainwright-Hsu equations \\eqref{eq:ode-unpacked}. \n \nIn Section \\ref{sect:equations}, we will introduce the Wainwright-Hsu ordinary differential equations and various auxiliary quantities and definitions, and provide a rough summary of their dynamics.\nThen we transform the Wainwright-Hsu equations into polar coordinates in Section \\ref{sect:polar-coords}, which are essential for the analysis in Section \\ref{sect:near-taub}. \n\n\nThere are multiple equivalent formulations of the Wainwright-Hsu equations in use by different authors, which differ in sign and scaling conventions, most importantly the direction of time. This work uses reversed time, such that the big bang singularity is at $t=+\\infty$.\nThe relation of the Wainwright-Hsu equations to the Einstein equations of general relativity will be relegated to Section \\ref{sect:append:derive-eq}. The relation between properties of solutions to the Wainwright-Hsu equations and physical properties of the corresponding spacetimes is discussed in Section \\ref{sect:append:derive-eq}.\n\\paragraph{General Notations.}\nIn this work, we will often use the notation $\\bpt x = (x_1,\\ldots,x_n)$ in order to emphasize that a variable $\\bpt x$ refers to a point and not to a scalar quantity. If we consider a curve $\\bpt x(t)$ into a space where different coordinates have names, e.g.~$\\bpt x: {\\mathbb{R}}\\to {\\mathbb{R}}^5=\\{(\\Sigma_+,\\Sigma_-,N_1,N_2,N_3)\\}$, then we will in an abuse of notation write $N_1(t)=N_1(\\bpt x(t))$ in order to refer to the $N_1$-coordinate of $\\bpt x(t)$.\n\nWe will use $\\pm$ to refer to either $+1$ or $-1$, and different occurrences of $\\pm$ are always unrelated, such that e.g.~$(\\pm,\\pm,\\pm)\\in\\{(+,+,+),(-,+,+), (+,-,+), (-,-,+),\\ldots\\}$. We will use $*$ to refer to either $+1$,$-1$ or $0$, also such that different occurrences of $*$ are unrelated. \n\n\n\\subsection{Spatially Homogeneous Spacetimes}\\label{sect:equations}\nWe study the behaviour of homogeneous spacetimes, also called Bianchi-models. These are Lorentz four-manifolds, foliated by space-like hypersurfaces on which a group of isometries acts transitively, subject to the vacuum Einstein Field equations. That is, we assume that we have a frame of four linearly independent vectorfields $e_0=\\partial_t, e_1,e_2,e_3$, where $e_1,e_2,e_3$ are Killing fields, with dual co-frame ${\\mathrm{d}} t, \\omega_1,\\omega_2,\\omega_3$, such that the metric has the form\n\\[\ng = g_{00}(t){\\mathrm{d}} t\\otimes {\\mathrm{d}} t + g_{11}(t)\\omega_1\\otimes \\omega_1 + g_{22}(t)\\omega_2\\otimes \\omega_2 + g_{33}(t)\\omega_3\\otimes \\omega_3,\n\\]\nand the commutators (i.e.~the Lie-algebra of the spatial homogeneity) has the form\n\\[\n[e_i,e_j]=\\sum_k \\gamma_{ij}^k e_k \\qquad \\gamma_{ij}^k={{\\hat{n}}}_k \\epsilon_{ijk},\n\\]\nwhere $\\epsilon_{ijk}$ is the usual Levi-Civita symbol ($\\epsilon_{ijk}=+1$ if $(ijk)\\in \\{(123), (231), (312)\\}$, $\\epsilon_{ijk}=-1$ if $(ijk)\\in \\{(132), (321), (213)\\}$ and $\\epsilon_{ijk}=0$ otherwise). The signs ${{\\hat{n}}}_i\\in \\{+1,-1,0\\}$ determine the Bianchi Type of the cosmological model, according to Table \\ref{table:bianchi-types}.\n\nThe metric is described by the seven Hubble-normalized variables $H,\\widetilde N_i, \\Sigma_i$, with $i\\in\\{1,2,3\\}$, according to \n\\begin{equation}\\label{eq:metric-in-wsh}\ng_{00}= - \\frac 1 4 H^{-2}\\qquad g_{ii}=\\frac 1 {48}\\frac{H^{-2}}{\\widetilde N_j \\widetilde N_k},\n\\end{equation}\nwhere $(i,j,k)$ is always assumed to be a permutation of $\\{1,2,3\\}$, and subject to the linear and sign constraints\n\\[\\numberthis\\label{eq:gauge-constraints}\n\\Sigma_1+\\Sigma_2+\\Sigma_3=0,\\qquad H < 0, \\qquad \\widetilde N_i >0 \\quad\\text{for all }i\\in\\{1,2,3\\}.\\]\nThe variable $H$ corresponds to the Hubble scalar, i.e.~the expansion speed of the cosmological model, i.e.~ the mean curvature of the surfaces $\\{t = \\textrm{const}\\}$ of spatial homogeneity. The ``shears'' $\\Sigma_i$ correspond to the trace-free Hubble-normalized principal curvatures (hence, the linear ``trace-free'' constraint). The condition $H<0$ corresponds to our choice of the direction of time: We choose to orient time such that the universe is shrinking, i.e.~the  singularity (big bang) lies in the future; this unphysical choice of time-direction is just for convenience of notation. \n\nThe vacuum Einstein Field equations state that the space-time is Ricci-flat. If we express the normalized trace-free principal curvatures $\\Sigma_i$ as time-derivatives of the metric variables $\\widetilde N_i$, then the the Einstein Field equations become the Wainwright-Hsu equations \\eqref{eq:ode-from-gr}, which are a system of seven ordinary differential equations, subject to one linear constraint equation ($\\Sigma_1+\\Sigma_2+\\Sigma_3=0$) and one algebraic equation, called the Gauss-constraint $G=1$ \\eqref{eq:gauge-constraints}): \n\n\\[\\numberthis\\label{eq:ode-from-gr}\\begin{aligned}\nH'&=\\frac 1 2 (1+2\\Sigma^2)H\\\\\n\\Sigma_i' &= (1-\\Sigma^2)\\Sigma_i +\\frac 1 2 S_i\\\\\n\\widetilde N_i'&=-(\\Sigma^2+\\Sigma_i)\\widetilde N_i\\\\\n 1 &\\overset{!}{=} \\Sigma^2+N^2 := G,\n\\end{aligned}\\]\nwhere we used the shorthands\n\\[\\begin{aligned}\nN_i &:= \\hat n_i \\widetilde N_i \\\\\n\\Sigma^2 &:= \\frac{1}{6}(\\Sigma_1^2+\\Sigma_2^2+\\Sigma_3^2) \\\\\nN^2&:= N_1^2+N_2^2+N_3^2 -2(N_1N_2+N_2N_3+N_3N_1)\\\\\nS_i &:= 4\\left(N_i(2N_i - N_j -N_k)-(N_j-N_k)^2\\right).\n\\end{aligned}\\]\n\\begin{table}\n\\fbox{ \\[\\begin{array}{cclcrclc}\n{{\\hat{n}}}_1 & {{\\hat{n}}}_2 & {{\\hat{n}}}_3 \\qquad & \\text{Bianchi Type} &\\qquad\n{{\\hat{n}}}_1 & {{\\hat{n}}}_2 & {{\\hat{n}}}_3 \\qquad & \\text{Bianchi Type}\\\\\n+&+&+\\qquad& \\text{\\textsc{IX}}&\\qquad\n+&-&+\\qquad& \\text{\\textsc{VIII}}\\\\\n+&+&0&\\text{$\\textsc{VII}_0$} &\n+&-&0& \\text{$\\textsc{VI}_0$}\\\\\n+&0&0& \\text{\\textsc{II}} &\n0&0&0& \\text{\\textsc{I}}.\n\\end{array}\\]}\n\\caption[Bianchi Types]{Bianchi Types of Class A, depending on ${{\\hat{n}}}_i$ (up to permutation and simultaneous sign-reversal)}\n\\label{table:bianchi-types}\n\\end{table}\n\\subsection{The Wainwright-Hsu equations}\nThe equation for $H$ is decoupled from the remaining equations. Thus, we can drop the equation for $H$, solve the remaining equations, and afterwards integrate to obtain $H$. Likewise, we can stick with the equations for $N_i$ instead of $\\widetilde N_i$, such that $\\hat n_i = \\mathrm{sign}\\, N_i$; for Bianchi-types \\textsc{VIII} and \\textsc{IX} this already determines the metric, and for the lower Bianchi types we can again integrate afterwards. This yields a standard form of the Wainwright-Hsu equations from \\eqref{eq:ode-from-gr}, as used in \\cite{heinzle2009mixmaster}, \\cite{heinzle2009new}, up to constant factors. The most useful equations are also summarized in Section \\ref{sect:eq-cheat-sheet}.\n\nIt is useful to solve for the linear constraint $\\Sigma_1+\\Sigma_2+\\Sigma_3=0$, introducing $\\bpt \\Sigma=(\\Sigma_+,\\Sigma_-)$ by\n\\[\\numberthis \\label{eq:taub-def}\\begin{gathered}\n\\taubp_1 = (-1,0) \\qquad\n\\taubp_2 = \\left(\\frac 1 2, -\\frac 1 2 \\sqrt{3}\\right) \\qquad\n\\taubp_3 = \\left(\\frac 1 2, \\frac 1 2 \\sqrt{3} \\right)\\\\\n\\Sigma_i = 2\\langle \\taubp_i, \\bpt \\Sigma \\rangle \\qquad \\Sigma_+ = -\\frac 1 2 \\Sigma_1 \\qquad \\Sigma_- = \\frac 1 {2\\sqrt 3}(\\Sigma_3-\\Sigma_2),\n\\end{gathered}\\]\nwhich turns the vacuum Wainwright-Hsu differential equations into a system of five ordinary differential equations on ${\\mathbb{R}}^5=\\{(\\Sigma_+,\\Sigma_-, N_1,N_2,N_3)\\}=\\{\\bpt{\\Sigma}, \\bpt{N}\\}$, with one algebraic constraint equation \\eqref{eq:constraint}. The three points $\\taubp_1,\\taubp_2,\\taubp_3$ are called Taub-points. We will, in an abuse of notation, consider the Taub-points both as points in ${\\mathbb{R}}^2$, and as points in ${\\mathbb{R}}^5$ (where all three $N_i$ vanish). The Wainwright-Hsu equations are then given by the differential equations\n\\begin{subequations\n\\label{eq:ode}\\begin{align}\nN_i' &=  -(\\Sigma^2 +2 \\langle \\taubp_i,\\bpt\\Sigma\\rangle)N_i\\label{eq:ode-ni} \\\\\n&=-\\left(\\left|\\bpt\\Sigma+\\taubp_i\\right|^2 -1\\right)N_i\\label{eq:ode2-ni}\\\\\n\\bpt\\Sigma' &= N^2 \\bpt\\Sigma + 2 \\left(N_1^2 \\taubp_1 + N_2^2\\taubp_2 + N_3^2\\taubp_3 + N_1N_2\\taubp_3 + N_2N_3\\taubp_1 + N_3N_1\\taubp_2\\right)\\label{eq:ode2-sigma}\\\\\n&=N^2 \\bpt\\Sigma + 2 \\threemat{\\taubp_1}{\\taubp_2}{\\taubp_3}{\\taubp_3}{\\taubp_1}{\\taubp_2}[\\bpt N, \\bpt N],\n\\end{align}\\end{subequations}\nand the Gauss constraint equation\n\\begin{equation}\\label{eq:constraint}\n1\\overset{!}{=} \\Sigma^2 + N^2 =:G(\\bpt x),\n\\end{equation}\nwhere we used the shorthands\n\\[\n\\Sigma^2=\\Sigma_+^2+\\Sigma_-^2,\\qquad N^2 = N_1^2+N_2^2+N_3^2 - 2(N_1N_2+N_2N_3+N_3N_1).\n\\]\nWe can unpack these equations with unambiguous notation into \n\\begin{subequations}\\label{eq:ode-unpacked}\\begin{align}\nN_1' &=  -(\\Sigma^2 -2\\Sigma_+)N_1 \\\\\nN_2' &=  -(\\Sigma^2 +\\Sigma_+ - \\sqrt{3}\\Sigma_-)N_2 \\\\\nN_3' &=  -(\\Sigma^2 +\\Sigma_+ + \\sqrt{3}\\Sigma_-)N_3 \\\\\n\\Sigma_+' &= N^2 \\Sigma_+ -2 N_1^2  +N_2^2 + N_3^2 + N_1N_2 -2 N_2N_3 + N_1N_3 \\\\\n\\Sigma_-' &= N^2\\Sigma_- +\\sqrt{3}\\left(- N_2^2 +N_3^2 + N_1N_2 -N_1N_3  \\right),\n\\end{align}\\end{subequations}\nwhich is, up to constant factors, the form of the Wainwright-Hsu equations used in \\cite{ringstrom2001bianchi}, \\cite{liebscher2011ancient} and \\cite{beguin2010aperiodic}.\n\nIt is occasionally useful to fully tensorize the Wainwright-Hsu equations, yielding the form\n\\begin{equation}\\label{eq:ode-fulltensor}\\begin{aligned}\n\\bpt N' &= -\\langle\\bpt \\Sigma, \\bpt \\Sigma\\rangle \\bpt N - \\bpt D[\\bpt \\Sigma,\\bpt N]\\\\\n\\bpt \\Sigma'&= Q[\\bpt N, \\bpt N]\\bpt \\Sigma + \\bpt T[\\bpt N, \\bpt N]\\\\\nG(\\bpt \\Sigma, \\bpt N)&= \\langle \\bpt \\Sigma, \\bpt \\Sigma\\rangle + Q[\\bpt N, \\bpt N]\\overset{!}{=}1,\n\\end{aligned}\\end{equation}\nwhere $Q:{\\mathbb{R}}^3\\times {\\mathbb{R}}^3\\to {\\mathbb{R}}$ and $\\bpt T: {\\mathbb{R}}^3\\times {\\mathbb{R}}^3\\to{\\mathbb{R}}^2$ and $\\bpt D: {\\mathbb{R}}^2\\times {\\mathbb{R}}^3 \\to {\\mathbb{R}}^3$. \nWe write $Q$ as a $3\\times 3$-matrix with entries in ${\\mathbb{R}}$ such that $Q[\\bpt N,\\bpt M]=\\bpt N^T Q\\bpt N=N^2$; we write $\\bpt T$ as a similar $3\\times 3$-matrix with entries in ${\\mathbb{R}}^2$. We write $\\bpt D$ as a $3\\times 3$-matrix with entries in ${\\mathbb{R}}^2$ such that $\\bpt D[\\bpt \\Sigma, \\bpt N] = (\\bpt D\\bpt N)\\cdot \\bpt \\Sigma$, where $\\bpt D\\bpt N$ is the usual matrix product (with entries in ${\\mathbb{R}}^2$) and the dot-product is evaluated component wise. Then the tensors $Q,\\bpt T, \\bpt D$ can be written as\n\n\\begin{equation}\\label{eq:ode-fulltensor-tensors}\\begin{aligned}\nQ= \\sixmat{1 & -2 & -2\\\\  &1 & -2 \\\\ & & 1}\\qquad\n\\bpt T = \\sixmat{2\\taubp_1 & 2\\taubp_3 & 2\\taubp_2\\\\ & 2\\taubp_2 & 2\\taubp_1\\\\  &  & 2\\taubp_3} \\qquad\n\\bpt D = \\sixmat{2\\taubp_1 &  & \\\\  & 2\\taubp_2 & \\\\  &  & 2\\taubp_3}.\n\\end{aligned}\\end{equation}\n\n\n\n\\paragraph{Permutation Equivariance.}\nThe equations \\eqref{eq:ode-from-gr} are equivariant under permutations $\\sigma:\\{1,2,3\\}\\to\\{1,2,3\\}$ of the three indices. This permutation invariance also applies to \\eqref{eq:ode} as\n\\[\n(\\bpt \\Sigma, N_1,N_2,N_3)\\to (A_\\sigma\\bpt\\Sigma, N_{\\sigma(1)}, N_{\\sigma(2)}, N_{\\sigma(3)}),\n\\]\nwhere $A_\\sigma:{\\mathbb{R}}^2\\to{\\mathbb{R}}^2$ is the linear isometry with  $A_\\sigma^{T}\\taubp_i = \\taubp_{\\sigma(i)}$. The equations are also equivariant under $(\\bpt\\Sigma, \\bpt N)\\to (\\bpt \\Sigma, -\\bpt N)$, as can be seen directly from \\eqref{eq:ode-fulltensor}.\n\n\\paragraph{Invariance of the Constraint.}\nThe signs of the $N_i$ are preserved under the flow, because $N_i'$ is a multiple of $N_i$ and therefore $N_i=0$ implies $N_i'=0$. The quantity $G$ is preserved under the flow. This can best be seen from \\eqref{eq:ode-fulltensor}:\n\\[\\begin{aligned}\n{\\mathrm{D}}_t G &= 2\\langle \\bpt \\Sigma, \\bpt \\Sigma'\\rangle + Q[\\bpt N, \\bpt N'] + Q[\\bpt N', \\bpt N]\\\\\n&= 2Q[\\bpt N, \\bpt N]\\langle \\bpt \\Sigma, \\bpt \\Sigma\\rangle + 2\\bpt \\Sigma \\cdot \\bpt T[\\bpt N, \\bpt N] - 2 \\langle \\bpt \\Sigma, \\bpt \\Sigma\\rangle Q[\\bpt N, \\bpt N] \\\\\n&\\qquad - Q[\\bpt D[\\bpt \\Sigma, \\bpt N], \\bpt N] - Q[\\bpt N, \\bpt D[\\bpt \\Sigma,\\bpt N]]\\\\\n&= \\Sigma\\cdot \\bpt N^T \\left[2 \\bpt T - \\bpt D^T Q - Q \\bpt D\\right]\\bpt N.\n\\end{aligned}\\]\nUsing $\\taubp_1+\\taubp_2+\\taubp_3=0$ and \\eqref{eq:ode-fulltensor-tensors}, it is a simple matter of matrix multiplication to verify that $2\\bpt T-\\bpt D^TQ-Q\\bpt D=0$ and hence ${\\mathrm{D}}_t G=0$. Therefore, sets of the form $\\{\\bpt x\\in{\\mathbb{R}}^5: G(\\bpt x) = c\\}$ are invariant for any $c\\in{\\mathbb{R}}$ and especially for the physical $c=1$.\n\nThe set $\\mathcal M=\\{\\bpt x \\in {\\mathbb{R}}^5: G(\\bpt x)=1\\}$ is a smooth embedded submanifold; this is apparent from the implicit function theorem, since (if $\\bpt x\\neq 0$)\n\\[\\begin{multlined}\n\\frac{1}{2}{\\mathrm{d}} G = \\Sigma_+{\\mathrm{d}}\\Sigma_+ + \\Sigma_-{\\mathrm{d}} \\Sigma_- \\\\+ (N_1-N_2-N_3){\\mathrm{d}} N_1 + (N_2-N_3-N_1){\\mathrm{d}} N_2 + (N_3-N_1-N_2){\\mathrm{d}} N_1\\neq 0.\n\\end{multlined}\\]\n\n\n\\paragraph{Named invariant sets.}\nThere are several recurring important sets, which require names and are listed in Table \\ref{table:inv-sets}.\n\\begin{table}\n\\fbox{\\small \\[\\begin{aligned}\n\\mathcal M &= \\{\\bpt x\\in {\\mathbb{R}}^5: \\; G(\\bpt x)=1\\}&&\\qquad\\text{the physically relevant Phase-space}\\\\\n\\mathcal M_{{{\\hat{n}}}} &= \\{\\bpt x\\in \\mathcal M: \\; \\mathrm{sign}\\,N_i = {{\\hat{n}}}_i\\}&&\\qquad\\text{a specific octant of the Phase-space}\\\\\n\\mathcal K &= \\mathcal M_{000}=\\{\\bpt x\\in\\mathcal M: \\; \\bpt N=0\\} &&\\qquad\\text{the Kasner circle}\\\\\n\\mathcal A &= \\{\\bpt x\\in\\mathcal M: \\; \\text{at most one }N_i\\neq 0\\} && \\qquad\\text{the Mixmaster attractor}\\\\\n\\mathcal A_{{{\\hat{n}}}} &= \\overline{\\mathcal M_{{{\\hat{n}}}}}\\cap \\mathcal A && \\qquad\\text{a specific octant of $\\mathcal A$}\\\\\n\\mathcal T_i &=\\{\\bpt x\\in\\mathcal M: N_j = N_k,\\, \\langle \\taubp_j,\\bpt \\Sigma\\rangle = \\langle \\taubp_k,\\bpt \\Sigma\\rangle\\}&&\\qquad\\text{a Taub-space}\\\\\n\\mathcal T &=\\mathcal T_1 \\cup \\mathcal T_2 \\cup\\mathcal T_3 && \\qquad\\text{all three Taub-spaces}\\\\\n\\mathcal{TL}_i &=\\{\\bpt x\\in\\mathcal M: N_j = N_k,\\,N_i=0,\\,\\bpt \\Sigma = \\taubp_i\\}\\subseteq \\mathcal T_i&& \\qquad\\text{a Taub-line}\\\\\n\\mathcal{T}^G_i&=\\{\\bpt x\\in\\mathcal M: |N_j| = |N_k|,\\, \\langle \\taubp_j,\\bpt \\Sigma\\rangle = \\langle \\taubp_k,\\bpt \\Sigma\\rangle\\}&&\\qquad\\text{a generalized Taub-space; }\\\\\n&&&\\qquad\\text{  only invariant if $\\mathrm{sign}\\,N_j=\\mathrm{sign}\\,N_k$}\n\\end{aligned}\\]}\n\\caption[Named Subsets]{Named subsets. Here $(i,j,k)$ stands for a permutation of $\\{1,2,3\\}$ and ${{\\hat{n}}}\\in\\{+,0,-\\}^3$.\nAll of these sets, except for $\\mathcal{T}^G_i$, are invariant.}\n\\label{table:inv-sets}\n\\end{table}\n\nThe set $\\mathcal M$ is invariant because $G$ is a constant of motion. The Taub-space $\\mathcal T_i$ is invariant because of the equivariance under exchange of the two other indices $j$ and $k$. The invariance of the Taub-lines $\\mathcal{TL}_i$ can be seen by considering \\eqref{eq:ode-unpacked} for $i=1$ and applying the permutation invariance for $\\mathcal{TL}_2$ and $\\mathcal{TL}_3$. The generalized Taub-spaces $\\mathcal T^G_i$ are not invariant if ${{\\hat{n}}}_j\\neq{{\\hat{n}}}_k$. The other sets are invariant because the signs ${{\\hat{n}}}_i=\\mathrm{sign}\\,N_i$ are fixed.\n\nRecall that the signs of the $N_i$ correspond to the Bianchi Type of the Lie-algebra associated to the homogeneity of the cosmological model and are given in Table \\ref{table:bianchi-types} (up to index permutations).\n\n\n\n\\paragraph{Auxiliary Quantities.}\nThe following quantities turn out to be useful later on (where $(i,j,k)$ is a permutation of $(1,2,3)$):\n\\begin{subequations}\\label{eq:delta-r-def}\n\\begin{align} \n\\delta_i &= 2\\sqrt{|N_jN_k|} \\label{eq:delta-r-def-delta}\\\\\nr_i &= \\sqrt{(|N_j|-|N_k|)^2 + \\frac{1}{3}\\langle\\taubp_j-\\taubp_k, \\bpt\\Sigma\\rangle^2}\\label{eq:delta-r-def-r}\\\\\n\\nonumber \\psi_i &\\quad\\text{such that:} \\\\\nr_1\\cos\\psi_1&=\\frac{1}{\\sqrt{3}}\\langle\\taubp_{3}-\\taubp_{2}, \\bpt \\Sigma\\rangle=\\Sigma_- \n&r_1\\sin\\psi_1 &= |N_2|-|N_3| \\\\\nr_2\\cos\\psi_2&=\\frac{1}{\\sqrt{3}}\\langle\\taubp_{1}-\\taubp_{3}, \\bpt \\Sigma\\rangle=-\\frac{\\sqrt 3}{2}\\Sigma_+ -\\frac{1}{2}\\Sigma_-\n&r_2\\sin\\psi_2 &= |N_3|-|N_1| \\\\\nr_3\\cos\\psi_3&=\\frac{1}{\\sqrt{3}}\\langle\\taubp_{2}-\\taubp_{1}, \\bpt \\Sigma\\rangle=\\frac{\\sqrt 3}{2}\\Sigma_+ -\\frac{1}{2}\\Sigma_-\n&r_3\\sin\\psi_3 &= |N_1|-|N_2|. \n\\end{align}\\end{subequations}\nThe auxiliary products $\\delta_i^2$ can be used to measure the distance from the Mixmaster attractor $\\mathcal A = \\{\\bpt x:\\, \\max_i \\delta_i(\\bpt x)=0\\}$. The $r_i$ and can be used to measure the distance from the generalized Taub-space $\\mathcal T_i^G = \\{\\bpt x:\\,r_i(\\bpt x)=0\\}$, and the $(r_i,\\psi_i)$-pairs form polar coordinates around the generalized Taub-spaces.\n\nThe products $\\delta_i$ obey an especially geometric differential equation, similar to \\eqref{eq:ode2-ni}:\n\\begin{equation}\\label{eq:ode2-delta}\n\\delta_i' =  -\\left(\\left|\\bpt\\Sigma-\\frac{\\taubp_i}{2}\\right|^2-\\frac{1}{4}\\right)\\delta_{i}.\n\\end{equation}\n\n\\subsection{The Wainwright-Hsu equations in polar coordinates}\\label{sect:polar-coords}\nNear the generalized Taub-spaces $\\mathcal T^G_i$, it is possible to use polar coordinates \\eqref{eq:delta-r-def}. Without loss of generality we will only transform \\eqref{eq:ode} into these coordinates around the Taub-space $\\mathcal T^G_1$ (the other ones can be obtained by permuting the indices and rotating or reflecting $\\bpt \\Sigma$).\n\nThe use of polar coordinates near the Taub-spaces $\\mathcal T_i$ for Bianchi \\textsc{IX}, i.e. $\\mathcal M_{+++}$, is by no means novel (c.f.~e.g.~\\cite{ringstrom2001bianchi}, \\cite{heinzle2009mixmaster}). However, to the best of our knowledge, polar coordinates around the generalized Taub-spaces $\\mathcal T^G_i$ have not been used previously in the case where the generalized Taub-space fails to be invariant.\n\nWe only use use polar coordinates on $\\mathcal M=\\{\\bpt x:\\,\\,G(\\bpt x)=1\\}$.\n\\paragraph{Polar Coordinates around the invariant Taub-spaces.}\nConsider the case $\\mathcal M_{*++}$, where $N_2,N_3>0$ are positive and we are interested in a neighborhood of $\\mathcal T_1 = \\{\\bpt x:\\, \\Sigma_-=0,\\,N_2-N_3=0\\}$. The sign of $N_1$ does not significantly matter.\n\n\nWe use the additional shorthands\n\\[ N_- = N_2-N_3\\qquad N_+=N_2+N_3,\n\\]\nsuch that (with \\eqref{eq:delta-r-def}):\n\\[\\begin{aligned}\nr_1\\ge 0 &:& r_1^2 &= \\Sigma_-^2 + N_-^2\\\\\n\\psi &:& N_-&= r_1\\sin\\psi&\\Sigma_-&=r_1\\cos\\psi\\\\\n&&N_+^2 &= N_-^2+\\delta_1^2 &N^2&= N_-^2+N_1(N_1-2N_+).\n\\end{aligned}\\]\n\n\n\\noindent This gives us the differential equations (using $\\Sigma^2+N^2=1$):\n\\[\\begin{aligned}\nN_-' &= (N^2 -1 - \\Sigma_+)N_- +\\sqrt{3}\\Sigma_-N_+\\\\\nN_+'&= (N^2 -1 -\\Sigma_+)N_+ +\\sqrt{3}\\Sigma_-N_-\\\\\n\\Sigma_-' &= N^2\\Sigma_- - \\sqrt{3}N_-\\left(N_+ - N_1\\right),\n\\end{aligned}\\]\nallowing us to further compute\n\\begin{subequations}\\label{eq:neartaub-b9-q}\n\\begin{align}\n\\frac{r_1'}{r_1}&= \\frac{\\Sigma_-\\Sigma_-' + N_-N_-'}{r_1^2} \n= N^2 - (\\Sigma_++1) \\frac{N_-^2}{r_1^2} +\\sqrt{3} N_1 \\frac{\\Sigma_- N_-}{r_1^2}\\label{eq:neartaub-b9-q:r}\\\\\n\\psi' &=\\frac{\\Sigma_-N_-' - N_-\\Sigma_-'}{r_1^2}\n=\\sqrt{3}N_+ - (\\Sigma_++1)\\frac{N_-\\Sigma_-}{r_1^2} -\\sqrt{3}N_1\\frac{N_-^2}{r_1^2}\\label{eq:neartaub-b9-q:psi}\\\\\n\\frac{\\delta_1'}{\\delta_1} &= N^2-(\\Sigma_++1)\\label{eq:neartaub-b9-q:delta}\\\\\n\\partial_t\\log \\frac{\\delta_1}{r_1}&= -(\\Sigma_+ + 1)\\frac{\\Sigma_-^2}{r_1^2} -\\sqrt{3}N_1\\frac{\\Sigma_- N_-}{r_1^2}.\\label{eq:neartaub-b9-q:delta-r}\n\\end{align}\\end{subequations}\nNear $\\taubp_1$, i.e. for $\\Sigma_+\\approx -1$, we can use $1+\\Sigma_+=\\frac{\\Sigma_-^2+N^2}{1-\\Sigma_+}$ in order to rearrange some terms in \\eqref{eq:neartaub-b9-q}:\n\\begin{subequations}\\label{eq:neartaub-b9-t}\n\\begin{align}\n\\frac{r_1'}{r_1} &= r_1^2\\sin^2\\psi \\frac{-\\Sigma_+}{1-\\Sigma_+} + N_1\\, h_r\\label{eq:neartaub-b9-t:r}\\\\\n\\frac{\\delta_1'}{\\delta_1}&= \\frac{-1}{1-\\Sigma_+}r_1^2\\cos^2\\psi + \\frac{-\\Sigma_+}{1-\\Sigma_+}r_1^2\\sin^2\\psi + N_1\\,h_\\delta \\label{eq:neartaub-b9-t:delta}\\\\\n\\partial_t \\log \\frac{\\delta_1}{r_1}&=\\frac{-1}{1-\\Sigma_+}r_1^2\\cos^2\\psi +N_1(h_\\delta-h_r) \\label{eq:neartaub-b9-t:delta-r}\\\\\n\\psi'&= \\sqrt{3}r_1 \\sqrt{\\sin^2\\psi + \\frac{\\delta_1^2}{r_1^2}} - \\frac{r_1^2}{1-\\Sigma_+} \\sin\\psi \\cos\\psi + N_1\\sin\\psi\\, h_{\\psi}, \\label{eq:neartaub-b9-t:psi}\n\\end{align}\\end{subequations}\nwhere\n\\begin{subequations}\\label{eq:neartaub-b9-t-h}\\begin{align}\nh_r &=+\\sqrt{3}\\frac{\\Sigma_-N_-}{r_1^2} + (N_1-2N_+)\\left(1+\\frac{N_-^2}{(1-\\Sigma_+)r_1^2}\\right)\\\\\nh_\\delta &= (N_1-2N_+)\\frac{-\\Sigma_+}{1-\\Sigma_+}\\\\\nh_\\psi &= -\\sqrt{3}\\sin\\psi - \\cos\\psi \\frac{N_1-2N_+}{1-\\Sigma_+}.\n\\end{align}\\end{subequations}\nLet us point out that $|h_r|,|h_\\delta|, |h_\\psi| \\le 5$ if $|N_1|, |N_+|, |N_-|\\le \\frac 1 2$ and $\\Sigma_+<0$.\n\\paragraph{Polar Coordinates around the non-invariant generalized Taub-spaces.}\nWithout loss of generality, assume that we are in $\\mathcal M_{*+-}$, i.e.~$N_2>0>N_3$, and that we are interested in a neighborhood of $\\mathcal T^G_1 = \\{\\bpt x:\\, \\Sigma_-=0,\\,|N_2|-|N_3|=0\\}$.\n\nThe assumption $N_2>0>N_3$ is incompatible with $N_3=N_2$; hence, $\\mathcal M_{*+-}\\cap \\mathcal T_1=\\emptyset$, which is why we have to work with $\\mathcal T^G_1$. Since $\\mathcal T^G_1$ is not invariant, we expect the corresponding equations for $r_1'$ and $\\psi'$ to become singular near $r_1=0$.\n\nThere is a possible symmetry-based motivation for expecting usable polar coordinates around $\\mathcal T^G_1$ that will be given at the end of this section. Regardless of this motivation, the usability of such polar coordinates is proven by our use of them.\n\nWe will proceed analogous to the case of $\\mathcal T_1$, using the same names for quantities which fulfill the same function in this work, such that the definitions of e.g.~$N_+,N_-$ will depend on the signs ${{\\hat{n}}}_2,{{\\hat{n}}}_3$. Hence, we introduce shorthands\n\\[ N_- = |N_2|-|N_3|=N_2+N_3\\qquad N_+=|N_2|+|N_3|=N_2-N_3,\\]\nsuch that (with \\eqref{eq:delta-r-def}):\n\\[\\begin{aligned}\nr_1\\ge 0 &:& r_1^2 &= \\Sigma_-^2 + N_-^2\\\\\n\\psi &:& N_-&= r_1\\sin\\psi&\\Sigma_-&=r_1\\cos\\psi\\\\\n&&N_+^2 &= N_-^2+\\delta_1^2 &N^2&= N_+^2+N_1(N_1-2N_-).\n\\end{aligned}\\]\n\nThis gives us the differential equations (using $\\Sigma^2+N^2=1$):\n\\[\\begin{aligned}\nN_-' &= (N^2 -1 - \\Sigma_+)N_- +\\sqrt{3}\\Sigma_-N_+\\\\\nN_+'&= (N^2 -1 -\\Sigma_+)N_+ +\\sqrt{3}\\Sigma_-N_-\\\\\n\\Sigma_-' &= N^2\\Sigma_- - \\sqrt{3}\\left(N_-N_+ - N_+N_1\\right),\n\\end{aligned}\\]\nallowing us to further compute\n\\begin{subequations}\\label{eq:neartaub-b8-q}\n\\begin{align}\n\\frac{r_1'}{r_1}&= \\frac{\\Sigma_-\\Sigma_-' + N_-N_-'}{r_1^2} \n= N^2 - (\\Sigma_++1) \\frac{N_-^2}{r_1^2} +\\sqrt{3} N_1 \\frac{\\Sigma_- N_+}{r_1^2} \\label{eq:neartaub-b8-q:r}\\\\\n\\psi' &=\\frac{\\Sigma_-N_-' - N_-\\Sigma_-'}{r_1^2}\n=\\sqrt{3}N_+ - (\\Sigma_++1)\\frac{N_-\\Sigma_-}{r_1^2} -\\sqrt{3}N_1\\frac{N_-N_+}{r_1^2} \\label{eq:neartaub-b8-q:psi}\\\\\n\\frac{\\delta_1'}{\\delta_1} &= N^2-(\\Sigma_++1) \\label{eq:neartaub-b8-q:delta}\\\\\n\\partial_t\\log \\frac{\\delta_1}{r_1}&= -(\\Sigma_+ + 1)\\frac{\\Sigma_-^2}{r_1^2} -\\sqrt{3}N_1\\frac{\\Sigma_- N_+}{r_1^2}.\\label{eq:neartaub-b8-q:delta-r}\n\\end{align}\\end{subequations}\n\\noindent\nNear $\\taubp_1$, i.e. for $\\Sigma_+\\approx -1$, we can use $1+\\Sigma_+=\\frac{\\Sigma_-^2+N^2}{1-\\Sigma_+}$ in order to rearrange some terms in \\eqref{eq:neartaub-b8-q}:\n\n\n\\begin{subequations}\\label{eq:neartaub-b8-t}\n\\begin{align}\n\\frac{r_1'}{r_1} &= \\frac{-\\Sigma_+}{1-\\Sigma_+}r_1^2\\sin^2\\psi +\\delta_1^2\\frac{\\cos^2\\psi-\\Sigma_+}{1-\\Sigma_+} + N_1\\, h_r \\label{eq:neartaub-b8-t:r}\\\\\n\\frac{\\delta_1'}{\\delta_1}&= \\frac{-1}{1-\\Sigma_+}r_1^2\\cos^2\\psi + \\frac{-\\Sigma_+}{1-\\Sigma_+}r_1^2\\sin^2\\psi +\\frac{-\\Sigma_+}{1-\\Sigma_+}\\delta_1^2+ N_1\\,h_\\delta \\label{eq:neartaub-b8-t:delta}\\\\\n\\partial_t \\log \\frac{\\delta_1}{r_1}&=\\frac{-1}{1-\\Sigma_+}r_1^2\\cos^2\\psi - \\delta_1^2\\frac{\\cos^2\\psi}{1-\\Sigma_+} + N_1(h_\\delta-h_r) \\label{eq:neartaub-b8-t:delta-r}\\\\\n\\psi'&= \\sqrt{3}r_1 \\sqrt{\\cos^2\\psi + \\frac{\\delta_1^2}{r_1^2}} - \\frac{r_1^2+\\delta_1^2}{1-\\Sigma_+} \\cos\\psi \\sin\\psi + N_1\\sin\\psi\\, h_{\\psi},\\label{eq:neartaub-b8-t:psi}\n\\end{align}\\end{subequations}\nwhere\n\\begin{subequations}\\label{eq:neartaub-b8-t-h}\\begin{align}\nh_r &=-\\sqrt{3}\\frac{\\Sigma_-N_+}{r_1^2} + (N_1-2N_-)\\left(1-\\frac{N_-^2}{(1-\\Sigma_+)r_1^2}\\right)\\\\\nh_\\delta &= (N_1-2N_-)\\frac{-\\Sigma_+}{1-\\Sigma_+}\\\\\nh_\\psi &= -\\sqrt{3}\\sqrt{\\sin^2\\psi + \\frac{\\delta_1^2}{r_1^2}} - \\sin\\psi \\frac{N_1-2N_-}{1-\\Sigma_+}.\n\\end{align}\\end{subequations}\nLet us point out that, if $|N_1|, |N_+|, |N_-|\\le \\frac 1 2$ and $\\Sigma_+<0$ and $\\frac{\\delta_1}{r_1}\\le 1$, then $\\frac{N_+}{r_1}\\le \\sqrt{2}$ and hence $|h_r|,|h_\\delta|, |h_\\psi| \\le 5$.\n\n\\paragraph{Motivation for polar coordinates around $\\mathcal T_1^G$.}\nOne possible motivation for a priori expecting useful equations from this approach is by a symmetry argument: The Taub-space is invariant since it is the fixed point space of the reflection $\\sigma:(\\Sigma_+,\\Sigma_-, N_1,N_2,N_3)\\to (\\Sigma_+,-\\Sigma_-, N_1, N_3, N_2)$, and \\eqref{eq:ode} is equivariant under $\\sigma$. This transformation $\\sigma:\\mathcal M_{\\pm +-}\\to \\mathcal M_{\\pm-+}$ does not map the quadrant $N_2>0>N_3$ into itself. Instead, we have $\\mathcal T^G_1$ as the fixed point space of $\\widetilde\\sigma:(\\Sigma_+,\\Sigma_-, N_1,N_2,N_3)\\to (\\Sigma_+,-\\Sigma_-, N_1, -N_3, -N_2)$.\nThe Wainwright-Hsu equations are not equivariant under this reflection $\\widetilde\\sigma$, and we therefore have no reason to expect the fixed-point space $\\mathcal T^G_1$ of $\\widetilde \\sigma$ to be invariant. However, considering \\eqref{eq:ode}, equivariance is only spoiled by terms of the form $N_2N_1$ and $N_3N_1$ changing their signs; hence, we expect $\\mathcal T^G_1=\\{\\bpt x:\\, \\Sigma_-=0,\\,|N_2|-|N_3|=0\\}$ to be invariant up to terms of order $|N_2N_1|$ and $|N_3N_1|$. Such terms can be well controlled, as it will turn out in Section \\ref{sect:near-taub}.\n\n\n\\section{Description of the Dynamics} \\label{sect:farfromA\nWe will now give an overview of the behaviour of trajectories of \\eqref{eq:ode}. This overview will contain most of the classic results about Bianchi cosmological models. \n\nOur overview will be organized by first describing the simplest subsets named in Table \\ref{table:inv-sets} and then progressing to the higher dimensional subsets, finally describing Bianchi Type \\textsc{IX} ($\\mathcal M_{+++}$) and Bianchi Type \\textsc{VIII} ($\\mathcal M_{+-+}$) solutions. Our approach in this section is very similar to \\cite{ringstrom2001bianchi} and \\cite{heinzle2009new}; unless explicitly otherwise stated, all observations in this section can be found therein.\n\n \n\\paragraph{A very short summary of relevant dynamics.}\nThe Kasner circle $\\mathcal K$ is actually a circle and consists entirely of equilibria. The so-called Mixmaster attractor $\\mathcal A$ consists of three $2$-spheres $\\{\\bpt x:\\, \\Sigma_+^2+\\Sigma_-^2+N_i^2=1\\,N_j=N_k=0\\}$, which intersect in $\\mathcal K$. Only half of these spheres are accessible for any trajectory, since the $\\mathrm{sign}\\,N_i$ are fixed. For this reason, these half-spheres are also called ``Kasner-caps'', i.e.~$\\mathcal A_{+00}$ is the $N_1>0$-cap.\n\nThe dynamics on the Kasner-caps will be discussed in Section \\ref{sect:lower-b-types}; each orbit in a Kasner-cap is a heteroclinic orbit connecting two equilibria on the Kasner-circle $\\mathcal K$.\n\nThe long-time behaviour of the lower dimensional Bianchi-types (at least one $N_i=0$) is well-understood: All such solutions converge to an equilibrium $\\bpt p \\in\\mathcal K$ as $t\\to\\infty$. The behavior in the highest-dimensional Bianchi Types \\textsc{IX} and \\textsc{VIII} is not yet fully understood, and is therefore of most interest in this work.\n\n\nIt is known (c.f.~\\cite{ringstrom2001bianchi}) that Bianchi Type \\textsc{IX} solutions that do not lie in a Taub-space converge to the Mixmaster attractor as $t\\to+\\infty$, i.e.~towards the big bang singularity. \nIt has been conjectured that generic Bianchi Type \\textsc{VIII} solutions share this behaviour; this will be proven in this work (Theorem \\ref{thm:b8-attractor-global} and \\ref{thm:b8-attractor-global-genericity}).\n\n\nThe question of particle horizons was already mentioned in the introduction and is further discussed from a physical viewpoint in Section \\ref{sect:gr-phys-interpret}. In terms of the Wainwright-Hsu equations, the question can be formulated as (see Section \\ref{thm:b8-attractor-global-genericity}, or c.f. e.g. \\cite{heinzle2009future}):\n\n\\[\n\\text{Is }\\quad\nI(\\bpt x) =   \\max_i \\int_{0}^\\infty \\delta_i(\\phi(\\bpt x, t)){\\mathrm{d}} t = 2\\max_i\\int_0^\\infty \\sqrt{|N_jN_k|}(t){\\mathrm{d}} t<\\infty\\,?\n\\]\nHere $\\phi$ is the flow to \\eqref{eq:ode}. The space-time associated to the solution $\\phi(\\bpt x, \\cdot)$ forms a particle horizon if and only if $I<\\infty$.\n\n\nIt is known that there exist solutions in Bianchi \\textsc{IX} and \\textsc{VIII}, where $I<\\infty$ (c.f.~\\cite{liebscher2011ancient}). It is not known, whether there exist any nontrivial solutions with $I=\\infty$ (of course solutions starting in $\\mathcal T$ that do not converge to $\\mathcal A$ must have $I=\\infty$). We prove that, in both Bianchi \\textsc{IX} and \\textsc{VIII} and for Lebesgue almost every initial condition, particle horizons develop ($I<\\infty$) (Theorem \\ref{thm:horizon-formation}).\n\n\n\n\\subsection{Lower Bianchi Types}\\label{sect:lower-b-types\n\\paragraph{Bianchi Type \\textsc{I}: The Kasner circle.}\nThe smallest, i.e.~lowest dimensional, Bianchi-type is Type \\textsc{I}, $\\mathcal M_{000}=\\mathcal K$, where all three $N_i$ vanish (see Table \\ref{table:inv-sets}). By the constraint $N^2+\\Sigma^2=1$, we can see that $\\mathcal K$ is the unit circle in the $(\\Sigma_+,\\Sigma_-)$-plane, and consists entirely of equilibria.\n\nThe linear stability of these equilibria is given by the following\n\\begin{lemma}\\label{farfromA:lemma:kasnermap-stability}\nLet $\\bpt p=(\\Sigma_+,\\Sigma_-,0,0,0)\\in \\mathcal K$; first consider the case $\\bpt p\\neq \\taubp_i$ and without loss of generality $|\\bpt p+\\taubp_1|<1$. Then the vectorfield has one central direction given by $\\partial_{\\mathcal K}=(-\\Sigma_-, \\Sigma_+,0,0,0)$, one unstable direction given by $\\partial_{N_1}=(0,0,1,0,0)$, and two stable directions given by $\\partial_{N_2}=(0,0,0,1,0)$ and $\\partial_{N_3}=(0,0,0,0,1)$.\n\nThe three Taub-points $\\taubp_i$ have each one stable direction given by $\\partial_{N_i}$ and three center directions given by $\\partial_{\\mathcal K}$, $\\partial_{N_j}$ and $\\partial_{N_k}$.\n\\end{lemma}\n\\begin{proof}\nWe first note that the four vectors $\\partial_{\\mathcal K}$ and $\\partial_{N_i}$ form a basis of the tangent space  $T_{\\bpt p}\\mathcal M = \\mathrm{ker}\\,{\\mathrm{d}} G$. \n\nThe stability of an equilibrium is determined by the eigenvalues and eigenspaces of the Jacobian of the vector field; a generalized eigenspace is \\emph{central}, if its eigenvalue has vanishing real part, it is \\emph{stable} if its eigenvalue has negative real part, and it is \\emph{unstable} if its eigenvalue has positive real part.\nThe Jacobian ${\\mathrm{D}} f$ of the vector field $f$ given by \\eqref{eq:ode} at $\\bpt p\\in\\mathcal K$ is diagonal with three entries of the form ${\\mathrm{D}} f = \\lambda_1 \\partial_{N_1} \\otimes {\\mathrm{d}} N_1 + \\lambda_{N_2} \\partial_2 \\otimes {\\mathrm{d}} N_2 + \\lambda_3 \\partial_{N_3}\\otimes {\\mathrm{d}} N_3$; we can read off the stability from \\eqref{eq:ode2-ni} and Figure \\ref{fig:n-discs}.\n\\end{proof}\n\n\\begin{figure}[hbpt]\n \\centering\n        \\begin{subfigure}[b]{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=\\textwidth]{.\/graphics\/classa-short-het}\n                \\caption{The three discs, where $N_i'\/N_i>0$ and a short heteroclinic chain.}\\label{fig:short-het}\\label{fig:n-discs}\n        \\end{subfigure}~~%\n        %\n        \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{.\/graphics\/classa-stab-products.pdf}\n        \\caption{The three discs, where $\\delta_i'\/\\delta_i >0$. Also, the Kasner-map $K$ is a double cover.}\\label{fig:double-cover}\\label{fig:delta-discs}\n        \\end{subfigure\n        \\caption{Stability properties and the Kasner map. All figures are in $(\\Sigma_+, \\Sigma_-)$-projection.}\\label{fig:kasner-segments}        \n\\end{figure}\n\n\\paragraph{The Taub-line.}\nThere exists another structure of equilibria, given by $\\mathcal{TL}_i:= \\{\\bpt x\\in\\mathcal M: N_j = N_k,\\,N_i=0,\\,\\bpt \\Sigma = \\taubp_i\\}$. Up to index permutations and $\\bpt N\\to -\\bpt N$, this set has the form $\\mathcal{TL}_1\\cup\\mathcal M_{++0}=\\{\\bpt p:\\, \\bpt p =(-1,0,0,n,n),\\, n>0\\}$. We observe that this is a line of equilibria. Each such equilibrium has one stable direction (corresponding to $N_1$) and three center directions. \n\n\\paragraph{Bianchi Type \\textsc{II}: The Kasner caps.}\nConsider without loss of generality the set $\\mathcal M_{+00}$. The constraint $G=1$ then reads\n\\[1=N_1^2 + \\Sigma_+^2+\\Sigma_-^2,\\]\ni.e.~the so-called ``Kasner-cap'' $\\mathcal M_{+00}$ forms a half-sphere with $\\mathcal K$ as its boundary. Considering \\eqref{eq:ode}, we can see that $\\bpt \\Sigma' = (\\bpt \\Sigma+2\\taubp_1)N_1^2$ is a scalar multiple of $\\bpt \\Sigma+2\\taubp_1$; hence, the $\\bpt\\Sigma$-projection of the trajectory stays on the same line through $-2\\taubp_1$. Since $N_1^2\\ge 0$, this trajectory is heteroclinic and converges in forward and backward times to the two intersections of this line with the Kasner circle $\\mathcal K$, where the $\\alpha$-limit is closer to $-2\\taubp_1$. Such a trajectory is depicted in Figure \\ref{fig:short-het}.\n\n\\paragraph{The Mixmaster-Attractor $\\mathcal A$.}\nThe most relevant set for the long-time behavior of the Bianchi system is the Mixmaster attractor $\\mathcal A$.\nIt consists of the union of all six Bianchi $\\textsc{II}$ pieces and the Kasner circle. Since the signs of the $N_i$ stay constant along trajectories, it is useful to only study a piece of $\\mathcal A$, given without loss of generality by:\n\n\\[\n\\mathcal A_{+++}=\\mathcal M_{+00}\\cup \\mathcal M_{0+0}\\cup \\mathcal M_{00+}\\cup \\mathcal K.\n\\]\nThe set $\\mathcal A_{+++}$ is given by the union of three perpendicular half-spheres (the three Kasner-caps), which intersect in the Kasner-circle. The dynamics in the Mixmaster-Attractor now consists of the Kasner circle $\\mathcal K$ of equilibria and the three caps, consisting of heteroclinic orbits to $\\mathcal K$. It is described in detail by the so-called Kasner-map.\n\n\n\\paragraph{The Kasner-map $K:\\mathcal K\\to\\mathcal K$.}\nWe wish to describe which equilibria in $\\mathcal K$ are connected by heteroclinic orbits. We can collect this in a relation $K\\subseteq \\mathcal K\\times \\mathcal K$, i.e.~we write $\\bpt p_- K\\bpt p_+$ if either $\\bpt p_-=\\bpt p_+=\\taubp_i$ or there exists a heteroclinic orbit $\\gamma:{\\mathbb{R}}\\to \\mathcal A$ such that $\\bpt p_- = \\lim_{t\\to-\\infty}\\gamma(t)$ and $\\bpt p_+=\\lim_{t\\to+\\infty}\\gamma(t)$.\n\nWe can see by Figure \\ref{fig:n-discs} (or Lemma \\ref{farfromA:lemma:kasnermap-stability}) that each non-Taub point $\\bpt p_-$ has a one-dimensional unstable manifold, i.e. one trajectory in $\\mathcal A_{{{\\hat{n}}}}$, which converges to $\\bpt p_-$ in backwards time. Therefore, the relation $K$ can be considered as a (single-valued, everywhere defined) map.\n\n\nThis map is depicted in Figure \\ref{fig:double-cover}, and has a simple geometric description in the $(\\Sigma_+,\\Sigma_-)$-projection: \nGiven some $\\bpt p_-\\in\\mathcal K$, we draw a straight line through $\\bpt p_-$ and the nearest of the three points $-2\\taubp_i$. This line has typically two intersections $\\bpt p_-$ and $\\bpt p_+\\in\\mathcal K$ with the Kasner-circle, one of which is nearer to $-2\\taubp_i$, which is $\\bpt p_-$, and one which is further away, which is $\\bpt p_+$. At a $\\taubp_i$, there are two possible choices of nearest $-2\\taubp_j$ and $-2\\taubp_k$ and the lines through these points are tangent to $\\mathcal K$; we just set $K(\\taubp_i)=\\taubp_i$. We see from Figure \\ref{fig:double-cover} that this map $K:\\mathcal K\\to\\mathcal K$ is continuous and a double cover (i.e.~each point $\\bpt p_+$ has two preimages $\\bpt p_{-}^1$ and $\\bpt p_-^2$, which both depend continuously on $\\bpt p_+$).\n\n\nLooking at Figure \\ref{fig:short-het}, we can also see that the Kasner-map is expanding. Hence it is $C^0$-conjugate to either $[z]_{{\\mathbb{Z}}}\\to [2z]_{{\\mathbb{Z}}}$ or $[z]_{{\\mathbb{Z}}}\\to [-2z]_{{\\mathbb{Z}}}$; since it has three fixed points the latter case must apply. Hence we have\n\n\\begin{proposition}\\label{prop:kasnermap-homeomorphism-class\nThere exists a homeomorphism $\\psi:\\mathcal K \\to {\\mathbb{R}}\/3{\\mathbb{Z}}$, such that $\\psi(\\taubp_i)=\\left[i\\right]_{3{\\mathbb{Z}}}$ and\n\\[\n\\psi(K(\\bpt p))=[-2 \\psi(\\bpt p)]_{3{\\mathbb{Z}}}\\qquad\\forall \\bpt p\\in\\mathcal K.\n\\]\n\\end{proposition}\nA formal proof of Proposition \\ref{prop:kasnermap-homeomorphism-class} is a digression; for this reason, it is deferred until Section \\ref{sect:appendix-kasner-map}, where we give a more detailed description of the Kasner map.\n\n\\paragraph{Basic Heuristics near the Mixmaster Attractor.}\nHeuristically, the Kasner-map determines the behavior of solutions near $\\mathcal A$: \nConsider an initial condition $\\bpt x_0\\in \\mathcal M_{\\pm\\pm\\pm}$ near $\\mathcal A$, i.e.~an initial condition where none of the $N_i$ vanish. Then the trajectory $\\bpt x(t)$ will closely follow the heteroclinic solution $\\gamma_1$ passing near $\\bpt x_0$. Let $\\bpt p_1$ be the end-point of this heteroclinic; $\\bpt x(t)$ will follow $\\gamma_1$ and stay for some time near $\\bpt p_1$ (since it is an equilibrium). However, if $\\bpt p_1\\neq \\taubp_i$, then one of the $N_i$ directions is unstable; therefore, $\\bpt x(t)$ will leave the neighborhood of $\\bpt p_1$ along the unique heteroclinic emanating from $\\bpt p_1$, and follow it until it is near $\\bpt p_2=K(\\bpt p_1)$. This should continue until $\\bpt x(t)$ leaves the vicinity of $\\mathcal A$, which should not happen at all (at least if the name ``Mixmaster Attractor'' is well deserved).\n\nThe expansion of the Kasner-map is the source of the (so far heuristic) chaoticity of the dynamics of the Bianchi system: The expansion along the Kasner-circle supplies the sensitive dependence on initial conditions, while the remaining two directions are contracting. Since we study a flow in four-dimensions, Lyapunov exponents need only account for three dimensions, the last one corresponding to the time-evolution.\n\n\n\n\n\\paragraph{Bianchi-Types $\\textsc{VII}_0$ and $\\textsc{VI}_0$.}\nThere are two Bianchi-types, where exactly one of the three $N_i$ vanishes: Types $\\textsc{VII}_0$ and $\\textsc{VI}_0$.\n In these Bianchi-Types, we have monotone functions (Lyapunov functions), which suffice to almost completely determine the long-time behaviour of trajectories. Without loss of generality, we focus on the case where $N_1=0$. Then we can write\n\\[\n\\Sigma_+' = (1-\\Sigma^2)(\\Sigma_++1)\\qquad 1= \\Sigma^2 + (N_2-N_3)^2.\n\\]\nWe can immediately see that $\\Sigma_+$ is non-decreasing along trajectories; indeed, we must have $\\lim_{t\\to \\pm \\infty}\\Sigma^2(t) = 1$ for all trajectories. Considering \\eqref{eq:ode2-ni} and Figure \\ref{fig:n-discs}, we can see that, for $t\\to +\\infty$ we must either have $\\Sigma_+ \\ge \\frac 1 2$, since both $N_2$ and $N_3$ must be stable, or $\\Sigma_+=-1$ all along; then $N_2=N_3$ and the trajectory lies in the Taub-line $\\mathcal {TL}_1$. In backward time, we must have $\\Sigma_+\\to -1$: All other points on the Kasner-circle have one of the two $N_2,N_3$ unstable. These statements can be formalized as\n\\begin{lemma}\\label{lemma:farfromA:b6}\nConsider an initial condition $\\bpt x_0$ with $\\bpt x_0\\in\\mathcal M_{0-+}$. Then, in forward time, the trajectory $\\bpt x(t)=\\phi(\\bpt x_0, t)$ converges to a point on the Kasner-circle $\\lim_{t\\to \\infty}\\bpt x(t)=\\bpt p_+\\in\\mathcal K$ with $\\Sigma_+(\\bpt p_+)\\ge \\frac 1 2$. In backwards time, the trajectory converges to the Taub-point $\\taubp_1=(-1,0,0,0,0)=\\lim_{t\\to -\\infty}\\bpt x(t)$.\n\\end{lemma}\n\\begin{lemma}\\label{lemma:farfromA:b7}\nConsider an initial condition $\\bpt x_0$ with $\\bpt x_0\\in\\mathcal M_{0++}\\setminus \\mathcal{TL}_1$. Then, in forward time, the trajectory $\\bpt x(t)=\\phi(\\bpt x_0, t)$ converges to a point on the Kasner-circle $\\lim_{t\\to \\infty}\\bpt x(t)=\\bpt p_+\\in\\mathcal K$ with $\\Sigma_+(\\bpt p_+)\\ge \\frac 1 2$. In backwards time, the $\\bpt\\Sigma$-projection of the trajectory converges to the Taub-point $\\taubp_1=(-1,0)=\\lim_{t\\to -\\infty}\\bpt\\Sigma(\\bpt x(t))$. No claim about the dynamics of the $N_i(t)$ for $t\\to-\\infty$ is made.\n\\end{lemma}\n\\begin{proof}\nContained in the preceding paragraph.\n\\end{proof}\nIt can be shown that, in the case of Bianchi $\\textsc{VII}_0$, i.e.~$\\bpt x_0\\in\\mathcal M_{0++}\\setminus \\mathcal{TL}_1$, the limit $\\lim_{t\\to -\\infty}\\bpt x(t)=\\bpt p\\in \\mathcal{TL}_1\\setminus\\{\\taubp_1\\}$ exists and does not lie on the Kasner circle. This claim follows directly from Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub}; however, the proof of Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub} is rather lengthy and not required for our main results.\n\n\n\\subsection{Bianchi-Types \\textsc{VIII} and \\textsc{IX} for large $N$\nAs we have seen, the lower Bianchi types do not support recurrent dynamics. This is different in the two top-dimensional Bianchi-types \\textsc{VIII} and \\textsc{IX}. This section is devoted to describing the behaviour far from $\\mathcal A$. \n\n\\begin{lemma}[Long-Time Existence]\nEvery solution $\\bpt x:[0,T)\\to \\mathcal M$ of $\\eqref{eq:ode}$ has bounded $\\Sigma^2(t)<C(\\bpt x_0)$  for all $t\\in[0,T)$ and has unbounded forward existence time (i.e.~no finite-time blow-up occurs towards the future, i.e.~towards the big bang singularity).\n\nThe product $|N_1N_2N_3|$ is non-increasing along solutions $\\bpt x(t)$, since (from \\eqref{eq:ode-unpacked}):\n\\[ \\frac{{\\mathrm{d}}}{{\\mathrm{d}} t} \\log|N_1N_2N_3| = -3\\Sigma^2 \\le 0. \\]\n\\end{lemma}\n\n\\begin{proof}\nThe monotonicity of $|N_1N_2N_3|$ is already proven in the statement.\n\nThe only way that long-term existence can fail is finite-time blow-up, i.e.~$\\lim_{t\\to T_{\\max}}|\\bpt x(t)|=\\infty$ for some $0<T_{\\max}<\\infty$. We cannot exclude this possibility a priori, since the vectorfield given by \\eqref{eq:ode} is polynomial. However, it suffices to estimate $|\\bpt x'(t)|\\le C + C|\\bpt x|$, with constants independent of of $t\\ge 0$ (but possibly depending on $\\bpt x_0$).\n\nWe first consider the case of Bianchi Type \\textsc{VIII}, without loss of generality $\\mathcal M_{-++}$.\nConsider a maximal solution $\\bpt x:[0,T_{\\max})\\to \\mathcal M_{-++}$. Since $N^2=(N_1-N_2+N_3)^2 -4 N_1N_3 >0$, we can see that $N^2>0$ and (from $1=\\Sigma^2+N^2$) that $\\Sigma^2<1$ for all times $t<T_{\\max}$. From this, we can easily estimate $|\\bpt x'|\\le C+C|\\bpt x|$; therefore, solutions exist for all positive times (i.e.~$T_{\\max}=\\infty$).\n \nNext, we consider the case of Bianchi Type \\textsc{IX}, without loss of generality $\\bpt x:[0,T_{\\max})\\to \\mathcal M_{+++}$. If $N_2\\ge \\max(N_1,N_3)$, then we can see\n\\[N^2=(N_1-N_2+N_3)^2 -4 N_1N_3 \\ge -4N_1N_3\\ge -4 \\left(N_1N_2N_3\\right)^{\\frac{2}{3}}\\left(\\frac{N_1N_3}{N_2N_2}\\right)^{\\frac 1 3}\\ge-4 \\left(N_1N_2N_3\\right)^{\\frac{2}{3}}. \\]\nBy permutation symmetry, the above inequality holds regardless of which $N_i$ is largest.\nTherefore, we have for all times $t>0$:\n\\[\nN^2(t) \\ge -4 \\left(N_1N_2N_3\\right)^{\\frac{2}{3}}(0),\\qquad \\Sigma^2(t) = 1-N^2(t) \\le 1+4 \\left(N_1N_2N_3\\right)^{\\frac{2}{3}}(0).\n\\]\nUnbounded time of existence follows as in the case of Bianchi \\textsc{VIII}.\n\\end{proof}\nNext, we show that $|N_1N_2N_3|\\to 0$ as $t\\to \\infty$, and that this convergence is essentially uniformly exponential:\n\\begin{lemma}[Essentially exponential convergence of $|N_1N_2N_3|$]\\label{farfrom-A:lemma:uniform}\nFor every $C_{N}>0$, there exist constants $C_1,C_0>0$ such that for all trajectories $\\bpt x:[t_1,t_2]\\to \\mathcal M$ with $|N_1N_2N_3|(t_1)< C_N$ we have\n\\[\n|N_1N_2N_3|(t_2) \\le |N_1N_2N_3|(t_1) \\exp\\left(C_0 - C_1(t_2-t_1)\\right).\n\\]\n\\end{lemma}\n\\begin{proof}\nConsider the function $s:\\mathcal M\\to [0,\\infty)$\n\\[s(\\bpt x) = \\int_0^1 \\Sigma^2(\\phi(\\bpt x, t)){\\mathrm{d}} t,\\]\nwhere $\\phi:\\mathcal M\\times {\\mathbb{R}}\\to \\mathcal M$ is the flow associated to \\eqref{eq:ode}.\nFix some $C_N>0$. We will show that we find some constant $\\widetilde C>0$ such that $s(\\bpt x)> \\widetilde C$ whenever $|N_1N_2N_3|(\\bpt x)\\le C_N$. From this, we can conclude\n\\[\n\\log\\frac{|N_1N_2N_3|(t_2)}{|N_1N_2N_3|(t_1)} = -3\\int_{t_1}^{t_2}\\Sigma^2(t){\\mathrm{d}} t \\le -3 (s(t_1)+ s(t_1+1) +\\ldots) \\le -3 (t_2-t_1-1)\\widetilde C.\n\\]\nWe will prove the estimate $s(\\bpt x)> \\widetilde C(C_N)$ by contradiction. Assume that we had a sequence $\\{\\bpt x_n\\}$, such that $s(\\bpt x_n)\\to 0$ and $|N_1N_2N_3|(\\bpt x_n)$ is bounded as $n\\to\\infty$.\n\nAt first, we show that this cannot happen in bounded regions of phase-space: We can see from \\eqref{eq:ode} that there do not exist any invariant sets with $\\Sigma^2=0$ (because then $N^2=1$ and $\\Sigma'\\neq 0$). Therefore $s(\\bpt x)>0$ for all $\\bpt x\\in\\mathcal M$. Since $s$ is continuous, the sequence $\\bpt x_n$ cannot converge, and hence cannot be bounded (otherwise, there would be some convergent subsequence).\n\nWe can assume without loss of generality that $\\Sigma^2(\\bpt x_n)\\to 0$ (since we assumed $\\int_0^1 \\Sigma^2(\\phi(\\bpt x_n, t)){\\mathrm{d}} t\\to 0$). In order to avoid convergent subsequences, we must have $\\max_{i}|N_i|(\\bpt x_n)\\to \\infty$ as $n\\to\\infty$.\n\n\nConsider first the case of Bianchi \\textsc{VIII} with $\\bpt x\\in\\mathcal M_{-++}$. Then \n\\[\nN^2 = (N_2-N_3)^2+N_1^2 -2 N_1 (N_2+N_3) = 1-\\Sigma^2.\n\\]\nAll three terms in the middle are non-negative and hence bounded; therefore, we must have $|N_2|,|N_3|\\to\\infty$ and $|N_1|\\to 0$. \nWe can write\n\\[{\\mathrm{D}}_t\\Sigma_+ = (1-\\Sigma^2)(\\Sigma_++1) + 3N_1(N_2+N_3-N_1)\\]\nWe can estimate the terms involving $N_i$ as \n\\[|N_1|(N_2+N_3+|N_1|) \\le \\frac{C_N}{\\max(N_2,N_3)}\\to 0\\qquad\\text{as $n\\to\\infty$}.\\]\nSince ${\\mathrm{D}}_t\\log{|N_i|} \\le |\\bpt \\Sigma|(2+|\\bpt \\Sigma|)$ is bounded, the estimate $N_2,N_3 \\gg 1 \\gg |N_1|$ and hence $|N_1|(N_2+N_3+|N_1|)\\ll 1$ stay valid for at least one unit of time. \nWe therefore cannot have $\\lim_{n\\to\\infty}s(\\bpt x_n)=0$: $s(\\bpt x_n)\\approx 0$ is only possible if $\\max_{t\\in[0,1]}\\Sigma^2(\\phi(\\bpt x_n, t))\\approx 0$. This is, however, impossible since then $\\Sigma_+'\\approx 1$.\n\nConsider now the case of Bianchi \\textsc{IX}, i.e.~$\\bpt x\\in\\mathcal M_{+++}$. Assume without loss of generality that $N_3\\ge N_2\\ge N_1$. Then we can write\n\\[\n1-\\Sigma^2=N^2 = (N_1+N_2-N_3)^2 -4 N_1N_2 \\ge  (N_1+N_2-N_3)^2 -4 C_N^{\\frac 2 3}.\n\\]\nTherefore, we must have $N_2,N_3\\to \\infty$ and $N_1\\to 0$. Apart from this, the same arguments as for Bianchi \\textsc{VIII} apply.\n\\end{proof}\nThis result, i.e.~Lemma \\ref{farfrom-A:lemma:uniform}, is not as explicitly stated in the previous works \\cite{ringstrom2001bianchi, heinzle2009new}, and certainly not as extensively used, but is not a novel insight either. It directly proves that metric coefficients stay bounded, see Section \\ref{sect:gr-phys-interpret}.\n\nUsing Lemma \\ref{farfrom-A:lemma:uniform}, we can quickly see the following:\n\\begin{lemma}[Existence of $\\omega$-limits]\\label{lemma:farfromtaub:omega-existence}\nFor any initial condition $\\bpt x_0 \\in \\mathcal M$, the $\\omega$-limit set is nonempty, $\\omega(\\bpt x_0)\\neq \\emptyset$, i.e.~there exists a sequence of times $(t_n)_{n\\in{\\mathbb{N}}}$ with $\\lim_{n\\to\\infty}t_n=\\infty$ such that the limit $\\lim_{n\\to\\infty}\\bpt x(t_n)$ exists.\n\\end{lemma}\n\\begin{proof}\nWe begin again by considering the Bianchi \\textsc{VIII} case of $\\bpt x_0\\in\\mathcal M_{-++}$. The only way of avoiding the existence of an $\\omega$-limit is to have $\\lim_{t\\to \\infty}|\\bpt x(t)|=\\infty$. As in the proof of Lemma \\ref{farfrom-A:lemma:uniform}, this is only possible via $N_2,N_3\\to \\infty$ and $N_1\\to 0$. Then\n\\[\\begin{aligned}\n\\Sigma_+'(t) &= (1-\\Sigma^2)(\\Sigma_++1) + 3N_1(N_2+N_3-N_1) \\\\\n& \\ge  (1-\\Sigma^2)(\\Sigma_++1) -9 |N_1N_2N_3|\\\\\n&\\ge (1-\\Sigma^2)(\\Sigma_++1) - C_1 e^{-C_2 t}\\\\\n\\frac{\\delta_1'}{\\delta_1}(t) &= -(\\Sigma^2 +\\Sigma_+),\n\\end{aligned}\\]\nwhere $\\delta_1=2\\sqrt{|N_2N_3|}$ as in \\eqref{eq:delta-r-def} and \\eqref{eq:ode2-delta}.\nThere are basically two possibilities with regards to the dynamics of $\\Sigma_+$: If $\\Sigma_+\\to -1$ as $t\\to+\\infty$, then this convergence must happen exponentially, since $(1-\\Sigma^2)(\\Sigma_++1)\\ge 0$. That is, if $\\Sigma_+\\to -1$, then we must have $|\\Sigma_++1|(t) \\le \\frac{C_1}{C_2}e^{-C_2 t}$ for all sufficiently large times $t>t_0>0$. Then \n\\[\n\\int_{t_0}^{\\infty}\\partial_t\\log \\delta_1(t){\\mathrm{d}} t = \\int_0^\\infty -\\Sigma_-^2 - \\Sigma_+(1+\\Sigma_+){\\mathrm{d}} t\\le 2\\frac{C_1}{C_2^2}e^{-C_2 t_0} < \\infty,\n\\]\nand hence $\\lim_{t\\to\\infty}\\delta_1(t)<\\infty$, contradicting our assumption. \n\nThe other option is to have $1+\\Sigma_+\\not\\to-1$ as $t\\to\\infty$. Then we must have\n$1+\\Sigma_+(t) > \\epsilon$ for some $\\epsilon>0$ for all sufficiently large times. Informally, we can see from Figure \\ref{fig:delta-discs} that this contradicts $\\delta_1\\to\\infty$. \nFormally, we can say: Since $\\Sigma_+$ is bounded, $\\int (1-\\Sigma^2)\\epsilon{\\mathrm{d}} t<\\int \\Sigma_+'{\\mathrm{d}} t + C < \\infty$, and hence $\\int(1-\\Sigma^2){\\mathrm{d}} t <\\infty$. On the other hand, ${\\mathrm{D}}_t\\log(\\delta_1) = -\\Sigma^2 -\\Sigma_+ \\le 1-\\Sigma^2 -\\epsilon$ and integration shows $\\lim_{t\\to\\infty}\\delta_1(t) = 0$, contradicting our assumption $N_2,N_3\\to\\infty$.\n\nNext, we consider the Bianchi \\textsc{IX} case of $\\bpt x_0\\in\\mathcal M_{+++}$. Again, the only way of avoiding the existence of an $\\omega$-limit is to have $\\lim_{t\\to \\infty}|\\bpt x(t)|=\\infty$. As in the proof of Lemma \\ref{farfrom-A:lemma:uniform}, this is only possible via $N_2,N_3\\to \\infty$ and $N_1\\to 0$ for some permutation of indices. Using $1-\\Sigma^2 \\ge |1-\\Sigma^2|-8|N_1N_2N_3|^{\\frac 2 3}$ and replacing $1-\\Sigma^2$ by $|1-\\Sigma^2|$, we can use the same arguments as in the Bianchi \\textsc{VIII} case.\n\\end{proof}\n\\subsection{The Bianchi \\textsc{IX} Attractor Theorem}\\label{sect:farfromA:attract}\nThe Mixmaster Attractor was named in the 60s. However, the first proof that $\\mathcal A$ actually is an attractor was given in \\cite[Theorem $19.2$, page 65]{ringstrom2001bianchi}, and simplified in \\cite{heinzle2009new}. We shall state this important result:\n\\begin{thm}[Classical Bianchi \\textsc{IX} Attractor Theorem]\\label{farfromA:thm:b9-attract}\nLet $\\bpt x_0\\in \\mathcal M_{+++}\\setminus \\mathcal T$. Then \\[\\lim_{t\\to\\infty} \\mathrm{dist}(\\bpt x(t), \\mathcal A)=0.\\] Also, the $\\omega$-limit set $\\omega(\\bpt x_0)$ does not consist of a single point.\n\\end{thm}\nThe proofs of Theorem \\ref{farfromA:thm:b9-attract} given in \\cite{ringstrom2001bianchi} and \\cite{heinzle2009new} require some subtle averaging arguments (summarized as Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub}), which are lengthy and fail to generalize to the case of Bianchi \\textsc{VIII} initial data. We will now give the first steps leading to the proof of Theorem \\ref{farfromA:thm:b9-attract}, up to the missing averaging estimates for Bianchi \\textsc{IX} solutions. Then, we will state the missing estimates and show how they prove Theorem \\ref{farfromA:thm:b9-attract}. Afterwards, we will give a high-level overview of how we replace Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub} in this work. Nevertheless, for the sake of completeness, we provide a proof of Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub} in Section \\ref{sect:neartaub:forbidden-cones}.\n\\paragraph{Rigorous steps leading to Theorem \\ref{farfromA:thm:b9-attract}.}\nWe first show that solutions cannot converge to the Taub-line $\\mathcal{TL}_i$, if they do not start in the Taub-space $\\mathcal T_i$:\n\\begin{lemma}[Taub Space Instability]\\label{lemma:farfromA:taub-instability}\nLet $\\epsilon>0$ small enough. Then there exists a constant $C_{r,\\epsilon}\\in(0,1)$, such that the following holds:\n\nRecall the definition of $r_1$, which measures the distance to $\\mathcal T_1$ (see \\eqref{eq:delta-r-def}). For any piece of trajectory $\\bpt x:[t_1,t_2]\\to \\{\\bpt x\\in\\mathcal M_{*++}: |\\bpt \\Sigma(\\bpt x)-\\taubp_1|\\le \\epsilon\\}$, the following estimate holds:\n\\[\\begin{aligned}\n r_1(\\gamma(t_2))&\\ge C_{r,\\epsilon}h(\\gamma(t_1)) r_1(\\gamma(t_1)),\\qquad \\text{ where}\\\\\n h(\\bpt x) &= |N_1| + |N_1|^2 + |N_1N_2N_3|.\n \\end{aligned}\\]\n\\end{lemma}\n\\begin{proof}\nThis Lemma uses the polar coordinates introduced in Section \\ref{sect:polar-coords}. We use \\eqref{eq:neartaub-b9-t} to see\n\\[\\begin{aligned}\n{\\mathrm{D}}_t \\log r_1&\\ge r_1^2\\sin^2\\psi \\frac{-\\Sigma_+}{1-\\Sigma_+} - C|N_1| - C |N_1^2| - C|N_1N_2N_3|,\\\\\n{\\mathrm{D}}_t \\log |N_1|&\\approx -3 < -1,\\\\\n{\\mathrm{D}}_t \\log|N_1N_2N_3|&\\approx -3 < -1.\n\\end{aligned}\\]\nThe desired estimate follows by integration.\n\\end{proof}\n\nTogether with Lemma \\ref{lemma:farfromtaub:omega-existence}, this allows us to see that there exist $\\omega$-limit points on $\\mathcal K$:\n\\begin{lemma}\\label{farfromA:lemma:limitpoint-on-A}\nLet $\\bpt x_0\\in\\mathcal M_{+++}\\setminus \\mathcal T$. Then there exists at least one $\\omega$-limit point \n$\\bpt p\\in \\left(\\mathcal K\\setminus\\{\\taubp_1, \\taubp_2,\\taubp_3\\}\\right)\\cap \\omega(\\bpt x_0)$.\n\nLet $\\bpt x_0\\in\\mathcal M_{+-+}\\setminus \\mathcal T_2$. Then there exists at least one $\\omega$-limit point \n$\\bpt p\\in \\left(\\mathcal K\\setminus\\{\\taubp_2\\}\\right)\\cap \\omega(\\bpt x_0)$.\n\\end{lemma}\n\\begin{proof}\nWe already know that there exists an $\\omega$-limit point $\\bpt y\\in\\omega(\\bpt x_0)$; this point must have $|N_1N_2N_3|=0$ and hence be of a lower Bianchi type. In view of Lemma \\ref{lemma:farfromA:b6} and Lemma \\ref{lemma:farfromA:b7} and the fact that both $\\alpha(\\bpt y)\\subseteq \\omega(\\bpt x_0)$ and $\\omega(\\bpt y)\\subseteq \\omega(\\bpt x_0)$, it suffices to exclude the case where $\\omega(\\bpt x_0)\\subseteq \\mathcal T\\mathcal L_i\\setminus \\mathcal K$ for some $i$. This possibility is excluded by Lemma \\ref{lemma:farfromA:taub-instability}.\n\\end{proof}\nTherefore, we know that $\\liminf_{t\\to\\infty}\\max_i\\delta_i(t)=0$ and hence $\\liminf_{t\\to\\infty}\\mathrm{dist}(\\bpt x(t), \\mathcal A)=0$ for initial conditions in $\\mathcal M_{\\pm\\pm\\pm}\\setminus \\mathcal T$. While we presently lack the necessary estimates to prove the missing part of the attractor theorem, $\\limsup_{t\\to\\infty}\\max_i\\delta_i(t)=0$, we can at least describe how this may fail: Each $\\delta_i$ can only grow by a meaningful factor in the vicinity of a Taub-point $\\taubp_i$:\n\\begin{lemma}\\label{lemma:farfromtaub:delta-increase-only-near-taub}\nThere exists a constant $C>0$ such that, given $\\epsilon_+>0$, we find $\\epsilon_{123}=\\epsilon_{123}(\\epsilon_+)>0$, the following holds:\n\nSuppose we have a piece of trajectory $\\bpt x:[t_1,t_2]\\to \\mathcal M_{\\pm\\pm\\pm}$, such that:\n\\begin{enumerate}\n\\item We have the product bound $|N_1N_2N_3|(t)<\\epsilon_{123}$ for all $t\\in[t_1,t_2]$\n\\item The first point of the partial trajectory is bounded away from the Taub-line $\\mathcal {TL}_1$, i.e.~$1+\\Sigma_+(t_1)>\\epsilon_+$\n\\item The piece of trajectory has comparatively large $\\delta_1$, in the sense $1\\ge\\delta_1^4(t)\\ge|N_1N_2N_3|(t)$ for all $t\\in[t_1,t_2]$, i.e.~$|N_1|\\le 4\\delta_1^2=16|N_2N_3|$ for all $t\\in[t_1,t_2]$.\n\\end{enumerate}\nThen $\\delta_1$ can only increase by a bounded factor along this piece of trajectory, i.e.\n\\[\\delta_1(t_2)\\le \\exp \\left(\\frac{C}{\\epsilon_+}\\right)\\delta_1(t_1).\\]\n\\end{lemma}\n\\begin{proof}\nRecall the proof of Lemma \\ref{lemma:farfromtaub:omega-existence}. From $\\delta_1^4\\ge|N_1N_2N_3|$, we know that $|N_1|\\le \\delta_1^2$ and therefore \n\\[\\begin{aligned}\n\\Sigma_+' &= (1-\\Sigma^2)(\\Sigma_++1) + 3N_1(N_2+N_3-N_1) \\\\\n& \\ge  |1-\\Sigma^2|(\\Sigma_++1) -C \\sqrt{|N_1N_2N_3|}.\n\\end{aligned}\\]\nIf $\\epsilon_{123}>0$ is small enough, this allows us to see that $1+\\Sigma_+(t)>\\frac 1 2 \\epsilon_+$ for all $t\\in [t_0,t_1]$. \nSince $|\\Sigma_+|\\le 2$ is bounded, we see that $\\int|1-\\Sigma^2|\\epsilon_+{\\mathrm{d}} t < C$  and hence $\\int |1-\\Sigma^2|{\\mathrm{d}} t < \\frac{C}{\\epsilon_+}$. On the other hand, ${\\mathrm{D}}_t \\log \\delta_1 < (1-\\Sigma^2) - \\frac 1 2\\epsilon_+$, which yields the claim upon integration.\n\\end{proof}\n\\paragraph{Sketch of classic proofs of Theorem \\ref{farfromA:thm:b9-attract}.}\nThe previous proofs of Theorem \\ref{farfromA:thm:b9-attract}, both in \\cite{ringstrom2001bianchi} and \\cite{heinzle2009new}, rely on the following estimate (Lemma $3.1$ in \\cite{heinzle2009new}, Section $15$ in \\cite{ringstrom2001bianchi}):\n\\begin{hlemma}{\\ref{lemma:farfromtaub:no-delta-increase-near-taub}}\\label{hlemma:no-delta-increase-near-taub}\nWe consider without loss of generality the neighborhood of $\\mathcal T_1$. Let $\\epsilon>0$ small enough. Then there exists a constant $C_{\\delta,\\epsilon}\\in(1,\\infty)$, such that, for any piece of trajectory $\\gamma:[t_1,t_2]\\to \\{\\bpt x\\in\\mathcal M_{*++}: |\\bpt \\Sigma(\\bpt x)-\\taubp_1|\\le \\epsilon,\\,|N_1|\\le 10, |\\delta_1|\\le 10\\}$, the following estimate holds:\n\\[ \\delta_1(\\gamma(t_2))\\le C_{\\delta,\\epsilon}\\delta_1(\\gamma(t_1)). \\]\n\\end{hlemma}\nThe proof of this Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub} requires some lengthy averaging arguments and will be deferred until Section \\ref{sect:averaging-unneeded}, page \\pageref{lemma:farfromtaub:no-delta-increase-near-taub}. We stress that Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub} is not actually needed for any of the results in this work, and is proven only for the sake of completeness of the literature review.\n\\begin{remark}\nThe above formulation of Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub} includes the Bianchi \\textsc{VIII} case of $N_2,N_3>0>N_1$. The versions stated in \\cite{heinzle2009new} and \\cite{ringstrom2001bianchi} only consider the Bianchi \\textsc{IX} case $N_1,N_2,N_3>0$; however, their proofs extend to this case virtually unchanged.\n\\end{remark}\n\\begin{proof}[Proof of Theorem \\ref{farfromA:thm:b9-attract} using Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub}]\nWe begin by showing $\\lim_{t\\to\\infty}\\delta_1(t)=0$.\nBy Lemma \\ref{farfromA:lemma:limitpoint-on-A}, we have $\\liminf_{t\\to\\infty}\\delta_1(t)=0$. Suppose $\\limsup_{t\\to\\infty}\\delta_1(t)>0$.\nThen $\\delta_1$ must increase from arbitrarily small values up to some finite nonzero infinitely often, and hence we find arbitrarily late subintervals $[T^L,T^R]\\subseteq [0,\\infty)$ such that $\\delta_1^4>|N_1N_2N_3|$ and $\\delta_1$ increases by an arbitrarily large factor. This contradicts Lemma \\ref{lemma:farfromtaub:delta-increase-only-near-taub} and Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub}, which basically say that large increases of $\\delta_1$ can neither happen with $1+\\Sigma_+>\\epsilon_+$ nor with $1+\\Sigma_+<\\epsilon_+$.\n\nThe same applies for $\\delta_2$ and $\\delta_3$. \n\nSuppose there was only a single $\\omega$-limit point. This point cannot lie in $\\mathcal K\\setminus \\{\\taubp_i\\}$, since at each of these points, at least one of the $N_i$ is unstable. The only remaining possibility is $\\omega(\\bpt x_0)=\\taubp_i$ for some $i\\in\\{1,2,3\\}$, which is excluded by Lemma \\ref{lemma:farfromA:taub-instability}.\n\\end{proof}\n\n\\begin{remark}\nThe above proof also shows that $N_2N_3\\to 0$ in the Bianchi \\textsc{VIII} case $\\bpt x_0\\in\\mathcal M_{-++}\\setminus \\mathcal T_1$. This generalization is directly possible while keeping \\cite{heinzle2009new} virtually unchanged, even though it has not been explicitly noted therein.\n\n\nThe above proof also shows that in Bianchi $\\textsc{VII}_0$, i.e.~for any $\\bpt x_0\\in\\mathcal M_{0++}$,\n we must have $\\lim_{t\\to-\\infty}\\bpt x(t)=\\bpt p_-$ with $\\bpt p_-=(-1,0,0,N,N)$ for some $N>0$. This is false in the case of Bianchi $\\textsc{VI}_0$: There we have for any $\\bpt x_0\\in\\mathcal M_{0+-}$ that $\\lim_{t\\to-\\infty}\\bpt x(t)=(-1,0,0,0,0)$. Hence, $\\delta_1$ can grow by an arbitrarily large factor near $\\taubp_1$ in $\\mathcal M_{*+-}$, and\nno analogue of Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub} can hold in the Bianchi \\textsc{VIII} models $\\mathcal M_{*+-}$ and $\\mathcal M_{*-+}$.\n\nThis difficulty is partially responsible for the fact that, for $\\bpt x_0\\in\\mathcal M_{+-+}$, it was previously unknown whether $\\lim_{t\\to\\infty}N_2N_3(t)\\overset{?}{=}0$ and $\\lim_{t\\to\\infty}N_1N_3(t)\\overset{?}{=}0$. \n\\end{remark}\n\\paragraph{Sketch of our replacement for Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub}.}\\label{paragraph:farfromA:sketch}\nIn this work, we will replace the rather subtle averaging estimates from Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub} by the program outlined in this paragraph. Let us first repeat the reasons, why we want to avoid Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub}:\n\\begin{enumerate}\n\\item The analogue statement of Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub} in Bianchi \\textsc{VIII} is wrong. Lemma \\ref{lemma:farfromA:b6} shows that counterexamples to such a generalization can be found by taking any sequence $\\{\\bpt x_n\\}$ of initial data converging to any point in $\\mathcal M_{0-+}$. Therefore, any argument relying on Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub} has no chance of carrying over to the Bianchi \\textsc{VIII} case.\n\\item The proof of Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub} is lengthy and requires subtle averaging arguments.\n\\item The complexity of the proof of Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub} in not unavoidable: Most of the effort is spent trying to understand asymptotic regimes \\emph{that do not occur anyway}.\n\\end{enumerate}\nOur replacement is described by the following program:\n\\begin{enumerate}\n\\item At first, we study pieces of trajectories $\\bpt x:[0,T]\\to\\mathcal M_{\\pm\\pm\\pm}$, which start near $\\mathcal A$ and stay bounded away from the generalized Taub-spaces $\\mathcal T^G_i$, i.e.~have all $r_i>\\epsilon$. Along such partial solutions, all $\\delta_i$ decay essentially exponentially (Proposition \\ref{prop:farfromtaub-main}).\n\\item Next, consider how solutions near $\\mathcal A$ can enter the neighborhood of the generalized Taub-spaces. This can only happen near some $-\\taubp_i$ (Proposition \\ref{prop:farfromtaub-main2}).\n\\item For such solutions entering the vicinity of $-\\taubp_i$, the quotient $\\frac{\\delta_i}{r_i}$ is initially small, and stays small near $-\\taubp_i$ and along the heteroclinic leading to $+\\taubp_i$ (Proposition \\ref{prop:near-taub:qc}).\n\\item Next, we study solutions near $\\taubp_i$ for which $\\frac{\\delta_i}{r_i}$ is initially small. Then, $\\frac{\\delta_i}{r_i}$ stays small. This additional condition ($\\delta_i \\ll r_i$) allows us to describe solutions with easier averaging arguments and stronger conclusions than Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub}. Bianchi \\textsc{VIII} solutions can be analyzed same way. This is done in Section \\ref{sect:neartaub:neartaub}, leading to the conclusion that $\\delta_i$ decays essentially exponentially, with nonuniform rate (Proposition \\ref{prop:neartaub:main}).\n\\item Finally, we combine the previous steps in Section \\ref{sect:global-attract} in order to prove Theorems \\ref{thm:local-attractor}, \\ref{thm:b9-attractor-global} and \\ref{thm:b8-attractor-global}. These extend Theorem \\ref{farfromA:thm:b9-attract} with somewhat finer control over solutions and provide an analogue in Bianchi \\textsc{VIII}.\n\\end{enumerate}\n\n\\section{Attractor Theorems}\\label{sect:global-attract}\nThe goal of this section is to prove that typical initial conditions converge to $\\mathcal A$. We have already seen Theorem \\ref{farfromA:thm:b9-attract}, which is however somewhat unsatisfactory: It tells nothing about the speed and the details of the convergence; it relies on Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub}, which has a rather lengthy proof (page \\pageref{lemma:farfromtaub:no-delta-increase-near-taub}f) mainly discussing the case $\\delta_1 \\gg r_1$, \\emph{which is not supposed to happen anyway}; lastly, the proof of Theorem \\ref{farfromA:thm:b9-attract} has no chance of generalizing to the case of Bianchi \\textsc{VIII}.\n\nIn this section, we will combine the analysis of the previous Sections \\ref{sect:far-from-taub} and \\ref{sect:near-taub} in order to prove a \\emph{local} attractor result, holding both in Bianchi \\textsc{VIII} and \\textsc{IX}. Together with the results from Section \\ref{sect:farfromA} and some minor calculation, this will yield a ``global attractor theorem'', i.e.~a classification of solutions failing to converge to $\\mathcal A$, which happen to be rare; in the case of Bianchi \\textsc{IX}, this recovers and extends Theorem \\ref{farfromA:thm:b9-attract}, and in the case of Bianchi \\textsc{VIII}, this answers a longstanding conjecture.\n\\paragraph{Statement of the attractor Theorems.}\nThe local attractor theorem is given by the following:\n\\begin{thm}[Local Attractor Theorem]\\label{thm:local-attractor}\nThere exist constants $C_1,C_2,C_3>0$ and $\\epsilon>0$ such that the following holds:\n\nLet $\\bpt x_0\\in \\mathcal M_{\\pm\\pm\\pm}$ be an initial condition in either Bianchi \\textsc{VIII} or \\textsc{IX}, with \n\\begin{equation}\\label{eq:local-attractor-condition}\n\\frac{\\delta_i}{r_i} <\\epsilon\\quad\\text{and}\\quad \\delta_i <\\epsilon\\qquad\\forall i\\in\\{1,2,3\\}.\n\\end{equation}\n\nThen, for all $i\\in \\{1,2,3\\}$ and $t_2\\ge t_1\\ge 0$:\n\\begin{subequations}\n\\begin{align}\n\\delta_i(t_2)&\\le C_1 {\\delta}(t_1)\\label{eq:local-attract:delta}\\\\\n\\frac{\\delta_i}{r_i}(t_2)&\\le C_2\\frac{\\delta_i}{r_i}(t_1) \\label{eq:local-attract:delta-r}.\n\\end{align}\nFurthermore, for all $i\\in \\{1,2,3\\}$,\n\\begin{align}\n\\lim_{t\\to \\infty}\\delta_i(t)&=0 \\label{eq:local-attract:delta-lim}\\\\\n\\lim_{t\\to\\infty} \\frac{\\delta_i}{r_i}(t)&=0\\label{eq:local-attract:delta-r-lim}\\\\\n\\int_0^{\\infty}\\delta_i^2(t){\\mathrm{d}} t &< C_3 \\frac{\\delta_i^2}{r_i^2}(\\bpt x_0),\\label{eq:local-attract:integral}\n\\end{align}\nand the $\\omega$-limit set $\\omega(\\bpt x_0)$ must contain at least three points in $\\mathcal K$.\n\\end{subequations}\n\\end{thm}\nThe name ``local attractor theorem'' is descriptive: We describe a subset of the basin of attraction, i.e.~an open neighborhood of $\\mathcal A\\setminus\\mathcal T$ which is attracted to $\\mathcal A$ and given by\n\\begin{equation}\n\\textsc{Basin}_{{\\hat{n}}}[\\epsilon] = \\{\\bpt x\\in \\mathcal M_{{\\hat{n}}}:\\, \\delta_i\\le\\epsilon,\\,\\frac{\\delta_i}{r_i}\\le \\epsilon\\quad\\forall i\\in\\{1,2,3\\}\\}.\n\\end{equation}\nThe integral estimate \\eqref{eq:local-attract:integral} tells us that the convergence to $\\mathcal A$ must be reasonably fast.\n\nWe can combine the local attractor Theorem \\ref{thm:local-attractor} with the discussion in Section \\ref{sect:farfromA} in order to prove a global attractor theorem. Since some trajectories fail to converge to $\\mathcal A$, most notably trajectories in the Taub-spaces, a global attractor theorem must necessarily take the form of a classification of all exceptions. In this view, the global result for the case of Bianchi Type \\textsc{IX} models is the following:\n\\begin{thm}[Bianchi \\textsc{IX} global attractor Theorem]\\label{thm:b9-attractor-global}\nConsider $\\mathcal M_{+++}$, i.e.~Bianchi \\textsc{IX}. Then, for any initial condition $\\bpt x_0\\in \\mathcal M_{+++}$, the long-time behaviour of $\\bpt x(t)$ falls into exactly one of the following mutually exclusive classes ($i\\in\\{1,2,3\\}$):\n\\begin{description}\n\\item[\\textsc{Attract.}] For large enough times, Theorem \\ref{thm:local-attractor} applies.\n\\item[$\\textsc{Taub}_i$.] We have $\\bpt x_0\\in \\mathcal T_i$, and hence $\\bpt x(t)\\in\\mathcal T_i$ for all times.\n\\end{description}\nThe set of initial conditions for which \\textsc{Attract} applies is ``generic'' in the following sense: It is open and dense in $\\mathcal M_{+++}$ and its complement has Lebesgue-measure zero (evident from the fact that $\\mathcal T_i$ are embedded lower dimensional submanifolds, of both dimension and codimension two).\n\\end{thm}\nThe analogous, novel result for the case of Bianchi \\textsc{VIII} is the following:\n\\begin{thm}[Bianchi \\textsc{VIII} global attractor Theorem]\\label{thm:b8-attractor-global}\nConsider $\\mathcal M_{+-+}$, i.e.~Bianchi \\textsc{VIII}. Then, for any initial condition $\\bpt x_0\\in \\mathcal M_{+-+}$, the long-time behaviour of $\\bpt x(t)$ falls into exactly one of the following mutually exclusive classes:\n\\begin{description}\n\\item[\\textsc{Attract.}] For large enough times, Theorem \\ref{thm:local-attractor} applies. \n\\item[$\\textsc{Taub}_2$.] We have $\\bpt x_0\\in \\mathcal T_2$, and hence $\\bpt x(t)\\in\\mathcal T_2$ for all times. \n\\item[$\\textsc{Except}_1$.] For large enough times, the trajectory follows the heteroclinic object\n\\[ -\\taubp_1 \\to W^u(-\\taubp_1) \\to \\taubp_1 \\to W^s(-\\taubp_1)\\to -\\taubp_1,\\]\nwhere $W^s(-\\taubp_1)$ and $W^u(-\\taubp_1)$ are the two-dimensional stable and one-dimensional unstable manifolds of $-\\taubp_1$. We have $\\lim_{t\\to\\infty}\\max(\\delta_2,\\delta_3)(t)=0$ and $\\limsup_{t\\to\\infty}\\delta_1(t)>0=\\liminf_{t\\to\\infty}\\delta_1(t)$.\n\\item[$\\textsc{Except}_3$.] The analogue of $\\textsc{Except}_1$ applies, with the indices $1$ and $3$ exchanged.\n\\end{description}\n\\end{thm}\nThis theorem should be read in conjunction with the following, the proof of which will be deferred until Section \\ref{sect:volume-form}, page \\pageref{proof:thm:b8-attractor-global-genericity}:\n\\begin{hthm}{\\ref{thm:b8-attractor-global-genericity}}[Bianchi \\textsc{VIII} global attractor Theorem genericity]\nIn Theorem \\ref{thm:b8-attractor-global}, \nthe set of initial conditions $\\bpt x_0$ for which \\textsc{Attract} applies is ``generic'' in the following sense: It is open and dense in $\\mathcal M_{+-+}$ and its complement has Lebesgue-measure zero. \n\\end{hthm}\n\\begin{question}\\label{quest:except}\nIt is currently unknown, whether the case \\textsc{Except} in Bianchi \\textsc{VIII} is possible at all.\n\nWe expect that solutions in $\\textsc{Except}_1$ actually converge to the heteroclinic cycle, where $\\taubp_1 \\to -\\taubp_1$ is realized by the unique connection in $\\mathcal T_1^G$, i.e.~$\\lim_{t\\to\\infty}r_1(t)=0$, instead of following any other heteroclinic in $W^u(\\taubp_1)\\cap W^s(-\\taubp_1)$.\n\nFor topological reasons, we expect that the set of initial conditions, where \\textsc{Except} applies, is nonempty, and is of dimension and codimension two.\n\\end{question}\n\\paragraph{Proof of the attractor Theorems.}\nThe remainder of this section is devoted to proving these theorems. We begin with the local attractor result:\n\\begin{proof}[Proof of the local attractor Theorem \\ref{thm:local-attractor}]\nWe already gave a rough outline of the proof at the end of Section \\ref{sect:farfromA:attract}, page \\pageref{paragraph:farfromA:sketch}. We now have all ingredients to complete this program.\n\nWe apply Propositions \\ref{prop:farfromtaub-main2}, \\ref{prop:near-taub:qc} and \\ref{prop:neartaub:main}. If the constants have been arranged appropriately, then each of these propositions describes a piece of the trajectory, which ends in the domain of the (cyclically) next proposition, and at least one of the three propositions is applicable.\n``Appropriate'' means at least $\\varepsilon_T^{\\ref{prop:neartaub:main}}> C_T^{\\ref{prop:farfromtaub-main2}}\\varepsilon_T^{\\ref{prop:farfromtaub-main2}}$.\n\nEach $\\delta_i$ shrinks by an arbitrarily large factor during each time interval, where Proposition \\ref{prop:farfromtaub-main2} applies, and can at most grow by a bounded factor in each remaining time-interval, which directly yields \\eqref{eq:local-attract:delta}, \\eqref{eq:local-attract:delta-lim} (``arbitrary'' if we adjust $\\epsilon$ in \\eqref{eq:local-attractor-condition}). \n\nIn order to see the estimates \\eqref{eq:local-attract:delta-r} and \\eqref{eq:local-attract:delta-r-lim}, we note that each $r_i$ can change only by a factor bounded by $\\varepsilon_T^{-1}$ in the region, where \\ref{prop:farfromtaub-main2} applies; however, the $\\delta_i$ must shrink by an arbitrarily large factor, and the quotient $\\frac{\\delta_i}{r_i}$ is directly controlled in the regions where Propositions \\ref{prop:near-taub:qc} and \\ref{prop:neartaub:main} apply.\n\nThe remaining estimate \\eqref{eq:local-attract:integral} follows by integration: The contribution away from the Taub-spaces can be trivially bounded by Proposition \\ref{prop:farfromtaub-main}, eq.~\\eqref{eq:nearA:exponential}, and Proposition \\ref{prop:neartaub:main}, eq.~\\eqref{eq:neartaub:main:delta}, bounds the contribution near the Taub-spaces by $\\frac{\\delta^2}{r^2}$. The long-time behaviour of $\\frac{\\delta_i}{r_i}$ is controlled by Proposition \\ref{prop:near-taub:qc}. \n\\end{proof}\nNext, we prove the global result for Bianchi \\textsc{IX}. Theorem \\ref{thm:b9-attractor-global} is trivially equivalent to Theorem \\ref{farfromA:thm:b9-attract}, so we could cite \\cite{ringstrom2001bianchi} instead; the novel results are contained in the local Theorem \\ref{thm:local-attractor}. The following is a novel proof of the global part of the result, which avoids Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub}:\n\\begin{proof}[Proof of the global Bianchi \\textsc{IX} attractor Theorem \\ref{thm:b9-attractor-global}]\nSuppose that there is an initial condition $\\bpt x_0\\in\\mathcal M_{+++}$, such that neither \\textsc{Attract} nor \\textsc{Taub} applies. We know that $\\omega(\\bpt x_0)\\cap \\mathcal A \\subseteq \\mathcal T$ (because otherwise \\textsc{Attract} would apply). By Lemma \\ref{farfromA:lemma:limitpoint-on-A}, there must exist at least one $\\omega$-limit point $\\bpt y\\in\\omega(\\bpt x_0)\\cap \\mathcal K$. The $\\omega$-limit set $\\omega(\\bpt x_0)$ must be connected or unbounded (this holds for general dynamical systems).\n\nNow consider $\\omega(\\bpt x_0)\\cap \\mathcal T$. We claim that $\\omega(\\bpt x_0)\\cap \\mathcal T\\subseteq \\mathcal T_i$ for some $i\\in\\{1,2,3\\}$: If it was otherwise, there would need to exist a heteroclinic orbit $\\gamma\\subseteq \\omega(\\bpt x_0)$ which connects two Taub-spaces (since $|N_1N_2N_3|\\to 0$ we can apply Lemma \\ref{lemma:farfromA:b7}). Without loss of generality, such a heteroclinic orbit can only connect $\\{\\bpt x: \\bpt \\Sigma=\\taubp_1\\}$ to $-\\taubp_2$ with $\\gamma\\subseteq \\mathcal M_{0++}$. Along $\\gamma$, $\\delta_2$ and $\\delta_3$ and hence also $\\frac{\\delta_2}{r_2}$ and $\\frac{\\delta_3}{r_3}$ vanish; near the end of this heteroclinic, $\\delta_1$ and $\\frac{\\delta_1}{r_1}$ must become arbitrarily small. In other words, we have some $\\bpt p\\in\\gamma$ near $-\\taubp_2$ such that $\\frac{\\delta_1}{r_1}(\\bpt p)<\\widetilde \\epsilon\/3$.\nHence, for $\\epsilon_2=\\epsilon_2(\\bpt p)>0$ small enough, $B_{\\epsilon}(\\bpt p)\\cap \\mathcal M_{+++}\\subseteq\\textsc{Basin}_{+++}[\\widetilde \\epsilon]$; this contradicts our assumptions $\\bpt p\\in\\omega(\\bpt x_0)$ and $\\omega(\\bpt x_0)\\cap \\textsc{Basin}_{+++}[\\widetilde \\epsilon]=\\emptyset$.\n\nWe also know by Lemma \\ref{lemma:farfromA:taub-instability} that $\\omega(\\bpt x_0)$ cannot be contained in a Taub-Line $\\mathcal{TL}_i=\\{\\bpt x: \\bpt \\Sigma(\\bpt x)=\\taubp_i\\}$. Therefore, the only remaining case is that the trajectory $\\phi(\\bpt x_0, \\cdot)$ follows the heteroclinic cycle\n\\[\n-\\taubp_i \\to W^u(-\\taubp_i) \\to \\taubp_i \\to W^s(-\\taubp_i)\\to -\\taubp_i,\n\\]\nwhere $W^u(-\\taubp_i)$ is the one-dimensional unstable manifold of $-\\taubp_i$ and $W^s(-\\taubp_i)$ is its two-dimensional stable manifold.\n\nWe now could use Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub} to finish our proof, since it prevents $\\taubp_i\\to W^s(-\\taubp_i)$.\nHowever, we prefer to avoid relying on the subtle averaging arguments needed to prove Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub}, substituting them by the simpler Lemma \\ref{neartaub:lemma:joint-increase-in-forbidden}.\nAssume without loss of generality that our trajectory follows this dynamics with $i=1$ and hence has $\\frac{\\delta_1}{r_1}\\ge \\widetilde \\epsilon$ for all sufficiently large times, while $\\frac{\\delta_2}{r_2}\\to 0$ and $\\frac{\\delta_3}{r_3}\\to 0$. We will follow $\\frac{\\delta_1}{r_1}$ and show that $\\frac{\\delta_1}{r_1}\\to 0$, which contradicts our assumptions. \n\nWe first estimate the variation of $\\frac{\\delta_1}{r_1}$ away from $\\taubp_1$. Using\n\\[\n{\\mathrm{D}}_t\\log \\frac{\\delta_1}{r} \\le C|N_1N_2N_3|^{\\frac 2 3} + C |N_1|,\n\\]\nwe can conclude that $\\int \\max(0,{\\mathrm{D}}_t\\log\\frac{\\delta_1}{r_1}){\\mathrm{d}} t$ is bounded along every loop. However, by Lemma \\ref{neartaub:lemma:joint-increase-in-forbidden}, $\\frac{\\delta_1}{r_1}$ must decrease by an arbitrarily large factor near $\\taubp_1$ along every loop, contradicting the assumptions$\\frac{\\delta_1}{r_1}>\\widetilde \\epsilon$. This is because Lemma \\ref{neartaub:lemma:joint-increase-in-forbidden} states that $\\frac{\\delta_1}{r_1}$ must decrease by a comparable factor as the increase of $r_1$, which  must increase by an arbitrarily large factor near $\\taubp_1$, since we know that $r_1$ must eventually become larger than, say, $0.1$.\n\\end{proof}\n\n\\begin{proof}[Proof of the global Bianchi \\textsc{VIII} attractor Theorem \\ref{thm:b8-attractor-global}]\nThe proof works like the above proof of Theorem \\ref{thm:b9-attractor-global}: We use Lemma \\ref{lemma:no-b8-taub-convergence} in order to prevent convergence to $\\taubp_1$ and $\\taubp_3$ (instead of Lemma \\ref{lemma:farfromA:taub-instability}, which is still used near $\\mathcal {TL}_2$). \n\nPreventing convergence to the heteroclinic cycle near $\\mathcal T_2$ works as before. This leaves us with the possibility of convergence to one of the two remaining heteroclinic loops described in \\textsc{Except}. Since we cannot exclude this case, we allow it in the conclusions of the theorem.\n\\end{proof}\n\n\n\n\n\\section{Dynamics near the Mixmaster-Attractor $\\mathcal A$}\\label{sect:near-A}\\label{sect:far-from-taub}\nOur previous arguments in Section \\ref{sect:farfromA} about the dynamics of \\eqref{eq:ode} have been of a rather qualitative and global character. We have established that there exist $\\omega$-limit points on the Mixmaster-attractor $\\mathcal A$.\n\nWe have also sketched the classical proof that trajectories converge to $\\mathcal A$ in the case of Bianchi Type \\textsc{IX} (Theorem \\ref{farfromA:thm:b9-attract}) (where we deferred the proof of the crucial estimate Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub} to a later point). \n\nIn this section, we will give a more precise description of the behaviour near $\\mathcal A$. The goal of this section is to show that pieces of trajectories $\\gamma:[0,T]\\to \\mathcal M$ near $\\mathcal A$ converge to $\\mathcal A$ essentially exponentially, at least as long as they stay bounded away from the Taub-points $\\taubp_i$. \n\n\n\n\\begin{figure}[hbpt]\n \\centering\n        \\begin{subfigure}[b]{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=\\textwidth]{.\/graphics\/classa-circles-ni.pdf}\n                \\caption{The stability defining discs for the $N_i$}\\label{fig:n-discs2}\n        \\end{subfigure}~~%\n        \\begin{subfigure}[b]{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=\\textwidth]{.\/graphics\/classa-circles-delta.pdf}\n                \\caption{The stability defining discs for the $\\delta_i$}\\label{fig:delta-discs2}\n        \\end{subfigure}\\\\%        \n        %\n        \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{.\/graphics\/nuclear-evil.pdf}\n        \\caption{A trajectory leaving the controlled regions}\\label{fig:evil-traj}\\label{fig:nuclear-evil}\n        \\end{subfigure}~~%\n        %\n        %\n        \\begin{subfigure}[b]{0.4\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{.\/graphics\/nuclear-cap.pdf}\n        \\caption{The region \\textsc{Cap}}\\label{fig:nuclear-cap}\n        \\end{subfigure}\\\\%        \n        %\n        %\n        \\begin{subfigure}[b]{0.4\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{.\/graphics\/nuclear-circle.pdf}\n        \\caption{The region \\textsc{Circle}}\\label{fig:nuclear-circle}        \n        \\end{subfigure}~\n        \\begin{subfigure}[b]{0.4\\textwidth}\n                \\centering\n                \\includegraphics[width=\\textwidth]{.\/graphics\/nuclear-hyp.pdf}\n                \\caption{The region \\textsc{Hyp}}\n                \\label{fig:nuclear-hyp}\n        \\end{subfigure\n        \\caption{The relevant regions are colored in gray and are not up to scale (except for Fig.\\ref{fig:n-discs} and Fig.\\ref{fig:delta-discs}). See also Fig.\\ref{fig:kasner-segments}.}\\label{fig:nuclear}        \n\\end{figure}\n\n\nThe goal of this section is to prove the following two propositions \\ref{prop:farfromtaub-main} and \\ref{prop:farfromtaub-main2}:\n\\begin{proposition}[Essentially uniform exponential convergence to $\\mathcal A$ away from the Taub points]\\label{prop:farfromtaub-main}\nFor any $\\varepsilon_T>0$ small enough, there exist constants $\\epsilon_d, \\epsilon_N, \\epsilon_s, C_0>0$ (depending on $\\varepsilon_T$) and $C_T=5$ such that the following holds:\n\n\\begin{subequations}\nConsider a trajectory $\\bpt x:[0,T^*)\\to\\mathcal M_{\\pm\\pm\\pm}$, such that, for all $t\\in [0,T^*)$ the following inequalities hold:\n\\begin{align}\n\\max_i\\delta_i(t)&< \\epsilon_d\\label{eq:nearA:capdist}\\\\\n\\min_i d(\\bpt x(t), \\taubp_i)&> \\varepsilon_T\\label{eq:nearA:taubdist}.\n\\end{align}\nAssume further that for the initial condition $\\bpt x_0=\\bpt x(0)$, the following stronger estimate holds:\n\\begin{equation}\n\\max_i\\delta_i(\\bpt x_0)< C_0^{-1}\\epsilon_d \n\\end{equation}\n\\end{subequations}\\begin{subequations}\nThen, each $\\delta_i$ is essentially uniformly exponentially decreasing in $[0,T^*)$, i.e.\n\\begin{equation}\n\\delta_i(t_2) \\le C_0 e^{-\\epsilon_s(t_2-t_1)}\\delta_i(t_1)\\qquad\\forall\\,0\\le t_1\\le t_2<T^*,\\quad i\\in\\{1,2,3\\}.\\label{eq:nearA:exponential}\n\\end{equation}\nHence, if $T^*<\\infty$ and one of the inequalities \\eqref{eq:nearA:taubdist}, \\eqref{eq:nearA:capdist} is violated at time $T^*$, it must be \\eqref{eq:nearA:taubdist}, and \\eqref{eq:nearA:capdist} must still hold at $T^*$.\n\\end{subequations}\n\\end{proposition}\nInformally, this proposition states that trajectories near $\\mathcal A$ converge exponentially to $\\mathcal A$, as long as they stay bounded away from the Taub points.\n\\begin{proposition}\\label{prop:farfromtaub-main2}\nAssume the setting of Proposition \\ref{prop:farfromtaub-main}. There are constants $\\epsilon_N=\\epsilon_N(\\varepsilon_T)>0$ and $C_T=5$ such that additionally the following holds:\n\n\\begin{subequations}\nAssume that $T^*<\\infty$, and that initially \n\\begin{equation}\\max_i d(\\bpt x_0, \\mathcal T_i) > C_T \\varepsilon_T.\\label{eq:nearA:init-far-from-taub}\\end{equation}\n\\end{subequations}\\begin{subequations}\n Then the final part of the trajectory preceding $T^*$ must have the form depicted in Figure \\ref{fig:evil-traj}, i.e.~there is $\\ell\\in\\{1,2,3\\}$ and there are times $0<T_1<T_2<T_3 \\le T^*$ (typically $T_3=T^*$) such that\n\\begin{align}\nd(\\bpt x(t), -\\taubp_\\ell)&\\le C_T \\varepsilon_T &\\forall t\\in [T_1,T_2]\\label{eq:farfromtaub-thm-12}\\\\\n\\max_i |N_i(t)|&\\ge \\epsilon_N &\\forall t\\in [T_2,T_3]\\label{eq:farfromtaub-thm-23}\\\\\nd(\\bpt x(t), \\taubp_\\ell)&\\le C_T\\varepsilon_T &\\forall t\\in [T_3, T^*]\\label{eq:farfromtaub-thm-3star}.\n\\end{align}\n\\end{subequations}\n\\end{proposition}\nInformally, this proposition states that the only way for trajectories near $\\mathcal A$ to reach the vicinity of the Taub-points is via the heteroclinic connection $-\\taubp_i\\to\\taubp_i$.\n\nWe first give an informal outline of the proofs:\n\\begin{proof}[Informal proof of Proposition \\ref{prop:farfromtaub-main}]\nWe split the trajectory into time intervals where it is either near $\\mathcal K$ (i.e.~$\\max_i |N_i|\\le \\epsilon_N$) or away from $\\mathcal K$ (i.e.~$\\max_i |N_i|\\ge \\epsilon_N$). \n\nNear $\\mathcal K$, we can see from \\eqref{eq:ode2-ni} and \\eqref{eq:ode2-delta} that each  ${\\mathrm{D}}_t \\log |N_i|$ and ${\\mathrm{D}}_t \\log \\delta_i$ depends only on the $\\bpt \\Sigma$-coordinates and is positive only on some disc in ${\\mathbb{R}}^2$. These six discs are plotted in Figure \\ref{fig:n-discs}. We can observe that these discs only touch and intersect $\\mathcal K$ at the three Taub-points and that near each point $\\bpt p\\in \\mathcal K\\setminus \\{\\taubp_i\\}$, exactly one of the $\\log|N_i|$ is increasing and the two remaining $\\log |N_j|$ and all three $\\log \\delta_i$ are decreasing. Under our assumption $\\min_i d(\\gamma(t), \\taubp_i)>\\varepsilon_T$, this increase and decrease is uniform when $\\max_i |N_i|\\le \\epsilon_N$ and if $\\epsilon_N=\\epsilon_N(\\varepsilon_\\taubp)>0$ is small enough. Hence, for any small piece of trajectory $\\bpt x:[t_1,t_2]\\to \\{\\bpt x\\in\\mathcal M:\\,\\min_i d(\\bpt x,\\taubp_i)\\ge\\varepsilon_T,\\,\\max_i|N_i|\\le \\epsilon_N\\}$, one of the $|N_i|$ is uniformly exponentially increasing, while all three $\\delta_i$ and the remaining two $|N_j|,|N_k|$ are uniformly exponentially decreasing, with some rate $2\\epsilon_s=2\\epsilon_s(\\varepsilon_T, \\epsilon_N)>0$.\n\nEventually the trajectory will leave the neighborhood of $\\mathcal K$; since we assumed that we are near $\\mathcal A$, i.e.~$\\max_i \\delta_i < \\epsilon_d$, this can only happen near one of the Kasner caps (Bianchi type \\textsc{II}). By continuity of the flow, the trajectory will follow a heteroclinic orbit until it is near $\\mathcal K$ again, and will spend only bounded amount of time $T < C(\\epsilon_N)$ for this transit. Hence, all $|N_i|$ and $\\delta_i$ can only change by a bounded factor during such a heteroclinic transit.\n\nThe time spent near $\\mathcal K$ between two heteroclinic transits is bounded below by $\\mathcal O(|\\log \\epsilon_d|)$ (for fixed $\\epsilon_N$): Consider an interval $[t_1,t_2]$ spent near $\\mathcal K$, where $t_1 >0$. Suppose without loss of generality that initially $|N_1|(t_1)=\\epsilon_N$ and that $|N_2|$ is uniformly exponentially increasing, such that $|N_2|(t_2)=\\epsilon_N$. Then, we must have $|N_2|(t_1)= \\frac{4|N_1|}{\\delta_3^2}(t_1)\\le C \\epsilon_N \\epsilon_d^{-2}$, and we must have $t_2-t_1 \\ge C |\\log \\epsilon_d|$. Hence, if $\\epsilon_d \\ll \\epsilon_N$ is small enough, the exponential decrease of the three $\\delta_i$ will dominate all contributions from the heteroclinic transits and we obtain an estimate of the form \\eqref{eq:nearA:exponential}. \n\\end{proof}\n\\begin{proof}[Informal proof of Proposition \\ref{prop:farfromtaub-main2}]\nWe again use continuity of the flow: Each small heteroclinic ``bounce'' near one of the $\\taubp_i$ must increase the distance from $\\taubp_i$ by at least some $C_u=C_u(\\varepsilon_T)$. By continuity of the flow, each episode with $\\max_i |N_i|\\ge\\epsilon_N$ therefore must increase the distance from $\\taubp_i$ by $C_u\/2$; near $\\mathcal K$, the trajectory is almost constant and $d(\\bpt x_0, \\taubp_i)$ cannot shrink by more that $C_u\/3$. Hence the only way to reach the vicinity of a Taub point is by following the heteroclinic $-\\taubp_i\\to\\taubp_i$.\n\\end{proof}\n\n\nThe remainder of this section is devoted to making these informal proofs rigorous, i.e.~filling all the gaps and replacing the hand-wavy arguments by formal ones. We begin by naming the regions of the phase-space, where the various estimates hold:\n\n\\begin{definition}\\label{def:farfromtaub-base-sets}\nGiven $\\varepsilon_\\taubp, \\epsilon_N, \\epsilon_d>0$ (later chosen in this order) we define:\n\\[\\begin{aligned}\n{\\textsc{Cap}}[\\epsilon_N, \\epsilon_d]&=\\{\\bpt x\\in \\mathcal M: \\max|N_i|\\ge \\epsilon_N, \\max_i\\delta_i \\le \\epsilon_d\\}\\\\\n{\\textsc{Circle}}[\\epsilon_N, \\epsilon_d]&= \\{\\bpt x\\in \\mathcal M: \\max |N_i|\\le\\epsilon_N,\\, \\max_i\\delta_i \\le \\epsilon_d\\}\\\\\n{\\textsc{Hyp}}[\\varepsilon_\\taubp, \\epsilon_N, \\epsilon_d]&= {\\textsc{Circle}}[\\epsilon_N, \\epsilon_d] \\setminus \\left[B_{\\varepsilon_\\taubp}(\\taubp_1)\\cup B_{\\varepsilon_\\taubp}(\\taubp_2)\\cup B_{\\varepsilon_\\taubp}(\\taubp_3) \\right].\n\\end{aligned}\\numberthis\n\\]\n\\end{definition}\nThese sets are sketched in Figure \\ref{fig:nuclear} (not up to scale). They are constructed such that for appropriate parameter choices:\n\\begin{enumerate}\n\\item The union $\\textsc{Cap}\\cup\\textsc{Circle}$ contains an entire neighborhood of $\\mathcal A$ (by construction).\n\\item The region $\\textsc{Circle}$ is a small neighborhood of the Kasner circle. This is because of the constraint $1=\\Sigma^2+N^2$ and $|N^2|< C \\epsilon_N^2$ (see Figure \\ref{fig:nuclear-circle}).\n\\item The region $\\textsc{Cap}$ has three connected components, where one of the three $|N_i|\\gg 0$, because by $\\max_i |N_i|\\ge \\epsilon_N$ at least one $N_i$ must be bounded away from zero and by $\\max_{i}2\\sqrt{|N_jN_k|} \\le \\epsilon_d$ at most one $N_i$ can be bounded away from zero (this only works if $\\epsilon_d$ is small enough, depending on $\\epsilon_N$).\n\nThe region $\\textsc{Cap}$ is bounded away from the Kasner circle (see Figure \\ref{fig:nuclear-cap}). \nBy continuity of the flow, the dynamics in $\\textsc{Cap}$ can be approximated by pieces of heteroclinic orbits in $\\mathcal A$, up to uniformly small errors (Lemma \\ref{lemma:farfromtaub-cap}).\n\\item The region $\\textsc{Hyp}$ has three connected components. In each connected component, one of the $|N_i|$ is uniformly exponentially increasing and the remaining two $|N_j|$, $|N_k|$ are uniformly exponentially decreasing. All three products $\\delta_i$ are uniformly exponentially decreasing in $\\textsc{Hyp}$ (Lemma \\ref{lemma:farfromtaub-uniform-hyp}; this only works if $\\epsilon_N$ is small enough, depending on $\\varepsilon_\\taubp$).\n\\item The remaining part of the neighborhood of $\\mathcal A$, i.e.~$\\textsc{Circle}\\setminus\\textsc{Hyp}$, consists of the neighborhoods of the three Taub points. The analysis of the dynamics in these neighborhoods is deferred until Section \\ref{sect:near-taub}.\n\\end{enumerate}\n\\begin{lemma}[Uniform Hyperbolicity Estimates]\\label{lemma:farfromtaub-uniform-hyp}\nGiven any $\\varepsilon_T>0$ small enough, we find $\\epsilon_N>0$ and $\\epsilon_s>0$ small enough such that, for any $\\bpt x\\in \\textsc{Hyp}[\\varepsilon_{\\taubp}, \\epsilon_N,\\infty]$, we find one $i\\in\\{1,2,3\\}$ such that ${\\mathrm{D}}_t \\log |N_i|>2\\epsilon_s$, and the remaining two ${\\mathrm{D}}_t\\log|N_j|< -2\\epsilon_s$ and all three ${\\mathrm{D}}_t \\log \\delta_j <-2\\epsilon_s$.\n\nLet $\\varepsilon_\\taubp, \\epsilon_N, \\epsilon_s>0$ as above.\nFor any piece of trajectory $\\bpt x:(t_1,t_2)\\to \\textsc{Hyp}[\\epsilon_N,\\infty]$, we can conclude\n\\[\\numberthis\\begin{aligned}\n\\int_{t_1}^{t_2}|N_i|{\\mathrm{d}} t &< \\frac{\\epsilon_N}{2\\epsilon_s}\\\\\n\\mathrm{diam}\\, \\gamma \\le \\int_{t_1}^{t_2}|\\gamma'(t)|{\\mathrm{d}} t &\\le C_\\Sigma\\int_{t_1}^{t_2}\\max_i|N_i|(t){\\mathrm{d}} t\\le C_\\Sigma\\frac{\\epsilon_N}{2\\epsilon_{s}},\n\\end{aligned}\\]\nwhere we can choose $C_\\Sigma= 2$ if $\\epsilon_N<0.1$ (from \\eqref{eq:ode}).\n\\end{lemma}\n\\begin{proof}\nThe first part of the lemma consists of choosing $\\epsilon_s$ and $\\epsilon_N$ dependent on $\\varepsilon_T$. \nFrom Equations \\eqref{eq:ode2-ni} and \\eqref{eq:ode2-delta}, we see that each  ${\\mathrm{D}}_t \\log |N_i|$ and ${\\mathrm{D}}_t \\log \\delta_i$ depends only on the $\\bpt \\Sigma$-coordinates and is positive only on some disc in ${\\mathbb{R}}^2$. These six discs are plotted in Figure \\ref{fig:n-discs}, and only touch or intersect $\\mathcal K$ near the three Taub-points, a neighborhood of which is excluded. By the constraint $1-\\Sigma^2 = N^2$ and $|N^2|\\le 9 \\epsilon_N^2$, the set \\textsc{Hyp}, depicted in Figure \\ref{fig:nuclear}, is near $\\mathcal K$, and the desired uniformity estimates hold.\n\n\nThe second part follows from the uniform hyperbolicity in \\textsc{Hyp}: In each component of \\textsc{Hyp}, exactly one $N_i$ is unstable (see Figure \\ref{fig:nuclear} and Figure \\ref{fig:n-discs}). Suppose without loss of generality that $N_1$ is the unstable direction; then we can estimate for $t\\in (t_1,t_2)$:\n\\[\n|N_1(t)| \\le e^{-2\\epsilon_s (t_2-t)}|N_1(t_2)|,\\qquad |N_2(t)|\\le e^{-2\\epsilon_s(t-t_1)}|N_2(t_1)|.\n\\]\nFor $|N_3|$, the analogous estimate as for $N_2$ holds.\nUsing $|N_1(t_2)|\\le \\epsilon_N$ and $|N_2(t_1)|\\le \\epsilon_N$ and integrating yields the claim about $\\int |N|{\\mathrm{d}} t$. From \\eqref{eq:ode}, we see $|\\bpt x'|\\le C_\\Sigma \\max_i|N_i|$ (if $\\max_i|N_i|\\le 1$).\n\\end{proof} \n\nContinuity of the flow allows us to approximate solutions in \\textsc{Cap} by heteroclinic solutions in $\\mathcal A$, up to any desired precision $\\varepsilon_c$, if we only chose the distance from $\\mathcal A$ (i.e.~$\\epsilon_d>0$) small enough. More precisely:\n\n\\begin{lemma}[Continuity of the flow near \\textsc{Cap}]\\label{lemma:farfromtaub-cap}\nLet $\\epsilon_N>0$ and $\\varepsilon_c>0$. Then there exists $\\epsilon_d=\\epsilon_d(\\epsilon_N, \\varepsilon_c)$ small enough and $\\hat C_0=\\hat C_0(\\epsilon_N)>0$ large enough, such that the following holds:\n\nLet $\\bpt x:(t_1,t_2)\\to \\textsc{Cap}[\\epsilon_N, \\epsilon_d]$ be a piece of a trajectory. \nThen $t_2-t_1 < \\hat C_0$ and there exists $y\\in \\mathcal A$ such that \n\\[\\numberthis\\label{eq:farfromtaum-cap-continuity} d(\\bpt x(t), \\phi(y, t-t_1))<\\varepsilon_c\\qquad \\text{for all}\\quad t\\in(t_1,t_2),\\]\nwhere $\\phi:\\mathcal M\\times {\\mathbb{R}}\\to\\mathcal M$ is the flow corresponding to \\eqref{eq:ode}.\n\\end{lemma}\n\\begin{proof}Follows from continuity of the flow and the fact that all trajectories in $\\mathcal A$ are heteroclinic and must leave \\textsc{Hyp} at some time.\n\\end{proof}\n \nWe now have collected all the ingredients to formally prove the two main results from this section. At first, we \ncombine Lemma \\ref{lemma:farfromtaub-uniform-hyp} and Lemma \\ref{lemma:farfromtaub-cap} in order to show \nthat each $\\delta_i$ is uniformly essentially exponentially decreasing in $\\textsc{Cap}\\cup \\textsc{Hyp}$:\n\\begin{proof}[Formal proof of Proposition \\ref{prop:farfromtaub-main}]\nGiven $\\varepsilon_T>0$, find $\\epsilon_N, \\epsilon_d, \\epsilon_s, \\hat C_0>0 $ such that Lemma \\ref{lemma:farfromtaub-uniform-hyp} and Lemma \\ref{lemma:farfromtaub-cap} hold (for arbitrary $\\varepsilon_c$).\n\nSet $\\mu_i = \\delta_i'\/\\delta_i + \\epsilon_s$; it suffices to show that $\\int_{t_1}^{t_2}\\mu(t) {\\mathrm{d}} t < \\log C_0$ is bounded above, independently of $\\gamma$, $i\\in\\{1,2,3\\}$ and $t_1$,$t_2$. Fix $\\gamma$ and $t_1<t_2$.\n\nDecompose $[t_1,t_2]$ into intervals $S_k < T_k <S_{k+1}$ corresponding to the preimages of the regions \\textsc{Cap} and \\textsc{Hyp}, i.e. such that\n\\[\n\\bpt x([S_k,T_k])\\subseteq {\\textsc{Hyp}} \\qquad\\text{and}\\qquad \\bpt x([T_k, S_{k+1}])\\subseteq {\\textsc{Cap}}.\n\\]\n\nWe begin by considering the contribution from \\textsc{Cap}, i.e.,~an interval $[T_k, S_{k+1}]$.\nBy Lemma \\ref{lemma:farfromtaub-cap} we have $S_{k+1}-T_k< \\hat C_0$; since $\\mu$ is bounded, we get $\\int_{T_k}^{S_{k+1}}\\mu(t){\\mathrm{d}} t < \\log C_0$ for some $C_0$.\n\nNext, we consider the contributions from \\textsc{Hyp}. In this region, $\\mu < -\\epsilon_s$. Take an interval $[S_k, T_k]$, which is not the initial or final interval, i.e.~$t_1 < S_k < T_k < t_2$.\nAssume without loss of generality that $|N_1(S_k)|=\\epsilon_N$ and $|N_2(T_k)|=\\epsilon_N$. \nThen $|N_2(S_k)|= 0.25 \\delta_3^2\/|N_1|(S_k)< \\epsilon_d^2\/\\epsilon_N$. Since $|N_2'\/N_2|<3$, we obtain $T_k - S_k >  - \\frac{2}{3}\\log \\frac{\\epsilon_d}{\\epsilon_N}$.\nAdjust $\\epsilon_d>0$ to be so small, that $C_s(T_k-S_k) > 2\\log C_0$. Then such an interval gives us a contribution of $\\int_{S_k}^{T_k}\\mu(t){\\mathrm{d}} t < -\\log C_0$. \n\n For the complete interval $(t_1,t_2)$, sum over $k$; two disjoint intervals in the \\textsc{Cap}-region must always enclose an interval in the \\textsc{Hyp}-section, which cancels the contribution of its preceding \\textsc{Cap}-region. Therefore, at most the last \\textsc{Cap}-region stays unmatched and we obtain\n \\(\n \\int_{t_1}^{t_2}\\mu(t){\\mathrm{d}} t < \\log C_0.\n \\)\n\\end{proof}\nNext, we adjust the constants from Lemma \\ref{lemma:farfromtaub-cap} in order to show that trajectories near $\\mathcal A$ can only enter the vicinity of Taub-points via the heteroclinic $-\\taubp_\\ell\\to\\taubp_\\ell$:\n\\begin{proof}[Formal proof of Proposition \\ref{prop:farfromtaub-main2}]\nWe find some $C_u>0$ such that $d(K(p), T_i)> d(p,T_i)+C_u$ for every $p\\in\\mathcal K$ with $d(p,\\taubp_i)\\in (\\varepsilon_T, 0.5]$. It is evident from Figure \\ref{fig:short-het} (or, formally, Proposition \\ref{prop:kasnermap-homeomorphism-class}) that this is possible.\n\nBy Lemma \\ref{lemma:farfromtaub-uniform-hyp}, we can make $\\epsilon_N$ small enough that $\\mathrm{diam}\\, \\gamma < C_u\/8$ for pieces of trajectories $\\gamma:(t_1,t_2)\\to{\\textsc{Hyp}}$. \nUsing the continuity of the flow, i.e. Lemma \\ref{lemma:farfromtaub-cap}, we can make $\\epsilon_d$ small enough such that pieces $\\bpt x:(t_1,t_2)\\to {\\textsc{Cap}}$ are approximated by heteroclinic orbits up to distance $C_u\/4$. \n\nNow suppose $T^*<\\infty$ and $\\min_i d(\\bpt x_0, \\mathcal T_i)>5\\varepsilon_T$.\nWe cannot have $\\max_i\\delta_i(\\phi(x_0,T^*))\\ge \\epsilon_d$; hence, $d(\\bpt x(T^*), \\taubp_\\ell)=\\varepsilon_T$ for some $\\ell\\in\\{1,2,3\\}$. Set\n\\[T^3 = \\sup\\left\\{t<T^*: \\bpt x(t)\\not\\in \\overline{B_{1.5 \\varepsilon_T}(\\taubp_\\ell)}\\cap \\textsc{Circle}[\\epsilon_n, \\epsilon_d]\\right\\}.\\]\nBy the assumption \\eqref{eq:nearA:init-far-from-taub}, we have $T^3>0$.\nWe cannot have $d(\\taubp_\\ell, \\bpt x(T^3))=1.5\\varepsilon_\\taubp$, since we already know $\\mathrm{diam}\\, \\bpt x([T^3,T^*])\\le C_u\/8 < 0.5 \\varepsilon_T$ (since, by construction, $\\bpt x([T^3, T^*])\\subseteq \\textsc{Hyp}(\\varepsilon_\\taubp, \\epsilon_N, \\epsilon_d)$). This proves \\eqref{eq:farfromtaub-thm-3star}, as well as \n\\[\\bpt x(T^3)\\in\\partial \\textsc{Cap}[\\epsilon_N, \\epsilon_d]\\cap \\partial \\textsc{Circle}[\\epsilon_N,\\epsilon_d]\\cap B_{1.5\\varepsilon_T}(\\taubp_\\ell).\\]\nNext, we set \n\\[T^2 = \\sup\\left\\{t\\in [0,T^3): \\bpt x([t,T_3))\\subseteq\\textsc{Cap}[\\epsilon_n, \\epsilon_d]\\right\\}.\\]\nIn the interval $t\\in [T_2,T_3]$, the trajectory is in one of the three \\textsc{Cap} regions; this must be the $|N_\\ell|\\ge \\epsilon_N$ cap, since otherwise $d(\\bpt x(t), \\taubp_\\ell)$ would be decreasing (see Figure \\ref{fig:short-het}). \nWe set \n\\[T^1 = \\sup\\left\\{t\\in [0,T^2): \\bpt x(t)\\not\\in \\textsc{Circle}[\\epsilon_n, \\epsilon_d]\\right\\}.\\]\nSimilar arguments yield the remaining claim \\eqref{eq:farfromtaub-thm-12}.\n\\end{proof}\n\n\\begin{remark}\nThe constants generated in this section are sub-optimal (at least doubly exponentially so). If one cared at all about their numerical values, then one would need to replace Lemma \\ref{lemma:farfromtaub-cap} and Proposition \\ref{prop:kasnermap-homeomorphism-class} by explicit estimates.\n\\end{remark}\n\n\n\\section{Analysis near the generalized Taub-spaces $\\mathcal T_i^G$}\\label{sect:near-taub}\nIn this section, we will study the dynamics in the vicinity of the generalized Taub-spaces, without loss of generality $\\mathcal T_1^G$, using the polar coordinates from Section \\ref{sect:polar-coords}. This section is structured in the following way:\n\nIn section \\ref{sect:neartaub:motivation}, we will give a highly informal motivation for the general form of our estimates. This part can be safely skipped by readers who are uncomfortable with its hand-wavy nature. In section \\ref{sect:neartaub:heteroclinic}, we will study the behaviour of trajectories near the heteroclinic orbit $-\\taubp_1\\to\\taubp_1$, which come from either the $|N_2|\\gg 0$ or the $|N_3|\\gg 0$ cap. In section \\ref{sect:neartaub:neartaub}, we will study the further behaviour near $\\taubp_1$ of such trajectories. In section \\ref{sect:neartaub:forbidden-cones}, we will study the behaviour of trajectories near $\\taubp_1$ which do \\emph{not} necessarily have the prehistory described in section \\ref{sect:neartaub:heteroclinic}, and especially provide the deferred proof of Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub}. This section is mostly optional for our main results: Any trajectory which ever leaves the region where Section \\ref{sect:neartaub:forbidden-cones} is necessary will never revisit this region, a fact which is proven without referring to any results from Section \\ref{sect:neartaub:forbidden-cones}.\n\n\n\n\n\\subsection{Informal Motivation}\\label{sect:neartaub:motivation}\nWe already alluded to the motivation for the estimates in this section in the introductory Section \\ref{sect:intro} , page \\pageref{paragraph:intro:strategy:quotients}: \nFrom Proposition \\ref{prop:farfromtaub-main}, by varying $\\varepsilon_\\taubp$, we can control the behaviour of trajectories near $\\taubp_1$ if $\\delta_1 \\ll r_1$ and obtain estimates of the following type for partial trajectories $\\gamma:(t_1,t_2)\\to B_{\\epsilon}(\\taubp_1)$ and continuous monotonous functions $\\rho:(0,1]\\to (0,1]$:\n\\begin{quote}\nSuppose $\\delta_1(t_1)<\\rho_0(r_1(t_1))$;\\\\ then $\\delta_1(t_2) < C_0 e^{-\\rho_1(r_1(t_1))}\\delta_1(t_1)$ and  $r_1(t_2)\\ge \\rho_2(r_1(t_1))$.\n\\end{quote}\nThe bounds will take the specific form $\\rho_0(r)=Cr$ and $\\rho_2(r) = C r$ and $\\rho_1(r)= \\frac{C}{r^2}$ (Proposition \\ref{prop:neartaub:main}).\n\nWe know some prehistory of trajectories entering the vicinity of $\\taubp_1$ (by Proposition \\ref{prop:farfromtaub-main2}), which allows us to track backwards the condition $\\delta_1<\\rho_0(r_1)$. Further tracking back this condition, \nit is clear that trajectories entering the vicinity of $-\\taubp_1$ must have $\\delta_1 \\le \\epsilon_d \\ll r_1\\sim \\epsilon_N$. \n\nAt least in the Bianchi \\textsc{IX}-like case, where $\\mathrm{sign}\\,N_2=\\mathrm{sign}\\,N_3$, the set $\\{\\bpt x:\\,r_1=0\\}$ is invariant, allowing us to get some $\\rho_3:(0,1]\\to (0,1]$ such that $\\delta_1 < \\rho_3(r_1)$ for \\emph{any} trajectory entering the vicinity of $\\taubp_1$ via the route in Proposition \\ref{prop:farfromtaub-main2}, i.e.~via $-\\taubp_1$ and some cap before.\n\nThese estimates combine well if we can make $\\rho_3 < \\rho_0$.\nThe estimates will take the specific form $\\rho_3(r)= \\epsilon r$, for arbitrarily small $\\epsilon>0$ (Proposition \\ref{prop:near-taub:qc}), which is precisely the required estimate at $\\taubp_1$. \nIn Bianchi \\textsc{VIII}, we have no qualitative a-priori reason to expect bounds of the same form. Nevertheless, we will prove that they hold, which allows us to control any solution entering $\\taubp_1$ as in Proposition \\ref{prop:farfromtaub-main2}.\n\nFor the sake of brevity of arguments, we will present our analysis in the reverse order: We chronologically follow a trajectory from $-\\taubp_1$ to $+\\taubp_1$ and then until it leaves the vicinity of $+\\taubp_1$, instead of tracking estimates backwards.\n\n\\subsection{Analysis near $-\\taubp_1$ and near the heteroclinic $-\\taubp_1\\to\\taubp_1$}\\label{sect:neartaub:heteroclinic}\nThe behaviour of trajectories away from $\\taubp_1$ is already partially described by Proposition \\ref{prop:farfromtaub-main}; we only need to additionally estimate the quotient $\\frac{\\delta_1}{r_1}$ in this region. The necessary estimates can be summarized in the following:\n\\begin{proposition}\\label{prop:near-taub:qc}\nLet $\\varepsilon_{T}\\in (0,0.1)$. We can chose $\\epsilon_N, \\epsilon_d>0$ small enough, such that Propositions \\ref{prop:farfromtaub-main} and \\ref{prop:farfromtaub-main2} hold, as well as choose constants $C_1,\\ldots,C_5>0$ large enough, such that the following holds:\n\nLet $0< T_1 \\le T_2$ and $\\bpt x:[0,T_2)\\to M_{\\pm,\\pm,\\pm}$ be piece of trajectory, such that\n\\begin{subequations}\n\\begin{align}\n\\bpt x(0)&\\in \\partial \\textsc{Circle}[\\epsilon_N, \\epsilon_d],\\label{eq:near-taub-main-ass-initial} \\\\\n\\nonumber &\\quad\\text{i.e.}\\quad \\max_i\\delta_i(0)<\\epsilon_d,\\ \\max(|N_2|,|N_3|)(0)=\\epsilon_N  \\\\\n\\bpt x([0,T_1))&\\subseteq B_{2\\varepsilon_{T}}(-\\taubp_1)\\cap \\textsc{Circle}[\\epsilon_N, \\epsilon_d],\\label{eq:near-taub-main-ass-q}\\\\\n\\nonumber &\\quad\\text{i.e.}\\quad \\max_i\\delta_i(t)<\\epsilon_d,\\ \\max_i|N_i|(t)\\le\\epsilon_N,\\ d(\\gamma(t), \\taubp_1)\\le 2\\varepsilon_T\\qquad\\forall\\,t\\in [0,T_1]\\\\\n\\bpt x([T_1,T_2))&\\subseteq\\textsc{Cap}[\\epsilon_N, \\epsilon_d]\\label{eq:near-taub-main-ass-c},\\\\\n\\nonumber &\\quad\\text{i.e.}\\quad \\max_i\\delta_i(t)<\\epsilon_d,\\ |N_1|(t)\\ge \\epsilon_N \\quad \\forall\\,t\\in [T_1,T_2),\n\\end{align}\\end{subequations}\ni.e.~we are in the situation of the conclusion of Proposition \\ref{prop:farfromtaub-main2}. \n\nThen the following estimates hold:\n\\begin{subequations}\\begin{align}\n\\frac{\\delta_1}{r_1}(0) &\\le  C_1\\epsilon_d&&\\label{eq:near-taub-main-conc-initial}\\\\\n\\frac{\\delta_1}{r_1}(t_2) &\\le C_2 \\frac{\\delta_1}{r_1}(t_1)\\qquad&\\forall& 0 \\le t_1 \\le t_2 < T_2\\label{eq:near-taub-main-conc-delta-r-qc}\\\\\n\\delta_1(t_2) &\\le C_3 e^{- C_4^{-1}(t_2-t_1)}\\delta_1(t_1)\\qquad&\\forall&0 \\le t_1 \\le t_2 < T_2\\label{eq:near-taub-main-conc-delta-qc}\\\\\nr_1(t_2) &\\ge C_5^{-1} r_1(t_1)\\qquad&\\forall& T_1 \\le t_1 \\le t_2 < T_2\\label{eq:near-taub-main-conc-r-c}.\n\\end{align}\\end{subequations}\nAlternatively to the assumption \\eqref{eq:near-taub-main-ass-initial}, we can assume \\eqref{eq:near-taub-main-conc-initial}; then the case $0=T_1$ is valid as well.\n\\end{proposition}\n\\begin{proof}\nThe claim \\eqref{eq:near-taub-main-conc-delta-qc} is already proven in Proposition \\ref{prop:farfromtaub-main}.\nAssume without loss of generality that $|N_3|(0)=\\epsilon_N$. Assuming that $\\epsilon_d$ is small enough compared to $\\epsilon_N$, we have $|N_2|(0)\\le \\delta_1^2(0)\/\\epsilon_N \\le 0.5 \\epsilon_N$ and hence $r_1(0) \\ge |N_3|-|N_2|\\ge 0.5\\epsilon_N$. Therefore $\\frac{\\delta_1}{r_1}(0)\\le C\\delta_1(0)\\le C\\epsilon_d$ and claim \\eqref{eq:near-taub-main-conc-initial} holds.\nFirst, consider the Bianchi \\textsc{IX}-like case ${{\\hat{n}}}_2={{\\hat{n}}}_3$ in polar coordinates, i.e.~equations \\eqref{eq:neartaub-b9-q}. We can immediately estimate $\\partial_t\\log\\frac{\\delta_1}{r_1}\\le C|N_1|$. We already established in Section \\ref{sect:far-from-taub} that $\\int |N_1|{\\mathrm{d}} t$ is bounded for $t_1,t_2\\in [0,T_1]$ (Lemma \\ref{lemma:farfromtaub-uniform-hyp}) and that $T_2-T_1$ is bounded (Lemma \\ref{lemma:farfromtaub-cap}), yielding \\eqref{eq:near-taub-main-conc-delta-r-qc}.\n\nNext, consider the Bianchi \\textsc{VIII}-like case ${{\\hat{n}}}_2=+1$ and ${{\\hat{n}}}_3=-1$, i.e.~equations \\eqref{eq:neartaub-b8-q:delta-r}.\nAssume that we have for all $t\\in [0,T_2]$,\n\\begin{equation}\\label{eq:delta-r-one}\\frac{\\delta_1}{r_1}(t) < 1.\\end{equation}\nUnder the assumption \\eqref{eq:delta-r-one}, we can estimate \\[\\frac{N_+}{r_1}=\\frac{\\sqrt{N_-^2+\\delta_1^2}}{r_1}=\\sqrt{\\frac{N_-^2}{N_-^2+\\Sigma_-^2}+\\frac{\\delta_1^2}{r_1^2}}\\le \\sqrt{2},\\] and hence $\\partial_t\\log\\frac{\\delta_1}{r_1}\\le C|N_1|$, yielding \\eqref{eq:near-taub-main-conc-delta-r-qc}.\nWhen we adjust $\\varepsilon_d$ such that $C_1C_2\\varepsilon_d<1$, then this argument bootstraps to prove \\eqref{eq:near-taub-main-conc-delta-r-qc}, without assuming a priori \\eqref{eq:delta-r-one} (proof: Assume there was a time $T\\in (0,T_2)$ such that \\eqref{eq:delta-r-one} was violated; then $\\frac{\\delta_1}{r_1}(T)\\le C_{2}\\frac{\\delta_1}{r_1}(0)\\le C_2C_1\\varepsilon_d<1$)\n\nNext, we prove the remaining claim \\eqref{eq:near-taub-main-conc-r-c}.\nSince we know that $\\frac{\\delta_1}{r_1}(t) < 1$, we can estimate ${\\mathrm{D}}_t \\log r < C$ for for all $t\\in [T_1,T_2)$, both in Bianchi \\textsc{VIII} and \\textsc{IX}. Since $T_2-T_1$ is bounded, we obtain \\eqref{eq:near-taub-main-conc-r-c}.\n\nConsidering the above proof, it is obvious that we can alternatively replace the assumption \\eqref{eq:near-taub-main-ass-initial}  by \\eqref{eq:near-taub-main-conc-initial}, and then also allow $0=T_1$.\n\\end{proof}\n\n\\begin{remark}\nWe excluded the set $\\forbidden[\\epsilon_v]$ from our analysis, given by\n\\[\n\\forbidden[\\epsilon_v] = \\{\\bpt x \\in {\\mathbb{R}}^5: \\delta_1 \\ge \\epsilon_v r_1 \\}, \n\\]\ni.e.~we described the dynamics outside of $\\forbidden$ and showed that the set $\\forbidden$ cannot reached by initial conditions described by Proposition \\ref{prop:farfromtaub-main2}. \n\nIgnoring the constraint $G=1$, the set $\\forbidden$ looks like a linear cone times ${\\mathbb{R}}^2$, since both $r_1$ and $\\delta_1$ are homogeneous of first order in $N_2,N_3,\\Sigma_-$ and independent of $N_1$ and $\\Sigma_+$. \n\nEven though Bianchi \\textsc{VIII} lacks an explicit invariant Taub-space, the $\\forbidden$-cone around the generalized Taub-space $\\mathcal T_1^G$ is a suitable ``morally backwards invariant'' replacement.\n\\end{remark}\n\n\\subsection{Analysis near $\\taubp_1$}\\label{sect:neartaub:neartaub}\nOur analysis of the neighborhood of $\\taubp_1$ can be summarized in the following\n\\begin{proposition}\\label{prop:neartaub:main}\nFor any $\\varepsilon_T\\in (0,0.1)$, there exist constants $\\epsilon_v>0$, $C_1,C_r,C_{\\delta,r}>0$ and $C_e=10$ such that the following holds:\n\nLet $\\gamma:[0,T^*)\\to B_{2\\varepsilon_T}(\\taubp_1)$ be a partial trajectory with\n\\begin{equation}\n\\frac{\\delta_1}{r_1}(0)< \\epsilon_v.\n\\end{equation}\nThen, for all $0\\le t_1\\le t_2 <T^*$:\n\\begin{subequations}\n\\begin{align}\n(|N_1|, \\delta_2,\\delta_3) (t_2) &\\le C_1 e^{-C_e^{-1} (t_2-t_1)}\\,(|N_1|, \\delta_2,\\delta_3)(t_1) \\label{eq:neartaub:main:others}\\\\\n\\frac{\\delta_1}{r_1}(t_2)&\\le C_{\\delta,r}\\frac{\\delta_1}{r_1}(t_1) \\label{eq:neartaub:main:q}\\\\\nr_1(t_2)&\\ge C_r^{-1}r_1(t_1)\\label{eq:neartaub:main:r}\\\\\n\\delta_1(t_2)&\\le C_0\\exp\\left(-\\frac{C_e^{-1}}{r_1^{2}(t_1)}(t_2-t_1)\\right)\\,\\delta_1(t_1)\\label{eq:neartaub:main:delta}\\\\\nT^*&<\\infty\\label{eq:neartaub:main:T}.\n\\end{align}\n\\end{subequations}\n\\end{proposition}\nWe begin by proving the first three of the claims, in a way analogous to the proof of Proposition \\ref{prop:near-taub:qc}:\n\\begin{proof}[Proof of Proposition \\ref{prop:neartaub:main}, conclusions \\eqref{eq:neartaub:main:others}, \\eqref{eq:neartaub:main:q}, \\eqref{eq:neartaub:main:r}]\nThe exponential decay \\eqref{eq:neartaub:main:others} follows trivially from \\eqref{eq:ode2-ni}, see e.g.~Figures \\ref{fig:n-discs2} and \\ref{fig:n-discs}. \n\nIn order to see \\eqref{eq:neartaub:main:q}, we again need to bootstrap: First consider $T_1=\\sup\\{t\\in [0,T^*): \\frac{\\delta_1}{r_1}(t') < 1\\,\\forall t'\\in [0,t)\\}$.\nWe can estimate $\\partial_t\\log \\frac{\\delta_1}{r_1} < C|N_1|$ for all $t\\in [0,T_1)$, using \\eqref{eq:neartaub-b9-t:delta-r} and \\eqref{eq:neartaub-b8-t:delta-r} and the fact that the higher order terms $|h_r|,|h_\\delta|, |h_\\psi|$ are bounded (by $C=5$). By the exponential decay of $N_1$, this yields \\eqref{eq:neartaub:main:q} upon integration, for all $t_2\\le T_1$. By adjusting $\\epsilon_v<C_{\\delta,r}^{-1}$ we can then conclude $T_1=T^*$.\n\nUsing $\\frac{\\delta_1}{r_1}<1$, we can estimate $\\partial_t r_1 > -C|N_1|$, which upon integration yields the claim \\eqref{eq:neartaub:main:r}.\n\\end{proof}\nThe next estimate \\eqref{eq:neartaub:main:delta} requires a slightly more involved averaging-style argument, similar to the proof of Proposition \\ref{prop:farfromtaub-main}:\n\\begin{proof}[Proof of Theorem \\ref{prop:neartaub:main}, conclusion \\eqref{eq:neartaub:main:delta}]\nWe set\n\\[\\mu = {\\mathrm{D}}_t\\log\\delta_1 + 0.1 r_1^2.\\]\nIt suffices to prove that $\\int_{t_1}^{t_2}\\mu(t){\\mathrm{d}} t\\le \\log C_0$ for some $C_0>0$.\n\n\\paragraph{Strategy.}\nWe will first consider times where $|N_1|\\not\\ll r_1$; the integral $\\int \\mu {\\mathrm{d}} t$ over these times will be bounded by $\\int |N_1|{\\mathrm{d}} t$. Next we will split $\\mu$ into a nonpositive and a nonnegative part; the nonnegative (bad) part will have a contribution for every $\\psi$ rotation, which is bounded by $C r_1$, while the nonpositive (good) part will have a negative contribution for every $\\psi$-rotation which scales with $r_1\\log\\frac{\\delta_1}{r_1}$. Adjusting $\\epsilon_v$ will then yield the desired estimate (after summing over $\\psi$-rotations).\n\n\\paragraph{Estimates for large $|N_1|$.}\nChoose $\\widetilde T$ (possibly $\\widetilde T= 0$) such that $|N_1(t)|\\ge C r_1^2(t)$ (with $C=0.05$) for $t\\in (0, \\widetilde T]$ and $|N_1(t)|\\le 0.1 r_1^2(t)$ for $t \\in [\\widetilde T, T]$. This is possible, since ${\\mathrm{D}}_t \\log |N_1| < 2{\\mathrm{D}}_t \\log r_1$. \nThen $|\\mu(t)| \\le C \\sqrt{|N_1(t)|}$ for all $t\\in [0,\\widetilde T]$ and hence $\\int_{0}^{\\widetilde T}|\\mu(t)|{\\mathrm{d}} t< C$.\n\n\\paragraph{Averaging Estimates.}\nConsider without loss of generality $t_1,t_2\\ge \\widetilde T$. Using $\\Sigma_+\\approx -1$ and $\\delta_1 \\le 0.1 r_1$, we can estimate\n\\[\\mu \\le -0.4 r_1^2 \\cos^2\\psi + 0.6 r_1^2\\sin^2\\psi + 0.01 r_1^2 + 0.1 r_1^2 + C|N_1| \\le -0.25 r_1^2 + r_1^2\\sin^2\\psi.\\]\nWe can also estimate $\\psi'$\n\\[r_1 |\\sin\\psi| \\le \\psi' \\le 2r_1\\sqrt{\\sin^2\\psi + \\frac{\\delta_1^2}{r_1^2}}.\\]\nLet $\\mu_+ = r_1^2\\sin^2\\psi$ be the positive (bad) part of $\\mu$; take times $t_1<t_L<t_R<t_2$ with $|\\psi(t_R)-\\psi(t_L)|\\le 2 \\pi$. Then\n\\[\\int_{t_L}^{t_R}\\mu_+(t){\\mathrm{d}} t \n\\le \\int_{\\psi(t_L)}^{\\psi(t_R)}\\frac{\\mu_+(t)}{\\psi'(t)}{\\mathrm{d}} t\n\\le C_{r}^{-1}r_1(t_R)\\int_{0}^{2\\pi}|\\sin\\psi|{\\mathrm{d}} \\psi \\le C_+ r_1(t_R),\\]\nfor some $C_+>0$.\nOn the other hand, let $\\mu_-=-0.25 r_1^2$ be the negative (good) part of $\\mu$. Take times $t_1<t_L<t_R<t_2$ with $\\psi(t_L)=k\\pi-0.1$ and $\\psi(t_R)=k\\pi +0.1$ for some $k\\in{\\mathbb{Z}}$; then we can estimate\n\\[\\begin{aligned}\n\\int_{t_L}^{t_R}\\mu_-(t){\\mathrm{d}} t &\\le \\int_{-0.1}^{+0.1}\\frac{-0.25\\, r_1^2}{\\psi'}{\\mathrm{d}} \\psi\\le \\int_{-0.1}^{+0.1}\\frac{-0.125\\, r_1}{\\sqrt{\\sin^2\\psi + \\frac{\\delta_1^2}{r_1^2}}}{\\mathrm{d}} \\psi\\\\\n&\\le -0.2\\, C_{r}^{-1}r_1(t_L)\\int_{0}^{0.1}\\frac{1}{\\sqrt{x^2 + C_{\\delta,r}^2 \\epsilon_v^2}}{\\mathrm{d}} x\n\\end{aligned}\\]\nWe can immediately see that for any $C_+>0$, we find $\\epsilon_v>0$ such that $\\frac{\\delta_1}{r_1}(t_1)<\\epsilon_v$ implies \n$\\int_{t_L}^{t_R}\\mu_-(t){\\mathrm{d}} t < -C_+ r_1(t_L)$. Hence, by summing over $\\psi$-rotations (and adjusting $\\epsilon_v$), we can conclude the assertion \\eqref{eq:neartaub:main:delta}.\n\\end{proof}\n\\begin{proof}[Proof of Proposition \\ref{prop:neartaub:main}, conclusion \\eqref{eq:neartaub:main:T}]\nWe need to show that solutions with small quotient $\\frac{\\delta_1} {r_1}$ cannot stay near $\\taubp_1$ forever.\n\nAssuming without loss of generality $|N_1|\\ll r_1^2$ we have ${\\mathrm{D}}_\\psi \\log r_1 > C r_1 |\\sin\\psi|$. This shows that the only way never leaving the vicinity of $\\taubp_1$ is for the angle $\\psi\\in {\\mathbb{R}}$ to stay bounded, i.e.~$\\lim_{t\\to\\infty}\\psi(t)=\\psi^{**}$ and $\\lim_{t\\to\\infty}\\delta_1(t)=0$ (since otherwise $r_1$ increases by a too large amount during each rotation). This is impossible, since the possible limit-points lie on the Kasner-circle $\\mathcal K\\setminus \\{\\taubp_1\\}$ and are not $\\taubp_1$; hence, either $N_2$ or $N_3$ is unstable and since initially $N_2\\neq 0\\neq N_3$, the trajectory cannot converge to such a point.\n\\end{proof}\n\\subsection{Analysis in the \\forbidden-cones}\\label{sect:neartaub:forbidden-cones}\\label{sect:averaging-unneeded}\nOur whole approach aims at avoiding the much more tricky analysis of the dynamics in the \\forbidden-cones, where possibly $\\delta_1\\ge r_1$: Since trajectories starting outside of these cones never enter them, it is unnecessary to know what happens in the \\forbidden-cones. However, for various global questions, it is useful to collect at least some results inside of these cones. \n\nWe already know that solutions in Bianchi \\textsc{IX} cannot converge to the Taub-points; the same holds in Bianchi \\textsc{VIII}, even for solutions in \\forbidden:\n\\begin{lemma}\\label{lemma:no-b8-taub-convergence}\nFor an initial condition $\\bpt x_0\\in \\mathcal M_{+-+}$, it is impossible to have $\\lim_{t\\to\\infty}\\bpt x(t)=\\taubp_1$.\n\\end{lemma}\n\\begin{proof}\nSuppose we have such a solution.\nUsing equation \\eqref{eq:neartaub-b8-q}, we can write near $\\taubp_1$:\n\\[\n{\\mathrm{D}}_t \\log |N_1|\\frac{\\delta_1}{r_1} \\le \\sqrt{3}\\frac{|\\Sigma_-|}{r_1} |N_1|\\frac{\\sqrt{N_-^2 + \\delta_1^2} }{r_1} - 2.5.\n\\]\nHence, if ever $|N_1|\\frac{\\delta_1}{r_1}<1$, this inequality is preserved and $|N_1|\\frac{\\delta_1}{r_1}$ decays exponentially. Then we can estimate ${\\mathrm{D}}_t \\log r_1 \\ge -C |N_1| -C |N_1|\\frac{\\delta_1}{r_1}$; all the terms on the right hand side have bounded integral and $r_1\\to 0$ is impossible.\n\nOn the other hand, if $r_1 < |N_1|\\delta_1$ for all sufficiently large times, we can estimate ${\\mathrm{D}}_t\\log \\delta_1\\ge -C r_1^2 - C |N_1| \\ge -C |N_1|$, which has bounded integral and thus contradicts $\\delta_1\\to 0$.\n\\end{proof}\n\nUnfortunately, this is all we can presently say in the \\forbidden{} cone in Bianchi \\textsc{VIII}. \n\n\\noindent \nIn the case of Bianchi \\textsc{IX}, we can still average over $\\psi$-rotations in order to show that $\\frac{\\delta_1}{r_1}$ decays, even in the \\forbidden{} region:\n\\begin{lemma}\\label{neartaub:lemma:joint-increase-in-forbidden}\nLet $h>0$. There exists constants $\\epsilon, C>0$ such that the following holds:\n\nLet $\\gamma:[0,T]\\to \\{\\bpt x\\in\\mathcal M_{*++}:\\, |N_1|\\le r_1^5,\\,r_1<\\epsilon,\\,r_1\\le h\\delta_1,\\, d(\\bpt x, \\taubp_1)<0.1\\}$ be a partial trajectory near $\\taubp_1$. Then \n\\[\n\\log \\frac{\\delta_1}{r_1}(0) -\\log \\frac{\\delta_1}{r_1}(T) \\le C (\\log r_1(T)-\\log r_1(0)),\n\\]\ni.e.~the increase of $r_1$ and the decrease of $\\frac{\\delta_1}{r_1}$ have comparable rates.\n\\end{lemma}\n\\begin{proof}\nWe can estimate\n\\[\\begin{aligned}\n{\\mathrm{D}}_t \\log r_1 &= r_1^2\\sin^2\\psi \\frac{-\\Sigma_+}{1-\\Sigma_+} + \\mathcal O(|N_1|)\\\\\n{\\mathrm{D}}_t \\log \\frac{\\delta_1}{r_1}&=\\frac{-1}{1-\\Sigma_+}r_1^2\\cos^2\\psi +\\mathcal O(|N_1|)\\\\\n\\psi'&= \\sqrt{3}r_1 \\sqrt{\\sin^2\\psi + \\frac{\\delta_1^2}{r_1^2}} - \\frac{r_1^2}{1-\\Sigma_+} \\cos\\psi \\sin\\psi + \\mathcal O(|N_1\\sin\\psi|).\n\\end{aligned}\\]\nIt suffices to show an estimate of the form\n\\[\n\\int_0^{2\\psi}r_1^2\\frac{\\sin^2\\psi}{\\psi'}{\\mathrm{d}} \\psi >C  \\int_0^{2\\psi}\\frac{r_1^2\\cos^2\\psi}{\\psi'}{\\mathrm{d}} \\psi.\n\\]\nUsing $\\delta_1 \\ge h r_1$ and $r_1\\le \\epsilon$, we can directly estimate $\\delta_1 \\le \\psi'\\le 2 \\sqrt{r_1^2+\\delta_1^2}$ (for $\\epsilon>0$ small enough). This allows us to see that $r_1, \\delta_1, \\frac{\\delta_1}{r_1}$ can all change only by a bounded factor during each rotation; hence it suffices to show for $\\psi(t_2)-\\psi(t_1)\\le 2\\pi$ that \n\\(\n\\min_{t\\in [t_1,t_2]}\\psi'(t) > C \\max_{t\\in [t_1,t_2]}\\psi'(t).\n\\)\n\n\\noindent \nHowever,\n\\[\n\\min_{t\\in [t_1,t_2]}\\psi'(t) \\ge \\min_t \\delta_1(t)\\qquad\n\\max_{t\\in [t_1,t_2]}\\psi'(t) \\ge 2 \\max_{t} \\sqrt{r_1^2(t) + \\delta_1^2(t)}\\le \\max_t \\delta_1(t) \\sqrt{1+ h^{-2}},\n\\]\nand we know that $\\delta_1$ can only change by a bounded factor during each rotation; hence\nthe desired estimate follows.\n\\end{proof}\n\nWith a more subtle averaging argument than the previous ones, we can also show the deferred Lemma \\ref{lemma:farfromtaub:no-delta-increase-near-taub}. However, this proof is only given for the sake of completeness and is nowhere used in this work, except for completing the literature review in Section \\ref{sect:farfromA:attract}.\n\\begin{lemma}\\label{lemma:farfromtaub:no-delta-increase-near-taub}\nWe consider without loss of generality the neighborhood of $\\mathcal T_1$. Let $\\epsilon>0$ small enough. Then there exists a constant $C_{\\delta,\\epsilon}\\in(1,\\infty)$, such that, for any piece of trajectory $\\gamma:[t_1,t_2]\\to \\{\\bpt x\\in\\mathcal M_{*++}: |\\bpt \\Sigma(\\bpt x)-\\taubp_1|\\le \\epsilon,\\,|N_1|\\le 10,\\,\\delta_1\\le 10\\}$, the following estimate holds:\n\\[ \\delta_1(\\gamma(t_2))\\le C_{\\delta,\\epsilon}\\delta_1(\\gamma(t_1)). \\]\n\\end{lemma}\n\\begin{proof}\nWe can assume without loss of generality that $\\delta_1 > h r_1$ for all $t\\in [t_1,t_2]$ and for some $h=\\hat\\epsilon>0$; otherwise Proposition \\ref{prop:neartaub:main} applies. We can also assume without loss of generality that $|N_1| \\le C r_1^2$ for all $t\\in [t_1,t_2]$ because of a similar argument as in the proof of Proposition \\ref{prop:neartaub:main}.\n\nWe will use the letters \n\\defC{C}\n\\defc{c}\n$C \\gg 1$ and $0<c\\ll 1 $ for unspecified constants. Recall \\eqref{eq:neartaub-b9-t}.\nWe use the auxiliary variable $\\zeta=\\frac{\\delta_1}{r_1}\\ge \\hat\\epsilon$. First, we note that \n\\[\\begin{array}{rccl}\n c r_1 \\zeta &\\le& \\psi' &\\le C r_1 \\zeta\\\\\nc r_1^2 \\zeta^{-1}\\cos^2\\psi - C |N_1|\\frac{1}{\\psi'}&\\le& {\\mathrm{D}}_\\psi r &\\le C r_1^2 \\zeta^{-1}\\\\\nc r_1\\cos^2\\psi - C|N_1|\\frac{1}{\\psi'} &\\le&-{\\mathrm{D}}_\\psi \\zeta &\\le c r_1.\n\\end{array}\\]\nWe see that $r_1$ and $\\zeta$ can change only by a bounded factor (and also a bounded amount) during a single rotation $[\\psi_1, \\psi_1+2\\pi]$; therefore, we can average the equation for $r_1'$ to see that\n\\[\\numberthis \\label{eq:average:rprime-intbound} \\int_{\\psi(t_1)}^{\\psi(t_2)}r_1^2\\zeta^{-1}{\\mathrm{d}} \\psi<C|r_1(t_1)-r_1(t_2)|<C.\\] \nAlso, $r_1$ is non-decreasing (except for the terms bounded by $|N_1|$); therefore, its total variation is bounded $\\int_{\\psi(t_1)}^{\\psi(t_2)}r_1'{\\mathrm{d}} t < C$.\n\n\\noindent \nOur goal is to bound $\\int_{t_1}^{t_2}\\frac{\\delta_1'}{\\delta_1} {\\mathrm{d}} t < C$. We first split the terms into\n\\[\\begin{aligned}\n{\\mathrm{D}}_t \\log \\delta_1 &= r_1^2 (\\sin^2\\psi-\\cos^2\\psi) \\frac{1}{1-\\Sigma_+} \n\\quad-\\quad r_1^2 \\cos^2\\psi\\frac{1+\\Sigma_+}{1-\\Sigma_+} \\quad+\\quad  N_1 h_\\delta.\n\\end{aligned}\\]\nThe last term is bounded by $|N_1|$ and hence has bounded integral. \nSince $1+\\Sigma_+ \\ge -C |N_1N_2N_3|^{\\frac 2 3} \\ge -C |N_1|^{\\frac 2 3}$, the second term has its integral bounded above. Therefore, our goal now is to estimate the integral of the first term:\n\\[\\begin{aligned}\n&\\int_{\\psi(t_1)}^{\\psi(t_2)} F(\\psi, \\zeta, r_1, N_1)(\\sin^2\\psi -\\cos^2\\psi){\\mathrm{d}} \\psi\\overset{!}{\\le}C,\\quad\\text{where}\\\\\nF(\\psi, \\zeta, r_1, N_1)&=\\frac{r_1^2}{1-\\Sigma_+}\\frac{1}{\\psi'}\\\\\n&=\\frac{r_1}{1-\\Sigma_+}\\frac{1}{\\sqrt{3}\\sqrt{\\sin^2\\psi +\\zeta^2} + \\frac{r_1}{1-\\Sigma_+} \\cos\\psi \\sin\\psi + N_1\\sin\\psi\\, h_{\\psi}},\\quad \\text{where}\\\\\nh_\\psi &= -\\sqrt{3}\\sin\\psi + \\cos\\psi \\frac{N_1-2r_1\\sqrt{\\sin^2\\psi + \\zeta^2}}{1-\\Sigma_+}.\n\\end{aligned}\\]\nBy the constraint $(1-\\Sigma_+)(1+\\Sigma_+)= r_1^2 + N_1(N_1-2r_1\\sqrt{\\sin^2\\psi + \\zeta^2})$, we know $\\Sigma_+ = \\Sigma_+(r_1, \\zeta, \\psi, N_1)\\approx -1$. The quantity $|F|$ is bounded by $|F| \\le C r_1\\zeta^{-1}$. We set\n\\[\n\\widetilde \\psi = \\psi+\\frac{\\pi}{4},\\qquad \\left(\\widetilde r_1,\\widetilde \\zeta,\\widetilde N_1\\right)(\\psi)=(r_1,\\zeta,N_1)\\left(\\psi+\\frac{\\pi}{4}\\right).\n\\]\nWe will then estimate\n\\begin{align*}\n&\\int_{\\psi(t_1)}^{\\psi(t_2)-\\frac{\\pi}{4}}F(\\psi, \\zeta, r_1, N_1)(\\sin^2\\psi -\\cos^2\\psi){\\mathrm{d}} \\psi + \n\\int_{\\psi(t_1)+\\frac{\\pi}{4}}^{\\psi(t_2)}F(\\psi, \\zeta, r_1, N_1)(\\sin^2\\psi -\\cos^2\\psi){\\mathrm{d}} \\psi\\\\\n&\\quad= \\int_{\\psi(t_1)}^{\\psi(t_2)-\\frac{\\pi}{4}}\\left[F(\\psi, \\zeta, r_1, N_1)-F(\\widetilde\\psi, \\widetilde\\zeta, \\widetilde r_1, \\widetilde N_1) \\right](\\cos^2\\psi -\\sin^2\\psi){\\mathrm{d}} \\psi\\\\\n&\\quad=\\int_{\\psi(t_1)}^{\\psi(t_2)-\\frac{\\pi}{4}}\\left[F(\\psi, \\zeta, r_1, 0)-F(\\widetilde\\psi, \\zeta,  r_1, 0) \\right](\\sin^2\\psi -\\cos^2\\psi){\\mathrm{d}} \\psi\\numberthis\\label{eq:average-negative-term}\\\\\n&\\quad+ \\mathcal O \\left[ \\int_{\\psi(t_1)}^{\\psi(t_2)-\\frac{\\pi}{4}} |\\partial_{\\zeta} F|\\,|\\zeta-\\widetilde\\zeta| + |\\partial_{r_1} F|\\,|r_1-\\widetilde r_1| +  |\\partial_{N_1} F|\\,|N_1+\\widetilde N_1|\\,{\\mathrm{d}} \\psi\\right]\\quad \\overset{!}{\\le}C.\n\\end{align*}\nIt is clear that this estimate will suffice for our claim. The crucial estimate is that the term \\eqref{eq:average-negative-term} is nonpositive; this can be seen by considering both cases $\\sin^2\\psi\\ge \\cos^2\\psi$ and $\\sin^2\\psi \\le \\cos^2\\psi$.\n\nThe remaining estimates including derivatives of $F$ are lengthy but straightforward. \nThe easiest derivative to estimate is by $N_1$; we can see that $|\\partial_{N_1}\\Sigma_+| \\le C$ (since $\\delta_1=\\zeta r_1 \\le C$) and then $|\\partial_{N_1} h_\\psi |\\le C$; then $|\\partial_{N_1}F| \\le C$. This implies that $\\int |\\partial_{N_1}F||N_1+\\widetilde N_1|{\\mathrm{d}} \\psi \\le C$. \n\nThe next derivative to estimate is ${\\mathrm{D}}_{r_1}F$; we can see that $|\\partial_{r_1}\\Sigma_+| \\le C \\zeta$ and $|\\partial_{r_1}h_\\psi| \\le C \\zeta$; then $|\\partial_{r_1}F|\\le C+C r_1\\zeta\\le C$. We already know that $r_1$ can only increase by small amounts; hence\n\\[\n\\int_{\\psi(t_1)}^{\\psi(t_2)-\\frac{\\pi}{4}}|r_1-\\widetilde r_1|{\\mathrm{d}} \\psi \\le -C + \\int_{\\psi(t_1)}^{\\psi(t_2)-\\frac{\\pi}{4}}(\\widetilde r_1-r_1){\\mathrm{d}} \\psi \\le C.\n\\]\n\nThe last derivative to estimate is by $\\zeta$. We can estimate $|\\partial_\\zeta \\Sigma_+|\\le C |N_1|r_1 \\zeta^{-2}$ and $|\\partial_\\zeta h_\\psi|\\le C r_1\\zeta^{-2}$. This allows us to estimate $|\\partial_\\zeta F|\\le C r_1 \\zeta^{-2}$. Now $|\\zeta-\\widetilde\\zeta|\\le C r_1$ and therefore\n\\[\n\\int_{\\psi(t_1)}^{\\psi(t_2)-\\frac{\\pi}{4}}|\\partial_\\zeta F|\\,|\\zeta-\\widetilde\\zeta| {\\mathrm{d}}\\psi \\le C \\int_{\\psi(t_1)}^{\\psi(t_2)-\\frac{\\pi}{4}} r_1^2\\zeta^{-2}{\\mathrm{d}}\\psi \\le C \\int_{\\psi(t_1)}^{\\psi(t_2)-\\frac{\\pi}{4}} r_1^2\\zeta^{-1}{\\mathrm{d}} \\psi \\le C,\n\\]\nwhere the last estimate was due to \\eqref{eq:average:rprime-intbound}.\n\\end{proof}\n\n\n\\section{Phase-Space Volume and Integral Estimates}\\label{sect:volume-form}\nThis section is devoted to proving the last three main Theorems of this work. The first one is the already stated genericity of the attracting case in Theorem \\ref{thm:b8-attractor-global}:\n\\begin{thm}[Bianchi \\textsc{VIII} global attractor genericity Theorem]\\label{thm:b8-attractor-global-genericity}\nIn Theorem \\ref{thm:b8-attractor-global}, \nthe set of initial conditions $\\bpt x_0$ for which \\textsc{Attract} applies is ``generic'' in the following sense: It is open and dense in $\\mathcal M_{+-+}$ and its complement has Lebesgue-measure zero. \n\\end{thm}\nThe second theorem of this section answers affirmatively the ``locality'' part of the longstanding BKL conjecture for spatially homogeneous Bianchi class A vacuum spacetimes, for measure theoretic notions of genericity, and can be considered the main result of this work:\n\\begin{thm}[Almost sure formation of particle horizons]\\label{thm:horizon-formation}\nFor Lebesgue almost every initial condition in $\\mathcal M_{\\pm\\pm\\pm}$ (with respect to the induced measure from ${\\mathbb{R}}^5=\\{(\\Sigma_+,\\Sigma_-, N_1,N_2,N_3)\\}$), the following holds:\n\\begin{equation}\n\\int_0^\\infty \\max_i \\delta_i (t){\\mathrm{d}} t = 2\\int_{0}^\\infty \\max_{i\\neq j}\\sqrt{|N_iN_j|}(t)<\\infty \\qquad \\text{for Lebesgue a.e. }\\,\\,\\bpt x_0\\in \\mathcal M_{\\pm\\pm\\pm}.\n\\end{equation}\nThis means that almost every Bianchi \\textsc{VIII} and \\textsc{IX} vacuum spacetime forms particle horizons towards the big bang singularity. This physical interpretation is described in Section \\ref{sect:gr-phys-interpret}, Lemma \\ref{lemma:horizon-integral} or \\cite{heinzle2009future}.\n\\end{thm}\nThis result about Lebesgure a.e.~solutions immediately raises the question for counterexamples:\n\\begin{question}\nIt is currently not known, whether there exist any solutions, which are attracted to $\\mathcal A$ and have infinite integral $\\int_0^\\infty\\delta_i(t)=\\infty$. \n\n\\begin{center}Do such solutions exist? Is it possible to describe an example of such a solution?\\end{center}\n\n\\noindent We very strongly expect that such solutions \\emph{do} exist, for reasons which will be explained in future work.\n\\end{question}\n\nThe third and last theorem of this section strengthens and extends the previous result:\n\\begin{thm}[$L^p$ estimates for the generalized localization integral]\\label{thm:horizon-formation-alpha-p}\nLet either $\\alpha\\in (0,2)$ and $p\\in (0,1)$ such that $\\alpha p > 2p-1$, i.e.~$p< \\frac{1}{2-\\alpha}$.\nLet $M\\subset \\mathcal M_{\\pm\\pm\\pm}$ be a compact subset such that Theorem \\ref{thm:local-attractor} holds for any initial condition in $\\bpt x_0\\in M$. Then $I_\\alpha \\in L^p(M)$, where\n\\begin{equation}\nI_\\alpha (\\bpt x_0) := \\int_0^\\infty \\max_i \\delta_i^\\alpha(t){\\mathrm{d}} t,\n\\end{equation}\ni.e., using $\\phi$ for the flow to \\eqref{eq:ode} and ${\\mathrm{d}}^4 \\bpt x$ for the (four dimensional) Lebesgue measure on $\\mathcal M_{\\pm\\pm\\pm}$,\n\\[\n\\int_{M} \\left[\\int_0^\\infty \\delta_i^\\alpha(\\phi(\\bpt x, t)){\\mathrm{d}} t\\right]^p {\\mathrm{d}}^4 \\bpt x<\\infty\\qquad\\forall\\,i\\in\\{1,2,3\\}.\n\\]\nIf instead $\\alpha\\ge 2$, we already know from Theorem \\ref{thm:local-attractor} that $I_\\alpha \\in L^\\infty(M)$.\n\\end{thm}\nTheorem \\ref{thm:horizon-formation-alpha-p} makes a much stronger claim than Theorem \\ref{thm:horizon-formation}, even for $\\alpha=1$: Local $L^p$-integrability is a sufficient condition for a.e.~finiteness, but very much not necessary. On the other hand, we are aware of no immediate physical interpretation of Theorem \\ref{thm:horizon-formation-alpha-p}.\n\nThe proof of Theorem \\ref{thm:horizon-formation-alpha-p} won't rely on Theorem \\ref{thm:horizon-formation}. Even though the proof of Theorem \\ref{thm:horizon-formation} is therefore entirely optional, we nevertheless choose to state and prove Theorem \\ref{thm:horizon-formation} separately, because we view it as more important, and can give a more geometric proof than for Theorem \\ref{thm:horizon-formation-alpha-p}.\n\\begin{question}\nUnfortunately, the case $\\alpha=1=p=1$, i.e.~$I_1 \\in L^1_{\\textrm{loc}}$, is maddeningly out of reach of Theorem \\ref{thm:horizon-formation-alpha-p}, which only provides $I_1\\in L^{1-\\epsilon}_{\\textrm{loc}}$ for any $\\epsilon>0$. An extension to $\\alpha=p=1$ would imply \\emph{finite expectation} of the particle horizon integral.\n\nIs it possible to say something about $\\alpha=p=1$? Maybe for special compact subsets $M\\subset \\mathcal M_{\\pm\\pm\\pm}$?\n\\end{question}\n\n\\paragraph{Outline of the proofs.}\nLet us now give a short overview over the remainder of this section. Our primary tool will be a volume-form $\\omega_4$, which is \\emph{expanded} under the flow $\\phi$ of \\eqref{eq:ode}. An alternative description would be to say that we construct a density function, such that $\\phi$ is volume-expanding. The volume-form will be constructed and discussed in Section \\ref{sect:volume-construct}. We will use some very basic facts from the intersection of differential forms, measure theory and dynamical systems theory, which are given in Appendix \\ref{sect:general-vol-app} for the convenience of the reader.\n\nWe will use this expanding volume-form in order to prove our three Theorems in Section \\ref{sect:volume:proofs}.\n\n\\subsection{Volume Expansion}\\label{sect:volume-construct}\nThis section studies the evolution of phase-space volumes. Using logarithmic coordinates $\\beta_i=-\\log|N_i|$, the equations differential equations \\eqref{eq:ode} \\emph{without} the constraint \\eqref{eq:constraint} yield the remarkably simple and controllable formula ${\\mathrm{D}}_t \\omega_5 = 2N^2 \\omega_5$ for the evolution of the five-dimensional Lebesgue-measure $\\omega_5$ (with respect to $\\Sigma_\\pm, \\beta_i$). \nThis formula shows that the flow $\\phi$ expands the volume $\\omega_5$. Such a volume expansion is impossible for systems living on a manifold with finite volume; it is possible with logarithmic coordinates because these coordinates have pushed the attractor to infinity, and typical solutions escape to infinity in these coordinates. \n\n\\paragraph{Volume expansion for the extended system (without constraint $G=1$).}\nConsider coordinates $\\beta_i$ given by\n\\[\\begin{aligned}\n\\beta_i &= -\\log |N_i|&                  N_i &= {{\\hat{n}}}_i e^{-\\beta_i}\\\\\n{\\mathrm{d}} \\beta_i &= -\\frac{{\\mathrm{d}} N_i}{N_i}& {\\mathrm{d}} N_i &= -{{\\hat{n}}}_i e^{-\\beta_i}{\\mathrm{d}}\\beta_i\\\\\n\\partial_{\\beta_i} &= -N_i \\partial_{N_i} & \\partial_{N_i}&=-{{\\hat{n}}}_i e^{\\beta_i}\\partial_{\\beta_i},\n\\end{aligned}\\]\nand consider the Lebesgue-measure with respect to the $\\beta_i$ coordinates:\n\\begin{equation}\\label{eq:omega5-def}\n\\omega_5 = |{\\mathrm{d}} \\Sigma_+ \\land {\\mathrm{d}} \\Sigma_- \\land {\\mathrm{d}} \\beta_1 \\land {\\mathrm{d}} \\beta_2 \\land {\\mathrm{d}} \\beta_3|,\n\\end{equation}\nwhich is given in $N_i$ coordinates by\n\\[\\begin{aligned}\n\\omega_5 &= \\left|\\frac{-1}{N_1N_2N_3}{\\mathrm{d}} \\Sigma_+ \\land {\\mathrm{d}} \\Sigma_- \\land {\\mathrm{d}} N_1 \\land {\\mathrm{d}} N_2 \\land {\\mathrm{d}} N_3\\right|.\n\\end{aligned}\\]\nLet $\\lambda(\\bpt x, t)$ denote the volume expansion for $\\phi(\\bpt x, t)$, i.e.\n\\[\n\\phi^*(\\bpt x, t)\\omega_5 = \\lambda(\\bpt x, t)\\omega_5,\n\\]\nwhere $\\phi^*$ is the pull-back acting on differential forms. Hence, in $(\\bpt \\Sigma,\\bpt \\beta)$-coordinates, $\\lambda(\\bpt x, t)=\\det \\partial_x \\phi(\\bpt x, t)$, and, with $f$ denoting the vectorfield corresponding to \\eqref{eq:ode}:\n\\[\\begin{aligned}\n{\\mathrm{D}}_t \\phi^*(\\bpt x, t)\\omega_5 &= \\left[\\mathrm{tr}\\,\\partial_x \\partial_t \\phi(\\bpt x, t)\\right] \\phi^*(\\bpt x, t)\\omega_5 \\\\\n&= \\left[\\partial_{\\Sigma_+}f_{\\Sigma_+}+\\ldots+\\partial_{\\beta_3}f_{\\beta_3}\\right] \\lambda(\\bpt x, t)\\omega_5=2N^2(\\phi(\\bpt x, t))\\lambda(\\bpt x, t)\\omega_5\\\\\n\\lambda(\\bpt x, t) &= \\exp\\left[ 2\\int_0^t N^2(\\phi(\\bpt x, s)){\\mathrm{d}} s\\right].\n\\end{aligned}\\]\nThe volume is really expanding: In most of the phase space, $N^2>0$, and always $N^2> -4|N_1N_2N_3|^{\\frac 2 3}$, which has bounded integral.\n\n\\paragraph{Volume expansion on $\\mathcal M$.}\nWe are not really interested in the behaviour of $\\phi$ on ${\\mathbb{R}}^5$, and the measure $\\omega_5$. Instead, we are interested in dynamics and measures on the set $\\mathcal M=\\{\\bpt x\\in {\\mathbb{R}}^5:\\, G(\\bpt x)=1\\}$. We can get an induced measure on $\\mathcal M$ by choosing a vectorfield $X:{\\mathbb{R}}^5\\to T{\\mathbb{R}}^5$ such that ${\\mathrm{D}}_X G =1$ in a neighborhood of $\\mathcal M$. Then we set\n\\begin{equation}\\label{eq:omega4-def}\n\\begin{multlined}\n\\omega_4=\\iota_{X}\\omega_5,\\\\\n\\text{i.e.}\\qquad\\omega_4[X_1,\\ldots, X_4]=\\omega_5[X,X_1,\\ldots X_4]\\qquad\\text{for}\\quad X_1,\\ldots, X_4\\in T\\mathcal M.\n\\end{multlined}\\end{equation}\nThis induced volume is independent of the choice of $X$ (as long as ${\\mathrm{D}}_X G=1$), and fulfills (see Section \\ref{sect:general-vol-app}\n\\[\n\\phi^*(\\bpt x, t)\\omega_4 = \\lambda(\\bpt x, t)\\omega_4 = \\frac{\\phi^*(\\bpt x, t)\\omega_5}{\\omega_5}\\omega_4.\n\\]\n\n\\paragraph{Volume expansion for Poincar\\'e-maps.}\nWe consider the volume-form $\\omega_3=\\iota_f \\omega_4$, where $f$ is the vectorfield \\eqref{eq:ode}. By invariance of $f$ under $\\phi$, we again have $\\phi^*(\\bpt x, t)\\omega_3=\\lambda(\\bpt x, t)\\omega_3$. \n\nLet $U\\subseteq \\mathcal M$ open and $T:U\\to {\\mathbb{R}}$ differentiable. Then the map $\\Phi_T:U\\to \\mathcal M$ with $\\Phi_T(\\bpt x)=\\phi(\\bpt x, T(\\bpt x))$ has $\\Phi_T^*\\omega_3=\\lambda(\\bpt x, T(\\bpt x))\\omega_3|U$ (also see Section \\ref{sect:general-vol-app}).\n\nThis especially holds when $S\\subset M$ is a Poincar\\'e-section and $\\Phi_S=\\Phi_{T_S}$ is the corresponding Poincar\\'e-map, i.e.~when $S\\subset \\mathcal M$ is a submanifold of codimension one, which is transverse to $f$, and $T=T_S(\\bpt x)= \\inf\\{t>0:\\,\\phi(\\bpt x, t)\\in S\\}$.\n\nIf $S\\subseteq \\mathcal M$ is a Poincar\\'e-section and $K\\subseteq S$ is a set with $|K|_{\\omega_3}=\\int_{S}\\mathrm{id}_{K}\\omega_3 =0$, then, by Fubini's Theorem, $|\\phi(K, {\\mathbb{R}})|_{\\omega_4}=0$. The measure $\\omega_4$ is absolutely bi-continuous with respect to the ordinary Lebesgue measure, i.e.~the notions of sets of measure zero coincide for the ordinary Lebesgue-measure and $\\omega_4$.\n\nThis especially applies to the boundary $\\partial S$ with $|\\partial S|=0$. Therefore, if $S_0,S$ are two Poincar\\'e-sections and $K\\subseteq S_0\\subseteq M$ is a set of initial conditions, such that $T_S(\\bpt x)<\\infty$ for almost every $\\bpt x\\in K$, then\n\\[\n\\int_{\\Phi_S(K)\\subseteq S} f(\\bpt x)\\omega_3 = \\int_{K\\subseteq S_0}f(\\Phi_S^{-1}(\\bpt x))\\lambda(\\bpt x, T_S(\\bpt x))\\omega_3,\n\\]\nfor any $L^\\infty$ function $f$. This is because the map $\\Phi_S$ is, by assumption, sufficiently smooth almost everywhere. \n\nOur primary source of ``almost everywhere'' statements is the following:\n\\begin{lemma}\\label{lemma:vol-dichotomy}\nLet $S\\subset \\mathcal M$ be a Poincar\\'e-section with $N^2>\\epsilon>0$, and let $K\\subseteq S \\subset \\mathcal M$ be forward invariant, i.e. $\\Phi_S$ is well-defined in $K$ and $\\Phi_S(K)\\subseteq K$. \n\nThen either $|K|_{\\omega_3}=0$ or $|K|_{\\omega_3}=\\infty$. If $|K|_{\\omega_3}=0$, then $|\\phi(K,{\\mathbb{R}})|_{\\omega_4}=0$.\n\\end{lemma}\n\\begin{proof}\nSince $\\Phi_S(K)\\subseteq K$, we have\n\\[|K|_{\\omega_3}\\ge |\\Phi_S(K)|_{\\omega_3} = \\int_S \\mathrm{id}_{\\Phi_S(K)}\\omega_3 = \\int_S \\mathrm{id}_K \\phi_S^*\\omega_3 = \\int_S \\mathrm{id}_K \\lambda \\omega_3 \\ge q |K|_{\\omega_3},\\]\nwhere $q=\\inf_{\\bpt x \\in K}\\lambda(\\bpt x, T_S(\\bpt x))>1$. Therefore, either $|K|_{\\omega_3}=0$ or $|K|_{\\omega_3}=\\infty$.\nIf $|K|_{\\omega_3}=0$ then this implies $|\\phi(K,[-h,h])|_{\\omega_4}=0$ for small $h>0$ by Fubini's Theorem and hence $|\\phi(K,{\\mathbb{R}})|_{\\omega_4}=0$.\n\\end{proof}\n\\subsection{Proofs of the main Theorems}\\label{sect:volume:proofs}\nWe will now use the $\\omega_3$-expansion between Poincar\\'e-sections in order give proofs of the main results.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:b8-attractor-global-genericity} (Bianchi \\textsc{VIII} global attractor genericity)]\\label{proof:thm:b8-attractor-global-genericity}\n\\textsc{Attract} holds for an open set of initial conditions (by virtue of continuity of the flow). Since \\textsc{Taub} can only happen on an embedded submanifold of lower dimension, it suffices to prove that \\textsc{Except} happens only for a set of initial conditions with Lebesgue measure zero.\n\nLet \\textsc{Except} also denote the set of initial conditions for which the case \\textsc{Except} holds. Without loss of generality, we will consider the case $\\mathcal M_{+-+}$ and trajectories bouncing between $+\\taubp_1$ and $-\\taubp_1$.\n\n\nWe choose a small Poincar\\'e-section \n\\[S\\subseteq\\{\\bpt x\\in\\mathcal M:\\,N_1 = \\textrm{const}=h,\\, r_1 \\le \\epsilon,\\, \\delta_1\\le \\epsilon\\},\\]\nintersecting the unique heteroclinic $W^u(-\\taubp_1)$, which connects $-\\taubp_1$ to $\\taubp_1$, near $\\taubp_1$, such that $S$ is a graph over $(\\Sigma_-, N_2, N_3)$. Let $T_S:S\\to (0,\\infty]$ be the (partially defined) recurrence time and $\\Phi:S\\to S$ with $\\Phi(\\bpt x)= \\phi(\\bpt x, T(\\bpt x))$ be the (partially defined) Poincar\\'e-map to $S$.\n\n\n\n\\textsc{Except} is (by definition) invariant under the flow and $\\Phi(\\textsc{Except}\\cap S)\\subseteq \\textsc{Except}\\cap S$. Hence $\\textsc{Except}\\cap S$ has either vanishing or infinite $\\omega_3$-volume, by Lemma \\ref{lemma:vol-dichotomy}. Therefore, it suffices to prove $|\\textsc{Except}\\cap S|_{\\omega_3}<\\infty$.\n\n\nRecalling the local attractor Theorem \\ref{thm:local-attractor}, we can note $\\delta_1 > \\widetilde \\epsilon r_1$ for every $\\bpt x\\in \\textsc{Except}\\cap S$. This allows us to estimate\n\\begin{subequations}\n\\label{eq:bad-finite-measure}\\begin{align}\n|\\textsc{Except}\\cap S|_{\\omega_3} &\\le \\left|\\{\\bpt x\\in S:\\, \\delta_1 > \\widetilde \\epsilon r_1\\}\\right|_{\\omega_3}\n\\le \\left|\\{\\bpt x\\in S:\\, \\delta_1 > \\widetilde \\epsilon |\\Sigma_-|\\}\\right|_{\\omega_3}\\\\\n&\\le \\left|\\{\\bpt x\\in S:\\,\\beta_2+\\beta_3 > C+C|\\log|\\Sigma_-||\\}\\right|_{\\omega_3}\\\\\n&= \\int_{\\{\\bpt x\\in S:\\, \\beta_2+\\beta_3 > C+C|\\log|\\Sigma_-||\\}} \\omega_3[\\ldots] \\ |{\\mathrm{d}} \\Sigma_-\\land {\\mathrm{d}} \\beta_2 \\land{\\mathrm{d}} \\beta_3|\\label{eq:bad-finite-measure:last-s}\\\\\n&\\le C \\int_{\\{\\bpt x\\in S:\\, \\beta_2+\\beta_3 > C+C|\\log|\\Sigma_-||\\}}|{\\mathrm{d}} \\Sigma_-\\land {\\mathrm{d}} \\beta_2 \\land{\\mathrm{d}} \\beta_3|\\label{eq:bad-finite-measure:last-m1}\\\\\n&\\le C+ C \\int\\left|\\log |\\Sigma_-|\\right|^2 {\\mathrm{d}} \\Sigma_- <\\infty\\label{eq:bad-finite-measure:last},\n\\end{align}\n\\end{subequations}\nwhere we integrated $\\beta_2,\\beta_3$, using $\\beta_2,\\beta_3\\ge 0$ from \\eqref{eq:bad-finite-measure:last-m1} to \\eqref{eq:bad-finite-measure:last}. In order to go from \\eqref{eq:bad-finite-measure:last-s} to \\eqref{eq:bad-finite-measure:last-m1}, we used that (in $S$) $N_1=\\mathrm{const}=h$ and $\\Sigma_+=\\Sigma_+(\\Sigma_-, \\beta_2,\\beta_3)$ and \n\\[\\begin{aligned}\n\\omega_3 &= |{\\mathrm{d}} \\Sigma_-\\land {\\mathrm{d}} \\beta_2 \\land {\\mathrm{d}} \\beta_3|\\\\\n&\\qquad\\cdot\\,\\left|\\omega_5\\left[f,\n|\\partial_{\\beta_1} G|^{-1} \\partial_{\\beta_1},\n\\partial_{\\Sigma_-}+\\partial_{\\Sigma_-}\\Sigma_+\\partial_{\\Sigma_-}, \n\\partial_{\\beta_2}+\\partial_{\\beta_2}\\Sigma_+\\partial_{\\Sigma_+},\n\\partial_{\\beta_3}+\\partial_{\\beta_3}\\Sigma_+\\partial_{\\Sigma_+}\\right]\\right|\\\\\n&\\le C |{\\mathrm{d}} \\Sigma_-\\land {\\mathrm{d}} \\beta_2 \\land {\\mathrm{d}} \\beta_3|.\n\\end{aligned}\\]\nThis can be seen by noting $\\partial_{\\beta_1} G  = 2 N_1^2 - 2 N_1N_- \\ge C>0$ if $\\epsilon>0$ is small enough and likewise $f_{\\Sigma_+}> C >0$, $|\\partial_{\\Sigma_-, \\beta_2,\\beta_3}\\Sigma_+| \\le \\mathcal O(\\epsilon)$.\n\\end{proof}\nWe will now give the first, more geometric proof of Theorem \\ref{thm:horizon-formation}. Readers who prefer the arithmetic variant can skip ahead to the proof of Theorem \\ref{thm:horizon-formation-alpha-p}, page \\pageref{proof:thm:horizon-formation-alpha-p}. The theorem follows from the following two Lemmas:\n\n\\begin{lemma}\\label{lemma:badrecurrent-zeroset}\nFix $i\\in\\{1,2,3\\}$, without loss of generality $i=1$. \nConsider a small Poincar\\'e-section $S$ as in the proof of Theorem \\ref{thm:b8-attractor-global-genericity}, i.e. $S\\subseteq \\mathcal M_{{{\\hat{n}}}}$ intersecting the unique heteroclinic connecting $-\\taubp_1$ to $\\taubp_1$,  near \n$\\taubp_1$ such that Proposition \\ref{prop:neartaub:main} holds and $S$ is a smooth graph over $\\Sigma_-, N_2, N_3$.\n\nDenote\n\\[\\begin{aligned}\n\\textsc{Bad}&=\\{\\bpt x\\in S:\\, \\delta_1 > |\\Sigma_-|^4\\}\\\\\n\\textsc{BadRecurrent}&=\n\\{\\bpt x\\in S:\\, \\bpt \\Phi_S^k(x)\\in\\textsc{Bad}\\, \\text{for infinitely many $k$}\\},\n\\end{aligned}\\]\nwhere $\\Phi_S$ is the Poincar\\'e-map to $S$. Then\n\\(\n\\left|\\textsc{BadRecurrent}\\right|_{\\omega_3}=0.\n\\)\n\\end{lemma}\n\\begin{lemma}\\label{lemma:badrecurrent-small-int}\nConsider the setting of Lemma \\ref{lemma:badrecurrent-zeroset}.\n\nLet $\\bpt x_0\\in S\\cap \\textsc{Basin}$ such that $\\bpt x_0\\not\\in \\textsc{BadRecurrent}$. Then $\\int_0^\\infty \\delta_1(\\phi(\\bpt x_0, t)){\\mathrm{d}} t<\\infty$.\n\\end{lemma}\n\\begin{proof}[Proof of Theorem \\ref{thm:horizon-formation} (almost sure formation of particle horizons)]\nFollows trivially from Lemma \\ref{lemma:badrecurrent-small-int} and Lemma \\ref{lemma:badrecurrent-zeroset}.\n\\end{proof}\n\\begin{proof}[Proof of Lemma \\ref{lemma:badrecurrent-small-int}]\nLet $\\bpt x_0\\not\\in \\textsc{BadRecurrent}$, and let $(T_n)_{n\\in{\\mathbb{N}}}\\subseteq {\\mathbb{R}}$ and $(\\bpt x_n)_{n\\in{\\mathbb{N}}}\\subseteq S$ be the recurrence times and points in $S$, i.e.~$\\bpt x_{n+1}=\\Phi_S(\\bpt x_n)=\\phi(\\bpt x_{n}, T_n)$, $T_n = T_S(\\bpt x_{n-1})$. By assumptions, we have some $N>0$ such that, for all $n\\in{\\mathbb{N}}$, $\\bpt x_{N+k}\\not\\in\\textsc{Bad}$. We can estimate\n\\[\\begin{aligned}\n\\int_0^\\infty\\delta_1(\\phi(\\bpt x_0, t)){\\mathrm{d}} t &= \\int_{0}^{T_n}\\delta_1(\\phi(\\bpt x_0, t)){\\mathrm{d}} t + \\sum_{n=N}^{\\infty}\\int_{0}^{T_n}\\delta_1(\\phi(\\bpt x_{n}, t )){\\mathrm{d}} t\\\\\n&\\le C(\\bpt x_0) + \\sum_{n=N}^{\\infty}\\int_{0}^{T_n}C\\exp\\{-\\frac{C}{r_1^2(\\bpt x_n)}\\}\\delta_1(\\bpt x_n){\\mathrm{d}} t\\\\\n& \\le C(\\bpt x_0)+ \\sum_{n=N}^{\\infty}C\\frac{\\delta_1}{r_1^2}(\\bpt x_n)\\\\\n& \\le C(\\bpt x_0)+ \\sum_{n=N}^{\\infty}C\\sqrt{\\delta_1}(\\bpt x_n) <\\infty,\n\\end{aligned}\\]\nwhere we used Proposition \\ref{prop:neartaub:main} and $\\delta_1 < r_1^4$ and the fact that $\\delta_1(\\bpt x_{n+1})< \\frac 1 2 \\delta_1(\\bpt x_n)$.\n\\end{proof}\n\\begin{proof}[Proof of Lemma \\ref{lemma:badrecurrent-zeroset}]\nAnalogously to the proof of Theorem \\ref{thm:b8-attractor-global-genericity}, we have $|\\textsc{Bad}|_{\\omega_3}<\\infty$. Then, we can write\n\\[\\begin{aligned}\n\\left|\\textsc{BadRecurrent}\\right|_{\\omega_3}&=\\left|\\bigcap_{n\\in{\\mathbb{N}}}\\bigcup_{k\\ge n} \\Phi^{-k}(\\textsc{Bad})\\right|_{\\omega_3}\\\\\n&\\le \\lim_{n\\to 0} \\sum_{k\\ge n}\\left|\\Phi^{-k}(\\textsc{Bad})\\right|_{\\omega_3} \n\\le \\lim_{n\\to 0} \\sum_{k\\ge n}q^{-k}\\left|\\textsc{Bad}\\right|_{\\omega_3}<\\infty,\n\\end{aligned}\\]\nwhere $q=\\inf \\{\\lambda(\\bpt x, T_S(\\bpt x)): \\bpt x\\in S\\}>1$.\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{thm:horizon-formation-alpha-p} ($L^p$ estimates for the generalized localization integral)]\\label{proof:thm:horizon-formation-alpha-p}\nThe claim for $\\alpha\\in [2,\\infty)$ follows trivially from Theorem \\ref{thm:local-attractor}. For the other case, we restrict our attention without loss of generality to $\\delta_1$. Furthermore, we can without loss of generality assume that we start with $C\\subseteq S$ for some Poincar\\'e section and estimate the integral with respect to $\\omega_3$. Recall the construction in the proof of Theorem \\ref{thm:b8-attractor-global-genericity} with a Poincar\\'e-section $S_1$ near the heteroclinic $-\\taubp_1\\to \\taubp_1$. We can estimate, for some positive $s\\in (2p-1, \\alpha p)$: \n\\begin{subequations}\n\\begin{align*}\n\\int_{\\bpt x\\in C} \\biggl[\\int_0^\\infty \\delta_i^\\alpha(&\\phi(\\bpt x, t)){\\mathrm{d}} t\\biggr]^p \\omega_3\n= \\int_{\\bpt x\\in C} \\left[\\sum_{n} \\int_{T_{n}}^{T_{n+1}}\\delta_i^\\alpha(\\phi(\\bpt x, t)){\\mathrm{d}} t\\right]^p \\omega_3\\\\\n&\\le \\sum_{n} \\int_{\\bpt x\\in C} \\left[\\int_{T_{n}}^{T_{n+1}}\\delta_i^\\alpha(\\phi(\\bpt x, t)){\\mathrm{d}} t\\right]^p \\omega_3\\numberthis\\label{eq:foobar:p}\\\\\n&\\le C\\sum_{n} \\int_{\\bpt x\\in C} \\left[\\frac{\\delta_i^\\alpha}{r_{i}^{2}}(T_n)\\right]^p \\omega_3\\\\\n&= C\\sum_{n} \\int_{\\bpt x\\in C}  \\left[ \\delta_i^{\\alpha p -s} r_i^{-2p +s} \\left(\\frac{\\delta_i}{r_i}\\right)^{s}\\right](T_n)\\omega_3\\\\\n&\\le C\\sum_{n}\\, \\sup_{\\bpt x\\in C} \\left(\\frac{\\delta_i}{r_i}\\right)^{s}(T_n)\\,\\,\\cdot\\,\\,  \\int_{\\bpt x\\in C} \\left[\\delta_i^{\\alpha p -s} r_i^{-2p +s}\\right] (T_n)\\omega_3.\\numberthis\\label{eq:foobar:hoelder}\n\\end{align*}\n\\end{subequations}\n\nWe have used $p<1$ in order to split the integral in \\eqref{eq:foobar:p} and the H\\\"older inequality in \\eqref{eq:foobar:hoelder}.\n We continue the estimates by noting that $\\sup_{\\bpt x\\in C}\\frac {\\delta_i} {r_i}(T_n)$ decreases exponentially in $n$. Hence we only need to bound the second factor. This can be done by using $\\alpha p -s>0$ and $-2p+s > -1$ in order to see\n\\[\\begin{aligned}\n\\int_{\\bpt x\\in C} \\left[\\delta_i^{\\alpha p -s} r_i^{-2p +s}\\right] (T_n)\\omega_3 \n&\\le C \\int_{S_1} \\left[\\delta_i^{\\alpha p -s} r_i^{-2p +s}\\right] \\omega_3\\\\\n&\\le C \\int_{-0.1}^{0.1} \\left[\\int_{\\beta_2,\\beta_3\\ge 0} e^{-\\frac{(\\beta_2+\\beta_3)(\\alpha p-s)}{2}}|\\Sigma_-|^{-2p+s} {\\mathrm{d}} \\beta_2\\land{\\mathrm{d}} \\beta_3  \\right] {\\mathrm{d}} \\Sigma_-\\\\\n&\\le C \\int_{-0.1}^{0.1}|\\Sigma_-|^{-2p+s} {\\mathrm{d}} \\Sigma_-  < \\infty.\n\\end{aligned}\\]\n\\end{proof}\n\n\n\n\\section{Physical Properties of Solutions for Bianchi \\textsc{VIII} and \\textsc{IX}}\\label{sect:gr-phys-interpret}\nWe will now use the results of this work in order to describe some physical properties of Bianchi spacetimes. Recall Section \\ref{sect:equations}, where we describe Bianchi spacetimes in terms of the Wainwright-Hsu equations. In Section \\ref{sect:append:derive-eq}\n\n\n\\paragraph{Bounded life-time.}\nSince the mean curvature $H$ corresponds to the time-derivative of the spatial volume form $\\sqrt{g_{11}g_{22}g_{33}}$, the universe described by such a metric is contracting for $H<0$. Physically, we are interested in the behaviour towards the initial (big bang) singularity; this setting is time-reversed to physical time-variables, and we should look at the behaviour of solutions for $t\\to +\\infty$. Since $|H|$ is at least uniformly exponentially growing, we can immediately see \\( \\textsc{EigenFuture} = \\int_{0}^\\infty \\sqrt{-g_{00}}{\\mathrm{d}} t <\\infty,\\) that is, the universe has only a finite (eigen-) lifetime until $H$ blows up and a singularity occurs. In our coordinates, this singularity is placed at $t=+\\infty$.\n\nA priori, we cannot know whether this singularity is a physical singularity (with curvature blow-up) or just a coordinate singularity (it might be that it is just our coordinate system, which blows up, while the actual space-time remains regular). In the case of Bianchi \\textsc{VIII} and \\textsc{IX}, the singularity is physical (with curvature blow-up). \nIt has been proven in \\cite{ringstrom2001bianchi} that the singularity is physical and curvature blows up. This is done by considering the so-called Kretschmann scalar\n$\\kappa = \\sum_{\\alpha,\\beta,\\gamma,\\delta}R_{\\alpha,\\beta}^{\\ \\ \\delta,\\gamma} R_{\\delta,\\gamma}^{\\alpha,\\beta}$ and showing that $\\lim_{t\\to\\infty}\\kappa(t)=\\infty$. We refer to \\cite{ringstrom2001bianchi} for the details.\n\\paragraph{Bounded spatial metric coefficients.}\nNow we restrict our attention to the case of Bianchi \\textsc{IX} and \\textsc{VIII}, where all $\\hat n_i\\neq 0$. \n\n\\noindent The coefficients $g_{ii}=\\frac 1 {48}\\frac {|N_i|}{H^2 |N_1N_2N_3|}$ stay bounded: We know, ssing the global attractor Theorems \\ref{thm:b9-attractor-global} and \\ref{thm:b8-attractor-global}, that the $|N_i|$ stay bounded. We can compute:\n\\[\n{\\mathrm{D}}_t \\log |H^2N_1N_2N_3|= -3\\Sigma^2 + 1 + 2\\Sigma^2 = N^2 \\ge -4|N_1N_2N_3|^{\\frac 2 3},\n\\]\nwhich shows that $|H^2N_1N_2N_3|$ stays bounded away from zero, using Lemma \\ref{farfrom-A:lemma:uniform}. Hence all three $g_{ii}$ stay bounded for $t\\to+\\infty$. Indeed, since $N^2>\\epsilon>0$ for large amounts of time, we can conclude $\\lim_{t\\to+\\infty} g_{ii}(t)=0$.\n\n\\paragraph{Particle Horizons.}\nRecall question of particle horizons from the introduction, and the definition of communication cones \\eqref{eq:intro:comcone}, which we here adjust to match our convention that the big bang singularity is situated in the future:\n\\[\\begin{aligned}\n&\\text{Singularity directed light cone of $\\bpt p$:}\\\\\n&\\quad J^-(\\bpt p) = \\{\\bpt q:\\, \\text{there is } \\gamma:[0,1]\\to M\\,\\,\\text{with}\\, \\gamma(0)=\\bpt p, \\gamma(1)=\\bpt q,\\,\\text{time-like future directed}\\}\\\\\n&\\text{Non-singularity directed light cone of $\\bpt p$:}\\\\\n&\\quad J^+(\\bpt p) = \\{\\bpt q:\\, \\text{there is } \\gamma:[0,1]\\to M\\,\\,\\text{with}\\, \\gamma(0)=\\bpt p, \\gamma(1)=\\bpt q,\\,\\text{time-like past directed}\\}\\\\\n&\\text{Communication cone of $\\bpt p$:}&\\\\\n&\\quad \\phantom{\\partial}J^+(J^-(\\bpt p)) = \\bigcup_{\\bpt q \\in J^-(\\bpt p)}J^+(\\bpt q)\\\\\n&\\text{Cosmic horizon of $\\bpt p$:}&\\\\\n&\\quad \\partial J^+(J^-(\\bpt p)) = \\text{the topological boundary of the past communication cone.}\n\\end{aligned}\\]\nWe can now relate the question of particle horizons with our estimates on $\\int \\sqrt{|N_iN_j|}(t) {\\mathrm{d}} t$. This gives the physical interpretation of Theorem \\ref{thm:horizon-formation}. Remember that time is oriented such that the big bang singularity is situated in the future at $t=\\infty$.\n\\begin{lemma}\\label{lemma:horizon-integral}\nThere is a constant $C>0$, such that, for Bianchi \\textsc{IX} and \\textsc{VIII} vacuum spacetimes $M$, we can estimate for $\\bpt p\\in M$ and $t_0\\ge t(\\bpt p)$\n\\begin{equation}\\label{eq:comcone-estimate}\\begin{aligned}\n\\mathrm{diam}_h \\left[J^-(\\bpt p)) \\cap \\{\\bpt q\\in M: t(\\bpt q)=t_0\\}\\right] &\\le C\\int_{t(\\bpt p)}^{t_0}  \\max_{j\\neq k} \\sqrt{|N_jN_k|}(t){\\mathrm{d}} t\\\\\n\\mathrm{diam}_h \\left[J^+(J^-(\\bpt p)) \\cap \\{\\bpt q\\in M: t(\\bpt q)=t(\\bpt p)\\}\\right] &\\le C \\int_{t(\\bpt p)}^\\infty \\max_{j\\neq k} \\sqrt{|N_jN_k|}(t){\\mathrm{d}} t,\n\\end{aligned}\\end{equation}\nwhere the diameter is measured with the symmetry metric $h$ given on the surfaces of homogeneity $\\{t=\\mathrm{const}\\}$ by $h=\\omega_1 \\otimes \\omega_1 + \\omega_2\\otimes \\omega_2 + \\omega_3\\otimes \\omega_3$. \nFurthermore, suppose that $M$ is a spacetime corresponding to a Bianchi \\textsc{VIII} or \\textsc{IX} solution with $\\int_0^\\infty \\delta_i(t){\\mathrm{d}} t <\\infty$ for all $i\\in\\{1,2,3\\}$, as in the conclusion of Theorem \\ref{thm:horizon-formation}. Use the shorthand $J^+(J^-(t_0))=J^+(J^-(\\bpt p))$ for some $\\bpt p\\in M$ with $t(\\bpt p)=t_0$.\nThen the following holds:\n\\begin{enumerate}\n\\item $\\lim_{t_0\\to\\infty} \\mathrm{diam}_h\\left[J^+(J^-(t_0))\\cap \\{\\bpt q\\in M: t(\\bpt q)=t_0\\}\\right]=0$.\n\\item $\\lim_{t_0\\to\\infty} \\mathrm{diam}_g\\left[J^+(J^-(t_0))\\cap \\{\\bpt q\\in M: t(\\bpt q)=t_0\\}\\right]=0$.\n\\item For $t_0>0$ large enough, $\\partial J^+(J^-(t_0))\\neq \\emptyset$.\n\\item For $t_0>0$ large enough, the communication cone is homeomorphic to $(0,1)$ times the three-dimensional unit-ball with boundary. In other words, the following manifolds with boundary are homeomorphic, where $B_1^{{\\mathbb{R}}^3}(0)$ is the three-dimensional unit-ball:\n\\begin{multline}\\nonumber\n\\left[J^+(J^-(t_0)) \\cap \\{\\bpt q\\in M: t(q) > t_0\\}\\,,\\, \\partial J^+(J^-(t_0)) \\cap \\{\\bpt q\\in M: t(q) > t_0\\}\\right]\\\\\n\\sim \\left[(0,1) \\times B_1^{{\\mathbb{R}}^3}(0)\\,,\\, (0,1)\\times \\partial B_1^{{\\mathbb{R}}^3}(0)\\right]\\end{multline}\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof\nAny time-like singularity-directed curve $\\gamma$ starting in $\\bpt p$ must fulfill $|{\\mathrm{D}}_t \\gamma_i| \\le \\sqrt{-g_{00} g^{ii}}=\\sqrt{12}\\sqrt{\\widetilde N_j \\widetilde N_k}$. It is clear that the $h$-length of such a curve must be bounded by $C\\max_{j\\neq k} \\int_{t(\\bpt p)}^\\infty \\sqrt{\\widetilde N_j \\widetilde N_k}(t){\\mathrm{d}} t$ (parametrized over the time $t$ corresponding to \\eqref{eq:ode-from-gr}), if the curve only accesses times later than $t_0$. This proves the estimate on $J^-(\\bpt p)$. The estimate on $J^+(J^-(\\bpt p))$ follows.\n\nIf $\\int_{t_0}^{\\infty}\\max_{j\\neq k}\\sqrt{|N_jN_k|}(t){\\mathrm{d}} t <\\infty$, then $\\lim_{t_0\\to\\infty}\\int_{t_0}^{\\infty}\\max_{j\\neq k}\\sqrt{|N_jN_k|}(t){\\mathrm{d}} t=0$, which proves $(1)$. $(2)$ follows because the metric coefficients $g_{ij}$ are bounded. $(4)$ follows because the injectivity radius $\\mathrm{inj}_h(t)$ of the hypersurfaces of homogeneity is independent of the time $t$, if we measure it with respect to the (time-independent) $h$-metric. $(3)$ follows trivially from $(4)$.\n\\end{proof}\nHence, Theorem \\ref{thm:horizon-formation} really shows that almost every Bianchi \\textsc{VIII} and \\textsc{IX} vacuum solution forms particle horizons.\n\n\n\\begin{figure}[hbt]\n\\centering\n\\begin{subfigure}[t]{0.48\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{.\/graphics\/recombination-cone-comm.pdf}\n\\caption{A sketched space-time, where homogeneity of the observable universe could be explained by mixing between the big bang and recombination.}\n\\end{subfigure}~~\n\\begin{subfigure}[t]{0.48\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{.\/graphics\/recombination-cone-noncomm.pdf}\n\\caption{A sketched space-time, where homogeneity of the observable universe cannot be explained by mixing between the big bang and recombination.}\n\\end{subfigure}\n\\caption{Whether observed homogeneity could be explained by mixing depends on the relation of the conformal distance between recombination and singularity versus the conformal distance between the present time and recombination.}\\label{fig:recomb-cone}\n\\end{figure}\nIn the introduction, we promised to give some more details on the relation of the formation of particle horizons to homogenization of the universe by mixing. Astronomical observations show that the universe appears to be mostly homogeneous at large scales. However, optical astronomical observations go back only to the recombination, the moment where the primordial plasma condensed to a gas and became transparent to light. Hence, there is only a bounded region of space-time in our past, which is optically accessible. A possible explanation for the observed homogeneity, c.f.~e.g.~\\cite{misner1969mixmaster}, might be that the universe, i.e.~matter, radiation, etc, mixed in the time-frame between the initial singularity and recombination. This is only possible if the outer parts of our optical past light-cone have a joint causal past (see Figure \\ref{fig:recomb-cone}). This is a reason that the formation of particle horizons does not necessarily spell doom for attempts to explain the observed homogeneity through mixing \n(apart from the actual universe not beeing a homogeneous, anisotropic vacuum spacetime of Bianchi type $\\textsc{VIII}$ or $\\textsc{IX}$).\n\nIndeed, the same problem is present in the ``standard model'' of cosmology, which is a (homogeneous isotropic) FLRW model. There, the observed homogeneity is typically explained by inflation, i.e.~one postulates a phase of rapid expansion that increases the conformal distance between recombination and singularity, mediated by an exotic, hitherto unobserved matter field.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDeep learning is revolutionizing big-data applications, after being responsible for groundbreaking improvements in image classification \\cite{krizhevsky2012imagenet} and speech recognition \\cite{hinton2012deep}.  With the recent upsurge in data, at a much faster rate than our computing capabilities, neural networks are growing larger to process information more effectively. In 2012, state-of-the-art convolutional neural networks contained at most 10 layers. Afterward, each successive year has brought deeper architectures with greater accuracy. Last year, Microsoft's deep residual network \\cite{he2015deep}, which won the ILSVRC 2015 competition with a 3.57\\% error rate, had 152 layers and 11.3 billion FLOPs. To handle such large neural networks, researchers usually train them on high-performance graphics cards or large computer clusters. \n\nThe basic building block of a neural network is a neuron. Each neuron activates when a specific feature appears in its input. The magnitude of a neuron's activation is a measure of the neuron's confidence for the presence or absence of a feature. In the first layer of deep network, the neurons detect simple edges in an image. Each successive layer learns more complex features. For example, the neurons will find parts of a face first, such as the eyes, ears, or mouth, before distinguishing between different faces. For classification tasks, the final layer is a classic linear classifier - Softmax or SVM. Multiple layers combine to form a complex, non-linear function capable of representing any arbitrary function \\cite{cybenko1989approximation}. Adding more neurons to a network layers increases its expressive power. This enhanced expressive power has made massive-sized deep networks a common practice in large scale machine learning systems, which has shown some very impressive boost over old benchmarks.  \n\nDue to the growing size and complexity of networks, efficient algorithms for training massive deep networks in a distributed and parallel environment is currently the most sought after problem in both academia and the commercial industry. For example, Google \\cite{dean2012large} used a 1-billion parameter neural network, which took three days to train on a 1000-node cluster, totaling over 16,000 CPU cores. Each instantiation of the network spanned 170 servers. In distributed computing environments, the parameters of giant deep networks are required to be split across multiple nodes. However, this setup requires costly communication and synchronization between the parameter server and processing nodes in order to transfer the gradient and parameter updates. The sequential and dense nature of gradient updates prohibits any efficient splitting (sharding) of the neural network parameters across computer nodes. There is no clear way to avoid the costly synchronization without resorting to some ad-hoc breaking of the network. This ad-hoc breaking of deep networks is not well understood and is likely to hurt performance. Synchronization is one of the major hurdles in scalability. Asynchronous training is the ideal solution, but it is sensitive to conflicting, overlapping parameter updates, which leads to poor convergence.\n\nWhile deep networks are growing larger and more complex, there is also push for greater energy efficiency to satisfy the growing popularity of machine learning applications on mobile phones and low-power devices. For example, there is recent work by \\cite{mcmahan2016federated} aimed at leveraging the vast data of mobile devices. This work has the users train neural networks on their local data, and then periodically transmit their models to a central server. This approach preserves the privacy of the user's personal data, but still allows the central server's model to learn effectively. Their work is dependent on training neural networks locally. Back-propagation~\\cite{rumelhart1988learning} is the most popular algorithm for training deep networks. Each iteration of the back-propagation algorithm is composed of giant matrix multiplications. These matrices are very large, especially for massive networks with millions of nodes in the hidden layer, which are common in industrial practice. Large matrix multiplications are parallelizable on GPUs, but not energy-efficient. Users require their phones and tablets to have long battery life. Reducing the computational costs of neural networks, which directly translates into longer battery life, is a critical issue for the mobile industry.\n\nThe current challenges for deep learning illustrate a great demand for algorithms that reduce the amount of computation and energy usage. To reduce the bottleneck matrix multiplications, there has been a flurry of works around reducing the amount of computations associated with them. Most of them revolve around exploiting low-rank matrices or low precision updates. We review these techniques in details in Section~\\ref{sec:related_work}. However, updates with these techniques are hard to parallelize making them unsuitable for distributed and large scale applications. On the contrary, our proposal capitalizes on the sparsity of the activations to reduce computations. To the best of our knowledge, this is the first proposal that exploits sparsity to reduce the amount of computation associated with deep networks. We further show that our approach admits asynchronous parallel updates leading to perfect scaling with increasing parallelism. \n\nRecent machine learning research has focused on techniques for dealing with the famous problem of over-fitting with deep networks. A notable line of work \\cite{ba2013adaptive, makhzani2013k, makhzani2015winner} improved the accuracy of neural networks by only updating the neurons with the highest activations. Adaptive dropout \\cite{ba2013adaptive} sampled neurons in proportion to an affine transformation of the neuron's activation. The Winner-Take-All (WTA) approach \\cite{makhzani2013k, makhzani2015winner} kept the top-k\\% neurons, by calculating a hard threshold from mini-batch statistics. It was found that such a selective choice of nodes and sparse updates provide a natural regularization \\cite{srivastava2014dropout}. However, these approaches rely on inefficient, brute-force techniques to find the best neurons. Thus, these techniques are equally as expensive as the standard back-propagation method, leading to no computational savings. \n\nWe propose a hashing-based indexing approach to train and test neural networks, which capitalizes on the rich theory of randomized sub-linear algorithms from the database literature to perform adaptive dropouts \\cite{ba2013adaptive} efficiently while requiring significantly less computation and memory overhead. Furthermore, hashing algorithms are embarrassingly parallel and can be easily distributed with small communications, which is perfect for large-scale, distributed deep networks.\n\nOur idea is to index the neurons (the weights of each neuron as a vector) into a hash table using locality sensitive hashing. Such hash tables of neurons allow us to select the neurons with the highest activations, without computing all of the activations, in the sub-linear time leading to significant computational savings. Moreover, as we show, since our approach results in a sparse active set of neurons randomly, the gradient updates are unlikely to overwrite. Such updates are ideal for asynchronous and parallel gradient updates. It is known that asynchronous stochastic gradient descent\\cite{recht2011hogwild} (ASGD) will converge if the number of simultaneous parameter updates is small. We heavily leverage this sparsity which unique to our proposal. On several deep learning benchmarks, we show that our approach outperforms standard algorithms including vanilla dropout \\cite{srivastava2014dropout} at high sparsity levels and matches the performance of adaptive dropout \\cite{ba2013adaptive} and winner-take-all \\cite{makhzani2013k, makhzani2015winner} while needing less computation (only 5\\%).\n\n\\subsection{Our Contributions}\n\\begin{enumerate} \n    \\item We present a scalable and sustainable algorithm for training and testing of fully-connected neural networks. Our idea capitalized on the recent successful technique of adaptive dropouts combined with the smart data structure (hash tables) based on recently found locality sensitive hashing (LSH) for maximum inner product search (MIPS) \\cite{shrivastava2014asymmetric}. We show significant reductions in the computational requirement for training deep networks, without any significant loss in accuracy (within 1\\% of the accuracy of the original model on average). In particular, our method achieves the performance of other state-of-the-art regularization methods such as Dropout, Adaptive Dropout, and Winner-Take-All when using only 5\\% of the neurons in a standard neural network.\n    \\item Our proposal reduces computations associated with both the training and testing (inference) of deep networks by reducing the multiplications needed for the feed-forward operation. \n    \\item The key idea in our algorithm is to associate LSH hash tables~\\cite{gionis1999similarity,Proc:Indyk_STOC98} with every layer. These hash tables support constant-time $O(1)$ insertion and deletion operations.\n    \\item Our scheme naturally leads to sparse gradient updates. Sparse updates are ideally suited for massively parallelizable asynchronous training \\cite{recht2011hogwild}. We demonstrate that this sparsity opens room for truly asynchronous training without any compromise with accuracy. As a result, we obtain near-linear speedup when increasing number of processors.\n\\end{enumerate}\n\n\\section{Related Work}\n\\label{sec:related_work}\nThere have been several recent advances aimed at improving the performance of neural networks. \\cite{lin2015neural} reduced the number of floating point multiplications by mapping the network's weights stochastically to \\{-1, 0, 1\\} during forward propagation and performing quantized back-propagation that replaces floating-point multiplications with simple bit-shifts. Reducing precision is an orthogonal approach, and it can be easily integrated with any other approaches. \n\n\\cite{sindhwani2015structured} uses structured matrix transformations with low-rank matrices to reduce the number of parameters for the fully-connected layers of a neural network. This low-rank constraint leads to a smaller memory footprint. However, such an approximation is not well suited for asynchronous and parallel training limiting its scalability. We instead use random but sparse activations, leveraging database advances in approximate query processing, because they can be easily parallelized. [See Section~\\ref{sec:lowrank} for more details] \n\nWe briefly review dropouts and its variants, which are popular sparsity promoting techniques relying on sparse activations.  Although such randomized sparse activations have been found favorable for better generalization of deep networks, to the best of our knowledge, this sparsity has not been adequately exploited to make deep networks computationally cheap and parallelizable. We provide first such evidence. \n\nDropout \\cite{srivastava2014dropout} is primarily a regularization technique that addresses the issue of over-fitting randomly dropping half of the nodes in a hidden layer while training the network. The nodes are independently sampled for every stochastic gradient descent epoch~\\cite{srivastava2014dropout}. We can reinterpret Dropout as a technique for reducing the number of multiplications during forward and backward propagation phases, by ignoring nodes randomly in the network which computing the feed-forward pass. It is known that the network's performance becomes worse when too many nodes are dropped from the network. Usually, only 50\\% of the nodes in the network are dropped when training the network. At test time, the network takes the average of the thinned networks to form a prediction from the input data, which uses full computations.  \n\nAdaptive dropout \\cite{ba2013adaptive} is an evolution of the dropout technique that adaptively chooses the nodes based on their activations. The methodology samples few nodes from the network, where the sampling is done with probability proportional to the node activations which are dependent on the current input.  Adaptive dropouts demonstrate better performance than vanilla dropout~\\cite{srivastava2014dropout}. A notable feature of adaptive dropouts was that it was possible to drop significantly more nodes compared to dropouts while still retaining superior performance. Winner-Take-All \\cite{makhzani2013k, makhzani2015winner} is an extreme form of adaptive dropouts that uses mini-batch statistics to enforce a sparsity constraint. With this technique, only the k\\% largest, non-zero activations are used during the forward and backward phases of training. This approach requires computing the forward propagation step before selecting the k\\% nodes with a hard threshold. Unfortunately, all these techniques require full computation of the activations to selectively sample nodes. Therefore, they are only intended for better generalization and not reducing computations.  Our approach uses the insight that selecting a very sparse set of hidden nodes with the highest activations can be reformulated as dynamic approximate query processing problem, which can solve efficiently using locality sensitive hashing. The differentiating factor between adaptive dropout and winner-take-all and our approach is we use sub-linear time randomized hashing to determine the active set of nodes instead computing the inner product for each node individually. \n\nThere is also another orthogonal line of work which uses hashing to reduce memory. \\cite{chen2015compressing} introduced a new type of deep learning architecture called Hashed Nets. Their objective was to decrease the number of parameters in the given neural network by using a universal random hash function to tie node weights. Network connections that map to the same hash value (hash collisions) are restricted to have same values. The architecture has the virtual appearance of a regular network while maintaining only a small subset of real weights. We point out that hashed nets are complementary to our approach because we focus on reducing the computational cost involved in using deep nets rather than its size in memory. \n\n\\section{Low-Rank vs Sparsity}\n\\label{sec:lowrank}\nThe low-rank (or structured) assumption is very convenient for reducing the complexity of general matrix operations. However, low-rank and dense updates do not promote sparsity and are not friendly for distributed computing. The same principle holds with deep networks. We illustrate it with a simple example. \n\nConsider a layer of the first network (left) shown in Figure~\\ref{fig:structure_matrix}.  The insight is that if the weight matrix $W \\in \\mathbb{R}^{mxn}$ for a hidden layer has low-rank structure where rank $r \\ll min(m,n)$, then it has a representation $W=UV$ where $U \\in \\mathbb{R}^{mxr}$ and $V \\in \\mathbb{R}^{rxn}$. This low-rank structure improves the storage requirements and matrix-multiplication time from $O(mn)$ to $O(mr + rn)$. As shown in Figure \\ref{fig:structure_matrix}, there is an equivalent representation of the same network using an intermediate hidden layer that contains $r$ nodes and uses the identity activation function. The weight matrices for the hidden layers in the second network (right) map to the matrix decomposition, $U$ and $V$. \\cite{sindhwani2015structured} uses structured matrix transformations with low-rank matrices to reduce the number of parameters for the fully-connected layers of a neural network. \n\nThe new network structure and the equivalent structure matrices require a dense gradient update, which is not ideally suited for data parallelism~\\cite{recht2011hogwild}. For example, they need sequential gradient descent updates. In this work, we move away from the low-rank assumption and instead make use of randomness and sparsity to reduce computations. We will later show that due to sparsity, our approach is well suited for Asynchronous Stochastic Gradient Descent (ASGD), leading to near-linear scaling.  \n\nIt should be noted that our methodology is very different from the notions of making networks sparse by thresholding (or sparsifying) node weights permanently \\cite{srivastava2014dropout, makhzani2013k, makhzani2015winner}. Our approach makes use of every node weight but picks them selectively for different inputs. On the contrary, permanent sparsification only has a static set of weights which are used for every input and rest are discarded. \n\n\\begin{figure} [ht]\n\\begin{center}\n  \\includegraphics[width=.50\\textwidth]{Structured_Matrix.pdf}\n\\end{center}\n\\begin{caption}\n{\\bf Illustration of why the Low-Rank assumption for neural networks naturally leads to fewer parameters:} \n(1) This network contains two layers with m and n neurons respectively. The weighted connections between the layers are characterized by the weight matrix $W \\in R^{m x n}$ of constrained rank $r$ such that $W = UV$ with $U \\in \\mathbb{R}^{m x r}$ and $V \\in \\mathbb{R}^{r x n}$.\n\n(2) An equivalent network contains three layers each with m, r, and n neurons. The top two layers are represented with the matrix $U \\in \\mathbb{R}^{m x r}$, while the bottom two layers are characterized with the matrix $V \\in \\mathbb{R}^{r x n}$. The intermediate layer uses the identity activation function $I$ such that the output from network 2 equals that of network 1. $a = f(W^T x) = f((U V)^T x) = f((V^T U^T) x) = f(V^T I(U^T x))$ where $a$ is the layer's output and f is a non-linear activation function.\n\\end{caption}\n\\label{fig:structure_matrix}\n\\end{figure}\n\n\\section{Background}\\subsection{Artificial Neural Networks}\nNeural networks are built from layers of neurons. The combination of multiple hidden layers creates a complex, non-linear function capable of representing any arbitrary function \\cite{cybenko1989approximation}. The forward propagation equation is $$a^{l} = f(W^{l} a^{l-1} + b^{l})$$ where $a^{l-1}$ is the output from the previous layer $l-1$, $W^{l}$ is an $[N x M]$ weight matrix for layer $l$, $b^{l}$ is the bias term for layer $l$, and $f$ is the activation function. Common non-linear activation functions are sigmoid, tanh, ReLU. Neural networks are trained using stochastic gradient descent (SGD) - $\\theta_{t+1} = \\theta_t - \\eta \\nabla J(\\theta)$. The gradient for the neural network $\\nabla J(\\theta)$ is calculated layer-by-layer using the back-propagation algorithm. The Momentum \\cite{polyak1964momentum} technique calculates the parameter update from a weighted sum of the previous momentum term and the current gradient - $\\nu_{t+1} = \\gamma \\nu_t + \\eta \\nabla J(\\theta)$. It reduces the training time of neural networks by mitigating the stochastic nature of SGD. Adaptive gradient descent (Adagrad) \\cite{duchi2011adaptive} is a variant of SGD that adapts the learning rate for each parameter. The global learning rate is normalized by the sum of squared gradients. It favors larger learning rates for infrequent or small parameter updates.\n\n\\subsection{Adaptive Dropout}\n\\label{sec:adaptive_dropout}\nAdaptive Dropout \\cite{ba2013adaptive} is a recently proposed dropout technique that selectively samples nodes with probability proportional to some monotonic function of their activations. To achieve this, authors in \\cite{ba2013adaptive} chose a Bernoulli distribution. The idea was that given an input $a_{l-1}$ to a layer $l$, we generate a Bernoulli random variable for every node in the layer with Bernoulli probability $q$ being a monotonic function of the activation. Then, the binary variable sampled from this distribution is used to determine if the associated neuron's activation is kept or set to zero. This adaptive selectivity allows fewer neurons to be activated in the hidden layer and to improve neural network performance over standard Dropout \\cite{srivastava2014dropout}. The Winner-Take-All approach \\cite{makhzani2015winner} is an extreme case of Adaptive Dropout that takes only the top k\\% activations in a hidden layer, which also performs very well.\n\n\\subsection{Locality-Sensitive Hashing (LSH)}\nLocality-Sensitive Hashing (LSH) \\cite{gionis1999similarity, sundaram2013streaming, huang2015query, gao2014dsh, shinde2010similarity} is a popular, sublinear time algorithm for approximate nearest-neighbor search. The high-level idea is to map similar items into the same bucket with high probability. An LSH hash function maps an input data vector to an integer key - $h: x \\in \\mathbb{R}^D \\mapsto [0, 1, 2,\\dots, N]$. A collision occurs when the hash values for two data vectors are equal - $h(x) = h(y)$. The probability of collision for an LSH hash function is proportional to the similarity distance between the two data vectors - $Pr[h(x) = h(y)] \\propto sim(x,y)$. Essentially, similar items are more likely to collide with the same hash fingerprint and vice versa. \n\n\\subsubsection*{Sub-linear Search with $(K,L)$ LSH Algorithm} To be able to answer approximate nearest-neighbor queries in sub-linear time, the idea is to create hash tables, (see Figure~\\ref{fig:algorithm}). Given the collection $\\mathcal{C}$, which we are interested in querying for the nearest-neighbor items, the hash tables are generated using the locality sensitive hash (LSH) family.  We assume that we have access to the appropriate locality sensitive hash (LSH) family $\\mathcal{H}$ for the similarity of interest. In the classical $(K,L)$ parameterized LSH algorithm, we generate $L$ different hash functions given by  $B_j(x)  = [h_{1j}(x);h_{2j}(x);...;h_{{Kj}}(x)]$. Here $h_{ij}, i \\in \\{1,2,...,K \\}$ and $j \\in  \\{1,2,...,L \\}$, are $KL$ different evaluations of the appropriate locality sensitive hash (LSH) function.  Each of these hash functions is formed by concatenating $K$ sampled hash values from $\\mathcal{H}$.\n\nThe overall algorithm works in two phases (See~\\cite{Report:E2LSH} for details):\n\\begin{enumerate}\n    \\itemsep0em\n    \\item {\\bf Pre-processing Phase:} We construct $L$ hash tables from the data by storing all elements $x \\in \\mathcal{C}$, at location $B_j(x)$ in hash-table $j$ (See Figure~\\ref{fig:algorithm} for an illustration). We only store pointers to the vector in the hash tables, as storing data vectors is very memory inefficient.\n    \\item {\\bf Query Phase:} Given a query $Q$, we will search for its nearest-neighbors. We report the union of all the points in the buckets $B_j(Q)$ $\\forall j \\in \\{1,2,...,L\\}$, where the union is over $L$ hash tables.  Note, we do not scan all the elements in $\\mathcal{C}$, we only probe $L$ different buckets, one from each hash table. \n\\end{enumerate}\n\n\\cite{Proc:Indyk_STOC98} shows that having an LSH family for a given similarity measure and an appropriate choice of $K$ and $L$, the above algorithm is provably sub-linear. In practice, it is near-constant time because the query only requires few bucket probes.  \n\n\\subsubsection*{Hashing Inner Products} It was recently shown that by allowing asymmetric hash functions, the same algorithm can be converted into a sub-linear time algorithm for Maximum Inner Product Search (MIPS) \\cite{shrivastava2014asymmetric}. In locality sensitive hashing, self-similarity $Pr[h(x) = h(x)] = 1$ is the closest nearest-neighbor for a query $x$. However, self-similarity does not necessarily correspond with the highest inner product $x^Tx = ||x||^2_2$. Therefore, an asymmetric transformation is necessary to map maximum inner product search (MIPS) to the standard nearest-neighbor search (LSH). Our work heavily leverages these algorithms. For the purpose of this paper, we will assume that we can create a very efficient data structure, which can be queried for large inner products. See~\\cite{shrivastava2014asymmetric} for details.\n\n\\subsubsection*{Multi-Probe LSH} One common complaint with classical LSH algorithm is that it requires a significant number of hash tables. Large $L$ increases the hashing time and memory cost. A simple solution was to probe multiple \"close-by\" buckets in every hash table rather than probing only one bucket~\\cite{lv2007multi}. Thus, for a given query $Q$, in addition to probing  $B_j(Q)$ in hash table $j$, we also generate several new addresses to probe by slightly perturbing values of $B_j(Q)$. This simple idea significantly reduces the number of tables needed with LSH, and we can work with only a few hash tables. See~\\cite{lv2007multi} for more details. \n\n\\section{Proposed Methodology}\n\\subsection{Intuition}\nWinner-Take-All~\\cite{makhzani2013k, makhzani2015winner} technique shows that we should only consider a few nodes (top k\\%) with large activations for a given input and ignore the rest while computing the feed-forward pass. Furthermore, the back-propagation updates should only be performed on those few chosen weights. Let us denote $AS$ (Active Set) to be the top k\\% nodes having significant activations. Let $n$ denote the total number of nodes in the neural network with $AS \\ll n$. For one gradient update, Winner-Take-All needs to perform first $O(n \\log{} n)$ work to compute $AS$, followed by updating $O(AS)$ weights.  $O(n log n)$ seems quite wasteful. In particular, given the input, finding the active set $AS$ is a search problem, which can be solved efficiently using smart data structures. Furthermore, if the data structure is dynamic and efficient, then the gradient updates will also be efficient. \n\nFor a node $i$ with weight $w_i$ and input $x$, its activation is a monotonic function of the inner product $w_i^Tx$. Thus, given $x$, selecting nodes with large activations (the set $AS$) is equivalent to searching $w_i$, from a collection of weight vectors, have large inner products with $x$. Equivalently from query processing perspective, if we treat input $x$ as a query, then the search problem of selecting top k\\% nodes can be done efficiently, in sub-linear time in the number of nodes, using the recent advances in maximum inner product search \\cite{shrivastava2014asymmetric}.  Our proposal is to create hash tables with indexes generated by asymmetric locality sensitive hash functions tailored for inner products. With such hash tables, given the query input $x$, we can very efficiently approximate the active set $AS$. \n\nThere is one more implementation challenge. We also have to update the nodes (weights associated with them) in $AS$ during the gradient update. If we can perform these updates in $O(AS)$ instead of $O(n)$, then we save significant computation. Thus, we need a data structure where updates are efficient as well. With years of research in data structures, there are many choices which can make update efficient. We describe our system in details in Section~\\ref{sec:system}.  \n\n\\subsection{Equivalence with Adaptive Dropouts}\nIt turns out that using randomized asymmetric locality sensitive hashing \\cite{shrivastava2014asymmetric} for finding nodes with large inner products is formally, in a statistical sense, equivalent to adaptive dropouts with a non-trivial sampling distribution. \n\nAdaptive Dropout technique uses the Bernoulli distribution (Section~\\ref{sec:adaptive_dropout}) to adaptively sample nodes with large activation. In theory, any distribution assigning probabilities in proportion to nodes activation is sufficient. We argue that Bernoulli is a sub-optimal choice. There is another non-intuitive, but a very efficient choice. This choice comes from the theory of Locality-Sensitive Hashing (LSH) \\cite{Proc:Indyk_STOC98}, which is primarily considered a black box technique for fast sub-linear search. Our core idea comes from the observation that, for a given search query $q$, the $(K,L)$ parametrized LSH algorithm \\cite{Report:E2LSH} inherently samples, in sub-linear time, points from datasets with probability proportional to $1 - (1-p^K)^L$. Here $p$, is the collision probability, a monotonic function of the similarity between the query and the retrieved point. See \\cite{Report:E2LSH} for details.  The expression holds for any $K$ and $L$. $K$ controls the sparsity of buckets. With a reasonable choice of $K$, we can always make buckets arbitrarily sparse. Even if the buckets are heavy, we can always do a simple random sampling, in the buckets, and adding only a constant factor in the probability $p$, keeping the monotonicity intact. \n\n\\begin{theorem} \n {\\bf Hash-Based Sampling} - For a given input $x$ to any layer, any $(K,L)$ parametrized hashing algorithm selects a node $i$, associated with weight vector $w_i$, with probability proportional to  $1-(1-p^K)^L$. Here $p = Pr(h(I) = h(w_i))$ is the collision probability of the associated locality sensitive hash function. The function $1- (1-p^K)^L$ is monotonic in $p$.\n\\end{theorem}\n\n\\begin{proof}\nThe probability of finding node $i$ in any hash table is $p^K$ (all $K$ bits collide where each bit occurs with probability $p$), so the probability of missing node $i$ in $L$ independent hash tables is $(1-p^K)^L$. If we further choose to sample the buckets randomly by another constant factor $r < 1$, then effective collision probability becomes $p' = rp$.\n\\end{proof}\n\nWith recent advances in hashing inner products, we can make $p$ a monotonic function of inner products \\cite{shrivastava2014asymmetric, shrivastava2014improved, shrivastava2015asymmetric,shrivastava2015probabilistic}. Translating into deep network notation, given the activation vector of previous layer $a^{l-1}$ as the query and layer's weights $W^{l}$ as the search space, we can sample with probability proportional to $$g(W^{l}) = 1-(1-f(W^{l} a^{l-1} )^K)^L$$ which is a monotonic function of node activation. Thus, we can naturally integrate adaptive dropouts with hashing. Overall, we are simply choosing a specific form of adaptive dropouts, which leads to efficient sampling in time sub-linear in the number of neurons (or nodes). \n\n\\begin{corollary}\n{\\bf Adaptive Dropouts are Efficient} - There exists an efficient family of sampling distributions for adaptive dropouts, where the sampling and updating cost is sub-linear in the number of nodes. In particular, we can construct an efficient sampling scheme such that for any two nodes $i$ and $j$, in any given layer $l$, the probability of choosing node $i$ is greater than the probability of choosing node $j$ if and only if the current activation of node $i$ is higher than that of $j$.\n\\end{corollary}\n\n\\begin{proof}\nSince asymmetric LSH functions \\cite{shrivastava2014asymmetric, shrivastava2014improved, shrivastava2015asymmetric} have a collision probability $p$ that is a monotonic function of the inner product, it is also a montonic function of the neuron's activation. All activation functions including sigmoid \\cite{funahashi1989approximate}, tanh \\cite{lecun2012efficient}, and ReLU \\cite{glorot2011deep} are monotonic functions of inner products. Composing monotonic functions retains monotonicity. Therefore, the corollary follows because $1-(1-p^K)^L$ is monotonic with respect to $p$. In addition, monotonicity goes in both direction as the inverse function exists and it is also monotonic.\n\\end{proof}\n\n\\begin{figure}[ht]\n\\mbox{\n  \\includegraphics[width=.50\\textwidth]{algorithm}} \\vspace{-0.3in}\n\\caption{{\\bf A visual representation of a neural network using randomized hashing}\n(1) Build the hash tables by hashing the weights of each hidden layer (one time operation)\n(2) Hash the layer's input using the layer's randomized hash function\n(3) Query the layer's hash table(s) for the active set $AS$\n(4) Only perform forward and back-propagation on the neurons in the active set. The solid-colored neurons in the hidden layer are the active neurons.\n(5) Update the $AS$ weights and the hash tables by rehashing the updated weights to new hash locations. }\n  \\label{fig:algorithm}\n\\end{figure}\n\n\\subsection{Hashing Based Back-Propagation}\n\\label{sec:system}\nAs argued, our approach is simple. We use randomized hash functions to build hash tables from the nodes in each hidden layer. We sample nodes from the hash table with probability proportional to the node's activation in sub-linear time. We then perform forward and back propagation only on the active nodes retrieved from the hash tables. We later update the hash tables to reorganize only the modified weights. \n\nFigure \\ref{fig:algorithm} illustrates an example neural network with two hidden layers, five input nodes, and two output nodes. Hash tables are built for each hidden layer, where the weighted connections for each node are hashed to place the node in its corresponding bucket. Creating hash tables to store all the initial parameters is a one-time operation which requires cost linear in the number of parameters. \n\nDuring a forward propagation pass, the input to the hidden layer is hashed with the same hash function used to build the hidden layer's hash table. The input's fingerprint is used to collect the active set $AS$ nodes from the hash table. The hash table only contains pointers to the nodes in the hidden layer. Then, forward propagation is performed only on the nodes in the active set $AS$. The rest of the hidden layer's nodes, which are not part of the active set, are ignored and are automatically switched off without even touching them. On the back propagation pass, the active set is reused to determine the gradient and to update the parameters. We rehash the nodes in each hidden layer to account for the changes in the network during training. \n\nIn detail, the hash function for each hidden layer is composed of $K$ randomized hash functions. We use the sign of an asymmetrically transformed random projection (see~\\cite{shrivastava2014improved} for details) to generate the $K$ bits for each data vector. The $K$ bits are stored together efficiently as an integer, forming a fingerprint for the data vector. We create a hash table of $2^K$ buckets, but we only keep the nonempty buckets to minimize the memory footprint (analogous to hash-maps in Java). Each bucket stores pointers to the nodes whose fingerprints match the bucket's id instead of the node itself. In figure \\ref{fig:algorithm}, we showed only one hash table, which is likely to miss valuable nodes in practice. In our implementation, we generate $L$ hash tables for each hidden layer, and each hash table has an independent set of $K$ random projections. Our final active set from these $L$ hash tables is the union of the buckets selected from each hash table. For each layer, we have $L$ hash tables. Effectively, we have two tunable parameters, $K$ bits and $L$ tables to control the size and the quality of the active sets. The $K$ bits increase the precision of the fingerprint, meaning the nodes in a bucket are more likely to generate higher activation values for a given input. The $L$ tables increase the probability of finding useful nodes that are missed because of the randomness in the hash functions.\n\n\\begin{algorithm}[tb]\n   \\caption{Deep Learning with Randomized Hashing}\n   \\label{alg:algorithm}\n\\begin{algorithmic}\n   \\STATE \/\/ HF$_l$ - Layer $l$ Hash Function\n   \\STATE \/\/ HT$_l$ - Layer $l$ Hash Tables\n   \\STATE \/\/ AS$_l$ - Layer $l$ Active Set\n   \\STATE \/\/ $\\theta^{l}_{AS} \\in W^{l}_{AS}$, $b^{l}_{AS}$ - Layer $l$ Active Set parameters\n   \\STATE Randomly initialize parameters $W^l$, $b^l$ for each layer $l$\n   \\STATE HF$_l$ = constructHashFunction($k$, $L$)\n   \\STATE HT$_l$ = constructHashTable($W^l$, HF$_l$)\n   \\WHILE {not stopping criteria}\n   \\FOR {{\\bf each} training epoch}\n   \\STATE \/\/ Forward Propagation\n   \\FOR {layer $l = 1 \\dots N$}\n   \\STATE fingerprint$_l$ = HF$_l(a_l)$\n   \\STATE AS$_l$ = collectActiveSet(HT$_l$, fingerprint$_l$)\n   \\FOR {{\\bf each} node $i$ in AS$_l$}\n   \\STATE $a^{l+1}_{i} = f(W^{l}_{i}a^{l}_{i} + b^{l}_{i})$\n   \\ENDFOR\n   \\ENDFOR\n   \\STATE \/\/ Backpropagation\n   \\FOR {layer $l = 1 \\dots N$}\n   \\STATE $\\Delta J(\\theta^{l}_{AS})$ = computeGradient($\\theta^{l}_{AS}$, AS$_l$)\n   \\STATE $\\theta^{l}_{AS}$ = updateParameters($\\theta^{l}_{AS}$, $\\Delta J(\\theta_{AS}$))\n   \\ENDFOR\n   \\STATE {\\bf for each} Layer $l$ -> updateHashTables(HF$_l$, HT$_l$, $\\theta^{l}$)\n   \\ENDFOR\n   \\ENDWHILE\n\\end{algorithmic}\n\\end{algorithm}\n\\subsection{Efficient Query and Updates}\n\nOur algorithm critically depends on the efficiency of the query and update procedure. The hash table is one of the most efficient data structures, so this is not a difficult challenge. Querying a single hash table is a constant time operation when the bucket size is small. Bucket size can be controlled by $K$ and by sub-sampling the bucket. There is always a possibility of crowded buckets due to bad randomness or because of too many near-duplicates in the data. These crowded buckets are not very informative and can be safely ignored or sub-sampled. \n\nWe never need to store weights in the hash tables. Instead, we only store references to the weight vectors, which makes hash tables a very light entity. Further, we reduce the number of different hash tables required by using multi-probe LSH \\cite{lv2007multi}.\n\nUpdating a weight vector $w_i$ associated with a node $i$ can require changing the location of $w_i$ in the hash table, if the updates lead to a change in the hash value. With the hash table data structure, the update is not an issue when the buckets are sparse.  Updating a weight only requires one insertion and one deletion in the respective buckets. There are plenty of choices for efficient insert and delete data structure for buckets. In theory, even if buckets are not sparse, we can use a red-black-tree \\cite{cormen2009introduction} to ensure both insertion and deletion cost is logarithmic in the size of the bucket. However, using simple arrays when the buckets are sparse is preferable because they are easy to parallelize. With arrays insertion is O(1) and deletion is $O(b)$, where $b$ is the size of buckets. Controlling the size of $b$ can be easily achieved by sub-sampling the bucket. Since we create $L$ independent hash tables, even for reasonably large $L$, the process is quite robust to cheap approximations. We further reduce the size of $L$ using multi-probing \\cite{lv2007multi}. Multi-probe with binary hash function is quite straightforward. We just have to randomly flip few bits of the $K$-bit hash to generate more addresses. \n\nThere are plenty of other choices than can make hashing significantly faster, such as cheap re-ranking~\\cite{shrivastava2012fast}. See~\\cite{Shrivastava:SOCC_16}, where authors show around 500x reduction in computations for image search by incorporating different algorithmic and systems choices. \n\n\\subsection{Overall Cost}\nIn every layer, during every Stochastic Gradient Descent (SGD) update, we compute $K \\times L$ hashes of the input, probe around $10L$ buckets, and take their union. In our experiments, we use $K=6$ and $L=5$, i.e. 30 hash computations only. There are many techniques to further reduce this hashing cost~\\cite{achlioptas2001database,li2006very,Proc:OneHashLSH_ICML14,Proc:Shrivastava_UAI14,Shrivastava:NIPS_16}. We probe around 10 buckets in each hash tables for obtaining 5\\% of active nodes, leading to the union of 50 buckets in total. The process gives us the active set $AS$ of nodes, which is usually significantly smaller when compared to the number of nodes $n$. During SGD, we update all of the weights in $AS$ along with the hash table. Overall, the cost is of the order of the number of nodes in $AS$, which in our experiments show can go as low as $5\\%$. For 1000 node in the layer, we have to update around 10-50 nodes only. The bottleneck cost is calculations of activations (actual inner products) of these nodes in the $AS$ which for every node is around $1000$ floating point multiplications each. The benefits will be even more significant for larger networks. \n\n\\subsection{Bonus: Sparse Updates can be Parallelized}\nAs mentioned, we only need to update the set of weights associated with nodes in the active set $AS$. If the $AS$ is very sparse, then it is unlikely that multiple SGD updates will overwrite the same set of weights. Intuitively, assuming enough randomness in the data vector, we will have a small active set $AS$ chosen randomly from among all the nodes. It is unlikely that multiple active sets of randomly selected data vectors will have significant overlaps. Small overlaps imply fewer conflicts while updating. Fewer conflicts while updating is an ideal ground where SGD updates can be parallelized without any overhead. In fact, it was both theoretically and experimentally that random and sparse SGD updates can be parallelized without compromising with the convergence ~\\cite{recht2011hogwild}. Parallel SGD updates is one the pressing challenges in large-scale deep learning systems~\\cite{chen2016revisiting}. Vanilla SGD for deep networks is sequential, and parallel updates lead to poor convergence due to significant overwrites. Our experimental results, in Section~\\ref{sec:ASGD}, support these known phenomena. Exploiting this unique property,  we show near linear scaling of our algorithm with increasing processors without hurting convergence. \n\n\\section{Evaluations}\nWe design experiments to answer the following six important questions: \n\\begin{enumerate}\n    \\item How much computation can we reduce without affecting the vanilla network's accuracy? \n    \\item How effective is adaptive sampling compared to a random sampling of nodes?\n    \\item How does approximate hashing compare with expensive but exact approaches of adaptive dropouts~\\cite{ba2013adaptive} and Winner-Takes-all~\\cite{makhzani2013k, makhzani2015winner} in terms of accuracy? In particular, is hashing working as intended? \n    \\item How is the network's convergence affected by increasing number of processors as we perform parallel SGD updates using our approach?\n    \\item Is sparse update necessary? Is there any deterioration in performance, if we perform vanilla dense updates in parallel?\n    \\item  What is the wall clock decrease in training time, as a function of increasing number of processors?\n\\end{enumerate}\n\nFor evaluation, we implemented the following five approaches to compare and contrast against.\n\\begin{itemize}\n    \\item Standard (NN) : A full-connected neural network\n    \\item Dropout (VD) \\cite{srivastava2014dropout}: A neural network that disables the nodes of a hidden layer using a fixed probability threshold\n    \\item Adaptive Dropout (AD) \\cite{ba2013adaptive}: A neural network that disables the nodes of a hidden layer using a probability threshold based on the inner product of the node's weights and the input.\n    \\item Winner Take All (WTA) \\cite{makhzani2013k, makhzani2015winner}: A neural network that sorts the activations of a hidden layer and selects the k\\% largest activations\n    \\item Randomized Hashing (LSH): A neural network that selects nodes using randomized hashing. A hard threshold limits the active node set to k\\% sparsity\n\\end{itemize}\n\n\\subsection{Datasets}\nTo test our neural network implementation, we used four publicly available datasets - MNIST8M \\cite{loosli-canu-bottou-2006}, NORB \\cite{lecun2004learning}, CONVEX, and RECTANGLES \\cite{larochelle2007empirical}. The statistics of these datasets are summarized in Table~\\ref{fig:dataset_size}. The MNIST8M, CONVEX, and RECTANGLES datasets contain 28x28 images, forming 784-dimensional feature vectors. The MNIST8M task is to classify each handwritten digit in the image correctly. It is derived by applying random deformations and translations to the MNIST dataset. The CONVEX dataset objective is to identify if a single convex region exists in the image. The goal for the RECTANGLES dataset is to discriminate between tall and wide rectangles overlaid on a black and white background image. The NORB dataset \\cite{lecun2004learning} contains images of 50 toys, belonging to 5 categories under various lighting conditions and camera angles. Each data point is a 96x96 stereo image pair. We resize the image from 96x96 to 32x32 and concatenate the image pairs together to form a 2048-dimensional feature vector.\n\n\\begin{figure}\n    \\begin{center}\n    \\begin{tabular}{ |c|c|c| } \n    \\hline\n    Dataset & Train Size & Test Size \\\\ \n    \\hline\n    MNIST8M & 8,100,000 & 10,000 \\\\ \n    NORB & 24,300 & 24,300 \\\\\n    Convex & 8,000 & 50,000 \\\\\n    Rectangles & 12,000 & 50,000 \\\\\n    \\hline\n    \\end{tabular}\n    \\end{center}\n    \\caption{Dataset - Training + Test Size}\n    \\label{fig:dataset_size}\n\\end{figure}\n\n\\begin{figure*}[ht]\n\\begin{center}\n\\mbox{\\hspace{-0.185in}\n\\includegraphics[width=1.9in]{MNIST8M_784_1000_1000_10} \\hspace{-0.15in}\n\\includegraphics[width=1.9in]{NORB_2048_1000_1000_5} \\hspace{-0.19in}\n\\includegraphics[width=1.9in]{Convex_784_1000_1000_2} \\hspace{-0.19in}\n\\includegraphics[width=1.9in]{Rectangles_784_1000_1000_2} \\hspace{-0.19in}}\n\\mbox{\\hspace{-0.185in}\n\\includegraphics[width=1.9in]{MNIST8M_784_1000_1000_1000_10} \\hspace{-0.15in}\n\\includegraphics[width=1.9in]{NORB_2048_1000_1000_1000_5} \\hspace{-0.19in}\n\\includegraphics[width=1.9in]{Convex_784_1000_1000_1000_2} \\hspace{-0.19in}\n\\includegraphics[width=1.9in]{Rectangles_784_1000_1000_1000_2} \\hspace{-0.19in}}\n\\end{center}\n\\caption{Classification accuracy under different levels of active nodes with networks on the MNIST (1st), NORB (2nd), Convex (3rd) and Rectangles (4th) datasets. The standard neural network (dashed black line) is our baseline accuracy. We can clearly see that adaptive sampling with hashing (LSH) is significantly more effective than random sampling (VD). {\\bf 1. Top Panels:} 2 hidden Layers. {\\bf 1. Bottom Panels:} 3 hidden Layers}\n  \\label{LSH_VD_STD}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n\\begin{center}\n\\mbox{\\hspace{-0.185in}\n\\includegraphics[width=1.9in]{MNIST8M_784_1000_1000_10_AD_WTA} \\hspace{-0.15in}\n\\includegraphics[width=1.9in]{NORB_2048_1000_1000_5_AD_WTA} \\hspace{-0.19in}\n\\includegraphics[width=1.9in]{Convex_784_1000_1000_2_AD_WTA} \\hspace{-0.19in}\n\\includegraphics[width=1.9in]{Rectangles_784_1000_1000_2_AD_WTA} \\hspace{-0.19in}}\n\\mbox{\\hspace{-0.185in}\n\\includegraphics[width=1.9in]{MNIST8M_784_1000_1000_1000_10_AD_WTA} \\hspace{-0.15in}\n\\includegraphics[width=1.9in]{NORB_2048_1000_1000_1000_5_AD_WTA} \\hspace{-0.19in}\n\\includegraphics[width=1.9in]{Convex_784_1000_1000_1000_2_AD_WTA} \\hspace{-0.19in}\n\\includegraphics[width=1.9in]{Rectangles_784_1000_1000_1000_2_AD_WTA} \\hspace{-0.19in}}\n\\end{center}\n\\caption{Classification accuracy under different levels of active nodes with networks on the MNIST (1st), NORB (2nd), Convex (3rd) and Rectangles (4th) datasets. The standard neural network (dashed black line) is our baseline accuracy. WTA and AD (dashed red + yellow lines) perform the same amount of computation as the standard neural network. Those techniques select nodes with high activations, after full computations, to achieve better accuracy. We compare our LSH approach to determine whether our randomized algorithm achieves comparable performance while reducing the total amount of computation. We do not have data for adaptive dropout at the 5\\% and 10\\% computation levels because those models diverged when the number of active nodes dropped below 25\\%. {\\bf 1. Top Panels:} 2 hidden Layers. {\\bf 1. Bottom Panels:} 3 hidden Layers }\n  \\label{LSH_AD_WTA_STD}\n\\end{figure*}\n\n\\subsection{Sustainability}\n\\label{sec:dropout}\n\\subsubsection{Experimental Setting}\nAll of the experiments for our approach and the other techniques were run on a 6-core Intel i7-3930K machine with 16 GB of memory. Our approach uses stochastic gradient descent with Momentum and Adagrad \\cite{dean2012large}. Since our approach uniquely selects an active set of nodes for each hidden layer, we focused on a CPU-based approach to simplify combining randomized hashing with neural networks. The ReLU activation function was used for all methods. The learning rate for each approach was set using a standard grid search and ranged between $10^{-2}$ and $10^{-4}$. The parameters for the randomized hash tables were K = 6 bits and L = 5 tables with multi-probe LSH~\\cite{lv2007multi} creating a series of probes in each hash tables. We stop early if we find that we have samples enough nodes even before exhausting all buckets. Since we evaluate all levels of selection, in order to increase the percentage of nodes retrieved we increase the number of probes in the buckets. For the experiments, we use a fixed threshold to cap the number of active nodes selected from the hash tables to guarantee the amount of computation is within a certain level. \n\n\\subsubsection{Effect of computation levels}\nFigures \\ref{LSH_VD_STD}, \\ref{LSH_AD_WTA_STD} show the accuracy of each method on neural networks with 2 (Top panel) and 3 (Bottom Panel) hidden layers with the percentage of active nodes ranging from [0.05, 0.10, 0.25, 0.5, 0.75, 0.9]. The standard neural network is our baseline in these experiments and is marked with a dashed black line. Each hidden layer contains 1000 nodes. The x-axis represents the average percentage of active nodes per epoch selected by each technique. Our approach only performs the forward and back propagation steps on the nodes selected in each hidden layer. The other baseline techniques except for Dropout (VD) perform the forward propagation step for each node first to compute all the activations, before setting node activations to zero based on the corresponding algorithm. Thus on VD and our proposal requires a lesser number of multiplications compared to a standard neural network training procedure. \n\nFigures~\\ref{LSH_VD_STD} and \\ref{LSH_AD_WTA_STD} summarizes the accuracy of different approaches at various computations levels. From the figures, we conclude the following. \n\\begin{itemize}\n    \\item The plots are consistent across all data-sets and architectures with different depths.\n    \\item Our method (LSH) gives the best overall accuracy with the fewest number of active nodes. The fact that our approximate method is even slightly better than WTA and adaptive dropouts is not surprising, as it is long known that a small amount of random noise leads to better generalization. For examples, see~\\cite{srivastava2014dropout}.\n    \\item As the number of active nodes decreases from 90\\% to 5\\%, LSH experiences the smallest drop in performance and less performance volatility.\n    \\item VD experiences the greatest drop in performance when reducing the number of active nodes from 50\\% to 5\\%.\n    \\item As the number of hidden layers increases in the network, the performance drop for VD becomes steeper.\n    \\item WTA performed better than VD when the percentage of active nodes is less than 50\\%\n    \\item The number of multiplications is a constant multiple of the percentage of active nodes in a hidden layer. The parameters, alpha, and beta determine how many nodes are kept active. We used alpha $\\alpha$ = 1.0 and beta $\\beta$ = [-1.5, -1.0, 0, 1.0, 3.5] for our experiments. Our model diverged when the number of active nodes dropped below 25\\%, so we do not have data for Adaptive Dropout at the 5\\% and 10\\%.\n    \\item The performance for each method stabilizes, as the computation level approaches 100\\%.\n\\end{itemize}\n\nLowering the computational cost of running neural networks by running fewer operations reduces the energy consumption and heat produced by the processor. However, large neural networks provide better accuracy and arbitrarily reducing the amount of computation hurts performance. Our experiments show that our method (LSH) performs well at low computation levels, providing the best of both worlds - high performance and low processor computation. This approach is ideal for mobile phones, which have a thermal power design (TDP) of 3-4 Watts, because reducing the processor's load directly translates into longer battery life.\n\n\\begin{figure*}[ht]\n\\begin{center}\n\\mbox{\\hspace{-0.185in}\n\\includegraphics[width=1.9in]{MNIST8M_784_1000_1000_10_epoch} \\hspace{-0.15in}\n\\includegraphics[width=1.9in]{NORB_2048_1000_1000_1000_5_epoch} \\hspace{-0.19in}\n\\includegraphics[width=1.9in]{Convex_784_1000_1000_1000_2_epoch} \\hspace{-0.19in}\n\\includegraphics[width=1.9in]{Rectangles_784_1000_1000_1000_2_epoch} \\hspace{-0.19in}\n}\n\\end{center}\n\\caption{The convergence of our randomized hashing approach (LSH-5\\%) over several training epochs using asynchronous stochastic gradient with 1, 8, 56 threads. We used a (3 hidden layer) network on the MNIST (1st), NORB (2nd), Convex (3rd) and Rectangles (4th) datasets. Only 5\\% of the standard network's computation was performed in this experiment.}\n  \\label{asgd_accuracy}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n\\begin{center}\n\\mbox{\\hspace{-0.185in}\n\\includegraphics[width=1.9in]{MNIST8M_784_1000_1000_1000_10_compare} \\hspace{-0.15in}\n\\includegraphics[width=1.9in]{NORB_2048_1000_1000_1000_5_compare}  \\hspace{-0.19in}\n\\includegraphics[width=1.9in]{Convex_784_1000_1000_1000_2_compare} \\hspace{-0.19in}\n\\includegraphics[width=1.9in]{Rectangles_784_1000_1000_1000_2_compare} \\hspace{-0.19in}\n}\n\\end{center}\n\\caption{Performance comparison between our randomized hashing approach and a standard network using asynchronous stochastic gradient descent on an Intel Xeon ES-2697 machine with 56-cores. We used (3 hidden layer) networks on MNIST (1st), NORB (2nd), Convex (3rd) and Rectangles (4th). All networks were initialized with the same settings for this experiment.}\n  \\label{asgd_lsh_std}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n\\begin{center}\n\\mbox{\\hspace{-0.185in}\n\\includegraphics[width=1.9in]{MNIST8M_784_1000_1000_1000_10_wct_lg_xy} \\hspace{-0.15in}\n\\includegraphics[width=1.9in]{NORB_2048_1000_1000_1000_5_wct} \\hspace{-0.19in}\n\\includegraphics[width=1.9in]{Convex_784_1000_1000_1000_2_wct} \\hspace{-0.19in}\n\\includegraphics[width=1.9in]{Rectangles_784_1000_1000_1000_2_wct} \\hspace{-0.19in}\n}\n\\end{center}\n\\caption{The wall-clock per epoch for our approach (LSH-5\\%) gained by using asynchronous stochastic gradient descent. We used a (3 hidden layer) network on the MNIST (1st), NORB (2nd), Convex (3rd) and Rectangles (4th). We see smaller performance gains with the Convex and Rectangles datasets because there are not enough training examples to use of all of the processors effectively. Only 5\\% of the standard network's computation was performed in this experiment.}\n\\label{asgd_performance}\n\\end{figure*}\n\n\\subsection{Scalability}\n\\label{sec:ASGD}\n\\subsubsection{Experimental Setting}\nWe now show experiments to demonstrate the scalability of our approach to large-scale, distributed computing environments. Specifically, we are testing if our approach maintains accuracy and improves training time, as we increase the number of processors. We use asynchronous stochastic gradient descent with momentum and adagrad \\cite{dean2012large, recht2011hogwild}. Our implementation runs the same model, with similar initializations, on multiple training examples concurrently. The gradient is applied without synchronization or locks to maximize performance. We run all of the experiments on an Intel Xeon ES-2697 machine with 56 cores and 256 GB of memory. The ReLU activation was used for all models, and the learning rate ranged between $10^{-2}$ and $10^{-3}$.\n\n\\subsubsection{Results with different numbers of processors}\nFigure \\ref{asgd_accuracy} shows how our method performs with asynchronous stochastic gradient descent (ASGD) using only 5\\% of the neurons of a full-sized neural network. The neural network has three hidden layers, and each hidden layer contains 1000 neurons. The x-axis represents the number of epochs completed, and the y-axis shows the test accuracy for the given epoch. We compare how our model converges with multiple processors working concurrently. Since our ASGD implementation does not use locks, it depends on the sparsity of the gradient to ensure the model converges and performs well \\cite{recht2011hogwild}. From our experiments, we see that our method converges at a similar rate and obtains the same accuracy regardless of the number of processors running ASGD.\n\nFigure \\ref{asgd_performance} illustrates how our method scales with multiple processors. The inherent sparsity of our randomized hashing approach reduces the number of simultaneous updates and allows for more asynchronous models without any performance penalty. We show the corresponding drop in wall-clock computation time per epoch while adding more processors. We achieve roughly a 31x speed up while running ASGD with 56 threads.\n\n\\subsubsection{ASGD Performance Comparison with Standard Neural Network}\nFigure \\ref{asgd_lsh_std} compares the performance of our LSH approach against a standard neural network (STD) when running ASGD with 56-cores. We used a standard network with a mini-batch of size 32. We clearly out-perform the standard network for all of our experimental datasets. However, since there is a large number of processors applying gradients to the parameters, those gradients are constantly being overridden, preventing ASGD from converging to an optimal local minimum. Our approach produces a spare gradient that reduces the conflicts between the different processors, while keeping enough valuable nodes for ASGD to progress towards the local minimum efficiently. \n\nFrom Figures \\ref{asgd_accuracy}, \\ref{asgd_lsh_std} and \\ref{asgd_performance}, we conclude the following. \n\\begin{enumerate}\n\\item The gradient updates are quite sparse with 5\\% LSH and running multiple ASGD updates in parallel does not affect the convergence rate of our hashing based approach. Even when running 56 cores in parallel, the convergence is indistinguishable from the sequential update (1 core) on all the four datasets.\n\\item If we instead run vanilla SGD in parallel, then the convergence is effected. The convergence is in general slower compared the sparse 5\\% LSH. This slow convergence is due to dense updates which leads to overwrites. Parallelizing dense updates affects the four datasets differently. For convex dataset, the convergence is very poor.\n\\item As expected, we obtain near-perfect decrease in the wall clock times with increasing the number of processors with LSH-5\\%. Note, if there are too many overwrites, then atomic overwrites are necessary, which will create additional overhead and hurt the parallelism. Thus, near-perfect scalability also indicates fewer gradient overwrites. \n\\item On the largest dataset - MNIST8M, the running time per epoch for the 1-core implementation is 50,254 seconds. The 56-core implementation runs in 1,637 seconds. Since the convergence is not affected, this amounts to about a 31x speedup in the training process with 56 processors. \n\\item We see that the gains of parallelism become flat with the Convex and Rectangle datasets, especially while using a large number of processors. This poor scaling is because the two datasets have fewer training examples, and so there is less parallel work for a large number of processor, especially when working with 56 cores. We do not see such behaviors with MNIST8M which has around 8 million training examples.\n\\end{enumerate}\n\n\\section{Discussions}\nMachine learning with a huge parameter space is becoming a common phenomenon. SGD remains the most promising algorithm for optimization due its effectiveness and simplicity. SGD that updates from a single data point is unlikely to change the entire parameter space significantly. Each SGD iteration is expected to change only a small set (the active set) of parameters, depending on the current sample. This sparse update occurs because there is not enough information in one data point. However, identifying that small set of active parameters is a search problem, which typically requires computations on the order of parameters. We can exploit the rich literature in approximate query processing to find this active set of parameters efficiently. Of course, the approximate active set contains a small amount of random noise, which is often good for generalization. Sparsity and randomness enhance data parallelism because sparse, and random SGD updates are unlikely to overwrite each other. \n\nApproximate query processing already sits on decades of research from the systems and database community, which makes the algorithm scalable over a variety of distributed systems. Thus, we can forget about the systems challenges by reformulating the machine learning problem into an approximate query processing problem and levering the ideas and implementation from a very rich literature. We have demonstrated one concrete evidence by showing how deep networks can be scaled-up. We believe that such integration of sparsity with approximate query processing, which is already efficient over different systems, is the future of large-scale machine learning.\n\n\\section{Conclusion and Future Work}\nOur randomized hashing approach is designed to minimize the amount of computation necessary to train a neural network effectively. Training neural networks requires a significant amount of computational resources and time, limiting the scalability of neural networks and its sustainability on embedded, low-powered devices. Our results show that our approach performs better than the other methods when using only 5\\% of the nodes executed in a standard neural network. The implication is that neural networks using our approach can run on mobile devices with longer battery life while maintaining performance. We also show that due to inherent sparsity our approach scales near-linearly with asynchronous stochastic gradient descent running with multiple processors.\n\nOur future work is to optimize our approach for different mobile processors. We have many choices to choose from including the Nvidia Tegra and Qualcomm Snapdragon platforms. Recently, Movidius released plans for a specialized deep learning processor called the Myriad 2 VPU (visual processing unit) \\cite{barry2015always}. The processor combines the best aspects of GPUs, DSPs, and CPUs to produce 150 GFLOPs and while using about 1 W of power. We plan to leverage the different platform architectures to achieve greater efficiency with our randomized hashing approach.\n\n\\section{Acknowledgments}\nThe work of Ryan Spring was supported from NSF Award 1547433. This work was supported by Rice Faculty Initiative Award 2016.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIn text games, which are also known as Interactive Fiction (IF) \\cite{Montfort:2005}, the player is given a description of the game state in natural language and then chooses one of the actions which are also given by textual descriptions.\nThe executed action results in a change of the game state, producing new state description and waiting for the player's input again.\nThis process repeats until the game is over.\n\nThe engine that is responsible for running IF games is called simply a \\emph{game simulator} and it provides an interface for player (human or bot) interaction.\n\nWhile the output of the game simulator is almost always a text description, the form of the input that the game expects --- the game-player interface --- does vary.\nThis is one of the most common criteria for classifying IF games (for examples, see Figure \\ref{fig:if-types}):\n\n\\begin{itemize}\n\n\\item \\emph{parser-based}, where the player types in any text input freely,\n\n\\item \\emph{choice-based}, where multiple actions to choose from are typically available, \\emph{in~addition} to the state description,\n\n\\item \\emph{hypertext-based}, where multiple actions are present as clickable links \\emph{inside} the state description.\n\\end{itemize}\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=\\textwidth]{if-types.png}\n\n\\caption{Parser, choice and hypertext-based games \\cite{DBLP:journals\/corr\/HeCHGLDO15}.}\n\\label{fig:if-types}\n\\end{figure*}\n\nThe largest source of information on IF games and their variants is the Interactive Fiction Database\\footnote{IFDB: \\url{http:\/\/ifdb.tads.org\/}\\label{ftn:ifdb}.} which features useful filters, tags and lists for finding specific kinds of games.\nFor our purposes, though, the differences in game genre or storytelling elements largely do not matter.\n\nText games come in virtually all genres, often with different minigames or puzzles incorporated into them.\nWe mostly deal with games without any additional special parts that do not fit into the general and simple input-output loop or we omit them from the selected games.\n\nIn this work, we only deal with \\emph{choice-based} and \\emph{hypertext-based} games but both variants are referred to as \\emph{choice-based}, as the hyperlinks are simply considered additional choices.\nIn fact, all parser-based games with finite number of actions can be converted to choice-based games by simply enumerating all the actions accepted by the interpreter at different time steps.\nThis trick was employed by \\cite{DBLP:journals\/corr\/NarasimhanKB15} where a subset of all possible action combinations was presented as choices to the agent.\n\n\nMore formally speaking, text games are sequential decision making tasks with both state and action spaces given in natural language.\nIn fact, a text game can be seen as a dialogue between the game and the player and the task is to find an optimal policy (a mapping from states to actions) that would maximise the player's total reward.\nConsequently, the task of learning to play text games is very similar to learning how to correctly answer in a dialogue, making this task relevant for real-world applications.\n\nUsually, text games have multiple endings and countless paths that the player can take.\nIt is often clear that some of the paths or endings are better than others and different rewards can be assigned to them.\nAvailability of these feedback signals makes text games an interesting platform for using reinforcement learning (RL), where one can make use of the feedback provided by the game in order to try and infer the optimal strategy of responding in the given game, and potentially, in a dialogue. \n\nOur aim is to:\n\n\\begin{enumerate}\n\\item Provide a platform which would enable researchers to easily add new text games. Importantly, this library unifies the interface of different types of text games, enabling researchers to conduct large-scale experiments.\n\\item Design a minimalistic agent capable of playing basic text games well. This agent will also implement the interface of the text-game library and could be used as a starting point for more complex models in the future.\n\\end{enumerate}\n\n\n\n\\section{Related Work}\n\nRL algorithms have a long history of being successfully applied to solving games, for example to Backgammon \\cite{DBLP:journals\/cacm\/Tesauro95} or the more recent general AlphaZero algorithm that achieved superhuman performance in Chess and Go \\cite{DBLP:journals\/corr\/abs-1712-01815}.\n\nThere have also been successful attempts \\cite{DBLP:journals\/corr\/NarasimhanKB15},  \\cite{DBLP:journals\/corr\/HeCHGLDO15} at playing IF games using RL agents.\nHowever, the selected games were quite limited in terms of their difficulty and even more importantly, the resulting models had mostly not been tested on games that had not been seen during learning.\n\nWhile being able to learn to play a text game is a success in itself, the resulting model must generalise to previously unseen data in order to be useful.\nIn other words, we can merely hypothesise that a successful IF game agent can at least partly understand the underlying game state and potentially transfer the knowledge to other, previously unseen, games, or even natural dialogues.\nAnd for the most part, it remains to be seen how the RL agents presented in \\cite{DBLP:journals\/corr\/NarasimhanKB15} and \\cite{DBLP:journals\/corr\/HeCHGLDO15} perform in terms of generalisation in the domain of IF games.\n\nPrior relevant work includes learning to play the game Civilization by leveraging text manuals \\cite{DBLP:journals\/corr\/BranavanSB14} and achieving human-like performance on Atari video games \\cite{DBLP:journals\/nature\/MnihKSRVBGRFOPB15}.\n\nAs our agent is partly inspired by the architectures employed in \\cite{DBLP:journals\/corr\/NarasimhanKB15} and \\cite{DBLP:journals\/corr\/HeCHGLDO15}, these models are next described in more detail.\n\nThe LSTM-DQN \\cite{DBLP:journals\/corr\/NarasimhanKB15} agent uses an LSTM network \\cite{hochreiter1991untersuchungen} for representation generation, followed by a variant of a Deep Q-Network (DQN) \\cite{DBLP:journals\/nature\/MnihKSRVBGRFOPB15} used for scoring the generated state and action vectors. The underlying task of playing parser-based games is effectively reduced to playing choice-based games by presenting a subset of all possible verb-object tuples as available actions to the agent.\n\nIn the framework, a single action consists of a verb and an object (e.g. \\emph{eat apple}) and the model computes Q-values for objects --- $Q(s, o)$ --- and actions --- $Q(s, a)$ --- separately. The final Q-value is obtained by averaging these two values.\n\nThe Deep Reinforcement Relevance Network (DRRN) \\cite{DBLP:journals\/corr\/HeCHGLDO15} was introduced for playing hypertext-based games.\nThe agent learns to play \\emph{Saving John} and \\emph{Machine of Death} games and used a simple bag-of-words (BOW) representation of the input texts.\n\nThe learning algorithm is also a variant of DQN; DRRN refers to the network that estimates the Q-value, using separate embeddings for states and actions. DRRN then uses a variable number of hidden layers and makes use of softmax action-selection.\nThe final Q-value is obtained by computing an inner product of the inner representation vectors of states and actions.\n\n\n\n\\section{Background}\n\nWe formally state the problem of learning to play the games as solving a Markov decision process.\nThen, we briefly review common machine learning methods that will be employed, most notably neural networks and Q-learning.\n\n\\subsection{Formulating the learning task}\n\\label{sec:task}\n\nText game is a sequential decision-making task with both input and output spaces given in natural language.\n\n\\begin{definition}[Text game]\n\\label{def:text-game}\nLet us define a text game as a tuple $G = \\langle H, H_t, S, A, \\mathcal{D}, \\mathcal{T}, \\mathcal{R} \\rangle$, where\n\\begin{itemize}\n\\item $H$ is a set of game states, $H_t$ is a set of terminating game states, $H_t \\subseteq H$,\n\\item $S$ is a set of possible state descriptions,\n\\item $A$ is a set of possible action descriptions,\n\\item $\\mathcal{D}$ is a function generating text descriptions, $\\mathcal{D}: H \\to (S \\times 2^A)$,\n\\item $\\mathcal{T}$ is a transition function, $\\mathcal{T}: (H \\times A) \\to H$,\n\\item $\\mathcal{R}$ is a reward function, $\\mathcal{R}: (S_t, A_t, S_{t+1}) \\to \\mathbb{R}$.\n\\end{itemize}\n\\end{definition}\n\n\n\nGenerally speaking, both the transition function $\\mathcal{T}$ and the description function $\\mathcal{D}$ may be stochastic and the properties of the game and its functions have a great impact on the task difficulty\\footnote{For a more thorough discussion, please refer to \\cite{zelinka}.}.\n\n\nIn particular, if both $\\mathcal{D}$ and $\\mathcal{T}$ are deterministic, the \\emph{whole game} is deterministic.\nConsequently the problem is then reduced to a simple graph search problem that can be solved by graph search techniques and there is no need --- from the perspective of finding optimal rewards --- to employ reinforcement learning methods.\nHowever, from the generalisation perspective, it still might be useful to attempt to learn simple games using RL techniques as the agents could potentially be able to generalise or transfer their knowledge to other, more difficult problems.\n\nWe define the task of learning to play a text game as the process of finding an optimal policy for the respective text game Markov decision process (MDP) \\cite{DBLP:books\/lib\/SuttonB98}, where $\\mathcal{M}~=~\\langle S_\\mathcal{M}, A_\\mathcal{M}, \\mathcal{P}, \\mathcal{R}, \\gamma_\\mathcal{M} \\rangle$.\n\nStates, actions and the reward function $S$, $A$, $\\mathcal{R}$ of the game correspond to the MDP states, actions and the reward function  $S_\\mathcal{M}$, $A_\\mathcal{M}$, $\\mathcal{R}$.\nThe MDP probability function $\\mathcal{P}(s' | s, a)$ is realised by the game transition function $\\mathcal{T}$ and is crucially unknown to the agent.\n\nNotice that not all IF games necessarily have the Markov property, i.e. in text games, there can be long-term dependencies.\nIn fact, it is even difficult to determine whether a text game is or is not Markovian.\nHowever, even problems with non-Markovian characteristics are commonly represented and modelled as MDPs while still giving good results \\cite{DBLP:books\/lib\/SuttonB98}.\n\nFollowing the common notation, the agent tries to maximise the  \\textbf{discounted cumulative reward} $\\sum^{\\infty}_{t=0} {\\gamma^t \\mathcal{R} (s_t, a_t, s_{t+1})}$ while following the policy $\\pi(s)$ by selecting actions $a \\gets \\pi(s)$.\nThe goal is to learn the \\textbf{optimal policy $\\pi^*$} by maximising the following term:\n\n\\begin{equation}\n\\label{eq:optimal-policy}\n\\sum^{\\infty}_{t=0} {\\gamma^t \\mathcal{R} (s_t, a_t, s_{t+1})}\\text{, where } a_t = \\pi^{*}(s_t).\n\\end{equation}\n\nUsing this approach, we have now formally reduced the task of learning to play the text-based games to estimating an optimal policy in a text-game MDP.\nThe Q-learning algorithm and its neural network estimator for solving the problem are described next.\n\n\n\\subsection{Deep Reinforcement Learning}\n\nFor finding the optimal policy in the text-game MDP, we employ Deep Reinforcement Learning (DRL) \\cite{DBLP:journals\/nature\/MnihKSRVBGRFOPB15}. This term is commonly used to describe RL algorithms that use (deep) neural networks as a part of their value function estimation.\n\nReinforcement learning is an area of machine learning methods and problems based on the notion of learning from numerical rewards obtained by an agent through interacting with an environment \\cite{DBLP:books\/lib\/SuttonB98}.\n\nIn any given step, the agent observes a \\emph{state} of the environment and receives a~\\emph{reward signal}.\nBased on the current state and the agent's behaviour function --- the \\emph{policy} --- the~agent chooses an~\\emph{action} to take.\nThe action is then sent to the environment which is updated and the loop repeats. See Figure \\ref{fig:rl-loop} for illustration of the agent-environment interface.\n\n\\begin{figure}[!ht]\\centering\n\\includegraphics[]{rl-loop.png}\n\\caption{Interaction between the agent and the environment \\cite{DBLP:books\/lib\/SuttonB98}.}\n\\label{fig:rl-loop}\n\\end{figure}\n\nThere are different approaches to learning the optimal policy.\nHere, we focus on a method for model-free control that has been shown to perform well on a variety of game-related tasks \\cite{DBLP:journals\/cacm\/Tesauro95}, \\cite{DBLP:journals\/nature\/MnihKSRVBGRFOPB15},  Q-learning.\n\nQ-learning makes use of the concept of a \\emph{value function} which evaluates how good a given state under a given policy is --- what reward we can expect to obtain in the long run if we follow the policy from the specific state.\nAction value function that determines the value of taking action $a$ in state $s$ using policy $\\pi$ is defined as $Q^\\pi(s, a) = \\mathbb{E}_{\\pi} \\left[ \\sum_{t=0}^{\\infty} \\gamma^t r_{t+1}\\ \\middle|\\ s, a \\right]$.\n\n\n\nNow the optimal policy, denoted $\\pi^*$, can be also characterised by the optimal action-value function $Q^*(s, a) = \\underset{\\pi}{\\max}\\ Q^\\pi(s,a)$.\n\n\nIn other words, if we know the optimal action-value function $Q^*$, we can obtain the optimal policy $\\pi^*$ by simply choosing the actions with maximum Q-values: $\\pi^{*}(s) = \\underset{a}{\\max}\\ Q^{*}(s,a)$.\n\n\nTypically, the agent's policy is also $\\epsilon$-greedy with relation to the Q-function, where $0 \\leq \\epsilon \\leq 1$ is a parameter that corresponds to the probability of choosing a random action.\n\nSince Q-learning attempts to find the optimal Q-values which obey the Bellman equation \\cite{bellman2013dynamic}, the update rule for improving the estimate of the Q-function is as follows \\cite{DBLP:books\/lib\/SuttonB98}:\n\\begin{equation} \\label{eq:q}\n\\begin{split}\n Q(s_{t},a_{t}) \\gets Q(s_{t},a_{t}) + \\phantom{=}\\phantom{=}\\phantom{=}\\phantom{=}\\phantom{=}\\phantom{=}\\phantom{=}\\phantom{=}\\phantom{=}\\phantom{=}\\phantom{=}\\phantom{=}\\phantom{=}\\phantom{.}\\\\\n\\alpha_{t} \\cdot \\left( r_{t+1} + \\gamma \\cdot \\max_{a_{t+1}}Q(s_{t+1}, a_{t+1}) - Q(s_{t},a_{t}) \\right),\n\\end{split}\n\\end{equation}\nwhere $0 \\leq \\alpha_t \\leq 1$ is the learning rate parameter.\n\nThe Q-learning algorithm is guaranteed to converge towards the optimal solution \\cite{DBLP:books\/lib\/SuttonB98}. In simpler problems and by default, it is assumed that the Q-values for all state-action pairs are stored in a table.\n\nThis approach is, however, not feasible for more complex problems such as the Atari platform games task \\cite{DBLP:journals\/nature\/MnihKSRVBGRFOPB15} or our text-game task, where the state and action spaces are simply too large to store.\nIn text games, for example, the spaces are infinite.\nWe address this problem by approximating the optimal Q-function by a function approximator in the form of a neural network. The Q-function is parametrised as\n\\begin{equation}\n\\label{eq:nn-q}\nQ^*(s_t, a_t) \\approx Q(s_t, a_t, \\theta_t) = \\theta_t(s_t, a_t),\n\\end{equation}\nwhere the $\\theta$ function is realised by a neural network.\n\nThe advantage of this non-tabular approach is that even in infinite spaces, the neural network can generalise to previously unobserved inputs and consequently cover the whole search space with reasonable accuracy.\nIn contrast to linear function approximators, though, non-linear approximators such as neural networks do not guarantee convergence in this context.\n\n\\section{Goals}\n\nOur goal is to introduce a minimalistic architecture serving as a proof of concept, with the ability to capture important sentence-level features and ideally capable of reasonable generalisation to previously unseen data.\nFirst, we highlight some of the aspects of the LSTM-DQN and DRRN models that could be improved upon in terms of these requirements.\n\nThe main drawback of DRRN is its use of BOW for representation generation.\nConsequently, the model is incapable of properly handling state aliasing and differentiating simple yet important nuances in the input, such as the difference between \\emph{``There is a treasure chest to your left and a dragon to your right.''} and \\emph{``There is a treasure chest to your right and a dragon to your left.''}.\n\nMoreover,\nHe et al. \\cite{DBLP:journals\/corr\/HeCHGLDO15} claim that separate embeddings for state and action spaces lead to faster convergence and to a better solution. However, since both state and action spaces really contain the same data --- at least in most games and especially in hypertext games where actions are a subset of states --- we aim to employ a joint embedding representation of states and actions.\n\nWe also believe that a joint representation of states and actions should eventually lead to stronger generalisation capabilities of the model, since such model should be able to transfer knowledge between state and action descriptions as their representation would be shared.\n\nThe LSTM-DQN agent, on the other hand, utilises an LSTM network that can theoretically capture more complex sentence properties such as the word order.\nHowever, its architecture only accepts actions consisting of two words.\n\nAdditionally, the two action Q-values are finally averaged, which would arguably be problematic if the verbs and objects were not independent.\nFor example, the value of the verb ``\\emph{drink}'' varies highly based on the object; consider the difference between the values of ``\\emph{drink}'' when followed by either ``\\emph{water}'' or ``\\emph{poison}'' objects.\n\nWe thus aim to utilise a minimalistic architecture that should:\n\\begin{itemize}\n\\item be able to capture dependencies on sentence level such as word order,\n\\item accept text input of any length for both states and action descriptions,\n\\item accept and evaluate any number of actions,\n\\item use a powerful interaction function between states and actions.\n\\end{itemize}\n\n\\section{Methods}\nWe present the \\emph{pyfiction} platform and specify the relevant learning tasks. Then we describe the architecture of our agent capable of learning the games, leveraging the general game interface of the platform. Finally, we describe our agent architecture in detail.\n\n\\subsection{Platform}\n\nThe \\emph{pyfiction}{\\footnote{\\emph{pyfiction}: \\url{https:\/\/github.com\/MikulasZelinka\/pyfiction}.} platform is a library for universal access to different kinds of IF games.\nIts interface is identical to the general RL interface (see Figure \\ref{fig:rl-loop}).\nCurrently, it supports the \\emph{Glulxe}, \\emph{Z-machine} and general HTML simulators.\n\nThere are eight games present as of now, however, adding new games is straightforward.\nApart from the game files, it is only necessary to provide a mapping from states to actions and to numerical rewards for the game to be playable by AI agents.\n\nAny agent compatible with the simple RL interface can directly play all games present in \\emph{pyfiction}. Pyfiction can also integrate to OpenAI Gym \\cite{DBLP:journals\/corr\/BrockmanCPSSTZ16}.\n\n\n\\subsection{The SSAQN architecture}\n\\label{sec:slsn}\n\nOur neural network model is inspired by both LSTM-DQN and DRRN. For the sake of clarity, it is referred to as \\textbf{SSAQN} (Siamese State-Action Q-Network).\n\nSimilarly to \\cite{DBLP:journals\/corr\/NarasimhanKB15} and \\cite{DBLP:journals\/corr\/HeCHGLDO15}, we employ a variant of DQN \\cite{DBLP:journals\/nature\/MnihKSRVBGRFOPB15} with experience replay and prioritised sampling which uses a neural network (SSAQN) for estimating the DQN's Q-function.\n\nSSAQN uses a siamese network architecture \\cite{DBLP:conf\/nips\/BromleyGLSS93}, where two branches --- a state branch and an action branch --- share most of the layers that effectively generate useful representation features.\nThis is best illustrated by visualising the network's computational graph; see Figure \\ref{fig:architecture}.\n\nAs we are using a siamese architecture, the weights of the embedding and LSTM layers are shared between state and action data passing through.\nStates and actions are only differentiated in the dense layers whose outputs are then fed into the similarity interaction function.\n\nThe most important differences to LSTM-DQN and DRRN are:\n\n\\begin{itemize}\n\\item the network accepts two text descriptions (state and action) of arbitrary length as an input,\n\\item the embedding and LSTM layers are the same for states and action, i.e. their weights are shared,\n\\item the interaction function of inner vectors of states and actions is realised by a normalised dot product, commonly called cosine similarity (see Section \\ref{sec:interaction-function}).\n\\end{itemize}\n\nThe output of the SSAQN is the estimated Q-value for the given state-action pair, i.e. the network with parameters $\\theta$ realises a function $\\theta(s, a) =  Q(s, a, \\theta) \\approx Q^*(s, a)$ where the input variables $s$ and $a$ contain preprocessed text\\footnote{For more details, see the \\emph{pyfiction} library.}.\n\n\nTo compute the $Q(s, a^i)$ for different $i$ --- for multiple actions --- we simply run the forward pass of the action branch multiple times.\n\nNext, the SSAQN architecture is described layer-by-layer in more detail.\n\n\\begin{figure}[!ht]\\centering\n\\includegraphics[width=0.5\\textwidth]{img\/nn_2_.pdf}\n\\caption[Architecture of the SSAQN model and its data flow.]{\nArchitecture of the SSAQN model and its data flow. Grey boxes represent data values in different layers; the bold text corresponds to the shape of the data~tensors.}\n\\label{fig:architecture}\n\\end{figure}\n\n\\subsubsection{Word embeddings}\n\\label{sec:emb-layer}\n\nAfter preprocessing the text, we convert the words to their vector representation using word embeddings \\cite{DBLP:conf\/nips\/MikolovSCCD13}.\nSince our dataset is comparatively small, we use a relatively low dimensionality for the word representations and work with \\cd{embedding\\_dim} of~$16$.\nThe weights of the word embeddings are initialised to small random values and trained as a part of the gradient descent algorithm. Using pre-trained vector models is also supported.\n\nAs the training is done in batches, we pad the input sequences of words for each batch so that they are all aligned to the same length. Still, note that in general, the length of the output of the embedding layer is variable in length.\n\n\\subsubsection{LSTM layer}\n\\label{sec:lstm-layer}\n\nThe inputs of the LSTM layer are the word embedding vectors of variable length.\nSimilarly to embeddings, the weights of the LSTM units are also initialised to small random values and their weights are shared between states and actions.\n\nThe role of the LSTM layer is to successively look at the input words, changing its inner state in the process and thus detecting time-based features in its input.\nIt is also at this layer that we go from having a data shape of arbitrary length to having a fixed-length output vector.\nThe output size is equal to the number of LSTM units in the layer and in our experiments, we use \\cd{lstm\\_dim} of~$32$.\n\n\n\n\n\\subsubsection{Dense layer}\n\\label{sec:dense-layer}\n\nFollowing the shared LSTM layer, we now have two dense (fully-connected) layers, one for states and one for actions.\nAgain, we initialise the weights randomly and we also apply the hyperbolic tangent activation function $\\mathrm{tanh}(x) = \\frac{1 - e^{-2x}} {1 + e^{-2x}}$.\n\n\nAs the dense layers for states and actions are the only layers to not necessarily share weights between the two network branches, they do play an important role in building differentiated representations for state and action input data.\n\nNote that as the interaction layer uses a dot product, we require both outputs of the dense layers to be of the same dimension and we set \\cd{dense\\_dim} of both branches to~$8$.\nHowever, theoretically, it would be both possible and interesting to use different layer dimensions for states and actions at this level, as usually, the original state text descriptions carry more information than action descriptions in IF games.\nThus, a possible extension of the network would be to use two or more hidden dense layers in the  state branch of the network and to only reduce the dimension to the action dimension in the last hidden dense state layer.\n\n\n\\subsubsection{Interaction function}\n\\label{sec:interaction-function}\n\nLastly, we apply the cosine similarity interaction function to the state and action dense activations, resulting in the final Q-value.\nIf the input are two vectors $\\mathbold{x}$ and $\\mathbold{y}$ of~$\\cd{dense\\_dim} = n$ elements, we define their cosine similarity as a dot product of their L2-normalised (and consequently unit-sized) equivalents:\n\\begin{equation}\n\\label{eq:cos}\n\\mathrm{cs}(\\mathbold{x}, \\mathbold{y}) = {\\frac{\\mathbold{x} \\cdot \\mathbold{y}} {\\|\\mathbold{x}\\|_2 \\|\\mathbold{y}\\|_2}} = \\frac{ \\sum\\limits_{i=1}^{n}{\\mathbold{x}_i  \\mathbold{y}_i} }{ \\sqrt{\\sum\\limits_{i=1}^{n}{\\mathbold{x}_i^2}}  \\sqrt{\\sum\\limits_{i=1}^{n}{\\mathbold{y}_i^2}} },\n\\end{equation}\nwhich corresponds to the cosine of the angle between the input vectors.\n\nCosine similarity is commonly used for determining document similarity \\cite{huang2008similarity}.\nHere, we apply it to the two hidden vectors of dense layer values that should meaningfully represent the condensed information that was originally received as a text input by the network and we interpret the resulting value as an indicator of mutual compatibility of the original state-action pair.\nSince the range of values of the $\\cos$ function is $[-1, 1]$ and since the original rewards that we aim to estimate have arbitrary values, we scale the approximated Q-values accordingly.\n\n\\subsubsection{Loss function and gradient descent}\n\nRecall the Q-learning rule (see equation \\ref{eq:q}). We define the loss function at time~$t$ as\n\\begin{equation}\n\\label{eq:loss}\n\\mathcal{L}_t = \\left( r_{t} + \\gamma \\cdot \\max_{a}Q(s_{t+1}, a) - Q(s_{t},a_{t}) \\right)^2,\n\\end{equation}\nwhich is simply a mean squared error (MSE) of the last estimated Q-value and the target Q-value.\n\nFor gradient descent, we make use of the RMSProp optimiser \\cite{rmsprop} that has been shown to perform well in numerous practical applications, especially when applied in LSTM networks \\cite{DBLP:journals\/corr\/DauphinVCB15}.\n\n\n\\subsection{Action selection}\n\\label{sec:action-selection}\n\nGiven an SSAQN $\\theta$, where $Q(s,a) \\gets \\theta(s, a)$, the agent selects an action by following the $\\epsilon$-greedy policy $\\pi_\\epsilon(s)$ \\cite{DBLP:books\/lib\/SuttonB98} realised by the following algorithm:\n\n\\begin{algorithm}[!ht]\n\\caption{Action selection}\\label{alg:action-selection}\n\\begin{algorithmic}[1]\n\\Statex \n\\Statex   $\\epsilon$: probability of choosing a random action\n\\Statex   $h(s,a)$: number of times the agent selected $a$ in $s$ in the current run\n\\Statex \n\\Function{Act}{$s$, $actions$, $\\epsilon$, $h$}\n  \\If{$random() < \\epsilon$}\n    \\Return {random action} \\EndIf\n  \\State $q\\_values = \\theta(s, a_i)$ \\textbf{for} $a_i$ in $actions$ \\label{row:4}\n  \\State $q\\_values = (q\\_values + 1) \/ 2$ \\Comment{from $[-1, 1]$ to~$[0, 1]$}\n  \\State $q_i = q_i^{h(s, a_i) + 1}$ \\textbf{for} $q_i$ in $q\\_values$ \\Comment{history function} \\label{row:history}\n  \\State $q\\_values = (q\\_values \\cdot 2) - 1$ \\Comment{from $[0, 1]$ to~$[-1, 1]$}\n  \\State \\textbf{return} $a_i$ with $\\max q_i$ \\label{row:8}\n\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\n\nThe $\\epsilon$ is the exploration parameter.\nFor sampling in training phase (see Algorithm \\ref{alg:dqn}), $\\epsilon$ is gradually decayed from the starting value of~$1$, i.e. at first, the agent's policy is completely random.\nIn testing phase, the agent is greedy, i.e. $\\epsilon$ is set to~$0$ and the agent always chooses the action with the maximum Q-value for the given state.\n\n\nThe only important difference between the standard $\\epsilon$-greedy control algorithm and our action selection policy is that we additionally employ a \\emph{history function}, $h(s, a)$.\nThe scope of the history function is a single run of the agent on a single game, i.e. it is reset every time a game ends.\n\nThe function $h(s,a)$ returns a value equal to the number of times the agent selected action $a$ in state $s$ in the current run.\nThat is, if the agent never selects an action twice in the same state during a run, the history function has no impact on action selection --- and it penalises the already visited state-action pairs, as seen on line \\ref{row:history} of Algorithm \\ref{alg:action-selection}.\n\nThe history function serves as a very simple form of intrinsic motivation \\cite{DBLP:conf\/nips\/SinghBC04}.\nIt is similar to optimistic initialisation \\cite{DBLP:books\/lib\/SuttonB98} in that it leads the agent to select previously unexplored state-action tuples.\nAdditionally, note that the history function is not Markovian in the sense that it takes the whole game episode into account.\nIn practice, the history function greatly helps the agent to avoid infinite loops, since for many games, it is likely to get stuck in an infinite loop when following a randomly chosen deterministic Markovian policy.\n\n\\subsection{Training loop}\n\nPutting together all the parts introduced above, we can now formally describe the agent's learning algorithm.\n\nWe use a variant of DQN \\cite{DBLP:journals\/nature\/MnihKSRVBGRFOPB15} with experience replay and prioritised sampling of experience tuples with positive rewards \\cite{DBLP:journals\/corr\/SchaulQAS15}. For more details, see Algorithm \\ref{alg:dqn}.\n\nNote that the agent inherently supports playing and learning multiple games at once.\n\n\\begin{algorithm}\n\\caption{Training algorithm (a variant of DQN)}\\label{alg:dqn}\n\\begin{algorithmic}[1]\n\\Statex \n\\Statex   $b$: batch size, $p$: prioritised fraction\n\\Statex   $\\epsilon$: exploration parameter, $\\epsilon\\_decay$: rate at which $\\epsilon$ decreases\n\\Statex \n\\Function{Train}{$episodes$, $b$, $p$, $\\epsilon = 1$, $\\epsilon\\_decay$}\n\n  \\State Initialise experience replay memory $\\mathcal{D}$\n  \\State Initialise the neural network $\\theta$ with random weights\n  \\State Initialise all game simulators and load the vocabulary\n\n  \\For {$e \\in {0, \\ldots, episodes - 1}$}\n    \\State Sample each game once using $\\pi_\\epsilon$, store experiences into~$\\mathcal{D}$\n    \\State $batch$ $\\gets$ $b$ tuples $(s_t, a_t, r_t, s_{t+1}, a_{t+1})$ from $\\mathcal{D}$, where a fraction of~$p$ have $r_t > 0$\n   \n    \\For {$i, (s_t^i, a_t^i, r_t^i, s_{t+1}^i, a_{t+1}^i)$ in $batch$}\n      \\State $target^i \\gets r_t^i$\n      \\If {$a_{t+1}^i$} \\Comment{$s_{t+1}^i$ is not terminal}\n        \\State $ target^i \\mathrel{{+}{=}} \\gamma  \\cdot \\mathrm{max}_{a_{t+1}^i}\\theta(s_{t+1}^i, a_{t+1}^i)$ \n      \\EndIf\n    \\EndFor\n    \n    \\State Define loss as $\\mathcal{L}_e(\\theta) \\gets (target^i - \\theta(s_t^i, a_t^i))^2$ \n    \\State Perform gradient descent on $\\mathcal{L}_e(\\theta)$\n    \\State $\\epsilon \\gets \\epsilon \\cdot \\epsilon\\_decay$ \n    \n  \\EndFor\n\\EndFunction  \n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Experiments}\n\nWe conduct experiments on six games, \\emph{Saving John} (SJ), \\emph{Machine of Death} (MoD), \\emph{Cat Simulator 2016} (CS), \\emph{Star Court} (SC), \\emph{The Red Hair} (TRH) and \\emph{Transit} (TT).\nAll of these games are available as a part of \\emph{pyfiction}.\nSince SJ and MoD are also present in \\cite{DBLP:journals\/corr\/HeCHGLDO15}, results on these two games for both agents are directly comparable.\n\nMore details about the games including the annotated endings, are available in the \\emph{pyfiction} repository.\nFor basic statistics, see Table \\ref{tab:games-summary}.\n\n\\subsection{Setup}\n\nIn all experiments, we use the following SSAQN layer dimensions: Embedding: $16$, LSTM: $32$, Dense: $8$. The RMSProp optimiser is used with learning rate between $0.001$ and $0.00001$, batch size of $256$ with the prioritised fraction of $25\\%$, $\\gamma$ of $0.95$, $\\epsilon$ of $1$ and $\\epsilon$-decay of $0.99$.\nWe run each experiment five times.\n\nThe different testing scenarios are as follows.\n\n\\subsubsection{Single-game task} The agent is trained and evaluated on the same game.\n\\subsubsection{Transfer learning} We pre-train the agent on five games except the one it is then trained and evaluated on. \n\\subsubsection{Generalisation} Same as transfer learning, except the agent is not trained on the final game but only evaluated on the game instead.\n\\subsubsection{Playing multiple games at once} We train and evaluate a single instance of the agent on all six games at once. In each training step, all games are presented successively to the agent.\n\n\nIn all of the scenarios, the agents are mainly compared to the random agent baseline.\nFor SJ and MoD, we can also compare the results with the DRRN agent as well as the baselines given in \\cite{DBLP:journals\/corr\/HeCHGLDO15}.\n\n\\subsection{Results}\n\nFinal rewards from all tasks can be seen in Table \\ref{tab:games-summary} and the learning progress is depicted in Figure \\ref{fig:results} for most of the tasks.\n\nIn the single-game task, the SSAQN agent learns to play all deterministic games optimally.\nMoD is not learned optimally, but the agent outperforms the DRRN agent (see Figure \\ref{fig:comparison_drrn}).\nThe agent doesn't significantly outperform the random baseline on SC, however, it is unknown if it is possible to do so.\n\n\\begin{figure}[ht!]\n  \\centering\n  \n  \\subfloat[DRRN on SJ \\cite{DBLP:journals\/corr\/HeCHGLDO15}]{\\includegraphics[width=0.25\\textwidth]{drrn_sj}\\label{fig:drrn_sj}}\n  \\subfloat[DRRN on MoD \\cite{DBLP:journals\/corr\/HeCHGLDO15}]{\\includegraphics[width=0.25\\textwidth]{drrn_mod}\\label{fig:drrn_mod}}\n\n  \n  \\subfloat[SSAQN on SJ]{\\includegraphics[width=0.25\\textwidth]{random_sj}\\label{fig:random_sj}}\n  \\subfloat[SSAQN on MoD]{\\includegraphics[width=0.25\\textwidth]{random_mod}\\label{fig:random_mod}}\n  \n\\caption{Comparison of the DRRN and the SSAQN agent Saving John (left) and Machine of Death (right). SSAQN converges considerably faster on both games and achieves higher performance but it has higher variance on MoD. }\n  \\label{fig:comparison_drrn}\n\\end{figure}\n\nTransfer learning resulted in a slower convergence rate but similar results on deterministic games and slightly worse results on MoD and SC.\nThe agent unfortunately but unsurprisingly didn't generalise to unseen games in the generalisation task and a much larger scale of experiments would be needed to safely conclude if it's capable of doing so.\nWe attribute it mostly to the lack of relevant words and formulations between the games.\nTable \\ref{tab:games-summary} also depicts how many words from a given game are also present in other games.\n\nEven if this percentage is comparatively high, note that it is just a statistic of single words, meaning it is very unlikely that a word phrases, which are actually important, match between the games.\nAt any rate, more experiments in different scales are needed to verify this hypothesis.\n\nIn the multiple-games tasks, the agent also learned deterministic games optimally, this time however, it was able to also play MoD in an almost optimal way, significantly outperforming DRRN.\nCuriously, the result on MoD is better than in the single-game setting, suggesting that information learned from other games might have been useful.\n\n\n\\begin{figure*}[!ht]\n  \\centering\n  \n  \\subfloat[Saving John]{\\includegraphics[width=0.33\\textwidth]{universal_sj.pdf}\\label{fig:sj}}\n  \\hfill\n  \\subfloat[Machine of Death]{\\includegraphics[width=0.33\\textwidth]{universal_mod.pdf}\\label{fig:mod}}\n  \\hfill\n    \\subfloat[Cat Simulator 2016]{\\includegraphics[width=0.33\\textwidth]{universal_cs.pdf}\\label{fig:cs}}\n\n  \n  \\subfloat[Star Court]{\\includegraphics[width=0.33\\textwidth]{universal_sc.pdf}\\label{fig:sc}}\n  \\hfill\n  \\subfloat[The Red Hair]{\\includegraphics[width=0.33\\textwidth]{universal_trh.pdf}\\label{fig:trh}}\n  \\hfill\n    \\subfloat[Transit]{\\includegraphics[width=0.33\\textwidth]{universal_t.pdf}\\label{fig:tt}}\n\n  \n  \\caption{Results of the SSAQN agent on all six games. Blue: random agent. Orange: train and evaluate on the same game \\emph{X}. Green: pre-train on all but \\emph{X}, then train and evaluate on \\emph{X}. Purple: train and evaluate on all games at once. Standard deviation is shown for the random agent and random games.}\n  \\label{fig:results}\n\\end{figure*}\n\n\n\n\\begin{table*}[!t]\n\\vspace{2pt}\n\\centering\n\\begin{tabular}{llrrrrrr}\n\\toprule\n& & \\textbf{Saving John} & \\textbf{Machine of Death} & \\textbf{Cat Simulator 2016} & \\textbf{Star Court} & \\textbf{The Red Hair} & \\textbf{Transit} \\\\\n\n\\midrule\n\\multirow{6}{*}{\\rotatebox[origin=c]{90}{\\makecell{\\textsc{Properties}}}}\n                                                               \n&\\# tokens                             &  1119  &   2055  &   364 &  3929  &  155   &  575  \\\\\n&\\# states                             & 70   &  $\\geq$ 200   &  37  & $\\geq$ 420  &  18   &  76  \\\\\n&\\# endings                            &  5  &  $\\geq$ 14  &  8  & $\\geq$ 17   &    7 &  10  \\\\\n&Avg. words\/description                &  73.9  & 71.9    & 74.4   &  66.7  &  28.7   & 87.0   \\\\\n&Deterministic transitions             &  Yes  & No    & Yes   & No   &   Yes  & Yes    \\\\\n&Deterministic descriptions            &   Yes & Yes    &  Yes  &  No  &  Yes   & Yes  \\\\\n&\\% of tokens present in other games    &  68.4 & 56.0  & 79.4   &  33.7  &  92.3   & 72.7  \\\\\n\n\\midrule\n\\multirow{6}{*}{\\rotatebox[origin=c]{90}{\\makecell{\\textsc{Final reward}}}}\n\n& Random agent (average)            &      -8.6       &  -10.8  &      -0.6    &     -11.6  &   -11.4  &     -10.1    \\\\\n& Individual game         &        19.4      & 15.4 &    19.4     &    -2.2    &     19.3   &    19.5     \\\\\n& Generalisation          &         -11.2     & -15.1 &     5.7       &   -13.2 &        -10.0      &    -10.2     \\\\\n& Transfer learning       &           19.4   &   8.7 &        19.4            &     -13.3       &     19.3 &      19.5   \\\\\n& Multiple games           &       19.4      &     21.0       &     19.4       &     -8.2       &      19.3 &     19.5    \\\\\n& Optimal                  &    19.4     &  $\\approx$ 21.4  &        19.4       &     ?       &      19.3    &    19.5 \\\\\n\n\\bottomrule\n\\end{tabular}\n\\caption[Game properties and agent performance summary.]{Summary of game statistics and performance comparison of the SSAQN agent on different tasks.}\n\\label{tab:games-summary}\n\\vspace{3pt}\n\\end{table*}\n\n\n\\section{Conclusion and Future Work}\n\nWe presented a minimalistic text-game playing agent capable of learning to play multiple games at once.\nThe agent uses a twin-like SSAQN architecture and outperforms the previously suggested DRRN architecture while also being considerably simpler.\n\nWe also test the transfer learning and generalisation capabilities of the agent, concluding that it unfortunately doesn't transfer its knowledge or generalise in the limited scope of six games.\n\nTo this end, however, we present \\emph{pyfiction}, an easily extensible library for universal access to various text games that could, together  with our agent, serve as a baseline for future research.\n\nFuture work should mainly focus on expanding the text-game domain and on conducting experiments at a much larger scale.\nThis should be made much easier thanks to the presented library and we hypothesise that events agents as simple as the SSAQN agent should show some generalisation capabilities given enough data.\n\n\\section*{Acknowledgement}\n\nThe author would like to thank Rudolf Kadlec for his ideas and guidance during author's work on this topic as a part of Master studies, and to Martin Pil\\'{a}t for helpful suggestions and comments.\n\nThis research was supported by Charles University under SVV (project number 260 453) and by the Czech Science Foundation (project number 17-17125Y).\n\n\n\n\n\\pagebreak\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\label{sec_intro}\n\nFor about two decades, the most accurate realization of the SI second has been obtained from caesium fountain clocks \\cite{Wynands2005}. Nowadays the best fountain clocks offer uncertainties at the low $10^{-16}$ level, while at the same time their instability allows the reaching of statistical uncertainties at the same level after averaging times of $\\approx$10\\,000\\,s only \\cite{Guena2012}. Such frequency instabilities are enabled by sufficiently increasing the signal-to-noise ratio ($\\text{SNR}$) by the efficient loading of high numbers of atoms from a cold atom beam \\cite{Guena2012,Dobrev2016}. Furthermore, an instability degradation due to the phase noise of the interrogation oscillator (Dick effect, \\cite{Santarelli1998}) needs to be avoided. The latter has been achieved by the utilization of a cryogenic sapphire instead of a quartz-oscillator-based frequency synthesis \\cite{Santarelli1999,Guena2012}. In this case, a drawback is the necessary regular costly liquid helium refill. Recently, a new generation of pulse-tube cryocoolers has been successfully tested for the microwave synthesis of fountain clocks, avoiding liquid helium refills \\cite{Takamizawa2014,Abgrall2016}.\n\nIn an alternative approach, the required low oscillator phase noise level can be obtained by transferring the frequency stability of a cavity stabilized laser via a frequency comb to the microwave spectral range. For tight locking, modern combs are equipped with fast actuators (high bandwidth pump diode current controllers, intracavity electro-optic modulators) \\cite{Puppe2016,Hudson2005,Zhang2012}, so that the frequency stability of the comb repetition rate and its harmonics almost reach the level of stability of the cavity stabilized laser frequency. \nConsequently, the low-noise microwave signal can be obtained directly from the femtosecond laser locked to an ultrastable optical cavity \\cite{Millo2009}. Lacking the option of a fast actuator in our setup, we have chosen a differing approach by using the frequency comb as a transfer oscillator \\cite{Telle2002,Lipphardt2009,Weyers2009,Tamm2014}. \nIn this case, the beat frequencies between an ultrastable laser and the frequency comb, and the microwave oscillator and the comb are measured simultaneously. Since the generation of the control signal for the microwave oscillator is based on the evaluation of the frequency difference of the two beat notes, the noise contributions of the frequency comb, acting as the transfer oscillator, are suppressed within the bandwidth of the transfer. For the frequency comb stabilization, it is only required that all relevant beat frequencies are reliably kept within the bandwidths of the respective filters. \n\nTo realize this approach, we have extended our setup for absolute frequency measurements of optical clock frequencies (e.g. \\cite{Tamm2014},\\cite{Huntemann2014}). We added a commercial 1.5\\,$\\mu$m fiber laser, locked to a high-finesse optical cavity, and transferred the laser frequency stability via the fiber laser femtosecond frequency comb to a 9.6\\,GHz microwave oscillator, which is used for the synthesis of the caesium fountain interrogation signal at 9.2\\,GHz. At the same time, the 9.6\\,GHz microwave oscillator frequency is measured with reference to the 100\\,MHz output frequency of a hydrogen maser. Based on this measurement, the long-term drift of the optical cavity and the correlated 1.5\\,$\\mu$m fiber laser frequency drift are compensated for by an acousto-optic modulator (AOM), which ensures the locking of the microwave oscillator to the hydrogen maser frequency in the long term.\n\nIn Section~\\ref{sec_scheme} we describe our setup in detail and then present our results in Section~\\ref{sec_results}.\n\n\n\\section{Microwave oscillator stabilization scheme}\n\\label{sec_scheme}\n\nOur setup for absolute optical frequency measurements (Fig.~\\ref{fig1_Dromaser}) has been extended by two modules (gray shaded boxes in Fig.~\\ref{fig1_Dromaser}): the setup for the cavity stabilized fiber laser, and the stabilization scheme for the microwave oscillator, a dielectric resonator oscillator (DRO) at 9.6\\,GHz. The core of the whole setup is a frequency comb (FC1500 from Menlo Systems), based on an erbium-doped fiber laser, with a repetition rate $\\nu_\\mathrm{rep}$ of about 252\\,MHz at a wavelength of 1.5\\,$\\mu$m. The oscillator of the frequency comb generates pulses of 100\\,fs in length. The corresponding optical comb has a Fourier limited spectrum width of 30\\,nm. In principle, optical detection of the 100\\,fs laser pulses generates a ``second frequency comb'' consisting of the comb repetition rate and its harmonics extending well into the microwave spectral range up to 4\\,THz, given by the Fourier limit. Absolute frequency measurements of optical frequencies and the transfer of stability of optical frequencies into the microwave regime are enabled, because of the phase coherence between the optical comb frequencies and the comb repetition rate.\n\nTo fulfill the requirements of our microwave oscillator stabilization scheme regarding optical power and bandwidth, we have chosen a fast InGaAs-PIN photodiode (DSC 40S, BW\\,=\\,12\\,GHz) for the detection of the repetition rate and its harmonics. The 38$^{\\mathrm{th}}$ harmonic (at $\\approx 9.6$\\,GHz) is subsequently selected by a band-pass filter (bandwidth 400\\,MHz) and amplified by a microwave amplifier with phase noise specification (HMC-C050).  \n\n\\begin{figure}[t]\n\\centering\n  \\includegraphics[width=\\columnwidth]{Fig1_DROmaser}\n\t\\caption{Setup for the generation of the optically stabilized 9.6\\,GHz microwave signal and absolute optical frequency measurements. The two extra modules for microwave generation are indicated by the gray shaded boxes. DRO: dielectric resonator oscillator; AOM: acousto-optic modulator.}\n\t\\label{fig1_Dromaser} \n\\end{figure}\n\nThe detection and amplification constitute a fundamental limit for the short-term frequency stability transfer from the optical to the microwave domain in our setup. An optical power of 0.5\\,mW at the photodiode results in $-27$\\,dBm electrical power of the 38$^\\mathrm{th}$ harmonic and thus in a calculated $\\text{SNR}$ of 144\\,dB\/Hz. The amplifier limits the $\\text{SNR}$ to 140\\,dB\/Hz close to the carrier at 100\\,Hz and 142\\,dB\/Hz at frequencies larger than 1\\,kHz. \n\nOne part of the 38$^{\\mathrm{th}}$ harmonic's signal of the repetition rate is used for the fast locking of the microwave oscillator (see below). The other part of the signal is mixed with a 9.6\\,GHz frequency synthesis output signal $\\nu_\\mathrm{synth}$, referenced to the maser. The resulting beat signal $\\nu_\\mathrm{y}=38\\times \\nu_\\mathrm{rep}-\\nu_\\mathrm{synth}$ is kept constant via an offset lock for locking the repetition rate of the frequency comb to the 9.6\\,GHz signal with a bandwidth of about 1\\,kHz (not shown in Fig.~\\ref{fig1_Dromaser}). \n\nThree erbium-doped fiber amplifiers (not shown in Fig.~\\ref{fig1_Dromaser}), including matched nonlinear optical components, are employed to spread and optimize the femtosecond laser spectrum. One of these amplifiers generates the octave-spanning spectral range of 1--2\\,$\\mu$m to provide the carrier envelope offset frequency $\\nu_\\mathrm{ceo}$ of the frequency comb, which is realized in a $f-2f$ scheme \\cite{Jones2000, Holzwarth2000}. The offset frequency ($\\text{SNR}$\\,=\\,42\\,dB at RBW\\,=\\,300\\,kHz) is locked to an intermediate frequency of 20\\,MHz with a bandwidth of a few kHz (not shown in Fig.~\\ref{fig1_Dromaser}). The spectrum core areas of the two other amplifiers are tuned to optimize the beat notes $\\nu_\\mathrm{x1},\\nu_\\mathrm{x2}$ with the optical clock transition frequencies of our quadrupole ({\\em E}\\\/2) and octupole ({\\em E}\\\/3) $^{171}$Yb$^+$ frequency standards \\cite{Tamm2014}, \\cite{Huntemann2016}. For optical frequency measurements, the  frequencies $\\nu_\\mathrm{rep}$, $\\nu_\\mathrm{ceo}$ and the difference frequencies $\\nu_\\mathrm{x1},\\nu_\\mathrm{x2}$ between the frequencies of the optical frequency standards and the individual comb teeth are registered by synchronous multichannel counters (K+K FXE \\cite{Kramer2004}), free of dead times. From the counter results, absolute optical frequencies, which are referenced to the maser and the caesium fountains, are calculated in a data processing unit (Fig.~\\ref{fig1_Dromaser}).\n\nTo provide the short-term stability of the optically stabilized DRO, a commercial fiber laser (Koheras BASIK E15) at a wavelength of 1.54\\,$\\mu$m is locked to an optical cavity made of ultra low expansion (ULE) glass with highly reflecting ULE mirrors by means of the Pound-Drever-Hall technique. With a cavity length of 75\\,mm (FSR\\,=\\,2\\,GHz), a finesse of 320\\,000 is obtained. The control bandwidth of this servo loop is limited by an AOM to 100\\,kHz. Low-frequency variations are compensated for by changing the effective fiber length of the laser using a piezo element adjusted by an integral element in the control loop. As a result, a frequency reference with a short-term stability of $10^{-15}$ for averaging times between 1 and 10\\,s is obtained with a prevailing linear drift of 40\\,mHz\/s. \n\nTo realize the transfer oscillator concept \\cite{Telle2002}, we need to simultaneously track the optical beat signals $\\nu_\\mathrm{x}$ and $\\nu_\\mathrm{ceo}$ and the microwave beat signal $\\nu_\\mathrm{z}$ between the DRO and the frequency comb. While the optical and microwave beat notes are from different spectral regions, they are based on strongly phase coupled modes of the same oscillator. Processing of the beat notes enables their subtraction from each other, which effectively eliminates the noise of the frequency comb. \n\nThe cavity stabilized laser light and the light from the frequency comb are combined via single-mode fibers. The beat frequency $\\nu_\\mathrm{x}$ (27\\,MHz, $\\text{SNR}$\\,=\\,43\\,dB at RBW\\,=\\,300\\,kHz) from the laser light (frequency $\\nu_\\mathrm{L}$) and the $m^\\mathrm{th}$ mode of the frequency comb ($m=769\\,794$) is detected by a standard InGaAs photodiode. An absolute frequency measurement with reference to the hydrogen maser frequency is reduced to the counting of the three radio frequencies $\\nu_\\mathrm{rep}$, $\\nu_\\mathrm{ceo}$ and $\\nu_\\mathrm{x}$:\n\n\\begin{equation}\n\\nu_\\mathrm{L}=m \\times \\nu_\\mathrm{rep}  + \\nu_\\mathrm{ceo} + \\nu_\\mathrm{x}. \n\\label{laserf}\n\\end{equation}\n\nThe DRO frequency $\\nu_\\mathrm{MW}$ is mixed with the 38$^\\mathrm{th}$ harmonic of the repetition rate, which yields $\\nu_\\mathrm{MW}=38 \\times \\nu_\\mathrm{rep} +\\nu_\\mathrm{z}$. Eliminating $\\nu_\\mathrm{rep}$ by utilizing (\\ref{laserf}), we obtain the microwave frequency $\\nu_\\mathrm{MW}$ as\n\n\\begin{equation}\n\\nu_\\mathrm{MW}=\\frac{38}{m} \\times \\nu_\\mathrm{L} + [\\nu_\\mathrm{z}-\\frac{38}{m} (\\nu_\\mathrm{ceo}+\\nu_\\mathrm{x})]\n\\label{MW}\n\\end{equation}\n\n\\noindent which establishes a fixed relation between $\\nu_\\mathrm{MW}$ and $\\nu_\\mathrm{L}$, if the term within the square brackets is kept constant. The latter contains beat frequencies generated in the microwave ($\\nu_\\mathrm{z}$\\,=\\,5.9\\,MHz) and in the optical spectral region ($\\nu_\\mathrm{ceo}+\\nu_\\mathrm{x}$\\,=\\,47\\,MHz), which are separated by a large scaling factor of $m\/38$.  \n\nThe spectral range adaptation of the beat notes is realized by a multiplication of $\\nu_\\mathrm{z}$ and a division of $\\nu_\\mathrm{ceo}+\\nu_\\mathrm{x}$. For this purpose, an extra module, an analogue data processor with a tracking bandwidth of about 1\\,MHz, is set up: An overtone oscillator generates the 8$^\\mathrm{th}$ harmonic of $\\nu_\\mathrm{z}$, which reduces the further requirements of the $\\text{SNR}$ and also of the counter resolution for data processing. Furthermore, $\\nu_\\mathrm{ceo}$ and $\\nu_\\mathrm{x}$ are added in a mixer, filtered by a tracking oscillator, amplified and divided by $c=m\/(38 \\times 8)$.\nThe division is undertaken in a four-step process: In three identical stages 100\\,MHz is added each time, before the sum frequency is divided by a factor of 8. Before the last of these stages another stage is inserted, in which 100\\,MHz is also added, before the sum frequency is divided by a scaling factor of 4.9457\\ldots which is obtained from a direct digital synthesizer with a 48-bit accumulator (AD9956). \n\nThe output frequency of the division process ($\\approx$15\\,MHz) is mixed with the overtone frequency of $\\nu_\\mathrm{z}$. As a final result, we obtain a transfer beat frequency $\\nu_\\mathrm{T}$ equal to the eightfold of the term in square brackets of (\\ref{MW}):\n\n\\begin{equation}\n\\nu_\\mathrm{T}=\\nu_\\mathrm{z} \\times 8 -\\frac{\\nu_\\mathrm{ceo}+\\nu_\\mathrm{x}}{c} = \\nu_\\mathrm{MW} \\times 8 - \\frac{\\nu_\\mathrm{L}}{c}. \n\\end{equation}\n\n\\noindent Since $\\nu_\\mathrm{T}$ is independent of $\\nu_\\mathrm{rep}$ and $\\nu_\\mathrm{ceo}$, fluctuations of these frequencies are effectively suppressed, and $\\nu_\\mathrm{T}$ comprises the direct phase comparison between the DRO and the stabilized fiber laser. The comparison result is used to control the DRO with a bandwidth of $\\approx$50\\,kHz using an offset lock (``fast lock'' in Fig.~\\ref{fig1_Dromaser}).\n\nFinally, to ensure the locking of the DRO to the maser, we need to compensate for the frequency drift of the optical cavity by locking $\\nu_\\mathrm{L}$ to the maser. For this purpose, we employ a slow lock (time constant $\\approx$50\\,s), which is based on a comparison of the maser frequency and the DRO frequency in the data processing unit, using an AOM for the adjustment of $\\nu_\\mathrm{L}$ (see Fig.~\\ref{fig1_Dromaser}).\n\nThe utilized ``state-of-the-art'' DRO (PSI, $-$\\,110\\,dBc at 10\\,kHz) has a tuning bandwidth of a few 100\\,kHz. The 100\\,MHz hydrogen maser reference frequency is delivered to our setup via a semi-rigid coaxial cable by a commercial distribution amplifier (SDI\u2026) with galvanic isolation. All synthesizers and counters of the setup are referenced to the maser frequency. A homemade frequency synthesis \\cite{Gupta2007} is utilized to generate the 96$^\\mathrm{th}$ harmonic $\\nu_\\mathrm{synth}$ of the 100\\,MHz signal. \n\nThe 9.6\\,GHz optically stabilized microwave signal from the maser-referenced DRO (Fig.~\\ref{fig1_Dromaser}) is split, amplified and delivered to the two syntheses of the PTB fountain clocks CSF1 \\cite{Weyers2001,Weyers2001b} and CSF2 \\cite{Gerginov2010,Weyers2012} to generate the 9.193\\,GHz signal needed for the atom interrogation. The syntheses are provided with electronic switches which enable either the utilization of the optically stabilized 9.6\\,GHz signal or alternatively a 9.6\\,GHz signal from another DRO stabilized to a low-noise quartz oscillator \\cite{Gupta2007}. In the case of a failure of the optically stabilized microwave signal, the fountain control software detects the resultant exceeding of the limits for the measured atom number, the transition probability or the frequency deviations, and arranges the automatic switching to the quartz-based 9.6\\,GHz signal.\n\n\n\\section{Results}\n\\label{sec_results}\n\n\\subsection{Characterization of the optically stabilized microwave oscillator signal}\n\\label{sec_CharDdro}\n\nFor the characterization of the optically stabilized microwave signal, a second frequency comb independently stabilized to the same laser was utilized. The short-term characterization by a phase noise measurement yielded a phase noise level of $-$117\\,dBc\/Hz at 10\\,Hz from the carrier frequency [curve (a) in Fig.~\\ref{fig2_Pn}] \\cite{Tamm2014}. Noise integration up to 30\\,kHz results in an Allan frequency deviation of $4 \\times 10^{-15}$ (1\\,s), dominated by the increase of the noise level at Fourier frequencies below 100\\,Hz. Also the noise analysis of the transfer beat frequency signal $\\nu_\\mathrm{T}$ (in-loop) showed the same limitation [curve (b) in Fig.~\\ref{fig2_Pn}], which was given by the noise properties of the utilized synthesizer for the offset lock to keep the transfer beat frequency $\\nu_\\mathrm{T}$ constant along with the limited selectivity of the utilized spectrum analyzer. \n\n\\begin{figure}\n\\centering\n  \\includegraphics[width=\\columnwidth]{Fig2_DROPN}\n\t\\caption{Single-sideband phase noise power spectral density $L$ relative to the carrier at 9.6\\,GHz. (a) red: measured phase noise spectrum of the employed dielectric resonator oscillator (DRO) for the case that it is locked to an ultrastable laser using the transfer oscillator scheme. (b) gray: measured phase noise spectrum of the transfer beat frequency signal $\\nu_\\mathrm{T}$. (c) green: measured phase noise spectrum of the transfer beat frequency signal $\\nu_\\mathrm{T}$ after modification of the offset lock of the transfer beat (see text). (d) blue: specified phase noise of the unstabilized DRO.}\n\t\\label{fig2_Pn} \n\\end{figure}\n\nRecently, the limitation has been overcome by a modification of the offset lock of the transfer beat avoiding additional noise from the offset synthesizer. Moreover, the servo loop bandwidth of the fast lock was increased to more than 100\\,kHz by utilizing filters with smaller delays. The resulting transfer beat spectrum is depicted in Fig.~\\ref{fig2_Pn} [curve (c)]. The expectation that the previous phase noise limitation at low Fourier frequencies has been overcome is supported by a frequency ratio measurement of the DRO and the clock laser of the $^{171}$Yb$^+$ single-ion frequency standard with the same frequency comb, which exhibited a significant improvement of the frequency instability. However, a real out-of-loop noise measurement would be needed to definitely confirm the removal of the former limitation.\n\nThe long-term performance of the optically stabilized microwave signal is characterized in terms of the Allan standard deviation. We measured the frequency instability of the hydrogen maser and of the cavity stabilized fiber laser locked to the hydrogen maser (Fig.~\\ref{fig3_DROADEV}). In both cases, the output frequency of the $^{171}$Yb$^+$ frequency standard served as a reference, which only contributes at a negligible level to the measured standard deviations. In the long term ($\\tau >$~100\\,s), both Allan deviations agree very well. Since the DRO is locked to the cavity stabilized fiber laser with a bandwidth of 100\\,kHz, the microwave signal exhibits the same long-term behavior. \n\n\\begin{figure}\n\\centering\n  \\includegraphics[width=\\columnwidth]{Fig3_RegDROmaserADEV}\n\t\\caption{Measured Allan standard deviations $\\sigma_y (\\tau)$ of the frequencies of the hydrogen maser (solid line, red) and the cavity stabilized 1.5\\,$\\mu$m fiber laser (dashed line, blue), locked to the hydrogen maser. For both measurements, the output frequency of the $^{171}$Yb$^+$ single-ion frequency standard served as a reference, which only contributes at a negligible level to the measured standard deviations.}\n\t\\label{fig3_DROADEV} \n\\end{figure}\n\nIf required, the short-term stability of the microwave signal could be further improved: On the one hand, superior radio-frequency amplifiers with lower phase noise could be utilized. Secondly, the amplitude to phase conversion in the fast InGaAs-PIN photodiode for the detection of the repetition rate and its harmonics can be reduced by stabilizing the optical power \\cite{Taylor2011,Zhang2012a} and selecting the operating point of the photodiode \\cite{Zhang2010}. Finally, the $\\text{SNR}$ can be increased by ``multiplication of the pulse rate'' \\cite{Haboucha2011}.\n\n\n\\subsection{Resulting frequency instabilities of the fountain clocks CSF1 and CSF2}\n\\label{sec_InstabCSF}\n\nIn the case of the operation of a fountain clock which is quantum projection noise limited, the $\\text{SNR}$ is given by $\\text{SNR}=N_{\\mathrm{\\mathrm{at}}}^{1\/2}$, with $N_{\\mathrm{\\mathrm{at}}}$ the total detected number of atoms in the F\\,=\\,3 and F\\,=\\,4 hyperfine components of the Cs ground state \\cite{Santarelli1999}. The frequency instability expressed by the Allan deviation is given by: \n\n\\begin{equation}\n\\label{sigmay}\n\t\\sigma_y\\left(\\tau\\right)=\\frac{1}{\\pi}\\frac{\\Delta\\nu}{\\nu_0}\\frac{1}{\\text{SNR}}\\sqrt{\\frac{T_c}{\\tau}}\n\\end{equation}\n\n\\noindent where $\\Delta\\nu$ is the full-width-at-half-maximum of the Ramsey fringe, $\\nu_0$\\,=\\,9\\,192\\,631\\,770\\,Hz is the clock transition frequency, $T_c$ is the cycle time, and $\\tau$ the measurement time. The precondition is that an instability degradation due to the phase noise of the interrogation oscillator (Dick effect, \\cite{Santarelli1998}) can be avoided. The instability described by (\\ref{sigmay}) is reduced with increased detected atom numbers (from increased loading times) as long as the factor with the square root of $T_c$ (also containing the loading time) does not become too large. This consideration gives a first criterion for the best choice of the employed loading time (and for the resulting detected atom number $N_{\\mathrm{\\mathrm{at}}}$ and cycle time $T_c$). To achieve the best compromise for the combined statistical and systematic uncertainties, one has to furthermore take into account that in general the systematic part of the uncertainty due to cold atom collisions scales with $N_{\\mathrm{\\mathrm{at}}}$ (e.g. see \\cite{Gerginov2010}).  \n\nIn the fountain CSF1, where magneto-optical trap (MOT) loading is utilized, the collisional frequency shift is evaluated by simply varying the microwave field power in the state selection cavity to operate the fountain at different atomic densities. This gives a 10\\% systematic uncertainty estimate for the collisional shift \\cite{Weyers2001b}. As a result, the usable $N_{\\mathrm{\\mathrm{at}}}$ is limited to keep the related uncertainty contribution at a reasonable level. We use an MOT loading time of 163.2\\,ms ($T_c$\\,=\\,1.1145\\,s), which optimizes the overall uncertainty. In the fountain CSF2 with molasses loading, the method of rapid adiabatic passage is employed for the collisional shift determination \\cite{Kazda2013}. Since the systematic uncertainty of the collisional shift is below 0.5\\%, much higher atom numbers are compatible with systematic collisional shift uncertainties below the $10^{-16}$ level. Molasses loading from a cold atom beam \\cite{Dobrev2016} enables a relatively short loading time of 340\\,ms ($T_c$\\,=\\,1.2345\\,s) for the minimization of the overall uncertainty. Increasing the loading time to 690\\,ms ($T_c$\\,=\\,1.5845\\,s), however, optimizes the statistical uncertainty.\n\nWith both fountains, $\\text{SNR}$ measurements utilizing either two $\\pi\/2$ pulses (at frequency detuning of half of the full linewidth) or two $\\pi\/4$ pulses (at zero frequency detuning) yield the same $\\text{SNR}$. Since the $\\text{SNR}$ is sensitive to frequency noise only in the first case, this finding is the first proof that the noise contribution by the optically stabilized microwave signal is negligible. The measured $\\text{SNR}$ values are given in Table~\\ref{table_instabilities} together with the resultant calculated frequency instabilities $\\sigma_\\mathrm{y} (\\mathrm{1\\,s})$ using (\\ref{sigmay}).\n\n\\begin{table*}[t]\n\\renewcommand{\\arraystretch}{1.0}\n\\caption{Loading and Cycle Times, Signal-to-Noise Ratios and Frequency Instabilities of the Fountain Clocks CSF1 and CSF2}\n\\label{table_instabilities}\n\\centering\n\\begin{tabular}{lcccccc}\n\\hline\n\\rule[-3mm]{0mm}{8mm} & Loading Time & Cycle Time & $\\text{SNR}$ & $\\sigma_\\mathrm{y} (\\mathrm{1\\,s})$ & $\\sigma_\\mathrm{y, DRO} (\\mathrm{1\\,s})$ & $\\sigma_\\mathrm{y, measured} (\\mathrm{1\\,s})$\\\\\n\\hline\\hline\n\\rule[-3mm]{0mm}{8mm}\\bfseries CSF1 & 163.2\\,ms & 1.1145\\,s & 350 & $9.2 \\times 10^{-14}$ & $4.8 \\times 10^{-15}$ & $9.1 \\times 10^{-14}$\\\\\n\\hline\n\\rule[-3mm]{0mm}{8mm}\\bfseries CSF2 & 340\\,ms & 1.2345\\,s & 1060 & $3.3 \\times 10^{-14}$ & $5.7 \\times 10^{-15}$ & $3.4 \\times 10^{-14}$\\\\\n\\rule[-3mm]{0mm}{5 mm}& 690\\,ms & 1.5845\\,s & 1660 & $2.4 \\times 10^{-14}$ & $7.8 \\times 10^{-15}$ & $2.5 \\times 10^{-14}$\\\\\n\\hline\n\\multicolumn{7}{l}{The frequency instabilities $\\sigma_\\mathrm{y}$ scale with $(\\tau\/1\\,\\mathrm{s})^{-1\/2}$. The instability $\\sigma_\\mathrm{y} (\\mathrm{1\\,s})$ is calculated from the}\\rule[0mm]{0mm}{4mm}\\\\\n\\multicolumn{7}{l}{measured $\\text{SNR}$ (see text) by using (\\ref{sigmay}). The instability contribution $\\sigma_\\mathrm{y, DRO} (\\mathrm{1\\,s})$ is caused by the Dick}\\\\\n\\multicolumn{7}{l}{effect \\cite{Santarelli1998} and calculated from the measured DRO phase noise (Fig.~\\ref{fig2_Pn}).}\\\\\n\\end{tabular}\n\\end{table*}\n\nTaking into account the measured phase noise level of the optically stabilized DRO (Fig.~\\ref{fig2_Pn}), we calculated the frequency instability contribution caused by the Dick effect \\cite{Santarelli1998} for the different cycle times of the two fountain clocks ($\\sigma_\\mathrm{y, DRO} (\\mathrm{1\\,s})$ in Table~\\ref{table_instabilities}). It becomes obvious that even with the former measured phase noise level shown in Fig.~\\ref{fig2_Pn}, which is a worst case estimate as explained in Section~\\ref{sec_CharDdro}, the resulting instability contribution of the optically stabilized DRO is negligible compared to the calculated instabilities $\\sigma_\\mathrm{y} (\\mathrm{1\\,s})$.\n\nIn Fig.~\\ref{fig4_CSFADEV}, the measured Allan standard deviations for fountain frequency measurements with cycle times $T_c$\\,=\\,1.1145\\,s (CSF1) and $T_c$\\,=\\,1.5845\\,s (CSF2) are presented. The results $\\sigma_\\mathrm{y, measured} (\\tau)=9.1 \\times 10^{-14} (\\tau\/1\\,\\mathrm{s})^{-1\/2}$ for CSF1 and $\\sigma_\\mathrm{y, measured} (\\tau)=2.5 \\times 10^{-14} (\\tau\/1\\,\\mathrm{s})^{-1\/2}$ for CSF2 agree very well with the instabilities calculated from the $\\text{SNR}$ data (see Table~\\ref{table_instabilities}) and prove again that the instability contributions of the optically stabilized DRO are negligible at the current fountain performance levels.\n\n\\begin{figure}\n\\centering\n  \\includegraphics[width=\\columnwidth]{Fig4_AllanCSF}\n\t\\caption{Measured Allan standard deviations $\\sigma_y (\\tau)$ of the CSF1 [full data points (green)] and CSF2 [open data points (red)] frequencies. For the CSF1 measurement, a hydrogen maser served as the reference while the CSF2 frequency was referenced to the output frequency of the  $^{171}$Yb$^+$ single-ion frequency standard [quadrupole (E2) transition], whose instability level is indicated by the dashed line. Subtraction of the noise levels of the maser and the single-ion frequency standard, respectively, yields the instability levels indicated by the full lines (CSF1: $9.1 \\times 10^{-14} (\\tau\/1\\,\\mathrm{s})^{-1\/2}$, CSF2: $2.5 \\times 10^{-14} (\\tau\/1\\,\\mathrm{s})^{-1\/2}$).}\n\t\\label{fig4_CSFADEV} \n\\end{figure}\n\n\n\\subsection{Operation of the optically stabilized microwave oscillator signal}\n\\label{sec_FrequMeas}\n\nIn March~2014, the first continuous 10-day measurement (dead time $\\approx $1\\%) with the fountains and the presented optically stabilized DRO scheme was performed to contribute to the monthly calibration of International Atomic Time (TAI) by the Bureau International des Poids et Mesures (BIPM). Both fountain frequencies agreed at the $10^{-16}$ level, which was very well within their combined uncertainty. \n\nSince June~2015, the optically stabilized DRO scheme has been routinely used for fountain operation. A number of TAI calibrations as well as optical frequency measurements have been performed with both fountains. The results demonstrate very good agreement with other fountain clocks and previously published optical frequency measurement data at the low $10^{-16}$ level. High duty cycles of the optically stabilized microwave oscillator signal, close to 100\\%, have been achieved during weeks of operation. In both fountains, the quartz-based microwave synthesis is only used when maintenance work needs to be done on the frequency comb or the cavity stabilized fiber laser, or in those rare cases when the DRO locking fails.\n\n\n\\section{Conclusion}\n\\label{sec_conclusion}\n\nThe technology of cavity stabilized lasers and femtosecond lasers is well established in many laboratories. Optically stabilized microwave setups as presented are therefore relatively straightforward to realize. The focus of our design is not the ultimate performance in terms of phase noise, but rather reliable continuous operation at a phase noise level which is compatible with fountain clock requirements. Our setup has proven to enable fountain clock frequency instabilities at a level otherwise only accessible by utilizing a cryogenic sapphire oscillator for providing the ultrastable microwave signal \\cite{Vian2005}.\n\nWe present phase noise measurements of the stabilized microwave oscillator and of the transfer beat frequency signal, and the measured Allan standard deviation of the cavity stabilized fiber laser, locked to the hydrogen maser. We demonstrate a quantum projection noise limited frequency instability of $2.5 \\times 10^{-14} (\\tau\/1\\,\\mathrm{s})^{-1\/2}$ and continuous long-term operation, typically required by fountain clocks. \n\nImprovements of fountain clock frequency instabilities not only improve the achievable statistical uncertainty in a given measurement time, but also enable more accurate investigations of systematic effects, which in the end may lead to reduced systematic uncertainties.\n\n\n\\section*{Acknowledgment}\n\nThe authors would like to thank N.~Huntemann for providing the $^{171}$Yb$^+$ single-ion frequency standard reference signal, G.~Dobrev for his contributions to enhancing the output of the cold atom beam source of CSF2, M.~Kazda for providing the necessary interface in the fountain frequency syntheses to incorporate the optically stabilized microwave signal, and N.~Nemitz for the required fountain control software adaptation.\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\n\\section{\\@startsection{section}{1}{0pt}\n{-3.5ex plus -1ex minus -.2ex}{1.0ex plus .2ex}{\\large\\bf}}\n\\renewcommand\\subsection{\\@startsection{subsection}{1}{0pt}\n{2.5ex plus 1ex minus .2ex}{-1em}{\\bf}}\n\\makeatother\n\n\\setlength{\\textwidth}{165mm}\n\\setlength{\\oddsidemargin}{-1.5mm}\n\\setlength{\\textheight}{230mm}\n\\setlength{\\topmargin}{0mm}\n\\setlength{\\headheight}{0mm}\n\\setlength{\\headsep}{0mm}\n\n\\newcommand{\\vspace{1.5mm}}{\\vspace{1.5mm}}\n\\newcommand{\\vspace{3mm}}{\\vspace{3mm}}\n\n\\theoremstyle{plain}\n\\newtheorem{thm}{Theorem}[subsection]\n\\newtheorem{lem}[thm]{Lemma}\n\\newtheorem{prop}[thm]{Proposition}\n\\newtheorem{cor}[thm]{Corollary}\n\n\\newtheorem{claim}{Claim}[thm]\n\\renewcommand{\\theclaim}{\\arabic{claim}}\n\n\\newtheorem*{claim*}{Claim}\n\n\\newtheorem{ithm}{Theorem}\n\\newtheorem*{iprop}{Proposition}\n\n\\theoremstyle{definition}\n\\newtheorem{dfn}[thm]{Definition}\n\n\\newtheorem*{question}{Question}\n\n\\theoremstyle{remark}\n\\newtheorem{rem}[thm]{Remark}\n\\newtheorem{ex}[thm]{Example}\n\n\\newtheorem*{irem}{Remark}\n\n\\begin{document}\n\n\\setlength{\\baselineskip}{18pt}\n\n\\title{\\Large\\bf \nPolytopal Estimate of Mirkovi\\'c-Vilonen polytopes \\\\\nlying in a Demazure crystal}\n\\author{\n Syu Kato%\n \\footnote{Supported in part by JSPS Research Fellowships \n           for Young Scientists (No.\\,20740011).} \\\\\n \\small Research Institute for Mathematical Sciences, Kyoto University, \\\\\n \\small Kitashirakawa-Oiwake-cho, Sakyo, Kyoto 606-8502, Japan \\\\\n \\small (e-mail: {\\tt kato@kurims.kyoto-u.ac.jp}) \\\\[5mm]\n Satoshi Naito%\n \\footnote{Supported in part by Grant-in-Aid for Scientific Research \n           (No.\\,20540006), JSPS.} \\\\ \n \\small Institute of Mathematics, University of Tsukuba, \\\\\n \\small Tsukuba, Ibaraki 305-8571, Japan \\ \n (e-mail: {\\tt naito@math.tsukuba.ac.jp})\n \\\\[2mm] and \\\\[2mm]\n Daisuke Sagaki%\n \\footnote{Supported in part by JSPS Research Fellowships \n           for Young Scientists (No.\\,19740004).} \\\\ \n \\small Institute of Mathematics, University of Tsukuba, \\\\\n \\small Tsukuba, Ibaraki 305-8571, Japan \\ \n (e-mail: {\\tt sagaki@math.tsukuba.ac.jp})\n}\n\\date{}\n\\maketitle\n\n\\begin{abstract} \\setlength{\\baselineskip}{16pt}\nIn this paper, we give a polytopal estimate of \nMirkovi\\'c-Vilonen polytopes lying in a Demazure crystal \nin terms of Minkowski sums of extremal Mirkovi\\'c-Vilonen polytopes.\nAs an immediate consequence of this result, \nwe provide a necessary (but not sufficient) \npolytopal condition for a Mirkovi\\'c-Vilonen polytope \nto lie in a Demazure crystal.\n\\end{abstract}\n\\section{Introduction.}\n\\label{sec:intro}\nThis paper is a continuation of our previous one \\cite{NS-dp}, and \nour purpose is to give a polytopal estimate of \nMirkovi\\'c-Vilonen polytopes lying in a Demazure crystal \nin terms of Minkowski sums of extremal Mirkovi\\'c-Vilonen polytopes.\nIt should be mentioned that \nas an immediate consequence of this result, \nwe can provide an affirmative answer to \na question posed in \\cite[\\S4.6]{NS-dp}.\n\nFollowing the notation and terminology of \\cite{NS-dp}, \nwe now explain our results more precisely.\nLet $G$ be a complex, connected, semisimple algebraic group \nwith Lie algebra $\\mathfrak{g}$, $T$ a maximal torus \nwith Lie algebra (Cartan subalgebra) $\\mathfrak{h}$, \n$B$ a Borel subgroup containing $T$, \nand $U$ the unipotent radical of $B$;  \nby our convention, the roots in $B$ are the negative ones. \nLet $X_{*}(T)$ denote the coweight lattice $\\mathop{\\rm Hom}\\nolimits(\\mathbb{C}^{*},\\,T)$ \nfor $G$, which we regard as an additive subgroup of \na real form $\\mathfrak{h}_{\\mathbb{R}}:=\\mathbb{R} \\otimes_{\\mathbb{Z}} X_{*}(T)$ of $\\mathfrak{h}$. \nDenote by $W$ the Weyl group of $\\mathfrak{g}$, \nwith $e$ the identity element and $w_{0}$ \nthe longest element of length $m$.\nAlso, let $\\mathfrak{g}^{\\vee}$ denote\nthe (Langlands) dual Lie algebra of $\\mathfrak{g}$ \nwith Weyl group $W$, and let $U_{q}(\\mathfrak{g}^{\\vee})$ \nbe the quantized universal enveloping algebra \nof $\\mathfrak{g}^{\\vee}$ over $\\mathbb{C}(q)$. \n\nFor each dominant coweight \n$\\lambda \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$, \nlet us denote by $\\mathcal{MV}(\\lambda)$ \nthe set of Mirkovi\\'c-Vilonen (MV for short) polytopes \nwith highest vertex $\\lambda$ that is contained in \nthe convex hull $\\mathop{\\rm Conv}\\nolimits (W \\cdot \\lambda)$ in $\\mathfrak{h}_{\\mathbb{R}}$\nof the Weyl group orbit $W \\cdot \\lambda$ through $\\lambda$, \nand by $\\mathcal{B}(\\lambda)$ the crystal basis of \nthe irreducible highest weight \n$U_{q}(\\mathfrak{g}^{\\vee})$-module $V(\\lambda)$ \nof highest weight $\\lambda$.\nRecall that Kamnitzer \\cite{Kam1}, \\cite{Kam2} \nproved the existence of an isomorphism of crystals \n$\\Psi_{\\lambda}$ from the crystal basis $\\mathcal{B}(\\lambda)$ \nto the set $\\mathcal{MV}(\\lambda)$ of MV polytopes, \nwhich is endowed with the Lusztig-Berenstein-Zelevinsky \n(LBZ for short) crystal structure; he also \nproved the coincidence of this LBZ crystal structure on \n$\\mathcal{MV}(\\lambda)$ with the Braverman-Finkelberg-Gaitsgory \n(BFG for short) crystal structure on $\\mathcal{MV}(\\lambda)$.\n\nIn \\cite{NS-dp}, for each $x \\in W$, \nwe gave a combinatorial description, \nin terms of the lengths of edges of an MV polytope, \nof the image $\\mathcal{MV}_{x}(\\lambda) \\subset \\mathcal{MV}(\\lambda)$ \n(resp., $\\mathcal{MV}^{x}(\\lambda) \\subset \\mathcal{MV}(\\lambda))$ of \nthe Demazure crystal $\\mathcal{B}_{x}(\\lambda) \\subset \\mathcal{B}(\\lambda)$ \n(resp., opposite Demazure crystal \n$\\mathcal{B}^{x}(\\lambda) \\subset \\mathcal{B}(\\lambda)$) \nunder the isomorphism \n$\\Psi_{\\lambda} : \\mathcal{B}(\\lambda) \\rightarrow \\mathcal{MV}(\\lambda)$ of crystals.\nFurthermore, in \\cite{NS-dp}, \nwe proved that for each $x \\in W$, \nan MV polytope $P \\in \\mathcal{MV}(\\lambda)$ lies in \nthe opposite Demazure crystal \n$\\mathcal{MV}^{x}(\\lambda)$ if and only if \nthe MV polytope $P$ contains (as a set) \nthe extremal MV polytope \n$P_{x \\cdot \\lambda}$ of weight $x \\cdot \\lambda$, \nwhich is identical to the convex hull \n$\\mathop{\\rm Conv}\\nolimits(W_{\\le x} \\cdot \\lambda)$ in $\\mathfrak{h}_{\\mathbb{R}}$\nof a certain subset $W_{\\le x} \\cdot \\lambda$ of $W \\cdot \\lambda$ \n(see \\S\\ref{subsec:extpoly} for details). \nHowever, we were unable to prove \nan analogous statement for Demazure crystals \n$\\mathcal{B}_{x}(\\lambda)$, $x \\in W$.\nThus, we posed the following question in \n\\cite[\\S4.6]{NS-dp}:\n\n\\begin{question}\nLet us take an arbitrary $x \\in W$. \nAre all the MV polytopes lying \nin the Demazure crystal $\\mathcal{MV}_{x}(\\lambda)$ \ncontained (as sets) in the extremal MV polytope \n$P_{x \\cdot \\lambda} = \\mathop{\\rm Conv}\\nolimits(W_{\\le x} \\cdot \\lambda)$ ?\n\\end{question}\n\n\\noindent\nNote that the converse statement fails to hold, \nas mentioned in \\cite[Remark~4.6.1]{NS-dp}.\n\nIn this paper, we provide an affirmative answer to this question. \nIn fact, we considerably sharpen the polytopal estimate above of \nMV polytopes lying in a Demazure crystal as follows. \nIn what follows, for each dominant coweight \n$\\lambda \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$, \nwe denote by $W_{\\lambda} \\subset W$ \nthe stabilizer of $\\lambda$ in $W$, and \nby $W^{\\lambda}_{\\min} \\subset W$ \nthe set of minimal (length) coset representatives modulo \nthe subgroup $W_{\\lambda} \\subset W$.\n\\begin{ithm}[$=$ Theorem~\\ref{thm:main} combined \nwith Proposition~\\ref{prop:N3}] \\label{ithm1}\nLet $\\lambda \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ \nbe a dominant coweight, and \nlet $x \\in W^{\\lambda}_{\\min} \\subset W$. \nIf an MV polytope $P \\in \\mathcal{MV}(\\lambda)$ lies \nin the Demazure crystal $\\mathcal{MV}_{x}(\\lambda)$, \nthen there exist a positive integer $N \\in \\mathbb{Z}_{\\ge 1}$ \nand minimal coset representatives \n$x_{1},\\,x_{2},\\,\\dots,\\,x_{N} \\in \n W^{\\lambda}_{\\min} \\subset W$ \nsuch that \n\\begin{equation*}\n\\begin{cases}\nx \\ge x_{k} \\quad \n \\text{\\rm for all $1 \\le k \\le N$;} \\\\[3mm]\nN \\cdot P \\subseteq\nP_{x_1 \\cdot \\lambda} + \nP_{x_2 \\cdot \\lambda} + \\cdots + \nP_{x_N \\cdot \\lambda},\n\\end{cases}\n\\end{equation*}\nwhere $N \\cdot P := \n\\bigl\\{N v \\mid v \\in P \\bigr\\} \\subset \\mathfrak{h}_{\\mathbb{R}}$ \nis an MV polytope in $\\mathcal{MV}(N \\lambda)$, and \n$P_{x_1 \\cdot \\lambda} + P_{x_2 \\cdot \\lambda} + \\cdots + \n P_{x_N \\cdot \\lambda}$ is the Minkowski sum of \nthe extremal MV polytopes $P_{x_1 \\cdot \\lambda}, \\,\nP_{x_2 \\cdot \\lambda},\\,\\dots,\\,P_{x_N \\cdot \\lambda}$.\n\\end{ithm}\n\n\\begin{irem}\nWe see from Remark~\\ref{rem:MV-LS} and \nTheorem~\\ref{thm:main} below that \nthe elements $x_{1}$, $x_{2}$, $\\dots$, $x_{N} \\in \nW^{\\lambda}_{\\min} \\subset W$ can be chosen \nin such a way that the vectors \n$x_1 \\cdot \\lambda$, $x_2 \\cdot \\lambda$, $\\dots$, \n$x_N \\cdot \\lambda \\in W \\cdot \\lambda \\subset \\mathfrak{h}_{\\mathbb{R}}$\ngive the directions of the Lakshmibai-Seshadri path of shape $\\lambda$ \nthat corresponds to the MV polytope $P \\in \\mathcal{MV}(\\lambda)$ \nunder the (inexplicit) bijection via the crystal basis $\\mathcal{B}(\\lambda)$.\nHence it also follows that \n$x_1 \\ge x_2 \\ge \\dots \\ge x_N$ \nin the Bruhat ordering on $W$.\n\\end{irem}\n\nFrom the theorem above, we can deduce immediately that \nfor an arbitrary $P \\in \\mathcal{MV}_{x}(\\lambda)$, \nthere holds $N \\cdot P \\subset N \\cdot P_{x \\cdot \\lambda}$ and \nhence $P \\subset P_{x \\cdot \\lambda}$. Indeed, this follows from \nthe inclusion \n$P_{x_k \\cdot \\lambda}=\\mathop{\\rm Conv}\\nolimits(W_{\\le x_k} \\cdot \\lambda) \n \\subset \\mathop{\\rm Conv}\\nolimits(W_{\\le x} \\cdot \\lambda)=P_{x \\cdot \\lambda}$\nfor each $1 \\le k \\le N$, and the fact that the Minkowski sum \n$P_{x \\cdot \\lambda}+P_{x \\cdot \\lambda}+\\cdots+P_{x \\cdot \\lambda}$ \n($N$ times) is identical to $N \\cdot P_{x \\cdot \\lambda}$ \n(see Remark~\\ref{rem:N1}). \n\nThe main ingredient in our proof of the theorem is \nthe following polytopal estimate of \ntensor products of MV polytopes.\nLet $\\lambda_{1},\\,\\lambda_{2} \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ \nbe dominant coweights. \nSince $\\mathcal{MV}(\\lambda) \\cong \\mathcal{B}(\\lambda)$ as crystals \nfor every dominant coweight $\\lambda \\in X_{\\ast}(T)$, \nthe tensor product \n$\\mathcal{MV}(\\lambda_{2}) \\otimes \\mathcal{MV}(\\lambda_{1})$ of \nthe crystals $\\mathcal{MV}(\\lambda_{1})$ and $\\mathcal{MV}(\\lambda_{2})$ \ndecomposes into a disjoint union of \nconnected components as follows:\n\\begin{equation*}\n\\mathcal{MV}(\\lambda_{2}) \\otimes \\mathcal{MV}(\\lambda_{1}) \\cong \n \\bigoplus_{\n   \\begin{subarray}{c}\n   \\lambda \\in X_{\\ast}(T) \\\\[0.5mm]\n   \\text{$\\lambda$ : dominant}\n   \\end{subarray}\n   } \n\\mathcal{MV}(\\lambda)^{\\oplus m_{\\lambda_{1},\\lambda_{2}}^{\\lambda}},\n\\end{equation*}\nwhere $m_{\\lambda_{1},\\lambda_{2}}^{\\lambda} \\in \\mathbb{Z}_{\\ge 0}$ denotes \nthe multiplicity of $\\mathcal{MV}(\\lambda)$ in \n$\\mathcal{MV}(\\lambda_{2}) \\otimes \\mathcal{MV}(\\lambda_{1})$. \nFor each dominant coweight \n$\\lambda \\in X_{\\ast}(T)$ \nsuch that $m_{\\lambda_{1},\\lambda_{2}}^{\\lambda} \\ge 1$, \nwe take (and fix) an arbitrary embedding \n$\\iota_{\\lambda}:\\mathcal{MV}(\\lambda) \\hookrightarrow \n \\mathcal{MV}(\\lambda_{2}) \\otimes \\mathcal{MV}(\\lambda_{1})$ \nof crystals that maps $\\mathcal{MV}(\\lambda)$ \nonto a connected component of \n$\\mathcal{MV}(\\lambda_{2}) \\otimes \\mathcal{MV}(\\lambda_{1})$, \nwhich is isomorphic to $\\mathcal{MV}(\\lambda)$ as a crystal. \n\\begin{ithm}[$=$ Theorem~\\ref{thm:tensor}] \\label{ithm2}\nKeep the notation above.\nLet $P \\in \\mathcal{MV}(\\lambda)$, and \nwrite $\\iota_{\\lambda}(P) \\in \n\\mathcal{MV}(\\lambda_{2}) \\otimes \\mathcal{MV}(\\lambda_{1})$ as\\,{\\rm:} \n$\\iota_{\\lambda}(P)=P_{2} \\otimes P_{1}$ for some \n$P_{1} \\in \\mathcal{MV}(\\lambda_{1})$ and \n$P_{2} \\in \\mathcal{MV}(\\lambda_{2})$. \nWe assume that the MV polytope \n$P_{2} \\in \\mathcal{MV}(\\lambda_{2})$ is an extremal MV polytope \n$P_{x \\cdot \\lambda_{2}}$ for some $x \\in W$. \nThen, we have\n\\begin{equation*}\nP \\subset P_{1} + P_{2},\n\\end{equation*}\nwhere $P_{1} + P_{2}$ is the Minkowski sum of \nthe MV polytopes $P_{1} \\in \\mathcal{MV}(\\lambda_{1})$ and \n$P_{2} \\in \\mathcal{MV}(\\lambda_{2})$.\n\\end{ithm}\n\n\nWe have not yet found a purely combinatorial proof \nof the theorem above. In fact, our argument is a geometric one, \nwhich is based on results of Braverman-Gaitsgory in \\cite{BrGa}, \nwhere tensor products of highest weight crystals are \ndescribed in terms of MV cycles in the affine Grassmannian; \nhere we should remark that the convention on the tensor product\nrule for crystals in \\cite{BrGa} is opposite to ours, i.e., to \nthat of Kashiwara \\cite{Kasoc}, \\cite{Kasb}. \nAlso, it seems likely that the theorem above still holds \nwithout the assumption of extremality on \nthe MV polytope $P_{2} \\in \\mathcal{MV}(\\lambda_{2})$.\n\nThis paper is organized as follows.\nIn \\S\\ref{sec:MVda}, we first recall \nthe basic notation and terminology concerning MV polytopes and \nDemazure crystals, and also review the relation between \nMV polytopes and MV cycles in the affine Grassmannian. \nFurthermore, we obtain \na few new results on extremal MV polytopes and MV cycles, \nwhich will be used in the proof of Theorem~\\ref{ithm2} \n($=$ Theorem~\\ref{thm:tensor}).\nIn \\S\\ref{sec:multi}, we introduce \nthe notion of $N$-multiple maps \nfrom $\\mathcal{MV}(\\lambda)$ to $\\mathcal{MV}(N \\lambda)$ \nfor a dominant coweight $\\lambda \\in X_{\\ast}(T)$ and \n$N \\in \\mathbb{Z}_{\\geq 1}$, which is given explicitly \nby: $P \\mapsto N \\cdot P$ in terms of MV polytopes, \nand also show that for each MV polytope \n$P \\in \\mathcal{MV}(\\lambda)$, there exists some $N \\in \\mathbb{Z}_{\\geq 1}$ \nsuch that $N \\cdot P \\in \\mathcal{MV}(N \\lambda)$ \ncan be written as the tensor product of \ncertain $N$ extremal MV polytopes in $\\mathcal{MV}(\\lambda)$.\nFurthermore, assuming Theorem~\\ref{ithm2}, \nwe prove Theorem~\\ref{ithm1} above, \nwhich provides an answer to \nthe question mentioned above.\nIn \\S\\ref{sec:tensor}, after revisiting results of \nBraverman-Gaitsgory on tensor products of \nhighest weight crystals\nin order to adapt them to our situation, \nwe prove Theorem~\\ref{ithm2} by using \nthe geometry of the affine Grassmannian.\nIn the Appendix, we give a brief account of \nwhy Theorem~\\ref{thm:BG} below is \na reformulation of results of Braverman-Gaitsgory. \n\n\\section{Mirkovi\\'c-Vilonen polytopes and Demazure crystals.}\n\\label{sec:MVda}\n\n\\subsection{Basic notation.}\n\\label{subsec:notation}\n\nLet $G$ be a complex, connected, reductive algebraic group, \n$T$ a maximal torus, $B$ a Borel subgroup containing $T$, and \n$U$ the unipotent radical of $B$; \nwe choose the convention that \nthe roots in $B$ are the negative ones. \nLet $X_{*}(T)$ denote the (integral) coweight lattice \n$\\mathop{\\rm Hom}\\nolimits(\\mathbb{C}^{*},\\,T)$ for $G$, and $X_{*}(T)_{+}$ \nthe set of dominant (integral) coweights for $G$; \nwe regard the coweight lattice \n$X_{*}(T)$ as an additive subgroup of \na real form $\\mathfrak{h}_{\\mathbb{R}}:=\\mathbb{R} \\otimes_{\\mathbb{Z}} X_{*}(T)$ \nof the Lie algebra $\\mathfrak{h}$ of the maximal torus $T$. \nWe denote by $G^{\\vee}$ \nthe (complex) Langlands dual group of $G$. \n\nFor the rest of this paper, \nexcept in \\S\\ref{subsec:geom}, \\S\\ref{subsec:BG}, and \nthe Appendix, we assume that $G$ is semisimple. \nDenote by $\\mathfrak{g}$ the Lie algebra of $G$, \nwhich is a complex semisimple Lie algebra. \nLet \n\\begin{equation*}\n\\Bigl(A=(a_{ij})_{i,j \\in I}, \\, \n \\Pi:=\\bigl\\{\\alpha_{j}\\bigr\\}_{j \\in I}, \\, \n \\Pi^{\\vee}:=\\bigl\\{h_{j}\\bigr\\}_{j \\in I}, \\, \n \\mathfrak{h}^{\\ast},\\,\\mathfrak{h}\n \\Bigr)\n\\end{equation*}\nbe the root datum of $\\mathfrak{g}$, where \n$A=(a_{ij})_{i,j \\in I}$ is the Cartan matrix, \n$\\mathfrak{h}$ is the Cartan subalgebra, \n$\\Pi:=\\bigl\\{\\alpha_{j}\\bigr\\}_{j \\in I} \\subset \n \\mathfrak{h}^{\\ast}:=\\mathop{\\rm Hom}\\nolimits_{\\mathbb{C}}(\\mathfrak{h},\\,\\mathbb{C})$ \nis the set of simple roots, and \n$\\Pi^{\\vee}:=\\bigl\\{h_{j}\\bigr\\}_{j \\in I} \\subset \\mathfrak{h}$ \nis the set of simple coroots; note that \n$\\pair{h_{i}}{\\alpha_{j}}=a_{ij}$ for $i,\\,j \\in I$, \nwhere $\\pair{\\cdot}{\\cdot}$ denotes the canonical pairing \nbetween $\\mathfrak{h}$ and $\\mathfrak{h}^{\\ast}$.\nWe set $\\mathfrak{h}_{\\mathbb{R}}:= \\sum_{j \\in I} \\mathbb{R} h_{j} \\subset \\mathfrak{h}$, \nwhich is a real form of $\\mathfrak{h}$; \nwe regard the coweight lattice \n$X_{*}(T)=\\mathop{\\rm Hom}\\nolimits(\\mathbb{C}^{\\ast},\\,T)$ \nas an additive subgroup of $\\mathfrak{h}_{\\mathbb{R}}$. \nAlso, for $h,\\,h' \\in \\mathfrak{h}_{\\mathbb{R}}$, we write \n$h' \\ge h$ if $h'-h \\in Q^{\\vee}_{+}:=\\sum_{j \\in I}\\mathbb{Z}_{\\ge 0}h_{j}$. \nLet $W:=\\langle s_{j} \\mid j \\in I \\rangle$ \nbe the Weyl group of $\\mathfrak{g}$, where $s_{j}$, $j \\in I$, are \nthe simple reflections, with length function \n$\\ell:W \\rightarrow \\mathbb{Z}_{\\ge 0}$, \nthe identity element $e \\in W$, and \nthe longest element $w_{0} \\in W$; \nwe denote by $\\le$ the (strong) Bruhat ordering on $W$. \nLet $\\mathfrak{g}^{\\vee}$ be the Lie algebra of \nthe Langlands dual group $G^{\\vee}$ of $G$, \nwhich is the complex semisimple Lie algebra\nassociated to the root datum \n\\begin{equation*}\n\\Bigl({}^{t}A=(a_{ji})_{i,j \\in I}, \\, \n \\Pi^{\\vee}=\\bigl\\{h_{j}\\bigr\\}_{j \\in I}, \\, \n \\Pi=\\bigl\\{\\alpha_{j}\\bigr\\}_{j \\in I}, \\, \n \\mathfrak{h},\\,\\mathfrak{h}^{\\ast}\n \\Bigr);\n\\end{equation*}\nnote that the Cartan subalgebra of \n$\\mathfrak{g}^{\\vee}$ is $\\mathfrak{h}^{\\ast}$, not $\\mathfrak{h}$. \nLet $U_{q}(\\mathfrak{g}^{\\vee})$ be \nthe quantized universal enveloping algebra of \n$\\mathfrak{g}^{\\vee}$ over $\\mathbb{C}(q)$, and $U_{q}^{+}(\\mathfrak{g}^{\\vee})$ \nits positive part. \nFor a dominant coweight \n$\\lambda \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$, \ndenote by $V(\\lambda)$ the integrable highest \nweight $U_{q}(\\mathfrak{g}^{\\vee})$-module \nof highest weight $\\lambda$, and by \n$\\mathcal{B}(\\lambda)$ the crystal basis of $V(\\lambda)$. \n\nNow, let $\\lambda \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ \nbe a dominant coweight, and $x \\in W$. \nThe Demazure module $V_{x}(\\lambda)$ is defined to be \nthe $U_{q}^{+}(\\mathfrak{g}^{\\vee})$-submodule of $V(\\lambda)$ \ngenerated by the one-dimensional weight space \n$V(\\lambda)_{x \\cdot \\lambda}  \\subset V(\\lambda)$ of \nweight $x \\cdot \\lambda \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$. \nRecall from \\cite{Kas} that \nthe Demazure crystal $\\mathcal{B}_{x}(\\lambda)$ is \na subset of $\\mathcal{B}(\\lambda)$ such that \n\\begin{equation} \\label{eq:dem-mod}\nV(\\lambda) \\supset \nV_{x}(\\lambda) = \n \\bigoplus_{b \\in \\mathcal{B}_{x}(\\lambda)}\n \\mathbb{C}(q) G_{\\lambda}(b),\n\\end{equation}\nwhere $G_{\\lambda}(b)$, $b \\in \\mathcal{B}(\\lambda)$, \nform the lower global basis of $V(\\lambda)$.\n\\begin{rem} \\label{rem:demcos}\nIf $x,\\,y \\in W$ satisfies \n$x \\cdot \\lambda=y \\cdot \\lambda$, then \nwe have $V_{x}(\\lambda)=V_{y}(\\lambda)$ since \n$V(\\lambda)_{x \\cdot \\lambda}=\n V(\\lambda)_{y \\cdot \\lambda}$. \nTherefore, it follows from \\eqref{eq:dem-mod} that \n$\\mathcal{B}_{x}(\\lambda)=\\mathcal{B}_{y}(\\lambda)$. \n\\end{rem}\n\nWe know from \\cite[Proposition~3.2.3]{Kas} that \nthe Demazure crystals $\\mathcal{B}_{x}(\\lambda)$, \n$x \\in W$, are characterized by the inductive relations: \n\\begin{align}\n& \\mathcal{B}_{e}(\\lambda)=\n   \\bigl\\{u_{\\lambda}\\bigr\\}, \\label{eq:demint1} \\\\[1.5mm]\n& \\mathcal{B}_{x}(\\lambda)=\n    \\bigcup_{k \\in \\mathbb{Z}_{\\ge 0}} f_{j}^{k} \n     \\mathcal{B}_{s_{j}x}(\\lambda) \n     \\setminus \\{0\\}\n  \\quad \\text{for $x \\in W$ and $j \\in I$ with $s_{j}x < x$},\n  \\label{eq:demint2}\n\\end{align}\nwhere $u_{\\lambda} \\in \\mathcal{B}(\\lambda)$ denotes \nthe highest weight element of $\\mathcal{B}(\\lambda)$, and \n$f_{j}$, $j \\in I$, denote the lowering Kashiwara operators\nfor $\\mathcal{B}(\\lambda)$. \n\n\\subsection{Mirkovi\\'c-Vilonen polytopes.}\n\\label{subsec:MV}\n\nIn this subsection, following \\cite{Kam1}, we recall \na (combinatorial) characterization of \nMirkovi\\'c-Vilonen (MV for short) polytopes; \nthe relation between this characterization and \nthe original (geometric) definition of MV polytopes \ngiven by Anderson \\cite{A} will be explained \nin \\S\\ref{subsec:geom}. \n\nAs in (the second paragraph of) \\S\\ref{subsec:notation}, \nwe assume that $\\mathfrak{g}$ is a complex semisimple Lie algebra. \nLet $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ be \na collection of elements of \n$\\mathfrak{h}_{\\mathbb{R}}=\\sum_{j \\in I} \\mathbb{R} h_{j}$.\nWe call $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ a \nGelfand-Goresky-MacPherson-Serganova \n(GGMS) datum if it satisfies \nthe condition that \n$w^{-1} \\cdot \\mu_{w'} - w^{-1} \\cdot \\mu_{w} \n \\in Q^{\\vee}_{+}$ for all $w,\\,w' \\in W$. \nIt follows by induction \nwith respect to the (weak) Bruhat ordering on $W$ \nthat $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ is a GGMS datum \nif and only if \n\\begin{equation} \\label{eq:length}\n\\mu_{ws_{i}}-\\mu_{w} \\in \\mathbb{Z}_{\\ge 0}\\,(w \\cdot h_{i}) \n\\quad \n\\text{for every $w \\in W$ and $i \\in I$}. \n\\end{equation}\n\\begin{rem} \\label{rem:GGMS-sum}\nLet $\\mu_{\\bullet}^{(1)}=(\\mu_{w}^{(1)})_{w \\in W}$ and \n$\\mu_{\\bullet}^{(2)}=(\\mu_{w}^{(2)})_{w \\in W}$ be GGMS data. \nThen, it is obvious from the definition of GGMS data \n(or equivalently, from \\eqref{eq:length}) that \nthe (componentwise) sum \n\\begin{equation*}\n\\mu_{\\bullet}^{(1)}+\\mu_{\\bullet}^{(2)}:=\n (\\mu_{w}^{(1)}+\\mu_{w}^{(2)})_{w \\in W}\n\\end{equation*}\nof $\\mu_{\\bullet}^{(1)}$ and $\\mu_{\\bullet}^{(2)}$ is \nalso a GGMS datum. \n\\end{rem}\n\nFollowing \\cite{Kam1} and \\cite{Kam2}, \nto each GGMS datum $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$, \nwe associate a convex polytope \n$P(\\mu_{\\bullet}) \\subset \\mathfrak{h}_{\\mathbb{R}}$ by:\n\\begin{equation} \\label{eq:poly}\nP(\\mu_{\\bullet})=\n\\bigcap_{w \\in W}\n \\bigl\\{ \n v \\in \\mathfrak{h}_{\\mathbb{R}} \\mid \n w^{-1} \\cdot v- w^{-1} \\cdot \\mu_{w} \\in \n \\textstyle{\\sum_{j \\in I}\\mathbb{R}_{\\ge 0} h_{j}}\n \\bigr\\}; \n\\end{equation}\nthe polytope $P(\\mu_{\\bullet})$ is called \na pseudo-Weyl polytope with GGMS datum $\\mu_{\\bullet}$. \nNote that the GGMS datum $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ is \ndetermined uniquely by the convex polytope $P(\\mu_{\\bullet})$. \nAlso, we know from \\cite[Proposition~2.2]{Kam1} that \nthe set of vertices of the polytope $P(\\mu_{\\bullet})$ \nis given by the collection $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ \n(possibly, with repetitions). In particular, we have\n\\begin{equation} \\label{eq:conv}\nP(\\mu_{\\bullet}) = \n\\mathop{\\rm Conv}\\nolimits\\,\\bigl\\{\\mu_{w} \\mid w \\in W\\bigr\\},\n\\end{equation}\nwhere for a subset $X$ of $\\mathfrak{h}_{\\mathbb{R}}$, \n$\\mathop{\\rm Conv}\\nolimits X$ denotes the convex hull in $\\mathfrak{h}_{\\mathbb{R}}$ of $X$. \n\nWe know the following proposition from \\cite[Lemma~6.1]{Kam1}, \nwhich will be used later. \n\\begin{prop} \\label{prop:Minkowski}\nLet $P_{1}=P(\\mu_{\\bullet}^{(1)})$ and \n$P_{2}=P(\\mu_{\\bullet}^{(2)})$ be pseudo-Weyl polytopes \nwith GGMS data \n$\\mu_{\\bullet}^{(1)}=(\\mu_{w}^{(1)})_{w \\in W}$ and \n$\\mu_{\\bullet}^{(2)}=(\\mu_{w}^{(2)})_{w \\in W}$, respectively. \nThen, the Minkowski sum\n\\begin{equation*}\nP_{1}+P_{2}:=\n \\bigl\\{v_{1}+v_{2} \\mid v_{1} \\in P_{1},\\,v_{2} \\in P_{2}\\bigr\\}\n\\end{equation*}\nof the pseudo-Weyl polytopes $P_{1}$ and $P_{2}$ is identical to \nthe pseudo-Weyl polytope $P(\\mu_{\\bullet}^{(1)}+\\mu_{\\bullet}^{(2)})$ \nwith GGMS datum $\\mu_{\\bullet}^{(1)}+\\mu_{\\bullet}^{(2)}=\n (\\mu_{w}^{(1)}+\\mu_{w}^{(2)})_{w \\in W}$ \n(see Remark~\\ref{rem:GGMS-sum}). \n\\end{prop}\n\nLet $R(w_{0})$ denote the set of all reduced words for $w_{0}$, that is, \nall sequences $(i_{1},\\,i_{2},\\,\\dots,\\,i_{m})$ of elements of $I$ \nsuch that $s_{i_{1}}s_{i_{2}} \\cdots s_{i_{m}}=w_{0}$, where \n$m$ is the length $\\ell(w_{0})$ of the longest element $w_{0}$.\nLet $\\mathbf{i}=(i_{1},\\,i_{2},\\,\\dots,\\,i_{m}) \\in R(w_{0})$ \nbe a reduced word for $w_{0}$. \nWe set $\\wi{l}:=\\si{1}\\si{2} \\cdots \\si{l} \\in W$ \nfor $0 \\le l \\le m$. For a GGMS datum \n$\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$, define integers \n(called lengths of edges) \n$\\Ni{l}=\\Ni{l}(\\mu_{\\bullet}) \\in \\mathbb{Z}_{\\ge 0}$, \n$1 \\le l \\le m$, via the following \n``length formula'' (see \\cite[Eq.(8)]{Kam1} and \n\\eqref{eq:length} above): \n\\begin{equation} \\label{eq:n}\n\\mu_{\\wi{l}}-\\mu_{\\wi{l-1}}=\\Ni{l} \\wi{l-1} \\cdot h_{i_{l}}.\n\\end{equation}\n\n\\vspace{3mm}\n\n{\\small \n\\hspace{55mm}\n\\input{dempoly2-fig003.tex}\n}\n\n\\vspace{3mm}\n\nNow we recall a (combinatorial) characterization of \nMirkovi\\'c-Vilonen (MV) polytopes, due to Kamnitzer \\cite{Kam1}. \nThis result holds for an arbitrary complex semisimple Lie algebra $\\mathfrak{g}$, \nbut we give its precise statement only in the case that \n$\\mathfrak{g}$ is simply-laced since we do not make use of it \nin this paper; we merely mention that \nwhen $\\mathfrak{g}$ is not simply-laced, \nthere are also conditions on the lengths \n$\\Ni{l}$, $1 \\le l \\le m$, $\\mathbf{i} \\in R(w_{0})$, \nfor the other possible values of $a_{ij}$ and $a_{ji}$\n(we refer the reader to \\cite[\\S3]{BeZe} for explicit formulas).\n\\begin{dfn} \\label{dfn:MV}\nA GGMS datum $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ is \nsaid to be a Mirkovi\\'c-Vilonen (MV) datum if \nit satisfies the following conditions: \n\n(1) If $\\mathbf{i}=(i_{1},\\,i_{2},\\,\\dots,\\,i_{m}) \\in R(w_{0})$ and \n$\\mathbf{j}=(j_{1},\\,j_{2},\\,\\dots,\\,j_{m}) \\in R(w_{0})$ are related by \na $2$-move, that is, if there exist indices $i,\\,j \\in I$ \nwith $a_{ij}=a_{ji}=0$ and an integer $0 \\le k \\le m-2$ \nsuch that $i_{l}=j_{l}$ for all $1 \\le l \\le m$ with \n$l \\ne k+1,\\,k+2$, and such that \n$i_{k+1}=j_{k+2}=i$, $i_{k+2}=j_{k+1}=j$, then \nthere hold\n\\begin{equation*}\n\\begin{cases}\n\\Ni{l}=\\Nj{l} \\quad \n  \\text{for all $1 \\le l \\le m$ with $l \\ne k+1,\\,k+2$, and} \\\\[1.5mm]\n\\Ni{k+1}=\\Nj{k+2}, \\quad \\Ni{k+2}=\\Nj{k+1}.\n\\end{cases}\n\\end{equation*}\n\n\\vspace{5mm}\n\n\\newcommand{\\vertexa}{\n  $\\mu_{\\wi{k+2}}=\\mu_{\\wi{k}s_is_j}=\n   \\mu_{\\wj{k}s_js_i}=\\mu_{\\wj{k+2}}$\n}\n\n{\\small \n\\hspace*{-20mm}\n\\input{dempoly2-fig001.tex}\n}\n\n\\vspace{3mm}\n\n(2) If $\\mathbf{i}=(i_{1},\\,i_{2},\\,\\dots,\\,i_{m}) \\in R(w_{0})$ and \n$\\mathbf{j}=(j_{1},\\,j_{2},\\,\\dots,\\,j_{m}) \\in R(w_{0})$ are \nrelated by a $3$-move, that is, \nif there exist indices $i,\\,j \\in I$ \nwith $a_{ij}=a_{ji}=-1$ and an integer $0 \\le k \\le m-3$ \nsuch that $i_{l}=j_{l}$ for all $1 \\le l \\le m$ with \n$l \\ne k+1,\\,k+2,\\,k+3$, and such that\n$i_{k+1}=i_{k+3}=j_{k+2}=i$, \n$i_{k+2}=j_{k+1}=j_{k+3}=j$, then \nthere hold \n\\begin{equation*}\n\\begin{cases}\n\\Ni{l}=\\Nj{l} \\quad\n  \\text{for all $1 \\le l \\le m$ with $l \\ne k+1,\\,k+2,\\,k+3$, and} \\\\[1.5mm]\n\\Nj{k+1}=\\Ni{k+2}+\\Ni{k+3}-\\min \\bigl(\\Ni{k+1},\\, \\Ni{k+3}\\bigr), \\\\[1.5mm]\n\\Nj{k+2}=\\min \\bigl(\\Ni{k+1},\\, \\Ni{k+3}\\bigr), \\\\[1.5mm]\n\\Nj{k+3}=\\Ni{k+1}+\\Ni{k+2}-\\min \\bigl(\\Ni{k+1},\\, \\Ni{k+3}\\bigr).\n\\end{cases}\n\\end{equation*}\n\n\\vspace{5mm}\n\n\\newcommand{\\vertexb}{\n  $\\mu_{\\wi{k+3}}=\\mu_{\\wi{k}s_is_js_i}=\n   \\mu_{\\wj{k}s_js_is_j}=\\mu_{\\wj{k+3}}$\n}\n\n{\\small \n\\hspace*{-22.5mm}\n\\input{dempoly2-fig002.tex}\n}\n\\end{dfn}\n\nThe pseudo-Weyl polytope $P(\\mu_{\\bullet})$ \nwith GGMS datum $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ \n(see \\eqref{eq:poly}) is \na Mirkovi\\'c-Vilonen (MV) polytope if and only if \nthe GGMS datum $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ is \nan MV datum (see the proof of \\cite[Proposition~5.4]{Kam1} and \nthe comment following \\cite[Theorem~7.1]{Kam1}).\nAlso, for a dominant coweight \n$\\lambda \\in X_{*}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ and a coweight \n$\\nu \\in X_{*}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$, \nan MV polytope $P=P(\\mu_{\\bullet})$ with GGMS datum \n$\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ is \nan MV polytope of highest vertex $\\lambda$ and lowest vertex $\\nu$ \nif and only if $\\mu_{w_{0}}=\\lambda$, $\\mu_{e}=\\nu$, and \n$P$ is contained in the convex hull $\\mathop{\\rm Conv}\\nolimits(W \\cdot \\lambda)$ of \nthe $W$-orbit $W \\cdot \\lambda \\subset \\mathfrak{h}_{\\mathbb{R}}$ \n(see \\cite[Proposition~7]{A}); \nwe denote by $\\mathcal{MV}(\\lambda)_{\\nu}$ the set of MV polytopes \nof highest vertex $\\lambda$ and lowest vertex $\\nu$. \nFor each dominant coweight \n$\\lambda \\in X_{*}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$, we set\n\\begin{equation*}\n\\mathcal{MV}(\\lambda):=\\bigsqcup_{\n \\nu \\in X_{*}(T)} \\mathcal{MV}(\\lambda)_{\\nu}.\n\\end{equation*}\n\n\\subsection{Relation between MV polytopes and MV cycles.}\n\\label{subsec:geom}\n\nIn this subsection, we review the relation between MV polytopes \nand MV cycles in the affine Grassmannian. \n\nLet us recall the definition of MV cycles \nin the affine Grassmannian, following \\cite{MV2} (and \\cite{A}). \nLet $G$ be a complex, connected, reductive algebraic group \nas in (the beginning of) \\S\\ref{subsec:notation}. \nLet $\\mathcal{O} = \\mathbb{C}[[t]]$ denote the ring of formal power series, \nand $\\mathcal{K} = \\mathbb{C}((t))$ the field of formal Laurent series \n(the fraction field of $\\mathcal{O}$). \nThe affine Grassmannian $\\CG r$ for $G$ over $\\mathbb{C}$ is \ndefined to be the quotient space $G(\\mathcal{K})\/G(\\mathcal{O})$, \nequipped with the structure of a complex, algebraic ind-variety, \nwhere $G(\\mathcal{K})$ denotes the set of $\\mathcal{K}$-valued points of $G$, and \n$G(\\mathcal{O}) \\subset G(\\mathcal{K})$ denotes the set of $\\mathcal{O}$-valued points of $G$; \nwe denote by $\\pi:G(\\mathcal{K}) \\twoheadrightarrow \\CG r=G(\\mathcal{K})\/G(\\mathcal{O})$ \nthe natural quotient map, which is locally trivial \nin the Zariski topology. In the following, \nfor a subgroup $H \\subset G(\\mathcal{K})$ \nthat is stable under the adjoint action of $T$ and \nfor an element $w$ of the Weyl group $W \\cong N_{G}(T)\/T$ of $G$, \nwe denote by ${}^{w}H$ the $w$-conjugate $\\dot{w}H\\dot{w}^{-1}$ of $H$, \nwhere $\\dot{w} \\in N_{G}(T)$ is a lift of $w \\in W$. \n\nSince each coweight $\\nu \\in X_{*}(T)=\\mathop{\\rm Hom}\\nolimits(\\mathbb{C}^{\\ast},\\,T)$ \nis a regular map from $\\mathbb{C}^{\\ast}$ to $T \\subset G$, \nit gives a point $t^{\\nu} \\in G (\\mathcal{K})$, \nwhich in turn, descends to a point \n$[t^{\\nu}] \\in \\CG r=G(\\mathcal{K})\/G(\\mathcal{O})$. \nThe following simple lemma will be used in \nthe proof of Lemma~\\ref{lem:tran}. \n\\begin{lem} \\label{lem:emb}\nLet $L \\subset G$ be a complex, connected, \nreductive algebraic group containing the maximal torus $T$ of $G$. \nThen, for each $\\nu \\in X_{*}(T)$, the inclusion \n$L(\\mathcal{K}) [t^{\\nu}] \\hookrightarrow \\CG r$ gives an embedding of \nthe affine Grassmannian for $L$ into $\\CG r$.\n\\end{lem}\n\n\\begin{proof}\nObserve that \n$\\bigl(t^{\\nu} G (\\mathcal{O}) t^{-\\nu}\\bigr) \\cap L (\\mathcal{K}) \n = t^{\\nu} L (\\mathcal{O}) t^{-\\nu}$.\nHence the map $i_{L}:L(\\mathcal{K}) \\rightarrow \\CG r$, \n$g \\mapsto g[t^{\\nu}]$, is factored through \n$L(\\mathcal{K}) \/ (t^{\\nu} L (\\mathcal{O}) t^{-\\nu})$. \nSince $t^{\\nu} \\in L (\\mathcal{K})$, we conclude that \nthe map $i_{L}:L(\\mathcal{K}) \\rightarrow \\CG r$ descends to a map between \nthe affine Grassmannian for $L$ and $\\CG r$, as desired. \n(This construction is only at the level of sets, \nbut we can indeed show that the map above commutes \nwith the ind-variety structures.)\n\\end{proof}\n\nFor each $\\nu \\in X_{*}(T)$, we set\n\\begin{equation*}\n\\CG r^{\\nu}:=G(\\mathcal{O})[t^{\\nu}] \\subset \\CG r,\n\\end{equation*}\nthe $G(\\mathcal{O})$-orbit of $[t^{\\nu}]$, which is a smooth \nquasi-projective algebraic variety over $\\mathbb{C}$. \nAlso, for each $\\nu \\in X_{*}(T)$ and $w \\in W$, \nwe set \n\\begin{equation*}\nS_{\\nu}^{w} := {}^{w}U(\\mathcal{K})[t^{\\nu}] \\subset \\CG r,\n\\end{equation*}\nthe ${}^{w}U(\\mathcal{K})$-orbit of $[t^{\\nu}]$, which is a (locally closed) \nind-subvariety of $\\CG r$; we write simply $S_{\\nu}$ for $S_{\\nu}^{e}$.\nThen, we know the following two kinds \nof decompositions of $\\CG r$ into orbits. \nFirst, we have\n\\begin{equation*}\n\\CG r=\\bigsqcup_{\\lambda \\in X_{*}(T)_{+}} \\CG r^{\\lambda} \\qquad \n\\text{(Cartan decomposition)},\n\\end{equation*}\nwith $\\CG r^{w \\cdot \\lambda}=\\CG r^{\\lambda}$ \nfor $\\lambda \\in X_{*}(T)_{+}$ and $w \\in W$; \nnote that (see, for example, \\cite[\\S2]{MV2}) \nfor each $\\lambda \\in X_{*}(T)_{+}$, the quasi-projective \nvariety $\\CG r^{\\lambda}$ is simply-connected, and of dimension \n$2\\pair{\\lambda}{\\rho}$, where $\\rho$ denotes the half-sum \nof the positive roots $\\alpha \\in \\Delta_{+}$ for $G$, i.e., \n$2\\rho=\\sum_{\\alpha \\in \\Delta_{+}}\\alpha$. \nSecond, we have for each $w \\in W$, \n\\begin{equation*}\n\\CG r=\\bigsqcup_{\\nu \\in X_{*}(T)} S_{\\nu}^{w} \\qquad\n\\text{(Iwasawa decomposition)}.\n\\end{equation*}\nMoreover, the (Zariski-) closure relations among these orbits are \ndescribed as follows (see \\cite[\\S2 and \\S3]{MV2}): \n\\begin{equation} \\label{eq:Grlam}\n\\ol{\\CG r^{\\lambda}}=\n \\bigsqcup_{\n   \\begin{subarray}{c} \n   \\lambda' \\in X_{\\ast}(T)_{+} \\\\[1mm]\n   \\lambda' \\le \\lambda\n   \\end{subarray}\n } \\CG r^{\\lambda'}\n\\qquad \\text{for $\\lambda \\in X_{*}(T)_{+}$};\n\\end{equation}\n\\begin{equation} \\label{eq:Snuw}\n\\ol{S_{\\nu}^{w}}=\n \\bigsqcup_{\n   \\begin{subarray}{c} \n   \\gamma \\in X_{\\ast}(T) \\\\[1mm]\n   w^{-1} \\cdot \\gamma \\ge w^{-1} \\cdot \\nu\n   \\end{subarray}\n } S_{\\gamma}^{w}\n\\qquad \\text{for $\\nu \\in X_{\\ast}(T)$ and $w \\in W$}.\n\\end{equation}\n\\begin{rem} \\label{rem:Snuw}\nLet $\\mathcal{X} \\subset \\CG r$ be an irreducible algebraic subvariety, \nand $\\nu \\in X_{*}(T)$, $w \\in W$. Then, \nit follows from \\eqref{eq:Snuw} that the intersection \n$\\mathcal{X} \\cap S_{\\nu}^{w}$ is an open dense subset of $\\mathcal{X}$ \nif and only if $\\mathcal{X} \\cap S_{\\nu}^{w} \\ne \\emptyset$ and \n$\\mathcal{X} \\cap S_{\\gamma}^{w} = \\emptyset$ \nfor every $\\gamma \\in X_{*}(T)$ with \n$w^{-1} \\cdot \\gamma \\not\\ge w^{-1} \\cdot \\nu$. \n\\end{rem}\n\nFor $\\lambda \\in X_{*}(T)_{+}$, \nlet $L(\\lambda)$ denote the irreducible finite-dimensional \nrepresentation of the (complex) Langlands dual \ngroup $G^{\\vee}$ of $G$ with highest weight $\\lambda$, and \n$\\Omega(\\lambda) \\subset X_{*}(T)$ \nthe set of weights of $L(\\lambda)$. \nWe know from \\cite[Theorem~3.2 and Remark~3.3]{MV2} that \n$\\nu \\in X_{*}(T)$ is an element of $\\Omega(\\lambda)$ \nif and only if $\\CG r^{\\lambda} \\cap S_{\\nu} \\ne \\emptyset$, \nand then the intersection $\\CG r^{\\lambda} \\cap S_{\\nu}$ \nis of pure dimension $\\pair{\\lambda-\\nu}{\\rho}$. \n\nNow we come to the definition of MV cycles \nin the affine Grassmannian.\n\\begin{dfn}[{\\cite[\\S3]{MV2}; see also \\cite[\\S5.3]{A}}] \\label{dfn:MVcycle}\nLet $\\lambda \\in X_{*}(T)_{+}$ and $\\nu \\in X_{*}(T)$ be such that \n$\\CG r^{\\lambda} \\cap S_{\\nu} \\ne \\emptyset$, \ni.e., $\\nu \\in \\Omega(\\lambda)$. \nAn MV cycle of highest weight $\\lambda$ and weight $\\nu$ is \ndefined to be an irreducible component of the (Zariski-) closure of \nthe intersection $\\CG r^{\\lambda} \\cap S_{\\nu}$. \n\\end{dfn}\n\nWe denote by $\\mathcal{Z}(\\lambda)_{\\nu}$ the set of MV cycles of \nhighest weight $\\lambda \\in X_{*}(T)_{+}$ and weight $\\nu \\in X_{*}(T)$. \nAlso, for each $\\lambda \\in X_{*}(T)_{+}$, we set \n\\begin{equation*}\n\\mathcal{Z}(\\lambda) := \n \\bigsqcup_{\\nu \\in X_{*}(T)} \\mathcal{Z}(\\lambda)_{\\nu},\n\\end{equation*}\nwhere $\\mathcal{Z}(\\lambda)_{\\nu} := \\emptyset$ if \n$\\CG r^{\\lambda} \\cap S_{\\nu}=\\emptyset$. \n\\begin{ex}[{cf.~\\cite[Eq.(3.6)]{MV2}}] \\label{ex:MVc}\nFor each $\\lambda \\in X_{\\ast}(T)_{+}$, we have\n\\begin{equation*}\n\\mathcal{Z}(\\lambda)_{\\lambda} = \\bigl\\{\\,[ t^{\\lambda}]\\,\\bigr\\}, \n\\quad \\text{and} \\quad\n\\mathcal{Z}(\\lambda)_{w_{0} \\lambda} = \\bigl\\{\\,\\ol{\\CG r^{\\lambda}}\\,\\bigr\\}.\n\\end{equation*}\n\\end{ex}\n\\begin{rem}[{\\cite[Lemma~5.2]{NP}, \\cite[Eq.(3.6)]{MV2}}] \n\\label{rem:extcyc}\nLet $\\lambda \\in X_{*}(T)_{+}$. If $\\nu \\in X_{*}(T)$ is \nof the form $\\nu=x \\cdot \\lambda$ for some $x \\in W$, then \n\\begin{equation*}\n\\mathbf{b}_{x \\cdot \\lambda}:=\n\\ol{ U(\\mathcal{O})[t^{x \\cdot \\lambda}] } \\subset \n\\ol{ G(\\mathcal{O})[t^{\\lambda}] \\cap U(\\mathcal{K})[t^{x \\cdot \\lambda}] }=\n\\ol{ \\CG r^{\\lambda} \\cap S_{x \\cdot \\lambda} }\n\\end{equation*}\nis the unique MV cycle of highest weight $\\lambda$ \nand weight $x \\cdot \\lambda$ \n(extremal MV cycle of weight $x \\cdot \\lambda$). \nFor an explicit (combinatorial) description \nof the corresponding extremal MV polytope, \nsee \\S\\ref{subsec:extpoly} below. \n\\end{rem}\n\nMotivated by the discovery of MV cycles in the affine Grassmannian, \nAnderson \\cite{A} proposed considering the ``moment map images'' \nof MV cycles as follows: Let $\\lambda \\in X_{*}(T)_{+}$. \nFor an MV cycle $\\mathbf{b} \\in \\mathcal{Z}(\\lambda)$, we set \n\\begin{equation*}\nP(\\mathbf{b}):=\\mathop{\\rm Conv}\\nolimits \\bigl\\{\n  \\nu \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}} \\mid \n  [t^{\\nu}] \\in \\mathbf{b} \\bigr\\},\n\\end{equation*}\nand call $P(\\mathbf{b}) \\subset \\mathfrak{h}_{\\mathbb{R}}$ the moment map \nimage of $\\mathbf{b}$\\,; note that $P(\\mathbf{b})$ is indeed \na convex polytope in $\\mathfrak{h}_{\\mathbb{R}}$. \n\nFor the rest of this paper, except in \\S\\ref{subsec:BG} and the Appendix, \nwe assume that $G$ (and hence its Lie algebra $\\mathfrak{g}$) is semisimple. \nThe following theorem, due to Kamnitzer \\cite{Kam1}, \nestablishes an explicit relationship between \nMV polytopes and MV cycles. \n\\begin{thm} \\label{thm:Kam1}\n{\\rm (1)} \nLet $\\lambda \\in X_{*}(T)_{+}$ and $\\nu \\in X_{*}(T)$ be \nsuch that $\\CG r^{\\lambda} \\cap S_{\\nu} \\ne \\emptyset$. \nIf $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ denotes the GGMS datum \nof an MV polytope $P \\in \\mathcal{MV}(\\lambda)_{\\nu}$, that is, \n$P=P(\\mu_{\\bullet}) \\in \\mathcal{MV}(\\lambda)_{\\nu}$, then \n\\begin{equation*}\n\\mathbf{b}(\\mu_{\\bullet}):=\n\\ol{ \\bigcap_{w \\in W} S^{w}_{\\mu_{w}} } \\subset \\ol{\\CG r^{\\lambda}}\n\\end{equation*}\nis an MV cycle that belongs to $\\mathcal{Z}(\\lambda)_{\\nu}$. \n\n{\\rm (2)} \nLet $\\lambda \\in X_{*}(T)_{+}$. \nFor an MV polytope $P=P(\\mu_{\\bullet}) \\in \\mathcal{MV}(\\lambda)$ \nwith GGMS datum $\\mu_{\\bullet}$, \nwe set $\\Phi_{\\lambda}(P):=\\mathbf{b}(\\mu_{\\bullet})$. Then, \nthe map $\\Phi_{\\lambda}:\\mathcal{MV}(\\lambda) \\rightarrow \\mathcal{Z}(\\lambda)$, \n$P \\mapsto \\Phi_{\\lambda}(P)$, is a bijection \nfrom $\\mathcal{MV}(\\lambda)$ onto $\\mathcal{Z}(\\lambda)$ \nsuch that $\\Phi_{\\lambda}(\\mathcal{MV}(\\lambda)_{\\nu})=\n\\mathcal{Z}(\\lambda)_{\\nu}$ for all $\\nu \\in X_{\\ast}(T)$\nwith $\\CG r^{\\lambda} \\cap S_{\\nu} \\ne \\emptyset$. \nIn particular, for each MV cycle $\\mathbf{b} \\in \\mathcal{Z}(\\lambda)$, \nthere exists a unique MV datum $\\mu_{\\bullet}$ \nsuch that $\\mathbf{b}=\\mathbf{b}(\\mu_{\\bullet})$, and in this case, \nthe moment map image $P(\\mathbf{b})$ of the MV cycle $\\mathbf{b}=\\mathbf{b}(\\mu_{\\bullet})$\nis identical to the MV polytope $P(\\mu_{\\bullet}) \\in \\mathcal{MV}(\\lambda)$. \n\\end{thm}\n\\begin{rem}[{\\cite[\\S2.2]{Kam1}}] \\label{rem:moment}\nFor $\\nu \\in X_{*}(T)$ and $w \\in W$, \nthe ``moment map image'' $P(\\ol{S_{\\nu}^{w}})$ of \n$\\ol{S_{\\nu}^{w}}$ is, by definition, \nthe convex hull in $\\mathfrak{h}_{\\mathbb{R}}$ of the set\n$\\bigl\\{\n  \\gamma \\in X_{*}(T) \\subset \\mathfrak{h}_{\\mathbb{R}} \\mid \n  [t^{\\gamma}] \\in \\ol{S_{\\nu}^{w}}\n\\bigr\\} \\subset \\mathfrak{h}_{\\mathbb{R}}$,\nwhich is identical to the (shifted) convex cone\n$\\bigl\\{\n v \\in \\mathfrak{h}_{\\mathbb{R}} \\mid \n w^{-1} \\cdot v - w^{-1} \\cdot \\nu \\in \n \\textstyle{\\sum_{j \\in I}\\mathbb{R}_{\\ge 0}h_{j}}\n\\bigr\\}$. \n\\end{rem}\n\n\\subsection{Lusztig-Berenstein-Zelevinsky (LBZ) crystal structure.}\n\\label{subsec:cry-MV}\n\nWe keep the notation and assumptions of \\S\\ref{subsec:MV}.\nFor an MV datum $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ and $j \\in I$, \nwe denote by $f_{j}\\mu_{\\bullet}$ \n(resp., $e_{j}\\mu_{\\bullet}$ if $\\mu_{e} \\ne \\mu_{s_{j}}$; \nnote that $\\mu_{s_{j}}-\\mu_{e} \\in \n\\mathbb{Z}_{\\ge 0}h_{j}$ by \\eqref{eq:length})\na unique MV datum $\\mu_{\\bullet}'=(\\mu_{w}')_{w \\in W}$ \nsuch that $\\mu_{e}'=\\mu_{e}-h_{j}$ \n(resp., $\\mu_{e}'=\\mu_{e}+h_{j}$) and \n$\\mu_{w}'=\\mu_{w}$ for all $w \\in W$ with $s_{j}w < w$ \n(see \\cite[Theorem~3.5]{Kam2} and its proof); \nnote that $\\mu_{w_{0}}'=\\mu_{w_{0}}$ and \n$\\mu_{s_j}'=\\mu_{s_j}$. \n\nLet $\\lambda \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ \nbe a dominant coweight. \nFollowing \\cite[\\S6.2]{Kam2}, we endow $\\mathcal{MV}(\\lambda)$ \nwith the Lusztig-Berenstein-Zelevinsky (LBZ) \ncrystal structure for $U_{q}(\\mathfrak{g}^{\\vee})$\nas follows. \nLet $P=P(\\mu_{\\bullet}) \\in \\mathcal{MV}(\\lambda)$ be \nan MV polytope with GGMS datum \n$\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$.\nThe weight $\\mathop{\\rm wt}\\nolimits(P)$ of $P$ is, by definition, \nequal to the vertex $\\mu_{e} \\in \\lambda-Q^{\\vee}_{+}$. \nFor each $j \\in I$, \nwe define the lowering Kashiwara operator\n$f_{j}:\\mathcal{MV}(\\lambda) \\cup \\{\\mathbf{0}\\} \\rightarrow \n \\mathcal{MV}(\\lambda) \\cup \\{\\mathbf{0}\\}$ and \nthe raising Kashiwara operator\n$e_{j}:\\mathcal{MV}(\\lambda) \\cup \\{\\mathbf{0}\\} \\rightarrow \n \\mathcal{MV}(\\lambda) \\cup \\{\\mathbf{0}\\}$ by: \n\\begin{align*}\ne_{j}\\mathbf{0}=f_{j}\\mathbf{0} & :=\\mathbf{0}, \\\\[3mm]\nf_{j}P=f_{j}P(\\mu_{\\bullet}) & :=\n\\begin{cases}\n P(f_{j}\\mu_{\\bullet}) \n & \\text{if $P(f_{j}\\mu_{\\bullet}) \\subset \n         \\mathop{\\rm Conv}\\nolimits (W \\cdot \\lambda)$}, \\\\[1.5mm]\n \\mathbf{0} & \\text{otherwise}, \n \\end{cases} \\\\[3mm]\ne_{j}P=e_{j}P(\\mu_{\\bullet}) & :=\n\\begin{cases}\n P(e_{j}\\mu_{\\bullet}) & \n \\text{if $\\mu_{e} \\ne \\mu_{s_{j}}$ \n (i.e., $\\mu_{s_{j}}-\\mu_{e} \\in \\mathbb{Z}_{> 0}h_{j}$)}, \\\\[1.5mm]\n \\mathbf{0} & \\text{otherwise}, \n \\end{cases}\n\\end{align*}\nwhere $\\mathbf{0}$ is an additional element, \nnot contained in $\\mathcal{MV}(\\lambda)$. \nFor $j \\in I$, we set \n$\\varepsilon_{j}(P):=\n \\max \\bigl\\{k \\in \\mathbb{Z}_{\\ge 0} \\mid e_{j}^{k}P \\ne \\mathbf{0} \\bigr\\}$ and \n$\\varphi_{j}(P):=\n \\max \\bigl\\{k \\in \\mathbb{Z}_{\\ge 0} \\mid f_{j}^{k}P \\ne \\mathbf{0} \\bigr\\}$;\nnote that for each $j \\in I$, we have\n\\begin{equation} \\label{eq:ax}\n\\varphi_{j}(P)=\n\\pair{\\mathop{\\rm wt}\\nolimits(P)}{\\alpha_{j}}+\\varepsilon_{j}(P) \n \\qquad \\text{for all $P \\in \\mathcal{MV}(\\lambda)$.}\n\\end{equation}\n\\begin{rem} \\label{rem:ve}\nLet $P=P(\\mu_{\\bullet}) \\in \\mathcal{MV}(\\lambda)$ be \nan MV polytope with GGMS datum \n$\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$. \nThen, we deduce from the definition of \nthe raising Kashiwara operators $e_{j}$ \n(or, the MV datum $e_{j}\\mu_{\\bullet}$) that \n$\\mu_{s_{j}}-\\mu_{e}=\\varepsilon_{j}(P) h_{j}$ \nfor $j \\in I$.\n\\end{rem}\n\n\\begin{thm}[{\\cite[Theorem~6.4]{Kam2}}] \\label{thm:kam-int}\nThe set $\\mathcal{MV}(\\lambda)$, equipped with the maps \n$\\mathop{\\rm wt}\\nolimits$, $e_{j},\\,f_{j} \\ (j \\in I)$, and \n$\\varepsilon_{j},\\,\\varphi_{j} \\ (j \\in I)$ above, is \na crystal for $U_{q}(\\mathfrak{g}^{\\vee})$. \nMoreover, there exists a unique isomorphism \n$\\Psi_{\\lambda}:\\mathcal{B}(\\lambda) \n \\stackrel{\\sim}{\\rightarrow} \n \\mathcal{MV}(\\lambda)$ of crystals for $U_{q}(\\mathfrak{g}^{\\vee})$. \n\\end{thm}\n\n\\begin{rem}\nKamnitzer \\cite{Kam2} proved that \nfor each $\\lambda \\in X_{*}(T)_{+}$, the bijection \n$\\Phi_{\\lambda}:\\mathcal{MV}(\\lambda) \\rightarrow \\mathcal{Z}(\\lambda)$ in \nTheorem~\\ref{thm:Kam1}\\,(2) also intertwines the LBZ crystal \nstructure on $\\mathcal{MV}(\\lambda)$ and the crystal structure on $\\mathcal{Z}(\\lambda)$ \ndefined in \\cite{BrGa} (and \\cite{BFG}). \n\\end{rem}\n\nFor each $x \\in W$, we denote by \n$\\mathcal{MV}_{x}(\\lambda) \\subset \\mathcal{MV}(\\lambda)$ \nthe image $\\Psi_{\\lambda}(\\mathcal{B}_{x}(\\lambda))$ \nof the Demazure crystal \n$\\mathcal{B}_{x}(\\lambda) \\subset \\mathcal{B}(\\lambda)$ \nassociated to $x \\in W$ under the isomorphism \n$\\Psi_{\\lambda}:\\mathcal{B}(\\lambda) \n \\stackrel{\\sim}{\\rightarrow} \n \\mathcal{MV}(\\lambda)$ in Theorem~\\ref{thm:kam-int};\nfor a combinatorial description of $\\mathcal{MV}_{x}(\\lambda)$ \nin terms of the lengths $\\Ni{l} \\in \\mathbb{Z}_{\\ge 0}$, \n$\\mathbf{i} \\in R(w_{0})$, $0 \\le l \\le m$, \nof edges of an MV polytope, see \\cite[\\S3.2]{NS-dp}. \n\n\\subsection{Extremal MV polytopes.}\n\\label{subsec:extpoly}\n\nLet $\\mathfrak{g}$ be a complex semisimple Lie algebra as in \n(the second paragraph of) \\S\\ref{subsec:notation}. \nLet $\\lambda \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ be \na dominant coweight. \nFor each $x \\in W$, we denote by \n$P_{x \\cdot \\lambda}$ the image of the extremal element \n$u_{x \\cdot \\lambda} \\in \\mathcal{B}(\\lambda)$ of weight \n$x \\cdot \\lambda \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ \nunder the isomorphism \n$\\Psi_{\\lambda}:\\mathcal{B}(\\lambda) \n \\stackrel{\\sim}{\\rightarrow} \n \\mathcal{MV}(\\lambda)$\nin Theorem~\\ref{thm:kam-int}; \nwe call $P_{x \\cdot \\lambda} \\in \\mathcal{MV}(\\lambda)$ \nthe extremal MV polytope of weight $x \\cdot \\lambda$.\nWe know the following polytopal description of \nthe extremal MV polytopes \nfrom \\cite[Theorem~4.1.5\\,(2)]{NS-dp}.\n\\begin{prop} \\label{prop:ext}\nLet $\\lambda \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ \nbe a dominant coweight, and $x \\in W$. \nThe extremal MV polytope \n$P_{x \\cdot \\lambda}$ of weight $x \\cdot \\lambda$ \nis identical to the convex hull \n$\\mathop{\\rm Conv}\\nolimits(W_{\\le x} \\cdot \\lambda)$ \nin $\\mathfrak{h}_{\\mathbb{R}}$ of the set $W_{\\le x} \\cdot \\lambda$, \nwhere $W_{\\le x}$ denotes the subset \n$\\bigl\\{z \\in W \\mid z \\le x\\bigr\\}$ of $W$. \n\\end{prop}\n\n\\begin{rem}\nIn \\cite{NS-dp}, we proved Proposition~\\ref{prop:ext} above and \nTheorem~\\ref{thm:GGMS-Ext} below in the case that \n$\\mathfrak{g}$ is simply-laced. However, \nthese results hold also in the case that \n$\\mathfrak{g}$ is not simply-laced; for example, we can use \na standard technique of ``folding'' by diagram automorphisms \n(see \\cite{NS-fm}, \\cite{Hong}, and also \\cite{Lu08}). \n\\end{rem}\n\n\\begin{rem} \\label{rem:ext}\nIt follows from Theorem~\\ref{thm:Kam1} \nthat for each $\\lambda \\in X_{*}(T)_{+} \\subset \\mathfrak{h}_{\\mathbb{R}}$ \nand $x \\in W$, the extremal MV polytope $P_{x \\cdot \\lambda}$ \nis identical to the moment map image $P(\\mathbf{b}_{x \\cdot \\lambda})$ \nof the extremal MV cycle $\\mathbf{b}_{x \\cdot \\lambda}$ \n(see Remark~\\ref{rem:extcyc}). \nIn particular, the highest weight element \n$P_{e \\cdot \\lambda}=P_{\\lambda}$ of $\\mathcal{MV}(\\lambda)$ \nis identical to the set $P([t^{\\lambda}])=\\bigl\\{\\lambda\\bigr\\}$, \nand the lowest weight element $P_{w_{0} \\cdot \\lambda}$ of \n$\\mathcal{MV}(\\lambda)$ is identical to the set \n$P(\\ol{\\CG r^{\\lambda}})=\\mathop{\\rm Conv}\\nolimits(W \\cdot \\lambda)$. \n\\end{rem}\n\nThe GGMS datum of an extremal MV polytope is given \nas follows (see \\cite[\\S4.1]{NS-dp}). \nLet us fix a dominant coweight \n$\\lambda \\in X_{*}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ \nand $x \\in W$ arbitrarily. \nLet $p$ denote the length $\\ell(xw_{0})$ of $xw_{0} \\in W$. \nFor each $\\mathbf{i}=(i_{1},\\,i_{2},\\,\\dots,\\,i_{m}) \\in R(w_{0})$, \nwith $m=\\ell(w_{0})$, we set \n\\begin{equation*}\nS(xw_{0},\\,\\mathbf{i})=\\left\\{\n\\begin{array}{l|l}\n (a_{1},\\,a_{2},\\,\\dots,\\,a_{p}) \\in [1,m]_{\\mathbb{Z}}^{p} \\ & \\ \n\\begin{array}{l}\n1 \\le a_{1} < a_{2} < \\cdots < a_{p} \\le m, \\\\[1.5mm]\n\\si{a_1}\\si{a_2} \\cdots \\si{a_p}=xw_{0}\n\\end{array}\n\\end{array}\n\\right\\},\n\\end{equation*}\nwhere $[1,m]_{\\mathbb{Z}}:=\\bigl\\{a \\in \\mathbb{Z} \\mid 1 \\le a \\le m\\bigr\\}$.\nWe denote by $\\min S(xw_{0},\\,\\mathbf{i})$ \nthe minimum element of the set $S(xw_{0},\\,\\mathbf{i})$ \nin the lexicographic ordering;\nrecall that the lexicographic ordering $\\succeq$ on \n$S(xw_{0},\\,\\mathbf{i})$ is defined as follows: \n$(a_{1},\\,a_{2},\\,\\dots,\\,a_{p}) \\succ\n (b_{1},\\,b_{2},\\,\\dots,\\,b_{p})$ \nif there exists some integer $1 \\le q_0 \\le p$ \nsuch that $a_{q}=b_{q}$ for all $1 \\le q \\le q_0-1$ \nand $a_{q_0} > b_{q_0}$. \nNow we define a sequence \n$\\yi{0},\\,\\yi{1},\\,\\dots,\\,\\yi{m}$ of elements of $W$ \ninductively by the following formula (see \\cite[\\S4.2]{NS-dp}): \n\\begin{equation} \\label{eq:yi}\n\\yi{m}=e, \\qquad \n\\yi{l-1}=\n \\begin{cases}\n \\yi{l} & \\text{if $l$ appears in $\\min S(xw_{0},\\,\\mathbf{i})$}, \\\\[1.5mm]\n s_{\\bti{l}}\\yi{l} & \\text{otherwise}\n \\end{cases}\n\\end{equation}\nfor $1 \\le l \\le m$, \nwhere we set $\\bti{l}:=\\wi{l-1} \\cdot \\alpha_{i_{l}}$ \nfor $1 \\le l \\le m$, and denote \nby $s_{\\beta} \\in W$ the reflection \nwith respect to a root $\\beta$. \n\\begin{rem} \\label{rem:zi}\nThe element $\\yi{l} \\in W$ above does not depend \non the dominant coweight $\\lambda \\in X_{*}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$. \n\\end{rem}\n\\begin{rem} \\label{rem:vi}\nLet $\\mathbf{i}=(i_{1},\\,i_{2},\\,\\dots,\\,i_{m}) \\in R(w_{0})$. \nWe define a sequence \n$\\vi{0},\\,\\vi{1},\\,\\dots,\\,\\vi{m}$ of elements of $W$ \ninductively by the following formula: \n\\begin{equation*}\n\\vi{m}=e, \\qquad \n\\vi{l-1}=\n \\begin{cases}\n s_{i_{l}}\\vi{l} & \\text{if $l$ appears in $\\min S(xw_{0},\\,\\mathbf{i})$}, \\\\[1.5mm]\n \\vi{l} & \\text{otherwise}\n \\end{cases}\n\\end{equation*}\nfor $1 \\le l \\le m$; we see from the definition of \nthe set $S(xw_{0},\\,\\mathbf{i})$ that \n$\\ell(\\vi{l-1})=\\ell(\\vi{l})+1$ \nif $l$ appears in $\\min S(xw_{0},\\,\\mathbf{i})$. \nThen we know from \\cite[Lemma~4.2.1]{NS-dp}\nthat $\\yi{l}=\\wi{l}\\vi{l}w_{0}^{-1}$ for every $0 \\le l \\le m$. \n\\end{rem}\n\\begin{thm} \\label{thm:GGMS-Ext}\nKeep the notation and assumptions above. \nLet $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ be the GGMS datum of \nthe extremal MV polytope $P_{x \\cdot \\lambda}$, i.e., \n$P_{x \\cdot \\lambda}=P(\\mu_{\\bullet})$. \nLet $w \\in W$ be such that $w=\\wi{l}$ for some \n$\\mathbf{i} \\in R(w_{0})$ and $0 \\le l \\le m$. Then, we have\n$\\mu_{w}=\\mu_{\\wi{l}}=\\yi{l} \\cdot \\lambda$.\n\\end{thm}\n\nThe following results on extremal MV polytopes and extremal MV cycles \nplay an important role in the proof of Theorem~\\ref{thm:tensor}\ngiven in \\S\\ref{subsec:prf-tensor}. \n\\begin{lem} \\label{lem:ext1}\nKeep the notation and assumptions of \nTheorem~\\ref{thm:GGMS-Ext}.\nFor each $w \\in W$ and $j \\in I$ with $w < ws_{j}$, \nwe have either \n{\\rm (a)} $\\mu_{ws_{j}}=\\mu_{w}$, or \n{\\rm (b)} $\\mu_{ws_{j}}=ws_{j}w^{-1} \\cdot \\mu_{w}$.\nMoreover, in both of the cases {\\rm (a)} and {\\rm (b)}, \nwe have $\\pair{\\mu_{ws_{j}}}{w \\cdot \\alpha_{j}} \\ge 0$. \n\\end{lem}\n\n\\begin{proof}\nTake $\\mathbf{i}=(i_{1},\\,i_{2},\\,\\dots,\\,i_{m}) \\in R(w_{0})$ such that \n$\\wi{l-1}=w$ and $\\wi{l}=ws_{j}$ for some $1 \\le l \\le m$; \nnote that $i_{l}=j$, $\\bti{l}=\\wi{l-1} \\cdot \\alpha_{i_{l}}=\nw \\cdot \\alpha_{j}$, and hence $s_{\\bti{l}}=ws_{j}w^{-1}$. \nSince $\\mu_{w}=\\mu_{\\wi{l-1}}=\\yi{l-1} \\cdot \\lambda$ and \n$\\mu_{ws_{j}}=\\mu_{\\wi{l}}=\\yi{l} \\cdot \\lambda$ \nby Theorem~\\ref{thm:GGMS-Ext}, and since \n$\\yi{l-1}$ is equal to $\\yi{l}$ or \n$s_{\\bti{l}}\\yi{l}=ws_{j}w^{-1}\\yi{l}$ \nby definition, it follows immediately that \neither (a) $\\mu_{ws_{j}}=\\mu_{w}$ or \n(b) $\\mu_{ws_{j}}=ws_{j}w^{-1} \\cdot \\mu_{w}$ holds. \n\nWe will show that \n$\\pair{\\mu_{ws_{j}}}{w \\cdot \\alpha_{j}} \\ge 0$. \nFirst, let us assume that \n$l$ does not appear in $\\min S(xw_{0},\\,\\mathbf{i})$.\nThen, we have $\\yi{l-1}=s_{\\bti{l}}\\yi{l}$ by definition, \nand hence $\\mu_{\\wi{l-1}}=s_{\\bti{l}} \\cdot \\mu_{\\wi{l}}$ \nby Theorem~\\ref{thm:GGMS-Ext}. Also, \nit follows from the length formula \\eqref{eq:length} \n(or \\eqref{eq:n}) that \n$\\mu_{\\wi{l}}-\\mu_{\\wi{l-1}} \\in \n \\mathbb{Z}_{\\ge 0}(\\wi{l-1} \\cdot h_{i_{l}})=\n \\mathbb{Z}_{\\ge 0}(\\bti{l})^{\\vee}$, where \n$(\\bti{l})^{\\vee}$ denotes the coroot corresponding to \nthe root $\\bti{l}$. \nCombining these, we obtain \n\\begin{equation*}\n\\mathbb{Z}_{\\ge 0}(\\bti{l})^{\\vee} \\ni \n\\mu_{\\wi{l}}-\\mu_{\\wi{l-1}}=\n\\mu_{\\wi{l}}-s_{\\bti{l}} \\cdot \\mu_{\\wi{l}}=\n\\pair{\\mu_{\\wi{l}}}{\\bti{l}}(\\bti{l})^{\\vee}, \n\\end{equation*}\nand hence $\\pair{\\mu_{\\wi{l}}}{\\bti{l}} \\ge 0$. \nThis implies $\\pair{\\mu_{ws_{j}}}{w \\cdot \\alpha_{j}} \\ge 0$ \nsince $\\wi{l}=ws_{j}$ and $\\bti{l}=w \\cdot \\alpha_{j}$. \n\nNext, let us assume that \n$l$ appears in $\\min S(xw_{0},\\,\\mathbf{i})$. \nBecause $\\mu_{\\wi{l}}=\\yi{l} \\cdot \\lambda=\n\\wi{l}\\vi{l}w_{0}^{-1} \\cdot \\lambda$ \nby Theorem~\\ref{thm:GGMS-Ext} and Remark~\\ref{rem:vi}, \nwe see, by noting $\\wi{l}=ws_{j}$ and $i_{l}=j$, that \n\\begin{align*}\n\\pair{\\mu_{ws_{j}}}{w \\cdot \\alpha_{j}} & =\n\\pair{\\mu_{\\wi{l}}}{w \\cdot \\alpha_{j}} =\n\\pair{\\wi{l}\\vi{l}w_{0}^{-1} \\cdot \\lambda}{w \\cdot \\alpha_{j}} \\\\\n& =\n \\pair{ws_{j}\\vi{l}w_{0}^{-1} \\cdot \\lambda}{w \\cdot \\alpha_{j}}=\n -\\pair{\\vi{l}w_{0}^{-1} \\cdot \\lambda}{\\alpha_{j}} \\\\\n&= -\\pair{\\lambda}{ w_{0}(\\vi{l})^{-1} \\cdot \\alpha_{j} }\n = -\\pair{\\lambda}{ w_{0}(\\vi{l})^{-1} \\cdot \\alpha_{i_{l}} }.\n\\end{align*}\nAlso, since $l$ appears in $\\min S(xw_{0},\\,\\mathbf{i})$ by assumption, \nwe have \n$\\vi{l-1}=s_{i_{l}}\\vi{l}$ with $\\ell(\\vi{l-1})=\\ell(\\vi{l})+1$ \n(see Remark~\\ref{rem:vi}). \nIt follows from the exchange condition that \n$(\\vi{l})^{-1} \\cdot \\alpha_{i_{l}}$ is a positive root, \nand hence $w_{0}(\\vi{l})^{-1} \\cdot \\alpha_{i_{l}}$ \nis a negative root. Therefore, we conclude that \n\\begin{equation*}\n\\pair{\\mu_{ws_{j}}}{w \\cdot \\alpha_{j}}=\n- \\underbrace{\\pair{\\lambda}\n { w_{0}(\\vi{l})^{-1} \\cdot \\alpha_{i_{l}} }}_{\\le 0} \\ge 0\n\\end{equation*}\nsince $\\lambda \\in X_{*}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ is \na dominant coweight, This proves the lemma. \n\\end{proof}\n\nLet $G$ be a complex, connected, semisimple algebraic group \nwith Lie algebra $\\mathfrak{g}$. \nTake $\\lambda \\in X_{*}(T)_{+}$ and $x \\in W$ arbitrarily, and \nlet $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ denote the GGMS datum of \nthe extremal MV polytope $P_{x \\cdot \\lambda} \\in \\mathcal{MV}(\\lambda)$ of \nweight $x \\cdot \\lambda$, i.e., \n$P_{x \\cdot \\lambda}=P(\\mu_{\\bullet})$; \nrecall from Theorem~\\ref{thm:GGMS-Ext} that \n$\\mu_{w} \\in W \\cdot \\lambda$ for all $w \\in W$. \nNow, for each $w \\in W$, we consider the irreducible \nalgebraic variety \n\\begin{equation} \\label{eq:bbw}\n\\mathbf{b}^{w} := \\ol{\\CG r^{\\lambda} \\cap S_{\\mu_{w}}^{w}},\n\\end{equation}\nwhich is the $\\dot{w}$-translate of \nthe extremal MV cycle $\\mathbf{b}_{w^{-1} \\cdot \\mu_{w}}$ \nof weight $w^{-1} \\cdot \\mu_{w}$ \nsince $\\CG r^{w^{-1} \\cdot \\lambda}=\\CG r^{\\lambda}$ \n(see Remark~\\ref{rem:extcyc}); \nnote that $\\mathbf{b}^{e} = \\mathbf{b}_{x \\cdot \\lambda}$ \nsince $\\mu_{e}=x \\cdot \\lambda$. \n\nFor each $j \\in I$, \nwe set $P_{j} := B \\sqcup (B \\dot{s_{j}} B)$, \nwhich is the minimal parabolic subgroup \n(containing $B$) of $G$ corresponding to $s_{j} \\in W$. \nAlso, let $P_{j} = L_{j} U_{j}$ be its Levi decomposition \nsuch that $T \\subset L_{j}$. \n\\begin{lem} \\label{lem:tran}\nKeep the notation above. \nFor each $w \\in W$ and $j \\in I$ with $ws_{j} < w$, \nwe have ${}^{w} L_{j}(\\mathcal{O}) \\mathbf{b}^{ws_{j}} \\subset \\mathbf{b}^{w}$.\n\\end{lem}\n\n\\begin{proof}\nFor simplicity of notation, we write \n$N_{+}$ for ${}^{w}(L_{j} \\cap U)$; \nthe root in $N_{+}$ is $-w \\cdot \\alpha_{j}$ \nby our convention. \nBecause $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ is \nthe GGMS datum of the extremal MV polytope $P_{x \\cdot \\lambda}$, \nit follows from Lemma~\\ref{lem:ext1} that \nwe have either (a) $\\mu_{ws_{j}}=\\mu_{w}$ or \n(b) $\\mu_{ws_{j}}=ws_{j}w^{-1} \\cdot \\mu_{w}$, and that \nin both of the cases (a) and (b), we have\n$\\pair{\\mu_{w}}{w \\cdot \\alpha_{j}} \\le 0$. \nConsequently, by taking into account \nLemma~\\ref{lem:emb}, applied to ${}^{w}L_{j} \\subset G$ \nand $\\mu_{w} \\in X_{*}(T)$, we deduce from Example~\\ref{ex:MVc} \nthat\n\\begin{equation} \\label{eq:tran1}\n\\ol{N_{+}(\\mathcal{O}) [t^{\\mu_{w}}]} = \n\\ol{{}^{w} L_{j}(\\mathcal{O}) [t^{\\mu_{w}}]}.\n\\end{equation}\nAlso, by Remark~\\ref{rem:extcyc}, applied to \nthe extremal MV cycle $\\dot{w}^{-1} \\cdot \\mathbf{b}^{w}$ of \nweight $w^{-1} \\cdot \\mu_{w}$, we obtain \n$\\dot{w}^{-1} \\cdot \\mathbf{b}^{w}=\n\\ol{U(\\mathcal{O})[t^{w^{-1} \\cdot \\mu_{w}}]}$, \nand hence\n\\begin{equation} \\label{eq:tran2}\n\\mathbf{b}^{w}=\\ol{{}^{w}U(\\mathcal{O})[t^{\\mu_{w}}]}.\n\\end{equation}\nSimilarly, we obtain\n\\begin{equation} \\label{eq:tran3}\n\\mathbf{b}^{ws_{j}}=\\ol{{}^{ws_{j}}U(\\mathcal{O})[t^{\\mu_{ws_{j}}}]}.\n\\end{equation}\nHere we note that \n${}^{w}L_{j} {}^{ws_{j}}U = {}^{w}U {}^{w}L_{j}$ \nsince $L_{j}U=UL_{j}$ and $\\dot{s}_{j} \\in L_{j}$.\nIt follows that \n\\begin{equation*}\n{}^{w}L_{j}(\\mathcal{O}) {}^{ws_{j}}U(\\mathcal{O}) [t^{\\mu_{ws_{j}}}]\n= {}^{w}U(\\mathcal{O}) {}^{w}L_{j}(\\mathcal{O})  [t^{\\mu_{ws_{j}}}].\n\\end{equation*}\nBecause \n$[t^{\\mu_{ws_{j}}}] = [t^{\\mu_{w}}]$ in case (a), and \n$[t^{\\mu_{ws_{j}}}] = \\dot{w} \\dot{s}_{j} \\dot{w}^{-1} [t^{\\mu_{w}}]$ \nin case (b) as above, we deduce that in both of the cases (a) and (b), \n\\begin{equation*}\n{}^{w}U(\\mathcal{O}) {}^{w}L_{j}(\\mathcal{O})  [t^{\\mu_{ws_{j}}}]=\n{}^{w}U(\\mathcal{O}) {}^{w}L_{j}(\\mathcal{O})  [t^{\\mu_{w}}].\n\\end{equation*}\nThus, we get\n\\begin{equation*}\n{}^{w}L_{j}(\\mathcal{O}) {}^{ws_{j}}U(\\mathcal{O}) [t^{\\mu_{ws_{j}}}]=\n{}^{w}U(\\mathcal{O}) {}^{w}L_{j}(\\mathcal{O})  [t^{\\mu_{w}}].\n\\end{equation*}\nIn addition, we have \n\\begin{align*}\n{}^{w}U(\\mathcal{O}) {}^{w}L_{j}(\\mathcal{O})  [t^{\\mu_{w}}]\n& \\subset {}^{w}U(\\mathcal{O}) \\ol{{}^{w}L_{j}(\\mathcal{O})  [t^{\\mu_{w}}]}\n  = {}^{w}U(\\mathcal{O}) \\ol{N_{+}(\\mathcal{O}) [t^{\\mu_{w}}]}\n  \\quad \n  \\text{by \\eqref{eq:tran1}} \\\\\n& \\subset {}^{w}U(\\mathcal{O}) \\ol{{}^{w}U(\\mathcal{O}) [t^{\\mu_{w}}]}\n  \\quad\n  \\text{since $N_{+} \\subset {}^{w}U$ by definition} \\\\\n& \\subset \\ol{ {}^{w}U(\\mathcal{O}){}^{w}U(\\mathcal{O}) [t^{\\mu_{w}}]} \\\\\n& = \\ol{{}^{w}U(\\mathcal{O}) [t^{\\mu_{w}}]}=\\mathbf{b}^{w}\n  \\quad \\text{by \\eqref{eq:tran2}}.\n\\end{align*}\nHence we obtain \n${}^{w}L_{j}(\\mathcal{O}) \n {}^{ws_{j}}U(\\mathcal{O}) [t^{\\mu_{ws_{j}}}] \\subset \\mathbf{b}^{w}$. \nFrom this, we conclude, by using \\eqref{eq:tran3}, that\n\\begin{align*}\n{}^{w} L_{j}(\\mathcal{O}) \\mathbf{b}^{ws_{j}} & =\n{}^{w}L_{j}(\\mathcal{O}) \\ol{{}^{ws_{j}}U(\\mathcal{O}) [t^{\\mu_{ws_{j}}}]} \\\\\n& \\subset \n\\ol{{}^{w}L_{j}(\\mathcal{O}) {}^{ws_{j}}U(\\mathcal{O}) [t^{\\mu_{ws_{j}}}]} \\\\\n& \\subset \\mathbf{b}^{w}.\n\\end{align*}\nThis proves the lemma.\n\\end{proof}\n\n\\section{$N$-multiple maps for MV polytopes and their applications.}\n\\label{sec:multi}\n\nAs in (the second paragraph of) \\S\\ref{subsec:notation}, \nwe assume that $\\mathfrak{g}$ is a complex semisimple Lie algebra. \nLet $\\lambda \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ be \nan arbitrary (but fixed) dominant coweight. \n\n\\subsection{$N$-multiple maps for MV polytopes.}\n\\label{subsec:multiple}\n\nLet $N \\in \\mathbb{Z}_{\\ge 1}$. \nFor a collection $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ of \nelements of $\\mathfrak{h}_{\\mathbb{R}}$, we set\n\\begin{equation*}\nN \\cdot \\mu_{\\bullet}:=(N\\mu_{w})_{w \\in W}.\n\\end{equation*}\nAlso, for a subset $P \\subset \\mathfrak{h}_{\\mathbb{R}}$, we set \n\\begin{equation*}\nN \\cdot P:=\\bigl\\{Nv \\mid v \\in P\\bigr\\} \\subset \\mathfrak{h}_{\\mathbb{R}}.\n\\end{equation*}\nThe next lemma follows immediately \nfrom the definitions. \n\\begin{lem} \\label{lem:N1}\nLet $N \\in \\mathbb{Z}_{\\ge 1}$ be a positive integer. \n\n{\\rm (1)} If $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ \nis a GGMS datum, then $N \\cdot \\mu_{\\bullet}$ is \nalso a GGMS datum.\n\n{\\rm (2)} If $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ \nis an MV datum, then $N \\cdot \\mu_{\\bullet}$ is \nalso an MV datum.\n\n{\\rm (3)} Let $P=P(\\mu_{\\bullet}) \\in \\mathcal{MV}$ be an MV polytope \nwith GGMS datum $\\mu_{\\bullet}$. Then, \n$N \\cdot P$ is the MV polytope\nwith GGMS datum $N \\cdot \\mu_{\\bullet}$, that is, \n$N \\cdot P=P(N \\cdot \\mu_{\\bullet})$. Moreover, \nif $P \\in \\mathcal{MV}(\\lambda)$, \nthen $N \\cdot P \\in \\mathcal{MV}(N\\lambda)$. \n\\end{lem}\n\\begin{rem} \\label{rem:N1}\nLet $N \\in \\mathbb{Z}_{\\ge 1}$ be a positive integer. \nIf $P=P(\\mu_{\\bullet})$ is a pseudo-Weyl polytope \nwith GGMS datum $\\mu_{\\bullet}$, then \nthe set $N \\cdot P$ is identical to the Minkowski sum \n$P+P+ \\cdots +P$ ($N$ times). Indeed, we see that \n\\begin{align*}\nN \\cdot P & = \n P(N \\cdot \\mu_{\\bullet}) \\quad \\text{by Lemma~\\ref{lem:N1}\\,(3)} \\\\\n& = \n P(\\underbrace{\\mu_{\\bullet}+\\mu_{\\bullet}+ \\cdots +\\mu_{\\bullet}}_{\\text{$N$ times}})\n \\quad \\text{(see Remark~\\ref{rem:GGMS-sum})} \\\\[3mm]\n& \n =\\underbrace{P(\\mu_{\\bullet})+P(\\mu_{\\bullet})+ \\cdots +P(\\mu_{\\bullet})}_{\\text{$N$ times}}\n \\quad \\text{by Proposition~\\ref{prop:Minkowski}} \\\\[3mm]\n& = \\underbrace{P+P+ \\cdots +P}_{\\text{$N$ times}}. \n\\end{align*}\n\\end{rem}\n\nBy Lemma~\\ref{lem:N1}\\,(3), \nwe obtain an injective map \n$S_{N}:\\mathcal{MV}(\\lambda) \\hookrightarrow \\mathcal{MV}(N\\lambda)$ \nthat sends $P \\in \\mathcal{MV}(\\lambda)$ to $N \\cdot P \\in \\mathcal{MV}(N\\lambda)$; \nwe call the map $S_{N}$ an $N$-multiple map. \nNote that $S_{N}(P_{\\lambda})=P_{N\\lambda}$ \n(see Remark~\\ref{rem:ext}).\n\n\\begin{prop} \\label{prop:N2}\nLet $N \\in \\mathbb{Z}_{\\ge 1}$. \nFor $P \\in \\mathcal{MV}(\\lambda)$, we have\n\\begin{align*}\n& \\mathop{\\rm wt}\\nolimits (S_{N}(P))=N \\mathop{\\rm wt}\\nolimits (P), \\\\\n& \nS_{N}(e_{j}P)=e_{j}^{N}(S_{N}(P)), \\qquad \nS_{N}(f_{j}P)=f_{j}^{N}(S_{N}(P)) \\qquad \n\\text{\\rm for $j \\in I$}, \\\\\n& \n\\varepsilon_{j}(S_{N}(P))=N \\varepsilon_{j}(P), \\qquad \n\\varphi_{j}(S_{N}(P))=N \\varphi_{j}(P) \\qquad \n\\text{\\rm for $j \\in I$},\n\\end{align*}\nwhere it is understood \nthat $S_{N}(\\mathbf{0})=\\mathbf{0}$. \n\\end{prop}\n\\begin{proof}\nLet $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ be \nthe GGMS datum of $P \\in \\mathcal{MV}(\\lambda)$. \nIt follows from the definition of $\\mathop{\\rm wt}\\nolimits$ and Lemma~\\ref{lem:N1}\\,(3) \nthat $\\mathop{\\rm wt}\\nolimits(P)=\\mu_{e}$ and $\\mathop{\\rm wt}\\nolimits(S_{N}(P))=\\mathop{\\rm wt}\\nolimits(N \\cdot P)=N\\mu_{e}$. \nHence we have $\\mathop{\\rm wt}\\nolimits (S_{N}(P))=N \\mathop{\\rm wt}\\nolimits (P)$. \n\nNext, let us show that \n$\\varepsilon_{j}(S_{N}(P))=N \\varepsilon_{j}(P)$ and \n$\\varphi_{j}(S_{N}(P))=N \\varphi_{j}(P)$ for $j \\in I$. \nLet $j \\in I$. By Remark~\\ref{rem:ve}, we have\n$\\mu_{s_{j}}-\\mu_{e}=\\varepsilon_{j}(P)h_{j}$. Also, \nsince $S_{N}(P)=N \\cdot P=P(N \\cdot \\mu_{\\bullet})$ \nby Lemma~\\ref{lem:N1}\\,(3), \nit follows from Remark~\\ref{rem:ve} that\n$N\\mu_{s_{j}}-N\\mu_{e}=\n\\varepsilon_{j}(N \\cdot P)h_{j}=\n\\varepsilon_{j}(S_{N}(P))h_{j}$. \nCombining these equations, we have\n\\begin{equation*}\n\\varepsilon_{j}(S_{N}(P))h_{j}=\nN\\mu_{s_{j}}-N\\mu_{e}=\nN(\\mu_{s_{j}}-\\mu_{e})=\nN\\varepsilon_{j}(P)h_{j}, \n\\end{equation*}\nwhich implies that \n$\\varepsilon_{j}(S_{N}(P))=N \\varepsilon_{j}(P)$. \nIn addition, we have \n\\begin{align*}\n\\varphi_{j}(S_{N}(P)) \n& =\n \\pair{\\mathop{\\rm wt}\\nolimits (S_{N}(P))}{\\alpha_{j}}+\\varepsilon_{j}(S_{N}(P))\n \\qquad \\text{by \\eqref{eq:ax}} \\\\\n& =\\pair{N \\mathop{\\rm wt}\\nolimits (P)}{\\alpha_{j}}+N\\varepsilon_{j}(P)\n \\qquad \\text{by the equations shown above} \\\\\n& =N\\bigl(\\pair{\\mathop{\\rm wt}\\nolimits (P)}{\\alpha_{j}}+\\varepsilon_{j}(P)\\bigr) \\\\\n& =N\\varphi_{j}(P)\n \\qquad \\text{by \\eqref{eq:ax}}.\n\\end{align*}\n\nFinally, let us show that \n$S_{N}(e_{j}P)=e_{j}^{N}(S_{N}(P))$ and \n$S_{N}(f_{j}P)=f_{j}^{N}(S_{N}(P))$ for $j \\in I$; \nwe give a proof only for the equality \n$S_{N}(e_{j}P)=e_{j}^{N}(S_{N}(P))$ \nsince the equality $S_{N}(f_{j}P)=f_{j}^{N}(S_{N}(P))$ \ncan be shown similarly. Let $j \\in I$. \nFirst observe that \n\\begin{align*}\nS_{N}(e_{j} P)=\\mathbf{0}\n & \\quad \\Longleftrightarrow \\quad \n   e_{j} P=\\mathbf{0}\n \\quad \\text{by Remark~\\ref{rem:ve}} \n \\\\\n & \\quad \\Longleftrightarrow \\quad \n   \\varepsilon_{j}(P)=0\n \\quad \\text{by the definition of $\\varepsilon_{j}(P)$} \\\\\n & \\quad \\Longleftrightarrow \\quad \n   e_{j}^{N}(S_{N}(P))=\\mathbf{0}\n \\quad \\text{since $\\varepsilon_{j}(S_{N}(P))=N \\varepsilon_{j}(P)$}.\n\\end{align*}\nNow, assume that $S_{N}(e_{j} P) \\ne \\mathbf{0}$, \nor equivalently, $e_{j} P \\ne \\mathbf{0}$. Recall \nfrom the definition of the raising Kashiwara operator \n$e_{j}$ that the GGMS datum of \nthe MV polytope $e_{j} P \\in \\mathcal{MV}(\\lambda)$ is equal to \n$e_{j}\\mu_{\\bullet}=(\\mu_{w}')_{w \\in W}$, \nwhich is the unique MV datum \nsuch that $\\mu_{e}'=\\mu_{e}+h_{j}$ and \n$\\mu_{w}'=\\mu_{w}$ for all $w \\in W$ with $s_{j}w < w$. \nHence we see from Lemma~\\ref{lem:N1}\\,(3) that \nthe GGMS datum of the MV polytope \n$S_{N}(e_{j}P)=N \\cdot (e_{j} P) \\in \\mathcal{MV}(N\\lambda)$ is equal to \nthe unique MV datum $\\mu_{\\bullet}''=(\\mu_{w}'')_{w \\in W}$ \nsuch that $\\mu_{e}''=N\\mu_{e}+Nh_{j}$ and \n$\\mu_{w}''=N\\mu_{w}$ for all $w \\in W$ with $s_{j}w < w$. \nBecause the GGMS datum of the MV polytope \n$S_{N}(P)=N \\cdot P \\in \\mathcal{MV}(N\\lambda)$ \nis equal to $N \\cdot \\mu_{\\bullet}=(N\\mu_{w})_{w \\in W}$, \nwe deduce from the definition of \nthe raising Kashiwara operator $e_{j}$ that \nthe GGMS datum $\\mu_{\\bullet}'''=(\\mu_{w}''')_{w \\in W}$ of \n$e_{j}^{N}(S_{N}(P))=e_{j}^{N}(N \\cdot P) \\in \\mathcal{MV}(N\\lambda)$ \nalso satisfies the condition that \n$\\mu_{e}'''=N\\mu_{e}+Nh_{j}$ and \n$\\mu_{w}'''=N\\mu_{w}$ for all $w \\in W$ with $s_{j}w < w$. \nHence, by the uniqueness of such an MV datum, we obtain \n$\\mu_{\\bullet}''=\\mu_{\\bullet}'''$, which implies that \n$S_{N}(e_{j}P)=P(\\mu_{\\bullet}'')=\n P(\\mu_{\\bullet}''')=e_{j}^{N}(S_{N}(P))$.\nThis completes the proof of Proposition~\\ref{prop:N2}.\n\\end{proof}\n\n\\subsection{Application of $N$-multiple maps.}\n\\label{subsec:app-multiple}\n\nLet $\\lambda \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ be \nan arbitrary (but fixed) dominant coweight \nas in \\S\\ref{subsec:multiple}. \nWe denote by $W_{\\lambda} \\subset W$ \nthe stabilizer of $\\lambda$ in $W$, and \nby $W^{\\lambda}_{\\min} \\subset W$ \nthe set of minimal (length) coset representatives \nmodulo the subgroup $W_{\\lambda} \\subset W$. \n\nFor a positive integer $N \\in \\mathbb{Z}_{\\ge 1}$, \nlet us denote by \n\\begin{equation*}\nK_{N}: \\mathcal{MV}(\\lambda) \\hookrightarrow \n \\mathcal{MV}(\\lambda)^{\\otimes N}=\n \\underbrace{\n   \\mathcal{MV}(\\lambda) \\otimes \n   \\mathcal{MV}(\\lambda) \\otimes \\cdots \\otimes \n   \\mathcal{MV}(\\lambda)}_{\\text{$N$ times}}\n\\end{equation*}\nthe composite of the $N$-multiple map \n$S_{N}:\\mathcal{MV}(\\lambda) \\hookrightarrow \\mathcal{MV}(N\\lambda)$ with \nthe canonical embedding \n$G_{N}:\\mathcal{MV}(N\\lambda) \\hookrightarrow \\mathcal{MV}(\\lambda)^{\\otimes N}$ \nof crystals that sends the highest weight element \n$P_{N\\lambda}$ of $\\mathcal{MV}(N\\lambda)$ to \nthe highest weight element \n$P_{\\lambda}^{\\otimes N}=\n P_{\\lambda} \\otimes P_{\\lambda} \\otimes \\cdots \\otimes P_{\\lambda}$ \n($N$ times) of $\\mathcal{MV}(\\lambda)^{\\otimes N}$.\n\\begin{prop} \\label{prop:N3}\nLet $\\lambda \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ be \na dominant coweight, and \nlet $x \\in W^{\\lambda}_{\\min}$. \nIf an MV polytope $P \\in \\mathcal{MV}(\\lambda)$ lies \nin the Demazure crystal $\\mathcal{MV}_{x}(\\lambda)$, \nthen there exist a positive integer $N \\in \\mathbb{Z}_{\\ge 1}$ \nand minimal coset representatives \n$x_{1},\\,x_{2},\\,\\dots,\\,x_{N} \\in W^{\\lambda}_{\\min}$ \nsuch that \n\\begin{equation} \\label{eq:N3}\n\\begin{cases}\nx \\ge x_{k} \\quad \n \\text{\\rm for all $1 \\le k \\le N$;} \\\\[3mm]\nK_{N}(P)=\n P_{x_{1} \\cdot \\lambda} \\otimes \n P_{x_{2} \\cdot \\lambda} \\otimes \\cdots \\otimes \n P_{x_{N} \\cdot \\lambda}.\n\\end{cases}\n\\end{equation}\n\\end{prop}\n\n\n\\begin{rem} \\label{rem:MV-LS}\nKeep the notation and assumption above. \nA positive integer $N \\in \\mathbb{Z}_{\\ge 1}$ and \nminimal coset representatives \n$x_{1},\\,x_{2},\\,\\dots,\\,x_{N} \\in W^{\\lambda}_{\\min}$ \nsatisfying the conditions \\eqref{eq:N3}\nare, in a sense, determined by \nthe ``turning points'' and ``directions'' of \nthe Lakshmibai-Seshadri path of \nshape $\\lambda$ that corresponds to \nthe MV polytope $P \\in \\mathcal{MV}(\\lambda)$ \nunder the (inexplicit) bijection \nvia the crystal basis $\\mathcal{B}(\\lambda)$.\n\nWe will explain the remark above more precisely. \nLet $\\mathbb{B}(\\lambda)$ denote \nthe set of Lakshmibai-Seshadri (LS for short) paths\nof shape $\\lambda$, which is endowed with \na crystal structure for $U_{q}(\\mathfrak{g}^{\\vee})$ \nby the root operators $e_{j}$, $f_{j}$, $j \\in I$ \n(see \\cite{L-inv} and \\cite{L-ann} for details). \nWe know from \\cite[Theorem~4.1]{Kassim} \n(see also \\cite[Th\\'eor\\`eme~8.2.3]{Kasb}) and \n\\cite[Corollary~6.4.27]{Jo} that \n$\\mathbb{B}(\\lambda)$ is isomorphic to \nthe crystal basis $\\mathcal{B}(\\lambda)$ \nas a crystal for $U_{q}(\\mathfrak{g}^{\\vee})$.\nThus, we have $\\mathbb{B}(\\lambda) \\cong \n\\mathcal{B}(\\lambda) \\cong \\mathcal{MV}(\\lambda)$ \nas crystals for $U_{q}(\\mathfrak{g}^{\\vee})$. \n\nNow, take an element \n$P \\in \\mathcal{MV}_{x}(\\lambda) \\subset \\mathcal{MV}(\\lambda)$, \nand assume that a positive integer $N \\in \\mathbb{Z}_{\\ge 1}$ \nand elements $x_{1},\\,x_{2},\\,\\dots,\\,x_{N} \n\\in W^{\\lambda}_{\\min}$ satisfy \nconditions \\eqref{eq:N3}. Consider \na piecewise linear, continuous map \n$\\pi:[0,1] \\rightarrow \\mathfrak{h}_{\\mathbb{R}}$ by:\n\\begin{equation*}\n\\pi(t)=\n \\sum_{l=1}^{k-1} \\frac{1}{N}\\,x_{l} \\cdot \\lambda +\n \\left(t-\\frac{k-1}{N}\\right)x_{k} \\cdot \\lambda\n \\qquad\n \\text{for $t \\in \n \\left[\\frac{k-1}{N},\\,\\frac{k}{N}\\right]$, \\ \n $1 \\le k \\le N$};\n\\end{equation*}\nnote that for each $1 \\le k \\le N$, \nthe vector $x_{k} \\cdot \\lambda \\in W \\cdot \\lambda \\subset \\mathfrak{h}_{\\mathbb{R}}$ \ngives the direction of $\\pi$ on the interval \n$[(k-1)\/N,\\,k\/N]$, and the point $t=k\/N$ is \na turning point of $\\pi$ if \n$x_{k-1} \\cdot \\lambda \\ne x_{k} \\cdot \\lambda$. \nThen, we deduce from the proof of \n\\cite[Theorem~4.1]{Kassim} \n(or \\cite[Th\\'eor\\`eme~8.2.3]{Kasb}), together with\nthe commutative diagram \\eqref{CD:multi} below, that \nthis map $\\pi:[0,1] \\rightarrow \\mathfrak{h}_{\\mathbb{R}}$ is \nprecisely the LS path of shape $\\lambda$ \nthat corresponds to the $P \\in \\mathcal{MV}(\\lambda)$ under \nthe isomorphism $\\mathbb{B}(\\lambda) \\cong \n\\mathcal{B}(\\lambda) \\cong \\mathcal{MV}(\\lambda)$ of crystals. \nThe argument above implies, in particular, that \nfor a fixed positive integer $N \\in \\mathbb{Z}_{\\ge 1}$, \nthe elements $x_{1},\\,x_{2},\\,\\dots,\\,x_{N} \\in W^{\\lambda}_{\\min}$ \nare determined uniquely by the MV polytope $P$ via \nthe corresponding LS path. \nAlso note that by the definition of LS paths, \nif a positive integer $N \\in \\mathbb{Z}_{\\ge 1}$ \nand elements $x_{1},\\,x_{2},\\,\\dots,\\,x_{N} \n\\in W^{\\lambda}_{\\min}$ satisfy \nconditions \\eqref{eq:N3}, then \nwe necessarily have \n$x_{1} \\ge x_{2} \\ge \\cdots \\ge x_{N}$. \n\\end{rem}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:N3}]\nLet $N \\in \\mathbb{Z}_{\\ge 1}$. \nWe know from \\cite[Theorem~3.1]{Kassim} and \n\\cite[Corollarie~8.1.5]{Kasb} that there exists \nan injective map $S_{N}:\\mathcal{B}(\\lambda) \\hookrightarrow \\mathcal{B}(N\\lambda)$ \nwhich sends the highest weight element $u_{\\lambda} \\in \\mathcal{B}(\\lambda)$ to \nthe highest weight element $u_{N\\lambda} \\in \\mathcal{B}(N\\lambda)$, and \nwhich has the same properties as the $N$-multiple map \n$S_{N}:\\mathcal{MV}(\\lambda) \\hookrightarrow \\mathcal{MV}(N\\lambda)$ \ngiven in Proposition~\\ref{prop:N2}.\nLet us denote by \n$K_{N}: \\mathcal{B}(\\lambda) \\hookrightarrow \n \\mathcal{B}(\\lambda)^{\\otimes N}$\nthe composite of \n$S_{N}:\\mathcal{B}(\\lambda) \\hookrightarrow \\mathcal{B}(N\\lambda)$ \nwith the canonical embedding \n$G_{N}:\\mathcal{B}(N\\lambda) \\hookrightarrow \\mathcal{B}(\\lambda)^{\\otimes N}$ \nof crystals that sends the highest weight element \n$u_{N\\lambda} \\in \\mathcal{B}(N\\lambda)$ to \nthe highest weight element \n$u_{\\lambda}^{\\otimes N} \\in \\mathcal{B}(\\lambda)^{\\otimes N}$ \n(see \\cite[p.~181]{Kassim} and \\cite[\\S8.3]{Kasb}). \nThen, it is easily shown that the following diagram commutes:\n\\begin{equation} \\label{CD:multi}\n\\begin{CD}\n\\mathcal{B}(\\lambda) @>{K_{N}}>> \\mathcal{B}(\\lambda)^{\\otimes N} \\\\\n@V{\\Psi_{\\lambda}}VV @VV{\\Psi_{\\lambda}^{\\otimes N}}V  \\\\\n\\mathcal{MV}(\\lambda) @>{K_{N}}>> \\mathcal{MV}(\\lambda)^{\\otimes N}.\n\\end{CD}\n\\end{equation}\nIndeed, take an element $b \\in \\mathcal{B}(\\lambda)$, and write it as: \n$b=f_{j_{1}}f_{j_{2}} \\cdots f_{j_{r}}u_{\\lambda}$ \nfor some $j_{1},\\,j_{2},\\,\\dots,\\,j_{r} \\in I$; \nfor simplicity of notation, we set \n$f_{\\ast}:=f_{j_{1}}f_{j_{2}} \\cdots f_{j_{r}}$ and \n$f_{\\ast}^{N}:=f_{j_{1}}^{N}f_{j_{2}}^{N} \\cdots f_{j_{r}}^{N}$. \nWe have \n\\begin{align*}\nK_{N}(\\Psi_{\\lambda}(b))\n & = K_{N}(\\Psi_{\\lambda}(f_{\\ast}u_{\\lambda})) \n   = K_{N}(f_{\\ast} P_{\\lambda}) \n   = G_{N}(S_{N}(f_{\\ast} P_{\\lambda})) \\\\\n & = G_{N}(f_{\\ast}^{N} P_{N\\lambda})\n     \\quad \\text{by Proposition~\\ref{prop:N2}} \\\\\n & = f_{\\ast}^{N}P_{\\lambda}^{\\otimes N}.\n\\end{align*}\nSimilarly, we have \n\\begin{align*}\n\\Psi_{\\lambda}^{\\otimes N}(K_{N}(b))\n & = \\Psi_{\\lambda}^{\\otimes N}(K_{N}(f_{\\ast}u_{\\lambda}))\n   = \\Psi_{\\lambda}^{\\otimes N}\\bigl(G_{N}(S_{N}(f_{\\ast} u_{\\lambda}))\\bigr) \\\\\n & = \\Psi_{\\lambda}^{\\otimes N}\\bigl(G_{N}(f_{\\ast}^{N} u_{N\\lambda})\\bigr)\n   = \\Psi_{\\lambda}^{\\otimes N}(f_{\\ast}^{N} u_{\\lambda}^{\\otimes N})\n   = f_{\\ast}^{N} P_{\\lambda}^{\\otimes N}.\n\\end{align*}\nThus, we obtain $K_{N}(\\Psi_{\\lambda}(b))=\n\\Psi_{\\lambda}^{\\otimes N}(K_{N}(b))$ \nfor all $b \\in \\mathcal{B}(\\lambda)$. \n\nNow, let $P \\in \\mathcal{MV}(\\lambda)$. \nApplying \\cite[Proposition~8.3.2]{Kasb} to \n$\\Psi_{\\lambda}^{-1}(P) \\in \\mathcal{B}(\\lambda)$, we see that \nif $N \\in \\mathbb{Z}_{\\ge 1}$ contains ``sufficiently many'' divisors, \nthen \n\\begin{equation} \\label{eq:c1}\nK_{N}(\\Psi_{\\lambda}^{-1}(P))=\n u_{x_{1} \\cdot \\lambda} \\otimes \n u_{x_{2} \\cdot \\lambda} \\otimes \\cdots \\otimes \n u_{x_{N} \\cdot \\lambda}\n\\end{equation}\nfor some $x_{1},\\,x_{2},\\,\\dots,\\,x_{N} \\in W^{\\lambda}_{\\min}$. \nThen, we have \n\\begin{align*}\nK_{N}(P) & = \n \\Psi_{\\lambda}^{\\otimes N}\\bigl(K_{N}(\\Psi_{\\lambda}^{-1}(P))\\bigr)\n \\quad \\text{by \\eqref{CD:multi}} \\\\\n& =\n \\Psi_{\\lambda}^{\\otimes N}(u_{x_{1} \\cdot \\lambda} \\otimes \n u_{x_{2} \\cdot \\lambda} \\otimes \\cdots \\otimes \n u_{x_{N} \\cdot \\lambda}) \\\\\n& = \n P_{x_{1} \\cdot \\lambda} \\otimes \n P_{x_{2} \\cdot \\lambda} \\otimes \\cdots \\otimes \n P_{x_{N} \\cdot \\lambda}. \n\\end{align*}\nIt remains to show that \n$x \\ge x_{k}$ for every $1 \\le k \\le N$. \nIn view of \\cite[Proposition~4.4]{Kas} \n(see also \\cite[\\S9.1]{Kasb}), \nit suffices to show that \n$u_{x_{k} \\cdot \\lambda} \\in \\mathcal{B}_{x}(\\lambda)$ \n for every $1 \\le k \\le N$. \nLet $x=s_{j_{1}}s_{j_{2}} \\cdots s_{j_{r}}$ \nbe a reduced expression of $x \\in W$. \nWe know from \\cite[Proposition~9.1.3\\,(2)]{Kasb} that \n\\begin{equation} \\label{eq:dem-f}\n\\mathcal{B}_{x}(\\lambda)=\n \\bigl\\{\n   f_{j_{1}}^{c_{1}}\n   f_{j_{2}}^{c_{2}} \\cdots \n   f_{j_{r}}^{c_{r}}u_{\\lambda} \\mid \n c_{1},\\,c_{2},\\,\\dots,\\,c_{r} \\in \\mathbb{Z}_{\\ge 0}\n \\bigr\\} \\setminus \\{0\\}.\n\\end{equation}\nSince $P \\in \\mathcal{MV}_{x}(\\lambda)$ by our assumption, \n$\\Psi_{\\lambda}^{-1}(P)$ is contained in $\\mathcal{B}_{x}(\\lambda)$, \nand hence $\\Psi_{\\lambda}^{-1}(P)$ can be written as:\n\\begin{equation*}\n\\Psi_{\\lambda}^{-1}(P)=\n   f_{j_{1}}^{c_{1}}\n   f_{j_{2}}^{c_{2}} \\cdots \n   f_{j_{r}}^{c_{r}}u_{\\lambda}\n\\quad \n\\text{for some \n$c_{1},\\,c_{2},\\,\\dots,\\,c_{r} \\in \\mathbb{Z}_{\\ge 0}$}.\n\\end{equation*}\nTherefore, we have \n\\begin{align*}\nK_{N}(\\Psi_{\\lambda}^{-1}(P)) & = \n K_{N}(f_{j_{1}}^{c_{1}}\n   f_{j_{2}}^{c_{2}} \\cdots \n   f_{j_{r}}^{c_{r}}u_{\\lambda})=\n G_{N}(S_{N}(f_{j_{1}}^{c_{1}}\n   f_{j_{2}}^{c_{2}} \\cdots \n   f_{j_{r}}^{c_{r}}u_{\\lambda})) \\\\\n& = G_{N}(f_{j_{1}}^{Nc_{1}}\n   f_{j_{2}}^{Nc_{2}} \\cdots \n   f_{j_{r}}^{Nc_{r}}u_{N\\lambda}) \n  =f_{j_{1}}^{Nc_{1}}\n   f_{j_{2}}^{Nc_{2}} \\cdots \n   f_{j_{r}}^{Nc_{r}}u_{\\lambda}^{\\otimes N}. \n\\end{align*}\nIt follows from the tensor product rule for crystals that \n\\begin{align*}\n& f_{j_{1}}^{Nc_{1}}\n  f_{j_{2}}^{Nc_{2}} \\cdots \n  f_{j_{r}}^{Nc_{r}}u_{\\lambda}^{\\otimes N} = \\\\\n& \n  \\bigl(f_{j_{1}}^{b_{1,1}}\n  f_{j_{2}}^{b_{1,2}} \\cdots \n  f_{j_{r}}^{b_{1,r}}u_{\\lambda}\\bigr) \\otimes\n  \\bigl(f_{j_{1}}^{b_{2,1}}\n  f_{j_{2}}^{b_{2,2}} \\cdots \n  f_{j_{r}}^{b_{2,r}}u_{\\lambda}\\bigr) \\otimes \\cdots \\otimes\n  \\bigl(f_{j_{1}}^{b_{N,1}}\n  f_{j_{2}}^{b_{N,2}} \\cdots \n  f_{j_{r}}^{b_{N,r}}u_{\\lambda}\\bigr)\n\\end{align*}\nfor some $b_{k,t} \\in \\mathbb{Z}_{\\ge 0}$, \n$1 \\le k \\le N$, $1 \\le t \\le r$, \nwith $\\sum_{k=1}^{N}b_{k,t}=Nc_{t}$ \nfor each $1 \\le t \\le r$.\nCombining these equalities with \\eqref{eq:c1}, \nwe obtain\n\\begin{align*}\n& u_{x_{1} \\cdot \\lambda} \\otimes \n  u_{x_{2} \\cdot \\lambda} \\otimes \\cdots \\otimes \n  u_{x_{N} \\cdot \\lambda}=K_{N}(\\Psi_{\\lambda}^{-1}(P))=\\\\\n& \n  \\bigl(f_{j_{1}}^{b_{1,1}}\n  f_{j_{2}}^{b_{1,2}} \\cdots \n  f_{j_{r}}^{b_{1,r}}u_{\\lambda}\\bigr) \\otimes\n  \\bigl(f_{j_{1}}^{b_{2,1}}\n  f_{j_{2}}^{b_{2,2}} \\cdots \n  f_{j_{r}}^{b_{2,r}}u_{\\lambda}\\bigr) \\otimes \\cdots \\otimes\n  \\bigl(f_{j_{1}}^{b_{N,1}}\n  f_{j_{2}}^{b_{N,2}} \\cdots \n  f_{j_{r}}^{b_{N,r}}u_{\\lambda}\\bigr),\n\\end{align*}\nfrom which it follows that \n$u_{x_{k} \\cdot \\lambda}=\n  f_{j_{1}}^{b_{k,1}}\n  f_{j_{2}}^{b_{k,2}} \\cdots \n  f_{j_{r}}^{b_{k,r}}u_{\\lambda}$ \nfor $1 \\le k \\le N$. \nThis implies that \n$u_{x_{k} \\cdot \\lambda} \\in \\mathcal{B}_{x}(\\lambda)$ \nfor each $1 \\le k \\le N$ since \n$f_{j_{1}}^{b_{k,1}}\n f_{j_{2}}^{b_{k,2}} \\cdots \n f_{j_{r}}^{b_{k,r}}u_{\\lambda} \\in \\mathcal{B}_{x}(\\lambda)$ \nby \\eqref{eq:dem-f}.\nThus, we have proved Proposition~\\ref{prop:N3}. \n\\end{proof}\n\\subsection{Main results.}\n\\label{subsec:main}\n\nIn this subsection, we prove the following theorem, \nby using a polytopal estimate \n(Theorem~\\ref{thm:tensor} below) of tensor products of \nMV polytopes. We use the setting of \\S\\ref{subsec:app-multiple}.\n\\begin{thm} \\label{thm:main}\nLet $\\lambda \\in X_{*}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ be a dominant coweight, \nand let $x \\in W^{\\lambda}_{\\min}$. \nIf a positive integer $N \\in \\mathbb{Z}_{\\ge 1}$ and \nminimal coset representatives \n$x_{1},\\,x_{2},\\,\\dots,\\,x_{N} \\in W^{\\lambda}_{\\min}$ \nsatisfy the condition \\eqref{eq:N3} \nin Proposition~\\ref{prop:N3}, then \n\\begin{equation*}\nN \\cdot P \\subseteq\nP_{x_1 \\cdot \\lambda} + \nP_{x_2 \\cdot \\lambda} + \\cdots + \nP_{x_N \\cdot \\lambda},\n\\end{equation*}\nwhere \n$P_{x_1 \\cdot \\lambda} + P_{x_2 \\cdot \\lambda} + \\cdots + \n P_{x_N \\cdot \\lambda}$ is the Minkowski sum of \nthe extremal MV polytopes $P_{x_1 \\cdot \\lambda}$, \n$P_{x_2 \\cdot \\lambda}$, $\\dots$, $P_{x_N \\cdot \\lambda}$.\n\\end{thm}\n\n\\begin{proof}\nBy our assumption, we have \n\\begin{equation*}\n\\begin{cases}\nx \\ge x_{k} \\quad \n \\text{\\rm for all $1 \\le k \\le N$;} \\\\[3mm]\nK_{N}(P)=G_{N}(N \\cdot P)=\n P_{x_{1} \\cdot \\lambda} \\otimes \n P_{x_{2} \\cdot \\lambda} \\otimes \\cdots \\otimes \n P_{x_{N} \\cdot \\lambda}.\n\\end{cases}\n\\end{equation*}\nHere we recall from \\S\\ref{subsec:app-multiple} \nthat $G_{N}:\\mathcal{MV}(N\\lambda) \\hookrightarrow \\mathcal{MV}(\\lambda)^{\\otimes N}$ \ndenotes the canonical embedding of crystals that \nsends $P_{N\\lambda} \\in \\mathcal{MV}(N\\lambda)$ to \n$P_{\\lambda}^{\\otimes N} \\in \\mathcal{MV}(\\lambda)^{\\otimes N}$.\nTherefore, by using Theorem~\\ref{thm:tensor} \n(or rather, Corollary~\\ref{cor:tensor}) successively, \nwe can show that\n\\begin{equation*}\nN \\cdot P \\subset \n P_{x_{1} \\cdot \\lambda}+\n P_{x_{2} \\cdot \\lambda}+\\cdots+\n P_{x_{N} \\cdot \\lambda}.\n\\end{equation*}\nThis completes the proof of Theorem~\\ref{thm:main}.\n\\end{proof}\n\nThis theorem, together with Proposition~\\ref{prop:N3}, yields \nTheorem~\\ref{ithm1} in the Introduction. \nAs an immediate consequence, \nwe can provide an affirmative answer to \na question posed in \\cite[\\S4.6]{NS-dp}.\n\\begin{cor} \\label{cor:main}\nLet $\\lambda \\in X_{*}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ be a dominant coweight, \nand let $x \\in W$. \nAll the MV polytopes lying in the Demazure crystal \n$\\mathcal{MV}_{x}(\\lambda)$ are contained (as sets) in \nthe extremal MV polytope $P_{x \\cdot \\lambda}=\n \\mathop{\\rm Conv}\\nolimits(W_{\\le x} \\cdot \\lambda)$ of weight $x \\cdot \\lambda$. \nNamely, for all $P \\in \\mathcal{MV}_{x}(\\lambda)$, there holds \n\\begin{equation*}\nP \\subset P_{x \\cdot \\lambda}=\n \\mathop{\\rm Conv}\\nolimits(W_{\\le x} \\cdot \\lambda).\n\\end{equation*}\n\\end{cor}\n\n\\begin{rem}\n(1) The assertion of Theorem~\\ref{thm:main} \nis not obvious, as explained in \n\\cite[Remark~4.6.2 and Example~4.6.3]{NS-dp}.\n\n(2) The converse statement fails to hold; \nsee \\cite[Remark~4.6.1]{NS-dp}.\n\\end{rem}\n\n\\begin{proof}[Proof of Corollary~\\ref{cor:main}]\nWe know from Remark~\\ref{rem:demcos} that \nif $x,\\,y \\in W$ satisfies \n$x \\cdot \\lambda=y \\cdot \\lambda$, \nthen $\\mathcal{MV}_{x}(\\lambda)=\\mathcal{MV}_{y}(\\lambda)$ and \n$P_{x \\cdot \\lambda}=P_{y \\cdot \\lambda}$. \nHence we may assume that $x \\in W^{\\lambda}_{\\min}$. \nLet us take an arbitrary $P \\in \\mathcal{MV}_{x}(\\lambda)$. \nBy Proposition~\\ref{prop:N3}, \nthere exist $N \\in \\mathbb{Z}_{\\ge 1}$ and \n$x_{1},\\,x_{2},\\dots,\\,x_{N} \\in W^{\\lambda}_{\\min}$ \nsatisfying the condition \\eqref{eq:N3}. \nAlso, for each $1 \\le k \\le N$, \nit follows from Proposition~\\ref{prop:ext} and \nthe inequality $x \\ge x_{k}$ that \n\\begin{equation} \\label{eq:ext-inc02}\nP_{x_{k} \\cdot \\lambda}=\n \\mathop{\\rm Conv}\\nolimits(W_{\\le x_{k}} \\cdot \\lambda) \\subset\n \\mathop{\\rm Conv}\\nolimits(W_{\\le x} \\cdot \\lambda)=P_{x \\cdot \\lambda}. \n\\end{equation}\nTherefore, we have \n\\begin{align*}\nN \\cdot P \n & \\subset  \n   P_{x_{1} \\cdot \\lambda}+\n   P_{x_{2} \\cdot \\lambda}+\\cdots+\n   P_{x_{N} \\cdot \\lambda} \n   \\quad \\text{by Theorem~\\ref{thm:main}} \\\\\n & \\subset \n   \\underbrace{\n    P_{x \\cdot \\lambda}+\n    P_{x \\cdot \\lambda}+\\cdots+\n    P_{x \\cdot \\lambda}\n   }_{\\text{$N$ times}} \n   \\quad \\text{by \\eqref{eq:ext-inc02}} \\\\[3mm]\n & =N \\cdot P_{x \\cdot \\lambda}\n   \\quad \\text{by Remark~\\ref{rem:N1}}.\n\\end{align*}\nConsequently, we obtain \n$N \\cdot P \\subset N \\cdot P_{x \\cdot \\lambda}$, \nwhich implies that $P \\subset P_{x \\cdot \\lambda}$. \nThis proves Corollary~\\ref{cor:main}. \n\\end{proof}\n\n\n\\section{Polytopal estimate of tensor products of MV polytopes.}\n\\label{sec:tensor}\n\nThe aim of this section is to state and prove \na polytopal estimate of tensor products of MV polytopes.\n\\subsection{Polytopal estimate.}\n\\label{subsec:tensor}\n\nAs in (the second paragraph of) \\S\\ref{subsec:notation}, \nwe assume that $\\mathfrak{g}$ is a complex semisimple Lie algebra. \nLet $\\lambda_{1},\\,\\lambda_{2} \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ \nbe dominant coweights. \nBecause $\\mathcal{MV}(\\lambda) \\cong \\mathcal{B}(\\lambda)$ as crystals \nfor every dominant coweight $\\lambda \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$, \nthe tensor product \n$\\mathcal{MV}(\\lambda_{2}) \\otimes \\mathcal{MV}(\\lambda_{1})$ of \nthe crystals $\\mathcal{MV}(\\lambda_{1})$ and $\\mathcal{MV}(\\lambda_{2})$ \ndecomposes into a disjoint union of connected components \nas follows:\n\\begin{equation*}\n\\mathcal{MV}(\\lambda_{2}) \\otimes \\mathcal{MV}(\\lambda_{1}) \\cong \n \\bigoplus_{\n   \\begin{subarray}{c}\n   \\lambda \\in X_{\\ast}(T) \\\\[0.5mm]\n   \\text{$\\lambda$ : dominant}\n   \\end{subarray}\n   } \n\\mathcal{MV}(\\lambda)^{\\oplus m_{\\lambda_{1},\\lambda_{2}}^{\\lambda}},\n\\end{equation*}\nwhere $m_{\\lambda_{1},\\lambda_{2}}^{\\lambda} \\in \\mathbb{Z}_{\\ge 0}$ denotes \nthe multiplicity of $\\mathcal{MV}(\\lambda)$ in \n$\\mathcal{MV}(\\lambda_{2}) \\otimes \\mathcal{MV}(\\lambda_{1})$. \nFor each dominant coweight $\\lambda \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ \nsuch that $m_{\\lambda_{1},\\lambda_{2}}^{\\lambda} \\ge 1$, \nwe take (and fix) an arbitrary embedding \n$\\iota_{\\lambda}:\\mathcal{MV}(\\lambda) \\hookrightarrow \n \\mathcal{MV}(\\lambda_{2}) \\otimes \\mathcal{MV}(\\lambda_{1})$ of crystals \nthat maps $\\mathcal{MV}(\\lambda)$ onto a connected component of \n$\\mathcal{MV}(\\lambda_{2}) \\otimes \\mathcal{MV}(\\lambda_{1})$, \nwhich is isomorphic to $\\mathcal{MV}(\\lambda)$ as a crystal. \n\\begin{thm} \\label{thm:tensor}\nKeep the notation above.\nLet $P \\in \\mathcal{MV}(\\lambda)$, and \nwrite $\\iota_{\\lambda}(P) \\in \n\\mathcal{MV}(\\lambda_{2}) \\otimes \\mathcal{MV}(\\lambda_{1})$ as\\,{\\rm:} \n$\\iota_{\\lambda}(P)=P_{2} \\otimes P_{1}$ for some \n$P_{1} \\in \\mathcal{MV}(\\lambda_{1})$ and \n$P_{2} \\in \\mathcal{MV}(\\lambda_{2})$. \nWe assume that the MV polytope \n$P_{2} \\in \\mathcal{MV}(\\lambda_{2})$ is an extremal MV polytope \n$P_{x \\cdot \\lambda_{2}}$ for some $x \\in W$. \nThen, we have\n\\begin{equation} \\label{eq:Minkowski}\nP \\subset P_{1} + P_{2},\n\\end{equation}\nwhere $P_{1} + P_{2}$ is the Minkowski sum of \nthe MV polytopes $P_{1} \\in \\mathcal{MV}(\\lambda_{1})$ and \n$P_{2} \\in \\mathcal{MV}(\\lambda_{2})$.\n\\end{thm}\n\n\\begin{rem}\nIt should be mentioned that in the theorem above, \nthe $\\iota_{\\lambda}(P)$ may lie in an arbitrary \nconnected component of \n$\\mathcal{MV}(\\lambda_{2}) \\otimes \\mathcal{MV}(\\lambda_{1})$ \nthat is isomorphic to $\\mathcal{MV}(\\lambda)$ as a crystal. \n\\end{rem}\n\nThe proof of this theorem will be given \nin \\S\\ref{subsec:prf-tensor} below; \nit seems likely that this theorem still holds \nwithout the assumption of extremality \non the MV polytope $P_{2} \\in \\mathcal{MV}(\\lambda_{2})$.\n\nFor dominant coweights \n$\\lambda_{1},\\,\\lambda_{2} \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$, \nthere exists a unique embedding\n\\begin{equation*}\n\\iota_{\\lambda_{1},\\lambda_{2}}:\n \\mathcal{MV}(\\lambda_{1}+\\lambda_{2}) \\hookrightarrow \n \\mathcal{MV}(\\lambda_{1}) \\otimes \\mathcal{MV}(\\lambda_{2})\n\\end{equation*}\nof crystals, which maps $\\mathcal{MV}(\\lambda_{1}+\\lambda_{2})$ \nonto the unique connected component of \n$\\mathcal{MV}(\\lambda_{1}) \\otimes \\mathcal{MV}(\\lambda_{2})$ \n(called the Cartan component) \nthat is isomorphic to $\\mathcal{MV}(\\lambda_{1}+\\lambda_{2})$ \nas a crystal; note that \n$\\iota_{\\lambda_{1},\\lambda_{2}}(P_{\\lambda_{1}+\\lambda_{2}})=\n P_{\\lambda_{1}} \\otimes P_{\\lambda_{2}}$. \n %\nApplying Theorem~\\ref{thm:tensor} to the case \n$\\lambda=\\lambda_{1}+\\lambda_{2}$, we obtain \nthe following corollary; notice that the ordering of the \ntensor factors $\\mathcal{MV}(\\lambda_{1})$, $\\mathcal{MV}(\\lambda_{2})$ is reversed. \n\\begin{cor} \\label{cor:tensor}\nLet $\\lambda_{1},\\,\\lambda_{2} \\in X_{\\ast}(T) \n\\subset \\mathfrak{h}_{\\mathbb{R}}$ be dominant coweights. \nLet $P \\in \\mathcal{MV}(\\lambda_{1}+\\lambda_{2})$, and \nwrite $\\iota_{\\lambda_{1},\\lambda_{2}}(P) \\in \n\\mathcal{MV}(\\lambda_{1}) \\otimes \\mathcal{MV}(\\lambda_{2})$ as\\,{\\rm:} \n$\\iota_{\\lambda_{1},\\lambda_{2}}(P)=P_{1} \\otimes P_{2}$ for some \n$P_{1} \\in \\mathcal{MV}(\\lambda_{1})$ and \n$P_{2} \\in \\mathcal{MV}(\\lambda_{2})$. \nWe assume that the MV polytope \n$P_{1} \\in \\mathcal{MV}(\\lambda_{1})$ is an extremal MV polytope \n$P_{x \\cdot \\lambda_{1}}$ for some $x \\in W$. \nThen, we have\n\\begin{equation} \\label{eq:Minkowski2}\nP \\subset P_{1} + P_{2}. \n\\end{equation}\n\\end{cor}\n\nThe following is a particular case in which \nthe equality holds in \\eqref{eq:Minkowski2}. \n\\begin{prop} \\label{prop:Min-Ext}\nLet $\\lambda_{1},\\,\\lambda_{2} \\in X_{\\ast}(T) \\subset \\mathfrak{h}_{\\mathbb{R}}$ \nbe dominant coweights, and let $x \\in W$. Then, \n$\\iota_{\\lambda_{1},\\lambda_{2}}(P_{x \\cdot (\\lambda_{1}+\\lambda_{2})})=\nP_{x \\cdot \\lambda_{1}} \\otimes P_{x \\cdot \\lambda_{2}}$, and \n$P_{x \\cdot (\\lambda_{1}+\\lambda_{2})}=\n P_{x \\cdot \\lambda_{1}}+P_{x \\cdot \\lambda_{2}}$. \n\\end{prop}\n\n\\begin{proof}\nFirst we show the equality \n$\\iota_{\\lambda_{1},\\lambda_{2}}(P_{x \\cdot (\\lambda_{1}+\\lambda_{2})})=\n P_{x \\cdot \\lambda_{1}} \\otimes P_{x \\cdot \\lambda_{2}}$ \n(which may be well-known to experts) by induction on $\\ell(x)$. \nIf $x=e$, then we have \n$\\iota_{\\lambda_{1},\\lambda_{2}}(P_{\\lambda_{1}+\\lambda_{2}})=\n P_{\\lambda_{1}} \\otimes P_{\\lambda_{2}}$ as mentioned above. \nAssume now that $\\ell(x) > 0$. We take $j \\in I$ \nsuch that $\\ell(s_{j}x) < \\ell(x)$. Set\n\\begin{equation*}\nk:=\\pair{s_{j}x \\cdot (\\lambda_{1}+\\lambda_{2})}{\\alpha_{j}}, \\quad\nk_{1}:=\\pair{s_{j}x \\cdot \\lambda_{1}}{\\alpha_{j}}, \\quad\nk_{2}:=\\pair{s_{j}x \\cdot \\lambda_{2}}{\\alpha_{j}};\n\\end{equation*}\nnote that we have $k=k_{1}+k_{2}$, \nwith $k_{1},\\,k_{2},\\,k \\in \\mathbb{Z}_{\\ge 0}$, \nsince $\\ell(s_{j}x) < \\ell(x)$. We see from \n\\cite[Lemme~8.3.1]{Kasb} that \n$f_{j}^{k}P_{s_{j}x \\cdot (\\lambda_{1}+\\lambda_{2})}$\nis equal to $P_{x \\cdot (\\lambda_{1}+\\lambda_{2})}$. \nHence we have \n\\begin{align*}\n\\iota_{\\lambda_{1},\\lambda_{2}}\n (P_{x \\cdot (\\lambda_{1}+\\lambda_{2})}) & \n =\\iota_{\\lambda_{1},\\lambda_{2}}\n (f_{j}^{k}P_{s_{j}x \\cdot (\\lambda_{1}+\\lambda_{2})}) \n =f_{j}^{k}\\iota_{\\lambda_{1},\\lambda_{2}}\n (P_{s_{j}x \\cdot (\\lambda_{1}+\\lambda_{2})}) \\\\\n& = f_{j}^{k}(\n  P_{s_{j}x \\cdot \\lambda_{1}} \\otimes \n  P_{s_{j}x \\cdot \\lambda_{2}})\n\\quad \\text{by the induction hypothesis}.\n\\end{align*}\nHere, by the tensor product rule \nfor crystals, \n\\begin{equation*}\nf_{j}^{k}(\n  P_{s_{j}x \\cdot \\lambda_{1}} \\otimes \n  P_{s_{j}x \\cdot \\lambda_{2}}\n)\n= (f_{j}^{l_1}P_{s_{j}x \\cdot \\lambda_{1}}) \\otimes \n  (f_{j}^{l_2}P_{s_{j}x \\cdot \\lambda_{2}})\n\\end{equation*}\nfor some $l_{1},\\,l_{2} \\in \\mathbb{Z}_{\\ge 0}$ with $k=l_{1}+l_{2}$.\nIt follows from \\cite[Lemme~8.3.1]{Kasb} that \n$l_{1}=k_{1}$ and $l_{2}=k_{2}$. \nTherefore, we deduce that \n\\begin{align*}\n\\iota_{\\lambda_{1},\\lambda_{2}}\n (P_{x \\cdot (\\lambda_{1}+\\lambda_{2})})\n& = \nf_{j}^{k}(\n  P_{s_{j}x \\cdot \\lambda_{1}} \\otimes \n  P_{s_{j}x \\cdot \\lambda_{2}})\n= (f_{j}^{k_1}P_{s_{j}x \\cdot \\lambda_{1}}) \\otimes \n  (f_{j}^{k_2}P_{s_{j}x \\cdot \\lambda_{2}}) \\\\\n& = \nP_{x \\cdot \\lambda_{1}} \\otimes P_{x \\cdot \\lambda_{2}}\n\\quad \\text{by \\cite[Lemme~8.3.1]{Kasb}}.\n\\end{align*}\nThis proves the first equality. \n\nNext we show the equality\n$P_{x \\cdot (\\lambda_{1}+\\lambda_{2})}=\n P_{x \\cdot \\lambda_{1}}+P_{x \\cdot \\lambda_{2}}$.\nLet us denote by\n\\begin{equation*}\n\\mu_{\\bullet}^{x \\cdot (\\lambda_{1}+\\lambda_{2})}=\n (\\mu_{w}^{x \\cdot (\\lambda_{1}+\\lambda_{2})})_{w \\in W}, \\quad\n\\mu_{\\bullet}^{x \\cdot \\lambda_{1}}=\n (\\mu_{w}^{x \\cdot \\lambda_{1}})_{w \\in W}, \\quad \\text{and} \\quad\n\\mu_{\\bullet}^{x \\cdot \\lambda_{2}}=\n (\\mu_{w}^{x \\cdot \\lambda_{2}})_{w \\in W}\n\\end{equation*}\nthe GGMS data of the extremal MV polytopes \n$P_{x \\cdot (\\lambda_{1}+\\lambda_{2})} \\in \\mathcal{MV}(\\lambda_{1}+\\lambda_{2})$, \n$P_{x \\cdot \\lambda_{1}} \\in \\mathcal{MV}(\\lambda_{1})$, and \n$P_{x \\cdot \\lambda_{2}} \\in \\mathcal{MV}(\\lambda_{2})$, respectively.\nWe verify the equality \n\\begin{equation} \\label{eq:me1}\n\\mu_{w}^{x \\cdot (\\lambda_{1}+\\lambda_{2})} = \n\\mu_{w}^{x \\cdot \\lambda_{1}}+\\mu_{w}^{x \\cdot \\lambda_{2}}\n\\quad \\text{for every $w \\in W$}.\n\\end{equation}\nLet $w \\in W$, and \ntake $\\mathbf{i} \\in R(w_{0})$ such that \n$w=\\wi{l}$ for some $0 \\le l \\le m$. \nThen it follows from \nTheorem~\\ref{thm:GGMS-Ext} that \n\\begin{align*}\n& \\mu_{w}^{x \\cdot (\\lambda_{1}+\\lambda_{2})}=\n  \\mu_{\\wi{l}}^{x \\cdot (\\lambda_{1}+\\lambda_{2})}=\n  \\yi{l} \\cdot (\\lambda_{1}+\\lambda_{2}), \\quad \\text{and} \\\\\n& \\mu_{w}^{x \\cdot \\lambda_{1}}=\n  \\mu_{\\wi{l}}^{x \\cdot \\lambda_{1}}=\n  \\yi{l} \\cdot \\lambda_{1}, \\qquad\n  \\mu_{w}^{x \\cdot \\lambda_{2}}=\n  \\mu_{\\wi{l}}^{x \\cdot \\lambda_{2}}=\n  \\yi{l} \\cdot \\lambda_{2},\n\\end{align*}\nwhere $\\yi{l}$ is defined as \\eqref{eq:yi}; \nrecall from Remark~\\ref{rem:zi} that \n$\\yi{l} \\in W$ does not depend \non the dominant coweights $\\lambda_{1}+\\lambda_{2}$, \n$\\lambda_{1}$, and $\\lambda_{2}$. \nTherefore, we deduce that \n\\begin{equation*}\n\\mu_{w}^{x \\cdot (\\lambda_{1}+\\lambda_{2})} = \n\\yi{l} \\cdot (\\lambda_{1}+\\lambda_{2})=\n\\yi{l} \\cdot \\lambda_{1}+\\yi{l} \\cdot \\lambda_{2}=\n\\mu_{w}^{x \\cdot \\lambda_{1}}+\\mu_{w}^{x \\cdot \\lambda_{2}},\n\\end{equation*}\nas desired. Hence it follows that \n\\begin{equation} \\label{eq:GGMS-Ext}\n\\mu_{\\bullet}^{x \\cdot (\\lambda_{1}+\\lambda_{2})}=\n (\\mu_{w}^{x \\cdot (\\lambda_{1}+\\lambda_{2})})_{w \\in W}=\n(\\mu_{w}^{x \\cdot \\lambda_{1}}+\\mu_{w}^{x \\cdot \\lambda_{2}})_{w \\in W}=\n\\mu_{\\bullet}^{x \\cdot \\lambda_{1}}+\n\\mu_{\\bullet}^{x \\cdot \\lambda_{2}}.\n\\end{equation}\nConsequently, we have \n\\begin{align*}\nP_{x \\cdot \\lambda_{1}}+\nP_{x \\cdot \\lambda_{2}} & =\nP(\\mu_{\\bullet}^{x \\cdot \\lambda_{1}})+\nP(\\mu_{\\bullet}^{x \\cdot \\lambda_{2}}) = \nP(\\mu_{\\bullet}^{x \\cdot \\lambda_{1}}+\n  \\mu_{\\bullet}^{x \\cdot \\lambda_{2}}) \\qquad\n\\text{by Proposition~\\ref{prop:Minkowski}} \\\\\n& =\nP(\\mu_{\\bullet}^{x \\cdot (\\lambda_{1}+\\lambda_{2})}) \\qquad\n\\text{by \\eqref{eq:GGMS-Ext}} \\\\\n& = P_{x \\cdot (\\lambda_{1}+\\lambda_{2})}.\n\\end{align*}\nThis proves the second equality, \nthereby completes the proof of the proposition. \n\\end{proof}\n\n\\subsection{Reformation of Braverman-Gaitsgory's result on tensor products.}\n\\label{subsec:BG}\nIn this subsection, we revisit results of Braverman-Gaitsgory on \ntensor products of highest weight crystals, and provide \na reformation of it, which is needed in our proof of \nTheorem~\\ref{thm:tensor} given in \\S\\ref{subsec:prf-tensor}. \n\nWe now recall the construction of \ncertain twisted product varieties. \nLet $G$ be a complex, connected, reductive algebraic group \nas in (the beginning of) \\S\\ref{subsec:geom}. \nThe twisted product variety $\\CG r \\,\\ti{\\times}\\, \\CG r$ is defined to be \nthe quotient space\n\\begin{equation*}\nG(\\mathcal{K}) \\times^{G(\\mathcal{O})} \\CG r = \\bigl(G(\\mathcal{K}) \\times \\CG r\\bigr)\/\\sim, \n\\end{equation*}\nwhere $\\sim$ is an equivalence relation on \n$G(\\mathcal{K}) \\times \\CG r$ given by: \n$(a,\\,bG (\\mathcal{O})) \\sim (ah^{-1},\\,hb G(\\mathcal{O}))$ for \n$a,\\,b \\in G(\\mathcal{K})$, and $h \\in G (\\mathcal{O})$; \nfor $(a,\\,bG(\\mathcal{O})) \\in G(\\mathcal{K}) \\times \\CG r$, we denote by \n$[(a,\\,bG(\\mathcal{O}))]$ the equivalence class of $(a,\\,bG(\\mathcal{O}))$. \nThis variety can be thought of as a fibration over $\\CG r$ \n(the first factor) with its typical fiber $\\CG r$ (the second factor). \nLet $\\pi_{1} : \\CG r \\,\\ti{\\times}\\, \\CG r \\twoheadrightarrow \\CG r$, \n$[(a,\\,bG(\\mathcal{O}))] \\mapsto a G(\\mathcal{O})$, \nbe the projection onto the first factor, and \n$\\mathsf{m}:\\CG r \\,\\ti{\\times}\\, \\CG r \\rightarrow \\CG r$, \n$[(a,\\,bG(\\mathcal{O}))] \\mapsto ab G(\\mathcal{O})$, the multiplication map. \nIf $\\mathcal{X} \\subset \\CG r$ is an algebraic subvariety and \n$\\mathcal{Y} \\subset \\CG r$ is an algebraic subvariety that \nis stable under the left $G(\\mathcal{O})$-action on $\\CG r$, \nthen we can form an algebraic subvariety\n\\begin{equation*}\n\\mathcal{X} \\,\\ti{\\times}\\, \\mathcal{Y} := \\ti{\\mathcal{X}} \\times^{G(\\mathcal{O})} \\mathcal{Y} \\subset \\CG r \\,\\ti{\\times}\\, \\CG r,\n\\end{equation*}\nwhere $\\ti{\\mathcal{X}}:=\\pi_{1}^{-1}(\\mathcal{X})$ is the pullback of \n$\\mathcal{X} \\subset \\CG r=G(\\mathcal{K})\/G(\\mathcal{O})$ to $G(\\mathcal{K})$.\n\\begin{prop}[{\\cite{Lu-Ast}, \\cite[Lemma~4.4]{MV2}}] \\label{prop:mv-44}\nThe multiplication map \n$\\mathsf{m}:\\CG r \\,\\ti{\\times}\\, \\CG r \\rightarrow \\CG r$, when restricted to \n$\\ol{\\CG r^{\\lambda_{1}}} \\,\\ti{\\times}\\, \\ol{\\CG r^{\\lambda_{2}}}$ \nfor $\\lambda_{1},\\,\\lambda_{2} \\in X_{*}(T)_{+}$, is \nprojective, birational, and semi-small \nwith respect to the stratification by \n$G(\\mathcal{O})$-orbits. In particular, \nfor $\\lambda_{1},\\,\\lambda_{2} \\in X_{*}(T)_{+}$ and \n$\\lambda \\in X_{*}(T)_{+}$, \n\\begin{equation*}\n\\mathsf{m} \\left(\\ol{\\CG r^{\\lambda_{1}}} \\,\\ti{\\times}\\, \\ol{\\CG r^{\\lambda_{2}}} \\right) = \n\\ol{\\CG r^{\\lambda_{1} + \\lambda_{2}}},\n\\end{equation*}\n\\begin{equation*}\n\\mathsf{m}^{-1} (\\CG r^{\\lambda}) \\cap \n\\left(\n \\ol{\\CG r^{\\lambda_{1}}} \\,\\ti{\\times}\\, \n \\ol{\\CG r^{\\lambda_{2}}}\\right) \\neq \\emptyset\n\\quad \\text{\\rm if and only if} \\quad\n\\lambda_{1} + \\lambda_{2} \\ge \\lambda;\n\\end{equation*}\nif $\\lambda_{1} + \\lambda_{2} \\ge \\lambda$, then \n\\begin{equation*}\n\\dim \\left( \n \\mathsf{m}^{-1} (\\CG r^{\\lambda}) \\cap \n \\left(\\ol{\\CG r^{\\lambda_{1}}} \\,\\ti{\\times}\\, \\ol{\\CG r^{\\lambda_{2}}}\\right)\n\\right) \n\\le \\dim \\CG r^{\\lambda} + \n\\pair{\\lambda_{1}+\\lambda_{2}-\\lambda}{\\rho}.\n\\end{equation*}\n\\end{prop}\n\nLet us introduce another kind of twisted products. \nFor each $\\nu_{1},\\,\\nu_{2} \\in X_{*}(T)$ and $w \\in W$, \nwe define $S^{w}_{\\nu_{1},\\,\\nu_{2}}$ to be the quotient space\n\\begin{equation*}\n{}^{w}U(\\mathcal{K}) t^{\\nu_{1}} \\times^{{}^{w}U(\\mathcal{O})} {}^{w}U(\\mathcal{K})[t^{\\nu_{2}}]\n=\n\\bigl(\n {}^{w}U(\\mathcal{K}) t^{\\nu_{1}} \\times {}^{w}U(\\mathcal{K})[t^{\\nu_{2}}]\n\\bigr)\/\\sim,\n\\end{equation*}\nwhere $\\sim$ is an equivalence relation on \n${}^{w}U(\\mathcal{K}) t^{\\nu_{1}} \\times {}^{w}U(\\mathcal{K})[t^{\\nu_{2}}]$ \ngiven by: \n$(a,\\,bG (\\mathcal{O})) \\sim (au^{-1},\\,ub G(\\mathcal{O}))$ for \n$a \\in {}^{w}U(\\mathcal{K}) t^{\\nu_{1}} \\subset G(\\mathcal{K})$, \n$b \\in {}^{w}U(\\mathcal{K})[t^{\\nu_{2}}] \\subset \\CG r$, and \n$u \\in {}^{w}U(\\mathcal{O}) \\subset G(\\mathcal{O})$. \nSince ${}^{w}U(\\mathcal{O})=G(\\mathcal{O}) \\cap {}^{w}U(\\mathcal{K})$ and \n$t^{\\nu} ({}^{w}U(\\mathcal{K})) = ({}^{w}U(\\mathcal{K})) t^{\\nu}$ \nfor $\\nu \\in X_{*}(T)$, we have a canonical embedding\n\\begin{equation} \\label{eq:emb-s}\nS^{w}_{\\nu_{1},\\,\\nu_{2}} \\hookrightarrow \nG(\\mathcal{K}) \\times^{G(\\mathcal{O})} \\CG r=\\CG r \\,\\ti{\\times}\\, \\CG r.\n\\end{equation}\n\\begin{lem} \\label{lem:invS}\nFor each $\\nu \\in X_{*}(T)$ and $w \\in W$, we have\n\\begin{equation*}\n\\mathsf{m}^{-1} (S^{w}_{\\nu}) = \n \\bigsqcup_{\n   \\begin{subarray}{c}\n   \\nu_{1},\\,\\nu_{2} \\in X_{*}(T) \\\\[1.5mm]\n   \\nu_{1} + \\nu_{2} = \\nu\n   \\end{subarray}\n } S^{w}_{\\nu_{1},\\,\\nu_{2}}\n\\end{equation*}\nunder the canonical embedding \\eqref{eq:emb-s}.\n\\end{lem}\n\n\\begin{proof}\nIt follows from the Iwasawa decomposition\n$\\CG r=\\bigsqcup_{\\nu_{1} \\in X_{*}(T)} S_{\\nu_{1}}^{w}$ that \n\\begin{equation*}\n\\CG r \\,\\ti{\\times}\\, \\CG r=\n\\bigsqcup_{\\nu_{1} \\in X_{*}(T)} \n \\pi_{1}^{-1}(S_{\\nu_{1}}^{w}),\n\\end{equation*}\nwhere $\\pi_{1}^{-1}(S_{\\nu_{1}}^{w})=\n {}^{w}U(\\mathcal{K})t^{\\nu_{1}}G(\\mathcal{O}) \\times^{G(\\mathcal{O})}\\CG r$. \nTherefore it suffices to show that\n\\begin{equation*}\n\\pi_{1}^{-1}(S_{\\nu_{1}}^{w}) \\cap \n\\mathsf{m}^{-1} (S^{w}_{\\nu}) =\nS^{w}_{\\nu_{1},\\,\\nu-\\nu_{1}}\n\\quad\n\\text{for each $\\nu_{1} \\in X_{*}(T)$}.\n\\end{equation*}\nNow, for each $\\nu_{1} \\in X_{*}(T)$, \nlet us take $[y] \\in \n\\pi_{1}^{-1}(S_{\\nu_{1}}^{w}) \\cap \n\\mathsf{m}^{-1} (S^{w}_{\\nu})$, \nwhere $y \\in \n {}^{w}U(\\mathcal{K})t^{\\nu_{1}}G(\\mathcal{O}) \\times \\CG r$, and \nwrite it as: $y=(u_{1}t^{\\nu_{1}}g_{1},\\,g_{2}G(\\mathcal{O}))$ \nfor $u_{1} \\in {}^{w}U(\\mathcal{K})$, $g_{1} \\in G(\\mathcal{O})$, and \n$g_{2} \\in G(\\mathcal{K})$. \nSince $\\mathsf{m}([y])=u_{1}t^{\\nu_{1}}g_{1}g_{2}G(\\mathcal{O}) \\in S^{w}_{\\nu}$, \nwe have \n\\begin{equation*}\ng_{1}g_{2}G(\\mathcal{O}) \\in \n (u_{1}t^{\\nu_{1}})^{-1}S^{w}_{\\nu}=\n {}^{w}U(\\mathcal{K})t^{\\nu-\\nu_{1}}G(\\mathcal{O}).\n\\end{equation*}\nConsequently, using the equivalence relation $\\sim$ \non $G(\\mathcal{K}) \\times \\CG r$, we see that\n\\begin{equation*}\n[y]\n =\\bigl[ (u_{1}t^{\\nu_{1}}g_{1},\\,g_{2}G(\\mathcal{O})) \\bigr]\n =\\bigl[ (u_{1}t^{\\nu_{1}},\\,g_{1}g_{2}G(\\mathcal{O})) \\bigr]\n =\\bigl[ (u_{1}t^{\\nu_{1}},\\,u_{2}t^{\\nu-\\nu_{1}}G(\\mathcal{O})) \\bigr]\n\\end{equation*}\nfor some $u_{2} \\in {}^{w}U(\\mathcal{K})$. \nThis implies that \n\\begin{equation*}\n[y] \\in \n  {}^{w}U(\\mathcal{K})t^{\\nu_{1}} \n  \\times^{{}^{w}U(\\mathcal{O})} \n  {}^{w}U(\\mathcal{K})[t^{\\nu-\\nu_{1}}]=\nS^{w}_{\\nu_{1},\\,\\nu-\\nu_{1}},\n\\end{equation*}\nand hence \n$\\pi_{1}^{-1}(S_{\\nu_{1}}^{w}) \\cap \n \\mathsf{m}^{-1} (S^{w}_{\\nu}) \\subset \nS^{w}_{\\nu_{1},\\,\\nu-\\nu_{1}}$.\nThe opposite inclusion is obvious. \nThus, we obtain \n$\\pi_{1}^{-1}(S_{\\nu_{1}}^{w}) \\cap \n \\mathsf{m}^{-1} (S^{w}_{\\nu})=\n S^{w}_{\\nu_{1},\\,\\nu-\\nu_{1}}$.\nThis proves the lemma. \n\\end{proof}\n\nFor $\\nu_{1},\\,\\nu_{2} \\in X_{*}(T)$, we set\n$S_{\\nu_{1},\\,\\nu_{2}} := S_{\\nu_{1},\\,\\nu_{2}}^{e}$. \nIf we take (and fix) an element $t \\in T(\\mathbb{R})$ such that\n\\begin{equation*}\n\\lim_{k \\to \\infty} \\mathop{\\rm Ad}\\nolimits(t^{k}) u = e\n\\quad \\text{for all $u \\in U$}, \n\\end{equation*}\nthen we have (by \\cite[Eq.(3.5)]{MV2})\n\\begin{equation*}\nS_{\\nu}=\n \\left\\{\n [y] \\in \\CG r \\ \\biggm| \\ \n \\lim_{k \\to \\infty} t^{k}[y]=[t^{\\nu}]\n \\right\\}\n\\quad \\text{for $\\nu \\in X_{*}(T)$}.\n\\end{equation*}\nFrom this, by using Lemma~\\ref{lem:invS}, \nwe have\n\\begin{equation} \\label{eq:Snn}\nS_{\\nu_{1},\\,\\nu_{2}} = \n \\left\\{ [y] \\in \\CG r \\,\\ti{\\times}\\, \\CG r \\ \\biggm| \\ \n \\lim_{k \\to \\infty} t^{k}[y]=\n (t^{\\nu_{1}},\\,[t^{\\nu_{2}}])\n \\right\\}\n\\quad \\text{for $\\nu_{1},\\,\\nu_{2} \\in X_{*}(T)$};\n\\end{equation}\nin particular, these strata of $\\CG r \\,\\ti{\\times}\\, \\CG r$ \nare simply-connected. \n\nFor each $\\nu \\in X_{*}(T)$ and $w \\in W$, \nwe set\n\\begin{equation*}\n\\mathbb{S}^{w}_{\\nu} := {}^{w}U(\\mathcal{K})t^{\\nu}({}^{w}U(\\mathcal{O})) \/ {}^{w}U(\\mathcal{O}), \n\\end{equation*}\nwhich is canonically isomorphic to \n${}^{w}U(\\mathcal{K}) t^{\\nu} G (\\mathcal{O}) \/ G (\\mathcal{O}) = S^{w}_{\\nu}$ \nsince ${}^{w}U(\\mathcal{K}) \\cap G(\\mathcal{O})={}^{w}U(\\mathcal{O})$; \nnote that ${}^{w}U(\\mathcal{K})t^{\\nu}({}^{w}U(\\mathcal{O}))={}^{w}U(\\mathcal{K})t^{\\nu}$\nsince $t^{\\nu}({}^{w}U(\\mathcal{K}))=({}^{w}U(\\mathcal{K}))t^{\\nu}$. \nAlso, for a subset $\\mathcal{X} \\subset \\CG r$, \nwe define the intersection $\\mathcal{X} \\cap \\mathbb{S}^{w}_{\\nu}$ \nto be the image of $\\mathcal{X} \\cap S^{w}_{\\nu} \\subset S_{\\nu}^{w}$ \nunder the identification $\\mathbb{S}^{w}_{\\nu} = S^{w}_{\\nu}$ above. \n\nIn the sequel, for an algebraic variety $\\mathcal{X}$, \nwe denote by $\\mathop{\\rm Irr}\\nolimits(\\mathcal{X})$ the set of irreducible components of $\\mathcal{X}$.\nLet $\\lambda \\in X_{*}(T)_{+}$ and $\\nu \\in X_{*}(T)$ be \nsuch that $\\CG r^{\\lambda} \\cap S_{\\nu} \\ne \\emptyset$, \ni.e., $\\nu \\in \\Omega(\\lambda)$; note that \n$\\CG r^{\\lambda} \\cap S_{\\nu} \\ne \\emptyset$ if and only if \n$\\CG r^{\\lambda} \\cap S_{w^{-1} \\cdot \\nu} \\ne \\emptyset$, i.e., \n$w^{-1} \\cdot \\nu \\in \\Omega(\\lambda)$ for each $w \\in W$. \nLet us take an arbitrary $w \\in W$. \nBecause $\\CG r^{\\lambda} \\cap S_{\\nu}^{w}=\n\\dot{w}(\\CG r^{\\lambda} \\cap S_{w^{-1} \\cdot \\nu})$ \nfor $w \\in W$, we have a bijection\n\\begin{equation*}\n\\mathop{\\rm Irr}\\nolimits\\left(\\ol{\\CG r^{\\lambda} \\cap S_{w^{-1} \\cdot \\nu}}\\right)\n\\rightarrow\n\\mathop{\\rm Irr}\\nolimits\\left(\\ol{\\CG r^{\\lambda} \\cap S_{\\nu}^{w}}\\right), \\quad\n\\mathbf{b} \\mapsto \\dot{w}\\mathbf{b}. \n\\end{equation*}\nThus, each element of \n$\\mathop{\\rm Irr}\\nolimits\\left(\\ol{\\CG r^{\\lambda} \\cap S_{\\nu}^{w}}\\right)$ \ncan be written in the form $\\dot{w}\\mathbf{b}$ for a unique \nMV cycle $\\mathbf{b} \\in \\mathcal{Z}(\\lambda)_{w^{-1} \\cdot \\nu}=\n\\mathop{\\rm Irr}\\nolimits\\left(\\ol{\\CG r^{\\lambda} \\cap S_{w^{-1} \\cdot \\nu}}\\right)$.\nThe variety $\\mathbf{b}^{w}$ defined by \\eqref{eq:bbw} is a special case of \nsuch elements, in which $\\mathbf{b} \\in \\mathcal{Z}(\\lambda)$ is an extremal \nMV cycle with GGMS datum $\\mu_{\\bullet}$ and $\\nu=\\mu_{w}$. \n\\begin{lem} \\label{lem:stable}\nWith the notation as above, \nlet us take an arbitrary element \n\\begin{equation*}\n\\dot{w}\\mathbf{b} \\in \n\\mathop{\\rm Irr}\\nolimits\\left(\\ol{\\CG r^{\\lambda} \\cap S_{\\nu}^{w}}\\right), \\quad\n\\text{\\rm where} \\quad \n\\mathbf{b} \\in \\mathcal{Z}(\\lambda)_{w^{-1} \\cdot \\nu}.\n\\end{equation*} \nThen, the intersection $\\dot{w}\\mathbf{b} \\cap S_{\\gamma}^{w}$ \n(and hence $\\dot{w}\\mathbf{b} \\cap \\mathbb{S}_{\\gamma}^{w}$) is \nstable under the action of ${}^{w}U(\\mathcal{O})$ \nfor all $\\gamma \\in X_{*}(T)$. \n\\end{lem}\n\n\\begin{proof}\nBy definition, each MV cycle is an irreducible component of \nthe (Zariski-) closure of the intersection of a $G(\\mathcal{O})$-orbit \nand a $U(\\mathcal{K})$-orbit. Since $U(\\mathcal{O})=G(\\mathcal{O}) \\cap U(\\mathcal{K})$ is \nconnected, such an irreducible component is stable under \nthe action of $U(\\mathcal{O})$. Therefore, the variety $\\dot{w}\\mathbf{b}$ \n(and hence its intersection with an arbitrary ${}^{w}U(\\mathcal{K})$-orbit) \nis stable under the action of ${}^{w}U(\\mathcal{O})$. \nThis proves the lemma. \n\\end{proof}\n\nLet $\\lambda_{1},\\,\\lambda_{2} \\in X_{*}(T)_{+}$ and \n$\\nu_{1},\\,\\nu_{2} \\in X_{*}(T)$ be such that \n$\\CG r^{\\lambda_{i}} \\cap S_{\\nu_{i}} \\ne \\emptyset$ \nfor $i=1,\\,2$, and let $w \\in W$. \nLet us take $\\mathbf{b}_{1} \\in \\mathcal{Z}(\\lambda_{1})_{\\nu_{1}}$, \n$\\dot{w}\\mathbf{b}_{2} \\in \n\\mathop{\\rm Irr}\\nolimits\\left(\\ol{\\CG r^{\\lambda_{2}} \\cap S_{\\nu_{2}}^{w}}\\right)$, \nand $\\gamma_{1},\\,\\gamma_{2} \\in X_{*}(T)$. \nThen, by virtue of the lemma above, \nwe can form the twisted product \n\\begin{equation*}\n(\\mathbf{b}_{1} \\cap \\mathbb{S}^{w}_{\\gamma_{1}})^{\\sim}\n \\times^{{}^{w}U(\\mathcal{O})} \n(\\dot{w}\\mathbf{b}_{2} \\cap \\mathbb{S}^{w}_{\\gamma_{2}}) \n\\subset \n{}^{w}U(\\mathcal{K})t^{\\gamma_{1}} \n \\times^{{}^{w}U(\\mathcal{O})} S^{w}_{\\gamma_{2}}=\nS^{w}_{\\gamma_{1},\\,\\gamma_{2}},\n\\end{equation*}\nwhere $(\\mathbf{b}_{1} \\cap \\mathbb{S}^{w}_{\\gamma_{1}})^{\\sim}$\ndenotes the pullback of \n\\begin{equation*}\n\\mathbf{b}_{1} \\cap \\mathbb{S}^{w}_{\\gamma_{1}} \\subset \\mathbb{S}^{w}_{\\gamma_{1}}=\n{}^{w}U(\\mathcal{K})t^{\\gamma_{1}}({}^{w}U(\\mathcal{O}))\/{}^{w}U(\\mathcal{O})\n\\end{equation*}\nto ${}^{w}U(\\mathcal{K})t^{\\gamma_{1}}({}^{w}U(\\mathcal{O}))=\n{}^{w}U(\\mathcal{K})t^{\\gamma_{1}} \\subset G(\\mathcal{K})$. \nBy $\\mathbf{b}_{1} \\str{w}{\\gamma_{1}}{\\gamma_{2}} \\dot{w}\\mathbf{b}_{2}$, \nwe denote the image of this algebraic subvariety of \n$\\CG r \\,\\ti{\\times}\\, \\CG r$ under the map \n$\\mathsf{m}:\\CG r \\,\\ti{\\times}\\, \\CG r \\rightarrow \\CG r$; note that \n$\\mathbf{b}_{1} \\str{w}{\\gamma_{1}}{\\gamma_{2}} \\dot{w}\\mathbf{b}_{2} \n \\subset \\mathsf{m}(S^{w}_{\\gamma_{1},\\,\\gamma_{2}})\n =S^{w}_{\\gamma_{1}+\\gamma_{2}}$. \n\nThe following is a reformulation of \nBraverman-Gaitsgory's result on tensor products of \nhighest weight crystals in \\cite{BrGa} (see also \\cite{BFG});\nwe will give a brief account of the relationship \nin the Appendix. Here we should warn the reader that \nthe convention on the tensor product rule for crystals \nin \\cite{BrGa} is opposite to ours, i.e., to that of Kashiwara \n(see, for example, \\cite{Kasoc} and \\cite{Kasb}). \n\\begin{thm} \\label{thm:BG}\nLet $\\lambda_{1},\\,\\lambda_{2} \\in X_{*}(T)_{+}$. \nThere exists a bijection\n\\begin{equation*}\n\\Phi_{\\lambda_{1},\\,\\lambda_{2}} : \n \\mathcal{MV} (\\lambda_{1}) \\times \\mathcal{MV} (\\lambda_{2})\n\\rightarrow\n \\bigsqcup_{\\nu \\in X_{*}(T)} \n   \\mathop{\\rm Irr}\\nolimits \\left(\n      \\ol{\\mathsf{m}^{-1} (S_{\\nu}) \\cap \n      \\bigl(\\CG r^{\\lambda_{1}} \\,\\ti{\\times}\\, \\CG r^{\\lambda_{2}}}\\bigr) \n   \\right)\n\\end{equation*}\ngiven as follows\\,{\\rm:}\nfor $P_{1} \\in \\mathcal{MV} (\\lambda_{1})$ and $P_{2} \\in \\mathcal{MV}(\\lambda_{2})$, \n\\begin{equation} \\label{eq:Phi-ll}\n\\Phi_{\\lambda_{1},\\,\\lambda_{2}}\n (P_{1},\\,P_{2})=\n\\ol{\n \\bigl(\\Phi_{\\lambda_{1}}(P_{1}) \\cap \\mathbb{S}_{\\nu_{1}}\\bigr)^{\\sim}\n \\times^{U(\\mathcal{O})}\n \\bigl(\\Phi_{\\lambda_{2}}(P_{2}) \\cap \\mathbb{S}_{\\nu_{2}}\\bigr)\n}, \n\\end{equation}\nwhere $\\nu_{1}:=\\mathop{\\rm wt}\\nolimits (P_{1})$, $\\nu_{2}:=\\mathop{\\rm wt}\\nolimits(P_{2})$, and \n$\\bigl(\\Phi_{\\lambda_{1}}(P_{1}) \\cap \\mathbb{S}_{\\nu_{1}}\\bigr)^{\\sim}$ denotes \nthe pullback of $\\Phi_{\\lambda_{1}}(P_{1}) \\cap \\mathbb{S}_{\\nu_{1}} \\subset S_{\\nu_{1}}$ \nto $U(\\mathcal{K})t^{\\nu_{1}} \\subset G(\\mathcal{K})$. \nMoreover, the bijection $\\Phi_{\\lambda_{1},\\,\\lambda_{2}}$ \nhas the following properties. \n\n{\\rm (i)} \nFor each $P_{1} \\in \\mathcal{MV}(\\lambda_{1})$ and \n$P_{2} \\in \\mathcal{MV}(\\lambda_{2})$, \nthe image of \n$\\Phi_{\\lambda_{1},\\,\\lambda_{2}}(P_{1},\\,P_{2})$ \nunder the map $\\mathsf{m}$ is equal to $\\Phi_{\\lambda}(P)$ \nfor a unique $\\lambda \\in X_{*}(T)_{+}$ and \na unique $P \\in \\mathcal{MV}(\\lambda)$ such that \n$\\iota_{\\lambda}(P)=P_{2} \\otimes P_{1}$, \nwhere $\\iota_{\\lambda}:\\mathcal{MV}(\\lambda) \\hookrightarrow \n \\mathcal{MV}(\\lambda_{2}) \\otimes \\mathcal{MV}(\\lambda_{1})$ is \nan embedding of crystals\\,{\\rm;}\n\n{\\rm (ii)}\n$\\pi_{1} (\\Phi_{\\lambda_{1},\\,\\lambda_{2}} (P_{1},\\,P_{2})) = \n \\Phi_{\\lambda_{1}} (P_{1})$ \nfor each $P_{1} \\in \\mathcal{MV} (\\lambda_{1})$ and \n$P_{2} \\in \\mathcal{MV} (\\lambda_{2})$\\,{\\rm;}\n\n{\\rm (iii)} \n$[(t^{\\nu_{1}},\\,\\Phi_{\\lambda_{2}} (P_{2}))] \\subset \n \\Phi_{\\lambda_{1},\\,\\lambda_{2}} (P_{1},\\,P_{2})$ \nfor each $P_{1} \\in \\mathcal{MV} (\\lambda_{1})$ with $\\nu_{1}=\\mathop{\\rm wt}\\nolimits (P_{1})$ \nand $P_{2} \\in \\mathcal{MV} (\\lambda_{2})$. \n\\end{thm}\n\n\\subsection{Proof of the polytopal estimate.}\n\\label{subsec:prf-tensor}\n\nThis subsection is devoted to the proof of Theorem~\\ref{thm:tensor}. \nLet $G$ be a complex, connected, semisimple algebraic group \nwith Lie algebra $\\mathfrak{g}$. We keep the setting of \\S\\ref{subsec:tensor}. \nLet $\\mu^{(1)}_{\\bullet}=(\\mu^{(1)}_{w})_{w \\in W}$ and \n$\\mu^{(2)}_{\\bullet}=(\\mu^{(2)}_{w})_{w \\in W}$ be the GGMS data of \n$P_{1} \\in \\mathcal{MV}(\\lambda_{1})$ and $P_{2} \\in \\mathcal{MV}(\\lambda_{2})$, \nrespectively. Also, let $\\mu_{\\bullet}=(\\mu_{w})_{w \\in W}$ be \nthe GGMS datum of $P \\in \\mathcal{MV}(\\lambda)$; \nnote that\n\\begin{equation*}\n\\mu_{e}=\\mathop{\\rm wt}\\nolimits(P)=\n\\mathop{\\rm wt}\\nolimits(P_{1})+\\mathop{\\rm wt}\\nolimits(P_{2})=\n\\mu^{(1)}_{e}+\\mu^{(1)}_{e}.\n\\end{equation*}\nRecall from Proposition~\\ref{prop:Minkowski} that \nthe Minkowski sum $P_{1}+P_{2}$ is a pseudo-Weyl polytope \n$P(\\mu^{(1)}_{\\bullet}+\\mu^{(2)}_{\\bullet})$ with GGMS datum \n$\\mu^{(1)}_{\\bullet}+\\mu^{(1)}_{\\bullet}\n =(\\mu^{(1)}_{w}+\\mu^{(2)}_{w})_{w \\in W}$. \nTherefore, it follows from equation \\eqref{eq:poly} \ntogether with Remark~\\ref{rem:moment} that \n\\begin{align*}\nP_{1}+P_{2} & = \n\\bigcap_{w \\in W}\n \\bigl\\{ \n v \\in \\mathfrak{h}_{\\mathbb{R}} \\mid \n w^{-1} \\cdot v- w^{-1} \\cdot \\bigl(\\mu^{(1)}_{w}+\\mu^{(2)}_{w}\\bigr) \n \\in \\textstyle{\\sum_{j \\in I}\\mathbb{R}_{\\ge 0}h_{j}}\n \\bigr\\} \\\\[3mm]\n& = \n\\bigcap_{w \\in W}\n \\mathop{\\rm Conv}\\nolimits \\biggl\\{\n  \\gamma \\in X_{*}(T) \\subset \\mathfrak{h}_{\\mathbb{R}} \\ \\biggm| \\ \n  [t^{\\gamma}] \\in \\ol{S^{w}_{\\mu^{(1)}_{w}+\\mu^{(2)}_{w}}}\n\\biggr\\}.\n\\end{align*}\nAlso, recall from Theorem~\\ref{thm:Kam1} that \n$\\Phi_{\\lambda}(P) \\in \\mathcal{Z}(\\lambda)$ and \n\\begin{equation*}\nP=\\mathop{\\rm Conv}\\nolimits \\bigl\\{\n  \\gamma \\in X_{*}(T) \\subset \\mathfrak{h}_{\\mathbb{R}} \\mid \n  [t^{\\gamma}] \\in \\Phi_{\\lambda}(P)\n\\bigr\\}.\n\\end{equation*}\nHence, in order to prove that $P \\subset P_{1}+P_{2}$, \nit suffices to show that\n\\begin{equation} \\label{eq:goal}\n\\Phi_{\\lambda} (P) \\subset \n \\ol{S^{w}_{\\mu_{w}^{(1)} + \\mu_{w}^{(2)}}}\n\\quad \\text{for all $w \\in W$}.\n\\end{equation}\n\nWe set $\\mathbf{b}^{(1)} := \\Phi_{\\lambda_{1}} (P_{1}) \\in \\mathcal{Z}(\\lambda_{1})$ \nand $\\mathbf{b}^{(2)} := \\Phi_{\\lambda_{2}} (P_{2}) \\in \\mathcal{Z}(\\lambda_{2})$.\nBecause $P_{2}=P(\\mu^{(2)}_{\\bullet})$ is the extremal MV polytope \nof weight $x \\cdot \\lambda$ for some $x \\in W$ by our assumption, \nwe know from Theorem~\\ref{thm:GGMS-Ext} that \n$\\mu^{(2)}_{w} \\in W \\cdot \\lambda_{2}$ for all $w \\in W$. \nHence the algebraic variety $\\mathbf{b}^{(2),w}:=\n\\ol{\\CG r^{\\lambda_{2}} \\cap S^{w}_{\\mu^{(2)}_{w}}}$ is \nirreducible, and is the $\\dot{w}$-translate of \nthe extremal MV cycle $\\mathbf{b}_{w^{-1} \\cdot \\mu^{(2)}_{w}}$ \nof weight $w^{-1} \\cdot \\mu^{(2)}_{w}$ \n(see Remark~\\ref{rem:extcyc}); note that \n$\\mathbf{b}^{(2),e}=\\mathbf{b}^{(2)}$. \n\nNow suppose, contrary to our assertion \\eqref{eq:goal}, \nthat \n$\\Phi_{\\lambda} (P) \\not\\subset \n \\ol{S^{w}_{\\mu_{w}^{(1)} + \\mu_{w}^{(2)}}}$\nfor some $w \\in W$; we take and fix such a $w \\in W$. \nThen, by equation \\eqref{eq:Snuw} (see also Remark~\\ref{rem:Snuw}), \nthere exists some $\\nu \\in X_{*}(T)$ such that\n\\begin{equation} \\label{eq:prf1}\n\\Phi_{\\lambda}(P) \\cap S_{\\nu}^{w} \\ne \\emptyset\n\\quad \\text{and} \\quad\nw^{-1} \\cdot \\nu \\not\\ge \nw^{-1} \\cdot (\\mu^{(1)}_{w} + \\mu^{(2)}_{w}). \n\\end{equation}\n\n\\begin{claim*}\nFor the (fixed) $w \\in W$ above, \nwe have the following inclusion of varieties when \nthey are regarded as subvarieties of $\\CG r$\\,{\\rm:}\n\\begin{equation*}\n\\mathbf{b}^{(1)} \\stra{e} \\mathbf{b}^{(2)}\n\\subset \n\\mathbf{b}^{(1)} \\stra{w} \\mathbf{b}^{(2),w}.\n\\end{equation*}\n\\end{claim*}\n\n\\noindent\n{\\it Proof of Claim.} \nLet $w = s_{i_{1}} s_{i_{2}} \\cdots s_{i_{\\ell}}$ be \na reduced expression, and set \n$w_{k} = s_{i_{1}} s_{i_{2}} \\cdots s_{i_{k}}$ \nfor $0 \\le k \\le \\ell$. \nFor simplicity of notation, we set for $0 \\le k \\le \\ell$\n\\begin{equation*}\n\\mathbf{b}^{(2)}_{k}:=\\mathbf{b}^{(2),w_{k}}, \\qquad\n\\mathbf{b}^{(1)} \\star_{k} \\mathbf{b}^{(2)}:=\n\\mathbf{b}^{(1)} \\stra{w_{k}} \\mathbf{b}^{(2), w_{k}}, \n\\quad \\text{and}\n\\end{equation*}\n\\begin{equation*}\n\\mu^{(1)}_{k}:=\\mu^{(1)}_{w_{k}}, \\quad\n\\mu^{(2)}_{k}:=\\mu^{(2)}_{w_{k}}; \n\\end{equation*}\nnote that\n\\begin{equation*}\n\\mathbf{b}^{(1)} \\star_{0} \\mathbf{b}^{(2)}=\n\\mathbf{b}^{(1)} \\stra{e} \\mathbf{b}^{(2),e}=\n\\mathbf{b}^{(1)} \\stra{e} \\mathbf{b}^{(2)}. \n\\end{equation*}\nIf we can show the inclusion\n\\begin{equation} \\label{eq:incl}\n\\mathbf{b}^{(1)} \\star_{k} \\mathbf{b}^{(2)} \\subset \n\\ol{ \\mathbf{b}^{(1)} \\star_{k+1} \\mathbf{b}^{(2)} }\n\\quad \\text{for each $0 \\le k \\le \\ell-1$},\n\\end{equation}\nthen we will obtain the following sequence of \ninclusions: \n\\begin{align*}\n& \\mathbf{b}^{(1)} \\stra{e} \\mathbf{b}^{(2)}=\n  \\mathbf{b}^{(1)} \\star_{0} \\mathbf{b}^{(2)} \\subset \n  \\ol{ \\mathbf{b}^{(1)} \\star_{1} \\mathbf{b}^{(2)} } \\subset \n  \\ol{ \\mathbf{b}^{(1)} \\star_{2} \\mathbf{b}^{(2)} } \\subset \\cdots \\\\\n& \\hspace*{20mm} \\cdots \\subset \n\\ol{ \\mathbf{b}^{(1)} \\star_{\\ell} \\mathbf{b}^{(2)} } =\n\\ol{ \\mathbf{b}^{(1)} \\stra{w_{\\ell}} \\mathbf{b}^{(2),w_{\\ell}} } =\n\\ol{ \\mathbf{b}^{(1)} \\stra{w} \\mathbf{b}^{(2),w} },\n\\end{align*}\nas desired. In order to show the inclusion \\eqref{eq:incl}, \ntake an element \n$[(y,\\,gG(\\mathcal{O}))] \\in \\mathbf{b}^{(1)} \\star_{k} \\mathbf{b}^{(2)}$, \nwhere \n\\begin{equation*}\ny \\in \n (\\mathbf{b}^{(1)} \\cap \\mathbb{S}^{w_{k}}_{\\mu^{(1)}_{k}})^{\\sim}\n \\subset {}^{w_{k}}U(\\mathcal{K})t^{\\mu^{(1)}_{k}}, \\qquad\ngG(\\mathcal{O}) \\in \n  \\mathbf{b}^{(2)}_{k} \\cap S^{w_{k}}_{\\mu^{(2)}_{k}} \\cong \n  \\mathbf{b}^{(2)}_{k} \\cap \\mathbb{S}^{w_{k}}_{\\mu^{(2)}_{k}}, \n\\end{equation*}\nand write the element $y \\in {}^{w_{k}}U(\\mathcal{K})t^{\\mu^{(1)}_{k}}$ \nas: $y =u_{k}t^{\\mu^{(1)}_{k}}$ for $u_{k} \\in {}^{w_{k}}U(\\mathcal{K})$. \nSince $\\mathbf{b}^{(1)}=\\ol{\\bigcap_{z \\in W} S^{z}_{\\mu^{(1)}_{z}}}$ \nby Theorem~\\ref{thm:Kam1}, we may (and do) assume that \n$yG(\\mathcal{O}) \\in S^{w_{k}}_{\\mu^{(1)}_{k}} \\cap \n S^{w_{k+1}}_{\\mu^{(1)}_{w_{k+1}}}$ to show \nthe inclusion \\eqref{eq:incl}. Therefore, we can take \n$u_{k+1} \\in {}^{w_{k+1}}U(\\mathcal{K})$ and $g_{k+1} \\in G(\\mathcal{O})$\nsuch that $y=u_{k}t^{\\mu^{(1)}_{k}}=\nu_{k+1}t^{\\mu^{(1)}_{k+1}}g_{k+1}$; note that \n\\begin{equation*}\ng_{k+1} \\in \n T(\\mathcal{K})({}^{w_{k+1}}U(\\mathcal{K}))({}^{w_{k}}U(\\mathcal{K}))T(\\mathcal{K}).\n\\end{equation*}\nHere, since $w_{k+1}=w_{k}s_{i_{k+1}}$ by definition, \nit follows that\n\\begin{equation*}\nT(\\mathcal{K})({}^{w_{k+1}}U(\\mathcal{K}))({}^{w_{k}}U(\\mathcal{K}))T(\\mathcal{K})\n = \\dot{w}_{k}\n \\bigl(\n  \\dot{s}_{i_{k+1}} U(\\mathcal{K})\\dot{s}_{i_{k+1}}U(\\mathcal{K})\n \\bigr) \\dot{w}_{k}^{-1}\n \\subset {}^{w_{k}}P_{i_{k+1}}(\\mathcal{K}),\n\\end{equation*}\nand hence that $g_{k+1} \\in {}^{w_{k}}P_{i_{k+1}}(\\mathcal{K})$. \nMoreover, since $g_{k+1} \\in G(\\mathcal{O})$, we get \n$g_{k+1} \\in {}^{w_{k}}P_{i_{k+1}}(\\mathcal{K}) \\cap G(\\mathcal{O})=\n{}^{w_{k}}P_{i_{k+1}}(\\mathcal{O})$. Therefore, we obtain \n\\begin{equation*}\ng_{k+1}\\mathbf{b}^{(2)}_{k} \\subset \n{}^{w_{k}}P_{i_{k+1}}(\\mathcal{O})\\mathbf{b}^{(2)}_{k}=\n{}^{w_{k}}L_{i_{k+1}}(\\mathcal{O})\n{}^{w_{k}}U_{i_{k+1}}(\\mathcal{O})\\mathbf{b}^{(2)}_{k}. \n\\end{equation*}\nSince the extremal MV cycle $\\dot{w}_{k}^{-1}\\mathbf{b}^{(2)}_{k}$ is stable under \n$U_{i_{k+1}}(\\mathcal{O}) \\subset U(\\mathcal{O})=G(\\mathcal{O}) \\cap U(\\mathcal{K})$ \n(see the proof of Lemma~\\ref{lem:stable}), we have\n${}^{w_{k}}U_{i_{k+1}}(\\mathcal{O})\\mathbf{b}^{(2)}_{k} \\subset \\mathbf{b}^{(2)}_{k}$, \nand hence\n\\begin{equation*}\ng_{k+1}\\mathbf{b}^{(2)}_{k} \\subset \n{}^{w_{k}}L_{i_{k+1}}(\\mathcal{O})\\mathbf{b}^{(2)}_{k}. \n\\end{equation*}\nAlso, we see that \n\\begin{align*}\n{}^{w_{k}}L_{i_{k+1}}(\\mathcal{O})\n\\mathbf{b}^{(2)}_{k} \n& ={}^{w_{k+1}}L_{i_{k+1}}(\\mathcal{O})\\mathbf{b}^{(2)}_{k}\n  \\quad \\text{since $w_{k+1}=w_{k}s_{i_{k+1}}$ and \n  $\\dot{s}_{i_{k+1}} \\in L_{i_{k+1}}$} \\\\[3mm]\n& \\subset \\mathbf{b}^{(2)}_{k+1}\n  \\quad \\text{by Lemma~\\ref{lem:tran} \n  since $w_{k} < w_{k}s_{i_{k+1}}=w_{k+1}$}.\n\\end{align*}\nCombining these, we obtain \n$g_{k+1}\\mathbf{b}^{(2)}_{k} \\subset \\mathbf{b}^{(2)}_{k+1}$, \nand hence $g_{k+1}gG(\\mathcal{O}) \\in g_{k+1}\\mathbf{b}^{(2)}_{k} \n\\subset \\mathbf{b}^{(2)}_{k+1}$. \nFurthermore, we have \n\\begin{align*}\n[(y,\\,gG(\\mathcal{O}))] & =\n[(u_{k}t^{\\mu^{(1)}_{k}},\\,gG(\\mathcal{O}))]=\n[(u_{k+1}t^{\\mu^{(1)}_{k+1}}g_{k+1},\\,gG(\\mathcal{O}))] \\\\ \n& = [(u_{k+1}t^{\\mu^{(1)}_{k+1}},\\,g_{k+1}gG(\\mathcal{O}))]\n\\end{align*}\nby the equivalence relation \n$\\sim$ on $G(\\mathcal{K}) \\times \\CG r$, and \n\\begin{equation*}\nyG(\\mathcal{O})=u_{k+1}t^{\\mu^{(1)}_{k+1}}G(\\mathcal{O}) \n \\in \\mathbf{b}^{(1)} \\cap S^{w_{k+1}}_{\\mu^{(1)}_{k+1}}\n\\end{equation*}\nby the choice above of $y$. \nConsequently, we conclude that \n\\begin{align*}\n[(y,\\,gG(\\mathcal{O}))] & \\in \n\\ol{ \\mathsf{m} \\Bigl(\n     \\bigl(\\mathbf{b}^{(1)} \\cap \\mathbb{S}^{w_{k+1}}_{ \\mu^{(1)}_{k+1} }\\bigr)^{\\sim}\n     \\times^{{}^{w_{k+1}}U(\\mathcal{O})} \n     \\mathbf{b}^{(2)}_{k+1}\n     \\Bigr)\n} \\\\[3mm]\n& = \n\\ol{ \\mathsf{m} \\Bigl(\n     \\bigl(\\mathbf{b}^{(1)} \\cap \\mathbb{S}^{w_{k+1}}_{ \\mu^{(1)}_{k+1} }\\bigr)^{\\sim}\n     \\times^{{}^{w_{k+1}}U(\\mathcal{O})} \n     \\bigl(\\mathbf{b}^{(2)}_{k+1} \\cap \\mathbb{S}^{w_{k+1}}_{ \\mu^{(2)}_{k+1} }\\bigr)\n     \\Bigr) \n} \\\\[3mm]\n& = \\ol{ \\mathbf{b}^{(1)} \\star_{k+1} \\mathbf{b}^{(2)} },\n\\end{align*}\nsince $\\ol{ \\mathbf{b}^{(2)}_{k+1} \\cap S^{w_{k+1}}_{ \\mu^{(2)}_{k+1} } }=\n\\mathbf{b}^{(2)}_{k+1}$ (see Remark~\\ref{rem:extcyc}).\nThis proves the inclusion \\eqref{eq:incl}, \nand hence the claim. \\quad \\hbox{\\rule[-0.5pt]{3pt}{8pt}}\n\n\\vspace{3mm}\n\nFinally, we complete the proof of Theorem~\\ref{thm:tensor}. \nBy the claim above, we obtain\n\\begin{equation} \\label{eq:prf2}\nS^{w}_{\\nu} \\cap \n\\bigl(\\ol{\\mathbf{b}^{(1)} \\stra{e} \\mathbf{b}^{(2)}}\\bigr)\n\\subset \nS^{w}_{\\nu} \\cap \n\\bigl(\\ol{\\mathbf{b}^{(1)} \\stra{w} \\mathbf{b}^{(2),w}}\\bigr). \n\\end{equation}\nAlso, we have \n\\begin{equation} \\label{eq:prf4}\nS^{w}_{\\nu} \\cap \n\\bigl(\\ol{ \\mathbf{b}^{(1)} \\stra{w} \\mathbf{b}^{(2),w} }\\bigr)\n\\subset \t\nS^{w}_{\\nu} \\cap \n\\ol{S^{w}_{\\mu^{(1)}_{w}+\\mu^{(2)}_{w}}}\n\\end{equation}\nby the inclusion $\\mathbf{b}^{(1)} \\stra{w} \\mathbf{b}^{(2),w} \\subset \nS^{w}_{\\mu^{(1)}_{w}+\\mu^{(2)}_{w}}$, which is \nan immediate consequence of the definition. \nSince $w^{-1} \\cdot \\nu \\not\\ge w^{-1} \\cdot \n(\\mu^{(1)}_{w}+\\mu^{(2)}_{w})$ by \\eqref{eq:prf1}, \nit follows from equation \\eqref{eq:Snuw} that\n$S^{w}_{\\nu} \\cap \n\\ol{S^{w}_{\\mu^{(1)}_{w}+\\mu^{(2)}_{w}}}=\\emptyset$,\nand hence by \\eqref{eq:prf2} together with \\eqref{eq:prf4} that \n\\begin{equation} \\label{eq:prf3}\nS^{w}_{\\nu} \\cap \n\\bigl(\\ol{\\mathbf{b}^{(1)} \\stra{e} \\mathbf{b}^{(2)}}\\bigr)=\\emptyset.\n\\end{equation}\n\nNow, we know from Theorem~\\ref{thm:BG} that\n\\begin{equation*}\nS_{\\mu^{(1)}_{e}+\\mu^{(2)}_{e}} \\cap \n\\mathsf{m}(\\Phi_{\\lambda_{1},\\,\\lambda_{2}}(P_{1},\\,P_{2})) \n\\subset \\Phi_{\\lambda}(P)\n\\end{equation*}\nis an open dense subset, where \n$\\mathop{\\rm wt}\\nolimits(P)=\\mu^{(1)}_{e}+\\mu^{(2)}_{e}$; \nfor an MV polytope $P \\in \\mathcal{MV}(\\lambda)$, choosing \nan embedding $\\iota_{\\lambda}:\\mathcal{MV}(\\lambda) \\hookrightarrow \n\\mathcal{MV}(\\lambda_{2}) \\otimes \\mathcal{MV}(\\lambda_{1})$ of crystals \nso that $\\iota_{\\lambda}(P)=P_{2} \\otimes P_{1}$ \nfor some $P_{1} \\in \\mathcal{MV}(\\lambda_{1})$ and \n$P_{2} \\in \\mathcal{MV}(\\lambda_{2})$ corresponds, \nvia Theorems~\\ref{thm:Kam1} and \\ref{thm:BG}, to choosing \nan irreducible component \n$\\mathcal{X} \\in \\mathop{\\rm Irr}\\nolimits(\\CG r^{\\lambda_{1},\\,\\lambda_{2}} \\cap S_{\\nu_{1},\\,\\nu_{2}})$ \nsuch that $\\Phi_{\\lambda}(P) \\in \\mathcal{Z}(\\lambda)$ is the (Zariski-) closure \nof the image of $\\mathcal{X}$ under the map $\\star$ \nin the commutative diagram of Theorem~\\ref{thm:BGc}\nin the Appendix, where $\\nu_{1}=\\mathop{\\rm wt}\\nolimits(P_{1})=\\mu^{(1)}_{e}$ and \n$\\nu_{2}=\\mathop{\\rm wt}\\nolimits(P_{2})=\\mu^{(2)}_{e}$.\nAlso, we see from the explicit construction of \n$\\Phi_{\\lambda_{1},\\,\\lambda_{2}}(P_{1},\\,P_{2})$ \ngiven in Theorem~\\ref{thm:BG} that \n\\begin{equation*}\n\\mathbf{b}^{(1)} \\stra{e} \\mathbf{b}^{(2)} \n\\subset\nS_{\\mu^{(1)}_{e}+\\mu^{(2)}_{e}} \\cap \n\\mathsf{m}(\\Phi_{\\lambda_{1},\\,\\lambda_{2}}(P_{1},\\,P_{2}))\n\\end{equation*}\nis an open dense subset. Therefore, \n$\\mathbf{b}^{(1)} \\stra{e} \\mathbf{b}^{(2)} \\subset \\Phi_{\\lambda}(P)$ \nis an open dense subset. \nCombining this fact with \\eqref{eq:prf3}, we conclude that \n$\\Phi_{\\lambda}(P) \\cap S^{w}_{\\nu}=\\emptyset$, \nwhich contradicts \\eqref{eq:prf1}. Thus, we have completed \nthe proof of Theorem~\\ref{thm:tensor}. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTo quote from \\citet{benjamini2014discussion},\n\\begin{quote}\nSignificance testing is an effort to address the selection of an\ninteresting finding regarding a single parameter from the background\nnoise. Modern science faces the problem of selection of promising\nfindings from the noisy estimates of many.\n\\end{quote}\nThis paper is about the latter. In contemporary studies, geneticists\nmay have measured hundreds of thousands of genetic variants and wish\nto know which of these influence a trait \\citep{sesia2019gene,\n  sesia2020multi}. Scientists may be interested in discovering which\ndemographic and clinical variables influence the susceptibility to\nParkinson's disease \\citep{gao2018model}. Economists study which\nvariables from individual employment and wage histories affect future\nprofessional careers \\citep{klose2020pipeline}. In all these examples\nand countless others, we have hundreds or even thousands of\nexplanatory variables and are interested in determining which of these\ninfluence a response of interest. The problem is to select\nassociations which are replicable, that is, without having too many\nfalse positives.\n\nFormally, let $Y \\in \\mathbb{R}$ be the response we wish to study, and\n$X = (X_1, X_2, \\ldots, X_p) \\in \\mathbb{R}^p$ be the vector of\nexplanatory variables. We call variable $j$ a null variable if $X_j$\nis conditionally independent of $Y$ given the other $X$'s. This says\nthat the $j$-th variable does not provide information about the\nresponse beyond what is already provided by all the other variables\n(roughly, if it is not in the Markov blanket of $Y$).  Expressed\ndifferently, a variable is null if and only if the hypothesis\n\\begin{equation}\n\\label{eqn:indep}\n\\mathcal{H}_j: X_j \\independent Y | X_{-j}, \n\\end{equation}\nis true. (Throughout, $X_{-j}$ is a shorthand for all $p$ variables\nexcept the $j$th.) Likewise, a variable $j$ is nonnull if\n$\\mathcal{H}_j$ is false.  Let\n$\\mathcal{H}_0 \\subset \\cb{1, \\dots, p}$ be the subset of\nnulls. Suppose now we have $n$ independent samples assembled in a data\nmatrix $\\mathbf{X} \\in \\mathbb{R}^{n \\times p}$ and a response vector\n$\\mathbf{Y} \\in \\mathbb{R}^n$. The goal is to identify the nonnull\nvariables with some form of type-I error control. Specifically, we\nconsider in this paper the false discovery rate (FDR)\n\\citep{benjamini1995controlling}, namely, the expected fraction of\nfalse positives defined as\n\\begin{equation}\n\\operatorname{FDR} = \\EE{\\frac{|\\hat{\\mathcal{S}} \\cap \\mathcal{H}_0|\n  }{|\\hat{\\mathcal{S}}| \\vee 1  }},\\footnote{Here and below, $a \\vee b = \\max(a,b)$ and\n  $a\\wedge b = \\min(a,b)$}.\n  \\end{equation}\nwhere $\\hat{\\mathcal{S}}$ is the selected set of variables.\n\n\n\n\\subsection{The conditional randomization test}\nNaturally, in order to identify the nonnull variables, one could test\nthe hypotheses $\\mathcal{H}_j$ in \\eqref{eqn:indep}.\n\\citet{candes2018panning} proposed to achieve this via the conditional\nrandomization test (CRT). To run the CRT, we resample $\\mathbf{X}_j$---the\n$j$th column of the matrix $\\mathbf{X}$---conditional on the other variables,\ncalculate the value of a test statistic, and compare it to the test\nstatistic computed on the true $\\mathbf{X}_j$. When the statistic computed on\nthe true $\\mathbf{X}_j$ has a high rank when compared with those obtained\nfrom imputed values, this is evidence against the null. Details of the\nCRT are given in Algorithm~\\ref{alg:crt}. There, the output p-value is\nvalid in the sense that under the null, it is stochastically larger\nthan a uniform variable.\n\\begin{algorithm}[h]\n\\caption{Conditional Randomization Test (CRT)}\n\\label{alg:crt}\n\\begin{algorithmic}\n\\REQUIRE Data $(\\mathbf{X},\\mathbf{Y})$, test statistic $T(\\cdot)$, number of randomizations $B$\n\\FOR{$b \\in \\cb{1, \\dots, B}$}\n\t\\STATE Sample $\\mathbf{X}_j^{(b)}$ from the distribution of $\\mathbf{X}_j | \\mathbf{X}_{-j}$, independently of $\\mathbf{X}_j$ and $\\mathbf{Y}$.\n\\ENDFOR\n\\ENSURE  The $p\\operatorname{-value}$\n\\[ p_j=\\frac{1}{B+1}\n\t\\p{1 + \\sum_{b = 1}^B \\one{\\cb{T(\\mathbf{X}_j, \\mathbf{X}_{-j}, \\mathbf{Y}) \\leq T(\\mathbf{X}_j^{(b)}, \\mathbf{X}_{-j}, \\mathbf{Y})}}}.  \\footnotemark\n\t\\] \n\\end{algorithmic}\n\\end{algorithm}\n\\footnotetext{If some values of the test statistics are the same, we break ties randomly. }\n\nInformally, under the null hypothesis that the variable $X_j$ is\nindependent of $Y$ conditional on $X_{-j}$, each one of the new\nsamples $\\mathbf{X}^{(b)}_j$ has the same distribution as $\\mathbf{X}_j$, and they are\nall independent conditionally on $\\mathbf{Y}$ and $\\mathbf{X}_{-j}$. As a\nconsequence, each $T(\\mathbf{X}^{(b)}_j, \\mathbf{X}_{-j}, \\mathbf{Y})$ has the same distribution\nas $T(\\mathbf{X}_j, \\mathbf{X}_{-j}, \\mathbf{Y})$. Thus the rank of\n$T(\\mathbf{X}_j, \\mathbf{X}_{-j}, \\mathbf{Y})$ among $\\cb{T(\\mathbf{X}^{(b)}_j, \\mathbf{X}_{-j}, \\mathbf{Y})}$ will\nbe uniform in $\\cb{1, \\dots, B}$ assuming we break ties at\nrandom. Formally, we have:\n\\begin{theo}[\\citet{candes2018panning}]\n If $X_j \\independent Y | X_{-j}$, then the $p\\operatorname{-values}$ from Algorithm\n \\ref{alg:crt} satisfy  $\\PP{p_j \\leq \\alpha} \\leq \\alpha$, for any\n $\\alpha \\in [0,1]$. This holds regardless of the test statistic\n $T(\\cdot)$. \n\\end{theo}\nThe validity of the procedure does not rely on any assumptions on the\ndistribution of $Y|X$, parametric or not. Yet it requires knowledge of\nthe distribution of the covariates $X$. This is known as the Model-X\nframework, and is an appropriate assumption in many important\napplications, including genetic and economics studies, where either\nknowledge about the exact covariate distribution or a large amount of\nunsupervised data of covariates is available \\citep{cong2013multiplex,\n  haldane1931inbreeding, peters2016comprehensive, saltelli2008global,\n  tang2006reconstructing}. Rapid progress has been made on\nmethodological advances in this framework \\citep{candes2018panning,\n  tansey2018holdout, sesia2019gene, bates2020metropolized,\n  liu2020fast, romano2020deep, ren2020derandomizing} and in applications\nto genetic studies \\citep{sesia2019gene, sesia2020multi,\n  bates2020causal}.\n\n\\subsection{Selective SeqStep+}\nWe still need a selection procedure that transforms the CRT $p\\operatorname{-values}$\ninto a selected set with FDR control guarantees. A natural choice of\nvariable selection procedure is the Benjamini-Hochberg procedure (BHq)\n\\citep{benjamini1995controlling}. If we were to apply BHq, we would\nneed to compare the $p\\operatorname{-values}$ with critical thresholds of the form\n$\\alpha_i = i q\/p$, where $q$ is the nominal FDR level.  When $p$ is\nlarge, $q$ is $0.1$, and $i = 1,2, \\dots$, this requires $p\\operatorname{-values}$ on\nan extremely fine scale.  The $p\\operatorname{-values}$ defined in Algorithm\n\\ref{alg:crt}, however, can only take values in\n$\\{1\/(B+1), 2\/(B+2), \\dots, 1\\}$, thus a huge number of randomizations\nin CRT is required.  This makes the combination of CRT and BHq\ncomputationally expensive or even infeasible. This is the motivation\nfor this paper: can we find a\nselection procedure that does not require any of the $p\\operatorname{-values}$ to be\nvery small and works well with discrete $p\\operatorname{-values}$?\n\n\\begin{algorithm}[t]\n\\caption{Selective SeqStep+}\n\\label{alg:seqstep}\n\\begin{algorithmic}\n\\REQUIRE A sequence of $p\\operatorname{-values}$ $p_1, \\dots p_p$\n\\STATE \nLet\n\\begin{equation}\n\\label{eqn:seqstep}\n\\hat{k} =\\max \\left\\{k \\in \\cb{1,\\dots, p}: \\frac{1+\\#\\left\\{j \\leq k: p_{j}>c\\right\\}}{\\#\\left\\{j \\leq k: p_{j} \\leq c\\right\\} \\vee 1} \\leq \\frac{1-c}{c} \\cdot q\\right\\}. \n\\end{equation}\n\\ENSURE  Selected set of nonnulls $\\hat{\\mathcal{S}} = \\cb{ j \\leq \\hat{k}: p_{j} \\leq c}$. \nIf the set in $\\eqref{eqn:seqstep}$ is empty, $\\hat{\\mathcal{S}}$ is the empty set as well.  \n\\end{algorithmic}\n\\end{algorithm}\n\nTo this end, we consider SeqStep+, a sequential testing procedure\nfirst introduced by \\citet{barber2015controlling}. We consider a\nspecific version, namely, Selective SeqStep+, which takes\na sequence of $p\\operatorname{-values}$ $p_1, \\dots p_p$ as input, and outputs a\nselected set $\\hat{\\mathcal{S}}$. The procedure starts by finding an\ninteger $\\hat{k}$ such that among the $p\\operatorname{-values}$\n$\\cb{p_1, \\dots, p_{\\hat{k}}}$, few are greater than a user-specified\nthreshold $c$. In details, $\\hat{k}$ is the largest $k$ in\n$\\cb{1, \\dots p}$ such that the ratio between\n$1+\\#\\left\\{j \\leq k: p_{j}>c\\right\\}$ and\n$\\#\\left\\{j \\leq k: p_{j} \\leq c\\right\\} \\vee 1$ is no greater than\n$(1-c)q\/c$. The procedure then selects all $j$'s, such that\n$j \\leq \\hat{k}$ and $p_j \\leq c$. We include details of the procedure\nin Algorithm~\\ref{alg:seqstep}.\n\nTo understand why the procedure works, assume that the null $p\\operatorname{-values}$ are $\\text{i.i.d.}\\operatorname{Unif}[0,1]$. Then, the ratio of $\\#\\left\\{\\text {null } j \\leq \\hat{k}: p_{j} \\leq c\\right\\}$ to $\\#\\left\\{\\text {null } j \\leq \\hat{k}: p_{j}>c\\right\\}$ is roughly $c\/(1-c)$. Hence \n\\begin{align*}\n\\operatorname{FDP} &= \\frac{\\#\\left\\{\\text {null } j \\leq \\hat{k}: p_{j} \\leq c\\right\\}}{\\#\\left\\{j \\leq \\hat{k}: p_{j} \\leq c\\right\\} \\vee 1}  \n\\approx \\frac{c}{1-c} \\cdot \\frac{\\#\\left\\{\\text {null } j \\leq \\hat{k}: p_{j}>c\\right\\}}{\\#\\left\\{j \\leq \\hat{k}: p_{j} \\leq c\\right\\} \\vee 1} \\\\\n&  \\leq \\frac{c}{1-c} \\cdot \\frac{\\#\\left\\{j \\leq \\hat{k}: p_{j}>c\\right\\}}{\\#\\left\\{j \\leq \\hat{k}: p_{j} \\leq c\\right\\} \\vee 1}\n\\leq \\frac{c}{1-c} \\cdot \\frac{1-c}{c}q = q. \n\\end{align*}\nFormally, we have the following result. \n\\begin{theo}[\\citet{barber2015controlling}]\n\\label{theo:indep_FDR}\nAssume that the ordering of the $p\\operatorname{-values}$ is fixed. If all null $p\\operatorname{-values}$ are independent with $p_j \\geq \\operatorname{Unif} [0,1]$, and are independent from the nonnulls, then Selective SeqStep+ controls the FDR at level $q$. \n\\end{theo}\n\nA close look at \\eqref{eqn:seqstep} shows that the only information\nSelective SeqStep+ uses from the $p\\operatorname{-values}$ is whether or not\n$p_j \\leq c$. This means that unlike BHq, Selective SeqStep+ does not\nrequire some of the $p\\operatorname{-values}$ to be very small to make rejections, and\nhence would require a much smaller number of randomizations $B$.\nSelective SeqStep+ would, therefore, be computationally far less\nintensive.\n\n\\subsection{Challenges and our contribution}\n\\label{subsection:introduction_challenges}\n\nIn this paper, we study variable selection procedures with the\nconditional randomization test and Selective SeqStep+. There are three challenges and we address them all. \n\nThe first challenge is in the dependency of the $p\\operatorname{-values}$. The $p\\operatorname{-values}$\nfrom CRT are not independent in general, hence Theorem\n\\ref{theo:indep_FDR} does not apply. In response, we will develop\ntheory in Section \\ref{section:theory} showing how we can make\nSeqStep+ valid under dependence. In particular, we will show examples\nof approximate FDR control when the $p\\operatorname{-values}$ are weakly dependent or\nwhen they are exchangeable in distribution.\n\n\\begin{figure}\n\\begin{subfigure}{\\textwidth}\n\\caption{A good ordering}\n\\includegraphics[width = \\textwidth]{plots\/good_ordering.pdf}\n\\label{fig:good_ordering}\n\\end{subfigure}\n\\begin{subfigure}{ \\textwidth}\n\\caption{A bad ordering}\n\\includegraphics[width = \\textwidth]{plots\/bad_ordering.pdf}\n\\label{fig:bad_ordering}\n\\end{subfigure}\n\\caption{An illustration of the ordering of the $p\\operatorname{-values}$ in Selective SeqStep+. The null $p\\operatorname{-values}$ are sampled from $\\operatorname{Unif}[0,1]$ and the nonnull $p\\operatorname{-values}$ from $\\operatorname{Unif}[0,0.2]$. We take $c = 0.2$ and $q = 0.1$. \n The $p\\operatorname{-values}$ are the same in the two plots, but they are ordered in two different ways. In (a), the nonnulls appear early in the sequence. In (b), the order of the $p\\operatorname{-values}$ is random. In terms of power, Selective SeqStep+ discovers all the nonnulls in (a) but only a subset of them in (b). }\n\\label{fig:ordering}\n\\end{figure}\n\nThe second challenge concerns the ordering of the $p\\operatorname{-values}$.  Unlike\nthe Benjamini-Hochberg procedure, which takes as input the $p\\operatorname{-values}$\nonly, Selective SeqStep+ essentially requires two inputs: the $p\\operatorname{-values}$\nand an ordering of the $p\\operatorname{-values}$. In other words, if we change the\norder of the input $p\\operatorname{-values}$, we could end up selecting a very\ndifferent set of variables. To illustrate this, consider the example\nin Figure \\ref{fig:ordering}, which fixes the $p\\operatorname{-values}$ and compare two\ndifferent orderings. The ``good\" ordering has the nonnulls appear\nearly in the sequence and the ``bad\" ordering randomly permutes the\n$p\\operatorname{-values}$. With the good ordering, the output set contains all the\nnonnulls; but with the bad ordering, only a fraction of the nonnulls is \ndiscovered. When the nonnull $p\\operatorname{-values}$ appear early in the sequence,\nthe proportion of $p\\operatorname{-values}$ greater than $c$ will be smaller, thus the\nquantity\n$\\frac{1+\\#\\left\\{j \\leq k: p_{j}>c\\right\\}}{\\#\\left\\{j \\leq k: p_{j}\n    \\leq c\\right\\} \\vee 1}$ in \\eqref{eqn:seqstep} will tend to be\nsmaller. Therefore a larger $\\hat{k}$ will be obtained and hence the\npower of the procedure will be higher. In general, to make the\nvariable selection procedure more powerful, it is important to look\nfor an informative ordering that places nonnulls early in the\nsequence.\n\nAnother requirement for the ordering is that it needs to be independent of the $p\\operatorname{-values}$. FDR is in general not controlled when the $p\\operatorname{-values}$ and the ordering are dependent. \n As a simple example, assume a researcher obtains independent $p\\operatorname{-values}$ and naively orders them by magnitude. Then the input sequence of $p\\operatorname{-values}$ into Selective SeqStep+ would be an ordered sequence $p_{(1)} \\leq p_{(2)} \\dots \\leq  p_{(p)}$. In this case, the null $p\\operatorname{-values}$ that appear early in the sequence will tend to be smaller and hence no longer uniform. In the case of the global null (all hypotheses are null) with independent $p\\operatorname{-values}$, we would expect to make around $c p$ false discoveries. This is because there are approximately $c p$ $p\\operatorname{-values}$ that are smaller than or equal to $c$, and they all appear early in the sequence, hence for $k = (c + (1-c)q)p$, \n\\[ \\frac{1 + \\#\\left\\{j \\leq k: p_{j}>c\\right\\}}{\\#\\left\\{j \\leq k: p_{j} \\leq c\\right\\} \\vee 1} \\approx \\frac{(c + (1-c)q)p - cp}{cp} = \\frac{(1-c)q}{c}.\\]\n\nIn Section \\ref{section:method}, we will present two methods to obtain\nthe ordering: the {\\em split} version and the {\\em symmetric\n  statistic} version.  The former splits the data into two parts,\nobtaining $p\\operatorname{-values}$ from one fold, and the ordering from the\nother. This makes the ordering and the $p\\operatorname{-values}$ stochastically\nindependent. No data-splitting is required for the symmetric statistic\nversion; we obtain both the $p\\operatorname{-values}$ and the ordering from the whole\ndataset. To obtain the ordering, we compute a statistic $z_j$ for each\nvariable $j$, and sort the $z_j$'s. The statistic $z_j$ is obtained in\nsuch a way that $z_j$ is marginally independent of the $p\\operatorname{-value}$ $p_j$.\nIn theory, this notion of independence is not sufficient for FDR\ncontrol; we however tested this method in many different empirical\nsettings and always controlled the $\\operatorname{FDR}$.  In terms of power, the\nsymmetric statistic version is more powerful than the split\nversion. Thus in practice, we would recommend the symmetric statistic\nversion.\n\nThe third challenge is computational in nature. Recall that with the\nCRT (Algorithm \\ref{alg:crt}), we need to compute the test statistic\n$T_j^{(b)}$ for each $j$ and each $b$. Each statistic $T_j^{(b)}$ is\nobtained by sampling $\\mathbf{X}_j^{(b)}$ and running a machine learning\nalgorithm with $\\mathbf{Y}$ as a response and $\\mathbf{X}_j^{(b)}, \\mathbf{X}_{-j}$ as\npredictors. It is computationally expensive to run the machine\nlearning algorithm $B$ times to get a single $p\\operatorname{-value}$. In Section\n\\ref{section:computation}, we will present a faster way of obtaining\nthe test statistics and, hence, the $p\\operatorname{-values}$.\n\n\\section{Selective SeqStep+ under dependence}\n\\label{section:theory}\n\n\\subsection{Almost independent $p$-values}\n\\label{subsection:cond_prob}\nWhen employing SeqStep+, it is natural to ask whether the FDR is still\ncontrolled when the $p\\operatorname{-values}$ are ``close'' to being independent. This\nsection derives an bound upper bound on the FDR, which depends on the\nvalue of\n$\\max_{j \\in \\mathcal{H}_0} \\mathbb{P}[p_j \\leq c \\mid \\one\\cb{p_{-j} \\leq\n  c}]$,\\footnote{For a set\n  $S = \\cb{j_1, \\dots, j_K} \\subset \\cb{1, \\dots, p}$, we define\n  $\\one\\cb{p_S \\leq c}$ as the Boolean vector\n  $(\\one\\cb{p_{j_1} \\leq c}, \\dots, \\one\\cb{p_{j_K} \\leq c}) $.} the\nmaximum of the probability that $p_j$ is at most $c$ conditional on the boolean sequence of whether other $p\\operatorname{-values}$ are smaller than or equal to $c$. \nUnder independence of the $p\\operatorname{-values}$, it holds that\n$\\max_{j \\in \\mathcal{H}_0} \\PP{p_j \\leq c \\mid \\one\\cb{p_{-j} \\leq c}} \\leq c$\nsince marginally, $\\PP{p_j \\leq c} \\leq c$ for $j \\in \\mathcal{H}_0$. Our first\nresult states that if\n$\\max_{j \\in \\mathcal{H}_0} \\PP{p_j \\leq c \\mid \\one\\cb{p_{-j} \\leq c}}$ is\nclose to $c$ with high probability, then the FDR inflation cannot be\nlarge.\n\\begin{theo}[Almost independent $p$-values]\n\\label{theo:cond_prob}\nSuppose the ordering of the $p\\operatorname{-values}$ is fixed. Set\n$a_j = \\PP{p_j \\leq c \\mid \\one\\cb{p_{-j} \\leq c}}$ and assume the\n$p\\operatorname{-values}$ satisfy\n$ \\PP{ \\max_{j \\in \\mathcal{H}_0} a_{j} \\leq c + \\delta} \\geq 1 -\n\\epsilon$. Then the output from Algorithm \\ref{alg:seqstep} obeys\n\\begin{equation}\n\\label{eqn:theo_cond_prob}\n \\operatorname{FDR} \\leq q \\frac{c+\\delta}{c}\\frac{1-c}{1-c-\\delta} + \\epsilon. \n\\end{equation}\n\\end{theo}\n\n\n\nAs an illustration, we describe two examples where the FDR bound can\nbe computed numerically. Consider data $(\\mathbf{X}, \\mathbf{Y})$, where each row of\n$\\mathbf{X}$ is generated independently from a multivariate Gaussian\ndistribution with block diagonal covariance; that is, $X_j$ is only\ndependent on nearby variables. Recalling that the $p\\operatorname{-values}$ are\nobtained from Algorithm \\ref{alg:crt}, the first example takes the\nmarginal test statistic to be\n$T(\\mathbf{X}_j, \\mathbf{X}_{-j}, \\mathbf{Y}) = \\abs{\\Corr{\\mathbf{X}_j, \\mathbf{Y}}}$, whereas in the\nsecond example, we regress $\\mathbf{Y}$ on $\\mathbf{X}_j$ and $\\mathbf{X}_{N(j)}$, and take\nthe test statistic $T(\\mathbf{X}_j, \\mathbf{X}_{-j}, \\mathbf{Y})$ to be the absolute value\nof the fitted coefficient of $\\mathbf{X}_j$. Here, the elements of $N(j)$ are\nthe ``neighbors\" of $j$, i.e. the indices in the same block as $j$.\nAdditional details of the simulation settings are included in Appendix\n\\ref{subsection:simulation_details_cond_prob}.\n\nBefore proceeding with the computation, we note that if $a_j$ were\ndefined conditional on additional information,\ne.g.~$a_j = \\PP{p_j \\leq c \\mid \\mathbf{Y}, \\one\\cb{p_{-j} \\leq c}}$, then\nTheorem \\ref{theo:cond_prob} would still hold. The specific block\ndiagonal structure of the covariance of $X$ implies that $X_i$ and\n$X_j$ are independent conditionally on $Y$ if $i$ and $j$ are not in\nthe same block.  Thus,\n$a_j = \\PP{p_j \\leq c \\mid\\mathbf{Y}, \\one\\cb{p_{-j} \\leq c}} = \\PP{p_j \\leq\n  c \\mid \\mathbf{Y}, \\one\\cb{p_{N(j)} \\leq c}}$. For a block of size $K$, the\nvariable $\\one\\cb{p_{N(j)} \\leq c}$ can take at most $2^K$ distinct\nvalues. In practice, the conditional probability\n$\\PP{p_j \\leq c \\mid \\mathbf{Y}, \\one\\cb{p_{N(j)} \\leq c}}$ can therefore be\nestimated using sample proportions. One can fix $\\mathbf{Y}$, sample $\\mathbf{X}$\nfrom the distribution of $X \\mid Y$, compute the corresponding\n$p\\operatorname{-values}$, and compute the frequency of the event $\\cb{p_j \\leq c}$\nconditional on the value of $\\one\\cb{p_{N(j)} \\leq c}$. This is the\nreason why the block diagonal structure of the covariance is used\nhere; this structure makes computations tractable since we are dealing\nwith $2^K$ rather than $2^{p-1}$ possible configurations.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width = 0.9\\textwidth]{plots\/hist_max_aj.pdf}\n\\caption{Histogram of $\\max_{j \\in \\mathcal{H}_0} a_{j}$ from 500\n  samples. In (a), the test statistic of CRT is taken to be the\n  absolute value of the correlation. In (b), the test statistic of CRT\n  is taken to be the absolute value of the regression coefficient. We\n  plot the threshold $c$ (blue dashed line). To apply the bound from\n  Theorem \\ref{theo:cond_prob}, one possible choice of $\\delta$ is\n  shown as the distance between the red and blue lines.}\n\\label{fig:hist_max_aj}\n\\end{figure}\n\nIn Figure \\ref{fig:hist_max_aj}, we plot the histogram of\n$\\max_{j \\in \\mathcal{H}_0} a_{j} = \\max_{j \\in \\mathcal{H}_0} \\PP{p_j \\leq c \\mid \\mathbf{Y}, \\one\\cb{p_{N(j)} \\leq c}}$ from 500 samples and show a possible choice of $\\delta$\nand $\\epsilon$.  Here, the FDR threshold $q$ is set to be 0.1 and $c$\nis chosen to be $0.3$.  In the example where the test statistic is the\nabsolute value of correlation between $\\mathbf{X}$ and $\\mathbf{Y}$, we can take\n$\\delta = 0.0893$ and $\\epsilon = 0.002$. The FDR bound in\n\\eqref{eqn:theo_cond_prob} is thus\n$q \\frac{c+\\delta}{c}\\frac{1-c}{1-c-\\delta} + \\epsilon = 0.1508$. In\nthe other example where the test statistic is the absolute value of\nthe fitted regression coefficient, we can take $\\delta = 0.11$ and\n$\\epsilon = 0.006$. The FDR bound in \\eqref{eqn:theo_cond_prob} is\nthus $q \\frac{c+\\delta}{c}\\frac{1-c}{1-c-\\delta} + \\epsilon = 0.1682$.\n\n\n\n\n\\iffalse\n\nA corresponding lower bound of achieving $ \\operatorname{FDR} \\approx q \\frac{1-c}{1-c-\\delta} $, when the null $p\\operatorname{-values}$ satisfy $p_{j} \\geq \\operatorname{Unif} [0,1]$ marginally. \n\n\\paragraph{An example: lower bound}\n\nLet $n_1$ be the number of nonnulls, $n_0$ be the number of nulls. $n = n_0 + n_1$. Consider an example where the variables $1, \\dots n_1$ are nonnulls and $n_1 + 1, \\dots n$ are nulls. Assume the nonnull $p\\operatorname{-values}$ are all 0. For the null $p\\operatorname{-values}$, they have a joint distribution of:\n\\begin{enumerate}\n\\item \nWith probability $\\delta\/(c+\\delta)$, they are i.i.d. $\\operatorname{U}[c,1]$.\n\n\\item \nWith probability $c\/(c+\\delta)$, they are i.i.d. Each one of the distribution is $\\operatorname{U}[0,c]$ with probability $c+\\delta$, and $\\operatorname{U}[c,1]$ with probability $1-c-\\delta$. \n\\end{enumerate}\n\nWe can easily verify that $a_j \\leq c+\\delta$ and hence we have $\\epsilon = 0$ as in proposition \\ref{prop:FDR_bound}. The marginal distribution of each null $p\\operatorname{-value}$ is $\\operatorname{U}[0,1]$. \n\nUnder the first scenario, the FDR is always 0 as we reject nothing. Under the second scenario, all the inequality in the proof in proposition \\ref{prop:FDR_bound} become equality except for the one that states \n\\[\\frac{ \\#\\cb{j\\leq \\hat{k}: p_j \\leq c, a_j \\leq c+ \\delta}}{1 +   \\#\\cb{j\\leq \\hat{k}: p_j > c} }  \\leq \\frac{1-c}{c}q. \n\\]\nThis will become very close to equality when both $n_1$ and $n_0$ goes to infinity. With all these, we show that the FDR can be very close to\n\\[ \\frac{c}{c+ \\delta}  q \\frac{c+\\delta}{c}\\frac{1-c}{1-c-\\delta} = q \\frac{1-c}{1-c-\\delta}. \\] \n\n\\fi\n\n\\subsection{Under exchangeability}\nIn this section, we study whether additional structure on the $p\\operatorname{-values}$\ncan be helpful in obtaining sharper FDR bounds.  To this end, consider\nthe assumption of exchangeability. We say that the random variables\n$A_1, A_2, \\dots, A_m$ are \\emph{exchangeable} conditional on a random\nvariable $B$ if\n$(A_{1}, \\dots, A_{m} ) \\mid B \\stackrel{d}{=} (A_{\\pi(1)}, \\ldots,\nA_{\\pi(m)} ) \\mid B$ for any permutation $\\pi$. With this, this section makes use of the following assumption:\n\\begin{assu}\n\\label{assu:exch}\nThe null $p\\operatorname{-values}$ are exchangeable conditional on the nonnull $p\\operatorname{-values}$.\n\\end{assu}\n\nTo understand Assumption \\ref{assu:exch}, we study examples where it\nholds. Consider $p\\operatorname{-values}$ obtained from the CRT. A sufficient set of\nconditions is that the variables $X_j$'s are exchangeable and that the\ntest statistic $T(\\cdot)$ in the CRT (Algorithm \\ref{alg:crt}) is\nsymmetric in $\\mathbf{X}_{-j}$.  As a concrete example, imagine $X$ follows a\n$\\mathcal{N}(\\mathbf{\\mu}, \\Sigma)$ distribution, where all entries in\n$\\mu$ are the same and all off-diagonal terms in $\\Sigma$ are the\nsame. Then if $T(\\mathbf{X}_j, \\mathbf{X}_{-j}, \\mathbf{Y})$ is obtained by running a lasso\nregression of $\\mathbf{Y}$ on $\\mathbf{X}$ and taking the regression coefficient\nof $j$, then the null $p\\operatorname{-values}$ are exchangeable conditional on the\nnonnulls.\n\nUnder the assumption of exchangeability, we can show that FDR\ninflation will not be large. In particular, if the $p\\operatorname{-values}$ are weakly\ncorrelated with each other, we get a sharper upper bound.\n\\begin{theo}[Under exchangeability]\n\\label{theo:exchangeable}\nSuppose the ordering of the $p\\operatorname{-values}$ is fixed and that the nulls are\nmarginally stochastically larger than uniform. Under Assumption\n\\ref{assu:exch}, Algorithm \\ref{alg:seqstep} gives\n\\begin{equation}\n\\label{eqn:exch_bound1}\n\\operatorname{FDR} \\leq q + c(1-q).\n\\end{equation}\nIf, in addition, the $p\\operatorname{-values}$ satisfy $\\Corr{\\one\\cb{p_i \\leq c}, \\one\\cb{p_j \\leq c}} \\leq \\rho$ for any nulls $i \\neq j$, then\n\\begin{equation}\n\\label{eqn:exch_bound2}\n\\operatorname{FDR} \\leq q + \\varepsilon(c, q, \\rho),\n\\end{equation}\nwhere\n$$ \\varepsilon(c, q, \\rho) = \\p{\\frac{\\delta}{1 + \\beta \\delta}\n  \\sqb{\\frac{c}{1-c} - \\frac{c - c\\delta}{1 - (c-c\\delta)}q} } \\wedge\nc(1-q), \\quad \\beta = \\frac{c + (1-c)q}{(1-c)(1-q)}, \\quad \\delta =\n\\rho \\frac{c(1-q) + q}{c(1-q)}.$$ The two bounds \\eqref{eqn:exch_bound1} and\n\\eqref{eqn:exch_bound2} are sharp asymptotically. For illustration,\nthe bound \\eqref{eqn:exch_bound2} is plotted in\nFigure~\\ref{fig:exch_bounds}.\n\\end{theo}\n\n\\begin{proof}\nWe will show the asymptotic sharpness of \\eqref{eqn:exch_bound1} here. Specifically, we will show an example where the $\\operatorname{FDR}$ converges to $q + c(1-q)$ as $p \\to \\infty$. We include a proof of the two upper bounds and the asymptotic sharpness of \\eqref{eqn:exch_bound2} in Appendix \\ref{subsection:proof_exch}.\n\nAssume we are under the global null, i.e., all variables are\nnulls. Set $m_0 = 1 + \\lceil \\frac{c p}{q + c(1-q)} \\rceil$ and\nconsider null $p\\operatorname{-values}$ sampled as follows:\n\\begin{enumerate}\n\\item With probability $c p\/m_0$, pick $m_0$ indices uniformly at\n  random from $\\cb{1, \\dots, p}$, and sample the corresponding\n  $p\\operatorname{-values}$ as $\\text{i.i.d.}\\operatorname{Unif}[0,c]$; sample the other $p\\operatorname{-values}$\n  independently from $\\operatorname{Unif}[c,1]$.\n\n\\item With probability $1 - c p\/m_0$, sample all $p\\operatorname{-values}$ as\n  $\\text{i.i.d.}\\operatorname{Unif}[c,1]$.\n\\end{enumerate}\nOne can easily verify that each $p\\operatorname{-value}$ marginally follows a\n$\\operatorname{Unif}[0,1]$ distribution. On the first event, we always reject all\nthe variables because\n\\[ \\frac{1+\\#\\left\\{j \\leq p: p_{j}>c\\right\\}}{\\#\\left\\{j \\leq p:\n      p_{j} \\leq c\\right\\} \\vee 1} = \\frac{1 + p - m_0}{m_0} \\leq\n  \\frac{1 + p - m_0}{m_0 - 1} \\leq \\frac{1-c}{c} \\cdot q. \\] Thus\n$\\operatorname{FDP} = 1$. On the second event, we reject none of the variables, thus\n$\\operatorname{FDP} = 0$. Combining the two cases, we get\n$\\operatorname{FDR} = cp\/m_0 \\to q + c(1-q)$ as $p \\to \\infty$.\n\\end{proof}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.7\\textwidth]{plots\/bound.pdf}\n\\caption{The bound on FDR inflation $\\varepsilon(c, q, \\rho)$. Here we set $c= 0.1$, vary $\\rho$ from 0 to 0.5, and vary the FDR threshold $q$ from $0.05$ to $0.2$. }\n\\label{fig:exch_bounds}\n\\end{figure}\n\n\nCompared to the FDR bound in Theorem \\ref{theo:cond_prob}, Theorem\n\\ref{theo:exchangeable} is neither weaker nor stronger. Theorem\n\\ref{theo:exchangeable} holds when the null $p\\operatorname{-values}$ are exchangeable,\nwhereas Theorem \\ref{theo:cond_prob} holds when the $p\\operatorname{-values}$ are close\nto being independent. When the $p\\operatorname{-values}$ are exchangeable and highly\ncorrelated, for example in the most extreme case where all the\n$p\\operatorname{-values}$ are the same, then \\eqref{eqn:exch_bound1} in Theorem\n\\ref{theo:exchangeable} gives that $\\operatorname{FDR} \\leq q + c(1 - q)$, whereas\n\\eqref{eqn:theo_cond_prob} in Theorem \\ref{theo:cond_prob} would not\nbe informative at all. In a different setting where the $p\\operatorname{-values}$ are\nindependent but follow different distributions, Theorem\n\\ref{theo:cond_prob} can be used to show that $\\operatorname{FDR} \\leq q$, whereas\nTheorem \\ref{theo:exchangeable} cannot be applied.\n\n\n\n\\subsection{Beyond exchangeability or almost independence of the $p$-values}\n\nIn general, when the $p\\operatorname{-values}$ have an arbitrary dependence structure,\nwe can bound the FDR with a logarithmic inflation; the sharpness of\nthe bound below is an open question.\n\\begin{theo}[Arbitrary dependence]\n  \\label{theo:no_assu}\n  Suppose the ordering of the $p\\operatorname{-values}$ is fixed and that the nulls are\n  marginally stochastically larger than uniform. If $(1-c)q < c$, then Algorithm\n  \\ref{alg:seqstep} yields\n\\begin{equation}\n\\label{eqn:no_assu_bound}\n \\operatorname{FDR} \\leq (q + c(1-q)) \\sum_{j \\in \\mathcal{H}_0} \\frac{1}{j+1} \\leq (q + c(1-q))\\log p. \n\\end{equation}\n\\end{theo}\nWhen we have a good ordering of the $p\\operatorname{-values}$, i.e., when the null\n$p\\operatorname{-values}$ tend to have larger indices, then the right-hand side\n$(q + c(1-q)) \\sum_{j \\in \\mathcal{H}_0} \\frac{1}{j+1}$ is\nsmaller. Comparing to the case with exchangeability, we observe a\npotential logarithmic inflation on the FDR. A similar phenomenon has\nbeen observed for the BHq procedure, where an arbitrary dependence\namong the $p\\operatorname{-values}$ also brings a possible logarithmic inflation\n\\citep{benjamini2001control}.\n\n\n\\section{Methods to order hypotheses} \n\\label{section:method}\nWhen performing variable selection with CRT and Selective SeqStep+, it\nis important to have a good ordering of the hypotheses\/CRT $p\\operatorname{-values}$. A naive way\nof obtaining the ordering is as follows: apply any machine learning\nalgorithm to $(\\mathbf{X},\\mathbf{Y})$, compute a statistic $z_j$ providing evidence\nagainst the hypothesis that $j$ is null, sort the CRT $p\\operatorname{-values}$ by\ndecreasing order of the $z_j$'s, and apply Selective SeqStep+.  As\nargued in Section \\ref{subsection:introduction_challenges}, despite\nthe intuitive structure of this procedure, the dependence between the\n$p\\operatorname{-values}$ and the ordering will, in general, imply a loss of FDR\ncontrol.\n\n\n\\subsection{Splitting}\n\\label{subsection:exact}\n\n\\floatname{algorithm}{Procedure}\n\\begin{algorithm}[h]\n\\caption{The Sequential CRT (Split version)}\n\\label{alg:exact}\n\\begin{algorithmic}\n\\REQUIRE Data $\\mathcal{D} = (\\mathbf{X}, \\mathbf{Y})$, number of randomizations $B$, test statistic $T(\\cdot)$, score function $Z$, $\\operatorname{FDR}$ threshold $q$, SeqStep threshold $c$. \n\\end{algorithmic}\n\n\\begin{enumerate}\n\\item \n\\begin{algorithmic}\n\\STATE Split the data into two folds $\\mathcal{D}_{\\operatorname{pval}} = (\\mathbf{X}^{\\operatorname{pval}}, \\mathbf{Y}^{\\operatorname{pval}})$ and $\\mathcal{D}_{\\operatorname{ordering}} = (\\mathbf{X}^{\\operatorname{ordering}}, \\mathbf{Y}^{\\operatorname{ordering}})$. \n\\end{algorithmic}\n\n\\item \n\\begin{algorithmic}\n\\STATE Obtain $p\\operatorname{-values}$ $p_1 \\dots, p_p$ on $\\mathcal{D}_{\\operatorname{pval}}$ from CRT (Algorithm \\ref{alg:crt}). \n\\end{algorithmic}\n\n\\item \n\\begin{algorithmic}\n\\STATE Compute statistics $z_j = Z\\p{\\mathbf{X}_j^{\\operatorname{ordering}},\\mathbf{X}_{-j}^{\\operatorname{ordering}},  \\mathbf{Y}^{\\operatorname{ordering}}}$ on $\\mathcal{D}_{\\operatorname{ordering}}$ for each $j \\in \\cb{1, \\dots p}$, and obtain ordering $\\pi$ by sorting the statistics: $z_{\\pi(1)} \\geq z_{\\pi(2)} \\dots \\geq z_{\\pi(p)}$. \n\\end{algorithmic}\n\n\\item \n\\begin{algorithmic}\n\\STATE \nApply Selective SeqStep+ (Algorithm \\ref{alg:seqstep}) to $p_{\\pi(1)}, p_{\\pi(2)}, \\dots, p_{\\pi(p)}$. \n\\end{algorithmic}\n\\end{enumerate}\n\n\\begin{algorithmic}\n\\ENSURE Discoveries from Selective SeqStep+. \n\\end{algorithmic}\n\n\\end{algorithm}\nThe split version of the sequential CRT (Procedure \\ref{alg:exact}) makes the $p\\operatorname{-values}$ and\nordering independent through data splitting: the data is split into\ntwo folds; the $p\\operatorname{-values}$ are obtained from the CRT on the first fold;\nand the ordering is obtained on the second fold. Independence ensures\nthat Theorem~\\ref{theo:cond_prob} holds for this procedure.  The\ndownside is that this suffers from a power loss as is the case for\nmany other data splitting procedures. This motivates us to look for\nprocedures that use the full data to obtain both the $p\\operatorname{-values}$ and the\nordering.\n\n\\subsection{Symmetric statistics}\n\\label{subsection:inexact}\n\nAs seen in Section \\ref{subsection:introduction_challenges}, the correlation between the null $p\\operatorname{-value}$ $p_j$ and the statistic $z_j$, which is sorted to obtain the ordering, largely accounts for the FDR inflation. It is thus natural to seek procedures that make $p_j$ and $z_j$ independent for nulls. To this end, recall that the $p\\operatorname{-value}$ $p_j$ is defined as \n\\[p_j=\\frac{1}{B+1} \\p{1 + \\sum_{b = 1}^B \\one \\cb{T(\\mathbf{X}_j, \\mathbf{X}_{-j},\n      \\mathbf{Y}) \\geq T(\\mathbf{X}_j^{(b)}, \\mathbf{X}_{-j}, \\mathbf{Y})}}. \\] We propose a\nmethod with $p_j$ as above and each $z_j$ constructed as follows:\nconsider a function $Z$ that is symmetric in its first $B+1$\narguments,\\footnote{We say a function $h(x_1, \\dots, x_n)$ is\n  symmetric in its first $m$ arguments if for any permutation $\\pi$ of\n  $\\cb{1,2,\\dots, m}$,\n  $h(x_1, x_2, \\dots, x_m, x_{m+1}, \\dots x_n) = h(x_{\\pi(1)},\n  x_{\\pi(2)}, \\dots, x_{\\pi(m)}, x_{m+1}, \\dots x_n)$.} and define \n\\[z_j = Z\\p{\\mathbf{X}_j, \\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)}, \\mathbf{X}_{-j}, \\mathbf{Y}}.\\]\nThis definition leads to Procedure \\ref{alg:inexact}.\n\n\\begin{algorithm}[h]\n\\caption{The Sequential CRT (Symmetric statistic version)}\n\\label{alg:inexact}\n\\begin{algorithmic}\n\\REQUIRE Data $\\mathcal{D} = (\\mathbf{X}, \\mathbf{Y})$, number of randomizations $B$, test statistic $T(\\cdot)$, score function $Z(\\cdot)$ (symmetric in its first $B+1$ arguments), $\\operatorname{FDR}$ threshold $q$, SeqStep threshold $c$. \n\\end{algorithmic}\n\n\\begin{enumerate}\n\\item \n\\begin{algorithmic}\n\\STATE Obtain $p\\operatorname{-values}$ $p_1 \\dots, p_p$ on $\\mathcal{D}$ from CRT (Algorithm \\ref{alg:crt}). \n\\end{algorithmic}\n\n\\item \n\\begin{algorithmic}\n  \\STATE Compute statistics:\n  $z_j = Z\\p{\\mathbf{X}_j, \\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)}, \\mathbf{X}_{-j}, \\mathbf{Y}}$\n  for each $j \\in \\cb{1, \\dots p}$, and obtain an ordering $\\pi$ by\n  sorting the statistics:\n  $z_{\\pi(1)} \\geq z_{\\pi(2)} \\dots \\geq z_{\\pi(p)}$.\n\\end{algorithmic}\n\n\\item \n\\begin{algorithmic}\n\\STATE Apply Selective SeqStep+ (Algorithm \\ref{alg:seqstep}) to $p_{\\pi(1)}, p_{\\pi(2)}, \\dots, p_{\\pi(p)}$. \n\\end{algorithmic}\n\\end{enumerate}\n\n\\begin{algorithmic}\n\\ENSURE Discoveries from Selective SeqStep+. \n\\end{algorithmic}\n\n\\end{algorithm}\n\nIntuitively, the $p\\operatorname{-value}$ $p_j$ is capturing the relative rank of\n$\\mathbf{X}_j$ among $\\cb{\\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)}}$, yet, $z_j$ is\nsymmetric in $\\cb{\\mathbf{X}_j, \\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)}}$. The\nsymmetry allows us to permute elements in\n$\\Big\\{\\mathbf{X}_j, \\mathbf{X}_j^{(1)}, \\dots,$ $\\mathbf{X}_j^{(B)}\\Big\\}$ while keeping\n$z_j$ fixed. This means that $z_j$ is not providing information\nregarding the relative rank of $\\mathbf{X}_j$ among\n$\\cb{\\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)}}$ and is hence independent of\n$p_j$. Put formally:\n\\begin{prop}\n\\label{theo:pj_zj}\nThe $p\\operatorname{-value}$ $p_j$ and the statistic $z_j$ defined in Procedure \\ref{alg:inexact} obey\n$p_j \\independent z_j$ for any null $j$. In addition, $p_j \\independent z_j |\\mathbf{X}_{-j}, \\mathbf{Y}$,  for any null $j$. \n\\end{prop}\nThis result is a special case of Proposition~\\ref{prop:one_shot} that will be presented later. \n\nNote that Proposition \\ref{theo:pj_zj} is not sufficient to guarantee\nFDR control. Even though $p_j$ is independent of $z_j$, $p_j$ could,\nin principle, still have a complicated relationship with\n$z_{-j}$. This makes the $p\\operatorname{-values}$ not entirely independent of the\nordering. This however does not appear to lead to FDR inflation in\npractice. We indeed observe FDR control in various simulation studies\nin Section \\ref{section:simulation}.\n\nThe statistic $z_j$ can be computed using complicated machine learning\nmethods. For example, one can run a gradient boosting algorithm with regression trees as base learners, $\\mathbf{Y}$ as a response, and\n$\\mathbf{X}_j, \\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)}, \\mathbf{X}_{-j}$ as predictors,\nobtain feature importance statistics of\n$\\mathbf{X}_j, \\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)}$, and take $z_j$ to be the\nmaximum of the statistics. One can easily verify that with this\nspecific construction, $z_j$ is symmetric in\n$\\p{\\mathbf{X}_j, \\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)}}$.\n\n\\section{Towards faster computation: one-shot CRT}\n\\label{section:computation}\nIn the original CRT (Algorithm \\ref{alg:crt}), to compute each $p\\operatorname{-value}$\n$p_j$ one runs a machine learning algorithm $B$ times to obtain the\ntest statistics $T_j^{(b)}$ for $b \\in \\cb{1, \\dots, B}$. This quickly\ngets computationally expensive when the machine learning algorithm is\nrun on a large dataset. To save computation time, another way of\ncomputing the statistics is to run the machine learning algorithm\nonce, with $\\mathbf{Y}$ as a response and\n$\\mathbf{X}_j, \\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)}, \\mathbf{X}_{-j}$ as the predictors.\nFormally, we consider a procedure $T$ that takes\n$\\p{\\mathbf{X}_j^{(0)}, \\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)}, \\mathbf{X}_{-j}, \\mathbf{Y}}$ as\ninput, and outputs importance statistics $T_j^{(b)}$ for each\n$b \\in \\cb{1, \\dots, B}$, i.e.,\n\\begin{equation}\n\\label{eqn:one_shot_procedure}\n \\p{T_j^{(0)}, \\dots, T_j^{(B)}} = T\\p{\\mathbf{X}_j^{(0)}, \\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)}, \\mathbf{X}_{-j}, \\mathbf{Y}}.  \n\\end{equation}\nWe restrict attention to procedures obeying the following symmetry\nproperty: for all permutations $\\pi$,\n\\begin{equation}\n\\label{eqn:one_shot_symmetric}\n T\\p{ \\sqb{\\mathbf{X}_j^{(0)}, \\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)} }_{\\operatorname{perm}(\\pi)} , \\mathbf{X}_{-j}, \\mathbf{Y}  } \n =  \\sqb{T\\p{\\mathbf{X}_j^{(0)}, \\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)}  , \\mathbf{X}_{-j}, \\mathbf{Y}  } }_{\\operatorname{perm}(\\pi)}. \n\\end{equation}\nThis is saying that if we permute the input $\\mathbf{X}_j^{(b)}$, this has the effect of permuting the output statistics. \nWith these statistics, we obtain $p\\operatorname{-values}$ via \n\\begin{equation}\n\\label{eqn:one_shot_pvalue}\np_j=\\frac{1}{B+1}\n\t\\p{1 + \\sum_{b = 1}^B \\one{\\cb{T_j^{(0)} \\leq T_j^{(b)} }}}.  \\footnote{We break ties randomly. }\n\\end{equation}\n We call this procedure \\textit{one-shot CRT}. \n\n As a concrete example, consider a case where the lasso is used to\n compute the test statistic $T_j^{(b)}$. To compute each $p\\operatorname{-value}$\n $p_j$, the original CRT runs the lasso by regressing $\\mathbf{Y}$ on\n $\\mathbf{X}_j^{(b)}, \\mathbf{X}_{-j}$ for each $b \\in \\cb{0, 1, \\dots, B}$, and\n takes $T_j^{(b)}$ to be the absolute value of the fitted coefficient\n for $\\mathbf{X}_j^{(b)}$. In total, we run $B+1$ regressions. In contrast, the one-shot\n CRT runs the lasso only once by regressing $\\mathbf{Y}$ on\n $\\mathbf{X}_j^{(0)}, \\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(b)} , \\mathbf{X}_{-j}$ and takes\n $T_j^{(b)}$ to be the corresponding $|\\hat{\\beta}|$ for\n $\\mathbf{X}_j^{(b)}$ (this obeys \\eqref{eqn:one_shot_symmetric}).\n \n\nThe symmetry in \\eqref{eqn:one_shot_symmetric} ensures that the theoretical properties of the CRT $p\\operatorname{-values}$ still hold for the one-shot CRT $p\\operatorname{-values}$.\n\\begin{prop}\n\\label{prop:one_shot}\nConsider a null variable $j$. Assume that the $p\\operatorname{-value}$ $p_j$ is obtained from \\eqref{eqn:one_shot_procedure} and \\eqref{eqn:one_shot_pvalue}, and that \\eqref{eqn:one_shot_symmetric} holds. Then $p_j$ satisfies $\\PP{p_j \\leq \\alpha} \\leq \\alpha$, for any $\\alpha \\in [0,1]$, and $p_j \\independent z_j|\\mathbf{X}_{-j}, \\mathbf{Y}$, where $z_j$ is defined in Procedure \\ref{alg:inexact}.\n\\end{prop}\n\\begin{proof}\n  For the sake of notation, set $\\mathbf{X}_j^{(0)} = \\mathbf{X}_j$. Consider a null\n  $j$. By construction of $\\mathbf{X}_j^{(b)}$, all the $\\mathbf{X}_j^{(b)}$'s are\n  i.i.d.~conditional on $\\mathbf{X}_{-j}$ and $\\mathbf{Y}$. Thus for any permutation\n  $\\rho$ of $\\cb{0, \\dots, B}$,\n  $\\p{\\mathbf{X}_j^{(0)}, \\dots, \\mathbf{X}_j^{(B)}} \\stackrel{d}{=}\n  \\p{\\mathbf{X}_j^{(\\rho(0))}, \\dots, \\mathbf{X}_j^{(\\rho(B))}} \\Big| \\mathbf{X}_{-j},\n  \\mathbf{Y}$. The symmetry of $Z$ in its first $B+1$ arguments further\n  ensures that\n  $z_j = Z\\p{\\mathbf{X}_j^{(0)}, \\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)}, \\mathbf{X}_{-j},\n    y} = Z\\p{\\mathbf{X}_j^{(\\rho(0))}, \\mathbf{X}_j^{(\\rho(1))}, \\dots,\n    \\mathbf{X}_j^{(\\rho(B))}, \\mathbf{X}_{-j}, \\mathbf{Y}}$. Combining these facts, we have\n\\[ \\p{\\mathbf{X}_j^{(0)}, \\dots, \\mathbf{X}_j^{(B)}} \\stackrel{d}{=} \\p{\\mathbf{X}_j^{(\\rho(0))}, \\dots, \\mathbf{X}_j^{(\\rho(B))}} \\Big| \\mathbf{X}_{-j}, \\mathbf{Y}, z_j.\\]\nBy property \\eqref{eqn:one_shot_symmetric}, \n\\[ \\p{T_j^{(\\rho(0))}, \\dots, T_j^{(\\rho(B))}} =\n  T\\p{\\mathbf{X}_j^{(\\rho(0))}, \\mathbf{X}_j^{(\\rho(1))}, \\dots, \\mathbf{X}_j^{(\\rho(B))},\n    \\mathbf{X}_{-j}, \\mathbf{Y}}. \\] This term has the same distribution as\n$T\\p{\\mathbf{X}_j^{(0)}, \\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)}, \\mathbf{X}_{-j}, \\mathbf{Y}}$\nconditional on $\\mathbf{X}_{-j}, \\mathbf{Y}$ and $z_j$.  Since\n$T\\p{\\mathbf{X}_j^{(0)}, \\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)}, \\mathbf{X}_{-j}, \\mathbf{Y}}=\n\\p{T_j^{(0)}, \\dots, T_j^{(B)}}$, we have\n\\[ \\p{T_j^{(\\rho(0))}, \\dots, T_j^{(\\rho(B))}} \\stackrel{d}{=}\n  \\p{T_j^{(0)}, \\dots, T_j^{(B)}} \\Big| \\mathbf{X}_{-j}, \\mathbf{Y}, z_j.\\] This\nimplies that conditional on $\\mathbf{X}_j, \\mathbf{Y}, z_j$,\n$p_j \\sim \\operatorname{Unif}\\cb{\\frac{1}{B+1}, \\frac{2}{B+1}, \\dots,\n  1}$. Note that the same holds without conditioning on $z_j$, i.e.,\nconditioning on $\\mathbf{X}_j, \\mathbf{Y}$ only. Hence\n$p_j \\independent z_j | \\mathbf{X}_{-j}, \\mathbf{Y}$. Finally, the claim that\n$\\PP{p_j \\leq \\alpha} \\leq \\alpha$ follows from the fact that the\ndistribution of\n$\\operatorname{Unif}\\cb{\\frac{1}{B+1}, \\frac{2}{B+1}, \\dots, 1}$ is\nstochastically greater than $\\operatorname{Unif}[0,1]$.\n\\end{proof}\nIn terms of FDR control, the theorems from Section\n\\ref{section:theory} still hold for either the split or symmetric statistic\nversion of our variable selection method applied with the one-shot CRT\n$p\\operatorname{-values}$.\n\nAs an illustration, we show the average computation time of the\none-shot CRT and the original CRT (on all variables together) on\nsynthetic datasets in Table \\ref{table:time}. The number of\nrandomizations is set to $B = 9$, and other details of the simulation\nstudy are included in Appendix\n\\ref{subsection:simulation_details_table}. Compared with the original\nCRT, the one-shot CRT reduces the computation time by a factor of\nroughly $1\/B$ as expected.\n\\begin{table}[h]\n\\centering\n\\caption{Average computation times (seconds)}\n\\label{table:time}\n\\begin{tabular}{|p{2.8cm} |p{2.8cm}| p{2.8cm}| p{2.95cm}| p{2.95cm}|}\n\\hline\nSetting   & Linear & Logistic & Non-linear 1 & Non-linear 2 \\\\ \n\\hline\nDimension &$n = 300, p = 300$ &$n = 300, p = 300$ &$n = 500, p = 200$ & $n = 500, p = 200$     \\\\ \n\\hline\nStatistics are  computed with& lasso  & glmnet & gradient boosting  & gradient boosting  \\\\\n\\hhline{|=|=|=|=|=|}\nOriginal CRT   & 694 & 709          & 2811       & 2672    \\\\\n \\hline\nOne-shot CRT     & 89  & 91             & 248         & 277     \\\\\n \\hline\n\\end{tabular}\n\\end{table}\n\nAdding $B$ predictors into a machine learning\nalgorithm cannot be used if we are combining CRT with BHq. As BHq\nrequires much finer $p\\operatorname{-values}$, $B$ needs to be much larger. Adding many\nirrelevant predictors into a regression problem is not generally a\nwise move.\n\nMore generally, the computational problem posed by the combination of\nthe CRT and BHq has been considered by \\citet{tansey2018holdout} and\n\\citet{liu2020fast}, which propose separate methods to reduce the\nrunning time.  \\citet{tansey2018holdout} consider data splitting: the\nalgorithm trains a complicated machine learning model on the first\npart of the data and obtains $p\\operatorname{-values}$ on the second part of the data\nmaking use of the trained model. The data splitting trick ensures that\nthe complicated machine learning model will be fitted only once, and\nhence makes the algorithm much faster and computationally\nfeasible. \\citet{liu2020fast} propose a technique called\n\\emph{distillation}. Their proposed algorithm distills all the\nhigh-dimensional information in $\\mathbf{X}_{-j}$ about $\\mathbf{Y}$ into a\nlow-dimensional representation, computes the test statistic as a\nfunction of $\\mathbf{X}_j$, $\\mathbf{Y}$, and the low-dimensional representation,\nand obtains the $p\\operatorname{-values}$ based on the test statistics. The computation\ntime is much lower since the expensive model fitting takes place in\nthe distillation step, which is performed only once for each $j$.  The\ntwo methods both give marginally valid $p\\operatorname{-values}$, but the $p\\operatorname{-values}$\nwould not be independent in general, and there is no theoretical\nguarantee on FDR control. (Both papers confirm in their simulations\nthat the FDR of each method is well controlled empirically.)\n\n\\section{Simulations}\n\\label{section:simulation}\n\nIn this section, we demonstrate the performance of our methods on synthetic data.\nSoftware for our method is available from \\url{https:\/\/github.com\/lsn235711\/sequential-CRT}, along with code to reproduce the analyses. We include in Appendix \\ref{section:simulation_details} implementation details and additional simulation studies. \n\n\\subsection{Comparison of the original CRT and the one-shot CRT}\n\\label{subsection:simulation_small}\nWe compare the proposed symmetric statistic version of the sequential CRT (with one-shot CRT) and the sequential CRT (with the original CRT). We also\ncompare our methods with Model-X knockoffs as a benchmark.  We\nconsider a few different settings: linear\/non-linear(tree like)\nmodels, and Gaussian\/binomial responses. In all settings, the number of true nonnulls is set to be 20. For the distribution of $X$,\nwe consider a Gaussian autoregressive model and a hidden Markov\nmodel. To compute test statistics, we consider algorithms including\n$L_1$-regularized regression (glmnet) and gradient boosting with\nregression trees as base learners. The statistic $z_j$ in Procedure\n\\ref{alg:inexact} is taken to be\n$z_j = \\max_{b \\in \\cb{0, \\dots, B}}T_j^{(b)}$, where $T_j^{(b)}$ is\nthe test statistic computed in the CRT. Knockoffs are constructed with\nthe Gaussian semi-definite optimization algorithm\n\\citep{candes2018panning} for the Gaussian autoregressive model, and\nwith Algorithm 3 from \\citep{sesia2019gene} for the hidden Markov\nmodel. Details of the simulation study are included in Appendix\n\\ref{subsection:simulation_details_small}. Figure\n\\ref{fig:fdr_power_small} compares the performance of the above\nmethods in terms of empirical false discovery rate and power averaged\nover 100 independent replications. In all settings, the sequential CRT appears\nto control the FDR around the desired level $q = 0.1$.  In terms of\npower, the performance of the one-shot CRT appears to be similar to\nthat of the original CRT in most of the settings.  Compared to\nknockoffs, the sequential CRT (both original CRT and\none-shot CRT) is more powerful.\n\n\\begin{figure}\n\\begin{subfigure}{\\textwidth}\n\\caption{$X$ follows a Gaussian AR model}\n\\includegraphics[width = \\textwidth]{plots\/small_experiment_ar.pdf}\n\n\\end{subfigure}\n\\noindent\\rule{\\textwidth}{0.4pt}\n\\begin{subfigure}{\\textwidth}\n\\caption{$X$ follows an HMM}\n\\includegraphics[width = \\textwidth]{plots\/small_experiment_hmm.pdf}\n\\end{subfigure}\n\\caption{Performance of the sequential CRT with the original CRT and one-shot CRT (both use the symmetric statistic) and knockoffs on small synthetic datasets. The nominal false discovery rate level is 10\\%. Results are averaged over 100 independent experiments.}\n\\label{fig:fdr_power_small}\n\\end{figure}\n\n\\subsection{Comparison of the sequential CRT with knockoffs}\n\\label{subsection:simulation_large}\n We compare the proposed split version (Procedure \\ref{alg:exact}) and symmetric statistic version (Procedure \\ref{alg:inexact}) of the sequential CRT with Model-X knockoffs. We run the sequential CRT with one-shot CRT. \nWe consider similar settings as in the above Section \\ref{subsection:simulation_small}. Since the computation time of one-shot CRT is much lower compared to the original CRT, here we run the experiments on larger datasets. In all settings in this section, the number of nonnulls is set to be 50.\nOther details can be found in Section \\ref{subsection:simulation_small} and Appendix \\ref{subsection:simulation_details_large}. Figure \\ref{fig:fdr_power_large} compares the performance of the above methods in terms of empirical false discovery rate and power averaged over 100 independent replications. In all settings, the sequential CRT appears to control the FDR around the desired level $q = 0.1$. In terms of power, the symmetric statistic version is comparable to knockoffs. \n\n\\begin{figure}\n\\begin{subfigure}{\\textwidth}\n\\caption{$X$ follows a Gaussian AR model}\n\\includegraphics[width = \\textwidth]{plots\/large_experiment_ar.pdf}\n\n\\end{subfigure}\n\\noindent\\rule{\\textwidth}{0.4pt}\n\\begin{subfigure}{\\textwidth}\n\\caption{$X$ follows an HMM}\n\\includegraphics[width = \\textwidth]{plots\/large_experiment_hmm.pdf}\n\\end{subfigure}\n\\caption{Performance of the proposed split version and symmetric statistic version of the sequential CRT compared to knockoffs on larger synthetic datasets. The nominal false discovery rate level is 10\\%. Results are averaged over 100 independent experiments.}\n\\label{fig:fdr_power_large}\n\\end{figure}\n\n\n\\subsection{The role of the number of potential discoveries}\n\\label{subsection:simulation_num_nonnull}\n\\begin{figure}\n\\includegraphics[width = \\textwidth]{plots\/num_nonnull_experiment_ar.pdf}\n\\caption{Performance of the sequential CRT compared to knockoffs as a\n  function of the number of non-nulls. The nominal false discovery\n  rate level is 10\\%. Results are averaged over 100 independent\n  experiments.}\n\\label{fig:fdr_power_num_nonnull}\n\\end{figure}\n\nComparing the results from Sections \\ref{subsection:simulation_small}\nand \\ref{subsection:simulation_large}, we observe that the power gain\nof the sequential CRT vis-a-vis model-X knockoffs is more noticeable\nwhen the number of nonnulls is small.  To understand this phenomenon,\nreturn to the connection between the knockoff filter and Selective\nSeqStep+. It was shown by \\citet{barber2015controlling} that the\nknockoff filter can be cast as a special case of the Selective\nSeqStep+ applied to ``one-bit'' p-values with $c$ chosen to be 0.5.\nWhen $q = 0.1$ and $c = 0.5$, the selected set of the Selective\nSeqStep+ becomes\n$\\hat{\\mathcal{S}} = \\cb{ j \\leq \\hat{k}: p_{j} \\leq c}$, where\n\\begin{equation}\n\\label{eqn:seqstep_knockoffs}\n\\hat{k} =\\max \\left\\{k \\in \\cb{1,\\dots, p}: \\frac{1+\\#\\left\\{j \\leq k: p_{j}>0.5\\right\\}}{\\#\\left\\{j \\leq k: p_{j} \\leq 0.5\\right\\} \\vee 1} \\leq 0.1 \\right\\}. \n\\end{equation}\nWhen the number of nonnulls is small, the set above may be\nempty. Consider an example where the number of nonnulls is 8. Even in\nthe ideal case where all the nonnulls have vanishing $p\\operatorname{-values}$ and the\nnonnulls appear early in the sequence, for any $k \\geq 8$, the left\nhand side in the inequality \\eqref{eqn:seqstep_knockoffs} becomes\n\\begin{equation}\n\\frac{1 + \\# \\cb{ \\text{null } 9\\leq j\\leq k: p_j \\geq 0.5}}{8 + \\# \\cb{ \\text{null } 9\\leq j \\leq k: p_j < 0.5}} \\gtrsim \\frac{1}{8} > 0.1.\n\\end{equation}\nTherefore, most of the time there is no $k$ satisfying the inequality \\eqref{eqn:seqstep_knockoffs}, thus we make no rejections. \nHence the power will be low. \n\nThe sequential CRT, however, will not suffer from the same problem. We recall that throughout this paper, we take $c=0.1$ in the sequential CRT. With $c = 0.1$, the definition of $\\hat{k}$ becomes\n\\begin{equation}\n\\hat{k} =\\max \\left\\{k \\in \\cb{1,\\dots, p}: \\frac{1+\\#\\left\\{j \\leq k: p_{j}>0.1\\right\\}}{\\#\\left\\{j \\leq k: p_{j} \\leq 0.1\\right\\} \\vee 1} \\leq 0.9 \\right\\}. \n\\end{equation}\nWith a good ordering the $p\\operatorname{-values}$, the left hand side of the\ninequality can easily become lower than 0.9, a much less stringent\nthreshold.\n\nWe run simulations varying the number of nonnulls.  We compare\nthe proposed symmetric statistic version of the sequential CRT with\nmodel-X knockoffs. We run the sequential CRT with one-shot CRT.  We\nconsider settings as in Section\n\\ref{subsection:simulation_small}; details are  in Appendix\n\\ref{subsection:simulation_details_num_nonnull}. Figure\n\\ref{fig:fdr_power_num_nonnull} compares the performance of the above\nmethods in terms of empirical false discovery rate and power averaged\nover 100 independent replications. In all settings, the sequential CRT\nappears to control the FDR around the desired level $q = 0.1$. In\nterms of power, we see that the sequential CRT overcomes ``the threshold\nphenomenon'' discussed earlier. \n\n\\subsection{Choice of the threshold $c$ and the number $B$ of randomizations}\n\\label{subsection:simulation_Bc}\nWe here study the effect on power of the threshold $c$ and of the\n  number $B$ of randomizations. We focus on the sequential CRT\n  (symmetric statistics version with one-shot CRT). Intuitively, we\n  expect the procedure with a smaller $c$ and a smaller $B$ to be more\n  powerful. With a smaller $c$, our procedure is more likely to overcome ``the threshold phenomenon'' as discussed in Section \\ref{subsection:simulation_num_nonnull}. When using a smaller value of $B$, we make sure\n  that we are not including too many irrelevant predictors in the\n  machine learning algorithm while running the one-shot CRT\n  algorithm. In addition, when we compute the statistics $z_j$ in\n  Algorithm \\ref{alg:inexact}, we take the maximum (or the difference\n  between the maximum and the median) of the feature importance\n  statistics of $\\mathbf{X}_j, \\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)}$; thus it is\n  helpful to have a smaller $B$ so that the signal, i.e., the feature\n  importance statistics of $\\mathbf{X}_j$ has a chance of standing out. If we\n  use an extremely large value of $B$, there is a chance that the\n  maximum of the feature importance statistics of\n  $\\mathbf{X}_j^{(1)}, \\dots, \\mathbf{X}_j^{(B)}$ exceeds that of $\\mathbf{X}_j$.  That\n  said, $c$ and $B$ cannot be too small at the same time. At the very\n  least, in order for our procedure to make any rejection, we need to\n  have some $p\\operatorname{-values}$ no larger than $c$. Since the $p\\operatorname{-values}$ are\n  bounded below by $1\/(B+1)$, a necessary condition for not being\n  powerless is to have $c \\geq 1\/(B+1)$.\n\n\\begin{figure}\n\\centering\n\\begin{subfigure}[b]{0.9\\textwidth}\n      \\caption{Small datasets}\n         \\centering\n         \\includegraphics[width = \\textwidth]{plots\/experiment_300_Bc.pdf}\n         \\label{fig:Bc_small}\n \\end{subfigure}\n\\begin{subfigure}[b]{0.9\\textwidth}\n\\noindent\\rule{\\textwidth}{0.4pt}\n  \\caption{Large datasets}\n\\centering\n \\includegraphics[width = \\textwidth]{plots\/experiment_1000_Bc.pdf}\n \\label{fig:Bc_large}\n\\end{subfigure}\n\\caption{Power of the sequential CRT (symmetric statistics version with one-shot CRT) with different choices of threshold $c$ and number of randomizations $B$. The nominal false discovery\n  rate level is 10\\%. Results are averaged over 100 independent\n  experiments.}\n\\label{fig:Bc}\n\\end{figure}\n\nThe above heuristic arguments are confirmed in simulation\nstudies. Figure \\ref{fig:Bc} compares the performance of the\nsequential CRT with different values of $c$ and $B$.\\footnote{In our\n  simulation studies, we take $B = 10k + 9$ for some\n  $k \\in \\mathbb{Z}$ because we want to make $B+1$ a multiple of 10,\n  and thus make it possible for $\\ell\/(B+1) = c$ to hold for some\n  integer $\\ell$. } We consider several settings: linear\/logistic\nmodels, small\/large synthetic datasets, low\/high signal to noise\nratios. We observe the same phenomenon in all settings. Namely, power\nincreases as $B$ decreases and $c$ decreases with the caveat that\nthey cannot both be small at the same time. It appears that the pair\n$(B, c) = (9, 0.1)$ is the most powerful in all settings, justifying\nthe choices we made in earlier simulation studies. In Appendix\n\\ref{subsection:simulation_details_Bc}, we provide implementation\ndetails to reproduce Figure \\ref{fig:Bc} and additionally show that\nthe FDR is controlled at the nominal level for all choices of $B$ and\n$c$.\n\n\\section{Real data application}\n\\label{section:application}\nWe now apply our method to a breast cancer dataset to identify gene expressions on which the cancer stage depends. The\ndataset is from \\cite{curtis2012genomic}, which consists of\n$n = 1,396$ staged cases of breast cancer. For each case, the data\nconsists of expression level (mRNA) and copy number aberration (CNA)\nof $p = 164$ genes. The goal is to identify genes whose expression\nlevel is not independent of the cancer stage, conditioning on all\nother genes and CNAs.  The response variable, the progression stage of\nbreast cancer, is binary.  We take the dataset from\n\\citep{liu2020fast} and pre-process the data as in their work. We\nrefer to Section~5 and Section~E of \\citep{liu2020fast} for further\ndetails. Following \\citep{liu2020fast}, we model the distribution of\nexpression levels using a multivariate Gaussian.  The nominal false\ndiscovery rate is set to be 10\\%.\n\nBelow, we compare the following  methods:\n\\begin{enumerate}\n\\item \\emph{Sequential CRT}: We consider Procedure \\ref{alg:inexact} with one-shot CRT. We take the SeqStep threshold $c$ to be 0.1, and the number of randomizations to be 9. We take the importance statistics to be the absolute values of the coefficient of a cross-validated $L_1$-penalized logistic regression. \n\\item \\emph{Distilled CRT} \\citep{liu2020fast}. We consider both $\\operatorname{d}_0$CRT and $\\operatorname{d}_{\\operatorname{I}}$CRT; we refer to Section 2.3 and 2.4 of \\citep{liu2020fast} for specific constructions of the dCRT. Since the response variable is binary, the distillation step is done by a cross-validated $L_1$-penalized logistic regression. \n\\item \\emph{HRT} \\citep{tansey2018holdout}. Algorithm 1 of \\citep{tansey2018holdout} is implemented with a logistic model fitted by a cross-validated $L_1$-penalized logistic regression and a data split of 50\\%-50\\%. \n\\item \\emph{Knockoffs} \\citep{candes2018panning}. Knockoffs are constructed with the Gaussian semi-definite optimization algorithm. We take the\nfeature importance statistic to be the glmnet coefficient difference. \n\\end{enumerate}\nFor \\emph{Distilled CRT} and \\emph{HRT}, we reproduce the analysis from \\citet{liu2020fast}. All methods considered above are randomized procedures, i.e., different runs of the same algorithm produce possibly different sets of discoveries. We run each method 100 times and compare the number of discoveries.\n Figure \\ref{fig:real_data_num_disc} is a boxplot showing the number of discoveries across random seeds. Our procedure appears to make more discoveries on average than the other methods. Compared to knockoffs, our procedure has less variability. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.8 \\textwidth]{plots\/real_data.pdf}\n\\caption{Number of discoveries on the breast cancer dataset}\n\\label{fig:real_data_num_disc}\n\\end{figure}\n\nWe present the full list of genes discovered by the sequential CRT in Table\n\\ref{table:discoveries_validate}. \nNote here that the sequential CRT is random, thus different runs could produce different results. Following \\cite{candes2018panning} and \\cite{sesia2019gene}, to make the discoveries more ``reliable\", we run the proposed method multiple times and only show the genes that are\nselected by our procedure more than 10\\% of the time. The 10\\% level is somewhat arbitrary and we do not make any claims about the discoveries exceeding this threshold. We leave the study of setting a threshold achieving theoretical error control guarantees to future research. We however observe that all discoveries above the 10\\% threshold were shown in other independent studies to be related to the development of cancer. \n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{ |c |c |c|c|} \n\\hline\nSelection frequency & Gene &  Discovered by dCRT? &Confirmed in?  \\\\\n\\hline\n99\\% & {\\em HRAS} & Yes&  \\citet{geyer2018recurrent}\\\\\n\\hline\n99\\% & {\\em RUNX1} & Yes & \\citet{li2019runx1}\\\\\n\\hline\n96\\% & {\\em FBXW7} & Yes &  \\citet{liu2019fbxw7}\\\\ \n\\hline\n95\\% & {\\em GPS2} & Yes & \\citet{huang2016g}\\\\ \n\\hline\n95\\% & {\\em NRAS} & & \\citet{galie2019ras}  \\\\ \n\\hline\n82\\% & {\\em FANCD2} & & \\citet{rudland2010significance}  \\\\ \n\\hline\n78\\% & {\\em MAP3K13} &Yes & \\citet{han2016microrna}\\\\ \n\\hline\n76\\% & {\\em AHNAK} &  & \\citet{chen2017ahnak}\\\\ \n\\hline\n67\\% & {\\em MAP2K4} &  &\\citet{liu2019map2k4}\\\\ \n\\hline\n58\\% & {\\em CTNNA1} &  &\\citet{clark2020loss} \\\\ \n\\hline\n35\\% & {\\em NCOA3} & & \\citet{gupta2016ncoa3}\\\\ \n\\hline\n13\\% & {\\em LAMA2} & & \\citet{liang2018targeted}\\\\ \n\\hline\n10\\% & {\\em GATA3} & & \\citet{mehra2005identification}\\\\ \n\\hline\n\\end{tabular}\n\\caption{Discoveries made by Procedure \\ref{alg:inexact} on the breast cancer dataset}\n\\label{table:discoveries_validate}\n\\end{table}\n\n\n\\section{Discussion}\n\\label{section:future_work}\n\\paragraph{Comparison with knockoffs}\nIn this paper, we proposed a variable selection procedure, the sequential CRT. In comparison with model-X knockoffs, the proposed sequential CRT is generally more powerful than model-X knockoffs as shown in the simulation studies in Section \\ref{section:simulation}. Specifically, we observe a much more noticeable power gain of the sequential CRT than is observed by model-X knockoffs when the number of nonnulls is small. In addition to the power gain, we note that the sequential CRT has another advantage over model-X knockoffs: it is usually easier to sample from the conditional distribution of $X_j \\mid X_{-j}$ than to generate knockoffs, especially for complicated joint distribution of $X$. \n\n\\paragraph{Derandomizing the sequential CRT}\nLike many other Model-X procedures (e.g.~Model-X knockoffs,\n  distilled CRT, etc), the sequential CRT is a randomized\n  procedure. In other words, different runs of the method produce\n  might produce different selected sets. When the method is applied in\n  practice, one would report those features whose selection frequency\n  exceeds a threshold along with the corresponding\n  frequencies. \\citet{ren2020derandomizing} studies the problem of\n  derandomizing knockoffs. It will be interesting to study whether it\n  is possible to derandomize the proposed method so that results are\n  more consistent across different runs.\n\n\\paragraph{Theoretically validating power gains}\nWhile this paper demonstrates enhanced statistical power through\nsimulations, it would be interesting to theoretically validate power\ngains. Intuitively, compared to Model-X knockoffs, the proposed method\neffectively reduces the number of covariates by a factor of 2 when\ncomputing feature importance statistics. It would be of interest to\nunderstand theoretically how important such reduction is in terms of\nstatistical power.\n\n\\paragraph{Robustness to misspecification in the distribution of the covariates}\nAnother interesting direction for future work is to study the\n    robustness of the sequential CRT to misspecification in the\n    distribution of the covariates. When the distribution of $X$ is\n    known only approximately, \\citet{barber2020robust} quantifies the\n    possible FDR inflation of model-X knockoffs; and\n    \\citet{berrett2020conditional} bounds the inflation in type-I\n    error of the CRT. It will be interesting to evaluate the FDR\n    inflation of the sequential CRT both empirically and\n    theoretically.\n\n\\section*{Acknowledgements}\n\nE. C. was supported by Office of Naval Research grant N00014-20-12157,\nby the National Science Foundation grants OAC 1934578 and DMS 2032014,\nand by the Simons Foundation under award 814641. S. L. was supported\nby the National Science Foundation grant OAC 1934578.\n\n\\bibliographystyle{unsrtnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $M$ be compact manifold with boundary with a chosen basepoint on the boundary. Denote by ${\\operatorname{map}}_*(M,M)$ the space of pointed maps with the compact open topology. Denote by ${\\operatorname{map}}_\\partial(M,M)$ the subspace of maps, fixing the boundary point-wise. We denote by ${\\operatorname{aut}}_\\partial(M)$ the sub-monoid of homotopy self-equivalences fixing the boundary point-wise.\n\nThe first object we study in this paper is the rational homology of the classifying space $B{\\operatorname{aut}}_\\partial(M).$ More precisely, we show that it satisfies homological stability for certain families of manifolds. For example, for the $g$-fold connected sum $$N_g=\\big (\\mathlarger{\\#}_g (S^p\\times S^q) \\big ) \\smallsetminus {\\operatorname{int}} (D^{p+q})\\text{, where }3\\leq p<q<2p-1,$$ we show that $H_i(B{\\operatorname{aut}}_\\partial(N_g);{\\mathbb Q})$ is independent of $g$ for $g>2i+2.$ Moreover we show that $H_i(B{\\operatorname{aut}}_\\partial(M\\#N_g);{\\mathbb Q})$ is independent of $g$ in the same range, where $M$ is some other connected sum of products of spheres. To make this statement precise, we introduce the following notation. \nLet $I$ be a finite indexing set for pairs of positive natural numbers $p_i,q_i$ and $n$ be a positive natural number, such that $3\\leq p_i\\leq q_i<2p_i-1$ and $p_i+q_i=n$ for all $i\\in I$. Note that this implies that necessarily $n\\geq 6.$ We define a smooth $n$-dimensional manifold with boundary diffeomorphic to $S^{n-1}$ $$N_I=\\left(\\mathlarger{\\#}_{i\\in I} (S^{p_i}\\times S^{q_i})\\right)\\smallsetminus {\\operatorname{int}} (D^n)$$ and we assume a basepoint chosen on the boundary.\n\nFor a given $p\\in {\\mathbb N}$ we define the ``generalized genus'' as $$g_p=\\begin{cases} \\operatorname{rank}(H_p(N_I))\/2 &\\text{ if } 2p=n \\\\ \\operatorname{rank}(H_p(N_I)) &\\text{ otherwise,}  \\end{cases}$$ i.e. $g_p$ is the number of $S^p\\times S^q$ summands of $N_I,$ where $q=n-p.$ In other words we see $N_I$ as the connected sum of the manifolds\n$$N_I= \\big(\\mathlarger{\\#}_{i\\in \\{j\\in I|p_j\\neq p  \\}} S^{p_i}\\times S^{q_i} \\big) \\# (\\mathlarger{\\#}_{g_p}S^{p}\\times S^{q} \\big) \\smallsetminus {\\operatorname{int}}{D^{n}}.$$\n\nDenote by $$V^{p,q}=S^p\\times S^q\\smallsetminus {\\operatorname{int}} (D_1^n\\sqcup D_2^n), \\text{ where $3\\leq p\\leq q<2p-1$ and $p+q=n.$}$$ We define a new manifold \n$$N'=N_I\\cup_{\\partial_1} V^{p,q},$$ by identifying one boundary component of $V^{p,q}$ with $\\partial N_I.$ Note that $N'$ is canonically diffeomorphic to the manifold $N_{I'}$ with $I'=I\\cup \\{i'\\}$, where $p_{i'}=p$ and  $q_{i'}=q.$ Using this, we define the stabilization map $$\\sigma:{\\operatorname{aut}}_\\partial( N_I) \\rightarrow {\\operatorname{aut}}_\\partial( N_I\\cup_{\\partial_1} V^{p,q})\\xrightarrow{\\cong} {\\operatorname{aut}}_\\partial( N_{I'}),  $$ by extending a self-map of $N_I$ by the identity on $V^{p,q}.$ In this paper we study the induced map on homology of the classifying spaces\n$$\\sigma_*:H_*(B{\\operatorname{aut}}_\\partial (N_I);{\\mathbb Q})\\rightarrow H_*(B{\\operatorname{aut}}_\\partial (N_{I'});{\\mathbb Q}).$$ The first main theorem of this paper is:\n\n\\begin{THMA}\nThe map\n$$H_i(B{\\operatorname{aut}}_\\partial(N_I);{\\mathbb Q})\\rightarrow H_i(B{\\operatorname{aut}}_\\partial(N_{I'});{\\mathbb Q})$$ \ninduced by the stabilization map with respect to $V^{p,q},$ where $3\\leq p\\leq q < 2p-1$, is an isomorphism for $g_p>2i+2$ when $2p\\neq n$ and $g_p>2i+4$ if $2p=n$ and an epimorphism for $g_p\\geq 2i+2$ respectively $g_p\\geq 2i+4$.\n\\end{THMA}\n\nThe block diffeomorphisms ${\\widetilde{\\operatorname{Diff}}}_\\partial (X)$ are the realization of the $\\Delta$-group, i.e. a simplicial group without degeneracies, with $k$-simplices, face preserving diffeomorphisms $$\\varphi:\\Delta^k\\times X\\rightarrow \\Delta^k\\times X,$$ such that $\\varphi$ is the identity on a neighborhood of $\\Delta^k\\times \\partial X.$ We map a $k$-simplex $\\varphi$ to the $k$-simplex in ${\\widetilde{\\operatorname{Diff}}}_\\partial(N_{I'})$ $$\\Delta^k\\times (N_I\\cup_{\\partial_1} V^{p,q})\\rightarrow \\Delta^k\\times (N_I\\cup_{\\partial_1} V^{p,q}),$$ given by $\\varphi$ on $\\Delta^k\\times N_I$ and the identity on $\\Delta^k\\times V^{p,q}$, where we use that $N_{I'}\\cong N_I\\cup_{\\partial_1} V^{p,q}.$ This induces the stabilization map $B{\\widetilde{\\operatorname{Diff}}}_\\partial(N_I)\\rightarrow B{\\widetilde{\\operatorname{Diff}}}_\\partial(N_{I'}).$\n\n\n\n\\begin{THMB}\nThe map\n$$H_i(B{\\widetilde{\\operatorname{Diff}}}_\\partial(N_I);{\\mathbb Q})\\rightarrow H_i(B{\\widetilde{\\operatorname{Diff}}}_\\partial(N_{I'});{\\mathbb Q})$$\ninduced by the stabilization map with respect to $V^{p,q},$ where $3\\leq p\\leq q < 2p-1,$ is an isomorphisms for $g_p>2i+2$ when $2p\\neq n$ and $g_p>2i+4$ if $2p=n$ and an epimorphism for $g_p\\geq 2i+2$ respectively $g_p\\geq 2i+4$.\n\\end{THMB}\n\nThe argument is based on \\cite{berg12,berg13}, where homological stability for the automorphism spaces of the manifolds $\\mathlarger{\\#_g}(S^d\\times S^d) \\smallsetminus {\\operatorname{int}}(D^{2d})$ is shown. Berglund and Madsen moreover determine the stable homology of the homotopy automorphisms and the block diffeomorphisms. The argument to determine the stable homology relies on \\cite{galatiusrandalwilliams} where the stable cohomology of the moduli space of many even-dimensional smooth manifolds is determined. There is not an analogue of these results for odd-dimensional manifolds yet, but the homological stability results \\cite{galatiusrandalwilliamshomstab} have been generalized to odd-dimensional manifolds \\cite{perlmutter,perlmutter2}.\n\\subsection*{Outline of the argument}\nThe main idea of the proof is to consider the universal covering $$B{\\operatorname{aut}}_\\partial (N_I)\\langle 1 \\rangle \\rightarrow B{\\operatorname{aut}}_\\partial (N_I)$$ or rather, the universal covering spectral sequence with $E^2$-page\n$$H_s(\\pi_1(B{\\operatorname{aut}}_\\partial (N_I));H_t(B{\\operatorname{aut}}_\\partial (N_I)\\langle 1 \\rangle;{\\mathbb Q}))\\Rightarrow H_{s+t}(B{\\operatorname{aut}}_\\partial (N_I);{\\mathbb Q}).$$\nIf we can show that the stabilization map induces isomorphisms for large generalized genus $g_p$ in a range of $s$ and $t$, the spectral sequence comparison theorem implies homological stability for the homotopy automorphisms. So we reduced the problem to showing homological stability for the groups $\\pi_1(B{\\operatorname{aut}}_\\partial (N_I))$ with certain twisted coefficients.\n\nThe first step is to determine the homotopy classes of homotopy automorphisms (Section \\ref{on mapping class groups}), or rather the quotient acting non-trivially on the homology of the universal covering. More precisely, we determine the image and kernel of the ``reduced homology'' map $$\\tilde H:\\pi_0({\\operatorname{aut}}_\\partial(N_I))\\rightarrow {\\operatorname{Aut}} (\\tilde H_*(N_I))$$ to the automorphisms of the reduced homology as a graded group. For the manifolds $N_I$ the kernel is finite. This is in fact a consequence of the connectivity assumption $3\\leq p_i\\leq q_i < 2p_i-1$. The image which we call $\\Gamma_I\\subset {\\operatorname{Aut}} (\\tilde H_*(N_I))$ is the subgroup respecting the intersection form and when the dimension of $N_I$ is even, a certain tangential invariant. In particular we show later that elements in the kernel of $\\tilde H_*$ act trivially on $H_t(B{\\operatorname{aut}}_\\partial (N_I)\\langle 1 \\rangle;{\\mathbb Q}).$ Thus we only have to study the groups $\\Gamma_I.$\n\nIn Section \\ref{automorphisms of hyperbolic modules} we review hyperbolic modules and give a slight generalization in order to describe the $\\Gamma_I$ as automorphisms of an object with underlying graded ${\\mathbb Z}$-module $\\tilde H_*(N_I)$.\n\nIn Section \\ref{van der kallens and charneys homological stab results} we use homological stability results by Van der Kallen and Charney to show a homological stability result for the groups $\\Gamma_I$ with certain twisted coefficient systems of ``finite degree''.\nIn Section \\ref{On the rational homotopy type of homotopy automorphisms} we review results of Berglund and Madsen in order to show that the $H_t(B{\\operatorname{aut}}_\\partial (N_I)\\langle 1 \\rangle;{\\mathbb Q})$ are in fact a twisted coefficient system satisfying homological stability for the $\\Gamma_I.$ A crucial tool for showing twisted homological stability are Schur multifunctors (defined in Section \\ref{Polynomial functors and Schur multifunctors}), which we use to determine the degree of the coefficient systems. In fact developing the Schur multifunctors as a tool to handle the different degrees coming from the homology $\\tilde H_*(N_I)$ was one the the main hurdles in generalizing Berglund and Madsen's result.\n\nTo show the homological stability for the block diffeomorphisms, we consider the homotopy fibration $${\\operatorname{aut}}_\\partial (N_I)\/{\\widetilde{\\operatorname{Diff}}}_\\partial (N_I) \\to B{\\widetilde{\\operatorname{Diff}}}_\\partial (N_I)\\to B{\\operatorname{aut}}_\\partial (N_I).$$ We can determine the homology of a component of the homotopy fiber using surgery theoretic methods applied by Berglund and Madsen, which suffices to show homological stability using similar arguments as for the homotopy automorphisms.\n\n\n\n\n\n\\subsection*{Acknowledgments}\nThis paper is based on the authors PhD thesis, which was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).The author wants to thank his advisor Ib Madsen for suggesting this project and his advice throughout working on it. The author moreover wants to thank Alexander Berglund for comments on the presentation of this article.\n\n\\section{Polynomial functors and Schur multifunctors}\\label{Polynomial functors and Schur multifunctors}\n\nWe start by recalling the definition of polynomial functors in the sense of \\cite[Section 9]{eilenberg} or rather slightly modified as in \\cite[Section 3]{dwyer}. \nLet $T:\\mathcal{A}\\rightarrow \\mathcal{B}$ be a (not necessarily additive) functor between abelian categories. The first cross-effect functor is defined to be $$T^1(X)=\\operatorname{kernel}(T(X)\\to T(0)),$$ where $X\\rightarrow 0$ is the natural map to the zero object in $\\mathcal{A}$. For $k>1$, the $k$-th cross-effect functor $$T^k:\\mathcal{A}^k\\rightarrow \\mathcal{B},$$ is uniquely defined up to isomorphism given $T^l$ for $l<k$ by the properties:\n\\begin{enumerate} \n \\item $T^k(A_1,...,A_k)=T(0)$ if $A_i=0$ for some $i$.\n \\item There is a natural isomorphism $$T(A_1\\oplus ...\\oplus A_k)\\cong T(0)\\oplus \\underset{\\{i_1,...,i_r\\}}\\bigoplus T^r(A_{i_1},...,A_{i_r}),$$ where the sum runs over all non-empty subsets $\\{ i_1,...,i_r \\}\\subset\\{ 1,...,k \\}$.\n\\end{enumerate}\n\n\\begin{defi}\nA functor $T$ is polynomial of degree $\\leq k$, if $T^{l}$ is the constant zero functor for $l>k$. \n\\end{defi}\n\nAn immediate consequence is that a functor is of degree $\\leq 0$ if and only if it is constant. The higher cross-effects can be defined using deviations. The $k$-fold deviation of $k$ maps $f_1,...,f_k:A\\rightarrow B$ in $\\mathcal{A}$ is the map\n$$T(f_1\\top ... \\top f_k):T(A)\\rightarrow T(B),$$ given by $$T(f_1\\top ... \\top f_k)= T(0)\\oplus \\underset{\\{ i_1,...,i_r \\}}\\bigoplus (-1)^{k-r} T(f_{i_1},...,f_{i_r} ) ,$$ where the sum runs over all non-empty subsets $\\{ i_1,...,i_r \\}\\subset\\{ 1,...,k \\}$ and $0$ denotes the canonical map $A\\rightarrow 0 \\rightarrow B.$ Setting $A=A_1\\oplus...\\oplus A_k$ and denoting by $\\pi_i:A\\rightarrow A_i$ the projections and by $\\iota_i:A_i\\rightarrow A$ the inclusions, the $k$-th cross-effect functor is given on objects by $$T^k(A_1,...,A_k)=\\operatorname{Image}(T(\\iota_1\\circ\\pi_1\\top ... \\top \\iota_k\\circ\\pi_k)).$$\n\nThe following properties of Polynomial functors are direct consequences of the definition.\n\n\\begin{prop}[See e.g. \\cite{dwyer}]\\mbox{}\n\\begin{enumerate}\n \\item An additive functor is of degree $\\leq 1$.\n \\item The composition of functors of degree $\\leq k$ and $\\leq l$ is a functor of degree $\\leq kl$.\n \\item Let $T:\\mathcal{A}\\rightarrow\\mathcal{B}$ and $R:\\mathcal{C}\\rightarrow\\mathcal{B}$ be of degree $\\leq k$ and $\\leq l$, respectively. The level-wise sum $$T\\oplus R :\\mathcal{A}\\times \\mathcal{C}\\rightarrow \\mathcal{B}$$ is polynomial of degree  $\\leq \\operatorname{max}\\{k,l \\}$\n\\end{enumerate}\n\\end{prop}\n\n\\begin{example}Let $\\operatorname{Mod}({\\mathbb Z})$ be the category of finitely generated ${\\mathbb Z}$-modules. An example of a degree $\\leq k$ functor is the $k$-fold tensor product\n\\begin{displaymath}\n\\begin{aligned}\n\\otimes: &\\operatorname{Mod}({\\mathbb Z})^k &&\\rightarrow &&&\\operatorname{Mod}({\\mathbb Z})\\\\\n&(A_1,...,A_k) &&\\mapsto &&&\\bigotimes_i A_i.\n\\end{aligned}\t\n\\end{displaymath}\\end{example}Schur functors give examples of polynomial functors. Note that the following definitions also make sense for general commutative rings, but we are going to restrict our presentation to the category of graded rational vector spaces ${Vect_*}({\\mathbb Q}).$ Schur functor are treated for example in \\cite{algop}. We could not find any literature on Schur multifunctors and hence state the facts we need here. \n\nLet $\\mathscr{M}=\\mathscr{M}(n)\\in {Vect_*}({\\mathbb Q})$, $n\\geq 0$ be a sequence of ${\\mathbb Q}[\\Sigma_n]$-modules. We refer to them as $\\Sigma_n$-modules but implicitly use the ${\\mathbb Q}[\\Sigma_n]$-modules structure, in particular $\\otimes_{\\Sigma_n}$ refers to the tensor product over ${\\mathbb Q}[\\Sigma_n]$. The \\emph{Schur functor} given by $\\mathscr{M}$ is defined to be the endofunctor of ${Vect_*}({\\mathbb Q})$ induced by\n$$\\mathscr{M}(V)=\\bigoplus_k \\mathscr{M}(k)\\otimes_{\\Sigma_k}V^{\\otimes k} \\text{ for all }V\\in {Vect_*}({\\mathbb Q}),$$\nwhere $V^{\\otimes k}$ is the left $\\Sigma_k$-module with action induced by the permutation of the factors by the inverse (with sign according to the Koszul sign convention). Note that $\\mathscr{M}(0)$ is just a constant summand. A Schur functor $\\mathscr{M}$ with $\\mathscr{M}(l)$ trivial for $l>k$ is a polynomial functor of degree $\\leq k$.\n\nLet $\\eta=(n_1,...,n_l),$ $n_1,...,n_l\\geq 0$ be a multiindex. Throughout this article we will assume all multiindices to have non-negative entries. We use the following conventions:\\begin{displaymath}\n\\begin{aligned}\n\t|\\eta| &=\\sum_{i=1}^l n_i,\\\\\n\t\\ell(\\eta) &=l,\\\\\n\t\\mu+\\eta &=(m_1+n_1,...,m_l+n_l) \\text{, for $\\mu=(m_1,...,m_l)$},\\\\\n\t(V_i)^{\\otimes \\eta} &=V_1^{\\otimes n_1}\\otimes ... \\otimes V_l^{\\otimes n_l} \\text{, where }(V_i)\\in ({Vect_*}({\\mathbb Q}))^l,\\\\\n\t\\Sigma_\\eta &= \\Sigma_{n_1}\\times ... \\times \\Sigma_{n_l}.\n\\end{aligned}\n\\end{displaymath}\nConsider a sequence of ${\\mathbb Q}[\\Sigma_\\eta]$-modules $\\mathscr{N}=\\mathscr{N}(\\eta)\\in {Vect_*}({\\mathbb Q})$, $\\ell(\\eta)=l.$ As before, we refer to them as $\\Sigma_\\eta$-modules. We define the \\emph{Schur multifunctor} given by $\\mathscr{N}$ on objects by\n$$\\mathscr{N}(V_1,...,V_l)=\\bigoplus_{\\ell(\\eta)=l} \\mathscr{N}(\\eta)\\otimes_{\\Sigma_\\eta}(V_i)^{\\otimes \\eta}\\text{ for all }(V_i)\\in ({Vect_*}({\\mathbb Q}))^l.$$\n Similarly a Schur multifunctor $\\mathscr{N}$ is a polynomial of degree $\\leq k$ if $\\mathscr{N}(\\eta)$ is trivial for $|\\eta|>k$.\n\\begin{example}\nConsider Schur functors $\\mathcal{N}_i:Vect_*({{\\mathbb Q}})\\rightarrow Vect_*({{\\mathbb Q}}),$ $i=1,...,l$ The tensor product $$\\bigotimes \\mathcal{N}_i:Vect_*({{\\mathbb Q}})^l\\rightarrow Vect_*({\\mathbb Q})$$ is a Schur multifunctor with $(\\bigotimes \\mathcal{N}_i)(\\eta)=\\bigotimes\\mathcal{N}_i(n_i).$\n\\end{example}\nWe define the tensor product of $\\mathscr{M}=\\mathscr{M}(\\mu)$ and $\\mathscr{N}=\\mathscr{N}(\\eta)$ as $$\\mathscr{M}\\otimes \\mathscr{N}(\\nu):= \\bigoplus_{\\substack{\\mu'+\\eta'=\\nu}}\\operatorname{Ind}^{\\Sigma_\\nu}_{\\Sigma_{\\mu'}\\times \\Sigma_{\\eta'}}\\mathscr{M}(\\mu')\\otimes \\mathscr{N}(\\eta').$$\nThe Schur functor defined by this tensor product is indeed (up to natural isomorphism) the level-wise tensor product of the two functors, as we see by the isomorphisms for $\\mu'+\\eta'=\\nu$ with $\\ell(\\mu')=\\ell(\\eta')=\\ell(\\nu)$\n\\begin{displaymath}\n\\begin{aligned}\n&\\left( \\operatorname{Ind}^{\\Sigma_\\nu}_{\\Sigma_{\\mu'}\\times \\Sigma_{\\eta'}}\\mathscr{M}(\\mu')\\otimes \\mathscr{N}(\\eta') \\right)\\otimes_{\\Sigma_\\nu}(V_i)^{\\otimes \\nu} \\\\&= \\left( \\mathscr{M}(\\mu')\\otimes \\mathscr{N}(\\eta') \\otimes_{\\Sigma_{\\mu'}\\times \\Sigma_{\\eta'}} {\\mathbb Q}[\\Sigma_\\nu] \\right)\\otimes_{\\Sigma_\\nu}(V_i)^{\\otimes \\nu} \\\\\n&\\cong \\left( \\mathscr{M}(\\mu')\\otimes \\mathscr{N}(\\eta') \\right)\\otimes_{\\Sigma_{\\mu'}\\times \\Sigma_{\\eta'}}\\left( (V_i)^{\\otimes \\mu'}\\otimes (V_i)^{\\otimes \\eta'}\\right)\\\\\n&\\cong \\left( \\mathscr{M}(\\mu')\\otimes_{\\Sigma_{\\mu'}}(V_i)^{\\otimes \\mu'} \\right) \\otimes \\left( \\mathscr{N}(\\eta') \\otimes_{\\Sigma_{\\eta'}} (V_i)^{\\otimes \\eta'}\\right).\n\\end{aligned}\t\n\\end{displaymath}\n\nThe tensor powers of a $\\mathscr{N}=\\mathscr{N}(\\eta)$ are (up to natural isomorphism) explicitly described by\n\\begin{equation}\\label{schurtensorp}\n\\mathscr{N}^{\\otimes r}(\\nu)=\\bigoplus \\operatorname{Ind}_{\\Sigma_{\\eta_1}\\times...\\times \\Sigma_{\\eta_r}}^{\\Sigma_\\nu}\\mathscr{N}(\\eta_1)\\otimes ... \\otimes \\mathscr{N}(\\eta_r),\n\\end{equation}\nwhere the sum runs over all $r$-tuples $(\\eta_1,...,\\eta_r),$ where $\\ell(\\eta_i)=l,$ such that $\\Sigma_{i=1}^r \\eta_i=\\nu$.\n\nNow consider a Schur functor $\\mathscr{M}=\\mathscr{M}(m)$ and a Schur multifunctor $\\mathscr{N}=\\mathscr{N}(\\eta)$. The composition is a Schur multifunctor isomorphic to the Schur multifunctor given by\n\\begin{equation}\\label{schurcomp}\n(\\mathscr{M}\\circ \\mathscr{N})(\\nu)=\\bigoplus_{r} \\mathscr{M}(r) \\otimes_{\\Sigma_r} \\bigoplus \\operatorname{Ind}_{\\Sigma_{\\eta_1}\\times...\\times \\Sigma_{\\eta_r}}^{\\Sigma_\\nu}\\mathscr{N}(\\eta_1)\\otimes ... \\otimes \\mathscr{N}(\\eta_r),  \n\\end{equation}\nwhere the second sum runs over all $r$-tuples $(\\eta_1,...,\\eta_r),$ where $\\ell(\\eta_i)=l,$ such that $\\Sigma_{i=1}^r \\eta_i=\\nu$. The action of $\\Sigma_r$ is by permuting the tuples $(\\eta_1,...,\\eta_r)$ by the inverse. Indeed as we check using (\\ref{schurtensorp}):\n\\begin{displaymath}\n\\begin{aligned}\n(&\\mathscr{M}\\circ \\mathscr{N})(V_i) = \\bigoplus_r \\mathscr{M}(r) \\otimes_{\\Sigma_r} \\mathscr{N}(V_i)^{\\otimes r}\\cong \\bigoplus_r \\mathscr{M}(r) \\otimes_{\\Sigma_r} \\mathscr{N}^{\\otimes r}(V_i)\\\\\n&\\cong \\bigoplus_{r,\\nu} \\mathscr{M}(r) \\otimes_{\\Sigma_r} \\left( \\mathscr{N}^{\\otimes r}(\\nu)\\otimes_{\\Sigma_\\nu} (V_i)^{\\otimes \\nu} \\right)\\\\\n&\\cong \\bigoplus_{r,\\nu}  \\mathscr{M}(r) \\otimes_{\\Sigma_r} \\left( \\bigoplus_{\\Sigma_{i=1}^r \\eta_i=\\nu} \\operatorname{Ind}_{\\Sigma_{\\eta_1}\\times...\\times \\Sigma_{\\eta_r}}^{\\Sigma_\\nu}\\mathscr{N}(\\eta_1)\\otimes ... \\otimes \\mathscr{N}(\\eta_r)\\otimes_{\\Sigma_\\nu} (V_i)^{\\otimes \\nu} \\right)\n\\\\\n&\\cong \\bigoplus_\\nu  \\bigoplus_r\\mathscr{M}(r) \\otimes_{\\Sigma_r} \\left( \\bigoplus_{\\Sigma_{i=1}^r \\eta_i=\\nu} \\operatorname{Ind}_{\\Sigma_{\\eta_1}\\times...\\times \\Sigma_{\\eta_r}}^{\\Sigma_\\nu}\\mathscr{N}(\\eta_1)\\otimes ... \\otimes \\mathscr{N}(\\eta_r) \\otimes_{\\Sigma_\\nu} (V_i)^{\\otimes \\nu}\\right) .\n\\end{aligned}\n\\end{displaymath}\n\n\\begin{rem}\nWe will later use Schur (multi)functors with domain the category of rational vector spaces - just consider them as graded rational vector spaces concentrated in degree $0$.\n\\end{rem}\n\n\n\n\\section{Automorphisms of hyperbolic modules over the integers}\\label{automorphisms of hyperbolic modules}\n\nIn this section we review hyperbolic modules in the sense of \\cite{MR632404} in the special case with ground ring the integers. Fix a $\\lambda\\in \\{+1,-1\\}.$ Let $\\Lambda\\subset {\\mathbb Z},$ be an additive subgroup, called the \\emph{form parameter}, such that \\begin{equation}\\label{Lambda}\n\\{ z - \\lambda z | z \\in {\\mathbb Z}\\}\\subset \\Lambda \\subset \\{ z\\in {\\mathbb Z}|z=-\\lambda z\\}. \n\\end{equation}\n\\begin{defi}[\\cite{MR632404}]\nA $\\Lambda$-quadratic module is a pair $(M,\\mu)$, where $M$ is a ${\\mathbb Z}$-module and $\\mu$ is a bilinear form, i.e. a homomorphism $$\\mu:M\\otimes M\\rightarrow {\\mathbb Z}.$$\\end{defi}\nTo a $\\Lambda$-quadratic module $(M,\\mu)$ we associate a \\emph{$\\Lambda$-quadratic form} $$q_\\mu:M\\rightarrow {\\mathbb Z}\/\\Lambda\\text{, } q_\\mu(x)=[\\mu(x,x)]$$ and a \\emph{$\\lambda$-symmetric bilinear form} $$\\langle - , - \\rangle_\\mu:M\\otimes M\\rightarrow {\\mathbb Z},$$ defined by $\\langle x , y \\rangle_\\mu=\\mu(x,y)+\\lambda \\mu(y,x).$ ($\\lambda$-symmetric bilinear form like this are called even.) We call a finitely generated projective $\\Lambda$-quadratic module $(M,\\mu)$ \\emph{non-degenerate}, if the map $$M\\rightarrow M^*\\text{, }x\\mapsto \\langle x,-\\rangle_\\mu $$ is an isomorphism.\n\nDenote by $Q^\\lambda({\\mathbb Z},\\Lambda)$ the category non-degenerate $\\Lambda$-quadratic modules and morphisms linear maps respecting the associated $\\lambda$-symmetric bilinear form and the associated $\\Lambda$-quadratic form.\n\nGiven a finitely generated ${\\mathbb Z}$-module $M$, we define a non-degenerate $\\Lambda$-quadratic module $H(M)=(M\\oplus M^*,\\mu_M),$ where $\\mu_M((x,f),(y,g))=f(y).$ We call $H(M)$ the \\emph{hyperbolic module} on $M.$ A $\\Lambda$-quadratic module is called \\emph{hyperbolic}, if it is isomorphic to $H(N)$ for some finitely generated ${\\mathbb Z}$-module $N$. Let $\\{e_i\\}$ be the standard basis for ${\\mathbb Z}^g$ and $\\{f_i\\}$ the dual basis of $({\\mathbb Z}^g)^*.$ Using this basis we consider the automorphisms of the $\\Lambda$-quadratic module $H({\\mathbb Z}^g)$ as a subgroup of $Gl_{2g}({\\mathbb Z}).$ The subgroups can be described as follows:\n\n\\begin{prop}[{\\cite[Cor. 3.2.]{MR632404}}]\nThe automorphisms of $H({\\mathbb Z}^g)$ in $Q^{\\lambda}({\\mathbb Z},\\Lambda)$ are isomorphic to the subgroup of $Gl_{2g}({\\mathbb Z})$ consisting of matrices $\\begin{pmatrix}\nA & B   \\\\\nC & D \\end{pmatrix}$ such that\n\\begin{displaymath}\n\t\\begin{aligned}\n\t\t&D^TA + \\lambda B^TC=1 \\\\\n\t\t&D^TB+ \\lambda B^T D=0 \\\\\n\t\t&A^t C + \\lambda C^T A=0\\\\\n\t\t&\\text{$C^T A$ and $D^TB$ have diagonal entries in $\\Lambda.$}\n\t\\end{aligned}\n\\end{displaymath}\n\n\\end{prop}\n\nNote that if $\\lambda=1$ we necessarily have $\\Lambda=0.$ When $\\lambda=-1,$ the condition (\\ref{Lambda}) implies that $2{\\mathbb Z}\\subset \\Lambda \\subset {\\mathbb Z},$ thus we have the two cases $\\Lambda={\\mathbb Z}$ and $\\Lambda=2{\\mathbb Z}.$ Thus we can list the automorphisms of hyperbolic modules:\n\\begin{enumerate}\\item[]\n\t\\item When $\\lambda=1$ and $\\Lambda=0,$ then ${\\operatorname{Aut}}(H({\\mathbb Z}^g))=O_{g,g}({\\mathbb Z})$ in $Q^{1}({\\mathbb Z},0).$ \n\\item When $\\lambda=-1$ and $\\Lambda={\\mathbb Z},$ then ${\\operatorname{Aut}}(H({\\mathbb Z}^g))=Sp_{2g}({\\mathbb Z})$ in $Q^{-1}({\\mathbb Z},{\\mathbb Z}).$\n\\item When $\\lambda=-1$ and $\\Lambda=2{\\mathbb Z},$ then ${\\operatorname{Aut}}(H({\\mathbb Z}^g))$ in $Q^{-1}({\\mathbb Z},2{\\mathbb Z})$ is the subgroup of $Sp_{2g}({\\mathbb Z})$ described as:\n$$\\left \\{\\begin{pmatrix}\nA & B   \\\\\nC & D \\end{pmatrix}\\in Sp_{2g}({\\mathbb Z})| C^TA \\text{ and } D^TB \\text{ have even entries at the diagonal} \\right \\}.$$\n\\end{enumerate}\n\n\nLet $N$ be a ($d-1$)-connected $2d$-manifold. Wall \\cite{MR0145540} has shown that the automorphisms of the homology realized by diffeomorphisms are the automorphisms of a $\\Lambda$-quadratic module with underlying ${\\mathbb Z}$-module $H_d(N).$ Later we show a similar statement for connected sums of products of spheres. For this we need a slight variation of $\\Lambda$-quadratic modules.\n\nLet $n\\in{\\mathbb N}$ and $\\Lambda\\subset {\\mathbb Z},$ be an additive subgroup, such that \\begin{equation*}\n\\{ z - (-1)^{\\frac{n}{2}}z | z \\in {\\mathbb Z}\\}\\subset \\Lambda \\subset \\{ z\\in {\\mathbb Z}|z=-(-1)^{\\frac{n}{2}} z\\}. \n\\end{equation*} Note that in the case that $n$ is odd $\\Lambda$ has to be the trivial group, in fact in that case it will not be part of the following definitions at all, but we keep it in the notion for convenience.\n\\begin{defi}A \\emph{graded $\\Lambda$-quadratic module} is a pair $(M,\\mu)$, where $M$ is a graded ${\\mathbb Z}$-modules and $\\mu$ bilinear $n$-pairing, i.e. a degree $0$ homomorphism \n$$\\mu: M\\otimes M \\rightarrow {\\mathbb Z}[n].$$\n\\end{defi}\nWe associate to $(M,\\mu)$ a \\emph{symmetric bilinear $n$-pairing}\n$$\\langle - , - \\rangle_\\mu:M\\otimes M\\rightarrow {\\mathbb Z}[n],$$ defined by $\\langle x , y \\rangle_\\mu=\\mu(x,y)+(-1)^{|x||y|} \\mu(y,x).$\nWhen $n=2d$, we associate a \\emph{$\\Lambda$-quadratic form} $$q_\\mu:M_d\\rightarrow {\\mathbb Z}\/\\Lambda\\text{, } q_\\mu(x)=[\\mu(x,x)]$$  We call a finitely generated projective graded $\\Lambda$-quadratic module $(M,\\mu)$ non-degenerate, if the map $$M\\rightarrow {\\operatorname{Hom}} (M,{\\mathbb Z}[n])\\text{, }x\\mapsto \\langle x,-\\rangle_\\mu $$ is an isomorphism.\n\nFor $n$ even, we define $Q_*^{n}({\\mathbb Z},\\Lambda)$ to be the category whose objects are non-degenerate graded $\\Lambda$-quadratic modules $(M,\\mu)$, where $(M,\\mu)$ is finitely generated as a ${\\mathbb Z}$-module and the morphisms respect $q_\\mu$ and $\\langle-,-\\rangle_\\mu.$\n\nFor $n$ odd, we define $Q_*^{n}({\\mathbb Z},\\Lambda)$ to be the category whose objects are non-degenerate graded $\\Lambda$-quadratic modules $(M,\\mu)$, where $(M,\\mu)$ is finitely generated as a ${\\mathbb Z}$-module and the morphisms respect $\\langle-,-\\rangle_\\mu.$\n\nLet $Q_+^{n}({\\mathbb Z},\\Lambda)$ be the full subcategory with objects concentrated in positive degrees and hence necessarily concentrated in degrees $1,...,n-1$.\n\nFor $0<p_i\\leq q_i,$ $i\\in I$ such that $p_i+q_i=n$ and $|I|$ finite, we define a graded $\\Lambda$-quadratic module $$H_I=H({\\mathbb Z}^{g_1}[1]\\oplus ...\\oplus {\\mathbb Z}^{g_{\\lfloor n\/2 \\rfloor}}[\\lfloor n\/2 \\rfloor]\\oplus {\\operatorname{Hom}}_{{\\mathbb Z}}({\\mathbb Z}^{g_{\\lfloor n\/2 \\rfloor}}[\\lfloor n\/2 \\rfloor ],{\\mathbb Z}[n])\\oplus ...\\oplus{\\operatorname{Hom}}_{{\\mathbb Z}}({\\mathbb Z}^{g_{1}}[1],{\\mathbb Z}[n])),$$ where $$g_k = \\{i\\in I|p_i=k\\}.$$ Denote by $\\{a_i\\}$ standard basis for ${\\mathbb Z}^{g_1}[1]\\oplus ...\\oplus {\\mathbb Z}^{g_{\\lfloor n\/2 \\rfloor}}[\\lfloor n\/2\\rfloor]$ and by $\\{b_i\\}$ the dual basis. The pairing $\\mu_{H_I}=\\mu_I$ then given by $$\\mu_I(a_i,b_j)=b_j(a_i)=\\delta_{i,j}\\text{ and } \\mu_I(a_i,a_j)=\\mu_I(b_i,b_j)=0.$$\n\n\nDenote by $\\Gamma_I={\\operatorname{Aut}}(H_I)$ in $Q_+^{n}({\\mathbb Z},\\Lambda).$ We get the following cases:\n\\begin{enumerate}\\item[]\n\t\\item When $n$ is odd we get $$\\Gamma_I\\cong\\prod_{k=1}^{\\lfloor n\/2\\rfloor}Gl_{g_k}({\\mathbb Z}).$$\n\t\\item When $n=2d$ and $d$ is even we necessarily have $\\Lambda=0$ and $$\\Gamma_I\\cong O_{g_d,g_d}({\\mathbb Z})\\times \\prod_{k=1}^{n\/2-1}Gl_{g_k}({\\mathbb Z}).$$\n\t\\item Similarly for $n=2d$ with $d$ odd the only cases are $\\Lambda={\\mathbb Z},2{\\mathbb Z}$ and we just get a products of general linear groups and $Sp_{2g_d}({\\mathbb Z})$ respectively the subgroup described in the list of automorphism groups above under point 3.\n\\end{enumerate}\n\n\nBerglund and Madsen call group $G$ \\emph{rationally} perfect, if $H^1(G;V)=0$ for any finite dimensional rational $G$-representation $V.$ We later need that the automorphism groups of graded hyperbolic modules are rationally perfect.\n\n\\begin{lem}\\label{gammaIh1}The groups $\\Gamma_I$ are rationally perfect.\n\\end{lem}\n\n\\begin{proof}We begin by observing that being rationally perfect is stable under group extensions, i.e. if in a group extension $$0\\rightarrow K \\rightarrow G \\rightarrow C \\rightarrow 0,$$ $K$ and $C$ are rationally perfect, then so is $G.$ This follows from the Lyndon spectral sequence, since $H^1(C;H^0(K;V))$ and $H^0(C;H^1(K;V))$ are trivial for any finite dimensional rational $G$-representation $V.$ In particular products of rationally perfect groups are rationally perfect. Moreover we observe that finite groups are rationally perfect.\n\nIt follows from Borel's work on the cohomology of arithmetic groups that the automorphism groups ${\\operatorname{Aut}}(H({\\mathbb Z}^g))$ of the hyperbolic modules $H({\\mathbb Z}^g)$ in $Q^\\lambda({\\mathbb Z},\\Lambda)$ are rationally perfect for $g\\geq 2$ (see e.g. \\cite{berg13}).\n\nIn \\cite{bassmilnorserre} it is shown that $Sl_g({\\mathbb Z})$ is rationally perfect for $g\\geq 3$. Since $Sl_g({\\mathbb Z})$ is an index two normal subgroup of $Gl_g({\\mathbb Z}),$ this now also implies that $Gl_g({\\mathbb Z})$ is rationally perfect for $g\\geq 3.$\n\nRecall that the $\\Gamma_I$ are products of ${\\operatorname{Aut}}(H({\\mathbb Z}^k))$ and $Gl_l({\\mathbb Z})$-s. Since $Gl_1({\\mathbb Z})$ is finite and hence rationally perfect, to finish the proof we have to show that ${\\operatorname{Aut}}(H({\\mathbb Z}))$ and $Gl_2({\\mathbb Z})$ are rationally perfect.\n\nThe group $Sl_2({\\mathbb Z})$ is an extension $$0\\rightarrow C_2\\rightarrow Sl_2({\\mathbb Z}) \\rightarrow C_2 * C_3 \\rightarrow 0$$ and $C_2 * C_3$ can be seen to be rationally perfect using a Mayer-Vietoris argument. Hence $Sl_2({\\mathbb Z})$ and also $Gl_2({\\mathbb Z})$ are rationally perfect.\n\nThe group ${\\operatorname{Aut}}(H({\\mathbb Z}))$ in $Q^{-1}({\\mathbb Z},{\\mathbb Z})$ is $Sp_2({\\mathbb Z}),$ which is isomorphic to $Sl_2({\\mathbb Z}).$ For $\\Lambda=2{\\mathbb Z}$ it follows that ${\\operatorname{Aut}}(H({\\mathbb Z}))$ is rationally perfect because it is a finite index subgroup of $Sp_2({\\mathbb Z}).$\n\nRecall that for $\\lambda=1,$ we necessarily have $\\Lambda=0$ and ${\\operatorname{Aut}}(H({\\mathbb Z}))\\cong O_{1,1}({\\mathbb Z})\\cong C_2\\times C_2$ and hence we are done.\n\\end{proof}\n\n\n\\section{Van der Kallen's and Charney's homological stability results}\\label{van der kallens and charneys homological stab results}\nIn this section we recall van der Kallen's homological stability for general linear groups and Charney's homological stability for automorphisms of hyperbolic quadratic modules. We combine them to homological stability for the $\\Gamma_I$ defined above with certain coefficient systems induced by polynomial functors.\n\\begin{rem}\nCharney's results hold for Dedekind domains with involutions and van der Kallen's for associative rings with finite stable range, but we restrict our presentation to ${\\mathbb Z}$ (with trivial involution).\t\n\\end{rem}\n\nWe begin by reviewing the notion of coefficient systems as discussed in \\cite{dwyer}.\nA coefficient system for $\\{ Gl_g({\\mathbb Z}) \\}_{g \\geq 1}$ is a sequence of $Gl_g({\\mathbb Z} )$-modules $\\{ \\rho_g\\}_{g \\geq 1}$ together with $ Gl_g({\\mathbb Z}) $-maps $F_g:\\rho_g\\rightarrow I^* (\\rho_{g+1})$, where $I^*$ denotes the restriction via the upper inclusion $I:Gl_g({\\mathbb Z})\\hookrightarrow Gl_{g+1}({\\mathbb Z})$. We denote the system by $\\rho$ and call the maps $F_g$ structure maps. A map of coefficient systems $\\rho$ and $\\rho'$ is a collection of $Gl_g({\\mathbb Z})$-maps $\\{\\tau_g\\}_{g \\geq 1}$ such that they commute with the structure maps.  The level-wise kernels and  cokernerls are again coefficient systems with the obvious structure maps.\nDenote by $J:Gl_g({\\mathbb Z})\\rightarrow Gl_{g+1}({\\mathbb Z})$ the lower inclusion map. For a coefficient system $\\rho$ we define the shifted system $\\Sigma \\rho$ by $\\Sigma \\rho_g := J^* (\\rho_{g+1})$ with structure maps $\\Sigma F_g:= J^* F_{g+1}: J^* (\\rho_{g+1})\\rightarrow I^*J^* (\\rho_{g+2})$ . Denote by $s_g\\in Gl_g({\\mathbb Z})$ the element permuting the last two standard basis elements.  We call a coefficient system central, if $s_{g+2}$ acts trivially on the image of $F_{g+1}F_g:\\rho_g\\rightarrow \\rho_{g+2}$. Denote by $e_{g-1,g}\\in Gl_g({\\mathbb Z})$ the element sending all but the $g$-th standard basis element to itself and the $g$-th $e_g$ to $e_{g-1}+e_g$. We call a central coefficient system strongly central, if $e_{g+1,g+2}$ acts trivially on the image of $F_{g+1}F_g:\\rho_g\\rightarrow \\rho_{g+2}.$\n\nLet $c_g\\in Gl_g({\\mathbb Z})$ $(g>1)$ be the element sending the i-th standard basis element to the $(i+1)$-st and the $g$-th to the first. \n\nDenote by $\\mu(c_g)$ the multiplication from the left by $c_g$. Then the following holds:\n\\begin{lem}[{\\cite[Lemma 2.1.]{dwyer}}]\nLet $\\rho$ be a central coefficient system. Then we have a map of coefficient systems $\\tau:\\rho\\rightarrow\\Sigma\\rho$, defined by\n$$\\tau_g:\\rho_g\\xrightarrow{F_g} I^*(\\rho_{g+1})\\xrightarrow{\\mu (c_{g+2})} {J^*}(\\rho_{g+1})=\\Sigma \\rho_g.$$\n\\end{lem}\nWe say that a central coefficient system $\\rho$ splits, if $\\Sigma \\rho$ is isomorphic to $\\rho\\oplus \\text{coker}(\\tau)$ via $\\tau$. We then denote $\\text{coker}(\\tau)$ by $\\Delta \\rho$. We now define the notion of degree of a strongly central coefficient system $\\rho$ inductively. We say it has degree $k<0$, if it is constant and for $k\\geq 0$ we say that it has degree $\\leq k$, if $\\Sigma \\rho$ splits and $\\Delta \\rho$ is a strongly central coefficient system of degree $k-1$.\n\n\\begin{thm}[{Van der Kallen \\cite[p. 291]{vdk}}]\\label{vdkstab}\n Let $\\rho$ be a strongly central coefficient system of degree $\\leq k$, then $$H_i(Gl_g({\\mathbb Z}),\\rho_g))\\rightarrow H_i(Gl_{g+1}({\\mathbb Z}),\\rho_{g+1})$$ is an isomorphism for $g>2i+k+2$ and en epimorphism for $g\\geq2i+k+2$.\n\\end{thm}\n\nDenote by $\\lambda_g$ the standard representation of $Gl_g({\\mathbb Z})$ on ${\\mathbb Z}^g$ and by $\\bar \\lambda_g$ the action by the inverse transposed on ${\\mathbb Z}^g.$ Let $\\mathcal{A}$ be an abelian category. Given a functor $$T:\\operatorname{Mod}({\\mathbb Z})\\times\\operatorname{Mod}({\\mathbb Z})\\rightarrow \\mathcal{A},$$ we define a coefficient system $\\{T(\\lambda_g,\\bar \\lambda_g)\\}_{g\\geq 1}$ with structure maps induced by the standard inclusions and actions induced by $\\lambda_g$ and $\\bar \\lambda_g$.\n\n\\begin{lem}[{Compare \\cite[5.5.]{vdk} and \\cite[Lemma 3.1.]{dwyer}}]\nIf $$T:\\operatorname{Mod}({\\mathbb Z})\\times\\operatorname{Mod}({\\mathbb Z})\\rightarrow \\mathcal{A}$$ is a polynomial functor of degree $\\leq k$, then $\\{ T(\\lambda_g,\\bar \\lambda_{g})\\}_{g\\geq 1}$ is a strongly central coefficient system of degree $\\leq k$. \n\\end{lem} \n\nDenote now $G_g$ the automorphisms of $H({\\mathbb Z}^g)$ in $Q^{\\lambda}({\\mathbb Z},\\Lambda).$ And denote by $e_1,...,e_g$ the standard basis for ${\\mathbb Z}^g$ and by $f_1,...,f_g$ the dual basis of $({\\mathbb Z}^g)^*.$ We see $G_g$ as a subgroup of $Gl_{2g}({\\mathbb Z})$, by considering the elements of $G_g$ as $2g\\times 2g$-matrices acting on $H({\\mathbb Z}^g)\\cong {\\mathbb Z}^{2g}.$ We define the upper inclusion $$I:G_g\\rightarrow G_{g+1}\\text{, }\\begin{pmatrix}\nA & B   \\\\\nC & D \\end{pmatrix}\\mapsto \\begin{pmatrix}\nA & 0 & B & 0\\\\\n0 & 1 & 0 & 1\\\\\nC & 0 & D & 0\\\\\n0 & 1 & 0 & 1\\\\\n\\end{pmatrix} $$ and similarly the lower inclusion $J:G_g\\rightarrow G_{g+1}.$ The definition of a coefficient system is very similar to the one for $GL_g({\\mathbb Z})$ and we only briefly summarize it.\nA coefficient system for $\\{ G_g \\}_{g \\geq 1}$ is a sequence of $G_g$-modules $\\{ \\rho_g\\}_{g \\geq 1}$ together with $ G_g $-maps $F_g:\\rho_g\\rightarrow I^* (\\rho_{g+1}).$ We denote a coefficient system again by $\\rho$ and let maps of coefficient systems be as above. The shifted coefficient system $\\Sigma \\rho$ is the restriction via the lower inclusion as above. A coefficient system is called central if $c_{g+2}\\circ c_{2g+4}$ acts trivially on the image of $F_{g+1}F_g:\\rho_g\\rightarrow \\rho_{g+2}$. For a central coefficient system we define the map of coefficient systems $\\tau:\\rho\\rightarrow\\Sigma\\rho$, by\n$$\\tau_g:\\rho_g\\xrightarrow{F_g} I^*(\\rho_{g+1})\\xrightarrow{\\mu (c_{g+2}\\circ c_{2g+4})} {J^*}(\\rho_{g+1})=\\Sigma \\rho_g.$$ We call a central coefficient system $\\rho$ split, if $\\tau$ is injective and $\\Sigma \\rho \\cong \\tau(\\rho)\\oplus \\operatorname{coker}(\\tau).$ For a central coefficient system we define the degree inductively: We say it has degree $k<0$, if it is constant and for $k\\geq 0$ we say that it has degree $\\leq k$, if $\\Sigma \\rho$ splits and $\\Delta  \\rho=\\operatorname{coker}(\\tau)$ is a strongly central coefficient system of degree $\\leq k-1$.\n\n\\begin{thm}[{Charney \\cite[Theorem 4.3.]{MR885099}}]\\label{charneystab}\n Let $\\rho$ be a central coefficient system of degree $\\leq k$, then $$H_i(G_g,\\rho_g))\\rightarrow H_i(G_{g+1},\\rho_{g+1})$$ is an isomorphism for $g>2i+k+4$ and an epimorphism for $g\\geq 2i+k+4$.\n\\end{thm}\n\nAgain we get a central coefficient system of degree $\\leq k$ by considering the standard $G_g$-action $\\lambda_{g,g}$ on $H({\\mathbb Z}^g)\\cong {\\mathbb Z}^{2g}$, induced by the inclusion $G_g\\subset Gl_{2g}({\\mathbb Z}).$ Let $\\mathcal{A}$ be an abelian category. Given a polynomial functor $$T:\\operatorname{Mod}({\\mathbb Z})\\rightarrow \\mathcal{A},$$ of degree $\\leq k$ then $\\{T(\\lambda_{g,g})\\}_{g\\geq 1}$ is a central coefficient system of degree $\\leq k$ for $\\{G_g \\}_{g\\geq 1}$.\n\nNow we combine the homological stability results above to a result for the groups $\\Gamma_I$ defined in the previous section.\n\nDenote by $\\lambda_I$ the \\emph{standard representation of $\\Gamma_I$} on $({\\mathbb Z}^{g_1},...,{\\mathbb Z}^{2 g_{n\/2}},...,{\\mathbb Z}^{g_{n-1}})$ induced by the inclusion $\\Gamma_I\\subset \\prod_{i=1}^{\\lfloor n\/2\\rfloor} Gl_{r_i}({\\mathbb Z}),$ where we consider ${\\mathbb Z}^{2 g_{n\/2}}$ to be the empty set if $n$ is odd and \n$$r_k = \\begin{cases} 2g_k=2|\\{i\\in I|p_i=k\\}| &\\text{ if }k=n\/2\\\\ g_k=|\\{i\\in I|p_i=k\\}| &\\text{ if }k <n\/2.\\end{cases}$$ More explicitly, we assume that an automorphisms $$A=(A_1,...,A_{n\/2})\\in \\Gamma_I\\subset \\prod_{i=1}^{\\lfloor n\/2\\rfloor} Gl_{r_i}({\\mathbb Z}),$$ acts by matrix multiplication of $A_i$ on ${\\mathbb Z}^{g_i}$ for $i\\leq n\/2$ and by multiplication by the inverse transposed of $A_i$ on ${\\mathbb Z}^{g_{n-i}}$ for $i>n\/2.$\n\nGiven a functor to an abelian category $T:Mod({\\mathbb Z})^{n-1}\\rightarrow \\mathcal{A}$, we get a $\\Gamma_I$-module $T({\\mathbb Z}^{g_1},...,{\\mathbb Z}^{2g_{n\/2}},...,{\\mathbb Z}^{g_{n-1}})$ with the induced action. We denote this $\\Gamma_I$-module by $T(\\lambda_I).$ For a fixed $p\\in{\\mathbb N}$, such that $0<p\\leq n\/2,$ denote by $\\Gamma_{I'}$ the automorphisms of $H_{I'},$ where $I'=I\\cup \\{i'\\}$ with $p_{i'}=p$ and $q_{i'}=n-p$. We define the \\emph{stabilization map} $$\\sigma_{p,n-p}:H_i(\\Gamma_I,T(\\lambda_I))\\rightarrow H_i(\\Gamma_{I'},T(\\lambda_{I'}))$$ to be the map induced by the obvious upper inclusion $I_{p,q}:\\Gamma_I\\rightarrow\\Gamma_{I'}$ and $T(I_{p,q}).$\n\n\\begin{prop}\\label{general algebraic homological stability result}Let $\\mathcal{A}$ be some abelian category, $T:Mod({\\mathbb Z})^{n-1}\\rightarrow \\mathcal{A}$ be a polynomial functor of degree $\\leq k$. The stabilization map $$\\sigma_{p,n-p}:H_i(\\Gamma_I,T(\\lambda_I))\\rightarrow H_i(\\Gamma_{I'},T(\\lambda_{I'}))$$ induces an isomorphism for $g_p>2i+k+2$ when $2p\\neq n$ and $g_p>2i+k+4$ if $2p=n$ and an epimorphism for $g_p\\geq 2i+k+2$ respectively $g_p\\geq2i+k+4.$\n\\end{prop}\n\n\\begin{proof} Denote by $\\Gamma_{g_p}$ the summand of $\\Gamma_I\\subset \\prod_{i=1}^{\\lfloor n\/2\\rfloor} Gl_{r_i}({\\mathbb Z})$ that sits in $Gl_{r_p}({\\mathbb Z})$. Let $\\Gamma={\\operatorname{Aut}}(H_{\\bar I}),$ where $\\bar I= I\\smallsetminus \\{i\\in I| p_i=p\\}.$ Note that $\\Gamma\\oplus \\Gamma_{g_p}= \\Gamma_I$ and $\\Gamma_{I'}=\\Gamma_{g_p+1}\\oplus \\Gamma,$ where $\\Gamma_{g_p+1}$ is defined analogous to $\\Gamma_{g_p}.$\nConsider the functor $$\\mathfrak{I}_{p,n-p}:\\begin{cases}\n\\operatorname{Mod}({\\mathbb Z})\\rightarrow Mod({\\mathbb Z})^{n-1}\\text{ if $2p=n$}\\\\\n\\operatorname{Mod}({\\mathbb Z})\\times\\operatorname{Mod}({\\mathbb Z}) \\rightarrow Mod({\\mathbb Z})^{n-1}\t\\text{ otherwise,}\n\\end{cases}$$ defined by sending a module $M$ to $({\\mathbb Z}^{g_1},...,M,...,{\\mathbb Z}^{g_{n-1}})$, where the $M$ sits at the $(n\/2)$-th summand and a pair $(M,N)$ to $({\\mathbb Z}^{g_1},...,M,...,N,...,{\\mathbb Z}^{g_{n-1}})$, where the $M$ sits at the $p$-th summand the $M$ sits at the $(n-p)$-th summand. This functor is clearly additive and hence of degree $\\leq 1.$ This implies that the composition $T\\circ \\mathfrak{I}_{p,n-p}$ is of degree $\\leq k$ and we get a (strongly) central coefficient system of degree $\\leq k$ for $$\\Gamma_{g_p}=\\begin{cases} G_{g_p} & \\text{ if $2 p=n$}  \\\\\n\tGl_{g_p}({\\mathbb Z}) & \\text{otherwise.} \n\\end{cases}$$\nThis implies that the stabilization maps\n$$H_i(\\Gamma_{g_p},T\\circ \\mathfrak{I}_{n\/2,n\/2}(\\lambda_{g_p,g_p}))\\rightarrow H_i(\\Gamma_{g_{p}+1},T\\circ \\mathfrak{I}_{n\/2,n\/2}(\\lambda_{g_p+1,g_p+1}))$$ respectively\n$$H_i(\\Gamma_{g_p},T\\circ \\mathfrak{I}_{p,n-p}(\\lambda_{g_p},\\bar \\lambda_{g_p}))\\rightarrow H_i(\\Gamma_{g_{p}+1},T\\circ \\mathfrak{I}_{p,n-p}(\\lambda_{g_p+1},\\bar \\lambda_{g_p+1}))$$ are isomorphism and epimorphism in the ranges in the statement of the Proposition. Observing that the $T\\circ \\mathfrak{I}_{n\/2,n\/2}(\\lambda_{g_p,g_p})$ respectively $T\\circ \\mathfrak{I}_{p,n-p}(\\lambda_{g_p},\\bar \\lambda_{g_p})$ are precisely the restrictions of the $\\Gamma_I$-representation to the subgroup $\\Gamma_{g_p}$ and using Lyndon spectral sequence $$H_k(\\Gamma_{\\bar I},H_l(\\Gamma_{g_p},T(\\lambda_I))) \\Rightarrow H_i(\\Gamma_I,T(\\lambda_I)),$$ the results follows by comparing spectral sequences.\n\\end{proof}\n\n\\section{On Mapping Class Groups}\\label{on mapping class groups}\nDenote by $$N=N_I=\\left(\\#_{i\\in I} (S^{p_i}\\times S^{q_i})\\right)\\smallsetminus {\\operatorname{int}} (D^n),$$ where $|I|<\\infty,$ $3\\leq p_i\\leq q_i<2p_i-1$ and $p_i+q_i=n$ for $i\\in I$.\n\nDenote by ${\\operatorname{Aut}}(\\tilde{H}_*(N_I))$ the automorphisms of the graded group $\\tilde{H}_*(N_I).$\nIn this section we study the map $$H_*:\\pi_0 \\; {\\operatorname{aut}}_\\partial (N_I)\\rightarrow {\\operatorname{Aut}}(\\tilde{H}_*(N_I)).$$ In particular we are going to determine its image and show that the kernel is finite.\n\nDenote by $in:\\partial N\\hookrightarrow N$ the inclusion of the boundary. We observe that $V_I=\\bigvee_{i\\in I} (S^{p_i}\\vee S^{q_i})\\subset N$ is a deformation retract and denote by $$\\alpha_j:S^{p_j} \\hookrightarrow \\bigvee_{i\\in I} (S^{p_i}\\vee S^{q_i}) \\text{ and } \\beta_j:S^{q_j} \\hookrightarrow \\bigvee_{i\\in I} (S^{p_i}\\vee S^{q_i})$$ the canonical inclusions. We consider $in$ as an element of $\\pi_{n-1}(\\bigvee_{i\\in I} (S^{p_i}\\vee S^{q_i}))$ and we observe that it is given by $\\sum_{i\\in I} [\\alpha_i,\\beta_i]$. Denote by $$\\begin{aligned}\\langle-,-\\rangle:&H_*(N)\\otimes H_{n-*}(N) \\rightarrow {\\mathbb Z}\\\\ &x\\otimes y \\mapsto (PD^{-1}(x)\\cup PD^{-1}(y))([N,\\partial N])\\end{aligned}$$ the intersection form, where $PD^{-1}:H_*(N)\\rightarrow H^{n-*}(N,\\partial N)$ denotes the Poincar\\'e duality isomorphisms and we evaluate on the fundamental class $[N,\\partial N].$ The $\\{ \\alpha_i \\}$ and $\\{ \\beta_i \\}$ define a basis for $\\tilde H_*(N)$ via the Hurewicz homomorphism, which we denote by $\\{ a_i\\}$ respectively $\\{b_i\\}$. Note that $b_i$ is dual to $a_i.$\n\nIn the case $n=2d$ is even we need to recall a further piece of structure from Wall's classification of highly connected even-dimensional manifolds \\cite{MR0145540}. The elements $x\\in H_{d}(N)$ can be represented by embedded $S^{d}.$ Denote by $\\nu_x\\in \\pi_{d-1}(SO(d))$ the clutching function of the normal bundle of this embedding. It is independent of the choice of embedding, since homotopic embeddings are isotopic in this case. This defines a function $$q:H_{d}(N)\\rightarrow \\pi_{d-1}(SO(d))\\text{, }x\\mapsto [\\nu_x] .$$ Denote by $\\iota_{d}$ the class of the identity in $\\pi_{d}(S^{d})$ and by $$\\partial:\\pi_d(S^d)\\rightarrow \\pi_{d-1}(SO(d))$$ the boundary map in the fibration $SO(d)\\rightarrow SO(d+1)\\rightarrow S^d.$ The function $q$ satisfies:\n$$\\langle x,x \\rangle=HJq(x) \\text{ and } q(x+y)=q(x)+q(y)+\\langle x,y \\rangle\\partial \\iota_d,$$ where $\\pi_{d-1}(SO(d))\\xrightarrow{J} \\pi_{2d-1}(S^d)\\xrightarrow{H}{\\mathbb Z}$ denote the $J$-homomorphism and the Hopf invariant. There is also a purely homotopy theoretic description of $Jq$ in \\cite[Section 8]{MR0148075}. \n\nNote that for $a_i,b_j\\in H_{d}(N),$ we have $q(a_i)=q(b_j)=0.$ Hence $\\operatorname{Image}(q)$ is contained in the subgroup $\\langle\\partial\\iota_d\\rangle$ generated by $\\partial\\iota_d.$ The $J$-homomorphism restricts to an isomorphism $$J|\\langle\\partial\\iota_d\\rangle:\\langle\\partial\\iota_d\\rangle\\rightarrow J(\\langle\\partial\\iota_d\\rangle)\\cong \\begin{cases} \\langle [\\iota_d,\\iota_d]\\rangle \\xrightarrow[H]{\\cong} 2{\\mathbb Z} &\\text{if $d$ even}\\\\\n0 &\\text{if $d=1,3,7$ }\\\\\n\\langle [\\iota_d,\\iota_d]\\rangle\\cong {\\mathbb Z}\/2{\\mathbb Z} &\\text{if $d$ is odd and not $1,3$ or $7$,}\n\\end{cases}$$ where the second isomorphism is induced by the Hopf invariant.\nLet $${\\operatorname{Aut}}(\\tilde H_*(N),\\langle-,-\\rangle, Jq)\\text{ and }{\\operatorname{Aut}}(\\tilde H_*(N),\\langle-,-\\rangle, q)$$ be the automorphisms of the reduced homology respecting the intersection form and the function $Jq$ ($q$ respectively). Note that $$\\langle x, y \\rangle=\\mu(x,y)+(-1)^{|x||y|}\\mu(y,x),$$ where $\\mu(-,-)$ is determined by $$\\mu(a_i,b_j)=\\delta_{i,j} \\text{ and } \\mu(b_i,a_j)=\\mu(a_i,a_j)=\\mu(b_i,b_j)= 0.$$ Now let $$\\Lambda=\\begin{cases}0 & \\text{ if $n=2d$ and $d$ is even}\\\\\n{\\mathbb Z} & \\text{ if $n=2d$ and $d$ is $3$ or $7$}\\\\\n2{\\mathbb Z} &\\text{ if $n=2d$ and $d$ is odd and not $3$ or $7$.} \n\n\\end{cases}$$ \n\nMoreover $Jq=q_\\mu,$ where $q_\\mu$ is the $\\Lambda$-quadratic form associated to $\\mu,$ where we identify $\\langle [\\iota_d,\\iota_d]\\rangle$ with ${\\mathbb Z}$ and ${\\mathbb Z}\/2{\\mathbb Z}$ respectively. It suffices to check this for the elements $a_i+b_i$ and for these $Jq(a_i+b_i)=[\\iota_d,\\iota_d]$ and $q_\\mu(a_i+b_i)=1.$ By the discussion above we see that \n$${\\operatorname{Aut}}(\\tilde H_*(N),\\langle-,-\\rangle, q)\\cong {\\operatorname{Aut}}(\\tilde H_*(N),\\langle-,-\\rangle, Jq)\\cong\\Gamma_I= {\\operatorname{Aut}}(H_I) \\text{ in } Q_*^n({\\mathbb Z},\\Lambda).$$\n\nFor a representative $f$ of $[f]\\in \\pi_0 ({\\operatorname{aut}}_\\partial (N))$ it is clear that $\\tilde H_*(f)\\in \\Gamma_I$. We are now going to show that all elements of $\\Gamma_I$ can be realized by a homotopy self-equivalence, fixing the boundary pointwise.\n\n\n\\begin{prop}\\label{hmcg}\nThe group homomorphism\n\\begin{equation*}\n\\pi_0 ({\\operatorname{aut}}_\\partial (N))\\xrightarrow{\\tilde H_*} {\\operatorname{Aut}} (\\tilde H_* (N),\\langle-,-\\rangle, Jq )\n\\end{equation*}\nis surjective and has finite kernel.\n\\end{prop}\n\\iffalse\nBefore we prove the Proposition, we need the following Lemma. \n\n\\begin{lem}\\label{splitting}\nLet $A,B$ be pointed topological spaces, such that $\\tilde{H}_i(A)$ and $\\tilde{H}_i(B)$ are not both nontrivial for all $i.$ Then the image of $$\\tilde{H}_*:{\\operatorname{aut}}_* (A\\vee B)\\rightarrow {\\operatorname{Aut}}(\\tilde{H}_*(A\\vee B))$$ splits as $$\\operatorname{Image}(\\tilde{H}_*:{\\operatorname{aut}}_* (A)\\rightarrow {\\operatorname{Aut}}(\\tilde{H}_*(A)))\\oplus \\operatorname{Image}(\\tilde{H}_*:{\\operatorname{aut}}_* (B)\\rightarrow {\\operatorname{Aut}}(\\tilde{H}_*(B)))$$\n\\end{lem}\n\n\\begin{proof}\nWe observe that ${\\operatorname{Aut}}(\\tilde{H}_*(A\\vee B))\\cong {\\operatorname{Aut}}(\\tilde{H}_*(A))\\oplus {\\operatorname{Aut}}(\\tilde{H}_*(B))$ where the isomorphism is induced by the canonical isomorphism $\\tilde{H}_*(A\\vee B)\\cong \\tilde{H}_*(A) \\oplus \\tilde{H}_*(B).$ The lemma now follows.\n\\end{proof}\n\\fi\n\\begin{proof}Compare \\cite[Proof of Theorem 2.10]{berg12}. The cofibration $in:\\partial N\\hookrightarrow N$ induces a fibration \n$${\\operatorname{map}}_\\partial (N,N)\\rightarrow{\\operatorname{map}}_*(N,N) \\rightarrow {\\operatorname{map}}_*(\\partial N, N),$$ where ${\\operatorname{map}}_\\partial (N,N)$ is the fiber over $in$. Let ${\\operatorname{map}}_\\partial (N,N)$ and ${\\operatorname{map}}_*(N,N)$ be based at the identity. Restricting the total space to invertible elements, we also get the following fibration: $${\\operatorname{aut}}_\\partial (N)\\rightarrow{\\operatorname{aut}}_*(N) \\rightarrow {\\operatorname{map}}_*(\\partial N, N).$$\n\nWe are going to analyze the long exact homotopy sequences $$\\xymatrix{\n...\\ar[r]& \\pi_1 (map_*(\\partial N, N),in)\\ar[r]\\ar@{=}[d]& \\pi_0( {\\operatorname{aut}}_\\partial(N) )\\ar[r]\\ar@{^{(}->}[d]& \\pi_0 ({\\operatorname{aut}}_*(N))\\ar[r]\\ar@{^{(}->}[d]& [\\partial N,N]_*\\ar@{=}[d]\\\\\n...\\ar[r]& \\pi_1 (map_*(\\partial N, N),in)\\ar[r]& [N,N]_\\partial \\ar[r]& [N,N]_*\\ar[r]& [\\partial N,N]_* .}$$ We consider the monoid homomorphism $$\\tilde H_*:[N,N]_*\\rightarrow \\operatorname{End}(\\tilde H_*(N))$$ and show that it is onto and with finite kernel. Using the relative Hurewicz isomorphism, it is easy to see that $V_I\\hookrightarrow \\prod_{i\\in I} (S^{p_i}\\times S^{q_i})$ is $(2\\operatorname{min}_{i\\in I}\\{ p_i \\}-1)$-connected and hence more than $\\operatorname{max}_{i\\in I}\\{ q_i \\}$-connected. Thus we get an isomorphism of sets \\begin{align*}&[N,N]_* \\cong [V_I,V_I]_* \\cong [V_I,\\prod_{i\\in I} (S^{p_i}\\times S^{q_i})]_* \\\\\n&\\cong \\prod_{(i,j)\\in I\\times I} [S^{p_i},S^{p_j}]_* \\times \\prod_{(i,j)\\in I\\times I} [S^{q_i},S^{q_j}]_* \\times \\prod_{(i,j)\\in I\\times I} [S^{q_i},S^{p_j}]_* \\times \\prod_{(i,j)\\in I\\times I} [S^{p_i},S^{q_j}]_* .\n\\end{align*} We write $I=\\bigcup_l I_l,$ where $I_l=\\{i|p_i=l\\}.$\nThe only non-finite factors of the product above are \\begin{equation}\\label{nonfinitefactors}\n\t\\prod_{l}\\prod_{(i,j)\\in I_l\\times I_l} \\left([S^{p_i},S^{p_j}]_*\\times[S^{q_i},S^{q_j}]_*\\right).\n\\end{equation} We identify $\\operatorname{End}(\\tilde H_*(N))\\cong \\prod \\operatorname{Mat}_{r_l}({\\mathbb Z}) ,$ where $r_l=\\operatorname{rank}(H_l(N)),$ using the basis $\\{a_i\\}\\cup\\{b_i\\}$. Note that for $l=n\/2$ a $b_i$ becomes a $(r_l\/2+i)$-th basis element. Denote by $\\alpha^l_1,...,\\alpha^l_{r_l}$ and $\\beta^l_1,...,\\beta^l_{r_l}$ the inclusions $S^l\\hookrightarrow V_I$ and $S^{n-l}\\hookrightarrow V_I$ respectively. There is a multiplicative section of $\\tilde H_*$ \\begin{equation}\\label{section} \\prod \\operatorname{Mat}_{r_i}({\\mathbb Z})\\rightarrow [N,N]_*,\\;\n\t(M^l)=(m^l_{i,j})\\mapsto f_{(M^l)}= \\bigvee_{l=1}^{\\lfloor n\/2\\rfloor} f_{M^l},\n\\end{equation} where $f_{M^l}:\\bigvee_{i\\in I_l} S^{p_i}\\vee S^{q_i}\\rightarrow V_I\\text{, }$ is given by $$f_{M^l}= \\begin{cases}\n\\bigvee_{i=1}^{r_l} (\\sum^{r_l}_{j=1} m^l_{i,j}\\alpha^l_j \\vee \\sum^{r_l}_{j=1} m^{n-l}_{i,j}\\beta^l_j ) &\\text{ if } l\\neq n\/2\\\\ \\\\\n\\bigvee_{i=1}^{r_l\/2} (\\sum^{r_l\/2}_{j=1} m^l_{i,j}\\alpha^l_j+\\sum^{r_l}_{j=r_l\/2+1} m^l_{i,j}\\beta^l_{(j-r_l\/2)})\\\\ \\vee \\bigvee_{i=r_l\/2+1}^{r_l} (\\sum^{r_l\/2}_{j=1} m^l_{i,j}\\alpha^l_j+\\sum^{r_l}_{j=r_l\/2+1} m^l_{i,j}\\beta^l_{(j-r_l\/2)}) &\\text{ if } l=n\/2.\n\\end{cases}$$ We observe that the image of this section is precisely the sub-monoid of $[N,N]_*$ corresponding to the non-finite factors (\\ref{nonfinitefactors}). Hence we get that $\\tilde H_*:[N,N]_*\\rightarrow {\\operatorname{Aut}}(\\tilde H_*(N))$ is surjective and has finite kernel.\nRestricting to the submonoids of invertible elements this implies upon using the section (\\ref{section}) that $\\pi_0({\\operatorname{aut}}_*(N))\\rightarrow \\operatorname{Aut}(\\tilde H_*(N))$ is surjective with finite kernel. The image of $\\pi_0({\\operatorname{aut}}_\\partial(N))\\rightarrow \\pi_0({\\operatorname{aut}}_*(N))$ consists of the elements $[f]\\in \\pi_0({\\operatorname{aut}}_*(N)),$ such that $f\\circ in\\simeq in$ (we assume all homotopy equivalences in this proof to be pointed). Since (\\ref{section}) restricts to a section $\\operatorname{Aut}(\\tilde H_*(N))\\rightarrow \\pi_0({\\operatorname{aut}}_*(N))$ we get that the image of $$\\tilde H_*:\\pi_0({\\operatorname{aut}}_\\partial(N))\\rightarrow \\operatorname{Aut}(\\tilde H_*(N))$$ is given by the $(M^l)$ such that $f_{(M^l)}\\circ in\\simeq in.$ Using the Hilton-Milnor Theorem we identify \\begin{equation}\\label{hiltonmilnoridentification}\n \t\\pi_{n-1}(N)\\cong \\pi \\oplus \\bigoplus_{l} \\pi_{n-1}(\\bigvee_{i\\in I_l}(S^{p_i}\\vee S^{q_l})),\n\\end{equation}where $\\pi$ is some subgroup of $\\pi_{n-1}(N).$ We observe that $$ f_{(M^l)}\\circ in \\simeq \\sum_{i\\in I} [f_{(M^l)}\\circ\\alpha_i,f_{(M^{l})}\\circ\\beta_i]\\simeq \\sum_{l=1}^{\\lfloor n\/2\\rfloor}\\sum_{i\\in I_l} [f_{M^l}\\circ\\alpha^l_i,f_{M^l}\\circ\\beta^l_i],$$ i.e. that the action of $f_{(M^l)}$ respects the summands of the identification (\\ref{hiltonmilnoridentification}). Thus it suffices to check that $f_{M^l}\\circ \\sum_{i\\in I_l} [\\alpha_i^l,\\beta_i^l]\\simeq \\sum_{i\\in I_l}[\\alpha_i^l,\\beta_i^l]$ for all $l.$ We use that left homotopy composition is distributive for suspensions \\cite[p. 126]{MR0041435}, i.e. $(x+y)\\circ \\Sigma z\\simeq x\\circ \\Sigma z+ y\\circ \\Sigma z $. For $l\\neq n\/2$ we calculate \\begin{align*} f_{M^l}\\circ \\sum_{i=1}^{r_l} [\\alpha_i^l,\\beta_i^l] \\simeq & \\sum_{i=1}^{r_l} [f_{M^l}\\circ\\alpha^l_i,f_{M^l}\\circ\\beta^l_i] \\simeq   \\sum_{i,j,k} [m^l_{i,j}\\alpha^l_j,m^{n-l}_{i,k}\\beta^l_k] \\\\\n  \\simeq &    \\sum_{i,j,k} m^l_{i,j}m^{n-l}_{i,k} [\\alpha^l_j,\\beta^l_k] \\simeq    \\sum_{j,k} ((M^l)^TM^{n-l})_{j,k} [\\alpha^l_j,\\beta^l_k].\n \\end{align*} This expression is homotopic to $\\sum_{i=1}^{r_l} [\\alpha_i^l,\\beta_i^l] $ if \\begin{equation}\\label{mcgconsition1}\n(M^l)^TM^{n-l}={\\operatorname{id}}_{Mat_{r_l}{({\\mathbb Z})}}.\t\n\\end{equation} For $l= n\/2$ we write $$M^l=\\begin{pmatrix}\nA^l & B^l   \\\\\nC^l & D^l \n\\end{pmatrix}.$$ We calculate:\\begin{align*} &f_{M^l}  \\circ \\sum_{i=1}^{r_l\/2} [\\alpha_i^l,\\beta_i^l] \\simeq  \\sum_{i=1}^{r_l\/2} [f_{M^l}\\circ\\alpha^l_i,f_{M^l}\\circ\\beta^l_i] \\\\ \n \\simeq & \\sum_{i=1}^{r_l\/2}\\left[\\sum^{r_l\/2}_{j=1} (a^l_{i,j}\\alpha^l_j + b^l_{i,j}\\beta^l_{j}) ,\\sum^{r_l\/2}_{k=1} (c^l_{i,k}\\alpha^l_k+ d^l_{i,k}\\beta^l_{k})\\right]\\\\\n \\simeq& \\sum_{i=1}^{r_l\/2}\\sum^{r_l\/2}_{j=1}\\sum^{r_l\/2}_{k=1} a^l_{i,j} d^l_{i,k} [\\alpha^l_j,\\beta^l_{k}]            +\\sum_{i=1}^{r_l\/2}\\sum^{r_l\/2}_{j=1}\\sum^{r_l\/2}_{k=1} b^l_{i,j}c^l_{i,k} [\\beta^l_{j},\\alpha^l_k] \\\\\n +& \\sum_{i=1}^{r_l\/2}\\sum^{r_l\/2}_{j=1}\\sum^{r_l\/2}_{k=1}  a^l_{i,j}c^l_{i,k} [\\alpha^l_j ,\\alpha^l_k]            +\\sum_{i=1}^{r_l\/2}\\sum^{r_l\/2}_{j=1}\\sum^{r_l\/2}_{k=1}  b^l_{i,j}d^l_{i,k}[\\beta^l_{j},\\beta^l_{k}]\\\\\n \\simeq& \\sum^{r_l\/2}_{j=1}\\sum^{r_l\/2}_{k=1} ( (A^l)^TD^l +(-1)^{n\/2}((C^l)^TB^l ))_{j,k} [\\alpha^l_j,\\beta^l_{k}] \\\\\n +& \\sum^{r_l\/2}_{j=1}\\sum^{r_l\/2}_{k=1}  ((A^l)^TC^l)_{j,k} [\\alpha^l_j ,\\alpha^l_k]            +\\sum^{r_l\/2}_{j=1}\\sum^{r_l\/2}_{k=1}  ((B^l)^T D^l)_{j,k}[\\beta^l_{j},\\beta^l_{k}]\n \\end{align*} This expression is homotopic to $\\sum_{i=1}^{r_l\/2} [\\alpha_i^l,\\beta_i^l] $, if\n$$\\begin{aligned}\n\t& (A^l)^TD^l +(-1)^{n\/2}(C^l)^TB^l=1  \\\\\n\t& (A^l)^TC^l+(-1)^{n\/2} (C^l)^TA^l=0\\\\\n\t& (B^l)^TD +(-1)^{n\/2}(D^l)^T B^l =0\\\\\n\t& (A^l)^TC^l \\text{ and } (B^l)^TD^l \\text{ have diagonal entries in $\\Lambda,$}\n\\end{aligned}$$\nwhere $$\\Lambda=\\begin{cases} 2{\\mathbb Z} &\\text{ if $n=2d$ and $d$ is odd and not $3$ or $7$} \\\\\n{\\mathbb Z} & \\text{ if $n=2d$ and $d$ is $3$ or $7$}\\\\\n0 &\\text{ if $n=2d$ and $d$ is even.}\\end{cases}$$ Where the diagonal entries of $(A^l)^TC^l$ and $(B^l)^TD^l$ have to be in $\\Lambda$ to kill the elements $[\\alpha^l_i ,\\alpha^l_i]$ and $[\\beta^l_i ,\\beta^l_i].$ These are exactly the conditions to be an automorphisms of $H({\\mathbb Z}^{g_{n\/2}})$ in $Q^{(-1)^{n\/2}}({\\mathbb Z},\\Lambda).$\nCombining this with the condition in (\\ref{mcgconsition1}) we see that the image of $\\tilde H_*$ in ${\\operatorname{Aut}}(\\tilde H_*(N))$ is given by $$\\Gamma_I \\subset \\prod^{\\lfloor n\/2\\rfloor}_{k=1} Gl_{r_k}({\\mathbb Z}).$$ Thus we proved that $$\\tilde H_*:\\pi_0({\\operatorname{aut}}_\\partial (N)) \\rightarrow {\\operatorname{Aut}}(\\tilde H_*(N),\\langle-,-\\rangle,Jq)$$ is surjective.\nTo show that the kernel is finite it suffices to check that $\\pi_0({\\operatorname{aut}}_\\partial (N))\\rightarrow \\pi_0({\\operatorname{aut}}_* (N))$ has finite kernel. This follows from the fact that $$\\pi_1(in^*):\\pi_1({\\operatorname{aut}}_*(N),\\operatorname{id}_N)\\otimes {\\mathbb Q}\\rightarrow \\pi_1({\\operatorname{map}}_*(\\partial N, N),in)\\otimes {\\mathbb Q}$$ is surjective as we will see in Remark $\\ref{ratsur}.$\n\n\\end{proof}\n\nIn fact it suffices to know $\\Gamma_I$ for our purposes, since the action of elements of the kernel of $\\tilde H_*$ by conjugation is trivial up to homotopy.\n\n\\begin{lem}\\label{gamma I action on universal cover}\nLet $f$ represent an element of the kernel of $$\\tilde H_*:\\pi_0( {\\operatorname{aut}}_\\partial(N_I))\\to \\Gamma_I,$$ then $f^{-1}\\circ g\\circ f\\simeq g$ for all $g\\in {\\operatorname{aut}}_\\partial(N_I).$\n\\end{lem}\n\n\\begin{proof}\nNote that if $[f]$ is in the kernel of $\\tilde H_*$ then it is given by an element $\\partial \\alpha,$ where $\\alpha\\in \\pi_1({\\operatorname{map}}_*(\\partial N_I,N_I,in))\\cong \\pi_n(N_I).$ We represent $\\alpha$ as a map $$\\alpha(x,t):\\partial N_I\\times I\\rightarrow N_I, \\text{ such that } \\alpha(x,0)=\\alpha(x,1)={\\operatorname{id}}_{\\partial N_I}.$$ Choosing a collar neighborhood of $\\partial N_I$ allows us to identify $$N_I\\xrightarrow{\\cong}N_I\\cup_{{\\operatorname{id}}_{\\partial N_I}}\\partial N_I\\times I.$$ Now we represent $f$ by the composite $$N_I\\xrightarrow{\\cong}N_I\\cup_{{\\operatorname{id}}_{\\partial N_I}}\\partial N_I\\times I\\xrightarrow{{\\operatorname{id}}_{N_I}\\cup \\alpha} N_I.$$ We represent $f^{-1}$ similarly using $-\\alpha$. If we now represent $f^{-1}\\circ g\\circ f$ by the composition $$\\begin{aligned}\nN_I\\xrightarrow{\\cong}N_I\\cup_{{\\operatorname{id}}_{\\partial N_I}}\\partial N_I\\times I\\xrightarrow{{\\operatorname{id}}_{N_I}\\cup \\alpha} N_I \\xrightarrow{\\cong}N_I\\cup_{{\\operatorname{id}}_{\\partial N_I}}\\partial N_I\\times I\\\\ \\xrightarrow{g\\cup {\\operatorname{id}}_{\\partial N_I\\times I}}  N_I\\cup_{{\\operatorname{id}}_{\\partial N_I}}\\partial N_I\\times I\\xrightarrow{{\\operatorname{id}}_{N_I}\\cup -\\alpha} N_I,\n\\end{aligned}$$ and see that it is homotopic to $$N_I\\xrightarrow{\\cong}N_I\\cup_{{\\operatorname{id}}_{\\partial N_I}}\\partial N_I\\times I\\cup_{{\\operatorname{id}}_{\\partial N_I}}\\partial N_I\\times I \\xrightarrow{g\\cup\\alpha \\cup -\\alpha}N_I.$$ This is homotopic (rel. $\\partial N_I$) to $g,$ since $$\\partial N_I\\times I\\cup_{{\\operatorname{id}}_{\\partial N_I}}\\partial N_I\\times I \\xrightarrow{\\alpha \\cup -\\alpha}N_I$$ is homotopic to the inclusion of $\\partial N_I\\times I\\cup_{{\\operatorname{id}}_{\\partial N_I}}\\partial N_I\\times I$ as a collar.\n\n\n\\end{proof}\n\nDenote by ${\\operatorname{Diff}}_\\partial(N)$ the space of self-diffeomorphism of $N$ fixing a collar neighborhood of the boundary point-wise with the Whitney $C^\\infty$-topology. Let $J:{\\operatorname{Diff}}_\\partial(N) \\rightarrow{\\operatorname{aut}}_\\partial(N)$ be the canonical inclusion. To show homological stability for block diffemorphism the following fact about the mapping class group suffices.\n\n\\begin{prop}\\label{mcg} \n\\begin{itemize} \\item[]\n\\item[$\\operatorname{(1)}$] The map $\\tilde H_*: \\pi_0 ({\\operatorname{Diff}}_\\partial(N))\\rightarrow {\\operatorname{Aut}}({\\tilde H_*(N),\\langle-,-\\rangle,q})$ is surjective.\n\\item[$\\operatorname{(2)}$] The image of $\\pi_0(J):\\pi_0 ({\\operatorname{Diff}}_\\partial(N))\\rightarrow \\pi_0({\\operatorname{aut}}_\\partial(N))$ has finite index.\n\\end{itemize} \n\\end{prop}\n\n\\begin{proof}\nThe first part follows from \\cite{MR561244} and \\cite[Lemma 17]{MR0156354}. Kreck shows that all elements of ${\\operatorname{Aut}}({\\tilde H_{n\/2}}(N),\\langle-,-\\rangle,q)$ can be realized as self-diffeomorphisms of $$(\\#_{g_{n\/2}}(S^{n\/2}\\times S^{n\/2}))\\smallsetminus {\\operatorname{int}}(D^n)$$ fixing the boundary pointwise. Wall shows that for manifolds $\\natural_g (D^{q+1}\\times S^p)$, where $3\\leq p\\leq q$ and $\\natural_g$ denotes the $g$-fold boundary connected sum, all automorphisms of the homology are realized by diffeomorphisms. Hence it follows for manifolds $\\#_{g_i} (S^{p_i}\\times S^{q_i}).$ Since we can assume that a diffeomorphism fixes a disk up to isotopy, we get it in particular for $(\\#_{g_i} (S^{p_i}\\times S^{q_i}))\\smallsetminus{\\operatorname{int}}(D^n).$ Using the diffeomorphisms above and extending them by the identity on the complement of the manifolds above the claim follows. The second part follows from the following commutative diagram\n\\begin{equation*}\n\\xymatrix{\n0 \\ar[r]& \\pi_0 (\\operatorname{S}{\\operatorname{Diff}}_\\partial (N)  )\\ar[r]\\ar[d]  & \\pi_0 ({\\operatorname{Diff}}_\\partial (N)) \\ar[r]^<<<<<{\\tilde H_*} \\ar[d]_{\\pi_0(J)}& {\\operatorname{Aut}}({\\tilde H_*(N),\\langle-,-\\rangle, q}) \\ar[r] \\ar[d]^\\cong & 0 \\\\\n0 \\ar[r]& \\pi_0 (\\operatorname{S}{\\operatorname{aut}}_\\partial (N)) \\ar[r] & \\pi_0 ({\\operatorname{aut}}_\\partial (N)) \\ar[r]^<<<<<{\\tilde H_*}& {\\operatorname{Aut}}({\\tilde H_*(N),\\langle-,-\\rangle , Jq}) \\ar[r]& 0,  }\n\\end{equation*}\nwhere $\\pi_0 (\\operatorname{S}{\\operatorname{Diff}}_\\partial (N))$ and $\\pi_0 (\\operatorname{S}{\\operatorname{aut}}_\\partial (N))$ denote the kernels of the maps $\\tilde H_*$ and the fact that $\\pi_0 (\\operatorname{S}{\\operatorname{aut}}_\\partial (N))$ is finite by Proposition \\ref{hmcg}.\n\n\\end{proof}\n\n\\begin{rem}\\label{mcglitremark}\nThere is a lot of literature on the groups of components of mapping spaces of closed manifolds in different categories. Highly connected even-dimensional manifolds are for example studied in \\cite{MR561244} and \\cite{MR0245031}. Products of spheres are studied in \\cite{MR0248848,MR0253369,MR0250323}. Homotopy self-equivalences of manifolds and in particular of connected sums of products of spheres are treated in \\cite{MR1340168}. \n\n\\end{rem}\n\nFor later use we need the following Lemma.\n\n\\begin{lem}\\label{H1 triv pi0} The groups \n$\\pi_{0}{\\operatorname{aut}}_\\partial(N_I)$ and $\\operatorname{Im}(\\pi_0(J))$ are rationally perfect.\n\\end{lem}\n\n\\begin{proof} This follows from Lemma \\ref{gammaIh1} and the fact that the groups are finite extension of $\\Gamma_I.$\n\\end{proof}\n\\section{On the rational homotopy type of homotopy automorphisms}\\label{On the rational homotopy type of homotopy automorphisms}\n\nIn the last section we determined the group $$\\pi_I=\\pi_1(B{\\operatorname{aut}}_\\partial (N_I),{\\operatorname{id}}_{N_I})\\cong \\pi_0({\\operatorname{aut}}_\\partial (N_I))$$ up to finite extensions. It acts on the simply-connected covering $X_I=B{\\operatorname{aut}}_\\partial (N_I){\\langle 1 \\rangle}$ by deck transformations. This section has two goals:\n\\begin{enumerate}\n\\item Describe the $\\pi_I$-modules $H_*(X_I;{\\mathbb Q})$ algebraically (Proposition \\ref{iso Hce and H as piI-modules}).\n\\item Make sure the algebraic model is appropriate for showing homological stability using the results in Section \\ref{van der kallens and charneys homological stab results} (Proposition \\ref{derivation as schurfunctor}).\n\\end{enumerate}\n\nAll the results in this section are either contained in \\cite{berg13} or straightforward generalizations.  \n\nWe assume some familiarity with Quillen's approach to rational homotopy theory \\cite{MR0258031}, i.e. the functor $$\\lambda:\\operatorname{Top}_1\\rightarrow \\operatorname{dgL}_0$$ from the category of simply connected based topological spaces to the category of reduced differential graded (dg) Lie algebras. It induces an equivalence of homotopy categories, where the weak equivalences in $\\operatorname{Top}_1$ are isomorphisms in rational homotopy groups and in $\\operatorname{dgL}_0$ quasi isomorphisms. The homology of $\\lambda(X)$ allows us to recover the rational homotopy groups of $X.$ More precisely, there is an isomorphism of graded Lie algebras $$H_*(\\lambda(X))\\cong \\pi_*(\\Omega X)\\otimes {\\mathbb Q},$$ where the Lie bracket on the right hand side is given by the Samelson product. The rational homology of $X$ is given by the Chevalley-Eilenberg homology of $\\lambda(X),$ which we will explain later.\n\nFor a given simply connected space $X$ the value $\\lambda(X)$ is in general very complicated and one considers dg Lie models instead. A \\emph{dg Lie model} for a simply connected topological space $X$ is a free differential graded Lie algebra $({\\mathbb L}(V),\\partial),$ together with a quasi isomorphism $$({\\mathbb L}(V),\\partial) \\xrightarrow{\\simeq} \\lambda(X).$$ When $\\pi_*(\\Omega X)\\otimes {\\mathbb Q}$ is quasi isomorphic to $\\lambda(X)$, the space $X$ is called \\emph{coformal}.\n\n\\subsection{On a dg Lie model for the simply-connected covering of the homotopy automorphisms}\n\nSince $N_I\\simeq \\bigvee_{i\\in I} (S^{p_i}\\vee S^{q_i})$, the free Lie algebra ${\\mathbb L}(s^{-1}\\tilde H_*(N_I,{\\mathbb Q}))$ with trivial differential is a dg Lie model for $N_I,$ where $s^{-1}$ denotes the desuspension.\nWe are going to denote: $${\\mathbb L}_I={\\mathbb L}(s^{-1}\\tilde H_*(N_I,{\\mathbb Q})).$$ Recall that we denoted the homology classes represented by the inclusions $$\\alpha_i:S^{p_i}\\hookrightarrow N_I,\\; \\beta_i: S^{q_i}\\hookrightarrow N_I,$$ by $a_i$ respectively $b_i$. Denote by $$\\omega_I =\\sum_{i\\in I} -(-1)^{|a_i|}[ s^{-1}a_i, s^{-1}b_i ].$$ We model the inclusion of the boundary $in:\\partial N_I\\rightarrow N_I$ by $${\\mathbb L}(\\gamma)\\rightarrow {\\mathbb L}_I\\text{, }\\gamma\\mapsto \\omega_I,$$ where ${\\mathbb L}(\\gamma)$ is generated by a single generator of degree $n-2.$ \n\nLet $f:L\\rightarrow K$ be a map of differential graded Lie algebras. We say that a degree $n$ linear map $\\theta\\in {\\operatorname{Hom}}_n(L,K)$, is an \\emph{$f$-derivations} of degree $n$, if $$\\theta [x,y]=[\\theta(x),f(y)]+(-1)^{n|x|}[f(x),\\theta(y)]\\text{, for all }x,y\\in L.$$ The $f$-derivations form a differential graded vector space ${\\operatorname{Der}}_f(L,K),$ with differential given by $$D(\\theta)=d_K\\circ \\theta - (-1)^{|\\theta |}\\theta\\circ d_L.$$ The \\emph{derivations} of a differential graded Lie algebra $L$ is the special case $${\\operatorname{Der}}(L)= {\\operatorname{Der}}_{\\text{id}_L}(L,L).$$ We define a bracket for $\\theta,\\eta\\in{\\operatorname{Der}} (L)$, by $$[\\theta,\\eta]=\\theta \\circ \\eta -(-1)^{|\\theta||\\eta|}\\eta\\circ \\theta,$$ which makes ${\\operatorname{Der}} (L)$ into a differential graded Lie algebra.\n\nDenote by ${\\operatorname{Der}}_\\omega({\\mathbb L}_I)$ the derivation Lie algebra annihilating $\\omega_I,$ i.e. the kernel of the evaluation at $\\omega_I$ $ev_{\\omega_I}:{\\operatorname{Der}}({\\mathbb L}_I)\\to {\\mathbb L}_I.$ The \\emph{positive truncation} $L^+$ of a dg Lie algebra $L$ is given by $$  L_i^+ = \\begin{cases} L_i&  \\text{for } i\\geq 2 \\\\\n     \\text{ker}(d_L:L_1\\rightarrow L_0)&  \\text{for } i=1 \\\\\n    0 & \\text{for } i\\leq 0 \\\\\n   \\end{cases}$$ with its obvious differential and Lie bracket.\n\n\\begin{prop}[{Special case of \\cite[Corollary 3.11]{berg13}}]\\label{der isomorphism of gLie}\nThe simply-connected covering $B{\\operatorname{aut}}_\\partial (N_I){\\langle 1 \\rangle}$ is coformal and there is an isomorphism of graded Lie algebras $$\\pi_*(\\Omega B{\\operatorname{aut}}_\\partial (N_I){\\langle 1 \\rangle})\\otimes {\\mathbb Q}\\cong{\\operatorname{Der}}^+_\\omega({\\mathbb L}_I)$$\n\\end{prop}\n\n\\begin{proof} The manifolds $N_I$ are formal (in the sense of Sullivan's rational homotopy theory) and have trivial reduced rational cohomology rings since they are homotopy equivalent to wedges of spheres. Thus we can apply \\cite[Corollary 3.11]{berg13}.\n\\end{proof}\n\n\n\n\n\\subsection{Derivations and the cyclic Lie operad}\n\nIn this section we collect the results from \\cite{berg13} Section 6.1 and 6.2. Let $V$ be a graded finite dimensional rational vector space. By an inner product of degree $D$ we mean a degree $-D$ morphism $$\\langle-,-\\rangle:V\\otimes V\\rightarrow {\\mathbb Q}$$ that is non-degenerate in the sense that the adjoint $$V\\to {\\operatorname{Hom}}(V,{\\mathbb Q}),\\; v\\mapsto \\langle x,-\\rangle,$$ is an isomorphism of graded vector spaces. The inner product is graded anti-symmetric if $$\\langle x,x\\rangle = -(-1)^{|x||y|} \\langle y,x\\rangle \\text{ for all } x,y\\in V.$$ \nDenote by $Sp^D$ the category with objects graded finite dimensional rational vector spaces $V$ together with a graded anti-symmetric inner product $\\langle-,-\\rangle_V$. The morphisms in $Sp^D$ are linear maps that respect the inner-product. For a morphisms $f:V\\rightarrow W,$ there is a unique linear map $f^!$ such that $$\\langle x,y \\rangle_V=\\langle f(x),f(y) \\rangle_W\\text{, for all }x,y\\in V.$$ Since $f^!f={\\operatorname{id}}_V$ we get that morphisms in $Sp^D$ are injective.\n\n\n\nGiven a object $V$ of $Sp^D$, consider the inner product $\\langle -,- \\rangle_V$ as an element of ${\\operatorname{Hom}}(V^{\\otimes 2},{\\mathbb Q}).$ We identify $V^{\\otimes 2} \\cong {\\operatorname{Hom}}(V^{\\otimes 2},{\\mathbb Q})$ using the inner product on $V^{\\otimes 2}$ defined by $$\\langle x\\otimes y, x'\\otimes y'\\rangle = (-1)^{|x'||y|}\\langle x,x'\\rangle\\langle y,y'\\rangle.$$ Thus $\\langle -,- \\rangle_V$ gives rise to an element $\\omega_V\\in V^{\\otimes 2}.$ The anti-symmetry of $\\langle -,- \\rangle $ implies that we can consider $\\omega_V$ as an element $\\omega_V \\in {\\mathbb L}(V).$ When we choose a graded basis $\\iota_1,...,\\iota_r$ with dual basis $\\iota_1^\\#,...,\\iota_r^\\#$ for $V$ then $$\\omega_V=\\pm\\frac{1}{2}\\sum_i [\\iota_i^\\#,\\iota_i].$$\n\\begin{example} We consider $s^{-1}H_I\\otimes {\\mathbb Q}$ as an element of $Sp^{(n-2)}.$ Then $\\omega_{s^{-1}H_I\\otimes {\\mathbb Q}}$ is equal to $\\omega_I$ up to sign.\n\\end{example}\n\nA $Sp^D$-module in a category $\\mathcal{V}$ is a functor $Sp^D\\rightarrow \\mathcal{V}.$ Our goal in this section is to show that we can describe ${\\operatorname{Der}}_\\omega ({\\mathbb L}(-))$ as a $Sp^D$-module in a category of graded Lie algebras $gL$. Moreover we are going to see that this functor is in fact naturally equivalent to a Schur functor.\n\nIt is clear that ${\\mathbb L}(-):Sp^D\\to gL$ defines a functor. Moreover using the adjoint $f^!$ of a morphisms $f:V\\to W$ in $Sp^D,$ it follows that we can consider ${\\operatorname{Der}}({\\mathbb L}(-)):Sp^D\\to gL$ as a functor, where ${\\operatorname{Der}}({\\mathbb L}(f))(\\theta)$ for $\\theta \\in {\\operatorname{Der}} ({\\mathbb L}(V))$is given by the unique derivation defined by $$ {\\operatorname{Der}}({\\mathbb L}(f))(\\theta)(x)={\\mathbb L}(f)\\theta(f^!(x)) \\text{ for }x \\in W.$$\n(see Proposition \\cite[Proposition 6.1]{berg13}, where it is also shown that ${\\operatorname{Der}}({\\mathbb L}(f))(\\theta)$ is injective). Proposition 6.2 in \\cite{berg13} now states that for a morphisms $f:V\\to W$ in $Sp^D,$ the diagram \\begin{displaymath}\n\\xymatrix{{\\operatorname{Der}} ({\\mathbb L}(V)) \\ar[r]^{ev_{\\omega_V}}\\ar[d]_{{\\operatorname{Der}}({\\mathbb L}(f))}  & {\\mathbb L}(V) \\ar[d]^{{\\mathbb L}(f)} \\\\\n{\\operatorname{Der}} ({\\mathbb L}(W)) \\ar[r]^{ev_{\\omega_W}} &{\\mathbb L}(W)}\\end{displaymath} commutes. This implies in particular, that we get a functor $${\\operatorname{Der}}_\\omega ({\\mathbb L}(-)):Sp^D\\to gL,$$ given by the kernel ${\\operatorname{Der}}_\\omega ({\\mathbb L}(V))$ of the evaluation map $ev_{\\omega_V}:{\\operatorname{Der}}({\\mathbb L}(V))\\rightarrow {\\mathbb L}(V)$ for $V\\in Sp^D$.\n\nWe identify the $Sp^D$-module ${\\operatorname{Der}}({\\mathbb L}(-))$ with the $Sp^D$-module ${\\mathbb L}(V)\\otimes V,$ upon using the map $$\\theta_{-,-}:{\\mathbb L}(V)\\otimes V\\to {\\operatorname{Der}}({\\mathbb L}(-)),\\; \\theta_{\\chi,x}(y)=\\chi \\langle x,y\\rangle\\text{, for $x\\in V$ and $\\theta\\in {\\mathbb L}(V)$  }$$ (see \\cite[Proposition 6.3]{berg13}). Under this identification the evaluation map becomes the map induced by the Lie bracket, i.e. \\begin{equation*}\n\\xymatrix{\n{\\mathbb L}(V)\\otimes V \\ar[rr]^-{[-,-]}\\ar[d]_{\\theta_{-,-}}  && {\\mathbb L}(V) \\ar@{=}[d] \\\\\n{\\operatorname{Der}}({\\mathbb L}(V)) \\ar[rr]^{ev_{\\omega_V}} &&{\\mathbb L}(V)}\n\\end{equation*} commutes. Denote by ${\\mathfrak{g}}(V)$ the kernel of $[-,-]$ and observing that $[-,-]$ surjects onto the sub graded Lie algebra ${\\mathbb L}^{\\geq 2}(V)$ of elements of bracket length $\\geq 2,$ we get a commutative diagram of $Sp^D$-modules\n\\begin{equation}\\label{gnat}\n\\xymatrix{\n0\\ar[r]&{\\mathfrak{g}}(V)\\ar[r]\\ar@{.>}[d]_\\cong &{\\mathbb L}(V)\\otimes V \\ar[rr]^-{[-,-]}\\ar[d]_{\\theta_{-,-}}  && {\\mathbb L}^{\\geq 2}(V) \\ar@{=}[d]\\ar[r] & 0\\\\\n0\\ar[r]&{\\operatorname{Der}}_{\\omega}({\\mathbb L}(V))\\ar[r] &{\\operatorname{Der}}({\\mathbb L}(V)) \\ar[rr]^{ev_{\\omega_V}} &&{\\mathbb L}^{\\geq 2}(V)\\ar[r] & 0. } \\end{equation} \nNote that in the top row we do not use the inner product thus it defines in fact a functor ${\\mathfrak{g}}$ from the category of graded vector spaces.\n\nDenote by $\\Lie= \\{\\Lie(n)\\}_{n\\geq 0}$ the graded Lie operad and by $\\Lie((n))$ the cyclic lie operad, i.e. the Lie operad in arity $n-1$ $\\Lie(n-1)$ seen as a $\\Sigma_{n}$-modules. Denote by $t=(123...n)\\in \\Sigma_n$ the cyclic permutation and denote by $t*\\xi$ the action of $t$ on $\\xi \\in \\Lie((n))$. There are short exact sequences of $\\Sigma_n$-modules $$0\\to \\Lie((n)) \\xrightarrow{\\mu} {\\mathbb Q}[\\Sigma_n]\\otimes_{\\Sigma_{n-1}}\\Lie(n-1)\\xrightarrow{\\epsilon} \\Lie(n)\\rightarrow 0,$$ where $\\mu(\\xi)=\\sum_i t^i\\otimes t^{-i} * \\xi$ and $\\epsilon(\\sigma\\otimes \\zeta)=\\sigma[\\zeta,x_n]$ (see \\cite[Proposition 6.4]{berg13}).\n\nUsing the exact sequence we identify the rows of (\\ref{gnat}) with $$ s^{-D} \\bigoplus_{n\\geq 2} \\Lie((n))\\otimes_{\\Sigma_n} V^{\\otimes n}\\rightarrow s^{-D}\\bigoplus_{n\\geq 2}\\Lie(n)\\otimes_{\\Sigma_n} V^{\\otimes n} \\rightarrow \\bigoplus_{n\\geq 2}\\Lie(n) \\otimes_{\\Sigma_n} V^{\\otimes n}.$$ Motivated by this we define $$\\Lie((V))=s^{-D} \\bigoplus_{n\\geq 2} \\Lie((n))\\otimes_{\\Sigma_n} V^{\\otimes n}.$$ The Lie algebra structure on $\\Lie((V))$ can be in fact be explicitly described using the cyclic operad structure of $\\Lie((n))$ and the one on $V^{\\otimes n}$ given by contractions, but we are not going to need it. We summarize the above as:\n\\begin{prop}[{\\cite[Proposition 6.6.]{berg13}}]\\label{derivation as schurfunctor} There is an isomorphism of $Sp^D$-modules in $gLie$ $$\\Lie((-))\\cong {\\operatorname{Der}}_\\omega ({\\mathbb L}(-)).$$\n\\end{prop}\n\n\\begin{rem}\nAs a composition of the Schur functors $s^{-D}$ and the one given by the $\\Sigma$-module $\\{\\Lie((n))\\}_{n\\geq1}$, we see that $\\Lie((-))$ is also a Schur functor.\n\\end{rem}\n\n\\subsection{The action of the homotopy mapping class group}\n\n\n\n\nTo identify the action induced by deck transformations on ${\\operatorname{Der}}^+_\\omega({\\mathbb L}_I)$ we begin by noting that the Lie algebras (both with Samelson product) $\\pi_*(\\Omega B{\\operatorname{aut}}_\\partial (N_I))\\otimes {\\mathbb Q}$ and $\\pi_*({\\operatorname{aut}}_\\partial (N_I))\\otimes {\\mathbb Q}$ are naturally isomorphic since ${\\operatorname{aut}}_\\partial (N_I)$ is a group-like monoid. Moreover the deck transformation action can be described in terms of the Samelson product, i.e. when we identify $\\pi_*(B{\\operatorname{aut}}_\\partial (N_I))\\cong\\pi_{*-1}({\\operatorname{aut}}_\\partial (N_I))$, the deck transformation action of $\\pi_1(B{\\operatorname{aut}}_\\partial (N_I){\\langle 1 \\rangle})$ corresponds to the action of $\\pi_0({\\operatorname{aut}}_\\partial (N_I))$ on $\\pi_{*-1}({\\operatorname{aut}}_\\partial (N_I))\\otimes {\\mathbb Q}$ given by conjugation. The main tool to identify the action is the following theorem.\n\n\\begin{thm}[{\\cite{lupton} stated as in \\cite[Theorem 3.6]{berg13}}]\\label{lsiso}\n Let $f:X\\rightarrow Y$ be a map of simply connected CW-complexes with $X$ finite and $\\phi_f:{\\mathbb L}_X\\rightarrow {\\mathbb L}_Y$ a Quillen model. There are natural isomorphisms of sets \n\\begin{equation*}\n  \\pi_k ({\\operatorname{map}}_* (X,Y),f)\\otimes {\\mathbb Q} \\cong H_k({\\operatorname{Der}}_{\\phi_f}({\\mathbb L}_X,{\\mathbb L}_Y)), \\text{ for }k\\geq 1,\n\\end{equation*} which are vector space isomorphisms for $k>1.$ In the case $X=Y$ and $f=$id$_X,$ there are isomorphisms of vectorspaces \\begin{equation*}\\label{natiso}\n  \\pi_k ({\\operatorname{aut}}_* (X),\\text{id}_X)\\otimes {\\mathbb Q} \\cong H_k({\\operatorname{Der}}({\\mathbb L}_X)), \\text{ for }k\\geq 1,\n\\end{equation*} and the Samelson product corresponds to the Lie bracket.\n\\end{thm}\n\n\\begin{prop}[{Compare \\cite[Proposition 5.5]{berg13}}]\\label{der iso comp with action} There is a $\\pi_0({\\operatorname{aut}}_\\partial(N_I))$-equivariant isomorphism of graded Lie algebras\n$$\\pi_{*}^+({\\operatorname{aut}}_\\partial (N_I))\\otimes {\\mathbb Q}\\cong{\\operatorname{Der}}^+_\\omega({\\mathbb L}_I),$$ where the action on the right hand side is through the canonical action of $\\Gamma_I$ on $H_I$\n\n\\end{prop}\n\n\\begin{proof}\n\nWe are going to study the long exact sequence of rational homotopy groups of the fibration\n$${\\operatorname{aut}}_\\partial(N_I) \\rightarrow {\\operatorname{aut}}_*(N_I)\\rightarrow {\\operatorname{map}}_*(\\partial N_I,N_I).$$ Denote by $\\varphi:{\\mathbb L}(\\omega)\\rightarrow {\\mathbb L}_I$ the inclusion of the sub graded Lie algebra of ${\\mathbb L}_I$ generated by $\\omega_I.$  Using Theorem \\ref{lsiso} we see that the map $$\\pi_k({\\operatorname{aut}}_*(N_I))\\otimes {\\mathbb Q}\\rightarrow \\pi_k({\\operatorname{map}}_*(\\partial N_I,N_I))\\otimes {\\mathbb Q}$$ is given by \\begin{equation*}\n{\\operatorname{Der}}({\\mathbb L}_{I})_k  \\xrightarrow{\\varphi^*_k}{\\operatorname{Der}}_{\\varphi}({\\mathbb L} (\\omega_I) ,{\\mathbb L}_{I})_k,\n\\end{equation*}where $\\varphi^*_k$ is the restriction to ${\\mathbb L}(\\omega_I).$ Note that ${\\operatorname{Der}}_{\\varphi}({\\mathbb L} (\\omega_I) ,{\\mathbb L}_{I})\\cong s^{(n-2)}{\\mathbb L}_I.$ Under this identifications the map $\\varphi^*$ becomes the evaluation map which is surjective as discussed for the diagram (\\ref{gnat}). Hence the long exact sequence of rational homotopy groups splits as \\begin{equation*}\n0\\rightarrow {\\operatorname{Der}}_\\omega({\\mathbb L}_{I})_*\\rightarrow {\\operatorname{Der}}({\\mathbb L}_{I})_*  \\xrightarrow{\\varphi^*_k}{\\operatorname{Der}}_{\\varphi}({\\mathbb L} (\\omega_I) ,{\\mathbb L}_{I})_*\\rightarrow 0,\n\\end{equation*} where we use that ${\\operatorname{Der}}_\\omega({\\mathbb L}_{I})$ is the kernel of the evaluation map. The resulting isomorphism $$\\pi_*^+({\\operatorname{aut}}_{\\partial}({N_I}),{\\operatorname{id}}_{N_I})\\otimes {\\mathbb Q}\\cong {\\operatorname{Der}}^+_{{\\omega} }({\\mathbb L}_{I})$$ is in fact an isomorphism of graded Lie algebras. Indeed, since the inclusion ${\\operatorname{aut}}_\\partial ({N_I})\\rightarrow  {\\operatorname{aut}}_*({N_I})$ is a map of topological monoids, the induced maps on rational homotopy group respect the Samelson product and ${\\operatorname{Der}}^+({\\mathbb L}_{I})\\cong\\pi^+_*({\\operatorname{aut}}(N_I))\\otimes {\\mathbb Q}$ is an isomorphism of Lie algebras. Hence we can calculate the Samelson product of $\\pi^+_*({\\operatorname{aut}}_\\partial(N_I))$ in ${\\operatorname{Der}}^+({\\mathbb L}_{I}).$\n\nNow let $f,g\\in {\\operatorname{aut}}_*({N_I})$. The action of $[f]\\in\\pi_0({\\operatorname{aut}}_*({N_I}))$ on $\\pi_k({\\operatorname{aut}}_*({N_I}))$ is induced by pointwise conjugation $g\\mapsto fgf^{-1},$ where $f^{-1}$ is some choice of homotopy inverse. Let $\\phi_f$ be a Quillen model for $f$ and $\\theta\\in {\\operatorname{Der}} ({{\\mathbb L}_I})_k.$ The action of $[f]$ on ${\\operatorname{Der}}({\\mathbb L}_{I})_k$ is given by \n\\begin{equation*}\n\\theta\\mapsto \\phi_f\\circ \\theta\\circ \\phi_f^{-1},\n\\end{equation*} by the naturality of the identification $\\pi_k({\\operatorname{aut}}_*({N_I}))\\otimes {\\mathbb Q} \\cong {\\operatorname{Der}}({\\mathbb L}_{I})_k.$ \nFor a homotopy self-equivalence $f$ consider the induced map $f_*\\in {\\operatorname{Aut}} (\\tilde{H_*}(N_I))$. The map ${\\mathbb L}(s^{-1}(f_*\\otimes {\\mathbb Q}))$ is in fact a Lie model for $f$, which shows that we can identify the conjugation action with the induced action of ${\\operatorname{Aut}} (\\tilde{H_*}(N_I))$ on ${\\operatorname{Der}}({\\mathbb L}_I)_k$.\\\\\nUsing that ${\\operatorname{Der}}^+_\\omega({\\mathbb L}_{I})_k\\rightarrow {\\operatorname{Der}}^+({\\mathbb L}_{I})_k$ is injective, we calculate the conjugation action of $\\pi_0({\\operatorname{aut}}_{\\partial}({N_I}))$ on $\\pi_k({\\operatorname{aut}}_{\\partial}({N_I}),{\\operatorname{id}}_{N_I})$ in terms of ${\\operatorname{Der}}_{{\\omega} }({\\mathbb L}_{I})_k$. Let $f$ be an element of ${\\operatorname{aut}}_{\\partial}({N_I})$, it is in particular also an element of ${\\operatorname{aut}}_*({N_I})$ and we know that its homotopy class $[f]$ in $\\pi_0 ({\\operatorname{aut}}_*({N_I}))$ gives us an element in $\\Gamma_I$. Considering $\\theta\\in {\\operatorname{Der}}_{{\\omega}}({\\mathbb L}_{I})_k$ as an element in ${\\operatorname{Der}}({\\mathbb L}_{I})_k$ we see that $[f]$ acts by the action induced by $f_*\\in \\Gamma_I$ on $\\tilde H_*(N_I).$\n\\end{proof}\n\n\\begin{rem}\\label{ratsur} Observe that we discussed in the proof that the map $$\\pi_1({\\operatorname{aut}}_* (N_I)) \\otimes {\\mathbb Q}\\rightarrow \\pi_1({\\operatorname{map}}_*(\\partial N_I,N_I))\\otimes {\\mathbb Q}$$ is surjective hence the kernel of $\\pi_0({\\operatorname{aut}}_\\partial(N_I))\\rightarrow \\Gamma_I$ is finite.\n\\end{rem}\n\nWe need to identify the maps induced by the stabilization map on rational homotopy groups. Given an element of $\\theta\\in {\\operatorname{Der}}^+_{\\omega}({\\mathbb L}_I)$ we define an element $\\theta'\\in{\\operatorname{Der}}^+_{\\omega}({\\mathbb L}_{I'}),$ by letting $\\theta'=\\theta$ on generators $\\iota_i,\\kappa_i,$ where $i\\in I$ and $\\theta(\\iota_{i'})=\\theta(\\kappa_{i'})=0.$ Using that ${\\mathbb L}_{I'}$ is free we get a derivation $\\theta',$ which is indeed an element of ${\\operatorname{Der}}^+_{\\omega}({\\mathbb L}_{I'}),$ since $\\omega_{I'}=\\omega_{I}+(-1)^{|\\iota_{i'}|}[\\iota_{i'},\\kappa_{i'}].$ We refer to this map again as stabilization map.\n\n\\begin{prop}\\label{der iso compatible with stab maps}\nThe isomorphism $$\\pi_*({\\operatorname{aut}}_\\partial (N_I){\\langle 1 \\rangle})\\otimes {\\mathbb Q}\\cong{\\operatorname{Der}}^+_\\omega({\\mathbb L}_I)$$ is compatible with the stabilization maps. \n\\end{prop}\n\n\\begin{proof} This works exactly as in \\cite[Proposition 7.7]{berg13}.\n\\end{proof}\n\nUltimately we describe the rational homology $H_*(B{\\operatorname{aut}}_\\partial(N_I){\\langle 1 \\rangle}),{\\mathbb Q})$ as $\\pi_I$-modules. The link between a dg Lie model and the rational homology of a space is given by the Chevalley-Eilenberg homology. The Chevalley-Eilenberg complex of a dg Lie algebra $L$ with differential $d_L$ is the chain complex $C_*^{CE}(L) = \\Lambda_* sL$ with differential $\\delta^{CE}=\\delta_0^{CE} + \\delta_1^{CE}$, where  $s$ denotes the suspension and $\\Lambda_*$ the free graded commutative algebra. In the simplest cases the differentials are given by\\begin{equation*}\n\\delta_0^{CE}(sx) =-s d_L x\n\\end{equation*}\\begin{equation*}\n\\delta_1^{CE}(sx_1 \\wedge sx_2) =(-1)^{|x_1|}s[x_1,x_2],\n\\end{equation*}\nwhere $x,x_1,x_2\\in L$.\n\nQuillen showed that for a dg Lie model $L_X$ of a space $X$ that the Chevalley-Eilenberg homology gives the rational homology groups of $X,$ i.e. that $H^{CE}_*(L_X)\\cong H_*(X;{\\mathbb Q}).$ \n\nGrade the Chevalley-Eilenberg chains by word length, i.e. let $(\\Lambda^{p}(L))_q$ be the elements of word length $p$ and degree $q$. Denote by $H_{p,q}^{CE} (L)$ the homology of the chain complex $$...\\to (\\Lambda^{p+1}(L))_q\\xrightarrow{\\delta_1}(\\Lambda^{p}(L))_q\\xrightarrow{\\delta_1}(\\Lambda^{p-1}(L))_q\\to ...$$\nThe Quillen Spectral Sequence is the spectral sequence coming from the filtration by word length. In case that the dg Lie algebra $L_X$ is a model for a space $X,$ we can identify the $E^2$-page with $$E^2(L)_{p,q} = H_{p,q}^{CE} (L_X)\\cong H_{p,q}^{CE} (\\pi_*(\\Omega X)\\otimes {\\mathbb Q})\\Rightarrow H_*^{CE}(L_X)\\cong H_*(X;{\\mathbb Q}).$$  \n\nThe Quillen spectral sequence collapses on the $E^2$-page for coformal spaces, hence we get isomorphisms $$H_r(B{\\operatorname{aut}}_\\partial(N_I){\\langle 1 \\rangle}),{\\mathbb Q})\\cong \\bigoplus_{p+q=r} H_{p,q}^{CE} (\\pi_*(\\Omega B{\\operatorname{aut}}_\\partial(N_I){\\langle 1 \\rangle})\\otimes {\\mathbb Q}).$$ The Quillen spectral sequence is in fact natural with respect to unbased maps of simply connected spaces (\\cite[Proposition 2.1.]{berg13}). Since $\\pi_I$ is rationally perfect (see Lemma \\ref{H1 triv pi0}), we do not have any extension problems, hence the isomorphism above is in fact an isomorphism of $\\pi_I$-modules (see \\cite[Proposition 2.3]{berg13}). \n\n\\begin{prop}\\label{iso Hce and H as piI-modules}\nThere are isomorphisms of $\\pi_I$-modules $$H^{CE}_r({\\operatorname{Der}}_\\omega({\\mathbb L}_I))\\cong H_r(B{\\operatorname{aut}}_\\partial(N_I){\\langle 1 \\rangle};{\\mathbb Q})$$ compatible with the stabilization maps\n\\end{prop}\n\n\\begin{proof}We use the isomorphism of $\\pi_I$-modules\n$$H_r(B{\\operatorname{aut}}_\\partial(N_I){\\langle 1 \\rangle}),{\\mathbb Q})\\cong \\bigoplus_{p+q=r} H_{p,q}^{CE} (\\pi_*(\\Omega B{\\operatorname{aut}}_\\partial(N_I){\\langle 1 \\rangle})\\otimes {\\mathbb Q})$$ constructed above. By Proposition \\ref{der iso comp with action}, we identify $$H_{p,q}^{CE} (\\pi_*(\\Omega B{\\operatorname{aut}}_\\partial(N_I){\\langle 1 \\rangle})\\otimes {\\mathbb Q})\\cong H_{p,q}^{CE} ({\\operatorname{Der}}_\\omega ({\\mathbb L}_I))$$ as $\\pi_I$-modules. This in turn gives the isomorphism in the claim.\n\nThe compatibility with the stabilization maps follows from Proposition \\ref{der iso compatible with stab maps}.\n\\end{proof}\n\n\\section{Homological Stability}\n\n\\subsection{An algebraic homological stability result}\n\n\nRecall the graded hyperbolic modules $H_I\\cong \\tilde H_*(N_I)$ from Section \\ref{automorphisms of hyperbolic modules}, which have the groups $\\Gamma_I$ as their automorphism groups. Recall that we denoted by $\\lambda_I$ the $\\Gamma_I$-module $H_I$ with standard action which we considered as an object $$((H_I)_1,...,(H_I)_{n-1})\\in {\\operatorname{Mod}}({\\mathbb Z})^{n-1}.$$\nMoreover recall that we defined\n\\begin{displaymath}\n\t\\begin{aligned}\n\t\t&g_p=\\begin{cases}\\operatorname{rank} ((H_I)_p) &\\text{if } 2p<n \\\\  \\operatorname{rank} ((H_I)_p)\/2 &\\text{if } 2p=n.\\end{cases}\\\\\n\t\t&H_{I'}=H_I\\oplus {\\mathbb Z}[p]\\oplus {\\operatorname{Hom}}({\\mathbb Z}[p],{\\mathbb Z}[n]) &&\\text{for fixed $p<\\lfloor n\/2 \\rfloor$}  \\\\\n\t\t&I_{p,n-p}:H_I\\rightarrow H_{I'} &&\\text{the upper inclusion}\\\\\n\t\t&\\mathfrak{g}_I={\\operatorname{Der}^+_\\omega}({\\mathbb L}_I)={\\operatorname{Der}^+_\\omega}({\\mathbb L}(s^{-1}H_I\\otimes {\\mathbb Q})).\n\t\\end{aligned}\n\\end{displaymath}\n\nWe begin by showing:\n\n\\begin{prop}\\label{polynomial functor degree}\nLet $n>3$. For all $l\\geq 0$ there are polynomial functors $$\\mathscr{C}_l:{\\operatorname{Mod}}({\\mathbb Z})^{n-1}\\rightarrow {\\operatorname{Vect}}({\\mathbb Q})$$ of degree $\\leq l\/2$ and isomorphisms of $\\Gamma_I$-modules\n$$\\mathscr{C}_l(\\lambda_I)\\cong C_l^{CE}(\\mathfrak{g}_I)$$ compatible with the maps induced by inclusions.\n\\end{prop}\n\n\n\n\\begin{proof}\n\nIn Proposition \\ref{derivation as schurfunctor} we described the derivations ${\\operatorname{Der}}_\\omega({\\mathbb L}_I)$ as a Schur functor $$\\Lie((-)):Sp^{n-2}\\to gLie,\\; V\\mapsto s^{-n+2}\\bigoplus_{k\\geq 2} \\Lie((k))\\otimes_{\\Sigma_k}V^{\\otimes k},$$ that extended to the category of graded vector spaces $Vect_*({\\mathbb Q})$. Consider the inclusion $$\\mathcal{I}:\\prod_{i=0}^{n-2} Vect({\\mathbb Q})\\rightarrow Vect_*({\\mathbb Q}), \\; (V_i)_{i=0}^{n-2}\\mapsto \\bigoplus_{i=0}^{n-2} V_i[i].$$ The composition $\\Lie((-))\\circ \\mathcal{I}$ is a Schur multifunctor, which we are denoting by $ \\widetilde {\\mathscr{U}},$ with $$\\widetilde {\\mathscr{U}}(\\mu)=s^{[1m_1+2m_2+...+(n-2)m_{n-2}-n+2]}\\Lie((|\\mu|)),$$ where $\\mu=(m_0,m_1,...,m_{n-2}),$ given the $\\Sigma_\\mu$-module structure induced by the inclusion $\\Sigma_\\mu\\subset \\Sigma_{|\\mu|}.$ Thus the positive degree derivations are given by the Schur multifunctor $$\\mathscr{U}:\\prod_{i=0}^{n-2} Vect({\\mathbb Q})\\to gLie,$$ where $\\mathscr{U}(\\mu)=\\widetilde{\\mathscr{U}}(\\mu),$ when $$1m_1+2m_2+...+(n-2)m_{n-2}-n+2\\geq 1\\Leftrightarrow \\sum_{i=1}^{n-2} m_i\\frac{i}{n-1}\\geq 1,$$ and $0$ otherwise. The Chevalley-Eilenberg chains are given by the Schur functor $\\Lambda$ with $\\Lambda(r)$ the trivial $\\Sigma_r$-module concentrated in degree $r$. \nThe composition $\\widetilde{\\mathscr{C}}=\\Lambda\\circ\\mathscr{U}$ is now given by (using (\\ref{schurcomp})) $$\\widetilde{\\mathscr{C}}_r(\\mu)\\cong  \\bigoplus_r  \\Lambda(r) \\otimes_{\\Sigma_r} \\bigoplus\\operatorname{Ind}_{\\Sigma_{\\mu_1}\\times...\\times \\Sigma_{\\mu_r}}^{\\Sigma_\\mu}\\mathscr{U}(\\mu_1)\\otimes ... \\otimes \\mathscr{U}(\\mu_r),   $$where the second sum runs over all $r$-tuples $(\\mu_1,...,\\mu_r),$ such that $\\Sigma_{i=1}^r \\mu_i=\\mu$. For a fixed r we get that the summand corresponding to $(\\mu_1,...,\\mu_r),$ where $\\mu_s=(m_{1,s},...m_{r,s})$, is only non-zero, if $\\sum_{i=1}^{n-2} (m_{i,s} \\frac{i}{n-1}) \\geq 1 $ for all $s=1,...,r$. That implies that\n\\begin{equation}\\label{r}\n\\sum_{i=1}^{n-2} m_i \\frac{i}{n-1}= \\sum_{s=1}^r (\\sum_{i=1}^{n-2} m_{i,s} \\frac{i}{n-1})\\geq r.\n\\end{equation}\nIf the summand is non-zero it is of degree\n\\begin{equation*}\n\\begin{aligned}\nr+ \\sum_{s=1}^r \\left(\\left(\\sum_{i=1}^{n-2} m_{i,s} i\\right)-n+2\\right) &=r+ \\left(\\sum_{i=1}^{n-2} m_ii\\right) -nr + 2r\\\\\n&=\\left(\\sum_{i=1}^{n-2} m_ii\\right) + r(3-n)\\\\\n&\\geq\\left(\\sum_{i=1}^{n-2} m_ii\\right)+  \\left(\\sum_{i=1}^{n-2} m_i \\frac{i}{n-1}\\right)(3-n) \\\\\n&=\\frac{2}{n-1}\\left(\\sum_{i=1}^{n-2} m_ii\\right),\n\\end{aligned}\n\\end{equation*} where we used that $3-n$ is negative and (\\ref{r}). This implies now that the Chevalley-Eilenberg $l$-chains are a Schur multifunctor $\\widetilde{\\mathscr{C}}_l,$ where $\\widetilde{\\mathscr{C}}_l(\\mu)$ vanishes for \n$$l<\\frac{2}{n-1}\\left(\\sum_{i=1}^{n-2} m_ii\\right)\\leq 2\\sum_{i=1}^{n-2}m_i=2|\\mu|.$$\nHence it is of degree $\\leq l\/2.$\nThe functor $\\mathscr{C}_l$ in the statement is given by the pre-composition with the additive functor $$-\\otimes {\\mathbb Q}: {\\operatorname{Mod}}({\\mathbb Z})^{n-1}\\to \\prod_{i=0}^{n-2} Vect({\\mathbb Q}).$$ The compatibility with the action follows from the fact the functor lifts to $$Q^n_+({\\mathbb Z},\\Lambda)\\rightarrow Sp^{(n-2)}, \\; M\\mapsto s^{-1}(M\\otimes {\\mathbb Q}).$$\n\n\\end{proof}\n\nAs an immediate consequence of Proposition \\ref{polynomial functor degree} and Proposition \\ref{general algebraic homological stability result} we get:\n\n\\begin{cor}\\label{homstab cor}\nThe stabilization map\n$$H_k(\\Gamma_I, C^{CE}_l (\\mathfrak{g}_{I})\\rightarrow H_k(\\Gamma_{I'},C^{CE}_l (\\mathfrak{g}_{I'}))$$\nis an isomorphism for $g_p> 2k+2l+2$ when $2p\\neq n$ and $g_p> 2k+2l+4$ if $2p=n$ and an epimorphism for $g_p\\geq 2k+2l+2$ respectively $g_p\\geq 2k+2l+4.$\n\\end{cor}\n\n\n\nBefore we proof the main proposition of this section, i.e. deduce homological stability for the Chevalley-Eilenberg homology from the stability for the chains, we need the following observation about the Chevalley-Eilenberg chains.\n\\begin{lem}\\label{hyperhom lemma}\nThere exists a chain homotopy equivalences $C^{CE}_* (\\mathfrak{g}_I)\\xrightarrow{\\simeq}H^{CE}_* (\\mathfrak{g}_I)$ sending cycles $z\\mapsto[z]$ such that\n\\begin{equation*}\n\\xymatrix{\nC^{CE}_* (\\mathfrak{g}_I) \\ar[rr]^{\\sigma_*}\\ar[d]_{\\simeq}  && C^{CE}_* (\\mathfrak{g}_{I'}) \\ar[d]_{\\simeq} \\\\\nH^{CE}_* (\\mathfrak{g}_I) \\ar[rr]^{\\sigma_*} && H^{CE}_* (\\mathfrak{g}_{I'})}\n\\end{equation*}\ncommutes up to chain homotopy of ${\\mathbb Q}[\\Gamma_I]$-chain complexes.\n\\end{lem}\n\\begin{proof}\nThis is is true for all degree-wise finite dimensional ${\\mathbb Q}[G]$-chain complexes for $G$ rationally perfect groups by \\cite[Lemma B.1]{berg13} and \\cite[Proposition B.5]{berg13}. The groups $\\Gamma_I$ are rationally perfect (see Lemma \\ref{gammaIh1}) and the the $C^{CE}_*(\\mathfrak{g}_I)$ are degree-wise finite dimensional, since the ${\\mathfrak{g}}_I$ are. \n\\end{proof}\n\n\n\n\\begin{prop}\\label{algebraic homological stability result}\nThe stabilization map\n$$H_k(\\Gamma_I, H^{CE}_l (\\mathfrak{g}_I))\\rightarrow H_k(\\Gamma_{I'},H^{CE}_l (\\mathfrak{g}_{I'}))$$\nis an isomorphism for $g_p> 2k+2l+2$ when $2p\\neq n$ and $g_p> 2k+2l+4$ if $2p=n$ and an epimorphism for $g_p\\geq 2k+2l+2$ respectively $g_p\\geq 2k+2l+4.$\n\\end{prop}\n\n\n\n\\begin{proof}\nConsider the first hyperhomology spectral sequence with $E^1$-page:\n$$ E^1_{k,l}(I)=H_l((\\Gamma_I; C^{CE}_k (\\mathfrak{g}_I)))\\Rightarrow {\\mathbb H}_{k+l}(\\Gamma_I; C^{CE}_* (\\mathfrak{g}_I)).$$ By Corollary \\ref{homstab cor} $E^1_{k,l}(I)\\rightarrow E^1_{k,l}(I')$ is an isomorphism for $$g_p>\\begin{cases} 2k+2l+4 \\geq k+2l+4 & \\text{ if } p=n\/2  \\\\ 2k+2l+2 \\geq k+2l+2 & \\text{ otherwise} \\end{cases}$$ and an epimorphism for $\"\\geq\"$. By the spectral sequence comparison theorem we get that the map\n$${\\mathbb H}_i(\\Gamma_I, C^{CE}_* (\\mathfrak{g}_I))\\rightarrow {\\mathbb H}_i(\\Gamma_{I'},C^{CE}_* (\\mathfrak{g}_{I'}))$$ induced by the stabilization map\nis an isomorphism for $g_p>2i+2$ when $2p\\neq n$ and $g_p>2i+4$ if $2p=n$ and an epimorphism for $\"\\geq\"$.\nUpon using Lemma \\ref{hyperhom lemma} and the chain homotopy invariance of hyperhomology we get that the map\n$${\\mathbb H}_i(\\Gamma_I, C^{CE}_* (\\mathfrak{g}_I))\\rightarrow {\\mathbb H}_i(\\Gamma_{I'},C^{CE}_* (\\mathfrak{g}_{I'}))$$ induced by the stabilization map\nis an isomorphism and epimorphism in the same rage as above. Ultimately we use the natural splitting for hyperhomology groups with coefficients in a chain complex with trivial differential\n\\begin{equation*}\n\\xymatrix{\n{\\mathbb H}_i(\\Gamma_I;H^{CE}_* (\\mathfrak{g}_I)) \\ar[r]^-{\\sigma_i}\\ar[d]_{\\cong}  & {\\mathbb H}_i(\\Gamma_{I'}; H^{CE}_* (\\mathfrak{g}_{I'})) \\ar[d]_{\\cong} \\\\\n\\bigoplus_{k+l=i}H_k(\\Gamma_I;H^{CE}_l (\\mathfrak{g}_I)) \\ar[r]^-{\\sigma_{k,l}} & \\bigoplus_{k+l=i} H_k(\\Gamma_{I'};H^{CE}_l (\\mathfrak{g}_{I'})).}\n\\end{equation*}\nHence we see that the maps $\\sigma_{k,l}$ are isomorphisms and epimorphisms in the range in the statement of Proposition \\ref{algebraic homological stability result}.\n\n\\end{proof}\n\\subsection{Homological stability for homotopy automorphisms}\n\nThe first main result of this article now easily follows from the previous results.\n\n\\begin{THMA}\nThe map $$H_i(B{\\operatorname{aut}}_\\partial(N_I);{\\mathbb Q})\\rightarrow H_i(B{\\operatorname{aut}}_\\partial(N_{I'});{\\mathbb Q})$$ induced by the stabilization map\nis an isomorphism for $g_p> 2i+2$ when $2d\\neq n$ and $g_p> 2i+4$ if $2d=n$ and an epimorphism for $g_p\\geq 2i+2$ respectively $g_p\\geq 2i+4.$\n\\end{THMA}\n\n\\begin{proof} We begin by observing that  $$H_k(\\Gamma_I, H^{CE}_l (\\mathfrak{g}_I))\\cong H_k(\\pi_I, H^{CE}_l (\\mathfrak{g}_I)),$$ since the action of $\\pi_I$ is through $\\tilde H :\\pi_I\\to \\Gamma_I$ and the kernel of $\\tilde H_*$ is finite (Propositions \\ref{hmcg} and \\ref{der iso comp with action}).\nThe result now follows from Proposition \\ref{algebraic homological stability result} combined with Proposition \\ref{iso Hce and H as piI-modules} upon using the spectral sequence comparison theorem.\n\\end{proof}\n\n \n\\subsection{Homological stability for block diffeomorphisms}\n  \nDenote by ${\\widetilde{\\operatorname{aut}}}_\\partial(X)$ the $\\Delta$-monoid of block homotopy equivalences, with $k$-simplices face preserving homotopy equivalences $$\\varphi:\\Delta^k\\times X\\rightarrow \\Delta^k\\times X,$$ s.t. $\\varphi|\\Delta^k\\times \\partial X$ is the identity. The block diffeomorphisms ${\\widetilde{\\operatorname{Diff}}}_\\partial (X)$ are the sub-$\\Delta$-group with $k$-simplices, face preserving diffeomorphisms $$\\varphi:\\Delta^k\\times X\\rightarrow \\Delta^k\\times X,$$ s.t.$\\varphi$ is the identity on a neighborhood of $\\Delta^k\\times \\partial X.$ We do not distinguish between $\\Delta$-objects and their realizations. Denote the inclusion ${\\widetilde{\\operatorname{Diff}}}_\\partial(X)\\hookrightarrow{\\widetilde{\\operatorname{aut}}}_\\partial(X)$ by $\\tilde J$. The inclusion $${\\operatorname{aut}}_\\partial (X) \\hookrightarrow {\\widetilde{\\operatorname{aut}}}_\\partial (X)$$ is a homotopy equivalences and hence we are going to consider then as identified. The block diffeomorphisms ${\\widetilde{\\operatorname{Diff}}}_\\partial (X)$ and the diffeomorphisms ${\\operatorname{Diff}}_\\partial (X)$ with the Whitney C$^\\infty$-topology on the other hand are not homotopy equivalent - the difference is related to algebraic K-theory (see \\cite{MR1818774}). The homogeneous space ${\\widetilde{\\operatorname{aut}}}_\\partial(X)\/{\\widetilde{\\operatorname{Diff}}}_\\partial(X)$ is by definition the homotopy fiber of the map $\\tilde J:B{\\widetilde{\\operatorname{Diff}}}_\\partial (X)\\rightarrow B{\\widetilde{\\operatorname{aut}}}_\\partial (X).$ It is related to Surgery theory as we explain now.\n\nLet $X$ be a smooth manifolds of dimension $\\geq 5$ with boundary $\\partial X.$ Quinn \\cite{quinn} shows that there is a quasi-fibration of Kan $\\Delta$-sets $$\\mathcal{S}^{G\/O}_\\partial(X)\\rightarrow {\\operatorname{map}}_*(X\/\\partial X,G\/O)\\rightarrow \\mathbb{L}(X) $$ and that its homotopy exact sequence is the surgery exact sequence. $\\mathcal{S}^{G\/O}_\\partial(X)$ is the realization of a $\\Delta$-set with $k$-simplexes pairs $(W,f),$ where $W$ is a smooth $(k+3)$-ad and $f:W\\rightarrow \\Delta^k\\times X$ is a face preserving homotopy equivalence, such that $f$ restricts to a diffeomorphisms $f|\\partial_{k+1}W:\\partial_{k+1}W\\rightarrow \\Delta^k\\times \\partial X.$ There is a map $${\\widetilde{\\operatorname{aut}}}_\\partial(X)\/{\\widetilde{\\operatorname{Diff}}}_\\partial(X)\\rightarrow \\mathcal{S}^{G\/O}_\\partial(X), $$ which by the h-cobordism theorem induces a weak homotopy equivalence $${\\widetilde{\\operatorname{aut}}}_\\partial(X)\/{\\widetilde{\\operatorname{Diff}}}_\\partial(X)_{(1)}\\simeq_{w.e.}\\mathcal{S}^{G\/O}_\\partial(X)_{(1)}$$ of the identity components (see \\cite[Section 3.2.]{berg12}). Now assume that $X$ is simply connected. Since $$G\/O\\simeq_{{\\mathbb Q}}BO \\simeq_{{\\mathbb Q}} \\prod_{i\\geq 1} K({\\mathbb Q},4i),$$ we understand the rational homotopy groups $$\\pi_*({\\operatorname{map}}_*(X\/\\partial X,G\/O))\\otimes {\\mathbb Q} \\cong H^*(X,\\partial X;{\\mathbb Q})\\otimes \\pi_*(G\/O).$$ Note that if $X$ is simply connected $$\\pi_i(\\mathbb{L}(X))\\otimes {\\mathbb Q}\\cong L_{\\operatorname{dim}(X)+i}(X) \\cong\\begin{cases}{\\mathbb Q}  &\\text{if} \\operatorname{dim}(X) + i \\equiv 0 \\text{ mod }4\\\\\n0  &\\text {otherwise.}\n\\end{cases}$$\n\nWe now specialize to $N_I.$\n\\begin{lem}[{\\cite[Lemma 3.5.]{berg12}}]\\label{plumbinglemma} The surgery obstruction map induces an isomorphism\n$$H^{n}(N_I,\\partial N_I;{\\mathbb Q})\\otimes \\pi_{n+k}(G\/O)\\rightarrow L_{n+k}({\\mathbb Z})\\otimes{\\mathbb Q}$$ for $n+k\\equiv 0 \\text{ mod } 4.$\n\\end{lem}\n\n\\begin{proof}\nConsider the smooth and topological surgery exact sequences:\n\\begin{displaymath}\n\\xymatrix{...\\ar[r] &N_\\partial^{G\/O} (N_I\\times D^k)\\otimes {\\mathbb Q} \\ar[d] \\ar[r] & L_{n+k}({\\mathbb Z})\\otimes {\\mathbb Q}\\ar@{=}[d] \\ar[r]&...\\\\\n...\\ar[r] & N_\\partial^{G\/Top} (N_I\\times D^k)\\otimes {\\mathbb Q}\\ar[r] & L_{n+k}({\\mathbb Z})\\otimes {\\mathbb Q}\\ar[r]&...}  \n\\end{displaymath}\nThe left hand vertical map is an isomorphism since $\\pi_i(Top\/O)$ is finite (see e.g. \\cite{MR0645390}). Milnor's Plumbing construction ensures that for $k+n$ even there is an element in $N_\\partial^{G\/Top} (N_I\\times D^k)$ with non-trivial surgery obstruction. Since $$N_\\partial^{G\/Top} (N_I\\times D^k)\\cong \\pi_k(map_* (N_I\/\\partial N_I,G\/Top) \\text{ and }$$\n$$\\pi_k(map_* (N_I\/\\partial N_I,G\/O)\\otimes {\\mathbb Q}\\cong H^{n}(N_I,\\partial N_I;{\\mathbb Q})\\otimes \\pi_{n+k}(G\/O)), \\text{ for $n+k\\equiv 0 \\text{ mod } 4,$}$$ this implies the claim because both sides are just one dimensional rational vectorspaces.\n\\end{proof}\n\nThis now implies that we have a natural isomorphism\n\\begin{equation}\\label{naturalmapandH^*}\n\\pi_k(\\mathcal{S}^{G\/O}_\\partial(N_I))\\otimes {\\mathbb Q}\\cong H^n(N_I;{\\mathbb Q})\\otimes \\pi_{n+k}(G\/O) \\text { for } *>0\n \\end{equation}\n(compare \\cite[Corollary 4.6.]{berg13}).\n\n\nRecall that $J_0\\pi_0({\\operatorname{Diff}}_\\partial (N_I))$ has finite index in $\\pi_0({\\operatorname{aut}}_\\partial (N_I))$ (Proposition \\ref{mcg}). By Cerf's pseudo-isotopy theorem thus also $\\tilde J_1\\pi_1(B{\\widetilde{\\operatorname{Diff}}}_\\partial (N_I))$ in $\\pi_1(B{\\operatorname{aut}}_\\partial (N_I)).$ Denote by $\\bar B{\\operatorname{aut}}_\\partial (N_I)$ the finite cover corresponding to $\\operatorname{Image}(\\tilde J_1).$ Note that it has the same (higher) rational homotopy groups. By construction $\\tilde J$ lifts to a map $B{\\widetilde{\\operatorname{Diff}}}_\\partial (N_I)\\rightarrow \\bar B{\\operatorname{aut}}_\\partial (N_I).$ Instead of ${\\widetilde{\\operatorname{aut}}}_\\partial(N_I)\/{\\widetilde{\\operatorname{Diff}}}_\\partial(N_I)$ we consider $$\\mathcal{F}_I=\\operatorname{hofib}(B{\\widetilde{\\operatorname{Diff}}}_\\partial (N_I)\\rightarrow \\bar B{\\operatorname{aut}}_\\partial (N_I)).$$ We reduced the problem of showing rational homological stability for the block diffeomorphisms to the study of the Serre spectral sequence of the homotopy fibration above. The only missing ingredient is now to understand the rational homology groups of $\\mathcal{F}_I$ as $\\bar\\pi_I =\\pi_1(\\bar B{\\operatorname{aut}}_\\partial (N_I))$-modules. Observe that by Proposition \\ref{mcg} there is a surjection $$\\pi_1(\\bar B{\\operatorname{aut}}_\\partial (N_I))\\rightarrow \\Gamma_I$$ induced by $\\tilde H_*.$\n\nDenote by $\\eta: S^{G\/O}_\\partial(X) \\rightarrow \\pi_0{\\operatorname{map}}_*(X\/\\partial X,G\/O)$ the normal invariant and by $\\sigma:\\pi_0{\\operatorname{map}}_*(X\/\\partial X,G\/O) \\rightarrow L_{\\operatorname{dim}(X)}(X)$ the surgery obstruction. Using the surgery exact sequence, we see that for $n$ odd $\\pi_1(\\mathcal{F}_I)$ is abelian, since it is a subgroup of the abelian group $[\\Sigma (N_I\/\\partial N_I),G\/O]_*$. For $n$ even it is a finite extension of the abelian group $[\\Sigma (N_I\/\\partial N_I),G\/O]_*$ by a finite cyclic group (in case $L_{n+2}({\\mathbb Z})\\cong {\\mathbb Z}$, the proof of Lemma \\ref{plumbinglemma} makes sure that the map to $L_{n+2}({\\mathbb Z})$ is non zero and hence the kernel of $\\sigma$ is a finite cyclic group). Denote by $$\\pi_k^{ab}(\\mathcal{F}_I)=\\begin{cases} \\pi_1(\\mathcal{F}_I)\/ \\operatorname{Image}(L_{n+2}({\\mathbb Z})\\rightarrow \\pi_1(\\mathcal{F}_I)) & \\text{ if } k=1 \\\\ \\pi_k(\\mathcal{F}_I) & \\text{ if } k>1.\\end{cases} $$\n\n\\begin{prop}There are isomorphisms of $\\bar\\pi_I$-modules compatible with the stabilization maps\n\\begin{enumerate} \n\t\\item[$\\operatorname{(1)}$] $\\pi^{ab}_k(\\mathcal{F}_I)\\otimes {\\mathbb Q}\\cong (\\tilde H_{*}(N_I,{\\mathbb Q}) \\otimes \\pi_*(G\/O))_{k}$, where $|a\\otimes \\alpha|=|\\alpha|-|a|$, $k\\geq 1$\n\t\\item[$\\operatorname{(2)}$] $H_*(\\mathcal{F}_I,{\\mathbb Q}) \\cong \\Lambda(\\pi^{ab}_*(\\mathcal{F}_I)\\otimes {\\mathbb Q})$,\n\\end{enumerate} where the actions on the left hand side are induced by the standard actions of $\\Gamma_I$ on $\\tilde H_*(N_I).$ \n\\end{prop}\n\n\\begin{proof} Compare \\cite[p.26 and Theorem 3.6]{berg12} and \\cite[Proposition 7.15.]{berg13}. Observe that the rationalization $(\\mathcal{F}_I)_{{\\mathbb Q}}$ has rational homotopy groups $\\pi^{ab}_k(\\mathcal{F}_I)\\otimes {\\mathbb Q}.$\nConsider the splitting of the homotopy exact sequence of the surgery fibration as\n$$0\\rightarrow L_{n+k+1}({\\mathbb Z})\/\\operatorname{Image}(\\sigma)\\rightarrow \\pi_k (\\mathcal{S}_\\partial^{G\/O}(N_I)) \\rightarrow \\operatorname{Image}(\\eta)\\rightarrow 0.$$ By Lemma \\ref{plumbinglemma} we get $$L_{n+k+1}({\\mathbb Z})\/\\operatorname{Image}(\\sigma)\\otimes {\\mathbb Q}\\cong 0\\text{ and }\\operatorname{Image}(\\eta)\\otimes {\\mathbb Q} \\cong (\\tilde H^*(N_I,{\\mathbb Q}) \\otimes \\pi_*(G\/O))_{k}.$$ Using the isomorphism $$\\tilde H_*(N_I;{\\mathbb Q})\\cong {\\operatorname{Hom}}_{\\mathbb Q}(\\tilde H_*(N_I;{\\mathbb Q});{\\mathbb Q})\\cong \\tilde H^*(N_I;{\\mathbb Q})$$  we get the isomorphism $(1).$ We see that the action on the right hand side is induced by the standard action of $\\Gamma_I$ as follows: Use the identification $$\\pi_k (\\mathcal{S}^{G\/O}_\\partial(X))\\cong S_\\partial^{G\/O}( N_I\\times D^k).$$ An element of $S_\\partial^{G\/O}( N_I\\times D^k)$ is represented by a manifold $(X,\\partial X)$ together with a homotopy equivalence $f:X\\rightarrow N_I\\times  D^k,$ such that $f|\\partial X:\\partial X\\rightarrow\\partial (N_I\\times D^k)$ is a diffeomorphism. The action of a $$[\\phi]\\in\\pi_1(\\bar B{\\widetilde{\\operatorname{aut}}}_\\partial(N_I))\\cong \\operatorname{Image}(\\tilde J_1) \\cong \\operatorname{Image}(J_0)\\subset \\pi_0 ({\\operatorname{aut}}_\\partial (N_I))$$ on $f$ is given by the composition $$X \\xrightarrow{f}N_I\\times D^k\\xrightarrow{ \\phi\\times \\operatorname{id}_{D^k}}N_I\\times D^k,$$ where $\\phi$ is a diffeomorphism representing $[\\phi]$ considered as an element of $\\operatorname{Image}(J_0)$. \\cite[Lemma 3.3.]{berg12} now implies that $$\\eta((\\phi\\times \\operatorname{id}_{D^k})\\circ f)=((\\phi\\times \\operatorname{id}_{D^k})^*)^{-1}(\\eta(f))+\\eta(\\operatorname{id}_{D^k})=((\\phi\\times \\operatorname{id}_{D^k})^*)^{-1}(\\eta(f)),$$ using that the normal invariant of a diffeomorphism is trivial. This implies that $[\\phi]$ acts on $\\tilde H^*(N;{\\mathbb Q})\\otimes \\pi_*(G\/O)_k$ via $(\\phi^{-1})^*\\otimes {\\operatorname{id}}_{\\pi_*(G\/O)}.$ But this exactly corresponds to the standard action under the isomorphism $$\\tilde H^*(N_I;{\\mathbb Q})\\cong {\\operatorname{Hom}}_{\\mathbb Q}(\\tilde H_*(N_I;{\\mathbb Q});{\\mathbb Q})\\cong \\tilde H_*(N_I;{\\mathbb Q}).$$ If $\\phi$ lies in the kernel of the map $$\\pi_1(\\bar B{\\operatorname{aut}}_\\partial(N_I))\\rightarrow \\Gamma_I,$$ then it is in the kernel of $\\tilde H_*$ and a similar argument as before show that it acts trivially on $\\pi^{ab}_k(\\mathcal{F}_I)$ (compare Lemma \\ref{gamma I action on universal cover}). The compatibility with the stabilization maps follows from the fact that the isomorphisms (\\ref{naturalmapandH^*}) is natural. \n\nThe statement (2) follows from (1) by using the fact that $G\/O$ and hence also the mapping-space ${\\operatorname{map}}_*(N_I\/\\partial N_I,G\/O)$ are infinite loop spaces. Thus all rational k-invariants vanish for ${\\operatorname{map}}_*(N_I\/\\partial N_I,G\/O).$ This is equivalent to: \\\\\nFor all elements $\\alpha\\in \\pi_k({\\operatorname{map}}_*(N_I\/\\partial N_I,G\/O))\\otimes {\\mathbb Q}$ there exists an element $c\\in H^k({\\operatorname{map}}_*(N_I\/\\partial N_I,G\/O);{\\mathbb Q}),$ such that $c(h(\\alpha))\\neq 0,$ where $h$ denotes the rational Hurewicz homomorphism. Since $$\\pi_k((\\mathcal{F}_I)_{{\\mathbb Q}})\\otimes {\\mathbb Q}\\rightarrow \\pi_k({\\operatorname{map}}_*(N_I\/\\partial N_I,G\/O))\\otimes {\\mathbb Q}$$ is injective, it follows that all rational k-invariants also vanish for $(\\mathcal{F}_I)_{{\\mathbb Q}}$. This shows that $(\\mathcal{F}_I)_{{\\mathbb Q}}$ is a product of Eilenberg-Maclane spaces and hence its homology is given by the free graded commutative algebra on its homotopy groups. Moreover the $\\pi_1(\\bar B{\\operatorname{aut}}_\\partial (N_I))$-action is induced by the standard action.\n\\end{proof}\n\nWe use the previous proposition to give a Schur multifunctor description of $H_r(\\mathcal{F}_I;{\\mathbb Q}).$ For a multiindex $\\mu$ with $\\ell(\\mu)=n-1,$ consider the $\\Sigma_{\\mu}$-modules\n$\\Pi(\\mu)$ given by $$\\Pi(0,...,1,...,0)=s^{-i}\\pi_{*}(G\/O)\\otimes {\\mathbb Q},$$ where the $1$ sits in the $i$-th position and $0$ otherwise. The corresponding Schur multifunctor $$\\Pi:Mod({\\mathbb Z})^{n-1}\\rightarrow Vect_*({\\mathbb Q}),$$ has the property that there is an isomorphism of the induced $\\Gamma_I$-modules\n$$\\Pi(H_I)\\cong (\\tilde H_{*}(N_I,{\\mathbb Q}) \\otimes \\pi_*(G\/O))^+.$$\n\nIt follows now that we get an isomorphisms of $\\Gamma_I$-modules $$\\Lambda\\circ \\Pi(H_I)\\cong \\Lambda( (\\tilde H_{*}(N_I,{\\mathbb Q}) \\otimes \\pi_*(G\/O)^+))\\cong H_*(\\mathcal{F}_I,{\\mathbb Q}),$$ where the left-hand $\\Lambda$ denotes the free graded commutative algebra endofunctor of $Vect_*({\\mathbb Q}).$ Recall that $\\Lambda$ is given as the Schur functor with $\\Lambda(n)={\\mathbb Q}[n]$ and trivial $\\Sigma_n$-action. Now setting $\\mathscr{H}_r=\\Lambda_r\\circ \\Pi$ and observing that $\\Lambda_r$ is of degree $\\leq r$ and $\\Pi$ additive, we get the following:\n\n\\begin{prop}\\label{Fschur}\nThere is an isomorphism of $\\Gamma_I$-modules\n$$H_r(\\mathcal{F}_I;{\\mathbb Q})\\cong \\bigoplus_\\mu \\mathscr{H}_r(\\mu) \\otimes_\\mu H_I^{\\otimes_\\eta},$$ compatible with the stabilization maps, where the $\\mathscr{H}_r(\\mu)$ are trivial for $|\\mu|>r.$\n\\end{prop}\n\nNow we proof the second main theorem of this paper.\n\n\\begin{THMB}\nThe stabilization map $$H_i(B{\\widetilde{\\operatorname{Diff}}}_\\partial(N_I);{\\mathbb Q})\\rightarrow H_i(B{\\widetilde{\\operatorname{Diff}}}_\\partial(N_{I'});{\\mathbb Q})$$\n is an isomorphism for $g_p> 2i+2$ when $2p\\neq n$ and $g_p> 2i+4$ id $2p=n$ and an epimorphism for $g_p\\geq 2i+2$ respectively $g_p\\geq 2i+4.$\n\\end{THMB}\n\n\\begin{proof}Denote by $Y_I=B{\\widetilde{\\operatorname{Diff}}}_\\partial(N_I)$ and $\\bar X_I=\\bar B{\\widetilde{\\operatorname{aut}}}_\\partial(N_I).$\nConsider the Serre spectral sequences of the homotopy fibration $$\\mathcal{F}_I\\rightarrow Y_I\\rightarrow  \\bar  X_I$$ and the analog for $I'.$ The stabilization map induces maps on the $E_2$-pages \n$$\\sigma_*:H_k(\\bar X_I;H_l(\\mathcal{F}_I;{\\mathbb Q}))\\rightarrow H_k(\\bar X_{I'};H_l(\\mathcal{F}_{I'};{\\mathbb Q})).$$ The theorem follows upon showing that these are isomorphisms for $g_p> 2k+2l+2$ $(+4$ if $p=n\/2)$ and epimorphisms for $g_p\\geq2k+2l+2$ $(+4$ if $p=n\/2).$\nFor this we consider the universal coving spectral sequence $$H_r(\\pi_1(\\bar X_I);H_s(\\bar  X_I{\\langle 1 \\rangle};H_l(\\mathcal{F}_I;{\\mathbb Q})))\\Rightarrow H_{r+s}(\\bar X_I;H_l(\\mathcal{F}_I,{\\mathbb Q})).$$ The condition above would follow upon showing that maps induced by the stabilization map on the $E^2$-page are isomorphisms for $g_p> 2r+2s+2l+2$ $(+4$ if $p=n\/2)$ and epimorphisms for $g_p\\geq2r+2s+2l+2$ $(+4$ if $p=n\/2).$ To show this we observe that there are isomorphism of $\\Gamma_I$-modules compatible with the stabilization maps:\n$$H_s(\\bar  X_I{\\langle 1 \\rangle};H_l(\\mathcal{F}_I;{\\mathbb Q}))\\cong H_s(\\bar  X_I{\\langle 1 \\rangle})\\otimes H_l(\\mathcal{F}_I;{\\mathbb Q}))\\cong H^{CE}_s({\\mathfrak{g}}_I)\\otimes H_l(\\mathcal{F}_I;{\\mathbb Q})),$$ where $\\Gamma_I$ acts on the $2$-nd and $3$-rd term diagonally. Note that $\\bar X_I$ and $X_I$ have the same universal cover, which is moreover naturally homotopy equivalent to $B{\\operatorname{aut}}_\\partial(N_I){\\langle 1 \\rangle}.$ The stability for $$H_r(\\pi_1(\\bar  X_I);H^{CE}_s({\\mathfrak{g}}_I)\\otimes H_l(\\mathcal{F}_I;{\\mathbb Q})))$$ follows from stability for $$H_r(\\pi_1(\\bar X_I);C^{CE}_s({\\mathfrak{g}}_I)\\otimes H_l(\\mathcal{F}_I;{\\mathbb Q}))),$$ exactly like in Proposition \\ref{algebraic homological stability result} upon using the two hyperhomology spectral sequences and that $\\bar \\pi_I$ is rationally perfect (Lemma \\ref{H1 triv pi0}). Hence we are left with showing that the stabilization maps \n$$H_r(\\pi_1(\\bar X_I);C^{CE}_s({\\mathfrak{g}}_I)\\otimes H_l(\\mathcal{F}_I;{\\mathbb Q})))\\rightarrow H_r(\\pi_1(\\bar X_{I'});C^{CE}_s({\\mathfrak{g}}_{I'})\\otimes H_l(\\mathcal{F}_{I'};{\\mathbb Q})))$$ are isomorphisms for $g_p> 2r+2s+2l+2$ $(+4$ if $p=n\/2)$ and epimorphisms for $g_p\\geq2r+2s+2l+2$ $(+4$ if $p=n\/2).$ The Lyndon spectral sequence reduces this to the corresponding statement for $$H_r(\\Gamma_I;C^{CE}_s({\\mathfrak{g}}_I)\\otimes H_l(\\mathcal{F}_I;{\\mathbb Q})))\\rightarrow H_r(\\Gamma_{I'};C^{CE}_s({\\mathfrak{g}}_{I'})\\otimes H_l(\\mathcal{F}_{I'};{\\mathbb Q}))).$$\nPropositions \\ref{Fschur} and \\ref{der iso compatible with stab maps} give us isomorphisms of $\\Gamma_I$-modules compatible with the stabilization map $C^{CE}_s({\\mathfrak{g}}_I)\\otimes H_l(\\mathcal{F}_I;{\\mathbb Q}))\\cong \\mathscr{C}_s(H_I) \\otimes \\mathscr{H}_l(H_I).$ The functor $\\mathscr{C}_s$ is polynomial of degree $\\leq s\/2$ and the functor $\\mathscr{H}_l$ is polynomial of degree $\\leq l.$ The tensor product (in the sense of Schur multifunctors) $\\mathscr{C}_s\\otimes \\mathscr{H}_l$ is of degree $\\leq s\/2+l.$ By Proposition \\ref{general algebraic homological stability result} the stabilization maps\n$$H_r(\\Gamma_I;\\mathscr{C}_s\\otimes \\mathscr{H}_l(H_I))\\rightarrow H_r(\\Gamma_{I'};\\mathscr{C}_s\\otimes \\mathscr{H}_l(H_{I'}))$$ are isomorphisms for $g_p> 2r+s\/2+l+2$ $(+4$ if $p=n\/2)$ and epimorphisms for $g_p\\geq 2r+s\/2+l+2$ $(+4$ if $p=n\/2),$ which finishes the proof.\n\\end{proof}\n\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\nImage quality has become a critical evaluation metric in most image-processing applications, including image denoising, image super-resolution, compression artifacts reduction, \\etc. Directly acquiring perceptual quality scores from human observers is accurate. However, this requires time-consuming and costly subjective experiments. The goal of Image Quality Assessment (IQA) is to allow computers to simulate the Human Visual System (HVS) through algorithms to score the perceptual quality of images. In this case, the images to be evaluated are often degraded during compression, acquisition, and post-processing. \n\nIn recent years, the invention of Generative Adversarial Networks (GANs)~\\cite{goodfellow2014generative} has greatly improved the image processing ability, especially image generation~\\cite{gu2020image,xia2021tedigan} and image restoration~\\cite{wang2018esrgan}, while it also brings new challenges to image quality assessment. GAN-based methods can fabricate seemingly realistic but fake details and textures~\\cite{jinjin2020pipal}. In detail, it is hard for the HVS to distinguish the misalignment of the edges and texture decreases in the region with dense textures. As long as the semantics of textures are similar, the HVS will ignore part of the subtle differences of textures. Most IQA methods for traditional distortion images assess image quality through pixel-wise comparison, which will lead to underestimation for GAN-generated images~\\cite{wang2004image}. To deal with the texture misalignment, recent studies~\\cite{bosse2017deep} introduce patch-wise prediction methods. \nSome following studies~\\cite{shi2021region,jinjin2020pipal} further propose different spatially robust comparison operations into the CNN-based IQA network. However, they take each patch as an independent input and separately calculate their score and weight, which will lead to the loss of context information and the inability to model the relationship between patches. \n\nTherefore, on the basis of patch-level comparison, we need to better model the interrelationship between patches. To this end, we use Vision Transformer (ViT)~\\cite{dosovitskiy2020image} as a feature extractor, which can effectively capture long-range dependencies among patches through a multi-head attention mechanism. However, the vanilla ViT uses a large convolution kernel to down-sample the input images in spatial dimension before entering the network; some details that should be considered are lost, which are also crucial to image quality assessment. Based on the observation, we found that a shallow CNN is a good choice to provide detailed spatial information. \nThe features extracted by a shallow CNN contains unwanted noises and merging ViT features with them would decrease the performance. To alleviate the impact of noise, we propose to mimic the characteristic of the HVS that human always pay attention to the salient regions of images. \nInstead of injecting the complete features from a shallow CNN into those from ViT, we only use those that convey spatial details of the salient regions for image quality assessment, thereby alleviating the aforementioned noise.\nFurthermore, using max-pooling or average-pooling to directly predict the score of an image will lose crucial information.\nTherefore, we use an adaptive weighted strategy to predict the score of an image.\n\n\nIn this work, we introduce an effective hybrid architecture for image quality assessment, which leverages local details from a shallow CNN and global semantic information captured by ViT to further improve IQA accuracy. Specifically, we first adopt a two-branch feature extractor. Then, we use semantic information captured by ViT to find the salient region in images through deformable convolution~\\cite{dai2017deformable}. Based on the consideration that each pixel in the deep feature map corresponds to different patches of the input image, we introduce the patch-wise prediction module, which contains two branches, one to calculate a score for each image patch, the other one to calculate the weight of each score.\n\nExtensive experiments show that our method outperforms current approaches in four benchmark image quality assessment datasets~\\cite{sheikh2006statistical,larson2010most,ponomarenko2015image,jinjin2020pipal}. The scatter diagram of the correlation between predicted scores and MOS is shown in \\cref{fig:scatter} where the plot for IQT is from our own implementation. Visualization experiments reveal that the proposed method is almost linear with MOS, which means that we can better imitate human image perception. Our primary contributions can be summarized as follows:\n\\begin{itemize}\n\\item We propose an effective hybrid architecture for image quality assessment, which compares images at the patch level, adds spatial details as a supplement, and scores images patch by patch, considering the relationship between patches and different contributions from each patch.\n\n\\item Our method outperforms the state-of-the-art approaches on four benchmark image quality assessment datasets. In particular, the proposed architecture achieves outstanding performance on the PIPAL dataset with various GAN-based distortion and ranked first in the NTIRE 2022 challenge on perceptual image quality assessment.\n\\end{itemize}\n\n\\section{Related Works}\n\\label{sec:related}\n\\subsection{Image Quality Assessment}\nThe goal of IQA is to mimic the HVS to rate the perceived quality of an image accurately. Although it's easy for human beings to assess an image's perceptual quality, IQA is considered to be difficult for machines. Depending on the scenarios and conditions, current IQA methods can be divided into three categories: full-reference (FR) ,and no-reference (NR) IQA. FR-IQA methods take the distortion image and the corresponding reference image as inputs to measure their perceptual similarity. The most widely-used FR-IQA metrics are PSNR and SSIM~\\cite{wang2004image} which are conventional and easy to optimize.\n\\begin{figure*}[th]\n  \\centering\n   \\includegraphics[scale=0.48]{architecture_final.pdf}\n   \\caption{Overview of AHIQ. The proposed model takes a pair of the reference image and distortion image as input and then obtains feature maps through ViT~\\cite{dosovitskiy2020image} and CNN, respectively. The feature maps of reference image from ViT are used as global information to obtain the offset map of the deformable convolution\\cite{dai2017deformable}. After the feature fusion module which fuses the feature maps, we use a patch-wise prediction module to predict a score for each image patch. The final output is the weighted sum of the scores.}\n   \\label{fig1:arch}\n\\end{figure*}\nApart from the conventional IQA methods, various learning-based FR-IQA methods~\\cite{zhang2018unreasonable,bosse2017deep,prashnani2018pieapp} have been proposed to address the limitations of conventional IQA methods recently. Zhang \\etal~\\cite{zhang2018unreasonable} proposed to use the learned perceptual image patch similarity (LPIPS) metric for FR-IQA and proved that deep features obtained through pre-trained DNNs outperform previous classic metrics by large margins. WaDIQaM~\\cite{bosse2017deep} is a general end-to-end deep neural network that enables jointly learning of local quality and local weights. PieAPP~\\cite{prashnani2018pieapp} is proposed to learn to rank rather than learn to score, which means the network learns the probability of preference of one image over another. IQT~\\cite{cheon2021perceptual} applies an encoder-decoder transformer architecture with trainable extra quality embedding and ranked first place in NTIRE 2021 perceptual image quality assessment challenge. In addition, common CNN-based NR-IQA methods~\\cite{su2020blindly,wu2020end,xia2020domain} directly extract features from the low-quality images and outperform traditional handcrafted approaches. You \\etal~\\cite{ you2021transformer} introduced transformer architecture for the NR-IQA recently.\n\n\n\n\n\\subsection{Vision Transformer}\nTransformer architecture based on self-attention mechanism~\\cite{vaswani2017attention} was first proposed in the field of Natural Language Processing (NLP) and significantly improved the performances of many NLP tasks thanks to its representation capability. Inspired by its success in NLP, efforts are made to apply transformers to vision tasks such as image classification~\\cite{dosovitskiy2020image}, object detection~\\cite{carion2020end, zhu2020deformable}, low-level vision~\\cite{yang2020learning}, \\etc. Vision Transformer (ViT) introduced by Dosovitskiy~\\etal~\\cite{dosovitskiy2020image} is directly inherited from NLP, but takes raw image patches as input instead of word sequences. ViT and its follow-up studies have become one of the mainstream feature extraction backbones except for CNNs.\n\nCompared with the most commonly used CNNs, transformer can derive global information while CNNs mainly focus on local features. In IQA tasks, global and local information are both crucial to the performance because when human beings assess image quality, both the information are naturally taken into account. Inspired by this assumption, we propose to combine long-distance features and local features captured by ViT and CNNs, respectively. To fulfill this goal, we use a two-branch feature extraction backbone and feature fusion modules, which will be detailed in~\\cref{sec:method}. \n\n\n\\subsection{Deformable Convolution}\nDeformable convolution~\\cite{dai2017deformable} is an efficient and powerful mechanism which is first proposed to deal with sparse spatial locations in high-level vision tasks such as object detection~\\cite{dai2017deformable,bertasius2018object, zhu2019deformable }, semantic segmentation~\\cite{zhu2019deformable}, and human pose estimation~\\cite{sun2018integral}. By using deformed sampling locations with learnable offsets, deformable convolution enhances the spatial sampling locations and improves the transformation modeling ability of CNNs. Recently, deformable convolution continues its strong performance in low-level vision tasks including video deblurring~\\cite{wang2019edvr}, video super-resolution~\\cite{chan2021understanding}. It is first combined with IQA methods by Shi~\\etal~\\cite{shi2021region} to perform a reference-oriented deformable convolution in the full-reference scenario.\n\n\n\n\\section{Methodology}\n\\label{sec:method}\n\nIn this section, we introduce the overall framework of the Attention-based Hybrid Image Quality Assessment Network (AHIQ). As shown in Fig \\ref{fig1:arch}, the proposed network takes pairs of reference images and distortion images as input, and it consists of three key components: a feature extraction module, a feature fusion module, and a patch-wise prediction module. \n\nFor the reason that GAN-based image restoration methods~\\cite{wang2018esrgan,gu2020image} often fabricate plausible details and textures, it is difficult for the network to distinguish GAN-generated texture from noise and real texture by pixel-wise image difference. Our proposed model aims to deal with it. We employ the Vision Transformer to model the relationship and capture long-range dependencies among patches. Shallow CNN features are introduced to add detailed spatial information. In order to help CNN focus on the salient region, we use deformable convolution guided by semantic information from ViT. We use an adaptive weighted scoring mechanism to give a comprehensive assessment.\n\n\n\n\\subsection{Feature Extraction Module}\n\\label{subsec:extraction}\n\\begin{figure}[th]\n\\centering\n\\includegraphics[scale=0.41]{architecture_feat.pdf}\n\\caption{The illustration of vision Transformer for feature extraction module. The class token (orange) is regarded when the feature maps are extracted.\n}\n\\label{fig2:feat}\n\\end{figure}\nAs is depicted in \\cref{fig1:arch}, the front part of the architecture is a two-branch feature extraction module that consists of a ViT branch and a CNN branch. The transformer feature extractor mainly focuses on extracting global and semantic representations. Self-attention modules in transformer enable the network to model long-distance features and encode the input image patches into feature representations. Patch-wise encoding is helpful to assess the output image quality of GAN-based image restoration because it enhances the tolerance of spatial misalignment. Since humans also pay attention to details when judging the quality of an image, so detailed and local information is also important. To this end, we introduce another CNN extraction branch apart from the transformer branch to add more local textures.\n\nIn the forward process, a pair of the reference image and distortion image are fed into the two branches, respectively, and we then take out their feature maps in the early stages. For the transformer branch, as illustrated in~\\cref{fig2:feat}, output sequences from Vision Transformer~\\cite{dosovitskiy2020image} are reshaped into feature maps $f_{T}\\in \\mathbb{R}^{p\\times p \\times 5c}$ discarding the class token, where $p$ represent the size of the feature map. For the CNN branch, we extract shallow feature map from ResNet~\\cite{he2016deep} $f_{C}\\in \\mathbb{R}^{4p\\times 4p \\times C}$ where $C=256\\times 3$. Finally, we put the obtained feature maps into the feature fusion module, which will be specified next.\n \n\n\\subsection{Feature Fusion Module}\n\\label{subsec:fusion}\nWe argue that feature maps from the early stages of CNN provide low-level texture details but bring along some noise. To address this problem, we take advantage of transformer architecture to capture global and semantic information. In our proposed network, feature maps from ViT with rich semantic information are used to find the salient region of the image. This perception procedure is performed in a content-aware manner and allows the network better mimic the way humans perceive image quality. Particularly, the feature maps from ViT are used to learn an offset map for deformable convolution as is shown in~\\cref{fig2:feat}. Then we perform this deformable convolution~\\cite{dai2017deformable} operation on feature maps from CNN, which we elaborate on previously. In this way, features from a shallow CNN can be better modified and utilized for further feature fusion. Obviously, in the previous description, feature maps from the two branches differ from each other in spatial dimension and need to be aligned. Therefore, a simple 2-layer convolution network is applied to project the feature maps after deformable convolution to the same width~\\textit{W} and height~\\textit{H} with ViT. The whole process can be formulated as follows:\n\\begin{align}\n    && \\Delta p &= \\text{Conv1}(f_{T})), \\\\\n    &&  f_{C} &= \\text{DConv}(f_{org},\\Delta p), \\\\\n    &&  f_{C}^{'} &= \\text{Conv2}(\\text{ReLU}(\\text{Conv2}(f_{C}))),   \\\\\n    &&  f^{u} &= \\text{Concat}[f_{T},f_{C}^{'}],   \\\\\n    &&  f_{all} &= \\text{Concat}[f^{u}_{dis},f^{u}_{ref},f^{u}_{dis}-f^{u}_{ref}],  \\\\\n    &&  f_{out} &= \\text{Conv3}(\\text{ReLU}(\\text{Conv3}(f_{all}))),\n\\end{align}\nwhere $f_T$ denotes feature maps from the transformer branch, $\\Delta p$ denotes offset map, $f_{org}$ and $f_C$ denote feature maps from CNN, DConv means deformable convolution. Note that Conv2 is a convolution operation with a stride of 2, downsampling $f_C \\in \\mathbb{R}^{4p \\times 4p \\times C}$ by four times to $f_C^{'} \\in \\mathbb{R}^{p \\times p \\times C}$. \n\n\\subsection{Patch-wise Prediction Module}\n\\label{subsec:pooling}\nGiven that each pixel in the deep feature map corresponds to a different patch of the input image and contains abundant information, the information in the spatial dimension is indispensable. However, in previous works, spatial pooling methods such as max-pooling and average-pooling are applied to obtain a final single quality score. This pooling strategy loses some information and ignores the relationships between image patches. Therefore, we introduce a two-branch patch-wise prediction module which is made up of a prediction branch and a spatial attention branch, as illustrated in \\cref{fig:pixel}. The prediction branch calculates a score for each pixel in the feature map, while the spatial attention branch calculates an attention map for each corresponding score. Finally, we can obtain the final score by weighted summation of scores. The weighted sum operation helps to model the significance of the region to simulate the human visual system. This can be expressed as follows:\n\\begin{equation}\n    s_f = \\frac{\\textbf{s} * \\textbf{w}}{\\sum \\textbf{w}},\n\\end{equation}\nwhere $\\textbf{s}\\in \\mathbb{R}^{H\\times W \\times 1}$ denotes score map, $\\textbf{w}\\in \\mathbb{R}^{H\\times W \\times 1}$ denotes the corresponding attention map, $*$ means Hadamard product and $s_f$ means the final predicted score. MSE loss between the predicted score and the ground truth score is utilized for the training process in our proposed method.\n\n\\begin{figure}[th]\n\\centering\n\\includegraphics[scale=0.43]{architecture_patch.pdf}\n\\caption{The pipeline of the proposed patch-wise prediction module. This two-branch module takes feature maps as input, then generates a patch-wise score map and its corresponding attention map to obtain the final prediction by weighted average.}\n\\label{fig:pixel}\n\\end{figure}\n\n\\begin{table*}[th]\n\\centering\n\\renewcommand{\\arraystretch}{1.08}\n\\caption{IQA datasets for performance evaluation and model training.}\n\\begin{tabular}{cccccccc}\n\\toprule[1.2pt]\nDatabase  & \\# Ref & \\# Dist & Dist. Type & \\# Dist. Type & Rating & Rating Type & Env. \\\\ \\hline\nLIVE~\\cite{sheikh2006statistical}      & 29    & 779    & traditional       & 5            & 25k    & MOS         & lab           \\\\\nCSIQ~\\cite{larson2010most}      & 30    & 866    & traditional       & 6            & 5k     & MOS         & lab           \\\\\nTID2013~\\cite{ponomarenko2015image}   & 25    & 3,000  & traditional       & 25           & 524k   & MOS         & lab           \\\\\nKADID-10k~\\cite{lin2019kadid} & 81    & 10.1k  & traditional       & 25           & 30.4k  & MOS         & crowdsourcing \\\\\nPIPAL~\\cite{jinjin2020pipal}     & 250   & 29k    & trad.+alg.outputs & 40           & 1.13m  & MOS         & crowdsourcing \\\\ \\toprule[1.2pt]\n\\label{tab:data}\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}[th]\n\\centering\n\\caption{Performance comparisons on LIVE, CSIQ, and TID2013 Databases. Performance scores of other methods are as reported in the corresponding original papers and~\\cite{ding2020image}. The best scores are~\\textbf{bolded} and missing scores are shown as ``\u2013'' dash.}\n\\setlength{\\tabcolsep}{6mm}{\n\\begin{tabular}{ccccccc}\n\\toprule[1.2pt]\n\\multirow{2}{*}{Method} & \\multicolumn{2}{c}{LIVE} & \\multicolumn{2}{c}{CSIQ} & \\multicolumn{2}{c}{TID2013} \\\\ \\cline{2-7} \n                        & PLCC        & SROCC      & PLCC        & SROCC      & PLCC         & SROCC        \\\\ \\hline\nPSNR                    & 0.865       & 0.873      & 0.819       & 0.810      & 0.677       & 0.687        \\\\\nSSIM~\\cite{wang2004image}                    & 0.937       & 0.948      & 0.852       & 0.865      & 0.777        & 0.727        \\\\\nMS-SSIM~\\cite{wang2003multiscale}                 & 0.940       & 0.951      & 0.889       & 0.906      & 0.830        & 0.786        \\\\\nFSIMc~\\cite{zhang2011fsim}                   & 0.961       & 0.965      & 0.919       & 0.931      & 0.877        & 0.851        \\\\\nVSI~\\cite{zhang2014vsi}                     & 0.948       & 0.952      & 0.928       & 0.942      & 0.900          & 0.897        \\\\\nMAD~\\cite{larson2010most} &0.968      &0.967      &0.950       &0.947     &0.827      &0.781      \\\\\nVIF~\\cite{sheikh2006image} &0.960      &0.964      &0.913      &0.911      &0.771      &0.677      \\\\\nNLPD~\\cite{laparra2016perceptual}    &0.932      &0.937      &0.923      &0.932      &0.839      &0.800      \\\\\nGMSD~\\cite{xue2013gradient}    &0.957      &0.960      &0.945      &0.950      &0.855      &0.804\\\\    \nSCQI~\\cite{bae2016novel}                    & 0.937       & 0.948      & 0.927       & 0.943      & 0.907        & 0.905        \\\\ \\hline\nDOG-SSIMc~\\cite{pei2015image}               & 0.966       & 0.963      & 0.943       & 0.954      & 0.934        & 0.926        \\\\\nDeepQA~\\cite{kim2017deep}                  & 0.982       & 0.981      & 0.965       & 0.961      & 0.947        & 0.939        \\\\\nDualCNN~\\cite{varga2020composition}                 & -           & -          & -           & -          & 0.924        & 0.926        \\\\\nWaDIQaM-FR~\\cite{bosse2017deep}              & 0.98        & 0.97       & -           & -          & 0.946        & 0.94         \\\\\nPieAPP~\\cite{prashnani2018pieapp}                  & 0.986       & 0.977      & 0.975       & 0.973      & 0.946        & 0.945        \\\\\nJND-SalCAR~\\cite{seo2020novel} & 0.987          & \\textbf{0.984} & 0.977          & \\textbf{0.976} & 0.956          & 0.949          \\\\\nAHIQ (ours) & \\textbf{0.989} & \\textbf{0.984} & \\textbf{0.978} & 0.975          & \\textbf{0.968} & \\textbf{0.962}       \\\\ \\toprule[1.2pt]\n\n\\end{tabular}}\n\\label{tab:sota}\n\\end{table*}\n\n\\section{Experiment}\n\\subsection{Datasets}\n\nWe employ four datasets that are commonly used in the research of perceptual image quality assessment, including LIVE~\\cite{sheikh2006statistical}, CSIQ~\\cite{larson2010most}, TID2013~\\cite{ponomarenko2015image}, and PIPAL~\\cite{jinjin2020pipal}. \\cref{tab:data} compares the listed datasets in more detail. In addition to PIPAL, the other datasets only include traditional distortion types, while PIPAL includes a large number of distorted images including GAN-generated images. \n\nAs recommended, we randomly split the datasets into training (60\\%), validation (20\\%), and test set (20\\%) according to reference images. Therefore, the test data and validation data will not be seen during the training procedure. We use the validation set to select the model with the best performance and use the test set to evaluate the final performance.\n\n\\subsection{Implementation Details}\nSince we use ViT~\\cite{dosovitskiy2020image} and ResNet~\\cite{he2016deep} models pre-trained on ImageNet~\\cite{ILSVRC15}, we normalize all input images and randomly crop them into $224\\times224$. We use the outputs of five intermediate blocks $\\{0,1,2,3,4\\}$ in ViT, each of which consists of a self-attention module and a Feed-Forward Network (FFN). The feature map from one block $f\\in \\mathbb{R}^{p\\times p \\times c}$, where $c=768, p=14 \\ \\text{or} \\ 28$, are concatenated into $f_{T}\\in \\mathbb{R}^{p\\times p \\times 6c}$. We also take out the output feature maps from all the 3 layers in stage 1 of ResNet and concatenate them together to get $f_{C}\\in \\mathbb{R}^{56\\times 56 \\times C}$ where $C=256\\times 3$. And random horizontal flip rotation is applied during the training. The training loss is computed using a mean squared error (MSE) loss function. During the validation phase and test phase, we randomly crop each image 20 times and the final score is the average score of each cropped image. It should be noted that we use pretrained ViT-B\/16 as the backbone in all experiments on traditional datasets including LIVE, CSIQ and TID2013, while ViT-B\/8 is utilized in PIPAL.\n\nFor optimization, we use the AdamW optimizer with an initial learning rate $lr$ of $10^{-4}$ and weight decay of $10^{-5}$. We set the minibatch size as 8. Set the learning rate of each parameter group using a cosine annealing schedule, where $\\eta_{max}$ is set to the initial $lr$ and the number of epochs $T_{cur}$ is 50. \nWe implemented our proposed model AHIQ in Pytorch and trained using a single NVIDIA GeForce RTX2080 Ti GPU. The practical training runtimes differ across datasets as the number of images in each dataset is different. Training one epoch on the PIPAL dataset requires thirty minutes.\n\n\\subsection{Comparison with the State-of-the-art Methods}\nWe assess the performance of our model with Pearson's linear\ncorrelation coefficient (PLCC) and Spearman's rank-order correlation coefficient (SROCC). \nPLCC assesses the linear correlation between ground truth and the predicted quality scores, whereas SROCC describes the level of monotonic correlation.\n\n\\vspace{2pt}\n\\noindent\\textbf{Evaluation on Traditional Dataset.} We evaluate the effectiveness of AHIQ on four benchmark datasets. For all our tests, we follow the above experimental setup. It can be shown in~\\cref{tab:sota} that AHIQ outperforms or is competitive with WaDIQaM~\\cite{bosse2017deep}, PieAPP~\\cite{prashnani2018pieapp}, and JND-SalCAR~\\cite{seo2020novel} for all tested datasets. Especially on the more complex dataset TID2013, our proposed model achieved a solid improvement over previous work. This shows that the AHIQ can cope well with different types of distorted images.\n\\begin{table}[th]\n\\centering\n\\caption{Performance comparison after training on the entire KADID dataset~\\cite{lin2019kadid}, then test on LIVE, CSIQ, and TID2013 Databases. Part of the performance scores of other methods are borrowed from~\\cite{ding2020image}. The best scores are~\\textbf{bolded} and missing scores are shown as ``\u2013'' dash.}\n\\scalebox{0.82}{\n\\begin{tabular}{cccc}\n\\toprule[1.2pt]\n\\multirow{2}{*}{Method} & LIVE                 & CSIQ                 & TID2013              \\\\ \\cline{2-4} \n& PLCC\/SROCC           & PLCC\/SROCC           & PLCC\/SROCC           \\\\ \\hline\n\nWaDIQaM~\\cite{bosse2017deep}              & 0.940\/0.947       & 0.901\/0.909          & 0.834\/0.831         \\\\\nPieAPP~\\cite{prashnani2018pieapp}                  & 0.908\/0.919      & 0.877\/0.892      & 0.859\/0.876        \\\\\nLPIPS~\\cite{zhang2018unreasonable}              & 0.934\/0.932  &0.896\/0.876  &0.749\/0.670           \\\\\nDISTS~\\cite{ding2020image}               & \\textbf{0.954}\/0.954        & 0.928\/0.929      & 0.855\/0.830     \\\\\nIQT~\\cite{cheon2021perceptual}                 & -\/\\textbf{0.970}      & -\/0.943      & -\/0.899      \\\\\nAHIQ (ours) & 0.952\/\\textbf{0.970} & \\textbf{0.955\/0.951}        & \\textbf{0.899\/0.901}       \\\\ \\toprule[1.2pt]\n\\end{tabular}\n}\n\\label{tab:kadid}\n\\end{table}\n\n\\vspace{2pt}\n\\noindent\\textbf{Evaluation on PIPAL.} We compare our models with the state-of-the-art FR-IQA methods on the NTIRE 2022 IQA challenge validation and testing datasets. As shown in \\cref{tab:pipal}, AHIQ achieves outstanding performance in terms of PLCC and SROCC compared with all previous work. In particular, our method substantially outperforms IQT, which is recognized as the first transformer-based image quality assessment network, through the effective feature fusion from the shallow CNN and ViT as well as the proposed patch-wise prediction module. This verifies the effectiveness of our model for GAN-based distortion image quality assessment.\n\n\\begin{table}[th]\n\\centering\n\\caption{\nPerformance comparison of different IQA methods on PIPAL dataset. AHIQ-C is the ensemble version we used for the NTIRE 2022 Perceptual IQA Challenge.\n}\n\\begin{tabular}{ccccc}\n\\toprule[1.2pt]\n\\multirow{2}{*}{Method} & \\multicolumn{2}{c}{Validation} & \\multicolumn{2}{c}{Test} \\\\ \\cline{2-5} \n & PLCC & SROCC & PLCC & SROCC \\\\ \\hline\nPSNR & 0.269 & 0.234 & 0.277 & 0.249 \\\\\nNQM~\\cite{damera2000image} & 0.364 & 0.302 & 0.395 & 0.364 \\\\\nUQI~\\cite{wang2002universal} & 0.505 & 0.461 & 0.450 & 0.420 \\\\\nSSIM~\\cite{wang2004image} & 0.377 & 0.319 & 0.391 & 0.361 \\\\\nMS-SSIM~\\cite{wang2003multiscale} & 0.119 & 0.338 & 0.163 & 0.369 \\\\\nRFSIM~\\cite{zhang2010rfsim} & 0.285 & 0.254 & 0.328 & 0.304 \\\\\nGSM~\\cite{liu2011image} & 0.450 & 0.379 & 0.465 & 0.409 \\\\\nSRSIM~\\cite{zhang2012sr} & 0.626 & 0.529 & 0.636 & 0.573 \\\\\nFSIM~\\cite{zhang2011fsim} & 0.553 & 0.452 & 0.571 & 0.504 \\\\\nVSI~\\cite{zhang2014vsi} & 0.493 & 0.411 & 0.517 & 0.458 \\\\\nNIQE~\\cite{mittal2012making} & 0.129 & 0.012 & 0.132 & 0.034 \\\\\nMA~\\cite{ma2017learning} & 0.097 & 0.099 & 0.147 & 0.140 \\\\\nPI~\\cite{blau2018perception} & 0.134 & 0.064 & 0.145 & 0.104 \\\\\nBrisque~\\cite{mittal2011blind} & 0.052 & 0.008 & 0.069 & 0.071 \\\\ \\hline\nLPIPS-Alex~\\cite{zhang2018unreasonable} & 0.606 & 0.569 & 0.571 & 0.566 \\\\\nLPIPS-VGG~\\cite{zhang2018unreasonable} & 0.611 & 0.551 & 0.633 & 0.595 \\\\\nDISTS~\\cite{ding2020image} & 0.634 & 0.608 & 0.687 & 0.655 \\\\\nIQT~\\cite{cheon2021perceptual} & 0.840 & 0.820 & 0.799 & 0.790 \\\\ \\hline\nAHIQ (ours) & 0.845 & 0.835 & 0.823 & 0.813 \\\\\nAHIQ-C (ours) & \\textbf{0.865} & \\textbf{0.852} & \\textbf{0.828} & \\textbf{0.822}  \\\\ \\toprule[1.2pt]\n\\end{tabular}\n\\label{tab:pipal}\n\\end{table}\n\n\\vspace{2pt}\n\\noindent\\textbf{Cross-Database Performance Evaluation.} \nTo evaluate the generalization of our proposed AHIQ, we conduct the cross-dataset evaluation on LIVE, CSIQ, and TID2013. We train the model on KADID and the training set of PIPAL respectively. Then we test it on the full set of the other three benchmark datasets. As shown in \\cref{tab:kadid} and \\cref{tab:cross}, AHIQ achieves satisfactory generalization ability. \n\n\\begin{table}[th]\n\\centering\n\\caption{Performance comparison for cross-database evaluations.}\n\\scalebox{0.82}{\n\\begin{tabular}{cccc}\n\\toprule[1.2pt]\n\\multirow{2}{*}{Method} & LIVE                 & CSIQ                 & TID2013              \\\\ \\cline{2-4} \n                        & PLCC\/SROCC           & PLCC\/SROCC           & PLCC\/SROCC           \\\\ \\hline\nPSNR                    & 0.865\/0.873          & 0.786\/0.809          & 0.677\/0.687          \\\\\nWaDIQaM~\\cite{bosse2017deep}                 & 0.837\/0.883          & -\/-                  & 0.741\/0.698          \\\\\nRADN~\\cite{shi2021region}                    & 0.878\/0.905          & -\/-                  & 0.796\/0.747 \\\\\nAHIQ (ours)              & \\textbf{0.911\/0.920} & \\textbf{0.861\/0.865} & \\textbf{0.804\/0.763} \\\\ \\toprule[1.2pt]\n\\end{tabular}\n}\n\\label{tab:cross}\n\\end{table}\n\n\n\n\\subsection{Ablation Study}\nIn this section, we analyze the effectiveness of the proposed network by conducting ablation studies on the NTIRE 2022 IQA Challenge testing datasets~\\cite{gu2022ntire}. With different configuration and implementation strategies, we evaluate the effect of each of the three major components: feature extraction module, feature fusion module, and patch-wise prediction module.\n\\begin{table}[ht]\n\\centering\n\\caption{Comparison of different feature fusion strategies on the NTIRE 2022 IQA Challenge testing datasets. CNN refers to Resnet50 and ViT refers to ViT-B\/8 in this experiment. }\n\n\\begin{tabular}{cccccc}\n\\toprule[1.2pt]\n\\multirow{2}{*}{No.} & \\multicolumn{2}{c}{Feature} & \\multirow{2}{*}{Fusion Method} & \\multirow{2}{*}{PLCC} & \\multirow{2}{*}{SROCC}  \\\\ \\cline{2-3}\n & CNN & ViT &  &  & \\\\ \\hline\n1 & \\checkmark & \\checkmark & deform+concat & \\textbf{0.823} & \\textbf{0.813}  \\\\\n2 & \\checkmark & \\checkmark & concat & 0.810 & 0.799 \\\\\n3 & \\checkmark &  & - & 0.792 & 0.789 \\\\\n4 &  & \\checkmark & - & 0.799 & 0.788 \\\\ \\toprule[1.2pt]\n\\end{tabular}\n\\label{tab:fusion}\n\\end{table}\n\n\n\\vspace{2pt}\n\\noindent\\textbf{Feature Extraction Backbone.} We experiment with different representative feature-extraction backbones and the comparison result is provided in \\cref{tab:backbone}. The CNN backbones used for comparison include ResNet50, ResNet101, ResNet152~\\cite{he2016deep}, HRNet~\\cite{wang2020deep}, Inception-ResNet-V2~\\cite{szegedy2017inception}, and the transformer backbones include ViT-B\/16 and ViT-B\/8~\\cite{dosovitskiy2020image}. It is noteworthy that ViT-B consists of 12 transformer blocks and the sizes of the image patches are $16\\times 16$ and $8\\times 8$ respectively with an input shape of $224\\times 224$. \n\nIt can be found that the network using ResNet50 and ViT-B\/8 ends up performing the best. The experimental results demonstrate that deeper and wider CNN is unnecessary for AHIQ. We believe this is because CNN plays the role of providing shallow and local feature information in AHIQ. We only take out the intermediate layers from the first stage, so shallow features will contain less information when the network is too deep or too complicated.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Comparison of different feature extraction backbones on the NTIRE 2022 IQA Challenge testing datasets.}\n\\scalebox{0.95}{\n\\begin{tabular}{ccccc}\n\\toprule[1.2pt]\nCNN & ViT & PLCC & SROCC & Main Score \\\\ \\hline\nResnet50 & \\multirow{5}{*}{ViT-B\/8} & \\textbf{0.823} & \\textbf{0.813} & \\textbf{1.636} \\\\\nResnet101 &  & 0.802 & 0.788 & 1.590 \\\\\nResnet152 &  & 0.807 & 0.793 & 1.600 \\\\\nHRnet &  & 0.806 & 0.796 & 1.601 \\\\\nIncepResV2 &  & 0.806 & 0.793 & 1.599 \\\\ \\hline\nResnet50 & ViT-B\/16 & 0.811 & 0.803 & 1.614 \\\\ \\toprule[1.2pt]\n\\end{tabular}}\n\\label{tab:backbone}\n\\end{table}\n\n\\vspace{2pt}\n\\noindent\\textbf{Fusion strategy.} We further examine the effect of features from CNN and ViT as well as the feature fusion strategies. As is tabulated in \\cref{tab:fusion}, the first two experiments adopt different methods for feature fusion. The first one is the method we adopt in our AHIQ. For the second experiment, the features from transformer and ViT are simply concatenated together. The first method outperforms the second one by a large margin which demonstrates that using deformable convolution to modify CNN feature maps is well-effective. This further illustrates the power of global and semantic information in transformer to guide the shallow features by paying more attention to the salient regions.\n\nWe also conduct ablation studies on using features from ViT and from CNN separately. Results are at the last two rows in~\\cref{tab:fusion}. One can observe that only using one of the CNN and Transformer branches results in a dramatic decrease in performance. This experimental result shows that both global semantic information brought by ViT and local texture information introduced by CNN is very crucial in this task, which is well consistent with our previous claim.\n\\begin{figure}[th]\n\\centering\n\\includegraphics[scale=0.26]{optical_flow2.pdf}\n\\caption{The visualization of learned offsets from deformable convolution. For each case, the vector flow which displays the learned offsets and zoomed-in details are included.}\n\\label{fig:optical}\n\\end{figure}\n\n\\vspace{2pt}\n\\noindent\\textbf{Visualization of Learned Offset.} We visualize the learned offsets from deformable convolution in \\cref{fig:optical}. It can be observed that the learned offsets indicated by arrows mainly affect edges and salient regions. In addition, most of the offset vectors point from the background to the salient regions, which means that in the process of convolution, the sampling locations moves to the significant region by the learned offsets. This visualization results illustrate the argument we made earlier that semantic information from ViT help CNN see better by deformable convolution.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Comparison of different pooling strategy on the NTIRE 2022 IQA Challenge testing datasets. Note that ``Patch'' denotes the patch-wise prediction and ``Spatial'' denotes the spatial pooling.}\n\\begin{tabular}{cccc}\n\\toprule[1.2pt]\nPooling Strategy & PLCC & SROCC & Main Score \\\\ \\hline\nPatch & \\textbf{0.823} & \\textbf{0.813} & \\textbf{1.636} \\\\\nSpatial & 0.794 & 0.795 & 1.589 \\\\\nPatch + Spatial & 0.801 & 0.791 &  1.593\\\\ \\toprule[1.2pt]\n\\end{tabular}\n\\label{tab:pooling}\n\\end{table}\n\n\n\\vspace{2pt}\n\\noindent\\textbf{Pooling Strategy.} Experiments on different pooling strategies are conducted, and the results are shown in \\cref{tab:pooling}. We first perform patch-wise prediction, which is elaborated in \\cref{subsec:pooling}. For comparison, we follow WaDIQaM~\\cite{bosse2017deep} and IQMA~\\cite{guo2021iqma} to use spatial pooling that combines max-pooling and average-pooling in spatial dimension to obtain a score vector $S\\in \\mathbb{R}^{1\\times 1 \\times C}$. The final score is the weighted sum of the score vector and the final result is shown in the second row of \\cref{subsec:pooling}. Then we try to combine the previous two pooling method and propose to use the average of the output score from patch-wise prediction and spatial pooling in the third experiment. Patch-wise prediction module proposed in AHIQ performs better than the other two, and experimental results further prove the validity of the patch-wise prediction operation. It confirms our previous claim that different regions should contribute differently to the final score.\n\n\n\\subsection{NTIRE 2022 Perceptual IQA Challenge}\nThis work is proposed to participate in the NTIRE 2022 perceptual image quality assessment challenge~\\cite{gu2022ntire}, the objective of which is to propose an algorithm to estimate image quality consistent with human perception. The final results of the challenge in the testing phase are shown in~\\cref{tab:final}. Our ensemble approach won the first place in terms of PLCC, SROCC, and main score.\n\\begin{table}[ht]\n\\centering\n\\caption{The results of NTIRE 2022 challenge FR-IQA track on the testing dataset. This table only shows part of the participants and best scores are~\\textbf{bolded}.}\n\\begin{tabular}{cccc}\n\\toprule[1.2pt]\nMethod & PLCC & SROCC & Main Score \\\\ \\hline\nOurs\n& \\textbf{0.828} & \\textbf{0.822} & \\textbf{1.651} \\\\\n\\nth{2} \n& 0.827 & 0.815 & 1.642 \\\\\n\\nth{3} & 0.823 & 0.817 & 1.64 \\\\\n\\nth{4} & 0.775 & 0.766 & 1.541 \\\\\n\\nth{5} & 0.772 & 0.765 & 1.538 \\\\ \\toprule[1.2pt]\n\\end{tabular}\n\\label{tab:final}\n\\end{table}\n\n\n\\section{Conclusion}\nIn this paper, we propose a novel network called Attention-based Hybrid Image Quality Assessment Network (AHIQ), for the full-reference image quality assessment task. The proposed hybrid architecture takes advantage of the global semantic features captured by ViT and local detailed textures from a shallow CNN during feature extraction. To help CNN pay more attention to the salient region in the image, semantic information from ViT is adopted to guide deformable convolution so that model can better mimic how humans perceive image quality. Then we further propose a feature fusion module to combine different features. We also introduce a patch-wise prediction module to replace spatial pooling and preserve information in the spatial dimension. Experiments show that the proposed method not only outperforms the state-of-the-art methods on standard datasets, but also has a strong generalization ability on unseen samples and hard samples, especially GAN-based distortions. The ensembled version of our method ranked first place in the FR track of the NTIRE 2022 Perceptual Image Quality Assessment Challenge.\n\n\\vspace{2pt}\n\\noindent\\textbf{Acknowledgment.} This work was supported by the Key Program of the National Natural Science Foundation of China under Grant No. U1903213 and the Shenzhen Key Laboratory of Marine IntelliSense and Computation under Contract ZDSYS20200811142605016.\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nGraphene, a single atomic layer of carbon arranged in a honeycomb lattice, holds great promise for numerous applications due to its remarkable mechanical, optical, and electronic properties and serves as a powerful material platform for studying relativistic Dirac fermions due to its linearly dispersing bands\\cite{novoselov_two-dimensional_2005,geim_rise_2007, castro_neto_electronic_2009}. Graphene was also the first material in which a member of the striking class of macroscopic quantum phenomena\\cite{anderson_observation_1995,osheroff_evidence_1972,willett_observation_1987,bardeen_theory_1957} -- the quantum Hall effect (QHE)\\cite{klitzing_new_1980} -- could be observed at room temperature, when subject to large magnetic fields\\cite{novoselov_room-temperature_2007}. In the quantum Hall state, charge carriers are forced into cyclotron orbits with quantized radii and energies known as Landau levels (LLs), once subjected to the influence of a magnetic field. In order to observe this effect, certain conditions must be met: The magnetic field must be large enough that the resulting spacing between LLs is larger than the thermal energy $(\\Delta E_{LL}>k_BT)$; the charge carrier lifetime between scattering events must be longer than the characteristic time of the cyclotron orbit $(t_\\textrm{life}>1\/\\omega _c)$; and the magnetic field must be uniform on length scales greater than the LL orbit. This typically mandates the need for cryogenic temperatures, clean materials, and large applied magnetic fields. Dirac fermions in graphene provide a way to lift these restrictions: Under certain strain patterns, graphene's electrons behave as if they were under the influence of large magnetic fields, without applying an actual field from outside the material\\cite{guinea_energy_2010,levy_strain-induced_2010,liu_tailoring_2018,rechtsman_strain-induced_2013}. These so-called pseudomagnetic fields only couple to the relativistic electrons around the Dirac point and, under the QHE conditions above, lead to the formation of flat, quantized LLs. This has been successfully observed using a range of methods\\cite{levy_strain-induced_2010,liu_tailoring_2018,rechtsman_strain-induced_2013}, but was so far restricted to small regions, which severely limits its applicability.\n\n\\section*{Results}\n\n\\begin{figure}\n\\makebox[\\textwidth]{\\includegraphics[width=183mm]{F2_AFM_v8.png}}\n\\caption{\\textbf{Identification of strained nanoprisms.} (\\textbf{A}) Horizontal derivative AFM topography image of our monolayer graphene grown on a SiC substrate. Triangular nanoprisms are dispersed on the surface. Inset: AFM topography image of the same area. Substrate terrace steps are about $10\\  \\textrm{nm}$ in height. (\\textbf{B}) Top: Close-up AFM topography of the area indicated by the black box in (\\textbf{A}). Bottom: Line cut through the AFM data marked by the purple line in the close-up. (\\textbf{C}) Overview STM topography image (200 nm x 200 nm, $V_{sample}=100\\ \\textrm{mV}$, $I_{tun.}=2\\ \\textrm{pA}$) showing a single nanoprism. (\\textbf{D}) Schematic structure of 6H-SiC, showing its layered ABCACB stacking order with epitaxial graphene on top (yellow). Inside the nanoprism a single layer \\emph{within} the unit cell is missing, exposing the graphene to a different substrate surface termination, as illustrated in the top view. The carbon buffer layer is not shown for clarity. (\\textbf{E}) Atomically resolved STM images (10\\,nm x 10\\,nm, $V_{sample}=30\\ \\textrm{mV}$, $I_{tun.}=2\\ \\textrm{pA}$) inside (top) and outside (bottom) of the nanoprism. (\\textbf{F}) Difference map of the two Fourier transformed (FT) images in (\\textbf{E}) visualizing the strain pattern inside the nanoprism.}\n\\label{fig:f2_afm}\n\\end{figure}\n\nHere, we directly visualize the formation of flat LLs close to the Fermi energy induced by pseudomagnetic fields on wafer-scale semiconductor samples. By measuring the hallmark $\\sqrt{n}$ energy spacing and momentum dependence of the ensuing pseudo-LLs with angle-resolved photoemission spectroscopy (ARPES), and with the aid of model calculations, we confirm their quantum hall nature and extract a pseudomagnetic field strength of B = 41\\,T. This is made possible by the presence of a distribution of triangular nanoprisms underneath the monolayer graphene in our samples based on the well-established platform of epitaxial graphene on SiC substrates\\cite{emtsev_towards_2009,riedl_quasi-free-standing_2009,bostwick_observation_2010,zhou_substrate-induced_2007}, as revealed by a combination of atomic force microscopy (AFM) and scanning tunneling microscopy (STM) measurements.\\par\nOur topographic images of these samples (Fig.\\,\\ref{fig:f2_afm}A inset) exhibit the well-known terraces and step edges of graphene grown on 6H-SiC\\cite{emtsev_towards_2009}, which are due to a miscut of the wafers from the $(0001)$ direction of up to $0.1^\\circ$. A population of triangular-shaped nanoscale features are identified on the terraces of our samples (Fig.\\,\\ref{fig:f2_afm}A), that appear similar to those reported on similar substrates\\cite{bolen_graphene_2009,momeni_pakdehi_homogeneous_2019}. These nanoprisms appear during the growth process of graphene on 6H-SiC and are controllable by the Argon flow in the chamber\\cite{momeni_pakdehi_homogeneous_2019}. They cover between 5\\% and 10\\% of the terraces and are completely covered by monolayer graphene, the latter being demonstrated by our AFM adhesion images (see Supplementary Material Figs.\\,S8 and S9B). They are equilateral, have a narrow size distribution around $300\\,\\textrm{nm}$ side length, are oriented in the same direction, and are about $(2.7\\pm 0.7)\\, \\textrm{\\AA}$ deep (Fig.\\,\\ref{fig:f2_afm}B), which corresponds to a single missing SiC double layer or $\\frac{1}{6}$ of the 6H-SiC unit cell. This leads to a change in the registry between the silicon atoms in the top layer of the substrate and the graphene as illustrated in Fig.\\,\\ref{fig:f2_afm}D. The strain created inside the nanoprisms cannot be relieved, because the nanostructures are continuously covered by monolayer graphene without additional grain boundaries, as corroborated by our STM images across the edge (see Supplementary Material Fig.\\,S9A). To get a more detailed view of the strain pattern we perform additional detailed atomic resolution STM measurements. The images taken inside and outside the nanoprisms (Fig.\\,\\ref{fig:f2_afm}E) show the expected $(6\\sqrt{3}\\times6\\sqrt{3})\\text{R}30^\\circ$ modulation with respect to SiC on top of the carbon honeycomb lattice\\cite{riedl_structural_2010}. However, taking the difference of the two Fourier transformed images (Fig.\\,\\ref{fig:f2_afm}F) reveals a shear strain pattern inside the nanoprism, with a maximal observed strain of roughly $3^\\circ$.\n\n\\begin{figure}\n\\makebox[\\textwidth]{\\includegraphics[width=184mm]{F1_ARPES_v9.png}}\n\\caption{\\textbf{Momentum-resolved visualization of Landau levels.} (\\textbf{A}) ARPES cut through the Dirac cone at the K point at 300\\,K. The data have been divided by the Fermi function and symmetrized to compensate for matrix element effects\\cite{shirley_brillouin-zone-selection_1995}. (\\textbf{B}) Cut along the energy axis integrated around the K point in (\\textbf{A}). (\\textbf{C}) Second derivative of the data in (\\textbf{A})\\cite{zhang_precise_2011}. (\\textbf{D}) Inverted second derivative of the data shown in (\\textbf{B}) after smoothing. (\\textbf{A})--(\\textbf{D}) Landau levels (LLs) are indicated by arrows. (\\textbf{E}) Summary of LL data sets, with model fit according to Eqn.\\,\\ref{eq:landaulevels} shown in black; the 95\\% confidence interval of the fit is shown in grey. Different symbols indicate different samples and temperatures: sample A (6\\,K) [hexagons], sample B (6\\,K) [squares], sample B 2nd data set (6\\,K) [stars], sample B (300\\,K) [diamonds], sample C (6\\,K) [circles], and sample C 2nd data set (6\\,K) [triangles]. ARPES data for the additional samples can be found in the supplementary Fig.\\,S7. Inset: Same data plotted versus $\\sqrt{n}$, giving the expected linear behaviour for LLs in a Dirac material. (\\textbf{F}) Sketch of various mechanisms which may lead to ARPES intensity inside the cone. Neither electron-phonon coupling nor contamination from bilayer graphene can explain the experimental findings.}\n\\label{fig:f1_arpes}\n\\end{figure}\n\nIn order to confirm if the induced strain pattern indeed leads to flat Landau levels close to the Fermi energy, we perform a series of high-resolution ARPES measurements. ARPES is a momentum- and energy-resolved technique that has proven to be a powerful tool in directly studying the electronic band structures of a vast variety of quantum phases of matter, from strongly-correlated electron systems and high-T$_{c}$ superconductors\\cite{damascelli_probing_2004} to topological insulators and semimetals\\cite{hsieh_topological_2008,chen_experimental_2009,lv_observation_2015}. Yet no study of quantum Hall states has been performed, since ARPES is strictly incompatible with the application of magnetic fields, as essential crystal momentum information carried by the photoemitted electrons would be lost through interaction with the field. This however is different for pseudomagnetic fields, as they only interact with the Dirac electrons inside the material. We note that, while a recently developed momentum-resolved technique amenable to magnetic fields has been reported\\cite{jang_full_2017}, it necessarily requires sophisticated heterostructures, physically accessible fields, and is limited to a small sector of the Brillouin zone.\n\nOur ARPES data, which -- due to the $\\sim$1\\,mm spot size of the photon source -- correspond to the spatial average over unstrained and strained regions of the sample, show the expected Dirac cone as well as new flat bands that gradually merge with the linear dispersion (Figs.\\,\\ref{fig:f1_arpes}A and \\ref{fig:f1_arpes}C). The unequal energy spacing of these newly observed bands can be extracted from cuts along the energy direction at the K point (Fig.\\,\\ref{fig:f1_arpes}B) and their second derivative (Fig.\\,\\ref{fig:f1_arpes}D). By plotting the positions of these bands (Fig.\\,\\ref{fig:f1_arpes}E), we observe the distinct $\\sqrt{n}$ energy spacing which is a hallmark of LLs for graphene's massless Dirac charge carriers\\cite{geim_rise_2007}, where $n$ is the integer LL index. The spectrum of LLs in graphene is given by\\cite{castro_neto_electronic_2009}\n\\begin{equation}\n    E_n=\\text{sgn}(n)\\sqrt{2v_F^2\\hbar eB\\cdot |n|}+E_{DP}\n    \\label{eq:landaulevels}\n\\end{equation}  \nwhere $v_F$ is the velocity of the electrons at the Fermi level, $\\hbar$ the reduced Planck constant, $e$ the electron charge, $B$ the magnitude of the (pseudo-)magnetic field, and $E_{DP}$ the binding energy of the Dirac point. Using the ARPES dispersion map in Fig.\\,\\ref{fig:f1_arpes}A, the Fermi velocity is determined to be $v_F=(9.50\\pm0.08)\\times10^5\\ \\textrm{ms}^{-1}$ (see Supplementary Material Fig.\\,S1). Fitting our experimental data to Eqn.\\,\\ref{eq:landaulevels} as done in Fig.\\,\\ref{fig:f1_arpes}E, we extract the magnitude of the pseudomagnetic field, which yields $B=(41\\pm 2)\\ \\textrm{T}$. Remarkably, this pseudomagnetic field value is consistent between several samples from cryogenic temperatures (6\\,K) up to room temperature. The model fit also consistently pinpoints the binding energy of the Dirac point to $E_{DP}=(460\\pm 10)\\ \\textrm{meV}$ relative to the Fermi level, which agrees well with previous reports on this sample system\\cite{bostwick_quasiparticle_2007,emtsev_towards_2009} and is attributed to charge transfer from the SiC substrate to the graphene layer. Additionally, the LLs are only resolved in the upper part of the Dirac cone, closer to the Fermi level\\cite{levy_strain-induced_2010, song_high-resolution_2010}. We attribute this effect to the increased scattering phase space as one moves away from the Fermi level, which manifests itself in our ARPES data by an increased line width of the bands (Supplementary Material Fig.\\,S1).\n\nAs for other alternative explanations of the data, we note that while previous ARPES studies of graphene on SiC have shown a rich variety of features\\cite{ludbrook_evidence_2015,ohta_controlling_2006}, the signature $\\sqrt{n}$ spacing of the levels (Fig.\\,\\ref{fig:f1_arpes}E and inset) allows us to unambiguously distinguish the observed effect from other possibilities (Fig.\\,\\ref{fig:f1_arpes}F). For example, if spectral weight inside the Dirac cone arose from coupling of electrons to phonons\\cite{ludbrook_evidence_2015}, it would be limited to characteristic vibrational energies. Similarly, contributions from bilayer and higher order graphene layers, which can appear in small quantities near step edges of the substrate during the growth process\\cite{ohta_controlling_2006} (see also AFM adhesion image in the Supplementary Material Fig.\\,S9B), would lead to a manifold of bands, but would not reproduce the observed band structure\\cite{ohta_interlayer_2007,marchenko_extremely_2018}. Furthermore, previously reported plasmaronic interactions in samples with higher electronic doping \\cite{bostwick_observation_2010} can also be excluded. They lead to renormalizations of electronic bands around the Dirac point, but show a distinctly different spectrum than what is observed in our experiments. Finally, the effects of different defect geometries in graphene and their influence on the Dirac cone dispersion have recently been discussed \\cite{kot_band_2018}, but do not lead to flat bands around the Dirac point.\n\n\\begin{figure}\n\\makebox[\\textwidth]{\\includegraphics[width=183mm]{F3_THEORY_v8.png}}\n\\caption{\\textbf{Model calculation of strain-induced Landau levels.} (\\textbf{A}) Top: Honeycomb lattice, with the two sublattices A (red) and B (yellow). The black arrows indicate the symmetry of the strain pattern. Bottom: Triangular flake with strain-induced pseudomagnetic field $B = 41$\\,T. The colour scale indicates the relative bond stretching. (\\textbf{B}) Spectral function for the gapless case with Semenoff mass $M = 0\\ \\textrm{meV}$. (\\textbf{C}) Energy cut through the Dirac point (K) of the spectral function in (\\textbf{B}). The dashed grey lines indicate the position of the Landau levels (LL) predicted by Eqn.\\,\\ref{eq:landaulevels}. (\\textbf{D}) Spectral function averaged over a uniform distribution of Semenoff masses $M \\in [-135, 135]$\\,meV. (\\textbf{E}) Energy cut through the Dirac point (K) of the spectral function in (\\textbf{D}). The shaded grey area indicates the broadening of the Landau levels predicted by Eqns.\\,\\ref{eq:landaulevels} and \\ref{eq:landaulevels_mass}.}\n\\label{fig:f3_theory}\n\\end{figure}\n\nTo gain deeper insights on the origin of the observed LLs, we model a region of graphene experiencing a uniform strain-induced pseudomagnetic field. We use the simplest such strain pattern, worked out by Guinea et al.\\cite{guinea_energy_2010}, which exhibits the triangular symmetry of the underlying honeycomb lattice. Using a tight-binding approach, we directly simulate a finite-size strained region with open boundary conditions and armchair edges (further details in Methods section). We find that the observed LL spectra can be well reproduced by a triangular flake of side length $L = 56$\\,nm (Fig.\\,\\ref{fig:f3_theory}A), subject to a uniform pseudomagnetic field $B = 41$\\,T over the entire flake (Fig.\\,\\ref{fig:f3_theory}A). The maximal strain (or relative bond stretching) reaches around $3\\%$, which is in good agreement with our STM measurements. The ARPES data can be simulated by calculating the energy and momentum-resolved spectral function $A(\\bm{k},\\omega)$ of this triangular flake, here shown in Figs.\\,\\ref{fig:f3_theory}B and \\ref{fig:f3_theory}C. Our simulation clearly reproduces the main features of the ARPES data, namely levels that: (i) follow $\\sqrt{n}$ spacing in energy; (ii) are flat \\emph{inside} the Dirac cone and merge with the linearly dispersing bands; (iii) become less clearly resolved with increasing index $n$.\n\nFeatures (ii) and (iii) can be understood by comparing the characteristic size of a Landau orbit $\\propto\\sqrt{n}\\,l_B$ (with the magnetic length $l_B=\\sqrt{\\frac{\\hbar}{eB}}$) to the length scale $\\lambda$ on which the pseudomagnetic field is uniform. For LLs to exist, an electron on a given Landau orbit must experience a uniform pseudomagnetic field\\cite{settnes_pseudomagnetic_2016}, leading to the condition $\\sqrt{n}\\,l_B \\ll \\lambda$. Hence, for large fields $B$ or large $\\lambda$, flat bands are expected across the entire Brillouin zone, whereas Dirac cones are recovered in the opposite limit (see Supplementary Material Fig.\\,S2). The bands observed in the ARPES data can thus be understood as LLs, where the orbit size is only somewhat smaller than $\\lambda$: by comparing the experimental data and the model calculation, we estimate $l_B \\sim 4$~nm and $\\lambda \\sim 30$\\,nm (see Methods section). Furthermore, since the size of Landau orbits grows as $ \\sim \\sqrt{|n|}$, eventually it becomes comparable to $\\lambda$, explaining why levels with higher index $n$ are less clearly resolved.\n\nHowever, our simple model (Figs.\\,\\ref{fig:f3_theory}B and \\ref{fig:f3_theory}C) consistently exhibits a sharp zeroth Landau level (LL0), which is absent in the ARPES data. This discrepancy is surprising, since LL0 is known to be stable against inhomogeneities of the magnetic field as well as against disorder, as long as the latter preserves the chiral symmetry of graphene\\cite{aharonov_ground_1979}. Below, we provide a possible mechanism that broadens LL0 without substantially affecting the higher LLs. It has been argued that graphene grown on SiC is subject to a sublattice-symmetry-breaking potential arising from the interaction with the substrate\\cite{zhou_substrate-induced_2007}. The minimal theoretical model describing this effect, which acts as a staggered potential between sublattices A and B, is the so-called Semenoff mass $M$\\cite{semenoff_condensed-matter_1984}. This mass term opens a gap at the Dirac point and shifts the LL spectrum for $n\\neq0$ to (for a more detailed discussion, and the particular case of $n=0$, see Supplementary Material)\\cite{hunt_massive_2013}:\n\\begin{equation}\n    E_n = \\text{sgn}(n) \\sqrt{ 2 v_F^2  \\hbar e B\\cdot|n| + M^2}+E_{DP}.\n    \\label{eq:landaulevels_mass}\n\\end{equation}\nHowever, a \\emph{uniform} mass term $M$ cannot explain the ARPES data. Indeed, a fit of the observed LL spectrum to Eqn.\\,\\ref{eq:landaulevels_mass} returns $M=(150\\pm 5)\\ \\textrm{meV}$, but places the Dirac point at an unrealistic binding energy of $E_{DP}=390\\ \\textrm{meV}$ (see Supplementary Material Fig.\\,S3 and Fig.\\,S6). Therefore, we postulate that the mass term $M$ varies on a length scale much greater than the magnetic length $l_B \\sim$\\,4\\,nm, but smaller than the ARPES spot size ($\\sim$\\,1\\,mm). In that situation, our ARPES measurements would simply average over spectral functions described by different mass terms.  This is shown in Figs.\\,\\ref{fig:f3_theory}D and \\ref{fig:f3_theory}E for a uniform distribution in the interval $M \\in [-135, 135]$\\,meV. As evident from Eqn.\\,\\ref{eq:landaulevels_mass}, the distribution of mass terms affects LL0 most, while merely contributing an additional broadening to the higher levels. Note that, as observed experimentally, the variation of the mass term is not limited to the strained areas, but instead is a property of the whole sample; as a result, ARPES always picks up a spatial average of strained areas with LLs and unstrained areas with the usual Dirac cone dispersion, both having the same distribution of mass terms and corresponding Dirac point gaps.  This phenomenological model is in good agreement with the experimental data and may renew interest in the variation of the mass term in this sample system\\cite{zhou_substrate-induced_2007}.\n\n\\section*{Discussion}\n\nThis study provides the first demonstration of the room temperature strain-induced quantum Hall effect in graphene on a wafer-scale platform, as well as the first direct momentum-space visualization of graphene electrons in the strain-induced quantum Hall phase by ARPES, whereby the linear Dirac dispersion collapses into a ladder of quantized LLs. This opens a path for future momentum-resolved studies of strain-induced, room temperature-stable topological phases in a range of materials including Dirac and Weyl semimetals\\cite{pikulin_chiral_2016,cortijo_elastic_2015,liu_quantum_2017}, monolayer transition metal dichalcogenides\\cite{rostami_theory_2015}, and even nodal superconductors\\cite{massarelli_pseudo-landau_2017,nica_landau_2018}, all under large, potentially controllable pseudomagnetic fields. Importantly, these systems will feature time reversal invariant ground states -- otherwise impossible with a true magnetic field -- and may act as future building blocks for pseudo spin- or valley-tronic based technologies\\cite{low_strain-induced_2010}. In light of the recently discovered unconventional superconductivity in 'magic angle' bilayer graphene\\cite{cao_correlated_2018,cao_unconventional_2018}, strain-induced pseudomagnetic fields likewise raise the possibility of engineering exotic variants of correlated states including superconductivity in LLs\\cite{uchoa_superconducting_2013} and fractional topological phases\\cite{ghaemi_fractional_2012}. Our results lay the foundations for bottom-up strain-engineering of novel quantum phases at room temperature and on a technologically relevant wafer-scale platform.\n\n\\section*{Materials and Methods}\n\n\\subsection*{Sample growth and characterization}\n\nGraphene samples with a carbon buffer layer were epitaxially grown on commercial 6H-SiC substrates. The substrates were hydrogen-etched prior to the growth under argon atmosphere. Details are described by \\emph{S. Forti and U. Starke}\\cite{forti_epitaxial_2014}. AFM characterization measurements were taken at the Max Planck Institute in Stuttgart. Adhesion images correspond to the force necessary to retract the tip from the sample. Adhesion is sensitive to the graphene coverage on the sample and can thus distinguish between zero layer, monolayer and bilayer graphene with sensitivity to grain boundaries.\n\n\\subsection*{ARPES measurements}\n\nExperiments were performed at UBC in a ultra-high vacuum chamber equipped with a SPECS Phoibos 150 analyzer with $\\Delta E=6\\,\\textrm{meV}$ and $\\Delta k=0.01\\,\\textrm{\\AA}$ optimum energy and momentum resolutions, respectively, at a base pressure of better than $p=7\\times10^{-11}\\,\\textrm{Torr}$. Photons with an energy of $21.2$\\,eV were provided by a SPECS UVS300 monochromatized gas discharge lamp. Our homebuilt six-axis cryogenic manipulator allows for measurements between 300\\,K and 3.5\\,K. Additional data sets were taken at UBC with a second ARPES setup equipped with a Scienta R4000 analyzer and a Scienta VUV5000 UV source with $\\Delta E=1.5\\,\\textrm{meV}$ and $\\Delta k=0.01\\,\\textrm{\\AA}^{-1}$ optimum energy and momentum resolutions, respectively, for $21.2\\ \\textrm{eV}$ photons. The samples were annealed at 600$^\\circ$C for about 2\\,h at $p=1\\times10^{-9}\\,\\textrm{Torr}$ and then at 500$^\\circ$C for about 10\\,h at $p=5\\times10^{-10}\\,\\textrm{Torr}$ immediately before the ARPES measurements.\n\n\\subsection*{STM measurements}\n\nExperiments were performed at UBC under ultra-high vacuum conditions ($<5\\times10^{-12}$\\,mbar) using a low-temperature scanning tunnelling microscope (Scienta Omicron) at liquid helium temperatures ($\\sim$\\,4.2\\,K). All images were acquired in constant-current mode using a cut\\linebreak platinum-iridium tip, which was conditioned by voltage pulsing and gentle indentation into a Ag(111) crystal. The samples were annealed at 550$^\\circ$C overnight with a final pressure of $p=3\\times10^{-10}\\,\\textrm{mbar}$ \\emph{in situ} prior to the STM measurements.\n\n\\subsection*{Model calculation}\n\nWe considered a minimal tight-binding model on the honeycomb lattice with nearest-neighbour hoppings and a sublattice-symmetry breaking Semenoff\\cite{semenoff_condensed-matter_1984} mass term $M$:\n\\begin{equation}\nH = -t \\sum_{<\\bm{r}, \\bm{r'}>} \\left( c_A^\\dagger(\\bm{r)} c_B(\\bm{r'}) + \\text{H.c.} \\right) + M \\left( \\sum_{\\bm{r}} c_A^\\dagger(\\bm{r}) c_A(\\bm{r}) - \\sum_{\\bm{r'}}  c_B^\\dagger(\\bm{r'}) c_B(\\bm{r'}) \\right)\n\\label{eq:tight_binding}\n\\end{equation}\nwhere $c^\\dagger_A(\\bm{r})$ ($c^\\dagger_B(\\bm{r'})$) creates an electron in the $p_z$ orbital at lattice site $\\bm{r}$ ($\\bm{r'}$) on the sublattice A (B) of the honeycomb lattice, $t = 2.7\\ \\text{eV}$ and the nearest-neighbour distance is $a_0 =0.142$\\,nm. We neglected the electron spin, and thus considered effectively spinless fermions.\n\nWe constructed a flake in the shape of an equilateral triangle of side length $L \\sim 56$\\,nm. The use of armchair edges avoids the zero-energy edge modes appearing for zigzag edges\\cite{castro_neto_electronic_2009}. We applied the simplest strain pattern respecting the triangular symmetry of the problem at hand, namely, the pattern introduced by \\textit{Guinea et al.}\\cite{guinea_energy_2010} which gives rise to a uniform (out-of-plane) pseudomagnetic field\n\\begin{equation}\n    \\bm{B} = 4 u_0 \\frac{\\hbar \\beta}{e a_0} \\hat{\\mathbf{z}}\n\\end{equation}\nwhere $\\beta \\approx 3.37$ in graphene\\cite{settnes_pseudomagnetic_2016}, and the corresponding displacement field is given by\n\\begin{equation}\n    \\mathbf{u}(r,\\theta) = \n    \\begin{pmatrix}\n    u_r \\\\\n    u_\\theta\n    \\end{pmatrix} \n    =\n    \\begin{pmatrix}\n    u_0 r^2 \\sin(3\\theta) \\\\\n    u_0 r^2 \\cos(3\\theta)\n    \\end{pmatrix}.\n\\end{equation}\nThe hopping parameter renormalization induced by this displacement field is calculated using the simple prescription:\n\\begin{align}\nt \\rightarrow t_{ij} = t \\exp \\left[ -\\frac{\\beta}{a_0^2} \\left( \\epsilon_{xx} x_{ij}^2 + \\epsilon_{yy} y_{ij}^2 + 2 \\epsilon_{xy} x_{ij} y_{ij} \\right) \\right]\n\\label{eq_hoppings_strain}\n\\end{align}\nwhere $(x_{ij}, y_{ij}) \\equiv \\bm{r}_i - \\bm{r}_j$ is the vector joining the original (unstrained) sites $i$ and $j$, and\n\\begin{equation}\n    \\epsilon_{ij} = \\frac{1}{2} \\left[ \\partial_j u_i + \\partial_i u_j \\right]\n\\end{equation}\nis the strain tensor corresponding to the (in-plane) displacement field $\\mathbf{u}$. Outside the strained region (which we take as a triangle of slightly smaller length $L_S \\sim 48$\\,nm), we allowed the strain tensor to relax: $\\bm{\\epsilon} \\rightarrow e^{-\\frac{r^2}{2\\sigma^2}} \\bm{\\epsilon}$, where $r$ is the perpendicular distance to the boundary of the strained region, and $\\sigma \\sim 1$\\,nm. We defined the length scale of the homogeneous magnetic field $\\bm{B}$ to be the diameter of the largest inscribed circle in the triangle of side $L_S$: $\\lambda \\equiv L_S\/\\sqrt{3} \\sim 28$\\,nm. We stress here that our simulated flakes are much smaller than the experimentally observed triangular features of size $\\sim 300$\\,nm. The fact that we nevertheless reproduce the experimental features underlines how the number of observable LLs is limited by the length scale of the homogeneous pseudomagnetic field $\\lambda$, rather than by the size $L$ of the nanoprisms themselves. This length scale could be caused by the more complicated strain pattern present in the nanoprisms or be induced by disorder.\n\nWe then diagonalized the Hamiltonian (Eqn.\\,\\ref{eq:tight_binding}) with hopping parameters given by Eqn.\\,\\ref{eq_hoppings_strain} to obtain the full set of eigenstates $\\ket{n}$ with energies $E_n$, and computed the momentum-resolved, retarded Green's function using the Lehman representation\n\\begin{align}\n    G_\\alpha^R(\\bm{k},\\omega) &= \\sum_n \\frac{ | \\bra n c^\\dagger_\\alpha({\\bm{k}}) \\ket{0} |^2}{\\omega - (E_n - E_0) -i \\eta}\n\\end{align}\nwhere $\\alpha = A,B$ is a sublattice (band) index, and $\\eta \\sim 20$\\,meV is a small broadening parameter comparable to the experimental resolution. \nWe then compute the one-particle spectral function,\n\\begin{align}\nA(\\bm{k}, \\omega) = -\\frac{1}{\\pi} \\sum_\\alpha \\text{Im}\\left[ G_\\alpha^R(\\bm{k}, \\omega)\\right]    \n\\end{align}\nwhich is proportional to the intensity measured in ARPES (modulo the Fermi-Dirac distribution and dipole matrix elements). We note that using a finite system introduces two main effects in the momentum-resolved spectral function: the appearance of a small finite-size gap at the Dirac points (in the absence of a magnetic field) and a momentum broadening of the bands.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nUnderstanding the origin of the Ultra High Energy Cosmic Rays (UHECR) with energy above $\\sim 5 \\; 10^{19}$ eV\nconstitute a real challenge for theoretical models, since their acceleration requires extreme conditions hardly fulfilled by known astrophysical objects.\nPrevious studies considered a number of potential sources, including gamma ray bursts, active galactic nuclei, large scale jets and neutron stars, but results are still inconclusive \\cite{reviews}.  \n\nIf, as many believe, UHECRs have an extragalactic origin, a suppression, known as the Greisen-Zatsepin-Kuzmin (GZK) cutoff, should be observed in the spectrum at an energy\n$\\sim 7 \\; 10^{19} {\\rm eV}$\n\\cite{GZK}.\nThis is because during propagation energetic protons undergo $p\\gamma$ interactions (mainly photo-pion production) in the\nCosmic Microwave Background (CMB)\nwith an energy loss length which decreases dramatically from $\\sim 650$ to $\\sim 20 \\, {\\rm Mpc}$ between energies of $7 \\; 10^{19}$ to $3 \\; 10^{20} {\\rm eV}$ \\cite{pgamma}.\nThe detection of a number of events above $7 \\, 10^{19}$ eV\nraised interest in non-acceleration\nmodels\nthat\ninvolve new physics and do not require the existence of a cutoff in the spectrum \\cite{reviews}.\nHowever, the issue of the presence\nof the GZK feature\nis still debated \\cite{daniel}.\nIn the following we assume that astrophysical objects capable of accelerating UHECRs\ndo exist and we discuss possible ways to identify them.\n\nIf UHECRs are\nnot significantly \ndeflected by the\nintergalactic magnetic field (IGMF)\ntheir arrival direction should point back to the position of their accelerators.\nIn principle, this fact can\nbe used to identify accelerators located at distances smaller than the proton loss length, that constitutes a sort of horizon for CRs. However, even in this case, the small event statistics and the high uncertainty\nin the determination of the CR arrival direction\nwould make any identification problematic. \nThe better angular resolution and\ncollection area of AUGER will hopefully provide \nadequate data to address this issue.\n\n\n\nThe interactions between UHECRs and CMB photons generate secondary gamma rays and electron-positron pairs (hereafter referred to as {\\it electrons}) that in turn initiate an electromagnetic (EM) cascade in the universal photon background \\cite{cascade}.\nSuch a cascade would appear to a distant observer as a flux of GeV\/TeV photons.\nRecently, Ferrigno et al. \\cite{carlo} suggested to\nidentify the sources of UHECRs\nby searching for this radiation.\nTheir calculations show that, in an unmagnetized Universe ($B_{IGMF} = 0$ G), a steady source emitting isotropically $2 \\; 10^{43}$ erg\/s in form of UHECRs ($E > 10^{19}$ eV) can be detected by a Cherenkov telescope like HESS up to a distance of $\\sim 100$ Mpc.\nIn this\ncase the source would be point-like, because in the absence of IGMF the EM cascade is one-dimensional and propagates radially away from the accelerator. \nUnfortunately, the scenario changes dramatically if\nthe more realistic case of a\nmagnetized Universe is considered. This is because low energy electrons produced during the last steps of the cascade are effectively deflected and eventually isotropized if their Larmor radius is smaller or comparable with the Compton cooling length. This condition is satisfied when the IGMF is above\n$\nB_{iso} \\sim 10^{-12} (E_{\\gamma}\/TeV) {\\rm G}\n$,\n$E_{\\gamma}$ being the energy of the Compton photon. This implies that \\textit{unless the IGMF is extremely weak} ($<< B_{iso}$), \nelectrons emit Compton photons after being fully isotropized. Thus, \\textit{an extended halo of emitting pairs forms around UHECR sources}. The radiation from the halo is emitted isotropically, resulting in a \\textit{very extended, and thus hard to be detected, gamma ray source} \\cite{cascade}.\n\nHowever, if the IGMF close to the accelerator is at the ${\\rm nG}$ level, first generation electrons ($E \\sim 10^{19}$ eV) generated during $p\\gamma$ interactions cool rapidly by emitting ${\\rm GeV}$ synchrotron photons and the development of the cascade is strongly inhibited. \nWe propose the possibility to detect these synchrotron photons from UHECR sources \\cite{noi}.  \nThis is of great interest because of the following reasons.\nBoth synchrotron emitting electrons and parent protons are extremely energetic and not appreciably deflected by the IGMF, at least on the first Mpcs distance scale.\nFor this reason, \\textit{synchrotron photons are emitted in the same direction of parent protons}. Thus, they move away from the source almost radially, and \\textit{the observed radiation is expected to be point-like}, and thus easily detectable and distinguishable from the extended cascade component.\nRemarkably, a detection of these sources would allow to infer the value of the IGMF close to the accelerator, constraining it in the range $\\approx 10^{-10} \\div 10^{-8}$ G.\nFinally, since the Universe is transparent to ${\\rm GeV}$ photons, powerful UHECR accelerators located outside the CR horizon might be identified in this way.  \n\n\\section{Development of the electromagnetic cascade}\n\nIn this section we describe the development of the EM cascade initiated by a UHECR.\nConsider a proton with energy $E_{p,20} = E_p\/10^{20}$ eV. The typical energies of photons and electrons produced in $p\\gamma$ interactions are $\\sim 10^{19} E_{p,20}$ and $\\sim 5 \\; 10^{18} E_{p,20}$ eV respectively \\cite{pgamma}.\nIn the absence of IGMF, such electrons and photons interact via Compton and pair production processes with photons in the CMB and radio background.\nIn general, this would lead to the development of an EM cascade, in which the number of electrons and photons increases rapidly.\nIn fact, due to the extremely high energy of the particles considered here, each interaction occurs in the limit $\\Gamma = \\epsilon_b E \\gg 1$, where $\\epsilon_b$ and $E$ are the energies of the background photon and of the energetic electron (photon) respectively, both calculated in units of the electron rest mass energy.\nUnder this condition the Compton scattering is in the extreme Klein-Nishina limit, namely, the upscattered photon carries away most of the energy of the incoming electron.\nThe same happens during a pair production event, in which most of the energy goes to one of the two outgoing electrons.\nTherefore the problem reduces essentially to a single-particle problem, in which a leading particle loses continuously energy and changes state from electron to photon and back\ndue to alternate Compton\/pair production interactions.\nThus, the effective loss length of an energetic electron can be identified with the loss length of the leading particle \\cite{gould}. \nWhen the leading particle loses its energy until $\\Gamma \\approx 1$,\nthe cascade enters the particle multiplication phase, in which the particle energy is roughly divided in half in every collision. This phase ends up with a large number of low energy electrons and photons.\nOn the other hand, if a IGMF is present, electrons also lose energy via synchrotron emission, subtracting energy to the cascade.\nIn Fig.~\\ref{eloss} we show the effective electron loss length for Compton\/pair production (solid), together with the synchrotron loss length for a IGMF equal to $0.1$, $1$ and $10 ~ {\\rm nG}$ (dashed lines).\nIt can be seen that if the IGMF is at the level of 1 nG or more,\nall the electrons \nwith energy above $\\sim 10^{18}$ eV cool fast via synchrotron losses and the development of the cascade is strongly suppressed.\n\nSummarizing, three different regimes, corresponding to different values of the IGMF, can be distinguished:\n\n\\begin{itemize}\n\\item[$\\bullet$]{{\\bf Regime I:} $B_{IGMF} \\ll B_{iso} \\sim 10^{-12}$ G. The EM cascade is not affected at all by the\nIGMF.}\n\\item[$\\bullet$]{{\\bf Regime II:} $B_{iso} \\le B_{IGMF} \\ll B_{syn} \\sim 10^{-9}$ G. No energy is subtracted to the EM cascade due to synchrotron losses, but low energy electrons are effectively isotropized by the IGMF.}\n\\item[$\\bullet$]{{\\bf Regime III:} $B_{IGMF} \\ge B_{syn}$. The development of the EM cascade is strongly suppressed since its very first steps due to strong synchrotron losses.}\n\\end{itemize}\n\n\\section{Regime I: one-dimensional cascade}\n\nIf the IGMF strength is much less than $B_{iso} \\sim 10^{-12}$ G, electrons in the cascade do not suffer synchrotron losses, nor are they deflected. Thus, the EM cascade develops along a straight line. In this case, the calculations by Ferrigno et al. apply, and nearby and powerful UHECR sources might be detected as point like TeV sources by currently operational Cherenkov telescopes \\cite{carlo}.\n\nIn principle, since the strength of the large scale IGMF is basically unknown \\cite{Bfield}, such a low values of the field cannot be ruled out. However, this is probably not a good assumption in the vicinity of UHECR accelerators, where the IGMF is expected to be appreciable, especially if such accelerators are, as it seems reasonable to believe, correlated with the structures in the Universe. \nThe cascade might still be one-dimensional if its last steps develop sufficiently far away from the source, in a region of very low IGMF. Another necessary condition is that the IGMF close to the source must be small enough ($\\ll 10^{-9}$ G) to avoid a suppression of the cascade due to synchrotron losses of first generation electrons. \n\n\\section{Regime II: extended pair halos}\n\nIf the IGMF is strong enough to deflect the electrons in the cascade, but not enough to make synchrotron losses relevant (namely, $10^{-12} {\\rm G} \\le B_{IGMF} \\ll 10^{-9}$ G), then the EM cascade fully develops, low energy electrons are isotropized, and a very extended pair halo forms around the UHECR source. For an isotropic source, the size of the halo can be roughly estimated as follows. Let $E_{\\gamma}^{obs}$ be the energy of the gamma ray photons observed from the Earth. Such photons are CMB photons Compton-upscattered by electrons with energy $E_e \\sim 20 (E_{\\gamma}^{obs}\/{\\rm TeV})^{1\/2}$ TeV. These are the electrons forming the pair halo. \nSince electrons are rapidly isotropized in the IGMF, one can assume that they do not propagate away from the sites in which they are created.\nElectrons in the halo are in turn produced by parent photons with energy $E_{\\gamma}^{par} \\lesssim E_e$. \nSince the photon mean free path against pair production in the infrared background $\\lambda_{pp}$ decreases rapidly with increasing energy \\cite{IRabs}, we can safely neglect the contribution to the halo size from older generation (higher energy) photons.  \nThus, the size of the halo \ncan be roughly estimated as $l_{halo} \\sim \\lambda_{pp}(E_{\\gamma}^{par})$ \\cite{cascade}.\nFor a $\\sim 20$ TeV photon the mean free path is about a few tens of megaparsecs (see Fig. 2 in \\cite{felixICRC}). In fact, for the situation considered here, the size of the halo is even bigger, since the UHECR protons and the first generation electrons\npropagate $\\sim 10 \\div 20$ Mpc before\ninitiating the EM cascade (see Fig. 2).\n    Thus, a conservative estimate of the apparent angular size of the halo at 1 TeV can be given by: $\\vartheta \\sim (l_{halo}\/D) \\approx 10^o (l_{halo}\/20Mpc) (D\/100Mpc)^{-1}$, where $D$ is the distance of the source.\nThis indicates that \\textit{pair halos are extremely extended}, bigger than the field of view of Cherenkov telescopes (the HESS field of view is $\\sim 5^o$) \\textit{and thus hardly detectable}. The problem becomes even worse if one considers also protons with energy below $10^{20}$ eV, which have much longer loss length, increasing up to $\\sim 1$ Gpc for proton energies equal to $\\sim 5 \\; 10^{19}$ eV. On the other hand, such protons are likely to contribute only to the TeV flux of very distant sources.\n\nIn the recent work by Armengaud et al. \\cite{sigl} the deflection of electrons in the IGMF has been neglected, even when a IGMF stronger than $B_{iso} \\sim 10^{-12}$ G was assumed. On the other hand, the authors considered the deflection of $\\sim 10^{20}$ eV protons, which is in fact totally negligible if compared with the full isotropization of electrons. \nTherefore their claim\n about the detectability of cascade gamma-rays \nseems to us over-optimistic.\n\nThe cascade emission peaks at TeV energies \\cite{carlo}, making a detection by GLAST problematic.\n\n\\section{Regime III: synchrotron gamma rays}\n\nIf the IGMF close to the UHECR accelerator is at the level of 1 nG or above, the development of the cascade is strongly suppressed, since the very high energy electrons produced during $p\\gamma$ interactions cool rapidly via synchrotron losses before undergoing Compton scattering. It is evident from Fig. 2 that for a $\\sim nG$ IGMF this is true for electron energies well in excess of $E_e \\sim 10^{18}$ eV. Such electrons emit synchrotron photons with energy \n$E_{syn} \\approx 2 \\, (B\/{\\rm nG}) (E\/10^{19}{\\rm eV})^2 {\\rm GeV}$, detectable by GLAST.\nIt is important to stress that our results are sensitive only to the value of the IGMF {\\it close} to the source, while they are unaffected by the value of the field on much larger scales. \nThis is because synchrotron emitting electrons are produced within a proton interaction length $l_{p\\gamma} \\approx 10$ Mpc from the accelerator.\nAs a consequence, the only assumption required\nis that the size of the magnetized region surrounding the accelerator must be greater or comparable with $l_{p\\gamma}$\nSuperclusters of galaxies constitute an example of large and magnetized regions satisfying our requirement \\cite{Bfield}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth]{gabiciposter_fig1.ps}\n\\caption{\\label{eloss}Effective electron loss length. Solid: Compton\/pair-production losses in the CMB and radio background (with a cutoff at $2 {\\rm MHz}$ \\cite{radiobackground}). Dashed: synchrotron losses.\n}\n\\end{figure}\n\n\nWe now estimate the angular size of the synchrotron emission.\nAfter propagating over an interaction length, a proton of energy $E_p$ is deflected by an angle\n$\\vartheta_p \\approx 0.8^o (10^{20}{\\rm eV}\/E_p) (B\/{\\rm nG}) \\sqrt{(l_{p\\gamma}\/10{\\rm Mpc})} \\sqrt{(l_c\/{\\rm Mpc})}$  \nwhere $l_c$ is the IGMF coherence length \\cite{deflection}.\nDue to the high energies considered, secondary electrons produced in $p\\gamma$ interactions move in the same direction of the parent protons.\nIn a cooling time electrons are deflected by an angle $\\vartheta_e \\sim \\alpha \\lambda \/ R_L$,\n$R_L$ being the electron Larmor radius and $\\alpha$ a number of order unity representing the probability that the leading particle is actually an electron \\cite{gould}.\nRemarkably, if expressed as a function of the synchrotron photon energy, \nthe deflection angle is independent on the magnetic field strength and reads: $\\vartheta_e \\approx 0.5^o (E_{syn}\/10 GeV)^{-1}$.\nThus, an observer at a distance $D$ would see a source with angular size: \n$\\vartheta_{obs} \\approx \\sqrt{\\vartheta_p^2+\\vartheta_e^2} \\left(\\frac{l_{p\\gamma}}{D}\\right)$ that, for $D = 100$ Mpc and for photon energies of $1 \\div 10$ GeV is of the order of a fraction of a degree.\nThis is comparable with the angular resolution of GLAST, that would classify these sources as point-like if they are\nlocated at a distance of $\\sim 100 \\; {\\rm Mpc}$ or more. \nThis leads to the important conclusion that, \\textit{even if synchrotron photons are produced in an extended region of size $\\sim l_{p\\gamma}$ surrounding the accelerator, the resulting gamma ray source would appear point-like to a distant observer}.\n\nIf the proton spectrum extends well above $10^{20}$ eV, these sources might also be detected by \nImaging Atmospheric Cherenkov Telescope arrays (IACT) operating at energies above $100 \\; {\\rm GeV}$. The angular resolution of these instruments is a few arcminutes, and thus the sources will appear extended. However, it has been proven that IACT are powerful instruments to image extended gamma ray sources \\cite{SNR}, and thus they might still detect and map the emission from UHECR sources. Finally, powerful UHECR accelerators located at a distance of 1 Gpc or more would appear as point sources. In the next section we discuss the energy requirement for a detection.\n\n\\begin{figure}\n\\includegraphics[width=0.4\\textwidth]{gabiciposter_fig2.ps}\n\\caption{\\label{spectra}Spectra for a source \nlocated at\n$100{\\rm Mpc}$.\nThe luminosity in UHECRs is $2 \\; 10^{44}{\\rm erg\/s}$, with spectral index is $\\delta = 2$. TOP: $B_{IGMF} = 0.5$(curve 1), $5$ (2), $50{\\rm nG}$(3), $E_{cut} = 10^{21} {\\rm eV}$. BOTTOM: $E_{cut} = 5 \\, 10^{20}$, $10^{21}$, $5 \\, 10^{21} {\\rm eV}$, $B_{IGMF} = 1{\\rm nG}$. Dotted: intrinsic spectra. Solid: spectra after absorption.}\n\\end{figure}\n\n\\section{Detectability and energetics\n\nFig.~\\ref{spectra} shows synchrotron spectra\nfor a steady source\nat a distance of $100 ~ {\\rm Mpc}$ in a uniformly magnetized region of size 20 Mpc.\nSince the radiation is produced in a region about one interaction length away from the source, the actual structure of the field is not crucial.\nSteady state proton and electron spectra have been calculated taking into account all the relevant energy losses and proton escape from the magnetized region.\nSolid lines have been computed \ntaking into account the opacity of the Universe to very high energy photons due to pair production in the cosmic infrared background \\cite{IRabs},\nwhile dotted lines show the unabsorbed spectra.\nThe total luminosity in UHECRs with energy above $10^{19} {\\rm eV}$ is $L_{UHE} = 2 ~ 10^{44} {\\rm erg\/s}$, with a differential energy distribution $\\propto E^{-\\delta} exp(-E\/E_{cut})$, with \n$\\delta =2$.\nResults are quite insensitive to the slope of the spectrum.\n\nIn the top panel of Fig.~\\ref{spectra} $E_{cut} = 10^{21} {\\rm eV}$\nand the IGMF\nis equal to $0.5$, $5$ and $50 \\; {\\rm nG}$ (curves $1$, $2$ and $3$).\nIf the IGMF is significantly greater than $\\sim 50 \\; {\\rm nG}$, the peak of the emission falls at TeV energies, where absorption is very strong.\nOn the other hand, if the field is well below $\\sim 0.5 \\; {\\rm nG}$, synchrotron emission becomes unimportant and the cascade contribution dominates.\nHowever, \\textit{for the broad interval of values of the IGMF strength between $0.5$ and $50 \\; {\\rm nG}$, the formation of a synchrotron point-like gamma ray source seems to be unavoidable}. \n\nIn the bottom panel of Fig.~\\ref{spectra}, our predictions are compared with the sensitivities of GLAST and of a generic\nIACT such as HESS or VERITAS. \nA IGMF of $1 \\; {\\rm nG}$ is assumed and the different curves refer to values of the cutoff energy in the proton spectrum equal to $5 \\; 10^{20}$, $10^{21}$ and  $5 \\; 10^{21} {\\rm eV}$ (top to bottom).\nFor such a field, the condition for the detectability of a point source by GLAST is roughly $L_{UHE} \\ge 8 \\; 10^{43} \\div 2 \\; 10^{44} (D\/100{\\rm Mpc})^2 {\\rm erg\/s}$ for $\\delta = 2.0 \\div 2.6$.\nIn contrast, for IACT arrays the minimum detectable luminosity is roughly 2 orders of magnitude higher, since the source has to be located at a distance of $\\sim 1 {\\rm Gpc}$ in order to appear point-like. However, less powerful accelerators can still be detected as extended sources.\nSince the peak of the emission falls at $\\sim 10 ~ {\\rm GeV}$, future IACT operating in the energy range $10 \\div 100$ GeV would be powerful tools to search for these sources. \n\nIf the CR spectrum smoothly \nextends down to GeV energies with slope $\\delta = 2$, the required total CR luminosity for a source to be detected by GLAST is $L_{CR} \\ge 5 \\, 10^{44} (D\/100{\\rm Mpc})^2 {\\rm erg\/s}$.\nFor a beamed source, the required luminosity is reduced by a factor $f_b \\sim 0.02 (\\vartheta_b\/10^o)$, $\\vartheta_b$ being the beaming angle,\nand the detectability condition reads: $L_{CR} \\ge 10^{43} (f_b\/0.02) (D\/100{\\rm Mpc})^2 {\\rm erg\/s}$.\nThis luminosity is small if compared, for example, with the\npower of an AGN jet, that can be as high as\n$10^{47} {\\rm erg\/s}$ \\cite{ghisella}.\nThus,\nastrophysical objects that can in principle satisfy the energy requirement for a detection do exist.\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n~~~~~The purpose of this paper is to bring a new tool in the\nstudy of the spectral type of rank one transformations. Rank one\ntransformations have simple spectrum and in \\cite{Ornstein} D.S.\nOrnstein, using a random procedure, produced a family of mixing\nrank one transformations. It follows that the Ornstein's class of\ntransformations may possibly contain a candidate for Banach's well-known\nproblem whether there exists a dynamical system $(\\Omega,{\\mathcal\n{A}},\\mu,T)$ with simple Lebesgue spectrum. But, in 1993, J.\nBourgain in \\cite{Bourgain} proved that almost surely Ornstein's\ntransformations have singular spectrum. Subsequently, using the\nsame method, I. Klemes \\cite{Klemes1} and I. Klemes \\& K. Reinhold\n  \\cite{Klemes2} obtain that mixing \nstaircase transformations of Adams \\cite{Adams1} and Adams \\&\nFriedman \\cite{Adams2} have singular spectrum. They conjecture\nthat all rank one transformations have singular spectrum.\\par\n\nHere we shall exhibit a new class of rank one transformations with\nsingular spectrum. Our assumption include some new class of\nOrnstein transformations and a class of Creutz-Silva rank one\ntransformations \\cite{Creutz-silva}. Our proof is based on\ntechniques introduced by J. Bourgain\n\\cite{Bourgain} in the context of rank one transformations and\ndeveloped by Klemes \\cite{Klemes1}, Klemes-Rienhold\n\\cite{Klemes2}, Dooley-Eigen \\cite{Eigen}, together with some ideas from the proof of \nthe central limit theorem for trignometric sums. The fundamental key, as noted by Klemes\n\\cite{Klemes1}, is the estimation of the $L^1$-norm of a certain\n trigonometric polynomial ${{(|P_m|^2-1)}}$\n. We shall use the method of central limit theorem for trignometric sums to produce an\nestimate of this $L^1$-norm.\n\n\\section*{2. Rank One Transformation by Construction}\n\nUsing the cutting and stacking method described in [Fr1], [Fr2],\none can construct inductively a family of measure preserving\ntransformations, called rank one transformations, as follows\n\\vskip 0.1cm Let $B_0$ be the unit interval equipped with the\nLebesgue measure. At stage one we divide $B_0$ into $p_0$ equal\nparts, add spacers and form a stack of height $h_{1}$ in the usual\nfashion. At the $k^{th}$ stage we divide the stack obtained at the\n$(k-1)^{th}$ stage into $p_{k-1}$ equal columns, add spacers and\nobtain a new stack of height $h_{k}$. If during the $k^{th}$ stage\nof our construction  the number of spacers put above the $j^{th}$\ncolumn of the $(k-1)^{th}$ stack is $a^{(k-1)}_{j}$, $ 0 \\leq\na^{(k-1)}_{j} < \\infty$,  $1\\leq j \\leq p_{k}$, then we have\n\n$$h_{k} = p_{k-1}h_{k-1} +  \\sum_{j=1}^{p_{k-1}}a_{j}^{(k-1)}.$$\n\\vskip 3 cm \\hskip 3.5cm\n  \\tower\n\\vskip 3.0cm\n\\noindent{}Proceeding in this way we get a rank one transformation\n$T$ on a certain measure space $(X,{\\mathcal B} ,\\nu)$ which may\nbe finite or\n$\\sigma-$finite depending on the number of spacers added. \\\\\n\\noindent{} The construction of a rank one transformation thus\nneeds two parameters, $(p_k)_{k=0}^\\infty$ (parameter of cutting\nand stacking), and $((a_j^{(k)})_{j=1}^{p_k})_{k=0}^\\infty$\n(parameter of spacers). Put\n\n$$T \\stackrel {def}= T_{(p_k, (a_j^{(k)})_{j=1}^{p_k})_{k=0}^\\infty}$$\n\n\\noindent In \\cite{Nadkarni1} and \\cite{Klemes2} it is proved that\nthe spectral type of this transformation is given (upto possibly some discrete measure) by\n\n\\begin{eqnarray}\\label{eqn:type1}\nd\\sigma  =W^{*}\\lim \\prod_{k=1}^n\\left| P_k\\right| ^2d\\lambda,\n\\end{eqnarray}\n\\noindent{}where\n\\begin{eqnarray*}\n&&P_k(z)=\\frac 1{\\sqrt{p_k}}\\left(\n\\sum_{j=0}^{p_k-1}z^{-(jh_k+\\sum_{i=1}^ja_i^{(k)})}\\right),\\nonumber  \\\\\n\\nonumber\n\\end{eqnarray*}\n\\noindent{}$\\lambda$ denotes the normalized Lebesgue measure on the\ncircle group ${\\sym T}$ and $W^{*} \\lim$ denotes weak*limit in the space of\nbounded Borel measures on ${{\\sym T}}$.\\\\ \n\n\n\\noindent{}The principal result of this paper is the following:\\\\\n\n\n\\noindent {\\bf Theorem 2.1. }{\\it Let $T=T_{(p_k,\n(a_j^{(k)})_{j=1}^{p_k})_{k=0}^\\infty}$ be a rank one\ntransformation such that,\n\n\\begin{eqnarray}{\\label{eq:eqt}}\n&&(i)~~a_{j+1}^{(k)}\\geq 2s_k(j),j=0,\\cdots, p_k-1,\\nonumber\\\\\n&& {\\rm {~~with~~~~}} s_k(j)=a_1^{(k)}+\\ldots +a_j^{(k)},\ns_k(0)=0,\\nonumber\\\\\n&&(ii)~~\\frac{s_k{(p_k)}}{h_k}<\\frac12 \\nonumber\n\\end{eqnarray} then the spectrum of $T$ is singular.}\n\\\\\n\n\\noindent{}We remark that the\nspectrum of rank one transformation is always singular if the\ncutting parameter $p_k$ is bounded. In fact, Klemes-Reinhold\nprove that if $\\displaystyle \\sum_{k=0}^{\\infty}\n\\frac{1}{{p_k}^2}=\\infty$ then the associated rank one\ntransformation has singular spectrum. Henceforth, we assume that the series\n$\\displaystyle \\sum_{k=0}^{\\infty} \\frac{1}{{p_k}^2}$\nconverges.\\\\\nWe point out also that the condition $(ii)$ of\ntheorem 2.1 holds in the case of rank one transformations satisfying a\nrestricted growth condition  provided that $\\displaystyle \\min_{1 \\leq j \\leq  p_k}(a_j^{(k)})=0$. Following Creutz-Silva \\cite{Creutz-silva},\nwe say that a rank one transformation $T=T_{(p_k, (a_j^{(k)})_{j=1}^{p_k})_{k=0}^\\infty}$ has\nrestricted growth if\n\\[\n\\displaystyle \\frac{\\displaystyle \\sum_{i=1}^{p_k} \\left( a_j^{(k)}- \\displaystyle \\min_{1 \\leq j \\leq  p_k}(a_j^{(k)})\\right) }{h_k+\\displaystyle \\min_{1 \\leq j \\leq  p_k}(a_j^{(k)}) }\\tend{k}{+\\infty}0,\n\\]\n\\noindent{}The proof of our main result is based on the method of\nJ. Bourgain in \\cite{Bourgain}. We shall recall the main ideas of\nthis\nmethod.\\\\\n\n\n\\noindent {\\bf Proposition 2.2. }{\\it \\ The following are\nequivalent  }\n\n\\begin{enumerate}\n\\item [(i)]  {\\it $\\sigma \\perp \\lambda$ }\n\n\\item[(ii)]  {\\it $\\inf \\{\\displaystyle \\displaystyle \\int \\displaystyle \\int\n\\prod_{l=1}^k\\left| {P_{n_l}(z)}\\right| d\\lambda,~k\\in\n{{\\sym N}},~n_1<n_2<\\ldots <n_k\\}=0.$ }\n\\end{enumerate}\n\n\\noindent{}Now fix some subsequence ${\\mathcal {N}}=\\left\\{\nn_1<n_2<\\ldots <n_k\\right\\}$ ,\\linebreak[0]$\\ k \\in {{\\sym N}}$, $m>n_k$\nand put\n\\[\nQ\\left(z\\right) =\\prod_{i=1}^k\\left |{P_{n_i}(z)}\\right |.\n\\]\n\n\\noindent{}One can show, using the same arguments as in\n\\cite{elabdal1}, the following lemma.\\\\\n\n\\noindent {\\bf Lemma 2.3. }{\\it\n\\begin{eqnarray*}\n  &&\\displaystyle \\int  Q(z) \\left| P_m(z)\\right|d\\lambda(z)\n\\leq \\\\\n  &&\\frac 12\\left( \\displaystyle \\int Q d\\lambda +\\displaystyle \\int\nQ(z) \\left| P_m(z)\\right| ^2 d\\lambda(z)\\right) -\\frac 18\\left(\n\\displaystyle \\int  Q \\left| \\left| P_m(z)\\right| ^2-1\\right|\nd\\lambda(z)\\right) ^2. \\end{eqnarray*}}\\\\\n\n\\noindent {\\bf Proposition 2.4. }{\\it $\\displaystyle \\lim_{m\n\\rightarrow\n  \\infty}\\displaystyle \\int  Q \\left|\nP_m(z)\\right|^2d\\lambda(z) {\\Bbb {=}} \\displaystyle \\displaystyle\n\\int  Q $ $d\\lambda(z)  .$}\\\\\n\\\\\n\\begin{proof}\n\\noindent{}The\nsequence of probability measures $\\left| {P_m(z)}%\n\\right|^2d\\lambda(z)$ converges weakly to the Lebsegue measure.\n   \\end{proof}\n   \\\\\n\n\n\\noindent We have also the following proposition: \\\\\n\n\\noindent {\\bf Proposition 2.5. }{\\it There exist a subsequence of\nthe sequence $\\left (\\left |\\left| P_m(z)\\right|-1 \\right|\\right)$\nwhich converge weakly to some non-negative function $\\phi$ which\nsatisfies  $ \\phi \\leq $2, almost surely with respect to the Lebesgue\nmeasure.}\n\\\\\n\n\\begin{proof}\n\\noindent{}The\nsequence $\\left |\\left| {P_m(z)}%\n\\right|-1 \\right |$ is bounded in $L^2$. It follows that there\nexist a subsequence which converges weakly to some non-negative\n$L^2$ function $\\phi$. Let $\\omega$ be a non-negative continuous\nfunction, then we have\n\n\\begin{eqnarray*}\n\\int \\omega\\left |\\left| {P_m(z)} \\right|-1 \\right |d\\lambda(z)\n&&\\leq \\int \\omega\\left| {P_m(z)} \\right| d\\lambda(z)+\\int \\omega\nd\\lambda(z)\\\\\n&&\\leq {(\\int \\omega d\\lambda(z))}^{\\frac12} {(\\int \\omega \\left |\n{P_m(z)} \\right|^2d\\lambda(z))}^{\\frac12}+\\int \\omega d\\lambda(z).\n\\end{eqnarray*}\n\\noindent{}Hence\n\\[\n\\int \\omega \\phi d\\lambda \\leq 2 \\int \\omega d\\lambda.\n\\]\n\nSince this holds for all non-negative continuous $\\omega$, we have $\\phi \\leq 2$ a.e. \n\\end{proof}\n\\\\\n\n\\noindent Put\n\\[\n\\alpha = \\phi~~ d\\lambda.\n\\]\n\\noindent We shall prove the following:\\\\\n\n\\noindent {\\bf Proposition 2.6. }{$\\alpha \\bot \\sigma$}.\\\\\n\n\\noindent For the proof of the proposition 2.6 we need the\nfollowing classical lemma \\cite{Kilmer-Seaki}.\\\\\n\n\\noindent {\\bf Lemma 2.7. }{\\it Let $\\rho, \\tau$ be two\nnonnegative finite measure on a measurable space $X$. Then the\nfollowing properties are equivalent :\n\\begin{enumerate}\n\\item[(i)]  {\\it $\\rho \\perp \\tau$ }\n\n\\item[(ii)]  {\\it Given $\\varepsilon>0$, there exists a\nnonnegative measurable function $f$ on $X$ such that $ f >0,$\n$\\tau-a.e.$ and such that\n\\[\n\\left(\\int f d\\rho) \\right) \\left (\\int \\frac{d\\tau}{f} \\right)\n<\\varepsilon.\n\\] }\n\\end{enumerate}\n}\n\n\\begin{proof}{of Proposition 2.6}\nLet $\\displaystyle \\beta_1=\\inf\\{\\int Q\nd\\alpha,~:~Q=\\prod_{i=1}^k\\left | {P_{n_i}(z)}\\right|, k \\in {\\sym N},\nn_1<n_2<\\cdots<n_k\\}$ and $\\displaystyle \\beta_2=\\inf\\{\\int Q\nd\\lambda,~:~Q=\\prod_{i=1}^k\\left | {P_{n_i}(z)}\\right|, k \\in {\\sym N},\nn_1<n_2<\\cdots<n_k\\}$. Then, for any $Q$, we have\n\n\\begin{eqnarray*}\n    \\int Q\nd\\alpha \\geq \\beta_1,~~~ \\liminf \\int Q |P_m| d\\lambda \\geq\n\\beta_2\n\\end{eqnarray*}\n\n\\noindent{} Combine lemma 2.3, proposition 2.4 and \nproposition 2.5 and take the inf over all $Q$ to obtain:\n\\begin{eqnarray*}\n\\beta_2 \\leq \\beta_2-\\frac18\\beta_1^2\n\\end{eqnarray*}\n\\noindent It follows that\n\\[\n\\beta_1=0.\n\\]\n\\noindent{}We claim: $\\beta_1=0$ implies $\\displaystyle \\int\n\\prod_{j=1}^{k}|P_j|d\\alpha \\tend{k}{\\infty}0$. In fact, let\n${\\mathcal{N}}=\\{n_1<n_2<\\cdots<n_k\\}\\subset {\\sym N}^*$ and $N>n_k$.\nThen, by Cauchy-Schwarz inequality, we have\n\n\\begin{eqnarray}{\\label{eq:Bo1}}\n\\int \\prod_{j=1}^{N}|P_j|d\\alpha & =&  \\int \\sqrt{\\prod_{j \\in\n\\mathcal{N}}|P_j|} \\sqrt{\\prod_{j \\in \\mathcal{N}}|P_j|} \\prod_{j\n\\not \\in \\mathcal{N}}|P_j| d\\alpha \\nonumber\\\\\n&\\leq& {\\left(\\int \\prod_{j \\in \\mathcal{N}}|P_j|d\\alpha\\right\n)}^{\\frac12} {\\left(\\int \\prod_{j \\in \\mathcal{N}} |P_j| d\\alpha\n\\prod_{j \\not \\in \\mathcal{N}}|P_j|^2 d\\alpha\\right )}^{\\frac12}\n\\end{eqnarray}\n\n\\noindent{}But\n\n\\begin{eqnarray}{\\label{eq:B2}} \\int \\prod_{j \\in \\mathcal{N}}\n|P_j|  \\prod_{j \\not \\in \\mathcal{N}}|P_j|^2 d\\alpha &\\leq & 2\n\\int \\prod_{j \\in \\mathcal{N}} |P_j|  \\prod_{j \\not \\in\n\\mathcal{N}}|P_j|^2 d\\lambda \\nonumber \\\\\n&\\leq&  2\\int \\prod_{j=1}^{N} |P_j|  \\prod_{j \\not \\in\n\\mathcal{N}}|P_j| d\\lambda \\nonumber\\\\\n  &\\leq& 2{\\left(\\int \\prod_{j=1}^{N} |P_j|^2 d\\lambda \\right)}^{\\frac12}\n{\\left(\\int\\prod_{j \\not \\in \\mathcal{N}}|P_j|^2\nd\\lambda\\right)}^{\\frac12} \\nonumber \\\\\n&\\leq& 2\n\\end{eqnarray}\n\n\\noindent{}Combine (\\ref{eq:Bo1}) and (\\ref{eq:B2}) to get the\nclaim. The proof of the proposition follows from lemma 2.7.\n\\end{proof}\n\n\n\\section*{3. Estimation of $L_1$-Norm of ($\\left |P_m(z)\\right |-1$)\nand the Central Limit Theorem.}\n\nIn this section we assume that for $m$\nsufficiently large\n\\begin{eqnarray}{\\label{eq:rg}}\n  &&(i)~~a_{j+1}^{(m)}\\geq 2s_m(j),j=0,\\cdots, p_m-1,\\nonumber\n  \\\\ &&{\\rm {~~with~~~~}} s_m(j)=a_1^{(m)}+\\ldots +a_j^{(m)},\ns_m(0)=0,\\nonumber\\\\\n&&(ii)~~\\frac{s_m{(p_m)}}{h_m}<\\frac12 \\nonumber\n\\end{eqnarray}\n\n\\noindent{}Under the above assumptions, we shall proof the following:\\\\\n\n\\noindent {\\bf Proposition 3.1. }{ $\\alpha \\geq K \\lambda$, for\nsome positive\nconstant $K$}.\\\\\n\n\\noindent{} The proof of the proposition is based on an\nestimate of $\\displaystyle \\int_A ||P_m|-1|d\\lambda$,   where\n$A$ is a Borel set with $\\lambda(A)>0$. More precisely we shall\nstudy the stochastic behavior of the sequence $ |P_m|$. For that\nour principal strategy is based on the method of the central\nlimit theorem for trigonometric sums. A nice account can be\nfound in \\cite{Kac}. It is well-known that\nHadamard lacunary trigonometric seires satisfies the central limit\ntheorem \\cite{Zygmund}. The central limit theorem for\ntrigonometric sums has been studied by many authors, Zygmund and\nSalem \\cite{Zygmund}, Erd\\\"os \\cite{Erdos},\n  J.P. Kahane \\cite{Kahane}, Berkers \\cite{Berkes}, Murai \\cite{Murai},\nTakahashi \\cite{Takahashi}, Fukuyama and  Takahashi\n\\cite{Fukuyama}, and many others. The same method is used to\nstudy the asymptotic behavior of Riesz-Raikov sums \\cite{petit}.\\\\\n\n\\noindent Here, we shall prove the following:\\\\\n\n\\noindent {\\bf Proposition 3.2. } {If $(i)$ and $(ii)$ hold then\nfor any Borel subset $A$ of ${\\sym T}$ with $\\lambda(A)>0$, the distribution of the\nsequence of random variables\n$\\frac{\\sqrt{2}}{\\sqrt{p_m}}\\sum_{j=0}^{p_m-1} \\cos((jh_m+s_m(j))t)$\nconverges to the Gauss distribution. That is\n\n\\begin{eqnarray}{\\label{eq:eq6}}\n\\frac1{\\lambda(A)}\\left \\{t \\in A~~:~~\\frac{\\sqrt{2}}{\\sqrt{p_m}}\n\\sum_{j=0}^{p_m-1} \\cos((jh_m+s_m(j))t) \\leq x \\right \\}\\nonumber \\\\\n\\tend{m}{\\infty}\\frac1{\\sqrt{2\\pi}}\\int_{-\\infty}^{x}\ne^{-\\frac12t^2}dt\\stackrel{\\rm {def}}{=}{\\mathcal {N}}\\left( \\left] -\\infty,x \\right] \\right) .\n\\end{eqnarray}\n}\\\\\nThe proof of proposition 3.2 is based on the idea of the proof of\nmartingale central limit theorem due to McLeish \\cite{Mcleish}.\nThe main ingredient is the following .\\\\\n\n\\noindent {\\bf Lemma 3.3. } {For $n \\geq 1$, let $U_n,T_n$ be\nrandom variables such that\n\\begin{enumerate}\n\\item  $U_n\\longrightarrow a$ in probability.\n\\item $\\{T_n\\}$ is uniformly integrable.\n\\item $\\{|T_nU_n|\\}$ is uniformly integrable.\n\\item ${\\sym E}(T_n) \\longrightarrow 1$.\n\\end{enumerate}\nThen ${\\sym E}(T_nU_n)\\longrightarrow a$.}\\\\\n\n\\begin{proof}\nWrite $T_nU_n=T_n(U_n-a)+aT_n$. As ${\\sym E}(T_n) \\longrightarrow\n1$, we need to show that ${\\sym E}(T_n(U_n-a))\\longrightarrow 0$.\\\\\nSince $\\{T_n\\}$ is uniformly integrable, we have\n$T_n(U_n-a)\\longrightarrow 0$ in probability. Also, both  $T_nU_n$\nand $aT_n$ are uniformly integrable, and so the combination\n$T_n(U_n-a)$ is uniformly integrable. Hence,\n${\\sym E}(T_n(U_n-a))\\longrightarrow 0$.\n\\end{proof}\\\\\n\n\\noindent{}Let us recall the following expansion\n\\[\n\\exp(ix)=(1+ix)\\exp(-\\frac{x^2}2+r(x)),\n\\]\n\\noindent{}where $|r(x)| \\leq |x|^3$, for real $x$.\\\\\n\n\\noindent {\\bf Theorem 3.4. }{Let $\\{X_{nj}~~:~~ 1 \\leq j \\leq\nk_n, n\\geq 1\\}$ be a triangular array of random variables.\n$\\displaystyle S_n=\\sum_{j=1}^{k_n}X_{nj}$, $\\displaystyle T_n=\\prod_{j=1}^{k_n}(1+itX_{nj})$,\nand $\\displaystyle U_n=\\exp \\left (-\\frac{t^2}2 \\sum_j\nX_{nj}^2+\\sum_{j}r(tX_{nj})\\right )$. Suppose that\n\\begin{enumerate}\n\\item $\\{T_n\\}$ is uniformly integrable.\n\\item ${\\sym E}(T_n) \\longrightarrow 1$.\n\\item $\\displaystyle \\sum_j X_{nj}^2 \\longrightarrow 1$ in probability.\n\\item $\\max |X_{nj}|\\longrightarrow 0$ in probability.\n\\end{enumerate}\nThen ${\\sym E}(\\exp(it S_n))\\longrightarrow \\exp(-\\displaystyle\\frac{t^2}2).$}\\\\\n\n\\begin{proof}\nLet $t$ be fixed. From condition $(3)$ and $(4)$,\n\\[\n|\\sum_j r(tX_{nj})| \\leq |t|^3 \\sum_{j}|X_{nj}|^3 \\\\\n\\leq |t|^3 \\max_j|X_{nj}|  \\sum_j X_{nj}^2\n\\tend{n}{\\infty}0 {\\rm {~in~probability}}.\n\\]\n\\noindent{}$U_n=\\exp\\left (-\\displaystyle \\frac{t^2}2 \\sum_j\nX_{nj}^2+\\sum_{j}r(tX_{nj})\\right ) \\longrightarrow  \\exp(-\\displaystyle\n\\frac{t^2}2)$ in probability as $n \\longrightarrow +\\infty$. This verifies the condition $(1)$ of Lemma\n3.3 with $a=\\exp(-\\displaystyle \\frac{t^2}2).$ It is easy to check\nthat the conditions $(2)$ ,$(3)$ and $(4)$ of the lemma 3.3 hold.\nThus $E\\left( \\exp(itS_n)\\right) =E\\left( T_nU_n\\right) \\longrightarrow\n\\exp(-\\displaystyle \\frac{t^2}2)$.\n\\end{proof}\\\\\n\n\\noindent{}Let $m$ be a positive integer and put\n\n\\begin{eqnarray*}\nW_m \\stackrel {\\rm {def}}{=}&& \\big \\{ \\big ({\\sum_{i \\in\nI}\\varepsilon_i p_i}\\big ) h_m+\\sum_{i \\in I}\\varepsilon_i\ns_m(p_i) ~~:~~\n\\varepsilon_i \\in \\{0,-1,1\\}, I \\subset \\{0,\\cdots,p_m-1\\},\\\\\n&&p_i \\in \\{0,\\cdots,p_m-1\\} \\}.\n\\end{eqnarray*}\n\n\\noindent{} The element $w =(\\sum_{i \\in\nI}\\varepsilon_ip_i)h_m+\\sum_{i \\in I}\\varepsilon_is_m(p_i)$ is\ncalled a word.\\\\\n\nWe shall need the following combinatorial lemma.\\\\\n\n\\noindent {\\bf Lemma 3.5.}{ Under the\nassumptions (i) and (ii) of theorem  2.1, all the words of $W_m$ are distinct.}\\\\\n\n\\begin{proof}{}\nLet $w,w' \\in W_m$, write\n\\begin{eqnarray*}\nw &=&(\\sum_{i \\in I}\\varepsilon_ip_i)h_m+\\sum_{i \\in\nI}\\varepsilon_is_m(p_i)\\\\\nw' &=&(\\sum_{i \\in I'}\\varepsilon'_ip'_i)h_m+\\sum_{i \\in\nI}\\varepsilon'_is_m(p'_i).\n\\end{eqnarray*}\n\\noindent{}Then $w=w'$ implies\n\\[\n\\{(\\sum_{i \\in I}\\varepsilon_ip_i)-(\\sum_{i \\in\nI'}\\varepsilon'_ip'_i)\\}h_m=\\sum_{i \\in\nI'}\\varepsilon'_is_m(p'_i)-\\sum_{i \\in I}\\varepsilon_is_m(p_i)\n\\]\n\\noindent{}But the modulus of LHS is greater than $h_m$ and the\nmodulus of RHS is less than $2 \\sum_{j=0}^{p_m-1}a_j^{(m)}$. It\nfollows that\n\\begin{eqnarray*}\n\\sum_{i \\in I}\\varepsilon_ip_i&=& (\\sum_{i \\in\nI'}\\varepsilon'_ip'_i){\\rm {~~~and }} \\\\\\sum_{i \\in\nI}\\varepsilon_is_m(p_i)&=&\\sum_{i \\in I'}\\varepsilon'_is_m(p'_i).\n\\end{eqnarray*}\nSince from (i) we have  $s_m(p+1) \\geq 3 s_m(p)$, we get that the\nrepresentation in the form $\\sum_{i \\in I}\\varepsilon_is_m(p_i)$\nis unique and the proof of the lemma is complete.\n\\end{proof}\n\\\\\n\n\\begin{proof}{of proposition 3.2} Let $A$ a Borel set,\n$\\lambda(A)>0$. Using the Helly theorem we may assume that the\nsequence $\\displaystyle \\frac{\\sqrt{2}}{\\sqrt{p_m}}\n\\sum_{j=0}^{p_m-1} \\cos((jh_m+s_m(j))t)$ converge in distribution.\nAs is well-known, it is sufficient to show that for every real\nnumber $x$,\n\n\\begin{eqnarray}{\\label {eq:eq7}}\n  \\displaystyle \\frac1{\\lambda(A)}\\int_A \\exp\\left \\{-ix\\frac{\\sqrt{2}}{\\sqrt{p_m}}\n  \\sum_{j=0}^{p_m-1} \\cos((jh_m+s_m(j))t) \\right \\} dt\n  \\tend{m}{\\infty} \\exp(-\\frac{x^2}2). \\nonumber\n\\end{eqnarray}\n\n\\noindent{} To this end we apply theorem 3.4. in the following\ncontext. The measure space is the given Borel set $A$ of positive Lebesgue\nmeasure in the circle with the normalised measure and the random\nvariables are given by\n$$\nX_{mj}=\\frac{\\sqrt{2}}{\\sqrt{p_m}} \\cos((jh_m+s_m(j))t),~~~~{\\rm {where}}~~~0 \\leq j \\leq p_m-1,~\nm \\in {\\sym N}.\n$$\n\n\\noindent{}It is easy to check that the variables $\\{X_{mj}\\}$\nsatisfy conditions (1) and (4). Further, condition (3) follows from fact that\n$$\n\\int_0^{2\\pi} \\left |\\sum_{j=0}^{p_m-1} X_{mj}^2-1 \\right |^2 dt\n\\tend{m}{\\infty} 0.\n$$\n\\noindent{} It remains to verify comdition (2) of theorem 3.4. it is sufficient to show that\n\\begin{eqnarray}{\\label {eq8}}\n\\int_A \\prod_{j=0}^{p_m-1}\\left (\n1-ix\\frac{\\sqrt{2}}{\\sqrt{p_m}}\\cos((jh_m+s_m(j))t)\\right)dt\n\\tend{m}{\\infty}\\lambda(A).\n\\end{eqnarray}\n\n\n\\noindent{}Write\n\\begin{eqnarray*}\n\\Theta_m(x,t)&=&\\prod_{j=0}^{p_m-1}\n\\left(1-ix\\frac{\\sqrt{2}}{\\sqrt{p_m}}\\cos((jh_m+s_m(j))t\\right)\\\\\n&=&1+\\sum_{w=1}^{N_m}{\\rho_w}^{(m)}(x) \\cos(wt),\n\\end{eqnarray*}\n\n\\noindent{}where $\\rho_w=0$ if $w$ is not of the form $(\\sum_{i\n\\in I}\\varepsilon_ip_i)h_m+\\sum_{i \\in I}\\varepsilon_is_m(p_i)$,\n$N_m=\\displaystyle \\left \\{\\frac{p_m(p_m-1)}2\\right\n\\}h_m+s_m(p_m-1)+s_m(p_m-2)+\\cdots+1$.\\\\\n\n\\noindent{}We claim that it is sufficient to prove the following:\n\n\\begin{eqnarray}{\\label{density}}\n\\int_{0}^{2\\pi} R(t) \\prod_{j=0}^{p_m-1} \\left\n(1-ix\\frac{\\sqrt{2}}{\\sqrt{p_m}}\\cos((jh_m+s_m(j))t)\\right) dt\n\\tend{m}{\\infty}\\int_{0}^{2\\pi} R(t) dt,\n\\end{eqnarray}\n\n\\noindent{}for any trigonometric polynomial $R$. In fact, assume\nthat (\\ref{density}) holds and let $\\epsilon>0$.\nThen, by the density of trigonometric polynomials, one can find a\ntrigonometric polynomial $R_\\epsilon$ such that\n\\[\n||\\chi_A-R_\\epsilon||_1 <\\epsilon,\n\\] where\n\\noindent{}$\\chi_A$  is indicator function of $A$. But\n\n\\begin{eqnarray*}\n\\left |\\prod_{j=0}^{p_m-1}\\left (\n1-ix\\frac{\\sqrt{2}}{\\sqrt{p_m}}\\cos((jh_m+s_m(j))t)\\right)\\right|  \\leq\n\\left \\{\\prod_{j=0}^{p_m-1}\\left ( 1+\\frac{2x^2}{p_m}\\right) \\right\n\\}^{\\frac12},\n\\end{eqnarray*}\n\n\\noindent{}Since $1+u \\leq e^u$, we get\n\n\\begin{eqnarray}{\\label{eq:eq11}}\n\\mid\\Theta_m(x,t)\\mid \\leq e^{x^2}.\n\\end{eqnarray}\n\n\\noindent{}Hence, according to (\\ref{density}), for $m$\nsufficiently large, we have\n\n\\begin{eqnarray*}{\\label{eq :eq12}}\n&&\\left |\\int_A \\Theta_m(x,t) dt -\\lambda(A)\\right|\n=|\\int_A\\Theta_m(x,t)dt -\\int_{0}^{2\\pi} \\Theta_m(x,t) R_\\epsilon(t) dt+\n\\\\&&\\int_{0}^{2\\pi} \\Theta_m(x,t) R_\\epsilon(t) dt-\\int_{0}^{2\\pi}  R_\\epsilon(t) dt + \\int_{0}^{2\\pi}\nR_\\epsilon(t) dt-\\lambda(A)|< e^{x^2}\\epsilon+2\\epsilon.\n\\end{eqnarray*}\n\n\\noindent{}The proof of the claim is complete. It still remains to prove\n(\\ref{density}). Observe that\n\n$$\\int_{0}^{2\\pi} \\Theta_m(x,t)R(t) dt =\n  \\int_{0}^{2\\pi} R(t) dt+\\sum_{w=1}^{N_m}{\\rho_w}^{(m)}(x) \\int_{0}^{2\\pi} R(t)\n  \\cos(wt)dt$$\n\n\\noindent{}and for\n$w=p_{i_1}h_m+s_m(p_{i_1})+\\sum_{j=1}^{r}\\varepsilon_j\n\\{(p_{i_j}h_m)+s_m(p_{i_j})\\}$, we have\n\n\\[\n|{\\rho_w}^{(m)}(x)| \\leq \\frac{2^{1-r}|x|^r}{p_m^{\\frac{r}2}},\n\\]\n\n\\noindent{}hence\n\\[\n\\max_{w \\in W}|{\\rho_w}^{(m)}(x)| \\leq\n\\frac{|x|}{p_m^{\\frac{1}2}}\\tend{m}{\\infty}0.\n\\]\n\\noindent{}Since $\\displaystyle \\sum_{w \\in{\\sym Z}}|\\int_{0}^{2\\pi} e^{-iwt}\nR(t)dt|$ is bounded, we deduce\n\\[\n|\\sum_{w=1}^{N_m}{\\rho_w}^{(m)}(x) \\int_{0}^{2\\pi} R(t)\n  \\cos(wt)dt|\\leq \\frac{|x|}{p_m^{\\frac{1}2}}\n  \\sum_{w \\in{\\sym Z}}|\\int_{0}^{2\\pi} e^{-iwt} R(t)dt|\\tend{m}{\\infty}0.\n\\]\n\\noindent{}The proof of the proposition 3.2. is complete.\n\\end{proof}\\\\\n\n\\begin{proof}{of proposition 3.1}\nLet $A$ be a Borel subset of ${\\sym T}$, and $x \\in ]1,+\\infty[$, then, for any\npositive integer $m$, we have\n\n\\begin{eqnarray*}{\\label{eq :eqfin}}\n\\int_A ||P_m(\\theta)|-1| d\\lambda(\\theta) &\\geq& \\int_{\\{\\theta\n\\in A\n~~~:~~~~|P_m(\\theta)|>x\\}}||P_m|-1| d\\lambda(\\theta)\\\\\n&\\geq&(x-1)\\lambda\\{\\theta \\in A~:~|P_m(\\theta)|>x\\}\\\\\n&\\geq& (x-1) \\lambda\\{\\theta \\in A ~~:~~|\\Re({P_m(\\theta)})|>x\\}\n\\end{eqnarray*}\n\\noindent{} Let $m$ go to infinity and use  propositions\n2.5 and 3.2 to get\n\n\\[\n\\int_A \\phi d\\lambda \\geq (x-1)\\{1-{\\mathcal\n{N}}([-\\sqrt{2}x,\\sqrt{2}x])\\}\\lambda(A).\n\\]\n\n\\noindent{}Put $K=(x-1)\\{1-{\\mathcal {N}}([-\\sqrt{2}x,\\sqrt{2}x])\\}$. Hence $\n\\alpha(A) \\geq K \\lambda(A)$, for any Borel subset $A$ of ${\\sym T}$ which proves the proposition.\n\\end{proof}\\\\\n\n\\noindent{}Now, We give the proof of our main result.\\\\\n\n\\begin{proof}{of theorem 2.1} Follows easily from the proposition 2.6 combined with\nproposition 3.1. \n\\end{proof}\\\\\n\n\\noindent{}Let us mention that the same proof works for the\nfollowing more general statement.\\\\\n\n\\noindent {\\bf Theorem 3.6. }{\\it Let $T=T_{(p_k,\n(a_j^{(k)})_{j=1}^{p_k})_{k=0}^\\infty}$ be a rank one\ntransformation such that,\n\n\\begin{eqnarray}{\\label{eq:eqt2}}\n&&(i)~~a_{j+1}^{(k)}\\geq 2s_k(j),j=0,\\cdots, p_k-1,\\nonumber\\\\\n&& {\\rm {~~with~~~~}} s_k(j)=a_1^{(k)}+\\ldots +a_j^{(k)},\ns_k(0)=0,\\nonumber\\\\\n&&(ii)~~\\frac{s_k{(p_k)}-p_k\\min_{1 \\leq j \\leq\np_k}(a_j^{(k)})}{h_k+\\min_{1 \\leq j \\leq  p_k}(a_j^{(k)})}<\\frac12\n\\nonumber\n\\end{eqnarray} then the spectrum of $T$ is singular.}\n\\\\\n\n\n\\noindent{}{\\bf Remark.} We note that, in \\cite{Bourgain} and\n\\cite{Klemes1}, the strategy of the authors is to show that the\nabsolutely continuous measure  $\\beta$, obtained as the limit of\nsome subsequence of the sequence $(||P_m|^2-1| d\\lambda)_{m \\geq\n0}$, is equivalent to Lebesgue measure, in fact the authors prove\nthat\n$$ \\beta \\geq K \\lambda,~~~~~~~~~~~~~~~~ (E)$$\n\\noindent{}for some $K>0$. In the case of Ornstein\ntransformations, the relation (E) hold almost surely.\\\\\n\n\n\n\\section*{4. Simple Proof Of Bourgain Theorem.}\n\nBourgain Theorem deals with Ornstein transformations. In\nOrnstein's construction, the $p_k$'s are rapidly increasing, and\nthe number of spacers, $a_i^{(k)}$, $1 \\leq i\\leq p_k-1$, are\nchosen randomly. This may be organized in different ways as\npointed out by J. Bourgain in \\cite{Bourgain}. Here we suppose that we are given\ntwo sequences $(t_k)$, $(p_k)$ of positive integers and a sequence ($\\xi_k$) \n of probability measure such that the support of each\n$\\xi_k$ is a subset of $X_k = \\{-\\displaystyle\n\\frac{t_k}{2},\\cdots,\\displaystyle \\frac{t_k}{2}\\}$. We choose now\nindependently, according to $\\xi_k$, the numbers\n$(x_{k,i})_{i=1}^{p_k-1}$, and $x_{k,p_k}$ is chosen\ndeterministically in ${\\sym N}$. We put, for $1 \\leq i \\leq p_k$,\n\n$$a_i^{(k)} = t_{k} + x_{k,i} - x_{k,i-1}, ~~{\\rm with} ~~x_{k,0}\n= 0.$$\\\\\n\n\\noindent{} We have\n\n$$h_{k+1} = p_k(h_k + t_{k}) + x_{k,p_k}.$$\n\n\\noindent{}So the deterministic sequences of positive integers\n$(p_k)_{k=0}^\\infty$, $(t_k)_{k=0}^\\infty $ and\n$(x_{k,p_k})_{k=0}^\\infty$ determine completely the sequence of\nheights $(h_k)_{k=0}^\\infty$. The total measure of the resulting\nmeasure space is finite if\n\\begin{eqnarray}\\label{eqn:fini}\n\\sum_{k=0}^{\\infty}\\frac{t_k}{h_k}+\\sum_{k=0}^\\infty\n\\frac{x_{k,p_k}}{p_kh_k} < \\infty.\n\\end{eqnarray}\n  \\noindent{}We will assume that this\nrequirement is satisfied.\\\\\nWe thus have a probability space of Ornstein transformations\n$\\Omega=\\prod_{l=0}^\\infty X_l^{p_l-1}$ equipped with the natural\nprobability measure ${\\sym P} \\stackrel {\\rm def}\n{=}\\otimes_{l=1}^{\\infty} P_l$, where $P_l\\stackrel {\\rm def}\n{=}\\otimes_{j=1}^{p_l-1}{\\xi_j}$; ${\\xi_j}$ is the probability\nmeasure on $X_j$. We denote this space by $(\\Omega, {\\mathcal\n{A}}, {{\\sym P}})$. So $x_{k,i}$, $1 \\leq i \\leq p_k -1$, is the\nprojection from $\\Omega$ onto the $i^{th}$ co-ordinate space of\n$\\Omega_k \\stackrel {\\rm def} {=} X_k^{p_k-1}$, $1 \\leq i \\leq\np_k-1$. Naturally each point $\\omega =(\\omega_k =\n(x_{k,i}(\\omega))_{i=1}^{p_k-1})_{k=0}^\\infty$ in $\\Omega$ defines\nthe spacers and therefore a rank one transformation\n$T_{\\omega,x}$, where $x=(x_{k,p_k})$.\n\n\\noindent{}This construction is more general than the\n construction due to Ornstein \\cite{Ornstein} which corresponds to the case \n$t_k=h_{k-1}$, $\\xi_k$ is the uniform distribution on $X_k$ and  $p_k >> h_{k-1}$. \\\\\n\n\n\\noindent{} We recall that Ornstein in \\cite{Ornstein} proved that\nthere exists a sequence ${(p_k,x_{k,p_k})}_{k \\in {\\sym N}}$ such that,\n$T_{\\omega,x}$ is almost surely mixing. Later in \\cite{Prikhodko}\nPrikhod'ko obtains the same result for some special choice of the\nsequence of the distribution ${(\\xi_m)}$ and recently, using the\nidea of D. Creutz and C. E. Silva \\cite{Creutz-silva} one can\nextend this result to a large family of probability measures\nassociated to Ornstein construction. In our general construction,\naccording to (\\ref{eqn:type1}) the spectral type of each\n$T_{\\omega}$ , up to a discrete measure, is given by\n   \\[\n\\sigma_{T_\\omega }=\\sigma^{(\\omega)}\n_{\\chi_{B_0}}=\\sigma^{(\\omega)} =W ^{*}\\lim \\prod_{l=1}^N\\frac\n1{p_l}\\left| \\sum_{p=0}^{p_l-1}z^{p(h_l+t_l)+x_{l,p}}\\right|\n^2d\\lambda.\n\\]\n\n\\noindent{} With the above notations, we state Bourgain theorem in\nthe following\nform:\\\\\n\n\\bigskip\n\\noindent {\\bf Theorem 4.1 (\\cite{elabdal2}). }{\\it  For every\nchoice of $(p_k), (t_k), (x_{k,p_k})$ and for any family of\nprobability measures ${\\xi_k}$ on $X_k=\\{-t_k,\\cdots,t_k\\}$ of\n${\\sym Z}$, $ {k \\in {\\sym N}^*}$, for which\n$$\\sum_{s \\in X_m}\\xi(s)^2 \\tend{m}{\\infty}0,$$ \\noindent{}the\nassociated generalized Ornstein transformations has almost surely\nsingular spectrum. i.e.,\n\\[\n{\\sym P}\\{\\omega~:~ \\sigma^{(\\omega)} ~\\bot ~\\lambda\\}=1.\n\\]\n\\noindent{}where ${\\sym P} \\stackrel {\\rm def} {=}\\otimes_{l=0}^{\\infty}\n\\otimes_{j=1}^{p_l-1}{\\xi_l}$ is the probability measure on\n$\\Omega=\\prod_{l=0}^\\infty X_l^{p_l-1}$. } \\vskip0.5cm\n\n\n\\noindent{}In the context of Ornstein's constrution, we state\nproposition 2.5 in the following form:\\\\\n\n\\noindent {\\bf Proposition 4.2. }{\\it There exist a subsequence of\nthe sequence $\\left (\\left |\\left| P_m(z)\\right|-1 \\right|\\right)$\nwhich converge weakly to some non-negative function\n$\\phi(\\omega,\\theta)$ which satisfies  $ \\phi \\leq $2, almost surely\nwith respect the ${\\sym P} \\bigotimes \\lambda$.}\n\\\\\n\n\\begin{proof}\nEasy exercise.\n\\end{proof}\\\\\n\n\n\\noindent Put, for any $\\omega\\in \\Omega$\n\\[\n\\alpha_{\\omega}=\\phi(\\omega,\\theta)d\\lambda.\n\\]\nWe shall prove that $\\alpha_{\\omega}$ is equivalent to the\nLebesgue measure for almost all $\\omega$. In fact, we have the\nfollowing proposition:\n\\\\\n\n\\noindent {\\bf Proposition 4.3. }{\\it There exist an absolutely\npositive constant $K$ such that\n  for almost all $\\omega$ we have\n$$\\alpha_{\\omega} \\geq K \\lambda.$$}\n\\\\\n\n\\noindent The proof is based on the following two lemmas.\n\n\\noindent {\\bf Lemma 4.4. }{\\it $\\displaystyle lim \\int \\int\n||P_m|-|P_m'|| d\\theta d{\\sym P} \\tend{n} {\\infty} 0$, where $\\displaystyle\nP_m'(\\theta)=P_m(\\theta)-\\int_{\\Omega}P_m(\\theta)d{\\sym P}.$ }\n\n\n\\begin{proof} For any $m \\in {\\sym N}$, we have\n\\begin{eqnarray*}\n\\int \\int ||P_m|-|P_m'|| d\\theta d{\\sym P} &\\leq& \\int \\int |P_m-P_m'|\nd\\theta d{\\sym P} \\\\ & = &\n\\int |\\int P_m d{\\sym P}| d\\theta\\\\\n&\\leq& \\int |\\sum_{p=0}^{p_m-1}\\frac1{\\sqrt{p_m}} z^{p(h_m+t_m))}| |\\sum_{s\\in\nX_{m}}\\xi_m(s)z^s| dz.\n\\end{eqnarray*}\nHence by Cauchy-Schwarz inequality\n\\begin{eqnarray*}\n\\int \\int ||P_m|-|P_m'|| d\\theta d{\\sym P} \\leq \\sum_{s \\in X_m}\\xi(s)^2\n\\tend{m}{\\infty}0.\n\\end{eqnarray*}\nThe proof of the lemma is complete.\n\\end{proof}\\\\\n\\vskip0.5cm \\noindent Now observe that we have\n\\[\n\\int_T |\\sum_{s\\in X_{m}}\\xi_m(s)z^s|^2 dz=\n\\int_T |\\sum_{s\\in X_{m}}\\xi_m(s)z^{2s}|^2 dz=\\sum_{s \\in X_m}\\xi(s)^2\n\\tend{m}{\\infty}0.\n\\]\nSo, we may extract a subsequence $(m_k)$ for which, for almost all\n$t \\in [0,2\\pi)$, we have\n\\[\n\\sum_{s\\in X_{m_k}}\\xi_{m_k}(s)e^{ i s t} \\tend{k}{\\infty}0\n{\\rm {~~and~~}} \\sum_{s\\in X_{m_k}}\\xi_{m_k}(s)e^{ i  s 2t} \\tend{k}{\\infty}0\n\\]\n\\noindent Define\n\\[\n\\Theta\n\\stackrel{def}{=}\\{\\theta ~:~ \\sum_{s\\in X_{m_k}}\\xi_{m_k}(s)e^{i\n s \\theta}\\tend{k}{\\infty}0 {\\rm {~and~}} \\sum_{s\\in X_{m_k}}\\xi_{m_k}(s)e^{ i  s 2t}\n \\tend{k}{\\infty}0\\}\n\\]\n\\noindent Choose $m \\in \\{m_k\\}$, $t \\in \\Theta$ and put, for $j \\in\n\\{0,\\cdots,p_m-1\\}$,\n\\begin{eqnarray*}\nY_{m,j}(\\omega)&=&\\cos((j(h_m+t_m)+x_{m,j}(\\omega))t)-\\int\n\\cos((j(h_m+t_m)+x_{m,j}(\\omega))t) d{\\sym P},\\\\\nZ_{m,j}(\\omega)&=&\\sqrt{\\frac{2}{p_m}}Y_{m,j}(\\omega).\n\\end{eqnarray*}\n\\\\\n\\noindent {\\bf Lemma 4.5. }{\\it For any fixed $t \\in \\Theta$, the\ndistribution of the sequence of random variables $\\displaystyle\n\\sum_{j=0}^{p_m-1} Z_{m,j}(\\omega)$ converges\nto the Gauss distribution. That is,\n\\\\\n\\begin{eqnarray}{\\label{eq:beq6}}\n{\\sym P} \\left \\{\\omega \\in \\Omega~~:~~\n\\sum_{j=0}^{p_m-1} Z_{m,j}(\\omega) \\leq x\n\\right \\} \\tend{m}{\\infty}\\frac1{\\sqrt{2\\pi}}\\int_{-\\infty}^{x}\ne^{-\\frac12u^2}du\\stackrel{\\rm {def}}{=}{\\mathcal {N}}(]-\\infty,x]).\n\\nonumber\n\\end{eqnarray}\n}\n\\begin{proof}\nSince the random variables are independent, centred and uniformly bounded by $\\displaystyle\n\\frac{2\\sqrt{2}}{\\sqrt{p_m}}$, the conditions (1), (2) and (4) of the\ntheorem 3.4 are satisfied. We have also the following:\n\\begin{eqnarray*}\n&&{\\sym E} \\left \\{ \\left(\\sum_{j=1}^{p_m-1}{Z_{m,j}^2-1}\\right) ^{2} \\right\\rbrace \\\\\n&& = \\frac{4(p_m-1)}{p_m^2}{\\sym E} ( Y_{m,1}^4)+\\frac{(p_m-1)(p_m-2)}{p_m^2}\n({\\sym E} (2Y_{m,1}^2))^2-2\\frac{p_m-1}{p_m} {\\sym E} (2 Y_{m,1}^2)+1.\n\\end{eqnarray*}\n\\noindent{}But ${\\sym E}(2Y_{m,1}^2) \\tend{m}{\\infty}1$, it follows\nthat the variables $\\{Z_{m,j}\\}$ satisfy  condition (3) of theorem 3.4.\nThus all the conditions of theorem 3.4 hold and we conclude that the distribution of\n$\\displaystyle \\sum_{j=0}^{p_m-1} Z_{m,j}(\\omega)$\nconverges to normal distribution.\n\\end{proof}\\\\\n\n\\begin{proof}{of proposition 4.3}\nLet $A$ be a Borel subset of ${\\sym T}$, $C$ a cylinder set in\n$\\Omega$, and $x \\in ]1,+\\infty[$, then, for any positive integer\n$m$, we have\n\n\\begin{eqnarray*}{\\label{eq :beqfin}}\n&&\\int_{A \\times C} ||P_m(\\theta)|-1| d\\lambda(\\theta) d{\\sym P} \\\\\n&\\geq& {\\sym P}(C) \\int_{A \\times \\Omega} ||P'_m(\\theta)|-1|\nd\\lambda(\\theta) d{\\sym P}\n-\\int||P_m|-|P_m'||d{\\sym P} d\\lambda\\\\\n&\\geq& {\\sym P}(C) \\int_{ \\{|P'_m|>x\\} \\bigcap A \\times\n\\Omega}||P'_m|-1| d\\lambda(\\theta)d{\\sym P}\n-\\int||P_m|-|P_m'||d{\\sym P} d\\lambda\\\\\n&\\geq&{\\sym P}(C) (x-1)\\int_A{\\sym P}\\{|\\Re\n{(P'_m(\\theta))}|>x\\}d\\lambda-\\int||P_m|-|P_m'||d{\\sym P} d\\lambda\n\\end{eqnarray*}\n\\noindent{} Let $m$ go to infinity and combine lemmas 4.4 and\n4.5 to get\n\n\\[\n\\int_{A \\times C} \\phi d\\lambda d{\\sym P} \\geq (x-1)\\{1-{\\mathcal\n{N}}([-\\sqrt{2}x,\\sqrt{2}x])\\}\\lambda(A) {\\sym P}(C).\n\\]\n\n\\noindent{}Put $K=(x-1)\\{1-{\\mathcal {N}}([-\\sqrt{2}x,\\sqrt{2}x])\\}$. Hence, for\nalmost all $\\omega$, we have, for any Borel set $A \\subset {\\sym T}$,  $ \\alpha_{\\omega}(A) \\geq K\n\\lambda(A)$, and the proof of the\nproposition is complete.\n\\end{proof}\\\\\n\n\\begin{proof}{of theorem 4.1}\nFollow easily from the proposition 4.3 combined with the\nproposition 2.6.\n\\end{proof}\n\\\\\n\n\\noindent{}{\\bf Remark.} We point out that there exist a rank one \nmixing transformations on the space with finite measure satisfying\nthe condition of theorem 3.6, In fact, following the notations of section 4, one may define\nthe spacers in the Ornstein construction by\n\\[\n {a_j^{(k)}}={3^j} t_k+x_{k,j}-x_{k,j-1},\n\\]\n\n\\noindent{}and choose the sequence $(t_k)_{k \\in {\\sym N}}$ such that the measure of dynamical system is finite.\nThus the condition of theorem 3.6 hold and the class is\nmixing almost surely.\\\\\n\n\\begin{center} {\\bf Acknowledgements}\n\\end{center}\nI would like to express my thanks to J-P. Thouvenot who posed to me\nthe problem of the singularity of the spectrum of the rank one\ntransformations, and also to B. Host, F. Parreau and F. Bassam for\nmany conversations on the subject I have had with them. My thanks\nalso to Andrew Granville and the organizers of SMS-NATO ASI 2005\nSummer School.\\\\\nI am also grateful to the referee for a number of valuable suggestions and remarks and for \nhaving correted the mistakes in the first version of this work.\n\n\\bibliographystyle{nyjplain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nLet $G\\subset \\mathrm{SU(2)}$ be a finite subgroup. According to McKay correspondence,\nsuch a subgroup gives rise to a graph $Q$ which turns out to be an affine\nDynkin diagram of ADE type. Let $\\g$ be the corresponding affine Lie algebra. \n\n\n\nThere are several approaches allowing one to  construct $\\g$ from $G$: \n\\begin{enumerate}\n\\item There is a geometric construction, due to Nakajima and Lusztig,  of $\\g$\n  and its representations in  terms of the quiver varieties associated to graph\n  $Q$. These varieties are closely related to the moduli spaces of instantons\n  on the resolution of singularities $\\widetilde{\\CC^2\/G}$. \n\n\\item There is an algebraic construction, due to Ringel \\cite{ringel2}, which\n  allows one to  get the universal enveloping algebra $U\\n_+$ of the positive\n  part of $\\g$ as the Hall algebra of the category $\\Rep Q_h$, where $h$ is an\n  orientation of $Q$ and $Q_h$ is the corresponding quiver. It was shown by \n  Peng and Xiao \\cite{peng-xiao} that replacing $\\Rep Q_h$ by the\n  quotient  $R=D^b(\\Rep Q_h)\/T^2$ of the corresponding derived category, we\n  can get a description of all of $\\g$.  Different choices of orientation\n  give rise to equivalent derived categories  $R$, with equivalences given\n  by  Bernstein-Gelfand-Ponomarev reflection functors; in\n  terms of $\\g$, these functors correspond to the action of the braid group\n  of the corresponding Weyl group.\n  \n  \\end{enumerate}\nRecently, a third approach was suggested by Ocneanu \\cite{ocneanu}\n(unpublished), in a closely related setting of a subgroup in quantum\n$\\mathrm{SU(2)}$, for  $q$ being a root of unity. His approach is based on\nstudying essential paths in ${\\widehat{Q}}=Q\\times \\Z\/h\\Z$, where $h$ is the\norder of the root of unity. This approach is purely combinatorial: all\nconstructions are done using this finite graph and vector spaces of\nessential paths between points in this graph, without involving any\ncategories at all. \n\nThis paper grew out of the author's attempt to understand Ocneanu's\nconstruction and in particular, find the appropriate\ncategorical interpretation of  his combinatorial constructions;\nhowever, for simplicity we  do it for subgroups in classical $\\mathrm{SU(2)}$,\nleaving the analysis of the subgroups in quantum $\\mathrm{SU(2)}$ for future\npapers. We show that Ocneanu's essential paths in ${\\widehat{Q}}$ have a\nnatural interpretation in terms of the category  ${Coh_G(\\PP)}$ of\n$G$--equivariant $\\OO$-modules on $\\PP$ (or, rather,\nits ``even part'' $\\C={Coh_G(\\PP)}_0$, for certain natural $\\Z_2$ grading on\n${Coh_G(\\PP)}$). This also provides a relation with Ringel--Lusztig\nconstruction: for (almost) any choice of\norientation $h$ of $Q$, we construct an equivalence of triangulated\ncategories \n$$\nR\\Ph_h\\colon D^b( {Coh_G(\\PP)}_0)\\simeq D^b(\\Rep Q_h)\n$$\nThese equivalences agree with the equivalences of $D^b(\\Rep Q_h)$ for\ndifferent choices of $h$ given by reflection functors. As a corollary, we\nsee that the Grothendieck group $L=K(\\C)$ is an affine root lattice, and\nthe set $\\Delta$ of classes of indecomposable modules is an affine root\nsystem. \n\nThis construction of the affine root system via equivariant\nsheaves has a number of remarkable properties,  namely:\n\\begin{enumerate}\n  \\item This does not require a choice of orientation of\n    $Q$ (unlike   the category $D^b(\\Rep Q_h)$, where we first choose\n    an  orientation and then   prove that the resulting\n    derived category is independent of orientation).\n  \\item The indecomposable objects in the category ${Coh_G(\\PP)}$ can be\n    explicitly described. Namely, they  are the\n    sheaves $\\OO(n)\\ttt X_i$, where $X_i$ are irreducible\n    representations of $G$ (these sheaves and their translations\n    correspond to real roots of $\\g$) and\n    torsion sheaves, whose support is a $G$--orbit in $\\PP$ (in\n    particular, torsion sheaves whose support is an orbit of a\n    generic  point correspond to imaginary roots of $\\g$).\n  \\item This construction of the affine root system does not give \n    a natural  polarization into negative and positive roots.\n    Instead, it gives a canonical Coxeter element in the corresponding\n    affine Weyl group, which is given by $C[\\F]=[\\F\\ttt \\OO(-2)]$\n    (this  also corresponds to the  Auslander--Reiten functor\n    $\\tau$).\n \n  \\item This construction gives a bijection of the vertices of the\n    affine Dynkin diagram $Q$ and (some of) the $C$-orbits in\n    $\\Delta$ (as opposed to the quiver construction, where vertices\n      of $Q$ are in bijection with the simple positive roots). \n          \n\\end{enumerate}\n\n\n\nFor $G=\\{1\\}$ (which is essentially equivalent to $G=\\{\\pm 1\\}$),\nthese results were first obtained in the paper \\cite{baumann-kassel},\nwhere it was shown that the corresponding Hall algebra contains the\nsubalgebra isomorphic to $U\\widehat{\\slt}$. This  paper in turn was\ninspired by an  earlier paper of Kapranov \\cite{kapranov}. \n\n\nIt should be noted that many of the results obtained here have already\nbeen proved in other ways. Most importantly, it had been proved by\nLenzing \\cite{geigle-lenzing,lenzing} that the derived category  of\nequivariant sheaves on $\\PP$ is equivalent to the derived category of\nrepresentation of the corresponding quiver; this result has been used by\nSchiffmann \\cite{schiffmann,schiffmann2} for construction of a subalgebra\nin the corresponding affine Lie algebra via Hall algebra of the category of\nequivariant sheaves. However, the construction of\nequivalence in \\cite{lenzing} is different than the one suggested here. The\nprimary difference is that in Lenzing's construction, the Dynkin diagram\n$Q$ is constructed as the star diagram, with lengths of branches determined\nby the branching points of the cover $\\PP\\to X=\\PP\/G$, and he uses\na standard orientation of this diagram. In the construction presented\nhere, the diagram is $Q$ is defined in a more standard way, using the set\n$I$ of irreducible representations of $G$; more importantly, we construct\nnot a single equivalence but a collection of equivalences, one for  each\nadmissible orientation. In Lenzing's construction, torsion sheaves\nnaturally play a major role; in our construction, we concentrate on\nthe study of locally free sheaves. \n\nLenzing's results   apply not only to $\\PP\/G$ but to a much larger\nclass of ``non-commutative projective curves''. However, the downside of\nthis is that the language he uses is rather technical, making his papers\nsomewhat hard for non-experts. For this reason, we had chosen to  give\nindependent proofs of some of the results, thus saving the reader the\nnecessity of learning the language of non-commutative  curves. Of course,\nwe tried to clearly mark the results which had already been known.\n\n\n{\\bf Acknowledgments.} The author would like to thank P.~Etingof,\nV.~Ostrik, and O.~Schiffmann for fruitful discussions. Special thanks to\nA.~Ocneanu, whose talk inspired this paper. \n\n\\section{Basic setup}\\label{s:basic}\nThroughout the paper, we work over the base field $\\CC$ of complex\nnumbers. $G$ is a finite subgroup in $\\mathrm{SU(2)}$; for simplicity, we assume\nthat $\\pm I\\in G$ and denote  $\\Gbar=G\/\\{\\pm I\\}$ (this excludes\n$G=\\Z_n$, $n$ odd; this case can also be included, but some of the\nconstructions of this paper will require minor changes). We denote by\n$V$ the standard 2-dimensional space, considered as a tautological\nrepresentation of $\\mathrm{SU(2)}$ (and thus of $G$) and \n$$\nV_k=S^k V\n$$\nis the $k+1$--dimensional irreducible representation of $\\mathrm{SU(2)}$ (and\nthus a representation, not necessarily irreducible, of $G$). \n\nWe denote by $\\Rep G$ the category of finite-dimensional\nrepresentations of $G$ and by $I=\\Irr G$ the set of isomorphism\nclasses of simple representations; for $i\\in I$, we denote by $X_i$\nthe corresponding representation of $G$. Since $G\\supset \\{\\pm I\\}$,\nthe category $\\Rep G$ is naturally $\\Z_2$--graded:\n\\begin{equation}\\label{e:even-odd}\n\\Rep G=\\Rep_0 G\\oplus \\Rep_1 G,\\quad \n\\Rep_p G=\\{V\\in \\Rep G\\st (-I)|_V=(-1)^p\\id \\}\n\\end{equation}\nFor homogeneous object $X$, we define its ``parity'' $p(X)\\in \\Z_2$\nby  \n\\begin{equation}\\label{e:parity}\np(X)=p\\text{ if } X\\in \\Rep_p G\n\\end{equation}\nin particular, $p(V)=1$. We will also define, for $i\\in I$, its\nparity $p(i)=p(X_i)$. This  gives a decomposition \n\\begin{equation}\\label{e:decomposition_I}\nI=I_0 \\sqcup I_1.\n\\end{equation}\n\n\nWe define the graph $Q$ with the set of vertices $V(Q)=I$ and\nfor every two vertices $i,j$, the number of edges connecting them is\ndefined by \n$$\nn(i,j)=\\dim \\Hom_G(X_i,X_j\\ttt V)\n$$\n\nNote that decomposition \\eqref{e:decomposition_I} shows that this\ngraph is bipartite: $V(Q)=V_0(Q)\\sqcup V_1(Q)$, and all edges connect\nvertices of different parities. \n\n\nIt is well-known that  one can construct an isomorphism\n$$\n\\{\\text{Paths of length $l$ in $Q$ connecting $i,j$}\\}\n=\\Hom_G(X_i, V^{\\ttt l}\\ttt X_j)\n$$\nand that $\\Hom_G(X_i, V_l \\ttt X_j)$ can be described as the space of\n``essential paths'' in $Q$, which is naturally a direct summand in\nthe space of all paths (see \\cite{coq-garcia}). The algebra of essential\npaths is also known as the preprojective algebra of $Q$. \n\n\nBy McKay correspondence, $Q$ must be an affine Dynkin diagram. \nWe denote by $\\Delta(Q)$ the  corresponding affine root system; it has a\nbasis of simple roots $\\alpha_i, i\\in I$. We denote \n$$\nL(Q)=\\bigoplus_{i\\in I} \\Z\\alpha_i =\\Z^I\n$$\nthe corresponding root lattice. It has a natural  bilinear\nform given by $(\\alpha_i,\\alpha_i)=2$ and $(\\alpha_i,\\alpha_j)=-n(i,j)$; as is\nwell-known, this form is positive semidefinite. The kernel of this\nform is $\\Z\\delta$, where $\\delta$ is the imaginary root of $\\Delta(Q)$. \n\nWe denote by $s_i\\colon L(Q)\\to L(Q)$ the reflection around root\n$\\alpha_i$. \n\nFinally, let $K(G)$ be the Grothendieck group of the category $\\Rep G$.\nIt is  freely generated by classes $[X_i], i\\in I$; thus, we have a\nnatural isomorphism \n\\begin{equation}\\label{e:K(G)}\n\\begin{aligned}\nK(G)&\\isoto L(Q)\\\\\n[X_i]&\\mapsto \\alpha_i\n\\end{aligned}\n\\end{equation}\n\n\n\n\\section{Quiver $Q_h$}\n\nWe will consider a special class of orientations of $Q$. \n\n\\begin{definition}\\label{d:height_function}\nA {\\em height function} $h$ is a map $I\\to \\Z$ such that \n$h(i)-h(j)=\\pm 1$ if $i,j$ are connected by an edge in $Q$, and\n$h(i)\\equiv  p(i) \\pmod 2$. \n\\end{definition}\n\nEvery height function gives rise to orientation of edges of $Q$: if\n$h(j)=h(i)-1$ then all edges connecting $i$ and $j$ are directed\ntowards $j$:\n$$\ni\\longrightarrow j \\quad\\text{ if } h(j)=h(i)-1\n$$\n\n\nWe will denote by $Q_h$ the quiver given by this orientation. We will\nwrite $i\\to j$ if there exists an edge whose tail is $i$ and head is\n$j$. The notation $\\sum_{j\\colon j\\to i}$ will mean the sum over all\nvertices  $j$  connected with $i$ by an edge $j\\to i$; if there are\nmultiple edges, the corresponding vertex $j$ will be taken more than\nonce. \n\nOrientations obtained in this way will be called {\\em\nadmissible} (note: our use of this word is slightly different from the\nuse in other sources). It is easy to check that if $Q$ has no loops,\nthen any orientation of $Q$ is admissible. For type $A$, an\norientation is admissible if the total number of clockwise arrows is\nequal to the number of counterclockwise ones (which again rules out\ntype $\\widehat{A}_n$, $n$ even, corresponding to $G=\\Z_{n+1}$).  It is\nalso obvious that $Q_h$ has no oriented loops, and that\nadding a constant to $h$ gives the same orientation. \n\nGiven a height function $h$, we will draw $Q$ in the plane so that\n$h$ is the $y$--coordinate; then all edges of $Q_h$ are directed\ndown. \\firef{f:Qh} shows an example of a height function for\nquiver of type $D$.\n\\begin{figure}[ht]\n\\fig{Qh.eps}\n\\caption{Example of  a height function for Dynkin diagram\n$\\widehat{D}_7$}\\label{f:Qh}\n\\end{figure}\n\n\\begin{definition}\\label{d:reflection_functors1}\nLet $h$ be a height function on $Q$, and $i\\in I$ be a sink in $Q_h$:\nthere are no edges of the form $i\\to j$ (in terms of $h$, it is\nequivalent to saying that $h$ has a local minimum at $i$). We define\nnew height function \n$$\ns_i^+h (j)=\\begin{cases}\n              h(j)+2, & j=i\\\\\n              h(j),   & j\\ne\\i\n           \\end{cases}\n$$\nSimilarly, if $i$ is a source, i.e., there are no vertices of the\nform $j\\to i$ (equivalently, $h$ has a local maximum at $i$), then \nwe define \n$$\ns_i^-h (j)=\\begin{cases}\n              h(j)-2, & j=i\\\\\n              h(j),   & j\\ne\\i\n           \\end{cases}\n$$\n\\end{definition}\nOne easily sees that $s_i^+h$, $s_i^-h$ are again height functions;\nthe quiver $Q_{s_i^\\pm h}$ is obtained from $Q_h$ by reversing\norientation of all edges adjacent to $i$. We will refer to $s_i^\\pm$\nas (elementary) orientation reversal operations. \n\nThe following lemma is known; however, for the benefit of the reader\nwe included the proof. \n\n\\begin{lemma}\\label{l:reflection_functors1}\nAny two height functions $h, h'$ can be obtained one from another by\na sequence of orientation reversal operations $s_i^\\pm$. \n\\end{lemma}\n\\begin{proof}\nDefine the ``distance'' between two height functions by \\\\\n$d(h,h')=\\sum_I |h(i)-h'(i)|\\in \\Z_+$. We will show that if $d(h,h')>0$, then\none can\nfind $s_i^\\pm$ such that $s_i^\\pm$ can be applied to $h$, and\n$d(s_i^\\pm h, h')<d(h, h')$; from this the theorem clearly follows. \n\nLet $I_+=\\{i\\st h(i)>h'(i)\\}$. One easily sees that if $i\\in I_+$, and\n$j\\to i$ in $Q_h$, then $j\\in I_+$. Thus, if $I_+$ is non-empty, it \nmust contain at least one source $i$ for $Q_h$.  But then $d(s_i^- h,\nh')=d(h, h')-2$. \n\nSimilarly, let $I_-=\\{i\\st h(i)<h'(i)\\}$. If $I_-$ is non-empty,\nsimilar\nargument shows it  must contain at least one sink $i$ for $Q_h$. \nBut then $d(s_i^+ h,h')=d(h, h')-2$. \n\\end{proof}\n\n\n\\section{Representations of quivers}\nFor readers convenience, we recall here the known results about the\nrelation between representations of quiver $Q_h$ and root system\n$\\Delta(Q)$ (see, e.g., \\cite{drab-ringel}, \\cite{kraft-riedtmann}). \n\nLet $h$ be a height function on $Q$ and $Q_h$ the corresponding quiver.\nConsider the category $\\Rep Q_h$ of finite-dimensional representations of\n$Q_h$; similarly, let $\\D^b(Q_h)$ be the bounded derived category of $\\Rep\nQ_h$. We denote by $K(Q_h)$ the Grothendieck group of $\\Rep Q_h$ which can also\nbe described as the Grothendieck group of $\\D^b(Q_h)$. It is well-known that\nthe category $\\Rep Q_h$ is a hereditary abelian  category: \n$$\n\\Ext^i(X,Y)=0\\text{ for all }i>1. \n$$\n\n For a representation  $M=(M_i)_{i\\in I}$ of $Q_h$, we define its\ndimension $\\dim X\\in L(Q)$ by  $\\dim M=\\sum (\\dim M_i)\\alpha_i$ (recall that\n$L(Q)=\\Z[I]$ is the root lattice of the root system $\\Delta(Q)$, see\n\\seref{s:basic}). \n \nThe following theorem summarizes some of the known results about\nrepresentations of quivers and root system $\\Delta(Q)$. \n\n\n\\begin{theorem}\\label{t:reps_of_quivers}\n\\par\\indent\n\\begin{enumerate}\n\\item The map $[X]\\mapsto \\dim X$ gives an isomorphism $K(Q_h)\\isoto\nL(Q)$. Under this  isomorphism, the bilinear form on $L(Q)$ is identified\nwith the following   bilinear form on $K(Q_h)$: \n  $$\n    (x,y)=\\<x,y\\>+\\<y,x\\>\n  $$\n  where by definition \n  $$\n   \\<[X],[Y]\\>= \\dim \\RHom (X,Y)=\\dim \\Hom (X,Y) -\\dim \\Ext^1(X,Y).\n  $$\n  \\item The set of dimensions of indecomposable modules is exactly the\n    set $\\Delta_+(Q)$ of positive roots in $\\Delta(Q)$. For real roots\n    $\\alpha$, there is exactly one  up to isomorphism indecomposable\n    module $M_\\alpha$ of dimension $\\alpha$; for\n    imaginary root $\\alpha$, there are infinitely many pairwise\n    non-isomorphic modules of dimension $\\alpha$. \n  \n\\end{enumerate}\n\\end{theorem}\n\nThere is also an explicit description of indecomposables in\n$\\D^b(Q_h)$ (see \\cite[Lemma I.5.2]{happel}). \n\\begin{theorem}\\label{t:indecompos_derived}\n  Indecomposable objects in $\\D^b(Q_h)$ are of the form $M[n]$, where\n  $M$ is an indecomposable object in $\\Rep Q_h$, $n\\in \\Z$. \n\\end{theorem}\n\n\nWe will also need reflection functors of Bernstein--Gelfand--Ponomarev.\nRecall that if $i$ is a sink in $Q_h$, then one has a natural functor \n\\begin{equation}\\label{e:reflection_2}\n  S_i^+\\colon \\Rep(Q_h)\\to \\Rep(Q_{s_i^+h})\n\\end{equation}\nsimilarly, if $i$ is a source, one has a natural functor \n\\begin{equation}\\label{e:reflection_3}\n  S_i^-\\colon \\Rep(Q_h)\\to \\Rep(Q_{s_i^-h})\n\\end{equation}\n(see definition in \\cite{drab-ringel},\n\\cite{kraft-riedtmann}). \n\nThe following result is known and easy to prove, so we skip the\nproofs. \n\\begin{theorem}\\label{t:coxeter2}\n\\par\\indent \n  \\begin{enumerate}\n  \\item Functor $S_i+$ is left exact and $S_i^-$ is right exact. \n  \n  \n  We will denote by $RS_i^+, LS_i^-\\colon \\D^b(Q_h)\\to\n    \\D^b(Q_{s_i^\\pm h})$ the  corresponding derived functors. \n  \\item\n    The functors $RS_i^+$, $LS_i^-$ are equivalences of categories\n    $\\D^b(Q_h)\\to\\D^b(Q_{s_i^\\pm h})$, which induce the usual\n    reflections $s_i$ on the  Grothendieck group: \n    $$\n    \\dim RS_i^+ (X)=s_i(\\dim X), \\quad \\dim LS_i^- (X)=s_i(\\dim X)\n    $$\n    \\item If $i,j$ are not neighbors in $Q$, then $RS_i^+,\n    RS_j^+$ commute (i.e., compositions in different orders are\n    isomorphic) and similarly for $LS_i^-$. \n  \\end{enumerate}\n\\end{theorem}\n\n\n\nIn particular, for a given height function $h$ let $s^+_{i_1}\\dots\ns^+_{i_r}$ be a sequence of elementary orientation\nreversal operations such that  \n$$\ns_{i_1}^+\\dots s_{i_r}^+ (h) =h+2\n$$\nOne easily sees that this condition is equivalent to requiring that every\nindex $i\\in I$ appears in the sequence $\\{i_1,\\dots, i_r\\}$ exactly once;\nit follows from \\leref{l:reflection_functors1}  that for every height\nfunction $h$, such sequences of orientation reversal operations exist. \nFor such a sequence,  the corresponding element of the  Weyl group\n\\begin{equation}\\label{e:coxeter}\nc^+_h=s_{i_1}\\dots s_{i_r}\n\\end{equation}\nis called the {\\em Coxeter element}, and\nthe corresponding composition of reflection functors \n\\begin{equation}\\label{e:C+}\nRC^+_h=RS_{i_1}^+\\dots RS_{i_r}^+\\colon \\D^b(Q_h)\\to\\D^b(Q_{h+2})\n\\end{equation}\nwill be called the {\\em Coxeter functor}. Note that since $Q_h\\simeq\nQ_{h+2}$ as a quiver, we can consider$RC_h^+$ as an autoequivalence of\n$\\D^b(Q_h)$. \n\nIt is easy to show (see \\cite{drab-ringel}, \\cite{shi}) that the\nCoxeter element $c^+_h$ only depends on $h$ and not on the choice of\nthe sequence $i_1,\\dots,i_r$; moreover, the proof of this only uses\nthe fact that $s_i, s_j$ commute if $i,j$ are not connected in $Q$\nand does not use the braid relations. Thus, by \\thref{t:coxeter2},\nthis implies that up to an isomorphism, $RC^+_h$ also depends only on\n$h$; this justifies the notation $RC_h^+$.\n\nSimilarly, we can define  functors \n\\begin{equation}\\label{e:coxeter2}\nLC^-_h\\colon \\D^b(Q_h)\\to\\D^b(Q_{h-2})\\simeq\\D^b(Q_h)\n\\end{equation}\nusing sequences of orientation reversals  $s_{i_1}^-\\dots s_{i_r}^- h\n=h-2$; the corresponding element of the Weyl group will be denoted by\n$c_h^-$. As before, it can be shown that $LC_h^-$, $c_h^-$ only depend\non $h$\n\n\nFor readers familiar with the theory of Auslander--Reiten sequences\n(see \\cite{auslander-reiten}, \\cite{happel}), we add\nthat the category $\\Rep Q_h$ has Auslander--Reiten sequences, and the\nAuslander--Reiten functor $\\tau$ is given by $\\tau=C_h^-$.\n\n\n\\section{Equivariant sheaves}\nIn this section, we introduce the main object of this paper, the\ncategory of equivariant sheaves on $\\PP$. Most of  the results of\nthis section  are well-known and given here only for the convenience\nof references. \\leref{l:frobenius} does not seem to be easily\navailable in the literature, but is very easy to prove. \n\nLet $V^*$ be the dual of the tautological representation $V$ of\n$\\mathrm{SU(2)}$. Since $G$ is a finite subgroup in $\\mathrm{SU(2)}$, it naturally acts on\n$\\PP=\\mathbb{P}(V^*)$, and the structure sheaf $\\OO$ has a standard\n$\\mathrm{SU(2)}$- (and thus $G$-) equivariant structure. Moreover, all twisted\nsheaves $\\OO(n)$ also have a standard equivariant structure, so that\nthe space of global sections $\\Gamma(\\OO(n))$ is a representation of \n$G$: \n\\begin{equation}\\label{e:H0}\n\\Gamma(\\OO(n))=\\begin{cases}\n                  S^nV=V_n, &n\\ge 0\\\\\n                  0, &n<0\n                  \\end{cases}\n\\end{equation}\nSimilarly, the higher homology spaces are naturally representations of $G$: it\nis well-known that $H^i(\\PP, \\OO(n))=0$ for $i>1$, and \n\\begin{equation}\\label{e:H1}\nH^1(\\PP,\\OO(n))=\\begin{cases}\n                  S^{-n-2}V^*=V^*_{-n-2}, &n\\le -2\\\\\n                  0, &n\\ge -1\n                  \\end{cases}\n\\end{equation}\n\nLet ${Qcoh_G(\\PP)}$, ${Coh_G(\\PP)}$ be the categories of $G$--equivariant\nquasi-coherent (respectively, coherent) $\\OO$--modules on $\\PP$\n(see, e.g., \\cite[Section 4]{bkr} for definitions). Note that we are\nconsidering isomorphisms $\\lambda_g\\colon \\F\\to g^*\\F$ as part of the\nstructure of the $G$--equivariant sheaf. For brevity, we will denote \nmorphisms and $\\Ext$ groups in ${Qcoh_G(\\PP)}$ by $\\Hom_G(X,Y)$, $\\Ext_G(X,Y)$.\nFor an equivariant  sheaf $\\F$ we will denote \n$$\n\\F(n)=\\OO(n)\\ttt_\\OO \\F\n$$\nwith the obvious equivariant structure. Similarly, for a\nfinite-dimensional representation $X$ of $G$, we denote \n$$\nX(n)=\\OO(n)\\ttt_\\CC X.\n$$\n\n\nWe list here some of the basic properties of equivariant\nsheaves; proofs can be found in \\cite[Section 4]{bkr}.\n\\begin{theorem}\\label{t:Gcoh}\n\\par\\noindent\n  \\begin{enumerate}\n    \\item ${Qcoh_G(\\PP)}$ is an abelian category, and a sequence $0\\to\n    \\F_1\\to \\F_2\\to \\F_3\\to 0$ is exact in ${Qcoh_G(\\PP)}$ iff it is exact in\n    $Qcoh(\\PP)$. \n\n  \\item For any $\\F,\\G\\in {Qcoh_G(\\PP)}$, the space $\\Hom_\\OO(\\F,\\G)$ is a\n    representation of $G$, and $\\Hom_G(\\F,\\G)=(\\Hom_\\OO(\\F,\\G))^G$.\n    Similarly, \\\\\n    $\\Ext_G^i(\\F,\\G)=(\\Ext^i_\\OO(\\F,\\G))^G$; in\n    particular, $\\Ext_G^i(\\F, \\G)=0$ for $i>1$.\n    \n   \\item For any $\\F,\\G\\in {Coh_G(\\PP)}$, the spaces  $\\Hom_G(\\F,\\G)$,\n    $\\Ext_G^1(\\F,\\G)$ are finite-dimensional. \n\n  \\item For any $\\F,\\G\\in {Coh_G(\\PP)}$, one has \n  \\begin{align*}\n  \\Hom_G(\\F, \\G(n))=\\Hom_G(\\F(-n),\\G)=0\\quad \\text{for }n\\ll 0,\\\\\n  \\Ext_G^1(\\F, \\G(n))=\\Hom_G(\\F(-n),\\G)=0\\quad \\text{for }n\\gg 0,\n  \\end{align*}\n \n\\end{enumerate}\n\\end{theorem}\n\n\nAs an immediate corollary of this, we see that the category ${Coh_G(\\PP)}$ has the \nKrull--Schmidt property: every object $\\F\\in {Coh_G(\\PP)}$ can be written as a direct\nsum of indecomposable modules, and the multiplicities do not depend on the\nchoice of such decomposition. It is also a hereditary category (recall that a\ncategory is called hereditary if $\\Ext^2(A,B)=0$ for any objects $A,B$; in\nparticular, this implies that a quotient of an injective object is injective,\nas can be easily seen from the long exact sequence of $\\Ext$ groups).  \n\nWe say that $\\F\\in{Coh_G(\\PP)}$ is locally free if it is locally free as an\n$\\OO$-module. \n\\begin{theorem}\\label{t:locally_free}\n\\par\\noindent\n\\begin{enumerate}\n  \\item Every locally free $\\F\\in{Coh_G(\\PP)}$ is isomorphic to direct sum of\n    sheaves of the  form \n    \\begin{equation}\n    X_i(n)=\\OO(n)\\ttt X_i, \\quad X_i \\text{ -- irreducible representation of\n$G$}    \\end{equation}\n  \\item If $\\F\\in{Coh_G(\\PP)}$ is locally free, then the functor $\\ttt \\F\\colon\n    {Coh_G(\\PP)}\\to {Coh_G(\\PP)}$ is exact. \n   \n \\item Every coherent $G$-equivariant sheaf has a resolution consisting of\n    locally free sheaves. \n \\item \\textup{(}Serre Duality\\textup{)} \n   For any two locally free sheaves $\\F, \\G$ we have isomorphisms \n   $$\n   \\Ext_G^1(\\F, \\G(-2))=\\Ext^1(\\F(2),\\G)=\\Hom_G(\\G, \\F)^* \n   $$\n\\end{enumerate}\n\\end{theorem}\n\nIt immediately follows from computation of homology of $\\OO(n)$ as coherent\nsheaves that \n\\begin{equation}\\label{e:hom1}\n  \\Hom_G ( X_i(n), X_j(k))\n  =\\begin{cases}\n     \\Hom_G( X_i, V_{k-n}\\ttt X_j), &k\\ge n\\\\\n     0, & k<n\n      \\end{cases}\n\\end{equation}\n\\begin{equation}\\label{e:hom2}\n  \\Ext_G^1 (X_i(n), X_j(k))\n  =\\begin{cases}\n     \\Hom_G(X_i, V^*_{n-k-2}\\ttt X_j), &k\\le n-2\\\\\n     0, & k\\ge n-1\n      \\end{cases}\n\\end{equation}\n\nFinally, we note that the action of $(-I)\\in \\mathrm{SU(2)}$ gives a decomposition\nof ${Coh_G(\\PP)}$ into even and odd part. Namely, since $-I$ acts trivially on\n$\\PP$, structure of $G$-equivariant sheaf gives an isomorphism\n$\\lambda_{-I}\\colon  \\F\\to (-I)^*\\F=\\F$. We define \n$$\n{Coh_G(\\PP)}_p=\\{\\F\\in{Coh_G(\\PP)}\\st \\lambda_{-I}=(-1)^p\\}, \n\\quad p\\in \\Z_2\n$$\n\n\nFrom now on, the main object of our study will be the category \n\\begin{equation}\\label{e:C}\n\\C={Coh_G(\\PP)}_0.\n\\end{equation}\nIn particular, a locally free sheaf $X_i(n)\\in \n\\C$ iff $n+p(X_i)\\equiv 0 \\mod 2$, where $p(X_i)$ is the parity defined by\n\\eqref{e:parity}. \n\nOne easily sees that $\\C$ is a full subcategory in ${Coh_G(\\PP)}$ closed under\nextensions. Equivalently, it can be described as follows. \n\n\\begin{lemma}\nLet $\\Gbar=G\/\\{\\pm I\\}$.Then $\\C$ is naturally equivalent to the\ncategory ${Coh_{\\Gbar}(\\PP)}$ of $\\Gbar$-equivariant coherent sheaves on $\\PP$.\n\\end{lemma}\n\n\nWe will denote the bounded  derived category of $\\C$ by  $\\D^b(\\C)$.\n\nFor future use, we will also need an analogue of induction functor.\nRecall that for any finite subgroup $H\\subset G$, we have an\ninduction functor $\\Ind_H^G\\colon \\Rep H\\to \\Rep G$. In particular,\nfor  $H=\\{1\\}$ and the trivial representation $\\CC$ it gives the\nregular representation of $G$:\n\\begin{equation}\\label{e:regular_rep}\nR=\\Ind_{\\{1\\} }^G\\CC\\simeq \\bigoplus_{i\\in I}d_i X_i,\n\\qquad d_i=\\dim X_i\n\\end{equation}\nAs any representation of $G$, $R$ can be decomposed into even and odd\npart (cf. \\eqref{e:even-odd}): $R=R_0\\oplus R_1$, where \n\\begin{equation}\\label{e:R_p}\nR_p=\\bigoplus_{i\\in I_p}d_iX_i\n\\end{equation}\n\nIt is easy to check that $R_0$ is exactly the regular representation\nof $\\Gbar$. \n\nSimilarly, we have an induction functor from the category of\n$H$-equivariant sheaves to $G$--equivariant sheaves; in particular,\nfor $H=\\{1\\}$, we get a functor $Coh(\\PP)\\to{Coh_G(\\PP)}$. However, it will be\nmore convenient to consider the functor \n\\begin{equation}\\label{e:ind}\n  \\begin{aligned}\n  \\Ind\\colon Coh(\\PP)&\\to\\C={Coh_{\\Gbar}(\\PP)}\\\\\n            \\F&\\mapsto \\bigoplus_{g\\in\\Gbar}g^*\\F \n  \\end{aligned}\n\\end{equation}\nThe following lemma lists some of the properties of this functor; the\nproof is straightforward and left to the reader.\n\\begin{lemma}\\label{l:frobenius}\nLet $\\Ind\\colon Coh(\\PP)\\to\\C$ be defined by \\eqref{e:ind}. Then for even\n$n$, $\\Ind \\OO(n)$ is naturally isomorphic as an \n    equivariant sheaf to $R_0\\ttt \\OO(n)$; for odd $n$, $\\Ind \\OO(n)$\n      is    naturally isomorphic as an \n    equivariant sheaf to $R_1\\ttt \\OO(n)$.\n  \n\\end{lemma}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Auslander--Reiten relations}\nThe following result will play a key role in the study of the category\n${Coh_G(\\PP)}$. \n\n\\begin{theorem}\\label{t:auslander_reiten}\nFor any $n\\in \\Z$, $i\\in I$, there is short exact sequence in ${Coh_G(\\PP)}$: \n\\begin{equation}\\label{e:aus-reiten1}\n0\\to X_i(n)\\to \\sum_j X_j(n+1)\\to X_i(n+2)\\to 0\n\\end{equation}\nwhere the sum is over all neighbors $j$ of $i$ in $Q$. \n\\end{theorem}\nWe will call sequences of the form \\eqref{e:aus-reiten1}  {\\em\nAuslander--Reiten sequences}.\n\n\n\\begin{proof}\nNotice first that we have a short exact sequence of $\\OO$-modules:\n$$\n0\\to \\OO\\to \\OO(1)\\ttt V\\to \\OO(2)\\to 0\n$$ \nLet us tensor it with $X_i(n)$. Since tensoring with a locally free sheaf\nis an exact functor (\\thref{t:locally_free}), this  gives\nshort exact sequence \n\\begin{equation}\\label{e:aus-reiten2}\n0\\to X_i(n)\\to  (V\\ttt X_i)(n+1)\\to X_i(n+2)\\to 0\n\\end{equation}\nSince $V\\ttt X_i=\\sum_{j} X_j$, we get the statement of the theorem. \n\\end{proof}\n\n\\begin{remark}\nFor readers familiar with the general theory of Auslander--Reiten (AR)\nsequences (see \\cite{auslander-reiten} for the overview of the\ntheory), we point out that our use of this name is justified: it can\nbe shown, using Serre duality that these sequences do satisfy the\nusual definition of AR sequences (see \\cite{reiten-bergh}). This, in\nparticular, shows that the Auslander--Reiten functor $\\tau$ for locally\nfree equivariant sheaves is given by \n$$\n\\tau(\\F)=\\F(-2).\n$$\nHowever, we are not going to use the general theory of AR sequences in\nthis paper.\n\\end{remark}\n\\begin{corollary}\n  For any $n\\in \\Z, i\\in I$ we have the following relations in\n  the Grothendieck group $K({Coh_G(\\PP)})$\n    \\begin{equation}\\label{e:aus-reiten3}\n    [X_i(n)]-\\sum_{j}[X_j(n+1)]+[X_i(n+2)]=0\n    \\end{equation}\n  where the sum is over all neighbors $j$ of $i$ in $Q$.\n\\end{corollary}\nLater we will study the Grothendieck group $K({Coh_G(\\PP)})$ in more detail,\nin particular showing that relations \\eqref{e:aus-reiten3} is the full\nset of relations among classes $[X_i(n)]$ (see\n\\coref{c:generators_of_K(C)}). \n\n\n\n\\section{Indecomposable objects in ${Coh_G(\\PP)}$}\nIn this section, we describe the indecomposable sheaves in ${Coh_G(\\PP)}$.\nAgain, these results are not new; they  follow from results of\n\\cite{lenzing}, \\cite{schiffmann} who considers more general setting of\n``non-commutative projective curves''. So this section is included just for\nthe reader's convenience. \n\n\n\nWe start by describing torsion sheaves. \n\nFor every point $x\\in \\PP$, $n\\in \\Z_+$, denote $\\OO_{n[x]}=\\OO\/m_x^n$,\nwhere $m_x$ is the ideal of functions vanishing at $x$. This sheaf is\nsupported at point $x$; choosing a local coordinate $z$ at $x$, we can\nidentify  the stalk of this sheaf at $x$ with $\\CC[z]\/z^n\\CC[z]$. It\nis well known that sheaves $\\OO_{n[x]}$ are indecomposable, and that\nevery  coherent $\\OO$-module is isomorphic to a direct\nsum of a locally free sheaf and sheaves of the form $\\OO_{n[x]}$. \n\nAssume now that  $x\\in \\PP$ is such that $\\Stab_G(x)=\\{\\pm I\\}$ and\nthus $\\Stab_\\Gbar x=\\{1\\}$ (recall that $\\Gbar=G\/\\{\\pm I\\}$). Such\npoints will be called  {\\em generic}. Define\n\\begin{equation}\\label{e:torsion_sheaf}\n\\OO_{n[Gx]}=\\Ind \\OO_{n[x]}=\\bigoplus_{g\\in\n\\Gbar}\\OO_{n[gx]}\n\\end{equation}\nwhere the functor  $\\Ind$ is defined by \\eqref{e:ind}.\nThis sheaf is supported on the orbit $Gx$ and has a canonical\n$\\Gbar$--equivariant structure which can be lifted to a\n$G$-equivariant structure. It is easy to see that the space of global\nsections $\\Gamma(\\OO_{n[Gx]})$ is isomorphic to $n$ copies of the regular\nrepresentation of $\\Gbar$.\n\nThis construction can be generalized to non-generic points.\n\n\\begin{theorem}\\label{t:torsion}\nLet $x\\in \\PP$ be non-generic: $H=\\Stab_\\Gbar x\\ne \\{1\\}$. Then \n\n\\begin{enumerate}\n\\item $H$ is  a cyclic group. \n\\item For suitable choice of the generator $\\sigma$ of $H$, its action in\n  the one-dimensional  vector space $\\OO_{[x]}=\\OO\/m_x$ is given by\n  $\\sigma\\mapsto e^{2\\pi i \/|H|}$.  \n\\item Consider the completed local ring  $\\widehat{\\OO}_{\\PP,x}$;\n  choosing a local coordinate $z$ at $x$ identifies this algebra with\n  $\\CC[[z]]$. It has a natural action of $H$. Then we have a natural\n  equivalence of categories\n  \\begin{align*}\n  &\\{\\Gbar\\text{-equivariant coherent sheaves supported on }Gx\\}\\\\\n  &\\quad \\simeq\n   \\{H\\text{-equivariant finite-dimensional modules over }\n     \\widehat{\\OO}_{\\PP,x}\\}             \n  \\end{align*}\n  given by \n  $$\n  \\F\\mapsto\\text{stalk of $\\F$ at $x$}\n  $$\n\n\\end{enumerate}\n\\end{theorem}\nThe proof of this theorem is straightforward and  left to the reader.\n\n\\begin{definition}\nLet $H=\\Stab_\\Gbar x$, and $Y$ --- a finite-dimensional representation\nof $H$. Then we denote by $\\OO_{n[Gx],Y}$ the $\\Gbar$-equivariant sheaf\nsupported on the orbit  $Gx$ and whose\n    stalk at $x$ is isomorphic as an $H$-module to   $Y\\otimes\n    (\\OO_{\\PP, x}\/m_x^n)$. \n\n\\end{definition}\nIt is easy to show that the space of global sections of $\\OO_{n[Gx],Y}$\nis isomorphic to \n  $$\n  \\Ind_H^G (Y\\otimes (\\OO\/m_x^n)).\n  $$\n\n\n\\begin{theorem}\\label{t:indecomposable}\n\\par\\noindent\n\\begin{enumerate}\n  \\item\n    The following is the full list of indecomposable objects in\n  $\\C$: \n    \\begin{itemize}\n      \\item Torsion sheaves $\\OO_{n[Gx]}$, $x\\in \\PP$,\n      $\\Stab_{\\Gbar}x=\\{1\\}$, $n\\in \\Z_+$\n      \\item Torsion sheaves $\\OO_{n[Gx],Y}$, $x\\in \\PP$,\n        $H=\\Stab_{\\Gbar }x\\ne\\{1\\}$, \n        $Y$-- an irreducible representation of $H$.\n      \\item Free sheaves $X_i(n)$, $n\\in \\Z$, $i\\in I$, $n+p(i)\\equiv\n      0\\mod $.  \n    \\end{itemize}\n  \\item Indecomposable objects in $\\D^b(\\C)$ are of the form $M[n]$,\n    $M$---an indecomposable object in $\\C$, $n\\in \\Z$. \n  \\item There are no injective and no projective indecomposable objects in\n    $\\C$. \n\\end{enumerate}\n\\end{theorem}\n\\begin{remark}\n  Of course, sheaves $\\OO_{n[Gx]}$, $H=\\Stab_{\\Gbar}x=\\{1\\}$, can be\n  considered as special case of sheaves $\\OO_{n[Gx],Y}$. However, we\n  choose to list cases $H=\\{1\\}$ and $H\\ne\\{1\\}$ separately. \n\\end{remark}\n\n\\begin{proof}\nIt is known that every coherent sheaf $\\F$ has a maximal torsion\nsubsheaf $\\TT$ so that $\\F\/\\TT$ is locally free, and short exact\nsequence $0\\to\\TT\\to \\F\\to \\F\/\\TT\\to 0$ splits. If we additionally\nassume that $\\F$ is a $\\Gbar$-equivariant sheaf, then $\\TT$ (and\nthus $\\F\/\\TT$) must also be $\\Gbar$-equivariant. It easily follows from\n\\thref{t:Gcoh} that then $0\\to\\TT\\to\\F\\to\\F\/\\TT\\to 0$ splits in the\ncategory of $G$-equivariant sheaves; thus, every coherent\n$G$-equivariant sheaf is a direct sum of a free sheaf and a torsion\nsheaf. \n\nBy \\thref{t:locally_free}, indecomposable free sheaves are of the form\n$X_i(n)$. For torsion sheaves, it is easy to see that an indecomposable\ntorison sheaf must be supported on an orbit $Gx$. Now the\nclassification of indecomposable torsion sheaves follows from\n\\thref{t:torsion}.\n\n\n  \n  Classification of indecomposable objects in $\\D^b$ follows from \n\\cite[Lemma I.5.2]{happel}. His argument was given for the category of\nrepresentations of a hereditary algebra, but actually works in any\nhereditary category:  it only  uses that a quotient of an injective\nobject is injective, which easily follows from long exact sequence of\n$\\Ext$ spaces and the fact that $\\Ext^2(\\F,\\G)=0$ for any $\\F,\\G$.\n\\end{proof}\n\n\\section{Auslander--Reiten quiver ${\\widehat{Q}}$}\nWe can now define a combinatorial structure which will play a crucial\nrole in our paper. Let ${\\widehat{Q}}$ be the set of isomorphism classes of\nlocally free indecomposable objects in $\\C$;\nby \\thref{t:indecomposable}, we can write \n\\begin{equation}\\label{e:qhat}\n{\\widehat{Q}}=\\{(i,n)\\st n\\in \\Z, i\\in I, n+p(i)\\equiv 0 \\mod 2\\}\n\\end{equation}\nWe will frequently use the notation \n$$\nX_q=X_i(n)=\\OO(n)\\ttt X_i, \\quad q=(i,n)\\in {\\widehat{Q}}.\n$$\n\nWe turn ${\\widehat{Q}}$ into a  quiver by defining, for $q=(i,n)$,\n$q'=(j,n+1)$, \n\\begin{align*}\n&(\\text{Number of edges }q\\to q')=\\dim \\Hom_\\C(X_q,X_{q'})\\\\\n&\\qquad =\\dim \\Hom_G(X_i, X_j\\ttt V)=n(i,j)\n\\end{align*}\n(recall that $n(i,j)$ is the number of edges between $i$ and $j$ in\n$Q$ and $V=\\Gamma(\\PP,\\OO(1))$ is the tautological representation of\n$\\mathrm{SU(2)}$).\n\nNote that the edges only connect vertices with $n$ differing by one,\nand all edges are directed ``up'' (i.e., so that $n$ increases). The\nfigure below shows an example of the quiver ${\\widehat{Q}}$ when $G$ is of\ntype $\\widehat{D_7}$. \n\\begin{figure}[ht]\n\\fig{Qhat.eps}\n\\caption{Auslander--Reiten quiver ${\\widehat{Q}}$ for $Q=\\widehat{D_7}$}\n\\end{figure}\n\n\\begin{remark}\nAgain, for users familiar with the Auslander--Reiten theory, we note that\n${\\widehat{Q}}$ is exactly the stable AR quiver of  the category ${Coh_G(\\PP)}$. As\nwas mentioned in the introduction, it is the picture of this quiver in\nOcneanu's work (who defined it purely combinatorially, without\nreference to the category of sheaves) that inspired the current paper. \n\\end{remark}\n\n\nThe following easy lemma is left as an exercise to the reader. \n\\begin{lemma}\\label{l:qhat_connected}\n${\\widehat{Q}}$ is a connected quiver. \n\\end{lemma}\nThis is actually the reason for considering the even part $\\C={Coh_G(\\PP)}_0$\nrather than all of ${Coh_G(\\PP)}$: otherwise, the resulting quiver would not be\nconnected. \n\n\n\nNote that  we have a natural pairing \n$$\n\\Hom_G(X_i, X_j\\ttt V)\\ttt\\Hom_G(X_j, X_i\\ttt V)\\to \\CC\n$$\ngiven by composition\n$$\nX_i\\xxto{\\varphi}X_j\\ttt V\\xxto{\\psi\\ttt 1}\n X_i\\ttt V^{\\ttt 2}\\xxto{1\\ttt ev}X_i \n$$\nwhere $ev\\colon V^{\\ttt 2}\\to \\CC$ is a $G$-equivariant pairing (which\nis defined uniquely up to a constant).  This\ngives rise to duality \n\\begin{equation}\\label{e:pairing}\n\\Hom_\\C(X_i(n), X_j(n+1))\n=\\Hom_\\C(X_j(n+1), X_i(n+2))^*\n\\end{equation}\n\n\nAs usual in quiver theory, we define path algebra of ${\\widehat{Q}}$ by \n\\begin{equation}\\label{e:path_algebra}\n\\begin{aligned}\n&\\Path(q,q')=\\text{Span(paths from $q$ to $q'$ in ${\\widehat{Q}}$)},\\\\\n&\\Path=\\bigoplus_{q,q'\\in {\\widehat{Q}}}\\Path(q,q')\n\\end{aligned}\n\\end{equation}\nOne easily sees that $\\Path( (i,k), (j,l))$ coincides with the space of\npaths in $Q$ of length $l-k$ between $j$ and $j$ (if $l-k\\ge 0$; otherwise,\n$\\Path((i,k), (j,l))=0$ ). \n\n\nWe also define an analogue of preprojective algebra by \n\\begin{equation}\\label{e:preproj_algebra}\n\\begin{aligned}\n&\\Pi_{\\widehat{Q}}=\\Path\/(\\theta_q)\\\\\n&\\theta_q=\\sum_{e\\colon q\\to q'}  e_i e^i\\colon q\\to q(2)\n\\end{aligned}\n\\end{equation}\nwhere for $q=(i,n)$, we denote $q(2)=(i,n+2)$, the sum is over all edges\noriginating at $q$, and $e_i, e^i$ are dual bases in $\\Hom_\\C(X_q, X_{q'})$\nand $\\Hom_\\C(X_{q'}, X_{q(2)})$ with respect to the pairing\n\\eqref{e:pairing}. (This algebra is isomorphic to\nOcneanu's algebra of essential paths, or, in the terminology of AR\nquivers,  to  the mesh algebra of ${\\widehat{Q}}$ considered as a polarized\ntranslation quiver.)\n\n\\begin{theorem}\\label{t:homs=paths}\nLet $q,q'\\in {\\widehat{Q}}$. Then \n$$\n\\Hom_\\C(X_q,X_{q'})=\\Pi_{\\widehat{Q}}(q,q')\n$$ \n\\end{theorem}\n\\begin{proof}\nLet $\\Pi_Q$ be the preprojective algebra of the graph $Q$, and $\\Pi_Q^l$\nsubspace of paths of length $l$. Then for $q=(i,k)$, $q'=(j,l)$ it follows\nfrom the definition that for $l<k$ we have \n$$\n\\Pi_{\\widehat{Q}} (q,q')=0=\\Hom_\\C(X_i(k),X_j(l))\n$$ \nand for $l\\ge k$, \n$$\n\\Pi_{\\widehat{Q}} (q,q')=\\Pi_Q^{l-k}(X_i,X_j)=\n\\Hom_G(X_i,V^{\\ttt (l-k)}\\ttt X_j)\n=\\Hom_\\C(X_i(k),X_j(l)).\n$$\n\\end{proof}\n\nThis theorem shows that morphisms between free indecomposable\nmodules in $\\C$ are described by essential paths in ${\\widehat{Q}}$, so the\nstructure of the subcategory $Free(\\C)$, consisting of locally free sheaves,\ncan be recovered from ${\\widehat{Q}}$. Note however that  realization of\n$Free(\\C)$ in terms of equivariant sheaves gives much more than\n$\\Hom$ spaces: it embeds $Free(\\C)$ into an\nabelian category with sufficiently many injectives, allowing one to\ndefine $\\Ext$ functors and derived category $\\D^b(\\C)$. Both of\nthese would be difficult to define in terms of ${\\widehat{Q}}$.  \n\n\n\\section{Equivalence of categories}\nLet $Q$, ${\\widehat{Q}}$ be as defined above, and let $h$ be a height\nfunction on $Q$ (see \\deref{d:height_function}). Recall that such \na height function defines a choice of orientation on $Q$; the\ncorresponding quiver is denoted $Q_h$. Then this height function\ndefines an embedding \n\\begin{equation}\\label{e:embedding}\n\\begin{aligned}\ni_h\\colon Q_h^{opp}& \\to {\\widehat{Q}}\\\\\n    i&\\mapsto (i,h_i)\n\\end{aligned}\n\\end{equation}\nwhere $Q_h^{opp}$ is the quiver obtained from $Q_h$ by reversing all\narrows. The figure below shows an example of such an embedding.\n\n\\begin{figure}[ht]\n\\fig{embedding.eps}\n\\end{figure}\n\n\\begin{remark}\nNote that once we have embedded $Q_h\\subset {\\widehat{Q}}$, we can \nidentify the quiver ${\\widehat{Q}}$ with $\\Z Q_h$ (see, e.g., \\cite{ringel1}\nfor the definition).\n\\end{remark}\n\n\nFrom now on, we will consider $Q_h^{opp}$ as a subset of ${\\widehat{Q}}$,\nomitting $i_h$ in our notation. \n\n\n\\begin{definition}\nLet $q_1,q_2\\in {\\widehat{Q}}$.  We say that $q_1$ is a predecessor of $q_2$\n(notation: $q_1\\prec q_2$) if there exists a path $q_1\\to\n\\bullet\\to\\dots\\to \\bullet \\to q_2$ in ${\\widehat{Q}}$. In this case, we also\nsay that $q_2$ is a successor of $q_1$ and write $q_2\\succ q_1$. \n\\end{definition}\n\n\\begin{lemma}\\label{l:technical1}\n  Let $q=(i,n)\\in {\\widehat{Q}}$. Then \n  \\begin{enumerate}\n    \\item $n>h_i \\iff (q\\notin Q_h, q\\succ q'$ for some $q'\\in Q_h)$. In\n    this case,  we will say that $q$ is above $Q_h$ and write\n    $q\\succ Q_h$.\n    \\item $n<h_i \\iff (q\\notin Q_h, q\\prec q'$ for some $q'\\in Q_h)$. In\n      this case,  we will say that $q$ is below $Q_h$ and write\n      $q\\prec Q_h$.\n  \\end{enumerate}\n\\end{lemma}\n\\begin{lemma}\\label{l:vanishing_of_homs}\n\\par\\indent\n  \\begin{enumerate}\n    \\item If $q_1\\succeq Q_h,  q_2\\prec Q_h$, then\n        $\\Hom_\\C(X_{q_1},X_{q_2})=0$\n    \\item If   $q_1\\preceq Q_h,  q_2\\succeq Q_h$, then\n        $\\Ext^1_\\C(X_{q_1},X_{q_2})=0$. In particular, if $q_1, q_2\\in\n        Q_h$, then $\\Ext^1_\\C(X_{q_1},X_{q_2})=0$. \n  \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n  It follows from $\\Hom_\\C(X_{q_1},X_{q_2})=\\Pi_{{\\widehat{Q}}}(X_{q_1},X_{q_2})$\n  that  $\\Hom_\\C(X_{q_1},X_{q_2})$ can be nonzero only if $q_1\\preceq q_2$.\n  But  then  $q_2\\succeq q_1\\succeq Q_h$, which proves part (a). \n  \n  Part (b) follows from part (a) and Serre duality\n  (\\thref{t:locally_free}) \n\\end{proof}\n\n\n\n\\begin{lemma}\\label{l:generators}\nThe sheaves $X_q$, $q\\in Q_h$, generate the category $\\D^b(\\C)$ as a\ntriangulated category: the smallest full triangulated subcategory in\n$\\D^b(\\C)$ containing all $X_q$, $q\\in Q_h$, is $\\D^b(\\C)$. \n\\end{lemma}\n\\begin{proof}\nLet $\\D$ be the smallest triangulated subcategory in\n$\\D^b(\\C)$ containing all $X_q$, $q\\in Q_h$. If $i$ is a sink for $Q_h$,\nthen it follows from AR exact sequence \\eqref{e:aus-reiten1} that\n$X_i(h_i+2)\\in \\D$. Thus, all $X_q$, $q\\in Q_{s_i^+ h}$, are in $\\D$.\nSimilarly, if $i$ is a source for $Q_h$, then all $X_q$, $q\\in Q_{s_i^-\nh}$, are in $\\D$. By \\leref{l:reflection_functors1}, this implies that\nall $X_q$, $q\\in {\\widehat{Q}}$, are in $\\D$, so $\\D$ contains all locally free\nequivariant sheaves. Since every coherent sheaf admits a locally free\nresolution, this completes the proof. \n\\end{proof}\n\nDefine now a functor \n\\begin{equation}\\label{e:Ph}\n\\begin{aligned}\n\\Ph_h\\colon \\C\\to \\Rep(Q_h)\n\\end{aligned}\n\\end{equation}\nby \n$$\n\\Ph_h(\\F)(i)=\\Hom_\\C(X_i(h_i), \\F)\n$$\nand for an edge $e\\colon j\\to j$ in $Q_h$, we define the corresponding\nmap by  \n\\begin{align*}\n\\Hom_\\C(X_j(h_j), \\F)&\\to \\Hom_\\C(X_i(h_i), \\F)\\\\\n                  \\varphi&\\mapsto \\varphi\\circ h(e)\n\\end{align*}\nwhere $h(e)\\colon (i,h_i)\\to (j,h_j)$ is the edge in ${\\widehat{Q}}$ corresponding to\n$e$ under embedding \\eqref{e:embedding}. \n\n\n\\begin{lemma}\\label{l:Ph}\n\\par\\indent\n\\begin{enumerate}\n\\item $\\Ph_h$ is left exact.\n\\item $R^i\\Ph_h=0$ for $i>1$, and \n$$\n(R^1\\Ph_h(\\F))(i)=\\Ext_\\C^1(X_i(h_i),\\F)\n$$\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nFollows from left exactness of functor $\\Hom_\\C(X_i(h_i), -)$ and definitions. \n\\end{proof}\n\n\n\\begin{example}\\label{x:standard}\nLet $h$ be the ``standard'' height function: $h(i)=0$ for $i\\in I_0$,\n$h(i)=1$ for $i\\in I_1$. Then the map $\\Ph_h$ can also be described\nas follows. Consider the functor $\\Rep Q_h\\to \\Rep G$ defined by\n$\\{V_i\\}\\mapsto \\bigoplus V_i\\ttt X_i$. Then the composition\n$$\n\\C\\xxto{\\Ph_h} \\Rep Q_h\\to \\Rep G\n$$\nis given by \n$$\n\\F\\mapsto \\Gamma(\\F)\\oplus \\Gamma(\\F(-1)).\n$$\n\nIndeed, for each representation $M$ of $G$ we have \n$$\nM\\simeq\\bigoplus \\Hom_G(X_i,M)\\ttt X_i\n$$\nApplying it to $\\Gamma(\\F)=\\Hom_\\OO(\\OO,\\F)$ we get \n$$\n\\Gamma(\\F)=\\bigoplus_{i\\in I} \\Hom_\\OO(X_i\\ttt \\OO,\\F)^G\\ttt X_i\n=\\bigoplus_{i\\in I} \\Hom_G (X_i\\ttt \\OO,\\F)\\ttt X_i\n$$\nSince $\\F\\in \\C={Coh_G(\\PP)}_0$, we have $\\Hom_G (X_i\\ttt \\OO,\\F)=0$ for\n$i\\in I_1$; thus, \n$$\n\\Gamma(\\F)=\\bigoplus_{i\\in I_0} \\Hom_G (X_i\\ttt \\OO,\\F)\\ttt X_i\n$$\nSimilar argument shows that \n$$\n\\Gamma(\\F(-1))=\\bigoplus_{i\\in I_1} \\Hom_G (X_i\\ttt \\OO(1),\\F)\\ttt X_i\n$$\nThus, \n$$\n\\Gamma(\\F)\\oplus \\Gamma(\\F(-1))=\n\\left(\\bigoplus_{i\\in I_0} \\Hom_\\C (X_i,\\F)\\ttt X_i\\right)\n\\oplus\n\\left(\\bigoplus_{i\\in I_1} \\Hom_\\C (X_i(1),\\F)\\ttt X_i\\right)\n$$\nas desired. \n\\end{example}\n\nSince $\\Ph_h$ is left exact, we can define the associated derived functor\n$R\\Ph_h\\colon \\D^b(\\C)\\to \\D^b(\\Rep Q_h)$. \nThe following two theorems are the main results of this paper. \n\n\\begin{theorem}\\label{t:main2}\nFor any height function $h$, the functor $R\\Ph_h\\colon \\D^b(\\C)\\to\n\\D^b(Q_h)$ defined by \\eqref{e:Ph} is an equivalence of triangulated\ncategories.\n\\end{theorem}\n\\begin{proof}\nIt follows from \\leref{l:vanishing_of_homs} that the object \n$$\nT=\\bigoplus_{q\\in Q_h}X_q\n$$\nsatisfies $\\Ext^1(T,T)=0$. By \\leref{l:generators}, direct summands of\n $T$ generate $\\D^b(\\C)$ as a triangulated category. Therefore, $T$ is\nthe tilting object in $\\D^b(\\C)$ as defined in \\cite{geigle-lenzing}.\nNow the statement of the theorem follows from the general result of\n\\cite{geigle-lenzing}.\n\\end{proof}\n\n\n\n\\begin{theorem}\\label{t:main1}\nLet $i$ be a sink for $h$, and $S_i^+$ the reflection functor  defined by \n\\eqref{e:reflection_2}. Then the following diagram is commutative:\n$$\n\\xymatrixrowsep{4pt}\n\\xymatrixcolsep{40pt}\n\\xymatrix{\n                  &\\D^b(Q_{s_i^+h})\\\\\n\\D^b(\\C)\\ar[ur]^{R\\Ph_{s_i^+h}}\\ar[dr]^{R\\Ph_{h}}&\\\\\n                 &\\D^b(Q_{h})\\ar[uu]_{RS_i^+}\n} \n$$\nand similarly for $S_i^-$.  \n\\end{theorem}\n\\begin{proof}\n\n\n\nLet us first prove that if $q\\succ Q_h$, then \n$$\nR\\Ph_{s_i^+h}(X_q)=RS_i^+\\circ R\\Ph_h (X_q).\n$$ \nIndeed, in this case it follows\nfrom \\leref{l:Ph}, \\leref{l:vanishing_of_homs} that $R^i\\Ph_h(X_q)=0$\nfor $i>0$,\nso $R\\Ph_h(X_q)=\\Ph_h(X_q)$; similarly, \n$R\\Ph_{s_i^+h}(X_q)=\\Ph_{s_i^+h}(X_q)$.\n\nLet $i$ be a sink for $Q_h$.  Then we have the Auslander--Reiten exact\nsequence\n$$\n0\\to X_i(h_i)\\to \\bigoplus_{j} X_j(h_j)\\to X_i(h_i+2)\\to 0\n$$\nApplying to this sequence functor $\\Hom_\\C(-, X_q)$ and using\n\\leref{l:vanishing_of_homs} which gives  vanishing of\n$\\Ext^1$,  we get a short exact\nsequence \n$$\n0\\to \\Hom_\\C(X_i(h_i+2), X_q)\\to \\bigoplus_{j} \\Hom_\\C(X_j(h_j), X_q)\\to\n\\Hom_\\C(X_i(h_i), X_q)\\to 0\n$$\nComparing this with definition of $S_i^+$, we see that \n$\\Ph_{s_i^+h}(X_q)=S_i^+(\\Ph_h(X_q))$. \n\nThus, we have shown that for $q\\succ Q_h$, we\nhave $R\\Ph_{s_i^+h}(X_q)=RS_i^+\\circ R\\Ph_h (X_q)$. \n\n\nTo complete the proof, we now use the following easy result.\n\n\\begin{lemma}\\label{l:technical4}\n   Let $\\D$, $\\D'$ be triangulated categories, $\\Ph_1,\\Ph_2\\colon\n  \\D\\to   \\D'$ triangle functors, and $\\alpha\\colon\n  \\Ph_1\\to\\Ph_2$ a morphism of  functors. Assume that  there is a\n  collection of objects $X_q\\in \\D$ which generate $\\D$ as a\n  triangulated category in the sense of \\leref{l:generators} and such\n  that for each $X_q$, \n  $$\n  \\alpha\\colon \\Ph_1(X_q)\\to \\Ph_2(X_q)\n  $$\n  is an isomorphism. Then $\\alpha$ is an isomorphism of functors. \n\\end{lemma}  \n\nApplying this lemma to $\\D=\\D^b(\\C)$, $\\D'=\\D^b(Q_{s_i^+h})$,\n$\\Ph_1=R\\Ph_{s_i^+ h}$, $\\Ph_2=RC_i^+\\circ R\\Ph_h$ and the \ncollection of objects $X_q, q\\succ Q_h$ (which\ngenerate $\\D^b(\\C)$ by \\leref{l:generators}), we get the statement of\nthe theorem. \n\\end{proof}\n\n\n\\begin{corollary}\\label{c:coxeter}\nWe have the following commutative diagram\n$$\n\\xymatrixrowsep{20pt}\n\\xymatrixcolsep{30pt}\n\\xymatrix{\n  \\D^b(\\C)\\ar[r]^-{R\\Ph_h}                       \n                      &\\D^b(Q_h)\\simeq \\D^b(Q_{h+2})\\\\\n  \\D^b(\\C)\\ar[r]^{R\\Ph_h}\\ar[u]_{\\ttt \\OO(-2)}   \n                      &\\D^b(Q_h)\\ar[u]_{RC_h^+}\n}\n$$\nwhere $RC_h^+$ is the Coxeter element \\eqref{e:C+}, and similarly for\n$LC_h^-$. \n\\end{corollary}\n\n\n\\begin{remark}\nIt should be noted that while $R\\Ph_h$ is an equivalence of derived\ncategories, it is definitely not true that $\\Ph_h$ is an equivalence\nof abelian categories. For example, there are no injective or\nprojective objects in $\\C$ while there are enough injectives and\nprojectives in $\\Rep Q_h$. Similarly, the set of simple modules is\nvery much different in these two categories. However, the sets of\nindecomposable modules are effectively the same. \n\\end{remark}\n\n\n\\section{The root system and the Grothendieck group}\nIn this section, we list some important corollaries of the equivalence of\ncategories constructed in the previous section. Again, many of these\nresults can also be obtained form the equivalence of categories\nconstructed by Lenzing; however, we choose to provide an independent\nexposition. \n\n\nThoroughout this section, we let $L=K(\\C)$ be the Grothendieck group of\ncategory $\\C$. We define the inner product on $L$ by \n\\begin{equation}\\label{e:euler_form}\n    (x,y)=\\<x,y\\>+\\<y,x\\>\n\\end{equation}\n where by definition \n$$\n    \\<[X],[Y]\\>=\\dim \\RHom (X,Y)=\\dim \\Hom (X,Y) -\\dim \\Ext^1(X,Y)\n$$\n(compare with \\thref{t:reps_of_quivers}). We define $\\Delta\\subset L$ by \n\\begin{equation}\n\\Delta=\\{[X], X \\text{--- a non-zero indecomposable object in\n}\\D^b(\\C)\\}\n\\end{equation}\n\nFinally, we define the map $C\\colon \\Delta\\to\\Delta$ by \n$$\nC([X])=[X(-2)]\n$$\n\n\n\n\\begin{theorem}\\label{t:main3}\n\\par\\indent\n\\begin{enumerate}\n  \\item The set $\\Delta\\subset L$ is an affine root system, and $C$ is\n    a Coxeter element. \n  \\item Recall the lattice $L(Q)$ and root system $\\Delta(Q)$ from\n    \\seref{s:basic}. Then for any height function $h$ on $Q$, the map\n    \\begin{equation}\n    \\begin{aligned}\n    R\\Ph_h\\colon L&\\to L(Q)\\\\\n              [\\F]&\\mapsto \\bigoplus \n              \\biggl(\\dim \\RHom_\\C(X_i(h_i), \\F)\\biggr) \\alpha_i\n    \\end{aligned}\n    \\end{equation}\n    is an isomorphism of abelian groups which identifies\n    $\\Delta\\subset L$  with $\\Delta(Q)\\subset L(Q)$ and $C$ with the\n    Coxeter element  $c_h^+$ defined by \\eqref{e:coxeter}. \n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n  The first part follows from the second one; the\n  second part follows from the fact that $R\\Ph_h$ is an equivalence of\n  categories (\\thref{t:main2} and \\coref{c:coxeter}). \n\\end{proof}\n\n\n\n\\begin{corollary}\\label{c:generators_of_K(C)}\n  \\par\\noindent\n  \\begin{enumerate}\n  \\item \n    For any height function $h$, the classes $[X_q]$, $q\\in Q_h$, are free\n      generators of   the Grothendieck group $K(\\C)$. \n   \\item \n    $K(\\C)$ is generated by the classes $[X_i(n)], n\\equiv p(i)\\mod 2$,\n    and AR relations \\eqref{e:aus-reiten3} is the full list of all\n    relations among classes $[X_i(n)]$.\n  \\end{enumerate}\n\\end{corollary}\n\\begin{proof}\nBy \\leref{l:generators}, classes $[X_q]$, $q\\in Q_h$, generate $K(\\C)$. On\nthe other hand, by \\thref{t:main3}, $K(\\C)$ is isomorphic to $\\Z^I$ and\nthus has rank $|I|$, so these generators must be linearly independent.  \n\nTo prove the second part, denote temporarily $L'=Span([X_i(n)])\/J$, where\n$J$ is the subgroup generated by AR relations \\eqref{e:aus-reiten3}.\nSince $[X_i(n)]$ generate $K(\\C)$ and AR relations hold in $K(\\C)$, we\nsee that $K(\\C)$ is a quotient of $L'$. \n\nNow choose some height function $h$. It follows from \n\\leref{l:reflection_functors1} that $[X_q]$, $q\\in Q_h$, generate $L'$, so\nit has rank at most $|I|$. On the other hand, $K(\\C)$ has rank $|I|$.\nThus, $L'$ has rank $|I|$ and $L'=K(\\C)$. \n\\end{proof}\n\n\n\\begin{example}\\label{x:standard2}\n  Let $h$ be the ``standard'' height function as defined in\n\\exref{x:standard}.  Then the map\n$R\\Ph_h\\colon K(\\C)\\to  L(Q)\\simeq K(G)$\n  is given by \n  \\begin{align*}\n  &[X_i]\\mapsto \\alpha_i, \\quad i \\in I_0\\\\\n  &[X_i(1)]\\mapsto  \\alpha_i+\\sum_{j-i}\\alpha_j, \\quad i \\in I_1\n  \\end{align*}\n  and the corresponding Coxeter element is $C=(\\prod_{i\\in\n    I_1}s_i)(\\prod_{i\\in I_0}s_i)$. \n  \n  This also implies that for this $h$, $[X_i(-1)]\\mapsto -\\alpha_i$, $i\\in\n    I_1$; in particular, we see that classes\n  $\\alpha_i, i\\in I_0$, and $-\\alpha_i, i\\in I_1$, form a set of representatives\n  of $C$-orbits on on ${\\widehat{Q}}$. An analogous statement for finite root\n  system has been proved in \\cite{kostant2}. \n\\end{example} \n\n\n\nFor completeness, we also describe here the indecomposable sheaves\ncorresponding to imaginary roots of $\\Delta$.\n\n\n\\begin{theorem}\\label{t:imaginary_root}\nLet $x\\in \\PP$ be a generic point: $\\Stab_\\Gbar x=\\{1\\}$, and let \n$$\n\\delta=[\\OO_{[Gx]}]\\in \\Delta\n$$\n\\textup{(}see \\eqref{e:torsion_sheaf}\\textup{)}. Then: \n\\begin{enumerate}\n\\item $\\delta$ does not depend on the choice of point $x$\n\\item $\\delta=\\delta_0-\\delta_1$, where \n\\begin{align*}\n\\delta_0&=\\sum_{i\\in I_0}d_i[X_i]=[R_0]\\\\\n\\delta_1&=\\sum_{i\\in I_1}d_i[X_i(-1)]=[R_1(-1)],\n\\end{align*}\n where $d_i=\\dim X_i$ and $R_0, R_1$ are even and odd parts of the\nregular representation defined by \\eqref{e:R_p}. \n\n\\item $C\\delta=\\delta$; $C\\delta_0=\\delta_0-2\\delta$; $C\\delta_1=\\delta_1-2\\delta$\n\\item For any $\\alpha\\in L$, $(\\delta,\\alpha)=0$\n\\item $\\delta$ is a generator of the set of imaginary roots of $\\Delta$:\n$$\n\\Delta^{im}=\\Z\\delta\n$$\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nWe start by proving (2), by explicitly constructing a resolution of \n$\\OO_{[Gx]}$. \n\nLet $\\varphi_x$ be a holomorphic section of $\\OO(1)$ which has a simple\nzero at $x$. Then we have a short exact sequence of sheaves (not\nequivariant):\n$$\n0\\to\\OO\\xxto{\\varphi_x}\\OO(1)\\to \\OO_{[x]}\\to 0\n$$\nTensoring it with $\\OO(-1)$, we get a sequence\n$$\n0\\to\\OO(-1)\\xxto{\\varphi_x}\\OO\\to \\OO_{[x]}\\to 0\n$$\nNow let us apply to this sequence functor $\\Ind$ defined by\n\\eqref{e:ind}. Using \\leref{l:frobenius}, we see that it gives a\n$\\Gbar$-equivariant short exact sequence \n$$\n0\\to R_1\\ttt \\OO(-1)\\to R_0\\ttt\\OO\\to \\OO_{[Gx]}\\to 0\n$$\nwhich gives equality $\\delta=\\delta_0-\\delta_1$, thus proving part (2) of the\ntheorem.\n\n\n\nPart (1) follows from (2).\n\nSince $\\OO_{[Gx]}\\ttt \\OO(-2)\\simeq \\OO_{[Gx]}$, we get $C\\delta=\\delta$.\nTo compute $C\\delta_0$, note that the same argument as in the proof of\npart (2) also gives a short exact sequence \n$$\n0\\to R_0\\ttt \\OO\\to R_1\\ttt\\OO(1)\\to \\OO_{[Gx]}\\to 0\n$$\nthus giving $C^{-1}\\delta_1=\\delta+\\delta_0=2\\delta+\\delta_1$. \n\nTo prove part (4), recall notation\n$\\<[X],[Y]\\>=\\dim\\Hom(X,Y)-\\dim\\Ext^1(X,Y)$. Then Serre duality\nimmediately gives \n$$\n\\<x,y\\>=-\\<y,Cx\\>.\n$$\nSince $C\\delta=\\delta$, we get \n$$\n(\\delta,x)=\\<x,\\delta\\>+\\<\\delta,x\\>=\\<x,\\delta\\>-\\<x,C\\delta\\>=0\n$$\n\n\nPart (4) implies that $\\delta$ is an imaginary root. Moreover, since by\n\\coref{c:generators_of_K(C)} classes $[X_i]$, $[X_i(-1)]$ are free\ngenerators of $L$. Since some of $d_i$ are equal to 1,  it follows\nfrom part (2) that for any $k>1$, $\\delta\/k\\notin L$; thus, $\\delta$ must be\na generator of the set of imaginary roots. \n\\end{proof}\n\nSince every indecomposable object in $\\D^b(\\C)$ is of the form\n$\\F[n]$, where $\\F$ is an indecomposable $G$-equivariant sheaf\n(\\thref{t:indecomposable}), it follows that every root $\\alpha\\in\\Delta$\ncan be written as either $\\alpha=[\\F]$ or $\\alpha=-[\\F]$, thus giving some\nsplitting of $\\Delta$ into positive and negative roots. This polarization\ncan be described explicitly. \n\nRecall from algebraic geometry (see, e.g., \\cite[Exercises II.6.10,\nII.6.12]{hartshorne}) that for any coherent sheaf, we can\ndefine two integer numbers, its {\\em rank} and {\\em degree}. In\nparticular, for a locally free sheaf $\\F=X\\ttt \\OO(n)$ (where $X$ is a\nfinite-dimensional vector space), we have \n\\begin{align*}\n&\\rk (X\\ttt \\OO(n))=\\dim X,\\\\\n&\\deg (X\\ttt \\OO(n))= n\\dim X.\n\\end{align*}\nDegree and rank give  well-defined linear maps $K\\to \\Z$, where $K$\nis the  Grothendieck group of the category of coherent sheaves. \n\nIn particular, we can define rank and degree for a $G$-equivariant\nsheaf, ignoring the equivariant structure; this gives linear maps\n$K(\\C)\\to\\Z$, which we also denote by $\\rk, \\deg$. \n\n\n\\begin{lemma}\\label{l:rank}\n\\par\\indent\n  \\begin{enumerate}\n    \\item If $\\F\\in\\C$ is a non-zero free sheaf, then $\\rk \\F>0$.\n    \\item If $\\F\\in\\C$ is a non-zero torsion sheaf, then $\\rk \\F=0$,\n          $\\deg\\F>0$.  \n    \\item For any $x\\in K(\\C)$, we have \n    $$\n    \\rk (x)=(x,\\delta_0)=(x,\\delta_1)\n    $$\n    where $\\delta_0$, $\\delta_1$ are defined in \\thref{t:imaginary_root}.\n  \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n  The first two parts are well-known. \n   \n  To check  $\\rk (x)=(x,\\delta_0)=(x,\\delta_1)$, it suffices to check it\n  for  $x=[X_i]$, $i\\in I_0$ and $x=[X_i(-1)], i\\in I_1$ (by\n  \\exref{x:standard2},  they generate $L$). If $i\\in I_0$,\n  then  \n  $$\n  ([X_i], \\delta_0)=\\dim \\Hom_G(X_i,\\sum_{j\\in I_0} d_jX_j)=d_i=\\rk X_i\n  $$\n  thus proving $\\rk (x)=(x,\\delta_0)$. Since $\\delta_0-\\delta_1=\\delta$ is in the\n  kernel of $(\\, , \\,)$ (\\thref{t:imaginary_root}), this implies\n  $(x,\\delta_0)=(x,\\delta_1)$. \n    \n  If $i\\in I_1$, then \n $$\n  ([X_i(-1)], \\delta_1)=\\dim \\Hom_G(X_i(-1),\\sum_{j\\in I_1}\n  d_jX_j(-1))=d_i=\\rk X_i\n $$\n  thus proving $\\rk (x)=(x,\\delta_1)$. Since $\\delta_0-\\delta_1=\\delta$ is in the\n  kernel of $(\\, , \\,)$ (\\thref{t:imaginary_root}), this implies\n  $(x,\\delta_0)=(x,\\delta_1)$.\n\n\\end{proof}\n\nNote that the functional $\\deg x$ can not be written in terms of the\nform $(\\, , \\,)$: indeed, $\\deg \\delta =|\\Gbar|=|G|\/2$, but $(\\delta,\n\\cdot)=0$. \n\n\\begin{theorem}\nLet $\\alpha\\in \\Delta$.\n\\begin{enumerate}\n  \\item $\\alpha=[\\F]$ for some indecomposable free sheaf $\\F\\in \\C$ \n    iff $\\rk(\\alpha)=(\\alpha,\\delta_0)>0$\n  \\item $\\alpha=-[\\F]$ for some indecomposable free sheaf $\\F\\in \\C$ \n    iff $\\rk(\\alpha)=(\\alpha,\\delta_0)<0$\n  \\item $\\alpha=[\\F]$ for some indecomposable torsion sheaf $\\F\\in \\C$ \n    iff $\\rk(\\alpha)=(\\alpha,\\delta_0)=0$, $\\deg(\\alpha)>0$\n  \\item $\\alpha=-[\\F]$ for some indecomposable torsion sheaf $\\F\\in \\C$ \n    iff $\\rk(\\alpha)=(\\alpha,\\delta_0)=0$, $\\deg(\\alpha)<0$\n\\end{enumerate}\n\\end{theorem}\nThus, we see that we have a triangular decomposition of $\\Delta$:\n\\begin{equation}\\label{e:polarization}\n  \\begin{aligned}\n  &\\Delta=\\Delta_+\\sqcup\\Delta_0\\sqcup \\Delta_-\\\\\n  &\\Delta_+=\\{\\alpha\\in \\Delta\\st \\rk(\\alpha)>0\\}=\\{[\\F],\\ \\F\\text{ --- a free\n    indecomposable sheaf}\\}\\\\\n  &\\quad \\text{(note that $\\Delta_+$ is exactly the set of vertices of\n    ${\\widehat{Q}}$)}\\\\\n  &\\Delta_-=\\{\\alpha\\in \\Delta\\st \\rk(\\alpha)<0\\}=\\{-[\\F],\\ \\F\\text{ --- a free\n    indecomposable sheaf}\\}\\\\\n  &\\Delta_0=\\{\\alpha\\in \\Delta\\st \\rk(\\alpha)=0\\}=\\{\\pm[\\F],\\ \\F\\text{ --- a free\n    torsion sheaf}\\}\n  \\end{aligned}\n\\end{equation}\nThe set $\\Delta_+$ has been discussed by Schiffmann \\cite{schiffmann2}, who\nused notation ${\\widehat{Q}}_+$ and denoted the corresponding subalgebra in the\nloop algebra by $\\mathcal{L}\\mathfrak{n}$. Note, however, that this\nnotation is somewhat misleading:  $\\mathcal{L}\\mathfrak{n}$ is not the\nloop algebra of a positive part of the finite dimensional algebra\n$\\bar\\g$, which easily follows from the fact that there are real roots in \n$\\Delta_0$ (see example in the next section).\n \n\\begin{theorem}\\label{t:C^g}\nLet $g=|\\Gbar|=|G|\/2$. Then for any $x\\in L$ we have \n\\begin{equation}\\label{e:C^g}\nC^g (x)=x-2(\\rk x)\\delta \n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nLet $\\varphi_x$ be a section of the sheaf $\\OO(1)$ which has a single zero at\ngeneric point $x$. Then we have a short exact sequence \n$$\n0\\to \\OO\\xxto{\\varphi_x} \\OO(1)\\to \\OO_{[x]}\\to 0\n$$\nwhich, however, is not equivariant even under the action of $\\{\\pm\nI\\}\\subset \\mathrm{SU(2)}$. To fix it we consider $\\varphi_x^2$ which gives the following\n$\\Z_2$-equivariant sequence\n$$\n0\\to \\OO(-2)\\xxto{\\varphi^2_x} \\OO\\to \\OO_{2[x]}\\to 0\n$$\nNow let us take product of pullbacks of $\\varphi_x^2$ under all $g\\in\\Gbar$\n$$\n0\\to \\OO(-2g)\\xxto{\\prod_{g\\in\\Gbar}g^*\\varphi^2_x} \\OO\\to \\OO_{2[Gx]}\\to 0\n$$\n\nTensoring it with any locally free sheaf $\\F$, we get \n$$\n0\\to \\F(-2g)\\xxto{\\prod_{g\\in\\Gbar}g^*\\varphi^2_x} \\F\\to\n\\OO_{2[Gx]}\\ttt \\F_x\\to 0\n$$\nwhich implies\n$C^g[\\F]-[\\F]+2(\\rk \\F)\\delta=0$.\n\\end{proof}\n\nThis result --- that $C^g$ is a translation --- was known before and\ncan be proved without the use of equivariant sheaves, see e.g.\nSteinberg \\cite{steinberg}. However, the approach via equivariant\nsheaves also provides a nice interpretation for the corresponding\nfunctional as the rank of the sheaf.\n\nComparing \\eqref{e:C^g} with the description of the action of Coxeter\nelement in the language of representations of the quiver, we see that rank\nis closely related to the notion of defect $\\partial_c(x)$ as defined in\n\\cite{drab-ringel}, namely \n$$\n\\rk(x)=-\\frac{1}{2}\\partial_c x\n$$\nTherefore, $\\Delta_0$ is exactly the set of indecomposable objects of defect\nzero, which shows that torsion sheaves correspond to regular\nrepresentations. \n\n\n\n\\begin{corollary}\n  \\par\\noindent\n  \\begin{enumerate}\n    \\item For any $\\alpha\\in \\Delta_0$, $C^gx=x$; in particular, $C$-orbit\n      of $\\alpha$ is finite.\n    \\item $C$ acts freely on $\\Delta_+$, and the set of orbits $\\Delta_+\/C$\n      is naturally in bijection with $Q$. \n  \\end{enumerate}\n\\end{corollary}\n\n\n\\section{Example: $\\widehat{A}_n$}\nIn this section, we consider the example of the cyclic group of even\norder: $G=\\Z_n$, $n=2k$. \n\nThe irreducible representations of this group are $X_i$, $i\\in \\Z_n$;\nall of them are one-dimensional. The corresponding Dynkin diagram $Q$\nis the cycle of length $n$. \n\nThe root system $\\Delta(Q)$ can be described as follows. Let $V$ be a\nreal vector space of dimension $n+1$, with basis $\\delta, e_i$, $i\\in\n\\Z_n$. Define inner product in $V$ by $(e_i,e_j)=\\delta_{ij}$,\n$(v,\\delta)=0$. Then \n\\begin{equation}\n  \\Delta(Q)=\\{e_i-e_j+a\\delta, i\\ne j\\in \\Z_n, a\\in \\frac{j-i}{n}+\\Z\\}\n\\end{equation}\nThe simple roots are \n$$\n\\alpha_i=e_i-e_{i+1}+\\frac{1}{n}\\delta,\\quad i\\in \\Z_n\n$$\nso that $\\sum_i\\alpha_i=\\delta$. The simple reflections $s_i$ are given by \n\\begin{align*}\n&s_i(e_i)=e_{i+1}-\\frac{1}{n}\\delta\\\\\n&s_i(e_{i+1})=e_{i}+\\frac{1}{n}\\delta\\\\\n&s_i(e_j)=e_j,\\quad j\\ne i,i+1\n\\end{align*}\n\n\nIt is easy to see that this description of\n$\\Delta(Q)$, while unusual, is  equivalent to the standard\ndescription of the affine root system $\\widehat{A}_{n-1}$.\n\nWe choose standard height function $h$:\n\\begin{equation}\nh(i)=\\begin{cases}0,& i\\text{ even}\\\\\n                  1, &i\\text{ odd}\n     \\end{cases}\n\\end{equation}\nThe corresponding Coxeter  element $C$ is \n$$\nC=\\left(\\prod_{i \\text{  odd}} s_i\\right)\n  \\left(\\prod_{i \\text{ even}} s_i\\right)\n$$\nThe action of $C$ on $\\Delta(Q)$ is given by \n\\begin{equation*}\nC(e_i)=\\begin{cases}e_{i+2}-\\frac{2}{n}\\delta,& i\\text{ even}\\\\\n                    e_{i-2}+\\frac{2}{n}\\delta,& i\\text{ odd}\n       \\end{cases}\n\\end{equation*}\nThus, we have \n\\begin{equation*}\nC^{n\/2}(e_i)=\\begin{cases}e_{i}-\\delta,& i\\text{ even}\\\\\n                    e_{i}+\\delta,& i\\text{ odd}\n       \\end{cases}\n\\end{equation*}\nwhich implies\n\\begin{equation}\n\\begin{aligned}\n&C^{n\/2}(\\alpha)=\\alpha-(\\alpha,\\varepsilon)\\delta,\\\\\n&\\quad \\varepsilon = \\sum_{i \\text{ even}}e_i-\\sum_{i \\text{ odd}}e_i\\equiv \n\\sum_{i \\text{ even}}\\alpha_i\\equiv -\\sum_{i \\text{ odd}}\\alpha_i\\mod \\Z\\delta\n\\end{aligned}\n\\end{equation}\n(compare with \\thref{t:C^g}, \\leref{l:rank} ). \n\nExplicitly, we can write \n\\begin{align*}\n&C^{n\/2}(\\alpha)=\\begin{cases}\n                 \\alpha, &i\\equiv j\\mod 2 \\\\\n                 \\alpha-2\\delta, &(i,j)\\equiv (0,1)\\mod 2 \\\\\n                 \\alpha+2\\delta, &(i,j)\\equiv (1,0)\\mod 2 \n       \\end{cases},\\\\\n&\\quad \\alpha=e_i-e_j+a\\delta\n\\end{align*}\nThus, in this case we have \n\\begin{align*}\n\\Delta_0=\\{e_i-e_j+a\\delta\\}, \\quad i\\equiv j\\mod 2\\\\\n\\Delta_+=\\{e_i-e_j+a\\delta\\}, \\quad i\\text{ even}, j\\text{ odd}\\\\\n\\Delta_-=\\{e_i-e_j+a\\delta\\}, \\quad i\\text{ odd}, j\\text{ even}\\\\\n\\end{align*}\n\n\n\n\\firef{f:An_graph} shows a segment of the AR graph ${\\widehat{Q}}$ for\nthis root system.\n\n\n\\begin{figure}[ht]\n$$ \n\\xy\n(0,0)*+{X_{i-2}}=\"A\"+(0,-3)*+{\\scriptstyle\n                      e_{i-2}-e_{i-1}+\\frac{1}{n}\\delta};\n(30,0)*+{X_{i}}=\"B\"+(0,-3)*{\\scriptstyle\n                            e_{i}-e_{i+1}+\\frac{1}{n}\\delta};\n(60,0)*+{X_{i+2}}=\"C\"+(0,-3)*{\\scriptstyle\n                           e_{i+2}-e_{i+3}+\\frac{1}{n}\\delta};\n(90,0)*+{X_{i+4}}=\"D\"+(0,-3)*{\\scriptstyle\n                           e_{i+4}-e_{i+5}+\\frac{1}{n}\\delta};\n(15,15)*+{X_{i-1}(1)}=\"E\"+(0,-3)*{\\scriptstyle\n                           e_{i-2}-e_{i+1}+\\frac{3}{n}\\delta}=\"EE\";\n(45,15)*+{X_{i+1}(1)}=\"F\"+(0,-3)*{\\scriptstyle\n                           e_{i}-e_{i+3}+\\frac{3}{n}\\delta}=\"FF\";\n(75,15)*+{X_{i+3}(1)}=\"G\"+(0,-3)*{\\scriptstyle\n                           e_{i+2}-e_{i+5}+\\frac{3}{n}\\delta}=\"GG\";\n(0,30)*+{X_{i-2}(2)}=\"H\"+(0,-3)*+{\\scriptstyle\n                      e_{i-4}-e_{i+1}+\\frac{5}{n}\\delta}=\"HH\";\n(30,30)*+{X_{i}(2)}=\"I\"+(0,-3)*{\\scriptstyle\n                            e_{i-2}-e_{i+3}+\\frac{5}{n}\\delta}=\"II\";\n(60,30)*+{X_{i+2}(2)}=\"J\"+(0,-3)*{\\scriptstyle\n                           e_{i}-e_{i+5}+\\frac{5}{n}\\delta}=\"JJ\";\n(90,30)*+{X_{i+4}(2)}=\"K\"+(0,-3)*{\\scriptstyle\n                           e_{i+2}-e_{i+7}+\\frac{5}{n}\\delta}=\"KK\";\n{\\ar \"A\";\"EE\"};\n{\\ar \"B\";\"EE\"};\n{\\ar \"B\";\"FF\"};\n{\\ar \"C\";\"FF\"};\n{\\ar \"C\";\"GG\"};\n{\\ar \"D\";\"GG\"};\n{\\ar \"E\";\"HH\"};\n{\\ar \"E\";\"II\"};\n{\\ar \"F\";\"II\"};\n{\\ar \"F\";\"JJ\"};\n{\\ar \"G\";\"JJ\"};\n{\\ar \"G\";\"KK\"};\n\\endxy\n$$\n \\caption{Fragment of graph ${\\widehat{Q}}$ for root system of type $A$.\n Here $i$ is an even number. The figure also shows, for each $X_q\\in\n {\\widehat{Q}}$, the image of  $R\\Ph_h[X_q]$ for the standard choice of height\n function $h$ as in \\exref{x:standard}.} \\label{f:An_graph}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Introduction}\n\nFerromagnetic films with perpendicular magnetic anisotropy (PMA) are of wide interest for applications in established and nascent technologies such as ultrahigh density magnetic hard drives \\cite{doi:10.1063\/1.2750414}, MRAM\\cite{wong2015memory}, superconducting spintronics \\cite{linder2015superconducting}, and energy efficient spin-orbit torque memory \\cite{RevModPhys.91.035004}. PMA can be achieved in Pt\/Co\/Pt multilayer systems as a result of the interfacial anisotropy, however at a critical Co thickness, typically about 1~nm, the anisotropy will fall in-plane. Increasing the total ferromagnetic layer thickness further therefore involves adding additional interfaces to the multilayer. Having a multilayer structure introduces interfacial resistance and interfacial spin-flip scattering \\cite{bass2007spin}, which are disadvantageous for applications such as ours which require the transport current perpendicular-to-plane  \\cite{birge2018spin, doi:10.1063\/1.5140095 ,satchell2021pt}. Alternately, robust intrinsic PMA can be achieved in certain Co$_x$Pt$_{100-x}$ alloys and compounds at any thickness, without increasing the number of interfaces.   \n\nHere, we study the equiatomic Co$_{50}$Pt$_{50}$ alloy, hereafter referred to as CoPt. Through growth at elevated temperatures it is possible to form the $L1_1$ and $L1_0$ chemically ordered compounds of CoPt as epitaxial films. Previous experimental studies of such compounds tend to use MgO substrates as the basis for high temperature epitaxial growth. Visokay and Sinclair report $L1_0$ crystal structure on MgO [001] substrates for growth temperatures above 520$^{\\degree}$C \\cite{doi:10.1063\/1.113895}. Iwata \\textit{et al.} report growth of $L1_1$ crystal structure on MgO [111] substrates for a growth temperature of 300$^{\\degree}$C \\cite{619533}.\n\nEarly thin film studies of the chemically ordered CoPt (and the related FePt and FePd) compounds were motivated by the large out-of-plane anisotropy and narrow domain wall widths being candidate for high density storage mediums \\cite{619533, PhysRevB.50.3419, PhysRevB.52.13419, doi:10.1063\/1.362122, doi:10.1063\/1.361368, doi:10.1063\/1.113895, doi:10.1063\/1.364504, caro1998structure, doi:10.1063\/1.368158, doi:10.1063\/1.368831, doi:10.1063\/1.371397, doi:10.1126\/science.287.5460.1989, zeng2002orientation, PhysRevB.66.024413, chen2003effect, zhao2003promotion, doi:10.1021\/jp027314o, LAUGHLIN2005383,  doi:10.1063\/1.1991968, LIAO2006e243, doi:10.1063\/1.2176306, PhysRevB.76.174435, doi:10.1063\/1.2730568, Sato2008, chen2008review, 4492859, doi:10.1063\/1.3153513, Seemann_2010, yuan2010effect, Antje_Dannenberg_2010, doi:10.1063\/1.3561115,  doi:10.1063\/1.3672856, ohtake2012structure, sun2013formation, 6416998, ohtake2014structure, VARVARO2014415, doi:10.1063\/1.4960554, zygridou2017exploring, PhysRevApplied.10.054015, GAO2019406, doi:10.1021\/acsanm.9b01192, PhysRevMaterials.6.024403, spencer2022characterization}. Recently, renewed interest in these compounds has been driven by the discovery of self-induced spin-orbit torque switching in these materials, which can be used as the switching mechanism for a low-dissipation magnetic memory \\cite{PhysRevB.101.220402,https:\/\/doi.org\/10.1002\/adfm.202005201,doi:10.1063\/5.0028815,liu2021symmetry,9433557,DONG2022,doi:10.1063\/5.0077465, liu2022current}.\n\nOur motivation for studying [111] CoPt is to incorporate this PMA ferromagnet in an all epitaxial heterostructure suitable for superconducting Josephson devices\\cite{birge2018spin, doi:10.1063\/1.5140095 ,satchell2021pt} or MRAM\\cite{wong2015memory}. For these applications growth in the [111] direction is favourable. In Josephson junctions, the superconductor of choice is Nb, which can be grown epitaxially as a bcc structure in the [110] direction\\cite{WILDES20017}. In MRAM, the seed layer of choice is Ta, which has almost identical structural properties to Nb. On Ta or Nb [110] layers, we know that Pt and Co will grow with [111] orientation\\cite{PhysRevB.97.214509}. We therefore choose Al$_2$O$_3$ [0001] substrates for this study and use thin Pt [111] seed layers to grow CoPt [111]\\cite{yuan2010effect}. As far as we are aware, no previous works have reported the properties of [111] ordered films of $L1_0$ and $L1_1$ CoPt on Al$_2$O$_3$ [0001] substrates. We are also unaware of any measurements of the spin polarisation of CoPt. \n\n\nWe fabricate and report the properties of three sets of samples. The first set is designed to determine the optimal growth temperature. Therefore, we fix the thickness $d_{CoPt} = 40$~nm and vary the substrate heater temperature in the range from 27${\\degree}$C to 850${\\degree}$C. The next two sample sets are thickness series grown by varying the growth time at a fixed temperature and magnetron powers. The temperatures chosen for the thickness series are to produce either the $L1_1$ (350${\\degree}$C) or $L1_0$ (800${\\degree}$C) crystal structures. These temperatures are guided by the results of the first temperature series of samples. Thicknesses are varied over the range 1~nm$ \\leq d_\\text{CoPt} \\leq 128$~nm.\n\nOn each sample set we report systematically on the structural and magnetic properties of the CoPt. Additionally, on the thickest 128~nm $L1_1$ and $L1_0$ samples we perform point contact Andreev reflection (PCAR) measurements with a Nb tip at 4.2~K to determine the spin polarisation. The use of Nb as the tip, the temperature of this measurement, and the ballistic transport regime probed are relevant for our proposed application of the CoPt in Josephson devices\\cite{birge2018spin, doi:10.1063\/1.5140095 ,satchell2021pt}.   \n\n\n\n\n\\section*{Results and Discussion}\n\n\\subsection*{\\label{Growth}CoPt properties as a function of growth temperature}\n\n We expect that as the growth temperature is increased the CoPt will form initially a chemically disordered $A1$ alloy phase, followed by a chemically ordered $L1_1$ crystal structure, an intermediate temperature $A1$ phase, and finally a chemically ordered $L1_0$ crystal structure respectively. In order to map out the growth parameters we report on Al$_2$O$_3$(sub)\/Pt(4~nm)\/CoPt(40~nm)\/Pt(4~nm) sheet films grown at different set temperatures in the range from room temperature (RT) to 850$^{\\degree}$C. \n \n \n\\subsubsection*{Structure}\n\nIn order to understand the magnetic phases of sputter deposited CoPt, it is necessary to understand the underpinning structure and the influence of the growth temperature. Therefore, the structural characteristics and film quality were investigated using x-ray diffraction (XRD) as a function of growth temperature. Fig.~\\ref{fig:XRD} (a-c) shows the acquired XRD for selected growth temperatures around the main CoPt [111] structural peak. Further XRD data are available in the Supplemental Information online. The CoPt [111] structural peak is clearly observable for all growth temperatures. The presented data at growth temperatures of 350$^{\\degree}$C and 800$^{\\degree}$C show additional Pendellosung fringes which bracket the primary CoPt [111] peak. In Fig.~\\ref{fig:XRD} (a), for a growth temperature of 350$^{\\degree}$C, the additional feature at $2\\theta\\approx40^{\\degree}$ corresponds to the superimposed Pendellosung fringes and the Pt [111] structural peak (bulk Pt has a lattice constant of 3.92\\AA). This additional Pt structural peak is present for growths up to 550$^{\\degree}$C, see Supplementary Information online, which is within the expected optimal temperature range for sputter deposited Pt films on Al$_2$O$_3$ \\cite{4856395}.\n \n\nUsing the Scherrer equation and the full widths at half maxima (FWHM) of the Gaussian fits to the CoPt [111] structural peaks, an estimation of the CoPt crystallite sizes can be made. The FWHM for the main [111] structrual peak is given in Figure~\\ref{fig:XRD} (e). In the expected range of the ordered $L1_1$ crystal structure between 200$^{\\degree}$C and 400$^{\\degree}$C, the estimated CoPt crystallite size is 38~nm, which compared to the nominal thickness of 40~nm indicates that the CoPt has high crystallinity. On the other hand, at RT growth and intermediate temperatures between 400$^{\\degree}$C and 800$^{\\degree}$C, the estimated CoPt crystallite size is much lower, showing a minimum value of 22~nm at 700$^{\\degree}$C where we expect $A1$ growth. Interestingly, the disorder in the $A1$ growth appears to affect both the chemical disorder (random positions of the Co and Pt atoms in the unit cell) and a poorer crystallite size compared to the chemically ordered crystal structures. Finally, upon increasing the temperature further to 800$^{\\degree}$C and 850$^{\\degree}$C the crystallite size reaches a maximum of 52~nm. This value corresponds approximately to the entire thickness of the Pt\/CoPt\/Pt trilayer. \n\nFig.~\\ref{fig:XRD} (f) shows the CoPt $c$-plane space calculated from the center of the fitter Gaussian to the main [111] structural peak. At all temperatures the measured $c$-plane spacing is very close to the expected value based on single crystal studies \\cite{mccurrie1969magnetic}, indicating low out-of-plane strain. The trend with temperature, however, is non-monotonic and shows a discrete increase between the samples grown at 550$^{\\degree}$C and 650$^{\\degree}$C. The transition associated with the $L1_0$ crystal structure is expected to result in a transition from cubic to tetragonal crystal with a $c\/a=0.979$ in the single crystal \\cite{mccurrie1969magnetic}. In the single crystal study, however, the $c$-axis is in the [001] orientation. In our [111] orientated film, the small structural transition from cubic to tetragonal crystal structure is not expected to be visible in the [111] peak studied here. Nonetheless, Figure~\\ref{fig:XRD} (f) shows features consistent with the CoPt undergoing structural transitions.\n\n\n\n\n\n\n\n\\subsubsection*{Magnetic characterisation}\n\n\n\nThe magnetisation versus field data are shown in Fig. \\ref{fig:Magnetization_Growth_T} for Al$_2$O$_3$(sub)\/Pt(4 nm)\/CoPt(40 nm)\/Pt(4 nm) sheet film samples. Further magnetisation data for all samples in this study are available in the Supplemental Information online. The 350${\\degree}$C, 550${\\degree}$C, and 800${\\degree}$C samples are plotted here as they are representative of the magnetic response of the $L1_1$ crystal structure, the chemically disordered $A1$ phase, and $L1_0$ crystal structure respectively. Magnetisation is calculated from the measured total magnetic moments, the areas of the sample portions, and the nominal thicknesses of the CoPt layer. \n\nFor the chemically ordered $L1_1$ crystal structure shown in Fig. \\ref{fig:Magnetization_Growth_T} (a), the OOP hysteresis loop shows a wasp waisted behavior associated with the formation of magnetic domains at remanence. Such behavior is common in CoPt alloys and multilayer thin films \\cite{doi:10.1063\/1.113895}. The wasp waisted OOP hysteresis loop along with the low IP remanence and higher IP saturation field indicates that the 40~nm CoPt samples with the $L1_1$ crystal structure have strong PMA. We can estimate the uniaxial anisotropy using the expression $K_u = \\mu_0M_sH_s\/2$, where $\\mu_0$ is the vacuum permeability, $M_s$ the saturation magnetisation, and $H_s$ is the saturation magnetic field. From the data presented in Fig. \\ref{fig:Magnetization_Growth_T} (a) we find $K_u = 5\\pm1$ MJ\/m$^2$. \n\nFor samples grown at intermediate temperatures with the chemically disordered $A1$ structure, the magnetism favours IP anisotropy at 40~nm, shown for growth at 550${\\degree}$C in Fig. \\ref{fig:Magnetization_Growth_T} (b).\n\nFor the chemically ordered $L1_0$ crystal structure shown in Fig. \\ref{fig:Magnetization_Growth_T} (c), there is a significant increase in the coercivity and squareness ratio ($M_r\/M_s$) for both the IP and OOP field orientations. The increased coercive field suggests that the $L1_0$ CoPt is magnetically hard compared to the $L1_1$ and $A1$ samples. The magnetisation of the $L1_0$ 40~nm CoPt sample does not show clear IP or OOP anisotropy from these measurements. \n\nThe magnetisation vs growth temperature is shown in Fig. \\ref{fig:Magnetization_Growth_T} (d). At growth temperatures below 550${\\degree}$C the magnetisation remains approximately constant, however at higher temperatures the magnetisation begins to decrease with increasing temperature. The possible cause of this decrease is the higher growth temperature contributing to interdiffusion between the Pt and CoPt layers, creating magnetic dead layers.\n\nThe saturation field and squareness ratio vs growth temperature are shown in Fig. \\ref{fig:Magnetization_Growth_T} (e) and (f) respectively. The general trends can be seen in the differences observed in the hysteresis loops of Fig. \\ref{fig:Magnetization_Growth_T} (a-c). These trends allow us to characterise the growth temperatures corresponding to the different crystal growths in our samples, further supporting our conclusions from the XRD: the $L1_1$ crystal structure driving the magnetic response of samples grown from 200${\\degree}$C to 400${\\degree}$C, the $L1_0$ crystal structure for samples grown above 750${\\degree}$C, and the intermediate temperatures being the chemically disordered $A1$ phase. \n\n\nIn the $L1_0$ crystal structure, Fig. \\ref{fig:Magnetization_Growth_T} (c), the high squareness ratio for both field orientations suggests that the anisotropy axis of the material is neither parallel or perpendicular to the film. Instead, it is possible that the magnetic anisotropy is perpendicular to the layer planes, which are stacked in the [100] direction. \n\nTo further investigate the anisotropy, we pattern our $L1_1$ 350${\\degree}$C and $L1_0$ 800${\\degree}$C samples into Hall bars and perform angular dependent Hall resistivity, R$_\\text{xy}$($\\theta$), measurements, Fig~\\ref{fig:Hallbar}. The fabricated Hall bar and measurement geometry is shown in Fig~\\ref{fig:Hallbar} (a). R$_\\text{xy}$($\\theta$) for the $L1_1$ 350${\\degree}$C and $L1_0$ 800${\\degree}$C Hall bars are shown in  Fig~\\ref{fig:Hallbar} (b) and (c) respectively. For the $L1_1$ 350${\\degree}$C sample with out-of-plane anisotropy, R$_\\text{xy}$($\\theta$) shows a plateau close to out-of-plane field and a uniform response for angles in between. The plateau is interpreted as an angle forming between the magnetisation and applied field because of the anisotropy axis \\cite{doi:10.1063\/1.3262635}. In comparison, the $L1_0$ 800${\\degree}$C sample also shows a  R$_\\text{xy}$($\\theta$) plateau for out-of-plane applied field plus an additional plateau for applied field angles between about 45${\\degree}$ and 60${\\degree}$. We interpret the additional plateau in R$_\\text{xy}$($\\theta$) for the $L1_0$ 800${\\degree}$C CoPt sample as evidence for an additional anisotropy axis, which we propose is perpendicular the [100] direction. The [100] plane has a dihedral angle of 54.75${\\degree}$ with the [111] growth plane. Additional sources of anisotropy in our samples are interface anisotropy at the Pt\/CoPt interfaces, which would favour out-of-plane magnetisation for thin layers, and shape anisotropy, which for our thin films would favour in-plane magnetisation.  \n\nFigure \\ref{fig:IP_Rotation} shows the extracted coercive field as a function of in-plane rotator angle for the $L1_0$ 800${\\degree}$C sample. The data shows a clear six-fold symmetry. This is consistent with an easy axis for the plurality tetragonal $L1_0$ phase of $[001]$ when grown on $\\{111\\}$ planes - the $<100>$ directions of the parent cubic structure are inclined at $\\pm 45\\degree$ from the plane and are coupled with the three fold symmetry of $\\{111\\}$. The magnetometry data therefore strongly suggests that over the sample the $[001]_{L1_0}$ can be found in any of the three possible $<100>$ of the parent cubic structure without a strong preference for which of the possible twins grow.\n\n\n\n\n\n\\subsection*{CoPt properties as a function of thickness for $L1_1$ and $L1_0$ crystal structures}\n\nHaving obtained optimal growth conditions for CoPt with chemically ordered $L1_1$ and $L1_0$ crystal structures, in this section, we report on the properties of samples Al$_2$O$_3$ (sub)\/Pt (4~nm)\/CoPt ($d_\\text{CoPt}$)\/Pt (4~nm) with $d_\\text{CoPt}$ varied between 1~nm and 128~nm for both the $L1_1$ and $L1_0$ crystal structures.\n\n\n\n\n\\subsubsection*{\\label{MagL11}Magnetisation of $L1_1$ and $L1_0$ CoPt}\n\nThe hysteresis loops for Al$_2$O$_3$(sub)\/Pt(4~nm)\/CoPt(4~nm)\/Pt(4~nm) sheet films is shown in Fig.~\\ref{fig:Magnetization_d} (a) and (b) for the $L1_1$ (350${\\degree}$C) and $L1_0$ (800${\\degree}$C) crystal structures respectively. Hysteresis loops over the full thickness range are given in the Supplementary Information online. The moment\/area at saturation (or 6~T) verses nominal CoPt thickness are presented in Fig~\\ref{fig:Magnetization_d} (c) and (d).\n\nWe calculate the magnetisation ($M$) by fitting the moment\/area versus nominal CoPt thickness data. In order to account for interfacial contributions to the magnetisation of the CoPt, we model the system as a magnetic slab with possible magnetic dead layers and\/or polarised adjacent layers. Magnetic dead layers can form as a result of interdiffusion, oxidation, or at certain interfaces with non-ferromagnetic layers. At some ferromagnet\/non-ferromagnet interfaces, the ferromagnetic layer can create a polarisation inside the non-ferromagnetic layer by the magnetic proximity effect. Polarisation is particularly common at interfaces with Pt \\cite{doi:10.1063\/1.344903, PhysRevB.65.020405, PhysRevB.72.054430,  rowan2017interfacial, PhysRevB.100.174418}. To take these into account, we fit to the expression,\n\\begin{equation}\n\\label{M2}\n    \\text{moment\/area}= M  (d_\\text{CoPt} - d_i),\n\\end{equation}\n\\noindent where $d_i$ is the contribution to $M$ from any magnetic dead layers or polarisation. The resulting best fit and the moment\/area versus the nominal CoPt thickness is shown in Fig.~\\ref{fig:Magnetization_d}.\n\nFor $L1_1$ growth at 350${\\degree}$C, the result of fitting Eqn.~\\ref{M2} shown in Fig.~\\ref{fig:Magnetization_d} (c) gives $M=750\\pm 50$~emu\/cm$^3$. For $L1_0$ growth at 800${\\degree}$C, the thinnest 1~nm and 2~nm films did not display any magnetic response and are excluded from the analysis in Fig.~\\ref{fig:Magnetization_d}. The result of fitting Eqn.~\\ref{M2} shown in Fig.~\\ref{fig:Magnetization_d} (d) for the samples thicker than 2~nm gives $M=520\\pm 50$~emu\/cm$^3$. Interestingly, we find a significant difference in $M$ between the two crystal structures. It is possible that the differences in $M$ corresponds to a true difference in the saturation magnetisation of the two crystal structures. An alternative scenario is that the 6~T applied field is not large enough to fully saturate the $L1_0$ samples, leading to a reduced measured $M$.\n\nThe thickness dependence of the magnetic switching of the $L1_1$ samples is well summarised by the squareness ratio shown in Fig~\\ref{fig:Magnetization_d} (e). The thickest 40 and 128~nm samples are wasp waisted, as presented in Fig.~\\ref{fig:Magnetization_Growth_T}. At reduced thicknesses, between 2 and 8~nm, the $L1_1$ CoPt no longer displays the wasp waisted switching for out-of-plane field orientation, and now has a square loop shown in Fig.~\\ref{fig:Magnetization_d} (a). The 16~nm sample showed an intermediate behaviour. At 1~nm, the magnetic switching showed \"S\" shaped hysteresis loops for both in- and out-of-plane applied fields with small remanent magnetisation, see Supplementary Information online. \n\nThe thickness dependence of the $L1_0$ crystal structure samples is significantly different to the $L1_1$. In the thinnest films of 1 and 2~nm there is no evidence of ferromagnetic ordering in the hystersis loops, see Supplementary Information online. For $L1_0$ growth at 800${\\degree}$C, the high growth temperature may cause interdiffusion at the Pt\/CoPt interfaces. The interdiffusion may account for magnetic dead layers, which in the thinnest samples may prevent ferromagnetic ordering. Upon increasing the thickness to 4~nm, a ferromagnetic response was recovered, however the hysteresis loops and extracted squareness ratio (Fig.~\\ref{fig:Magnetization_d}) (f)) indicate that the 4~nm and 8~nm $L1_0$ samples have in-plane magnetisation. The in-plane magnetisation in the thinner films suggests that the long-range $L1_0$ ordering may have not established at those thicknesses. \n\nThe thickness dependence in both crystal structures suggest that the Pt\/CoPt\/Pt trilayers grown on Al$_2$O$_3$ substrates are not suitable for applications where ultrathin magnetic layers are required. To improve the magnetic properties of the thinnest samples in this study our future work will focus on replacing the Pt layers with seed and capping layers where interdiffusion may be less. \n\n\n\n\n\n\\subsection*{Spin polarisation}\n\nTo estimate the spin polarisation in the chemically ordered $L1_0$ and $L1_1$ CoPt samples, we perform Point Contact Andreev Reflection (PCAR) spectroscopy  experiments \\cite{doi:10.1126\/science.282.5386.85, baltz2009conductance, PhysRevB.85.064410, Seemann_2010, 6971749, PhysRevB.76.174435}. In the PCAR technique, spin polarisation in the ballistic transport regime can be determined from fitting the bias dependence of the conductance with the a modified Blonder-Tinkham-Klapwijk (BTK) model \\cite{PhysRevB.63.104510}. \n\nWe measure the Al$_2$O$_3$(sub)\/Pt(4~nm)\/CoPt(128~nm)\/Pt(4~nm) samples grown at 350${\\degree}$C, corresponding to $L1_1$ crystal structure, and 800${\\degree}$C, corresponding to $L1_0$ crystal structure. The PCAR experiment was performed with a Nb wire tip at $4.2\\,$K. Exemplar conductance spectra with fits to the BTK model are given in Figure \\ref{fig:PCAR} (a). The interpretation of PCAR data is rife with difficulties\\cite{Yates2018} and a common issue with the PCAR technique is the presence of degenerate local fitting minima. To ensure that a global best fit is obtained, the fitting code makes use of a differential-evolution algorithm and we then consider the spin polarisation and barrier strength parameter for a large number of independent contacts to the same sample.\n\nFigure \\ref{fig:PCAR} (c) shows the dependence of the polarisation as a function of the square of the barrier strength, Z$^2$. The dashed lines in Figure \\ref{fig:PCAR} are linear fits to the data. The value of the true spin polarisation is is often taken to correspond to $Z=0$, however this is strictly nonphysical. Nevertheless, in an all metal system it is possible to produce contacts approaching an ideal case and extrapolating to $Z=0$ is close to the (finite) minimum. We find that P = 47$\\pm3$\\% for both $L1_1$ and $L1_0$ CoPt samples. This compares to $\\approx42$\\% for $L1_0$ FePt \\cite{PhysRevB.76.174435} and $\\approx50$\\% for $L1_0$ FePd \\cite{Seemann_2010}.\n\n\n\n\n \n\n\n\n\\section*{Conclusions}\n\nThe major conclusions of this work may be summarised as follows. On $c$-plane Al$_2$O$_3$ with thin Pt [111] buffer layers, the Co$_{50}$Pt$_{50}$ grows following the [111] ordering. Through growth at elevated temperatures, Co$_{50}$Pt$_{50}$ is grown epitaxially in the chemically ordered $L1_1$ and $L1_0$ crystal structures or the chemically disordered $A1$ phase. The $L1_1$ Co$_{50}$Pt$_{50}$ grown between 200$^\\circ$C and 400$^\\circ$C shows perpendicular magnetic anisotropy for thicknesses $\\geq2$~nm. The $L1_0$ Co$_{50}$Pt$_{50}$ grown above 800$^\\circ$C shows significantly harder magnetic anisotropy, which is perpendicular to the [100] direction for thicknesses $\\geq40$~nm. For growth at intermediate temperatures, $450<800^\\circ$C, the Co$_{50}$Pt$_{50}$ shows a disordered structure and in-plane magnetic anisotropy associated with the $A1$ phase. The spin-polarisation of the $L1_1$ and $L1_0$ Co$_{50}$Pt$_{50}$ is determined by the PCAR technique to be 47$\\pm3$\\%.\n\n\n\n\n\\section*{Methods}\n\n\n\n\n\\subsection*{Epitaxial growth}\n\nSamples are dc sputter deposited in the Royce Deposition System \\cite{Royce}. The magnetrons are mounted below, and confocal to, the substrate with a source-substrate distance of 134~mm. The base pressure of the vacuum chamber is $1\\times10^{-9}$~mbar. The samples are deposited at elevated temperatures, with an Ar (6N purity) gas pressure of $4.8\\times10^{-3}$~mbar.\n\nFor alloy growth, we use the co-sputtering technique. To achieve as close to a Co$_{50}$Pt$_{50}$ stoichiometry as possible, first, single layer samples of Co or Pt are grown at room temperature on 10x10~mm thermally oxidised Si substrates varying the magnetron power. From this initial study, it is found that a growth rate of 0.05~nm s$^{-1}$ is achieved for a Co power of 45W and a Pt power of 25W. These growth powers are fixed for the rest of the study.\n\nFor the growth of the CoPt samples, 20x20~mm $c$-plane sapphire substrates are used. The substrates are heated by a ceramic substrate heater mounted directly above the substrate holder. The measured substrate heater temperature is reported. We note that the temperature on the substrate surface is most likely to be below the reported heater temperature. The substrate heater is ramped up from room temperature to the set temperature at a rate of 3-5${\\degree}$C min$^{-1}$. Once at the set temperature, the system is given 30~min to reach equilibrium before starting the sample growth.\n\nOnce the system is ready for growth, 4~nm Pt seed layer is deposited. The seed layer is immediately followed by the  CoPt layer, which is deposited at a rate of 0.1~nm s$^{-1}$ by co-sputtering from the two targets at the determined powers. Finally, a 4~nm Pt capping layer is deposited to prevent the samples from oxidising. The final sample structure is Al$_2$O$_3$(sub)\/Pt(4~nm)\/CoPt($d_{CoPt}$)\/Pt(4~nm). Following deposition, the samples are post growth annealed for 10~min at the growth temperature before the substrate heater is ramped down to room temperature at 10${\\degree}$C min$^{-1}$.\n\n\n\n\\subsection*{Characterisation}\n\n\n Magnetisation loops are measured using a Quantum Design MPMS 3 magnetometer. Angular dependent magnetization measurements are performed using the Quantum Design Horizontal Rotator option. X-ray diffraction and reflectivity is performed on a Bruker D8 diffractometer with an additional four-bounce monochromator to isolate Cu K-$\\alpha$ at a wavelength of 1.5406~\\AA. Sheet films are patterned into Hall bars of 5~$\\mu$m width using conventional photolithography and Ar ion milling. Resulting devices are measured using 4-point-probe transport to measure the Hall resistance of the films using a combined Keithley 6221-2182A current source and nano-voltmeter.\n\n\\subsection*{PCAR}\n\nOur experimental setup for performing PCAR measurements is described elsewhere \\cite{baltz2009conductance, PhysRevB.85.064410, Seemann_2010, 6971749, PhysRevB.76.174435}. The Nb tips are prepared from commercial 99.9\\% pure Nb wires with a diameter of 0.5~mm. An AC lock-in detection technique using Stanford Research Systems SR830 lock-in amplifiers is used for the differential conductance measurements. The tip position is mechanically adjusted by a spring-loaded rod driven by a micrometer screw. The experiment is carried out in liquid He at a fixed temperature of $4.2\\,$K and at zero applied magnetic field.\n\n\n\n\\section*{Data Availability}\n\nThe datasets generated during the current study are available in the University of Leeds repository, https:\/\/doi.org\/xx.xxxx\/xxx.\n\n\n\n\\section*{X-ray Diffraction Data}\n\nFigures \\ref{fig:XRD_ALL_1}, \\ref{fig:XRD_ALL_2}, \\ref{fig:XRD_ALL_3} show a complete set of X-ray diffraction scans for samples grown at each temperature. The very sharp peak at $2\\theta\\approx42\\degree$ is from the c-plane sapphire $(0\\,0\\,0\\,6)$ plane with $\\{1\\,1\\,1\\}$ peaks from the CoPt phase clearly visible to left. The substrate $(0\\,0\\,0\\,12)$ and CoPt $\\{2\\,2\\,2\\}$ peaks are also present close to $2\\theta$ of $90\\degree$. Samples grown at $200-450\\degree\\,$C and also at $800\\degree\\,$C clearly exhibit Pendellosung fringes, indicative of good crystal ordering from the L$1_1$ and L$1_0$ phases. The peak intensity and FWHM shown in the main text are extracted from a Gaussian peak fit to the CoPt $\\{1\\,1\\,1\\}$ peak data, although  similar data could be obtained from the CoPt $\\{2\\,2\\,2\\}$ data. \n\n\\section*{Magnetometry}\n\nMagnetic hysteresis loops were measured for a set of samples of common thickness grown at temperatures from room temperature to $850\\degree\\,$C with the field applied in the plane of the substrate and perpendicular to the substrate plane - shown in Figure \\ref{fig:stream}. For the data shown here, a diamagnetic background resulting from the substrate and sample stick has been removed. The saturation magnetisation is determine by fitting a line to the outer section of the upper and lower branches and taking the difference in intercepts with the $H=0$ axis. The uncertainty is dominated by the determination of the sample area. The saturation field is determined from the point at which the measured data has deviated from the saturation magnetisation by more than twice the scatter in the data in the saturated regions of the loop. Figures \\ref{fig:stream2} and \\ref{fig:stream3} show similar magnetic hysteresis loops for samples of increasing thickness grown at $350\\degree\\,$C and $800\\degree\\,$C respectively.\n\n\\section*{PCAR}\n\nThe Point Contact Andreev Reflection spectra are measured by an AC lockin technique where the contact is subjected to a controlled DC bias with a small AC modulation which is sensed across the contact and across a control resistor to allow a differential conductance to be determined. The DC bias is separately monitored as it is adjusted to give a tip bias typically in the range of $\\approx\\pm30\\,$meV. The DC bias is swept over 6 quarters from 0 to maximum and then to minimum and back to maximum and finally to zero. We only analyse the parts of the trace from maximum to minimum and back to maximum. The measured DC bias is subject to a small instrumentation offset which is removed at this first stage of data processing. Since all the features of interest in the spectra should be symmetrical about zero bias, we decompose the data into symmetric and anti-symmetric parts and retain only the symmetric part. The BTK model is usually considered in terms of the normalised conductance of the contact -- normalised to the conductance at high bias where transfer between electron\/hole states in the ferromagnetic and electron\/hole-like quasiparticles in the superconductor. In this high bias regime it is possible in some contact to get parabolic conductance due to tunnelling or Joule heating effects\\cite{baltz2009conductance} and so we normalise the experimental data by dividing through a parabola fitted to the outer 20\\% of the spectra.\n\nThe modified BTK model used in the analysis of the PCAR data has 4 fitting parameters. Firstly, an interfacial barrier strength $Z$ that is dimensionless. This effectively determines a scattering potential at the interface and so even in a perfect contact will take a small, but finite, value. Secondly, the spin polarisation $P=\\frac{N_\\uparrow v_{f\\uparrow}-N_\\downarrow v_{f\\downarrow}}{N_\\uparrow v_{f\\uparrow}+N_\\downarrow v_{f\\downarrow}}$ which thus ranges from 0 for non-spin-polarised materials to 1 for fully spin-polarised materials. Thirdly, the superconducting energy gap $\\Delta$ which might reasonably be expected not to be fully bulk like at the tip of the contact. Finally there is a `smearing parameter` $\\omega$ that encompasses both thermal effects and athermal scattering processes not otherwise accounted for in the model.\n\nThe modified BTK model is then fitted to the now normalised data. We start with an initial set of fitting values of $Z=0.5$, $\\omega=0.5\\,$meV (c.f. $4.2\\,$K $\\equiv 0.36\\,$meV), $\\Delta=1.5\\,$meV and $P=0.5$. We use a differential-evolution algorithm to identify a global minimum $\\chi^2$ value to locate the best fit. This best-fit location is then used to seed a non-linar least squares fit using the Levenberg\u2013Marquardt algorithm, allowing us to determine estimated standard errors for the fitting parameters.\n\nWe then discard fits with non-physical values of $\\Delta$, excessibley large $\\omega$ -- both of which would indicate an unaccounted spreading resistance, before considering all the fitted $(Z,P)$ combinations. As noted in the main text, the extrapolation back to $Z^2=0$ is not strictly physical, however, in practice in an all-metal system, very low values of $Z$ with uncertainties that encompass $Z=0$ can be obtained.\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{XRD_ALL_1}\n\\caption{X-ray diffraction for the growth temperature set of samples reported in the main text. Left column: Full data range. Right column: Cropped region around the 111 structural peak.}\n\\label{fig:XRD_ALL_1}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{XRD_ALL_2}\n\\caption{X-ray diffraction for the growth temperature set of samples reported in the main text. Left column: Full data range. Right column: Cropped region around the 111 structural peak.}\n\\label{fig:XRD_ALL_2}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{XRD_ALL_3}\n\\caption{X-ray diffraction for the growth temperature set of samples reported in the main text. Left column: Full data range. Right column: Cropped region around the 111 structural peak.}\n\\label{fig:XRD_ALL_3}\n\\end{figure}\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\linewidth]{Magnetization_Growth_T_ALL}\n\\caption{Volume magnetisation for the growth temperature set of samples reported in the main text. The thickness of all samples is 40~nm.}\n\\label{fig:stream}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\linewidth]{Magnetization_350_ALL}\n\\caption{Volume magnetisation for the thickness series set of samples reported in the main text for growth at 350$^\\circ$C.}\n\\label{fig:stream2}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\linewidth]{Magnetization_800_ALL}\n\\caption{Volume magnetisation for the thickness series set of samples reported in the main text for growth at 800$^\\circ$C. For the 1~nm and 2~nm samples which did not show ferromagnetism, the uncorrected moment vs field is presented.}\n\\label{fig:stream3}\n\\end{figure}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWith the establishment of the standard model (SM) by the discovery of the Higgs boson, searching for physics \nbeyond the SM (BSM) and understanding the electroweak phase transition have become main topics in  particle physics.\nAs one scenario of BSM, the gauge-Higgs unification (GHU) scenario has been studied\n\\cite{Manton,Hosotani1983,Hosotani1988,\nDavies:1987,Davies:1988,\nHIL,Hatanaka1999,Kubo:2001zc,Burdman:2002se,\nCsaki:2002,\nScrucca2003,\nAgashe,\nCacciapaglia2006,\nMedina,\nFalkowski}.\nIn GHU, the Higgs boson is a part of the extra-dimensional \ncomponent of gauge potentials, appearing as a fluctuation mode of an \nAharonov-Bohm (AB) phase $\\theta_H$ in the fifth dimension.\nMany GHU models are proposed for electroweak unification \\cite{\nHosotani:2007qw,\nHosotani:2008by,\nFunatsu:2013ni,\nFunatsu:2014fda,\nFunatsu:2016uvi,\nFunatsu:2017nfm,\nHatanaka:2013iya,\nFunatsu:2019xwr,\nFunatsu:2019fry,\nFunatsu:2020haj,\nFunatsu2019a,\nFunatsu-FT,\nFunatsu:2021yrh,\nYoon2017,Yoon2018,Yoon2019,\nKurahashi:2014jca,\nMatsumoto:2016okl,\nHasegawa:2018jze,\nKakizaki:2021kof\n},\nand  GHU models for grand unification are also proposed \\cite{\nLim2007,\nHosotani:2015hoa,\nFurui,\nHosotani:2017edv,\nEnglert:2019xhz,\nEnglert:2020eep,\nKakizaki:2013eba,\nKojima:2017qbt,\nMaru:2019bjr,\nAngelescu:2021nbp\n}. \nAmong them, two types of $\\mathrm{SU}(3)_{C} \\times \\mathrm{SO}(5) \\times \\mathrm{U}(1)_{X}$ GHU models,\nthe A-model\\cite{\nHosotani:2007qw,\nHosotani:2008by,\nFunatsu:2013ni,\nFunatsu:2014fda,\nFunatsu:2016uvi,\nFunatsu:2017nfm,\nHatanaka:2013iya}\nand B-model\\cite{Funatsu:2019xwr,Funatsu:2019fry,Funatsu:2020haj,\nFunatsu2019a,\nFunatsu-FT,\nFunatsu:2021yrh},  in warped spacetime have been extensively studied. \nAt low energies below the electroweak scale the mass spectrum and gauge and Higgs couplings of\nSM particles are nearly the same as in the SM.\n\nCouplings of the first Kaluza-Klein (KK) neutral gauge bosons to right-handed SM fermions are large in the A-model,\nwhereas those to left-handed SM fermions become large in the B-model.\nIn proton-proton collisions KK bosons of photons and $Z$ bosons appear as huge broad resonances\nof $Z'$ bosons in the Drell-Yan process, and can be seen in current and future hadron collider experiments\n \\cite{Funatsu:2014fda,Funatsu:2016uvi,Funatsu:2021yrh}.\nThe KK-excited states of the $W$ boson are also seen as resonances of $W'$ bosons. \nIn the A-model the couplings of the first KK $W$ boson to \nthe SM fermions are small.\nIn the B-model the couplings of the first KK $W$ boson to right-handed fermions are negligible, while couplings to left-handed fermions are much larger than the $W$ boson couplings.\nTherefore at the LHC the first KK $W$ boson appears as a  narrow resonance of $W'$ boson in the A-model, \nbut appears as a broad resonance in the B-model\\cite{Funatsu:2016uvi,Funatsu:2021yrh}.  \n\n\nIn  $\\mathrm{e}^{+}\\mathrm{e}^{-}$ collider experiments,   effects of GHU can be examined\nby exploring interference effects among photon, $Z$ boson and $Z'$ bosons.\nIn the previous papers we have studied  effects of new physics (NP) on\nsuch observable quantities  as cross section, forward-backward asymmetry and left-right asymmetry \nin $\\mathrm{e}^{+}\\mathrm{e}^{-} \\to f\\bar{f}$ ($f\\ne \\mathrm{e}$) with polarized and unpolarized $\\mathrm{e}^{+}\\mathrm{e}^{-}$ beams\n\\cite{Funatsu:2017nfm,Funatsu:2020haj,\nYoon2018,\nYoon2019,\nBilokin2017,\nIrles2019,\nIrles2020}.\nIn Ref.\\ \\cite{Funatsu:2017nfm} we compared such observable quantities of GHU with those of the SM in LEP experiments\nat $\\sqrt{s} = M_Z$, and LEP2 experiments for $130 \\text{GeV} \\le \\sqrt{s} \\le 207 \\,\\text{GeV}$\n\\cite{ALEPH:2005ab,ALEPH:2013dgf}.\nIn Refs.~\\cite{Funatsu:2017nfm,Funatsu:2020haj} we also gave predictions of several  signals of $Z'$ bosons in \nGHU in  $\\mathrm{e}^{+}\\mathrm{e}^{-} $ collider experiments \ndesigned  for future  with collision energies $\\sqrt{s} \\ge 250 \\text{ GeV}$ with polarized electron and positron beams.\nIn the $\\mathrm{e}^{+}\\mathrm{e}^{-} \\to f\\bar{f}$ ($f\\ne \\mathrm{e}$) modes,\nthe deviations of total cross sections become large for\nright-polarized electrons in the A-model, whereas\nin the B-model the deviations are large for left-polarized electrons.\n\nDeviations from the SM can be seen in the Higgs couplings as well.\n$HWW$, $HZZ$ and Yukawa couplings deviate from those in the SM in a universal manner\\cite{Hosotani:2007qw,Hosotani:2008by,Kurahashi:2014jca}.\nThey are suppressed by a common factor\n\\begin{eqnarray}\n\\frac{g_{HWW}^{\\text{GHU}}}{g_{HWW}^{{\\rm SM}}},\\,\n\\frac{g_{HZZ}^{\\rm \\text{GHU}}}{g_{HZZ}^{{\\rm SM}}}\n\\simeq\n\\cos(\\theta_H) \n\\end{eqnarray}\nfor $W$ and $Z$ bosons, and \n\\begin{eqnarray}\n\\frac{y_{\\bar{f}f}^{\\text{GHU}}}{y_{\\bar{f}f}^{{\\rm SM}}}\n\\simeq\n\\begin{cases}\n\\cos(\\theta_{H}) & \\text{A-model \\cite{Hosotani:2008by}} \\\\\n\\cos^{2}(\\theta_{H}\/2) & \\text{B-model \\cite{Funatsu:2019fry}} \n\\end{cases}\n\\label{Hcoupling1}\n\\end{eqnarray}\nfor SM fermions $f$.\nIn the analysis of both $Z'$ and $W'$ bosons in hadron colliders\n\\cite{Funatsu:2014fda,Funatsu:2016uvi,\nFunatsu:2021yrh},\nit is found that the AB phase is constrained as $\\theta_{H} \\lesssim 0.1$.\nFor $\\theta_H = \\mathcal{O}(0.1)$,  probable values in the model,    \nthe deviation of the couplings amounts to $(1-\\cos\\theta_H) = \\mathcal{O}(0.005)$, and is small.\nAt the International Linear Collider (ILC) at $\\sqrt{s} = 250\\text{GeV}$, the $ZZH$ coupling can be measured \nin the $0.6 \\%$ accuracy with $2\\text{ ab}^{-1}$ data \\cite{Barklow:2017suo}.\nSince the masses and Higgs couplings of the SM fields in the GHU models are very close to those in the SM, \nthe electroweak phase transition (EWPT) occurs\nat $T_{C} \\sim 160 \\text{GeV}$ and appears very weak first order\\cite{Hatanaka:2013iya,Funatsu-FT}\nin both A- and B-models, which is very similar to EWPT in the SM\\cite{Senaha:2020mop}.\n\nIn this paper we study  effect of $Z'$ bosons in GHU models\non the $\\mathrm{e}^{+} \\mathrm{e}^{-} \\to \\mathrm{e}^{+} \\mathrm{e}^{-}$ Bhabha scattering.\nMeasurements of  Bhabha scattering at  linear colliders have\ncontributed to the establishment of the SM\\cite{\nAbe:1994sh,\nAbe:1994wx,\nAbe:1996nj\n}.\nBhabha scattering is also useful to explore NP \\cite{\nRichard:2018zhl,\nDas:2021esm\n}.\nUnlike $\\mathrm{e}^{+} \\mathrm{e}^{-} \\to f\\bar{f}$ ($f\\ne \\mathrm{e}$) scattering, \nin Bhabha scattering not only $s$-channel but also  $t$-channel contribution enter the process.\nSince $t$-channel contribution of photon exchange dominates in   forward scattering amplitudes,\nthe cross section becomes very large for  small scattering angles,\nwhich improves the statistics of experiments.\nIt will be seen below that effects of $Z'$ bosons on cross sections can be measured \nwith large significances.\n\nBhabha scattering at very small scattering angle\nis used for the determination of the luminosity of the $\\mathrm{e}^{+}\\mathrm{e}^{-}$ beams.\nSince cross sections of  all other scattering processes depend on the luminosity, \none needs to know whether or not effects of $Z'$ bosons on the $\\mathrm{e}^{+}\\mathrm{e}^{-} \\to \\mathrm{e}^{+}\\mathrm{e}^{-}$ cross section\nare sufficiently small.\nForward-backward asymmetry $A_{FB}$ of the cross section in Bhabha scattering  is no longer a good quantity \nfor searching NP, since the backward scattering cross section is much smaller than the forward scattering cross section.\nWe will propose a new quantity $A_X$ to measure with polarized $\\mathrm{e}^{+}\\mathrm{e}^{-}$ beams,\nwhich can be used for seeing NP effects instead of $A_{FB}$.\n\nIn Section 2 we briefly review the GHU A- and B-models  and \ndiscuss the $\\mathrm{e}^{+}\\mathrm{e}^{-} \\to \\mathrm{e}^{+}\\mathrm{e}^{-}$ scattering \nin both the SM and GHU models. In Section 3, we show the formulas of\n$\\mathrm{e}^{+}\\mathrm{e}^{-}\\to \\mathrm{e}^{+}\\mathrm{e}^{-}$ scattering cross sections for longitudinally polarized $\\mathrm{e}^{\\pm}$ beams, \nand numerically evaluate the effects\nof $Z'$ bosons in GHU models on  differential cross sections and left-right asymmetries.\nWe also show that effects of $Z'$ bosons on the cross section\nare very small at the very small scattering angle.\n\n\\section{Gauge-Higgs unification}\n\nIn GHU A- and B-models the electroweak $\\mathrm{SU}(2)\\times\\mathrm{U}(1)$ symmetry\nis embedded in $\\mathrm{SO}(5)\\times\\mathrm{U}(1)_{X}$ symmetry in the Randall-Sundrum warped space \\cite{Randall:1999ee},\nwhose metric is given by\n\\begin{align}\nds^{2} &= \\frac{1}{z^{2}} \\left[\\eta_{\\mu\\nu}dx^{\\mu}dx^{\\nu} \n+ \\frac{dz^{2}}{k^{2}}\n\\right],\n\\quad\n1 \\le z \\le z_{L} = e^{kL}\n\\end{align}\nwhere $\\eta_{\\mu\\nu} = \\diag(-1,+1,+1,+1)$ and $k$ is the AdS-curvature.\n We refer two 4D hyperplanes at $z=1$ and $z=z_{L}$ as the UV and IR branes, respectively.\nThe $\\mathrm{SO}(5)$ symmetry is broken to $\\mathrm{SO}(4)\\simeq \\mathrm{SU}(2)_{L} \\times \\mathrm{SU}(2)_{R}$ by the boundary conditions at $z=1,\\,z_{L}$ and\nthe $\\mathrm{SU}(2)_{R} \\times \\mathrm{U}(1)_{X}$ symmetry is broken to $\\mathrm{U}(1)_{Y}$ by a scalar field \nlocalized on the UV brane.\n$\\mathrm{SU}(2)_{L}\\times\\mathrm{U}(1)_{Y}$ symmetry is broken to the \nelectromagnetic $\\mathrm{U}(1)_{\\rm EM}$ symmetry by\nthe VEV of the $z$-component gauge fields of $\\mathrm{SO}(5)\/\\mathrm{SO}(4)$. \nThe VEV is related to the gauge-invariant AB phase $\\theta_{H}$. \n\nThe difference between the A- and B-models lies in the content of fermions.\nIn the A-model, quarks and leptons in the SM are embedded in the $\\mathrm{SO}(5)$-vector representation. \nIn the B-model quarks and leptons are embedded in the $\\mathrm{SO}(5)$-spinor, vector and singlet representations.\nThese fields are naturally derived from spinor and vector multiplets in the $\\mathrm{SO}(11)$ \ngauge-Higgs grand unification\\cite{Hosotani:2015hoa, Furui}. \n\nInteractions of the electron and gauge bosons are given by\n\\begin{align}\n\\int d^{4}x \\int_{1}^{z_{L}} \\frac{dz}{k}\n\\biggl\\{\n\\bar{\\check{\\Psi}} \\gamma^{\\mu} (\\partial_{\\mu} - i g A_{\\mu}) \\check{\\Psi}\n\\biggr\\}\n\\end{align}\nwhere $A_{\\mu}(x^{\\mu},z)$ is a four-dimensional component of the 5D gauge field and\n$\\Psi(x^{\\mu},z) = z^2 \\check{\\Psi}$ is the 5D electron field.\nThe electron corresponds to the zero mode of the $\\check{\\Psi}(x^{\\mu},z)$.\nIn the A-model  the left-handed electron is localized in the vicinity of the UV brane and the right-handed \ncomponent is localized near the IR brane.\nIn the B-model the right-handed electron is localized in the vicinity of the UV brane and the left-handed \ncomponent is localized near the IR brane.\n$A_{\\mu}(x^{\\mu},z)$ has a KK expansion which contains the photon, $Z$ boson and their KK excited modes. \nThe wave function of the photon is constant in the fifth dimension coordinate $z$.\nThe wave function of  the $Z$ boson is almost constant in $z$, but has nontrivial dependence near the IR brane. \nCouplings of the electron to  $Z$ boson are very close to those in the SM.\nWave functions of the first KK-excited modes of gauge bosons  are localized near the IR brane so that \nthe first KK-excited gauge bosons  couple  strongly with   fermions localized near the IR brane.\nIn the A-model  right-handed electrons have large couplings with first KK-excited gauge bosons \nwhereas in the B-model left-handed electrons couple strongly with the first KK-excited gauge bosons. \n\nIn Tables~\\ref{tbl:model} and \\ref{tbl:couplings}, parameters and couplings in the A- and B-models are tabulated.\nHere, model parameters ($\\theta_H,m_{{\\rm KK}}$ and $z_{L}$),\nmasses, widths and couplings of $Z'$-bosons\nare referred from Refs.~\\cite{Funatsu2019a,Funatsu:2020haj}.\n\\begin{table}[t]\n\\caption{Parameters in GHU models.}\\label{tbl:model}\n\\vspace{5mm}\n\\centering\n\\begin{tabular}{c|ccc|cccccc}\n\\hline\\hline\nModel & $\\theta_{H}$ & $m_{{\\rm KK}}$ & $z_{L}$ \n& $m_{\\gamma^{(1)}}$ & $\\Gamma_{\\gamma^{(1)}}$  \n& $m_{Z^{(1)}}$ & $\\Gamma_{Z^{(1)}}$  \n& $m_{Z_{R}^{(1)}}$ & $\\Gamma_{Z_{R}^{(1)}}$   \n\\\\\n& [rad] & [TeV] & & [TeV] & [TeV] & [TeV] & [TeV] & [TeV] & [TeV] \n\\\\\n\\hline\nA & $0.08$ & $9.54$ & $1.01\\times10^{4}$\n& $7.86$ & $0.99$\n& $7.86$ & $0.53$\n& $7.31$ & $1.01$\n\\\\\nB & $0.10$ & $13.0$ & $3.87\\times10^{11}$ \n& $10.2$ & $3.25$ \n& $10.2$ & $7.84$ \n& $9.95$ & $0.816$  \n\\\\\n\\hline\n\\end{tabular}\n\\end{table}\nThe difference in the magnitude  of $z_{L}$  in the A- and B-models\noriginates from the formulas of top-quark mass.\nIn the A-model \n$\nm_{\\rm top}^{\\rm A} \\simeq (m_{{\\rm KK}}\/(\\sqrt{2}\\pi)) \\sqrt{1 - 4c_{\\rm top}^{2}} \\sin\\theta_{H}\n$ \\cite{Funatsu:2013ni}\nwhereas in the B-model \n$\nm_{\\rm top}^{\\rm B} \\simeq (m_{{\\rm KK}}\/\\pi) \\sqrt{1-4c_{\\rm top}^{2}} \n\\sin\\frac{1}{2}\\theta_{H}$ \\cite{Funatsu:2019xwr}.\nIn both models $W$ boson mass is given by\n$m_{W} \\simeq  m_{{\\rm KK}}\/(\\pi \\sqrt{kL}) \\sin\\theta_{H}$\nso that the lower bounds of $z_{L}$ becomes\n$z_{L} \\gtrsim 8\\times10^{3}$ in the A-model  and $z_{L} \\gtrsim 7 \\times 10^{7}$ in the B-model.\n\\begin{table}[t]\n\\caption{\nLeft-handed and right-handed couplings of the electron, \n$\\ell_{V},r_{V}$ ($V = Z,Z^{(1)}$, $Z_{R}^{(1)}$ and $\\gamma^{(1)}$), in unit of $g_{w}\\equiv g_{A}\/\\sqrt{L}$ (see text).\nRatios of $e$ and $g_{w}$ are shown at the second column.}\\label{tbl:couplings}\n\\vspace{5mm}\n\\centering\\small\n\\begin{tabular}{c|cccccccccc}\n\\hline\\hline\nModel & $(e\/g_{w})^{2}$ \n & $\\ell_{Z}$ & $r_{Z}$ \n & $\\ell_{Z^{(1)}}$ & $r_{Z^{(1)}}$ \n & $\\ell_{Z_{R}^{(1)}}$ & $r_{Z_{R}^{(1)}}$ \n & $\\ell_{\\gamma^{(1)}}$ & $r_{\\gamma^{(1)}}$ & \n\\\\\n\\hline\nA & $0.2312$\n& $-0.3066$ & $0.2638$\n& $0.1195$ & $0.9986$\n& $0.0000$ & $-1.3762$ \n& $0.1879$ & $-1.8171$ \\\\\nB  & $0.2306$ \n& $-0.3058$ & $0.2629$ \n& $-1.7621$ & $-0.0584$ \n& $-1.0444$ & $0.0000$ \n& $-2.7587$ & $0.1071$\n\\\\\n\\hline\n\\end{tabular}\n\\end{table}\nIn Table~\\ref{tbl:couplings}, the left- and right-handed electron couplings to $Z'$ bosons, $r_{V}, \\ell_{V}$ ($V = Z,Z^{(1)},Z_{R}^{(1)},\\gamma^{(1)}$), are tabulated.\nIn the table $g_{w}\\equiv g_{A}\/\\sqrt{L}$ is the 4D gauge coupling of the $\\mathrm{SO}(5)$ where\n$g_{A}$ is the 5D $\\mathrm{SO}(5)$ coupling.\nIn terms of $g_{A}$ and the 5D $\\mathrm{U}(1)_{X}$ coupling $g_{B}$, a mixing parameter is defined as \\cite{Funatsu:2014fda,Funatsu:2020haj}\n\\begin{align}\ne\/g_{w} = \\sin\\theta_{W}^{0} &\\equiv \n\\frac{s_{\\phi}}{\\sqrt{1 + s_{\\phi}^{2}}},\n\\quad\ns_{\\phi} \\equiv g_{B}\/\\sqrt{ g_{A}^{2} + g_{B}^{2}}.\n\\end{align}\nThe value of $\\sin\\theta_{W}^{0}$ is determined so as to reproduce \nthe experimental value of the forward-backward asymmetry in \n$\\mathrm{e}^{+}\\mathrm{e}^{-}\\to \\mu^{+}\\mu^{-}$ scattering at the $Z$-pole.\nIn the A-model electron's right-handed couplings to $Z'$-bosons are larger than left-handed couplings.\nIn the B-model electron's left-handed couplings to $Z'$-bosons are larger than right-handed couplings.\n\n\n\\section{Bhabha scattering in $\\mathrm{e}^{+}\\mathrm{e}^{-}$ colliders}\n\nWe consider the $\\mathrm{e}^{+} \\mathrm{e}^{-} \\to \\mathrm{e}^{+} \\mathrm{e}^{-}$ scattering in the center-of-mass frame.\nIn this frame, the Mandelstam variables $(s,t,u)$ are given by\n\\begin{align}\ns &= 4 E^{2},\n\\nonumber \\\\\nt &=-\\frac{s}{2}(1-\\cos\\theta) = -s \\sin^{2}\\frac{\\theta}{2},\n\\nonumber \\\\\nu &=-\\frac{s}{2}(1+\\cos\\theta) = -s \\cos^{2} \\frac{\\theta}{2}\n\\end{align}\nwhere $E$  is the energy of initial electron and positron, and $\\theta$ is the scattering angle of the electron.\nSince the $\\mathrm{e}^{+}\\mathrm{e}^{-} \\to \\mathrm{e}^{+}\\mathrm{e}^{-}$ scattering process consists\nboth $s-$ and $t-$channel processes,\nthe scattering amplitude is written in terms of the following six building blocks:\n\\begin{align}\nS_{LL} = S_{LL}(s) \n&\\equiv \n\\sum_{i} \\frac{\\ell_{V_{i}}^{2}}{s - M_{V_{i}}^{2} + iM_{V_{i}}\\Gamma_{V_{i}}},\n\\nonumber \\\\\nS_{RR}= S_{RR}(s) \n&\\equiv\n\\sum_{i} \\frac{r_{V_{i}}^{2}}{s - M_{V_{i}}^{2} + iM_{V_{i}}\\Gamma_{V_{i}}},\n\\nonumber \\\\\nS_{LR} = S_{LR}(s) &\\equiv\n\\sum_{i} \\frac{\\ell_{V_{i}} r_{V_{i}}}{s - M_{V_{i}}^{2} + iM_{V_{i}}\\Gamma_{V_{i}}},\n\\nonumber \\\\\nT_{LL} = T_{LL}(s,\\theta) &\\equiv\n\\sum_{i} \\frac{\\ell_{V_{i}}^{2}}{t - M_{V_{i}}^{2} + iM_{V_{i}}\\Gamma_{V_{i}}},\n\\nonumber \\\\\nT_{RR} = T_{RR}(s,\\theta) &\\equiv\n\\sum_{i} \\frac{r_{V_{i}}^{2}}{t - M_{V_{i}}^{2} + iM_{V_{i}}\\Gamma_{V_{i}}},\n\\nonumber \\\\\nT_{LR} = T_{LR}(s,\\theta) &\\equiv \n\\sum_{i} \\frac{\\ell_{V_{i}} r_{V_{i}} }{t - M_{V_{i}}^{2} + iM_{V_{i}}\\Gamma_{V_{i}}},\n\\label{eq:blocks}\n\\end{align}\nwhere $M_{V_{i}}$ and $\\Gamma_{V_{i}}$ are mass and width of the vector boson $V_{i}$.\n$\\ell_{V_{i}}$ and $r_{V_{i}}$ are left- and right-handed couplings of electrons to the vector boson $V_{i}$ ($V_{0} =\\gamma$, $V_{1} = Z$), respectively. \nIn particular, we have\n$\\ell_{\\gamma} = r_{\\gamma} = Q_{\\mathrm{e}} e$, $Q_{\\mathrm{e}} = -1$,\n$\\ell_{Z} = \\frac{e}{\\sin\\theta_{W}\\cos\\theta_{W}} [I_{\\mathrm{e}}^{3} - Q_{e}\\sin^{2}\\theta_{W}]$,\n$r_{Z} = \\frac{e}{\\sin\\theta_{W}\\cos\\theta_{W}} [- Q_{e}\\sin^{2}\\theta_{W}]$,\n $I^{3}_{\\mathrm{e}} = -\\frac{1}{2}$ in the SM. \nHere $e$, $I_{\\mathrm{e}}^{3}$ and $\\theta_{W}$ are the electromagnetic coupling, weak isospin of the electron and Weinberg angle, respectively.\n\nWhen  initial state electrons and positrons are longitudinally polarized, \nthe differential cross section is given by\n\\begin{align}\n\\frac{d\\sigma}{d\\cos\\theta}(P_{\\mathrm{e}^{-}}, P_{\\mathrm{e}^{+}})\n&= \\frac{1}{4} \\biggl\\{\n(1+ P_{\\mathrm{e}^{-}}) (1+P_{\\mathrm{e}^{+}}) \n\\frac{d\\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{R} }}{d\\cos\\theta}\n+\n(1 - P_{\\mathrm{e}^{-}}) (1 - P_{\\mathrm{e}^{+}}) \n\\frac{d\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{L} }}{d\\cos\\theta}\n\\nonumber\\\\& \\qquad\n+\n(1 + P_{\\mathrm{e}^{-}}) (1 - P_{\\mathrm{e}^{+}}) \n\\frac{d\\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{L} }}{d\\cos\\theta}\n+\n(1 - P_{\\mathrm{e}^{-}}) (1 + P_{\\mathrm{e}^{+}}) \n\\frac{d\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{R} }}{d\\cos\\theta}\n\\biggr\\},\n\\end{align}\nwhere $P_{\\mathrm{e}^{-}}$ and $P_{\\mathrm{e}^{+}}$ are the polarization of the electron and positron beam, respectively. $P_{\\mathrm{e}^{-}}=+1$ ($P_{\\mathrm{e}^{+}}=+1$) denotes purely right-handed electrons (positrons) \\cite{MoortgatPick:2005cw,Arbuzov:2020ghr}.\n$\\sigma_{\\mathrm{e}_{X}^{-} \\mathrm{e}_{Y}^{+}}$ ($X,Y=L,R$) denotes the cross section \nfor left-handed or right-handed electron and positron. \nWhen the electron mass is neglected, these cross sections \nare given by\n\\begin{align}\n\\frac{d\\sigma_{\\mathrm{e}^-_L \\mathrm{e}^+_R}}{d\\cos\\theta}\n&= \\frac{1}{8\\pi s}\n\\left(  u^2 |S_{LL} + T_{LL}|^2 + t^2 |S_{LR}|^2 \\right),\n\\nonumber \\\\\n\\frac{d\\sigma_{\\mathrm{e}^-_R \\mathrm{e}^+_L}}{d\\cos\\theta}\n&= \\frac{1}{8\\pi s}\n\\left(  u^2 |S_{RR} + T_{RR}|^2 + t^2 |S_{LR}|^2 \\right),\n\\nonumber \\\\\n\\frac{d\\sigma_{\\mathrm{e}^-_L \\mathrm{e}^+_L}}{d\\cos\\theta}\n&= \\frac{d\\sigma_{\\mathrm{e}^-_R \\mathrm{e}^+_R}}{d\\cos\\theta}\n= \\frac{1}{8\\pi s} \\cdot \\left(s^2 |T_{LR}|^2 \\right).\n\\end{align}\n\nWhen $s,t \\ll M_{Z}^{2}$, \nthe cross section is  approximated by the one at the QED level,\nwhere we obtain $S_{LL} = S_{RR} = S_{LR} = e^2\/s$ and\n$T_{LL} = T_{RR} = T_{LR} = e^2\/t$, and \n\\begin{align}\n\\frac{d\\sigma_{\\rm QED}}{d\\cos\\theta}(P_{e^-},P_{e^+})\n&= \n\\frac{e^{4}}{16\\pi s}\\left\\{\n(1 - P_{\\mathrm{e}^{-}}P_{\\mathrm{e}^{+}} ) \\frac{t^{4}+u^{4}}{s^{2}t^{2}}\n+ (1 + P_{\\mathrm{e}^{-}} P_{\\mathrm{e}^{+}}) \\frac{s^{2}}{t^{2}}\n\\right\\}.\n\\end{align}\nFor unpolarized electron or positron beams, the above expression reduces to\na familiar formula\n\\begin{align}\n\\frac{d\\sigma^{\\rm unpolarized}_{\\rm QED}}{ d\\cos\\theta }\n&=\n \\frac{e^4}{16\\pi s} \\frac{s^4 + t^4 + u^4}{s^2 t^2}.\n\\end{align}\nWe also note that in terms of building blocks \\eqref{eq:blocks} we \ncan write down components of s-, t-channels, and interference terms as\n\\begin{align}\n\\frac{d\\sigma}{d\\cos\\theta}\n&=\\frac{d\\sigma^{\\text{s-channel}}}{d\\cos\\theta}\n+ \\frac{d\\sigma^{\\text{t-channel}}}{d\\cos\\theta}\n+ \\frac{d\\sigma^{\\text{interference}}}{d\\cos\\theta},\n\\end{align}\nwhere each component is given by\n\\begin{align}\n\\frac{d\\sigma^{\\text{s-channel}}}{d\\cos\\theta}(P_{\\mathrm{e}^{-}},P_{\\mathrm{e}^{+}})\n&= \\frac{1}{32\\pi s} \\biggl\\{\n  (1+P_{\\mathrm{e}^{-}})(1-P_{\\mathrm{e}^{+}}) \\left[ u^{2} |S_{RR}|^{2} + t^{2} |S_{LR}|^{2} \\right]\n\\nonumber\\\\& \\qquad\n+ (1-P_{\\mathrm{e}^{-}})(1+P_{\\mathrm{e}^{+}}) \\left[ u^{2} |S_{LL}|^{2} + t^{2} |S_{LR}|^{2} \\right]\n\\biggr\\},\n\\nonumber \\\\\n\\frac{d\\sigma^{\\text{t-channel}}}{d\\cos\\theta}(P_{\\mathrm{e}^{-}},P_{\\mathrm{e}^{+}})\n&= \\frac{1}{32\\pi s}\\biggl\\{\n  (1+P_{\\mathrm{e}^{-}})(1+P_{\\mathrm{e}^{+}}) s^{2} |T_{LR}|^{2}\n+ (1-P_{\\mathrm{e}^{-}})(1-P_{\\mathrm{e}^{+}}) s^{2} |T_{LR}|^{2}\n\\nonumber\\\\& \\qquad\n+ (1+P_{\\mathrm{e}^{-}})(1-P_{\\mathrm{e}^{+}}) u^{2} |T_{RR}|^{2}\n+ (1-P_{\\mathrm{e}^{-}})(1+P_{\\mathrm{e}^{+}}) u^{2} |T_{LL}|^{2}\n\\biggr\\},\n\\nonumber \\\\\n\\frac{d\\sigma^{\\text{interference}}}{d\\cos\\theta}(P_{\\mathrm{e}^{-}},P_{\\mathrm{e}^{+}})\n&= \\frac{1}{16\\pi s} u^{2} \\biggl\\{\n (1+P_{\\mathrm{e}^{-}})(1-P_{\\mathrm{e}^{+}}) \\Re(S_{RR} T_{RR}^{*})\n\\nonumber\\\\& \\qquad\n+(1-P_{\\mathrm{e}^{-}})(1+P_{\\mathrm{e}^{+}}) \\Re(S_{LL} T_{LL}^{*})\n\\biggr\\}.\n\\end{align}\n\n\nWhen initial electrons and\/or positrons are longitudinally polarized, one can measure left-right asymmetries.\nThe left-right asymmetry of polarized cross sections is given by\n\\begin{align}\nA_{\\text{LR}}(P_{-},P_{+})\n&\\equiv \\frac{\n\\sigma(P_{\\mathrm{e}^{-}} = -P_{-}, P_{\\mathrm{e}^{+}}= - P_{+})\n- \\sigma(P_{\\mathrm{e}^{-}} = +P_{-}, P_{\\mathrm{e}^{+}}= + P_{+})\n}{\n\\sigma(P_{\\mathrm{e}^{-}} = -P_{-}, P_{\\mathrm{e}^{+}}= - P_{+})\n+ \\sigma(P_{\\mathrm{e}^{-}} = +P_{-}, P_{\\mathrm{e}^{+}}= + P_{+})\n}\n\\nonumber \\\\\n&= (P_{-} -  P_{+}) \\cdot \n\\frac{\n\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{R}} - \\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{L}}\n}{\n(1 + P_{-} P_{+}) (\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{L}} + \\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{R}})\n+\n(1 - P_{-} P_{+}) (\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{R}} + \\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{L}})\n}\n,\n\\nonumber \\\\ \n& 1\\ge P_{-} \\ge 0,\\quad 1 \\ge P_{+} \\ge -1,\n\\label{eq:ALR}\n\\end{align}\nwhere the cross section in a given bin $[\\theta_{1},\\theta_{2}]$ is given by\n$\n\\sigma \\equiv \\int_{\\cos\\theta_{1}}^{\\cos\\theta_{2}} \\frac{d\\sigma}{d\\cos\\theta} d\\cos\\theta\n$.\nWe have used $\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{L}} = \\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{R}}$\nbecause $\\frac{d\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{L}}}{d\\cos\\theta} - \n \\frac{d\\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{R}}}{d\\cos\\theta} = 0$.\nWe can also define the left-right asymmetry of the differential cross section as\n\\begin{align}\n\\lefteqn{A_{\\text{LR}}(P_{-},P_{+},\\cos\\theta)}\n\\quad\n\\nonumber\\\\\n&\\equiv\n\\frac{\n\\dfrac{d\\sigma}{d\\cos\\theta}(P_{\\mathrm{e}^{-}} = -P_{-}, P_{\\mathrm{e}^{+}}= - P_{+})\n- \\dfrac{d\\sigma}{d\\cos\\theta}(P_{\\mathrm{e}^{-}} = +P_{-}, P_{\\mathrm{e}^{+}}= + P_{+})\n}{\n\\dfrac{d\\sigma}{d\\cos\\theta}(P_{\\mathrm{e}^{-}} = -P_{-}, P_{\\mathrm{e}^{+}}= - P_{+})\n+ \\dfrac{d\\sigma}{d\\cos\\theta}(P_{\\mathrm{e}^{-}} = +P_{-}, P_{\\mathrm{e}^{+}}= + P_{+})\n}\n\\nonumber \\\\\n&= \\frac{\n(P_{-} +  P_{+}) \\left(\\dfrac{d\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{L}}}{d\\cos\\theta} \n- \\dfrac{d\\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{R}}}{d\\cos\\theta} \\right)\n+\n(P_{-} -  P_{+}) \\left(\\dfrac{d\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{R}}}{d\\cos\\theta}\n - \\dfrac{d\\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{L}}}{d\\cos\\theta} \\right)\n}{\n(1 + P_{-} P_{+}) \\left(\\dfrac{d\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{L}}}{d\\cos\\theta}\n + \\dfrac{d\\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{R}}}{d\\cos\\theta} \\right)\n+\n(1 - P_{-} P_{+}) \\left(\\dfrac{d\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{R}}}{d\\cos\\theta}\n + \\dfrac{d\\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{L}}}{d\\cos\\theta} \\right)\n}\n\\nonumber \\\\\n&= (P_{-} - P_{+}) \\cdot\n\\frac{\n \\Sigma_{LR-RL}\n}{(1 + P_{-} P_{+}) \\Sigma_{LL+RR} + (1-P_{-} P_{+}) \\Sigma_{LR+RL}},\n\\end{align}\nwhere we have used $d\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{L}}\/d\\cos\\theta =d\\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{R}}\/d\\cos\\theta  $ and defined\n\\begin{align}\n\\Sigma_{LL+RR} &\\equiv 2s^{2} |T_{LR}|^{2},\n\\nonumber \\\\\n\\Sigma_{LR+RL} &\\equiv u^{2} (|S_{LL} + T_{LL}|^{2} + |S_{RR} + T_{RR}|^{2}) + 2t^{2} |S_{LR}|^{2},\n\\nonumber \\\\\n\\Sigma_{LR-RL} &\\equiv  u^{2} ( |S_{LL} + T_{LL}|^{2} - |S_{RR} + T_{RR}|^{2}  ).\n\\end{align}\n\nIn $\\mathrm{e}^+ \\mathrm{e}^- \\to \\mathrm{e}^+ \\mathrm{e}^-$ scatterings we have\n$A_{LR}(P_{-},+P_{+},\\cos\\theta)$ and $A_{LR}(P_{-},-P_{+},\\cos\\theta)$ as independent observables\nand one may define the following non-trivial quantity:\n\\begin{align}\n&A_{X}(\\cos\\theta) \\equiv \n\\frac{\\Sigma_{LL+RR} - \\Sigma_{LR+RL}}{\\Sigma_{LL+RR} + \\Sigma_{LR+RL}}\n\\nonumber \\\\\n&\\quad\n= \\frac{1}{P_{-} P_{+}} \\cdot \\dfrac{\n (P_{-} - P_{+}) A_{\\text{LR}}(P_{-},-P_{+},\\cos\\theta) - (P_{-} + P_{+}) A_{\\text{LR}}(P_{-}, +P_{+}\\cos\\theta)\n}{\n (P_{-} - P_{+}) A_{\\text{LR}}(P_{-},-P_{+},\\cos\\theta) + (P_{-} + P_{+}) A_{\\text{LR}}(P_{-}, +P_{+},\\cos\\theta)\n}.\\label{eq:AX}\n\\end{align} \nThis quantity may be used to explore NP beyond the SM as discussed below.\n\nSince $\\mathrm{e}^+ \\mathrm{e}^- \\to \\mathrm{e}^+ \\mathrm{e}^-$ scattering contains $t$-channel processes,\nforward scatterings dominate. Therefore unlike the $\\mathrm{e}^+\\mathrm{e}^- \\to f\\bar{f}$ ($f\\ne \\mathrm{e}^{-}$) scattering, \nthe forward-backward asymmetry of $\\mathrm{e}^{+} \\mathrm{e}^{-} \\to \\mathrm{e}^{+} \\mathrm{e}^{-} $ scattering is a less-meaningful quantity.\n\nWe note that all of the above formulas can be applied\nto $\\ell^{+}\\ell^{-} \\to \\ell^{+}\\ell^{-}$  ($\\ell = \\mu,\\tau$) scatterings.\n\n\n\n\\section{Numerical Study}\n\nIn the followings we calculate $\\mathrm{e}^{+} \\mathrm{e}^{-} \\to \\mathrm{e}^{+}\\mathrm{e}^{-}$\ncross sections both in the SM and GHU models,\nand we evaluate  effects of $Z'$ bosons in GHU models on observables\ngiven in the previous section.\nAs benchmark points, we have chosen typical parameters of the A- and B-models\nin Tables \\ref{tbl:model} and \\ref{tbl:couplings}. \nFor experimental parameters we choose\n$\\sqrt{s}=250$ GeV and $L_{\\mathrm{int}} = 250$ fb$^{-1}$ as typical value of linear $\\mathrm{e}^{+}\\mathrm{e}^{-}$ colliders like ILC\\cite{ILC}. \nWe also choose $L_{\\mathrm{int}} = 2$ ab$^{-1}$, which will be achieved\nat circular $\\mathrm{e}^{+}\\mathrm{e}^{-}$ colliders like FCC-ee\\cite{Blondel:2021ema} and CEPC\\cite{CEPCStudyGroup:2018ghi}. \nFor the new asymmetry $A_{X}(\\cos\\theta)$ in \\eqref{eq:AX},\n we consider $\\sqrt{s} = 3$ TeV for future linear colliders like  CLIC\\cite{CLICdesign}.\nAs for the longitudinal polarization, we set the parameter ranges\n$- 0.8 \\le P_{\\mathrm{e}^{-}} \\le +0.8$ and $-0.3 \\le P_{\\mathrm{e}^{+}} \\le 0.3$,\nwhich can be achieved at ILC\\cite{ILC}.\n\nIn Figure~\\ref{fig:dsigma}   $\\mathrm{e}^{+}\\mathrm{e}^{-}\\to \\mathrm{e}^{+}\\mathrm{e}^{-}$ differential cross sections in the SM are plotted.\nIn the forward-scattering region ($\\cos\\theta>0$),\n the magnitudes of cross sections of $t$-channel and the interference\nparts are much larger than those of the $s$-channel part.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{dsigma-ee-SM-P00.pdf\n\\includegraphics[width=0.47\\textwidth]{dsigma-ee-SM-P00-Linear.pdf}\n\\caption{%\nDifferential cross sections for unpolarized $\\mathrm{e}^{+}\\mathrm{e}^{-}$ initial\nstates in the SM.\n(a) Log-scale plot with $-0.9 \\le \\cos\\theta \\le 0.9$.\n(b) Linear plot with $-0.7 \\le \\cos\\theta \\le 0.7$.\nIn both plots, red-solid lines indicate the total of $s$-, $t$-channels and interferences. \nThe $s$-channel and $t$-channel contributions are drawn as blue-dashed and purple-dotted lines, respectively. \nIn (b), the negative contribution from interferences is plotted with the green-dashed line.\n}\\label{fig:dsigma}\n\\end{figure}\n\n\n\n\\begin{figure}[H]\n\\includegraphics[width=0.48\\textwidth]{Delta-dsigma-ee-GHU-A4-P00.pdf}\\hspace{0.2cm}\n\\includegraphics[width=0.48\\textwidth]{Delta-dsigma-ee-GHU-B1-P00.pdf}\n\\caption{\nDeviations of differential cross sections \nof GHU from those in the SM,\n$\n\\frac{d\\sigma^{\\text{GHU}}}{d\\cos\\theta}\/\n\\frac{d\\sigma^{{\\rm SM}}}{d\\cos\\theta}-1\n$,\nfor unpolarized $\\mathrm{e}^{+}\\mathrm{e}^{-}$ beams\nare plotted.  \nThe left plot is for the A-model and the right plot is for the B-model. In each plot,\nthe red-solid curve represents the deviation  of the sum of all the components of the differential cross section.\nBlue-dashed, purple-dotted and green dot-dashed curves correspond to\n the deviations of $s$-, $t$-channels and interference components of differential cross sections, respectively.\n Error-bars are estimated for $L_{\\mathrm{int}} = 250 \\text{ fb}^{-1}$. \n}\\label{fig:Delta-dsigma}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{Delta-dsigma-ee-GHU-A4-P+8-3.pdf}\\hspace{0.2cm}\n\\includegraphics[width=0.48\\textwidth]{Delta-dsigma-ee-GHU-A4-P-8+3.pdf}\\\\\n\\includegraphics[width=0.48\\textwidth]{Delta-dsigma-ee-GHU-B1-P-8+3.pdf}\\hspace{0.2cm}\n\\includegraphics[width=0.48\\textwidth]{Delta-dsigma-ee-GHU-B1-P+8-3.pdf}%\n\\caption{%\nDeviations of differential cross sections\nof GHU models from the SM \nfor polarized $\\mathrm{e}^{+}\\mathrm{e}^{-}$ beams.\nGHU-A [(a) and (b)] and GHU-B [(c) and (d)].\n(a) and (c) are for $(P_{e^{-}},P_{e^{+}}) = (-0.8,+0.3)$.\n(b) and (d) are for$(P_{e^{-}},P_{e^{+}}) = (+0.8,-0.3)$.\nMeanings of the curves and error-bars are the same as in Figure~\\ref{fig:Delta-dsigma}.\n}\\label{fig:Delta-dsigma0}\n\\end{figure}\nIn Figures~\\ref{fig:Delta-dsigma} and \\ref{fig:Delta-dsigma0}, \nthe differences of differential cross sections of the GHU from the SM for unpolarized and polarized beams are plotted, respectively.\nIn the figures, differences of $s$-channel, $t$-channel and interference contributions are also plotted.\nIn the $s$-channel, the NP effects contribute destructively in the forward scattering.\nOn the other hand, in the $t$-channel NP effects contribute constructively.\nSince the cross section is dominated by $t$-channel, \nthe total of \n$s$-, $t$- and interference channels \nincreases due to the NP effects.\n\nIn the A-model $Z'$ bosons have larger couplings to right-handed electrons than to left-handed electrons.\nTherefore\nthe cross section of the $\\mathrm{e}_{R}^{-}\\mathrm{e}_{L}^{+}$ initial states  becomes larger than that of $\\mathrm{e}_{L}^{-}\\mathrm{e}_{R}^{+}$. \nOn the other hand,\nin the B-model $Z'$ bosons have larger couplings to left-handed electrons than to right-handed electrons,\nand the cross section of $\\mathrm{e}_{L}^{-}\\mathrm{e}_{R}^{+}$ initial states becomes larger.\n\nNP effects become smaller when $\\theta$ becomes smaller.\nThe statistical uncertainty, however, also becomes small since the cross section\nbecomes very large.\nTherefore deviations of the cross section relative to statistical uncertainties may become large.\n\nFor unpolarized $\\mathrm{e}^{+}\\mathrm{e}^{-}$ beams (Figure~\\ref{fig:Delta-dsigma0}),\nthe new physics effect in  both models tends to enhance the cross section at\nforward scattering with almost the same magnitude.\nIn the B-model the suppression of NP effects due to larger $Z'$ masses \nis compensated by larger couplings of $Z'$ bosons than the couplings  in the A-model.\nThe enhancement of the differential cross section due to the NP effects at $\\cos\\theta \\sim 0.3$ is around 1\\%.\n\nFor polarized beams  deviations can be much larger.\nIn the A-model, electrons have large right-handed couplings to $Z'$ bosons and for right-handed polarized beam relative deviations of the cross-section \nbecome as much as $2\\,\\%$ [Figure~\\ref{fig:Delta-dsigma}-(b)], whereas for right-handed beams relative deviations are around $0.1\\,\\%$ [Figure~\\ref{fig:Delta-dsigma}-(a)].\nContrarily, in the B-model a left-handed electron has large couplings to $Z'$ bosons. Therefore in the B-model deviations become large for left-polarized beam [Figures~\\ref{fig:Delta-dsigma}-(c) and (d)].\nIn Figure \\ref{fig:Delta-dsigma},\nwe have also shown statistical 1 $\\sigma$ relative errors at $L_{\\mathrm{int}} =250$ fb$^{-1}$ \nfor bins\n$[\\cos\\theta_{0}-0.05,\\cos\\theta_{0}+0.05]$ ($\\cos\\theta_{0} = -0.90, -0.80,\\dots, +0.90$) as vertical bars.\nIn each bin, the observed number of events and statistical uncertainty are given by $N$ and $\\sqrt{N}$, respectively.\nTherefore relative error of the cross section is \nestimated as the inverse of the square of the number of events $N$.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.58\\textwidth]{sigdsigma.pdf}\n\\caption{Estimated significances on differential cross sections of GHU models\nwith unpolarized $\\mathrm{e}^{+} \\mathrm{e}^{-}$ beam with the integrated luminosity $L_{\\mathrm{int}} = 2\\text{ ab}^{-1}$.\nA statistical significance of 0.1\\% non-statistical error is plotted with a purple dot-dashed line.}\n\\label{fig:sigdsigma}\n\\end{figure}\n\nIn Figure~\\ref{fig:sigdsigma}, the statistical significances in the GHU models are plotted.\nAn estimated significance of the deviation of the cross section \nin a bin is given by\n\\begin{align}\n\\frac{|N_{\\text{GHU}} - N_{{\\rm SM}}|}{N_{{\\rm SM}}} \\bigg\/ \\frac{\\sqrt{N_{{\\rm SM}}}}{N_{{\\rm SM}}}\n= \\frac{|N_{\\text{GHU}} - N_{{\\rm SM}}|}{\\sqrt{N_{{\\rm SM}}}},\n\\end{align} \nwhere $N_{\\text{GHU}}$ and $N_{{\\rm SM}}$ are observed numbers of events in a bin.\nIn Figure \\ref{fig:sigdsigma},\nsignificances are larger than 5 $\\sigma$\nfor $\\cos\\theta\\gtrsim0.1$. \nSignificances are very large for forward scatterings,\nbut are very small for backward scatterings.\nIn Figure \\ref{fig:sigdsigma}, relative 0.1\\% errors are also shown. Errors due to the NP effects become very small and are around $0.1\\%$ for $\\cos\\theta \\simeq 0.9$. \nA similar analysis has been given in Ref.\\cite{Richard:2018zhl}.\n\nFor small scattering angles $\\theta$, the scattering amplitude is dominated by\n$t$-channel contributions which are constructed with the blocks $T_{LL}$, $T_{RR}$ and $T_{LR}$.\nWhen $|t| \\simeq s\\theta^{2}\/4 \\ll M_{Z}^{2}, M_{Z'}^{2}$,\nwe can approximate the SM and NP contributions to the block $T_{LL}$ in the scattering amplitude as \n\\begin{align}\nT_{LL}^{{ Nucl.\\ Phys.} } &\\equiv \n\\sum_{Z'} \\frac{\\ell_{Z'}^{2}}{t - M_{Z'}^{2} + iM_{Z'}\\Gamma_{Z'}}\n\\simeq \n\\sum_{Z'} \\frac{\\ell_{Z'}^{2} }{- M_{Z'}^{2}+ i M_{Z'}\\Gamma_{Z'}}.\n\\end{align}\nWhen $s\\theta^{2}\/4 \\ll M_{Z}^{2}$, \n$T_{LL}$ is dominated by the QED part $T_{LL}^{\\rm QED} \\simeq -4e^{2}\/(\\theta^{2}s)$ and the NP effects are estimated as\n\\begin{align}\n\\frac{T_{LL}^{{ Nucl.\\ Phys.} }}{T_{LL}^{\\rm QED}} &\\simeq \\theta^{2} \\sum_{Z'} \\frac{\n(\\ell_{Z'}^{2}\/4e^{2})s }{ M_{Z'}^{2} - i M_{Z'}\\Gamma_{Z'}}\n= \\theta^{2} \\cdot \\mathcal{O}(s\/M_{Z'}^{2}),\n\\end{align}\nand similar analysis is applied to $T_{LR}$ and $T_{RR}$.\nConsequently, this correction arises not only in amplitudes but also in\ndifferential cross sections.\nFor $\\sqrt{s} = 250\\,\\text{GeV}$ and  $\\theta \\lesssim 300\\text{ mrad}$,\nthe QED $t$-channel contribution dominates \nand corrections due to $Z'$ bosons are suppressed by a factor $\\theta^{2}s\/M_{Z'}^{2}$.\nIn Figure~\\ref{fig:farforward}, deviations of differential cross sections of GHU from the SM for $\\theta<300\\text{ mrad}$ are plotted.\nDeviations of cross sections from the SM are proportional to the square of \n$\\theta$ and become smaller than $0.1$\\% when $\\theta < 250\\text{ mrad}$.\nThe measurement of Bhabha scatterings at small scattering angle is used for the determination of the luminosity of $\\mathrm{e}^{+} \\mathrm{e}^{-}$ collision and uncertainties of the luminosity should be smaller than $0.1$\\%. In GHU models the NP effects on such uncertainty are well suppressed when $\\theta \\lesssim 100\\text{ mrad}$.\nAt ILC, the luminosity calorimeter in ILD (SiD) operates between 43 and 68 (40 and 90) mrad \\cite{ILC}, where the influence of the $Z'$ bosons is below 0.1 \\%.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.55\\textwidth]{farforward.pdf}\n\\caption{Deviations of differential cross sections of GHU models from the SM in the forward-scattering region ($\\theta \\le 300$\\text{ mrad}).}\\label{fig:farforward}\n\\end{figure}\n\nWhen the initial electron and positron beams are longitudinally polarized,\nthe left-right asymmetry $A_{\\text{LR}}$ \\eqref{eq:ALR} can be measured.\nIn Figure~\\ref{fig:ALR}, the left-right asymmetries of the SM and GHU models are plotted.\nThe measured asymmetries become larger when $|P_{\\mathrm{e}^{-}} - P_{\\mathrm{e}^{+}}|$ are larger. \nAs seen in Figure~\\ref{fig:Delta-dsigma}, in the A-model cross section of $\\mathrm{e}_{R}^{-}\\mathrm{e}_{L}^{+}$ initial states becomes large whereas\nin the B-model cross section of $\\mathrm{e}_{L}^{-}\\mathrm{e}_{R}^{+}$ initial states is enhanced\ndue to the large left-handed $Z'$ couplings.\nTherefore $A_{\\text{LR}}$ of B-models are larger than the SM, whereas $A_{\\text{LR}}$ of A-models are smaller. \n Since the $A_{\\text{LR}}$ is proportional to $|P_{\\mathrm{e}^{-}} - P_{\\mathrm{e}^{+}}|$,\nthe asymmetries in Figure~\\ref{fig:ALR}-(a) are almost twice as large as in (b).\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{ALR-ee-GHU-s=250-Pmppm.pdf}\n\\includegraphics[width=0.48\\textwidth]{ALR-ee-GHU-s=250-Pmpmp.pdf}\n\\caption{Left-right asymmetries. In both plots,\nblue-dotdashed, and red-dashed and black-solid curves correspond to\n the GHU-A, GHU-B and the SM, respectively. \n Error-bars are indicated for $L_{\\mathrm{int}} = 250\\,\\mathrm{fb}^{-1}$ in each polarization. \n (a) Asymmetries for $(P_{\\mathrm{e}^{-}},P_{\\mathrm{e}^{+}}) = (\\mp0.8,\\pm0.3)$.\n (b) Asymmetries for $(P_{\\mathrm{e}^{-}},P_{\\mathrm{e}^{+}}) = (\\mp0.8,\\mp0.3)$. }\\label{fig:ALR}\n\\end{figure}\nIn Figure \\ref{fig:ALR}, an asymmetry $A_{\\text{LR}}$ in a bin and the \nstatistic error $\\Delta A_{\\text{LR}}$ are also shown. \nHere\n\\begin{align}\nA_{\\text{LR}} &= \\frac{N_{L} - N_{R}}{N_{L}+N_{R}},\n&\n\\Delta A_{\\text{LR}} &= \n\\sqrt{\\frac{2 (N_{L}^{2} + N_{R}^{2})}{ (N_{L} + N_{R})^{3}}} \n\\end{align} \nwith $N_{L}$ and $N_{R}$ being observed number of events for the left-handed  ($P_{\\mathrm{e}^{-}}<0$) and right-handed ($P_{\\mathrm{e}^{-}}>0$) electron beams, respectively.\nFor small scattering angle $\\cos\\theta \\gtrsim 0.8$, both \n$A_{\\text{LR}}^{\\text{GHU}}$ and $A_{\\text{LR}}^{{\\rm SM}}$ \nbecome close to each other.\n\nTo see how NP effects against statistical uncertainty grow for small $\\theta$,\nwe plotted in Figure~\\ref{fig:sigALR} the averages and statistical significances of\nleft-right asymmetries in GHU models in each bin which are estimated as  \n\\begin{align}\n\\frac{A_{LR}^{\\text{GHU}} - A_{LR}^{{\\rm SM}} }{\\Delta A_{\\text{LR}}}.\n\\end{align}\nFor the forward scattering with $\\cos\\theta \\gtrsim 0.2$,\nthe deviations are bigger than several times of standard deviations.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.58\\textwidth]{sigALR.pdf}\n\\caption{%\nEstimated statistical significance on left-right asymmetries in GHU models.\nThe blue-dotdashed, red-dashed and\nblack-dotted lines indicate\nGHU-A, GHU-B and\n 0.1\\% non-statistical errors, respectively.}\\label{fig:sigALR}\n\\end{figure}\nIn Figure~\\ref{fig:sigALR}, \nthe significance is larger than $5\\,\\sigma$ for $\\cos\\theta\\gtrsim0.2$. \nBoth models are well distinguished from the SM.\nUsing the magnitude and sign of deviations, the A-model\nand B-model can be distinguished.\n\nIn Figure~\\ref{fig:AX},\nthe asymmetry $A_{X}$ defined in Eq.~\\eqref{eq:AX} is plotted\nfor $\\sqrt{s} = 250\\,\\text{GeV}$ and \n$\\sqrt{s} = 3\\,\\text{TeV}$.\nAt $\\sqrt{s} = 250\\,\\text{GeV}$, the NP effect on $A_{X}$ is very small.\nFor $\\sqrt{s} = 3\\,\\text{TeV}$, the asymmetry $A_{X}$ of the SM and GHU models is clearly different and may be discriminated experimentally.\nIn the present analysis of NP effects, only first KK excited states of neutral bosons are taken into account. At $\\sqrt{s} \\sim  3\\,\\text{TeV}$, effects of second KK modes on $A_{X}$ are estimated as a few percent. These effects are much smaller than the effects of the first KK modes.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{AX-ee-GHU-A4B1-s=250.pdf}\\hspace{0.2cm}\n\\includegraphics[width=0.48\\textwidth]{AX-ee-GHU-A4B1_s=3000.pdf}\n\\caption{Plot of an asymmetry $A_{X}$ in Eq.~\\eqref{eq:AX}.\nThe left plot (a) is for $\\sqrt{s} = 250\\,\\text{GeV}$ with $L_{\\mathrm{int}} = 250\\,\\mathrm{fb}^{-1}$ for each polarizations.\nThe right plot (b) is for $\\sqrt{s} = 3\\,\\text{TeV}$ with $L_{\\mathrm{int}} = 3\\,\\mathrm{ab}^{-1}$ for each polarizations.\nThe red-dashed, blue-dotdashed and black-solid curves correspond to the A-model, B-model and the SM, respectively.\n Error-bars are indicated for $L_{\\mathrm{int}} = 250\\,\\mathrm{fb}^{-1}$ in each polarization.}\\label{fig:AX}\n\\end{figure}\n\n\n\\section{Summary}\nIn this paper we examined the effects of $Z'$ bosons\nin the gauge-Higgs unification (GHU) models in the $\\mathrm{e}^{+}\\mathrm{e}^{-}\\to \\mathrm{e}^{+}\\mathrm{e}^{-}$ (Bhabha) scatterings.\nWe first formulated  differential cross sections in Bhabha scattering including  $Z'$ bosons.\nWe then numerically evaluated the deviations of differential cross sections in the two \n$\\mathrm{SO}(5) \\times \\mathrm{U}(1) \\times \\mathrm{SU}(3)$ GHU models (the A- and B-models) at $\\sqrt{s} = 250\\,\\text{GeV}$. \nWe found that at $L_{\\mathrm{int}} = 2\\,\\mathrm{ab}^{-1}$ with unpolarized $\\mathrm{e}^{+}\\mathrm{e}^{-}$ beams, the deviation due to $Z'$ bosons \nin the GHU models from the SM can be clearly seen.\nWe also found that for $80\\%$-longitudinally polarized electron and $30\\%$-polarized positron beams, \ndeviations of the differential cross sections from the SM \nbecome as large as a few percent for $\\cos\\theta \\sim 0.2$, and that\nthe A-model and the B-model are well distinguished\nat more than $3\\,\\sigma$ significance\nat $L_{\\mathrm{int}} =250\\,\\mathrm{fb}^{-1}$.\nWe also checked the effects of $Z'$ bosons are negligible\nfor the scattering angle smaller than $100$ mrad at $\\sqrt{s} = 250\\,\\text{GeV}$. Therefore Bhabha scatterings at very small $\\theta$ \ncan be safely used as the measurements of the luminosity in the $\\mathrm{e}^{+}\\mathrm{e}^{-}$ collisions.  \nFinally we introduced the new observable $A_{X}$. \nWe numerically evaluated it at $\\sqrt{s} = 250\\,\\text{GeV}$ and\n$3\\,\\text{TeV}$. \nEffects of the GHU models on $A_{X}$\ncan be seen at future TeV-scale $\\mathrm{e}^{+}\\mathrm{e}^{-}$ colliders.\n\nIn this paper the effects of $Z'$ bosons are calculated at the Born level. Higher-order QED effects should be taken \ninto account for more precise evaluation \\cite{Bardin:1990,Bardin:2017}.\n\n\n\\section*{Acknowledgements}\n\nThis work was supported in part \nby European Regional Development Fund-Project Engineering Applications of \nMicroworld Physics (No.\\ CZ.02.1.01\/0.0\/0.0\/16\\_019\/0000766) (Y.O.), \nby the National Natural Science Foundation of China (Grant Nos.~11775092, \n11675061, 11521064, 11435003 and 11947213) (S.F.), \nby the International Postdoctoral Exchange Fellowship Program (IPEFP) (S.F.), \nand by Japan Society for the Promotion of Science, \nGrants-in-Aid  for Scientific Research, Nos.\nJP19K03873 (Y.H.) and JP18H05543 (N.Y.).\n\n\n\\vskip 1.cm\n\n\\def\\jnl#1#2#3#4{{#1}{\\bf #2},  #3 (#4)}\n\n\\def{ Z.\\ Phys.} {{ Z.\\ Phys.} }\n\\def{ J.\\ Solid State Chem.\\ }{{ J.\\ Solid State Chem.\\ }}\n\\def{ J.\\ Phys.\\ Soc.\\ Japan }{{ J.\\ Phys.\\ Soc.\\ Japan }}\n\\def{ Prog.\\ Theoret.\\ Phys.\\ Suppl.\\ }{{ Prog.\\ Theoret.\\ Phys.\\ Suppl.\\ }}\n\\def{ Prog.\\ Theoret.\\ Phys.\\  }{{ Prog.\\ Theoret.\\ Phys.\\  }}\n\\def{ Prog.\\ Theoret.\\ Exp.\\  Phys.\\  }{{ Prog.\\ Theoret.\\ Exp.\\  Phys.\\  }}\n\\def{ J. Math.\\ Phys.} {{ J. Math.\\ Phys.} }\n\\def Nucl.\\ Phys. \\textbf{B}{ Nucl.\\ Phys. \\textbf{B}}\n\\def{ Nucl.\\ Phys.} {{ Nucl.\\ Phys.} }\n\\def{ Phys.\\ Lett.} \\textbf{B}{{ Phys.\\ Lett.} \\textbf{B}}\n\\def{ Phys.\\ Lett.} {{ Phys.\\ Lett.} }\n\\defPhys.\\ Rev.\\ Lett. {Phys.\\ Rev.\\ Lett. }\n\\defPhys.\\ Rev. \\textbf{B}{Phys.\\ Rev. \\textbf{B}}\n\\defPhys.\\ Rev. \\textbf{D}{Phys.\\ Rev. \\textbf{D}}\n\\defPhys.\\ Rep. {Phys.\\ Rep. }\n\\def{Ann.\\ Phys.\\ (N.Y.)}{{Ann.\\ Phys.\\ (N.Y.)}}\n\\defRev.\\ Mod.\\ Phys. {Rev.\\ Mod.\\ Phys. }\n\\defZ.\\ Phys. \\textbf{C}{Z.\\ Phys. \\textbf{C}}\n\\defScience{Science}\n\\defComm.\\ Math.\\ Phys. {Comm.\\ Math.\\ Phys. }\n\\defMod.\\ Phys.\\ Lett. \\textbf{A}{Mod.\\ Phys.\\ Lett. \\textbf{A}}\n\\defEur.\\ Phys.\\ J. \\textbf{C}{Eur.\\ Phys.\\ J. \\textbf{C}}\n\\defJHEP {JHEP }\n\\def{\\it ibid.} {{\\it ibid.} }\n\n\n\\renewenvironment{thebibliography}[1]\n         {\\begin{list}{[$\\,$\\arabic{enumi}$\\,$]} \n         {\\usecounter{enumi}\\setlength{\\parsep}{0pt}\n          \\setlength{\\itemsep}{0pt}  \\renewcommand{\\baselinestretch}{1.2}\n          \\settowidth\n         {\\labelwidth}{#1 ~ ~}\\sloppy}}{\\end{list}}\n\n\n\n\\leftline{\\Large \\bf References}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\\vskip -0.25cm\nCompressed sensing \\cite{Donoho2006,Candes2008} is a mathematical framework that defines the conditions and tools for the recovery of a signal from a small number of its linear projections (i.e. measurements). In the CS framework, the measurement device acquires the signal in the linear projections domain, and the full signal is reconstructed by convex optimization techniques. CS has diverse applications including image acquisition \\cite{Romberg2008}, radar imaging \\cite{5420035}, Magnetic Resonance Imaging (MRI) \\cite{4472246, 6153065}, spectrum sensing \\cite{6179814}, indoor positioning \\cite{6042868}, bio-signals acquisition \\cite{6184345}, and sensor networks \\cite{6159081}. In this paper we address the problem of block-based CS (BCS) \\cite{Fowler-NOW}, which employs CS on distinct low-dimensional segments of a high-dimensional signal. BCS is mostly suitable for processing very high-dimensional images and video, where it operates on distinct local patches. Our approach is based on a deep neural network \\cite{Bengio-2009}, which simultaneously learns the linear sensing matrix and the non-linear reconstruction operator.\\\\\nThe contributions of this paper are two-fold: (1) It presents for the first time, to the best knowledge of the authors, the utilization of a fully-connected deep neural network for the task of BCS; and (2) The proposed network performs both the linear sensing and non-linear reconstruction operators, and during training these operators are \\emph{jointly} optimized, leading to a significant advantage compared to state-of-the-art.\\\\\nThis paper is organized as follows: section \\ref{Problem Formulation} introduces CS concepts, and motivates the utilization of BCS for very high-dimensional images and video. Section \\ref{The Proposed Approach} presents the deep neural network approach, and discusses structure and training aspects. Section \\ref{Results} evaluates the performance of the proposed approach for compressively sensing and reconstructing natural images, and compares it with state-of-the-art BCS methods and full-image Total Variation-based CS. Section \\ref{Conclusions} concludes the paper and discusses future research directions. \\vskip -0.25cm\n\\section{Compressed Sensing Overview}\n\\vskip -0.25cm\n\\label{Problem Formulation}\n\\subsection{Full-Signal Compressed Sensing}\nGiven a signal $\\signal \\in \\mathbf{R}^N$, an $M \\times N$ sensing matrix $\\Phi$ (such that $M \\ll N$) and a measurements vector $\\measurement = \\Phi \\signal$, the goal of CS is to recover the signal from its measurements. The sensing rate is defined by $R=M\/N$, and since $R \\ll 1$ the recovery of $\\signal$ is not possible in the general case. According to CS theory \\cite{Donoho2006,Candes2008}, signals that have a sparse representation in the domain of some linear transform can be exactly recovered with high probability from their measurements: let $\\signal = \\Psi \\reps $, where $\\Psi$ is the inverse transform, and $\\reps$ is a sparse coefficients vector with only $S \\ll N$ non-zeros entries, then the recovered signal is synthesized by $ \\hat{\\signal} = \\Psi \\repsHat$, and $\\repsHat$ is obtained by solving the following convex optimization program:\n\\begin{equation}\n \\repsHat  = \\argmin_{\\reps{'}} \\left\\|\\reps{'}\\right\\|_1 \\text{ subject to } \\measurement = \\Phi \\Psi \\reps{'},\n\\end{equation}\n\\vskip -0.25cm\n\\noindent where $\\left\\|\\alpha\\right\\|_1$ is the $l_1$-norm, which is a convex relaxation of the $l_0$ pseudo-norm that counts the number of non-zero entries of $\\alpha$. The exact recovery of $\\signal$ is guaranteed with high probability if $\\reps$ is sufficiently sparse and if certain conditions are met by the sensing matrix and the transform.\n\n\\begin{table*}[]\n  \\caption{Average reconstruction PSNR [dB] and SSIM vs. sensing rate (R=M\/N): for each method and sensing rate, the result is displayed as PSNR | SSIM  (each result is the average over the 10 test images).}\n  \\label{Reconstruction-Quality}\n  \\centering\n\\resizebox{\\textwidth}{!}{\n  \\begin{tabular}{lccccc}\n   \\hline\n    Method                        & R = 0.1  & R = 0.15 & R = 0.2 & R = 0.25 &  R = 0.3\\\\\n   \\hline\n   Proposed (block-size = 16$\\times$16)     & \\textbf{28.21} | \\textbf{0.916} & \\textbf{29.73} | \\textbf{0.948} & \\textbf{31.03} | \\textbf{0.965}  & \\textbf{32.15} | \\textbf{0.976} & \\textbf{33.11} | \\textbf{0.983}\\\\\n   BCS-SPL-DDWT (16$\\times$16)\\cite{Fowler2009}  & 24.92 | 0.789 & 26.12 | 0.834 & 27.17 | 0.873 & 28.16 | 0.898 & 29.02 | 0.917 \\\\\n   BCS-SPL-DDWT (32$\\times$32)\\cite{Fowler2009}   & 24.99 | 0.781 & 26.40 | 0.833 & 27.46 | 0.868 & 28.43 | 0.894 & 29.29 | 0.914 \\\\\n   MH-BCS-SPL (16$\\times$16)  \\cite{MH-Fowler}    & 26.01 | 0.827 & 27.92 | 0.888 & 29.46 | 0.919 & 30.69 | 0.939 & 31.69 | 0.952 \\\\\n   MH-BCS-SPL (32$\\times$32)  \\cite{MH-Fowler}    & 26.79 | 0.845 & 28.51 | 0.895 & 29.81 | 0.923 & 30.77 | 0.938 & 31.73 | 0.950 \\\\\n   MS-BCS-SPL \\cite{MS-Fowler}                      & 27.32 | 0.883 & 28.77 | 0.909 & 30.04 | 0.934 & 31.15 | 0.956 & 32.05 | 0.974 \\\\\n   MH-MS-BCS-SPL \\cite{MH-Fowler}                     & 27.74 | 0.889 & 29.10 | 0.919 & 30.78 | 0.947 & 31.38 | 0.960 & 32.82 | 0.979 \\\\\n   TV (Full Image) \\cite{Romberg2008}               & 27.41 | 0.867 & 28.57 | 0.890 & 29.62 | 0.909 & 30.63 | 0.926 & 31.59 | 0.939 \\\\\n   \\hline\n  \\end{tabular}}\n\\end{table*}\n\\subsection{Block-based Compressed Sensing}\n\\label{Block-based Compressed Sensing}\nConsider applying CS to an image of $L \\times L$ pixels: the technique described above can be employed by column-stacking (or row-stacking) the image to a vector $\\signal \\in \\mathbb{R}^{L^2}$, and the dimensions of the measurement matrix $\\Phi$ and the inverse transform $\\Psi$ are $M \\times {L^2}$ and $ {L^2} \\times {L^2}$, respectively. For modern high-resolution cameras, a typical value of $L$ is in the range of $2000$ to $4000$, leading to overwhelming memory requirements for storing $\\Phi$ and $\\Psi$: for example, with $L=2000$ and a sensing rate $R=0.1$ the dimensions of $\\Phi$ are $400,000 \\times 4,000,000$ and of $\\Psi$ are $4,000,000 \\times 4,000,000$. In addition, the computational load required to solve the CS reconstruction problem becomes prohibitively high. Following this line of arguments, a BCS framework was proposed in \\cite{Gan2007}, in which the image is decomposed into non-overlapping blocks (i.e. patches) of $B \\times B$ pixels, and each block is compressively sensed independently. The full-size image is obtained by placing each reconstructed block in its location within the reconstructed image canvas, followed by full-image smoothing. The dimensions of the block sensing matrix $\\Phi_B$ are ${B^2}R\\times{B^2}$, and the measurement vector of the \\emph{i}-th block is given by:\n\\begin{equation}\n\\label{block-based sensing}\n\\measurement_i = \\Phi_B \\signal_i,\n\\end{equation}\n\\noindent where $\\signal_i \\in \\mathbb{R}^{B^2}$ is the column-stacked block, and $\\Phi_B$ was chosen in \\cite{Gan2007} as an orthonormalized i.i.d Gaussian matrix. Following a per-block minimum mean squared error reconstruction stage, a full-image iterative hard-thresholding algorithm is employed for improving full-image quality. An improvement to the performance of this approach was proposed by \\cite{Fowler2009}, which employed the same BCS approach as \\cite{Gan2007} and evaluated the incorporation of directional transforms such as the Contourlet Transform (CT) and the Dual-tree Discrete Wavelet Transform (DDWT) in conjunction with a Smooth Projected Landweber (SPL) \\cite{bertero1998introduction} reconstruction of the full image. The conclusion of the experiments conducted in \\cite{Fowler2009} was that in most cases the DDWT transform offered the best performance, and we term this method as BCS-SPL-DDWT. A multi-scale approach was proposed by \\cite{MS-Fowler}, termed MS-BCS-SPL, which improved the performance of BCS-SPL-DDWT by applying the block-based sensing and reconstruction stages in multiple scales and sub-bands of a discrete wavelet transform. A different block dimension was employed for each scale and with a 3-level transform, dimensions of $B=64,32,16$ were set for the high, medium and low resolution scales, respectively. A multi-hypothesis approach was proposed in \\cite{MH-Fowler} for images and videos, which is suitable for either spatial domain BCS (termed MH-BCS-SPL) or multi-scale BCS (termed MH-MS-BCS-SPL). In this approach, multiple predictions of a block are computed from neighboring blocks in an initial reconstruction of the full image, and the final prediction of the block is obtained by an optimal linear combination of the multiple predictions. For video frames, previously reconstructed adjacent frames provide the sources for multiple predictions of a block. The multi-scale version of this approach provides the best performance among all above mentioned BCS methods. A detailed survey of BCS theory and performance is provided in \\cite{Fowler-NOW}, which describes additional applications such as BCS of multi-view images and video, and motion-compensated BCS of video.\n\\section{The Proposed Approach}\n\\vskip -0.25cm\n\\label{The Proposed Approach}\nIn this paper we propose to employ a deep neural network that performs BCS by processing each block independently\\footnote{In this paper we treat only block-based processing, and a full-image post-processing stage is not performed.} as described in section \\ref{Block-based Compressed Sensing}. Our choice is motivated by the outstanding success of deep neural networks for the task of full-image denoising \\cite{Burger} in which a 4-layer neural network achieved state-of-the-art performance by block-based processing. In our approach, the first hidden layer performs the linear block-based sensing stage (\\ref{block-based sensing}) and the following hidden layers perform the non-linear reconstruction stage. The advantage and novelty of this approach is that during training, the sensing matrix and the non-linear reconstruction operator are \\emph{jointly} optimized, leading to better performance than state-of-the-art at a fraction of the computation time. \\\\ The proposed fully-connected network includes the following layers: (1) an input layer with $B^2$ nodes; (2) a compressed sensing layer with $B^{2}R$ nodes, $R\\ll1$ (its weights form the sensing matrix); (3) $K\\ge1$ reconstruction layers with $B^{2}T$ nodes, each followed by a ReLU \\cite{icml2010_NairH10} activation unit, where $T>1$ is the redundancy factor; and (4) an output layer with $B^2$ nodes. Note that the performance of the network depends on the block-size $B$, the number of reconstruction layers $K$, and their redundancy $T$. We have evaluated\\footnote{The network was implemented using Torch7 \\cite{Collobert_NIPSWORKSHOP_2011} scripting language, and trained on NVIDIA Titan X GPU card.} these parameters by a set of experiments that compared the average reconstruction PSNR of 10 test images, depicted in Figure \\ref{test_images}, and by training the network with 5,000,000 distinct patches, randomly extracted from 50,000 images in the LabelMe dataset \\cite{LabelMe}. The chosen optimization algorithm was AdaGrad \\cite{AdaGrad} with a learning rate of $0.005$ (100 epochs), and batch size of 16. Our study revealed that best\\footnote{Note that by increasing significantly the training set, slightly different values of $B$, $K$, and $T$ may provide better results, as discussed in \\cite{Burger}.} performance were achieved with a block size $B\\times B = 16\\times16$, $K=2$ reconstruction layers and redundancy $T=8$, leading to a total of 4,780,569 parameters ($R=0.1$). Table \\ref{Reconstruction-Quality-vs-Size} provides a comparison for varying the block size between $8\\times8$ to $20\\times20$ (with 2 reconstruction layers and redundancy of 8), and indicates that block size of $16\\times16$ provides the best results. Table \\ref{Reconstruction-Quality-vs-Redundancy} provides a comparison for varying the redundancy between 2 to 12 (with 2 reconstruction layers and block size of $16\\times 16$), and indicates that a redundancy of 8 provides the best results. Table \\ref{Reconstruction-Quality-vs-Layers} provides a comparison for varying the number of hidden reconstruction layers between 1 to 8 (with redundancy of 8 and block size of $16\\times 16$), and indicates that two reconstruction layers provided the best performance.\n\\begin{table}[]\n \\vskip -0.25cm\n  \\caption{Reconstruction PSNR [dB] vs. block size ($B\\times B$)}\n  \\label{Reconstruction-Quality-vs-Size}\n  \\centering\n  \\begin{tabular}{cllll}\n    \\hline\n   \n   \n    Training Examples & $B=8$        & $B=12$     & $B=16$      & $B=20$ \\\\\n    \\hline\n   \n    $5 \\times 10^6$   &     27.21  &  27.66   & \\textbf{28.21}   &  27.73 \\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\n\\begin{table}[]\n \\vskip -0.25cm\n  \\caption{Reconstruction PSNR [dB] vs. network redundancy}\n  \\label{Reconstruction-Quality-vs-Redundancy}\n  \\centering\n  \\begin{tabular}{cllll}\n    \\hline\n   \n   \n    Training Examples & $T=2$ & $T=4$ & $T=8$ & $T=12$ \\\\\n    \\hline\n     $5 \\times 10^6$   & 27.99  &  28.11 & \\textbf{28.21}   &  28.15 \\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\n\\begin{table}[htb]\n \\vskip -0.25cm\n  \\caption{Reconstruction PSNR [dB] vs. no. of reconstruction layers}\n  \\label{Reconstruction-Quality-vs-Layers}\n  \\centering\n  \\begin{tabular}{ccccc}\n    \\hline\n   \n   \n    Training Examples & $K=1$ & $K=2$ & $K=4$ & $K=8$ \\\\\n    \\hline\n     $5 \\times 10^6$   & 27.98  &  \\textbf{28.21} & 28.18   &  27.07 \\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\n\\begin{table}[t]\n  \\caption{Computation time at R=0.25 ($512 \\times 512$ images):}\n  \\label{Comp-Time}\n   \\centering\n  \\begin{tabular}{lc}\n  \\hline\n    Method    &  Time [seconds]\\\\\n     \\hline\n   \n    Proposed  & 0.80\\\\\n    BCS-SPL-DDWT ($16 \\times 16$) \\cite{Fowler2009}   & 13.57\\\\\n    BCS-SPL-DDWT ($32 \\times 32$) \\cite{Fowler2009}   & 13.10\\\\\n    MH-BCS-SPL ($16 \\times 16$) \\cite{MH-Fowler}   &  144.61\\\\\n    MH-BCS-SPL ($32 \\times 32$) \\cite{MH-Fowler}   &  69.73\\\\\n    MS-BCS-SPL \\cite{MS-Fowler}   & 6.39\\\\\n    MH-MS-BCS-SPL \\cite{MH-Fowler} & 207.32\\\\\n    TV (Full Image) \\cite{Romberg2008} & 1675.09\\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\n\\section{Performance Evaluation}\n\\vskip -0.25cm\n\\label{Results}\nThis section provides performance evaluation results of the proposed approach\\footnote{A MATLAB package implementing the proposed approach is available at: \\url{http:\/\/www.cs.technion.ac.il\/~adleram\/BCS_DNN_2016.zip}} vs. the leading BCS approaches: BCS-SPL-DDWT \\cite{Fowler2009}, MS-BCS-SPL \\cite{MS-Fowler}, MH-BCS-SPL \\cite{MH-Fowler} and MH-MS-BCS-SPL \\cite{MH-Fowler}, using the original code provided by their authors. The proposed approach was employed with block size $16 \\times 16$, BCS-SPL-DDWT with block sizes of $16 \\times 16$ and $32 \\times 32$ (the optimal size for this method), MH-BCS-SPL with block sizes of $16 \\times 16$ and $32 \\times 32$ (the optimal size for this method). MS-BCS-SPL and MH-MS-BCS-SPL utilized a 3-level discrete wavelet transform with block sizes as indicated in section \\ref{Block-based Compressed Sensing} (their optimal settings). In addition, we also compared to the classical full-image Total Variation (TV) CS approach of \\cite{Romberg2008} that utilizes a sensing matrix with elements from a discrete cosine transform and Noiselet vectors. Reconstruction performance was evaluated for sensing rates in the range of $R=0.1$ to $R=0.3$, using the average PSNR and SSIM \\cite{SSIM} over the collection of 10 test images ($512 \\times 512$ pixels), depicted in Figure \\ref{test_images}. Reconstruction results are summarized in Table \\ref{Reconstruction-Quality}, and reveal a consistent advantage of the proposed approach vs. all BCS methods as well as the full-image TV approach. Visual quality comparisons (best viewed in the electronic version of this paper) are provided in Figures \\ref{results_comp_01}-\\ref{results_comp_025_2}, and demonstrate the high visual quality of the proposed approach. Computation time comparison at a sensing rate $R=0.25$, with a MATLAB implementation of all tested methods (without GPU), is provided in Table \\ref{Comp-Time} and demonstrates that the proposed approach is over 200-times faster than state-of-the-art (MH-MS-BCS-SPL), and over 1600-times faster than full-image TV CS.\n\\begin{figure*}[]\n\\begin{minipage}{\\linewidth}\n\\vskip -1cm\n \\makebox[\\linewidth]{\n\\centering\n\\includegraphics[width=200mm, scale=0.55]{Test_Images.eps}}\n\\vskip -1cm\n \\caption{Test images ($512 \\times 512$): 'lena', 'bridge', 'barbara', 'peppers', 'mandril', 'houses', 'woman', 'boats', 'cameraman' and 'couple'.}\n\\label{test_images}\n \\end{minipage}\n\\end{figure*}\n \\begin{figure*}[]\n \\begin{minipage}{\\linewidth}\n \\vskip -1cm\n \\makebox[\\linewidth]{\n\\centering\n\\includegraphics[width=200mm,scale=0.5]{compare_reconst_rate_01_couple.eps}}\n\\vskip -1cm\n  \\caption{Reconstruction of 'couple' at R = 0.1 (PSNR [dB] | SSIM): (a) Original; (b) Full image TV (27.1691 | 0.8812); (c) MS-BCS-SPL (26.8429 | 0.8756); (d) MH-MS-BCS-SPL (27.1804 | 0.8877); and (e) Proposed (28.5902 | 0.9414).}\n \\label{results_comp_01}\n \\end{minipage}\n\\end{figure*}\n \\begin{figure*}[]\n \\begin{minipage}{\\linewidth}\n \\vskip -1cm\n \\makebox[\\linewidth]{\n\\centering\n\\includegraphics[width=200mm,scale=0.5]{compare_reconst_rate_02_houses.eps}}\n\\vskip -1cm\n \\caption{Reconstruction of 'houses' at R = 0.2 (PSNR [dB] | SSIM): (a) Original; (b) Full image TV (31.0490 | 0.9304); (c) MS-BCS-SPL (31.4317 | 0.9497); (d) MH-MS-BCS-SPL (31.7030 | 0.9544); and (e) Proposed (32.9328 | 0.9766).}\n \\label{results_comp_02}\n  \\end{minipage}\n\\end{figure*}\n \\begin{figure*}[]\n \\begin{minipage}{\\linewidth}\n \\vskip -1cm\n \\makebox[\\linewidth]{\n\\centering\n\\includegraphics[width=200mm,scale=0.5]{compare_reconst_rate_025.eps}}\n\\vskip -1cm\n \\caption{Reconstruction of 'lena' at R = 0.25 (PSNR [dB] | SSIM): (a) Original; (b) Full image TV (35.4202 | 0.9718); (c) MS-BCS-SPL (36.5555 | 0.9861); (d) MH-MS-BCS-SPL (35.7346 | 0.9825; and (e) Proposed (36.3734 | 0.9910).}\n \\label{results_comp_025_1}\n  \\end{minipage}\n\\end{figure*}\n \\begin{figure*}[]\n \\begin{minipage}{\\linewidth}\n \\vskip -1cm\n \\makebox[\\linewidth]{\n\\centering\n\\includegraphics[width=200mm,scale=0.5]{compare_reconst_rate_03_boats.eps}}\n\\vskip -1cm\n \\caption{Reconstruction of 'boats' at R = 0.3 (PSNR[dB] | SSIM): (a) Original; (b) Full image TV (32.5986 | 0.9598); (c) MS-BCS-SPL (32.5063 | 0.9835); (d) MH-MS-BCS-SPL (32.7697 | 0.9847); and (e) Proposed (34.0065 | 0.9914).}\n \\label{results_comp_025_2}\n  \\end{minipage}\n\\end{figure*}\n\\section{Conclusions}\n\\vskip -0.25cm\n\\label{Conclusions}\nThis paper presents a deep neural network approach to BCS, in which the sensing matrix and the non-linear reconstruction operator are jointly optimized during the training phase. The proposed approach outperforms state-of-the-art both in terms of reconstruction quality and computation time, which is two orders of magnitude faster than the best available BCS method. Our approach can be further improved by extending it to compressively sense blocks in a multi-scale representation of the sensed image, either by utilizing standard transforms or by deep learning of new transforms using convolutional neural networks.\n\\vskip -0.25cm\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe scalar sectors of the supersymmetric field theories especially\nthe supergravity theories which are the theories that govern the\nmassless sector low energy background coupling of the relative\nsuperstring theories \\cite{kiritsis} can be formulated as\nprincipal sigma models or non-linear sigma models whose target\nspaces are group manifolds or coset spaces. In particular a great\nmajority of the supergravity scalar sectors are constructed as\nsymmetric space sigma models\n\\cite{sm1,sm2,sm3,nej1,nej2,nej3,sssm1}. When the Abelian\n(Maxwell) vector multiplets are coupled to the graviton multiplets\nin these theories the scalar sector which has the non-linear sigma\nmodel interaction with in itself is coupled to the Abelian gauge\nfields through a kinetic term in the Lagrangian\n\\cite{julia1,julia2,ker1,ker2}.\n\nIn this work, we will focus on the gauge field equations of the\nprincipal sigma model and the Abelian gauge field couplings\nmentioned above. Bearing in mind that the non-linear sigma model\ncan be obtained from the principal sigma model by imposing extra\nrestrictions for the sake of generality we will consider the\ngeneric form of the principal sigma model as the non-linear\ninteraction of the scalars. We will simply show that the gauge\nfield equations can be locally integrated so that they can be\nexpressed as first-order equations containing arbitrary locally\nexact differential forms. The first-order form of the gauge field\nequations will be used to show that there exists a one-sided\ndecoupling between the scalars and the gauge fields in a sense\nthat the scalar field equations do not contain the gauge fields in\nthem whereas the scalars enter as sources in the gauge field\nequations. Thus the scalar solution space of the coupled theory\ncoincides with the general solution space of the pure sigma model.\nFurthermore we will also discuss that this hidden on-shell\nscalar-matter decoupling results in a number of coupled Maxwell\ntheories with sources whose currents contain the general solutions\nof the principal sigma model which is completely decoupled from\nthe Maxwell sector. Therefore we will show that when the general\nsolutions of the pure principal sigma model are obtained and when\none fixes the sector of the solution space of the coupled theory\nby fixing the field dependence or the independence of the locally\nexact differential forms appearing in the first-order gauge field\nequations one may determine the currents of the coupled Maxwell\nfields. In this respect one may solve the gauge fields from the\nMaxwell sector field equations. Consequently the solution space of\nthe scalar-matter coupling can be entirely generated by the\ngeneral solutions of the pure principal sigma model and the\narbitrary choice of the locally exact differential forms. This\nfact is a consequence of the solution methodology which is based\non the implicit on-shell decoupling between the matter fields and\nthe scalar sector which provides current sources to the former.\n\\section{Hidden Decoupling Structure of the Gauge Fields and Their Sources}\nIn a $D$-dimensional spacetime $M$ the Lagrangian which gives the\ninhomogeneous Maxwell equations can be given as\n\\begin{equation}\\label{de1}\n {\\mathcal{L}}=-\\frac{1}{2}dA\\wedge\n \\ast dA-A\\wedge\\ast J,\n\\end{equation}\nwhere $A$ is the $U(1)$ electromagnetic gauge potential one-form\nand $F=dA$ is the field strength of it. Also in the units where\nthe speed of light is unity the current one-form in a local\ncoordinate basis $\\{dt,dx^{a}\\}$ is\n\\begin{equation}\\label{de2}\n J=-\\rho dt+\\mathbf{J}_{a}dx^{a},\n\\end{equation}\nwhere in the temporal component $\\rho$ is the charge density and\nthe spatial components $\\mathbf{J}_{a}$ are the current densities.\nThe Lagrangian \\eqref{de1} defines a media in which the charge\ndensity and the currents are not influenced by the electromagnetic\nfield. The current one-form is predetermined and static that is to\nsay although it acts as a source for the electromagnetic field it\ndoes not interact with it dynamically. From \\eqref{de1} the\ninhomogeneous Maxwell equations read\n\\begin{equation}\\label{de2.5}\n d\\ast F=-\\ast J.\n\\end{equation}\nIn this section we will consider the coupling of $N$ $U(1)$ gauge\nfield one-forms $A^{i}$ to the principal sigma model whose target\nspace is a group manifold $G$. The sigma model Lagrangian can be\ngiven as\n\\begin{equation}\\label{de3}\n {\\mathcal{L}}=\\frac{1}{2}\\, tr(\\ast dg^{-1}\\wedge\n dg).\n\\end{equation}\nHere we take a differentiable map\n\\begin{equation}\\label{de4}\n h: M\\longrightarrow G,\n\\end{equation}\nwe also consider a representation $f$ of $G$ in $Gl(N,\\Bbb{R})$\n\\begin{equation}\\label{de5}\n f: G\\longrightarrow Gl(N,\\Bbb{R}),\n\\end{equation}\nwhich may be taken as a differentiable homomorphism. Then the map\n$g$ can be given as\n\\begin{equation}\\label{de6}\n g=f\\circ h: M\\longrightarrow Gl(N,\\Bbb{R}),\n\\end{equation}\nwhich is a matrix-valued function on $M$\n\\begin{equation}\\label{de7}\ng(p)=\\left(\\begin{array}{ccc}\n  \\varphi^{11}(p) & \\varphi^{12}(p) & \\cdots \\\\\n  \\varphi^{21}(p) & \\varphi^{22}(p) & \\cdots \\\\\n  \\vdots & \\vdots & \\vdots \\\\\n\\end{array}\\right),\n\\end{equation}\n$\\forall p\\in M$. Due to the presence of the scalar fields\n$\\{\\varphi^{ij}\\}$ the theory can be considered to be a scalar\nfield theory. In \\eqref{de3} we have a matrix multiplication with\nthe wedge product used between the components and the trace is\nover the representation chosen in \\eqref{de5}. Depending on the\nrestrictions on the sigma model and the nature of $G$ the scalars\nin \\eqref{de7} can be all independent or not. This model also\ncovers the non-linear (coset) sigma models \\cite{westsugra,tanii}\nand in particular the symmetric space sigma models\n\\cite{sm1,sm2,sm3} which shape the scalar sectors of the\nsupergravities thus the low energy effective string theories. The\nconstruction of the Lagrangian of the symmetric space sigma model\nin which an internal metric substitutes the map $g$ can be found\nin \\cite{nej1,nej2,nej3}.\n\nBy generalizing the supersymmetric coupling of the supergravity\nmatter multiplets with the graviton multiplets\n\\cite{julia1,julia2,ker1,ker2,sssugradivdim} we can write down the\ncoupling of $N$ $U(1)$ gauge field one-forms $A^{i}$ to the\nprincipal sigma model whose target space is a group manifold as\n\\begin{equation}\\label{de8}\n \\mathcal{L}_{tot}=\\frac{1}{2}tr( \\ast dg^{-1}\\wedge dg)\n -\\frac{1}{2}\\ast F^{T}g\\wedge\n F,\n\\end{equation}\nwhere we define the column vector $F$ whose components are $F^{i}$\nthus the coupling term can be explicitly written as\n\\begin{equation}\\label{de9}\n-\\frac{1}{2}\\ast F^{T}g\\wedge\n F =-\\frac{1}{2}g^{i}_{\\:\\:\\:j} \\ast F_{i}\\wedge\n F^{j}.\n\\end{equation}\n Now if we\nvary the Lagrangian \\eqref{de8} with respect to $A^{i}$ we find\nthe corresponding field equations as\n\\begin{equation}\\label{de10}\nd(\\mathcal{T}^{i}_{\\:\\:\\:k}\\ast F_{i})=0,\n\\end{equation}\nwhere\n\\begin{equation}\\label{de11}\n\\mathcal{T}^{i}_{\\:\\:\\:k}=g^{i}_{\\:\\:\\:k}+(g^{T})^{i}_{\\:\\:\\: k}.\n\\end{equation}\nThe expression \\eqref{de10} defines a closed form. Since locally\nany closed form is equal to an exact form we can integrate\n\\eqref{de10} to write\n\\begin{equation}\\label{de12}\n\\mathcal{T}^{i}_{\\:\\:\\:k}\\ast F_{i}=dC_{k},\n\\end{equation}\nwhere $\\{C_{k}\\}$ are arbitrary $N$ $(D-3)$-forms on $M$. They may\nbe chosen to depend on the fields $\\{\\varphi^{ij},A^{i}\\}$ or they\nmay be fixed. The former case is the most general one. The general\nsolutions of the field equations of the Lagrangian \\eqref{de8}\nmust satisfy \\eqref{de12} on-shell in which one can freely change\nthe set of $(D-3)$-forms $\\{C_{k}\\}$ which may be functions of\n$\\{\\varphi^{ij},A^{i}\\}$ or not. To generate the entire solution\nspace one can first choose a set of field-dependent or\nfield-independent $\\{C_{k}\\}$ and solve the system of field\nequations of \\eqref{de8} to find the corresponding solutions then\none can repeat this procedure for a different set of $\\{C_{k}\\}$.\nIn this respect the field-independent or the fixed $(D-3)$-forms\n$\\{C_{k}\\}$ can be considered as the integration constants coming\nfrom the reduction of the degree of the gauge field equations\nwhich are second-order differential equations. Now we will follow\nthe methodology described above to show that there exists an\non-shell decoupling between the scalars and the gauge fields. Let\nus first take a look at the more restricted case of\nfield-independent $\\{C_{k}\\}$ and let us fix a set of\nfield-independent $\\{C_{k}\\}$. In this case if we take the partial\nderivative of both sides of \\eqref{de12} with respect to the\nscalar fields $\\{\\varphi^{ml}\\}$ we immediately see that\n\\begin{equation}\\label{de13}\n\\frac{\\partial\\mathcal{T}^{i}_{\\:\\:\\:k}}{\\partial\\varphi^{ml}}\\ast\nF_{i}=0,\n\\end{equation}\nsince the right hand side of \\eqref{de12} is a fixed $(D-2)$-form\nand it does not depend on the scalar fields $\\{\\varphi^{ml}\\}$.\nThese conditions must be satisfied on-shell by a sector of the\nsolution space of $\\{\\varphi^{ml},A^{i}\\}$ which is restricted to\nthe condition of choosing fixed $\\{C_{k}\\}$. After fixing the set\nof $(D-3)$-forms $\\{C_{k}\\}$ if we use \\eqref{de12} in \\eqref{de8}\nwe can obtain the on-shell Lagrangian as\n\\begin{equation}\\label{de14}\n \\mathcal{L}_{tot}=\\frac{1}{2}tr( \\ast dg^{-1}\\wedge dg)\n -\\frac{1}{2}dC_{j}\\wedge\n F^{j}.\n\\end{equation}\nSince we have made use of the gauge field equations varying this\non-shell Lagrangian with respect to $A^{i}$ yields an identity. On\nthe other hand by varying the above Lagrangian with respect to the\nscalars to find the solutions which satisfy \\eqref{de12} for a\nchosen fixed set of $\\{C_{k}\\}$ we find that the scalar field\nequations are the same ones which can be obtained directly from\nthe pure sigma model Lagrangian \\eqref{de3} since the coupling\npart in \\eqref{de14} does not depend on the scalars. This result\ncan also be obtained by directly varying \\eqref{de8} and by using\nthe conditions \\eqref{de13} which are satisfied by the sector of\nthe general solutions of the theory which we have restricted\nourselves in by fixing $\\{C_{k}\\}$. The same result would be\nobtained if one chooses another field-independent set of\n$\\{C_{k}\\}$. Thus we observe that the scalar solutions of a\nsub-sector of the scalar-gauge field theory defined by the\ncoupling Lagrangian \\eqref{de8} are the same with the general\nsolution space of the pure principal sigma model. This is a\nconsequence of the local integration given in \\eqref{de12} and\nfixing the arbitrary $\\{C_{k}\\}$.\n\nOn the other hand a similar result with a minor difference can\nalso be derived for the rest of the solution space of the theory.\nIf we consider the most general case of field-dependent\ndifferential forms $\\{C_{k}(\\varphi^{ml},A^{i})\\}$ and use\n\\eqref{de12} in the Lagrangian \\eqref{de8} we get\n\\begin{equation}\\label{de14.5}\n \\mathcal{L}_{tot}=\\frac{1}{2}tr( \\ast dg^{-1}\\wedge dg)\n -\\frac{1}{2}dC_{j}(\\varphi^{ml},A^{i})\\wedge\n F^{j}.\n\\end{equation}\nThis on-shell Lagrangian gives us the same decoupling conditions\ndiscussed above for the restricted sub-sector case. To see this we\nshould realize that the second term in the Lagrangian\n\\eqref{de14.5} which is written in an on-shell form is a closed\ndifferential form. Locally any closed differential form is an\nexact one. Thus the second term in the Lagrangian \\eqref{de14.5}\ncan be written as an exact differential form as\n\\begin{equation}\\label{de14.6}\n -\\frac{1}{2}dC_{j}(\\varphi^{ml},A^{i})\\wedge\n F^{j}=dB(\\varphi^{ml},A^{i}).\n\\end{equation}\nIn fact we may simply calculate $B(\\varphi^{ml},A^{i})$ as\n\\begin{equation}\\label{de14.65}\n B(\\varphi^{ml},A^{i})=-\\frac{1}{2}C_{j}(\\varphi^{ml},A^{i})\\wedge\n F^{j}.\n\\end{equation}\n Thus the Lagrangian \\eqref{de14.5} becomes\n\\begin{equation}\\label{de14.7}\n \\mathcal{L}_{tot}=\\frac{1}{2}tr( \\ast dg^{-1}\\wedge dg)\n +d(-\\frac{1}{2}C_{j}(\\varphi^{ml},A^{i})\\wedge\n F^{j}).\n\\end{equation}\nIf one varies the above Lagrangian one immediately sees that the\nsecond term does not contribute to the field equations as by using\nthe Stoke's theorem we have\n\\begin{equation}\\label{de14.75}\n\\int\\limits_{M}d\\delta B(\\varphi^{ml},A^{i})=\\int\\limits_{\\partial\nM}\\delta B(\\varphi^{ml},A^{i}).\n\\end{equation}\nIf $M$ does not posses a boundary the right hand side of the above\nequation is automatically zero whereas if it has a boundary then\nthe usual variation principles demand that the variation of the\nfields on the boundary are chosen to be zero in which case again\nthe right hand side of \\eqref{de14.75} becomes zero. Therefore we\nconclude that the on-shell Lagrangian \\eqref{de14.5} which is\nresponsible for the most general structure of the solution space\ngives us the scalar field equations which are the same with the\npure sigma model field equations since the coupling part in\n\\eqref{de14.5} does not contribute to the scalar field equations\nat all as we have proven as an on-shell condition above. Also like\nwe have already encountered for the non-field-dependent\n$\\{C_{k}\\}$ sub-sector case varying \\eqref{de14.5} does not give\nany information (which may also mean an identity) about the gauge\nfields $A^{i}$ since we have already made use of their field\nequations in writing it.\n\nIn summary, we have shown that the solutions of the theory must\nobey \\eqref{de12} for arbitrary right hand sides. If one\ndetermines the right hand sides in \\eqref{de12} one restricts him\nor herself to a layer of solutions. In this case \\eqref{de12}\nwhich are the descendants of the gauge field equations become\non-shell conditions. We have proven that if we use these\nfirst-order field equations which are on-shell conditions back in\nthe general Lagrangian \\eqref{de8} then the scalars of this\nparticular layer of solutions do not have gauge fields in their\nfield equations. Therefore the scalars of this particular layer\nbelong to the general solutions of the pure principal sigma model.\nBy running the right hand sides in \\eqref{de12} over the entire\nlocal field-dependent exact $(D-2)$-forms one can generalize this\nresult to the whole solution space. Thus in this way we have\nproven that the entire scalar solutions of the coupled theory\ncoincide with the pure sigma model solution space. Now from\n\\eqref{de12} we can write down the gauge field strengths as\n\\begin{equation}\\label{de15}\nF_{l}=(-1)^{s}(\\mathcal{T}^{-1})^{k}_{\\:\\:\\:l}\\ast\ndC_{k}(\\varphi^{mn},A^{i}),\n\\end{equation}\nwhere $s$ is the signature of the spacetime. We observe that if we\nconsider the sector of the solution space generated by the\nfield-independent $\\{C_{k}\\}$ then after obtaining the general\nsolutions of the pure principal sigma model and after choosing a\nfixed set $\\{C_{k}\\}$ one can use these in \\eqref{de15} to find\nthe corresponding $U(1)$ gauge field strengths. Also for the more\ngeneral field-dependent case of $\\{C_{k}(\\varphi^{ml},A^{i})\\}$\none again obtains the general solutions of the pure principal\nsigma model then one inserts these solutions in \\eqref{de15} to\nsolve for the gauge fields. We can say that in general we have a\npartial decoupling between the scalars and the gauge fields. The\nscalars are not affected by the presence of the gauge fields on\nthe other hand as we will show next they act as sources for the\ngauge fields. Now after multiplying by the Hodge star operator if\nwe take the exterior derivative of both sides of \\eqref{de15} we\nobtain\n\\begin{equation}\\label{de16}\nd\\ast F_{l}=(d\\mathcal{T}^{-1})^{k}_{\\:\\:\\:l}\\wedge\ndC_{k}(\\varphi^{mn},A^{i}).\n\\end{equation}\nWhen we compare this result with the inhomogeneous Maxwell\nequations \\eqref{de2.5} we observe that the current one-forms\nbecome\n\\begin{equation}\\label{de17}\nJ_{l}=(-1)^{(D+s)}\\ast((d\\mathcal{T}^{-1})^{k}_{\\:\\:\\:l}\\wedge\ndC_{k}(\\varphi^{mn},A^{i})).\n\\end{equation}\nOne can furthermore verify that the currents in \\eqref{de17} obey\nthe current conservation law\n\\begin{equation}\\label{de18}\nd\\ast J_{l}=0,\n\\end{equation}\nwhich guarantees that the equations \\eqref{de16} have solutions.\nIn a local moving co-frame field $\\{e^{\\alpha}\\}$ on the spacetime\n$M$ if we introduce the components of the one-forms\n$(d\\mathcal{T}^{-1})^{k}_{\\:\\:\\:l}$, the $(D-2)$-forms\n$dC_{k}(\\varphi^{ml},A^{i})$, and the one-forms $J_{l}$ as\n\\begin{subequations}\\label{de19}\n\\begin{align}\n(d\\mathcal{T}^{-1})^{k}_{\\:\\:\\:l}&=(\\mathcal{T}^{-1k}_{l})_{\\alpha}e^{\\alpha},\\notag\\\\\n\\notag\\\\\ndC_{k}(\\varphi^{ml},A^{i})&=\\frac{1}{(D-2)!}(\\mathcal{C}_{k})_{\\alpha_{1}\\cdots\\alpha_{(D-2)}}e^{\\alpha_{1}\\cdots\n\\alpha_{(D-2)}},\\notag\\\\\n \\notag\\\\\nJ_{l}&=\\mathcal{J}_{l\\beta}e^{\\beta},\\tag{\\ref{de19}}\n\\end{align}\n\\end{subequations}\nfrom \\eqref{de17} we can calculate the components of the current\none-forms as\n\\begin{equation}\\label{de20}\n\\mathcal{J}_{l\\beta}=\\frac{(-1)^{s}\\sqrt{|detH|}}{(D-2)!}(\\mathcal{T}^{-1k}_{l})_{\\alpha}\n(\\mathcal{C}_{k})_{\\alpha_{1}\\cdots\\alpha_{(D-2)}}H^{\\alpha_{1}\\beta_{1}}\\cdots\nH^{\\alpha\\beta_{(D-1)}}\\varepsilon_{\\beta_{1}\\cdots\\beta_{(D-1)}\\beta},\n\\end{equation}\nwhere we have introduced the metric $H$ on $M$ and the Levi-Civita\nsymbol $\\varepsilon$. In \\eqref{de20} $\n\\alpha,\\beta,\\alpha_{i},\\beta_{j}=1,\\cdots,D$. Also for\n$\\alpha_{i}$ $i=1,\\cdots,D-2$ and for $\\beta_{j}$\n$j=1,\\cdots,D-1$.\n\nIf one cancels an exterior derivative (performs integration) on\nboth sides of \\eqref{de16} one finds\n\\begin{equation}\\label{de21}\n\\ast\nF_{l}=(\\mathcal{T}^{-1})^{k}_{\\:\\:\\:l}dC_{k}(\\varphi^{mn},A^{i})+dC^{\\prime}_{l}(\\varphi^{mn},A^{i}).\n\\end{equation}\nNow if we compare this with the first-order equations \\eqref{de12}\noriginating from \\eqref{de8} we see that our model is a sub-sector\nof the one defined in \\eqref{de21} with the choice of\n$dC^{\\prime}_{l}(\\varphi^{mn},A^{i})=0$. One may also inspect\nwhich scalar-gauge field coupling kinetic term would result in a\nfirst-order formulation of gauge fields in the form \\eqref{de21}.\n\nWe should finally state that by using the local first-order\nformulation \\eqref{de12} of the gauge field equations of\n\\eqref{de8} we have shown that there exists a decoupling between\nthe $U(1)$ gauge fields and the scalar fields of the theory. The\nscalars are proven to be the general solutions of the pure\nprincipal sigma model and they generate current sources for the\ngauge fields as can be explicitly seen in \\eqref{de16}. For the\nsub-sector of the solution space which is generated by the\nfield-independent $\\{C_{k}\\}$ we end up with a decoupled set of\n$N$ non-interacting Maxwell theories with prescribed and known\ncurrents whose sources are predetermined by the principal sigma\nmodel scalar fields. These currents interact with each other via\nthe sigma model which is completely decoupled from the Maxwell\nsector and they define a media which does not interact with the\nMaxwell fields. For this sub-sector the $N$ decoupled and\nnon-interacting Maxwell theories can alternatively be formulated\nby the Lagrangian\n\\begin{equation}\\label{de22}\n {\\mathcal{L}}^{\\prime}=\\sum\\limits_{i=1}^{N}\\bigg(-\\frac{1}{2}dA^{i}\\wedge\n \\ast dA^{i}+A^{i}\\wedge (d\\mathcal{T}^{-1})^{k}_{\\:\\:\\:i}\\wedge\n dC_{k}\\bigg).\n\\end{equation}\nWe see that for a single scalar field the above Lagrangian drops\nto be the ordinary Maxwell Lagrangian with a known current\none-form. We also realize that the Maxwell fields in this\nrestricted case are coupled to each other only by means of\nintegration constants. Each of these decoupled Maxwell theories is\nan embedding into the ordinary Maxwell theory with known sources.\nWe should remark that \\eqref{de22} must only be used to\nderive\\footnote{By integrating the field equations to first-order\nand by choosing $dC^{\\prime}_{l}=0$.} the field equations of the\ngauge fields which belong to a restricted sector of the solution\nspace generated by fixing $\\{C_{k}\\}$ and in this case the field\nequations of the scalars must again be derived from \\eqref{de3}.\nOn the other hand for the most general solution space elements\ngenerated by choosing field-dependent\n$\\{C_{k}(\\varphi^{ml},A^{i})\\}$ we can again adopt the general\nsolutions of the pure sigma model as the general scalar solutions\nof our theory however in this case we have $N$ coupled Maxwell\ntheories whose potentials may enter in the currents. For the\nsub-sector generated by $\\{C_{k}(\\varphi^{ml})\\}$ which are not\ndependent on the gauge fields we have a similar situation of\ndecoupled Maxwell theories with prescribed currents discussed\nabove.\n\nFinally before concluding we should summarize the local general\nsolution methodology of the principal sigma model and $U(1)$ gauge\nfields coupling defined in \\eqref{de8}. The method has three steps\nfirst find the general scalar field solutions of the pure\nprincipal sigma model, secondly define field-dependent or\nfield-independent $\\{C_{k}\\}$ and use the general scalar solutions\nin them then insert these in \\eqref{de15} to solve the\ncorresponding gauge field strengths. By combining the general pure\nprincipal sigma model solutions with a different set of\n$\\{C_{k}\\}$ each time one can generate the entire set of\nsolutions. Such a solution methodology of the general solution\nspace of \\eqref{de8} is a consequence of the first-order\nformulation of the gauge field equations in \\eqref{de12} and the\non-shell decoupling between the scalars and the gauge fields which\nwe have described in detail for both the restricted sector of the\nsolution space and for the entire solution space in its most\ngeneral generation process. From the scalars point of view there\nis a complete decoupling so that the scalars of the coupled theory\ncoincide with the pure sigma model solution space. However the\ngauge fields are not decoupled from the scalars since we have\nshown that the scalars act in the current part of the gauge field\nequations.\n\\section{Conclusion}\nBy integrating the gauge field equations of the principal sigma\nmodel and Abelian gauge field coupling Lagrangian we have\nexpressed these equations in a first-order form. Then we have\nshown that when these first-order and on-shell expressions which\ncontain local $(D-3)$-forms in them are used in the scalar-matter\nLagrangian one reveals an implicit decoupling between the scalars\nand the matter gauge fields. We have proven that this decoupling\nstructure exists for the entire solution space. Therefore it is\nshown that the scalar solutions of the coupled theory are the\ngeneral solutions of the pure principal sigma model. We should\nstate here that a similar result is derived in \\cite{consist2} for\nthe heterotic string. However in that work the decoupling occurs\ndue to the existence of a dilatonic field and one derives the\ndecoupling structure of the coset scalars from the field equation\nof the dilaton. On the other hand in the present work we prove\nthat such a decoupling exists for a generic principal sigma model\nwith an arbitrary number of Abelian gauge field couplings.\n\nWe have also mentioned that the on-shell hidden scalar-matter\ndecoupling mentioned above generates a theory composed of the pure\nsigma model and a number of coupled Maxwell theories with sources\ninduced by the scalars. In particular we have discriminated a\nsub-sector of the general theory generated by the\nfield-independent integration constants. This sub-sector contains\na number of separate and dynamically non-interacting Maxwell\ntheories whose current sources are drawn from the general scalar\nsolutions of the pure principal sigma model and the integration\nconstants of the first-order formulation.\n\nIn this work, we prove that the scalars of the coupled theory are\nnot affected by the presence of the $U(1)$ gauge fields however\nthey take role in the currents of the gauge fields. Thus one may\ncall such a coupling between the scalars and the gauge fields a\none-sided or a one and a half coupling. We also see that for the\nrestricted sub-sector of the theory which is obtained by fixing\nthe integration constants the currents of the electromagnetic\nfields interact with in each other via the sigma model but they do\nnot interact with the corresponding gauge fields. Thus for this\nsub-sector the currents form up a subset of the general currents\nof the prescribed current-Maxwell theory. In this sub-sector the\nscalar-gauge field decoupling induces another decoupling scheme\nalso among the gauge fields and one obtains $N$ non-interacting\nMaxwell theories whose sources come from the pure principal sigma\nmodel. For each $U(1)$ gauge field we have an embedded sector of\nthe general predetermined current-Maxwell theory which has the\nmost general form of currents. In our case these currents are\nrestricted to the solutions of the pure principal sigma model. The\n$N$ $U(1)$ gauge fields in this case probe each other only by the\npresence of the integration constants however this does not\ncorrespond to a dynamic coupling. The coupling among the gauge\nfields occur only at the level of determining the integration\nconstants from the boundary conditions. Since we have a static and\nunchanged current structure which emerges from the pure principal\nsigma model where the currents interact with each other and since\nthese currents are not affected by the presence of the gauge\nfields this sub-sector of the general theory defines a theory of a\nstatic media with $N$ non-dynamically interacting Maxwell fields.\nOne can inspect the other sectors of the general theory based on\nvarious choice of field dependent $\\{C_{k}\\}$ which define dynamic\nmedia with current-gauge field interactions. For example one can\nstudy the sectors that result in gauge field equations that define\ndynamic media with specially defined conducting properties\n\\cite{thring}. The analysis of this work can furthermore be\nextended on another scalar-gauge field coupling which might have\nbroader dynamic media sub-sectors namely on the gauged sigma\nmodel.\n\nIn conclusion, we may state that the decoupling structure studied\nin this work is an important remark on the solutions of the\nscalar-gauge field interactions which form up a basic sector of\nthe Bosonic dynamics of the supergravities and the effective\nstring theories. Therefore the scalar-gauge field decoupling\nrevealed here points out a simplification in seeking solutions to\nthese theories.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWe wrote a small class library to render\nwith computer graphics images of\nthe highly mathematical structures\ncreated by the artist Elias Wakan \\cite{wakan06}.\nThe \\hbox{{\\tt C++}}\\ classes we wrote will be extended\nto output the type of description\nrequired by a rendering package such as POV-Ray \\cite{povRay06}.\nWe were surprised that this thoroughly practical enterprise\nled us to two fascinating fundamental issues:\nthe limits of computability\nand the abstract nature of axiomatic geometry.\nHence our title ``Computational Euclid''.\n\n\\vspace{0.05in}\nTo accommodate the limits of computability,\nwe use interval arithmetic.\nWe use it in such a way that when we pose the question\nwhether two straight lines intersect, \nthe answer ``no'' has the force of a mathematical proof they do not.\nThe other possible answer is a box in 3-space, typically very small.\nThis answer means that the intersection is in this box,\n\\emph{if there is an intersection}.\nThis proviso is probably essential\nbecause we sense that a decision procedure\nfor the intersection of two lines can be used\nto implement a decision procedure\nfor the equality of any two real numbers,\nwhich has been shown to be impossible by Turing \\cite{trng37,aberth98}.\nHowever, we have not pursued the details of such a problem reduction.\n\nThe other fundamental issue\nis the abstract nature of an axiomatic approach to geometry.\n\nEuclid is widely credited with inaugurating the axiomatic method in which\naxioms contain references to undefined concepts of which the meaning is\nonly constrained by the axioms. However, we should not look to the Elements\nfor a literal embodiment of the axiomatic method.\nIt fell to Hilbert in 1902 \\cite{hlbrt02} to cast these in a form that\nis recognized today as an axiomatic treatment of the subject.\n\nLack of space prevents us here to go into a detailed analysis of the concepts\nof Euclid's geometry.\nSuffice it to say that Hilbert's formulation contains as undefined concepts,\namong others, the following that we found useful in our work:\n\\emph{point},\n\\emph{line},\n\\emph{segment},\n\\emph{plane},\n\\emph{angle}.\n\nIn the modern conception of the axiomatic method these are undefined.\nTheir meaning is only constrained by the relations between them as\nasserted by the axioms. In logic this is formalized by the axioms being\na theory, which, if consistent, can have a variety of models.\nIt is only the model that says what a point \\emph{is}.\nFor example, in one type of model,\npoints, lines, and planes are solution spaces\nof sets of linear equations in three variables.\n\nWhat we find surprising is how well the way in which the axiomatic method,\nas realized in formal logic, combines with the most widely accepted principles\nof object-oriented software design \\cite{wbww90}.\nAccording to it, one looks for the \\emph{nouns} in an informal specification\nof the software to be written.\nThese are candidates for the \\emph{classes} of an object-oriented program.\n\nWe used Hilbert's axioms as specification.\nThe recipe of \\cite{wbww90} has, of course, to be taken with a grain of salt:\nonly the \\emph{important} nouns are candidates for classes.\nUsually, informal specifications contain a majority of not-so-important nouns.\nTo our delight,\nwe found that Hilbert's axioms\ncontain an unusually small number\nof not-so-important nouns.\n\n\\section{The structure of our class library}\n\nOf the nouns occurring in Hilbert's axioms,\n{\\tt Point}, {\\tt Line}, and {\\tt Plane}\\ are a special subset.\nThey are special in the sense\nthat any unordered pair of these determine\nan object in this trinity,\nunless a specific condition prevails.\nIn the case of a point and a line determining a  plane,\nthe condition is ``unless the point is on the line''.\nNote that these conditions are called \\emph{predicates}\nby some authors~\\cite{effpred02,itar98}.\nThe table in Figure~\\ref{pointsLinesPlanes}\nsummarizes the operations for all unordered pairs,\neach with the attendant disabling condition.\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{l|l}\n\\hline\n Construction &  Disabling Condition   \\\\\n\\hline\n{\\tt Point}\\ $\\times$ {\\tt Point}\\ $\\rightarrow$ {\\tt Line}\\ & equal                          \\\\\n{\\tt Point}\\ $\\times$ {\\tt Line}\\ $\\rightarrow$ {\\tt Plane}\\ & on                             \\\\\n{\\tt Point}\\ $\\times$ {\\tt Plane}\\ $\\rightarrow$ {\\tt Line}\\ & (none)                         \\\\\n{\\tt Line}\\ $\\times$ {\\tt Line}\\ $\\rightarrow$ {\\tt Line}\\ & parallel or intersect          \\\\\n{\\tt Line}\\ $\\times$ {\\tt Plane}\\ $\\rightarrow$ {\\tt Point}\\ & in                             \\\\\n{\\tt Plane}\\ $\\times$ {\\tt Plane}\\ $\\rightarrow$ {\\tt Line}\\ & parallel                       \\\\\n \\hline\n\\end{tabular}\n\\caption{Operations for all unordered pairs formed from {\\tt Point}, {\\tt Line}\\ and {\\tt Plane}.\nThe operations cannot be performed if the condition listed holds between\nthe input arguments of the construction.\n}\n\\label{pointsLinesPlanes}\n\\end{center}\n\\end{figure*}\n\nTwo of the constructions involve perpendiculars.\nThe line determined by the point\nand the plane is the perpendicular\nto the plane through the point.\nThe line determined by two lines in general position\nlikewise is a perpendicular:\nthe unique one that is perpendicular to both given lines.\n\nIn this way, an object-oriented reading\nof Hilbert's axioms determines that the class {\\tt Line}\\\ncontains constructors with parameters\n({\\tt Point}, {\\tt Point}), with ({\\tt Point}, {\\tt Plane}), and with ({\\tt Plane}, {\\tt Plane}).\nThe class {\\tt Point}\\ contains a constructor\nwith parameters ({\\tt Line}, {\\tt Plane}).\nThe class {\\tt Plane}\\ contains a constructor\nwith arguments ({\\tt Point}, {\\tt Line}).\n\nThe constructors cannot be invoked\nwhen the conditions noted in Figure~\\ref{pointsLinesPlanes} hold\nbetween the arguments.\nFor example, if a {\\tt Point}\\ is on a {\\tt Line}, then these do not determine a plane.\nThese conditions are semi-decidable: they either determine that the\ncondition does not hold, or that the condition \\emph{may} hold.\nHowever, in rare cases it can be determined that, say, two\ninstances of {\\tt Point}\\ are equal.\nThe conditions therefore return the truth values of a 3-valued logic.\n\n\nThe abstract nature of an axiomatic approach to geometry\nrequires that the {\\tt Point}, {\\tt Line}\\ and {\\tt Plane}\\ \nare left undefined.\nThis abstraction is not only essential in \nthe axiomatic treatment of mathematical theories,\nbut it is also the essence of object-oriented design.\n\nIn object-oriented design one may distinguish two forms of abstraction.\nThe weaker form is achieved by any class in which the variables\nare private.\nOne can then modify the representation of the objects without\nconsequences for the code using the class.\nThere is also a stronger form of abstraction in which\npolymorphism makes it possible\nto use more than one implementation of the same abstraction simultaneously.\nThe concept is then represented by an abstract class for the concept in\nwhich the representation-dependent methods are virtual.\nFor each representation there is a separate derived class of\nwhich the methods are dispatched at run time.\nWe have found this stronger form of abstraction\nadvantageous in our suite of \\hbox{{\\tt C++}} classes.\n\nA UML class diagram summarizing the classes \nand the conditions above is shown in Figure~\\ref{geomuml}. \nThe purpose of the extra classes in the diagram is explained later.\n\\begin{figure}[!htbp]\n\\begin{center}\n\\epsfxsize=3.5in\n\\leavevmode\n\\epsfbox{Figures\/euclclassdiagram.eps}\n\\caption{\n\\label{geomuml}\nUML class diagram for our system.}\n\\end{center}\n\\end{figure}\n\n\\section{Consequences of computability limitations}\n\nWhatever computer representation is chosen,\nthere will only be finitely many points, lines, and planes\nthat can be represented.\nThe conventional method of mapping\nthe infinity of abstract objects\nto the finitely many representable ones\nis to choose a representation in terms of reals\nand then to map each real to a nearby floating-point number.\nWhen this method is followed,\nit has so far not been found possible\nto give precise meaning to the outcomes\nof tests such as whether a point is on a line.\nThe outcomes have to be interpreted\nas ``probably not'' and ``possibly'',\ndepending on whether the computed distance\n(subject to an unknown error)\nis greater than a certain tolerance.\n\nIt may seem that this degree of uncertainty\nis inherent in the limitation\nto a finite number of representations.\nThis is not case.\nEven when restricted to floating-point numbers,\nit is possible to represent the point $p$\nby a set $P$ of points containing $p$;\nlikewise, the line $l$ can be represented by a set $L$.\nThese sets are specified in terms of floating-point numbers,\nso there are only finitely many of these.\nBecause of this finiteness it is decidable\nwhether the set of points contains any\nthat is on any in the set of lines.\nIt may seem computationally formidable\nto make such a determination.\nActually, the techniques of interval constraints\nmake this perfectly feasible \\cite{hcqvn99},\nand this is what we use.\n\nIf it is determined that no point in $P$ is on any line in $L$,\nthen it is clear that $p$ is not on $l$.\nIf, on the other hand, some point in $P$ is on some line in $L$,\nthis says nothing about  whether $p$ is on $l$.\nHowever, if $P$ and $L$ are, in a suitable sense, small,\nthen it follows that $p$ is close to $l$.\nIt is this asymmetry that is a consequence of the fact that the test for a point\non a line can at best be a \nsemi-decision algorithm.\nSimilarly, the other tests\nin Figure~\\ref{pointsLinesPlanes} are semi-decision algorithms.\n\nIt is worth mentioning that to cope with the computability limitations \nin the area of computational geometry, the \\emph{exact geometric computing} \nparadigm was proposed \\cite{yap95}. This paradigm encompasses all techniques \nfor which the outcomes are correct. As shown in \\cite{itar98}, \ninterval arithmetic can be used to do exact geometric computing.\nThis paper is also classified under this paradigm.\n\n\n\n\\section{Our implementation}\nIn the previous section we explained the need for interval methods\nto ensure that in most cases where a test should have a negative\noutcome, this is indeed proved numerically.\nInterval methods can do this in several ways.\nIn \\cite{itar98}, Br{\\\"o}nnimann {\\it et al.} used \ninterval arithmetic to dynamically bound \narithmetic errors when computing tests (i.e. to compute dynamic filters).\nIn our case, we use interval constraints\nnot only to compute tests but also to implement\ngeometrical constructions. \nThis means that the representations\nof {\\tt Point}, {\\tt Line}, and {\\tt Plane}\\ are in the form of constraint\nsatisfaction problems.\nFor example, a plane is represented by the constraint\n$ax+by+cz+d=0$, where\n$a$, $b$, $c$ and $d$ are real-valued constants and\n$x$, $y$, and $z$ are real-valued variables.\nDue to computability limitations\ndiscussed in the previous section,\nthe coefficients $a$, $b$, $c$ and $d$\nare implemented as floating-point intervals.\nFor each point with coordinates\nin these intervals, the constraint has a different\nplane as solution. In this way our concrete representation is\na set of planes in the abstract sense.\nThe reader may refer to the following papers \n\\cite{rthpra01}, \\cite{bhlgfg99}, \\cite{hckmdw98}, \nand \\cite{hckvnmdn01} for more information\non constraints, propagation algorithms, interval constraints,\ncorrectness and implementation of interval constraints.\n\nAs shown in Figure~\\ref{geomuml},\nthe abstract classes {\\tt Point}, {\\tt Line}, and {\\tt Plane}\\ are extended \nand modelled using intervals and constraints.\nThe abstract class {\\tt Constraint}\\ represents the constraint class,\nwhich can be extended to implement primitive constraints\nsuch as {\\tt Sum} and {\\tt Prod}. \nEach of these primitive constraints has \na \\emph{domain reduction operator} (DRO), \nrepresented by {\\tt shrink}() method,\nwhich removes inconsistent values from \nthe domains of the variables in the constraint.\nThe DROs of primitive constraints are computed based on \ninterval arithmetic. As an example, \nthe {\\tt Sum} constraint defined \nby $x+y=z$ has the\nfollowing DRO\n\\begin{eqnarray}\n&&X^{new}=X^{old} \\cap (Z^{old} - Y^{old})\\nonumber \\\\\n&&Y^{new}=Y^{old} \\cap (Z^{old} - X^{old})\\nonumber \\\\\n&&Z^{new}=Z^{old} \\cap (X^{old} + Y^{old})\\nonumber \n\\end{eqnarray}\nwhere the intervals $X^{old}$, $Y^{old}$, \nand $Z^{old}$ are the domains of $x$, $y$, and $z$\nrespectively before applying the DRO,\nand $X^{new}$, $Y^{new}$ and $Z^{new}$ are the\ndomains of $x$, $y$, and $z$ respectively after applying the DRO.\nFor a non-primitive constraint, such as {\\tt Line}\\ and {\\tt Plane}, \nwe first decompose it into primitive constraints and then\nuse the propagation algorithm to implement the {\\tt shrink}() method.\nA simple version of this algorithm is shown in Figure~\\ref{gpa}.\n\\begin{center}\n\\begin{figure}\n\\begin{center}\n\\fbox{\n\\parbox{1 cm}{\n\\small{\n\\begin{tabbing}\nmake $A$ the set of primitive constraints;\\\\\nwhile \\=( $A \\neq \\emptyset$) $\\{$\\\\  \n         \\>choose a constraint $C$ from $A$ and apply its DRO;\\\\\n         \\> if one of the domains becomes empty, then stop;\\\\\n         \\>add \\=to $A$ all constraints involving variables whose\\\\\n         \\>\\> domains have changed, if any;\\\\\n         \\>remove $C$ from $A$; $\\}$\n\\end{tabbing}\n}\n}\n}\n\\end{center}\n\\caption{\n\\label{gpa}\nPropagation algorithm.}\n\\end{figure}\n\\end{center}\nIn what follows, we present some examples\nin two dimensions illustrating the use of our implementation.\nWe ran the examples on a Pentium II machine with a CPU rate of 400 MHz, \nand with 128 MB of memory. \n\\paragraph{Are two points in the same side of a line?}\nLet $L$ be a line represented by $[2.0,2.5]*x - [0.5,1.0]*y=[1.0,1.05]$.\nLet $P$ and $Q$ be the two points represented respectively by\n$([0.0,0.0],[0.0,0.0])$ and $([0.5,0.5],[0.5,0.5])$.\nThe question we are interested in is to\ndetermine whether $P$ and $Q$ are on the same side of $L$.\nUsing the function \\emph{sameSide(Point, Point)},\nwhich checks whether two points are in the same side of a line,\nour system outputs the following results:\n\\begin{verbatim}\nDuration (musec): 179\nTrue: the points are in the same side\n\\end{verbatim}\nThis means that our system was able to prove that\n$P$ and $Q$ are in the same side of $L$.\n\nNow suppose that $Q$ is represented by $([1 , 1],[0.5 , 0.5])$.\nIn this case, our system returned the following output:\n\\begin{verbatim}\nDuration (musec): 133\nFalse: the points are not in the same side\n\\end{verbatim}\nIf, somehow, the point $Q$ is only known to be represented by\n$([0.75 , 1],[0.25 , 0.5])$ (note that the intervals are not singletons),\nthen our system was not able to prove that\nthe points $P$ and $Q$ are in the same side.\nThe output in this case is:\n\\begin{verbatim}\nDuration (musec): 265\nUndetermined\n\\end{verbatim}\n\\paragraph{Circumcenter of a triangle}\nGiven three points $P$, $Q$ and $R$ represented respectively by\n$([0.0,0.0],[0.0,0.0])$, $([1.0,1.0],[0.5,0.5])$ and $([0.5,0.5],[1.0,1.0])$\nwe wish to find the center of the circle passing through $P$, $Q$ and $R$.\nThis example is taken from \\cite{fnpr02}.\nSince this center is given by the intersection of $L_1$ and $L_2$,\nwhere $L_1$ is the line that passes through the middle of the segment $PQ$ and is\nperpendicular to the line passing through $P$ and $Q$, and $L_2$\nis the line that passes through the middle of the segment $QR$ and\nis perpendicular to the line passing through $Q$ and $R$.\nUsing the \\emph{intersect (Line)} function that\nchecks whether a line intersects with another\nline, our system returned the following output:\n\\begin{verbatim}\nDuration (msec):  6\nTrue: the lines intersect at\nx = [0.41666666666666663 , \n     0.41666666666666669]\ny = [0.41666666666666663 , \n     0.41666666666666669]\n\\end{verbatim}\nWe then checked whether the point\n$(x,y)$\nis on the line $L_3$\nthat passes through the middle of the segment $PR$\nand is perpendicular to the line passing through $P$ and $R$.\nThe output of our system indicates it is possible:\n\\begin{verbatim}\nDuration (musec): 112\nUndetermined\n\\end{verbatim}\n\n\\section*{Acknowledgements}\nThis research was supported by the University of Victoria and by\nthe Natural Science and Engineering Research Council of Canada.\n\n\\section*{For further information please contact us:}\n\\begin{flushleft}\n\\begin{tabular}{l@{\\quad}l@{\\hspace{3mm}}l@{\\quad}l}\n$\\bullet$&\\multicolumn{3}{@{}l}{\\bfseries Springer-\\kern-2pt Verlag\nHeidelberg}\\\\[1mm]\n&\\multicolumn{3}{@{}l}{Department New Technologies\/Product\nDevelopment}\\\\\n&\\multicolumn{3}{@{}l}{Springer-Verlag, Postfach 105280, D-69042\nHeidelberg\n1, FRG}\\\\[0.5mm]\n & Telefax:   & (0\\,62\\,21)487688\\\\\n &            & (0\\,62\\,21)487366\\\\\n & Internet:  & \\tt lncs@springer.de    & for editorial questions\\\\\n &            & \\tt texhelp@springer.de & for \\TeX{} problems\n\\end{tabular}\n\\end{flushleft}\n\\rule{\\textwidth}{1pt}\n\\section*{Acceptable formats of your disk\/magnetic tape and output:}\nThe following formats are acceptable: 5.25$^{\\prime\\prime}$ diskette\nMS-DOS, 5.25$^{\\prime\\prime}$ CP\/M, 3.5$^{\\prime\\prime}$ diskette\nMS-DOS, 3.5$^{\\prime\\prime}$ diskette Apple MacIntosh, 9-track 1600\nbpi magnetic tape VAX\/VMS, 9-track 1600 bpi magnetic tape ANSI with\nlabel, SUN-Streamer Tape.\n\nOnce you have completed your work using this macro package,\nplease submit your own printout of the {\\em final\nversion together with the disk or magnetic tape}, containing your\n\\LaTeX{} input (source) file und the final DVI-file and make sure\nthat the text is {\\em identical in both cases.}\n\n\\bigskip\nThis macro package, as well as all other macro packages, style\nfiles, and document classes that Springer distributes, are also\navailable through our mailserver (for people with only e-mail access).\n\n{\\tt svserv@vax.ntp.springer.de}\\hfil first try the \\verb|help|\ncommand.\n\n\\noindent We are also reachable through the world wide web:\n\\begin{flushleft}\n\\tt\n\\begin{tabular}{@{}l@{\\quad}r@{\\tt:}l}\n\\rmfamily URLs are &   http&\/\/www.springer.de       \\\\\n             & gopher&\/\/ftp.springer.de \\\\\n             &    ftp&\/\/ftp.springer.de\n\\end{tabular}\n\\end{flushleft}\n\n\n\n\n\\newpage\n\\tableofcontents\n\\newpage\n\\section{Introduction}\nAuthors wishing to code their contribution\nwith \\LaTeX{}, as well as those who have already coded with \\LaTeX{},\nwill be provided with a document class that will give the text the\ndesired layout. Authors are requested to\nadhere strictly to these instructions; {\\em the class\nfile must not be changed}.\n\nThe text output area is automatically set within an area of\n12.2\\,cm horizontally  and 19.3\\,cm vertically.\n\nIf you are already familiar with \\LaTeX{}, then the\nLLNCS class should not give you any major difficulties.\nIt will change the layout to the required LLNCS style\n(it will for instance define the layout of \\verb|\\section|).\nWe had to invent some extra commands,\nwhich are not provided by \\LaTeX{} (e.g.\\\n\\verb|\\institute|, see also Sect.\\,\\ref{contbegin})\n\nFor the main body of the paper (the text) you\nshould use the commands of the standard \\LaTeX{} ``article'' class.\nEven if you are familiar with those commands, we urge you to read\nthis entire documentation thoroughly. It contains many suggestions on\nhow to use our commands properly; thus your paper\nwill be formatted exactly to LLNCS standard.\nFor the input of the references at the end of your contribution,\nplease follow our instructions given in Sect.\\,\\ref{refer} References.\n\nThe majority of these hints are not specific for LLNCS; they may improve\nyour use of \\LaTeX{} in general.\nFurthermore, the documentation provides suggestions about the proper\nediting and use\nof the input files (capitalization, abbreviation etc.) (see\nSect.\\,\\ref{refedit} How to Edit Your Input File).\n\\section{How to Proceed}\nThe package consists of the following files:\n\\begin{flushleft}\n\\begin{tabular}{@{}p{2.5cm}l}\n{\\tt history.txt}& the version history of the package\\\\[2pt]\n{\\tt llncs.cls}  & class file for \\LaTeX{}\\\\[2pt]\n{\\tt llncs.dem}  & an example showing how to code the text\\\\[2pt]\n{\\tt llncs.doc}  & general instructions (source of this document),\\\\\n        & {\\tt llncs.doc} means {\\itshape l\\\/}atex {\\itshape doc\\\/}umentation for\\\\\n        & {\\itshape L\\\/}ecture {\\itshape N}otes in {\\itshape C\\\/}omputer {\\itshape S\\\/}cience\\\\\n{\\tt llncsdoc.sty}  & class modifications to help for the instructions\\\\\n{\\tt llncs.ind} & an external (faked) author index file\\\\\n{\\tt subjidx.ind} & subject index demo from the Springer book package\\\\\n{\\tt llncs.dvi}    & the resultig DVI file (remember to use binary transfer!)\\\\[2pt]\n{\\tt sprmindx.sty} & supplementary style file for MakeIndex\\\\\n                   & (usage: {\\tt makeindex -s sprmindx.sty <yourfile.idx>})\n\\end{tabular}\n\\end{flushleft}\n\\subsection{How to Invoke the LLNCS Document Class}\nThe LLNCS class is an extension of the standard \\LaTeX{} ``article''\ndocument class. Therefore you may use all ``article'' commands for the\nbody of your contribution to prepare your manuscript.\nLLNCS class is invoked by replacing ``article'' by ``llncs'' in the\nfirst line of your document:\n\\begin{verbatim}\n\\documentclass{llncs}\n\\begin{document}\n  <Your contribution>\n\\end{document}\n\\end{verbatim}\n\\subsection{Contributions Already Coded with \\protect\\LaTeX{} without\nthe LLNCS document class}\nIf your file is already coded with \\LaTeX{} you can easily\nadapt it a posteriori to the LLNCS document class.\n\nPlease refrain from using any \\LaTeX{} or \\TeX{} commands\nthat affect the layout or formatting of your document (i.e. commands\nlike \\verb|\\textheight|, \\verb|\\vspace|, \\verb|\\headsep| etc.).\nThere may nevertheless be exceptional occasions on which to\nuse some of them.\n\nThe LLNCS document class has been carefully designed to produce the\nright layout from your \\LaTeX{} input. If there is anything specific you\nwould like to do and for which the style file does not provide a\ncommand, {\\em please contact us}. Same holds for any error and bug you\ndiscover (there is however no reward for this -- sorry).\n\\section{General Rules for Coding Formulas}\nWith mathematical formulas you may proceed as described\nin Sect.\\,3.3 of the {\\em \\LaTeX{} User's Guide \\& Reference\nManual\\\/} by Leslie Lamport (2nd~ed. 1994), Addison-Wesley Publishing\nCompany, Inc.\n\nEquations are automatically numbered sequentially throughout your\ncontribution using arabic numerals in parentheses on the right-hand\nside.\n\nWhen you are working in math mode everything is typeset in italics.\nSometimes you need to insert non-mathematical elements (e.g.\\\nwords or phrases). Such insertions should be coded in roman\n(with \\verb|\\mbox|) as illustrated in the following example:\n\\begin{flushleft}\n{\\itshape Sample Input}\n\\end{flushleft}\n\\begin{verbatim}\n\\begin{equation}\n  \\left(\\frac{a^{2} + b^{2}}{c^{3}} \\right) = 1 \\quad\n  \\mbox{ if } c\\neq 0 \\mbox{ and if } a,b,c\\in \\bbbr \\enspace .\n\\end{equation}\n\\end{verbatim}\n{\\itshape Sample Output}\n\\begin{equation}\n  \\left(\\frac{a^{2} + b^{2}}{c^{3}} \\right) = 1 \\quad\n  \\mbox{ if } c\\neq 0 \\mbox{ and if } a,b,c\\in \\bbbr \\enspace .\n\\end{equation}\n\nIf you wish to start a new paragraph immediately after a displayed\nequation, insert a blank line so as to produce the required\nindentation. If there is no new paragraph either do not insert\na blank line or code \\verb|\\noindent| immediately before\ncontinuing the text.\n\nPlease punctuate a displayed equation in the same way as other\nordinary text but with an \\verb|\\enspace| before end punctuation.\n\nNote that the sizes of the parentheses or other delimiter\nsymbols used in equations should ideally match the height of the\nformulas being enclosed. This is automatically taken care of by\nthe following \\LaTeX{} commands:\\\\[2mm]\n\\verb|\\left(| or \\verb|\\left[| and\n\\verb|\\right)| or \\verb|\\right]|.\n\\subsection{Italic and Roman Type in Math Mode}\n\\begin{alpherate}\n\\item\nIn math mode \\LaTeX{} treats all letters as though they\nwere mathematical or physical variables, hence they are typeset as\ncharacters of their own in\nitalics. However, for certain components of formulas, like short texts,\nthis would be incorrect and therefore coding in roman is required.\nRoman should also be used for\nsubscripts and superscripts {\\em in formulas\\\/} where these are\nmerely labels and not in themselves variables,\ne.g. $T_{\\mathrm{eff}}$ \\emph{not} $T_{eff}$,\n$T_{\\mathrm K}$ \\emph{not} $T_K$ (K = Kelvin),\n$m_{\\mathrm e}$ \\emph{not} $m_e$ (e = electron).\nHowever, do not code for roman\nif the sub\/superscripts represent variables,\ne.g.\\ $\\sum_{i=1}^{n} a_{i}$.\n\\item\nPlease ensure that {\\em physical units\\\/} (e.g.\\ pc, erg s$^{-1}$\nK, cm$^{-3}$, W m$^{-2}$ Hz$^{-1}$, m kg s$^{-2}$ A$^{-2}$) and\n{\\em abbreviations\\\/} such as Ord, Var, GL, SL, sgn, const.\\\nare always set in roman type. To ensure\nthis use the \\verb|\\mathrm| command: \\verb|\\mathrm{Hz}|.\nOn p.\\ 44 of the {\\em \\LaTeX{} User's Guide \\& Reference\nManual\\\/} by Leslie Lamport you will find the names of\ncommon mathe\\-matical functions, such as log, sin, exp, max and sup.\nThese should be coded as \\verb|\\log|,\n\\verb|\\sin|, \\verb|\\exp|, \\verb|\\max|, \\verb|\\sup|\nand will appear in roman automatically.\n\\item\nChemical symbols and formulas should be coded for roman,\ne.g.\\ Fe not $Fe$, H$_2$O not {\\em H$_2$O}.\n\\item\nFamiliar foreign words and phrases, e.g.\\ et al.,\na priori, in situ, brems\\-strah\\-lung, eigenvalues should not be\nitalicized.\n\\end{alpherate}\n\\section{How to Edit Your Input (Source) File}\n\\label{refedit}\n\\subsection{Headings}\\label{headings}\nAll words in headings should be capitalized except for conjunctions,\nprepositions (e.g.\\ on, of, by, and, or, but, from, with, without,\nunder) and definite and indefinite articles (the, a, an) unless they\nappear at the beginning. Formula letters must be typeset as in the text.\n\\subsection{Capitalization and Non-capitalization}\n\\begin{alpherate}\n\\item\nThe following should always be capitalized:\n\\begin{itemize}\n\\item\nHeadings (see preceding Sect.\\,\\ref{headings})\n\\item\nAbbreviations and expressions\nin the text such as  Fig(s)., Table(s), Sect(s)., Chap(s).,\nTheorem, Corollary, Definition etc. when used with numbers, e.g.\\\nFig.\\,3, Table\\,1, Theorem 2.\n\\end{itemize}\nPlease follow the special rules in Sect.\\,\\ref{abbrev} for referring to\nequations.\n\\item\nThe following should {\\em not\\\/} be capitalized:\n\\begin{itemize}\n\\item\nThe words figure(s), table(s), equation(s), theorem(s) in the text when\nused without an accompanying number.\n\\item\nFigure legends and table captions except for names and abbreviations.\n\\end{itemize}\n\\end{alpherate}\n\\subsection{Abbreviation of Words}\\label{abbrev}\n\\begin{alpherate}\n\\item\nThe following {\\em should} be abbreviated when they appear in running\ntext {\\em unless\\\/} they come at the beginning of a sentence: Chap.,\nSect., Fig.; e.g.\\ The results are depicted in Fig.\\,5. Figure 9 reveals\nthat \\dots .\\\\\n{\\em Please note\\\/}: Equations should usually be referred to solely by\ntheir number in parentheses: e.g.\\ (14). However, when the reference\ncomes at the beginning of a sentence, the unabbreviated word\n``Equation'' should be used: e.g.\\ Equation (14) is very important.\nHowever, (15) makes it clear that \\dots .\n\\item\nIf abbreviations of names or concepts are used\nthroughout the text, they should be defined at first occurrence,\ne.g.\\ Plurisubharmonic (PSH) Functions, Strong Optimization (SOPT)\nProblem.\n\\end{alpherate}\n\\section{How to Code the Beginning of Your Contribution}\n\\label{contbegin}\nThe title of a single contribution (it is mandatory) should be coded as\nfollows:\n\\begin{verbatim}\n\\title{<Your contribution title>}\n\\end{verbatim}\nAll words in titles should be capitalized except for conjunctions,\nprepositions (e.g.\\ on, of, by, and, or, but, from, with, without,\nunder) and definite and indefinite articles (the, a, an) unless they\nappear at the beginning. Formula letters must be typeset as in the text.\nTitles have no end punctuation.\n\nIf a long \\verb|\\title| must be divided please use the code \\verb|\\\\|\n(for new line).\n\nIf you are to produce running heads for a specific volume the standard\n(of no such running heads) is overwritten with the \\verb|[runningheads]|\noption in the \\verb|\\documentclass| line. For long titles that do not\nfit in the single line of the running head a warning is generated.\nYou can specify an abbreviated title for the running head on odd pages\nwith the command\n\\begin{verbatim}\n\\titlerunning{<Your abbreviated contribution title>}\n\\end{verbatim}\n\nThere is also a possibility to change the text of the title that goes\ninto the table of contents (that's for volume editors only -- there is\nno table of contents for a single contribution). For this use the\ncommand\n\\begin{verbatim}\n\\toctitle{<Your changed title for the table of contents>}\n\\end{verbatim}\n\nAn optional subtitle may follow then:\n\\begin{verbatim}\n\\subtitle{<subtitle of your contribution>}\n\\end{verbatim}\n\nNow the name(s) of the author(s) must be given:\n\\begin{verbatim}\n\\author{<author(s) name(s)>}\n\\end{verbatim}\nNumbers referring to different addresses or affiliations are\nto be attached to each author with the \\verb|\\inst{<no>}| command.\nIf there is more than one author, the order is up to you;\nthe \\verb|\\and| command provides for the separation.\n\nIf you have done this correctly, this entry now reads, for example:\n\\begin{verbatim}\n\\author{Ivar Ekeland\\inst{1} \\and Roger Temam\\inst{2}}\n\\end{verbatim}\nThe first name\\footnote{Other initials are optional\nand may be inserted if this is the usual\nway of writing your name, e.g.\\ Alfred J.~Holmes, E.~Henry Green.}\nis followed by the surname.\n\nAs for the title there exist two additional commands (again for volume\neditors only) for a different author list. One for the running head\n(on odd pages) -- if there is any:\n\\begin{verbatim}\n\\authorrunning{<abbreviated author list>}\n\\end{verbatim}\nAnd one for the table of contents where the\naffiliation of each author is simply added in braces.\n\\begin{verbatim}\n\\tocauthor{<enhanced author list for the table of contents>}\n\\end{verbatim}\n\nNext the address(es) of institute(s), company etc. is (are) required.\nIf there is more than one address, the entries are numbered\nautomatically with \\verb|\\and|, in the order in which you type them.\nPlease make sure that the numbers match those placed next to\nto the authors' names to reflect the affiliation.\n\\begin{verbatim}\n\\institute{<name of an institute>\n\\and <name of the next institute>\n\\and <name of the next institute>}\n\\end{verbatim}\n\nIn addition, you can use\n\\begin{verbatim}\n\\email{<email address>}\n\\end{verbatim}\nto provide your email address within \\verb|\\institute|. If you need to\ntypeset the tilde character -- e.g. for your web page in your unix\nsystem's home directory -- the \\verb|\\homedir| command will happily do\nthis.\n\n\\medskip\nIf footnote like things are needed anywhere in the contribution heading\nplease code\n(immediately after the word where the footnote indicator should be\nplaced):\n\\begin{verbatim}\n\\thanks{<text>}\n\\end{verbatim}\n\\verb|\\thanks| may only appear in \\verb|\\title|, \\verb|\\author|\nand \\verb|\\institute| to footnote anything. If there are two or more\nfootnotes or affiliation marks to a specific item separate them with\n\\verb|\\fnmsep| (i.e. {\\itshape f}oot\\emph note \\emph mark\n\\emph{sep}arator).\n\n\\medskip\\noindent\nThe command\n\\begin{verbatim}\n\\maketitle\n\\end{verbatim}\nthen formats the complete heading of your article. If you leave\nit out the work done so far will produce \\emph{no} text.\n\nThen the abstract should follow. Simply code\n\\begin{verbatim}\n\\begin{abstract}\n<Text of the summary of your article>\n\\end{abstract}\n\\end{verbatim}\nor refer to the demonstration file {\\tt llncs.dem} for an example or\nto the {\\em Sample Input\\\/} on p.~\\pageref{samppage}.\n\n\\subsubsection{Remark to Running Heads and the Table of Contents}\n\\leavevmode\\\\[\\medskipamount]\nIf you are the author of a single contribution you normally have no\nrunning heads and no table of contents. Both are done only by the editor\nof the volume or at the printers.\n\\section{Special Commands for the Volume Editor}\nThe volume editor can produce a complete camera ready output including\nrunning heads, a table of contents, preliminary text (frontmatter), and\nindex or glossary. For activating the running heads there is the class\noption \\verb|[runningheads]|.\n\nThe table of contents of the volume is printed wherever\n\\verb|\\tableofcontents| is placed. A simple compilation of all\ncontributions (fields \\verb|\\title| and \\verb|\\author|) is done. If you\nwish to change this automatically produced list use the commands\n\\begin{verbatim}\n\\titlerunning  \\toctitle\n\\authorrunning \\tocauthor\n\\end{verbatim}\nto enhance the information in the specific contributions. See the\ndemonstration file \\verb|llncs.dem| for examples.\n\nAn additional structure can be added to the table of contents with the\n\\verb|\\addtocmark{<text>}| command. It has an optional numerical\nargument, a digit from 1 through 3. 3 (the default) makes an unnumbered\nchapter like entry in the table of contents. If you code\n\\verb|\\addtocmark[2]{text}| the corresponding page number is listed\nalso, \\verb|\\addtocmark[1]{text}| even introduces a chapter number\nbeyond it.\n\\section{How to Code Your Text}\nThe contribution title and all headings should be capitalized\nexcept for conjunctions, prepositions (e.g.\\ on, of, by, and, or, but,\nfrom, with, without, under) and definite and indefinite articles (the,\na, an) unless they appear at the beginning. Formula letters must be\ntypeset as in the text.\n\nHeadings will be automatically numbered by the following codes.\\\\[2mm]\n{\\itshape Sample Input}\n\\begin{verbatim}\n\\section{This is a First-Order Title}\n\\subsection{This is a Second-Order Title}\n\\subsubsection{This is a Third-Order Title.}\n\\paragraph{This is a Fourth-Order Title.}\n\\end{verbatim}\n\\verb|\\section| and \\verb|\\subsection| have no end punctuation.\\\\\n\\verb|\\subsubsection| and \\verb|\\paragraph|\nneed to be punctuated at the end.\n\nIn addition to the above-mentioned headings your text may be structured\nby subsections indicated by run-in headings (theorem-like environments).\nAll the theorem-like environments are numbered automatically\nthroughout the sections of your document -- each with its own counter.\nIf you want the theorem-like environments to use the same counter\njust specify the documentclass option \\verb|envcountsame|:\n\\begin{verbatim}\n\\documentclass[envcountsame]{llncs}\n\\end{verbatim}\nIf your first call for a theorem-like environment then is e.g.\n\\verb|\\begin{lemma}|, it will be numbered 1; if corollary follows,\nthis will be numbered 2; if you then call lemma again, this will be\nnumbered 3.\n\nBut in case you want to reset such counters to 1 in each section,\nplease specify the documentclass option \\verb|envcountreset|:\n\\begin{verbatim}\n\\documentclass[envcountreset]{llncs}\n\\end{verbatim}\n\nEven a numbering on section level (including the section counter) is\npossible with the documentclass option \\verb|envcountsect|.\n\n\\section{Predefined Theorem like Environments}\\label{builtintheo}\nThe following variety of run-in headings are at your disposal:\n\\begin{alpherate}\n\\item\n{\\bfseries Bold} run-in headings with italicized text\nas built-in environments:\n\\begin{verbatim}\n\\begin{corollary} <text> \\end{corollary}\n\\begin{lemma} <text> \\end{lemma}\n\\begin{proposition} <text> \\end{proposition}\n\\begin{theorem} <text> \\end{theorem}\n\\end{verbatim}\n\\item\nThe following generally appears as {\\itshape italic} run-in heading:\n\\begin{verbatim}\n\\begin{proof} <text>    \\qed    \\end{proof}\n\\end{verbatim}\nIt is unnumbered and may contain an eye catching square (call for that\nwith \\verb|\\qed|) before the environment ends.\n\\item\nFurther {\\itshape italic} or {\\bfseries bold} run-in headings with roman\nenvironment body may also occur:\n\\begin{verbatim}\n\\begin{definition} <text> \\end{definition}\n\\begin{example} <text> \\end{example}\n\\begin{exercise} <text> \\end{exercise}\n\\begin{note} <text> \\end{note}\n\\begin{problem} <text> \\end{problem}\n\\begin{question} <text> \\end{question}\n\\begin{remark} <text> \\end{remark}\n\\begin{solution} <text> \\end{solution}\n\\end{verbatim}\n\\end{alpherate}\n\n\\section{Defining your Own Theorem like Environments}\nWe have enhanced the standard \\verb|\\newtheorem| command and slightly\nchanged its syntax to get two new commands \\verb|\\spnewtheorem| and\n\\verb|\\spnewtheorem*| that now can be used to define additional\nenvironments. They require two additional arguments namely the type\nstyle in which the keyword of the environment appears and second the\nstyle for the text of your new environment.\n\n\\verb|\\spnewtheorem| can be used in two ways.\n\\subsection{Method 1 {\\itshape (preferred)}}\nYou may want to create an environment that shares its counter\nwith another environment, say {\\em main theorem\\\/} to be numbered like\nthe predefined {\\em theorem\\\/}. In this case, use the syntax\n\\begin{verbatim}\n\\spnewtheorem{<env_nam>}[<num_like>]{<caption>}\n{<cap_font>}{<body_font>}\n\\end{verbatim}\n\n\\noindent\nHere the environment with which the new environment should share its\ncounter is specified with the optional argument \\verb|[<num_like>]|.\n\n\\paragraph{Sample Input}\n\\begin{verbatim}\n\\spnewtheorem{mainth}[theorem]{Main Theorem}{\\bfseries}{\\itshape}\n\\begin{theorem} The early bird gets the worm. \\end{theorem}\n\\begin{mainth} The early worm gets eaten. \\end{mainth}\n\\end{verbatim}\n\\medskip\\noindent\n{\\em Sample Output}\n\n\\medskip\\noindent\n{\\bfseries Theorem 3.}\\enspace {\\em The early bird gets the worm.}\n\n\\medskip\\noindent\n{\\bfseries Main Theorem 4.} The early worm gets eaten.\n\n\\bigskip\nThe sharing of the default counter (\\verb|[theorem]|) is desired. If you\nomit the optional second argument of \\verb|\\spnewtheorem| a separate\ncounter for your new environment is used throughout your document.\n\n\\subsection[Method 2]{Method 2 {\\itshape (assumes {\\tt[envcountsect]}\ndocumentstyle option)}}\n\\begin{verbatim}\n\\spnewtheorem{<env_nam>}{<caption>}[<within>]\n{<cap_font>}{<body_font>}\n\\end{verbatim}\n\n\\noindent\nThis defines a new environment \\verb|<env_nam>| which prints the caption\n\\verb|<caption>| in the font \\verb|<cap_font>| and the text itself in\nthe font \\verb|<body_font>|. The environment is numbered beginning anew\nwith every new sectioning element you specify with the optional\nparameter \\verb|<within>|.\n\n\\medskip\\noindent\n\\paragraph{Example} \\leavevmode\n\n\\medskip\\noindent\n\\verb|\\spnewtheorem{joke}{Joke}[subsection]{\\bfseries}{\\rmfamily}|\n\n\\medskip\n\\noindent defines a new environment called \\verb|joke| which prints the\ncaption {\\bfseries Joke} in boldface and the text in roman. The jokes are\nnumbered starting from 1 at the beginning of every subsection with the\nnumber of the subsection preceding the number of the joke e.g. 7.2.1 for\nthe first joke in subsection 7.2.\n\n\\subsection{Unnumbered Environments}\nIf you wish to have an unnumbered environment, please\nuse the syntax\n\\begin{verbatim}\n\\spnewtheorem*{<env_nam>}{<caption>}{<cap_font>}{<body_font>}\n\\end{verbatim}\n\n\\section{Program Codes}\nIn case you want to show pieces of program code, just use the\n\\verb|verbatim| environment or the \\verb|verbatim| package of \\LaTeX.\n(There also exist various pretty printers for some programming\nlanguages.)\n\\noindent\n\\subsection*{Sample Input {\\rmfamily(of a simple\ncontribution)}}\\label{samppage}\n\\begin{verbatim}\n\\title{Hamiltonian Mechanics}\n\n\\author{Ivar Ekeland\\inst{1} \\and Roger Temam\\inst{2}}\n\n\\institute{Princeton University, Princeton NJ 08544, USA\n\\and\nUniversit\\'{e} de Paris-Sud,\nLaboratoire d'Analyse Num\\'{e}rique, B\\^{a}timent 425,\\\\\nF-91405 Orsay Cedex, France}\n\n\\maketitle\n\\begin{abstract}\nThis paragraph shall summarize the contents of the paper\nin short terms.\n\\end{abstract}\n\\section{Fixed-Period Problems: The Sublinear Case}\nWith this chapter, the preliminaries are over, and we begin the\nsearch for periodic solutions \\dots\n\\subsection{Autonomous Systems}\nIn this section we will consider the case when the Hamiltonian\n$H(x)$ \\dots\n\\subsubsection*{The General Case: Nontriviality.}\nWe assume that $H$ is\n$\\left(A_{\\infty}, B_{\\infty}\\right)$-subqua\\-dra\\-tic\nat infinity, for some constant \\dots\n\\paragraph{Notes and Comments.}\nThe first results on subharmonics were \\dots\n\\begin{proposition}\nAssume $H'(0)=0$ and $ H(0)=0$. Set \\dots\n\\end{proposition}\n\\begin{proof}[of proposition]\nCondition (8) means that, for every $\\delta'>\\delta$, there is\nsome $\\varepsilon>0$ such that \\dots \\qed\n\\end{proof}\n\\begin{example}[\\rmfamily (External forcing)]\nConsider the system \\dots\n\\end{example}\n\\begin{corollary}\nAssume $H$ is $C^{2}$ and\n$\\left(a_{\\infty}, b_{\\infty}\\right)$-subquadratic\nat infinity. Let \\dots\n\\end{corollary}\n\\begin{lemma}\nAssume that $H$ is $C^{2}$ on $\\bbbr^{2n}\\backslash \\{0\\}$\nand that $H''(x)$ is \\dots\n\\end{lemma}\n\\begin{theorem}[(Ghoussoub-Preiss)]\nLet $X$ be a Banach Space and $\\Phi:X\\to\\bbbr$ \\dots\n\\end{theorem}\n\\begin{definition}\nWe shall say that a $C^{1}$ function $\\Phi:X\\to\\bbbr$\nsatisfies \\dots\n\\end{definition}\n\\end{verbatim}\n{\\itshape Sample Output\\\/} (follows on the next page together with\nexamples of the above run-in headings)\n\\newcounter{save}\\setcounter{save}{\\value{section}}\n{\\def\\addtocontents#1#2{}%\n\\def\\addcontentsline#1#2#3{}%\n\\def\\markboth#1#2{}%\n\\title{Hamiltonian Mechanics}\n\n\\author{Ivar Ekeland\\inst{1} \\and Roger Temam\\inst{2}}\n\n\\institute{Princeton University, Princeton NJ 08544, USA\n\\and\nUniversit\\'{e} de Paris-Sud,\nLaboratoire d'Analyse Num\\'{e}rique, B\\^{a}timent 425,\\\\\nF-91405 Orsay Cedex, France}\n\n\\maketitle\n\\begin{abstract}\nThis paragraph shall summarize the contents of the paper\nin short terms.\n\\end{abstract}\n\\section{Fixed-Period Problems: The Sublinear Case}\nWith this chapter, the preliminaries are over, and we begin the search\nfor periodic solutions \\dots\n\\subsection{Autonomous Systems}\nIn this section we will consider the case when the Hamiltonian\n$H(x)$ \\dots\n\\subsubsection{The General Case: Nontriviality.}\nWe assume that $H$ is\n$\\left(A_{\\infty}, B_{\\infty}\\right)$-subqua\\-dra\\-tic at\ninfinity, for some constant \\dots\n\\paragraph{Notes and Comments.}\nThe first results on subharmonics were \\dots\n\\begin{proposition}\nAssume $H'(0)=0$ and $ H(0)=0$. Set \\dots\n\\end{proposition}\n\\begin{proof}[of proposition]\nCondition (8) means that, for every $\\delta'>\\delta$, there is\nsome $\\varepsilon>0$ such that \\dots \\qed\n\\end{proof}\n\\begin{example}[{{\\rmfamily External forcing}}]\nConsider the system \\dots\n\\end{example}\n\\begin{corollary}\nAssume $H$ is $C^{2}$ and\n$\\left(a_{\\infty}, b_{\\infty}\\right)$-subquadratic\nat infinity. Let \\dots\n\\end{corollary}\n\\begin{lemma}\nAssume that $H$ is $C^{2}$ on $\\bbbr^{2n}\\backslash \\{0\\}$\nand that $H''(x)$ is \\dots\n\\end{lemma}\n\\begin{theorem}[Ghoussoub-Preiss]\nLet $X$ be a Banach Space and $\\Phi:X\\to\\bbbr$ \\dots\n\\end{theorem}\n\\begin{definition}\nWe shall say that a $C^{1}$ function $\\Phi:X\\to\\bbbr$ satisfies \\dots\n\\end{definition}\n}\\setcounter{section}{\\value{save}}\n\\section{Fine Tuning of the Text}\nThe following should be used to improve the readability of the text:\n\\begin{flushleft}\n\\begin{tabular}{@{}p{.19\\textwidth}p{.79\\textwidth}}\n\\verb|\\,|   & a thin space, e.g.\\ between numbers or between units\n              and num\\-bers; a line division will not be made\n              following this space\\\\\n\\verb|--|   & en dash; two strokes, without a space at either end\\\\\n\\verb*| -- |& en dash; two strokes, with  a space at either end\\\\\n\\verb|-|    & hyphen; one stroke, no space at either end\\\\\n\\verb|$-$|  & minus, in the text {\\em only} \\\\[8mm]\n{\\em Input} & \\verb|21\\,$^{\\circ}$C etc.,|\\\\\n            &  \\verb|Dr h.\\,c.\\,Rockefellar-Smith \\dots|\\\\\n            & \\verb|20,000\\,km and  Prof.\\,Dr Mallory \\dots|\\\\\n            & \\verb|1950--1985 \\dots|\\\\\n            & \\verb|this -- written on a computer -- is now printed|\\\\\n            & \\verb|$-30$\\,K \\dots|\\\\[3mm]\n{\\em Output}& 21\\,$^{\\circ}$C etc., Dr h.\\,c.\\,Rockefellar-Smith \\dots\\\\\n            & 20,000\\,km and  Prof.\\,Dr Mallory \\dots\\\\\n            & 1950--1985 \\dots\\\\\n            & this -- written on a computer -- is now printed\\\\\n            & $-30$\\,K \\dots\n\\end{tabular}\n\\end{flushleft}\n\\section {Special Typefaces}\nNormal type (roman text) need not be coded. {\\itshape Italic}\n(\\verb|{\\em <text>}| better still \\verb|\\emph{<text>}|) or, if\nnecessary, {\\bfseries boldface} should be used for emphasis.\\\\[6pt]\n\\begin{minipage}[t]{\\textwidth}\n\\begin{flushleft}\n\\begin{tabular}{@{}p{.25\\textwidth}@{\\hskip6pt}p{.73\\textwidth}@{}}\n\\verb|{\\itshape Text}|   & {\\itshape Italicized Text}\\\\[2pt]\n\\verb|{\\em Text}|   & {\\em Emphasized Text --\n   if you would like to emphasize a {\\em definition} within an\n   italicized text (e.g.\\ of a {\\em theorem)} you should code the\n   expression to be emphasized by} \\verb|\\em|.\\\\[2pt]\n\\verb|{\\bfseries Text}|& {\\bfseries Important Text}\\\\[2pt]\n\\verb|\\vec{Symbol}| & Vectors may only appear in math mode. The default\n   \\LaTeX{} vector symbol has been adapted\\footnotemark\\\n to LLNCS conventions.\\\\[2pt]\n & \\verb|$\\vec{A \\times B\\cdot C}| yields $\\vec{A\\times B\\cdot C}$\\\\\n & \\verb|$\\vec{A}^{T} \\otimes \\vec{B} \\otimes|\\\\\n & \\verb|\\vec{\\hat{D}}$|yields $\\vec{A}^{T} \\otimes \\vec{B} \\otimes\n\\vec{\\hat{D}}$\n\\end{tabular}\n\\end{flushleft}\n\\end{minipage}\n\n\\footnotetext{If you absolutely must revive the original \\LaTeX{}\ndesign of the vector symbol (as an arrow accent), please specify the\noption \\texttt{[orivec]} in the \\texttt{documentclass} line.}\n\\newpage\n\\section {Footnotes}\nFootnotes within the text should be coded:\n\\begin{verbatim}\n\\footnote{Text}\n\\end{verbatim}\n{\\itshape Sample Input}\n\\begin{flushleft}\nText with a footnote\\verb|\\footnote{The |{\\tt footnote is automatically\nnumbered.}\\verb|}| and text continues \\dots\n\\end{flushleft}\n{\\itshape Sample Output}\n\\begin{flushleft}\nText with a footnote\\footnote{The footnote is automatically numbered.}\nand text continues \\dots\n\\end{flushleft}\n\\section {Lists}\nPlease code lists as described below:\\\\[2mm]\n{\\itshape Sample  Input}\n\\begin{verbatim}\n\\begin{enumerate}\n  \\item First item\n  \\item Second item\n  \\begin{enumerate}\n    \\item First nested item\n    \\item Second nested item\n  \\end{enumerate}\n  \\item Third item\n\\end{enumerate}\n\\end{verbatim}\n{\\itshape Sample Output}\n \\begin{enumerate}\n\\item First item\n\\item Second item\n  \\begin{enumerate}\n    \\item First nested item\n    \\item Second nested item\n  \\end{enumerate}\n\\item Third item\n\\end{enumerate}\n\\section {Figures}\nFigure environments should be inserted after (not in)\nthe  paragraph in which the figure is first mentioned.\nThey will be numbered automatically.\n\nPreferably the images should be enclosed as PostScript files -- best as\nEPS data using the epsfig package.\n\nIf you cannot include them into your output this way and use other\ntechniques for a separate production,\nthe figures (line drawings and those containing halftone inserts\nas well as halftone figures) {\\em should not be pasted into your\nlaserprinter output}. They should be enclosed separately in camera-ready\nform (original artwork, glossy prints, photographs and\/or slides). The\nlettering should be suitable for reproduction, and after a\nprobably necessary reduction the height of capital letters should be at\nleast 1.8\\,mm and not more than 2.5\\,mm.\nCheck that lines and other details are uniformly black and\nthat the lettering on figures is clearly legible.\n\nTo leave the desired amount of space for the height of\nyour figures, please use the coding described below.\nAs can be seen in the output, we will automatically\nprovide 1\\,cm space above and below the figure,\nso that you should only leave the space equivalent to the size of the\nfigure itself. Please note that ``\\verb|x|'' in the following\ncoding stands for the actual height of the figure:\n\\begin{verbatim}\n\\begin{figure}\n\\vspace{x cm}\n\\caption[ ]{...text of caption...}          (Do type [ ])\n\\end{figure}\n\\end{verbatim}\n\\begin{flushleft}\n{\\itshape Sample Input}\n\\end{flushleft}\n\\begin{verbatim}\n\\begin{figure}\n\\vspace{2.5cm}\n\\caption{This is the caption of the figure displaying a white\neagle and a white horse on a snow field}\n\\end{figure}\n\\end{verbatim}\n\\begin{flushleft}\n{\\itshape Sample Output}\n\\end{flushleft}\n\\begin{figure}\n\\vspace{2.5cm}\n\\caption{This is the caption of the figure displaying a white eagle and\na white horse on a snow field}\n\\end{figure}\n\\section{Tables}\nTable captions should be treated\nin the same way as figure legends, except that\nthe table captions appear {\\itshape above} the tables. The tables\nwill be numbered automatically.\n\\subsection{Tables Coded with \\protect\\LaTeX{}}\nPlease use the following coding:\\\\[2mm]\n{\\itshape Sample Input}\n\\begin{verbatim}\n\\begin{table}\n\\caption{Critical $N$ values}\n\\begin{tabular}{llllll}\n\\hline\\noalign{\\smallskip}\n${\\mathrm M}_\\odot$ & $\\beta_{0}$ & $T_{\\mathrm c6}$ & $\\gamma$\n  & $N_{\\mathrm{crit}}^{\\mathrm L}$\n  & $N_{\\mathrm{crit}}^{\\mathrm{Te}}$\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n 30 & 0.82 & 38.4 & 35.7 & 154 & 320 \\\\\n 60 & 0.67 & 42.1 & 34.7 & 138 & 340 \\\\\n120 & 0.52 & 45.1 & 34.0 & 124 & 370 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\end{verbatim}\n\n\\medskip\\noindent{\\itshape Sample Output}\n\\begin{table}\n\\caption{Critical $N$ values}\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.4}\n\\setlength\\tabcolsep{3pt}\n\\begin{tabular}{llllll}\n\\hline\\noalign{\\smallskip}\n${\\mathrm M}_\\odot$ & $\\beta_{0}$ & $T_{\\mathrm c6}$ & $\\gamma$\n  & $N_{\\mathrm{crit}}^{\\mathrm L}$\n  & $N_{\\mathrm{crit}}^{\\mathrm{Te}}$\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n 30 & 0.82 & 38.4 & 35.7 & 154 & 320 \\\\\n 60 & 0.67 & 42.1 & 34.7 & 138 & 340 \\\\\n120 & 0.52 & 45.1 & 34.0 & 124 & 370 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nBefore continuing your text you need an empty line. \\dots\n\n\\vspace{3mm}\nFor further information you will find a complete description of\nthe tabular environment\non p.~62~ff. and p.~204 of the {\\em \\LaTeX{} User's Guide \\& Reference\nManual\\\/} by Leslie Lamport.\n\\subsection{Tables Not Coded with \\protect\\LaTeX{}}\nIf you do not wish to code your table using \\LaTeX{}\nbut prefer to have it reproduced separately,\nproceed as for figures and use the following coding:\\\\[2mm]\n{\\itshape Sample Input}\n\\begin{verbatim}\n\\begin{table}\n\\caption{text of your caption}\n\\vspace{x cm}    \n\\end{table}\n\\end{verbatim}\n\\subsection{Signs and Characters}\n\\subsubsection*{Special Signs.}\nYou may need to use special signs.  The available ones are listed in the\n{\\em \\LaTeX{} User's Guide \\& Reference Manual\\\/} by Leslie Lamport,\npp.~41\\,ff.\nWe have created further symbols for math mode (enclosed in \\$):\n\\begin{center}\n\\begin{tabular}{l@{\\hspace{1em}yields\\hspace{1em}}\nc@{\\hspace{3em}}l@{\\hspace{1em}yields\\hspace{1em}}c}\n\\verb|\\grole| & $\\grole$ & \\verb|\\getsto| & $\\getsto$\\\\\n\\verb|\\lid|   & $\\lid$   & \\verb|\\gid|    & $\\gid$\n\\end{tabular}\n\\end{center}\n\\subsubsection*{Gothic (Fraktur).}\nIf gothic letters are {\\itshape necessary}, please use those of the\nrelevant \\AmSTeX{} alphabet which are available using the amstex\npackage of the American Mathematical Society.\n\nIn \\LaTeX{} only the following gothic letters are available:\n\\verb|$\\Re$| yields $\\Re$ and \\verb|$\\Im$| yields $\\Im$. These should\n{\\itshape not\\\/} be used when you need gothic letters for your contribution.\nUse \\AmSTeX{} gothic as explained above. For the real and the imaginary\nparts of a complex number within math mode you should use instead:\n\\verb|$\\mathrm{Re}$| (which yields Re) or \\verb|$\\mathrm{Im}$| (which\nyields Im).\n\\subsubsection*{Script.}\nFor script capitals use the coding\n\\begin{center}\n\\begin{tabular}{l@{\\hspace{1em}which yields\\hspace{1em}}c}\n\\verb|$\\mathcal{AB}$| & $\\mathcal{AB}$\n\\end{tabular}\n\\end{center}\n(see p.~42 of  the \\LaTeX{} book).\n\\subsubsection*{Special Roman.}\nIf you need other symbols than those below, you could use\nthe blackboard bold characters of \\AmSTeX{},  but there might arise\ncapacity problems\nin loading additional \\AmSTeX{} fonts. Therefore  we created\nthe blackboard bold characters listed below.\nSome of them are not esthetically\nsatisfactory. This need not deter you from using them:\nin the final printed form they will be\nreplaced by the well-designed MT (monotype) characters of\nthe phototypesetting machine.\n\\begin{flushleft}\n\\begin{tabular}{@{}ll@{ yields }\nc@{\\hspace{1.em}}ll@{ yields }c}\n\\verb|\\bbbc| & (complex numbers)   & $\\bbbc$\n  & \\verb|\\bbbf| & (blackboard bold F) & $\\bbbf$\\\\\n\\verb|\\bbbh| & (blackboard bold H) & $\\bbbh$\n  & \\verb|\\bbbk| & (blackboard bold K) & $\\bbbk$\\\\\n\\verb|\\bbbm| & (blackboard bold M) & $\\bbbm$\n  & \\verb|\\bbbn| & (natural numbers N) & $\\bbbn$\\\\\n\\verb|\\bbbp| & (blackboard bold P) & $\\bbbp$\n  & \\verb|\\bbbq| & (rational numbers)  & $\\bbbq$\\\\\n\\verb|\\bbbr| & (real numbers)      & $\\bbbr$\n  & \\verb|\\bbbs| & (blackboard bold S) & $\\bbbs$\\\\\n\\verb|\\bbbt| & (blackboard bold T) & $\\bbbt$\n  & \\verb|\\bbbz| & (whole numbers)     & $\\bbbz$\\\\\n\\verb|\\bbbone| & (symbol one)      & $\\bbbone$\n\\end{tabular}\n\\end{flushleft}\n\\begin{displaymath}\n\\begin{array}{c}\n\\bbbc^{\\bbbc^{\\bbbc}} \\otimes\n\\bbbf_{\\bbbf_{\\bbbf}} \\otimes\n\\bbbh_{\\bbbh_{\\bbbh}} \\otimes\n\\bbbk_{\\bbbk_{\\bbbk}} \\otimes\n\\bbbm^{\\bbbm^{\\bbbm}} \\otimes\n\\bbbn_{\\bbbn_{\\bbbn}} \\otimes\n\\bbbp^{\\bbbp^{\\bbbp}}\\\\[2mm]\n\\otimes\n\\bbbq_{\\bbbq_{\\bbbq}} \\otimes\n\\bbbr^{\\bbbr^{\\bbbr}} \\otimes\n\\bbbs^{\\bbbs_{\\bbbs}} \\otimes\n\\bbbt^{\\bbbt^{\\bbbt}} \\otimes\n\\bbbz \\otimes\n\\bbbone^{\\bbbone_{\\bbbone}}\n\\end{array}\n\\end{displaymath}\n\\section{References}\n\\label{refer}\nThere are three reference systems available; only one, of course,\nshould be used for your contribution. With each system (by\nnumber only, by letter-number or by author-year) a reference list\ncontaining all citations in the\ntext, should be included at the end of your contribution placing the\n\\LaTeX{} environment \\verb|thebibliography| there.\nFor an overall information on that environment\nsee the {\\em \\LaTeX{} User's Guide \\& Reference\nManual\\\/} by Leslie Lamport, p.~71.\n\nThere is a special {\\sc Bib}\\TeX{} style for LLNCS that works along\nwith the class: \\verb|splncs.bst|\n-- call for it with a line \\verb|\\bibliographystyle{splncs}|.\nIf you plan to use another {\\sc Bib}\\TeX{} style you are customed to,\nplease specify the option \\verb|[oribibl]| in the\n\\verb|documentclass| line, like:\n\\begin{verbatim}\n\\documentclass[oribibl]{llncs}\n\\end{verbatim}\nThis will retain the original \\LaTeX{} code for the bibliographic\nenvironment and the \\verb|\\cite| mechanism that many {\\sc Bib}\\TeX{}\napplications rely on.\n\\subsection{References by Letter-Number or by Number Only}\nReferences are cited in the text -- using the \\verb|\\cite|\ncommand of \\LaTeX{} -- by number or by letter-number in square\nbrackets, e.g.\\ [1] or [E1, S2], [P1], according to your use of the\n\\verb|\\bibitem| command in the \\verb|thebibliography| environment. The\ncoding is as follows: if you choose your own label for the sources by\ngiving an optional argument to the \\verb|\\bibitem| command the citations\nin the text are marked with the label you supplied. Otherwise a simple\nnumbering is done, which is preferred.\n\\begin{verbatim}\nThe results in this section are a refined version\nof \\cite{clar:eke}; the minimality result of Proposition~14\nwas the first of its kind.\n\\end{verbatim}\nThe above input produces the citation: ``\\dots\\ refined version of\n[CE1]; the min\\-i\\-mality\\dots''. Then the \\verb|\\bibitem| entry of\nthe \\verb|thebibliography| environment should read:\n\\begin{verbatim}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nMany dynamical properties, such as hyperbolicity, are robust in\n$C^r$-topology of diffeomorphisms. That is, the property holds under any appropriate small\nperturbation of the dynamical system.  However, many others\ninteresting phenomena, non-hyperbolic strange attractors for\ninstance, are not stable in that sense. Hence, the question that\narises is whether such dynamical properties could be survive if not for\nall perturbations but, at least, for most. For one-dimensional\ndynamics the Malliavin-Shavgulidze measure has been recently\nproposed as a good analogy to the Lebesgue measure in order to\nquantify this abundance in a probabilistic\nsense~\\cite{Triestino}. However, in higher dimensions, it is not\nknown how to introduce a good notion of a measure in the space of\ndynamical systems. Kolmogorov in his plenary talk ending the ICM 1954\nproposed to consider finite dimensional parametric families taking into\naccount the Lebesgue measure in the parameter space\n(see~\\cite{hunt2010prevalence}). A parametric family $(f_a)_a$\nexhibits \\emph{persistently} a property $\\mathscr{P}$ if\nit is observed for $f_a$ in a set of parameter values\n$a$ with positive Lebesgue measure. Furthermore, the property\n$\\mathscr{P}$ is called \\emph{typical} (in the sense of\nKolmogorov) if there is a Baire (local) generic set of parametric\nfamilies exhibiting the property $\\mathscr{P}$\npersistently with full Lebesgue measure. In this direction, a\nmilestone in recent history of dynamical systems has been the\npaper of Berger~\\cite{Ber16} (see also~\\cite{Ber17}) where it was\nproven that the coexistence of infinitely many periodic sinks is\nKolmogorov typical in parametric families of endomorphisms in\ndimension two and diffeomorphisms in higher dimensions.\nThe work of Berger extends, in a measurable sense according to\nKolmogorov, the important results in the 70's due to\nNewhouse~\\cite{New74,New79} (see\nalso~\\cite{robinson1983bifurcation,PT93,PV94,GST93a}) on the local\ngenericity of the coexistence of infinitely many hyperbolic\nattractors (sinks) in $C^r$-topology. This celebrated result was coined as\n\\emph{Newhouse phenomena}. Mimicking this terminology we will refer\nto the Kolmorov typical coexistence\nof infinitely many attractors as \\emph{Berger phenomena}.\n\nNewhouse phenomena has been showen to occur in open sets of\ndiffeomorphisms having a dense subset of systems displaying\nhomoclinic tangencies associated with saddle periodic points. Such\nan open set of dynamical systems is called a \\emph{Newhouse\ndomain}. In certain cases, these open sets are also the support of\nmany other interesting phenomena such as the coexistence of\ninfinitely many attracting invariant circles~\\cite{GST08} and\ninfinitely many strange\nattractors~\\cite{colli1998infinitely,leal2008high}, or wandering\ndomains~\\cite{KS17} among others. Berger phenomena also occurs\nwith respect to some open set but now in the topology of\nparametric families. Namely, in open sets where the families\nhaving persistent homoclinic tangencies are dense. As before,\nmimicking the terminology, we will refer to these open sets of\nparametric families as \\emph{Berger domains}. In the original\npaper of Berger~\\cite{Ber16,Ber17}, these open sets were\nimplicitly constructed for sectional dissipative dynamics. In this\npaper, we will introduce formally the notion of a Berger domain\nand construct new examples, not necessarily for sectional\ndissipative dynamics. As an application, we will prove Berger\nphenomena for a certain type of non-sectional dissipative Berger\ndomains and obtain that the coexistence of infinitely many\nattracting invariant circles is also Kolmogorov typical.\n\n\n\n\n\\subsection{Degenerate unfoldings}\n\nA $C^r$-diffeomorphism $f$ of a manifold $\\mathcal{M}$ has a\n\\emph{homoclinic tangency} if there is a pair of points ${P}$ and\n$Q$, in the same transitive hyperbolic set, so that the unstable\ninvariant manifold of ${P}$ and the stable invariant manifold of\n$Q$ have a non-transverse intersection at a point $Y$. The\ntangency is said to be of codimension $c>0$ if\n\\begin{equation*}\nc= c_Y(W^u(P),W^s(Q)) \\eqdef \\dim M - \\dim (T_YW^u(P)+T_YW^s(Q)).\n\\end{equation*}\nThis number measures how far from being transverse is the intersection between the invariant\nmanifolds at $Y$. Since the codimension\nof $W^u(P)$ coincides with the dimension of $W^s(Q)$ we have, in\nthis case, that the codimension $c$ at $Y$ coincides with $\\dim T_Y\nW^u(P)\\cap T_Y W^s(Q)$.\nA homoclinic tangency can be unfolded by considering a\n$k$-parameter family in the $C^{d,r}$-topology with $1\\leq d\\leq\nr$. That is, a  $C^d$-family $(f_a)_a$ of $C^r$-diffeomorphisms\npa\\-ra\\-me\\-te\\-ri\\-zed by $a\\in \\mathbb{I}^k$ with $f_{a_0}=f$\nwhere $\\mathbb{I}=[-1,1]$ and $k\\geq 1$ (see~\\S\\ref{sec:topology}\nfor a more precise definition).\nThe unfolding of {a\ntangency $Y$ of codimension $c$} is said to be\n\\emph{$C^d$-{degenerate}} at $a=a_0$ if there are points $p_a \\in\nW^u(P_a)$, $q_a \\in W^s(Q_a)$ and $c$-dimensional subspaces $E_a$,\n$F_a$ of $T_{p_a}W^u(P_a)$ and $T_{q_a}W^s(Q_a)$ respectively\nsuch that\n$$\n d(p_a,q_a)=o(\\|a-a_0\\|^{d}) \\ \\ \\text{and} \\ \\\nd(E_a,F_a)={o(\\|a-a_0\\|^{d})} \\ \\ \\text{at $a=a_0$}.\n$$\nHere $P_a$ and $Q_a$ are the continuations of $P_{a_0}=P$ and\n$Q_{a_0}=Q$ for $f_a$. Also $p_{a_0}=q_{a_0}=Y$ and $(p_a,E_a)$,\n$(q_a,F_a)$ vary $C^d$-continuously with respect to the parameter\n$a\\in \\mathbb{I}^k$. Observe that in this case it is necessary to\nassume that $d<r$ because the above definition involves the\ndynamics of the family $(f_a)_a$ in the tangent bundle (in fact,\nin certain Grassmannian bundles). In~\\cite{Ber16}, the notion of\n$C^d$-degenerate unfoldings of homoclinic tangencies were\nintroduced for short under the name of\n\\emph{$C^d$-paratangencies}.\n\n\\subsection{Berger domains}\n\\label{BergerDomain} {{\nLet us remind the reader the notion of Newhouse domains.\nFollowing~\\cite{BD12}, we say that a $C^r$-open set $\\mathcal{N}$\nof diffeomorphisms is a \\emph{$C^r$-Newhouse domain} (of\ntangencies of codimension $c>0$) if there exists a dense set\n$\\mathcal{D}$ in $\\mathcal{N}$ such that every $g\\in \\mathcal{D}$\nhas a homoclinic tangency (of codimension $c>0$) associated with\nsome hyperbolic periodic saddle. A $C^r$-Newhouse domain\n$\\mathcal{N}$ ($r\\geq 1$) of homoclinic tangencies (of codimension\none) associated with sectional dissipative periodic points gives\nrise to the $C^r$-Newhouse phenomenon. Namely, there exists a\nresidual subset $\\mathcal{R}$ of $\\mathcal{N}$ where every $g\\in\n\\mathcal{R}$ has infinitely many hyperbolic\nperiodic attractors\n\\cite{New74,New79,PT93,GST93a,PV94,Ro95,GST08}. As Berger showed\nin~\\cite{Ber16}, open sets of families displaying degenerate\nunfoldings play the same role for parametric families as Newhouse\ndomains do for the case free of parameters.\nFor this reason\nmimicking the above terminology,\none could say that:\n\\begin{enumerate}[label={}, rightmargin=2em, leftmargin=2em]\n\\item \\it An open set $\\mathscr{U}$ of $k$-parameter $C^d$-families of\n$C^r$-diffeomorphisms is called a \\emph{$C^{d,r}$-Berger domain}\nof paratangencies (of codimension $c>0$) if  the following holds.\nThere exists a dense set $\\mathscr{D}\\subset \\mathbb{I}^k\\times\n\\mathscr{U}$ such that for any $(a_0, f)\\in \\mathscr{D}$, the\nfamily $f=(f_a)_a$ displays a $C^d$-degenerate unfolding at\n$a=a_0$ of a homoclinic tangency (of codimension $c>0$) associated\nwith a hyperbolic periodic~saddle.\n\\end{enumerate}\nFor codimension $c=1$, this definition appears implicitly\nin~\\cite{Ber16} where\nit is proven that the\ncoexistence of infinitely many hyperbolic periodic attractors is\nKolmogorov typical.\n\nActually, by modifying the\ninitial construction Berger showed a stronger result\nin~\\cite{Ber16,Ber17} that we will refer to as\n\\emph{$C^{d,r}$-Berger phenomena}: the existence of a residual set\nin a $C^{d,r}$-open set of parametric families where each family\nhas\ninfinitely many sinks at\n\\emph{any} parameter.\nThe\nfollowing\nstronger version of the above tentative\ndefinition allowed Berger to prove such a result:\n\\begin{defi} \\label{def:Berger-domain}\nAn open set $\\mathscr{U}$ of $k$-parameter $C^d$-families of\n$C^r$-diffeomorphisms is called \\emph{$C^{d,r}$-Berger domain} of\npersistent homoclinic tangencies\n(of codimension $c>0$) if there exists a dense subset\n$\\mathscr{D}$ of $\\mathscr{U}$ such that for any $f=(f_a)_a \\in\n\\mathscr{D}$ there is a covering of $\\mathbb{I}^k$ by open balls\n$J_i$\nhaving the following property:\nthere is a continuation of a\nsaddle periodic point $Q_a$ having a homoclinic tangency $Y_{a}$\n(of codimension $c>0$) which depends $C^d$-continuously on the\nparameter $a\\in J_i$.\n\\end{defi}\n\n\nObserve that the first tentative definition above requires $d<r$\nbecause of the notion of the $C^d$-paratangency. However,\ndefinition~\\ref{def:Berger-domain} admits\n$d\\leq r$\nsince it deals with the\nnotion of a $C^d$-persistent homoclinic tangency.\nThe following result shows the existence of Berger\ndomains of large codimension for families of diffeomorphisms:\n\n\n\\begin{mainthm}\n\\label{thmD} Any manifold of dimension $m> c^2+c$ admits an open\nset $\\mathscr{U}$ of $k$-parameter $C^d$-families of\n$C^r$-diffeomorphisms with {$0<d< r-1$}, so that  $\\mathscr{U}$ is\na \\emph{$C^{d,r}$-Berger domain} of persistent homoclinic\ntangencies of codimension $c>0$.\n\\end{mainthm}\n}\n\nThe proof of Theorem~\\ref{thmD} is based on the notion of a\n$C^d$-degenerate unfolding of homoclinic tangencies and previous\nresults from~\\cite{BR20}. For this reason, we have only been able\nto show the existence of $C^{d,r}$-Berger  for families of\ndiffeomorphisms with $d<r-1$ and manifolds of dimension $m\\geq 3$.\nRecall that, in the case of codimension $c=1$, Berger, in his\noriginal papers~\\cite{Ber16} and~\\cite{Ber17}, constructed this\nkind of open sets for $C^d$-families of $C^r$-endomorphisms in any\nsurface with $1\\leq d\\leq r$. Afterwards this construction is lifted\nto $C^d$-families of sectionally dissipative\n$C^r$-diffeomorphims in manifolds of dimension $m\\geq 3$. It is\nunknown if Berger domains exist for families of diffeomorphisms\nin dimension $m=2$.\n\nThe persistent homoclinic tangencies obtained in the above theorem\ncan be associated with a finite collection of saddle periodic\npoints, $Q_{ja}$, having unstable index $c$ and\nthe same type of multipliers. For instance, we can take these\npoints to be sectionally dissipative (as in the original\nconstruction of Berger) but also these saddles can be of\ntype~$(1,1)$, $(2,1)$, $(1,2)$ or $(2,2)$ according to the\nnomenclature introduced in~\\cite{GST08}.\nWe remark that in the codimension one case we may assume the homoclinic tangencies are simple\\footnote{The tangency is called simple if it is\nquadratic, of codimension one and in the case the ambient manifold has dimension\n$m>3$, any extended unstable manifold is transverse to the leaf of\nthe strong stable foliation which passes through the tangency\npoint. Observe that these conditions are generic.}, also in the sense of ~\\cite{GST08}.\n\n\n\n\n\n\\subsection{Berger phenomena:}\nThe $C^{d,r}$-Berger phenomena was shown in~\\cite{Ber16,Ber17} for\nsectionally dissipative families in dimension\n$m\\geq 3$. We will obtain similar results for\nfamilies that are not sectionally dissipative\nby working with a\n$C^{d,r}$-Berger domain\n$\\mathscr{U}$ of type $(2,1)$ with unstable index one. That is,\nthe persistent homoclinic tangencies are simple and associated with\nhyperbolic periodic points having multipliers\n$\\lambda_1,\\dots,\\lambda_{m-1}$ and $\\gamma$ satisfying\n\\begin{equation} \\label{eq1}\n   |\\lambda_j|<|\\lambda|<1<|\\gamma| \\ \\ \\ \\text{and \\ \\ \\ $|\\lambda^2\\gamma|<1<|\\lambda\\gamma|$} \\quad \\text{for $j\\not= 1,2$ where\n   $\\lambda_{1,2}=\\lambda e^{\\pm i \\varphi}$ with $\\varphi\\not=0,\\pi$.}\n\\end{equation}\nIn the following result we obtain Berger phenomena with respect to\nattracting invariant circles and hyperbolic sinks\nfor these new types of\nBerger domains.\n\n\n\\begin{mainthm} \\label{thm-thmA} Let $\\mathscr{U}$ be a\na $C^{d,r}$-Berger domain whose persistent homoclinic tangencies\nare simples and associated with hyperbolic periodic points having multipliers\nsatisfying~\\eqref{eq1}.  Then there exists a residual set\n$\\mathcal{R}\\subset \\mathscr{U}$ such that for every family\n$f=(f_a)_a \\in \\mathcal{R}$ and every $a\\in \\mathbb{I}^k$, the\ndiffeomorphism $f_a$ has simultaneously\n\\begin{enumerate}\n\\item[-] infinitely many normally hyperbolic attracting invariant circles and\n\\item[-] infinitely many hyperbolic periodic sinks\n\\end{enumerate}\n\\end{mainthm}\n\n\n\n\n\n\n\\subsection{Topology of families of diffeomorphisms}\n\\label{sec:topology} Set $\\mathbb{I}=[-1,1]$. Given $0<d \\leq\nr\\leq \\infty$, $k\\geq 1$ and a compact manifolds $\\mathcal{M}$ we\ndenote by $C^{d,r}(\\mathbb{I}^k,\\mathcal{M})$ the space of\n$k$-parameter $C^d$-families $f=(f_a)_a$ of $C^r$-diffeomorphisms\n$f_a$ of $\\mathcal{M}$ parameterized by $a\\in\\mathbb{I}^k$ such\nthat\n\\begin{equation*}\n  \\partial^i_a \\partial^j_x f_a(x) \\ \\ \\text{exists continuously\n  for all $0\\leq i\\leq d$, \\ \\   $0\\leq i+j\\leq r$  \\ \\  and \\ \\\n   $(a,x)\\in \\mathbb{I}^k\\times \\mathcal{M}.$}\n\\end{equation*}\nWe endow this space with the topology given by the $C^{d,r}$-norm\n$$\n\\|f\\|_{{C}^{d,r}}=\\max\\{\\sup \\|\\partial^i_a\\partial_x^j f_a (x):\n\\, 0\\leq i \\leq d, \\  0\\leq i + j \\leq r\\} \\quad \\text{where \\\n$f=(f_a)_a \\in {C}^{d,r}(\\mathbb{I}^k,\\mathcal{M})$.}\n$$\n\n\\subsection{Structure of the paper}\nSection~\\S\\ref{sec:berger-domain} contains the proof of\nTheorem~\\ref{thmD}. Independently\nin section~\\S\\ref{sec:thm-thmA} we prove Theorem~\\ref{thm-thmA}.\nActually, the proof of Theorem~\\ref{thm-thmA} only requires\nDefinition~\\ref{def:Berger-domain}.\n\n\\section{Berger domains: Proof of Theorem~\\ref{thmD}}\n\\label{sec:berger-domain} In this section we will prove the\nexistence of $C^{d,r}$-Berger domains\nof\ncodimension $u>0$ for families of diffeomorphisms with $d<r-1$ and\nmanifolds of dimension $m> u^2+u \\geq 2$ (see \\S\\ref{BergerDomain}\nand Definition~\\ref{def:Berger-domain}).\n\nNow we will introduce the family that will be ``the organizing\ncenter'' of Berger domains. To do this, we need some results\nfrom~\\cite{BR20}. In~\\cite[Thm.~B]{BR20} we construct an open set\n$\\mathscr{U}\\subset C^{d,r}(\\mathbb{I}^k,\\mathcal{M})$\nfor $0<d<r-1$  and $\\dim \\mathcal{M} \\geq 3$ where any family\n$f=(f_a)_a\\in \\mathscr{U}$ has a $C^d$-degenerate unfolding of a\nhomoclinic tangency of codimension $u$ at $a=0$ (actually at any\nparameter $a_0\\in \\mathbb{I}^k$). The construction of this open\nset is local and only requires two ingredients: a family of\nblenders (a certain type of a hyperbolic basic set)\n $\\Gamma=(\\Gamma_a)_a$ and a family of folding manifolds\n$(S_a)_a$ (a certain type of manifold that folds along some\ndirection). We refer to~\\cite{BR20} for a precise definition of\nthese objets.  To be more specific, the main result could be\nstated as follows:\n\n\\begin{thm}[{\\cite[Thm.~7.5, Rem.~7.6]{BR20}}] \\label{thm-BR20}\nFor any $0<d<r-1$ and $k\\geq 1$, there exists a $C^d$-family\n$\\Phi=(\\Phi_a)_a$ of locally defined $C^r$-diffeomorphisms of\n$\\mathcal{M}$ of dimension $m>u+u^2$ having a family of\n$cs$-blenders $\\Gamma=(\\Gamma_a)_a$ with unstable dimension $u\\geq\n1$\nand a family of folding manifolds $\\mathcal{S}=(\\mathcal{S}_a)_a$\nof dimension $m-u$ satisfying the following:\n\nFor any $a_0\\in \\mathbb{I}^k$, any family $g=(g_a)_a$ close enough\nto $f$ in the $C^{d,r}$-topology and any $C^{d,r}$-perturbation\n$\\mathcal{L}=(\\mathcal{L}_a)_a$ of $\\mathcal{S}$ there exists\n$z=(z_a)_a\\in C^d(\\mathbb{I}^k,\\mathcal{M})$ such that\n {\n\\begin{enumerate}\n\\item $z_a\\in\\Gamma_{g,a}$, where $\\Gamma_{g,a}$ denotes the\ncontinuation for $g_a$ of the blender $\\Gamma_a$,\n\\item the family of local unstable manifolds\n$\\mathcal{W}=(W^u_{loc}(z_a;g_a))_a$ and $\\mathcal{L}$ have a\ntangency\n of dimension $u$ at $a=a_0$ which unfolds $C^d$-degenerately.\n\\end{enumerate}\n}\n\\end{thm}\n\n\n\nLet us consider the family $\\Phi=(\\Phi_a)_a$  given in the above\ntheorem.  Assume in addition the next hypothesis:\n\\begin{enumerate}[label=(H\\arabic*)]\n\\item \\label{H1}\n$\\Phi_a$ has a equidimensional cycle between\nsaddle periodic points $P_a$ and $Q_a$,\n\\item \\label{H2} $P_a$ belongs to $\\Gamma_a$ and the folding manifold $S_a$\nis contained in $W^s(Q_a,\\Phi_a)$.\n\\end{enumerate}\nTheorem~\\ref{thm-BR20} implies that the family $\\Phi=(\\Phi_a)_a$\nunder the above assumptions~\\ref{H1} and~\\ref{H2} defines a\n$C^{d,r}$-open set $\\mathscr{U}=\\mathscr{U}(\\Phi)$ of\n$k$-parameter $C^d$-families of $C^r$-diffeomorphisms\nsuch that any $g=(g_a)_a \\in \\mathscr{U}$ is a $C^d$-degenerate\nunfolding at any parameter $a=a_0$ of a tangency of dimension $u$.\nThe tangency is between $W^s(Q_{a_0},g_{a_0})$ and the local\nunstable manifold of some point in the blender $\\Gamma_{a_0}$ of\n$g_{a_0}$.\nSince the codimension of $W^s(Q_{a_0},g_{a_0})$ and the\ndimension of the local unstable manifolds of $\\Gamma_{a_0}$\ncoincide, the tangency also has codimension $u$. We will prove\nthat the open set $\\mathscr{U}$ is a $C^{d,r}$-Berger domain. To\ndo this, we will first need the following result,\nsee~\\cite[Lemma~3.7]{Ber16} and~\\cite[Lemma~3.2]{Ber17}.\n\n\n\n\\begin{prop}[Parametrized Inclination Lemma] \\label{thm-Parametrized-Inclination-Lemma}\nLet $g=(g_a)_a$ be a $C^{d,r}$-family of diffeomorphisms having a\nfamily $K=(K_a)_a$ of transitive hyperbolic sets $K_a$ with\nunstable dimension $d_u$. Let $C_a$ be a $C^r$-submanifold of\ndimension $d_u$ that intersects transversally a local stable\nmanifold $W^s_{loc}(x_a,g_a)$ with $x_a\\in K_a$ at a point $z_a$\nwhich we assume depends $C^d$-continuously on $a  \\in\n\\mathbb{I}^k$. Then, for any $P_a\\in K_a$\nthere exists a $d_u$-dimensional disc $D_a \\subset C_a$\ncontaining $z_a$ such that the family of discs $D=(g^n_a(D_a))_a$\nis $C^{d,r}$-close to $W=(W^u_{loc}(P_a,g_a))_a$, for $n$ sufficiently large.\n\\end{prop}\n\n\n\n\nUsing the parameterized inclination lemma, the following\nproposition proves that the above open set $\\mathscr{U}$ is a\nBerger domain according to the first tentative (weaker) definition given\nin~\\S\\ref{BergerDomain}.\n\n\\begin{prop} \\label{prop-dense-of-paratangencies}\nFor any $a_0\\in \\mathbb{I}^k$ and $g\\in \\mathscr{U}$, there is\n$C^{d,r}$-arbitrarily close to $g$ a family $f=(f_a)_a$ such that\n$f_{a}=g_a$ for any parameter far from a small neighborhood of\n$a_0$ and {which displays a $C^d$-degenerate unfolding at $a=a_0$\nof a homoclinic tangency  of codimension $u$ associated with the\nperiodic point $Q_{a_0}(f)$.}\n\\end{prop}\n\\begin{proof}\nBy construction, any $g=(g_a)_a\\in \\mathscr{U}$ is a\n$C^d$-degenerate unfolding at $a=a_0$ of a tangency between\n$W^s(Q_{a_0},g_{a_0})$ and some local unstable manifold\n$W^u_{loc}(x_{a_0},g_{a_0})$ of a point $x_{a_0}\\in \\Gamma_{a_0}$.\nFrom the assumptions~\\ref{H1} and~\\ref{H2} we have that both $x_a$\nand $Q_a$ belongs to the homoclinic class $H(P_a,g_a)$ of $P_a$\nfor $g_a$. Moreover, we get that $W^{u}(Q_a,g_a)$ intersects\ntransversally $W^s_{loc}(x_a,g_a)$ at a point $z_a$ which depends\n$C^d$-continuously on $a\\in\\mathbb{I}^k$. Then\nProposition~\\ref{thm-Parametrized-Inclination-Lemma} implies the\nexistence of discs $D_a$ in $W^u(Q_a,g_a)$ containing $z_a$ such\nthat the family $D_n=(g^n_a(D_a))_a$ is $C^{d,r}$-close to\n$W=(W^u_{loc}(x_a,g_a))$ when $n$ is large.\nBy a small perturbation, we now will find a new family\n$C^{d,r}$-close to $g$, which is a $C^d$-degenerate unfolding at\n$a=a_0$ of a homoclinic tangency of codimension $u$ associated\nwith the continuation of the periodic point $Q_{a_0}$.\n\n\nWe  take local coordinates denoted by $x$ in a neighborhood of\n$x_{a_0}$\nwhich correspond to the origin.\nAlso denote by $y$ the tangency point between\n$W^s(Q_{a_0},g_{a_0})$ and the local unstable manifold\n$W^u_{loc}(x_{a_0},g_{a_0})$.\nSince the tangency (of dimension $u$) unfolds $C^d$-degenerately,\nwe have $\\vec{p}=(p_a,E_a)_a, \\vec{q}=(q_a,F_a)_a\\in\nC^d(\\mathbb{I}^k,G_u(\\mathcal{M}))$  such that\n\\begin{align*}\nq_a \\in W^s(Q_{a},g_{a}), \\quad &\\text{and} \\quad E_a \\subset\nT_{q_{a}}W^s(Q_{a},g_{a}), \\\\\np_a \\in W^u_{loc}(x_{a},g_{a}) \\quad &\\text{and} \\quad  F_a\n\\subset\nT_{p_a}W^u_{loc}(x_{a},g_{a}), \\\\\np_{a_0}=q_{a_0}=y \\quad &\\text{and} \\quad J(\\vec{p})=J(\\vec{q}).\n\\end{align*}\nTake $\\delta>0$ such that, in these local coordinates, the\n$2\\delta$-neighborhoods of $y$ and its iterations by $g_{a_0}$ and\n$g^{-1}_{a_0}$ are pairwise disjoint. Let\n$U$ be the $2\\delta$-neighborhood of $y$\nand assume that $p_a$ and $q_a$ belong to $U$ for all $a$ close\nenough to $a_0$. Call $C_a$ the local disc in $W^s(Q_{a},g_{a})$\ncontaining the point $q_a$ and we may suppose that $U$ is such that\nthe forward iterates of $C_a$, with respect to $g_a$,\nare disjoint from each other and from $U$.\nSince $D_n=(g^n_a(D_a))_a$ is $C^{d,r}$-close to $W$, we obtain a\n$C^{d,r}$-family $\\tau_n=(\\tau_{n,a})_a$ of diffeomorphisms of\n$\\mathbb{R}^m$ which sends, in local coordinates,\n$g^n(D_a)$ onto $W^u_{loc}(x_a,g_a)$,\nis equal to the identity\noutside of $U$ and is $C^{d,r}$-close to the constant family\n$I=(\\mathrm{id}_{\\mathbb{R}^m})_a$ as $n\\to \\infty$.\nLet $t_a$ be the point in $D_n\\subset W^u(Q_{a},g_{a})$ so that $\\tau_{n,a}(t_a)=p_a$.\n\nConsider a $C^\\infty$-bump function $ \\phi:\\mathbb{R}\\to\n\\mathbb{R}$ with support in $[-1,1]$ and equal to 1 on\n$[-1\/2,1\/2]$. Let\n$$\n\\rho: a=(a_1,\\dots,a_k) \\in \\mathbb{I}^k \\mapsto\n\\phi(a_1)\\cdot\\ldots\\cdot \\phi(a_k) \\in \\mathbb{R}.\n$$\nFor a fixed $\\alpha>0$, define  the perturbation $g_n=(g_{n,a})_a$\nof $g=(g_a)_a$ by\n$$\n  g_{n,a}= H_{n,a} \\circ g_a \\quad \\text{for all $a\\in \\mathbb{I}^k$,}\n$$\nwhere $H_{n,a}$  in the above local coordinates takes the form\n\\begin{align*}\n  H_{n,a}(x)=x+ \\theta \\cdot (\\tau_{n,a}(x)-x)\n  \\quad \\text{where} \\ \\\n  \\theta=\\rho\\left(\\frac{a-a_0}{2\\alpha}\\right)\\phi\\left(\\frac{\\|x-y\\|}{2\\delta}\\right)\n\\end{align*}\nand is the identity otherwise. Observe that if $a\\not\\in a_0 +\n(-2\\alpha,2\\alpha)^k$ or $x\\not\\in U$ then $H_{n,a}(x)=x$. In\nparticular, $g_{n,a}(x)=g_a(x)$ if $a\\not\\in a_0 +\n(-2\\alpha,2\\alpha)^k$ or $x\\not\\in g_a^{-1}(U)$.\n\nOn the other hand, if $a\\in a_0 + (-\\alpha,\\alpha)^k$ and $x\\in\nU$, then $H_{n,a}(x)=\\tau_{n,a}(x)$. This implies that for $a\\in\na_0+(-\\alpha,\\alpha)^k$, the point $g_a^{-1}(t_a)$ that belongs to\n$W^u(Q_a,g_a)$ is sent by $g_{n,a}$ to $p_a=\\tau_{n,a}(t_a)$ and\ntherefore $p_a \\in W^u(Q_a,g_{n,a})$. Moreover, since $\\tau_{n,a}$\nis a $C^r$-diffeomorphism ($r\\geq 2$) we also have that $F_a\n\\subset T_{p_a}W^u(Q_a,g_{n,a})$.\n\n\nAt $a=a_0$ we have that $\\tau_{n,a_0}(t_{a_0})=p_{a_0}=q_{a_0}\\in\nW^s(Q_{a_0}, g_{n,a_0})$ and so the stable and unstable manifolds\nof $Q_{a_0}$ for $g_{n,a_0}$ meet at this point. Moreover, since\n$\\tau_{n,a_0}$ is a $C^r$-diffeomorphism ($r\\geq 2$) this\nintersection is still non-transverse, i.e., we have a homoclinic\ntangency of codimension~$u$. Observe that the perturbed family\n$g_{n,a}$ does not affect the disc $C_{a}$ of the stable manifold\nof $W^s(Q_a, g_a)$ in $U$. That is, $C_{a}\\subset W^s(Q_a,\ng_{n,a})$ and $E_a \\subset T_{q_a}W^s(Q_a,g_{n,a})$.  Hence, since\nfrom the initial hypothesis $J(\\vec{p})=J(\\vec{q})$  we get  a\n$C^d$-degenerate unfolding at $a=a_0$ of a homoclinic tangency of\ncodimension $u$ associated with  the hyperbolic periodic point\n$Q_{a_0}(g_n)$. Finally, observe\n$$\n  \\|f-g_n\\|_{C^{d,r}}=\\|(I-H)\\circ f\\|_{C^{d,r}} \\leq \\|f\\|_{C^{d,r}}\n  \\|\\theta (\\tau_n -\n  I)\\|_{C^{d,r}} \\leq \\|f\\|_{C^{d,r}} o_{\\alpha\\to 0}(\\alpha^{-d}) o_{\\delta\\to\n  0}(\\delta^{-r}) \\|\\tau_n-I\\|_{C^{d,r}}.\n$$\nSince $\\|\\tau_n-I\\|_{C^{d,r}}$ goes to zero as $n \\to \\infty$, we\ncan obtain that for a given $\\epsilon>0$, there are $n$ large\nenough so that $\\|f-g_n\\|_{C^{d,r}}<\\epsilon$.\n\\end{proof}\n\n\\begin{rem} \\label{rem-finite-points}\nNotice that the perturbation in the previous proposition,\nto create the degenerate homoclinic  unfolding at $a_0$, is local\n(in the parameter space and in the manifold). Thus, fixing\n$N\\in\\mathbb{N}$ and a finite number of points $a_1,\\dots,a_N\\in\n\\mathbb{I}^k$, we can perform the same type of perturbation\ninductively and obtain a\ndense set of families in $\\mathscr{U}$ having degenerate unfoldings at any\n$a=a_i$ for\n$i=1,\\dots,N$.\n\\end{rem}\n\n\n\n\n\nThe following proposition is an adaptation to the context of\ndiffeomorphisms of~\\cite[Lemma~5.4]{Ber16}. Roughly speaking, this\nproposition explains how it is possible to ''stop'' a tangency for\nan interval of parameters using a degenerate unfolding,\ni.e, how to create a persistent homoclinic\ntangency in the language of~\\cite{Ber17}.\n\n\\begin{prop} \\label{prop-paratangecia}\nLet $f=(f_a)_a$ be a $k$-parameter $C^d$-family of\n$C^r$-diffeomorphisms of a manifold of dimension $m\\geq 2$.\nSuppose that $f$ is a $C^d$-degenerate unfolding at $a=a_0$ of a\nhomoclinic tangency of codimension $u>0$  associated with a\nhyperbolic periodic point $Q$. Then, for any $\\epsilon>0$ there\nexists $\\alpha_0>0$ such that for every $0<\\alpha<\\alpha_0$ there\nis a $C^{d}$-family $h=(h_a)_a$ of $C^r$-diffeomorphisms such~that\n\\begin{enumerate}\n\\item $h$ is $\\epsilon$-close to $f$ in the $C^{d,r}$-topology,\n\\item $h_a=f_a$ for every $a \\not \\in a_0 + (-2\\alpha,2\\alpha)^k$,\n\\item $h_a$ has a homoclinic tangency $Y_a$ of codimension $u$ associated with the\ncontinuation  $Q_a$ of $Q$ for all $a\\in a_0+ (-\\alpha,\\alpha)^k$\nand which depend $C^d$-continuously on the parameter $a$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nBy assumption $f$ is a $C^d$-degenerate unfolding at $a=a_0$ of a\nhomoclinic tangency $Y$ associated with a hyperbolic periodic\npoint $P$. By the definition of degenerate unfolding we have\npoints $p_a\\in W^u(Q_a,f_a)$, $q_a\\in W^s(Q_a,f_a)$ and\n$c$-dimensional subspaces $E_a$ and $F_a$ of $T_{p_a}W^u(Q_a,f_a)$\nand $T_{q_a}W^s(Q_a,f_a)$ respectively such that\n\\begin{equation} \\label{eq:o(ad)}\n d(p_a,q_a)=o(\\|a-a_0\\|^{d}) \\ \\ \\text{and} \\ \\\nd(E_a,F_a)={o(\\|a-a_0\\|^{d})} \\ \\ \\text{at $a=a_0$}.\n\\end{equation}\nHere $Q_a$ is the continuation of $Q_{a_0}=P$ for $f_a$. Also\n$p_{a_0}=q_{a_0}=Y$ and $(p_a,E_a)$, $(q_a,F_a)$ vary\n$C^d$-continuously with respect to the parameter $a\\in\n\\mathbb{I}^k$. We  take local coordinates $x$  in a neighborhood\nof $Q$ which correspond to the origin.\nBy considering an iteration if necessary, we\nassume that the tangency point $Y$ belongs to\nthis neighborhood of local coordinates. Take $\\delta>0$ so that\nthe $2\\delta$-neighborhoods of $Y$ and its iterations by $f_{a_0}$\nand $f^{-1}_{a_0}$ are pairwise disjoint.\nNamely, we will denote by $U$ the $2\\delta$-neighborhood of $Y$.\nAssume that\n$p_a$ and $q_a$ belong to $U$ for all $a$ close enough to $a_0$.\nFrom~\\eqref{eq:o(ad)} it follows that\n\\begin{equation} \\label{eq:orden1}\n  p_a-q_a=o(\\|a-a_0\\|^d) \\quad \\text{at \\  $a=a_0$.}\n\\end{equation}\nObserve that the Grasmannian distance between $E_a$ and $F_a$\n is given by the norm of $I-R_a$ restricted\nto $F_a$, where $I$ denotes the identity and $R_a$ is the\northogonal projection onto $E_a$. Then we obtain\nfrom~\\eqref{eq:o(ad)} that\n\\begin{equation} \\label{eq:orden2}\nI-R_a =o(\\|a-a_0\\|^d) \\quad \\text{at \\ $a=a_0$}.\n\\end{equation}\nWe would like that the rotation occurs around the point $p_a$\nand so consider\n$\\tilde{R}_a x =R_a (x-p_a) + p_a$.\nSince $(I-\\tilde{R}_a)x=(I-R_a)x+(R_a-I)p_a$ then\nfrom~\\eqref{eq:orden2} we still have\n\\begin{equation} \\label{eq:orden3}\n I-\\tilde{R}_a =o(\\|a-a_0\\|^d) \\quad \\text{at \\ $a=a_0$}.\n\\end{equation}\n\nConsider a $C^\\infty$-bump function $ \\phi:\\mathbb{R}\\to\n\\mathbb{R}$ with support in $[-1,1]$ and equal to 1 on\n$[-1\/2,1\/2]$. Let $$\\rho: a=(a_1,\\dots,a_k) \\in \\mathbb{I}^k\n\\mapsto \\phi(a_1)\\cdot\\ldots\\cdot \\phi(a_k) \\in \\mathbb{R}.\n$$\nFor a fixed $\\alpha>0$, define the perturbation $h=(h_a)_a$ of\n$f=(f_a)_a$ by the relation\n$$\n  h_{a}= H_{a} \\circ f_a \\quad \\text{for all $a\\in \\mathbb{I}^k$}.\n$$\nHere $H_{a}$  in the above local coordinates takes the form\n\\begin{align*}\n  \\bar{x}=\\big(I-\\theta\\cdot(I-\\tilde{R}_a)\\big)\\big(x- \\theta \\cdot\n  (q_a-p_a)\\big) \\quad \\text{where} \\ \\ \\theta=\\rho\\left(\\frac{a-a_0}{2\\alpha}\\right)\\phi\\left(\\frac{\\|x-Y\\|}{2\\delta}\\right)\n\\end{align*}\nand is the identity otherwise. Observe that if $a\\not\\in a_0 +\n(-2\\alpha,2\\alpha)$ or $x\\not\\in U$ then $H_a(x)=x$. In\nparticular, $h_a(x)=f_a(x)$ if $a\\not\\in a_0 + (-2\\alpha,2\\alpha)$\nor $x\\not\\in f_a^{-1}(U)$. On the other hand, if $a\\in a_0 +\n(-\\alpha,\\alpha)$ and $x\\in U$ then\n$$H_a(x)=\\tilde{R}_a\\big(x-(q_a-p_a)\\big).$$\nSince $\\tilde{R}_a$ fixes the point $p_a$, we have that\n$H_a(q_a)=p_a$. This implies that the point\n$q_a^{-1}=f_a^{-1}(q_a)$ that belongs to $W^u(Q_a,f_a)$ is sent by\n$h_a$ to $p_a$ which belongs to $W^s_{loc}(Q_a,f_a)$. As the orbit\nof $f^{n}_a(p_a)$ for $n\\geq 0$ and $f_a^{-n}(q^{-1}_a)$ for $n>0$\nnever goes through $f_a^{-1}(U)$, then $p_a \\in\nW^s_{loc}(P_a,h_a)$ and $q_a^{-1}\\in W^u(Q_a,h_a)$. Thus, the\nstable and unstable manifolds of $Q_a$ for $h_a$ meet at\n$p_a=h_a(q_a^{-1})$. Moreover, $F_a^{-1}=Df_a(q^{-1}_a)F_a$ is a\nsubspace of $T_{q_a^{-1}}W^u(Q_a,f_a)=T_{q_a^{-1}}W^u(Q_a,h_a)$\nand\n$$\nDh_a(q_a^{-1})F_a^{-1}=\nDH_a(q_a)Df_a(q_a^{-1})F_a^{-1}=DH_a(q_a)F_a=D\\tilde{R}_a(q_a)\nF_a= R_aF_a=E_a.\n$$\nSince $E_a$ is a subspace of\n$T_{p_a}W^s(Q_a,f_a)=T_{p_a}W^s(Q_a,h_a)$, then the intersection\nbetween the stable and unstable manifolds of $Q_a$ for $h_a$ is\ntangencial.\n\nTo conclude the proposition we only need to prove that for a given\n$\\epsilon>0$ we can find $\\alpha_0$ such that for any\n$0<\\alpha<\\alpha_0$ the above perturbation $h=h(\\alpha)$ of $f$ is\nactually $\\epsilon$-close in the $C^{d,r}$-topology. Notice that\nthe $C^{d,r}$-norm satisfies\n$$\n \\|h-f\\|_{C^{d,r}}=\\|(H-I) \\circ f \\|_{C^{d,r}}  \\leq\n  \\|I-H\\|_{C^{d,r}} \\, \\|f\\|_{C^{d,r}}.\n$$\nThus we only need to calculate the $C^{d,r}$-norm of the family\n$(I-H_{a})_a$. Since $H_{a}=I$ if $a\\not \\in a_0 +\n(-2\\alpha,2\\alpha)^k$ or $x\\not \\in U$ then\n\\begin{align*}\n  \\|I-H_{a}\\|_{C^{d,r}}\\leq\\left\\| \\theta \\cdot (I-\\tilde{R}_a)(x-\\theta\\cdot (q_a-p_a))+\\theta\\cdot (q_a-p_a)\\right\\|_{C^{d,r}}.\n\\end{align*}\nSince the $C^{d,r}$-norm of $\\phi(\\|x-Y\\|\/2\\delta)$ is bounded\n(depending only on $\\delta$), we can disregard this function from\nthe estimate. Then, to bound the $C^{d,r}$-norms from above it is\nenough to show that for $a \\in  a_0+(-2\\alpha,2\\alpha)^k$ the\nfunctions\n$$\n  F_{\\alpha}(a)= \\rho(\\frac{a-a_0}{2\\alpha})(p_a-q_a) \\quad\n  \\text{and} \\quad G_{\\alpha}(a)  = \\rho(\\frac{a-a_0}{2\\alpha})(I-\\tilde{R}_a)\n$$\nhave $C^d$-norm small when $\\alpha$ is small enough. But this is\nclear from~\\eqref{eq:orden1} and \\eqref{eq:orden3}, as having into\naccount that $\\|a-a_0\\|\\leq \\alpha$, it follows that\n$$\n F_a(a)=\\alpha^{-d}\\cdot o_{a\\to a_0}(\\|a-a_0\\|^d)=o_{\\alpha\\to 0}(1) \\quad \\text{and}\n \\quad G_\\alpha(a)=\\alpha^{-d}\\cdot o_{a\\to a_0}(\\|a-a_0\\|^d)=o_{\\alpha \\to 0}(1).\n$$\nThis completes the proof.\n\\end{proof}\n\n\\begin{rem} \\label{rem-paratangency}\nObserve that the positive constant $\\alpha_0$ in\nthe above proposition depends initially on the family $f=(f_a)_a$,\nthe constants $\\epsilon>0$, $\\delta>0$ and the parameter $a_0\\in \\mathbb{I}^k$.\nThe dependence of $a_0$ comes from the function $o_{a\\to\na_0}(\\|a-a_0\\|^d)$ in~\\eqref{eq:o(ad)}. However, one can bound\nthis function by $\\nu(\\|a-a_0\\|)\\cdot \\|a-a_0\\|^d$ where\n$\\lim_{t\\to 0}\\nu(t)=0$, controlling the modulus of continuity\n$\\nu$ of the derivatives of the unfolding. In this form we can get\nthat $\\alpha_0$ does not depend on the parameter $a_0$. Also,\nsimilarly to what was done in the previous proposition, the surgery\nusing bump functions around a neighborhood of\nthe initial paratangency point $Y$ can actually be done around any point\n$f^N_{a_0}(Y)$\nbelonging to $W^s_{loc}(P_{a_0},f_{a_0})$.\nThis allows us to fix a\npriori a uniform $\\delta>0$ because we only need to control\nthe distance between one forward\/backward iterate of $f^N_{a_0}(Y)$.\nThus,\n$\\alpha_0$ also does not depend on\n$\\delta$. Finally, if\n$f$ belongs to $\\mathscr{U}$\nthen one can obtain a uniform bound on the continuity modulus\nusing the fact that we are dealing with compact families of local\nstable and unstable manifolds. This proves that in this construction\n$\\alpha_0$ only depends on $\\epsilon$ and $\\mathscr{U}$ (i.e, on\nthe dynamics of the organizing family $\\Phi$).\n\\end{rem}\n\n\\begin{rem} \\label{rem:simples}\nIn the case of codimension $u=1$, the tangency $Y_a$ obtained in\nthe previous proposition could be assumed \\emph{simple} in the\nsense of~\\cite{GST08}.\n\\end{rem}\n\nFinally, in the next theorem we will show the\nexistence of Berger domains as stated in\nDefinition~\\ref{def:Berger-domain}. The idea behind the proof is\nthe replication of the arguments coming from~\\cite[Sec.~6.1]{Ber16}\nand~\\cite[Sec.~7]{Ber17}.\n\\begin{thm} \\label{thm-berger-domain}\nThere exists a dense subset $\\mathscr{D}$ of $\\mathscr{U}$ such\nthat for any $h=(h_a)_a \\in \\mathscr{D}$ there is a covering of\n$\\mathbb{I}^k$ by open sets $J_i$\nhaving a persistent  homoclinic tangency\nof codimension $u$. That is, $\\mathscr{U}$ is a\n$C^{d,r}$-Berger domain (of paratangencies of codimension $u$).\n\\end{thm}\n\\begin{proof}\nFora a fixed family $g=(g_a)_a\\in \\mathscr{U}$ and $\\epsilon>0$,\nPropositions~\\ref{prop-dense-of-paratangencies},~\\ref{prop-paratangecia}\nand Remarks~\\ref{rem-finite-points},~\\ref{rem-paratangency} imply\nthe following. We obtain $\\alpha_0=\\alpha_0(\\epsilon)>0$ so that for a fixed\n$\\alpha$ with $0<\\alpha<\\alpha_0$ there are points $a_1,\\dots,a_N$ in\n$\\mathbb{I}^k$ such that\n\\begin{itemize}\n \\item[-] the open sets $a_i+(-2\\alpha,2\\alpha)^k$ for $i=1,\\dots,N$\n are pairwise disjoint;\n \\item[-] the union of $(a_i+(-2\\alpha,2\\alpha)^k)\\cap \\mathbb{I}^k$\n for $i=1,\\dots,N$ is dense in $\\mathbb{I}^k$;\n\\item[-] there is a $C^{d,r}$-family\n $h=(h_a)_a$, $\\epsilon$-close to $g$, having a persistent homoclinic tangency associated with the continuation of $Q_a$\n for all $a\\in (a_i+(\\alpha,\\alpha)^k)\\cap \\mathbb{I}^k$ and $i=1,\\dots,N$.\n\\end{itemize}\nHowever this result does not provide an open cover of\n$\\mathbb{I}^k$. We need to perturbe again $h$ without destroying\nthe persistent homoclinic tangencies associated with $Q_a$ and\nat the same time provide new persistent homoclinic tangencies in the complement\nof the union of $(a_i+(\\alpha,\\alpha)^k)\\cap \\mathbb{I}^k$\nfor $i=1,\\dots,N$.\nTo do this, we will need a finite set of different\nperiodic points $Q^j_a$ to replicate the above argument and ensure\nthat each new perturbation does not modify the previous one (i.e, there\nexists a common $\\delta>0$ so that the supports of all the peturbations\nare disjoint).\n\\begin{lem}\nThere exists a set $\\{Q^j_a\\}_{j=1,\\dots, 3^k}$ of $3^k$\nhyperbolic periodic points  satisfying the assumptions~\\ref{H1} and~\\ref{H2}.\n\\end{lem}\n\\begin{proof}\nSince the\nhomoclinic class  $H(Q_a,\\Phi_a)$ is not trivial (contains $P_a$),\nthere exists a horseshoe $\\Lambda_a$ containing $Q_a$. Thus,\nthere are infinitely many different hyperbolic periodic points of $\\Phi_a$\nwhose stable manifold intersect transversely the unstable\nmanifold of $Q_a$. Then by the inclination lemma, and the\nrobustness of the folding manifold $S_a$, the stable manifold of\nthese periodic points also contains a folding manifold.\n\\end{proof}\nAssociated with each point $Q^j_a$, as in Proposition~\\ref{prop-paratangecia},\nthere is the paratangency point $Y_j$ where the size of the perturbation is governed by\n$\\delta_j$. Thus, we can obtain a uniform $\\delta$ by taking the infimum over $\\delta_j$.\nAlso corresponding to each $Q^j_a$, there exists a lattice of points\n$\\{a_i^j\\}_{i=1,\\dots, N}$ in $\\mathbb{I}^k$ such that\nthe union of the $\\alpha$-neighborhoods $a_i^j+(\\alpha,\\alpha)^k$\nin $\\mathbb{I}^k$ cover $\\mathbb{I}^k$.\nThat is,\n$$\\mathbb{I}^k\\subset \\bigcup_{j=1}^{3^k}\\bigcup_{i=1}^N a_i^j+(\\alpha,\\alpha)^k.$$\nThen we can apply Proposition~\\ref{prop-paratangecia} independently in $j$ to obtain\nthe family with the required properties, concluding the proof the the theorem.\n\\end{proof}\n\\begin{rem}\nThe persistent homoclinic tangencies in the open set $J_i$\nobtained in the above theorem can be associated with a collection\nof saddle periodic points, $Q_a^j$ for $j=1,\\dots,3^k$, where all\nof them have the same type of multipliers. For instance, we can\ntake these points being sectionally dissipative or of type~$(1,1)$,\n$(2,1)$, $(1,2)$ or $(2,2)$ according to the nomenclature\nintroduced in~\\cite{GST08}.\nAlso, according\nto Remark~\\ref{rem:simples}, in the codimension one case, the\npersistent homoclinic tangency can be assumed \\emph{simple} in the\nsense of~\\cite{GST08}.\n\\end{rem}\n\n\n\\section{Proof of Theorem~\\ref{thm-thmA}:\n periodic sinks and invariant circles} \\label{sec:thm-thmA}\nIn this section we will prove the $C^{d,r}$-Berger phenomenena of\nthe coexistence of infinitely many normally hyperbolic attracting\ninvariant circles and also obtain a similar result for hyperbolic periodic sinks.\nFor short, we will refer to\nboth of these types of attractors as \\emph{periodic attractors}.\n\nThe next proposition claims that every family $f=(f_a)_a$ in\n$\\mathscr{U}$ can be approximated by a family $g=(g_a)_a$ having a\nperiodic attractor for every parameter $a$ in $\\mathbb{I}^k$.\nMoreover, the period of the attractors can be chosen arbitrarily\nlarge. To prove this, since $\\mathscr{U}$ is a Berger domain, it\nis enough to restrict our attention to the dense set $\\mathcal{D}$\nof $\\mathscr{U}$ having persistent tangencies.\n\n\\begin{prop} \\label{main-lema}\nFor any $\\epsilon>0$, and every $f=(f_a)_a  \\in \\mathcal{D}$ there\nexists $n_0=n_0(\\varepsilon,f)\\in\\mathbb{N}$ such that for any\n$n\\geq n_0$ there is a $\\epsilon$-close family $g=(g_a)_a$ to\n$f=(f_a)_a$ in the $C^{d,r}$-topology satisfying that $g_a$ has a\nperiodic attractor of period $n$  for all $a\\in \\mathbb{I}^k$.\nMoreover, the attractor obtained for $g_a$ is the continuation of\na (hyperbolic or normally hyperbolic) $n$-periodic attractor\nobtained for a map $g_{a_0}$, where $a_0$ belongs to a finite\ncollection of parameters.\n\\end{prop}\n\nBefore proving the above proposition, we will conclude first from\nthis result the next main theorem of the section, which in particular\nproves Theorem~\\ref{thm-thmA}.\n\n\\begin{thm} For any $m\\in \\mathbb{N}$, there exists an open and\ndense set $\\mathcal{O}_m$ in $\\mathscr{U}$ such that for any\nfamily $g=(g_a)_a$ in $\\mathcal{O}_m$ there exist positive\nintegers $n_1<\\dots<n_m$ so that $g_a$ has a periodic attractor of\nperiod $n_\\ell$ for all $a\\in \\mathbb{I}^k$ and $\\ell=1,\\dots,m$.\nMoreover, there is a residual subset $\\mathcal{R}$ of\n$\\mathscr{U}$ such that  any $g=(g_a)_a \\in \\mathcal{R}$ satisfies\nthat $g_a$ has both infinitely many hyperbolic periodic sinks and\ninfinitely many normally hyperbolic attracting invariant\ncircles for all $a\\in \\mathbb{I}^k$.\n\\end{thm}\n\\begin{proof}  First of all consider\nthe sequence $\\epsilon_i=1\/i$ for $i\\geq 1$. We will prove the\nresult by induction. To do this, we are going first to construct\n$\\mathcal{O}_m$ for $m=1$.\n\nBy applying Proposition~\\ref{main-lema}, for each $f=(f_a)_a$ in\n$\\mathcal{D}$ taking a sequences of integers $n(i) \\geq\nn_0(\\epsilon_i,f)$ for all $i\\geq 1$, we find a $\\epsilon_i$-close\nfamily $g=(g_a)_a$ to $f$ such that $g_a$ has a $n(i)$-periodic\nattractor for all $a\\in \\mathbb{I}^k$. Actually, for any parameter\n$a$, the attractor that we obtain for $g_a$ is the continuation of\na (hyperbolic or normally hyperbolic) $n(i)$-periodic attractor\nobtained for a map $g_{a_0}$ where $a_0$ belongs to a finite\ncollection of parameters. Thus, from the hyperbolicity of the\nattractor, this property persists under perturbations and then we\nhave a sequence of open sets $\\mathcal{O}_1(\\epsilon_i,f)$\nconverging to $f$ where the same conclusion holds for any family\nin these open sets. By taking the union of all these open sets for\nany $f$ in $\\mathcal{D}$ and $\\epsilon_i>0$ for $i\\geq 1$, we get\nan open and dense set $\\mathcal{O}_1$ in $\\mathscr{U}$ where for\nany $g=(g_a)_a\\in \\mathcal{O}_1$  there exists a positive integer\n$n$ such that $g_a$ has an $n$-periodic attractor  for all\nparameters $a\\in \\mathbb{I}^k$.\n\n\nNow we will assume that $\\mathcal{O}_m$ was constructed and show how to\nobtain $\\mathcal{O}_{m+1}$. Since $\\mathcal{O}_m$ is open\nand dense set in $\\mathscr{U}$ we can start now by taking\n$f=(f_a)_a \\in \\mathcal{O}_m\\cap \\mathcal{D}$. Hence, there exist\npositive integers $n_1<\\dots<n_m$ so that $f_a$ has a persistent\n$n_\\ell$-periodic attractor (a sink or an invariant circle) for\nall $a\\in \\mathbb{I}^k$ for $\\ell=1,\\dots,m$. As before,\nthese attractors are the smooth\ncontinuation of periodic attractors centered at a finite collection of parameters.\nTherefore, there exists $\\epsilon'=\\epsilon'(f)>0$\nsuch that any $\\epsilon'$-close family $g=(g_a)_a$ still\nhas the same properties. Then, for any $\\epsilon_i<\\epsilon'\/2$,\nwe can apply again Proposition~\\ref{main-lema}, taking integers\n$n(i)_{m+1} > \\max\\{n_0(\\epsilon_i,f),n_{m}\\}$ in order to obtain\nan $\\epsilon_i$-perturbation $g=(g_a)_a$ of $f$ such that $g_a$ has\nalso a $n(i)_{m+1}$-periodic attractor  for all $a\\in\n\\mathbb{I}^k$. As before, from the persistence of these\nattractors, we have a sequence of open sets\n$\\mathcal{O}_{m+1}(\\epsilon_i,f)\\subset \\mathcal{O}_m$  converging\nto $f$ where the same conclusion holds for any family in these\nopen sets. By taking the union of all these open sets for any\n$f\\in \\mathcal{O}_m\\cap \\mathcal{D}$ and\n$\\epsilon_i<\\epsilon'(f)$, we get an open and dense set\n$\\mathcal{O}_{m+1}$ in $\\mathscr{U}$ where for any $g=(g_a)_a\\in\n\\mathcal{O}_{m+1}$  there exist positive integers\n $n_1<\\dots<n_{m+1}$  such that  $g_a$ has\na $n_\\ell$-periodic attractor  for all $a\\in \\mathbb{I}^k$ for all\n$\\ell=1,\\dots,m+1$.\n\n\nTo conclude the proof of the theorem observe that if $g=(g_a)_a$ belongs to the\nresidual set $\\mathcal{R}=\\cap \\mathcal{O}_m$ then $g_a$ has\ninfinitely many of attractors for all $a\\in \\mathbb{I}^k$.\n\\end{proof}\n\n\nNow we will prove Proposition~\\ref{main-lema}. To do this we need\nthe following lemma.\n\n\n\\begin{lem} \\label{lema2}\nGiven $\\alpha>0$, let $g=(g_a)_a$ be a $C^{d,r}$-family and assume\nthat $g_a$ has a simple homoclinic tangency at a point~$Y_a$\n(depending $C^d$-continuously on~$a$) associated with a saddle\n$Q_a$ satisfying~\\eqref{eq1} for any parameter $a\\in\na_0+(-\\alpha,\\alpha)^k$.  Then there exists a sequence of families\n$g_n=(g_{na})_a$ approaching $g$ in the $C^{d,r}$-topology such\nthat $g_{na}=g_a$ if $a\\not\\in a_0 + (-2\\alpha,2\\alpha)^k$ and\n$g_{na}$ has an $n$-periodic sink or invariant circle\nfor all $a \\in a_0\n+ (-\\alpha,\\alpha)^k$. Moreover, for $n$ large enough\n$$\n  \\|g_n-g\\|_{C^{d,r}}=O\\left(\\frac{\\alpha^{-d}}{n}\\right).\n$$\n\\end{lem}\n\n\nBefore proving this result, let us show how to get\nProposition~\\ref{main-lema} from the above lemma.\n\n\n\n\\begin{proof}[Proof of Proposition~\\ref{main-lema}]\n Given $f=(f_a)_a$ in $\\mathcal{D}$,\nrelabeling and resizing if necessary, we can assume that the cover\nof $\\mathbb{I}^k$ by the open balls $J_i$ that appears in\nDefinition~\\ref{def:Berger-domain}   is of the form\n$$\n \\mathbb{I}^k  \\subset \\bigcup_{j=1}^M \\bigcup_{\\ell=1}^{N_j}\n J_{j\\ell} \\quad \\text{with} \\quad\n  J_{j\\ell}=a_{j\\ell}+(-\\alpha_{j\\ell},\\alpha_{j\\ell})^k\n  \\quad a_{j\\ell}\\in \\mathbb{I}^k \\ \\ \\text{and} \\ \\ \\alpha_{j\\ell}>0.\n$$\nMoreover, the persistent homoclinic tangency $Y_a$ of $f_a$ on\n$J_{j\\ell}\\cap \\mathbb{I}^k$ is simple in the sense of~\\cite{GST08} and is associated with a saddle $Q_{ja}$\nfor $j=1,\\dots,M$, where for each $j$, the sets\n$\\overline{J_{j\\ell}}$ for $\\ell=1,\\dots,N_j$ are pairwise\ndisjoint. To avoid unnecessary complications in the\nnotation, we can assume that $\\alpha_{j\\ell}=\\alpha$ for all\n$j=1,\\dots,M$ and $\\ell=1,\\dots N_j$. Moreover, for each $j$,\nthe intervals $2J_{j\\ell}=a_{j\\ell}+(-2\\alpha,2\\alpha)^k$ are pairwise disjoint with respect to $\\ell$.\n\n\nOn the other hand, given $\\epsilon>0$, according to\nLemma~\\ref{lema2}, we can control the approximation by a function\n$F(\\alpha,n)$ of order $O(\\alpha^{-d}n^{-1})$. We take\n$n_0=n_0(\\epsilon,f)\\in \\mathbb{N}$ where\n$F(\\alpha,n_0)=O(\\alpha^{-d}n_0^{-1})<\\epsilon$. Now, consider an\ninteger $n\\geq n_0$. We want to find an $\\epsilon$-close family\n$g=(g_a)_a$ having an $n$-periodic attractor at any parameter $a\n\\in \\mathbb{I}^k$. Having into account that for each $j$ the\nintervals $2J_{j\\ell}$ are pairwise disjoint,\nwe can apply Lemma~\\ref{lema2} inductively to obtain an\n$\\epsilon$-close family $g=(g_{a})_a$ to $f$ such that $g_a$ has\nan $n$-periodic attractor  for all $a\\in J_{j\\ell}$. This\nconcludes the proof.\n\\end{proof}\n\nFinally to complete the proof we will show Lemma~\\ref{lema2}.\nHowever, in order to understand better how periodic sinks\nand invariant circles appear in the unfolding of homoclinic\ntangencies associated with saddles of the form~\\eqref{eq1}, we need\nsome preliminaries on the theory of rescaling lemmas\nfrom~\\cite{GST08}.\n\n\\subsection{Rescaling lemma: Generalized Henon map} \\label{sec:rescaling-lemma}\nLet $f$ be a $C^r$-diffeomorphism of a manifold of dimension\n$m\\geq 3$ with a homoclinic tangency\n  associated with a hyperbolic\nperiodic point $Q$ with multipliers satisfying the assumptions~\\eqref{eq1}. We\nassume that the tangency is \\emph{simple} in the sense\nof~\\cite[sec.~1, pg. 928]{GST08}. That is, the tangency is\nquadratic, of codimension one and, in the case that the dimension\n$m>3$, any extended unstable manifold is transverse to the leaf of\nthe strong stable foliation which passes through the tangency\npoint. We need to consider a two-parameter unfolding\n$f_\\varepsilon$ of $f=f_0$ where\n$\\varepsilon=(\\mu,\\varphi-\\varphi_0)$ being $\\mu$ the parameter\nthat controls the splitting of the tangency and $\\varphi$ the\nparameter related to the eigenvalues of $Q$ (here $\\varphi_0$ is the value\nof $\\varphi$ at $\\varepsilon=0$). Let $T_0=T_0(\\varepsilon)$\ndenote the local map and in this case this map\ncorresponds to $f^q_{\\varepsilon}$ defined in a neighborhood $W$ of $Q$, where $q$ is the period of $Q$.\nBy\n$T_1=T_1(\\varepsilon)$ we denote the map $f_\\varepsilon^{n_0}$\ndefined from a neighborhood $\\Pi^-$ of a tangent point $Y^-\\in\nW^u_{loc}(Q,f_0)\\cap W$ of $f_0$ to a neighborhood $\\Pi^+$ of\n$Y^+=f_0^{n_0}(Y^-)\\in W^s_{loc}(Q,f_0)\\cap W$. Then, for $n$\nlarge enough, one defines the first return map $T_n=T_1\\circ\nT_0^n$ on a subset $\\sigma_n=T_0^{-n}(\\Pi^-)\\cap \\Pi^+$ of $\\Pi^+$\nwhere $\\sigma_n\\to W^s_{loc}(Q)$ as $n\\to\\infty$. According\nto~\\cite[Lemma~1 and~3]{GST08} we have the following result:\n\n\n\n\\begin{lem}\\label{lema-GHM} There exists a sequence of open sets $\\Delta_n$ of parameters\nconverging to $\\varepsilon=0$ such that for this values the\nfirst-return $T_n$ has a two-dimensional attracting invariant\n$C^r$-manifold $\\mathcal{M}_n \\subset \\sigma_n$\nso that after a $C^r$-smooth transformation of\ncoordinates, the restriction of the map is given by the\nGeneralized H\\'enon map\n\\begin{equation} \\label{eq-GHM}\n   \\bar{x}=y, \\qquad \\bar{y}=M-Bx-y^2-R_n(xy+ o(1)).\n\\end{equation}\n The rescaled parameters $M$, $B$ and $R_n$ are functions of\n$\\varepsilon \\in \\Delta_n$ such that $R_n$ converges to zero as\n$n\\to \\infty$ and $M$ and $B$ run over asymptotically large\nregions which, as $n\\to \\infty$, cover all finite values. Namely,\n\\begin{align*}\nM \\sim \\gamma^{2n}(\\mu + O(\\gamma^{-n}+\\lambda^n)), \\quad B \\sim\n(\\lambda \\gamma)^{n}\\cos(n\\varphi+O(1)) \\quad \\text{and} \\quad R_n\n= \\frac{2J_1}{B}(\\lambda^2\\gamma)^n\n\\end{align*}\nwhere $J_1\\not=0$ is the Jacobian of the global map $T_1$\ncalculated at the homoclinic point $Y^-$ for $\\varepsilon=0$. The\n$o(1)$-terms tend to zero as $n\\to \\infty$ along with all the\nderivatives up to order $r$ with respect to the coordinates\nand up to order $r-2$ with respect to the rescaled parameters\n$M$ and $B$. Moreover, the limit family is the Henon map.\n\\end{lem}\n\nThe dynamics of the following generalized H\\'enon map\n\\begin{equation} \\label{eq-GHM-simplified}\n   \\bar{x}=y, \\qquad \\bar{y}=M-Bx-y^2-R_nxy\n\\end{equation}\nwas studied in~\\cite{GG00,GG04,GKM05} (see also \\cite{GGT07}). We\npresent here the main results for the case of small $R_n$ with\nemphasis on the stable dynamics (stable periodic orbits and\ninvariant circles) in order to apply the corresponding results to\nthe first return maps coming from \\eqref{eq-GHM}. Observe that the difference between equations \\eqref{eq-GHM-simplified} (generalized H\\'enon map) and \\eqref{eq-GHM} (perturbed map) has order $O(R_n)$. Then the existence of stable periodic orbits and\ninvariant circles for \\eqref{eq-GHM} can be inferred from the bifurcation diagram of \\eqref{eq-GHM-simplified}. In Figure~\\ref{fig-MB} we show the\nbifurcation curves for the generalized H\\'enon map in \\eqref{eq-GHM-simplified}\nin the parameter space $(M,B)$.\n\n\n\\begin{figure}\\vspace{0.3cm}\n\\labellist \\small\\hair 2pt \\pinlabel $L^+_n$ at 100 470 \\pinlabel\n$L^-_n$ at 405 770 \\pinlabel $B$ at 490 530 \\pinlabel $M$ at 60\n790 \\pinlabel $\\mathrm{HT}_n$ at 180 686 \\pinlabel $\\mathrm{BT}_n$\nat 255 440 \\pinlabel $L^\\varphi_n$ at 210 570 \\pinlabel $L_n$ at\n340 570\n\\endlabellist\n\\includegraphics[scale=0.6]{GHM-MB}\n\\caption{Bifurcation curves for the generalized H\\'enon\nmap~\\eqref{eq-GHM-simplified} with $R_n > 0$.\n   The case $R_n=0$ corresponds with the bifurcation diagram of the H\\'enon map.\n   In that case, $L_n^\\varphi$ collapses with $L_n$ at $B=1$.\n   The diagram for the case $R_n<0$ is similar, changing the position of the curves\n   $L_n^\\varphi$ and $L_n$ and the stability of the invariant circle.      }\n\\label{fig-MB}\n\\end{figure}\n\n\nThe map~\\eqref{eq-GHM-simplified} has in the parameter plane $(M,B)$\nthe following three bifurcation curves\n\\begin{align*}\nL^+_n \\ &: \\quad  M = -\\frac{(1 +B)^2}{4(1 + R_n)} \\\\\nL^-_n \\ &: \\quad  M = \\frac{1}{4}(1 +B)^2(3 + R_n) \\\\\nL^\\varphi_n  \\ &: \\quad M=\\frac{\\cos^2 \\omega- \\cos \\omega(2 +\nR_n)}{(1+R_n\/2)^2}, \\quad B = 1+ \\frac{R_n\\cos \\omega}{1 + R_n\/2}.\n\\end{align*}\nThese correspond to the existence of fixed points having multipliers\non the unit circle: $+1$ at $(M,B) \\in L^+_n$; $-1$ at $(M,B)\\in\nL^-_n$; and $e^{\\pm i\\omega}$ at $(M,B)\\in L^\\varphi_n$. Note that\nthe curve $L^\\varphi_n$ is written in a parametric form such that\nthe  argument $\\omega$ ($0 < \\omega < \\pi$) of the complex\nmultipliers is the parameter.\nWe also point out here the points\n\\begin{equation} \\label{eq:pontos}\n\\begin{aligned}\n\\mathrm{BT}_n \\ &: \\quad  M = \\frac{-1 -R_n}{(1 + R_n\/2)^2},  \\quad B=1+\\frac{R_n}{1+R_n\/2} \\\\\n\\mathrm{HT}_n \\ &: \\quad  M = \\frac{3 +R_n}{(1 + R_n\/2)^2}, \\quad\n\\ \\ \\ B=\\frac{1-R_n\/2}{1+R_n\/2}.\n\\end{aligned}\n\\end{equation}\nThe are called as follows: \\emph{Bogdanov-Takens point} for\n$\\mathrm{BT}_n$ and the \\emph{Horozov-Takens point} for\n$\\mathrm{HT}_n$. Also denoted by $L_n$ is an interesting curve\n(nonbifurcational) starting at the point $\\mathrm{BT}_n$, which\ncorresponds to the existence of a saddle fixed point\nof~\\eqref{eq-GHM-simplified} of neutral type (i.e., the fixed\npoint has positive multipliers whose product is equal to one).\nThis curve is drawn in Figure~\\ref{fig-MB} as the dotted line and\nits equation is given by the same expression of\n\\mbox{$L^\\varphi_n$ replacing $\\cos \\omega$ by $\\alpha>1$.}\n\n\\begin{figure}\n\\labellist \\small\\hair 2pt \\pinlabel $K$ at 414 638 \\pinlabel $H$\nat 310 690 \\pinlabel $L_n^-$ at 120 590 \\pinlabel $L^\\varphi_n$ at\n230 445 \\pinlabel {\\footnotesize$\\mathrm{HT}_n$} at 262 575\n\\endlabellist\n  \\includegraphics[scale=0.6]{GHM-Bminus}\n\\caption{Bifurcation diagrams near a Horozov-Takens point for\n$R_n<0$. For $R_n=0$ (H\\'enon map) the curves $K$ and $H$ collapse\nat $B=1$. The diagram for the  case $R_n>0$ is similar but changes\nthe position of $K$, $H$ and the stability of both invariant\ncurves. The open domain between $K$ and $H$ is  rather small,\nhaving  the size of a finite order along the $M$-direction and\norder $O(R_n)$ along the $B$-direction.} \\label{fig-Bminus}\n\\end{figure}\n\nThere is an open domain $D^s_n$ , bounded by the curves $L^+_n$,\n$L^-_n$ $L^\\varphi_n$ with vertices\n$\\mathrm{BT}_n$, $\\mathrm{HT}_n$ (see\nFigure~\\ref{fig-MB}), such that map~\\eqref{eq-GHM-simplified} has\na stable fixed point for parameters in $D^s_n$.\nThe bifurcations of periodic points\nwith multipliers $e^{\\pm i\\omega}$\ncan lead\nto asymptotically stable or\/and unstable invariant circles.\nThe first return map $T_n$ has an\ninvariant circle which is either stable if $R_n>0$ or unstable if\n$R_n<0$. Observe that the sign of $R_n$ actually only depends on\n$J_1\\not=0$. Thus, if $J_1>0$ we obtain parameter values where we\nhave a stable close invariant curve. In the case $J_1<0$, the\nexistence of a stable closed invariant curve\nin~\\eqref{eq-GHM-simplified} follows from the bifurcation analysis\nof the Horozov-Takens point $\\mathrm{HT}_n$. Some\nof the elements that appear in this non-degenerate bifurcation are\nshowen in Figure~\\ref{fig-Bminus}. Actually, when $J_1\\not=0$ (i.e,\n$R_n\\not =0$), near the point $\\mathrm{HT}_n$\nthere are open domains parameter values where stable and\nunstable closed invariant curves coexist.\n\nMoreover, for any map that\nis $O(R_n)$-close to to~\\eqref{eq-GHM-simplified} in the\n$C^3$-topology the corresponding bifurcations still remain\nnon-degenerate and preserve the same stability. Thus, we can\nobtain the same type of results for the pertubed map~\\eqref{eq-GHM}.\nTo summarize for future reference,  for $n$ large\nenough we find open sets $A^1_n$, $A^2_n$ of $(M,B)$-parameters accumulating at\n$\\mathrm{BT}=(-1,1)$ and $\\mathrm{HT}=(3,1)$ respectively as $n\\to\n\\infty$ such that the following holds. If\n$(M,B)\\in A^1_n$ (resp.~$(M,B)\\in A^2_n$) then $T_n$ has a\nhyperbolic attracting periodic point (resp.~a normally hyperbolic\nattracting invariant circle).\n\n\n\n\n\n\\subsection{Proof of Lemma~\\ref{lema2}}\nBy assumption the map $g_a$  has a homoclinic tangency at a point\n$Y_a$ associated with a sectional dissipative periodic point $Q_a$\nfor all $a\\in a_0+(-\\alpha,\\alpha)^k$. Actually, the tangency must\nbe smoothly continued until $\\|a-a_0\\|_{\\infty}=\\alpha$. We\nconsider a two-parameter unfolding $g_{a,\\varepsilon}$ of the\nhomoclinic tangency $Y_a$ of $g_a$ for $a\\in a_0+\n[-\\alpha,\\alpha]^k$,\nwhere $\\varepsilon=(\\mu,\\varphi)$.\nHere $\\mu$ is the parameter that controls the splitting of the\ntangency and $\\varphi$ is the perturbation of the argument of the complex\neigenvalues of $Q_a$.\nWe can take local coordinates $(x,u,y)$  with $x\\in \\mathbb{C}$,\n$u\\in\\mathbb{R}^{m-3}$ and $y\\in \\mathbb{R}$ in a neighborhood of\n$Q_a$ which corresponds to the origin, such that $W^s_{loc}(Q_a)$\nand  $W^u_{loc}(Q_a)$ acquire the form $\\{y=0\\}$ and $\\{x=0,u=0\\}$\nrespectively. Moreover, the complex eigenvalues related to the stable\nmanifold of $Q_a$ correspond to the $x$-variable and\nthe tangency point $Y_a$\nis represented by $(x^+,u^+,0)$.\n\nLet us consider a $C^\\infty$-bump\nfunction $ \\phi:\\mathbb{R}\\to \\mathbb{R}$ with support in $[-1,1]$\nand equal to 1 on $[-1\/2,1\/2]$. Let $$\\rho: a=(a_1,\\dots,a_k) \\in\n\\mathbb{I}^k \\mapsto \\phi(a_1)\\cdot\\ldots\\cdot \\phi(a_k) \\in\n\\mathbb{R}.$$ Take $\\delta>0$ so that the $\\delta$-neighborhoods\nin local coordinates of $Q_a$ and $g_a^{-1}(Y_a)$ are disjoint.\nObserve that\nthese two neighborhoods, call $U$ and $V$ respectively, can be taken\nindependent of $a$.  We can write\n$$g_{a,\\varepsilon}= H_{a,\\varepsilon}\n\\circ g_a,$$ where $H_{a,{\\varepsilon}}$  in these local coordinates\ntakes the form\n\\begin{align*}\n   \\bar{x}&=\\left(1-\\rho\\left(\\frac{a-a_0}{2\\alpha}\\right)\\phi\\left(\\frac{\\|(x,u,y)\\|}{2\\delta}\\right)\n  \n   (1-e^{i\\varphi})\\right) x\n   \\\\\n   \\bar{u}&=u \\\\\n   \\bar{y}&=y+\\rho\\left(\\frac{a-a_0}{2\\alpha}\\right)\\phi\\left(\\frac{\\|(x,u,y)-(x^+,u^+,0)\\|}{2\\delta}\\right)\\mu,\n\\end{align*}\nand is the identity otherwise.\nObserve that if $a\\not\\in a_0 + (-2\\alpha,2\\alpha)^k$ then\n$g_{a,\\varepsilon}=g_a$ and if $(x,u,y)\\not \\in U\\cup V$ then\n$g_{a,\\varepsilon}=g_a$.\n\nRecall in Section~\\ref{sec:rescaling-lemma} the definition of\nthe first return map associated with the unfolding of a simple\nhomoclinic tangency. Since the tangency point $Y_a$ depends\n$C^d$-continuously on $a_0+[-\\alpha,\\alpha]^k$, we may assume that the\nfirst-return map $T_n=T_n(a,\\varepsilon)$ also depends smoothly as\na function of the parameter $a$ on $a_0+[-\\alpha,\\alpha]^k$.\n\n\\begin{lem}[Parametrized rescaling lemma]\\label{para-lema-GHM}\nThere exist\nfamilies of open sets $(\\Delta_n(a))_a$ of parameters converging to\n$\\varepsilon=0$ as $n\\to\\infty$ such that for any $\\varepsilon \\in\n\\Delta_n(a)$ the map $T_n=T_n(a,\\varepsilon)$ has an attracting\n$C^r$-manifold $\\mathcal{M}_n=\\mathcal{M}_n(a,\\varepsilon)$ for any $a\\in a_0+[-\\alpha,\\alpha]^k$.\nMoreover, there exists a $C^{d,r}$-family of transformations of coordinates which bring the first-return map $T_n$ restricted to $\\mathcal{M}_n$ to the form\ngiven by~\\eqref{eq-GHM}\n\\begin{equation} \\label{eq-GHM2}\n   \\bar{x}=y, \\qquad \\bar{y}=M-Bx-y^2-R_n(xy+ o(1)).\n\\end{equation}\nHere the rescaled parameters $M=M_n(a,\\varepsilon)$ and $B=B_n(a,\\varepsilon)$ are at least $C^d$-smooth\nfunctions (recall that $d\\leq r-2$) on\n$$\n\\Delta_n=\\left\\{(a,\\varepsilon):  a\\in a_0+(-\\alpha,\\alpha)^k \\\n\\text{and} \\ \\varepsilon \\in \\Delta_n(a)\\right\\}.\n$$\nThe same\nproperty holds for the coefficient $R_n=R_n(a,\\varepsilon)$ and the $o(1)$-terms. More\nspecifically,\n\\begin{align}  \\label{eq:MB}\nM \\sim \\gamma^{2n}_a(\\mu + O(\\gamma_a^{-n}+\\lambda_a^n)), \\quad\nB \\sim (\\lambda_a\n\\gamma_a)^{n}\\cos(n\\varphi+O(1))\n \\quad \\text{and}\n\\quad R_n = \\frac{2J_{1a}}{B}(\\lambda^2_a\\gamma_a)^n\n\\end{align}\nwhere $\\lambda_a=\\lambda_a(g)$ and $\\gamma_a=\\gamma_a(g)$ are the\neigenvalues of $Q_a=Q_a(g)$  satisfying~\\eqref{eq1} for $g_a$.\n\\end{lem}\n\n\\begin{proof}\nLet us analyze the proof of the rescaling lemma in~\\cite{GST08}, more specifically\nthe change of coordinates for the first return maps given in\nSection 3.2~\\cite{GST08} for the case $(2,1)$ with $\\lambda\\gamma>1$.\nFrom equations ~\\cite[Eq.~(3.12)-(3.16)]{GST08} we can observe the all the transformations of coordinates can be performed smoothly on the parameter\n$a \\in a_0+[-\\alpha,\\alpha]^k$. The exponents that will appear in the orders of convergence will not depend on the parameter $a$ but only on $n$. On the other hand the constants in the $O$-terms will depend on the parameter but these can be uniformely bounded due to the compactness of the parameter space.\nThese considerations allow us basically to apply\nthe rescaling Lemma~\\ref{lema-GHM} smoothly on the parameter $a \\in\na_0+[-\\alpha,\\alpha]^k$.\n\\end{proof}\n\n\\begin{lem} \\label{Delta_expand}\nFor\n$n$ large enough, $\\Delta_{n}(a)$ can be taken in such a way that\n$$\n \\phi_{n,a}: \\Delta_{n}(a) \\to (-10,10)^2 \\setminus B(0,r), \\qquad \\phi_{n,a}(\\epsilon)\n=(M_n(a,\\epsilon), B_n(a,\\epsilon))\n$$\nis a diffeomorphism where $M_n$ and $B_n$ are the functions given\nin~\\eqref{eq:MB} for fixed $n$ and $a$. Here $B(0,r)$ is a closed\nball of some small fixed radius $r$ around the origin.\n\\end{lem}\n\n\\begin{proof}\nLet us introduce the parameter value $\\varepsilon^0_n(a)=(\\mu_n^0(a),\\varphi^0_n(a))$,\nwhich by definition satisfies $\\phi_{n,a}(\\varepsilon^0_n(a))=0$. It can be seen from\nthe expressions of $M$ and $B$ in~\\eqref{eq:MB} that $\\mu^0_n(a)=O(\\gamma^{-n}_a+\\lambda^n_a)$ and $\\varphi^0_n(a)= \\frac{\\pi}{2n} +O(1\/n)$. We take out $B(0,r)$ around the origin in $(-10,10)^2$ so that $B$ is bounded away from zero and $R_n$ is well-defined. Similar values were considered in~\\cite[pg.~946]{GST08} and as is\nclaimed there, the rescaled functions $M$ and\n$B$ can take arbitrarily finite values when $\\mu$ varies close to\n$\\mu^0_n(a)$ and $\\varphi$ near\n$\\varphi^0_n(a)$. Let us explain this.\nAlthough the $O$-functions in~\\eqref{eq:MB}  depend on\n$\\varepsilon$, the functions $M$ and $B$ basically only depend on\n$\\mu$ and $\\varphi$ respectively for $n$ large enough. Actually\non~\\cite[pg.~946]{GST08} are giving explicit expressions for\n$M_{n,a}$ and $B_{n,a}$.\nUsing them one can\nobserve that\n$$\n \\partial_\\mu M_{n,a} \\sim \\gamma_a^{2n} \\not = 0 \\quad \\text{and} \\quad\n \\partial_\\varphi B_{n,a} \\sim n(\\lambda_a\\gamma_a)^n\n \\sin(n\\varphi+O(1)) \\not = 0\n$$\nfor all $\\varepsilon=(\\mu,\\varphi)$ close to $\\varepsilon^0_n(a)$.\nThen the Jacobian of $\\phi_{n,a}$ converges to infinity, uniformly\non $a\\in a_0+[-\\alpha,\\alpha]^k$  as $n\\to \\infty$. The rate of\ngrowth is exponential. This implies that $\\phi_{n,a}$ is an\ninvertible expanding map with arbitrarily large uniform expansion\non $a\\in a_0 + [-\\alpha,\\alpha]^k$. On the other hand, the size of\n$\\Delta_{n}(a)$ (coming mainly from considerations on the angle\n$\\varphi$) where the expanding map $\\phi_{n,a}$ is defined has\ndecay of order $O(1\/n)$. Thus, for $n$ large enough we get that a\nneighborhood of $\\varepsilon^0_n(a)$ can be taken so that its\nimage under $\\phi_{n,a}$ is $(-10,10)^2$. In particular we can\ntake $\\Delta_n(a)$ being diffeomorphic to $(-10,10)^k\\setminus\nB(0,r)$.\n\\end{proof}\nConsequently,\n$$\n\\Phi_n(a,\\varepsilon)=(a,\\phi_{n,a}(\\varepsilon))\n$$\nis a diffeomorphism between the set $\\Delta_n$ defined above and\n$(a_0+[-\\alpha,\\alpha]^k)\\times (-10,10)^2\\setminus B(0,r)$.\nOn the other hand, although the coefficient $R_n$ depends on $B$,\nnote that the range of values it takes is negligible when $B$ is\nbounded from zero and $n$ is large enough. Actually from the\nrelations in ~\\eqref{eq:MB} it follows that $R_n=o(1)$. Thus, the\nbifurcation diagram of~\\eqref{eq-GHM2} can be studied from the\nresults described in Section~\\ref{sec:rescaling-lemma} assuming\n$R_n=o(1)$ independent of $B$.\n\nLet us remind the reader of the Bogdanov-Takens $\\mathrm{BT}_n(a)$\nand the Horozov-Takens $\\mathrm{HT}_n(a)$ points given\nin~\\eqref{eq:pontos}, which now also depend on $a$ and accumulate\nat $\\mathrm{BT}=(-1,1)$ and $\\mathrm{HT}=(3,1)$ respectively as\n$n\\to \\infty$. Hence, according to\nSection~\\ref{sec:rescaling-lemma}  for each $n$ large enough we\nfind open subsets $A^1_n(a)$, $A^2_n(a)$ in the $(M,B)$-parameter\nplane such that if $(M,B)\\in A^1_n(a)$ (resp.~$(M,B)\\in\nA^2_n(a)$), then $T_n$ has a hyperbolic attracting periodic point\n(resp.~\nattracting\nsmooth invariant circle) for all $a \\in a_0+[-\\alpha,\\alpha]^k$.\nMoreover, we can assume the points $\\mathrm{BT}_n(a)$ and\n$\\mathrm{HT}_n(a)$\nbelong to the boundary of $A^1_n(a)$ and\n$A^2_n(a)$ respectively. Since these sets vary $C^d$-continuously\nwith respect to the parameter $a \\in [-\\alpha,\\alpha]^k$, we can\nchoose $C^d$-continuously $(M^*_n(a),B^*_n(a)) \\in A^*_n(a)$ for\n$*=1,2$ arbitrarily close to $\\mathrm{BT}_n(a)$ and\n$\\mathrm{HT}_n(a)$ respectively.\n\nSince $\\Phi_n$ is a diffeomorphism, we may find\na $C^{d}$-function\n$\\varepsilon^*_n(a)=(\\mu_n^*(a),\\varphi^*_n(a))$ for $a \\in\na_0+[-\\alpha,\\alpha]^k$, $*=1,2$, defined as\n$\\Phi_n^{-1}(a,(M^*_n(a),B^*_n(a))=(a,\\varepsilon^*_n(a))$. In\nparticular,\n\\begin{equation} \\label{eq2}\n\\begin{aligned}\nM^*_n(a)&=M_{n,a}(\\varepsilon^*_n(a))\\sim \\gamma^{2n}_a(\\mu_n^*(a) +\nO(\\gamma_a^{-n}+\\lambda_a^n)) \\\\\nB^*_n(a)&=B_{n,a}(\\varepsilon^*_n(a)) \\sim (\\lambda_a\n\\gamma_a)^{n}\\cos(n\\varphi^*_n(a)+O(1)).\n\\end{aligned}\n\\end{equation}\nExtending smoothly $\\varepsilon^*_n(a)$ to $\\mathbb{I}^k$ we can\nconsider the sequence of families\n$\\tilde{g}_n=(\\tilde{g}_{n,a})_a$ where\n$$\n   \\tilde{g}_{n,a}=g_{a,\\varepsilon^*_n(a)} \\qquad \\text{for  $a\\in\n   \\mathbb{I}^k$ and $n$ large enough}.\n$$\nObserve that $\\tilde{g}_{n,a}=g_a$ for $a \\not \\in a_0 +\n(-2\\alpha,2\\alpha)^k$ and we may assume\nhas an $n$-periodic attractor (a sink or an invariant circle) for\nall $a\\in a_0+(-\\alpha,\\alpha)^k$.\n\nTo conclude the proof of the lemma  we only need to show that\n$\\tilde{g}_n$ converges to $g$ in the $C^{d,r}$-topology. To do\nthis, notice that the $C^{d,r}$-norm satisfies\n$$\n \\|\\bar{g}_n-g\\|=\\|(I-H_{a,\\varepsilon^*_n(a)})\\circ g_a \\|  \\leq\n  \\|I-H_{a,\\varepsilon^*_n(a)}\\| \\, \\|g\\|\n$$\nwhere $I$ denotes the identity and  the $C^{d,r}$-norm of any $g=(g_a)_a$ in\nthe Berger domain $\\mathscr{U}$ is bounded.\nThus, we\nonly need to calculate the $C^{d,r}$-norm of the family\n$(I-H_{a,\\varepsilon^*_n(a)})_a$. Since\n$H_{a,\\varepsilon^*_n(a)}=I$ if $a\\not \\in a_0 +\n(-2\\alpha,2\\alpha)^k$ or $(x,u,y)\\not \\in U \\cup V$, therefore\n$\\|I-H_{a,\\varepsilon^*_n(a)}\\|$ is less or equal than\n\\begin{align*}\n\n  \\left\\|\\rho\\left(\\frac{a-a_0}{2\\alpha}\\right)\\phi\\left(\\frac{\\|(x,u,y)\\|}{2\\delta}\\right)\n  (1-e^{i\\varphi^*_n(a)})x\\right\\| +\n  \\left\\|\\rho\\left(\\frac{a-a_0}{2\\alpha}\\right)\\phi\\left(\\frac{\\|(x,u,y)-(x^+,u^+,0)\\|}{2\\delta}\\right)\\mu^*_n(a)\\right\\|.\n\\end{align*}\nTo estimate the $C^{d,r}$-norms above it suffices to show\nthat the functions\n$$\n  F_{n}(a)= \\rho(\\frac{a-a_0}{2\\alpha})(1-e^{i\\varphi^*_n(a)}) \\quad\n  \\text{and} \\quad G_{n}(a)  = \\rho(\\frac{a-a_0}{2\\alpha})\\mu^*_n(a)\n   \n$$\nhave $C^d$-norm small when $n$ is large and $a \\in  a_0+(-2\\alpha,2\\alpha)^k$. Actually we will prove\nthe following:\n\\begin{equation} \\label{eq:order}\n\\|F_{n}\\|_{C^d}= O\\left(\\frac{\\alpha^{-d}}{n}\\right) \\quad\n\\text{and} \\quad \\|G_{n}\\|_{C^d}=\nO\\left(\\frac{\\alpha^{-d}}{n}\\right)\n\\end{equation}\nObserve that this assertion completes the proof.\nTo prove this we will need the following derivative estimates on the functions $\\mu_n^*(a)$ and $\\varphi_n^*(a)$. Here the symbol $\\partial_a^j$ is used to denote the $j$-th partial\nderivative with respect to  the coordinates $a_i$ of $a$ using the multi-index notation.\n\n\\begin{lem} \\label{Est_der} For $1\\leq |j|\\leq d$\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mu_n^*(a)=O(\\gamma_a^{-n}+\\lambda_a^n) \\ \\ \\text{and}  \\ \\ \\partial_a^j\\mu_n^*(a)=O(n^{|j|}(\n\\gamma_a^{-n}+\\lambda_a^n)).$ \\\\[-0.3cm]\n\\item $\\varphi_n^*(a)=O\\left(\\frac{1}{n}\\right), \\ \\\n\\partial_a^j\\varphi_n^*(a)=O\\left(\\frac{n^{{|j|-1}}}{(\\gamma_a\\lambda_a)^{n}}\\right) \\ \\ \\text{and}\n\\ \\\n\\partial_a^j(e^{i\\varphi_n^*(a)})=O\\left(\\frac{n^{{|j|-1}}}{(\\gamma_a\\lambda_a)^{n}}\\right).$\n\\end{enumerate}\n\n\\end{lem}\n\nAssuming the above lemma let us now prove the estimates in~\\eqref{eq:order}, starting with the second one. To do\nthis, using the Leibniz formula,\n$$\n\\partial^\\ell_a G_{n}(a)=\\sum_{j:j\\leq\\ell} \\binom{\\ell}{j} \\ \\partial_a^{\\ell-j}\n\\rho\\left(\\frac{a-a_0}{2\\alpha}\\right) \\cdot \\partial_a^j\\mu_n^*(a).\n$$\nSubstituting the estimate from Lemma \\ref{Est_der} (i) in the above expression we\nobtain that\n$$\\partial_a^\\ell G_{n,\\alpha}(a)=O(\\alpha^{-d}\\cdot n^{\\left|\\ell \\right|}\n(\\gamma_a^{-n}+\\lambda_a^n))$$\nwhich, in fact, implies a better\nestimate than~\\eqref{eq:order}.\n\nTo prove  the first estimate in~\\eqref{eq:order} for $\\|F_{n}\\|_{C^d}$, using again the Leibniz formula\n$$\n\\partial_a^\\ell F_{n,\\alpha}(a) =\n(1-e^{i\\varphi_n^*(a)}) \\ \\partial_a^\\ell\\rho\\left(\\frac{a-a_0}{2\\alpha}\\right)\n-\\sum_{j:\\ 0<j\\leq\\ell} \\binom{\\ell}{j}\\, \\\n \\partial_a^{j}(e^{i\\varphi_n^*(a)}) \\ \\partial_a^{\\ell-j}\\rho\\left(\\frac{a-a_0}{2\\alpha}\\right).\n $$\nApplying Lemma~\\ref{Est_der} (ii),\n$$\n\\partial_a^\\ell F_{n,\\alpha}(a) =O\\left(\\frac{\\alpha^{-\\left|\\ell\\right|}}{n}\\right)\n+\\sum_{j:\\ 0<j\\leq\\ell} \\binom{\\ell}{j}\\, O\\left(\n\\frac{n^{{|j|-1}}}{(\\gamma_a\\lambda_a)^{n}} \\cdot\n\\alpha^{-\\left|\\ell-j\\right|} \\right).\n$$\nThus, $\\partial_a^\\ell F_{n,\\alpha}(a)=\nO\\left(\\alpha^{-d}n^{-1}\\right) $. To complete the proof of Lemma\n\\ref{lema2} we have to show the estimates of Lemma \\ref{Est_der}.\n\\begin{proof} [Proof of Lemma \\ref{Est_der}]\nFirst we will prove the estimates on $\\mu_n^*(a)$ and\n$\\partial_a^j\\mu_n^*(a)$. Let us first observe that $M$, up to a\nmultiplicative factor, is actually equal to  $\\gamma^{2n}_a(\\mu +\nO(\\gamma_a^{-n}+\\lambda_a^n))$ (see the exact expressions for $M$\nin ~\\cite[Section 3.2]{GST08}). Although this multiplicative\nfactor depends on the perturbation $\\varepsilon_n$, to avoid\nnotational clutter we will assume in what follows and without loss\nof generality that this factor is always $1$. Then $M^*_n(a)$ can\nbe written as\n\\begin{equation} \\label{Eq_M}\nM^*_n(a)=\\gamma^{2n}_a\\mu_n^*(a) + h_n(a)\n\\end{equation}\nwhere $h_n(a)$ is independent of $\\mu_n^*(a)$ and\n$h_n(a)=O(\\gamma_a^{-n}+\\lambda_a^n)$. Again using the explicit\nexpressions for $M$ in ~\\cite[Section 3.2]{GST08}, one can see\nthat $\\partial_a^j h_n(a)=O(n^{|j|}(\\gamma_a^{-n}+\\lambda_a^n))$.\n\n\\begin{claim} \\label{claim_M}\n\\begin{equation}  \\label{covergence_M}\nM^*_n=O(1) \\quad \\text{and} \\quad \\partial_a^j M^*_n=o(1) \\ \\\n\\text{for all $1\\leq |j|\\leq d$}\n\\end{equation}\n\\end{claim}\n\\begin{proof}[Proof of Claim~\\ref{claim1}]\nThe expressions of $\\mathrm{BT}_n(a)$ and $\\mathrm{HT}_n(a)$  in\n\\eqref{eq:pontos} imply that these functions have order $O(R_n)$\nand analogously their respective derivatives have order\n$O(\\partial_a^j(R_n))$. On the other hand $M^*_n(a)$\n is\n$O(R_n)$-close to $\\mathrm{BT}_n(a)$ or $\\mathrm{HT}_n(a)$, see Section~\\ref{sec:rescaling-lemma}. Also the derivatives of $\\partial_a^j M^*_n$ are $O(\\partial_a^j(R_n))$-close. Therefore,\n$$M^*_n=O(1)+O(R_n) \\quad \\text{and} \\quad \\partial_a^j M^*_n=O(\\partial_a^j R_n).$$\nFinally, from the equation for $R_n$ given in Lemma~\\ref{para-lema-GHM}, we have that\n$R_n=o(1)$ and $\\partial_a^j R_n=o(1)$. Combining this with the previous estimates proves the claim.\n\\end{proof}\nIn particular, from~\\eqref{Eq_M} we obtain that\n$$\\gamma^{2n}_a \\mu_n^*(a)=O(1) + O(\\gamma_a^{-n}+\\lambda_a^n),$$\nwhich implies\n$\\mu_n^*(a)=O(\\gamma^{-2n}_a)=O(\\gamma_a^{-n}+\\lambda_a^n)$. To\nshow the estimates on $\\partial_a^j\\mu_n^*(a)$ we will proceed by\nan inductive argument on $|j|$. In the case $|j|=1$, taking the\nderivate of \\eqref{Eq_M} gives\n$$\n\\partial_a^j M^*_n(a)=\\partial_a^j(\\gamma^{2n}_a)\\mu_n^*(a) +\\gamma^{2n}_a \\partial_a^j\\mu_n^*(a)+\n\\partial_a^j h_n(a).\n$$\nWe have that $\\mu_n^*(a)=O(\\gamma_a^{-n}+\\lambda_a^n)$,\n$\\partial_a^j M^*_n(a)=o(1)$ from Claim~\\ref{claim_M} and\n$\\partial_a^j h_n(a)=O(n(\\gamma_a^{n}+\\lambda_a^n))$. Also\n$\\partial_a^j(\\gamma^{2n}_a)=2n\\gamma^{2n-1}_a\\cdot\n\\partial_a^j(\\gamma_a)=O(n\\gamma^{2n}_a)$. Combining all these\nestimates with the previous equation implies\n$$\\gamma^{2n}_a \\partial_a^j\\mu_n^*(a)=O(n\\gamma^{2n}_a)\\cdot O(\\gamma_a^{-n}+\\lambda_a^n)\n+O(n(\\gamma_a^{-n}+\\lambda_a^n))+o(1).$$ Then\n$\\partial_a^j\\mu_n^*(a)=O(n( \\gamma_a^{-n}+\\lambda_a^n))$, which\nproves the formula for $|j|=1$. To prove the necessary expression\nfor any multi-index $\\ell$, we will proceed by induction assuming\nthat $\\partial_a^j\\mu_n^*(a)=O(n^{|j|}(\n\\gamma_a^{-n}+\\lambda_a^n))$ for any $j$ with $1\\leq |j|\n<\\left|\\ell\\right|$ and will show the same estimate for $\\ell$.\nFrom~\\eqref{Eq_M} and using the Leibniz formula we obtain\n\\begin{equation} \\label{eq_M2}\n \\partial_a^{\\ell}M^*_n(a) =\n\\sum_{j:j\\leq\\ell} \\binom{\\ell}{j} \\ \\partial_a^{\\ell-j}\n(\\gamma^{2n}_a) \\cdot \\partial_a^j\\mu_n^*(a)+\\partial_a^{\\ell}h_n(a).\n\\end{equation}\nNow $\\partial_a^{\\ell-j}\n(\\gamma^{2n}_a)=O(n^{\\left|\\ell-j\\right|}\\gamma^{2n}_a)$ and so\nthe order of $\\partial_a^{\\ell-j}(\\gamma^{2n}_a) \\cdot \\partial_a^j\\mu_n^*(a)$ is\n$$O(n^{\\left|\\ell-j\\right|}\\gamma^{2n}_a)\\cdot O(n^{|j|}( \\gamma_a^{-n}+\\lambda_a^n))\n=O(n^{\\left|\\ell\\right|}\\gamma^{2n}_a(\n\\gamma_a^{-n}+\\lambda_a^n)).$$ Also\n$\\partial_a^{\\ell}M^*_n(a)=o(1)$ by Claim~\\ref{claim_M} and\n$\\partial_a^{\\ell}h_n(a)=O(n^{\\left|\\ell\\right|}(\\gamma_a^{n}+\\lambda_a^n))$.\nThen isolating the term with\n$j=\\ell$ from the sum in~\\eqref{eq_M2} we get that\n$$\\gamma^{2n}_a\\cdot\\partial_a^\\ell\\mu_n^*(a)=\nO(n^{\\left|\\ell\\right|}\\gamma^{2n}_a( \\gamma_a^{-n}+\\lambda_a^n))\n+O(n^{\\left|\\ell\\right|}(\\gamma_a^{-n}+\\lambda_a^n))+o(1).$$ Then\nwe may conclude that\n$\\partial_a^\\ell\\mu_n^*(a)=O(n^{\\left|\\ell\\right|}(\n\\gamma_a^{-n}+\\lambda_a^n))$ proving item~(i) of\nLemma~\\ref{Est_der}.\n\nNow we will prove the second part of the lemma, that is, the\nestimates on $\\varphi_n^*(a)$ and $\\partial_a^j\\varphi_n^*(a)$. As\nwas mentioned before in Lemma~\\ref{Delta_expand} because of the\nexpressions in~\\cite[pg.~946]{GST08}, we have that\n$\\varphi^0_n(a)= \\frac{\\pi}{2n} +O(1\/n)$ and the size of\n$\\Delta_{n}(a)$ has decay of order $O(1\/n)$. Thus, we may also\nconclude that $\\varphi_n^*(a)=O(n^{-1})$. Now assuming\n$\\partial_a^j\\varphi_n^*(a)=O\\left(n^{{|j|-1}}(\\gamma_a\\lambda_a)^{-n}\\right)$,\nit is not hard to see by an inductive argument on the derivatives\nthat also\n$\\partial_a^j\\left(e^{i\\varphi_n^*(a)}\\right)=O\\left(n^{{|j|-1}}(\\gamma_a\\lambda_a)^{-n}\\right)$.\n\nIn what follows we will prove that $\\partial_a^j\\varphi_n^*(a)\nO(n^{|j|-1}(\\gamma_a\\lambda_a)^{-n})$ for $j\\geq 1$. This will be\ndone by induction in $|j|$, starting with $|j|=1$. To do this,\naccording to~\\eqref{eq2} and writting\n$h_n(a)=n\\varphi_n^*(a)+O(1)$, we have that $B^*_n(a)\n\\sim(\\lambda_a\\gamma_a)^{n}\\cos(h_n(a))$. Having in mind the exact\nexpression for the function $B$ given in~\\cite[pg.~946]{GST08},\n$B_n(a)$ differs from $(\\lambda_a\\gamma_a)^{n}\\cos(h_n(a))$ by a\nmultiplicative constant $b_0$, that depends on $\\varepsilon_n$.\nSimilarly to what was done for $M^*_n(a)$ and to avoid unnecessary\nnotational complications we will assume, without loss of\ngenerality, that this factor is always $1$. Then,\n\\begin{equation} \\label{B*}\nB^*_n(a)=(\\lambda_a\\gamma_a)^{n}\\cos(h_n(a))\n\\end{equation} and\nderivating both sides of the equation with respect to some index\n$j$, $|j|=1$, we obtain\n\\begin{align}\n\\label{eq:O1}  \\partial_a^j B^*_n(a) &= n \\cdot B^*_n(a) \\cdot \\partial_a^j (\\log\n\\lambda_a \\gamma_a) + (\\lambda_a \\gamma_a)^{n}\n  \\sin(h_n(a)) \\cdot \\partial_a^j h_n(a).\n\\end{align}\nNow to estimate the order of $\\partial_a^j B^*_n(a)$, we have the following claim\nwhose proof we omit as it is analogous to that of Claim~\\ref{claim_M}.\n\\begin{claim} \\label{claim1}\n\\begin{equation}  \\label{covergence}\nB^*_n=O(1) \\quad \\text{and} \\quad \\partial_a^j B^*_n=o(1) \\ \\\n\\text{for all $1\\leq |j|\\leq d$}\n\\end{equation}\n\\end{claim}\nSince the norms of the functions $B^*_n(a),\\ \\partial_a^j (\\log\n\\lambda_a \\gamma_a)$ are bounded from above and using that $\\partial_a^j B^*_n=o(1)$ due to Claim~\\ref{claim1} we get from~\\eqref{eq:O1} that\n\\begin{align}\n(\\lambda_a \\gamma_a)^{n}\n  \\sin(h_n(a)) \\cdot \\partial_a^j h_n(a)=O(n)+o(1).\n\\end{align}\nSince $\\sin(h_n(a))$ is bounded away from zero by the choice of\n$\\varphi_n$, resolving  the previous equation for $\\partial_a^j\nh_n(a)$\nwe obtain that $\\partial_a^j h_n(a)=O(n(\\lambda_a\\gamma_a)^{-n})$.\nThis implies that\n$\\partial_a^j\\varphi_n^*(a)=O((\\lambda_a\\gamma_a)^{-n})$ and\nproves the formula for $|j|=1$. To prove the expression for any\nindex $\\ell$, we will proceed by induction assuming that\n$\\partial_a^j h_n(a)= O(n^{|j|}(\\gamma_a\\lambda_a)^{-n})$ for any\n$j$ with $1\\leq |j| <\\left|\\ell\\right|$ and will show that\n$\\partial_a^\\ell h_n(a)=\nO(n^{\\left|\\ell\\right|}(\\gamma_a\\lambda_a)^{-n})$. This will imply\nthat\n$\\partial_a^\\ell\\varphi_n^*(a)=O(n^{\\left|\\ell\\right|-1}(\\lambda_a\\gamma_a)^{-n})$,\nas it is required to complete the proof of the lemma.\n\nTake $\\ell$ with $\\left|\\ell\\right|> 1$, and fix $\\iota<\\ell$ such\nthat $\\left|\\iota\\right|=1$. Since\n$\\partial_a^{\\ell}B^*_n(a)=\\partial_a^{\\ell-\\iota}(\\partial_a^{\\iota}B^*_n(a))$\nwe get from~\\eqref{B*}, \\eqref{eq:O1} and by using the Leibniz\nformula,\n\\begin{equation} \\label{eq:final}\n \\partial_a^{\\ell}B^*_n(a) = n\\cdot\n\\partial_a^{\\ell-\\iota}(B^*_n(a)\\cdot \\partial_a(\\log \\lambda_a\\gamma_a)) +\n  \\sum_{j:j\\leq (\\ell-\\iota)} \\binom{\\ell-\\iota}{j}\\,\n \\partial_a^{(\\ell-\\iota)-j}((\\lambda_a\\gamma_a)^{n}\\sin(h_n(a)))\\cdot\n \\partial_a^{j+\\iota} h_n(a).\n\\end{equation}\nNow we will determine the order of the terms in the above equation.\n\\begin{claim} \\label{claim2}\n\\begin{equation}\n\\partial_a^j\n((\\lambda_a\\gamma_a)^n\\sin(h_n(a)))=O(n^{|j|}(\\lambda_a\\gamma_a)^n)\n\\quad \\text{for all $1\\leq |j|\\leq d$}\n\\end{equation}\n\\end{claim}\n\\begin{proof}[Proof of Claim~\\ref{claim2}]\nObserve from~\\eqref{B*} that\n$[(\\lambda_a\\gamma_a)^n\\sin(h_n(a))]^2=(\\lambda_a\\gamma_a)^{2n}-(B^*_n)^2$.\nSince   $\\partial_a^j B^*_n=o(1)$ from Claim~\\ref{claim1}, we get\nthat\n$$\\partial_a^j\n((\\lambda_a\\gamma_a)^n\\sin(h_n(a)))=O(\\partial_a^j(\\lambda_a\\gamma_a)^n).$$\nOn the other hand it can be seen by an inductive argument that\n$\\partial_a^j(\\lambda_a\\gamma_a)^n=O(n^{|j|}(\\lambda_a\\gamma_a)^n)$,\nwhich gives the required estimate.\n\\end{proof}\n\nFor $j<(\\ell-\\iota)$, by the induction hypothesis\n$\\partial_a^{j+\\iota} h_n(a)=\nO(n^{\\left|j+\\iota\\right|}(\\gamma_a\\lambda_a)^{-n})$. Combining\nthis with Claim~\\ref{claim2} we obtain that for $j<(\\ell-\\iota)$\n$$\\partial_a^{(\\ell-\\iota)-j}((\\lambda_a\\gamma_a)^{n}\\sin(h_n(a)))\\cdot\n \\partial_a^{j+\\iota} h_n(a)= O(n^{\\left|(\\ell-\\iota)-j\\right|}(\\lambda_a\\gamma_a)^n) \\cdot O(n^{\\left|j+\\iota\\right|}(\\gamma_a\\lambda_a)^{-n})=O(n^{\\ell}).$$\nOn the other hand $\\partial_a^{\\ell}B^*_n(a) = o(1)$ and $n\\cdot\n\\partial_a^{\\ell-\\iota}(B^*_n(a)\\cdot \\partial_a(\\log \\lambda_a\\gamma_a))=O(n)$. Thus, putting all these estimates together in~\\eqref{eq:final} and isolating the term corresponding with the index $j=\\ell-\\iota$ we obtain\n$$(\\lambda_a\\gamma_a)^{n}\\sin(h_n(a)))\\cdot\n \\partial_a^{\\ell} h_n(a)=O(n)+O(n^{\\left|\\ell\\right|})+o(1).$$\nThis implies that $\\partial_a^\\ell h_n(a)=\nO(n^{\\left|\\ell\\right|}(\\gamma_a\\lambda_a)^{-n})$ concluding the\nproof.\n\\end{proof}\n\n\n\\bibliographystyle{alpha2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nThe use of identically prepared systems, or ensembles, has been\nessential to our understanding of equilibrium and far-from-equilibrium\nproperties of classical and quantum systems. Traditionally, three\ntypes of ensembles are used---(a) the microcanonical ensemble, which\ninvolves systems with fixed energy and particle number, (b) the\ncanonical ensemble (CE), which involves systems with fixed particle number\nin contact with a large reservoir (at temperature $T$) with which they\ncan exchange energy, and (c) the grand canonical ensemble (GE), which\ninvolves systems in contact with a reservoir with which they can\nexchange energy and particles (in equilibrium, the average particle\nnumber is determined by the chemical potential $\\mu$). Whereas these\nthree ensembles pose fundamentally different physical constraints, \nit can be shown that they are equivalent in the thermodynamic limit\n(provided, of course, that temperatures and chemical potentials are\nselected appropriately). Being technically easier to deal with, the\ncanonical and grand canonical ensembles are the most commonly used\nensembles in the literature. Several texts on statistical mechanics\ncover these topics in detail; see e.g.,~Ref.~\\cite{huang87}.\n\nIn finite systems, differences appear between calculations carried \nout using the three ensembles. These differences, dubbed finite-size\neffects, have to do with the effect of energy and particle number\nfluctuations, and with boundary effects. For example, to describe an\nisolated system with mean energy $E$, it is most appropriate to \nuse the microcanonical ensemble with that energy. However, \none can also use a canonical ensemble at a temperature $T$ for which \nthe mean energy is $E$. Since the systems used to construct the \ncanonical ensemble have different energies from the ones used to \nconstruct the microcanonical ensemble, one finds differences in the\npredictions of each ensemble.  Remarkably, one can show that energy\nfluctuations in the canonical ensemble typically scale as the square\nroot of the volume of the system, whereas the average energy scales \nas the volume of the system.  Hence, the ratio between the energy\nfluctuations and the average energy scales as the inverse of the\nsquare root of the volume, and vanishes in the thermodynamic\nlimit. One then finds that differences between the predictions of \neach ensemble decrease polynomially with increasing volume\n(at fixed density). The same applies if one considers the grand\ncanonical ensemble, where particle number fluctuations typically \nscale with the square root of the volume of the system. Indeed, \nexplicit calculations in one dimensional (1D) lattices have shown \nthat the differences between the predictions of the canonical and\ngrand-canonical ensembles for various observables decrease with the\ninverse of the number of particles (or lattice sites) in the system\n\\cite{lebowitz61,rigol_05}.\n\nExperiments usually deal with thermodynamically large systems, whereas\nnumerical analyses of many-body interacting systems can generally be\ndone for only (relatively) much smaller system sizes.  Hence, when\ntrying to theoretically predict\/reproduce the outcome an experimental\nmeasurement, a question of much relevance is: \\emph{which ensemble should \none use to minimize finite size effects and obtain the ``thermodynamic \nlimit'', or, experimental result?} From the previous discussion about \nthe differences between ensembles, one might naively conclude that finite \nsize effects always scale polynomially with system size and that, therefore, \nthe best one can do theoretically is to optimize exponents and prefactors. \n\nIn this article we show that this is not the case. There is a\npreferred ensemble (the grand canonical ensemble) and preferred boundary \nconditions (periodic boundary conditions, so that the system is translationally \ninvariant) for which finite-size effects are exponentially small in the \nsystem size. This holds if the system of interest is in an unordered \n(i.e., without long or quasi-long range order) phase at finite temperature. \nWe also consider a different approach to calculating finite-temperature \nproperties of many-particle systems, namely, numerical linked cluster \nexpansions (NLCEs) \\cite{rigol_bryant_06,rigol_bryant_07,rigol_bryant_07b}. \nWe show that NLCEs not only exhibit exponential convergence with\nincreasing system size but generally outperform grand canonical\nensemble calculations in systems with periodic boundary conditions.\n\nThe paper is organized as follows. In Sec.~\\ref{sec:gener-cons}\nwe argue, based on a high temperature expansion of the partition\nfunction, that grand canonical ensemble calculations in\ntranslationally invariant systems have exponentially small finite-size\nerrors. In Sec.~\\ref{sec:verification}, we discuss analytically\nsolvable examples, the 1D and 2D Ising models, that substantiate the\narguments in Sec.~\\ref{sec:gener-cons}. In Sec.~\\ref{sec:pert-th}, we present \na proof that finite-size errors are indeed exponentially small in the grand\ncanonical ensemble for translationally invariant noninteracting systems \nand that, within perturbation theory, the same scaling applies to interacting \nsystems. We then study numerically, in Sec.~\\ref{sec:numerical-tests}, \nthree examples where we systematically compare results from canonical and \ngrand canonical ensemble calculations, each for open boundary conditions (OBC) \nand periodic boundary conditions (PBC), and NLCEs. We\nsummarize our results and conclude in Sec.~\\ref{sec:conclusions}.\n\n\n\n\\section{General considerations}\n\\label{sec:gener-cons}\n\nIn this section, we argue that the GE for a translationally invariant\nsystem (we abbreviate the CE [GE] with open and\n  periodic boundary conditions as CE-O [GE-O] and CE-P [GE-P]\n  respectively) has exponentially small finite size corrections.  For\nthat, we make use of a $\\beta$ expansion of the free energy, where\n$\\beta=(k_{B}T)^{-1}$ is the inverse temperature and $k_{B}$ is the\nBoltzmann constant. This kind of expansion has been used extensively\nin the literature to compute partition functions for various\nmodels \\cite{domb60a,domb60b,oitmaa_hamer_book_06}.\n\nConsider the Taylor expansion of the grand partition function $Z\\equiv\n\\Tr e^{-\\beta \\hat{H}}$ (we set $\\mu=0$ for brevity; all the arguments\nbelow are valid for nonzero $\\mu$, which will be required for bosons\nto prevent Bose-Einstein condensation):\n\\begin{equation}\n  Z(\\beta) = \\Tr(1) - \\beta \\Tr(\\hat{H}) + \\frac{\\beta^{2}}{2!}\\Tr(\\hat{H}^{2}) + \\ldots\n\\end{equation}\nWe are interested in $\\ln Z$, from which thermodynamic quantities can\nbe obtained by taking suitable $\\beta$ or $\\mu$ derivatives,\n\\begin{multline}\n  \\label{eq:9}\n  \\ln Z(\\beta) = \\ln\\Tr(1) - \\beta \\frac{\\Tr(\\hat{H})}{\\Tr(1)} + \\\\\n  + \\frac{\\beta^{2}}{2}\n  \\left[\\frac{\\Tr(\\hat{H}^{2})}{\\Tr(1)}-\\frac{\\Tr(\\hat{H})^{2}}{\\Tr(1)^{2}}\\right]\n  + \\cdots\n\\end{multline}\nWe note that\n\\begin{equation}\n  \\label{eq:8}\n  \\frac{\\Tr(\\hat{H}^{n})}{\\Tr(1)} = \\frac{\\Tr(\\hat{H}^{n}e^{-0\\cdot \\hat{H}})}\n  {\\Tr(e^{-0\\cdot \\hat{H}})},\n\\end{equation}\nis an infinite temperature expectation value. At infinite temperature\nall unconnected parts of the system, however close to each other, are\nuncorrelated. Therefore, the expansion in Eq.~\\eqref{eq:9} reduces to\na sum over only the connected graphs that can be embedded in the\nfinite system \\cite{domb60a,sykes66,huang87}. In the CE, the particle\nnumber constraint, i.e., that the total particle number is fixed,\nimplicitly correlates unconnected pieces of a graph, and therefore\nthis simplification does not occur.\n\nFor a system that has no ordered phase or, equivalently, where order\nappears only at $T=0$, the above expansion must converge beyond a\ncertain order (because the correlation length is finite).  Since this\nmust occur for any temperature $\\beta < \\infty$, the convergence must\ncome from the coefficients of the $\\beta$-expansion, i.e., from the\ntraces in Eq.~\\eqref{eq:9}.  In other words, for the expansion $(\\ln\nZ)\/N = a_{0}+a_{1}\\beta + a_{2}\\beta^{2}+\\cdots$, the coefficient\n$a_{n}$ must fall faster than $e^{-n}$ for the series to converge for\nany $\\beta$. The convergence cannot come from cancellation of terms of\nopposite sign, since any such cancellation can work only at some\nfine-tuned value of $\\beta$.\n\nFor a system that has a phase transition between an unordered high\ntemperature phase and an ordered low temperature phase at a finite\ncritical temperature $\\beta_{c}$, the coefficients do not exhibit this\nbehavior -- in the critical phase the correlation length is infinite\nand all orders of this expansion are relevant. The convergence of the\nseries for $\\beta<\\beta_{c}$ instead comes from the fact that\n$\\beta\/\\beta_{c} < 1$.  We verify these arguments in the 1D and 2D\nIsing models.\n\nFrom here on, we assume that we are in a phase where the\n$\\beta$-expansion converges. We will now show that with PBC (when the\nsystem is translationally invariant), all orders of the expansion\nEq.~\\eqref{eq:9} up to the system size (to be properly defined below)\nare identical to those in the thermodynamic limit. We will further\nshow that this is not the case with open boundary conditions.\n\n\\subsection{Periodic boundary conditions}\n\nConsider a system with $N$ sites and periodic boundary conditions (a\nsystem that is translationally invariant). The $\\beta$ expansion of\n$\\ln Z$ is shown in Eq.~\\eqref{eq:9}, in which each term can be\nrepresented by a graph embedded on the finite system. We will call\nthese graphs clusters \\cite{huang87}.  First note that each cluster\nhas $N$ equivalent positions on the lattice since the system is\ntranslationally invariant.  That gives a factor of $N$ that we move to\nthe left hand size in Eq.~\\eqref{eq:9} to get $(\\ln Z)\/N$, i.e., an\nintensive quantity. Let us now consider a cluster with $c$\nsites. First, since we have a cumulant expansion, as discussed above,\nonly connected clusters enter (see, e.g.,\nRefs.~\\cite{huang87,domb60a,oitmaa_hamer_book_06} for details).  If\nthe extent of our cluster in each direction is less than the system\nsize in that directions (say $L$), then the cluster has open boundary\nconditions. Furthermore, even if the system size is increased in any\ndirection, this cluster is present. Hence, this cluster is present in\nthe thermodynamic limit. In general, \\emph{every cluster with $c$\n  sites that, in the finite lattice with $N$ sites, does not wrap\n  around any boundary appears in the infinite system, and vice versa}.\nTherefore the contribution of this cluster in a finite system is\nexactly the same as its contribution in the thermodynamic limit. On\nthe other hand, a cluster with $L$ sites in any given direction wraps\naround a boundary, i.e., it does not appear in the thermodynamic\nlimit. As a result, clusters that wrap around boundaries give\ncontributions that are not present in the thermodynamic limit\n\\cite{lebowitz61}. Hence, the difference between results in the\nthermodynamic limit and in finite-size periodic systems is\n$O(\\beta^{L-p})$, $p$ being determined by the Hamiltonian.  We note\nthat $p$ is $O(1)$ for local Hamiltonians, which are the ones of\ninterest here. Further, based on the earlier argument, we must have\nthat the coefficient at this order falls faster than $e^{-(L-p)}$ or\nthat the expansion parameter ($\\beta\/\\beta_{c}$) is smaller than one.\nTherefore, finite-size errors in a GE calculation of a translationally\ninvariant system at any temperature in the unordered phase are\nsmaller than $O(e^{-L})$, for systems with linear dimension $L$.\n\n\\subsection{Open boundary conditions}\n\nFor a system with OBC, one immediately realizes that clusters do not\nhave $N$ equivalent positions on the lattice. As a result, even if a\ngiven cluster in the finite system appears in the thermodynamic limit,\nits contribution in the finite system will differ from the\nthermodynamic limit. For example, for a lattice model in which the\nHamiltonian is a sum of terms involving only nearest neighbor sites,\nthe term linear in $\\beta$ in Eq.~\\eqref{eq:9} for a system with OBC\nhas a correction $O(A\/2N)$ relative to the result for PBC, where $A$\nis the number of sites in the boundary.  This correction vanishes as\n$1\/L$ with increasing system size.  Complicated geometric and\ncombinatorial factors appear at higher orders, all of which approach\nthe thermodynamic limit result with increasing system size. Hence,\nnone of the coefficients of a $\\beta$ expansion for a finite system\nwith OBC match the result in the thermodynamic limit, and\nfinite size errors in $(\\ln Z)\/N$ are $O(1\/L)$.\n\n\\subsection{Numerical linked cluster expansions}\n\nRather than making calculations of finite systems with periodic or\nopen boundary conditions, and then extrapolating the results to the\nthermodynamic limit, another way to calculate finite-temperature\nproperties of lattice systems in the thermodynamic limit is to use\nNLCEs\n\\cite{rigol_bryant_06,rigol_bryant_07,rigol_bryant_07b,tang_khatami_13}.\nThe idea in this case is to directly use the linked cluster expansion\nof the infinite (translationally invariant) system, for which any\nextensive quantity ${\\cal O}$ per site can be computed as the sum\n\\begin{equation}\n  \\label{eq:nlce1}\n  \\frac{\\cal O}{N} = \\sum_{c}M(c)\\times W_{\\cal O}(c),\n\\end{equation}\nover all connected clusters $c$ that can be embedded in the infinite\nlattice. In Eq.~\\eqref{eq:nlce1}, $M(c)$ is the \\emph{multiplicity} of\ncluster $c$, namely, the number of ways per site in which cluster $c$\ncan be embedded on the lattice, and $W_{\\cal O}(c)$ is the\n\\emph{weight} of the cluster $c$ for observable ${\\cal O}$.  \n$W_{\\cal O}(c)$ is calculated by an inclusion-exclusion principle, one\nsystematically subtracts contributions from the connected subclusters \nof $c$ \\cite{sykes66}\n\\begin{equation}\\label{eq:nlce2}\n  W_{\\cal O}(c) = \\mathcal{O}(c) - \\sum_{s\\subset c}W_{\\cal O}(s).\n\\end{equation} \n${\\cal O}(c)$ is the value of the observable evaluated on the cluster\n$c$. In NLCEs, ${\\cal O}(c)$ is obtained using a full exact\ndiagonalization of the Hamiltonian for cluster $c$.\n\nDue to computational limitations, only a finite number of clusters can\nultimately be calculated in Eq.~\\eqref{eq:nlce1}. Nevertheless, as\nshown in\nRefs.~\\cite{rigol_bryant_06,rigol_bryant_07,rigol_bryant_07b}, NLCEs\ncan converge at lower temperatures than high temperature expansions,\nand sometimes all the way to the ground state for systems with\nunordered ground states. Also, NLCEs can provide very accurate\nresults for temperatures at which exact diagonalization results for\nsystems with periodic boundary conditions suffer from very large\nfinite-size effects. A pedagogical introduction to implementing NLCEs\ncan be found in Ref.~\\cite{tang_khatami_13}.\n\nIn what follows, we compare NLCE results with those obtained in\ncalculations in finite systems with different boundary conditions.\nOur goal is to find how each of them converges to the thermodynamic\nlimit result and which converges the fastest. For NLCEs, the accuracy\nof the results is determined by the size of the largest clusters\nconsidered in the sum in Eq.~\\eqref{eq:nlce1} and the model under\nconsideration.\n\n\\section{Verification in Ising models}\n\\label{sec:verification}\n\nIn this section, we verify the arguments given in\nSec.~\\ref{sec:gener-cons} in the 1D and 2D Ising models, both of which\ncan be solved analytically.\n\n\\subsection{1D Ising model}\n\\label{sec:1d-ising}\n\nIn the thermodynamic limit, the log of the partition function per\nsite, $\\Omega$, can be obtained using the transfer matrix method\n\\cite{huang87}, and is given by (we set $J=1$)\n\\begin{equation}\n  \\label{eq:1}\n  \\Omega(\\beta) = \\ln(e^{\\beta} + e^{-\\beta}).\n\\end{equation}\nThe expansion in powers of $\\beta$ of this result is\n\\begin{equation}\n  \\label{eq:4}\n  \\Omega(\\beta) = \\ln2 + \\frac{\\beta^{2}}{2}-\\frac{\\beta^{4}}{12} \n  + \\frac{\\beta^{6}}{45}-\\frac{17\\beta^{8}}{2520}\n  + \\frac{31\\beta^{10}}{14175}+ \\cdots\n\\end{equation}\n\nThe result for finite systems with periodic boundary conditions is\ngiven by\n\\begin{equation}\n  \\label{eq:2}\n  \\begin{split}\n    \\Omega_L(\\beta) &= \\frac{1}{L}\\ln \\left[(e^{\\beta} +\n      e^{-\\beta})^{L}\n      + (e^{\\beta} - e^{-\\beta})^{L}\\right] \\\\\n    &= \\Omega(\\beta) + \\frac{\\tanh^{L}\\beta}{L} + \\cdots\n  \\end{split}\n\\end{equation}\nSince $0\\leq \\tanh\\beta<1$ for $0\\leq \\beta<\\infty$, the finite size\nerror is indeed faster than exponential. Quantities like the energy,\nwhich are derivatives of the free energy, converge exponentially fast\nwith $L$.\n\nExpanding Eq.~\\eqref{eq:2} in powers of $\\beta$ for different values\nof $L$, we get\n\\begin{equation}\n  \\label{eq:3}\n  \\begin{split}\n    \\Omega_2(\\beta) &= \\ln2 + \\beta^{2} + \\ldots \\\\\n    \\Omega_3(\\beta) &= \\ln2 + \\frac{\\beta^{2}}{2} + \\frac{\\beta^{3}}{3} + \\ldots \\\\\n    \\Omega_4(\\beta) &= \\ln2 + \\frac{\\beta^{2}}{2} + \\frac{\\beta^{4}}{6} + \\ldots \\\\\n    \\Omega_5(\\beta) &= \\ln2 + \\frac{\\beta^{2}}{2} -\n    \\frac{\\beta^{4}}{12} +\n    \\frac{\\beta^{5}}{5} + \\ldots \\\\\n    \\Omega_6(\\beta) &= \\ln2 + \\frac{\\beta^{2}}{2} -\n    \\frac{\\beta^{4}}{12} + \\frac{17\\beta^{6}}{90} + \\ldots\n  \\end{split}\n\\end{equation}\nAs one can see, the results are exact to $O(\\beta^{L-1})$. Naturally,\nas one goes to lower temperatures, the correlation length increases\nand larger systems are required to capture the relevant powers of\n$\\beta$.  It is easy to verify that, with open boundary conditions,\nthe corrections are always $O(1\/L)$.\n\nAs discussed in Sec.~\\ref{sec:gener-cons}, although the coefficients\nin Eq.~\\eqref{eq:3} are exact up to $O(\\beta^{L-1})$, the convergence\nfor \\emph{all} temperatures comes from the fact that they fall off\nrapidly (faster than exponential) with increasing expansion\norder. Figure \\ref{fig:ising-fit} shows a plot of the coefficients of\nexpansion in Eq.~\\eqref{eq:4} along with a fit to $a b^{-L}\/L$ that\ndemonstrates the faster-than-exponential behavior.\n\nTo conclude our discussion of the 1D Ising model, we evaluate the\nfirst few orders of the NLCE for this model (instead of numerical\nexact diagonalization of the clusters, we obtain these results\nanalytically).  First, we evaluate the partition function [$\\ln\nZ_L(\\beta)$] on finite clusters with OBC\n\\begin{equation}\n  \\label{eq:ising-nlce-prop}\n  \\begin{split}\n    \\ln Z_1(\\beta) &= \\ln2, \\\\\n    \\ln Z_2(\\beta) &= \\ln2 + \\ln\\left(e^{\\beta} + e^{-\\beta}\\right),\\\\\n    \\ln Z_3(\\beta) &= \\ln2 + \\ln\\left(e^{2\\beta}+e^{-2\\beta} +\n      2\\right),\n  \\end{split}\n\\end{equation}\nand then carry out the subtractions. The weights are given by [see\nEq.~\\eqref{eq:nlce2}]\n\\begin{equation}\n  \\label{eq:ising-nlce-weights}\n  \\begin{split}\n    W_{1} &= \\ln Z_{1}(\\beta)  = \\ln2, \\\\\n    W_{2} &= \\ln Z_{2}(\\beta)-2W_{1} = \\ln\\left(e^{\\beta} + e^{-\\beta}\\right)- \\ln2,\\\\\n    W_{3} &= \\ln Z_{3}(\\beta)-2W_{2}-3W_{1}= 0.\n  \\end{split}\n\\end{equation}\nHence, the result for $\\Omega$ obtained in calculations including up\nto $n$ sites, $(\\Omega)_n$, is given by [see Eq.~\\eqref{eq:nlce1}]\n\\begin{equation}\n  \\label{eq:ising-nlce-sums}\n  \\begin{split}\n    (\\Omega)_1 &= W_{1}=\\ln2, \\\\\n    (\\Omega)_2 &= W_{1}+W_{2}= \\ln\\left(e^{\\beta} + e^{-\\beta}\\right), \\\\\n    (\\Omega)_3 &= W_{1}+W_{2}+W_{3} = \\ln\\left(e^{\\beta} +\n      e^{-\\beta}\\right).\n  \\end{split}\n\\end{equation}\nIt can be verified that the last result is valid at all higher orders\nin the ``NLCE''.  The thermodynamic limit result is therefore obtained\nby just considering clusters with one and two sites. This is an\ninfinite improvement over the use of the grand canonical ensemble with\nperiodic boundary conditions.  Whereas this infinite gain is specific\nto the 1D Ising model --- the model can, after all, be solved using a\ntwo dimensional transfer matrix, we show in what follows that the fact\nthat NLCEs outperform exact calculations in finite systems appears to\nbe generic.\n\n\\begin{figure}[!tb]\n  \\centering\n  \\includegraphics[width=0.95\\columnwidth]{ising-new.eps}\n  \\caption{(Color online) Coefficients (absolute value) of the\n    $\\beta$-expansion of the free energy vs. the expansion order\n    $l$. We show the coefficients in Eq.~\\eqref{eq:4} for the 1D Ising\n    model, and a fit to the function $a b^{-l}\/l$ with $a=2.0000$ and\n    $b=1.5708$ (the difference between the exact result and\n    $2(\\pi\/2)^{-l}\/l$ vanishes exponentially fast with $l$), and from\n    Eq.~\\eqref{eq:7} for the 2D Ising model. The coefficients in the\n    latter case do not fall off exponentially fast. Because of this,\n    the $\\beta$-expansion only converges for $\\beta < \\beta_{c}$. It\n    is only in this regime that finite-size errors in grand-canonical\n    ensemble calculations of translationally invariant systems are\n    exponentially small in system size. \\textbf{Inset:}\n      Shows the rational part of the coefficients in\n      Eq.~\\eqref{eq:7}. They decrease at first but increase after\n      $O(\\eta^{24})$. See text for further discussion.}\n  \\label{fig:ising-fit}\n\\end{figure}\n\n\\subsection{2D Ising model}\n\\label{sec:2d-ising}\n\nFor the 2D Ising model, $\\Omega$ is given by \\cite{huang87,Onsager_44}\n\\begin{multline}\n  \\label{eq:5}\n  \\Omega(\\beta) = \\ln[2\\cosh(2\\beta)] + \\\\ +\n  \\int_{0}^{\\pi}\\frac{d\\phi}{2\\pi}\n  \\ln\\left[\\frac{1+\\sqrt{1-\\frac{4\\sin^{2}\\phi}{\\cosh^{2}(2\\beta)\\coth^{2}(2\\beta)}}}{2}\n  \\right].\n\\end{multline}\nThe $\\beta$-expansion of this result is given by\n\\begin{equation}\n  \\label{eq:6}\n  \\Omega(\\beta) = \\ln2 + \\beta^{2} + \\frac{5\\beta^{4}}{6} + \\frac{32\\beta^{6}}{45} \n  + \\frac{425\\beta^{8}}{252} + \\ldots\n\\end{equation}\nOne can see that the coefficients become larger than 1 for higher\norders. The correct expansion parameter for models with a\nfinite-temperature transition is $\\beta\/\\beta_{c}$.  For the classical\n2D Ising model on a square lattice, $\\beta_{c} = \\ln(1+\\sqrt{2})\/2$.\nThis gives (with $\\eta \\equiv \\beta\/\\beta_{c}$),\n\\begin{equation}\n  \\label{eq:7}\n  \\Omega(\\beta) = \\ln2 + \\frac{a^{2}\\eta^{2}}{4} + \\frac{5a^{4}\\eta^{4}}{96} + \n  \\frac{a^{6}\\eta^{6}}{90} + \\frac{425a^{8}\\eta^{8}}{64512} + \\ldots\n\\end{equation}\nwhere $a\\equiv\\ln(1+\\sqrt{2})<1$. The coefficients of the $\\eta$-expansion \nare plotted versus the order of the expansion in Fig.~\\ref{fig:ising-fit}.  \nThey do not fall off faster than exponential, or exponentially. \nWe note that the rational part of the coefficients of the \nfirst few orders of the $\\eta$-expansion reported in Eq.~\\eqref{eq:7} is \ndeceiving. They decrease with increasing order of  \nexpansion. This, together with the fact that $a<1$, suggests that the \ncoefficients of the $\\eta$-expansion should fall off \nfaster than exponentially. However, as shown in the inset in \nFig.~\\ref{fig:ising-fit}, the aforementioned rational part \\emph{increases} \nwith increasing order of expansion after $O(\\eta^{24})$. Because of this,\nconvergence in the $\\eta$ expansion is only expected for $\\eta<1$ and \ndoes not come from the coefficients.\n\nWe also calculate $\\ln Z_{L\\times L}$ for small systems with $N=L\\times\nL$ sites in the GE for both periodic and open boundary\nconditions. With PBCs\n\\begin{eqnarray}\n  \\label{eq:2d-ising-g-p}\n  \\ln Z^{\\rm 2D-P }_{1\\times1} &=& \\ln2, \\nonumber\\\\\n  \\ln Z^{\\rm 2D-P}_{2\\times 2} &=& \\ln2 + \\ln\\left(e^{4\\beta} + e^{-4\\beta}+2\\right), \\\\\n  \\ln Z^{\\rm 2D-P}_{3\\times 3} &=& \\ln2 + \\ln\\bigl(e^{-18 \\beta\n  }+9 e^{-10 \\beta }+24 e^{-6 \\beta }+ \\nonumber\\\\\n  && +99 e^{-2 \\beta }+72 e^{2\n    \\beta }+51 e^{6 \\beta }\\bigr),\\nonumber\n\\end{eqnarray}\nwhereas, with OBCs,\n\\begin{eqnarray}\n  \\label{eq:2d-ising-g-o}\n  \\ln Z^{\\rm 2D-O}_{1\\times1} &=& \\ln2 \\nonumber\\\\\n  \\ln Z^{\\rm 2D-O}_{2\\times 2} &=& \\ln2 + \\ln\\left(e^{4\\beta} + e^{-4\\beta}+2\\right) \\\\\n  \\ln Z^{\\rm 2D-O}_{3\\times 3} &=& \\ln2 + \\ln\\bigl(e^{-12 \\beta\n  }+4 e^{-8 \\beta }+16 e^{-6 \\beta }+\\nonumber\\\\ \n  &&+23 e^{-4 \\beta }+48 e^{-2\n    \\beta }+\n  48 e^{2 \\beta }+23 e^{4 \\beta }+\\nonumber\\\\\n  &&+ 16 e^{6 \\beta }+4 e^{8\n    \\beta }+ e^{12 \\beta }+72 \\bigr)\\nonumber.\n\\end{eqnarray}\n\nFor the $\\beta$-expansion of the $3\\times 3$ systems, up to the first\nterm that differs from Eq.~\\eqref{eq:6}, we obtain\n\\begin{equation}\n  \\label{eq:2d-ising-finite-expansion}\n  \\begin{split}\n    \\Omega^{\\rm 2D-O}_{3\\times 3} &= \\ln2 + \\frac{2\\beta^{2}}{3} +\n    \\ldots \\\\\n    \\Omega^{\\rm 2D-P}_{3\\times 3} &= \\ln2 + \\beta^{2} +\n    \\frac{2\\beta^{3}}{3} + \\ldots\n  \\end{split}\n\\end{equation}\nWe see that whereas for OBC the coefficient of the second order term is\nincorrect, for PBC it is correct, i.e., once again GE-P gives results\nthat are correct to $O(\\beta^{L-1})$, with $L=3$ in this case.\n\nFor the NLCE calculation with clusters with up to four sites, we\nobtain\n\\begin{eqnarray}\n  \\label{eq:2d-ising-nlce}\n  &&(\\Omega)_4 = 20 \\ln \\left(e^{-\\beta }+e^{\\beta }\\right)+  54\n  \\ln \\left(e^{- \\beta } e^{2 \\beta }+1\\right)\\nonumber\\\\\n  && \\ -38\n  \\ln \\left(e^{-2 \\beta }+e^{2 \\beta }+2\\right)+\\ln \\left(e^{-4\n      \\beta }+e^{4 \\beta\n    }+6\\right).\\\n\\end{eqnarray}\nExpanding in powers of $\\beta$, and reporting terms up to the first\none that differs from Eq.~\\eqref{eq:6}, we get\n\\begin{equation}\\label{eq:2dinlcee}\n  (\\Omega)_4 = \\ln2 + \\beta^{2} + \\frac{5\\beta^{4}}{6} -\n  \\frac{58\\beta^{6}}{45} + \\ldots\n\\end{equation}\nThe above result is correct to $O(\\beta^{5})$. We must stress that\nEq.~\\eqref{eq:2dinlcee} was obtained in an expansion in which the\nlargest cluster has $N=4$, while\nEqs.~\\eqref{eq:2d-ising-finite-expansion} are for systems with\n$N=9$. The gain is evident.\n\n\\section{Finite temperature perturbation theory to all orders}\n\\label{sec:pert-th}\n\nIn the case of bosons or fermions on a lattice that can be treated by\nfinite-temperature perturbation theory (with a noninteracting theory\nas the unperturbed starting point), a proof that finite-size errors are \nexponentially small to all orders in perturbation theory can be made based \non the momentum-space representation of the Hamiltonian. The proof is \nessentially identical for bosons and fermions, so we focus on the former.\n\nWe consider a generic massive scalar field theory in a 1D lattice,\nwith unit lattice spacing, $L$ sites, and PBC:\n\\begin{equation}\n  \\label{H}\n  \\hat{H}_{s}=\\frac12 \\sum_{j=1}^L\\left[\\hat{\\pi}_j^2\n    +(\\hat{\\varphi}_{j+1}-\\hat{\\varphi}_j)^2 + m^2\\hat{\\varphi}_j^2\\right],\n\\end{equation}\nwhere $[\\hat{\\varphi}_j,\\hat{\\pi}_{j'}]=i\\delta_{jj'}$ and\n$\\hat{\\varphi}_{j+L}\\equiv\\hat{\\varphi}_j$,\n$\\hat{\\pi}_{j+L}\\equiv\\hat{\\pi}_j$.  \nThis Hamiltonian is diagonalized via\n\\begin{equation}\n  \\begin{split}\n    \\hat{\\varphi}_j &= \\frac{1}{\\sqrt{2L}}\\sum_{n=0}^{L-1}\\omega_n^{1\/2}\n    \\left[e^{\\frac{2\\pi i n j}{L}}\\hat{a}_n + e^{-\\frac{2\\pi i n\n          j}{L}}\\hat{a}^\\dagger_n\\right],\\\\\n    \\hat{\\pi}_j &= -\\frac{i}{\\sqrt{2L}}\\sum_{n=0}^{L-1}\\omega_n^{-1\/2}\n    \\left[e^{\\frac{2\\pi i n j}{L}}\\hat{a}_n - e^{-\\frac{2\\pi i n j}{L}}\\hat{a}^\\dagger_n\\right],\n  \\end{split}\n\\end{equation}\nso that\n\\begin{equation} \\label{Hk} \\hat{H}_{s} = \\sum_{n=0}^{L-1}\\omega_n\\hat{a}^\\dagger_n\n  \\hat{a}_n + \\text{constant}\\hfill,\n\\end{equation}\nwhere $[\\hat{a}^{}_n,\\hat{a}^\\dagger_{n'}] = \\delta_{nn'}$, $\\omega_n =\\omega(k_n)$, $k_n = 2\\pi n\/L$, \n$n\\in[0,L-1]$, and\n\\begin{equation}\n  \\label{wk}\n  \\omega(k) = \\sqrt{2(1-\\cos k)+m^2} =\n  \\sqrt{4\\sin^2(k\/2)+m^2}.\n\\end{equation}\n\nWe now want to compute the grand canonical partition function\n\\begin{equation}\n  Z(\\beta,\\mu)\\equiv \\mathop{\\rm Tr}e^{-\\beta (\\hat{H} -\\mu \\hat{N})}\\,,\n  \\label{Z}\n\\end{equation} \nwhere $\\hat{N}=\\sum_n \\hat{a}^\\dagger_n \\hat{a}_n$ is the total number operator.\nWe take the trace in the Fock basis of eigenstates of each $\\hat{a}^\\dagger_n a_n$,\n\\begin{equation}\n  Z_L(\\beta,\\mu) = \\prod_{n=0}^{L-1}\\sum_{N_n=0}^\\infty\n  e^{-\\beta(\\omega_n-\\mu)N_n} = \\prod_{n=0}^{L-1} \\frac{1}{1-e^{-\\beta(\\omega_n-\\mu)}}.\n  \\label{Z1}\n\\end{equation} \nEquivalently,\n\\begin{equation}\n  \\Omega_L(\\beta,\\mu) = \\frac{1}{L}\\ln Z_L(\\beta,\\mu) = \n  \\frac{1}{L}\\sum_{n=0}^{L-1} F(k_n),\n  \\label{OL}\n\\end{equation} \nwhere we have defined\n\\begin{equation}\n  \\label{F}\n  F(k) \\equiv -\\ln[1-e^{-\\beta(\\omega(k)-\\mu)}].\n\\end{equation}\nIn the thermodynamic limit, the sum over $n$ becomes an integral,\n\\begin{equation}\n  \\Omega(\\beta,\\mu)\\equiv\\lim_{L\\to\\infty}\\Omega_L(\\beta,\\mu)= \n  \\frac{1}{2\\pi}\\int_{0}^{2\\pi}dk\\,F(k)\\,.\n  \\label{Oi}\n\\end{equation}\n\nWe now wish to show that $|\\Omega_L(\\beta,\\mu)-\\Omega(\\beta,\\mu)|$ is\nexponentially small in $L$.\n\nWe first note that $F(k)$ is periodic in $k$ with period $2\\pi$.\nTherefore its Fourier expansion takes the form\n\\begin{equation}\n  F(k)=\\sum_{j=-\\infty}^{+\\infty}\\tilde F_j\\,e^{i jk}\\,,\n  \\label{fsum}\n\\end{equation} \nwhere the Fourier coefficients are given by\n\\begin{equation}\n  \\tilde F_j = \\frac{1}{2\\pi} \\int_{0}^{2\\pi}dk\\,e^{-i jk}F(k)\\,.\n  \\label{tildef}\n\\end{equation} \nUsing \\eqs{Oi} and (\\ref{tildef}), we get\n\\begin{equation}\n  \\Omega(\\beta,\\mu) = \\tilde F_0\\,.\n  \\label{Oi1}\n\\end{equation} \nUsing \\eqs{OL} and (\\ref{fsum}), we get\n\\begin{eqnarray}\n  \\Omega_L(\\beta,\\mu) &=& \\frac{1}{L}\\sum_{n=0}^{L-1} F(2\\pi n\/L)\n  = \\frac{1}{L}\\sum_{n=0}^{L-1}\\sum_{j=-\\infty}^{+\\infty}\\tilde\n  F_j\\,e^{2\\pi i jn\/L} \\nonumber\\\\\n  &=& \\sum_{j=-\\infty}^{+\\infty}\\tilde F_j\\biggl[\\frac{1}{L}\n  \\sum_{n=0}^{L-1}e^{2\\pi i jn\/L}\\biggr]  \\label{OL5}\n  \\\\& =& \\sum_{j=-\\infty}^{+\\infty}\\tilde\n  F_j\\delta_{j\\,\\rm{mod}\\,L,\\,0}\n  = \\sum_{j'=-\\infty}^{+\\infty}\\tilde F_{j'\\!L}\\,.\\nonumber\n\\end{eqnarray}\nSubtracting \\eq{Oi1} from \\eq{OL5}, we get \\begin{equation}\n  O_L(\\beta,\\mu)-\\Omega(\\beta,\\mu) = \\sum_{j\\ne 0}\\tilde F_{jL}\\,.\n  \\label{Odiff}\n\\end{equation} \nExamining \\eqs{wk} and (\\ref{F}), we see that if $m>0$ and $\\mu<m$\n(necessary to avoid Bose condensation), then $F(k)$ is continuous and\ninfinitely differentiable for all real $k$.  It then follows from a\ngeneral theorem of Fourier series \\cite{tolstov76} that $\\tilde F_{jL}$ goes\nto zero faster than any power of $|j|L$ as $L\\to\\infty$.  The sum over\n$j$ in \\eq{Odiff} will then be dominated by the $j=\\pm1$ terms.  We\nconclude that $|O_L(\\beta,\\mu)-\\Omega(\\beta,\\mu)|$ is exponentially\nsmall in $L$ if $m>0$ and $\\mu<m$.\n\nWe can verify this explicitly. Again examining \\eqs{wk} and (\\ref{F}),\nwe see that $F(k)$ is changing most rapidly near $k=0$. For $m>0$ and\nsmall enough $k$, $F(k)$ can be approximated via\n\\begin{equation}\n  F(k)\\simeq -\\ln[1-e^{-\\beta(m-\\mu)}e^{-\\beta k^2\/2m}].\n  \\label{F0}\n\\end{equation} \nAssuming $\\mu<m$, we have\n\\begin{equation}\n  F(k)\\simeq \\sum_{n=1}^\\infty\\frac{1}{n}e^{-n\\beta(m-\\mu)}e^{-n \\beta k^2\/2m}\\,.\n  \\label{F2}\n\\end{equation} \nIn this approximation, we get\n\\begin{equation}\n  \\tilde F_{jL} \\simeq\n  \\sqrt{\\frac{m}{2\\pi\\beta}} \\sum_{n=1}^\\infty \\frac{1}{n^{3\/2}}\n  e^{-n\\beta(m-\\mu)}e^{-j^2 L^2 m\/2 \\beta n}\\,.\n  \\label{tF2}\n\\end{equation} \nThe sum over $n$ can be approximated by steepest descent.  For\n$j^2L^2\\gg1\/m(m-\\mu)$, we find\n\\begin{equation}\n  \\tilde F_{jL} \\simeq  \\frac{1}{|j|L} \\exp\\Bigl[-[2m(m-\\mu)]^{1\/2}|j|L\\Bigr]\\,,\n  \\label{tF3}\n\\end{equation} \nThis is exponentially small in $L$, as expected from the general\ntheorem.  The sum over $j$ in \\eq{Odiff} is then dominated by\n$j=\\pm1$, and we have\n\\begin{equation}\n  |\\Omega_L(\\beta,\\mu)-\\Omega(\\beta,\\mu)|\n  \\simeq \\frac{2}{L}\\exp\\Bigl[-[2m(m-\\mu)]^{1\/2}L\\Bigr]\\,,\n  \\label{Odiff2}\n\\end{equation}\nwhich is the behavior observed for the 1D Ising model in\nFig.~\\ref{fig:ising-fit}.\n\nThis proof extends straightforwardly to higher dimensions, assuming\nperiodic boundary conditions in each dimension.\n\nNow consider adding an interaction term to $\\hat{H_s}$, such as\n$g\\sum_j\\varphi_j^4$.  We can compute $\\Omega_L(\\beta,\\mu)$ order by order\nin finite temperature perturbation theory.  Each term is represented by \na connected Feynman diagram \\cite{fetter_bk_03}. The expression for a diagram with $p$ \npropagators and $v$ vertices takes the form \n\\begin{equation} \\frac{1}{L}\\sum_{n_1=0}^{L-1}\n  \\ldots \\frac{1}{L}\\sum_{n_{p-v+1}=0}^{L-1} g^v\n  F(k_{n_1},\\ldots,k_{n_{p-v+1}})\\,.\n  \\label{feyn}\n\\end{equation} \nFor $m>0$ and $\\mu<m$, $F$ is infinitely differentiable in each\n$k_{n_i}$. Hence, one can once again apply the general theorem that says\nthat the difference between \\eq{feyn} and the $L\\to\\infty$ limit must\nbe exponentially small in $L$.\n\nOther models of bosons and\/or fermions can be analyzed in exactly the same\nway, with only the form of $F(k)$ changing. As long as $F(k)$ is infinitely \ndifferentiable, the argument holds.  \n\n\\section{Numerical tests}\n\\label{sec:numerical-tests}\n\nIn this section, we discuss numerical results that support the\ngenerality of the conclusions reached so far. The results are obtained\nusing full exact diagonalization calculations for fermionic systems in\none dimension. We study spinless fermions with nearest-neighbor\nhoppings that are either noninteracting or that have nearest and\nnext-nearest-neighbor interactions. They are described by the\nfollowing generic Hamiltonian\n\\begin{multline}\n  \\label{eq:gen_f_ham}\n  \\hat{H}_{\\rm f} =\n  \\sum_{i}\\Biggl[-t(\\hat{c}^{\\dag}_{i}\\hat{c}^{}_{i+1}+\\text{H.c.})\n  + V \\left(\\hat{n}_{i}-\\frac12\\right)\\left(\\hat{n}_{i+1}-\\frac12\\right) \\\\\n  + V'\\left(\\hat{n}_{i}-\\frac12\\right)\\left(\\hat{n}_{i+2}-\\frac12\\right)\\Biggl],\n  \\qquad\\quad\n\\end{multline}\nwhere $\\hat{n}^{}_{i}=\\hat{c}^{\\dag}_{i}\\hat{c}^{}_{i}$ is the number\noperator.  We present results for different values of the parameters\n$V$ and $V'$, and using both the canonical and grand canonical\nensembles with open and periodic boundary conditions. We also report\nNLCE results.\n\n\\begin{figure}[!t]\n  \\centering\n  \\includegraphics[width=0.98\\columnwidth]{FF_v02.eps}\n  \\caption{(Color online) Comparison of results of canonical and grand canonical\n    ensembles with open and periodic boundary conditions, and NLCEs,\n    for noninteracting fermions hopping on a one dimensional lattice\n    (corresponds to Eq.~\\eqref{eq:gen_f_ham} with $t=1$,\n    $V=V'=0$). The energy difference $\\delta E_l$ (see text) is\n    plotted for two different temperatures in (a) and (b). Panel (c)\n    shows the occupation of the $k=0$ mode, or, equivalently, the\n    correlation $L^{-1}\\sum_{ij}\\langle c^{\\dag}_{i}c_{j}\\rangle$.\n    $\\delta n_{k=0}$ is zero for the grand-canonical ensemble with\n    PBCs, see text.}\n  \\label{fig:free-ferm}\n\\end{figure}\n\nIn all cases, we calculate the energy per site $E_{l}$. For\nnoninteracting fermions, we also compute the occupation of the\nmomentum $k=0$ mode, $n^l_{k=0}\\equiv \\sum_{i,j}\\langle\n\\hat{c}^{\\dag}_{i}\\hat{c}^{}_{j}\\rangle$.  For the interacting models,\nwe compute the the nearest-neighbor single-particle correlation\nfunction $K_l = \\sum_{i}\\langle\n\\hat{c}^{\\dag}_{i}\\hat{c}^{}_{i+1}+{\\rm H.c.}\\rangle$. For the results\nreported, by $l$ we mean the number of sites of the finite system or\nthe number of sites of the largest cluster in the NLCE. We plot the\nfinite size errors $\\delta E_{l}$, $\\delta n^l_{k=0}$, and $\\delta\nK_{l}$, with $\\delta O_{l}\\equiv (O_{l}-O)\/O$ where $O$\n($E,\\,n_{k=0},\\,K$) is either the exact analytic result in the\nthermodynamic limit, when known, or the highest order result from a\nnumerical linked cluster expansion.\n\n\\begin{figure}[!t]\n  \\centering\n  \\includegraphics[width=0.98\\columnwidth]{t1V1_v01.eps}\n  \\caption{(Color online) (a) $\\delta E_{l}$ and (b) $\\delta K_{l}$ for interacting\n    fermions [Eq.~\\eqref{eq:gen_f_ham}] with $t=1$ (unit of energy),\n    $V=1.0,\\,V'=0$, and $T=1.0$.}\n  \\label{fig:t1v1}\n\\end{figure}\n\\begin{figure}[!b]\n  \\centering\n  \\includegraphics[width=0.98\\columnwidth]{t1V1Vp1_v01.eps}\n  \\caption{(Color online) (a) $\\delta E_{l}$ and (b) $\\delta K_{l}$ for interacting\n    fermions [Eq.~\\eqref{eq:gen_f_ham}] with $t=1$ (unit of energy),\n    $V=1.0,\\,V'=1.0$, and $T=1.0$.}\n  \\label{fig:t1v1vp1}\n\\end{figure}\n\nFigure \\ref{fig:free-ferm} reports results for $\\delta E_{l}$ and\n$\\delta n^l_{k=0}$ for noninteracting fermions\n[Eq.~\\eqref{eq:gen_f_ham} with $t=1$ and $V=V'=0$].  Panels (a) and\n(b) show the finite-size errors of the energy at two temperatures, and\n(c) shows the finite-size errors of the zero-mode occupation (the sum\nof all one-particle correlations). The results for the grand-canonical\nenergy in finite systems can be obtained analytically\n\\begin{equation}\n  \\label{eq:10}\n  E_{l} = \\sum_{n=0}^{l-1} \\frac{\\epsilon_{n} e^{-\\beta\\epsilon_{n}}}\n  {1+e^{-\\beta\\epsilon_{n}}},\n\\end{equation}\nwhere $\\epsilon_{n}$ are the single-particle eigenenergies.  The\nmomentum distribution function with periodic boundary conditions is\ngiven by the Fermi-Dirac distribution\n\\begin{equation}\n  \\label{eq:11}\n  n_{k} = \\frac{1}{1+e^{-\\beta\\epsilon_{k}}}.\n\\end{equation}\nWe note that $n_{k}$ for finite systems (for the values of $k$ allowed)\nis exactly the same as in the thermodynamic limit. Hence, $\\delta\nn^l_{k=0}=0$ for the grand canonical ensemble with PBC.  Therefore,\nno error is reported for the GE-P in Fig.~\\ref{fig:free-ferm}(c).  The results in\nthe canonical ensemble are obtained as described in\nRef.~\\cite{rigol_05}.\n\nFor the two temperatures and two quantities reported in\nFig.~\\ref{fig:free-ferm}, one can see that the errors of the GE-O,\nCE-P, and CE-O results decrease as $1\/l$. This is expected and is made\napparent in the plots by comparing those results to the $1\/l$ plots\n(dashed lines) depicted for reference. On the other hand, the GE-P\n(for the energy) and NLCE errors can be seen to decrease exponentially\nwith increasing $l$. For the energy, the NLCE errors are much smaller\nthan the GE-P ones, showing once again that NLCEs generally ($n_{k}$\nfor noninteracting fermions being a counterexample) outperform\ngrand canonical calculations in finite translationally invariant\nsystems.\n\n\\begin{figure}[!t]\n  \\centering\n  \\includegraphics[width=0.96\\columnwidth]{2dmodels_v01.eps}\n  \\caption{(Color online) $\\delta E_{l}$ in NLCE calculations of the Heisenberg model\n    in four different two-dimensional lattices, as indicated. The\n    approach to the thermodynamic limit result is again exponentially\n    fast as one increases the order of the expansion. The results presented \n    here were taken from \n    Refs.~\\cite{rigol_bryant_07,tang_khatami_13,tang_paiva_12,tang_paiva_13}.\n    $J=1$ is taken to be the unit of energy.}\n  \\label{fig:2dmodels}\n\\end{figure}\n\nFigures \\ref{fig:t1v1} and \\ref{fig:t1v1vp1} show the results for: (a)\n$\\delta E_{l}$ and (b) $\\delta K_{l}$ for interacting fermions\n[Eq.~\\eqref{eq:gen_f_ham}] with $(t,V,V')=(1,1,0)$ and $(1,1,1)$,\nrespectively.  In all cases, it is apparent that only the GE-P and\nNLCE errors decrease exponentially fast with $l$, while the GE-O and\nCE errors decrease as $1\/l$. Once again, these results (now for\ninteracting systems) show that the NLCE errors are significantly\nsmaller than those of the GE-P. Here, we have used the highest order\nof the NLCE ($l=18$) as the estimate for the\nthermodynamic limit \\cite{rigol_14}. All results for the interacting\nmodels were obtained using full exact diagonalization of the\nHamiltonian.\n\n\nFinally, although we cannot make an analysis in 2D equivalent to that\npresented for interacting models in 1D (because of the exponential\nscaling of the computational cost combined with the fast increase of $N$), \nwe can still verify that NLCE calculations\nhave exponentially small errors. In Fig.~\\ref{fig:2dmodels}, we show\nresults for $\\delta E_{l}$ for the 2D Heisenberg model in four\ndifferent lattice geometries --- square \\cite{tang_khatami_13}, \nhoneycomb \\cite{tang_paiva_12,tang_paiva_13}, kagome, and triangular\n\\cite{rigol_bryant_07}. In all cases, the error is once again seen to\ndecrease exponentially fast with increasing the number of sites in the\nclusters considered.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe have shown that grand canonical ensemble calculations in\ntranslationally invariant systems that are in unordered phases at\nfinite temperatures have exponentially small finite size errors, whereas\ncanonical ensemble calculations have errors that are power law in system \nsize. Hence, while full exact diagonalization calculations in the\ncanonical ensemble are computationally less demanding than grand canonical \nones, if one is interested in accurately computing quantities in an \nunordered finite-temperature phase, using the grand canonical ensemble is\npreferable. The additional computational cost incurred by\ndiagonalizing all particle sectors (around a factor two in fermionic \nsystems at half filling) for the GE is far less than the speed up\ngained by having to study smaller systems (exponential). Furthermore, we have shown\nthat numerical linked cluster expansions generally have even smaller\nerrors than grand canonical ensemble calculations in translationally\ninvariant systems. The additional cost incurred in the diagonalization\nof many clusters with a given size is far less than the speed up\ngained by having to study clusters that are much smaller than finite\nsystems with periodic boundary conditions. The benefit of using NLCEs\nis most striking in two-dimensional lattices. \n\nThe specific system sizes (cluster sizes) \nrequired to observe exponential convergence in grand canonical calculations \nof systems with periodic boundary conditions (in numerical linked cluster expansions) \ndepend on details such as the model under consideration and the temperatures \nof interest, which set the relevant correlation length. Otherwise, our \nconclusions are completely general.\n\n\\section*{Acknowledgments}\nWe thank Anders Sandvik and Cristian Batista for stimulating\ndiscussions.  This work was supported by the Office of Naval Research\n(D.I. and M.R.), and by NSF Grant PHY13-16748 (M.S.).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe consider the following system of weakly coupled equations of obstacle type\n\\begin{equation}  \\label{system2}\n\\begin{cases}\n\\min (-\\Delta u^1+f^1, u^1-u^2+\\psi^1)=0 \\\\\n\\min (-\\Delta u^2+f^2, u^2-u^1+\\psi^2)=0,\n\\end{cases}\n\\end{equation}\nwith given Dirichlet boundary conditions $ u^i=g^i \\textrm{ on } \\partial\\Omega$.\nThese type of systems arise in optimal switching problems with two switching modes.\nHere $f^1$ and $f^2 $ are the running cost functions corresponding to the switching modes.\nThe functions $\\psi^1$ and $\\psi^2$ are the costs of switching \nfrom one mode to the other. More details on the optimal switching problem are provided in Section 2.1.\n\n\n\n\\par\nThe uniqueness and $C^{1,1}$-regularity of the solutions to such systems \nhave been studied in the literature under the assumption \nthat the switching costs are nonnegative constants, \\cite{EF79}, \\cite{LB83}, \\cite{BL86}. \nIn the paper \\cite{CGT13}\nthe regularity of the solutions to an obstacle type weakly coupled system with first \norder Hamiltonians\nis studied using adjoint methods under the assumption that \neach of the switching costs \nis bounded from below by a positive constant.\n\n\\par\nIn our paper we make only the nonnegative loop assumption. This is a necessary \ncondition for the system to be well-defined. Indeed, let $(u^1, u^2)$ be a solution to \\eqref{system2}, then\n$u^1-u^2+\\psi^1 \\geq 0$ and $u^2-u^1+\\psi^2 \\geq 0 $, which implies\n\\begin{equation} \\label{loop}\n \\psi^1 (x) + \\psi^2 (x) \\geq 0.\n\\end{equation}\n\n\\par\nIn the optimal switching setting, the condition \\eqref{loop}\nprevents the agent from making arbitrary gains by looping, in the sense that\n$\\psi^1 (x) + \\psi^2 (x)$\nis the cost of switching from one mode to the other and immediately\nswitching back. We denote the set where it is possible to switch for \nfree by \n\\begin{equation*}\n  \\mathscr{L}=\\{x \\in \\Omega \\mid \\psi^1 (x) + \\psi^2 (x) = 0\\},\n\\end{equation*}\nand call it free switching or zero loop set.\n\n\n\\par\nBy using the penalization\/regularization method we derive the existence of solutions,\nshowing that through a subsequence the solutions of the penalized system converge to the minimal solution\n$ ( u^1_0,u^2_0)$ to \\eqref{system2}.\nThen we see that the solution $u^i_0 \\in C^{1,\\gamma}$,\nfor every $0<\\gamma <1$ and\n\\begin{equation} \\label{bddlap}\n \\lVert \\Delta u^i_0 \\rVert_{L^\\infty(\\Omega)} \\leq \n \\max_i \\Arrowvert \\Delta \\psi^i \\rVert_{L^{\\infty}(\\Omega)}+\n 3 \\max_i \\Arrowvert f^i \\rVert_{L^{\\infty}(\\Omega)}.\n\\end{equation}\n\n\\par\nThe aim of the paper is to investigate if the solutions are $C^{1,1}$, which is the best regularity\nthat we can hope that the solutions achieve. The structure of our system shows that at some subdomains of $\\Omega$, \nthe regularity of the solutions can be derived by already known $C^{1,1}$-regularity results for the\nobstacle problem. In our discussion we see that the main point is to describe the regularity \nat so called meeting points lying on $\\partial \\mathscr{L}$, the boundary of the zero loop set. \n\n\n\n\\par\nIn the main theorem, Theorem 4, we show that at the meeting points  $x_0 \\in \\partial \\mathscr{L}^0 \\cap \\Omega$\nthe solutions are $C^{2, \\alpha}$, \nunder the assumption that $f^i \\in C^\\alpha $ and $\\psi^i \\in C^{2, \\alpha}$. \nBy $ \\mathscr{L}^0$ we denote the interior of the set $ \\mathscr{L}$, and\nby pointwise $C^{2,\\alpha}$\nregularity we mean uniform approximation with a second order polynomial with the speed $r^{2+\\alpha}$.\n\n\\par\nThe idea of the proof is the same as in deriving the optimal regularity for the no-sign obstacle problem\nin  \\cite{ALS13}.\nThe proof is based on the $BMO$-estimates for $D^2 u^1_0$ and $D^2 u^2_0$ following from \nthe estimate  \\eqref{bddlap}. At the point $x_0$, we consider $r^{2+\\alpha}$-th order rescalings\nof $ u^i_0$ denoted by $v^i_r$, \nand show that these are uniformly bounded in $W^{2,2}(B_1)$. Then, looking at the corresponding system \nfor $ (v^1_r, v^2_r)$,\nwe conclude that the rescalings\nare uniformly bounded in the ball $B_1$.\n\n\\par \nIn the end we justify our assumption $0\\in \\partial \\mathscr{L}^0 $ with a counterexample:\nWe consider a particular system in $\\mathbb{ R }^2$,\nwhere the zero loop set $\\mathscr{L}= \\{0\\}$, then we find an explicit solution,\nthat is not $C^{1,1}$.\n\n\n\n\n\\par\nThe paper is structured as follows: In Section 2 we provide some background material.\nIn Section 3 we use the penalization method to derive the existence of strong solutions, and\nobserve that these are  actually minimal solutions.\nThe main results are presented in the last section, where we prove that the minimal solution is locally\n$C^{1,1}$ if  the zero loop set is the closure of its interior, and provide a\ncounterexample to $C^{1,1}$-regularity when $\\psi^1+\\psi^2 $ has an isolated zero.\n\n\n\n\n\\begin{subsection}{Acknowledgements}\n I would like to thank Prof John Andersson, Prof Diogo Gomes and Prof Henrik Shahgholian\n for several fruitful discussions. In particular, I would like to thank John Andersson\n for the idea of the proof of $C^{2, \\alpha}$-regularity at the meeting points.\n\\end{subsection}\n\n\n\n\n\\begin{section}{Background material}\nIn this section we state some known results, which we use in our discussion, without \ngiving any proofs.\n\n\n\n\\begin{subsection}{Optimal switching problem}\nLet $\\Omega\\subset \\mathbb{R}^n$ be a bounded domain with a smooth boundary.\nWe consider an agent that can be anywhere in $\\Omega$ and in one of\na finite number $m$ of states. \nFor every $1\\leq i\\leq m $, the agent moves in $\\Omega$ according to a diffusion\n\\begin{equation*}\ndx=b_i(x) dt +\\sigma_i(x) dW_t,\n\\end{equation*}\nwhere $W_t$ is a Brownian motion in a suitable probability space, $b_i:\\Omega\\to \\mathbb{R}^n$ and\n$\\sigma_i: \\Omega\\to \\mathbb{R}^{n\\times m}$ are smooth functions. The generator of the diffusions\nis denoted by\n$ L^iv=\\frac{1}{2}  \\sigma_i \\sigma^T_i:D^2 v+b_i \\cdot Dv $.\n\n\n\\par\nThe agent can switch from any diffusion mode to another. \nAt every instant $t $ the agent pays a running cost $f^{i(t)}(x)$, depending on the present state $i(t)$\nand position $x$. Additionally, when changing state $i$ to state $j$ he incurs in a switching cost $-\\psi^{ij}(x)$. \nFinally, when the diffusion reaches the boundary and the agent is in state $i$,\nthe process is stopped and a cost $-g^i(x)$ is incurred.\nAs it is traditional in optimal switching setting, we consider the problem of maximizing a\ncertain profit (the negative of the cost) functional\n\\begin{align*}\nu_i(x)=&\\max_{i(t), i(0)=i}\nE \\Big[\\int_0^{T_{\\partial \\Omega}} f_{i(t)}(x(t))dt \\\\\n&-\\sum_{t\\leq T_{\\partial \\Omega}} \\psi_{i(t^-),i(t^+)}(x(t))+\ng_{i(T_{\\partial \\Omega})}(x(T_{\\partial \\Omega})\\Big], \n\\end{align*}\nwhere \n$T_{\\partial \\Omega}$ denotes the exit time  of $\\Omega$. Additionally, \nthe convention  \n$\\psi_{ii}=0$ is assumed. \n\n\n\\par\nAs it has been discussed in the literature  \\cite{LB83}, \\cite{BL86}, the corresponding value function $u^i$\nsolves the following\nsystem: \n\\begin{equation} \\label{ms}\n\\min_{i} (-L^i u^i+f^i, \\min_j (u^i-u^j+\\psi^{ij}))=0. \n\\end{equation}\nwith boundary conditions $u^i=g^i \\textrm{ on } \\partial \\Omega$.\n\n\\par\nFor the optimal switching problem to be well defined, we need to impose the\nnonnegative loop condition: Let \n $i_{0},i_{1},\\hdots , i_{l}=i_{0}$ be any loop of length $l$, i.e. including $l$ number of states.\n Assume that $ (u^1, u^2,...,u^m)$ is a solution to system \\eqref{ms}, then \n $ u^i-u^j+\\psi^{ij} \\geq 0 $ for any $ i,j \\in \\{ 1,2,..., m \\}$, then after summing the equations\n over the loop, we get\n\\begin{equation*}\n\\sum_{j=1}^{l}\\psi _{i_{j-1},i_{j}}\\geq 0.\n\\end{equation*}\nThis condition is a \nnecessary assumption for the existence of a solution to (3), and \nit prevents the agent from making arbitrary gains by looping.\n\n\n\\par\nIn this paper we consider a system, arising in a model optimal switching problem with only two states.\n\\end{subsection}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{subsection}{The Poisson equation, Calderon-Zygmund estimates}\n\n \n\n\n\\par\nWe start by recalling the definition of the H\\\"{o}lder space $C^{k,\\gamma}$.\nLet us denote the continuity norm \n\\begin{equation*}\n \\lVert u \\rVert _{C(\\overline{ \\Omega})}= \\sup_{x\\in \\Omega }\\lvert u(x)\\rvert,\n\\end{equation*}\nand the H\\\"{o}lder seminorm \n\\begin{equation*}\n [ u ] _{C^{0,\\gamma}(\\overline{ \\Omega })}= \\sup_{x,y\\in \\Omega, x \\neq y}\n \\frac{ \\lvert u(x)-u(y)\\rvert}{ \\lvert x-y \\rvert^\\gamma}.\n\\end{equation*}\n\n\n\n\n\n\\begin{definition}\n The H\\\"{o}lder space $C^{k,\\gamma}(\\Omega)$ consists of all functions $u\\in C^k(\\overline{\\Omega})$\n such that \n\\begin{equation*}\n  \\Arrowvert u \\rVert_{C^{k,\\gamma}(\\Omega)}:= \\sum_{\\lvert \\alpha \\rvert \\leq k}\n   \\Arrowvert D^\\alpha u \\rVert_{C(\\overline{ \\Omega })}+\n   \\sum_{\\lvert \\alpha \\rvert = k}\n   [ D^\\alpha u ]_{C^{0,\\gamma}(\\overline{ \\Omega })} < \\infty.\n\\end{equation*}\n\\end{definition}\n\nThe next theorem states the known regularity of the solutions to the Poisson equation $\\Delta u =f$,\nunder the assumption that $f$ is H\\\"{o}lder continuous, and can be found in the book\n\\cite{GT98} .\n\n\n\\begin{theorem}\n Assume that $f\\in C^{\\gamma} $, then there exists a classical solution to the \n Poisson equation \n \\begin{equation*}\n  \\Delta u= f  \\textrm{ in } \\Omega.\n \\end{equation*}\nMoreover, the solution is locally $ C^{2,\\gamma}(\\Omega)$, and for every $ \\Omega' \\Subset \\Omega$\n \\begin{equation*}\n\\Arrowvert u \\rVert_{C^{2,\\gamma}(\\Omega')} \\leq\nC_{n,\\gamma}(\\Omega') \\left( \\lVert u \\rVert_{C(\\overline{ \\Omega})} + \n\\Arrowvert f \\rVert_{C^{0,\\gamma}(\\Omega)} \\right),\n \\end{equation*}\n where the constant $C_{n,\\gamma}(\\Omega') $ depends on $\\textrm{diam} \\Omega'$ and\n $ \\textrm{dist}(\\Omega', \\partial \\Omega) $.\n\\end{theorem}\n\n \n\\par\nNext let us recall the definition of $BMO $ spaces, and then state the Calderon-Zygmund estimates\nfor the Poisson equation $\\Delta u =f$, when $f\\in L^p$, $1<p \\leq \\infty$.\n\n\\begin{definition}\n We say that a function $u\\in L^2(\\Omega)$ is in $BMO(\\Omega)$ if\n \\begin{equation*}\n  \\lVert f \\rVert^2_{BMO(\\Omega)}:=\n  \\sup_{x\\in U, r>0} \\frac{1}{r^n} \\int_{B_r(x) \\cap \\Omega } \\lvert f(y)-(f)_{r,x}\\rvert^2 dy\n  +\\lVert f \\rVert^2_{L^2(\\Omega)}<\\infty,\n \\end{equation*}\nwhere $(f)_{r,x}$ is the average of $f$ in $ B_r(x) \\cap \\Omega$.\n\\end{definition}\n\n\\par\nThe proofs of the following results can be found in \\cite{GT98} when $p<\\infty$\nand in \\cite{S93} when $p=\\infty$.\n\n\n\\begin{theorem}\n Consider the equation \n \\begin{equation*}\n  \\Delta u= f  \\textrm{ in } B_{2R}.\n \\end{equation*}\n If $f \\in L^p(B_{2R})$ for $ 1< p <\\infty $, then the solution $u\\in W^{2,p}(B_R)$, and\n \\begin{equation*}\n  \\lVert D^2 u\\rVert_{L^p(B_R)}\\leq C_{p,n} \\left( \\lVert f \\rVert_{L^p(B_{2R})} \n  +  \\lVert u\\rVert_{L^1(B_{2R})} \\right)\n \\end{equation*}\n If $f \\in L^\\infty (B_{2R})$, then in general $ u \\notin W^{2,\\infty}(B_R)$, but\n\\begin{equation*}\n  \\lVert D^2 u\\rVert_{BMO(B_R)}\\leq C_{\\infty,n} \\left( \\lVert f \\rVert_{L^\\infty(B_{2R})} \n  +  \\lVert u\\rVert_{L^1(B_{2R})} \\right),\n \\end{equation*}\nhere $ C_{p,n}$, $ C_{\\infty,n} $ are dimensional constants.\n\\end{theorem}\n\n\\end{subsection}\n\n\n\n\n\n\n\n\n\n\n\\begin{subsection}{The obstacle problem} \n\nIn this section we state the regularity of the solution to the following obstacle problem,\n\\begin{equation*}\n \\min (-\\Delta u+f, u-\\psi)= 0 \\textrm{ in } \\Omega\n\\end{equation*}\nwith boundary conditions $u-g\\in W_0 ^{1,2}(\\Omega)$.\n\n\\par\nHere we will omit the variational formulation of the problem, the first\nregularity results and will state the $ C^{1,1}$-regularity of the solutions \nreferring to the book \\cite{PSU12}.\n\n\\par\nIn order to be consistent with the assumptions in our paper, we will \nassume that $f\\in C^\\alpha$ and the obstacle  $\\psi \\in C^{2,\\alpha}$,\nalthough these assumptions can be weekened.\n\n\\begin{theorem}\nAssume that $f\\in C^\\alpha$ and $\\psi \\in C^{2,\\alpha}$, and \n $ u $ solves the obstacle problem\n \\begin{equation*}\n \\min (-\\Delta u+f, u-\\psi)= 0 \\textrm{  a.e. in } \\Omega.\n\\end{equation*}\n Then $ u \\in C^{1,1}(\\Omega')$ for every $ \\Omega' \\Subset \\Omega$, and\n \\begin{equation*}\n  \\lVert u \\rVert_{C^{1,1}(\\Omega')}\\leq C\\left( \\lVert u \\rVert_{L^\\infty(\\Omega)}+\n  \\lVert f \\rVert_{C^{0,\\alpha}(\\Omega)}+ \\lVert \\psi \\rVert_{C^{2,\\alpha}(\\Omega)}\\right),\n \\end{equation*}\nwhere the constant $C$ depends on the dimension and on the \nsubset $ \\Omega' \\Subset \\Omega$.\n \n\\end{theorem}\n\n\n\\end{subsection}\n\n\n\\end{section} \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Existence of $C^{1,\\alpha}$ solutions}\n\n\\par\nWe consider the system \\eqref{system2} with boundary conditions $ u^i=g^i \\textrm{ on } \\partial \\Omega $,\n$g^i \\in C^2$.\nThen we also need to impose the following compatibility condition on the boundary data:\n\\begin{equation} \\label{bc}\n g^1-g^2+\\psi^1 \\geq 0, \\textrm{ and }  g^2-g^1+\\psi^2 \\geq 0 \\textrm{ on } \\partial \\Omega.\n\\end{equation}\nClearly, without the compatibility conditions, there are no solutions to \\eqref{system2}\nachieving the boundary data.\n\n\\par\nWe are interested in deriving $C^{1,1}$-regularity for the solutions to our system,\nwhich is the best regularity one can expect.\nThroughout our discussion we will assume that \n\\begin{equation} \\label{costass}\n f^1, f^2 \\in  C^\\alpha (\\Omega), \\textrm{ and } \\psi^1, \\psi^2 \\in  C^{2,\\alpha} (\\Omega),\n\\end{equation}\nfor some $0 <\\alpha <1$.\nThese are natural assumptions, since $f$ being bounded or continuous, \nis not enough for its Newtonian potential to be $C^{1,1}$. \nWe also  provide a one-dimensional counterexample to the existence of solutions in case\nthe switching costs are not smooth.\n\n\n\\begin{example}[Diogo Gomes]\nConsider the following system in the interval $(-1,1)$ with zero Dirichlet boundary conditions,\n\\begin{equation*}  \n\\begin{cases}\n\\min\\left(-(u_{1})_{xx}, u_1-u_2+(1-|x|) \\cos\\left( \\frac \\pi\n{1-|x|}\\right)\\right)=0 , \\\\\n\\min\\left(-(u_2)_{xx}, u_2-u_1+(1-|x|)(1- \\cos\\left( \\frac \\pi\n{1-|x|}\\right)\\right)=0.\n\\end{cases}\n\\end{equation*}\n Then the value function\nof the corresponding optimal control problem is not finite.\n\\end{example}\n\n\n\\begin{proof}\nIn our example the running costs are identically zero, the switching costs\nsatisfy the nonnegative loop assumption $ \\psi_{1}(x)+\\psi_{2}(x)>0 $ in $ (-1,1)$, and the\ncompatibility condition on the boundary  $ \\psi^{1}(\\pm1)=\\psi^{2}(\\pm1)=0 $.\n\n\\par\nThe example illustrates that when the switching costs are not smooth, then the negative values give infinity\ngrowth to the value function of the corresponding optimal control problem.\nIn order to show this, we choose optimal controls $ i(t)$ as follows: the switching occurs at times $ t_{k} $ where\n$ \\frac \\pi {1-|x(t_{k})|}=\\pi k $: When $ \\frac \\pi {1-|x(t_{k})|}=\\pi k = \\pi (2n+1) $, $n\\in \\mathbb{N}_0$ ,\nwe switch from\nregime 1 to regime 2 gaining $ \\frac 1 {2n+1} $ and for the values $ \\frac \\pi {1-|x(t_{k})|}=\\pi k = 2\\pi n $\nwe switch back from regime 2 to 1 paying zero cost, and so\n\\begin{equation*}\nu_i(x)\\geq - \\sum_{0\\leq t\\leq T_{\\partial \\Omega}} \\psi_{i(t_k),\ni(t_{k+1})}(x(t)) = \\sum \\frac{1}{2n+1}.\n\\end{equation*}\nThen the conclusion follows from the divergence of harmonic series.\n \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Penalization method}\n\n\n\nIn this section we approximate the system \\eqref{system2} with a smooth penalized system.\nLet us take any smooth nonpositive function $\\beta : \\mathbb{R} \\rightarrow (-\\infty,0]$,\nsuch that \n\\begin{align*}\n \\beta (s)=0 \\textrm{ for } s\\geq 0, \\\\\n \\beta (s)<0 \\textrm{ for } s<0 \\textrm{ and } \\\\\n 0<\\beta'(s)\\leq 1 \\textrm{ for } s <0, \\\\\n \\lim_{s \\rightarrow -\\infty}\\beta (s)=-\\infty\n\\end{align*}\nNext we consider the following penalization function $\\beta_\\varepsilon (s)= \\beta (s\/ \\varepsilon)$, \nfor  $ s\\in \\mathbb{R}, \\varepsilon >0$, and the corresponding penalized system\n\\begin{equation} \\label{pensystem}\n\\begin{cases}\n-\\Delta u^1_{\\varepsilon}+f^1+\\beta_\\varepsilon (u^1_\\varepsilon-u^2_\\varepsilon+\\psi^1)=0 \\\\\n-\\Delta u^2_{\\varepsilon}+f^2+\\beta_\\varepsilon (u^2_\\varepsilon-u^1_\\varepsilon+\\psi^2)=0 ,\n\\end{cases}\n\\end{equation}\nwith boundary conditions $ u^i_\\varepsilon=g^i \\textrm{ on } \\partial\\Omega $. \n\n\n\\par\nFor $\\varepsilon >0$ fixed, the penalized system \\eqref{pensystem}\ncan be solved by several methods. In the paper \n\\cite{EF79} the authors use nonlinear functional analysis methods in order to derive the existence \nof classical solutions, that is $u^i_\\varepsilon \\in C^2(\\Omega)$, assuming that the switching costs are positive constants. \nThe proof is rather technical, however it works line for line in our case with variable switching costs,\ntherefore we omit it.\n\n\n\\begin{lemma} \\label{penlemma}\nUnder the assumptions \\eqref{bc} and \\eqref{costass} the solutions to the penalized system \n\\eqref{pensystem}, $u^i_{\\varepsilon}$ satisfy the\nfollowing estimates for every $\\varepsilon>0$\n\\begin{enumerate}\n\n\\item [ i.) ]\n \\begin{equation*}  \n - \\max_i  \\Arrowvert f^i \\rVert_{L^{\\infty}} \\leq -\\Delta u^i_{\\varepsilon} \n \\leq \\max_i \\Arrowvert \\Delta \\psi^i \\rVert_{L^{\\infty}}+\n 3 \\max_i \\Arrowvert f^i \\rVert_{L^{\\infty}}.\n \\end{equation*}\n \n \\item[ ii.) ]\n \\begin{equation*}\n u^1_\\varepsilon-u^2_\\varepsilon+\\psi^1 \\geq -C \\varepsilon \n \\textrm{ and }\n u^2_\\varepsilon-u^1_\\varepsilon+\\psi^2 \\geq -C \\varepsilon \n\\end{equation*}\n\\end{enumerate}\nIn $ ii.) $ the constant $C>0$ depends only on the given data and can \nbe computed explicitly in terms of $\\beta $.\n\\end{lemma}\n\n\\begin{proof}\n For our convenience, let us denote $\\theta^1_\\varepsilon=u^1_\\varepsilon-u^2_\\varepsilon+\\psi^1$ and \n  $\\theta^2_\\varepsilon=u^2_\\varepsilon-u^1_\\varepsilon+\\psi^2$, and observe that $\\theta^1_\\varepsilon $\n  and $\\theta^2_\\varepsilon $ cannot be \n  negative at the same time according to\n  the nonnegative loop assumption. \n  \n  \\par\n  Now let us fix $\\varepsilon >0$, \n  and consider the function $\\beta_\\varepsilon( \\theta^i_\\varepsilon(x))$, $x \\in \\Omega$. \n  It is bounded from above by $0$, our aim is to prove that $\\beta_\\varepsilon( \\theta^i_\\varepsilon(x))$ is\n  bounded from below.\n  Let $x_0=x_0(\\varepsilon)$ be a point of minimum for the function $\\beta_\\varepsilon( \\theta^1_\\varepsilon(x))$,\n  moreover without loss of generality, we may assume that \n  \\begin{equation*}\n   \\min_{i=1,2;x\\in \\overline{\\Omega}} \\beta_\\varepsilon( \\theta^i_\\varepsilon(x))=\n   \\beta_\\varepsilon( \\theta^1_\\varepsilon(x_0))<0.\n  \\end{equation*}\nIf $x_0 \\in \\partial \\Omega$, then $ \\beta_\\varepsilon( \\theta^1_\\varepsilon(x_0))=0$ \naccording to \\eqref{bc}. Therefore $ x_0 \\in \\Omega $ is an interior point, and $ \\beta_\\varepsilon( \\theta^1_\\varepsilon(x_0))<0$.\nThen $ \\theta^1_\\varepsilon(x_0) <0$, and since $\\theta^1_\\varepsilon+ \\theta^2_\\varepsilon \\geq 0$, we get\n$ \\theta^2_\\varepsilon(x_0) \\geq 0 $ consequently $ \\beta_\\varepsilon( \\theta^2_\\varepsilon(x_0))=0$.\nSince $\\beta_\\varepsilon$ is nondecreasing and $\\beta_\\varepsilon(t)<0$ if and only if $t<0$, we \nget that \n\\begin{equation*}\n   \\min_{i=1,2;x\\in \\overline{\\Omega}}  \\theta^i_\\varepsilon(x)=\n   \\theta^1_\\varepsilon(x_0).\n \\end{equation*}\n This implies that $\\theta ^1_\\varepsilon= u^1_\\varepsilon-u^2_\\varepsilon+\\psi^1 $ \n achieves its minimum at an interior point $x_0$,\n hence $\\Delta u^1_\\varepsilon-\\Delta u^2_\\varepsilon+\\Delta \\psi^1\\geq 0$\nat $x_0$. The last inequality together with $-\\Delta u^2_\\varepsilon(x_0)+f^2(x_0)=0$ shows that \n\\begin{equation*}\n \\beta_\\varepsilon( \\theta^1_\\varepsilon(x_0))= \\Delta u^1_{\\varepsilon}(x_0)-f^1(x_0)=\n\\end{equation*}\n\\begin{equation*}\n \\Delta u^1_{\\varepsilon}(x_0)-\\Delta u^2_{\\varepsilon}(x_0) +f^2(x_0)-f^1(x_0)\n \\geq -\\Delta \\psi^1 (x_0)+f^2(x_0)-f^1(x_0).\n\\end{equation*}\nThe estimate above is true for any $\\varepsilon >0$, and therefore it proves the right inequality in $ i.) $. \nThe left inequality in $ i.) $ is a direct consequence of $ -\\beta_\\varepsilon \\geq 0$.\n\n\n\\par \nIn order to prove  $ ii.) $, we\nrecall that $\\lim_{s\\rightarrow -\\infty} \\beta(s)=-\\infty$, and\n$\\beta_\\varepsilon(s)=\\beta(s\/\\varepsilon)$, hence $\\beta_\\varepsilon(\\theta^i_\\varepsilon)$ is\nbounded imples that $ \\frac{\\theta^i_\\varepsilon}{\\varepsilon}$ is uniformly bounded from below \nby a negative constant $-C\\leq 0$. This finishes the proof of point $ ii.) $ in our lemma.\n\\end{proof}\n\n\n\\par\nUsing the Sobolev embedding theorem and Calderon-Zygmund estimates, we can conclude that\nthe functions $u^i_\\varepsilon$ are uniformly bounded in $W^{2,p}$ for every $1<p<\\infty$.\nTherefore through a subsequence $u^i_{\\varepsilon}$ converges to a function $u^i_0$\nlocally weakly in $W^{2,p}$ and strongly in $C^{1,\\gamma}$ for every $0<\\gamma<1$. \n\n\\par\nNow we proceed to prove the existence of solutions to system \\eqref{system2}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{proposition}\nLet $ (u^1_0,u^2_0)= \\lim_{\\varepsilon \\rightarrow 0} (u^1_\\varepsilon,u^2_\\varepsilon)$ \nthrough a subsequence weakly in $W^{2,p}$ and strongly in $C^{1,\\gamma}$.\n Then $(u^1_0,u^2_0)$ solves the following system \n \\begin{equation} \\label{system3}\n\\begin{cases}\n\\min (-\\Delta u^1_0+f^1, u^1_0-u^2_0+\\psi^1)=0, \\\\\n\\min (-\\Delta u^2_0+f^2, u^2_0-u^1_0+\\psi^2)=0,  \\\\\n\\min (-\\Delta u^1_0+f^1, -\\Delta u^2_0+f^2)=0\n\\end{cases}\n\\end{equation}\nin a strong sense, i.e.\n$ u^i-u^j+\\psi^i \\geq 0$ \nand if we have a strict inequality at some point then $u^i$ satisfies $ \\Delta u^i_0=f^i$ in a\nneighborhood of that point, and  $ -\\Delta u^i_0+f^i \\geq 0 \\textrm{ a.e. } $.\n\\end{proposition}\n\n\n\n\\begin{proof}\nThe property $ ii.) $ in Lemma 1, together with the strong convergence in $C^{1,\\gamma}$ shows that\n$ u^1_0-u^2_0+\\psi^1\\geq 0$ and $u^2_0-u^1_0+\\psi^2 \\geq 0$. If $ u^1(x_0)-u^2(x_0)+\\psi^1(x_0) >0$,\nthen the strict inequality\n$u^1_\\varepsilon-u^2_\\varepsilon+\\psi^1>0$   holds in a small ball $B_r(x_0)$, centered at $x_0$\nfor $\\varepsilon>0$ small enough. Then it follows that  $ -\\Delta u^1_\\varepsilon+f^1= 0$ in $B_r(x_0)$, \nand we know that $ \\Arrowvert  \\Delta u^1_\\varepsilon \\rVert_{L^\\infty}$ is uniformly bounded, \ntherefore through a subsequence, \n$ \\Delta u^1_\\varepsilon \\rightarrow \\Delta u^1_0 $  a.e. as $\\varepsilon \\rightarrow 0$, \nconsequently\n  $ -\\Delta u^1+f^1= 0$  a.e. in $B_r(x_0)$. Moreover, since $f^1 \\in C^\\alpha$, we that $u^1$ is\n  a classical solution to\n  $ -\\Delta u^1+f^1= 0$ in the ball $B_r(x_0)$.\n\n\\par\nThe solutions of the penalized system satisfy the equation\n\\begin{equation*}\n \\min(-\\Delta u^1_\\varepsilon+f^1, -\\Delta u^2_\\varepsilon+f^2)=0.\n\\end{equation*}\nAfter passing to a limit through a subsequence, we get the following\n\\begin{equation*}\n \\min(-\\Delta u^1_0+f^1, -\\Delta u^2_0+f^2)=0 \\textrm{ a.e.}.\n\\end{equation*}\n\\end{proof}\n\n\n\\par\nProposition 1 shows that there exists $(u^1_0, u^2_0), u^i_0 \\in W^{2,p}, \\forall p <\\infty$ solving \n\\eqref{system2} in a strong sense. According to Lemma 1, $u^i_0$ has the following property\n\n\\begin{equation}\\label{keyest}\n\\Arrowvert \\Delta u^i_0 \\rVert_{L^\\infty} \\leq\n\\max_i \\Arrowvert \\Delta \\psi^i \\rVert_{L^{\\infty}}+\n 3 \\max_i \\Arrowvert f^i \\rVert_{L^{\\infty}},\n\\end{equation}\nwhich will be relevant for deriving further regularity of solutions.\n\n\n\n\n\\par\n Furthermore, Proposition 1 tells us that the solution we get via the penalization method, \n solves an extra equation,\n which turns out to be very important in the discussion of the uniqueness.\n \n \n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\\subsection{Uniqueness}\nIt has been shown in the paper \\cite{LB83} that if there are no zero loops, then the\nsolution to the system \\eqref{system2} is unique. Here we give a counterexample showing\nthat the uniqueness does not hold in case \nthere are zero loops.\n\n\\begin{example}[Diogo Gomes] \nThe following system\n\\begin{equation} \\label{exun}\n\\begin{cases}\n\\min (-\\Delta u^1-M, u^1-u^2+\\psi)=0 \\\\\n\\min (-\\Delta u^2+M, u^2-u^1-\\psi)=0,\n\\end{cases}\n\\end{equation}\nwith given boundary conditions $u^i=g^i, g^1-g^2+\\psi =0 $ on $\\partial \\Omega $,\nadmits infinitely many solutions, provided $ 2 M > \\Arrowvert \\Delta \\psi \\rVert_{L^{\\infty}} $.\n\\par\nMoreover, \\eqref{exun} admits solutions $u^1, u^2 \\notin C^{1,1}$.\n\\end{example}\n\n\n\\begin{proof}\nLet $ (u^1, u^2)$ be a solution to the system \\eqref{exun}.\nSince both $u^1-u^2+\\psi \\geq 0$  and $u^2-u^1-\\psi \\geq 0$, it follows that\n $u^1-u^2+\\psi \\equiv 0$, therefore $ -\\Delta u^1= -\\Delta u^2+ \\Delta \\psi$.\n \n \\par\n Now let us take any $u^1 \\in W^{2,p}$, $p>n$,  $u^1=g^1$ on $\\partial \\Omega $, such that $ -\\Delta u^1-M \\geq 0$  a.e..\n Then the function $u^2=u^1+\\psi $ satisfies the boundary conditions\n$ u^2=g^2$ on $\\partial \\Omega $, and  $ -\\Delta u^2+M \\geq 0$ a.e. \n since $ 2 M > \\Arrowvert \\Delta \\psi \\rVert_{L^{\\infty}} $. \nThus we get infinitely many solutions of the form $(u^1, u^1+\\psi)$, which may not be $C^{1,1}$.\n\\end{proof}\n\n\n\n\\par\nWe observe that if the zero loop set is empty, then the equation \n$  \\min(-\\Delta u^1+f^1, -\\Delta u^2+f^2)=0$ is satisfied automatically. \nUnder the nonnegative loop assumption, we saw that there exists a \nsolution to system \\eqref{system2} also solving system \\eqref{system3}. Next we show that \nthe system \\eqref{system3} has a unique solution, which is actually the minimal solution to (1).\n\n\\begin{proposition}\n The system \\eqref{system3} has a unique solution $(u^1_0,u^2_0)$ in $W^{2,p}$ for every $p<\\infty$.\n\\end{proposition}\n\n\\begin{proof}\nLet us assume that $(u^1,u^2)$ is a solution to system \\eqref{system3}, then \nthe difference $ U= u^1-u^2 $ solves the following double-obstacle\nproblem in $\\Omega $:\n\\begin{equation*}\n\\begin{cases}\n-\\Delta U +f^1-f^2 \\leq 0 \\textrm{ a.e.   if  } U >-\\psi^1 \\\\\n-\\Delta U +f^1-f^2 \\geq 0 \\textrm{ a.e.   if  } U <\\psi^2  \\\\\n-\\psi^1 \\leq U \\leq \\psi^2,\n\\end{cases}\n\\end{equation*}\nwith boundary conditions $ U= g^1-g^2 $\non $\\partial \\Omega $.\n\n\\par\nIt is well-known that the solution to the double-obstacle problem with given boundary data \nis unique in $W^{2,p}$. Indeed, let $V $ be another solution, then \nwithout loss of generality, we may assume that \n$ \\max_{x\\in \\Omega } (U-V)= U(x_0)-V(x_0)>0$.\nThen in a small ball $B_r(x_0)$, one has $ U-V>0$, and $U-V$\n has a maximum at $x_0$. The inequality $ U>V \\geq -\\psi^1$ imples that\n $U > -\\psi^1$ in $B_r(x_0)$, hence $ -\\Delta U -f^1+f^2 \\leq 0$.\n Similarly, $V < \\psi^2 $ in $B_r(x_0)$ and therefore $ -\\Delta V -f^1+f^2 \\geq 0$.\n After combining the inequalities $ -\\Delta U -f^1+f^2 \\leq 0$ and\n $ -\\Delta V -f^1+f^2 \\geq 0$, we see that $U-V$ is a subharmonic function in the \n ball $B_r(x_0)$. Recalling that $U-V $ has a maximum at an interior point $x_0$, we get a \n contradiction to the maximum principle for subharmonic functions.\n \n\n\n\n\n\n\\par\nNow let us assume that $(v^1,v^2)$ \nis another solution to system \\eqref{system3}, then $ u^1-u^2 \\equiv v^1- v^2 $ in $\\Omega$.\nDenote $ h= u^1-v^1 \\equiv u^2- v^2$ in $\\Omega$, then $h=0$ on $\\partial \\Omega$.\n\n\\par\nNow let us plugg-in $ v^1= u^1-h $ and $ v^2= u^2-h $   to the equation\n \\begin{align*}\n 0=\\min(-\\Delta v^1+f^1, -\\Delta v^2+f^2)=\\\\\n \\min(-\\Delta u^1+f^1, -\\Delta u^2+f^2)+ \\Delta h= \\Delta h \\textrm{        a.e. }.\n \\end{align*}\n Then it follows that $\\Delta h = 0  \\textrm{    a.e. } $ in $\\Omega$, $h\\in W^{2,p}(\\Omega)$,\n for every $ 1 <p<\\infty$, hence\n $ h $ is a harmonic function.  Then the difference $u^i-v^i$ is a harmonic function in \n $\\Omega $, vanishing on the boundary, therefore\n $u^i-v^i \\equiv 0$, according to the maximum principle for harmonic functions.\n\n\\end{proof}\n\n\n\\begin{corollary}\nThe solution to the system \\eqref{system3} is the minimal solution to system \\eqref{system2}, that is\n if $(v^1, v^2)$ solves \\eqref{system2}, then $u^1_0\\leq v^1$ and $u^2_0 \\leq v^2$.\n\\end{corollary}\n\\begin{proof}\n Assume $(v^1, v^2)$ solves \\eqref{system2}\nwith given boundary conditions, and let \n$ \\omega = \\min (-\\Delta v^1+f^1, -\\Delta v^2+f^2)$ , then $ \\omega \\geq 0 \\textrm{ a.e.}$, $ \\omega \\in L^{\\infty}$.\nLet $h$ be the solution to  $\\Delta h = \\omega $ in $\\Omega $ with zero Dirichlet\nboundary conditions on $\\partial \\Omega$. Then according to the weak maximum principle\nfor subharmonic functions, we get that $h \\leq 0 $ in $\\Omega$. \n\n\\par\nNow we note that \nthe pair $(v^1+h, v^2+h)$ solves the system \\eqref{system3} with the same boundary conditions as \n$(u^1_0, u^2_0)$, hence\n$v^1+h=u^1_0 \\textrm{ and }  v^2+h=u^2_0 $. Then the minimality of $(u^1_0, u^2_0)$\nfollows from nonpositivity of $h$.\n\\end{proof}\n\nFrom now on we will be interested in studying the regularity for the minimal solutions.\nAs the Example 2 shows, there is no hope to get $C^{1,1}$-regularity for non-minimal solutions.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Optimal regularity of the solutions}\n\nIn this section we prove that the solution to the system \\eqref{system3} is locally $C^{1,1}$,\n if $\\mathscr{L}= \\overline{\\mathscr{L}^0}$. In particular \nwe study the regularity of the solutions on $ \\partial \\mathscr{L}$, the boundary of the zero-loop set.\n\n\\par\nBefore proceeding to the discussion of $C^{1,1}$-regularity, let us rewrite our system in a more \nconvenient way. We have assumed that $ f^1, f^2 \\in  C^\\alpha$, therefore there exist\n$ v^1, v^2 \\in C^{2,\\alpha}_{loc}$ solving the Poisson equation $\\Delta v^i= f^i$ in $\\Omega$.\nRecall that $ (u^1_0, u^2_0)$ is the solution to system \\eqref{system3}, and define \n$ u^i= u^i_0-v^i $, then $u^i_0$ is as regular as  $u^i$ up to $C^{2,\\alpha}$, and\n $ (u^1, u^2 )$ solves the following system \n\\begin{equation} \\label{system}\n\\begin{cases}\n\\min (-\\Delta u^1, u^1-u^2+\\varphi^1)=0,\\\\\n\\min (-\\Delta u^2, u^2-u^1+\\varphi^2)=0,\\\\\n\\min (-\\Delta u^1,-\\Delta u^2)=0.\n\\end{cases}\n\\end{equation}\nHere $ \\varphi^1= v^1-v^2+\\psi^1$ \nand $ \\varphi^2= v^2-v^1+\\psi^2$ are the new switching cost functions preserving\nthe loop condition $\\varphi^1+\\varphi^2 \\equiv \\psi^1+\\psi^2$, and\n$ \\varphi^1, \\varphi^2 \\in C^{2,\\alpha}_{loc}$. \n\n\\par\nFrom now on we will be focused on studying the regularity  of $(u^1, u^2)$ solving\nthe system \\eqref{system}.\n\n\n\\par\nWe define the open set $ \\Omega_1:= \\cup B_r(x_0, u^1)$, where the union \nis taken over the balls $ B_r(x_0, u^1)$, such that $ -\\Delta u^1 > 0$ a.e. \nin $B_r(x_0, u^1)$. Similarly we define the set $\\Omega_2$ corresponding to the function \n$u^2$, and let $\\Omega_{12}= \\Omega \\setminus \\overline{ \\Omega_1 \\cup \\Omega_2}$.\nThen $\\Omega_1$, $\\Omega_2 $ and $\\Omega_{12}$\nare disjoint open sets, and since $\\varphi^1, \\varphi^2 \\in C^{2,\\alpha}$,\n\\begin{align*}\n -\\Delta u^1= \\Delta \\varphi^1 >0, -\\Delta u^2= 0  \\textrm{  in  } \\Omega_1, \\\\\n -\\Delta u^2= \\Delta \\varphi^2 >0, -\\Delta u^1= 0  \\textrm{  in  } \\Omega_2,  \\\\\n -\\Delta u^1=0, -\\Delta u^2=0 \\textrm{  in  } \\Omega_{12}.\n\\end{align*}\n\n\\par\nIn the set $\\Omega \\setminus \\overline{\\Omega_1}$ we get \n$-\\Delta u^1=0$, and the function $u^2$ solves the\n obstacle problem $\\min(-\\Delta u^2, u^2-u^1+\\varphi^2)= 0$, with a $C^{2, \\alpha}$ obstacle\n $ u^1-\\varphi^2$. Therefore $u^2$ is locally $C^{1,1}$ in $  \\Omega \\setminus \\overline{\\Omega_1} $.\n Similarly we get that $u^2$ is locally $C^{1,1}$ in $  \\Omega \\setminus \\overline{\\Omega_2} $.\n\n \n \n \\par\n Next we need to study the regularity of the solution in a neighborhood of the set \n $\\partial \\Omega_1 \\cap \\partial \\Omega_2 \\cap \\Omega$. Let us note that it is contained in\n the zero loop set, $\\partial \\Omega_1 \\cap \\partial \\Omega_2 \\subset \\mathscr{ L}$, since \n $ u^1-u^2+\\varphi^1 \\equiv 0 \\textrm{ in } \\Omega_1$ and\n $ u^2-u^1+\\varphi^2 \\equiv 0 \\textrm{ in } \\Omega_2$.\n In the interior of the zero loop set the system \\eqref{system} reduces to the equation \n \\begin{equation} \\label{Leq}\n -\\Delta u^1 = (\\Delta \\varphi^1)^+, u^2=u^1+\\varphi^1 \\textrm{ in } \\mathscr{ L }^0.\n \\end{equation}\n From the classical theory, solutions to the equation \\eqref{Leq} are locally $C^{2, \\alpha}$\n if $\\Delta \\varphi^1  \\in C^{ \\alpha}$. So in a neighborhood of the points \n $x \\in \\partial \\Omega_1 \\cap \\partial \\Omega_2 \\textrm{ and } x \\in \\mathscr{ L}^0 $,\n the solution is $C^{2, \\alpha} $.\n \n \\par\n It remains to study the regularity of $(u^1, u^2)$ at the points\n $ x_0 \\in \\partial \\Omega_1 \\cap \\partial \\Omega_2 \\cap \\partial \\mathscr{ L}$, called a \n ''meeting'' point. In this section we show that $u^1$ and $u^2$ are actually $C^{2,\\alpha}$-regular at such points.\n \n \\par\n For simplicity, let us study the system locally in the unit ball $B_1$, assuming that\n $0\\in \\partial \\mathscr{L} \\cap \\partial \\Omega_1 \\cap \\partial \\Omega_2$. \n We can always come to such a situation with a change of variables.\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{Blow-up procedure}\nAssume that $(u^1, u^2)$ solves system \\eqref{system} in the unit ball $B_1$, and\n$0\\in \\partial \\mathscr{L} = \\partial \\mathscr{L}^0 $ is a meeting point, and let us study the regularity of the \nfunctions $u^1$ and $u^2$ at $0$.\n\n\n\\begin{definition} \\label{appol}\nFor a function $u \\in W^{2,2}$, define $\\Pi(u(x),r)=p_r(x) $, where\n$p_r(x)= x \\cdot A_r\\cdot x+b_r\\cdot x+c_r$ is a second order polynomial with the matrix $A_r$, vector $b_r$ \nand scalar $c_r$ minimizing the following expression\n\\begin{equation*}\n \\int_{B_r}\\left( |D^2u-2A_r|^2+|\\nabla u -b_r|^2 +|u-c_r|^2 \\right)dx.\n\\end{equation*}\n\\end{definition}\n\nThen $ \\Pi (u(rx),1)=p_r(rx)$, and\nit is easy to see that \n\\begin{equation*}\n p_r(x)=\\frac{1}{2} x \\cdot (D^2u)_r \\cdot x+ (\\nabla u)_r\\cdot x + (u)_r,\n\\end{equation*}\nwhere $ (u)_r:=(u)_{r,0}$, and $(u)_{r,x_0}$ is the average of $ u $ over the ball\n$B_r(x_0)= \\{ x \\in \\mathbb{R}^n \\mid \\lvert x-x_0 \\rvert <r \\}$,\n\\begin{equation*}\n(u)_{r,x_0}=\\frac{1}{\\lvert B_r\\rvert} \\int_{B_r(x_0)}u.\n\\end{equation*}\n\n\\par\nNow let $0 <r<1 $ be a real number, and define \n\\begin{equation*}\n v^i_r(x)= \\frac{u^i(r x)-\\Pi(u^i(r x),1)}{S(r)},\n\\end{equation*}\nwhere $S(r)$ is chosen such that $\\max_i \\Arrowvert D^2v^i_r \\rVert_{L^2(B_1)}=1$.\nOur aim is to describe the rate of convergence of $S(r)$ as $r$ goes to zero.\n\n\n\n\n\n\n\n\n\\par\nIt follows immediately from our definition of $ S(r)$, and $BMO$-estimates \nthat  $\\frac{S(r)}{r^2}$ is uniformly bounded from above.\nIn order to show this, let us recall that\n $ \\Arrowvert \\Delta u^i \\rVert_{L^\\infty} \\leq \\max_ i \\Arrowvert \\Delta \\varphi^i \\rVert_{L^\\infty}$,\nhence\n$D^2 u^i \\in BMO $ locally, with the following estimate\n\\begin{equation*}\n \\Arrowvert D^2 u^i \\rVert_ {BMO(B_{\\frac{1}{2}})}\\leq C(\\max_ i \\Arrowvert \\Delta \\varphi^i \\rVert_{L^\\infty}\n +\\Arrowvert u^i \\rVert_{L^2(B_1)}).\n\\end{equation*}\n\nWithout loss of generality, we may assume that $\\Arrowvert D^2 v^1_r \\rVert_{L^2(B_1)}=1$, for a fixed $r>0$, then \n\\begin{equation*}\n \\frac{S(r)}{r^2}=\\Arrowvert D^2 u^1(r x)-2A^1_{r} \\rVert_{L^2(B_1)}.\n\\end{equation*}\n\n\\par\nA change of variable will give us\n\\begin{align*}\n \\left(\\frac{S(r)}{r^2} \\right)^2=\\frac{1}{r^n} \\int_{B_{r}}|D^2u^1(y)-2A^1_{r}|^2dy =\\\\\n \\frac{1}{r^n} \\int_{B_{r}}|D^2u^1(y)-(D^2u^1)_{r}|^2 dy\\leq \n C \\left( \\max_ i \\Arrowvert \\Delta \\varphi^i \\rVert_{L^\\infty}+\\Arrowvert u^1 \\rVert_{L^2(B_1)}\\right)^2,\n\\end{align*}\ntherefore $ \\forall r<\\frac{1}{2} $\n\\begin{equation}\n \\frac{S(r)}{r^2}\n  \\leq C \\left(\\max_ i \\Arrowvert \\Delta \\varphi^i \\rVert_{L^\\infty}+\n\\max_i \\Arrowvert u^i \\rVert_{L^2(B_1)} \\right) :=C_0.\n\\end{equation}\n\n \\par\nSo $S(r)$ has at least quadratic decay  as $r\\rightarrow 0$.\n Next we improve the estimate, showing that actually \n $S(r) \\leq C_0 r^{2+\\alpha }$ for $r>0$ small enough.\n \n \n \n \n \n \\begin{proposition}\n Let $\\varphi^1, \\varphi^2 \\in C^{2, \\alpha}$ for some $0<\\alpha<1$, and $\\mathscr{L}= \\overline {\\mathscr{L}^0}$,\n then the function $\\frac{S(r)}{r^{2+\\alpha}}$ is uniformly bounded as $r$ goes to zero\n \\begin{equation} \\label{main}\n \\frac{S(r)}{r^{2+\\alpha}}  \\leq C \\left( \\max_ i \\Arrowvert \\Delta \\varphi^i \\rVert_{L^\\infty}+\n\\max_i \\Arrowvert u^i \\rVert_{L^2(B_1)} \\right),\n\\end{equation}\nwhere $C$ is a dimensional constant.\n\\end{proposition} \n\n\\begin{proof}\nLet us start with an important observation: The assumptions $ 0\\in \\partial \\mathscr{L} \\cap \\partial \\Omega_1 \\cap\n\\partial \\Omega_2$,\n$\\mathscr{L}=\\{ \\varphi^1+\\varphi^2 =0 \\}$, $\\mathscr{ L}= \\overline{\\mathscr{ L}^0}$ and \n$ \\varphi^1, \\varphi^2 \\in C^{2, \\alpha}$ imply that $ \\Delta \\varphi^1(0)+\\Delta \\varphi^2(0)=0 $. \nOn the other hand $ \\Delta \\varphi^1>0 $ in $\\Omega_1$ and \n$ \\Delta \\varphi^2>0 $ in $\\Omega_2$, therefore $ \\Delta \\varphi^1(0) =\n\\Delta \\varphi^2(0)=0$.\n\n\\par\nNext we show that  $\\varphi^i \\in C^{2, \\alpha}$ together with $\\Delta \\varphi^i(0)=0$, provide the growth \nestimate \\eqref{main}. The proof is based on an argument of contradiction, assume that $\\frac{S(r)}{r^{2+\\alpha}}$\nis not bounded, \nthen there exists a sequence\n$r_k\\rightarrow 0$ as $ k\\rightarrow\\infty $, such that $S(r_k)=kr_k^{2+\\alpha}$ and $S(r)\\leq kr^{2+\\alpha}$ for\nall $r \\geq  r_k$. Our aim is to study the convergence of the sequence $ v^i_{k}:=v^i_{r_k}$ as $k\\rightarrow\\infty$.\nFor that we will need some basic properties of the functions $v^i_r$, where $ 0<r<1$.\n\n\n\n\n\n\n\\par\nAccording to the definition of $S(r)$, $ \\Arrowvert D^2 v^i_r \\rVert_{L^2(B_1)}\\leq1$ for $r<1$.\nThen applying Poincare's inequality for  the function $\\nabla v^i_r$ in the unit ball $B_1$,\nwe get $ \\Arrowvert \\nabla v^i_r \\rVert_{L^2(B_1)}\\leq C_n$ for every $0<r<1$, since\n $(\\nabla v^i_r)_1= 0$. Next we study the average of $v^i_r$ in the unit ball\n \\begin{align*}\n(v^i_r)_1=\\frac{(u^i)_{r}-\\frac{n}{2} r^2 \\cdot tr(D^2 u^i)_{r} \\fint_{B_1}x_1^2 dx -\nr \\fint_{B_1} x \\cdot (\\nabla u^i)_{r}dx\n-(u^i)_{r}}{S(r)} \\\\\n= -\\alpha_n \\frac{r^2 \\left( \\Delta u^i (r x) \\right)_1}{S(r)},\n\\end{align*}\nwhere $\\alpha_n= \\frac{n}{2}\\fint_{B_1}x_1^2 dx$ is a dimensional constant.\nNow let us recall that $ (u^1, u^2)$ solves \\eqref{system},\nand therefore \n\\begin{align*}\n0 \\leq -\\Delta u^i(r x) \\leq \\max (0, \\Delta \\varphi^i (r x))\\leq \n\\Arrowvert \\varphi^i \\rVert_{C^{2,\\alpha}} r^\\alpha \\lvert x \\rvert^\\alpha, \n\\end{align*}\nsince $ \\Delta \\varphi^i (0) =0$, $\\varphi^i \\in C^{2, \\alpha}$. \nHence we get $ 0\\leq (v^i_r)_1 \\leq C_n \\frac{r^{2+\\alpha}}{S(r)} \\Arrowvert \\varphi^i \\rVert_{C^{2,\\alpha}}$,\nwhere $C_n>0$ is a dimensional constant.\nNext we apply Poincare's inequality one more time,\n$ \\Arrowvert v^i_r- (v^i_r)_1 \\rVert_{L^2(B_1)} \\leq C_n \\Arrowvert \\nabla v^i_r \\rVert_{L^2(B_1)}$.\n Therefore  we may conclude that  \n\\begin{equation} \\label{w22}\n \\Arrowvert \\nabla v^i_r \\rVert_{L^{2}(B_1)}\\leq C,\n \\Arrowvert v^i_r \\rVert_{L^{2}(B_1)}\\leq C\\left(1+  \n \\lVert \\varphi \\rVert_{C^{2,\\alpha}}\\frac{r^{2+\\alpha}}{S(r)}\\right),\n\\end{equation}\nfor every $ 0<r<1$, where the constant $C>0$ depends only on the dimension.\n\n\n\\par\nNext by using \\eqref{w22} we want to estimate the  $\\Arrowvert v^i_k \\rVert_{W^{2,2}(B_R)} $ \nfor $ R < 1\/r_k $ as $k\\rightarrow\\infty$.\nLet us start by looking at the expressions $ \\lvert A^i_{2^lr_k}-A^i_{2^{l-1}r_k} \\rvert$,\nwhere $l\\in \\mathbb{N}$ and $r_k<1$ are chosen such that $s=r_k 2^{l-1}\\leq \\frac{1}{4}$.\nIt follows from Minkowski's inequality, that\n \\begin{align*}\n \\lvert A^i_{2s}-A^i_s \\rvert \\leq {\\left( \\fint _{B_1} |D^2u^i(xs)-2A^i_s|^2\\right)}^{\\frac{1}{2}}+\n   \\left(\\fint _{B_1} |D^2u^i(xs)-2A^i_{2s}|^2 \\right)^{\\frac{1}{2}}\\\\\n \\leq \\frac{S(s)}{s^2}+2^{\\frac{n}{2}} \\left(\\fint _{B_{\\frac{1}{2}}} |D^2u^i(2xs)-2A^i_{2s}|^2 \\right)^{\\frac{1}{2}} \n\\leq \\frac{S(s)}{s^2}+2^{\\frac{n}{2}}\\frac{S(2s)}{4s^2}.\n \\end{align*}\nHence\n\\begin{equation*}\n \\lvert A^i_{2^lr_k}-A^i_{2^{l-1}r_k} \\rvert \\leq  k(1+2^{\\frac{n}{2}+\\alpha})(r_k 2^{l-1})^\\alpha,\n\\end{equation*}\nprovided $r_k 2^{l-1} \\leq \\frac{1}{4}$.\n\n\\par\nNow let us take any $m \\in \\mathbb{ N }$ such that $ 2^{m+1} r_k \\leq 1$, then\n\\begin{align*}\n  \\left(\\int _{B_{2^m}}|D^2 v^i_k(x)|^2dx \\right)^{\\frac{1}{2}}=\\frac{r_k^2}{S(r_k)}\n  \\left(\\int _{B_{2^m}}|D^2 u^i(r_k x)-2A^i_{r_k}|^2dx \\right)^{\\frac{1}{2}}\\\\  \n  \\leq \\frac{ 2^{\\frac{mn}{2}}}{k r_k^{\\alpha}}\n  \\left(\\int _{B_1}|D^2 u^i(2^m r_k x)-2A^i_{r_k}|^2dx \\right)^{\\frac{1}{2}} \\\\\n   \\leq \\frac{ 2^{\\frac{mn}{2}}}{k r_k^{\\alpha}}\n  \\left(   \\left(\\int _{B_1}|D^2 u^i(2^m r_k x)-2A^i_{2^m r_k}|^2dx \\right)^{\\frac{1}{2}}+\n  \\lvert A^i_{2^mr_k}-A^i_{r_k} \\rvert \\right)\\\\\n    \\leq \\frac{ 2^{\\frac{mn}{2}}}{k r_k^{\\alpha}}\n    \\left(  \\frac{S(2^m r_k)}{(2^m r_k)^2} +\\sum_{j=1}^{m}\\lvert A^i_{2^jr_k}-A^i_{2^{j-1}r_k} \\rvert \\right)\\\\\n\\leq \\frac{ 2^{\\frac{mn}{2}}}{k r_k^{\\alpha}} \\left(k 2^{m \\alpha }r_k^\\alpha\n+  k 2^n r_k^\\alpha \\sum_{j=1}^{m} 2^{\\alpha (j-1)} \\right) \\leq\n2^{n+1} 2^{m\\left( \\frac{n}{2}+\\alpha \\right)}.\n \\end{align*}\n\n \\par\nFor every $R<\\frac{1}{2r_k}$ we can find an $m \\in \\mathbb{N }$ such that \n$2^{m-1}\\leq R<2^m $, and then applying the estimates above, we get \n\\begin{equation*}\n \\int _{B_R}|D^2 v^i_k(x)|^2dx\\leq C_n R^{n+2\\alpha},\n\\end{equation*}\nfor every $R<\\frac{1}{2r_k}$, where $C_n$ is a dimensional constant.\n Then we can also show that $ \\Arrowvert \\nabla v^i_{r_k} \\rVert_{L^2(B_R)}$ and\n $ \\Arrowvert v^i_{r_k} \\rVert_{L^2(B_R)}$ are bounded by a constant depending only on $R $.\n Indeed, applying the corresponding estimates for $A^i_{r_k}$, and the first inequality in \\eqref{w22}, we get\n \\begin{equation*}\n  \\lvert b^i_{2^l r_k}- b^i_{2^{l-1}r_k}\\rvert \\leq C_{n,\\alpha} k(r_k 2^{l})^{1+\\alpha},\n \\end{equation*}\nand therefore\n\\begin{equation*}\n   \\lvert b^i_{R r_k}- b^i_{r_k}\\rvert \\leq C_{n,\\alpha} k (r_k R )^{1+\\alpha}.\n\\end{equation*}\n\nThen the Poincare's inequality in a ball $B_R$ implies that\n\\begin{equation*}\n \\lVert \\nabla v^i_k -( \\nabla v^i_k)_R \\rVert_{L^2(B_R)} \\leq C_n R \\lVert D^2 v^i_k \\rVert_{L^2(B_R)},\n\\end{equation*}\nand\n\\begin{equation*}\n \\lVert  v^i_k -( v^i_k)_R \\rVert_{L^2(B_R)} \\leq C_n R \\lVert \\nabla v^i_k \\rVert_{L^2(B_R)},\n\\end{equation*}\nwhere \n\\begin{align*}\n ( \\nabla v^i_k)_R= \\frac{r_k}{S(r_k)} ( \\nabla u^i_{r_k}-r_k A^i_{r_k} \\cdot x -b^i_{r_k} )_R \\\\\n= \\frac{r_k}{S(r_k)} \\left(  (\\nabla u^i )_{R r_k}- b^i_{r_k}\\right)\n = \\frac{r_k}{k r_k^{2+\\alpha}} \\left(  b^i_{ R r_k}- b^i_{r_k} \\right),\n\\end{align*}\nand\n\\begin{align*}\n (v^i_k)_R=\\frac{1}{S(r_k)} \\left( c^i_{R r_k} -c^i_{r_k} +\\frac{n}{2} r_k^2(\\Delta u^i)_{r_k}(x_1^2)_R \\right).\n\\end{align*}\n\n\\par\nNext let us observe that the second inequality in \\eqref{w22}, with the corresponding estimates for \n$A^i_r$ and $b^i_r$ imply that\n\\begin{equation*}\n \\lvert c^i_{Rr_k}-c^i_{r_k}\\rvert \\leq C k(R r_k)^{2+\\alpha}.\n\\end{equation*}\n\n\\par\nThen it follows from the triangle's inequality that\n\\begin{equation*}\n \\lVert \\nabla v^i_k \\rVert_{L^2(B_R)} \\leq  C \\left(  R^{\\frac{n}{2}+1+\\alpha}+\n R^{\\frac{n}{2}}\\frac{r_k}{k r_k^{2+\\alpha}}\\lvert b^i_{Rr_k}-b^i_{r_k}\\rvert \\right) \\leq C_{n}R^{\\frac{n}{2}+1+\\alpha},\n\\end{equation*}\nand also\n\\begin{align*}\n \\lVert v^i_k\\rVert_{L^2(B_R)}\\leq C \\left( R \\lVert \\nabla v^i_k \\rVert_{L^2(B_R)}+\n R^{\\frac{n}{2}}(v^i_k)_R \\right) \\leq \\\\\n C\\left( R^{\\frac{n}{2}+2+\\alpha} +R^{\\frac{n}{2}} \\frac{\\lvert c^i_{Rr_k}-c^i_{r_k}\\rvert}{S(r_k)} +\n  \\lVert \\varphi \\rVert_{C^{2,\\alpha}} R^{\\frac{n}{2}+2} \\frac{r_k^{2+\\alpha}}{S(r_k)}\\right)\n \\leq C' R^{\\frac{n}{2}+2+\\alpha}.\n\\end{align*}\n\n\n \\par\nTherefore we have shown that the sequence $v^i_k$ is locally uniformly bounded in $W^{2,2}$, hence \n through a subsequence, $v^i_k$ converges weakly in $W^{2,2}(B_R)$, and strongly in $W^{1,2}(B_R)$, denote \n $ v^i_0= \\lim_{k\\rightarrow\\infty}v^i_k$ for $i=1,2$.\n Then the weak convergence of the second order derivatives implies that \n \\begin{equation*}\n   \\int _{B_R}|D^2 v^i_0(x)|^2dx \\leq \\limsup_{k \\rightarrow \\infty} \n \\int _{B_R}|D^2 v^i_k(x)|^2dx ,\n \\end{equation*}\n and therefore\n \\begin{equation} \\label{impest}\n \\int _{B_R}|D^2 v^i_0(x)|^2dx \\leq C_n R^{n+2\\alpha}.\n\\end{equation}\n\n\n\\par\nNext we describe further properties of the limit functions, $v^1_0$ and  $v^2_0$.\nRecall that \n\\begin{equation*}\n v^i_k(x)=\\frac{u^i(r_k x)-\\Pi(u^i(r_k x),1)}{S(r_k)},\n\\end{equation*}\nthen we have\n\\begin{equation*}\n -\\Delta v^i_k (x)=\\frac{r_k^2}{S(r_k)} \\left( -\\Delta u^i(r_k x)+tr A^i_{r_k} \\right).\n\\end{equation*}\nLet us denote\n\\begin{align*}\n q^1_k(x)=\\frac{p^1_{r_k}(r_k x)-p^2_{r_k}(r_k x)+\\varphi^1(r_k x)}{S(r_k)}, \\textrm{ and } \\\\\n  q^2_k(x)=\\frac{p^2_{r_k}(r_k x)-p^1_{r_k}(r_k x)+\\varphi^2(r_k x)}{S(r_k)}.\n\\end{align*}\n\nThen $ (v^1_k, v^2_k)$ is a strong solution to the following system\n\\begin{equation*}\n\\begin{cases}\n\\min (-\\Delta v^1_k-\\frac{tr A^1_{r_k}}{k r_k^\\alpha}, v^1_k-v^2_k+q^1_k)=0 \\\\\n\\min (-\\Delta v^2_k-\\frac{tr A^2_{r_k}}{k r_k^\\alpha}, v^2_k-v^1_k+q^2_k)=0 \\\\\n\\min (-\\Delta v^1_k-\\frac{tr A^1_{r_k}}{k r_k^\\alpha},-\\Delta v^2_k-\\frac{tr A^2_{r_k}}{k r_k^\\alpha})=0,\n\\end{cases}\n\\end{equation*}\ntherefore\n\\begin{equation*}\n -\\Delta v^1_k(x)=\n \\begin{cases}\n  \\frac{tr A^1_{r_k}}{k r_k^\\alpha}+\\frac{\\Delta \\varphi^1(r_kx)}{k r_k^\\alpha},  \\textrm{    if }\n  r_k x \\in \\Omega_1 \\\\\n \\frac{tr A^1_{r_k}}{k r_k^\\alpha},  \\textrm{   otherwise },\n \\end{cases}\n\\end{equation*}\nand\n\\begin{equation*}\n -\\Delta v^2_k(x)=\n \\begin{cases}\n   \\frac{tr A^2_{r_k}}{k r_k^\\alpha}+\\frac{\\Delta \\varphi^2(r_kx)}{k r_k^\\alpha},\n   \\textrm{    if } r_k x \\in \\Omega_2 \\\\\n \\frac{tr A^2_{r_k}}{k r_k^\\alpha},  \\textrm{    otherwise }.\n \\end{cases}\n\\end{equation*}\n\n\n\\par\nThen $ \\Delta \\varphi^i(0) =0$, for $i=1,2$ \ntogether with $\\varphi^i \\in C^{2,\\alpha}$, implies that \n\\begin{equation*}\n \\frac{\\Arrowvert\\Delta \\varphi^i(r_k\\cdot)\\rVert_{L^2(B_R)}}{kr_k^\\alpha} \\leq \n\\frac{C_n}{k} R^{\\frac{n}{2}+\\alpha} \\Arrowvert \\varphi^i \\rVert_{C^{2,\\alpha}} \n\\rightarrow 0, \\textrm{ as } k\\rightarrow\\infty,\n\\end{equation*}\nfor $i=1,2$, and for any fixed $1\\leq R<\\infty$.\n\n\\par\nWe have that $v^i_k \\rightharpoonup v^i_0 $  weakly in $W^{2,2}(B_R)$ and $ v^i_k \\rightarrow v^i_0$ \nin $W^{1,2}(B_R)$, therefore\n$ \\Delta v^i_k \\rightharpoonup \\Delta v^i_0$ weakly in $L^{2}(B_R)$, but\n$ \\Delta v^i_k = \\frac{tr A^i_{r_k}}{k r_k^\\alpha}\n+\\frac{\\Delta \\varphi^i(r_kx)}{k r_k^\\alpha} \\chi_{\\Omega_i}(r_kx)$, and\n$\\Arrowvert \\frac{\\Delta \\varphi^i(r_kx)}{k r_k^\\alpha} \\chi_{\\Omega_i}(r_kx) \\rVert_{L^2(B_R)} \\rightarrow 0$.\nThus we may conclude that the sequence of numbers \n$\\frac{tr A^i_{r_k}}{k r_k^\\alpha}$ converges, and denote \n$ a^i:= \\lim_ {k \\rightarrow\\infty} \\frac{tr A^i_{r_k}}{k r_k^\\alpha}$.\nThen $ \\Arrowvert \\Delta v^i_k -a^i \\rVert_{L^2(B_R)} \\rightarrow 0 $ as $k\\rightarrow\\infty$ \nfor every $1\\leq R <\\infty$. Therefore\nboth $ -\\Delta v^1_0 -a^1 \\equiv 0 $ and $ -\\Delta v^2_0 -a^2 \\equiv 0 $ in $\\mathbb{R}^n$.\n\n\n\\par\nWe have shown that $ v^i_0(x)-a^i \\frac{\\lvert x \\rvert^2}{2n}$ is a harmonic functions in $\\mathbb{ R}^n $.\nHence the matrix $ D^2v^i_0$ has harmonic entries $ D^k v^i_0$, where $k$ is a multiindex, $ \\lvert k \\rvert=2$.\nNext we can apply the estimates of the \nderivatives for harmonic functions and inequality \\eqref{impest}, to get \n\\begin{align*}\n \\lvert \\nabla D^k v^i_0(x_0) \\rvert \\leq  R^{ -\\frac{n}{2}-1}\\lVert D^k v^i_0 \\rVert_{L^2(B_R(x_0))} \\\\\n \\leq R^{ -\\frac{n}{2}-1} \\lVert  D^k v^i_0 \\rVert_{L^2(B_{2R})} \n \\leq C' R^{-1+\\alpha},\n\\end{align*}\nprovided $ R > \\lvert x_0\\rvert$.\nLetting $ R \\rightarrow \\infty$, we see that \nthe derivatives of $D^k v^i_0$ are vanishing, hence $ D^2 v^i_0$ is a constant matrix, and therefore\n$v^i_0$, $i \\in \\{1,2\\} $ is a second order polynomial. \n\n\n\\par\nAccording to our construction, \n$v^i_0$  are  orthogonal to the second order polynomials in $L^2(B_1)$-sense, hence\nboth $ v^1_0$ and $ v^2_0$ must be identically zero. Then the constants $a^1=a^2=0$, and\n$ \\Arrowvert \\Delta v^i_k \\rVert_{L^2(B_1)} \\rightarrow 0$ as $k\\rightarrow\\infty$, the latter \ncontradicts to the condition $\\max_i \\Arrowvert D^2v^i_k \\rVert_{L^2(B_1)}=1$.\n \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{$C^{2, \\alpha}$- regularity at the meeting points}\n\nWe start by showing that the approximating polynomials $p^i_r$ \nconverge to a polynomial $p^i_0$, and describe the rate of convergence. \n\n\n\n\\begin{lemma}\nLet $(u^1, u^2)$ be a solution to \\eqref{system}, and assume that $\\varphi^i \\in C^{2, \\alpha}$.\nLet the polynomials $p^i_r$ be as in the Definition 3, then there exists a polynomial $p^i_0$ such that \n\\begin{equation} \\label{pol}\n \\sup _{x \\in B_r}\\lvert p^i_r(x)-p^i_0(x)\\rvert \\leq C r^{2+\\alpha}.\n\\end{equation}\n\\end{lemma}\n\n\n\n\\begin{proof}\nThe condition $\\Arrowvert D^2v^i_r \\rVert _{L^2(B_1)}\\leq 1$ with the inequality \n\\eqref{main} implies that \n\\begin{equation} \\label{1}\n \\left( \\int_{B_1} \\lvert D^2u^i(rx)-D^2p^i_r \\rvert ^2 dx \\right) ^{\\frac{1}{2}} \\leq  \\frac{S(r)}{r^2}\n \\leq C_0 r^{\\alpha}\n\\end{equation}\n\nRecall that $A^i_r=D^2p^i_r$, then using the triangle inequality, and that $A^i_r$ is minimizing \n$ \\Arrowvert D^2 u^i(rx) - A \\rVert_{L^2(B_1)}$ over matricies $A \\in \n\\mathbb{R}^n  \\times \\mathbb{R}^n$, we get \n$ \\lvert A^i_r-A^i_{\\frac{r}{2}} \\rvert \\leq C_0 r^{\\alpha}$ for all $0<r<1$. \nBy taking $r=2^{-n}$, we see that $A^i_{2^{-n}}$ is a Cauchy sequence;\n\\begin{equation*}\n \\lvert A^i_{2^{-n}} - A^i_{2^{-n-m}}\\rvert \\leq  \\Sigma_{k=0}^{m-1} \n \\lvert A^i_{2^{-n-k}} - A^i_{2^{-n-k-1}}\\rvert \\leq\n\\end{equation*}\n\\begin{equation*}\n  \\Sigma_{k=0}^{m-1}(2^{-\\alpha})^{n+k}= 2^{-\\alpha n} \\Sigma_{k=0}^{m-1}2^{-\\alpha k},\n\\end{equation*}\nfrom the convergence of the series $ \\Sigma(2^{-\\alpha})^{n}$, it follows that \n$A^i_{2^{-n}}$ converges to some matrix $A^i_0$ as $n$ goes to infinity.\nThe inequality also provides the rate of convergence; for a fixed $n$, by letting $m$\ngo to infinity, we see that $ \\lvert A^i_{2^{-n}} - A^i_0 \\rvert \\leq C_0 2^{-n\\alpha}$.\n\n\\par\nMoreover, we get the estimate \n\\begin{equation*}\n \\lvert A^i_{r} - A^i_0 \\rvert \\leq C r^{\\alpha},\n\\end{equation*}\nfor $0<r < 1$, by choosing $n$ so that $2^{-n-1} < r \\leq 2^{-n}$.\n\n\n\\par\nNext we proceed to describe the rate of convergence of $b^i_r$ and $c^i_r$.\nWe know that $ b^i_r=(\\nabla u ^i )_r$, and $c^i_r=(u^i)_r$, taking into account that $u^i \\in C^{1, \\gamma}$,\nwe see that $ b^i_r \\rightarrow \\nabla u^i(0)$  and $c^i_r\\rightarrow u(0)$ as $r \\rightarrow 0$. \nOur aim is to show that\nactually\n\\begin{equation*}\n\\lvert b^i_{r} - \\nabla u^i(0) \\rvert \\leq C r^{1+\\alpha}  \\textrm{  and }\n\\lvert c^i_r- u^i(0) \\rvert \\leq C r^{2+\\alpha}.\n\\end{equation*}\n\n\\par\nLet us recall that $ (\\nabla v^i_r)_1= 0 $, and therefore Poincare's inequality implies that \n\\begin{align*}\n\\Arrowvert \\nabla v^i_r \\rVert _{L^2(B_1)} \\leq C \\Arrowvert D^2 v^i_r \\rVert _{L^2(B_1)}\\leq C'.\n\\end{align*}\nHence\n\\begin{equation} \\label{2}\n \\left( \\int_{B_1} \\lvert \\nabla u^i(rx)-\\nabla p^i_r(rx) \\rvert ^2 dx \\right) ^{\\frac{1}{2}} \\leq \n   C' \\frac{S(r)}{r}\n \\leq C r^{1+\\alpha}.\n\\end{equation}\n\n\\par\nTaking into account that $b^i_r$ is minimizing $ \\lVert \\nabla u^i(rx)- r x \\cdot A^i_r -b \\rVert_{L^2(B_1)}$ over \n$ b \\in \\mathbb{R}^n$, and applying triangle's inequality we get \n$ \\lvert b^i_{r} - \\nabla u^i(0) \\rvert \\leq C r^{1+\\alpha}$.\n\n\n\\par\nFurthermore, using Poincare's inequality once again, we see that\n\\begin{equation} \\label{3}\n \\left( \\int_{B_1} \\lvert u^i(rx)- p^i_r(rx)+r^2 (\\Delta u^i)_r \\rvert ^2 dx \\right) ^{\\frac{1}{2}} \\leq \n   C' S(r)\n \\leq C r^{2+\\alpha},\n\\end{equation}\nthen \n\\begin{equation*}\n\\lvert c^i_{r} - u^i(0) \\rvert \\leq C r^{2+\\alpha},\n\\end{equation*}\nby using that $c^i_r$\nis minimizing $ \\Arrowvert u^i(rx)+ r^2(\\Delta u^i)_r-\\frac{1}{2} r^2 x \\cdot A^i_r \\cdot x -\nr b^i_r \\cdot x -c \\rVert_{L^2(B_1)}$ over $c \\in \\mathbb{ R}$.\n\n\n\\par\nFinally, after combining our estimates for $ A^i_r, b^i_r \\textrm{ and } c^i_r $, we get \\eqref{pol},\nwhere \n\\begin{equation*}\np^i_0(x)= \\frac{1}{2} x \\cdot A^i_0 \\cdot x + \\nabla u^i(0) \\cdot x+ u^i(0),\n\\end{equation*}\nfor $i= 1,2$.\n\n\n\\end{proof}\n\n\n\n\\begin{corollary}\nUnder the assumptions of Lemma 2 it follows that\n\\begin{equation}\n\\Arrowvert u^i(rx)-p^i_r(rx) \\rVert_{W^{2,2}(B_1)} \\leq C_{n,\\alpha}\n\\left( \\max_i \\Arrowvert \\varphi^i \\rVert_{C^{2,\\alpha}}+ \\max_i \\Arrowvert u^i \\rVert_{L^2}\\right)r^{2+\\alpha},\n\\end{equation}\nwhere $ C_{n,\\alpha}$ is a dimensional constant.\n\\end{corollary}\n\n\\begin{proof}\nLet us recall that $ \\Arrowvert \\Delta u^i(rx)\\rVert_{L^2(B_1)} \\leq C \\Arrowvert \\Delta \\varphi^i \\rVert_{C^\\alpha}r^\\alpha$,\nthen the statement follows from the inequalities \\eqref{1}, \\eqref{2} and \\eqref{3}.\n\\end{proof}\n\n\n\\par\nNow we are ready to prove the main theorem.\n\\begin{theorem}\n Assume $\\varphi^1, \\varphi^2 \\in C^{2,\\alpha }$, and $\\mathscr{L}=\\overline{\\mathscr{L}^0}$\n then the solution to the system \\eqref{system}, $(u^1, u^2)$\n is $C^{2, \\alpha}$-regular on $ \\partial \\Omega_1 \\cap \\partial \\Omega_2 \\cap \\partial \\mathscr{L} \\cap \\Omega$,\n in the sense that for every $x_0 \\in \\partial \\Omega_1 \\cap \\partial \\Omega_2 \\cap \\partial \\mathscr{L}\\cap \n \\Omega$,\n there exist second order polynomials $ p^1_{x_0} $, $ p^2_{x_0} $,  such that\n \\begin{equation}\n \\sup _{x \\in B_r(x_0)}\\lvert u^i(x)-p^i_{x_0}(x) \\rvert \\leq C r^{2+\\alpha}\n\\end{equation}\n where the constant $C>0$ depends only on the given data.\n\\end{theorem}\n\n\\begin{proof}\nWithout loss of generality, we may assume $x_0=0$,\nand consider the following rescalings\n\\begin{equation*}\n v^i_r(x)= \\frac{u^i(rx) -p^i_0(rx)}{r^{2+\\alpha}}, \n\\end{equation*}\nthen according to Lemma 2 and Corollary 2, $ \\Arrowvert v^i_r \\rVert_ {W^ {2,2}(B_1)} \\leq C$.\n\n\\par\nThe pair $(v^1_r, v^2_r)$ solves the following system\n\n\\begin{equation*}\n\\begin{cases}\n\\min (-\\Delta v^1_r-\\frac{tr A^1_0}{r^\\alpha}, v^1_r-v^2_r+q^1_r)=0 \\\\\n\\min (-\\Delta v^2_r- \\frac{tr A^2_0}{r^\\alpha}, v^2_r-v^1_r+q^2_r)=0 \\\\\n\\min (-\\Delta v^1_r-\\frac{tr A^1_0}{r^\\alpha}, -\\Delta v^2_r- \\frac{tr A^2_0}{r^\\alpha} )=0,\n\\end{cases}\n\\end{equation*}\n\nwhere\n\\begin{equation*}\n q^1_r(x)= \\frac{p^1_0(rx)-p^2_0(rx) +\\varphi^1(rx)}{r^{2+\\alpha}},\n q^2_r(x)= \\frac{p^2_0(rx)-p^1_0(rx) +\\varphi^2(rx)}{r^{2+\\alpha}}.\n\\end{equation*}\n\nThen\n\\begin{equation*}\n -\\Delta v^i_r=\\begin{cases}\n    \\frac{tr A^i_0}{r^\\alpha}+ \\frac{\\Delta \\varphi^i(rx)}{r^\\alpha}  \\textrm{    if  } rx \\in \\Omega_i  \\\\\n    \\frac{tr A^i_0}{r^\\alpha} \\textrm{    otherwise.  }\n   \\end{cases}\n\\end{equation*}\n\n\\par\nWe assumed that $0 \\in \\partial \\Omega_1 \\cap \\partial \\Omega_2 \\cap \\partial \\mathscr{L}$, then \n$ \\Delta \\varphi^i (0)=0$, $i=1,2$. Hence \n$ \\lvert \\frac{\\Delta \\varphi^1(rx)}{r^\\alpha}\\rvert \\leq\n\\Arrowvert \\varphi^1_r \\rVert_{C^{2,\\alpha}(B_1)} \\lvert x \\rvert^\\alpha$.\nWe know that $ \\Arrowvert \\Delta v^i_r\\rVert_{L^2(B_1)}$\nis bounded, therefore $tr A^i_0=0$, and $ \\Delta v^i_r (x)$ is uniformly bounded.\n\n\n\n\\par\nWe have that $ \\Arrowvert v^i_r \\rVert_{L^2(B_1)} \\leq C$\nand we saw that $ \\Arrowvert \\Delta v^i_r \\rVert_{L^\\infty(B_1)} \\leq \n\\Arrowvert \\varphi^i \\rVert_{C^{2,\\alpha}(B_1)}$. \nUsing the Calderon-Zygmund estimates, we conclude that \n$ \\Arrowvert v^i_r \\rVert_{C^{1,\\gamma}}$\nis uniformly bounded. In particular,  $ \\lvert v^i_r(x) \\rvert \\leq \nC'(C_0+\\Arrowvert \\varphi^i \\rVert_{C^{2,\\alpha}(B_1)} ) $ for every\n$ x \\in B_1$ and $r\\leq \\frac{1}{2} $. \n\n\\par\nRecall that we set  $C_0=C \\left(\\max_ i \\Arrowvert \\Delta \\varphi^i \\rVert_{L^\\infty}+\n\\max_i \\Arrowvert u^i \\rVert_{L^2(B_1)} \\right) $, and $ v^i_r(x)= \\frac{u^i(rx) -p^i_0(rx)}{r^{2+\\alpha}} $.\nThen we get the desired inequality\n \\begin{equation*}\n \\sup _{x \\in B_r}\\lvert u^i(x)-p^i_0(x) \\rvert \\leq C\n \\left(\\max_{i \\in \\{1,2\\}} \\Arrowvert u^i \\rVert_{L^2(B_1)} +\\max_{i\\in \\{1,2\\}}\\Arrowvert \\varphi^i \\rVert_{C^{2,\\alpha}(B_1)} \\right)\n r^{2+\\alpha}.\n\\end{equation*}\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{A counterexample in case the zero-loop set has an isolated point}\nHere we give a counterexample, showing that if the zero loop set has an isolated point,\nthen the solution may not be $C^{1,1}$.\n\n\n\\par\nWe consider the following system in $\\mathbb{R}^2$\n\\begin{equation} \\label{countex}\n\\begin{cases}\n\\min (-\\Delta u^1, u^1-u^2+\\varphi)=0 \\\\\n\\min (-\\Delta u^2, u^2-u^1+\\varphi)=0,\n\\end{cases}\n\\end{equation}\nwith $ \\varphi= \\frac{1}{4} \\lvert x \\rvert ^2 $.\n\n\n\n\n\\par\nThen the difference $U=u^1-u^2$ solves the following double-obstacle problem \nin $\\mathbb{ R }^2$\n\n\\begin{equation} \\label{do}\n-\\Delta U = \\begin{cases}\n1, \\textrm{   if } U=-\\varphi \\\\\n-1, \\textrm{   if }  U= \\varphi \\\\\n0 , \\textrm{   if }  -\\varphi< U< \\varphi,\n\\end{cases}\n\\end{equation}\nand $-\\Delta u^1 =(-\\Delta U )^+$ and $ -\\Delta u^2 =(\\Delta U )^+ $.\n\n\\par\nNow let us consider a function $ w $ defined as follows\n\\begin{equation*}\nw = \\begin{cases}\n- \\frac{1}{4} \\lvert x \\rvert^2 , \\textrm{   if } x_1> 0, x_2>0 \\\\\n\\frac{1}{4} (x_1^2-x_2^2), \\textrm{   if }  x_1<0, x_2 >0 \\\\\n\\frac{1}{4} (x_2^2-x_1^2), \\textrm{   if }  x_1>0, x_2 <0 \\\\ \n \\frac{1}{4} \\lvert x \\rvert^2 , \\textrm{   if } x_1< 0, x_2<0.\n\\end{cases}\n\\end{equation*}\nThen $w \\in C^{1,1}$ also solves the double-obstacle problem \\eqref{do}, therefore we \nchoose $ U \\equiv w $. \n\n\\par\nNext we write $u^1$ explicitly in polar coordinates\n\\begin{equation*}\nu^1(r,\\theta) =\n\\begin{cases}\n -\\frac{1}{4}r^2-\\frac{1}{2 \\pi } r^2 \\theta \\cos 2\\theta -\\frac{1}{2 \\pi} r^2 \\ln r \\sin 2\\theta,\n \\textrm{   if } 0<\\theta \\leq \\frac{\\pi }{2}\\\\\n-\\frac{1}{4}r^2 \\cos 2 \\theta +\\frac{1}{2 \\pi } r^2 \\theta \\cos 2\\theta +\\frac{1}{2 \\pi} r^2 \\ln r \\sin 2\\theta, \\textrm{   otherwise  }, \n\\end{cases}\n\\end{equation*}\nhere the function   $ r^2 \\theta \\cos 2\\theta + r^2 \\ln r \\sin 2\\theta \\in C^{1, \\gamma} $   \nfor every $0<\\gamma<1$, solves the Laplace equation in $ \\mathbb{ R }^2 \\backslash \\{0\\}$,\nbut is not $C^{1,1}$ near the origin.\n\n\n\\par\nTherefore $u^1$ is a $C^{1,\\gamma}$ function in the unit ball in $\\mathbb{ R }^2$\nbut it is not $C^{1,1}$\nin the neighborhood of the origin, since $ \\lvert \\frac{\\partial^2 u^1}{\\partial r^2}\\rvert \n\\approx \\lvert \\ln r \\rvert \\rightarrow\\infty $ as $r\\rightarrow 0$.\nMoreover, \n  $ -\\Delta u^1(r, \\theta) = \\chi _{\\{0<\\theta \\leq \\frac{\\pi }{2} \\}}=  \\chi _{\\{ u^1 > 0 \\}}$,\nprovided $r> 0 $ is small enough.\n \n\n\n\n\\par\nNext we take $ u^2(r, \\theta )= u^1(r, \\theta)-w$, then \n\\begin{equation*}\nu^2(r,\\theta) =\n\\begin{cases}\n-\\frac{1}{2 \\pi } r^2 \\theta \\cos 2\\theta -\\frac{1}{2 \\pi} r^2 \\ln r \\sin 2\\theta,\n \\textrm{   if } 0<\\theta \\leq \\frac{\\pi }{2}\\\\\n - \\frac{1}{2}r^2 \\cos 2\\theta+\\frac{1}{2 \\pi } r^2 \\theta \\cos 2\\theta +\\frac{1}{2 \\pi} r^2 \\ln r \\sin 2\\theta,\n \\textrm{  if } \\frac{\\pi }{2} <\\theta \\leq \\pi \\\\\n - \\frac{1}{4}r^2- \\frac{1}{4}r^2 \\cos 2\\theta+\\frac{1}{2 \\pi } r^2 \\theta \\cos 2\\theta +\\frac{1}{2 \\pi} r^2 \\ln r \\sin 2\\theta,\n \\textrm{   if } \\pi <\\theta \\leq \\frac{3 \\pi }{2}\\\\\n \\frac{1}{2 \\pi } r^2 \\theta \\cos 2\\theta +\\frac{1}{2 \\pi} r^2 \\ln r \\sin 2\\theta,\n \\textrm{   if }\\frac{3 \\pi }{2} <\\theta \\leq 2 \\pi \n\\end{cases}\n\\end{equation*}\n\nNeither $u^1$ nor  $u^2 $ is a $C^{1,1}$ function. However, it is easy to see that $(u^1,u^2) $\nsolves \\eqref{countex}.\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAfter the Higgs boson discovery in 2012~\\cite{Chatrchyan:2012ufa} and\nsubsequent analyses of its properties~\\cite{Hcoup}, evidence for physics\nbeyond the Standard Model (BSM) remains elusive. Although consistency\nwith SM Higgs properties is expected in many BSM\nscenarios, current measurements do not fully constrain\nthe Higgs sector. One coupling which is currently unconstrained and has recently been subject of much interest is the Higgs\nself-interaction $\\sim \\eta$, which is responsible for the spontaneous\nbreaking of electroweak gauge symmetry in the SM via the potential\n\\begin{equation}\n  \\label{eq:higgspot}\n  V(H^\\dagger H) =\n  \\mu^2 H^\\dagger H + \\eta (H^\\dagger H)^2 \\, , \n\\end{equation}\nwith $\\mu^2<0$, where $H=(0,v+h)^T\/\\sqrt{2}$ in unitary gauge. The\nHiggs self-coupling manifests itself primarily in a destructive\ninterference in gluon fusion-induced di-Higgs\nproduction~\\cite{nigel,uli2,Maltoni:2014eza} through feeding into the\ntrilinear Higgs interaction with strength\n$\\lambda_{\\text{SM}}=m_h\\sqrt{\\eta\/2}=gm_h^2\/(4 m_W)$ in the SM. The\nlatter relation can be altered in BSM scenarios, e.g. the SM coupling\npattern can be distorted by the presence of a dimension six operator\n$\\sim (H^\\dagger H)^3$, and di-Higgs production is the only channel\nwith direct sensitivity to this interaction~\\cite{Azatov:2015oxa}. A\n modification solely of the Higgs trilinear coupling, which is typically\ninvoked in di-Higgs feasibility studies, is predicted in models of\n$\\mu^2$-less electroweak symmetry breaking,~e.g.~\\cite{womu2}.\n\n\\begin{figure*}[!t]\n  \\centering\n  \\subfigure[\\label{fig:pth}]{\\includegraphics[width=0.45\\textwidth]{gf_ptmaxhiggs.eps}}\n  \\hfill\n  \\subfigure[\\label{fig:ptj}]{\\includegraphics[width=0.45\\textwidth]{gf_pt0.eps}}\n  \\caption{\\label{fig:pt} Maximum Higgs and jet transverse momenta\n    for gluon fusion-induced $hhjj$ production, including the ratio of\n    full theory to the effective theory calculation for three\n    different values of the Higgs trilinear coupling $\\lambda$.}\n\\end{figure*}\n\n\n\\begin{figure}[t!]\n  \\centering\n  \\vspace{0.3cm}\n  \\parbox{0.45\\textwidth}{\n \\includegraphics[angle=-90,width=0.22\\textwidth]{ggHHgg_diag1.eps}\n \\includegraphics[angle=-90,width=0.22\\textwidth]{ggHHgg_diag2.eps}\\\\\n \\includegraphics[angle=-90,width=0.22\\textwidth]{ggHHuu_diag1.eps}\n \\includegraphics[angle=-90,width=0.22\\textwidth]{uuHHdd_diag1.eps}\n \\caption{Sample Feynman diagrams contributing to $pp\\to hhjj$ via gluon\n   fusion.}\n \\label{fig:sample_gf}\n  \\vspace{-0.5cm}\n}\n\\end{figure}\n\nAfter the Higgs discovery, analyses of the di-Higgs final state at the\nhigh-luminosity LHC and beyond have experienced a renaissance, and\ndi-Higgs final states such as the $b\\bar b \\gamma\n\\gamma$~\\cite{Baur:2003gp,Barger:2013jfa,Azatov:2015oxa,Barr:2014sga},\n$b\\bar b \\tau^+\\tau^-$~\\cite{Baur:2003gpa,us,us2}, $b \\bar b\nW^+W^-$~\\cite{Baur:2003gpa,Papaefstathiou:2012qe,us} and $b\\bar b b\n\\bar b$~\\cite{Baur:2003gpa,deLima:2014dta,us} channels have been\nstudied phenomenologically, often relying on boosted jet substructure\ntechniques~\\cite{bdrs} (see also an\ninvestigation~\\cite{Papaefstathiou:2015iba} of rare decay channels\nrelevant for a 100 TeV collider). Recent analyses by ATLAS and\nCMS~\\cite{atlasecfa,cmsecfa} have highlighted the complexity of these\nanalyses and the necessity to explore different production mechanisms\nto formulate constraints on the Higgs self-interactions in the\nfuture. This program has already been initiated by feasibility\nanalyses of the $hhj$, $hhjj$ and $t\\bar t hh$ production modes in\nRefs.~\\cite{us,Barr:2014sga,hhjj,tthh,Liu:2014rva}.\n\nDi-Higgs production in association with two jets is a particularly\nimportant channel in this regard since this final state receives\ncontributions from the weak boson fusion (WBF) production mode. The\nphenomenological appeal of the WBF mode is twofold. Firstly, the weak\nboson fusion component of $pp\\to hh jj$ is sensitive to modifications\nof the gauge-Higgs sector~\\cite{hhjj,Brooijmans:2014eja}, which can lead to \nlarge cross-section enhancements. Secondly, the QCD uncertainties for\nthe WBF topologies are known and under theoretical\ncontrol~\\cite{hhjj2,Frederix:2014hta}, such that a search for BSM\nelectroweak-induced deviations is not hampered by QCD\nsystematics. This situation is very different from QCD-induced\nproduction~\\cite{qcd}, and can be attributed to the particular\nphenomenology of WBF-like processes~\\cite{Figy:2003nv,hhjj1}\n\nHowever, an additional source of uncertainty that was neglected until\nrecently~\\cite{hhjj} is the correct inclusion of the gluon fusion\ncontribution to $pp\\to hh jj$ analyses. In contrast to single Higgs\nphenomenology, the correct inclusion of massive fermion\nthresholds is crucial to a reliable prediction of QCD-induced $pp \\to\nhh jj$~\\cite{us}.\n\nGiven that the cross sections in WBF $hhjj$ production are very\nsuppressed compared to WBF $hjj$ production (the WBF $hhjj$ cross\nsection is $\\sim 750$ times smaller), we have to rely on the dominant\nhadronic Higgs decay modes to be able to observe this final state.\nThis rules out one of the most crucial single Higgs WBF selection\ntools - the central jet veto~\\cite{dieter}. The observation of\nWBF-induced $pp\\to hhjj$ production is further hampered by the top\nthreshold in the QCD-mediated process. Since the top threshold sets\nthe scale of the di-Higgs subsystem, an analysis that tries to retain\nas many low $p_T$ Higgs bosons as possible leads to a QCD contribution\nthat dominates over the WBF component when minimal WBF-like cut\nrequirements are imposed~\\cite{hhjj}.\n\nIn this paper we extend the discussion of Ref.~\\cite{hhjj} in a number\nof directions. We first perform a detailed comparison of\nEFT-approaches to QCD-mediated $pp\\to hhjj$ against a calculation\nkeeping the full mass dependencies of top and bottom quarks in\nSec.~\\ref{sec:gf}. We compare the QCD-induced $pp\\to hhjj$\nphenomenology to the WBF signature in Sec.~\\ref{sec:wbf} before we\ndiscuss general approaches to isolate the signal from the dominant top\nbackgrounds in a hadron level analysis in Sec.~\\ref{sec:tt}. This sets\nthe stage for a discussion about the prospects to isolate the WBF and\nGF components in Secs.~\\ref{sec:isolgf} and \\ref{sec:isolwbf},\nfollowed by a study on constraining $VV hh$ couplings using the WBF\ninduced signal in Section~\\ref{sec:limitzeta}. We focus on collisions\nwith $14$ TeV throughout.\n\n\n\\section{The gluon fusion contribution}\n\\label{sec:gf}\n\\subsection{Finite top mass effects}\n\nIt is well known that effective field theory approximations in the\n$m_t\\to \\infty$ limit cannot be invoked to study di-Higgs final states\nat colliders in a reliable way due to the effects of top-quark\nthreshold~\\cite{uli2,Dawson:2015oha}. Further, the breakdown of the $m_t\\to\n\\infty$ approximation is worsened in the presence of additional jet\nemission~\\cite{us,Maierhofer:2013sha}. Finite $m_t$ effects must\ntherefore be considered for all QCD di-Higgs production channels,\nwhich will be required to set the best limits on the Higgs\nself-coupling or formulate a realistic estimate of the GF contribution\nin a WBF-like selection.\n\n\\begin{table}[!t]\n \\begin{tabular}{lccc}\n  \\toprule\n  $\\lambda$ \\hspace{1cm}  & \\hspace{0.25cm}\t$0 \\cdot \\lambda_\\text{SM}$ [fb] \\hspace{0.25cm}  & \\hspace{0.25cm} $1 \\cdot \\lambda_\\text{SM}$ [fb]\\hspace{0.25cm}  & \\hspace{0.25cm} $2 \\cdot \\lambda_\\text{SM}$ [fb] \\\\\n  \\botrule\n   GF\t&\t10.73\t\t\t\t\t     & 5.502\t\t\t\t\t    & 2.669\t\t \\\\\n   WBF  &\t4.141\t\t\t\t\t     & 2.010\t\t\t\t\t    & 0.9648\t\t\\\\\n   \\botrule\n \\end{tabular}\n \\caption{Cross section normalisations for the GF and WBF\n   samples at 14~TeV, for details see text. The WBF normalisation follows\n   from~\\cite{hhjj2} and includes higher order QCD effects.}\n \\label{tab:xsnormalisation}\n\\end{table}\n\nThe computational challenges in QCD-mediated $hhjj$ production are\nsignificant, with the gluon-fusion channels particularly time\nconsuming even when using state-of-the-art techniques. The standard\nmethod of simulating a differential cross section from unweighted\nevents is not feasible in this case, and we instead use a reweighting\ntechnique that is exploited in higher order calculations and\nexperimental analyses~(see e.g. \\cite{Aad:2013nca}).\n\nWe generate GF $hhjj$ events by implementing the\nrelevant higher dimensional operators in the $m_t\\to \\infty$ limit\nobtained by expanding the low-energy effective theory~\\cite{kniehl}\n\\begin{equation}\n  \\mathcal{L}_\\text{eff} = -\\frac{1}{4} \\frac{\\alpha_S}{3 \\pi} \n  G^a_{\\mu \\nu} G^{a\\, \\mu \\nu} \\log(1 + h\/v)\n  \\label{eq:efft}\n\\end{equation}\nin \\textsc{MadEvent} v5.1~\\cite{Alwall:2011uj} using the\n\\textsc{FeynRules\/Ufo}~\\cite{Degrande:2011ua} framework.\\footnote{The\n  effective theory implementation can be modified in the sense that\n  only one effective vertex insertion is allowed. This is gives only a\n  mild $\\sim 10\\%$ effect in the tail of the distribution, and is not\n  relevant for an order one EFT\/full theory rescaling, see below.}\nThis allows us to sample a weighted set of events that we subsequently\nfeed into our analysis solely depending on their final state\nkinematics. If an event passes the selection requirements of a certain\nsearch region, we correct for the full mass dependence using a\nreweighting library based on \\textsc{GoSam} package~\\cite{gosam} at\nthis stage. The reweighting employs exactly the same matrix elements\nused for the event generation and the trilinear coupling is steered\nthrough a modification of the {\\sc{GoSam}} matrix element,\ni.e. variations of the trilinear coupling are part of the\nreweighting. A selection of Feynman diagrams which contribute to the\ngluon fusion signal are illustrated in Fig.~\\ref{fig:sample_gf}. The\n{\\sc{GoSam}} code used for the reweighting is based on a Feynman\ndiagrammatic approach using {\\sc{QGRAF}}~\\cite{qgraf} and\n{\\sc{FORM}}~\\cite{form} for the diagram generation, and\n{\\sc{Spinney}}~\\cite{spinney}, {\\sc{Haggies}}~\\cite{haggies} and\n{\\sc{FORM}} to write an optimised fortran output. The reduction of the\none-loop amplitudes was done using {\\sc{Samurai}}~\\cite{samurai},\nwhich uses a $d$-dimensional integrand level decomposition based on\nunitarity methods~\\cite{unitarity}. The remaining scalar integrals\nhave been evaluated using {\\sc{OneLoop}}~\\cite{oneloop}. Alternative\nreduction techniques can be used employing {\\sc{Ninja}}~\\cite{ninja}\nor {\\sc{Golem95}}~\\cite{golem95}. To validate the reweighting\nprocedure we regenerated the code that has been used in~\\cite{hhjj} \nwith the improvements that became available within {\\sc{GoSam 2.0}},\nin particular improvements in code optimisation and in the reduction\nof the amplitudes. For the reduction we used\n{\\sc{Ninja}}, which employs an improved reduction algorithm based\non an Laurent expansion of the integrand. This leads to substantial\nimprovements in both speed and numerical stability compared to the \nprevious version. We combined the\ncode with a phase space integration provided by\n\\textsc{MadEvent}~\\cite{Maltoni:2002qb}.  Further substantial speed-up\nhas been obtained by Monte Carlo sampling over the helicities rather\nthen performing the helicity sum. This enabled us to perform a full\nphase space integration and we found full agreement within the \nstatistical uncertainties between the result obtained from reweighting\nand the result from the full phase space integration.\n\n\n\n\n\\begin{figure*}[t!]\n  \\subfigure[\\label{fig:mjj}]{\\includegraphics[width=0.45\\textwidth]{gf_jjmass.eps}}\n  \\hfill\n  \\subfigure[\\label{fig:etajj}]{\\includegraphics[width=0.44\\textwidth]{gf_deta12.eps}}\n  \\caption{\\label{fig:jets} Invariant mass and pseudo-rapidity\n    distributions of the jet system in QCD-mediated $hhjj$\n    production. We show the  effective theory and full theory results for\n    three values of the trilinear Higgs coupling, applying only generator-level cuts of $p_{T,j} \\ge 20$ GeV\n    and $|\\eta_j| < 5$.}\n\\end{figure*}\n\n\\begin{figure*}[t!]\n  \\subfigure[\\label{fig:mhh}]{\\includegraphics[width=0.45\\textwidth]{gf_hhmass.eps}}\n  \\hfill\n  \\subfigure[\\label{fig:etahj}]{\\includegraphics[width=0.45\\textwidth]{gf_detahiggsj.eps}}\n  \\caption{\\label{fig:hjet} Invariant mass and pseudo-rapidity distributions of the leading jet and di-Higgs system in QCD-mediated $hhjj$ production. We show the  effective theory and full theory results for\n    three values of the trilinear Higgs coupling, applying only generator-level cuts of $p_{T,j} \\ge 20$ GeV\n    and $|\\eta_j| < 5$.}\n\\end{figure*}\n\n\\subsection{Phenomenology of QCD-mediated $hhjj$ production}\n\nTop thresholds are particularly prominent in the di-Higgs invariant\nmass distribution, which is thus well suited\nto benchmark the relation of the finite $m_t$ limit to the effective theory of Eq.~\\gl{eq:efft}. Other\nobservables constructed from the six particle final state are also\nrelevant when performing a targeted phenomenological analysis, and we\ndiscuss both these and the  phase space dependent parton-level reweighting in detail\nin the following.\n\nIn Figures~\\ref{fig:pt}, \\ref{fig:jets}, and \\ref{fig:hjet} we show a\nselection of $hhjj$ final state observables for inclusive cuts\n$p_{T,j}>20~\\text{GeV}$ and $|\\eta_j|< 5$, no cuts on Higgs bosons are\nimposed. We label Higgs bosons and jets according to their hardness,\ni.e. $p_{T,h1}>p_{T,h2}$ and $p_{T,j1}>p_{T,j2}$. The cross sections\nare given in Tab.~\\ref{tab:xsnormalisation}. The inclusive gluon\nfusion cross section is about 2.5 times larger than the WBF cross\nsection approximately independent of the value of the Higgs trilinear\ncoupling.\n\nAs previously established in~\\cite{uli2,us,hhjj} the di-Higgs system\nis badly modelled by the effective theory which under- and overshoots\nthe full theory cross section at low and high momenta\nrespectively. For $pp\\to hhjj$ this is a qualitatively similar\nbehaviour compared the $pp\\to hh(j)$ production: The $m_{hh}$\ndistribution is the crucial observable which parametrises the finite\ntop quark mass effects. The EFT describes the low maximum transverse\nHiggs momenta $p_{T,h1}$ reasonably well, as shown in\nFig.~\\ref{fig:pth}. The jet emission on the other hand integrates over\na considerable range of $m_{hh}$, and the ratio of full theory vs\neffective theory smaller than one for the finite $m_t$ limit produces\na smaller integrated cross section than the $m_t\\to \\infty$ limit for\nthe jet kinematics.\n\nConsidering just the dijet system in Fig.~\\ref{fig:jets}, we observe\nthat the jet kinematics is not severely impacted by the reweighting\nprocedure upon marginalising over the di-Higgs kinematics. The phase\nspace dependence of the dijet invariant mass Fig.~\\ref{fig:mjj} is\nrelatively mild aside from the total rescaling of the inclusive cross\nsections, and the ratio for the pseudo-rapidity distribution of\nthe jets is nearly flat, Fig.~\\ref{fig:etajj}. This is also true for\nthe azimuthal angle difference $\\Delta\\phi_{jj}$. The angular\ndistributions of the leading members of the jet-Higgs system are\nrelatively mildly impacted by the reweighting too\nFig.~\\ref{fig:etahj}. This agrees with the $m_{hh}$ being the\nobservable most sensitive to the top threshold (as in $pp\\to hh(j)$),\nand is also supported by the larger impact of the reweighting of\n$m_{hh}$ in Fig.~\\ref{fig:mhh}. A reweighting based on $m_{hh}$ to\n  correct for finite top mass effects suggests itself for future\n  analyses as a time-saving approach with reasonable accuracy.\n\n\n\\section{The weak boson fusion contribution}\n\\label{sec:wbf}\n\nThe weak boson fusion contribution to $pp\\to hhjj$ has received\nconsiderable attention recently and precise higher-order QCD\ncorrections have been provided\nin~\\cite{hhjj1,hhjj2,Frederix:2014hta}. Due to the sensitivity of the\nWBF contribution to both the trilinear coupling and the quartic $VVhh$\n($V=W,Z,\\gamma$), as shown in the Feynman diagrams in\nFig.~\\ref{fig:sample_wbf}, weak boson fusion to two Higgs bosons can,\nin principle, provide complementary information about BSM physics\nwhich remains uncaptured in $pp\\to hh(j)$ and $pp\\to t\\bar t\nhh$~\\cite{Brooijmans:2014eja}.\n\nWe generate WBF samples with varying $\\lambda$ using \\textsc{MadEvent}\nv4~\\cite{Alwall:2007st} and normalise the cross section to NLO\naccuracy~\\cite{hhjj2}.  The WBF $hhjj$ contribution shares the QCD properties\nof WBF $hjj$ production \\cite{Figy:2003nv} which means it shares\nthe distinctive $\\Delta \\eta(j1,j2)$ distribution shown in\nFig.~\\ref{fig:wbfeta}: To produce the heavy di-Higgs pair we probe the\ninitial state partons at large momentum fractions. This together with\na colour-neutral $t$-channel exchange of the electroweak\nbosons~\\cite{gaporig} (see also \\cite{gaporig2}) leads to energetic\nback-to-back jet configurations at large rapidity separation and\nmoderate transverse momenta with a centrally produced Higgs pair. The\nproduction of an additional Higgs boson in comparison to single Higgs\nproduction via WBF leads to a cross section reduction by three orders\nof magnitude (see Tab.~\\ref{tab:xsnormalisation}) in the SM. Such a\nsmall inclusive production cross section highlights the necessity of\nconsidering dominant Higgs decay channels such as $h\\to b \\bar b$ and\n$h\\to \\tau^+\\tau^-$ and the non-availability of central jet vetos\n\\cite{dieter} as a means to control the background {\\it{and}} GF\ncontribution in a targeted analysis as a consequence.\n\n\\begin{figure}[!t]\n   \\vspace{1.2cm}\n  \\parbox{0.45\\textwidth}{\n    \\includegraphics[width=0.22\\textwidth]{vbf_diag1.eps}\n    \\includegraphics[width=0.22\\textwidth]{vbf_diag2.eps}\\\\\n    \\includegraphics[width=0.22\\textwidth]{vbf_diag3.eps}\n    \\includegraphics[width=0.22\\textwidth]{vbf_diag4.eps}\n  }\n  \\caption{Sample Feynman diagrams contributing to $pp \\to hhjj$ in weak boson\n    fusion. \\label{fig:sample_wbf}}\n\\end{figure}\n\nThe gluon fusion contribution is bigger by a factor of 2.5 than the\nWBF component of $hhjj$ production, however, with increasing invariant\ndi-Higgs mass the WBF contribution is enhanced relative to GF\nproduction as a consequence of the suppression above the $2m_t$\nthreshold, as shown in Fig.~\\ref{fig:gfwbfmhh}.\n\n \\begin{figure}[t!]\n\\centering\n\\subfigure[\\label{fig:wbfeta}]{\\includegraphics[width=0.45\\textwidth]{wbf_deta12.eps}} \\\\[0.1cm]\n\\subfigure[\\label{fig:gfwbfmhh}]{\\includegraphics[width=0.45\\textwidth]{mhh_comp_partonlvl.eps}}\n\\caption{The $\\Delta \\eta(j1,j2)$ distribution of the weak boson fusion\n  contribution at parton-level (a) and the $m_{hh}$ distribution of\n  the weak boson fusion and gluon fusion contributions compared with\n  correct cross section normalisation (b), both satisfying generator-level\n  cuts of $p_{T,j} \\ge 20$ GeV and $|\\eta_j| < 5$.}\n\\label{fig:detapartonlvl}\n\\end{figure}\n\n\n\n\\begin{table*}[!t]\n\\renewcommand\\arraystretch{1.1}\n    \\begin{tabular}{llllr}\n    \\toprule\n    Cut setup      \t\t\t\t\t&  Base Selection   [fb]\\hspace{0.1cm}\t& GF Selection [fb]\\hspace{0.1cm}& WBF Selection [fb] \\hspace{0.1cm} &   Normalisation$*$ [fb] \\\\\n    \\botrule\n    GF\t($\\lambda = 1 \\cdot \\lambda_\\text{SM}$)\t& \\hspace{0.1cm} \\num{0.01396}\t\t& \\hspace{0.1cm}  \\num{0.005722} \t\t& \\hspace{0.1cm}  \\num{0.0005378}      &\t\\num{0.4013}\t\\\\\n    GF\t($\\lambda = 0 \\cdot \\lambda_\\text{SM}$)\t& \\hspace{0.1cm} \\num{0.02562}       \t& \\hspace{0.1cm}  \\num{0.008122}    \t\t& \\hspace{0.1cm}  \\num{0.0008767}      &\t\\num{0.7831}\t\\\\\n    GF\t($\\lambda = 2 \\cdot \\lambda_\\text{SM}$)\t& \\hspace{0.1cm} \\num{0.007167}      \t& \\hspace{0.1cm}  \\num{0.003906}    \t\t& \\hspace{0.1cm}  \\num{0.0003034}     &\t\\num{0.1947}\t\\\\\n    WBF\t($\\lambda = 1 \\cdot \\lambda_\\text{SM}$)\t& \\hspace{0.1cm} \\num{0.003292}      \t& \\hspace{0.1cm}  \\num{0.0004999}   \t\t& \\hspace{0.1cm}  \\num{0.001485}      & \t\\num{0.1466}\t\\\\\n    WBF\t($\\lambda = 0 \\cdot \\lambda_\\text{SM}$)\t& \\hspace{0.1cm} \\num{0.007706}      \t& \\hspace{0.1cm}  \\num{0.0007154}   \t\t& \\hspace{0.1cm}  \\num{0.002820}      &\t\\num{0.3020}\t\\\\\n    WBF\t($\\lambda = 2 \\cdot \\lambda_\\text{SM}$)\t& \\hspace{0.1cm} \\num{0.001103}      \t& \\hspace{0.1cm}  \\num{0.0001815}   \t\t& \\hspace{0.1cm}  \\num{0.0003912}     &\t\\num{0.07037}   \\\\\n    $t\\bar{t}jj$\t\t\t\t\t& \\hspace{0.1cm} \\num{5.712}\t      \t& \\hspace{0.1cm}  \\num{0.03390}   \t\t& \\hspace{0.1cm}  \\num{0.01801}       &\t\\num{10130}\t\\\\\n    $t\\bar{t}h$\t\t\t\t\t\t& \\hspace{0.1cm} \\num{0.06229}       \t& \\hspace{0.1cm}  \\num{0.007047}   \t\t& \\hspace{0.1cm}  \\num{0.00005658}     & \t\\num{38.62}\t\\\\\n    $Zhjj$\t\t\t\t\t\t& \\hspace{0.1cm} \\num{0.005118}      \t& \\hspace{0.1cm}  \\num{0.001278}   \t\t& \\hspace{0.1cm}  \\num{0.0001026}     & \t\\num{47.37}\t\\\\\n    $ZZjj$\t\t\t\t\t\t& \\hspace{0.1cm} \\num{0.001171}  \t& \\hspace{0.1cm}  \\num{0.00006659}   \t\t& \\hspace{0.1cm}  \\num{0.0000007639}\t&\t\\num{225.7}\t\\\\\n    $ZWWjj$\t\t\t\t\t\t& \\hspace{0.1cm} \\num{0.00001888}       & \\hspace{0.1cm}  \\num{0.000005461}   \t\t& \\hspace{0.1cm}  \\num{0.0000002039}\t&\t\\num{0.5368}  \\\\\n    total background\t\t\t\t\t& \\hspace{0.1cm} \\num{5.781}       \t& \\hspace{0.1cm}  \\num{0.04230}   \t\t& \\hspace{0.1cm}  \\num{0.01817}\t&\t-\t\\\\\n    \\botrule    \n    $S\/B$\t($\\lambda = 1 \\cdot \\lambda_\\text{SM}$) \t\t\t& \\hspace{0.3cm} 1\/335.1       \t& \\hspace{0.3cm} 1\/6.799\t& \\hspace{0.3cm}  1\/8.983          \\\\\n    $S\/B$\tGF$^\\dagger$\t ($\\lambda = 1 \\cdot \\lambda_\\text{SM}$)\t& \\hspace{0.3cm} 1\/414.3       \t& \\hspace{0.3cm} 1\/7.480 \t& \\hspace{0.3cm}  1\/36.55         \\\\\n    $S\/B$\tWBF$^\\dagger$\t($\\lambda = 1 \\cdot \\lambda_\\text{SM}$)\t\t& \\hspace{0.3cm} 1\/1760       \t& \\hspace{0.3cm} 1\/96.06 \t& \\hspace{0.3cm}  1\/12.60          \\\\\n \\botrule\n    $S\/\\sqrt{\\text{B}}$ (3 ab$^{-1}$, $\\lambda = 1 \\cdot \\lambda_\\text{SM}$)\t\t& \\hspace{0.3cm} 0.3930       & \\hspace{0.3cm} 1.657\t& \\hspace{0.3cm}  0.8219        \\\\\n \\botrule\n $*$ branchings included in normalisation \\\\\n $^\\dagger$ considering only this as signal\n     \\end{tabular}\n     \\caption{Cross sections for the two sources of signal, and backgrounds, after the various selections described in the text are applied, together with various measures of significance in the bottom four rows.}\n \\label{tab:xstable}\n\\end{table*}\n\nSince we cannot rely on vetoing hadronic activity in the central part\nof the detector, a potential discrimination of GF from WBF needs to be\nbuilt on the following strategy, which we will investigate in\nSec.~\\ref{sec:tt}:\n\n\\begin{itemize}\n  \\item To isolate the di-Higgs (WBF+GF) signal we can exploit the relative\n    hardness of the di-Higgs pair which peaks around $\\sim 2m_t$. Such\n    hard events are less likely to be produced by (ir)reducible\n    backgrounds.\n\n  \\item Focussing on large $m_{hh}$ we can enhance WBF over GF by\n    stringent cuts on the jet rapidity separation. This will also\n    imply a significant decrease of QCD-dominated backgrounds.\n\n  \\item By explicitly allowing central jet activity, we can exploit\n    the colour correlation differences in WBF vs GF to further purify\n    our selection. Since colour flow is tantamount to energy flow in the\n    detector,  event shapes are particularly well-suited observables\n    for unravelling the colour correlations in the final state once\n    the reconstructed di-Higgs pair has been removed\\footnote{A\n      detailed discussion of event shapes at hadron colliders can be\n      found in~\\cite{Banfi:2010xy}.}. This strategy was first proposed\n    for single Higgs\n    analyses in \\cite{Englert:2012ct} (see also\n    \\cite{plehn}).\n\\end{itemize}\n\n\n\n\n\\section{Taming the background}\n\\label{sec:tt}\nFor our hadron-level analysis we shower our signal samples with\n\\textsc{Herwig++}~\\cite{Bahr:2008pv} and generate backgrounds as\nfollows: $t\\bar{t}jj$, $t\\bar{t}h$, $Zhjj$, and $ZZjj$ with\n\\textsc{Sherpa}~\\cite{Gleisberg:2008ta}, and $ZWWjj$ with\n\\textsc{MadEvent} v5. We find the dominant backgrounds to be\n$t\\bar t jj$ and $t\\bar t h$ production, for which next-to-leading\norder results are available~\\cite{ttjjnlo,tthnlo}\nand we use inclusive $K$ factors $K_{t\\bar t jj}\\simeq 1$ and $K_{t\\bar t\n  h}\\simeq 1.5$ to estimate the higher order contributions\nto these backgrounds. Higgs branching ratios are set to the values\nagreed upon by the Higgs Cross Section Working\nGroup~\\cite{Heinemeyer:2013tqa}.\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{hadronlvl_deta12.eps}\n\\hfill\n\\includegraphics[width=0.45\\textwidth]{hadronlvl_hhmass.eps}\n\\caption{Shape comparison of $\\Delta \\eta(j1,j2)$ and $m_{hh}$\n  distributions for our two sources of signal (GF and WBF), the dominant background\n  $t\\bar{t}jj$ and the rest of the backgrounds (stacked scaled by\n  relative cross sections), after the Base Selection of\n  Section~\\ref{sec:tt} has been applied.}\n\\label{fig:basehadronlvl}\n\\end{figure*}\n\n\n\nWe begin the hadron-level analysis implemented in\nRivet~\\cite{Buckley:2010ar} by recreating a base selections similar\nto~\\cite{hhjj}:\\footnote{Our analysis has been validated with two\n  independent implementations.}\n\\begin{enumerate}[1.)]\n\\item We require two tau leptons using a two tau-trigger based on\n  staggered transverse momentum selection cuts $p_T \\ge\n  29,20~\\text{GeV}$ in $|\\eta_\\tau|<2.5$ and assume a flat tau tagging\n  efficiency of 70\\% with no fakes.\n\n  Jets are constructed by clustering $R = 0.4$ anti-$k_T$ jets using\n  \\textsc{FastJet}~\\cite{Cacciari:2011ma} with $p_{T,j} \\ge 25$ GeV\n  and $| \\eta_j | \\le 4.5$.\n\\item The two leading jets are $b$-tagged with an acceptance of 70\\%\n  and fake rate of 1\\% \\cite{btagging} in the central part of the\n  detector $|\\eta_j|<2.5$. We remove events if either of the two\n  leading jets overlaps with a tau. Any additional jets which do not\n  overlap with a tau are considered as potential ``tagging jets'', of\n  which we require at least two.\n\\item As a final step of this base selection we require the $b$ jet\n  and tau pairs to reproduce the Higgs mass of 125 GeV within $\\pm 15$\n  and $\\pm 25~\\text{GeV}$ respectively.\\footnote{A high mass\n    resolution is a crucial cornerstone of any successful di-Higgs\n    analysis to assure a minimum pollution of $Z$ boson decay\n    backgrounds~\\cite{us2}.} \n\\end{enumerate}\nThe signal and background cross sections after these cuts\nare presented in the Base Selection column of\nTable~\\ref{tab:xstable}.  We find that the background contribution of\n$t\\bar{t}jj$ dominates with $t\\bar{t}h$ also providing a larger-than-signal\nbackground resulting in $S\/B\\sim~1\/300$, making a study based only on these \nselections extremely challenging. Since we only have $\\sim 40$ expected \ngluon fusion and $\\sim~10$ expected weak boson fusion events at 3 ab$^{-1}$ \nluminosity, additional selections must also be careful to retain enough signal\ncross section to allow statistically meaningful statements to be made\nwith a finite amount of data.\n\nShape comparisons for the rapidity and dihiggs invariant mass distributions  as\nmotivated in the previous section are shown in\nFig.~\\ref{fig:basehadronlvl}. Indeed, as expected, cutting on the\nangular distance of the jets will serve to both purify towards a\nWBF-only selection at a reduced background rate. The dominant\nbackgrounds are unlikely to produce a large invariant mass\n$m_{hh}$. However the WBF contribution, due to the lack of the $2m_t$\nthreshold peaks at a considerably lower invariant mass, leading to\nsignificant decrease of the WBF contribution for a reasonably strong\ncut on $m_{hh}$, which is required to observe the $hhjj$ signal at the\ngiven low signal yield, even at 3 ab$^{-1}$ luminosity.\n\n\\begin{figure*}[t!]\n  \\centering\n  \\subfigure[\\label{fig:nosystematics}]{\\includegraphics[width=0.45\\textwidth]{zetalimits_nosystematics.eps}}\n  \\hfill\n  \\subfigure[\\label{fig:20systematics}]{\\includegraphics[width=0.45\\textwidth]{zetalimits_20systematics.eps}}\n  \\caption{Expected limits on the gauge-Higgs quartic couplings $\\zeta=g_{VV hh}\/g_{VV hh}^{\\rm{SM}}$ under the assumption of no systematic \n    uncertainties~(a) and 20\\% systematic uncertainties~(b).}\n  \\label{fig:zetalimits}\n\\end{figure*}\n\n\\begin{figure*}[t!]\n  \\centering\n  \\subfigure[\\label{fig:gfjetti32}]{\\includegraphics[width=0.45\\textwidth]{hadronlvl_GFnjettiness32.eps}}\n  \\hfill\n  \\subfigure[\\label{fig:wbfjetti32}]{\\includegraphics[width=0.45\\textwidth]{hadronlvl_WBFnjettiness32.eps}}\\\\[0.2cm]\n  \\subfigure[\\label{fig:gfthrustmaj}]{\\includegraphics[width=0.45\\textwidth]{hadronlvl_GFthrustmajor.eps}}\n  \\hfill\n  \\subfigure[\\label{fig:wbfthrustmaj}]{\\includegraphics[width=0.45\\textwidth]{hadronlvl_WBFthrustmajor.eps}}\n  \\caption{Shape comparisons of $N$-jettiness and thrust calculated in the\n    major direction after the gluon fusion selection of Sec.~\\ref{sec:isolgf} (a,c)\n    and WBF Selection of Sec.~\\ref{sec:isolwbf} (b,d) have been applied.}\n   \\label{fig:evtshape}\n\\end{figure*}\n\n\n\n\\subsection{Prospects to isolate gluon fusion}\n\\label{sec:isolgf}\nWe can extend the analysis outlined in Sec.~\\ref{sec:tt} with the aim\nto purify the selection towards the GF component.\\footnote{Following\n  the analysis of \\cite{Bredenstein:2008tm}, we can expect negligible\n  interference between WBF and GF and which allows us to make this\n  distinction.} We make use of the hard Higgs candidates to greatly\nreduce the backgrounds by requiring $m_{hh} \\ge 500$ GeV and\nadditionally require $\\Delta \\eta(j1,j2) \\le 5$ to minimise the weak\nboson fusion contribution.  The signal and background cross sections\nafter these cuts are applied are presented in the `GF Selection'\ncolumn of Table~\\ref{tab:xstable}.\n\nThe total background is reduced by a factor of $\\sim~100$ while the\ngluon fusion contribution only is reduced by a factor of $\\sim 2.5$\nwhich allows for an encouraging $S\/\\sqrt{\\text{B}}~\\sim~1.7$ with\n3 ab$^{-1}$ of data. The weak boson fusion contribution is \nalso suppressed compared to GF which allows for a clean probe of the \nphysics accessible in the gluon fusion contribution.\n\n\\subsection{Prospects to isolate weak boson fusion}\n\\label{sec:isolwbf}\nSimilarly we can extend the analysis towards isolating the WBF\ncomponent. Since it has slightly softer Higgs candidates we require\n$m_{hh} \\ge 400$ GeV and $\\Delta \\eta(j1,j2) \\ge 5$ to reduce both the\ngluon fusion and background contributions.  The signal and background\ncross sections after these cuts are applied are presented in the `WBF\nSelection' column of Table~\\ref{tab:xstable}.\n\nThe total background is reduced by a factor of $\\sim~300$ while three\ntimes more of the weak boson fusion contribution is retained compared\nto the GF selection, resulting in $S\/\\sqrt{{B}}~\\sim~0.8$ with 3\nab$^{-1}$ of data due to the large reduction in the gluon fusion\ncontribution. However even so the WBF selection is composed of one-to-three parts\nGF to WBF, which means measurements of physics that only enters the weak\nboson fusion contribution will need to take this gluon fusion\n``pollution'' into account.\n\n\\subsection{Constraining the quartic $VV hh$ contribution}\n\\label{sec:limitzeta}\nAs mentioned in Section~\\ref{sec:wbf} there is a contribution from\nquartic $VV hh$ vertices to the WBF induced signal, and modifications\nof the corresponding $g_{VV hh}$ couplings away from their SM values\nusing the Higgs Cross Section Working Group $\\kappa$ framework\n\\cite{Heinemeyer:2013tqa} will greatly enhance the signal cross section. This\nallows us to constrain $\\zeta$ defined by $g_{VV hh} = \\zeta \\times\ng_{VV hh}^\\text{SM}$.  To achieve this we have generated events with\nvarying $\\zeta$ using \\textsc{MadEvent} v5 and applied the WBF\nselections described in Section~\\ref{sec:isolwbf} to estimate the\nenhancement of the signal, which is compared to expected cross section\nlimits on the signal with 3 ab$^{-1}$ of data in the WBF selection\nunder the assumptions of no systematic uncertainties and 20\\% total\nsystematic uncertainties for comparison. The results are presented in\nFigure~\\ref{fig:zetalimits}.  We find that in the more realistic\nscenario of 20\\% systematic uncertainties the expected constraint on\nthe $g_{VV hh}$ couplings is $0.55 < \\zeta < 1.65$ at 95\\% confidence\nlevel. A measurement of $pp\\to hhjj$ is therefore crucial to constrain\nnew physics which enters predominantly through enhancements to $g_{VV\n  hh}$.\n\n\\subsection{Event shapes of the tagging jets system}\nThe analysis strategies outlined so far have mainly relied on\nexploiting correlations in the di-Higgs system, with only $\\Delta\n\\eta(j1,j2)$ carrying information about the tagging jets. Following similar applications in the context of single Higgs\nproduction \\cite{Englert:2012ct}, we investigate a range of event\nshapes in the tagging jets system in the following, which could offer\nadditional discriminating power through capturing colour correlations\nin the different signal contributions beyond angular\ndependencies. More specifically, we will focus on $N$-jettiness\n\\cite{jettiness,Thaler:2010tr} and thrust major which provided \nthe best results.\n\nWe calculate $N$-jettiness by minimising\n\\begin{equation}\n \\tau_N = C \\sum_k p_{T,k} \\min(\\Delta R_{k,1},\\dots, \\Delta R_{k,N})\n\\end{equation}\nwhere $C$ is a normalisation which cancels when taking the ratio of\ntwo $\\tau$s, the sum is taken over all visible momenta which do not\nbelong to one of the identified Higgs candidates within $|\\eta| < 5$,\nand $\\Delta R_{k,n}$ is the distance in the $\\eta - \\phi$ plane \nbetween the $k$-th momentum and the $n$-th reference vector. \n$\\tau_{3\/2}$ is then explicitly given by $\\tau_{3}\/\\tau_{2}$.\n\nThrust major is defined by\n\\begin{subequations}\n  \\begin{equation}\n    T_{\\text{maj}}=\\max_{{\\vec{n}}\\cdot{\\vec{n}}_T=0} \\frac{\\sum_k\n      |{\\vec{p}}_k\\cdot {\\vec{n}}|}{\\sum_k |{\\vec{p}}_k|}\n  \\end{equation}\n  where ${\\vec{n}}_T$ is the normalised thrust vector\n  \\begin{equation}\n    {\\vec{n}}_T=\\max_{\\vec{n}}\\frac{\\sum_k\n      |{\\vec{p}}_k\\cdot {\\vec{n}}|}{\\sum_k |{\\vec{p}}_k|}\\,,\n  \\end{equation}\n\\end{subequations}\nAgain the sums run over all visible momenta which do not belong\nto one of the identified Higgs candidates within $|\\eta| < 5$.\n\nWe find $\\tau_{3\/2}$ and $T_\\text{maj}$ show promise for improving the\nWBF selection, but the signal cross section is already too low for us\nto be able to make meaningful use of this insight. The $\\tau_{3\/2}$\nand $T_\\text{maj}$ distributions after the GF and WBF selections have\nbeen applied are presented in Fig.~\\ref{fig:evtshape}. Cutting, e.g.,\non $T_{\\text{maj}}<0.05$, the gluon fusion contribution is reduced by\n80\\%, while the WBF contribution is reduced by only 55\\% amounting to\na total of 2 expected WBF and 0.3 expected GF events, with backgrounds \nvery strongly suppressed. This means \nthat WBF can in principle be observed at a small rate that can be used \nto set constraints on new physics in an almost GF-free selection with \ngreatly reduced backgrounds.\n\nThe event shape distributions can also be used to greatly reduce the\nbackground in the GF selection, Fig.~\\ref{fig:gfthrustmaj}. It should\nbe noted that these improvements of GF vs WBF vs background ultimately\ndepend on underlying event and pile up conditions and have to be taken\nwith a grain of salt at this stage early in run 2. However the clear\nseparation that can be achieved with these observables indicate that\nan analysis employing MVA techniques could, at least in theory,\nsignificantly improve the results presented here. These techniques may\nalso prove useful at a 100~TeV collider where the dihiggs production\ncross-section is substantially higher~\\cite{Barr:2014sga}.\n\n\\section{Summary and Conclusions}\nAfter discovering single Higgs production at the Large\nHadron Collider, new analysis strategies need to be explored to\nfurther constrain the presence of new physics beyond the Standard\nModel. Higgs pair production is pivotal in this regard as constraints\nfrom multi-Higgs production contain complementary information, in\nparticular with respect to the Higgs boson's self-interaction. Cross\nsections for di-Higgs production are generically small at the LHC,\nwhich highlights the necessity to explore other viable channels than\n$pp\\to hh$ to enhance sensitivity in a combined fit at high\nluminosity. To this end, we have investigated $pp\\to hh jj$ production\nin detail in this paper. Keeping the full top and bottom mass\ndependencies, we find sensitivity of $pp\\to hhjj$ searches at the LHC\nfor production in the SM and beyond. The gluon fusion contribution\nremains important at high invariant di-Higgs masses where the dominant\nbackgrounds can be suppressed to facilitate a reasonable signal vs\nbackground discrimination. Unfortunately, the gluon fusion\ncontribution remains large even for selections that enhance the weak\nboson fusion fraction of $pp\\to hhjj$ events. This ``pollution'' is\nimportant when such selections are employed to set constraints on new\nphysics effects that enter in the WBF contribution exclusively. Large\nnew physics effects in the WBF contribution can still be constrained,\nwhich we have illustrated through an investigation of the constraints\nthat can be set on deviations of the quartic $VV hh$ couplings from\ntheir SM values with the HL-LHC, demonstrating that a measurement of\n$pp\\to hhjj$ will provide a powerful probe of these.  Employing\nobservables which are intrinsically sensitive to the different colour\ncorrelation of WBF compared to GF, the discrimination between GF, WBF,\nand background can be further improved.  However, the signal cross\nsection is typically already too small to use such a strategy to\nconstrain the presence of new physics if those effects are only a\nsmall deviation around the SM. If new physics effects are sizable,\nsuch an approach will remain a well-adapted strategy to minimise GF\ntowards a pure WBF selection.\n\n\n\\vskip 1\\baselineskip\n\\noindent {\\emph{Acknowledgements.}} CE and MS are grateful to the\nMainz Institute for Theoretical Physics (MITP) for its hospitality and\nits partial support during the workshop ``Higgs Pair Production at\nColliders'' where part of this work was completed. CE is supported in\nparts by the Institute for Particle Physics Phenomenology\nAssociateship programme. KN thanks the University of Glasgow College\nof Science \\& Engineering for a PhD scholarship. This research was\nsupported in part by the European Commission through the 'HiggsTools'\nInitial Training Network PITN-GA-2012-316704.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe study of B physics is essential to determine the flavor\nstructure of the Standard Model, through knowledge of the \nCabibbo-Kobayashi-Maskawa (CKM) matrix describing quark mixing \nand CP violation, which may be associated with the lack of \nsymmetry between matter and anti-matter in the Universe. \nIn fact, since the amount of baryons in the Universe predicted using \nthe CKM mechanism is several orders of magnitude smaller\nthan what is observed by astronomers, extensions of the \nStandard Model propose additional sources of CP violation,  \nwhich must be tested against Standard-Model predictions.\nB mesons provide the ideal environment for such tests.\\cite{Antonelli:2009ws}\nIn particular, high-precision theoretical inputs are needed\nfor hadronic matrix elements, which may be computed starting\nfrom the gauge theory itself using numerical simulations of lattice QCD.\nAt present, however, it is not yet feasible to perform simulations \non lattices that can simultaneously represent the two relevant scales \nof B physics:\nthe low energy scale $\\Lambda_{\\rm QCD}$, requiring\nlarge physical lattice size, and the high energy scale of\nthe b-quark mass $m_b$, requiring very small lattice spacing $a$.\nAn approximate framework is therefore needed, but one should strive\nto achieve sufficiently precise results, otherwise the task of\noverconstraining the parameters of the Standard Model is compromised.\n\nA promising such framework is to consider (lattice) heavy-quark effective \ntheory (HQET), which allows for an elegant theoretical treatment, with \nthe possibility of fully nonperturbative \nrenormalization.\\cite{Heitger:2003nj,Sommer:2006sj}\nThe approach is briefly described as follows.\nHQET provides a valid low-momentum description for systems with one\nheavy quark, with manifest heavy-quark symmetry in the limit \n$m_b \\to \\infty$. The heavy-quark flavor \nand spin symmetries are broken at finite values of $m_b$\nrespectively by kinetic and spin terms, with first-order corrections \nto the static Lagrangian parametrized by $\\omega_{\\rm kin}$ and \n$\\omega_{\\rm spin}$\n\\begin{equation}\n{\\cal L}^{\\rm HQET}\n\\;=\\; \\overline{\\psi}_{\\!h}(x)\\,D_0\\,\\psi_h(x)\n\\,-\\, \\omega_{\\rm kin}\\, {\\cal O}_{\\rm kin} \n\\,-\\, \\omega_{\\rm spin}\\,{\\cal O}_{\\rm spin}\\,,\n\\end{equation}\nwhere \n\\begin{equation}\n{\\cal O}_{\\rm kin}\\;=\\; \\overline{\\psi}_{\\!h}(x)\\,{\\bf D}^2\\,\\psi_h(x)\\,,\n\\quad\\quad\n{\\cal O}_{\\rm spin}\\;=\\; \n\\overline{\\psi}_{\\!h}(x)\\,{\\bf \\sigma}\\cdot{\\bf B}\\,\\psi_h(x)\\,.\n\\end{equation}\nThese ${\\rm O}(1\/m_b)$ corrections are incorporated by an expansion \nof the statistical weight in $1\/m_b$\nsuch that ${\\cal O}_{\\rm kin}$, ${\\cal O}_{\\rm spin}$ are\ntreated as insertions into static correlation functions.\nThis guarantees the existence of a continuum limit, with results that\nare independent of the regularization, provided that the renormalization\nbe done nonperturbatively. \n\nAs a consequence, expansions for masses and decay constants \nare given respectively by\n\\begin{equation}\nm_B \\;=\\; m_{\\rm bare} \\,+\\,E^{\\rm stat}\n\\,+\\, \\omega_{\\rm kin} \\, E^{\\rm kin} \\,+\\,\n\\omega_{\\rm spin} \\, E^{\\rm spin}\n\\end{equation}\nand\n\\begin{equation}\nf_B\\,\\sqrt{\\frac{m_B}{2}} \\;=\\; Z_A^{\\rm HQET}\\,p^{\\rm stat}\\,\n(1\\,+\\, c_A^{\\rm HQET} \\, p^{\\delta A}\n\\,+\\, \\omega_{\\rm kin} \\, p^{\\rm kin} \\,+\\,\n\\omega_{\\rm spin} \\, p^{\\rm spin}) \\,,\n\\end{equation}\nwhere the parameters $m_{\\rm bare}$ and $Z_A^{\\rm HQET}$ are written as\nsums of a static and an ${\\rm O}(1\/m_b)$ term (denoted respectively with the \nsuperscripts ``${\\rm stat}$'' and ``$1\/m_b$'' below), and \n$c_A^{\\rm HQET}$ is of order $1\/m_b$.\nBare energies ($E^{\\rm stat}$, etc.) and matrix elements \n($p^{\\rm stat}$, etc.) are computed in the numerical \nsimulation.\n\nThe divergences (with inverse powers of $a$) in the above parameters \nare cancelled through the nonperturbative renormalization, \nwhich is based on a matching of HQET parameters to QCD on lattices of \nsmall physical volume\n--- where fine lattice spacings can be considered --- and extrapolation \nto a large volume by the step-scaling method.\nSuch an analysis has been recently completed for the quenched \ncase.\\cite{Blossier:2010jk}\nIn particular, there are nonperturbative (quenched) \ndeterminations of the static coefficients $m_{\\rm bare}^{\\rm stat}$\nand $Z_{A}^{\\rm stat}$ for HYP1 and HYP2 static-quark \nactions\\cite{DellaMorte:2003mn} at the physical b-quark mass, and similarly\nfor the ${\\rm O}(1\/m_b)$ parameters\n$\\omega_{\\rm kin}$, $\\omega_{\\rm spin}$,\n$\\,m_{\\rm bare}^{1\/m_b}$, $\\,Z_{A}^{1\/m_b}$ and $c_A^{\\rm HQET}$.\n\nThe newly determined HQET parameters are very precise (with errors of \na couple of a percent in the static case) and show the expected behavior \nwith $a$. They are used in our calculations reported here, to perform\nthe nonperturbative renormalization of the (bare) observables computed\nin the simulation. Of course, in order to keep a high precision, also \nthese bare quantities have to be accurately determined. This is accomplished\nby an efficient use of the generalized eigenvalue problem\n(GEVP) for extracting energy levels $E_n$ and matrix elements, as\ndescribed below.\n\n\nA significant source of systematic errors in the determination of\nenergy levels in lattice simulations is the contamination from\nexcited states in the time correlators \n\\begin{equation}\nC(t) \\;=\\; \\,\\langle O(t)\\,O(0)\\rangle\\,  \\;=\\;\n\\sum_{n=1}^{\\infty}\\, |\\langle n|\\,\\hat{O}\\,|0 \\rangle |^2\\;e^{-E_n t}\n\\end{equation}\nof fields $\\,O(t)$ with the quantum numbers of a given bound state. \n\nInstead of starting from simple local fields $O$\nand getting the (ground-state) energy from an effective-mass plateau in \n$C(t)$ as defined above, it is then advantageous to consider all-to-all \npropagators\\cite{Foley:2005ac} and to solve, instead, the GEVP \n\\begin{equation}\nC(t)\\,v_n(t,t_0)\\;=\\;\n\\lambda_n(t,t_0)\\,C(t_0)\\,v_n(t,t_0)\\,,\n\\end{equation}\nwhere $t>t_0$ and $C(t)$ is now a matrix of correlators, given by\n\\begin{equation}\nC_{ij}(t)\\,\\;=\\;\\,\\langle O_i(t) O_j(0)\\rangle\n\\,\\;=\\;\\, \\sum_{n=1}^\\infty {\\rm e}^{-E_n t}\\, \\Psi_{ni}\\Psi_{nj}\\,,\\quad\ni, j = 1,\\ldots, N\\,.\n\\end{equation}\nThe chosen interpolators $O_i$ are taken (hopefully) \nlinearly independent, e.g.\\ they may be built from the \nsmeared quark fields using $N$ different smearing levels. \nThe matrix elements $\\Psi_{ni}$ are defined by\n\\begin{equation}\n\\Psi_{ni} \\;\\equiv\\; (\\Psi_n)_i \\;=\\; \\langle n|\\hat O_i|0\\rangle\\;,\n\\quad\\; \\langle m|n\\rangle \\,=\\,\\delta_{mn}\\,.\n\\end{equation}\n\nOne thus computes $C_{ij}$ for the interpolator basis $O_i$ from \nthe numerical simulation, then gets effective energy levels $E_n^{\\rm eff}$ \nand estimates for the matrix elements $\\Psi_{ni}$ from the\nsolution $\\lambda_n(t,t_0)$ of the GEVP at large $t$. For the energies\n\\begin{equation}\nE_n^{\\rm eff}(t,t_0)\\;\\equiv\\; \n{1\\over a} \\, \\log{\\lambda_n(t,t_0) \\over \\lambda_n(t+a,t_0)}\n\\end{equation}\nit is shown\\cite{Luscher:1990ck}\nthat $E_n^{\\rm eff}(t,t_0)$ converges exponentially as $t\\to\\infty$ \n(and fixed $t_0$) to the true energy $E_n$. However, since the\nexponential falloff of higher contributions may be slow,\nit is also essential to study the convergence as a function of\n$t_0$ in order to achieve the required efficiency for the method.\nThis has been recently done,\\cite{Blossier:2009kd} by explicit application \nof (ordinary) perturbation theory to a hypothetical truncated problem \nwhere only $N$ levels contribute. The solution in this case is exactly \ngiven by the true energies, and corrections due to the higher states \nare treated perturbatively. We get\n\\newcommand{\\hat {\\cal A}_n^{\\rm eff}}{\\hat {\\cal A}_n^{\\rm eff}}\n\\newcommand{\\hat {\\cal Q}_n^{\\rm eff}}{\\hat {\\cal Q}_n^{\\rm eff}}\n\\newcommand{\\varepsilon_{n}}{\\varepsilon_{n}}\n\\newcommand{\\pi_{n}}{\\pi_{n}}\n\\begin{equation}\nE_n^{\\rm eff}(t,t_0) \\;=\\; \nE_n \\,+\\, {\\varepsilon_{n}(t,t_0)}\\,\n\\end{equation}\nfor the energies and\n\\begin{equation}\n{\\rm e}^{-\\hat H t}(\\hat {\\cal Q}_n^{\\rm eff}(t,t_0))^\\dagger|0\\rangle \\;=\\;\n|n\\rangle \\,+\\, \\sum_{n'=1}^\\infty  \\pi_{nn'}(t,t_0)\n\\, |n'\\rangle\n\\end{equation}\nfor the eigenstates of the Hamiltonian, which may be estimated through\n\\begin{eqnarray}\n    \\hat {\\cal Q}_n^{\\rm eff}(t,t_0) &=& R_n \\,(\\hat O\\,,\\,v_n(t,t_0)\\,) \\,, \\\\[2mm]\nR_n &=& {\\left(v_n(t,t_0)\\,,\\, C(t)\\,v_n(t,t_0)\\right)}^{-1\/2}\n\\; \\left[{\\lambda_n(t_0+a,t_0) \\over \\lambda_n(t_0+2a,t_0)}\\right]^{t\/2}\\,.\n\\end{eqnarray}\n\nIn our analysis we see that, due to cancellations of $t$-independent \nterms in the effective energy, the first-order corrections in \n${\\varepsilon_{n}(t,t_0)}$ are independent of $t_0$ and very strongly \nsuppressed at large $t$. We identify two regimes:\n1) for $t_0 \\,<\\, t\/2$, the 2nd-order corrections dominate and\ntheir exponential suppression is given by the smallest energy gap \n$\\,|E_m-E_n|\\equiv \\Delta E_{m,n}\\,$ between level $n$ and its neighboring \nlevels $m$; and 2) for $t_0 \\,\\geq\\, t\/2$,\nthe 1st-order corrections dominate and the suppression is given by the \nlarge gap $\\Delta E_{N+1,n}$. \nAmplitudes $\\,\\pi_{nn'}(t,t_0)\\,$ get main contributions\nfrom the first-order corrections. For fixed $t-t_0$ these are also \nsuppressed with $\\Delta E_{N+1,n}$.\nClearly, the appearance of large energy gaps in the second regime\nimproves convergence significantly. We therefore work with $t$, $t_0$\ncombinations in this regime.\n\n\n\\def1\/m_b{1\/m_b}\n\\def\\rm {stat}{\\rm {stat}}\n\nA very important step of our approach is to realize that the same \nperturbative analysis may be applied to get the $1\/m_b$ \ncorrections in the HQET correlation functions mentioned previously\n\\begin{equation}\n  C_{ij}(t) \\;=\\; C_{ij}^{\\rm {stat}}(t) \\,+\\,\n  \\omega \\,C_{ij}^{1\/m_b}(t) \\,+\\, {\\rm O}(\\omega^2)\\,,\n\\end{equation}\nwhere the combined ${\\rm O}(1\/m_b)$ corrections are symbolized by \nthe expansion parameter $\\omega$.\nFollowing the same procedure as above, we get similar exponential\nsuppressions (with the static energy gaps) for static and ${\\rm O}(1\/m_b)$ \nterms in the effective theory. We arrive at\n\\begin{equation}\n    E_n^{\\rm eff}(t,t_0) \\;=\\; E_n^{{\\rm eff},{\\rm stat}}(t,t_0)\n     +\\omega E_n^{{\\rm eff},{1\/m_b}}(t,t_0) +{\\rm O}(\\omega^2)\n\\end{equation}\nwith\n\\begin{eqnarray}\n    E_n^{{\\rm eff},{\\rm stat}}(t,t_0) &=&\n    E_n^{\\rm {stat}} \\,+\\, \\beta_n^{\\rm {stat}} \\,\n    {\\rm e}^{-\\Delta E_{N+1,n}^{\\rm {stat}}\\, t}+\\ldots\\,, \n\\label{effenergies}\n\\\\[2mm]\n    E_n^{\\rm eff,1\/m_b}(t,t_0) &=&\n    E_n^{1\/m_b} \\,+\\, [\\,\\beta_n^{1\/m_b}\n    \\,-\\, \\beta_n^{\\rm {stat}}\\,t\\;\\Delta E_{N+1,n}^{1\/m_b}\\,]\n    {\\rm e}^{-\\Delta E_{N+1,n}^{\\rm {stat}}\\, t}+\\ldots\\, .\n\\end{eqnarray}\nand similarly for matrix elements.\nPreliminary results of our application of the methods described \nin this section were presented recently\\cite{Blossier:2009mg}\nand are summarized in the next section. A more detailed version of this \nstudy will be presented elsewhere.\\cite{inprep}\n\n\n\\section{Results}\n\nWe carried out a study of static-light B$_{\\rm s}$-mesons in \nquenched HQET with the nonperturbative parameters described in the\nprevious section,\nemploying the HYP1 and HYP2 lattice actions for the static quark\nand an ${\\rm O}(a)$-improved Wilson action for the strange quark\nin the simulations. The lattices considered were of the form\n$L^3 \\times 2L$ with periodic boundary conditions. We took\n$L\\approx1.5$ fm and lattice spacings $0.1$ fm, $0.07$ fm \nand $0.05$ fm, corresponding respectively to\n$\\beta=6.0219$, $6.2885$ and $6.4956$.\nWe used all-to-all strange-quark propagators constructed from\napproximate low modes, with 100 configurations.  \nGauge links in interpolating fields were smeared with 3 iterations \nof (spatial) APE smearing, whereas Gaussian smearing (8 levels) was used\nfor the strange-quark field.\nA simple $\\gamma_0\\gamma_5$ structure in Dirac space was taken\nfor all 8 interpolating fields. \nAlso, the local field (no smearing) was included in order\nto compute the decay constant.\n\n\nThe resulting ($8\\times8$) correlation matrix may be\nconveniently truncated to an $N\\times N$ one and the GEVP\nsolved for each $N$, so that results can be studied as a function of $N$.\nWe then pick a basis from unprojected interpolators, by\nsampling the different smearing levels (from 1 to 7) as\n$\\{1,7\\}$,$\\{1,4,7\\}$, etc. We perform fits of the various energy \nlevels and values of $N$ to the behavior in Eq.\\ (\\ref{effenergies})\nand extract our results from the predicted plateaus.\nNext, we take the continuum limit, extrapolating our results to \n$a\\to 0$. We see that the correction to the ground-state energy\ndue to terms of order $1\/m_b$, which is positive for finite $a$,\nis quite small (consistent with zero) in the continuum limit.\nOur results for the pseudoscalar meson \ndecay constant, both in the static limit and including ${\\rm O}(1\/m_b)$\ncorrections, are shown in terms of the combination \n$\\Phi^{\\rm HQET}\\equiv F_{\\rm PS}\\,\\sqrt{m_{\\rm PS}}\/C_{\\rm PS}$, \nwhere $C_{\\rm PS}(M\/\\Lambda_{\\rm QCD})$ \nis a known matching function and $\\Phi^{\\rm RGI}$ denotes the\nrenormalization-group-invariant matrix element of the static \naxial current.\\cite{DellaMorte:2007ij}\nThese two continuum extrapolations are shown in comparison with\nfully relativistic heavy-light (around charm-strange) \ndata\\cite{DellaMorte:2007ij} in Fig.\\ \\ref{fig:fBs_comp} below.\nNote that, up to perturbative corrections of order $\\alpha^3$\nin $C_{\\rm PS}$, HQET predicts a behavior $const. + {\\rm O}(1\/r_0 m_{\\rm PS})$\nin this graph. Surprisingly no $1\/(r_0 m_{\\rm PS})^2$ terms are \nvisible, even with our rather small errors.\n\n\n\\begin{figure}\n\\ \\vspace*{-6mm}\n\\begin{center}\n\\includegraphics[width=10truecm]{fb_new3.eps}\n\\ \\vspace*{-3mm}\n\\caption{Comparison of the continuum values for the pseudoscalar\nmeson decay constant from Fig.\\ 4\nto fully relativistic data in the charm region. The solid line is a\nlinear interpolation between the static limit and the points around\nthe charm-quark mass, which corresponds to $\\,1\/r_0\\, m_{\\rm PS}\\approx 0.2$.\n}\n\\label{fig:fBs_comp}\n\\end{center}\n\\ \\vspace*{-6mm}\n\\end{figure}\n\n\n\\section{Conclusions}\n\nThe combined use of nonperturbatively determined HQET parameters \n(in action and currents) and efficient GEVP allows us to reach \nprecisions of a few percent in matrix elements and of a few MeV \nin energy levels, even with only a moderate number of configurations. \nThe method is robust with respect to the choice of interpolator basis.\nAll parameters have been determined nonperturbatively and in \nparticular power divergences are completely subtracted.\nWe see that HQET plus ${\\rm O}(1\/m_b)$ corrections at the b-quark \nmass agrees well with an interpolation between the static point and the\ncharm region, indicating that linearity in $1\/m$ extends even to \nthe charm point. A corresponding study for $N_f=2$ is in progress.\n\n\n\\vskip 3mm\n\\noindent\n{\\bf Acknowledgements.}\nThis work is supported by the  DFG\nin the SFB\/TR 09, and by the EU Contract \nNo.\\ MRTN-CT-2006-035482, ``FLAVIAnet''. \nT.M. thanks the A. von Humboldt Foundation;\nN.G. thanks the MICINN grant FPA2006-05807, the\nComunidad Aut\\'onoma de Madrid programme HEPHACOS P-ESP-00346 and\nthe Consolider-Ingenio 2010 CPAN (CSD2007-00042).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIn a fundamental paper on the coarse geometry of Banach spaces (\\cite{kaltonpropertyq}), N. Kalton introduced a property of metric spaces that he named property $\\mathcal Q$. In particular,\nits absence served as an obstruction to coarse embeddability into reflexive Banach spaces. This property is related to the behavior of Lipschitz maps defined on a particular family of metric graphs that we shall denote $([\\N]^k,d_{\\K}^k)_{k \\in \\N}$. We will recall the precise definitions of these graphs and of property $\\mathcal Q$ in section \\ref{subsection:Q}. Let us just say, vaguely speaking for the moment, that a Banach space $X$ has property $\\mathcal Q$ if for every Lipschitz map $f$ from $([\\N]^k,d_{\\K}^k)$ to $X$, there exists a full subgraph $[\\M]^k$ of $[\\N]^k$, with $\\M$ infinite subset of $\\N$, on which $f$ satisfies a strong concentration phenomenon. It is then easy to see that if a Banach space $X$ has property $\\mathcal Q$, then the family of graphs $([\\N]^k,d_{\\K}^k)_{k \\in \\N}$ does not equi-coarsely embed into $X$ (see the definition in section \\ref{s:coarse}). One of the main results in \\cite{kaltonpropertyq} is that any reflexive Banach space has property $\\mathcal Q$. It then readily follows that a reflexive Banach space cannot contain a coarse copy of all separable metric spaces, or equivalently does not contain a coarse copy of the Banach space $c_0$. In fact, with a sophistication of this argument, Kalton proved an even stronger result in \\cite{kaltonpropertyq}: if a separable Banach space contains a coarse copy of $c_0$, then there is an integer $k$ such that the dual of order $k$ of $X$ is non separable. In particular, a quasi-reflexive Banach space does not contain a coarse copy of $c_0$.\nHowever, Kalton proved that the most famous example of a quasi-reflexive space, namely the James space $\\J$, as well as its dual $\\J^*$, fail property $\\mathcal Q$.\n\nThe main purpose of this paper is to show that, although they do not obey the concentration phenomenon described by property $\\mathcal Q$, neither $\\J$ nor $\\J^*$ equi-coarsely contains the family of graphs $([\\N]^k,d_{\\K}^k)_{k \\in \\N}$ (Corollary \\ref{James}). This provides a coarse invariant, namely ``not containing equi-coarsely the Kalton graphs'', that is very close to but different from property $\\mathcal Q$. This could allow to find obstructions to coarse embeddability between seemingly close Banach spaces. Our result is actually more general. We prove in Theorem \\ref{general} that a quasi-reflexive Banach space $X$ such that both $X$ and $X^*$ admit an equivalent $p$-asymptotically uniformly smooth norm (see the definition in section \\ref{asymptotics}), for some $p$ in $(1,\\infty)$, does not equi-coarsely contain the Kalton graphs.\n\nWe conclude this note by showing that if the James tree space $\\J\\T$ or its predual coarsely embeds into a separable Banach space $X$, then there exists $k\\in \\N$ so that the dual of order $k$ of $X$ is non separable. This extends slightly Theorem 3.5 in \\cite{kaltonpropertyq}.\n\n\\section{Metric notions}\n\n\\subsection{Coarse embeddings}\\label{s:coarse}\n\nLet $M$, $N$ be two metric spaces and $f \\colon M \\to N$ be a map. We define the compression modulus $\\rho_f$ and the expansion modulus $\\omega_f$ as follows. For $t\\in [0,\\infty)$, we set\n\\begin{eqnarray*}\n&&\\rho_f (t) = \\inf \\{ d_N(f(x),f(y)) \\, : \\, d_M(x,y) \\geq t \\},\\\\\n&&\\omega_f (t) = \\sup \\{ d_N(f(x),f(y)) \\, : \\, d_M(x,y) \\leq t \\}.\n\\end{eqnarray*}\nWe adopt the convention $\\sup(\\emptyset)=0$ and $\\inf(\\emptyset)=\\infty$.\nNote that for every $x,y \\in M$,\n$$\\rho_f (d_M(x,y)) \\leq d_N(f(x),f(y)) \\leq \\omega_f (d_M(x,y)).$$\nWe say that $f$ is a \\emph{coarse embedding}  if $\\omega_f (t) < \\infty$ for every $t \\in [0,+\\infty)$ and $\\lim_{t \\to \\infty} \\rho_f (t) = \\infty$.\n\nNext, let $(M_i)_{i \\in I}$ be a family of metric spaces. We say that the family $(M_i)_{i \\in I}$ \\emph{equi-coarsely embeds} into a metric space $N$ if there exist two maps $\\rho, \\, \\omega \\colon [0,+\\infty) \\to [0,+\\infty)$ and maps $f_i \\colon M_i \\to N$ for $i \\in I$ such that:\n\\begin{enumerate}[(i)]\n\\item $\\lim_{t \\to \\infty} \\rho(t) = \\infty$,\n\\item $\\omega(t) < \\infty$ for every $t \\in [0,+\\infty)$,\n\\item $\\rho(t) \\leq \\rho_{f_i}(t)$ and $\\omega_{f_i}(t) \\leq \\omega(t)$ for every $i \\in I$ and $t \\in [0,\\infty)$.\n\\end{enumerate}\n\n\n\n\\subsection{The Kalton interlaced graphs and property Q}\n\\label{subsection:Q}\n\nFor $k\\in \\N$  and $\\M$ an infinite subset of $\\N$, we put $[\\M]^{\\le k}=\\{ S\\subset \\M: |S|\\le k\\}$,   $[\\M]^{ k}=\\{ S\\subset \\M: |S|=k\\} $,  $[\\M]^\\omega=\\{ S\\subset \\M: S\\text{ is infinite}\\} $, and $[\\M]^{< \\omega}=\\{ S\\subset \\M: S\\text{ is finite}\\}$.\nWe always list the elements of some $\\m$ in $[\\N]^{< \\omega}$ or in $[\\N]^{ \\omega}$ in increasing order, meaning that if we write $\\m=(m_1,m_2,\\ldots, m_l)$ or  $\\m=(m_1,m_2,m_3, \\ldots )$, we tacitly assume that $m_1<m_2<\\cdots $.\n\nFor $\\m=(m_1,m_2,\\ldots, m_r)\\in [\\N]^{<\\omega}$ and $\\n=(n_1,n_2,\\ldots, n_s)\\in [\\N]^{<\\omega}$, we write $\\m \\prec \\n$, if $r<s\\le k$ and  $m_i=n_i$, for $i=1,2,\\ldots, r$, and we write $\\m\\preceq \\n$ if $\\m \\prec \\n$ or $\\m=\\n$. Thus $\\m \\preceq \\n$ if $\\m$ is an initial segment of $\\n$.\n\nFollowing Kalton \\cite{kaltonpropertyq}, for $\\M \\in [\\N]^\\omega$, we equip $[\\M]^k$ with a graph structure by declaring $\\m\\neq\\n\\in [\\M]^k$ adjacent if and only if\n$$\nn_1 \\leq m_1 \\leq n_2  \\ldots \\leq n_k \\leq m_k\\ \\  \\text{or}\\ \\  m_1 \\leq n_1 \\leq m_2  \\ldots \\leq m_k \\leq n_k .\n$$\nFor any $\\m,\\n \\in [\\M]^k$, the distance $\\dk(\\m,\\n)$ is then defined as the shortest path distance in the graph $[\\M]^k$.\n\n\\begin{remark}The distance $\\dk$ is independent of the set $\\M$ and therefore $[\\M_1]^k$ is a metric subspace of $[\\M_2]^k$ whenever $\\M_1 \\in [\\M_2]^\\omega$.\n\\end{remark}\nThis last claim is an immediate consequence of the following explicit formula for the distance.\n\\begin{prop}\\label{p:KaltonDistanceFormula}\nLet $k \\in \\N$ and $\\M \\in [\\N]^\\omega$.\nThen $\\dk(\\n,\\m)=d(\\n,\\m)$ for all $\\n,\\m\\in [\\M]^k$ where\n$d(\\n , \\m) = \\sup \\{\\big| |\\n\\cap S| - |\\m\\cap S| \\big| \\; : \\; S \\text{ segment of } \\N \\}.$\n\\end{prop}\n\\begin{proof}\nIt is easily seen that $d$ is a metric on $[\\M]^k$.\nSince $\\dk$ is a graph metric on $[\\M]^k$, in order to show $\\dk=d$ it is enough to verify that $\\dk(\\n,\\m)=1$ if and only if $d(\\n,\\m)=1$ and that $d$ is a graph metric.\n\n\nFor $A\\subset \\N$ let us denote $\\indicator{A}:\\N \\to \\set{0,1}$ the indicator function of $A$ and let us first observe the following fact.\\\\\n\\medskip\n\\textbf{Fact:} \\emph{For every $\\n,\\m\\in [\\M]^k$,\n$$d(\\n,\\m)=\\max_{i}F(i)-\\min_{i}F(i)$$\nwhere $F(i)=F_{\\n,\\m}(i)=\\sum_{j=1}^i\\indicator{\\n}(j)-\\indicator{\\m}(j)$ (and $F(0)=0$).}\n\n\\medskip\nIndeed, we have for any segment\n$S=[a,b]$ that\n$$\n\\abs{S\\cap \\n}-\\abs{S\\cap\\m}=\\sum_{j\\in S}\\Big(\\indicator{\\n}(j)-\\indicator{\\m}(j)\\Big)=F(b)-F(a-1).\n$$\nIn particular $\\max_S\\big||S\\cap \\n|-|S\\cap\\m|\\big|\\leq \\max F-\\min F$.\nOn the other hand if $S=[a,b]$ is such that $\\{F(a-1),F(b)\\}=\\{\\max F,\\min F\\}$ then\n$\\big||S\\cap \\n|-|S\\cap\\m|\\big|\\geq \\max F-\\min F$ which finishes the proof of the fact.\n\nIt is clear that $\\dk(\\n,\\m)=1$ if and only if $\\max F-\\min F=1$.\nThus it only remains to prove that $d$ is a graph metric.\nNow given $\\n,\\m$ such that $d(\\n,\\m)\\geq 2$ we are looking for $\\lbar\\in [\\M]^k\\setminus\\set{\\m,\\n}$ such that $d(\\m,\\n)=d(\\n,\\lbar)+d(\\lbar,\\m)$.\nWithout loss of generality we will assume that $\\max F_{\\n,\\m}>0$. \nNotice that the sets $\\arg\\max(F)$ and $\\arg\\min(F)$ are disjoint.\nWe select inductively $\\{a_1<\\ldots<a_p\\}\\subset \\arg\\max(F)$ and $\\{b_1<\\ldots<b_q\\}\\subset \\arg\\min(F)$ (with $p\\geq 1$ and $q\\geq 0$) with the property that\n\n\\begin{itemize}\n\\item $a_1=\\min \\arg\\max(F)$,\n\\item For $i\\ge 1$,\\ $b_i=\\min \\left(\\{n>a_i\\} \\cap \\arg\\min(F)\\right)$, if this is not empty.\n\\item $a_{i+1}=\\min\\left( \\{n>b_i\\} \\cap \\arg\\max(F)\\right)$, if this set is not empty.\n\\end{itemize}\nNotice that $\\{a_1,\\ldots,a_p\\} \\subset \\n \\setminus \\m$ and $\\{b_1,\\ldots,b_q\\} \\subset \\m\\setminus \\n$.\nNotice also that either $p=q$ or $p=q+1$.\nIn the latter case we define $b_p:=r$ for some $r$ such that $r>a_p$ and $F(r-1)>F(r)$.\nSuch $r$ must exist since $F(\\max\\{n_k,m_k\\})=0$.\nAlso we have $r \\in \\m \\setminus \\n$.\nWe will set\n$$\n\\lbar=\\n \\cup \\{b_1,\\ldots,b_p\\} \\setminus \\{a_1,\\ldots,a_p\\}.\n$$\nIt is clear that $\\lbar \\in [\\M]^k$.\nWe also have $\\max F_{\\lbar,\\m}=\\max F_{\\n,\\m}-1$ and $\\min F_{\\lbar,\\m}=\\min F_{\\n,\\m}$.\nIndeed, the point $\\lbar$ is constructed in such a way that when $F_{\\n,\\m}$ attains its maximum for the first time  (going from the left), $F_{\\lbar,\\m}$ is reduced by one and stays reduced by 1 until the next time the minimum of $F_{\\n,\\m}$ is attained (or until the point $r$) where this reduction is corrected back; and so on.\nThus $d(\\lbar,\\m)=d(\\n,\\m)-1$.\nAlso, since the sets $\\set{a_1,\\ldots,a_p}$ and $\\set{b_1,\\ldots,b_p}$ are interlaced we have $F_{\\n,\\m}-1\\leq F_{\\lbar,\\m}\\leq F_{\\n,\\m}$.\nTherefore, since $F_{\\n,\\m}=F_{\\n,\\lbar}+F_{\\lbar,\\m}$, we have that $0\\leq F_{\\n,\\lbar}\\leq 1$ and so finally $d(\\n,\\lbar)=1$, since it is clear that $\\n \\neq \\lbar$.\n\\end{proof}\nNote that if $X$ is a Banach space and $f \\colon ([\\M]^k,d_{\\K}^k) \\to X$ is a map with finite expansion modulus $\\omega_f$, then $\\omega_f(1)$ is actually the Lipschitz constant of $f$ as $d_{\\K}^k$ is a graph distance on $[\\M]^k$.\n\nIn \\cite{kaltonpropertyq} the property $\\mathcal Q$ is defined in the setting of metric spaces. For homogeneity reasons, its definition can be simplified for Banach spaces. Let us  recall it here.\n\n\\begin{defi}\nLet $X$ be a Banach space. We say that $X$ has \\emph{property $\\mathcal{Q}$} if there exists $C\\ge 1$ such that for every $k \\in \\N$ and every Lipschitz map $f \\colon ([\\N]^k,d_{\\K}^k) \\to X$, there exists an infinite subset $\\M$ of $\\N$ such that:\n$$\\forall \\, \\overline{n},\\overline{m} \\in [\\M]^k,\\  \\|f(\\overline{n})-f(\\overline{m})\\| \\leq  C\\omega_f(1).$$\n\\end{defi}\n\nThe following proposition should be clear from the definitions. We shall however include its short proof.\n\\begin{prop} \\label{QandGk}\nLet $X$ be a Banach space. If $X$ has property $\\mathcal Q$, then the family of graphs $([\\N]^k,d_{\\K}^k)_{k\\in \\N}$ does not equi-coarsely embed into $X$.\n\\end{prop}\n\n\\begin{proof}\nLet $C\\ge 1$ be given by the definition of property $\\mathcal Q$. Aiming for a contradiction, assume that the family $([\\N]^k,d_{\\K}^k)_{k\\in \\N}$ equi-coarsely embeds into $X$. That is, there are maps $f_k \\colon ([\\N]^k,d_{\\K}^k) \\to X$ and two functions $\\rho, \\omega \\colon [0,+\\infty)\\to[0,+\\infty)$  such that $\\lim_{t \\to \\infty} \\rho (t)=\\infty$ and\n$$\\forall k\\in \\N\\ \\ \\forall t>0\\ \\ \\rho(t) \\leq \\rho_{f_k}(t)\\ \\ \\text{and}\\ \\ \\omega_{f_k}(t) \\leq \\omega(t)<\\infty.$$\nThus, for every $k\\in \\N$, there exists an infinite subset $\\mathbb M_k$ of $\\N$ such that $\\diam(f([\\M_k]^k)))\\le C\\omega(1)$. Since $\\diam([\\M_k]^k)=k$, this implies that for all $k\\in \\N$, $\\rho(k) \\le C\\omega(1)$. This contradicts the fact that $\\lim\\limits_{t \\to \\infty} \\rho(t)=\\infty$.\n\\end{proof}\n\nA concrete bi-Lipschitz copy of the metric spaces $([\\N]^k,d_{\\K}^k)$ in $c_0$ is given by the following proposition.\n\n\\begin{prop}\\label{Kgraphsinc0} Let $(s_n)_{n=1}^\\infty$ be the summing basis of $c_0$, that is\\\\\n$s_n=\\sum_{i=1}^ne_i$, where $(e_i)_{i=1}^\\infty$ is the canonical basis of $c_0$.\\\\\nFor $k\\in \\N$, define  $f_k:([\\N]^k,d_{\\K}^k)\\to c_0$ by $f_k(\\n)=\\sum_{i=1}^k s_{n_i}$. Then\n$$\\frac12d_{\\K}^k(\\n,\\m)\\le \\|f_k(\\n)-f_k(\\m)\\|_\\infty\\le d_{\\K}^k(\\n,\\m)$$\nfor all $\\n,\\m \\in [\\N]^k$.\n\\end{prop}\n\n\\begin{proof}\nSince $\\dk=d$, one can show (as in the Fact in the proof of Proposition~\\ref{p:KaltonDistanceFormula}) that\n$\\dk(\\n,\\m)=\\max(f_k(\\n)-f_k(\\m))-\\min(f_k(\\n)-f_k(\\m)).$\nThe result then follows easily since $\\min(f_k(\\n)-f_k(\\m))\\leq 0\\leq \\max(f_k(\\n)-f_k(\\m))$ for all $\\n,\\m\\in [\\N]^k$.\n\\end{proof}\n\n\\begin{remark} We already explained that $c_0$ cannot coarsely embed into any Banach space with property $\\mathcal Q$ (in particular into any reflexive Banach space) and that Kalton even showed with additional arguments that if $c_0$ coarsely embeds into a separable Banach space $X$, then one of the iterated duals of $X$ has to be non separable. An inspection of his proof shows that \nthe uniformly discrete metric spaces\n\\[\n M_k=\\set{\\sum_{i=1}^k s_{n_i} \\times \\indicator{A}: (n_1,\\ldots,n_k) \\in [\\N]^k, A \\in [\\N]^\\omega} \\subset c_0\n\\]\ndo not equi-coarsely embed into any Banach space $X$ such that $X^{(r)}$ is separable for all $r$.\nSee Theorem \\ref{kalton} below for more on this subject.\n\\end{remark}\n\nStudying further the property $\\mathcal Q$ in \\cite{kaltonpropertyq}, Kalton exhibited non reflexive quasi-reflexive spaces with the property $\\mathcal Q$ but showed that $\\J$ and $\\J^*$ fail property $\\mathcal Q$. It is worth noticing that a theorem of Schoenberg  \\cite{Schoenberg} implies that $\\ell_1$ coarsely embeds into $\\ell_2$, and therefore $\\ell_1$ provides a simple example of a non-reflexive Banach space with property $\\mathcal Q$.\n\n\\medskip\nWe conclude this section with two propositions that we state here for future reference. We start with a classical version of Ramsey's theorem.\n\\begin{prop}[Corollary 1.2 in \\cite{Gowers}] \\label{ramsey}\nLet $(K,d)$ be a compact metric space, $k \\in \\N$ and $f \\colon [\\N]^k \\to K$. Then for every $\\ep >0$, there exists an infinite subset $\\M$ of $\\N$ such that $d(f(\\overline{n}),f(\\overline{m}))< \\ep$ for every $\\overline{n},\\overline{m} \\in [\\M]^k$.\n\\end{prop}\n\nFor a Banach space $X$, we call \\emph{tree of height $k$} in $X$ any family $(x(\\n))_{\\n\\in[\\N]^{\\le k}}$, with $x(\\n)\\in X$. Then, if $\\M \\in [\\N]^\\omega$, $(x(\\n))_{\\n\\in[\\M]^{\\le k}}$ will be called a \\emph{full subtree} of $(x(\\n))_{\\n\\in[\\N]^{\\le k}}$. A tree $(x^*(\\n))_{\\n\\in[\\M]^{\\le k}}$ in $X^*$ is called \\emph{weak$^*$-null} if for any $\\n \\in [\\M]^{\\le k-1}$, the sequence $(x^*(n_1,\\ldots,n_{k-1},t))_{t>n_{k-1},t\\in \\M}$ is weak$^*$-null.\n\nThe next  proposition is\nbased on a weak$^*$-compactness argument and will be crucial for our proofs. Although the distance considered on $[\\N]^k$ is different, the proof follows the same lines as Lemma 4.1 in \\cite{blms}. We therefore state it now without further detail.\n\n\\begin{prop}\\label{nulltree}  Let $X$ be a separable Banach space,  $k\\in\\N$,  and $f:([\\N]^k,d_{\\K}^k)\\to X^*$  a Lipschitz map. Then there exist $\\M \\in [\\N]^\\omega$ and a weak$^*$-null tree $(x^*(\\m))_{\\m\\in[\\M]^{\\le k}}$ in $X^*$ with $\\|x^*_{\\m}\\| \\leq \\omega_f(1)$ for all $\\m\\in [\\M]^{\\leq k}\\setminus\\{\\emptyset\\}$ and so that\n$$\\forall \\n \\in [\\M]^k,\\ f(\\n)=\\sum_{i=0}^k x^*(n_1,\\ldots,n_i)=\\sum_{\\m \\preceq \\n} x^*(\\m).$$\n\\end{prop}\n\n\n\n\n\\section{Uniform asymptotic properties of norms and related estimates}\n\\label{asymptotics}\n\nWe recall the definitions that will be considered in this paper. For a Banach space $(X,\\|\\ \\|)$ we\ndenote by $B_X$ the closed unit ball of $X$ and by $S_X$ its unit\nsphere. The following definitions are due to V. Milman \\cite{Milman} and we adopt the notation from \\cite{JohnsonLindenstraussPreissSchechtman2002}. For $t\\in [0,\\infty)$ we define $$\\overline{\\rho}_X(t)=\\sup_{x\\in S_X}\\inf_{Y}\\sup_{y\\in S_Y}\\big(\\|x+t y\\|-1\\big),$$\nwhere $Y$ runs through all closed subspaces of $X$ of finite codimension. Then, the norm $\\|\\ \\|$ is said to be {\\it asymptotically uniformly smooth} (in short AUS) if\n$$\\lim_{t \\to 0}\\frac{\\overline{\\rho}_X(t)}{t}=0.$$\nFor $p\\in (1,\\infty)$ it is said to be {\\it $p$-asymptotically uniformly smooth} (in short $p$-AUS) if there exists $c>0$ such that for all $t\\in [0,\\infty)$, $\\overline{\\rho}_X(t)\\le ct^p$.\n\nWe will also need the dual modulus defined \nby\n$$ \\overline{\\delta}_X^*(t)=\\inf_{x^*\\in S_{X^*}}\\sup_{E}\\inf_{y^*\\in S_E}\\big(\\|x^*+ty^*\\|-1\\big),$$\nwhere $E$ runs through all finite-codimensional weak$^*$-closed subspaces of~$X^*$. \nThe norm of $X^*$ is said to be {\\it weak$^*$ asymptotically uniformly convex} (in short AUC$^*$) if $\\overline{\\delta}_X^*(t)>0$ for all $t$ in $(0,\\infty)$. If there exists $c>0$ and $q\\in [1,\\infty)$ such that for all $t\\in [0,1]$ $\\overline{\\delta}_X^*(t)\\ge ct^q$, we say that the norm of $X^*$ is $q$-AUC$^*$.\nThe following proposition is elementary.\n\n\\begin{prop}\\label{as-sequences} Let $X$ be a Banach space. For any $t\\in (0,1)$, any weakly null sequence $(x_n)_{n=1}^\\infty$ in $B_{X}$ and any  $x \\in S_X$ we have:\n$$ \\limsup_{n \\rightarrow \\infty} \\| x+tx_n \\| \\leq 1 + \\overline{\\rho}_{X}(t).$$\n\nFor any weak$^*$-null sequence $(x^*_n)_{n=1}^\\infty \\subset X^*$ and for any $x^* \\in X^*\\setminus \\set{0}$ we have\n\\[\n \\limsup_{n\\to \\infty} \\norm{x^*+x^*_n} \\geq \\norm{x^*}\\left(1+\\overline{\\delta}_X^*\\left(\\frac{\\limsup \\norm{x_n^*}}{\\norm{x^*}}\\right)\\right).\n\\]\n\n\\end{prop}\n\nWe will also need the following refinement (see Proposition 2.1 in \\cite{LancienRaja2018}).\n\n\\begin{prop}\\label{waus}\nLet $X$ be a Banach space. \nThen the bidual norm on $X^{**}$ has the following property. \nFor any $t\\in (0,1)$, any weak$^*$-null sequence $(x^{**}_n)_{n=1}^\\infty$ in $B_{X^{**}}$ and any  $x \\in S_X$ we have:\n$$ \\limsup_{n \\rightarrow \\infty} \\| x+tx^{**}_n \\| \\leq 1 + \\overline{\\rho}_{X}(t).$$\n\\end{prop}\n\n\n\\medskip Let us now recall the following classical duality result concerning these moduli (see for instance \\cite{DKLR} Corollary 2.3 for a precise statement).\n\n\\begin{prop}\\label{duality} Let $X$ be a Banach space. \nThen $\\|\\ \\|_X$ is AUS if and and only if $\\|\\ \\|_{X^*}$ is AUC$^*$.\n\nIf $p,q\\in (1,\\infty)$ are conjugate exponents, then $\\|\\ \\|_X$ is $p$-AUS if and and only if $\\|\\ \\|_{X^*}$ is $q$-AUC$^*$.\n\\end{prop}\n\nWe conclude this section with a list of a few classical properties of Orlicz functions and norms that are related to these moduli.\nA map $\\varphi:[0,\\infty) \\to [0,\\infty)$ is called an \\emph{Orlicz function} if it is continuous, non decreasing, convex and so that $\\varphi(0)=0$ and $\\lim_{t \\to \\infty}\\varphi(t)=\\infty$. \nThe \\emph{Orlicz norm} $\\|\\ \\|_{\\ell_\\varphi}$, associated with  $\\varphi$ is defined on $c_{00}$, the space of finitely supported sequences, as follows:\n$$\\forall x=(x_n)_{n=1}^\\infty \\in c_{00},\\ \\ \\|x\\|_{\\ell_\\varphi}=\\inf\\big\\{r>0,\\ \\sum_{n=1}^\\infty \\varphi(x_n\/r)\\le 1\\big\\}.$$\nThe following is immediate from the definition.\n\n\\begin{lem}\\label{Orlicz-lp} Let $\\varphi:[0,\\infty) \\to [0,\\infty)$ be an Orlicz function and $p\\in [1,\\infty)$.\n\\begin{enumerate}[(i)]\n\\item If there exists $C>0$ such that $\\varphi(t)\\le Ct^p$, for all $t\\in [0,1]$, then there exists $A>0$ such that $\\|x\\|_{\\ell_\\varphi} \\le A\\|x\\|_{\\ell_p}$, for all $x\\in c_{00}$.\n\\item If there exists $c>0$ such that $\\varphi(t)\\ge ct^p$, for all $t\\in [0,1]$, then there exists $a>0$ such that $\\|x\\|_{\\ell_\\varphi} \\ge a\\|x\\|_{\\ell_p}$, for all $x\\in c_{00}$.\n\\end{enumerate}\n\\end{lem}\n\nAssume now that $\\varphi:[0,\\infty) \\to [0,\\infty)$ is an Orlicz function which is 1-Lipschitz and such that $\\lim_{t\\to \\infty}\\varphi(t)\/t=1$. \nConsider for $(s,t) \\in \\R^2$, \n\\[\nN_2^\\varphi(s,t)=\n\\begin{cases}|s|+|s|\\varphi(|t|\/|s|) & \\text{ if }s\\neq 0,\\\\\n \\abs{t}& \\text{ if }s=0.\n\\end{cases}\n\\]\nThen define by induction for all $n\\geq 3$:\n$$\\forall (s_1,\\ldots,s_n)\\in \\R^n,\\ N_n^\\varphi(s_1,\\ldots,s_n)=\nN_2^\\varphi\\big(N_{n-1}^\\varphi(s_1,\\ldots,s_{n-1}),s_n\\big).$$\nThe following is proved in \\cite{KaltonTAMS2013} (see Lemma 4.3 and its preparation).\n\n\n\\begin{lemma}\\label{Orlicz-Kalton}\\\n\\begin{enumerate}[(i)]\n\\item For any $n \\ge 2$, the function $N_n^\\varphi$ is an absolute (or lattice) norm on $\\R^n$, meaning that $N_n(s_1,\\ldots,s_n)\\le N_n(t_1,\\ldots,t_n)$, whenever $|s_i|\\le |t_i|$ for all $i\\le n$.\n\\item For any $n\\in \\N$ and any $x\\in \\R^n$:\n$$\\frac12 \\|s\\|_{\\ell_\\varphi} \\le N_n^\\varphi(s) \\le e\\|s\\|_{\\ell_\\varphi}.$$\n\\end{enumerate}\n\\end{lemma}\n\nWhen $X$ is a Banach space, it is easy to see that $\\overline{\\rho}_X$ is a 1-Lipschitz Orlicz function such that $\\lim_{t\\to \\infty}\\rho(t)\/t=1$.\nBut due to its lack of convexity, $\\overline{\\delta}_X^*$ is not an Orlicz function and we need to modify it. Following \\cite{KaltonTAMS2013}, we define\n$$\\delta(t)=\\int_0^t \\frac{\\overline{\\delta}_X^*(s)}{s}\\,ds.$$\nIt is easy to  see that $\\overline{\\delta}_X^*(t)\/{t}$ is increasing and tends to $1$ as $t$ tends to $\\infty$. Therefore, $\\delta$ is an Orlicz function which is 1-Lipschitz, such that $\\lim_{t\\to \\infty}\\delta(t)\/t=1$ and satisfying:\n$$\\forall t\\in [0,\\infty),\\ \\ \\overline{\\delta}_X^*(t\/2) \\le \\delta(t) \\le \\overline{\\delta}_X^*(t).$$\nThe following statement is now a direct consequence of Lemmas \\ref{Orlicz-lp} and \\ref{Orlicz-Kalton}.\n\n\\begin{lem}\\label{Nnorm-lp} Let $X$ be a Banach space and $p\\in [1,\\infty)$.\n\\begin{enumerate}[(i)]\n\\item If there exists $C>0$ such that $\\overline{\\rho}_X(x)\\le Ct^p$, for all $t\\in [0,1]$, then there exists $A>0$ such that\n$$\\forall n\\in \\N\\ \\forall x\\in \\R^n,\\ \\ N_n^{\\overline{\\rho}_X}(x)\\le A\\|x\\|_{\\ell_p^n}.$$\n\\item If there exists $c>0$ such that $\\overline{\\delta}_X^*(t)\\ge ct^p$, for all $t\\in [0,1]$, then there exists $a>0$ such that\n$$\\forall n\\in \\N\\ \\forall x\\in \\R^n,\\ \\ N_n^{\\delta}(x)\\ge a\\|x\\|_{\\ell_p^n}.$$\n\\end{enumerate}\n\\end{lem}\n\n\nWe will also use the following reformulation of Propositions~\\ref{as-sequences} and~\\ref{waus} in terms of the norms $N_2^\\delta$ and $N_2^{\\overline{\\rho}_X}$.\n\\begin{lem}\\label{l:asymptotic-Nnorm}\n Let $X$ be a Banach space. \n \\begin{enumerate}[(i)]\n \\item  Let $(x_n^*) \\subset X^*$ be weak$^*$-null. Then for  any $x^* \\in X^*$ we have \n \\[\n  \\limsup_{n\\to \\infty} \\norm{x^*+x_n^*}\\geq N_2^\\delta(\\norm{x^*},\\limsup \\norm{x_n^*}).\n \\]\n \\item Similarly, if $(x_n^{**}) \\subset X^{**}$ is weak$^*$-null and $x\\in X$, then \n$$ \\liminf_{n \\rightarrow \\infty} \\| x+x^{**}_n \\| \\leq N_2^{\\overline{\\rho}_X}(\\norm{x},\\liminf \\norm{x_n^{**}}).$$\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n If $x^*=0$ there is nothing to do, so we may assume that $x^*\\neq 0$.\n By application of Proposition~\\ref{as-sequences} we see that \n \\[\n \\begin{aligned}\n  \\limsup_{n\\to\\infty}\\norm{x^*+x_n^*}&\\geq \\norm{x^*}\\left(1+\\overline{\\delta}^*_X\\left(\\frac{\\limsup\\norm{x_n^*}}{\\norm{x^*}}\\right)\\right)\\\\\n  &\\geq \\norm{x^*}\\left(1+\\delta\\left(\\frac{\\limsup\\norm{x_n^*}}{\\norm{x^*}}\\right)\\right)=N_2^\\delta(\\norm{x^*},\\limsup \\norm{x_n^*})\n  \\end{aligned}\n \\]\nThe proof of the second claim is even simpler so we leave it to the reader.\n\\end{proof}\n\n\n\n\n\\section{The general result}\n\nLet us first recall that a Banach space is said to be {\\it quasi-reflexive} if the image of its canonical embedding into its bidual is of finite codimension in its bidual. We can now state our main result.\n\n\\begin{thm}\\label{general} Let $X$ be a quasi-reflexive Banach space, let $p\\in (1,\\infty)$ and denote $q$ its conjugate exponent. Assume that $X$ admits an equivalent $p$-AUS norm and that $X^*$ admits an equivalent $q$-AUS norm. Then the family $([\\N]^k,d_{\\K}^k)_{k\\in \\N}$ does not equi-coarsely embed into $X^{**}$.\n\\end{thm}\n\nWe immediately deduce the following.\n\n\\begin{cor}\\label{generalcorollary} \nLet $X$ be a quasi-reflexive Banach space, let $p\\in (1,\\infty)$ and denote $q$ its conjugate exponent. \nAssume that $X$ admits an equivalent $p$-AUS norm and that $X^*$ admits an equivalent $q$-AUS norm. \nThen the family $([\\N]^k,d_{\\K}^k)_{k\\in \\N}$ does not equi-coarsely embed into $X$, nor does it equi-coarsely embed into any iterated dual $X^{(r)}$ ($r\\geq 0$) of $X$.\n\\end{cor}\n\n\\begin{proof} \nSince $X$ is quasi reflexive we infer that $X^{(r)}$ admits an equivalent $p$-AUS norm when $r$ is even and it admits an equivalent $q$-AUS norm when $r$ is odd. \nIndeed, note that when $r$ is even $X^{(r)}$ is isomorphic to $X\\oplus_p F$ where $F$ is finite-dimensional (resp. $X^{(r)}\\simeq X^*\\oplus_q F$ when $r$ is odd).\nNow it is obvious from Theorem~\\ref{general} that $([\\N]^k)_{k\\in \\N}$ do not equi-coarsely embed into $X^{(r)}$ when $r$ is even. \nWhen $r$ is odd, we just exchange the roles of $p$ and $q$.\n\\end{proof}\n\nBefore going into the detailed proof of Theorem~\\ref{general} let us briefly indicate the main idea. We assume that there is an equi-coarse family of embeddings $(f_k)$ of $[\\N]^k$ into $X^{**}$ with moduli $\\rho$ and $\\omega$.\nWe fix $k$ sufficiently large and observe that, up to passing to a subgraph, $f_k$ can be represented as the sum along the branches of a weak$^*$-null countably branching tree of height $k$, say $(z_{\\n})_{\\n\\in [N]^{\\leq k}}$.\nMoreover the norms of the elements of this tree stabilize on each level towards values $(K_i)_{i=1}^k \\subset [0,\\omega(1)]$.\nApplying the existence of a $q-AUS$ norm on $X^*$ one can show that $\\sum_{i=1}^k K_i^p \\leq c^p\\omega(1)^p$ where $c$ is a constant depending only on $X$.\nThe benefit of this observation is twofold.\nOn one hand we will be able to construct two elements $\\n_0,\\m_0 \\in [\\N]^{l}$ (with $l\\leq k$) such that $\\sum_{i=1}^l z_{(n_1,\\ldots,n_i)}-z_{(m_1,\\ldots,m_i)}$ is small in norm (say less than $2c\\omega(1)$) while $d_{\\K}^l(\\n_0,\\m_0)$ is large (say $\\rho(d_{\\K}^l(n_0,\\m_0))> 3c\\omega(1)$). \nOn the other hand the $p-AUS$ renormability of $X$ together with the quasi-reflexivity allows to extend these elements to elements $\\n,\\m \\in [\\N]^k$ such that $\\dk(\\n,\\m)$ is still large and \n\\[\\begin{aligned}\n\\norm{\\sum_{i=l+1}^k z_{(n_1,\\ldots,n_i)}-z_{(m_1,\\ldots,m_i)}} &\\sim \\left(\\sum_{i=l+1}^k\\norm{z_{(n_1,\\ldots,n_i)}-z_{(m_1,\\ldots,m_i)}}^p\\right)^{1\/p}\\\\ &\\sim (\\sum_{i=l+1}^k K_i^p)^{1\/p}\\leq c\\omega(1)   .\n\\end{aligned}\n\\]\nEventually, summing the tree from $1$ to $k$ over the branches ending by $\\n$ and $\\m$ we get the desired contradiction\n\\[\n 3c\\omega(1)<\\rho(\\dk(\\n,\\m))\\leq \\norm{f_k(\\n)-f_k(\\m)}\\leq 3c\\omega(1).\n\\]\n\n\\begin{proof}[Proof of Theorem \\ref{general}] Let us assume that there are two maps $\\rho, \\, \\omega \\colon [0,+\\infty) \\to [0,+\\infty)$ and maps $f_k ([\\N]^k,d_{\\K}^k) \\colon \\to (X^{**},\\|\\ \\|)$ for $k \\in \\N$ such that:\n\\begin{enumerate}[(i)]\n\\item $\\lim_{t \\to \\infty} \\rho(t) = \\infty$,\n\\item $\\omega(t) < \\infty$ for every $t \\in (0,+\\infty)$,\n\\item $\\rho(t) \\leq \\rho_{f_k}(t)$ and $\\omega_{f_k}(t) \\leq \\omega(t)$ for every $k \\in \\N$ and $t \\in (0,\\infty)$.\n\\end{enumerate}\nNote that all $f_k$'s are $\\omega(1)$-Lipschitz for $\\|\\ \\|$ and so $\\omega(1)>0$. Since all the sets $[\\N]^k$ are countable, we may and will assume that $X$ and therefore, by the quasi-reflexivity of $X$, that all its iterated duals are separable.\\\\\nLet us fix $N\\in \\N$. Pick $\\alpha\\in \\N$ such that $\\alpha\\ge \\frac{p}{q}$ and set $k=N^{1+\\alpha} \\in \\N$. We also fix $\\eta>0$. We shall provide at the end of our proof a contradiction if $N$ is chosen large enough and $\\eta$ small enough. We denote $\\|\\ \\|$ the original norm on $X$, as well as its dual and bidual norms. Let us assume, as we may, that $\\|\\ \\|$ is $p$-AUS on $X$. We denote its modulus of asymptotic uniform smoothness $\\overline{\\rho}_{\\|\\ \\|}$ or simply $\\overline{\\rho}_{X}$.\n\n\\medskip\nFor the first step of the proof we shall exploit the existence of an equivalent $q$-AUS norm $|\\ |$ on $X^*$ (we also denote $|\\ |$ its dual norm on $X^{**}$). \nIt is worth mentioning that if $X$ is not reflexive, $|\\ |$ cannot be the dual norm of an equivalent norm on $X$ (see for instance Proposition 2.6 in \\cite{CauseyLancien}). \nAssume also that there exists $b>0$ such that\n\\begin{equation}\\label{equivalence}\n\\forall z \\in X^{**}\\ \\ b\\|z\\|\\le |z|\\le \\|z\\|.\n\\end{equation}\nThen we have that all $f_k$'s are also $\\omega(1)$-Lipschitz for $|\\ |$.\\\\\nBy Proposition \\ref{duality}, we have that there exists $c>0$ such that for all $t\\in [0,1]$, $\\overline{\\delta}_{|\\ |}^*(t)\\ge ct^p$. \nWe denote again\n$$\\delta(t)=\\int_0^t \\frac{\\overline{\\delta}_{|\\ |}^*(s)}{s}\\, ds.$$\nRecall that Lemma \\ref{Nnorm-lp} ensures the existence of $a>0$ such that for all $n\\in \\N$, $N_n^\\delta \\ge 2a\\|\\ \\|_{\\ell_p^n}$.\n\nFirst, using the separability of $X^*$ and Proposition \\ref{nulltree}, we may assume by passing to a full subtree, that there exist a weak$^*$-null tree $(z(\\m))_{\\m\\in[\\N]^{\\le k}}$ in $X^{**}$ with $|z_{\\m}| \\leq \\omega(1)$ for all $\\m\\in [\\N]^{\\leq k}\\setminus\\{\\emptyset\\}$ and so that\n$$\\forall \\n \\in [\\N]^k,\\ f_k(\\n)=\\sum_{i=0}^k z(n_1,\\ldots,n_i)=\\sum_{\\m \\preceq \\n} z(\\m).$$\n\nFor $r\\in \\N$ we denote $E_r=\\{\\m=(m_1,\\ldots,m_j)\\in [\\N]^{\\le k}\\setminus \\{\\emptyset\\},\\ m_j=r\\}$ and $F_r=\\bigcup_{u=1}^r E_u$. \nFix a sequence $(\\lambda_r)_{r=1}^\\infty$ in $(0,1)$ such that $\\prod_{r=1}^\\infty \\lambda_r > \\frac12$. \nWe now use Lemma~\\ref{l:asymptotic-Nnorm}~(i) and the fact that $(z(\\m))_{\\m\\in[\\N]^{\\le k}}$ is a weak$^*$-null tree to build inductively $n_1<\\ldots<n_r$ so that for all $\\n^1,\\ldots,\\n^L \\in F_{n_{r}-1}$, for all $\\eps_1,\\ldots,\\eps_L \\in \\{-1,1\\}$ and all $\\n \\in E_{n_r}$, we have\n$$\\big|z(\\n)+\\sum_{l=1}^L\\eps_l z(\\n^l)\\big|\\ge \\lambda_r N_2^{\\delta}\\Big(\\big|\\sum_{l=1}^L\\eps_l z(\\n^l)\\big|, \\big|z(\\n)\\big|\\Big).$$\nTherefore, using the fact that $N_2^\\delta$ is an absolute norm and after passing to a full subtree, we may assume that for all $r_1<\\cdots<r_L$ in $\\N$, all $\\eps_1,\\ldots,\\eps_L \\in \\{-1,1\\}$ and  all $\\n^1,\\ldots,\\n^L$ so that $\\n^l \\in E_{r_l}$ for $1\\le l\\le L$, we have\n\\begin{equation}\\label{treelower-lp}\n\\big|\\sum_{l=1}^L\\eps_l z(\\n^l)\\big| \\ge \\frac12 N_L^\\delta\\big(|z(\\n^1)|,\\ldots,|z(\\n^L)|\\big)  \\ge a\\Big(\\sum_{i=1}^L \\big|z(\\n^l)\\big|^p\\Big)^{1\/p}.\n\\end{equation}\nAssume now that $\\n=(n_1,\\ldots,n_k)\\in \\N^k$ is such that $n_1<\\cdots<n_k$ are even and choose $\\m=(m_1,\\ldots,m_k)$ so that $n_1<m_1<\\cdots<n_k<m_k$. It follows from (\\ref{treelower-lp}) that\n\\begin{eqnarray*}\n\\big|f(\\n)-f(\\m)|=&&\\Big|\\sum_{i=1}^k z(n_1,\\ldots,n_i)-z(m_1,\\ldots,m_i)\\Big| \\\\\n&&\\ge a\\Big(\\sum_{i=1}^k \\big|z(n_1,\\ldots,n_i)\\big|^p+\\big|z(m_1,\\ldots,m_i)\\big|^p\\Big)^{1\/p}.\n\\end{eqnarray*}\nWe now use the fact that $d_\\K^k(\\n,\\m)=1$ and $f$ is $\\omega(1)$-Lipschitz, to deduce\n$$\\Big(\\sum_{i=1}^k \\big|z(n_1,\\ldots,n_i)\\big|^p\\Big)^{1\/p}\\le \\frac{1}{a}\\omega(1).$$\nSo replacing $\\N$ with $2\\N$ and setting $A=1\/a$, we may assume that\n\\begin{equation}\\label{endAUC}\n\\forall \\n\\in [\\N]^k,\\ \\ \\Big(\\sum_{i=1}^k \\big|z(n_1,\\ldots,n_i)\\big|^p\\Big)^{1\/p}\\le A\\omega(1).\n\\end{equation}\n\nBy Ramsey's theorem (Proposition \\ref{ramsey}), we may also assume, after passing again to a full subtree, that for all $i\\in\\{1,\\ldots,k\\}$ there exists $K_i\\in [0,\\omega(1)]$ such that\n$$\\forall (n_1,\\ldots,n_i) \\in [\\N]^i,\\ K_i\\le |z(n_1,\\dots,n_i)|\\le K_i+\\eta.$$\nThe estimate \\eqref{endAUC} yields\n\\begin{equation}\\label{p-concentration}\n\\sum_{i=1}^k K_i^p \\le A^p\\omega(1)^p.\n\\end{equation}\nTherefore, since $k=N^{1+\\alpha}$, there exists $j\\in \\{0,N,\\ldots,N(N^{\\alpha}-1)\\}$ such that\n$$\\sum_{i=j+1}^{j+N} K_i^p \\le \\frac{A^p\\omega(1)^p}{N^\\alpha}.$$\nThen we deduce from H\\\"{o}lder's inequality that\n\\begin{equation}\\label{Holder}\n\\sum_{i=j+1}^{j+N} K_i \\le N^{1\/q}\\frac{A\\omega(1)}{N^{\\alpha\/p}} \\le A\\omega(1).\n\\end{equation}\n\n\nWe now use the assumption that $X$ is quasi-reflexive, so that $X^{**}=X \\oplus F$, where $F$ is of finite dimension. Thus, for each $(n_1,\\dots,n_i) \\in [\\N]^{\\le k}$, we can decompose $z(n_1,\\dots,n_i)=x(n_1,\\dots,n_i)+e(n_1,\\dots,n_i)$, with $x(n_1,\\dots,n_i)\\in X$ and $e(n_1,\\dots,n_i)\\in F$. Then, the compactness of bounded sets in $F$ and another application of Proposition \\ref{ramsey} allows us to assume, after passing to a full subtree, that\n$$\\forall i\\in \\{1,\\ldots,k\\}\\  \\forall \\n,\\overline{v} \\in [\\N]^{i},\\ \\ \\|e(\\n)-e(\\overline{v})\\| < \\eta,$$\nWhich implies that for all $i\\in \\{1,\\ldots,k\\}$ and all $\\n,\\overline{v} \\in [\\N]^{i}$ we have\n\\begin{equation}\\label{stabilize}\n\\big| \\|z(\\n)-z(\\overline{v})\\|-\\|x(\\n)-x(\\overline{v})\\|\\big| < \\eta.\n\\end{equation}\n\n\nWe are now ready for the last step of the proof, where we shall build $\\m$ and $\\overline{u}$ in $[\\N]^k$ so that $d_\\K^k(\\m,\\overline{u})=N$, but $|f(\\m)-f(\\overline{u})|$ is bounded by a constant depending only on $\\omega(1)$ and on $X$. This will yield a contradiction with the fact $\\lim_{N\\to \\infty}\\rho(N)=\\infty$.\n\nFirst, we set $m_i=u_i=i$, for all $1\\le i\\le j$. \nThen, for $j+1\\le i\\le j+N$, we set $m_i=i$ and $u_i=i+N$. \nFinally, we shall build $m_i=u_i$ inductively, for $j+N<i\\le k$. \nNote, that when this will be done, we will indeed have $d_\\K^k(\\m,\\overline{u})=N$.\n\nFirst, we obviously have\n\\begin{equation}\\label{samedebut}\n\\sum_{i=1}^{j} z(m_1,\\ldots,m_i)-z(u_1,\\ldots,u_i)=0.\n\\end{equation}\nThe next estimate follows from (\\ref{Holder}).\n\\begin{equation}\\label{smallblock}\n\\Big|\\sum_{i=j+1}^{j+N} z(m_1,\\ldots,m_i)-z(u_1,\\ldots,u_i)\\Big| \\le \\sum_{i=j+1}^{j+N} 2(K_i+\\eta) \\le 3A\\omega(1),\n\\end{equation}\nif $\\eta$ was initially chosen small enough.\n\nWe now select the remaining coordinates of  $\\m$ and $\\overline{u}$ inductively using the fact that $\\|\\ \\|$ is $p$-AUS. \nTo shorten the notation for the end of the proof, we shall now denote $x_i=x(m_1,\\ldots,m_i)$, $z_i=z(m_1,\\ldots,m_i)$, $x'_i=x(u_1,\\ldots,u_i)$ and $z'_i=z(u_1,\\ldots,u_i)$. \nFirst, we simply set $m_{j+N+1}=u_{j+N+1}=j+2N+1$. \nWe now use the fact that the tree $(z(\\m))_{\\m \\in [\\N]^{\\le k}}$ is weak$^*$-null and Lemma~\\ref{l:asymptotic-Nnorm}~(ii) to find $m_{j+N+2}=u_{j+N+2}>j+2N+1$ such that\n\\begin{multline*}\n\\|x_{j+N+1}-x'_{j+N+1}+z_{j+N+2}-z'_{j+N+2}\\| \\\\\n\\le N_2^{\\overline{\\rho}_{X}}\\big(\\|x_{j+N+1}-x'_{j+N+1}\\|,\n\\|z_{j+N+2}-z'_{j+N+2}\\|\\big)+\\eta\n\\end{multline*}\nIt follows from (\\ref{stabilize}) that\n\\begin{multline*}\n\\|z_{j+N+1}-z'_{j+N+1}+z_{j+N+2}-z'_{j+N+2}\\| \\\\\n\\begin{aligned}\n&\\le N_2^{\\overline{\\rho}_{X}}\\big(\\|z_{j+N+1}-z'_{j+N+1}\\|+\\eta,\n\\|z_{j+N+2}-z'_{j+N+2}\\|\\big)+2\\eta\\\\\n&\\le N_2^{\\overline{\\rho}_{X}}\\Big(\\frac{2}{b}\\big(K_{j+N+1}+\\eta\\big)+\\eta,\\frac{2}{b}\\big(K_{j+N+2}+\\eta\\big)\\Big)+2\\eta.\n\\end{aligned}\n\\end{multline*}\nSimilarly, we can inductively find $m_{j+N+2}=u_{j+N+2}<\\cdots<m_{k}=u_{k}$ such that,\n$$\n\\Big\\|\\sum_{i=j+N+1}^k (z_i-z'_i)\\Big\\| \\le \\frac{2}{b}N_{k-j-N}^{\\overline{\\rho}_{X}}\\big(K_{j+N+1},\\ldots,K_k)+\\omega(1)\n$$\nprovided $\\eta$ is chosen small enough.\nSince Lemma \\ref{Nnorm-lp} ensures the existence of $C>0$ such that $N_n^{\\overline{\\rho}_{X}}\\le C\\|\\ \\|_{\\ell_p^n}$ for all $n\\in \\N$ the above inequality yields \n$$\\Big\\|\\sum_{i=j+N+1}^k (z_i-z'_i)\\Big\\| \\le \\frac{2C}{b}\\Big(\\sum_{i=j+N+1}^k K_i^p\\Big)^{1\/p} +\\omega(1)\\le \\Big(\\frac{2CA}{b}+1\\Big)\\omega(1).$$\nFinally, combining the above estimate with (\\ref{samedebut}) and (\\ref{smallblock}), we get that\n$$\\|f(\\m)-f(\\overline{u})\\|\\le \\frac{3A+2CA+b}{b}\\omega(1).$$\nAs announced at the beginning of the proof, this yields a contradiction if $N$ was initially chosen, as it was possible, so that $\\rho(N)>\\frac{3A+2CA+b}{b}\\omega(1).$\n\\end{proof}\n\n\n\n\n\nUnlike reflexivity, quasi-reflexivity itself is not enough to prevent the Kalton graphs from embedding into a Banach space. We thank P. Motakis for showing us the next example.\n\n\\begin{prop}[Motakis]\\label{Pavlos} There exists a quasi-reflexive Banach space $X$ such that the family of graphs $([\\N]^k,d_\\K^k)_{k\\in \\N}$ equi-Lipschitz embeds into $X$.\n\\end{prop}\n\n\\begin{proof} The proof relies on the existence of a quasi-reflexive Banach space $X$ of order one which admits a spreading model, generated by a basis of $X$ that is equivalent to the summing basis $(s_n)_{n=1}^\\infty$ of $c_0$. This is shown in \\cite{FOSZ} (Proposition 3.2) and based on a construction given in \\cite{BHO}. We refer the reader to \\cite{BeauzLap} for the necessary definitions. Consequently, there exists a sequence $(x_n)_{n=1}^\\infty$ in $S_X$ and constants $A,B>0$ such that for all $k\\le n_1<\\cdots<n_k$ and all $\\eps_1,\\ldots,\\eps_k$ in $\\{-1,0,1\\}$ one has\n\\begin{equation}\\label{spreadingsumming}\nA\\big\\|\\sum_{i=1}^k \\eps_i s_i\\big\\|_{c_0} \\le \\big\\|\\sum_{i=1}^k \\eps_i x_{n_i}\\big\\|_X \\le B\\big\\|\\sum_{i=1}^k \\eps_i s_i\\big\\|_{c_0}.\n\\end{equation}\nFor $k\\in \\N$ and $\\n=(n_1,\\ldots,n_k) \\in [\\N]^k$ we define\n$$g_k(\\n)=\\sum_{i=1}^k x_{2k+n_i}.$$\nIt follows easily from Proposition \\ref{Kgraphsinc0}, the inequality (\\ref{spreadingsumming})  and the fact that $(s_n)_{n=1}^\\infty$ is a spreading sequence that \n$$\\frac{A}{2}d_\\K^k(\\n,\\m) \\le \\|g_k(\\n)-g_k(\\m)\\|_X \\le B d_\\K^k(\\n,\\m)$$\nfor all $\\n,\\m \\in [\\N]^k$.\n\n\\end{proof}\n\n\\begin{remark} Let us mention that, more generally, it is proved in \\cite{AMS} that for any conditional normalized spreading sequence $(e_n)_{n=1}^\\infty$, there exists\na quasi-reflexive Banach space $X$ of order 1  with a normalized basis $(x_i)_{i=1}^\\infty$ which generates $(e_n)_{n=1}^\\infty$ as a spreading model.\n\\end{remark}\n\n\n\n\n\n\\section{The James sequence spaces}\n\\label{s:james}\n\nLet $p\\in (1,\\infty)$. We now recall the definition and some basic properties of the James space $\\J_p$. We refer the reader to \\cite{albiackalton}(Section 3.4) and references therein for more details on the classical case $p=2$. The James space $\\J_p$ is the real Banach space of all sequences $x=(x(n))_{n\\in \\N}$ of real numbers with finite $p$-variation and verifying $\\lim_{n \\to \\infty} x(n) =0$. The space $\\J_p$ is endowed with the following norm\n$$\\|x\\|_{\\J_p} = \\sup  \\Big \\{  \\big (\\sum_{i=1}^{k-1} |x(p_{i+1})-x(p_i)|^p \\big )^{1\/p}     \\; \\colon \\; 1 \\leq p_1 < p_2 < \\ldots < p_{k} \\Big \\}. $$\nThis is the historical example, constructed for $p=2$ by R.C. James in \\cite{James1}, of a quasi-reflexive Banach space which is isomorphic to its bidual. In fact $\\J_p^{**}$ can be seen as the space of all sequences $x=(x(n))_{n\\in \\N}$ of real numbers with finite $p$-variation, which is $\\J_p \\oplus \\R \\car$, where $\\car$ denotes the constant sequence equal to $1$.\\\\\nThe standard unit vector basis $(e_n)_{n=1}^{\\infty}$ ($e_n(i)=1$ if $i = n$ and $e_n(i)=0$ otherwise) is a monotone shrinking basis for $\\J_p$. Hence, the sequence $(e_n^*)_{n=1}^{\\infty}$ of the associated coordinate functionals is a basis of its dual $\\J_p^*$. Then the weak$^*$ topology $\\sigma(\\J_p^*,\\J_p)$ is easy to describe. A sequence $(x^*_n)_{n=1}^{\\infty}$ in $\\J_p^*$ converges to $0$ in the $\\sigma(\\J_p^*,\\J_p)$ topology if and only if it is bounded and $\\lim_{n \\to \\infty} x^*_n(i)=0$ for every $i \\in \\N$.\\\\\nFor $x\\in \\J_p$, we define $\\supp x = \\{i \\in \\N \\, : \\, x(i) \\neq 0 \\}$. For $x,y \\in \\J_p$, we denote: $x \\prec y$ whenever $\\max \\supp x < \\min \\supp y$.\\\\\nSimilarly, an element $x^*$ of $\\J_p^*$ will be written $x^*=\\sum_{n=1}^\\infty x^*(n)e_n^*$ and $\\supp x^* = \\{i \\in \\N \\, : \\, x^*(i) \\neq 0 \\}$ and we shall denote $x^* \\prec y^*$ whenever $\\max \\supp x^* < \\min \\supp y^*$.\n\nThe detailed proof of the following proposition can be found in \\cite{Netillard} (Proposition 2.3). This a consequence of the following fact: there exists $C\\ge 1$ such that $\\|\\sum_{i=1}^n x_i\\|_{\\J_p}^p\\le C\\sum_{i=1}^n \\|x_i\\|_{\\J_p}^p$, for all $x_1 \\prec \\ldots \\prec x_n$ in $\\J_p$.\n\n\\begin{prop}\\label{Jsmooth} There exists an equivalent norm $|\\ |$ on $\\J_p$ such that its dual norm  $|\\ |_*$ has the following property. For any $x^*,y^* \\in J_p^*$ such that $x^* \\prec y^*$, we have that\n$$|x^*+y^*|_*^q\\ge |x^*|_*^q+|y^*|_*^q.$$\nIn particular, $|\\ |_*$ is $q$-AUC$^*$ for the weak$^*$ topology induced by $\\J_p$ and therefore $|\\ |$ is $p$-AUS on $\\J_p$.\n\\end{prop}\n\nThere is also a natural weak$^*$ topology on $\\J_p$. Indeed, the summing basis $(s_n)_{n=1}^{\\infty}$ ($s_n(i)=1$ if $i \\leq n$ and $s_n(i)=0$ otherwise) is a monotone and boundedly complete basis for $\\J_p$. Thus, $\\J_p$ is naturally isometric to a dual Banach space: $\\J_p = X^*$ with $X$ being the closed linear span of the biorthogonal functionals $(e_n^* - e_{n+1}^*)_{n=1}^{\\infty}$ in $\\J_p^*$ associated with $(s_n)_{n=1}^{\\infty}$. Note that $X=\\{x^*\\in \\J_p^*,\\ \\sum_{n=1}^\\infty x^*(n)=0\\}$. Thus, a sequence $(x_n)_{n=1}^{\\infty}$ in $\\J_p$ converges to $0$ in the $\\sigma(\\J_p,X)$ topology if and only if it is bounded and $\\lim_{n \\to \\infty} \\big(x_n(i) - x_n(j)\\big) = 0$ for every $i \\neq j \\in \\N$. The next proposition is easy (see Proposition 2.3 in \\cite{LancienPMB} for the case $p=2$).\n\n\\begin{prop}\\label{Jconvex} The usual norm on $\\J_p$ is p-AUC$^*$ for the weak$^*$ topology induced by $X$. In other words, the restriction to $X$ of the usual norm on $\\J_p^*$ is $q$-AUS.\n\\end{prop}\n\n\n\nThen, since $X$ is one codimensional in $\\J_p^*$, we have that $\\J_p^*$ is isomorphic to $X\\oplus \\R$ and therefore also admits an equivalent $q$-AUS norm.\n\n\n\\medskip\nThe above remarks combined with Corollary \\ref{generalcorollary} immediately yield the following.\n\n\\begin{cor}\\label{James} Let $p\\in (1,\\infty)$. Then, the family $([\\N]^k,d_{\\K}^k)_{k\\in \\N}$ does not equi-coarsely embed into $\\J_p$, nor does it equi-coarsely embed into $\\J_p^*$.\n\\end{cor}\n\n\n\n\n\\section{A Remark on the James tree space}\\label{s:coarsejamestree}\n\nLet us recall the construction of the James tree space $\\J\\T$. We denote $T=2^{<\\omega}$ the tree of all finite sequences with coefficients in $\\{0,1\\}$ equipped with its natural order: for $s,t \\in T$, we say that $s \\le t$ if the sequence $t$ extends $s$. The set of all infinite sequences with coefficients in $\\{0,1\\}$ will be denoted $2^\\omega$. For $s\\in T$, the length of $s$ is denoted $|s|$. We call $\\emph{segment}$ of $T$ any set of the form $\\{s\\in T,\\ t\\le s \\le t'\\}$ with $t\\le t'$ in $T$. For a map $x:T\\to \\R$, we define\n$$\\|x\\|_{\\J\\T}=\\sup\\Big\\{\\Big(\\sum_{i=1}^n\\big(\\sum_{s\\in S_i} x(s)\\big)^2\\Big)^{1\/2} \\Big\\},$$\nwhere the supremum is taken over all pairwise disjoint segments $S_1,\\ldots,S_n$ of $T$. Then the \\emph{James tree} space is the space $\\J\\T=\\{x:T \\to \\R,\\ \\|x\\|_{\\J\\T}<\\infty\\}$ equipped with the norm $\\|\\ \\|_{\\J\\T}$. For $s\\in T$, we denote $e_s:T\\to \\R$ defined by $e_s(t)=\\delta_{s,t},\\ t\\in T$. If $\\psi:\\N \\to T$ is a bijection such that $|\\psi(n)|\\le |\\psi(m)|$ whenever $n\\le m$, then $(e_{\\psi(n)})_{n=1}^\\infty$ is a normalized, monotone and boundedly complete basis of $\\J\\T$. For $s\\in T$, the coordinate functional $e^*_s$ is defined by $e^*_s(x)=x(s)$, $x\\in \\J\\T$. Then the closed linear span of $\\{e^*_s,\\ s\\in T\\}$ in $\\J\\T^*$ is denoted $\\B$ and $\\B^*$ is isometric to $\\J\\T$.\nThe space $\\J\\T$ was built by R.C. James in \\cite{James2} to serve as the first example of a separable Banach space with non separable dual, which does not contain an isomorphic copy of $\\ell_1$.\n\nIn \\cite{kaltonpropertyq} it is shown that if a Banach space $X$ coarsely contains $c_0$ then there exists $k\\in \\N$ such that $X^{(k)}$, the dual of order $k$ of $X$, is non separable. A close look at the proof of Theorem 3.5 in \\cite{kaltonpropertyq} allows to state the following.\n\n\\begin{thm}[Kalton]\\label{kalton} Let $X$ and $Y$ be two Banach spaces such that $X$ coarsely embeds into $Y$. Assume moreover that there exist an uncountable set $I$ and for every $i\\in I$ and $k\\in \\N$, a 1-Lipschitz map $f_i^k:([\\N]^k,d_{\\K}^k))\\to X$ such that\n$$\\lim_{k\\to \\infty}\\ \\inf_{i\\neq j \\in I}\\ \\inf_{\\M\\in [\\N]^\\omega}\\ \\sup_{\\n\\in [\\M]^k} \\|f_i^k(\\n)-f_j^k(\\n)\\|=\\infty.$$\nThen there exists $r\\in \\N$ such that $Y^{(r)}$ is not separable.\n\\end{thm}\n\n\nAs an application, we can show the following.\n\n\\begin{thm}\\label{kalton+} Let $Y$ be a Banach space such that $\\B$ or $\\J\\T$ coarsely embeds into $Y$. Then there exists $r\\in \\N$ such that $Y^{(r)}$ is not separable.\n\\end{thm}\n\n\\begin{proof} For $\\sigma \\in 2^\\omega$, we denote $\\sigma_{|n}=(\\sigma_1,\\ldots,\\sigma_n)$.\nThen, for $k\\in \\N$, we define $f_\\sigma^k: [\\N]^k \\to \\B$ as follows. For $\\n=(n_1,\\ldots,n_k)\\in [\\N]^k$ let\n$$f_\\sigma^k(\\n)=\\frac{1}{\\sqrt{k}}\\sum_{i=1}^k \\sum_{s \\le \\sigma_{|n_i}} e^*_s.$$\nAssume for instance that $n_1\\le m_1 \\le \\cdots n_k\\le m_k$. Then we can write\n$$f_\\sigma^k(\\m)-f_\\sigma^k(\\n)= \\frac{1}{\\sqrt{k}} \\sum_{i=1}^k \\sum_{s \\in S_i}e^*_s,$$\nwhere $S_1,\\ldots,S_k$ are pairwise disjoint segments in $T$. \nNote that for any segment $S_i$ the sum $\\sum_{s \\in S_i}e^*_s$ belongs to the unit ball of $\\J\\T^*$. It then follows from the Cauchy-Schwarz inequality that $f_\\sigma^k$ is 1-Lipschitz on $([\\N]^k,d_{\\K}^k)$. \nAssume now that $\\sigma\\neq \\tau \\in 2^\\omega$. Pick $r\\in \\N$ such that $\\sigma_r\\neq \\tau_r$. Then for any $\\M\\in [\\N]^\\omega$ and any $\\n=(n_1,\\dots,n_k)\\in [\\M]^k$ with $n_1\\ge r$, we have\n$$\\|f_\\sigma^k(\\n)-f_\\tau^k(\\n)\\|_{\\B}\\ge \\Big|\\langle f_\\sigma^k(\\n)-f_\\tau^k(\\n), e_{\\sigma_{|n_1}} \\rangle \\Big| \\ge \\sqrt{k}.$$\nBy Theorem \\ref{kalton} and the uncountability of $2^\\omega$, this finishes our proof for $\\B$.\n\n\nFor $\\sigma \\in 2^\\omega$ and $k\\in \\N$ define now $g_\\sigma^k: [\\N]^k \\to \\J\\T$ by\n$$\\forall \\n=(n_1,\\ldots,n_k)\\in [\\N]^k,\\ g_\\sigma^k(\\n)=\\frac{1}{\\sqrt{2k}} \\sum_{i=1}^k e_{\\sigma_{|n_i}}.$$\nIt is easily checked that $g_\\sigma^k$ is 1-Lipschitz on $([\\N]^k,d_{\\K}^k)$.\nAssume now that $\\sigma\\neq \\tau \\in 2^\\omega$. Pick $r\\in \\N$ such that $\\sigma_r\\neq \\tau_r$. Then for any $\\M\\in [\\N]^\\omega$ and any $\\n=(n_1,\\dots,n_k)\\in [\\M]^k$ with $n_1\\ge r$, denote $S=\\{s\\in T, \\sigma_{|n_1} \\le s \\le \\sigma_{|n_k}\\}$. The set $S$ is a segment in $T$ and $x^*=\\sum_{s\\in S}e^*_s$ is in the unit ball of $\\J\\T^*$. Therefore\n$$\\|g_\\sigma^k(\\n)-g_\\tau^k(\\n)\\|_{\\J\\T}\\ge \\langle g_\\sigma^k(\\n)-g_\\tau^k(\\n), x^* \\rangle \\ge \\frac{\\sqrt{k}}{\\sqrt{2}}.$$\nThis concludes our proof for $\\J\\T$.\n\\end{proof}\n\n\n\\noindent{\\bf Acknowledgements.} We are grateful to P. Motakis for allowing us to include Proposition \\ref{Pavlos} and for very useful discussions. We also wish to mention that the formula describing the interlaced distance given in Proposition \\ref{p:KaltonDistanceFormula} has been independently obtained by \nF. Baudier, P. Motakis and Th. Schlumprecht.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{bibsection}\n\\begin{biblist}\n\n\\bibitem{albiackalton} F. Albiac and N.J. Kalton, Topics in Banach Space Theory, Second Edition, volume 233 of {\\it Graduate Texts in Mathematics}, Springer, New York, (2006-2016).\n\n\\bibitem{AMS} S. Argyros, P. Motakis and B. Sari, A study of conditional spreading sequences, preprint arXiv: 1611.04443v1.\n\n\\bibitem{blms} F. Baudier, G. Lancien, P. Motakis and Th. Schlumprecht, A new coarsely rigid class of Banach spaces, preprint arXiv:1806.00702.\n\n\\bibitem{bls} F. Baudier, G. Lancien and Th. Schlumprecht, The coarse geometry of Tsirelson's space and applications, {\\it J. Amer. Math. Soc.}, {\\bf 31} (2018), no. 3, 699--717.\n\n\\bibitem{BeauzLap} B. Beauzamy and J.T. Laprest\\'e, Mod\\`eles \\'etal\\'es des espaces de {B}anach, {\\it Travaux en Cours}, Hermann, Paris, (1984).\n\n\\bibitem{BHO} S.F. Bellenot, R. Haydon, and E. Odell, Quasi-reflexive and tree spaces constructed in\nthe spirit of R.C. James, {\\it Contemp. Math., Amer. Math. Soc., Providence, RI}, {\\bf 85} (1989), 19--43.\n\n\\bibitem{CauseyLancien} R.M. Causey and G. Lancien, Prescribed Szlenk index of separable Banach spaces, to appear in {\\it Studia Math.}, preprint arXiv: 1710.01638.\n\n\\bibitem{DKLR} S. Dilworth, D. Kutzarova, G. Lancien and L. Randrianarivony, Equivalent norms with the property $(\\beta)$ of Rolewicz, {\\it Rev. R. Acad. Cienc. Exactas, F\\'{\\i}s.  Nat. Ser. A Mat. RACSAM},  {\\bf 111} (2017), no. 1, 101--113.\n\n\\bibitem{FOSZ} D. Freeman, E. Odell, B. Sari, and B. Zheng, On spreading sequences and asymptotic structures, to appear {\\it in Trans. Amer. Math. Soc.}, preprint arXiv: 1607.03587v1.\n\n\\bibitem{Gowers} W.T. Gowers, Ramsey Methods in Banach Spaces, in Handbook of the Geometry of Banach Spaces vol. 2, 2003, 1071--1097.\n\n\\bibitem{James1} R.C. James, Bases and reflexivity of Banach spaces, {\\it Ann. of Math. (2)}, {\\bf 52}, (1950). 518--527.\n\n\\bibitem{James2} R.C. James, A separable somewhat reflexive Banach space with nonseparable dual, {\\it Bull. Amer. Math. Soc.}, {\\bf 80}, (1974), 738--743.\n\n\\bibitem{JohnsonLindenstraussPreissSchechtman2002} W.B. Johnson, J. Lindenstrauss, D. Preiss and G. Schechtman, Almost Fr\\'echet differentiability of Lipschitz mappings between infinite-dimensional Banach spaces, {\\it Proc. London Math. Soc. (3)} {\\bf 84} (2002), no. 3, 711-746.\n\n\\bibitem{kaltonpropertyq} N.J. Kalton, Coarse and uniform embeddings into reflexive spaces, {\\it Quart. J. Math. (Oxford)}, {\\bf 58} (2007), 393--414.\n\n\\bibitem{KaltonRandrianarivony2008} N.J. Kalton and L. Randrianarivony, The coarse Lipschitz geometry of $\\ell_p\\oplus\\ell_q$, {\\it Math. Ann.} {\\bf 341} (2008), no. 1, 223--237.\n\n\\bibitem{KaltonTAMS2013} N.J Kalton, Uniform homeomorphisms of Banach spaces and asymptotic structure, {\\it Trans. Amer. Math. Soc.}, {\\bf 365}  (2013), 1051--1079.\n\n\\bibitem{LancienPMB} G. Lancien, R\\'eflexivit\\'e et normes duales poss\\`{e}dant la propri\\'et\\'e uniforme de Kadec-Klee, {\\it Publications Math\\'ematiques de Besan\\c{c}on}, {\\bf 14}, 1993\/94.\n\n\\bibitem{LancienRaja2018} G. Lancien and M. Raja, Asymptotic and coarse Lipschitz structures of quasi-reflexive Banach spaces, {\\it Houston J. of Math}, {\\bf 44} (2018) no 3, 927-940.\n\n\\bibitem{Milman} V.D. Milman, Geometric theory of Banach spaces. II. Geometry of the unit ball (Russian), {\\it Uspehi Mat. Nauk} {\\bf 26} (1971), 73-149. English translation: {\\it Russian Math. Surveys} {\\bf 26} (1971), 79-163.\n\n\n\\bibitem{Netillard} F. N\\'etillard, Coarse Lipschitz embeddings of James spaces, {\\it Bull. Belg. Math. Soc. Simon Stevin}, {\\bf 25} (2018), 1-14.\n\n\\bibitem{Schoenberg} I. J. Schoenberg, Metric spaces and positive definite functions, {\\it Trans. Am. Math. Soc.} {\\bf 44} (1938), 522-536.\n\n\n\n\\end{biblist}\n\\end{bibsection}\n\n\n\n\n\n\\end{document} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nConceptNet 5 \\cite{speer2013conceptnet} is a multilingual, domain-general\nknowledge graph that connects words and phrases of natural language\n(\\emph{terms}) with labeled, weighted edges. Compared to other knowledge\ngraphs, it avoids trying to be a large gazetteer of named entities. It aims most\nof all to cover the frequently-used words and phrases of every language, and to\nrepresent generally-known relationships between the meanings of these terms.\n\nThe paper describing ConceptNet 5.5 \\cite{speer2017conceptnet} showed that it\ncould be used in combination with sources of distributional semantics,\nparticularly the word2vec Google News skip-gram embeddings\n\\cite{mikolov2013word2vec} and GloVe 1.2 \\cite{pennington2014glove}, to produce\nnew embeddings with state-of-the-art performance across many word-relatedness\nevaluations. The three data sources are combined using an extension of the\ntechnique known as retrofitting \\cite{faruqui2015retrofitting}.  The result is\na system of pre-computed word embeddings we call ``ConceptNet Numberbatch''.\n\nThe system we submitted to SemEval-2017 Task 2, ``Multilingual and\nCross-lingual Semantic Word Similarity'', is an update of that system,\ncoinciding with the release of version 5.5.3 of ConceptNet\\footnote{Data and\ncode are available at \\url{http:\/\/conceptnet.io}.}.  We added multiple fallback\nmethods for assigning vectors to out-of-vocabulary words. We also experimented\nwith, but did not submit, systems that used additional sources of word\nembeddings in the five languages of this SemEval task.\n\nThis task \\cite{semeval2017task2} evaluated systems at their ability to rank\npairs of words by their semantic similarity or relatedness. The words are in\nfive languages: English, German, Italian, Spanish, and Farsi.  Subtask 1\ncompares pairs of words within each of the five languages; subtask 2 compares\npairs of words that are in different languages, for each of the ten pairs of\ndistinct languages.\n\nOur system took first place in both subtasks. Detailed results for our system appear in\nSection~\\ref{sec:results}.\n\n\\section{Implementation}\n\nThe way we built our embeddings is based on retrofitting\n\\cite{faruqui2015retrofitting}, and in particular, the elaboration of it we\ncall ``expanded retrofitting'' \\cite{speer2017conceptnet}. Retrofitting, as\noriginally described, adjusts the values of existing word embeddings based on a\nnew objective function that also takes a knowledge graph into account.  Its\noutput has the same vocabulary as its input. In expanded retrofitting, on the\nother hand, terms that are only present in the knowledge graph are added to the\nvocabulary and are also assigned vectors.\n\n\\subsection{Combining Multiple Sources of Vectors}\n\nAs described in the ConceptNet 5.5 paper \\cite{speer2017conceptnet}, we apply\nexpanded retrofitting separately to multiple sources of embeddings (such as\npre-trained word2vec and GloVe), then align the results on a unified vocabulary\nand reduce its dimensionality.\n\nFirst, we make a unified matrix of embeddings, $M_1$, as follows:\n\n\\begin{itemize}\n    \\item Take the subgraph of ConceptNet consisting of nodes whose degree is at least 3. Remove\n        edges corresponding to negative relations (such as \\emph{NotUsedFor} and \\emph{Antonym}).\n        Remove phrases with 4 or more words.\n    \\item Standardize the sources of embeddings by case-folding their terms and $L_1$-normalizing\n        their columns.\n    \\item For each source of embeddings, apply expanded retrofitting over that source with the subgraph\n        of ConceptNet. In each case, this provides vectors for a vocabulary of terms that includes\n        the ConceptNet vocabulary.\n    \\item Choose a unified vocabulary (described below), and look up the vectors\n        for each term in this vocabulary in the expanded retrofitting outputs. If a vector is missing\n        from the vocabulary of a retrofitted output, fill in zeroes for those components.\n    \\item Concatenate the outputs of expanded retrofitting over this unified vocabulary to give $M_1$.\n\\end{itemize}\n\n\\subsection{Vocabulary Selection}\n\nExpanded retrofitting produces vectors for all the terms in its knowledge graph\nand all the terms in the input embeddings. Some terms from outside the\nConceptNet graph have useful embeddings, representing knowledge we would like\nto keep, but using all such terms would be noisy and wasteful.\n\nTo select the vocabulary of our term vectors, we used a heuristic that takes\nadvantage of the fact that the pre-computed word2vec and GloVe embeddings we\nused have their rows (representing terms) sorted by term frequency.\n\nTo find appropriate terms, we take all the terms that appear in the first\n500,000 rows of both the word2vec and GloVe inputs, and appear in the first\n200,000 rows of at least one of them. We take the union of these with the terms\nin the ConceptNet subgraph described above. The resulting vocabulary, of\n1,884,688 ConceptNet terms plus 99,869 additional terms, is the vocabulary we\nuse in the system we submitted and its variants.\n\n\\subsection{Dimensionality Reduction}\n\nThe concatenated matrix $M_1$ has $k$ columns representing features that may be\nredundant with each other. Our next step is to reduce its dimensionality to a\nsmaller number $k'$, which we set to 300, the dimensionality of the largest\ninput matrix. Our goal is to learn a projection from $k$ dimensions to $k'$\ndimensions that removes the redundancy that comes from concatenating multiple\nsources of embeddings.\n\nWe sample 5\\% of the rows of $M_1$ to get $M_2$, which we will use to find the\nprojection more efficiently, assuming that its vectors represent approximately\nthe same distribution as $M_1$.\n\n$M_2$ can be approximated with a truncated SVD: $M_2 \\approx U \\Sigma^{1\/2}\nV^T$, where $\\Sigma$ is truncated to a $k' \\times k'$ diagonal matrix of the $k'$\nlargest singular values, and $U$ and $V$ are correspondingly truncated to have\nonly these $k'$ columns.\n\n$U$ is a matrix mapping the same vocabulary to a smaller set of features.\nBecause $V$ is orthonormal, $U \\Sigma$ is a rotation and truncation of the\noriginal data, where each feature contributes the same amount of variance as it\ndid in the original data.  $U \\Sigma^{1\/2}$ is a version that removes some of\nthe variance that came from redundant features, and also is analogous to the\ndecomposition used by \\citet{levy2015embeddings} in their SVD process.\n\nWe can solve for the operator that projects $M_2$ into $U \\Sigma^{1\/2}$:\n$$U \\Sigma^{1\/2} \\approx M_2 V \\Sigma^{-1\/2}$$\n\n$V \\Sigma^{-1\/2}$ is therefore a $k \\times k'$ operator that, when applied on\nthe right, projects vectors from our larger space of features to our smaller\nspace of features. It can be applied to any vector in the space of $M_1$, not\njust the ones we sampled. $M_3 = M_1 V \\Sigma^{-1\/2}$ is the projection of the\nselected vocabulary into $k'$ dimensions, which is the matrix of term\nvectors that we output and evaluate.\n\n\\subsection{Don't Take ``OOV'' for an Answer}\n\nPublished evaluations of word embeddings can be inconsistent about what to do\nwith out-of-vocabulary (OOV) words, those words that the system has learned no\nrepresentation for. Some evaluators, such as \\citet{fasttext}, discard all\npairs containing an OOV word. This makes different systems with different vocabularies\ndifficult to compare. It enables gaming the evaluation by limiting the\nsystem's vocabulary, and gives no incentive to expand the vocabulary.\n\nThis SemEval task took a more objective position: no word pairs may be\ndiscarded. Every system must submit a similarity value for every word pair, and\n``OOV'' is no excuse. The organizers recommended using the midpoint of the\nsimilarity scale as a default.\n\nIn our previous work with ConceptNet, we eliminated one possible cause of OOV\nterms. A term that is outside of the selected vocabulary, perhaps because its\ndegree in ConceptNet is too low, can still be assigned a vector. When we\nencounter a word with no computed vector, we look it up in ConceptNet, find its\nneighbors, and take the average of whatever vectors those neighboring terms\nhave. This approximates the vector the term would have been assigned if it had\nparticipated in retrofitting. If the term has no neighbors with vectors, it\nremains OOV.\n\nFor this SemEval task, we recognized the importance of minimizing OOV terms,\nand implemented two additional fallback strategies for the terms that are still\nOOV.\n\nIt is unavoidable that training data in non-English languages will be harder\nto come by and sparser than data in English. It is also true that some words\nin non-English languages are borrowed directly from English, and are therefore\nexact cognates for English words.\n\nAs such, we used a simple strategy to further increase the coverage of our\nnon-English vocabularies: if a term is not associated with a vector in matrix\n$M_3$, we first look up the vector for the term that is spelled identically in\nEnglish. If that vector is present, we use it.\n\nThis method is in theory vulnerable to false cognates, such as the German word\n{\\em Gift} (meaning ``poison''). However, false cognates tend to appear among\ncommon words, not rare ones, so they are unlikely to use this fallback\nstrategy.  Our German embeddings do contain a vector for ``Gift'', and it is\nsimilar to English ``poison'', not English ``gift''.\n\nAs a second fallback strategy, when a term cannot be found in its given\nlanguage or in English, we look for terms in the vocabulary that have the given\nterm as a prefix. If we find none of those, we drop a letter from the end of\nthe unknown term, and look for that as a prefix. We continue dropping letters\nfrom the end until a result is found. When a prefix yields results, we use the\nmean of all the resulting vectors as the word's vector.\n\n\\section{Results}\n\nIn this task, systems were scored by the harmonic mean of their Pearson and\nSpearman correlation with the test set for each language (or language pair in\nSubtask 2).  Systems were assigned aggregate scores, averaging their top 4\nlanguages on Subtask 1 and their top 6 pairs on Subtask 2.\n\n\\subsection{The Submitted System: ConceptNet + word2vec + GloVe}\n\nThe system we submitted applied the retrofitting-and-merging process described\nabove, with ConceptNet 5.5.3 as the knowledge graph and two well-regarded\nsources of English word embeddings. The first source is the word2vec Google\nNews embeddings\\footnote{\\url{https:\/\/code.google.com\/archive\/p\/word2vec\/}},\nand the second is the GloVe 1.2 embeddings that were trained on 840 billion\ntokens of the Common\nCrawl\\footnote{\\url{http:\/\/nlp.stanford.edu\/projects\/glove\/}}.\n\nBecause the input embeddings are only in English, the vectors in other\nlanguages depended entirely on propagating these English embeddings via the\nmultilingual links in ConceptNet.\n\nThis system appears in the results as ``Luminoso-run2''. Run 1 was similar, but\nit was looking up neighbors in an unreleased version of the ConceptNet graph\nwith fewer edges from DBPedia in it.\n\nThis system's aggregate score on subtask 1 was 0.743. Its combined score on\nsubtask 2 (averaged over its six best language pairs) was 0.754.\n\n\\subsection{Variant A: Adding Polyglot Embeddings}\n\nInstead of relying entirely on English knowledge propagated through ConceptNet,\nit seemed reasonable to also include pre-calculated word embeddings in other\nlanguages as inputs. In Variant A, we added inputs from the Polyglot embeddings\n\\cite{alrfou2013polyglot} in German, Spanish, Italian, and Farsi as four\nadditional inputs to the retrofitting-and-merging process.\n\nThe results of this variant on the trial data were noticeably lower, and when\nwe evaluate it on the test data in retrospect, its test results are lower as well.\nIts aggregate scores are .720 on subtask 1 and .736 on subtask 2.\n\n\\subsection{Variant B: Adding Parallel Text from OpenSubtitles}\n\nIn Variant B, we calculated our own multilingual distributional embeddings from\nword co-occurrences in the OpenSubtitles2016 parallel corpus\n\\cite{lison2016opensubtitles}, and used this as a third input alongside\nword2vec and GloVe.\n\nFor each pair of aligned subtitles among the five languages, we combined the\nlanguage-tagged words into a single set of $n$ words, then added $1\/n$ to the\nco-occurrence frequency of each pair of words, yielding a sparse matrix of word\nco-occurrences within and across languages. We then used the SVD-of-PPMI\nprocess described by \\citet{levy2015embeddings} to convert these sparse\nco-occurrences into 300-dimensional vectors.\n\nOn the trial data, this variant compared inconclusively to Run 2. We submitted\nRun 2 instead of Variant B because Run 2 was simpler and seemed to perform\nslightly better on average.\n\nHowever, when we run variant B on the released test data, we note that it would\nhave scored better than the system we submitted. Its aggregate scores are .759\non subtask 1 and .767 on subtask 2.\n\n\\subsection{Comparison of Results}\n\\label{sec:results}\n\n\\begin{table}[t]\n\\begin{small}\n\\begin{tabular}{lrrrrr}\n\\hline\nEval.       & Base & Ours & $-$OOV & Var. A & Var. B \\\\\n\\hline\n\\em en      & .683 & .789 & .747 & .778 & \\bf .796 \\\\\n\\em de      & .513 & .700 & .599 & .673 & \\bf .722 \\\\\n\\em es      & .602 & .743 & .611 & .716 & \\bf .761 \\\\\n\\em it      & .597 & .741 & .606 & .711 & \\bf .756 \\\\\n\\em fa      & .412 & .503 & .363 & .506 & \\bf .541 \\\\\n\\bf Score 1 & .598 & .743 & .641 & .720 & \\bf .759 \\\\\n\\hline\n\\em en-de   & .603 & .763 & .696 & .749 & \\bf .767 \\\\\n\\em en-es   & .636 & .761 & .675 & .752 & \\bf .778 \\\\\n\\em en-it   & .650 & .776 & .677 & .759 & \\bf .786 \\\\\n\\em en-fa   & .519 & .598 & .502 & .590 & \\bf .634 \\\\\n\\em de-es   & .550 & .728 & .620 & .704 & \\bf .747 \\\\\n\\em de-it   & .565 & .741 & .612 & .722 & \\bf .757 \\\\\n\\em de-fa   & .464 & .587 & .501 & .586 & \\bf .610 \\\\\n\\em es-it   & .598 & .753 & .613 & .732 & \\bf .765 \\\\\n\\em es-fa   & .493 & .627 & .482 & .623 & \\bf .646 \\\\\n\\em it-fa   & .497 & .604 & .474 & .599 & \\bf .635 \\\\\n\\bf Score 2 & .598 & .754 & .649 & .736 & \\bf .767 \\\\\n\\hline\n\\end{tabular}\n\\end{small}\n\\caption{\n    Evaluation scores by language. ``Score 1'' and ``Score 2'' are\n    the combined subtask scores. ``Base'' is the Nasari baseline,\n    ``Ours'' is Luminoso-Run2 as submitted, ``$-$OOV'' removes our OOV\n    strategy, and ``Var. A'' and ``Var. B'' are the variants we describe\n    in this paper.\n}\n\\label{table:results-table}\n\\end{table}\n\nThe released\nresults\\footnote{\\url{http:\/\/alt.qcri.org\/semeval2017\/task2\/index.php?id=results}}\nshow that our system, listed as Luminoso-Run2, got the highest aggregate score\non both subtasks, and the highest score on each test set except the monolingual\nFarsi set.\n\nTable~\\ref{table:results-table} compares the results per language of the system\nwe submitted, the same system without our OOV-handling strategies, variants A\nand B, and the baseline Nasari \\cite{camacho2016nasari} system.\n\nVariant B performed the best in the end, so we will incorporate parallel text\nfrom OpenSubtitles in the next release of the ConceptNet Numberbatch system.\n\n\\section{Discussion}\n\nThe idea of producing word embeddings from a combination of distributional and\nrelational knowldedge has been implemented by many others, including\n\\citet{iacobacci2015sensembed} and various implementations of retrofitting\n\\cite{faruqui2015retrofitting}. ConceptNet is distinguished by the large\nimprovement in evaluation scores that occurs when it is used as the source of\nrelational knowledge. This indicates that ConceptNet's particular blend of\ncrowd-sourced, gamified, and expert knowledge is providing valuable information\nthat is not learned from distributional semantics alone.\n\nThe results transfer well to other languages, showing ConceptNet's usefulness\nas ``multilingual glue'' that can combine knowledge in multiple languages into\na single representation.\n\nOur submitted system relies heavily on inter-language links in ConceptNet that\nrepresent direct translations, as well as exact cognates. We suspect that this\nmakes it perform particularly well at directly-translated English. It would\nhave more difficulty determining the similarity of words that lack direct\ntranslations into English that are known or accurate.  This is a weak point of\nmany current word-similarity evaluations: The words that are vague when\ntranslated, or that have language-specific connotations, tend not to appear.\n\nOn a task with harder-to-translate words, we may have to rely more on observing\nthe distributional semantics of corpus text in each language, as we did in the\nunsubmitted variants.\n\n\\bibliographystyle{acl_natbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Ch:Intro}\n\nThroughout this paper, let $\\mms$ be a nonempty set endowed with a proper, thus separable metric $\\met$. All topological and measure-theoretic properties will be understood with respect to the topology induced by $\\met$ and its induced Borel $\\sigma$-algebra. The completeness of $\\met$ is \\emph{not} assumed; every subsequent result which requires this re\\-gu\\-larity is phrased relative to compact subsets of $\\mms$. Moreover, let $\\meas$ be a given Radon measure on $\\mms$ with full support, symbolically $\\supp\\meas=\\mms$.\n\n\\subsection*{Background} Mathematical general relativity has recently developed a promising novel research direction.  McCann \\cite{mccann2020} pioneered lower boundedness of the Ricci tensor $\\Ric$ of a smooth Lorentzian spacetime $\\mms$ in  timelike directions to be equivalent to convexity properties of the Boltzmann entropy $\\Ent_\\meas$, for $\\meas$ a possibly weighted Lorentzian volume measure, along ``chronological geodesics'' on the space of probability measures ${\\ensuremath{\\mathscr{P}}}(\\mms)$ in the entropic sense \\cite{erbar2015}. This includes the \\emph{strong energy condition} of Hawking and Penrose \\cite{hawking1966, hawking1970,penrose1965} which, for a vanishing cosmological constant, reduces to timelike nonnegativity of $\\Ric$. Mondino and Suhr \\cite{mondinosuhr2018} pro\\-vided a similar description of the Einstein equations, i.e.~timelike lower and upper Ricci curvature bounds, through synthetic optimal transport means. In a dif\\-ferent direction, Kunzinger and S\u00e4mann \\cite{kunzinger2018}, see also \\cite{sormani2016} for a related approach, set up the theory of Lorentzian (pre-length, length, and geodesic) spaces. These are singular analogues of smooth Lorentzian spacetimes, comparable to metric spaces generalizing Riemannian manifolds. These discoveries let Caval\\-letti and Mondino \\cite{cavalletti2020}  introduce the \\emph{entropic timelike curvature-dimension condition} $\\smash{\\mathrm{TCD}_p^e(K,N)}$ and the \\emph{entropic timelike measure-contraction property} $\\smash{\\mathrm{TMCP}_p^e(K,N)}$, $p\\in (0,1)$, $K\\in \\R$, and $N\\in [1,\\infty)$. Both are ways to concretize the meaning of a Lorentzian space measured by $\\meas$ to admit  timelike Ricci curvature bounded from below by $K$ and dimension bounded from above by $N$. As indicated above, the inherent for\\-mu\\-lations are phrased by convexity properties of $\\Ent_\\meas$ along \\emph{$\\ell_p$-geodesics}, thus make sense a priori in far more general frameworks than smooth Lorentzian spacetimes. Here, $\\smash{\\ell_p}$ is the so-called \\emph{$p$-Lorentz--Wasserstein distance} on ${\\ensuremath{\\mathscr{P}}}(\\mms)$, see \\eqref{Eq:LORWAS},  first introduced in \\cite{eckstein2017} and studied further in \\cite{cavalletti2020,kellsuhr2020,mccann2020,mondinosuhr2018,\nsuhr2018}. Recently \\cite{braun2022}, we have furthermore introduced an ``infinite-dimensional'' Lorentzian analogue of \\cite{sturm2006a}.\n\n\nThis line of thought to axiomatize synthetic Ricci curvature bounds is not new. In metric geometry, results for Riemannian manifolds \\cite{cordero2001,otto2000,vonrenesse2005} analogous to \\cite{mccann2020,mondinosuhr2018} lead Sturm \\cite{sturm2006a,sturm2006b}, independently Lott and Villani \\cite{lott2009}, to paraphrase lower Ricci curvature and upper dimension bounds for metric measure spaces by convexity of certain entropy functionals along geodesics in ${\\ensuremath{\\mathscr{P}}}(\\mms)$ with respect to the $2$-Wasserstein distance $W_2$. This was the starting point of an intense activity on so-called $\\CD$ or $\\RCD$ spaces, vivid to date, which produced remarkable outcomes and singular generalizations of known smooth results. Among others, we mention the splitting theorem \\cite{gigli2013}, structure theory \\cite{brue2020,mondino2019}, a first and second order Sobolev calculus \\cite{ambrosio2014a,ambrosio2014b,ambrosio2015,gigli2015,\ngigli2018, savare2014}, rigidity \\cite{erbar2020,ketterer2015a, ketterer2015b}, and stability \\cite{ambrosio2017,giglimondino2015,suzuki2019}.\n\nSynthetic timelike Ricci curvature bounds for Lorentzian spaces  through optimal transport as initiated by the mentioned  works  \\cite{cavalletti2020,mccann2020,mondinosuhr2018} are expected to lead to an equally flourishing theory. There are many smooth  results, both classical and recent, that might (or, very occasionally, already do) admit generalizations to such spaces,  notably singularity theorems \u00e0 la Hawking and Penrose \\cite{alexander2019, cavalletti2020, graf2020, hawking1973, kunzinger2015, lu2021, woolgar2016} and \u00e0 la Gannon and Lee \\cite{gannon1975, gannon1976, lee1976, schinnerl2021}, as well as splitting theorems \\cite{beem1985, eschenburg1988, galloway1989, lu2021b}. See the recent survey article \\cite{steinbauer}. Already from the conceptional viewpoint, \\cite{cavalletti2020} sheds a novel light to recent efforts to understand Lorentzian spacetimes with low regularity metrics, where curvature bounds are phrased distributionally thus far \\cite{geroch1987,graf2020,steinbauer2009}; the setting of \\cite{cavalletti2020,kunzinger2018} allows not only the potential metric tensor, but also the base space to be irregular. This extension is desirable since general relativity generally \\emph{predicts} the presence of singularities under extreme physical conditions, e.g.~in black holes, causal Fermion systems \\cite{finster2016, finster2018}, or causal sets \\cite{bombelli1987} which, even worse,  might not be approximable by smooth structures (even if so, they can well be irregular). See also \\cite{kunzinger2018,martin2006} and the introduction of \\cite{cavalletti2020}. Moreover, by what we learned from  \\cite{mondinosuhr2018} such a theory is a further step to understand the Einstein equations \\cite{klainerman2015,rendall2002} in more singular, yet physically relevant settings \\cite{burtscher2014, griffiths2015, lichnerowicz1955, mars1993, penrose1972, vickers1990, vickers2000}, which also relates to the \\emph{censorship conjectures} \\cite{christodoulou2009,dafermos2014}. For recently introduced synthetic notions of timelike \\emph{upper} Ricci curvature bounds, see \\cite{cavalletti2022}. \n\n\\subsection*{Objective} The goal of this paper is to introduce and investigate new timelike cur\\-vature-dimension conditions and timelike measure-contraction properties for measured Lorentzian spaces by optimal transport, different from \\cite{cavalletti2020}: instead of the Boltzmann entropy, we  use the \\emph{R\u00e9nyi entropy}, cf.~\\eqref{Eq:RN} below. Our treatise is the Lorentzian counterpart of the groundbreaking works \\cite{bacher2010,sturm2006b}, as is  \\cite{cavalletti2020} relative to \\cite{erbar2015}. It sets up an especially geometrically valuable viewpoint to synthetic timelike lower Ricci curvature and upper dimension bounds. In fact, even in the smooth case, we are the first to study the R\u00e9nyi entropy within general relativity.\n\n\\subsection*{Setting} Before making all above indicated objects, notions, and results precise, we outline our framework, see \\autoref{Sub:Lorentzian nonsmooth} and \\autoref{Sec:OT Lorentzian} for details. \n\nLet $(\\mms,\\met,\\ll,\\leq,\\uptau)$ be a Lorentzian pre-length space in the sense of \\cite{kunzinger2018}. Here, $\\ll$ and $\\leq$ are generalized notions of \\emph{chronology} and \\emph{causality} between points in $\\mms$. Moreover, $\\uptau\\colon \\mms^2\\to[0,\\infty]$ is the so-called \\emph{time separation function}. Roughly speaking, $\\uptau(x,y)$ measures the maximal proper-time a point $x\\in\\mms$ in spacetime needs to travel to $y\\in\\mms$. Compared to \\cite{bacher2010, sturm2006b}, $\\uptau$ takes over the role of the metric in the formulation of our timelike curvature-dimension conditions and time\\-like measure-contraction properties, yet this analogy should be handled carefully; for instance, $\\uptau$ satisfies  the \\emph{reverse} triangle inequality \\eqref{Eq:Reverse tau}, it is symmetric only in pathological cases, and $\\uptau(x,y)=0$ as soon as $y$ is not in the chronological future of $x$. With few exceptions, we naturally assume $(\\mms,\\met,\\ll,\\leq,\\uptau)$ to be causally closed, ${\\ensuremath{\\mathscr{K}}}$-globally hyperbolic, and geodesic in addition, cf.~\\autoref{Ass:ASS}. Inter alia, this hypothesis ensures that $\\uptau$ is finite and continuous. When endowed with a reference measure $\\meas$ as initially fixed, for terminological cleanness, in this introduction we call $(\\mms,\\met,\\meas,\\ll,\\leq,\\uptau)$ \\emph{measured Lorentzian space}.\n\nFor $p\\in(0,1]$, define the total transport cost $\\smash{\\ell_p\\colon{\\ensuremath{\\mathscr{P}}}(\\mms)^2 \\to [0,\\infty]\\cup\\{-\\infty\\}}$ by\n\\begin{align}\\label{Eq:LORWAS}\n\\ell_p(\\mu,\\nu) := \\sup \\Vert \\uptau\\Vert_{\\Ell^p(\\mms^2,\\pi)}.\n\\end{align}\nThe supremum is taken over all \\emph{causal} couplings $\\pi\\in\\Pi(\\mu,\\nu)$, i.e.~$x\\leq y$ for $\\pi$-a.e. $(x,y)\\in\\mms^2$, subject to the usual convention $\\sup \\emptyset := -\\infty$. (If such a $\\pi$ satisfies $x\\ll y$ for $\\pi$-a.e.~$(x,y)\\in\\mms^2$, it is called \\emph{chronological}.) A causal coupling attaining this supremum is called \\emph{$\\smash{\\ell_p}$-optimal}.\n Similarly as for $\\uptau$, we interpret the quantity $\\smash{\\ell_p}$ as measuring the maximal proper-time a spacetime mass distribution $\\mu$ needs to be moved to a mass distribution $\\nu$ in a future-directed, causal manner. Moreover, $\\smash{\\ell_p}$ satisfies the reverse triangle inequality \\eqref{Eq:Reverse triangle lp}.\n\nThe convexity properties making up our main definitions are understood along \\emph{timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics} in ${\\ensuremath{\\mathscr{P}}}(\\mms)$, axiomatized as now described in \\autoref{Sub:GEO}, \\autoref{Sub:Geodesics}, and \\autoref{App:B}. A timelike geodesic on $\\mms$ is a curve $\\gamma\\in \\Cont([0,1]; \\mms)$ which is $\\met$-Lipschitz continuous [sic], obeys $\\gamma_s \\ll \\gamma_t$ for every $s,t\\in[0,1]$ with $s<t$, and maximizes the $\\uptau$-length \\eqref{Eq:Length tau def}, symbolically $\\gamma\\in\\mathrm{TGeo}(\\mms)$. As for maximizing geodesics in metric geometry \\cite{burago2001}, every timelike geodesic has a proper-time reparametrization $\\eta\\in\\Cont([0,1];\\mms)$, i.e.\n\\begin{align}\\label{Eq:tau s t equ}\n\\uptau(\\eta_s,\\eta_t) = (t-s)\\,\\uptau(\\eta_0,\\eta_1)>0\n\\end{align}\nfor every $s,t\\in[0,1]$ with $s<t$ \\cite{kunzinger2018}. The inherent reparametrization procedure in\\-duces a map ${\\ensuremath{\\mathsf{r}}}\\colon \\mathrm{TGeo}(\\mms) \\to \\Cont([0,1];\\mms)$ which is continuous, cf.~\\autoref{Le:Cty reparametrization}. Given any $\\mu_0,\\mu_1\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$, let $\\smash{\\mathrm{OptTGeo}_{\\ell_p}(\\mu_0,\\mu_1)}$ be the set of all ${\\ensuremath{\\boldsymbol{\\pi}}}\\in{\\ensuremath{\\mathscr{P}}}(\\mathrm{TGeo}(\\mms))$ such that $\\smash{\\pi := (\\eval_0,\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}}$ --- where $\\eval_t$ is the usual evaluation map at $t\\in[0,1]$ --- is a chronological $\\smash{\\ell_p}$-optimal coupling of $\\mu_0$ and $\\mu_1$. (Our geodesy hypothesis from \\autoref{Ass:ASS} below  ensures that every $x,y\\in\\mms$ with $x\\ll y$ can be connected by an element of $\\mathrm{TGeo}(\\mms)$.) Set  $\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1) := {\\ensuremath{\\mathsf{r}}}_\\push\\mathrm{OptTGeo}_{\\ell_p}(\\mu_0,\\mu_1)}$, and observe that all measures belonging to this set are concentrated on $\\mathrm{TGeo}^\\uptau(\\mms) := {\\ensuremath{\\mathsf{r}}}(\\mathrm{TGeo}(\\mms))$ whose elements, in turn, obey \\eqref{Eq:tau s t equ} for every $s,t\\in[0,1]$ with $s< t$.\n\n\\begin{definition}\\label{Def:Termin} A weakly continuous curve $(\\mu_t)_{t\\in[0,1]}$ in ${\\ensuremath{\\mathscr{P}}}(\\mms)$ is called\n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item \\emph{timelike $\\smash{\\ell_p}$-geodesic} if there exists a measure $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}(\\mu_0,\\mu_1)}$ such that $\\mu_t = (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}$ for every $t\\in[0,1]$, and\n\\item \\emph{timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic} provided there is a measure ${\\ensuremath{\\boldsymbol{\\pi}}}\\in\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that $\\mu_t= (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}$ for every $t\\in[0,1]$.\n\\end{enumerate}\n\\end{definition}\n\nWe point out that our definition of a bare $\\smash{\\ell_p}$-geodesic differs from \\cite{mccann2020}. Indeed, every timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ satisfies\n\\begin{align}\\label{Eq:FBHAHB}\n\\ell_p(\\mu_s,\\mu_t) = (t-s)\\,\\ell_p(\\mu_0,\\mu_1)>0\n\\end{align}\nfor every $s,t\\in[0,1]$ with $s<t$ and is thus an $\\smash{\\ell_p}$-geodesic as in \\cite{mccann2020}. On the other hand, timelike $\\smash{\\ell_p}$-geodesics in our sense are implicit maximizers of some length functional, cf.~\\autoref{Pr:Max}. Thus, \\autoref{Def:Termin} reflects how geodesics are set up at the base space terminologically and conceptionally \\cite{kunzinger2018}. In general, it is more restrictive, as \\cite{mccann2020} does not even assume weak con\\-tinuity of the considered paths in ${\\ensuremath{\\mathscr{P}}}(\\mms)$. (It is remarkable that this regularity is implied solely by \\eqref{Eq:FBHAHB} in all relevant smooth cases, cf.~\\autoref{Re:Compat}.) Our approach seems right for $\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ to admit good compactness properties, cf.~\\autoref{Le:Villani lemma for geodesic}. In\\-deed, proper-time reparametrizations of timelike geodesics in $\\mms$ are \\emph{not} Lipschitz continuous \\cite{kunzinger2018}, which makes it difficult to find compact subsets of $\\Cont([0,1];\\mms)$ satisfying \\eqref{Eq:tau s t equ}. On the other hand, by \\autoref{Ass:ASS} --- more precisely, a property termed \\emph{non-total imprisonment} ---  all elements of $\\mathrm{TGeo}(\\mms)$ passing through a fixed causal diamond have uniformly bounded Lipschitz constants. This entails excellent compactness properties of $\\mathrm{TGeo}(\\mms)$ and $\\smash{\\mathrm{OptTGeo}_{\\ell_p}(\\mu_0,\\mu_1)}$ which transfer to $\\smash{\\mathrm{TGeo}^\\uptau(\\mms)}$ and $\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ by continuity of ${\\ensuremath{\\mathsf{r}}}$ and $\\smash{{\\ensuremath{\\mathsf{r}}}_\\push}$, respectively.\n\nLastly, the basic object which will be hypothesized to have some convexity along the named curves in ${\\ensuremath{\\mathscr{P}}}(\\mms)$ is the $N$-R\u00e9nyi entropy\n\\begin{align}\\label{Eq:RN}\n{\\ensuremath{\\mathscr{S}}}_N(\\mu) := -\\int_\\mms \\rho^{-1\/N}\\d\\mu = -\\int_\\mms \\rho^{1-1\/N}\\d\\meas,\n\\end{align}\nwhere $\\mu = \\rho\\,\\meas + \\mu_\\perp$ is the Lebesgue decomposition of $\\mu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$ with respect to $\\meas$. Its properties are summarized in \\autoref{Subsub:Renyi}. It only relates to $\\Ent_\\meas$ in the limit case $N\\to\\infty$, see \\eqref{Eq:Ent SN limit}.\n\n\n\\subsection*{Contributions} Now we come to the communication of our novelties. The respective advantages of our different conditions are outlined along the way.\n\n$\\blacktriangleright$ Given $p\\in (0,1)$, in \\autoref{Def:TCD*} and \\autoref{Def:TCD} we introduce the sub\\-se\\-quent synthetic notions of the timelike Ricci curvature of $(\\mms,\\met,\\meas,\\ll,\\leq,\\uptau)$ being bounded from below by $K\\in \\R$, and its dimension being bounded from above by $N\\in [1,\\infty)$, respectively:\n\\begin{itemize}\n\\item the reduced timelike curvature-dimension condition $\\smash{\\mathrm{TCD}_p^*(K,N)}$,\n\\item the weak reduced timelike curvature-dimension condition $\\smash{\\mathrm{wTCD}_p^*(K,N)}$,\n\\item the timelike curvature-dimension condition $\\smash{\\mathrm{TCD}_p(K,N)}$, and\n\\item the weak timelike curvature-dimension condition $\\smash{\\mathrm{wTCD}_p(K,N)}$.\n\\end{itemize} \nThese are Lorentzian versions of \\cite{bacher2010,sturm2006b}  and are all formulated via  convexity properties of the R\u00e9nyi entropy along timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics $(\\mu_t)_{t\\in[0,1]}$ between pairs $\\smash{(\\mu_0,\\mu_1)\\in{\\ensuremath{\\mathscr{P}}}(\\mms)^2}$ of compactly supported, $\\meas$-absolutely continuous mass distributions, symbolically $\\smash{\\mu_0,\\mu_1\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$. The inherent  inequalities de\\-fining $\\smash{\\mathrm{TCD}_p^*(K,N)}$ and $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ use the distortion coefficients $\\smash{\\sigma_{K,N}^{(r)}}$, while $\\smash{\\mathrm{TCD}_p(K,N)}$ and $\\smash{\\mathrm{wTCD}_p(K,N)}$ employ the quantities $\\smash{\\tau_{K,N}^{(r)}}$; we refer to \\autoref{Def:Dist coeff} below. The strong versions require the relevant estimate for R\u00e9nyi's entropy to hold for all \\emph{time\\-like $p$-dualizable} endpoint pairs $(\\mu_0,\\mu_1)$, meaning that there exists an appropriate chronological $\\smash{\\ell_p}$-optimal coupling $\\pi\\in\\Pi(\\mu_0,\\mu_1)$. The weak versions require $(\\mu_0,\\mu_1)$ to be \\emph{strongly timelike $p$-dualizable}, which essentially asks \\emph{all} $\\smash{\\ell_p}$-optimal couplings of $\\mu_0$ and $\\mu_1$ to be chronological. The latter notions have been introduced in \\cite{cavalletti2020}, see \\autoref{Def:TL DUAL} for their  precise definition.\n\nIn \\autoref{Pr:TCD and TCD*}, we establish the following general relations between these conditions, where we set $K^* := K(N-1)\/N$:\n\\begin{align}\\label{RKB}\n\\begin{split}\n\\mathrm{TCD}_p(K,N) &\\ \\Longrightarrow \\ \\mathrm{TCD}_p^*(K,N),\\\\\n\\mathrm{wTCD}_p(K,N) &\\ \\Longrightarrow \\ \\mathrm{wTCD}_p^*(K,N),\\\\\n\\mathrm{TCD}_p^*(K,N) &\\ \\Longrightarrow \\ \\mathrm{TCD}_p(K^*,N),\\\\\n\\mathrm{wTCD}_p^*(K,N) &\\ \\Longrightarrow \\ \\mathrm{wTCD}_p(K^*,N).\n\\end{split}\n\\end{align}\n\n$\\blacktriangleright$ Our first major result is the compatibility of the above notions with timelike lower Ricci curvature bounds in the smooth framework, cf.~\\autoref{Th:Equiv}. In fact, we also cover the weighted case which has recently lead to singularity and splitting theorems for so-called Bakry--\u00c9mery spacetimes \\cite{woolgar2016}.\n\n\\begin{theorem}\\label{Th:Smothh} Let $p\\in(0,1)$, $K\\in\\R$, and $N\\in[n,\\infty)$, where $\\smash{n\\in\\N_{\\geq 2}}$. For the Lo\\-rentz\\-ian space induced by a smooth, $n$-dimensional Lorentzian spacetime $\\mms$ and measured by the Lorentzian volume measure, any of $\\smash{\\mathrm{TCD}_p^*(K,N)}$, $\\smash{\\mathrm{wTCD}_p^*(K,N)}$, $\\smash{\\mathrm{TCD}_p(K,N)}$, and $\\smash{\\mathrm{wTCD}_p(K,N)}$ holds if and only if the Ricci curvature of $\\mms$ is bounded from below  by $K$ in all timelike directions of the tangent bundle $T\\mms$.\n\\end{theorem}\n\nIn particular, for $K=0$ this gives a new characterization of the strong energy condition \u00e0 la Hawking and Penrose, paralleling \\cite{mccann2020}, and a new description of one inequality in the Einstein equations \\cite{mondinosuhr2018}.\n\n$\\blacktriangleright$ The conditions $\\smash{\\mathrm{wTCD}_p(K,N)}$ and $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ are proven to imply important timelike geometric estimates \u00e0 la\n\\begin{itemize}\n\\item Brunn--Minkowski, see \\autoref{Pr:Brunn-Minkowski} and \\autoref{Propos},\n\\item Bonnet--Myers, see \\autoref{Cor:Bonnet-Myers} and \\autoref{Cor:Reduced BM}, as well as\n\\item Bishop--Gromov, see \\autoref{Th:BG} and \\autoref{Th:Reduced BG}.\n\\end{itemize}\nHere the key advantage of $\\smash{\\mathrm{wTCD}_p(K,N)}$ is that it yields these estimates in \\emph{sharp} form. Indeed, for a continuous, globally hyperbolic, $n$-dimensional spacetime  $\\mms$,  $\\smash{n\\in\\N_{\\geq 2}}$, which is additionally  causally plain \\cite{grant2012}, the timelike Bishop--Gromov inequality obtained from $\\smash{\\mathrm{wTCD}_p(K,N)}$ yields $n = \\dim^\\uptau\\mms \\leq N$ for the so-called geometric dimension $\\smash{\\dim^\\uptau\\mms}$ recently introduced in \\cite{mccann2021}, see \\autoref{Cor:HD}. As inferred in \\autoref{Re:HD2}, the same reasoning starting from $\\mathrm{wTCD}_p^*(K,N)$ only gives $\\smash{n=\\dim^\\uptau\\mms \\leq N+1}$; compare with \\eqref{RKB}. Corresponding geometric inequalities have been obtained from the weak entropic timelike curvature-dimension condition $\\smash{\\mathrm{wTCD}_p^e(K,N)}$ in \\cite{cavalletti2020} as well, but do not yield sharp versions either.\n\n$\\blacktriangleright$ Another important general feature of our timelike curvature-dimension conditions is their weak stability according to a natural Lorentzian notion of convergence introduced in \\cite{cavalletti2020} following \\cite{giglimondino2015}, concretized in \\autoref{Th:Stability TCD}.\n\n\\begin{theorem}\\label{Th:Asddd} Let $(\\mms_k,\\met_k,\\meas_k,\\ll_k,\\leq_k,\\uptau_k)_{k\\in\\N}$ be a sequence of measured Lorentz\\-ian spaces obeying $\\smash{\\mathrm{TCD}_p^*(K,N)}$ for fixed $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$ which converges to a measured Lorentzian  space $(\\mms_\\infty,\\met_\\infty,\\meas_\\infty,\\ll_\\infty,\\leq_\\infty,\\uptau_\\infty)$ in the sense of \\autoref{Def:Convergence}.  Then the latter satisfies $\\smash{\\mathrm{wTCD}_p^*(K,N)}$. \n\nThe same conclusion holds true when  replacing $\\smash{\\mathrm{TCD}_p^*(K,N)}$ and $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ by $\\smash{\\mathrm{TCD}_p(K,N)}$ and $\\smash{\\mathrm{wTCD}_p(K,N)}$, respectively.\n\\end{theorem}\n\nHence, limits according to \\autoref{Def:Convergence}  of sequences of smooth Lo\\-rentz\\-ian spacetimes satisfying the hypothesis of \\autoref{Th:Smothh} are $\\smash{\\mathrm{wTCD}_p(K,N)}$.\n\n\\autoref{Th:Asddd} is where the a priori distinction between weak and strong versions of our timelike curvature-dimension conditions, as in its entropic counterpart from \\cite{cavalletti2020}, comes into play. There are essentially two technical reasons which arise from the proof of \\autoref{Th:Stability TCD}. First, given a strongly timelike $p$-dualizable pair in $\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms_\\infty,\\meas_\\infty)^2}$, the members $\\smash{\\mu_{k,0},\\mu_{k,1}\\in {\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms_k,\\meas_k)}$, $k\\in\\N$, of correspon\\-ding recovery sequences between which $\\smash{\\mathrm{TCD}_p^*(K,N)}$ applies can only be constructed to be timelike $p$-dua\\-li\\-zable in general. Second, in the passage as $k\\to\\infty$, our timelike curvature-dimension condition requires the involved limit objects (namely, timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics and $\\smash{\\ell_p}$-optimal couplings), obtained by a compactness argument, to be chronological. However, unlike causality, chronology of either objects is not stable under weak limits in general, unless we know that \\emph{every} $\\smash{\\ell_p}$-optimal coupling of the limit marginals is chronological.\n\n$\\blacktriangleright$ In \\cite{cavalletti2020}, the most powerful consequences of $\\smash{\\mathrm{TCD}_p^e(K,N)}$ ---  e.g.~uniqueness of $\\smash{\\ell_p}$-optimal couplings and $\\smash{\\ell_p}$-geodesics, localization, and singularity theorems --- were obtained under the assumption of \\emph{timelike nonbranching}. Roughly speaking, the latter means that elements of $\\smash{\\mathrm{TGeo}^\\uptau(\\mms)}$ do not admit forward or backward branching at intermediate times. A drawback of this definition is that it is expected to fail e.g.~for spacetimes with regularity less than $\\smash{\\Cont^{1,1}}$ \\cite{cavalletti2020}. Hence, it may well be lost in limit cases, e.g.~in the framework of \\autoref{Th:Asddd}.\n\nThus, we newly introduce the following notion called \\emph{timelike $p$-essential non\\-branching}. See \\autoref{Def:Essentially nonbranching} for the precise definition.\n\n\\begin{definition}\\label{Def:pENB definition} Given $p\\in (0,1]$, we term the space $(\\mms,\\met,\\ll,\\leq,\\uptau)$ \\emph{timelike $p$-essentially nonbranching} if every element of $\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau({\\ensuremath{\\mathscr{P}}}(\\mms)^2)}$ is concentrated on a timelike nonbranching subset of $\\smash{\\mathrm{TGeo}^\\uptau(\\mms)}$.\n\\end{definition}\n\nEvery timelike nonbranching Lorentzian pre-length space is timelike $p$-essentially nonbranching for every $p\\in (0,1]$, cf.~\\autoref{Re:reversed}.\n\n\\autoref{Def:pENB definition} is leaned on the notion of \\emph{essentially nonbranching} metric measure spaces  \\cite{rajala2014}. In \\cite{rajala2014}, every $\\RCD$ space was proven to be  essentially nonbranching. Since the $\\RCD$ condition is stable with respect to measured Gromov--Hausdorff (mGH) convergence  \\cite{giglimondino2015, sturm2006a}, this implicitly gives a stability result for the essential  nonbranching condition. For this reason and the suspected issues for low regularity spacetimes, we believe \\autoref{Def:pENB definition} to be relevant for the setting of nonsmooth general relativity as well, given that a suitable  Lorentzian version of infinitesimal Hilbertianity \\cite{ambrosio2014a,ambrosio2014b,gigli2015} is found.\n\nHere, we obtain a local-to-global property for the $\\smash{\\mathrm{TCD}_p^*(K,N)}$ condition under timelike $p$-essential nonbranching, cf.~\\autoref{Th:Local to global}. This constitutes a Lorentzian variant of \\cite{bacher2010} and is a feature of the reduced timelike curvature-dimension condition; indeed, as detailed in \\autoref{Eq:Blurt}, such a property seems subtle for $\\smash{\\mathrm{TCD}_p(K,N)}$. Furthermore, after \\cite[Thm.~3.14]{erbar2015} we expect such a local-to-global property for $\\smash{\\mathrm{TCD}_p^e(K,N)}$ to hold \\emph{a priori} only in a strong form.\n\n$\\blacktriangleright$ Another cornerstone result of our paper is the equivalence proof of \\emph{all} our reduced timelike curvature-dimension conditions with \\emph{all} entropic ones from \\cite{cavalletti2020} under timelike $p$-essential nonbranching as follows, see \\autoref{Th:Equivalence TCD* and TCDe}. Remarkably, none of these conditions is \\emph{a priori} weaker than the other.\n\n\\begin{theorem}\\label{Zums} Assume $(\\mms,\\met,\\meas,\\ll,\\leq,\\uptau)$ to be timelike $p$-essentially nonbranching for some $p\\in (0,1)$. Then the following are equivalent.\n\\begin{enumerate}[label=\\textnormal{\\textcolor{black}{(}\\roman*\\textcolor{black}{)}}]\n\\item The condition $\\smash{\\mathrm{TCD}_p^*(K,N)}$ holds.\n\\item The condition $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ holds.\n\\item The condition $\\smash{\\mathrm{TCD}_p^e(K,N)}$ holds.\n\\item The condition $\\smash{\\mathrm{wTCD}_p^e(K,N)}$ holds.\n\\item\\label{joh} For every timelike $p$-dualizable pair $(\\mu_0,\\mu_1) = (\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)^2$, there exists some $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that for every $t\\in[0,1]$, the measure $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}$ is $\\meas$-absolutely continuous, and its density $\\rho_t$ with respect to $\\meas$ obeys, for every $N'\\geq N$, the inequality\n\\begin{align*}\n\\rho_t(\\gamma_t)^{-1\/N'}\\geq \\sigma_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_0(\\gamma_0)^{-1\/N'} + \\sigma_{K,N'}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_1(\\gamma_1)^{-1\/N'}\n\\end{align*}\nfor ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$.\n\\end{enumerate}\n\\end{theorem}\n\nEven for the timelike nonbranching case and apart from the smooth case \\cite{mccann2020,mondinosuhr2018}, the equivalence of $\\smash{\\mathrm{TCD}_p^e(K,N)}$ and $\\smash{\\mathrm{wTCD}_p^e(K,N)}$ has not been proven in \\cite{cavalletti2020}. In fact, we establish both inherent equivalences of $\\smash{\\mathrm{TCD}_p^*(K,N)}$ and $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ as well as $\\smash{\\mathrm{TCD}_p^e(K,N)}$ and $\\smash{\\mathrm{wTCD}_p^e(K,N)}$ only through their respective one-to-one correspondence with \\ref{joh}. The latter point constitutes a \\emph{pathwise} form of our reduced time\\-like curvature-dimension conditions. Contrary to the entropic case, such an interpretation being \\emph{a priori} meaningful --- even when  timelike branching occurs --- thus makes our notions especially geometrically valuable. Moreover, point  \\ref{joh} provides density bounds for interpolating timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics which, under $\\smash{\\mathrm{wTCD}_p^e(K,N)}$, were obtained in \\cite{braun2022} using variational methods. (In \\cite{braun2022}, we have not assumed timelike essential nonbranching, though.) Compare with \\autoref{Sub:Good TCD} and \\autoref{Sub:Good} below, where we use similar methods to construct ``good geodesics''.\n\nAn analogous pathwise form of the condition $\\smash{\\mathrm{TCD}_p(K,N)}$ and $\\smash{\\mathrm{wTCD}_p(K,N)}$ --- however, with no direct link to \\cite{cavalletti2020} --- is established in \\autoref{Th:Equiv TCD with geo}.\n\n$\\blacktriangleright$ Lastly, we introduce the synthetic notions of\n\\begin{itemize}\n\\item the reduced timelike measure-contraction property $\\smash{\\mathrm{TMCP}_p^*(K,N)}$, and\n\\item the timelike measure-contraction property $\\smash{\\mathrm{TMCP}_p(K,N)}$.\n\\end{itemize}\nThese are versions of the respective timelike curvature-dimension conditions where convexity of the R\u00e9nyi entropy is not required along the chronological $\\smash{\\ell_p}$-optimal transport between two $\\meas$-absolutely continuous measures, but rather from an $\\meas$-absolutely continuous distribution to a Dirac mass. For metric measure spaces, the corresponding notion was introduced in \\cite{ohta2007, sturm2006b}; the entropic Lorentzian version called $\\smash{\\mathrm{TMCP}_p^e(K,N)}$ is due to \\cite{cavalletti2020}.\n\nIn \\autoref{Pr:TMCP to TMCP*} and  \\autoref{Pr:TMCP to TCD} we prove the following correspondences, were $K^* := K(N-1)\/N$ is defined as for \\eqref{RKB}:\n\\begin{align*}\n\\mathrm{TMCP}_p(K,N) &\\ \\Longrightarrow \\ \\mathrm{TMCP}_p^*(K,N),\\\\\n\\mathrm{TMCP}_p^*(K,N) &\\ \\Longrightarrow \\ \\mathrm{TMCP}_p(K^*,N),\\\\\n\\mathrm{wTCD}_p^*(K,N) &\\ \\Longrightarrow\\ \\mathrm{TMCP}_p^*(K,N),\\\\\n\\mathrm{wTCD}_p(K,N) &\\ \\Longrightarrow\\ \\mathrm{TMCP}_p(K,N).\n\\end{align*}\nMoreover, we extend the above mentioned geometric inequalities, cf.~\\autoref{Re:Geom inequ TMCP}, we prove the full stability of both properties under the notion of convergence from \\autoref{Def:Convergence}, cf.~\\autoref{Th:Stability TMCP}, and we show the equivalence of $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ with $\\smash{\\mathrm{TMCP}_p^e(K,N)}$ under timelike $p$-essential nonbranching, cf.~\\autoref{Th:Equivalence TMCP* and TMCPe}. We also provide pathwise characterizations of $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ and $\\smash{\\mathrm{TMCP}_p(K,N)}$, see \\autoref{Th:Equivalence TMCP* and TMCPe} and \\autoref{Th:Equivalence TMCP}. \n\nLastly, in \\autoref{Th:Uniqueness couplings} and \\autoref{Th:Uniqueness geodesics} we establish  the uniqueness of chronological $\\smash{\\ell_p}$-optimal couplings as well as  timelike proper-time parametrized $\\smash{\\ell_p}$-optimal geodesics for large classes of initial measures, again under timelike essential nonbranching. (In fact, this was our main motivation to introduce the named weaker timelike measure-contraction properties.) This extends the smooth outcomes from  \\cite{mccann2020}. Our reasoning follows \\cite{cavalletti2020}; in fact, we extend the corresponding uniqueness results obtained therein under $\\smash{\\mathrm{TMCP}_p^e(K,N)}$ from the timelike nonbranching to the timelike essentially nonbranching case, cf.~\\autoref{Re:From TNB to TENB}. However, let us stress that  even in the timelike nonbranching case, our uniqueness results do \\emph{not} follow by combining \\autoref{Zums} and  \\cite{cavalletti2020}. Rather, \\autoref{Th:Uniqueness couplings} and \\autoref{Th:Uniqueness geodesics} are \\emph{prerequisites} for the proof of \\autoref{Zums}, cf.~\\autoref{Re:Uniq}.\n\n\\subsection*{Outlook} Finally, we outline various possible directions the notions and results of our work could lead to in the future.\n\nFrom the viewpoint of general relativity, it is natural to ask whether singularity theorems \u00e0 la Hawking and Penrose, such as those from \\cite{cavalletti2020} under $\\smash{\\mathrm{TMCP}_p^e(K,N)}$ and the assumption of timelike nonbranching, hold in our setting as well, possibly under the weaker hypothesis of timelike essential nonbranching. It is also worth seeking for nonsmooth analogues of the above mentioned singularity theorems \u00e0 la Gannon and Lee. Note that the universal cover of a spacetime appears in regular formulations of these, cf.~\\cite[Thm.~2.6]{schinnerl2021} for the most recent result in this direction for $\\smash{\\Cont^1}$-metrics. To this aim, in \\autoref{Th:TCD universal cover} we make a first step by showing that $\\smash{\\mathrm{TCD}_p^*(K,N)}$ lifts to the universal cover of a measured Lorentzian space, in case of its existence. Coupled with an extension of the Lorentzian localization technique, initiated in \\cite{cavalletti2020}, to the timelike essentially nonbranching case, this will be the goal of our forthcoming work \\cite{braun+}.\n\nSecond, our work is a further contribution to our long-term goal already outlined in \\cite{braun2022}, namely to obtain a  splitting theorem for measured Lorentzian spaces. In the corresponding result for metric measure spaces \\cite{gigli2013}, R\u00e9nyi's entropy plays a crucial role, and we expect a similar importance of it in our relativistic  framework. As mentioned, what is still missing to date in order to get started is a Lorentzian notion of infinitesimal Hilbertianity. \n\nAnalogously to \\cite{brue2020, mondino2019}, such a splitting theorem could then provide insights into the structure of $\\mathrm{TCD}$ spaces, e.g.~giving that they look like Minkowski space $\\meas$-a.e.\n\nWe  learned from \\eqref{RKB} that $\\smash{\\mathrm{TCD}_p^*(K,N)}$ is a priori weaker than the condition $\\smash{\\mathrm{TCD}_p(K,N)}$. Recently, for essentially nonbranching metric measure spaces, Ca\\-valletti and Milman \\cite{cavallettimilman2021} proved the \\emph{equivalence} of $\\CD^*(K,N)$ \\cite{bacher2010} and $\\CD(K,N)$ \\cite{sturm2006b} by using sophisticated localization tools. The analogous step from $\\smash{\\mathrm{TCD}_p^*(K,N)}$ to $\\smash{\\mathrm{TCD}_p(K,N)}$, possibly under timelike $p$-essential nonbranching, together with \\autoref{Zums} would close the circle in \\eqref{RKB} and show the equivalence of all (finite-dimensional) $\\mathrm{TCD}$ conditions known thus far.\n\nA further possible direction is to compare different notions of convergence for measured Lorentzian spaces, different from \\autoref{Def:Convergence}, and to establish their equivalence in  analogy to \\cite{giglimondino2015} for metric measure spaces. Our proof of \\autoref{Th:Asddd} suggests a Lorentzian adaptation of Sturm's $\\boldsymbol{{\\ensuremath{\\mathsf{D}}}}$-convergence \\cite{sturm2006b} to be potentially interesting, see \\autoref{Re:A remarkable} (and also \\autoref{RE:POTEN}). In general, finding a good definition of convergence of Lorentzian spaces is a central question in relativity theory; for instane, what it should provide is a Lorentzian precompactness theorem \u00e0 la Gromov \\cite{gromov}, which is missing thus far. In particular, it remains open whether any of the classes of spaces obeying $\\smash{\\mathrm{TCD}^e_p(K,N)}$, $\\smash{\\mathrm{TCD}_p^*(K,N)}$, or $\\smash{\\mathrm{TCD}_p(K,N)}$ (or suitable subsets thereof) are compact in any reasonable topology.\n\nLastly, leaned on a similar question asked in \\cite{cavalletti2022} and the results from \\cite{akdemir}, it is natural to expect that $\\smash{\\mathrm{TCD}_p(K,N)}$ and $\\smash{\\mathrm{TCD}_q(K,N)}$ are equivalent if the space $(\\mms,\\met,\\ll,\\leq,\\uptau)$ is timelike $p$- and $q$-essentially nonbranching, $p,q\\in (0,1)$. Con\\-cep\\-tionally, this would be  important in Lorentzian geometry where, unlike the metric setting \\cite{lott2009,sturm2006a,sturm2006b}, there is a priori no exponent $p\\in (0,1)$ which seems most natural to ground the theory of synthetic timelike Ricci curvature bounds on.\n\n\n\n\n\n\n\\subsection*{Organization} In \\autoref{Ch:prel}, we collect all necessary background material about measured Lorentzian spaces and optimal transport theory on these. \\autoref{Ch:TCD conditions} is devoted to a detailed investigation of our timelike curvature-dimension conditions. \\autoref{Ch:TMCP} studies the timelike measure-contraction property. \\autoref{App:Smooth} contains the smooth proof of \\autoref{Th:Smothh}, and \\autoref{App:B} collects some technical results evolving around \\autoref{Def:Termin}.\n\n\\addtocontents{toc}{\\protect\\setcounter{tocdepth}{2}}\n\n\\section{Preliminaries}\\label{Ch:prel}\n\n\\subsection{Volume distortion coefficients} We start by introducing the \\emph{volume distortion coefficients} that  make up our main definitions, and by listing their basic  properties. We refer to \\cite{bacher2010, erbar2015,sturm2006b} for details.\n\nGiven $\\kappa\\in\\R$, $t\\in[0,1]$, and $\\vartheta\\geq 0$,  we define the quantities\n\\begin{align*}\n\\mathfrak{S}_\\kappa(\\vartheta) &:= \\begin{cases}\n\\dfrac{\\sin(\\!\\sqrt{\\kappa}\\,\\vartheta)}{\\sqrt{\\kappa}\\,\\vartheta} & \\text{if }\\kappa >0,\\\\\n1 & \\text{if } \\kappa =0,\\\\\n\\dfrac{\\sinh(\\!\\sqrt{-\\kappa}\\,\\vartheta)}{\\sqrt{-\\kappa}\\,\\vartheta} & \\text{otherwise},\n\\end{cases}\\\\\n\\sigma_\\kappa^{(t)}(\\vartheta) &:= \\begin{cases} \\infty & \\textnormal{if }\\kappa\\,\\vartheta^2 \\geq \\pi^2,\\\\\nt\\,\\dfrac{\\mathfrak{S}_\\kappa(t\\,\\vartheta)}{\\mathfrak{S}_\\kappa(\\vartheta)} & \\textnormal{otherwise},\n\\end{cases}\n\\end{align*}\n\n\\begin{definition}\\label{Def:Dist coeff} For $K\\in\\R$ and $N\\in[1,\\infty)$, slightly abusing notation we set\n\\begin{align*}\n\\sigma_{K,N}^{(t)}(\\vartheta) &:= \\sigma_{K\/N}^{(t)}(\\vartheta),\\\\\n\\tau_{K,N}^{(t)}(\\vartheta) &:= t^{1\/N}\\,\\sigma_{K,N-1}^{(t)}(\\vartheta)^{1-1\/N}.\n\\end{align*}\n\\end{definition}\n\nNote that for every $t\\in(0,1)$ and every $\\vartheta > 0$, $\\smash{\\sigma_{K,N}^{(t)}(\\vartheta)}$ is continuous in $(K,N)\\in \\R\\times[1,\\infty)$, nondecreasing in $K$, and nonincreasing in $N$ \\cite[Rem.~2.2]{bacher2010}. Analogous claims apply to the quantity $\\smash{\\tau_{K,N}^{(t)}(\\vartheta)}$ \\cite[p.~138]{sturm2006b}. Furthermore, for every $t\\in [0,1]$, every $\\vartheta\\geq 0$, and every $\\kappa\\in (-\\infty,\\pi^2\/\\vartheta^2)$,\n\\begin{align}\\label{Eq:Distortion coeff property}\n\\sigma_\\kappa^{(t)}(\\vartheta) = \\sigma_{\\kappa\\vartheta^2}^{(t)}(1).\n\\end{align}\n\n\\begin{remark}\\label{Re:Lower bounds sigma} We recall the following elementary inequality from \\cite[Rem.~2.3]{cavalletti2017}: for every $K\\in\\R$, every $N\\in[1,\\infty)$, every $t\\in[0,1]$, and every $\\vartheta\\geq 0$,\n\\begin{align*}\n\\sigma_{K,N}^{(t)}(\\vartheta) \\geq t\\,{\\ensuremath{\\mathrm{e}}}^{-(1-t)\\vartheta\\sqrt{K^-\/N}}.\n\\end{align*}\n\\end{remark}\n\nLastly, in view of \\autoref{Th:Equivalence TCD* and TCDe} and \\autoref{Th:Equivalence TMCP* and TMCPe}, given any $t\\in[0,1]$ let us define  ${\\ensuremath{\\mathrm{G}}}_t\\colon \\R^2 \\times (-\\infty,\\pi^2)\\to (-\\infty,\\infty]$ and ${\\ensuremath{\\mathrm{H}}}_t\\colon \\R\\times (-\\infty,\\pi^2)\\to (-\\infty,\\infty]$ by\n\\begin{align}\\label{Eq:GtHt}\n\\begin{split}\n{\\ensuremath{\\mathrm{G}}}_t(x,y,\\kappa) &:= \\log\\!\\big[\\sigma_\\kappa^{(1-t)}(1)\\,{\\ensuremath{\\mathrm{e}}}^x + \\sigma_\\kappa^{(t)}(1)\\,{\\ensuremath{\\mathrm{e}}}^y\\big]\\\\\n{\\ensuremath{\\mathrm{H}}}_t(x,\\kappa) &:= \\log\\!\\big[\\sigma_\\kappa^{(1-t)}(1)\\,{\\ensuremath{\\mathrm{e}}}^x\\big] = \\log\\sigma_\\kappa^{(1-t)}(1) + x.\n\\end{split}\n\\end{align}\nThen the functions ${\\ensuremath{\\mathrm{G}}}_t$ and ${\\ensuremath{\\mathrm{H}}}_t$ are jointly convex \\cite[Lem.~2.11]{erbar2015}.\n\n\\subsection{Nonsmooth Lorentzian spaces}\\label{Sub:Lorentzian nonsmooth} We continue with a concise digression on the theory of nonsmooth Lorentzian (pre-length, length, and geodesic) spaces. We refer to \\cite{cavalletti2020, kunzinger2018} for  details, proofs, and examples about the corresponding notions.\n\n\\subsubsection{Chronology and causality} Let us fix a preorder $\\leq$  and a transitive relation $\\ll$, contained in $\\leq$, on $\\mms$. The triple $(\\mms,\\ll,\\leq)$ is called \\emph{causal space} \\cite[Def. 2.1]{kunzinger2018}. We say that $x,y\\in\\mms$ are \\emph{timelike} or \\emph{causally} related if $x\\ll y$ or $x\\leq y$, respectively. We write $x<y$ whenever $x\\leq y$ yet $x\\neq y$. Define the sets $\\smash{\\mms_\\ll^2}$ and $\\smash{\\mms_\\leq^2}$ to consist of all pairs $(x,y)\\in\\mms^2$ with $x\\ll y$  and $x\\leq y$, respectively.\n\n\n\\begin{definition} We term the triple $(\\mms,\\ll,\\leq)$ \n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item \\emph{causally closed} if the relation $\\leq$ is closed, i.e.~$\\smash{\\mms_\\leq^2}$ is closed in $\\mms^2$, and\n\\item \\emph{locally causally closed} if every point in $\\mms$ has some open neighborhood $U\\subset\\mms$ such that $\\smash{\\bar{U}^2\\cap\\mms_\\leq^2}$ is relatively  closed in $\\smash{\\bar{U}^2}$.\n\\end{enumerate}\n\\end{definition}\n\nGiven any  $A\\subset\\mms$, we  define  the \\emph{chro\\-no\\-logical future} $I^+(A)\\subset\\mms$ and the \\emph{causal future} $J^+(A)\\subset\\mms$ of $A$ \\cite[Def.~2.3]{kunzinger2018} through\n\\begin{align*}\nI^+(A) &:= \\{y\\in\\mms : x\\ll y\\textnormal{ for some }x\\in A\\},\\\\\nJ^+(A) &:= \\{y\\in\\mms : x\\leq y\\textnormal{ for some }x\\in A\\}.\n\\end{align*}\nAnalogously, we define the \\emph{chronological past} $I^-(A)$ and the \\emph{causal past} $J^-(A)$ of $A$. Employing a slight abuse of notation, given  $x\\in\\mms$ we write $\\smash{I^\\pm(x) := I^\\pm(\\{x\\})}$ and $\\smash{J^\\pm(x) := J^\\pm(\\{x\\})}$. For a Borel measure $\\mu$ on $\\mms$, we shall write $\\smash{I^\\pm(\\mu) := I^\\pm(\\supp\\mu)}$ and $\\smash{J^\\pm(\\mu)} := \\smash{J^\\pm(\\supp\\mu)}$. For all these objects, set $I(A,B) := I^+(A) \\cap I^-(B)$, and define $I(x,y)$, $I(\\mu,\\nu)$, $J(A,B)$, $J(x,y)$, and $J(\\mu,\\nu)$ in an analogous way. \n\n\\subsubsection{Lorentzian pre-length spaces}  A function $\\uptau\\colon\\mms^2 \\to [0,\\infty]$ is called \\emph{time separation function} \\cite[Def.~2.8]{kunzinger2018} if it is lower semicontinuous, and for every $x,y,z\\in\\mms$ it satisfies the properties\n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item $\\uptau(x,y) = 0$ if $x\\not\\leq y$,\n\\item $\\uptau(x,y)>0$ if and only if $x\\ll y$, and\n\\item if $x\\leq y\\leq z$ we have the \\emph{reverse triangle inequality}\n\\begin{align}\\label{Eq:Reverse tau}\n\\uptau(x,z) \\geq \\uptau(x,y) + \\uptau(y,z).\n\\end{align}\n\\end{enumerate}\nThe existence of such a $\\uptau$ implies that $\\ll$ is an \\emph{open} relation \\cite[Prop.~2.13]{kunzinger2018}; in particular, the set $\\smash{I^\\pm(A)}$ is open for every $A\\subset\\mms$ \\cite[Lem.~2.12]{kunzinger2018}.\n\n\\begin{definition}\\label{Def:LLSSP} A \\emph{Lorentzian pre-length space} is a quintuple $(\\mms,\\met,\\ll,\\leq,\\uptau)$ which consists of a causal space $(\\mms,\\ll,\\leq)$ endowed with a proper metric $\\met$ and a time separation function $\\uptau$ as introduced above.\n\\end{definition}\n\n\n\\subsubsection{Length of curves}\\label{Sub:Length curves} Let $(\\mms,\\met,\\ll,\\leq,\\uptau)$ be a given Lorentzian pre-length space. A \\emph{path} designates a  map $\\gamma\\colon [a,b]\\to\\mms$, where $a,b\\in\\R$ with $a<b$; if $\\gamma$ is continuous, we call $\\gamma$ a \\emph{curve}. Such a curve $\\gamma$ is \\emph{\\textnormal{(}future-directed\\textnormal{)} timelike} or \\emph{\\textnormal{(}future-directed\\textnormal{)} causal} if it is $\\met$-Lipschitz continuous [sic], and $\\gamma_s \\ll \\gamma_t$ or $\\gamma_s \\leq \\gamma_t$ for every $s,t\\in[a,b]$ with $s< t$, respectively.\nIt is  \\emph{null} if it is causal and $\\gamma_a \\not\\ll\\gamma_b$. Similar notions evidently make sense for  \\emph{past-directed} curves and their \\emph{causal character} (that is, their property of being chronological, causal, or null). Unless explicitly stated otherwise, any curve of a specified causal character is assumed future-directed.\n\nThe \\emph{length} with respect to $\\uptau$ of a path $\\gamma\\colon [a,b]\\to \\mms$ is \n\\begin{align}\\label{Eq:Length tau def}\n\\mathrm{Len}_\\uptau(\\gamma) := \\inf\\{\\uptau(\\gamma_{t_0},\\gamma_{t_1}) + \\dots + \\uptau(\\gamma_{t_{n-1}},\\gamma_{t_n})\\}, \n\\end{align}\nwhere the infimum is taken over all integers $n\\in\\N$ and all times $t_0,\\dots,t_n \\in[a,b]$ with $t_0=a$, $t_n=b$, and $t_i < t_{i+1}$ for every $i\\in\\{0,\\dots,n-1\\}$ \\cite[Def.~2.24]{kunzinger2018}. It is additive \\cite[Lem.~2.25]{kunzinger2018}, and reparametrizations do neither change causal characters \\cite[Lem.~2.27]{kunzinger2018} nor the $\\uptau$-length itself \\cite[Lem.~2.28]{kunzinger2018}.\n\n\\subsubsection{Geodesics}\\label{Sub:GEO} A causal curve $\\gamma\\colon[0,1]\\to\\mms$ is called \\emph{geodesic} (or \\emph{maximal}) provided $\\mathrm{Len}_\\uptau(\\gamma) = \\uptau(\\gamma_0,\\gamma_1)$ \\cite[Def.~2.33]{kunzinger2018}. The space of all such curves is denoted $\\Geo(\\mms)$, and its subset of timelike curves is written $\\mathrm{TGeo}(\\mms)$.\n\n\\begin{definition}\\label{Def:Geod base space} We call $(\\mms,\\met,\\ll,\\leq,\\uptau)$ \\emph{geodesic} if for every $x,y\\in\\mms$ with $x<y$, there exists a geodesic $\\gamma\\in\\Geo(\\mms)$ starting at $x$ and ending at $y$.\n\\end{definition}\n\nEvery \\emph{timelike} geodesic has a weak parametrization \\cite[Def.~3.31]{kunzinger2018} by $\\uptau$-arclength provided  $\\uptau$ is continuous as well as $\\uptau(x,x) = 0$ for every $x\\in\\mms$ \\cite[Cor.~3.35]{kunzinger2018}. More precisely, given $\\gamma\\in\\mathrm{TGeo}(\\mms)$ define $\\psi_\\gamma\\in \\Cont([0,1];[0,1])$ by\n\\begin{align*}\n\\psi_\\gamma(t) := \\frac{\\mathrm{Len}_\\uptau(\\gamma\\big\\vert_{[0,t]})}{\\uptau(\\gamma_0,\\gamma_1)} = \\frac{\\uptau(\\gamma_0,\\gamma_t)}{\\uptau(\\gamma_0,\\gamma_1)}.\n\\end{align*}\nConsider the \\emph{reparametrization map} ${\\ensuremath{\\mathsf{r}}}\\colon \\mathrm{TGeo}(\\mms) \\to \\Cont([0,1];\\mms)$ given by\n\\begin{align*}\n{\\ensuremath{\\mathsf{r}}}(\\gamma)_t := \\gamma_{\\psi_\\gamma^{-1}(t)};\n\\end{align*}\nsee also \\autoref{App:B}. Then for every $\\gamma\\in\\mathrm{TGeo}(\\mms)$, the curve $\\eta := {\\ensuremath{\\mathsf{r}}}(\\gamma)$ obeys\n\\begin{align}\\label{Eq:Proper time par}\n\\uptau(\\eta_s,\\eta_t) = (t-s)\\,\\uptau(\\eta_0,\\eta_1)>0\n\\end{align}\nfor every $s,t\\in[0,1]$ with $s<t$. Note that elements of $\\smash{\\mathrm{TGeo}^\\uptau(\\mms) := {\\ensuremath{\\mathsf{r}}}(\\mathrm{TGeo}(\\mms))}$ might not be $\\met$-Lipschitz continuous any more \\cite[p.~424]{kunzinger2018}. On the other hand, as shown in \\autoref{Le:Cty reparametrization}, ${\\ensuremath{\\mathsf{r}}}$ is con\\-tinuous with respect to the uniform topology, which will be important for our axiomatization of $\\smash{\\ell_p}$-geodesics and the ``transference'' of compactness properties in $\\mathrm{TGeo}(\\mms)$ to $\\smash{\\mathrm{TGeo}^\\uptau(\\mms)}$, see \\autoref{Sub:Geodesics} below. In order to not interrupt the exposition, we have deferred corresponding definitions and results to \\autoref{App:B}.\n\n\\subsubsection{Global hyperbolicity} Following \\cite[Sec.~1.1]{cavalletti2020}, we term $(\\mms,\\met,\\ll,\\leq,\\uptau)$  \\emph{non-to\\-tally imprisoning} if for every compact $C\\subset \\mms$ there exists a constant  $c>0$ such that the $\\met$-arclength of any causal curve contained in $C$ is no larger than $c$.\n\n\\begin{definition} The space $(\\mms,\\met,\\ll,\\leq,\\uptau)$ is \n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item  \\emph{globally hyperbolic} if it is non-totally imprisoning and  the causal diamond $J(x,y)$ is compact for every $x,y\\in\\mms$, and\n\\item  \\emph{${\\ensuremath{\\mathscr{K}}}$-globally hyperbolic} if it is non-totally imprisoning and the causal diamond $J(C_0,C_1)$ is compact for all compact $C_0,C_1\\subset\\mms$.\n\\end{enumerate}\n\\end{definition}\n\nWe list some properties of (${\\ensuremath{\\mathscr{K}}}$-)globally hyperbolic Lorentzian pre-length spaces which will lead to our standing   \\autoref{Ass:ASS} below as well as useful consequences thereof. If $(\\mms,\\met,\\ll,\\leq,\\uptau)$ is locally causally closed, globally hyperbolic, and $I^\\pm(x) \\neq \\emptyset$ for every $x\\in \\mms$, then ${\\ensuremath{\\mathscr{K}}}$-global hyperbolicity holds \\cite[Lem.~1.5]{cavalletti2020}. On the other hand, every locally causally closed, ${\\ensuremath{\\mathscr{K}}}$-globally hyperbolic Lorentzian geodesic space is in fact causally closed \\cite[Lem.~1.6]{cavalletti2020}. By \\cite[Def.~3.25, Thm.~3.26]{kunzinger2018}, global hyperbolicity implies the nonsmooth analogue of the \\emph{strong causality condition} for smooth Lorentzian spacetimes, cf.~e.g.~\\cite[Def.~14.11]{oneill1983}. On every globally hyperbolic Lorentzian length space (see  \\cite[Def.~3.22]{kunzinger2018} for the corresponding  definition, we will not really need it), the time separation function $\\uptau$ is finite and continuous \\cite[Thm.~3.28]{kunzinger2018}; in particular, we have $\\uptau(x,x) = 0$ for every $x\\in\\mms$. Furthermore, every such space is geodesic by the nonsmooth analogue of Avez--Seifert's  theorem \\cite[Thm.~3.30]{kunzinger2018}. In \\cite{burtscher2021}, a singular analogue of Geroch's characterization \\cite{geroch1970} of global hyperbolicity via Cauchy time functions is proven.\n\n\n\\subsection{Optimal transport on Lorentzian spaces}\\label{Sec:OT Lorentzian} Next, let us briefly review the   theory of optimal transport on the class of spaces introduced above \\cite{cavalletti2020}. We refer to \\cite{eckstein2017,kellsuhr2020, mccann2020, mondinosuhr2018, suhr2018} for prior developments in the smooth case.\n\n\\subsubsection{Basic probabilistic notation} Let ${\\ensuremath{\\mathscr{P}}}(\\mms)$ denote the set of all Borel probability measures on $\\mms$. Let ${\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$ and ${\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)$ be its subsets consisting of all compactly supported and $\\meas$-absolutely continuous elements, respectively; set $\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)} := \\smash{{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)\\cap{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$. Given any $\\mu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$, by $\\mu_\\perp$ we mean the $\\meas$-singular part in the corresponding Lebesgue decomposition of $\\mu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$.\n\nFor a Borel map $F\\colon \\mms\\to \\mms'$ into a metric space $(\\mms',\\met')$, given any $\\mu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$ the measure $F_\\push\\mu \\in{\\ensuremath{\\mathscr{P}}}(\\mms')$ designates the usual \\emph{push-forward}  of $\\mu$ under $F$, defined by the formula $F_\\push\\mu[B] := \\smash{\\mu\\big[F^{-1}(B)\\big]}$ for every Borel set $B\\subset\\mms'$.\n\nGiven  $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$, let $\\Pi(\\mu,\\nu)$ denote the set of all \\emph{couplings} of $\\mu$ and $\\nu$, i.e.~all $\\pi\\in{\\ensuremath{\\mathscr{P}}}(\\mms^2)$ with $\\pi[\\, \\cdot\\times\\mms] = \\mu$ and $\\pi[\\mms\\times\\cdot\\, ] = \\nu$.\n\nWith $\\Cont([0,1];\\mms)$ denoting the set of all curves $\\gamma\\colon [0,1]\\to\\mms$, endowed with the uniform topology, for $t\\in [0,1]$ the so-called \\emph{evaluation map} $\\eval_t\\colon \\Cont([0,1];\\mms) \\to \\mms$ is defined through $\\eval_t(\\gamma) := \\gamma_t$.\n\n\\subsubsection{Chronological and causal couplings} Let $(\\mms,\\met,\\ll,\\leq,\\uptau)$ be a Lorentzian pre-length space. We define the (possibly empty) set $\\Pi_\\ll(\\mu,\\nu)$ of all \\emph{chronological couplings} of two probability measures $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$ to consist of all $\\pi\\in\\Pi(\\mu,\\nu)$ such that $\\smash{\\pi[\\mms_\\ll^2]=1}$. The set $\\Pi_\\leq(\\mu,\\nu)$ of all \\emph{causal couplings} of $\\mu$ and $\\nu$ is defined analogously. Under causal closedness, clearly $\\pi\\in\\Pi_\\leq(\\mu,\\nu)$ if and only if $\\pi\\in\\Pi(\\mu,\\nu)$ as well as $\\smash{\\supp\\pi\\subset\\mms_\\leq^2}$; an analogous statement  holds for the locally causally closed case if $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$.\n\nA chronological or causal coupling of $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$  intuitively describes a way of transporting an infinitesimal mass portion ${\\ensuremath{\\mathrm{d}}}\\mu(x)$ to an infinitesimal mass portion ${\\ensuremath{\\mathrm{d}}}\\nu(y)$ as to guarantee $x\\ll y$ or $x\\leq y$, respectively.\n\nWe call $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$ \\emph{chronologically related} if $\\Pi_\\ll(\\mu,\\nu)\\neq\\emptyset$.\n\n\\subsubsection{The $\\smash{\\ell_p}$-optimal transport problem} Given an exponent $p\\in (0,1]$ and following the expositions in  \\cite{cavalletti2020,mccann2020} we will adopt the conventions\n\\begin{align*}\n\\sup\\emptyset &:= -\\infty,\\\\\n(-\\infty)^p &:= (-\\infty)^{1\/p} := -\\infty,\\\\\n\\infty -\\infty &:=-\\infty. \\textcolor{white}{a^{1\/p}}\n\\end{align*}\n\nLet the total transport cost function $\\ell_p\\colon{\\ensuremath{\\mathscr{P}}}(\\mms)^2\\to [0,\\infty]\\cup\\{-\\infty\\}$ be given by\n\\begin{align*}\n\\ell_p(\\mu,\\nu) &:=  \\sup\\{\\Vert \\uptau\\Vert_{\\Ell^p(\\mms^2,\\pi)} : \\pi \\in \\Pi_\\leq(\\mu,\\nu)\\}\\\\\n&\\textcolor{white}{:}= \\sup\\{\\Vert l\\Vert_{\\Ell^p(\\mms^2,\\pi)} : \\pi\\in\\Pi(\\mu,\\nu)\\},\n\\end{align*}\nwhere $l\\colon \\mms^2 \\to [0,\\infty]\\cup\\{-\\infty\\}$ is defined through\n\\begin{align*}\nl^p(x,y)  := \\begin{cases} \\uptau^p(x,y) & \\textnormal{if }x\\leq y,\\\\\n-\\infty & \\textnormal{otherwise}.\n\\end{cases}\n\\end{align*}\n\n\\begin{remark}\nThe sets of maximizers for both suprema defining $\\smash{\\ell_p(\\mu,\\nu)}$ coincide, including the case $\\smash{\\Pi_\\leq(\\mu,\\nu)=\\emptyset}$. One advantage of the  formulation via $l$ is that under (local) causal closedness and global hyperbolicity, $l^p$ is upper semicontinuous. In this case, customary optimal transport techniques \\cite{ambrosiogigli2013, villani2009} are applicable to study the second problem, which in turn yields corresponding results for the first  \\cite[Rem.~2.2]{cavalletti2020}. Note that the preimages $l^{-1}([0,\\infty))$ and $l^{-1}((0,\\infty))$ encode causality and chronology of points in $\\mms^2$, respectively.\n\\end{remark}\n\nA coupling $\\pi\\in\\Pi(\\mu,\\nu)$ of $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$ is called \\emph{$\\ell_p$-optimal} if $\\pi\\in\\Pi_\\leq(\\mu,\\nu)$ and \n\\begin{align*}\n\\ell_p(\\mu,\\nu) = \\Vert\\uptau\\Vert_{\\Ell^p(\\mms^2,\\pi)} = \\Vert l \\Vert_{\\Ell^p(\\mms^2,\\pi)}.\n\\end{align*}\nFor existence of such couplings, for our work it will suffice to keep the following in mind. If $(\\mms,\\met,\\ll,\\leq,\\uptau)$ is locally causally closed and globally hyperbolic, and if $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$ with $\\Pi_\\leq(\\mu,\\nu)\\neq\\emptyset$, existence of an $\\smash{\\ell_p}$-optimal coupling $\\pi$ of $\\mu$ and $\\nu$ holds, and $\\ell_p(\\mu,\\nu) <\\infty$ \\cite[Prop.~2.3]{cavalletti2020}.\n\nA key property of $\\smash{\\ell_p}$ is the \\emph{reverse triangle inequality} \\cite[Prop. 2.5]{cavalletti2020} somewhat  reminiscent of \\eqref{Eq:Reverse tau} and of $l$ satisfying the reverse triangle inequality for \\emph{every}, i.e.~not necessarily causally related, $x,y,z\\in\\mms$: for \\emph{every} $\\mu,\\nu,\\sigma\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$,\n\\begin{align}\\label{Eq:Reverse triangle lp}\n\\ell_p(\\mu,\\sigma) \\geq \\ell_p(\\mu,\\nu) + \\ell_p(\\nu,\\sigma).\n\\end{align}\n\n\\subsubsection{Timelike $p$-dualizability}\\label{Sub:Timelike dual} The concept of so-called \\textit{\\textnormal{(}strong\\textnormal{)} timelike $p$-dualiza\\-bi\\-lity}, where $p\\in(0,1]$, of pairs $(\\mu,\\nu)\\in{\\ensuremath{\\mathscr{P}}}(\\mms)^2$ originates in \\cite{cavalletti2020} and generalizes the notion of \\emph{$p$-separation} introduced in \\cite[Def.~4.1]{mccann2020}. Pairs satisfying this condition admit good duality properties \\cite[Prop.~2.19, Prop.~2.21, Thm.~2.26]{cavalletti2020}, which leads to a characterization of $\\smash{\\ell_p}$-geodesics (see \\autoref{Sub:Geodesics} below) in the smooth case \\cite[Thm.~4.3, Thm.~5.8]{mccann2020}, cf.~\\autoref{Th:Equiv}.\n\nIn view of the following \\autoref{Def:TL DUAL}, cf.~\\cite[Def.~2.18, Def.~2.27]{cavalletti2020}, we refer to \\cite[Def.~2.6]{cavalletti2020} for the definition of \\emph{cyclical monotonicity} of a set in $\\smash{\\mms_\\leq^2}$ with respect to $l^p$, see also \\cite[Def.~5.1]{villani2009}. It will not be relevant in our work.\n\nGiven $a,b\\colon\\mms\\to\\R$ we define  $a\\oplus b\\colon\\mms^2\\to\\R$ by $(a\\oplus b)(x,y) := a(x) + b(y)$.\n\n\\begin{definition}\\label{Def:TL DUAL} Given any $p\\in (0,1]$, a pair $(\\mu,\\nu)\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$ is called \n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item  \\emph{timelike $p$-dua\\-lizable} by $\\pi\\in \\Pi_\\ll(\\mu,\\nu)$ if  $\\pi$ is $\\smash{\\ell_p}$-optimal, and there exist Borel functions $a,b\\colon\\mms\\to\\R$ such that $a\\oplus b\\in\\Ell^1(\\mms^2,\\mu\\otimes\\nu)$ as well as $l^p\\leq a\\oplus b$ on $\\supp\\mu\\times\\supp\\nu$,\n\\item  \\emph{strongly timelike $p$-dualizable} by $\\pi\\in\\Pi_\\ll(\\mu,\\nu)$ if the pair $(\\mu,\\nu)$ is timelike $p$-dualizable by $\\pi$, and there is an $l^p$-cyclically monotone Borel set $\\Gamma\\subset \\smash{\\mms_\\ll^2\\cap(\\supp\\mu_0\\times\\supp\\mu_1)}$ such that every given  $\\sigma\\in \\Pi_\\leq(\\mu_0,\\mu_1)$ is $\\ell_p$-optimal if and only if $\\sigma[\\Gamma]=1$ \\textnormal{(}in particular, $\\sigma$ is chronological\\textnormal{)}, and\n\\item \\emph{timelike $p$-dualizable} if $(\\mu,\\nu)$ is timelike $p$-dualizable by some coupling $\\pi\\in\\Pi_\\ll(\\mu,\\nu)$; analogously for strong timelike $p$-dualizability.\n\\end{enumerate}\n\nMoreover, any $\\pi$ as in the above items is called \\emph{timelike $p$-dualizing}. \n\\end{definition}\n\n\n\\begin{remark}\\label{Re:Strong timelike} If $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$ then the pair $(\\mu,\\nu)$ is timelike $p$-dualizable if and only if there is an $\\ell_p$-optimal coupling $\\smash{\\pi\\in\\Pi_\\leq(\\mu,\\nu)}$ which is concentrated on $\\smash{\\mms_\\ll^2}$.\n\nIf $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$ on a causally closed, globally hyperbolic Lorentzian geodesic space $(\\mms,\\met,\\ll,\\leq,\\uptau)$ with $\\smash{\\supp\\mu\\times\\supp\\nu\\subset\\mms_\\ll^2}$, then  $(\\mu,\\nu)$ is automatically strongly timelike $p$-dualizable, $p\\in (0,1]$ \\cite[Cor.~2.29]{cavalletti2020}, see also \\cite[Lem.~4.4, Thm.~7.1]{mccann2020}. \n\\end{remark}\n\n\\subsubsection{Geodesics revisited}\\label{Sub:Geodesics} The definitions we consider need the notion of an \\emph{$\\smash{\\ell_p}$-geodesic}, $p\\in (0,1]$, to be made precise. We propose \\autoref{Def:Lorentzian geodesic} as Lorentzian version of Wasserstein geodesics in metric spaces. Precise technical results will be deferred to \\autoref{App:B}.\n\nSolely in this subsection, for simplicity we assume compactness of $\\mms$; the results derived here and in \\autoref{App:B} will mostly be used by replacing $\\mms$ by a causal diamond of two compact sets, which is compact under ${\\ensuremath{\\mathscr{K}}}$-global hyperbolicity.\n\nRecall the continuous reparametrization map $\\smash{{\\ensuremath{\\mathsf{r}}}\\colon \\mathrm{TGeo}(\\mms) \\to \\mathrm{TGeo}^\\uptau(\\mms)}$ and the   definition $\\mathrm{TGeo}^\\uptau(\\mms) := {\\ensuremath{\\mathsf{r}}}(\\mathrm{TGeo}(\\mms))$ from \\autoref{Sub:GEO}. If $(\\mms,\\met,\\ll,\\leq,\\uptau)$ is causally closed, the set $\\Geo(\\mms)$ is Polish \\cite[Prop.~3.17]{kunzinger2018}. Since $\\mathrm{TGeo}(\\mms)$ is open, $\\mathrm{TGeo}^\\uptau(\\mms)$ is a Suslin set, hence universally measurable. \n\nGiven any $\\mu_0,\\mu_1\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$, we define\n\\begin{align*}\n\\OptGeo_{\\ell_p}(\\mu_0,\\mu_1) &:= \\{{\\ensuremath{\\boldsymbol{\\pi}}}\\in{\\ensuremath{\\mathscr{P}}}(\\Geo(\\mms)) : (\\eval_0,\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\Pi_\\leq(\\mu_0,\\mu_1)\\\\\n&\\qquad\\qquad \\textnormal{ is }\\ell_p\\textnormal{-optimal}\\},\\\\\n\\mathrm{OptTGeo}_{\\ell_p}(\\mu_0,\\mu_1) &:= \\{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\OptGeo_{\\ell_p}(\\mu_0,\\mu_1) : (\\eval_0,\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}[\\mms_\\ll^2]=1\\},\\\\\n\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1) &:= {\\ensuremath{\\mathsf{r}}}_\\push\\mathrm{OptTGeo}_{\\ell_p}(\\mu_0,\\mu_1).\n\\end{align*}\nWe  say that a measure ${\\ensuremath{\\boldsymbol{\\pi}}}\\in{\\ensuremath{\\mathscr{P}}}(\\Cont([0,1];\\mms))$ \\emph{represents} a curve $(\\mu_t)_{t\\in[0,1]}$ in ${\\ensuremath{\\mathscr{P}}}(\\mms)$ provided $\\mu_t=(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}$ for every $t\\in[0,1]$.\n\n\\begin{definition}\\label{Def:Lorentzian geodesic} We term a weakly continuous path $(\\mu_t)_{t\\in[0,1]}$ in ${\\ensuremath{\\mathscr{P}}}(\\mms)$\n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item \\emph{causal $\\ell_p$-geodesic} if it is represented by some $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\OptGeo_{\\ell_p}(\\mu_0,\\mu_1)}$,\n\\item \\emph{timelike $\\ell_p$-geodesic} if it is represented by some $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}(\\mu_0,\\mu_1)}$, and\n\\item\\label{Spans} \\emph{timelike proper-time parametrized $\\ell_p$-geodesic} if it is represented by some $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$.\n\\end{enumerate}\n\nA measure ${\\ensuremath{\\boldsymbol{\\pi}}}$ as in the last item will be called \\emph{$\\smash{\\ell_p}$-optimal geodesic plan}.\n\\end{definition}\n\n\\begin{remark}\\label{Re:Lip} If $(\\mms,\\met,\\ll,\\leq,\\uptau)$ is ${\\ensuremath{\\mathscr{K}}}$-globally hyperbolic, every causal or timelike $\\smash{\\ell_p}$-geodesic connecting compactly supported $\\mu_0$ and $\\mu_1$ is Lipschitz continuous with respect to the $q$-Wasserstein metric $W_q$ induced by $(\\mms,\\met)$ for every $q\\in[1,\\infty]$.\n\\end{remark}\n\nBy construction, we have ${\\ensuremath{\\boldsymbol{\\pi}}}[\\mathrm{TGeo}(\\mms)]=1$ for every $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}(\\mu_0,\\mu_1)}$. Consequently, ${\\ensuremath{\\boldsymbol{\\pi}}}[\\mathrm{TGeo}^\\uptau(\\mms)]=1$ for every $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$, and therefore ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\Cont([0,1];\\mms)$ obeys $\\uptau(\\gamma_s,\\gamma_t) = (t-s)\\,\\uptau(\\gamma_0,\\gamma_1)>0$ for every $s,t\\in [0,1]$ with $s<t$ by \\eqref{Eq:Proper time par}. Since $\\smash{{\\ensuremath{\\mathsf{r}}}(\\gamma)_r = \\gamma_r}$ for every $\\gamma\\in\\mathrm{TGeo}(\\mms)$ and every $r\\in\\{0,1\\}$ by definition of ${\\ensuremath{\\mathsf{r}}}$,  $(\\eval_0,\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\Pi_\\ll(\\mu_0,\\mu_1)$ is $\\smash{\\ell_p}$-optimal for $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in \\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$. A standard argument using the reverse triangle inequality \\eqref{Eq:Reverse tau} thus yields\n\\begin{align*}\n\\ell_p(\\mu_s,\\mu_t) = (t-s)\\,\\ell_p(\\mu_0,\\mu_1) > 0\n\\end{align*}\nfor every timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ and every $s,t\\in[0,1]$ with $s<t$. Hence every such curve $(\\mu_t)_{t\\in[0,1]}$ is an $\\smash{\\ell_p}$-geodesic in the sense of \\cite[Def.~2.13]{cavalletti2020} and \\cite[Def.~1.1]{mccann2020}, but the converse is unclear to us.\n\n\n\\subsubsection{R\u00e9nyi entropy}\\label{Subsub:Renyi} From now on, we draw our attention to \\emph{measured} Lorentzian pre-length spaces, defined as follows.\n\n\\begin{definition} We call a sextuple $(\\mms,\\met,\\meas,\\ll,\\leq,\\uptau)$ a \\emph{measured Lorentzian pre-length space} if $(\\mms,\\met,\\ll,\\leq,\\uptau)$ is a Lorentzian pre-length space according to \\autoref{Def:LLSSP}, and $\\meas$ is a fully supported Radon measure on $\\mms$ as in \\autoref{Ch:Intro}. Accordingly, \\emph{measured Lorentzian length spaces} and \\emph{measured Lorentzian geodesic spaces} are defined.\n\\end{definition}\n\nAll over the sequel, we use the convenient abbreviation\n\\begin{align}\\label{Eq:X}\n{\\ensuremath{\\mathscr{X}}} := (\\mms,\\met,\\meas,\\ll,\\leq,\\uptau).\n\\end{align}\nAll Lorentzian notions from \\autoref{Sub:Lorentzian nonsmooth} phrased for ${\\ensuremath{\\mathscr{X}}}$ will be evidently understood with respect to the Lorentzian structure $(\\mms,\\met,\\ll,\\leq,\\uptau)$.\n\nGiven any $N\\in [1,\\infty)$,  define the \\emph{$N$-R\u00e9nyi entropy} ${\\ensuremath{\\mathscr{S}}}_N\\colon{\\ensuremath{\\mathscr{P}}}(\\mms)\\to [-\\infty,0]$ with respect to $\\meas$ through\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_N(\\mu) := -\\int_\\mms \\rho^{-1\/N}\\d\\mu = - \\int_\\mms \\rho^{1-1\/N}\\d\\meas\n\\end{align*}\nsubject to the decomposition $\\mu = \\rho\\,\\meas + \\mu_\\perp$. This functional will play a central role in our work. Observe  that ${\\ensuremath{\\mathscr{S}}}_1(\\mu) = -\\meas[\\supp(\\rho\\,\\meas)]$. More generally, for every $N\\in[1,\\infty)$, by Jensen's inequality we have $\\smash{{\\ensuremath{\\mathscr{S}}}_N(\\mu) \\geq -\\meas[\\supp\\mu]^{1\/N}}$ for every $\\mu\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$. Moreover, ${\\ensuremath{\\mathscr{S}}}_N$ is weakly lower semicontinuous on ${\\ensuremath{\\mathscr{P}}}(C)$ for every closed $C\\subset\\mms$ with $\\meas[C]< \\infty$, see \\cite[Lem.~1.1]{sturm2006b}.\n\n\\subsubsection{Boltzmann entropy} Another functional on ${\\ensuremath{\\mathscr{P}}}(\\mms)$ we occasionally consider, cf.~\\autoref{Le:Stu}, \\autoref{Sub:Equiv TCDs}, and \\autoref{Sec:Equiv TMCP's}, and whose relativistic consequences have been studied deeply in \\cite{cavalletti2020, mccann2020, mondinosuhr2018}, is the \\emph{Boltzmann entropy} $\\Ent_\\meas\\colon {\\ensuremath{\\mathscr{P}}}(\\mms)\\to [-\\infty,\\infty]$ with respect to $\\meas$. It is defined by\n\\begin{align*}\n\\Ent_\\meas(\\mu) := \\begin{cases}\\displaystyle\\int_\\mms \\rho\\log\\rho\\d\\meas & \\textnormal{if }\\mu = \\rho\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas),\\, (\\rho\\log\\rho)^+\\in\\Ell^1(\\mms,\\meas),\\\\\n\\infty & \\textnormal{otherwise}.\n\\end{cases}\n\\end{align*}\nLet $\\Dom(\\Ent_\\meas)$ denote the usual finiteness domain of $\\Ent_\\meas$. \n\nFor every $\\mu\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$, we have $\\Ent_\\meas(\\mu)\\geq -\\log\\meas[\\supp\\mu]$. Moreover, as for the R\u00e9nyi entropy $\\Ent_\\meas$ is weakly lower semicontinuous on ${\\ensuremath{\\mathscr{P}}}(C)$ for every closed $C\\subset\\mms$ with $\\meas[C]<\\infty$ \\cite[Lem.~4.1]{sturm2006a}; in fact, for every $\\mu\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$,\n\\begin{align}\\label{Eq:Ent SN limit}\n\\Ent_\\meas(\\mu) = \\lim_{N\\to\\infty}\\big[N - N\\,{\\ensuremath{\\mathscr{S}}}_N(\\mu)\\big] = \\sup\\{N-N\\,{\\ensuremath{\\mathscr{S}}}_N(\\mu) : N\\in [1,\\infty)\\}.\n\\end{align}\n\n\\subsection{Timelike essential nonbranching} Finally, we introduce the following Lo\\-rentz\\-ian version of essential  nonbranching for metric measure spaces \\cite{rajala2014}, termed \\emph{timelike essential nonbranching}.\n\n\\begin{definition}\\label{Def:Essentially nonbranching} A subset $G\\subset\\mathrm{TGeo}^\\uptau(\\mms)$ is termed \\emph{timelike nonbranching} if the evaluation map $\\eval_t\\colon G\\to \\mms$ is injective for every $t\\in(0,1)$.\n\nWe call $(\\mms,\\met,\\ll,\\leq,\\uptau)$ \\emph{timelike $p$-essentially  nonbranching}, $p\\in (0,1]$, if every plan ${\\ensuremath{\\boldsymbol{\\pi}}}$ representing some timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic is concentrated on a time\\-like nonbranching subset of $\\mathrm{TGeo}^\\uptau(\\mms)$.\n\\end{definition}\n\n\\begin{remark}\\label{Re:reversed} A curve $(\\gamma_t)_{t\\in[0,1]}$ is an element of the set $\\mathrm{TGeo}^\\uptau(\\mms)$ with respect to $(\\mms,\\met,\\ll,\\leq,\\uptau)$ if and only if the curve $(\\gamma_{1-t})_{t\\in[0,1]}$ is a corresponding timelike proper-time parametrized geodesic with respect to the reversed causal structure of $\\smash{(\\mms,\\met,\\ll,\\leq,\\uptau)}$, cf.~\\cite[Sec.~1.1]{cavalletti2020} for the definition of the latter. Hence, $G:= \\mathrm{TGeo}^\\uptau(\\mms)$ is timelike nonbranching if $(\\mms,\\met,\\ll,\\leq,\\uptau)$ is timelike nonbranching as in \\cite[Def.~1.10]{cavalletti2020}. In particular, timelike nonbranching Lorentzian pre-length spaces are timelike $p$-essentially nonbranching for every $p\\in (0,1]$. \n\\end{remark}\n\nImportant results derived later which assume $p$-timelike essential nonbranching are the local-to-global property, \\autoref{Th:Local to global}, uniqueness of $\\smash{\\ell_p}$-optimal couplings and timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics, \\autoref{Th:Uniqueness couplings} and \\autoref{Th:Uniqueness geodesics}, and the equivalence of some of our curvature-dimension conditions and measure-contraction properties to their entropic versions set up by Cavalletti and Mondino, \\autoref{Th:Equivalence TCD* and TCDe} and \\autoref{Th:Equivalence TMCP* and TMCPe}.\n\nThe following result will be useful at many occasions. It follows directly from the definition of $p$-timelike essential nonbranching. See \\cite[Lem.~2.6]{bacher2010}, \\cite[Lem.~3.11]{erbar2015}, and \\cite[Lem.~2.11]{sturm2006a} for related results for metric measure spaces.\n\n\\begin{lemma}\\label{Le:Mutually singular} Assume timelike $p$-essential nonbranching for some $p\\in (0,1]$. Let ${\\ensuremath{\\boldsymbol{\\pi}}}$ be an $\\smash{\\ell_p}$-optimal geodesic plan which can be decomposed as $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}} = \\lambda_1\\,{\\ensuremath{\\boldsymbol{\\pi}}}^1 + \\dots +\\lambda_n\\,{\\ensuremath{\\boldsymbol{\\pi}}}^n}$ for some $n\\in\\N$, $\\lambda_1,\\dots,\\lambda_n \\in (0,1)$, and $\\smash{\\ell_p}$-optimal geodesic plans $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}^1,\\dots,{\\ensuremath{\\boldsymbol{\\pi}}}^n}$. If the measures $\\smash{(\\eval_0)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^1,\\dots,(\\eval_0)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^n}$ or $\\smash{(\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^1,\\dots,(\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^n}$ are mutually singular, so are $\\smash{(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^1,\\dots,(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^n}$ for every $t\\in (0,1)$.\n\\end{lemma}\n\n\\section{Timelike curvature-dimension condition}\\label{Ch:TCD conditions}\n\n\\begin{assumption}\\label{Ass:ASS}\nFrom now on, in the notation of \\eqref{Eq:X}, unless stated otherwise we assume ${\\ensuremath{\\mathscr{X}}}$ to be a causally closed, ${\\ensuremath{\\mathscr{K}}}$-globally hyperbolic, measured Lorentzian geodesic space. \n\\end{assumption}\n\nOf course, our main notions from  \\autoref{Def:TCD*}, \\autoref{Def:TCD}, and \\autoref{Def:TMCP} a priori make sense for general Lorentzian pre-length spaces. We  always mention explicitly if other results hold in greater generality than under \\autoref{Ass:ASS}.\n\n\\subsection{Definition and basic properties} \\autoref{Def:TCD*} is the Lorentzian counterpart of the \\emph{reduced} curvature-dimension condition \\cite[Def.~2.3]{bacher2010}. On the other hand, \\autoref{Def:TCD} is the natural Lorentzian adaptation of the $\\CD(K,N)$ condition for metric measure spaces from \\cite[Def.~1.3]{sturm2006b}. Both conditions deserve their names from their compatibility with smooth Lorentzian case, cf.~\\autoref{Th:Equiv}. \n\n\\begin{definition}\\label{Def:TCD*} Let $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$.\n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item We say that ${\\ensuremath{\\mathscr{X}}}$ satisfies the \\emph{reduced timelike curva\\-ture-di\\-men\\-sion condition} $\\smash{\\mathrm{TCD}_p^*(K,N)}$ if for every timelike $p$-dualizable pair $(\\mu_0,\\mu_1) = (\\rho_0\\,\\meas,\\rho_1\\,\\meas) \\in \\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)^2}$, there is a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in [0,1]}$ connecting $\\mu_0$ to $\\mu_1$ as well as a timelike $p$-dua\\-lizing coupling $\\pi\\in \\smash{\\Pi_\\ll(\\mu_0,\\mu_1)}$ such that for every $t\\in [0,1]$ and every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t) &\\leq -\\int_{\\mms^2} \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N'} \\d\\pi(x^0,x^1) \\\\\n&\\qquad\\qquad -\\int_{\\mms^2} \\sigma_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\,\\rho_1(x^1)^{-1\/N'} \\d\\pi(x^0,x^1).\n\\end{align*}\n\\item If the previous statement holds only for every strongly timelike $p$-dualizable  $\\smash{(\\mu_0,\\mu_1)=(\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)^2}$, we say that ${\\ensuremath{\\mathscr{X}}}$ obeys the \\emph{weak reduced timelike curvature-dimension condition} $\\smash{\\mathrm{wTCD}_p^*(K,N)}$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{definition}\\label{Def:TCD} Let $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$. \n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item We say that ${\\ensuremath{\\mathscr{X}}}$ obeys the \\emph{timelike curva\\-ture-di\\-men\\-sion condition} $\\smash{\\mathrm{TCD}_p(K,N)}$ if for every timelike $p$-dualizable pair $(\\mu_0,\\mu_1) = (\\rho_0\\,\\meas,\\rho_1\\,\\meas) \\in \\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)^2}$, there is a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in [0,1]}$  connecting $\\mu_0$ to $\\mu_1$ as well as a timelike $p$-dua\\-lizing coupling $\\pi\\in \\smash{\\Pi_\\ll(\\mu_0,\\mu_1)}$ such that for every $t\\in [0,1]$ and every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t) &\\leq -\\int_{\\mms^2} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N'} \\d\\pi(x^0,x^1) \\\\\n&\\qquad\\qquad -\\int_{\\mms^2} \\tau_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\,\\rho_1(x^1)^{-1\/N'} \\d\\pi(x^0,x^1).\n\\end{align*}\n\\item If the previous statement holds only for every strongly timelike $p$-dualizable $(\\mu_0,\\mu_1)=(\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)^2$, we say that ${\\ensuremath{\\mathscr{X}}}$ obeys the \\emph{weak timelike curvature-dimension condition} $\\smash{\\mathrm{wTCD}_p(K,N)}$.\n\\end{enumerate}\n\\end{definition}\n\n\n\n\\begin{remark} The conditions from  \\autoref{Def:TCD*}  and \\autoref{Def:TCD} are a priori formulated for compactly supported endpoints $\\mu_0$ and $\\mu_1$ only. In the timelike $p$-essentially nonbranching case, cf.~\\autoref{Def:Essentially nonbranching}, this does in fact imply the respective defining statements for more general $\\smash{\\mu_0,\\mu_1\\in {\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$ according to \\autoref{Th:Equiv TCD with geo} and \\autoref{Th:Equivalence TCD* and TCDe}.\n\\end{remark}\n\n\\begin{remark}\\label{Re:TCD 0} Note that $\\smash{\\mathrm{TCD}_p^*(0,N)}$ and $\\smash{\\mathrm{wTCD}_p^*(0,N)}$ coincide with $\\smash{\\mathrm{TCD}_p(0,N)}$ and $\\smash{\\mathrm{wTCD}_p(0,N)}$, respectively, for every $p\\in (0,1)$ and every $N\\in[1,\\infty)$, since\n\\begin{align*}\n\\sigma_{0,N'}^{(r)}(\\vartheta) = \\tau_{0,N'}^{(r)}(\\vartheta) = r\n\\end{align*}\nfor every $r\\in [0,1]$, every $N'\\in [1,\\infty)$, and every $\\vartheta\\geq 0$.\n\\end{remark}\n\nIn the remainder of this section, we examine elementary properties of the  conditions introduced above, and their respective relations. \\autoref{Pr:TCD and TCD*} as well as  \\autoref{Pr:Consistency TCD} hold on any measured Lorentzian pre-length space.\n\n\\begin{proposition}\\label{Pr:TCD and TCD*} The following  hold for every $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$; set $K^* := K(N-1)\/N$.\n\\begin{enumerate}[label=\\textnormal{(\\roman*)}]\n\\item The condition $\\smash{\\mathrm{TCD}_p(K,N)}$ implies $\\smash{\\mathrm{TCD}_p^*(K,N)}$.\n\\item The condition $\\smash{\\mathrm{wTCD}_p(K,N)}$ implies $\\smash{\\mathrm{wTCD}_p^*(K,N)}$.\n\\item The condition $\\smash{\\mathrm{TCD}_p^*(K,N)}$ implies $\\smash{\\mathrm{TCD}_p(K^*,N)}$.\n\\item The condition $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ implies $\\smash{\\mathrm{wTCD}_p(K^*,N)}$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof} The first two statements are a direct consequence of the inequality\n\\begin{align*}\n\\tau_{K,N'}^{(r)}(\\vartheta) \\geq \\sigma_{K,N'}^{(r)}(\\vartheta)\n\\end{align*}\nfor every $r\\in[0,1]$, every $N'\\geq N$, and every $\\vartheta\\geq 0$, cf.~the proof of \\cite[Prop.~2.5]{bacher2010}. The last two statements  follow similarly by noting that\n\\begin{align*}\n\\tau_{K^*,N'}^{(t)}(\\vartheta)  \\leq \\sigma_{K,N'}^{(t)}(\\vartheta).\\tag*{\\qedhere}\n\\end{align*}\n\\end{proof}\n\n\\begin{proposition}\\label{Pr:Consistency TCD} Assume $\\smash{\\mathrm{TCD}_p^*(K,N)}$ for $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$. Then the following statements hold.\n\\begin{enumerate}[label=\\textnormal{\\textcolor{black}{(}\\roman*\\textcolor{black}{)}}]\n\\item\\label{La:DieEINS} \\textnormal{\\textbf{Consistency.}} The condition $\\smash{\\mathrm{TCD}_p^*(K',N')}$ holds for every $K'\\leq K$ and every $N'\\geq N$.\n\\item\\label{La:DieZWEI} \\textnormal{\\textbf{Scaling.}} Given any $a,b,\\theta>0$, the rescaled measured Lorentzian pre-length space $(\\mms, a\\met, b\\meas,\\ll,\\leq, \\theta\\uptau)$ satisfies $\\smash{\\mathrm{TCD}_p^*(K\/\\theta^2,N)}$.\n\\end{enumerate}\n\nAnalogous statements are obeyed by the conditions $\\smash{\\mathrm{wTCD}_p^*(K,N)}$, $\\smash{\\mathrm{TCD}_p(K,N)}$, and $\\smash{\\mathrm{wTCD}_p(K,N)}$. \n\\end{proposition}\n\n\\begin{proof} Concerning  \\ref{La:DieEINS}, for all four conditions consistency in $K$ follows by nondecreasingness of $\\smash{\\sigma_{K,N}^{(r)}(\\vartheta)}$ and $\\smash{\\tau_{K,N}^{{(r)}}(\\vartheta)}$ in $K\\in\\R$ for fixed $r\\in[0,1]$, $N\\in [1,\\infty)$, and $\\vartheta\\geq 0$. Consistency in $N$ is clear  since the defining inequality, in either case, is asked to hold for every $N''\\geq N$, so is particularly satisfied for every  $N''\\geq N'$.\n\nItem \\ref{La:DieZWEI} follows from the scaling properties \n\\begin{align*}\n\\sigma_{K\/\\theta^2,N'}^{(r)}(\\theta\\,\\vartheta) &= \\sigma_{K,N'}^{(r)}(\\vartheta),\\\\\n\\tau_{K\/\\theta^2,N'}^{(r)}(\\theta\\,\\vartheta) &= \\tau_{K,N'}^{(r)}(\\vartheta).\\qedhere\n\\end{align*}\n\\end{proof}\n\nLastly, recall that the $N$-R\u00e9nyi entropy ${\\ensuremath{\\mathscr{S}}}_N(\\mu)$, $N\\in[1,\\infty)$, at $\\mu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$ only depends on the $\\meas$-absolutely continuous part of $\\mu$, which might be trivial. Along those timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics in \\autoref{Def:TCD*} or \\autoref{Def:TCD}, this does not happen; in fact, under mild additional hypotheses, these always consist of $\\meas$-absolutely continuous measures. This is addressed in \\autoref{Le:Stu} and will be useful at various occasions. (We  also use variants of it without explicit notice occasionally, e.g.~in \\autoref{Sub:Local global}.) Its proof uses \\eqref{Eq:Ent SN limit} and is analogous to the one of  \\cite[Prop.~1.6]{sturm2006b}, hence omitted. \n\n\\begin{lemma}\\label{Le:Stu} Fix $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$. Let $(\\mu_0,\\mu_1) = (\\rho_0\\,\\meas,\\mu_1 \\,\\meas)\\in({\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)\\cap \\Dom(\\Ent_\\meas))^2$,  let $\\smash{\\mu\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)}$, and let $\\smash{\\pi\\in\\Pi(\\mu_0,\\mu_1)}$ be a coupling of $\\mu_0$ and $\\mu_1$ such that for every $N'\\geq N$, \n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu)&\\leq -\\int_{\\mms^2} \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N'}\\d\\pi(x^0,x^1)\\\\\n&\\qquad\\qquad - \\int_{\\mms^2} \\sigma_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\,\\rho_1(x^1)^{-1\/N'}\\d\\pi(x^0,x^1).\n\\end{align*}\nThen $\\smash{\\mu = \\rho\\,\\meas\\in\\Dom(\\Ent_\\meas)}$ with\n\\begin{align*}\n\\Ent_\\meas(\\mu) \\leq (1-t)\\Ent_\\meas(\\mu_0) + t\\Ent_\\meas(\\mu_1) - \\frac{K}{2}\\,t(1-t)\\,\\big\\Vert \\uptau\\big\\Vert_{\\Ell^2(\\mms^2,\\pi)}^2.\n\\end{align*}\n\\end{lemma}\n\n\\begin{remark} Using \\autoref{Le:Villani lemma for geodesic}, the assumptions in \\autoref{Le:Stu} are satisfied for $\\mu := \\mu_t$, $t\\in[0,1]$, and $\\pi$ if these objects are coming from a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic as well as a timelike $p$-dualizing coupling, respectively, witnessing the inequality defining $\\smash{\\mathrm{TCD}_p^*(K,N)}$.\n\\end{remark}\n\n\\begin{corollary} For every $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$, the $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ condition implies  $\\smash{\\mathrm{wTCD}_p(K,\\infty)}$  in the sense of \\textnormal{\\cite[Def.~4.1]{braun2022}}.\n\\end{corollary}\n\n\n\\subsection{Geometric inequalities} Now we derive fundamental geometric inequalities from the  conditions introduced in \\autoref{Ch:TCD conditions}. We start with the more restrictive $\\smash{\\mathrm{wTCD}_p(K,N)}$-case which implies sharp versions of these facts, cf.~\\autoref{Cor:HD};   \\autoref{Sub:Versions reduced} provides nonsharp extensions for $\\smash{\\mathrm{wTCD}_p^*(K,N)}$-spaces. All proofs are  variations of standard arguments \\cite{sturm2006b}, see also \\cite{bacher2010,cavalletti2020}.\n\nSome of the geometric inequalities to follow hold in fact under weaker timelike measure-contraction properties, cf.~\\autoref{Pr:TMCP to TCD} and \\autoref{Re:Geom inequ TMCP}.\n\nMoreover, all results in this section also hold if ${\\ensuremath{\\mathscr{X}}}$ is a locally causally closed and globally hyperbolic measured Lorentzian geodesic space.\n\n\\subsubsection{Sharp timelike Brunn--Minkowski inequality}\n\nTo introduce the \\emph{timelike Brunn--Minkowski inequality} in the next \\autoref{Pr:Brunn-Minkowski}, given  $A_0,A_1\\subset \\mms$ and $t\\in [0,1]$,  define the set of timelike $t$-intermediate points between $A_0$ and $A_1$ as\n\\begin{align*}\nA_t := \\{\\gamma_t : \\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms),\\, \\gamma_0 \\in A_0,\\, \\gamma_1\\in A_1\\}.\n\\end{align*}\nFurthermore, define \n\\begin{align}\\label{Eq:THETA}\n\\Theta := \\begin{cases} \\sup\\uptau(A_0\\times A_1) & \\textnormal{if }K<0,\\\\\n\\inf\\uptau(A_0\\times A_1) & \\textnormal{otherwise}.\n\\end{cases}\n\\end{align}\n\n\\begin{proposition}\\label{Pr:Brunn-Minkowski}  Assume $\\smash{\\mathrm{wTCD}_p(K,N)}$ for some $p\\in (0,1)$, $K\\in\\R$, and $N\\in [1,\\infty)$. Let $A_0,A_1\\subset\\mms$ be relatively compact Borel sets with  $\\meas[A_0]\\,\\meas[A_1] >0$, and assume strong timelike $p$-dualizability of \n\\begin{align*}\n(\\mu_0,\\mu_1) := (\\meas[A_0]^{-1}\\,\\meas\\mres A_0,\\meas[A_1]^{-1}\\, \\meas\\mres A_1).\n\\end{align*}\nThen for every $t\\in [0,1]$ and every $N'\\geq N$, \n\\begin{align}\\label{Eq:BM}\n\\meas[A_t]^{1\/N'} \\geq \\tau_{K,N'}^{(1-t)}(\\Theta)\\,\\meas[A_0]^{1\/N'} + \\tau_{K,N'}^{(t)}(\\Theta)\\,\\meas[A_1]^{1\/N'}.\n\\end{align}\n\nAssuming $\\smash{\\mathrm{TCD}_p(K,N)}$ in place of $\\smash{\\mathrm{wTCD}_p(K,N)}$, the same conclusion holds if $(\\mu_0,\\mu_1)$ is merely timelike $p$-dualizable.\n\\end{proposition}\n\n\\begin{proof} Without loss of generality, we assume $N' > 1$ and $\\meas[A_t] < \\infty$. Let a time\\-like proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in [0,1]}$ connecting $\\mu_0$ to $\\mu_1$ and a timelike $p$-dualizing  $\\pi\\in\\smash{\\Pi_\\ll(\\mu_0,\\mu_1)}$ witness the semiconvexity inequality defining $\\smash{\\mathrm{wTCD}_p(K,N)}$. Note that $\\supp\\mu_t\\subset A_t$ and, by $\\smash{\\mathrm{wTCD}_p(K,N)}$, that $\\meas[A_t]>0$. Let $\\rho_t$ denote the density of the absolutely continuous part of $\\mu_t$ with respect to $\\meas$. Employing  Jensen's inequality,  and $\\smash{\\mathrm{wTCD}_p(K,N)}$ again we  obtain\n\\begin{align*}\n&\\tau_{K,N}^{(1-t)}(\\Theta)\\,\\meas[A_0]^{1\/N'} + \\tau_{K,N}^{(t)}(\\Theta)\\,\\meas[A_1]^{1\/N'}\\\\\n&\\qquad\\qquad\\leq \\int_{A_t}\\rho_t^{1-1\/N'}\\d\\meas \\leq \\meas[A_t]\\,\\Big[\\!\\fint_{A_t}\\rho_t\\d\\meas\\Big]^{1-1\/N'} \\leq \\meas[A_t]^{-1\/N'},\n\\end{align*}\nwhich is the desired claim.\n\nThe proof of the last statement is completely analogous.\n\\end{proof}\n\n\n\\begin{remark} Note that in the case $K\\geq 0$, \\eqref{Eq:BM} implies\n\\begin{align*}\n\\meas[A_t]^{1\/N'} \\geq (1-t)\\,\\meas[A_0]^{1\/N'} + t\\,\\meas[A_1]^{1\/N'}.\n\\end{align*}\n\\end{remark}\n\n\\begin{remark} Recall from \\autoref{Re:Strong timelike} that the strong timelike $p$-dualizability hy\\-pothesis for $(\\mu_0,\\mu_1)$ in  \\autoref{Pr:Brunn-Minkowski} is satisfied if $\\smash{A_0 \\times A_1 \\subset \\mms_\\ll^2}$ provided the space ${\\ensuremath{\\mathscr{X}}}$ is locally causally closed, globally hyperbolic, and geodesic.\n\\end{remark}\n\n\n\\subsubsection{Sharp timelike Bonnet--Myers inequality} Next, an immediate consequence of \\autoref{Pr:Brunn-Minkowski} is the subsequent \\emph{timelike Bonnet--Myers inequality}.\n\n\\begin{corollary}\\label{Cor:Bonnet-Myers}  Assume the $\\smash{\\mathrm{wTCD}_p(K,N)}$ condition  for some $p\\in(0,1)$, $K>0$, and $N\\in [1,\\infty)$. Then \n\\begin{align*}\n\\sup\\uptau(\\mms^2) \\leq \\pi\\sqrt{\\frac{N-1}{K}}.\n\\end{align*}\n\\end{corollary}\n\n\\begin{proof} Suppose to the contrapositive that for some $\\varepsilon > 0$, we have\n\\begin{align*}\n\\uptau(z_0,z_1) \\geq \\pi\\sqrt{\\frac{N-1}{K}}+4\\varepsilon\n\\end{align*}\nfor two given points $z_0,z_1\\in\\mms$. By continuity of $\\uptau$, we fix $\\delta > 0$ and $x_0,x_1\\in\\mms$ such that $\\smash{A_0 := {\\ensuremath{\\mathsf{B}}}^\\met(x_0,\\delta) \\subset I^+(z_0)}$, $\\smash{A_1 := {\\ensuremath{\\mathsf{B}}}^\\met(x_1,\\delta) \\subset I^-(z_1)}$, and \n\\begin{align*}\n\\inf\\uptau({\\ensuremath{\\mathsf{B}}}^\\met(x_0,\\delta) \\times {\\ensuremath{\\mathsf{B}}}^\\met(x_1,\\delta)) \\geq \\pi\\sqrt{\\frac{N-1}{K}}+\\varepsilon.\n\\end{align*}\nThis implies strong timelike $p$-dualizability of $\\smash{(\\meas[A_0]^{-1}\\,\\meas\\mres{A_0},\\meas[A_1]^{-1}\\,\\meas\\mres{A_1})}$ by \\autoref{Re:Strong timelike}. Hence \\autoref{Pr:Brunn-Minkowski} is applicable, and the inherent  inequality $\\smash{\\Theta \\geq \\pi\\sqrt{(N-1)\/K}+\\varepsilon}$ together with the definition of $\\smash{\\tau_{K,N}^{(r)}}$ gives $\\smash{\\meas[A_{1\/2}]=\\infty}$. On the other hand, the set $\\smash{A_{1\/2}}$ is relatively compact since $\\smash{A_{1\/2} \\subset J(z_0,z_1)}$ by global hy\\-per\\-bolicity, which makes the situation $\\meas[A_{1\/2}]=\\infty$ impossible.\n\\end{proof}\n\n\\begin{remark} \\autoref{Cor:Bonnet-Myers} implies  that, under its assumptions, the $\\uptau$-length of every causal curve in $\\mms$ is no larger than $\\smash{\\pi\\sqrt{(N-1)\/K}}$. Moreover, if $K>0$ and $N=1$ then $\\mms$ does not contain any chronologically related pair of points.\n\\end{remark}\n\n\\subsubsection{Sharp timelike Bishop--Gromov inequality}\\label{Sub:Sharp BG inequ} To prove  volume growth estimates \u00e0 la \\emph{Bishop--Gromov}, cf.~\\autoref{Th:BG} below, we  recall the notion of $\\uptau$-star-shaped sets from \\cite[Sec.~3.1]{cavalletti2020}; this is used to localize the volume of $\\uptau$-balls \n\\begin{align}\\label{Eq:TAU BALLS}\n{\\ensuremath{\\mathsf{B}}}^\\uptau(x,r) := \\{y\\in I^+(x) : \\uptau(x,y)< r\\}\\cup\\{x\\},\n\\end{align}\nwith $x\\in\\mms$ and $r>0$, which typically have infinite volume. \n\nA set $\\smash{E\\subset I^+(x)\\cup\\{x\\}}$  is termed \\emph{$\\uptau$-star-shaped} with respect to $x\\in\\mms$ if for every $\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$ with $\\gamma_0 = x$ and $\\gamma_1 \\in E$, we have $\\gamma_t\\in E$ for every $t\\in (0,1)$. For such $E$ and $x$, and given $r>0$, we define the quantities\n\\begin{align*}\n{\\ensuremath{\\mathrm{v}}}_r &:= \\meas\\big[\\bar{{\\ensuremath{\\mathsf{B}}}}^\\uptau(x,r)\\cap E\\big],\\\\\n{\\ensuremath{\\mathrm{s}}}_r &:= \\limsup_{\\delta \\to 0} \\delta^{-1}\\,\\meas\\big[(\\bar{{\\ensuremath{\\mathsf{B}}}}^\\uptau(x,r+\\delta)\\setminus {\\ensuremath{\\mathsf{B}}}^\\uptau(x,r))\\cap E\\big].\n\\end{align*}\nWhenever confusion is excluded, we avoid making notationally explicit the  dependency of ${\\ensuremath{\\mathrm{v}}}_r$ and ${\\ensuremath{\\mathrm{s}}}_r$ on $E$ and $x$. Lastly, we set\n\\begin{align}\\label{Eq:integral def}\n\\mathfrak{v}_{K,N}(r) := \\Big[\\!\\int_0^r \\mathfrak{s}_{K,N-1}(r)^{N-1}\\d r\\Big]^{1\/N}.\n\\end{align}\n\nIn the next result, we employ the convention $1\/0 := \\infty$.\n\n\\begin{theorem}\\label{Th:BG} Assume  $\\smash{\\mathrm{wTCD}_p(K,N)}$ for some $p\\in (0,1)$, $K\\in\\R$, and $N\\in (1,\\infty)$. Let $\\smash{E\\subset I^+(x)\\cup\\{x\\}}$ be a compact set which is $\\uptau$-star-shaped with respect to $x\\in\\mms$. Then for every $r,R > 0$ with $\\smash{r < R \\leq \\pi\\sqrt{(N-1)\/\\max\\{K,0\\}}}$,\n\\begin{align*}\n\\frac{{\\ensuremath{\\mathrm{s}}}_r}{{\\ensuremath{\\mathrm{s}}}_R} &\\geq \\Big[\\frac{\\mathfrak{s}_{K,N-1}(r)}{\\mathfrak{s}_{K,N-1}(R)}\\Big]^{N-1},\\\\\n\\frac{{\\ensuremath{\\mathrm{v}}}_r}{{\\ensuremath{\\mathrm{v}}}_R} &\\geq \\Big[\\frac{\\mathfrak{v}_{K,N}(r)}{\\mathfrak{v}_{K,N}(R)}\\Big]^N.\n\\end{align*}\n\\end{theorem}\n\n\\begin{proof} We assume $K>0$, the rest is argued similarly. Define $t := r\/R$,  let $\\delta > 0$, and let $\\varepsilon > 0$ be small enough such that $\\smash{A_0 \\times A_1 \\subset \\mms_\\ll^2}$, where $\\smash{A_0 := {\\ensuremath{\\mathsf{B}}}^\\uptau(x,\\varepsilon)\\cap E}$ and $A_1 := (\\bar{{\\ensuremath{\\mathsf{B}}}}^\\uptau(x,R+\\delta R) \\setminus {\\ensuremath{\\mathsf{B}}}^\\uptau(x,R))\\cap E$. Thus $(\\meas[A_0]^{-1}\\,\\meas\\mres{A_0},\\meas[A_1]^{-1}\\,\\meas\\mres{A_1})$ is strongly time\\-like $p$-dualizable according to  \\autoref{Re:Strong timelike}. Now we  observe  that $\\smash{A_t \\subset ({\\ensuremath{\\mathsf{B}}}^\\uptau(x,r+\\delta r)\\setminus {\\ensuremath{\\mathsf{B}}}^\\uptau(x,r))\\cap E}$. By the reverse triangle inequality for $\\uptau$, we get $\\Theta \\leq R+\\delta R$. Employing \\autoref{Pr:Brunn-Minkowski}, we therefore obtain\n\\begin{align*}\n&\\meas\\big[(\\bar{{\\ensuremath{\\mathsf{B}}}}^\\uptau(x,r+\\delta r)\\setminus {\\ensuremath{\\mathsf{B}}}^\\uptau(x,r))\\cap E\\big]^{1\/N}\\\\\n&\\qquad\\qquad \\geq \\tau_{K,N}^{(1-r\/R)}(R+\\delta R)\\,\\meas\\big[{\\ensuremath{\\mathsf{B}}}^\\uptau(x,\\varepsilon)\\cap E\\big]^{1\/N}\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\tau_{K,N}^{(r\/R)}(R+\\delta R)\\, \\meas\\big[(\\bar{{\\ensuremath{\\mathsf{B}}}}^\\uptau(x,R+\\delta R)\\setminus {\\ensuremath{\\mathsf{B}}}^\\uptau(x,R))\\cap E\\big]^{1\/N}\\\\\n&\\qquad\\qquad \\geq \\tau_{K,N}^{(r\/R)}(R+\\delta R)\\, \\meas\\big[(\\bar{{\\ensuremath{\\mathsf{B}}}}^\\uptau(x,R+\\delta R)\\setminus {\\ensuremath{\\mathsf{B}}}^\\uptau(x,R))\\cap E\\big]^{1\/N}.\n\\end{align*}\nWriting out the expression $\\smash{\\tau_{K,N}^{(r\/R)}(R+\\delta R)^N}$ and finally sending $\\delta\\to 0$ implies the claimed lower bound for ${\\ensuremath{\\mathrm{s}}}_r\/{\\ensuremath{\\mathrm{s}}}_R$.\n\nAs in the proof of \\cite[Thm.~2.3]{sturm2006b}, we argue that ${\\ensuremath{\\mathrm{v}}}$ is locally Lipschitz continuous on $(0,\\infty)$. Hence, it is  differentiable $\\smash{\\Leb^1}$-a.e.~on $(0,\\infty)$, and $\\smash{{\\ensuremath{\\mathrm{v}}}_r = \\int_0^r \\dot{{\\ensuremath{\\mathrm{v}}}}_s\\d s}$ for every $r > 0$. Therefore, the claimed lower bound on ${\\ensuremath{\\mathrm{v}}}_r\/{\\ensuremath{\\mathrm{v}}}_R$  follows from the previous part in com\\-bination with \\cite[Lem.~18.9]{villani2009}.\n\\end{proof}\n\n\\begin{remark} If $K=0$, the estimates from \\autoref{Th:BG}  become\n\\begin{align*}\n\\frac{{\\ensuremath{\\mathrm{s}}}_r}{{\\ensuremath{\\mathrm{s}}}_R} &\\geq \\Big[\\frac{r}{R}\\Big]^{N-1},\\\\\n\\frac{{\\ensuremath{\\mathrm{v}}}_r}{{\\ensuremath{\\mathrm{v}}}_R} &\\geq \\Big[\\frac{r}{R}\\Big]^N.\n\\end{align*}\nIn fact, these estimates hold for $N=1$ as well.\n\\end{remark}\n\nIncidentally, from \\autoref{Th:BG} we deduce the following natural and sharp upper bound on the $\\uptau$-Hausdorff dimension as introduced and studied in \\cite{mccann2021}. Our results improve \\cite[Thm.~5.2, Cor.~5.3]{mccann2021}. We refer to \\cite{mccann2021} for details about the notions presented in the subsequent \\autoref{Cor:HD}.\n\n\\begin{corollary}\\label{Cor:HD} Let $(\\mms,\\langle\\cdot,\\cdot\\rangle)$ be a continuous, globally hyperbolic, causally plain  spacetime of dimension $n\\in\\N$ \\textnormal{(}which is thus causally closed \\cite[Prop.~3.5]{kunzinger2018} and geodesic  \\cite[Thm.~3.30, Thm.~5.12]{kunzinger2018}\\textnormal{)}. Assume its induced Lorentzian geodesic space, with $\\meas$ being the Lorentzian volume measure induced by $\\langle\\cdot,\\cdot\\rangle$, to obey $\\smash{\\mathrm{wTCD}_p(K,N)}$ for some $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$. Then the geometric dimension $\\smash{\\dim^\\uptau\\mms}$ in the sense of \\textnormal{\\cite[Def.~3.1]{mccann2021}} satisfies\n\\begin{align*}\nn=\\dim^\\uptau\\mms \\leq N.\n\\end{align*}\n\\end{corollary}\n\n\\begin{remark}\\label{Re:HD2} As in \\cite[Thm.~5.2]{mccann2021}, which is proven under $\\smash{\\mathrm{wTCD}_p^e(K,N)}$, assuming $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ instead of $\\smash{\\mathrm{wTCD}_p(K,N)}$ in  \\autoref{Cor:HD} would only lead to the conclusion $n = \\dim^\\uptau\\mms \\leq N+1$ by \\autoref{Th:Reduced BG}.\n\\end{remark}\n\n\\subsubsection{Versions in the reduced case}\\label{Sub:Versions reduced} With evident modifications, the following nonsharp versions of \\autoref{Pr:Brunn-Minkowski}, \\autoref{Cor:Bonnet-Myers}, \\autoref{Th:BG}, and \\autoref{Cor:HD} are readily verified under the more general $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ condition.  The deduced inequalities are analogous to their counterparts drawn from the $\\smash{\\mathrm{wTCD}_p^e(K,N)}$ condition in  \\cite[Prop.~3.4, Prop.~3.5, Prop.~3.6]{cavalletti2020}.\n\nIn view of \\autoref{Th:Reduced BG} below, recall the definition of $\\mathfrak{v}_{K,N}$ from \\eqref{Eq:integral def}.\n\n\\begin{proposition}\\label{Propos} Assume $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ for some $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$. Let $A_0,A_1\\subset\\mms$ be relatively compact Borel sets which satisfy  $\\meas[A_0]\\,\\meas[A_1]>0$, and assume strong timelike $p$-dualizability of \n\\begin{align*}\n(\\mu_0,\\mu_1) := (\\meas[A_0]^{-1}\\,\\meas\\mres A_0, \\meas[A_1]^{-1}\\,\\meas\\mres A_1).\n\\end{align*}\nLet $\\Theta$ be as in \\eqref{Eq:THETA}. Then for every $t\\in [0,1]$ and every $N'\\geq N$,\n\\begin{align*}\n\\meas[A_t]^{1\/N'} \\geq \\sigma_{K,N'}^{(1-t)}(\\Theta)\\,\\meas[A_0]^{1\/N'} + \\sigma_{K,N'}^{(t)}(\\Theta)\\,\\meas[A_1]^{1\/N'}.\n\\end{align*}\n\nAssuming $\\smash{\\mathrm{TCD}_p^*(K,N)}$ in place of $\\smash{\\mathrm{wTCD}_p^*(K,N)}$, the same conclusion holds if $(\\mu_0,\\mu_1)$ is merely timelike $p$-dualizable.\n\\end{proposition}\n\n\\begin{corollary}\\label{Cor:Reduced BM} Assume the $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ condition for some $p\\in(0,1)$, $K>0$, and $N\\in [1,\\infty)$. Then \n\\begin{align*}\n\\sup\\uptau(\\mms^2) \\leq \\pi\\sqrt{\\frac{N}{K}}.\n\\end{align*}\n\\end{corollary}\n\n\\begin{theorem}\\label{Th:Reduced BG} Assume $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ for some $p\\in (0,1)$, $K\\in\\R$, and $N\\in (1,\\infty)$. Let $\\smash{E\\subset I^+(x)\\cup\\{x\\}}$ be a compact set which is $\\uptau$-star-shaped with respect to $x\\in\\mms$. Then for every $r,R > 0$ with $\\smash{r < R \\leq \\pi\\sqrt{N\/\\max\\{K,0\\}}}$,\n\\begin{align*}\n\\frac{{\\ensuremath{\\mathrm{s}}}_r}{{\\ensuremath{\\mathrm{s}}}_R} &\\geq \\Big[\\frac{\\mathfrak{s}_{K,N}(r)}{\\mathfrak{s}_{K,N}(R)}\\Big]^N,\\\\\n\\frac{{\\ensuremath{\\mathrm{v}}}_r}{{\\ensuremath{\\mathrm{v}}}_R} &\\geq \\Big[\\frac{\\mathfrak{v}_{K,N+1}(r)}{\\mathfrak{v}_{K,N+1}(R)}\\Big]^{N+1}.\n\\end{align*}\n\\end{theorem}\n\n\\subsection{Stability}\\label{Sec:Stability TCD} In this section, we prove a key property of our timelike curvature-dimension conditions, namely the weak stability of the notions from \\autoref{Def:TCD*} and \\autoref{Def:TCD}, cf.~\\autoref{Th:Stability TCD}. The relevant notion of convergence of measured Lorentzian pre-length spaces, see \\autoref{Def:Convergence}, is due to \\cite[Sec. 3.3]{cavalletti2020}.\n\n\\begin{definition}\\label{Def:Lor isometry} For Lorentzian pre-length spaces $\\smash{(\\mms^i,\\met^i,\\ll^i,\\leq^i,\\uptau^i)}$, $i\\in\\{0,1\\}$, we term a map $\\smash{\\iota\\colon \\mms^0\\to\\mms^1}$ a \\emph{Lorentzian isometric embedding} if $\\iota$ is a topolo\\-gical embedding such that for every $x,y\\in\\mms^0$,\n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item $x\\leq^0 y$ if and only if $\\iota(x) \\leq^1\\iota(y)$, and\n\\item $\\tau^1(\\iota(x),\\iota(y)) = \\tau^0(x,y)$.\n\\end{enumerate}\n\\end{definition}\n\nGiven any $k\\in\\N_\\infty := \\N\\cup\\{\\infty\\}$ let $(\\mms_k,\\met_k,\\meas_k,\\ll_k,\\leq_k,\\uptau_k)$ be a fixed measured Lorentzian pre-length space which, as in \\eqref{Eq:X}, we will abbreviate by ${\\ensuremath{\\mathscr{X}}}_k$.\n\n\\begin{definition}\\label{Def:Convergence} We define $({\\ensuremath{\\mathscr{X}}}_k)_{k\\in\\N}$ to converge to ${\\ensuremath{\\mathscr{X}}}_\\infty$ if the following holds.\n\\begin{enumerate}[label=\\textnormal{\\alph*\\textcolor{black}{.}}]\n\\item\\label{La:Blurr1reg} There exists a causally closed, ${\\ensuremath{\\mathscr{K}}}$-globally hyperbolic measured Lorentzian geodesic space $\\smash{(\\mms,\\met,\\ll,\\leq,\\uptau)}$ such that for every $k\\in\\N_\\infty$ there exists a Lorentzian isometric embedding $\\smash{\\iota_k\\colon \\mms_k\\to \\mms}$.\n\\item In terms of the maps $\\iota_k$ from \\ref{La:Blurr1reg}, the sequence $((\\iota_k)_\\push\\meas_k)_{k\\in\\N}$ converges to $(\\iota_\\infty)_\\push\\meas_\\infty$ in duality with $\\Cont_\\comp(\\mms)$, i.e.~for every $\\varphi\\in\\Cont_\\comp(\\mms)$ we have\n\\begin{align*}\n\\lim_{k\\to\\infty}\\int_\\mms \\varphi\\d(\\iota_k)_\\push\\meas_k = \\int_\\mms \\varphi\\d(\\iota_\\infty)_\\push\\meas_\\infty.\n\\end{align*}\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark} If $({\\ensuremath{\\mathscr{X}}}_k)_{k\\in\\N}$ converges to ${\\ensuremath{\\mathscr{X}}}_\\infty$ according to \\autoref{Def:Convergence},  ${\\ensuremath{\\mathscr{X}}}_k$ is  automatically causally closed, ${\\ensuremath{\\mathscr{K}}}$-globally hyperbolic, and geodesic for every $k\\in\\N_\\infty$.\n\\end{remark}\n\n\n\nThe following \\autoref{Le:CM lemma} is an implicit consequence from the proof of \\cite[Lem. 3.5]{cavalletti2020} and will be useful for the proof of \\autoref{Th:Stability TCD}. \n\n\\begin{lemma}\\label{Le:CM lemma} Assume that ${\\ensuremath{\\mathscr{X}}}$ is causally closed, globally hyperbolic, and geodesic. Let the pair $\\smash{(\\mu_0,\\mu_1)=(\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)^2}$ admit some $\\smash{\\ell_p}$-optimal coupling $\\smash{\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)}$, $p\\in (0,1]$. Then there exist sequences $(\\pi^n)_{n\\in\\N}$ in ${\\ensuremath{\\mathscr{P}}}(\\mms^2)$ and $(a_n)_{n\\in\\N}$ in $[1,\\infty)$ with the properties \n\\begin{enumerate}[label=\\textnormal{\\textcolor{black}{(}\\roman*\\textcolor{black}{)}}]\n\\item $(a_n)_{n\\in\\N}$ converges to $1$,\n\\item $\\pi^n[\\mms_\\ll^2]=1$ for every $n\\in\\N$,\n\\item $\\smash{\\pi^n = \\rho^n\\,\\meas^{\\otimes 2}\\in {\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms^2,\\meas^{\\otimes 2})}$ with $\\smash{\\rho^n\\in\\Ell^\\infty(\\mms^2,\\meas^{\\otimes 2})}$ for every $n\\in\\N$,\n\\item $(\\pi^n)_{n\\in\\N}$ converges weakly to $\\pi$, \n\\item\\label{DENS} the density $\\smash{\\rho_0^n}$ and $\\smash{\\rho_0^n}$ of the first and second marginal of $\\pi_n$ with respect to $\\meas$ is no larger than $a_n\\,\\rho_0$ and $a_n\\,\\rho_1$, respectively, for every $n\\in\\N$, and\n\\item $\\smash{\\rho_0^n\\to \\rho_0}$ and $\\smash{\\rho_1^n\\to\\rho_1}$ in $\\Ell^1(\\mms,\\meas)$ as $n\\to\\infty$.\n\\end{enumerate}\n\\end{lemma}\n\nFurthermore, in the notation of \\autoref{Le:CM lemma}, given any $K\\in\\R$, $N\\in[1,\\infty)$, and $t\\in [0,1]$, as well as a measure $\\pi\\in\\Pi(\\mu_0,\\mu_1)$ with marginals $\\mu_0 = \\rho_0\\,\\meas\\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$ and $\\mu_1 = \\rho_1\\,\\meas\\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$, for brevity we define\n\\begin{align}\\label{Eq:TDdef}\n\\begin{split}\n{\\ensuremath{\\mathscr{T}}}_{K,N}^{(t)}(\\pi) &:= -\\int_{\\mms^2} \\tau_{K,N}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N}\\d\\pi(x^0,x^1)\\\\\n&\\qquad\\qquad -\\int_{\\mms^2} \\tau_{K,N}^{(t)}(\\uptau(x^0,x^1))\\,\\rho_1(x^1)^{-1\/N}\\d\\pi(x^0,x^1).\n\\end{split}\n\\end{align}\n\nWe  note the following variant of \\cite[Lem.~3.2]{sturm2006b}. The latter proof easily carries over to the setting of \\autoref{Le:Const perturb}, where the marginal densities, instead of being assumed to be the same throughout the sequence as in \\cite{sturm2006b}, are allowed to be perturbed by constants vanishing in the limit.  Compare with \\autoref{Le:USC lemma}.\n\n\\begin{lemma}\\label{Le:Const perturb} Suppose  $\\meas\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$. Let $(a_k)_{k\\in\\N}$ be a sequence in $[1,\\infty)$ converging to $1$, and let $(b_k)_{k\\in\\N}$ be a sequence of nonnegative real numbers converging to $0$. Define $\\smash{\\nu_{k,0},\\nu_{k,1}\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$, $k\\in\\N$, by\n\\begin{align*}\n\\nu_{k,0} &:= (a_k+b_k)^{-1}\\,(a_k\\,\\mu_0 + b_k\\,\\meas) = (a_k+b_k)^{-1}\\,(a_k\\,\\rho_0 + b_k)\\,\\meas,\\\\\n\\nu_{k,1} &:= (a_k+b_k)^{-1}\\,(a_k\\,\\mu_1 + b_k\\,\\meas) = (a_k+b_k)^{-1}\\,(a_k\\,\\rho_1 + b_k)\\,\\meas.\n\\end{align*}\nThen for every sequence $(\\pi_k)_{k\\in\\N}$ of  measures $\\pi_k\\in \\Pi(\\nu_{k,0},\\nu_{k,1})$ which converges weakly to $\\pi\\in\\Pi(\\mu_0,\\mu_1)$,  every $K\\in\\R$, every $N\\in[1,\\infty)$, and every $t\\in[0,1]$, \n\\begin{align*}\n\\limsup_{k\\to\\infty}{\\ensuremath{\\mathscr{T}}}_{K,N}^{(t)}(\\pi_k) \\leq {\\ensuremath{\\mathscr{T}}}_{K,N}^{(t)}(\\pi).\n\\end{align*}\n\\end{lemma}\n\n\\begin{remark} Of course, the assertion from \\autoref{Le:Const perturb} remains valid if $\\smash{\\tau_{K,N}^{(1-t)}}$ and $\\smash{\\tau_{K,N}^{(t)}}$ are replaced by $\\smash{\\sigma_{K,N}^{(1-t)}}$ and $\\smash{\\sigma_{K,N}^{(t)}}$ in \\eqref{Eq:TDdef}, respectively.\n\\end{remark}\n\nThe subsequent proof of \\autoref{Th:Stability TCD} below  follows  \\cite[Thm.~3.1]{sturm2006b}. However, by the nature of our timelike curvature-dimension conditions, we additionally  have to ensure chronological relations of many measures under consideration. Hence, the proof becomes much longer and quite technical. It may be skipped at first reading; alternatively, the reader may consult the proof of the counterpart  \\autoref{Th:Stability TMCP} of \\autoref{Th:Stability TCD} below for a somewhat easier argument in which one does not have to trace chronological $\\smash{\\ell_p}$-optimal couplings.\n\n\n\n\n\\begin{theorem}\\label{Th:Stability TCD} Assume the convergence of $({\\ensuremath{\\mathscr{X}}}_k)_{k\\in\\N}$ to ${\\ensuremath{\\mathscr{X}}}_\\infty$ as in  \\autoref{Def:Convergence}. Moreover, let $(K_k,N_k)_{k\\in\\N}$ be a sequence in $\\R\\times [1,\\infty)$ converging to $(K_\\infty,N_\\infty)\\in\\R\\times[1,\\infty)$. Suppose the existence of $p\\in (0,1)$ such that ${\\ensuremath{\\mathscr{X}}}_k$ obeys $\\mathrm{TCD}_p(K_k,N_k)$ for every $k\\in\\N$. \nThen $\\smash{{\\ensuremath{\\mathscr{X}}}_\\infty}$ satisfies $\\smash{\\mathrm{wTCD}_p(K_\\infty,N_\\infty)}$. \n\nThe analogous statement in which $\\smash{\\mathrm{TCD}_p(K_k,N_k)}$ and $\\smash{\\mathrm{wTCD}_p(K_\\infty,N_\\infty)}$ are replaced by $\\smash{\\mathrm{TCD}_p^*(K_k,N_k)}$ and $\\smash{\\mathrm{wTCD}_p^*(K_\\infty,N_\\infty)}$, $k\\in\\N$, respectively, holds too.\n\\end{theorem}\n\n\\begin{proof} We only prove the first implication, the second is similar. To relax  notation, without restriction and further notice we identify $\\mms_k$ with its image $\\iota_k(\\mms_k)$ in $\\mms$, and the measure $\\meas_k$ with its push-forward $(\\iota_k)_\\push\\meas_k$ for every $k\\in\\N_\\infty$.\n\n\\textbf{Step 1.} \\textit{Reduction to compact $\\mms$.} Fix any  strongly timelike $p$-dualizable pair  $(\\mu_{\\infty,0},\\mu_{\\infty,1}) = (\\rho_{\\infty,0}\\,\\meas_\\infty,\\rho_{\\infty,1}\\,\\meas_\\infty)\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas_\\infty)$. By compactness of $\\supp\\mu_{\\infty,0}$ and $\\supp\\mu_{\\infty,1}$, and by ${\\ensuremath{\\mathscr{K}}}$-global hyperbolicity of $\\smash{\\mms}$, we restrict all  arguments below to a compact subset $\\smash{C\\subset \\mms}$ with $\\smash{\\meas_k[C]^{-1}\\,\\meas_k\\mres C \\to \\meas_\\infty[C]^{-1}\\,\\meas_\\infty\\mres C}$ weakly as $k\\to\\infty$. To relax notation, we will thus assume with no loss of generality  that $\\smash{\\mms}$ itself is compact, and that  $\\smash{\\meas_k\\in{\\ensuremath{\\mathscr{P}}}(\\mms)}$ for every $k\\in\\N_\\infty$. In particular, without notationally expressing this property all the time, all measures will henceforth be compactly supported.\n\n\\textbf{Step 2.} \\textit{Restriction of the assumptions on $\\mu_{\\infty,0}$ and $\\mu_{\\infty,1}$.} We will first assume that $\\rho_{\\infty,0},\\rho_{\\infty,1}\\in\\Ell^\\infty(\\mms,\\meas_\\infty)$. The general case is discussed in Step 8 below; we note for now that this conclusion will not conflict with our reductions from Step 1.\n\n\\textbf{Step 3.} \\textit{Construction of a chronological recovery sequence.} The goal of this step is the construction of a sequence $\\smash{(\\mu_{k,0},\\mu_{k,1})_{k\\in\\N}}$ of timelike $p$-dualizable pairs $\\smash{(\\mu_{k,0},\\mu_{k,1})\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)^2}$, $k\\in\\N$, such that $\\smash{\\mu_{k,0}\\to \\mu_{\\infty,0}}$ and $\\smash{\\mu_{k,1}\\to \\mu_{\\infty,1}}$ weakly as $k\\to\\infty$, up to a subsequence. The highly nontrivial part we address here is that such a pair can in fact be constructed to be chronological.\n\n\\textbf{Step 3.1.} Let $W_2$ denote the $2$-Wasserstein distance on $\\smash{(\\mms,\\met)}$. Since $(\\meas_k)_{k\\in\\N}$ converges weakly to $\\meas_\\infty$ and since $\\mms$ is compact, we have $W_2(\\meas_k,\\meas_\\infty)\\to 0$ as $k\\to\\infty$. Let $\\mathfrak{q}_k\\in{\\ensuremath{\\mathscr{P}}}(\\mms^2)$ be a $W_2$-optimal coupling of $\\meas_k$ and $\\meas_\\infty$, $k\\in\\N$. For later use, we disintegrate $\\mathfrak{q}_k$ with respect to $\\pr_1$ and $\\pr_2$, respectively, writing\n\\begin{align*}\n{\\ensuremath{\\mathrm{d}}}\\mathfrak{q}_k(x, y) = {\\ensuremath{\\mathrm{d}}}\\mathfrak{p}_{x}^k(y)\\d\\meas_k(x) = {\\ensuremath{\\mathrm{d}}}\\mathfrak{p}_{y}^\\infty(x)\\d\\meas_\\infty(y)\n\\end{align*}\nfor certain Borel maps $\\smash{\\mathfrak{p}^k \\colon \\mms\\to {\\ensuremath{\\mathscr{P}}}(\\mms)}$ and $\\smash{\\mathfrak{p}^\\infty \\colon \\mms\\to {\\ensuremath{\\mathscr{P}}}(\\mms)}$. (Although $\\mathfrak{p}^\\infty$  depends on $k$ as well, we do not make this explicit to not overload  our  notation.) These disin\\-te\\-grations induce nonrelabeled Borel maps $\\smash{\\mathfrak{p}^k\\colon {\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)\\to {\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)}$ and $\\smash{\\mathfrak{p}^\\infty}\\colon {\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)\\to {\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)$ defined by\n\\begin{align*}\n{\\ensuremath{\\mathrm{d}}}\\mathfrak{p}^k(f\\,\\meas_\\infty)(x) &:= \\Big[\\!\\int_\\mms  f(y)\\d\\mathfrak{p}_{x}^k(y) \\Big]\\d\\meas_k(x),\\\\\n{\\ensuremath{\\mathrm{d}}}\\mathfrak{p}^\\infty(g\\,\\meas_k)(y) &:= \\Big[\\!\\int_\\mms  g(x)\\d\\mathfrak{p}_{y}^\\infty(x) \\Big]\\d\\meas_\\infty(y).\n\\end{align*} \n\n\\textbf{Step 3.2.} Let $\\pi_\\infty\\in\\Pi_\\ll(\\mu_{\\infty,0},\\mu_{\\infty,1})$ be timelike $p$-dualizing, and let $(\\pi^n_\\infty)_{n\\in\\N}$ be an asso\\-ciated sequence in $\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms^2,\\meas_\\infty^{\\otimes 2})}$ as in \\autoref{Le:CM lemma}, where we write\n\\begin{align*}\n\\pi^n_\\infty &= \\rho_\\infty^n\\,\\meas_\\infty^{\\otimes 2},\\\\\n\\mu_{\\infty,0}^n := (\\pr_1)_\\push\\pi_\\infty^n &= \\rho_{\\infty,0}^n\\,\\meas_\\infty,\\\\\n\\mu_{\\infty,1}^n := (\\pr_2)_\\push\\pi_\\infty^n &= \\rho_{\\infty,1}^n\\,\\meas_\\infty.\n\\end{align*}\nFor $k\\in\\N$, define $\\smash{\\mu_{k,0}^n,\\mu_{k,1}^n\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)}$ by\n\\begin{align*}\n\\mu_{k,0}^n &:= \\mathfrak{p}^k(\\mu_{\\infty,0}^n) = \\rho_{k,0}^n\\,\\meas_k,\\\\\n\\mu_{k,1}^n &:= \\mathfrak{p}^k(\\mu_{\\infty,1}^n) = \\rho_{k,1}^n\\,\\meas_k\n\\end{align*}\nas well as  $\\smash{\\pi_k^n\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms^2,\\meas_k^{\\otimes 2})}$ by\n\\begin{align*}\n{\\ensuremath{\\mathrm{d}}} \\pi_k^n(x^0,x^1) &:= \\Big[\\!\\int_{\\mms^2}\\rho_\\infty^n(y^0,y^1)\\d(\\mathfrak{p}_{x^0}^k\\otimes\\mathfrak{p}_{x^1}^k)(y^0,y^1)\\Big]\\d\\meas_k^{\\otimes 2}(x^0,x^1),\n\\end{align*}\nor in other words,\n\\begin{align*}\n\\pi_k^n = (\\pr_1,\\pr_3)_\\push\\big[(\\rho_\\infty^n \\circ (\\pr_2,\\pr_4))\\,\\mathfrak{q}_k\\otimes\\mathfrak{q}_k\\big].\n\\end{align*}\nA straightforward computation entails $\\smash{\\pi_k^n\\in\\Pi(\\mu_{k,0}^n,\\mu_{k,1}^n)}$. Also, for given $n\\in\\N$, weak convergence of $(\\mathfrak{q}_k)_{k\\in\\N}$ to the unique $W_2$-optimal (diagonal) coupling of $\\meas_\\infty$ and $\\meas_\\infty$ by tightness \\cite[Lem.~4.3, Lem.~4.4]{villani2009}, up to a nonrelabeled subse\\-quence, imply  weak convergence of $\\smash{(\\pi_k^n)_{k\\in\\N}}$ to $\\smash{\\pi_\\infty^n}$ up to a nonrelabeled subsequence.\n\n\\textbf{Step 3.3.} By \\autoref{Le:CM lemma}, a tightness and a diagonal argument, passage to a subsequence, and relabeling of indices, we find $n\\colon\\N\\to\\N$ such that, setting\n\\begin{align*}\n\\tilde{\\pi}_k := \\pi_k^{n_k},\n\\end{align*}\nthe sequence $\\smash{(\\tilde{\\pi}_k)_{k\\in\\N}}$ converges weakly to $\\pi_\\infty$. By Portmanteau's theorem,\n\\begin{align}\\label{Eq:Portmanteau in stability}\n1 = \\pi_\\infty[\\mms_\\ll^2] \\leq \\liminf_{k\\to\\infty} \\tilde{\\pi}_k[\\mms_\\ll^2].\n\\end{align}\nHence, up passing to a  subsequence, we henceforth assume that $\\tilde{\\pi}_k[\\mms_\\ll^2] > 0$ for every $k\\in\\N$. We write the mar\\-ginals $\\smash{\\tilde{\\mu}_{k,0},\\tilde{\\mu}_{k,1}\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)}$ of $\\tilde{\\pi}_k$ as\n\\begin{align*}\n\\tilde{\\mu}_{k,0} &= \\tilde{\\rho}_{k,0}\\,\\meas_k = \\rho_{k,0}^{n_k}\\,\\meas_k,\\\\\n\\tilde{\\mu}_{k,1} &= \\tilde{\\rho}_{k,1}\\,\\meas_k = \\rho_{k,1}^{n_k}\\,\\meas_k.\n\\end{align*}\n\n\n\n\\textbf{Step 3.4.} Now we \ndefine $\\smash{\\hat{\\pi}_k\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms^2)}$ by\n\\begin{align*}\n\\hat{\\pi}_k := \\tilde{\\pi}_k[\\mms_\\ll^2]^{-1}\\,\\tilde{\\pi}_k\\mres \\mms^2_\\ll.\n\\end{align*}\nNote that $\\smash{(\\hat{\\pi}_k)_{k\\in\\N}}$ converges weakly to $\\smash{\\pi_\\infty}$, whence\n\\begin{align}\\label{Eq:Der Limes}\n\\lim_{k\\to\\infty}\\int_{\\mms^2}\\uptau^p\\d\\hat{\\pi}_k = \\int_{\\mms^2}\\uptau^p\\d\\pi_\\infty.\n\\end{align}\nLet $\\smash{\\hat{\\mu}_{k,0},\\hat{\\mu}_{k,0}\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)}$ denote the marginals of $\\smash{\\hat{\\pi}_k}$, with\n\\begin{align*}\n\\hat{\\mu}_{k,0} &= \\hat{\\rho}_{k,0}\\,\\meas_k,\\\\\n\\hat{\\mu}_{k,1} &= \\hat{\\rho}_{k,1}\\,\\meas_k.\n\\end{align*}\n\nThese measures are chronologically related, but we do not know whether they have a chronological \\emph{$\\smash{\\ell_p}$-optimal} coupling, i.e.~whether they are timelike $p$-dualizable. To achieve this,  note that $\\smash{\\ell_p(\\hat{\\mu}_{k,0},\\hat{\\mu}_{k,1})} \\in (0,\\infty)$ by chronology of $\\smash{\\hat{\\pi}_k}$ as well as compactness of $\\mms^2$. Let $\\smash{\\check{\\pi}_k\\in\\Pi_\\leq(\\hat{\\mu}_{k,0},\\hat{\\mu}_{k,1})}$ be an $\\smash{\\ell_p}$-optimal coupling. By weak convergence of $\\smash{(\\hat{\\pi}_k)_{k\\in\\N}}$, its marginal sequences $\\smash{(\\hat{\\mu}_{k,0})_{k\\in\\N}}$ and $\\smash{(\\hat{\\mu}_{k,1})_{k\\in\\N}}$ are tight, and so is $\\smash{(\\check{\\pi}_k)_{k\\in\\N}}$ \\cite[Lem.~4.4]{villani2009}. By Prokhorov's theorem, the latter  converges weakly to some $\\smash{\\check{\\pi}_\\infty\\in\\Pi(\\mu_{\\infty,0},\\mu_{\\infty,1})}$; causal closedness of $\\mms$ and Portmanteau's theorem imply $\\smash{\\check{\\pi}_\\infty[\\mms_\\leq^2]=1}$. Moreover,  \\eqref{Eq:Der Limes} gives\n\\begin{align}\\label{Eq:Lp calc}\n\\begin{split}\n\\ell_p(\\mu_{\\infty,0},\\mu_{\\infty,1})^p &= \\int_\\mms \\uptau^p\\d\\pi_\\infty = \\lim_{k\\to\\infty}\\int_\\mms \\uptau^p\\d\\hat{\\pi}_k\\\\ &\\leq \\lim_{k\\to\\infty} \\int_\\mms\\uptau^p\\d\\check{\\pi}_k = \\int_\\mms\\uptau^p\\d\\check{\\pi}_\\infty.\n\\end{split}\n\\end{align}\nHere the inequality follows from $\\smash{\\ell_p}$-optimality of $\\smash{\\check{\\pi}_k}$. Thus, $\\smash{\\check{\\pi}_\\infty}$ is an $\\smash{\\ell_p}$-optimal coupling of $\\smash{\\mu_{\\infty,0}}$ and $\\smash{\\mu_{\\infty,1}}$. Since the latter pair is  strongly timelike $p$-dualizable, every such coupling is chronological, whence $\\smash{\\check{\\pi}_\\infty[\\mms_\\ll^2]=1}$; thus,\n\\begin{align*}\n1 = \\check{\\pi}_\\infty[\\mms_\\ll^2] \\leq \\liminf_{k\\to\\infty} \\check{\\pi}_k[\\mms_\\ll^2].\n\\end{align*}\nAs usual, we may and will assume  $\\smash{\\check{\\pi}_k[\\mms_\\ll^2] > 0}$ for every $k\\in\\N$.\n\n\\textbf{Step 3.5.} Lastly, we define $\\smash{\\bar{\\pi}_k\\in{\\ensuremath{\\mathscr{P}}}(\\mms^2)}$ by\n\\begin{align*}\n\\bar{\\pi}_k := \\check{\\pi}_k[\\mms_\\ll^2]^{-1}\\,\\check{\\pi}_k\\mres \\mms_\\ll^2.\n\\end{align*}\nSince the restriction of an $\\smash{\\ell_p}$-optimal coupling is $\\smash{\\ell_p}$-optimal \\cite[Lem.~2.10]{cavalletti2020}, $\\smash{\\bar{\\pi}_k}$ is a chronological $\\smash{\\ell_p}$-optimal coupling of its marginals $\\smash{\\mu_{k,0},\\mu_{k,1}\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)}$ with\n\\begin{align*}\n\\mu_{k,0} &= \\rho_{k,0}\\,\\meas_k,\\\\\n\\mu_{k,1} &= \\rho_{k,1}\\,\\meas_k.\n\\end{align*}\nThe pair $\\smash{(\\mu_{k,0},\\mu_{k,1})}$ is timelike $p$-dualizable by $\\bar{\\pi}_k$ for every $k\\in\\N$, as desired.\n\n\\textbf{Step 4.} \\textit{Invoking the $\\mathrm{TCD}$ condition.} Fix  $K\\in\\R$ and $N\\in (1,\\infty)$ with $K < K_\\infty$ and $N > N_\\infty$. Then $K < K_k$ and $N > N_k$ for large enough $k\\in\\N$. Hence, again up to a subsequence, we may and will assume that the previous strict inequalities hold for every $k\\in\\N$, respectively.\n\nBy \\autoref{Pr:Consistency TCD}, for every $k\\in\\N$ there are a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_{k,t})_{t\\in[0,1]}$ con\\-necting $\\smash{\\mu_{k,0}}$ to $\\mu_{k,1}$ as well as a timelike $p$-dualizing coupling $\\smash{\\pi_k\\in\\Pi_\\ll(\\mu_{k,0},\\mu_{k,1})}$ such that for every $t\\in[0,1]$ and every $N'\\geq N$,\n\\begin{align}\\label{Eq:TCD COND INVOKING}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{k,t}) \\leq {\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\pi_k).\n\\end{align}\n\n\\textbf{Step 5.} \\textit{Estimating $\\smash{-{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\pi_k)}$ from below.} Tracing back the construction from Step 2 and Step 3, we infer the inequalities\n\\begin{align}\\label{Eq:Inequ rhok0}\n\\begin{split}\n\\rho_{k,0} &\\leq \\check{\\pi}_k[\\mms_\\ll^2]^{-1}\\,\\tilde{\\pi}_k[\\mms_\\ll^2]^{-1}\\,\\tilde{\\rho}_{k,0}\\quad \\meas\\textnormal{-a.e.},\\\\\n\\rho_{k,1} &\\leq \\check{\\pi}_k[\\mms_\\ll^2]^{-1}\\,\\tilde{\\pi}_k[\\mms_\\ll^2]^{-1}\\,\\tilde{\\rho}_{k,1} \\quad\\meas\\textnormal{-a.e.}\n\\end{split}\n\\end{align}\nHowever, when using these estimates to bound $\\smash{-{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\pi_k)}$ from below, we change the involved densities in the integrands, but not the coupling with respect to which it is integrated; we thus have to make up for this in the coupling as well. In turn, this requires us to take into account that both $\\smash{\\tilde{\\pi}_k}$ and $\\smash{\\check{\\pi}_k}$ are not chro\\-nological, but only close to being so for large $k\\in\\N$.\n\n\\textbf{Step 5.1.} We modify $\\pi_k$ into a  measure with marginals \n\\begin{align*}\n\\nu_{k,0} &:=  (1+\\delta_k+\\varepsilon_k+\\zeta_k)^{-1}\\,(\\tilde{\\rho}_{k,0}+ \\delta_k + \\varepsilon_k+\\zeta_k)\\,\\meas_k = \\varrho_{k,0}\\,\\meas_k,\\\\\n\\nu_{k,1} &:=  (1+\\delta_k+\\varepsilon_k+\\zeta_k)^{-1}\\,(\\tilde{\\rho}_{k,1}+ \\delta_k + \\varepsilon_k+\\zeta_k)\\,\\meas_k = \\varrho_{k,1}\\,\\meas_k,\n\\end{align*}\nwhere $\\delta_k,\\varepsilon_k \\in[0,1]$ and $\\zeta_k \\geq 0$ are defined by\n\\begin{align}\\label{Eq:DEFS}\n\\begin{split}\n\\delta_k &:= \\tilde{\\pi}_k[\\{\\uptau=0\\}],\\\\\n\\varepsilon_k &:= \\check{\\pi}_k[\\{\\uptau=0\\}],\\\\\n\\zeta_k &:= W_2(\\meas_k,\\meas_\\infty).\n\\end{split}\n\\end{align}\nNote that $(\\delta_k)_{k\\in\\N}$, $(\\varepsilon_k)_{k\\in\\N}$, and $(\\zeta_k)_{k\\in\\N}$ converge to $0$. Define  $\\varpi_k\\in{\\ensuremath{\\mathscr{P}}}(\\mms^2)$ by\n\\begin{align*}\n\\varpi_k &:= (1+\\delta_k+\\varepsilon_k+\\zeta_k)^{-1}\\,\\big[\\tilde{\\pi}_k[\\mms_\\ll^2]\\,\\check{\\pi}_k[\\mms_\\ll^2]\\,\\pi_k + \\tilde{\\pi}_k\\mres \\{\\uptau=0\\}\\\\\n&\\qquad\\qquad + (\\delta_k + \\varepsilon_k+\\zeta_k)\\,\\meas_k^{\\otimes 2}  + \\tilde{\\pi}_k[\\mms_\\ll^2]\\,\\check{\\pi}_k\\mres \\{\\uptau=0\\}\\big].\n\\end{align*}\nBy tracing back the definitions of $\\tilde{\\pi}_k$ and $\\check{\\pi}_k$ in Step 2, one verifies that $\\varpi_k$ is a coupling, not necessarily $\\smash{\\ell_p}$-optimal, of $\\nu_{k,0}$ and $\\nu_{k,1}$. \n\n\\textbf{Step 5.2.} By \\eqref{Eq:Inequ rhok0}, we have \n\\begin{alignat*}{3}\n\\rho_{k,0} &\\leq \\check{\\pi}_k[\\mms_\\ll^2]^{-1}\\,\\tilde{\\pi}_k[\\mms_\\ll^2]^{-1}\\,(1+\\delta_k + \\varepsilon_k+\\zeta_k)\\,\\varrho_{k,0} & \\quad & \\meas\\textnormal{-a.e.},\\\\\n\\rho_{k,1} &\\leq \\check{\\pi}_k[\\mms_\\ll^2]^{-1}\\,\\tilde{\\pi}_k[\\mms_\\ll^2]^{-1}\\,(1+\\delta_k + \\varepsilon_k+\\zeta_k)\\,\\varrho_{k,1} & \\quad & \\meas\\textnormal{-a.e.}\n\\end{alignat*}\nIn the sequel, to clear up notation, given $a,b\\in\\R$ we write  $a\\geq_k b$ if there exists a sequence $(c_k)_{k\\in\\N}$ of positive real numbers converging to $1$ with $a\\geq c_k\\,b$ for every $k\\in\\N$. By definition of the inherent distortion coefficients, we obtain\n\\begin{align*}\n-{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\pi_k) &= \\int_{\\mms^2}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_{k,0}(x^0)^{-1\/N'}\\d\\pi_k(x^0,x^1)\\\\\n&\\qquad\\qquad +\\int_{\\mms^2}\\tau_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\,\\rho_{k,1}(x^1)^{-1\/N'}\\d\\pi_k(x^0,x^1)\\\\\n&\\geq_k \\int_{\\mms^2} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\pi_k(x^0,x^1)\\\\\n&\\qquad\\qquad + \\int_{\\mms^2} \\tau_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\,\\varrho_{k,1}(x^1)^{-1\/N'}\\d\\pi_k(x^0,x^1)\\\\\n&\\geq_k\\int_{\\mms^2} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\varpi_k(x^0,x^1)\\\\\n&\\qquad\\qquad + \\int_{\\mms^2}\\tau_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\varpi_k(x^0,x^1)\\\\\n&\\qquad\\qquad - (1-t)\\int_{\\{\\uptau=0\\}}\\varrho_{k,0}(x^0)^{-1\/N'} \\d\\tilde{\\pi}_k(x^0,x^1)\\\\\n&\\qquad\\qquad -  t\\int_{\\{\\uptau=0\\}} \\varrho_{k,1}(x^1)^{-1\/N'}\\d\\tilde{\\pi}_k(x^0,x^1)\\\\\n&\\qquad\\qquad - (1-t)\\int_{\\{\\uptau=0\\}}\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\check{\\pi}_k(x^0,x^1)\\\\\n&\\qquad\\qquad - t\\int_{\\{\\uptau=0\\}}\\varrho_{k,1}(x^1)^{-1\/N'}\\d\\check{\\pi}_k(x^0,x^1)\\\\\n&\\qquad\\qquad - c\\,(\\delta_k + \\varepsilon_k+\\zeta_k)\\int_{\\mms} \\varrho_{k,0}(x^0)^{-1\/N'}\\d\\meas_k(x^0)\\\\\n&\\qquad\\qquad - c\\,(\\delta_k + \\varepsilon_k+\\zeta_k)\\int_{\\mms} \\varrho_{k,1}(x^1)^{-1\/N'}\\d\\meas_k(x^1),\n\\end{align*}\nwhere, recalling our choice $K < K_k$ for large enough $k\\in\\N$ and \\autoref{Cor:Bonnet-Myers},\n\\begin{align}\\label{Eq:c value}\nc := \\max\\{\\sup\\tau_{K,N'}^{(1-t)}\\circ\\uptau(\\mms^2),\\sup\\tau_{K,N'}^{(t)}\\circ\\uptau(\\mms^2)\\}.\n\\end{align}\nSince $\\varrho_{k,0} \\geq_k \\delta_k + \\varepsilon_k+\\zeta_k$ $\\meas_k$-a.e.~and $\\varrho_{k,1} \\geq_k \\delta_k+\\varepsilon_k+\\zeta_k$ $\\meas_k$-a.e., by definition of $\\delta_k$ and $\\varepsilon_k$ we obtain the estimates\n\\begin{align*}\n(\\delta_k + \\varepsilon_k+\\zeta_k)^{1-1\/N'}&\\geq_k (1-t)\\int_{\\{\\uptau=0\\}} \\varrho_{k,1}(x^0)^{-1\/N'}\\d\\tilde{\\pi}_k(x^0,x^1) \\\\\n&\\qquad\\qquad + t\\int_{\\{\\uptau=0\\}} \\varrho_{k,1}(x^0)^{-1\/N'}\\d\\tilde{\\pi}_k(x^0,x^1),\\\\\n(\\delta_k + \\varepsilon_k+\\zeta_k)^{1-1\/N'} &\\geq_k (1-t)\\int_{\\{\\uptau=0\\}} \\varrho_{k,1}(x^0)^{-1\/N'}\\d\\check{\\pi}_k(x^0,x^1)\\\\\n&\\qquad\\qquad  + t\\int_{\\{\\uptau=0\\}} \\varrho_{k,1}(x^0)^{-1\/N'}\\d\\check{\\pi}_k(x^0,x^1),\\\\\n2\\,(\\delta_k + \\varepsilon_k+\\zeta_k)^{1-1\/N'} &\\geq_k (\\delta_k+\\varepsilon_k+\\zeta_k)\\int_{\\mms}\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\meas_k(x^0)\\\\\n&\\qquad\\qquad + (\\delta_k+\\varepsilon_k+\\zeta_k)\\int_{\\mms}\\varrho_{k,1}(x^1)^{-1\/N'}\\d\\meas_k(x^1),\n\\end{align*}\nand therefore\n\\begin{align*}\n-{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\pi_k) &\\geq_k\\int_{\\mms^2} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\varpi_k(x^0,x^1)\\\\\n&\\qquad\\qquad + \\int_{\\mms^2}\\tau_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\varpi_k(x^0,x^1)\\\\\n&\\qquad\\qquad -2\\,(c+1)\\,(\\delta_k + \\varepsilon_k+\\zeta_k)^{1-1\/N'}.\\textcolor{white}{\\int}\n\\end{align*}\n\n\\textbf{Step 5.3.} Let $\\smash{\\mathfrak{r}^0,\\mathfrak{r}^1\\colon \\mms\\to {\\ensuremath{\\mathscr{P}}}(\\mms)}$   denote the ($k$-dependent)  disintegrations of $\\varpi_k$ with respect to $\\pr_1$ and $\\pr_2$, respectively, i.e.\n\\begin{align*}\n{\\ensuremath{\\mathrm{d}}} \\varpi_k(x^0,x^1) = {\\ensuremath{\\mathrm{d}}}\\mathfrak{r}^0_{x^0}(x^1)\\d\\nu_{k,0}(x^0) = {\\ensuremath{\\mathrm{d}}}\\mathfrak{r}^1_{x^1}(x^0)\\d\\nu_{k,1}(x^1),\n\\end{align*}\nand define\n\\begin{align}\\label{Eq:Disint tau}\n\\begin{split}\nv_0(x^0) &:= \\int_{\\mms} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\d\\mathfrak{r}_{x^0}^0(x^1),\\\\\nv_1(x^1) &:= \\int_{\\mms} \\tau_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\d\\mathfrak{r}_{x^1}^1(x^0).\n\\end{split}\n\\end{align}\nMoreover, define $\\smash{\\nu_{\\infty,0}^k,\\nu_{\\infty,1}^k\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)}$ by\n\\begin{align*}\n\\nu_{\\infty,0}^k &:= (1+\\delta_k + \\varepsilon_k+\\zeta_k)^{-1}\\,(\\rho_{\\infty,0}^{n_k} + \\delta_k +\\varepsilon_k+\\zeta_k)\\,\\meas_\\infty = \\varrho_{\\infty,0}^k\\,\\meas_\\infty,\\\\\n\\nu_{\\infty,1}^k &:= (1+\\delta_k + \\varepsilon_k+\\zeta_k)^{-1}\\,(\\rho_{\\infty,1}^{n_k} + \\delta_k +\\varepsilon_k+\\zeta_k)\\,\\meas_\\infty = \\varrho_{\\infty,1}^k\\,\\meas_\\infty,\n\\end{align*}\nand observe that\n\\begin{align}\\label{Eq:Observation!}\n\\begin{split}\n\\mathfrak{p}^k(\\nu_{\\infty,0}^k) &= \\nu_{k,0} = \\varrho_{k,0}\\,\\meas_k,\\\\\n\\mathfrak{p}^k(\\nu_{\\infty,1}^k) &= \\nu_{k,1} = \\varrho_{k,1}\\,\\meas_k.\n\\end{split}\n\\end{align}\nThen by Jensen's inequality,\n\\begin{align*}\n&\\int_{\\mms^2} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\varpi_k(x^0,x^1)\\textcolor{white}{\\sum_0^1}\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\int_{\\mms^2}\\tau_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\varpi_k(x^0,x^1)\\\\\n&\\qquad\\qquad = \\sum_{i=0}^1 \\int_{\\mms} \\varrho_{k,i}(x^i)^{1-1\/N'}\\,v_i(x^i)\\d\\meas_k(x^i)\\\\\n&\\qquad\\qquad  = \\sum_{i=0}^1 \\int_{\\mms}\\Big[\\!\\int_{\\mms} \\varrho_{\\infty,i}^k(y^i)\\d\\mathfrak{p}_{x^i}^k(y^i)\\Big]^{1-1\/N'}v_i(x^i)\\d\\meas_k(x^i)\\\\\n&\\qquad\\qquad \\geq \\sum_{i=0}^1 \\int_{\\mms^2} \\varrho_{\\infty,i}^k(y^i)^{1-1\/N'}\\,v_i(x^i)\\d\\mathfrak{p}_{x^i}^k(y^i)\\d\\meas_k(x^i)\\\\\n&\\qquad\\qquad = \\sum_{i=0}^1 \\int_{\\mms^2}  \\varrho_{\\infty,i}^k(y^i)^{1-1\/N'}\\,v_i(x^i)\\d\\mathfrak{p}_{y^i}^\\infty(x^i)\\d\\meas_\\infty(y^i).\n\\end{align*}\n\n\\textbf{Step 5.4.} Next, we estimate the latter sum from below. Using \\eqref{Eq:Disint tau} and \\eqref{Eq:Observation!},\n\\begin{align*}\n&\\int_{\\mms^2} \\varrho_{\\infty,0}^k(y^0)^{1-1\/N'}\\,v_0(x^0)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad =\\int_{\\mms^4}\\d\\mathfrak{p}_{x^1}^k(y^1)\\d\\mathfrak{r}_{x^0}^0(x^1)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad \\Big[\\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,1}(x^1)}\\Big].\n\\end{align*}\n\nLet $\\varepsilon > 0$. Since $\\smash{\\tau_{K,N'}^{(t)}\\circ\\uptau}$ is uniformly continuous by compactness of $\\smash{\\mms^2}$, our choice of $K$, and possibly invoking \\autoref{Cor:Bonnet-Myers}, we fix $\\delta> 0$ such that\n\\begin{align*}\n\\big\\vert \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1)) - \\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\big\\vert &\\leq \\varepsilon,\\\\\n\\big\\vert \\tau_{K,N'}^{(t)}(\\uptau(x^0,x^1)) - \\tau_{K,N'}^{(t)}(\\uptau(y^0,y^1))\\big\\vert &\\leq \\varepsilon\n\\end{align*}\nfor every  $(x^0,y^0,x^1,y^1)\\in A_\\delta$, where\n\\begin{align}\\label{Eq:A_delta}\nA_\\delta := \\{ z \\in\\mms^4 :  \\met(z_1,z_2) + \\met(z_3,z_4) \\leq \\delta\\}.\n\\end{align}\nLastly, somewhat suggestively, for $k\\in\\N$ we define a coupling $\\smash{\\eta_k\\in\\Pi(\\nu_{\\infty,0}^k,\\nu_{\\infty,1}^k)}$, which is notably independent of $\\varepsilon$, by\n\\begin{align*}\n{\\ensuremath{\\mathrm{d}}}\\eta_k(y^0,y^1) &:= \\int_{\\mms^2} \\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,0}(x^0)\\,\\varrho_{k,1}(x^1)}\\d\\mathfrak{p}_{x^0}^k(y^0)\\d\\mathfrak{p}_{x^1}^k(y^1)\\d\\varpi_k(x^0,x^1)\\\\\n&\\textcolor{white}{:}= \\int_{\\mms^2} \\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,1}(x^1)}\\d\\mathfrak{p}_{x^1}^k(y^1)\\d\\mathfrak{r}_{x^0}^0(x^1)\\d\\mathfrak{p}_{y^0}^\\infty(x^0) \\d\\meas_\\infty(y^0)\\\\\n&\\textcolor{white}{:}= \\int_{\\mms^2} \\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,0}(x^0)}\\d\\mathfrak{p}_{x^0}^k(y^0)\\d\\mathfrak{r}_{x^1}^1(x^0)\\d\\mathfrak{p}_{y^1}^\\infty(x^1)\\d\\meas_\\infty(y^1).\n\\end{align*}\nThe previous computations then entail\n\\begin{align*}\n&\\int_{\\mms^2} \\varrho_{\\infty,0}^k(y^0)^{1-1\/N'}\\,v_0(x^0)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad \\geq -\\varepsilon+\\int_{A_\\delta} \\d\\mathfrak{p}_{x^1}^k(y^1)\\d\\mathfrak{r}_{x^0}^0(x^1)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\Big[\\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\,\\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,1}(x^1)}\\Big]\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\int_{A_\\delta^{\\ensuremath{\\mathsf{c}}}} \\d\\mathfrak{p}_{x^1}^k(y^1)\\d\\mathfrak{r}_{x^0}^0(x^1)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\Big[\\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,1}(x^1)}\\Big]\\\\\n&\\qquad\\qquad = - \\varepsilon + \\int_{\\mms^2} \\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\d\\eta_k(y^0,y^1)\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\int_{A_\\delta^{\\ensuremath{\\mathsf{c}}}} \\d\\mathfrak{p}_{x^1}^k(y^1)\\d\\mathfrak{r}_{x^0}^0(x^1)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\Big[\\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\big[\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))- \\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\big]\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\times \\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,1}(x^1)}\\Big].\n\\end{align*}\n\n\\textbf{Step 5.5.} Let us estimate the latter error term. We first recall the definition $\\zeta_k := W_2(\\meas_k,\\meas_\\infty)$ from \\eqref{Eq:DEFS}. By definition of $A_\\delta$ and the value $c$ from \\eqref{Eq:c value}, the $\\meas$-a.e.~valid inequality for the density $\\smash{\\rho_{\\infty,1}^{n_k}}$ involved in the definition of $\\smash{\\varrho_{\\infty,1}^k}$ from \\autoref{Le:CM lemma}, as well as H\u00f6lder's inequality,\n\\begin{align*}\n&\\Big\\vert\\!\\int_{A_\\delta^{\\ensuremath{\\mathsf{c}}}} \\d\\mathfrak{p}_{x^1}^k(y^1)\\d\\mathfrak{r}_{x^0}^0(x^1)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad \\Big[\\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\big[\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))- \\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\big]\\\\\n&\\qquad\\qquad\\qquad\\qquad \\times \\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,1}(x^1)}\\Big]\\Big\\vert\\\\\n&\\qquad\\qquad \\leq 2\\,c\\,\\delta^{-1}\\!\\int_{\\mms^4} \\d\\mathfrak{p}_{x^1}^k(y^1)\\d\\mathfrak{r}_{x^0}^0(x^1)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad\\Big[\\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\big[\\met(x^0,y^0) + \\met(x^1,y^1)\\big]\\,\\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,1}(x^1)}\\Big]\\\\\n&\\qquad\\qquad \\leq_k 2\\,c\\,\\delta^{-1}\\!\\int_{\\mms^2}\\varrho_{\\infty,0}^k(y^0)^{1-1\/N'}\\,\\met(x^0,y^0)\\d\\mathfrak{q}_k(x^0,y^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad + 2\\,c\\,\\delta^{-1}\\,\\zeta_k^{-1\/N'}\\!\\int_{\\mms} \\rho_{\\infty,1}(y^1)\\,\\met(x^1,y^1)\\d\\mathfrak{q}_k(x^1,y^1)\\\\\n&\\qquad\\qquad \\leq_k 2\\,c\\,\\delta^{-1}\\,\\big\\Vert\\rho_{\\infty,0}\\big\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}^{1-1\/N'}\\,\\zeta_k + 2\\,c\\,\\delta^{-1}\\,\\Vert\\rho_{\\infty,1}\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}\\,\\zeta_k^{1-1\/N'}\\!.\\!\\!\\!\\textcolor{white}{\\int}\n\\end{align*}\n\n\\textbf{Step 5.6.} Taking the estimates from Step 5.4 and Step 5.5 together, we get\n\\begin{align*}\n&\\int_{\\mms^2}\\varrho_{\\infty,0}^k(y^0)^{1-1\/N'}\\,v_0(x^0)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad \\geq_k -\\varepsilon + \\int_{\\mms^2}\\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\d\\eta_k(y^0,y^1)\\\\\n&\\qquad\\qquad\\qquad\\qquad - 2\\,c\\,\\delta^{-1}\\,\\big\\Vert\\rho_{\\infty,0}\\big\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}^{1-1\/N'}\\,\\zeta_k\\textcolor{white}{\\int}\\\\\n&\\qquad\\qquad\\qquad\\qquad - 2\\,c\\,\\delta^{-1\/N'}\\,\\Vert\\rho_{\\infty,1}\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}\\,\\zeta_k^{1-1\/N'}\\!.\\textcolor{white}{\\int}\n\\end{align*}\nAnalogously, for the second summand in the last part of Step 5.3 we obtain\n\\begin{align*}\n&\\int_{\\mms^2} \\varrho_{\\infty,1}^k(y^1)^{1-1\/N'}\\,v_1(x^1)\\d\\mathfrak{p}_{y^1}^\\infty(x^1)\\d\\meas_\\infty(y^1)\\\\\n&\\qquad\\qquad \\geq_k -\\varepsilon + \\int_{\\mms^2}\\varrho_{\\infty,1}^k(y^1)^{-1\/N'}\\,\\tau_{K,N'}^{(t)}(\\uptau(y^0,y^1))\\d\\eta_k(y^0,y^1)\\\\\n&\\qquad\\qquad\\qquad\\qquad - 2\\,c\\,\\delta^{-1}\\,\\big\\Vert\\rho_{\\infty,1}\\big\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}^{1-1\/N'}\\,\\zeta_k\\textcolor{white}{\\int}\\\\\n&\\qquad\\qquad\\qquad\\qquad - 2\\,c\\,\\delta^{-1}\\,\\Vert\\rho_{\\infty,0}\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}\\,\\zeta_k^{1-1\/N'}\\!.\\textcolor{white}{\\int}\n\\end{align*}\n\n\\textbf{Step 6.} \\textit{Passage to the limit.} Let $(a_k)_{k\\in\\N}$ be a given sequence of normalization constants in $[1,\\infty)$ as provided by \\autoref{Le:CM lemma}, i.e. \n\\begin{align*}\n\\rho_{\\infty,0}^{n_k} &\\leq a_k\\,\\rho_{\\infty,0}\\quad\\meas\\textnormal{-a.e.},\\\\\n\\rho_{\\infty,1}^{n_k} &\\leq a_k\\,\\rho_{\\infty,1}\\quad\\meas\\textnormal{-a.e.}\n\\end{align*}\n\n\\textbf{Step 6.1.} For $k\\in\\N$, we define $\\smash{\\tilde{\\nu}_{\\infty,0}^k,\\tilde{\\nu}_{\\infty,1}^k}\\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)}$ by\n\\begin{align*}\n\\tilde{\\nu}_{\\infty,0}^k &:=(a_k^2+\\delta_k+\\varepsilon_k+\\zeta_k)^{-1}\\,(a_k^2\\,\\rho_{\\infty,0} + \\delta_k+\\varepsilon_k+\\zeta_k)\\,\\meas_\\infty = \\tilde{\\varrho}_{\\infty,0}^k\\,\\meas_\\infty,\\\\\n\\tilde{\\nu}_{\\infty,1}^k &:= (a_k^2+\\delta_k+\\varepsilon_k+\\zeta_k)^{-1}\\,(a_k^2\\,\\rho_{\\infty,1} + \\delta_k+\\varepsilon_k+\\zeta_k)\\,\\meas_\\infty = \\tilde{\\varrho}_{\\infty,1}^k\\,\\meas_\\infty.\n\\end{align*}\nWe turn $\\eta_k$ into a coupling $\\smash{\\alpha_k \\in\\Pi(\\nu_{\\infty,0}^k,\\nu_{\\infty,1}^k)}$ by setting\n\\begin{align*}\n\\alpha_k &:= (a_k^2+\\delta_k+\\varepsilon_k+\\zeta_k)^{-1} \\big[(1+\\delta_k+\\varepsilon_k+\\zeta_k)\\,\\eta_k\\\\\n&\\qquad\\qquad + a_k^2\\,\\mu_{\\infty,0}\\otimes\\mu_{\\infty,1} - \\tilde{\\mu}_{k,0}\\otimes\\tilde{\\mu}_{k,1}\\big].\n\\end{align*}\nThis construction yields\n\\begin{align*}\n&\\int_{\\mms^2}\\varrho_{k,0}^k(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\d\\eta_k(y^0,y^1)\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\int_{\\mms^2}\\varrho_{k,1}^k(y^1)^{-1\/N'}\\,\\tau_{K,N'}^{(t)}(\\uptau(y^0,y^1))\\d\\eta_k(y^0,y^1)\\\\\n&\\qquad\\qquad \\geq_k \\int_{\\mms^2} \\tilde{\\varrho}_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\d\\alpha_k(y^0,y^1)\\\\\n&\\qquad\\qquad\\qquad\\qquad - c\\int_\\mms \\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\big\\vert \\rho_{\\infty,0}(y^0) - \\tilde{\\rho}_{k,0}(y^0)\\big\\vert\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\int_{\\mms^2}\\tilde{\\varrho}_{\\infty,1}^k(y^1)^{-1\/N'}\\,\\tau_{K,N'}^{(t)}(\\uptau(y^0,y^1))\\d\\alpha_k(y^0,y^1)\\\\\n&\\qquad\\qquad\\qquad\\qquad - c\\int_\\mms \\varrho_{\\infty,1}^k(y^1)^{-1\/N'}\\,\\big\\vert \\rho_{\\infty,1}(y^1) - \\tilde{\\rho}_{k,1}(y^1)\\big\\vert\\d\\meas_\\infty(y^1).\n\\end{align*}\n\n\\textbf{Step 6.2.} Taking together the above estimates with those obtained in Step 5 and invoking the $\\Ell^1$-convergence asserted in  \\autoref{Le:CM lemma}, we obtain\n\\begin{align*}\n-\\limsup_{k\\to\\infty} {\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\pi_k) \\geq - \\limsup_{k\\to\\infty}{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\alpha_k) - 2\\varepsilon.\n\\end{align*}\nNow, by Prokhorov's theorem, the sequence $(\\alpha_k)_{k\\in\\N}$ converges weakly to some $\\alpha\\in\\Pi(\\mu_{\\infty,0},\\mu_{\\infty,1})$ up to a nonrelabeled subsequence. Since $\\varepsilon$ was arbitrary and did not influence the construction of $\\alpha_k$, by \\autoref{Le:Const perturb} we thus get\n\\begin{align}\\label{Eq:Eqeqeqeq}\n-\\limsup_{k\\to\\infty}{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\pi_k)\\geq -{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\alpha).\n\\end{align}\n\n\\textbf{Step 6.3.} Now we send $k\\to\\infty$ in \\eqref{Eq:TCD COND INVOKING}. For every $k\\in\\N$, there exists some plan $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_k\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_{k,0},\\mu_{k,1})}$ representing the curve $(\\mu_{k,t})_{t\\in[0,1]}$. \\autoref{Le:Villani lemma for geodesic} allows us to extract a nonrelabeled subsequence of $\\smash{({\\ensuremath{\\boldsymbol{\\pi}}}_k)_{k\\in\\N}}$ converging weakly to some $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_{\\infty,0},\\mu_{\\infty,1})}$. The assignment $\\mu_{\\infty,t} := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_\\infty$ produces a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_{\\infty,t})_{t\\in[0,1]}$ connecting $\\mu_{\\infty,0}$ to $\\mu_{\\infty,1}$. Lower semicontinuity of R\u00e9nyi's entropy on ${\\ensuremath{\\mathscr{P}}}(\\mms)$ by compactness of $\\mms$ and \\eqref{Eq:Eqeqeqeq} thus yield, for every $t\\in[0,1]$ and every $N'\\geq N$,\n\\begin{align}\\label{Eq:Renyi inequ}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{\\infty,t})\\leq \\limsup_{k\\to\\infty} {\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{k,t}) \\leq \\limsup_{k\\to\\infty}{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\pi_k) \\leq {\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\alpha).\n\\end{align}\n\n\\textbf{Step 7.} \\textit{Proof of the $\\smash{\\ell_p}$-optimality of $\\alpha$.} To conclude the desired property leading towards $\\smash{\\mathrm{TCD}_p(K,N)}$ for ${\\ensuremath{\\mathscr{X}}}_\\infty$, at least under the restrictions of Step 2, we have to prove the causality and the  $\\smash{\\ell_p}$-optimality of $\\alpha$. Both is argued similarly to Step 5.4; we concentrate on the proof of the estimate entailing $\\smash{\\ell_p}$-optimality later, and then outline how to modify this argument to prove $\\smash{\\alpha[\\mms_\\ll^2]=1}$.\n\nGiven any $\\varepsilon>0$, fix $\\delta > 0$ with the property\n\\begin{align*}\n\\big\\vert\\uptau(x^0,x^1) - \\uptau(y^0,y^1)\\big\\vert \\leq \\varepsilon\n\\end{align*}\nfor every $(x^0,y^0,x^1,y^1)\\in\\mms^4$  belonging to the set $\\smash{A_\\delta}$ from \\eqref{Eq:A_delta}. As in Step 5.5 and tracing back the definitions of all considered couplings, we obtain\n\\begin{align*}\n&\\int_{\\mms^2}\\uptau^p(y^0,y^1)\\d\\alpha(y^0,y^1) = \\lim_{k\\to\\infty}\\int_{\\mms^2}\\uptau^p(y^0,y^1)\\d\\alpha_k(y^0,y^1)\\\\\n&\\qquad\\qquad \\geq \\liminf_{k\\to\\infty} \\int_{\\mms^2}\\uptau^p(y^0,y^1)\\d\\eta_k(y^0,y^1)\\\\\n&\\qquad\\qquad \\geq \\liminf_{k\\to\\infty} \\int_{\\mms^2} \\uptau^p(x^0,x^1)\\d\\varpi_k(x^0,x^1) - \\varepsilon\\\\\n&\\qquad\\qquad\\qquad\\qquad -2\\,c\\,\\delta^{-1}\\,\\Vert\\rho_{\\infty,0}\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}\\limsup_{k\\to\\infty} W_2(\\meas_k,\\meas_\\infty)\\!\\!\\!\\textcolor{white}{\\int}\\\\\n&\\qquad\\qquad\\qquad\\qquad -2\\,c\\,\\delta^{-1}\\,\\Vert\\rho_{\\infty,1}\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}\\limsup_{k\\to\\infty} W_2(\\meas_k,\\meas_\\infty)\\!\\!\\!\\textcolor{white}{\\int}\\\\\n&\\qquad\\qquad \\geq \\liminf_{k\\to\\infty}\\int_{\\mms^2}\\uptau^p(x^0,x^1)\\d\\pi_k(x^0,x^1)-\\varepsilon\\\\\n&\\qquad\\qquad = \\liminf_{k\\to\\infty} \\int_{\\mms^2}\\uptau^p(x^0,x^1)\\d\\bar{\\pi}_k(x^0,x^1)-\\varepsilon\\\\\n&\\qquad\\qquad \\geq \\liminf_{k\\to\\infty}\\int_{\\mms^2}\\uptau^p(x^0,x^1)\\d\\check{\\pi}_k(x^0,x^1)-\\varepsilon\\geq \\ell_p(\\mu_{\\infty,0},\\mu_{\\infty,1})^p-\\varepsilon.\n\\end{align*}\nIn the equality in the third last step, we have used that both $\\smash{\\pi_k,\\bar{\\pi}_k\\in\\Pi_\\ll(\\mu_{k,0},\\mu_{k,1})}$ are $\\smash{\\ell_p}$-optimal; the last inequality is due to \\eqref{Eq:Lp calc}.\n\nThe relation $\\smash{\\alpha[\\mms_\\ll^2]=1}$ is argued similarly by replacing, given $\\varepsilon > 0$, $\\uptau$ by a nonnegative function $\\phi\\in\\Cont(\\mms^2)$ obeying $\\smash{\\phi(\\mms_\\ll^2)=\\{1\\}}$, $\\sup\\phi(\\mms)\\leq 1$, and\n\\begin{align*}\n\\alpha[\\mms_\\ll^2] \\geq \\int_{\\mms^2}\\phi\\d\\alpha - \\varepsilon.\n\\end{align*}\nBoth results together and the arbitrariness of $\\varepsilon$ yield the claim.\n\n\\textbf{Step 8.} \\textit{Relaxation of the assumptions on $\\mu_{\\infty,0}$ and $\\mu_{\\infty,1}$.} Now we outline how to get rid of the assumption $\\smash{\\rho_{\\infty,0},\\rho_{\\infty,1}\\in\\Ell^\\infty(\\mms,\\meas_\\infty)}$ from Step 2. \n\nIf $\\smash{\\rho_{\\infty,0}}$ and $\\smash{\\rho_{\\infty,1}}$ are not $\\meas$-essentially bounded, given any $i\\in\\N$ we set\n\\begin{align*}\nE_i := \\{\\rho_{\\infty,0} \\leq i,\\, \\rho_{\\infty,1}\\leq i\\}.\n\\end{align*}\nLet $\\smash{\\pi\\in\\Pi_\\ll(\\mu_{\\infty,0},\\mu_{\\infty,1})}$ be an $\\smash{\\ell_p}$-optimal coupling, and set\n\\begin{align*}\n\\pi_i := \\pi[E_i]^{-1}\\,\\pi\\mres E_i\n\\end{align*}\nprovided $\\pi[E_i]>0$. By restriction \\cite[Lem.~2.10]{cavalletti2020}, $\\pi_i$ is an $\\smash{\\ell_p}$-optimal coupling of its marginals $\\smash{\\lambda_{\\infty,0}^i,\\lambda_{\\infty,1}^i\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)}$; moreover, the pair $\\smash{(\\lambda_{\\infty,0}^i,\\lambda_{\\infty,1}^i)}$ is strongly timelike $p$-dualizable.  Let $\\smash{\\dot{\\pi}_i\\in\\Pi_\\ll(\\lambda_{\\infty,0}^i,\\lambda_{\\infty,1}^i)}$ be an $\\smash{\\ell_p}$-optimal coupling as constructed in the previous steps. Define $\\smash{\\beta_i\\in\\Pi_\\ll(\\mu_{\\infty,0},\\mu_{\\infty,1})}$ by\n\\begin{align*}\n\\beta_i := \\pi[E_i]\\,\\dot{\\pi}_i + \\pi\\mres E_i^{\\ensuremath{\\mathsf{c}}},\n\\end{align*}\nand note that $\\beta_i$ is an $\\smash{\\ell_p}$-optimal coupling of its marginals. Moreover, we observe that, for every $t\\in[0,1]$ and every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\dot{\\pi}_i) &\\leq_i -\\int_{\\mms^2} \\rho_{\\infty,0}(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\d\\dot{\\pi}_i(y^0,y^1)\\\\\n&\\qquad\\qquad -\\int_{\\mms^2} \\rho_{\\infty,1}(y^1)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\d\\dot{\\pi}_i(y^0,y^1)\\\\\n&\\leq_i -\\int_{\\mms^2} \\rho_{\\infty,0}(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\d\\beta_i(y^0,y^1)\\\\\n&\\qquad\\qquad + c \\int_{E_i^{\\ensuremath{\\mathsf{c}}}} \\rho_{\\infty,0}(y^0)^{-1\/N'}\\d\\pi(y^0,y^1)\\\\\n&\\qquad\\qquad -\\int_{\\mms^2}\\rho_{\\infty,1}(y^1)^{-1\/N'}\\,\\tau_{K,N'}^{(t)}(\\uptau(y^0,y^1))\\d\\beta_i(y^0,y^1)\\\\\n&\\qquad\\qquad + c\\int_{E_i^{\\ensuremath{\\mathsf{c}}}}\\rho_{\\infty,1}(y^1)^{-1\/N'}\\d\\pi(y^0,y^1).\n\\end{align*}\n\nBy Prokhorov's theorem, $(\\beta_i)_{i\\in\\N}$ converges weakly to some $\\smash{\\beta\\in\\Pi(\\mu_{\\infty,0},\\mu_{\\infty,1})}$ which, thanks to Portmanteau's theorem, is a causal coupling. By stability \\cite[Thm. 2.14]{cavalletti2020}, we thus infer the $\\smash{\\ell_p}$-optimality of $\\beta$. Together with \\autoref{Le:Const perturb} and H\u00f6lder's inequality, this addresses the right-hand side of the desired inequality; the left-hand side is treated with the  compactness argument from Step 6.3.\n\n\\textbf{Step 9.} \\textit{Passing from $K$ and $N$ to $K_\\infty$ and $N_\\infty$.} Lastly, the restrictions from Step 4 the previous arguments imply $\\smash{\\mathrm{TCD}_p(K,N)}$ for ${\\ensuremath{\\mathscr{X}}}_\\infty$ for every $K<K_\\infty$ and every $N>N_\\infty$. Given $\\smash{(\\mu_{\\infty,0},\\mu_{\\infty,1})\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)}$ strongly timelike $p$-dualizable, letting $K$ and $N$ approach $K_\\infty$ and $N_\\infty$, respectively, and using compactness arguments as in Step 6.2 and Step 6.3 gives the existence of a proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_{\\infty,t})_{t\\in[0,1]}$ from $\\mu_{\\infty,0}$ to $\\mu_{\\infty,1}$ and a timelike $p$-dualizing coupling $\\smash{\\pi\\in\\Pi_\\ll(\\mu_{\\infty,0},\\mu_{\\infty,1})}$ witnessing the desired inequality for the R\u00e9nyi entropy for every $K < K_\\infty$ and every $N' > N_\\infty$. The semicontinuity properties of both sides in the respective parameters yields $\\smash{\\mathrm{TCD}_p(K_\\infty,N_\\infty)}$. The proof is terminated.\n\\end{proof}\n\n\\begin{remark}\\label{RE:POTEN} Very roughly speaking, the relevant convergences  in the above proof are justified by \\emph{uniform continuity}, recall e.g.~Step 5.4. This is considerably weaker than \\emph{Lipschitz continuity} of the quantity inside the distortion coefficients, which has been used in the proof of \\cite[Thm.~3.1]{sturm2006b} in the metric case at a similar point, yet is useless here since $\\uptau$ is no distance. Our proof thus has the potential to admit straightforward adaptations to settings with continuous potentials other than $\\uptau$ which do not come from a distance either.\n\\end{remark}\n\n\\begin{remark}\\label{Re:A remarkable} A remarkable byproduct of the above proof is that both $\\smash{\\mathrm{TCD}_p^*(K,N)}$ and $\\smash{\\mathrm{TCD}_p(K,N)}$ are weakly stable under convergence of \\emph{normalized} Lorentzian pre-length spaces with respect to the following Lorentzian modification of Sturm's transport distance $\\boldsymbol{\\mathsf{D}}$ \\cite[Def. 3.2]{sturm2006a}: it is still defined in terms of $\\met$, but the inherent metric measure isometric embeddings should also respect the given Lorentzian structures, and are required to map into spaces obeying the regularity assumptions from item \\ref{La:Blurr1reg} of \\autoref{Def:Convergence}.\n\\end{remark}\n\n\\subsection{Good geodesics}\\label{Sub:Good TCD} In this section, following \\cite{braun2022}, see also  \\cite{cavalletti2017,rajala2012a,rajala2012b}, we show the existence of timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics with $\\meas$-densities uniformly in $\\Ell^\\infty$ in time under mild assumptions on their endpoints. This treatise does not require any nonbranching condition on the underlying space. The proofs of the results to follow are mostly analogous to those of \\cite{braun2022} and hence only outlined. \n\nWe regard the corresponding result from \\autoref{Th:Good geos TCD} as a key to develop the notion of \\emph{timelike weak gradients} on Lorentzian geodesic spaces by using so-called \\emph{test plans} --- many of which should exist by \\autoref{Th:Good geos TCD} --- following \\cite{ambrosio2014a}, and to prove a Lorentzian analogue of the \\emph{Sobolev-to-Lipschitz property} \\cite[Subsec.~4.1.3]{gigli2013}.\n\nThe crucial result providing the critical threshold for \\eqref{Eq:Thresh} is the following.\n\n\\begin{lemma}\\label{Le:Lalelu} Assume $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ for  $p\\in(0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$. Let $\\smash{(\\mu_0,\\mu_1)=(\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ be strongly timelike $p$-dualizable with $\\rho_0,\\rho_1\\in\\Ell^\\infty(\\mms,\\meas)$. Let  $D$ be any real number no smaller than $\\sup\\uptau(\\supp\\mu_0\\times\\supp\\mu_1)$. Then there exists a timelike proper-time parametrized  $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\mu_1$ such that $\\mu_t=\\rho_t\\,\\meas\\in\\Dom(\\Ent_\\meas)$ for every $t\\in[0,1]$, and\n\\begin{align*}\n\\meas\\big[\\{\\rho_{1\/2}>0\\}\\big] \\geq {\\ensuremath{\\mathrm{e}}}^{-D\\sqrt{K^-N}\/2}\\,\\max\\{ \\Vert\\rho_0\\Vert_{\\Ell^\\infty(\\mms,\\meas)},\\Vert\\rho_1\\Vert_{\\Ell^\\infty(\\mms,\\meas)}\\}^{-1}.\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof} Recall that $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ implies $\\smash{\\mathrm{wTCD}_p^*(K^-,N)}$ by \\autoref{Pr:Consistency TCD}. Let $\\smash{(\\mu_t)_{t\\in[0,1]}}$ be a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic from $\\mu_0$ to $\\mu_1$ and $\\smash{\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)}$ be some $\\smash{\\ell_p}$-optimal coupling obeying the inequality defining the condition $\\smash{\\mathrm{wTCD}_p^*(K^-,N)}$ as in  \\autoref{Def:TCD*}. Thanks to \\autoref{Le:Stu}, we obtain $\\mu_t\\in\\Dom(\\Ent_\\meas)$ for every $t\\in[0,1]$; in particular, we have  $\\smash{\\mu_t = \\rho_t\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$. Moreover, by \\autoref{Re:Lower bounds sigma},\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_N(\\mu_{1\/2})&\\leq -\\int_{\\mms^2} \\sigma_{K^-,N}^{(1\/2)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N}\\d\\pi(x^0,x^1)\\\\\n&\\qquad\\qquad -\\int_{\\mms^2} \\sigma_{K^-,N}^{(1\/2)}(\\uptau(x^0,x^1))\\,\\rho_1(x^1)^{-1\/N}\\d\\pi(x^0,x^1)\\\\\n&\\leq -{\\ensuremath{\\mathrm{e}}}^{-D\\sqrt{K^-\/N}\/2}\\,\\max\\{\\Vert\\rho_0\\Vert_{\\Ell^\\infty(\\mms,\\meas)}, \\Vert\\rho_1\\Vert_{\\Ell^\\infty(\\mms,\\meas)}\\}^{-1\/N}.\\!\\!\\!\\textcolor{white}{\\int}\n\\end{align*}\nOn the other hand, we have $\\smash{{\\ensuremath{\\mathscr{S}}}_N(\\mu_{1\/2}) \\geq -\\meas[\\{\\rho_{1\/2}>0\\}]^{1\/N}}$ by Jensen's inequality, and rearranging terms provides the claim.\n\\end{proof}\n\n\\begin{theorem}\\label{Th:Good geos TCD} Under the hypotheses of \\autoref{Le:Lalelu}, there exists a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\mu_1$ such that for every $t\\in[0,1]$, $\\smash{\\mu_t = \\rho_t\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ and\n\\begin{align}\\label{Eq:Thresh}\n\\Vert\\rho_t\\Vert_{\\Ell^\\infty(\\mms,\\meas)} \\leq {\\ensuremath{\\mathrm{e}}}^{D\\sqrt{K^-N}\/2}\\,\\max\\{\\Vert\\rho_0\\Vert_{\\Ell^\\infty(\\mms,\\meas)},\\Vert\\rho_1\\Vert_{\\Ell^\\infty(\\mms,\\meas)}\\},\n\\end{align}\nwhere\n\\begin{align*}\nD := \\sup\\uptau(\\supp\\mu_0\\times\\supp\\mu_1).\n\\end{align*}\n\\end{theorem}\n\n\\begin{proof} We construct the required geodesic at dyadic times in $[0,1]$ by a bisection argument and then define the rest of the geodesic by completion. By induction, given $n\\in\\N_0$ we assume that measures $\\smash{{\\ensuremath{\\boldsymbol{\\alpha}}}^0, \\dots, {\\ensuremath{\\boldsymbol{\\alpha}}}^n\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ have already been constructed with the following properties. For every $\\smash{k\\in\\{0,\\dots,2^n\\}}$, the pair $\\smash{((\\eval_{(k-1)2^{-n}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^n,(\\eval_{k2^{-n}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^n)\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)^2}$ is strongly timelike $p$-dualizable, and\n\\begin{align*}\n\\sup\\uptau(\\supp\\,(\\eval_{(k-1)2^{-n}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^n\\times\\supp\\,(\\eval_{k2^{-n}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^n) \\leq 2^{-n}\\,D.\n\\end{align*}\n\n\\textbf{Step 1.} \\textit{Minimization of an appropriate functional.} Set\n\\begin{align*}\nc_{n+1} := {\\ensuremath{\\mathrm{e}}}^{2^{-n-2}D\\sqrt{K^-N}}\\,\\max\\{\\Vert\\rho_0\\Vert_{\\Ell^\\infty(\\mms,\\meas)},\\Vert\\rho_1\\Vert_{\\Ell^\\infty(\\mms,\\meas)}\\}\n\\end{align*}\nand define the functional $\\smash{{\\ensuremath{\\mathscr{F}}}_{c_{n+1}}\\colon {\\ensuremath{\\mathscr{P}}}(\\mms)\\to [0,1]}$ by\n\\begin{align}\\label{Eq:Functional FC}\n{\\ensuremath{\\mathscr{F}}}_{c_{n+1}}(\\mu) := \\big\\Vert (\\rho- c_{n+1})^+\\big\\Vert_{\\Ell^1(\\mms,\\meas)} + \\mu_\\perp[\\mms].\n\\end{align}\nsubject to the decomposition $\\mu = \\rho\\,\\meas + \\mu_\\perp$. It measures how far $\\rho$ deviates from being bounded from above by $c$, and how much the input $\\mu$ fails to be $\\meas$-absolutely continuous. Let $\\smash{k\\in\\{1,\\dots,2^{n+1}-1\\}}$ be odd. As for \\cite[Le.~3.13]{braun2022}, we prove that $\\smash{{\\ensuremath{\\mathscr{F}}}_{c_{n+1}}\\circ \\eval_{1\/2}}$ admits a minimizer  $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_k^{n+1}\\in \\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_{(k-1)2^{-n-1}},\\mu_{(k+1)2^{-n-1}}}$). By gluing, we find a measure $\\smash{{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that\n\\begin{align*}\n(\\Restr_{(k-1)2^{-n-1}}^{(k+1)2^{-n-1}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1} = {\\ensuremath{\\boldsymbol{\\pi}}}_k^{n+1}\n\\end{align*}\nfor every $k$ as above. Here, given $s,t\\in [0,1]$ with $s<t$, $\\smash{\\Restr_s^t}$ is defined in \\eqref{Eq:Restr def}.\n\nFor every odd $k\\in\\{1,\\dots,2^{n+1}-1\\}$, the pairs \n\\begin{align*}\n&((\\eval_{(k-1)2^{-n-1}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1},(\\eval_{k2^{-n-1}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1}),\\\\\n&((\\eval_{k2^{-n-1}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1},(\\eval_{(k+1)2^{-n-1}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1})\n\\end{align*}\nare both strongly timelike $p$-dualizable \\cite[Lem.~3.1]{braun2022}. Furthermore, since $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_k^{n+1}}$ is concentrated on curves in $\\mathrm{TGeo}^\\uptau(\\mms)$ with $\\uptau$-length at most $2^{-n}\\,D$, we obtain\n\\begin{align*}\n\\sup\\uptau(\\supp\\,(\\eval_{(k-1)2^{-n-1}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1}\\times\\supp\\,(\\eval_{k2^{-n-1}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1}) &\\leq 2^{-n-1}\\,D,\\\\\n\\sup\\uptau(\\supp\\,(\\eval_{k2^{-n-1}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1}\\times\\supp\\,(\\eval_{(k+1)2^{-n-1}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1})&\\leq 2^{-n-1}\\,D.\n\\end{align*}\n\nAfter Step 2, this will allow us to iteratively produce a sequence $({\\ensuremath{\\boldsymbol{\\alpha}}}^n)_{n\\in\\N}$ in $\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ according to the preceding scheme.\n\n\\textbf{Step 2.} \\textit{Uniform density bound.} Given any $n\\in\\N_0$, any $c'>c_{n+1}$, and any odd  $k\\in\\smash{\\{1,\\dots,2^{n+1}-1\\}}$, arguing as in \\cite[Prop.~3.14]{braun2022} and using \\autoref{Le:Lalelu} we get\n\\begin{align}\\label{Eq:Fc inequality}\n\\begin{split}\n&\\inf\\{ {\\ensuremath{\\mathscr{F}}}_{c'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}) : {\\ensuremath{\\boldsymbol{\\pi}}}\\in \\mathrm{OptTGeo}_{\\ell_p}^\\uptau((\\eval_{(k-1)2^{-n-1}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1},\\\\\n&\\qquad\\qquad (\\eval_{(k+1)2^{-n-1}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1}) \\}=0.\n\\end{split}\n\\end{align}\nThe support of all considered measures belongs to the compact set $J(\\mu_0,\\mu_1)$; thus, using \\cite[Cor.~3.15]{braun2022} we get \\eqref{Eq:Fc inequality} for $c'$ replaced by $c_{n+1}$. Thus, $\\smash{(\\eval_{k2^{-n-1}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1}} = \\rho_{k2^{-n-1}}\\,\\meas\\in \\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ and, computing the geometric sum in the exponent,\n\\begin{align}\\label{Eq:DENS BD}\n\\Vert\\rho_{k2^{-n-1}}\\Vert_{\\Ell^\\infty(\\mms,\\meas)} \\leq {\\ensuremath{\\mathrm{e}}}^{D\\sqrt{K^-N}\/2}\\,\\max\\{\\Vert\\rho_0\\Vert_{\\Ell^\\infty(\\mms,\\meas)}, \\Vert\\rho_1\\Vert_{\\Ell^\\infty(\\mms,\\meas)}\\}.\n\\end{align}\n\n\\textbf{Step 3.} \\textit{Completion.} Let $\\smash{({\\ensuremath{\\boldsymbol{\\alpha}}}^n)_{n\\in\\N}}$ be the sequence in $\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ iteratively constructed according to Step 1 and Step 2. Using \\autoref{Le:Villani lemma for geodesic}, this sequence converges weakly, up to a  subsequence, to some $\\smash{\\ell_p}$-optimal geodesic plan $\\smash{{\\ensuremath{\\boldsymbol{\\alpha}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$. In particular, the assignment $\\mu_t := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}$ defines a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\mu_1$. Since the functional ${\\ensuremath{\\mathscr{F}}}_c$ given in terms of the threshold\n\\begin{align*}\nc := {\\ensuremath{\\mathrm{e}}}^{D\\sqrt{K^-N}\/2}\\,\\max\\{\\Vert\\rho_0\\Vert_{\\Ell^\\infty(\\mms,\\meas)}, \\Vert\\rho_1\\Vert_{\\Ell^\\infty(\\mms,\\meas)}\\}\n\\end{align*}\nanalogously to \\eqref{Eq:Functional FC} is weakly lower semicontinuous in ${\\ensuremath{\\mathscr{P}}}(J(\\mu_0,\\mu_1))$ \\cite[Lem.~3.13]{braun2022}, \\eqref{Eq:DENS BD} yields ${\\ensuremath{\\mathscr{F}}}_c(\\mu_t) =0$ for every $t\\in[0,1]$, which is the claim.\n\\end{proof}\n\n\\subsection{Equivalence with the entropic TCD condition}\\label{Sub:Equiv TCDs} In this section, we prove the equivalence of $\\smash{\\mathrm{TCD}_p^*(K,N)}$ with its entropic counterpart $\\smash{\\mathrm{TCD}_p^e(K,N)}$ introduced in \\cite{cavalletti2020}, provided the underlying space is timelike $p$-essentially nonbranching according to \\autoref{Def:Essentially nonbranching}. We also obtain equivalence of the respective weak with their strong versions; see \\autoref{Th:Equivalence TCD* and TCDe}. Yet another characterization will be obtained in \\autoref{Pr:MDPTS} below.\n\nIn this section, in addition to our standing assumptions on ${\\ensuremath{\\mathscr{X}}}$ we suppose the latter to be timelike $p$-essentially nonbranching for a fixed $p\\in (0,1)$. \n\nFor the convenience of the reader, let us recall the following notions \\cite[Def.~3.2, Prop.~3.3]{cavalletti2020}, for which we define ${\\ensuremath{\\mathscr{U}}}_N \\colon {\\ensuremath{\\mathscr{P}}}(\\mms)\\to [0,\\infty]$, $N\\in (0,\\infty)$, by\n\\begin{align}\\label{Eq:Expo Boltzmann}\n{\\ensuremath{\\mathscr{U}}}_N(\\mu) := {\\ensuremath{\\mathrm{e}}}^{-\\Ent_\\meas(\\mu)\/N}.\n\\end{align}\n\n\\begin{definition}\\label{Def:CD entropic} Let $K\\in\\R$ and $N\\in (0,\\infty)$. \n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item We say that ${\\ensuremath{\\mathscr{X}}}$ satis\\-fies the \\emph{entropic timelike curvature-dimension condition} $\\smash{\\mathrm{TCD}_p^e(K,N)}$ if for every timelike $p$-dualizable pair $(\\mu_0,\\mu_1)\\in\\Dom(\\Ent_\\meas)^2$, there exist a timelike proper-time parametrized $\\smash{\\ell_p}$-geo\\-desic $(\\mu_t)_{t\\in [0,1]}$ connecting $\\mu_0$ and $\\mu_1$  as well as a timelike $p$-dualizing coupling $\\pi\\in\\smash{\\Pi_\\ll(\\mu_0,\\mu_1)}$ such that for every $t\\in [0,1]$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{U}}}_N(\\mu_t) \\geq \\sigma_{K,N}^{(1-t)}\\big[\\Vert\\uptau\\Vert_{\\Ell^2(\\mms^2,\\pi)}\\big]\\,{\\ensuremath{\\mathscr{U}}}_N(\\mu_0) + \\sigma_{K,N}^{(t)}\\big[\\Vert\\uptau\\Vert_{\\Ell^2(\\mms^2,\\pi)}\\big]\\,{\\ensuremath{\\mathscr{U}}}_N(\\mu_1).\n\\end{align*}\n\\item If the previous claim holds for every strongly timelike $p$-dualizable  $(\\mu_0,\\mu_1)\\in({\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)\\cap\\Dom(\\Ent_\\meas))^2$, we say that ${\\ensuremath{\\mathscr{X}}}$ satisfies the \\emph{weak entropic timelike curvature-dimension condition} $\\smash{\\mathrm{wTCD}_p^e(K,N)}$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{theorem}\\label{Th:Equivalence TCD* and TCDe} The  following statements are equivalent for every given $K\\in\\R$ and $N\\in[1,\\infty)$.\n\\begin{enumerate}[label=\\textnormal{\\textcolor{black}{(}\\roman*\\textcolor{black}{)}}]\n\\item\\label{La:1} The condition $\\smash{\\mathrm{TCD}_p^*(K,N)}$ holds.\n\\item\\label{La:1.5} The condition $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ holds.\n\\item\\label{La:2} For every timelike $p$-dualizable  $(\\mu_0,\\mu_1) = (\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)^2$ there exists an $\\smash{\\ell_p}$-optimal geodesic plan ${\\ensuremath{\\boldsymbol{\\pi}}}\\in\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that for every $t\\in [0,1]$,  we have $\\smash{(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}=\\rho_t\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$, and for every $N'\\geq N$, the inequality\n\\begin{align*}\n\\rho_t(\\gamma_t)^{-1\/N'}&\\geq \\sigma_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_0(\\gamma_0)^{-1\/N'}  + \\sigma_{K,N'}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_1(\\gamma_1)^{-1\/N'}\n\\end{align*}\nholds for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$.\n\\item\\label{La:3} The condition $\\smash{\\mathrm{TCD}_p^e(K,N)}$ holds.\n\\item\\label{La:3.5} The condition $\\smash{\\mathrm{wTCD}_p^e(K,N)}$ holds.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark} The exceptional set in \\ref{La:2} may depend on $t$ and $N'$.\n\nBy \\autoref{Pr:iii to iv} and the proofs of \\autoref{Pr:ii to iii} and \\autoref{Pr:v to iii}, any of the claims in \\autoref{Th:Equivalence TCD* and TCDe} is equivalent to the following weaker version of \\ref{La:2}: for every pair $\\smash{(\\mu_0,\\mu_1)=(\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)^2}$ such that $\\smash{\\supp\\mu_0\\times\\supp\\mu_1\\subset\\mms_\\ll^2}$ and $\\rho_0,\\rho_1\\in\\Ell^\\infty(\\mms,\\meas)$, there exists $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that for every $t\\in[0,1]$, we have $\\smash{(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$ and\n\\begin{align*}\n\\rho_t(\\gamma_t)^{-1\/N}&\\geq \\sigma_{K,N}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_0(\\gamma_0)^{-1\/N}  + \\sigma_{K,N}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_1(\\gamma_1)^{-1\/N}\n\\end{align*}\nholds for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$, where $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas$.\n\nAn analogous note applies to \\autoref{Th:Equiv TCD with geo},  \\autoref{Th:Equivalence TMCP* and TMCPe}, and \\autoref{Th:Equivalence TMCP}.\n\\end{remark}\n\nIt is clear that \\ref{La:1} implies \\ref{La:1.5}, and that \\ref{La:3} yields \\ref{La:3.5}. Moreover, \\ref{La:2}  implies \\ref{La:1} by integration.  The proofs of \\ref{La:1.5} implying \\ref{La:2},  \\ref{La:2} implying \\ref{La:3}, and \\ref{La:3.5} implying \\ref{La:2} are, for the sake of clarity,  outsourced to \\autoref{Pr:ii to iii}, \\autoref{Pr:iii to iv}, and \\autoref{Pr:v to iii}, respectively. \n\n\\begin{remark}\\label{Re:Uniq} Below, we occasionally use two results which are only established later in \\autoref{Ch:TMCP}, namely  the following.\n\\begin{itemize}\n\\item First, the condition $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ implies the timelike measure-contraction property $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ according to \\autoref{Def:TMCP}, cf.~\\autoref{Pr:TMCP to TCD}.\n\\item Second, on timelike $p$-essentially nonbranching spaces satisfying the latter condition --- or $\\smash{\\mathrm{TMCP}_p^e(K,N)}$ according to \\autoref{Def:TMCPe} ---, chronological $\\smash{\\ell_p}$-optimal couplings and $\\smash{\\ell_p}$-optimal geodesic plans with sufficiently well-behaved endpoints are unique, cf.~\\autoref{Th:Uniqueness couplings}, \\autoref{Th:Uniqueness geodesics}, and \\autoref{Re:From TNB to TENB}. \n\\end{itemize}\nAs no result of this section is used in \\autoref{Ch:TMCP}, no circular reasoning occurs.\n\nThe key point of these results here is that the timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics and the $\\smash{\\ell_p}$-optimal couplings from \\autoref{Def:TCD*}, \\autoref{Def:TCD}, as well as  \\autoref{Def:TMCP}, which are a priori unrelated, are in fact induced by the same $\\smash{\\ell_p}$-optimal geodesic plan.\n\nThis means that we must have these uniqueness results at our disposal \\emph{a priori} to establish \\autoref{Th:Equivalence TCD* and TCDe}. In particular, these do \\emph{not} follow from \\autoref{Th:Equivalence TCD* and TCDe}, \\autoref{Re:From TNB to TENB}, and \\cite[Thm.~3.19, Thm.~3.20]{cavalletti2020}.\n\\end{remark}\n\n\\begin{proposition}\\label{Pr:ii to iii} Assume $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ for some $K\\in\\R$ and $N\\in [1,\\infty)$. Then for every timelike $p$-dualizable pair $(\\mu_0,\\mu_1) = (\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)^2$ there exists some ${\\ensuremath{\\boldsymbol{\\pi}}}\\in\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that for every $t\\in [0,1]$, we have $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}= \\rho_t\\, \\meas\\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$, and for every $N'\\geq N$,  the inequality\n\\begin{align*}\n\\rho_t(\\gamma_t)^{-1\/N'}\\geq \\sigma_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_0(\\gamma_0)^{-1\/N'} + \\sigma_{K,N'}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_1(\\gamma_1)^{-1\/N'}\n\\end{align*}\nholds for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$, where $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas$.\n\\end{proposition}\n\n\\begin{proof} \\textbf{Step 1.} \\emph{Reinforcement of the assumptions on $\\mu_0$ and $\\mu_1$.} We first assume that $\\smash{\\supp\\mu_0\\times\\supp\\mu_1\\subset\\mms_\\ll^2}$, that $\\mu_0,\\mu_1\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$, and that $\\rho_0,\\rho_1\\in\\Ell^\\infty(\\mms,\\meas)$. As mentioned above, timelike $p$-dualizable couplings and $\\ell_p$-optimal geodesic plans will then be unique all over Step 1, Step 2, and Step 3 without further notice. \n\n\n\\textbf{Step 2.} \\textit{Restriction.} Observe that the pair  $(\\mu_0,\\mu_1)$ is in fact strongly timelike $p$-dualizable thanks to  \\autoref{Re:Strong timelike}. Fix a $\\cap$-stable generator $\\{M_n : n\\in\\N\\}$  of the  Borel $\\sigma$-field of $\\mms$; to be precise, the $M_n$'s  should be chosen to generate the relative Borel $\\sigma$-field of the compact set $\\supp\\mu_0\\cup\\supp\\mu_1$, but we ignore this minor technicality by assuming compactness of $\\mms$ instead until Step 4. Given $n\\in\\N$ we cover $\\mms$ by the mutually disjoint Borel sets \n\\begin{align*}\nL_1 &:= M_1\\cap \\dots \\cap M_n,\\\\\nL_2 &:= M_1\\cap\\dots\\cap M_{n-1}\\cap M_n^{\\ensuremath{\\mathsf{c}}},\\,\\dots\\\\\nL_{2^n} &:= M_1^{\\ensuremath{\\mathsf{c}}} \\cap \\dots \\cap M_n^{\\ensuremath{\\mathsf{c}}}.\n\\end{align*}\nGiven the unique timelike $p$-dualizing coupling $\\smash{\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)}$ of $\\mu_0$ and $\\mu_1$, let us define $\\smash{\\mu_0^{ij},\\mu_1^{ij}\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)\\cap\\Dom(\\Ent_\\meas)}$ by \n\\begin{align*}\n\\mu_0^{ij}[A] &:= \\lambda_{ij}^{-1}\\,\\pi[(A\\cap  L_i) \\times L_j] = \\varrho_0^{ij}\\,\\meas,\\\\\n\\mu_1^{ij}[A] &:= \\lambda_{ij}^{-1}\\,\\pi[L_i\\times (A\\cap L_j)] = \\varrho_1^{ij}\\,\\meas\n\\end{align*}\nfor $i,j\\in\\{1,\\dots,2^n\\}$ provided $\\lambda_{ij} := \\pi[L_i\\times L_j]>  0$. Then \n\\begin{align}\\label{Eq:MSING}\n\\begin{split}\n\\mu_0^{ij} &\\perp \\mu_0^{i'j'},\\\\\n\\mu_1^{ij} &\\perp \\mu_1^{i'j'}\n\\end{split}\n\\end{align}\nfor every $i,i',j,j'\\in\\{1,\\dots,2^n\\}$ with $i\\neq i'$ or $j\\neq j'$. Also,  $\\smash{(\\mu_0^{ij},\\mu_1^{ij})}$ is strongly timelike $p$-dualizable by \\autoref{Re:Strong timelike}. By the consistency part in \\autoref{Pr:Consistency TCD} and the above mentioned uniqueness results, and $\\mathrm{wTCD}_p^*(K,N)$,  there exists some $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}^{ij}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0^{ij},\\mu_1^{ij})}$ with \n\\begin{align}\\label{Eq:THa inequ}\n\\begin{split}\n&\\int_{\\mathrm{TGeo}^\\uptau(\\mms)} \\sigma_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\varrho_0^{ij}(\\gamma_0)^{-1\/N'}\\d{\\ensuremath{\\boldsymbol{\\pi}}}^{ij}(\\gamma)\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\int_{\\mathrm{TGeo}^\\uptau(\\mms)} \\sigma_{K,N'}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\varrho_1^{ij}(\\gamma_1)^{-1\/N'}\\d{\\ensuremath{\\boldsymbol{\\pi}}}^{ij}(\\gamma)\\\\\n&\\qquad\\qquad \\leq \\int_{\\mathrm{TGeo}^\\uptau(\\mms)} \\varrho_t^{ij}(\\gamma_t)^{-1\/N'}\\d{\\ensuremath{\\boldsymbol{\\pi}}}^{ij}(\\gamma)\n\\end{split}\n\\end{align}\nfor every $t\\in [0,1]$ and every $N'\\geq N$; note that since $\\smash{\\mu_0^{ij},\\mu_1^{ij}\\in {\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)\\cap\\Dom(\\Ent_\\meas)}$ as a consequence of the assumption $\\rho_0,\\rho_1\\in\\Ell^\\infty(\\mms,\\meas)$, we have $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^{ij} = \\smash{\\rho_t^{ij}}\\,\\meas \\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$ thanks to \\autoref{Le:Stu}.\n\n\\textbf{Step 3.} \\textit{Pasting plans together and conclusion.}\nBy uniqueness, we get\n\\begin{align*}\n{\\ensuremath{\\boldsymbol{\\pi}}} = \\lambda_{11}\\,{\\ensuremath{\\boldsymbol{\\pi}}}^{11} + \\lambda_{12}\\,{\\ensuremath{\\boldsymbol{\\pi}}}^{12} + \\dots + \\lambda_{2^n2^n}\\,{\\ensuremath{\\boldsymbol{\\pi}}}^{2^n2^n}\n\\end{align*}\nfor every $n\\in\\N$, since the right-hand side belongs to $\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$. In particular, setting $\\smash{\\mu_t := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas}$, $t\\in[0,1]$ --- the $\\meas$-ab\\-solute continuity stemming from \\autoref{Le:Stu} and again uniqueness ---, we have\n\\begin{align*}\n\\rho_t = \\lambda_{11}\\,\\varrho_t^{11} + \\lambda_{12}\\,\\varrho_t^{12} + \\dots + \\lambda_{2^n2^n}\\,\\varrho_t^{2^n2^n}\\quad\\meas\\text{-a.e.}\n\\end{align*}\nMoreover, \\eqref{Eq:MSING} and \\autoref{Le:Mutually singular} imply\n\\begin{align*}\n(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^{ij} \\perp (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^{i'j'}\n\\end{align*}\nfor every $t\\in (0,1)$ and every $i,i',j,j'\\in\\{1,\\dots,2^n\\}$ with $i\\neq i'$ or $j\\neq j'$, i.e.\n\\begin{align*}\n\\meas\\big[\\{\\varrho_t^{ij} > 0\\} \\cap \\{\\varrho_t^{i'j'}>0\\}\\big] = 0.\n\\end{align*}\nSetting $G_{ij} := (\\eval_0,\\eval_1)^{-1}(L_i\\times L_j)$, by construction  we obtain from \\eqref{Eq:THa inequ} that\n\\begin{align*}\n&\\int_{G_{ij}} \\sigma_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_0(\\gamma_0)^{-1\/N'}\\d{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma)\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\int_{G_{ij}} \\sigma_{K,N'}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_1(\\gamma_1)^{-1\/N'}\\d{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma)\\\\\n&\\qquad\\qquad \\leq \\int_{G_{ij}} \\rho_t(\\gamma_t)^{-1\/N'}\\d{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma).\n\\end{align*}\nSince $i,j\\in\\{1,\\dots,2^n\\}$ and $n\\in\\N$ were arbitrary, the claim follows.\n\n\\textbf{Step 4.} \\textit{Removing the restrictions on $\\mu_0$ and $\\mu_1$.} We initially address the first two restrictions in Step 1, then the last one. \n\n\\textbf{Step 4.1.} If the pair $\\smash{(\\mu_0,\\mu_1) \\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)^2}$ with $\\rho_0,\\rho_1\\in\\Ell^\\infty(\\mms,\\meas)$ is merely timelike $p$-dualizable, let $\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)$ be a corresponding timelike $p$-dualizing coupling. Since $\\smash{\\mms_\\ll^2}$ is open and $\\met$ is separable, we cover $\\smash{\\supp\\pi\\cap \\mms_\\ll^2}$ by countably many (not necessarily disjoint) relatively compact rectangles $A_i\\times B_i$ with $\\smash{\\bar{A}_i\\times\\bar{B}_i \\subset \\mms_\\ll^2}$, where $A_i,B_i\\subset\\mms$ are open, $i\\in\\N$. Set $\\pi^0 := 0$, and define the measure $\\pi^i$ on $\\mms$ inductively by\n\\begin{align*}\n\\pi^i := (\\pi - \\pi^{i-1})\\mres (A_i\\times B_i).\n\\end{align*}\nThen $\\smash{\\pi = \\pi^1 + \\pi^2 + \\dots}$ by construction. Define $\\smash{\\hat{\\pi}^i\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms^2)}$ by\n\\begin{align*}\n\\hat{\\pi}^i := \\pi^i[\\mms^2]^{-1}\\,\\pi^i\n\\end{align*}\nprovided $\\smash{\\pi^i[\\mms^2]> 0}$, and consider its marginals\n\\begin{align*}\n\\mu_0^i := (\\pr_1)_\\push\\hat{\\pi}^i = \\varrho_0^i\\,\\meas,\\\\\n\\mu_1^i := (\\pr_2)_\\push\\hat{\\pi}^i = \\varrho_1^i\\,\\meas.\n\\end{align*} \nBy \\autoref{Re:Strong timelike}, the pair $\\smash{(\\mu_0^i,\\mu_1^i)}$ is strongly timelike $p$-dualizable for every $i\\in\\N$. Thus, the previous steps imply the existence of $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}^i\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0^i,\\mu_1^i)}$ such that for every $t\\in (0,1)$, we have $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^i = \\smash{\\varrho_t^i}\\,\\meas\\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$,  and for every $N'\\geq N$,\n\\begin{align}\\label{Eq:The Inequality rhoT}\n\\varrho_t^i(\\gamma_t)^{-1\/N'} \\geq \\sigma_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\varrho_0^i(\\gamma_0)^{-1\/N'} + \\sigma_{K,N'}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\varrho_1^i(\\gamma_1)^{-1\/N'}\n\\end{align}\nholds for $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}^i}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$. By construction,\n\\begin{align*}\n\\mu_0^i &\\perp \\mu_0^j,\\\\\n\\mu_1^i &\\perp \\mu_1^j\n\\end{align*}\nfor every $i,j\\in\\N$ with $i\\neq j$, and from  \\autoref{Le:Mutually singular} we derive\n\\begin{align*}\n(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^i \\perp (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^j.\n\\end{align*}\nNote that the measure\n\\begin{align*}\n{\\ensuremath{\\boldsymbol{\\pi}}} := \\pi^1[\\mms^2]\\,{\\ensuremath{\\boldsymbol{\\pi}}}^1 + \\pi^2[\\mms^2]\\,{\\ensuremath{\\boldsymbol{\\pi}}}^2 + \\dots\n\\end{align*}\nis an $\\smash{\\ell_p}$-optimal geodesic plan interpolating $\\mu_0$ and $\\mu_1$. It satisfies $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas\\in \\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$ for every $t\\in[0,1]$, and \n\\begin{align*}\n\\rho_t = \\pi^1[\\mms^2]\\,\\varrho_t^1 + \\pi^2[\\mms^2]\\,\\varrho_t^2 + \\dots\\quad\\meas\\textnormal{-a.e.}\n\\end{align*}\nTogether with \\eqref{Eq:The Inequality rhoT} this discussion implies, for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$,\n\\begin{align*}\n\\rho_t(\\gamma)^{-1\/N'} \\geq \\sigma_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_0(\\gamma_0)^{-1\/N'} + \\sigma_{K,N'}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_1(\\gamma_1)^{-1\/N'}.\n\\end{align*}\n\n\\textbf{Step 4.2.} Finally, given any timelike $p$-dualizable pair $(\\mu_0,\\mu_1)= (\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)^2}$, for every $i,j\\in\\N$ we set \n\\begin{align*}\nE_i &:= \\{i-1 \\leq \\rho_0^i < i\\},\\\\\nF_j &:= \\{j-1 \\leq \\rho_1^j < j\\}.\n\\end{align*}\nBy restriction \\cite[Lem. 2.10]{cavalletti2020}, $\\smash{(\\mu_0^i,\\mu_1^j)}$ is timelike $p$-dualizable, where \n\\begin{align*}\n\\mu_0^i &:= \\mu_0[E_i]^{-1}\\,\\mu_0\\mres E_i = \\rho_0^i\\,\\meas,\\\\\n\\mu_1^j &:= \\mu_1[F_j]^{-1}\\,\\mu_1\\mres F_j = \\rho_1^j\\,\\meas\n\\end{align*}\nprovided $\\mu_0[E_i] > 0$ as well as $\\mu_1[F_j]>0$. Clearly  $\\smash{\\rho_0^i,\\rho_1^j\\in\\Ell^\\infty(\\mms,\\meas)}$, and a similar argument as above  leads to the conclusion. We omit the details.\n\\end{proof}\n\n\\begin{proposition}\\label{Pr:iii to iv} Let $K\\in\\R$ and $N\\in [1,\\infty)$. Assume that for every timelike $p$-dualizable pair $(\\mu_0,\\mu_1)=(\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in\\Dom(\\Ent_\\meas)^2$, there is $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that for every $t\\in [0,1]$, we have $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)$ and\n\\begin{align}\\label{Eq:rhot in}\n\\rho_t(\\gamma_t)^{-1\/N}\\geq \\sigma_{K,N}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_0(\\gamma_0)^{-1\/N} + \\sigma_{K,N}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_1(\\gamma_1)^{-1\/N}\n\\end{align}\nholds for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$. Then ${\\ensuremath{\\mathscr{X}}}$ obeys $\\smash{\\mathrm{TCD}_p^e(K,N)}$.\n\\end{proposition}\n\n\\begin{proof} The assignment $\\mu_t := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}$ creates a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\mu_1$. Moreover, $\\smash{\\pi := (\\eval_0,\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\Pi_\\ll(\\mu_0,\\mu_1)}$ forms a timelike $p$-dualizing coupling of $\\mu_0$ and $\\mu_1$. Taking logarithms on both sides of \\eqref{Eq:rhot in} and recalling \\eqref{Eq:Distortion coeff property}, we get\n\\begin{align*}\n-\\frac{1}{N}\\log\\rho_t(\\gamma_t) \\geq {\\ensuremath{\\mathrm{G}}}_t\\Big[\\!-\\!\\frac{1}{N}\\log\\rho_0(\\gamma_0), -\\frac{1}{N}\\log\\rho_1(\\gamma_1), \\frac{K}{N}\\,\\uptau^2(\\gamma_0,\\gamma_1)\\Big]\n\\end{align*}\nfor ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$; recall that the function ${\\ensuremath{\\mathrm{G}}}_t$ is defined in \\eqref{Eq:GtHt}. Integrating this inequality with respect to ${\\ensuremath{\\boldsymbol{\\pi}}}$ and using Jensen's inequality together with joint convexity of ${\\ensuremath{\\mathrm{G}}}_t$ yields\n\\begin{align*}\n-\\frac{1}{N}\\Ent_\\meas(\\mu_t) \\geq {\\ensuremath{\\mathrm{G}}}_t\\Big[\\!-\\!\\frac{1}{N}\\Ent_\\meas(\\mu_0), -\\frac{1}{N}\\Ent_\\meas(\\mu_1), \\frac{K}{N}\\,\\big\\Vert\\uptau\\big\\Vert_{\\Ell^2(\\mms^2,\\pi)}^2\\Big].\n\\end{align*}\nHere, in the case $K>0$ and $\\smash{\\Vert\\uptau\\Vert_{\\Ell^2(\\mms^2,\\pi)} = \\infty}$ the right-hand side is consistently interpreted as $\\infty$, which implies  $\\Ent_\\meas(\\mu_t) = -\\infty$; for $K<0$ and $\\smash{\\Vert\\uptau\\Vert_{\\Ell^2(\\mms^2,\\pi)}}=\\infty$ the right-hand side has an evident interpretation. Exponentiating both sides now yields the desired $\\smash{\\mathrm{TCD}_p^e(K,N)}$ condition.\n\\end{proof}\n\n\\begin{proposition}\\label{Pr:v to iii} Assume $\\smash{\\mathrm{wTCD}_p^e(K,N)}$ for some $K\\in\\R$ and $N\\in [1,\\infty)$. Then for every timelike $p$-dualizable pair $(\\mu_0,\\mu_1) = (\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)^2$ there exists ${\\ensuremath{\\boldsymbol{\\pi}}}\\in\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that for every $t\\in [0,1]$, we have $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas$, and for every $N'\\geq N$, the inequality \n\\begin{align*}\n\\rho_t(\\gamma_t)^{-1\/N'}\\geq \\sigma_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_0(\\gamma_0)^{-1\/N'} + \\sigma_{K,N'}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_1(\\gamma_1)^{-1\/N'}\n\\end{align*}\nholds for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$, where $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas$. \n\\end{proposition}\n\n\\begin{proof} The proof is similar to the one of \\autoref{Pr:ii to iii} --- whose notation we retain here ---, whence we only outline the main differences. Again, it suffices to consider the case  $\\smash{\\supp\\mu_0\\times\\supp\\mu_1\\subset\\mms_\\ll^2}$,  $\\mu_0,\\mu_1\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$, and  $\\smash{\\rho_0,\\rho_1\\in\\Ell^\\infty(\\mms,\\meas)}$. In particular, these restrictions imply $\\mu_0,\\mu_1\\in\\Dom(\\Ent_\\meas)$. The parallel uniqueness results outlined in \\autoref{Re:Uniq} under $\\smash{\\mathrm{TCD}_p^e(K,N)}$ which are implicitly used below are due to \\cite[Thm.~3.19, Thm.~3.20]{cavalletti2020}, see also \\autoref{Re:From TNB to TENB}.\n\nLet $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ be an  $\\smash{\\ell_p}$-optimal geodesic plan interpolating $\\mu_0$ and  $\\mu_1$. Given any $i,j\\in\\{1,\\dots,2^n\\}$, we define\n\\begin{align*}\n\\mu_0^{ij} &:= \\lambda_{ij}^{-1}\\,(\\pr_1)_\\push\\big[(\\eval_0,\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}\\mres(L_i\\times L_j)\\big] = \\varrho_0^{ij}\\,\\meas\\\\\n\\mu_1^{ij} &:= \\lambda_{ij}^{-1}\\,(\\pr_2)_\\push\\big[(\\eval_0,\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}\\mres(L_i\\times L_j)\\big] = \\varrho_1^{ij}\\,\\meas\n\\end{align*}\nprovided $\\smash{\\lambda_{ij} := {\\ensuremath{\\boldsymbol{\\pi}}}[G_{ij}] > 0}$. Recall that $\\smash{\\mathrm{wTCD}_p^e(K,N)}$ implies $\\smash{\\mathrm{wTCD}_p^e(K,N')}$ by  \\cite[Lem.~3.10]{cavalletti2020}.  Arguing as for \\autoref{Pr:ii to iii}, invoking $\\smash{\\mathrm{wTCD}_p^e(K,N')}$ for the strongly timelike $p$-dualizable  pair $\\smash{(\\mu_0^{ij}, \\mu_1^{ij})} \\in \\smash{({\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)\\cap\\Dom(\\Ent_\\meas))^2}$, employing the uniqueness both of ${\\ensuremath{\\boldsymbol{\\pi}}}$ as well as the involved timelike $p$-dualizing coupling $\\pi^{ij}\\in\\smash{\\Pi_\\ll(\\mu_0^{ij},\\mu_1^{ij})}$ by \\autoref{Re:From TNB to TENB}, and taking logarithms on both sides of the resulting ``pasted inequality'' we infer that\n\\begin{align*}\n&-\\frac{\\lambda_{ij}^{-1}}{N'}\\int_{G_{ij}}\\log \\rho_t(\\gamma_t)\\d{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma)\\\\\n&\\qquad\\qquad \\geq {\\ensuremath{\\mathrm{G}}}_t\\Big[\\!-\\!\\frac{\\lambda_{ij}^{-1}}{N'}\\int_{G_{ij}} \\log\\rho_0(\\gamma_0)\\d{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma),-\\frac{\\lambda_{ij}^{-1}}{N'}\\int_{G_{ij}} \\log\\rho_1(\\gamma_1)\\d{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma),\\\\\n&\\qquad\\qquad\\qquad\\qquad \\lambda_{ij}^{-1}\\,\\frac{K}{N'}\\int_{G_{ij}}\\uptau^2(\\gamma_0,\\gamma_1)\\d{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma)\\Big]\\\\\n&\\qquad\\qquad\\geq \\lambda_{ij}^{-1}\\int_{G_{ij}} {\\ensuremath{\\mathrm{G}}}_t\\Big[\\!-\\!\\frac{1}{N'}\\log\\rho_0(\\gamma_0), -\\frac{1}{N'}\\log\\rho_1(\\gamma_1), \\frac{K}{N'}\\,\\uptau^2(\\gamma_0,\\gamma_1)\\Big]\\d{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma).\n\\end{align*}\nHere the deduction that $\\smash{(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$ comes directly from the definition of the functional ${\\ensuremath{\\mathscr{U}}}_N$, and the last inequality follows from joint convexity of ${\\ensuremath{\\mathrm{G}}}_t$ and Jensen's inequality. Since $i,j\\in\\{1,\\dots,2^n\\}$ and $n\\in\\N$ were arbitrary, the claim follows by taking exponentials of both integrands, respectively.\n\\end{proof}\n\nBy evident adaptations of the above arguments, we also obtain the following  result for the conditions from \\autoref{Def:TCD}.\n\n\\begin{theorem}\\label{Th:Equiv TCD with geo} The  following statements are equivalent for every given $K\\in\\R$ and $N\\in[1,\\infty)$.\n\\begin{enumerate}[label=\\textnormal{\\textcolor{black}{(}\\roman*\\textcolor{black}{)}}]\n\\item The condition $\\smash{\\mathrm{TCD}_p(K,N)}$ holds.\n\\item The condition $\\smash{\\mathrm{wTCD}_p(K,N)}$ holds.\n\\item For every timelike $p$-dualizable  $(\\mu_0,\\mu_1) = (\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)^2$  there exists an $\\smash{\\ell_p}$-optimal geodesic plan ${\\ensuremath{\\boldsymbol{\\pi}}}\\in\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that  for every $t\\in [0,1]$,   we have $\\smash{(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} =\\rho_t\\, \\meas\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$, and for every $N'\\geq N$, the inequality\n\\begin{align*}\n\\rho_t(\\gamma_t)^{-1\/N'}&\\geq \\tau_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_0(\\gamma_0)^{-1\/N'}  + \\tau_{K,N'}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_1(\\gamma_1)^{-1\/N'}\n\\end{align*}\nholds for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$.\n\\end{enumerate}\n\\end{theorem}\n\n\\subsection{From local to global}\\label{Sub:Local global} The goal of this section is to establish the equivalence of $\\smash{\\mathrm{TCD}_p^*(K,N)}$ to its corresponding local version in \\autoref{Def:TCD loc} below, i.e.~to establish a Lorentzian analogue of the local-to-global property from \\cite[Ch.~5]{bacher2010}.\n\nTo this aim, in addition to our standing assumptions on the base space, suppose again ${\\ensuremath{\\mathscr{X}}}$ to be timelike $p$-essentially nonbranching for  $p\\in (0,1)$.\n\n\n\\begin{definition}\\label{Def:TCD loc} Let $K\\in\\R$ and $N\\in[1,\\infty)$. \n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item We say that ${\\ensuremath{\\mathscr{X}}}$ satisfies $\\smash{\\mathrm{TCD}_p^*(K,N)}$ \\emph{locally}, briefly $\\smash{\\mathrm{TCD}_{p,\\loc}^*(K,N)}$, if there exists an open cover $(\\mms_i)_{i\\in I}$ of $\\mms$ with the following property. For every $i\\in I$ and every pair $(\\mu_0,\\mu_1)=(\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms_i,\\meas)^2}$ such that $\\smash{\\supp\\mu_0\\times\\supp\\mu_1\\subset (\\mms_i)_\\ll^2}$, there is a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ in  ${\\ensuremath{\\mathscr{P}}}(\\mms)$ from $\\mu_0$ to $\\mu_1$ and an $\\smash{\\ell_p}$-optimal coupling $\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)$ such that for every $t\\in[0,1]$ and every $N'\\geq N$,\n\\begin{align}\\label{Eq:RED}\n\\begin{split}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t) &\\leq -\\int_{\\mms^2} \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N'}\\d\\pi(x^0,x^1)\\\\\n&\\qquad\\qquad -\\int_{\\mms^2}\\sigma_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\,\\rho_1(x^1)^{-1\/N'}\\d\\pi(x^0,x^1).\n\\end{split}\n\\end{align}\n\\item If the previous statement holds for $\\smash{\\sigma_{K,N'}^{(1-t)}}$ and $\\smash{\\sigma_{K,N'}^{(t)}}$ replaced by $\\smash{\\tau_{K,N'}^{(1-t)}}$ and $\\smash{\\tau_{K,N'}^{(t)}}$, respectively, ${\\ensuremath{\\mathscr{X}}}$ is termed to satisfy $\\smash{\\mathrm{TCD}_p(K,N)}$ \\emph{locally}, briefly $\\smash{\\mathrm{TCD}_{p,\\loc}(K,N)}$.\n\\end{enumerate} \n\\end{definition}\n\n\n\\begin{proposition}\\label{Pr:MDPTS} Given any $K\\in\\R$ and $N\\in[1,\\infty)$, the property $\\smash{\\mathrm{TCD}_p^*(K,N)}$ holds if and only if the following does. For every $\\smash{\\mu_0,\\mu_1\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ with $\\supp\\mu_0\\times\\supp\\mu_1 \\subset\\smash{\\mms_\\ll^2}$ there  exists $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that for every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}) \\leq \\sigma_{K,N'}^{(1\/2)}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_0) + \\sigma_{K,N'}^{(1\/2)}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_1),\n\\end{align*}\nwhere\n\\begin{align*}\n\\theta := \\begin{cases} \\sup \\uptau(\\supp\\mu_0\\times\\supp\\mu_1) & \\textnormal{if }K<0,\\\\\n\\inf \\uptau(\\supp\\mu_0\\times\\supp\\mu_1) & \\textnormal{otherwise}.\n\\end{cases}\n\\end{align*}\n\\end{proposition}\n\n\\begin{proof} We only outline the main differences of the proof to its counterpart from \\cite[Prop.~2.10]{bacher2010}. The forward implication is clear. Conversely, similarly to the proof of \\autoref{Th:Good geos TCD}, by successively gluing plans corresponding to ``midpoints'' for which the R\u00e9nyi entropy obeys the given inequality as in the proof of \\cite[Prop.~2.10]{bacher2010}, we inductively construct a sequence $\\smash{({\\ensuremath{\\boldsymbol{\\alpha}}}^n)_{n\\in\\N}}$ in $\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that for every $n\\in\\N$ and every odd $k\\in\\{1,\\dots,2^n-1\\}$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{k2^{-n}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^n) &\\leq \\sigma_{K,N'}^{(1\/2)}(2^{-n+1}\\,\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{(k-1)2^{-n}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^n)\\\\\n&\\qquad\\qquad + \\sigma_{K,N'}^{(1\/2)}(2^{-n+1}\\,\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{(k+1)2^{-n}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^n)\\\\\n&\\leq \\sigma_{K,N'}^{(1\/2)}(2^{-n+1}\\,\\theta)\\,\\sigma_{K,N'}^{(1-(k-1)2^{-n})}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_0)\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\sigma_{K,N'}^{(1\/2)}(2^{-n+1}\\,\\theta)\\,\\sigma_{K,N'}^{((k-1)2^{-n})}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_1)\\\\\n&\\qquad\\qquad + \\sigma_{K,N'}^{(1\/2)}(2^{-n+1}\\,\\theta)\\,\\sigma_{K,N'}^{(1-(k+1)2^{-n})}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_0)\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\sigma_{K,N'}^{(1\/2)}(2^{-n+1}\\,\\theta)\\,\\sigma_{K,N'}^{((k+1)2^{-n})}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_1)\\\\\n&= \\sigma_{K,N'}^{(1-k2^{-n})}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_0) + \\sigma_{K,N'}^{(k2^{-n})}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_1).\n\\end{align*}\nholds for every $N'\\geq N$. The latter equality is a crucial property of the distortion coefficients $\\smash{\\sigma_{K,N'}^{(r)}}$ \\cite[Lem.~3.1]{rajala2012b} which does not hold for $\\smash{\\tau_{K,N'}^{(r)}}$. Moreover, note that to proceed by induction in this strategy, it is important to know that the respective ``midpoints'' come from $\\smash{\\ell_p}$-optimal geodesic plans, which ensures that all chronology re\\-lations are preserved through this process.\n\nAs in the proof of \\autoref{Th:Good geos TCD}, $\\smash{({\\ensuremath{\\boldsymbol{\\alpha}}}^n)_{n\\in\\N}}$ admits an accumulation point ${\\ensuremath{\\boldsymbol{\\alpha}}} \\in \\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$, giving rise to a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\mu_1$ via $\\mu_t := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}$. Thanks to the weak lower semicontinuity of the R\u00e9nyi entropy for probability measures with uniformly compact support, the latter satisfies, for every $t\\in[0,1]$ and every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t)\\leq \\sigma_{K,N'}^{(1-t)}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_0) + \\sigma_{K,N'}^{(t)}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_1).\n\\end{align*}\n\nAs for \\cite[Prop.~2.8]{bacher2010}, where the replacement of \\cite[Rem.~2.9]{bacher2010} is \\autoref{Le:Stu},  and \\autoref{Pr:ii to iii}, and using \\autoref{Th:Equivalence TCD* and TCDe} we obtain $\\smash{\\mathrm{TCD}_p^*(K,N)}$.\n\\end{proof}\n\n\\begin{definition}\\label{Def:Chron lp geod} We term $\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ \\emph{chronologically $\\smash{\\ell_p}$-geodesic} if  every $\\mu_0,\\mu_1\\in\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ with $\\supp\\mu_0\\times\\supp\\mu_1\\subset\\smash{\\mms_\\ll^2}$ are joined by a timelike proper-time parame\\-tri\\-zed $\\smash{\\ell_p}$-geodesic $(\\mu_r)_{r\\in[0,1]}$ which consists of $\\meas$-absolutely continuous measures.\n\\end{definition}\n\n\\begin{theorem}\\label{Th:Local to global} Let $K\\in\\R$ and $N\\in[1,\\infty)$. Then $\\smash{\\mathrm{TCD}_p^*(K,N)}$ holds if and only if $\\smash{\\mathrm{TCD}_{p,\\loc}^*(K,N)}$ is satisfied and $\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ is chronologically $\\smash{\\ell_p}$-geodesic.\n\\end{theorem}\n\nWe will prove \\autoref{Th:Local to global} only in the case $K\\geq 0$, the other situation follows  by analogous computations. To this aim, we formulate the following property in  \\autoref{Def:Cn} for which, given any $m\\in\\N_0$, we define\n\\begin{align*}\nI_m := \\{k\\,2^{-m} : k\\in\\{0,\\dots,2^m\\}\\}.\n\\end{align*}\nMoreover, given a collection $\\mu:= (\\mu_t)_{t\\in[0,1]}$ in ${\\ensuremath{\\mathscr{P}}}(\\mms)$, let $\\smash{G_\\mu^m}$ denote the set of all $\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$ with $\\gamma_t\\in\\supp\\mu_t$ for every $t\\in I_m$.\n\n\\begin{definition}\\label{Def:Cn} Let $C\\subset\\mms$ be a compact set.  Given $m\\in\\N_0$, we say that the property ${\\ensuremath{\\mathrm{P}}}_m(C)$ is satisfied if for every timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\mu:=(\\mu_r)_{r\\in[0,1]}$ consisting of $\\meas$-absolutely continuous measures with $\\mu_0,\\mu_1\\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$ with $\\smash{\\supp\\mu_0\\times\\supp\\mu_1\\subset\\mms_\\ll^2\\cap C^2}$, and every $s,t\\in I_n$ with $t-s =2^{-m}$ there is an $\\smash{\\ell_p}$-optimal geodesic plan $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_s,\\mu_t)}$ with\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}) \\leq \\sigma_{K,N'}^{(1\/2)}(\\theta_{s,t}^m)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_s) + \\sigma_{K,N'}^{(1\/2)}(\\theta_{s,t}^m)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t),\n\\end{align*}\nwhere\n\\begin{align}\\label{Eq:Thetat}\n\\theta_{s,t}^m := \\inf\\{\\uptau(\\gamma_s,\\gamma_t) : \\gamma\\in G_\\mu^m\\}.\n\\end{align}\n\\end{definition}\n\n\\begin{lemma}\\label{Le:PnC} Retaining the notation of \\autoref{Def:Cn}, ${\\ensuremath{\\mathrm{P}}}_m(C)$ implies ${\\ensuremath{\\mathrm{P}}}_{m-1}(C)$ for every $m\\in\\N$.\n\\end{lemma}\n\n\\begin{proof} Let a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\smash{\\mu:=(\\mu_r)_{r\\in[0,1]}}$ according to  ${\\ensuremath{\\mathrm{P}}}_n(C)$ be given. Let $s,t \\in I_{m-1}$ such that $\\smash{t-s = 2^{1-m}}$, and let $\\smash{\\theta:=\\theta_{s,t}^{m-1}}$ be defined with respect to $\\mu$. Inductively, we build a sequence $\\smash{(\\mu^i)_{i\\in\\N_0}}$ of timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics $\\smash{\\mu^i := (\\mu_r^i)_{r\\in[0,1]}}$ obeying $\\smash{\\mu^i_r = \\mu_r}$ for every $i\\in\\N_0$ and every $r\\in[0,s]\\cup[t,1]$ as follows. Initially, set $\\smash{\\mu^0 := \\mu}$. \n\n\\textbf{Step 1.} \\textit{Construction for odd $i\\in\\N_0$.} If the element $\\smash{\\mu^{2i}}$ has been constructed for a given $i\\in\\N_0$, as in the proof of \\cite[Thm.~2.10]{ambrosiogigli2013} and using that the respective ``midpoints'' result from an $\\smash{\\ell_p}$-optimal geodesic plan interpolating its endpoints, we construct a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\smash{\\mu^{2i+1}:= (\\mu_r^{2i+1})_{r\\in[0,1]}}$ with the subsequent properties. For every $r\\in[0,s]\\cup[t,1]$, we have $\\smash{\\mu_r^{2i+1}=\\mu_r}$, and moreover, for every $N'\\geq N$, \n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+2^{-m-1}}^{2i+1}) &\\leq \\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_s) + \\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+2^{-m}}^{2i}),\\\\\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+3\\times 2^{-m-1}}^{2i+1}) &\\leq \\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+2^{-m}}^{2i}) + \\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t).\n\\end{align*}\nHere we have used the property ${\\ensuremath{\\mathrm{P}}}_m(C)$, the inequalities\n\\begin{align*}\n2\\,\\theta_{s,s+2^{-m}}^m &\\geq \\theta,\\\\\n2\\, \\theta_{s+2^{-m},t}^m &\\geq \\theta, \n\\end{align*}\nwhere both quantities on the left-hand sides are defined with respect to $\\smash{\\mu^{2i}}$, and nonincreasingness of $\\smash{\\sigma_{K,N'}^{(1\/2)}(\\vartheta)}$ in $\\vartheta\\geq 0$ by our assumption $K\\geq 0$.\n\n\\textbf{Step 2.} \\textit{Construction for even $i\\in\\N_0$.} Given $\\smash{\\mu^{2i+1}}$ exhibited in Step 1 above, similarly to this step we construct a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\smash{\\mu^{2i+2} := (\\mu_r^{2i+2})_{r\\in[0,1]}}$ with the property that $\\smash{\\mu_r^{2i+2} = \\mu_r^{2i+1}}$ for every $r\\in[0,s+2^{-m-1}]\\cup\\smash{[s+3\\times 2^{-m-1},1]}$ such that for every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+2^{-m}}^{2i+2}) \\leq \\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+2^{-m-1}}^{2i+1}) + \\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+3\\times 2^{-m-1}}^{2i+1}).\n\\end{align*}\n\n\\textbf{Step 3.} \\textit{Conclusion.} Pasting together the inequalities from Step 1 and Step 2 yields, for every $i\\in\\N_0$ and every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu^{2i+2}_{s+2^{-m}}) &\\leq 2\\,\\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]^2\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+2^{-m}}^{2i})\\\\\n&\\qquad\\qquad + \\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]^2\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_s) + \\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]^2\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t). \n\\end{align*}\nIterating this inequality gives\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+2^{-m}}^{2i})&\\leq 2^i\\,\\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]^2\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+2^{-m}})\\\\\n&\\qquad\\qquad + \\frac{1}{2}\\sum_{k=1}^i 2^k\\,\\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]^{2k}\\,\\big[{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_s) + {\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t)\\big].\n\\end{align*}\n\nAs $\\smash{\\mu_0^{2i} = \\mu_0}$ and $\\smash{\\mu_1^{2i}= \\mu_1}$ for every $i\\in\\N_0$ by construction, using \\autoref{Le:Villani lemma for geodesic} we get the existence of a sequence $\\smash{({\\ensuremath{\\boldsymbol{\\pi}}}^{2i})_{i\\in\\N_0}}$ in $\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ with $\\smash{\\mu_r^{2i} = (\\eval_r)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^{2i}}$ for every $r\\in[0,1]$ which  converges weakly, up to a nonrelabeled subsequence, to an $\\smash{\\ell_p}$-optimal geodesic plan $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$. By ${\\ensuremath{\\mathscr{K}}}$-global hyperbolicity, lower semicontinuity of ${\\ensuremath{\\mathscr{S}}}_{N'}$ on ${\\ensuremath{\\mathscr{P}}}(J(\\mu_0,\\mu_1))$, and the same computations as for  \\cite[Clm.~5.2]{bacher2010} for the distortion coefficients on the right-hand side of the previous inequality, setting $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t} := (\\Restr_s^t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_s,\\mu_t)}$ we get\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}) \\leq \\sigma_{K,N'}^{(1\/2)}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_s) + \\sigma_{K,N'}^{(1\/2)}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t).\n\\end{align*}\nHere $\\smash{\\Restr_s^t}$ is defined in \\eqref{Eq:Restr def}. This accomplishes ${\\ensuremath{\\mathrm{P}}}_{n-1}(C)$.\n\\end{proof}\n\n\\begin{proof}[Proof of \\autoref{Th:Local to global}] The forward implication is a consequence of \\autoref{Le:Stu}. \n\nConcerning the backward implication, by the partition  procedure in the proof of \\autoref{Pr:ii to iii} it suffices to prove the desired inequality defining the condition $\\smash{\\mathrm{TCD}_p^*(K,N)}$ for every $\\smash{\\mu_0,\\mu_1\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ with $\\supp\\mu_0\\times\\supp\\mu_1\\subset\\smash{\\mms_\\ll^2}$. \n\nThanks to ${\\ensuremath{\\mathscr{K}}}$-global hyperbolicity, every member of any timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic from $\\mu_0$ to $\\mu_1$ has support in $J(\\mu_0,\\mu_1)$. By $\\smash{\\mathrm{TCD}_{p,\\loc}^*(K,N)}$ and by compactness of $J(\\mu_0,\\mu_1)$, there exist $\\delta > 0$, a disjoint cover $L_1,\\dots,L_n\\subset\\mms$ of $J(\\mu_0,\\mu_1)$, where $n\\in\\N$, and closed sets $\\smash{C_k \\subset\\mms}$ containing $\\smash{{\\ensuremath{\\mathsf{B}}}^\\met(L_k,\\delta)}$, $k\\in\\{1,\\dots,n\\}$, such that every two measures in $\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(C_k,\\meas)}$ are joined by a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic obeying \\eqref{Eq:RED}. Let $m\\in \\N$ with\n\\begin{align}\\label{Eq:m choice}\n2^{-m} \\leq \\delta.\n\\end{align}\n\nWe claim  $\\smash{{\\ensuremath{\\mathrm{P}}}_m(J(\\mu_0,\\mu_1))}$. Let $(\\mu_r)_{r\\in[0,1]}$ be a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic from $\\mu_0$ to $\\mu_1$ consisting of $\\meas$-absolutely continuous measures. Let ${\\ensuremath{\\boldsymbol{\\pi}}}\\in\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ represent  $(\\mu_r)_{r\\in[0,1]}$, and define $\\smash{\\varpi\\in{\\ensuremath{\\mathscr{P}}}(\\mms^{2^m+1})}$ by\n\\begin{align}\\label{Eq:PROJ}\n\\varpi := (\\eval_0,\\eval_{2^{-m}}, \\eval_{2^{-m+1}},\\dots,\\eval_{(2^m-1)2^{-m}},\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}.\n\\end{align}\nLet $s,t\\in I_m$ with $t-s = 2^{-m}$.\nFor $k\\in\\{1,\\dots,n\\}$, define $\\smash{\\nu_s^k,\\nu_t^k\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ by\n\\begin{align*}\n\\nu_s^k &:= \\alpha_k^{-1}\\,(\\pr_{2^ms})_\\push\\varpi\\mres L_k,\\\\\n\\nu_t^k &:= \\alpha_k^{-1}\\,(\\pr_{2^mt})_\\push\\big[(\\pr_{2^ms},\\pr_{2^mt})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}\\mres (L_k\\times \\mms)\\big],\n\\end{align*}\nprovided $\\smash{\\alpha_k := \\mu_s[L_k]>0}$. Note that $\\smash{\\supp\\nu_s^k\\subset \\bar{L}_k}$ and $\\smash{\\supp\\nu_s^k\\times\\supp\\nu_t^k\\subset\\mms_\\ll^2}$. Since $r:=\\inf\\uptau(\\supp\\mu_0\\times\\supp\\mu_1) > 0$, every curve in $\\mathrm{TGeo}^\\uptau(\\mms)$ starting in $\\supp\\mu_0$ and ending in $\\supp\\mu_1$ belongs to the set $G_r$ defined in \\autoref{Cor:Cptness and equicty}, which is uniformly equi\\-continuous.  Hence, using \\eqref{Eq:m choice} and \\eqref{Eq:PROJ} we get  $\\smash{\\supp\\nu_s^k, \\supp\\nu_t^k \\subset \\bar{{\\ensuremath{\\mathsf{B}}}}^\\met(L_k,\\delta)\\subset C_k}$. By $\\smash{\\mathrm{TCD}_{p,\\loc}^*(K,N)}$ in the form described above, there is some $\\smash{\\ell_p}$-optimal geodesic plan $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}^k\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\nu_s^k,\\nu_t^k)}$ such that\n\\begin{align}\\label{Eq:Mrg}\n{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}^k)\\leq \\sigma_{K,N'}^{(1\/2)}(\\theta_{s,t}^m)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\nu_s^k) + \\sigma_{K,N}^{(1\/2)}(\\theta_{s,t}^m)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\nu_t^k)\n\\end{align}\nfor every $N'\\geq N$, where $\\theta_{s,t}^m$ is defined as in \\eqref{Eq:Thetat} with respect to $\\mu := (\\mu_r)_{r\\in[0,1]}$.\n\nDefine $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_s,\\mu_t)}$ by\n\\begin{align*}\n{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t} := \\alpha_1\\,{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}^1 + \\dots + \\alpha_n\\,{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}^n.\n\\end{align*}\nBy construction and \\autoref{Le:Mutually singular}, respectively, $\\smash{(\\eval_r)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}^1,\\dots,(\\eval_r)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}^n}$ are mutually singular for every fixed $r\\in\\{0,1\/2\\}$, whence\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_s) &= \\alpha_1^{1-1\/N'}\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\nu_s^1) + \\dots + \\alpha_1^{1-1\/N'}\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\nu_s^n),\\\\\n{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}) &= \\alpha_1^{1-1\/N'}\\,{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}^1) + \\dots + \\alpha_n^{1-1\/N'}\\,{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}^n).\n\\end{align*} \nOn the other hand, since mutual singularity of $\\smash{\\nu_t^1,\\dots,\\nu_t^n}$ may fail, \n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t) \\geq \\alpha_1^{1-1\/N'}\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\nu_t^1) + \\dots + \\alpha_n^{1-1\/N'}\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\nu_t^n).\n\\end{align*}\nMerging these inequalities with \\eqref{Eq:Mrg} yields\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}) \\leq \\sigma_{K,N'}^{(1\/2)}(\\theta_{s,t}^m)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_s) + \\sigma_{K,N'}^{(1\/2)}(\\theta_{s,t}^m)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t),\n\\end{align*}\nand the desired property ${\\ensuremath{\\mathrm{P}}}_m(J(\\mu_0,\\mu_1))$ follows.\n\nTo close the proof of \\autoref{Th:Local to global}, let $(\\mu_r)_{r\\in[0,1]}$ be a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic from $\\mu_0$ to $\\mu_1$ which consists of $\\meas$-absolutely continuous measures, as hypothesized. Iteratively, the property ${\\ensuremath{\\mathrm{P}}}_m(J(\\mu_0,\\mu_1))$ proven above implies ${\\ensuremath{\\mathrm{P}}}_0(J(\\mu_0,\\mu_1))$ by \\autoref{Le:PnC}. By \\autoref{Propos}, this already implies the condition $\\smash{\\mathrm{TCD}_p^*(K,N)}$.\n\\end{proof}\n\nThe following \\autoref{Cor:K-} easily follows from \\autoref{Th:Local to global} and the stability result from \\autoref{Th:Stability TCD}. \\autoref{Pr:Pr} is proven along the same lines as \\cite[Prop.~5.5]{bacher2010} by using uniform continuity of $\\uptau$ on compact subsets of $\\smash{\\mms^2}$; the point  is that for small $\\vartheta\\geq 0$, the quantities  $\\smash{\\sigma_{K,N}^{(r)}(\\vartheta)}$ and $\\smash{\\tau_{K,N}^{(r)}(\\vartheta)}$ ``almost coincide''. See also the computations in \\cite{deng}.\n\n\\begin{corollary}\\label{Cor:K-} Let $K\\in\\R$ and $N\\in[1,\\infty)$. Then chronological $\\smash{\\ell_p}$-geodesy of $\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ and the condition $\\smash{\\mathrm{TCD}_{p,\\loc}^*(K',N)}$ hold for every $K' < K$ if and only if $\\smash{\\mathrm{TCD}_p^*(K,N)}$ is satisfied.\n\\end{corollary}\n\n\\begin{proposition}\\label{Pr:Pr} Assume $\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ to be chronologically $\\smash{\\ell_p}$-geodesic. Given any $K\\in\\R$ and $N\\in[1,\\infty)$,  $\\smash{\\mathrm{TCD}_{p,\\loc}^*(K',N)}$ holds for every $K'<K$ if and only if $\\smash{\\mathrm{TCD}_{p,\\loc}(K',N)}$ is satisfied for every $K'<K$.\n\\end{proposition}\n\n\\begin{remark}\\label{Eq:Blurt} We do not know whether \\autoref{Th:Local to global} holds for the stronger conditions $\\smash{\\mathrm{TCD}_{p,\\loc}(K,N)}$ and $\\smash{\\mathrm{TCD}_p(K,N)}$, respectively; compare with \\cite[Rem.~5.6]{bacher2010}. It would follow from \\autoref{Th:Local to global} if we succeeded in proving that $\\smash{\\mathrm{TCD}_p^*(K,N)}$ and $\\smash{\\mathrm{TCD}_p(K,N)}$ are \\emph{equivalent} on timelike $p$-essentially nonbranching spaces, similarly to the metric measure result \\cite[Thm.~1.1]{cavallettimilman2021}; recall \\autoref{Pr:TCD and TCD*}. \n\\end{remark}\n\n\\subsection{Universal cover} Now we discuss an application of \\autoref{Th:Local to global} to the universal cover of a Lorentzian pre-length space, assuming that the latter exists. By construction of the Lorentzian structure on the universal cover, the latter inherits the curvature properties of its base space locally, and hence globally if ${\\ensuremath{\\mathscr{X}}}$ is timelike $p$-essentially nonbranching; see \\autoref{Th:TCD universal cover} below for details. The universal cover plays a key role in the smooth \\emph{Gannon--Lee singularity theorems} \\cite{gannon1975, gannon1976, lee1976}. (See also the recent work \\cite{schinnerl2021} for spacetimes with $\\smash{\\Cont^1}$-metrics, and the references therein.) More elementary, our discussion also creates further examples of spaces satisfying the $\\smash{\\mathrm{TCD}_p^*(K,N)}$ condition.\n\n\\subsubsection{Covering spaces}  We start by recalling the following basic definition  due to \\cite[p.~62, p.~80]{spanier1966} (given for general topological spaces therein).\n\n\\begin{definition} Let $(\\mms,\\met)$ be a metric space. \n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item Given a topological space $\\smash{\\hat{\\mms}}$, a map $\\smash{\\textnormal{\\texttt{p}}\\colon\\hat{\\mms}\\to\\mms}$ is called \\emph{covering map} if every point in  $\\mms$ has an open neighborhood $U\\subset\\mms$ such that $\\smash{\\textnormal{\\texttt{p}}^{-1}(U)}$ is the disjoint union of open subsets of $\\smash{\\hat{\\mms}}$ on each of which $\\textnormal{\\texttt{p}}$ restricts to a homeomorphism.\n\\item A topological space $\\smash{\\hat{\\mms}}$ is called \\emph{covering space} for $\\mms$ if there exists a covering map $\\smash{\\textnormal{\\texttt{p}}\\colon\\hat{\\mms}\\to\\mms}$.\n\\item A topological space $\\smash{\\hat{\\mms}}$ is called \\emph{universal cover} for $\\mms$ if it is a covering space for $\\mms$, and for every covering space $\\smash{\\check{\\mms}}$ for $\\mms$ \\textnormal{(}with covering map $\\smash{\\textnormal{\\texttt{q}}}$\\textnormal{)} there exists a continuous map $\\smash{F \\colon \\hat{\\mms}\\to \\check{\\mms}}$ such that\n\\begin{align*}\n\\textnormal{\\texttt{p}} = F\\circ \\textnormal{\\texttt{q}}.\n\\end{align*}\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark} A universal cover might fail to exist in general \\cite[Ex.~2.5.17]{spanier1966}, but if it exists, it is unique up to homeomorphism. If $\\mms$ is connected, locally path-connected \\cite[p.~65]{spanier1966}, and semi-locally simply connected \\cite[p.~78]{spanier1966}, then it admits a simply connected universal cover \\cite[Cor.~2.5.14]{spanier1966}. If one drops semi-local simple  connectedness, a universal cover might still exist, but fail to be simply connected \\cite[Ex.~2.5.18]{spanier1966}.\n\\end{remark}\n\nEvery covering projection $\\smash{\\textnormal{\\texttt{p}}\\colon\\hat{\\mms}\\to\\mms}$ is a local homeomorphism \\cite[Lem.~2.1.8]{spanier1966}. Hence, defining $\\smash{\\hat{\\met}\\colon\\hat{\\mms}^2\\to[0,\\infty)}$ by \n\\begin{align}\\label{Eq:d hut}\n\\hat{\\met}(\\hat{x},\\hat{y}) := \\met(\\textnormal{\\texttt{p}}(\\hat{x}),\\textnormal{\\texttt{p}}(\\hat{y})),\n\\end{align}\nthe  covering space $\\smash{\\hat{\\mms}}$ of a metric space is automatically a metric space with respect to $\\smash{\\hat{\\met}}$, and $\\smash{\\hat{\\met}}$ induces the topology of $\\smash{\\hat{\\mms}}$. \n\n\\begin{remark}\\label{Re:Bruh} By definition of a covering map and \\eqref{Eq:d hut}, for every compact $C\\subset\\mms$ there exists an open subset $U\\subset\\mms$ containing $C$ such that $\\smash{\\textnormal{\\texttt{p}}\\big\\vert_{\\textnormal{\\texttt{p}}^{-1}(U)}}$ is an isometric isomorphism,  in particular bi-Lipschitz continuous. This implies that properness of $\\met$ lifts to $\\smash{\\hat{\\met}}$. Moreover, a curve $\\smash{\\hat{\\gamma}}\\colon[a,b]\\to\\smash{\\hat{\\mms}}$, $a,b\\in\\R$ with $a<b$, is $\\smash{\\hat{\\met}}$-rectifiable if and only if $\\smash{\\gamma := \\textnormal{\\texttt{p}}\\circ\\hat{\\gamma}}$ is $\\met$-rectifiable, and in either case $\\smash{\\mathrm{Len}_{\\hat{\\met}}(\\hat{\\gamma}) = \\mathrm{Len}_\\met(\\gamma)}$.\n\\end{remark}\n\n\\subsubsection{Lift of Lorentzian structures} In this subsection, we fix a Lorentzian pre-length space $(\\mms,\\met,\\ll,\\leq,\\uptau)$. In the remainder of this section, we assume the existence of a universal cover $\\smash{\\hat{\\mms}}$ for $\\mms$ with covering map $\\textnormal{\\texttt{p}}$. \n\nWe introduce a natural lifted Lorentzian pre-length structure on $\\smash{\\hat{\\mms}}$;  various  Lorentzian properties will lift to $\\smash{\\hat{\\mms}}$ as well. We confine ourselves to those introduced in \\autoref{Sub:Lorentzian nonsmooth} and needed for our work, although we expect  more of the vast list of properties from  \\cite{cavalletti2020,kunzinger2018} to lift to $\\smash{\\hat{\\mms}}$.\n\nThe following \\autoref{Pr:Lift} essentially follows from \\autoref{Re:Bruh} and the way $\\smash{\\mathbin{\\hat{\\ll}}}$, $\\smash{\\mathbin{\\hat{\\leq}}}$, and $\\smash{\\hat{\\uptau}}$ below are defined. (In the language of \\autoref{Def:Lor isometry}, $\\textnormal{\\texttt{p}}$ is a Lorentzian isometric embedding.) In particular, a curve $\\smash{\\hat{\\gamma}\\colon[a,b]\\to\\hat{\\mms}}$, $a,b\\in\\R$ with $a<b$, is causal or timelike if and only if $\\gamma:= \\textnormal{\\texttt{p}}\\circ\\hat{\\gamma}$ is causal or timelike, respectively, and in either case $\\smash{\\mathrm{Len}_{\\hat{\\uptau}}(\\hat{\\gamma}) = \\mathrm{Len}_\\uptau(\\gamma)}$. Moreover, the former is a geodesic if and only if the latter is. We omit the straightforward proof.\n\n\\begin{proposition}\\label{Pr:Lift} Define two relations $\\smash{\\hat{\\leq}}$ and $\\smash{\\hat{\\ll}}$ on $\\smash{\\hat{\\mms}}$ by $\\hat{x}\\mathbin{\\hat{\\leq}} \\hat{y}$ provided $\\textnormal{\\texttt{p}}(\\hat{x})\\leq \\textnormal{\\texttt{p}}(\\hat{y})$, and $\\hat{x} \\mathbin{\\hat{\\ll}}\\hat{y}$ provided $\\textnormal{\\texttt{p}}(\\hat{x})\\ll \\textnormal{\\texttt{p}}(\\hat{y})$. Moreover, define  $\\smash{\\hat{\\uptau}\\colon\\hat{\\mms}^2\\to [0,\\infty]}$ by \n\\begin{align*}\n\\hat{\\uptau}(\\hat{x},\\hat{y}) &:= \\uptau(\\textnormal{\\texttt{p}}(\\hat{x}),\\textnormal{\\texttt{p}}(\\hat{y})).\n\\end{align*}\nThen $\\smash{(\\hat{\\mms},\\hat{\\met},\\hat{\\ll},\\hat{\\leq},\\hat{\\uptau})}$ is a Lorentzian pre-length space. If $(\\mms,\\met,\\ll,\\leq,\\uptau)$ is\n\\begin{enumerate}[label=\\textnormal{\\textcolor{black}{(}\\roman*\\textcolor{black}{)}}]\n\\item\\label{La:Un} causally closed,\n\\item\\label{La:Deux} locally causally closed,\n\\item\\label{La:Trois} causally path-connected,\n\\item\\label{La:NT} non-totally imprisoning,\n\\item\\label{La:Quatre} localizable,\n\\item\\label{La:Six} globally hyperbolic,\n\\item\\label{La:Sept} ${\\ensuremath{\\mathscr{K}}}$-globally hyperbolic, \n\\item\\label{La:Len} length, or\n\\item\\label{La:Geo} geodesic,\n\\end{enumerate}\nthen the respective property holds for $\\smash{(\\hat{\\mms},\\hat{\\met},\\hat{\\ll},\\hat{\\leq},\\hat{\\uptau})}$ as well.\n\\end{proposition}\n\n\n\n\n\n\n\n\n\\subsubsection{Lift of curvature properties} Given a measured Lorentzian pre-length space ${\\ensuremath{\\mathscr{X}}}$, using the covering map $\\textnormal{\\texttt{p}}$ the Lorentzian structure on the universal cover from \\autoref{Pr:Lift} can be naturally endowed with a reference measure $\\smash{\\hat{\\meas}}$ as follows. Let $\\smash{\\hat{\\Sigma}}$ be  the family of all sets $\\smash{\\hat{B}\\subset \\hat{\\mms}}$ such that $\\smash{\\textnormal{\\texttt{p}}\\big\\vert_{\\hat{B}}}$ is a local isometry with Borel image $\\smash{\\textnormal{\\texttt{p}}(\\hat{B})\\subset\\mms}$. This family is closed under any number of intersections, and since $\\smash{\\hat{\\met}}$ is proper by \\autoref{Re:Bruh}, the $\\sigma$-algebra $\\smash{\\sigma(\\hat{\\Sigma})}$ coincides with the Borel $\\sigma$-algebra on $\\smash{\\hat{\\mms}}$. The definition $\\smash{\\hat{\\meas}[\\hat{B}] := \\meas[\\textnormal{\\texttt{p}}(\\hat{B})]}$, $\\smash{B\\in\\hat{\\Sigma}}$, thus evidently gives rise to the desired (lifted) Borel measure $\\smash{\\hat{\\meas}}$ on $\\smash{\\hat{\\mms}}$.\n\nFor the subsequent theorem, we add the hypothesis of timelike $p$-essential nonbranching, $p\\in (0,1)$, to our standing assumptions on ${\\ensuremath{\\mathscr{X}}}$.\n\n\\begin{theorem}\\label{Th:TCD universal cover} Let $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$. Assume $\\smash{\\mathrm{TCD}_p^*(K,N)}$ for ${\\ensuremath{\\mathscr{X}}}$. Then the causally closed, ${\\ensuremath{\\mathscr{K}}}$-globally hyperbolic measured Lorentzian geodesic space $\\smash{\\hat{{\\ensuremath{\\mathscr{X}}}} := (\\hat{\\mms},\\hat{\\met},\\hat{\\meas},\\mathbin{\\hat{\\ll}},\\mathbin{\\hat{\\leq}},\\hat{\\uptau})}$ according to \\autoref{Pr:Lift} is timelike $p$-essentially nonbranching, and it obeys $\\smash{\\mathrm{TCD}_p^*(K,N)}$.\n\\end{theorem}\n\n\\begin{proof} Composition with the covering map $\\textnormal{\\texttt{p}}$ yields a canonical nonrelabeled and continuous map $\\smash{\\textnormal{\\texttt{p}}\\colon \\mathrm{TGeo}^\\uptau(\\hat{\\mms})\\to \\mathrm{TGeo}^\\uptau(\\mms)}$. In an evident notation, given any $\\smash{\\hat{\\ell}_p}$-optimal geodesic plan $\\smash{\\hat{{\\ensuremath{\\boldsymbol{\\pi}}}}\\in{\\ensuremath{\\mathscr{P}}}(\\mathrm{TGeo}^\\uptau(\\hat{\\mms}))}$, the plan $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}} := \\textnormal{\\texttt{p}}_\\push\\hat{{\\ensuremath{\\boldsymbol{\\pi}}}}}$ lies in ${\\ensuremath{\\mathscr{P}}}(\\mathrm{TGeo}^\\uptau(\\mms))$. Given a timelike nonbranching set $\\smash{G\\subset\\mathrm{TGeo}^\\uptau(\\mms)}$ with ${\\ensuremath{\\boldsymbol{\\pi}}}[G]=1$, the set $\\smash{\\hat{G}} := \\smash{\\textnormal{\\texttt{p}}^{-1}(G)\\subset\\mathrm{TGeo}^\\uptau(\\hat{\\mms})}$ is timelike nonbranching as well, and $\\smash{\\hat{{\\ensuremath{\\boldsymbol{\\pi}}}}[\\hat{G}]=1}$. Hence, timelike $p$-essential nonbranching lifts to the Lorentzian structure from \\autoref{Pr:Lift}.\n\nThanks to the construction given in \\autoref{Pr:Lift} as well as \\autoref{Re:Bruh}, the spaces $\\smash{\\hat{{\\ensuremath{\\mathscr{X}}}}}$ and $\\smash{{\\ensuremath{\\mathscr{X}}}}$ are locally isometric as metric and Lorentzian pre-length spaces; recall \\autoref{Sec:Stability TCD}. Therefore, the condition $\\smash{\\mathrm{TCD}_p^*(K,N)}$ lifts and makes $\\smash{\\hat{{\\ensuremath{\\mathscr{X}}}}}$ obey $\\smash{\\mathrm{TCD}_p^*(K,N)}$ locally. By lifting of timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics witnessing the inequality defining $\\smash{\\mathrm{TCD}_p^*(K,N)}$ for ${\\ensuremath{\\mathscr{X}}}$, $\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\hat{\\mms},\\hat{\\meas})}$ is chronologically $\\smash{\\ell_p}$-geodesic according to \\autoref{Def:Chron lp geod}. Using \\autoref{Pr:Lift} again with \\autoref{Th:Local to global},  $\\smash{\\mathrm{TCD}_p^*(K,N)}$ thus holds globally on the universal cover.\n\\end{proof}\n\n\n\\section{Timelike measure-contraction property}\\label{Ch:TMCP}\n\nIn this chapter, we introduce and study Lorentzian analogues of the mea\\-sure-contraction property for metric measure spaces  \\cite{ohta2007,sturm2006b}. An entropic version has recently been introduced in \\cite{cavalletti2020}, cf.~\\autoref{Def:TMCPe} below. Again, for smooth space\\-times the respective conditions are tightly linked to timelike lower Ricci curvature bounds, cf.~\\autoref{Th:TMCP Smooth}.\n\n\\subsection{Definition and basic properties} \n\n\\begin{definition}\\label{Def:TMCP} Let $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$. \n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item We say that ${\\ensuremath{\\mathscr{X}}}$ obeys the \\emph{reduced timelike measure-contraction property} $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ if for every $\\smash{\\mu_0\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ and every $x_1\\in I^+(\\mu_0)$ there is a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\smash{\\mu_1 := \\delta_{x_1}}$ such that for every $t\\in[0,1)$ and every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t) \\leq -\\int_\\mms \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0, x_1))\\,\\rho_0(x^0)^{-1\/N'}\\d\\mu_0(x^0).\n\\end{align*}\n\\item If the previous statement holds for $\\smash{\\tau_{K,N'}^{(1-t)}}$ instead of $\\smash{\\sigma_{K,N'}^{(1-t)}}$, ${\\ensuremath{\\mathscr{X}}}$ is said to satisfy the \\emph{timelike measure-contraction property} $\\smash{\\mathrm{TMCP}_p(K,N)}$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark} As in \\autoref{Re:TCD 0}, the conditions  $\\smash{\\mathrm{TMCP}_p^*(0,N)}$ and $\\smash{\\mathrm{TMCP}_p(0,N)}$ coincide for every $p\\in (0,1)$ and every $N\\in[1,\\infty)$.\n\\end{remark}\n\n\\begin{remark}\\label{Re:mae x1} Under our standing assumptions on ${\\ensuremath{\\mathscr{X}}}$, $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ holds if and only if the statement from \\autoref{Def:TMCP} is  merely satisfied for $\\meas$-a.e. $x_1\\in \\mms$ thanks to  \\autoref{Le:Villani lemma for geodesic} and Fatou's lemma. The analogous statement holds for $\\smash{\\mathrm{TMCP}_p(K,N)}$ as well.\n\\end{remark}\n\nAs for \\autoref{Pr:TCD and TCD*} and \\autoref{Pr:Consistency TCD}, the statements from \\autoref{Pr:TMCP to TMCP*} and  \\autoref{Pr:Consistency TMCP} below are readily established (in all generality, the proofs work for general measured Lorentzian pre-length spaces). \n\n\\begin{proposition}\\label{Pr:TMCP to TMCP*} The following hold for every $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$; as in \\autoref{Pr:TCD and TCD*}, set $K^* := K(N-1)\/N$.\n\\begin{enumerate}[label=\\textnormal{(\\roman*)}]\n\\item  The condition $\\smash{\\mathrm{TMCP}_p(K,N)}$ implies $\\smash{\\mathrm{TMCP}_p^*(K,N)}$.\n\\item The condition $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ implies $\\smash{\\mathrm{TMCP}_p(K^*,N)}$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proposition}\\label{Pr:Consistency TMCP} Assume $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ for $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$. Then the following statements hold.\n\\begin{enumerate}[label=\\textnormal{(\\roman*)}]\n\\item \\textnormal{\\textbf{Consistency.}} The condition $\\smash{\\mathrm{TMCP}_p^*(K',N')}$ holds for every $K'\\leq K$ and every $N'\\geq N$.\n\\item \\textnormal{\\textbf{Scaling.}} Given any $a,b,\\theta>0$, the rescaled measured Lorentzian pre-length space $(\\mms, a\\met, b\\meas,\\ll,\\leq, \\theta\\uptau)$ satisfies $\\smash{\\mathrm{TMCP}_p^*(K\/\\theta^2,N)}$.\n\\end{enumerate}\n\nAnalogous statements hold for the $\\smash{\\mathrm{TMCP}_p(K,N)}$ condition.\n\\end{proposition}\n\n\\begin{remark}\\label{Re:Unlikely} An analogue of \\autoref{Le:Stu} is unlikely to hold under $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ or $\\smash{\\mathrm{TMCP}_p(K,N)}$. In particular, we do not know a priori whether the measures $\\mu_t$, $t\\in[0,1)$, of a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic as in \\autoref{Def:TMCP} are always --- or can be chosen  to be --- $\\meas$-absolutely continuous. Instead, we establish the existence of timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics consisting of $\\meas$-absolutely continuous measures whose densities are uniformly $\\Ell^\\infty$ in time under $\\smash{\\mathrm{TMCP}_p^*(K,N)}$, at least if $\\smash{\\mu_0\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ has $\\meas$-essentially bounded density, by a variational method following \\cite{braun2022}, see also \\cite{cavalletti2017,rajala2012a,rajala2012b} in \\autoref{Sub:Good}; cf.~\\autoref{Th:Good TMCP}. By uniqueness of timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics under timelike essential nonbranching conditions, see \\autoref{Th:Uniqueness geodesics}, this  suffices to show an equivalence result analogous to \\autoref{Th:Equivalence TCD* and TCDe} for our timelike measure-contraction property, see \\autoref{Th:Equivalence TMCP* and TMCPe} and \\autoref{Th:Equivalence TMCP}.\n\\end{remark}\n\nThe link of $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ and $\\smash{\\mathrm{TMCP}_p(K,N)}$ to their respective $\\mathrm{TCD}$ counterparts is more subtle. The proof of the corresponding \\autoref{Pr:TMCP to TCD} follows  \\cite[Prop.~3.11]{cavalletti2020}. We need the following result whose proof is a straigtforward adaptation of the argument for \\cite[Lem.~3.3]{sturm2006b}, recall also \\autoref{Le:Const perturb}. The difference to \\cite{sturm2006b} is that we only require the marginal whose density is contained in the respective integrand to be constant.\n\n\\begin{lemma}\\label{Le:USC lemma} Let $K\\in\\R$ and $N\\in[1,\\infty)$. Let  $\\rho\\colon\\mms\\to [0,\\infty)$ be a Borel function with $\\mu:= \\rho\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$. Moreover, let $(\\pi_n)_{n\\in\\N}$ be a sequence in ${\\ensuremath{\\mathscr{P}}}(\\mms^2)$ converging weakly to  $\\pi\\in{\\ensuremath{\\mathscr{P}}}(\\mms^2)$  such that for some $i\\in\\{1,2\\}$, we have \n\\begin{align*}\n(\\pr_i)_\\push\\pi_n = \\mu\n\\end{align*}\nfor every $n\\in\\N$. Then for every every $r\\in[0,1]$,\n\\begin{align*}\n&\\int_{\\mms^2} \\tau_{K,N}^{(r)}(\\uptau(x^0,x^1))\\,(\\rho\\circ\\pr_i)(x^0,x^1)^{-1\/N} \\d\\pi(x^0,x^1)\\\\\n&\\qquad\\qquad\\leq \\liminf_{n\\to\\infty}\\int_{\\mms^2} \\tau_{K,N}^{(r)}(\\uptau(x^0,x^1))\\,(\\rho\\circ\\pr_i)(x^0,x^1)^{-1\/N}\\d\\pi_n(x^0,x^1).\n\\end{align*}\n\nAn analogous assertion holds with $\\smash{\\tau_{K,N}^{(r)}}$ replaced by $\\smash{\\sigma_{K,N}^{(r)}}$.\n\\end{lemma}\n\n\\begin{proposition}\\label{Pr:TMCP to TCD}  The following  hold for every $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$.\n\\begin{enumerate}[label=\\textnormal{\\textcolor{black}{(}\\roman*\\textcolor{black}{)}}]\n\\item\\label{LART} The condition $\\smash{\\mathrm{wTCD}_p(K,N)}$ implies $\\smash{\\mathrm{TMCP}_p(K,N)}$.\n\\item\\label{LARTT} The condition $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ implies $\\smash{\\mathrm{TMCP}_p^*(K,N)}$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof} We only prove \\ref{LART}, the proof of \\ref{LARTT} is similar.\n\n\\textbf{Step 1.} \\textit{Approximation of $\\mu_0$ and $\\mu_1$.}  \nGiven any $\\varepsilon\\in (0,1)$, let $C_\\varepsilon \\subset\\mms$ be a compact set with $\\mu_0[C_\\varepsilon] \\geq 1-\\varepsilon$ and $\\smash{C_\\varepsilon \\times \\{x_1\\}\\subset\\mms_\\ll^2}$. Define $\\smash{\\mu_0^\\varepsilon\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ by\n\\begin{align*}\n\\mu_0^\\varepsilon := \\mu_0[C_\\varepsilon]^{-1}\\,\\mu_0\\mres C_\\varepsilon = \\rho_0^\\varepsilon\\,\\meas\n\\end{align*}\nMoreover, since $\\smash{\\mms_\\ll^2}$ is open while $C_\\varepsilon\\times\\{x_1\\}$ is compact, there exists $\\eta > 0$ such that  $\\smash{C_\\varepsilon\\times {\\ensuremath{\\mathsf{B}}}^\\met(x_1,\\eta)\\subset\\mms_\\ll^2}$. For $\\delta\\in (0,\\eta)$, we set \n\\begin{align*}\n\\mu_1^\\delta := \\meas\\big[{\\ensuremath{\\mathsf{B}}}^\\met(x_1,\\delta)\\big]^{-1}\\,\\meas\\mres {\\ensuremath{\\mathsf{B}}}^\\met(x_1,\\delta) = \\rho_1^\\delta\\,\\meas.\n\\end{align*}\n\nBy \\autoref{Re:Strong timelike}, the pair $\\smash{(\\mu_0^\\varepsilon,\\mu_1^\\delta)}$ is strongly  timelike $p$-dualizable, and using $\\smash{\\mathrm{wTCD}_p(K,N)}$ we find a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\smash{(\\mu_t^{\\varepsilon,\\delta})_{t\\in[0,1]}}$ connecting $\\smash{\\mu_0^\\varepsilon}$ to $\\smash{\\mu_1^\\delta}$ as well as a timelike $p$-dualizing $\\smash{\\pi^{\\varepsilon,\\delta}\\in\\Pi_\\ll(\\mu_0^\\varepsilon,\\mu_1^\\delta)}$, with support in $\\smash{C_\\varepsilon\\times {\\ensuremath{\\mathsf{B}}}^\\met(x_1,\\eta)}$, such that for every $t\\in[0,1]$ and every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t^{\\varepsilon,\\delta}) &\\leq -\\int_{\\mms^2} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0^\\varepsilon(x^0)^{-1\/N'}\\d\\pi^{\\varepsilon,\\delta}(x^0,x^1)\\\\\n&\\qquad\\qquad -\\int_{\\mms^2}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_1^\\delta(x^1)^{-1\/N'}\\d\\pi^{\\varepsilon,\\delta}(x^0,x^1)\\\\\n&\\leq -\\int_{\\mms^2}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0^\\varepsilon(x^0)^{-1\/N'}\\d\\pi^{\\varepsilon,\\delta}(x^0,x^1).\n\\end{align*}\n\n\\textbf{Step 2.} \\textit{Sending $\\delta \\to 0$.} Given any $\\varepsilon > 0$ and a fixed sequence $(\\delta_n)_{n\\in\\N}$ decreasing to $0$, from $\\smash{(\\mu_t^{\\varepsilon,\\delta_n})_{t\\in[0,1]}}$, $n\\in\\N$, we construct a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic from $\\mu_0^\\varepsilon$ to $\\mu_1$ as follows. Let $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}^{\\varepsilon,\\delta_n}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0^\\varepsilon,\\mu_1^{\\delta_n})}$ represent $\\smash{(\\mu_t^{\\varepsilon,\\delta_n})_{t\\in[0,1]}}$. As $\\smash{(\\mu_1^{\\delta_n})_{n\\in\\N}}$  converges weakly to $\\mu_1$, this sequence is tight, and so is $\\smash{({\\ensuremath{\\boldsymbol{\\pi}}}^{\\varepsilon,\\delta_n})_{n\\in\\N}}$ by \\autoref{Le:Villani lemma for geodesic}. Let $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}^\\varepsilon\\in \\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0^\\varepsilon,\\mu_1)}$ be a weak limit of a nonrelabeled subsequence. Then the assignment $\\smash{\\mu_t^\\varepsilon := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^\\varepsilon}$, $t\\in[0,1]$, gives rise to a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic connecting $\\smash{\\mu_0^\\varepsilon}$ to $\\mu_1$.\n\n\nEvery weak limit point of $\\smash{(\\pi^{\\varepsilon,\\delta_n})_{n\\in\\N}}$ equals $\\smash{\\mu_0^\\varepsilon\\otimes\\mu_1 = \\mu_0^\\varepsilon\\otimes\\delta_{x_1}}$.\nTherefore, ${\\ensuremath{\\mathscr{K}}}$-global hyperbolicity,  weak lower semicontinuity of $\\smash{{\\ensuremath{\\mathscr{S}}}_{N'}}$ on measures with uniformly bounded support, and \\autoref{Le:USC lemma} yield, for every such $t$ and every $N'\\geq N$, \n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t^\\varepsilon) &\\leq \\limsup_{n\\to\\infty} {\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t^{\\varepsilon,\\delta_n})\\\\\n&\\leq -\\liminf_{n\\to\\infty}\\int_{\\mms^2}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0^\\varepsilon(x^0)^{-1\/N'}\\d\\pi^{\\varepsilon,\\delta_n}(x^0,x^1)\\\\\n&\\leq -\\int_{\\mms^2} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_1))\\,\\rho_0^\\varepsilon(x^0)^{-1\/N'}\\d\\mu_0^\\varepsilon(x^0)\n\\end{align*}\n\n\\textbf{Step 3.} \\textit{Sending $\\varepsilon \\to 0$.} Given a sequence $(\\varepsilon_n)_{n\\in\\N}$ decreasing to $0$, note that $\\smash{(\\mu_0^{\\varepsilon_n})_{n\\in\\N}}$ converges weakly to $\\mu_0$.  Similarly to  Step 2, from $\\smash{(\\mu_t^{\\varepsilon_n})_{n\\in\\N}}$ we construct a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ connecting $\\mu_0$ to $\\mu_1$. As in Step 2 and using Fatou's lemma we get, for every $t\\in[0,1]$ and every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t) &\\leq \\limsup_{n\\to\\infty} {\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t^{\\varepsilon_n})\\\\\n&\\leq -\\liminf_{n\\to\\infty}\\int_{\\mms^2}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_1))\\,\\rho_0^{\\varepsilon_n}(x^0)^{-1\/N'}\\d\\mu_0^{\\varepsilon_n}(x^0)\\\\\n&\\leq -\\int_{\\mms^2}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_1))\\,\\rho_0(x^0)^{-1\/N'}\\d\\mu_0(x^0).\\qedhere\n\\end{align*}\n\\end{proof}\n\n\\begin{remark}\\label{Re:Geom inequ TMCP} With essentially identical proofs, the respective versions of the timelike Bonnet--Myers inequality, \\autoref{Cor:Bonnet-Myers} and \\autoref{Cor:Reduced BM}, and the timelike Bishop--Gromov inequality, \\autoref{Th:BG} and \\autoref{Th:Reduced BG}, hold as well under $\\smash{\\mathrm{TMCP}_p(K,N)}$ and $\\smash{\\mathrm{TMCP}_p^*(K,N)}$. We only need to assume ${\\ensuremath{\\mathscr{X}}}$ to be a causally closed and globally hyperpolic Lorentzian geodesic space.\n\\end{remark}\n\n\\subsection{Stability}  Next, we discuss the stability of the notions from \\autoref{Def:TMCP}. In contrast to the weak stability from \\autoref{Th:Stability TCD}, both timelike measure-contraction properties are stable under the convergence introduced in \\autoref{Def:Convergence}.\n\n\\begin{theorem}\\label{Th:Stability TMCP} Assume the convergence of $({\\ensuremath{\\mathscr{X}}}_k)_{k\\in\\N}$ to ${\\ensuremath{\\mathscr{X}}}_\\infty$ as in  \\autoref{Def:Convergence}. Moreover, let $(K_k,N_k)_{k\\in\\N}$ be a sequence in $\\R\\times [1,\\infty)$ converging to $(K_\\infty,N_\\infty)\\in\\R\\times[1,\\infty)$. Suppose the existence of $p\\in (0,1)$ such that ${\\ensuremath{\\mathscr{X}}}_k$ obeys $\\mathrm{TMCP}_p(K_k,N_k)$ for every $k\\in\\N$. \nThen $\\smash{{\\ensuremath{\\mathscr{X}}}_\\infty}$ satisfies $\\smash{\\mathrm{TMCP}_p(K_\\infty,N_\\infty)}$. \n\nThe analogous statement in which  $\\smash{\\mathrm{TMCP}_p(K_k,N_k)}$ is respectively replaced by $\\smash{\\mathrm{TMCP}_p^*(K_k,N_k)}$, $k\\in\\N_\\infty$, holds as well.\n\\end{theorem}\n\n\\begin{proof} It suffices to prove the first statement, the second  is argued analogously. \n\nIn this proof, we often adopt notations from the proof of \\autoref{Th:Stability TCD} without explicit notice. In particular, we again identify $\\mms_k$ with its image $\\iota_k(\\mms_k)$ in $\\mms$ and $\\meas_k$ with its push-forward $(\\iota_k)_\\push\\meas_k$ for every $k\\in\\N_\\infty$. \n\n\\textbf{Step 1.} \\textit{Reduction to compact $\\mms$.} Owing to \\autoref{Re:mae x1}, given any $\\mu_{\\infty,0} = \\rho_{\\infty,0}\\,\\meas_\\infty\\in {\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas_\\infty)$ and  $x_{\\infty,1}\\in I^+(\\mu_{\\infty,0}) \\cap \\supp\\meas_\\infty$, as for \\autoref{Th:Stability TCD} we use the compactness of $\\supp\\mu_{\\infty,0}$ and $\\supp\\mu_{\\infty,1}$, where $\\smash{\\mu_{\\infty,1} := \\delta_{x_{\\infty,1}}}$, to assume without restriction that $\\mms$ is compact, and that $\\meas_k\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$ for every $k\\in\\N_\\infty$. All  measures considered below will thus be compactly supported. Moreover, we may and will suppose that $W_2(\\meas_k,\\meas_\\infty)\\to 0$ as $k\\to\\infty$.\n\n\\textbf{Step 2.} \\textit{Restriction of the assumptions on $\\mu_{\\infty,0}$ and $\\mu_{\\infty,1}$.} We will first assume that $\\tau(\\cdot,x_{\\infty,1})$ is bounded away from zero on $\\supp\\mu_{\\infty,0}$, that $\\rho_{\\infty,0}\\in\\Ell^\\infty(\\mms,\\meas_\\infty)$, and that $x_{\\infty,1}$ can be approximated with respect to $\\met$ by a sequence $(x_{k,1})_{k\\in\\N}$ of points $x_{k,1}\\in \\supp\\meas_k$ such that $x_{\\infty,1} \\in I^-(x_{k,1})$ for every $k\\in\\N$. The general case is discussed in Step 7 below; we note for now that this conclusion will not conflict with our reductions from Step 1.\n\n\\textbf{Step 3.} \\textit{Construction of a chronological recovery sequence.}\nIn this step, given the sequence $(x_{k,1})_{k\\in\\N}$ from Step 2 we construct a sequence $(\\mu_{k,0})_{k\\in\\N}$ of measures $\\smash{\\mu_{k,0} = \\rho_{k,0}\\,\\meas_k\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)}$, $k\\in\\N$, such that $\\mu_{k,0}\\to\\mu_{\\infty,0}$ weakly as $k\\to \\infty$ possibly up to extracting a subsequence, and  $x_{k,1}\\in I^+(\\mu_{k,0})$ for every $k\\in\\N$. The constructed sequence will allow for the correct behavior of all  functionals under consideration, cf.~Step 5 below.\n\nGiven any $k\\in\\N$, let $\\mathfrak{q}_k\\in{\\ensuremath{\\mathscr{P}}}(\\mms^2)$ be a $W_2$-optimal coupling of $\\meas_k$ and $\\meas_\\infty$. We disintegrate $\\mathfrak{q}_k$ with respect to $\\pr_1$, writing\n\\begin{align*}\n{\\ensuremath{\\mathrm{d}}} \\mathfrak{q}_k(x,y) = {\\ensuremath{\\mathrm{d}}}\\mathfrak{p}_x^k(y)\\d\\meas_k(x).\n\\end{align*}\nLet $\\smash{\\mathfrak{p}^k\\colon {\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)\\to{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)}$ denote the canonically induced map.\n\nGiven any $k\\in\\N$, define $\\tilde{\\mu}_{k,0}\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)$ by\n\\begin{align*}\n\\tilde{\\mu}_{k,0} := \\mathfrak{p}^k(\\mu_{\\infty,0}) = \\tilde{\\rho}_{k,0}\\,\\meas_k.\n\\end{align*}\nNote that the measure $\\mathfrak{r}_k := (\\rho_{\\infty,0}\\circ\\pr_2)\\,\\mathfrak{q}_k\\in{\\ensuremath{\\mathscr{P}}}(\\mms^2)$  constitutes a coupling of $\\tilde{\\mu}_{k,0}$ and $\\mu_{\\infty,0}$. With the $W_2$-optimality of $\\mathfrak{q}_k$,  this  implies\n\\begin{align*}\nW_2(\\tilde{\\mu}_{k,0},\\mu_{\\infty,0}) \\leq  \\big\\Vert\\rho_{\\infty,0}\\big\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}^{1\/2}\\,W_2(\\meas_k,\\meas_\\infty),\n\\end{align*}\nand consequently $\\smash{\\tilde{\\mu}_{k,0}\\to \\mu_{\\infty,0}}$ weakly as $k\\to\\infty$. Hence, by our assumption on the se\\-quence $(x_{k,1})_{k\\in\\N}$ from Step 2 and Portmanteau's theorem,\n\\begin{align*}\n\\liminf_{k\\to\\infty} \\tilde{\\mu}_{k,0}[I^-(x_{k,1})] \\geq \\liminf_{k\\to\\infty} \\tilde{\\mu}_{k,0}[I^-(x_{\\infty,1})] \\geq \\mu_{\\infty,0}[I^-(x_{\\infty,1})] =1.\n\\end{align*}\nUp to passing to a subsequence we may and will thus assume $\\smash{\\tilde{\\mu}_{k,0}[I^-(x_{k,1})] >0}$ for every $k\\in\\N$. We then define $\\smash{\\mu_{k,0}\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)}$ by\n\\begin{align*}\n\\mu_{k,0} := \\tilde{\\mu}_{k,0}[I^-(x_{k,1})]^{-1} \\tilde{\\mu}_{k,0} \\mres I^-(x_{k,1}).\n\\end{align*}\nBy construction, the sequence $(\\mu_{k,0})_{k\\in\\N}$ converges to $\\mu_{\\infty,0}$ weakly as $k\\to\\infty$, and we have $x_{k,1}\\in I^+(\\mu_{k,0})$ for every $k\\in\\N$, as desired.\n\n\\textbf{Step 4.} \\textit{Invoking the $\\mathrm{TMCP}$ condition.} Fix $K\\in\\R$ and $N\\in (1,\\infty)$ such that $K< K_\\infty$ and $N>N_\\infty$. Up to passing to a subsequence if necessary, we may and will thus assume that $K<K_k$ and $N>N_k$ for every $k\\in\\N$.\n\nBy \\autoref{Pr:Consistency TMCP}, for every $k\\in\\N$ there exists a timelike proper-time para\\-metrized $\\smash{\\ell_p}$-geodesic $(\\mu_{k,t})_{t\\in[0,1]}$ connecting $\\mu_{k,0}$ and $\\smash{\\mu_{k,1} = \\delta_{x_{k,1}}}$ such that for every $t\\in [0,1)$ and every $N'\\geq N$,\n\\begin{align}\\label{Eq:TMCP cond}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{k,t}) \\leq -\\int_{\\mms} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{k,1}))\\,\\rho_{k,0}(x^0)^{-1\/N'} \\d\\mu_{k,0}(x^0).\n\\end{align}\n\n\\textbf{Step 5.} \\textit{Estimating the previous right-hand side.} We estimate the negative of the right-hand side of \\eqref{Eq:TMCP cond} from below, up to errors which become arbitrarily small.  By construction of $\\mu_{k,0}$, we find a sequence $(a_k)_{k\\in\\N}$ of normalization constants converging to $1$ such that $\\smash{\\rho_{k,0} \\leq a_k\\,\\tilde{\\rho}_{k,0}}$ $\\meas_k$-a.e.~for every $k\\in\\N$.\n\n\\textbf{Step 5.1.} First, we argue that $x_{k,1}$ can be replaced $x_{\\infty,1}$ up to a small error. By our choice of $K$ and $N$ and the timelike Bonnet--Myers inequality from  \\autoref{Re:Geom inequ TMCP},\n\\begin{align*}\nc := \\sup\\tau_{K,N'}^{(1-t)}\\circ\\uptau(\\mms^2)\n\\end{align*}\nis finite. Furthermore, the involved function is jointly uniformly continuous. Thus, given any $\\varepsilon> 0$, by uniform continuity we obtain, for sufficiently large $k\\in\\N$,\n\\begin{align*}\n&\\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{k,1}))\\,\\rho_{k,0}(x^0)^{-1\/N'}\\d\\mu_{k,0}(x^0)\\\\\n&\\qquad\\qquad \\geq \\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\rho_{k,0}(x^0)^{-1\/N'}\\d\\mu_{k,0}(x^0)-\\varepsilon.\n\\end{align*}\n\n\\textbf{Step 5.2.} We modify $\\smash{\\tilde{\\mu}_{k,0}}$ into the measure\n\\begin{align*}\n\\nu_{k,0} := (1+\\delta_k)^{-1}\\,(\\tilde{\\rho}_{k,0}+\\delta_k)\\,\\meas_k =\\varrho_{k,0}\\,\\meas_k,\n\\end{align*} \nwhere $\\delta_k\\in[0,1]$ is defined by\n\\begin{align*}\n\\delta_k = \\tilde{\\mu}_{k,0}[I^-(x_{k,1})^{\\ensuremath{\\mathsf{c}}}].\n\\end{align*}\nNote that $(\\delta_k)_{k\\in\\N}$ converges to $0$. Employing the inequality $\\rho_{k,0} \\leq a_k\\,\\varrho_{k,0}$ $\\meas_k$-a.e.,  the definition of $\\mu_{k,0}$, and a similar notation as in the proof of \\autoref{Th:Stability TCD}, \n\\begin{align}\\label{Eq:Integral rechnung}\n&\\int_{\\mms} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\rho_{k,0}(x^0)^{-1\/N'} \\d\\mu_{k,0}(x^0)\\nonumber\\\\\n&\\qquad\\qquad \\geq_k\\int_{\\mms}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\mu_{k,0}(x^0)\\nonumber\\\\\n&\\qquad\\qquad \\geq_k\\int_{\\mms}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\nu_{k,0}(x^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad - \\int_{I^-(x_{k,1})^{\\ensuremath{\\mathsf{c}}}} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\tilde{\\mu}_{k,0}(x^0)\\nonumber\\\\\n&\\qquad\\qquad\\qquad\\qquad - \\delta_k\\int_{\\mms}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\meas_k(x^0).\\nonumber\n\\end{align}\n\n\\textbf{Step 5.3.} Possibly invoking the timelike Bishop--Gromov inequality outlined in \\autoref{Re:Geom inequ TMCP}, we obtain the estimates\n\\begin{align*}\nc\\,\\delta_k^{1-1\/N'} &\\geq \\int_{I^-(x_{k,1})^{\\ensuremath{\\mathsf{c}}}}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\tilde{\\mu}_{k,0}(x^0)\\\\\nc\\,\\delta_k^{1-1\/N'} &\\geq \\delta_k\\int_{\\mms} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\meas_k(x^0).\n\\end{align*}\nHence, our main task is to estimate the integral in \\eqref{Eq:Integral rechnung} from below. To this aim, by  definition of $\\nu_{k,0}$ and Jensen's inequality, we obtain\n\\begin{align*}\n&\\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\nu_{k,0}(x^0)\\\\\n&\\qquad\\qquad = \\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\Big[\\!\\int_\\mms (\\rho_{\\infty,0}(y^0)+\\delta_k)\\d\\mathfrak{p}_{x^0}^k(y^0)\\Big]^{1-1\/N'}\\!\\d\\meas_k(x^0)\\\\\n&\\qquad\\qquad \\geq \\int_{\\mms^2} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\rho_{\\infty,0}(y^0)^{1-1\/N'}\\d\\mathfrak{q}_k(x^0,y^0).\n\\end{align*}\nLet $(\\phi_i)_{i\\in\\N}$ be a sequence in $\\Cont_\\bounded(\\mms)$ such that\n\\begin{align*}\n\\big\\Vert \\phi_i- \\rho_{\\infty,0}^{1-1\/N'}\\big\\Vert_{\\Ell^1(\\mms,\\meas_\\infty)} \\leq 2^{-i}\n\\end{align*}\nas well as $\\sup\\phi_i(\\mms) \\leq \\Vert\\rho_{\\infty,0} \\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}$ for every $i\\in\\N$. Then we get\n\\begin{align*}\n&\\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\nu_{k,0}(x^0)\\\\\n&\\qquad\\qquad \\geq \\int_{\\mms^2} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\phi_i(y^0)\\d\\mathfrak{q}_k(x^0,y^0) - 2^{-i}\\,c.\n\\end{align*}\n\n\\textbf{Step 5.4.} By tightness and stability of $W_2$-optimal couplings \\cite[Lem.~4.3, Lem. 4.4]{villani2009}, $(\\mathfrak{q}_k)_{k\\in\\N}$ converges weakly to the dia\\-gonal coupling of $\\meas_\\infty$ and $\\meas_\\infty$ along a nonrelabeled subsequence. In particular, by Lebesgue's theorem,\n\\begin{align*}\n&\\liminf_{k\\to\\infty} \\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\nu_{k,0}(x^0)\\\\\n&\\qquad\\qquad \\geq \\liminf_{i\\to\\infty} \\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\phi_i(x^0)\\d\\meas_\\infty(x^0) - c\\,\\limsup_{i\\to\\infty}2^{-i}\\\\\n&\\qquad\\qquad =\\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\rho_{\\infty,0}(x^0)^{-1\/N'}\\d\\mu_{\\infty,0}(x^0).\n\\end{align*}\n\n\\textbf{Step 6.} \\textit{Conclusion.} Let $({\\ensuremath{\\boldsymbol{\\pi}}}_k)_{k\\in\\N}$ be a sequence of $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_k\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_{k,0},\\mu_{k,1})}$ representing the $\\smash{\\ell_p}$-geodesic $(\\mu_{k,t})_{t\\in[0,1]}$ in Step 4, $k\\in\\N$. By compactness of $\\mms$ and \\autoref{Le:Villani lemma for geodesic}, this sequence converges to an $\\smash{\\ell_p}$-optimal geodesic plan $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_\\infty\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_{\\infty,0},\\mu_{\\infty,1})}$ along a nonrelabeled subsequence. In particular, the assignment $\\smash{\\mu_{\\infty,t} := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_\\infty}$ gives rise to a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_{\\infty,t})_{t\\in[0,1]}$ from $\\mu_{\\infty,0}$ to $\\mu_{\\infty,1}$; note that every optimal coupling of $\\mu_{\\infty,0}$ and $\\mu_{\\infty,1}$ is concentrated on $\\smash{\\mms_\\ll^2}$. Using  weak lower semicontinuity of the R\u00e9nyi entropy and \\eqref{Eq:TMCP cond}, given any $\\varepsilon > 0$ this yields, for every $t\\in[0,1)$ and every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{\\infty,t}) &\\leq \\limsup_{k\\to\\infty} {\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{k,t})\\\\\n&\\leq -\\liminf_{k\\to\\infty} \\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{k,1}))\\,\\rho_{k,0}(x^0)^{-1\/N'}\\d\\mu_{k,0}(x^0)\\\\\n&\\leq \\varepsilon - \\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\rho_{\\infty,0}(x^0)^{-1\/N'}\\d\\mu_{\\infty,0}(x^0).\n\\end{align*}\n\n\n\n\n\\textbf{Step 7.} \\textit{Relaxation of the assumptions on $\\mu_\\infty,0$ and $\\mu_{\\infty,1}$.} First, we argue how to construct the hypothesized sequence $(x_{k,1})_{k\\in\\N}$ from Step 2. The idea is to approximate points which lie ``in between'' $\\supp\\mu_{\\infty,0}$ and $x_{\\infty,1}$ from the future. For these points, the above discussion applies, and we will be able to conclude the desired $\\mathrm{TMCP}$ property by a tightness argument.\n\n\\textbf{Step 7.1.} As $\\inf\\tau(\\supp\\mu_{\\infty,0}, x_{\\infty,1})>0$, given any $i\\in \\N$ we may and will fix a sequence $\\smash{(x_{\\infty,1}^i)_{i\\in\\N}}$ of points $\\smash{x_{\\infty,1}^i}\\in I(\\mu_{\\infty,0},\\mu_{\\infty,1})\\cap\\supp\\meas_\\infty$ converging to $\\smash{x_{\\infty,1}}$. Since $I(x_{\\infty,1}^i,x_{\\infty,1})$ is open and nonempty,  for every $i\\in\\N$ we  construct a sequence $\\smash{(x_{\\infty,1}^{i,j})_{j\\in\\N}}$ such that $\\smash{x_{\\infty,1}^{i,j} \\in I(x_{\\infty,1}^i, x_{\\infty,1})\\cap\\supp\\meas_\\infty}$ and\n\\begin{align*}\nx_{\\infty,1}^{i,j} \\ll x_{\\infty,1}^{i,j-1}\n\\end{align*}\nfor every $j\\in\\N$. Since $\\smash{x_{\\infty,1}^{i,j}\\in\\supp\\meas_\\infty}$, the weak convergence of $(\\meas_k)_{k\\in\\N}$ implies the existence of a sequence $\\smash{(x_{k,1}^{i,j})_{k\\in\\N}}$ of points $\\smash{x_{k,1}^{i,j}\\in\\supp\\meas_k}$ which converges to $\\smash{x_{\\infty,1}^{i,j}}$. In particular, for a sufficiently large integer $k_j\\in \\N$, we have \n\\begin{align*}\nx_{\\infty,1}^i\\ll x_{\\infty,1}^{i,j} \\ll x_{k_j,1}^{i,j}\n\\end{align*}\nfor every $i,j\\in\\N$, as well as $\\smash{\\met(x_{k_j}^{i,j}, x_{\\infty,1}^i) \\to 0}$ as $j\\to\\infty$ for every $i\\in\\N$. \n\nThe above arguments applied for a fixed $i\\in\\N$ (along a suitable subsequence in $k$ which depends on $i$) yield the existence of a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\smash{(\\mu_{\\infty,t}^i)_{t\\in[0,1]}}$ from $\\mu_{\\infty,0}$ to $\\smash{\\mu_{\\infty,1}^i:= \\delta_{x_{\\infty,1}^i}}$ such that for every $t\\in[0,1)$ and every $N'\\geq N_\\infty$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{\\infty,t}^i) \\leq -\\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}^i))\\,\\rho_{\\infty,0}(x^0)^{-1\/N'}\\d\\mu_{\\infty,0}(x^0).\n\\end{align*}\nSimilarly to Step 6 and using lower semicontinuity of R\u00e9nyi's entropy and Fatou's lemma, we find a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\smash{(\\mu_{\\infty,t})_{t\\in[0,1]}}$ connecting $\\smash{\\mu_{\\infty,0}}$ to $\\smash{\\mu_{\\infty,1}}$ such that the previous inequality holds for $\\smash{\\mu_{\\infty,t}^i}$ and $\\smash{x_{\\infty,1}^i}$ replaced by $\\smash{\\mu_{\\infty,t}}$ and $\\smash{x_{\\infty,1}}$,  respectively, for every $t\\in[0,1)$ and every $N'\\geq N_\\infty$.\n\n\\textbf{Step 7.2.} Finally, we remove the two assumptions on $\\mu_{\\infty,0}$ from Step 2. Let $\\mu_{\\infty,0}\\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)}$ and $\\smash{x_{\\infty,1}\\in I^+(\\mu_{\\infty,0})\\cap\\supp\\meas_\\infty}$. Given  sufficiently large $n,m\\in\\N$, define $\\smash{\\mu_{\\infty,0}^n,\\mu_{\\infty,0}^{n,m}\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)}$ by\n\\begin{align*}\n\\mu_{\\infty,0}^n &:= \\mu_{\\infty,0}\\big[\\{\\uptau(\\cdot, x_{\\infty,1}) \\geq 2^{-n}\\}\\big]^{-1}\\,\\mu_{\\infty,0} \\mres \\{\\uptau(\\cdot, x_{\\infty,1}) \\geq 2^{-n}\\} = \\rho_{\\infty,0}^n\\,\\meas_\\infty,\\textcolor{white}{\\big\\vert^{0}}\\\\\n\\mu_{\\infty,0}^{n,m} &:= \\big\\Vert \\!\\min\\{\\rho_{\\infty,0}^n,m\\}\\big\\Vert_{\\Ell^1(\\mms,\\meas_\\infty)}^{-1}\\,\\min\\{\\rho_{\\infty,0}^n,m\\}\\,\\meas_\\infty.\n\\end{align*}\nBy continuity of $\\uptau$, $\\smash{\\mu_{\\infty,0}^{n,m}}$ obeys  the hypotheses from Step 2 for every $n,m\\in\\N$. A diagonal procedure gives rise to a map $m\\colon \\N\\to\\N$ such that  $\\smash{(\\tilde{\\mu}_{\\infty,0}^n)_{n\\in\\N}}$, where \n\\begin{align*}\n\\tilde{\\mu}_{\\infty,0}^n := \\mu_{\\infty,0}^{n,m_n},\n\\end{align*}\nconverges weakly to $\\smash{\\mu_{\\infty,0}}$ as $n\\to\\infty$.\n\nThe above discussion thus yields the desired inequality along some timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic connecting $\\smash{\\tilde{\\mu}_{\\infty,0}^n}$ to $\\smash{\\mu_{\\infty,1} := \\delta_{x_{\\infty,1}}}$. The usual tightness and lower semicontinuity argument directly gives the claim; note that the convergence of the corresponding right-hand sides is granted by Levi's theorem.\n\n\\textbf{Step 8.} \\textit{Passage from $K$ and $N$ to $K_\\infty$ and $N_\\infty$.} According to Step 4, all in all we thus infer $\\smash{\\mathrm{TMCP}_p(K,N)}$ for ${\\ensuremath{\\mathscr{X}}}_\\infty$ for every $K<K_\\infty$ and every $N>N_\\infty$, and we deduce  $\\smash{\\mathrm{TMCP}_p(K_\\infty,N_\\infty)}$ as in Step 9 in the proof of \\autoref{Th:Stability TCD}.\n\\end{proof}\n\n\\subsection{Good geodesics}\\label{Sub:Good} Now we prove an analogue of \\autoref{Th:Good geos TCD}, assuming the $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ condition. As indicated in \\autoref{Re:Unlikely}, this result will be used in the next \\autoref{Sec:Equiv TMCP's} to prove the equivalence of the latter condition with the entropic timelike measure-contraction property from \\cite{cavalletti2020}.\n\nAs in \\autoref{Sub:Good TCD}, we only outline the proof and refer to \\cite[Sec.~4.3]{braun2022}, see also \\cite{cavalletti2017}, for a similar discussion in the entropic case.\n\nThe following result is proven similarly to \\autoref{Le:Lalelu}.\n\n\\begin{lemma}\\label{Le:Lululu} Assume the $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ condition for some $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$. Let $\\smash{\\mu_0 = \\rho_0\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$, and let $x_1\\in I^+(\\mu_0)$. Finally, let $D$ be any real number no smaller than $\\sup\\uptau(\\supp\\mu_0\\times\\{x_1\\})$. Then there is a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\smash{\\mu_1 := \\delta_{x_1}}$ such that for every $t\\in [0,1)$, denoting by $\\rho_t$ the $\\meas$-ab\\-solutely continuous part of $\\mu_t$,\n\\begin{align*}\n\\meas\\big[\\{\\rho_t>0\\}\\big] \\geq (1-t)^N\\,{\\ensuremath{\\mathrm{e}}}^{-tD\\sqrt{K^-N}}\\,\\big\\Vert\\rho_0\\big\\Vert_{\\Ell^\\infty(\\mms,\\meas)}^{-1}.\n\\end{align*}\n\\end{lemma}\n\n\\begin{theorem}\\label{Th:Good TMCP} Under the hypotheses of \\autoref{Le:Lululu}, there exists a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\smash{\\mu_1:= \\delta_{x_1}}$ such that for every $t\\in[0,1]$, $\\smash{\\mu_t = \\rho_t\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$ and\n\\begin{align*}\n\\Vert\\rho_t\\Vert_{\\Ell^\\infty(\\mms,\\meas)} \\leq \\frac{1}{(1-t)^N}\\,{\\ensuremath{\\mathrm{e}}}^{Dt\\sqrt{K^-N}}\\,\\Vert\\rho_0\\Vert_{\\Ell^\\infty(\\mms,\\meas)},\n\\end{align*}\nwhere\n\\begin{align*}\nD:= \\sup\\uptau(\\supp\\mu_0\\times\\{x_1\\}).\n\\end{align*}\n\\end{theorem}\n\n\\begin{proof} As for \\cite[Thm.~4.18]{braun2022}, we first observe that the bisection argument from the proof of  \\autoref{Th:Good geos TCD} does not apply here since $\\smash{\\mu_1\\notin{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$, while we aim to establish that $\\smash{\\mu_t \\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$ for every $t\\in[0,1)$.\n\nGiven $n\\in\\N$ and $k\\in\\N_0$, we set $\\smash{s_n^k := (1-2^{-n})^k}$. Fix $n\\in\\N$, and assume that a measure $\\smash{{\\ensuremath{\\boldsymbol{\\beta}}}_n^k\\in \\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_1,\\mu_0)}$ [sic] has already been defined in such a way that for every $i\\in\\{1,\\dots,k\\}$, we have $\\smash{(\\eval_{s_n^i})_\\push{\\ensuremath{\\boldsymbol{\\beta}}}_n^k \\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ and\n\\begin{align*}\n\\sup\\uptau(\\supp\\,(\\eval_{s_n^i})_\\push{\\ensuremath{\\boldsymbol{\\beta}}}_n^k\\times\\{x_1\\})\\leq 2^{-n}\\,s_n^{i-1}\\,D.\n\\end{align*}\n\n\\textbf{Step 1.} \\textit{Minimization of an appropriate functional.} Set\n\\begin{align*}\nc_n^{k+1} := \\frac{1}{(1-2^{-n})^N}\\,{\\ensuremath{\\mathrm{e}}}^{2^{-n}s_n^kD\\sqrt{K^-N}}\\,\\Vert\\rho_0\\Vert_{\\Ell^\\infty(\\mms,\\meas)}\n\\end{align*}\nand let the functional $\\smash{{\\ensuremath{\\mathscr{F}}}_{c_n^{k+1}}}$ be defined as in \\eqref{Eq:Functional FC} above. As in \\cite[Le.~3.13]{braun2022}, the functional $\\smash{{\\ensuremath{\\mathscr{F}}}_{c_n^{k+1}}\\circ \\eval_{2^{-n}}}$ admits some minimizer $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_n^{k+1}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau((\\eval_{s_n^k})_\\push{\\ensuremath{\\boldsymbol{\\beta}}}_n^k,\\mu_1)}$. Let ${\\ensuremath{\\boldsymbol{\\sigma}}}_{k+1}^n\\in\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_1,(\\eval_{s_n^k})_\\push{\\ensuremath{\\boldsymbol{\\beta}}}_n^k)}$ be the $\\smash{\\ell_p}$-optimal geodesic plan obtained by ``time-reversal'' of $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_n^{k+1}}$. By gluing, we build $\\smash{{\\ensuremath{\\boldsymbol{\\beta}}}_n^{k+1}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_1,\\mu_0)}$ with\n\\begin{align*}\n(\\Restr_0^{s_n^k})_\\push{\\ensuremath{\\boldsymbol{\\beta}}}_n^{k+1} &= {\\ensuremath{\\boldsymbol{\\pi}}}_n^{k+1},\\\\\n(\\Restr_{s_n^k}^1)_\\push{\\ensuremath{\\boldsymbol{\\beta}}}_n^{k+1} &= (\\Restr_{s_n^k}^1)_\\push{\\ensuremath{\\boldsymbol{\\beta}}}_n^k.\n\\end{align*}\nHere, given any $s,t\\in[0,1]$ with $s<t$, the map $\\smash{\\Restr_s^t}$ is defined in \\eqref{Eq:Restr def}. Moreover, \n\\begin{align*}\n\\sup \\uptau(\\supp\\,(\\eval_{s_n^{k+1}})_\\push{\\ensuremath{\\boldsymbol{\\beta}}}_n^{k+1}\\times\\{x_1\\})\\leq 2^{-n}\\,s_n^k\\,D.\n\\end{align*}\n\n\\textbf{Step 2.} \\textit{Uniform density bound.} Given any $n\\in\\N$ and any $k\\in\\N_0$, arguing as in \\cite[Prop.~3.14, Cor.~3.15]{braun2022} and using \\autoref{Le:Lululu},\n\\begin{align}\\label{Eq:BLUBBB}\n\\inf\\{ {\\ensuremath{\\mathscr{F}}}_{c_n^{k+1}}((\\eval_{2^{-n}})_\\push{\\ensuremath{\\boldsymbol{\\pi}}} ) : {\\ensuremath{\\boldsymbol{\\pi}}} \\in \\mathrm{OptTGeo}_{\\ell_p}^\\uptau((\\eval_{s_n^k})_\\push{\\ensuremath{\\boldsymbol{\\beta}}}_n^k,\\mu_1)\\}= 0.\n\\end{align}\nNote that \\autoref{Le:Lululu} does \\emph{not}  assert absolute $\\meas$-continuity of the $t$-slice of the timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic under consideration. However, this does not affect the proof of \\cite[Prop.~3.14]{braun2022} as a possible further  $\\meas$-singular part gives an upper bound for the difference  ${\\ensuremath{\\mathscr{E}}}_{c'}({\\ensuremath{\\boldsymbol{\\pi}}}) - {\\ensuremath{\\mathscr{E}}}_{c'}({\\ensuremath{\\boldsymbol{\\alpha}}})$ in Step 3 therein. In particular, $\\smash{(\\eval_{s_n^{k+1}})_\\push{\\ensuremath{\\boldsymbol{\\beta}}}_n^{k+1} = \\rho_{s_n^{k+1}}\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ and, by geometric summation,\n\\begin{align}\\label{Eq:Blubbbb}\n\\big\\Vert\\rho_{s_n^{k+1}}\\big\\Vert_{\\Ell^\\infty(\\mms,\\meas)} \\leq \\frac{1}{(s_n^{k+1})^N}\\,{\\ensuremath{\\mathrm{e}}}^{(1-s_n^{k+1})D\\sqrt{K^-N}}\\,\\Vert\\rho_0\\Vert_{\\Ell^\\infty(\\mms,\\meas)}.\n\\end{align}\n\n\\textbf{Step 3.} \\textit{Completion.} Iterating the procedure in Step 1 and Step 2, we construct a family $\\smash{\\{{\\ensuremath{\\boldsymbol{\\beta}}}_n^k : n \\in\\N,\\, k\\in\\N_0\\}\\subset\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_1,\\mu_0)}$. Let $({\\ensuremath{\\boldsymbol{\\beta}}}^n)_{n\\in\\N}$ be an enumeration of the elements of the former set. By \\autoref{Le:Villani lemma for geodesic}, the latter admits an accumulation point $\\smash{{\\ensuremath{\\boldsymbol{\\beta}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_1,\\mu_0)}$. Let $\\smash{{\\ensuremath{\\boldsymbol{\\alpha}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ be the ``time-reversal'' of ${\\ensuremath{\\boldsymbol{\\beta}}}$, which induces a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\mu_1$. By weak lower semicontinuity of ${\\ensuremath{\\mathscr{F}}}_c$ in ${\\ensuremath{\\mathscr{P}}}(J(\\mu_0,\\mu_1))$ for appropriate values $c>0$, recall \\eqref{Eq:BLUBBB} and \\eqref{Eq:Blubbbb}, we get $\\mu_t = \\rho_t\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)$, and $\\Vert\\rho_t\\Vert_{\\Ell^\\infty(\\mms,\\meas)}$ obeys the desired upper bound for every $t\\in[0,1)$.\n\\end{proof}\n\n\\subsection{Uniqueness of $\\ell_p$-optimal couplings and $\\ell_p$-geodesics}\\label{Sub:Uniqueneess} In this section, we prove uniqueness of \\emph{chronological} $\\smash{\\ell_p}$-optimal couplings (if existent), \\autoref{Th:Uniqueness couplings}, as well as of $\\smash{\\ell_p}$-optimal geodesic plans,  \\autoref{Th:Uniqueness geodesics}. We follow \\cite[Sec.~3.4]{cavalletti2020}, see also \\cite{cavalletti2017}. A byproduct of our discussion is an extension of  the results from \\cite{cavalletti2020} from the timelike nonbranching to the timelike \\emph{essential}  nonbranching case, cf.~\\autoref{Re:From TNB to TENB}.\n\nIn this section, in addition to our standing assumptions, let ${\\ensuremath{\\mathscr{X}}}$ be timelike $p$-essentially nonbranching for some fixed $p\\in (0,1)$. \n\n\\begin{lemma}\\label{Le:Uniqueness Diracs} Assume $\\smash{\\mathrm{TMCP}^*_p(K,N)}$ for some  $K\\in\\R$ and $N\\in [1,\\infty)$. Let $\\mu_0 = \\rho_0\\,\\meas\\in\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$, and define $\\mu_1 := \\lambda_1\\,\\delta_{x_1} + \\dots + \\lambda_n\\,\\delta_{x_n}$ for  $\\lambda_1,\\dots,\\lambda_n\\in (0,1]$ with $\\lambda_1+\\dots+\\lambda_n=1$ and pairwise distinct $x_1,\\dots,x_n\\in\\mms$. Let $\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)$ be an  $\\smash{\\ell_p}$-optimal coupling. Then there is a $\\mu_0$-measurable map $T\\colon \\supp\\mu_0\\to\\mms$ with \n\\begin{align*}\n\\pi = (\\Id,T)_\\push\\mu_0,\n\\end{align*}\nand consequently,\n\\begin{align*}\n\\ell_p(\\mu_0,\\mu_1)^p = \\int_\\mms \\uptau(x,T(x))^p\\d\\mu_0(x).\n\\end{align*}\n\nIn particular, $\\pi$ is the unique chronological $\\smash{\\ell_p}$-optimal coupling of $\\mu_0$ and $\\mu_1$.\n\\end{lemma}\n\n\\begin{proof} By a standard argument, cf.~the proof of \\cite[Lem.~3.17]{cavalletti2020}, it suffices to prove the existence of a $\\mu_0$-measurable map $T\\colon \\supp\\mu_0\\to\\mms$ such that $\\pi=(\\Id,T)_\\push\\mu_0$. To this aim, define the compact set $E\\subset\\supp\\mu_0$ by\n\\begin{align*}\nE := \\{x\\in \\mms : \\#[(\\{x\\}\\times\\mms)\\cap \\supp\\pi] \\geq 2\\}.\n\\end{align*}\nWe claim that $\\mu_0[E] = 0$, which directly gives the desired $T$.\n\nAssume to the contrapositive that $\\mu_0[E]>0$. We first reduce the discussion to the uniform distribution on some subset of $\\mms$ as follows. Up to shrinking $E$, we assume without restriction that $\\varepsilon\\leq \\rho_0\\leq 1\/\\varepsilon$ $\\meas$-a.e.~on $E$ for some $\\varepsilon > 0$. A further possible shrinking of $E$ entails  the existence of  distinct points $z_1,z_2\\in\\{x_1,\\dots,x_n\\}$ and well-defined maps $T_1,T_2\\colon E\\to \\mms$ with $T_1(x) = z_1$ and $T_2(x)=z_2$ for every $x\\in E$. We may and will additionally assume that $\\smash{E\\times\\{z_1\\}, E\\times\\{z_2\\}\\subset \\mms_\\ll^2}$. By restric\\-tion \\cite[Lem.~2.10]{cavalletti2020}, the couplings\n\\begin{align*}\n\\pi_1 &:= \\meas[E]^{-1}\\,\\One_{E\\times\\{z_1\\}}\\,(\\rho_0\\circ\\pr_1)^{-1}\\,\\pi,\\\\\n\\pi_2 &:= \\meas[E]^{-1}\\,\\One_{E\\times\\{z_2\\}}\\,(\\rho_0\\circ\\pr_1)^{-1}\\,\\pi\n\\end{align*}\nare $\\smash{\\ell_p}$-optimal with $\\smash{\\pi_1\\in \\Pi_\\ll(\\nu_0,\\delta_{z_1})}$ and $\\smash{\\pi_2\\in\\Pi_\\ll(\\nu_0,\\delta_{z_2})}$, respectively, where\n\\begin{align*}\n\\nu_0 := \\meas[E]^{-1}\\,\\meas\\mres E = \\varrho_0\\,\\meas.\n\\end{align*}\n\nFrom now on, we will work with $\\nu_0$ instead of $\\mu_0$. Let $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_1\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\nu_0,\\delta_{z_1})}$ and $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_2\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\nu_0,\\delta_{z_2})}$ be $\\smash{\\ell_p}$-optimal geodesic plans which represent timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics witnessing the defining inequality of $\\smash{\\mathrm{TMCP}_p^*(K,N)}$. Note that $(\\eval_0)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_1$ and $(\\eval_0)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_2$ are mutually singular since\n\\begin{align*}\n{\\ensuremath{\\boldsymbol{\\pi}}}_i[\\{\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms) : \\gamma_1 = z_i\\}] = 1\n\\end{align*}\nfor every $i\\in\\{1,2\\}$, and $z_1\\neq z_2$. By \\autoref{Le:Mutually singular}, \n\\begin{align}\\label{Eq:Mutual singularity}\n(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_1 \\perp (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_2\n\\end{align}\nfor every $t\\in(0,1]$. Given $i\\in\\{1,2\\}$, let $\\smash{\\varrho_t^i}$ denote the density of the $\\meas$-absolutely continuous part of $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_1$, which is nontrivial by $\\smash{\\mathrm{TMCP}_p^*(K,N)}$. By \\autoref{Re:Lower bounds sigma}, \n\\begin{align*}\n\\int_\\mms (\\varrho_t^i)^{1-1\/N'}\\d\\meas &\\geq (1-t)\\,{\\ensuremath{\\mathrm{e}}}^{-tD\\sqrt{K^-\/N'}}\\int_\\mms \\varrho_0^{1-1\/N'}\\d\\meas\\\\\n&= (1-t)\\,{\\ensuremath{\\mathrm{e}}}^{-tD\\sqrt{K^-\/N'}}\\,\\meas[E]^{1\/N'}\\textcolor{white}{\\int}\n\\end{align*}\nfor some $N'>1$. On the other hand, by Jensen's inequality,\n\\begin{align*}\n\\int_\\mms (\\varrho_t^i)^{1-1\/N'}\\d\\meas &\\leq \\meas\\big[\\{\\varrho_t^i > 0\\}\\big]\\,\\meas\\big[\\{\\varrho_t^i > 0\\}\\big]^{-1}\\int_{\\{\\varrho_t^i > 0\\}} (\\varrho_t^i)^{1-1\/N'}\\d\\meas\\\\\n&\\leq \\meas\\big[\\{\\varrho_t^i > 0\\}\\big]\\,\\Big[\\meas\\big[\\{\\varrho_t^i > 0\\}\\big]^{-1}\\int_{\\{\\varrho_t^i>0\\}}\\varrho_t^i\\d\\meas\\Big]^{1-1\/N'}\\\\\n&\\leq \\meas\\big[\\{\\varrho_t^i > 0\\}\\big]^{1\/N'}.\\textcolor{white}{\\int}\n\\end{align*}\nThese  inequalities imply\n\\begin{align}\\label{Eq:mm}\n\\liminf_{t\\to 0} \\meas\\big[\\{\\varrho_t^i > 0\\}\\big] \\geq \\meas[E]= \\meas\\big[\\{\\varrho_0^i > 0\\}\\big].\n\\end{align}\n\nBy ${\\ensuremath{\\mathscr{K}}}$-global hyperbolicity, the sets\n\\begin{align*}\nF &:= \\{x\\in \\mms : x\\in\\supp\\,(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_i\\textnormal{ for some }t\\in[0,1],\\, i\\in\\{1,2\\}\\},\\\\\nG_\\delta &:= \\{x\\in F : \\uptau(y,x) \\leq \\delta\\textnormal{ for some }y\\in E\\}\n\\end{align*}\nare compact for every $\\delta>0$. Since $\\meas[G_\\delta]\\to \\meas[E]$ as $\\delta\\to 0$ by Lebesgue's theorem, there exists some $\\delta_0 \\in (0,1)$ such that\n\\begin{align}\\label{Eq:mmm}\n\\meas[G_{\\delta_0}] \\leq \\frac{3}{2}\\,\\meas[E].\n\\end{align}\nBy construction, for every $i\\in\\{1,2\\}$, every $t\\in (0,1)$, and $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_i$-a.e.~$x\\in \\mms$ there exists $\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$ such that $x=\\gamma_t$, $\\gamma_0\\in E$, and $\\smash{\\gamma_1 = z_i}$. By definition of $\\smash{G_{\\delta_0}}$, we thus obtain $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_i[G_{\\delta_0}] = 1$ for every $t\\in[0,\\delta_0]$. Hence, \\eqref{Eq:mm} and \\eqref{Eq:mm} imply\n\\begin{align*}\n\\meas\\big[\\{\\rho_s^1 > 0\\} \\cap \\{\\rho_s^2 > 0\\}\\big] > 0\n\\end{align*}\nfor some $s\\in (0,\\delta_0)$. However, this contradicts \\eqref{Eq:Mutual singularity}.\n\\end{proof}\n\n\\begin{lemma}\\label{Pr:More than TMCP} Let $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ hold for some $K\\in\\R$ and $N\\in[1,\\infty)$. Let $\\smash{\\mu_0 = \\rho_0\\,\\meas \\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ and $\\mu_1\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$ with $\\smash{\\supp\\mu_0\\times\\supp\\mu_1\\subset\\mms_\\ll^2}$. Then there exist a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ connecting $\\mu_0$ to $\\mu_1$ as well as an $\\smash{\\ell_p}$-optimal coupling $\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)$ such that for every $t\\in [0,1)$ and every $N'\\geq N$,\n\\begin{align}\\label{Eq:Claim conv TMCP}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t) \\leq -\\int_{\\mms^2} \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N'}\\d\\pi(x^0,x^1).\n\\end{align}\n\\end{lemma}\n\n\\begin{proof} \\textbf{Step 1.} \\textit{Approximation of $\\mu_1$.} Let $B_1,\\dots,B_n\\subset\\mms$, $n\\in\\N$, be a given Borel partition of $\\supp\\mu_1$ with $\\mu_1[B_i]>0$ for every $i\\in\\{1,\\dots,n\\}$. Given such an $i$, fix $\\smash{x_1^i\\in\\supp\\mu_1}$ and set $\\lambda_i := \\mu_1[B_i]$ as well as\n\\begin{align*}\n\\mu_1^n := \\lambda_1\\,\\delta_{x_1^1} + \\dots + \\lambda_n\\,\\delta_{x_1^n}.\n\\end{align*}\nSince every coupling of $\\mu_0$ and $\\smash{\\mu_1^n}$ is chronological, an $\\smash{\\ell_p}$-optimal coupling of these exists uniquely by \\autoref{Le:Uniqueness Diracs}. Let $\\smash{\\pi_n\\in \\Pi_\\ll(\\mu_0,\\mu_1^n)}$ be this coupling, given by a $\\mu_0$-measurable map $T_n\\colon \\supp\\mu_0\\to\\mms$. For $i\\in\\{1,\\dots,n\\}$, we define \n\\begin{align*}\nA_i := T_n^{-1}(\\{x_1^i\\})\\times\\{x_1^i\\},\n\\end{align*}\nand observe that $A_1,\\dots,A_n\\subset \\mms^2$ constitutes a Borel partition of $\\supp\\pi_n$. \n\nDefine $\\smash{\\pi_n^i\\in{\\ensuremath{\\mathscr{P}}}(\\mms^2)}$, $\\smash{\\nu_0^i\\in {\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$, and $\\smash{\\nu_1^i\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)}$ by\n\\begin{align*}\n\\pi_n^i &:= \\lambda_i^{-1}\\,\\pi_n\\mres A_i,\\\\\n\\nu_0^i &:= (\\pr_1)_\\push\\pi_n^i = \\varrho_0^i\\,\\meas,\\\\\n\\nu_1^i &:= (\\pr_2)_\\push\\pi_n^i = \\delta_{x_1^i}.\n\\end{align*}\nBy construction, we have $\\mu_0 = \\lambda_1\\,\\nu_0^i + \\dots + \\lambda_n\\,\\nu_0^n$ and thus\n\\begin{align*}\n\\rho_0 = \\lambda_1\\,\\varrho_0^i + \\dots + \\lambda_n\\,\\varrho_0^n\\quad\\meas\\textnormal{-a.e.}\n\\end{align*}\nMoreover $\\smash{\\supp\\nu_0^i \\cap \\supp\\nu_0^j = \\emptyset}$ for every $i,j\\in\\{1,\\dots,n\\}$ with $i\\neq j$, whence\n\\begin{align}\\label{Eq:MUT SING}\n\\nu_0^i\\perp\\nu_0^j.\n\\end{align}\n\n\\textbf{Step 2.} \\textit{Invoking the $\\mathrm{TMCP}$ condition.} As $\\smash{x_1^i\\in I^+(\\nu_0^i)}$ for every $i\\in\\{1,\\dots,n\\}$, using the  $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ condition there exists a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\smash{(\\nu_t^i)_{t\\in[0,1]}}$ from $\\smash{\\nu_0^i}$ to $\\smash{\\nu_1^i}$ such that for every $t\\in[0,1)$ and every $N'\\geq N$,\n\\begin{align}\\label{Eq:Interm TMCP}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\nu_t^i) \\leq -\\int_\\mms \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0, x_1^i))\\,\\rho_0^i(x^0)^{1-1\/N'}\\d\\meas(x^0).\n\\end{align}\n\nWith this information, we now build a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\smash{(\\mu_t^n)_{t\\in [0,1]}}$ from $\\mu_0$ to $\\smash{\\mu_1^n}$ for which \\eqref{Eq:Claim conv TMCP} holds with $\\pi$ replaced by $\\pi_n$. By definition, $\\smash{(\\nu_t^i)_{t\\in[0,1]}}$ is represented by some $\\smash{{\\ensuremath{\\boldsymbol{\\alpha}}}^i\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\nu_0^i,\\nu_1^i)}$. Thanks to  \\eqref{Eq:MUT SING} and \\autoref{Le:Mutually singular}, we obtain\n\\begin{align}\\label{Eq:MUT SING II}\n(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^i \\perp (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^j\n\\end{align}\nfor every $t\\in(0,1]$ and every $i,j\\in\\{1,\\dots,n\\}$ with $i\\neq j$. Then\n\\begin{align*}\n{\\ensuremath{\\boldsymbol{\\pi}}}_n := \\lambda_1\\,{\\ensuremath{\\boldsymbol{\\alpha}}}^1 + \\dots + \\lambda_n\\,{\\ensuremath{\\boldsymbol{\\alpha}}}^n\n\\end{align*}\nis an $\\smash{\\ell_p}$-optimal geodesic plan from $\\mu_0$ to $\\smash{\\mu_1^n}$, and the assignment $\\smash{\\mu_t^n := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_n}$ gives rise to a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic connecting these probability measures. Let $\\smash{\\varrho_t^i}$ denote the nontrivial density of the $\\meas$-absolutely continuous part of $\\smash{(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^i}$. Then we obtain that\n\\begin{align*}\n\\rho_t^n\\,\\meas := \\big[\\lambda_1\\,\\varrho_t^1 + \\dots + \\lambda_n\\,\\varrho_t^n\\big]\\,\\meas\n\\end{align*}\nis the $\\meas$-absolutely continuous part of $\\mu_t^n$, $t\\in[0,1]$. Now \\eqref{Eq:MUT SING II} ensures \n\\begin{align*}\n\\meas\\big[\\{\\varrho_t^i > 0\\} \\cap \\{\\varrho_t^j > 0\\}\\big] = 0.\n\\end{align*}\nfor every $t\\in [0,1]$ and every $i,j\\in\\{1,\\dots,n\\}$ with $i\\neq j$.  Hence,  using  \\eqref{Eq:Interm TMCP} and \\eqref{Eq:MUT SING} we get, for every $t\\in[0,1)$ and every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t^n) &=\n\\sum_{i=1}^n \\lambda_i^{1-1\/N'}\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t^i)\\\\\n&\\leq -\\sum_{i=1}^n\\lambda_i^{1-1\/N'}\\!\\int_\\mms \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0,x_1^i))\\,\\varrho_0^i(x^0)^{-1\/N'}(x^0)\\d\\nu_0^i(x^0)\\\\\n&= -\\sum_{i,i'=1}^n \\int_\\mms \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0,x_1^i))\\,\\big[\\lambda_{i'}\\,\\varrho_0^{i'}(x^0)\\big]^{-1\/N'}\\d\\big[\\lambda_i\\,\\nu_0^i\\big](x^0)\\\\\n&= -\\int_{\\mms^2} \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N'}\\d\\pi_n(x^0,x^1).\n\\end{align*}\n\n\\textbf{Step 3.} \\textit{Conclusion.} If $\\supp\\mu_1$ consists of finitely many points, the claim simply follows by choosing $n\\in\\N$ such that $\\smash{\\mu_1 = \\mu_1^n}$. \n\nTherefore, only the case $\\#\\supp\\mu_1=\\infty$ remains to be studied. By \\cite[Thm.~2.16]{cavalletti2020}, there exists a sequence $\\smash{(\\bar{\\mu}_1^n)_{n\\in\\N}}$ in ${\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$ which converges weakly to $\\mu_1$ such that $\\smash{\\bar{\\mu}_1^n\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\mu_1^n)}$ for every $n\\in\\N$, and for every sequence $(\\pi_n)_{n\\in\\N}$ of $\\smash{\\ell_p}$-optimal couplings $\\smash{\\pi_n \\in\\Pi_\\leq(\\mu_0,\\bar{\\mu}_1^n)}$, every weak limit of any subsequence of the latter is $\\smash{\\ell_p}$-optimal for $\\mu_0$ and $\\mu_1$. In particular, the previous discussion applies to $\\smash{\\bar{\\mu}_1^n}$ in place of $\\smash{\\mu_1^n}$, $n\\in\\N$, and yields sequences $\\smash{({\\ensuremath{\\boldsymbol{\\pi}}}_n)_{n\\in\\N}}$ and $(\\pi_n)_{n\\in\\N}$  of $\\smash{\\ell_p}$-optimal geodesic plans $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_n\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\bar{\\mu}_1^n)}$ and $\\smash{\\ell_p}$-optimal couplings $\\smash{\\pi_n\\in\\Pi_\\ll(\\mu_0,\\mu_1^n)}$ as in Step 2, respectively. By ${\\ensuremath{\\mathscr{K}}}$-global hyperbolicity, Prokhorov's theorem, and  \\autoref{Le:Villani lemma for geodesic} these sequences converge weakly to some $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ and an $\\smash{\\ell_p}$-optimal coupling $\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)$, respectively, up to a nonrelabeled subsequence. Define a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\mu_1$ by $\\mu_t := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}$. Employing weak lower semicontinuity of ${\\ensuremath{\\mathscr{S}}}_{N'}$ on measures with uniformly bounded support and \\autoref{Le:USC lemma}, we then obtain, for every $t\\in[0,1)$, every $N'\\geq N$, and with $\\smash{\\mu_t^n := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_n}$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t) &\\leq \\limsup_{n\\to\\infty} {\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t^n)\\\\\n&\\leq -\\liminf_{n\\to\\infty} \\int_{\\mms^2} \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N'}\\d\\pi_n(x^0,x^1)\\\\\n&\\leq -\\int_{\\mms^2} \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N'}\\d\\pi(x^0,x^1).\\qedhere\n\\end{align*}\n\\end{proof}\n\n\\begin{remark} Arguing as in Step 4 in the proof of \\cite[Prop.~3.18]{cavalletti2020} as well as Step 2 above, the conclusion of \\autoref{Pr:More than TMCP} holds under the  following weaker assumption: the measures $\\mu_0$ and $\\mu_1$ admit an $\\smash{\\ell_p}$-optimal coupling $\\varpi\\in\\Pi_\\ll(\\mu_0,\\mu_1)$ such that $\\smash{\\supp\\pi\\subset\\mms_\\ll^2}$. We will not need this extension in the sequel.\n\\end{remark}\n\nThe proof of the following \\autoref{Th:Uniqueness couplings} follows the lines of \\autoref{Le:Uniqueness Diracs} modulo some modifications we briefly discuss.\n\n\\begin{theorem}\\label{Th:Uniqueness couplings} Assume $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ for $K\\in\\R$ and $N\\in [1,\\infty)$. Suppose the pair $(\\mu_0,\\mu_1)\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)\\times{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$ to be timelike $p$-dualizable by $\\smash{\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)}$. Then there exists a $\\mu_0$-measurable map $T\\colon \\mms \\to \\mms$ such that\n\\begin{align*}\n\\pi = (\\Id,T)_\\push\\mu_0,\n\\end{align*}\nand consequently,\n\\begin{align*}\n\\ell_p(\\mu_0,\\mu_1)^p = \\int_\\mms \\uptau(x,T(x))^p\\d\\mu_0(x).\n\\end{align*}\n\nIn particular, $\\pi$ is the unique chronological $\\ell_p$-optimal coupling of $\\mu_0$ and $\\mu_1$.\n\\end{theorem}\n\n\\begin{proof} As in the proof of \\autoref{Le:Uniqueness Diracs}, it suffices to prove the existence of $T$. Let $\\Gamma\\subset\\mms_\\ll^2$ be an $\\smash{\\ell_p}$-cyclically monotone set with $\\pi[\\Gamma]=1$ \\cite[Prop.~2.8]{cavalletti2020}, and set\n\\begin{align*}\nE := \\{x\\in\\mms :  \\pr_2[\\Gamma \\cap (\\{x\\}\\times \\mms)] \\geq 2\\},\n\\end{align*}\nwhich is a Suslin set. We claim that $\\mu_0[S]=0$, which directly gives the desired $T$.\n\nAssume to the contrapositive that $\\mu_0[S]>0$. By the von Neumann selection theorem \\cite[Thm.~9.1.3]{bogachev2007b}, there exist $\\mu_0$-measurable maps $T_1,T_2\\colon E\\to \\mms$ whose graphs are both contained in $\\Gamma$ and such that $T_1(x) \\neq T_2(x)$ for every $x\\in E$. By Lusin's theorem, there exists a compact set $C\\subset E$ with $\\mu_0[C]>0$ such that the restrictions $\\smash{T_1\\big\\vert_C}$ and $\\smash{T_2\\big\\vert_C}$ are continuous. This yields\n\\begin{align*}\n\\min\\{\\met(T_1(x),T_2(x)) : x\\in C\\} > 0.\n\\end{align*}\nIn particular, there exist $z_1,z_2\\in\\mms$ and $r>0$ with $\\met(z_1,z_2)>r$, and a compact set $C'\\subset C$ with $\\mu_0[C]>0$ as well as $\\smash{T_1(C')\\subset{\\ensuremath{\\mathsf{B}}}^\\met(z_1,r\/2)}$ and $\\smash{T_2(C')\\subset{\\ensuremath{\\mathsf{B}}}^\\met(z_2,r\/2)}$. Up to possibly shrinking the radius $r$, we may and will assume without restriction that  $\\smash{C'\\times [\\bar{{\\ensuremath{\\mathsf{B}}}}^\\met(z_1,r\/2)\\cup \\bar{{\\ensuremath{\\mathsf{B}}}}^\\met(z_2,r\/2)] \\subset\\mms_\\ll^2}$.\n\nAs in the proof of \\autoref{Le:Uniqueness Diracs}, we may  further shrink $C'$, hence assume that $\\rho_0$ is bounded and bounded away from zero on $C'$. Consider $\\smash{\\nu_0\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ with\n\\begin{align*}\n\\nu_0 := \\meas[C']^{-1}\\,\\meas\\mres C'.\n\\end{align*}\nFurthermore, define $\\smash{\\mu_1^1,\\mu_1^2\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)}$ by\n\\begin{align*}\n\\mu_1^1 &:= (T_1)_\\push\\mu_0,\\\\\n\\mu_1^2 &:= (T_2)_\\push\\mu_0.\n\\end{align*}\nBy construction $\\smash{\\supp\\mu_1^1 \\cap \\supp\\mu_1^2 = \\emptyset}$, and the pairs $\\smash{(\\nu_0,\\mu_1^1)}$ and $\\smash{(\\nu_0,\\mu_1^2)}$ are both strongly $p$-timelike dualizable by \\autoref{Re:Strong timelike}.  \\autoref{Pr:More than TMCP} applies to both pairs; following the proof of \\autoref{Le:Uniqueness Diracs} from now on gives the claim.\n\\end{proof}\n\n\\begin{theorem}\\label{Th:Uniqueness geodesics} Assume $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ for some $K\\in\\R$ and $N\\in[1,\\infty)$. Suppose the pair $(\\mu_0,\\mu_1)\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)\\times{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$ to be timelike $p$-dualizable. Then for every ${\\ensuremath{\\boldsymbol{\\pi}}}\\in\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ there is a $\\mu_0$-measurable map $\\smash{\\mathfrak{T} \\colon \\supp\\mu_0\\to \\mathrm{TGeo}^\\uptau(\\mms)}$ with\n\\begin{align*}\n{\\ensuremath{\\boldsymbol{\\pi}}} = \\mathfrak{T}_\\push\\mu_0.\n\\end{align*}\n\nIn particular, the set $\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ is a singleton.\n\\end{theorem}\n\n\\begin{proof} By the usual argument outlined in the proof of \\autoref{Le:Uniqueness Diracs}, it suffices to prove that every $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ is induced by a map $\\mathfrak{T}$ as above.\n\nAssume to the contrapositive that one such ${\\ensuremath{\\boldsymbol{\\pi}}}$, henceforth fixed, is not given by a map. We disintegrate ${\\ensuremath{\\boldsymbol{\\pi}}}$ with respect to $\\eval_0$, writing, with some abuse of notation,\n\\begin{align*}\n{\\ensuremath{\\mathrm{d}}}{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma) = {\\ensuremath{\\mathrm{d}}}{\\ensuremath{\\boldsymbol{\\pi}}}_x(\\gamma)\\d\\mu_0(x)\n\\end{align*}\nfor a $\\mu_0$-measurable map ${\\ensuremath{\\boldsymbol{\\pi}}}\\colon \\supp\\mu_0\\to {\\ensuremath{\\mathscr{P}}}(\\mathrm{TGeo}^\\uptau(\\mms))$. By assumption on ${\\ensuremath{\\boldsymbol{\\pi}}}$, there is a compact set $C\\subset\\supp\\mu_0$ such that $\\mu_0[C]>0$ and $\\#\\supp {\\ensuremath{\\boldsymbol{\\pi}}}_x \\geq 2$ for every $x\\in C$.\nHence, for every $x\\in C$ there exists  $t_x\\in (0,1)$ such that $\\#\\supp\\,(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_x \\geq 2$; since $\\smash{\\mathrm{TGeo}^\\uptau(\\mms)\\subset\\Cont([0,1];\\mms)}$ and $\\smash{\\supp{\\ensuremath{\\boldsymbol{\\pi}}}_x \\subset \\mathrm{TGeo}^\\uptau(\\mms)}$, there exists an open interval $I_x\\subset (0,1)$ with $t_x\\in I_x$ such that $\\#\\supp\\,(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_x  \\geq 2$ for every $t\\in I_x$.\n\nGiven any $t\\in \\Q\\cap (0,1)$, we define\n\\begin{align*}\nC_t &:= \\{x \\in C : \\#\\supp\\,(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_x\\geq 2\\}.\n\\end{align*}\nSince $C_t$ unites to $C$ when ranging over $t\\in \\Q\\cup (0,1)$, we find $s\\in \\Q\\cap (0,1)$ with $\\smash{\\mu_0[C_s] >0}$. Finally, we define $\\smash{{\\ensuremath{\\boldsymbol{\\alpha}}}\\in{\\ensuremath{\\mathscr{P}}}(\\mathrm{TGeo}^\\uptau(\\mms))}$ by\n\\begin{align*}\n{\\ensuremath{\\mathrm{d}}}{\\ensuremath{\\boldsymbol{\\alpha}}}(\\gamma) = \\mu_0[C_s]^{-1}\\,\\One_{C_s}(x)\\d{\\ensuremath{\\boldsymbol{\\pi}}}_x(\\gamma)\\d\\mu_0(x).\n\\end{align*}\nBy restriction \\cite[Lem.~2.10]{cavalletti2020}, $(\\eval_0,\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}$ is an $\\smash{\\ell_p}$-optimal coupling of its marginals, the first one of which is $\\smash{\\mu_0[C_s]^{-1}\\,\\mu_0\\mres C_s\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$. Since ${\\ensuremath{\\boldsymbol{\\alpha}}}$ is concentrated on $\\mathrm{TGeo}^\\uptau(\\mms)$, a standard argument using the reverse triangle inequality \\eqref{Eq:Reverse tau} for $\\uptau$ implies that $(\\eval_0,\\eval_s)_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}$ is a chronological $\\smash{\\ell_p}$-optimal coupling of its marginals. By definition of $C_s$, it is not given by a map, which contradicts  \\autoref{Th:Uniqueness couplings}.\n\\end{proof}\n\n\\begin{remark}\\label{Re:From TNB to TENB} The above arguments --- which, as mentioned, are leaned on \\cite{cavalletti2020} --- show that the uniqueness  results of \\cite[Thm.~3.19, Thm.~3.20]{cavalletti2020} remain true if the timelike nonbranching assumption therein is replaced by the hypothesis of timelike $p$-essential nonbranching for the considered exponent $p\\in (0,1)$.\n\\end{remark}\n\n\\subsection{Equivalence with the entropic TMCP condition}\\label{Sec:Equiv TMCP's} In this section, additionally to our standing assumptions we suppose ${\\ensuremath{\\mathscr{X}}}$ to be timelike $p$-essentially nonbranching for some  fixed $p\\in (0,1)$. Building upon the results in \\autoref{Sub:Good} and \\autoref{Sub:Uniqueneess}, we establish the equivalence of $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ with the entropic timelike measure-contraction property from \\cite[Def.~3.7]{cavalletti2020}, restated in \\autoref{Def:TMCPe}. A byproduct of our argumentation is a pathwise characterization of $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ and $\\smash{\\mathrm{TMCP}_p^e(K,N)}$, cf.~\\autoref{Th:Equivalence TMCP* and TMCPe}, which is also available for $\\smash{\\mathrm{TMCP}_p(K,N)}$ according to \\autoref{Th:Equivalence TMCP}.\n\nRecall the definition \\eqref{Eq:Expo Boltzmann} of the exponentiated Boltzmann entropy ${\\ensuremath{\\mathscr{U}}}_N$.\n\n\\begin{definition}\\label{Def:TMCPe} Let $K\\in\\R$ and $N\\in (0,\\infty)$. We say that ${\\ensuremath{\\mathscr{X}}}$ satis\\-fies the \\emph{entropic timelike measure-contraction property} $\\smash{\\mathrm{TMCP}_p^e(K,N)}$ if for every $\\mu_0\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)$ and every $x_1\\in I^+(\\mu_0)$, there exists an $\\smash{\\ell_p}$-geo\\-desic $(\\mu_t)_{t\\in [0,1]}$ connecting $\\mu_0$ and $\\smash{\\mu_1:= \\delta_{x_1}}$ such that for every $t\\in [0,1]$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{U}}}_N(\\mu_t) \\geq \\sigma_{K,N}^{(1-t)}\\big[\\Vert\\uptau\\Vert_{\\Ell^2(\\mms^2,\\mu_0\\otimes\\mu_1)}\\big]\\,{\\ensuremath{\\mathscr{U}}}_N(\\mu_0).\n\\end{align*}\n\\end{definition}\n\n\\begin{theorem}\\label{Th:Equivalence TMCP* and TMCPe} The following statements are equivalent for every given $K\\in\\R$ and $N\\in[1,\\infty)$.\n\\begin{enumerate}[label=\\textnormal{\\textcolor{black}{(}\\roman*\\textcolor{black}{)}}]\n\\item\\label{LAAA} The condition $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ holds.\n\\item\\label{LAAAA} For every $\\mu_0 = \\rho_0\\,\\meas \\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)$ and every $x_1\\in I^+(\\mu_0)$ there is an $\\smash{\\ell_p}$-optimal geodesic plan $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$, where $\\smash{\\mu_1 := \\delta_{x_1}}$, such that for every $t\\in[0,1)$, we have $\\smash{(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas \\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$, and for every $N'\\geq N$, the inequality\n\\begin{align*}\n\\rho_t(\\gamma_t)^{-1\/N'} \\geq \\sigma_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,x_1))\\,\\rho_0(\\gamma_0)^{-1\/N'}\n\\end{align*}\nholds for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$.\n\\item\\label{LAAAAA} The condition $\\smash{\\mathrm{TMCP}_p^e(K,N)}$ holds.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof} We outline the necessary adaptations of the arguments in \\autoref{Sub:Equiv TCDs} and leave the details to the reader. \n\nBy integration, \\ref{LAAAA} implies \\ref{LAAA}. The implication of \\ref{LAAA} to \\ref{LAAAA} is argued analogously to \\autoref{Pr:ii to iii}. Note that to obtain the asserted  absolute continuity of $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}$ with respect to $\\meas$ for every $t\\in[0,1)$, we have combined \\autoref{Th:Good TMCP} with the uniqueness \\autoref{Th:Uniqueness geodesics}. The step from \\ref{LAAAA} to \\ref{LAAAAA} follows since\n\\begin{align*}\n-\\frac{1}{N}\\log\\rho_t(\\gamma_t) \\geq {\\ensuremath{\\mathrm{H}}}_t\\Big[\\!-\\!\\frac{1}{N}\\log\\rho_0(\\gamma_0),\\frac{K}{N}\\,\\uptau^2(\\gamma_0,x_1)\\Big]\n\\end{align*}\nfor ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$, using that the function ${\\ensuremath{\\mathrm{H}}}_t$ defined in \\eqref{Eq:GtHt} has similar convexity properties as the function ${\\ensuremath{\\mathrm{G}}}_t$ used to derive  \\autoref{Pr:iii to iv}, and then arguing as in the proof of the latter result. Similarly, the implication from \\ref{LAAAAA} to \\ref{LAAAA} is shown analogously to the proof of \\autoref{Pr:v to iii}.\n\\end{proof}\n\nEvident adaptations of the previous arguments give the following.\n\n\\begin{theorem}\\label{Th:Equivalence TMCP} The following statements are equivalent for every given $K\\in\\R$ and $N\\in[1,\\infty)$.\n\\begin{enumerate}[label=\\textnormal{\\textcolor{black}{(}\\roman*\\textcolor{black}{)}}]\n\\item The condition $\\smash{\\mathrm{TMCP}_p(K,N)}$ holds.\n\\item For every $\\mu_0 =\\rho_0\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)$ and every $x_1\\in I^+(\\mu_0)$ there is an $\\smash{\\ell_p}$-optimal geodesic plan $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$, where $\\smash{\\mu_1 := \\delta_{x_1}}$, such that for every $t\\in[0,1)$, we have $\\smash{(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$, and for every $N'\\geq N$, the inequality\n\\begin{align*}\n\\rho_t(\\gamma_t)^{-1\/N'} \\geq \\tau_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,x_1))\\,\\rho_0(\\gamma_0)^{-1\/N'}\n\\end{align*}\nholds for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$.\n\\end{enumerate}\n\\end{theorem}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOff-resonant Raman scattering is a robust approach to light-atom interfaces.\nOne of the methods is to induce spontaneous Stokes scattering in which\npairs of photons and collective atomic excitations - a two-mode squeezed\nstate - are created. These excitations can be stored and later retrieved\nin the anti-Stokes process \\cite{VanderWal2003,Bashkansky2012}. This approach is\ncommonly known to be a basic building block of the DLCZ protocol \\cite{Duan2001}.\n\nTypically rubidium and cesium have been used as atomic systems in\nboth warm and cold atomic ensembles \\cite{Chrapkiewicz2012,Michelberger2015,Dabrowski2014}.\nThese systems are coupled to light at near-IR wavelengths,\nsuch as 795 and 780 nm for rubidium D1 and D2 lines. Coupling to new wavelengths holds a promise to greatly extend capabilities of quantum memories. This\ncan be accomplished by non-linear frequency conversion in the four-wave\nmixing (4WM) process using strong resonant non-linearities in atoms\nthanks to transitions between excited states. Such processes\nhave been used to demonstrate frequency conversion in rubidium \\cite{Donvalkar2014,Becerra2008,Parniak2015,Akulshin2009a,Khadka2012} or to generate photon pairs in the\ncascaded spontaneous 4WM process \\cite{Willis2010,Srivathsan2013a,Chaneliere2006}. Multiphoton\nprocesses are also a developing method to interface light and Rydberg\natoms \\cite{Kondo2015,Huber2014a,Kolle2012}.\n\n\nChaneli\\`ere \\textit{et al.} \\cite{Chaneliere2006} proposed to combine the\nprocesses of Raman scattering and 4WM by first creating photon pairs\nin a cascaded spontaneous 4WM in one atomic ensemble and then storing\nphotons as collective atomic excitations in another cold ensemble,\nand an experiment was recently realized \\cite{Zhang2016}.\nAs a result they obtained a two-mode squeezed state of atomic excitations\nand telecom photons. Another approach was to frequency-convert light\ngenerated in quantum memory with 4WM \\cite{Radnaev2010a}, in order to create a frequency-converted\nquantum memory. In all cases one atomic ensemble was used for storage,\nand another for frequency conversion or photon generation. \n\n\\begin{figure}\n\\includegraphics[scale=1.02]{fig1}\\protect\\caption{(a) Configuration of atomic levels and fields we use to realize the\nfour-photon interface, (b) pulse sequence used in the experiment,\n(c) the central part of experimental setup demonstrating phase-matching\ngeometry and (d) trace of the 1 GHz FSR Fabry-P\\'erot interferometer signal showing the frequency of 4Ph field being different from what one would expect from the closed-loop process. Rows in (b) correspond to different beam paths presented\nin (c).}\n\\label{fig:schemat}\n\\end{figure}\n\n\nIn this paper we realize a Raman-like interface based\non 4WM in warm rubidium vapors driven by ground-state atomic coherence. The process we present may be in principle\nused to generate correlated pairs of collective atomic excitations\nand photons coupled to transition between two excited states in a\nsingle, four-photon process in a single atomic ensemble. Transition between two excited states corresponds to 776-nm light\nas illustrated in Fig.~\\hyperref[fig:schemat]{\\ref*{fig:schemat}(a)}. As the two intermediate states we use $5\\mathrm{P}_{3\/2}$\nand $5\\mathrm{D}_{5\/2}$.\n\nThe paper is organized as follows. In Sec. \\ref{sec:theory} we discuss the principles behind our idea. In Sec. \\ref{sec:experimental} we describe the experimental setup and methods we use to verify our findings. Finally, we give the results of our studies of the four-wave mixing interface, namely correlations and statistical properties in Sec. \\ref{sec:corr} and detuning dependencies in Sec. \\ref{sec:spectral}. We conclude the paper and give prospects for future developments in Sec. \\ref{sec:concl}.\n\n\\section{General idea}\\label{sec:theory}\nIn our experiment we generate ground-state atomic coherence $\\rho_{gh}$\nand light denoted by 2Ph in a two-photon stimulated Raman Stokes process, seeded by vacuum fluctuations. The advantage of this approach is the fact that it is a well-established and effective way to prepare atomic-ground state coherence. In particular, it may be used in different regimes, starting from the single-photon and single-excitation spontaneous regime as in the DLCZ protocol \\cite{Duan2001,Chrapkiewicz2012}, through the linear gain regime \\cite{Raymer1981,Duncan1990} and even in the nonlinear gain-saturation regime \\cite{Walmsley1985,Lewenstein1984,Trippenbach1985}. In the two latter cases, macroscopic ground-state coherence is generated \\cite{Zhang2014c}. The generated coherence and the number of scattered photons will be highly random but correlated. The atomic coherence is not averaged out to zero due to atomic motion, since the buffer gas makes the motion diffusive \\cite{Raymer2004}. Moreover, the generated Raman field remains coherent with the driving field, so phase fluctuations of the driving field do not disturb the process \\cite{Wu2010}. In particular, the generated macroscopic ground-state coherence may be probed \\cite{Chen2010}, read-out \\cite{VanderWal2003}, or may enhance further stimulated Raman process \\cite{Yuan2010}.\n\nIn this experiment, we observe concurrent generation of 776-nm light denoted by 4Ph in\na four-photon process analogous to stimulated Raman scattering driven by\nground-state coherence. It does not occur spontaneously in the macroscopic\nregime due to small gain. However, with macroscopic $\\rho_{gh}$ generated in the two-photon stimulated Raman process, the\ndriving fields $a$, $b$ and $c$ couple ground-state atomic coherence\nto the weak optical 4Ph field. In other words, the 4Ph process is stimulated by pre-existing atomic coherence. In the leading order in drive beam\nfields Rabi frequencies $\\Omega_{i}$ the atomic polarization resulting\nin emission of 4Ph signal field is:\n\n\\begin{equation}\n\\mathbf{P}_\\mathrm{4Ph}=-n\\mathbf{d}_{e_{3}e_{2}}\\rho_{gh}\\frac{{\\Omega_{a}\\Omega_{b}\\Omega_{c}^{*}}}{4\\Delta_{g}\\delta\\Delta_{h}},\n\\label{eq:P2ndrho}\n\\end{equation}\n\n\nwhere $n$ is the atom number density and $\\mathbf{d}_{ij}$ is the\ndipole moments of respective transition. From this formula it follows\nwhich detunings play role.\n\nFor the experimental observation \\textit{a priori} knowledge of polarization\nproperties is crucial. To find it, we add contributions from\nall possible paths through intermediate hyperfine states. Since the\ndetunings from intermediate $5\\mathrm{P}_{3\/2}$ state are much larger\nthan respective hyperfine splittings we ignored the latter. Same approximation\nis adopted for the highest excited state $5\\mathrm{D}_{5\/2}$.\nEven though respective detuning $\\delta$ it is of the order of several\nMHz, similar to the hyperfine splitting of the highest excited state\n$5\\mathrm{D}_{5\/2}$, the hyperfine structure is completely unresolved\ndue to significant pressure broadening \\cite{Zameroski2014} in 0.5~torr krypton as buffer gas we use. Consequently, we may omit any detuning\ndependance and calculate the unnormalized polarization vector $\\boldsymbol{\\epsilon}$\nof the signal light using path-averaging formalism we developed in\n\\cite{Parniak2015}, by decalling the definitions of Rabi frequencies\nin Eq.~\\hyperref[eq:P2ndrho]{(\\ref*{eq:P2ndrho})}:\n\n\\begin{equation}\n\\boldsymbol{\\epsilon}_\\mathrm{4Ph}\\propto\\sum_{F,\\:m_{F}}\\mathbf{d}_{e_{3}e_{2}}(\\mathbf{E}_{a}\\cdot\\mathbf{d}_{ge_{1}}\\mathbf{E}_{b}\\cdot\\mathbf{d}_{e_{1}e_{2}}\\mathbf{E}_{c}^{*}\\cdot\\mathbf{d}_{e_{3}h}^{*}),\n\\label{eq:polaryzacje}\n\\end{equation}\n\n\nwhere $\\mathbf{E}_{i}$ are electric fields of respective beams $i$.\nSummation is carried over all possible magnetic sublevels ($F_{i}$,\n$m_{F_{i}}$) of all intermediate states $|e_{1}\\rangle$, $|e_{2}\\rangle$\nand $|e_{3}\\rangle$. \n\\section{Experimental Methods}\\label{sec:experimental}\nThe experimental setup is built around the rubidium-87 vapor cell\nheated to 100$^{\\circ}$C, corresponding to atom number density $n=7.5\\times10^{12}\\ \\mathrm{cm^{-3}}$ and optical density of 1600 at D2 line for optically pumped ensemble. For the two lowest, long-lived states,\nwe use two hyperfine components of rubidium ground-state manifold,\nnamely $|5\\mathrm{S}_{1\/2},F=1\\rangle=|g\\rangle$ and $|5\\mathrm{S}_{1\/2},F=2\\rangle=|h\\rangle$.\nFor the $|e_{1}\\rangle$ and $|e_{3}\\rangle$ states we take hyperfine\ncomponents of the $5\\mathrm{P}_{3\/2}$ manifold, and for the highest\nexcited state $|e_{2}\\rangle$ we take the $5\\mathrm{D}_{5\/2}$ manifold.\nAtoms are initially pumped into $|g\\rangle$ state using 400 $\\mu$s\noptical pumping pulse (at 795 nm). Next, three square-shaped driving pulses of 4 $\\mu$s duration each\nare applied simultaneously, as shown in Fig.~\\hyperref[fig:schemat]{\\ref*{fig:schemat}(b)}. Inside\nthe cell, beams are collimated and have diameters of 400~$\\mu$m,\nbeing well-overlapped over a 2-cm-long cylindrical region. They intersect\nat a small angle of 14 mrad, with 780-nm beams nearly counter-propagating\nwith respect to the 776-nm beams, as presented in Fig.~\\hyperref[fig:schemat]{\\ref*{fig:schemat}(c)}.\nPowers of driving beams $a$, $b$ and $c$ are 10 mW, 45 mW and 8\nmW, respectively. \n\nThe 780-nm two-photon Raman signal 2Ph co-propagates with the driving\nfield $a$. It is separated using a Wollaston polarizer (Wol) and\ndetected on a fast photodiode (PD), with $10^4$ signal to driving field leakage ratio. The four-photon signal 4Ph, is\nemitted in the direction given by the phase matching condition $\\mathbf{k}_{a}+\\mathbf{k}_{b}=\\mathbf{k}_{\\mathrm{{4Ph}}}+\\mathbf{k}_{c}+\\mathbf{K}$.  The wavevector $\\mathbf{K}$ of spin-wave generated in the 2Ph process Raman scattering equals $\\mathbf{K}=\\mathbf{k}_a-\\mathbf{k}_{\\mathrm{2Ph}}$. We note that both longitidual and transverse components of the spin-wave $\\mathbf{K}$ are much smaller than that of light wavevectors, and consequently it has only an insignificant effect on the 4Ph signal emission direction. This particular, nearly co-propagating geometry allows us to couple the same spin-wave of wavevector $\\mathbf{K}$ to both the 2Ph and 4Ph processes.\nIn addition to the desired 4Ph signal, 776-nm light coming from the\nclosed-loop process in which field $c$ couples level $|e_{3}\\rangle$\ndirectly to $|g\\rangle$ is emitted in the same direction \\cite{Parniak2015}\nand at a frequency differing by only 6.8~GHz. However,\nwhen driving fields $a$, $b$ and $c$ are $x$-, $y$- and $y$-\npolarized, respectively, the 4Ph signal light is $x$-polarized, while\nthe closed-loop 4WM light is $y$-polarized, where $x\\perp y$. This arrangement enables\nfiltering out the 4Ph signal from both stray 776-nm driving light\nand the closed-loop 4WM light with a polarizing beam-splitter (PBS, $10^2$ extinction).\nTo suppress residual drive laser background at 780 nm 4Ph signal goes through\nan interference filter (transmission of 80$\\%$ for 776 nm and $10^{-3}$ for 780 nm) and is detected by an avalanche photodiode\n(APD). We were able to obtain signal to background ratio of up to $10^2$. \n\nBy rotating the half-wave\nplate ($\\lambda\/2$) we can easily switch between observing 4Ph\nand closed-loop signal. For detunings optimal for the 4Ph process, we register less than 10 nW of the closed-loop 4WM light.  The two signals display different temporal characteristics,\nalso only 4Ph is correlated with the 2Ph light. \n\nWe verify the frequencies of 4Ph and closed-loop light using\na scanning Fabry-P\\'erot confocal interferometer of 1 GHz free spactral range (FSR) inserted in the 4Ph beam path. Fig.~\\hyperref[fig:schemat]{\\ref*{fig:schemat}(d)} show the trace obtained by scanning the interferometer for detuning difference $\\Delta_g+\\Delta_h=2\\pi\\times3.43$ GHz. Leakage of driving field $b$, induced by slight misalignement, is used as frequency reference. The middle peak corresponds to the 4Ph signal, with a solid line indicating expected frequency. Dashed line corresponds the frequency one would expect from the closed-loop signal. \n\n\n\\section{Results}\n\\subsection{Statistics and Correlations}\\label{sec:corr}\n\\begin{figure}[b]\n\\begin{centering}\n\\includegraphics[scale=1.2]{fig2}\n\\par\\end{centering}\n\n\\protect\\caption{(a - 2Ph, b - 4Ph) Averaged signal intensities (solid lines) with several\nsingle realizations (dashed lines), denoted by same colors in (a)\nand (b), demonstrating visible strong correlations and (c) calculated\ncross-correlation $C(t)$ between signals of four-photon (4Ph) and\nthe two-photon processes (2Ph). }\n\\label{fig:corr}\n\\end{figure}\n\n\n\\begin{figure}[b]\n\\begin{centering}\n\\includegraphics[scale=0.7]{fig3}\n\\par\\end{centering}\n\n\\protect\\caption{Joint statistics $P(\\mathcal{E}_\\mathrm{2Ph},\\mathcal{E}_\\mathrm{4Ph})$ together with marginal distributions of registered pulse energies for (a) short driving time of 200 ns, yield nearly single-mode thermal statistics (note the logarythimc scale in this plot), and (b) longer driving time of 4 $\\mu$s with observable pulse energy stabilization well described by multimode thermal statistics with mode number $\\mathcal{M}$ of 4.1 for 2Ph pulses and 1.6 for 4Ph pulses. Solid curves correspond to fitted thermal distributions.}\n\\label{fig:corrmap}\n\\end{figure}\n\nIn our experiment we remain in the macroscopic scattered light intensity\nregime. When strong Raman driving field is present, atoms are transferred\nto $|h\\rangle$ simultaneously with scattering of\n2Ph photons and buildup of large atomic coherence $\\rho_{gh}$.\nThe temporal shape of the 2Ph pulse is an exponential only at the\nvery beginning, which we observe in Fig.~\\hyperref[fig:corr]{\\ref*{fig:corr}(a)}. Not only pulse energies, but also the shapes fluctuate significantly from shot to shot, as the process is\nseeded by vacuum fluctuations \\cite{Raymer1989}. However, the\n4Ph pulse, presented in Fig.~\\hyperref[fig:corr]{\\ref*{fig:corr}(b)}, nearly exactly follows\nthe 2Ph pulse. We calculate temporal correlations between the 2Ph\nsignal, which is known to be proportional to the ground-state atomic\ncoherence $\\rho_{gh}$, and the 4Ph signal. Normalized intensity\ncorrelation at time $t$ between the 2Ph signal $I_{2\\mathrm{Ph}}(t)$\nand the 4Ph signal $I_{4\\mathrm{{Ph}}}(t)$ is calculated according\nto the formula $C(t)=\\langle\\Delta I_{\\mathrm{{2Ph}}}(t)\\Delta I_{\\mathrm{{4Ph}}}(t)\\rangle\/\\sqrt{\\langle\\Delta I_{\\mathrm{{2Ph}}}^{2}(t)\\rangle\\langle\\Delta I_{\\mathrm{{4Ph}}}^{2}(t)\\rangle}$\nby averaging over 500 realizations, where the standard deviations\n$\\langle\\Delta I_{\\mathrm{{2Ph}}}^{2}(t)\\rangle$ and $\\langle\\Delta I_{\\mathrm{{4Ph}}}^{2}(t)\\rangle$\nare corrected for electronic noise. Figure \\hyperref[fig:corr]{\\ref*{fig:corr}(c)} presents\nthe cross-correlation $C(t)$. We observe that correlations are high\nduring the entire process, which proves that at any time both processes\ninteract with the same atomic coherence $\\rho_{gh}$, similarly as in some previous works in $\\Lambda$-level configurations \\cite{Chen2010,Yang2012b}.\nIn particular, we are able to measure high correlation >0.9 at the very\nbeginning of the pulses, were light intensities are low. This regime\nis quite promising for further quantum applications. To estimate the\nuncertainty of calculated correlations, we divided data into 10 equal\nsets of 50 repetitions and calculated the correlation inside each\nset to finally obtain the uncertainty by calculating standard\ndeviation of results from all sets. \n\nNext, we study statistics and correlations of pulse energies in detail. We consider short and long pulse duration regimes. Figure \\hyperref[fig:corrmap]{\\ref*{fig:corrmap}(a)} corresponds to short driving time of 200 ns. In this regime light generated in both 2Ph and 4Ph processes is well-characterized by single-mode thermal energy distribution $P(\\mathcal{E})=\\langle\\mathcal{E}\\rangle^{-1} \\exp(-\\mathcal{E}\/\\langle\\mathcal{E}\\rangle)$ \\cite{Raymer1982}, where $\\mathcal{E}$ is the total scattered light energy in a single realization. This observation shows that we excite only a single transverse-spatial mode, as intended by using an adequately small size of Raman driving beam $a$ \\cite{Raymer1985a,Duncan1990}. Thermal distribution yields very high pulse energy fluctuations (namely, mean energy is equal to standard deviation), that are due to vacuum fluctuations of electromagnetic field and quantum fluctuations of atomic state \\cite{Walmsley1985,Walmsley1983}. Still, we observe that energies of 2Ph and 4Ph pulses are highly correlated, which is demonstrated by  joint statistics $P(\\mathcal{E}_\\mathrm{2Ph},\\mathcal{E}_\\mathrm{4Ph})$. The residual spread is mainly due to detection noise of both signals.\n\nIn the second regime [Fig. \\hyperref[fig:corrmap]{\\ref*{fig:corrmap}(b)}] the driving pulses of 4~$\\mu$s are longer than in the previous scheme. The relative fluctuations become smaller due to gain saturation. We found the marginal statistics to be well described by multimode thermal distributions (with number of modes $\\mathcal{M}$) given by \n\\begin{equation}\nP(\\mathcal{E})=\\frac{\\mathcal{M}}{\\langle\\mathcal{E}\\rangle\\Gamma(\\mathcal{M})}\\left(\\frac{\\mathcal{E}\\mathcal{M}}{\\langle\\mathcal{E}\\rangle}\\right)^{\\mathcal{M}-1}\\exp(-\\mathcal{E}\\mathcal{M}\/\\langle\\mathcal{E}\\rangle).\n\\end{equation}\nThe joint statistics $P(\\mathcal{E}_\\mathrm{2Ph},\\mathcal{E}_\\mathrm{4Ph})$ demonstrate clear correlations, which are here less influenced by detection noise than previously, as the pulse energies are higher.\n\n\n\\begin{figure}[b]\n\\begin{centering}\n\\includegraphics[scale=0.75]{fig4}\n\\par\\end{centering}\n\\protect\\caption{(a) Scheme of experiment to demonstrate storage of ground-state atomic coherence for $\\tau=450$ ns. (b) Measured two-point correlation $C(t_1,t_2)$ between 2Ph and 4Ph signals with average registered signal powers. Off-diagonal elements of the correlation map demonstrate that light fields interact with the same atomic coherence before and after the time delay. }\n\\label{fig:corrmaptemporal}\n\\end{figure}\n\nFinally, we check that correlations are indeed mediated by ground-state atomic coherence by interrupting the scattering process for a dark period of $\\tau=450$ ns. This is proved by strong correlations between the intensities of light scattered before and after the dark period, observed in both 2Ph and 4Ph processes. \n\nAfter the atoms are optically pumped as in the original scheme, we drive the processes with 150 ns pulses of the three driving fields $a$, $b$ and $c$. After a dark period of $\\tau=450$ ns, we drive the process for another 200 ns. The coherence $\\rho_{gh}$ generated in the first stage induces optical polarization at both the 4Ph field frequency (as in Eq. \\ref{eq:P2ndrho}) and 2Ph field frequency:\n\\begin{equation}\n\\mathbf{P}_\\mathrm{2Ph}=-n\\mathbf{d}_{e_h e_1}\\rho_{gh}\\frac{\\Omega_a}{\\Delta_g},\n\\label{eq:2ph}\n\\end{equation}\nresulting in stimulated Raman emission. The full two-point correlation map presented in Fig.~\\hyperref[fig:corrmaptemporal]{\\ref*{fig:corrmaptemporal}(b)} is calculated as $C(t_1,t_2)=\\langle\\Delta I_{\\mathrm{{4Ph}}}(t_1)\\Delta I_{\\mathrm{{2Ph}}}(t_2)\\rangle\/\\sqrt{\\langle\\Delta I_{\\mathrm{{4Ph}}}^{2}(t_1)\\rangle\\langle\\Delta I_{\\mathrm{{2Ph}}}^{2}(t_2)\\rangle}$. Apart from the diagonal correlated areas at $t_1\\approx t_2 \\approx 100$ ns and 800 ns, we observe the anti-diagonal terms corresponding to correlations between the two pulses. Due to spontaneous emission and collisional dephasing all the excited atomic states decay quickly, with their lifetime limited to 20 ns. This proves that we store information in the ground-state atomic coherence. Storage time $\\tau$ is limited mainly by atomic motion. In fact, the stored ground-state coherence and in turn correlations decay in multiple ways, e.g. by diffusive spatial spread of atomic coherence \\cite{Chrapkiewicz2014b,Parniak2014} and by influx of optically pumped atoms into the interaction region.\nFinally, we mention that very similar results are obtained if the system is driven by field $a$ only in the first stage of the sequence and in turn only the 2Ph field and atomic coherence are generated. In the second stage of the sequence, where all driving field are present, we observe that both the 2Ph and 4Ph signals are correlated with the 2Ph signal emitted in the first stage.\n\nAgreement of the above measured statistics with theoretical predictions and multiple previous experiments proves that the correlations do arise from vacuum fluctuations. With driving power stable to within less than 1\\% and laser frequency stable to within 1 MHz, these external contributions to fluctuations can be neglected. Large magnitude of fluctuations, with nearly perfect correlations between 2Ph and 4Ph signals together with demonstrated storage of correlated signals allow to reject phase-noise to amplitude-noise conversion as a source of correlations \\cite{Cruz2006}.\n\nAll of the above results were measured for $\\Delta_g\/2\\pi=1000$ MHz, $\\Delta_h\/2\\pi=1200$ MHz and $\\delta\/2\\pi=-50$ MHz.\n\n\n\\subsection{Detuning dependance}\\label{sec:spectral}\n\n\\begin{figure}\n\\includegraphics[scale=1.03]{fig5}\\protect\\caption{(a) Averaged (over 500 realizations) pulse shapes\nfor the intensities of the 2Ph and the 4Ph for a set of single-photon\ndetunings $\\Delta_{g}$ for constant $\\delta\/2\\pi=-50$\nMHz and $\\Delta_h\/2\\pi=1500$ MHz, (b) pulses full-width at half-maxima (FWHM), (c) their energies and (d) pulse energy correlation between the two processes as\na function of field $a$ single photon detuning $\\Delta_{g}$. Subsequent\nplots of 2Ph (4Ph) signals in (a) are shifted by 120 $\\mu$W (40 nW).}\n\\label{fig:shapes}\n\\end{figure}\n\n\nNow we switch to verifying properties of 4Ph signal for various drive\nfield detunings. The influence of field $a$ detuning $\\Delta_{g}$\nis seen in Fig. \\ref{fig:shapes}. A number of pronounced effects\ncomes about as this laser drives the Raman scattering and produces\nground-state atomic coherence $\\rho_{gh}$. Initially the 2Ph signal grows exponentially. The corresponding\nRaman gain coefficient is strongly dependent on drive field detuning\n$\\Delta_{g}$. \nThe final effect is shortening of 2Ph pulses closer to resonance, as shown in Fig.~\\hyperref[fig:shapes]{\\ref*{fig:shapes}(b)}. The 4Ph pulse follows\nthe ground-state coherence $\\rho_{gh}$ and 2Ph pulse as shown\nin the previous section, however its maximum is somewhat delayed.\nWe attribute this effect to internal atom dynamics at high drive intensity\nlevels which is not captured by Eq.~\\hyperref[eq:P2ndrho]{(\\ref*{eq:P2ndrho})}. Energies\nof pulses are also higher closer to resonance, although the two-photon\nRaman pulse energy saturates due to absorption losses [see Fig.~\\hyperref[fig:shapes]{\\ref*{fig:shapes}(c)}].\n\nImportant insight is provided by calculating energy correlation between\nthe 2Ph and 4Ph light pulses, which fluctuate significantly from shot\nto shot. Correlation is calculated as $C=\\langle\\Delta \\mathcal{E}_{\\mathrm{{2Ph}}}\\Delta \\mathcal{E}_{\\mathrm{{4Ph}}}\\rangle\/\\sqrt{\\langle\\Delta \\mathcal{E}_{\\mathrm{{2Ph}}}^{2}\\rangle\\langle\\Delta \\mathcal{E}_{\\mathrm{{4Ph}}}^{2}\\rangle}$,\nwhere $\\mathcal{E}_{\\mathrm{{2Ph}}}$ and $\\mathcal{E}_{\\mathrm{{4Ph}}}$ are total energies\nof 2Ph and 4Ph light pulses, respectively, in a single realization.\nThe averaging $\\langle.\\rangle$ is done over 500 realizations of\nthe process and results are plotted in Fig.~\\hyperref[fig:shapes]{\\ref*{fig:shapes}(d)}.\nStrong correlations at various detunings reinforce the observation that indeed we are able\nto couple 4Ph optical field to the ground-state coherence, since number\nof photons in the 2Ph pulse is proportional to generated atomic coherence $|\\rho_{gh}|^2$.\nWe attribute the drop in correlations close to resonance line to absorption\nlosses. Finally, we estimate the efficiency of conversion from the ground-state atomic coherence $\\eta=\\mathcal{E}_\\mathrm{4Ph}\/\\mathcal{E}_\\mathrm{2Ph}$ to the 4Ph field to be $5\\times10^{-4}$. By comparing Eq.~(\\ref{eq:P2ndrho}) with analogous expression for the 2Ph process given by Eq. \\ref{eq:2ph}, we obtain $\\eta\\approx10^{-4}$ as well. This figure of merit could be improved by choosing different experimental geometries, or laser-cooling of the atomic ensemble.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[scale=1.15]{fig6} \n\\par\\end{centering}\n\n\\protect\\caption{Experimental average pulse energies of 2Ph and 4Ph signals as a function\nof field $a$ detuning $\\Delta_{g}$ (measured from $F=1\\rightarrow F'=1$\nresonance) and field $b$ detuning $\\delta-\\Delta_{g}$ around the\ntwo-photon resonance. As we change the field $a$ detuning both single\nphoton detuning $\\Delta_{g}$ and the two-photon detuning $\\delta$\nchange accordingly. Dots (squares) represent maxima (minima) of the 4Ph (2Ph)\nsignal, while $\\delta=0$ line indicates the two-photon absorption resonance.}\n\\label{fig:detuning}\n\\end{figure}\n\n\nTo capture the physics in the vicinity of the two-photon resonance\nwe study 4Ph and 2Ph pulse energies as a function of detunings of\nfields $a$ and $b$. First we scan the field $b$ detuning $\\delta-\\Delta_{g}$\nacross the two-photon resonance line (see Fig. \\ref{fig:detuning}).\nWe observed strong suppression of 2Ph signal (minima are marked by squares in Fig. \\ref{fig:detuning}) due to two-photon absorption\n(TPA) in ladder configuration \\cite{Moon2013}. As a consequence,\nless atomic coherence $\\rho_{gh}$ is generated and the 4Ph signal\nis also reduced. In turn, the 4Ph shifted maxima position (marked by dots) are due to a trade-off\nbetween TPA losses and 4WM efficiency, as the latter is highest at\nthe two-photon resonance (marked by $\\delta=0$ line) according to Eq.~\\hyperref[eq:P2ndrho]{(\\ref*{eq:P2ndrho})}. The peak appears only on one side of the resonance due to the phase-matching condition being influenced by atomic dispersion \\cite{Zibrov2002}. \nAdditionally, we observe expected broadening of 60 MHz of two-photon resonance due to buffer gas collisions.\nBy changing the field $a$ detuning $\\Delta_{g}$ we see expected\nshifting of the two-photon resonance.  We checked that even at sub-optimal two-photon detuning $\\delta$ the 2Ph and 4Ph signals are correlated, but the correlations become harder to measure as the 4Ph signals becomes very weak. \n\n\\begin{figure}\n\\centering{}\\includegraphics[scale=1.15]{fig7}\\protect\\caption{Four-photon signal pulse energy as a function of the detuning $\\Delta_{h}$\n(measured from $F=2\\rightarrow F'=2$ resonance) of field $c$ for\noptimal conditions of other lasers ($\\Delta_{g}\/2\\pi=900$~MHz and\n$\\delta\/2\\pi=-50$ MHz). Absorption line corresponds to the $F=2$\nhyperfine component of the ground state, where the right side of the\nplot is the red-detuned side. Drop at around $\\Delta_{h}\/2\\pi=$-900\nMHz corresponds exactly to 6.8 GHz detuning between the\ntwo 780-nm lasers, or $\\Delta_{g}+\\Delta_{h}=0$.}\n\\label{fig:deltah}\n\\end{figure}\n\n\nContrary to the above, changing the detuning $\\Delta_{h}$ of the\n780-nm driving field $c$ has only mild effect on the 4Ph signal.\nFig.~\\ref{fig:deltah} presents 4Ph signal pulse energy as a function\nof $\\Delta_{h}$ while other lasers were tuned for maximal signal\n($\\Delta_{g}\/2\\pi=900$ MHz and $\\delta\/2\\pi=-50$ MHz). Since the\n4Ph field frequency adapts to match the energy conservation for the\n$|h\\rangle\\rightarrow|e_{3}\\rangle\\rightarrow|e_{2}\\rangle$ two-photon\ntransition, the frequency of the driving field $c$ is not critical.\nThe laser must only be off-resonant, so the driving field is not absorbed\nand does not disturb the ground-state coherence. We also observe\t\na marked narrow drop in 4Ph process efficiency when the detuning between\nfields $a$ and $c$ is exactly the ground-state hyperfine splitting, or equivalently $\\Delta_{g}+\\Delta_{h}=0$,\nwhich is due to $\\Lambda$ configuration two-photon resonance yielding strong interaction with the ground-state coherence. \n\n\\section{Conclusions}\\label{sec:concl}\nThe experiment we performed is a proof-of-principle\nof a light atom interface that enables coupling of long-lived\nground-state atomic coherence and light resonant with transition between\nexcited states. The non-linear process we discussed is a novel type\nof process with typical characteristics of both Raman scattering and\n4WM. \nThe observation of inverted-Y type nonlinear four-photon process involving ground-state coherence, performed in a very different regime in cold atoms, has been so far reported only in \\cite{Ding2012}.\nHere, we generated ground-state atomic coherence via the well known\ntwo-photon process. We demonstrated ability to couple the\nvery same atomic coherence to optical field resonant with transition\nbetween two excited states via a four-photon process. This was verified\nby measuring high correlations between 2Ph and 4Ph fields, as well as frequency and\npolarization characteristics of the four-photon process. \n\nWe studied the behavior of pulse shapes as a function\nof driving laser detunings. Among many results, we found that maximum\nsignal is achieved when lasers are detuned from the two-photon resonance\nby approximately $\\delta\/2\\pi=-50$ MHz, which is a trade-off\nbetween TPA spoiling the generation of atomic coherence, and the efficiency\nof the four-photon process. This results demonstrate that we are able\nto control the 4WM process with ground-state coherence, which constitutes\na novel type of Raman scattering driven by three non-degenerate fields,\nin analogy to hyper-Raman scattering where the scattering process is driven\nby two degenerate fields.\n\nHere we used light at 776 nm coupled to $5\\mathrm{P}_{3\/2}\\rightarrow5\\mathrm{D}_{5\/2}$\ntransition. Using different states, such as $4\\mathrm{D}_{3\/2}$ as\nthe highest excited state, and $5\\mathrm{P}_{1\/2}$ and $5\\mathrm{P}_{3\/2}$\nas two intermediate states, would enable coupling telecom light (at 1475.7 nm or 1529.3 nm). Such process could be used as a building\nblock for a telecom quantum repeater or memory \\cite{Michelberger2015,Chaneliere2006,Zhang2016,Radnaev2010a}. By applying\nexternal weak quantum field as the 4Ph field, the system may serve\nas an atomic quantum memory, based on a highly non-linear process\nas in \\cite{DeOliveira2015}, but still linear in the input field\namplitude. It may also solve a variety of filtering problems \\cite{Michelberger2015,Dabrowski2014},\nsince many similar configurations exists (e.g. with $5\\mathrm{P}_{3\/2}$,\n$5\\mathrm{D}_{3\/2}$ and $5\\mathrm{P}_{1\/2}$ as intermediate states) in which all driving lasers operate at different wavelengths\nthan the signal. Even thought the 2Ph field was measured to be much stronger than the 4Ph field, we note that using a cold atomic ensemble would\noffer selectivity in intermediate states of the process and small\ndetuning. Thanks to selection rules, exclusively the 4Ph process could be driven. This would be the requirement for generating pairs of photons and collective atomic excitations in the 4Ph process only.  Additionally, 4WM character of the\nprocess enables engineering of phase-matching condition, namely changing\nangles between incident driving beams to address different spin-wave excitations \\cite{Chrapkiewicz2012},\nto explore spatially-multimode capabilities of the system. In future\nstudies of the process we propose to address patterns \nunachievable\nby typical Raman light-atom interface based on $\\Lambda$ level configuration.\n\n\n\\begin{acknowledgments}\nWe acknowledge R. Chrapkiewicz, M. D\\k{a}browski and J. Nunn for insightful discussions and  K. Banaszek and T. Stacewicz for their generous support.\nThis work was supported by Polish Ministry of Science and Higher Education ``Diamentowy Grant''\nProject No. DI2013 011943 and National Science Center Grant\nNo. 2011\/03\/D\/ST2\/01941.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{\\fontsize{10}{15}\\selectfont A. Details of STS spectra and DFT calculations}\n\n\\textbf{We provide additional information about the STS spectra, Mott states of the theoretical models and interlayer coupling effect.}\n\n\\renewcommand{\\figurename}{SFIG.}\n\n\\begin{figure} [h!]\n\\centering{ \\includegraphics[width=14.5cm]{Sfig1} }\n\\caption{ \\label{fig1}\nWell filtered STM image of the NC phase.\nDashed lines separate two different domains and domain wall.\nThe solid lines represent the same area as Fig. 2(c).\n}\n\\end{figure}\n\n\\begin{figure} [h!]\n\\centering{ \\includegraphics[width=16cm]{Sfig2} }\n\\caption{ \\label{fig1}\nSTS spectra of the NC phase and theoretical DOS of DW-1 structure. (a) STS spectra, (b) Total DOS as a function of the electronic temperature parameter $\\sigma$ (eV). (c) Local DOS of domain and domain wall regions ($\\sigma$ = 0.15 and 0.2 eV). \n}\n\\end{figure}\n\n\\begin{figure} [ht]\n\\centering{ \\includegraphics[width=16cm]{Sfig3} }\n\\caption{ \\label{fig2}\nLocal density of states at center Ta atom of David stars (a) DW-1 and (b) Hexagonal domain model.\n}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure} [h!]\n\\centering{ \\includegraphics[width=16cm]{Sfig4} }\n\\caption{ \\label{fig3}\nOne possible tri-layer stacking order in the C phase. (a) atomic structure, (b) and (c) are the band structure and total DOS without and with electron-electron correlations, respectively. The interlayer distance of 5.9 {\\AA} is fixed.\n}\n\\end{figure}\n\n\\begin{figure} [h!]\n\\centering{ \\includegraphics[width=16cm]{Sfig5} }\n\\caption{ \\label{fig4}\nTwo different bilayer stacking order in the DW-1 structure. (a) on-top stacking and (b) shifted stacking. Top and bottom panels are atomic structure and LDOS, respectively. All atoms are fixed at their atomic position in the single-layer DW-1 structure. The interlayer distance of 5.9 {\\AA} is fixed.\n}\n\\end{figure}\n\n\\clearpage\n\n\\section{\\fontsize{10}{15}\\selectfont B. Details of Honeycomb Network Model}\nIn this supplementary material, we present the details of the honeycomb network model. Here we apply the strategy of Ref. \\cite{Efimkin} to the honeycomb lattice.\n\n\\subsection{Construction of Network Model}\n\nWe theoretically model the system by a regular array of one-dimensional metals living on the links of a honeycomb lattice. The construction of this model is motivated from the following experimental observations present in the main text. \n\\begin{enumerate}\n\\item \\textbf{Emergent honeycomb lattice}: The domains of the NC-CDW state form a regular honeycomb lattice, and the domain walls are the links of this honeycomb lattice.\n\\item \\textbf{Metallic domain walls}: The domain walls trap finite local density of states near the Fermi level (Note that this is generically expected for any domain wall of a charge-density wave since the domain wall carries in-gap states whose origin are topological \\cite{su79}).   \n\\end{enumerate} \n\nMotivated from these, we consider a regular array of one-dimensional metals living on the links of a honeycomb lattice. Similar network models of one-dimensional metals have been studied to some degree in the context of quantum Hall plateau transitions, known as ``Chalker-Coddington model\" \\cite{chal88}, and recently have been revived to explain the physics of the twisted bilayer graphene at a small twisting angle \\cite{Efimkin}. We apply this latest theoretical progress to model the network on the honeycomb lattice. \n\nSome details of the model are following:  \n\\begin{enumerate}\n\\item \\textbf{Degrees of freedom}: On each link $a=x,y,z$ of the honeycomb lattice, we assign the two wavefunctions $\\psi_a$ and $\\psi_{\\bar{a}}$. Here $\\psi_a$ represent the chiral mode propagating from an A-sublattice to its neighboring B-sublattice and $\\psi_{\\bar{a}}$ for the mode propagating from a B-sublattice to its neighboring A-sublattice (See Fig. \\ref{fig1}).\nMicroscopically they correspond to the low-energy modes near the Fermi momentum of one-dimensional metals propagating along the links. (Here we suppress the spin index for the modes because we are mainly interested in the spectral properties of the network model.)\n\\item \\textbf{Scattering between wavefunctions}: We assume that the modes propagate coherently within each link and scatter only at the nodes of the honeycomb lattice. We further assume that there are three-fold rotation and two-fold mirror symmetries at each nodes, and the scattering between the modes respects the crystal symmetries. \n\\end{enumerate}\n\n\\begin{figure}[htbp]\n\\centering\\includegraphics[width=0.5\\textwidth]{Sfig6.pdf}\n\\caption{Pictorial Representation of Network Model: (A) labeling of the links by $x,y,z$ on real space. Each link has the two degrees of freedom. One is from the neighboring A-sublattice [See (B)] and another is from the neighboring B-sublattice [See (C)]. They are written as $\\psi_a$ and $\\psi_{\\bar{a}}$, and the arrows represent the propagating directions of the modes. Here $\\hat{e}_a, a= x,y,z$ is the vector connecting the A-sublattice to the B-sublattice.}\\label{fig1}\n\\end{figure}\n\nHaving these in mind, we have the following scattering problem at the A-sublattice.   \n\\begin{align}\n\\left[\n\\begin{array}{c}\n\\psi_x (\\bm{R}) \\\\\n\\psi_y (\\bm{R}) \\\\ \n\\psi_z (\\bm{R}) \n\\end{array} \\right] = e^{-i\\frac{E}{v_F \\hbar} L} \\cdot \\hat{T}_A \\cdot \\left[\n\\begin{array}{c}\n\\psi_{\\bar{x}} (\\bm{R} +\\hat{e}_x) \\\\\n\\psi_{\\bar{y}}  (\\bm{R} + \\hat{e}_y) \\\\ \n\\psi_{\\bar{z}}  (\\bm{R} +\\hat{e}_z) \n\\end{array} \\right]\n\\end{align}\nHere, the left-hand side $\\psi_a (\\bm{R}), a= x,y,z$ represents the out-going modes from the A-sublattice, which is related by a scattering matrix $\\hat{T}_A$ to the in-coming modes $\\psi_{\\bar{a}} (\\bm{R}), a = x,y,z$ appearing on the right-hand side (See Fig. \\ref{fig1}). The additional phase factor $\\sim \\exp (-i\\frac{E}{v_F \\hbar} L)$ is the phase accumulated by the incoming modes while it propagates coherently from the neighboring B-sublattices to the A-sublattice at $\\bm{R}$. Here $v_F$ is the Fermi velocity within the one-dimensional metal, which is expected to be similar to that of the bulk electron, and $L$ is the length of the link. The scattering matrix $\\hat{T}_A$ can be fixed by the three-fold rotation as well as the two-fold mirrors at the node. With the unitarity of the scattering matrix, we find \n\\begin{align}\n\\hat{T}_A = e^{i\\chi_A} \\left[\n\\begin{array}{ccc}\nT_A & t_A & t_A \\\\\nt_A & T_A & t_A \\\\ \nt_A & t_A & T_A  \n\\end{array} \\right], ~~ |T_A| \\in \\Big[\\frac{1}{3}, ~1\\Big], ~ t_A   = e^{i\\phi_A} \\sqrt{\\frac{1-|T_A|^2}{2}},  \n\\end{align}\nwith $\\phi_A = \\cos^{-1}(\\frac{|t_A|}{2|T_A|})$. Similarly we have the following scattering problem at the B-sublattice.\n\\begin{align}\n\\left[\n\\begin{array}{c}\n\\psi_{\\bar{x}} (\\bm{R}) \\\\\n\\psi_{\\bar{y}} (\\bm{R}) \\\\ \n\\psi_{\\bar{z}} (\\bm{R}) \n\\end{array} \\right] = e^{-i\\frac{E}{v_F \\hbar} L} \\cdot \\hat{T}_B \\cdot \\left[\n\\begin{array}{c}\n\\psi_{x} (\\bm{R} -\\hat{e}_x) \\\\\n\\psi_{y}  (\\bm{R} - \\hat{e}_y) \\\\ \n\\psi_{z}  (\\bm{R} -\\hat{e}_z) \n\\end{array} \\right],\n\\end{align}\nwhere $\\hat{T}_B$ has the same structure as the $\\hat{T}_A$. \n\nNow we can perform the Fourier transformation on $\\bm{R}$ and solve these scattering problems. \n\\begin{align}\n\\bm{\\Psi}_{\\bm{q}} = e^{-i\\frac{E_{\\bm{q}}}{v_F \\hbar} L} \\hat{T}_{\\bm{q}} \\cdot \\bm{\\Psi}_{\\bm{q}}, ~~ \\bm{\\Psi}_{\\bm{q}} = \\left[\n\\begin{array}{c}\n\\psi_{x} (\\bm{q}) \\\\\n\\psi_{y} (\\bm{q}) \\\\ \n\\psi_{z} (\\bm{q}) \\\\\n\\psi_{\\bar{x}} (\\bm{q}) \\\\\n\\psi_{\\bar{y}} (\\bm{q}) \\\\ \n\\psi_{\\bar{z}} (\\bm{q})  \n\\end{array} \\right], ~~ \n\\hat{T}_{\\bm{q}} = \\left[\n\\begin{array}{cc}\n0 & \\hat{T}_A \\cdot \\hat{V}_{\\bm{q}} \\\\ \n\\hat{T}_B \\cdot \\hat{V}_{\\bm{q}}^* & 0  \n\\end{array} \\right],\n\\label{Energy}\n\\end{align}\nwhere $\\hat{V}_{\\bm{q}} = $ diag $[\\exp (i\\bm{q}\\cdot \\hat{e}_x), ~\\exp (i\\bm{q}\\cdot \\hat{e}_y), ~\\exp (i\\bm{q}\\cdot \\hat{e}_z)]$. Hence, the energy spectrum can be obtained by diagonalizing $\\hat{T}_{\\bm{q}}$, which is again an unitary matrix. In terms of the eigenvalues $e^{i \\epsilon_{j} (\\bm{q})}, j= 1, 2, \\cdots 6$ of $\\hat{T}_{\\bm{q}}$, we have the energy spectrum: \n\\begin{align}\nE_{j, \\bm{q}}^n = 2\\pi \\frac{v_F \\hbar}{L} n + \\frac{v_F \\hbar}{L} \\epsilon_{j} (\\bm{q}), ~~j = 1,2, \\cdots 6.\n\\end{align}\nHere $n \\in \\mathbb{Z}$ and thus the minibands are repeating in the energy in period of $2\\pi \\frac{v_F \\hbar}{L}$. Mathematically this repetition in $n$ originates from the ambiguity of $\\epsilon_j (\\bm{q})$ by $2\\pi$ appearing in the eigenvalues $e^{i \\epsilon_{j} (\\bm{q})}, j= 1, 2, \\cdots 6$. Physically this repetition can be traced back to the excitation energy of the microscopic one-dimensional modes with the same momentum $\\bm{q}$, i.e., for a given $\\bm{q}$, there are multiple different one-dimensional modes with energy $2\\pi \\frac{v_F \\hbar}{L} n, ~ n\\in \\mathbb{Z}$. Thus we expect that the energy spectrum given by $\\frac{v_F \\hbar}{L}\\epsilon_j (\\bm{q})$ will repeat in energy with a period $2\\pi \\frac{v_F \\hbar}{L}$ and entirely fills up the bulk CDW gap. Below we will analyze only one period of the band spectrum. \n\n\\subsection{Band Spectrum}\nWe first consider the case where we have a full symmetry of the honeycomb lattice, i.e., $\\hat{T}_A = \\hat{T}_B$. As apparent from the Fig \\ref{fig2}, the spectrum features (i) Dirac cones at the $K$ and $K'$ points, (ii) flat bands, and (iii) quadratic band touchings at the $\\Gamma$ point. Now we discuss the stabilities of these features. \n\\begin{enumerate}\n\\item \\textbf{Dirac Cones at the $K$ and $K'$ points}: The Dirac cones are protected by the sublattice symmetry as in the graphene. It is easily removed by breaking the symmetry, i.e., $T_A \\neq T_B$. See the spectrum in Fig \\ref{fig2}.\n\\item \\textbf{Quadratic Band Touching at the $\\Gamma$ point}: The quadratic band touchings can be protected by the six-fold rotation symmetry \\cite{sun09}. However, even when the symmetry is broken (while keeping the three-fold rotation and mirror symmetries are kept), the band touchings are robust within our network model. See the Fig \\ref{fig2}.  \n\\item \\textbf{``Flat-ness\" of Flat bands}: The flat-ness of the bands cannot be protected. However, within our network model (with the three-fold rotation $C_3$ and mirror symmetries), we find that it is robust. See the Fig \\ref{fig2}. \n\\end{enumerate}\n\n\n\\begin{figure}[htbp]\n\\centering\\includegraphics[width=0.5\\textwidth]{Sfig7.pdf}\n\\caption{Energy Spectrum of Honeycomb Network Model. Here we plot the spectrum of $\\epsilon_j (\\bm{q})$ of the scattering matrix $\\hat{T}_{\\bm{q}}$ along $M \\to K \\to \\Gamma \\to M$. (A) $C_6$-symmetric spectrum $\\hat{T}_A = \\hat{T}_B$. It is straightforward to note the Dirac fermion at the $K$ point, quadratic band touching at $\\Gamma$ point, and also the flat bands. (B) $C_3$-symmetric spectrum $\\hat{T}_A \\neq \\hat{T}_B$. Here the Dirac band touching is removed.}\\label{fig2}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\\subsection{Symmetry Analysis of Quadratic Band Touching}  \nHere we perform the symmetry analysis of the quadratic band touchings at the $\\Gamma$ point. By explicitly diagonalizing Eq.\\eqref{Energy} at the $\\Gamma$ point (with $\\hat{T}_A = \\hat{T}_B$), we obtain the wavefunctions of the degenerate states. They are labeled as $|\\Psi_a \\rangle, a =1,2$ (with $\\langle \\Psi_a | \\Psi_b \\rangle = \\delta_{ab}$), from which we reconstruct the representations of the three-fold rotation $C_3$, the x-mirror $R_x: x \\to -x$, and the six-fold rotation $C_6$, i.e., $\\{[C_3]_p, [R_x]_p, [C_6]_p\\}$ within these bands. For example, we obtain the three-fold rotation $[C_3]_p$ within the two states by computing\n\\begin{align}\\label{Ex}\n[C_3]_p = \\left[\n\\begin{array}{cc}\n\\langle \\Psi_1 | \\hat{C}_3 |\\Psi_1 \\rangle & \\langle \\Psi_2 | \\hat{C}_3 |\\Psi_1 \\rangle \\\\ \n\\langle \\Psi_1 | \\hat{C}_3 |\\Psi_2 \\rangle & \\langle \\Psi_2 | \\hat{C}_3 |\\Psi_2 \\rangle\n\\end{array} \\right],\n\\end{align} \nin which $\\hat{C}_3$ is the representation of the three-fold rotation in the six-component $\\Psi_{\\bm{q}}$ in Eq.\\eqref{Energy}. \n\n\n\n\nBy computing these explicitly, we find\n\\begin{align}\n[C_3]_p = e^{\\frac{2\\pi i}{3} \\sigma^z}, ~ [R_x]_p = e^{-\\frac{\\pi i}{3} \\sigma^z} \\sigma^x, ~[C_6]_p = e^{-\\frac{2\\pi i}{3} \\sigma^z}, \n\\end{align}\nwhere $\\sigma^a, a= x,y,z$ is the Pauli matrix acting on the space spanned by $\\{|\\Psi_1 \\rangle, |\\Psi_2 \\rangle\\}$.\n\nWith these, we can write down symmetry-allowed Hamiltonian near the $\\Gamma$ point. First, by imposing $[C_3]_p$ and $[R_x]_p$, we find that no perturbation is allowed to split $|\\Psi_1 \\rangle$ and $|\\Psi_2 \\rangle$ at the $\\Gamma$ point.  \n\\begin{align}\n[C_3]_p^{\\dagger} H_0 [C_3]_p = H_0, ~~[R_x]_p^{\\dagger} H_0 [R_x]_p = H_0, ~ \\to H_0 \\propto \\mu \\sigma^0\n\\end{align}\nHence the degeneracy cannot be removed when $[C_3]_p$ and $[R_x]_p$ are imposed. On the other hand, near the $\\Gamma$ point, we find that the linear band touching is allowed. \n\\begin{align}\n[C_3]_p^{\\dagger} H (\\bm{k}) [C_3]_p = H (C_3^{-1}[\\bm{k}]), ~~[R_x]_p^{\\dagger} H(k_x, k_y) [R_x]_p = H(-k_x, k_y),\n\\end{align}\nallows $H(\\bm{k}) \\propto k_x \\hat{s}^x + k_y \\hat{s}^y$ (where $(\\hat{s}_x, \\hat{s}_y)$ are the Pauli matrices obtained by properly rotating $\\sigma^x$ and $\\sigma^y$.). Hence, the quadratic band touching cannot be protected by $[C_3]_p$ and $[R_x]_p$. Nevertheless, within our network model, the touching is found to be robust though the touching is not protected by the symmetries. \n\nWe can show that we need the six-fold rotation symmetry $[C_6]_p$ to protect the quadratic band touching and this is consistent with Ref. \\cite{sun09}. Thus, on imposing $[C_6]_p$, we can fix the Hamiltonian as \n\\begin{align}\nH = \\epsilon_0 (|\\bm{k}|) + \\Big( \\frac{k_x^2 - k_y^2}{2m} \\sigma^x + \\frac{2k_x k_y}{2m} \\sigma^y \\Big).\n\\end{align}\nTo match the band spectrum seen in the model, we have $\\epsilon_0 (|\\bm{k}|) = \\frac{\\bm{k}^2}{2m}$ and thus the lower band is completely flat and the density of state at the zero energy is divergent.  \n\n\\subsection{Comparison with Twisted Graphene Bilayer}\nHere we extend our discussion in the main text on the similarity between our network system and the theoretical models \\cite{Efimkin, bist11} for the twisted graphene bilayers. In particular, we compare ours with the network model in Ref. \\cite{Efimkin} and a continuum Dirac fermion model in Ref. \\cite{bist11}.\n\nTo start with, we find that our network system is close to the network model appeared in Ref \\cite{Efimkin}. In Ref. \\cite{Efimkin}, the twisted graphene bilayer at a small twisting angle has been considered. When the twisting angle is small, there is a periodic array of domain walls separating the locally AA-stacked regions and the locally AB-stacked regions. Ref. \\cite{Efimkin} argued that these domain walls trap localized one-dimensional metallic channels. These one-dimensional modes scatter at the nodes, which form a triangular lattice (in our case, the nodes form a honeycomb lattice). The structure of their model is quite similar to ours and indeed Ref. \\cite{Efimkin} obtained a similar spectrum as ours: Dirac fermions, nearly flat bands, as well as van-Hove singularities.   \n\nWe can also make a comparison of our network system with the continuum Dirac theory of magic-angle twisted bilayer graphene in Ref. \\cite{bist11}. In this approach, the Dirac fermions coming from the top and bottom layers interfere each other, and as a result, the bands become flat. Theoretically, this flat spectrum is speculated to be the source of surprising correlation-driven phenomena seen in the experiments \\cite{cao18, lian18, po18}. One may note that our network model appears to be different than the continuum Dirac theory of Ref. \\cite{bist11}. Despite of the difference in the theoretical treatments, we emphasize that our network model and the result of Ref. \\cite{bist11} share the strikingly-similar features in spectrum: Dirac fermions, flat bands and associated singularities in density of states, which are believed to play an essential role in the correlation physics.\n\nBoth the twisted bilayer graphene and our honeycomb network have weak disorders \\cite{brih12, raza16}. For example, there are some imperfect hexagons in our network and imperfect triangles in twisted bilayer graphene. Naively one expects that such weak disorders would immediately localize the flat bands and completely destroy associated many-body physics.  However, the previous study \\cite{chal10} surprisingly found that the flat bands do not get immediately localized but become critical. This implies that the flat bands are stronger against disorders than we naively expect. Though a more thorough investigation is desirable, we expect from the reference \\cite{chal10} that the flat bands retain relatively flat spectrum even with the weak disorders and hence is expected to remain very susceptible to many-body physics.\n\nIn summary, we have shown that the two systems, twisted graphene bilayer and our network system, share the surprising similarities including the flat bands and a large density of states, which are the key to the exotic correlation-driven phenomena.  \n\n\\subsection{Interlayer coupling}\n\nIn this subsection, we consider the effect of interlayer coupling to the electronic structures in the conducting network, and we will argue that, in general, the interlayer couplings between the layers will little affect the emergent electronic structures.\n\nFor the concrete-ness of our theoretical discussion, we first assume that the charge-density wave domains in the nearly commensurate phase remain insulating even after the inclusions of interlayer couplings [see SFig.5]. With this in hand, all the lowest-energy electronic states are in the domain walls in the conducting networks, and the interlayer couplings will introduce the coupling between these metallic modes inside the conducting networks living in different layers.\n\nAmong various possible couplings, the most important coupling, which can largely modify the band structure, is the electron hopping process between the layers. Note that this is proportional to the wavefunction overlap between the states of domain walls in different layers, and the states are highly localized within each domain walls. Hence, the effect of coupling will be strongly suppressed if not the networks are almost exactly overlapping to each other when seen from c-axis. From the available literature \\cite{cho16}, we note that the networks in different layers are not correlated to each other and thus we expect that the emergent band structure of the low-energy theory will not be affected much by the interlayer coupling.     \n \n\n\\newcommand{\\AP}[3]{Adv.\\ Phys.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\CMS}[3]{Comput.\\ Mater.\\ Sci. \\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\PR}[3]{Phys.\\ Rev.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\PRL}[3]{Phys.\\ Rev.\\ Lett.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\PRB}[3]{Phys.\\ Rev.\\ B\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\NA}[3]{Nature\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\NAP}[3]{Nat.\\ Phys.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\NAM}[3]{Nat.\\ Mater.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\NAC}[3]{Nat.\\ Commun.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\NAN}[3]{Nat.\\ Nanotechnol.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\NARM}[3]{Nat.\\ Rev.\\ Mater.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\NT}[3]{Nanotechnology {\\bf #1}, #2 (#3)}\n\\newcommand{\\JP}[3]{J.\\ Phys.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\JAP}[3]{J.\\ Appl.\\ Phys.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\JPSJ}[3]{J.\\ Phys.\\ Soc.\\ Jpn.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\JPC}[3]{J.\\ Phys.\\ C: \\ Solid State Phys.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\PNAS}[3]{Proc.\\ Natl.\\ Acad.\\ Sci. {\\bf #1}, #2 (#3)}\n\\newcommand{\\PRSL}[3]{Proc.\\ R.\\ Soc.\\ Lond. A {\\bf #1}, #2 (#3)}\n\\newcommand{\\PBC}[3]{Physica\\ B+C\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\PAC}[3]{Pure Appl.\\ Chem. \\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\SCI}[3]{Science\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\SCA}[3]{Sci.\\  Adv.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\RPP}[3]{Rep.\\ Prog.\\ Phys. \\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\SCR}[3]{Sci.\\  Ref.\\ {\\bf #1}, #2 (#3)}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\n The performance of direct-sequence code-division multiple-access (DS-CDMA)\ntransmission over mobile fading channels depends strongly on the\nreliability of channel parameter and quality of synchronization for each user: state-of-the-art detection algorithms that\nexploit multiple-access-interference and inter-symbol-interference\nrequire very powerful estimation algorithms.\n\n\nSubstantial amount of relevant references appeared in the\nliterature on  delay estimation. Namely, a new prospective is\npresented in \\cite{stuller97} for the maximum likelihood (ML) time-delay estimation. Code timing estimation in a near-far environment for DS-CDMA systems was introduced in \\cite{smith}.\n Joint symbol detection, time-delay and channel parameter estimation\nproblems for asynchronous DS-CDMA systems have been investigated in several previous works (e.g., \\cite{pelin,miller}). Most of these works either work on one signal at a time and treat the other signals as interference, or employ a training sequence to obtain a coarse\nestimate of the channel parameters which is consequently used to\ndetect data. It is clear that these approaches have disadvantages of\nhaving higher overhead and additional noise enhancement.\n\nSome other proposed approaches for joint blind multiuser detection and\nchannel estimation for DS-CDMA systems are  subspace-based  and\nlinear prediction-based methods. Subspace-based method usually\nrequire singular value decomposition or eigenvalue decomposition\nwhich is computationally costly and does not tolerate mismatched\nchannel parameters. Another drawback of this approach is that\naccurate rank determination may be difficult in a noisy\nenvironment \\cite{bensley,strom}. Moreover, it is not clear how these methods can be extended to include the estimation of the transmission delays jointly with the channel parameters.\n\nThe expectation maximization (EM) and space alternating EM (SAGE) algorithms are\nideally suited to these kind of problems as they are guaranteed to\nconverge in likelihood. Earlier  work related with delay estimation based on the EM algorithm  has appeared \\cite{Georg-90,Georg-91}.\nEfficient iterative receiver structures are presented in\n\\cite{kocian2003,kocian2007}, performing joint multiuser detection and\nchannel estimation  for synchronous as well as asynchronous coded\nDS-CDMA systems operating over quasi-static flat Rayleigh fading\nchannels, under the assumption that the transmissions delays are\nknown. The Bayesian EM\/SAGE algorithm can be used for joint\n\\emph{soft}-data estimation and channel estimation but the\ncomputational complexity of the resulting receiver architecture is non-polynomial in the number of\nusers \\cite{Gallo-04}. To overcome this draw-back, Hu \\emph{et al.}\napplied the Variational Bayesian EM\/SAGE algorithms to joint\nestimation of the distributions for channel coefficients, noise\nvariance, and information symbols for synchronous DS-CDMA in\n\\cite{Hu-08}. Our work may be considered to be a twofold extension of the\nwork by Gallo \\emph{et al.} in \\cite{Gallo-04}:\nFirst, the proposed receiver performs joint channel coefficient and\ntransmission delay estimation within the SAGE framework. Secondly, the\nimplication of the Monte-Carlo method in the SAGE framework makes it\npossible to compute \\emph{soft}-data estimates for all users at polynomial\ncomputational complexity, as well. Here, an\nefficient Markov chain Monte Carlo (MCMC) technique \\cite{gelfand} called\n{\\em Gibbs sampling} is used to compute the {\\em a posteriori}\nprobabilities (APP) of data symbols \\cite{doucet}. The APP's can be computed\nexactly with the MCMC algorithm, which is significantly less complex than\na standard hidden Markov model approach. \nThe resulting receiver architecture\nworks in principal fully blind and is guaranteed to converge. For\nuncoded transmission, a few\npilot bits must be inserted, though, to resolve the phase ambiguity problem.\n\n\nThe theoretical framework for the joint transmission delays and\nchannel estimation as well as the data detection algorithms can easily\nbe extended to coded transmission. \n\n\n\n\\section {System Description}\n\\label{sec:system}\n\nWe consider an asynchronous single-rate DS-CDMA system with $K$\nactive users using binary phase shift keying (BPSK) modulation\nsharing the same propagation channel. The signal transmitted by each\nuser experiences flat Rayleigh fading, which is assumed to be\nconstant over the observation frame of $L$ data symbols. Each user employs\na random signature  waveform for transmitting symbols of duration\n$T_{b}$, such that each symbol consists of $N_{c}$ chips with duration\n$T_{c}=T_{b}\/N_{c}$ where $N_{c}$ is an integer. The received signal\nis the noisy sum of all user's contribution, delayed by the propagation delays $\\tau_{k}\\in [0,T_{b}\/2)$, where the subscript $k$\ndenotes the label of  the $k$th user. After down-converting the\nreceived signal to baseband and passing it through an\nintegrate-and-dump filter with integration time $T_{s}=T_{c}\/Q$, $Q\n\\in \\mathbb{Q}^+$, $QN_c(L+1)$ samples over an observation frame of $L$ symbols  are\nstacked into a signal column vector ${\\boldsymbol{r}} \\in \\mathbb{C}^{Q N_{c}\n(L+1)-1}$. Note that sampling is chip-synchronous without\nknowledge of the individual transmission delays. It can\ntherefore be expressed as\n\\begin{equation}\n\\boldsymbol{r} = \n\\boldsymbol{S}(\\boldsymbol{\\tau})\\bf{A}\\bf{d}+\\boldsymbol{w}.\n\\label{sys:received}\n\\end{equation}\n\nIn this expression the matrix\n$\\boldsymbol{S} (\\boldsymbol{\\tau})\n\\in \\mathbb{C}^{Q N_{c} (L+1)-1 \\times LK}$ contains the signature\nsequences of all the users\n\\[\n\\boldsymbol{S}( \\boldsymbol \\tau)= \\left[ {\\begin{array}{*{20}c}\n   \\boldsymbol{S}_{1} (\\tau_1), &\n   \\boldsymbol{S}_{2}(\\tau_2),  & \\cdots &  ,\\boldsymbol{S}_{K} (\\tau_K) \\\\\n\\end{array}} \\right]\n\\]\nwhere $\\boldsymbol{S}_k (\\tau_k) \\in \\mathbb{C}^{Q N_{c} (L+1)-1\n\\times L}$ has the form\\\\\n\\[\n\\boldsymbol{S}_k( \\tau_k)=\n\\left[ {\\begin{array}{*{20}c}\n   | & | & & | \\\\\n   \\boldsymbol{S}_{k} (\\tau_k,0) &\n   \\boldsymbol{S}_{k}(\\tau_k,1)  & \\cdots &  \\boldsymbol{S}_{k} (\\tau_k,L-1) \\\\\n   | & | & & | \\\\\n\\end{array}} \\right]\n\\]\nand the spreading code vector $\\boldsymbol{S}_k(\\tau_k,\\ell)\n\\in \\mathbb{C}^{Q N_{c}(L+1)-1 \\times 1}$ is given by\n\\[\n\\boldsymbol{S}_k (\\tau_k,\\ell)=\n\\left[ {\\begin{array}{c}\n   \\boldsymbol{0}_{Q N_{c}\\ell+\\tau_{k} \\times 1} \\\\\n   | \\\\\n   \\boldsymbol{s}_{k}(\\tau_k,\\ell) \\\\\n   | \\\\\n   \\boldsymbol{0} \\\\\n\\end{array}} \\right].\n\\]\n\n The vector $\\boldsymbol{s}_k (\\tau_k,\\ell)$ contains the\n spreading code of user~$k$ having support\n $[\\ell N_{c}T_{c}, (\\ell+1) N_{c}T_{c}]$ with energy\n $\\boldsymbol{s}_k^{\\dag}(\\tau_k,\\ell)\\boldsymbol{s}_k\n (\\tau_k,\\ell)=1$.  \n  Finally, $\\boldsymbol{0}_{M \\times 1}$ denotes the $M \\times 1$-dim. all-zero\n  column vector.\n\n\nThe block diagonal channel matrix\n$\\boldsymbol{A}\\in \\mathbb{C}^{LK\\times\n  LK}$ in (1) is given by  $\\boldsymbol{A} =\n\\mbox{diag}\\{\\boldsymbol{A}_1,\\cdots, \\boldsymbol{A}_K\\}$. The channel\nmatrix for user~$k$,\n$\\boldsymbol{A}_k \\in \\mathbb{C}^{L\\times L}$,\n is given by\n$\\boldsymbol{A}_k = {\\bf I}_{L} \\otimes$ $a_{k}$ where  $\\bf{I}_{L}$ is the $L$-dim. identity matrix, and the symbol $\\otimes$\ndenotes the Kronecker product. The $k$th user's channel coefficient\n$a_{k}$ is a circularly symmetric complex\nGaussian random variable with zero mean and variance $\\sigma_k^2$. The\n$k$th user's transmission delay is assumed to be uniformly distributed.\n\nThe symbol vector $\\boldsymbol{d} \\in \\mathbb{C}^{LK}$ takes the form $\\boldsymbol{d} =\\mbox{col}\\{{\\boldsymbol{d}_1,\\cdots,\\boldsymbol{d} _K}\\}$\nwhere the vector $\\boldsymbol{d}_k  \\in\\mathbb{C}^{L} $ contains the\n$k$th user's symbols, i.e. $\\boldsymbol{d} _k =\n\\mbox{col}\\{d_{k}(0),\\cdots,d_{k}(L-1) \\}$\nwith $d_{k}(\\ell) \\in\n\\{-1,+1\\}$ denoting the symbol transmitted by the $k$th user during\nthe $\\ell$th signalling interval. Finally, the column vector\n$\\boldsymbol{w} \\in \\mathbb{C}^{QN_{c}(L+1)-1}$ contains complex,\ncircularly symmetric white Gaussian noise having  covariance matrix\n$N_{0}{\\bf I}$.\nWe assume that the vectors\n$\\boldsymbol{a}\\triangleq \\mbox{col}\\{a_{1},a_{2},\\cdots,a_{K}\\}$,\n$\\boldsymbol{\\tau}\\triangleq\n\\mbox{col}\\{\\tau_{1},\\tau_{2},\\cdots,\\tau_{K}\\}$, ${\\boldsymbol{d}}$ and\n$\\boldsymbol{w}$ and their components are independent.\nThe receiver does not know the data sequences, the (complex) channel coefficients, or the  transmission delays.\n\n\n\n\\section{Monte-Carlo SAGE Joint Parameter Estimation}\n\\label{sec:MCSAGE_appl}\n\n\\subsection{The SAGE Algorithm}\n\nIn previous applications, the SAGE algorithm \\cite{fessler} has been extensively used\nto iteratively approximate the ML\/MAP estimate of a parameter vector  $\\boldsymbol{\\theta}$\nwith respect to the observed data ${\\boldsymbol{r}}$.\n To obtain a receiver architecture that iterates between soft-data and\n channel estimation, one might choose the parameter vector as\n ${\\boldsymbol{\\theta}}=\\left\\{\\mathfrak{R}(a_{1}),\\cdots,\n   \\mathfrak{R}(a_{K}),\\mathfrak{I}(a_{1}),\\cdots,\\mathfrak{I}(a_{K}),\\tau_{1},\\cdots,\\tau_{K} \\right\\}$. The symbols $\\mathfrak{R}(\\cdot)$ and $\\mathfrak{I}(\\cdot)$ denote the real and imaginary parts of the complex argument, respectively. At iteration $i$,\nonly the parameter vector of user $k$, ${\\boldsymbol{\\theta}}_{k}$ are updated, while the\nparameter vectors of the other users ${{\\boldsymbol{\\theta}}}_{\\bar k}={\\boldsymbol{\\theta}}\n\\backslash {\\boldsymbol{\\theta}}_{k}$ are kept fixed. In the SAGE framework ${\\boldsymbol{r}}$ is\nreferred to as the \\emph{incomplete} data. The so-called  {\\em admissible hidden} data\n${\\boldsymbol{\\chi}}_k$ with respect to ${\\boldsymbol{\\theta}}$ is selected to be\n${\\boldsymbol{\\chi}}_k=\\{{\\boldsymbol{r}},{\\boldsymbol{d}}\\}$. Notice that ${\\boldsymbol{\\chi}}_k$ can only be partially\nobserved. Applying the SAGE algorithm to MAP parameter estimation, yields the expectation (E)-step\n\\begin{equation}\n\\label{alg:sage_estep}\nQ_k({\\boldsymbol{\\theta}}_k,{\\boldsymbol{\\theta}}^{[i]})=E_{{\\boldsymbol{d}}} \\left\\{ \\log\n  p\\left({\\boldsymbol{r}},{\\boldsymbol{d}},{\\boldsymbol{a}}_k,\\boldsymbol \\tau_k,{{\\boldsymbol{a}}}_{\\bar k}^{[i]},{\\boldsymbol \\tau}_{\\bar k}^{[i]} \\right) \\mid {\\boldsymbol{r}},{\\boldsymbol{a}}^{[i]},\\boldsymbol \\tau^{[i]}  \\right\\}.\n\\end{equation}\nThe maximization (M)-step computes a value of the argument $\\boldsymbol \\tau_k$\nin (\\ref{alg:sage_estep}) to obtain the update\n${\\boldsymbol{\\theta}}_k^{[i+1]}$. The objective function is non-decreasing at each\niteration.\n\n\\subsection{The Monte-Carlo SAGE algorithm}\n\nWe will see that direct computation of the expectation in\n(\\ref{alg:sage_estep}) requires a non-polynomial number of operations in the\nnumber of users $K$ and thus becomes prohibitive with increasing $K$. To make the computation of the expectation in\n(\\ref{alg:sage_estep}) feasible though,  we propose to use the technique of Markov\nchain Monte Carlo (MCMC) to obtain the Monte-Carlo SAGE\nalgorithm. MCMC is a statistical technique that allows generation of ergodic\npseudo-random samples ${\\boldsymbol{d}}^{[i,1]},\\ldots,{\\boldsymbol{d}}^{[i,N_t]}$ from the current\napproximation to the conditional pdf $p({\\boldsymbol{d}} | {\\boldsymbol{r}}, {\\boldsymbol{\\theta}}^{[i]})$. These samples are used to approximate the expectation in\n(\\ref{alg:sage_estep}) by the sample-mean.  The Gibbs sampler and the\nMetropolis-Hastings algorithm are widely used MCMC algorithms. Here we\ndescribe only the Gibbs sampler \\cite{Borunjeny,doucet}, as it is the\nmost commonly used in applications. Having initialized\n${\\boldsymbol{d}}^{[0,0]}$ randomly, the Gibbs sampler iterates the following loop\nat SAGE iteration $i$:\n\\begin{itemize}\n\\item Draw sample ${\\boldsymbol{d}}_1^{[i,t]}$ from\n  $p({\\boldsymbol{d}}_1|{\\boldsymbol{d}}_2^{[i,t-1]},\\ldots,{\\boldsymbol{d}}_K^{[i,t-1]}, {\\boldsymbol{r}}, {\\boldsymbol{\\theta}}^{[i]})$\\\\\n\\item Draw sample ${\\boldsymbol{d}}_2^{[i,t]}$ from\n  $p({\\boldsymbol{d}}_2|{\\boldsymbol{d}}_1^{[i,t]},{\\boldsymbol{d}}_3^{[i,t-1]}\\ldots,{\\boldsymbol{d}}_K^{[i,t-1]}, {\\boldsymbol{r}},\n  {\\boldsymbol{\\theta}}^{[i]})$\\\\\n\\vdots\n\\item  Draw sample ${\\boldsymbol{d}}_K^{[i,t]}$ from\n  $p({\\boldsymbol{d}}_K|{\\boldsymbol{d}}_1^{[i,t]},\\ldots,{\\boldsymbol{d}}_{K-1}^{[i,t]}, {\\boldsymbol{r}}, {\\boldsymbol{\\theta}}^{[i]})$\\\\\n\\end{itemize}\nFollowing this approach, we have\n\\begin{equation*}\n\\label{alg:mcmc_sage_estep}\nQ_k({\\boldsymbol{\\theta}}_k,{\\boldsymbol{\\theta}}^{[i]})=\\frac{1}{N_t} \\sum_{t=1}^{N_t} \\left\\{ \\log\n  p\\left({\\boldsymbol{r}},{\\boldsymbol{d}}^{[i,t]},{\\boldsymbol{a}}_k,\\boldsymbol \\tau_k,{{\\boldsymbol{a}}}_{\\bar\n      k}^{[i]},{\\boldsymbol \\tau}_{\\bar k}^{[i]} \\right) \\right\\}.\n\\end{equation*}\n\nNotice that with increasing $N_t$, the Monte-Carlo SAGE algorithm converges to the MAP\nsolution ${\\boldsymbol{\\theta}} = {\\boldsymbol{\\theta}}^\\star$ up to random fluctuations around ${\\boldsymbol{\\theta}}^\\star$ \\cite{Tanner-90}.\n\n\n\\subsection{Receiver design}\n\n\nThis subsection is devoted to the derivation of a receiver\narchitecture for joint estimation of parameters within the Monte-Carlo SAGE framework. Discarding terms\nindependent of ${\\boldsymbol{a}}$ and $\\boldsymbol \\tau$, we obtain\n\\begin{equation}\n\\log p({\\boldsymbol{r}},{\\boldsymbol{d}},{\\boldsymbol{a}},\\boldsymbol \\tau) =\n\\log p({\\boldsymbol{r}}|{\\boldsymbol{d}},{\\boldsymbol{a}},\\boldsymbol \\tau) +\n\\log p({\\boldsymbol{d}}) +\n\\log p({\\boldsymbol{a}}) + \\log p(\\boldsymbol \\tau).\n\\label{appl:likefkt}\n\\end{equation}\nFrom (\\ref{sys:received}), it follows that\n\\begin{equation}\n\\log p({\\boldsymbol{r}}|{\\boldsymbol{a}},\\boldsymbol \\tau,{\\boldsymbol{d}}) \\varpropto\n\\Re{\\{{\\boldsymbol{r}}^{\\dag}\\boldsymbol{S}\\boldsymbol{A}{\\boldsymbol{d}}}\\} -\\frac{1}{2}\n{\\boldsymbol{\\mu}}({\\boldsymbol{\\theta}},{\\boldsymbol{d}})^{\\dag}{\\boldsymbol{\\mu}}({\\boldsymbol{\\theta}},{\\boldsymbol{d}}),\n\\label{appl:likelihood}\n\\end{equation}\nwhere ${\\boldsymbol{\\mu}}({\\boldsymbol{\\theta}},{\\boldsymbol{d}}) \\triangleq\n\\sum_{k=1}^{K}\\sum_{\\ell=0}^{L-1}{\\boldsymbol{S}}_{k}(\\ell,\\tau_{k})a_{k}d_{k}(\\ell)$\nand $(.)^{\\dag}$ is the conjugate transpose of the argument.\n\\subsubsection{The E-step}\nSubstituting (\\ref{appl:likelihood}) into (\\ref{appl:likefkt}) yields\nafter some algebraic manipulations for the E-step of the Monte-Carlo SAGE algorithm:\n\\begin{eqnarray}\n\\lefteqn{Q_{k}({\\boldsymbol{\\theta}}_{k}|{\\boldsymbol{\\theta}}^{[i]})=}\\nonumber\\\\\n&& \\hspace*{-5ex}\n \\frac{2}{N_{0}} \\sum_{\\ell=0}^{L-1} \\Re\\left\\{a^{*}_{k}\n\\Psi(\\ell,\\tau_{k})\\right\\}-\\frac{L}{N_{0}}|a_{k}|^{2} -\n\\frac{1}{\\sigma_k^2}|a_k|^2 \\label{appl:Q-fct}\n\\end{eqnarray}\nwith the branch definition\n\\[\n\\Psi(\\ell,\\tau_{k}) \\triangleq \\boldsymbol{S}^{\\dag}_{k}(\\ell,\\tau_{k})\\left(\\exd{k}{\\ell}{\\boldsymbol{r}} -\n  \\mathcal{I}_k^{[i]}(\\ell)\\right)\n\\]\nand the interference term \n\\begin{eqnarray*}\n \\mathcal{I}_k^{[i]}(\\ell) &\\triangleq& \\sum_{k' \\neq\nk}a_k'^{[i]}\\bigg(\\boldsymbol{S}_{k'}(\\ell+1,\n\\tau^{[i]}_{k'})\\corrd{k}{\\ell}{k'}{\\ell+1}\\\\\n&&\\hspace*{1ex} +\n\\boldsymbol{S}_{k'}(\\ell,\\tau^{[i]}_{k'})\\corrd{k}{\\ell}{k'}{\\ell}\\\\\n&&\\hspace*{1ex}+\\boldsymbol{S}_{k'}\n(\\ell-1,\\tau^{[i]}_{k'})\n\\corrd{k}{\\ell}{k'}{\\ell-1}\\bigg).\n\\end{eqnarray*}\nMoreover,\n\\begin{equation}\n\\label{appl:softdata}\n \\exd{k}{\\ell}\\triangleq \\sum_{m\\in\n   \\mathcal{S}}m P(d_{k}(\\ell)=m| {\\boldsymbol{r}}, \\boldsymbol \\tau^{[i]},{\\boldsymbol{a}}^{[i]})\n\\end{equation}\nand\n\\begin{eqnarray}\n\\label{appl:softcorr}\n\\lefteqn{\\corrd{k}{\\ell}{k'}{\\ell'} \\triangleq\n\\sum_{m\\in \\mathcal{S}}\\sum_{n\\in \\mathcal{S}}~m~n}\\nonumber \\\\&&\n\\!\\!\\times P(d_{k}(\\ell)=m,d_{k'}(\\ell')=n \\mid {\\boldsymbol{r}},\n  \\boldsymbol \\tau^{[i]},{\\boldsymbol{a}}^{[i]}), \\mbox{   for  } k'\\neq k,~\n\\end{eqnarray}\n  where $\\mathcal{S}\\triangleq\\{-1,+1\\}$ is the signal constellation\n  and the lag is within range $\\ell' \\in \\{\\ell-1,\\ell,\\ell+1\\}$.\n\n\n\\subsubsection{The M-step}\n\nThe M-step of the SAGE algorithm is realized by first maximizing (\\ref{appl:Q-fct}) with respect to the transmission delays $\\tau_k$,\n\\begin{equation}\n\\label{appl:tau_update}\n\\tau^{(i+1)}_{k}=\\arg \\max_{\\tau_k} \\left|\\sum_{\\ell=0}^{L-1}\\Psi(\\ell,\\tau_k)\\right|.\n\\end{equation}\nThen by inserting  (\\ref{appl:tau_update}) into\n(\\ref{appl:Q-fct}), taking derivatives with respect to the $a_k$'s,\nsetting the results equal to zero, and solving yields\n\\[\na_k^{(i+1)} = \\frac{1}{L+N_0\/\\sigma_k^2}\\sum_{\\ell=0}^{L-1}\\Psi(\\ell,\\tau^{(i+1)}_{k}).\n\\]\n\n\\section{Monte-Carlo Implementation to the Computation of A Posteriori Probabilities}\n\n\\subsection{Computation of the soft-data symbols in (\\ref{appl:softdata})}\n\n\\label{sec:softdata}\n\nLet $\\overline{{\\boldsymbol{d}}_{k}(\\ell)}\\triangleq {\\boldsymbol{d}} \\backslash \\{d_{k}(\\ell)\\}$.\nFor notational simplicity we use $\\bar{{\\boldsymbol{d}}}\\triangleq\\overline{{\\boldsymbol{d}}_{k}(\\ell)}$ throughout this section. Then, the {\\em  a\nposteriori} probability of  $d_{k}(\\ell)$ in (\\ref{appl:softdata}) can be evaluated as\n\n\\setlength\\arraycolsep{1pt}\n\\begin{eqnarray}\n\\lefteqn{P(d_{k}(\\ell)=m \\mid {\\boldsymbol{r}},\n  \\boldsymbol \\tau^{[i]},{\\boldsymbol{a}}^{[i]})}\\nonumber\\\\&=&\\sum _{\\bar{{\\boldsymbol{d}}}}P(d_{k}(\\ell)=m \\mid \\bar{{\\boldsymbol{d}}}, {\\boldsymbol{r}}, \\boldsymbol \\tau^{[i]},{\\boldsymbol{a}}^{[i]})~P( \\bar{{\\boldsymbol{d}}}|{\\boldsymbol{r}}, \\boldsymbol \\tau^{[i]},{\\boldsymbol{a}}^{[i]}) \\nonumber\\\\\n   &\\approx&\\frac{1}{N_t}\\sum_{t=1}^{N_t}\n   P(d_{k}(\\ell)=m| \\bar{{\\boldsymbol{d}}}^{[i,t]},{\\boldsymbol{r}}, \\boldsymbol \\tau^{[i]},{\\boldsymbol{a}}^{[i]}).\n\\label{MC:APP}\n\\end{eqnarray}\nTo compute $P(d_{k}(\\ell)=m| \\bar{{\\boldsymbol{d}}}^{[i,t]},{\\boldsymbol{r}},\n\\boldsymbol \\tau^{[i]},{\\boldsymbol{a}}^{[i]})$ for this Markov chain Rao-Blackwellization technique, we define\n \\begin{equation}\n\\label{MC:LLR}\n \\lambda^{[i,t]}\\triangleq\\ln \\frac{P\\left(d_{k}(\\ell)= +1|\\bar{{\\boldsymbol{d}}}^{[i,t]},{\\boldsymbol{r}},\n\\boldsymbol \\tau^{[i]},{\\boldsymbol{a}}^{[i]}\\right)}{ P\\left(d_{k}(\\ell)=\n-1|\\bar{{\\boldsymbol{d}}}^{[i,t]},{\\boldsymbol{r}}, \\boldsymbol \\tau^{[i]},{\\boldsymbol{a}}^{[i]}\\right)},\n \\end{equation}\nFor uncoded transmission, the data symbols are i.i.d. and equally\nlikely. Therefore, it follows from (\\ref{MC:LLR}) that\n\\begin{equation}\n\\label{MC:LLR_exp}\n \\lambda^{[i,t]} = \\ln \\frac{P( {\\boldsymbol{r}} \\mid d_{k}(\\ell)= +1, \\bar{{\\boldsymbol{d}}}^{[i,t]},\n\\boldsymbol \\tau^{[i]},{\\boldsymbol{a}}^{[i]})}{ P( {\\boldsymbol{r}} \\mid d_{k}(\\ell)= -1,\n\\bar{{\\boldsymbol{d}}}^{[i,t]},\\boldsymbol \\tau^{[i]},{\\boldsymbol{a}}^{[i]})},\n\\end{equation}\nfrom which it can be easily seen that\n\\begin{equation*}\nP\\left(d_{k}(\\ell)= m\\mid \\bar{{\\boldsymbol{d}}}^{(t)},{\\boldsymbol{r}},\n\\boldsymbol \\tau^{[i]},{\\boldsymbol{a}}^{[i]}\\right)=\\frac{1}{1+\\exp\\left(-m\\lambda^{[i,t]}\\right)}.\n\\label{MC:logapp_lambda}\n\\end{equation*}\nFrom (\\ref{sys:received}),  we have $p({\\boldsymbol{r}}|{\\boldsymbol{D}})\\thicksim \\exp(-\\frac{1}{N_{0}}|{\\boldsymbol{r}}-{\\boldsymbol{G}}\n{\\boldsymbol{d}}|^{2})$, with ${\\boldsymbol{G}}\\triangleq\\bf{S}(\\boldsymbol \\tau){\\boldsymbol{A}}$ and\n${\\boldsymbol{d}}=\\mbox{col}\\{{\\boldsymbol{d}}_{1},{\\boldsymbol{d}}_{2},\\cdots,{\\boldsymbol{d}}_{K}\\},$. After some\nalgebra (\\ref{MC:LLR_exp}) can be expressed as\n\\begin{equation}\n\\label{MC:LLR_fin}\n\\lambda^{[i,t]}=\\frac{4}{N_{0}}\n\\Re\\left\\{({\\boldsymbol{g}}^{[i]}_{q})^{\\dag}({\\boldsymbol{r}}-{{\\boldsymbol{G}}}^{[i]}_{\\bar q} \\mbox{  }\n{{\\boldsymbol{d}}}^{[i,t]}_{\\bar q})\\right\\},\n\\end{equation}\nwhere $q\\triangleq kL+\\ell$, and ${{\\boldsymbol{G}}}_{\\bar q}$ is ${\\boldsymbol{G}}$ with its $q$th column\n${\\boldsymbol{g}}_{q}$ removed. Similarly, ${{\\boldsymbol{d}}}_{\\bar q}$ denotes the\nvector ${\\boldsymbol{d}}$ with its $q$th component removed.\n\nIn summary, for each $k=1,2,\\cdots,K$ and $\\ell=0,1,\\cdots,L-1$, to\nestimate the {\\em a posteriori} probabilities\n$P(d_{k}(\\ell)|{\\boldsymbol{r}},\\boldsymbol \\tau^{[i]},{\\boldsymbol{a}}^{[i]})$ in (\\ref{MC:APP}), the Gibbs\nsampler runs over all symbols $N_{t}$ times to generate a collection\nof vectors $\\left\\{\\bar{{\\boldsymbol{d}}}^{[i,t]}\\triangleq\n\\bar{{\\boldsymbol{d}}}^{[i,t]}_{k}(\\ell)\\right\\}_{t=1}^{N_{t}}$ which are used\nin (\\ref{MC:LLR_fin}) to estimate the desired quantities.\n\n\n\\subsection{Computation of  the soft-value for the product of two\n  data symbols in (\\ref{appl:softcorr})}\n\nSimilarly, a number of random samples\n$\\overline{\\overline{{\\boldsymbol{d}}}}^{{[i,t]}}\\triangleq\n\\overline{\\overline{{\\boldsymbol{d}}_{k,k'}(\\ell')}}^{[i,t]},\nt=1,2,\\cdots, N_t, \\ell'\n\\in \\{-1,0,+1\\}$ are drawn, using the Gibbs sampling technique,\nfrom the joint conditional posterior distribution,\n$P(\\overline{\\overline{{\\boldsymbol{d}}}} \\mid {\\boldsymbol{r}}, \\boldsymbol \\tau^{[i]},{\\boldsymbol{a}}^{[i]}).$ Based on the samples\n$\\overline{\\overline{{\\boldsymbol{d}}}}^{[i,t]}$,  $\\corrd{k}{\\ell}{k'}{\\ell'}$ in (\\ref{appl:softcorr}) can be evaluated by\n\\begin{eqnarray*}\n\\lefteqn{\\corrd{k}{\\ell}{k'}{\\ell'}\\approx\n(1\/N_{t})}\\nonumber\\\\&&\n\\times \\sum_{t=1}^{N_{t}}\n\\sum_{m,n\\in \\mathcal{S}}m n\nP\\left(d_{k}(\\ell)=m,d_{k'}(\\ell')=n \\mid \\overline{\\overline{{\\boldsymbol{d}}}}^{{[i,t]}}, {\\boldsymbol{r}}, \\boldsymbol \\tau^{[i]},\n{\\boldsymbol{a}}^{[i]}\\right).\n\\end{eqnarray*}\nWe need to evaluate the probability in the expression above. Following the same route taken as in the previous section and after some algebra, it can be expressed as\n\\[\nP\\left(d_{k}(\\ell)= m, d_{k'}(\\ell')= n \\mid \\overline{\\overline{{\\boldsymbol{d}}}}^{[i,t]}, {\\boldsymbol{r}}, \\boldsymbol \\tau^{[i]},\n{\\boldsymbol{a}}^{[i]}\\right)=\\]\n\\begin{equation}\\hspace*{3cm} \\frac{1}{1+\\exp\\left(-\\zeta^{[i,t]}\\right)}\\cdot\n\\frac{1}{1+\\exp\\left(-\\lambda^{[i,t]}\\right)}.\n\\label{MC:LLR_corr}\n\\end{equation}\n\nThe quantities $\\zeta^{[i,t]}$ and $\\lambda^{[i,t]}$  (\\ref{MC:LLR_corr}) are given by\n\\begin{eqnarray*}\n\\zeta^{[i,t]}&=&\\frac{4}{N_{0}}\n\\Re\\left\\{n({\\boldsymbol{g}}^{[i]}_{p})^{\\dag}({\\boldsymbol{r}}-{{\\boldsymbol{G}}}^{[i]}_{\\overline{p,q}}\n\\mbox{  }\n{{\\boldsymbol{d}}}^{[i,t]}_{\\overline{p,q}})-mn({\\boldsymbol{g}}^{[i]}_{p})^{\\dag}{\\boldsymbol{g}}^{[i]}_{q}\\right\\},\\\\\n\\lambda^{[i,t]}&=&\\frac{4}{N_{0}}\n\\Re\\left\\{m({\\boldsymbol{g}}^{[i]}_{q})^{\\dag}({\\boldsymbol{r}}-{{\\boldsymbol{G}}}^{[i]}_{\\bar q} \\mbox{  }\n{{\\boldsymbol{d}}}^{[i,t]}_{\\bar q})\\right\\},\n\\end{eqnarray*}\nwhere $p\\triangleq k'L+\\ell'$ and $q\\triangleq kL+\\ell$.\n${{\\boldsymbol{G}}}_{\\overline{p,q}}$ is ${\\boldsymbol{G}}$ with its $p$th and $q$th columns\n${\\boldsymbol{g}}_{p},{\\boldsymbol{g}}_{q}$ removed.  Similarly, ${{\\boldsymbol{d}}}_{\\overline{p,q}}$\ndenotes the vector ${\\boldsymbol{d}}$ with its $p$th and $q$th components\nremoved.\n\n\n\n\n\\section{Performance analysis}\n\n\\subsection{ Modified Cramer-Rao Bounds for the Estimated Parameters}\nWe now derive the modified Cramer-Rao lower bounds (MCRB) on the\nvariances of any unbiased estimates $\\widehat{{\\boldsymbol{\\theta}}}$ of the\nparameter vector ${\\boldsymbol{\\theta}}$. It is shown in \\cite{VanTrees} that for $\\theta_{p} \\in {\\boldsymbol{\\theta}}$, $\\mbox{var}(\\widehat{\\theta}_{p}-\\theta_{p})\\geq [{\\boldsymbol{I}}^{-1}({\\boldsymbol{\\theta}})]_{pp}$, where ${\\boldsymbol{I}}({\\boldsymbol{\\theta}})$ is the $3K\\times 3K$ Fisher information matrix whose $(p,q)$th component is defined by\n\\begin{equation*}\n[{\\boldsymbol{I}}({\\boldsymbol{\\theta}})]_{pq}\\triangleq - E_{{\\boldsymbol{r}},{\\boldsymbol{a}}} \\bigg\\{\\frac{\\partial^{2}\\ln p({\\boldsymbol{r}},{\\boldsymbol{a}} \\mid \\boldsymbol \\tau)}{\\partial \\theta_{p}\\partial \\theta_{q}} \\bigg\\}, \\mbox{   for   }p,q=1,2,\\cdots,3K.\n\\end{equation*}\nFor the joint likelihood function in (\\ref{appl:likelihood}),\nit is shown in \\cite{kay} that the Fisher information matrix can be computed by\n\\begin{equation}\n[{\\boldsymbol{I}}({\\boldsymbol{\\theta}})]_{pq}=\n\\frac{2}{N_{0}}E_{{\\boldsymbol{d}}}\\bigg\\{E_{{\\boldsymbol{a}}|{\\boldsymbol{d}}}\\bigg\\{\\Re\\bigg[\n\\frac{\\partial {\\boldsymbol{\\mu}}^{\\dag}({\\boldsymbol{\\theta}},{\\boldsymbol{d}})}{\\partial\n  \\theta_{p}}\\frac{\\partial {\\boldsymbol{\\mu}}({\\boldsymbol{\\theta}},{\\boldsymbol{d}})}{\\partial\n  \\theta_{q}}\\bigg]\\bigg\\}\\bigg\\},\n\\label{CRB_Fisher}\n\\end{equation}\n$p,q=1,2,\\cdots, 3K.$\n\nTaking the expectations with respect to channel coefficients ${\\boldsymbol{a}}$ and data ${\\boldsymbol{d}}$ after taking the partial derivatives in (\\ref{CRB_Fisher})  with respect to $\\theta_{p}$ and $\\theta_{q}$, for different regions of $p$ and $q$ values,  under the assumption that the data sequences are independent and equally likely and the fact that ${\\boldsymbol{S}}^{\\dag}(\\tau_{p},\\ell){\\boldsymbol{S}}(\\tau_{p},\\ell)=1,$ for $p=1,2\\cdots K; \\mbox{  }\\ell=0,1,\\cdots,L-1$, the Fisher information matrix becomes a diagonal matrix whose $(p,p)$th component can  be evaluated as\n\\begin{equation}\n[{\\boldsymbol{I}}({\\boldsymbol{\\theta}})]_{pp}=\\frac{2}{N_{0}}\\left\\{ \\begin{array}{ll}\n        L; & p=1,\\cdots,K\\\\\n        L; & p=K+1,\\cdots,2K\\\\\n        \\sigma_{p}^{2}\\sum_{\\ell=0}^{L-1} \\mid {\\boldsymbol{S}}'(\\ell) \\mid^{2}; & p=2K+1,\\cdots,3K.\\\\\n   \\end{array}\n   \\right.\n\\label{CRB_Fisher_eval}\n\\end{equation}\nwith the short-cut ${\\boldsymbol{S}}'[\\ell] \\triangleq\n\\frac{\\partial{\\boldsymbol{S}}_p(\\tau_{p},\\ell)}{\\partial\\tau_{p}} \\mid_{t=\\ell\n  T_b + \\widehat\\tau_p}$.\nThe final result for the MCRBs on the estimates of the channel coefficients and the transmission delays is obtained by inverting the diagonal matrix ${\\boldsymbol{I}}({\\boldsymbol{\\theta}})$ in (\\ref{CRB_Fisher_eval}) as follows.\n\\begin{eqnarray}\n\\mbox{var}(\\widehat{a}_{k})&\\geq & N_0\/L, \\label{MCRB:a} \\\\\n\\mbox{  }\\mbox{var}(\\widehat{\\tau}_{k})&\\geq & 1\/(8 \\pi^{2} L\\overline{\\gamma_{k}}\\mbox{  }B^{2}_{s_{k}} ) \\label{MCRB:tau},\n\\end{eqnarray}\n$k = 1,2,\\dots,K$. The symbol $\\overline{\\gamma_{k}}\\triangleq \\sigma^{2}_{k}\/N_{0}$\nis the average SNR, $B_{s_{k}}$ is the Gabor bandwidth of the $k$th user's\nspreading code waveform, $s_{k}(t)$ i.e., \n\\begin{equation*}\nB_{s_{k}}\\triangleq \\bigg(\\int_{-\\infty}^{+\\infty}f^{2} \\mid S_{k}(f)\\mid^{2} df\\bigg)^{1\/2},\n\\end{equation*}\nand $S_{k}(f)$ is the Fourier transform of $s_{k}(t), t\\in\n[0,T_{b}]$. Note that the Gabor bandwidth $B_{s_{k}}$ tends to infinity for rectangular-shaped (continuous-time)\nchip waveforms.\n\n\n\n\n\\subsection{Numerical Examples}\n\nTo assess the performance of the proposed (non-linear) Monte-Carlo SAGE scheme, an asynchronous uncoded DS-CDMA system with $K = 5$ users, rectangular chip waveforms with processing gain $N_c = 8$, and $L = 80$ transmitted symbols per block is considered. The receiver processes $Q = 12$ samples per chip. For each data block, Gibbs sampling is performed over $50$ iterations. \nA few, say $L_p=4$ pilot symbols are embedded in each block to overcome the phase ambiguity problem. Each user's strongest path from the MMSE\nestimate of ${\\boldsymbol{a}}$ given the $K~Q~(L_p+1)-1$ samples of ${\\boldsymbol{r}}$ and the pilot symbols, yield the initial estimates ${\\boldsymbol{a}}^{[0]}$ and $\\boldsymbol \\tau^{[0]}$. The MMSE estimate of $\\boldsymbol{d}$, given\n${\\boldsymbol{r}}$ and weighted by ${\\boldsymbol{a}}^{[0]}$ yields the initial symbol estimate\n${\\boldsymbol{d}}^{[0]}$. We refer to this method as MMSE-separate estimation (MMSE-SE). \nFor comparison purpose, the SAGE-scheme for joint data detection and channel estimation in\n\\cite{kocian2007} for known transmission delays and hard-decision\ndecoding has also been considered subsequently. We refer to these scheme as\n\"SAGE-JDE, $\\boldsymbol \\tau$ known\".\\\\\n\n\\begin{figure}\n\\begin{center}\n\\begin{psfrags}\n\\psfrag{0.10}[][][0.8]{$0.10$}\n\\psfrag{0.15}[][][0.8]{$0.15$}\n\\psfrag{0.20}[][][0.8]{$0.20$}\n\\psfrag{0.25}[][][0.8]{$0.25$}\n\\psfrag{0.30}[][][0.8]{$0.30$}\n\\psfrag{0.35}[][][0.8]{$0.35$}\n\\psfrag{0.40}[][][0.8]{$0.40$}\n\\psfrag{0.45}[][][0.8]{$0.45$}\n\\psfrag{0.50}[][][0.8]{$0.50$}\n\\psfrag{delay}[][][0.8]{$\\tau\/T_b$}\n\\psfrag{10}[][][0.8]{$10$}\n\\psfrag{-1}[][][0.6]{$-1$}\n\\psfrag{-2}[][][0.6]{$-2$}\n\\psfrag{-3}[][][0.6]{$-3$}\n\\psfrag{user1}[l][l][0.8]{MCMC-SAGE, user 1}\n\\psfrag{user3}[l][l][0.8]{MCMC-SAGE, user 3}\n\\psfrag{MCRLB(1)xxxxxxxxxxxx}[l][l][0.8]{MCRB (\\ref{MCRB:a}), user 1}\n\\psfrag{MCRLB(3)}[l][l][0.8]{MCRB (\\ref{MCRB:a}), user 3}\n\\psfrag{MSE(a)}[][][0.8]{$\\mbox{var}(\\widehat{a}_k)$}\n\\includegraphics[width=10cm]{MSEa33_MCMC.eps}\n\\end{psfrags}\n\\end{center}\n\\caption{$\\mbox{var}(\\widehat{a}_k)$ of the MCMC-SAGE in near-far scenario. \\label{fig:MMSE_a}}\n\\end{figure}\n\nTo study the behavior of the proposed MCMC-SAGE scheme, we consider communication over AWGN (not known to the receiver). The individual powers are given by\n\\begin{equation*}\n\\begin{array}{lll}\n\\sigma_1^2 = -4~\\mathrm{dB}, & \\hspace{2ex} \\sigma_2^2 = -2~\\mathrm{dB}, & \\hspace{2ex} \\sigma_3^2 = 0~\\mathrm{dB}, \\\\\n\\sigma_4^2 = +2~\\mathrm{dB}, & \\hspace{2ex} \\sigma_5^2 = +4~\\mathrm{dB}, \\\\  \n\\end{array}\n\\end{equation*}\nFig.~\\ref{fig:MMSE_a} shows the mean-square-error (MSE) of the channel estimates $\\widehat{\\boldsymbol a}_1$ (weakest user) and $\\widehat{\\boldsymbol a}_3$ (normal user) as a function of the normalized transmission delays $\\boldsymbol \\tau\/T_b$ which are uniformly distributed on the interval between zero and the value on the abscissa. It can be seen that the MCMC-SAGE performs close to the MCRB over the entire range of $\\boldsymbol \\tau$. Not shown in the plot, convergence is achieved after around 25 iterations i.e., every user's parameter vector is updated five times.\n\n\\begin{figure}\n\\begin{center}\n\\begin{psfrags}\n\\psfrag{0.10}[][][0.8]{$0.10$}\n\\psfrag{0.15}[][][0.8]{$0.15$}\n\\psfrag{0.20}[][][0.8]{$0.20$}\n\\psfrag{0.25}[][][0.8]{$0.25$}\n\\psfrag{0.30}[][][0.8]{$0.30$}\n\\psfrag{0.35}[][][0.8]{$0.35$}\n\\psfrag{0.40}[][][0.8]{$0.40$}\n\\psfrag{0.45}[][][0.8]{$0.45$}\n\\psfrag{0.50}[][][0.8]{$0.50$}\n\\psfrag{delay}[][][0.8]{$\\tau\/T_b$}\n\\psfrag{10}[][][0.8]{$10$}\n\\psfrag{-4}[][][0.6]{$-4$}\n\\psfrag{-5}[][][0.6]{$-5$}\n\\psfrag{-6}[][][0.6]{$-6$}\n\\psfrag{-7}[][][0.6]{$-7$}\n\\psfrag{user1xxxxxxxxxxxxx}[l][l][0.8]{MCMC-SAGE, user 1}\n\\psfrag{user3}[l][l][0.8]{MCMC-SAGE, user 3}\n\\psfrag{MSE(tau)}[][][0.8]{$\\mbox{var}(\\widehat{\\tau}_k)$}\n\\includegraphics[width=10cm]{MSEtau33_MCMC.eps}\n\\end{psfrags}\n\\end{center}\n\\caption{$\\mbox{var}(\\widehat{\\tau}_k)$ of the MCMC-SAGE in near-far scenario. \\label{fig:MMSE_tau}}\n\\end{figure}\n\nFig.~\\ref{fig:MMSE_tau} depicts the MSE of the delay estimates $\\widehat{\\boldsymbol \\tau}_1$ and $\\widehat{\\boldsymbol \\tau}_3$. Notice that the MCRB for $\\boldsymbol \\tau$ tends to zero for time-continuous signature waveforms. It can be seen that user~3 does not encounter delay estimation errors for small transmission delays i.e., $\\tau\/T_b \\leq 0.2$. This effect can be partially explained by the large number of samples per chip i.e., $Q=12$. Though for higher transmission delays, $\\mbox{var}(\\widehat{\\tau}_3)$ is finite, because of the increasing residual interference in the receiver.\n\n\\begin{figure}\n\\begin{center}\n\\begin{psfrags}\n\\psfrag{2}[][][0.8]{$2$}\n\\psfrag{4}[][][0.8]{$4$}\n\\psfrag{6}[][][0.8]{$6$}\n\\psfrag{8}[][][0.8]{$8$}\n\\psfrag{10}[][][0.8]{$10$}\n\\psfrag{12}[][][0.8]{$12$}\n\\psfrag{14}[][][0.8]{$14$}\n\\psfrag{16}[][][0.8]{$16$}\n\\psfrag{effective SNR}[][][0.8]{$\\frac{L-L_p}{L}\\bar{\\gamma}_1=\\frac{L-L_p}{L}\\bar{\\gamma}_2=\\ldots=\\frac{L-L_p}{L}\\bar{\\gamma}_K$ [dB]}\n\\psfrag{10}[][][0.8]{$10$}\n\\psfrag{0}[][][0.6]{$0$}\n\\psfrag{-1}[][][0.6]{$-1$}\n\\psfrag{-2}[][][0.6]{$-2$}\n\\psfrag{-3}[][][0.6]{$-3$}\n\\psfrag{-4}[][][0.6]{$-4$}\n\\psfrag{BER(user1)}[][][0.6]{${\\mathrm{BER}_1}$}\n\\psfrag{BER(user3)}[][][0.6]{${\\mathrm{BER}_3}$}\n\\psfrag{SU}[l][l][0.7]{SU, known channel}\n\\psfrag{SAGE(kc)}[l][l][0.7]{MCMC-SAGE, $\\boldsymbol \\tau$ known}\n\\psfrag{SAGE}[l][l][0.7]{MCMC-SAGE, $\\hat{\\boldsymbol \\tau}$}\n\\psfrag{MMSE-SDEXXXXXX}[l][l][0.7]{MMSE-SE}\n\\includegraphics[width=10cm]{BER33_MCMC.eps}\n\\end{psfrags}\n\\end{center}\n\\caption{BER-performance in near-far scenario. \\label{fig:BER}}\n\\end{figure}\n\nThe bit-error-rate ($\\overline{\\mathrm{BER}}$) of the proposed\nreceiver is plotted in Fig.~\\ref{fig:BER} versus the \\emph{effective}\nSNR $\\frac{L-L_p}{L}\\bar{\\gamma}_k$, $\\bar{\\gamma}_k \\triangleq\n\\sigma_k^2\/N_0$, $k=1,\\ldots,K$. The transmission delays are uniformly distributed on $[0,T_b\/2)$. It can be seen that the MMSE-SDE scheme cannot handle delay estimation errors at all due to high\ncorrelations between the users' signature sequences. The proposed\nMCMC-SAGE scheme and the \"SAGE-JDE, $\\boldsymbol \\tau$ known\" perform similar. The weakest user 1 performs close to the single-user (SU) bound. The normal user 3 has a multiuser efficiency\nof roughly 1~dB over the entire range of SNR values.\n\n\\section{Conclusions}\nA computationally efficient estimation algorithm has been proposed\nfor  estimating the transmission delays and the channel coefficients\njointly  in a non-data-aided fashion via the SAGE algorithm. The {\\em a\nposteriori} probabilities needed to implement the SAGE algorithm\nhave been  computed by means of the Gibbs sampling technique. Exact\nanalytical expression have been obtained for the estimates of\ntransmission delays and channel coefficients. At each iteration the\nlikelihood function is non-decreasing.\n\n\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Motivation}\n\nDESIDERATA: ``To translate the information contained on protein\ndatabases in terms of random variables in order to model a dynamics\nof folding and unfolding of proteins''.\n\nThe information on the planetary motion has been annotated on Astronomical\nalmanacs (Ephemerides) along centuries and can be derived and analyzed\nby Classical Dynamics and Deterministic Chaos as well as confirmed and\ncorrected by General Relativity.  The information which is accumulated\non Biological almanacs (Protein databases) on the last decades, is\nstill waiting for its first description to be done by a successful\ntheory of folding and unfolding of proteins.  We think that this study\nshould be started from the evolution of protein families and its\nassociation into Family clans as a necessary step of their development.\n\nThe first  fundamental idea to be developed here is that proteins do not\nevolute independently. We introduce several arguments and we have\ndone many calculations to span the bridge over the facts about\nprotein evolution in order to emphasize the existence of a protein\nfamily formation process (PFFP), a successful pattern recognition method\nand a coarse-grained protein dynamics are driven by optimal control\ntheory \\cite{mondaini1,mondaini2,mondaini3,mondaini4,mondaini5,mondaini6}.\nProteins, or their ``intelligent'' parts,\nprotein domains, evolute together as a family of protein domains. We then\nrealize that the exclusion of the evolution of an ``orphan'' protein is\nguaranteed by the probabilistic approach to be introduced in the present\ncontribution. We think that the elucidation of the nature of intermediate\nstages of the folding\/unfolding dynamics, in order to circumvent the\nLevinthal ``paradox'' \\cite{levinthal,karplus} as well as the determination\nof initial conditions, should be found from a detailed study of this PFFP\nprocess. A byproduct of this approach is the possibility of testing the\nhypothesis of agglutination of protein families into Clans by rigorous\nstatistical methods like ANOVA \\cite{deGroot,mondaini4}.\n\nWe take many examples of Entropy Measures as the generalized\nfunctions of random variables on our modelling. These are the\nprobabilities of occurrence of amino acids in rectangular arrays\nwhich are the representatives of families of protein domains.\nIn section 2, the sample space which is adequate for this\nstatistical approach is described in detail. We start from the\ndefinition of probabilities of occurrence and the restrictions\nimposed on the number of feasible families by the structure of\nthis sample space. Section 3 introduces the set of Sharma-Mittal\nEntropy Measures \\cite{sharma,mondaini4} to be adopted as the\nfunctions of probabilities of occurrence in the statistical\nanalysis to be developed. The Mutual Information measures\nassociated with the Sharma-Mittal set, as well as the normalized\nJaccard distance measures, are also introduced in this section.\nIn section 4, we present a naive sketch of assessing Protein\ndatabase, to set the stage for a more efficient approach of the\nfollowing sections. In section 5, we point out the inconvenience\nof the Maple computing system for the statistical calculations to\nbe done, by displaying tables with all CPU and real times\nnecessary to perform all necessary calculations. We have also\nprovided in this section, some adaptation of our methods in order\nto be used with the Perl computing system and we compare the new\ntimes of calculation with those by using Maple at the beginning\nof the section. We also include some comments on the use of Perl,\nespecially on its oddness to calculate with the input data given\nin arrays and the way of circumventing this. However, we also stress\nthat despite the fact that joint probabilities and their powers could be\nusually calculated, the output will come randomly distributed and the\nCPU and real times will increase too much to favour\nthe calculation of the entropy measures. This is due to the\nintrinsic ``Hash'' structure \\cite{cormen} of the Perl computing system.\nWe then introduce a modified array structure in order to calculate with\nPerl.\n\n\\section{The Sample Space for a Statistical Treatment}\nWe consider a rectangular array of \\textbf{\\emph{m}} rows (protein domains)\nand \\textbf{\\emph{n}} columns (amino acids). These arrays are organized\nfrom the protein database whose domains are classified into families and\nclans by the professional expertise of senior biologists \\cite{finn1,finn2}.\n\nThe random variable is the probability of occurrences of amino\nacids, $p_j(a)$, $j=1,2,\\ldots,n$, $a=A,C,D,\\ldots,W,Y$ (one-letter\ncode for amino acids), to be given by\n\\begin{equation}\n p_j(a) \\equiv \\frac{n_j(a)}{m} \\label{eq:eq1}\n\\end{equation}\n\n\\noindent where $n_j(a)$ is the number of occurrences of the amino acid\n\\textbf{\\emph{a}} in the $j$-th column. Eq.(\\ref{eq:eq1}) could be\nalso interpreted as the components of n vectors of 20 components\neach\n\\begin{equation}\n \\begin{pmatrix}\n  p_1(A) \\\\ \\relax\n  \\vdots \\\\ \\relax\n  p_1(Y)\n \\end{pmatrix}\n \\begin{pmatrix}\n  p_2(A) \\\\ \\relax\n  \\vdots \\\\ \\relax\n  p_2(Y)\n \\end{pmatrix}\n \\ldots\n \\begin{pmatrix}\n  p_n(A) \\\\ \\relax\n  \\vdots \\\\ \\relax\n  p_n(Y)\n \\end{pmatrix} \\label{eq:eq2}\n\\end{equation}\n\n\\noindent and we have\n\\begin{equation}\n \\sum_a n_j(a)=m\\,, \\,\\, \\forall j \\, \\Rightarrow \\,\n \\sum_a p_j(a)=1\\,, \\,\\, \\forall j \\label{eq:eq3}\n\\end{equation}\n\nAnalogously, we could also introduce the joint probability\nof occurrence of a pair of amino acids \\textbf{\\emph{a}}, \\textbf{\\emph{b}}\nin columns \\textbf{\\emph{j}}, \\textbf{\\emph{k}}, respectively $P_{jk}(a,b)$\nas the random variables. These are given by\n\\begin{equation}\n P_{jk}(a,b) = \\frac{n_{jk}(a,b)}{m} \\label{eq:eq4}\n\\end{equation}\n\n\\noindent where $n_{jk}(a,b)$ is the number of occurrences of the pair\nof amino acids \\textbf{\\emph{a}}, \\textbf{\\emph{b}} in columns\n\\textbf{\\emph{j}}, \\textbf{\\emph{k}}, respectively.\n\nA convenient interpretation of these joint probabilities could be\nthe elements of $\\frac{n(n-1)}{2}$, square matrices of $20 \\times 20$\nelements, to be written as\n\\begin{equation}\n P_{jk} =\n \\begin{pmatrix}\n  P_{jk}(A,A) & \\ldots & P_{jk}(A,Y) \\\\ \\relax\n  \\vdots & \\ddots & \\vdots \\\\ \\relax\n  P_{jk}(Y,A) & \\ldots & P_{jk}(Y,Y)\n \\end{pmatrix} \\label{eq:eq5}\n\\end{equation}\n\n\\noindent where $j=1,2,\\ldots,(n-1)$; $k=j+1,\\ldots,n$.\n\nWe can also write,\n\\begin{equation}\n P_{jk}(a,b)=P_{jk}(a|b)p_k(b) \\,, \\label{eq:eq6}\n\\end{equation}\nThis equation can be also taken as another definition of joint\nprobability. $\\!P\\!_{jk}(\\!a|b)$ is the Conditional probability of\noccurrence of the amino acid \\textbf{\\emph{a}} in column\n\\textbf{\\emph{j}} if the amino acid \\textbf{\\emph{b}} is already\nfound in column \\textbf{\\emph{k}}. We then have,\n\\begin{equation}\n \\sum_a P_{jk}(a|b) = 1 \\label{eq:eq7}\n\\end{equation}\nFrom eqs.(\\ref{eq:eq6}), (\\ref{eq:eq7}), we have:\n\\begin{equation}\n \\sum_a P_{jk}(a,b) = p_k(b) \\label{eq:eq8}\n\\end{equation}\nand from eq.(\\ref{eq:eq8}),\n\\begin{equation}\n \\sum_a\\sum_b P_{jk}(a,b) = 1 \\label{eq:eq9}\n\\end{equation}\n\n\\noindent which is an identity since $P_{jk}(a,b)$ is also a probability.\n\nEqs.(\\ref{eq:eq8}) and (\\ref{eq:eq9}) can be also derived from\n\\begin{equation}\n \\sum_a n_{jk}(a,b) = n_k(b) \\,;\\quad \\sum_a\\sum_b n_{jk}(a,b)\n = m  \\label{eq:eq10}\n\\end{equation}\n\n\\noindent and the definitions, eqs.(\\ref{eq:eq1}), (\\ref{eq:eq4}).\n\nWe now have from Bayes' law:\n\\begin{equation}\n P_{jk}(a|b)p_k(b) = P_{kj}(b|a)p_j(a) \\label{eq:eq10a}\n\\end{equation}\nand from eq.(\\ref{eq:eq10a}), the property of symmetry,\n\\begin{equation}\n P_{jk}(a,b) = P_{kj}(b,a) \\label{eq:eq10b}\n\\end{equation}\n\nThe matrices $P_{jk}$ can be organized in a triangular array:\n\\begin{equation}\n P =\n \\begin{matrix}\n  {} & P_{12} & P_{13} & P_{14} & \\ldots & P_{1\\,n-2} & P_{1\\,n-1}\n  & P_{1\\,n} \\\\ \\relax\n  {} & {} & P_{23} & P_{24} & \\ldots & P_{2\\,n-2} & P_{2\\,n-1}\n  & P_{2\\,n} \\\\ \\relax\n  {} & {} & {} & P_{34} & \\ldots & P_{3\\,n-2} & P_{3\\,n-1}\n  & P_{3\\,n} \\\\ \\relax\n  {} & {} & {} & {} & \\ddots & \\vdots & \\vdots & \\vdots \\\\ \\relax\n  {} & {} & {} & {} & {} & P_{n-3\\,n-2} & P_{n-3\\,n-1}\n  & P_{n-3\\,n} \\\\ \\relax\n  {} & {} & {} & {} & {} & {} & P_{n-2\\,n-1} & P_{n-2\\,n}\n  \\\\ \\relax\n  {} & {} & {} & {} & {} & {} & {} & P_{n-1\\,n}\n \\end{matrix} \\label{eq:eq11}\n\\end{equation}\nThe number of matrices until the $P_{jk}$-th one is given by\n\\begin{equation}\n C_{jk} = j(n-1) - \\frac{j(j-1)}{2} - (n-k) \\label{eq:eq12}\n\\end{equation}\n\n\\noindent These numbers can be also arranged as a triangular array:\n\\begin{equation}\n \\resizebox{.99\\hsize}{!}{$C =\n \\begin{matrix}\n  1 & 2 & 3 & 4 & 5 & 6 & \\ldots & (n-3) & (n-2) & (n-1) \\\\ \\relax\n  {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\\\ \\relax\n  {} & n & (n+1) & (n+2) & (n+3) & (n+4) & \\ldots & (2n-5) & (2n-4) &\n  (2n-3) \\\\ \\relax\n  {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\\\ \\relax\n  {} & {} & (2n-2) & (2n-1) & 2n & (2n+1) & \\ldots & (3n-8) & (3n-7) &\n  (3n-6) \\\\ \\relax\n  {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\\\ \\relax\n  {} & {} & {} & (3n-5) & (3n-4) & (3n-3) & \\ldots & (4n-12) & (4n-11) &\n  (4n-10) \\\\ \\relax\n  {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\\\ \\relax\n  {} & {} & {} & {} & (4n-9) & (4n-8) & \\ldots & (5n-17) & (5n-16) & (5n-15)\n  \\\\ \\relax\n  {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\\\ \\relax\n  {} & {} & {} & {} & {} & (5n-14) & \\ldots & (6n-23) & (6n-22) & (6n-21)\n  \\\\ \\relax\n  {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\\\ \\relax\n  {} & {} & {} & {} & {} & {} & \\ddots & \\vdots & \\vdots & \\vdots \\\\ \\relax\n  {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\\\ \\relax\n  {} & {} & {} & {} & {} & {} & {} & \\frac{1}{2}(n^2-n-10) & \\frac{1}{2}\n  (n^2-n-8) & \\frac{1}{2}(n+2)(n-3) \\\\ \\relax\n  {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\\\ \\relax\n  {} & {} & {} & {} & {} & {} & {} & {} & \\frac{1}{2}(n^2-n-4) &\n  \\frac{1}{2}(n+1)(n-2) \\\\ \\relax\n  {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\\\ \\relax\n  {} & {} & {} & {} & {} & {} & {} & {} & {} & \\frac{1}{2}\\,n(n-1)\n \\end{matrix}\n $} \\label{eq:eq13}\n\\end{equation}\n\nEq.(\\ref{eq:eq11}) should be used for the construction of a computational\ncode to perform all necessary calculations. We postpone to other publication\nthe presentation of some interesting results on the analysis of eq.(\\ref{eq:eq13}).\n\nThe calculation of the matrix elements $P_{jk}(a,b)$ from a rectangular array\n$m \\times n$ of amino acids is done by the ``concatenation'' process\nwhich is easily implemented on computational codes. We choose a pair of\ncolumns $j=\\bar{\\jmath}$, $k=\\bar{k}$ from the strings, $a=A$, $C$,\n$\\ldots$, $W$, $Y$, $b=A$, $C$, $\\ldots$, $W$, $Y$ and we look for the\noccurrence of the combinations $ab=AA$, $AC$, $\\ldots$, $AW$, $AY$, $CA$,\n$CC$, $\\ldots$, $CW$, $CY$, $\\ldots$, $WA$, $WC$, $\\ldots$, $WW$, $WY$,\n$\\ldots$, $YA$, $YC$, $\\ldots$, $YW$, $YY$. We then calculate their numbers\nof occurrences $n_{\\bar{\\jmath}\\bar{k}}(A,A)$,\n$n_{\\bar{\\jmath}\\bar{k}}(A,C)$, $\\ldots$, $n_{\\bar{\\jmath}\\bar{k}}(Y,W)$,\n$n_{\\bar{\\jmath}\\bar{k}}(Y,Y)$ and the corresponding probabilities\n$P_{\\bar{\\jmath}\\bar{k}}(A,A)$, $P_{\\bar{\\jmath}\\bar{k}}(A,C)$, $\\ldots$,\\\\\n$P_{\\bar{\\jmath}\\bar{k}}(Y,W)$, $P_{\\bar{\\jmath}\\bar{k}}(Y,Y)$ from\neq.(\\ref{eq:eq4}). We do the same for the other $\\frac{n^2-n-2}{2}$ pairs of\ncolumns.\n\nAs an example, let us suppose that we have the $3 \\times 4$ array:\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=0.25\\linewidth]{figure1}\n \\caption{\\small An example of a $3 \\times 4$ array with amino acids\n A, C, D. \\label{fig1}}\n\\end{figure}\n\nLet us choose the pair of columns 1,2. We look for the occurrence of the\ncombinations $AA$, $AC$, $AD$, $CA$, $CC$, $CD$, $DA$, $DC$, $DD$ on the\npair of columns 1,2 of the array above and we found $n_{12}(A,C) = 1$,\n$n_{12}(C,A) = 1$, $n_{12}(D,A) = 1$. The others $n_{12}(a,b) = 0$. From\neq.(\\ref{eq:eq4}) we can write for the matrices $P_{jk}$ of eq.(\\ref{eq:eq5}):\n\\begin{align}\n & P_{12} =\n \\begin{pmatrix}\n  0 & 1\/3 & 0 \\\\ \\relax\n  1\/3 & 0 & 0 \\\\ \\relax\n  1\/3 & 0 & 0\n \\end{pmatrix}\n \\!;\n P_{13} =\n \\begin{pmatrix}\n  1\/3 & 0 & 0 \\\\ \\relax\n  0 & 0 & 1\/3 \\\\ \\relax\n  0 & 1\/3 & 0\n \\end{pmatrix}\n \\!;\n P_{14} =\n \\begin{pmatrix}\n  0 & 0 & 1\/3 \\\\ \\relax\n  0 & 0 & 1\/3 \\\\ \\relax\n  0 & 1\/3 & 0\n \\end{pmatrix}\n \\nonumber \\\\\n & P_{23} =\n \\begin{pmatrix}\n  0 & 1\/3 & 1\/3 \\\\ \\relax\n  1\/3 & 0 & 0 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix}\n \\!;\n P_{24} =\n \\begin{pmatrix}\n  0 & 1\/3 & 1\/3 \\\\ \\relax\n  0 & 0 & 1\/3 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix}\n \\!;\n P_{34} =\n \\begin{pmatrix}\n  0 & 0 & 1\/3 \\\\ \\relax\n  0 & 1\/3 & 0 \\\\ \\relax\n  0 & 0 & 1\/3\n \\end{pmatrix} \\label{eq:eq14}\n\\end{align}\nThe Maple computing system ``recognizes'' the matricial structure through\nits Linear Algebra package. The Perl computing system ``operates'' only\nwith ``strings''. The results above are easily obtained in Maple, but in\nPerl we have to find alternative ways of calculating the joint probabilities.\nThe first method is to calculate the probabilities per row of the\n$3 \\times 4$ array. We have for the first row:\n\\begin{align}\n & \\Pi_{12}^{(1)} =\n \\begin{pmatrix}\n  0 & 1\/3 & 0 \\\\ \\relax\n  0 & 0 & 0 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix}\n \\!; \\,\\,\n \\Pi_{13}^{(1)} =\n \\begin{pmatrix}\n  1\/3 & 0 & 0 \\\\ \\relax\n  0 & 0 & 0 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix}\n \\!; \\,\\,\n \\Pi_{14}^{(1)} =\n \\begin{pmatrix}\n  0 & 0 & 1\/3 \\\\ \\relax\n  0 & 0 & 0 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix}\n \\nonumber \\\\\n & \\Pi_{23}^{(1)} =\n \\begin{pmatrix}\n  0 & 0 & 0 \\\\ \\relax\n  1\/3 & 0 & 0 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix}\n \\!; \\,\\,\n \\Pi_{24}^{(1)} =\n \\begin{pmatrix}\n  0 & 0 & 0 \\\\ \\relax\n  0 & 0 & 1\/3 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix}\n \\!; \\,\\,\n \\Pi_{34}^{(1)} =\n \\begin{pmatrix}\n  0 & 0 & 1\/3 \\\\ \\relax\n  0 & 0 & 0 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix} \\label{eq:eq15}\n\\end{align}\nFor the second row:\n\\begin{align}\n & \\Pi_{12}^{(2)} =\n \\begin{pmatrix}\n  0 & 0 & 0 \\\\ \\relax\n  1\/3 & 0 & 0 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix}\n \\!; \\,\\,\n \\Pi_{13}^{(2)} =\n \\begin{pmatrix}\n  0 & 0 & 0 \\\\ \\relax\n  0 & 0 & 1\/3 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix}\n \\!; \\,\\,\n \\Pi_{14}^{(2)} =\n \\begin{pmatrix}\n  0 & 0 & 0 \\\\ \\relax\n  0 & 0 & 1\/3 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix}\n \\nonumber \\\\\n & \\Pi_{23}^{(2)} =\n \\begin{pmatrix}\n  0 & 0 & 1\/3 \\\\ \\relax\n  0 & 0 & 0 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix}\n \\!; \\,\\,\n \\Pi_{24}^{(2)} =\n \\begin{pmatrix}\n  0 & 0 & 1\/3 \\\\ \\relax\n  0 & 0 & 0 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix}\n \\!; \\,\\,\n \\Pi_{34}^{(2)} =\n \\begin{pmatrix}\n  0 & 0 & 0 \\\\ \\relax\n  0 & 0 & 0 \\\\ \\relax\n  0 & 0 & 1\/3\n \\end{pmatrix} \\label{eq:eq16}\n\\end{align}\nFor the third row:\n\\begin{align}\n & \\Pi_{12}^{(3)} =\n \\begin{pmatrix}\n  0 & 0 & 0 \\\\ \\relax\n  0 & 0 & 0 \\\\ \\relax\n  1\/3 & 0 & 0\n \\end{pmatrix}\n \\!; \\,\\,\n \\Pi_{13}^{(3)} =\n \\begin{pmatrix}\n  0 & 0 & 0 \\\\ \\relax\n  0 & 0 & 0 \\\\ \\relax\n  0 & 1\/3 & 0\n \\end{pmatrix}\n \\!; \\,\\,\n \\Pi_{14}^{(3)} =\n \\begin{pmatrix}\n  0 & 0 & 0 \\\\ \\relax\n  0 & 0 & 0 \\\\ \\relax\n  0 & 1\/3 & 0\n \\end{pmatrix}\n \\nonumber \\\\\n & \\Pi_{23}^{(3)} =\n \\begin{pmatrix}\n  0 & 1\/3 & 0 \\\\ \\relax\n  0 & 0 & 0 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix}\n \\!; \\,\\,\n \\Pi_{24}^{(3)} =\n \\begin{pmatrix}\n  0 & 1\/3 & 0 \\\\ \\relax\n  0 & 0 & 0 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix}\n \\!; \\,\\,\n \\Pi_{34}^{(3)} =\n \\begin{pmatrix}\n  0 & 0 & 0 \\\\ \\relax\n  0 & 1\/3 & 0 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix} \\label{eq:eq17}\n\\end{align}\n\nWe stress that Perl does not recognize these matrix structures. This is just\nour arrangement in order to make comparison with Maple calculations. However,\nPerl ``knows'' how to sum the calculations done per rows to obtain:\n\\begin{equation}\n \\Pi_{12}^{(1)}+\\Pi_{12}^{(2)}+\\Pi_{12}^{(3)} =\n \\begin{pmatrix}\n  0 & 1\/3 & 0 \\\\ \\relax\n  1\/3 & 0 & 0 \\\\ \\relax\n  1\/3 & 0 & 0\n \\end{pmatrix}\n \\equiv P_{12} \\label{eq:eq18}\n\\end{equation}\n\\begin{equation}\n \\Pi_{13}^{(1)}+\\Pi_{13}^{(2)}+\\Pi_{13}^{(3)} =\n \\begin{pmatrix}\n  1\/3 & 0 & 0 \\\\ \\relax\n  0 & 0 & 1\/3 \\\\ \\relax\n  0 & 1\/3 & 0\n \\end{pmatrix}\n \\equiv P_{13} \\label{eq:eq19}\n\\end{equation}\n\\begin{equation}\n \\Pi_{14}^{(1)}+\\Pi_{14}^{(2)}+\\Pi_{14}^{(3)} =\n \\begin{pmatrix}\n  0 & 0 & 1\/3 \\\\ \\relax\n  0 & 0 & 1\/3 \\\\ \\relax\n  0 & 1\/3 & 0\n \\end{pmatrix}\n \\equiv P_{14} \\label{eq:eq20}\n\\end{equation}\n\\begin{equation}\n \\Pi_{23}^{(1)}+\\Pi_{23}^{(2)}+\\Pi_{23}^{(3)} =\n \\begin{pmatrix}\n  0 & 1\/3 & 1\/3 \\\\ \\relax\n  1\/3 & 0 & 0 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix}\n \\equiv P_{23} \\label{eq:eq21}\n\\end{equation}\n\\begin{equation}\n \\Pi_{24}^{(1)}+\\Pi_{24}^{(2)}+\\Pi_{24}^{(3)} =\n \\begin{pmatrix}\n  0 & 1\/3 & 1\/3 \\\\ \\relax\n  0 & 0 & 1\/3 \\\\ \\relax\n  0 & 0 & 0\n \\end{pmatrix}\n \\equiv P_{24} \\label{eq:eq22}\n\\end{equation}\n\\begin{equation}\n \\Pi_{34}^{(1)}+\\Pi_{34}^{(2)}+\\Pi_{34}^{(3)} =\n \\begin{pmatrix}\n  0 & 0 & 1\/3 \\\\ \\relax\n  0 & 1\/3 & 0 \\\\ \\relax\n  0 & 0 & 1\/3\n \\end{pmatrix}\n \\equiv P_{34} \\label{eq:eq23}\n\\end{equation}\n\nWe are then able to translate the Perl output in ``matrix language''.\nHowever, this output does not come as an ordered set of joint\nprobabilities, as we have done by arranging the output in the form of the\nmatrices $\\Pi_{jk}^{(l)}$, $j=1,2,3$, $k=2,3,4$, $l=1,2,3$. In order to\ncalculate functions of the probabilities as the entropy measures, it will\ntake too much time for the Perl computing system to collect the necessary\nprobability values. This is due to the ``Hash'' structure of the Perl as\ncompared to the usual ``array'' structure of the Maple. A new form of\narranging the strings to favour an a priori ordination will circumvent this\ninconvenience of the ``hash'' structure. Let us then write the following\nextended string associated to the $m \\times n$ rectangular array:\n\\begin{equation*}\n \\Big(\\underbrace{(\\overset{1}{A}\\overset{2}{C}\\overset{3}{D})}_{1}\n \\underbrace{(\\overset{1}{C}\\overset{2}{A}\\overset{3}{A})}_{2}\n \\underbrace{(\\overset{1}{A}\\overset{2}{D}\\overset{3}{C})}_{3}\n \\underbrace{(\\overset{1}{D}\\overset{2}{D}\\overset{3}{C})}_{4}\\Big)\n\\end{equation*}\nWe then get\n\\begin{align}\n &P_{12}(A,C) = 1\/3,\\quad P_{12}(C,A) = 1\/3,\\quad P_{12}(D,A) = 1\/3 \\nonumber \\\\\n &P_{13}(A,A) = 1\/3,\\quad P_{13}(C,D) = 1\/3,\\quad P_{13}(D,C) = 1\/3 \\nonumber \\\\\n &P_{14}(A,D) = 1\/3,\\quad P_{14}(C,D) = 1\/3,\\quad P_{14}(D,C) = 1\/3 \\nonumber \\\\\n &P_{23}(C,A) = 1\/3,\\quad P_{23}(A,D) = 1\/3,\\quad P_{23}(A,C) = 1\/3 \\nonumber \\\\\n &P_{24}(C,D) = 1\/3,\\quad P_{24}(A,D) = 1\/3,\\quad P_{24}(A,C) = 1\/3 \\nonumber \\\\\n &P_{34}(A,D) = 1\/3,\\quad P_{34}(D,D) = 1\/3,\\quad P_{34}(C,C) = 1\/3 \\label{eq:eq24}\n\\end{align}\nAll the other joint probabilities $P_{jk}(a,b)$ are equal to zero.\n\nThis is a feasible treatment for the ``hash'' structure. In the example\nsolved above the probabilities will come already ordered in triads. This\nwill save time in the calculations with the Perl system.\n\nIt should be stressed that the Perl computing system does not recognize any\nformal relations of Linear Algebra. However, it does quite well if these\nrelations are converted into products and sums. In order to give an example\nof working with the Perl system, we take a calculation with the usual Shannon\nEntropy measure. The calculation of the Entropy for the\ncolumns $j$,$k$ is done by\n\\begin{equation}\n S_{jk} = -\\sum_a\\sum_b P_{jk}(a,b)\\log P_{jk}(a,b) =\n -\\mathrm{Tr}\\Big(P_{jk}(\\log P_{jk})^{\\mathrm{T}}\\Big) \\label{eq:eq25}\n\\end{equation}\n\n\\noindent where $P_{jk}$ is the matrix given in eq.(\\ref{eq:eq5}) and\n$\\mathrm{Tr,T}$ stands for the operations of taking the trace and transposing\na matrix, respectively. The matrix $(\\log P_{jk})^{\\mathrm{T}}$ is given by\n\\begin{equation*}\n (\\log P_{jk})^{\\mathrm{T}} =\n \\begin{pmatrix}\n  \\log P_{jk}(A,A) & \\ldots & \\log P_{jk}(Y,A) \\\\ \\relax\n  \\vdots & {} & \\vdots \\\\ \\relax\n  \\log P_{jk}(A,Y) & \\ldots & \\log P_{jk}(Y,Y)\n \\end{pmatrix}\n\\end{equation*}\n\n\\noindent we also include for a useful reference, the matrix\n\\begin{equation*}\n p_j(p_k)^{\\mathrm{T}} =\n \\begin{pmatrix}\n  p_j(A)p_k(A) & \\ldots & p_j(A)p_k(Y) \\\\ \\relax\n  \\vdots & {} & \\vdots \\\\ \\relax\n  p_j(Y)p_k(A) & \\ldots & p_j(Y)p_k(Y)\n \\end{pmatrix}\n\\end{equation*}\n\nSince from eqs.(\\ref{eq:eq18})--(\\ref{eq:eq23}), we have\n\\begin{equation}\n P_{jk} = \\sum_{l=1}^m \\Pi_{jk}^{(l)} \\label{eq:eq26}\n\\end{equation}\nwe can write:\n\\begin{equation}\n S_{jk} = -\\mathrm{Tr}\\left(\\left(\\sum_{l=1}^m \\Pi_{jk}^{(l)}\\right)\n (\\log P_{jk})^{\\mathrm{T}}\\right) = -\\sum_{l=1}^m \\mathrm{Tr}\n \\Big(\\Pi_{jk}^{(l)}(\\log P_{jk})^{\\mathrm{T}}\\Big) \\label{eq:eq27}\n\\end{equation}\nThere is no problem for calculating in Perl, if we prepare eq.(\\ref{eq:eq26})\nby expressing previously all the products and sums to be done. The real\nproblem with Perl calculations is the arrangement of the output of values\n$P_{jk}(a,b)$, due to the ``hash'' structure as have been stressed above.\n\n\\section{Entropy Measures. The Sharma-Mittal set and the associated Jaccard\nEntropy measure}\nWe start this section with the definition of the two-parameter Sharma-Mittal\nentropies \\cite{sharma,mondaini1,mondaini2}\n\\begin{align}\n (SM)_{jk}(r,s) &= -\\frac{1}{1-r}\\left(1-\\left(\\sum_a\\sum_b\\big(P_{jk}(a,b)\n \\big)^s\\right)^{\\frac{1-r}{1-s}}\\right) \\label{eq:eq28} \\\\\n (SM)_{j}(r,s) &= -\\frac{1}{1-r}\\left(1-\\left(\\sum_a \\big(p_{j}(a)\n \\big)^s\\right)^{\\frac{1-r}{1-s}}\\right) \\label{eq:eq29}\n\\end{align}\n\n\\noindent where $p_j(a)$ and $P_{jk}(a,b)$ are the simple and joint probabilities\nof occurrence of amino acids as defined on eqs.(\\ref{eq:eq1}) and (\\ref{eq:eq4}),\nrespectively. \\textbf{\\emph{r}}, \\textbf{\\emph{s}} are non-dimensional parameters.\n\nWe can associate to the entropy measures above their corresponding\none parameter forms to be given by the limits:\n\\begin{align}\n H_{jk}(s) &= \\lim_{r \\to s} (SM)_{jk}(r,s) = -\\frac{1}{1-s}\n \\left(1-\\sum_a\\sum_b\\big(P_{jk}(a,b)\\big)^s\\right) \\label{eq:eq30} \\\\\n H_{j}(s) &= \\lim_{r \\to s} (SM)_{j}(r,s) = -\\frac{1}{1-s}\n \\left(1-\\sum_a\\big(p_{j}(a)\\big)^s\\right) \\label{eq:eq31}\n\\end{align}\n\n\\noindent These are the Havrda-Charvat Entropy Measures and they will be\nspecially emphasized in the present work. Other alternative proposals for\nthe single parameter entropies are given by\n\n\\noindent The Renyi's Entropy measures:\n\\begin{align}\n R_{jk}(s) &= \\lim_{r \\to 1} (SM)_{jk}(r,s) = \\frac{1}{1-s}\n \\log\\left(\\sum_a\\sum_b\\big(P_{jk}(a,b)\\big)^s\\right) \\label{eq:eq32} \\\\\n R_{j}(s) &= \\lim_{r \\to 1} (SM)_{j}(r,s) = \\frac{1}{1-s}\n \\log\\left(\\sum_a\\big(p_{j}(a)\\big)^s\\right) \\label{eq:eq33}\n\\end{align}\nThe Landsberg-Vedral Entropy measures:\n\\begin{align}\n L_{jk}(s) =\\! \\lim_{\\quad r \\to 2-s} (SM)_{jk}(r,s) &= \\frac{1}{1-s}\n \\left(1-\\Big(\\sum_a\\sum_b\\big(P_{jk}(a,b)\\big)^s\\Big)^{-1}\\right)\n \\label{eq:eq34} \\\\\n &= \\frac{H_{jk}(s)}{\\sum\\limits_a\\sum\\limits_b\\big(P_{jk}(a,b)\\big)^s}\n \\nonumber \\\\\n L_{j}(s) =\\! \\lim_{\\quad r \\to 2-s} (SM)_{j}(r,s) &= \\frac{1}{1-s}\n \\left(1-\\Big(\\sum_a\\big(p_{j}(a)\\big)^s\\Big)^{-1}\\right) \\label{eq:eq35} \\\\\n &= \\frac{H_j(s)}{\\sum\\limits_a\\big(p_j(a)\\big)^s} \\nonumber\n\\end{align}\nAll these Entropy measures have the free-parameter Shannon entropy\nin the limit $s \\to 1$.\n\\begin{align}\n \\lim_{s \\to 1} H_{jk}(s) &= \\lim_{s \\to 1} R_{jk}(s) = \\lim_{s \\to 1}\n L_{jk}(s) = S_{jk} \\label{eq:eq36} \\\\\n \\lim_{s \\to 1} H_j(s) &= \\lim_{s \\to 1} R_j(s) = \\lim_{s \\to 1} L_j(s)\n = S_j \\label{eq:eq37}\n\\end{align}\nwhere\n\\begin{align}\n S_{jk} &= -\\sum_a\\sum_b P_{jk}(a,b)\\log P_{jk}(a,b) \\tag{\\ref{eq:eq25}} \\\\\n S_j &= -\\sum_a p_j(a)\\log p_j(a) \\label{eq:eq38}\n\\end{align}\nare the Shannon entropy measures \\cite{mondaini6}.\n\nWe now introduce a convenient version of a Mutual Information measure:\n\\begin{equation}\n M_{jk}(r,s) = \\frac{1}{1-r}\\left(1-\\left(\\frac{\\sum\\limits_a\\sum\\limits_b\n \\big(P_{jk}(a,b)\\big)^s}{\\sum\\limits_a\\sum\\limits_b\\big(p_j(a)p_k(b)\n \\big)^s}\\right)^{\\frac{1-r}{1-s}}\\right) \\label{eq:eq39}\n\\end{equation}\n\n\\noindent We can see that $M_{jk}(r,0) = 0$ and if $\\exists\\, \\bar{\\jmath}, \\bar{k}$\nsuch that $P_{\\bar{\\jmath}\\bar{k}}(a,b) = p_{\\bar{\\jmath}}(a)p_{\\bar{k}}(b)$\n$\\Rightarrow$ $M_{\\bar{\\jmath}\\bar{k}}(r,s) = 0$. We also have,\n\\begin{equation}\n M_{jk}(1,s) = \\lim_{r \\to 1} M_{jk}(r,s) = -\\frac{1}{1-s}\\log\\left(\n \\frac{\\sum\\limits_a\\sum\\limits_b\\big(P_{jk}(a,b)\\big)^s}{\\sum\\limits_a\n \\sum\\limits_b\\big(p_j(a)p_k(b)\\big)^s}\\right) \\label{eq:eq40}\n\\end{equation}\nand in the limit $s \\to 1$\n\\begin{align}\n M_{jk} = \\lim_{s \\to 1} M_{jk}(1,s) =& \\sum_a\\sum_b P_{jk}(a,b)\\log\n P_{jk}(a,b) \\nonumber \\\\\n &- \\sum_a\\sum_b p_j(a)p_k(b)\\log\\big(p_j(a)p_k(b)\\big) \\label{eq:eq41}\n\\end{align}\nand from the identities:\n\\begin{align*}\n &\\sum_a p_j(a) = 1 \\,,\\, \\forall j \\,; \\quad \\sum_b p_k(b) = 1 \\,,\\,\n \\forall k \\\\\n &\\sum_a P_{jk}(a,b) = p_k(b) \\,,\\, \\forall j \\,; \\quad \\sum_b P_{jk}(a,b)\n = p_j(a) \\,,\\, \\forall k\n\\end{align*}\nobtained from eqs.(\\ref{eq:eq3}), (\\ref{eq:eq4}), (\\ref{eq:eq6}), (\\ref{eq:eq7}),\nwe can also write instead eq.(\\ref{eq:eq41}):\n\\begin{equation}\n M_{jk} = \\sum_a\\sum_b P_{jk}(a,b)\\log P_{jk}(a,b) - \\sum_a\\sum_b P_{jk}(a,b)\n \\log\\big(p_j(a)p_k(b)\\big) \\label{eq:eq42}\n\\end{equation}\nIt should be stressed that we are not assuming that $P_{jk}(a,b) \\equiv\np_j(a)p_k(b)$ above. This equality is assumed to be valid only for\n$j = \\bar{\\jmath}$, $k = \\bar{k}$.\n\nEq.(\\ref{eq:eq41}) or (\\ref{eq:eq42}) can be also written as:\n\\begin{equation}\n M_{jk} = -S_{jk} + S_j + S_k \\label{eq:eq43}\n\\end{equation}\n\n\\noindent where $S_{jk}$ and $S_j$, $S_k$ are the Shannon entropy measures for\njoint and single probabilities, respectively, eqs.(\\ref{eq:eq25}), (\\ref{eq:eq38}).\n\nAs an additional topic, we emphasize that the Mutual Information measure can\nbe also derived from the Kullback-Leibler divergence \\cite{mondaini6} which is\nwritten as\n\\begin{equation}\n (KL)_{jk}(b) = \\sum_a P_{jk}(a|b)\\log\\left(\\frac{P_{jk}(a|b)}{p_j(a)}\n \\right) \\label{eq:eq44}\n\\end{equation}\n\n\\noindent where $P_{jk}(a|b)$ is the Conditional probability, eq.(\\ref{eq:eq8}).\nWe then have,\n\\begin{equation}\n (KL)_{jk}(b) = \\sum_a \\frac{P_{jk}(a,b)}{p_k(b)}\\log\\left(\\frac{P_{jk}\n (a,b)}{p_j(a)p_k(b)}\\right) \\label{eq:eq45}\n\\end{equation}\n\n\\noindent and the $M_{jk}$ mutual information measure will be given by\n\\begin{equation}\n M_{jk} = \\sum_b p_k(b)(KL)_{jk}(b) = \\sum_a\\sum_b P_{jk}(a,b)\\log\\left(\n \\frac{P_{jk}(a,b)}{p_j(a)p_k(b)}\\right) \\label{eq:eq46}\n\\end{equation}\nwhich is the same as eq.(\\ref{eq:eq42}), q.e.d.\n\nAs the last topic of this section, we now introduce the concept of\nInformation Distance and we then derive the Jaccard Entropy measure as\nan obvious consequence. Let us write:\n\\begin{equation}\n d_{jk}(r,s) = H_{jk}(r,s) - M_{jk}(r,s) \\label{eq:eq47}\n\\end{equation}\nSince we are working with Entropy measures, we have to satisfy the\nnon-negativeness criteria:\n\\begin{equation}\n H_{jk}(r,s) \\geq 0 \\,;\\quad M_{jk}(r,s) \\geq 0 \\,;\\quad H_{jk}(r,s)\n -M_{jk}(r,s) \\geq 0 \\label{eq:eq48}\n\\end{equation}\nThis means that by satisfying the inequalities (\\ref{eq:eq48}), restrictions\non the $r$, $s$ parameters should be discovered and considered for\nthe description of the protein databases by Entropy measures like $H_{jk}(r,s)$.\n\nFrom inequalities (\\ref{eq:eq48}), we can write,\n\\begin{equation}\n 0 \\leq d_{jk}(r,s) = H_{jk}(r,s) - M_{jk}(r,s) \\leq H_{jk}(r,s)\n \\label{eq:eq49}\n\\end{equation}\nand\n\\begin{equation}\n 0 \\leq J_{jk}(r,s) \\leq 1 \\label{eq:eq50}\n\\end{equation}\nwhere\n\\begin{equation}\n J_{jk}(r,s) = 1-\\frac{M_{jk}(r,s)}{H_{jk}(r,s)} \\label{eq:eq51}\n\\end{equation}\n\n\\noindent is the normalized Jaccard Entropy Measure as obtained from the\nnormalized Information Distance. We then give below the results of checking\nthe inequalities (\\ref{eq:eq48}) for some families of the Pfam database. We\nshall take the limit $r \\to s$ and we work with the corresponding\none-parameter Entropy measures: $H_j(s)$, $H_{jk}(s)$, $M_{jk}(s)$,\n$J_{jk}(s)$. We then have to check:\n\\begin{align*}\n H_{jk}(s) \\geq 0 \\,&,\\quad M_{jk}(s) \\geq 0 \\,, \\quad H_{jk}(s)-M_{jk}(s)\n \\geq 0 \\,, \\\\\n &0 \\leq J_{jk}(s) = 1 - \\frac{M_{jk}(s)}{H_{jk}(s)} \\leq 1\n\\end{align*}\n\n\\begin{table}[H]\n \\begin{center}\n  \\caption{\\small Study of the non-negativeness of $H_{jk}(s)$, $M_{jk}(s)$\n  and $d_{jk}(s)$ values for the protein family PF06850. \\label{tab1}}\n  \\begin{tabular}{|c|c|c|c|}\n   \\hline\n   $\\mathbf{s}$ & $\\mathbf{H_{jk}(s)}$ & $\\mathbf{M_{jk}(s)}$ &\n   $\\mathbf{d_{jk}(s)}$ \\\\\n   \\hline\n   $0.1$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $0.3$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $0.5$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $0.7$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $0.9$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $1.0$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $1.2$ & $0$ & $718$ & $16$ \\\\\n   \\hline\n   $1.5$ & $0$ & $1708$ & $38$ \\\\\n   \\hline\n   $1.7$ & $0$ & $2351$ & $61$ \\\\\n   \\hline\n   $1.9$ & $0$ & $2898$ & $192$ \\\\\n   \\hline\n   $2.0$ & $0$ & $3139$ & $309$ \\\\\n   \\hline\n  \\end{tabular}\n \\end{center}\n\\end{table}\n\n\\noindent The $s$-values corresponding to negative $M_{jk}(s)$ values do not\nlead to a useful characterization of the Jaccard Entropy measure according\nto the inequality on eq.(\\ref{eq:eq49}) which is violated in this case and\nthese $s$-values will not be taken into consideration. Other studies of the\nEntropy values and specially those of the behaviour of the association of\nentropies, will give additional restrictions on the feasible $s$-range. The\nscope of the present work does not allow an intensive study of these\ntechniques of entropy association \\cite{mondaini2} which will then appear on\na forthcoming contribution.\n\n\n\\begin{table}[!ht]\n \\begin{center}\n  \\caption{\\small Study of the non-negativeness of $H_{jk}(s)$, $M_{jk}(s)$\n  and $d_{jk}(s)$ values for the protein family PF00135. \\label{tab2}}\n  \\begin{tabular}{|c|c|c|c|}\n   \\hline\n   $\\mathbf{s}$ & $\\mathbf{H_{jk}(s)}$ & $\\mathbf{M_{jk}(s)}$ &\n   $\\mathbf{d_{jk}(s)}$ \\\\\n   \\hline\n   $0.1$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $0.3$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $0.5$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $0.7$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $0.9$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $1.0$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $1.2$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $1.5$ & $0$ & $0$ & $467$ \\\\\n   \\hline\n   $1.7$ & $0$ & $0$ & $14509$ \\\\\n   \\hline\n   $1.9$ & $0$ & $0$ & $19026$ \\\\\n   \\hline\n   $2.0$ & $0$ & $0$ & $19451$ \\\\\n   \\hline\n  \\end{tabular}\n \\end{center}\n\\end{table}\n\\begin{table}[!hbt]\n \\begin{center}\n  \\caption{\\small Study of the non-negativeness of $H_{jk}(s)$, $M_{jk}(s)$\n  and $d_{jk}(s)$ values for the protein family PF00005. \\label{tab3}}\n  \\begin{tabular}{|c|c|c|c|}\n   \\hline\n   $\\mathbf{s}$ & $\\mathbf{H_{jk}(s)}$ & $\\mathbf{M_{jk}(s)}$ &\n   $\\mathbf{d_{jk}(s)}$ \\\\\n   \\hline\n   $0.1$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $0.3$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $0.5$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $0.7$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $0.9$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $1.0$ & $0$ & $0$ & $0$ \\\\\n   \\hline\n   $1.2$ & $0$ & $8$ & $5$ \\\\\n   \\hline\n   $1.5$ & $0$ & $33$ & $4741$ \\\\\n   \\hline\n   $1.7$ & $0$ & $55$ & $9679$ \\\\\n   \\hline\n   $1.9$ & $0$ & $65$ & $12442$ \\\\\n   \\hline\n   $2.0$ & $0$ & $69$ & $13203$ \\\\\n   \\hline\n  \\end{tabular}\n \\end{center}\n\\end{table}\n\nThe results on the previous three tables will clarify the idea of\nrestriction of the $s$-values of entropy measures for obtaining a\nsound classification of families and clans on the Pfam database.\nWe now announce that the non-negativeness of the values of\n$H_{jk}(s)$, $M_{jk}(s)$ and $d_{jk}(s)$ is actually guaranteed if\nwe restrict to $s \\leq 1$ for all 1069 families which are classified\ninto 68 clans and already characterized at section 2. In figures\n\\ref{fig2}, \\ref{fig3}, \\ref{fig4}, we present the histograms of the\nJaccard Entropy measures for some $s \\leq 1$ values of the\n$s$-parameter.\n\\begin{figure}[p]\n \\centering\n \\includegraphics[width=1\\linewidth]{figure2}\n \\caption{\\small Histograms of Jaccard Entropy for family PF06850.\n \\label{fig2}}\n\\end{figure}\n\\begin{figure}[p]\n \\centering\n \\includegraphics[width=1\\linewidth]{figure3}\n \\caption{\\small Histograms of Jaccard Entropy for family PF00135.\n \\label{fig3}}\n\\end{figure}\n\\begin{figure}[p]\n \\centering\n \\includegraphics[width=1\\linewidth]{figure4}\n \\caption{\\small Histograms of Jaccard Entropy for family PF00005.\n \\label{fig4}}\n\\end{figure}\n\n\\newpage\n\nWe also present the curves corresponding to the Average Jaccard Entropy\nMeasure (formula) for 09 families, a well-posed measure, with the\nrestriction $s \\leq 1$, which is given by\n\\begin{equation*}\n J(s,f) = \\frac{2}{n(n-1)}\\sum_j\\sum_k J_{jk}(s,f)\n\\end{equation*}\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=1\\linewidth]{figure5}\n \\caption{\\small Curves of the Average Jaccard Entropy measures for families\n PF00005, PF05673, PF13481, PF00135, PF06850, PF11339, PF02388, PF09924,\n PF13718. \\label{fig5}}\n\\end{figure}\n\n\\section{A First Assessment of Protein Databases with Entropy Measures}\nAs a motivation for future research to be developed in sections 6, 7, we\nnow introduce the first application of the formulae derived on the previous\nsections in terms of a naive analysis of averages and standard deviations of\nEntropy measure distributions. This will be also the first attempt at\nclassifying the distribution of amino acids in a generic protein database.\nA robust approach to this research topic will be introduced and intensively\nanalyzed on sections 6, 7 with the introduction of ANOVA statistics and the\ncorresponding Hypothesis testing.\n\nWe then consider a Clan with \\textbf{\\emph{F}} families. The Havrda-Charvat\nentropy measure associated to a pair of columns on the representative\n$m \\times n$ array of each family with a specified value of the\n\\textbf{\\emph{s}} parameter is given by\n\\begin{equation}\n H_{jk}(s;f) = -\\frac{1}{1-s} \\left(1-\\sum_a\\sum_b\\big(P_{jk}(a,b;f)\n \\big)^s\\right) \\label{eq:eq52}\n\\end{equation}\n\n\\noindent We can then define an average of these entropy measures for\neach family by\n\\begin{equation}\n \\langle H(s;f) \\rangle = \\frac{2}{n(n-1)} \\sum_j\\sum_k H_{jk}(s;f)\n \\label{eq:eq53}\n\\end{equation}\n\n\\noindent We also consider the average value of the averages over the\nset of \\emph{F} families:\n\\begin{equation}\n \\langle H(s) \\rangle_F = \\frac{1}{F} \\sum_{f=1}^F \\langle H(s;f) \\rangle\n \\label{eq:eq54}\n\\end{equation}\n\n\\noindent The Standard deviation of the Entropy measures $H_{jk}(s;f)$\nwith relation to the given average in eq.(\\ref{eq:eq53}) can be written\nas:\n\\begin{equation}\n \\sigma(s;f) = \\left(\\frac{1}{\\frac{n(n-1)}{2}-1}\\sum_j\\sum_k\\big(\n H_{jk}(s;f)-\\langle H(s;f) \\rangle\\big)^2\\right)^{1\/2} \\label{eq:eq55}\n\\end{equation}\n\n\\noindent and finally, the Standard deviation of the average\n$\\langle H(s;f)\\rangle$ with respect to the average\n$\\langle H(s)\\rangle_F$:\n\\begin{equation}\n \\sigma_F(s) = \\left(\\frac{1}{F-1} \\sum_{f=1}^F \\big(\\langle H(s;f)\\rangle -\n \\langle H(s)\\rangle_F\\big)^2\\right)^{1\/2} \\label{eq:eq56}\n\\end{equation}\n\nWe present in figs.\\ref{fig6}, \\ref{fig7} below the diagrams corresponding to\nformulae (\\ref{eq:eq53}) and (\\ref{eq:eq55}). We should stress that only\nClans with a minimum of five families are considered.\n\\begin{figure}[p]\n \\centering\n \\includegraphics[width=0.8\\textwidth]{figure6}\n \\caption{\\small The Average values of the Havrda-Charvat Entropy measures\n for the families of a selected set of Clans and eleven values of the\n $s$-parameter, eq.(\\ref{eq:eq53}). \\label{fig6}}\n\\end{figure}\n\n\n\\begin{figure}[p]\n \\centering\n \\includegraphics[width=0.8\\linewidth]{figure7}\n \\caption{\\small The standard deviation of the Havrda-Charvat Entropy\n measures with relation to the averages of these entropies for each family\n and eleven values of the  $s$-parameter, eq.(\\ref{eq:eq55}). \\label{fig7}}\n\\end{figure}\n\n\\newpage\n\nWe now present the values of $\\langle H(s)\\rangle_F$ and $\\sigma_F(s)$ for\na selected number of Clans and eleven values of the $s$-parameter, according\nto eqs.(\\ref{eq:eq54}) and (\\ref{eq:eq56}).\n\\begin{table}[H]\n \\begin{center}\n  \\caption{\\small The average values and the standard deviation of the Average\n  Havrda-Charvat Entropy measures for eleven values of the $s$-parameter and\n  a selected set of 8 Clans. \\label{tab4}}\n  \\begingroup\n  \\everymath{\\scriptstyle}\n  \\footnotesize\n   \\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n    \\hline\n    \\multicolumn{9}{|c|}{Clans from Pfam 27.0 --- Havrda-Charvat\n    Entropies} \\\\\n    \\hline\n    Clan number & $s$ & $\\langle H(s)\\rangle_F$ & $\\sigma(s)_F$ & {} &\n    Clan number & $s$ & $\\langle H(s)\\rangle_F$ & $\\sigma(s)_F$ \\\\\n    \\cline{1-4} \\cline{6-9}\n    {} & $0.1$ & $44.212$ & $6.396$ & {} &\n    {} & $0.1$ & $43.001$ & $4.667$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.3$ & $23.375$ & $3.057$ & {} &\n    {} & $0.3$ & $22.851$ & $2.295$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.5$ & $12.992$ & $1.492$ & {} &\n    {} & $0.5$ & $12.765$ & $1.161$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.7$ & $7.645$ & $0.746$ & {} &\n    {} & $0.7$ & $7.546$ & $0.605$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.9$ & $4.784$ & $0.384$ & {} &\n    {} & $0.9$ & $4.741$ & $0.326$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    CL0020 & $1.0$ & $3.875$ & $0.278$ & {} &\n    CL0123 & $1.0$ & $3.846$ & $0.242$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    (38 families) & $1.2$ & $2.661$ & $0.150$ & {} &\n    (06 families) & $1.2$ & $2.648$ & $0.137$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.5$ & $1.680$ & $0.063$ & {} &\n    {} & $1.5$ & $1.676$ & $0.062$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.7$ & $1.311$ & $0.038$ & {} &\n    {} & $1.7$ & $1.309$ & $0.038$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.9$ & $1.062$ & $0.023$ & {} &\n    {} & $1.9$ & $1.061$ & $0.024$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $2.0$ & $0.967$ & $0.018$ & {} &\n    {} & $2.0$ & $0.966$ & $0.019$ \\\\\n    %\n    \\cline{1-4} \\cline{6-9}\n    {} & $0.1$ & $39.235$ & $10.175$ & {} &\n    {} & $0.1$ & $46.084$ & $6.790$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.3$ & $20.908$ & $4.984$ & {} &\n    {} & $0.3$ & $24.260$ & $3.224$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.5$ & $11.733$ & $2.510$ & {} &\n    {} & $0.5$ & $13.417$ & $1.561$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.7$ & $6.982$ & $1.305$ & {} &\n    {} & $0.7$ & $7.853$ & $0.772$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.9$ & $4.422$ & $0.704$ & {} &\n    {} & $0.9$ & $4.886$ & $0.392$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    CL0023 & $1.0$ & $3.604$ & $0.525$ & {} &\n    CL0186 & $1.0$ & $3.949$ & $0.282$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    (119 families) & $1.2$ & $2.504$ & $0.302$ & {} &\n    (29 families) & $1.2$ & $2.701$ & $0.149$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.5$ & $1.606$ & $0.144$ & {} &\n    {} & $1.5$ & $1.696$ & $0.060$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.7$ & $1.263$ & $0.093$ & {} &\n    {} & $1.7$ & $1.320$ & $0.035$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.9$ & $1.030$ & $0.063$ & {} &\n    {} & $1.9$ & $1.067$ & $0.020$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $2.0$ & $0.940$ & $0.053$ & {} &\n    {} & $2.0$ & $0.970$ & $0.016$ \\\\\n    \\cline{1-4} \\cline{6-9}\n    %\n    {} & $0.1$ & $44.906$ & $7.996$ & {} &\n    {} & $0.1$ & $42.862$ & $4.566$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.3$ & $23.671$ & $3.906$ & {} &\n    {} & $0.3$ & $22.791$ & $2.190$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.5$ & $13.114$ & $1.961$ & {} &\n    {} & $0.5$ & $12.741$ & $1.072$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.7$ & $7.692$ & $1.015$ & {} &\n    {} & $0.7$ & $7.538$ & $0.537$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.9$ & $4.799$ & $0.545$ & {} &\n    {} & $0.9$ & $4.739$ & $0.275$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    CL0028 & $1.0$ & $3.882$ & $0.406$ & {} &\n    CL0192 & $1.0$ & $3.847$ & $0.199$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    (41 families) & $1.2$ & $2.660$ & $0.232$ & {} &\n    (26 families) & $1.2$ & $2.650$ & $0.107$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.5$ & $1.676$ & $0.109$ & {} &\n    {} & $1.5$ & $1.678$ & $0.044$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.7$ & $1.307$ & $0.070$ & {} &\n    {} & $1.7$ & $1.311$ & $0.026$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.9$ & $1.058$ & $0.047$ & {} &\n    {} & $1.9$ & $1.062$ & $0.015$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $2.0$ & $0.963$ & $0.039$ & {} &\n    {} & $2.0$ & $0.967$ & $0.012$ \\\\\n    %\n    \\cline{1-4} \\cline{6-9}\n    {} & $0.1$ & $42.312$ & $8.023$ & {} &\n    {} & $0.1$ & $43.251$ & $7.469$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.3$ & $22.454$ & $3.857$ & {} &\n    {} & $0.3$ & $22.905$ & $3.564$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.5$ & $12.534$ & $1.896$ & {} &\n    {} & $0.5$ & $12.757$ & $1.734$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.7$ & $7.411$ & $0.956$ & {} &\n    {} & $0.7$ & $7.524$ & $0.863$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.9$ & $4.660$ & $0.496$ & {} &\n    {} & $0.9$ & $4.719$ & $0.440$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    CL0063 & $1.0$ & $3.784$ & $0.362$ & {} &\n    CL0236 & $1.0$ & $3.828$ & $0.317$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    (92 families) & $1.2$ & $2.611$ & $0.198$ & {} &\n    (21 families) & $1.2$ & $2.636$ & $0.169$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.5$ & $1.658$ & $0.086$ & {} &\n    {} & $1.5$ & $1.669$ & $0.069$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.7$ & $1.297$ & $0.051$ & {} &\n    {} & $1.7$ & $1.304$ & $0.040$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.9$ & $1.053$ & $0.032$ & {} &\n    {} & $1.9$ & $1.058$ & $0.024$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $2.0$ & $0.960$ & $0.026$ & {} &\n    {} & $2.0$ & $0.964$ & $0.019$ \\\\\n    \\cline{1-4} \\cline{6-9}\n   \\end{tabular}\n   \\endgroup\n  \\end{center}\n\\end{table}\n\n\nIn figs.\\ref{fig8}a and \\ref{fig8}b below, we present the graphs\ncorresponding to Table \\ref{tab4}. These results just point out a more\nelaborate formulation of the problem.\n\n\\begin{figure}[!hbt]\n \\centering\n \\includegraphics[width=1\\linewidth]{figure8}\n \\caption{\\small (a) The average values of Havrda-Charvat Entropies for a\n set of 08 Clans. (b) The Standard Deviation of the averages for\n Havrda-Charvat Entropies for a set of 08 Clans.\\label{fig8}}\n\\end{figure}\n\nWe now proceed to analyze a proposal (naive) for testing the robustness of\nthe Clan concept. We will check if Pseudo-Clans which have the same number\nof families (a minimum of 05 families) of the corresponding Clans will have\nessentially different values of $\\langle H(s)\\rangle_F$ and $\\sigma_F(s)$.\nThe families to be associated with a Pseudo-Clan are obtained by sorting on\nthe set of 1069 families and by withdrawal of the families already sorted.\nIn Table \\ref{tab5} below we present the values $\\langle H(s)\\rangle_F$ and\n$\\sigma_F(s)$ for the Pseudo-Clans obtained by the procedure described above.\n\nThe Figures 9a, 9b, do correspond to the comparison of data of Table\n\\ref{tab4} (Clans) with those of Table \\ref{tab5} (Pseudo-Clans). Clans are\nin red, Pseudo-Clans in blue.\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=1\\linewidth]{figure9}\n \\caption{\\small (a) The comparison of Clans and Pseudo-Clans average values.\n (b) The comparison of Clans and Pseudo-Clans standard deviation values.\n \\label{fig9}}\n\\end{figure}\n\n\\begin{table}[!ht]\n \\begin{center}\n  \\caption{\\small The average values and the standard deviation of the\n  Average Havrda-Charvat Entropy measures for eleven values of the\n  $s$-parameter and a selected set of 8 Pseudo-Clans. \\label{tab5}}\n  \\begingroup\n  \\everymath{\\scriptstyle}\n  \\footnotesize\n   \\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n    \\hline\n    \\multicolumn{9}{|c|}{Pseudo-Clans \/ Pfam 27.0 --- Havrda-Charvat\n    Entropies} \\\\\n    \\hline\n    Pseudo-Clan & \\multicolumn{1}{c|}{\\multirow{2}{*}{$s$}} &\n    \\multicolumn{1}{c|}{\\multirow{2}{*}{$\\langle H(s)\\rangle_F$}} &\n    \\multicolumn{1}{c|}{\\multirow{2}{*}{$\\sigma(s)_F$}} &\n    \\multicolumn{1}{c|}{} & Pseudo-Clan &\n    \\multicolumn{1}{c|}{\\multirow{2}{*}{$s$}} &\n    \\multicolumn{1}{c|}{\\multirow{2}{*}{$\\langle H(s)\\rangle_F$}} &\n    \\multicolumn{1}{c|}{\\multirow{2}{*}{$\\sigma(s)_F$}} \\\\\n    number & \\multicolumn{1}{c|}{} & \\multicolumn{1}{c|}{} &\n    \\multicolumn{1}{c|}{} & \\multicolumn{1}{c|}{} & number &\n    \\multicolumn{1}{c|}{} & \\multicolumn{1}{c|}{} &\n    \\multicolumn{1}{c|}{} \\\\\n    \\cline{1-4} \\cline{6-9}\n    {} & $0.1$ & $42.381$ & $8.224$ & {} &\n    {} & $0.1$ & $42.042$ & $4.484$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.3$ & $22.480$ & $3.979$ & {} &\n    {} & $0.3$ & $22.359$ & $2.175$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.5$ & $12.542$ & $1.972$ & {} &\n    {} & $0.5$ & $12.508$ & $1.082$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.7$ & $7.410$ & $1.005$ & {} &\n    {} & $0.7$ & $7.409$ & $0.555$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.9$ & $4.657$ & $0.528$ & {} &\n    {} & $0.9$ & $4.667$ & $0.293$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    PCL0020 & $1.0$ & $3.781$ & $0.389$ & {} &\n    PCL0123 & $1.0$ & $3.791$ & $0.216$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    (38 families) & $1.2$ & $2.607$ & $0.216$ & {} &\n    (06 families) & $1.2$ & $2.618$ & $0.120$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.5$ & $1.655$ & $0.096$ & {} &\n    {} & $1.5$ & $1.663$ & $0.053$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.7$ & $1.295$ & $0.059$ & {} &\n    {} & $1.7$ & $1.301$ & $0.032$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.9$ & $1.051$ & $0.037$ & {} &\n    {} & $1.9$ & $1.056$ & $0.020$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $2.0$ & $0.958$ & $0.030$ & {} &\n    {} & $2.0$ & $0.962$ & $0.016$ \\\\\n    %\n    \\cline{1-4} \\cline{6-9}\n    {} & $0.1$ & $41.023$ & $9.007$ & {} &\n    {} & $0.1$ & $40.888$ & $7.719$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.3$ & $21.825$ & $4.360$ & {} &\n    {} & $0.3$ & $21.790$ & $3.768$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.5$ & $12.219$ & $2.163$ & {} &\n    {} & $0.5$ & $12.220$ & $1.891$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.7$ & $7.247$ & $1.104$ & {} &\n    {} & $0.7$ & $7.258$ & $0.981$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.9$ & $4.573$ & $0.582$ & {} &\n    {} & $0.9$ & $4.585$ & $0.529$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    PCL0023 & $1.0$ & $3.714$ & $0.427$ & {} &\n    PCL0186 & $1.0$ & $3.730$ & $0.394$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    (119 families) & $1.2$ & $2.573$ & $0.240$ & {} &\n    (29 families) & $1.2$ & $2.582$ & $0.227$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.5$ & $1.640$ & $0.109$ & {} &\n    {} & $1.5$ & $1.646$ & $0.109$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.7$ & $1.286$ & $0.068$ & {} &\n    {} & $1.7$ & $1.290$ & $0.071$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.9$ & $1.046$ & $0.044$ & {} &\n    {} & $1.9$ & $1.048$ & $0.049$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $2.0$ & $0.954$ & $0.037$ & {} &\n    {} & $2.0$ & $0.956$ & $0.041$ \\\\\n    \\cline{1-4} \\cline{6-9}\n    %\n    {} & $0.1$ & $42.716$ & $7.435$ & {} &\n    {} & $0.1$ & $42.756$ & $9.361$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.3$ & $23.658$ & $3.573$ & {} &\n    {} & $0.3$ & $22.666$ & $4.535$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.5$ & $12.641$ & $1.756$ & {} &\n    {} & $0.5$ & $12.635$ & $2.253$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.7$ & $7.468$ & $0.886$ & {} &\n    {} & $0.7$ & $7.458$ & $1.151$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.9$ & $4.692$ & $0.460$ & {} &\n    {} & $0.9$ & $4.681$ & $0.608$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    PCL0028 & $1.0$ & $3.808$ & $0.335$ & {} &\n    PCL0192 & $1.0$ & $3.797$ & $0.448$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    (41 families) & $1.2$ & $2.625$ & $0.183$ & {} &\n    (26 families) & $1.2$ & $2.615$ & $0.250$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.5$ & $1.664$ & $0.079$ & {} &\n    {} & $1.5$ & $1.657$ & $0.113$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.7$ & $1.302$ & $0.048$ & {} &\n    {} & $1.7$ & $1.296$ & $0.070$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.9$ & $1.056$ & $0.030$ & {} &\n    {} & $1.9$ & $1.051$ & $0.046$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $2.0$ & $0.962$ & $0.024$ & {} &\n    {} & $2.0$ & $0.958$ & $0.037$ \\\\\n    %\n    \\cline{1-4} \\cline{6-9}\n    {} & $0.1$ & $41.870$ & $8.134$ & {} &\n    {} & $0.1$ & $41.551$ & $7.476$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.3$ & $22.252$ & $3.914$ & {} &\n    {} & $0.3$ & $22.090$ & $3.591$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.5$ & $12.440$ & $1.929$ & {} &\n    {} & $0.5$ & $12.357$ & $1.763$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.7$ & $7.366$ & $0.977$ & {} &\n    {} & $0.7$ & $7.322$ & $0.887$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $0.9$ & $4.638$ & $0.510$ & {} &\n    {} & $0.9$ & $4.614$ & $0.459$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    PCL0063 & $1.0$ & $3.771$ & $0.375$ & {} &\n    PCL0236 & $1.0$ & $3.752$ & $0.334$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    (92 families) & $1.2$ & $2.603$ & $0.207$ & {} &\n    (21 families) & $1.2$ & $2.595$ & $0.181$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.5$ & $1.654$ & $0.092$ & {} &\n    {} & $1.5$ & $1.652$ & $0.077$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.7$ & $1.295$ & $0.057$ & {} &\n    {} & $1.7$ & $1.294$ & $0.046$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $1.9$ & $1.052$ & $0.036$ & {} &\n    {} & $1.9$ & $1.051$ & $0.028$ \\\\\n    \\cline{2-4} \\cline{7-9}\n    {} & $2.0$ & $0.959$ & $0.030$ & {} &\n    {} & $2.0$ & $0.959$ & $0.022$ \\\\\n    \\cline{1-4} \\cline{6-9}\n   \\end{tabular}\n   \\endgroup\n  \\end{center}\n\\end{table}\n\nFrom the figures and tables above we can see that the region $s \\leq 1$ leads\nto a better characterization of the Entropy measures distributions on protein\ndatabases.\n\nFor completeness, we list some useful formulae obtained from\neqs.(\\ref{eq:eq53}), (\\ref{eq:eq54}), (\\ref{eq:eq56}) which help to predict\nthe profile of the curves above:\n\\begin{equation}\n \\langle H(s;f) \\rangle = -\\frac{1}{1-s} \\left(1-\\frac{2}{n(n-1)}\n \\sum_{a,b,j,k} e^{-s \\lvert \\log P_{jk}(a,b;f)\\rvert}\\right) \\label{eq:eq57}\n\\end{equation}\n\\begin{equation}\n \\langle H(s) \\rangle_F = -\\frac{1}{1-s} \\left(1-\\frac{2}{Fn(n-1)}\n \\sum_{f,a,b,j,k} e^{-s \\lvert \\log P_{jk}(a,b;f)\\rvert} \\right)\n \\label{eq:eq58}\n\\end{equation}\n\\begin{equation}\n \\sigma_F(s) = \\frac{2(F-1)^{1\/2}}{Fn(n-1)}\\left(\\sum_{f=1}^F\\left(\\sum_{a,b,\n j,k} e^{-s \\lvert \\log P_{jk}(a,b;f)\\rvert}\\right)^2\\right)^{1\/2}\n \\label{eq:eq59}\n\\end{equation}\n\n\\section{The treatment of data with the Maple Computing system and its\ninadequacy for calculating Joint probabilities of occurrence. Alternative\nsystems.}\nIn this section, we specifically study the performance of the Maple system and\nan example of alternative computing system, the Perl system, for calculating\nthe simple and joint probabilities of occurrences of amino acids. We also use\nthese two systems for calculating 19 $s$-power values of these probabilities.\nWe now select the family PF06850 in order to get an idea of the CPU and real\ntimes which are necessary for calculating the probabilities and their powers\nfor the set of 1069 families. We start the calculation by adopting the Maple\nsystem version 18. There are some comments to be made on the construction of\na computational code for calculating joint probabilities. This will be done\nin detail at the end of CPU and real times for the calculation of the simple\nand joint probabilities by using the developed code. The table below will\nrepeat the times for calculating $200 \\times 20 = 4 \\times 10^3$ and $200\n\\times \\frac{200-1}{2} \\times 20 \\times 20 = 7.96 \\times 10^6$ of simple and\njoint probability values, respectively, for the PF06850 Pfam family.\n\n\\begin{table}[!ht]\n \\begin{center}\n  \\caption{\\small CPU time and real times for the calculation of the simple and\n  joint probabilities of occurrence associated with the protein family\n  PF06850. \\label{tab6}}\n  \\begin{tabular}{|c|c|c|}\n   \\hline\n   Maple System, version 18 & $t_{CPU}$ (sec) & $t_R$ (sec) \\\\\n   \\hline\n   Simple probabilities & $0.527$ & $0.530$ \\\\\n   \\hline\n   Joint probabilities & $5073.049$ & $4650.697$ \\\\\n   \\hline\n  \\end{tabular}\n \\end{center}\n\\end{table}\n\nAfter calculating all values of the probabilities $p_j(a)$ and $P_{jk}(a,b)$,\nwe can proceed to evaluate the powers $\\big(p_j(a)\\big)^s$ and $\\big(P_jk(a,\nb)\\big)^s$ for 19 $s$-values. Our aim will be to use these values for\ncalculating the Entropy Measures according to eqs.(\\ref{eq:eq30}),\n(\\ref{eq:eq31}). It should be noticed that the values of $p_j(a)$ and\n$P_{jk}(a,b)$, have to be calculated only once by using a specific\ncomputational code already referred on this work. Nevertheless, the use of\nthe code for calculating the joint probabilities associated to 1069 protein\nfamilies is the hardest of all calculations to be undertaken and it takes\ntoo much time. These probabilities once calculated should be grouped in sets\nof 400 values each corresponding to a pair of columns $j$, $k$ among the\n$\\frac{n(n-1)}{2}$ feasible ones and the calculating of\nentropy value associated to this pair of columns $j$, $k$.\nGiven a $s$-value and after calculating the entropy of this\nfirst pair $H_{1\\,2}$ as a function of the 400 variables\n$\\big(P_{jk}(a,b)\\big)^s$, $j \\neq 1$, $k \\neq 2$ and he\/she will proceeds to\ncalculate again all values of $\\frac{n(n-1)}{2} \\times (20)^2$ in order\nto extract another value of joint probability for calculating the\ncorresponding entropy value. This seems to be associated to\nthe unknowing of the concepts of a function of several variables,\nunfortunately. After circumventing these mistakes coming from a bad\neducational formation, we succeed at keeping all calculated values of the\nprobabilities and we then proceed to the calculation of the powers $\\big(\np_j(a)\\big)^s$, $\\big(P_{jk}(a,b)\\big)^s$ of these values and the\ncorresponding entropy measures. In tables \\ref{tab7}, \\ref{tab8}, \\ref{tab9},\n\\ref{tab10} below, we report all these calculations for 19 values of the\n$s$-parameter.\n\n\\begin{table}[!ht]\n \\begin{center}\n  \\caption{\\small CPU and real times for the calculation of 19 $s$-powers\n  of simple probabilities of occurrence associated with the protein family\n  PF06850. \\label{tab7}}\n  \\begin{tabular}{|c|c|c|}\n   \\hline\n   \\multicolumn{3}{|c|}{Maple System, version 18,} \\\\\n   \\multicolumn{3}{|c|}{$s$-powers of probability $\\big(p_j(a)\\big)^s$}\\\\\n   \\hline\n   $s$ & $t_{CPU}$ (sec) & $t_R$ (sec) \\\\\n   \\hline\n   $0.1$ & $0.263$ & $0.358$ \\\\\n   \\hline\n   $0.2$ & $0.137$ & $0.145$ \\\\\n   \\hline\n   $0.3$ & $0.268$ & $0.277$ \\\\\n   \\hline\n   $0.4$ & $0.139$ & $0.153$ \\\\\n   \\hline\n   $0.5$ & $0.240$ & $0.219$ \\\\\n   \\hline\n   $0.6$ & $0.144$ & $0.157$ \\\\\n   \\hline\n   $0.7$ & $0.276$ & $0.254$ \\\\\n   \\hline\n   $0.8$ & $0.144$ & $0.157$ \\\\\n   \\hline\n   $0.9$ & $0.264$ & $0.235$ \\\\\n   \\hline\n   $1.0$ & $0.088$\/$0.151$ & $0.095$\/$0.307$ \\\\\n   \\hline\n   $2.0$ & $0.153$ & $0.095$ \\\\\n   \\hline\n   $3.0$ & $0.128$ & $0.131$ \\\\\n   \\hline\n   $4.0$ & $0.148$ & $0.141$ \\\\\n   \\hline\n   $5.0$ & $0.096$ & $0.144$ \\\\\n   \\hline\n   $6.0$ & $0.148$ & $0.167$ \\\\\n   \\hline\n   $7.0$ & $0.148$ & $0.155$ \\\\\n   \\hline\n   $8.0$ & $0.181$ & $0.094$ \\\\\n   \\hline\n   $9.0$ & $0.104$ & $0.092$ \\\\\n   \\hline\n   $10.0$ & $0.104$ & $0.100$ \\\\\n   \\hline\n   Total & $3.173$ & $3.164$ \\\\\n   \\hline\n  \\end{tabular}\n \\end{center}\n\\end{table}\n\n\\begin{table}[!ht]\n \\begin{center}\n  \\caption{\\small CPU and real times for the calculation of 19 $s$-powers\n  of joint probabilities of occurrence associated with the protein family\n  PF06850. \\label{tab8}}\n  \\begin{tabular}{|c|c|c|}\n   \\hline\n   \\multicolumn{3}{|c|}{Maple System, version 18,} \\\\\n   \\multicolumn{3}{|c|}{$s$-powers of probability $\\big(P_{jk}(a,b)\\big)^s$}\n   \\\\\n   \\hline\n   $s$ & $t_{CPU}$ (sec) & $t_R$ (sec) \\\\\n   \\hline\n   $0.1$ & $390.432$ & $206.646$ \\\\\n   \\hline\n   $0.2$ & $382.887$ & $202.282$ \\\\\n   \\hline\n   $0.3$ & $401.269$ & $210.791$ \\\\\n   \\hline\n   $0.4$ & $416.168$ & $216.993$ \\\\\n   \\hline\n   $0.5$ & $427.572$ & $221.541$ \\\\\n   \\hline\n   $0.6$ & $430.604$ & $223.227$ \\\\\n   \\hline\n   $0.7$ & $421.904$ & $218.484$ \\\\\n   \\hline\n   $0.8$ & $434.888$ & $224.267$ \\\\\n   \\hline\n   $0.9$ & $431.948$ & $223.023$ \\\\\n   \\hline\n   $1.0$ & $442.933$\/$482.612$ & $224.731$\/$259.301$ \\\\\n   \\hline\n   $2.0$ & $176.212$ & $147.455$ \\\\\n   \\hline\n   $3.0$ & $234.100$ & $174.853$ \\\\\n   \\hline\n   $4.0$ & $289.184$ & $181.552$ \\\\\n   \\hline\n   $5.0$ & $327.740$ & $178.117$ \\\\\n   \\hline\n   $6.0$ & $334.800$ & $194.691$ \\\\\n   \\hline\n   $7.0$ & $349.064$ & $195.258$ \\\\\n   \\hline\n   $8.0$ & $361.304$ & $195.437$ \\\\\n   \\hline\n   $9.0$ & $386.217$ & $197.150$ \\\\\n   \\hline\n   $10.0$ & $397.276$ & $197.868$ \\\\\n   \\hline\n   Total & $7036.502$ & $3834.366$ \\\\\n   \\hline\n  \\end{tabular}\n \\end{center}\n\\end{table}\n\nThe last row in tables \\ref{tab7}, \\ref{tab8}, includes the times necessary\nfor calculating the probabilities of table \\ref{tab6}.\n\nWe are then able to proceed to the calculation of the corresponding\nHavrda-Charvat entropy measures, $H_j(s)$, $H_{jk}(s)$: The results for 19\n$s$-values are given in tables \\ref{tab9}, \\ref{tab10} below.\n\n\\begin{table}[!ht]\n \\begin{center}\n  \\caption{\\small CPU and real times for the calculation of the Entropy measures\n  $H_j(s)$ for the protein family PF06850. \\label{tab9}}\n  \\begin{tabular}{|c|c|c|}\n   \\hline\n   \\multicolumn{3}{|c|}{Maple System, version 18,} \\\\\n   \\multicolumn{3}{|c|}{Entropy Measures $H_j(s)$}\\\\\n   \\hline\n   $s$ & $t_{CPU}$ (sec) & $t_R$ (sec) \\\\\n   \\hline\n   $0.1$ & $0.148$ & $0.261$ \\\\\n   \\hline\n   $0.2$ & $0.084$ & $0.153$ \\\\\n   \\hline\n   $0.3$ & $0.120$ & $0.189$ \\\\\n   \\hline\n   $0.4$ & $0.124$ & $0.198$ \\\\\n   \\hline\n   $0.5$ & $0.160$ & $0.299$ \\\\\n   \\hline\n   $0.6$ & $0.092$ & $0.139$ \\\\\n   \\hline\n   $0.7$ & $0.159$ & $0.199$ \\\\\n   \\hline\n   $0.8$ & $0.137$ & $0.175$ \\\\\n   \\hline\n   $0.9$ & $0.120$ & $0.166$ \\\\\n   \\hline\n   $1.0$ & $0.192$\/$0.175$ & $0.339$\/$0.159$ \\\\\n   \\hline\n   $2.0$ & $0.144$ & $0.099$ \\\\\n   \\hline\n   $3.0$ & $0.147$ & $0.105$ \\\\\n   \\hline\n   $4.0$ & $0.084$ & $0.101$ \\\\\n   \\hline\n   $5.0$ & $0.136$ & $0.070$ \\\\\n   \\hline\n   $6.0$ & $0.096$ & $0.119$ \\\\\n   \\hline\n   $7.0$ & $0.115$ & $0.078$ \\\\\n   \\hline\n   $8.0$ & $0.120$ & $0.109$ \\\\\n   \\hline\n   $9.0$ & $0.133$ & $0.080$ \\\\\n   \\hline\n   $10.0$ & $0.132$ & $0.133$ \\\\\n   \\hline\n   Total & $2.443$ & $3.012$ \\\\\n   \\hline\n  \\end{tabular}\n \\end{center}\n\\end{table}\n\n\\begin{table}[!hb]\n \\begin{center}\n  \\caption{\\small CPU and real times for the calculation of the Entropy measures\n  $H_{jk}(s)$ for the protein family PF06850. \\label{tab10}}\n  \\begin{tabular}{|c|c|c|}\n   \\hline\n   \\multicolumn{3}{|c|}{Maple System, version 18,} \\\\\n   \\multicolumn{3}{|c|}{Entropy Measures $H_{jk}(s)$}\\\\\n   \\hline\n   $s$ & $t_{CPU}$ (sec) & $t_R$ (sec) \\\\\n   \\hline\n   $0.1$ & $156.332$ & $133.242$ \\\\\n   \\hline\n   $0.2$ & $160.797$ & $136.706$ \\\\\n   \\hline\n   $0.3$ & $169.024$ & $140.960$ \\\\\n   \\hline\n   $0.4$ & $176.824$ & $147.853$ \\\\\n   \\hline\n   $0.5$ & $184.120$ & $150.163$ \\\\\n   \\hline\n   $0.6$ & $190.304$ & $154.058$ \\\\\n   \\hline\n   $0.7$ & $196.633$ & $157.750$ \\\\\n   \\hline\n   $0.8$ & $205.940$ & $164.101$ \\\\\n   \\hline\n   $0.9$ & $215.559$ & $169.549$ \\\\\n   \\hline\n   $1.0$ & $253.648$\/$124.501$ & $204.634$\/$148.993$ \\\\\n   \\hline\n   $2.0$ & $141.148$ & $184.030$ \\\\\n   \\hline\n   $3.0$ & $158.536$ & $167.173$ \\\\\n   \\hline\n   $4.0$ & $173.136$ & $181.282$ \\\\\n   \\hline\n   $5.0$ & $197.680$ & $238.723$ \\\\\n   \\hline\n   $6.0$ & $215.000$ & $111.476$ \\\\\n   \\hline\n   $7.0$ & $145.257$ & $115.221$ \\\\\n   \\hline\n   $8.0$ & $156.848$ & $122.957$ \\\\\n   \\hline\n   $9.0$ & $157.300$ & $126.233$ \\\\\n   \\hline\n   $10.0$ & $166.399$ & $135.080$ \\\\\n   \\hline\n   Total & $3420.485$ & $2941.791$ \\\\\n   \\hline\n  \\end{tabular}\n \\end{center}\n\\end{table}\n\nThe total time for calculating all the Havrda-Charvat Entropy measure\ncontent of probabilities of occurrence of amino acids on a\nspecific family is given in table \\ref{tab11} below.\n\n\\begin{table}[!hb]\n \\begin{center}\n  \\caption{\\small Total CPU and real times for calculating the Entropy measure\n  content of a family PF06850 and approximations for Grand\n  Total of all sample space. \\label{tab11}}\n  \\begingroup\n  \\everymath{\\scriptstyle}\n  \\small\n  \\begin{tabular}{|c|c|c|}\n   \\hline\n   Maple System, & Entropy Measures &\n   Entropy Measures \\\\\n   version 18 & $H\\!_j(s)$ --- 19 $s$-values & $H\\!_{jk}(s)$ ---\n   19 $s$-values \\\\\n   \\hline\n   Total CPU time & $0.527+3.173+2.443$\n   & $5,073.049+7,036.502+$ \\\\\n   (family PF06850) & $=6.143$ sec & $3,420.485=15,530.036$ sec \\\\\n   \\hline\n   Grand Total CPU time & $6,566.867$ sec\n   & $16,601,608.484$ sec \\\\\n   (1069 families) & $=1.824$ hs & $=192.148$ days \\\\\n   \\hline\n   Total Real time & $0.530+3.164+3.012$\n   & $4,650.697+3,834.366+$ \\\\\n   (family PF06850) & $=6.706$ sec & $2,941.791=11,426.854$ sec \\\\\n   \\hline\n   Grand Total Real time & $7,168.714$ sec\n   & $12,215,306.926$ sec \\\\\n   (1069 families) & $=1.991$ hs & $=141.381$ days \\\\\n   \\hline\n  \\end{tabular}\n  \\endgroup\n \\end{center}\n\\end{table}\n\nFrom inspections of table \\ref{tab11}, we realize that the Total CPU\nand real times for calculating the Havrda-Charvat entropies $H_j(s)$, of\nsimple probabilities of occurrence $p_j(a)$ are obtained by summing up the\ntotal time results from tables \\ref{tab6}, \\ref{tab7} and \\ref{tab9}. For the\nHavrda-Charvat entropies $H_{jk}(s)$, we have to sum up the total times at table\n\\ref{tab6}, \\ref{tab8} and \\ref{tab10}. We take for granted that the times for\ncalculating the Entropy Measure content of each family will not differ too much\nand the results 3rd and 5th rows of table \\ref{tab11} are obtained by multiplying\nby 1069 --- the number of families in the sample space.\n\nThe results of table \\ref{tab11} suggest the inadequacy of the Maple computing\nsystem for analyzing the Entropy measure content of an example of protein database.\nWe have restricted ourselves to operate with usual operating systems, Linux\nor OSX, on laptops. We have also worked with the alternative Perl computing\nsystem. The Maple computing system has an ``array'' structure which is very\neffective for doing calculations which require a knowledge of mathematical\nmethods. On the contrary, the alternative Perl computing system has a\n``hash'' structure as was emphasized in the 2nd section and it operates very\nwell elementary operations with very large numbers. It is essential the\ncomparison of a senior erudite which is largely conversant with a large amount\nof mathematical methods versus a ``genius'' brought to fame by media,\nwho is able only to multiply in a very fast way, numbers of many digits.\n\nIn order to specify the probabilities of computational configurations with\nthe usual desktops and laptops, we list below some of them which have been\nused in the present work. The computing systems were the Maple ($M$) and Perl\n($P$), the operating systems, the Linux ($L$) and Mac OSX ($O$) and the\nstructures: The Array I ($A_I$), Array II ($A_{I\\!I}$) and Hash ($H$) (page 8,\nsection 2). The available computational configurations to undertake the task\nof assessment of protein databases with Entropy measures could be listed as:\n\\begin{enumerate}\n \\item $MLA_I$ --- Maple, Linux, Array I\n \\item $POH$ --- Perl, OSX, Hash\n \\item $PLA_{I\\!I}$ --- Perl, Linux, Array II\n \\item $POA_{I\\!I}$ --- Perl, OSX, Array II\n\\end{enumerate}\n\nThe following table will display a comparison of the CPU and real times for\nthe calculation of 19 $s$-values of the joint probabilities $\\big(P_{jk}(a,\nb)^s\\big)$ for the protein family PF06850 by the four configurations\nnominated above. It should be stressed that we are here comparing the times\nfor calculating the $s$-power with the values of the probabilities\nthemselves previously calculated and kept on a file.\n\n\\begin{sidewaystable}[p]\n \\begin{center}\n  \\caption{\\small A comparison of calculation times (CPU and real) for 19\n  $s$-powers of joint probabilities only of PF06850 protein family, using\n  the four configurations $MLA_I$, $POA_{I\\!I}$, $PLA_{I\\!I}$, $POH$. \\label{tab12}}\n  \\begingroup\n  \\everymath{\\scriptstyle}\n  \\small\n  \\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n   \\hline\n   \\multicolumn{1}{|c|}{\\multirow{2}{*}{$\\mathbf{s}$}} &\n   \\multicolumn{4}{c|}{$\\mathbf{t_{CPU}}$ (sec)} &\n   \\multicolumn{4}{c|}{$\\mathbf{t_R}$ (sec)} \\\\\n   \\cline{2-9}\n   \\multicolumn{1}{|c|}{} & $MLA_I$ & $POA_{I\\!I}$ & $PLA_{I\\!I}$ & $POH$ &\n   $MLA_I$ & $POA_{I\\!I}$ & $PLA_{I\\!I}$ & $POH$ \\\\\n   \\hline\n   $0.1$ & $390.432$ & $33.478$ & $\\mathbf{41.633}$ & $24.849$ & $206.646$ &\n   $88.527$ & $83.139$ & $26.862$ \\\\\n   \\hline\n   $0.2$ & $382.887$ & $31.711$ & $26.483$ & $25.093$ & $202.282$ & $80.624$\n   & $53.384$ & $26.904$ \\\\\n   \\hline\n   $0.3$ & $401.269$ & $31.726$ & $25.286$ & $24.751$ & $210.791$ & $80.519$\n   & $50.689$ & $27.305$ \\\\\n   \\hline\n   $0.4$ & $416.168$ & $31.448$ & $26.138$ & $26.345$ & $216.993$ & $79.032$\n   & $51.409$ & $27.687$ \\\\\n   \\hline\n   $0.5$ & $427.572$ & $32.860$ & $\\mathbf{25.822}$ & $26.726$ & $221.541$\n   & $93.048$ & $51.334$ & $27.528$ \\\\\n   \\hline\n   $0.6$ & $430.604$ & $33.444$ & $27.013$ & $25.021$ & $223.227$ & $102.317$\n   & $52.889$ & $25.408$ \\\\\n   \\hline\n   $0.7$ & $421.904$ & $31.053$ & $25.814$ & $25.011$ & $218.484$ & $79.255$\n   & $51.466$ & $26.414$ \\\\\n   \\hline\n   $0.8$ & $434.888$ & $31.526$ & $26.725$ & $25.183$ & $224.267$ & $80.469$\n   & $53.388$ & $25.668$ \\\\\n   \\hline\n   $0.9$ & $431.948$ & $31.482$ & $26.895$ & $25.409$ & $223.023$ & $80.002$\n   & $53.579$ & $25.536$ \\\\\n   \\hline\n   $1.0$ & $442.933$ & $32.056$ & $25.990$ & $25.096$ & $224.731$ & $80.917$\n   & $51.687$ & $25.640$ \\\\\n   \\hline\n   $2.0$ & $176.212$ & $32.638$ & $27.089$ & $26.012$ & $147.454$ & $80.751$\n   & $54.384$ & $26.960$ \\\\\n   \\hline\n   $3.0$ & $234.100$ & $31.892$ & $25.766$ & $24.498$ & $174.853$ & $85.853$\n   & $51.843$ & $24.717$ \\\\\n   \\hline\n   $4.0$ & $284.184$ & $31.662$ & $26.515$ & $25.251$ & $181.552$ & $91.837$\n   & $52.636$ & $25.718$ \\\\\n   \\hline\n   $5.0$ & $327.740$ & $32.295$ & $27.516$ & $24.925$ & $178.117$ & $87.486$\n   & $54.908$ & $25.814$ \\\\\n   \\hline\n   $6.0$ & $334.800$ & $32.674$ & $28.126$ & $25.440$ & $194.691$ & $86.569$\n   & $54.611$ & $25.847$ \\\\\n   \\hline\n   $7.0$ & $349.064$ & $31.674$ & $23.908$ & $26.389$ & $195.258$ & $86.215$\n   & $49.262$ & $27.745$ \\\\\n   \\hline\n   $8.0$ & $361.304$ & $33.105$ & $26.020$ & $25.106$ & $195.437$ & $116.601$\n   & $51.889$ & $26.735$ \\\\\n   \\hline\n   $9.0$ & $386.217$ & $31.881$ & $26.208$ & $24.783$ & $197.150$ & $81.372$\n   & $53.114$ & $25.155$ \\\\\n   \\hline\n   $10.0$ & $397.276$ & $32.269$ & $26.125$ & $24.979$ & $197.868$ & $87.963$\n   & $52.541$ & $26.504$ \\\\\n   \\hline \\hline\n   Total & $7036.502$ & $611.374$ & $515.072$ & $480.867$ & $3834.365$\n   & $1649.357$ & $1028.655$ & $500.147$ \\\\\n   \\hline \\hline\n   Total & $7,522,020.640$ & $653,588.806$ & $550,611.968$ & $514,046.813$ &\n   $4,098,936.190$ & $1,763,162.630$ & $1,099,632.200$ & $534,157.143$ \\\\\n   (1069) families & $=87.067$ days & $=7.565$ days & $=6.373$ days & $=5.950$ days &\n   $=47.441$ days & $=20.407$ days & $=12.727$ days & $=6.188$ days \\\\\n   \\hline\n  \\end{tabular}\n  \\endgroup\n \\end{center}\n\\end{sidewaystable}\n\nWe can check that the times on table \\ref{tab12} seem to be generically\nordered as\n\\begin{equation}\n t_{MLA_I} > t_{POA_{I\\!I}} > t_{PLA_{I\\!I}} > t_{POH} \\label{eq:eq60}\n\\end{equation}\nFrom this ordering of computing times, we are then able to consider that the\ninconvenience of using the ``hash'' structure which has been emphasized on\nsection 2, was not circumvented by working with a modified array structure\n($A_{I\\!I}$ instead of $H$), at least for the Mac Pro machine used in these\ncalculations. We do not also know if this machine has been even used with an\n``overload'' of programs from the assumed part-time job of the experimenter\n(maybe 99.99\\% time job!). Anyhow, the usual Hash structure of Perl computing\nsystem has delayed the calculation of the Entropy Measures and even with the\nhelp of the modified $A_{I\\!I}$ structure, it does not succeed at computing\nwith this structure if operated on a OSX computing system.\n\nOn the other hand, the configuration $MLA_I$ could be chosen for parallelizing\nthe respective adopted code in a work to be done with supercomputer facilities.\nIf we try to avoid this kind of computational facility, in the belief that the\nproblem of classifying the distribution of amino acids of a protein database\nin terms of Entropy Measures could be treated with less powerful but very\nobjective ``weapons'', we should try to look for very fast laptop machines\ninstead, by working with a Linux operating system, a Perl computing system and\na modified array structure. This means that it would be worthwhile the\ncontinuation of the present work with the $PLA_{I\\!I}$ configuration. This is\nnow in progress and will be published elsewhere.\n\nWe summarize the conclusions commented above on tables \\ref{tab13}--\\ref{tab16}\nbelow for the calculation of CPU and Real times of 19 $s$-powers of joint\nprobabilities $\\big(P_{jk}(a,b)\\big)^s$ and the corresponding values of\nHavrda-Charvat entropy measures. The necessary times for calculating the joint\nprobabilities themselves has not been taken into consideration. It would be\nvery useful to make a comparison of the results of table \\ref{tab10} with those\non tables \\ref{tab14}, \\ref{tab16}, and table \\ref{tab12} with tables \\ref{tab13},\n\\ref{tab15} as well.\n\nAs a last remark of this section, we shall take into consideration,\nthe restrictions of $s \\leq 1$ for working with Jaccard Entropy\nmeasures and we calculate the total CPU and real times for the set\nof $s$-values: $s =$ $0.1$, $0.2$, $0.3$, $0.4$, $0.5$, $0.6$, $0.7$,\n$0.8$, $0.9$, $1.0$. The results are presented in table \\ref{tab17}\nbelow for the $PLA_{I\\!I}$ configuration and the calculation of the\nHavrda-Charvat entropies.\n\n\\begin{sidewaystable}[p]\n \\begin{center}\n  \\caption{\\small Calculation of CPU and Real times of 19 $s$-powers of joint\n  probabilities $\\big(P_{jk}(a,b)\\big)^s$ measures for 06 families from 03\n  Clans with the $POA_{I\\!I}$ configuration. \\label{tab13}}\n  \\begingroup\n  \\everymath{\\scriptstyle}\n  \\footnotesize\n  \\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n   \\hline\n   {} & \\multicolumn{4}{c|}{CL0028} & \\multicolumn{4}{c|}{CL0023}\n   & \\multicolumn{4}{c|}{CL0257} \\\\\n   \\cline{2-13}\n   s & \\multicolumn{2}{c|}{PF06850} & \\multicolumn{2}{c|}{PF00135}\n   & \\multicolumn{2}{c|}{PF00005} & \\multicolumn{2}{c|}{PF13481}\n   & \\multicolumn{2}{c|}{PF02388} & \\multicolumn{2}{c|}{PF09924} \\\\\n   \\cline{2-13}\n   {} & $t_{CPU}$ (sec) & $t_R$ (sec) & $t_{CPU}$ (sec) & $t_R$ (sec)\n   & $t_{CPU}$ (sec) & $t_R$ (sec) & $t_{CPU}$ (sec) & $t_R$ (sec)\n   & $t_{CPU}$ (sec) & $t_R$ (sec) & $t_{CPU}$ (sec) & $t_R$ (sec) \\\\\n   \\hline\n   $0.1$ & $33.478$ & $88.527$ & $46.747$ & $130.014$ & $41.475$ & $169.679$\n   & $44.187$ & $171.636$ & $44.716$ & $242.376$ & $46.177$ & $288.259$ \\\\\n   \\hline\n   $0.2$ & $31.711$ & $80.624$ & $42.157$ & $90.687$ & $36.893$ & $78.893$\n   & $38.567$ & $88.754$ & $37.088$ & $88.091$ & $40.709$ & $148.371$ \\\\\n   \\hline\n   $0.3$ & $31.726$ & $80.519$ & $39.957$ & $82.240$ & $36.929$ & $97.270$\n   & $38.800$ & $81.613$ & $37.350$ & $88.664$ & $38.986$ & $119.781$ \\\\\n   \\hline\n   $0.4$ & $31.448$ & $79.032$ & $41.737$ & $87.934$ & $36.252$ & $80.428$\n   & $38.500$ & $78.757$ & $35.592$ & $80.337$ & $36.773$ & $87.217$ \\\\\n   \\hline\n   $0.5$ & $32.860$ & $93.048$ & $41.130$ & $89.997$ & $38.203$ & $93.595$\n   & $37.618$ & $80.179$ & $35.117$ & $78.476$ & $35.534$ & $82.528$ \\\\\n   \\hline\n   $0.6$ & $33.944$ & $102.317$ & $41.417$ & $120.420$ & $37.143$ & $81.289$\n   & $38.387$ & $81.667$ & $45.900$ & $501.222$ & $35.830$ & $83.439$ \\\\\n   \\hline\n   $0.7$ & $31.053$ & $79.255$ & $40.422$ & $78.531$ & $36.386$ & $84.421$\n   & $43.216$ & $562.659$ & $43.452$ & $259.861$ & $35.261$ & $72.605$ \\\\\n   \\hline\n   $0.8$ & $31.526$ & $80.469$ & $41.556$ & $118.519$ & $36.955$ & $79.543$\n   & $41.862$ & $148.028$ & $35.047$ & $78.469$ & $35.718$ & $85.142$ \\\\\n   \\hline\n   $0.9$ & $31.482$ & $80.002$ & $41.386$ & $81.747$ & $36.811$ & $81.025$\n   & $35.518$ & $78.547$ & $35.540$ & $74.570$ & $40.534$ & $204.673$ \\\\\n   \\hline\n   $1.0$ & $32.056$ & $80.917$ & $40.724$ & $80.958$ & $37.234$ & $80.587$\n   & $36.610$ & $87.848$ & $36.353$ & $78.767$ & $39.095$ & $125.292$ \\\\\n   \\hline\n   $2.0$ & $32.638$ & $80.751$ & $40.701$ & $79.667$ & $38.452$ & $79.336$\n   & $36.916$ & $111.199$ & $36.658$ & $81.415$ & $40.415$ & $182.552$ \\\\\n   \\hline\n   $3.0$ & $31.892$ & $85.853$ & $40.822$ & $79.293$ & $38.223$ & $80.688$\n   & $37.356$ & $84.312$ & $36.658$ & $81.415$ & $39.774$ & $157.090$ \\\\\n   \\hline\n   $4.0$ & $31.662$ & $91.837$ & $41.208$ & $79.825$ & $37.936$ & $98.905$\n   & $37.000$ & $86.906$ & $36.012$ & $77.741$ & $38.794$ & $140.857$ \\\\\n   \\hline\n   $5.0$ & $32.295$ & $87.486$ & $41.059$ & $82.191$ & $38.293$ & $83.289$\n   & $36.245$ & $80.873$ & $35.308$ & $77.225$ & $39.927$ & $147.368$ \\\\\n   \\hline\n   $6.0$ & $32.674$ & $86.569$ & $41.215$ & $86.311$ & $37.601$ & $82.375$\n   & $36.395$ & $97.307$ & $35.748$ & $76.573$ & $39.327$ & $154.829$ \\\\\n   \\hline\n   $7.0$ & $31.674$ & $86.215$ & $41.290$ & $88.714$ & $38.341$ & $80.991$\n   & $36.724$ & $95.148$ & $36.194$ & $75.341$ & $39.826$ & $153.204$ \\\\\n   \\hline\n   $8.0$ & $33.105$ & $116.601$ & $40.984$ & $83.157$ & $37.756$ & $81.073$\n   & $40.524$ & $105.466$ & $36.716$ & $82.921$ & $41.056$ & $175.943$ \\\\\n   \\hline\n   $9.0$ & $31.881$ & $81.372$ & $41.469$ & $113.561$ & $38.748$ & $82.499$\n   & $37.874$ & $94.981$ & $40.341$ & $121.311$ & $39.847$ & $144.595$ \\\\\n   \\hline\n   $10.0$ & $32.269$ & $87.963$ & $41.466$ & $89.366$ & $37.807$ & $81.043$\n   & $37.134$ & $88.005$ & $40.053$ & $136.243$ & $40.333$ & $171.395$ \\\\\n   \\hline \\hline\n   Total & $611.374$ & $1649.357$ & $787.447$ & $1743.132$ & $717.438$\n   & $1676.929$ & $729.433$ & $2303.885$ & $719.843$ & $2381.018$ & $743.916$\n   & $2725.140$ \\\\\n   \\hline\n  \\end{tabular}\n  \\endgroup\n \\end{center}\n\\end{sidewaystable}\n\n\\begin{sidewaystable}[p]\n \\begin{center}\n  \\caption{\\small Calculation of CPU and Real times of Havrda-Charvat Entropy\n  measures for 06 families from 03 Clans with the $POA_{I\\!I}$\n  configuration. \\label{tab14}}\n  \\begingroup\n  \\everymath{\\scriptstyle}\n  \\footnotesize\n  \\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n   \\hline\n   {} & \\multicolumn{4}{c|}{CL0028} & \\multicolumn{4}{c|}{CL0023}\n   & \\multicolumn{4}{c|}{CL0257} \\\\\n   \\cline{2-13}\n   s & \\multicolumn{2}{c|}{PF06850} & \\multicolumn{2}{c|}{PF00135}\n   & \\multicolumn{2}{c|}{PF00005} & \\multicolumn{2}{c|}{PF13481}\n   & \\multicolumn{2}{c|}{PF02388} & \\multicolumn{2}{c|}{PF09924} \\\\\n   \\cline{2-13}\n   {} & $t_{CPU}$ (sec) & $t_R$ (sec) & $t_{CPU}$ (sec) & $t_R$ (sec)\n   & $t_{CPU}$ (sec) & $t_R$ (sec) & $t_{CPU}$ (sec) & $t_R$ (sec)\n   & $t_{CPU}$ (sec) & $t_R$ (sec) & $t_{CPU}$ (sec) & $t_R$ (sec) \\\\\n   \\hline\n   $0.1$ & $19.451$ & $26.196$ & $22.620$ & $56.287$ & $25.411$ & $121.508$\n   & $18.968$ & $32.706$ & $23.380$ & $80.826$ & $22.502$ & $77.826$ \\\\\n   \\hline\n   $0.2$ & $19.245$ & $24.901$ & $22.851$ & $69.992$ & $23.362$ & $65.508$\n   & $18.810$ & $26.835$ & $23.366$ & $87.098$ & $23.330$ & $60.684$ \\\\\n   \\hline\n   $0.3$ & $19.582$ & $26.001$ & $23.963$ & $60.960$ & $22.598$ & $72.363$\n   & $18.602$ & $26.028$ & $20.012$ & $43.734$ & $21.929$ & $48.947$ \\\\\n   \\hline\n   $0.4$ & $20.194$ & $31.418$ & $22.211$ & $54.562$ & $23.448$ & $63.369$\n   & $19.176$ & $30.482$ & $21.218$ & $67.817$ & $22.182$ & $45.480$ \\\\\n   \\hline\n   $0.5$ & $20.162$ & $31.653$ & $23.163$ & $67.391$ & $22.153$ & $55.173$\n   & $19.281$ & $34.495$ & $21.037$ & $55.237$ & $23.200$ & $62.384$ \\\\\n   \\hline\n   $0.6$ & $20.941$ & $34.979$ & $23.961$ & $62.158$ & $22.342$ & $57.081$\n   & $20.794$ & $44.558$ & $21.087$ & $54.900$ & $22.477$ & $62.421$ \\\\\n   \\hline\n   $0.7$ & $20.805$ & $34.057$ & $23.679$ & $82.669$ & $19.587$ & $35.750$\n   & $19.982$ & $41.314$ & $20.375$ & $32.699$ & $22.398$ & $66.438$ \\\\\n   \\hline\n   $0.8$ & $20.900$ & $34.146$ & $22.787$ & $60.924$ & $19.081$ & $34.116$\n   & $18.890$ & $28.054$ & $20.167$ & $32.685$ & $23.056$ & $61.376$ \\\\\n   \\hline\n   $0.9$ & $20.909$ & $34.568$ & $22.808$ & $54.890$ & $19.030$ & $33.258$\n   & $18.894$ & $29.712$ & $19.422$ & $26.934$ & $23.543$ & $65.099$ \\\\\n   \\hline\n   $1.0$ & $21.353$ & $35.024$ & $22.860$ & $51.308$ & $19.789$ & $32.218$\n   & $19.869$ & $37.922$ & $21.076$ & $35.774$ & $22.528$ & $52.209$ \\\\\n   \\hline\n   $2.0$ & $20.505$ & $32.920$ & $21.085$ & $46.921$ & $19.672$ & $33.778$\n   & $19.083$ & $62.081$ & $22.530$ & $82.336$ & $21.783$ & $51.656$ \\\\\n   \\hline\n   $3.0$ & $23.923$ & $58.898$ & $22.020$ & $47.298$ & $18.788$ & $31.926$\n   & $19.097$ & $35.399$ & $24.210$ & $83.371$ & $21.636$ & $69.905$ \\\\\n   \\hline\n   $4.0$ & $24.622$ & $68.692$ & $21.954$ & $50.909$ & $18.556$ & $26.512$\n   & $19.208$ & $32.014$ & $24.300$ & $112.613$ & $21.458$ & $49.931$ \\\\\n   \\hline\n   $5.0$ & $24.221$ & $60.107$ & $22.131$ & $58.975$ & $19.424$ & $33.185$\n   & $18.231$ & $25.898$ & $24.199$ & $95.807$ & $22.011$ & $55.109$ \\\\\n   \\hline\n   $6.0$ & $24.475$ & $64.843$ & $22.911$ & $72.004$ & $20.582$ & $38.006$\n   & $18.330$ & $25.959$ & $22.741$ & $61.516$ & $21.857$ & $67.976$ \\\\\n   \\hline\n   $7.0$ & $24.593$ & $70.336$ & $22.779$ & $51.309$ & $20.564$ & $38.390$\n   & $19.583$ & $30.719$ & $23.222$ & $88.997$ & $23.018$ & $85.054$ \\\\\n   \\hline\n   $8.0$ & $25.613$ & $83.423$ & $20.058$ & $34.861$ & $20.460$ & $35.589$\n   & $19.977$ & $37.130$ & $23.825$ & $110.933$ & $24.474$ & $90.322$ \\\\\n   \\hline\n   $9.0$ & $24.139$ & $73.873$ & $21.418$ & $44.709$ & $19.274$ & $32.129$\n   & $18.871$ & $29.089$ & $24.207$ & $106.471$ & $23.349$ & $74.724$ \\\\\n   \\hline\n   $10.0$ & $25.785$ & $101.370$ & $22.183$ & $46.878$ & $23.625$ & $87.442$\n   & $22.652$ & $64.895$ & $23.779$ & $86.225$ & $21.802$ & $57.444$ \\\\\n   \\hline \\hline\n   Total & $421.418$ & $927.405$ & $427.442$ & $1075.005$ & $397.746$\n   & $927.301$ & $368.298$ & $675.290$ & $424.153$ & $1345.973$ & $428.533$\n   & $1204.985$ \\\\\n   \\hline\n  \\end{tabular}\n  \\endgroup\n \\end{center}\n\\end{sidewaystable}\n  \n\\begin{sidewaystable}[p]\n \\begin{center}\n  \\caption{\\small Calculation of CPU and Real times of 19 $s$-powers of joint\n  probabilities $\\big(P_{jk}(a,b)\\big)^s$ measures for 06 families from 03\n  Clans with the $PLA_{I\\!I}$ configuration. \\label{tab15}}\n  \\begingroup\n  \\everymath{\\scriptstyle}\n  \\footnotesize\n  \\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n   \\hline\n   {} & \\multicolumn{4}{c|}{CL0028} & \\multicolumn{4}{c|}{CL0023}\n   & \\multicolumn{4}{c|}{CL0257} \\\\\n   \\cline{2-13}\n   s & \\multicolumn{2}{c|}{PF06850} & \\multicolumn{2}{c|}{PF00135}\n   & \\multicolumn{2}{c|}{PF00005} & \\multicolumn{2}{c|}{PF13481}\n   & \\multicolumn{2}{c|}{PF02388} & \\multicolumn{2}{c|}{PF09924} \\\\\n   \\cline{2-13}\n   {} & $t_{CPU}$ (sec) & $t_R$ (sec) & $t_{CPU}$ (sec) & $t_R$ (sec)\n   & $t_{CPU}$ (sec) & $t_R$ (sec) & $t_{CPU}$ (sec) & $t_R$ (sec)\n   & $t_{CPU}$ (sec) & $t_R$ (sec) & $t_{CPU}$ (sec) & $t_R$ (sec) \\\\\n   \\hline\n   $0.1$ & $41.633$ & $83.139$ & $43.135$ & $83.901$ & $44.003$ & $84.655$\n   & $26.613$ & $52.365$ & $27.452$ & $53.350$ & $43.132$ & $83.249$ \\\\\n   \\hline\n   $0.2$ & $26.483$ & $53.384$ & $26.373$ & $50.442$ & $26.610$ & $52.904$\n   & $26.599$ & $51.822$ & $43.199$ & $83.497$ & $25.762$ & $50.203$ \\\\\n   \\hline\n   $0.3$ & $25.286$ & $50.689$ & $26.644$ & $50.992$ & $29.965$ & $53.250$\n   & $24.946$ & $48.846$ & $25.498$ & $50.303$ & $25.904$ & $50.737$ \\\\\n   \\hline\n   $0.4$ & $26.138$ & $51.409$ & $25.255$ & $48.576$ & $26.696$ & $52.482$\n   & $27.769$ & $53.837$ & $25.753$ & $51.013$ & $25.684$ & $50.371$ \\\\\n   \\hline\n   $0.5$ & $25.822$ & $51.837$ & $26.041$ & $50.492$ & $26.648$ & $52.171$\n   & $25.026$ & $48.927$ & $25.727$ & $49.887$ & $25.513$ & $48.793$ \\\\\n   \\hline\n   $0.6$ & $27.013$ & $52.889$ & $25.778$ & $50.324$ & $24.912$ & $49.340$\n   & $26.062$ & $51.312$ & $26.066$ & $51.417$ & $25.435$ & $49.686$ \\\\\n   \\hline\n   $0.7$ & $25.814$ & $51.466$ & $25.496$ & $49.954$ & $25.284$ & $50.268$\n   & $23.698$ & $48.134$ & $25.723$ & $50.760$ & $25.822$ & $51.122$ \\\\\n   \\hline\n   $0.8$ & $26.725$ & $53.388$ & $26.254$ & $50.865$ & $25.456$ & $50.870$\n   & $25.816$ & $51.196$ & $26.883$ & $52.511$ & $29.837$ & $57.367$ \\\\\n   \\hline\n   $0.9$ & $26.895$ & $53.579$ & $23.740$ & $47.296$ & $26.237$ & $51.725$\n   & $28.441$ & $54.743$ & $25.237$ & $50.532$ & $27.951$ & $54.289$ \\\\\n   \\hline\n   $1.0$ & $25.990$ & $51.687$ & $27.225$ & $54.384$ & $26.014$ & $51.969$\n   & $27.148$ & $52.701$ & $23.672$ & $48.547$ & $26.314$ & $51.932$ \\\\\n   \\hline\n   $2.0$ & $27.089$ & $54.384$ & $26.237$ & $50.335$ & $27.756$ & $54.761$\n   & $26.424$ & $52.503$ & $24.811$ & $49.859$ & $25.846$ & $51.125$ \\\\\n   \\hline\n   $3.0$ & $25.766$ & $51.843$ & $27.342$ & $52.508$ & $25.135$ & $50.954$\n   & $27.753$ & $53.650$ & $25.072$ & $49.142$ & $25.727$ & $50.990$ \\\\\n   \\hline\n   $4.0$ & $26.515$ & $52.636$ & $25.716$ & $50.531$ & $25.106$ & $50.449$\n   & $24.871$ & $50.279$ & $25.674$ & $51.536$ & $26.897$ & $52.227$ \\\\\n   \\hline\n   $5.0$ & $27.516$ & $54.908$ & $26.002$ & $50.390$ & $24.666$ & $49.463$\n   & $23.973$ & $47.554$ & $25.675$ & $51.032$ & $27.810$ & $53.820$ \\\\\n   \\hline\n   $6.0$ & $28.126$ & $54.611$ & $27.441$ & $53.032$ & $26.014$ & $52.209$\n   & $25.676$ & $50.426$ & $26.235$ & $51.375$ & $26.180$ & $51.346$ \\\\\n   \\hline\n   $7.0$ & $23.908$ & $49.262$ & $26.812$ & $51.556$ & $24.696$ & $49.783$\n   & $25.519$ & $50.741$ & $25.863$ & $50.995$ & $25.593$ & $50.611$ \\\\\n   \\hline\n   $8.0$ & $26.020$ & $51.889$ & $25.431$ & $49.135$ & $26.757$ & $52.518$\n   & $26.369$ & $49.668$ & $26.401$ & $52.682$ & $25.084$ & $50.596$ \\\\\n   \\hline\n   $9.0$ & $26.208$ & $53.114$ & $25.856$ & $50.090$ & $25.803$ & $51.253$\n   & $24.628$ & $47.880$ & $25.379$ & $51.014$ & $27.049$ & $52.891$ \\\\\n   \\hline\n   $10.0$ & $26.125$ & $52.541$ & $24.963$ & $47.402$ & $24.369$ & $48.065$\n   & $42.270$ & $83.281$ & $24.878$ & $49.926$ & $24.563$ & $48.616$ \\\\\n   \\hline \\hline\n   Total & $515.072$ & $1028.655$ & $511.741$ & $992.205$ & $509.127$\n   & $1009.089$ & $509.601$ & $999.865$ & $505.198$ & $999.378$ & $516.103$\n   & $1009.971$ \\\\\n   \\hline\n  \\end{tabular}\n  \\endgroup\n \\end{center}\n\\end{sidewaystable}\n\n\\begin{sidewaystable}[p]\n \\begin{center}\n  \\caption{\\small Calculation of CPU and Real times of Havrda-Charvat Entropy\n  measures for 06 families from 03 Clans with the $PLA_{I\\!I}$\n  configuration. \\label{tab16}}\n  \\begingroup\n  \\everymath{\\scriptstyle}\n  \\footnotesize\n  \\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n   \\hline\n   {} & \\multicolumn{4}{c|}{CL0028} & \\multicolumn{4}{c|}{CL0023}\n   & \\multicolumn{4}{c|}{CL0257} \\\\\n   \\cline{2-13}\n   s & \\multicolumn{2}{c|}{PF06850} & \\multicolumn{2}{c|}{PF00135}\n   & \\multicolumn{2}{c|}{PF00005} & \\multicolumn{2}{c|}{PF13481}\n   & \\multicolumn{2}{c|}{PF02388} & \\multicolumn{2}{c|}{PF09924} \\\\\n   \\cline{2-13}\n   {} & $t_{CPU}$ (sec) & $t_R$ (sec) & $t_{CPU}$ (sec) & $t_R$ (sec)\n   & $t_{CPU}$ (sec) & $t_R$ (sec) & $t_{CPU}$ (sec) & $t_R$ (sec)\n   & $t_{CPU}$ (sec) & $t_R$ (sec) & $t_{CPU}$ (sec) & $t_R$ (sec) \\\\\n   \\hline\n   $0.1$ & $21.219$ & $26.895$ & $18.698$ & $23.442$ & $19.383$ & $24.675$\n   & $21.351$ & $26.966$ & $19.059$ & $24.560$ & $21.451$ & $26.829$ \\\\\n   \\hline\n   $0.2$ & $22.604$ & $28.014$ & $21.109$ & $26.536$ & $20.336$ & $25.665$\n   & $19.555$ & $24.562$ & $19.370$ & $24.681$ & $22.504$ & $27.855$ \\\\\n   \\hline\n   $0.3$ & $19.056$ & $24.130$ & $20.692$ & $25.663$ & $21.934$ & $27.466$\n   & $21.084$ & $26.236$ & $19.317$ & $24.934$ & $19.681$ & $24.732$ \\\\\n   \\hline\n   $0.4$ & $21.552$ & $27.442$ & $19.981$ & $25.021$ & $18.794$ & $24.142$\n   & $21.065$ & $26.485$ & $20.878$ & $26.291$ & $21.947$ & $26.979$ \\\\\n   \\hline\n   $0.5$ & $31.952$ & $88.406$ & $21.161$ & $26.374$ & $19.123$ & $24.309$\n   & $22.534$ & $27.851$ & $19.729$ & $25.011$ & $21.888$ & $27.168$ \\\\\n   \\hline\n   $0.6$ & $20.195$ & $25.362$ & $21.006$ & $26.371$ & $21.426$ & $30.303$\n   & $20.640$ & $26.050$ & $19.424$ & $24.279$ & $20.750$ & $26.175$ \\\\\n   \\hline\n   $0.7$ & $19.168$ & $23.924$ & $20.989$ & $26.482$ & $21.440$ & $27.129$\n   & $19.370$ & $24.287$ & $19.274$ & $24.779$ & $21.545$ & $26.830$ \\\\\n   \\hline\n   $0.8$ & $21.982$ & $27.362$ & $19.760$ & $24.681$ & $20.662$ & $25.948$\n   & $21.324$ & $26.648$ & $21.685$ & $26.964$ & $21.644$ & $27.160$ \\\\\n   \\hline\n   $0.9$ & $21.356$ & $27.235$ & $21.251$ & $26.550$ & $19.518$ & $24.880$\n   & $21.276$ & $26.736$ & $21.283$ & $26.981$ & $19.217$ & $23.910$ \\\\\n   \\hline\n   $1.0$ & $20.206$ & $25.608$ & $20.914$ & $26.173$ & $20.726$ & $25.877$\n   & $21.430$ & $26.625$ & $20.102$ & $25.198$ & $19.810$ & $25.348$ \\\\\n   \\hline\n   $2.0$ & $19.435$ & $24.882$ & $20.660$ & $25.675$ & $20.798$ & $26.326$\n   & $20.301$ & $25.481$ & $19.588$ & $24.837$ & $20.446$ & $25.755$ \\\\\n   \\hline\n   $3.0$ & $20.549$ & $25.646$ & $19.988$ & $25.479$ & $20.629$ & $25.516$\n   & $19.914$ & $25.230$ & $20.438$ & $25.648$ & $19.393$ & $24.815$ \\\\\n   \\hline\n   $4.0$ & $19.528$ & $24.693$ & $20.649$ & $25.610$ & $19.428$ & $24.637$\n   & $20.505$ & $26.001$ & $20.868$ & $26.009$ & $20.060$ & $25.643$ \\\\\n   \\hline\n   $5.0$ & $19.698$ & $24.824$ & $21.076$ & $26.468$ & $19.865$ & $24.753$\n   & $19.120$ & $24.267$ & $19.466$ & $24.423$ & $21.760$ & $27.244$ \\\\\n   \\hline\n   $6.0$ & $20.809$ & $26.319$ & $20.352$ & $25.914$ & $19.507$ & $24.362$\n   & $20.110$ & $25.587$ & $20.708$ & $25.961$ & $21.240$ & $26.665$ \\\\\n   \\hline\n   $7.0$ & $20.287$ & $25.951$ & $21.073$ & $26.459$ & $19.482$ & $24.861$\n   & $18.272$ & $23.535$ & $19.015$ & $24.460$ & $20.646$ & $25.964$ \\\\\n   \\hline\n   $8.0$ & $21.427$ & $26.885$ & $19.494$ & $24.421$ & $20.107$ & $25.228$\n   & $20.645$ & $26.095$ & $21.039$ & $26.222$ & $20.014$ & $25.155$ \\\\\n   \\hline\n   $9.0$ & $21.623$ & $27.335$ & $21.554$ & $26.517$ & $19.239$ & $24.529$\n   & $20.629$ & $25.929$ & $21.035$ & $26.450$ & $21.086$ & $26.526$ \\\\\n   \\hline\n   $10.0$ & $20.815$ & $26.379$ & $21.127$ & $26.630$ & $19.927$ & $25.340$\n   & $19.781$ & $25.063$ & $21.438$ & $27.019$ & $17.286$ & $22.390$ \\\\\n   \\hline \\hline\n   Total & $403.461$ & $557.294$ & $391.524$ & $490.466$ & $382.324$\n   & $485.946$ & $388.906$ & $489.634$ & $383.716$ & $484.707$ & $392.368$\n   & $493.143$ \\\\\n   \\hline\n  \\end{tabular}\n  \\endgroup\n \\end{center}\n\\end{sidewaystable}\n\n\\begin{sidewaystable}[p]\n \\begin{center}\n  \\caption{\\small Calculation of CPU and Real times of Havrda-Charvat\n  Entropy measures for 06 families from 03 Clans with parameters\n  $0 < s \\leq 1$ and the $PLA_{I\\!I}$ configuration. \\label{tab17}}\n  \\begingroup\n  \\everymath{\\scriptstyle}\n  \\footnotesize\n  \\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n   \\hline\n   {} & \\multicolumn{4}{c|}{CL0028} & \\multicolumn{4}{c|}{CL0023}\n   & \\multicolumn{4}{c|}{CL0257} \\\\\n   \\cline{2-13}\n   s & \\multicolumn{2}{c|}{PF06850} & \\multicolumn{2}{c|}{PF00135}\n   & \\multicolumn{2}{c|}{PF00005} & \\multicolumn{2}{c|}{PF13481}\n   & \\multicolumn{2}{c|}{PF02388} & \\multicolumn{2}{c|}{PF09924} \\\\\n   \\cline{2-13}\n   {} & $t_{CPU}$ (sec) & $t_R$ (sec) & $t_{CPU}$ (sec) & $t_R$ (sec)\n   & $t_{CPU}$ (sec) & $t_R$ (sec) & $t_{CPU}$ (sec) & $t_R$ (sec)\n   & $t_{CPU}$ (sec) & $t_R$ (sec) & $t_{CPU}$ (sec) & $t_R$ (sec) \\\\\n   \\hline\n   $0.1$ & $21.219$ & $26.895$ & $18.698$ & $23.442$ & $19.383$ & $24.675$\n   & $21.351$ & $26.966$ & $19.059$ & $24.560$ & $21.451$ & $26.829$ \\\\\n   \\hline\n   $0.2$ & $22.604$ & $28.014$ & $21.109$ & $26.536$ & $20.336$ & $25.665$\n   & $19.555$ & $24.562$ & $19.370$ & $24.681$ & $22.504$ & $27.855$ \\\\\n   \\hline\n   $0.3$ & $19.056$ & $24.130$ & $20.692$ & $25.663$ & $21.934$ & $27.466$\n   & $21.084$ & $26.236$ & $19.317$ & $24.934$ & $19.681$ & $24.732$ \\\\\n   \\hline\n   $0.4$ & $21.552$ & $27.442$ & $19.981$ & $25.021$ & $18.794$ & $24.142$\n   & $21.065$ & $26.485$ & $20.878$ & $26.291$ & $21.947$ & $26.979$ \\\\\n   \\hline\n   $0.5$ & $31.952$ & $88.406$ & $21.161$ & $26.374$ & $19.123$ & $24.309$\n   & $22.534$ & $27.851$ & $19.729$ & $25.011$ & $21.888$ & $27.168$ \\\\\n   \\hline\n   $0.6$ & $20.195$ & $25.362$ & $21.006$ & $26.371$ & $21.426$ & $30.303$\n   & $20.640$ & $26.050$ & $19.424$ & $24.279$ & $20.750$ & $26.175$ \\\\\n   \\hline\n   $0.7$ & $19.168$ & $23.924$ & $20.989$ & $26.482$ & $21.440$ & $27.129$\n   & $19.370$ & $24.287$ & $19.274$ & $24.779$ & $21.545$ & $26.830$ \\\\\n   \\hline\n   $0.8$ & $21.982$ & $27.362$ & $19.760$ & $24.681$ & $20.662$ & $25.948$\n   & $21.324$ & $26.648$ & $21.685$ & $26.964$ & $21.644$ & $27.160$ \\\\\n   \\hline\n   $0.9$ & $21.356$ & $27.235$ & $21.251$ & $26.550$ & $19.518$ & $24.880$\n   & $21.276$ & $26.736$ & $21.283$ & $26.981$ & $19.217$ & $23.910$ \\\\\n   \\hline\n   $1.0$ & $20.206$ & $25.608$ & $20.914$ & $26.173$ & $20.726$ & $25.877$\n   & $21.430$ & $26.625$ & $20.102$ & $25.198$ & $19.810$ & $25.348$ \\\\\n   \\hline \\hline\n   Total & $219.290$ & $324.380$ & $205.561$ & $257.293$ & $203.342$\n   & $260.394$ & $209.629$ & $262.446$ & $200.121$ & $253.678$ & $210.437$\n   & $262.986$ \\\\\n   \\hline \\hline\n   Grand & \\multicolumn{1}{c|}{\\multirow{2}{*}{$1284.343$}}\n   & \\multicolumn{1}{c|}{\\multirow{2}{*}{$1895.873$}}\n   & \\multicolumn{1}{c|}{\\multirow{2}{*}{$1199.098$}}\n   & \\multicolumn{1}{c|}{\\multirow{2}{*}{$1727.300$}}\n   & \\multicolumn{1}{c|}{\\multirow{2}{*}{$1131.878$}}\n   & \\multicolumn{1}{c|}{\\multirow{2}{*}{$1693.572$}}\n   & \\multicolumn{1}{c|}{\\multirow{2}{*}{$1066.375$}}\n   & \\multicolumn{1}{c|}{\\multirow{2}{*}{$1511.927$}}\n   & \\multicolumn{1}{c|}{\\multirow{2}{*}{$1124.465$}}\n   & \\multicolumn{1}{c|}{\\multirow{2}{*}{$1681.087$}}\n   & \\multicolumn{1}{c|}{\\multirow{2}{*}{$1211.808$}}\n   & \\multicolumn{1}{c|}{\\multirow{2}{*}{$1772.106$}} \\\\\n   Total & \\multicolumn{1}{c|}{} & \\multicolumn{1}{c|}{}\n   & \\multicolumn{1}{c|}{} & \\multicolumn{1}{c|}{} & \\multicolumn{1}{c|}{}\n   & \\multicolumn{1}{c|}{} & \\multicolumn{1}{c|}{} & \\multicolumn{1}{c|}{}\n   & \\multicolumn{1}{c|}{} & \\multicolumn{1}{c|}{} & \\multicolumn{1}{c|}{}\n   & \\multicolumn{1}{c|}{} \\\\\n   \\hline\n  \\end{tabular}\n  \\endgroup\n \\end{center}\n\\end{sidewaystable}\n\n\\newpage\n\nThe corresponding times for calculating the joint probabilities and\n$s$-powers of these have been added up to report the results for the\nHavrda-Charvat entropies of the Grand total row. We think that these\ntimes are very well affordable indeed.\n\n\\begin{table}[H]\n \\begin{center}\n  \\caption{\\small Total CPU and real times for calculating the Entropy measure\n  content of a family PF06850 and approximations for Grand\n  Total of all sample space. \\label{tab18}}\n  \\begin{tabular}{|c|c|c|}\n   \\hline\n   \\multicolumn{1}{|c|}{\\multirow{2}{*}{$POA_{I\\!I}$}} & Entropy Measures &\n   Entropy Measures \\\\\n   {} & $H\\!_j(s)$ --- 19 $s$-values & $H\\!_{jk}(s)$ ---\n   19 $s$-values \\\\\n   \\hline\n   Total CPU time & $0.358+2.562+0.292$\n   & $550.129+611.374$ \\\\\n   (family PF06850) & $=3.212$ sec & $+421.418=1,582.921$ sec \\\\\n   \\hline\n   Grand Total CPU time & $3,433.628$ sec\n   & $1,692,142.549$ sec \\\\\n   (1069 families) & $=0.954$ hs & $=19.585$ days \\\\\n   \\hline\n   Total Real time & $1.062+9.300+0.332$\n   & $593.848+1,649.357+$ \\\\\n   (family PF06850) & $=10.694$ sec & $927.405=3.170.61$ sec \\\\\n   \\hline\n   Grand Total Real time & $11,431.886$ sec\n   & $3,389,382.090$ sec \\\\\n   (1069 families) & $=3.175$ hs & $=39.229$ days \\\\\n   \\hline\n  \\end{tabular}\n \\end{center}\n\\end{table}\n\n\\begin{table}[H]\n \\begin{center}\n  \\caption{\\small Total CPU and real times for calculating the Entropy measure\n  content of a family PF06850 and approximations for Grand\n  Total of all sample space. \\label{tab19}}\n  \\begin{tabular}{|c|c|c|}\n   \\hline\n   \\multicolumn{1}{|c|}{\\multirow{2}{*}{$PLA_{I\\!I}$}} & Entropy Measures &\n   Entropy Measures \\\\\n   {} & $H\\!_j(s)$ --- 19 $s$-values & $H\\!_{jk}(s)$ ---\n   19 $s$-values \\\\\n   \\hline\n   Total CPU time & $0.291+1.801+0.640$\n   & $787.254+515.072$ \\\\\n   (family PF06850) & $=2.732$ sec & $+403.461=1,705.787$ sec \\\\\n   \\hline\n   Grand Total CPU time & $2,920.508$ sec\n   & $1,823,486.303$ sec \\\\\n   (1069 families) & $=0.811$ hs & $=21.105$ days \\\\\n   \\hline\n   Total Real time & $0.642+7.261+1.291$\n   & $1,068.026+1,028.655$ \\\\\n   (family PF06850) & $=9.194$ sec & $+557.294=2,603.975$ sec \\\\\n   \\hline\n   Grand Total Real time & $9,828.386$ sec\n   & $2,783,649.275$ sec \\\\\n   (1069 families) & $=2.730$ hs & $=32.218$ days \\\\\n   \\hline\n  \\end{tabular}\n \\end{center}\n\\end{table}\n\n\n\\section{Concluding Remarks and Suggestions for Future Work}\nThe treatment of the distributions of probability of occur in protein\ndatabases is a twofold procedure. We intend to find a way of characterizing\nthe protein database by values of Entropy Measures in order to provide a\nsound discussion to be centered on the maximization of a convenient average\nEntropy Measure to represent the entire protein database. We also intend to\nderive a partition function in order to derive a thermodynamical theory\nassociated to the temporal evolution of the database. If the corresponding\nevolution of the protein families is assumed to be registered on the\nsubsequent versions of the database (Table \\ref{tab17}), we will then be\nable to describe the sought thermodynamical evolution from this theory as\nwell as to obtain from it the convenient description of all intermediate\nLevinthal's stages which seem to be necessary for describing the\nfolding\/unfolding dynamical process.\n\nWe summarize this approach by the need of starting from a thermodynamical\ntheory of the evolution of protein databases via Entropy measures to the\nconstruction of a successful dynamical theory of protein families. In other\nwords, from the thermodynamics of evolution of a protein database, we will\nderive a statistical mechanics to give us physical insight on the\nconstruction of a successful dynamics of protein families.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\n\nSemiflexible polymers is a term coined to understand a variety of physical systems that involve linear molecules. For instance, understanding the behaviors of such polymers serves as the basis to understand phenomena encountered in polymer industry, biotechnology, and mo\\-le\\-cu\\-lar processes in living cells \\cite{Fal-Sakaue2007}.  Indeed, biopolymers functionality are ruled by their conformation, which in turn is considerably modified in the geometrically confined or crowded environment inside the cell \\cite{Fal-Koster2008, Fal-Reisner2005,Fal-Cifra2010, Fal-Benkova2017}.  Beyond the most prominent polymer example being DNA compaction in the nucleus or  viral DNA  packed in capsids \\cite{Fal-Cifra2010, Fal-Locker2006}, there is also the important outstanding  example of DNA transcription and replication processes that are governed by the binding of specific proteins. These  mechanisms are strongly connected to polymer configuration \\cite{Fal-Gowers2003,Fal-Broek2008,Fal-Ostermeir2010}. Furthermore, a wide range of biophysical processes is derived from  DNA constrained to a ring enclosure and more general topologies \\cite{Fal-Witz2011}. \n\nOn the one hand, motivated by the packaging and coiling problems mentioned above, Mondescu and Muthukumar (MM) studied in \\cite{PDoi-Mondescu1998} the conformational states of an ideal Gaussian polymer \\cite{PDoi-Doi1988book} wrapping different curved surfaces, where they presented theoretical predictions for the mean-square end-to-end distance.  Later on, Spakowitz and Wang (SW) in \\cite{PSaito-Spakowitz2003} studied the conformational states of an ideal semiflexible polymer confined to a spherical surface based on the continuous Worm-Like Chain Model (WLC) \\cite{PSaito-Saito1967}. Unlike the conformational states of the Gaussian polymer, SW found the existence of a shape transition from an ordered to a disordered phase, where polymer roughly looks like cooked spaghetti  and a random walk, respectively. Moreover, in the appropriate limit, the behavior of the semiflexible polymer reduces to the one of the Gaussian polymer in the spherical case. Subsequently, the MM and SW results were confirmed through computer simulations, where the validity regimes for each theory were established. Additionally,  as a consequence of the excluded volume effect a helical state was found in \\cite{Fal-Cerda2005,Fal-Slosar2006} and a Tennis ball-like state in  \\cite{Psim-Zhang2011}; these states are absent in both MM and SW theories. The problem becomes richer when short-range and long-range electrostatic interactions are considered since they can induce a Tennis ball-like state  which is predominant over the helical conformation in the case of long-range electrostatic interactions \\cite{PSim-Angelescu2008}. A transition is also reported in a similar manner to that reported by SW  with slight corrections in the case of short-range interaction and more pronounced in the long-range ones. In the extreme limit of zero temperature, the conformational states are expected to be in the ordered phase. Indeed, by a variational principle consistent to the WLC model one can obtain an abundance of conformational states including those observed in the simulations mentioned above \\cite{Psim-Guven2012,Psim-Lin2007} . On the other hand, the confinement can induce a transition from a circular polymer to a figure eight shape \\cite{Fal-Ostermeir2010}; even when the conformation of the polymer is considered as a self-avoiding random walk, properties similar to that of a critical phenomenon are found \\cite{Fal-Berger2006,Fal-Sakaue2006,Fal-Sakaue2007,Fal-Usatenko2017}. Confinement can also occur in the three-dimensional volume enclosed by rigid or soft surfaces \\cite{Fal-Brochard2005, Fal-Cacciuto2006, Fal-Chen2007, Fal-Koster2008, PSaito-Morrison2009, Psim-Gao2014},  and in a crowded environment, for instance, modeled by a nanopost array \\cite{Fal-Benkova2017}. \n\nFurthermore, two-dimensional confinement in closed flat spaces can result in a order-disorder shape transition \\cite{Polcon-Liu2008} similar to that of SW. One advantage of confinement in the flat  two-dimensional case is that it can be compared with the experiment \\cite{Fal-Moukhtar2007,Fal-Witz2011, Fal-Japaridze2017, Fal-Frielinghaus2013}. However, in the literature as far as we know, even in the semiflexible ideal chain  there is not a systematic study of the SW transition in a flat and bounded region. In this work, we present a theoretical and numerical analysis of the conformational states of an ideal semiflexible polymer in a compact two-dimensional flat space. First, we deduce the Hermans-Ullman (HU) equation \\cite{PSaito-Hermans1952}  under the supposition that the conformational states correspond to stochastic realizations of paths defined by the Frenet equations and the assumption that stochastic ``curvature'' satisfies a fluctuation theorem given by a white noise distribution. This latter hypothesis is consistent with the continuous version of the WLC model as we will see below. Using the HU equation we shall perform a multipolar decomposition for the probability density function $P({\\bf R}, \\theta, {\\bf R}^{\\prime}, \\theta^{\\prime}, L)$  that gives the probability to find a polymer with length $L$ with endings ${\\bf R}$  and  ${\\bf R}^{\\prime}$, and directions $\\theta$ and $\\theta^{\\prime}$, respectively. This decomposition allows us to find a hierarchy of equations associated to the multipoles of $P({\\bf R}, \\theta, {\\bf R}^{\\prime}, \\theta^{\\prime}, L)$, namely, the positional  density distribution $\\rho({\\bf R}, {\\bf R}^{\\prime}, L)$, the dipolar distribution $\\mathbb{P}^{i}\\left({\\bf R}, {\\bf R}^{\\prime}, L\\right)$, the quadrupolar two-rank tensor distribution $\\mathbb{Q}_{ij}({\\bf R}, {\\bf R}^{\\prime}, L)$, and so on.  We shall show, for instance, that the positional density and the quadrupolar distributions are exactly related through the modified Telegrapher's Equation (MTE), \n\\begin{eqnarray}\n\\frac{\\partial^2\\rho}{\\partial s^2}+\\frac{1}{2\\ell_{p}}\\frac{\\partial\\rho}{\\partial s}=\\frac{1}{2}\\nabla^2\\rho+\\partial_{i}\\partial_{j}\\mathbb{Q}^{ij}.\n\\end{eqnarray}\nIn particular, using this equation and the traceless condition of  $\\mathbb{Q}_{ij}$, we are going to verify the well known exact result of Kratky-Porod for a semiflexible polymer in a two-dimensional space \\cite{PSaito-Kratky1949}.   Besides, we will show that as a consequence of the exponential decay of $\\mathbb{Q}_{ij}$ we are going to define a regime where quadrupolar distribution can be neglected in the MTE. In addition, we shall explore the conformational states for a semiflexible chain enclosed by a bounded compact two-dimensional domain through the mean-square end-to-end distance. In particular, for a square domain we will show the existence of a shape transition order-disorder of the same nature as the one found by SW \\cite{PSaito-Spakowitz2003}. Furthermore, we will develop a Monte Carlo algorithm for use in computer simulations in order to study the conformational states enclosed in a compact domain. Particularly, the algorithm shall be suited in the square domain which, additionally, will allow us to confirm the shape transition and validate the theoretical predictions. \n\n\nThis paper is organized as follows. In Sec. II, we present the stochastic version of the Frenet equations whose Fokker-Planck formalism  give us a derivation of  the Hermans-Ullman (HU) equation. In addition, we discuss a multipolar decomposition for the  HU equation.  In Sec. III, we provide an application of  the methods developed in Sec. II in order to study semiflexible polymer conformations enclosed in a compact domain. Particularly, we focus on  a square box domain. In Sec. IV, we present a  Monte Carlo algorithm to study the conformational states of a semiflexible polymer enclosed in a compact domain. In Sec. V,  we give the main results in  a square box domain and we provide a comparison with the theoretical predictions. In the final Sec. VI, we give our concluding remarks and perspectives on this work.\n\n\n\\section{Preliminary notation and semiflexible polymers  }\\label{theory}\n\nLet us consider a polymer on a two dimensional Euclidean space as a plane curve $\\gamma$,  ${\\bf R}: I\\subset \\mathbb{R}\\to\\mathbb{R}^2$, parametrized by an arc-length, $s$.  For each point $s\\in I$, a Frenet dihedral can be defined whose vector basis corresponds to the set\n $\\{{\\bf T}(s), {\\bf N}(s)\\}$, consisting of the tangent vector ${\\bf T}(s)={\\bf R}^{\\prime}(s)\\equiv d{\\bf R}\/ds$  and the normal vector ${\\bf N}(s)=\\boldsymbol{\\epsilon}{\\bf T}(s)$, where $\\boldsymbol{\\epsilon}$ is a rotation by an angle of $\\pi\/2$. Note that the components of the rotation correspond to the Levi-Civita antisymmetric tensor in two dimensions.  Both are unit vectors ($\\left|{\\bf T}(s)\\right|=\\left|{\\bf N}(s)\\right|=1$), and by construction are orthogonal to each other. It is well known that along the points of the curve these vectors satisfy the Frenet equations, ${\\bf T}^{\\prime}(s)=\\kappa(s){\\bf N}(s)$ and ${\\bf N}^{\\prime}(s)=-\\kappa(s){\\bf T}(s)$,  where $\\kappa(s)$ is the curvature of the curve \\cite{Fal-Montiel2009}. \n \nIn absence of thermal fluctuations, the conformations of the polymer are studied through different curve configurations determined by variational principles. For instance, one of the most successful models to des\\-cribe  configurations of a semiflexible polymer is\n\\begin{eqnarray}\nH[{\\bf R}]=\\frac{\\alpha}{2}\\int ds ~\\kappa^2\\left(s\\right) , \n\\label{funcional}\n\\end{eqnarray}\nwhere $H[{\\bf R}]$ is the bending energy, and $\\alpha$ the bending rigidity modulus. This energy functional (\\ref{funcional}) corresponds to the continuous form of the worm-like chain model (WLC) \\cite{PSaito-Saito1967}. In a rather different context, a classical problem originally proposed by D. Bernoulli, later by L. Euler, between the XVIII and XIX centuries, consists of finding the family of curves $\\{\\gamma\\}$ with a fixed length that minimizes the functional \\eqref{funcional}. The solution to this problem is composed of those  curves whose curvature satisfies the differential equation $k^{\\prime\\prime}+\\frac{1}{2}k^{3}-\\frac{\\lambda}{\\alpha}\\kappa=0$, where $\\lambda$ is a Lagrange multiplier introduced to constrain the curve length \\cite{Polcla-Miura2017}. This problem has been generalized to study elastic curves in manifolds \\cite{Polcla-Singer2008, Psim-Guven2012, Polcla-Manning1987}, that are nowadays relevant to understand the problem of DNA packaging and the winding problem of DNA around histone octamers \\cite{Fal-Hardin2012}. \n\n\nIn what follows, we shall develop an unusual approach  that incorporates the thermal fluctuations in the study of semiflexible polymers described by the bending energy  \\eqref{funcional}. \n\n\\subsection{Stochastic Frenet Equations Approach}\nIn this section, we propose an approach to study conformational states of a semiflexible polymer immersed in a thermal reservoir and confined to a two-dimensional Euclidean space. We start by postulating that each conformational realization of any polymer on the plane is described by a stochastic path satisfying the stochastic Frenet equations defined by\n\\begin{subequations}\n\\label{ecsestom1}\n\\begin{align}\n\\label{ecsesto0}\n\\frac{d}{ds}{\\bf R}\\left(s\\right)=&{\\bf T}(s),\\\\\n\\frac{d}{ds}{\\bf T}\\left(s\\right)=&{\\kappa}(s){\\boldsymbol{\\epsilon}}{\\bf T}(s),\n\\label{ecsesto}\n\\end{align}\n\\end{subequations}\nwhere  ${\\bf R}(s)$, ${\\bf T}(s)$ and $\\kappa(s)$ are now random variables. According to   this postulate, it can be show that $\\left|{\\bf T}(s)\\right|$ is a constant that can be fixed to unit. \n\nIn addition we postulate that $\\kappa(s)$ for semiflexible polymers is a random variable, named here stochastic curvature, and is distributed according to the following probability density\n\\begin{eqnarray}\n\\mathcal{P}[\\kappa]\\mathcal{D}\\kappa:=\\frac{1}{Z}\\exp\\left[-\\beta H\\right]\\mathcal{D}\\kappa, \n\\label{dft}\n\\end{eqnarray}\nwhere $H$ is given by Eq.~\\eqref{funcional},  $Z$ is an appropriate normalization constant, $\\mathcal{D}\\kappa$ is a functional measure, and $\\beta=1\/k_{B}T$   the inverse of the thermal energy $k_{B}T$, with $k_{B}$ and $T$ being the Boltzmann constant and the absolute temperature, respectively. It is also convenient to introduce the persistence length by $\\ell_{p}=\\beta\\alpha$. Note that due to the Gaussian structure of the probability density \\eqref{dft}, the stochastic curvature satisfies the following fluctuation theorem \\cite{Fal-Zinn1996}\n\\begin{subequations}\n\\label{flucm1}\n\\begin{align}\n\\label{fluc0}\n\\left<\\kappa(s)\\kappa(s^{\\prime})\\right>=&\\frac{1}{\\ell_{p}}\\delta(s-s^{\\prime}),\\\\\n\\left<\\kappa(s)\\right>=&0. \n\\label{fluc}\n\\end{align}\n\\end{subequations}\n \nSince the polymer is confined to a plane and  ${\\bf T}(s)$ is a unit vector, then it may be written as ${\\bf T}(s)=(\\cos\\theta(s), \\sin\\theta(s))$, where $\\theta(s)$ is another random variable.  In this way, the stochastic equations \\eqref{ecsestom1} can be rewritten in the following manner\n\\begin{subequations}\n\\label{stochasticecs}\n\\begin{align}\n\\label{stochasticecs0}\n\\frac{d}{ds}{\\bf R}\\left(s\\right)=&\\left(\\cos\\theta(s),\\sin\\theta(s)\\right),\\\\\n\\frac{d}{ds}\\theta\\left(s\\right)=&\\kappa(s).\n\\label{stochasticecs1}\n\\end{align}\n\\end{subequations}\nThe most important feature of these equations is their analogy with the Langevin equations for an active particle in the overdamped limit, where the noise is introduced through the stochastic curvature $\\kappa(s)$ \\cite{Fal-Sevilla2014}. Moreover, these equations can be studied through  traditional numerical methods, for example, using standard routines implemented in Brownian dynamics \\cite{Fal-Ermak1978}.  Here, from an analytical viewpoint, we find it is more convenient to use a Fokker-Planck formalism in order to extract information of the above stochastic equations  (\\ref{stochasticecs}).  \n\n\\subsection{From Frenet Stochastic Equations to the Hermans-Ullman Equation}\n\nIn this section, we present the Fokker-Planck formalism corresponding  to the stochastic equations \\eqref{stochasticecs}. This description consists of determining the equation that governs the probability density function defined by\n\\begin{eqnarray}\nP(\\left.{\\bf R}, \\theta\\right|{\\bf R}^{\\prime}, \\theta^{\\prime}; s)=\\left<\\delta({\\bf R}-{\\bf R}(s))\\delta(\\theta-\\theta(s))\\right>, \n\\end{eqnarray}\nwhere ${\\bf R}$ and ${\\bf R}^{\\prime}$ are the ending positions of the polymer, and the angles $\\theta$ and $\\theta^{\\prime}$ are their corresponding directions, respectively. The parameter $s$ is the polymer length.  \n\nApplying the standard procedure described in Refs. \\cite{Fal-Zinn1996, Fal-Gardiner1986} on the stochastic equations \\eqref{stochasticecs}, we obtain the corresponding Fokker-Planck equation\n\\begin{eqnarray}\n\\frac{\\partial P}{\\partial s}+\\nabla\\cdot\\left({\\bf t}\\left(\\theta\\right)P\\right)=\\frac{1}{2\\ell_{p}}\\frac{\\partial^2P}{\\partial\\theta^2},\n\\label{F-P}\n\\end{eqnarray}\nwhere ${\\bf t}(\\theta)=\\left(\\cos\\theta,\\sin\\theta\\right)$ and $\\nabla$ is the gradient operator with respect to ${\\bf R}$.  Let us look carefully at the last equation. Surprisingly, this equation is exactly the equation found by J. J. Hermans and R. Ullman in 1952 \\cite{PSaito-Hermans1952}. They derived it supposing that the conformation of a polymer is determined by Markovian walks, taking the mean values of $\\theta$ and $\\theta^2$ as phenomenological parameters. These are parameters based on the X-ray dispersion experiments performed by Kratky and Porod \\cite{PSaito-Kratky1949}.   For this reason, from now on, we name \\eqref{F-P}  the Hermans-Ullman (HU) equation. It must be mentioned that H. E. Daniels found an equivalent equation few months before Hermans and Ullman \\cite{PSaito-Daniels1952}. A revision of the methods used to obtain the HU equation can be found in Refs. \\cite{Wlc-Yamakawa2016, Polcon-Chen2016}. For instance, taking into account \\eqref{funcional}, HU can be derived through the Green formalism \\cite{Polcon-Chen2016}. In contrast, in the present work, we have deduced the Hermans-Ullman equation considering two postulates, namely, I) the conformation of the semiflexible polymer satisfy the Frenet stochastic equations \\eqref{ecsestom1}, and II) the stochastic curvature is distributed according to \\eqref{dft}, which is consistent with the worm-like chain model (\\ref{funcional}). As far as we know, this procedure has not been reported in the literature. \n \n To end this section, let us remark, as it is pointed  out in   \\cite{Wlc-Yamakawa2016, Polcon-Chen2016}, that\n    \\begin{eqnarray}\n  \\int d^{2}{\\bf R}d^{2}{\\bf R}^{\\prime}~P(\\left.{\\bf R}, \\theta\\right|{\\bf R}^{\\prime}, \\theta_{0}; s)\\propto\\mathbb{Z}\\left(\\theta,\\theta_{0}, s\\right),\n  \\end{eqnarray} \n where $\\mathbb{Z}\\left(\\theta,\\theta_{0}, s\\right)$ is  the marginal probability density function (see appendix \\ref{A}), which establishes a bridge to the formalism in  N. Sait$\\hat{\\rm o}$ et al. \\cite{PSaito-Saito1967} for the semiflexible polymer in the thermal bath.\n\n\\subsection{Multipolar decomposition for the Hermans-Ullman Equation} \\label{sect3}\nIt is necessary to emphasize that the HU equation naturally arises in the description of the motion of an active particle. Thus, being careful with the right interpretation, the methods developed in Refs. \\cite{Fal-Sevilla2014, Fal-Castro-Villarreal2018} to solve Eq.~\\eqref{F-P} can be applied in this context.  Particularly, we use the multipolar expansion approach to solve Eq.~\\eqref{F-P}, which in the orthonormal Cartesian basis $\\{1, 2t_{i}, 4(t_{i}t_{j}-\\frac{1}{2}\\delta_{ij}), \\cdots\\}$, takes the following form  \\cite{ Fal-Castro-Villarreal2018}\n\\begin{eqnarray}\nP({\\bf R},\\theta, s)&=&\\rho({\\bf R}, s)+2\\mathbb{P}_{i}({\\bf R}, s) {t_{i}}\\nonumber\\\\&+&4\\mathbb{Q}_{ij}\\left({\\bf R},s\\right)\\left(t_{i}t_{j}-\\frac{1}{2}\\delta_{ij}\\right)\\nonumber\\\\\n&+&8\\mathbb{R}_{ijk}\\left({\\bf R}, s\\right)\\left(t_{i}t_{j}t_{k}-\\frac{1}{4}\\delta_{\\left(ij\\right.}t_{\\left.k\\right)}\\right)+\\cdots,\\nonumber\\\\\n\\end{eqnarray}\nwhere we have adopted the Einstein summation convention, and the symbol $(ijk)$ means symmetrization on the indices $i,j,k$. The coefficients of the series are multipolar tensors given by\n \\begin{eqnarray}\n\\rho({\\bf R}, s)&=&\\int_{0}^{2\\pi} \\frac{d\\theta}{2\\pi}~P({\\bf R},\\theta, s)\\nonumber,\\\\ \n\\mathbb{P}_{i}({\\bf R}, s)&=&\\int_{0}^{2\\pi} \\frac{d\\theta}{2\\pi}~{t}_{i}~ P({\\bf R}, \\theta, s),\\nonumber\\\\ \n\\mathbb{Q}_{ij}({\\bf R}, s)&=&\\int_{0}^{2\\pi} \\frac{d\\theta}{2\\pi}~\\left(t_{i}t_{j}-\\frac{1}{2}\\delta_{ij}\\right)P({\\bf R}, \\theta, s), \\nonumber\\\\\n\\mathbb{R}_{ijk}({\\bf R}, s)&=&\\int_{0}^{2\\pi} \\frac{d\\theta}{2\\pi}~\\left(t_{i}t_{j}t_{k}-\\frac{1}{4}\\delta_{\\left(ij\\right.}t_{\\left.k\\right)}\\right)P({\\bf R}, \\theta, s), \\nonumber\\\\\n&\\vdots&.\n\\end{eqnarray}\nIn the latter coefficients, we have ignored the $\\theta$ dependence of the vector ${\\bf t}$ for reasons of notation. We also have obviated the dependence on ${\\bf R}^{\\prime}$ and $\\theta^{\\prime}$ to improve notation. The physical meaning of these tensors is as follows: $\\rho({\\bf R}, s)$ is the probability density function (PDF) of finding configurations with ends at ${\\bf R}$ and ${\\bf R}^{\\prime}$, $\\mathbb{P}({\\bf R}, s)$ \nis the local average of the polymer conformational direction, $\\mathbb{Q}_{ij}\\left({\\bf R}, s\\right)$ \nis the correlation between the components $i$ and $j$ of the polymer direction ${\\bf t}$, etc.\n\nFrom Hermans-Ullman Eq.~\\eqref{F-P}, it is possible to determine hierarchy equations for the multipolar tensors, which have already been shown for active particles in Refs. \\cite{Fal-Sevilla2014, Fal-Castro-Villarreal2018}.\nThe same hierarchy equations can also be found in the semiflexible polymer context. Integrating over the angle $\\theta$ in Eq.~\\eqref{F-P}, we obtain the following continuity-type equation\n\\begin{eqnarray}\n\\frac{\\partial \\rho({\\bf R}, s)}{\\partial s}=-\\partial_{i}\\mathbb{P}^{i}\\left({\\bf R}, s\\right). \n\\label{eq1}\n\\end{eqnarray}\nThe related equation for $\\mathbb{P}_{i}\\left({\\bf R},s\\right)$ is obtained by multiplying Eq.~\\eqref{F-P} by ${\\bf t}(\\theta)$, and using the definition of the tensor $\\mathbb{Q}_{ij}({\\bf R}, s)$. Thus, we found\n\\begin{eqnarray}\n\\frac{\\partial \\mathbb{P}_{i}({\\bf R}, s)}{\\partial s}=-\\frac{1}{2\\ell_{p}}\\mathbb{P}_{i}({\\bf R}, s)-\\frac{1}{2}\\partial_{i}\\rho({\\bf R},s)-\\partial^{j}\\mathbb{Q}_{ij}({\\bf R}, s).\\nonumber\\\\\n\\label{eq2}\n\\end{eqnarray}\nIn the same way, we obtain the equation for $\\mathbb{Q}_{ij}({\\bf R}, s)$,\n\\begin{eqnarray}\n\\frac{\\partial\\mathbb{Q}_{ij}({\\bf R}, s)}{\\partial s}=-\\frac{2}{\\ell_{p}}\\mathbb{Q}_{ij}({\\bf R}, s)-\\frac{1}{4}\\mathbb{T}_{ij}({\\bf R}, s)-\\partial^{k}\\mathbb{R}_{ijk}({\\bf R}, s), \\nonumber\\\\\n\\end{eqnarray}\nwhere $\\mathbb{T}_{ij}$ denotes the second rank tensor $-\\delta_{ij}\\partial_{k}\\mathbb{P}^{k}+\\left(\\partial^{i}\\mathbb{P}^{j}+\\partial^{j}\\mathbb{P}^{k}\\right)$. Similarly, the equations for the rest of tensorial fields can be computed recursively for consecutive ranks. Taking a combination of \\eqref{eq1} and \\eqref{eq2}, we observe that the PDF $\\rho({\\bf R}, s)$ and the two-rank tensor $\\mathbb{Q}_{ij}({\\bf R}, s)$ are involved in one equation given by\n\\begin{eqnarray}\n\\frac{\\partial^2\\rho({\\bf R}, s)}{\\partial s^2}+\\frac{1}{2\\ell_{p}}\\frac{\\partial\\rho({\\bf R}, s)}{\\partial s}=\\frac{1}{2}\\nabla^2\\rho({\\bf R}, s)+\\partial_{i}\\partial_{j}Q^{ij}({\\bf R}, s).\\nonumber\\\\\n\\label{eq3}\n\\end{eqnarray}\n\nIt is noteworthy to mention, that Eq. (\\ref{eq3}) is a modified version of Telegrapher's equation \\cite{Fal-Masoliver1989}, where the term $\\partial_{i}\\partial_{j}Q^{ij}({\\bf R}, s)$ makes the difference.  In the following, we use Eq. (\\ref{eq3}) for the case of a semiflexible polymer in the open Euclidean plane as a test case. This allows us to verify the famous experimental result of Kratky and Porod \\cite{PSaito-Kratky1949} using this procedure.\n\n\\subsubsection{Example: Testing the Kratky-Porod result.}\\label{sectII}\nIn this section, we study the case of a semiflexible polymer on the Euclidean plane. In order to reproduce the well known result of Kratky-Porod \\cite{PSaito-Kratky1949}, we apply the multipolar series method shown in the previous section to compute the mean square end-to-end distance.\nThe end-to-end distance is defined as $\\delta{\\bf R}:={\\bf R}-{\\bf R}^{\\prime}$, thus the mean square end-to-end distance is given by \n\\begin{equation}\n \\left<\\delta{\\bf R}^2\\right>\\equiv\\int_{\\mathbb{R}^{2}\\times\\mathbb{R}^{2}}\\rho({\\bf R}|{\\bf R}^{\\prime}; s)\\delta{\\bf R}^2~d^{2}{\\bf R}~d^2{\\bf R}^{\\prime}.\n \\label{MS}\n \\end{equation}\nTo compute this quantity, we use Eq.~\\eqref{eq3} to show  that l.h.s of (\\ref{MS}) satisfies \n  {\\small\\begin{eqnarray}\n\\frac{\\partial^2\\left<\\delta{\\bf R}^2\\right>}{\\partial s^2}+\\frac{1}{2\\ell_{p}}\\frac{\\partial\\left<\\delta{\\bf R}^2\\right>}{\\partial s}&=&\\int d^{2}{\\bf R}~d^2{\\bf R}^{\\prime}\\left(\\delta{\\bf R}\\right)^2\\times\\nonumber\\\\&&\\left[\\frac{1}{2}\\nabla^2\\rho({\\bf R}, s)+\\partial_{i}\\partial_{j}Q_{ij}({\\bf R}, s)\\right].\\nonumber\\\\\n\\label{eq4}\n\\end{eqnarray}}\nIntegrating by parts on the r.h.s. of \\eqref{eq4}  with respect to ${\\bf R}$, using that $\\nabla^2\\delta{\\bf R}^2=4$ and the traceless condition $\\delta_{ij}\\mathbb{Q}^{ij}=0$, we have that  $\\left<\\delta{\\bf R}^2\\right>$ satisfies the differential equation\n\n\\begin{eqnarray}\n\\frac{\\partial^2\\left<\\delta{\\bf R}^2\\right>}{\\partial s^2}+\\frac{1}{2\\ell_{p}}\\frac{\\partial\\left<\\delta{\\bf R}^2\\right>}{\\partial s}=2.\n\\label{eq5}\n\\end{eqnarray}\nNow, we solve this differential equation with the initial conditions, for $s=0$, $\\left<\\delta{\\bf R}^2\\right>=0$ and $\\frac{d}{ds}\\left<\\delta{\\bf R}^2\\right>=0$. The final polymer length is denoted by $L$. \n\nIn this way, we found that the mean square end-to-end distance is given by\n\\begin{eqnarray}\n\\left<\\delta{\\bf R}^2\\right>=4\\ell_{p}L-8\\ell_{p}^2\\left(1-\\exp\\left(-\\frac{L}{2\\ell_{p}}\\right)\\right),\n\\label{planoabierto}\n\\end{eqnarray}\nwhich is the standard Kratky-Porod result for semiflexible polymers confined to a plane \n\\cite{PSaito-Kratky1949, PSaito-Hermans1952}. The last result has two well-known asymptotic limits, namely, \n\\begin{eqnarray}\n\\left<\\delta{\\bf R}^2\\right>\\simeq\\left\\{\\begin{array}{cc}\n 4\\ell_{p}L, & {\\rm if }~L\\gg \\ell_{p},\\\\\n &\\label{asymptotics}\\\\ \n L^2, & {\\rm if}~L\\ll\\ell_{p}.\\\\\n \\end{array}\\right.\n\\end{eqnarray}\n\nIn the first case, the polymer conformations are equivalent to brownian trajectories. In this case, the polymer is called Gaussian polymer \\cite{PDoi-Doi1988book}. In the second case, the polymer takes only one configuration; it goes in a straight line, which is known as the ballistic limit. We remark that the result in Eq.~\\eqref{planoabierto} is usually obtained by using different analytical approaches (for example, see Appendix \\ref{A} and Refs. \\cite{MGD-Kamien2002, PSaito-Saito1967}).\n  \n  In the next section, we address the study of a confined polymer to a flat compact domain within the approach developed above.\n\n\\section{Semiflexible polymer in a compact plane domain }\n\n\\subsection{General expressions for a semiflexible polymer in an arbitrary compact domain}\\label{sectTE}\nIn this section, we apply the hierarchy equations developed in section \\ref{sect3} in order to determine the conformational states of a semiflexible polymer confined to a flat compact domain $\\mathcal{D}$.  Commonly, it is necessary to truncate the hierarchy equations at some rank. For instance, at first order, let us consider  $\\mathbb{P}_{i}({\\bf R}, s)$ as a constant vector field, then (\\ref{eq1}) implies  that  $\\rho({\\bf R}, s)$ is uniformly distributed, which clearly it is not an accurate description because otherwise it means that the mean square end-to-end distance would be a constant for all values of the polymer length $s$. An improved approximation consists of taking the truncation on the second hierarchy rank, which corresponds to assume that $\\mathbb{Q}_{ij}({\\bf R}, s)$ is uniformly distributed. Indeed, the truncation approximation gets better the larger the polymer length is, since as it is pointed out in \\cite{Fal-Castro-Villarreal2018} from Eqs. (\\ref{eq1}) and (\\ref{eq2}) one can conclude that the tensors $\\mathbb{P}_{i}({\\bf R}, s)$ and $\\mathbb{Q}_{ij}({\\bf R}, s)$ damp out as $e^{-L\/(2\\ell_{p})}$ and $e^{-2 L\/\\ell_{p}}$, respectively. From these expressions, clearly $\\mathbb{Q}_{ij}({\\bf R}, s)$ damps out more strongly than $\\mathbb{P}_{i}({\\bf R}, s)$ for larger polymer length. In the polymer context, it means that the tangent directions of the polymer are uniformly correlated.\n\n In the following, let us define a characteristic length $a$ associated to the size of the compact domain $\\mathcal{D}$, thus if we scale polymer length $s$ with $a$ one can consider $2a\/\\ell_{p}$ as a dimensionless attenuation coefficient associated to the damp out of $\\mathbb{Q}_{ij}({\\bf R}, s)$.  Thus as long as we consider cases when $2a\/\\ell_{p}$ far from 1, we may  neglect the contribution of $\\mathbb{Q}_{ij}({\\bf R}, s)$. Here, we are going to consider this latter case, therefore according to (\\ref{eq3}), the Telegrapher's equation is the one considered as the governing equation of the PDF $\\rho(\\left.{\\bf R}\\right|{\\bf R}^{\\prime}, s)$, that is, \n\\begin{eqnarray}\n\\frac{\\partial^2\\rho({\\bf R}, s)}{\\partial s^2}+\\frac{1}{2\\ell_{p}}\\frac{\\partial\\rho({\\bf R}, s)}{\\partial s}=\\frac{1}{2}\\nabla^2\\rho({\\bf R}, s),\n\\label{eq6}\n\\end{eqnarray}\nwith the initial conditions \n\\begin{subequations}\n\\label{IC}\n\\begin{align}\n\\label{ICa}\n\\lim_{s\\to 0}\\rho(\\left.{\\bf R}\\right|{\\bf R}^{\\prime}, s)=&\\delta^{\\left(2\\right)}({\\bf R}-{\\bf R}^{\\prime}),\\\\\n\\lim_{s\\to 0}\\frac{\\partial \\rho(\\left.{\\bf R}\\right|{\\bf R}^{\\prime}, s)}{\\partial s}=&0.\n\\label{ICb}\n\\end{align}\n\\end{subequations}\nThese conditions have the following physical meaning. Clearly, Eq.~\\eqref{ICa} means that the polymer ends coincide when the polymer length is zero, whereas Eq.~\\eqref{ICb} means that the polymer length does not change spontaneously. Since the polymer is confined to a compact domain, we also impose a Neumann boundary condition\n\\begin{eqnarray}\n\\left.\\nabla \\rho\\left(\\left.{\\bf R}\\right|{\\bf R}^{\\prime}, s\\right)\\right|_{{\\bf R}, {\\bf R}^{\\prime}\\in\\partial \\mathcal{D}}&=&0, ~~~~\\forall s, \n\\label{BC}\n\\end{eqnarray}\nwhere $\\partial \\mathcal{D}$ is the boundary of $\\mathcal{D}$. \nThis boundary condition means that the polymer does not cross the boundary coating the domain. \n\nTo solve the differential equation \\eqref{eq6}, we use the standard separation of variables \\cite{Fal-Feshbach1953}.  This method requires to solve the so-called Neumann eigenvalue problem. It consists of finding all possible real values $\\lambda$, for which there exists a non-trivial solution $\\psi\\in C^{2}(\\mathcal{D})$ that satisfies the eigenvalue equation $-\\nabla^2\\psi=\\lambda\\psi$, and the Neumann boundary condition \\eqref{BC}.  In this case, the set of eigenvalues is a sequence $\\lambda_{{\\bf k}}$ with ${\\bf k}$ in a numerable set $I$, and each associated eigenspace is finite dimensional. These latter eigenespaces are orthogonal to each other in the space of square-integrable functions $L^2(\\mathcal{D})$ \\cite{Fal-Chavel1984, Fal-Feshbach1953}. That is, the sequence $\\lambda_{\\bf k}$ is associated with the set of eigenfunctions $\\{ \\psi_{\\bf k}({\\bf R})\\}$ that satisfy the orthonormal relation\n\\begin{eqnarray}\n\\int_{\\mathcal{D}}\\psi_{\\bf k}\\left({\\bf R}\\right)\\psi_{\\bf k^{\\prime}}\\left({\\bf R}\\right)d^{2}{\\bf R}=\\delta_{{\\bf k}, {\\bf k}^{\\prime}}. \\label{ortogonal}\n\\end{eqnarray}\n\n Next, we expand the probability density function $\\rho(\\left.{\\bf R}\\right|{\\bf R}^{\\prime}, s)$ in a linear combination of those eigenfunctions $\\{ \\psi_{\\bf k}({\\bf R})\\}$, that is, a spectral decomposition  $\\rho(\\left.{\\bf R}\\right|{\\bf R}^{\\prime}; s)=\\sum_{{\\bf k}}g_{\\bf k}(s)\\psi_{\\bf k}({\\bf R})\\psi_{\\bf k}({\\bf R}^{\\prime})$. Substituting this series in the Telegrapher's  equation \\eqref{eq6}, we find that the functions $g_{\\bf k}(s)$ satisfy the following ordinary differential equation\n\\begin{eqnarray}\n\\frac{d^2g_{\\bf k}(s)}{ds^2}+\\frac{1}{2\\ell_{p}}\\frac{dg_{\\bf k}(s)}{ds}+\\frac{1}{2}\\lambda_{\\bf k}g_{\\bf k}(s)=0, \n\\label{eq8}\n\\end{eqnarray}\nwhere the initial conditions, (\\ref{IC}), imply $g_{\\bf k}(0)=1$, and $dg_{\\bf k}(0)\/ds=0$. Therefore, the solution is given by \n\\begin{eqnarray}\ng_{\\bf k}(s)=G\\left(\\frac{s}{4\\ell_{p}}, 8\\ell_{p}^2\\lambda_{\\bf k}\\right),\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n{\\scriptsize G(v, w)=e^{-v}\\left[\\cosh\\left(v\\sqrt{1-w}\\right)+ \\frac{\\sinh\\left(v\\sqrt{1-w}\\right)}{\\sqrt{1-w}}\\right]}.\\nonumber\\\\\n\\label{Gfunction}\n\\end{eqnarray}\n\nFinally, using the above information the probability density function is given by \n\\begin{eqnarray}\n\\rho(\\left.{\\bf R}\\right|{\\bf R}^{\\prime}, s)=\\frac{1}{A\\left(\\mathcal{D}\\right)}\\sum_{{\\bf k}\\in I}G\\left(\\frac{s}{4\\ell_{p}}, 8\\ell_{p}^2\\lambda_{\\bf k}\\right)\\psi_{\\bf k}({\\bf R})\\psi_{\\bf k}({\\bf R}^{\\prime}), \\nonumber\\\\\\label{pdf}\n\\end{eqnarray}\nwhere $A(\\mathcal{D})$ is the area of the domain $\\mathcal{D}$, which is needed in order to have a normalized probability density function in the space $\\mathcal{D}\\times{\\mathcal{D}}$. Then, we have that $\\rho(\\left.{\\bf R}\\right|{\\bf R}^{\\prime}, s)d^{2}{\\bf R}d^{2}{\\bf R}^{\\prime}$ is the probability of having a polymer in a conformational state with polymer length $s$, and ends at ${\\bf R}$ and ${\\bf R}^{\\prime}$. Additionally, using the expression (\\ref{pdf}), the mean square end-to-end distance can be computed in the standard fashion by\n\\begin{eqnarray}\n\\left<\\delta{\\bf R}^2\\right>_{\\mathcal{D}}=\\sum_{{\\bf k}\\in I}a_{k}~G\\left(\\frac{s}{4\\ell_{p}}, 8\\ell_{p}^2\\lambda_{\\bf k}\\right), \n\\label{gen-sol}\n\\end{eqnarray}\nwhere the coefficients $a_k$ are obtained by \n\\begin{eqnarray}\na_{k}=\\int_{\\mathcal{D}\\times{\\mathcal{D}}}\\left({\\bf R}-{\\bf R}^{\\prime}\\right)^2 \\psi_{\\bf k}({\\bf R})\\psi_{\\bf k}({\\bf R}^{\\prime})~d^{2}{\\bf R}~d^{2}{\\bf R}^{\\prime}.\n\\label{coef}\n\\end{eqnarray}\n\nIn the following, we shall discuss the specific case when the polymer is enclosed in a square box.\n\n\\subsection{Example: semiflexible polymer in a square domain}\\label{thdomain}\n\nIn this section, we study the case when the semiflexible polymer is enclosed  in a square box $\\mathcal{D}=\\left[0,a\\right]\\times\\left[0, a\\right]$. For this domain, it is well known that the eigenfunctions of the Laplacian operator $\\nabla^2$ correspond to a combination of products of trigonometric functions \\cite{Fal-Chavel1984}. That is, for each pair of positive integer numbers $(n, m)$ and positions ${\\bf R}=(x,y)\\in\\mathcal{D}$, it is not difficult to show  that the eigenfunctions of the Laplacian operator consistent with (\\ref{BC}) are \n\\begin{eqnarray}\n\\psi_{\\bf k}\\left({\\bf R}\\right)=\\frac{2}{a}\\cos\\left(\\frac{\\pi n}{a}x\\right)\\cos\\left(\\frac{\\pi m}{a}y\\right).\\nonumber\n\\end{eqnarray}\nThese functions constitute a complete orthonormal basis, that satisfy \\eqref{ortogonal}. The corresponding eigenvalues are  $\\lambda_{\\bf k}={\\bf k}^2$, with ${\\bf k}=\\left(\\frac{\\pi n}{a}, \\frac{\\pi m}{a}\\right)$. \n\nNow, we proceed to determine the coefficients $a_{\\bf k}$ \\-using Eq.~\\eqref{coef} in  order to give an expression for the mean square end-to-end distance. By straightforward calculation, the coefficients $a_{\\bf k}$ are given explicitly by \n\\begin{eqnarray}\na_{\\bf k}=\\left\\{\\begin{array}{cc}\n\\frac{1}{3}a^2,  & {\\bf k}=0,\\\\\n-\\frac{4a^2}{\\pi^4}\\left(\\frac{\\left(1-(-1)^{n}\\right)}{n^4}\\delta_{m,0}+ \\frac{\\left(1-(-1)^{m}\\right)}{m^4}\\delta_{n,0}\\right), & {\\bf k}\\neq 0.\n\\end{array}\\right.\\nonumber\\\\\n\\end{eqnarray}\nUpon substituting the latter coefficients in the general expression \\eqref{gen-sol}, we found that\n\\begin{eqnarray}\n\\frac{\\left<\\delta{\\bf R}^2\\right>_{\\mathcal{D}}}{a^2}=\\frac{1}{3}-\\sum_{n\\in 2\\mathbb{N}+1 }\\frac{32}{\\pi^4 n^4}G\\left(\\frac{L}{4\\ell_{p}}, 8\\pi^2\\left(\\frac{\\ell_{p}}{a}\\right)^2n^2\\right),\\nonumber\\\\\n\\label{sol}\n\\end{eqnarray}\nwhere $2\\mathbb{N}+1$ is the set of odd natural numbers.  Since the function $G(v,w)$ satisfy that $G(v, w)\\leq 1$ for all positive real numbers $v$ and $w$, the series in Eq.~\\eqref{sol} is convergent for all values of $L\/\\ell_{p}$ y $\\ell_{p}\/a$. Considering this last property, it is possible to prove the following assertions.\n\n\\begin{prop} \\label{result1}\nLet $L\/\\ell_{p}$  any positive non-zero real number, then the mean square end-to-end distance \\eqref{sol} obeys \n\\begin{eqnarray}\n \\lim_{\\ell_{p}\/a\\to 0}\\frac{\\left<\\delta{\\bf R}^2\\right>_{\\mathcal{D}}}{\\ell^2_{p}}=\\frac{4L}{\\ell_{p}}-8\\left(1-\\exp\\left(-\\frac{L}{2\\ell_{p}}\\right)\\right). \\nonumber\n\\end{eqnarray}\n\\end{prop}\n\n\\begin{prop}\\label{result2} Let $L\/\\ell_{p}$ and $\\ell_{p}\/a$ any positive non-zero real numbers and $c=2\/3-64\/\\pi^4$, then the mean square end-to-end distance \\eqref{sol} obeys\n\\begin{eqnarray}\n 0\\leq \\frac{\\left<\\delta{\\bf R}^2\\right>_{\\mathcal{D}}}{a^2}\\leq \\frac{2}{3},\\nonumber\\end{eqnarray}\nand\n\\begin{eqnarray}\n 0\\leq \\frac{\\left<\\delta{\\bf R}^2\\right>_{\\mathcal{D}}}{a^2}-\\left(\\frac{1}{3}-\\frac{1}{3}G\\left(\\frac{L}{4\\ell_{p}}, 8\\pi^2\\frac{\\ell^2_{p}}{a^2}\\right)\\right)\\leq c.\\nonumber\n\\end{eqnarray}\n\n\\end{prop}\n\nClaim {\\bf \\ref{result1}} recovers  the Kratky-Porod result about the mean square end-to-end distance (see Eq.~\\eqref{planoabierto}). Claim {\\bf\\ref{result2}} means that the mean-square end-to-end distance is bounded from below by $0$ and  is bounded from above by $2\/3 a^2$. In addition, this second claim also provides an approximation formula  for $\\left<\\delta{\\bf R}^2\\right>_{\\mathcal{D}}$, that is,  for all values of $L\/\\ell_{p}$ and $\\ell_{p}\/a$ such that the condition\n\\begin{eqnarray}\n1-G\\left(\\frac{L}{4\\ell_{p}}, 8\\pi^2\\frac{\\ell^2_{p}}{a^2}\\right)\\gg 3c\\label{cond}\n\\end{eqnarray}\nholds, one has the following approximation\n{\\small\\begin{eqnarray}\n\\frac{\\left<\\delta{\\bf R}^2\\right>_{\\mathcal{D}}}{a^2}&\\simeq&\\frac{1}{3}-\\frac{1}{3}\\exp\\left(-\\frac{L}{4\\ell_{p}}\\right)\\nonumber\\\\\n&\\times&\\left\\{ \\cosh\\left[\\frac{L}{4\\ell_{p}}\\left(1-8\\pi^2\\frac{\\ell^2_{p}}{a^2}\\right)^{\\frac{1}{2}}\\right]\\right.\\nonumber\\\\&+&\\left.\\left(1-8\\pi^2\\frac{\\ell^2_{p}}{a^2}\\right)^{-\\frac{1}{2}}\\sinh\\left[\\frac{L}{4\\ell_{p}}\\left(1-8\\pi^2\\frac{\\ell^2_{p}}{a^2}\\right)^{\\frac{1}{2}}\\right]\\right\\}.\\nonumber\\\\ \n\\label{approx2}\n\\end{eqnarray}}\nLet us point out,  the validity of this approximation occurs provided that the condition (\\ref{cond}) holds, that is, whenever one can neglect the value $c$ (See Appendix \\ref{coefficients} for proofs of claims 1 and 2). \n\nIn the following, let us remark that for any fixed value of $a$, the r.h.s. of (\\ref{approx2}), as a function of $L$, shows the existence of a critical persistence length, $\\ell^{*}_{p}\\equiv a\/(\\pi\\sqrt{8})$, such that for all values $\\ell_{p}>\\ell^{*}_{p}$ it exhibits an oscillating behavior, whereas for $\\ell_{p}<\\ell^{*}_{p}$, it is monotonically increasing. In addition, for each value of $\\ell_{p}$ the function (\\ref{approx2}) converges to $1\/3$ as long as $L\\gg a$. The critical persistence length, therefore,  distinguishes two conformational behaviors of the semiflexible polymer enclosed in the square box. \nIn Fig. \\ref{MSD-ex}, the mean-square end-to-end distance, Eq. (\\ref{sol}) and r.h.s. of (\\ref{approx2}),  have been shown  for the ratios $\\ell_{p}\/a=1\/32, 1\/16,1\/8, 1\/4,1\/2, 1$ where we can appreciate both conformational states. Furthermore, one of the most intriguing features of the above approximation (\\ref{approx2}) is its similar structure to the corresponding one in the case of a polymer wrapping a spherical surface. Indeed, let us remark that \\eqref{approx2} has the same mathematical structure that the mean square end-to-end distance found by Spakowitz and Wang in \\cite{PSaito-Spakowitz2003}, exhibiting both conformational states . \n\nIn the next section, we address the study of the semiflexible polymer  through a Monte Carlo algorithm in order to corroborate the results found here.\n\n\n\n\n\\section{Monte Carlo-Metropolis algorithm for semiflexible polymers  }\\label{mc}\n\nHere, we develop a Monte Carlo algorithm to be use in computer simulations in order to study the conformational states of a semiflexible polymer enclosed in a compact domain. Particularly, the algorithm shall be suited to the square domain which, additionally,  will allow us to validate our analytical approximations shown in the latest section.\n \nAs we have emphasized above, the worm-like chain model is the suitable framework to describe the spatial distribution of semiflexible polymers, which are modeled as $n$ beads consecutively connected by $n-1$ rigid bonds, called Kuhn's segments \\cite{Fal-Yamakawa1971}. Each bead works like a pivot allowing us to define an angle $\\theta_i$ between two consecutive bonds, where $i$ is the label of the $i-$th bead. This model requires a potential energy description where all possible contributions due to bead-bead, bead-bond and bond-bond interactions are taken into account. In a general setting, energies of bond-stretching, elastic bond angle, electrostatic interaction, torsional potential, etc., should be considered, such as in Refs.~\\cite{Wlc-Qiu2016, Psim-Chirico1994, Psim-Allison1989,Psim-DeVries2005,Psim-Jian1997,Psim-Jian1998}. However, here we are interested solely on the study of  possible spatial configurations of a single semiflexible polymer enclosed in a compact domain $\\mathcal{D}$,  such as the one shown in Fig. \\ref{frame}. Thus in our case we only take into account  two energetic contributions, namely, the elastic bond angle and the  wall-polymer interaction. The first contribution, that is, the elastic bond angle is given by\n\\begin{equation}\nE_b=\\frac{g}{2}\\sum_i \\theta_i^2,\n\\end{equation}\nwhere $\\theta_i$ is the angle between two consecutively bonds, and $g=\\alpha\/l_{0}$, where we recall that  $\\alpha$ is the bending rigidity, and $l_{0}$ is the Kuhn length. In addition, we must consider the  wall-polymer interaction given by\n\\begin{eqnarray}\n  E_{w}=\n  \\begin{cases}\n                                   0, & \\text{if all beads are in $\\mathcal{D}$}, \\\\\n                                   \\infty, & \\text{if there are beads outside of $\\mathcal{D}$}.\n  \\end{cases}\n\\end{eqnarray}\n\nIn the algorithm, the acceptance changes criteria of the polymer spatial configurations take the structure of a Gaussian distribution function. In this context, we generate random chains enclosed in $\\mathcal{D}$, constituted by $N$ bonds with constant Kuhn length, implementing a growth algorithm. Our computational realization consists of bead generation attending the following conditions:\n{\\it starting bead}, {\\it beads far from walls}, {\\it beads near to walls}, and {\\it selection problem}\nwhich are explained in the following subsections. \n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[scale=.45]{contornob}\n\\caption{Generic compact domain $\\mathcal{D}$ is shown, where the boundary wall $\\partial\\mathcal{D}$ is represented by the black continuous line. The condition defining the beads near to the wall is represented by the blue filled region of width $l_s$. }\n\\label{frame}\n\\end{figure}\n\n\n\\subsection*{Starting bead}\n\nThis condition  describes the process of initial bead generation and the acceptance criteria of the second bead. Let us regard, without loss of generality that the origin of $\\mathbb{R}^2$ belongs to $\\mathcal{D}$. We choose the initial bead  at $\\mathbf{x}_0$ as a uniformly distributed random point in the region $\\mathcal{D}$. We define the auxiliary vector $\\mathbf{R}_{l_0}=(l_0,0)$, which is parallel to the horizontal axis, it will allow us to determine if the next bead is inside $\\mathcal{D}$. Also, we consider the angle $\\theta_0$ as the one that is formed between the horizontal axis and the first bead, which is taken from a uniform distribution in the interval $[0,2\\pi]$. Now, we compute the following vector\n\\begin{equation}\n\\mathbf{R}'=\\mathcal{R}\n\\left(\\theta_0\\right){\\mathbf{R}_{l_0}}^T,\n\\end{equation}\nwhere $\\mathcal{R}\n\\left(\\theta_0\\right)$\ndenotes the two-dimensional rotational matrix by an angle $\\theta_0$, defined as\n\\begin{equation}\n\\mathcal{R}\\left(\\theta_0\\right)=\n\\begin{bmatrix}\n\\cos \\theta_0 & -\\sin \\theta_0 \\\\\n\\sin \\theta_0 & \\cos \\theta_0 \\\\\n\\end{bmatrix},\n\\end{equation}\nand the superscript denotes the transposition of $\\mathbf{R}_{l_0}$. The resultant vector $\\mathbf{x}_0+\\mathbf{R}'^T$ will be the position of the second bead only if this is inside $\\mathcal{D}$. If this happened,  the new bead will be denoted by $\\mathbf{x}_1$ as well as the vector $\\mathbf{R}_{l_0}=\\mathbf{R}'^T$ is actualized. On the contrary, we repeat the process until finding an angle that satisfies the condition that the second bead is enclosed in the domain.\n\n\\subsection*{Beads far from walls}\\label{bffw}\n\nThis condition describes the method of subsequence bead generation. We say that a bead is far from walls if the perpendicular distance between the boundary $\\partial\\mathcal{D}$ and the bead is greater than a particular distance $l_{s}$. In this case, if the $(k-1)$-th bead satisfies this condition, we generate the subsequent bead taking a random angle $\\theta_{k}$ distributed according to a Gaussian density function\n $\\mathcal{N}(0,l_{0}\/\\ell_{p})$, where we recall $\\ell_{p}$ as the polymer persistence length. As in the previous condition, we compute the vector $\\mathbf{R}'=\\mathcal{R}\\left(\\theta_0\\right){\\mathbf{R}_{l_0}}^T$ corresponding to the $k$-th rotation of $\\mathbf{R}_{l_0}$. If $l_0<l_{s}$,\n the resultant vector $\\mathbf{x}_{k-1}+\\mathbf{R}'^T$ is in $\\mathcal{D}$ for any angle. Therefore, all rotations are accepted, so we assign the position $\\mathbf{x}_{k}$ to the $k$-th bead. \n\n\\subsection*{Beads near walls}\n\nThis condition describes the interaction between the polymer and the walls. The most important problem to solve here is the smooth bending of the chain when the polymer is near the walls. In a similar fashion as in the previous condition, we say that a bead is near the walls if the perpendicular distance from $\\partial\\mathcal{D}$ to the bead is smaller than $l_{s}$.  This condition looks like a frame surrounding the boundary of $\\mathcal{D}$ (see blue filled region in Fig \\ref{frame}). The scalar product $\\mathbf{\\hat{n}}\\cdot\\mathbf{R'}^T $  is used to seek the generation of a new bead, where $\\mathbf{\\hat{n}}$ is the normal vector of $\\partial\\mathcal{D}$, and $\\mathbf{R'}^{T}$ is the orientation of the bead with position $\\mathbf{x}_{k}$. In this region, we favor the smooth bending of the polymer taking into account the following rules. If $\\mathbf{\\hat{n}}\\cdot\\mathbf{R'}^T \\geq0$, the generation of the new bead is going away from the wall. In this case, we generate the next bead according to second condition. If $\\mathbf{\\hat{n}}\\cdot\\mathbf{R'}^T <0$, the new bead generated approaches the wall. In this case, we promote the bending of the chain generating angles with the same Gaussian function as in the previous condition. However, we do the rotation by an angle $-\\text{sgn}(R'^T_2)|\\theta_k|$, where $\\text{sgn}$ is the sign function, and $R'^T_2$ is the projection of $\\mathbf{R'}^T$ over the perpendicular vector to $\\mathbf{\\hat{n}}$ (a $\\pi\/2$ rotation of $\\mathbf{\\hat{n}}$). The sign ``$-$'' appears there because we do counterclockwise rotations. Also, we need to check that the resultant vector $\\mathbf{x}_{k-1}+\\mathbf{R}'^T$ is  in $\\mathcal{D}$, if this is the case, we assign the position $\\mathbf{x}_k$ to the $k-$th bead. Otherwise, we generate angles $\\theta_{ki}$ with a Gaussian distribution function $\\mathcal{N}(0,l_0\/\\ell_p)$ until the next bead be in $\\mathcal{D}$, doing rotation by angles\n\\begin{equation}\n\\theta_k=-\\text{sgn}(R'^T_2)\\sum_{i=1}^m|\\theta_{ki}|,\n\\end{equation}\nwhere $m$ is the number of angles generated until the next bead is in $\\mathcal{D}$. In Fig.~\\ref{bend}, we show a schematic representation of this conditions. Dashed arrows (green and red for positive and negative rotations, respectively) are the promoted direction according to the sign of the projection of $\\mathbf{R}'^T$ over the perpendicular vector to  $\\mathbf{\\hat{n}}$ (blue arrow, denoted by $\\mathbf{\\hat{n}}_\\bot$).\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[scale=.6]{bending}\n\\caption{Local schematic representation of the bending conditions when the beads are near to walls and $\\mathbf{\\hat{n}}\\cdot\\mathbf{R'}^T <0$.}\n\\label{bend}\n\\end{figure}\n\nFinally, these steps are repeated $n-$times to generate a polymer of $n$ bonds (or $n+1$ beads) taking care of polymer-walls interaction. This algorithm allows us to generate the spatial configurations for confined polymers into a compact domain. It is clear that the total length of a polymer of $n$ bonds is $L=nl_0$. We denote the $\\mathbf{x}(L)$ of the last bead after $n$ bonds so the mean square end-to-end distance is computed as $\\left<{\\delta \\bf R}^2\\right>_{\\mathcal{D}}=\\langle (\\mathbf{x}(L)-\\mathbf{x}_0)^2 \\rangle$.  In Sec.~\\ref{results} we shall analyze the results obtained for $\\left<{\\delta \\bf R}^2\\right>_{\\mathcal{D}}$ using this algorithm as a function of the persistence length and the polymer length when $\\mathcal{D}$ is a square box.\n\n\\subsection*{Selection problem }\n\nThe selection problem consist of choosing the adequate value of $l_{s}$. This value should be suitable to avoid the over or under bending of the polymer. For instance, if $\\ell_p$ is comparable with the size of $\\mathcal{D}$, and $l_s$ is not appropriate to promote the chain bending when the polymer is near the boundary $\\partial\\mathcal{D}$, the generation of beads outside of the domain will be favorable.  Therefore, the polymer will present bendings with high values of angles where the chain meets the boundary. The selection problem is resolved using the dimensional analysis of $\\left<\\delta{\\bf R}^2\\right>_{\\mathcal{D}}$. Indeed, observe that in the continous limit of the chain, the mean-square end-to-end distance can be written as $\\left<\\delta{\\bf R}^2\\right>_{\\mathcal{D}}=a^2 g(\\ell_{p}\/a, L\/a)$, where $g(\\ell_{p}\/a, L\/a)$ is a dimensionless function. Then, we choose $l_{s}$ such that the mean-square end-to-end distance computed with the simulation data depends just on this combination $\\ell_{p}\/a$ and $L\/a$. In other words,  for $k$ we calculate $k$ profiles of $\\left<\\delta{\\bf R}^2\\right>_{\\mathcal{D}}\/a^2$ for a $k$ number of pairs $(\\ell^{1}_{p}, a^{1}),(\\ell^{2}_{p}, a^{2}), \\cdots, (\\ell^{k}_{p}, a^{k})$, with $\\ell^{i}_{p}\/a^{i}$  fixed for all $i=1,\\cdots, k$, then we choose $l_{s}$ such that all these profiles collapse in a unique curve. \n\n\\section{Semiflexible polymers enclosed in a square box: simulation vs analytical results}\\label{results}\nIn this section, we are going to implement the algorithm explained in the preceding section for the particular case of a polymer enclosed in a square box of side  $a$. In this case, let us first note that for beads near the corners, checking the conditions to promote the chain bending for both adjacent walls at the same time is needed. \nNext, in the simulation we set up our unit length  by $d=10^{2}~l_{0}$. Now, we have to present the selection of $l_{s}$ according to the last part of the general algorithm. Thus, for the fixed ratios $\\ell_{p}\/a=1\/50, 1\/32, 1\/16,1\/8, 1\/4,1\/2, 1$, we study three profiles corresponding to the values $a\/d=5,10,15$, respectively. In Table \\ref{tab1} the selection values $l_{s}$ are shown once we collapse the three profiles in a unique curve. \n\n\\begin{table}[h!]\n\\caption{\\label{tab1}Values of $l_s$ used in simulations for different values of persistence length $\\ell_p$.}\n\\begin{tabular}{|c| c|}\n\\hline\n$\\ell_p\/a$  & $ l_s\/a$\\\\\n\\hline\n1 & 0.085\\\\\n1\/2 &  0.065\\\\\n1\/4 &  0.050\\\\\n1\/8 &  0.040\\\\\n1\/16 &  0.010\\\\\n1\/32 &  0.005\\\\\n$\\leq$1\/50 & 0 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\nThe results shown in this section were computed as the average over $10^6$ spatial configuration of confined polymers, which were obtained using the algorithm described in the previous section. In particular, we shall study two respective regimes defined through the comparison between the box side $a$ and the polymer length $L$. The first one, polymers in weak confinement whenever $L\/a\\leq 1$ is discussed in subsection~\\ref{noconfinado}. The second one,  polymers in strong confinement, corresponding to the situation when the polymer length is larger than the box side, is  discussed in subsection ~\\ref{confinado}.\n\n\n\n\n\\subsection{Polymer in weak confinement}\\label{noconfinado}\n\nIn this regime, once we have solved the selection problem of $l_{s}$ we present the simulation results for polymers enclosed in a square box of side $a=10~d$ for different values of persistence lengths. In Fig.~\\ref{ex-short}, examples of semiflexible polymers in weak confinement are shown. Notice that for very short persistence length, $\\ell_p\\simeq l_0$, the chain looks like a very curly string  like random walks, whereas when $\\ell_p$ increases, the polymer adopts uncoiled configurations.\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[scale=1.]{ex-shortb}\n\\caption{Examples of semiflexible polymers, with length $L=a$, in the weak confinement regime for several values of persistence length. Solid black lines represent the walls of the box.}\n\\label{ex-short}\n\\end{figure} \n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[scale=1]{test-MSD-short2b}\n\\caption{Mean square end-to-end distance for confined polymers (point) with several persistence lengths. The deviation from the predictions for a polymer in a infinite plane (solid lines) given by Eq.~\\eqref{planoabierto} is because on average the chain meets the polymer at lengths $L\\simeq a$.}\n\\label{fig-MSDs}\n\\end{figure}\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[scale=1.]{MSD-short-weakb}\n\\caption{Mean square end-to-end distance for polymers in the gaussian chain limit generated by the algorithm described in Sec.~\\ref{mc}.}\n\\label{gchain}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[scale=1.]{MSD-Wangb}\n\\caption{Universal behavior of the Mean Square end-to-end distance in Fig.~\\ref{fig-MSDs} in the scaling $\\langle {\\delta \\bf R}^2 \\rangle\/ l_p^2$ vs $L\/l_p$ (solid lines). Dashed lines are references for the ballistic and diffusive regimes for polymers in weak confinement.}\n\\label{fig-MSDs-Wang}\n\\end{figure}\n\n\n\n\n\nThe mean-square end-to-end distance is shown in Fig. \\ref{fig-MSDs}, where  it is noted that it increases as long as $\\ell_p$ increases, allowing the polymer to explore more surface as a function of the total polymer length. Also, notice that for small polymer lengths, the mean square end-to-end distance is in an excellent agreement with the results for semiflexible polymers in an infinite plane given by Eq.~\\eqref{planoabierto}. Conversely, for polymer lengths around $L\\simeq a$ we observe an slight deviation between the mean square end-to-end distance and the infinite plane solution (\\ref{planoabierto}), because of the finite-size of the box. Notwithstanding, for small persistence lengths ($\\ell_p\\simeq 10^{-3}a$),  the mean square end-to-end distance is well fitted by Eq.~\\eqref{planoabierto}. In this case, the polymer does not seem to be affected by the walls, since the  area explored by the chain is too short, on average, to meet the walls.\n\nFurthermore, when the mean square end-to-end distance and the polymer length is scaled by $l_p^2$ and $l_p$, respectively, we found that the data shown in Fig.~\\ref{fig-MSDs} and Fig.~\\ref{gchain} are collapsed, respectively, into an unique plot shown in Fig.~\\ref{fig-MSDs-Wang}, evidencing a ballistic behavior for small values of $L\/\\ell_p$ followed by a ``diffusive'' regime for large values of $L\/\\ell_{p}$. These results correspond to the well-known asymptotic limits of the Kratky-Porod result (\\ref{asymptotics}). In addition, it is noteworthy to mention that these asymptotic limits are also  reported in \\cite{PSaito-Spakowitz2003} for a semiflexible polymer wrapping a spherical surface in the corresponding plane limit.  \n\n\\subsection{Polymer in strong confinement}\\label{confinado}\n\nIn this section, we discuss the case of a polymer enclosed in a square box when its length is large enough to touch the walls, and to interact several times with them.  We perform simulations in order to generate polymers of lengths up to $L\/a=10$ (chains of $10^4$ beads) for persistence lengths $\\ell_{p}\/a=1\/32, 1\/16,1\/8, 1\/4,1\/2, 1$ for the values of  box side $a\/d=5,10,$ and $15$, respectively. These simulations use the values of $l_s$ shown in Table~\\ref{tab1}. Examples of these polymers are shown in  Fig. \\ref{evo-ex} and in Fig.~\\ref{MSD-ex}. \n\\begin{figure}[h!]\n\\centering\n\\includegraphics[scale=1]{test10b}\n\\caption{Semiflexible polymer realizations for the value of persistence length $l_p=a$. Each column shows a polymer with a particular  length from the left to the right  with L\/a=2, L\/a=4, L\/a=6, L\/a=8, and L\/a=10, respectively. Black filled circles indicate the endings of the polymer. It is also shown how the polymer rolling up around the square box while the polymer length becomes larger. }\n\\label{evo-ex}\n\\end{figure}\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=.68]{MSD-ex4b}\n\\caption{\\small Examples of polymers, in the first and third columns, are shown as solid green lines, where the initial and final beads are represented by black filled circles, and solid black lines represent the walls of the box. The figure also shows, in the second and fourth columns, the mean-square end-to-end distance for polymers in strong confinement as a function of the polymer length: The bold black line represents the superposition of the theoretical prediction (Eqs. (\\ref{sol}) and (\\ref{approx2})) with the simulation results shown for different box sides $a\/d=5$ (blue triangles), $a\/d=10$ (red squares) and $a\/d=15$ (green circles). }\n\\label{MSD-ex}\n\\end{figure*}\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[scale=1]{msd-Wang-strong22b}\n\\caption{\\small Mean square end-to-end distance for polymers in strong confinement as a function of the persistence length. Note that $\\langle \\delta \\mathbf{R} ^2 \\rangle$ shows an oscillating behavior for values of persistence length satisfying the relation $\\ell_p\/a>8$, which is the same signature for the mean square end-to-end distance for polymers confined into a sphere. Dashed and doted lines has been plotted as references for the ballistic and diffusive behaviors, respectively.}\n\\label{MSD-Wang-strong}\n\\end{figure}\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[scale=1]{convb}\n\\caption{\\small Convergence of the mean square end-to-end distance $\\langle \\delta \\mathbf{R}^2\\rangle_{\\mathcal{D}}=a^2\/3$ for very large polymers in strong confinement as a function of the ratio $a\/l_p$.}\n\\label{MSD-conv}\n\\end{figure}\n\n\nIn Fig.~\\ref{MSD-ex}, we also report the mean square end-to-end distance scaled by $a^2$ as a function of $L\/a$ for the different box sides ($a=$ 5, 10 and 15). In addition,  in Fig. \\ref{MSD-Wang-strong} we report as well the mean square end-to-end distance scaled by $\\ell_{p}^2$ as a function of $L\/\\ell_{p}$.   By simple inspection, in both cases an oscillating behavior is exhibited for values $\\ell_p\/a>\\frac{1}{8}$, whereas a monotonically increasing behavior becomes evident for persistence lengths such that $\\ell_{p}\/a<\\frac{1}{8}$. A growth in the number of oscillations is observed while $\\ell_{p}\/a$ increases from $1\/8$. In addition, for values $\\ell_{p}\/a$ less than $1\/8$ the behavior of the mean-square end-to-end distance corresponds to the one of a Gaussian polymer enclosed in a box and the corresponding conformational realization of the polymer looks like a confined random walk. As was mentioned before this transition between the oscillating and monotonic behavior of the conformation of the semiflexible polymer is very similar to that described by Wang and Spakowitz in Ref.~\\cite{PSaito-Spakowitz2003} for semiflexible polymer confined to a spherical surface. Moreover, as noted in Fig. \\ref{MSD-Wang-strong}  in the small polymer lengths regime, the mean square end-to-end distance shows a ballistic behavior, followed by a brief interval with a ``diffusive'' behavior.  Furthermore, an interesting observation is that the mean square end-to-end distance exhibits asymptotic plateau behavior for large values of $L\/l_p$ as a function of $a\/l_p$. In Fig.~\\ref{MSD-conv} we show the value of the mean square end-to-end distance for the the polymer length $L=10~a$ in logarithmic scale as a function of $a\/l_p$ in binary logarithmic scale. In this conditions, the plateau behavior is well fitted by a linear function, where the slope, $m$, of the line satisfies $m\/\\log2=1.97\\sim 2$, whereas that the intercept takes the value $b=-0.473\\pm0.006$. This last fact leads us to a universal scaling law of the mean square end-to-end distance regarding the box side for very large polymers:\n\\begin{equation}\n\\frac{\\langle \\delta \\mathbf{R}(L=10a)^2\\rangle}{a^2}=10^b\\sim 0.336\\pm 0.004,\n\\label{eq-msd-a}\n\\end{equation}\nwhere the error has been computed as the  propagation error from the linear fit of the data in Fig.~\\ref{MSD-conv}. This result is the universal convergence of the rate $\\langle \\delta \\mathbf{R}^2\\rangle_{\\mathcal{D}}\/a^2$ to $1\/3$ for very large polymers, which becomes independent of the box side when the quotient $l_p\/a$ keeps fixed. This is comprised, if we consider all available space in the box occupied  so that  a uniform distribution of beads occurs. Indeed, through the definition (\\ref{MS}) with $\\rho(\\left. {\\bf R}\\right|{\\bf R}^{\\prime}; L\\to\\infty)=1\/a^4$ one can get the desired result. Finally, it is noticeable that by simple inspection of the five polymer realizations shown in Fig. \\ref{evo-ex}, for  $\\ell_{p}\/a=1$, all have the same conspicuous relation between the period of oscillation and the number of turns that the polymer performs. \n\nAll these features of the behavior of the mean-square end-to-end distance are reproduced by the theoretical prediction (\\ref{approx2}) and (\\ref{sol}). In particular, it is significant to note that the critical persistence length found in our earlier discussion (see section (\\ref{thdomain})) satisfies approximately $\\ell^{*}_{p}\/a=1\/(\\pi\\sqrt{8})\\approx 1\/8$. We also note that for $\\ell_{p}\/a=1$ there is a slight discrepancy between the simulations results and the theoretical prediction (\\ref{sol}) appearing in the three local minima shown in Fig. \\ref{MSD-ex}. This is  due to the fact that for values of $\\ell_{p}\/a$ near $2$ there is a breakdown of the Telegrapher approximation that we performed in the section \\ref{sectTE}.  In other words, the small disagreement appears for $\\ell_{p}\/a\\approx 1$ because the role of the tensor $\\mathbb{Q}_{ij}$ becomes important and it can not be neglected in Eq. (\\ref{eq3}). \n\n\n\n\n\n\\section{Concluding remarks and perspectives}\\label{conclusions}\n\nIn this paper, we have analyzed the conformational states of a semiflexible polymer enclosed in a compact domain. The approach followed rests on two postulates, namely, that the conformation of a semiflexible polymer satisfies the Frenet stochastic equations \\eqref{ecsestom1}, and the stochastic curvature is distributed according to \\eqref{dft}, which is consistent with the worm-like chain model. In addition, it turned out that the Fokker-Planck equation, corresponding to the stochastic Frenet equations,  is exactly the same as the Hermans-Ullman equation (see Eq. (\\ref{F-P})) \\cite{PSaito-Hermans1952}. Furthermore, taking advantage of the analogy between the Hermans-Ullman equation and the Fokker-Planck equation for a free active particle motion \\cite{Fal-Castro-Villarreal2018}, we establish a multipolar decomposition for the probability density function, $P(\\left.{\\bf R}, \\theta\\right|{\\bf R}^{\\prime}, \\theta^{\\prime}; L)$, that describes  the manner in which a polymer with length $L$ distributes in the domain with certain endings, ${\\bf R}$ and ${\\bf R}^{\\prime}$, and their associated directions $\\theta$ and $\\theta^{\\prime}$, respectively.  In consequence, exploiting this analogy we provide an approximation for the positional distribution $\\rho({\\bf R}, {\\bf R}^{\\prime}, L)$ through Telegrapher's Equation, which for a compact domain is a good approximation as long as $2a\/\\ell_{p}>1$, where $a$ is a characteristic length of the compact domain.  In particular, we derive results for a semiflexible polymer enclosed in a square box domain, where we can give a mathematical formula for the {\\it mean-square end-to-end distance}.  \n\nFurthermore, we have developed a Monte Carlo-Metropolis algorithm to study the conformational states of a semiflexible polymer enclosed in a compact domain. In particular, for the square box domain, we compare the results of the simulation with the theoretical predictions finding an excellent agreement. Particularly, we have considered two situations, namely, a {\\it polymer in weak confinement} and a {\\it polymer in strong confinement} corresponding to polymers with length lesser and greater than the box side, respectively. In the weak confinement case, we reproduce the two-dimensional solution of a free chain i.e. the  Kratky-Porod result for polymers confined in two-dimensions. In the strong confinement case, we showed the existence of a critical persistent length $\\ell^{*}_{p}\\simeq a\/8$ such that for all values $\\ell_{p}>\\ell^{*}_{p}$ the mean-square end-to-end distance exhibits an oscillating behavior, whereas for $\\ell_{p}<\\ell^{*}_{p}$, it is monotonically increasing. In addition, for each value of $\\ell_{p}$ the function converges to $1\/3$ as long as $L\\gg a$. The critical persistence length, thus,  distinguishes two conformational behaviors of the semiflexible polymer enclosed in the square box. As was mentioned above, this result is the same type to the one found by Wang and Spakowitz in \\cite{PSaito-Spakowitz2003} for a semiflexible polymer wrapping a spherical surface.  As a consequence of this resemblance, one can conclude that the shape transition from oscillating to monotonic conformational states provides evidence of a universal signature for a semiflexible polymer enclosed in a compact space. \n\nOur approach can be extended in various directions. For instance, the whole formulation can be extended easily to semiflexible polymers in three dimensions.  Although,  we must consider an stochastic version of the Frenet-Serret equations, now in this case, one would obtain the three dimensional case of the Hermans-Ullman equation, because the worm-like chain model involves just the curvature. In addition, the approach developed here can also be extended to the case where the semiflexible polymer wraps a curved surface.  \n\n\\section*{Acknowledgement}\nP.C.V. acknowledges financial support by CONACyT Grant No. 237425 and PROFOCIE-UNACH 2017. J.E.R. acknowledges financial support from VIEP-BUAP (grant no. VIEP2017-123). The computer simulations were performed at the LARCAD-UNACH.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSeveral experimental collaborations at $ep$ and $p\\overline{p}$ colliders\npresented data on the differential cross section $d^2\\sigma\/dy\\,dp_T$ for the\ninclusive production of $D^0$, $D^+$, and $D_s^+$ mesons, $\\Lambda_c^+$\nbaryons, and their charge-conjugate counterparts.\nAt DESY HERA, such data were collected by the ZEUS Collaboration\n\\cite{zeus,pad} in low-$Q^2$ $ep$ collisions, equivalent to photoproduction,\nand by the H1 Collaboration \\cite{h1} in deep-inelastic $ep$ scattering.\nAt the Fermilab Tevatron, such data were taken by the CDFII Collaboration\n\\cite{cdf} in $p\\overline{p}$ collisions.\n\nOn the theoretical side, fragmentation functions (FF's) for the transitions\n$c,b\\to X_c$, where $X_c$ denotes a generic charmed hadron, are needed as\nnonperturbative inputs for the calculation of all the cross sections mentioned\nabove.\nSuch FF's are preferably constructed by using precise information from\n$e^+e^-\\to X_c+X$ via $e^+e^-$ annihilation at the $Z$-boson resonance, where \n$X$ denotes the hadronic rest.\nIn this process, two mechanisms contribute with similar rates:\n(i) $Z\\to c\\overline{c}$ decay followed by $c\\to X_c$ (or\n$\\overline{c}\\to X_c$) fragmentation; and\n(ii) $Z\\to b\\overline{b}$ decay followed by $b\\to X_b$ (or\n$\\overline{b}\\to X_b$) fragmentation and weak $X_b\\to X_c+X$ decay of the\nbottom-flavored hadron $X_b$.\nThe latter two-step process is usually treated as a one-step fragmentation\nprocess $b\\to X_c$.\n\nUsing ALEPH \\cite{aleph} and OPAL \\cite{opal} data on inclusive $D^{*+}$\nproduction at the $Z$-boson resonance, we determined separate FF's for\n$c\\to D^{*+}$ and $b\\to D^{*+}$ in collaboration with Binnewies \\cite{bkk}.\nIt is the purpose of this work to extract nonperturbative FF's for\n$c,b\\to D^0,D^+,D_s^+,\\Lambda_c^+$ from the respective data samples collected\nby the OPAL Collaboration at LEP1 \\cite{opal1} using the same theoretical\nframework as in Ref.~\\cite{bkk}.\n\nThe work in Ref.~\\cite{bkk} is based on the QCD-improved parton model\nimplemented in the modified minimal-subtraction ($\\overline{\\mathrm{MS}}$)\nrenormalization and factorization scheme in its pure form with $n_f=5$\nmassless quark flavors, which is also known the as the massless scheme\n\\cite{spira} or zero-mass variable-flavor-number scheme.\nIn this scheme, the masses $m_c$ and $m_b$ of the charm and bottom quarks are\nneglected, except in the initial conditions of their FF's.\nThis is a reasonable approximation for center-of-mass (c.m.) energies\n$\\sqrt s\\gg m_c,m_b$ in $e^+e^-$ annihilation or transverse momenta\n$p_T\\gg m_c,m_b$ in $ep$ and $p\\overline{p}$ scattering, if the respective\nFF's are used as inputs for the calculation of the cross sections for these\nreactions.\nHence, we describe the $c,b\\to X_c$ transitions by nonperturbative FF's, as is\nusually done for the fragmentation of the up, down, and strange quarks into\nlight hadrons.\n\nThe outline of this paper is as follows.\nIn Sec.~\\ref{sec:two}, we briefly recall the theoretical framework underlying\nthe extraction of FF's from the $e^+e^-$ data, which has already been\nintroduced in Refs.~\\cite{bkk,bkk1}.\nIn Sec.~\\ref{sec:three}, we present the $D^0$, $D^+$, $D_s^+$, and\n$\\Lambda_c^+$ FF's we obtained by fitting the respective LEP1 data samples\nfrom OPAL \\cite{opal1} at leading order (LO) and next-to-leading order (NLO)\nin the massless scheme and discuss their properties.\nIn Sec.~\\ref{sec:four}, we present predictions for the inclusive production of\nthese $X_c$ hadrons in nonresonant $e^+e^-$ annihilation at lower c.m.\\\nenergies and compare them with data from other experiments.\nOur conclusions are summarized in Sec.~\\ref{sec:five}.\n\n\\section{Theoretical Framework}\n\\label{sec:two}\n\nOur procedure to construct LO and NLO sets of $D$ FF's has already been\ndescribed in Refs.~\\cite{bkk,bkk1}.\nAs experimental input, we use the LEP1 data from OPAL \\cite{opal1}.\n\nIn $e^+e^-$ annihilation at the $Z$-boson resonance, $X_c$ hadrons are\nproduced either directly through the hadronization of charm quarks produced \nby $Z\\to c\\overline{c}$ or via the weak decays of $X_b$ hadrons from\n$Z\\to b\\overline{b}$.\nIn order to disentangle these two production modes, the authors of\nRef.~\\cite{opal1} utilized the apparent decay length distributions and energy\nspectra of the $X_c$ hadrons.\nBecause of the relatively long $X_b$-hadron lifetimes and the hard $b\\to X_b$\nfragmentation, $X_c$ hadrons originating from $X_b$-hadron decays have\nsignificantly longer apparent decay lengths than those from primary production.\nIn addition, the energy spectrum of $X_c$ hadrons originating from\n$X_b$-hadron decays is much softer than that due to primary charm production. \n\nThe experimental cross sections \\cite{opal1} were presented as distributions\ndifferential in $x=2E(X_c)\/\\sqrt s$, where $E(X_c)$ is the measured energy of\nthe $X_c$-hadron candidate, and normalized to the total number of hadronic\n$Z$-boson decays.\nBesides the total $X_c$ yield, which receives contributions from $Z\\to c\\bar c$\nand $Z\\to b\\bar b$ decays as well as from light-quark and gluon fragmentation,\nthe OPAL Collaboration separately specified results for $X_c$ hadrons from\ntagged $Z\\to b\\bar b$ events.\nAs already mentioned above, the contribution due to charm-quark fragmentation\nis peaked at large $x$, whereas the one due to bottom-quark fragmentation has\nits maximum at small $x$.\n\nFor the fits, we use the $x$ bins in the interval $[0.15,1.0]$ and integrate\nthe theoretical cross sections over the bin widths used in the experimental\nanalysis.\nFor each of the four charmed-hadron species considered here,\n$X_c=D^0,D^+,D_s^+,\\Lambda_c^+$, we sum over the two charge-conjugate states as\nwas done in Ref.~\\cite{opal1}.\nAs a consequence, there is no difference between the FF's of a given quark\nand its antiquark.\nAs in Refs.~\\cite{bkk,bkk1}, we take the starting scales for the $X_c$ FF's of\nthe gluon and the $u$, $d$, $s$, and $c$ quarks and antiquarks to be\n$\\mu_0=2m_c$, while we take $\\mu_0=2m_b$ for the FF's of the bottom quark and\nantiquark.\nThe FF's of the gluon and the first three flavors are assumed to be zero at\ntheir starting scale.\nAt larger scales $\\mu$, these FF's are generated through the usual\nDokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) \\cite{dglap} evolution at LO\nor NLO.\nThe FF's of the first three quarks and antiquarks coincide with each other at\nall scales $\\mu$.\n\nWe employ two different forms for the parameterizations of the charm- and\nbottom-quark FF's at their respective starting scales.\nIn the case of charm, we use the distribution of Peterson {\\it et al.}\\\n\\cite{pet},\n\\begin{equation}\nD_c(x,\\mu_0^2)=N\\frac{x(1-x)^2}{[(1-x)^2+\\epsilon x]^2}.\n\\label{eq:peterson}\n\\end{equation}\nIn the case of bottom, we adopt the ansatz\n\\begin{equation}\nD_b(x,\\mu_0^2)=Nx^{\\alpha}(1-x)^{\\beta},\n\\label{eq:standard}\n\\end{equation}\nwhich is frequently used for the FF's of light hadrons.\nEquation~(\\ref{eq:peterson}) is particularly suitable for FF's that peak at\nlarge values of $x$, as is typically the case for $c\\to X_c$ transitions.\nSince the $b\\to X_c$ FF is a convolution of the $b\\to X_b$ fragmentation and\nthe subsequent $X_b\\to X_c+X$ decay, it has its maximum at small $x$ values.\nTherefore, Eq.~(\\ref{eq:peterson}) is less suitable in this case.\nWe apply Eqs.~(\\ref{eq:peterson}) and (\\ref{eq:standard}) for the FF's of all \nfour $X_c$-hadron species considered here.\n\nThe calculation of the cross section $(1\/\\sigma_{\\rm tot})d\\sigma\/dx$ for\n$e^+e^-\\to\\gamma\/Z\\to X_c+X$ is performed as described in Ref.~\\cite{bkk}, in\nthe pure $\\overline{\\mathrm{MS}}$ subtraction scheme, {\\it i.e.}, without the\nsubtraction terms $d_{Qa}(x)$ specified in Eq.~(2) of Ref.~\\cite{kks}.\nAll relevant formulas and references may be found in Ref.~\\cite{bkk1}.\nAs for the asymptotic scale parameter for five active quark flavors, we adopt\nthe LO (NLO) value $\\Lambda_{\\overline{\\rm MS}}^{(5)}=108$~MeV (227~MeV) from\nour study of inclusive charged-pion and -kaon production \\cite{bkk2}.\nThe particular choice of $\\Lambda_{\\overline{\\rm MS}}^{(5)}$ is not essential,\nsince other values can easily accommodated by slight shifts of the other fit\nparameters.\nAs in Refs.~\\cite{bkk,bkk1}, we take the charm- and bottom-quark masses to be  \n$m_c=1.5$~GeV and $m_b=5$~GeV, respectively.\n\n\\boldmath\n\\section{Determination of the $D^0$, $D^+$, $D_s^+$, and $\\Lambda_c^+$ FF's}\n\\label{sec:three}\n\\unboldmath\n\nThe OPAL Collaboration \\cite{opal1} presented $x$ distributions for their full\n$D^0$, $D^+$, $D_s^+$, and $\\Lambda_c^+$ samples and for their $Z\\to b\\bar b$\nsubsamples.\nWe received these data in numerical form via private communication\n\\cite{martin}.\nThey are displayed in Figs.~4 (for the $D^0$ and $D^+$ mesons) and 5 (for the\n$D_s^+$ meson and the $\\Lambda_c^+$ baryon) of Ref.~\\cite{opal1} in the form\n$(1\/N_{\\rm had})dN\/dx$, where $N$ is the number of $X_c$-hadron candidates\nreconstructed through appropriate decay chains.\nIn order to convert this into the cross sections\n$(1\/\\sigma_{\\rm tot})d\\sigma\/dx$, we need to divide by the branching \nfractions of the decays that were used in Ref.~\\cite{opal1} for the\nreconstruction of the various $X_c$ hadrons, namely,\n\\begin{eqnarray}\nB(D^0\\to K^-\\pi^+)&=&(3.84\\pm0.13)\\%,\\nonumber\\\\\nB(D^+\\to K^-\\pi^+\\pi^-)&=&(9.1\\pm0.6)\\%,\\nonumber\\\\\nB\\left(D_s^+\\to \\phi\\pi^+\\right)&=&(3.5\\pm0.4)\\%,\\nonumber\\\\\nB\\left(\\Lambda_c^+\\to pK^-\\pi^+\\right)&=&(4.4\\pm0.6)\\%,\n\\end{eqnarray}\nrespectively.\nThe experimental errors on these branching fractions are not included in our\nanalysis.\n\nThe values of $N$ and $\\epsilon$ in Eq.~(\\ref{eq:peterson}) and of $N$,\n$\\alpha$, and $\\beta$ in Eq.~(\\ref{eq:standard}) which result from our LO and\nNLO fits to the OPAL data are collected in Table~\\ref{tab:par}.\nFrom there, we observe that the parameters $\\alpha$ and $\\beta$, which\ncharacterize the shape of the bottom FF, take very similar values for the\nvarious $X_c$ hadrons, which are also similar to those for the $D^{*+}$\nmeson listed in Table~I of Ref.~\\cite{bkk}.\nOn the other hand, the values of the $\\epsilon$ parameter, which determines\nthe shape of the charm FF, significantly differ from particle species to\nparticle species.\nIn the $D^{*+}$ case \\cite{bkk}, our LO (NLO) fits to ALEPH \\cite{aleph} and\nOPAL \\cite{opal} data, which required separate analyses, yielded\n$\\epsilon=0.144$ (0.185) and 0.0851 (0.116), respectively.\nWe observe that, for each of the $X_c$-hadron species considered, the LO\nresults for $\\epsilon$ are considerably smaller than the NLO ones.\nFurthermore, we notice a tendency for the value of $\\epsilon$ to decrease as\nthe mass ($m_{X_c}$) of the $X_c$ hadron increases. \n\n\\begin{table}\n\\begin{center}\n\\caption{Fit parameters of the charm- and bottom-quark FF's for the various\n$X_c$ hadrons at LO and NLO.\nThe corresponding starting scales are $\\mu_0=2m_c=3$~GeV and \n$\\mu_0=2m_b=10$~GeV, respectively.\nAll other FF's are taken to be zero at $\\mu_0=2m_c$.}\n\\label{tab:par}\n\\begin{tabular}{ccccccc}\n\\hline\\hline\n$X_c$ & Order & $Q$ & $N$ & $\\alpha$ & $\\beta$ & $\\epsilon$ \\\\\n\\hline\n$D^0$         & LO  & $c$ & 0.998  & --   & --   & 0.163  \\\\\n              &     & $b$ & 71.8   & 1.65 & 5.19 & --     \\\\\n              & NLO & $c$ & 1.16   & --   & --   & 0.203  \\\\\n              &     & $b$ & 97.5   & 1.71 & 5.88 & --     \\\\\n$D^+$         & LO  & $c$ & 0.340  & --   & --   & 0.148  \\\\\n              &     & $b$ & 48.5   & 2.16 & 5.38 & --     \\\\\n              & NLO & $c$ & 0.398  & --   & --   & 0.187  \\\\\n              &     & $b$ & 64.9   & 2.20 & 6.04 & --     \\\\\n$D_s^+$       & LO  & $c$ & 0.0704 & --   & --   & 0.0578 \\\\\n              &     & $b$ & 40.0   & 2.05 & 4.93 & --     \\\\\n              & NLO & $c$ & 0.0888 & --   & --   & 0.0854 \\\\\n              &     & $b$ & 21.8   & 1.64 & 4.71 & --     \\\\\n$\\Lambda_c^+$ & LO  & $c$ & 0.0118 & --   & --   & 0.0115 \\\\\n              &     & $b$ & 44.1   & 1.97 & 6.33 & --     \\\\\n              & NLO & $c$ & 0.0175 & --   & --   & 0.0218 \\\\\n              &     & $b$ & 27.3   & 1.66 & 6.24 & --     \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIn Table~\\ref{tab:chi}, we list three values of $\\chi^2$ per degree of freedom\n($\\chi_{\\rm DF}^2$) for each of the fits from Table~\\ref{tab:par}:\none for the $Z\\to b\\overline{b}$ subsample, one for the total sample (sum of\ntagged-$c\\overline{c}$, tagged-$b\\overline{b}$, and gluon-splitting events),\nand an average one evaluated by taking into account the $Z\\to b\\overline{b}$\nsubsample and the total sample.\nThe actual $\\chi_{\\rm DF}^2$ values are rather small.\nThis is due to the sizeable errors and the rather limited number of data\npoints, especially for the $D_s^+$ and $\\Lambda_c^+$ data.\nIn each case, the $Z\\to b\\overline{b}$ subsample is somewhat less well\ndescribed than the total sample.\nThe NLO fits yield smaller $\\chi_{\\rm DF}^2$ values than the LO ones, except\nfor the $\\Lambda_c^+$ case.\n\n\\begin{table}\n\\begin{center}\n\\caption{$\\chi^2$ per degree of freedom achieved in the LO and NLO fits to the\nOPAL \\cite{opal1} data on the various $D$ hadrons.\nIn each case, $\\chi_{\\rm DF}^2$ is calculated for the $Z\\to b\\overline{b}$\nsample ($b$), the full sample (All), and the combination of both (Average).}\n\\label{tab:chi}\n\\begin{tabular}{ccccc}\n\\hline\\hline\n$X_c$ & Order & $b$ & All & Average \\\\\n\\hline\n$D^0$         & LO  & 1.16  & 0.688 & 0.924 \\\\\n              & NLO & 0.988 & 0.669 & 0.829 \\\\\n$D^+$         & LO  & 0.787 & 0.540 & 0.663 \\\\\n              & NLO & 0.703 & 0.464 & 0.584 \\\\\n$D_s^+$       & LO  & 0.434 & 0.111 & 0.273 \\\\\n              & NLO & 0.348 & 0.108 & 0.228 \\\\\n$\\Lambda_c^+$ & LO  & 1.05  & 0.106 & 0.577 \\\\\n              & NLO & 1.05  & 0.118 & 0.582 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nThe normalized differential cross sections $(1\/\\sigma_{\\rm tot})d\\sigma\/dx$\nfor $D^0$, $D^+$, $D_s^+$, and $\\Lambda_c^+$ hadrons (circles), extracted from\nRef.~\\cite{opal1} as explained above, are compared with our LO (upmost dashed\nlines) and NLO (upmost solid lines) fits in Figs.~\\ref{fig:xs}(a)--(d),\nrespectively.\nThe same is also done for the $Z\\to b\\overline{b}$ subsamples (squares).\nIn addition, our LO and NLO fit results for the $Z\\to c\\overline{c}$\ncontributions are shown.\nIn each case, the $X_c$ hadron and its charge-conjugate partner are summed\nover.\nFrom Figs.~\\ref{fig:xs}(a)--(d), we observe that the LO and NLO results are\nvery similar, except for very small values of $x$.\nThis is also true for the distributions at the starting scales, as may be seen\nby comparing the corresponding LO and NLO parameters in Table~\\ref{tab:par}.\nThe branching of the LO and NLO results at small values of $x$ indicates that,\nin this region, the perturbative treatment ceases to be valid. \nThis is related to the phase-space boundary for the production of $X_c$ hadrons\nat $x_{\\rm min}=2m_{X_c}\/\\sqrt s$.\nThese values are somewhat larger than the $x$ values where our NLO results turn\nnegative.\nSince our massless-quark approach is not expected to be valid in regions of\nphase space where finite-$m_{X_c}$ effects are important, our results should\nonly be considered meaningful for $x\\agt x_{\\rm cut}=0.1$, say.\nWe also encountered a similar small-$x$ behavior for the $D^{*+}$ FF's in\nRefs.~\\cite{bkk,bkk1}.\n\nAs mentioned above, we take the FF's of the partons\n$g,u,\\overline{u},d,\\overline{d},s,\\overline{s}$ to be vanishing at their\nstarting scale $\\mu_0=2m_c$.\nHowever, these FF's are generated via the DGLAP evolution to the high scale\n$\\mu=\\sqrt s$.\nThus, apart from the FF's of the heavy quarks $c,\\overline{c},b,\\overline{b}$,\nalso these radiatively generated FF's contribute to the cross section.\nAll these contributions are properly included in the total result for\n$(1\/\\sigma_{\\rm tot})d\\sigma\/dx$ shown in Figs.~\\ref{fig:xs}(a)--(d).\nAt LEP1 energies, the contribution from the first three quark flavors is still\nnegligible; it is concentrated at small values of $x$ and only amounts to a\nfew percent of the integrated cross section.\nHowever, the contribution from the gluon FF, which appears at NLO in \nconnection with $q\\overline{q}g$ final states, is numerically significant.\nAs in our previous works \\cite{bkk,bkk1}, motivated by the decomposition of\n$(1\/\\sigma_{\\rm tot})d\\sigma\/dx$ in terms of parton-level cross sections, we\ndistributed this contribution over the $Z\\to c\\bar c$ and $Z\\to b\\bar b$\nchannels in the ratio $e_c^2:e_b^2$, where $e_q$ is the effective electroweak\ncoupling of the quark $q$ to the $Z$ boson and the photon including propagator\nadjustments.\nThis procedure should approximately produce the quantities that are compared\nwith the OPAL data \\cite{opal1}.\n\nAs in Refs.~\\cite{bkk,bkk1}, we study the branching fractions for the\ntransitions\\break $c,b\\to D^0,D^+,D_s^+,\\Lambda_c^+$, defined by\n\\begin{equation}\nB_Q(\\mu)=\\int_{x_{\\rm cut}}^1dx\\,D_Q(x,\\mu^2),\n\\label{eq:br}\n\\end{equation}\nwhere $Q=c,b$, $D_Q$ are the appropriate FF's, and $x_{\\rm cut}=0.1$.\nThis allows us to test the consistency of our fits with information presented\nin the experimental paper \\cite{opal1} that was used for our fits.\nThe contribution from the omitted region $0<x<x_{\\rm cut}$ is small.\nTable~\\ref{tab:br} contains the values of $B_Q(\\mu)$ for all eight transitions\n$c,b\\to D^0,D^+,D_s^+,\\Lambda_c^+$ evaluated according to Eq.~(\\ref{eq:br}) in\nLO and NLO at the respective thresholds $\\mu=2m_Q$ and at the $Z$-boson\nresonance $\\mu=M_Z$.\nAs expected, the values of $B_Q(\\mu)$ change very little under the evolution\nfrom $\\mu=2m_Q$ to $\\mu=M_Z$, and they are rather similar for $Q=c,b$.\nLeaving aside the insignificant contribution due to strange charm baryons,\nthe values of $B_Q(\\mu)$ for $X_c=D^0,D^+,D_s^+,\\Lambda_c^+$ should\napproximately add up to unity for each heavy flavor $Q=c,b$ at any value of\n$\\mu$.\nAlthough we did not impose this sum rule as a constraint on our fits, it is\nwell satisfied for $B_c(M_Z)$ and $B_b(M_Z)$ at NLO.\nIn fact, from Table~\\ref{tab:br} one obtains 103\\% and 99.5\\%, respectively.\nThe corresponding LO values, being 108\\% and 105\\%, are somewhat too large, as\nmay be understood by observing the excess of the LO fits over the experimental\ndata at small values of $x$ in Figs.~\\ref{fig:xs}(a)--(d).\nThe corresponding sums of the LO and NLO entries for $B_c(2m_c)$ and\n$B_b(2m_b)$ in Table~\\ref{tab:br} range between 110\\% and 116\\%.\nIn view of the long evolution paths from the charm and bottom thresholds way\nup to the $Z$-boson resonance, such violations of the sum rule can be\nconsidered acceptable.\nThe situation is expected to improve once experimental data at lower c.m.\\\nenergies (see Sec.~\\ref{sec:four}) are included in our fits.\n\n\\begin{table}\n\\begin{center}\n\\caption{Branching fractions (in \\%) of $c,b\\to D^0,D^+,D_s^+,\\Lambda_c^+$\nevaluated according to Eq.~(\\ref{eq:br}) in LO and NLO at the respective\nstarting scales $\\mu=2m_Q$ and at the $Z$-boson resonance $\\mu=M_Z$.}\n\\label{tab:br}\n\\begin{tabular}{cccccc}\n\\hline\\hline\n$X_c$ & Order & $B_c(2m_c)$ & $B_c(M_Z)$ & $B_b(2m_b)$ & $B_b(M_Z)$ \\\\\n\\hline\n$D^0$         & LO  & 72.1  & 66.9  & 57.8 & 52.8 \\\\\n              & NLO & 69.5  & 63.9  & 55.2 & 49.8 \\\\\n$D^+$         & LO  & 26.6  & 24.7  & 19.4 & 17.9 \\\\\n              & NLO & 25.6  & 23.6  & 18.6 & 17.1 \\\\\n$D_s^+$       & LO  & 11.5  & 10.9  & 22.4 & 20.6 \\\\\n              & NLO & 10.8  & 10.1  & 21.6 & 19.6 \\\\\n$\\Lambda_c^+$ & LO  &  5.88 &  5.67 & 15.1 & 13.7 \\\\\n              & NLO &  5.74 &  5.48 & 14.5 & 13.0 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIt is interesting to compare our LO and NLO values of $B_c(M_Z)$ and $B_b(M_Z)$\nfor the $D^0$ ,$D^+$, $D_s^+$, and $\\Lambda_c^+$ hadrons with the respective\nresults determined by the OPAL Collaboration through Peterson model fits.\nThese results are presented in Table~9 of Ref.~\\cite{opal1} in the dressed form\n\\begin{equation}\np_{Q\\to X_c}=R_QB_Q(M_Z)B_{X_c},\n\\label{eq:pr}\n\\end{equation}\nwhere $R_Q=\\Gamma_{Q\\overline{Q}}\/\\Gamma_{\\rm had}$ are the production rates of\nthe quarks $Q=c,b$ in $e^+e^-$ annihilation on the $Z$-boson resonance and\n$B_{X_c}$ are the decay branching fractions of the four $X_c$ hadrons\n$X_c=D^0,D^+,D_s^+,\\Lambda_c^+$ into the channels considered in\nEq.~(\\ref{eq:br}).\nFor the reader's convenience, these results are copied to Table~\\ref{tab:pr},\nwhere they are compared with our results for $p_{Q\\to X_c}$, which are\nobtained from the appropriate entries in Table~\\ref{tab:br} through\nmultiplication with the branching fractions from Eq.~(\\ref{eq:br}) and the\nproduction rates $R_c=0.1689\\pm0.0047$ and $R_b=0.21643\\pm0.00072$ determined\nby the Particle Data Group \\cite{pdg} in the framework of the Standard Model.\nFor simplicity, the values deduced from Table~\\ref{tab:pr} do not include the\nerrors on $R_Q$ and $B_{X_c}$ and those on $B_Q(M_Z)$ resulting from our fits.\n\nIn Table~9 of Ref.~\\cite{opal1}, the OPAL Collaboration also presented the\ntotal rates $\\overline{n}(Z\\to X_c)B_{X_c}$, which include the estimated\ncontributions from gluon splitting $g\\to Q\\overline{Q}$; for further details,\nsee Ref.~\\cite{opal1}.\nIn Table~\\ref{tab:pr}, these results are quoted and compared with the\ncorresponding quantities $2(p_{c\\to X_c}+p_{b\\to X_c})$ resulting from our LO\nand NLO analyses.\nNotice that the experimental results are corrected to include the unmeasured\ncontributions from $x<0.15$, whereas our evaluations of Eq.~(\\ref{eq:br})\nexclude the contributions from $x<x_{\\rm cut}$.\nThis explains why the experimental results somewhat overshoot ours.\nThe agreement is worse at NLO, which may be understood by observing that our\nevaluations of $2(p_{c\\to X_c}+p_{b\\to X_c})$ do not include the\ncontributions from gluon fragmentation, which enters the stage at NLO.\nKeeping these caveats in mind, we find reasonable overall agreement between\nthe OPAL results for $\\overline{n}(Z\\to X_c)B_{X_c}$ and our results for\n$2(p_{c\\to X_c}+p_{b\\to X_c})$.\n\n{\\scriptsize\n\\begin{table}\n\\begin{center}\n\\caption{$X_c$-hadron production rates reported by OPAL \\cite{opal1} compared\nto results evaluated at LO and NLO from Eq.~(\\ref{eq:pr}) using the branching\nfractions from Table~\\ref{tab:br}.}\n\\label{tab:pr}\n\\begin{tabular}{ccccccc}\n\\hline\\hline\n$X_c$ & \\multicolumn{2}{c}{$p_{c\\to X_c}$ [\\%]} &\n\\multicolumn{2}{c}{$p_{b\\to X_c}$ [\\%]} &\n\\multicolumn{2}{c}{$\\overline{n}(Z\\to X_c)B_{X_c}$ [\\%]} \\\\\n & \\cite{opal1} & fit ${{\\rm LO}\\atop{\\rm NLO}}$ &\n\\cite{opal1} & fit ${{\\rm LO}\\atop{\\rm NLO}}$ &\n\\cite{opal1} & fit ${{\\rm LO}\\atop{\\rm NLO}}$ \\\\\n\\hline\n$D^0$ & $0.389\\pm0.027{+0.026\\atop-0.024}$ & 0.434 &\n$0.454\\pm0.023{+0.025\\atop-0.026}$ & 0.439 &\n$1.784\\pm0.066\\pm0.086$ & 1.746 \\\\\n & & 0.414 & & 0.414 & & 1.656 \\\\\n$D^+$ & $0.358\\pm0.046{+0.025\\atop-0.031}$ & 0.380 &\n$0.379\\pm0.031{+0.028\\atop-0.025}$ & 0.353 &\n$1.548\\pm0.082{+0.082\\atop-0.080}$ & 1.466 \\\\\n & & 0.363 & & 0.337 & & 1.400 \\\\\n$D_s^+$ & $0.056\\pm0.015\\pm0.007$ & 0.0644 &\n$0.166\\pm0.018\\pm0.016$ & 0.156 &\n$0.460\\pm0.036\\pm0.040$ & 0.441 \\\\\n & & 0.0597 & & 0.148 & & 0.415 \\\\\n$\\Lambda_c^+$ & $0.041\\pm0.019\\pm0.007$ & 0.0421 &\n$0.122\\pm0.023\\pm0.010$ & 0.130 &\n$0.345\\pm0.052\\pm0.029$ & 0.344 \\\\\n & & 0.0407 & & 0.124 & & 0.329 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n}\n\nOur LO and NLO values of $B_c(M_Z)$ and $B_b(M_Z)$ for the $D^0$, $D^+$,\n$D_s^+$, and $\\Lambda_c^+$ hadrons in Table~\\ref{tab:br} can also be compared\nwith experimental results published more recently by the ALEPH \\cite{aleph}\nand DELPHI \\cite{delphi} Collaborations.\nIn Ref.~\\cite{aleph}, $B_c(M_Z)$ are called $f(c\\to X_c)$ and may be found in\nSecs.~7.1 and 7.3.\nIn Ref.~\\cite{delphi}, $B_Q(M_Z)$ are called $P_{Q\\to X_c}$ and may be\nextracted for $Q=c$ from Table 13 (in connection with sum rule of Eq.~(12) and\ntaking into account the discussion of the contribution from the strange charm\nbaryons in Sec.~8.2) and for $Q=b$ from Table 15.\nFor simplicity, we add the three types of errors quoted in\nRefs.~\\cite{aleph,delphi} (from statistics, systematics, and decay branching\nfractions) in quadrature.\nIn 1999, Gladilin \\cite{gla} derived world-average values of $B_c(M_Z)$ for\nthe $D^0$ ,$D^+$, $D_s^+$, and $\\Lambda_c^+$ hadrons related to $e^+e^-$\nannihilation, which are also listed in Table~\\ref{tab:ad}.\n\nThe branching fractions of the $c\\to D^0,D^+,D_s^+,\\Lambda_c^+$ transitions\nwere also measured in $ep$ collisions at HERA, in photoproduction by the ZEUS\nCollaboration \\cite{zeus,pad} and in deep-inelastic scattering by the H1\nCollaboration \\cite{h1}.\nThese results are also included in Table~\\ref{tab:ad} for comparison.\nStrictly speaking, they do not correspond to $B_c(M_Z)$, but rather to\n$B_c(\\mu)$, where $\\mu$ is set by the average value of $p_T$ (in the case of\nphotoproduction) or $Q$ (in the case of deep-inelastic scattering).\nHowever, from Table~\\ref{tab:br} we know that the $\\mu$ dependence of\n$B_c(\\mu)$ is relatively mild.\n\nWe observe that the experimental results collected in Table~\\ref{tab:ad}, which\nare mostly independent from each other, are mutually consistent within errors.\nComparing them with the corresponding entries in the forth and sixth columns\nof Table~\\ref{tab:br}, we find resonable overall agreement. \n\n\\begin{table}\n\\begin{center}\n\\caption{Branching fractions (in \\%) of $c,b\\to D^0,D^+,D_s^+,\\Lambda_c^+$\nreported by ALEPH \\cite{aleph}, DELPHI \\cite{delphi}, Gladilin \\cite{gla},\nZEUS \\cite{pad}, and H1 \\cite{h1}.}\n\\label{tab:ad}\n\\begin{tabular}{ccccccc}\n\\hline\\hline\n$X_c$ & \\multicolumn{5}{c}{$B_c(M_Z)$}& $B_b(M_Z)$ \\\\\n & \\cite{aleph} & \\cite{delphi} & \\cite{gla} & \\cite{pad} & \\cite{h1} &\n\\cite{delphi} \\\\\n\\hline\n$D^0$ & $55.9\\pm2.2$ & $54.80\\pm4.78$ & $54.9\\pm2.6$ &\n$55.7{+2.0\\atop-2.3}$ & $65.8{+15.5\\atop-15.9}$ & $60.05\\pm4.39$ \\\\ \n$D^+$ & $23.79\\pm2.42$ & $22.70\\pm1.82$ & $23.2\\pm1.8$ &\n$24.9{+1.5\\atop-1.6}$ & $20.2{+5.7\\atop-4.4}$ & $23.01\\pm2.13$ \\\\ \n$D_s^+$ & $11.6\\pm3.6$ & $12.51\\pm2.97$ & $10.1\\pm2.7$ &\n$10.7\\pm1.0$ & $15.6{+7.5\\atop-7.2}$ & $16.65\\pm4.50$ \\\\ \n$\\Lambda_c^+$ & $7.9\\pm2.2$ & $8.76\\pm3.30$ & $7.6\\pm2.1$ &\n$7.6{+2.6\\atop-2.0}$ & -- & $8.90\\pm3.00$ \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nAnother quantity of interest, which can directly be compared with experiment,\nis the mean momentum fraction,\n\\begin{equation}\n\\langle x\\rangle_Q(\\mu)=\\frac{1}{B_Q(\\mu)}\\int_{x_{\\rm cut}}^1dx\\,xD_Q(x,\\mu).\n\\end{equation}\nIn Table~\\ref{tab:xav}, we present the values of $\\langle x\\rangle_Q(\\mu)$ for\n$Q=c,b$ evaluated at $\\mu=2m_Q,M_Z$ with the LO and NLO FF's of the $D^0$,\n$D^+$, $D_s^+$, and $\\Lambda_c^+$ hadrons.\nAt fixed value of $\\mu$, the differences between the LO and NLO sets are\ninsignificant.\nThe DGLAP evolution from $\\mu=2m_Q$ to $\\mu=M_Z$ leads to a significant \nreduction of $\\langle x\\rangle_Q(\\mu)$, especially in the case of $Q=c$.\nThe values of $\\langle x\\rangle_b(\\mu)$ are appreciably smaller than the\nvalues of $\\langle x\\rangle_c(\\mu)$, as is expected because the bottom-quark\nfragmentation into $X_c$ hadrons is much softer than the charm-quark one.\n\nOur values of $\\langle x\\rangle_c(M_Z)$ for the $D^0$ and $D^+$ mesons should\nbe compared with the respective results obtained by the OPAL Collaboration\n\\cite{opal1} in the framework of the Peterson model \\cite{pet}, which read\n\\begin{eqnarray}\n\\langle x\\rangle_c(M_Z)&=&0.487\\pm0.009{+0.011\\atop-0.009}\n\\qquad(D^0),\n\\nonumber\\\\\n\\langle x\\rangle_c(M_Z)&=&0.483\\pm0.015{+0.007\\atop-0.011}\n\\qquad(D^+)\n\\label{eq:xav}\n\\end{eqnarray}\nfor the $D^0$ and $D^+$ mesons, respectively.\nThe differences to the values obtained for three other fragmentation models\nare included in the systematical errors.\nComparing Eq.~(\\ref{eq:xav}) with the corresponding entries in\nTable~\\ref{tab:xav}, we observe that the latter are slightly smaller.\n\n\\begin{table}\n\\begin{center}\n\\caption{Average momentum fractions of $c,b\\to D^0,D^+,D_s^+,\\Lambda_c^+$\nevaluated according to Eq.~(\\ref{eq:xav}) in LO and NLO at the respective\nstarting scales $\\mu=2m_Q$ and at the $Z$-boson resonance $\\mu=M_Z$.}\n\\label{tab:xav}\n\\begin{tabular}{cccccc}\n\\hline\\hline\n$X_c$ & Order & $\\langle x\\rangle_c(2m_c)$ & $\\langle x\\rangle_c(M_Z)$ &\n$\\langle x\\rangle_b(2m_b)$ & $\\langle x\\rangle_b(M_Z)$ \\\\\n\\hline\n$D^0$         & LO  & 0.588 & 0.452 & 0.316 & 0.284 \\\\\n              & NLO & 0.568 & 0.431 & 0.300 & 0.270 \\\\\n$D^+$         & LO  & 0.596 & 0.458 & 0.341 & 0.303 \\\\\n              & NLO & 0.575 & 0.436 & 0.323 & 0.287 \\\\\n$D_s^+$       & LO  & 0.676 & 0.512 & 0.349 & 0.310 \\\\\n              & NLO & 0.644 & 0.482 & 0.332 & 0.296 \\\\\n$\\Lambda_c^+$ & LO  & 0.791 & 0.590 & 0.302 & 0.273 \\\\\n              & NLO & 0.750 & 0.553 & 0.288 & 0.261 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\boldmath\n\\section{Comparison with $e^+e^-$ data at lower energies}\n\\label{sec:four}\n\\unboldmath\n\nThe fractional energy spectra of inclusive $D^0$, $D^+$, $D_s^+$, and\n$\\Lambda_c^+$ production was also measured in nonresonant $e^+e^-$ annihilation\nat lower energies.\nSpecifically, the CLEO Collaboration took $D^0$, $D^+$ (Table~XII in\nRef.~\\cite{cleo04}), $D_s^+$ (Table~IV in Ref.~\\cite{cleo0}), and\n$\\Lambda_c^+$ (Table~V in Ref.~\\cite{cleo88}) data at LEPP CESR with\n$\\sqrt s=10.55$~GeV;\nthe HRS Collaboration took $D^0$, $D^+$ (Table~1 in Ref.~\\cite{hrs88}), and\n$D_s^+$ (Table~I in Ref.~\\cite{hrs85}) data at SLAC PEP with $\\sqrt s=29$~GeV;\nand the TASSO Collaboration took $D_s^+$ (Fig.~3 in Ref.~\\cite{tasso}) data\nat DESY PETRA with $\\sqrt s=34.7$~GeV.\nIt is instructing to confront these data with LO and NLO predictions based on\nour new FF's, so as to test the scaling violations predicted by the DGLAP\nevolution equations.\nAn especially interesting situation arises for the CLEO data\n\\cite{cleo04,cleo0,cleo88}, from which all $X_c$ hadrons coming from\n$X_b$-hadron decays are excluded by appropriate acceptance cuts, so that only\n$n_f=4$ quark flavors are active and a direct test of the charm-quark FF's is\nfeasible.\n\nThe $D^+$ and $\\Lambda_c^+$ data explicitly refer to the decay channels\n$D_s^+\\to\\phi\\pi^+\\to K^+K^-\\pi^+$ and $\\Lambda_c^+\\to pK^-\\pi^+$,\nrespectively, and we have to divide them by the corresponding branching\nfractions.\nFor this, we use the up-to-date values\n$B(D_s^+\\to\\phi\\pi^+)B(\\phi\\to K^+K^-)=(3.6\\pm0.9)\\%$ and\n$B(\\Lambda_c^+\\to pK^-\\pi^+)=(5.0\\pm1.3)\\%$ \\cite{pdg}, except for the $D_s^+$\ndata of Ref.~\\cite{tasso}.\nIn the latter case, for consistency, we adopt the value\n$B(D_s^+\\to\\phi\\pi^+)B(\\phi\\to K^+K^-)=0.13\\pm0.03\\pm0.04$ from\nRef.~\\cite{tasso} itself.\n\nThe differential cross sections $d\\sigma\/dx$ for the $D^0$, $D^+$, $D_s^+$,\nand $\\Lambda_c^+$ hadrons measured by CLEO \\cite{cleo04,cleo0,cleo88}\n(circles), HRS \\cite{hrs88,hrs85} (squares), and TASSO \\cite{tasso} (diamonds)\nare confronted with our LO (dashed lines) and NLO (solid lines) predictions in\nFigs.~\\ref{fig:xs1}(a)--(d), respectively.\n\nLet us first concentrate on our NLO predictions.\nAs for the $D^0$, $D^+$, and $D_s^+$ mesons, we observe that our NLO\npredictions generally lead to a satisfactory description of the experimental\ndata, both in normalization and shape.\nIn particular, the maxima of the measured $x$ distributions are approximately\nreproduced.\nHowever, in the case of the $\\Lambda_c^+$ baryon, the predicted $x$\ndistribution appears to be too hard, its peak being set off by approximately\n$+0.2$ relative to the one shown by the experimental data.\nIn particular, the data points at 0.55 and 0.65 are poorly described by the NLO\nprediction.\nAlthough the $\\Lambda_c^+$ baryon is 22\\% heavier than the $D^0$ and $D^+$\nmesons, and 16\\% heavier than the $D_s^+$ meson, mass effects are unlikely to\nbe responsible for this disagreement, since $\\sqrt s=10.55$~GeV is\nsufficiently far above the charm threshold.\n\nLet us now include the LO predictions in our considerations.\nThe CLEO data \\cite{cleo04,cleo0,cleo88}, which are most precise, clearly\nfavor the NLO predictions, while the LO predictions are too large at small\nvalues of $x$ and too small in the peak region.\nUnfortunately, the HRS \\cite{hrs88,hrs85} and TASSO \\cite{tasso} data do not\nreach the small-$x$ regime, where the LO and NLO predictions depart from each\nother, and their errors are too large in order to support this observation.\n\nActually, the CLEO data \\cite{cleo04,cleo0,cleo88} are considerably more\nprecise than the OPAL data \\cite{opal1}, which we fitted to, and it would be\ndesirable to also include them in ours fits.\nHowever, we refrain from doing so for the time being because their high\nprecision would make it necessary to properly treat finite-$m_Q$ effects,\nwhich are neglected altogether in the theoretical formalism employed here.\nThe general-mass variable-flavor-number (GM-VFN) scheme \\cite{acot}, which has\nrecently been extended to inclusive $X_Q$-hadron production in $\\gamma\\gamma$\n\\cite{ks1}, $ep$ \\cite{ks2}, and $p\\overline{p}$ \\cite{kkss} collisions,\nprovides a rigorous theoretical framework that retains the full finite-$m_Q$\neffects while preserving the indispensible virtues of the factorization\ntheorem \\cite{col}, namely the universality and the DGLAP \\cite{dglap} scaling\nviolations of the FF's entailing the resummation of dominant logarithmic\ncorrections.\nA global analysis of experimental data on inclusive $X_c$-hadron production in\nthe GM-VFN scheme is left for future work.\n\n\\section{Conclusions}\n\\label{sec:five}\n\nThe OPAL Collaboration presented measurements of the fractional energy spectra\nof inclusive $D^0$, $D^+$, $D_s^+$, and $\\Lambda_c^+$ production in $Z$-boson\ndecays based on their entire LEP1 data sample \\cite{opal1}.\nApart from the full cross sections, they also determined the contributions\narising from $Z\\to b\\bar b$ decays.\nThis enabled us to determine LO and NLO sets of FF's for these $X_c$ hadrons.\n\nAs in our previous analysis of $D^{*+}$ FF's \\cite{bkk}, we worked in the\nQCD-improved parton model implemented in the pure $\\overline{\\mathrm{MS}}$\nrenormalization and factorization scheme with $n_f=5$ massless quark flavors\n(zero-mass variable-flavor-number scheme).\nThis scheme is particularly appropriate if the characteristic energy scale of\nthe considered production process, {\\it i.e.}, the c.m.\\ energy $\\sqrt s$ in\nthe case of $e^+e^-$ annihilation and the transverse momentum $p_T$ of the\n$X_c$ hadron in other scattering processes, is large compared to the\nbottom-quark mass $m_b$.\nOwing to the factorization theorem, the FF's defined in this scheme satisfy\ntwo desirable properties:\n(i) their scaling violations are ruled by the timelike DGLAP equations; and\n(ii) they are universal.\nThus, this formalism is predictive and suitable for global data analyses.\n\nWe verified that the values of the branching and average momentum fractions of\nthe various $c,b\\to X_c$ transitions evaluated at LO and NLO\nusing our FF's are in reasonable agreement with the corresponding results from\nOPAL \\cite{opal1} and other experiments \\cite{pad,h1,aleph,delphi,gla}.\n\nWe tested the scaling violations of our FF's by comparing the fractional\nenergy spectra of inclusive $D^0$, $D^+$, $D_s^+$, and $\\Lambda_c^+$\nproduction measured in nonresonant $e^+e^-$ annihilation at $\\sqrt s=10.55$~GeV\n\\cite{cleo04,cleo0,cleo88}, 29~GeV \\cite{hrs88,hrs85}, and 34.7 \\cite{tasso}\nwith our LO and NLO predictions to find reasonable agreement.\nSince events of $X_c$-hadron production from $X_b$-hadron decay were excluded\nfrom the data samples at $\\sqrt s=10.55$~GeV, we obtained a clean test of our\ncharm-quark FF's.\n\nIt is important to bear in mind that the fit results for the input parameters\nin Eqs.~(\\ref{eq:peterson}) and (\\ref{eq:standard}), including the value of\nPeterson's $\\epsilon$ parameter, are highly scheme dependent at NLO, and must\nnot be na\\\"\\i vely compared without careful reference to the theoretical\nframework which they refer to.\n\n\\bigskip\n\\centerline{\\bf ACKNOWLEDGMENTS}\n\\smallskip\\noindent\nWe thank A. J. Martin for providing to us the OPAL data \\cite{opal1} in\nnumerical form.\nB.A.K. thanks the Department of Energy's Institute for Nuclear Theory at the\nUniversity of Washington for its hospitality and the Department of Energy for\npartial support during the completion of this work.\nThis work was supported in part by the Deutsche Forschungsgemeinschaft through\nGrant No.\\ KN~365\/3-1, and by the Bundesministerium f\\\"ur Bildung und Forschung\nthrough Grant No.\\ 05~HT1GUA\/4.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPattern avoidance is a central topic in graph theory and combinatorics \\cite{B, Fox, HM, Ki, KM, MT, SA, S, St}. In this paper, we study $d$-dimensional matrices (or arrays) with only two distinct entries, $0$ and $1$,  that avoid certain patterns. We are interested in the extremal problem of finding the largest number of ones in these matrices. \n\nThe extremal theory of matrices was initiated in papers \\cite{BG, F, M} around 1990 to study specific problems arising in computational and discrete geometry. Mitchell produced an algorithm for computing a shortest rectilinear path avoiding rectilinear obstacles in the plane \\cite{M}. He showed that the complexity of this algorithm is bounded above by the solution of the extremal problem of certain matrices. Bienstock and Gy\\\"ori \\cite{BG} found an upper bound on the solution of the extremal problem,  thus bounding the complexity of the algorithm. Mitchell's algorithm can be directly applied to motion planning in robotics and wire routing in VLSI circuit design \\cite{LYW}. F\\\"{u}redi \\cite{F} used the extremal problem to derive an upper bound on Erd\\H{o}s-Moser \\cite{EM} problem of determining the maximum number of unit distances in a convex polygon.\n\n\nMatrix extremal problems also find applications to graph theory and other areas of combinatorics. If we associate $2$-dimensional $0$-$1$ matrices with ordered bipartite graphs by relating rows and columns to the two ordered partite sets of vertices and  interpreting ones as edges, then this extremal problem can be viewed as the Tur\\'an extremal problem for ordered bipartite graphs \\cite{PT}.  The most recent surge in interest in the extremal theory of matrices is due to the resolution of the Stanley-Wilf conjecture in enumerative combinatorics using the extremal problem of matrices \\cite{Kl, MT}.\n\nWe denote a $d$-dimensional $n_1 \\times \\cdots \\times n_d$ matrix by $A = \\left(a_{i_1, \\ldots , i_d}\\right)$, where $1 \\le i_l \\le n_{\\ell}$ for $\\ell = 1, 2, \\ldots , d$. Matrix $A$ is called a $ 0$-$1$  matrix if all its entries are either $0$ or $1$. A multidimensional matrix is also called a multidimensional array in computer science. We may view a $d$-dimensional $0$-$1$ matrix $A = \\left( a_{i_1, \\ldots , i_d} \\right)$ geometrically as a $d$-dimensional rectangular box of lattice points with coordinates $(i_1, \\ldots, i_d)$.  An $\\ell$-cross section of matrix $A$ is the set of all the entries $a_{i_1,\\ldots , i_d}$ whose $\\ell^{\\text{th}}$ coordinates \nhave the same value. An $\\ell$-row of matrix $A$ is the collection of all the entries $a_{i_1,\\ldots , i_d}$ whose coordinates other than the \n$\\ell^{\\text{th}}$ coordinate have fixed values.\n\nA $d$-dimensional $k  \\times \\cdots \\times k$ zero-one matrix is a permutation matrix if each of its $\\ell$-cross sections contains exactly one nonzero entry for every $\\ell=1, \\ldots , d$.\nThe Kronecker product of two $d$-dimensional $0$-$1$ matrices $M$ and $N$, denoted by $M \\otimes N$, is a \n$d$-dimensional matrix obtained by replacing each 1-entry of $M$ with a copy of $N$ and each 0-entry of $M$ with a zero matrix the same size as $N$. \n\nWe say that a $d$-dimensional $0$-$1$ matrix $A$ contains another $0$-$1$ matrix $P$ if $A$ has a submatrix that can be transformed into $P$ by changing any number of ones to zeros. Otherwise, $A$ is said to avoid $P$.  Denote by $f(n, P, d)$ the maximum number of ones in a $d$-dimensional $n \\times \\cdots \\times n$ zero-one matrix that avoids a given $d$-dimensional $0$-$1$ matrix $P$. We are interested in the asymptotic behavior of the extremal function $f(n,P,d)$ for large $n$.\n  \nA motivation to study the extremal problem of multidimensional matrices comes from graph theory. A $2$-dimensional $n \\times n$ zero-one matrix \ncan be represented as an ordered bipartite graph with partite sets of size $n$. Similarly, a $d$-dimensional $0$-$1$ matrix corresponds  \nto  an ordered $d$-partite, $d$-uniform hypergraph. The extremal function $f(n, P, d)$ is the maximum \nnumber of hyperedges in an ordered $d$-partite, $d$-uniform hypergraph where each partite set has $n$ vertices. Avoiding a given pattern $P$ amounts to avoiding the\ncorresponding ordered hypergraph.\n\nIt is easy to obtain trivial lower and upper bounds on $f(n,P,d)$.\n\\begin{pro}\n\\label{Easy}\nIf $P$ is a $0$-$1$ matrix that contains at least two ones, then\n$n^{d-1} \\leq f(n,P,d) \\leq n^{d}$.\n\\end{pro}\n\\begin{proof}\nWe can always choose a $d$-dimensional  $n \\times \\cdots \\times n$ zero-one matrix $A$, with $1$-entries on a single $\\ell$-cross section for some $\\ell$ \nand $0$-entries elsewhere, such that $A$ avoids $P$. Matrix $A$ has exactly $n^{d -1}$ ones, so the left inequality follows.\n\nThe right inequality follows from the fact that every $d$-dimensional  $n \\times  \\cdots \\times n$ zero-one matrix has a total of $n^d$ entries and that the matrix hence has at most $n^d$ ones.\n\\end{proof}\n\nThe upper bound in Proposition \\ref{Easy} is one order higher than the lower bound. The main problem is to improve the lower and upper bounds on $f(n,P,d)$ so that their orders are as close as possible.\n\nThe two-dimensional case of $d=2$ has been well studied.  F\\\"{u}redi and Hajnal conjectured that $f(n,P,2)=O(n)$ for all permutation matrices $P$ \\cite{FH}.  Klazar showed that this conjecture implies the Stanley-Wilf conjecture \\cite{Kl}. Marcus and Tardos proved the  F\\\"{u}redi and Hajnal conjecture \\cite{MT} and hence settled the Stanley-Wilf conjecture.  Keszegh conjectured that $f(n, P, 2)=O(n)$ for all tuple permutation matrices $P$ \\cite{K}. Geneson proved that the conjecture is true \\cite{G}.\n\nWhen $R$ is a $k_1 \\times k_2$ matrix of all ones, the extremal problem for $f(n, R, 2)$ is the matrix version of the classical \nZarankiewicz problem.  K\\H{o}v\\'{a}ri, S\\'{o}s, and Tur\\'{a}n found an upper bound $O(n^{2 - {\\max({k_1, k_2}) \\over k_1 k_2}})$ on $f(n,R,2)$ \\cite{KST}.\nA lower bound $\\Omega(n^{2 - {k_1 + k_2 - 2 \\over k_1 k_2 -1}})$ was also known \\cite{ES}.  Hesterberg extended these bounds to $f(n, P \\otimes R, 2)$ where $P$ is a permutation matrix \\cite{HP, H}.\n\nPach and Tardos showed that $f(n, P, 2)$ is super-additive in $n$ \\cite{PT}.\nBy Fekete's Lemma on super-additive sequences \\cite{Fekete}, the sequence $\\{ {f(n, P, 2) \\over n} \\}$ is convergent. The limit is known as the F\\\"uredi-Hajnal limit and has recently attracted great attention \\cite{C, Fox, Fox2}. Cibulka \\cite{C} showed that this limit is always at least $2(k-1)$ when $P$ is a $k \\times k$ permutation matrix and that the limit is exactly $2(k-1)$ when $P$ is the identity matrix. \nFox showed that the F\\\"uredi-Hajnal limit has a lower bound $2^{\\Omega((k^{1 \\over 2})}$ for a family of $k \\times k$ permutation matrices \\cite{Fox2}.\nMarcus and Tardos \\cite{MT} showed that this limit has an upper bound  $2^{O(k \\log k)}$ for every $k \\times k$ permutation matrix $P$, and Fox \\cite{Fox} improved this upper bound to $2^{O(k)}$. \n\nLittle has been done on the multidimensional case. Klazar and Marcus \\cite{KM} studied the extremal function when the $d$-dimensional matrix $P$ is a permutation matrix of size \n$k \\times \\cdots \\times k$ and found $f(n,P,d)=O(n^{d-1})$, generalizing the $d=2$ results \\cite{ MT}. In particular, they showed that ${f(n, P, d) \\over n^{d-1}} = 2^{O(k \\log k)}$, which is the  \nmultidimensional generalization of the Marcus and Tardos upper bound on the F\\\"uredi-Hajnal limit \\cite{ MT}. \n\nIn this paper, we advance the extremal theory of matrices in two directions. In the first direction,\nwe study the extremal functions $f(n, R, d)$ and $f(n, P \\otimes R, d)$, where $P$ is a permutation matrix and $R$ is a\n$k_1 \\times  \\cdots \\times k_d$ matrix of ones only; matrix $P \\otimes R$ is called a block permutation matrix.\nWe show that both $f(n, R, d)$ and $f(n, P \\otimes R, d)$ have a lower bound\n$\\Omega(n^{d - \\beta})$ and \nan upper bound $O(n^{d - \\alpha})$, where  $\\alpha = {\\max({k_1, \\ldots , k_d}) \\over  k_1 \\cdot k_2 \\cdots k_d }$ and $\\beta = {k_1+ \\cdots +k_d - d \\over k_1 \\cdot k_2  \\cdots k_d-1}$. These \nbounds significantly improve the trivial ones given in Proposition \\ref{Easy}. \n\nBoth the lower bound $\\Omega(n^{d - \\beta})$ and upper bound $O(n^{d - \\alpha})$  are in the same order as \n$\\Theta(n^{d - 1})$ whenever $\\alpha = \\beta$. This is exactly when only one of $k_1, \\ldots , k_d$ of \n$R$ is larger than $1$; the corresponding $P \\otimes R$ is a tuple permutation matrix.\n\nWhen $\\alpha \\neq \\beta$, our results on $f(n,R,d)$ generalize the K\\H{o}v\\'{a}ri-S\\'{o}s-Tur\\'{a}n upper bound \\cite{KST} from two dimensions to higher dimensions. Our results on $f(n, P \\otimes R, d)$ also extend Hesterberg's results \\cite{HP} from $d=2$ to $d > 2$. \n\nWhen $\\alpha = \\beta$, our result that $f(n, P, d) = \\Theta(n^{d - 1})$ for every $d$-dimensional tuple permutation matrix $P$, on one hand, generalizes Geneson's result \\cite{G}\nfrom $d=2$ to $d \\geq 2$. On the other hand, even when $d=2$ our ideas improve some key calculations in Geneson's paper \\cite{G}.\nThese improvements are vital in our derivation of a new upper bound on the limit superior of the sequence $\\{ {f(n,P,d) \\over n^{d-1}}\\}$ that we discuss below. \n\nThe importance of our result $f(n, P, d) = \\Theta(n^{d - 1})$ for every $d$-dimensional tuple permutation matrix $P$ lies in the fact that, in view of Proposition \\ref{Easy}, $\\Theta(n^{d - 1})$ is the lowest possible order for the extremal function of any nontrivial $d$-dimensional matrix. \n\nIn the second direction, we study the limit inferior and limit superior of the sequence $\\{ {f(n,P,d) \\over n^{d-1}} \\}$ where $P$ satisfies $f(n, P, d) = \\Theta(n^{d - 1})$. These are the multidimensional analogues of the F\\\"uredi-Hajnal limit. We show that the limit inferior is at least $d(k-1)$ for $k \\times \\cdots \\times k$ permutation matrices, \ngeneralizing Cibulka's result \\cite{C} from $d=2$ to all $d \\ge 2$. \n\nWe observe that $f(n, P, d)$ is super-homogeneous in higher dimensions, i.e., $f(sn, P, d) \\geq K s^{d-1} f(n, P, d)$ for some positive constant $K$.\nThis super-homogeneity is key to our proof that \nthe limit inferior of $\\{ {f(n,P,d) \\over n^{d-1}} \\}$ has a lower bound $2^{\\Omega(k^{1 \/ d})}$ for a family of $k \\times \\cdots \\times k$ permutation matrices, generalizing Fox's result \\cite{Fox2} from $d=2$ to $d \\geq 2$.\n\nFinally, we show that the limit superior of  the sequence $\\{ {f(n,P,d) \\over n^{d-1}} \\}$ is bounded above by $ 2^{O(k)}$ for all $k \\times \\cdots \\times k$ permutation matrices $P$.\nThis is a substantial improvement of Klazar and Marcus upper bound $2^{O(k \\log k)}$ for $d > 2$ in \npaper \\cite{KM} and it also generalizes Fox's bound $2^{O(k)}$ on the F\\\"uredi-Hajnal limit in two dimensions \\cite{Fox}. We further show that this upper bound $2^{O(k)}$ is also true for every tuple permutation matrix $P$, which is a new result even for $d=2$. We are able to extend the new upper bound from permutation matrices to tuple permutation matrices mainly because of our improvement of Geneson's approach as mentioned above.\n\nThe rest of the paper is organized as follows. In Section 2, we study $f(n,P,d)$ when $P$ is a block permutation matrix but not a tuple permutation matrix. The more difficult case when $P$ is a tuple permutation matrix is analyzed in Section \\ref{tuple}. \nIn Section 4, we study the limit inferior and limit superior of the sequence $\\{ {f(n,P,d) \\over n^{d-1}}\\}$ for permutation and tuple permutation matrices $P$. We conclude the paper and discuss our future directions in Section \\ref{conclusion}.\n\n\n\n\n\n\n\\section{Block permutation matrices}\n\\label{Block}\n\\bigskip\n\nIn this section, we study the extremal function of a variant of $d$-dimensional permutation matrices. \nWe are interested in the forbidden matrices which can be written as the Kronecker product of a $d$-dimensional permutation \nmatrix and a $d$-dimensional matrix of $1$-entries only.\n\nLet $R^{k_1,...,k_d}$ be the $d$-dimensional $k_1 \\times \\cdots \\times k_d$ matrix of all ones. We \nstudy lower and upper bounds on the extremal function of block permutation matrix $P \\otimes R^{k_1,...,k_d}$, \nwhere $P$ is a $d$-dimensional permutation matrix.\n\nWe first study the extremal function of  $R^{k_1, \\ldots  , k_d}$. We use the probabilistic method to  obtain a lower \nbound on $f(n,R^{k_1,...,k_d},d)$. When $d=2$, this lower bound is classical \\cite{ES}.\n\\begin{theo}\n\\label{lowerbound}\nIf $k_1 \\cdot k_2 \\cdots k_d > 1$, then\n$f(n,R^{k_1, \\ldots , k_d},d) = \\Omega \\left( n^{d- \\beta(k_1, k_2, \\ldots , k_d)} \\right)$, where $\\beta = {k_1+ \\cdots +k_d - d \\over k_1 \\cdot k_2 \\cdots k_d-1}$.\n\\end{theo}\n\\begin{proof} Let each entry of a $d$-dimensional $n \\times \\cdots \\times n$ zero-one matrix \n$A$ be chosen to be $1$ with probability $p=n^{- \\beta(k_1, \\ldots , k_d)}$ and $0$ with probability $1 - p$. \nThe expected number of $1$-entries in $A$ is $pn^d$. There are ${n \\choose k_1} \\cdot {n \\choose k_2} \\cdots {n \\choose k_d}$ \npossible copies of $R^{k_1, \\ldots , k_d}$ in matrix $A$ and each has a probability of $p^{k_1 \\cdot k_2 \\cdots k_d}$ of occurring.\nThe expected number of copies of $R^{k_1, \\ldots , k_d}$ in $A$ is\n$$ {n \\choose k_1} \\cdot {n \\choose k_2} \\cdots {n \\choose k_d}p^{k_1 \\cdot k_2 \\cdots k_d} \\le Cn^{k_1+ \\cdots + k_d}p^{k_1 \\cdot k_2 \\cdots k_d} \\ ,$$\nwhere, since at least one of $k_1$, $\\ldots$ , $k_d$ is greater than one,  $C$ is a positive constant less than 1. \n\nLet $A'$ be the matrix formed by changing a single $1$-entry in each copy of $R^{k_1, \\ldots , k_d}$ in \n$A$ to a 0-entry. Then $A'$ avoids $R^{k_1,...,k_d}$ and the expected number of $1$-entries in $A'$ is at least\n$pn^d- Cn^{k_1+k_2+ \\cdots + k_d}p^{k_1 \\cdot k_2 \\cdots k_d}=(1 - C) \\ n^{d-\\beta(k_1, k_2, \\ldots, k_d) }$.\n As a consequence, there exists some matrix $A'$ that avoids $R^{k_1, \\ldots , k_d}$ \nand has at least so many $1$-entries.\n\\end{proof}\n\nWe now obtain an upper bound on the extremal function of $R^{k_1, \\ldots , k_d}$. When $d=2$,  this upper bound is due to K\\H{o}v\\'{a}ri, S\\'{o}s, and Tur\\'{a}n \\cite{KST}.\n\\begin{theo}\n\\label{upperbound}\n$f(n,R^{k_1, \\ldots , k_d},d)=O(n^{d- \\alpha(k_1, \\ldots , k_d)})$, where $\\alpha = {\\max({k_1, \\ldots , k_d}) \\over  k_1 \\cdot k_2 \\cdots k_d }$.\n\\end{theo}\n\\begin{proof}\nWe prove the theorem by induction on $d$. The base case of $d=1$ is trivial. Assuming that \n$f(n, R^{k_1, \\ldots, k_{d-1}},d-1)=O(n^{d-1- \\alpha(k_1, \\ldots, k_{d-1})})$ \nfor some $d \\geq 2$, we show that $f(n,R^{k_1, \\ldots , k_d},d)=O(n^{d- \\alpha(k_1, \\ldots , k_d)})$.\n\nThroughout the proof, we let  $A = (a_{i_1, \\ldots , i_{d}})$ be a $d$-dimensional $n \\times \\cdots \\times n$ matrix  \nthat avoids $R^{k_1, \\ldots , k_d}$ and has the maximum number, $f(n,R^{k_1, \\ldots , k_d},d)$, of  ones. We need the following  lemma on the number \nof $d$-rows that have $1$-entries in each of predetermined $k_d$ $d$-cross sections.\n\n\\begin{lem}\n\\label{d-row}\nFor any set of $k_d$ $d$-cross sections of $A$, there are $O\\left(n^{d-1-\\alpha(k_1, \\ldots , k_{d-1})} \\right)$ $d$-rows in $A$ which contain a 1-entry in each of these $d$-cross sections.\n\n\\end{lem}\n\\begin{proof}\nLet the $d^{\\text{th}}$ coordinates of these $d$-cross sections be $\\ell_1, \\ldots , \\ell_{k_d}$. Define a ($d-1$)-dimensional\n$n\\times \\cdots \\times n$ matrix $B=(b_{i_1, \\ldots , i_{d-1}})$ such that\n$b_{i_1, \\ldots , i_{d-1}}=1$ if $a_{ i_1, \\ldots , i_{d-1},\\ell_1}= \\cdots =a_{i_1, \\ldots , i_{d-1}, \\ell_{k_d}}=1$ \nand $b_{i_1, \\ldots , i_{d-1}}=0$ otherwise.\n\nWe claim that matrix $B$ must avoid $R^{k_1, \\ldots , k_{d-1}}$. Suppose to the contrary that \n$B$ contains $R^{k_1, \\ldots , k_{d-1}}$. Let $e_1, \\ldots , e_{k_1 \\cdot k_2 \\cdots k_{d-1}}$ be all the $1$-entries \nin $B$ that represent $R^{k_1,...,k_{d-1}}$. By the construction of $B$,\nthere are $k_d$ nonzero entries  with coordinates $(x_1, \\ldots , x_{d-1},\\ell_1), \\ldots , (x_1, \\ldots , x_{d-1}, \\ell_{k_d})$ in $A$ corresponding to\neach $e_i$ with coordinates $(x_1, \\ldots , x_{d-1})$ in $B$. All these $k_1 \\cdot k_2 \\cdots k_d$ nonzero entries \nform a copy of $ R^{k_1, \\ldots , k_{d}}$ in $A$, a contradiction. Thus $B$ must avoid \n$R^{k_1, \\ldots , k_{d-1}}$ and  by our inductive assumption, $B$ must have \n$O(n^{d-1-\\alpha(k_1, \\ldots , k_{d-1})})$ ones. The result follows.\n\\end{proof}\n\nSuppose all the $d$-rows of $A$ have $r_1, \\ldots , r_{n^{d-1}}$  non-zero entries, respectively. \nCounting the total number of sets of $k_d$ nonzero entries in the same $d$-row in two different ways yields\n\\begin{equation}\n\\label{two}\n\\sum_{i=1}^{n^{d-1}}{r_i \\choose k_d} = {n \\choose k_d}O\\left( n^{d-1-\\alpha(k_1, \\ldots , k_{d-1})} \\right) \\ ,\n\\end{equation}\nwhere we use Lemma \\ref{d-row} to obtain the right hand side.\n\nMatrix $A$ avoids $R^{k_1, \\ldots, k_d}$ and has the largest possible number of $1$-entries, so $r_i \\geq k_d - 1$ for $1 \\leq i \\leq  n^{d-1}$. \nSince ${r \\choose k}$ is a convex function of $r$ for $r \\geq k - 1$,  we apply Jensen's inequality to obtain\n\\begin{eqnarray*}\n\\sum_{i=1}^{n^{d-1}}{r_i \\choose k_d} \\ge n^{d-1} {{1\\over n^{d-1}}\\sum_{i=1}^{n^{d-1}} r_i \\choose k_d} \n= n^{d-1}{{1\\over n^{d-1}}f(n,R^{k_1, \\ldots , k_d},d) \\choose k_d} \\ ,\n\\end{eqnarray*}\nwhere, in the equality, we use the assumption that $A$ has $f(n,R^{k_1, \\ldots , k_d}, d)$ total $1$-entries.\nSubstituting this into equation (\\ref{two}) yields\n$$n^{d-1}{{1\\over n^{d-1}}f(n,R^{k_1, \\ldots , k_d},d) \\choose k_d} = {n \\choose k_d}O\\left(n^{d-1-\\alpha(k_1,\\ldots , k_{d-1})} \\right) \\ ,$$\n\\noindent\nwhich together with ${n \\choose k}=\\Theta(n^k)$ gives\n$$n^{d-1} \\left({1\\over n^{d-1}}f(n,R^{k_1, \\ldots , k_d},d)\\right)^{k_d} = O\\left( n^{ k_d}\\cdot n^{d-1-\\alpha(k_1,\\ldots , k_{d-1})} \\right) \\ . $$ \nThis implies\n$$f\\left( n,R^{k_1, \\ldots , k_d},d \\right) = O \\left( n^{d-{\\alpha(k_1, \\ldots , k_{d-1}) \\over k_d }} \\right) \\ . $$\nSimilarly, we have\n$$f(n,R^{k_1, \\ldots , k_d},d) = O\\left(n^{d-{\\alpha(k_2, \\ldots , k_d) \\over k_1}}\\right) \\ .$$ \nNote that $\\max\\left({\\alpha(k_2, \\ldots , k_d) \\over k_1}, {\\alpha(k_1, \\ldots , k_{d-1}) \\over k_d}\\right)=\\alpha(k_1, \\ldots , k_d)$. \nThus taking the smaller of the two upper bounds gives\n$$f(n,R^{k_1, \\ldots , k_d},d) = O\\left(n^{d-\\alpha(k_1, \\ldots , k_d)}\\right) $$ \nwhich completes the inductive step, and thus Theorem \\ref{upperbound} is proved.\n\\end{proof}\n\nWe make the following observation on $\\alpha(k_1, \\ldots , k_d)$ and $\\beta(k_1, \\ldots , k_d)$.\n\\begin{pro}\n\\label{alpha}\n Suppose $d > 1$ and $k_1, \\ldots , k_d$ be positive integers such that $k_1 \\cdot k_2 \\cdots k_d > 1$. If only one  of $k_1, \\ldots , k_d$ is greater than 1, then\n$\\alpha(k_1, \\ldots , k_d) = \\beta(k_1, \\ldots , k_d) = 1 $.\nOtherwise,\n$0 < \\alpha(k_1, \\ldots , k_d) <  \\beta(k_1, \\ldots , k_d) < 1 $.\n\\end{pro}\n\nWe omit the proof since it is straightforward. Proposition \\ref{alpha} implies that the lower bound of Theorem \\ref{lowerbound} and the upper bound of Theorem \\ref{upperbound} are significant improvements of the bounds in Proposition \\ref{Easy}.\n\nWe now study the extremal function of the Kronecker product $P \\otimes R^{k_1, \\ldots , k_d}$, where $P$ is a $d$-dimensional permutation matrix. We show that the extremal functions of $P \\otimes R^{k_1, \\ldots , k_d}$ and $R^{k_1, \\ldots , k_d}$ share the same lower and upper bounds.\n\n\\noindent\n\\begin{theo}\n\\label{super}\nIf $P$ is a $d$-dimensional permutation matrix and at least two of $k_1, \\ldots , k_d$ are greater than 1, then there exist constants $C_1$ and $C_2$ such that for all $n$,\n\\begin{equation}\n\\label{block}\nC_1 n^{d - \\beta(k_1, \\ldots , k_d)} \\leq f(n,P\\otimes R^{k_1, \\ldots , k_d}, d) \\leq C_2 n^{d - \\alpha(k_1, \\ldots , k_d)}\n\\end{equation}\n\\end{theo}\n\\noindent\n\\begin{proof} We first have \n\\begin{equation}\n\\label{R}\nf(n, R^{k_1, \\ldots , k_d}, d) \\leq  f(n,P\\otimes R^{k_1, \\ldots , k_d}, d).\n\\end{equation}\n This follows from the fact that any matrix that avoids $R^{k_1, \\ldots , k_d}$ must also avoid\n$P\\otimes R^{k_1, \\ldots , k_d}$. The left inequality of (\\ref{block}) is then the result of (\\ref{R}) and Theorem \\ref{lowerbound}.\n\nTo prove the right inequality of (\\ref{block}), we follow Hesterberg's idea for the $2$-dimensional case \\cite{H} to \nestimate $f(n,P\\otimes R^{k_1, \\ldots , k_d}, d)$ first for $n = c^m$, where $m$ is an arbitrary positive integer and $c$ \nis a positive integer to be determined, and then for all other positive integers $n$.  \n\nWe make use  of the upper bound in Theorem \\ref{upperbound}\n\\begin{equation}\n\\label{Kconstant}\nf(n, R^{k_1, \\ldots , k_d}, d) \\leq g(n) \\ , \n\\end{equation}\nwhere $g(n) = K n^{d - \\alpha(k_1, \\ldots , k_d)}$ for some positive constant $K$, and claim that\n\\begin{equation}\n\\label{cm}\n f(c^m,P\\otimes R^{k_1, \\ldots , k_d}, d) \\leq  2 c^d g(c^m).\n\\end{equation}\n\nWe justify the claim by induction. The base case of $m=0$ is trivially true. Suppose that \n\\begin{equation}\n\\label{indu}\nf(n,P\\otimes R^{k_1, \\ldots , k_d},d) \\le 2c^d g(n) \n\\end{equation}\nfor $n = c^m$. We show that \n$f(cn,P\\otimes R^{k_1, \\ldots , k_d},d)\\le 2c^d g(cn)$.\n\nLet $A$ be a $d$-dimensional $cn\\times \\cdots \\times cn$ matrix avoiding $P\\otimes R^{k_1, \\ldots , k_d}$ with $f(cn, P \\otimes R^{k_1, \\ldots , k_d}, d)$ total $1$-entries. We divide $A = (a_{i_1, \\ldots , i_d})$ into $c^d$ \ndisjoint submatrices of size $n \\times \\cdots \\times n$. We label these submatrices by $S(i_1, \\ldots , i_d) = (s_{j_1, \\ldots , j_d})$, where \n$$s_{j_1, \\ldots , j_d} =a_{j_1+n(i_1-1), \\ldots , j_d+n(i_d-1)} \\ .$$\nThese are called $S$ submatrices throughout the paper. \n\nLet $C$ be the $d$-dimensional $c \\times \\cdots \\times c$  matrix such that $c_{i_1, \\ldots , i_d}=1$ \nif submatrix $S(i_1, \\ldots , i_d)$ of $A$ contains $R^{k_1, \\ldots , k_d}$ and that $c_{i_1, \\ldots , i_d}=0$ otherwise.  \nSince any two $1$-entries of the  permutation matrix $P$ differ in all coordinates,  $C$ must avoid $P$ or else $A$ contains $P\\otimes R^{k_1, \\ldots , k_d}$.\n\nWe can classify all the $S$ submatrices of $A$ into two classes.\n\n\\medskip\n\n\\noindent\n{\\bf Case 1: $S$ contains $R^{k_1, \\ldots , k_d}$} \n\nSince $C$ avoids $P$, there are at most $f(c,P,d)$ such $S$ submatrices. Clearly each $S$ \nsubmatrix must avoid $P \\otimes R^{k_1, \\ldots , k_d}$, so it has at most $f(n,P\\otimes R^{k_1, \\ldots , k_d},d)$ \n$1$-entries. There are at most $f(c,P,d)f(n,P\\otimes R^{k_1, \\ldots , k_d},d)$ $1$-entries from this type of $S$ submatrices.\n\n\\medskip\n\n\\noindent\n{\\bf Case 2: $S$ avoids $R^{k_1, \\ldots , k_d}$} \n\nThere are at most $c^d$  such submatrices in total. Each has at most $f(n,R^{k_1, \\ldots , k_d},d)$ $1$-entries. \nThere are at most $c^df(n,R^{k_1, \\ldots , k_d},d)$ $1$-entries from the second type of $S$ submatrices.\n\n\\medskip\n\n\\noindent\nSumming the numbers of $1$-entries in both cases gives \n\\begin{equation}\nf(cn,P\\otimes R^{k_1,...,k_d},d)\\le f(c,P,d)f(n,P\\otimes R^{k_1, \\ldots , k_d},d)+c^df(n,R^{k_1, \\ldots , k_d},d)  . \\no\n\\end{equation}\nOn the right hand side of the inequality, $f(n,P\\otimes R^{k_1, \\ldots , k_d},d)$ has an upper bound $2c^d g(n)$ \nbecause of the inductive assumption (\\ref{indu}) and $f(n,R^{k_1, \\ldots , k_d},d)$ has an upper bound $g(n)$ by (\\ref{Kconstant}). Since $f(c,P,d) = O(c^{d-1})$ \nfor any permutation matrix $P$ \\cite{KM},  \nthere exists a constant $L$ such that $f(c,P,d) \\le Lc^{d-1}$. Because at least two of $k_1, k_2, \\ldots , k_d$ are greater than $1$, it follows from Proposition \\ref{alpha}\nthat $\\alpha < 1$. Hence, the integer $c$ can be chosen \nso large that $2 L c^{\\alpha - 1} \\leq  1$. Therefore,\n\\begin{eqnarray*}\nf(cn,P\\otimes  R^{k_1,...,k_d},d)\n\\le (L c^{d-1})(2c^d g(n))+c^dg(n)\n\\le [2L  c^{\\alpha(k_1, \\ldots, k_d) - 1}]c^{d}g(cn)+c^dg(cn)\n\\le2c^dg(cn) \\ ,\n\\end{eqnarray*}\nwhere we use $g(n) = K n^{d - \\alpha}$ in the second inequality.  This completes our induction and hence proves equation (\\ref{cm}).\n\nFinally,  we estimate $f(n,P\\otimes R^{k_1, \\ldots , k_d}, d)$ for all positive integers $n$.\n\\begin{eqnarray*}\nf(n, P \\otimes R^{k_1,...,k_d}, d) &=& f(c^{\\log_c n},P \\otimes R^{k_1, \\ldots , k_d},d) \\\\\n&\\le& f(c^{\\lceil\\log_c n \\rceil},P \\otimes R^{k_1, \\ldots , k_d},d) \\\\\n& \\le & 2c^dg(c^{\\lceil\\log_c n \\rceil}) \\\\\n&\\le& 2c^dg(c^{\\log_c n +1}) \\\\\n&=& 2c^dg(cn) \\\\\n&\\le& 2c^d c^dg(n),\n\\end{eqnarray*}\nwhere $\\lceil\\log_c n \\rceil$ is the smallest integer $\\ge \\log_c n$, \nand we use (\\ref{cm}) in the second inequality and $g(n) = K n^{d - \\alpha}$ in the last inequality.\nThis proves the right inequality of (\\ref{block}).\n\nThe proof of Theorem \\ref{super} is completed.\n\\end{proof}\n\nWe conclude this section with an observation. If only one of $k_1, \\ldots , k_d$ is greater than one, the matrix $P \\otimes R^{k_1, \\ldots, k_d}$\n is a tuple permutation matrix. By Proposition \\ref{alpha}, $\\alpha(k_1, \\ldots , k_d) = 1$. The proof of Theorem \\ref{super} fails in this case, but it can be \nmodified to show that $f(n, P \\otimes R^{k_1, \\ldots , k_d}, d) = O(n^{d - 1 + \\epsilon})$, where $\\epsilon$ is an arbitrarily small positive number.  To see this, we can replace $g(n)$ of (\\ref{Kconstant}) by $g(n) = K n^{d - 1 + \\epsilon}$ and choose $c$ so large that $2 L c^{- \\epsilon} \\leq  1$. In the next section, we improve this result and show that $f(n, P \\otimes R^{k_1, \\ldots , k_d}, d) = O(n^{d - 1})$. The  method is quite different from that of this section.\n\n\n\n\n\n\n\n\n\n\\section{Tuple permutation matrices}\n\\label{tuple}\nIn this section, we study the extremal function of an arbitrary tuple permutation matrix. As previously mentioned, a tuple permutation matrix is the Kronecker product of a $d$-dimensional permutation matrix and  $ R^{k_1, \\ldots , k_d}$, where only one of $k_1, \\ldots, k_d$ is larger than unity. We improve Geneson's ideas for $d=2$ case \\cite{G} and obtain a tight bound on the extremal function for $d \\geq 2$.\n \nSuppose $P$ is a permutation matrix. We call a matrix $P \\otimes R^{k_1, \\ldots , k_d}$ a $j$-tuple  permutation matrix generated by $P$ if one of $k_1, \\ldots , k_d$ is equal to $j$ and the rest are unity. In particular, a $j$-tuple permutation matrix is called a double permutation matrix if $j=2$.\n\nLet\n\\begin{displaymath}\nF(n,j,k,d)=\\max_{M} f(n,M,d) \\ ,\n\\end{displaymath}\nwhere $M$ ranges through all $d$-dimensional $j$-tuple permutations matrices generated by $d$-dimensional $k \\times \\cdots \\times k$ permutation matrices.\n\n\n\\begin{theo}\n\\label{main}\nFor all $j \\ge 2$,\n$F(n,j,k,d)=\\Theta(n^{d-1})$.\n\\end{theo}\n\n\\noindent\nThe proof of this theorem is based on a series of lemmas.\n\nSince $F(n,j,k,d)$ has $n^{d-1}$ as a  lower bound in view of Proposition \\ref{Easy}, it suffices to prove that it has upper bound $O(n^{d-1})$.\n\nWe first observe that $F(n,j,k,d)$ and $F(n,2,k,d)$ are bounded by each other.\n\\begin{lem}\n\\label{2j}\n$F(n, 2, k, d) \\leq F(n, j, k, d) \\leq (j-1) F(n, 2, k, d) ~~~~~ \\mbox{for $j > 2$} \\ .$\n\\end{lem}\n\\begin{proof}\nIt suffices to show that \n\\begin{equation}\n\\label{2j'}\nf(n, P, d) \\leq f(n, P', d) \\leq (j-1) f(n, P, d) ,\n\\end{equation}\nwhere $P$ is a double permutation $2k \\times k \\times \\cdots \\times k$ matrix, $P'$ is a $j$-tuple permutation $jk \\times k \\times \\cdots \\times k$ matrix, and  both $P$ and $P'$ are generated from the same arbitrary permutation matrix of size $k \\times \\cdots \\times k$.\n\nThe left inequality of (\\ref{2j'}) follows from the fact that a $d$-dimensional $n \\times \\cdots \\times n$ matrix that avoids $P$ must also avoid $P'$.\n\nTo prove the right inequality, we suppose $A$ is a $d$-dimensional $n \\times \\cdots \\times n$ matrix that avoids $P'$ and has $f(n, P', d)$ nonzero entries. \nIn each $1$-row of $A$, we list all the $1$-entries $e_1, e_2, \\ldots$ in the order of increasing first coordinates and then change all the $1$-entries in this $1$-row \nexcept $e_1, e_{j}, e_{2j-1}, \\ldots$ to $0$-entries. In this way, we obtain a new matrix $A'$, which avoids $P$ since $A$ avoids $P'$.\nThis together with $|A| \\leq (j-1)|A'|$, where $|M|$ denotes the number of $1$-entries in $M$, justifies the right inequality of (\\ref{2j'}).\n\\end{proof}\n\nIn view of Lemma \\ref{2j}, it suffices to study the upper bound on $f(n, P, d)$, where $P$ is a $d$-dimensional double permutation matrix of size \n$2k \\times k \\times \\cdots \\times k$.\n\nSuppose $A$ is an arbitrary $d$-dimensional $kn \\times \\cdots \\times kn$ matrix that avoids $P$. As in Section 2, we study the $S$ submatrices of $A$, which are constructed by dividing $A$ into $n^d$ disjoint submatrices of size $k \\times \\cdots \\times k$ and labeling these submatrices as $S(i_1, \\ldots , i_d)$.\n\nThe contraction matrix of $A$ is defined to be the $d$-dimensional $n \\times \\cdots \\times n$ matrix $C = \\left(c_{i_1,i_2, \\ldots , i_d}\\right)$ such that $c_{i_1,i_2, \\ldots ,i_d}=1$ if $S(i_1, i_2, \\ldots , i_d)$ is a nonzero matrix and $c_{i_1,i_2, \\ldots , i_d}=0$ if $S(i_1, i_2, \\ldots ,i_d)$ is a zero matrix.\n\nWe now construct a  $d$-dimensional $n \\times \\cdots \\times n$ zero-one matrix $Q=(q_{i_1, \\ldots , i_d})$.  Each entry $q_{i_1, \\ldots , i_d}$ is defined based on the $S$ submatrices of $A$.\n\\begin{enumerate}\n\\item  $q_{i_1, \\ldots , i_d}=0$  if $S(i_1, \\ldots , i_d)$ is a zero matrix.\n\n\\item  $q_{i_1, \\ldots , i_d}=1$ if $S(i_1, i_2, \\ldots , i_d)$ is a nonzero matrix and $S(1,i_2, \\ldots , i_d)$, $\\ldots$, $S(i_1-1,i_2, \\ldots , i_d)$ are all zero matrices.\n\n\\item Let $x$ be the largest integer less than $i_1$ for which $q_{x,i_2, \\ldots , i_d}=1$. Then define $q_{i_1, i_2, \\ldots , i_d}=1$ if the augmented matrix \nformed by submatrices $S(x, i_2, \\ldots , i_d)$, $\\ldots$ , $S(i_1, i_2, \\ldots , i_d)$ contains at least two $1$-entries in the same $1$-row, and  $q_{i_1, \\ldots , i_d}= 0$ otherwise.\n\n\\end{enumerate}\n\n\\noindent\n\\begin{lem}\n\\label{Q}\n$Q$ avoids $P$.\n\\end{lem}\n\\begin{proof}\nSuppose to the contrary that $Q$ contains $P$. Suppose the $1$-entries $e_1$, $e_2$, $\\ldots$ , $e_{2k}$, where $e_{2i-1}$ and  $e_{2i}$ are in the same $1$-row, form a copy of $P$ in $Q$. Denote $e_{2i-1}= q_{x_1, x_2, \\ldots , x_d}$ and $e_{2i}= q_{x_1',x_2, \\ldots, x_d}$, where $x_1 < x_1'$. Then, by the definition of matrix $Q$, the augmented matrix formed by $S(x_1, x_2, \\ldots , x_d), \\ldots , S(x_1', x_2, \\ldots , x_d)$ contains two $1$-entries, denoted by $f_{2i-1}$ and $f_{2i}$, in the same $1$-row of $A$. The one-entries\n$f_1, \\ldots , f_{2k}$ form a copy of $P$ in $A$, a contradiction.\n\\end{proof}\n\nWe now study those $S$ submatrices of $A$ which contain two nonzero entries in the same $1$-row. The next lemma is the key difference between our approach and Geneson's approach \\cite{G} even for $d=2$.\n\\begin{lem}\n\\label{wide}\n$A$ has at most $F(n, 1, k, d)$ total $S$ submatrices with two nonzero entries in the same $1$-row.\n\\end{lem}\n\\begin{proof} We assume to the contrary that $A$ has more than $F(n,1,k,d)$ such $S$ submatrices. Let $A'$ be formed by changing all $1$-entries in all other $S$ submatrices to $0$-entries in $A$. Suppose that the double permutation matrix $P$ is generated from the permutation matrix $P'$ and that $C'$ is the contraction matrix of $A'$. Matrix $C'$ has more than $F(n,1,k,d) \\ge f(n,P',d)$ $1$-entries, so it must contain $P'$. Denote by $e_1, \\ldots , e_k$  the $1$-entries in $C'$ forming a copy of $P'$.  Then each of $S(e_1), \\ldots, S(e_k)$ is a $S$ submatrix of $A'$ that has at least two nonzero entries in the same $1$-row. All of these pairs of nonzero entries in $S(e_1), \\ldots , S(e_k)$ form a copy of $P$ in $A'$.  Hence, $A'$ contains $P$ and so does $A$, a contradiction.\n\\end{proof}\n\nFor each 1-entry $q_{i_1, i_2, \\ldots  , i_d}=1$ of $Q$, we define a chunk $C^*(i_1, i_2, \\ldots , i_d)$, which is an augmented matrix formed by consecutive $S$ submatrices, as follows \\cite{G}. \n\n\\begin{enumerate}\n\n\\item If $q_{i_1, i_2, \\ldots , i_d}=1$ and $i_1'$ is the smallest integer greater than $i_1$ such that $q_{i_1', i_2, \\ldots , i_d}=1$,  then the chunk $C^*(i_1, i_2, \\ldots , i_d)$ is defined to be the augmented matrix formed by $S(i_1, i_2, \\ldots , i_d)$, $\\ldots$ , $S(i_1' - 1, i_2, \\ldots , i_d)$.\n\n\\item If $q_{i_1, i_2, \\ldots , i_d}=1$ and there is no $i_1'>i_1$ such that $q_{i_1', i_2, \\ldots , i_d}=1$, then $C^*(i_1, i_2, \\ldots  , i_d)$ is the augmented matrix formed by $S(i_1, i_2, \\ldots , i_d)$, $\\ldots$ , $S(n, i_2, \\ldots , i_d)$.\n\n\\end{enumerate}\n\nWe call a chunk {\\it $j$-tall}, where $j=2,3, \\ldots , d$, if each of its $j$-cross sections contains at least one 1-entry. The ($d-1$)-dimensional matrix $M' = (m'_{i_1, \\ldots, i_{j-1}, i_{j+1}, \\ldots, i_d})$ is called the $j$-remainder of a $d$-dimensional matrix $M = (m_{i_1, \\ldots , i_d})$ if $m'_{i_1, \\ldots , i_{j-1},i_{j+1}, \\ldots , i_d}$ is defined to be $1$\nwhen there exists $i_j$ such that $m_{i_1, \\ldots , i_d}=1$ and to be $0$ otherwise.\n\n\\begin{lem}\n\\label{tall}\nFor each $j= 2, 3, \\ldots, d$ and each $m=1, \\ldots , n$, $A$ has at most $F(n,1+k^{d-2},k,d-1)$ total $j$-tall chunks of the form $C^*(i_1, \\ldots , i_{j-1},m,i_{j+1}, \\ldots , i_d)$.\n\\end{lem}\n\\begin{proof}\nAssume to the contrary that $A$ has $r$ chunks $C^*_1, C^*_2, \\ldots , C^*_r$, where $r>F(n,1+k^{d-2},k,d-1)$,  of the form $C^*(i_1, \\ldots , i_{j-1},m,i_{j+1}, \\ldots , i_d)$ \nthat have $1$-entries in\nall their $j$-cross sections. Let $S_1, S_2, \\ldots, S_r$ be the starting $S$ submatrices of the chunks $C^*_1, C^*_2, \\cdots , C^*_r$, respectively. Let $A'$ be the matrix formed\nby changing all $1$-entries of $A$ that do not lie in the chunks $C^*_1, \\ldots , C^*_r$ to $0$-entries. We further change all the $1$-entries of $A'$ that do not sit in \n$S_1, \\ldots, S_r$ to $0$-entries and denote the resulting matrix by $A''$. \nDenote by $C$ the contraction matrix of the $j$-remainder of $A''$. Then $C$ is a $(d-1)$-dimensional $n \\times \\cdots \\times n$ matrix and it has $r$ ones so it contains every ($1+k^{d-2}$)-tuple \n($d-1$)-dimensional permutation matrix.\n\nWe now pick a $(d-1)$-dimensional $(1 + k^{d-2})$-tuple permutation matrix. Since $P$ is a $d$-dimensional double permutation matrix of size $2k \\times k \\times \\cdots \\times k$ and $j \\neq 1$, the $j$-remainder of $P$ is a $(d-1)$-dimensional double permutation matrix\nof size $2k \\times k \\times \\cdots \\times k$. We denote by $P'$ the $(d-1)$-dimensional $(1 + k^{d-2})$-tuple permutation matrix of size $(1+k^{d-2})k \\times k \\times \\cdots \\times k$ such that $P'$ and the $j$-remainder of $P$ are generated from the same $(d-1)$-dimensional permutation matrix.\n\nFor each pair of ones in a row of $P$ with coordinates $(x_1,x_2, \\ldots , x_d)$ and $(x_1 + 1,x_2, \\ldots , x_d)$, $P'$ has corresponding  ($1 + k^{d-2}$) \nones with coordinates $(\\tilde{x}_1, x_2, \\ldots, x_{j-1}, x_{j+1}, \\ldots , x_d)$,   $(\\tilde{x}_1 + 1, x_2, \\ldots , x_{j-1}, x_{j+1}, \\ldots , x_d)$,  $\\cdots$ , \n$(\\tilde{x}_1 + k^{d-2}, x_2, \\ldots , x_{j-1}, x_{j+1}, \\ldots, x_d)$  \nin a single $1$-row. Since $C$ contains $P'$, this set of ($1  + k^{d-2}$) ones is represented by $1$-entries with coordinates \n$(t_1(\\lambda), t_2, \\ldots , t_{j-1}, t_{j+1}, \\ldots , t_d)$, where $\\lambda = 1, 2, \\ldots , 1 + k^{d-2}$,  in the same $1$-row of $C$.\n\nLet $S(t_1(\\lambda), t_2, \\ldots , t_{j-1}, m, t_{j+1}, \\ldots , t_d)$,   $1 \\leq \\lambda \\leq 1 + k^{d-2}$, be the corresponding $S$ submatrices of $A'$. By the construction of $A'$, $A''$ and $C$, these $S$ submatrices are the starting $S$ submatrices of some of the chunks $C^*_1, \\ldots , C^*_r$. Each of these ($1 + k^{d-2}$) chunks has $1$-entries in every $j$-cross section; in particular each chunk has a nonzero entry \nwith the same $j^{\\text{th}}$ coordinate $(m-1)k+x_j$. There are at least $1+k^{d-2}$ nonzero entries with this given $j^{\\text{th}}$ coordinate in these chunks, but there are $k^{d-2}$ $1$-rows in a $j$-cross section of these chunks. By the pigeonhole principle,  there exist a pair of $1$-entries in the same $1$-row of $A'$.\n\nHence, for each pair of ones in the same $1$-row of $P$, we have a corresponding pair of ones in the same $1$-row of $A'$. Since two $1$-entries of $P$ not in the same \n$1$-row differ in all their coordinates, $A'$ contains $P$, and so does $A$; a contradiction.\n\\end{proof}\n\n\n\n\n\n\n\nWe can now derive a recursive inequality on $F(n, j, k, d)$, the resolution of which gives an upper bound on $F(n,j,k,d)$.\n\n\\begin{lem}\n\\label{Ine}\nLet $d$, $s$, $n$ be positive integers where $d \\ge 2$. Then\n\\begin{eqnarray}\nF(kn,2,k,d) &\\le& (d-1)nk^{d-1}F(n, 1+k^{d-2},k,d-1) + k^dF(n,1,k,d) \n +(k-1)^{d-1}F(n,2,k,d) . ~~~~~\n\\label{IN}\n\\end{eqnarray}\n\\end{lem}\n\\begin{proof} We count the maximum number of $1$-entries in $A$ by counting the number of ones in three types of chunks of $A$.\n\n\\medskip\n\n\\noindent\n{\\bf Case 1: chunk has two $1$-entries in the same $1$-row} \n\nIn view of the definitions of matrix $Q$ and a chunk, such a chunk has only one nonzero $S$ submatrix so it has at most $k^d$  nonzero entries. By Lemma \\ref{wide}, there are at most $F(n,1,k,d)$ such $S$ submatrices. Chunks of this type contain at most $k^d F(n, 1, k, d)$ nonzero entries.\n\n\\medskip\n\n\\noindent\n{\\bf Case 2: chunk is $j$-tall for some $j=2, 3, \\ldots, d$ and has no two $1$-entries in the same $1$-row}\n\nThere are $(d-1)$ choices for $j$-tall since $j=2, 3, \\ldots , d$. For each $j$, the integer $m$ of Lemma \\ref{tall} can be $1, \\ldots , n$. A $j$-tall chunk with no two $1$-entries in the same row has at most $k^{d-1}$ $1$-entries. For each pair of $j$ and $m$,  there are at most $F(n, 1 + k^{d-2}, k, d-1)$ such chunks in view of Lemma \\ref{tall}. In total, chunks of this type contain at most $(d-1)nk^{d-1}F(n, 1+k^{d-2},k,d-1)$ nonzero entries.\n\n\\medskip\n\n\\noindent\n{\\bf Case 3: chunk is not $j$-tall for any $j=2, 3, \\ldots , d$ and has no two $1$-entries in the same $1$-row}\n\nSuch a chunk has at most $(k-1)^{d-1}$ ones. By the definition of a chunk, the number of chunks is equal to the number of nonzero entries in matrix $Q$, which, by Lemma \\ref{Q}, has \nat most $F(n,2,k,d)$ nonzero entries. There are at most $(k-1)^{d-1} F(n,2,k,d)$ ones in chunks of this type.\n\nSumming all cases proves Lemma \\ref{Ine}.\n\\end{proof}\n\nWe are now ready to finish the proof of  Theorem \\ref{main}.\n\n\\begin{proof}[Proof of Theorem \\ref{main}]\nWe proceed by induction on $d$. The base case of $d=1$ is trivial. We then make the inductive assumption that \n\\begin{equation}\nF(n,j,k,d-1)=O(n^{d-2}) ~~ \\mbox{for some $d \\geq 2$}  \\label{inductive}\n\\end{equation}\nand prove that $F(n,j,k,d) = O(n^{d-1})$.\n\nWe first use Lemma \\ref{Ine} to show that \n\\begin{equation}\nF(n,2,k,d) \\leq k(c+dk)n^{d-1} \\label{j=2} ,\n\\end{equation}\nwhere $c$ is a positive constant to be determined.\n\nWe simplify inequality (\\ref{IN}) of Lemma \\ref{Ine}. Inductive assumption (\\ref{inductive}) implies that $F(n, 1+ k^{d-2},k,d-1)=O(n^{d-2})$.  \nWe also have $F(n,1,k,d)=O(n^{d-1})$, which was proven by Marcus and Tardos \\cite{MT} for $d=2$ and by  Klazar and Marcus \\cite{KM} for $d > 2$. Hence,  we can choose a sufficiently large constant $c$ \nsuch that the sum of the first two terms on the right hand side of (\\ref{IN}) is bounded by $c n^{d-1}$. Therefore,\n\\begin{equation}\n F(kn,2,k,d) \\le (k-1)^{d-1}F(n,2,k,d)+cn^{d-1}  ~~~~~ \\mbox{for all $n$.} \\label{sn}\n\\end{equation}\n\nWe then use another induction, which is a strong induction on $n$, to prove inequality (\\ref{j=2}). The base case of $n \\leq k$ is trivial.  Assuming that (\\ref{j=2}) is true for all $n < m$, we show that (\\ref{j=2}) also holds for $n =m$.\n\nLet $N$ be the maximum integer that is less than $m$ and divisible by $k$. A $d$-dimensional $m \\times \\cdots \\times m$ zero-one  matrix has at most $m^d-N^d \\le m^d-(m-k)^d \\le dkm ^{d-1}$ more entries than a $d$-dimensional $N \\times N \\times \\cdots \\times N$ matrix. Thus we have $F(m,2,k,d)\\le F(N,2,k,d)+dkm^{d-1}$. This together with (\\ref{sn}) gives\n\\begin{eqnarray*}\nF(m, 2, k, d) &\\le&  (k-1)^{d-1}F\\left({N \\over k}, 2,k,d\\right)+c\\left({N \\over k}\\right)^{d-1}+dkm^{d-1} \\\\\n&\\le&  (k-1)^{d-1}k(c+dk) \\left({N \\over k}\\right)^{d-1} + c\\left({N \\over k}\\right)^{d-1}+dkm^{d-1} \\\\\n&\\le& (k-1)(c+dk)N^{d-1}+(c+dk)m^{d-1} \\\\\n&\\le& k(c+dk)m^{d-1} \\ ,\n\\end{eqnarray*}\nwhere we use the strong inductive assumption in the second inequality. Hence, inequality (\\ref{j=2}) holds for $n=m$. The strong induction shows that (\\ref{j=2}) is true for all positive integers $n$.\n\nHaving verified the inequality (\\ref{j=2}), we continue to complete the induction on $d$ by showing that $F(n,j,k,d)=O(n^{d-1})$. This easily follows from inequality (\\ref{j=2}) and Lemma \\ref{2j}.\nWe have completed the induction.\n\nSince $F(n,j,k,d) = \\Omega(n^{d-1})$ in view of Proposition \\ref{Easy}, this together with  $F(n,j,k,d)=O(n^{d-1})$ completes the proof of Theorem \\ref{main}.\n\\end{proof}\n\nWe conclude this section with a remark. In the paragraph between two inequalities (\\ref{j=2}) and (\\ref{sn}), we use Klazar and Marcus' result \\cite{KM} $F(n, 1, k, d) = O(n^{d-1})$ to \nchoose the constant $c$ in (\\ref{j=2}). In fact, Klazar and Marcus gave a more refined upper bound ${F(n,1,k, d) \\over n^{d-1}} = 2^{O(k \\log k)}$. This allows us to improve the inductive assumption (\\ref{inductive}) to ${F(n,j,k,d-1) \\over n^{d-2}} = 2^{O( k \\log k)}$ and choose \n$c = 2^{O(k \\log k)}$.  In this way, we are able to prove ${F(n,j,k,d) \\over n^{d-1}} = 2^{O(k \\log k)}$. \n\nIn the next section, we improve Klazar and Marcus upper bound from $2^{O(k \\log k)}$ to $2^{O(k)}$. As a consequence, $c = 2^{O(k)}$ and hence ${F(n,j,k,d) \\over n^{d-1}} = 2^{O(k)}$. Lemma \\ref{wide} is crucial in making the extension from ${F(n,1,k,d) \\over n^{d-1}} = 2^{O(k)}$ to ${F(n,j,k,d) \\over n^{d-1}} = 2^{O(k)}$ possible.\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\\section{Limit inferior and limit superior }\n\\label{constant}\nIn this section, we consider matrices $P$ such that $f(n, P, d) = \\Theta(n^{d-1})$. This tight bound implies that $\\{{ f(n, P, d) \\over n^{d - 1}} \\}$ is a bounded sequence. We are interested in the limits of this sequence.\n\nWhen $d=2$, Pach and Tardos showed that $f(n, P, 2)$ is super-additive \\cite{PT}.\nBy Fekete's Lemma on super-additive sequences \\cite{Fekete}, the sequence $\\{ {f(n, P, 2) \\over n} \\}$ is convergent. The limit is known as the F\\\"uredi-Hajnal limit.\n\nWhen $d > 2$,  it is still an open problem to prove the convergence of the sequence $\\{ {f(n, P, d) \\over n^{d -1 }} \\}$. Instead, we consider the limit inferior and limit superior of the sequence and define\n\\begin{displaymath}\n\\label{ISd}\nI(P, d) = \\liminf_{n \\rightarrow \\infty} { f(n, P, d) \\over n^{d - 1}} \\ , ~~~~~   S(P, d) = \\limsup_{n \\rightarrow \\infty} { f(n, P, d) \\over n^{d - 1}} \\ .\n\\end{displaymath}\nWe derive lower bounds on $I(P,d)$ and an upper bound on $S(P,d)$. These bounds are written in terms of the size of $P$.\n\nThe main ideas in this section are Fox's interval minor containment \\cite{Fox} and our observation that the extremal function is super-homogeneous in higher dimensions.\n\n\n\\subsection{An improved upper bound}\n\nKlazar and Marcus \\cite{KM} showed that $S(P, d) =2^{O(k \\log k)}$ \nfor $k \\times \\cdots \\times k$ permutation matrices $P$. In this subsection, we extend Fox's ideas for the $d=2$ case \\cite{Fox} to improve this upper bound to $2^{O(k)}$ for $d \\geq 2$. We then show that the new upper bound also holds for tuple permutation matrices, which is a new result even for $d=2$.\n\n\n\\begin{theo} \n\\label{IS}\nIf $P$ is a $d$-dimensional $k \\times \\cdots \\times k$ permutation matrix or a tuple permutation matrix generated by such a permutation matrix, then\n$S(P,d) = 2^{O(k)}$.\n\\end{theo}\n\nThe proof uses the notion of cross section contraction and interval minor containment \\cite{Fox}. Contracting several consecutive $\\ell$-cross sections of a $d$-dimensional matrix means that we replace these $\\ell$-cross sections by a single $\\ell$-cross section, placing a one in an entry of the new cross section if at least one of the corresponding entries in the original $\\ell$-cross sections is a $1$-entry and otherwise placing a zero in that entry of the new cross section. The contraction matrix, as defined in Section 3, of an $sn \\times \\cdots \\times sn$ matrix $A$ can be obtained by contracting every $s$ consecutive $\\ell$-cross sections of $A$ uniformly for $1 \\leq \\ell \\leq d$.\n\nWe say that $A$ contains $B$ as an interval minor if we can use repeated cross section contraction to transform $A$ into a matrix which contains $B$. Matrix $A$ avoids $B$ as an interval minor if $A$ does not contain $B$ as an interval minor. \n\nEquivalently, a $k_1 \\times k_2 \\times \\cdots \\times k_d$ matrix $B =  (b_{i_1, i_2, \\ldots , i_d} )$ is an interval minor of a matrix $A$ if\n\\begin{itemize}\n\n\\item for each $i=1, \\ldots , d$, there are $k_i$ disjoint intervals, $W_{i, 1}, \\ldots , W_{i, k_i}$, which are sets of  consecutive positive integers,\n\n\\item and if $b_{i_1, \\ldots , i_d} = 1$ then the submatrix $W_{1, i_1} \\times \\cdots \\times W_{d, i_d}$ of $A$ contains a $1$-entry.\n\n\\end{itemize}\n\nThe containment in previous sections is generally stronger than containment as an interval minor. Indeed, $A$ contains $B$ implies that $A$ contains $B$ as an interval minor. However, since a permutation matrix has only one $1$-entry in every cross section, containment of a permutation matrix $P$ is equivalent to containment of $P$ as an interval minor.\n\nAnalogous to $f(n, P, d)$, we define $m(n, P, d)$ to be the maximum number of $1$-entries in a $d$-dimensional $n \\times \\cdots \\times n$ zero-one matrix that avoids $P$ as an interval minor.\n\nWe observe that\n\\begin{equation}\n\\label{fm}\nf(n, P, d) \\leq m(n, R^{k, \\ldots, k}, d) \n\\end{equation}\nfor every $k \\times \\cdots \\times k$ permutation matrix $P$. This follows from the fact that containment of $R^{k, \\ldots, k}$ as an interval minor implies containment of $P$. Hence, we seek an upper bound on $m(n, R^{k, \\ldots, k}, d)$. We denote by $f_{k_1, \\ldots , k_d}(n,t,s, d)$ the maximum number of $1$-rows that have at least $s$ nonzero entries in a $d$-dimensional $t \\times n \\times\\cdots \\times n$ matrix that avoids \n$R^{k_1, \\ldots , k_d}$ as an interval minor.\n\n\\begin{lem}\n\\label{mblock}\n\n\\begin{equation}\nm(tn,R^{k, \\ldots, k},d) \\le s^d m(n,R^{k, \\ldots, k},d)+dn t^df_{k, \\ldots, k}(n,t,s, d) ,\n\\end{equation}\n\n\\end{lem}\n\\begin{proof}\nLet $A$ be a $d$-dimensional $tn \\times \\cdots \\times tn$ matrix that avoids  $R^{k, \\ldots , k}$ as an interval minor and has $m(tn,R^{k, \\ldots , k},d)$ $1$-entries. Partition $A$ uniformly into $S$ submatrices of size $t \\times \\cdots \\times t$. Let $C$ be the contraction matrix of $A$ as defined in Section 3.\n\nWe do casework based on whether an $S$ submatrix of $A$ has $s$ nonzero $\\ell$-cross sections for some $\\ell$.\n\nWe first count the number of $1$-entries from the $S$ submatrices which do not have $s$ nonzero $\\ell$-cross sections for any $\\ell$. The contraction matrix $C$ has at most\n$m(n,R^{k, \\ldots, k},d)$ $1$-entries for, otherwise, $C$ contains $R^{k, \\ldots, k}$ as an interval minor, and thus $A$ contains $R^{k, \\ldots, k}$ as an interval minor as well, \na contradiction.\nHence,  $A$ has at most $m(n,R^{k, \\ldots, k},d)$ such $S$ submatrices, each of which contains at most $(s-1)^d <s^d$ $1$-entries. There are at most $s^dm(n,R^{k, \\ldots, k},d)$ $1$-entries from the $S$ submatrices of this type.\n\nWe next count the number of $1$-entries from the $S$ submatrices that have $s$ nonzero $\\ell$-cross sections for some $\\ell$. Let $A'$ be the matrix obtained from $A$ by changing all \nthe $1$-entries from the $S$ submatrices of $A$ that do not have $s$ nonzero $\\ell$-cross sections to $0$-entries. Without loss of generality we let $\\ell=1$.  Divide $A'$ into $n$ blocks, each of which is a $t \\times tn \\times \\cdots \\times tn$ submatrix of $A'$. For each block, contract every $t$ consecutive $j$-cross sections uniformly for \nall $j \\not =1$ to get  a $t \\times n \\times \\cdots \\times n$ matrix, which has at most $f_{k, \\ldots , k}(n,t,s, d)$ $1$-rows with at least $s$ nonzero entries in each $1$-row.  \nHence, in each block, there are at most $f_{k, \\ldots , k}(n,t,s,d )$  nonzero $S$ submatrices. As we range over \nall $n$ blocks and as $\\ell$ ranges from $1$ to $d$, we have at most $dnt^df_{k, \\ldots , k}(n,t,s, d)$ ones from $S$ submatrices of this type.\n\nSumming both cases proves Lemma \\ref{mblock}.\n\\end{proof}\n\n\n\n\nIt remains to find an upper bound on $f_{k, \\ldots , k}(n,t,s, d)$. We prove the following recursive inequality.\n\\begin{lem}\n\\label{f}\n$f_{k_1, \\ldots , k_d}(n,2t,2s, d) \\le 2 f_{k_1, \\ldots , k_d}(n,t,2s, d)+2f_{k_1 - 1, k_2, \\ldots , k_d}(n,t,s, d)$.\n\\end{lem}\n\\begin{proof}\nLet $A$ be a $d$-dimensional $2t \\times n \\times \\cdots \\times n$ matrix that avoids $R^{k_1, \\cdots, k_d}$ as an interval minor and has $f_{k_1, \\ldots , k_d}(n,2t,2s, d)$ $1$-rows, each of which has at least $2s$ ones.\n\nThe first type of these $1$-rows have all their $1$-entries among their first $t$ or last $t$ entries. There are clearly at most $2 f_{k_1, \\ldots , k_d}(n,t,2s, d)$ such $1$-rows in $A$.\n\nThe other type of these $1$-rows must have at least one $1$-entry among  both the first $t$ and the last $t$ entries. Since each $1$-row in question has at least $2s$ ones, \nthere are at least $s$ ones among either the first or last $t$ entries. Without loss of generality, we consider those $1$-rows in which the first $t$ entries contain at least $s$ $1$-entries. \nLet $A'$ be the matrix obtained from $A$ by changing all $1$-entries to $0$-entries in all other $1$-rows and then contracting the last $t$ $1$-cross sections. Hence, the last entry \nin each nonzero $1$-row of $A'$ is a $1$-entry.  The first $t$ $1$-cross sections of $A'$ must avoid $R^{k_1-1, \\ldots , k_d}$ as an interval minor for, otherwise, $A'$ \ncontains $R^{k_1, \\ldots , k_d}$ as an interval minor and so does $A$, a contradiction. Thus, there are at most $2f_{k_1-1, k_2, \\ldots , k_d}(n,t,s, d)$ $1$-rows in which \nboth the first $t$ and last $t$ entries include at least one $1$-entry.\n\nAdding up both cases gives the result.\n\\end{proof}\n\nThe recursive inequality of Lemma \\ref{f} allows us to get an upper bound on $f_{l, k, \\ldots , k}(n,t,s,d)$. \n\\begin{lem}\n\\label{closed}\nIf $s$, $t$ are powers of $2$ and $2^{\\ell -1} \\le s \\le t$, then \n$f_{\\ell ,k, \\ldots, k}(n,t,s, d) \\leq {2^{\\ell -1}t^2 \\over s} \\ m(n, R^{k, \\ldots, k}, d-1)$.\n\\end{lem}\n\\begin{proof}\nWe induct on $\\ell$. For $\\ell =1$, we show that $f_{1,k, \\ldots, k}(n,t,s, d) \\le m(n,R^{k, \\ldots, k},d-1)$. Suppose on the\ncontrary $f_{1,k, \\ldots, k}(n,t,s, d) > m(n,R^{k, \\ldots, k},d-1)$. Then there is a $t \\times n \\times \\cdots \\times n$ matrix $A$ which avoids $R^{1,k,\\ldots,k}$\nas an interval minor and has more than $m(n,R^{k, \\ldots, k},d-1)$ $1$-rows with at least $s$ $1$-entries in each $1$-row. Let $B$ be\nthe $1 \\times n \\times \\cdots \\times n$ matrix obtained from $A$ by contracting all the $1$-cross sections. Then $B$, which can be viewed as a $(d-1)$-dimensional matrix, has over $m(n,R^{k, \\ldots, k},d-1)$ $1$-entries \nand thus contains the $(d-1)$-dimensional matrix $R^{k, \\ldots, k}$ as an interval minor. Consequently, $A$ contains the $d$-dimensional $R^{1, k, \\ldots, k}$ as\nan interval minor, a contradiction. Therefore, $$f_{1,k, \\ldots, k}(n,t,s, d) \\le m(n,R^{k, \\ldots, k},d-1)  \\leq {2^{1-1}t^2 \\over s} \\ m(n, R^{k, \\ldots, k}, d - 1) \\ , $$ \nwhich proves the base case.\n\nAssuming that for all $s$ and $t$ that are powers of $2$ satisfying $2^{\\ell-2} \\le s \\le t$ we have \n\\begin{equation}\n\\label{f_i}\nf_{\\ell-1,k, \\ldots, k}(n,t,s, d) \\leq {2^{\\ell-2}t^2 \\over s}\\ m(n, R^{k, \\ldots, k}, d - 1)\n\\end{equation}\n for some $\\ell \\geq 2$, we need to show that \n\\begin{equation}\n\\label{f_}\nf_{\\ell ,k, \\ldots, k}(n,t,s, d) \\leq {2^{\\ell -1}t^2 \\over s} \\ m(n, R^{k, \\ldots, k}, d - 1) \n\\end{equation}\nfor all $s$ and $t$ that are powers of $2$ satisfying $2^{\\ell -1} \\le s \\le t$. \n\nWe use another induction on $t$ to show that (\\ref{f_}) is true for all $t \\geq s$ that are powers of $2$. The base case of $t=s$ is trivial.  If $f_{\\ell,k, \\ldots, k}(n,t,s, d) \n\\leq {2^{\\ell-1}t^2 \\over s} m(n, R^{k, \\ldots, k}, d - 1) $ for some $t \\geq s$ that is a power of $2$, we prove the same inequality for $2t$.\nBy Lemma \\ref{f}, we have \n\\begin{eqnarray*}\nf_{\\ell,k, \\ldots, k}(n,2t,s, d) &\\le& 2f_{\\ell,k, \\ldots, k}(n,t,s, d)+2f_{\\ell-1,k, \\ldots, k}(n,t,s\/2, d) \\\\\n&\\le & 2{2^{\\ell-1}t^2 \\over s} m(n, R^{k, \\ldots, k}, d - 1) +2{2^{\\ell-2}t^2 \\over s\/2} m(n, R^{k, \\ldots, k}, d - 1)  \\\\ \n&=& {2^{\\ell-1}(2t)^2 \\over s} m(n, R^{k, \\ldots, k}, d - 1) \\ ,\n\\end{eqnarray*}\nwhere we use the two inductive assumptions in the second inequality. \nThus our induction on $t$ is complete and (\\ref{f_}) is proved. As a result, our induction on $l$ is also complete.\n\\end{proof}\n\nWe are now ready to prove Theorem \\ref{IS}.\n\\begin{proof}[Proof of Theorem \\ref{IS}]\nWe first bound the right hand side of inequality (\\ref{fm}). \nWe claim that\n\\begin{equation}\n\\label{mO}\n{m(n,R^{k, \\ldots, k},d) \\over n^{d -1}} =2^{O(k)} \\ .\n\\end{equation}\n\nThe base case of $d=1$ is trivial. Assuming that (\\ref{mO}) is true for $(d-1)$, we\ncombine Lemmas \\ref{mblock} and \\ref{closed} to get\n$$m(tn,R^{k, \\ldots , k},d) \\le s^dm(n,R^{k, \\ldots, k}, d)+dt^d{2^{k-1}t^2 \\over s}2^{O(k)}n^{d-1} .$$\nChoosing $t=2^{dk }$ and $s=2^{k-1}$ yields\n$$m(2^{dk}n,R^{k, \\ldots , k},d) \\le 2^{(k-1)d}m(n,R^{k, \\ldots, k},d)+d  2^{kd(d + 2)} 2^{O(k)}n^{d-1} .$$\nIn particular, if $n$ is a positive integer power of $2^{dk}$, iterating this inequality yields\n\\begin{eqnarray*}\n\\lefteqn{m((2^{dk})^L, R^{k, \\ldots, k}, d)} \\\\\n& \\leq& 2^{(k-1)d} m((2^{dk})^{L - 1}, R^{k, \\ldots, k}, d)  \n + d 2^{kd(d+2)} 2^{O(k)} (2^{dk})^{(L-1)(d-1)} \\\\\n&\\leq & 2^{2(k-1)d}  m((2^{dk})^{L - 2}, R^{k, \\ldots, k}, d)  \n + \\  d 2^{kd(d+2)} 2^{O(k)} \\left( 1 + {1 \\over 2^{d(dk - 2k +1)}} \\right) (2^{dk})^{(L-1)(d-1)} \\\\\n&\\leq& 2^{L(k-1)d}  m(1, R^{k, \\ldots, k}, d)  \n + \\ d 2^{kd(d+2)} 2^{O(k)} \\left(  1 + \n{1 \\over 2^{d(dk - 2k +1)}} + {1 \\over 2^{2d(dk - 2k +1)}} + \\cdots \\right) (2^{dk})^{(L-1)(d-1)} \\\\\n&=& 2^{O(k)} (2^{dk})^{(L-1)(d-1)} \\ .\n\\end{eqnarray*}\nHence, if $(2^{kd})^{L-1} \\leq n < (2^{kd})^L$, then\n$$m(n, R^{k, \\ldots, k}, d) \\leq m((2^{dk})^L, R^{k, \\ldots, k}, d) = 2^{O(k)} (2^{dk})^{(L-1)(d-1)} \\leq 2^{O(k)} n^{d-1} \\ .$$\nThis completes the induction on $d$, and hence (\\ref{mO}) is proved.\n\nIt follows from (\\ref{fm}) and (\\ref{mO}) that Theorem \\ref{IS} is true for every permutation matrix $P$. By the remark at the end of Section 3, this result can be extended to tuple permutation matrices. The proof of Theorem \\ref{IS} is completed.\n\\end{proof}\n\n\\subsection{\nLower bounds and super-homogeneity}\n\nWe first use Cibulka's method in \\cite{C} to show that $I(P, d) \\geq d(k-1)$ for all permutation matrices of size $k \\times \\cdots \\times k$ and extend this  lower bound to tuple permutation matrices. \n\\begin{theo}\n\\label{llb}\nIf $P$ is a $d$-dimensional $k \\times \\cdots \\times k$ permutation matrix or a tuple permutation matrix generated by such a permutation matrix, then $I(P, d) \\geq d(k-1)$. Furthermore, if $P$ is the identity matrix, then $I(P, d) = S(P, d) = d(k-1)$.\n\\end{theo}\n\\begin{proof} We first show that, for all $n\\ge k-1$, we have\n\\begin{equation}\n\\label{lb}\nf(n, P, d) \\geq n^d - (n - k +1)^d\n\\end{equation}\nfor every permutation matrix $P$. \nPick one nonzero entry $p_{i_1, \\ldots , i_d}=1$ of $P$. Construct a $d$-dimensional $n \\times \\cdots \\times n$ matrix $A$ with entries such that $a_{j_1, \\ldots , j_d}=0$ if $i_l \\le j_l \\le n-k+i_l$ for all $1 \\leq l \\leq d$ and $a_{j_1, \\ldots , j_d}=1$ otherwise. We first show that $A$ avoids $P$. Suppose to the contrary that $A$ contains $P$. Let the special nonzero entry \n$p_{i_1, \\ldots, i_d}=1$ of $P$ be represented by entry $a_{y_1, \\ldots, y_d}$ of $A$.  By the construction of $A$, we must have \neither $y_l\\le i_l-1$ or $y_l \\ge n-k+i_l+1$. If $y_l \\le i_l -1$, since $A$ contains $P$,\n$A$ has $i_l -1$ other nonzero entries whose $l^{\\text{th}}$ coordinates are smaller than $y_l \\leq i_l -1$ to represent $1$-entries of $P$, an impossibility. If $y_l \\geq n - k + i_l +1$, a similar argument leads to another impossibility. Counting the number of $1$-entries in $A$ proves (\\ref{lb}).\n\nWe next show that \n\\begin{equation}\n\\label{ub}\nf(n,P,d) \\le n^d-(n-k+1)^d\n\\end{equation}\nwhen $P$ is the identity matrix, i.e.,   $p_{i_1, \\ldots, i_d}$ is one on the main diagonal $i_1=\\cdots=i_d$ and zero otherwise. \nIf $A$ is a matrix that avoids $P$,  each diagonal of $A$, which is parallel to the main diagonal, has at most $k-1$ nonzero entries. Summing over the maximum numbers of $1$-entries in all diagonals\nproves (\\ref{ub}). \n\nThe second part of Theorem \\ref{llb} follows immediately from (\\ref{lb}) and (\\ref{ub}). The first part is obvious for a permutation matrix $P$ because of (\\ref{lb}). \nThe first part is also true \nfor a tuple permutation matrix $P'$ since $f(n, P, d) \\leq f(n, P', d)$ if $P'$ is generated by a permutation matrix $P$.\n\\end{proof}\n\nThe lower bound given in Theorem \\ref{llb} is linear in $k$. One may ask how large a lower bound on $I(P,d)$ can be for some $P$. In the the rest of this section, we extend Fox's idea for the $d=2$ case \\cite{Fox, Fox2} to show that a lower bound can be as large as an exponential function in $k$ in multiple dimensions. The crucial part in our approach is our observation that $f(n,P,d)$ is super-homogeneous.\n\n\\begin{theo}\n\\label{lower}\nFor each large $k$, there exists a family of $d$-dimensional $k\\times \\cdots \\times k$ permutation matrices $P$ such that \n$I(P,d)=2^{\\Omega(k^{1 \/ d})}$.\n\\end{theo}\nThe proof uses the super-homogeneity of extremal functions. In dimension two, the extremal function \nwas shown to be super-additive \\cite{PT}, i.e., $f(m+ n, P, 2) \\geq f(m, P, 2) + f(n, P, 2)$. This was the key in showing the convergence of the sequence\n$\\{ {f(n,P, 2) \\over n} \\}$ for those  matrices $P$ whose extremal functions are $\\Theta(n)$.  The  limit is the well-known F\\\"uredi-Hajnal limit \\cite{FH}.\n\nWe note that the super-additivity of $f(n, P, 2)$ implies super-homogeneity, i.e., $f(s n, P, 2) \\geq s f(n, P, 2)$ for every positive integer $s$. In higher dimensions, \nwe show that $f(n, P, d)$ is\nsuper-homogeneous of a higher degree.\n\nA corner entry of a $k_1 \\times \\cdots \\times k_d$ matrix $P = (p_{i_1, \\ldots, i_d})$ is defined to be an  entry $p_{i_1, \\ldots, i_d}$ located at a corner of $P$, i.e.,  where $i_{\\tau}=1$ or $k_{\\tau}$ for $1 \\leq \\tau \\leq d$.\n\n\\begin{lem}\n\\label{homo}\nIf $P$ is a $d$-dimensional matrix with a corner $1$-entry, then $f(sn,P,d) \\ge {s^{d-1} \\over (d-1)!}f(n,P,d)$.\n\\end{lem}\n\\begin{proof}\n Without loss of generality, we assume that $p_{1, \\ldots, 1}=1$ is the corner $1$-entry in $P$. Let $M$ be an $s \\times \\cdots\\times s$ matrix with $1$-entries \nat the coordinates $(i_1, \\ldots, i_d)$ where $i_1+ \\cdots+i_d=s+d-1$ and $0$-entries everywhere else, so $M$ has ${s+d-2 \\choose d-1} \\ge {s^{d-1} \\over (d-1)!}$ $1$-entries. \nLet $N$ be an $n \\times \\cdots \\times n$ matrix that avoids $P$ and has $f(n,P,d)$ $1$-entries. It then suffices to prove that $M \\otimes N$  avoids $P$.\n\nAssume for contradiction that the Kronecker product $M \\otimes N$ contains $P$. Pick an arbitrary $1$-entry $p^*$ in $P$ other than $p_{1, \\ldots, 1}$.  Suppose that $p_{1, \\ldots, 1}$ and $p^*$ are represented by $e_1$ and $e_2$  in $M \\otimes N$, respectively.\nWe consider the $n \\times \\cdots \\times n$ $S$ submatrices of $M \\otimes N$. We may assume that $e_1$ and $e_2$ are in\nthe $S$-submatrices $S(i_1, \\ldots, i_d)$ and $S(j_1, \\ldots, j_d)$, respectively. Note that $i_1+ \\cdots +i_d = j_1+ \\cdots + j_d$. Since $p^*$\nhas larger coordinates than $p_{1, \\ldots, 1}$ in $P$, entry $e_2$ must also have larger coordinates than $e_1$ in $M \\otimes N$ and hence $i_{\\tau} \\leq j_{\\tau}$ for $\\tau = 1, 2, \\ldots, d$. It then follows from $i_1+ \\cdots +i_d = j_1+ \\cdots + j_d$\nthat $i_{\\tau} = j_{\\tau}$ for $\\tau= 1, 2, \\ldots, d$, i.e., \nthe two entries $e_1$ and $e_2$ must be in the same $S$ submatrix in $M \\otimes N$. Since $p^*$ is an arbitrary $1$-entry other than $p_{1, \\ldots , 1}$ in $P$, \nthe $S$ submatrix contains $P$. But this is a contradiction since each nonzero $S$ submatrix in $M \\otimes N$ is an exact copy of $N$, \nwhich avoids $P$. Thus $M \\otimes N$ avoids $P$.\n\\end{proof}\n\nJust as super-additivity leads to the F\\\"uredi-Hajnal limit in dimension two, super-homogeneity also produces an interesting result on limits.\n\\begin{lem}\n\\label{converge}\nIf $P$ is a $d$-dimensional matrix which contains a corner $1$-entry, then for any positive integer $m$,\n$$I(P,d) \\ge {1 \\over (d-1)!} \\ {f(m,P,d) \\over m^{d-1}}.$$\n\\end{lem}\n\\begin{proof}\nFor each fixed positive integer $m$, we write $n$ as $n=sm+r$, where $0 \\le r<m$. Then \n$${f(n,P,d) \\over n^{d-1}} = {f(sm+r,P,d) \\over (sm+r)^{d-1}} \n \\ge  {f(sm,P,d) \\over (sm+r)^{d-1}} \n\\ge {s^{d-1} \\over (d-1)!}{f(m,P,d) \\over (sm+r)^{d-1}} \\ ,$$\nwhere we use Lemma \\ref{homo} in the second inequality. Now we take the limit inferior of the left side as $n$ goes to $\\infty$. Since $m$ is fixed and $r$ is bounded, $s={n-r \\over m}$ goes to $\\infty$ as well. Hence,\n$$ \\liminf_{n \\to \\infty}{f(n,P,d) \\over n^{d-1}} \\ge \\lim_{s \\to \\infty}{s^{d-1} \\over (d-1)!}{f(m,P,d) \\over (sm+r)^{d-1}} \n= {1 \\over (d-1)!}{f(m,P,d) \\over m^{d-1}} \\ . $$\n\\end{proof}\n\nThe following lemma gives a lower bound on the right hand side of the inequality in Lemma \\ref{converge} for a particular $m = N$. The proof is based on Fox's ideas in his lecture \\cite{Fox2} for $d=2$ case.\n\\begin{lem}\n\\label{exists}\nThere exists a $d$-dimensional  $N \\times \\cdots \\times N$ matrix $A$, where $N=2^{\\Omega(\\ell)}$, that has $\\Theta(N^{d-1\/2})$ $1$-entries and avoids $R^{\\ell,\\ldots, \\ell}$ as an interval minor.\n\\end{lem}\n\\begin{proof}\n\nWe prove the lemma for $\\ell$ that are multiples of $20$ and the result can then be easily extended to all $\\ell$. Let $r={\\ell \\over 20}$, $q={\\ln 2 \\over 2r^{d-1}}$, and $N=2^r$. \n\nDefine a dyadic interval to be a set of consecutive integers of the form $\\{ (s-1)2^t+1, \\ldots, s2^t \\}$ for nonnegative integers $s$ and $t$, and a dyadic hyper-rectangle to be the Cartesian product of $d$ dyadic intervals. We consider only dyadic intervals which are subsets of $\\{1, \\ldots, N \\}$.\nNote that there are exactly $\\sum_{t=0}^{r}2^t=2N-1$ such dyadic intervals and $(2N-1)^d$ such dyadic hyper-rectangles. There are $(r+1)^d$ dyadic hyper-rectangles \ncontaining each lattice point $(i_1, \\ldots , i_d)$, where $i_j = 1, 2, \\ldots , N$ and $j= 1, 2, \\ldots , d$,  since each $i_j$ is contained in exactly $r+1$ dyadic intervals. \n\nLet $\\cal R$ be a random collection of these dyadic hyper-rectangles, each included independently with probability $q$. Define $A$ to be the $N\\times \\cdots \\times N$ $d$-dimensional matrix such that $a_{i_1, \\ldots, i_d}=1$ if $(i_1, \\ldots, i_d)$ is not contained in any dyadic hyper-rectangle of $\\cal R$ and $a_{i_1, \\ldots, i_d}=0$ otherwise. The expected number of $1$-entries in $A$ is \n$$(1-q)^{(r+1)^d}N^d =[(1 - q)^{1\/q}]^{q (r+1)^d} N^d = \\Theta(e^{-qr^d}N^d)=\\Theta(2^{-r\/2}N^d)=\\Theta(N^{d-1\/2}),$$\nwhere we use $(1 - q)^{1\/q} = \\Theta(e^{-1})$ for small $q$ in the second  equality.\n\nDenote by $X$ and $Y$ the events that $A$ contains and avoids $R^{\\ell, \\ldots, \\ell}$ as an interval minor, respectively. We estimate the probability $\\mathbb{P}(X)$. If $B$ is a set of dyadic intervals, let $\\chi(B)$ be the number of dyadic intervals that contain at least one interval in $B$ as a subset. Then we define $h(x)$ to be the  number of sets $B$ containing $\\ell$ dyadic intervals such that $\\chi(B)=x$. \n\nIf $A$ contains $R^{\\ell, \\ldots, \\ell}$ as an interval minor, then there are  intervals of consecutive integers, denoted by $W_{i,1}, \\ldots, W_{i, \\ell}$, partitioning the set $\\{ 1, 2, \\ldots , N \\}$  in the $i$th dimension such that every submatrix $W_{1, j_1} \\times W_{2, j_2} \\times \\cdots \\times W_{d, j_d}$ of $A$ contains at least one $1$-entry. We denote by $I_{i,j}$ the unique smallest-length dyadic interval which contains $W_{i,j}$. If there are $x_i$ dyadic intervals which contain at least one of the dyadic intervals in $B_i=\\{I_{i,1}, \\ldots, I_{i,\\ell} \\}$, then $x_i \\geq \\ell$ and there are $h(x_i)$ possible sets $B_i$. Thus there are $h(x_1) \\cdots h(x_d)$ choices for $B_1, \\ldots, B_d$ and there are $x_1 \\cdots x_d$ dyadic hyper-rectangles which contain at least one of the $\\ell^d$ dyadic hyper-rectangles of the form $I_{1, i_1}\\times \\cdots \\times I_{d, i_d}$. Since these $x_1 \\cdots x_d$ dyadic hyper-rectangles contain $1$-entries of $A$, none of them is in $\\cal R$. Hence, the probability \n\\begin{equation}\n\\label{P(X)}\n\\mathbb{P}(X)  \\leq \\sum_{x_1,\\ldots,x_d\\ge \\ell} (1-q)^{x_1\\cdots x_d}h(x_1)\\cdots h(x_d).\n\\end{equation}\n\nTo find a bound on $h(x)$, we estimate the number of sets $U=\\{u_1, \\ldots, u_{\\ell}\\}$ of dyadic intervals such that $\\chi(U)=x$. Let $v_1, \\ldots, v_\\ell$ be the integers such that $v_1=\\chi(\\{u_1 \\})$ and $v_i=\\chi(\\{u_1, \\ldots, u_i\\})-\\chi(\\{u_1, \\ldots, u_{i-1}\\})$, for $i = 2, \\ldots , \\ell$. Since $v_1+\\cdots+v_{\\ell}=x$, there are at most ${x+\\ell-1 \\choose \\ell-1}$ possible values for $v_1, \\ldots, v_{\\ell}$. Given $v_1, \\ldots, v_{\\ell}$, there are at most $2^{v_1}$ choices for $u_1$ and at most $2^{v_{i+1}}\\chi(\\{u_1, \\ldots, u_i\\})$ choices for $u_{i+1}$. Calculating the number of possible choices for $U$ gives\n$$h(x) \\le{{x+\\ell-1 \\choose \\ell-1}2^{v_1}\\prod_{i=1}^{\\ell-1} 2^{v_{i+1}}\\chi(\\{u_1, \\ldots, u_i\\}) \\over \\ell !} \\le {{x+\\ell-1 \\choose \\ell-1}2^x x^{\\ell}  \\over \\ell !}.$$ \nSubstituting this into (\\ref{P(X)}) yields\n\\begin{equation}\n\\mathbb{P}(X)  \\leq \\sum_{x_1,\\ldots,x_d\\ge \\ell} (1-q)^{x_1\\cdots x_d} \\prod_{i=1}^{d} {{x_i+\\ell-1 \\choose \\ell-1}2^{x_i} {x_i}^{\\ell}  \\over \\ell !} \\ .  \\nonumber\n\\end{equation}\nThe summand \n $$r(x_1,\\ldots,x_d)=(1-q)^{x_1\\cdots x_d}\\prod_{i=1}^{d}{{x_i+\\ell-1 \\choose \\ell-1}2^{x_i} x_i^{\\ell}  \\over \\ell !}$$\nis a symmetric function of its variables.\nNote that the ratio ${r(x_1+1,\\ldots, x_d) \/ r(x_1,\\ldots,x_d)}$ is equal to \n$$2(1-q)^{x_2 \\cdots x_d}{(x_1+\\ell) \\over x_1+1}(1+{1\\over x_1})^{\\ell} \\le 4 e^{-qx_2\\cdots x_d}(1+{1\\over x_1})^{x_1}\\le e^{-q\\ell^{d-1}}4e=2^{-20^{d-1} \/ 2}4e \\le 1\/2 \\ ,$$\nwhere we use $(1 - q)^{1\/q} \\leq 1\/e$ for small $q > 0$ and $(1 + 1\/x_1)^{x_1} \\leq e$ for $x_1 \\geq 1$. We also note that $$r(\\ell, \\ldots , \\ell) = (1-q)^{\\ell^d}\\left[{2\\ell-1\\choose \\ell-1}2^{\\ell}{\\ell^{\\ell} \\over \\ell!} \\right]^d  \\le  e^{-q\\ell^d}(2^{2\\ell-1}2^{\\ell}e^{\\ell})^d \\le \n2^{-20^{d-1}\\ell\/2 } (2^{3\\ell - 1}e^{\\ell})^d \\le (2N)^{-d}, $$\nwhere we use Stirling's inequality and ${2\\ell-1 \\choose \\ell-1}\\le 2^{2\\ell-1}$.\nWe now use the \nsymmetry of $r(x_1, \\cdots, x_d)$ to obtain\n$$ \\mathbb{P}(X)  \\le \\sum_{x_1, \\ldots, x_d\\ge \\ell} r(x_1,\\ldots, x_d)  \\le \\left(\\sum_{i=0}^{\\infty}(1\/2)^i \\right)^d r(\\ell,\\ldots,\\ell) \n\\le 2^d  (2N)^{-d} = N^{-d}. $$\n\nWe now estimate conditional expectation $\\mathbb{E}( \\xi | Y)$, where $\\xi = |A|$.  Note that $\\mathbb{E}(\\xi | Y)\\mathbb{P}(Y)=\\mathbb{E}(\\xi)-\\mathbb{E}(\\xi | X) \\mathbb{P}(X) \\ge \\Theta(N^{d-1\/2})-N^d N^{-d} = \\Theta(N^{d-1\/2})$ so $\\mathbb{E}(\\xi | Y)=\\Theta(N^{d-1\/2})$.\nThus, there exists an $A$ that avoids $R^{l, \\ldots,l}$ as an interval minor and has at least $\\Theta(N^{d-1\/2})$ $1$-entries.\n\\end{proof}\n\n\nWe are now ready to prove Theorem \\ref{lower}.\n\\begin{proof} [Proof of Theorem \\ref{lower}]\n\nLet $\\ell=\\lfloor k^{1\/d} \\rfloor$ be the largest integer less than or equal to $ k^{1\/d}$. There is a family  of $d$-dimensional permutation matrices of size $\\ell^d \\times \\cdots \\times \\ell^d$ that contain $R^{\\ell, \\ldots , \\ell}$ as an interval minor and have at least one corner $1$-entry. To see this, \nmany such permutation matrices can be constructed so that they have exactly one $1$-entry in each of their $S$-submatrices of size $\\ell^{d-1} \\times \\cdots \\times \\ell^{d-1}$, including a corner $1$-entry.\n\nSince $k \\geq \\ell^d$, there is a family of  permutation matrices $P$ of size $k \\times \\cdots \\times k$ that contain $R^{\\ell, \\ldots , \\ell}$ as an interval minor and have at least one corner $1$-entry.\nEach $P$ has a corner $1$-entry, so we can apply Lemma \\ref{converge} to obtain\n\\begin{equation}\n\\label{key}\nI(P,d) \\ge {1 \\over (d-1)!} \\ {f(N,P,d) \\over N^{d-1}} ,\n\\end{equation}\nwhere $N$ can be chosen to be the positive integer given in Lemma 4.9. \n\nMatrix $P$ contains $R^{\\ell,\\ldots,\\ell}$ as an interval minor, \nso $f(N,P,d)\\ge m(N,R^{\\ell, \\ldots, \\ell},d)$,\nwhich along with (\\ref{key}) and Lemma \\ref{exists} yields \n$$I(P,d)  \\ge {1 \\over (d-1)!} \\ {m(N, R^{\\ell, \\ldots, \\ell}, d) \\over N^{d - 1}} \n \\geq \\Theta(N^{1 \\over 2}) \n= 2^{\\Omega(\\ell)}\n= 2^{\\Omega(k^{1 \/ d})}.\n$$\nThis completes the proof of Theorem \\ref{lower}.\n\\end{proof}\n\n\\section{Conclusions and future directions}\n\\label{conclusion}\nWe obtained non-trivial lower and upper bound on $f(n,P,d)$ when $n$ is large for block permutation matrices $P$. In particular, we established the tight bound $\\Theta(n^{d-1})$ on $f(n,P,d)$ for every $d$-dimensional tuple permutation matrix $P$. \nWe improved the previous upper bound on the limit superior of the sequence $\\{ {f(n,P,d) \\over n^{d-1}}\\}$ for \nall permutation and tuple permutation matrices. We used the super-homogeneity of the extremal function to show that \nthe limit inferior is exponential in $k$ for a family of $k \\times \\cdots \\times k$ permutation matrices. Our results substantially advance the extremal theory of matrices. We believe that super-homogeneity is fundamental to pattern avoidance in multidimensional matrices.\n\nOne possible direction for future research would be to strengthen the super-homogeneity as expressed in Lemma \\ref{homo} to\n$f(sn, P, d) \\geq s^{d-1} f(n, P, d)$. We have successfully tested this super-homogeneity on the identity matrix and the matrices whose $1$-entries are on rectilinear paths. If this super-homogeneity is true \nfor permutation matrices $P$, we can then use a Fekete-like lemma to show the convergence of the sequence $\\{ {f(n,P,d) \\over n^{d-1}} \\}$.\n\nAnother possible direction would be to extend Theorem \\ref{lower} from a family of permutation matrices to almost all permutation matrices. We think this becomes possible if  the corner $1$-entry condition is removed in Lemmas \\ref{homo} and \\ref{converge}.\n\n\\section{Acknowledgments}\n\nWe would like to thank Professor Jacob Fox for valuable discussions about this research. The first author was supported by the NSF graduate fellowship under grant number 1122374. The research of the second author was supported in part by the Department of Mathematics, MIT through PRIMES-USA 2014. It was also supported in part  by the Center for Excellence in Education and the Department of Defense through RSI 2014. \n\n\n\\bibliographystyle{amsplain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThis paper examines split-improvement feature importance scores for tree-based methods.  Starting with Classification and Regression Trees \\citep[CART;][]{breiman1984classification} and C4.5 \\citep{quinlan2014c4}, decision trees have been a workhorse of general machine learning, particularly within ensemble methods such as Random Forests \\citep[RF;][]{breiman2001random} and Gradient Boosting Trees \\citep{friedman2001greedy}. They enjoy the benefits of computational speed, few tuning parameters and natural ways of handling missing values. Recent statistical theory for ensemble methods \\citep[e.g.][]{denil2014narrowing,scornet2015consistency,mentch2016quantifying,wager2018estimation,zhou2018boulevard} has provided theoretical guarantees and allowed formal statistical inference. \\textcolor{black}{Variants of these models have also been proposed such as Bernoulli Random Forests \\citep{yisen2016bernoulli,wang2017novel} and Random Survival Forests \\citep{ishwaran2008random}.} For all these reasons, tree-based methods have seen broad applications including in protein interaction models \\citep{meyer2017interactome} in product suggestions on Amazon \\citep{sorokina2016amazon} and in financial risk management \\citep{khaidem2016predicting}.\n\nHowever, in common with other machine learning models, large ensembles of trees act as ``black boxes'', providing predictions but little insight as to how they were arrived at. There has thus been considerable interest in providing tools either to explain the broad patterns that are modeled by these methods, or to provide justifications for particular predictions. This paper examines variable or feature\\footnote{We use ``feature'', ``variable'' and ``covariate'' interchangeably here to indicate individual measurements that act as inputs to a machine learning model from which a prediction is made.} importance scores that provide global summaries of how influential a particular input dimension is in the models' predictions. These have been among the earliest diagnostic tools for machine learning and have been put to practical use  as screening tools, see for example \\citet{diaz2006gene} and \\citet{menze2009comparison}. Thus, it is crucial that these feature importance measures reliably produce well-understood summaries.\n\nFeature importance scores for tree-based models can be broadly split into two categories. Permutation methods rely on measuring the change in value or accuracy when the values of one feature are replaced by uninformative noise, often generated by a permutation. These have the advantage of being applicable to any function, but have been critiqued by \\citet{hooker2007generalized,strobl2008conditional,hooker2019please} for forcing the model to extrapolate. By contrast, in this paper we study the alternative split-improvement scores \\textcolor{black}{(also known as Gini importance, or mean decrease impurity)} that are specific to tree-based methods. These naturally aggregate the improvement associated with each note split and can be readily recorded within the tree building process \\citep{breiman1984classification,friedman2001greedy}. In Python, split-improvement is the default implementation for almost every tree-based model, including \\textbf{RandomForestClassifier}, \\textbf{RandomForestRegressor}, \\textbf{GradientBoostingClassifier} and \\textbf{GradientBoostingRegressor} from \\textbf{scikit-learn} \\citep{pedregosa2011scikit}.\n\nDespite their common use, split-improvement measures are biased towards features that exhibit more potential splits and in particular towards continuous features or features with large numbers of categories. \\textcolor{black}{This weakness was already noticed in \\citet{breiman1984classification} and \\citet{strobl2007bias} conducted thorough experiments followed by more discussions in \\citet{boulesteix2011random} and \\citet{nicodemus2011letter}\\footnote{See \\underline{https:\/\/explained.ai\/rf-importance\/} for a popular demonstration of this.}.} While this may not be concerning when all covariates are similarly configured, in practice it is common to have a combination of categorical and continuous variables in which emphasizing more complex features may mislead any subsequent analysis. For example, gender will be a very important binary predictor in applications related to medical treatment; whether the user is a paid subscriber is also central to some tasks such as in Amazon and Netflix. But each of these may be rated as less relevant to age \\textcolor{black}{which is a more complex feature} in either case. \\textcolor{black}{In the task of ranking single nucleotide polymorphisms with respect to their ability to predict a target phenotype, researchers may overlook rare variants as common ones are systematically favoured by the split-improvement measurement. \\citep{boulesteix2011random}}.\n\nWe offer an intuitive rationale for this phenomenon and design a simple fix to solve the bias problem. The observed bias is similar to overfitting in training machine learning models, where we should not build the model and evaluate relevant performance using the same set of data. To fix this, split-improvement calculated from a separate test set is taken into consideration. We further demonstrate that this new measurement is unbiased in the sense that features with no predictive power for the target variable will receive an importance score of zero in expectation. \\textcolor{black}{These measures can be very readily implemented in tree-based software packages.} We believe the proposed measurement provides a more sensible means for evaluating feature importance in practice.\n\nIn the following, we introduce some background and notation for tree-based methods in Section \\ref{tree}. In Section \\ref{FI}, split-improvement is described in detail and its bias and limitations are presented.  The proposed unbiased measurement is introduced in Section \\ref{USI}. Section \\ref{real} applies our idea to a simulated example and three real world data sets. We conclude with some discussions and future directions in Section \\ref{dis}. Proofs and some additional simulation results are collected in Appendix \\ref{proof} and \\ref{sim} respectively. \n\n\\section{Tree-Based Methods}\\label{tree}\n\nIn this section, we provide a brief introduction and mathematical formulation of tree-based models \\textcolor{black}{that will also serve to introduce our notation}. We refer readers to relevant chapters in \\citet{friedman2001elements} for a more detailed presentation. \n\n\\subsection{Tree Building Process}\\label{cart}\n\nDecision trees are a non-parametric machine learning tool for constructing prediction models from data. They are obtained by recursively partitioning feature space by axis-aligned splits and fitting a simple prediction function, usually constant, within each partition. The result of this partitioning procedure is represented as a binary tree. Popular tree building algorithms, such as CART and C4.5, may differ in how they choose splits or deal with categorical features. Our introduction in this section mainly reflects how decision trees are implemented in \\textbf{scikit-learn}.\n\nSuppose our data consists of $p$ inputs and a response, denoted by $z_i = (x_i, y_i)$ for $i = 1, 2, \\ldots, n$, with $x_i = (x_{i1}, x_{i2}, \\ldots, x_{ip})$. For simplicity we assume our inputs are continuous\\footnote{Libraries in different programming languages differ on how to handle categorical inputs. \\textbf{rpart} and \\textbf{randomForest} libraries in \\textbf{R} search over every possible subsets when dealing with categorical features. However, tree-based models in \\textbf{scikit-learn} do not support categorical inputs directly. Manually transformation is required to convert categorical features to integer-valued ones, such as using dummy variables, or treated as ordinal when applicable.}. Labels can be either continuous (regression trees) or categorical (classification trees). Let the data at a node $m$ represented by $Q$. Consider a splitting variable $j$ and a splitting point $s$, which results in two child nodes: \n$$\nQ_{l} = \\{(x, y)|x_j \\leq s \\}\n$$\n$$\nQ_{r} = \\{(x, y)|x_j > s \\}.\n$$\nThe impurity at node $m$ is computed by a function $H$, which acts as a measure for goodness-of-fit \\textcolor{black}{and is invariant to sample size}. Our loss function for split $\\theta = (j, s)$ is defined as the weighted average of the impurity at two child nodes:\n$$\nL(Q, \\theta) = \\frac{n_{l}}{n_m}H(Q_{l}) + \\frac{n_{r}}{n_m}H(Q_{r}),\n$$\nwhere $n_m, n_l, n_r$ are the number of training examples falling into node $m, l, r$ respectively.\nThe best split is chosen by minimizing the above loss function:\n\\begin{equation}\\label{best}\n    \\theta^* = \\argmin_{\\theta}L(Q, \\theta).\n\\end{equation}\nThe tree is built by recursively splitting child nodes until some stopping criterion is met. For example, we may want to limit tree depth, or keep the number of training samples above some threshold within each node.\n\nFor regression trees, $H$ is usually chosen to be mean squared error, using average values as predictions within each node. At node $m$ with $n_m$ observations, $H(m)$ is defined as:\n$$\n\\bar{y}_m = \\frac{1}{n_m}\\sum_{x_i \\in m}y_i,\n$$\n$$\nH(m) = \\frac{1}{n_m}\\sum_{x_i \\in m}(y_i - \\bar{y}_m)^2.\n$$\nMean absolute error can also be used depending on specific application.\n\nIn classification, there are several different choices for the impurity function $H$. Suppose for node $m$, the target $y$ can take values of $1, 2, \\ldots, K$, define\n$$\np_{mk} = \\frac{1}{n_m}\\sum_{x_i \\in m}\\mathbbm{1}(y_i = k)\n$$\nto be the proportion of class $k$ in node $m$, for $k = 1, 2, \\ldots, K$. Common choices are:\n\\begin{enumerate}\n\\item Misclassification error:\n$$\nH(m) = 1 - \\max_{1 \\leq k \\leq K} p_{mk}.\n$$\n\\item Gini index:\n$$\nH(m) = \\sum_{k \\neq k'}p_{mk}p_{mk'} = 1 - \\sum_{k = 1}^K p_{mk}^2.\n$$\n\\item Cross-entropy or deviance:\n$$\nH(m) = -\\sum_{k=1}^K p_{mk}\\log p_{mk}.\n$$\n\\end{enumerate}\n\nThis paper will focus on mean squared error for regression and Gini index for classification. \n\n\\subsection{Random Forests and Gradient Boosting Trees}\n\nThough intuitive and interpretable, there are two major drawbacks associated with a single decision tree: they suffer from high variance and in some situations they are too simple to capture complex signals in the data. Bagging \\citep{breiman1996bagging} and boosting \\citep{friedman2001greedy} are two popular techniques used to improve the performance of decision trees.\n\nSuppose we use a decision tree as a base learner $t(x; z_1, z_2, \\ldots, z_n)$, where $x$ is the input for prediction and $z_1, z_2, \\ldots, z_n$ are training examples as before. Bagging aims to stabilize the base learner $t$ by resampling the training data. In particular, the bagged estimator can be expressed as:\n$$\n\\hat{t}(x) = \\frac{1}{B}\\sum_{b=1}^Bt(x; z^*_{b1}, z^*_{b2}, \\ldots, z^*_{bn})\n$$\nwhere $z^*_{bi}$ are drawn independently with replacement from the original data (bootstrap sample), and $B$ is the total number of base learners. Each tree is constructed using a different bootstrap sample from the original data. Thus approximately one-third of the cases are left out and not used in the construction of each base learner. We call these \\textit{out-of-bag} samples.\n\nRandom Forests \\citep{breiman2001random} are a popular extension of bagging with an additional randomness injected. At each step when searching for the best split, only $p_0$ features are randomly selected from all $p$ possible features and the best split $\\theta^*$ must be chosen from this subset. When $p_0 = p$, this reduces to bagging. Mathematically, the prediction is written as \n$$\n\\hat{t}^{RF}(x) = \\frac{1}{B}\\sum_{b=1}^Bt(x; \\xi_b, z^*_{b1}, z^*_{b2}, \\ldots, z^*_{bn})\n$$\nwith $\\xi_b \\stackrel{\\mbox{iid}}{\\sim} \\Xi$ denoting the additional randomness for selecting from a random subset of available features.\n\nBoosting is another widely used technique by data scientists to achieve state-of-the-art results on many machine learning challenges \\citep{chen2016xgboost}. Instead of building trees in parallel as in bagging, it does this sequentially, allowing the current base learner to correct for any previous bias. In \\citet{ghosal2018boosting}, the authors also consider boosting RF to reduce bias. We will skip over some technical details on boosting and restrict our discussion of feature importance in the context of decision trees and RF. Note that as long as tree-based models combine base learners in an additive fashion, their feature importance measures are naturally calculated by (weighted) average across those of individual trees. \n\n\n\\section{Measurement of Feature Importance}\\label{FI}\n\nAlmost every feature importance measures used in tree-based models belong to two classes: split-improvement or permutation importance. Though our focus will be on split-improvement, permutation importance is introduced first for completeness. \n\n\\subsection{Permutation Importance}\n\nArguably permutation might be the most popular method for assessing feature importance in the machine learning community. Intuitively, if we break the link between a variable $X_j$ and $y$, the prediction error increases then variable $j$ can be considered as important. \n\nFormally, we view the training set as a matrix $X$ of size $n \\times p$, where each row $x_i$ is one observation. Let $X^{\\pi, j}$ be a matrix achieved by permuting the $j^{th}$ column according to some mechanism $\\pi$. If we use $l(y_i, f(x_i))$ as the loss incurred when predicting $f(x_i)$ for $y_i$, then the importance of $j^{th}$ feature is defined as:\n\\begin{equation} \\label{p}\n\\text{VI}^{\\pi}_j = \\sum_{i=1}^n l(y_i, f(x_i^{\\pi, j}) - l(y_i, f(x_i)))\n\\end{equation}\nthe increase in prediction error when the $j^{th}$ feature is permuted. Variations include choosing different permutation mechanism $\\pi$ or evaluating Equation (\\ref{p}) on a separate test set. In Random Forests, \\citet{breiman2001random} suggest to only permute the values of the $j^{th}$ variable in the \\textit{out-of-bag} samples for each tree, and final importance for the forest is given by averaging across all trees. \n\nThere is a small literature analyzing permutation importance in the context of RF. \\citet{ishwaran2007variable} studied paired importance.\n\\citet{hooker2007generalized, strobl2008conditional, hooker2019please} advocated against permuting features by arguing it emphasizes behavior in regions where there is very little data. More recently, \\citet{gregorutti2017correlation} conducted a theoretical analysis of permutation importance measure for an additive regression model. \n\n\\subsection{Split-Improvement}\n\nWhile permutation importance measures can generically be applied to any prediction function, \nsplit-improvement is unique to tree-based methods, and can be calculated directly from the training process. Every time a node is split on variable $j$,  the combined impurity for the two descendent nodes is less than the parent node. Adding up the weighted impurity decreases for each split in a tree and averaging over all trees in the forest yields an importance score for each feature.\n\nFollowing our notation in Section \\ref{cart}, the impurity function $H$ is either mean squared error for regression or Gini index for classification. The best split at node $m$ is given by $\\theta^*_m$ which splits at $j^{th}$ variable and results in two child nodes denoted as $l$ and $r$. Then the decrease in impurity for split $\\theta^*$ is defined as:\n\\begin{equation} \\label{di}\n\\Delta(\\theta^*_m) =  \\omega_mH(m) - (\\omega_lH(l) + \\omega_rH(r)),\n\\end{equation}\nwhere $\\omega$ is the proportion of observations falling into each node, i.e.,  $\\omega_m = \\frac{n_m}{n}$, $\\omega_l = \\frac{n_l}{n}$ and $\\omega_r = \\frac{n_r}{n}$. Then, to get the importance for $j^{th}$ feature in a single tree, we add up all $\\Delta(\\theta^*_m)$ where the split is at the $j^{th}$ variable:\n\\begin{equation} \\label{t}\n\\text{VI}^{\\text{T}}_j = \\sum_{m, j \\in \\theta^*_m}\\Delta(\\theta^*_m).\n\\end{equation}\nHere the sum is taken over all non-terminal nodes of the tree, and we use the notation $j \\in \\theta^*_m$ to denote that the split is based on the $j^{th}$ feature. \n\nThe notion of split-improvement for decision trees can be easily extended to Random Forests by taking the average across all trees. Suppose there are $B$ base learners in the forest, we could naturally define\n\\begin{equation} \\label{si}\n\\text{VI}^{\\text{RF}}_j = \\frac{1}{B}\\sum_{b = 1}^B\\text{VI}^{\\text{T(b)}}_j = \\frac{1}{B}\\sum_{b = 1}^B\\sum_{m, j \\in \\theta^*_m}\\Delta_b(\\theta^*_m).\n\\end{equation}\n\n\n\\subsection{Bias in Split-Improvement}\n\n\\citet{strobl2007bias} pointed out that the split-improvement measure defined above is biased towards increasing the importance of continuous features or categorical features with many categories. This is because of the increased flexibility afforded by a larger number of potential split points. We conducted a similar simulation to further demonstrate this phenomenon. All our experiments are based on Random Forests which gives more stable results than a single tree.\n\nWe generate a simulated dataset so that $X_1 \\sim N(0, 1)$ is continuous, and $X_2, X_3, X_4, X_5$ are categorically distributed with $2, 4, 10, 20$ categories respectively. The probabilities are equal across categories within each feature. In particular, $X_2$ is Bernoulli distribution with $p=0.5$. In classification setting, the response $y$ is also generated as a Bernoulli distribution with $p=0.5$, but independent of all the $X$'s. For regression, $y$ is independently generated as $N(0, 1)$. We repeat the simulation 100 times, each time generating $n=1000$ data points and fitting a Random Forest model\\footnote{Our experiments are implemented using \\textbf{scikit-learn}. Unless otherwise noted, default parameters are used.} using the data set. Here categorical features are encoded into dummy variables, and we sum up importance scores for corresponding dummy variables as final measurement for a specific categorical feature. In Appendix \\ref{sim}, we also provide simulation results when treating those categorical features as (ordered) discrete variables. \n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{Split_Improvement_for_Classification_Dummy.png}\n        \\caption{Classification}\n        \\label{fig:si_cls}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{Split_Improvement_for_Regression_Dummy.png}\n        \\caption{Regression}\n        \\label{fig:si_regr}\n    \\end{subfigure}\n    \\hfill\n    \n    \\caption{Split-improvement measures on five predictors. Box plot is based on 100 repetitions. 100 trees are built in the forest and maximum depth of each tree is set to 5.}\n    \\label{fig:si}\n    \n\\end{figure}\n\nBox plots are shown in Figure \\ref{fig:si_cls} and \\ref{fig:si_regr} for classification and regression respectively. The continuous feature $X_1$ is frequently given the largest importance score in regression setting, and among the four categorical features, those with more categories receive larger importance scores. Similar phenomenon is observed in classification as well, while $X_5$ appears to be artificially more important than $X_1$. Also note that all five features get positive importance scores, though we know that they have no predictive power for the target value $y$.\n\nWe now explore how strong a signal is needed in order for the split-improvement measures to discover important predictors. We generate $X_1, X_2, \\ldots, X_5$ as before, but in regression settings set $y = \\rho X_2 + \\epsilon $ where $\\epsilon \\sim N(0, 1)$. We choose $\\rho$ to range from 0 to 1 at step size 0.1 to encode different levels of signal. For classification experiments, we first make $y = X_2$ and then flip each element of $y$ according to $P(U > \\frac{1 + \\rho}{2})$ where $U$ is Uniform$[0,1]$. This way, the correlation between $X_2$ and $y$ will be approximately $\\rho$. We report the average ranking of all five variables across 100 repetitions for each $\\rho$. The results are shown in Figure \\ref{fig:rank}.\n\nWe see that $\\rho$ needs to be larger than 0.2 to actually find $X_2$ is the most important predictor in our classification setting, while in regression this value increases to 0.6. And we also observe that a clear order exists for the remaining (all unimportant) four features. \n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{Average_Ranking_for_Classification_Dummy.png}\n        \\caption{Classification}\n        \\label{fig:c_rank}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{Average_Ranking_for_Regression_Dummy.png}\n        \\caption{Regression}\n        \\label{fig:r_rank}\n    \\end{subfigure}\n    \\hfill\n    \n    \\caption{Average feature importance ranking across different signal strengths over 100 repetitions. 100 trees are built in the forest and maximum depth of each tree is set to 5.}\n    \\label{fig:rank}\n    \n\\end{figure}\n\n\nThis bias phenomenon could make many statistical analyses based on split-improvement invalid. For example, gender is a very common and powerful binary predictor in many applications, but feature screening based on split-improvement might think it is not important compared to age. In the next section, we explain intuitively why this bias is observed, and provide a simple but effective adjustment. \n\n\\subsection{Related Work}\n\n\\textcolor{black}{Before presenting our algorithm, we review some related work aiming at correcting the bias in split-improvement. Most of the methods fall into two major categories: they either propose new tree building algorithms by redesigning split selection rules, or perform as a post hoc approach to debias importance measurement.}  \n\n\\textcolor{black}{There has been a line of work on designing trees which do not have such bias as observed in classical algorithms such as CART and C4.5. For example, Unbiased and Efficient Statistical Tree \\citep[QUEST;][]{loh1997split} removed the bias by using F-tests on ordered variables and contingency table chi-squared tests on categorical variables. Based on QUEST, CRUISE \\citep{kim2001classification} and GUIDE \\citep{loh2009improving} were developed. We refer readers to \\citet{loh2014fifty} for a detailed discussion in this aspect. In \\citet{strobl2007bias}, the authors resorted to a different algorithm called cforest \\citep{hothorn2010party}, which was based on a conditional inference framework \\citep{hothorn2006unbiased}. They also implemented a stopping criteria based on multiple test procedures.} \n\n\\textcolor{black}{\\citet{sandri2008bias} expressed split-improvement as two components: a heterogeneity reduction and a positive bias. Then the original dataset $( \\mathbf{X}, Y)$ is augmented with pseudo data $\\mathbf{Z}$ which is uninformative but shares the structure of $\\mathbf{X}$ \\citep[this idea of generating pseudo data is later formulated in a general framework termed  ``knockoffs'';][]{barber2015controlling}. The positive bias term is estimated by utilizing the pseudo variables $\\mathbf{Z}$ and subtracted to get a debiased estimate. \\citet{nembrini2018revival} later modified this approach to shorten computation time and provided empirical importance testing procedures. Most recently, \\citet{li2019debiased} derived a tight non-asymptotic bound on the expected bias of noisy features and provided a new debiased importance measure. However, this approach only alleviates the issue and still yields biased results.}\n\n\\textcolor{black}{Our approach works as a post hoc analysis, where the importance scores are calculated after a model is built. Compared to previous methods, it enjoys several advantages: \n\\begin{itemize}\n    \\item It can be easily incorporated into any existing framework for tree-based methods, such as Python or R.\n    \\item It does not require generating additional pseudo data or computational repetitions as in \\citet{sandri2008bias, nembrini2018revival}.\n    \\item Compared to \\citet{li2019debiased} which does not have a theoretical guarantee, our method is proved to be unbiased for noisy features.\n\\end{itemize}}\n\n\\section{Unbiased Split-Improvement}\\label{USI}\n\nWhen it comes to evaluating the performance of machine learning models, we generally use a separate test set to calculate generalization accuracy. The training error is usually smaller than the test error as the algorithm is likely to \"overfit\" on the training data. This is exactly why we observe the bias with split-improvement. Each split will favor continuous features or those features with more categories, as they will have more flexibility to fit the training data. The vanilla version of split-improvement is just like using train error for evaluating model performance. \n\nBelow we propose methods to remedy this bias phenomenon by utilizing a separate test set, and prove that for features with no predictive power, we're able to get an importance score of 0 in expectation for both classification and regressions settings. Our method is entirely based on the original framework of RF, requires barely no additional computational efforts, and can be easily integrated into any existing software libraries.\n\nThe main ingredient of the proposed method is to calculate the impurity function $H$ using additional information provided from test data. In the context of RF, we can simply take \\textit{out-of-bag} samples for each individual tree. \\textcolor{black}{Our experiments below are based on this strategy. In the context of the honest trees proposed in \\citet{wager2018estimation} that divide samples into a partition used to determine tree structures and a partition used to obtain leaf values, the latter could be used as our test data below.  In boosting, it is common not to sample, but to keep a test set separate to determine a stopping time.} Since the choice of impurity function $H$ is different for classification and regression, in what follows we will treat them separately. \n\nFigure \\ref{fig:test} and \\ref{fig:urank} shows the results on previous classification and regression tasks when our unbiased method is applied\\footnote{Relevant codes can be found at \\href{https:\/\/github.com\/ZhengzeZhou\/unbiased-feature-importance}{https:\/\/github.com\/ZhengzeZhou\/unbiased-feature-importance}.}. Feature scores for all variables are spread around $0$, though continuous features and categorical features with more categories tend to exhibit more variability. In the case where there is correlation between $X_2$ and $y$, even for the smallest $\\rho = 0.1$, we can still find the most informative predictor, whereas there are no clear order for the remaining noise features. \n\n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{Unbiased_Split_Improvement_for_Classification_Dummy.png}\n        \\caption{Classification}\n        \\label{fig:test_cls}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{Unbiased_Split_Improvement_for_Regression_Dummy.png}\n        \\caption{Regression}\n        \\label{fig:test_regr}\n    \\end{subfigure}\n    \\hfill\n    \n    \\caption{Unbiased split-improvement. Box plot is based on 100 repetitions. 100 trees are built in the forest and maximum depth of each tree is set to 5. Each tree is trained using bootstrap samples and \\textit{out-of-bag} samples are used as test set.}\n    \\label{fig:test}\n    \n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{Unbiased_Average_Ranking_for_Classification_Dummy.png}\n        \\caption{Classification}\n        \\label{fig:uc_rank}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{Unbiased_Average_Ranking_for_Regression_Dummy.png}\n        \\caption{Regression}\n        \\label{fig:ur_rank}\n    \\end{subfigure}\n    \\hfill\n    \n    \\caption{Unbiased feature importance ranking across different signal strengths averaged over 100 repetitions. 100 trees are built in the forest and maximum depth of each tree is set to 5. Each tree is trained using bootstrap samples and \\textit{out-of-bag} samples are used as test set.}\n    \\label{fig:urank}\n    \n\\end{figure}\n\n\n\\subsection{Classification}\n\nConsider a root node $m$ and two child nodes, denoted by $l$ and $r$ respectively. The best split $\\theta^*_m = (j, s)$ was chosen by Formula (\\ref{best}) and Gini index is used as impurity function $H$. \n\nFor simplicity, we focus on binary classification. Let $p$ denote class proportion within each node. For example, $p_{r, 2}$ denotes the proportion of class 2 in the right child node. Hence the Gini index for each node can be written as:\n$$\nH(m) = 1 - p_{m,1}^2 - p_{m,2}^2,\n$$\n$$\nH(l) = 1 - p_{l,1}^2 - p_{l,2}^2,\n$$\n$$\nH(r) = 1 - p_{r,1}^2 - p_{r,2}^2.\n$$\nThe split-improvement for a split at $j^{th}$ feature when evaluated using only the training data is written as in Equation (\\ref{di}). This value is always positive no matter which feature is chosen and where the split is, which is exactly why a selection bias will lead to overestimate of feature importance.\n\nIf instead, we have a separate test set available, the predictive impurity function for each node is modified to be:\n\\begin{equation}\n  \\label{pi}\n  \\begin{aligned}\n    H'(m) &= 1 - p_{m,1}p'_{m,1} - p_{m,2}p'_{m,2}, \\\\        \n    H'(l) &= 1 - p_{l,1}p'_{l,1} - p_{l,2}p'_{l,2}, \\\\\n    H'(r) &= 1 - p_{r,1}p'_{r,1} - p_{r,2}p'_{r,2},\n  \\end{aligned}\n\\end{equation}\nwhere $p'$ is class proportion evaluating on the test data.\nAnd similarly, \n\\begin{equation}\n  \\label{delta_c}\n  \\begin{aligned}\n\\Delta'(\\theta^*_m) & = \\omega_mH'(m) - (\\omega_lH'(l) + \\omega_rH'(r)) \\\\\n& = \\omega_l(H'(m) - H'(l)) + \\omega_r(H'(m) - H'(r)).\n  \\end{aligned}\n\\end{equation}\n\\textcolor{black}{Using these definitions, we first demonstrate that an individual split is unbiassed in the sense that if $y$ has no bivariate relationship with $X_j$, $\\Delta'(\\theta^*_m)$ will have expectation 0.}\n\\begin{lemma}\\label{l_c}\nIn classification settings, for a given feature $X_j$, if $y$ is marginally independent of $X_j$ within the region defined by node $m$, then \n$$\nE\\Delta'(\\theta^*_m) = 0\n$$\nwhen splitting at the $j^{th}$ feature.\n\\end{lemma}\n\n\\begin{proof}\nSee Appendix \\ref{proof}.\n\\end{proof}\nSimilar to Equation (\\ref{t}), split-improvement of $x_j$ in a decision tree is defined as:\n\\begin{equation} \\label{ut}\n\\text{VI}^{\\text{T,C}}_j = \\sum_{m, j \\in \\theta^*_m}\\Delta'(\\theta^*_m).\n\\end{equation}\n\\textcolor{black}{We can now apply Lemma \\ref{l_c} to provide a global result so long as $X_j$ is always irrelevant to $y$.}\n\\begin{theorem}\nIn classification settings, for a given feature $X_j$, if $y$ is independent of $X_j$ in every hyper-rectangle subset of the feature space, then we always have\n$$\nE\\text{VI}^{\\text{T,C}}_j = 0.\n$$\n\\end{theorem}\n\n\\begin{proof}\nThe result follows directly from Lemma \\ref{l_c} and Equation (\\ref{ut}). \n\\end{proof}\n\nThis unbiasedness result can be easily extended to the case of RF by (\\ref{si}), as it's an average across base learners. \\textcolor{black}{We note here that our independence condition is designed to account for relationships that appear before accounting for splits on other variables, possibly due to relationships between $X_j$ and other features, and afterwards. It is trivially implied by the independence of $X_j$ with both $y$ and the other features. Our condition may also be stronger than necessary, depending on the tree-building process. We may be able to restrict the set of hyper-rectangles to be examined, but only by analyzing specific tree-building algorithms.}\n\n\\subsection{Regression}\n\nIn regression, we use mean squared error as the impurity function $H$:\n$$\n\\bar{y}_m = \\frac{1}{n_m}\\sum_{x_i \\in m}y_i,\n$$\n$$\nH(m) = \\frac{1}{n_m}\\sum_{x_i \\in m}(y_i - \\bar{y}_m)^2.\n$$\nIf instead the impurity function $H$ is evaluated on a separate test set, we define \n$$\nH'(m) = \\frac{1}{n'_m}\\sum_{i=1}^{n'_m}(y'_{m,i} - \\bar{y}_m)^2 \n$$\nand similarly\n$$\n\\Delta'(\\theta^*_m) =  \\omega_mH'(m) - (\\omega_lH'(l) + \\omega_rH'(r)).\n$$\nNote that here $H'(m)$  measures mean squared error within node $m$ on test data with the fitted value $\\bar{y}_m$ from training data. If we just sum up $\\Delta'$ as feature importance, it will end up with negative values as $\\bar{y}_m$ will overfit the training data and thus make mean squared error much larger deep in the tree. In other words, it \\textit{over-corrects} the bias. For this reason, our unbiased split-improvement is defined slightly different from the classification case (\\ref{ut}):\n\\begin{equation} \\label{utr}\n\\text{VI}^{\\text{T,R}}_j = \\sum_{m, j \\in \\theta^*_m} (\\Delta(\\theta^*_m)  + \\Delta'(\\theta^*_m)).\n\\end{equation}\n\nNotice that although Equation (\\ref{ut}) and (\\ref{utr}) are different, they originates from the same idea by correcting bias using test data. Unlike Formula (\\ref{pi}) for Gini index, where we could design a predictive impurity function by combining train and test data together, it's hard to come up with a counterpart in regression setting. \n\nJust as in the classification case, we could show the following unbiasedness results:\n\n\\begin{lemma}\\label{l_r}\nIn regression settings, for a given feature $X_j$, if $y$ is marginally independent of $X_j$ within the region defined by node $m$, then \n$$\nE(\\Delta(\\theta^*_m) + \\Delta'(\\theta^*_m)) = 0\n$$\nwhen splitting at the $j^{th}$ feature.\n\\end{lemma}\n\n\\begin{proof}\nSee Appendix \\ref{proof}.\n\\end{proof}\n\\begin{theorem}\nIn regression settings, for a given feature $X_j$, if $y$ is independent of $X_j$ in every hyper-rectangle subset of the feature space, then we always have\n$$\nE\\text{VI}^{\\text{T,R}}_j = 0.\n$$\n\\end{theorem}\n\n\n\n\\section{Empirical Studies}\\label{real}\n\n\\textcolor{black}{In this section, we apply our method to one simulated example and three real data sets. We compare our results to three other algorithms: the default split-improvement in \\textbf{scikit-learn}, cforest \\citep{hothorn2006unbiased} in R package \\textbf{party} and bias-corrected impurity \\citep{nembrini2018revival} in R package \\textbf{ranger}. We did not include comparison with \\citet{li2019debiased} since their method does not enjoy the unbiased property. In what follows, we use shorthand SI for the default split-improvement, UFI for our method (unbiased feature importance).}\n\n\\subsection{Simulated Data}\n\n\\textcolor{black}{The data has 1000 samples and 10 features, where $X_i$ takes values in $0, 1, 2, \\ldots, i - 1$ with uniform probability for $1 \\leq i \\leq 10$. Here, we assume only $X_1$ contains true signal and all remaining nine features are noisy features. The target value $y$ is generated as follows:\n\\begin{itemize}\n    \\item Regression: $y = X_1 + 5 \\epsilon$, where $\\epsilon \\sim \\mathcal{N}(0, 1)$.\n    \\item Classification: $P(y = 1 | X) = 0.55$ if $X_1 = 1$, and $P(y = 1 | X) = 0.45$ if $X_1 = 0$.\n\\end{itemize}\nNote that this task is designed to be extremely hard by choosing the binary feature as informative, and adding large noise (regression) or setting the signal strength low (classification). To evaluate the results, we look at the ranking of all features based on importance scores. Ideally $X_1$ should be ranked $1^{st}$ as it is the only informative feature. Table \\ref{table:sum} shows the average ranking of feature $X_1$ across 100 repetitions. The best result of each column is marked in \\textbf{bold}. Here we also compare the effect of tree depth by constructing shallow trees (with tree depth 3) and deep trees (with tree depth 10). Since cforest does not provide a parameter for directly controlling tree depth, we change the values of mincriterion as an alternative. }\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n\\multirow{2}{*}{} & \\multicolumn{2}{c|}{Tree depth = 3} & \\multicolumn{2}{c|}{Tree depth = 10} \\\\ \\cline{2-5} \n                  & R                & C                & R                 & C                \\\\ \\hline\nSI                & 3.71             & 4.10             & 10.00             & 10.00            \\\\ \\hline\nUFI               & \\textbf{1.47}    & 1.39             & \\textbf{1.55}     & \\textbf{1.69}    \\\\ \\hline\ncforest           & 1.57             & \\textbf{1.32}    & 1.77              & 1.88             \\\\ \\hline\nranger            & 1.54             & 1.64             & 2.46              & 1.93             \\\\ \\hline\n\\end{tabular}\n\\caption{Average importance ranking of informative feature $X_1$. R stands for regression and C for classification. The result averages over 100 repetitions. Lower values indicate better abilities in identifying informative features. In cforest, we set mincriterion to be 2.33 (0.99 percentile of normal distribution) for shallow trees and 1.28 (0.9 percentile) for deep trees.}\n\\label{table:sum}\n\\end{table}\n\n\\textcolor{black}{We can see that our method UFI achieves the best results in three situations except the classification case for shallow trees, where it is only slightly worse than cforest. Another interesting observation is that deeper trees tend to make the task of identifying informative features harder when there are noisy ones, since it is more likely to split on noisy features for splits deep down in the tree. This effect is most obvious for the default split-improvement, where it performs the worst especially for deep trees: the informative feature $X_1$ is consistently ranked as the least important ($10^{th}$ place). UFI does not seem to be affected too much from tree depth.}\n\n\\subsection{RNA Sequence Data}\nThe first data set examined is the prediction of C-to-U edited sites in plant mitochondrial RNA. This task was studied statistically in \\citet{cummings2004simple}, where the authors applied Random Forests and used the original split-improvement as feature importance. Later, \\citet{strobl2007bias} demonstrated the performance of cforest on this data set. \n\nRNA editing is a molecular process whereby an RNA sequence is\nmodified from the sequence corresponding to the DNA\ntemplate.  In the mitochondria of land plants, some cytidines\nare converted to uridines before translation \\citep{cummings2004simple}. \n\nWe use the \\textit{Arabidopsis thaliana} data file\\footnote{The data set can be downloaded from \\href{https:\/\/bmcbioinformatics.biomedcentral.com\/articles\/10.1186\/1471-2105-5-132}{https:\/\/bmcbioinformatics.biomedcentral.com\/articles\/10.1186\/1471-2105-5-132}.} as in \\citet{strobl2007bias}. The features are based on the nucleotides surrounding the edited\/non-edited sites and on the estimated folding energies of those regions. After removing missing values and one column which will not be used, the data file consists of 876 rows and 45 columns:\n\\begin{itemize}\n    \\item the response (binary).\n    \\item 41 nucleotides at positions -20 to 20 relative to the\nedited site (categorical, one of A, T, C or G).\n    \\item the codon position (also 4 categories).\n    \\item two continuous variables based on on the estimated folding energies.\n\\end{itemize} \n\nFor implementation, we create dummy variables for all categorical features, and build forest using 100 base trees. The maximum tree depth for this data set is not restricted as the number of potential predictors is large. We take the sum of importance across all dummy variables corresponding to a specific feature for final importance scores. All default parameters are used unless otherwise specified.\n\nThe results are shown in Figure \\ref{fig:ctou}. Red error bars depict one standard deviation when the experiments are repeated 100 times. From the default split-improvement (Figure \\ref{fig:si_ctou}), we can see that except several apparently dominant predictors (nucleotides at position -1 and 1, and two continuous features \\textit{fe} and \\textit{dfe}), the importance for the remaining nearly 40 features are indistinguishable. The feature importance scores given by UFI (Figure \\ref{fig:ufi_ctou}) and cforest (Figure \\ref{fig:cf_ctou}) are very similar. Compared with SI, although all methods agree on top three features being the nucleotides at position -1 and 1, and the continuous one \\textit{fe}, there are some noticeable differences. Another continuous feature \\textit{dfe} is originally ranked at the fourth place in Figure \\ref{fig:si_ctou}, but its importance scores are much lower by UFI and cforest. The result given by ranger (Figure \\ref{fig:ranger_ctou}) is slightly different from UFI and cforest, where it seems to have more features with importance scores larger than 0. In general, we see a large portion of predictors with feature importance close to 0 for three improved methods, which makes subsequent tasks like feature screening easier. \n\n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{0.495\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{SI_on_CtoU.png}\n        \\caption{SI}\n        \\label{fig:si_ctou}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.495\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{UFI_on_CtoU.png}\n        \\caption{UFI}\n        \\label{fig:ufi_ctou}\n    \\end{subfigure}\n    \\hfill\n\n    \\begin{subfigure}[b]{0.495\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{cforest_on_CtoU.png}\n        \\caption{cforest}\n        \\label{fig:cf_ctou}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.495\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{ranger_on_CtoU.png}\n        \\caption{ranger}\n        \\label{fig:ranger_ctou}\n    \\end{subfigure}\n    \n    \\caption{Feature importance for RNA sequence data. 100 trees are built in the forest. Red error bars depict one standard deviation when the experiments are repeated 100 times.}\n    \\label{fig:ctou}\n    \n\\end{figure}\n\n\n\\subsection{Adult Data}\n\nAs a second example, we will use the Adult Data Set from UCI Machine Learning Repository\\footnote{\\href{https:\/\/archive.ics.uci.edu\/ml\/datasets\/adult}{https:\/\/archive.ics.uci.edu\/ml\/datasets\/adult}}. The task is to predict whether income exceeds \\$50K\/yr based on census data. We remove all entries including missing values, and only focus on people from Unites States. In total, there are 27504 training samples and Table \\ref{table:adult} describes relevant feature information. Notice that we add a standard normal random variable, which is shown in the last row. We randomly sample 5000 entries for training. \n\n\\begin{table}[]\n\\centering\n\\begin{tabular}{|c|c|}\n\\hline\nAttribute      & Description      \\\\ \\hline\nage            & continuous       \\\\ \\hline\nworkclass      & categorical (7)  \\\\ \\hline\nfnlwgt         & continuous       \\\\ \\hline\neducation      & categorical (16) \\\\ \\hline\neducation-num  & continuous       \\\\ \\hline\nmarital-status & categorical (7)  \\\\ \\hline\noccupation     & categorical (14) \\\\ \\hline\nrelationship   & categorical (6)  \\\\ \\hline\nrace           & categorical (5)  \\\\ \\hline\nsex            & binary           \\\\ \\hline\ncapital-gain   & continuous       \\\\ \\hline\ncapital-loss   & continuous       \\\\ \\hline\nhours-per-week & continuous       \\\\ \\hline\nrandom         & continuous       \\\\ \\hline\n\\end{tabular}\n\\caption{Attribute description for adult data set.}\n\\label{table:adult}\n\\end{table}\n\n\nThe results are shown in Figure \\ref{fig:adult}. UFI (\\ref{fig:ufi_adult}), cforest(\\ref{fig:cf_adult}) and ranger (\\ref{fig:ranger_adult}) display similar feature rankings which are quite different from the original split-improvement (\\ref{fig:si_adult}). Notice the random normal feature we added (marked in black) is actually ranked the third most important in \\ref{fig:si_adult}. This is not surprising as most of the features are categorical, and even for some continuous features, a large portion of the values are actually 0 (such as \\textit{capital-gain} and \\textit{capital-loss}). For UFI, cforest and ranger, the random feature is assigned an importance score close to 0. Another feature with big discrepancy is \\textit{fnlwgt}, which is ranked among top three originally but is the least important for other methods. \\textit{fnlwgt} represents final weight, the number of units in the target population that the responding unit represents. Thus it is unlikely to have strong predictive power for the response. For this reason, some analyses deleted this predictor before fitting models\\footnote{\\href{http:\/\/scg.sdsu.edu\/dataset-adult\\_r\/}{http:\/\/scg.sdsu.edu\/dataset-adult\\_r\/}}.\n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{SI_on_adult_data.png}\n        \\caption{SI}\n        \\label{fig:si_adult}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{UFI_on_adult_data.png}\n        \\caption{UFI}\n        \\label{fig:ufi_adult}\n    \\end{subfigure}\n    \\hfill\n    \n    \\begin{subfigure}[b]{0.49\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{cforest_on_adult_data.png}\n    \\caption{cforest}\n    \\label{fig:cf_adult}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{ranger_on_adult_data.png}\n        \\caption{ranger}\n        \\label{fig:ranger_adult}\n    \\end{subfigure}\n    \n    \\caption{Feature importance for adult data. 20 trees are built in the forest. Red error bars depict one standard deviation when the experiments are repeated 100 times.}\n    \\label{fig:adult}\n\\end{figure}\n\n\n\\subsection{Boston Housing Data}\n\n\\textcolor{black}{We also conduct analyses on a regression example using the Boston Housing Data\\footnote{\\href{https:\/\/archive.ics.uci.edu\/ml\/machine-learning-databases\/housing\/}{https:\/\/archive.ics.uci.edu\/ml\/machine-learning-databases\/housing\/}}, which has been widely studied in previous literature \\citep{bollinger1981book, quinlan1993combining}. The data set contains 12 continuous, one ordinal and one binary features and the target is median value of owner-occupied homes in \\$1000's. We add a random feature distributed as $\\mathcal{N}(0, 1)$ as well.}\n\n\\begin{figure}[h]\n    \\centering\n    \\begin{subfigure}[b]{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{SI_on_boston_data.png}\n        \\caption{SI}\n        \\label{fig:si_boston}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{UFI_on_boston_data.png}\n        \\caption{UFI}\n        \\label{fig:ufi_boston}\n    \\end{subfigure}\n    \\hfill\n    \n    \\begin{subfigure}[b]{0.49\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{cf_on_boston_data.png}\n    \\caption{cforest}\n    \\label{fig:cf_boston}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{ranger_on_boston_data.png}\n        \\caption{ranger}\n        \\label{fig:ranger_boston}\n    \\end{subfigure}\n    \n    \\caption{Feature importance for Boston housing data. 100 trees are built in the forest. Red error bars depict one standard deviation when the experiments are repeated 100 times.}\n    \\label{fig:boston}\n\\end{figure}\n\n\\textcolor{black}{All four methods agree on two most important features: RM (average number of rooms per dwelling) and LSTAT (\\% lower status of the population). In SI, the random feature still appears to be more important than several other features such as INDUS (proportion of non-retail business acres per town) and RAD (index of accessibility to radial highways), though the spurious effect is much less compared to Figure \\ref{fig:si_adult}. As expected, the importance of random feature is close to zero in UFI. In this example, the SI did not seem to provide misleading result as most of the features are continuous, and the only binary feature CHAS (Charles River dummy variable) turns out to be not important. }\n\n\\subsection{Summary}\n\n\\textcolor{black}{Our empirical studies confirm that the default split-improvement method is biased towards increasing the importance of features with more potential splits. The bias is more severe in deeper trees. Compared to three other approaches, our proposed method performs the best in a difficult task to identify the only important feature from 10 noisy features. For real world data sets, though we do not have a ground truth for feature importance scores, our method gives similar and meaningful outputs as two state-of-the-art methods cforest and ranger.}\n\n\\section{Discussions}\\label{dis}\n\nTree-based methods are widely employed in many applications. One of the many advantages is that these models come naturally with feature importance measures, which practitioners rely on heavily for subsequent analysis such as feature ranking or screening. It is important that these measurements are trustworthy.\n\nWe show empirically that split-improvement, as a popular measurement of feature importance in tree-based models, is biased towards continuous features, or categorical features with more categories. This phenomenon is akin to overfitting in training any machine learning model. We propose a simple fix to this problem and demonstrate its effectiveness both theoretically and empirically. Though our examples are based on Random Forests, the adjustment can be easily extended to any other tree-based model. \n\nThe original version of split-improvement is the default and only feature importance measure for Random Forests in \\textbf{scikit-learn}, and is also returned as one of the measurements for \\textbf{randomForest} library in R. Statistical analyses utilizing these packages will suffer from the bias discussed in this paper. Our method can be easily integrated into existing libraries, and require almost no additional computational burden.  \\textcolor{black}{As already observed, while we have used \\textit{out-of-bag} samples as a natural source of test data,  alternatives such as sample partitions -- thought of as a subsample of \\textit{out-of-bag} data for our purposes -- can be used in the context of honest trees, or a held-out test set will also suffice.   The use of subsamples fits within the methods used to demonstrate the asymptotic normality of Random Forests developed in \\citet{mentch2016quantifying}. This potentially allows for formal statistical tests to be developed based on the unbiased split-improvement measures proposed here.}  Similar approaches have been taken in \\citet{zhou2018approximation} for designing stopping rules in approximation trees. \n\n\n\nHowever, feature importance itself is very difficult to define exactly, with the possible exception of linear models, where the magnitude of coefficients serves as a simple measure of importance. There are also considerable discussion on the subtly introduced when correlated predictors exist, see for example \\cite{strobl2008conditional, gregorutti2017correlation}. We think \\textcolor{black}{that clarifying the relationship between split-improvement and the topology of the resulting function represents an important} future research direction. \n\n\\subsection*{Acknowledgements}\nThis work was supported in part by NSF grants DMS-1712554, DEB-1353039 and TRIPODS 1740882. \n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn environmental epidemiology,  we are often interested in making inference about the relationships between spatially-referenced exposures and spatially-referenced health outcomes. Spatial structure in exposures can arise from underlying environmental processes and human activities, while health outcomes can have spatial structure derived from both the exposure of interest and other spatially-varying factors. For example, coronary heart disease is associated with socioeconomic status \\citep{DiezRoux2001}, which can have geographic structure and is associated with air pollution \\citep{Brochu2011, Jerrett2004, Mensah2005}.  When not measured (or measured incompletely), these factors can cause \\emph{unmeasured spatial confounding}, which is the inability to distinguish the effect of a spatially-varying exposure from residual spatial variation in a health outcome,  resulting in biased point estimates and standard errors \\citep{Paciorek2010}.\n\nEarly discussion of spatial confounding in the literature includes \\citet{Clayton1993}, who described the `confounding effect due to  location' in regression models for ecological studies, which they attributed  to unmeasured confounding factors. \\citet{Clayton1993} advocated for the inclusion of a spatially-correlated error term in hierarchical models for spatial data \\citep{Clayton1987, Besag1991} and claimed this would account for  confounding bias but might result in conservative inference.\nSince then, the  approach  of adding spatially structured error terms has frequently been used in spatial  models for areal data \\citep{Reich2006, Wakefield2007, Hodges2010, Hughes2013, Hanks2015, Page2017}.  In a causal inference framework, confounding due to geography has been recently addressed using spatial propensity scores \\citep{Papadogeorgou2019, Davis2019}, however in those settings the exposure of interest was not explicitly spatial.\n\nFor point-referenced data, \\citet{Paciorek2010} provided a rigorous discussion of spatial confounding and \n the importance of the spatial scales of variability in the exposure and outcome.\n\\cite{Paciorek2010} demonstrated that reductions in bias could be obtained when the scale of unconfounded variability in the exposure, which he quantified by the spatial range parameter in a Matern covariance function,  is smaller than the scale of confounded variability. \n\nOne approach in the literature for adjusting for spatial confounding at broad scales with point-referenced data is to estimate the coefficient of interest from a semi-parametric model that includes spatial splines \\citep{Paciorek2010, Chan2015}.  This approach is a natural extension of time series studies, where flexible, one-dimensional basis functions can account for confounding due to unmeasured temporal variation \\citep{Burnett1991,Schwartz1994, Dominici2003a, Dominici2004, SzpiroSheppard2014}.  Thin-plate regression splines (TPRS) \\citep{Wood2003} are a commonly used basis and the amount of adjustment can be tuned using the degrees of freedom ($df$) parameter \\citep{Paciorek2010}. However, particular values of $df$ do not have clear spatial scale or interpretation. Generalizing inference about associations between exposures and health outcomes is difficult when the extent of spatial confounding adjustment is not easily quantified.\n\n\\subsection{Air Pollution in the NIEHS Sister Study}\n\\label{sec:spatconf_motivatingexmp}\nThis work is motivated by an analysis of systolic blood pressure and fine particulate matter (PM$_{2.5}$) in the NIEHS Sister Study.  \\citet{Chan2015} found that a 10 $\\upmu$g\/m$^3$ difference in long-term exposure to ambient fine particulate matter (PM$_{2.5}$) was associated with 1.4 mmHg (95\\% Confidence Interval [CI]: 0.6, 2.3) higher systolic blood pressure (SBP) at baseline. \nTo account for spatial confounding from unmeasured regional differences in socioeconomic and health patterns,  \\citet{Chan2015} included TPRS with 10 $df$ in their model. \n\n\nHere we consider a re-analysis of this cohort using PM$_{2.5}$ exposures at  grid locations, rather than at subject residences, to accommodate the methods we describe below. \nWe use predictions of the 2006 annual average ambient concentration from the universal kriging model of \\citet{Sampson2013}, made on a  25km by 25km grid across the contiguous United States.\nFor each of the 47,206  Sister Study subjects, we assign exposure based upon the closest grid cell center.\n  Using the same measured confounders as \\citet{Chan2015},  but no spatial adjustment, we find that a difference of 10 $\\upmu$g\/m$^3$ in PM$_{2.5}$ is associated with 0.26 mmHg higher SBP (95\\% CI: -0.14, 0.66). However, when TPRS with 10 $df$ are added to the model then the estimated difference in SBP is 0.77 mmHg (95\\% CI: 0.14, 1.41). \n  Figure~\\ref{fig:sister_tprsdf} shows the estimates for other amounts of adjustment. \nThe change in the estimates as a function of $df$\n suggests that some form of spatial confounding is present, but it is not clear from Figure~\\ref{fig:sister_tprsdf} on what scales the confounding is occurring,  how much adjustment should be done, and which estimate should be reported.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{Sister_Est_TPRS_Semipar.pdf}\n\\caption{Estimated difference in SBP (in mmHg) associated with a difference of 10 $\\upmu$g\/m$^3$ in annual average ambient PM$_{2.5}$,   when adjusting for TPRS with varying values of $df$ in the outcome model. The square marker (\\mysquare{black}) at $df=0$ is the estimate without spatial adjustment and has error bars indicating a 95\\% confidence interval.  The thick black curve (\\drawline{ultra thick}) represents estimates for different choices of $df$ and the thin black curves (\\drawline{}) represent point-wise 95\\% confidence intervals.}\n\\label{fig:sister_tprsdf}\n\\end{center}\n\\end{figure}\n\n\\subsection{Manuscript Outline}\nIn the remainder of this paper, we address the question of how to interpret and select the spatial scale  of adjustment.\nIn Section~\\ref{sec:adjustment} we  introduce the statistical framework and  procedures for spatial confounding adjustment.\nIn Section~\\ref{sec:spatial_scales} we describe three choices of spatial basis (TPRS,  a Fourier basis, and a wavelet basis) and present a method for assigning variation in these bases to a spatial distance. In Section~\\ref{sec:choosing_m} we describe approaches for selecting the amount of adjustment and compare the approaches in simulations in Section~\\ref{sec:simulations}. \nIn Section~\\ref{sec:sisterfullapp} we  apply the adjustment approaches to  the motivating Sister Study cohort. Section~\\ref{sec:discussion} provides a concluding discussion.\n\n\\section{Methods for Spatial Confounding Adjustment}\n\\label{sec:adjustment}\n\n\\subsection{Statistical Framework}\n\\label{sec:framework}\nWe consider a cohort study of $n$ subjects with measured health outcomes $y_i$, exposures $x_i$, and observed confounder covariates ${\\bm z}_i$. \nWe partition the spatial domain into a rectangular grid  $\\mathcal{S}$ with distinct locations $\\{1, \\dots, S\\}$. \nEach subject is assigned to a location ${\\bm s}_i \\in \\{1, \\dots, S\\}$.  We assume \nthat the subjects are distributed evenly across the locations; we discuss  non-uniform sampling of subject locations in Section~\\ref{sec:pop_spatial_dist}.\n\nExtending the framework of \\cite{SzpiroSheppard2014} to a spatial setting, \nwe assume that the exposure for each individual is measured without error and  takes the form\n\\begin{equation}\nx_i = x({\\bm s}_i) = g({\\bm s}_i) + \\xi({\\bm s}_i),\n\\end{equation}\nwhere $g({\\bm s}_i)$ is a smooth spatial surface and $\\xi_i$ is (fixed) residual spatial variation.  We decompose the observed  confounders  in a similar manner as ${\\bm z}_i = \\bm w({\\bm s}_i) + \\bm\\zeta_i$, except that $\\bm\\zeta_i$ is not a function of space and is modelled as $N(0, \\sigma_\\zeta^2{\\bm I})$. The stochasticity of $\\bm\\zeta$ arises from the sampling of subjects.\n\nWe assume that the observed health outcomes $y_i$ arise from the model\n\\begin{equation}\n\\label{eq:dgm}\ny_i = \\beta_0 + \\beta x_i +  {\\bm z}_i^\\ensuremath{\\mathsf{T}} \\bm\\gamma + f({\\bm s}_i) + \\epsilon_i,\n\\end{equation}\nwhere $f({\\bm s}_i)$ is an unknown, fixed spatial surface and $\\epsilon_i \\stackrel{iid}{\\sim} N(0, \\sigma_\\epsilon^2)$.  The target of inference is the parameter $\\beta \\in \\mathbb{R}$, which represents the association between exposure $x_i$ and the outcome $y_i$ \\emph{conditional} on $\\bm z$ and $f$. The $f({\\bm s}_i)$ term represents unmeasured spatial variability in $y_i$, which we assume to be fixed and unknown. \n\n\nIn contrast to the approach taken by ``restricted spatial regression'', which targets the unconditional effect of exposure on outcome \\citep{Hodges2010,Hughes2013}, our parameter of interest here is the exposure-outcome association conditional on the unmeasured confounder $f$. In observational air pollution epidemiology studies, the conditional association is targeted in order to provide information that is generalizable beyond the specific time and location of the study cohort \\citep[e.g.][]{Peng2006}.\n\nTo model the spatially-varying terms in \\eqref{eq:dgm}, we employ a set of hierarchical spatial basis functions $\\{h_1(\\cdot), h_2(\\cdot), \\dots. \\}$ that are in order of non-increasing resolution.\n For compactness of notation, let ${H}_m  = \\begin{bmatrix} \\bm h_1 & \\dots & \\bm h_m\\end{bmatrix}$ denote the $n \\times m$ matrix of the first $m$ basis functions  evaluated at the $n$ subject locations.\nFollowing \\citet{Dominici2004} and \\citet{SzpiroSheppard2014}, we decompose $g(\\cdot)$, $\\bm w(\\cdot)$, and $f(\\cdot)$ as:\n  \\begin{align}\n\\notag   g({\\bm s}_i)  &= \\sum_{k=1}^{m_1} \\alpha_kh_k({\\bm s}_i)  = \\bm e_i^\\ensuremath{\\mathsf{T}}{H}_{m_1}\\bm\\alpha\\\\\n  \\label{eq:gwf_basis} w_j({\\bm s}_i) & = \\sum_{k=1}^{m_2} \\delta_{jk}h_k({\\bm s}_i)  = \\bm e_i^\\ensuremath{\\mathsf{T}}{H}_{m_2}\\bm\\delta_j\\\\\n\\notag   f({\\bm s}_i) &=  \\sum_{k=1}^{m_3} \\theta_kh_k({\\bm s}_i) = \\bm e_i^\\ensuremath{\\mathsf{T}}{H}_{m_3}\\bm\\theta,\n  \\end{align}\n  where $\\bm e_i$ is the unit vector with 1 as the $i$th element.\nWe assume\n$m_1 > m_3$. This assumption means that there is finer scale variability in the exposure surface than in the unobserved confounding surfaces, which \\citet{Paciorek2010} identified as a necessary condition to achieve reductions in bias via adjustment.\nIn Section~\\ref{sec:spatial_scales}, we consider three different choices of $h_j$: a regression spline basis, a Fourier basis, and a wavelet basis.\n\nThe ordinary least squares estimator from the model ${\\rm E}[y_i] = {\\beta}_0 + {\\beta}_1x_i + {\\gamma} {\\bm z}_i$\n will be biased due to the correlation of the observed $g({\\bm s}_i)$ and  $\\bm w({\\bm s}_i)$ with the unobserved $f({\\bm s}_i)$. In the literature, this correlation is sometimes referred to  as\n \\textit{concurvity} \\citep{Hastie1990, Ramsay2003,Ramsay2003a}.\n We describe two strategies for adjusting for this correlation in order to eliminate the confounding bias.\n  \n  \\subsection{Adjustment in Outcome Model}\n\\label{sec:adjust_health}\nA straightforward way to adjust for the unmeasured spatial structure $f({\\bm s}_i)$ is to fit the semiparametric regression model\n\\begin{equation}\n\\label{eq:adj_in_health_model}\n{\\rm E}[y_i] = \\breve{\\beta}_0 + \\breve{\\beta}_1x_i +  {\\bm z}_i^\\ensuremath{\\mathsf{T}}\\breve{\\gamma} + {H}_m\\breve{\\bm \\theta},\n\\end{equation}\nfor a chosen value of $m$.\nThis is a direct extension of confounding-adjustment approaches used in time series methods \\citep{Dominici2004,Peng2006} and was the approach taken by \\cite{Chan2015} and in Figure~\\ref{fig:sister_tprsdf}. For basis functions such as TPRS that are easily computed, fitting model \\eqref{eq:adj_in_health_model} in standard statistical software is straightforward. \n\nIf $m \\ge m_3$, then the ordinary least squares estimator $\\widehat{\\breve{\\beta}}_1$ from model \\eqref{eq:adj_in_health_model} will be unbiased, because the inclusion of ${H}_m$ fully adjusts for variation in its column space, which includes $f$. \nIf $m < m_3$, then $\\widehat{\\breve{\\beta}}_1$ will remain biased because of correlation between $x({\\bm s})$ and the $m_3 - m$ terms of $f({\\bm s})$ (from the decomposition in \\eqref{eq:gwf_basis}) that are not within the column space of ${H}_m$.\n \n  \n\\subsection{Pre-adjusting Exposure}\n\\label{sec:preadjust}\n\nA second approach to confounding adjustment is to first `pre-adjust' the  exposure, to remove variation in the exposure that is correlated with the confounding surface.  In comparison to the semi-parametric adjustment approach described above, the pre-adjustment approach does not necessarily require the explicit construction of ${H}_m$, which allows for additional choices of spatial basis.\n\\cite{SzpiroSheppard2014} outlined how preadjustment can be done for cohort studies with time series exposures, and here we present an extension of that approach to spatial settings. This approach can be considered a form of spatial filtering \\citep[e.g.][]{Haining1991, Tiefelsdorf2007}\n\nWe decompose the vector of observed exposures ${\\bm x} = \\begin{bmatrix} x({\\bm s}_1) & \\cdots &x({\\bm s}_n)\\end{bmatrix}^\\ensuremath{\\mathsf{T}} $ into two components, ${\\bm x}_1$ and ${\\bm x}_2$, such that ${\\bm x}_2$ is orthogonal to ${H}_m$ for some $m$ (we discuss methods for selecting $m$ in Section~\\ref{sec:choosing_m}). \nIn settings when ${H}_m$ can be explicitly computed, this decomposition can be accomplished by taking ${\\bm x}_1$ to be the projection of ${\\bm x}$ onto ${H}_m$ ($\\bm x_1 = {H}_m({H}_m^\\ensuremath{\\mathsf{T}}{H}_m)^{-1}{H}_m^\\ensuremath{\\mathsf{T}}{\\bm x}$) and ${\\bm x}_2$ to be its complement ($\\bm x_2 = {\\bm x} - {\\bm x}_1)$.\nWhen it is not practical to explicitly construct ${H}_m$, as in the Fourier and wavelet approaches we discuss in Section~\\ref{sec:spatial_scales}, we use a filtering procedure in the frequency domain to achieve this decomposition. \n\n\n\nOnce ${\\bm x}$ has been partitioned into ${\\bm x}_1$ and ${\\bm x}_2$,  we fit the model\n\\begin{equation}\n\\label{eq:preadj_model}\ny_i = \\tilde{\\beta}_0 + \\tilde{\\beta}_1x_{1i} + \\tilde{\\beta}_2x_{2i}+ \\tilde{\\gamma} {\\bm z}_i + \\tilde{\\epsilon}_i\n\\end{equation}\nwhere $\\tilde\\epsilon_i = \\epsilon_i + f(s_i)$ and $\\tilde\\beta_1, \\tilde\\beta_2 \\in \\mathbb{R}$.  Because $\\sum x_1x_2 = 0$, the bias of $\\hat{\\tilde{\\beta}}_2$ is proportional to\n\\begin{equation}\n\\label{eq:beta2bias}\n\\sum x_2f\\left(\\sum x_1z^2 - \\sum x_1^2\\sum z^2\\right) + \\sum x_2z\\left(\\sum x_1^2\\sum zf - \\sum x_1z\\sum x_1f\\right),\n\\end{equation}\nwhere  the summations are over $i$. If $m \\ge m_3$, then ${\\bm x}_2$ is by construction orthogonal to $\\bm f$ and so $\\sum x_2 f = 0$. If $m \\ge m_2$, then ${\\bm x}_2$ is orthogonal to $\\bm z$ and so $\\sum x_2z=0$. Under these two conditions, the bias of $\\hat{\\tilde{\\beta}}_2$ given in \\eqref{eq:beta2bias} is zero.\nIf either $m < m_2$ or $m < m_3$, then $\\hat{\\tilde{\\beta}}_2$ will be biased.  Forcing $f(s_i)$ into the error term creates correlation between the $\\tilde\\epsilon_i$, but this can be accounted for using `sandwich' standard error estimates \\citep{White1980}. \n\n\n\\subsection{Non-Uniform Spatial Distributions of People}\n\\label{sec:pop_spatial_dist}\n\nSo far we have assumed  that all locations $s \\in \\mathcal{S}$ have an observed subject and that the number of subjects was that same at each location. This assumption was necessary to achieve the orthogonality of the pre-adjusted exposure (${\\bm x}_2$) and the basis functions matrix $H_m$ (and thus ${\\bm x}_1$, ${\\bm z}$, and $\\bm f$). \nAdjustment in the outcome model or exposure pre-adjustment in which $H_m$ can be explicitly constructed do not require that the locations be spatially uniform. However, pre-adjustment using the filtering approaches does require that all locations be observed the same number of times.\n\nTo implement pre-adjustment in the presence of duplicate locations, first the duplicated locations are removed (and unobserved locations added) to create the vector $\\bm x'$. The filtering pre-adjustment is done on $\\bm x'$ to generate $\\bm x_1'$ and $\\bm x_2'$. By construction $\\bm x_2'^\\ensuremath{\\mathsf{T}} \\bm h_j' = 0 $, where $\\bm h_j'$ is the $j$th basis function evaluated at all locations in $\\mathcal{S}$ once.\nThe duplicated locations are then added back in by repeating the relevant entries in ${\\bm x}_2'$ to form $\\bm x_2$ (and similarly for $\\bm x_1$). However, the duplicated locations means that in general $\\bm x_2^\\ensuremath{\\mathsf{T}} \\bm h_j \\ne 0$. To recover the orthogonality necessary to eliminate the bias term \\eqref{eq:beta2bias}, the outcome model can use re-weighting.\n\nSuppose that subject locations are distributed according to a spatial distribution $\\Phi({\\bm s})$, which has positive support across all of $\\mathcal{S}$. \nTo recover orthogonality of  ${\\bm x}_2$ and $\\bm h_j$ ($j=1, \\dots, m$), we reweight the study population by its spatial distribution. Specifically, we fit a weighted least squares (WLS) version of \\eqref{eq:preadj_model} where the observation weights are proportional to the inverse of $\\frac{d}{d {\\bm s}} \\Phi({\\bm s}) = \\phi({\\bm s})$.  \nThis assures orthogonality of ${\\bm x}_2$ and $\\bm h_j$ as $n\\to \\infty$, since\n\\begin{equation*}\n \\frac{S}{n} \\sum_{i=1}^n \\frac{1}{\\phi({\\bm s}_i)}x_2({\\bm s}_i)h_j({\\bm s}_i) \\rightarrow \\sum_{{\\bm s} \\in \\mathcal{S}} \\phi({\\bm s}) \\left(\\frac{1}{\\phi({\\bm s})} x_2'({\\bm s})h_j'({\\bm s})\\right)  = \\sum_{{\\bm s} \\in \\mathcal{S}} x_2'({\\bm s}) h_j'({\\bm s}) = \\bm x_2'^\\ensuremath{\\mathsf{T}}\\bm h_j' = 0.\n\\label{eq:reweighting}\n \\end{equation*}\n\n\nThis WLS approach essentially downweights the contribution of people living in highly-populated areas and gives added weight to those who live in more sparsely populated areas.  The parameter being estimated remains the same. For applications in air pollution epidemiology, the correlation between high exposures and population density means that this has the effect of weighting subjects with different exposure levels differently. This is similar in spirit to propensity score weighting, in which groups are re-weighted by their probability of exposure in order to remove bias due to confounding \\citep{Stuart2010}.\n\n\n\n\n\\section{Interpreting Spatial Scales of Adjustment}  \n\\label{sec:spatial_scales}\n\n\nHere we present specific details for three different choices of spatial basis functions: thin-plate regression splines, Fourier basis, and wavelet basis.  We describe how to implement adjustment for each basis and how to interpret the amount of adjustment in terms of physical distance. \nWe will describe methods for selecting the amount of adjustment in Section~\\ref{sec:choosing_m}.\n\n\\subsection{Effective Bandwidth $\\hat k$}\nFor each basis, we describe how to relate the amount of confounding adjustment (i.e.  $m$) to a physical spatial scale. We call this scale, denoted by $\\hat k$, the \\emph{effective bandwidth} of the adjustment. Heuristically, the effective bandwidth is the width of the smoothing kernel induced by the choice of basis function.   This allows for adjustment using $m$ basis functions to be interpreted as adjusting for confounding by smoothing at $\\hat k$ distance. In the context of particular application,  $\\hat k$ may represent a national, regional, or metropolitan scales of variation.  \n\n\\subsection{Thin-plate regression splines}\n\\label{sec:tprs}\nThin-plate regression splines (TPRS) are low-rank approximations to  thin-plate smoothing splines  \\citep{Wood2003}.  The full-rank thin-plate splines are derived from a set of radial basis functions, and TPRS achieve computational benefits by using a truncated eigenbasis  to approximate the radial basis. TPRS are based upon the spatial locations of observed points, eliminating the need for knot-selection.\n\nFor problems of inference, unpenalized TPRS are indexed by the degrees of freedom ($df$), which controls the dimension of the basis and  can be fixed to yield a set of hierarchical basis functions $h_1(s), h_2(s),  \\dots, h_{df}(s)$ where higher values of $df$ correspond to adjustment at a finer scale.   The \\texttt{mgcv} package in R \\citep{Wood2003} makes explicit construction of the TPRS basis straightforward, so both the semi-parametric adjustment and pre-adjustment approaches can be used with this basis.\n\nWe propose the following procedure for computing the effective bandwidth for a chosen number of TPRS basis functions. The smoothing induced by TPRS can be represented in the smoothing matrix  $ S = {H}_{df}({H}_{df}^\\ensuremath{\\mathsf{T}}{H}_{df})^{-1}{H}_{df}^\\ensuremath{\\mathsf{T}}$.  Unlike the frequency-based methods discussed below, the amount of smoothing, which is given by the columns of $ S$, varies at each location due to the location of neighboring points, which is also impacted by the geometry of the domain. We compute $\\hat k$ by finding the  distance at which the ``average'' value of the smoothing matrix values first cross zero. For each observation $i$, we first compute the vector of distances $\\bm d_i$ to all other points. We then fit a loess smooth to ${\\bm s}_i$, the $i$th column of $S$, as a function of $\\bm d_i$.  The procedure is repeated across all locations, and the pointwise median of the loess smooths computed. The distance at which this median smooth first crosses zero is $\\hat k$. A detailed description of this procedure is provided in Supplementary Material Section A. The publicly-available R package  \\texttt{spconf} implements this procedure for any provided spline or smoothing matrix values.\n\n\n\\subsection{Fourier decomposition and Frequency-based filtering}\n\\label{sec:fourier}\nA second choice for  basis $\\{h_j\\}$ is the set of sinusoidal Fourier basis functions.  Without loss of generality, let $u=0, \\dots, M-1$ and $v=0, \\dots, N-1$  denote coordinates of $\\mathcal{S}$. \nThe value of the function $g(u, v)$ can be written as a linear combination of sine and cosine functions, whose coefficients are determined by the Discrete Fourier Transform (DFT) of $g(u, v)$ \\citep{Gonzalez2008}.  \nSpectral methods have been widely used in spatial statistics, often as a means to approximate covariance functions \\citep[e.g.][]{Guinness2017, Royle2005,Fuentes2007}.\n  \nAllowing for ties as necessary, we can use the spectral coordinates $(p, q)$ to order the spatial basis functions by \\emph{effective frequency}: $\\omega_{(p, q)} = \\sqrt{\\left(\\frac{p}{M}\\right)^2 + \\left(\\frac{q}{N}\\right)^2}$  \\citep{Burger2009}.\nBecause each basis function is oriented to a particular angle, there may be multiple basis functions with the same frequency $\\omega$ but different orientation. \n\nTo avoid explicit construction of all $MN$ basis functions, we apply the pre-adjustment approach described in Section~\\ref{sec:preadjust} for a selected frequency $\\omega$. We decompose  ${\\bm x}$ into its projection onto $H_{\\omega}$ and corresponding complement by applying a high-pass filter in the frequency domain.  The values of the filter\n are\n\\begin{equation}\nF_{\\omega}(p, q) = \\begin{cases} 1 & \\text{ if } \\omega_{(p, q)}> \\omega \\\\ 0 & \\text{ if } \\omega_{(p, q)} \\le \\omega \\end{cases}.\n\\end{equation}\nTo implement pre-adjustment, we first compute the DFT $\\mathcal{X}(p,q)$ of $\\bm x'$, which is the gridded exposure surface ${\\bm x}({\\bm s})$ with duplicated locations removed.  We then define $\\mathcal{X}_{\\omega}$ to be the element-wise product of $\\mathcal{X}$ and $F_{\\omega}$. The inverse DFT of $\\mathcal{X}_{\\omega}$ provides ${\\bm x}_2'$. Duplicated locations are assigned the same value from ${\\bm x}_2'$, yielding ${\\bm x}_2$.\n\nOne interpretation of removing variation described by frequencies $\\omega$ and lower is that it removes periodic variation with periods $1\/\\omega$ and higher.  %\nThe analogue of $\\hat k$ for the Fourier basis is the effective bandwidth of the kernel smoother in the spatial domain that corresponds to the frequency filter $F_{\\omega}$. \nThis smoother is given by the inverse Fourier transform of $F_{\\omega}$, which is analytically a 'sinc' function. However, edge effects from finite grids mean the  inverse DFT of $F_{\\omega}$ does not exactly follow the expected analytic form. Therefore, we compute the effective bandwidth $\\hat k$ of $F_{\\omega}$ empirically in a manner similar to the approach for TPRS: Let $F'_\\omega(u, v)$ denote the inverse DFT of $F_{\\omega}$. We take $\\hat k$ to be the minimum value of $\\sqrt{u^2 + v^2}$, scaled by grid size, for which $F'_\\omega(u, v)=0$. \n\n\\subsection{Wavelet Basis and Thresholding}\n\\label{sec:wavelets}\nA third choice of spatial basis  $\\{h_j\\}$ is a set of  wavelets. Like Fourier basis functions, wavelets are localized in frequency, but they are also localized in space \\citep{Nason2008}. This means that wavelets can compactly describe variation at different frequencies in different areas of a spatial domain. There are many different sets of wavelets, and here we use the smooth Daubechies wavelets \\citep{Daubechies1988}.\n Wavelet basis functions are indexed by \\emph{level}, based upon successive halving of the spatial domain. Within each level $L$ (and thus, within each frequency $2^{L}$), there are multiple wavelet basis functions with different orientations and spatial positions. The effective bandwidth that corresponds to each level is $\\hat k = 2^{-L}$.\n \n Multi-resolution approaches such as wavelets have recently been used to quantify spatial variation of air pollution exposures \\citep{Antonelli2017}. However, \\cite{Antonelli2017} only described exposures with variation at ``low\" and ``high'' frequencies, and did not attempt to use wavelets for explicit confounding adjustment nor did they provide a specific distance to these labels.  De-correlation of the exposure and outcome via wavelets was presented in an ecological context by  \\citet{Carl2008}. However,  they applied the thresholding to the exposure and the outcome jointly and performed regression on the wavelet coefficients, which is not practical for the large grids and when we have other confounders in the model.\n \n For finite data on a discrete grid, the Discrete Wavelet Transform (DWT) maps any surface to a set of wavelet coefficients.  \n Unlike the DFT, the DWT requires that the grid have length equal to a power of two.  Data on a non-square grid can be embedded within  a larger grid with dyadic dimension to apply the DWT. We use an implementation of the DWT available in the R package \\texttt{wavethresh} \\citep{Nason2008}.\n\nTo pre-adjust exposure using wavelets, we apply a filtering approach similar to that used for a Fourier basis. We first compute the DWT $\\mathcal{W}$ of ${\\bm x}'$. We then threshold  all coefficients at levels $\\ell = 0, \\dots, L$ to get a modified wavelet transform $\\mathcal{W}_L$.\n We then apply the inverse wavelet transform to $\\mathcal{W}_L$ to get the pre-adjusted exposure ${\\bm x}_2'$.\nAlthough we use a global threshold of all wavelet coefficients up to a particular level, the extent of thresholding could be varied across the exposure domain if desired.\n\n\n\\section{Selecting the amount of adjustment}\n\\label{sec:choosing_m}\n\nA critical question for data analysis is how much adjustment to do or, equivalently, what value of $\\hat k$ or $m$ should be chosen. In both approaches presented in Section~\\ref{sec:adjustment}, the estimates of $\\beta$ are unbiased only if $m \\ge m_3$ (with an additional condition of $m \\ge m_2$ for the pre-adjustment approach) and if the basis for adjustment is correctly chosen. \nBut in most practical settings, the true values of $m_1$, $m_2$, and $m_3$ and the correct choice of basis $\\{h(\\cdot)\\}$ are unknown. \nIn this section, we describe different approaches to choosing the amount of adjustment ($m$ or $\\hat k$).\n\nIf there is specific  knowledge of assumed unmeasured confounders or other external, content-area knowledge, then $m$ could be selected \\emph{a priori}. Exact knowledge of $m_3$ is unlikely to be known, but a choice of $m$ or $\\hat k$ might be based upon a combination of known scale of variation in the exposure (e.g. if it is predicted from a spatial model with known parameter values) and the approximate scale of the possible confounders (e.g. regional variation in socioeconomic status). This choice might also be influenced by estimates of $m_1$ and $m_2$, which could be estimated from the data. While an \\emph{a priori} choice of $m$ may not necessarily lead to an unbiased estimate, it does provide a way to pre-specify the extent and interpretation of adjustment, without risking overfitting the model to the outcome data $\\bm y$.\n\nIn the absence of external knowledge, the amount of adjustment $m$ can be selected in a data-driven manner. Estimating $m_3$ directly is challenging, since it requires knowing the amount of variation in $y$ that is due to $\\bm x$ and $\\bm z$--exactly what we are trying to estimate. However, measures of model fit (for either \\eqref{eq:adj_in_health_model} or \\eqref{eq:preadj_model}) can be used to calculate the scale of spatial variation in $\\bm y$. Specifically, after fitting the outcome model with a range of choices for $m$, the model that minimizes the Akaike Information Criterion (AIC) \\citep{Akaike1973} or Bayesian Information Criterion (BIC)  \\citep{Schwarz1978} can provide a selection of the model that best fits the data. However, selecting the model that best fits $\\bm y$ is not guaranteed to reduce error in the estimate of $\\beta$.\n\nA variation of this procedure for estimating the scale of relevant spatial variation in $\\bm y$ is to fit outcome models that include the adjustment basis but do not include the exposure.  The amount of adjustment that leads to the smallest AIC or BIC can be chosen as $m$ for the primary model. Observed confounders can either be included in the model, or first projected out from $\\bm y$. Here we consider the former approach and refer to is as the  ``AIC-NE'' approach for choosing $m$, respectively, where ``NE'' stands for ``No Exposure''.  The BIC alternative is named in the analogous manner.\nThe rationale of this approach is that if the unmeasured confounders have a strong impact on the values of $\\bm y$ and the association of interest ($\\beta$) is relatively weak (a relatively common occurrence in environmental epidemiology), this approach can yield a choice of $m$ without overfitting the relationship between $\\bm y$ and $\\bm x$. Because it involves fitting a model with the outcome $\\bm y$, this approach is mostly limited to adjustment bases such as TPRS that can be explicitly computed. If all locations on the grid had a single outcome value, then this approach could be used for Fourier and wavelet filtering, but that setting is unlikely to occur in practice. \n\nAn approach  that targets estimation error directly is to choose $m$ to minimize the estimated MSE of $\\hat{\\beta}(m)$.  Let $m'$ be some large value for which it is assumed the estimator $\\hat{\\beta}$ is unbiased. Then estimate the bias of $\\hat{\\beta}(m)$ as $\\hat{\\beta}(m) - \\hat{\\beta}(m')$ and estimate ${\\rm Var}(\\hat{\\beta}(m))$ via the `sandwich' estimator \\citep{White1980} for the fitted model.  Together these provide a method for selecting $m$:\n\\begin{equation}\n\\label{eq:pick_m_mse}\n\\hat{m}_{MSE} = \\argmin_{\\tilde{m}} \\widehat{\\text{MSE}}(\\hat{\\beta}(\\tilde{m})) = \n\\argmin_{\\tilde{m}} \\left\\{\\left[\\hat{\\beta}(\\tilde{m}) - \\hat{\\beta}(m')\\right]^2 + \\widehat{\\rm Var}(\\hat{\\beta}(\\tilde{m}))\\right\\}\n\\end{equation}\nThis approach, however, has the clear drawback that $\\hat{\\beta}(m')$ may not be an unbiased estimator (see Section~\\ref{sec:overadj}).\n\nApproaches that are more \\emph{ad hoc} are also possible. One such option is to choose $m$ by selecting a point right after a ``knee'' in the plot of $\\hat{\\beta}$ against $m$ (for example, Figure~\\ref{fig:sister_tprsdf}). While visually intuitive, this approach requires a cumbersome formal definition. We define $\\hat{m}_{knee}$ to be:\n\\begin{equation}\n\\label{eq:pick_m_knee}\n\\hat{m}_{knee} = \\argmin_{\\tilde m} \\left\\{|D^1(\\hat{\\beta}(\\tilde m))| \\Big| |D^1(\\hat{\\beta}(\\tilde m))| < |D^1(\\hat{\\beta}(\\tilde m + 1)), \\tilde{m}  \\in \\mathcal{M}\\right\\},\n\\end{equation} \nwhere  $D^k(\\cdot)$ is a $k$-order difference operator and $\\mathcal{M} = \\{m |  m >  \\argmax_\\omega D^2(\\hat{\\beta}(\\omega))\\}$.\nThe set $\\mathcal{M}$  limits to values of $m$ that are beyond the largest second-order difference, an approximation of curvature, in the $\\hat{\\beta}(m)$ sequence. Within this set, the value of $m$ that gives the first minimum in the first-order differences is selected.  Importantly, this approach does not account for differences in the resolution of $m$ and can be greatly impacted by the noise in the finite differences.\n\n\n  \n\n\\subsection{Impact of over-adjustment}\n\\label{sec:overadj}\nIf the choice of basis functions exactly matches the underlying confounding surface $f(s)$, then sufficiently rich adjustment will remove bias. However, if the choice of basis functions does not match the underlying confounding surface $f(s)$, then residual bias may remain regardless of the choice of $k$. This could lead to increases in bias, since the adjustment basis ($H_m$) then becomes a near-instrumental variable (IV) that is highly correlated with the exposure $x(s)$ but only weakly related to the outcome $y$. Adjustment for a near-IV in contexts with residual confounding can amplify bias from the residual confounders at a rate greater than any bias reduction due to confounding by the near-IV \\citep{Pearl2011}. This suggests that the MSE approach presented above may perform quite poorly if the choice of basis functions is incorrect.\n\n\n\n\\section{Simulations}\n\\label{sec:simulations}\n\\subsection{Primary Setup}\nWe conducted a set of simulations to compare the different adjustment approaches, demonstrating both different choices of spatial basis and the different methods for selecting the scale of adjustment.\nThe data for the simulations are created at points $(u,v)$ lying on a $512 \\times 512$ grid over the unit square $[0, 1) \\times [0, 1)$.  \nFor each simulation, we constructed a fixed exposure surface $x(u, v)$ and a fixed unmeasured confounder surface $f(u, v)$ on this grid. \n\nWe considered six different unmeasured confounder surfaces, $f_1, \\dots, f_6$, constructed to have ``large'' and ``fine'' scale variation for each of three choices of basis: TPRS, sinusoidal functions, and a spatial Gaussian process (GP). Table~\\ref{tab:simf} summarizes these surfaces, and mathematical detail on their definitions is provided in the Supplemental Material, Section B.1. \n\n\\begin{table}\n\\caption{Description of the different confounder surfaces in the simulations.\n\\label{tab:simf}}\n\\centering\n\\fbox{\n\\begin{tabular}{ll}\nSurface & Description\\\\\n\\hline\n$f_1$ & TPRS with 10 df\\\\ %\n$f_2$ & TPRS with 50 df\\\\\n$f_3$ & Sinusoidal up to frequency $\\sqrt{40}$\\\\ %\n$f_4$ & Sinusoidal up to frequency $\\sqrt{500}$\\\\ %\n$f_5$ & Exponential GP with range 0.5 \\\\ %\n$f_6$ & Exponential GP with range 0.15\\\\ %\n\\hline\n\\end{tabular}\n}\n\\end{table}\n\n\n\nWe constructed the exposure surface as $x(u, v) = \\theta f(u, v) + g(u, v)$,  where $g(u, v)$ is a fixed realization of a Gaussian process with exponential covariance structure.\nThis structure for $x$ allows for the amount of correlation between $x$ and $f$ to be controlled, without requiring that they are generated in the same manner (e.g. correlated Gaussian process as used by \\cite{Page2017}).\n For Simulations 1 and 2, we considered two values for the range parameter in the covariance function used to generate $g(u,v)$: 0.05 and 0.50, respectively. The value of 0.05 results in smaller scale spatial variation than the candidate confounders, which is needed to eliminate bias.  The value of 0.50 results in spatial variation at a coarser scale than some of the confounder surfaces, which may lead to the persistence of bias but is also a situation that could reasonably arise in practice.  The parameter $\\theta$ controls the correlation between $x$ and $f$ and thus the amount of bias due to unmeasured confounding. Because the correlations between $g(u, v)$ and $f(u, v)$ differ for each choice of $f(u, v)$, we calculated $\\theta$ so that the amount of bias was equal for all confounder surfaces, which makes comparison of the results more straightforward. Plots of the surfaces $g, f_1, \\dots, f_6$ and the formula for $\\theta$ are provided in Supplemental Material Section B.2. The values of $x(u, v)$ and $f(u, v)$ were standardized to have mean zero and unit variance.\n\nFor each simulation replication, the outcome was constructed as\n$y(u,v) = \\beta x(u, v) +  f(u, v) + \\epsilon(u, v),$\n where $\\epsilon(u, v) \\stackrel{iid}{\\sim} N(0, 16)$.  This setup is designed to reflect a feature of the Sister Study application in that the residual variation in the outcome, even after accounting for unmeasured confounding, is large relative to the exposure variation.\n We redrew a sample of  $n=2,000$ observation locations (selected from the grid of points) for 1,000 replications of each simulation.\nFor Simulations 1 and 2, we set $\\beta = 1$ and chose $\\theta$ such that the uncorrected bias would be 0.2. For each simulation we also conducted a variation with no effect ($\\beta = 0$) to estimate Type 1 error at the nominal $\\alpha=0.05$ level.\n \nWe compare estimates from four sets of models: \n (i) a model with semi-parametric adjustment via TPRS  in the outcome model and models with pre-adjustment of exposure (ii) by TPRS (at observed locations),  (iii) via a high-pass Fourier filter, and (iv) via wavelet thresholding. We present estimates for a sequence of adjustment amounts and using the selection methods from Section~\\ref{sec:choosing_m}.\n\n\\subsection{Effective bandwidth } \n The effective bandwidths $\\hat k$ for TPRS and a high-pass filter on the unit square are shown in Figure~\\ref{fig:eff_band_unit_square}. The difference in ranges of $\\hat k$ between the two bases is clear, with the high-pass frequency filter spanning a much larger range of spatial scales than TPRS. The slight ``hiccup'' in Figure~\\ref{fig:eff_bandwidth_tprs_unit_square} at $df= 10$ is an artifact of the shape of that particular basis function (and the constraints of the gridded locations). For the high-pass filter, the spatial smoother corresponding to high-pass filters $F_1$ and $F_2$ never cross zero, which is why points are only shown for $\\omega \\ge 3$. For both adjustment bases, there is an approximately linear relationship between tuning parameter ($df$ or $\\omega$) and $\\hat k$ on the log-log scale.\n\n\n\\begin{figure}\n\\begin{center}\n\\subfloat[\\label{fig:eff_bandwidth_tprs_unit_square}]{\n\\includegraphics[width=0.45\\textwidth]{Grid_TPRS_Median_Effective_Bandwidth.pdf}\n}\n\\subfloat[\\label{fig:eff_bandwidth_hpf_unit_square}]{\n\\includegraphics[width=0.45\\textwidth]{Grid_HPF_Effective_Bandwidth.pdf}\n}\n\\caption{Effective bandwidth (a) by df for TPRS and (b) by frequency for HPF on a $512 \\times 512$ grid over the unit square.}\n\\label{fig:eff_band_unit_square}\n\\end{center}\n\\end{figure}\n \n \\subsection{Simulation Results}\n The point estimates for the different choices of adjustment basis and unobserved confounder surfaces are provided in \nFigure~\\ref{fig:sim_S1_g1_results} for Simulation 1 (exposure with fine scale variation) and Figure~\\ref{fig:sim_S1_g2_results} for Simulation 2 (exposure with larger scale variation).  In each figure, each panel shows the mean estimate for the estimators that automatically select $\\hat m$ and across a range of fixed values of $m$. For both simulations, the bias of the unadjusted estimator that ignores the unmeasured confounding is 1.2, by construction.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{Simulation_S1_M512_Results_g1_Paper.pdf}\n\\caption{Estimates (\\drawline{thick})  of $\\beta$ in Simulation 1 when pre-adjusting exposure with different choices of spatial basis (panel columns) and  different underlying confounding surfaces (panel rows). The far left column shows the unadjusted estimate. The dashed lines (\\drawline{dashed}) and error bars indicate 2x the standard error. The true parameter value $\\beta =1$ is plotted as a dotted line (\\drawline{thick, blue,densely dotted}).}\n\\label{fig:sim_S1_g1_results}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{Simulation_S1_M512_Results_g2_Paper.pdf}\n\\caption{Estimates (\\drawline{thick})  of $\\beta$ in Simulation 2 when pre-adjusting exposure with different choices of spatial basis (panel columns) and  different underlying confounding surfaces (panel rows). The far left column shows the unadjusted estimate. The dashed lines (\\drawline{dashed}) and error bars indicate 2x the standard error. The true parameter value $\\beta =1$ is plotted as a dotted line (\\drawline{thick, blue,densely dotted}).}\n\\label{fig:sim_S1_g2_results}\n\\end{center}\n\\end{figure}\n\n\n \\subsubsection{Results for fixed $m$}\n \nWhen the adjustment approach matches the basis used to generate the confounder, we see that the bias can be removed after sufficient adjustment. For Simulation 1 when confounding is due to $f_1$ and $f_2$, this occurs when there is TPRS adjustment of at least  $df=10$ and $df=50$, respectively. In addition to leading to low bias (top-left panels of Figures~\\ref{fig:sim_S1_g1_results}), this leads to low MSE that matches or beats all other estimators (Supplemental Materials Table~1), and nearly correct coverage rates (94\\% for nominal 95\\% confidence interval [CI]; see Supplemental Materials Table~2).\nFor $f_3$ and $f_4$, complete adjustment occurs when there is a high pass filter with cutoff at a frequency of 7 and 23, respectively. However the variance of the estimates from the Fourier filtering approach increases greatly with the amount of adjustment, so the MSE remains relatively large despite the lack of bias (Supplemental Materials Table~1).\n\nWhen the adjustment approach does not match the basis generating the confounder, we see that in some cases most bias can still be eliminated but in other cases bias can persist. \nWhen the unmeasured confounder has large scale variation ($f_1$, $f_3$, and $f_5$), we see that all approaches are able to eliminate most bias after sufficient adjustment.\nHowever, for the confounder surfaces that have small scale variation, we see the persistence of bias regardless of the amount of adjustment ($f_6$) and bias amplification with increasing adjustment ($f_4$). This bias amplification occurs even when the correct adjustment basis is chosen, which can seen by the increase in bias for confounder surface $f_4$ when adjusting with Fourier filtering up to $\\omega = 22$.\n\nIn general, we see that TPRS can successfully adjust for confounding even when the unobserved confounder is generated from a different basis. For example, when $f=f_3$,  adjustment with TPRS is able to remove the confounding bias for $df>100$. However, when $f=f_4$, the bias is not eliminated even for very high choices of $df$ relative to the sample size of $n=2,000$. This difference is due to the smaller range of  effective bandwidths of TPRS compared to those of the high-pass filter approach (Figure~\\ref{fig:eff_band_unit_square}). By construction, variation in $f_2$ and $f_3$ extends up to $\\omega =\\sqrt{40} \\approx 6.3$ and $\\omega = \\sqrt{500} \\approx 22.4$, respectively. Using the relationship in Figure~\\ref{fig:eff_bandwidth_hpf_unit_square}, this corresponds to values of $\\hat{k}$ of approximately 0.12 and 0.03, respectively. A value of $df = 85$ corresponds to approximately $\\hat k = 0.12$, but there is not value of $df$ for which TPRS can achieve $\\hat k = 0.03$. \n\n\nFor Simulation 2, in which the exposure has larger scale spatial variation, we see similar trends to most of the Simulation 1 results. However, we do see greater effects of bias amplification, which now appears when the confounder is $f_2$ as well as when the confounder is $f_4$ (Figure~\\ref{fig:sim_S1_g2_results}). Unlike in Simulation 1, the bias induced by  $f_6$ is not reduced at all.\n\n\nIn both sets of simulations, empirical Type I error when $\\beta = 0$ followed the same patterns as the coverage rates when $\\beta = 1$. Specifically, the error rates were at or close to 0.05 when the correct amount of adjustment was done using the correct basis (Supplemental Materials Table~3).\n\n\n\n\n\n\n \\subsubsection{Results when selecting $m$ automatically}\n\nThe estimators that automatically select $\\hat m$ using information criteria perform generally well.  For adjustment using TPRS in the outcome model in Simulation 1, the model selected by minimizing AIC from the outcome model without exposure (AIC-NE) performs the best in almost all settings. For confounders other than $f_4$, a large amount of adjustment is selected, which leads to reduced bias (Figure~\\ref{fig:sim_S1_g1_results}). The MSE of the estimator with adjustment selected by AIC-NE is smaller than the MSE for estimators using  fixed values of $m$ (Supplemental Materials Table~1). For some surfaces, the amount of adjustment selected by BIC-NE yielded slightly smaller MSE, but coverage rates were always better for selection by AIC-NE (Supplemental Materials Table~2). \nIn Simulation 2, selection using BIC-NE  performs best in bias (Figure~\\ref{fig:sim_S1_g2_results}) and  MSE (Supplemental Materials Table~4)  for all settings except when the confounder is $f_2$. This reflects the benefit of greater penalization for model complexity (which leads to less adjustment) in settings with extensive bias amplification. When the confounder is $f_2$, selection using BIC from the full outcome model performs the best.  However, coverage rates are slightly better using AIC-NE for selection in most cases.\n\nWhen pre-adjusting with TPRS at observed locations in Simulation 1, the amount of adjustment selected by minimizing AIC-NE was also better in most settings. Again while MSE was slightly lower for the estimator with amount of adjustment selected by BIC-NE, coverage rates were always better for selection by AIC-NE. Performance in all settings was similar to using adjustment for TPRS in the outcome model.\n  In Simulation 2, minimizing BIC-NE does better in most settings, having lower variance since it tends to select a smaller model. \n\nFor adjustment using Fourier or wavelet filtering, the minimally-adjusted model is almost always selected when minimizing AIC and BIC since the number of parameters estimated for filtering increases exponentially. This results in lower MSE and better coverage than the unadjusted models, but poorer performance than adjusting with TPRS (Supplemental Materials Tables~1 through 5). The exception is in Simulation 2 for confounders $f_2$ and $f_4$, when the models selected by AIC and BIC do much worse than the unadjusted model.\n\nSelecting the amount of adjustment by the estimated-MSE criterion (equation \\eqref{eq:pick_m_mse}) reduced most of the bias in Simulation 1. However, it let to standard errors that were much larger than the other approaches, as can be seen in Figure~\\ref{fig:sim_S1_g1_results}. For Fourier and wavelet filtering, the standard errors were so large that a single estimate provided little information. The corresponding MSE was much larger than all other estimators (Supplemental Materials Table~1). In Simulation 2, the bias amplification led to estimates that were as bad or worse.\nThe \\emph{ad hoc} ``knee'' based approach (equation \\eqref{eq:pick_m_knee}) performed similarly poorly to the estimated-MSE approach, although had better coverage.\n\n\\subsection{Additional Simulations}\nWe conducted additional simulations that included measured confounders $\\bm z$, and the results were not substantively different from Simulation 1 and 2 so are not shown here.\nWe also conducted simulations in which the number of distinct locations was smaller than the number of subjects, leading to repeated locations. However, little difference was observed between those results compared to the primary results presented here. Approaches to reweighting had some benefit in terms of reducing variance in settings with extreme bias (small variance in $\\epsilon$ and $\\theta$ chosen for large relative bias), but no practical impact on the amount of bias.\n\n\n\\subsection{Simulation Conclusions}\n\\label{sec:sim_conclusions}\nBased on the results of Simulation 1 and 2, the best overall approach to reducing confounding bias is adjustment with TPRS at observed locations, either in the outcome model or using exposure pre-adjustment. The best approach for selecting the amount of adjustment is to  minimize AIC-NE or BIC-NE (AIC or BIC from an outcome model without the exposure). In cases where bias amplification may be a concern, due to either only large scale variation in the exposure surface or very fine scale confounding, preference should be given to BIC-NE over AIC-NE. \n\n\n\n\\section{Sister Study Application}\n\\label{sec:sisterfullapp}\nWe now compare these different spatial confounding adjustment approaches in the Sister Study example of the association between SBP and PM$_{2.5}$.\nThe estimates from performing exposure pre-adjustment using TPRS across all observed grid locations, with each location repeated for the number of subjects at the same location, are provided  \n in Figure~\\ref{fig:sister_obsgrid}.\nThese results are similar to those in Figure~\\ref{fig:sister_tprsdf}, as the adjustment basis is the same and the pre-adjustment that leads to Figure~\\ref{fig:sister_obsgrid} repeated locations as needed.  Based upon the simulations (Section~\\ref{sec:sim_conclusions}), the preferred method for selecting the appropriate amount of adjustment is to minimize AIC-NE  or BIC-NE. Because the exposure surface in this application is quite coarse (25km grid), we use BIC-NE for selecting our primary result to reduce the impact of bias amplification from over-adjustment. This choice results in adjustment using $m=3$ TPRS basis functions. When adjustment is done in the health model, the estimated association is a difference of 0.96 mmHg (95\\% CI: 0.50, 1.42) in SBP for each 10 $\\mu g\/m^3$ difference in PM$_{2.5}$ (Table~\\ref{tab:sister_tprs_res}). When the exposure is pre-adjusted, the estimated association is a difference of 0.92 mmHg (95\\% CI: 0.46, 1.41) in SBP for each 10 $\\mu g\/m^3$ difference in PM$_{2.5}$. \n\n\n\\begin{figure}[t]\n\\begin{center}\n\n\\subfloat[\\label{fig:sister_obsgrid}]{\n\\includegraphics[width=0.47\\textwidth]{Sister_Est_TPRS_PW_obsgrid.pdf}\n}\n\\subfloat[\\label{fig:sister_obs1grid}]{\n\\includegraphics[width=0.47\\textwidth]{Sister_Est_TPRS_PW_obs1grid.pdf}\n}\\\\\n\\subfloat[\\label{fig:sister_tprs_grid}]{\n\\includegraphics[width=0.47\\textwidth]{Sister_Est_TPRS_PW_grid.pdf}\n}\n\\caption{Estimates (\\drawline{ultra thick})  and pointwise confidence intervals (\\drawline{black}) of the association between SBP and PM$_{2.5}$ for different amounts of confounding adjustment using TPRS.  The horizontal line (\\drawline{blue, dashed}) is at zero.}\\label{fig:sister_tprs}\n\\end{center}\n\\end{figure}\n\n\n \\begin{table}\n\\caption{\\label{tab:sister_tprs_res}Point estimates and 95\\% confidence intervals of the difference in SBP (in mmHg) associated with a difference of 10 $\\upmu$g\/m$^3$ in PM$_{2.5}$ exposure in the Sister Study, corresponding  to different scales of adjustment using TPRS.}\n\\centering\n\\fbox{\n\\begin{tabular}{l c c c c cc}\n&  \\multicolumn{2}{c}{Outcome Model}  & \\multicolumn{2}{c}{Pre-Adjustment}   \\\\\n&  \\multicolumn{2}{c}{Adjustment} &\\multicolumn{2}{c}{Observed Locations} \\\\\n& $m$ & $\\hat \\beta$ (95\\% CI) &  $m$ & $\\hat \\beta$ (95\\% CI) \\\\\n  \\hline\n    BIC-NE & 3 & $0.96$ ($0.50$, 1.42) & 3&0.92 (0.46, 1.41) \\\\\n  AIC-NE & 401 & $-0.49$ ($-1.44$, 0.45) & 401  & $-0.43$ ($-1.38$, 0.52) \\\\\n        BIC & 3 & $0.96$ ($0.50$, 1.42) & 3 &0.92 (0.46, 1.41) \\\\\n  AIC & 401 & $-0.49$ ($-1.44$, 0.45) & 4  &0.95 (0.49, 1.41) \\\\\n  $\\hat k = $ 1,000 km & 12 &  1.05 (0.37, 1.73) & 12 &  0.87 (0.20, 1.55) \\\\\nMSE & 352 & $-0.59$ ($-1.52$, 0.33) & 113 & $-0.37$ ($-1.18$, 0.44) \\\\\nKnee & 11&  $1.10$  (0.43, 1.78) &  11 & $0.93$ ($0.26$, 1.60)  \\\\\n\\hline\n\\hline\n& \\multicolumn{2}{c}{Pre-Adjustment}  & \\multicolumn{2}{c}{Pre-Adjustment} \\\\\n& \\multicolumn{2}{c}{Observed Locations (1 time  each) } &  \\multicolumn{2}{c}{All Locations (1 time each)}\\\\\n& $m$ & $\\hat \\beta$ (95\\% CI) &  $m$ & $\\hat \\beta$ (95\\% CI) \\\\\n  \\hline\nAIC & 4 & 0.88 (0.43, 1.33) & 5 & 1.34 (0.81, 1.86)\\\\\nBIC & 3 & 0.80 (0.35, 1.26) & 4 & 0.81 (0.36, 1.26) \\\\\n$\\hat k = $1,000 km &   12 & 1.32 (0.68, 1.96) & 12 &  1.34 (0.73, 1.95) \\\\\nMSE & 268 &0.23 ($-0.58$, 1.03) & 3 & 0.62 (0.17, 1.07) \\\\\nKnee &  5 & 1.41 (0.86, 1.96) & 5 & 1.34 (0.81, 1.86) \\\\\n\\end{tabular}}\n\\end{table}\n\n\nFor comparison, Table~\\ref{tab:sister_tprs_res} also includes the results for the other approaches to selecting $m$. Using AIC-NE or AIC  results in a large amount of adjustment being selected ($m=401$). This appears to be extensive overfitting of the model; the increases in the value of the log-likelihood as $m$ increases are small relative to the large sample size ($n=47,206$). The MSE approach also chooses a large amount of adjustment: $m=352$ for outcome model adjustment and $m=113$ for exposure pre-adjustment. All of these approaches with large adjustment result in negative point estimates due to the downward trend observed in Figures~\\ref{fig:sister_tprsdf} and \\ref{fig:sister_obsgrid}. This downward trend is likely due to bias amplification from residual confounding, since PM$_{2.5}$ only explains a small fraction of the variation in SBP and the measured confounders, while numerous, likely cannot capture all of the confounding relationships. Furthermore, the coarse spatial scale of the exposure means that variation due to unmeasured confounders is likely finer-scale  than the exposure.\n\n\n\nResults from two alternative approaches to exposure pre-adjustment are presented in Figures~\\ref{fig:sister_obs1grid} and \\ref{fig:sister_tprs_grid}. These figures correspond, respectively, to pre-adjustment at observed locations, with duplicated locations removed, and pre-adjustment over all locations in the domain, with each location included once. \nThe point estimates corresponding to the different selection methods for these adjustment approaches are provided in the bottom half of Table~\\ref{tab:sister_tprs_res}. The amount of adjustment selected by AIC and BIC (using the full outcome model) is small, ranging from $m=3$ to $m=5$. These choices of adjustment yield point estimates ($0.80$ to $1.34$) similar to the main results from the models that adjust at all observed locations ($0.96$ and $0.92$).\nThe difference in estimates between Figures~\\ref{fig:sister_obs1grid} and \\ref{fig:sister_tprs_grid} is relatively small, suggesting that the restriction of the pre-adjustment to the observed locations does not have a large impact on the results. However, there is a notable qualitative difference between these results and those that included duplicate locations for the adjustment (Figures~\\ref{fig:sister_tprsdf} and \\ref{fig:sister_obsgrid}). While all four approaches show a negative trend in the point estimates for large values of $m$, the approaches that adjust using observed locations including duplicates decrease at smaller values of $m$ and yield negative point estimates.  \nWe explored using reweighting to correct for the impact of the duplicated locations in the pre-adjustment approaches and obtained results that were either qualitatively similar to the results without reweighting or were highly unstable (see Supplemental Materials Section C.2). The magnitude of the difference in the point estimates from these alternative approaches using TPRS (including or excluding duplicate locations) suggests that residual confounding is likely occurring in this setting.\n\nEstimates for a fixed choice of $\\hat k = $ 1,000 km are also provided in  Table~\\ref{tab:sister_tprs_res}. This corresponds to adjustment using $12$ degrees of freedom (Supplemental Materials Section C.1).\n A bandwidth of this size smooths  large scale variation across the contiguous United States (which is approximately 4,500 km by 2,900 km). Because there are well-established large-scale trends in health outcomes across the United States \\citep{Mensah2005}, this value was chosen so that the analysis accounts for long-scale trends in systolic blood pressure. \n\n \nBased on the simulation results and duplicated observed locations, we recommend adjustment using TPRS as described above instead of using the Fourier or wavelet approaches. However, we present results for those approaches here for illustration.\nTo apply the pre-adjustment approaches with Fourier and wavelet basis functions, we embed the gridded locations within a larger square grid.  \nBecause the predictions of 2006 annual average  PM$_{2.5}$  of \\citet{Sampson2013} are only defined over the contiguous United States, \nthe added points are assigned exposure concentration of zero. This results in a grid of size 184 $\\times$ 184 for the Fourier approach and 256 $\\times $ 256 for the wavelet approach.  We see from Figures~\\ref{fig:sister_hpf} and \\ref{fig:sister_wave} that the point estimates are relatively stable for all amounts of adjustment. This is probably due in part to the sinusoidal and wavelet basis functions not being representative of the spatial structure of the real-world unmeasured confounders, so increased adjustment removes little variation from the outcome or exposure. Additionally, because these adjustment methods rely on each location being present only a single time, we expect trends similar to Figures~\\ref{fig:sister_obs1grid} and \\ref{fig:sister_tprs_grid}.\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\subfloat[\\label{fig:sister_hpf}]{\n\\includegraphics[width=0.47\\textwidth]{Sister_Est_HPF_PW.pdf}\n}\n\\subfloat[\\label{fig:sister_wave}]{\n\\includegraphics[width=0.47\\textwidth]{Sister_Est_Wave_PW.pdf}\n}\n\\caption{Estimates (\\drawline{ultra thick})  and pointwise confidence intervals (\\drawline{black}) of the association between SBP and PM$_{2.5}$ for different amounts of confounding adjustment using Fourier and wavelet filtering. The horizontal line (\\drawline{blue, dashed}) is at zero.\n\\label{fig:sister_all}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\nWe have examined different approaches to adjusting for unmeasured spatial confounding and quantifying and selecting the scale of adjustment.  These approaches are motivated as extensions of  time series methods for temporal confounding \\citep{Peng2006,SzpiroSheppard2014}. \nWe presented a method for comparing the spatial scales across choices of basis using the effective bandwidth $\\hat k$. We showed examples using  TPRS, a Fourier basis, and wavelets, each of which indexes variability in different forms and on different scales.\nWe showed that adjustment with TPRS is limited to a smaller range of spatial scales than the Fourier approach. However, the smaller bandwidth range of TPRS can have more flexibility at these scales. But when the true variation is beyond the scales TPRS can reach, it fails to remove confounding bias.\n\nFor a particular application, a choice of the amount of adjustment must be made. We identified selecting the amount of adjustment using AIC-NE and BIC-NE as the preferred approaches for most settings.\nWhen there is substantive knowledge available about the scales of variation involved, $\\hat k$ can be chosen \\emph{a priori}.\n\\emph{Ad hoc} approaches such as the ``knee'' and estimated MSE methods performed poorly.\n\nPre-adjusting the exposure by Fourier filtering or wavelet thresholding is attractive due to the orthogonality of those bases. However, they are both severely limited by the requirement of a square grid, assumptions of periodicity, and the need to reweight the population. For wavelets, the thresholding at dyadic levels also leads to limited available intervals for adjustment. Although in settings where there is a priori knowledge about confounding relationships,  non-uniform thresholding of the wavelet coefficients could add flexibility. \nFor these reasons, the more flexible TPRS are likely to be preferred over the Fourier or wavelet basis for settings where TPRS can represent variation at the scale of interest, such as the example study of PM$_{2.5}$ in the Sister Study.\nThe Fourier basis ccould be chosen when very-fine scale adjustment is needed or there is external evidence to support periodic variation in the confounders.\n\nOne of the key results from our simulations is that the addition of spatial basis functions, or an equivalent pre-adjustment procedure, does not necessarily reduce bias in the point estimate.  In fact, it can lead to substantial bias amplification. \nThis is an important contrast to the time series settings, in which more aggressive adjustment is recommended to reduce bias from unmeasured confounding \\citep{Peng2006}. The exposure in a time series settings are typically measured directly, which means that there is substantial residual variation in the exposure at each time point, whereas the use of an exposure prediction model in a spatial context means that spatial exposures are often quite smooth.\nThe potential for bias amplification means that using changes in the point estimates is often not a good approach to assessing the extent of confounding bias.\n\nWe have presented results here in the context of a linear health model. However, these approaches to spatial confounding adjustment can be applied in generalized linear model settings. The patterns of bias reduction (or increase) for a sequence of adjustment values $m$ will differ by context and choice of link function, but the connection between adjustment basis, spatial scale, and overall interpretation remains the same as the linear case. For example, \\cite{Keet2018} used TPRS to adjust for large-scale confounding across the contiguous United States in an analysis of particulate matter and asthma-related outcomes. Automated selection could be done using extensions of information criteria such as QIC \\citep{Pan2001}.\n\nIn summary, we have presented  methods for describing and selecting the spatial scale of spatial confounding adjustment using different choices of basis. \nThese methods can be used to more accurately describe the extent of spatial confounding adjustment in future studies with spatial exposures.\n\n\n\\section*{Acknowledgements}\nThis work was supported by grants T32ES015459 and R21ES024894 from the the National Institute for Environmental Health Sciences (NIEHS). The Sister Study supported in part by the Intramural Research Program of the NIH, NIEHS (Z01ES044005). This work was also supported in part by the US Environmental Protection Agency (EPA) through awards RD83479601 and RD835871. This was work has not been formally reviewed by the EPA. The views expressed in this document are solely those of the authors and do not necessarily reflect those of the Agency. EPA does not endorse any products or commercial services mentioned in this publication.\n\n\n\\bibliographystyle{chicago}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\@startsection {section}{1}{\\z@\n                                   {-3.5ex \\@plus -1ex \\@minus -.2ex\n                                   {2.3ex \\@plus.2ex\n                                   {\\normalfont\\large\\bfseries}}\n\\renewcommand\\subsection{\\@startsection{subsection}{2}{\\z@\n                                     {-3.25ex\\@plus -1ex \\@minus -.2ex\n                                     {1.5ex \\@plus .2ex\n                                     {\\normalfont\\bfseries}}\n\n\\parskip 6 pt\n\n \\marginparwidth 0pt\n \\oddsidemargin  -0.3cm\n \\evensidemargin  -0.3cm\n \\marginparsep 0pt\n \\topmargin   -0.4in\n \\textwidth   7.1in\n \\textheight  9.0 in\n\n\\newcommand{\\be}{\\begin{equation}}\n\\newcommand{\\ee}{\\end{equation}}\n\\newcommand{\\bea}{\\begin{eqnarray}}\n\\newcommand{\\eea}{\\end{eqnarray}}\n\\newcommand{\\bse}{\\begin{subequations}}\n\\newcommand{\\ese}{\\end{subequations}}\n\\newcommand{\\beqa}{\\begin{eqnarray}}\n\\newcommand{\\eeqa}{\\end{eqnarray}}\n\\newcommand{\\beqar}{\\begin{eqnarray*}}\n\\newcommand{\\eeqar}{\\end{eqnarray*}}\n\\newcommand{\\bi}{\\begin{itemize}}\n\\newcommand{\\ei}{\\end{itemize}}\n\\newcommand{\\bn}{\\begin{enumerate}}\n\\newcommand{\\en}{\\end{enumerate}}\n\\newcommand{\\labell}[1]{\\label{#1}\\qquad_{#1}}\n\\newcommand{\\fixme}[1]{\\textbf{FIXME: }$\\langle$\\textit{#1}$\\rangle$}\n\\newcommand{\\note}[1]{\\textbf{NOTE: }$\\langle$\\textit{#1}$\\rangle$}\n\\newcommand{\\ba}{\\begin{array}}\n\\newcommand{\\ea}{\\end{array}}\n\\newcommand{\\bc}{\\begin{center}}\n\\newcommand{\\ec}{\\end{center}}\n\\newcommand{\\nnr}{\\nonumber \\\\}\n\\newcommand{\\nn}{\\nonumber}\n\n\\newcommand{\\cf}{{\\em cf.}\\ }\n\\newcommand{\\ie}{{\\em i.e.}\\ }\n\\newcommand{\\eg}{{\\em e.g.} }\n\\newcommand{\\viz}{{\\em viz.}\\ }\n\\newcommand{\\nb}{{\\em N.B.}\\ }\n\\newcommand{\\etal}{{\\em et al.}\\ }\n\n\\newcommand{\\tcb}{\\textcolor{blue}}\n\\newcommand{\\tcr}{\\textcolor{red}}\n\\definecolor{darkgreen}{rgb}{0,0.3,0}\n\\definecolor{darkblue}{rgb}{0,0,0.3}\n\\definecolor{darkred}{rgb}{0.7,0,0}\n\\definecolor{VioletRed4}{rgb}{0.55,0.13,0.32}\n\\definecolor{VioletRed}{rgb}{0.82,0.13,0.56}\n\\definecolor{VioletRed2}{rgb}{0.93,0.23,0.55}\n\\newcommand{\\tcdr}{\\textcolor{darkred}}\n\\newcommand{\\tcdg}{\\textcolor{darkgreen}}\n\\newcommand{\\tcdb}{\\textcolor{darkblue}}\n\n\n\n\\makeatother\n\n\\begin{document}\n\n\\newcommand{\\email}[1]{\\footnote{\\href{mailto:#1}{#1}}}\n\n\n\n\\title{\\bf\\Large{One-loop Photon's Effective Action in the Noncommutative Scalar QED$_{3}$}}\n\n\\author{\\bf{M.~Ghasemkhani}\\email{ghasemkhani@ipm.ir} $^{a}$, \\bf{R.~Bufalo}\\email{rodrigo.bufalo@ufla.br} $^{b}$,  \\bf{V.~Rahmanpour}\\email{v.rahmanpour@mail.sbu.ac.ir} $^{a}$ and M.~Alipour\\email{moj.alipour@yahoo.com} $^{a}$ \\\\\\\\\n\\textit{\\small$^a$  Department of Physics, Shahid Beheshti University,  G.C., Evin, Tehran 19839, Iran}\\\\\n\\textit{\\small $^b$   Departamento de F\\'isica, Universidade Federal de Lavras,}\\\\\n\\textit{\\small Caixa Postal 3037, 37200-000 Lavras, MG, Brazil}\\\\\n}\n\\maketitle\n\n\\begin{abstract}\nIn this paper, we consider the evaluation of the effective action for photons coupled to charged scalar fields in the framework of a $(2+1)$-dimensional noncommutative spacetime.\nIn order to determine the noncommutative Maxwell Lagrangian density, we follow a perturbative approach, by integrating out the charged scalar fields, to compute the respective graphs for the vev's $\\left\\langle AA \\right\\rangle$, $\\left\\langle AAA \\right\\rangle$ and $\\left\\langle AAAA \\right\\rangle$.\nSurprisingly, it is shown that these contributions are planar and that, in the highly noncommutative limit, correspond to the Maxwell effective action and its higher-derivative corrections.\nIt is explicitly verified that the one-loop effective action is gauge invariant, as well as under discrete symmetries: parity, time reversal, and charge conjugation.\nMoreover, a comparison of the main results with the noncommutative QED$_{3}$ is established.\nIn particular, the main difference is the absence of parity violating terms in the photon's effective action coming from integrating out the charged scalar fields.\n\n\\end{abstract}\n\n\n\n\\setcounter{footnote}{0}\n\\renewcommand{\\baselinestretch}{1.05} \n\n\\newpage\n\\tableofcontents\n\\section{Introduction}\n\\label{sec1}\n\nIn recent years a great amount of attention has been paid in the analysis and calculation of covariant effective action for different types of quantum fields, exploring the diversity of new interactions that mainly depend on the spin of the fields involved as well as the spacetime dimensionality \\cite{Bonora:2016otz, Bonora:2017ykb, Quevillon:2018mfl}.\nOne may say that the canonical example of a complete analysis is the Euler-Heisenberg effective action \\cite{Heisenberg:1935qt}, where quantum effects from QED are responsible to induce nonlinear interactions among photons.\nMoreover, the effective action framework has served as an important tool to explore different point of views about the quantum gravitational theory, where the Einstein-Hilbert action is augmented by metric and\/or torsion fields higher-order terms \\cite{Buchbinder:1992rb,Masud}.\n\nNaturally, since the framework of effective action is a powerful tool, there is a great expectation that this approach can be used to make contact with modern phenomenology of physics beyond the standard model.\nThe main idea behind this formulation is that at energies below some cutoff scale $\\mu$, \\footnote{That may signal symmetry violation, for instance Lorentz symmetry violation.} all the effects of the massive degrees of freedom above $\\mu$ can be encoded as new interactions among the fields remaining active below $\\mu$.\nThe effective action approach has been extensively used to the study of Lorentz violating field theories, where the energy scale $\\mu$ is related to the Planck energy scale $E_{\\rm Pl}$ (or length $\\ell_{\\rm Pl}$) where our notion of smooth geometry is expected to break \\cite{ref53,Bluhm:2005uj}.\nIn this case, the current understanding is that the low energy Lorentz violating terms come as quantum corrections from heavy modes \\cite{Borges:2013eda,Borges:2016uwl}.\n\n\nAlthough the majority of studies of Lorentz violating field theories is developed in a four-dimensional spacetime, there are considerable interests in the description of\nthree-dimensional (3D) ones \\cite{Charneski:2008hy, Nascimento:2014owa, Casana:2015hda}.\nBesides the algebraic richness of odd dimensional spacetimes, one may say that the most appealing aspect of 3D field theories is the UV finiteness in some models.\nThis feature might provide an ambiguity free description of Lorentz violation, allowing thus a close contact of violating effects with physical planar phenomena.\nIn particular, it is worth recall the example of the description of quantum hall fluids in terms of noncommutative geometry \\cite{Susskind:2001fb,Douglas:2001ba}.\n\nOver the past two decades, field theories defined in a noncommutative (NC) geometry have been considered as one of the most prominent candidates presenting Lorentz violation to make contact with quantum gravity phenomenology \\cite{Douglas:2001ba,AmelinoCamelia:2008qg}.\nWithin this description, the noncommutativity measurement parameter is related to a length scale $\\ell_{\\rm nc} \\sim \\sqrt{\\theta}$.\nIn one hand, this length scale can be seen as a manifestation of the discreteness of the spacetime, presenting a smooth profile in the UV region \\cite{Arzano:2017uuh}.\nOn the other hand, this same scale is responsible for introducing instabilities in the dispersion relations of the fields, the so-called UV\/IR mixing \\cite{Matusis:2000jf}.\n\nNC field theories have been studied through the effective action approach, where the behavior of the new couplings was deeply analyzed \\cite{Vassilevich:2005vk}, where the presence of UV\/IR mixing in the 1PI functions signals that applying the usual Wilsonian field theory notions and techniques to NC QFT's one should be careful.\nThis type of analysis was also developed to two and three-dimensional NC models\n\\cite{Ghasemkhani:2013bs, Ghasemkhani:2017bjp, Chu:2000bz, Banerjee:2007ua, Bufalo:2014ooa}.\nThese studies of effective action in 3D models were exclusive to the coupling of gauge and fermion fields, no much attention has been paid to the case involving scalar fields, in particular the case of spinless charged fields interacting with photons.\n\n\nIn one hand, it is of physical significance to study scalars in 3D field theories independently of fermions in condensed matter systems, as in Quantum Hall systems, since we have scalar quasiparticle excitations.\nOn the other hand, recently 3D versions of fermionization\/bosonization have also been introduced \\cite{Hsin:2016blu,Benini:2017dus}.\nIn these studies, it was discussed the duality between nonspin\nChern-Simons theory and a spin Chern-Simons theory, exploring precisely the spin structure of the given models.\nIn this sense, the present work could be the first step in extending such analysis to the NC case.\nMotivated by these facts, we will analyze throughout the paper to what extent the spin of the matter fields can change the effective action when charged scalar and fermion fields are considered in the presence of the spacetime noncommutativity.\nA straightforward result is that in the case of the 3D scalar quantum electrodynamics (scalar QED$_{3}$), it is not possible to generate the parity odd Chern-Simons terms, showing thus that the dynamics of the 3D gauge field is significantly different in the presence of either charged scalar or fermion fields. It is well known that the presence of the degree of freedom associated with the spin changes in most of the cases only the magnitude of physical quantities, e.g. the beta function \\cite{Ghasemkhani:2016zjy} and electron's magnetic moment \\cite{Panigrahi:2004cf}.\n\nIn this paper we discuss the effective action for the photon in the scalar QED in the noncommutative three-dimensional spacetime.\nIn Sec.~\\ref{sec2} we present an overview of the scalar QED, where the charged scalar fields are minimally coupled with the photons.\nThere we define the main aspects regarding the Moyal product used in our analysis,\\footnote{The noncommutativity we will be using in the paper is\ndefined by the algebra $[\\hat{x}_\\mu,\\hat{x}_\\nu] = i \\theta_{\\mu\\nu}$. So in order to construct a noncommutative field theory, using the Weyl-Moyal (symbol)\ncorrespondence, the ordinary product is replaced\nby the Moyal star product as defined below.}\nwe also discuss the content of discrete symmetries in the NC 3D spacetime.\nIn addition, all the Feynman rules are presented for the propagators and 1PI vertices.\nSection \\ref{sec4} is focused in the perturbative computation of the relevant graphs corresponding to the one-loop effective action for the photon gauge field.\nIt is also discussed the generation of higher-derivative terms, similarly to the\nAlekseev-Arbuzov-Baikov effective Lagrangian for non-Abelian fields.\nIn Sec.~\\ref{sec5} we establish a comparison of the obtained results for the effective action in the NC-scalar QED to those of ordinary NC-QED, exploring the part played by the spin in these cases.\nWe present our final remarks in Sec.~\\ref{conc}.\n\n\n\n\\section{The model}\n\\label{sec2}\nIn this section, we introduce the model and fix our notation.\nThe noncommutative extension of the bosonic electrodynamics is described by the following action\n\\begin{equation}\nS=\\int d^{3}x\\Big[\\left(D_{\\mu}\\phi\\right)^{\\dagger}\\star D^{\\mu}\\phi-m^{2}\\phi^{\\dagger}\\star \\phi\\Big],\n\\label{eq:a1}\n\\end{equation}\nthis functional action consists of the interaction of charged scalar fields minimally coupled with an external gauge field.\nWe consider the covariant derivative form in the fundamental representation $D_{\\mu}\\phi=\\partial_{\\mu}\\phi+ieA_{\\mu}\\star\\phi$.\nThis action is invariant under the infinitesimal gauge transformation\n\\begin{equation}\n\\delta A_{\\mu}=\\partial_{\\mu}\\lambda+ie[A_{\\mu},\\lambda]_{\\star}~,\\quad \\delta\\phi=ie\\lambda\\star\\phi,\n\\label{eq:a2}\n\\end{equation}\nwhere $[\\,\\, ,\\,]_{\\star}$ is the Moyal bracket.\nMoreover, the Moyal star product between the functions $f$ and $g$ is defined as\n\\begin{equation}\nf\\left( x\\right)\\star g\\left(x\\right) = f\\left( x\\right) \\exp \\left(\\frac{i}{2}\\theta ^{\\mu \\nu}\n\\overleftarrow{\\partial_\\mu}\n \\overrightarrow{\\partial_\\nu}\\right) g\\left( x\\right),\n \\label{eq:a3}\n\\end{equation}\nwhere $\\theta^{\\mu\\nu}=-\\theta^{\\nu\\mu}$ are constant parameters that measure the noncommutative structure of the space-time.\nIn order to avoid unitarity violation, we assume that $\\theta^{0i}=0$, hence we have only one nonzero independent component $\\theta^{12}$ in our model.\n\nIt is worth mentioning that although the couplings \\eqref{eq:a1} are simply modified by the presence of a nonplanar phase due to the Moyal product, the noncommutativity of spacetime coordinates shows its importance in the computation of the one-loop effective action for the gauge field, where nonlinear self-couplings are present solely due to the NC framework.\nThe one-loop effective action for the gauge field\ncan be readily obtained by integrating out the charged scalar fields of \\eqref{eq:a1}\n\\begin{equation}\ne^{i\\Gamma_{\\rm eff}[A]}=\\int D\\phi^{\\dagger} D\\phi~e^{-i\\int d^{3}x~\\phi^{\\dagger}\\star(D^{2}+m^{2})\\star\\phi}.\n\\end{equation}\nUsing the Gaussian functional integration formulas for the case of interacting charged scalar fields, we can write the noncommutative 1PI effective action as below\n\\begin{equation}\ni\\Gamma_{\\rm eff}[A]={\\rm Tr}\\ln\\left[\\frac{(i\\partial_{\\mu}-eA_{\\mu})\\star(i\\partial^{\\mu}-eA^{\\mu})\\star-m^{2}}{-\\partial^{2}-m^{2}}\\right],\n\\end{equation}\nwhere ${\\rm Tr}$ is a sum over eigenvalues of the operator inside the bracket which can also be evaluated in momentum space.\nSimilarly to the description of one-loop effective action for the gauge field in the case of NC-QED \\cite{Bufalo:2014ooa}, one can show that $\\Gamma_{\\rm eff}[A]$ has a convergent series expansion in coupling constant $e$. From a diagrammatic point of view, it includes the one-loop graphs contributing to the gauge field $n$-point functions which is considered as\n\\begin{equation}\n\\Gamma_{\\rm eff}[A]={\\cal{S}}_{\\rm eff}[AA]+{\\cal{S}}_{\\rm eff}[AAA]+{\\cal{S}}_{\\rm eff}[AAAA]+\\cdots.\n\\label{eq:aa}\n\\end{equation}\nHowever, the functional $\\Gamma_{\\rm eff}[A]$, in comparison to the NC-QED case, has more graphs due to the presence of an additional interacting vertex.\n\nMoreover, it is important to emphasize that as we will show in our model,\nsimilarly to the case of NC-QED \\cite{Ghasemkhani:2017bjp}, the one-loop effective action for the  photons is completely planar.\nExplicitly, in the evaluation of the one-loop diagrams with an arbitrary number of external legs of photons, for energies below the mass scale $m$, only planar diagrams contribute. This means the absence of IR\/UV mixing.\n\n\\subsection{Discrete symmetries}\n\\label{sec2.1}\n\nSince we are interested in computing the one-loop effective action for the photon, it is useful to analyze the behavior of the original action \\eqref{eq:a1} under discrete symmetries: parity, charge conjugation and time reversal.\nThis study will allow us to determine which of them may be anomalous in the obtained results for the one-loop order effective action.\n\n\\begin{itemize}\n\\item{\\emph{Parity}}\n\nParity transformation in $d=2+1$ is defined as $x_{1} \\rightarrow -x_{1}$ and $x_{2}\\rightarrow x_{2}$, in this case we have that the field $\\phi$ is even under parity, and the components of the gauge field $A_{\\mu}$ behave as $A_{0}\\rightarrow A_{0}$, $A_{1}\\rightarrow -A_{1}$ and $A_{2}\\rightarrow A_{2}$.\nMoreover, we observe from the NC algebra that the $\\theta$ parameter changes under this transformation as $\\theta^{12}\\rightarrow -\\theta^{12}$ .\nWith these considerations, it is easy to show that the whole of the action \\eqref{eq:a1} is parity invariant.\n\\item{\\emph{Time Reversal}}\n\nUnder time reversal, we have that $x_{0} \\rightarrow -x_{0}$. In this case, the components of the gauge field behave as $\\left( A_{0}, A_i\\right) \\rightarrow \\left( A_0,-A_i\\right)$.\nBy demanding that the scalar field does not change $\\phi \\to \\phi $, and that necessarily the NC parameter transforms as $\\theta^{12}\\rightarrow -\\theta^{12}$ under time reversal, we are left with a $T$-invariant action.\n\\item{\\emph{Charge Conjugation}}\n\nAs we know, the behavior of the gauge field under charge conjugation is given by $A_{\\mu}\\rightarrow -A_{\\mu}$ for any space-time dimensionality. Taking the scalar field to be unchanged under $C$, and the transformation for the NC parameter $\\theta\\rightarrow-\\theta$, we conclude that the action \\eqref{eq:a1} is $C$-invariant.\n\\end{itemize}\n\n\n\\subsection{Propagators and vertex functions}\n\\label{sec3}\n\nIn order to discuss the computation of the perturbative effective action, we must determine the basic propagators and 1PI vertex functions.\nFrom the functional action described in \\eqref{eq:a1}, we can obtain the bosonic propagator\n\\begin{equation}\n{\\cal{D}}(p) = \\frac{i}{{p^{2}-m^{2}}},\n \\label{eq:b1}\n\\end{equation}\nthe cubic vertex $\\left \\langle A \\,\\phi \\, \\phi^\\dagger \\right \\rangle$\n\\begin{align}\n\\Gamma^\\mu(p,q)= -ie\\left(p+q\\right)^{\\mu}\\exp\\Big(\\frac{i}{2} p \\wedge q\\Big),\n \\label{eq:b2}\n\\end{align}\nand the quartic vertex $\\left \\langle AA\\,\\phi \\,\\phi^\\dagger \\right \\rangle$\n\\begin{align}\n\\Lambda^{\\mu\\nu}(p,q,s)=2ie^{2}\\eta^{\\mu\\nu}\\exp{\\Big(\\frac{i}{2}k \\wedge s\\Big)}\\cos\\Big(\\frac{p \\wedge  q}{2}\\Big),\n \\label{eq:b3}\n\\end{align}\nwhere we have introduced the notation $p \\wedge q = p_\\mu \\theta^{\\mu\\nu}q_\\nu$.\nA straightforward difference of the scalar and fermionic electrodynamics is the presence of the quartic vertex $\\left \\langle AA\\phi \\phi^\\dagger \\right \\rangle$, which increases significantly the number of the one-loop graphs.\nMoreover, the scalar vertices are rather simpler due to the absence of the Dirac $\\gamma$ matrices, resulting in a much simpler algebraic analysis.\n\n\n\\section{Perturbative Effective Action}\n \\label{sec4}\n\nNow that we have determined the basic Feynman rules for the 1PI functions, we shall proceed to the computation of the one-loop diagrams related to the effective action for the gauge field.\nFor this purpose, we shall compute along this section the respective contributions: the free part of the effective action $\\left\\langle AA \\right\\rangle$, and the interacting parts for the cubic vertex $\\left\\langle AAA \\right\\rangle$ and quartic vertex $\\left\\langle AAAA \\right\\rangle$.\nIn general, the final results of our analysis related to the graphs contributing to $\\left\\langle AA \\right\\rangle$, $\\left\\langle AAA \\right\\rangle$ and $\\left\\langle AAAA \\right\\rangle$ vertices shall be a function of $e$, $\\tilde{p}_\\mu = \\theta_{\\mu \\nu } p^\\nu$ and $p^2\/m^2$.\n\nTo highlight the effects of noncommutativity in the low energy effective   action and the photon two, three and four-point functions, while we take the external momenta $p$ such that $p^2\/m^2\\ll 1$, we also take the highly noncommutative limit, i.e., the low-energy regime $p^2\/m^2 \\to 0$ while $\\tilde{p}$ is kept finite.\nIn this limit the noncommutative (planar) phase factors, which are a function of $\\tilde{p}$, remain finite.\nMoreover, we shall focus our attention on those terms of order $m^{-1}$.\nWe present by complementarity, at the next to leading order, the terms of order $m^{-3}$ that correspond to higher-derivative corrections.\n \\subsection{One-loop $\\left\\langle AA \\right\\rangle$ part}\n\nFrom the Feynman rules we can compute the one-loop contribution to the $AA$-term corresponding to the free part of the photon effective action.\nThe two diagrams contributing at this order are depicted in Fig.~\\ref{oneloop1}, which the respective expressions have the form\n\\begin{align}\n\\Pi^{\\mu\\nu}_{(a)}(p)  &= e^{2} \\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{(p+2k)^{\\mu}(p+2k)^{\\nu}}{[(p+k)^{2}-m^{2}][k^{2}-m^{2}]}, \\nonumber \\\\\n\\Pi^{\\mu\\nu}_{(b)}(p) &= -e^{2} \\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{2\\eta^{\\mu\\nu}[(p+k)^{2}-m^{2}]}{[(p+k)^{2}-m^{2}][k^{2}-m^{2}]},\n \\label{eq:c1}\n\\end{align}\nso that the full contribution is written as\n\\begin{align}\n\\Pi^{\\mu\\nu}(p) &= e^{2}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{(p+2k)^{\\mu}(p+2k)^{\\nu}-2\\eta^{\\mu\\nu}[(p+k)^{2}-m^{2}]}{[(p+k)^{2}-m^{2}][k^{2}-m^{2}]}.\n \\label{eq:c2}\n\\end{align}\n\nA first comment is that this piece is completely planar, carrying no noncommutative effects.\nThe explicit computation is straightforward using dimensional regularization.\nAfter some algebraic calculation, we can consider the low-energy limit, $p^2\/m^2 \\to 0$, resulting into\n\\begin{align}\n\\Pi^{\\mu\\nu}(p) =\\frac{ie^{2}}{48\\pi m}\\left(p^{\\mu}p^{\\nu}-\\eta^{\\mu\\nu}p^{2}\\right).\n \\label{eq:c3}\n\\end{align}\nMoreover, for the next to leading order contribution, ${\\cal{O}}(m^{-3})$, we find that\n\\begin{equation}\n\\Pi^{\\mu\\nu}_{\\rm hd}(p)=\\frac{ie^{2}}{960\\pi m^{3}}\\left(p^{\\mu}p^{\\nu}-\\eta^{\\mu\\nu}p^{2}\\right)p^{2}.\n \\label{eq:c4}\n\\end{equation}\nThese two terms Eqs.~\\eqref{eq:c3} and \\eqref{eq:c4} satisfy straightforwardly the Ward identity, $p_\\mu \\Pi^{\\mu\\nu}=0$, as we expected.\nWe can determine the respective contribution to the effective action by means of\n\\begin{equation}\ni{\\cal{S}}_{\\rm eff}[AA]=\\int\\int d^{3}x_{1}d^{3}x_{2}~A_{\\mu}(x_{1})\\Gamma^{\\mu\\nu}(x_{1},x_{2})A_{\\nu}(x_{2}),\n \\label{eq:c5}\n\\end{equation}\nwhere ${\\cal{S}}_{\\rm eff}[AA]$ is the quadratic part of the effective action $\\Gamma[A]$ in \\eqref{eq:aa}. Here, we have defined by simplicity\n\\begin{equation}\n\\Gamma^{\\mu\\nu}(x_{1},x_{2})=\\int\\frac{d^{3}p}{(2\\pi)^{3}} e^{-i p\\cdot (x_{1}-x_{2})}\\Pi^{\\mu\\nu}(p ).\n \\label{eq:c6}\n\\end{equation}\n\\begin{figure}[t]\n\\vspace{-1.2cm}\n\\includegraphics[height=6\\baselineskip]{2-point.eps}\n \\centering\\caption{Relevant graphs for the induced $AA$-term.}\n\\label{oneloop1}\n\\end{figure}\n\nAfter some algebra, the quadratic part of the induced effective action for the photon, considering \\eqref{eq:c3} and \\eqref{eq:c4}, is given by\n\\begin{align}\ni{\\cal{S}}_{\\rm eff}[AA]&=-\\frac{ie^{2}}{48\\pi m}\\int d^{3}x\\Big(\\partial_{\\mu}A_{\\nu}\\partial^{\\mu}A^{\\nu}-\\partial^{\\mu}A_{\\nu}\\partial^{\\nu}A_{\\mu}\\Big) \\nonumber \\\\\n&+\\frac{ie^{2}}{960\\pi m^{3}}\\int d^{3}x\\Big(\\partial_{\\mu}A_{\\nu}\\Box\\partial^{\\mu}A^{\\nu}-\\partial^{\\mu}A_{\\nu}\\Box\\partial^{\\nu}A_{\\mu}\\Big).\n \\label{eq:c7}\n\\end{align}\nAs we have previously mentioned, the first term of the expression \\eqref{eq:c7} corresponds to the kinetic part of the noncommutative Maxwell action, ${\\cal O}(m^{-1})$, while the second term is the higher-derivative correction to the kinetic term, of order ${\\cal O}(m^{-3})$.\nMoreover, the obtained result does not contain any noncommutativity effect, since the produced phase factors cancel for $n=2$.\nIt is worth noticing the absence of the parity odd Chern-Simons term in the scalar QED$_{3}$, which in turn is generated in the fermionic electrodynamics due to the algebraic structure of the two-dimensional  realization of $\\gamma$ matrices.\n\n\n\\subsection{One-loop $\\left\\langle A AA\\right\\rangle$ vertex}\n\nThe relevant graphs for the $\\left\\langle A AA\\right\\rangle$ part of the effective action are shown in Fig.~\\ref{oneloop2}.\nHowever, in order to determine correctly the full contribution to the effective action, it is necessary to consider all different permutations of the external bosonic lines of the given graphs.\nIt is easy to see that the diagram (a) has an additional contribution (b), corresponding to a permutation of the external photon legs, which has an equivalent structure but with a reversed momentum flow, which comes exactly from the S-matrix expansion at the order of $e^{3}$.\nWith help of the Feynman rules, we can easily write the relevant expression for the sum of the graphs (a) and (b)\n\\begin{equation}\n\\Pi^{\\mu\\nu\\rho}_{(a+b)}(p,q)=2i e^{3}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{(p+2k)^{\\mu}(2p+2k+q)^{\\nu}(p+q+2k)^{\\rho}}{[(p+k)^{2}-m^{2}][(p+q+k)^{2}-m^{2}][k^{2}-m^{2}]} \\sin\\big(\\frac{p \\wedge q}{2}\\big),\n \\label{eq:c8}\n\\end{equation}\nwhich is a planar quantity, we can see that its integrand is independent of the noncommutativity.\nThe contribution from the graph (c) also has a simple planar structure, which is given by the expression\n\\begin{equation}\n\\Pi^{\\mu\\nu\\rho}_{(c)}(p,q)= -e^{3}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{\\eta^{\\mu\\nu}(p+q+2k)^{\\rho}}{[(p+q+k)^{2}-m^{2}][k^{2}-m^{2}]} \\cos\\big(\\frac{p \\wedge q}{2}\\big).\n \\label{eq:c9}\n\\end{equation}\n\nSince the graph (c) is planar, one can perform straightforward manipulations to show that this contribution is identically zero, i.e. $\\Pi^{\\mu\\nu\\rho}_{(c)}=0$, for any value of the external momenta. Hence the full contribution for the $\\left\\langle A AA\\right\\rangle$ vertex reads\n\\begin{align}\n\\Pi^{\\mu\\nu\\rho}(p,q)=2i e^{3}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{(2k+p)^{\\mu}(2k+p+s)^{\\nu}(s+2k)^{\\rho}}{[(p+k)^{2}-m^{2}][(s+k)^{2}-m^{2}][k^{2}-m^{2}]} \\sin\\big(\\frac{p \\wedge q}{2}\\big).\n \\label{eq:c10}\n\\end{align}\n\\begin{figure}[t]\n\\vspace{-1.2cm}\n\\includegraphics[height=8\\baselineskip]{3-point.eps}\n \\centering\\caption{Relevant graphs for the induced $AAA$-term.}\n\\label{oneloop2}\n\\end{figure}\n\nThe computation of the loop integral is lengthy but straightforward using dimensional regularization, and in the low-energy limit $p^{2},q^{2}\\ll m^{2}$, we find that\n\\begin{equation}\n\\Pi^{\\mu\\nu\\rho}(p,q)=\\frac{e^{3}}{12\\pi m}\\bigg[\\big(p-q\\big)^{\\rho}\\eta^{\\mu\\nu}-\\big(2p+ q\\big)^{\\nu}\\eta^{\\mu\\rho} +\\big(p+2 q\\big)^{\\mu}\\eta^{\\nu\\rho}\\bigg] \\sin\\big(\\frac{p \\wedge  q}{2}\\big).\n \\label{eq:c11}\n\\end{equation}\nHere, we notice that the Eq.~\\eqref{eq:c11} corresponds exactly to the standard Feynman vertex of the 3-photon interaction term in the NC spacetime.\nMoreover, in the next to leading order, ${\\cal{O}}(m^{-3})$, we have the contribution from the higher-derivative terms\n\\begin{align}\n\\Pi^{\\mu\\nu\\rho}_{\\rm hd}(p,q)=-\\frac{e^3}{240\\pi m^3}\n\\bigg\\{&\\eta^{\\mu\\nu}\\Big[ p^2(2q-p)^{\\rho}+q^2(q-2p)^{\\rho}+(p.q)(q-p)^{\\rho}\\Big]\\nonumber\\\\\n+&\\eta^{\\mu\\rho}\\Big[p^2(4p+2q)^{\\nu}+q^2(3p+q)^{\\nu}+(p.q)(4p+q)^{\\nu}\\Big]\\nonumber\\\\\n-& \\eta^{\\nu\\rho}\\Big[p^2(p+3q)^{\\mu}+q^2(2p+4q)^{\\mu}+(p.q)(p+4q)^{\\mu}\\Big]\\nonumber\\\\\n+&p^{\\mu}q^{\\rho}(q-p)^{\\nu}+p^{\\rho}q^{\\mu}(q-p)^{\\nu}-p^{\\mu}\np^{\\rho}(2p+q)^{\\nu}+q^{\\mu}q^{\\rho}(p+2q)^{\\nu}\\bigg\\}\\sin\\big(\\frac{p\\wedge q}{2}\\big).\n \\label{eq:c12}\n\\end{align}\nThis expression corresponds to the higher-derivative correction to the 3-photon vertex.\n\nIt is important to observe that in the commutative limit, the graphs (a) and (b) cancel each other, so that the induced 3-photon vertex is completely removed in the scalar QED.\nWe can understand this result from the charge conjugation invariance of the scalar QED in any space-time dimension, known as Furry's theorem, that forbids the presence of an odd number of photon lines in the case of commutative theory.\nAnother important aspect from our analysis is the absence of the Chern-Simons self-coupling $\\epsilon_{\\mu\\nu\\lambda}A^{\\mu}\\star A^{\\nu}\\star A^{\\lambda}$ for the noncommutative scalar QED$_{3}$ effective action \\eqref{eq:c11}, which is only generated in the case of fermionic electrodynamics \\cite{Bufalo:2014ooa}.\n\n\n\\subsection{One-loop $\\left\\langle AA AA \\right\\rangle$ vertex}\n\nThe full contribution to the $\\left\\langle AA AA \\right\\rangle$ part is determined by considering three different types of diagrams that are depicted in Fig.~\\ref{oneloop3}. Since all of these graphs have 4 external bosonic legs, 24 different permutations for each graph must be considered in order to obtain the fully symmetrized contribution.\nHence, the full contribution can be formally written as\n\\begin{align}\n \\Gamma^{\\mu\\nu\\rho\\sigma}_{\\rm total}= \\Gamma^{\\mu\\nu\\rho\\sigma}_{(a)} +\n \\Gamma^{\\mu\\nu\\rho\\sigma}_{(b)} +\\Gamma^{\\mu\\nu\\rho\\sigma}_{(c)} =\\sum_{i=1}^{24}\\Gamma^{\\mu\\nu\\rho\\sigma}_{(a,i)}+\n  \\sum_{i=1}^{24}\\Gamma^{\\mu\\nu\\rho\\sigma}_{(b,i)}+\n   \\sum_{i=1}^{24}\\Gamma^{\\mu\\nu\\rho\\sigma}_{(c,i)}.\n    \\label{eq:c17}\n\\end{align}\nWe shall present next the explicit discussion for the first contribution of each graph, whereas the remaining graphs are obtained by a direct permutation of momenta and spacetime indices.\n\\begin{figure}[t]\n\\vspace{-1.2cm}\n\\includegraphics[height=8\\baselineskip]{4-point.eps}\n \\centering\\caption{Relevant graphs for the induced $AAAA$-term.}\n\\label{oneloop3}\n\\end{figure}\n\nThe box diagram contribution represented in graph (a) has the following expression\n\\begin{align}\n\\Pi^{\\mu\\nu\\rho\\sigma}_{(a,1)}&=e^{4}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{(p+2k)^{\\mu}(2p+q+2k)^{\\nu}(2p+2q+s+2k)^{\\rho}(p+q+s+2k)^{\\sigma}}{[(p+q+s+k)^{2} -m^{2}][(p+q+k)^{2}-m^{2}][(p+k)^{2}-m^{2}][k^{2}-m^{2}]}\ne^{\\frac{i}{2}p \\wedge q}e^{\\frac{i}{2}(p+q)\\wedge s},\n \\label{eq:c13}\n\\end{align}\nwhere we have labeled the momenta $(p,q,s,r)$ accordingly to the spacetime indices of the external legs $(\\mu,\\nu,\\rho,\\sigma)$.\nMoreover we have adopted the notation, in order to satisfy the energy-momentum conservation, where the momenta flow satisfies the relation $r = p+q+s$.\nThe remaining contributions from the other 23 box diagrams, coming from the S-matrix expansion, can easily be obtained from the equation  \\eqref{eq:c13} by considering the respective permutation.\nNext, we have the contribution from the bubble diagram represented in (b), which is given by\n\\begin{align}\n\\Pi^{\\mu\\nu\\rho\\sigma}_{(b,1)} =e^{4} \\int  \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}}{[(p+q+k)^{2}-m^{2}][k^{2}-m^{2}]}\\cos\\big(\\frac{p \\wedge q}{2}\\big)\\cos\\big(\\frac{s\\wedge (p+q)}{2}\\big).\n \\label{eq:c14}\n\\end{align}\nAt last, the triangle contribution shown in graph (c) is written as\n\\begin{align}\n\\Pi^{\\mu\\nu\\rho\\sigma}_{(c,1)} =-e^{4}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{\\eta^{\\mu\\nu}(2p+2q+s+2k)^{\\rho}(p+q+s+2k)^{\\sigma}}{[(p+q+s+k)^{2}-m^{2}][(p+q+k)^{2}-m^{2}][k^{2}-m^{2}]}\\cos\\big( \\frac{p \\wedge q}{2}\\big) e^{\\frac{i}{2} (p+q) \\wedge s}.\n \\label{eq:c15}\n\\end{align}\nThe Feynman expressions of the graphs (a), (b) and (c) show that all of them are planar, making the evaluation of the momentum integration easier by dimensional regularization.\nHence the resulting expressions from the contributions \\eqref{eq:c13} to \\eqref{eq:c15}, evaluated in the highly noncommutative limit, where $p^2,q^2,s^2 \\ll m^{2}$, are written as follows\n\\begin{align}\n\\Gamma^{\\mu\\nu\\rho\\sigma}_{(a,1)}&=\\frac{1}{4}\\times \\frac{ie^{4}}{12\\pi m}\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\n~e^{\\frac{i}{2}p\\wedge q}e^{\\frac{i}{2}r\\wedge s},\\nonumber\\\\\n\\Gamma^{\\mu\\nu\\rho\\sigma}_{(b,1)}&= \\frac{1}{2}\\times\\frac{ie^{4}}{8\\pi m}\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\n\\cos\\big(\\frac{p\\wedge q}{2}\\big)\\cos\\big(\\frac{r\\wedge s}{2}\\big),\\nonumber\\\\\n\\Gamma^{\\mu\\nu\\rho\\sigma}_{(c,1)}&=1\\times \\frac{-ie^{4}}{8\\pi m} ~\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\cos\\big(\\frac{p\\wedge q}{2}\\big)\\cos\\big(\\frac{r\\wedge s}{2}\\big),\n \\label{eq:c16}\n\\end{align}\nwhere the coefficients $\\frac{1}{4}$, $\\frac{1}{2}$ and $1$ are the symmetry factors for the graphs (a), (b) and (c) of Fig.~\\ref{oneloop3}, respectively.\nWe then apply to the results \\eqref{eq:c16} all the 24 permutations, necessary to evaluate \\eqref{eq:c17}, yielding\n\\begin{align}\n\\Gamma^{\\mu\\nu\\rho\\sigma}_{(a)}&=\\frac{ie^{4}}{6\\pi m}\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\n\\Big[\\cos\\big(12\\big)\\cos\\big(34\\big)+\\cos\\big(13\\big)\\cos\\big(24\\big)+\\cos\\big(14\\big)\\cos\\big(23\\big)\\Big],\n\\nonumber\\\\\n\\Gamma^{\\mu\\nu\\rho\\sigma}_{(b)}&=\\frac{ie^{4}}{2\\pi m}\n\\Big[\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\cos\\big(12\\big)\\cos\\big(34\\big)+\\eta^{\\mu\\rho}\n\\eta^{\\nu\\sigma}\\cos\\big(13\\big)\\cos\\big(24\\big)+\n\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\cos\\big(14\\big)\\cos\\big(23\\big)\\Big],\n\\nonumber\\\\\n\\Gamma^{\\mu\\nu\\rho\\sigma}_{(c)}&=- \\frac{ie^{4}}{\\pi m}\n\\Big[\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\cos\\big(12\\big)\\cos\\big(34\\big)+\\eta^{\\mu\\rho}\n\\eta^{\\nu\\sigma}\\cos\\big(13\\big)\\cos\\big(24\\big)+\n\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\cos\\big(14\\big)\\cos\\big(23\\big)\\Big],\n \\label{eq:c18}\n\\end{align}\nwhere we have introduced, by simplicity of the upcoming analysis, the following notation for the NC momenta product: $\\left(12 \\right) \\equiv \\left(\\frac{p\\wedge q}{2}\\right)$, $\\left(13 \\right) \\equiv \\left(\\frac{p\\wedge s}{2}\\right)$, $\\left(14 \\right) \\equiv \\left(\\frac{p\\wedge r}{2}\\right)$, $\\left(23 \\right) \\equiv \\left(\\frac{q\\wedge s}{2}\\right)$, $\\left(24 \\right) \\equiv \\left(\\frac{q\\wedge r}{2}\\right)$, and $ \\left(34\\right) \\equiv  \\left(\\frac{s\\wedge r}{2}\\right)$.\nFinally, we substitute the results \\eqref{eq:c18} into the Eq.~\\eqref{eq:c17} to obtain the total one-loop contribution expression corresponding to the photon 4-point function\n\\begin{align}\n\\Gamma^{\\mu\\nu\\rho\\sigma}_{\\rm total}=\\frac{ie^{4}}{\\pi m}\\Bigg\\{\n&\\frac{1}{6}\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big) \\Big[\\cos(12)\\cos(34)+\\cos(14)\\cos(23)+\\cos(13)\\cos(24)\\Big]\\nonumber\\\\\n-&\\frac{1}{2} \\Big[\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\cos(12)\n\\cos(34)+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}\\cos(13)\\cos(24)+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\cos(14)\\cos(23)\\Big]\\Bigg\\}.\n \\label{eq:c19}\n\\end{align}\nWe can verify whether the quartic vertex \\eqref{eq:c19} satisfy the Ward identity in scalar QED. First, we consider the commutative limit, i.e. $\\theta\\rightarrow 0$, so that the one-loop contribution \\eqref{eq:c19} is reduced to\n\\begin{equation}\n\\lim_{\\theta\\rightarrow 0}\\Gamma^{\\mu\\nu\\rho\\sigma}_{\\rm total}=\\frac{ie^{4}}{\\pi m}\n\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big) \\Big(\\frac{1}{2}+\\frac{1}{2}-1\\Big)=0,\n \\label{eq:c20}\n\\end{equation}\nshowing that the photon quartic self-coupling in the order ${\\cal{O}}(m^{-1})$ is absent in the Abelian theory; however higher order contributions could be nonvanishing, corresponding to nonlinear Euler-Heisenberg-like terms.\nMoreover, in the case of the violation of the Ward identity, a nonvanishing result for the contribution \\eqref{eq:c20} would generate a four photon interaction term in the effective action of the type\n\\begin{align}\n\\lim_{\\theta\\rightarrow 0}{\\cal{S}}_{\\rm eff}[AAAA]\\sim\\int d^{3}x~\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)A_{\\mu}(x) A_{\\nu}(x) A_{\\rho}(x)A_{\\sigma}(x),\n \\label{eq:c21}\n\\end{align}\nthat is not explicitly gauge invariant. Hence, with the result \\eqref{eq:c20} we conclude that the gauge invariance is satisfied in our analysis of ${\\cal{O}}(m^{-1})$ terms at the one-loop approximation.\n\nOn the other hand, in the noncommutative case, as it is well known, we expect to find the relevant Feynman rule corresponding to the 4-photon interaction term at this order.\nTo accomplish that, we start working separately each one of the tensor terms present in the function $\\Gamma^{\\mu\\nu\\rho\\sigma}_{\\rm total}$  Eq.~\\eqref{eq:c19}.\nWe shall illustrate the analysis for the terms proportional to $\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}$, the remaining terms can be evaluated in the same fashion.\nHence, by picking the pieces that are proportional to $\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}$ in \\eqref{eq:c19}, we have that\n\\begin{equation}\n\\mathcal{I}^{\\mu\\nu,\\rho\\sigma}\\equiv\n\\frac{ie^{4}}{6\\pi m}\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\Big[-2\\cos(12)\\cos(34)+ \\cos(14)\\cos(23)+\\cos(13)\\cos(24)\\Big]. \\label{eq:c22}\n\\end{equation}\nThe major work here consist in simplifying the trigonometric part of this function by making use of the energy-momentum conservation  $r = p+q+s$, which in the new notation is renamed as $4=1+2+3$, together with the manipulation of some trigonometric identities, e.g. $\\cos\\alpha\\cos\\beta =\\cos(\\alpha+\\beta)+\\sin\\alpha\\sin\\beta$.\nAfter some laborious but straightforward calculation, we arrive at the desired expression\n\\begin{equation}\n\\mathcal{I}^{\\mu\\nu,\\rho\\sigma}=\\frac{ie^{4}}{6\\pi m}\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\Big[\\sin(14)\\sin(23)+\\sin(13)\\sin(24)\\Big].\n \\label{eq:c24}\n\\end{equation}\nSimilarly, we can apply the same process in order to simplify the remaining terms, proportional to $\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}$ and $\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}$, so  that it yields to\n\\begin{align}\n\\mathcal{I}^{\\mu\\rho,\\nu\\sigma}&=\\frac{ie^{4}}{6\\pi m}\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}\\Big[\\sin(12)\\sin(34)-\\sin(14)\\sin(23)\\Big],\n\\nonumber\\\\\n\\mathcal{I}^{\\mu\\sigma,\\nu\\rho}&=- \\frac{ie^{4}}{6\\pi m}\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big[\\sin(12)\\sin(34)+\\sin(13)\\sin(24)\\Big].\n \\label{eq:c25}\n\\end{align}\n\nHence, by considering the results from our manipulations, Eqs.~\\eqref{eq:c24} and \\eqref{eq:c25}, we can rewrite \\eqref{eq:c19} in a convenient form as the following\n\\begin{align}\n\\Gamma^{\\mu\\nu\\rho\\sigma}_{\\rm total}=\\frac{ie^{4}}{6\\pi m}\\bigg[&\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}-\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\\sin\\big(\\frac{p\\wedge s}{2}\\big)\\sin\\big(\\frac{q\\wedge r}{2}\\big)\\nonumber\\\\\n+&\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}-\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}\\Big)\\sin\\big(\\frac{p\\wedge r}{2}\\big)\\sin\\big(\\frac{q\\wedge s}{2}\\big)\\nonumber\\\\\n+&\\Big(\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}-\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\\sin\\big(\\frac{p\\wedge q}{2}\\big)\n\\sin\\big(\\frac{s\\wedge r}{2}\\big)\\bigg],\n \\label{eq:c27}\n\\end{align}\nwhere we have reintroduced the notation in terms of the external momenta $p,q,s,r$.\nAs we can observe, the expression inside the bracket corresponds exactly to the Feynamn vertex of the 4-photon interaction within the noncommutative $U_{\\star}(1)$ gauge theory.\n\nFinally, we can gather the leading ${\\cal{O}}(m^{-1})$ contributions from the one-loop order parts related to the two, three and four-point functions, Eqs.~\\eqref{eq:c3}, \\eqref{eq:c11} and \\eqref{eq:c27}, respectively, so that we can write the complete expression of the NC Maxwell action as\n\\begin{equation}\ni{\\cal{S}}_{\\rm eff}\\bigg|_{{\\cal{O}}(m^{-1})}=-\\frac{ie^{2}}{96\\pi m}\\int~d^{3}x~F_{\\mu\\nu}\\star F^{\\mu\\nu},\n\\label{eq:c28}\n\\end{equation}\nin which the field strength tensor in the NC framework is defined as $F_{\\mu\\nu}=\\partial_{\\mu}A_{\\nu}-\\partial_{\\nu}A_{\\mu}+ie[A_{\\mu},A_{\\nu}]_{\\star}$.\nAs we have previously discussed, this action is manifestly $U_{\\star}(1)$  gauge invariant under the transformation $U=e^{ie\\lambda}_{\\star}$, where the field strength has the following transformation law $F_{\\mu\\nu}\\rightarrow U\\star F_{\\mu\\nu}\\star U^{-1}$.\n\nRegarding the higher-derivative corrections to the 4-photon vertex \\eqref{eq:c27}, corresponding to the next to leading order ${\\cal{O}}(m^{-3})$ terms, we arrive at a result involving a long expression which can be found in the Appendix \\ref{sec-appA}.\nThis ${\\cal{O}}(m^{-3})$ result can be seen as the 3D version of the Euler-Heisenberg Lagrangian.\nWe notice that in the commutative limit, the gauge invariant field strength is defined as $f_{\\mu\\nu}=\\partial_{\\mu}A_{\\nu}-\\partial_{\\nu}A_{\\mu}$, so that the commutative version of the effective action \\eqref{eq:c28} only receives contribution from the one-loop $\\langle AA\\rangle$ part, the remaining contributions are vanishing.\nThus, the commutative one-loop effective action in the presence of the higher-derivative term, at the next to leading order, is described as\n \\begin{equation}\n \\lim\\limits_{\\theta\\rightarrow 0}i{\\cal{S}}_{\\rm eff}\\bigg|_{{\\cal{O}}(m^{-3})}=\n -\\frac{ie^{2}}{96\\pi m}\\int d^{3}x~f_{\\mu\\nu}f^{\\mu\\nu}+\\frac{ie^{2}}{1920\\pi m^{3}}\\int d^{3}x~\nf_{\\mu\\nu}\\Box f^{\\mu\\nu}\n\\end{equation}\n\nSome comments about the result \\eqref{eq:c28} are now in place.\nRegarding discrete symmetries, following the aforementioned discussion in Sec. \\ref{sec2.1}, it is easy to show that the above one-loop effective action is also invariant under all of the discrete symmetries and therefore we have not faced any anomalous symmetry at this order.\n\nSince the effective action \\eqref{eq:c28}, arising from the 2, 3 and 4-point functions at the order ${\\cal{O}}(m^{-1})$, is exactly gauge invariant, it is possible to conclude that no further ${\\cal{O}}(m^{-1})$ terms are generated from higher-order graphs with $n>4$ external photon legs.\nAccording to this reasoning, we can also discuss the gauge invariance of the higher-derivative terms generated by considering the next to leading order ${\\cal{O}}(m^{-3})$ of our expansion.\nActually, it is possible to make use of a dimensional analysis, based on arguments of gauge invariance, to establish the perturbative generation of all possible gauge invariant higher-derivative terms in the one-loop effective action \\cite{Bufalo:2014ooa}.\nAs an example, if we consider all of the ${\\cal{O}}(m^{-3})$ contributions up to the diagrams with $n=6$ external photon legs, we can generate the following effective higher-derivative Lagrangian\n\\begin{equation}\n\\mathcal{L}_{\\rm hd} = \\frac{1}{6\\mu^2} \\nabla _\\mu F^{\\mu \\nu }\\star \\nabla^\\lambda F_{\\lambda \\nu }\n+\\frac{1}{6\\mu^2} \\nabla _\\lambda F^{\\mu \\nu }\\star \\nabla^\\lambda F_{\\mu \\nu }\n- \\frac{e}{18\\mu^2} F^{\\mu \\nu }\\star  F_{\\nu \\lambda}\\star  F^{\\lambda } _{\\;\\;\\;  \\mu},\n\\end{equation}\nwhere $\\nabla _\\mu  = \\partial_\\mu +i e \\left[A_\\mu, \\,\\right]_{\\star} $ is the covariant derivative in the adjoint representation and $\\mu \\sim m$ is the mass scale of the theory.\nThis expression can been seen as a noncommutative extension of the Alekseev-Arbuzov-Baikov effective Lagrangian \\cite{Quevillon:2018mfl,Alekseev:1981fu}.\n\nOne last comment about the photon effective action is in regard of some of the nonlinear contributions.\nIt is well known that either fermionic or scalar electrodynamics generate nonlinear corrections of quantum character to the photon dynamics \\cite{Quevillon:2018mfl}.\nIn the case of a $(2+1)$ spacetime the effective Euler-Heisenberg Lagrangian density has the  appearance of fractional powers of the field strength \\cite{Redlich:1983dv}\n\\begin{equation}\n\\mathcal{L}_{\\rm EH} \\sim  \\left( e  \\sqrt{B^2-E^2}  \\right)^{\\frac{3}{2}}.\n\\end{equation}\nHence, it is reasonable to expect that the coordinates noncommutativity would also present corrections to this nonlinear coupling term.\n\n \\section{Comparison with NC-QED$_{3}$}\n\\label{sec5}\n\nIn this section, we shall present a comparative discussion of the 2,3 and 4-point functions in the case of fermionic and bosonic matter fields coupled to the photon.\nIt is notable that the presence of the trace of $\\gamma$ matrices in fermionic QED$_{3}$ leads to two sectors characterized as: odd and even in regard to parity symmetry.\nThe former is related to the induced Chern-Simons (CS) terms, appearing with odd powers of the fermion mass $m_e$, while the latter sector contributes to the induced Maxwell (M) terms, with even powers of $m_e$.\nNow in the scalar framework, odd parity terms are absent, and only parity preserving terms are present in the induced effective action, which can be understood as the main difference between these two matter fields.\nThese results can be briefly described in terms of a mass expansion as the following:\n\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n{\\color{blue}Order} & {\\color{blue}Induced action} & {\\color{blue}NC-QED$_{3}$}& {\\color{blue}NC-scalar QED$_{3}$} \\\\\n\\hline\n${\\cal{O}}(m^{0})$ & ordinary~NC-CS & $\\checkmark$ & $\\times$ \\\\\n\\hline\n~${\\cal{O}}(m^{-1})$ & ordinary~NC-M~ & $\\checkmark$ & $\\checkmark$ \\\\\n\\hline\n~${\\cal{O}}(m^{-2})$& higher-derivative~NC-CS & $\\checkmark$ & $\\times$ \\\\\n\\hline\n~${\\cal{O}}(m^{-3})$& higher-derivative~NC-M~ & $\\checkmark$ & $\\checkmark$\\\\\n\\hline\n\\vdots& \\vdots& \\vdots& \\vdots\\\\\n\\hline\n~~${\\cal{O}}(m^{-2\\ell})$& higher-derivative~NC-CS & $\\checkmark$ & $\\times$\\\\\n\\hline\n~~~~~${\\cal{O}}(m^{-2\\ell-1})$& higher-derivative~NC-M~ & $\\checkmark$ & $\\checkmark$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\nHere some important comments are in order.\nIn NC-QED$_{3}$, the structure of the kinetic part, coming from the $n=2$ photon external legs analysis, has the contribution of two types of terms:\na CS-type parity violating term $e^{2}A \\partial\\Box^{\\ell} A$, at the order ${\\cal{O}}(m^{-2\\ell})$, and a M-type parity preserving term $e^{2}A\\partial\\partial \\Box^{\\ell} A$, at the order ${\\cal{O}}(m^{-2\\ell-1})$.\nWe can observe that the mass dimension of the CS and M-type terms is given by $3+2\\ell$ and $4+2\\ell$, respectively.\nThus, we can conclude that in order to have a gauge invariant CS-type action at the order ${\\cal{O}}(m^{-2\\ell})$, it is necessary to consider all contributions originating from the graphs with $n=2,3,\\ldots, 3+2\\ell$ photon legs.\nOn the other hand, to obtain a gauge invariant M-type action at the order ${\\cal{O}}(m^{-2\\ell-1})$, it is necessary to consider all contributions arising from the relevant graphs with $n=2,3,\\ldots, 4+2\\ell$ photon legs.\n\n\nFurthermore, in $(2+1)$ dimensions, the number of degrees of freedom of the charged boson and Dirac fermion (in the 2 dimensional representation) is equal and hence it is easy to see that the numerical coefficient appearing in the 2-point function  \\eqref{eq:c7} would be the same as in the fermionic QED$_{3}$.\nNow for the $n=3$ graphs, in the case of NC-QED$_{3}$, there are only the contribution of two triangle graphs in the one-loop order.\nThese contributions are the same as in the NC-scalar QED$_{3}$ because the additional graph is identically zero.\nThus, it is easy to realize that the final result in the parity preserving sector for both cases is the same \\cite{Bufalo:2014ooa}.\nThe last type of diagrams is for $n=4$ legs, that for the fermionic case we have the contribution of the box diagram only, i.e. type (a).\nThe expression for this diagram at the leading order of ${\\cal{O}}(m^{-1})$ is given by\n\\begin{equation}\n\\Gamma^{\\mu\\nu\\rho\\sigma}_{(a,1)}\\bigg|_{\\tiny\\mbox{NC-QED}}=-\\frac{ie^{4}}{3\\pi m}\\Big(\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}-2\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+ \\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\Big)e^{\\frac{i}{2}p\\wedge q}e^{\\frac{i}{2}r\\wedge s}.\n\\end{equation}\nBy considering all of the 24 permutations and some manipulations we obtain the following\n\\begin{align}\n\\Gamma^{\\mu\\nu\\rho\\sigma}_{\\rm total}\\bigg|_{\\tiny\\mbox{NC-QED}}=\\frac{4ie^4}{3\\pi m}\\bigg[ \\Big(&\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}-\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}\\Big)\\sin{\\big(\\frac{p\\wedge r}{2}\\big)}\\sin{\\big(\\frac{q\\wedge s}{2}\\Big)}\\nonumber\\\\\n+\\Big(&\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}-\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\\sin{\\big(\\frac{q\\wedge r}{2}\\big)}\\sin{\\big(\\frac{p\\wedge s}{2}\\big)}\\nonumber\\\\\n+\\Big(&\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}-\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\\sin{\\big(\\frac{p\\wedge q}{2}\\big)}\\sin{\\big(\\frac{s\\wedge r}{2}\\big)}\\bigg],\n\\end{align}\nwhich has the same tensor and momenta structure as the standard 4-photon vertex \\eqref{eq:c27}, but with a different numerical coefficient.\nIn this case, we can conclude that the commutative limit of this function is also satisfied, and have a vanishing result as we would expect.\n\\section{Final remarks}\n\\label{conc}\n\nIn this paper we have considered the perturbative evaluation of the effective action for photon in the context of scalar QED in the $(2+1)$ noncommutative spacetime.\nOur main interest was to determine to what extent the spin of the matter fields can change the effective action in the presence of the spacetime noncommutativity.\nSince the number of degrees of freedom of the charged boson and Dirac fermion (in the 2 dimensional representation) is equal, the main difference between these fields is solely to the well known presence of different couplings in the case of scalar QED, implying in new types of graphs.\nAn important drawback from the scalar QED in terms of the induced effective action for the photon is the absence of parity violating terms, showing that no Chern-Simons terms are generated when charged scalar fields are considered.\n\nThe perturbative analysis followed the computation of the $\\left\\langle AA  \\right\\rangle$, $\\left\\langle AA A \\right\\rangle$ and $\\left\\langle AA AA \\right\\rangle$ vertex functions.\nA more detailed and careful analysis was necessary to the computation of the 4 point vertex, where the process of symmetrization is rather intricate.\nThe evaluation of the noncommutative Maxwell action $\\int F_{\\mu\\nu}\\star F^{\\mu\\nu} $ was done by considering the highly noncommutative limit of these $1PI$ functions at order ${\\cal{O}}(m^{-1})$.\n\nIn addition, we have considered the generation of ${\\cal{O}}(m^{-3})$ terms, which are higher-derivative terms for the photon fields, and can be seen as the noncommutative generalization of the phenomenological Alekseev-Arbuzov-Baikov effective Lagrangian.\nAnother possible terms to be present in the gauge invariant photon's effective action are those nonlinear couplings, e.g. analogous to the Euler-Heisenberg action, thus within our study of NC-scalar QED, we would find noncommutative corrections to the fractional powers of the field strength that appear in the $(2+1)$ dimensional Euler-Heisenberg action.\n\n \\subsection*{Acknowledgements}\n\n We are grateful to M.M. Sheikh-Jabbari for fruitful discussions and comments on the manuscript. R.B. acknowledges partial support  from Conselho Nacional de Desenvolvimento Cient\\'ifico e Tecnol\\'ogico (CNPq Projects No. 304241\/2016-4 and 421886\/2018-8 ) and Funda\\c{c}\\~{a}o de Amparo \\`a Pesquisa do Estado de Minas Gerais (FAPEMIG Project No. APQ-01142-17).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAn important part in the design of non-player characters (NPCs) in video-games has to do \nwith the artificial intelligence (AI) of the characters in terms of their actions in the \ngame. This amounts to handling a variety of problems that also involve low-level aspects \nof the game, including for example pathfinding, detection of conditions, execution \nmonitoring for the actions of the characters, and many more. \n\nWhile many of these aspects have been studied extensively separately, for instance \npathfinding has been traditionally a very active research topic with significant impact on \nthe video-game industry, others such as execution monitoring are typically treated in per \ncase basis as part of the implementation of the underlying game engine or the particular\ncharacter in question. As NPCs become more of real autonomous entities in the game, \nhandling such components in a more principled way becomes very important. This is relevant\nboth from the designers and developers point of view that benefit from an architecture \nthat is easier to maintain and reuse, but also from the point of view of the more refined \ninteractions with the game-world that NPCs can achieve.\n\nIn this paper we propose an architecture for specifying the interaction of NPCs in the \ngame-world in a way that abstracts common tasks in four main conceptual components, \nnamely \\emph{perception, deliberation, control, action}. The architecture is inspired by \nAI research on autonomous agents and robots, in particular the notion of \\emph{high-level\ncontrol} for \\emph{cognitive robotics} as it is used in the context of deliberative\nagents and robots such as \\cite{levesque98highlevel,Shanahan01Highlevel}. \nThe motivation is that by adopting a clear role for each component and specifying a simple \nand clean interface between them, we can have several benefits as we discuss next. \n\nFirst, as there are many techniques for specifying how an NPC decides on the next action \nto pursue in the game world, an architecture that abstracts this part in an appropriate \nway could allow to easily switch between approaches. For example, it could facilitate \nexperimentation with a finite-state machine \\cite{Rabin02FSM}, a behavior tree\n\\cite{Isla05BehaviorTrees} or a goal-oriented action planning approach \\cite{orkin06fear} \nfor deciding on NPC actions, in a way that keeps all other parts of the architecture \nagnostic to the actual method for deliberation.\n\nSecond, this can provide the ground for a thorough investigation among the different ways\nof combining the available methodologies for different components, possibly leading \nto novel ways of using existing approaches. Also, this clear-cut separation of roles can \nencourage the development of modules that encapsulate existing approaches that have not \nbeen abstracted out of their application setting before, increasing re-usability of \ncomponents.\n\nWe also believe that this type of organization is a necessary prerequisite for enabling \nmore advanced behaviors that rely on each NPC holding a \\emph{personalized view} of the \ngame-world that is separated from the current (updated and completely specified) state \nof affairs. In particular, we argue that it is important for believable NPCs to adopt a \nhigh-level view of relevant aspects of the game-world including the topology and \nconnectivity of available areas in the world. In such cases then, when the deliberation \nis more tightly connected with low-level perception and action, we believe that a \nwell-principled AI architecture becomes important for maintaining and debugging the \ndevelopment process.\n\nFor example, typically the game engine includes a pathfinding module that is used by all\nNPCs in order to find their way in the game world. But what happens when one NPC knows \nthat one path to the target destination is blocked while another NPC does not possess this\ninformation? The approach of handling all requests with a single pathfinding module cannot \nhandle this differentiation unless the pathfinder takes into account the personalized \nknowledge of each NPC. This type of mixing perception, deliberation, and action can\nbe handled in the low-level of pathfinding using various straightforward tricks, but \nmaking a separation of the high-level knowledge for each NPC and the low-level game-world\nstate can be very useful.\n\nAdopting this separation, each NPC may keep a personalized high-level view of the \ngame-world in terms of available areas or zones in the game and their connectivity, and\nuse the low-level pathfinding module as a service only for the purpose of finding paths \nbetween areas that are \\emph{known to the NPC} to be connected or points in the same area. \nA simple coarse-grained break-down of the game-world in large areas can ensure that the \ndeliberation needed in the high-level is very simple and does not require effort that is\nat all similar to the low-level pathfinding. Alternatively, a more sophisticated approach\nwould be to model this representation based on the actual one used by a hierarchical \npathfinding approach such as \\cite{Botea04HPAstar}, so that the high-level personalized \ndeliberation is also facilitated by the pathfinding module but using the NPCs version of\nthe higher level of the map.\n\nThe rest of the paper is organized as follows. We continue  with the \ndescription of our proposed \\emph{CogBot} architecture. We then introduce a motivating \nexample that shows the need for personalized knowledge of the game-world for NPCs, and\nreport on an implementation of CogBots in the popular game engine of \nUnity.\\footnote{\\url{www.unity3d.com}} \nThen we continue with a discussion on the \nstate-of-the-art for approaches that deal with AI for NPCs wrt to actions in the \ngame-world, and finally, we conclude with our view on what are \ninteresting future directions to investigate. \n\n\\section{The CogBot NPC AI architecture}\n\nThe proposed architecture formalizes the behavior of NPCs as far as their actions in the \ngame-world is concerned, in terms of four basic components as follows. \n\n\\medskip\n\\centerline{\\includegraphics[width=0.95\\linewidth]{img\/GeneralArchitecture.png}}\n\\smallskip\n\n\\begin{itemize}\n\\item The \\emph{perception} component (\\emph{CogBotPerception}) is responsible for \nidentifying objects and features \nof the game-world in the field of view of the NPC, including conditions or events that \noccur, which can be useful for the deliberation component. \n\\item The \\emph{deliberation} component (\\emph{CogBotDeliberation}) is responsible for \ndeciding the immediate action that should be performed by the NPC by taking into account\nthe input from the perception component as well as internal representations. This component \nmay be used to abstract the logic or strategy that the NPC should follow which could be \nfor instance expressed in terms of reactive or proactive behavior following any of \nthe existing approaches.\n\\item The \\emph{control} component (\\emph{CogBotControl}) is responsible for going \nover a loop that passes information between the perception and deliberation components,\nand handling the execution of actions as they are decided. In particular, the controller is \nagnostic of the way that perception, deliberation and action is implemented, but is responsible\nfor coordinating the information between the components while handling exceptions, monitoring \nconditions and actions, and allocating resources to the deliberator accordingly.\n\\item The \\emph{action} component (\\emph{CogBotAction}) is responsible for realizing the \nconceptual actions that are decided by the deliberator in the game-world and provide \ninformation about the state of action execution, e.g., success or failure.\n\\end{itemize}\n\nEssentially, the control component acts as a mediator that distributes information between\nthe other components. Note that more than one instance of each component may be used in an\nNPC architecture. This particular specification of components should be seen as a means for\nstructuring the number of various processes that need to operate and coordinate so that an \nNPC can perform challenging tasks in a game-world environment. One may think of cases where\na single controller is used as a hub that manages many perception, deliberation, and action\ncomponents, or other cases where networks of one instance of each these four components are\nused to mange different aspects of the NPC behavior.\n\n\n\\subsection{Perception}\n\nThe perception component is the main information source for the NPC control component. In \nthe typical case it is attached to a mesh object surrounding the NPC and it provides instant\ninformation about all the game objects that lie in the area of the mesh object, e.g., a \nsight cone positioned on the head of the NPC. Also, the perception component may be \nmonitoring the field of view for conditions or events which are also propagated to the\ncontrol component.\n\nThe communication with the control component is asynchronous as the perception component\npushes information to the control component by calling appropriate callback functions as\nfollows. \n\n\\begin{itemize}\n\\item An ``Object Entering FoV'' and ``Object Leaving FoV'' callback function is \ncalled when an object enters or leaves the field of view of the NPC.\n\\item A ``Notify Object Status'' is called when the internal state of an object in the \nfield of view is changed. \n\\item A ``Notify Event'' callback function is called whenever an implemented monitoring\ncondition check is triggered.\n\\end{itemize}\n\n\\medskip\n\\centerline{\\includegraphics[width=.95\\linewidth]{img\/PerceptionControl.png}}\n\\medskip\n\nMoreover, a the perception component provides a filtering mechanism based on the type of\nobjects and conditions so that only the ones for which the NPC registers for will be reported \nand others will be ignored. In general an NPC may have more than one instance of a perception \ncomponent, each of which may have a different range and can be used to track different \nobject types. A simple example is an NPC having one perception mesh for sight and one \nfor hearing, which are both set up to communicate with the same control component.\n\nFinally, note that conditions that may be triggered in the environment are propagated as\nnotification of events. This type of information is also used in other parts of the \ncommunication between components. This is adopted as a means to provide a simple uniform\nmechanism for communication between components.\n\n\\subsection{Deliberation}\n\nThe deliberation component is the bridge between the low-level space of perception and\naction in the game-world and the high-level space of conceptual representation of the\nstate of affairs as far as the NPC is concerned. The deliberation component exposes an\ninterface for the control component to asynchronously invoke communication as follows. \n\\begin{itemize}\n\\item A ``Get Next Action'' function abstracts the decision by the deliberation component\nwith respect to the next immediate action to be performed by the NPC.\n\\item A ``Notify Object'' function notifies the deliberation component about relevant \nobjects that become visible or have changed their state.\n\\item A ``Notify Event'' function notifies the deliberation component about relevant \nconditions received by the perception component that may affect the internal \nrepresentation of the state or knowledge of the NPC.\n\\end{itemize}\n\n\\medskip\n\\centerline{\\includegraphics[width=0.95\\linewidth]{img\/DeliberatorControl.png}}\n\\medskip\n\nThe deliberation component is responsible for modeling the decision making of the \nNPC with respect to action executing in the game-world, using the information that\nis provided by the control component and other internal representations that are\nappropriate for each approach or specific implementation.\n\nIn particular, one useful view is to think of the deliberation component as maintaining\nan internal model of the game-world that is initiated and updated by the low-level input \ncoming from the perception component, and using this model along with other models of \naction-based behavior in order to specify the course of action of the NPC. The actual\nimplementation or method for the model of the game-world and the model of desired \nbehavior is not constrained in any way other than the type of input that is provided \nand the type of action descriptions that can be passed to the action component.\n\nUnder this abstraction an NPC may for example just keep track of \nsimple conditions in the game-world and a representation of its internal state in\nterms of health and inventory, and use a finite-state machine approach to specify\nthe immediate next action to be performed at any time. Similarly, but following a\ntotally different methodology, an NPC may keep track of a propositional logic\nliteral-based model of the current state of the game-world, and use a automated planning\ndecision making process for deciding on the next immediate action to be performed as\npart of a longer plan that achieves a desired goal for the NPC.\n\nObserve that in the general case this level of abstraction allows the decision making of\nthe NPC to possess information that is different that the true state of affairs in the \ngame-world. This may be either because some change happened for which the NPC was not\ninformed by the perception component, for example because the NPC was simply not able\nto perceive this information, or even because the perception component itself is \nimplemented in such way as to provide filtered or altered information, serving some \naspect of the game design and game-play.\n\n\n\\subsection{Control}\n\nAs we mentioned earlier, the control component acts as a mediator that distributes \ninformation between the other components, including notifications about the state of \nobjects, notifications about conditions in the game-world, as well as feedback about \naction execution. In order to do so it propagates a callback invocation from one \ncomponent to the other (as for example in the case of the perception and deliberation \ncomponent). The way in which this is performed depends on the particular\ncase and implementation choices. For example different types of information may be \ndelivered to different instances of the same component as we discussed earlier in \nthis section.\n\nAlso, the controller component goes over a loop that handles action execution. In its\nmost simple form this could be just repeatedly calling in sequence the corresponding\nfunction of the deliberation component that informs about the next action to be\nperformed, and then pass on this information to the action component so that the\naction is actually executed in the game-world. The architecture does not limit or \nprescribe the way that this loop should be implemented and indeed should be done in\na variable way depending on the characteristics of the game. Nonetheless, the intention\nis that under this abstraction the control component can encourage more principled \napproaches for action execution monitoring, handling exceptions, and reviving from \nerrors.\n\n\n\\subsection{Action}\n\nThe action component abstracts the actions of the NPC in the game-world, allowing the \nrest of the architecture to work at a symbolic level and the other components be\nbe agnostic as per the implementation details for each action. Note that the architecture\nview we adopt does not prescribe the level of detail that these actions should be \nabstracted to. For some cases an action could be an atomic low-level task in the game-world\nas for example the task of turning the face of the NPC toward some target, while at other\ncases a more high-level view would be appropriate, essentially structuring NPC action \nbehavior in terms of strategies or macro-actions. In the latter case, the action component\nmay be used for example to connect a conceptual high-level view of low-level \nimplementations of behaviors as it would be typically done when a finite-state machine\nis used for reactive behavior.\n\n\nIn terms of the communication of the action component with the control component, again\na very simple interface is adopted for asynchronous interaction as follows.\n\\begin{itemize}\n\\item An ``Invoke Action'' function abstracts the action execution from the point of view\nof the architecture, initiating an implemented internal function that operates in the \ngame-world.\n\\item A ``Notify Event'' callback function is called to inform the control component about\ninformation related to action execution, such as that the action has finished with success\nor that some error occurred, etc.\n\\end{itemize}\n \n\\medskip\n\\centerline{\\includegraphics[width=0.95\\linewidth]{img\/ActionControl.png}}\n\\medskip\n\nAs in the deliberation component an internal representation is assumed to model the\npersonalized view of the game-world for the NPC, in the action component a similar \nconceptual representation needs to be utilized in order to capture the available actions\nand their characteristics. As in this case the representation can be more straightforward,\nwe also assume the following simple schema for registering actions in the architecture.\nAn new action can be registered by means of calling an internal ``Register Action'' \nfunction which requires i) a string representing the action name (for instance move-to, \npick-up, etc, and ii) a reference to a function that implements the action. An appropriate\naccount for additional parameters for actions is needed but this is an implementation \ndetail.\n\n\nWe now proceed do discuss a concrete scenario that shows how the CogBot architecture and an\nappropriate separation between the low-level state of affairs in the game-world and a personalized\nconceptual view of the game-world can enable novel behaviors for NPCs.\n \n\n\\section{A motivating example}\n\nAs a motivating running example we will consider a simple case of a room-based \ngame-world in which the human player and some NPCs can move between rooms, pickup and \ndrop objects, and use some of them to block pathways or set traps. Suppose also that the \ngoal of the game is to eventually get hold of a special item and deliver it at the a \ndesignated spot. \n\nNow consider the scenario according to which the player decides to block a passage that \nworks as a shortcut to some room, by putting some obstacle in the way after he passes \nthrough. This means that the characters will have to go round using a different route in\norder to get close to the human player. This is an important piece of information that\nactually should affect greatly how the NPCs will decide to move in the game-world. \n\nFor example, at first, no NPC knows that the passage is blocked so perhaps an NPC that \nwants to go to the room in question will have to go through the normal (shortest) route \nthat goes through the shortcut, see that it is blocked, and then go round using the \nalternative route. From then on and until the NPC sees or assumes that the passage is \ncleared, this is what the NPC would be expected to do in order to act in a believable way, \nand this line of reasoning should be adopted for each one of the NPCs separately.\n\nHow would this behavior be realized in a game though? This simple scenario suggests that\nthe pathfinding module of the game-engine should somehow keep track of which obstacles \neach NPC is aware and return a different \\emph{personalized path} for each one that is \nconsistent with their knowledge. To see why simpler ways to handle this would hurt the \nbelievability of NPCs consider that the pathfinding module follows an NPC-agnostic \napproach for finding a route such that i) does not take into account object obstacles \nand ii) does take into account object obstacles. In both cases, NPCs are assumed to \nreplan if the planned route turns out to be unrealizable.\n\nIt is easy to see that both approaches are problematic. In the first case, an NPC may\ntry to go through the shortcut more than once, in a way that exposes that they has no\nway of remembering that the shortcut is blocked. In the second case, an NPC that did \nnot first try to use the shortcut will nonetheless immediately choose to follow the \nalternative route even though they did not observe that the shortcut is blocked, probably\nruining the player's plan to slow down other NPCs. \n\nEssentially, in order to maintain \nthe believability of NPCs, each one should be performing pathfinding based on the \ninformation they possess which needs to be updated accordingly from their observations.\nNote how this account of knowledge of topology sets the ground for other more advanced\nfeatures for NPCs as for example the capability of exchanging information in the sense\nthat if a player sees that NPCs gathered together and talked his little trick is no\nlonger able to slow down other NPCs any more.\n\nWe now continue to discuss an application of the CogBot architecture in an implemented\nversion of this scenario, which is able to handle the intended behavior for NPCs \nmaking use of an appropriate separation of the actual state of the game-world and\nthe personalized view of the NPCs. \n\n\\section{CogBots in GridWorld}\n\n\\emph{GridWorld} is a grid-based environment developed in the Unity game engine modeled after the \nmotivating example we introduced in the previous section. It was specifically designed to be used\nfor performing automated tests with respect to NPC action-based behavior, therefore all of the\ncomponents of the game-world are procedurally generated from an input text file that specifies\nthe topology as well as the available objects and their state. For example we can design a map \nconsisting of rooms interconnected through doors and use it to observe the behavior of one (or \nmore) NPCs navigating in the environment while the user opens or closes the doors. In particular\nwe will experiment with a prototype implementation of our proposed architecture.\n\n\n\n\\subsection{GridWorld}\n\nThe main component of GridWorld is the \\emph{map} generated from an ASCII file that is taken as\ninput. Each element in the grid including wall sections, floor tiles, and doors, is a low-level\ngame-object of Unity with a component containing some basic information about the object such as:\na flag that indicates whether the object is static or not and its type. Non-static objects, that \nis, objects that may be in more than one states, have an additional component for handle the\ninternal state representation and the way it changes by means of interactions in the game-world.\n\nA simple A* heuristic search path-finding procedure is implemented based on the grid representation \nin order to provide a low-level navigation system. Moreover, in the initialization phase a processing \nof the map takes place that decomposes the map into interconnected areas using the standard method of \nconnected-component labeling \\cite{Dillencourt92Component} from the area of image processing. The \nresulting high-level topology is represented as:\n\\begin{itemize} \n\\item A list of \\emph{areas}. An area is a fully connected set of tiles in which the movement from \nany tile to any tile is guaranteed to succeed.\n\\item A list of \\emph{way-points} between areas. In our case these way-points are explicitly\nrepresented by \\emph{doors}. Each door connects two adjacent areas and has an internal state that\ncan be open or close.\n\\item A list of \\emph{points of interest} that can be used to model the tiles in the map that are\npotential target destinations for the NPC with respect to the scenario in question.\n\\end{itemize}\n\nThis type of information will be maintained by our prototype CogBot NPC, which as we will see \nshortly, combined with the low-level pathfinding procedure can address the challenges raised by\nthe motivating example we introduced earlier.\nThe map data ground truth is stored in a global object in the root of the game scene graph. This \nobject can be accessed by any other object in the game to ask for world information such as the\nsize of the map, the object type in some $(i,j)$ position, etc.\n\n\n\\subsection{CogBots}\n\nA prototype NPC in GridWorld is implemented following the CogBot architecture described in\nthe previous section. The CogBot NPC consists of a standard game character object with a\nmesh, colliders, and animations. In addition to these the NPC has the following components:\n\n\\begin{itemize}\n\\item A \\emph{conic collider} representing the field of view of the NPC. This collider is attached \nto the main body of the NPC positioned in such way as to simulate its sight cone.\n\\item A \\emph{CogBotPerception} component attached to the conic collider. The perception component\nis implemented so as to pass information about all visible objects in the field of view of the NPC.\nIn this example, no conditions are raised by means of events by the perception component.\n\\item A \\emph{CogBotController} component that simply acts as a bridge for communication between\nperception, deliberation, and action.\n\\item A \\emph{PlayerAction} component. This component is a collection of actions of the human\nplayer\/spectator in GridWorld that instruct the NPC to perform some activity. \nIn this simple example we only have actions from the PlayerAction component that invoke\nmoving actions for the NPC in order to reach a target destination tile.\n\\item A \\emph{CogBotAction} component that implements the moving actions of the NPC.\n\\item A \\emph{ManualDeliberator} component.  This is an implementation\nof the CogBotDeliberator interface that provides the immediate action to be performed by the NPC\nbased on an internal representation and an model for activity that we will describe with an example\nnext.\n\\end{itemize}\n\n\\subsection{A simple example}\n\nConsider the following setting for the map of GridWorld.\\footnote{\nOur prototype implementation of the GridWorld test-bed and the CogBots architecture are \navailable at \\url{https:\/\/github.com\/THeK3nger\/gridworld} and\n\\url{https:\/\/github.com\/THeK3nger\/unity-cogbot}. The simple example reported here can \nbe found in the ``KBExample'' folder in the GridWorld repository.}\n\n\\medskip\n\\centerline{\\includegraphics[width=0.8\\linewidth]{img\/GridWorldShortcutMap.png}}\n\\medskip\n\nAfter the decomposition of the map we have three areas: area 1 where the NPC is located, area 2 that is\nsymmetrical to area 1 connected through a door, and area 3,  a little room that intervenes between areas \n1 and 2 and connects to both with a door. In particular we call $D_{1,2}$ the door connect areas 1 and 2, and\n$D_{1,3}$, $D_{3,2}$ the doors that connect the little room depicted as area 3 with the other two areas.\n\nIn this example the human player\/observer can instruct the NPC to move to particular tiles by clicking on\nthem. Suppose for example that the player asks the NPC to go from the initial position in area 1 into area \n2 in a symmetrical position. The actual path to follow would be different depending on which doors are open\nallowing the NPC to go through. \n\nSimilarly, assuming an internal representation for the connectivity of \n areas, and a personalized internal representation of the state of the doors, the NPC could\n\\emph{deliberate in the level of areas} about the path to take. In other words, the NPC can first build a\ncoarse-grained high-level plan of the form $\\{$``move to area X'', ``move to area Y''$\\}$, and then can use\nthe low-level pathfinding procedure to generate paths for each part of the plan. Apart from achieving a\nhierarchical form of pathfinding that could be beneficial for various reasons, this approach actually \nhandles the type of believability property that we discussed in the motivating example.\n\nAt the beginning of the simulation the internal knowledge of the NPC assumes all doors to be closed.\nIn this situation instructing the NPC to move to area 2, the deliberation component returns no plan as \naccording to the NPC's knowledge it is impossible to go there as all doors are closed.\nIf we open the doors between 1 and 2 by clicking on them, the deliberation component is still unable to\nfind a plan because the NPC was not able to get this information through the perception component and its\nfield of view. Even though the pathfinding procedure would return a valid path, this is irrelevant for the\npersonalized view of the NPC.\n\nNow assume that we move the NPC close to the doors and the internal representation is updated to reflect\nthe real world state and take it back to the starting point. If we instruct the NPC to move to area 2, the \ndeliberation component produces the straightforward plan which is executed using the pathfinding procedure. \nSimilarly, if we open the door $D_{1,3}$, the deliberate component would still return the same plan, as in\norder to get the shortest path that goes through area 3 the NPC needs to see that the door is open. \n\n\n \n\n\\section{Challenges and related work}\n\nThere is a variety of work that aims for action-based AI for NPCs.\nTraditionally, a combination of scripts and finite state machine are used in interactive games for controlling NPCs. These methods, even if fairly limited, allow the game designer to control every aspect of the NPCs actions. This approach has been employed in different types of succesfull videogames such as the Role Playing Game (RPG) \\textit{Never Winter Nights} or the First Person Shooter (FPS) \\textit{Unreal Tournament}. Scripts are written off-line in a high-level language and are used to define simple behaviors for NPCs. Procedural script generation has been proposed in \\cite{Mcnaughton04scriptease:generative} by using simple patter templates which are tuned and combined by hand. Complex behaviors can be developed in a short amount of time but many of the intricacies of the classical scripting approach are not solved and it remains difficult to manage the NPCs as the complexity of the virtual world increases.\n\nState machines are still the preferred way to control NPCs in modern games, from FPS like the \\textit{Quake} series to RTS like Blizzard's \\textit{Warcraft III}. A problem with state machines is that they allow little reusability and they must often be rebuilt for every different case \\cite{Orkin2003}. Furthermore, the number of the states grows exponentially if the behavior of the character becomes slightly more sophisticated which is problematic in the design, maintenance, and debugging of NPC behavior.\nIn Bungie's \\textit{Halo 2}, a form of hierarchical state machines or behavior trees are used \\cite{Isla05BehaviorTrees}.\nIn the FPSs from Monolith \\textit{F.E.A.R.} and Epic's \\textit{Unreal Tournament}, STRIPS and HTN planning have been used to define complex behaviors such as NPCs able to coordinate as squads and perform advanced strategies as sending for backup or flanking \\cite{orkin06fear}.\nIn Bethesda's \\textit{Elder Scrolls IV Oblivion}, NPCs are controlled by  \ngoals that involve scheduling and are given for NPCs to achieve. \nThis allows to define behaviors and tasks that depends on preconditions and scheduled times. \nNonetheless, there is much less effort on standardizing approaches for action-based\nAI in a framework that would allow better comparison or collaboration between the existing\ntechniques. To that end, our proposed architecture allows to:\n\\begin{itemize}\n\\item Try out different existing approaches and decision algorithms by abstracting them in the \nas different deliberation compoments that can be easily switched, allowing a comparative AI \nperformance analysis.\n\\item Build NPC that are able to dynamically change the underlying AI method depending on\nthe state of the game or the vicinity of the player.\nFor example a simple FSM can be used while an NPC is idle or far away from the player,\nand then switch to BT or GOAP when it must defend a location or attack the player.\n\\item Build NPC that are able to use a combination of existing decision algorithms. For example\nwe could think of a system that combines BTs for low level decisions and GOAP for high level \ntactical reasoning. The high level planner then would return a sequence of actions, each of \nwhich would invoke a corresponding BT.\n\\item Develop NPCs with a \\emph{personalized} conceptual representation of the game-world that\nenables rich and novel behaviors. The motivating example we examined is just a very simple \ncase which to the best of our knowledge has not been handled by approaches in the literature.\nMoreover this view can lead to more interesting cases involving also communication between NPC \nand their personalized knowledge.\n\\end{itemize}\n\n\n\n\n\n \n\\section{Conclusions}\n\nIn this paper we have introduced a robotics-inspired architecture for\nhandling non-player character artificial intelligence for the purposes of specifying\naction in the game world. We demonstrated that certain\nbenefits can come out of the principled approach for decomposing the artificial \nintelligence\neffort in components, in particular with respect to a separation between the \nlow-level ground truth for the \nstate of the game-world and a personalized conceptual representation of the world for each NPC.\n\nOur proposed architecture provides modularity allowing each of the four main \ncomponents of the architecture, namely, perception, deliberation, control, and\naction, to encapsulate a self-contained independent functionality through a clear\ninterface. We expect that this way of developing characters can enable better code \nreusability and speed up prototyping, testing and debugging.\nMoreover our proposed architecture provides the ground for revisiting problems and\ntechniques that have been extensively studied, such as pahtfinding, and arrive to\nfeasible methods for developing believable characters with their own view of the\ntopology and connectivity of the game-world areas.\n\n \n\n\\clearpage\n\n\\bibliographystyle{aaai}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Overview}\n\nThe concept of a $G$-\\emph{principal bundle} for a topological or\nLie group $G$ is fundamental in \nclassical topology and differential\ngeometry, e.g. \\cite{Husemoeller}. More generally, for $G$ a \\emph{geometric}\ngroup in the sense of a \\emph{sheaf of groups} over some site, the notion of \n$G$-principal bundle or \\emph{$G$-torsor} is fundamental in \\emph{topos theory}\n\\cite{Johnstone,Moerdijk}.\nIts relevance rests in the fact that \n$G$-principal bundles constitute natural geometric representatives \nof cocycles in degree 1 nonabelian cohomology\n$H^1(-,G)$  and that general fiber bundles are \n\\emph{associated} to principal bundles.\n\nIn recent years it has become clear that\nvarious applications, notably in \n``\\emph{String-geometry}'' \\cite{SSS,Schreiber},  involve a notion of principal bundles where\ngeometric groups $G$ are generalized to \\emph{geometric grouplike $A_\\infty$-spaces},\nin other words\n\\emph{geometric $\\infty$-groups}: geometric objects that are equipped with a group\nstructure up to higher coherent homotopy. The resulting \\emph{principal $\\infty$-bundles}\nshould be natural geometric representatives of geometric nonabelian \\emph{hypercohomology}:\n{\\v C}ech cohomology with coefficients in arbitrary positive degree. \n\nIn the \\emph{absence} of geometry, \nthese principal $\\infty$-bundles are essentially just the classical \\emph{simplicial} principal bundles  \nof simplicial sets \\cite{May} (this we discuss in Section 4.1 of \\cite{NSSb}). However,\nin the presence of non-trivial geometry the situation is both more subtle and richer, and plain \nsimplicial principal bundles can only serve as a specific \\emph{presentation}\nfor the general notion (section 3.7.2 of \\cite{NSSb}).\n\nFor the case of \\emph{principal 2-bundles}, \nwhich is the first step after ordinary principal bundles,\naspects of a geometric definition and theory have been proposed and developed by\nvarious authors, see section 1 of \\cite{NSSb} for references and see \n\\cite{NikolausWaldorf} for a comprehensive discussion.\nNotably the notion of a \\emph{bundle gerbe} \\cite{Mur}\nis, when regarded as an extension of a {\\v C}ech-groupoid, \nalmost manifestly that of a principal 2-bundle, \neven though this perspective is not prominent in the respective literature.\nWe discuss these relations in detail in \\cite{NSSc}.\nThe oldest definition of geometric 2-bundles is conceptually different, but closely related:\nGiraud's \\emph{$G$-gerbes} \\cite{Giraud} are by definition not principal 2-bundles but are \nfiber 2-bundles \\emph{associated} to $\\mathbf{Aut}(\\mathbf{B}G)$-principal 2-bundles,\nwhere $\\mathbf{B}G$ is the \\emph{geometric moduli stack} of $G$-principal bundles.\nThis means that $G$-gerbes provide the universal \\emph{local coefficients}, in the sense\nof \\emph{twisted cohomology}, for $G$-principal bundles.\n\nFrom the definition of principal 2-bundles\/bundle gerbes it is fairly clear that these\nought to be just the first step (or second step) in an infinite tower of higher analogs. Accordingly, \ndefinitions of \\emph{principal 3-bundles} have been considered in the literature, \nmostly in the guise of\n\\emph{bundle 2-gerbes} \\cite{Stevenson} (we discuss the relation in \\cite{NSSc}). \nThe older notion\nof Breen's \\emph{$G$-2-gerbes} \\cite{Breen} (also discussed by Brylinski-MacLaughlin),  \nis, as before, not that of a principal \n3-bundle, but that of a fiber 3-bundle which is\n\\emph{associated} to an $\\mathbf{Aut}(\\mathbf{B}G)$-principal 3-bundle, \nwhere now\n$\\mathbf{B}G$ is the \\emph{geometric moduli 2-stack} of $G$-principal 2-bundles\n(details are also in \\cite{NSSc}).\n\nGenerally, for every $n \\in \\mathbb{N}$ and every geometric $n$-group $G$, \nit is natural to consider the theory of \n$G$-principal $n$-bundles \\emph{twisted} by an $\\mathbf{Aut}(\\mathbf{B}G)$-principal\n$(n+1)$-bundle, hence by the associated \\emph{$G$-$n$-gerbe}. \nA complete theory of principal bundles therefore needs to involve \nthe notion of principal $n$-bundles \nand also that of twisted principal $n$-bundles\nin the limit as $n \\to \\infty$. \n\nAs $n$ increases, the  piecemeal \nconceptualization of principal $n$-bundles quickly becomes tedious \nand their structure opaque, without a general theory of higher geometric\nstructures. In recent years such a theory -- long conjectured and with many precursors -- \nhas materialized in a comprehensive and elegant form, now known as \\emph{$\\infty$-topos theory} \n\\cite{ToenVezzosiToposb, Rezk, Lurie}. \nWhereas an ordinary topos is a category of \\emph{sheaves} over \nsome site\\footnote{Throughout \\emph{topos} here stands for \\emph{Grothendieck topos},\nas opposed to the more general notion of \\emph{elementary topos}.}, an \n$\\infty$-topos is an \\emph{$\\infty$-category} of \\emph{$\\infty$-sheaves} or equivalently\nof \\emph{$\\infty$-stacks} over some \\emph{$\\infty$-site}, where the prefix ``$\\infty-$''\nindicates that all these notions are generalized to structures up to \n\\emph{coherent higher homotopy} (as in the older terminology of \n$A_\\infty$-, $C_\\infty$-, $E_\\infty$- and $L_\\infty$-algebras, all of which re-appear\nas algebraic structures in $\\infty$-topos theory).  \nIn as far as an ordinary topos is a context\nfor general \\emph{geometry}, an $\\infty$-topos is a context for what is called\n\\emph{higher geometry} or \\emph{derived geometry}: the pairing of the notion of\n\\emph{geometry} with that of \\emph{homotopy}.\n(Here ``derived'' alludes to \n``derived category'' and ``derived functor'' in homological algebra, \nbut refers in fact to a nonabelian generalization of these concepts.) \n\nAs a simple instance of this pairing, \none observes that\nfor any geometric abelian group (sheaf of abelian groups) \n$A$, the higher degree (sheaf) cohomology $H^{n+1}(-,A)$ in ordinary geometry\nmay equivalently be understood as the degree-1 cohomology\n$H^1(-,\\mathbf{B}^{n}A)$ in higher geometry, where \n$\\mathbf{B}^{n}A$ is the geometric $\\infty$-group obtained by\nsuccessively \\emph{delooping} $A$ geometrically. \nMore generally, there are geometric $\\infty$-groups $G$ not of this abelian form.\nThe general degree-1 geometric cohomology\n$H^1(X,G)$ is a nonabelian and simplicial generalization of \\emph{sheaf hypercohomology},\nwhose cocycles are morphisms $X \\to \\mathbf{B}G$ into the geometric delooping of $G$. \nIndeed, delooping plays a central role \nin $\\infty$-topos theory;  \na fundamental fact of $\\infty$-topos theory (recalled as \nTheorem \\ref{DeloopingTheorem} below) says that, quite generally, under internal \nlooping and delooping, $\\infty$-groups $G$ in an $\\infty$-topos $\\mathbf{H}$ are equivalent\nto connected and \\emph{pointed} objects in $\\mathbf{H}$:\n$$\n  \\xymatrix{\n    \\left\\{\n\t  \\mbox{\n\t  groups in $\\mathbf{H}$\n\t  }\n\t\\right\\}\n\t\t\\ar@{<-}@<+5pt>[rrr]^<<<<<<<<<<<<<{\\mbox{\\small  looping}\\; \\Omega}\n\t\\ar@<-5pt>[rrr]_<<<<<<<<<<<<<{\\mbox{\\small delooping}\\; \\mathbf{B}}^<<<<<<<<<<<<<\\simeq\n    &&&\n\t\\left\\{\n\t  \\mbox{\\begin{tabular}{c} pointed connected \\\\ objects in $\\mathbf{H}$ \\end{tabular}}\n\t\\right\\}\n  }\n  \\,.\n$$\nWe will see that this fact plays a key role in the theory of \nprincipal $\\infty$-bundles. \n \nTopos theory is renowned for providing a general convenient context for the development\nof geometric structures. In some sense, $\\infty$-topos theory \nprovides an even more convenient context, due to the fact that\n\\emph{$\\infty$-(co)limits} or \\emph{homotopy (co)limits} in an $\\infty$-topos\nexist, and refine or \\emph{correct} the corresponding naive (co)limits. \nThis convenience manifests itself in the central definition of \nprincipal $\\infty$-bundles (Definition~\\ref{principalbundle} below): whereas the \ntraditional definition of a $G$-principal bundle over $X$ as a quotient map\n$P \\to P\/G \\simeq X$ requires the additional clause that the quotient be\n\\emph{locally trivial}, $\\infty$-topos theory comes pre-installed with the correct homotopy quotient for\nhigher geometry, and as a result the local triviality of $P \\to P\/\\!\/G =: X$ is automatic;\nwe discuss this in more detail in Section \\ref{PrincBund-Intro} below.\nHence \nconceptually, $G$-principal $\\infty$-bundles are in fact simpler than their\ntraditional analogs, and so their theory is stronger. \n\n\n\nA central theorem of topos theory is \\emph{Giraud's theorem}, which \nintrinsically characterizes  \ntoposes as those presentable categories that satisfy three simple \nconditions: 1.\\  coproducts are disjoint, 2.\\  colimits are preserved by pullback, and\n3.\\  quotients are effective. The analog of this characterization turns out\nto remain true essentially verbatim in $\\infty$-topos theory: \nthis is the \\emph{Giraud-To{\\\"e}n-Vezzosi-Rezk-Lurie} characterization\nof $\\infty$-toposes, recalled as Definition \\ref{GiraudRezkLurieAxioms} below. \nWe will show that given an $\\infty$-topos $\\mathbf{H}$, the second\nand the third of these axioms \nlead directly to the {\\em classification theorem} for principal $\\infty$-bundles \n(Theorem~\\ref{PrincipalInfinityBundleClassification} below) which states that \nthere is an equivalence of $\\infty$-groupoids \n\\[\n\\begin{array}{cccc}\nG \\mathrm{Bund}(X)\n\t  &\n\t  \\simeq\n\t  &\n\t  \\mathbf{H}(X, \\mathbf{B}G)\n\\\\\n\\\\\n  \t\\left\\{\n\t  \\begin{array}{c}\n\t  \\mbox{$G$-principal $\\infty$-bundles}\n\t  \\\\\n\t  \\mbox{over $X$}\n\t  \\end{array}\n\t  \\raisebox{20pt}{\n\t  \\xymatrix{\n\t     P \\ar[d] \\\\ X\n\t  }}\n\t\\right\\}\n    &\\simeq&\n  \t\\left\\{\n\t  \\mbox{cocycles}\\;\\;\\;\n\t   g : X \\to \\mathbf{B}G\n\t\\right\\}\n\t  \t\\end{array}\n\\]\t\nbetween the $\\infty$-groupoid of $G$-principal $\\infty$-bundles on $X$, \nand the mapping space $\\mathbf{H}(X,\\mathbf{B}G)$.   \n\nThe mechanism underlying the proof of this theorem is summarized \nin the following diagram, which is supposed to indicate that the geometric $G$-principal $\\infty$-bundle corresponding \nto a cocycle is nothing but the\ncorresponding homotopy fiber:\n $$\n  \\raisebox{50pt}{\n  \\xymatrix@C=-7pt@R=2pt{\n    \\vdots && \\vdots\n    \\\\\n    P \\times G \\times G \n\t  \\ar[rr] \\ar@<-4pt>[ddd] \\ar@<+0pt>[ddd] \\ar@<+4pt>[ddd] \n\t  &&\n\tG \\times G\n      \\ar@<-4pt>[ddd] \\ar@<+0pt>[ddd] \\ar@<+4pt>[ddd]\t\n    \\\\\n\t\\\\\n    \\\\\n    P \\times G \\ar[rr] \\ar@<-3pt>[ddd]_{p_1} \\ar@<+3pt>[ddd]^\\rho \n\t&& G\n\t\\ar@<-3pt>[ddd]_{} \\ar@<+3pt>[ddd]\n    \\\\\n\t\\\\ && & \\mbox{\\small $G$-$\\infty$-actions}\n\t\\\\\n    P \\ar[rr] \\ar[ddd] && {*} \\ar[ddd] & \\mbox{\\small total objects}\n\t\\\\\n\t\\\\ & \\mbox{\\small $\\infty$-pullback} && \n\t\\\\\n\tX \\ar[rr]_{g } &&  \\mathbf{B}G &  \\mbox{\\small quotient objects}\n\t\\\\\n\t\\mbox{\\small \\begin{tabular}{c} $G$-principal \\\\ $\\infty$-bundle \\end{tabular}}\n\t\\ar@{}[rr]_{\\mbox{\\small cocycle}}\n\t&& \\mbox{\\small \\begin{tabular}{c} universal \\\\ $\\infty$-bundle \\end{tabular}}\n  }\n  }\n$$\nThe fact that all geometric $G$-principal $\\infty$-bundles arise this\nway, up to equivalence, is quite useful in applications, and also sheds helpful light \non various existing constructions and provides more examples; we discuss this in \\cite{NSSc}. \n\nNotably, the implication\nthat \\emph{every} geometric $\\infty$-action $\\rho : V \\times G \\to V$ \nof an $\\infty$-group $G$ on an object $V$ \nhas a classifying morphism $\\mathbf{c} : V\/\\!\/G \\to \\mathbf{B}G$,\ntightly connects the theory of associated $\\infty$-bundles with that of\nprincipal $\\infty$-bundles (Section~\\ref{StrucRepresentations} below):\nthe fiber sequence\n$$\n  \\raisebox{20pt}{\n  \\xymatrix{\n    V \\ar[r] & V\/\\!\/G \n\t\\ar[d]^{\\mathbf{c}}\n\t\\\\\n\t& \\mathbf{B}G\n  }\n  }\n$$\nis found to be the $V$-fiber $\\infty$-bundle which is \\emph{$\\rho$-associated} \nto the universal $G$-principal $\\infty$-bundle $* \\to \\mathbf{B}G$.  \nAgain, using the $\\infty$-Giraud axioms, \nan $\\infty$-pullback of $\\mathbf{c}$ along a cocycle $g_X : X \\to \\mathbf{B}G$ \nis identified with the $\\infty$-bundle $P \\times_G V$ that is \\emph{$\\rho$-associated}\nto the principal $\\infty$-bundle $P \\to X$ classified by $g_X$ \n(Proposition \\ref{RhoAssociatedToUniversalIsUniversalVBundle}) and \nevery $V$-fiber $\\infty$-bundle arises this way, associated to an \n$\\mathbf{Aut}(V)$-principal $\\infty$-bundle (Theorem \\ref{VBundleClassification}).\n\nUsing this, we may observe that the space $\\Gamma_X(P \\times_G V)$ \nof \\emph{sections} of $P \\times_G V$ is equivalently\nthe space $\\mathbf{H}_{\/\\mathbf{B}G}(g_X, \\mathbf{c})$ \nof cocycles $\\sigma : g_X \\to \\mathbf{c}$ in the slice $\\infty$-topos \n$\\mathbf{H}_{\/\\mathbf{B}G}$:\n$$\n \\begin{array}{cccc}\n  \\Gamma_X(P \\times_G V)\n\t  &\n\t  \\simeq\n\t  &\n\t  \\mathbf{H}_{\/\\mathbf{B}G}(g_X, \\mathbf{c})\n\t   \\\\\n\t  \\\\\n  \t\\left\\{\n\t  \\raisebox{20pt}{\n\t  \\xymatrix{\n\t    P \\times_G V \\ar[r] \\ar[d] & V \/\\! \/ G\n\t\t  \\ar[d]^\\rho\n\t    \\\\\n\t    X \\ar[r]_-{g_X} \n\t\t\\ar@\/^1pc\/@{-->}[u]^{\\sigma}\n\t\t& \\mathbf{B}G\n\t  }\n\t  }\n\t\\right\\}\n    &\\simeq&\n  \t\\left\\{\n\t  \\raisebox{20pt}{\n\t  \\xymatrix{\n\t    & V \/\\! \/ G\n\t\t  \\ar[d]^\\rho\n\t    \\\\\n\t    X \\ar[r]_-{g_X} \n\t\t\\ar@{-->}[ur]^{\\sigma}\n\t\t& \\mathbf{B}G\n\t  }\n\t  }\n\t\\right\\}\n\t\\end{array}\n$$\nMoreover, by the above classification theorem of $G$-principal $\\infty$-bundles, \n$g_X$ trivializes over some\ncover $\\xymatrix{U \\ar@{->>}[r] &  X}$, \nand so the universal property of the $\\infty$-pullback implies that\n\\emph{locally} a section $\\sigma$ is a $V$-valued function\n$$\n \\raisebox{20pt}{\n  \\xymatrix{\n    & V \\ar[r] & V \/\\! \/ G\n\t  \\ar[d]^{\\mathbf{c}}\n    \\\\\n    U \\ar@{->>}[r]_<<<{\\mathrm{cover}}\\ar@{-->}[ur]^{\\sigma|_U} & X \\ar[r]^{g_X} \n\t& \\mathbf{B}G\\, .\n  }\n  }\n$$\nFor $V$ an ordinary space, hence a \\emph{0-truncated} object in the $\\infty$-topos,\nthis is simply the familiar statement about sections of associated bundles. \nBut in higher geometry $V$ may more generally itself be a higher \nmoduli $\\infty$-stack, which makes the general theory of sections more\ninteresting.\nSpecifically, if $V$ is a pointed connected object, this\nmeans that it is locally a cocycle for an $\\Omega V$-principal $\\infty$-bundle, \nand so globally is a \\emph{twisted $\\Omega V$-principal $\\infty$-bundle}.\nThis identifies $\\mathbf{H}_{\/\\mathbf{B}G}(-, \\mathbf{c})$ as the \n\\emph{twisted cohomology} induced by the \\emph{local coefficient bundle $\\mathbf{c}$}\nwith \\emph{local coefficients} $V$.\nThis yields a geometric and unstable analogue \nof the \npicture of twisted cohomology discussed in \\cite{AndoBlumbergGepner}.\n\nGiven $V$, the most general \ntwisting group is the \\emph{automorphism $\\infty$-group} \n$\\mathbf{Aut}(V) \\hookrightarrow [V,V]_{\\mathbf{H}}$, \nformed in the $\\infty$-topos (Definition \\ref{InternalAutomorphismGroup}). If \n$V$ is pointed connected and hence of the form $V = \\mathbf{B}G$, this means that the\nmost general universal local coefficient bundle is\n$$\n  \\raisebox{20pt}{\n  \\xymatrix{\n    \\mathbf{B}G \\ar[r] \n\t  & \n\t (\\mathbf{B}G)\/\\!\/\\mathbf{Aut}(\\mathbf{B}G) \n\t \\ar[d]^{\\mathbf{c}_{\\mathbf{B}G}}\n\t \\\\\n\t  & \\mathbf{B}\\mathbf{Aut}(\\mathbf{B}G).\n  }}\n$$\nThe corresponding associated twisting $\\infty$-bundles are \n\\emph{$G$-$\\infty$-gerbes}: fiber $\\infty$-bundles with typical fiber \nthe moduli $\\infty$-stack $\\mathbf{B}G$. These are the\nuniversal local coefficients for twists of $G$-principal $\\infty$-bundles.\n\nWhile twisted cohomology in $\\mathbf{H}$ is hence identified simply with ordinary cohomology \nin a slice of $\\mathbf{H}$, the corresponding geometric representatives, the \n$\\infty$-bundles, do not translate to the \nslice quite as directly. The reason is that a universal local coefficient bundle\n$\\mathbf{c}$ as above is rarely a pointed connected object in the slice\n(if it is, then it is essentially trivial) and so the theory of principal \n$\\infty$-bundles does not directly apply to these coefficients. \nIn Section~\\ref{ExtensionsOfCohesiveInfinityGroups} we show that what  \ndoes translate is a notion of\n\\emph{twisted $\\infty$-bundles}, a generalization of the twisted bundles known from \ntwisted K-theory: given a section $\\sigma : g_X \\to \\mathbf{c}$ as above, \nthe following pasting diagram of $\\infty$-pullbacks\n$$\n  \\xymatrix{\n    Q \n\t\\ar[d] \\ar[r] & {*} \\ar[d]\n\t&&\n\t\\mbox{$P$-twisted $\\Omega V$-principal $\\infty$-bundle}\n    \\\\\n    P \\ar[d] \\ar[r] & V \\ar[r]  \\ar[d] & {*} \\ar[d] & \\mbox{$G$-principal $\\infty$-bundle}\n    \\\\\n    X \\ar[r]^\\sigma \\ar@\/_2pc\/[rr]_{g_X} & V\/\\!\/G \n    \\ar[r]^{\\mathbf{c}} & \\mathbf{B}G & \\mbox{section of $\\rho$-associated $V$-$\\infty$-bundle}\n  }\n$$\nnaturally identifies an $\\Omega V$-principal $\\infty$-bundle $Q$ on the total space $P$\nof the twisting $G$-principal $\\infty$-bundle, and since this is classified by \na $G$-equivariant morphism $P \\to V$ it enjoys itself a certain twisted $G$-equivariance\nwith respect to the defining $G$-action on $P$. We call such $Q \\to P$ the\n\\emph{$[g_X]$-twisted $\\Omega V$-principal bundle} classified by $\\sigma$.\nAgain, a special case of special importance is that where \n$V = \\mathbf{B}A$ is pointed connected, \nwhich identifies the universal $V$-coefficient bundle with an \n\\emph{extension of $\\infty$-groups}\n$$\n  \\raisebox{20pt}{\n  \\xymatrix{\n    \\mathbf{B}A \\ar[r] & \\mathbf{B}\\hat G\n\t\\ar[d]\n\t\\\\\n\t& \\mathbf{B}G.\n  }\n  }\n$$\nAccordingly, $P$-twisted $A$-principal $\\infty$-bundles are equivalently\n\\emph{extensions} of $P$ to $\\hat G$-principal $\\infty$-bundles.\n\nA direct generalization of the previous theorem yields the \nclassification Theorem \\ref{ClassificationOfTwistedGEquivariantBundles}, \nwhich identifies $[g_X]$-twisted $A$-principal $\\infty$-bundles with cocycles in twisted cohomology\n$$\n \\begin{array}{cccc}\n \t  A \\mathrm{Bund}^{[g_X]}(X)\n\t  &\n\t  \\simeq\n\t  &\n\t  \\mathbf{H}_{\/\\mathbf{B}G}(g_X, \\mathbf{c}_{\\mathbf{B}G})\n      \\\\\n      \\\\\n  \t\\left\\{\n\t  \\begin{array}{c}\n\t   \\mbox{$[g_X]$-twisted}\n\t   \\\\\n\t  \\mbox{$A$-principal $\\infty$-bundles}\n\t  \\\\\n\t  \\mbox{over $X$}\n\t  \\end{array}\n\t  \\raisebox{40pt}{\n\t  \\xymatrix{\n\t     Q \\ar[d] \\\\ P = (g_X)^* {*} \\ar[d] \\\\ X\n\t  }}\n\t\\right\\}\n    &\\simeq&\n  \t\\left\\{\n\t  \\mbox{twisted cocycles}\\;\\;\\;\n\t   \\sigma : g_X \\to \\mathbf{c}_{\\mathbf{B}G}\n\t\\right\\}\n\t\\end{array}\n$$\nFor instance if $\\mathbf{c}$ is the connecting homomorphism\n$$\n  \\xymatrix{\n    \\mathbf{B}\\hat G \\ar[r] & \\mathbf{B}G \\ar[d]^{\\mathbf{c}}\n\t\\\\\n\t& \\mathbf{B}^2 A\n  }\n$$\nof a central \nextension of ordinary groups $A \\to \\hat G \\to G$, then\nthe corresponding twisted $\\hat G$-bundles are those known from geometric models of\ntwisted K-theory (discussed in \\cite{NSSc}).\n\nWhen the internal \\emph{Postnikov tower} of a coefficient object is regarded as \na sequence of local coefficient bundles as above, the induced \ntwisted $\\infty$-bundles are decompositions of nonabelian principal $\\infty$-bundles\ninto ordinary principal bundles together with\nequivariant abelian hypercohomology cocycles on their total spaces. \nThis construction identifies much of equivariant cohomology theory\nas a special case of higher nonabelian cohomology. \nSpecifically, when applied to a Postnikov stage of the delooping of an\n$\\infty$-group of internal automorphisms, the corresponding \ntwisted cohomology reproduces\nthe notion of Breen \\emph{$G$-gerbes with band} (Giraud's \\emph{lien}s); \nand the corresponding \ntwisted $\\infty$-bundles are their incarnation as \nequivariant \\emph{bundle gerbes} over principal bundles. \n\n\\medskip\n\n\nThe classification statements for principal and fiber $\\infty$-bundles in this article, \nTheorems \\ref{PrincipalInfinityBundleClassification} \nand \\ref{VBundleClassification} are not surprising, they say exactly what one would\nhope for. It is however useful to see how they flow naturally from the abstract\naxioms of $\\infty$-topos theory, and to observe that they immediately imply \na series of classical as well as recent theorems as special cases, \nsee Remark \\ref{ReferencesOnClassificationOfVBundles}. \nAlso the corresponding long exact sequences in (nonabelian) cohomology,\nTheorem \\ref{LongExactSequenceInCohomology}, reproduce classical theorems, \nsee Remark \\ref{ReferencesOnLongSequences}.\nSimilarly the\ndefinition and classification of lifting of principal $\\infty$-bundles, \nTheorem \\ref{ExtensionsAndTwistedCohomology}, and of twisted principal $\\infty$-bundles \nin Theorem \\ref{ClassificationOfTwistedGEquivariantBundles} flows naturally\nfrom the $\\infty$-topos theory and yet it immediately implies various\nconstructions and results in the literature as special cases, see \nRemark \\ref{ReferencesOnLiftings} and Remark \\ref{ReferencesTwistedBundles}, respectively.\nIn particular the notion of nonabelian twisted cohomology itself is elementary\nin $\\infty$-topos theory,\nSection \\ref{TwistedCohomology}, and yet it sheds light on  a wealth of \napplications, see Remark \\ref{ReferencesOnTwisted}.\n\nThis should serve to indicate that the theory of (twisted) principal $\\infty$-bundles\nis rich and interesting.  \nThe present article is intentionally written in general abstraction only, aiming to present\nthe general theory of (twisted) principal $\\infty$-bundles as elegantly as possible, \ntrue to its roots in \nabstract higher topos theory. We believe that this serves to usefully make transparent \nthe overall picture. In the companion article \n\\cite{NSSb} we give a complementary discussion and construct \nexplicit presentations of the structures appearing here that lend themselves\nto explicit computations. \nFinally in \\cite{NSSc} we use the combination of the general abstract \nformulation and its explicit presentations to discuss a list of \ninteresting examples and applications.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Preliminaries}\n\\label{structures}\n\nThe discussion of principal $\\infty$-bundles in \nSection~\\ref{Principal infinity-bundles general abstract} \nbelow builds on \nthe concept of an \\emph{$\\infty$-topos} and on a handful\nof basic structures and notions that are present in any $\\infty$-topos,\nin particular the notion of \\emph{group objects} and of \\emph{cohomology} with coefficients\nin these group objects. The relevant theory has been\ndeveloped in \\cite{ToenVezzosiToposb, Rezk, Lurie, LurieAlgebra}. \nWhile we assume the reader to be familiar with basic ideas of this theory, \nthe purpose of this section\nis to recall the main aspects that we need, to establish our notation, and to \nhighlight some aspects of the general theory \nthat are relevant to our discussion and which have perhaps \nnot been highlighted in this way in the\nexisting literature. \n\n\\medskip\n\nFor many purposes the notion of \\emph{$\\infty$-topos} is best thought of as \na generalization of the notion of a sheaf topos --- the category \nof sheaves over some site is replaced by an \n\\emph{$\\infty$-category of $\\infty$-stacks\/$\\infty$-sheaves}\nover some $\\infty$-site (there is also supposed to be a more general notion of an \n\\emph{elementary $\\infty$-topos}, which however we do not consider here).  \nIn this context the $\\infty$-topos $\\mathrm{Gpd}_\\infty$ of $\\infty$-groupoids\nis the natural generalization of the punctual topos $\\mathrm{Set}$ to \n$\\infty$-topos theory.  A major achievement\nof \\cite{ToenVezzosiToposb}, \\cite{Rezk} and \\cite{Lurie} was to provide a more intrinsic characterization of\n$\\infty$-toposes, which generalizes the classical characterization of \nsheaf toposes (Grothendieck toposes) originally given by Giraud. \nWe will show that the theory of \nprincipal $\\infty$-bundles is naturally expressed in terms of these intrinsic\nproperties, and therefore we here take these \\emph{Giraud-To{\\\"e}n-Vezzosi-Rezk-Lurie axioms}\nto be the very definition of an $\\infty$-topos \n(\\cite{Lurie}, Theorem 6.1.0.6, the main ingredients will be recalled below):\n\\begin{definition}[$\\infty$-Giraud axioms]\n  \\label{GiraudRezkLurieAxioms}\n  An \\emph{$\\infty$-topos} is a presentable $\\infty$-category $\\mathbf{H}$ that satisfies the\n  following properties.\n  \\begin{enumerate}\n    \\item {\\bf Coproducts are disjoint.}  For every two objects \n    $A, B \\in \\mathbf{H}$, the intersection of $A$ and \n    $B$ in their coproduct is the initial object: in other words the diagram\n\t$$\n\t  \\xymatrix{\n\t    \\emptyset \\ar[r] \\ar[d] & B \\ar[d]\n\t\t\\\\\n\t\tA \\ar[r] & A \\coprod B\n\t  }\n\t$$\n\tis a pullback.\n\t\n\t\\item {\\bf Colimits are preserved by pullback.} \n\t  For all morphisms $f\\colon X\\to B$ in $\\mathbf{H}$ and all \n\t  small diagrams $A\\colon I\\to \\mathbf{H}_{\/B}$, there is an \n\t  equivalence \n\t  $$\n\t    \\varinjlim_i f^*A_i \\simeq f^*(\\varinjlim_i A_i)\n\t  $$\t  \n\t  between the pullback of the colimit and the colimit over the pullbacks of its\n\t  components.\n\t\\item \n\t  {\\bf Quotient maps are effective epimorphisms.} Every simplicial object\n\t  $A_\\bullet : \\Delta^{\\mathrm{op}} \\to \\mathbf{H}$ that satisfies the\n\t  groupoidal Segal property (Definition~\\ref{GroupoidObject}) is the {\\v C}ech nerve of its quotient projection:\n\t  $$\n\t    A_n \\simeq \n\t\tA_0 \\times_{\\varinjlim_n A_n} A_0 \\times_{\\varinjlim_n A_n} \\cdots \\times_{\\varinjlim_n A_n} A_0\n\t\t\\;\\;\\;\n\t\t\\mbox{($n$ factors)}\n\t\t\\,.\n\t  $$\n  \\end{enumerate}\n\\end{definition}\nRepeated application of the second and third axiom provides the proof of \nthe classification of principal $\\infty$-bundles, \nTheorem \\ref{PrincipalInfinityBundleClassification} and the\nuniversality of the universal associated $\\infty$-bundle, \nProposition \\ref{UniversalAssociatedBundle}.\n\nAn ordinary topos is famously characterized by the existence of a classifier object\nfor monomorphisms, the \\emph{subobject classifier}. \nWith hindsight, this statement already carries in it the seed \nof the close relation between topos theory and bundle theory, for we may think of \na monomorphism $E \\hookrightarrow X$ as being a \\emph{bundle of $(-1)$-truncated fibers}\nover $X$. The following axiomatizes the existence of arbitrary universal bundles\n\\begin{definition}\n  An \\emph{$\\infty$-topos} $\\mathbf{H}$ is a presentable $\\infty$-category with the following\n  properties.\n  \\begin{enumerate}\n\t\\item {\\bf Colimits are preserved by pullback.} \n\t\\item {\\bf There are universal $\\kappa$-small bundles.}\n\t  For every sufficiently large regular cardinal $\\kappa$, there exists \n\t  a morphism $\\widehat {\\mathrm{Obj}}_\\kappa \\to \\mathrm{Obj}_\\kappa$ in $\\mathbf{H}$, \n\t  such that for every other object $X$, pullback along morphisms $X \\to \\mathrm{Obj}$\n\t  constitutes an equivalence\n\t  $$\n\t    \\mathrm{Core}(\\mathbf{H}_{\/_{\\kappa}X})\n\t\t\\simeq\n\t\t\\mathbf{H}(X, \\mathrm{Obj}_\\kappa)\n\t  $$\n\t  between the $\\infty$-groupoid of bundles (morphisms) $E \\to X$ which are $\\kappa$-small over $X$\n\t  and the $\\infty$-groupoid of morphisms from $X$ into $\\mathrm{Obj}_\\kappa$.\t  \n  \\end{enumerate}\n  \\label{RezkCharacterization}\n\\end{definition}\nThese two characterizations of $\\infty$-toposes, Definition \\ref{GiraudRezkLurieAxioms}\nand Definition \\ref{RezkCharacterization} are equivalent;\nthis is due to Rezk and Lurie, appearing as Theorem 6.1.6.8 in \\cite{Lurie}.\nWe find that the second of these axioms \ngives the equivalence between $V$-fiber bundles and \n$\\mathbf{Aut}(V)$-principal $\\infty$-bundles\nin Proposition \\ref{VBundleIsAssociated}.\n\n\nIn addition to these axioms, a basic property of $\\infty$-toposes\n(and generally of $\\infty$-categories with pullbacks) which we will \nrepeatedly invoke, is the following.\n\\begin{proposition}[pasting law for pullbacks]\n Let $\\mathbf{H}$ be an $\\infty$-category with pullbacks.  If \n  $$\n    \\xymatrix{\n\t  A \\ar[r]\\ar[d] & B \\ar[r]\\ar[d] & C \\ar[d]\n\t  \\\\\n\t  D \\ar[r] & E \\ar[r] & F\n\t}\n  $$\n  is a diagram in $\\mathbf{H}$ such that the right \n  square is an $\\infty$-pullback, then the left square is an $\\infty$-pullback precisely\n  if the outer rectangle is.\n  \\label{PastingLawForPullbacks}\n\\end{proposition}\nNotice that here and in all of the following\n\\begin{itemize}\n  \\item all square diagrams are filled by a 2-cell, even if we do not indicate this\n    notationally;\n  \\item\n    all limits are $\\infty$-limits\/homotopy limits\n\t(hence all pullbacks are $\\infty$-pullbacks\/homotopy pullbacks), and so on;\n\\end{itemize}\nthis is the only consistent way of speaking about $\\mathbf{H}$ in generality.\nOnly in the followup article \\cite{NSSb} do we consider presentations of \n$\\mathbf{H}$ by 1-categorical data; there we will draw a careful \ndistinction between 1-categorical\nlimits and $\\infty$-categorical\/homotopy limits.\n\n\n\\subsection{Epimorphisms and monomorphisms}\n\\label{StrucEpi}\n\nIn an $\\infty$-topos there is an infinite tower of notions of epimorphisms and monomorphisms:\nthe $n$-connected and $n$-truncated morphisms for all $-2 \\leq n \\leq \\infty$\n\\cite{Rezk, Lurie}. \nThe case when $n = -1$ is the most direct generalization of the 1-categorical notion, and this\nis what we need in the following. Here we briefly recall the main definitions and properties.\n\n\\medskip\n\n\n\\begin{definition}\nLet $\\mathbf{H}$ be an $\\infty$-topos.  \nFor $X \\to Y$ any morphism in $\\mathbf{H}$, there is a\n  simplicial object $\\check{C}(X \\to Y)$ in $\\mathbf{H}$ (the {\\em \\v{C}ech \n  nerve} of $f\\colon X\\to Y$) which in degree $n$ is the \n  $(n+1)$-fold $\\infty$-fiber product of $X$ over $Y$ with itself\n  $$\n    \\check{C}(X \\to Y) : [n] \\mapsto X^{\\times^{n+1}_Y}\n  $$\n  A morphism $f : X \\to Y$ in $\\mathbf{H}$ is an \\emph{effective epimorphism}\n  if it is the colimiting cocone under its own {\\v C}ech nerve:\n  $$\n    f : X \\to \\varinjlim \\check{C}(X\\to Y)\n\t\\,.\n  $$  \n  Write $\\mathrm{Epi}(\\mathbf{H}) \\subset \\mathbf{H}^I$ for the collection of \n  effective epimorphisms.\n  \\label{EffectiveEpi}\n\\end{definition}\n\\begin{proposition}\n  A morphism $f : X \\to Y$ in the $\\infty$-topos $\\mathbf{H}$\n  is an effective epimorphism precisely if its 0-truncation\n  $\\tau_0 f : \\tau_0 X \\to \\tau_0 Y$ is an epimorphism (necessarily effective)\n  in the 1-topos $\\tau_{\\leq 0} \\mathbf{H}$.\n  \\label{EffectiveEpiIsEpiOn0Truncation}\n\\end{proposition}\nThis is Proposition 7.2.1.14 in \\cite{Lurie}.\n\\begin{proposition}\n  The classes $( \\mathrm{Epi}(\\mathbf{H}), \\mathrm{Mono}(\\mathbf{H}) )$\n  constitute an orthogonal factorization system.\n  \\label{EpiMonoFactorizationSystem}\n\\end{proposition}\nThis is Proposition 8.5 in \\cite{Rezk} and Example 5.2.8.16 in \\cite{Lurie}.\n\\begin{definition}\n  For $f : X \\to Y$ a morphism in $\\mathbf{H}$, we write its\n  epi\/mono factorization given by Proposition \\ref{EpiMonoFactorizationSystem}\n  as\n  $$\n    f : \n    \\xymatrix{\n\t  X \\ar@{->>}[r] & \\mathrm{im}(f)\\  \\ar@{^{(}->}[r] & Y \n\t}\n  $$\n  and we call $\\xymatrix{\\mathrm{im}(f)\\ \\ar@{^{(}->}[r] & Y}$ the \\emph{$\\infty$-image}\n  of $f$.\n  \\label{image}\n\\end{definition}\n\n\n\n\\subsection{Groupoids}\n\\label{StrucInftyGroupoids}\n\nIn any $\\infty$-topos $\\mathbf{H}$ we may consider groupoids \\emph{internal}\nto $\\mathbf{H}$, in the sense of internal category theory \n(as exposed for instance in the introduction of \\cite{Lurie2}). \n\nSuch a \\emph{groupoid object} \n$\\mathcal{G}$ in $\\mathbf{H}$ is an $\\mathbf{H}$-object $\\mathcal{G}_0$ ``of $\\mathcal{G}$-objects''\ntogether with an $\\mathbf{H}$-object $\\mathcal{G}_1$ ``of $\\mathcal{G}$-morphisms''\nequipped with source and target assigning morphisms $s,t : \\mathcal{G}_1 \\to \\mathcal{G}_0$,\nan identity-assigning morphism $i : \\mathcal{G}_0 \\to \\mathcal{G}_1$ and a composition\nmorphism $\\mathcal{G}_1 \\times_{\\mathcal{G}_0} \\mathcal{G}_1 \\to \\mathcal{G}_1$\nwhich together satisfy all the axioms of a groupoid (unitality, associativity, existence of\ninverses) up to coherent homotopy in $\\mathbf{H}$. One way to formalize what it\nmeans for these axioms to hold up to coherent homotopy is as follows.\n\nOne notes that\nordinary groupoids, i.e.\\  groupoid objects internal to $\\mathrm{Set}$, are \ncharacterized by the fact that their nerves are simplicial sets  \n$\\mathcal{G}_\\bullet : \\Delta^{\\mathrm{op}} \\to \\mathrm{Set}$ \nwith the property that the groupoidal Segal maps  \n\\[\n\\mathcal{G}_n\\to \\mathcal{G}_1\\times_{\\mathcal{G}_0} \n\\mathcal{G}_1\\times_{\\mathcal{G}_0} \\cdots \\times_{\\mathcal{G}_0} \n\\mathcal{G}_1 \n\\]\nare isomorphisms for all $n\\geq 2$.  This last condition \nis stated precisely in Definition~\\ref{GroupoidObject} below, \nand clearly gives a characterization of groupoids that makes sense more generally, in\nparticular it makes sense internally to higher categories: \na groupoid object in $\\mathbf{H}$ is an $\\infty$-functor \n$\\mathcal{G} : \\Delta^{\\mathrm{op}} \\to \\mathbf{H}$ such that all groupoidal \nSegal morphisms are equivalences in $\\mathbf{H}$.\nThese $\\infty$-functors $\\mathcal{G}$ form the \nobjects of an $\\infty$-category $\\mathrm{Grpd}(\\mathbf{H})$\nof groupoid objects in $\\mathbf{H}$.\n\nHere a subtlety arises that is the source of a lot of interesting structure\nin higher topos theory: the objects of $\\mathbf{H}$ are themselves\n ``structured $\\infty$-groupoids''. Indeed, there is a \nfull embedding $\\mathrm{const} : \\mathbf{H} \\hookrightarrow \\mathrm{Grpd}(\\mathbf{H})$\nthat forms constant simplicial objects and thus regards every object $X \\in \\mathbf{H}$\nas a groupoid object which, even though it has a trivial object of morphisms, already\nhas a structured $\\infty$-groupoid of objects. This embedding is in fact \nreflective, with the reflector given by forming the $\\infty$-colimit\nover a simplicial diagram, the ``geometric realization''\n$$\n  \\xymatrix{\n    \\mathbf{H}\n      \\ar@{<-}@<+1.25ex>[rr]^-{\\varinjlim}\n\t  \\ar@{^{(}->}@<-1.25ex>[rr]_-{\\mathrm{const}}^-{\\perp}\n\t  &&\n\t  \\mathrm{Grpd}(\\mathbf{H})\n  }\n  \\,.\n$$\nFor $\\mathcal{G}$ a groupoid object in $\\mathbf{H}$, the object \n$\\varinjlim \\mathcal{G}_\\bullet$ in $\\mathbf{H}$ \nmay be thought of as the \n$\\infty$-groupoid obtained by ``gluing together the object of objects of\n$\\mathcal{G}$ along the object of morphisms of $\\mathcal{G}$''. \nThis idea that groupoid objects in an $\\infty$-topos are \nlike structured $\\infty$-groupoids together with gluing information \nis formalized by the statement recalled as \nTheorem~\\ref{NaturalThirdGiraud} below, which says that groupoid objects in \n$\\mathbf{H}$ are equivalent to the \\emph{effective epimorphisms} \n$\\xymatrix{Y \\ar@{->>}[r] & X}$ in $\\mathbf{H}$, the intrinsic notion of\n\\emph{cover} (of $X$ by $Y$) in $\\mathbf{H}$. The effective epimorphism\/cover\ncorresponding to a groupoid object $\\mathcal{G}$ is the colimiting cocone\n$\\xymatrix{\\mathcal{G}_0 \\ar@{->>}[r] & \\varinjlim \\mathcal{G}_\\bullet}$.\n\n\\medskip\n\nAfter this preliminary discussion we state the \nfollowing definition of groupoid object in \nan $\\infty$-topos (this definition appears in \\cite{Lurie}\nas Definition 6.1.2.7, using Proposition 6.1.2.6).\n\n\\begin{definition}[\\cite{Lurie}, Definition 6.1.2.7] \n  \\label{GroupoidObject}\n  A \\emph{groupoid object} in an $\\infty$-topos $\\mathbf{H}$ is \n  a simplicial object\n  $$\n    \\mathcal{G} : \\Delta^{\\mathrm{op}} \\to \\mathbf{H}\n  $$\n  all of whose groupoidal Segal maps are equivalences: \n  in other words, for every\n  $n \\in \\mathbb{N}$\n  and every partition $[k] \\cup [k']  =  [n]$ into two subsets \n  such that $[k] \\cap [k'] = \\{*\\}$, the canonical diagram\n  \\[\n    \\xymatrix{\n      \\mathcal{G}_n \\ar[r] \\ar[d] & \\mathcal{G}_k \\ar[d]\n      \\\\\n      \\mathcal{G}_{k'} \\ar[r] & \\mathcal{G}_{0}\n    }\n  \\]\n  is an $\\infty$-pullback diagram.  We write\n  \\[\n    \\mathrm{Grpd}(\\mathbf{H}) \\subset \\mathrm{Func}(\\Delta^{\\mathrm{op}}, \\mathbf{H})\n  \\]\n  for the full subcategory of the $\\infty$-category of simplicial\n  objects in $\\mathbf{H}$ on the groupoid objects. \n\\end{definition}\nThe following example is fundamental. In fact the third $\\infty$-Giraud axiom\nsays that up to equivalence, all groupoid objects are of this form. \n\\begin{example}\n  For $X \\to Y$ any morphism in $\\mathbf{H}$, the \\v{C}ech \n  nerve $\\check{C}(X\\to Y)$ of $X\\to Y$ (Definition~\\ref{EffectiveEpi}) is a\n  groupoid object $\\check{C}(Y \\to X)$.  This example appears in \\cite{Lurie} as Proposition 6.1.2.11.\n\\end{example}\n \n\nThe following statement refines\nthe third $\\infty$-Giraud axiom, Definition \\ref{GiraudRezkLurieAxioms}.\n\\begin{theorem}\n  \\label{NaturalThirdGiraud}\n   There is a natural equivalence of $\\infty$-categories\n  $$\n    \\mathrm{Grpd}(\\mathbf{H})\n     \\simeq\n    (\\mathbf{H}^{\\Delta[1]})_{\\mathrm{eff}}\n    \\,,\n  $$ \n  where $(\\mathbf{H}^{\\Delta[1]})_{\\mathrm{eff}}$ \n  is the full sub-$\\infty$-category of the \n  arrow category $\\mathbf{H}^{\\Delta[1]}$ \n  of $\\mathbf{H}$ on the effective epimorphisms, Definition \\ref{EffectiveEpi}.\n\\end{theorem}\nThis appears below Corollary 6.2.3.5 in \\cite{Lurie}.\n\n\n\\subsection{Groups}\n\\label{StrucInftyGroups}\n\nEvery $\\infty$-topos \ncomes with a notion of \\emph{$\\infty$-group objects} that generalize both the\nordinary notion of group objects in a topos as well as that of\ngrouplike $A_\\infty$-spaces in $\\mathrm{Top} \\simeq \\mathrm{Grpd}_{\\infty}$.\n\n\\medskip\n\nThroughout the following, let $\\mathbf{H}$ be an $\\infty$-topos.\nAn explicit definition of group objects in $\\mathbf{H}$ is the following \n(this appears as Definition 5.1.3.2 together with\nRemark 5.1.3.3 in \\cite{LurieAlgebra}).  \n\\begin{definition}[Lurie \\cite{LurieAlgebra}] \n\\label{inftygroupinootopos}\n  An \\emph{$\\infty$-group} in $\\mathbf{H}$\n  is an $A_\\infty$-algebra $G$ in $\\mathbf{H}$ such that\n  the sheaf of connected components\n  $\\pi_0(G)$ is a group object in $\\tau_{\\leq 0} \\mathbf{H}$. \n  Write\n  $\\mathrm{Grp}(\\mathbf{H})$ for the $\\infty$-category\n  of $\\infty$-groups in $\\mathbf{H}$.\n\\end{definition}\n\nWe will mostly conceive group objects in $\\mathbf{H}$ as loop space\nobjects of connected objects.\n\\begin{definition}\n   Write \n  \\begin{itemize}\n    \\item $\\mathbf{H}^{*\/}$ for the\n  $\\infty$-category of pointed objects in $\\mathbf{H}$;\n    \\item  $\\mathbf{H}_{\\geq 1}$  \n    for the full sub-$\\infty$-category of $\\mathbf{H}$ on the \n    connected objects;\n    \\item\n      $\\mathbf{H}^{*\/}_{\\geq 1}$ for the full sub-$\\infty$-category\n      of the pointed objects on the connected objects.\n   \\end{itemize}\n\\end{definition}\n\\begin{definition}\n  \\label{loop space object}\n  Write \n  $$\n    \\Omega : \\mathbf{H}^{*\/} \\to \\mathbf{H}\n  $$\n  for the $\\infty$-functor that sends a pointed object $* \\to X$\n  to its \\emph{loop space object}, i.e.\\  the $\\infty$-pullback\n  $$\n   \\raisebox{20pt}{\n   \\xymatrix{\n      \\Omega X \\ar[r]\\ar[d] & {*} \\ar[d]\n      \\\\\n      {*} \\ar[r] & X\\,.\n   }\n   }\n  $$\n\\end{definition}\n\\begin{theorem}[Lurie]\n  \\label{DeloopingTheorem}\n  \\label{delooping}\n  Every loop space object canonically has the structure of an \n  $\\infty$-group, and this construction extends to an \n  $\\infty$-functor\n  $$\n    \\Omega : \\mathbf{H}^{*\/} \\to \\mathrm{Grp}(\\mathbf{H})\n    \\,.\n  $$\n  This $\\infty$-functor constitutes part of an equivalence of $\\infty$-categories\n  $$\n    (\\Omega \\dashv \\mathbf{B})\n    :\n    \\xymatrix{\n      \\mathrm{Grp}(\\mathbf{H})\n       \\ar@{<-}@<+5pt>[r]^<<<<<<{\\Omega}\n       \\ar@<-5pt>[r]_<<<<<<{\\mathbf{B}}^<<<<<<\\simeq\n      &\n      \\mathbf{H}^{*\/}_{\\geq 1}\\, .\n    }  \n  $$  \n\\end{theorem}\nThis is Lemma 7.2.2.1 in \\cite{Lurie}. \n(See also Theorem 5.1.3.6 of \\cite{LurieAlgebra} \nwhere this is the equivalence denoted $\\phi_0$ in the proof.) For \n$\\mathbf{H} = \\mathrm{Grpd}_{\\infty} \\simeq \\mathrm{Top}$ this \nreduces to various classical\ntheorems in homotopy theory, for instance the \nconstruction of classifying spaces (Kan and Milnor) and de-looping theorems (May and Segal).\n\\begin{definition}\nWe call the inverse \n$\\mathbf{B} : \\mathrm{Grp}(\\mathbf{H}) \\to \\mathbf{H}^{*\/}_{\\geq 1}$\nin Theorem~\\ref{DeloopingTheorem} above the \n\\emph{delooping} functor of $\\mathbf{H}$. By convenient abuse\nof notation we write $\\mathbf{B}$ also for the composite\n$\\mathbf{B} : \\mathrm{Grpd}(\\mathbf{H}) \\to \\mathbf{H}^{*\/}_{\\geq 1} \n\\to \\mathbf{H}$ with the functor that forgets the basepoint and the\nconnectivity.\n\\end{definition}\n\\begin{remark}\n  Even if the connected objects involved admit an essentially\n  unique point,\n  the homotopy type of the full hom-$\\infty$-groupoid \n  $\\mathbf{H}^{*\/}(\\mathbf{B}G, \\mathbf{B}H)$ \n  of pointed objects in general differs\n  from the hom $\\infty$-groupoid $\\mathbf{H}(\\mathbf{B}G, \\mathbf{B}H)$\n  of the underlying unpointed objects. \n  For instance let $\\mathbf{H} := \\mathrm{Grpd}_{\\infty}$ and let $G$ be \n  an ordinary group, regarded as a group object in $\\mathrm{Grpd}_{\\infty}$.\n  Then $\\mathbf{H}^{*\/}(\\mathbf{B}G, \\mathbf{B}G) \\simeq \\mathrm{Aut}(G)$\n  is the ordinary automorphism group of $G$, but\n  $\\mathbf{H}(\\mathbf{B}G, \\mathbf{B}G) = \\mathrm{Aut}(\\mathbf{B}G)$ is the \n  automorphism 2-group of $G$, we discuss this \n  further around Example~\\ref{automorphism2GroupAbstractly} below.\n\\end{remark}\n\\begin{proposition}[Lurie]\n  \\label{InfinityGroupObjectsAsGroupoidObjects}\n  $\\infty$-groups $G$ in $\\mathbf{H}$ are equivalently \n  those groupoid objects $\\mathcal{G}$ in $\\mathbf{H}$ (Definition~\\ref{GroupoidObject}) \n  for which $\\mathcal{G}_0 \\simeq *$. \n\\end{proposition}\nThis is the statement of the compound equivalence\n$\\phi_3\\phi_2\\phi_1$ in the proof of Theorem 5.1.3.6 in \n\\cite{LurieAlgebra}.\n\\begin{remark}\n  \\label{PointIntoBGIsEffectiveEpimorphism}\n  \\label{Cech nerve of * -> BG}\n  This means that for $G$ an $\\infty$-group object, \n  the {\\v C}ech nerve extension of its delooping fiber\n  sequence $G \\to * \\to \\mathbf{B}G$ is the simplicial \n  object\n  $$\n    \\xymatrix{\n       \\cdots\n       \\ar@<+6pt>[r] \\ar@<+2pt>[r] \\ar@<-2pt>[r] \\ar@<-6pt>[r]\n       &\n       G \\times G\n       \\ar@<+4pt>[r] \n       \\ar[r] \n        \\ar@<-4pt>[r] \n         & \n         G \n         \\ar@<+2pt>[r] \\ar@<-2pt>[r] \n       & \n       {*}\n       \\ar@{->>}[r]\n       &\n       \\mathbf{B}G\n    }\n  $$\n  that exhibits $G$ as a groupoid object over $*$.\n  In particular it means that for $G$ an $\\infty$-group, the\n  essentially unique morphism $* \\to \\mathbf{B}G$\n  is an effective epimorphism.\n  \\end{remark}\n\n\n\n\n\\subsection{Cohomology}\n\\label{StrucCohomology}\n\\label{section.Cohomology}\n\nThere is an intrinsic notion of \\emph{cohomology} \nin every $\\infty$-topos $\\mathbf{H}$: it is simply given by the \nconnected components of mapping spaces. Of course such mapping spaces\nexist in every $\\infty$-category, but we need some extra conditions on \n$\\mathbf{H}$ in order for them to behave like cohomology sets. For\ninstance, if $\\mathbf{H}$ has pullbacks then there is a notion of\nlong exact sequences in cohomology. \nOur main theorem (Theorem~\\ref{PrincipalInfinityBundleClassification} below) will show that the second\nand third $\\infty$-Giraud axioms imply that this intrinsic\nnotion of cohomology has the property that it \\emph{classifies} \ncertain geometric structures in the $\\infty$-topos.\n\n\\begin{definition}\n \\label{cohomology}\nFor $X,A \\in \\mathbf{H}$ two objects, we say that\n$$\n  H^0(X,A) := \\pi_0 \\mathbf{H}(X,A)\n$$\nis the \\emph{cohomology set}\\index{cohomology!general abstract} of $X$ with coefficients in $A$. \nIn particular if $G$ is an  $\\infty$-group we write\n$$\n  H^1(X,G) := H^0(X,\\mathbf{B}G) = \\pi_0 \\mathbf{H}(X, \\mathbf{B}G)\n$$\nfor cohomology with coefficients in the delooping \n$\\mathbf{B}G$ of $G$. \nGenerally, if  $K \\in \\mathbf{H}$ has a $n$-fold delooping \n$\\mathbf{B}^nK$ for some non-negative integer $n$, we write\n$$\n  H^n(X,K) := H^0(X,\\mathbf{B}^n K) = \\pi_0 \\mathbf{H}(X, \\mathbf{B}^n K)\n  \\,.\n$$\n\\end{definition}\nIn the context of cohomology on $X$ wth coefficients in $A$ we say that\n\\begin{itemize}\n\\item\n the hom-space $\\mathbf{H}(X,A)$ is the \\emph{cocycle $\\infty$-groupoid}\\index{\n!cocycle};\n\\item\n  an object $g : X \\to A$ in $\\mathbf{H}(X,A)$ is a \\emph{cocycle};\n\\item \n  a morphism: $g \\Rightarrow h$ in $\\mathbf{H}(X,A)$ is a \\emph{coboundary} between cocycles.\n\\item \n  a morphism $c : A \\to B$ in $\\mathbf{H}$ \n  represents the \\emph{characteristic class}\\index{characteristic class!general abstract}\n  $$\n    [c] : H^0(-,A) \\to H^0(-,B)\n    \\,.\n  $$\n\\end{itemize}\nIf $X\\simeq Y\/\\!\/G$ is a homotopy quotient, then the cohomology of $X$ is \nequivariant cohomology of $Y$. Similarly, for general $X$ this notion of cohomology\nincorporates various local notions of equivariance.\n\\begin{remark}\n \\label{CohomologyOverX}\nOf special interest is the cohomology defined by a slice  \n$\\infty$-topos \n$$\n  \\mathcal{X} := \\mathbf{H}_{\/X}\n$$ \nover some $X \\in \\mathbf{H}$.\nSuch a slice is canonically equipped with the \n{\\'e}tale geometric morphism (\\cite{Lurie}, Remark 6.3.5.10)\n$$\n  (X_! \\dashv X^* \\dashv X_*)\n  :\n  \\xymatrix{\n     \\mathbf{H}_{\/X}\n\t \\ar@<1.3ex>[rr]^-{X_!}\n\t \\ar@{<-}[rr]|-{X^*}\n\t \\ar@<-1.3ex>[rr]_-{X_*}\n\t &&\n\t \\mathbf{H}\n  }  \n  \\,,\n$$\nwhere $X_!$ simply forgets the morphism to $X$ and \nwhere $X^* = X \\times (-)$ forms the product with $X$. \nAccordingly $X^* (*_{\\mathbf{H}}) \\simeq *_{\\mathcal{X}} =: X$ \nand $X_! (*_{\\mathcal{X}}) = X \\in \\mathbf{H}$. Therefore \ncohomology over $X$ with coefficients of the form $X^* A$ is \nequivalently the cohomology in $\\mathbf{H}$ of $X$ with coefficients in $A$:\n$$\n  \\mathcal{X}(X, X^* A) \\simeq \\mathbf{H}(X,A)\n  \\,.\n$$ \nBut for a general coefficient object $A \\in \\mathcal{X}$ the \n$A$-cohomology over $X$ in $\\mathcal{X}$ is a \n\\emph{twisted} cohomology of $X$ in $\\mathbf{H}$.\nThis we discuss below in Section~\\ref{TwistedCohomology}.\n\\end{remark}\nTypically one thinks of a morphism $A \\to B$ in \n$\\mathbf{H}$ as presenting a \\emph{characteristic class} of $A$ if\n$B$ is ``simpler'' than $A$, notably if $B$ is an Eilenberg-MacLane object $B = \\mathbf{B}^n K$ for\n$K$ a 0-truncated abelian group in $\\mathbf{H}$. In this case the characteristic class may\nbe regarded as being in the degree-$n$ $K$-cohomology of $A$\n$$\n  [c] \\in H^n(A,K)\n  \\,.\n$$\n\n\\begin{definition}\n  For $f : Y \\to Z$ any morphism in $\\mathbf{H}$\n  and $z : * \\to Z$ a point, the \\emph{$\\infty$-fiber} or\n  \\emph{homotopy fiber} of $f$ over this point is the $\\infty$-pullback \n  $ X := {*} \\times_Z Y$\n  $$\n    \\raisebox{20pt}{\n    \\xymatrix{\n        X \\ar[r] \\ar[d]& {*} \\ar[d]\n        \\\\\n        Y \\ar[r]^f & Z\\,.\n    }\n\t}\n  $$\n\\end{definition}\n\\begin{observation}\n  Let $f\\colon Y\\to Z$ in $\\mathbf{H}$ \n  be as above.  Suppose that $Y$ is pointed and $f$ is a morphism of pointed objects.\n  Then the $\\infty$-fiber of an $\\infty$-fiber is the loop object of the base.\n\\end{observation}\nThis means that we have a diagram\n  $$\n    \\xymatrix{\n        \\Omega Z  \\ar[d] \\ar[r] & X \\ar[r] \\ar[d]& {*} \\ar[d]\n        \\\\\n        {*} \\ar[r] & Y \\ar[r]^f & Z\n    }\n  $$\nwhere the outer rectangle is an $\\infty$-pullback if the left square is an \n$\\infty$-pullback. This follows from the pasting law, Proposition~\\ref{PastingLawForPullbacks}.\n\\begin{definition}\n \\label{fiber sequence}\n \\label{LongFiberSequence}\nFor every morphism $c : \\mathbf{B}G \\to \\mathbf{B}H \\in \\mathbf{H}$ define the \n\\emph{long fiber sequence to the left}\n$$\n   \\cdots \n   \\to \n   \\Omega G\n  \\to \n  \\Omega H \n   \\to \n   F\n    \\to \n   G \n     \\to  \n   H \n     \\to \n    \\mathbf{B} F \n     \\to \n    \\mathbf{B}G\n      \\stackrel{c}{\\to} \n   \\mathbf{B}H\n$$\nby the consecutive pasting diagrams of $\\infty$-pullbacks\n$$\n  \\xymatrix{\n      & \\ar@{..}[d] & \\ar@{..}[d]\n      \\\\\n     \\ar@{..}[r] &  F \\ar[d]\\ar[r]& G \\ar[r] \\ar[d] & {*} \\ar[d]\n      \\\\\n      & {*} \\ar[r] & H \\ar[r] \\ar[d] & \\mathbf{B}F \\ar[r] \\ar[d] & {*} \\ar[d]\n      \\\\\n      & & {*} \\ar[r] & \\mathbf{B}G \\ar[r]^c & \\mathbf{B}H\n  }\n$$\n\\end{definition}\n\\begin{theorem}\n\\begin{enumerate}\n\\item The long fiber sequence to the left of \n$c : \\mathbf{B}G \\to \\mathbf{B}H$ becomes constant on the point after $n$ iterations if $H$ is $n$-truncated. \n\\item For every object $X \\in \\mathbf{H}$ we have a long exact sequence of pointed cohomology sets\n  $$  \n    \\cdots \\to H^0(X,G) \\to H^0(X,H) \\to H^1(X,F) \\to H^1(X,G) \\to H^1(X,H)\n   \\,.\n  $$\n\\end{enumerate}\n \\label{LongExactSequenceInCohomology}\n\\end{theorem}\n\\proof\nThe first statement follows from the observation that a \nloop space object $\\Omega_x A$ is a fiber of the free loop space object $\\mathcal{L} A$\nand that this may equivalently be computed by the \n$\\infty$-powering $A^{S^1}$, where $S^1 \\in \\mathrm{Top} \\simeq \\mathrm{Grpd}_{\\infty}$ is the circle. \n\nThe second statement follows by observing that the \n$\\infty$-hom-functor \n$$\n  \\mathbf{H}(X,-) : \\mathbf{H} \\to \\mathrm{Grpd}_\\infty\n$$\npreserves all $\\infty$-limits, so that we have $\\infty$-pullbacks in $\\mathrm{Grpd}_{\\infty}$ of the form \n$$ \n  \\xymatrix{\n    \\mathbf{H}(X,F) \\ar[r] \\ar[d] & {*} \\ar[d]\n    \\\\\n    \\mathbf{H}(X,G) \\ar[r] & \\mathbf{H}(X,H)\n  }\n$$\nat each stage of the fiber sequence. \nThe statement then follows from the familiar long exact sequence for homotopy groups \nin $\\mathrm{Top} \\simeq \\mathrm{Grpd}_{\\infty}$.\n\\hfill{$\\square$}\\\\\n\\begin{remark}\n  For the special case that $G$ is a 1-truncated $\\infty$-group (or\n  \\emph{2-group}) Theorem \\ref{LongExactSequenceInCohomology} is \n  a classical result due to \\cite{BreenBitorseurs}. The first and\n  only nontrivial\n  stage of the internal Postnikov tower \n  $$\n    \\xymatrix{\n\t  \\mathbf{B}^2 A \\ar[r] & \\mathbf{B}G \\ar[d]\n\t  \\\\\n\t  & \\mathbf{B} H\n\t}\n  $$  \n  of the delooped 2-group (with $H := \\tau_0 G\\in \\tau_{\\leq 0} \\mathrm{Grp}(\\mathbf{H})$ \n  an ordinary group object and $A := \\pi_1 G \\in \\tau_{\\leq 0} \\mathrm{Grp}(\\mathbf{H})$\n  an ordinary abelian group object) yields the long exact sequence of pointed cohomology\n  sets\n  $$\n    0 \\to H^1(-,A) \\to H^0(-,G) \\to H^0(-,H) \\to H^2(-,A) \\to H^1(-,G) \\to \n\tH^1(-,H) \\to H^3(-,A)\n  $$\n  (see also \\cite{NikolausWaldorf2}.)\n  Notably, the last morphism gives the obstructions against lifting traditional\n  nonabelian cohomology $H^1(-,H)$ to nonabelian cohomology $H^1(-,G)$ with values\n  in the 2-group. This we discuss further in Section \\ref{ExtensionsOfCohesiveInfinityGroups}.\n  \\label{ReferencesOnLongSequences}\n\\end{remark}\n\nGenerally, to every cocycle $g : X \\to \\mathbf{B}G$ is \ncanonically associated its $\\infty$-fiber \n$P \\to X$ in $\\mathbf{H}$, the $\\infty$-pullback\n$$\n  \\raisebox{20pt}{\n  \\xymatrix{\n    P \\ar[r] \\ar[d]& {*} \\ar[d]\n    \\\\\n    X \\ar[r] ^g & \\mathbf{B}G\n    \\,.\n  }\n  }\n$$\nWe discuss now that each such $P$ canonically has the structure of a \n\\emph{$G$-principal $\\infty$-bundle} and that $\\mathbf{B}G$ is the \n\\emph{fine moduli object} (the \\emph{moduli $\\infty$-stack}) for \n$G$-principal $\\infty$-bundles.\n\n\n\n\\section{Principal bundles}\n\\label{Principal infinity-bundles general abstract}\n\\label{PrincipalInfBundle}\n\n\nWe define here $G$-principal $\\infty$-bundles in any $\\infty$-topos\n$\\mathbf{H}$, discuss their basic properties and show that they are classified\nby the intrinsic $G$-cohomology in $\\mathbf{H}$, as discussed in \nDefinition~\\ref{cohomology}.\n\n\\subsection{Introduction and survey}\n\\label{PrincBund-Intro}\n\nLet $G$ be a topological group, or Lie group or \nsome similar such object. The traditional \ndefinition of \\emph{$G$-principal bundle} is the following:  \nthere is a map  \n$$\n  P \\to X := P\/G\n$$ \nwhich is the quotient projection\ninduced by a \\emph{free} action \n$$\n  \\rho : P \\times G \\to P\n$$ \nof $G$ on a space (or manifold, depending on context) $P$,\nsuch that there is a cover $U \\to X$ over which the quotient projection is isomorphic\nto the trivial one $U \\times G \\to U$.\n \nIn higher geometry, if $G$ is a topological or smooth \n$\\infty$-group, the quotient projection must be \nreplaced by the $\\infty$-quotient (homotopy quotient) \nprojection \n\\[\nP\\to X := P\/\\!\/ G\n\\]\nfor the action of $G$ on a topological or smooth $\\infty$-groupoid \n(or $\\infty$-stack) $P$.  It is a remarkable fact that this \nsingle condition on the map $P\\to X$ \nalready implies that $G$ acts freely on $P$ and that $P\\to X$ \nis locally trivial, when the latter notions are understood in the \ncontext of higher geometry.  We will therefore define \na $G$-principal $\\infty$-bundle to be such a map $P\\to X$.  \n\n   \n\nAs motivation for this, notice that if a Lie group $G$ acts properly, \nbut not freely, then the quotient $P \\to X := P\/G$ differs from the homotopy quotient.\nSpecifically, if precisely the subgroup $G_{\\mathrm{stab}} \\hookrightarrow G$ acts trivially, then  \nthe homotopy quotient is \ninstead the \\emph{quotient stack} $X \/\\!\/ G_{\\mathrm{stab}}$ \n(sometimes written $[X\/\\!\/G_{\\mathrm{stab}}]$, \nwhich is an orbifold if $G_{\\mathrm{stab}}$ is finite). The ordinary \nquotient coincides with the homotopy quotient if and only if the stabilizer subgroup\n$G_{\\mathrm{stab}}$ is trivial, and hence if and only if the action of $G$ is free.\n\nConversely this means that in the context of higher geometry a non-free action\nmay also be principal: with respect not to a base space, but with respect to a base groupoid\/stack.\nIn the example just discussed, we have that the projection $P \\to X\/\\!\/ G_{\\mathrm{stab}}$\nexhibits $P$ as a $G$-principal bundle over the action groupoid \n$P \/\\!\/ G \\simeq X\/\\!\/ G_{\\mathrm{stab}}$. For instance if $P = V$ is a\nvector space equipped with a $G$-representation, then $V \\to V\/\\!\/ G$ is a\n$G$-principal bundle over a groupoid\/stack.\nIn other words, the traditional requirement of freeness in a principal action is not so much\na characterization of principality as such, as rather a condition that ensures that the\nbase of a principal action is a 0-truncated object in higher geometry.\n\nBeyond this specific class of 0-truncated examples, this means that we have the following \nnoteworthy general statement: in higher geometry \\emph{every} $\\infty$-action \nis principal with respect to\n\\emph{some} base, namely with respect to its $\\infty$-quotient. In this sense\nthe notion of principal bundles is (even) more fundamental to higher geometry than\nit is to ordinary geometry. \nAlso, several constructions in ordinary geometry that are\ntraditionally thought of as conceptually different from the notion of principality\nturn out to be special cases of principality in higher geometry. For instance\na central extension of groups $A\\to \\hat G \\to G$ turns out to be \nequivalently a higher principal bundle, namely a $\\mathbf{B}A$-principal 2-bundle\nof moduli stacks $\\mathbf{B}\\hat G \\to \\mathbf{B}G$. Following this\nthrough, one finds that the topics \nof principal $\\infty$-bundles, of $\\infty$-group extensions (\\ref{ExtensionsOfCohesiveInfinityGroups}), \nof $\\infty$-representations (\\ref{StrucRepresentations}), \nand of $\\infty$-group cohomology\nare all different aspects of just one single concept in higher geometry.\n\nMore is true: in the context of an $\\infty$-topos \nevery $\\infty$-quotient projection of an $\\infty$-group action \nis locally trivial, with respect to\nthe canonical intrinsic notion of cover, hence of locality. Therefore\nalso the condition of local triviality in the classical definition of principality\nbecomes automatic. This is a direct consequence of the third\n$\\infty$-Giraud axiom, Definition~\\ref{GiraudRezkLurieAxioms}\nthat ``all $\\infty$-quotients are effective''.\nThis means that the projection map $P \\to P \/\\!\/ G$ is always a cover\n(an \\emph{effective epimorphism}) and so, since every $G$-principal $\\infty$-bundle\ntrivializes over itself, it exhibits a local trivialization of itself; \neven without explicitly requiring it to be locally trivial.\n\nAs before, this means that the local triviality clause appearing in the\ntraditional definition of principal bundles is not so much a characteristic of\nprincipality as such, as rather a condition that ensures that a given quotient \ntaken in a category of geometric spaces coincides with the ``correct'' quotient\nobtained when regarding the situation in the ambient $\\infty$-topos. \n\nAnother direct consequence of the $\\infty$-Giraud axioms\nis the equivalence of the definition of principal bundles as quotient maps, which \nwe discussed so far, with the other main definition of principality: the condition \nthat the ``shear map'' $ (\\mathrm{id}, \\rho) :  P \\times G \\to P \\times_X P$ is an equivalence. \nIt is immediate to verify in traditional 1-categorical contexts that this is \nequivalent to the action being properly free and exhibiting $X$ as its quotient\n(we discuss this in detail in \\cite{NSSc}). \nSimple as this is, one may observe, in view of the above discussion, \nthat the shear map being an equivalence is much more fundamental even: notice\nthat $P \\times G$ is the first stage of the \\emph{action groupoid object}\n$P\/\\!\/G$, and that $P \\times_X P$ is the first stage of the \\emph{{\\v C}ech nerve groupoid object}\n$\\check{C}(P \\to X)$ of the corresponding quotient map. Accordingly, the shear map equivalence\nis the first stage in the equivalence of groupoid objects in the $\\infty$-topos\n$$\n  P \/\\!\/G \\simeq \\check{C}(P \\to X)\n  \\,.\n$$\nThis equivalence is just the explicit statement of the fact mentioned before: the groupoid object\n$P\/\\!\/G$ is effective -- as is any groupoid object in an $\\infty$-topos -- and, equivalently,\nits principal $\\infty$-bundle map $P \\to X$ is an effective epimorphism.\n\nFairly directly from this fact, finally, springs the classification theorem of \nprincipal $\\infty$-bundles. For we have a canonical morphism of groupoid objects\n$P \/\\!\/G \\to * \/ \\!\/G$ induced by the terminal map $P \\to *$. By the $\\infty$-Giraud\ntheorem the $\\infty$-colimit over this sequence of morphisms of groupoid objects\nis a $G$-cocycle on $X$ (Definition~\\ref{cohomology}) canonically induced by $P$:\n$$\n  \\varinjlim  \\left(\\check{C}(P \\to X)_\\bullet \\simeq (P \/\\!\/G)_\\bullet \\to (* \/\\!\/G)_\\bullet \\right) \n    = \n  (X \\to \\mathbf{B}G) \n  \\;\\;\\;\n  \\in \\mathbf{H}(X, \\mathbf{B}G)\n  \\,.\n$$\nConversely, from any such $G$-cocycle one finds that one obtains a $G$-principal \n$\\infty$-bundle simply by forming its $\\infty$-fiber: the $\\infty$-pullback of\nthe point inclusion ${*} \\to \\mathbf{B}G$. We show in \\cite{NSSb} that in presentations\nof the $\\infty$-topos theory by 1-categorical tools, the computation of this homotopy\nfiber is \\emph{presented} by the ordinary pullback of a big resolution of the point, \nwhich turns out to be nothing but the universal $G$-principal bundle. \nThis appearance of the universal $\\infty$-bundle as just \na resolution of the point inclusion may be understood in light of the above discussion \nas follows.   \nThe classical characterization of the \nuniversal $G$-principal bundle $\\mathbf{E}G$ is as a space that is homotopy equivalent\nto the point and equipped with a \\emph{free} $G$-action. But by the above, freeness of the\naction is an artefact of 0-truncation and not a characteristic of principality in higher\ngeometry. Accordingly, in higher geometry the universal $G$-principal $\\infty$-bundle\nfor any $\\infty$-group $G$ may be taken to \\emph{be} the point, equipped with the \ntrivial (maximally non-free) $G$-action. As such, it is a bundle not over the \nclassifying \\emph{space} $B G$ of $G$, but over the full moduli $\\infty$-stack $\\mathbf{B}G$.\n\nThis way we have natural assignments of $G$-principal $\\infty$-bundles to \ncocycles in $G$-nonabelian cohomology, and vice versa. We find (see \nTheorem~\\ref{PrincipalInfinityBundleClassification} below) that\nprecisely the second $\\infty$-Giraud axiom of Definition~\\ref{GiraudRezkLurieAxioms}, \nnamely the fact that in an $\\infty$-topos $\\infty$-colimits are preserved by\n$\\infty$-pullback, \nimplies that these \nconstructions constitute an equivalence of $\\infty$-groupoids, hence \nthat $G$-principal $\\infty$-bundles are classified by $G$-cohomology.\n\nThe following table summarizes the relation between\n$\\infty$-bundle theory and the $\\infty$-Giraud axioms as indicated above, and as \nproven in the following section.\n\n\\medskip\n\\begin{center}\n\\begin{tabular}{c|c}\n {\\bf $\\infty$-Giraud axioms} & {\\bf principal $\\infty$-bundle theory}\n \\\\\n \\hline\n \\hline\n quotients are effective & \n   \\begin{tabular}{c} \\\\ every $\\infty$-quotient $P \\to X := P\/\\!\/ G$ \\\\is principal \\\\ \\ \\end{tabular}\n \\\\\n \\hline\n  colimits are preserved by pullback & \n  \\begin{tabular}{c}\\\\ $G$-principal $\\infty$-bundles \\\\ are classified by $\\mathbf{H}(X,\\mathbf{B}G)$\\\\\n   \\end{tabular}\n\\end{tabular}\n\\end{center}\n\n\\subsection{Definition and classification}\n\\label{DefintionAndClassification}\n\n\n\\begin{definition} \n  \\label{ActionInPrincipal}\n  For $G \\in \\mathrm{Grp}(\\mathbf{H})$ a group object,\n  we say a \\emph{$G$-action} on an object $P \\in \\mathbf{H}$\n  is a groupoid object $P\/\\!\/G$ (Definition~\\ref{GroupoidObject}) of the form\n  $$\n    \\xymatrix{\n       \\cdots\n       \\ar@<+6pt>[r] \\ar@<+2pt>[r] \\ar@<-2pt>[r] \\ar@<-6pt>[r]\n       &\n       P \\times G \\times G\n       \\ar@<+4pt>[r] \n       \\ar[r] \n        \\ar@<-4pt>[r] \n         & \n         P \\times G \n         \\ar@<+2pt>[r]^<<<<<{\\rho := d_0 } \\ar@<-2pt>[r]_{d_1} \n       & P\n    }\n  $$\n  such that $d_1 : P \\times G \\to P$ is the projection, and such that \n  the degreewise projections \n  $P \\times G^n \\to G^n $ constitute a morphism of groupoid\n  objects\n  $$\n    \\xymatrix{\n       \\cdots\n       \\ar@<+6pt>[r] \\ar@<+2pt>[r] \\ar@<-2pt>[r] \\ar@<-6pt>[r]\n       &\n       P \\times G \\times G\n       \\ar[d]\n       \\ar@<+4pt>[r] \n       \\ar[r] \n        \\ar@<-4pt>[r] \n         & \n         P \\times G \n         \\ar[d]\n         \\ar@<+2pt>[r] \\ar@<-2pt>[r] \n       & P\n       \\ar[d]\n       \\\\\n       \\cdots\n       \\ar@<+6pt>[r] \\ar@<+2pt>[r] \\ar@<-2pt>[r] \\ar@<-6pt>[r]\n       &\n       G \\times G\n       \\ar@<+4pt>[r] \n       \\ar[r] \n        \\ar@<-4pt>[r] \n         & \n         G \n         \\ar@<+2pt>[r] \\ar@<-2pt>[r] \n       & {*}       \n    }\n  $$\nwhere the lower simplicial object exhibits $G$ as a groupoid \nobject over $\\ast$ (see Remark~\\ref{Cech nerve of * -> BG}). \n  \n  With convenient abuse of notation we also write\n  $$ \n    P\/\\!\/G := \\varinjlim (P \\times G^{\\times^\\bullet})\\;\\; \n    \\in \\mathbf{H}\n  $$\n  for the corresponding $\\infty$-colimit object, the \\emph{$\\infty$-quotient} of this \n  action.   \n  \n  Write\n  $$\n    G \\mathrm{Action}(\\mathbf{H}) \\hookrightarrow \\mathrm{Grpd}(\\mathbf{H})_{\/({*}\/\\!\/G)}\n  $$\n  for the full sub-$\\infty$-category of groupoid objects over $*\/\\!\/G$\n  on those that are $G$-actions. \n\\end{definition}\n\\begin{remark}\n  \\label{ActionMapEncodedInGBundle}\n  The remaining face map $d_0$\n  $$\n    \\rho := d_0 : P \\times G \\to P\n  $$\n  is the action itself.\n\\end{remark}\n\\begin{remark}\n  Using this notation in Proposition~\\ref{InfinityGroupObjectsAsGroupoidObjects}\n  we have\n  $$\n    \\mathbf{B}G \\simeq *\/\\!\/G\n    \\,.\n  $$\n\\end{remark}\nWe list examples of $\\infty$-actions below as Example \\ref{ExamplesOfActions}.\nThis is most conveniently done after astablishing the theory of \nprincipal $\\infty$-actions, to which we now turn.\n\n\\begin{definition}\n  \\label{principalbundle}\n  Let $G \\in \\infty \\mathrm{Grp}(\\mathbf{H})$ be an \n  $\\infty$-group and let $X$ be an object of $\\mathbf{H}$.   \n  A {\\em $G$-principal $\\infty$-bundle} over $X$ \n  (or \\emph{$G$-torsor over $X$}) \n  is \n  \\begin{enumerate}\n    \\item a morphism $P \\to X$ in $\\mathbf{H}$;\n\t\\item together with a $G$-action on $P$;\n  \\end{enumerate}  \n  such that $P \\to X$ is the colimiting cocone exhibiting the quotient map\n  $X \\simeq P\/\\!\/G$ (Definition \\ref{ActionInPrincipal}).\n    \n  A \\emph{morphism} of $G$-principal $\\infty$-bundles over $X$ is a morphism of $G$-actions \n  that fixes $X$; the $\\infty$-category of $G$-principal $\\infty$-bundles over $X$\n  is the homotopy fiber of $\\infty$-categories\n  $$\n    G \\mathrm{Bund}(X) := G \\mathrm{Action}(\\mathbf{H}) \\times_{\\mathbf{H}} \\{X\\}\n  $$\n  over $X$ of the quotient map\n  $$\n    \\xymatrix{\n      G \\mathrm{Action}(\\mathbf{H}) \n\t  \\ar@{^{(}->}[r] & \\mathrm{Grpd}(\\mathbf{H})_{\/(*\/\\!\/G)} \n\t  \\ar[r] &\n\t  \\mathrm{Grpd}(\\mathbf{H}) \n\t  \\ar[r]^-{\\varinjlim}\n\t  &\n\t  \\mathbf{H}\n\t}\n\t\\,.\n  $$\n\\end{definition}\n\\begin{remark}\n  \\label{GBundlesAreEffectiveEpimorphisms}\n  By the third $\\infty$-Giraud axiom (Definition~\\ref{GiraudRezkLurieAxioms})\n  this means in particular that a \n  $G$-principal $\\infty$-bundle $P \\to X$ is an \n  effective epimorphism in $\\mathbf{H}$.\n\\end{remark}\n\\begin{remark}\n  Even though $G \\mathrm{Bund}(X)$ is by definition a priori an $\\infty$-category,\n  Proposition \\ref{MorphismsOfInfinityBundlesAreEquivalences} below says\n  that in fact it happens to be $\\infty$-groupoid: all\n  its morphisms are invertible.\n\\end{remark}\n\\begin{proposition}\n  \\label{PrincipalityCondition}\n  A $G$-principal $\\infty$-bundle $P \\to X$ satisfies the\n  \\emph{principality condition}: the canonical morphism\n  $$\n    (\\rho, p_1)\n\t:\n    \\xymatrix{\n\t  P \\times G \n\t  \\ar[r]^{\\simeq}\n\t  &\n\t  P \\times_X P\n\t}\n  $$\n  is an equivalence, where $\\rho$ is the $G$-action.\n\\end{proposition}\n\\proof\n  By the third $\\infty$-Giraud axiom (Definition~\\ref{GiraudRezkLurieAxioms}) the groupoid object\n  $P\/\\!\/G$ is effective, which means that it is equivalent\n  to the   {\\v C}ech nerve of $P \\to X$. In first degree this implies\n  a canonical equivalence $P \\times G \\to P \\times_X P$. Since\n  the two face maps $d_0, d_1 : P \\times_X P \\to P$ in the\n  {\\v C}ech nerve are simply the projections out of the fiber product, \n  it follows that the two components of this canonical equivalence\n  are the two face maps $d_0, d_1 : P \\times G \\to P$ of $P\/\\!\/G$.\n  By definition, these are the projection onto the first factor\n  and the action itself.   \n\\hfill{$\\square$}\\\\\n\\begin{proposition}\n  \\index{principal $\\infty$-bundle!construction from cocycle}\n  \\label{BundleStructureOnInfinityFiber}\n  \\label{PrincipalBundleAsHomotopyFiber}\n  For $g : X \\to \\mathbf{B}G$ any morphism, its homotopy fiber\n  $P \\to X$ canonically carries the structure of a \n  $G$-principal $\\infty$-bundle over $X$.\n\\end{proposition}\n\\proof\n  That $P \\to X$ is the fiber of $g : X \\to \\mathbf{B}G$\n  means that we have an $\\infty$-pullback diagram\n  $$\n    \\raisebox{20pt}{\n    \\xymatrix{\n      P \\ar[d]\\ar[r] & {*}\\ar[d]\n      \\\\\n      X \\ar[r]^g & \\mathbf{B}G.\n    }\n\t}\n  $$\n  By the pasting law for $\\infty$-pullbacks, Proposition~\\ref{PastingLawForPullbacks},\n  this induces a compound diagram\n  $$\n    \\xymatrix{\n       \\cdots\n       \\ar@<+6pt>[r] \\ar@<+2pt>[r] \\ar@<-2pt>[r] \\ar@<-6pt>[r]\n       &\n       P \\times G \\times G\n       \\ar[d]\n       \\ar@<+4pt>[r] \n       \\ar[r] \n        \\ar@<-4pt>[r] \n         & \n         P \\times G \n         \\ar[d]\n         \\ar@<+2pt>[r] \\ar@<-2pt>[r] \n       & \n       P\n       \\ar[d]\n       \\ar@{->>}[r]\n       &\n       X\n       \\ar[d]^g\n       \\\\\n       \\cdots\n       \\ar@<+6pt>[r] \\ar@<+2pt>[r] \\ar@<-2pt>[r] \\ar@<-6pt>[r]\n       &\n       G \\times G\n       \\ar@<+4pt>[r] \n       \\ar[r] \n        \\ar@<-4pt>[r] \n         & \n         G \n         \\ar@<+2pt>[r] \\ar@<-2pt>[r] \n       & \n       {*}       \n       \\ar@{->>}[r]\n       &\n       \\mathbf{B}G\n    }\n  $$\n  where each square and each composite rectangle is an \n  $\\infty$-pullback. \n  This exhibits the $G$-action on $P$. \n  Since $* \\to \\mathbf{B}G$\n  is an effective epimorphism, so is its $\\infty$-pullback\n  $P \\to X$. Since, by the $\\infty$-Giraud theorem, $\\infty$-colimits are preserved\n  by $\\infty$-pullbacks we have that $P \\to X$ exhibits the \n  $\\infty$-colimit $X \\simeq P\/\\!\/G$.  \n\\hfill{$\\square$}\\\\\n\\begin{lemma}\n  For $P \\to X$ a $G$-principal $\\infty$-bundle\n  obtained as in Proposition~\\ref{BundleStructureOnInfinityFiber}, and for\n  $x : * \\to X$ any point of $X$, we have \n  a canonical equivalence\n  $$\n    \\xymatrix{\n\t    x^* P \\ar[r]^{\\simeq} & G\n\t}\n  $$\n  between the fiber $x^*P$ and the $\\infty$-group object $G$.\n\\end{lemma}\n\\proof\n  This follows from the pasting law for $\\infty$-pullbacks, which \n  gives the diagram\n  $$\n    \\xymatrix{\n      G \\ar[d] \\ar[r] & P \\ar[d]\\ar[r] & {*} \\ar[d]\n      \\\\\n      {*} \\ar[r]^x & X \\ar[r]^g & \\mathbf{B}G\n    }\n  $$\n  in which both squares as well as the total rectangle are\n  $\\infty$-pullbacks.\n\\hfill{$\\square$}\\\\\n\\begin{definition}\n  \\label{TrivialGBundle}\n  The \\emph{trivial} $G$-principal $\\infty$-bundle $(P \\to X) \\simeq (X \\times G \\to X)$\n  is, up to equivalence, the one obtained via Proposition~\\ref{BundleStructureOnInfinityFiber}\n  from the morphism $X \\to * \\to \\mathbf{B}G$. \n\\end{definition}\n\\begin{observation}\n  \\label{PullbackOfInfinityTorsors}\n  For $P \\to X$ a $G$-principal $\\infty$-bundle and $Y \\to X$ any morphism, the\n  $\\infty$-pullback $Y \\times_X P$ naturally inherits the structure of \n  a $G$-principal $\\infty$-bundle.\n\\end{observation}\n\\proof\n  This uses the same kind of argument as in Proposition~\\ref{BundleStructureOnInfinityFiber}\n  (which is the special case of the pullback of what we will see is the\n  universal $G$-principal $\\infty$-bundle $*\\to \\mathbf{B}G$ below in \n  Proposition~\\ref{LocalTrivialityImpliesCocycle}).\n\\hfill{$\\square$}\\\\\n\\begin{definition}\n  \\label{GInfinityBundleAndMorphism}\n  \\label{GPrincipalInfinityBundle}\n  A $G$-principal $\\infty$-bundle $P \\to X$ is called\n  \\emph{locally trivial}\n  if there exists an effective epimorphism $\\xymatrix{U \\ar@{->>}[r] & X}$\n  and an equivalence of $G$-principal $\\infty$-bundles\n  $$\n    U \\times_X P \\simeq U \\times G\n  $$\n  from the pullback of $P$ (Observation~\\ref{PullbackOfInfinityTorsors})\n  to the trivial $G$-principal $\\infty$-bundle over $U$ (Definition~\\ref{TrivialGBundle}).\n\\end{definition}\n\\begin{proposition}\n  \\label{EveryGBundleIsLocallyTrivial}\n  Every $G$-principal $\\infty$-bundle is locally trivial.\n\\end{proposition}\n\\proof\n  For $P \\to X$ a $G$-principal $\\infty$-bundle, it is, by \n  Remark~\\ref{GBundlesAreEffectiveEpimorphisms}, itself an effective\n  epimorphism. The pullback of the $G$-bundle to its \n  own total space along this morphism is trivial,\n   by the principality condition\n  (Proposition~\\ref{PrincipalityCondition}). Hence setting $U := P$ \n  proves the claim.\n\\hfill{$\\square$}\\\\\n\\begin{remark}\n  This means that every $G$-principal $\\infty$-bundle is in particular a\n  $G$-fiber $\\infty$-bundle (in the evident sense of Definition~\\ref{FiberBundle} below).\n  But not every $G$-fiber bundle is $G$-principal, since the local trivialization\n  of a fiber bundle need not respect the $G$-action. \n\\end{remark}\n\\begin{proposition}\n  \\label{LocalTrivialityImpliesCocycle}\n  For every $G$-principal $\\infty$-bundle $P \\to X$ the square\n  $$\n    \\xymatrix{\n       & P \\ar[d] \\ar[r] & {*} \\ar[d]\n       \\\\\n       X \\ar@{}[r]|<<<\\simeq\n       & \\varinjlim_n (P \\times G^{\\times_n})\n       \\ar[r]\n       &\n       \\varinjlim_n G^{\\times_n}\n       \\ar@{}[r]|\\simeq\n       &\n       \\mathbf{B}G\n    }\n  $$\n  is an $\\infty$-pullback diagram.\n\\end{proposition}\n\\proof\n  Let $U \\to X$ be an effective epimorphism\n  such that $P \\to X$ pulled back to $U$ becomes the trivial $G$-principal\n  $\\infty$-bundle. By Proposition~\\ref{EveryGBundleIsLocallyTrivial} this exists.\n  By definition of morphism of $G$-actions and \n  by functoriality of the $\\infty$-colimit, this induces \n  a morphism in ${\\mathbf{H}^{\\Delta[1]}}_{\/(* \\to \\mathbf{B}G)}$ \n  corresponding to the diagram\n  $$\n    \\raisebox{20pt}{\n    \\xymatrix{\n\t  U \\times G \\ar@{->>}[r] \\ar@{->>}[d] & P \\ar[r] \\ar@{->>}[d] & {*} \\ar@{->>}[d]^{\\mathrm{pt}}\n\t  \\\\\n\t  U \\ar@{->>}[r] & X \\ar[r] & \\mathbf{B}G\n\t}\n\t}\n\t\\;\\;\n\t\\simeq\n\t\\;\\;\n    \\raisebox{20pt}{\n    \\xymatrix{\n\t  U \\times G \\ar@{->>}[rr] \\ar@{->>}[d] & & {*} \\ar@{->>}[d]^{\\mathrm{pt}}\n\t  \\\\\n\t  U \\ar[r] & {*} \\ar[r]^{\\mathrm{pt}} & \\mathbf{B}G\n\t}\n\t}\n  $$\n  in $\\mathbf{H}$.  \n  By assumption, in this diagram the outer rectangles and the square on the very left \n  are $\\infty$-pullbacks. We need to show that\n  the right square on the left is also an $\\infty$-pullback.\n  \n  Since $U \\to X$ is an effective epimorphism by assumption, and since these are\n  stable under $\\infty$-pullback, $U \\times G \\to P$ is also an effective epimorphism,\n  as indicated. This means that \n  $$\n    P \\simeq {\\varinjlim_n}\\, (U \\times G)^{\\times^{n+1}_P}\n\t\\,.\n  $$\n  We claim that for all $n \\in \\mathbb{N}$ the fiber products in the colimit on the right\n  are naturally equivalent to $(U^{\\times^{n+1}_X}) \\times G$. For $n = 0$ this is\n  clearly true. Assume then by induction that \n  it holds for some $n \\in \\mathbb{N}$. Then\n  with the pasting law (Proposition~\\ref{PastingLawForPullbacks}) we find  an \n  $\\infty$-pullback diagram of the form\n  $$\n    \\raisebox{30pt}{\n    \\xymatrix{\n\t  (U^{\\times^{n+1}_X}) \\times G \t  \n\t  \\ar@{}[r]|\\simeq\n\t  & \n\t  (U \\times G)^{\\times^{n+1}_P}\n\t  \\ar[r] \\ar[d] \n\t  &  \n\t  (U \\times G)^{\\times^n_P}\n      \\ar[d]\n\t  \\ar@{}[r]|{\\simeq}\n\t  &\n\t  (U^{\\times^{n}_X}) \\times G \n\t  \\\\\n\t  & U \\times G \\ar[r] \\ar[d] & P \\ar[d]\n\t  \\\\\n\t  & U \\ar[r] & X.\n\t}\n\t}\n  $$\n  This completes the induction.\n  With this the above expression for $P$ becomes\n  $$\n    \\begin{aligned}\n      P & \\simeq {\\varinjlim_n}\\,  (U^{\\times^{n+1}_X}) \\times G\n\t   \\\\\n\t    & \\simeq {\\varinjlim_n}  \\,\\mathrm{pt}^* \\, (U^{\\times^{n+1}_X}) \n\t   \\\\\n\t    & \\simeq \\mathrm{pt}^* \\, {\\varinjlim_n}\\,    (U^{\\times^{n+1}_X}) \t  \n\t  \\\\\n\t    & \\simeq \\mathrm{pt}^* \\, X,\n\t\\end{aligned}\n  $$\n  where we have used that by the second $\\infty$-Giraud axiom (Definition~\\ref{GiraudRezkLurieAxioms})\n  we may take the $\\infty$-pullback out of the $\\infty$-colimit and \n  where in the last step we \n  used again the assumption that $U \\to X$ is an effective epimorphism.\n\\hfill{$\\square$}\\\\\n\\begin{example}\n  The fiber sequence \n  $$\n    \\xymatrix{\n\t  G \\ar[r] & {*} \\ar[d]\n\t  \\\\\n\t  & \\mathbf{B}G\n\t}\n  $$\n  which exhibits the delooping $\\mathbf{B}G$ of $G$ according to \n  Theorem \\ref{DeloopingTheorem} is a $G$-principal $\\infty$-bundle\n  over $\\mathbf{B}G$, with \\emph{trivial} $G$-action on its total space\n  $*$. Proposition \\ref{LocalTrivialityImpliesCocycle} says that this is \n  the \\emph{universal $G$-principal $\\infty$-bundle} in that every\n  other one arises as an $\\infty$-pullback of this one. \n  In particular, $\\mathbf{B}G$ is a classifying object for $G$-principal\n  $\\infty$-bundles. \n  \n  Below in Theorem \\ref{ClassificationOfTwistedGEquivariantBundles} \n  this relation is strengthened:\n  every \\emph{automorphism} of a $G$-principal $\\infty$-bundle, and in fact\n  its full automorphism $\\infty$-group arises from pullback of the above\n  universal $G$-principal $\\infty$-bundle: $\\mathbf{B}G$ is the fine\n  \\emph{moduli $\\infty$-stack} of $G$-principal $\\infty$-bundles.\n  \n  The traditional definition of universal $G$-principal bundles in terms of  \n  contractible objects equipped with a free $G$-action has no intrinsic\n  meaning in higher topos theory. Instead this appears in \n  \\emph{presentations} of the general theory in model categories\n  (or categories of fibrant objects)\n  as \\emph{fibrant representatives}\n  $\\mathbf{E}G \\to \\mathbf{B}G$ of the above point inclusion.\n  This we discuss in \\cite{NSSb}.  \n  \\label{UniversalPrincipal}\n\\end{example}\nThe main classification Theorem \\ref{PrincipalInfinityBundleClassification} below\nimplies in particular that every morphism in $G\\mathrm{Bund}(X)$ is an equivalence.\nFor emphasis we note how this also follows directly:\n\\begin{lemma}\n  \\label{EquivalencesAreDetectedOverEffectiveEpimorphisms}\n  Let $\\mathbf{H}$ be an $\\infty$-topos and let $X$ be an \n  object of $\\mathbf{H}$.  A morphism \n  $f\\colon A\\to B$ in $\\mathbf{H}_{\/X}$ \n  is an equivalence if and only if $p^*f$ is an equivalence in \n  $\\mathbf{H}_{\/Y}$ for any effective epimorphism $p\\colon Y\\to X$ in \n  $\\mathbf{H}$.  \n\n  \n\\end{lemma}\n\\proof\n  It is clear, by functoriality, that $p^* f$ is a weak equivalence if $f$ is. \n  Conversely, assume that $p^* f$ is a weak equivalence. \n  Since effective epimorphisms as well as \n  equivalences are preserved by pullback \n  we get a simplicial diagram of the form\n  $$\n    \\raisebox{20pt}{\n    \\xymatrix{\n       \\cdots\n       \\ar@<+4pt>[r]\n       \\ar@<+0pt>[r]\n       \\ar@<-4pt>[r]\n       &\n       p^* A \\times_A p^* A\n       \\ar@<+2pt>[r]\n       \\ar@<-2pt>[r]\n       \\ar[d]^\\simeq\n       &\n       p^* A\n       \\ar[d]^\\simeq\n       \\ar@{->>}[r]\n       & \n       A\n       \\ar[d]^f\n       \\\\\n       \\cdots\n       \\ar@<+4pt>[r]\n       \\ar@<+0pt>[r]\n       \\ar@<-4pt>[r]\n       &\n       p^* B \\times_B p^* B\n       \\ar@<+2pt>[r]\n       \\ar@<-2pt>[r]\n       &\n       p^* B\n       \\ar@{->>}[r]\n       & \n       B\n    }\n\t}\n  $$\n  where the rightmost horizontal morphisms are effective epimorphisms, as indicated.\n  By definition of effective epimorphisms this exhibits\n  $f$ as an $\\infty$-colimit over equivalences, hence as\n  an equivalence.\n\\hfill{$\\square$}\\\\\n\\begin{proposition}\n  \\label{MorphismsOfInfinityBundlesAreEquivalences}\n  Every morphism between $G$-actions over $X$ that are \n  $G$-principal $\\infty$-bundles over $X$ is an equivalence.\n\\end{proposition}\n\\proof\n  Since a morphism of $G$-principal bundles\n  $P_1 \\to P_2$ is a morphism of {\\v C}ech nerves that fixes \n  their $\\infty$-colimit $X$, up to equivalence, \n  and since $* \\to \\mathbf{B}G$ is an effective \n  epimorphism,\n  we are, by Proposition~\\ref{LocalTrivialityImpliesCocycle}, in the situation of  \n  Lemma~\\ref{EquivalencesAreDetectedOverEffectiveEpimorphisms}.\n\\hfill{$\\square$}\\\\\n\\begin{theorem}\n  \\label{PrincipalInfinityBundleClassification}\n  For all $X, \\mathbf{B}G \\in \\mathbf{H}$ there is a natural\n  equivalence of $\\infty$-groupoids\n  $$\n    G \\mathrm{Bund}(X)\n    \\simeq\n    \\mathbf{H}(X, \\mathbf{B}G)\n  $$ \n  which on vertices is the construction of Definition~\\ref{BundleStructureOnInfinityFiber}:\n  a bundle $P \\to X$ is mapped to a morphism\n  $X \\to \\mathbf{B}G$ such that $P \\to X \\to \\mathbf{B}G$ is a fiber\n  sequence.\n\\end{theorem}\nWe therefore say\n\\begin{itemize}\n  \\item $\\mathbf{B}G$ is the \\emph{classifying object} \n  or \\emph{moduli $\\infty$-stack} for \n  $G$-principal $\\infty$-bundles;\n  \\item a morphism $c : X \\to \\mathbf{B}G$ is a \\emph{cocycle}\n  for the corresponding $G$-principal $\\infty$-bundle and its class \n  $[c] \\in \\mathrm{H}^1(X,G)$ is its \n \\emph{characteristic class}.\n\\end{itemize}\n\\proof\n  By Definitions~\\ref{ActionInPrincipal} \n  and~\\ref{principalbundle} and using \n  the refined statement of the third $\\infty$-Giraud axiom  \n  (Theorem~\\ref{NaturalThirdGiraud}), the\n  $\\infty$-groupoid of $G$-principal $\\infty$-bundles over $X$ is \n  equivalent to the fiber over $X$\n  of the\n  sub-$\\infty$-category \n  of the slice\n  of the arrow $\\infty$-topos \n  on those squares\n  $$\n    \\xymatrix{\n\t  P \\ar[r] \\ar@{->>}[d] & {*} \\ar@{->>}[d]\n\t  \\\\\n\t  X \\ar[r] & \\mathbf{B}G\n\t}\n  $$\n  that exhibit $P \\to X$ as a $G$-principal $\\infty$-bundle.  By \n  Proposition~\\ref{BundleStructureOnInfinityFiber} and\n  Proposition~\\ref{LocalTrivialityImpliesCocycle} these are \n  the $\\infty$-pullback squares \n      $\n    \\mathrm{Cart}({\\mathbf{H}^{\\Delta[1]}}_{\/{(* \\to \\mathbf{B}G)}})\n    \\hookrightarrow {\\mathbf{H}^{\\Delta[1]}}_{\/{(* \\to \\mathbf{B}G)}}\n  $, hence\n  $$\n    G \\mathrm{Bund}(X) \\simeq \n\t \\mathrm{Cart}({\\mathbf{H}^{\\Delta[1]}}_{\/{(* \\to \\mathbf{B}G)}}) \\times_{\\mathbf{H}} \\{X\\}\n\t \\,.\n  $$\n  By the universality of the $\\infty$-pullback\n  the morphisms between these are fully determined by their value on $X$,\n  so that the above is equivalent to \n  $$\n  \\mathbf{H}_{\/\\mathbf{B}G} \\times_{\\mathbf{H}} \\{X\\}\n\t\\,.\n  $$\n  (For instance in terms of model categories: choose a model structure for\n  $\\mathbf{H}$ in which all objects are cofibrant, choose a fibrant representative\n  for $\\mathbf{B}G$ and a fibration resolution $\\mathbf{E}G \\to \\mathbf{B}G$\n  of the universal $G$-bundle. Then the slice model structure of the arrow model structure\n  over this presents the slice in question and the statement follows from the analogous\n  1-categorical statement.)\n  This finally is equivalent to \n  $$\n\t\\mathbf{H}(X, \\mathbf{B}G)\n\t\\,.\n  $$\n  (For instance in terms of quasi-categories: the projection \n  $\\mathbf{H}_{\/\\mathbf{B}G} \\to \\mathbf{H}$ is a fibration by \n  Proposition 2.1.2.1 and 4.2.1.6 in \\cite{Lurie}, hence the homotopy fiber\n  $\\mathbf{H}_{\/\\mathbf{B}G} \\times_{\\mathbf{X}} \\{X\\}$ \n  is the ordinary fiber of quasi-categories. This is manifestly \n  the\n  $\\mathrm{Hom}^R_{\\mathbf{H}}(X, \\mathbf{B}G)$ from Proposition 1.2.2.3 of \\cite{Lurie}.\n  Finally, by Proposition 2.2.4.1 there, this is equivalent to $\\mathbf{H}(X,\\mathbf{B}G)$.)\n\\hfill{$\\square$}\\\\\n\\begin{corollary}\n  Equivalence classes of $G$-principal $\\infty$-bundles over $X$ are\n  in natural bijection with the degree-1 $G$-cohomology of $X$:\n  $$\n    G \\mathrm{Bund}(X)_{\/\\sim} \\simeq H^1(X, G)\n\t\\,.\n  $$\n\\end{corollary}\n\\proof\n  By Definition \\ref{cohomology} this is the restriction of \n  the equivalence $G \\mathrm{Bund}(X) \\simeq \\mathbf{H}(X, \\mathbf{B}G)$ to\n  connected components.\n\\hfill{$\\square$}\\\\\n\n\\section{Twisted bundles and twisted cohomology}\n\\label{StrucTwistedCohomology}\n\nWe show here how the general notion of cohomology in an \n$\\infty$-topos, considered above in Section~\\ref{StrucCohomology}, subsumes the notion of\n\\emph{twisted cohomology} and we discuss the corresponding\ngeometric structures classified by twisted cohomology: \n\\emph{extensions} of principal $\\infty$-bundles and \\emph{twisted $\\infty$-bundles}.\n \nWhereas ordinary cohomology is given by a derived hom-$\\infty$-groupoid, \ntwisted cohomology is given by the $\\infty$-groupoid of \n\\emph{sections of \na local coefficient bundle} in an $\\infty$-topos, \nwhich in turn is \nan \\emph{associated $\\infty$-bundle} induced via a representation \nof an $\\infty$-group $G$ from a $G$-principal $\\infty$-bundle\n(this is a geometric and unstable variant of the picture   \nof twisted cohomology developed in \\cite{AndoBlumbergGepner,MaySigurdsson}).\n\nIt is fairly immediate that, given a \\emph{universal} local coefficient bundle\nassociated to a universal principal $\\infty$-bundle,\nthe induced twisted cohomology is equivalently ordinary\ncohomology in the corresponding slice $\\infty$-topos. This\nidentification provides a clean formulation of the contravariance\nof twisted cocycles. However, a universal coefficient bundle\nis a pointed connected object in the slice $\\infty$-topos only\nwhen it is a trivial bundle, so that twisted cohomology does not classify\nprincipal $\\infty$-bundles in the slice. We show below that instead\nit classifies \\emph{twisted principal $\\infty$-bundles}, which are\nnatural structures that generalize the twisted bundles familiar from\ntwisted K-theory.\nFinally, we observe that twisted cohomology in an $\\infty$-topos \nequivalently classifies extensions of structure groups\nof principal $\\infty$-bundles. \n\nA wealth of structures turn out to be special cases of \nnonabelian twisted cohomology and of twisted\nprincipal $\\infty$-bundles and also turn out to be usefully informed by the\ngeneral theory of twisted cohomology, we will discuss some of these structures in \\cite{NSSc}.\n\n\n\n\\subsection{Actions and associated $\\infty$-bundles}\n\\label{StrucRepresentations}\n\nLet $\\mathbf{H}$ be an $\\infty$-topos, $G \\in \\mathrm{Grp}(\\mathbf{H})$\nan $\\infty$-group.\nFix an action $\\rho : V \\times G \\to V$ (Definition \\ref{ActionInPrincipal}) on an object $V\\in \\mathbf{H}$.\nWe discuss the induced notion of \\emph{$\\rho$-associated $V$-fiber $\\infty$-bundles}.\nWe show that there is a \\emph{universal} $\\rho$-associated $V$-fiber bundle over \n$\\mathbf{B}G$ and observe that under Theorem \\ref{PrincipalInfinityBundleClassification}\nthis is effectively identified with the action itself. Accordingly, we also further discuss\n$\\infty$-actions as such.\n\n\\medskip\n\n\\begin{definition}\n  For $V,X \\in \\mathbf{H}$ any two objects, \na \\emph{$V$-fiber $\\infty$-bundle} over $X$ is a morphism $E \\to X$, \nsuch that there is an effective epimorphism\n$\\xymatrix{U \\ar@{->>}[r] & X}$ and an $\\infty$-pullback of the form\n$$\n  \\raisebox{20pt}{\n  \\xymatrix{\n    U \\times V \\ar[r] \\ar[d] & E \\ar[d]\n\t\\\\\n\tU \\ar@{->>}[r] & X\\, .\n  }\n  }\n$$  \n \\label{FiberBundle}\n\\end{definition}\nWe say that $E \\to X$ locally trivializes with respect to $U$.\nAs usual, we often say \\emph{$V$-bundle} for short.\n\n\\begin{definition}\n  For $P \\to X$ a $G$-principal $\\infty$-bundle, we write\n  $$ \n    P \\times_G V := (P\\times V)\/\\!\/G \n  $$ \n  for the $\\infty$-quotient of the diagonal $\\infty$-action of $G$ on $P \\times V$.\n  Equipped with the canonical morphism\n  $P \\times_G V \\to X$ we call this the $\\infty$-bundle \\emph{ $\\rho$-associated} to $P$.\n  \\label{AssociatedBundle}\n\\end{definition}\n\\begin{remark}\n  The diagonal $G$-action on $P \\times V$ is the product in \n  $G \\mathrm{Action}(\\mathbf{H})$ of the given actions on $P$ and on $V$.\n  Since $G\\mathrm{Action}(\\mathbf{H})$ is a full sub-$\\infty$-category of a slice\n  category of a functor category, the product is given by a degreewise\n  pullback in $\\mathbf{H}$:\n  $$\n    \\raisebox{20pt}{\n    \\xymatrix{\n\t  P \\times V \\times G^{\\times_n} \n\t  \\ar[r]\n\t  \\ar[d]\n\t  &\n\t  V \\times G^{\\times_n}\n\t  \\ar[d]\n\t  \\\\\n\t  P \\times G^{\\times_n}\n\t  \\ar[r]\n\t  &\n\t  G^{\\times_n}\\,.\n\t}\n\t}\n  $$\n  and so\n  $$\n    P \\times_G V \\simeq \\varinjlim_n (P \\times V \\times G^{\\times_n})\n\t\\,.\n  $$\n  The canonical bundle morphism of the corresponding $\\rho$-associated\n  $\\infty$-bundle  is the realization of the left morphism of this diagram:\n  $$\n    \\raisebox{20pt}{\n    \\xymatrix{\n\t  P \\times_G V\n\t  \\ar@{}[r]|<<<<{:=}\n\t  \\ar[d]\n\t  &\n\t  \\varinjlim_n (P \\times V \\times G^{\\times_n})\n\t  \\ar[d]\n\t  \\\\\n\t  X \\ar@{}[r]|<<<<<<<<<{\\simeq} & \n\t  \\varinjlim_n (P \\times G^{\\times_n})\\,.\n\t}\n\t}\n  $$\n  \\label{ProductActionByPullback}\n\\end{remark}\n\\begin{example}\nBy Theorem \\ref{PrincipalInfinityBundleClassification} every $\\infty$-group action\n$\\rho : V \\times G \\to V$ has a classifying morphism $\\mathbf{c}$ defined on its homotopy\nquotient, which fits into a fiber sequence of the form\n$$\n  \\raisebox{20pt}{\n  \\xymatrix{\n     V \\ar[r] & V\/\\!\/G \\ar[d]^{\\mathbf{c}} \n     \\\\\t \n\t & \\mathbf{B}G\\,.\n  }\n  }\n$$\n\n  Regarded as an \n  $\\infty$-bundle, this is\n  $\\rho$-associated to the universal $G$-principal $\\infty$-bundle\n  $\\xymatrix{{*} \\ar[r] & \\mathbf{B}G}$ from Example \\ref{UniversalPrincipal}:\n  $$\n    V\/\\!\/G \\simeq {*} \\times_G V\n\t\\,.\n  $$\n  \\label{ActionGroupoidIsRhoAssociated}\n\\end{example}\n\\begin{lemma}\n  The realization functor $\\varinjlim : \\mathrm{Grpd}(\\mathbf{H}) \\to \\mathbf{H}$\n  preserves the $\\infty$-pullback of Remark \\ref{ProductActionByPullback}:\n  $$\n    P \\times_G V \\simeq \\varinjlim_n (P \\times V \\times G^{\\times_n})\n\t\\simeq\n\t(\\varinjlim_n P \\times G^{\\times_n}) \\times_{(\\varinjlim_n G^{\\times_n})} (\\varinjlim_n V \\times G^{\\times_n})\n\t\\,.\n  $$\n  \\label{RealizationPreservesProductOfRepresentations}\n\\end{lemma}\n\\proof\n  Generally, let $X \\to Y \\leftarrow Z \\in \\mathrm{Grpd}(\\mathbf{H})$ be a\n  diagram of groupoid objects, such that in the induced diagram\n  $$\n    \\xymatrix{\n\t  X_0 \\ar[r] \\ar@{->>}[d] & Y_0 \\ar@{<-}[r] \\ar@{->>}[d] & Z_0 \\ar@{->>}[d]\n\t  \\\\\n\t  \\varinjlim_n X_n \\ar[r] & \\varinjlim_n Y_n \\ar@{<-}[r] & \\varinjlim_n Z_n\n\t}\n  $$\n  the left square is an $\\infty$-pullback. By the third \n  $\\infty$-Giraud axiom (Definition~\\ref{GiraudRezkLurieAxioms}) the vertical \n  morphisms are effective epi, as indicated. \n  By assumption we have a pasting of $\\infty$-pullbacks as shown on the\n  left of the following diagram, and by\n  the pasting law (Proposition \\ref{PastingLawForPullbacks}) this is equivalent to\n  the pasting shown on the right:\n  $$\n    \\raisebox{38pt}{\n    \\xymatrix{\n\t  X_0 \\times_{Y_0} Z_0 \\ar[r] \\ar[d] & Z_0 \\ar[d]\n\t  \\\\\n\t  X_0 \\ar[r] \\ar[d] & Y_0 \\ar[d]\n\t  \\\\\n\t  \\varinjlim_n X_n \\ar[r] & \\varinjlim_n Y_n\n\t}\n\t}\n\t\\;\\;\\;\n\t\\simeq\n\t\\;\\;\\;\n    \\raisebox{38pt}{\n    \\xymatrix{\n\t  X_0 \\times_{Y_0} Z_0 \\ar[r] \\ar@{->>}[d] & Z_0 \\ar@{->>}[d]\n\t  \\\\\n\t  (\\varinjlim_n X_n) \\times_{(\\varinjlim_n Y_n)} (\\varinjlim_n Z_n) \\ar[r] \\ar[d] & \n\t  \\varinjlim_n Z_n \\ar[d]\n\t  \\\\\n\t  \\varinjlim_n X_n \\ar[r] & \\varinjlim_n Y_n.\n\t}\n\t}\n  $$\nSince effective epimorphisms are stable under $\\infty$-pullback, this identifies \nthe canonical morphism \n$$\n  X_0 \\times_{Y_0} Z_0\n  \\to \n  (\\varinjlim_n X_n) \\times_{(\\varinjlim_n Y_n)} (\\varinjlim_n Z_n)\n$$\nas an effective epimorphism, as indicated. \n\nSince $\\infty$-limits commute over each other, the {\\v C}ech nerve of this morphism \nis the groupoid object $[n] \\mapsto X_n \\times_{Y_n} Z_n$.\nTherefore the third $\\infty$-Giraud axiom now says that $\\varinjlim$ preserves the\n$\\infty$-pullback of groupoid objects:\n$$\n  \\varinjlim (X \\times_Y Z) \n   \\simeq  \n  \\varinjlim_n (X_n \\times_{Y_n} Z_n )\n   \\simeq\n  (\\varinjlim_n X_n) \\times_{(\\varinjlim_n Y_n)} (\\varinjlim_n Z_n)\n  \\,.\n$$\n\nConsider this now in the special case that $X \\to Y \\leftarrow Z$ is \n$(P \\times G^{\\times_\\bullet}) \\to G^{\\times_\\bullet} \\leftarrow (V \\times G^{\\times_\\bullet})$.\nTheorem \\ref{PrincipalInfinityBundleClassification} implies that the initial assumption above is \nmet, in that $P \\simeq (P\/\\!\/G) \\times_{*\/\\!\/G} {*} \\simeq X \\times_{\\mathbf{B}G} {*}$, \nand so the claim follows.\n\\hfill{$\\square$}\\\\\n\\begin{proposition}\n  For $g_X : X \\to \\mathbf{B}G$ a morphism and $P \\to X$\n  the corresponding $G$-principal $\\infty$-bundle according to Theorem \n  \\ref{PrincipalInfinityBundleClassification},  \n  there is a natural equivalence\n  $$\n    g_X^*(V\/\\!\/G) \\simeq P \\times_G V\n  $$\n  over $X$, between the pullback of the\n  $\\rho$-associated $\\infty$-bundle\n  $\\xymatrix{V\/\\!\/G \\ar[r]^{\\mathbf{c}} & \\mathbf{B}G}$\n  of Example \\ref{ActionGroupoidIsRhoAssociated}\n  and the $\\infty$-bundle $\\rho$-associated to $P$ by Definition \\ref{AssociatedBundle}.\n  \\label{UniversalAssociatedBundle}\n\\end{proposition}\n\\proof\n By Remark \\ref{ProductActionByPullback} the product action is given by the \n pullback \n $$\n   \\xymatrix{\n     P \\times V \\times G^{\\times_\\bullet}\n\t \\ar[r]\n\t \\ar[d]\n\t &\n\t V \\times G^{\\times_\\bullet}\n\t \\ar[d]\n\t \\\\\n\t P \\times G^{\\times_\\bullet} \\ar[r] & G^{\\times_\\bullet}\n   }\n $$\n in $\\mathbf{H}^{\\Delta^{\\mathrm{op}}}$. \n By Lemma $\\ref{RealizationPreservesProductOfRepresentations}$ the realization functor\n preserves this $\\infty$-pullback. By \n Remark \\ref{ProductActionByPullback} it sends the left morphism to the \n associated bundle, and by Theorem \\ref{PrincipalInfinityBundleClassification}\n it sends the bottom morphism to $g_X$. Therefore it produces an $\\infty$-pullback\n diagram of the form\n $$\n   \\raisebox{20pt}{\n   \\xymatrix{\n     V \\times_G P \\ar[r] \\ar[d] & V\/\\!\/G \\ar[d]^{\\mathbf{c}}\n\t \\\\\n\t X \\ar[r]^{g_X} & \\mathbf{B}G\\,.\n   }\n   }\n $$ \n\\hfill{$\\square$}\\\\\n\\begin{remark}\n  This says that $\\xymatrix{V\/\\!\/G \\ar[r]^{\\mathbf{c}} & \\mathbf{B}G}$ is both, \n  the $V$-fiber $\\infty$-bundle \n  $\\rho$-associated to the universal $G$-principal $\\infty$-bundle, Observation\n  \\ref{ActionGroupoidIsRhoAssociated},  \n  as well as the universal $\\infty$-bundle for $\\rho$-associated $\\infty$-bundles.\n  \\label{RhoAssociatedToUniversalIsUniversalVBundle}\n\\end{remark}\n\\begin{proposition}\n  Every $\\rho$-associated $\\infty$-bundle is a $V$-fiber $\\infty$-bundle, \n  Definition \\ref{FiberBundle}.\n  \\label{AssociatedIsFiberBundle}\n\\end{proposition}\n\\proof\n  Let $P \\times_G V \\to X$ be a $\\rho$-associated $\\infty$-bundle.\n  By the previous Proposition \\ref{UniversalAssociatedBundle} it is \n  the pullback $g_X^* (V\/\\!\/G)$ of the universal $\\rho$-associated bundle.\n  By Proposition \\ref{EveryGBundleIsLocallyTrivial} there exists an \n  effective epimorphism $\\xymatrix{U \\ar@{->>}[r] & X}$ over which\n  $P$ trivializes, hence such that $g_X|_U$ factors through the point, up\n  to equivalence. In summary and by the pasting law, Proposition \\ref{PastingLawForPullbacks},\n  this gives a pasting of $\\infty$-pullbacks of the form\n  $$\n    \\raisebox{20pt}{\n    \\xymatrix@R=8pt{\n\t  U \\times V\n\t  \\ar[dd]\n\t  \\ar[r]\n\t  &\n\t  P \\times_G V\n\t  \\ar[r]\n\t  \\ar[dd]\n\t  &\n\t  V\/\\!\/G\n\t  \\ar[dd]\n\t  \\\\\n\t  \\\\\n\t  U \\ar@{->>}[r] \n\t  \\ar[dr]\n\t  & \n\t  X\n\t  \\ar[r]^{g_X}\n\t  &\n\t  \\mathbf{B}G\n\t  \\\\\n\t  & {*}\n\t  \\ar[ur]\n\t}\n\t}\n  $$\n  which exhibits $P \\times_G V \\to X$ as a $V$-fiber bundle by a local trivialization\n  over $U$.\n\\hfill{$\\square$}\\\\\n\nSo far this shows that every $\\rho$-associated $\\infty$-bundle is a\n$V$-fiber bundle. We want to show that, conversely, every $V$-fiber bundle\nis associated to a principal $\\infty$-bundle.\n\\begin{definition}\n  Let $V \\in \\mathbf{H}$ be a $\\kappa$-compact object, for some regular cardinal $\\kappa$. \n  By the characterization of Definition \\ref{RezkCharacterization}, there exists\n  an $\\infty$-pullback square in $\\mathbf{H}$ of the form\n  $$\n    \\raisebox{20pt}{\n    \\xymatrix{\n\t  V \\ar[r] \\ar[d] & \\widehat {\\mathrm{Obj}}_\\kappa \\ar[d]\n\t  \\\\\n\t  {*} \\ar[r]^<<<<<<{\\vdash V} & \\mathrm{Obj}_\\kappa\n\t}\n\t}\n  $$\n  Write\n  $$\n    \\mathbf{B}\\mathbf{Aut}(V) := \\mathrm{im}(\\vdash V)\n  $$\n  for the $\\infty$-image, Definition \\ref{image}, \n  of the classifying morphism $\\vdash V$ of $V$. \n  By definition this comes with an effective epimorphism \n  $$\n    \\xymatrix{\n\t  {*} \\ar@{->>}[r] & \\mathbf{B}\\mathbf{Aut}(V)\n\t  \\ar@{^{(}->}[r] & \\mathrm{Obj}_\\kappa\n\t}\n\t\\,,\n  $$\n  and hence, by Proposition \\ref{InfinityGroupObjectsAsGroupoidObjects},\n  it is the delooping of an $\\infty$-group \n  $$\n    \\mathbf{Aut}(V) \\in \\mathrm{Grp}(\\mathbf{H})\n  $$\n  as indicated. We call this the \\emph{internal automorphism $\\infty$-group} of $V$.\n  \n  By the pasting law, Proposition \\ref{PastingLawForPullbacks}, \n  the image factorization gives a pasting\n  of $\\infty$-pullback diagrams of the form\n  $$\n    \\xymatrix{\n\t  V \\ar[r] \\ar[d] & V\/\\!\/\\mathbf{Aut}(V) \\ar[r] \\ar[d]^{\\mathbf{c}_V} & \n\t  \\widehat {\\mathrm{Obj}}_\\kappa \\ar[d]\n\t  \\\\\n\t  {*} \\ar@{->>}[r]^<<<<<<{\\vdash V} & \\mathbf{B}\\mathbf{Aut}(V) \\ar@{^{(}->}[r] & \n\t  \\mathrm{Obj}_\\kappa\n\t}\n  $$\n  By Theorem~\\ref{PrincipalInfinityBundleClassification} this defines a canonical \n  $\\infty$-action \n  $$\n    \\rho_{\\mathbf{Aut}(V)} : V \\times \\mathbf{Aut}(V) \\to V\n  $$\n  of $\\mathbf{Aut}(V)$ on $V$ with homotopy quotient $V\/\\!\/\\mathbf{Aut}(V)$\n  as indicated. \n  \\label{InternalAutomorphismGroup}\n\\end{definition}\n\\begin{proposition}\n  Every $V$-fiber $\\infty$-bundle is $\\rho_{\\mathbf{Aut}(V)}$-associated to an \n  $\\mathbf{Aut}(V)$-principal $\\infty$-bundle.\n  \\label{VBundleIsAssociated}\n\\end{proposition}\n\\proof\n  Let $E \\to V$ be a $V$-fiber $\\infty$-bundle. \n  By Definition \\ref{FiberBundle} there exists an effective epimorphism\n  $\\xymatrix{U \\ar@{->>}[r] & X}$  along which the bundle trivializes locally.\n  It follows \n  by the second Axiom in Definition \\ref{RezkCharacterization}\n  that on $U$ the morphism $\\xymatrix{X \\ar[r]^<<<<<{\\vdash E} & \\mathrm{Obj}_\\kappa}$\n  which classifies $E \\to X$ factors through the point\n  $$\n    \\raisebox{20pt}{\n    \\xymatrix@R=8pt{\n\t  U \\times V \\ar[r]\\ar[dd] & E \\ar [r] \\ar[dd] & \\widehat{\\mathrm{Obj}}_\\kappa \\ar[dd]\n\t  \\\\\n\t  \\\\\n\t  U \\ar@{->>}[r] \\ar[dr] & X \\ar[r]^<<<<<{\\vdash E} & \\mathrm{Obj}_\\kappa.\n\t  \\\\\n\t  & {*} \\ar[ur]_<<<<<{\\vdash V}\n\t}\n\t}\n  $$\n  Since the point inclusion, in turn, factors through its $\\infty$-image \n  $\\mathbf{B}\\mathbf{Aut}(V)$, Definition \\ref{InternalAutomorphismGroup},\n  this yields the outer commuting diagram of the following form\n  $$\n    \\raisebox{20pt}{\n    \\xymatrix{\n\t  U \\ar[r] \\ar@{->>}[d] & {*} \\ar[r] &  \\mathbf{B}\\mathbf{Aut}(V) \\ar@{^{(}->}[d]\n\t  \\\\\n\t  X \\ar[rr]_{\\vdash E} \n\t   \\ar@{-->}[urr]^{g}\n\t  && \\mathrm{Obj}_\\kappa\n\t}}\n  $$\n  By the epi\/mono factorization system of Proposition \\ref{EpiMonoFactorizationSystem}\n  there is a diagonal lift $g$ as indicated. Using again the \n  pasting law and by Definition \\ref{InternalAutomorphismGroup}\n  this factorization induces a pasting of $\\infty$-pullbacks of the form\n  $$\n    \\raisebox{20pt}{\n    \\xymatrix{\n\t  E \\ar[r] \\ar[d] & V\/\\!\/\\mathbf{Aut}(V) \\ar[r] \\ar[d]^{\\mathbf{c}_V} & \n\t  \\widehat {\\mathrm{Obj}}_\\kappa \\ar[d]\n\t  \\\\\n\t  X \\ar[r]^<<<<<g & \\mathbf{B}\\mathbf{Aut}(V) \\ar@{^{(}->}[r] & \\mathrm{Obj}_\\kappa\n\t}\n\t}\n  $$\n  Finally, by Proposition \\ref{UniversalAssociatedBundle}, this \n  exhibits $E \\to X$ as being $\\rho_{\\mathbf{Aut}(V)}$-associated to the \n  $\\mathbf{Aut}(V)$-principal $\\infty$-bundle with class $[g] \\in H^1(X,G)$.\n\\hfill{$\\square$}\\\\\n\\begin{theorem}\n  $V$-fiber $\\infty$-bundles over $X \\in \\mathbf{H}$ are classified by \n  $H^1(X, \\mathbf{Aut}(V))$.\n  \\label{VBundleClassification}\n\\end{theorem}\n  Under this classification, the $V$-fiber $\\infty$-bundle corresponding to \n  $[g] \\in H^1(X, \\mathbf{Aut}(V))$ is identified,\n  up to equivalence, with the $\\rho_{\\mathbf{Aut}(V)}$-associated $\\infty$-bundle \n  (Definition~\\ref{AssociatedBundle}) to the $\\mathbf{Aut}(V)$-principal $\\infty$-bundle\n  corresponding to $[g]$ by Theorem~\\ref{PrincipalInfinityBundleClassification}.\n \\\\\n\\proof\n  By Proposition~\\ref{VBundleIsAssociated} every morphism \n  $\\xymatrix{\n    X \\ar[r]^<<<<<{\\vdash E} & \\mathrm{Obj}_\\kappa\n  }$\n  that classifies a small $\\infty$-bundle $E \\to X$ \n  which happens to be a $V$-fiber $\\infty$-bundle factors via some $g$\n  through \n  the moduli for $\\mathbf{Aut}(V)$-principal $\\infty$-bundles\n  $$ \n    \\xymatrix{\n\t  X\n\t  \\ar[r]^<<<<<{g}\n\t  \\ar@\/_1pc\/[rr]_{\\vdash E}\n\t  &\n      \\mathbf{B}\\mathbf{Aut}(V) \n\t  \\ar@{^{(}->}[r]\n\t  & \\mathrm{Obj}_\\kappa\n     }\n\t \\,.\n   $$ \n   Therefore it only remains to show that also every homotopy \n   $(\\vdash E_1) \\Rightarrow (\\vdash E_2)$ factors \n   through a homotopy $g_1 \\Rightarrow g_2$. \n   This follows by applying the epi\/mono lifting property of \n   Proposition \\ref{EpiMonoFactorizationSystem} to the diagram\n   $$\n     \\xymatrix{\n\t  X \\coprod X \\ar[r]^<<<<<{(g_1, g_2)} \\ar@{->>}[d] & \\mathbf{B}\\mathbf{Aut}(V) \n\t  \\ar@{^{(}->}[d]\n\t  \\\\\n\t  X \\ar[r] \\ar@{-->}[ur] & \\mathrm{Obj}_\\kappa\n\t }\n   $$\n   The outer diagram exhibits the original homotopy. The left morphism is \n   an effective epi (for instance immediately by Proposition \\ref{EffectiveEpiIsEpiOn0Truncation}), \n   the right morphism is a monomorphism by construction. Therefore the dashed\n   lift exists as indicated and so the top left triangular diagram exhibits\n   the desired factorizing homotopy.\n\\hfill{$\\square$}\\\\\n\\begin{remark}\n  In the special case that $\\mathbf{H} = \\mathrm{Grpd}_{\\infty}$, the classification \n  Theorem \\ref{VBundleClassification}\n  is classical \\cite{Stasheff,May}, traditionally\n  stated in (what in modern terminology is) the \n  presentation of $\\mathrm{Grpd}_{\\infty}$ by simplicial sets\n  or by topological spaces. Recent discussions include \\cite{BlomgrenChacholski}.\n  For $\\mathbf{H}$ a general 1-localic $\\infty$-topos (meaning: with a 1-site of definition), \n  the statement of Theorem \\ref{VBundleClassification} appears in \\cite{Wendt}, \n  formulated there in terms of the presentation of $\\mathbf{H}$ by simplicial presheaves.\n  (We discuss the relation of these presentations to the above general abstract result \n  in \\cite{NSSb}.)\n  Finally, one finds that the classification of \\emph{$G$-gerbes} \\cite{Giraud} and \n  \\emph{$G$-2-gerbes} in \\cite{Breen} is the special case of the general statement,\n  for $V = \\mathbf{B}G$ and $G$ a 1-truncated $\\infty$-group. \n  This we discuss below in Section~\\ref{StrucInftyGerbes}.\n  \\label{ReferencesOnClassificationOfVBundles}\n\\end{remark}\n\nWe close this section with a list of some fundamental classes \nof examples of $\\infty$-actions, or equivalently, \nby Remark \\ref{RhoAssociatedToUniversalIsUniversalVBundle}, \nof universal associated $\\infty$-bundles. \nFor doing so we use again that, \nby Theorem \\ref{PrincipalInfinityBundleClassification}, to give an \n$\\infty$-action of $G$ on $V$ is equivalent to giving a fiber\nsequence of the form $V \\to V\/\\!\/G \\to \\mathbf{B}G$.\nTherefore the following list mainly serves to associate a tradition\n\\emph{name} with a given $\\infty$-action.\n\\begin{example}\n    \\label{ExamplesOfActions}\n  The following are $\\infty$-actions.\n  \\begin{enumerate}\n    \\item\n\t For every $G \\in \\mathrm{Grp}(\\mathbf{H})$, the fiber sequence\n\t $$\n\t   \\raisebox{20pt}{\n\t   \\xymatrix{\n\t     G \\ar[d]\n\t\t \\\\\n\t\t {*} \\ar[r] & \\mathbf{B}G\n\t   }\n\t   }\n\t $$\n\t which defines $\\mathbf{B}G$ by Theorem \\ref{DeloopingTheorem}\n\t induces the \\emph{right action of $G$ on itself}\n\t $$\n\t   * \\simeq G\/\\!\/G\n\t   \\,.\n\t $$\n\t At the same time\n\t this sequence, but now regarded as a bundle over $\\mathbf{B}G$, \n\t is the universal $G$-principal $\\infty$-bundle, Remark \\ref{UniversalPrincipal}.\n   \\item\n     For every object $X \\in \\mathbf{H}$ write \n\t $$\n\t   \\mathbf{L}X := X \\times_{X \\times X} X\n\t $$ \n\t for its \\emph{free loop space} object, the $\\infty$-fiber product of the \n\t diagonal on $X$ along itself\n\t $$\n\t   \\raisebox{20pt}{\n\t   \\xymatrix{\n\t     \\mathbf{L}X \\ar[r] \\ar[d]_{\\mathrm{ev}_{*}} & X \\ar[d]\n\t\t \\\\\n\t\t X \\ar[r] & X \\times X\n\t   }\n\t   }.\n\t $$\n     For every $G \\in \\mathrm{Grp}(\\mathbf{H})$ there is a fiber sequence\n\t $$\n\t   \\raisebox{20pt}{\n\t   \\xymatrix{\n\t     G \n\t\t \\ar[d]\n\t\t \\\\\n\t\t \\mathbf{L}\\mathbf{B}G\n\t\t \\ar[r]^{\\mathrm{ev}_{*}}\n\t\t &\n\t\t \\mathbf{B}G\n\t   }\n\t   }\n\t   \\,.\n\t $$\n\t This exhibits the \\emph{adjoint action of $G$ on itself}\n\t $$\n\t   \\mathbf{L}\\mathbf{B}G \\simeq G\/\\!\/_{\\mathrm{ad}} G\n\t   \\,.\n\t $$\n\t\\item\n\t  For every $V \\in \\mathbf{H}$ there is the canonical \n\t  $\\infty$-action of the \\emph{automorphism $\\infty$-group}\n\t  $$\n\t    \\raisebox{20pt}{\n\t    \\xymatrix{\n\t\t  V\n\t\t  \\ar[d]\n\t\t  \\\\\n\t\t  V\/\\!\/\\mathbf{Aut}(V)\n\t\t  \\ar[r]\n\t\t  &\n\t\t  \\mathbf{B}\\mathbf{Aut}(V)\n\t\t}\n\t\t}\n\t\t\\,,\n\t  $$\n\t  introduced in Definition \\ref{InternalAutomorphismGroup}, this exhibits the\n\t  \\emph{automorphism action}.\n  \\end{enumerate}\n\\end{example}\n\n\\subsection{Sections and twisted cohomology}\n\\label{TwistedCohomology}\n\nWe discuss a general notion of \\emph{twisted cohomology} or \n\\emph{cohomology with local coefficients} in any $\\infty$-topos $\\mathbf{H}$, \nwhere the \\emph{local coefficient $\\infty$-bundles} are \nassociated $\\infty$-bundles as discussed above, and where the cocycles are\n\\emph{sections} of these local coefficient bundles.\n\n\\medskip\n\n\\begin{definition}\n  Let $p : E \\to X$ be any morphism in $\\mathbf{H}$, to be regarded as an \n  $\\infty$-bundle over $X$. A \\emph{section} of $E$ is a diagram\n  $$\n    \\raisebox{20pt}{\n    \\xymatrix@!C=40pt@!R=30pt{\n\t   & E \\ar[d]^p\n\t   \\\\\n\t  X \\ar[r]_{\\mathrm{id}}^{\\ }=\"t\" \n\t   \\ar[ur]^{\\sigma}_{\\ }=\"s\"\n\t  & X\n\t  %\n\t  \\ar@{=>}^{\\simeq} \"s\"; \"t\"\n\t}\n\t}\n  $$\n  (where for emphasis we display the presence of the homotopy filling the diagram).\n  The \\emph{$\\infty$-groupoid of sections} of $E \\stackrel{p}{\\to} X$ is the homotopy fiber\n  $$\n    \\Gamma_X(E) := \\mathbf{H}(X,E) \\times_{\\mathbf{H}(X,X)} \\{\\mathrm{id}_X\\}\n  $$\n  of the space of all morphisms $X \\to E$ on those that cover the identity on $X$.\n  \\label{Sections}\n\\end{definition}\nWe record two elementary but important observations about spaces of sections.\n\\begin{observation}\n  There is a canonical identification\n  $$\n    \\Gamma_X(E) \\simeq \\mathbf{H}_{\/X}(\\mathrm{id}_X, p)\n  $$\n  of the space of sections of $E \\to X$ with the hom-$\\infty$-groupoid in the \n  slice $\\infty$-topos $\\mathbf{H}_{\/X}$ between the identity on $X$ and the bundle map $p$.\n  \\label{SectionBySliceMaps}\n\\end{observation}\n\\proof\n  For instance by Proposition 5.5.5.12 in \\cite{Lurie}.\n\\hfill{$\\square$}\\\\\n\\begin{lemma}\n  Let \n  $$\n    \\xymatrix{\n\t  E_1 \\ar[r] \\ar[d]^{p_1} & E_2 \\ar[d]^{p_2}\n\t  \\\\\n\t  B_1 \\ar[r]^{f} & B_2\n\t}\n  $$ \n  be an $\\infty$-pullback diagram in $\\mathbf{H}$\n  and let $\\xymatrix{X \\ar[r]^{g_X} & B_1}$ be any morphism. Then \n  post-composition with $f$ induces\n  a natural equivalence of hom-$\\infty$-groupoids\n  $$\n    \\mathbf{H}_{\/B_1}(g_X, p_1)\n\t\\simeq\n\t\\mathbf{H}_{\/B_2}(f\\circ g_X, p_2)\n\t\\,.\n  $$\n  \\label{SliceMapsIntoPullbacks}\n\\end{lemma}\n\\proof\n  By Proposition 5.5.5.12 in \\cite{Lurie}, the left hand side is given by the homotopy pullback\n  $$\n    \\raisebox{20pt}{\n    \\xymatrix{\n\t  \\mathbf{H}_{\/B_1}(g_X, p_1) \\ar[r] \\ar[d] & \\mathbf{H}(X, E_1) \n\t    \\ar[d]^{\\mathbf{H}(X,p_1)}\n\t  \\\\\n\t  \\{g_X\\} \\ar[r] & \\mathbf{H}(X,B_1)\\,.\n\t}\n\t}\n  $$\n  Since the hom-$\\infty$-functor\n  $\\mathbf{H}(X,-) : \\mathbf{H} \\to \\mathrm{Grpd}_{\\infty}$ \n  preserves the $\\infty$-pullback $E_1 \\simeq f^* E_2$, this\n  extends to a pasting of $\\infty$-pullbacks, which by the pasting law \n  (Proposition \\ref{PastingLawForPullbacks}) is\n  $$\n    \\raisebox{20pt}{\n    \\xymatrix{\n\t  \\mathbf{H}_{\/B_1}(g_X, p_1) \\ar[r] \\ar[d] \n\t     & \n\t  \\mathbf{H}(X, E_1) \\ar[d]^{\\mathbf{H}(X,p_1)} \\ar[r]\n\t  &\n\t  \\mathbf{H}(X, E_2)\n\t  \\ar[d]^{\\mathbf{H}(X, p_2)}\n\t  \\\\\n\t  \\{g_X\\} \\ar[r] & \\mathbf{H}(X,B_1)\n\t  \\ar[r]_-{\\mathbf{H}(X, f)} &\n      \\mathbf{H}(X, B_2)\t  \n\t}\n\t}\n\t\\;\\;\\;\n\t\\simeq\n\t\\;\\;\\;\n\t\\raisebox{20pt}{\n\t\\xymatrix{\n\t  \\mathbf{H}_{\/B_2}(f \\circ g_X, p_2)\n\t  \\ar[r]\n\t  \\ar[d]\n\t  &\n\t  \\mathbf{H}(X, E_2)\n\t  \\ar[d]^{\\mathbf{H}(X, p_2)}\n\t  \\\\\n\t  \\{f\\circ g_X\\}\n\t  \\ar[r]\n\t  &\n\t  \\mathbf{H}(X, B_2).\n\t}\n\t}\n  $$\n\\hfill{$\\square$}\\\\\nFix now an $\\infty$-group $G \\in \\mathrm{Grp}(\\mathbf{H})$ and an \n$\\infty$-action $\\rho : V \\times G \\to V$. Write \n$$\n  \\xymatrix{\n    V \\ar[r] & V\/\\!\/G \\ar[d]^{\\mathbf{c}}\n\t\\\\\n\t& \\mathbf{B}G\n  }\n$$\nfor the corresponding \\emph{universal $\\rho$-associated $\\infty$-bundle} \nas discussed in Section~\\ref{StrucRepresentations}.\n\\begin{proposition}\n  For $g_X : X \\to \\mathbf{B}G$ a cocycle and $P \\to X$ the corresponding\n$G$-principal $\\infty$-bundle according to Theorem \\ref{PrincipalInfinityBundleClassification},\nthere is a natural equivalence\n$$\n  \\Gamma_X(P \\times_G V) \\simeq \\mathbf{H}_{\/\\mathbf{B}G}(g_X, \\mathbf{c})\n$$\nbetween the space of sections of the corresponding $\\rho$-associated $V$-bundle \n(Definition~\\ref{AssociatedBundle}) and the hom-$\\infty$-groupoid of the slice\n$\\infty$-topos of $\\mathbf{H}$ over $\\mathbf{B}G$, between $g_X$ and $\\mathbf{c}$. \nSchematically:\n\\[\n\\left\\{ \n\t\\begin{xy} \n\t\t(10,10)*+{E}=\"1\";\n\t\t(-10,-10)*+{X}=\"2\";\n\t\t(10,-10)*+{X}=\"3\"; \n\t\t(5,0)*+{}=\"4\";\n\t\t(3,-5)*+{}=\"5\"; \n\t\t{\\ar^-{p} \"1\";\"3\"};\n\t\t{\\ar^-{\\sigma} \"2\";\"1\"}; \n\t\t{\\ar_-{\\mathrm{id}} \"2\";\"3\"};\t\n\t\t{\\ar@{=>}^-{\\simeq} \"4\";\"5\"};\n\t\\end{xy} \n\\right\\} \n \\;\\;\n    \\simeq\n  \\;\\;\n\\left\\{\n\\begin{xy} \n\t\t(10,10)*+{V\/\\!\/G}=\"1\";\n\t\t(-10,-10)*+{X}=\"2\";\n\t\t(10,-10)*+{\\mathbf{B}G}=\"3\"; \n\t\t(5,0)*+{}=\"4\";\n\t\t(3,-5)*+{}=\"5\"; \n\t\t{\\ar^-{\\mathbf{c}} \"1\";\"3\"};\n\t\t{\\ar^-{\\sigma} \"2\";\"1\"}; \n\t\t{\\ar_-{\\mathrm{g_X}} \"2\";\"3\"};\t\n\t\t{\\ar@{=>}^-{\\simeq} \"4\";\"5\"};\n\t\\end{xy} \n\\right\\} \n\\]\n\\label{SectionsAndSliceHoms}\n\\end{proposition}\n\\proof\n  By Observation \\ref{SectionBySliceMaps} and Lemma \\ref{SliceMapsIntoPullbacks}.\n\\hfill{$\\square$}\\\\\n\\begin{observation}\n  If in the above the cocycle $g_X$ is trivializable, in the sense that it factors through the\n  point $* \\to \\mathbf{B}G$ (equivalently if its class $[g_X] \\in H^1(X,G)$ is\n  trivial) then there is an equivalence\n  $$\n    \\mathbf{H}_{\/\\mathbf{B}G}(g_X, \\mathbf{c})\n\t\\simeq\n\t\\mathbf{H}(X,V)\n\t\\,.\n  $$\n  \\label{TwistedCohomologyIsLocallyVCohomology}\n\\end{observation}\n\\proof\n  In this case the homotopy pullback on the right in the proof of\n  Proposition \\ref{SectionsAndSliceHoms} is\n  $$\n\t\\raisebox{20pt}{\n\t\\xymatrix{\n\t  \\mathbf{H}_{\/\\mathbf{B}G}(g_X, \\mathbf{c})\n\t  \\ar@{}[r]|{\\simeq}\n\t  &\n\t  \\mathbf{H}(X, V)\n\t  \\ar[r]\n\t  \\ar[d]\n\t  &\n\t  \\mathbf{H}(X, V\/\\!\/G)\n\t  \\ar[d]^{\\mathbf{H}(X, \\mathbf{c})}\n\t  \\\\\n\t  \\{g_X\\}\n\t  \\ar@{}[r]|\\simeq\n\t  &\n\t  \\mathbf{H}(X, {*})\n\t  \\ar[r]\n\t  &\n\t  \\mathbf{H}(X, \\mathbf{B}G)\n\t}\n\t}\n  $$\n  using that $V \\to V\/\\!\/G \\stackrel{\\mathbf{c}}{\\to} \\mathbf{B}G$ is a fiber\n  sequence by definition, and that $\\mathbf{H}(X,-)$ preserves this fiber sequence.\n\\hfill{$\\square$}\\\\\n\\begin{remark}\n  Since by Proposition~\\ref{EveryGBundleIsLocallyTrivial} \n  every cocycle $g_X$ trivializes locally over some cover \n  $\\xymatrix{U \\ar@{->>}[r] & X}$\n  and equivalently, by Proposition \\ref{AssociatedIsFiberBundle}, every\n  $\\infty$-bundle $P \\times_G V$ trivializes locally, \n  Observation \\ref{TwistedCohomologyIsLocallyVCohomology} says that\n  elements $ \\sigma \\in \\Gamma_X(P \\times_G V) \\simeq \\mathbf{H}_{\/\\mathbf{B}G}(g_X, \\mathbf{c})$\n  \\emph{locally} are morphisms $\\sigma|_U : U \\to V$ with values in $V$.\n   They fail to be so \\emph{globally} to the extent that $[g_X] \\in H^1(X, G)$ is non-trivial,\n   hence to the extent that $P \\times_G V \\to X$ is non-trivial.\n\\end{remark}\nThis motivates the following definition.\n\\begin{definition}\n  We say that \n  the $\\infty$-groupoid\n  $\\Gamma_X(P \\times_G V) \\simeq \\mathbf{H}_{\/\\mathbf{B}G}(g_X, \\mathbf{c})$\n  from Proposition \\ref{SectionsAndSliceHoms}\n  is the $\\infty$-groupoid of \\emph{$[g_X]$-twisted cocycles} with values in $V$, with respect to the\n  \\emph{local coefficient $\\infty$-bundle} $V\/\\!\/G \\stackrel{\\mathbf{c}}{\\to} \\mathbf{B}G$.\n  \n  Accordingly, its set of connected components we call the \n  \\emph{$[g_X]$-twisted $V$-cohomology} with respect to the local coefficient bundle $\\mathbf{c}$\n  and write:\n  $$\n    H^{[g_X]}(X, V) := \\pi_0 \\mathbf{H}_{\/\\mathbf{B}G}(g_X, \\mathbf{c})\n\t\\,.\n  $$\n  \\label{TwistedCohomologyInOvertopos}\n\\end{definition}\n\\begin{remark}\n  The perspective that twisted cohomology is the theory of sections of \n  associated bundles whose fibers are classifying spaces is maybe most famous\n  for the case of twisted K-theory, where it was described in this form in \n  \\cite{Rosenberg}. But already the old theory of \n  \\emph{ordinary cohomology with local coefficients} is of this form, \n  as is made manifest in \\cite{BFG} (we discuss this in detail in \\cite{NSSc}).\n  \n  A proposal for a comprehensive theory in terms of bundles of topological spaces is\n  in \\cite{MaySigurdsson} and a systematic formulation \n  in $\\infty$-category theory and\n  for the case of multiplicative generalized cohomology theories is in \n  \\cite{AndoBlumbergGepner}. The formulation above refines this, unstably, \n  to geometric cohomology\n  theories\/(nonabelian) sheaf hypercohomology, \n  hence from bundles of classifying spaces to \n  $\\infty$-bundles of moduli $\\infty$-stacks. \n  \n  A wealth of examples and applications of such geometric nonabelian twisted cohomology\n  of relevance in quantum field theory and in string theory is discussed in \n  \\cite{SSS,Schreiber}.  We discuss further examples in \\cite{NSSc}.\n  \\label{ReferencesOnTwisted}\n\\end{remark}\n\\begin{remark}\n  Of special interest is the case where \n  $V$ is pointed connected, hence (by Theorem \\ref{DeloopingTheorem}) \n  of the form $V = \\mathbf{B}A$ for some $\\infty$-group $A$, and so \n  (by Definition \\ref{cohomology}) the coefficient for degree-1 $A$-cohomology, \n  and hence itself (by Theorem \\ref{PrincipalInfinityBundleClassification}) the moduli $\\infty$-stack for\n  $A$-principal $\\infty$-bundles. \n  In this case $H^{[g_X]}(X, \\mathbf{B}A)$ is\n  \\emph{degree-1 twisted $A$-cohomology}. Generally, if $V = \\mathbf{B}^n A$\n  it is \\emph{degree-$n$ twisted $A$-cohomology}. In analogy with Definition \n  \\ref{cohomology} this is sometimes written\n  $$\n    H^{n+[g_X]}(X,A) := H^{[g_X]}(X,\\mathbf{B}^n A)\n\t\\,.\n  $$\n  \n  Moreover, in this case $V\/\\!\/G$ is itself pointed connected, hence of the form\n  $\\mathbf{B}\\hat G$ for some $\\infty$-group $\\hat G$, and so the universal local \n  coefficient bundle\n  $$\n    \\xymatrix{\n\t  \\mathbf{B}A \\ar[r] & \\mathbf{B}\\hat G \\ar[d]^{\\mathbf{c}}\n\t  \\\\\n\t  & \\mathbf{B}G\n\t}\n  $$\n  exhibits $\\hat G$ as an \\emph{extension of $\\infty$-groups} of $G$ by $A$.\n  This case we discuss below in Section \\ref{ExtensionsOfCohesiveInfinityGroups}.\n  \\label{PointedConnectedLocalCoefficients}\n\\end{remark}\nIn this notation the local coefficient bundle $\\mathbf{c}$ is left implicit.\nThis convenient abuse of notation is justifed to some extent by the fact that there\nis a \\emph{universal local coefficient bundle}:\n\\begin{example}\n  The classifying morphism of the $\\mathbf{Aut}(V)$-action on some \n  $V \\in \\mathbf{H}$ from Definition \\ref{InternalAutomorphismGroup} according to \n  Theorem \\ref{PrincipalInfinityBundleClassification} yields a local coefficient \n  $\\infty$-bundle of the form\n  $$\n    \\raisebox{20pt}{\n    \\xymatrix{\n\t  V \\ar[r] & V\/\\!\/\\mathbf{Aut}(V)\\ar[d]\n\t  \\\\\n\t  & \\mathbf{B}\\mathbf{Aut}(V)\n\t}\n\t}\n  $$\n  which we may call the \\emph{universal local $V$-coefficient bundle}.\n  In the case that $V$ is pointed connected and hence of the form\n  $V = \\mathbf{B}G$\n  $$\n    \\raisebox{20pt}{\n    \\xymatrix{\n\t  \\mathbf{B}G \\ar[r] & (\\mathbf{B}G)\/\\!\/\\mathbf{Aut}(\\mathbf{B}G)\\ar[d]\n\t  \\\\\n\t  & \\mathbf{B}\\mathbf{Aut}(\\mathbf{B}G)\n\t}\n\t}\n  $$\n  the universal twists of the corresponding twisted $G$-cohomology are \n  the \\emph{$G$-$\\infty$-gerbes}. These we discuss below in section\n  \\ref{StrucInftyGerbes}.\n  \\label{UniversalTwistByAutomorphisms}\n\\end{example}\n\n\n\n\n\\subsection{Extensions and twisted bundles}\n\\label{ExtensionsOfCohesiveInfinityGroups}\n\n\nWe discuss the notion of \\emph{extensions} of $\\infty$-groups\n(see Section~\\ref{StrucInftyGroups}), generalizing the traditional notion of \ngroup extensions. \nThis is in fact a special case of the notion of \nprincipal $\\infty$-bundle, Definition~\\ref{principalbundle}, for base space objects that \nare themselves deloopings of $\\infty$-groups. \nFor every extension of $\\infty$-groups, there is the \ncorresponding notion of \n\\emph{lifts of structure $\\infty$-groups} of principal $\\infty$-bundles. \nThese are classified equivalently by trivializations of an \\emph{obstruction class}\nand by the twisted cohomology with coefficients in the extension itself, regarded\nas a local coefficient $\\infty$-bundle.\n\nMoreover, we show\nthat principal $\\infty$-bundles with an extended structure $\\infty$-group\nare equivalent to principal $\\infty$-bundles with unextended structure $\\infty$-group\nbut carrying a principal $\\infty$-bundle for the \\emph{extending} $\\infty$-group on their\ntotal space, which on fibers restricts to the given $\\infty$-group extension.\nWe formalize these \\emph{twisted (principal) $\\infty$-bundles}\nand observe that they are classified by twisted cohomology, \nDefinition~\\ref{TwistedCohomologyInOvertopos}.\n\n\\medskip  \n\n\\begin{definition}\n  \\label{ExtensionOfInfinityGroups}\nWe say a sequence of $\\infty$-groups (Definition~\\ref{inftygroupinootopos}),\n\\[\n  A \\to \\hat G \\to G\n\\]\nin $\\mathrm{Grp}(\\mathbf{H})$\n\\emph{exhibits $\\hat G$ as an extension of $G$ by $A$} if $\\hat G \\to G$ \nin $\\mathbf{H}$ is the\nquotient map $\\hat G \\to \\hat G \/\\!\/A$, such that the cocycle \n$G \\to \\mathbf{B}A$ in $\\mathbf{H}$ in $\\mathbf{H}$ corresponding\nto this by Theorem \\ref{PrincipalInfinityBundleClassification} is once deloopable.\n\nHence, by \nTheorem~\\ref{DeloopingTheorem}, the extension corresponds to a \nfiber sequence in $\\mathbf{H}$, Definition~\\ref{fiber sequence} of the form\n$$\n  \\xymatrix{\n    \\mathbf{B}A \\ar[r] &  \\mathbf{B}\\hat G \\ar[r]^{\\mathbf{p}} &  \\mathbf{B}G\n\t\\ar[r]^{\\mathbf{c}} & \\mathbf{B}^2 A\n\t\\,.\n  }\n$$\nWe write \n$$\n  \\mathrm{Ext}(G,A) := \\mathbf{H}(\\mathbf{B}G, \\mathbf{B}^2 A)\n  \\simeq (\\mathbf{B}A)\\mathrm{Bund}(\\mathbf{B}G)\n$$\nfor the \\emph{$\\infty$-groupoid of extensions} of $G$ by $A$.\n\\end{definition}\n\\begin{definition}\nGiven an $\\infty$-group extension  \n$\\xymatrix{A \\ar[r] & \\hat G \\ar[r]^{\\Omega \\mathbf{c}} & G}$ \nand given a $G$-principal $\\infty$-bundle $P \\to X$ in $\\mathbf{H}$, we say that a \n\\emph{lift} $\\hat P$ of $P$ to a $\\hat G$-principal $\\infty$-bundle is a lift \n$\\hat g_X$ of its classifying \ncocycle $g_X : X \\to \\mathbf{B}G$, under the equivalence of Theorem\n\\ref{PrincipalInfinityBundleClassification}, through the extension:\n$$\n  \\raisebox{20pt}{\n  \\xymatrix{\n      & \\mathbf{B} {\\hat G}\n      \\ar[d]^{\\mathbf{p}}\n    \\\\\n    X \\ar@{-->}[ur]^{\\hat g_X} \\ar[r]_-{g_X} & \\mathbf{B}G.\n  }\n  }\n$$\nAccordingly, the \\emph{$\\infty$-groupoid of lifts} of $P$ with respect to \n$\\mathbf{p}$ is \n$$\n  \\mathrm{Lift}(P,\\mathbf{p}) := \\mathbf{H}_{\/\\mathbf{B}G}(g_X, \\mathbf{p})\n  \\,.\n$$\n\\label{PrincipalInfinityBundleExtension}\n\\end{definition}\n\\begin{observation}\n  By the universal property of the $\\infty$-pullback, a lift exists precisely if the \n  cohomology class\n  $$\n    [\\mathbf{c}(g_X)] := [\\mathbf{c}\\circ g_X] \\in H^2(X, A)\n  $$\n  is trivial. \n  \\label{ObstructionClassObstructs}\n\\end{observation}\nThis is implied by Theorem \\ref{ExtensionsAndTwistedCohomology}, to which we turn after\nintroducing the following terminology.\n\\begin{definition}\n  In the above situation, \n  we call $[\\mathbf{c}(g_X)]$ the \\emph{obstruction class} to the extension;\n  and we call $[\\mathbf{c}] \\in H^2(\\mathbf{B}G, A)$ the\n  \\emph{universal obstruction class} of extensions through $\\mathbf{p}$.\n  \n  We say that a \\emph{trivialization} of the obstruction cocycle\n  $\\mathbf{c}(g_X)$ is a morphism $\\mathbf{c}(g_X) \\to *_X$ in $\\mathbf{H}(X, \\mathbf{B}^2 A)$,\n  where ${*}_X : X \\to * \\to \\mathbf{B}^2 A$ is the trivial cocycle. Accordingly, the\n  \\emph{$\\infty$-groupoid of trivializations of the obstruction} is\n  $$\n    \\mathrm{Triv}(\\mathbf{c}(g_X)) := \\mathbf{H}_{\/\\mathbf{B}^2 A}(\\mathbf{c}\\circ g_X, *_X)\n\t\\,.\n  $$\n  \\label{Obstructions}\n\\end{definition}\nWe give now three different characterizations of spaces of extensions of $\\infty$-bundles.\nThe first two, by spaces of twisted cocycles and by spaces of trivializations \nof the obstruction class, are immediate consequences of the previous discussion:\n\\begin{theorem}\n  Let $P \\to X$ be a $G$-principal $\\infty$-bundle corresponding by\n  Theorem \\ref{PrincipalInfinityBundleClassification} to a cocycle $g_X : X \\to \\mathbf{B}G$.\n  \\begin{enumerate}\n    \\item \n\t  There is a natural equivalence \n\t $$\n\t    \\mathrm{Lift}(P, \\mathbf{p}) \\simeq \\mathrm{Triv}(\\mathbf{c}(g_X))\n\t $$\n\t between the $\\infty$-groupoid of lifts of $P$ through $\\mathbf{p}$, \n     Definition \\ref{PrincipalInfinityBundleExtension},\t \n\t and the $\\infty$-groupoid of trivializations \n\t of the obstruction class, Definition \\ref{Obstructions}.\n\t \\item \n\t There is a natural equivalence\n\t $\\mathrm{Lift}(P, \\mathbf{p}) \\simeq \\mathbf{H}_{\/\\mathbf{B}G}(g_X, \\mathbf{p})$\n\t between the $\\infty$-groupoid of lifts and the $\\infty$-groupoid of \n\t $g_X$-twisted cocycles relative to $\\mathbf{p}$, Definition \\ref{TwistedCohomologyInOvertopos},\n\t hence a classification\n\t $$\n\t   \\pi_0 \\mathrm{Lift}(P, \\mathbf{P}) \\simeq H^{1+[g_X]}(X, A)\n\t $$\n\t of equivalence classs of lifts by the $[g_X]$-twisted $A$-cohomology of $X$\n\t relative to the local coefficient bundle\n    $$\n      \\raisebox{20pt}{\n      \\xymatrix{\n         \\mathbf{B}A \\ar[r] & \\mathbf{B}\\hat G \\ar[d]^{\\mathbf{p}}  \n    \t \\\\\n\t     & \\mathbf{B}G\\,.\n       }\n      }\n    $$  \n\\end{enumerate}\n \\label{ExtensionsAndTwistedCohomology}\n\\end{theorem}\n\\proof\n  The first statement is the special case of Lemma \\ref{SliceMapsIntoPullbacks}\n  where the $\\infty$-pullback $E_1 \\simeq f^* E_2$ in the notation there is identified\n  with $\\mathbf{B}\\hat G \\simeq \\mathbf{c}^* {*}$.\n  The second is evident after unwinding the definitions.\n\\hfill{$\\square$}\\\\\n\\begin{remark}\n  For the special case that $A$ is 0-truncated, we may, by the discussion in \n  \\cite{NikolausWaldorf, NSSc}, identify \n  $\\mathbf{B}A$-principal $\\infty$-bundles with $A$-\\emph{bundle gerbes},\n  \\cite{Mur}.\n  Under this identification the $\\infty$-bundle classified by the \n  obstruction class $[\\mathbf{c}(g_X)]$ above is what is called the\n  \\emph{lifting bundle gerbe} of the lifting problem, see for instance \n  \\cite{CBMMS} for a review. \n  In this case \n  the first item of Theorem \\ref{ExtensionsAndTwistedCohomology} reduces to\n  Theorem 2.1 in \\cite{Waldorf} and Theorem A (5.2.3) in \\cite{NikolausWaldorf2}. \n  The reduction of this statement to \n  connected components, \n  hence the special case of Observation \\ref{ObstructionClassObstructs},\n  was shown in \\cite{BreenBitorseurs}.\n  \\label{ReferencesOnLiftings}\n\\end{remark}\nWhile, therefore, the discussion of extensions of $\\infty$-groups and of \nlifts of structure $\\infty$-groups\nis just a special case of the discussion in the previous sections, this special case\nadmits geometric representatives of cocycles in the corresponding twisted cohomology by\ntwisted principal $\\infty$-bundles. This we turn to now. \n\\begin{definition}\n  \\label{TwistedBundle}\n  Given an extension of $\\infty$-groups $A \\to \\hat G \\xrightarrow{\\Omega \\mathbf{c}} G$\n  and given a $G$-principal $\\infty$-bundle $P \\to X$, with class $[g_X] \\in H^1(X,G)$, \n  a \\emph{$[g_X]$-twisted $A$-principal $\\infty$-bundle} on $X$ is an $A$-principal\n  $\\infty$-bundle $\\hat P \\to P$ such that the cocycle $q : P \\to\\mathbf{B}A$\n  corresponding to it under Theorem \\ref{PrincipalInfinityBundleClassification}\n  is a morphism of $G$-$\\infty$-actions.\n  \n  The \\emph{$\\infty$-groupoid of $[g_X]$-twisted $A$-principal $\\infty$-bundles on $X$} is \n  $$\n    A\\mathrm{Bund}^{[g_X]}(X)\n\t:=\n\tG \\mathrm{Action}( P, \\mathbf{B}A )\n\t\\subset\n\t\\mathbf{H}(P, \\mathbf{B}A)\n\t\\,.\n  $$\n\\end{definition}\n\\begin{observation}\n  \\label{BundleOnTotalSpaceFromExtensionOfBundles}\n  Given an $\\infty$-group extension $A \\to \\hat G \\stackrel{\\Omega \\mathbf{c}}{\\to} G$,\n  an extension of a $G$-principal $\\infty$-bundle $P \\to X$\n  to a $\\hat G$-principal $\\infty$-bundle, Definition \\ref{PrincipalInfinityBundleExtension}, induces\n  an $A$-principal $\\infty$-bundle $\\hat P \\to P$ \n  fitting into a pasting diagram of $\\infty$-pullbacks of the form\n$$\n  \\raisebox{30pt}{\n  \\xymatrix{\n    \\hat G \\ar[r] \\ar[d]^{\\Omega \\mathbf{c}} & \\hat P \\ar[r] \\ar[d] &  {*} \\ar[d]\n    \\\\\n    G \\ar[r] \\ar[d] & P \\ar[r]^{q} \\ar[d]& \\mathbf{B}A \\ar[r] \\ar[d] & {*} \\ar[d]\n    \\\\\n    {*} \\ar[r]^{x}&\n    X \\ar@\/_1pc\/[rr]_g \\ar[r]^{\\hat g} & \\mathbf{B}\\hat G\n    \\ar[r]^{\\mathbf{c}}& \\mathbf{B}G.\n  }\n  }\n$$\n  In particular, it has the following properties: \n  \\begin{enumerate}\n    \\item $\\hat P \\to P$ is a $[g_X]$-twisted $A$-principal bundle, Definition \\ref{TwistedBundle};\n\t\\item for all points $x : * \\to X$ the restriction of $\\hat P \\to P$ to the \n\tfiber $P_x$ is equivalent to the $\\infty$-group extension $\\hat G \\to G$.\n  \\end{enumerate}  \n\\end{observation}\n\\proof\nThis follows from repeated application of the pasting law for $\\infty$-pullbacks, \nProposition~\\ref{PastingLawForPullbacks}.\n\nThe bottom composite $g : X \\to \\mathbf{B}G$ is a cocycle for the given $G$-principal $\\infty$-bundle \n$P \\to X$ and it factors through $\\hat g : X \\to \\mathbf{B}\\hat G$ by assumption of the existence of \nthe extension $\\hat P \\to P$. \n\nSince also the bottom right square is an $\\infty$-pullback by the given $\\infty$-group extension, \nthe pasting law asserts that the square over $\\hat g$ is also an  $\\infty$-pullback, and then that so \nis the square over $q$. This exhibits $\\hat P$ as an $A$-principal $\\infty$-bundle over $P$ classified\nby the cocycle $q$ on $P$. By Proposition~\\ref{ClassificationOfTwistedGEquivariantBundles}\nthis $\\hat P \\to P$ is twisted $G$-equivariant.\n\nNow choose any point $x : {*} \\to X$ of the base space \nas on the left of the diagram. Pulling this back upwards \nthrough the diagram and using the pasting law and the \ndefinition of loop space objects $G \\simeq \\Omega \n\\mathbf{B}G \\simeq * \\times_{\\mathbf{B}G} *$ the diagram \ncompletes by $\\infty$-pullback squares on the left as indicated, which proves the claim.\n\\hfill{$\\square$}\\\\\n\n\\begin{theorem}\n  The construction of Observation \\ref{BundleOnTotalSpaceFromExtensionOfBundles}\n  extends to an equivalence of $\\infty$-groupoids\n  $$\n    A\\mathrm{Bund}^{[g_X]}(X)\n\t\\simeq\n\t\\mathbf{H}_{\/\\mathbf{B}G}(g_X, \\mathbf{c})\n  $$\n  between that of $[g_X]$-twisted $A$-principal bundles on $X$, \n  Definition \\ref{TwistedBundle}, and the cocycle $\\infty$-groupoid of\n  degree-1 $[g_X]$-twisted $A$-cohomology, Definition \\ref{TwistedCohomologyInOvertopos}.\n  \n  In particular the classification of $[g_X]$-twisted $A$-principal bundles is\n  $$\n    A\\mathrm{Bund}^{[g_X]}(X)_{\/\\sim}\n\t\\simeq\n\tH^{1+[g_X]}(X, A)\n\t\\,.\n  $$\n  \\label{ClassificationOfTwistedGEquivariantBundles}\n\\end{theorem}\n\\proof\n  For $G = *$ the trivial group, the statement reduces to  \n  Theorem~\\ref{PrincipalInfinityBundleClassification}. The general proof\n  works along the same lines as the proof of that theorem.\n  The key step is the generalization of the proof of Proposition~\\ref{LocalTrivialityImpliesCocycle}.\n  This proceeds verbatim as there, only with $\\mathrm{pt} : * \\to \\mathbf{B}G$ generalized to\n  $i : \\mathbf{B}A \\to \\mathbf{B}\\hat G$. \n  The morphism of $G$-actions\n  $P \\to \\mathbf{B}A$ and a choice of effective epimorphism $U \\to X$ over\n  which $P \\to X$ trivializes gives rise to a morphism in \n  $\\mathbf{H}^{\\Delta[1]}_{\/(* \\to \\mathbf{B}G)}$ which involves the diagram\n    $$\n    \\raisebox{20pt}{\n    \\xymatrix{\n\t  U \\times G \\ar@{->>}[r] \\ar[d] & P \\ar[r] \\ar[d] & \\mathbf{B}A \\ar[d]^{i}\n\t  \\\\\n\t  U \\ar@{->>}[r] & X \\ar[r] & \\mathbf{B}\\hat G\n\t}\n\t}\n\t\\;\\;\n\t\\simeq\n\t\\;\\;\n    \\raisebox{20pt}{\n    \\xymatrix{\n\t  U \\times G \\ar@{->>}[rr] \\ar[d] & & \\mathbf{B}A \\ar[d]^{i}\n\t  \\\\\n\t  U \\ar[r] & {*} \\ar[r]^{\\mathrm{pt}} & \\mathbf{B}\\hat G\n\t}\n\t}\n  $$\n  in $\\mathbf{H}$. (We are using that for the 0-connected object $\\mathbf{B}\\hat G$\n  every morphism $* \\to \\mathbf{B}G$ factors through $\\mathbf{B}\\hat G \\to \\mathbf{B}G$.)\n  Here the total rectangle and the left square on the left are $\\infty$-pullbacks,\n  and we need to show that the right square on the left is then also an \n  $\\infty$-pullback. Notice that by the pasting law the rectangle on the\n  right is indeed equivalent to the pasting of $\\infty$-pullbacks\n  $$    \n    \\raisebox{20pt}{\n    \\xymatrix{\n\t  U \\times G \\ar@{->}[r] \\ar[d] & G \\ar[r] \\ar[d] &  \\mathbf{B}A \\ar[d]^{i}\n\t  \\\\\n\t  U \\ar[r] & {*} \\ar[r]^{\\mathrm{pt}} & \\mathbf{B}\\hat G\n\t}\n\t}\n  $$\n  so that the relation\n  $$\n    U^{\\times^{n+1}_X} \\times G\n\t\\simeq\n\ti^* (U^{\\times^{n+1}_X})\n  $$\n  holds. With this the proof finishes as in the proof of \n  Proposition~\\ref{LocalTrivialityImpliesCocycle}, with $\\mathrm{pt}^*$\n  generalized to $i^*$. \n\\hfill{$\\square$}\\\\\n\n\\begin{remark}\n  Aspects of special cases of this theorem can be identified in the literature.\n  For the special case of ordinary extensions of ordinary Lie groups,\n  the equivalence of the corresponding extensions of a principal bundle with \n  certain equivariant structures on its total space is essentially the content\n  of \\cite{Mackenzie, Androulidakis}. \n  In particular the  twisted unitary bundles or \\emph{gerbe modules}\n  of twisted K-theory \\cite{CBMMS}\n  are equivalent to such structures.\n\n  For the case of $\\mathbf{B}U(1)$-extensions of Lie groups, such as the\n  $\\mathrm{String}$-2-group, the equivalence of the corresponding \n  $\\mathrm{String}$-principal 2-bundles, \n  by the above theorem, to certain bundle gerbes on the total spaces of \n  principal bundles underlies constructions such as in \\cite{Redden}.\n  Similarly the bundle gerbes on double covers considered in \n  \\cite{SSW} are $\\mathbf{B}U(1)$-principal 2-bundles on $\\mathbb{Z}_2$-principal\n  bundles arising by the above theorem from the extension \n  $\\mathbf{B}U(1) \\to \\mathbf{Aut}(\\mathbf{B}U(1)) \\to \\mathbb{Z}_2$,\n  a special case of the extensions that we consider in the next Section\n  \\ref{StrucInftyGerbes}.\n  \n  These and more examples we discuss in detail in \\cite{NSSc}.\n  \\label{ReferencesTwistedBundles}\n\\end{remark}\n\n\n\n\\subsection{Gerbes}\n\\label{section.InfinityGerbes}\n\\label{StrucInftyGerbes}\n\nRemark \\ref{PointedConnectedLocalCoefficients} above indicates that of special relevance are those\n$V$-fiber $\\infty$-bundles $E \\to X$ in an $\\infty$-topos $\\mathbf{H}$ \nwhose typical fiber $V$ is pointed connected,\nand hence is the moduli $\\infty$-stack $V = \\mathbf{B}G$ of $G$-principal \n$\\infty$-bundles for some $\\infty$-group $G$. \nDue to their local triviality, \nwhen regarded as objects in the slice $\\infty$-topos $\\mathbf{H}_{\/X}$,\nthese $\\mathbf{B}G$-fiber $\\infty$-bundles are themselves \\emph{connected objects}.\nGenerally, for $\\mathcal{X}$ an $\\infty$-topos regarded as an $\\infty$-topos\nof $\\infty$-stacks over a given space $X$, it makes sense to consider\nits connected objects as $\\infty$-bundles over $X$. Here we discuss these\n\\emph{$\\infty$-gerbes}.\n\n\\medskip\n\nIn the following discussion it is useful to consider two $\\infty$-toposes:\n\\begin{enumerate}\n  \\item an ``ambient'' $\\infty$-topos $\\mathbf{H}$ as before, to be thought of \n    as an $\\infty$-topos ``of all geometric homotopy types'' for a given notion of geometry,  \n\tin which $\\infty$-bundles are given by \\emph{morphisms} and the terminal \n\tobject plays the role of the geometric point $*$;\n  \\item an $\\infty$-topos $\\mathcal{X}$, to be thought of as the topos-theoretic\n   incarnation of a single geometric homotopy type (space) $X$, hence as  \n   an $\\infty$-topos of\n  ``geometric homotopy types {\\'e}tale over $X$'', in which an $\\infty$-bundle over $X$\n  is given by an \\emph{object} and the terminal object plays the role of the base space $X$. \n  \n  In practice, $\\mathcal{X}$ is the slice\n  $\\mathbf{H}_{\/X}$ of the previous  ambient $\\infty$-topos over $X \\in \\mathbf{H}$, or\n  the smaller $\\infty$-topos $\\mathcal{X} = \\mathrm{Sh}_\\infty(X)$ of (internal)\n  $\\infty$-stacks over $X$. \n\\end{enumerate}\nIn topos-theory literature the role of $\\mathbf{H}$ above is sometimes referred to as that of a \n\\emph{gros} topos and then the role of $\\mathcal{X}$ is referred to as that of a \\emph{petit} topos. \nThe reader should beware that much of the classical literature on gerbes is written\nfrom the point of view of only the \\emph{petit} topos $\\mathcal{X}$. \n\n\n\\medskip\n\nThe original definition of a \\emph{gerbe} \non $X$ \\cite{Giraud} is: a stack $E$ (i.e.\\  a \n1-truncated $\\infty$-stack) over $X$ that is \n1. \\emph{locally non-empty} and 2. \\emph{locally connected}. \nIn the more intrinsic language of higher topos theory, these \ntwo conditions simply say that $E$ is \na \\emph{connected object} (Definition 6.5.1.10 in \\cite{Lurie}): \n1. the terminal morphism $E \\to *$ is an \neffective epimorphism and 2. the 0th homotopy sheaf is trivial, \n$\\pi_0(E) \\simeq *$. This reformulation is \nmade explicit in the literature for instance in \nSection 5 of \\cite{JardineLuo} and in Section 7.2.2 of \\cite{Lurie}. Therefore:\n\\begin{definition}\n  For $\\mathcal{X}$ an $\\infty$-topos, a \\emph{gerbe} in $\\mathcal{X}$ is an object $E \\in \\mathcal{X}$ which is\n  \\begin{enumerate}\n    \\item connected;\n\t\\item 1-truncated.\n  \\end{enumerate}\n  For $X \\in \\mathbf{H}$ an object, a \\emph{gerbe $E$ over $X$} is a gerbe in the slice\n  $\\mathbf{H}_{\/X}$. This is an object $E \\in \\mathbf{H}$ together with an effective epimorphism $E \\to X$ such that $\\pi_i(E) = X$ for all $i \\neq 1$. \n  \\label{Gerbe}\n\\end{definition}\n\\begin{remark}\n  Notice that conceptually this is different from the\n  notion of \\emph{bundle gerbe} introduced in \\cite{Mur} \n  (see \\cite{NikolausWaldorf} for a review). \n  We discuss in \\cite{NSSc} that \n  bundle gerbes are presentations of \\emph{principal} $\\infty$-bundles \n  (Definition \\ref{principalbundle}).\n  But gerbes -- at least the \\emph{$G$-gerbes}\n  considered in a moment in Definition \\ref{G-Gerbe} -- \n  are $V$-fiber $\\infty$-bundles (Definition \\ref{FiberBundle})\n  hence \\emph{associated} to principal $\\infty$-bundles \n  (Proposition \\ref{VBundleIsAssociated}) with the special \n  property of having pointed connected fibers. \n  By Theorem \\ref{VBundleClassification}  \n  $V$-fiber $\\infty$-bundles may be identified with their underlying $\\mathbf{Aut}(V)$-principal \n  $\\infty$-bundles and so one may identify $G$-gerbes with nonabelian \n  $\\mathrm{Aut}(\\mathbf{B}G)$-bundle gerbes\n  (see also around Proposition~\\ref{ClassificationOfGGerbes} below), \n  but considered generally, neither of these\n  two notions is a special case of the other. Therefore the terminology is \n  slightly unfortunate, but it is standard.\n\\end{remark}\nDefinition \\ref{Gerbe} has various obvious generalizations. The following is considered in \\cite{Lurie}.\n\\begin{definition}\n  \\index{$\\infty$-gerbe!EM $n$-gerbe}\n For $n \\in \\mathbb{N}$, an \\emph{EM $n$-gerbe} is an object $E \\in \\mathcal{X}$ which is\n \\begin{enumerate}\n   \\item $(n-1)$-connected;\n   \\item $n$-truncated.\n \\end{enumerate}\n\\end{definition}\n\\begin{remark}\nThis is almost the definition of an \n\\emph{Eilenberg-Mac Lane object} in $\\mathcal{X}$, \nonly that the condition requiring a global section \n$* \\to E$ (hence $X \\to E$) is missing. Indeed, the \nEilenberg-Mac Lane objects of degree $n$ in \n$\\mathcal{X}$ are precisely the EM $n$-gerbes of \n\\emph{trivial class}, according to Proposition~\\ref{ClassificationOfGGerbes} below.\n\\end{remark}\nThere is also an earlier established definition of \n\\emph{2-gerbes} in the literature \\cite{Breen}, which is \nmore general than EM 2-gerbes. Stated in the above fashion it reads as follows.\n\\begin{definition}[Breen \\cite{Breen}]\n  \\index{$\\infty$-gerbe!2-gerbe}\n  \\index{2-gerbe}\n  A \\emph{2-gerbe} in $\\mathcal{X}$ is an object $E \\in \\mathcal{X}$ which is\n  \\begin{enumerate}\n    \\item connected;\n\t\\item 2-truncated.\n  \\end{enumerate}\n\\end{definition}\nThis definition has an evident generalization to arbitrary degree, which we adopt here.\n\\begin{definition}\n  \\label{nGerbe}\n  An \\emph{$n$-gerbe} in $\\mathcal{X}$ is an object $E \\in \\mathcal{X}$ which is \n   \\begin{enumerate}\n      \\item connected;\n\t  \\item $n$-truncated.\n   \\end{enumerate}\nIn particular an \\emph{$\\infty$-gerbe} is a connected object.\n\\end{definition}\nThe real interest is in those $\\infty$-gerbes which have a prescribed\ntypical fiber:\n\\begin{remark}\nBy the above, $\\infty$-gerbes (and hence EM $n$-gerbes \nand 2-gerbes and hence gerbes) are much like deloopings \nof $\\infty$-groups (Theorem \\ref{DeloopingTheorem}) \nonly that there is no requirement that there \nexists a global section. An $\\infty$-gerbe for which there exists a  \nglobal section $X \\to E$ is called \\emph{trivializable}. By \nTheorem~\\ref{DeloopingTheorem} trivializable $\\infty$-gerbes are equivalent to $\\infty$-group objects in $\\mathcal{X}$ \n(and the $\\infty$-groupoids of all of these are equivalent \nwhen transformations are required to preserve the canonical global section).\n\\end{remark}\nBut \\emph{locally} every $\\infty$-gerbe $E$ is of this form. For let\n$$\n  (x^* \\dashv x_*) :\n  \\xymatrix{\n     \\mathrm{Grpd}_{\\infty}\n\t    \\ar@{<-}@<+3pt>[r]^-{x^*}\n\t    \\ar@<-3pt>[r]_-{x_*}\n\t\t&\n     \\mathcal{X}\t \n  }\n$$\nbe a topos point. Then the stalk $x^* E \\in \\mathrm{Grpd}_{\\infty}$ \nof the $\\infty$-gerbe is connected: because inverse images \npreserve the finite $\\infty$-limits involved in the definition of homotopy sheaves, and preserve the terminal object. Therefore\n$$\n  \\pi_0\\,  x^* E \\simeq x^* \\pi_0 E \\simeq x^* * \\simeq *\n  \\,.\n$$\nHence for every point $x$ we have a stalk $\\infty$-group $G_x$ and an equivalence \n$$\n  x^* E \\simeq B G_x \n  \\,.\n$$\nTherefore one is interested in the following notion.\n\\begin{definition}\n  For $G \\in \\mathrm{Grp}(\\mathcal{X})$ an $\\infty$-group \n  object, a \\emph{$G$-$\\infty$-gerbe} is an $\\infty$-gerbe $E$ such that there exists \n  \\begin{enumerate}\n    \\item an effective epimorphism $\\xymatrix{U \\ar@{->>}[r] & X}$;\n\t\\item an equivalence $E|_U \\simeq \\mathbf{B} G|_U$.\n  \\end{enumerate}\n  Equivalently: a $G$-$\\infty$-gerbe is a $\\mathbf{B}G$-fiber $\\infty$-bundle,\n  according to Definition \\ref{FiberBundle}.\n  \\label{G-Gerbe}\n\\end{definition}\nIn words this says that a $G$-$\\infty$-gerbe is one that locally looks like the \nmoduli $\\infty$-stack of $G$-principal $\\infty$-bundles.\n\\begin{example}\n  For $X$ a topological space and $\\mathcal{X} = \n  \\mathrm{Sh}_\\infty(X)$ the $\\infty$-topos of $\\infty$-sheaves over it, these notions reduce to the following.\n  \\begin{itemize}\n    \\item a 0-group object $G \\in \\tau_{0}\\mathrm{Grp}(\\mathcal{X}) \n    \\subset \\mathrm{Grp}(\\mathcal{X})$ is a sheaf of groups on $X$ \n    (here $\\tau_0\\mathrm{Grp}(\\mathcal{X})$ denotes the 0-truncation of \n    $\\mathrm{Grp}(\\mathcal{X})$;\n\t\\item for $\\{U_i \\to X\\}$ any open cover, the canonical \n\tmorphism $\\coprod_i U_i \\to X$ is an effective epimorphism to the terminal object;\n\t\\item $(\\mathbf{B}G)|_{U_i}$ is the stack of \n\t   $G|_{U_i}$-principal bundles ($G|_{U_i}$-torsors).\n  \\end{itemize}\n\\end{example}\nIt is clear that one way to construct a $G$-$\\infty$-gerbe \nshould be to start with an $\\mathbf{Aut}(\\mathbf{B}G)$-principal \n$\\infty$-bundle, Remark \\ref{UniversalTwistByAutomorphisms},\nand then canonically \\emph{associate} a fiber $\\infty$-bundle to it.\n\\begin{example}\n  \\label{automorphism2GroupAbstractly}\n  For $G \\in \\tau_{0}\\mathrm{Grp}(\\mathrm{Grpd}_{\\infty})$ an \n  ordinary group, $\\mathbf{Aut}(\\mathbf{B}G)$ is usually called the \n  \\emph{automorphism 2-group} of $G$. Its underlying groupoid is equivalent to\n  \\[\n  \\mathbf{Aut}(G) \\times G\\rightrightarrows \n  \\mathbf{Aut}(G), \n  \\]\n  the action groupoid for the action of $G$ on $\\mathbf{Aut}(G)$ \n  via the homomorphism $\\mathrm{Ad}\\colon G\\to \\mathbf{Aut}(G)$.  \n\\end{example}\n\\begin{corollary}\n  Let $\\mathcal{X}$ be a  1-localic $\\infty$-topos \n  (i.e.\\ one that has  a 1-site of definition). \n  Then for $G \\in \\mathrm{Grp}(\\mathcal{X})$ any \n  $\\infty$-group object, $G$-$\\infty$-gerbes are classified by $\\mathbf{Aut}(\\mathbf{B}G)$-cohomology:\n  $$\n    \\pi_0 G \\mathrm{Gerbe}\n\t\\simeq\n\t\\pi_0 \\mathcal{X}(X,\\mathbf{B}\\mathbf{Aut}(\\mathbf{B}G))\n\t=:\n\tH^1_{\\mathcal{X}}(X,\\mathbf{Aut}(\\mathbf{B}G))\n\t\\,.\n  $$\n  \\label{ClassificationOfGGerbes}\n\\end{corollary}\n\\proof\n  This is the special case of Theorem \\ref{VBundleClassification}\n  for $V = \\mathbf{B}G$.\n\\hfill{$\\square$}\\\\\nFor the case that $G$ is 0-truncated (an ordinary group object) \nthis is the content of Theorem 23 in \\cite{JardineLuo}.\n\\begin{example}\n  For $G \\in \\mathrm{Grp}(\\mathcal{X}) \\subset \n  \\tau_{\\leq 0}\\mathrm{Grp}(\\mathcal{X})$ an ordinary 1-group object, \n  this reproduces the classical result of \\cite{Giraud}, \n  which originally motivated the whole subject: \n  by Example~\\ref{automorphism2GroupAbstractly} in this case $\\mathbf{Aut}(\\mathbf{B}G)$ is the traditional automorphism 2-group and \n  $H^1_{\\mathcal{X}}(X, \\mathbf{Aut}(\\mathbf{B}G))$\n  is Giraud's nonabelian $G$-cohomology that classifies $G$-gerbes\n  (for arbitrary \\emph{band}, see Definition \\ref{BandOfInfinityGerbe} below).\n  \n  For $G \\in \\tau_{\\leq 1}\\mathrm{Grp}(\\mathcal{X}) \\subset \n  \\mathrm{Grp}(\\mathcal{X})$ a 2-group, we recover the classification of 2-gerbes\n  as in \\cite{Breen,BreenNotes}.\n\\end{example}\n\\begin{remark}\n  In Section 7.2.2 of \\cite{Lurie} the special case that here\n  we called \\emph{EM-$n$-gerbes} is considered. Beware \n  that there are further differences: for instance the notion \n  of morphisms between $n$-gerbes as defined in \\cite{Lurie} is more \n  restrictive than the notion considered here. For instance with our definition \n  (and hence also that in \\cite{Breen}) each group automorphism of \n  an abelian group object $A$ induces an automorphism of the \n  trivial $A$-2-gerbe $\\mathbf{B}^2 A$. But, except for the identity, \n  this is not admitted in \\cite{Lurie} (manifestly so by the diagram \n  above Lemma 7.2.2.24 there). Accordingly, the classification \n  result in \\cite{Lurie} is different: it involves  the cohomology group  \n  $H^{n+1}_{\\mathcal{X}}(X, A)$. Notice that there is a canonical morphism\n  $$\n    H^{n+1}_{\\mathcal{X}}(X, A) \\to H^1_{\\mathcal{X}}(X, \\mathbf{Aut}(\\mathbf{B}^n A))\n  $$  \n  induced from the morphism $\\mathbf{B}^{n+1}A \\to \\mathbf{Aut}(\\mathbf{B}^n A)$.\n\\end{remark}\n\nWe now discuss how the $\\infty$-group extensions, Definition \\ref{ExtensionOfInfinityGroups},\ngiven by the Postnikov stages of $\\mathbf{Aut}(\\mathbf{B}G)$ induces the notion of\n\\emph{band} of a gerbe, and how the corresponding twisted cohomology,\naccording to Remark \\ref{ExtensionsAndTwistedCohomology}, reproduces \nthe original definition of nonabelian cohomology in \\cite{Giraud} and generalizes\nit to higher degree.\n\\begin{definition}\n \\label{outerAutomorphismInfinityGroup}\n Fix $k \\in \\mathbb{N}$. For $G \\in \\infty\\mathrm{Grp}(\\mathcal{X})$ a \n $k$-truncated $\\infty$-group object (a $(k+1)$-group), write\n $$\n   \\mathbf{Out}(G) := \\tau_{k}\\mathbf{Aut}(\\mathbf{B}G)\n $$\n for the $k$-truncation of $\\mathbf{Aut}(\\mathbf{B}G)$. (Notice \n that this is still an $\\infty$-group, since by Lemma 6.5.1.2 in \n \\cite{Lurie} $\\tau_n$ preserves all $\\infty$-colimits and additionally \n all products.) We call this the \\emph{outer automorphism $n$-group} of $G$.\n \n In other words, we write\n $$\n   \\mathbf{c} : \\mathbf{B}\\mathbf{Aut}(\\mathbf{B}G) \\to \\mathbf{B}\\mathbf{Out}(G)\n $$\n for the top Postnikov stage of $\\mathbf{B}\\mathbf{Aut}(\\mathbf{B}G)$.\n\\end{definition}\n\\begin{example}\n  Let $G \\in \\tau_0\\mathrm{Grp}(\\mathrm{Grpd}_{\\infty})$ be \n  a 0-truncated group object, an \n  ordinary group,.\n  Then by Example \\ref{automorphism2GroupAbstractly}, \n  $\\mathbf{Out}(G) = \\mathbf{Out}(G)$ is the coimage of \n  $\\mathrm{Ad} : G \\to \\mathrm{Aut}(G)$, which is the traditional group of outer automorphisms of $G$.\n\\end{example}\n\\begin{definition}\n  Write $\\mathbf{B}^2 \\mathbf{Z}(G)$ for the $\\infty$-fiber of \n  the morphism $\\mathbf{c}$ from Definition \\ref{outerAutomorphismInfinityGroup}, \n  fitting into a fiber sequence\n  $$\n    \\xymatrix{\n      \\mathbf{B}^2 \\mathbf{Z}(G) \\ar[r] & \n\t    \\mathbf{B}\\mathbf{Aut}(\\mathbf{B}G) \\ar[d]^{\\mathbf{c}}\n\t  \\\\\n\t  & \\mathbf{B}\\mathbf{Out}(G)\n\t}\n\t\\,.\n  $$\n  We call $\\mathbf{Z}(G)$ the \\emph{center} of the $\\infty$-group $G$.\n  \\label{Center}\n\\end{definition}\n\\begin{example}\n  For $G$ an ordinary group, so that $\\mathbf{Aut}(\\mathbf{B}G)$ \n  is the automorphism 2-group from Example \\ref{automorphism2GroupAbstractly},\n  $\\mathbf{Z}(G)$ is the center of $G$ in the traditional sense.\n\\end{example}\nBy theorem \\ref{ClassificationOfGGerbes} there is an  induced morphism\n$$\n  \\mathrm{Band} \n    : \n  \\pi_0 G \\mathrm{Gerbe} \\to H^1(X, \\mathbf{Out}(G))\n   \\,.\n$$\n\\begin{definition}\n  \\label{BandOfInfinityGerbe}\n  For $E \\in G \\mathrm{Gerbe}$ we call $\\mathrm{Band}(E)$ the \\emph{band} of $E$.\n\n  By using Definition \\ref{Center} in Definition \\ref{TwistedCohomologyInOvertopos}, \n  given a band $[\\phi_X] \\in H^1(X, \\mathbf{Out}(G))$, \n  we may regard it as a twist for twisted $\\mathbf{Z}(G)$-cohomology,\n  classifying $G$-gerbes with this band:\n  $$\n    \\pi_0 G \\mathrm{Gerbe}^{[\\phi_X]}(X) \\simeq H^{2+[\\phi_X]}(X, \\mathbf{Z}(G))\n\t\\,.\n  $$\n\\end{definition}\n\\begin{remark}\n  The original definition of \\emph{gerbe with band} in \\cite{Giraud} is slightly\n  more general than that of \\emph{$G$-gerbe} (with band) in \\cite{Breen}: in the former\n  the local sheaf of groups whose delooping is locally equivalent to the gerbe need not\n  descend to the base. These more general Giraud gerbes are 1-gerbes in the sense of\n  Definition \\ref{nGerbe}, but only the slightly more restrictive $G$-gerbes of Breen\n  have the good property of being connected fiber $\\infty$-bundles. From our\n  perspective this is the decisive property of gerbes, and the notion of band is \n  relevant only in this case.\n\\end{remark}\n\\begin{example}\n  For $G$ a 0-group this reduces to the notion of band as introduced in  \\cite{Giraud},\n  for the case of $G$-gerbes as in \\cite{Breen}.\n\\end{example}\n\n\\medskip\n\n\\noindent{\\bf Acknowledgements.}\nThe writeup of this article and the companions \\cite{NSSb, NSSc} was\ninitiated during a visit by the first two authors to the third author's institution, \nUniversity of Glasgow, in summer 2011.  It was completed in summer 2012\nwhen all three authors were guests at the \nErwin Schr\\\"{o}dinger Institute in Vienna.  \nThe authors gratefully acknowledge the support of \nthe Engineering and Physical Sciences Research Council \ngrant number EP\/I010610\/1 and the support of the ESI.\nU.S. thanks Domenico Fiorenza for inspiring discussion about\ntwisted cohomology.\n\n\n\n\n\\addcontentsline{toc}{section}{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe \\texttt{iopart-num} Bib\\TeX{} style is intended for use in\npreparing manuscripts for Institute of Physics Publishing journals,\nincluding Journal of Physics.  It provides numeric citation with\nHarvard-like formatting, based upon the specification in ``How to\nprepare and submit an article for publication in an IOP journal using\n\\LaTeXe'' by Graham Douglas (2005).\n\nThe \\texttt{iopart-num} package is available on the Comprehensive\n\\TeX{} Archive Network (CTAN) as \\texttt{\/biblio\/bibtex\/contrib\/iopart-num}.\n\n\\section{General instructions}\n\nTo use the \\texttt{iopart-num} style, include the command\n\\verb+\\bibliographystyle{iopart-num}+ in the\ndocument preamble.  The reference section is then inserted into the\ndocument with the command \\verb+\n\\section{Introduction}\n\nThe Compact Muon Solenoid (CMS) Experiment is a general purpose particle detector experiment located at the Large Hadron Collider (LHC) at CERN, in Geneva, Switzerland.\nThe design of the  CMS detector centers around a 3.8T iron-core superconducting solenoid.  Within the solenoid, at the very heart of the detector, is an all silicon tracker, consisting of 66M pixel and 10M strip detectors.  \nCalorimetry is performed by a high-resolution, high-granularity electromagnetic calorimeter (ECAL), comprised of more than 70k lead-tungstate crystals, and a brass-scintillator hadronic calorimeter (HCAL).\nJust outside the solenoid is a scintillator-based tail-catching hadronic outer calorimeter (HO).  In addition, a steel and quartz fibre forward hadronic calorimeter system (HF) is placed on both sides of the detector endcaps to ensure hermiticity.\nThe iron yoke of the magnet system is instrumented with a muon spectrometer, composed of 3 detector technologies: drift tubes (DT) in the central region and cathode strip chambers (CSC) in the endcaps, both complemented by resistive plate chambers (RPC).\nConsidering the late availability of the underground cavern, a modular detector design was chosen to allow for construction, installation, and testing before lowering underground.  \nFigure~\\ref{fig:cms} (left) shows a schematic drawing of the CMS Detector.\nFurther detail on the design and performance of the CMS detector can be found elsewhere~\\cite{cmsjinst,tdr}.\n\\begin{figure}[bt!]\n\\begin{center}\n\\includegraphics[height=5.5cm]{cms_complete_labelled.eps}\\hspace{1pc}\n\\includegraphics[height=5.0cm]{cms_photo.eps}\n\\caption{\\label{fig:cms}(left) Schematic of the Compact Muon Solenoid detector. (right) The CMS detector during installation of the silicon strip tracker, Dec 2007.  }\n\\end{center}\n\\end{figure}\n\nThe construction phase of the CMS detector is now complete, and the detector has been fully installed and commissioned in the underground cavern at LHC Point 5.   \nFigure~\\ref{fig:cms} (right) shows a photo of the CMS detector during installation of the tracker.\nSince May 2007, CMS has undergone frequent ``global runs'' devoted to global commissioning, exercising all of the installed detector subsystems together.\nIn addition to testing the global readout, global runs have also helped exercise event selection and reconstruction, data analysis, alignment and calibration, data quality monitoring, computing operations and data transfer.\nThe complexity and scope has increased with each global run, with a larger fraction of subdetectors joining at each interval.\nThe global runs culminated in data taking for over a week with first LHC beam in September 2008, and a two week Cosmic Run at Full Tesla (CRAFT) in October-November 2008.\n\n\n\n\\section{Performance with First LHC Beam Data}\\label{sec:beam}\n\nIn September 2008, the CMS detector collected first beam data from the LHC.  The first proton beams were circulated successfully on 10 September 2008.  Single beams were circulated through the LHC for several days, at injection energy of 450 GeV, until 19 September.  During this time, CMS triggered and recorded data successfully.\n\nDuring beam running, CMS triggered on beam halo muons, muons outside of the beam-pipe arising from decays of pions created when off-axis protons scrape collimators or other beam-line elements.  \nBeam halo events were particularly useful for testing the triggering and reconstruction of events in the endcap muon system.  \nFigure~\\ref{fig:csc} (left) shows the beam halo trigger rate as measured by one sector of the muon endcap CSCs.  The shaded region denotes the period of first successful LHC beam RF capture, which lasted for approximately 10 minutes and ended with a beam abort.  The halo rate measured by the CSCs was clearly reduced during and after the RF capture of the beam.  The HCAL endcap system also saw a ``cleaner'' beam with less energy deposition along the beam-line after the RF capture of the beam.\n\nFigure~\\ref{fig:csc} (right) shows the distribution of the angle with respect to the beam-line of muon tracks passing through the endcap CSC system.\nSince beam halo muons pass through the detector collinear with the beam-line, we expect this angle to be small for pure beam halo events (beam halo simulation, in blue).  Cosmic muons pass through the detector more orthogonally to the beam-line, and thus we expect this angle to be larger for cosmic muon ``beam off'' events (in black).  For ``beam on'' events, we expect a combination of beam halo and cosmic muon events (in orange filled).  Figure~\\ref{fig:csc} shows this trend validated with data and simulation events.\n\n\\begin{figure}[bt!]\n\\begin{center}\n\\includegraphics[height=4.5cm]{beamhalorate.eps}\n\\includegraphics[height=5.0cm]{cms_csc_halo_cosmics.eps}\n\\caption{\\label{fig:csc}(left) Beam halo rate (Hz) as a function of time (hour:min).  The shaded region denotes the ~10min period when the LHC beam was first RF captured.  (right) Angle of muon tracks with respect to beam line, as measured by the endcap muon cathode strip chambers, with beam data (orange filled), cosmics data (black), and beam halo simulation (blue). }\n\\end{center}\n\\end{figure}\n\nIn the days preceding September 10th, single beam shots were sent onto closed collimators near the CMS detector.  The interactions of the beam with the collimator and surrounding material created a large number of secondary particles which passed through the CMS detector.  These beam-on-collimator events are called ``beam splash'' events because of the spectacular signature they left in the detectors.  In particular, these events deposited an enormous amount of energy in the calorimeters;  in a single beam splash event, $\\sim100$ and $\\sim1000$ TeV of energy were deposited in the hadronic and electromagnetic calorimeters, respectively.  Figure~\\ref{fig:evtdisplay} shows an event display for such a ``beam splash'' event. In addition to leaving a spectacular signature in the detector, these events have been used to internally synchronize ECAL and HCAL, and also to validate the inter-calibration of the ECAL endcap channels.\n\n\\begin{figure}[b!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{splash_longitudinal_ecal.eps}\n\\includegraphics[width=0.49\\textwidth]{splash_longitudinal_hcal.eps}\n\\caption{Event display images from a single ``beam splash'' event.  \nThe left image shows the energy deposited in the electromagnetic calorimeter (pink).  The right image shows the energy deposited in the hadronic calorimeter (blue).\nIn a single beam splash event, $\\sim100$ and $\\sim1000$ TeV of energy were deposited in the hadronic and electromagnetic calorimeters, respectively. \n\n\\label{fig:evtdisplay}}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{Performance with Data from the Cosmic Run at Full Tesla (CRAFT)}\\label{sec:craft}\n\nThe aim of the Cosmic Run at Full Tesla (CRAFT) was to run CMS for four weeks, with all subsystems participating, collecting data continuously to further gain operational experience before data-taking with $p$-$p$ collisions.  In addition, a major goal of CRAFT was to operate CMS at full field (3.8T) for as much of the running period as possible and to study the effects of the magnetic field on the detector components.  Prior to CRAFT, the most extensive test of the CMS detector at full magnetic field was during the Magnet Test and Cosmic Challenge in 2006.\n\nDuring CRAFT, we collected more than 370M cosmic events during four weeks in the interval of 13 October to 11 November 2008.  In this period, CMS was operated with B=3.8T for 19 days.  290M of the 370M events had B=3.8T, with the silicon strip tracker and the muon drift tube system in the readout.  194M of the total had all subdetector components in the readout.  Figure~\\ref{fig:craft} (left) shows an event display for a CRAFT cosmic ray muon passing through the CMS detector.\nThe large CRAFT data set has given rise to many analyses, from subsystem and trigger performance studies, calibration and alignment, validation of the magnetic field map, measurements of cosmic muon flux and charge ratio, to cosmic muon background studies for LHC running.  Figure~\\ref{fig:craft} (right) shows the results from one such analysis, the study of energy deposition in the ECAL crystals.  We see good agreement between the measured $dE\/\\rho dx$ in ECAL and the expected stopped power for lead-tungstate crystals.\n\n\\begin{figure}[bt!]\n\\begin{center}\n\\includegraphics[height=14pc]{cms_craft.eps}\n\\includegraphics[height=14pc]{cms_ecal_dedx.eps}\n\\caption{\\label{fig:craft}(left) Event display for a CRAFT cosmic muon passing through the CMS Detector. In blue is the reconstructed muon track passing through the muon system, the hadronic and electromagnetic calorimeter, as well as the tracker. (right) The $dE\/\\rho dx$ stopping power of the CMS electromagnetic crystal calorimeter, as a function of muon momentum.  The blue points correspond to data from CRAFT cosmic muon events; the black curve is the expected stopping power of lead-tungstate crystals.  The red dashed line shows the contribution from collision loss; the blue dashed line shows the contribution from bremsstrahlung.}\n\\end{center}\n\\end{figure}\n\n\n\\section{Conclusions}\n\nAfter many years of design and construction, the CMS detector has been commissioned and has collected first data with LHC beams.  The detector performance has been proven with beam splash, beam halo, as well as cosmic ray muon events with and without B field.\nSince the beginning of September 2008, all installed subsystems of the CMS detector have routinely been in global readout, and the stability of running with all CMS components has been proven.  \nSince the CRAFT run at the end of 2008, CMS has continued global commissioning with cosmic ray muons, in preparation for upcoming LHC beams in 2009.\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDuring the 1960s and 1970s, with the advent of powerful numerical techniques for\nperforming non-LTE radiative transfer calculations, several groups vigorously\npursued modeling of critical spectral lines in an effort to understand the\nstructure and energy balance of the solar chromosphere.\nTwo striking problems quickly emerged.\nThe first concerned the wings of strong resonance lines\\footnote{Here we define line ``cores'' and ``wings'' in terms of observed profiles, and\nFigure~\\ref{fig:problem} shows the \\ion{Mg}{2} lines.\nThe core region lies roughly between the maxima in intensity marked historically\nas ``$k_2$'' in the figure, the wings lie outside of these maxima.},\n\\new{which arose under}  the simplifying\napproximation of complete redistribution (CRD): the wings were factors of\nseveral too bright\n\\citep{1967AnAp...30..861D, 1973A&A....22...61L}.\nThe wing problem was resolved using the more realistic approach of partial\nredistribution\n\\citep[PRD:][]{1973ApJ...185..709M,1974ApJ...192..769M,1976ApJ...205..874A}.\nIn this more realistic description, the emission and absorption processes in\nspectral lines allow for frequency-by-frequency coupling between these processes\nin the line wings, where scattering in the atomic reference frame is coherent.\nThe resolution of the wing discrepancy is illustrated by the qualitative\nagreement between observations and the standard VAL3-C 1D model\n\\citep[][``VAL\"]{Vernazza+Avrett+Loeser1981}\nin Fig.~\\ref{fig:problem}.\nIn this figure, the variations of the wing intensities with wavelength are\ncaptured by these traditional models.\n(The offset in intensity is within the calibration uncertainties of the rocket\nspectra).\n\n\\figproblem\n\nIn the second problem, the centers of line cores ($k_3$ in the figure),\ncontrolled by CRD, were computed to be too dim by factors of several, relative\nto the $k_2$ peaks.\nThese discrepancies remain today for some of the strongest lines in solar and\nstellar spectra, including H~L$\\alpha$, and to a lesser extent L$\\beta$\n\\citep{1978ApJ...225..655G,1979ApJ...230..924B};\nthe \\ion{Mg}{2} $h$ and $k$ lines\n\\citep{Vernazza+Avrett+Loeser1981};\nthe \\ion{Ca}{2} $H$ and $K$ lines\n\\citep{1981A&A...103..160L}.\n\nObserved profiles of these strong lines were recognized to be asymmetric.\nThe explanation proposed early concerns the relative bulk motion between the\nmiddle and upper chromosphere\n\\citep{1970SoPh...11..347A},\na result compatible with the upwardly-propagating radiating shocks calculated in\n1D models by\n\\citet{Carlsson+Stein1995}.\nThis remains a related problem of interest here, but we do not address it\ndirectly here.\n\nMore sophisticated calculations based upon 3D radiation-MHD (R-MHD) models\ncomputed with the Bifrost code\n\\citep{2011A&A...531A.154G}\nhave reduced core $k_3\/k_2$ \\ion{Mg}{2} intensity ratio discrepancies somewhat\n\\citep{2017A&A...597A..46S},\nand also for the $K_3\/K_2$ \\ion{Ca}{2} ratios\n\\citep{2018A&A...611A..62B}.\nBut issues in line cores \\new{remained, in particular \nthey found that observed peak separations implied that something was missing in their models.}\n\nPhysically, the cores of the strong lines generally correspond to wavelengths\nwhere Doppler-shifted thermal and other motions control the opacity and source function.\nTherefore in this ``Doppler core'', CRD is a reasonable approximation.\nWhile the Doppler broadening widths are not known \\textit{a priori}, the cores\nare considered to span about 3~times the r.m.s. Doppler width, before\nboth opacity and source function become controlled by coherent\nscattering.\nIn models and observations, this 3$\\times$Doppler width varies systematically\nfrom $\\pm 25$~$\\,$km$\\,$s$^{-1}${} for heavy elements such as calcium, to $\\pm 50$~$\\,$km$\\,$s$^{-1}${}\nfor hydrogen.\n\nGiven the importance of strong lines to the chromospheric energy balance, the\nonset of the solar corona, the irradiance effects in the solar system and the\nrecent work on the Hanle effect in the cores of these lines\n\\citep[see, for example,][]{2018cosp...42E1564I},\nwe re-analyse various models and data-sets to understand better the meaning of\nthe remaining discrepancies between observation and models.\n\n\\figval\n\nThe early 1D models remain a reasonable first physical approximation because of\nsteep hydrostatic stratification, where the dominant direction of transfer of\nradiative energy lies in the vertical direction\n\\citep[e.g.][]{2017ApJ...851....5J}.\nHowever, even though in the 3D R-MHD models the stratification is steep (isobars\nare more horizontal than vertical), there are many conditions where, owing to\nmagnetic and Reynolds stresses evolving in response to convection beneath, the\neffects of horizontal radiative transfer are important. \\new{Thus we study \n1D and \n3D~models, focussing on  the differences \narising between 1D~formal versus 3D~formal solutions.}\nOur attention is focused on the resonance lines of \\ion{H}{1}, \\ion{Mg}{2} and\n\\ion{Ca}{2}.\n\nLastly, a unique study by \\citet{1994A&A...287..233B} reported simultaneous spectra\nacquired with various $1\\arcsec$-wide image-plane slits and different spectrograph\nexit slits, in the resonance lines of \\ion{H}{1}, \\ion{Mg}{2} and \\ion{Ca}{2},\nfrom the LPSP instrument on the OSO-8 satellite.\nThey found that, while \\ion{Mg}{2} and \\ion{Ca}{2} data were similar, the\nL$\\alpha$ data appeared qualitatively different.\n\n\\section{Statement of the problem} \n\nFigures~\\ref{fig:problem} and \\ref{fig:val3c} illustrate our core problem of\ninterest: \\textit{the calculated profiles have ratios between peak and core\nbrightness that systematically exceed observations, even those with spectral\nresolutions of 200,000, by a factor of 1.5--2}.\nBelow we will examine in detail data of \\ion{Mg}{2} from the Interface Region\nImaging Spectrograph \\citep[IRIS,][]{2014SoPh..289.2733D} and from calculations\nwhich highlight spatially-resolved spectra in contrast to the comparison of\nspatial averages shown.\n\nArmed only with 1-dimensional calculations, early resolutions that were proposed\nto this problem were necessarily limited.\nWithin each calculation, the only option is to examine the assumed velocity\nfields  within the chromosphere.\nIf included as a formal ``microturbulence'' (random fluid velocities exist on\nscales below the photon mean free path), the effect is to increase the\nwavelength differences between the  intensity peaks.\nAs shown in Fig.~\\ref{fig:problem}, the separation-of-peaks for the $h$ and\n$k$ lines in standard 1D calculations are close to spatially-averaged observed\nvalues.\nThe well-known effect of variations in micro-turbulence is recalled in\nFigure~\\ref{fig:val3c}.\n(Curiously, using the distribution of micro-turbulence of VAL3-C, the peak\nintensities are split by similar amounts in wavelength, yet the line center\nwavelengths are in the ratio $1 : 2.3 : 3.3$).\n\nAnother option in 1D modeling is to include line broadening as macro-turbulence,\nthen within each 1D calculation, the velocity would be introduced as a\nmacroscopic, not microscopic flow.\nIn this case, the emergent profiles, for changes small in magnitude compared\nwith micro-scale motions, introduce shifts (and asymmetries if velocity\ngradients exist) of the entire core profiles.\nThis resolution, proposed by earlier authors\n\\citep[e.g.][]{1978ApJ...225..655G},\nsuggests that each observation sums spatially over intensities from several such\nadjacent atmospheres in each pixel.\nFigure~\\ref{fig:cartoon} shows a schematic picture of this proposed solution.\nEach atmosphere is a 1D solution to the non-LTE equations.\nIf each observation corresponds to a superposition of a set of 1D calculations\nthat are randomly distributed in line of sight velocity, to some degree the\ndifferences can be reconciled.\nEach observation then corresponds to the simple sum of individual theoretical\nprofiles, each shifted in wavelength according to a bulk velocity shift.\n\n\\cartoon\n\nSpectroscopic turbulence was studied in the context of 1D~models by\n\\citet{1985cdm..proc..137C}.\nThey found that the regimes of applicability of the micro- and macro- turbulent\nlimits were found to be ${<}25$~km and ${>}3000$~km respectively, using velocity\nfields with a variety of correlation lengths, for lines of \\ion{Ca}{2}.\nThese lengths can be compared with a photon mean free path of 120~km and\nthickness of the entire stratified chromosphere of 1500~km.\nThey showed that typical profiles could not be explained by a linear combination\nof micro- and macro-turbulent flows.  \n\nThree-dimensional radiation-MHD (R-MHD) models\n\\citep{2017A&A...597A..46S,2018A&A...611A..62B}\nare based not on \\textit{ad-hoc} explorations, but are solutions to the dynamic equations of motion and radiative transfer.\nThey are performed on grids fine enough to avoid too much dissipation, but for\nwhich solutions can be obtained in reasonable computing time.\nThe calculations examined here have grids with spacings of 49 and 34~km\nhorizontally and vertically respectively, and are therefore capable of exploring\nradiative transfer in different regimes of spectroscopic turbulence to the\nextent that their intrinsic numerical smoothing permits significant changes in\nvelocities to occur between neighboring slices through the 3D~atmosphere which\ndetermine the opacities and source functions.\nThese calculations were advanced in time using radiation transfer solutions based\nupon the short-characteristics method, which is encumbered with a large,\nunphysical diffusion.\n\\new{The source functions for strong lines are dominated by scattering.\nAs such, the modeled profiles, even computed with long  characteristics,}  must be treated with some caution.\nOur final goal is to explore how profiles of strong lines\ncomputed with the best 3D~MHD models compare with these observations.\n\n\\section{Observations and their analyses}\n\n\\subsection{Typical quiet Sun line profiles}\n\nFirst, we examine the highest dispersion spectra of the three lines from the\nliterature, of the quiet Sun, but with low spatial resolution.\nFigure~\\ref{fig:problem} shows data of both the $h$ and $k$ lines from a\n$7\\arcsec\\times130\\arcsec$ area of the quiet Sun\n\\citep{1977STIN...7831029A}.\nFigure~\\ref{fig:val3c} shows profiles from regions of quiet Sun, highlighting\nthe cores of \\ion{Ca}{2}, \\ion{Mg}{2} and \\ion{H}{1} resonance lines, from\nseveral instruments.\nThe figure compares the observations with calculations from the 1D VAL-3C model.\nSpectra of \\ion{Ca}{2} lines were taken from the atlas of\n\\citet{1973apds.book.....D}.\nThe spectrometer used had a native spectral resolution of\n${\\cal R}= 600,000$, so that instrumental broadening is negligible.\nThe \\ion{Mg}{2} spectra are taken from the ${\\cal R}= 200,000$ quiet Sun spectra\nshown in Fig.~\\ref{fig:problem}\n\\citep{1977STIN...7831029A}.\nSolar data for \\ion{H}{1}~L$\\alpha$ are all of a significantly lower spectral\nresolution.\nThose obtained from rocket flights or low Earth orbit are also affected by\nabsorption by neutral hydrogen in the Earth's upper atmosphere in the line core.\nThese include data from the HRTS-II experiment from\n\\citet{1991ApJS...75.1337B}\nwith ${\\cal R}= 24,000$ and the LPSP instrument on OSO-8, which obtained quiet\nSun spectra with ${\\cal R}= 60,000$.\nData from the SUMER instrument on SoHO have a lower spectral resolution of\n15,000, even though the instrument, on the SoHO spacecraft at the L1~Lagrange\npoint, is unaffected by the geocorona.\nL$\\alpha$ SUMER data also have been obtained, mostly behind a mesh at the edge\nof the detector, to avoid saturation.\n\\citet{2008A&A...492L...9C} obtained SUMER L$\\alpha$ data behind the\npartially-shut door of the telescope, of six quiet-Sun regions on\nJune 24--25, 2008.\nOwing to the apodization of the pupil plane by the partly-closed door,\nthe angular resolution is difficult to assess, but they found the characteristic\nasymmetry (blue peak brighter than red) which increased with brightness.\nThe central core was found to be typically 75\\%{} of the brightness of\nthe blue peak, independent of limb distance, at a spectral resolution of 15,000.\n\nQuiet Sun L$\\alpha$ profiles from all three instruments are compiled in the\nlower middle panel of Fig.~\\ref{fig:val3c}.\nThe SUMER data shown there are from \\citet{2001A&A...375..591C}.\nAll three lines show similar qualitative differences with 1D~calculations.\n1D~calculations tend to predict brighter peaks and a darker central intensity.\n\nA closer look at Figures~\\ref{fig:val3c} and \\ref{fig:cartoon}\nimmediately yields tight constraints on any macro-turbulence.\nThe right-most panels of Fig.~\\ref{fig:val3c}, plotted against Doppler velocity,\nshow that the \\ion{Ca}{2} $K$, \\ion{Mg}{2} $k$ and \\ion{H}{1} L$\\alpha$ line cores \nwould need systematically different macroturbulent velocity distributions to\nexplain the observed core-peak intensity ratios.\n\\textit{The \\ion{Ca}{2} $K$ line would require r.m.s. speeds of\n10~$\\,$km$\\,$s$^{-1}$, but \\ion{Mg}{2} $k$ and \\ion{H}{1} L$\\alpha$ would need\nr.m.s. values of 30 and 50~$\\,$km$\\,$s$^{-1}${} for the superposition shown in\nFig.~\\ref{fig:cartoon} to work.}\nIn stratified atmospheres, these three line cores form all within a region\ntermed the ``upper chromosphere'' (see, for example, Fig.~1 of\n\\citealp{2019A&A...623A..74D}\nand Fig.~3 of\n\\citealp{2015ApJ...803...65S}).\nWe note that the area coverage of spicules inferred from data above the limb is\ntoo small to contribute significantly to the spatially-averaged profiles,\n\\citep[e.g.][]{2010ApJ...719..469J,2016ARep...60..848M}.\nHowever, models do show that the $k_2$ and $K_2$ peaks form substantially lower\nin the atmosphere\n\\citep{Vernazza+Avrett+Loeser1981,2013ApJ...772...90L,2018A&A...611A..62B}.\n\nThe 3D~models have revealed that the peak separations of these lines can\nincrease in locations where the chromospheric temperature rise is located deep\nin the atmosphere (see Fig.~20 of \\citealp{2018A&A...611A..62B}).\nBut, no matter the details of the complex non-LTE formation of these\nlines, we know of no first-principles reasons nor data that are compatible with\nthe idea of a systematic gradient of macro-scale motions that can rescue the\nexplanation that the peak-to-core ratios are small because of\nmacro-turbulence.\nIndeed, below we show that the line profiles of \\ion{Mg}{2} lines obtained with\nthe IRIS instrument, obtained with a high cadence and at the highest angular\nresolution ever achieved, also can be used to reject this hypothesis.\n\n\\figsji\n\n\\subsection{Quiet Sun \\ion{Mg}{2} line  profiles in time and space}\n\nWe examine detailed profiles for the \\ion{Mg}{2} $h$ and $k$ lines from the IRIS\ninstrument\n\\citep{2014SoPh..289.2733D}.\nWhile IRIS does not measure the UV~spectra at the highest spectral resolution,\nthe linear properties of the detector combined with the high angular resolution\nmake it uniquely suited to address the problem of interest.\nMeasurements in the \\ion{Mg}{2} $h$ and $k$ lines have a spatial step along its\nslit of $0.168\\arcsec$, a critical sampling of the angular resolution of\n$0.33\\arcsec$.\nInspection of typical data (e.g. Figure~5 of \\citealp{2014SoPh..289.2733D})\nsuggests that quiet regions have only  modest significant spatial\nvariations of the $k_2$ and $h_2$ peaks observed along the projected spectrograph\nslit, as seen at the nominal angular resolution of IRIS.\nThe center-limb behavior shown in Fig.~6 of\n\\citet{2014SoPh..289.2733D},\nreveals that peak separations first increase, and finally disappear some\n$5\\arcsec$ above the limb seen in the neighboring continuum.\nThe line profiles become a narrow single peak $5\\arcsec$ higher.\n\nAll of the IRIS data were reduced to photometrically-calibrated spectra\nusing the IDL SolarSoft packages as well as deconvolved with the point spread\nfunction (PSF) of the IRIS telescope, which has been measured during a Mercury\ntransit by \\citet{2018SoPh..293..125C}.\n\nQuiet Sun \\ion{Mg}{2} data were obtained with IRIS close to disk center\nbeginning on February 27, 2014, at 5:39:28.850, in a ``sit and stare'' mode.\n290 frames of 774 0.16635$\\arcsec$-wide pixels (solar $y$) by\n554 0.0254~\\AA{}-wide wavelength pixels.\nWe will focus on spectra, but simultaneous IRIS slitjaw images were obtained in\nthe 1400~\\AA{} channel.\nFigure~\\ref{fig:sji} shows examples of these images obtained 1000 and\n3500 seconds into the time-series.\nEach has been median-filtered over time (90~seconds) to remove internetwork\noscillations and fast dynamics in the low-density transition region.\n\\figyt\nThe thick lines show three regions we highlight below: network, haze, and\ninter-network.\nThe color table saturates the highest DN to reveal UV~continuum emission that\nforms in the lower chromosphere.\n\n\\figirismacrotk\n\nFigure~\\ref{fig:irisyt} shows wavelength-integrated intensities of the\n\\ion{Mg}{2}~$k$ line core, spanning 1~\\AA{} (roughly the wavelength\nseparation between the $k_1$ minima), as a function of position along\nthe slit and time.\nThe data were acquired with a cadence of 16.8051 seconds, for a total duration\nof 81 minutes.\nThe total solar area rotating under the fixed slit was therefore roughly\n$13\\arcsec\\times 120\\arcsec$ (solar $x$ and $y$ respectively).\nThis area covers 1~part in 2000 of the area of the solar disk, below\n(Section~\\ref{subsec:lucia}) we therefore examine a larger region.\n\n\\figiris\n\nThe data, when deconvolved, have a sufficiently high angular resolution\n($0\\farcs33$) that the macro-turbulent picture can be directly addressed.\nEach resolution element (two pixels) corresponds to about 240~km on the solar\nsurface.\nIf a superposition such as that shown in Fig.~\\ref{fig:cartoon} were responsible\nfor filling-in the line cores, then some profiles adjacent in space should\nexhibit large variations between separate resolution elements, i.e. on scales\ndown to 240~km.\nBut such variations are absent, as shown in the typical, randomly chosen samples\nof line profiles of network, inter-network, and a hazy intermediate region.\nThese profiles are typical of the entire data-set, including the $h$-line.\n\n\\subsection{Spatial auto-correlations}\n\n\nThe leftmost panel \nof \nFigure~\\ref{fig:irismacrotk} \nhas not been spatially de-convolved, the others have.\nThe tick-marked line in the rightmost panel shows ten spatial pixels, \\new{each of} which\ncritically sample the convolved data.\nVery little structure below scales of about 5 pixels ($\\equiv 600$ km) is visible.\nThis figure prompted us to measure quantitatively the spatial structure along\nthe slit of the entire dataset, using auto-correlations.\nThus, we computed the spatial auto-correlation lengths along the IRIS slit at\neach wavelength, and time, for those columns in Fig.~\\ref{fig:irisyt}\ncorresponding to all three marked regions.\n\nThe figures below show autocorrelation lengths of the intensity images\n$I_\\lambda(y)$, which are the characteristic full-widths at half-maximum (FWHM)\nof the functions\n\\begin{equation}\n    c_\\lambda(\\ell) = \n         I_\\lambda(y) \\cdot I_\\lambda(y+\\ell),\n\\end{equation}\ncentered on the maximum ($\\ell=0$).\nSince $c_\\lambda(\\ell)$ depends on each wavelength $\\lambda$ across the line\nprofiles, any variations of line-of-sight velocity on resolvable scales\nwill be reflected by changes in the monochromatic intensities on the same length\nscales, thereby imprinting the velocities onto these correlation lengths.\nFigure~\\ref{fig:iris} shows these $c_\\lambda(\\ell)$ values.\nOver-plotted is the average line profile in blue, and the autocorrelation FWHM\nlengths averaged over space and time, in white, along the network, haze, and\ninter-network slit regions.\n\nSmall values of the FWHM of $c_\\lambda(\\ell)$ correspond to rapid changes in\nspace, and \\textit{vice versa}.\nAn obvious feature of Fig.~\\ref{fig:iris} are that in the wings of the\n$k$ line ($\\lambda$ outside of the $k_1$ minima) the three regions are not\neasily distinguishable, unlike the line cores which are radically different.\nThere are some extended periods where the structure in the wings of the network\ncontain consistently small FWHM values (between 1400 and 1600, or 1800 to 2200\nseconds for example), these upper photospheric signatures behave, statistically\nat least, in a similar fashion.\nThe average autocorrelation lengths outside of line cores are all close to\n400~km (the averages are shown as white lines in the figure).\n\nBut within the cores, the network and other regions behave quite differently.\nThe average intensity profiles in the haze and internetwork regions are similar\nin shape to the average correlation length profiles (white and blue lines).\nBut in the network, the very cores of the autocorrelation profiles possess a\npeak at the $k_3$ core that is absent in the intensity profiles.\nThe spatial autocorrelations are also larger farther from line center\n($\\Delta\\lambda > 0.1$~\\AA{} in the network region, with the exception of a short\nperiod around $t=2600$ seconds (left panel of Fig.~\\ref{fig:iris}).\nDuring the first half of the time series autocorrelation lengths can exceed\n${\\approx}2500$~km near the $k_2$ peaks, and almost symmetric about line center.\nLater, after about 3000 seconds the region of network shows an asymmetric\nautocorrelation profile in wavelength, line cores lengths exceeding 2000~km, but\nwith a coherent darker feature, i.e. smaller lengths between 0.1--0.2~\\AA{} to\nthe red of line center (Doppler redshifts of 10 and 20~$\\,$km$\\,$s$^{-1}$) between 3200 and\n4300 seconds.\n\nAll three regions have average spatial scales exceeding 800~km at all\nwavelengths which have significant core emission (within about ${\\pm}0.3$, 0.2,\nand 0.1~\\AA{} respectively for the network, haze, and internetwork).\nThe spatial auto-correlations within the cores (${\\pm}0.5$~\\AA) have structures\nwith a median of 1060~km, mean of 1140~km, and r.m.s. variation of\n440~km, for the haze and internetwork data, with 20\\%\\ higher typical values for\nthe network.\n\nIn contrast, the haze and internetwork regions possess a significant and\npersistent dark streak within 5--8~$\\,$km$\\,$s$^{-1}${} (Doppler shift) of $k_3$.\nThere, the spatial FWHM correlation lengths fall to 400~km, which is only about\ntwice the spatial resolution of IRIS.\nWe will speculate on the origin of these small structures below.\n\nIn summary, these IRIS data reveal that:\n\\begin{enumerate}\n\\item Away from the line cores (outside $k_1$ minima), all regions have\n    autocorrelation FWHM values close to 400~km, roughly twice the IRIS\n    resolution.\n\\item At and near wavelengths of intensity maxima ($k_2$), FWHM \nmeasurements \nexceed\n    1000~km, increasing to over 1500~km in network regions.\n\\item Outside of network regions, $k_3$ has a persistent small spatial FWHM of\n    $\\approx 400$~km, seen only in line center pixels and those two immediately\n    to the red side (${\\le}10$~$\\,$km$\\,$s$^{-1}${} from line center).\n\\item Over network regions, autocorrelation lengths are systematically larger,\n    extend further in $\\Delta\\lambda$, and have a broad minimum near 800--900~km\n    across the central ${\\pm}0.2$~\\AA.\n    Within ${\\pm}0.1$~\\AA{} the lengths increase producing a third, central small\n    peak in average autocorrelation profiles.\n\\end{enumerate}\nTo explain the small $k_3\/k_2$ ratios, the macro-turbulent picture requires\nlarge changes in line-of-sight velocities on scales down to the 200~km\nresolution limit of IRIS.\nOur analysis of typical IRIS data for \\ion{Mg}{2} rejects this hypothesis.\n\n\\subsection{Other relevant observations}\n\nOther spectral lines have been observed at a high angular resolution\nwhich can also shed light on conditions where the cores of strong solar\nresonance lines form.\nSpin-forbidden lines of \\ion{O}{1} at 1356 and 1358~\\AA{} have been observed\nrepeatedly with UV spectrometers.\nThe lines are optically thin across most of the chromosphere, the shared upper\nlevel is 9.15~eV above the ground level of \\ion{O}{1}.\nThe \\ion{O}{1} ionization is strongly tied to the \\ion{H}{1} level populations\nthrough a well-known resonant charge-transfer reaction, and so excitation by\ncollisions with electrons will occur under conditions common to these lines of\n\\ion{O}{1} and L$\\alpha$, whose upper level lies at 10.19~eV.\nNaturally, the L$\\alpha$ line source function and emergent intensity are\ndominated by photon scattering, unlike the spin-forbidden lines of \\ion{O}{1}.\nBut the two lines otherwise are excited under similar conditions within the\n chromosphere.\n\nThe HRTS instrument has provided spectra near 1350~\\AA{} \\new{at a spectral resolution \n(50~m\\AA) similar to IRIS, but a lower spatial (1$\\arcsec$) resolution.}\nHRTS data analyzed by\n\\citet{1989ApJ...346..514A}\nshow that this line's kinematics is unspectacular, with r.m.s. Doppler\nshifts of 2~$\\,$km$\\,$s$^{-1}$, and linewidths close to that of the instrument.\nThe unresolved motions are at most 40\\%{} of the sound speed, which is\n10--15~$\\,$km$\\,$s$^{-1}$.\nIt seems there is little power in motions, resolved or unresolved, across the\nquiet Sun's upper chromosphere.\n\\citet{2015ApJ...809L..30C}\nreported widths of ${\\sim}7$~km$\\,$s$^{-1}$ of the \\ion{O}{1} 1356~\\AA{} line in IRIS data\nin quiet and plage regions. \n\\new{Optically thin lines such as the 1356 line have contributions from many scale heights across the chromosphere, so their line widths reflect\nthe line-of-sight sum of many different heights.}\n\n\\subsection{A broader sample of \\ion{Mg}{2} IRIS data}\n\\label{subsec:lucia}\n\nThe above IRIS dataset was selected to represent a typical quiet Sun\nregion at disk center.\nBut these data sampled just 1 part in 2000 of the solar surface.\nWe therefore looked at statistical properties of the IRIS \\ion{Mg}{2} $h$ and\n$k$ profiles from a spatial (raster) scan, that of April 15, 2014,\nbetween 05:25:24 and 05:43:28 UT.\nThis spanned an area of $127\\arcsec \\times 119\\arcsec$, which is ten\ntimes larger than the sit-and-stare observation.\n\nThis observation included a little more active Sun, lying at the periphery of a\nsubstantial active region (NOAO 12036).\nWe found that only about 10\\%{} of the line profiles had $k_2\/k_3$ intensity\nratios between 4 and 6, spanning the value of ${\\approx}5$ for the VAL3-C\ncalculation.\nThe bright chromospheric network showed no more spatial correlation with these\ndeep line ratios than other pixels in the instrument's field of view.\nOn balance, these data from a larger area confirm that large $k_2\/k_3$ intensity\nratios above 4 are present at a 10\\%{} level.\nThe median ratio is 2.5. \n\n\\section{Re-analysis of recent 3D calculations}\n\nWe have examined model data from \\citet{2017A&A...597A..46S} and\n\\citet{2018A&A...611A..62B}.\nThe atmospheric structure in the models is the same in both articles and is\ndetermined by  R-MHD calculations, using the short\ncharacteristics method to solve for radiation gains and losses in the energy\nequation.\nAfter the calculations have been evolved, the authors then studied  solutions\nto the transfer equation using long characteristics for the 1D and 3D\ncalculations.\nHere we focus on the computations for the $k$ line by\n\\citet{2017A&A...597A..46S},\nand also comment on those for the \\ion{Ca}{2} $K$ line at 393.4nm.\nOur purpose here is to examine the hypothesis that the line cores of\nchromospheric resonance lines are filled in by horizontal radiative transfer.\n\nThe horizontal component of the numerical grid has pixels of width 47~km.\nWhile the MHD calculations upon which these radiative transfer calculations are\nbased are themselves highly diffusive compared with the real Sun, the thermal\nstructure in their calculations nevertheless has features down to these scales.\nThe computational data analyzed here have been re-binned to scales half of the native\nresolution, 98~km in the horizontal direction.\nThis scale is comparable or smaller than typical photon mean free paths through\nthe stratified chromosphere.\nWhile a finer grid is desirable, these calculations are fine enough to reveal\nthe effects explored below.\n\nThe modeled region is similar to  enhanced network on the Sun, with\nan unsigned magnetic field strength of 50~G passing through the photosphere\n\\citep{2016A&A...585A...4C,2017A&A...597A..46S,2018A&A...611A..62B}.\nA typical snapshot of a time-dependent 3D calculation is taken.\nFrom the source functions in each voxel, the long-characteristic method is used\nto produce the emergent intensities, including PRD.\nTherefore the solutions shown are a kind of hybrid: the source functions and\nthe optical depth scales are computed using the diffusive\nshort-characteristic method, but the final emergent intensities, given these\nparameters, are far less diffusive.\n\nFigure~\\ref{fig:mgii} shows emergent intensities from a small part\n($6\\times6$ Mm$^2$) of the calculated area ($24\\times24$ Mm$^2$), using the long\ncharacteristic method, for a region at disk center ($\\mu=1$).\nWhen compared with observations, we note firstly, that the computed features\n$k_{2V}$ and $k_{2R}$ are factors of between one and two brighter than the\naverage quiet Sun data shown.\nThis is to be expected given the higher concentration of the magnetic\nfield used in the computations.\nBut the $k_2\/k_3$ contrasts are also significantly higher, both in 3D and 1D,\nand the $k_{2V}$ and $k_{2R}$ separations are smaller than observed.\nQualitatively similar results (from \\citealp{2018A&A...611A..62B} but not shown\nhere) are seen for the \\ion{Ca}{2} $K$ line, where the average $K_{2R}$\nand $K_{2V}$ peaks are separated by 0.33~\\AA{} in both quiet and plage regions\n\\citep[compare Figures~3 and 5 of][]{Linsky+Avrett1970},\nwhereas the average over the computed enhanced network region is about half of\nthis (0.17~\\AA).\n\n\\figmgii\n\nSignificantly, in comparison with plage observations, these $k_3$ and $K_3$\ncomponents are far deeper in the computations.\nThis suggests that the source function is on average larger where this feature\nis formed than is captured in the calculations.\n\nIn Figure~\\ref{fig:autoc} we show spatial auto-correlation data computed from\nthe 3D and 1D intensities of the $k$ line, together with spatially-averaged\nintensity profiles.\nSolid lines show observations, dashed lines computations.\nThe auto-correlation characteristic lengths show three kinds of behavior\ndepending on the wavelengths relative to line center.\nIn the line wings, the images have the smallest spatial scales, with FWHM values\nof about 750 and 610~km.\nIn 3D, these scales steadily increase towards the line cores, peaking near the\nobserved peaks $k_{2V}$ and $k_{2R}$.\n(In 1D these lengths decrease towards the line center before showing a small\nincrease at $k_{2V}$ wavelengths).\nWithin the line core, the widths computed in 3D drop dramatically to below\n700~km.\n\n\\figautoc\n\nThe comparison of 3D and 1D images in Fig.~\\ref{fig:mgii} is instructive in ways\ncomplementary to the interesting points made by\n\\citet{2018A&A...611A..62B}.\nFigure~\\ref{fig:autoc} shows clearly that 3D~transfer effects are\nimportant within $\\pm 1$~\\AA{} of line center: the FWHM length scales are larger\non average by a factor of two for the $k$ line (15--20\\%\\ for the $K$ line) in\n3D than 1D.\nThe 3D features all appear ``fuzzier'' to the eye than their 1D counterparts, on\nscales of a few hundred kilometers in Fig.~\\ref{fig:mgii}.\nWe will speculate further on what is missing in these calculations in\nSection~\\ref{sec:discussion}.\n\nThe second and fourth rows of Fig.~\\ref{fig:mgii} show how the calculated model\nmight be observed through filter instruments, with FWHM values of 0.1, 1, and\n3~\\AA.\nWider FWHM instruments have been used in solar physics for many years to record\nthe \\ion{Ca}{2} lines, and other, narrower filter widths have been developed%\n\\footnote{see, e.g. \n\\url{https:\/\/www.su.se\/isf\/}}.\n\nThese panels suggest that 3D smearing effects are biggest at wavelengths within\nthe emission cores.\nThe differences are probably lower limits, given that the source functions for\nstrongly scattering lines are smeared by the use of the short characteristics,\nwhich will tend to underestimate differences between the 1D and 3D formal\nsolutions to the transfer equation.\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nWe have re-visited an old problem concerning the upper chromosphere of the Sun:\nis there a clear discrepancy between observed and computed profiles in the\nDoppler cores of the strongest lines?\nIf so, can we determine its origin?\nFirstly, we compiled data to re-examine literature from the 1970s when this\nproblem was first addressed.\nUsing these data, including optically thin lines generated in the same regions\nas the strong lines, we confirmed earlier work showing that micro-turbulence\ncannot account for the systematic behavior of the data and argued that\nmacro-turbulence also fails.\nOn this basis, we then examined the $k$ line of \\ion{Mg}{2} in detail, taking\nadvantage of the quality and angular resolution of the IRIS instrument.\nWe then explored radiation MHD simulations of an enhanced network\nregion of the $K$ line of \\ion{Ca}{2}.\nIn particular, we compared the line cores in 1D and 3D to explore the effects of\n horizontal radiative transfer on the emergent spectra generated\nthroughout the chromosphere.\n\nOur first conclusion is that \\textit{the small peak-core ratios in these\nresonance lines cannot be due to turbulence on any scale}.\nGiven the well-known result that micro-turbulence serves mostly to increase the\nseparation of peaks (see Fig.~\\ref{fig:val3c}), our conclusion rests on the\nrefutation of macro-turbulence.\nThree lines of argument all point to this result.\n\n\\begin{itemize}\n\\item The macro-turbulence limit requires the spatial averaging of profiles of\n    fluid elements, which are Doppler-shifted by a velocity distribution,\n    which spans the computed 1D separation of peaks.\n    If the distribution were narrower, little difference would be seen between\n    the 1D and average profiles.\n    If larger, then we would see multiple peaks for a small number of averaged\n    elements, and\/or a severely broadened feature.\n    But observations clearly show a steady increase of separation of peaks from\n    \\ion{Ca}{2} to \\ion{H}{1} by a factor of three in Doppler shift.\n    Yet the \\new{deep} cores of these features all form within the confines of the upper\n    chromosphere and higher. \\new{Whether or not the chromosphere is thermally isolated from the corona above by mostly horizontal magnetic fields, these line cores  likely form in regions of very steep gradients in electron  temperature where ionization causes them to become optically thin.}\n    \n    \n    \n\\item Optically thin UV lines observed with the HRTS instrument since the 1970s\n    also \\new{have contribnutions from} the upper chromosphere.\n    Spin-forbidden lines of \\ion{O}{1} near 1356~\\AA{} require almost the same\n    population of electron energies to be excited as H~L$\\alpha$, yet they are\n    spectrally unresolved by HRTS and show small subsonic deviations of just a\n    few $\\,$km$\\,$s$^{-1}${} along the instrument's slit\n    \\citep[e.g.,][]{1989ApJ...346..514A}.\n\\item Spatially de-convolved profiles of the \\ion{Mg}{2} resonance lines\n    observed by IRIS show very few features smaller than 600~km along the slit,\n    yet the spatial resolution is close to 240~km (Figures~\\ref{fig:irismacrotk}\n    and \\ref{fig:iris}).\n    Between the $k_1$ minima, average autocorrelation lengths vary between\n    800--1200~km, larger values being seen in brighter regions of the network.\n\\end{itemize}\n\nFor our second conclusion, based upon differences between the 1D and 3D\nautocorrelation calculations (Figure~\\ref{fig:mgii}), \\textit{at least some of\nthe peak-to-core ratio discrepancy is due to horizontal components of\nradiation transfer}.\nFigures~\\ref{fig:mgii} and \\ref{fig:autoc} unambiguously reveal the small-scale\nsmearing effects of horizontal radiative transfer in the emergent intensities.\nHorizontal radiative transport in space is the \\new{most likely way to explain the difference} between the\n1D and 3D calculations.\nThese differences cause the lowering of contrast across the line profiles in\nboth space and frequency, and larger the autocorrelation lengths in space.\n\n\n\nWe can also draw a third conclusion, namely that the upper chromosphere may well\nharbor \\textit{less ``turbulence'' than previously thought}.\nThe profiles of the three strongest lines formed in this region simply do not\nsupport supersonic motions within the uppermost layers of the stratified\natmosphere for the vast majority of the time.\nIn this picture, supersonic spicules and other features simply are not abundant\nenough and do not cover enough area of the surface to be significant, on\naverage, in affecting these line profiles.\n\n\\subsection{Further speculations}\n\nBeyond these conclusions, the confrontation between computed and observed\nprofiles necessarily becomes more speculative.\nArguably, modern MHD calculations capture the lower chromosphere better than the\nupper chromosphere and higher layers.\nAfter all, the energy in higher layers is modified and filtered by its\npropagation and dissipation through the lower layers, which themselves remain a\nsubject of active research.\nThe interface between the cool chromospheric and hot coronal plasma is not only\ndifficult to model accurately, but observations over the past century have\nconsistently revealed the complex thermal nature of the fine structure\nof the upper chromosphere.\nEven the latest radiation MHD calculations attempting to understand why the Sun\nmust produce spicules are based not on 3D but ``2.5D'' calculations in which\nspecial symmetries are imposed \\citep{2018ApJ...860..116M}.\nIt remains premature to state that we really know what spicules, and other\nproducts of the plasma dynamics of the chromosphere extending higher up, really\nare.\n\nThese and other structures entirely absent in 1D~calculations and\n3D~calculations may well influence in our analysis.\nAs made explicit by\n\\citet{2015SoPh..290..979J},\nsuch structures can have optical depths and source functions \\textit{radiatively\nde-coupled} from the chromosphere beneath.\nThese structures will simply absorb or emit radiation along the line of sight on\ntheir physical scales, essentially disconnected from the non-locally coupled\nradiative transfer solutions leading to the bulk of the line profiles beneath.\nThey can have structure on scales much smaller than the photon path\nlengths.\nOwing to the magnetic control of these plasmas owing to lower densities and\npressures, they might appear at almost any Doppler shift up to the (high)\nAlfv\\'en speed.\nThus we might expect to see absorbing and\/or emitting structures perhaps with\nlength scales down to the diffraction limit if such structures cover much of the\nchromosphere much of the time.\nThis kind of picture might explain the peculiar structures seen in the network\npanel in the upper part of Fig.~\\ref{fig:iris}.\nIn the internetwork and haze regions, we see the smallest structures in the line\ncores, mostly within $\\pm 0.05~\\text{\\AA} \\equiv \\pm 5$~$\\,$km$\\,$s$^{-1}${} Doppler shifts\n(Figure~\\ref{fig:iris}, seen in the images and the column-averaged white lines).\nThese are sub-sonic speeds.\nSuper-sonic motions associated with spicules and some fibrils are notable by\ntheir absence.\nFor a typical length of a spicule of 5--10$\\arcsec$, and assuming that $10^5$\nsuch spicules are present at any time on the Sun\n\\citep{2016ARep...60..848M},\nwe would expect to see one spicule cross a randomly oriented slit of this length\nat any time.\nThe small-scale features at line-center observed almost everywhere are therefore\nnot related to spicules.\nInterestingly, similar features are present in the numerical models for the\n\\ion{Ca}{2} $K$ line, outside of the chromospheric network.\n\nFinally, as noted earlier, if the location of the chromospheric temperature rise\nis deeper in the atmosphere than typical models suggest (see Fig.~20 of\n\\citet{2018A&A...611A..62B}),\nthen this problem should be re-visited.\nHints of a potential contribution of such a regime to a new understanding are\npresent in line widths computed from A--F models in\n\\citet{Vernazza+Avrett+Loeser1981},\nand the calculations of\n\\citet{2015ApJ...809L..30C}.\nThis is beyond the scope of the present paper but will be addressed in future.\n\n\\section{Conclusions}\n\nOur  results listed above  suggest that the amount of ``turbulence'' present in the\nupper chromosphere has been over-estimated in prior work.\nThis may have significant consequences for the energy budget that can be used to\nheat the overlying regions of the solar atmosphere.\n\nOur analysis is incomplete in many ways: for example, we have no pure solar\nspectra for H~L$\\alpha$ with sufficient spectral resolution and photometric\nquality to make comparisons with computations; neither have we any images with\nsufficiently narrow passbands (Doppler widths ${\\lta}10$~$\\,$km$\\,$s$^{-1}${}) to be able to\ncompare with predictions from computations including scattering such as\n\\andrii{in} Fig.~\\ref{fig:mgii}.\nEven so, it is clear that the various models in 3D as well as 1D are missing\nessential physical processes affecting the typical conditions in the upper solar\nchromosphere.\n\nThe consequences of revisiting of early work remain to be fully understood.\nThere are many observations relevant to this study, such as the remarkable\nimages of broad-band L$\\alpha$ light from the VAULT instrument\n\\citep[e.g.][]{Patsourakos+Gouttebroze+Vourlidas2007},\nof chromospheric fine structure in very \\andrii{narrowband} images in H$\\alpha$,\nthe \\ion{Ca}{2} infrared triplet lines and others\n\\citep[e.g.][]{2014ApJ...785..109L}.\nOur work is almost certainly important in terms of the amount of energy stored\nin and transported by the small-scale mass motions that we need to invoke, as\nwell as 3D transfer effects, to help explain the problem we have addressed.\n\n\\acknowledgements\n\nJ.L.\\ acknowledges support through the CHROMATIC project (2016.0019) funded by\nthe Knut and Alice Wallenberg foundation.\nThe Institute for Solar Physics is supported by a grant for research\ninfrastructures of national importance from the Swedish Research Council\n(registration number 2017-00625).\nA.V.S.\\ acknowledges financial support from the European Research Council (ERC)\nunder the European Union's Horizon 2020 research and innovation programme (ERC\nAdvanced Grant agreement No~742265) and from the Swiss National Science\nFoundation (SNSF) through Grant CRSII5\\_180238.\n\nNCAR is sponsored by the National Science Foundation.\n\\emph{IRIS} is a NASA small explorer mission developed and operated by LMSAL\nwith mission operations executed at NASA Ames Research center and major\ncontributions to downlink communications funded by ESA and the Norwegian Space\nCentre.\nComputations presented in this paper were performed on resources provided by the\nSwedish National Infrastructure for Computing (SNIC) at the National\nSupercomputer Centre (NSC) at Link\\\"oping University, at the PDC Centre for High\nPerformance Computing at the Royal Institute of Technology in Stockholm, and at\nthe High Performance Computing Center North (HPC2N) at Ume{\\aa} University.\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nAn important question on the mechanism of high-temperature superconductivity in iron pnictides concerns the relationship between the quantum criticality and superconductivity. In the vicinity of the range of the superconducting phase, antiferromagnetic order with a close-by tetragonal-orthorhombic structural transition is frequently found, and the anomalous normal-state properties near the endpoint of the order have been detected in BaFe$_2$As$_2$-based superconductors, implying the importance of quantum critical fluctuations \\cite{Shibauchi14,Kasahara10,Walmsley13,Nakai13,Yoshizawa12,Chu12,Gallais13}. These anomalies are mostly associated with the enhanced electron correlations, which manifest themselves as divergently increasing electron effective mass $m^*$ approaching a quantum critical point (QCP) \\cite{Walmsley13}. \n\nOn the other hand, in the high doping regime of the Ba$_{1-x}$K$_x$Fe$_2$As$_2$ system, an increasing trend of $m^*$ has been found with hole doping $x$; the Sommerfeld coefficient of the electronic specific heat $\\gamma$, a measure of $m^*$, reaches $\\sim 100$\\,mJ\/mol\\,K$^2$ in KFe$_2$As$_2$ ($x=1$) \\cite{Fukazawa11,Hardy13,Storey13}, and even exceeds $\\gamma$ of the quantum critical concentration $z=0.3$ in BaFe$_2$(As$_{1-z}$P$_z$)$_2$ system \\cite{Walmsley13}. Such a very large mass in KFe$_2$As$_2$, comparable to moderately heavy fermion materials, is also confirmed by the large Fermi-liquid coefficient of the resistivity \\cite{Hashimoto10}, and by quantum oscillations \\cite{Terashima13},  angle-resolved photoemission \\cite{Yoshida14}, and optical conductivity \\cite{Nakajima14}. It has been theoretically suggested that these strongly enhanced electron correlations for the $x=1$ system, where the number $N$ of $3d$ electrons per Fe site is 5.5, are related to the possible proximity to the Mottness of $N=5$ (half-filled for all $3d$ orbitals) \\cite{Medici14}, which is much more pronounced than the $N=6$ case of BaFe$_2$As$_2$ \\cite{Gerorges13,Yu13}. It has also been found that the isovalent substitution for K by larger alkali-ions Rb or Cs results in even larger  $\\gamma$ values up to  $\\sim 180$\\,mJ\/mol\\,K$^2$ \\cite{Wang13,Zhang15}. This suggests that the negative chemical pressure effect in $A$Fe$_2$As$_2$ ($A=$ K, Rb, and Cs) [see Fig.\\:\\ref{lambda}(a)] effectively reduces the band width with increasing alkali-ion radius, which leads to an approach to a QCP of another antiferromagnetically ordered phase \\cite{Zocco15}. \n\nOne of the key issues is how the quantum critical fluctuations affect the superconducting properties\n\\cite{Hashimoto12,Hashimoto13,Chowdhury13,Levchenko13,Nomoto13,Putzke14,Lamhot15}.  It has been demonstrated both experimentally \\cite{Hashimoto12,Lamhot15} and theoretically \\cite{Chowdhury13,Levchenko13,Nomoto13} that the enhanced electron correlations lead to an enhancement of the London penetration depth $\\lambda(0)$ in the superconducting phase. It has also been proposed \\cite{Hashimoto13,Nomoto13} that the low-energy quasiparticle excitations in nodal superconductors near an antiferromagnetic QCP are significantly affected in a non-trivial way, giving rise to an unusual power-law temperature dependence of $\\lambda(T)$ with a non-integer exponent close to 1.5, which differs distinctly from the $T$-linear dependence expected for line nodes of the superconducting energy gap $\\Delta(\\bm{k})$. In this viewpoint, clarifying the effect of the possible QCP on $\\lambda(T)$ in $A$Fe$_2$As$_2$ ($A=$ K, Rb, and Cs) is fundamentally important. Here, we report on such measurements of $\\lambda(T)$ down to low temperatures by using very clean single crystals. We find a systematic change in the exponent $\\alpha$ of its power-law temperature dependence with increasing alkali-ion radius, from $\\alpha\\sim 1$ for K \\cite{Hashimoto10} to $\\alpha\\sim 1.5$ for Cs. This supports the presence of quantum critical fluctuations in CsFe$_2$As$_2$, strongly affecting quasiparticle excitations as observed in other quantum critical superconductors.\n\n\n\nSingle crystals of $A$Fe$_2$As$_2$ were synthesized in alumina crucibles by a self-flux method \\cite{Zocco13}. In these crystals, quantum oscillations have been observed in magnetostriction measurements for $A=$ K \\cite{Zocco14}, Rb, and Cs \\cite{Zocco15}, indicating that the samples are very clean. In this study, the penetration depth measurements were performed for $A=$ Rb, and Cs using a self-resonant tunnel-diode oscillator with the resonant frequency $f$ of $\\sim 13$\\,MHz in a dilution refrigerator, and we compare the results with the previously reported data for $A=$ K taken in a similar setup \\cite{Hashimoto10}. The shift of the resonant frequency $\\Delta f$ is directly proportional to the change in the magnetic penetration depth, $\\Delta\\lambda(T)$ = $G\\Delta f(T)$, in the superconducting state below $T_c$. The geometric factor $G$ is determined from the geometry of samples and the coil. The samples are cleaved on all sides to avoid degradation due to the reaction with air and coated by Apiezon grease. The samples were mounted in the cryostat within 15 minutes after cleavage to minimize the exposure to air. Typical lateral size of crystals is $\\sim 500 \\times 500$\\,$\\mu$m$^2$ and the thickness is less than 50\\,$\\mu$m. \n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=1\\linewidth]{fig1.eps}\n\\vspace{-3mm}\n\\caption{(Color online). (a) Schematic phase diagrams of BaFe$_2$As$_2$-based superconductors with Co, K, and P substitutions \\cite{Shibauchi14}, combined with that of  $A$Fe$_2$As$_2$ ($A=$ K, Rb, and Cs). Only antiferromagnetic $T_N$ (blue circles) and superconducting transition temperatures $T_c$ (red circles) are shown for clarity. The isovalent  $A$Fe$_2$As$_2$ and BaFe$_2$(As$_{1-z}$P$_z$)$_2$ systems correspond to $3d$ electron numbers per Fe atom of $N=5.5$ and 6.0, respectively (green lines). (b) Temperature dependence of the frequency shift multiplied by the geometric factor $G$ in $A$Fe$_2$As$_2$ single crystals. The data for $A=$ K are taken from Ref.\\,\\cite{Hashimoto10}. (c)-(e) Low-temperature change of the penetration depth $\\Delta\\lambda$ below $T\/T_c=0.20$ plotted against $T\/T_c$ (c), $(T\/T_c)^{1.5}$ (d), and $(T\/T_c)^{2}$ (e). The data are vertically shifted for clarity. Dashed lines are the guides to the eyes.\n}\n\\label{lambda}\n\\end{figure}\n\n\nFigure\\:\\ref{lambda}(b) shows the temperature dependence of the resonant frequency shift $\\Delta f$ multiplied by $G$. The superconducting transition temperatures $T_c$ defined by the midpoint of the transition are 3.4, 2.5, and 2.2\\,K for $A=$ K, Rb, and Cs, respectively, consistent with the previous studies \\cite{Shermadini10, Hong13, Wang13, Tafti15, Wu15}. In the normal state above $T_c$, $G\\Delta f$ is determined by the skin depth $\\delta(T)$, which is related to the dc resistivity $\\rho(T)$ through $\\delta(T) = \\sqrt{2\\rho(T)\/\\mu\\omega}$. Here,  $\\mu$ is the permeability and $\\omega=2\\pi f$ is the angular frequency of the oscillator. From these relations we estimate $\\rho$ values just above $T_c$ as $\\sim 0.8$, $\\sim 1.0$, and $\\sim 1.2$\\,$\\mu\\Omega$\\,cm for the K, Rb, and Cs cases, respectively. These low resistivity values indicate the high quality of our samples.\n\nIn Figs.\\:\\ref{lambda}(c)-(e), the change in the penetration depth $\\Delta\\lambda(T) = \\lambda(T)-\\lambda(0)$ at low temperatures is plotted against $T\/T_c$, $(T\/T_c)^{1.5}$ and $(T\/T_c)^2$ for all samples. Two crystals are measured for both Rb and Cs systems and they exhibit almost identical $T$-dependence in each case. For all systems $\\Delta\\lambda(T)$ clearly exhibits a non-exponential $T$-dependence in a wide temperature range, which can be approximated by a power law. The exponent of the power-law $T$-dependence is less than 2, which is clearly demonstrated by the convex curvature against $(T\/T_c)^2$ [Fig.\\:\\ref{lambda}(e)]. Such small exponents $\\alpha<2$ found for all three compounds, which can hardly be explained by the pair-breaking scattering effect for fully opened gaps \\cite{Prozorov11}, immediately indicates the robust presence of low-energy quasiparticle excitations. This is consistent with nodal gap structure inferred from the observation of a residual term $\\kappa\/T$ for $T\\to 0$\\,K of the thermal conductivity $\\kappa$ in $A$Fe$_2$As$_2$ \\cite{Dong10,Reid12,Watanabe14,Zhang15,Hong13}, although measurements at lower temperature below $\\sim0.1$\\,K would be required for the ultimate determination of the presence of nodes or very small minima in $\\Delta(\\bm{k})$ \\cite{Hardy14,Carrington15}. \n\n\\begin{figure}[tbp]\n\\includegraphics[width=1.0\\linewidth]{fig2.eps}\n\\vspace{-3mm}\n\\caption{(Color online). (a) Band structures of $A$Fe$_2$As$_2$ ($A=$ K, Rb, and Cs) calculated within GGA-PBE. (b) Density of states as a function of energy near the Fermi level.\n}\n\\label{band}\n\\end{figure}\n\nA closer look at the data shows a systematic increase of the exponent in the order of K, Rb, and Cs. To discuss the origin of this change, we first examine the evolution of electronic structure in this series. The band-structure calculations are performed within the generalized gradient approximation (GGA-PBE \\cite{Perdew96})  for the experimentally obtained lattice parameters by using the Wien2k package \\cite{Blaha}. As shown in Fig.\\:\\ref{band}(a), the results indicate very similar structures for all three systems, which are consistent with the local density approximation results for $A=$ K \\cite{Terashima13} and Cs \\cite{Kong14}. The Fermi surfaces of the K, Rb, and Cs systems all have the same topology, which is confirmed by the quantum oscillation experiments mentioned above \\cite{Zocco15}. Furthermore, $T_c$ of these materials shows a universal V-shaped dependence on hydrostatic pressure \\cite{Tafti15}, suggesting that the superconducting gap structure is also essentially similar. \n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=1\\linewidth]{fig3.eps}\n\\vspace{-3mm}\n\\caption{(Color online). (a) Superfluid stiffness $\\rho_s(T) = \\lambda^2(0)\/\\lambda^2(T)^2$ obtained by using the reported values of $\\lambda(0)=203$ \\cite{Kawano11}, 267\\,nm \\cite{Shermadini10}, and the estimate of $\\lambda(0)=305$\\,nm (see text) for $A=$ K, Rb, and Cs, respectively.  (b)  Comparisons of $\\rho_s(T)$ for $T\/T_c<0.2$ with multigap calculations with and without considering the momentum-dependent effective Fermi velocity $v_F^*\\propto|\\Delta(\\bm{k})|^{1\/2}$. (c)-(e) Low-$T$ part of $\\rho_s(T)$ below $T\/T_c=0.15$ plotted against $T\/T_c$ for K (c), $(T\/T_c)^{1.4}$ for Rb (d), and $(T\/T_c)^{1.5}$ for Cs (e). The data are vertically shifted for clarity. Dashed lines are the guides for the eyes.\n}\n\\label{rho_s}\n\\end{figure}\n\n\nTo make a more quantitative analysis, we use the value of $\\lambda(0) \\sim 203(10)$\\,nm reported in the small-angle neutron scattering measurements for K \\cite{Kawano11} and $\\lambda(0) \\sim 267(5)$\\,nm obtained in the $\\mu$SR measurement for Rb \\cite{Shermadini10}. As there is no report on $\\lambda(0)$ for Cs so far, we roughly estimate $\\lambda(0) \\sim 305(30)$\\,nm on the assumption that $\\lambda^2(0)$ is proportional to $\\gamma$ \\cite{Walmsley13}. By using these values, we obtain the full temperature dependence of $\\lambda(T)$ and the normalized superfluid stiffness $\\rho_s(T) = \\lambda^2(0)\/\\lambda^2(T)$ [Fig.\\:\\ref{rho_s}(a)]. The overall $T$ dependence of $\\rho_s$ is quite similar in all samples, consistent with the very similar electronic and gap structures in these materials except for the effective mass, which is mainly reflected by the difference of the $\\lambda(0)$ values. \nIn contrast, $\\rho_s(T)$ at low temperatures shows a drastic change; compared with the data for K, the Rb and Cs data show more convex curvatures [Fig.\\:\\ref{rho_s}(b)], indicating a significant change in the amount of quasiparticle excitations at low energies. \n\nThe low-temperature part of $\\rho_s(T)$ plotted against $(T\/T_c)^\\alpha$ with different exponents $\\alpha$ for different systems [Figs.\\:\\ref{rho_s}(c)-(e)] confirms a power-law $T$-dependence with a systematic change in $\\alpha$. In particular, we find $1-\\rho_s(T)\\propto T^{\\alpha}$ with $\\alpha \\sim 1.5$ for Cs in a wide $T$ range, which is clearly different from $\\alpha \\sim 1$ for K. It should be noted that there is a small deviation from the power-law dependence at the lowest temperatures below $\\sim 0.15$\\,K, which may be due to impurity scattering. Such disorder effects can induce $T^2$ dependence of penetration depth at sufficiently low temperatures \\cite{Hirschfeld93,Bonn94}, and sometimes open a small gap in $s$-wave superconductors with accidental nodes \\cite{Mizukami14}, but these effects are limited to the correspondingly low temperature scale and will not affect the higher-$T$ behavior, which is determined by the intrinsic spectrum of quasiparticle excitations. To describe the disorder effect in superconductors with line nodes, the empirical formula $\\Delta\\lambda(T) \\propto (T\/T_c)^2\/\\{(T\/T_c)+(T^*\\!\/T_c)\\}$ that interpolates $T$-linear and $T^2$ dependences has been widely used, where $T^*$ is a measure of the impurity scattering rate \\cite{Hirschfeld93}. However, this fitting procedure to our data in a wide temperature range up to $T\/T_c=0.2$ gives considerably large values of $T^*\\!\/T_c$; the power-law dependence with $\\alpha\\sim1.5$ in CsFe$_{2}$As$_{2}$ can be approximated by the above equation with $T^*\\!\/T_c\\sim 0.9$ which is unphysically large for clean samples with low residual resistivity \\cite{Hashimoto13}. Such a large  $T^*\\!\/T_c$ is also incompatible with the nonlocality effect, in which $T^*\\!\/T_c \\sim \\xi(0)\/\\lambda(0) \\ll 1$ where $\\xi$ is the coherence length \\cite{Kosztin97}. Therefore we conclude that the anomalous power $\\alpha\\sim 1.5$ observed for the Cs system is an intrinsic property of the low-energy quasiparticles excited near the nodes or minima of $\\Delta(\\bm{k})$.  \n\nThe most pronounced change with the alkali-ion radius $R_{\\rm ion}$ in this series is the strong enhancement of the experimentally determined Sommerfeld coefficient $\\gamma_{\\rm exp}$, as shown in Fig.\\:\\ref{trends}(a). This $R_{\\rm ion}$ dependence of $\\gamma_{\\rm exp}$ is much more rapid than that of the band-structure calculations $\\gamma_{\\rm calc}$, which changes by only $\\sim 15$\\% as can be seen in the small increase in the density of states at the Fermi level [Fig.\\:\\ref{band}(b)]. This indicates that the mass renormalization factor averaged over the Fermi surface, $1\/Z=\\gamma_{\\rm exp}\/\\gamma_{\\rm calc}$, is strongly enhanced with increasing $R_{\\rm ion}$ [Fig.\\:\\ref{trends}(b)], which is consistent with the analysis of quantum oscillations \\cite{Zocco15}. The value of $1\/Z$ in CsFe$_2$As$_2$ reaches $\\sim13$, which is larger than the estimated value of $\\sim10$ for the quantum critical concentration $z=0.3$ in the BaFe$_2$(As$_{1-z}$P$_z$)$_2$ system \\cite{Walmsley13}. This result suggests that negative chemical pressure brings the $A$Fe$_2$As$_2$ system toward a QCP. The exponent $\\alpha$  obtained from the power-law fit to the $1-\\rho_s(T)$ data in the $T$ range up to $0.15 T_c$ also shows a systematic change with $R_{\\rm ion}$ as shown in Fig.\\:\\ref{trends}(b).  This highlights a close link between the unconventional exponent $\\sim1.5$ and quantum criticality. Considering the fact that $\\alpha\\sim1.5$ has been reported for several unconventional superconductors in the vicinity of QCPs \\cite{Hashimoto13}, we infer that this anomaly might have a common origin associated with quantum critical fluctuations. \n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=0.7\\linewidth]{fig4.eps}\n\\vspace{-3mm}\n\\caption{(Color online). (a) Sommerfeld coefficients $\\gamma_{\\rm exp}$ in single crystals of $A$Fe$_2$As$_2$ ($A=$ K, Rb, and Cs) \\cite{Fukazawa11,Hardy13,Wang13,Zhang15}, compared with the estimated $\\gamma_{\\rm calc}$ from the band-structure calculations, as a function of alkali-ion radius $R_{\\rm ion}$ in the eight-fold coordination. (b) Mass renormalization factor $1\/Z$ extracted from the comparisons of experimental and calculated $\\gamma$ values, and the exponent $\\alpha$ of the power-law $T$ dependence of $\\rho_s(T)$ in a temperature range up to $T\/T_c=0.15$.\n}\n\\label{trends}\n\\end{figure}\n\nIt has been proposed \\cite{Hashimoto13} that an unusual power-law dependence of $\\rho_s(T)$ with a non-integer exponent might arise from a strong momentum dependence of the effective Fermi velocity $v_F^*(\\bm{k})$ near the nodes of the superconducting gap $\\Delta(\\bm{k})$. Such a $\\bm{k}$ dependence of $v_F^*\\propto 1\/m^*$ requires that quantum critical fluctuations responsible for the mass renormalization also have strong momentum dependence associated with $\\Delta(\\bm{k})$. This can be naturally expected when the quantum critical fluctuations are quenched by opening the superconducting gap, the degree of which is determined by the gap magnitude $|\\Delta(\\bm{k})|$ \\cite{Hashimoto13}. Recent theoretical calculations for quantum critical superconductors with line nodes \\cite{Nomoto13} indicate that the current vertex corrections to the $T$-linear penetration depth can be significant if the antiferromagnetic hot spots are located near the nodal points such as is the case of electron-doped cuprates. \n\nNow we extend this consideration to the multigap systems, where $\\Delta(\\bm{k})$ is very different for different bands. The strong curvature found in $\\rho_s(T)$ near $T_c$ indicates the importance of the multiband effect \\cite{Prozorov11}. Indeed recent ARPES and specific-heat measurements have reported highly band-dependent gap structures in KFe$_{2}$As$_{2}$ \\cite{Okazaki12,Hardy14}. As shown in  Fig.\\:\\ref{rho_s}(b), we find that a set of multigap functions similar to that accounting for the specific-heat data \\cite{Hardy14} can reproduce the low-temperature $\\rho_s(T)$ of KFe$_2$As$_2$ fairly well. Here we calculate  $\\rho_s(T)=\\sum_i{w^i\\rho_s^i}(T)$ by assuming an extended $s$-wave state with a nodal gap in one band (the middle hole band $\\zeta$ around $\\Gamma$) and different constant gaps in other bands. The weight $w^i$ of each band $i$ is estimated from the quantum oscillation results \\cite{Terashima13} (see Table\\:\\ref{multigap}). Apparently, these weights do not change significantly for Rb and Cs \\cite{Zocco15}, and thus we need another ingredient to explain the enhanced exponent $\\alpha$. To include the effect of quantum criticality discussed above, we then calculate $\\rho_s(T)$ by applying the $\\bm{k}$-dependent effective velocity in the superconducting state of the form $v_F^*(\\bm{k})\\propto|\\Delta(\\bm{k})|^{1\/2}$ \\cite{Hashimoto13} to all bands, which affects the weight that includes the effective mass \\cite{SI}. For the $\\zeta$ band this $\\bm{k}$-dependence increases the low-$T$ power-law exponent of $\\rho_s^{\\zeta}(T)$. For the other bands, the $T$ dependence of each $\\rho_s^i$ is unchanged but the relative weights are modified by their gap magnitudes; for the small-gap band the weight is reduced because of the suppressed $v_F^*$. This simple analysis captures salient features of the systematic change in $\\rho_s(T)$ with $A$ [Fig.\\:\\ref{rho_s}(b)], although at higher temperatures there are some deviations which are possibly due to the interband coupling effect. The essence of this $|\\Delta(\\bm{k})|$-dependent mass renormalization is that it can naturally reduce the contribution of quasiparticles excited at the positions where the gap is small. Thus the observed positive correlation between the mass enhancement and the exponent of the superfluid stiffness supports this non-trivial effect of quantum critical fluctuations in the superconducting state. \n\n\n\n\n\n\n\\begin{table}[tbp]\n \\begin{center}\n  \\caption{Multigap parameters used in the calculations of $\\rho_s(T)$. The weight $w^i$ to $\\rho_s$ of band $i$ is estimated from the number of holes $n$ and the effective mass $m^*$ in units of the electron mass $m_e$ \\cite{Hashimoto10}, which are determined by the quantum oscillations in KFe$_2$As$_2$ \\cite{Terashima13}. The gap values of each band and the angle dependence of $\\Delta(\\phi)$ of $\\zeta$ band are assumed following Ref.\\,\\cite{Hardy14}.}\n   \\begin{tabular}{cccccc} \n\\hline \n\\hline\nband & $n$ & ${m^*}\/{m_e}$ & $w^i$ & $\\Delta(\\phi)\/k_BT_c$ \\\\\n\\hline\n$\\alpha$ (inner) & 0.17 & 6 & 0.31 & 0.61 \\\\\n$\\zeta$ (middle) & 0.26 & 13 & 0.23 &  $0.35+0.45\\cos(4\\phi)$ \\\\\n$\\beta$ (outer) & 0.48 & 18 & 0.31 & 0.25 \\\\\n$\\varepsilon$ (corner) & 0.09 & 7 & 0.15 & 1.8 \\\\\ntotal & 1.0 &  & 1.0 &  \\\\\n\\hline\n\\hline\n   \\end{tabular}\n   \\label{multigap}\n \\end{center}\n\\end{table}\n\n\n\n\n\nIn summary,  from the penetration-depth measurements of $A$Fe$_2$As$_2$ ($A=$ K, Rb, and Cs) in the heavily hole-doped regime with $3d$ electron number of $N=5.5$, we show that the superconducting gaps have robust strong anisotropies, consistent with the previous reports of thermal conductivity measurements  \\cite{Dong10,Reid12,Watanabe14,Zhang15,Hong13}. The exponent of the approximate power-law dependence of the superconducting stiffness changes from $\\alpha\\sim 1$ toward $\\alpha\\sim 1.5$ with negative chemical pressure, indicating a systematic decrease of the low-energy quaiparticle excitations which correlates with the enhancement of the effective mass toward the possible QCP in $A$Fe$_2$As$_2$. This observation entails a new relation between the momentum dependencies of quantum critical antiferromagnetic fluctuations and quasiparticle excitations of the superconducting condensate.\n\n\n\n\nWe thank A. Carrington, F. Eilers, F. Hardy, R. Heid, P.\\,J. Hirschfeld, and C. Meingast for valuable discussions. This work has been supported by\nBilateral Joint Research Project program and KAKENHI from JSPS.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\n\n\nEfficient photon-to-electron energy conversion is an important property of a nanomaterial, which has been receiving a \nlot of attention. One mechanism of increasing the conversion efficiency is called multiple-exciton generation (MEG), where \nabsorption of a high-energy photon results in the generation of multiple charge carriers. Fig. \\ref{MEGprocess} illustrates \nthe MEG process: absorption of a high-energy photon creates an electron-hole pair (exciton) with the energy exceeding twice \nthe bandgap. In MEG, this excess photon energy is diverted into generation of additional charge carriers instead of being   \nlost to generating atomic vibrations.  \n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.4\\textwidth]{process3.pdf}  \n  \\caption{Schematic illustration of the MEG process. Absorption of a photon with energy $\\hbar \\omega \\ge 2E_{gap}$ creates and exciton which can decay into a biexciton.  \nPictured here, a hot electron loses some of its energy to the creation of another electron-hole pair. } \n\\label{MEGprocess}   \n\\end{figure}\n\n\nIt has been shown that biexciton bound states play an important role in the excited-state properties of SWCNTs and that exciton-exciton binding energies are strongly dependent on the nanotube chirality \\cite{Matsunaga, Santos}. Experimental work aimed at addressing specifically the importance of exciton-exciton interactions in chiral SWCNTs has been performed by Colombier \\textit{et al.}, where biexciton binding energy of 106 meV in (9,7) SWCNT has been reported \\cite{Colombier}.  Previous theoretical work on exciton and biexciton binding energies in carbon nanotubes has been reported by Kammerlander \\textit{et al.} where the quantum Monte Carlo method has been combined with the tight-binding approximation and binding energies of $\\approx 120-150$ meV were reported \\cite{Kammerlander}.\nTheoretical efforts based on MBPT for calculating exciton-to-biexciton rates in MEG calculations in carbon nanotubes have been reported by Rabani \\textit{et al.}, although in their work, the final biexciton state is treated as a pair of two non-interacting excitons \\cite{Rabani}. {Additionally, there has been extensive theoretical work based on perturbation theory aimed at predicting carrier multiplication (CM) rates in semiconductor nanostructures \\cite{Voros, Marri, Marri2}, but exciton-exciton interactions in the final biexciton state are not included. \nExplicit, first-principles treatment of exciton-exciton interactions in semiconductor nanostructures is reported by Piryatinski \\textit{et al.} \\cite{Piryatinski}, \nwhere first-order perturbation theory is applied to biexciton states in type II core\/shell nanocrystals, where spatial separation of opposite charges and enhanced \nconfinement of like-charges leads to large ($\\sim$ 100 meV) positive shifts to the bi-exciton energies. However, the methodology developed in \\cite{Piryatinski} is not \napplicable to biexciton states where exciton separation distance is comparable to electron-hole separation distance, as can be expected in CNTs.}   \n\nRecently, Kryjevski \\textit{et al.} have developed several methods for a comprehensive description of MEG in a nanosctructure using \nDFT-based MBPT, including exciton effects \\cite{KryjevskiMEG2,KryjevskiMEG3,KryjevskiCNT,KryjevskiSF,KryjevskiMEG,KryjevskiCT}. \nFirst, one uses DFT-based MBPT technique to compute exciton-to-biexciton decay and recombination rates, {\\it i.e.}, the rates of \nthe inverse and direct Auger processes, respectively \\cite{KryjevskiMEG2, KryjevskiMEG3}. Next, one utilizes DFT software to compute \nphonon frequencies and normal modes, \nwhich are then used to compute one- and two-phonon exciton emission rates \\cite{KryjevskiMEG}. Finally, all these rates are incorporated \ninto the Boltzmann transport equation (BE) which provides comprehensive nonperturbative description of time evolution of the excited state \nincluding ``competition\" between different relaxation channels, such as MEG, phonon-mediated relaxation, {\\it etc.} \\cite{KryjevskiMEG}. \nIn particular, one can compute number of excitons generated from a single high-energy photon, which is the internal quantum efficiency (QE). \nThis method has been successfully applied to several chiral SWCNTs and efficient low-energy MEG was predicted, in a good agreement with the \navailable experimental data \\cite{MEGExperiment, MEGExperiment2}. Further, when augmented with the exciton transfer terms \nthis BE technique has been applied to the doped SWCNT-$Si$ nanocrystal system and formation of a long-lived charge transfer state was predicted \\cite{KryjevskiCT}. \nAdditionally, in \\cite{KryjevskiSF} MEG rates method for the singlet fission (SF) process, where a singlet exciton decays into two spin-one (triplet) excitons in \nthe overall singlet state, has been developed and applied to SWCNTs.  \n\nIn all the MEG work so far \nthe final biexciton state has been approximated as a non-interacting exciton pair. However, knowledge of the low-energy biexciton energy levels is essential \nfor the accurate prediction of the MEG threshold. So, here we investigate this issue by developing a DFT-based MBPT method for biexciton state energies by including \nresidual electrostatic (dipole-dipole) interactions between the excitons in the biexciton state. The technique is then applied to chiral SWCNTs (6,2), (6,5) and (10,5). Here, only biexciton states composed of two singlet excitons are included.\n\nThe paper is organized as follows. Section \\ref{theory} contains description of the methods and\napproximations employed in this work. Section \\ref{CompDetail} contains description of the atomistic\nmodels studied in this work and of DFT simulation details. Section \\ref{Results} contains discussion\nof the results obtained. Conclusions and Outlook are presented in Section \\ref{Conclusion}. \n\n\\section{Theoretical Methods and Approximations} \\label{theory}\n\n\\subsection{Microscopic Hamiltonian} \\label{Hamiltonian}\n\nFor completeness, let us review basics of the DFT-based MBPT approach that includes electron-exciton terms \\cite{KryjevskiSF}. \n\nThe electron field operator $\\psi_{\\sigma}(\\textbf{x})$ is related to the annihilation operator of the $i^{th}$ Kohn-Sham (KS) state $a_{i\\alpha}$ via\n\\begin{eqnarray}\n  \\psi_{\\sigma}(\\textbf{x}) = \\sum_{i, \\sigma} \\phi_{i\\sigma}(\\textbf{x})a_{i\\sigma}\n\\end{eqnarray} \nwhere $\\phi_{i\\sigma}(\\textbf{x})$ is the $i^{th}$ KS orbital with spin $\\sigma$ and $a_{i\\sigma}, a_{i\\sigma}^{\\dagger} $ obey fermion anti-commutation relations $\\{a_{i\\sigma}, a_{j\\nu}^{\\dagger}\\} = \\delta_{ij}\\delta_{\\sigma\\nu}, \\{a_{i\\sigma}, a_{j\\nu}\\} = 0$ \\cite{Fetter,Mahan}. In the spin nonpolarized case we consider here, \n$\\phi_{i\\uparrow}(\\textbf{x})=\\phi_{i\\downarrow}(\\textbf{x})\\equiv \\phi_{i}(\\textbf{x}).$ In terms of $a_{i\\sigma}, a_{i\\sigma}^{\\dagger}$, the electron Hamiltonian is\n\n\\begin{equation}\n  {H}= \\sum_{i\\sigma} \\varepsilon_i a_{i\\sigma}^{\\dagger} a_{i\\sigma} + H_C - H_V + H_{e-exciton} \\label{Hamiltonian} \n\\end{equation} \nwhere $\\varepsilon_{i\\uparrow} = \\varepsilon_{i\\downarrow} \\equiv \\varepsilon_{i}$ is the energy of the $i^{\\text{th}}$ KS orbital. In a periodic structure KS energies and orbitals \nare labeled by the band number and lattice wavevector but here, as explained in Sec. \\ref{CompDetail}, the state label is just an integer. The $H_C$ term is the microscopic Coulomb interaction operator  \n\\begin{eqnarray}\n  & H_C = \\frac{1}{2} \\sum_{ijkl \\sigma, \\nu} V_{ijkl} a^{\\dagger}_{i \\sigma} a^{\\dagger}_{j \\nu} a_{l \\nu} a_{k \\sigma} , \\\\\n  & V_{ijkl}  = \\int d\\textbf{x} d\\textbf{y} \\phi_i^*(\\textbf{x})\\phi_j^*(\\textbf{y}) \\frac{e^2}{| \\textbf{x} - \\textbf{y} |} \\phi_l(\\textbf{y})\\phi_k(\\textbf{x}), \n\\label{COULOMB}\n\\end{eqnarray}\nwhere KS indices $i,j,k,l,...$ can refer to both occupied and unoccupied states within the included range. The $H_V$ term prevents double-counting of electron interactions \n\\begin{equation}\n   H_V = \\sum_{ij, \\sigma} a^{\\dagger}_{i \\sigma} \\bigg( \\int d\\textbf{x} d\\textbf{y} \\phi_i^*(\\textbf{x}) V_{KS}(\\textbf{x}, \\textbf{y}) \\phi_j(\\textbf{y}) \\bigg) a_{j \\sigma} ,\n\\label{HV}\n\\end{equation}\nwhere $V_{KS}(\\textbf{x}, \\textbf{y})$ is the Kohn-Sham potential used in the DFT simulation \\cite{Onida,Kummel}.  \n\nThe electron-exciton coupling term $H_{e-exciton}$ appearing in Eq. \\eqref{Hamiltonian} is\n\\begin{eqnarray}\n& H_{e-exciton} = \\sum_{e h \\alpha} \\sum_{\\sigma} \\frac{1}{\\sqrt{2}} \\bigg( [\\varepsilon_{e} - \\varepsilon_h - E^{\\alpha}]\\Psi_{eh}^{\\alpha} a_{h \\sigma} a^{\\dagger}_{e \\sigma}({\\rm B}^{\\alpha} + {\\rm B}^{\\alpha \\dagger}) + h.c. \\bigg) + \\sum_{\\alpha} E^{\\alpha} {\\rm B}^{\\alpha \\dagger} {\\rm B}^{\\alpha}  \\label{EXCFERMSF} \n\\end{eqnarray}\nwhere $e \\ge$ LU (the lowest unoccupied KS state) and $h \\le HO$ (the highest occupied KS state), ${\\rm B}^{\\alpha \\dagger}$, $\\Psi_{eh}^{\\alpha}$ and  $E^{\\alpha}$ \nare the singlet exciton creation operator, wave function and energies respectively. Technically, the full electron Hamiltonian is comprised of only the first three \nterms in Eq. \\eqref{Hamiltonian}, however adding $H_{e-exciton}$ along with the rules to avoid double counting allows for the perturbative treatment of excitonic \neffects and is a standard approch to describing the coupling between excitons and electrons and holes \\cite{Beane, Spataru1, Perebeinos, Spataru2}. The $H_{e-exciton}$ term is a result of resummation of Coulomb perturbative corrections resulting from the Coulomb interaction operator $H_C$ (Eq. \\eqref{COULOMB}) to the electron-hole state, \nwhich is an implementation of a standard method to include bound states into the MBPT framework \\cite{Berestetskii, Beane, KryjevskiSF}.      \n\nExciton wave functions, $\\Psi_{eh}^{\\alpha},$ and energies, $E^{\\alpha},$ are solutions to the Bethe-Salpeter equation (BSE), which is (see, {\\it e.g.}, \\cite{Benedict})       \n\\begin{equation}\n   [\\varepsilon_{e} - \\varepsilon_h - E^{\\alpha}]\\Psi_{eh}^{\\alpha} + \\sum_{e' h'} (K_{C} + K_{D})(e,h;e',h')\\Psi_{e'h'}^{\\alpha} = 0, \\label{BSE}\n\\end{equation}\nwhere\n\\begin{equation}\n  K_{C} =\\frac{8 \\pi e^2}{V} \\sum_{\\textbf{k} \\neq 0} \\frac{\\rho_{eh}(\\textbf{k})\\rho_{e'h'}^*(\\textbf{k})}{k^2}, \\label{KCOUL}\n\\end{equation}\n\\begin{equation}\n  K_{D} = - \\frac{4 \\pi e^2}{V} \\sum_{\\textbf{k} \\neq 0} \\frac{\\rho_{ee'}(\\textbf{k})\\rho_{hh'}^*(\\textbf{k})}{|\\textbf{k}|^2 - \\Pi(0, -\\textbf{k}, \\textbf{k})}, \\label{KDIR}\n\\end{equation}\nwhere\n\\begin{equation}\n  \\rho_{ij}(\\textbf{k}) = \\sum_{\\textbf{p}} \\phi_j^*(\\textbf{p} - \\textbf{k}) \\phi_i(\\textbf{p})\n\\end{equation}\n\n{Medium screening is taken into account via the polarization function (Eq. \\eqref{PI}) which appears only in the direct term (Eq. \\eqref{KDIR}) of BSE.} In the random-phase approximation (RPA) \n\\begin{equation}\n   \\Pi(\\omega,\\textbf{k},\\textbf{p}) = \\frac{8 \\pi e^2}{\\hbar V} \\sum_{ij} \\rho_{ij}(\\textbf{k})\\rho_{ji}(\\textbf{p})\\bigg(\\frac{\\theta_{-j}\\theta_{i}}{\\omega - \\omega_{ij} + i\\delta} - \\frac{\\theta_{j}\\theta_{-i}}{\\omega - \\omega_{ij} - i\\delta} \\bigg), \\;\\;\\; \\omega_{ij} = \\frac{\\varepsilon_i - \\varepsilon_j}{\\hbar} \\label{PI}\n\\end{equation}\nwhere $V$ is the simulation cell volume, $\\delta$ is the width parameter, which will be set to $0.025~eV$ corresponding to the room temperature scale.  \n\nHere, we use static approximation, taking  $\\Pi(\\omega=0,\\textbf{k},\\textbf{p}).$ This is a widely-used approach for semiconductor nanostructures ({\\it e.g.}, \\cite{PhysRevLett.90.127401,PhysRevB.68.085310,PhysRevB.79.245106}), which is justified by the cancellations that appear when the electron-hole screening and the single-particle Green's functions are  \nboth treated dynamically \\cite{Bechstedt}. Next, the main simplifying approximation employed in this work is to retain only the diagonal elements of the polarization matrix, \n\\textit{i.e.}, $\\Pi(0,\\textbf{k},\\textbf{p}) \\simeq \\Pi(0,\\textbf{k},-\\textbf{k})\\delta_{\\textbf{k}, -\\textbf{p}} $, or $\\Pi(0,\\textbf{x},\\textbf{x}') \\simeq \\Pi(0,\\textbf{x} - \\textbf{x}')$ in position space, {\\it i.e.}, the system \nis approximated as a uniform medium. This is a valid approximation for quasi-one-dimensional systems, such as CNTs,  where one can expect $\\Pi(\\textbf{x},\\textbf{x}') \\simeq \\Pi(z-z')$, with $z,z'$ being the axial positions. This diagonal polarization matrix approximation, which is a an improvement on previous studies on CNTs in which screening has been approximated by a dielectric constant \\cite{Perebeinos2}, has been employed for the time being, as it significantly reduces computational costs. Calculations including full treatment of the polarization matrix, although unlikely to change qualitative conclusions, are left for future work. Also, in the DFT simulations one uses hybrid Heyd-Scuseria-Ernzerhof (HSE06) exchange-correlation functional \\cite{HSE06}   \nwhich is to substitute for the $G_0W_0$ calculation of single-particle energies - the second step in the standard three-step process in the electronic structure  \ncalculation \\cite{Rohlfing2,Hybertsen}. The HSE06 functional, which is significantly less computationally expensive than $G_0W_0$, has been shown to produce somewhat  reasonable \nresults for bandgaps in semiconducting nanostructures \\cite{Kummel, HSE06Muscat, LouieHSE06}. So, single-particle energies and orbitals are approximated by the KS $\\varepsilon_i$ and \n$\\phi_i({\\bf x})$ from the HSE06 DFT output. Therefore, in our approximation using HSE06 replaces ``dressing'' fermion \nlines in the Feynman diagrams, including subtraction of the compensating term  (\\ref{HV}). \n\nFor SWCNTs, the set of approximations stated above have been checked and shown to be reasonable by reproducing experimental results for the low-energy absorption peaks in (6,2), (6,5) and \n(10,5) SWCNTs within 5-13\\% error \\cite{KryjevskiMEG}.\n     \n\n\\section{Expressions for the Biexciton Self-Energy} \\label{Expressions}\n\nIn order to account for the exciton-exciton interactions one computes self-energy of the biexciton state $\\hbar\\Sigma_{\\alpha\\beta} (E)$. \nThen the biexciton energy is approximated as   \n\\begin{eqnarray}\n E_{\\alpha\\beta}=E^{\\alpha} + {E}^{\\beta}+\\hbar\\Sigma_{\\alpha\\beta} (E = E^{\\alpha} + E^{\\beta}) \\label{self_energy_corr}\n\\end{eqnarray}\nwhere $E^{\\alpha}$ and $E^{\\beta}$ are the energies of the two excitons in the biexciton state. $E^{\\alpha}$ and $E^{\\beta}$ are obtained by solving the Bethe-Salpeter equation (Eq. \\eqref{BSE}). $\\hbar\\Sigma_{\\alpha\\beta} (E = E^{\\alpha} + E^{\\beta})$ is the biexciton self-energy evaluated on the total energy of a state composed of two non-interacting excitons $\\ket{\\alpha}$ and $\\ket{\\beta}$. \n\nTo the leading order in the Coulomb interaction the relevant Feynman diagrams are shown in Fig. \\ref{BIEXCfig}. One only retains \ncorrections to the bare biexciton state where a particle or a hole from the exciton $\\ket{\\alpha}$ interacts with a particle or a hole \nfrom the other exciton $\\ket{\\beta}$; the interactions between particles and holes within the same exction are already included by the BSE. \nNote that including contributions with all possible arrow directions in each fermion loop is needed to include\nall the interactions between electrons and holes within the biexciton state.                                        \n\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.97\\textwidth]{biexciton_self_ab_alphabeta_2.pdf}\n  \\caption{Leading order Feynman diagrams representing contributions to the biexciton self-energy. Thin (fermion) lines are Kohn-Sham particle\/hole propagators, thick lines - excitons (see Fig. \\ref{EXCbound}), zigzag lines - the {screened} Coulomb interaction. Not shown for brevity are similar diagrams where arrow directions in one of the fermion loops are reversed, and diagrams where all arrows are reversed.  \n } \\label{BIEXCfig}\n\\end{figure}\n\\noindent\nNote that for each diagram in Fig. \\ref{BIEXCfig} there is an additional contribution upon exchange of the $\\alpha$ and $\\beta$ excitons on the right-hand-side of the fermion loop. These contributions are currently a subject of future work. The expressions resulting from each diagram in Fig. \\ref{BIEXCfig} are similar in general form, so for brevity, only one of them will be quoted here and the rest will be presented in the appendix. For instance, the diagram from Fig. \\ref{BIEXCfig} - \\textit{a)} results in the following expression\n\\begin{eqnarray}\n & \\hbar\\Sigma_{\\alpha\\beta} (E = E^{\\alpha} + E^{\\beta}) = \\frac{4}{5} \\sum_{ijklmn} \\frac{\\Theta_{i}\\Theta_n\\Theta_{l}\\Theta_{-k}\\Theta_{-j}\\Theta_{-m}}{(E^{\\alpha}-\\epsilon_{nk} - i\\delta)(E^{\\beta}-\\epsilon_{lm}+ i\\delta)(\\epsilon_{kj} - i\\delta)(\\epsilon_{li} + i\\delta)} \\times  \\nonumber  \\\\ \n& \\times (\\Psi_{im}^{\\beta})^*(E^{\\beta} - \\epsilon_{im})(\\Psi_{nk}^{\\alpha})^*(E^{\\alpha} - \\epsilon_{nk}) \\Psi_{lm}^{\\beta} (E^{\\beta} - \\epsilon_{lm})  \\Psi_{nj}^{\\alpha} (E^{\\alpha} - \\epsilon_{nj}) \\times W_{ijkl} \\label{expression}\n\\end{eqnarray}\nwhere $\\Psi_{ij}^{\\alpha}$ and $E^{\\alpha}$ are the exciton wave function and exciton energies respectively obtained by solving the Bethe-Salpeter equation (Eq. \\eqref{BSE}). $\\epsilon_{ij} = \\epsilon_i - \\epsilon_j$; $W_{ijkl}$ are the RPA-screened Coulomb matrix elements  \n\\begin{equation}\n  W_{ijkl} = \\frac{4 \\pi e^2}{V} \\sum_{\\textbf{k} \\neq 0} \\frac{\\rho_{il}^*(\\textbf{k})\\rho_{jk}(\\textbf{k})}{|\\textbf{k}|^2 - \\Pi(0, -\\textbf{k}, \\textbf{k})} \\label{W}\n\\end{equation}\nwhere $\\Pi(0, -\\textbf{k}, \\textbf{k})$ is defined in Eq. \\eqref{PI}. The theta-functions determine whether the KS indices $i,j,k,l$ are particles or holes\n\\begin{equation}\n\\Theta_i = \\sum_{i > \\text{HO} } ~~ , ~~ \\Theta_{-i} = \\sum_{i \\le \\text{HO}}.\n\\end{equation}\n In this work we use \n\\begin{eqnarray}\n\\frac{1}{x \\pm i \\delta}=\\frac{x}{x^2+\\delta^2} \\mp i \\frac{\\delta}{x^2+\\delta^2},\n\\label{1xidelta}\n\\end{eqnarray}\ni.e., both the principal value and delta function parts of $1\/(x \\pm i \\delta)$ factors are included.\n\nThe other three diagrams - \\ref{BIEXCfig}-$b),~c),~d)$ - produce similar expressions not shown for the sake of brevity.  \n\nNext, the possibility of including higher order perturbative corrections was explored. Naively including terms of second (or higher) order in the Coulomb interaction, i.e., \ndecorating diagrams in Fig. \\ref{BIEXCfig} with additional zigzag lines, would lead to prohibitively expensive calculations, even for a small system. However, certain classes \nof perturbative contributions can be summed to all orders, {such as the Coulomb interactions between electron and hole resulting in formation of an exciton bound state (see Fig. \\ref{EXCbound}). \nEq. \\ref{EXCFERMSF} describes the resulting exciton-electron coupling term. } \n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.89\\textwidth]{exc_bound_3.pdf}\n  \\caption{ {Summing perturbative contributions from Coulomb interactions (zigzag lines) between electron and hole (thin lines) is equivalent to including intermediate exciton bound state (thick line).} } \\label{EXCbound}\n\\end{figure}\n\\noindent\nThis summation of Coulomb corrections to all orders can be applied to modify the diagrams shown in Fig. \\ref{BIEXCfig}. In Fig. \\ref{BIEXCresum} it is illustrated how an intermediate exciton state \n$\\gamma$ appears from decorating the electron-hole lines in Fig. \\ref{BIEXCfig} with zigzag lines in all possible ways. Here, we work to the leading order in the electron-exciton coupling - Eq. \\ref{EXCFERMSF} - and, so, only decorate one pair of the electron-hole lines.  \n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.99\\textwidth]{biexciton_self_resummation_4.pdf}\n  \\includegraphics[width=0.99\\textwidth]{biexciton_self_resummation_5.pdf}\n  \\caption{ Summing perturbative contributions from Coulomb interactions (zigzag lines) between electron and hole (thin lines) from different excitons  - $\\alpha$ and $\\beta$ - results in appearance of the intermediate exciton $\\gamma$.} \\label{BIEXCresum} \n\\end{figure}\n\\noindent\nThe two distinct diagrams resulting from modifying Fig. \\ref{BIEXCfig} are shown in Fig. \\ref{BIEXCfig2}. \n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.99\\textwidth]{biexciton_self_with_excitons.pdf}\n  \\caption{ \\textit{a)} - diagram resulting from modification of diagrams \\ref{BIEXCfig}-\\textit{a)} and \\textit{b)}.  \\textit{b)} - diagram resulting from modification of diagrams \\ref{BIEXCfig}-\\textit{c)} and \\textit{d)}.  } \\label{BIEXCfig2}\n\\end{figure}\n\\noindent\nThe contributions to  the biexciton self-energy \nfrom the diagrams in Fig. \\ref{BIEXCfig2} are \n\\begin{eqnarray}\n & \\hbar\\Sigma_{\\alpha\\beta}^+ = \\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{a}} + \\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{b}} \\label{expression+} \\\\\n & \\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{a}}(E = E^{\\alpha} + E^{\\beta})  = \\frac{8}{45} \\sum_{ijklmn\\gamma} \\frac{\\Theta_{j}\\Theta_k\\Theta_{n}\\Theta_{-i}\\Theta_{-l}\\Theta_{-m}}{(E^{\\alpha} - \\epsilon_{ki} + i\\delta)(E^{\\beta} - \\epsilon_{nm}- i\\delta)(E^{\\gamma} - \\epsilon_{km} + i\\delta)(E^{\\gamma} - \\epsilon_{jl} + i\\delta)(\\epsilon_{jk} + i\\delta)} \\times  \\label{expression2} \\\\  \n& \\times (\\Psi_{ji}^{\\alpha})^*(E^{\\alpha} - \\epsilon_{ji})\\Psi_{ki}^{\\alpha}(E^{\\alpha} - \\epsilon_{ki}) \\Psi_{nl}^{\\beta} (E^{\\beta} - \\epsilon_{nl})  (\\Psi_{nm}^{\\beta})^* (E^{\\beta} - \\epsilon_{nm}) \\Psi_{jl}^{\\gamma} (E^{\\gamma} - \\epsilon_{jl})  (\\Psi_{km}^{\\gamma})^* (E^{\\gamma} - \\epsilon_{km}) \\nonumber  \n\\end{eqnarray}\n\\begin{eqnarray}\n & \\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{b}}(E^{\\alpha} + E^{\\beta})  = \\frac{8}{45} \\sum_{ijklmn\\gamma} \\frac{\\Theta_{j}\\Theta_k\\Theta_{n}\\Theta_{-i}\\Theta_{-l}\\Theta_{-m}}{(E^{\\alpha} - \\epsilon_{kl} + i\\delta)(E^{\\beta} - \\epsilon_{km}- i\\delta)(E^{\\gamma} - \\epsilon_{jl} + i\\delta)(E^{\\gamma} - \\epsilon_{nm} + i\\delta)(\\epsilon_{jn} + i\\delta)} \\times \\label{expression3} \\\\ \n& \\times (\\Psi_{ji}^{\\alpha})^*(E^{\\alpha} - \\epsilon_{ji})\\Psi_{kl}^{\\alpha}(E^{\\alpha} - \\epsilon_{kl}) \\Psi_{ni}^{\\beta} (E^{\\beta} - \\epsilon_{ni})  (\\Psi_{km}^{\\beta})^* (E^{\\beta} - \\epsilon_{km}) \\Psi_{jl}^{\\gamma} (E^{\\gamma} - \\epsilon_{jl})  (\\Psi_{nm}^{\\gamma})^* (E^{\\gamma} - \\epsilon_{nm}) \\nonumber \n\\end{eqnarray}\n\\noindent\nwhere the additional summation label $\\gamma$ is over the intermediate exciton state. Note that here the screened Coulomb interaction only appears implicitly through the exciton wavefunctions and energies. \n\nAdditionally, there are self-energy contributions from the repulsive electron-electron and hole-hole interactions. These come from, e.g., diagrams in Fig. \\ref{BIEXCfig} a) and d)\nbut with arrows in one of the fermion loops reversed. These corrections are treated to the leading order in the Coulomb interaction. There are eight distinct  expressions contributing to the biexciton self-energy due to the inter-exciton electron-electron or hole-hole interactions, two from each of the 4 diagrams in Fig. \\ref{BIEXCfig} with the appropriate fermion arrow flips. The expressions are \n\\begin{eqnarray}\n & \\hbar\\Sigma_{\\alpha\\beta}^- = \\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{ee},a} + \\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{hh},a}+\\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{ee},b} + \\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{hh},b}+\\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{ee},c} + \\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{hh},c} +\\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{ee},d} + \\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{hh},d} \\label{expression-}\\\\         \n & \\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{ee},a} (E^{\\alpha} + E^{\\beta}) = -\\frac{4}{5} \\sum_{ijklmn} \\frac{\\Theta_{i}\\Theta_{l}\\Theta_{k}\\Theta_{j}\\Theta_{-m}\\Theta_{-n}}{(E^{\\alpha}+\\epsilon_{kn} - i\\delta)(E^{\\beta}-\\epsilon_{lm}+ i\\delta)(\\epsilon_{kj} - i\\delta)(\\epsilon_{li} + i\\delta)} \\times  \\nonumber  \\\\ \n& \\times (\\Psi_{im}^{\\beta})^*(E^{\\beta} - \\epsilon_{im})(\\Psi_{kn}^{\\alpha})^*(E^{\\alpha} - \\epsilon_{kn}) \\Psi_{lm}^{\\beta} (E^{\\beta} - \\epsilon_{lm})  \\Psi_{jn}^{\\alpha} (E^{\\alpha} - \\epsilon_{jn}) \\times W_{ijkl} \\label{expression4} \\\\\n& \\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{hh},a} (E^{\\alpha} + E^{\\beta}) = -\\frac{4}{5} \\sum_{ijklmn} \\frac{\\Theta_{-i}\\Theta_{-l}\\Theta_{-k}\\Theta_{-j}\\Theta_{m}\\Theta_{n}}{(E^{\\alpha}+\\epsilon_{nk} - i\\delta)(E^{\\beta}-\\epsilon_{ml} + i\\delta)(\\epsilon_{kj} - i\\delta)(\\epsilon_{li} + i\\delta)} \\times  \\nonumber  \\\\ \n& \\times (\\Psi_{mi}^{\\beta})^*(E^{\\beta} - \\epsilon_{mi})(\\Psi_{nk}^{\\alpha})^*(E^{\\alpha} - \\epsilon_{nk}) \\Psi_{ml}^{\\beta} (E^{\\beta} - \\epsilon_{ml})  \\Psi_{nj}^{\\alpha} (E^{\\alpha} - \\epsilon_{nj}) \\times W_{ijkl} \\label{expression5} \\\\\n& \\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{ee},b} (E^{\\alpha} + E^{\\beta}) = -\\frac{4}{5} \\sum_{ijklmn} \\frac{\\Theta_{i}\\Theta_{l}\\Theta_{k}\\Theta_{j}\\Theta_{-m}\\Theta_{-n}}{(E^{\\alpha}+\\epsilon_{im} - i\\delta)(E^{\\beta}-\\epsilon_{ln}+ i\\delta)(\\epsilon_{il} - i\\delta)(\\epsilon_{jk} + i\\delta)} \\times  \\nonumber \\\\\n& \\times (\\Psi_{kn}^{\\beta})^*(E^{\\beta} - \\epsilon_{kn})(\\Psi_{jm}^{\\alpha})^*(E^{\\alpha} - \\epsilon_{jm}) \\Psi_{ln}^{\\beta} (E^{\\beta} - \\epsilon_{ln})  \\Psi_{im}^{\\alpha} (E^{\\alpha} - \\epsilon_{im}) \\times W_{ijkl} \\label{expression6} \\\\\n& \\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{hh},b} (E^{\\alpha} + E^{\\beta}) = -\\frac{4}{5} \\sum_{ijklmn} \\frac{\\Theta_{-i}\\Theta_{-l}\\Theta_{-k}\\Theta_{-j}\\Theta_{m}\\Theta_{n}}{(E^{\\alpha}-\\epsilon_{mi} - i\\delta)(E^{\\beta}-\\epsilon_{nl} + i\\delta)(\\epsilon_{il} - i\\delta)(\\epsilon_{jk} + i\\delta)} \\times  \\nonumber  \\\\ \n& \\times (\\Psi_{nk}^{\\beta})^*(E^{\\beta} - \\epsilon_{nk})(\\Psi_{mj}^{\\alpha})^*(E^{\\alpha} - \\epsilon_{mj}) \\Psi_{nl}^{\\beta} (E^{\\beta} - \\epsilon_{nl})  \\Psi_{mi}^{\\alpha} (E^{\\alpha} - \\epsilon_{mi}) \\times W_{ijkl} \\label{expression7} \\\\\n& \\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{ee},c} (E^{\\alpha} + E^{\\beta}) = -\\frac{4}{5} \\sum_{ijklmn} \\frac{\\Theta_{i}\\Theta_{l}\\Theta_{k}\\Theta_{j}\\Theta_{-m}\\Theta_{-n}}{(E^{\\alpha}+\\epsilon_{im} - i\\delta)(E^{\\beta}-\\epsilon_{jm}+ i\\delta)(\\epsilon_{il} - i\\delta)(\\epsilon_{jk} + i\\delta)} \\times  \\nonumber \\\\\n& \\times (\\Psi_{kn}^{\\beta})^*(E^{\\beta} - \\epsilon_{kn})(\\Psi_{ln}^{\\alpha})^*(E^{\\alpha} - \\epsilon_{ln}) \\Psi_{jm}^{\\beta} (E^{\\beta} - \\epsilon_{jm})  \\Psi_{im}^{\\alpha} (E^{\\alpha} - \\epsilon_{im}) \\times W_{ijkl} \\label{expression8} \\\\\n& \\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{hh},c} (E^{\\alpha} + E^{\\beta}) = -\\frac{4}{5} \\sum_{ijklmn} \\frac{\\Theta_{-i}\\Theta_{-l}\\Theta_{-k}\\Theta_{-j}\\Theta_{m}\\Theta_{n}}{(E^{\\alpha}-\\epsilon_{mi} - i\\delta)(E^{\\beta}+\\epsilon_{mj}+ i\\delta)(\\epsilon_{il} - i\\delta)(\\epsilon_{jk} + i\\delta)} \\times  \\nonumber \\\\\n& \\times (\\Psi_{nk}^{\\beta})^*(E^{\\beta} - \\epsilon_{nk})(\\Psi_{nl}^{\\alpha})^*(E^{\\alpha} - \\epsilon_{nl}) \\Psi_{mj}^{\\beta} (E^{\\beta} - \\epsilon_{mj})  \\Psi_{mi}^{\\alpha} (E^{\\alpha} - \\epsilon_{mi}) \\times W_{ijkl} \\label{expression9} \\\\\n& \\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{ee},d} (E^{\\alpha} + E^{\\beta}) = -\\frac{4}{5} \\sum_{ijklmn} \\frac{\\Theta_{i}\\Theta_{l}\\Theta_{k}\\Theta_{j}\\Theta_{-m}\\Theta_{-n}}{(E^{\\alpha}-\\epsilon_{im} - i\\delta)(E^{\\beta}+\\epsilon_{lm}+ i\\delta)(\\epsilon_{il} + i\\delta)(\\epsilon_{kj} - i\\delta)} \\times  \\nonumber \\\\\n& \\times (\\Psi_{kn}^{\\beta})^*(E^{\\beta} - \\epsilon_{kn})(\\Psi_{jn}^{\\alpha})^*(E^{\\alpha} - \\epsilon_{jn}) \\Psi_{lm}^{\\beta} (E^{\\beta} - \\epsilon_{lm})  \\Psi_{im}^{\\alpha} (E^{\\alpha} - \\epsilon_{im}) \\times W_{ijkl} \\label{expression10} \\\\\n& \\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{hh},d} (E^{\\alpha} + E^{\\beta}) = -\\frac{4}{5} \\sum_{ijklmn} \\frac{\\Theta_{-i}\\Theta_{-l}\\Theta_{-k}\\Theta_{-j}\\Theta_{m}\\Theta_{n}}{(E^{\\alpha}+\\epsilon_{mi} - i\\delta)(E^{\\beta}-\\epsilon_{ml}+ i\\delta)(\\epsilon_{il} + i\\delta)(\\epsilon_{kj} - i\\delta)} \\times  \\nonumber \\\\\n& \\times (\\Psi_{nk}^{\\beta})^*(E^{\\beta} - \\epsilon_{nk})(\\Psi_{nj}^{\\alpha})^*(E^{\\alpha} - \\epsilon_{nj}) \\Psi_{ml}^{\\beta} (E^{\\beta} - \\epsilon_{ml})  \\Psi_{mi}^{\\alpha} (E^{\\alpha} - \\epsilon_{mi}) \\times W_{ijkl}, \\label{expression11}\n\\end{eqnarray}\n{ where, for example, $\\hbar\\Sigma_{\\alpha\\beta}^{\\textrm{ee},a}$ stands for a self-energy contribution from a diagram representing electron-electron interactions obtained by flipping the arrows in one of the fermion loops in Fig. \\ref{BIEXCfig}-\\textit{a)}. Other symbols hold the same meaning as in Eq. \\eqref{expression}. Note that here, Coulomb matrix elements $W_{ijkl}$ have either all-electron or all-hole indices. Thus, the complete expression for the biexciton self-energy is\n\\begin{eqnarray}\n\\hbar\\Sigma_{\\alpha\\beta} = \\hbar\\Sigma_{\\alpha\\beta}^+ + \\hbar\\Sigma_{\\alpha\\beta}^-\n\\end{eqnarray}\nwhere $\\hbar\\Sigma_{\\alpha\\beta}^+$ is defined in \\eqref{expression+} and $\\hbar\\Sigma_{\\alpha\\beta}^-$ is defined in \\eqref{expression-}. }      \n\n\n\n\n\\section{Atomistic Models}\\label{CompDetail}\n\nCalculations of biexciton self-energies have been performed for chiral SWCNTs (6,5), (6,2) and (10,5), which are shown in Fig. \\ref{CNTall}. DFT with HSE06 functional, as implemented in VASP (Vienna \\textit{ab-initio} simulation package) \\cite{VASP}, was used to optimize geometries and obtain KS orbitals $\\phi_{i\\sigma}(\\textbf{x})$ and energies $\\varepsilon_i$. The momentum cutoff is defined as\n\\begin{equation}\n   \\frac{\\hbar^2 \\textbf{k}^2}{2m} \\le E_{max} , \\;\\;\\;\n   \\textbf{k} = 2\\pi \\bigg( \\frac{n_x}{L_x}, \\frac{n_y}{L_y}, \\frac{n_z}{L_z} \\bigg), \\;\\;\\; n_x, n_y, n_z = 0, \\pm 1, \\pm 2, \\cdots  \\label{cutoff}\n\\end{equation}\nwhere $m$ is the electron mass; we used $E_{max} = 300~ eV$. The number of orbitals used in the calculations was determined by the condition   \n$E_{i_{max}} - E_{HO} \\simeq E_{LU} - E_{i_{min}} \\ge 3.5~eV$ where $i_{max}$\/$i_{min}$ label the highest\/lowest orbital. \n\nPeriodic boundary conditions were used in the DFT simulations. In the axial direction the simulation cell length was chosen to accommodate \nan integer number of unit cells, while in the other two directions the SWCNT surfaces were separated by about 1 $nm$ of vacuum in order to avoid spurious interactions \nbetween their periodic images. For (6,2) and (10,5) the simulations have been performed including three unit cells. The rationale for this is that \nfuture work will involve SWCNTs with functionalized surfaces. So, including several unit cells will allow to keep the dopant concentration reasonably low. \nAlso, including three unit cells instead of one substitutes for the Brillouin zone sampling. So, here we perform calculations at the $\\Gamma$ point only. \nPreviously, it has been shown that the variations in the single particle energies over the Brillouin zone is reasonably small ($\\approx 10\\%$) when three \nunit cells have been included in the simulation instead of one \\cite{KryjevskiMEG}. For the (6,5) SWCNT only one unit cell was included due to high computational cost. \nHowever, the absorption spectrum for (6,5) was reproduced with the same accuracy as for the other two CNTs \\cite{KryjevskiMEG}. \n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{CNT_all.pdf} \n  \\caption{ Atomistic models of the three chiral SWCNTs: a) - (6,2), b) - (6,5) and c) - (10,5)  } \\label{CNTall} \n\\end{figure}\n\n\n\n\\section{Results and Discussion} \\label{Results} \nTable \\ref{table1} shows the results for the three SWCNTs. In all cases the resulting $\\Sigma_{\\alpha\\beta}({E}_{\\alpha} + {E}_{\\beta})$ come out to be real, with vanishingly small imaginary parts. This is as expected since in our approximation we only include the possibility of the biexciton state decaying into an exciton and an electron-hole pair, which is suppressed due to energy conservation.     \nThe biexciton self-energy correction to the biexciton gap was obtained by evaluating Eqs. \\eqref{expression} and \\eqref{expression2} for a biexciton state made of two lowest-energy excitons, \\textit{i.e.}, \n$\\hbar\\Sigma(E = 2 E_{\\rm gap}^{\\rm exc}),~E_{\\rm gap}^{\\rm exc}= {E}_1$.  For reference, the quasiparticle gap $E_{\\rm gap}=\\varepsilon_{LU}-\\varepsilon_{HO}$ obtained from the DFT simulation is included. Figure \\ref{plots1} shows plots of the density of states (DOS) for the excitons and biexcitons, including the shift due to exciton-exciton interactions. {The DOS was calculated using DOS$ = \\sum_i \\delta( E - E_i)$, where the delta function was approximated by a Gaussian function with a width corresponding to room temperature. For the exciton DOS, $E_i$ are the solutions to the Bethe-Salpeter equation - $E_{\\alpha}$ (Eq. \\eqref{BSE}). For biexciton DOS, $E_i = E_{\\alpha} + E_{\\beta}$ for a pair of non-interacting excitons and $E_i = E_{\\alpha\\beta}$ (Eq. \\eqref{self_energy_corr}) for the self-energy corrected pair of excitons.  }\n\\begin{table}[H]\n  \\centering\n  {\\begin{tabular}{|c|c|c|c| } \\hline\n Chirality & \\textbf{$E_{gap}$} & $E_{\\rm gap}^{\\rm exc}$  & $\\Sigma(2 E_{\\rm gap}^{\\rm exc})$   \\\\ \\hline\n$(6,2)$ & 1.33 & 0.98  & -0.045  \\\\ [1ex] \\hline\n$(6,5)$ & 1.22 & 1.09  & -0.041  \\\\ [1ex] \\hline\n$(10,5)$ & 0.91 & 0.835 & -0.036  \\\\ [1ex]\\hline \n  \\end{tabular}}\n   \\caption{ $E_{gap}$ is the HO-LU gap and $E_{\\rm gap}^{\\rm exc}=\\epsilon_1$ is the lowest exciton energy obtained from BSE. $\\Sigma(2 E_{\\rm gap}^{\\rm exc})$ is the biexciton self-energy for the lowest-energy biexciton.  All energies are in $eV$.} \n  \\label{table1}\n\\end{table}\n\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.8\\textwidth]{multiplot.pdf} \n  \\caption{ Exciton and biexciton DOS for the three nanotubes in units of exciton gap $E_{\\rm gap}^{\\rm exc}$ . Blue line is the exciton DOS, red line - biexciton DOS without self-energy corrections, green line - biexciton DOS with self-energy corrections. } \\label{plots1}  \n\\end{figure}\n\\noindent\nResults show a small redshift in the biexciton density of states for all the nanostructures under consideration. As a percentage of the non-interacting gap, these shifts are: -4.59\\% for (6,2),  -4.47\\% for (6,5) and -4.31\\% for (10,5) SWCNT. This is as expected since dipole-dipole electrostatic interactions are attractive. \n\n\nIn our approach it has been possible to sum perturbative contributions from the attractive interactions between the excitons in the biexciton state. However, the repulsive \ncontributions are only included to the leading order. A calculation where both types of interactions are included to the leading order, {\\it i.e.},  only including the contributions from \nthe four diagrams in Fig. \\ref{BIEXCfig}, has been performed. In this case the corrections to the biexciton gap as a fraction of \nthe non-interacting gap are:  -1.84\\% for (6,2),  -1.93\\% for (6,5) and -2.27\\% for (10,5) SWCNT. This suggests that better treatment of repulsive interactions by including \nhigher order perturbative corrections, which is prohibitively expensive, would only reduce the negative shift in the biexciton gap.  \n \n\n\n\n\\section{Conclusions and Outlook} \\label{Conclusion}  \nWe have developed a first-principles DFT-based MBPT method to compute biexciton state energies in a semiconductor nanostructure. \nFor that, we have computed self-energy of the biexciton state including (residual) electrostatic exciton-exciton interactions. In \nthis work we have only included spin-zero excitons. These biexciton energies are relevant for, {\\it e.g.}, accurate determination \nof the MEG threshold in a nanostructure. \n\n\nTo the first order in the Coulomb interaction there are four distinct Feynman diagrams contributing to the biexciton self-energy shown in Fig. \\ref{BIEXCfig}. \nHowever, it has been possible to perform partial resummation of the perturbative corrections, such as those included in the exciton-electron coupling term \nin Eq. \\ref{EXCFERMSF}. As a result, to the leading order in the electron-exciton coupling, just two distinct contributions that include intermediate exciton \nstate appear, as shown Fig. \\ref{BIEXCfig2}. Additionally, there are repulsive interactions between the like charges in the two excitons (electron-electron and hole-hole), \nwhich have been included to the leading order resulting in eight distinct contributions.\n \nCalculations have been performed for the chiral SWCNTs (6,2), (6,5) and (10,5). \nWe have found small negative corrections to the biexciton state gaps: -0.045 $eV$ in (6,2), which is 4.59\\% of the non-interacting biexciton gap; -0.041 $eV$ in \n(6,5), which is 4.47\\% of the non-interacting gap and -0.039 $eV$ in (10,5),  which is 4.31\\%.\nSmall magnitude of energy shifts confirms validity of the perturbative approach for the biexciton states, at least for the SWCNTs,\nand justifies simplified treatment of the biexcitons in SWCNTs as a pair of non-interacting excitons.\n\nAn important extension of the approach left to future work is to include the effects of triplet excitons, which would allow to compute energy of a biexciton \nmade of a pair of a triplets in the overall singlet state. This is needed for precise determination of the MEG threshold in the SF channel. \n\nAnother extension of this work would be to decorate both pairs of electron-hole lines in the diagrams in Fig. \\ref{BIEXCfig2} with interactions (zigzag lines) and \nperform resummation. This would result in the processes where two intermediate exciton states appear. But small magnitude of the first-order biexciton \nenergy corrections suggests that this modification of the technique is not likely to change the results significantly, at least for the SWCNTs.   \n\nImproving the overall precision of the calculations preformed here can be done by the use of $G_0W_0$ calculations for single particle energies instead of using the HSE06 functional. \nThis is expected to introduce an overall blueshift in both exciton and biexciton DOS, but not to alter our overall qualitative conclusions. Another step would be \ninclusion of the full, dynamically screened polarization function $\\Pi(\\omega,\\textbf{k},\\textbf{p})$ instead of the static, diagonal approximation $\\Pi(0,-\\textbf{k},\\textbf{k})$. This would greatly \nincrease computational cost, but it is not expected to alter the conclusions reached in this work.      \n\n\n\\section{Acknowledgements}       \n  Authors acknowledge financial support from the NSF grant CHE-1413614. \nThe authors acknowledge the use of computational resources at\nthe Center for Computationally Assisted Science and\nTechnology (CCAST) at North Dakota State University. The diagrams shown in Figs. 1-5 have been created with JaxoDraw \\cite{Jaxodraw}.     \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nConvolutional Neural Networks (CNN) have become a widely used architecture since they have proven to be an accurate universal function estimator in many domains, such as Image Classification and Natural Language Processing \\cite{CNN-lecun}. As a particular type of Neural Network (NN), a CNN consists of a collection of parameters, which represent weights and biases, that are tuned depending on the problem that is faced. These parameters, put together after long training sessions, have proven to be able to solve  complex problems even outperforming humans \\cite{GO-deepmind}. The tuning is performed automatically and the values that the parameters take are outside of the human comprehension, turning the models into black boxes. \n\nThe use of CNNs in custom applications demands high computational power, since there is no physical limitation to the complexity of the designed models. Simple augmentation of the size of the collection of trainable parameters has proven to work as long as new data keeps being fed to the model \\cite{hestness2017deep}. Furthermore, following the exponential evolution of the availability of computational power, the tendency of solving complex tasks with complex models is increasing \\cite{anthes2017artificial}. Language models are good examples of this, e.g. Mann et al. with the description of a NN with 175 billion parameters \\cite{gpt3}.\n\nAccording to this tendency, the most common deployment platforms used historically for DL-based software are servers that enable cloud computing. For the purpose of making use of a trained NN, a data pipeline is developed manually so that it links the pre-processing steps, computation of results, post-processing and eventually the display of the predictions. When all these steps are implemented in a server\/cloud, the requirement of an internet connection for using the model is a must. However, with the growth of the mobile applications market, approaches on the integration of NNs in lightweight devices are becoming popular given their ability to provide great services to the user. Within this setting, DL-based software consists of software-reliant systems that \\textbf{include data and components} that implement algorithms mimicking learning and problem solving \\cite{arpteg2018software}.\n\nIn this work, we aim to perform an exploratory and descriptive analysis on: {\\it (i)} what are the current challenges regarding the deployment of DL-based software as mobile applications; and {\\it (ii)} how can complexity be controlled while keeping a good level of accuracy. The subject is analysed by means of a practical study in which we perform all the required cycles from data acquisition, DL modelling, classification and application, and operation in real context. In this way, we study the relation between the DL-based software development and the accuracy when operating with it in the production settings \\cite{9238323}. Ensuring the generalization of the DL-based software in the operation environment is key and can be especially concerning in safety-critical applications \\cite{martinez-fernandez}.\n\nOur work is performed over the German Traffic Sign Recognition Benchmark dataset \\cite{Benchmarking-tsr}, created in order to standardize the traffic sign recognition literature. In the DL literature, this faced task is an instance of the Image Classification problem \\cite{doi:10.1080\/01431160600746456}. Each image in the data is from one and only one class so the models will be trained using classification accuracy as the success criteria. CNNs are the standard approach to Image Classification problems for their ability to fit in image processing. The CNN architecture has already shown that can achieve very good results in the literature but also for this specific dataset \\cite{li2014medical} \\cite{Benchmarking-tsr}.\n\nThis work has four main contributions: \n\\begin{itemize}\n    \\item Show experiences of the end-to-end DL lifecycle from a practical study on traffic sign recognition.\n    \\item  Discuss the challenges encountered in the deployment of CNNs.\n    \\item Discuss the trade-offs between accuracy and complexity in environments with limited computational power (e.g., mobile applications).\n    \\item An open science package of the developed DL-based software freely available on GitHub, following the \"Cookiecutter Data Science\" project structure \\footnote{https:\/\/drivendata.github.io\/cookiecutter-data-science\/} to foster correctness and reproducibility.\n\\end{itemize} \n\nThe document is structured as follows. In Section 2 we describe Related Work on both the challenges of integrating DL models in mobile applications and the focuses on optimizing the performance trade-off. In Section 3 we describe the Study Goal and Research Questions. In Section 4 we describe the study design and the end-to-end DL software lifecycle. In Section 5 we show the results of practical study that allow to answer the Research Questions. In Section 6 we discuss what is strictly analysed in our study and what specific parts of the whole topic are addressed.\nTo end with, in Section 7 we draw conclusions from the research performed and motivate future work.\n\n\\section{Related Work}\nIn this section, we respectively describe the challenges of deploying CNNs in DL-based software, and related work on the optimization of the performance trade-off in systems with limited computation power.\n\n\\subsection{Challenges in deployment}\\label{sec:sota-challenges}\nRegarding the recent interests in the deployment of DL-based software, Chen et al. state that mobile applications come with the strongest popularity trend over other platforms like servers\/clouds or browsers for integrating DL models \\cite{deployment-DL}. They also show that this increasing trend in the deployment of DL models in mobile applications comes with an inverse proportional relation with the knowledge that the community has about the subject. This way, they demonstrate that the deployment of DL-based software becomes the most challenging part in the life cycle of DL, and that this effect is even more critical when the target are mobile devices.\n\nWhen analysing the challenges of deployment of DL-based software, several studies focus on the quality attributes that are more relevant to this type of software. The importance of each quality attribute varies depending on the application but it is always the case that DL-based software implies taking care of measures that might not be considered when building traditional software systems. Remarkably, DL-based software effectiveness strongly depends on the structure of the data.\n\nIndeed, Pons and Ozkaya clearly state that, compared to software systems that do not integrate DL-based components, the deployment of DL-based software increases the risk in many quality attributes \\cite{quality-AI}. They identify Data centricity as the cause that affects the robustness, security, privacy and sustainability of these systems. Additionally, they analyse the methodology for architecting in Artificial Intelligence (AI) engineering. With the focus put on the software-data dependency, Ozkaya makes the difference between the development of software systems and DL-based software in the processes of building and sustaining \\cite{engineering-AI}. Also, Lwakatare et al. provide a list of challenges and solutions regarding the life cycle of DL-based software within industry settings. The challenges and solutions are synthesized into four quality attributes: adaptability (e.g. unstable data dependencies and quality problems), scalability (e.g. balancing efficiency and effectiveness in DL workflow), safety (e.g. explainability of DL models) and privacy (e.g. difficult data exploration in private datasets) \\cite{large-scale-AI}.\n\nThe identified challenges in the above related work that are objects of study in our work are: \n\\begin{itemize}\n    \\item (C1) \\textbf{Frameworks in early stage} of its development to support DL-based mobile applications.\n    \\item (C2) \\textbf{Software--Data dependency}.\n    \\item (C3) \\textbf{Explainability} of the models.\n    \\item (C4) \\textbf{Sustainability} of the models when deployed.\n\\end{itemize}\n   \n\\subsection{In the quest for optimizing the performance}\nIn the pursuit of optimizing the performance of DL-based software, there exist several key points that affect the overall system implementation. Important bottlenecks that involve limitations to the efficiency are found in both the modelling stage and in the usage of different frameworks to deploy the models. On the one hand, recent contributions \\cite{mobile-nets} show an optimized CNN architecture for enabling its usage in devices with less energy capacity. This allows to reduce the model's complexity (e.g. storage weight, computing power) while keeping the desired accuracy, hence optimizing the performance. In this case, the increase in the performance is due to the increase in the efficiency. \nOn the other hand, the availability of different frameworks supporting the implementation of DL-based software in all its stages has a strong influence on the accuracy of the integrated DL models, as shown in \\cite{empirical-frameworks}.\n\nCompared to the aforementioned studies, in our work we focus on analysing the challenges when deploying a specific type of NN, namely CNN, into mobile devices. Furthermore, we do so by means of a practical study. We relate the challenges found with the analysis of the performance, defined as the trade-off between two quality attributes: accuracy and complexity. We provide a comparative analysis between different configurations for each of the models in order to gather evidence of the implications of the accuracy and complexity. Moreover, the sustainability of the models once deployed is also studied to enhance the capabilities of the applications during the DL-based software lifecycle. Our work is developed under the PyTorch framework and Android operating system.\n\n\\section{Research Questions}\nIn this Section, we define the Research Questions (RQs), following the GQM guidelines \\cite{GQM}.\n\n\\begin{itemize}\n\\item RQ1 - What are the challenges of the creation and integration of complex CNNs in DL-based mobile applications?\n\n\\item RQ2 - What criteria are needed to reason about the trade-off between accuracy and complexity of DL models in mobile applications?\n\n\\end{itemize}\n\nThe motivation of RQ1 within this project is to study the \\textbf{challenges of the creation, training and integration of a CNN in a mobile application}, while identifying, exploring and documenting as many challenges as possible. The obtained conclusions aim to be generic for anyone whose desire is deploying complex models in quotidian software applications.\n\nFurthermore, RQ2 motivates a \\textbf{comparative analysis between different solutions to the problem of optimizing the performance trade-off}, since balancing the accuracy of the models and their complexity can be studied from different viewpoints (e.g. design of optimized operations or limitation of architecture's complexity).\n\n\\section{Study design}\\label{study-design}\nIn this section we describe the proposed pipeline for achieving a functional implementation of the traffic sign recognition mobile application. \n\nThere are four main cycles in the DL-based software lifecycle \\cite{9238323}. First, the \\textit{Data Management} which consists of the collection and processing of raw data. Second, the \\textit{DL Modelling} which consists of adjusting different models to obtain the best-performing possible solution. Third, the \\textit{Development} of the environment (mobile application in our work) and the integration of the DL model in it. Fourth, the \\textit{DL-based System Operation} with the application that integrates the DL model. This last phase enables sustainability of the model and experiments the performance trade-off within the production settings. \n\nA global overview of the study design and DL-based software lifecycle is shown in Figure \\ref{global_overview}. The goals required over the system are shown in the following list:\n\\begin{enumerate}\n    \\item Use of the device integrated camera\n    \\item Real-time response for a call to the model\n    \\item High accuracy in traffic sign recognition\n    \\item Data augmentation by the community\n    \\item Integration of updated models \n\\end{enumerate}\nWe implement these goals through iterations along the aforementioned DL-based software lifecycle phases. We also provide a solution for the \\textit{Send Data} operation in Figure \\ref{global_overview}. This allows the possibility that the user collects the generated data at run-time during the DL-based software system operation, which is then manually annotated and used for re-training the models on the performant computer. Then, this updated models can be integrated again for inference in the mobile application.\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{static\/GlobalProject.png}\n    \\caption{Software development lifecycle in our DL-based software}\n    \\label{global_overview}\n\\end{figure*}\n\n\\subsection{Platforms and technologies used}\nFor joining the DL modelling and the development of DL-based software, there exist recent DL frameworks, all of them under continuous development. These frameworks offer up to two different approaches to model transferring operations. On the one hand, \\emph{model conversion}. CoreML for iOS software and TensorFlow-Light for Android arguably are the most famous examples that offer this capability. On the other hand, \\emph{model export} using the Open Neural Network Exchange (.ONNX) file format, which was presented in 2017 by Microsoft and Facebook. This file format allows to store serialized versions of trained networks in compressed files. \n\nSince the .ONNX file format is widely used in the current state of the art, it is defined as a key part part of the proposed project pipeline for the practical study.\n\nWith all this, the focus is put on finding two different platforms that allow development-sided and operation-sided implementation of both the DL-component and the application that integrates it. The development-sided platform has to allow training and writing of a CNN into a .ONNX file. The operation-sided one has to be capable of reading and using the model and also of supporting the use of the camera in a mobile device. In this work, the development-sided platform is set to be PyTorch and the operation-sided platform is set to be Unity. We make use of the Unity Barracuda libraries for connecting the two platforms. Also, as observed in Figure \\ref{global_overview} the models built on PyTorch are trained on the Kaggle GPU, which is available freely for 40 hours a week.\n \nMoreover, it is also seen in Figure \\ref{global_overview} that only two of the five models that are trained are later deployed to the application. For an appropriate analysis of the performance of these two models when deployed, we built two applications that only vary in the integrated models.\n \n\\subsection{The Dataset}\nThe GTSRB traffic sign data consists of 144,769 labelled images of 2,416 traffic sign instances of 70 classes belonging to a portion of the real traffic signs that can be found in the roads of Germany. However, after following several criteria for guaranteeing the quality of data, explained in \\cite{Benchmarking-tsr}, the final dataset consists of a collection of 51,840 images from 43 classes fulfilling: {\\it (i)} images are of sizes from 15\u00d715 to 222\u00d7193 pixels, stored in PPM format; {\\it (ii)} it contains the definitions of the region of interest with a separating margin of the size equal to the 10\\% of the pixels; {\\it (iii)} it records the temporal order of the creation of the images; {\\it (iv)} it provides post-processed features like HOG descriptors, Haar-like features and Color Histograms. No usage of the post-processed features is made for the purpose of the deployment of a full-stack convolutional-based architecture for solving the image classification problem. \n\\subsubsection{The Pre-processing}\nWe applied several pre-processing steps that can be considered critical in the context of image processing model training: {\\it (i)} the images are resized to a fixed and squared size; {\\it (ii)} a center crop is applied to the images; {\\it (iii)} The images are encoded as tensors of the form \\textit{[B, C, H, W]}, where \\textit{B} is the batch size, \\textit{C} is the number of channels, \\textit{H} is height and \\textit{W} is width. The values of these tensors are normalized into the 0-1 range; {\\it (iv)} the tensors are standardized in the \\textit{H}x\\textit{W} channels with the pre-computed means and standard deviations of the 3 RGB channels of the images in the dataset.\n\n\\subsubsection{The Augmentation}\nFor the purpose of supporting a comparative analysis of the performance of different models, we implemented two variants for applying the technique of data augmentation to the pre-processed dataset: {\\it (i)} rotation in 90\u00ba degrees plus the vertical and horizontal flips to the images; {\\it (ii)} addition of manually tagged images by the users of the application during the system operation phase. \n\n\\subsection{The DL\/CNN models}\nThe models that are applied for solving the traffic sign recognition task are the standard CNN, the MobileNet v2 (MBv2) \\cite{mobile-nets} and the ResNet34 (RN34) \\cite{resnet}. For the first one, we test and evaluate different configurations in order to choose an optimal one that balances accuracy and complexity. For the last two, we test and compare two different ways of training the model: pre-training with feature extraction and fine-tuning, and full-training from scratch. The CNN works with standard convolutional blocks that consist of convolutions followed by batch normalization, an activation function, a max-pooling layer and a dropout layer. It can learn representations of images from scratch which are then passed to a classifier that is built on top of the convolutional blocks. The MBv2 is similar but works with separable depthwise convolutions. It is a form of factorized convolutions which factorize a standard convolution into a depthwise one and a 1x1 one called a pointwise convolution. The depthwise convolution applies a single filter to each input channel and the pointwise convolution then applies a 1x1 convolution to combine the outputs of the depthwise convolution \\cite{mobile-nets}. Finally, the RN34 is a more complex CNN that has been successfully applied in the image classification domain and has become a benchmark model for related tasks in the DL literature \\cite{resnet}. Both the MBv2 and the RN34 used in the practical study are downloaded from PyTorch and  are pre-trained on ImageNet data \\cite{deng2009imagenet}. \n\nThe results on the CNN models are shown in Table \\ref{tab:cnn-results}, where we compare the number of Convolutional Layers, the sizes of the two implemented Fully Connected layers (FC1 and FC2), the total number of parameters, and also the accuracy achieved in the test set with 10 epochs of training. Furthermore, we write down if the augmented dataset is used, the fixed image sizes chosen and the model weight in MegaBytes (MB). Note that the input size of FC1 is equal to the number of output features of the convolutional blocks. Also note that the output size of FC1 is the input size of FC2 and the output size of FC2 is 43, which is the number of unique classes of traffic signs in the data. The resulting optimal configuration from Table \\ref{tab:cnn-results} is promising because of the little complexity, the small storage weight and the great performance shown.\n\n\\begin{table*}[t]\n\\begin{center}\n\\caption{\\label{tab:cnn-results}Performance analysis of the different CNN configurations.}\n\\scalebox{1}{%\n    \\begin{tabular}{|c|c|r|c|c|c|r|r|r|}\n        \\hline\n        \\textbf{Data Aug.} & \\textbf{Image Size} & \\textbf{Batch Size} & \\textbf{Conv. Layers} & \\textbf{FC1 input} & \\textbf{FC2 input} \n        & \\textbf{Num. Parameters} & \\textbf{Test accuracy} &  \\textbf{Model Weight} \\\\\n        \\hline\n         NO & 256x256 & 256 & 5 & 8x8x256 & 4x4x64 & 17.21M & 88.02\\% & 65.68MB \\\\\n         NO & 512x512 & 64 & 6 & 8x8x512 & 4x4x128 & 67.13M & 95.00\\% & 259.70MB \\\\\n         YES & 256x256 & 64 & 5 & 8x8x256 & 4x4x64 & 17.21M & 89.31\\% & 65.68MB \\\\\n         YES & 512x512 & 64 & 6 & 8x8x512 & 4x4x128 & 67.13M & \\textbf{95.52\\%} & 259.70MB \\\\\n         NO & 256x256 & 64 & 6 & 4x4x512 & 2x2x128 & \\textbf{5.10M} & \\textbf{95.00\\%} & \\textbf{22.11MB} \\\\\n         \\hline\n    \\end{tabular}}\n\\end{center}\n\\end{table*}\n\nAs it can be seen in Table \\ref{tab:cnn-results}, no increase in performance appears when the augmented set of data is used. This might be because very fast training is happening, thanks to the non-saturating ReLU function in combination of the convolutional operations plus the use of minibatches. More importantly, it can be seen that the highest tier classification accuracy can be achieved with the lightest of the tested configurations.\n\nRegarding the MBv2, the experiments do not change its architecture but only its training methodology. Three techniques are distinguished: Feature Extraction (FE), which uses a pre-trained version of the model and just tunes the built classifier on top of it, training its weights and biases. Fine-Tuning (FT), which also uses the pre-trained version of the model but tunes the weights of all its layers. And Training from Scratch (TfS), which loads the architecture with a random initialization and fully trains it. The results on the training stage with the MBv2 for 10 epochs are presented in Table \\ref{tab:mobilenets-test}.\n\n\\begin{table}[h!]\n\\centering\n\\caption{Performance analysis of the Mobile Net v2}\n\\label{tab:mobilenets-test}\n\\scalebox{1}{%\n    \\begin{tabular}{|c|c|r|}\n        \\hline\n         \\textbf{Data Aug.} & \\textbf{Training method} & \n         \\textbf{Test accuracy} \\\\\n         \\hline\n         NO & FE & 4.00\\%  \\\\\n         NO & FT & 4.00\\%  \\\\\n         NO & TfS & 4.00\\% \\\\\n         YES & FE & 68.00\\%  \\\\\n         YES & FT & \\textbf{95.20\\%}  \\\\\n         YES & TfS & 92.50\\%  \\\\\n         \\hline\n    \\end{tabular}\n}\n\\end{table}\n\nAn important drawback of the MBv2 is spotted in Table \\ref{tab:mobilenets-test}, where it can be seen that without the augmented set of data the network is not capable of learning on the traffic sign recognition task. Anyway, when the augmented set of data is used it can be seen that the same performance as the best CNN is achieved with a much smaller number of parameters, showing that this architecture performs very efficiently. \n\nFinally, the results on the benchmark RN34 is shown in Table \\ref{tab:resnet-test}. Training experiments follow the same logic as with the MBv2 but without trying the augmented set of data, since it is rapidly seen that it is not needed for this architecture.\n\n\\begin{table}[h!]\n\\centering\n\\caption{Performance analysis of the RN34}\\label{tab:resnet-test}\n\\scalebox{1}{%\n    \\begin{tabular}{|c|c|r|}\n        \\hline\n         \\textbf{Data Aug.} & \\textbf{Training method} & \\textbf{Test accuracy}  \\\\\n         \\hline\n         NO & FE & 20.00\\%  \\\\\n         NO & FT & \\textbf{97.80\\%}   \\\\\n         NO & TfS & 94.10\\%   \\\\\n        \\hline\n    \\end{tabular}\n}\n\\end{table}\n\nAs it is seen in Table \\ref{tab:resnet-test} the training method of FT gets to the highest seen accuracy in the test split of the training set during the practical study. The output model of this training stage is the one deployed in the application.\n\n\\subsection{The DL-based software}\nThe architecture of the DL-based software system operation is shown in Figure \\ref{app_DL}, following design patterns from \\cite{washizaki2020machine}. We build two applications that have the same architecture but only differ in the DL model they integrate. The application that loads the SmallCNN has a total weight of 68.84MB and the one that loads the RN34 of 131MB. The baseline weight of the application bundle generated by Unity3D is of 46.73MB without the models.\n\nThe two built applications implement the goals 1,2 and 3 defined in Section \\ref{study-design}. However, for ensuring the sustainability of the models when deployed, and for satisfying the applications goals 4 and 5, an external server\/computer is needed for performing the re-training of the model with the new tagged data added by the community. This implementation still lets the user obtain recognition predictions without external calls. The user has this option because the application client will only make use of the server\/computer for downloading new updates on the model and for manually submit new tagged data in a personally controlled way, whenever it is desired. We consider allowing the user to annotate the data that it generates at run-time an efficient solution to the sustainability of the model when deployed. This could be done either by verifying correct model recognition predictions or by correcting wrong ones. However, the architecture that we implement in this practical study makes use of the mobile device storage to save the taken pictures, which are then manually annotated and sent to the local machine, where they are fed to the model again in the Kaggle GPU. This is not a scalable implementation but releasable for research purposes.\n\n\\subsection{Threats to validity}\nRegarding construct validit\n, we build two DL models following a small CNN and ResNet34 (RN34) \\cite{resnet}, in order to mitigate mono-operation bias. As for conclusion validit\n, due to the exploratory nature of the study we do not execute statistical tests. Regarding internal validit\n, the DL models are executed in the same context (e.g., computational power, mobile technologies), mitigating that the efficiency results are caused accidentally rather than by the DL models themselves. Finally, regarding external validit\n, our results are tied to the context of traffic sign recognition in mobile applications. Furthermore, the analysis of the performance in the operation-sided environment is carried out in a mobile device with the Qualcomm Adreno 618 GPU running over Android, which determines the extent to which the results obtained that regard the complexity can be generalized.\n\n\\begin{figure*}[h]\n    \\centering\n    \\scalebox{0.65}{%\n    \\includegraphics[width=\\linewidth]{static\/sysops.png}\n    }\n    \\caption{DL-based Software System Operation Architecture}\n    \\label{app_DL}\n\\end{figure*}\n\n\\section{Results}\nIn this section we present and discuss the results of our practical study for each RQ. \n\n\\subsection{RQ1: Challenges of the creation and integration of complex CNNs in DL-based mobile applications}\nFor answering the RQ1, we analyze both the identified challenges in the state-of-the-art (see Section \\ref{sec:sota-challenges}) and our results. This analysis is shown in Table \\ref{results-challenges}, where: \\textbf{v} indicates that the challenge has been verified to be real and concerning; \\textbf{\u00b1} indicates that the challenge has been identified but not faced in the practical study; \\textbf{new} defines newly spotted challenges up to our knowledge emerging from our study.\n\n\\begin{table}[h]\n\\begin{center}\n\\caption{\\label{results-challenges}Diagnoses of identified challenges}\n\\scalebox{1.15}{\n        \\begin{tabular}{|c|c|}\n        \\hline\n        \\textbf{Challenges} & \\textbf{Diagnostic}\\\\\n        \\hline\n        Frameworks in early stage (C1) & v \\\\\n        Software-Data dependency (C2) & v \\\\\n        Explainability (C3) & \u00b1 \\\\\n        Sustainability (C4) & v \\\\\n        Software dependencies (C5) & new \\\\\n        Model performance (C6) & new \\\\\n        \\hline\n    \\end{tabular}\n}\n\\end{center}\n\\end{table}\n\nWe verify C1 because we find few alternatives to develop DL-based mobile applications. Furthermore, we encounter difficulties in the support of DL architectures in these frameworks. In our work we experience limitations in the ONNX file format and in the Unity Barracuda package, so we motivate further development of these. From C1 we uncover C5, stating that \\textbf{we see a too marked separating line between the development-sided frameworks and the operation-sided ones}. C5 reveals a constraint we found in DL-based software defined as follows. \\textbf{No matter what framework one wishes to use, it will always need to wrap at least two different technologies}: one for the data management and model training outside the device, and the other for the model operation inside the user's device. This implies challenges regarding the mutual support between the two technologies. In the following paragraph, the faced challenges regarding C5 are listed.\n\nFirst, \\textbf{the pre-processing steps that are applied to both the initial dataset in the development phase and the ones in the operation phase (which are applied in real time to the images taken by the user) have to be the same despite being implemented in different frameworks}. DL-based software developers shall ensure that the inputs on both platforms are the same to the network, so it gives the same predictions for the same images.\nSecondly, \\textbf{the operation-sided framework must support model importing and must provide functional libraries that enable the appropriate usage of the model in its exported format}. In Unity Barracuda, a challenging limitation found is the inability to support a lot of the basic operations in the treatment of DL models (a reason for which we verify C1). For example, let's consider the \\textit{view()} function from PyTorch. For the hand-made CNNs, this operation has been replaced by \\textit{Flatten()} which does exactly the same, but for the pre-trained MBv2 this method is used and hence the architecture cannot be deployed directly to Unity. In conclusion, due to the limitations in the used operation-sided framework, a model architecture that is built for ensuring efficiency on mobile devices cannot be deployed to the mobile application.\n\nRegarding C2, \\textbf{we have experienced high volatility in the applications' effectiveness when testing them with real-world data outside of the training set}. We verify C2 because we experience that the DL-based mobile applications effectiveness strongly depends on the data that is fed to it. This fact makes sustainability (C4) \\textbf{become a critical quality attribute for reducing the data dependency and ensuring high performance of the models when deployed}. In the practical study, we implement a solution for the challenge of ensuring sustainability in DL-based software that consists of real world context data augmentation. By doing this, we smooth the differences between the development and operation phases of the DL-based software lifecycle by adapting the models to the real world environment. Hence, we also smooth the dependency between the applications' effectiveness and the data that is fed to it, which makes the DL-based mobile applications more sustainable and makes them generalize better. See the performance of the two deployed models in the two iterations of the operation phase in Table \\ref{tab:operation-performance} where it is shown how the data augmentation technique applied in Iteration 2 reduces the data-software dependency (C2) and hence increases the sustainability (C4) of the applications. \n\n\\begin{table}[ht]\n\\centering\n\\caption{Performance of the deployed models when receiving operation-sided data}\n\\label{tab:operation-performance}\n\\scalebox{1}{%\n    \\begin{tabular}{|c|c|r|r|}\n        \\hline\n         \\textbf{Model} & \\textbf{Iteration} & \n         \\textbf{Accuracy} & \\textbf{Storage Weight} \\\\\n         \\hline\n         Small CNN & 1 & 45.07\\% & 22.11MB \\\\\n         RN34 & 1 & 64.78\\% & 83.24MB \\\\\n         Small CNN & 2 & \\textbf{92.95\\%} & 22.11MB \\\\\n         RN34 & 2 & \\textbf{98.59\\%} & 83.24MB \\\\\n         \\hline\n    \\end{tabular}\n}\n\\end{table}\n\nRegarding C3, our approach for providing explainability in the built applications is based on providing standard confidence indicators of the models' predictions. These indicators consist of the output probabilities of class membership which have been scaled to become percentages of confidence for a given input. \\textbf{These simple indicators provide a level of explainability of the models} that is acceptable for the applications in the practical study, meaning that \\textbf{they allow some level of interpretation to the user}. We do not consider ensuring this quality attribute a critical challenge because it might not generalize to lots of applications that integrate a DL-based component. However, \\textbf{we recognize the importance of the presence of any measure that provides the user with information about why the model has given a result}.\n\nOur practical study uncovered C6. Compared to the usage of a server, the model exporting operation and its integration into the mobile application \\textbf{has the drawback of the inclusion of the model's storage weight in the application, hence, in the mobile storage}. For this reason the \\textbf{optimization of the performance trade-off is a must when deploying DL-based software to mobile devices}. We have shown in the practical study that not limiting the complexity of the model's architecture, as in the case of the RN34, yields a cost in performance in the mobile application in terms of real time response and storage weight. For the deployment of DL-based software to mobile devices, ensuring efficiency is not an option, it is a must, and facing it becomes a critical challenge.\n\n\\subsection{RQ2: Criteria needed to reason about the trade-off between accuracy and complexity of DL models in mobile applications}\nMost of the criteria applied to optimize the performance trade-off are approaches based on the reduction of the models' complexity. In an environment where the developer has few alternatives in the choice of both development-sided and operation-sided frameworks, the deployment of these systems becomes a bottleneck of the DL pipeline. This way, \\textbf{the evolution of the performance of DL-based software is strictly linked to the evolution of the current model conversion and exporting technologies}, a field that is in its very early stages, where few file formats can be used and with several limitations. For this reason, the deployment of DL-based software to the server cloud and the remote usage through internet connections is still a more flexible choice in terms of DL-based services that can be provided to a mobile user. However, this is not fault-tolerant, due to the dependency on hosting and internet connections. \n\n\\textbf{A fact that makes the deployment of DL-based software to mobile devices not always suitable is that it collides with the benefits of transfer learning}. The major capabilities of transfer learning include great results with little task-specific data, the ease in fine-tuning pre-built architectures and omitting the model design stage. The common available models for transferring learning are of massive weight, and the revolution of transfer learning has created a motivation to use these largest-scaled models for solving any task before designing task-specific architectures \\cite{transfer-learn}. This has never been a problem until now, when one wants the massive model to be deployed in an environment with limited computational power.\n\nWe show in Table \\ref{tab:operation-performance} that the most famous architecture amongst all the experimented ones, which is the RN34 \\cite{resnet}, is the one carrying the biggest drawbacks in terms of storage weight. In this way, the simple CNN architecture with the appropriate configuration outperforms the massive model in terms of performance because of the ability to provide almost the same accuracy when solving the traffic sign recognition task and because of its reduction in the complexity of the architecture.\n\nFinally, the MBv2 looks very promising in this field for its low complexity and high accuracy. Although it has not been deployed for the proposed practical study due to framework incompatibilities, it is concluded that the modifications in its architecture provide the best capabilities for the purpose of deploying DL-based software into environments with limited computational power, in this case the mobile devices. \n\nThis way, in order to lean towards a specific architecture for DL models that are going to be deployed in mobile applications we look for the ones that keep low complexity (e.g. fast real-time response, limited number of parameters, reasonable storage weight) and accomplish high accuracy and generalization. \n\n\\section{Discussion}\nIn the following we review our findings and discuss their implications. First, we have verified that the identified challenges that regard the software-data dependency and sustainability of the models deployed in the software are of major concern. To solve them, we have proposed the implementation of a data augmentation technique that adapts the models to the real world environment, hence providing sustainability. We have shown significant increases in the accuracy when sustainability is ensured in the two systems by means of the data augmentation technique (see Table \\ref{tab:operation-performance}). This step is represented by the \"Send Data\" operation in Figure \\ref{global_overview}. We have also verified the need of the evolution of frameworks that support the development of DL-based software, which is another challenge that has been identified in related work. Additionally, we have uncovered a new challenge related to the mutual dependencies of these frameworks, which enriches the identification of the framework capabilities challenge. Furthermore, we have provided identification and description of the criteria that can be applied to approach efficiency in the deployment of DL models in mobile applications. Also, we have related the reasoning of the accomplishment of efficiency to a newly identified challenge.\n\nTaking into account the differences between the performance of the two DL-based mobile applications according to our criteria, we now discuss a possible functional and efficient implementation. This implementation could consist of the integration of the two deployed models in the same DL-based mobile application. The default model could be set to the most efficient one, and in case the results given by this were not satisfactory the user would have the option to switch to the most robust and least efficient model.\n\n\n\\section{Conclusions}\nIn this work we have studied the DL-based software lifecycle in a practical manner. We have provided an analysis of the challenges found when developing this type of software. Specifically, we have put the focus on the deployment of DL-based software in mobile applications. Concretely, we have tested the performance of different CNN architectures under the PyTorch and Android frameworks. With all this, we have experienced, solved and documented many of the previously identified challenges in the related work and also have highlighted new ones. Furthermore, we have analysed the roles of the accuracy and the complexity in the DL-based software performance, relating this quality attribute to a newly identified challenge of the deployment of DL-based software.\n\nThis study motivates several key points of future work. First, the search for optimized model architectures that allow more efficient solutions to the deployment of DL-based components in mobile applications. These can both implement more efficient operations in its architecture or have limitations in its complexity when designed. The study also encourages the community to develop alternative solutions to the conversion and export of models together with software technologies that support mobile application development with DL-based components. The latter would reduce the severity of the challenges that are met while implementing this type of software. \nMoreover, the study motivates the design of more sophisticated confidence indicators that describe in a more precise way the uncertainty in the models predictions. The uncertainty of a model prediction given the input conditions is a measure that is hard to determine but useful to influence the decisions of autonomous DL-based systems. These can be key in many applications related to safety-critical environments. Also, sophisticated confidence indicators increase the quality of the systems since these enhance the explainability to the user. Last, we expect long evolution in the design of empowered GPUs for mobile devices which can relax the limitations of complexity in the architectures for achieving efficiency within the mobile device settings.  \n\n\\section{Data Availability}\nIn this section we provide a demo video of the developed and evolved DL-based software, available at \\url{https:\/\/www.youtube.com\/watch?v=yFsp6kxO5kI}. Additionally, we provide an open source code repository of the whole project in \\url{https:\/\/github.com\/yuyecreus\/CNN-in-mobile-device}.\n\n\\section*{Acknowledgment}\nWe thank Lisa J{\\\"o}ckel for having reviewed our work and for being supportive and useful to enhance it.\nThe research presented in this paper has been developed in the context of the CBI course at the GCED@FIB.\n\n{\\small\n\\bibliographystyle{IEEE\/IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\n\n\n\n\nDyson introduced his model of $N$ Brownian particles evolving in a constraining potential in 1962 \\cite{Dyson62}. The particle evolution described by Dyson with quadratic potential gives the evolution of the eigenvalues a Hermitian matrix Brownian motion $M_t$. For fixed time $t>0$ the matrix $M_t$ has Gaussian Unitary Ensemble (GUE) distribution. Dyson also formulated a more general particle flow which corresponds to a larger class of point processes called $\\beta$-ensembles. The $\\beta$-ensembles are finite point processes with $N$ points and joint density\n\\begin{equation}\n\\label{eq:betaensemble}\n  \\rho^{\\beta}_N(\\lambda) = \\frac{1}{Z^{\\beta}_N} e^{-\\frac{\\beta}{2}\n  \\left( -\\sum_{1 \\leq i \\neq j \\leq N} \\log|\\lambda_i - \\lambda_j| + \\sum_{i=1}^N V(\\lambda_i) \\right)\n},\n\\end{equation}\nwith $Z_N^{\\beta}$ a scaling factor $V(x)$ the potential, and $\\beta>0$. The case $V(x)=x^2\/2$ and $\\beta=1,2, $ and $4$ correspond to the original Gaussian ensembles and their Dyson evolution. These could be further generalized by changing the interaction term, but we restrict ourselves to this case. This joint density leads to the generalization of the Dyson Brownian motion. Let \n\\begin{equation}\n\\label{eq:dlambda}\n  d\\lambda_{i,t} = \\sqrt{\\frac{2}{\\beta}}dZ_{i,t} - \\biggl( \\frac{V'(\\lambda_i)}{2} - \\sum_{j: j \\neq i} \\frac{1}{\\lambda_i - \\lambda_j} \\biggr)\\,dt\n\\end{equation}\nwhere $\\left\\{ Z_{i,t} \\right\\}_1^n$ are independent standard Brownian motions. Then (\\ref{eq:betaensemble}) is the equilibrium measure for this process, and the flow is stationary with respect to this measure. \n\nThere is significant existing work on Dyson's model. Rogers and Shi showed convergence to the Wigner law in the $N\\to \\infty$ limit for the original quadratic potential \\cite{RogersShi}. Israelsson showed the fluctuations converged to a Gaussian process with later work by Bender giving the covariance structure \\cite{Israelsson, Bender}. Other work on the fluctuations of the trace moments was done by Perez-Abreu and Tudor \\cite{AbreuTudor}. Later work by Unterberger extended this to the general $V,\\beta$ case \\cite{Unterberger}. This work done in the asymptotic setting rests heavily on convergence of the Stieltjes transform but gives little information on local interactions.\n\nIn the case of the Gaussian ensembles where $V(x)=x^2\/2$ and $\\beta=1, 2,$ or $4$ the eigenvalues of the matrices form a finite determinantal or Pfaffian point process. The Dyson flow for $\\beta=2$ can also be described as a determinantal point process with the extended Hermite kernel $K(\\overline x,\\overline y)$, now with points in space-time. These descriptions give exact formulas for correlation functions and may be used to prove local limits. Forrester and Nagao give local point process limits at multiple times via determinantal methods for the $\\beta=2$ case \\cite{FN2}. Under the appropriate substitutions the extended Hermite kernel converges to the extended sine kernel. Similar results hold at the edge of the spectrum. \n\nWork in recent years has been focused on the more general Dyson model introduced in (\\ref{eq:dlambda}), but much remains to be done. Landon, Sosoe, and Yau give universality results relating eigenvalue statistics for a class of potentials $V$ to the classical Dyson Brownian motion \\cite{LSY}. Huang and Landon give results on rigidity and a mesoscopic CLT \\cite{HuangLandon}. In particular the question of local limits for the general $\\beta$ remain open even in the case of the quadratic potential. These questions are the motivation behind the current work. To see the connection we give a brief review of the work done for non-dynamic model.\n\n\nIn the case of the $\\beta$-Hermite ensembles (not time-evolving) for general $\\beta>0$ the proof of local limits rests on a tridiagonal matrix models introduced by Dumitriu and Edelman \\cite{DE}. The tridiagonal (Jacobi) model for the case $V(x) = 2 x^2$ is given by \n\\begin{equation}\n\\label{eq:tridiagonal}\nA_{n} = \\left[ \\begin{array}{ccccc}\nb_1 & a_1 & && \\\\\na_1 & b_2 & a_2 & &\\\\\n& \\ddots & \\ddots & \\ddots \\\\\n& & a_{n-2} & b_{n-1} & a_{n-1}\\\\\n&&& a_{n-1} & b_n\n\\end{array}\\right]\n\\end{equation}\nwith $a_i \\sim \\chi_{\\beta(n-k)\/4}$ and $b_i \\sim \\mathcal{N}(0,2)$ all independent. Notice that the potential in this case is off by a factor of 4 from the potential mentioned previously. This will be a convenient choice later. Dumitriu and Edelman showed that the eigenvalues of $A_n$ have joint density given by (\\ref{eq:betaensemble}) \\cite{DE}. They in fact go further, using a bijection between the entries of a Jacobi matrix and the eigenvalues together with the square of their spectral weights. \n\\begin{proposition}[Dumitriu-Edelman, \\cite{DE}]\nWe take $q_i$ to be the first entry of the vector $v_i$ that satisfies $A_n v_i = \\lambda_i v_i$ with $\\|v_i \\|= 1$, and $\\lambda_1< \\lambda_2<\\cdots < \\lambda_n$ to be the ordered eigenvalues of $A_n$. Then the map $(\\overline a, \\overline b) \\leftrightarrow (\\overline \\lambda, \\overline q^2)$ is a bijection, and moreover for $A_n$ defined as above we get \n\\begin{equation}\n\\label{eq:dirichlet}\n(q_1^2,q_2^2,...,q_n^2) \\sim \\operatorname{Dirichlet}(\\tfrac{\\beta}{2},...,\\tfrac{\\beta}{2}).\n\\end{equation}\n\\end{proposition}\nThis tridiagonal model was used by Valk\\'o and Vir\\'ag to give a point process limit in the bulk \\cite{BVBV}, and Ram\\'irez, Rider and Vir\\'ag to give a point process limit at the edge \\cite{RRV}. \n\n\n\\subsection{Notation}\n\n \nWe deal with matrix-valued random processes.  The subscripts of such processes are used both for referring to matrix and vector entries, as well as a time parameter.  So, we adopt the following convention.  A process $\\MP{X} = \\left( \\MP{X}{t}, t\\geq 0\\right)$ will be shown in bold whenever its time parameter is not shown.  Otherwise, it will be shown in normal weight font.  Matrix indexing subscripts will also appear in the subscript, but always first.  So for example, $\\MP{X}{t}[i,j]$ refers to the $i,j$ entry of the matrix process $\\MP{X}$ at time $t,$ while $\\MP{X}[i,j]$ refers to the scalar-valued process of the $i,j$ entry.  \n\nThere are many instances in this paper where $[A,B]=AB-BA$ denotes the commutator. In order to avoid confusion all quadratic variation and covariation terms will be written in the form $dX_t dY_t$ or $(dX_t)^2$.\n\n\n\n\\subsection{Goals and Results}\n\nThe goal of this paper is to give a description of the tridiagonal matrix evolution associated to the $\\beta$-Dyson Brownian motion flow. These models can in fact be completely characterized, but require a substantial amount of machinery to describe. In this paper we begin by showing the existence of such a tridiagonal flow and then developing the linear algebra tools needed to characterize the model. We then move on to characterizing the model and asymptotics for the case $V(x)= 2x^2$. \n\nRecall that the tridiagonal model is in bijection with the eigenvalues and the spectral weights. We have an explicit description of the eigenvalue flow, but we still have the freedom to choose the process by which the spectral weights evolve. The first partial description of the tridiagonal model is in Theorem \\ref{thm:martingales} which gives a description of the martingale terms for the general model. We then specialize to the case where the spectral weights remain fixed. The model here is given in its entirety by Theorem \\ref{thm:tridiagonalfrozen}. This model is particularly nice in the case where the eigenvalue flow is stationary. In this case we get that the tridiagonal model is also stationary. We describe two more cases, the first is the case where the spectral weights evolve according to some finite variation process. The description is given in Theorem \\ref{thm:tridiagonalFV}. The second is a model where the martingale terms in the evolution vanish on the first half of the matrix. \n\n \n\n\nIn addition to explicit descriptions of the evolution of the finite matrices we get the following asymptotic result on the evolution of the entries of $\\MP{A}$ as $n \\to \\infty$.  Define the operator $\\mathcal{F}$ on Jacobi matrices by\n  \\[\n    \\mathcal{F}(A)_{k,\\ell}\n    =\n    \\begin{cases}\n      -b_\\ell(4\\ell-2)+\n       4\\sum_{j=1}^{\\ell-1} b_j & \\text{ if } k = \\ell , \\\\\n      -2 (\\ell+1)a_\\ell- 2\\ell a_{\\ell-1} \n      + 4\\sum_{j=1}^{\\ell-2} a_j \n      & \\text{ if } k = \\ell + 1 ,\\\\\n    0, & \\text{ otherwise.}\n    \\end{cases}\n  \\]\n  and $D$ be the infinite Jacobi matrix\n\\[\n  D\n  =\n  \\begin{bmatrix}\n    0 & \\frac{1}{2} &   &  &  & \\\\\n    \\frac{1}{2} & 0 & \\frac{1}{2} &  &  & \\\\\n      & \\frac{1}{2} & 0 & \\frac{1}{2}&  & \\\\\n      &   & \\frac{1}{2} &  &  &  & \\\\\n      &   &   &  &  \\ddots &  & \\\\\n  \\end{bmatrix}.\n\\]\n\nLet $\\mathfrak{s}$ denote the semicircle density on $[-1,1],$\n\\[\n  \\mathfrak{s}(x) = \\frac{2}{\\pi}\\sqrt{1-x^2}\\,\\one[|x| \\leq 1],\n\\]\nand define for $z \\in {\\mathbb C} \\setminus [-1,1]$\n\\[\n  s^{\\mathfrak{s}}(z) =\n  \\int_{-1}^1 \\frac{\\mathfrak{s}(x)\\,dx}{x-z}\n  =\n  2(-z+\\sqrt{z^2-1})\n  .\n\\]\n\n\n \\begin{theorem}\n \\label{thm:asymptotics}\n Let $\\MP{A}$ satisfy the evolution for the case of frozen spectral weights in Theorem \\ref{thm:tridiagonalfrozen}, with $V(x)=2x^2.$  Suppose that at time $0,$ the eigenvalue distribution of $\\MP{A}{0}^{(n)}\/\\sqrt{n}$ satisfies\n \\[\n   n^{-1\/2}\\operatorname{tr}\\left( z - \\MP{A}{0}^{(n)}\/\\sqrt{n} \\right)^{-1}\n   - n^{1\/2} s^{\\mathfrak{s}}(z)\n   \\overset{\\Pr}{\\to} \\int_{-1}^1 \\frac{\\nu(dx)}{x-z}\n \\]\n for some signed measure $\\nu$\n uniformly on compact sets of $\\mathbb{C}\\setminus [-1,1].$  Suppose that the spectral weights of $\\MP{A}$ satisfy \\eqref{eq:dirichlet} and are independent of the eigenvalues of $\\MP{A}$.  If $\\MP{A}{0}$ is the tridiagonal in (\\ref{eq:tridiagonal}), then this is satisfied with $\\nu = 0.$ \n \nThen any bounded order principal submatrix of $\\MP{A}-D\\sqrt{n}$ converges to the solution of\n\\[\ndA_t = \\mathcal{F}(A_t) - { \\sqrt{\\beta}} \\mathcal{G}\n\\]\nwhere $\\mathcal{G}$ is a fixed (non-time dependent) Gaussian vector with an explicitly computable correlation structure. If we choose $\\overline q$ to be flat (that is $q_i^2 = \\frac{1}{n}$ for all $i$) then we get\n\\[\ndA_t = \\mathcal{F}(A_t).\n\\]\n\\end{theorem}\nIn the stationary case, or more generally when $\\nu$ is $0,$ the limiting evolution of the tridiagonal matrix is identically $0.$\n\n\n\\subsubsection{Derivatives of the Lanczos algorithm}\n\nFinding an explicit description for the tridiagonal evolution requires the us to solve for the tridiagonalization of a symmetric or Hermitian matrix after a small perturbation. This may be used to give an explicit derivative of the Lanczos algorithm. \n\nWe will use $\\mathscr{T}$ to denote the tridiagonalization operator on Hermitian matrices. Let $M$ be any $n \\times n$ Hermitian matrix so that $\\mathscr{T}(M)=A$ where the off diagonal entries are strictly positive. We will take $p_0(x),...,p_{k-1}(x)$ to the be orthogonal polynomials that are orthonormal with respect to the weights $\\{q_i\\}_{i=1}^n$. See Section \\ref{sec:OPs} for further information about the construction. We further take $p_{k}^{(\\ell)}(x)$ to denote the $\\ell$-th minor polynomials. We use a notation that may not be consistent with other notations choices for the minor polynomials, see Section \\ref{sec:minors} for details. \n\nWe make one further definition. We use $D_{G, \\mathscr{T}}$ to denote the derivative of the Lanczos algorithm in the direction of $G$ in the tridiagonal basis. In particular \n\\[\nD_{G,\\mathscr{T}} \\mathscr{T}(M) = \\lim_{\\varepsilon \\to 0} \\frac{1}{\\varepsilon} \\mathscr{T}(A+\\varepsilon G).\n\\]\nNotice that on the right hand side we are taking the tridiagonalization of a perturbed tridiagonal matrix rather than in the original basis for $M$.\n\n\\begin{theorem}\n\\label{thm:lanczos}\nLet $M$ be any $n \\times n$ Hermitian matrix so that $\\mathscr{T}(M)=A$ where the off diagonal entries are strictly positive, and let $W^{k,\\ell}$ be the matrix that is 0 everywhere except $(k,\\ell)$ and $(\\ell,k)$ where it is 1. Let $G$ be a matrix and $D_{G,\\mathscr{T}}$ denote the derivative in the direction of $G$ in the tridiagonal basis. Then \n\\[\nD_{W^{k,\\ell},\\mathscr{T}} \\mathscr{T}(M) = G^{k,\\ell}(A)\n\\]\nwhere \n  \\[\n    G^{k,\\ell}_{u,r}\n    =\n    \\frac{1}{2}\\sum_{i=1}^n\n    q_i^2\n    p_{k-1}(\\lambda_i)\n    \\begin{cases}\n    a_{r-1} p_{r-2}(\\lambda_i)p_{r-1}^{(\\ell-1)}(\\lambda_i) - a_r p_{r-1}(\\lambda_i) p_{r}^{(\\ell-1)}(\\lambda_i)&  \\\\\n    +a_{r-1} p_{r-1}(\\lambda_i)p_{r-2}^{(\\ell-1)}(\\lambda_i) - a_r p_{r}(\\lambda_i) p_{r-1}^{(\\ell-1)}(\\lambda_i),  & \\text{ if } u = r  < n, \\\\\n    a_r (p_{r-1}(\\lambda_i)p_{r-1}^{(\\ell-1)}(\\lambda_i) - p_r(\\lambda_i)p_r^{(\\ell-1)}(\\lambda_i)), & \\text{ if } u = r + 1 < n,\\\\\n    0, & \\text{ otherwise.}\n  \\end{cases}\n\\]\n\\end{theorem}\n\n\\begin{remark}\nThe previous theorem is enough to give the derivative for an arbitrary $G$. We can also consider the derivative in the original basis for $M$, this will be equivalent to $D_{O^{*}GO}$ where $O$ is the change of basis matrix $O^*MO=A$.\n\\end{remark}\n\n\n\n\\subsection{Organization}\n\nThe paper is organized in the following way: section \\ref{sec:iso} characterizes matrix diffusions that have the same spectral process, and in section \\ref{sec:dysoniso} specializes these matrix diffusions to the case of the $\\beta$-Dyson Brownian motion flow. These sections will essentially give existence of the tridiagonal model, but the description of the evolution itself will be implicit. Section \\ref{sec:commutator} will develop the linear algebra and orthogonal polynomial tools that will be needed to write down an explicit representation of the tridiagonal matrix evolution. In this section we also prove Theorem \\ref{thm:lanczos}. The proof is not explicitly identified, but this theorem will be a consequence of remark \\ref{rem:Gkl}. \n\nSection \\ref{sec:commutator} gives a brief introduction to discrete orthogonal polynomials (see section \\ref{sec:OPs}) and later defines the corresponding minor polynomials (see section \\ref{sec:minors}). The remainder of the sections is split into the derivation of several identities related to the commutator of $[A,M]$ for a tridiagonal $A$, and ends with a summary of the identities derived in the section. This final summary section \\ref{sec:summary} gives a summary of the relations needed to describe the tridiagonal model explicitly and the necessary pieces for Theorem \\ref{thm:lanczos}.\n\n\nThis explicit representation will be written down in several cases in section \\ref{sec:models}. The organization of this sections is pretty self-explanatory from the section heading, but it is worth noting that section on the tridiagonal model with frozen weights \\ref{sec:tridiagonalfrozen} will be necessary for reading the section on finite variation spectral weights \\ref{sec:tridiagonalfv}. Additionally section \\ref{sec:tridiagonalfrozen} contains a section where the finite variation terms are worked out and somewhat simplified. This explicit computation will be necessary for the asymptotic results in the next section, but is not essential otherwise. \n\nThe final section gives the proof of Theorem \\ref{thm:asymptotics}. A more detailed description about the organization may be found at the start of this section.\n\n\n\n\n\n\\section{Isospectral processes}\n\\label{sec:iso}\n\nWe begin by characterizing real matrix diffusions that carry the same eigenvalue distributions.  Suppose that $Z_t$ is a real symmetric semimartingale and $O_t$ is a real orthogonal semimartingale that are adapted to a common filtration.  Real orthogonal processes can be characterized by an associated process on the lie algebra associated to the orthogonal group, the real antisymmetric matrices.  Specifically, if $M_t$ is any real antisymmetric continuous semimartingale, the process\n\\[\n  dO_t^t = O_t^t( dM_t + \\frac{1}{2} (dM_t)^2)\n\\]\nis a real symmetric process.  Moreover, any real orthogonal continuous semimartingale arises in this way (see the proof sketch in Theorem~\\ref{thm:iso}).\n\nLet $Y_t=O_t Z_t O_t^t.$  Applying the stochastic product rule to $Y_t,$ we get\n\\begin{align*}\n  dY_t &= \n  dO_t Z_t O_t^t\n  + O_t dZ_t O_t^t\n  + O_t Z_t dO_t^t \\\\\n  &+ dO_t dZ_t O_t^t\n  + dO_t Z_t dO_t^t\n  + O_t dZ_t dO_t^t.\n\\end{align*}\nLet $W_t = \\int_0^t O_s dZ_s O_s^t.$  Then, we can write\n\\begin{align*}\n  dY_t &= \n  (dM_t + \\frac{1}{2} (dM_t)^2)^t Y_t \n  + dW_t  + Y_t( dM_t + \\frac{1}{2} (dM_t)^2) \\\\\n  &+ dM_t^t dW_t\n  +dM_t^t Y_tdM_t\n  +dW_t dM_t,\n\\end{align*}\nnoting that higher order terms disappear.  This can be rewritten as\n\\[\n  dY_t = dW_t + [Y_t, dM_t] + \\frac{1}{2}[2dW_t + [Y_t,dM_t], dM_t].\n\\]\nAlternatively, using that $[Y_t,dM_t] = dY_t - dW_t,$ up to finite variation terms,\n\\[\n  dY_t = dW_t + [Y_t, dM_t] + \\frac{1}{2}[dW_t + dY_t, dM_t].\n\\]\nWe have, in effect, proven the following.\n\\begin{theorem}\n  Suppose $\\MP{Z}$ is a real symmetric process adapted to a filtration $\\mathscr{F}$ and $\\MP{Y}=\\MP{O}\\MP{Z}\\MP{O}^t$ for some continuous orthogonal process $\\MP{O}$ adapted to $\\mathscr{F},$ then\n  \\begin{equation}\n    \\label{eq:prog}\n    \\begin{aligned}\n    dY_t &= dW_t + [Y_t, dM_t] + \\frac{1}{2}[dW_t + dY_t, dM_t] \\\\\n    dO_t^t &= O_t^t( dM_t + \\frac{1}{2} (dM_t)^2)\n    \\end{aligned}\n  \\end{equation}\n  for some real anti-symmetric process $\\MP{M}$ adapted to $\\mathscr{F}$ and where $\\MP{W}$ solves\n  $dW_t = O_t dZ_t O_t^t$ for all $t \\geq 0.$\n  \\label{thm:iso}\n\\end{theorem}\n\\begin{proof}\n  We need to demonstrate the existence of the process $\\MP{M}$ solving\n  \\[\n    dO_t^t = O_t^t( dM_t + \\frac{1}{2} (dM_t)^2).\n  \\]\n  On having done so, the product rule computations preceeding the theorem statement completes the proof.\n  Let $T > 0$ be arbitrary.\n  For any fixed time $t_0 \\in [0,T],$ we have that \n  \\(\n    \\lim_{t \\to t_0} O_t^t O_{t_0} = \\operatorname{Id}.\n  \\)\n  Hence, we can take the matrix logarithm of $\\tilde M_{t_0,t} = \\log(O_t^t O_{t_{0}})$ for time $t$ sufficiently close to $t_0,$ which as an analytic function of $O_t$ is adapted to $(\\mathscr{F}_t, t \\geq t_0).$  Further, we have that \n  \\(\n  O_t^t = O_{t_0}^t e^{\\tilde M_{t_0,t}}\n  \\)\n  for a sufficiently small window of time around $t_0.$  Hence, for a mesh $\\left\\{ t_i \\right\\}_0^d$ of $[0,T]$ of sufficiently fine spacing we may define for $t_i \\leq t < t_{i+1}$\n  \\(\n  M_t^d = \\tilde M_{t_i,t} + \\sum_{j=1}^i \\tilde M_{t_{j-1},t_{j}}.\n  \\)\n  On sending the mesh size to $0$ it is routine to check that $(M_t^{(d)}, 0 \\leq t \\leq T)$ converges to a solution of \n  \\(\n    dO_t^t = O_t^t( dM_t + \\frac{1}{2} (dM_t)^2).\n  \\)\n\n\\end{proof}\n\n\n\\section{The time evolving tridiagonal model for general $\\beta$}\n\\label{sec:dysoniso}\n\nSo, any time evolving tridiagonal model for which the eigenvector process is adapted will follow \\eqref{eq:prog}.  In particular, any adapted conjugation of $\\beta$-Dyson Brownian motion, which is a real diagonal semimartingale, has this form.  Since the eigenvalues and eigenvectors form locally differentiable processes in terms of the matrix entries, it follows that any tridiagonal model for $\\beta$-Dyson Brownian motion satisfies \\eqref{eq:prog}.\n\\begin{theorem}\n\\label{thm:dysoniso}\n  If $\\MP{A}$ is a tridiagonal model for $\\beta$-Dyson Brownian motion with $\\beta \\geq 1$ whose starting configuration is almost surely simple, i.e.\\;there is an orthogonal matrix process $\\MP{O}$ so that $\\MP{A} = \\MP{O} \\MP{\\Lambda} \\MP{O}^t$ for diagonal $\\MP{\\Lambda}$ satisfying for each $1 \\leq i \\leq n,$\n  \\[\n    d\\Lambda_{i,t} = \\sqrt{\\frac{2}{\\beta}}dB_{i,t} - \\biggl( \\frac{V'(\\lambda_i)}{2} - \\sum_{j: j \\neq i} \\frac{1}{\\lambda_i - \\lambda_j} \\biggr)\\,dt\n  \\]\n  then with respect to the filtration $(\\mathscr{F}_t, t\\geq 0) = (\\sigma(A_t),t\\geq 0),$\n  \\[\n    \\begin{aligned}\n    dA_t &= dW_t + [A_t, dM_t] + \\frac{1}{2}[dW_t + dA_t, dM_t] \\\\\n    dO_t^t &= O_t^t( dM_t + \\frac{1}{2} (dM_t)^2)\n  \\end{aligned}\n  \\]\n  for some real antisymmetric process $\\MP{M}$ where $\\MP{W}$ solves $dW_t = O_t d\\Lambda_t O_t^t.$ \n  \\label{thm:tri}\n\\end{theorem}\n\\begin{proof}\n  Since the eigenvalues of $\\MP{A}$ are always distinct almost surely, the eigenvector matrix $\\MP{O}$ is continuous and in fact differentiable as a function of the entries of $\\MP{A}.$  Hence $\\MP{O}$ is a continuous semimartingale adapted to $\\mathscr{F}$ and by Theorem~\\ref{thm:iso} the result follows.  \n \\end{proof}\n\nThe presence of the Dyson Brownian motion generator inside the theorem may at first sight appear disconcerting.  For example, we may wish to find a representation for the driving term $dW_t$ so that it is a local martingale.  Indeed this is possible:\n\\begin{theorem}\n  Suppose that $\\MP{W}$ is a symmetric matrix diffusion that solves\n  \\[\n    dW_t = \\sqrt{\\tfrac{2}{\\beta}} O_t dB_t O_t^t + O_t dZ_t O_t^t,\n  \\]\n  where $\\beta \\geq 1,$ $\\MP{B}$ is a diagonal matrix Brownian motion, $\\MP{Z}$ is a symmetric matrix Brownian motion with $0$-diagonal and $\\MP{O}$ is a continuous version of the eigenvector matrix of $\\MP{W}.$  Then, $\\MP{\\Lambda} = \\MP{O}^t \\MP{W} \\MP{O}$ has entries satisfying\n  \\begin{equation}\n    \\label{eq:dl1}\n    d\\Lambda_{i,t} = \\sqrt{\\frac{2}{\\beta}}dB_{i,t} +\\sum_{j: j \\neq i} \\frac{1}{\\lambda_i - \\lambda_j}\\,dt,\n  \\end{equation}\n  that is the eigenvalues of $\\MP{W}$ follow $\\beta$-Dyson Brownian motion with no potential.\n  Hence if $\\MP{A} = \\MP{U}\\MP{W}\\MP{U}^t$ is any tridiagonal matrix diffusion satisfying\n  \\[\n    \\begin{aligned}\n    dA_t &= U_tdW_tU_t^t + [A_t, dM_t] + \\frac{1}{2}[dW_t + dA_t, dM_t] \\\\\n    dU_t^t &= U_t^t( dM_t + \\frac{1}{2} (dM_t)^2),\n    \\end{aligned}\n  \\]\n  then the eigenvalues of $\\MP{A}$ evolve according to \\eqref{eq:dl1}.\n\n  \\label{thm:tri2}\n\\end{theorem}\nWe require some linear algebra before proving this theorem, which we give in Section~\\ref{sec:commutator}\nHowever, we note that in the case $\\beta=1,$ this driving noise simplifies:\n\\begin{corollary}\n  In the case $\\beta=1,$ the driving noise $\\MP{W} \\overset{\\mathscr{L}}{=} \\frac{\\MP{G} + \\MP{G}^t}{\\sqrt{2}}$ where $\\MP{G}$ is a matrix of i.i.d.\\;standard Brownian motions.\n\\end{corollary}\n\\noindent The previous Theorem and Corollary are related to work of Allez and Guionnet see \\cite{AllezGuionnet}.\n\n\\begin{remark}\n  Similar simplifications occur for $\\beta=2$ and $\\beta=4$ if we replace the process $\\MP{O}$ on the orthogonal group by appropriate unitary or symplectic processes, respectively. \n\\end{remark}\n\n\n\n\\section{Solving the commutator equation}\n\\label{sec:commutator}\n\nIntroduce the space $\\mathbb{T}$ of symmetric tridiagonal matrices.  \nIn this section, we consider the linear algebra problem of solving for antisymmetric $M$ so that \n\\begin{equation}\n  [A,M] + W \\in \\mathbb{T}.\n  \\label{eq:comm}\n\\end{equation}\nBy dimension counting, we can see that this problem does not have unique solutions.  Indeed there is possibly (and generically there is) an $(n-1)$-dimensional space of $M$ that produce the desired answer (see Theorem \\ref{thm:span}).\nThis is locally the problem that must be solved to describe the rotation differentials $dM_t$ that tridiagonalize $A_t$ when perturbed by $dW_t$ in \\eqref{eq:prog}.  Further, we will need to understand the resulting space of tridiagonal matrices that solve this equation for a given $W.$\nUnderstanding this algebra leads us to the theory of discrete orthogonal polynomials.  \n\n\n\n\\subsection{Discrete orthogonal polynomials}\n\\label{sec:OPs}\n\nLet\n\\begin{equation}\nA = \\left[ \\begin{array}{ccccc}\nb_1 & a_1 & && \\\\\na_1 & b_2 & a_2 & &\\\\\n& \\ddots & \\ddots & \\ddots \\\\\n& & a_{n-2} & b_{n-1} & a_{n-1}\\\\\n&&& a_{n-1} & b_n\n\\end{array}\\right]\n\\end{equation}\nbe a Jacobi matrix. We can define a sequence of polynomials $p_0(x), p_1(x),...,p_n(x)$ to be solutions to the three term recurrence generated by the equation $A\\underline{p}(x)=x\\underline{p}(x)$ with $\\underline{p}^T(x)=[p_0(x),p_1(x),...,p_{n-1}(x)]$. Taking $p_0(x)=1$ we get \n\\begin{align*}\nb_1 p_0(x)+ a_1 p_1(x) & =x p_0(x)\\\\\na_{k-1}p_{k-2}(x) + b_k p_{k-1}(x) + a_k p_k(x) & =x p_{k-1}(x)\\\\\na_{n-1} p_{n-2}(x) + (b_n-x)p_{n-1}(x) & = - p_n(x).\n\\end{align*}\nFor convenience we have completed the recurrence by taking $a_{n} =1$. This makes $p_n(x)$ a multiple of $\\det(x-A).$  If $\\lambda$ is an eigenvalue of $A$ the recurrence closes exactly and so $p_n(\\lambda) = 0$. In other words the zeros $\\lambda_1,...,\\lambda_n$ of $p_n(x)$ are the eigenvalues of $A$, and moreover they have associated eigenvectors $\\underline{p}(\\lambda_k)$. Because $A$ is self-adjoint it follows that the eigenvectors are orthogonal. Take $q_k^{-2} = \\| \\underline{p}(\\lambda_k)\\|^2$, then the matrix \n\\begin{equation}\n  \\label{eq:COB}\nO^t= \\left[ \\begin{array}{cccc}\nq_1 & q_1 p_1(\\lambda_1) & \\cdots & q_1 p_{n-1}(\\lambda_1)\\\\\nq_2 & q_2 p_1(\\lambda_2) & \\cdots & q_2 p_{n-1}(\\lambda_2)\\\\\n\\vdots & & & \\vdots \\\\\nq_n & q_n p_1(\\lambda_n) & \\cdots & q_n p_{n-1}(\\lambda_n)\n\\end{array}\n\\right]\n\\end{equation}\nis the orthogonal change of basis matrix that diagonalizes $A$ with $O^t A O = \\Lambda$. Moreover because $O$ is orthogonal we have the standard orthonormality conditions on both rows and columns. Therefore we get the following relationships:\n\\begin{align}\n\\sum_{i=1}^n q_i^2 p_{k}(\\lambda_i) p_{\\ell}(\\lambda_i) = \\delta_{k,\\ell},\\\\\n\\sum_{i=1}^n q_k q_\\ell p_{i-1}(\\lambda_k)p_{i-1}(\\lambda_\\ell) = \\delta_{k,\\ell}. \n\\end{align}\n\n\n\\subsection{Christoffel-Darboux Formula}\n\nThe Christoffel-Darboux formula is an automatic consequence of the three-term recurrence for $\\left\\{ p_k \\right\\}_{k=0}^n.$  As it is in symmetric form, this formula is\n\\begin{equation}\n  \\sum_{k=0}^m p_k(x)p_k(y) = a_{m+1}\\frac{p_{m+1}(x)p_m(y) - p_m(x)p_{m+1}(y)}{x-y}.\n  \\label{eq:CD}\n\\end{equation}\nMultiplying the left hand side through by $(x-y),$ and applying the recurrence, all but the terms on the right survive.  We can also take the limit in this formula as $y \\to x$ to get the \\emph{confluent} form of this formula:\n\\begin{equation}\n  \\sum_{k=0}^m p_k(x)^2 = a_{m+1}(p_{m+1}'(x)p_m(x) - p_m'(x)p_{m+1}(x) ).\n  \\label{eq:cCD}\n\\end{equation}\nThis allows us to represent the spectral weights $q_i^2$ by the following special case\n\\begin{equation}\n  \\frac{1}{q_i^2} = \\sum_{k=0}^{n-1} p_k(\\lambda_i)^2 = p_{n}'(\\lambda_i)p_{n-1}(\\lambda_i).\n  \\label{eq:qipn}\n\\end{equation}\nWe will also use the summation\n\\begin{align}\n  \\sum_{k=0}^m p_k(x)p'_k(y) \n  = &a_{m+1}\\partial_y\\biggl(\\frac{p_{m+1}(x)p_m(y) - p_m(x)p_{m+1}(y)}{x-y}\\biggr) \\nonumber\\\\\n  = \n  &a_{m+1}\\frac{p_{m+1}(x)p'_m(y) - p_m(x)p'_{m+1}(y)}{x-y} \\nonumber \\\\\n  + &a_{m+1}\\frac{p_{m+1}(x)p_m(y) - p_m(x)p_{m+1}(y)}{(x-y)^2}. \n  \\label{eq:dCD}\n\\end{align}\nSending $x$ to $y$ (or more directly, differentiating \\eqref{eq:cCD}), we also get\n\\begin{equation}\n  \\sum_{k=0}^m p_k(x)p_k'(x) = a_{m+1}(p_{m+1}''(x)p_m(x) - p_m''(x)p_{m+1}(x) ).\n  \\label{eq:dcCD}\n\\end{equation}\n\n\n\n\n\n\n\\subsection{Fundamental identity}\n\nThe commutator at entry $k,\\ell$ is given by \n\\begin{equation}\n[A,M]_{k,\\ell} = a_{k-1} m_{k-1,\\ell} + b_k m_{k,\\ell} +a_k m_{k+1,\\ell} - (a_{\\ell-1} m_{k,\\ell-1} + b_\\ell m_{k,\\ell}+ a_\\ell m_{k,\\ell+1}).\n\\label{eq:AM}\n\\end{equation}\nTerms for which $k$ or $\\ell$ exceed the allowable indices are taken to be $0.$\nThis makes a natural choice for $M$ to be built out of the orthogonal polynomials associated to $A$, $p_{k-1}(x)p_{\\ell-1}(y).$\nTo this end, define the matrix\n\\[\n  E(x,y)_{k,\\ell} = \n  p_{k-1}(x) p_{\\ell-1}(y) \\one[ k > \\ell]\n  +\\frac{1}{2}p_{k-1}(x) p_{k-1}(y) \\one[ k = \\ell].\n\\]\nIn terms of $E,$ let $E^+$ be the symmetric analogue and let $E^{-}$ be the antisymmetric analogue \n\\begin{equation}\n\\label{eq:E}\nE^{+}(x,y) = E(x,y) + E(x,y)^t \\qquad \\text{ and } \\qquad\nE^{-}(x,y) = E(x,y) - E(x,y)^t.\n\\end{equation}\n\nSome of these matrices, when evaluated at the eigenvalues of $A,$ have a nice expression when conjugated by the eigenvector change of basis matrix $O.$ These changes of basis are simple to check, and we do not verify them.  In what follows, we let $\\delta_{i,j}$ be the matrix which zero everywhere except in the $(i,j)$ position where it is $1.$\n\\begin{align}\n  &E^{+}(\\lambda_i,\\lambda_j) \n  + \n  E^{+}(\\lambda_j,\\lambda_i) \n  =\n  O^t \\biggl( \\frac{\\delta_{i,j} + \\delta_{j,i}}{q_iq_j} \\biggr) O,\n  \\label{eq:spec1} \\\\\n  &E^{+}(\\lambda_i,\\lambda_i) \n  =\n  O^t\\biggl(\n  \\frac{\\delta_{i,i}}{q_i^2}\n  \\biggr) O,\n  \\label{eq:spec2} \\\\\n  &E^{-}(\\lambda_i,\\lambda_j) \n  -\n  E^{-}(\\lambda_j,\\lambda_i) \n  =\n  O^t \\biggl( \\frac{\\delta_{i,j} - \\delta_{j,i}}{q_iq_j} \\biggr) O.\n  \\label{eq:spec3}\n\\end{align}\nWe note that \\eqref{eq:spec2} is really just a special case of \\eqref{eq:spec1}, but we include it for emphasis.  Note that, despite the importance of $E^{-}(\\lambda_i,\\lambda_i),$ which will become clear, it does not have any such simple representation.\n\n\nCommutation of $A$ with $E^{-}$ nearly gives a multiple of $E^+.$  In effect, save for boundary terms that result from the problem having finite size and the boundary terms that result from the necessary antisymmetry conditions, $[A,E^{-}(x,y)] = (x-y)E^{+}:$\n\\begin{proposition}\n  Define the matrix $\\tilde B(x,y) = [A,E^{-}(x,y)] - (x-y)E^{+}(x,y).$  Then, for any $k \\geq \\ell,$\n  \\[\n    \\tilde B(x,y)_{k,\\ell}\n    =\\begin{cases}\n      -a_{\\ell-1} p_{\\ell-2}(x)p_{\\ell-1}(y) + a_\\ell p_{\\ell-1}(x) p_{\\ell}(y)&  \\\\\n      -a_{\\ell-1} p_{\\ell-1}(x)p_{\\ell-2}(y) + a_\\ell p_{\\ell}(x) p_{\\ell-1}(y),  & \\text{ if } k = \\ell  < n, \\\\\n      -a_\\ell (p_{\\ell-1}(x)p_{\\ell-1}(y) - p_\\ell(x)p_\\ell(y)), & \\text{ if } k = \\ell + 1 < n,\\\\\n      -p_n(x)p_{\\ell-1}(y), & \\text{ if } k = n, \\ell < n-1, \\\\\n      -p_n(x)p_{\\ell-1}(y) \n      -a_\\ell(p_{\\ell-1}(x)p_{\\ell-1}(y) - p_\\ell(x)p_\\ell(y))\n      , & \\text{ if } k = n, {\\ell = n-1},\\\\\n      -a_{\\ell-1} p_{\\ell-2}(x)p_{\\ell-1}(y) + a_\\ell p_{\\ell-1}(x) p_{\\ell}(y)&  \\\\\n      -a_{\\ell-1} p_{\\ell-1}(x)p_{\\ell-2}(y) - a_\\ell p_{\\ell}(x) p_{\\ell-1}(y),\n      & \\text{ if } k = n, {\\ell = n}, \\text{ or } \\\\\n      0, & \\text{ otherwise.}\n    \\end{cases}\n  \\]\n  Observe that the matrix $[A,E^{-}(x,y)] - (x-y)E^{+}(x,y)$ is symmetric, which determines the $k > \\ell$ terms. \n  \\label{prop:commutator}\n\\end{proposition}\n\\begin{proof}\n  \\noindent If $1 < k < n$ and $\\ell + 1 < k$ there are no boundary effects.  Hence \\eqref{eq:AM} becomes\n  \\begin{align*}\n    [A,E^{-}(x,y)]_{k,\\ell} = \n    &(a_{k-1} p_{k-2}(x) + b_k p_{k-1}(x) + a_k p_{k}(x))p_{\\ell-1}(y) - \\\\\n    &p_{k-1}(x)(a_{\\ell-1} p_{\\ell-2}(y) + b_\\ell p_{\\ell-1}(y) + a_\\ell p_{\\ell}(y)) \\\\\n    =&(x-y)p_{k-1}(x)p_{\\ell-1}(y).\n  \\end{align*}\n  The formula additionally holds when $k = 1,$ as the orthogonal polynomial recurrence for this case holds with $a_0 = 0.$\n\n  \\vspace{1cm}\n  \\noindent $k = n:$  \n  When $k = n$ and $\\ell + 1 < n,$ due to the[A,M] + W border at the bottom of the matrix, we have\n  \\begin{align*}\n    [A,E^{-}(x,y)]_{n,\\ell} = \n    &(a_{n-1} p_{n-2}(x) + b_n p_{n-1}(x) + 0 \\cdot p_{n}(x))p_{\\ell-1}(y) - \\\\\n    &p_{n-1}(x)(a_{\\ell-1} p_{\\ell-2}(y) + b_\\ell p_{\\ell-1}(y) + a_\\ell p_{\\ell}(y)) \\\\\n    =&(x-y)p_{n-1}(x)p_{\\ell-1}(y) - p_n(x) p_{\\ell-1}(y),\n  \\end{align*}\n  using that $a_n$ is taken to be $1.$  \n\n  \\vspace{1cm}\n  \\noindent $k = \\ell+1:$  \n  When $k = \\ell+1$ and $\\ell + 1 < n,$ due to the diagonal of $E^{-}$ being $0,$ we have\n  \\begin{align*}\n    [A,E^{-}(x,y)]_{\\ell+1,\\ell} = \n    &(0 \\cdot p_{\\ell-1}(x) + b_{\\ell+1} p_{\\ell}(x) + a_{\\ell+1} p_{\\ell+1}(x))p_{\\ell-1}(y) - \\\\\n    &p_{\\ell}(x)(a_{\\ell-1} p_{\\ell-2}(y) + b_\\ell p_{\\ell-1}(y) + 0 \\cdot p_{\\ell}(y)) \\\\\n    =&(x-y)p_{\\ell}(x)p_{\\ell-1}(y) - a_\\ell (p_{\\ell-1}(x)p_{\\ell-1}(y) - p_\\ell(x)p_\\ell(y)).\n  \\end{align*}\n  If $\\ell+1 = n,$ then we must subtract from the expression above for $[A,E^{-}(x,y)]_{n,n-1}$ any terms containing $a_n.$\n\n  \\vspace{1cm}\n  \\noindent $k = \\ell:$ \n  When $k=\\ell,$ antisymmetry of $E^{-}$ plays a larger role.  Taking care that the terms above the diagonal are properly accounted, for $k=\\ell < n,$ \n  \\begin{align*}\n    [A,E^{-}(x,y)]_{\\ell,\\ell} = \n    &(0 \\cdot p_{\\ell-2}(x) + 0 \\cdot p_{\\ell-1}(x) + 2 a_\\ell p_{\\ell}(x))p_{\\ell-1}(y) - \\\\\n    &p_{\\ell-1}(x)(2a_{\\ell-1} p_{\\ell-2}(y) + 0 \\cdot p_{\\ell-1}(y) + 0 \\cdot p_{\\ell}(y)) \\\\\n    =&(x-y)p_{\\ell-1}(x)p_{\\ell-1}(y) \n    - a_{\\ell-1} p_{\\ell-2}(x)p_{\\ell-1}(y) \n    + a_{\\ell} p_{\\ell}(x)p_{\\ell-1}(y) \\\\\n    -&a_{\\ell-1} p_{\\ell-1}(x) p_{\\ell-2}(y)\n    + a_\\ell p_{\\ell-1}(x) p_{\\ell}(y). \n  \\end{align*}\n  If $k=\\ell = n,$ then we must subtract from the expression above for $[A,E^{-}(x,y)]_{n,n-1}$ any terms containing $a_n.$\n\\end{proof}\n\nIn the case that we pick $x=\\lambda_i$ and $y=\\lambda_j$ to be eigenvalues of $A,$ the error term $\\tilde B(\\lambda_i,\\lambda_j)$ will be tridiagonal.  We begin by characterizing the space of matrices whose commutator with $A$ is tridiagonal.  Recall that $\\left\\{ \\lambda_i : 1 \\leq i \\leq n \\right\\}$ are the eigenvalues of $A.$  Essentially everything we do here is predicated on working with tridiagonal matrices $A$ which have nonzero offdiagonal entries.  We call these tridiagonal matrices \\emph{nondegenerate}.\n\n\\begin{theorem}\n  Suppose that $A$ is \\emph{nondegenerate} and all $\\left\\{ \\lambda_i \\right\\}$ are distinct.\n  Let $V \\subset \\mathbb{M}_{n}$ be the space of antisymmetric matrices so that for all $v \\in V,$\n  \\[\n    [A,v] \\in \\mathbb{T}.\n  \\]\n  The space $V$ is spanned by $\\left\\{ E^{-}(\\lambda_i,\\lambda_i) : 1 \\leq i \\leq n\\right\\}.$ \n  \\label{thm:span}\n\\end{theorem}\n\\begin{proof}\n  The argument is by dimension counting.  We start by showing that the dimension of $V$ is $n-1.$  Observe first that there is a linear dependence among the $V$ that follows from the orthogonality of the polynomials. For any $\\ell > r$ in $[n],$\n  \\[\n    \\sum_{i=1}^n q_i^2 E^{-}(\\lambda_i,\\lambda_i)_{\\ell,r} \n    = \n    \\sum_{i=1}^n q_i^2 p_{\\ell-1}(\\lambda_i) p_{r-1}(\\lambda_i) = 0. \n  \\]\n  As the diagonal of $E^{-}$ is identically $0,$ this linear combination of these matrices is identically $0.$  On the other hand, looking at the first columns of $E^{-}(\\lambda_i,\\lambda_i),$ the span of these first columns is at least $n-1$ dimensional from the linear independence of $\\left\\{ p_k : k=1,2,\\dots,n-1 \\right\\}.$\n\n  On the other hand, by the lattice path solution, after picking the first column of $v,$ the constraint that $[A,v] \\in \\mathbb{T}$ determines the remainder of $v$ when $A$ has nonzero offdiagonal entries.  Hence, this space is at most $(n-1)$-dimensional, which completes the proof.\n\\end{proof}\n\n\\begin{remark}\n  The requirement that $A$ have nonzero offdiagonal entries is necessary for the previous theorem.  When $A$ is diagonal for example, the commutator of any tridiagonal matrix with $A$ will be tridiagonal.\n\\end{remark}\n\n\n\\subsection{Symmetrized form}\n\nDefine the symmetrized matrix\n\\[\n  H^{-}(x,y)\n  = \\frac{E^{-}(x,y) - E^{-}(y,x)}{x-y}.\n\\]\nDefine $L(x,y) = [A,H^{-}(x,y)] - E^{+}(x,y) - E^{+}(y,x).$\nBy Proposition~\\ref{prop:commutator} we have that\n\\begin{equation}\n  L(x,y)_{k,\\ell}\n    =\\begin{cases}\n      \\frac{-p_n(x)p_{\\ell-1}(y) + p_{\\ell-1}(x)p_n(y)}{x-y}, & \\text{ if } k = n, \\ell {\\leq} n-1, \\\\\n      \\frac{-2p_n(x)p_{n-1}(y) + 2p_{n-1}(x)p_n(y)}{x-y}, & \\text{ if } k = n, \\ell = n, \\\\\n      0, & \\text{ otherwise.}\n    \\end{cases}\n  \\label{eq:L}\n\\end{equation}\nObserve that this error vanishes entirely when $x$ and $y$ are chosen to be eigenvalues of $A.$  In short, this is because the matrix $H^{-}$ has a simple expression when conjugated by $O$ (recall \\eqref{eq:COB} and \\eqref{eq:spec3})\n\\begin{equation*}\n  O^t H^{-}(\\lambda_i, \\lambda_j) O = \\frac{\\delta_{i,j} + \\delta_{j,i}}{q_iq_j(\\lambda_i - \\lambda_j)},\n\\end{equation*}\n\nwith $\\delta_{i,j}$ the matrix with a single nonzero entry, equal to $1,$ in position $(i,j).$\nThis gives an alternative derivation of \\eqref{eq:L} for the case that $x$ and $y$ are eigenvalues by writing (and recalling \\eqref{eq:spec1})\n\\begin{equation}\n  \\label{eq:H-}\n  [A,H^{-}(\\lambda_i,\\lambda_j)]\n  = O\\biggl[\\Lambda, \\frac{\\delta_{i,j} - \\delta_{j,i}}{q_iq_j(\\lambda_i - \\lambda_j)}\\biggr]O^t\n  = O\\biggl(\\frac{\\delta_{i,j} + \\delta_{j,i}}{q_iq_j}\\biggr)O^{t}\n  = {E^{+}(\\lambda_i,\\lambda_j) + E^{+}(\\lambda_j,\\lambda_i)}.\n\\end{equation}\n\nThis will prove useful in what follows, but it will also be helpful to define another preimage of $E^{+}(\\lambda_i,\\lambda_j) + E^{+}(\\lambda_j,\\lambda_i) + \\mathbb{T}$ that vanishes in the first column.\nThe matrix $H^{-}$ in its first column has $k$-th entry\n  \\[\n    H^{-}(\\lambda_i, \\lambda_j)_{k,1}\n    =\\frac{p_{k-1}(\\lambda_i) - p_{k-1}(\\lambda_j)}{\\lambda_i - \\lambda_j}.\n  \\]\n  This is equal to the $k$-th entry of the first column of the matrix \n  \\begin{equation*}\n    \\frac{E^{-}(\\lambda_i,\\lambda_i) - E^{-}(\\lambda_j,\\lambda_j)}{\\lambda_i-\\lambda_j}.\n  \\end{equation*}\nHence, we define\n\\begin{equation}\nH_0^{-}(x,y)\n= \\frac{E^{-}(x,y) - E^{-}(y,x)-E^{-}(x,x) + E^{-}(y,y)}{x-y},\n\\label{eq:H0}\n\\end{equation}\nwhich satisfies $[A,H_0^{-}(\\lambda_i,\\lambda_j)] - E^{+}(\\lambda_i,\\lambda_j) - E^{+}(\\lambda_j,\\lambda_i) \\in \\mathbb{T}$ and $H_0^{-} e_1=0,$ with $e_1$ the first standard basis vector.\n\nRecapitulating: we would like to solve the equation \n\\[\n  [A,M] + W \\in \\mathbb{T}\n\\]\nwhere $W$ is an arbitrary symmetric matrix for $M$.  If $W$ is in the span of \n\\[\n\\left\\{ {E^{+}(\\lambda_i,\\lambda_j) + E^{+}(\\lambda_j,\\lambda_i), i < j} \\right\\},\n\\]\nthen we can do this exactly without the need for a tridiagonal error.  Thus it essentially remains to determine how to solve for \n\\[\n  [A,M] + E^{+}(\\lambda_i,\\lambda_i) \\in \\mathbb{T}\n\\]\nfor each $1\\leq i \\leq n.$\nNoting that $O^t E^{+}(\\lambda_i,\\lambda_i) O = q_i^{-2} \\delta_{i,i},$ the combination of these matrices and \n\\[\n  \\left\\{ {E^{+}(\\lambda_i,\\lambda_j) + E^{+}(\\lambda_j,\\lambda_i)}, i < j \\right\\},\n\\]\nclearly span all symmetric matrices, and so we will have a complete solution.\n\n\nUsing this discussion, we can give a quick proof of Theorems \\ref{thm:tri2}.\n\\begin{proof}[Proof of Theorem~\\ref{thm:tri2}]\n  We recall the setup for the theorem.  Suppose that $\\MP{W}$ is a stochastic matrix diffusion that satisfies \n  \\[\n  d W_t = \\sqrt{\\tfrac{2}{\\beta}} O_t dB_t O_t^t + O_t dZ_t O_t,\n  \\] \n  where $\\MP{O}$ is the eigenvector matrix of $\\MP{W}$. Let $\\MP{\\Lambda}$ be the eigenvalue matrix of $\\MP{W}$, we check that \n  \\begin{align*}\n  d \\Lambda_t &= O_t^t dW_t O_t + [\\Lambda_t, dN_t] + \\frac{1}{2} [ O_t^t dW_t O_t + d\\Lambda_t, dN_t]\\\\\n  & =\\sqrt{\\tfrac{2}{\\beta}}dB_t+dZ_t + [\\Lambda_t, dN_t] + \\frac{1}{2} [ \\sqrt{\\tfrac{2}{\\beta}}dB_t+dZ_t+ d\\Lambda_t, dN_t]\n  \\end{align*}\n  where $N_t$ is chosen so that $dN_{i,j,t}(\\lambda_i-\\lambda_j) = dZ_{i,j,t}$ which gives perfect cancelation of the $dZ_t$ term. This induces the evolution of $\\MP{O}$ and so the evolution of $\\MP{W}$. By independence the previous equation simplifies further to \n  \\[\n  d \\Lambda_t =\\sqrt{\\tfrac{2}{\\beta}}dB_t+ \\frac{1}{2} [ dZ_t, dN_t].\n  \\]\n  Further\n    \\[\n    [dZ_t, dN_t ]_{i,i}\n    = \\sum_{j \\neq i} \\frac{\n      dZ_{i,j,t}dZ_{j,i,t}\n    }{\\lambda_{j,t}-\\lambda_{i,t}}\n    -\\sum_{j \\neq i} \\frac{\n      dZ_{i,j,t}dZ_{j,i,t}\n    }{\\lambda_{i,t}-\\lambda_{j,t}}\n    = \\sum_{j \\neq i} \\frac{\n    -2dt}{\\lambda_{i,t}-\\lambda_{j,t}}.\n  \\]\n  This gives us that $\\Lambda$ evolves according to a Dyson Brownian motion with zero potential. To get the conclusion of the theorem we can take any tridiagonal evolution of $\\MP{W}$ and the diffusion equation follows from Theorem \\ref{thm:dysoniso}.\n\n\\end{proof}\n\n\\subsection{Confluent form}\n\nGoing forward, we will essentially always specialize $x$ and $y$ to be eigenvalues.  For eigenvalues the matrix $B$ simplifies.  So, we define\n\\begin{equation}\n    B(x,y)_{k,\\ell}\n    =\\begin{cases}\n      -a_{\\ell-1} p_{\\ell-2}(x)p_{\\ell-1}(y) + a_\\ell p_{\\ell-1}(x) p_{\\ell}(y)&  \\\\\n      -a_{\\ell-1} p_{\\ell-1}(x)p_{\\ell-2}(y) + a_\\ell p_{\\ell}(x) p_{\\ell-1}(y),  & \\text{ if } k = \\ell, \\\\\n      -a_\\ell (p_{\\ell-1}(x)p_{\\ell-1}(y) - p_\\ell(x)p_\\ell(y)), & \\text{ if } k = \\ell + 1,\\\\\n      0, & \\text{ otherwise,}\n    \\end{cases}\n    \\label{eq:B}\n\\end{equation}\nand observe that for eigenvalues $\\lambda_1$ and $\\lambda_2,$ $B(\\lambda_1,\\lambda_2) = \\tilde B(\\lambda_1,\\lambda_2).$\n\nOne possible choice for $x$ and $y$ is to take them equal.  Indeed, by symmetrizing the $E^{-},$ dividing by $x-y$ and sending $x \\to y$ we are also led to the existence of an approximate preimage to $E^{+}(y,y)$ that involves derivatives of $E^{-}.$  A more direct identity is possible simply by differentiating $E^{-}.$  Observe that\n\\[\n  \\partial_y [A,E^{-}(x,y)] = [A, \\partial_y E^{-}(x,y)],\n\\]\nby virtue of $A$ having no $y$ dependence.  Hence,\n\\[\n  [A, \\partial_y E^-(x,y)] \n  = \\partial_y [A, E^-(x,y)]\n  = -E^+(x,y) + (x-y)\\partial_y E^+(x,y) + \\partial_y\\tilde B(x,y).\n\\]\nThis operation can be iterated to produce a basis of derivatives, by using\n\\[\n  [A, \\partial_x^k \\partial_y^\\ell E^{-}(x,y)]\n  =\\partial_x^k \\partial_y^\\ell \\left( (x-y)E^{+}(x,y) + \\tilde B(x,y) \\right),\n\\]\nbut we do not pursue this further.\nIf we define $F(y) = -\\partial_y E^{-}(x,y) \\vert_{x=y},$ then\n\\begin{equation}\n  [A,F(y)] = E^{+}(y,y) - \\partial_y \\tilde B(x,y) \\vert_{x=y}.\n  \\label{eq:F}\n\\end{equation}\nWe write some entries of this error term:\n\\[\n  -\\partial_y \\tilde B(x,y) \\vert_{x=y}\n  =\\begin{cases}\n    a_{\\ell-1} p_{\\ell-2}(y)p_{\\ell-1}'(y) - a_\\ell p_{\\ell-1}(y) p_{\\ell}'(y)&  \\\\\n    +a_{\\ell-1} p_{\\ell-1}(y)p_{\\ell-2}'(y) - a_\\ell p_{\\ell}(y) p_{\\ell-1}'(y),  & \\text{ if } k = \\ell  < n, \\\\\n    a_\\ell (p_{\\ell-1}(y)p_{\\ell-1}'(y) - p_\\ell(y)p_\\ell'(y)), & \\text{ if } k = \\ell + 1 < n,\\\\\n    p_n(y)p_{\\ell-1}'(y), & \\text{ if } k = n, \\ell < n-1, \\\\\n    \\text{other}, & \\text{ if } k = n, {\\ell \\geq n-1}, \\text{ or } \\\\\n    0, & \\text{ otherwise.}\n  \\end{cases}\n\\]\n\nSpecializing to $x=y=\\lambda_i$ for a zero $\\lambda_i $ of $p_n$ will be especially useful here, as in this case the boundary term above is tridiagonal. In addition, in this case, the formulas for the tridiagonal that hold in the middle of the matrix also hold in the final entries.  Otherwise said, at an eigenvalue $\\lambda_i$ we define\n\\begin{equation}\n  \\label{eq:G}\n  G(\\lambda_i)\n  =\n  -\\partial_y \\tilde B(x,y) \\vert_{x=y=\\lambda_i}\n  = \n  -\\partial_y B(x,y)\\vert_{x=y=\\lambda_i} \n  = \n  -\\frac{1}{2}\\partial_y B(y,y)\\vert_{y=\\lambda_i}, \n\\end{equation}\nand we define $G(y) = -\\frac{1}{2}\\partial_y B(y,y)$ for general $y.$\n\n\n\n\\subsection{Difference quotients and minor polynomials}\n\\label{sec:minors}\n\nDefine the difference quotient operator of a polynomial as\n\\[\n  (\\Delta^y f)(x) = \\frac{f(x)-f(y)}{x-y}.\n\\]\nWe always consider $\\Delta^y$ as a map on polynomials.  In particular, for $x=y,$ the previous definition extends by continuity.\nFor a multivariate polynomial, let $\\Delta^y_x$ denote the corresponding partial difference quotient in the $x$ variable at $y.$\n\nFor any polynomial $p,$ the function  \n\\[\n  \\Delta^y p(x) = \\frac{p(x)-p(y)}{x-y}\n\\]\nis a polynomial in two variables, which as a polynomial in $x$ has degree one less than $p.$  By continuity, it takes the value $p'(x)$ at $y=x.$\nWith this in mind, we define the polynomials for any $0 \\leq k,\\ell \\leq n-1,$\n\\begin{equation}\n  p^{(k)}_{\\ell}(x) = \\sum_{j=1}^n q_j^2 \n  p_k(\\lambda_j)\n  \\frac{p_\\ell(x)-p_\\ell(\\lambda_j)}{x-\\lambda_j},\n  \\label{eq:minor}\n\\end{equation}\nfor $x$ not in the spectrum.  Observe that if $\\ell \\leq k,$ then by orthogonality, this polynomial is $0.$  On the other hand, observe that\n\\begin{equation*}\n  \\begin{aligned}\n    xp^{(k)}_{\\ell}(x) \n    &= \n    \\sum_{j=1}^n q_j^2 \n    p_k(\\lambda_j)\n    \\frac{xp_\\ell(x)-xp_\\ell(\\lambda_j)}{x-\\lambda_j} \\\\\n    &= \n    \\sum_{j=1}^n q_j^2 \n    p_k(\\lambda_j)\n    \\biggl\\{\n      \\frac{xp_\\ell(x)-\\lambda_j p_\\ell(\\lambda_j)}{x-\\lambda_j} \n      -p_\\ell(\\lambda_j)\n    \\biggr\\}\n    \\\\\n    &=\n    a_{\\ell+1}p^{(k)}_{\\ell+1}(x) \n    +b_{\\ell+1}p^{(k)}_{\\ell}(x) \n    +a_{\\ell}p^{(k)}_{\\ell-1}(x) \n    -\\delta_{k,\\ell}.\n  \\end{aligned}\n\\end{equation*}\nThis is to say that these polynomials satisfy the same $3$-term recurrence as $\\left\\{ p_\\ell \\right\\}_{\\ell}$ but started from different initial conditions.  This also implies that \n\\[\n  \\deg(p^{(k)}_{\\ell}) = (\\ell-k-1)_{+} \\quad \\text{and} \\quad p_{\\ell+1}^{(\\ell)}(x) = \\frac{1}{a_{\\ell+1}}.\n\\]\nOtherwise stated, they are multiples of the orthogonal polynomials associated to the lower-right-principal submatrix of $A$ of size $n-k-1.$  \nFurthermore, we have the identity that\n\\[\n  \\Delta^y p(x)\n  =\n  \\frac{p_\\ell(x)-p_\\ell(y)}{x-y}\n  =\n  \\sum_{k = 1}^\\ell p^{(k-1)}_{\\ell}(x) p_{k-1}(y),\n\\]\nwhich by continuity specializes to \n\\[\n  p_\\ell'(x) = \\sum_{k = 1}^\\ell p^{(k-1)}_{\\ell}(x) p_{k-1}(x).\n\\]\n\nWe also observe that\n\\begin{equation}\n  \\sum_{k = 1}^\\ell p^{(k-1)}_{\\ell}(x) p_{k-1}'(y)\n  =\n  \\partial_y \\Delta^y p_\\ell(x)\n  =\n  \\frac{(p_\\ell(x)-p_\\ell(y))-(x-y)p_\\ell'(y)}{(x-y)^2}\n  = \\Delta^y \\Delta^y p_\\ell(x),\n  \\label{eq:secondderiv}\n\\end{equation}\nwhich by continuity specializes to $\\frac{1}{2}p_\\ell''(y)$ when $x=y.$\n\n\\subsubsection*{Approximating the minor polynomials}\n\nBecause the minor polynomials satisfy the same recurrence as the original polynomials, it is possible to formulate a mixed Christoffel--Darboux formula using the two sets of polynomials, namely for $r < \\ell$\n\\begin{equation}\n  \\sum_{k=1}^{\\ell} p_{k-1}(x) p_{k-1}^{(r)}(y) (x-y) = \n  a_{\\ell} \\left( \n  p_{\\ell}(x) p_{\\ell-1}^{(r)}(y)\n  -p_{\\ell-1}(x) p_{\\ell}^{(r)}(y) \\right) + p_{r}(x),\n  \\label{eq:CDmixed}\n\\end{equation}\nwhere the extra $p_r(x)$ results from the defect in the recurrence for the minor polynomial in the bottom term.  Note that on taking $x=y$ in this formula, we are left with\n\\begin{equation}\np_{\\ell-1}(x) p_{\\ell}^{(r)}(x)\n-\np_{\\ell}(x) p_{\\ell-1}^{(r)}(x)\n=\\frac{p_{r}(x)}{a_\\ell},\n  \\label{eq:Wronskian}\n\\end{equation}\nwhich shows that $\\left\\{ ((p_\\ell), (p_\\ell^{(0)})), \\ell=1,2,\\dots \\right\\}$ forms a fundamental set of solutions to the three--term recurrence for any initial conditions.  In particular, \n\\begin{equation}\n  \\label{eq:MinorIdentity}\n  p^{(j)}_\\ell(x) = \n  \\begin{cases}\n  p_j(x) p_\\ell^{(0)}(x)\n  - \n  p_\\ell(x)p_j^{(0)}(x), & j \\leq \\ell, \\\\\n  0, & \\text{otherwise,}\n\\end{cases}\n\\end{equation}\nobserving that the $3$--term--recurrence is trivially satisfied, and that the initial conditions at $\\ell=j$ and $\\ell=j+1$ are satisfied.  \nThis identity also shows that $\\{p_{\\ell}^{(j)}\\}$ satisfies a recurrence in $j$ as well, specifically that for $j < \\ell$\n\\begin{equation}\n  \\label{eq:DualRecurrence}\n  x p^{(j)}_\\ell(x) = \n  a_{j+1}p^{(j+1)}_\\ell(x) \n  +b_{j+1}p^{(j)}_\\ell(x) \n  +a_{j}p^{(j-1)}_\\ell(x). \n\\end{equation}\nThis formula is also true for $j=0,$ taking $a_0 = 1$ and $p^{(-1)}_\\ell(x) = p_\\ell(x),$ as it is equivalent to \\eqref{eq:MinorIdentity} with $j=1.$\n\n\\subsubsection*{Perturbation theory for orthogonal polynomials}\n\nThe dual recurrence \\eqref{eq:DualRecurrence} is used in \\cite{Geronimo} to create a perturbation theory for orthogonal polynomials, one which shows that when two families of orthogonal polynomials have similar coefficients, they can be compared.  We present this perturbation theory here. \nSuppose that $\\{\\hat p_j(x)\\}$ are defined by\n\\begin{align*}\n\\hat b_1 \\hat p_0(x)+  \\hat a_1  \\hat p_1(x) & =x  \\hat p_0(x)\\\\\n\\hat a_{j} \\hat p_{j-1}(x) +  \\hat b_{j+1}  \\hat p_{j}(x) +  \\hat a_{j+1}  \\hat p_{j+1}(x) & =x  \\hat p_{j}(x),\n\\end{align*}\nfor $j = 1,2,3,\\dots.$  If we multiply \\eqref{eq:DualRecurrence} through by $\\hat p_j$ and subtract the previous equation multiplied by  $p^{(j)}_\\ell(x),$ we get\n\\[\n  0 =\n  (a_{j+1}p^{(j+1)}_\\ell(x) \n  +b_{j+1}p^{(j)}_\\ell(x) \n  +a_{j}p^{(j-1)}_\\ell(x))\\hat p_j(x)\n  -\n  (\\hat a_{j} \\hat p_{j-1}(x) +  \\hat b_{j+1}  \\hat p_{j}(x) +  \\hat a_{j+1}  \\hat p_{j+1}(x)\n  )p^{(j)}_\\ell(x). \n\\]\nHence, if we sum this identity for $j=0,1,\\dots,\\ell-1,$ we get\n\\begin{align*}\n  0\n  &=\n  \\sum_{j=0}^{\\ell-1} (b_{j+1} - \\hat b_{j+1})p^{(j)}_\\ell(x)\\hat p_j(x) \n  +\n  \\sum_{j=0}^{\\ell-2} (a_{j+1} - \\hat a_{j+1})\n  (\n  p^{(j+1)}_\\ell(x)\\hat p_j(x) \n  +p^{(j)}_\\ell(x)\\hat p_{j+1}(x) \n  ) \\\\\n  &+a_0 p^{(-1)}_\\ell(x)\\hat p_0(x)-\\hat a_\\ell \\hat p_\\ell(x) p_\\ell^{(\\ell-1)}(x).\n\\end{align*}\nIn summary,\n\\begin{equation}\n  \\begin{aligned}\n  \\frac{\\hat a_\\ell}{a_\\ell} \\hat p_\\ell(x)\n  &=\n  p_\\ell(x)\\hat p_0(x) \\\\\n  &+\n  \\sum_{j=1}^{\\ell} (b_{j} - \\hat b_{j})p^{(j-1)}_\\ell(x)\\hat p_{j-1}(x) \n  \\\\\n  &+\n  \\sum_{j=1}^{\\ell-1} (a_{j} - \\hat a_{j})\n  (\n  p^{(j)}_\\ell(x)\\hat p_{j-1}(x) \n  +p^{(j-1)}_\\ell(x)\\hat p_{j}(x) \n  ). \n\\end{aligned}\n  \\label{eq:Perturb}\n\\end{equation}\nHere we take $\\hat{p}_j(x) = p_{j+1}^{(0)}(x),$ so that $\\hat a_j = a_{j+1},$ $\\hat b_j = b_{j+1},$ and $\\hat p_0(x) = \\frac{1}{a_1}.$  Specializing these formulas, we get\n\\begin{equation}\n  \\begin{aligned}\n     p^{(0)}_{\\ell+1}(x)\n  &=\n  \\frac{a_\\ell}{a_{\\ell+1}a_1}p_\\ell(x) \\\\\n  &+\n  \\frac{a_\\ell}{a_{\\ell+1}}\\sum_{j=1}^{\\ell} (b_{j} - b_{j+1})p^{(j-1)}_\\ell(x)p_{j}^{(0)}(x) \n  \\\\\n  &+\n  \\frac{a_\\ell}{a_{\\ell+1}}\\sum_{j=1}^{\\ell-1} (a_{j} - a_{j+1})\n  (\n  p^{(j)}_\\ell(x)p_{j}^{(0)}(x) \n  +p^{(j-1)}_\\ell(x)p_{j+1}^{(0)}(x) \n  ). \n\\end{aligned}\n  \\label{eq:MinorPerturb}\n\\end{equation}\n\n\n\n\\subsection{Summary}\n\\label{sec:summary}\n\nThe summary of the previous discussion is that we have a complete description of the solutions of the commutator equation \\eqref{eq:comm} in terms of the orthogonal polynomials associated to $A.$\n\\begin{theorem}\n  A solution to the commutator equation\n  \\[\n    [A,M] - W \\in \\mathbb{T}\n  \\]\n  for antisymmetric $M$ in terms of a symmetric $W$ is given by $M = M_1 + M_2$ where\n  \\begin{align*}\n    M_1 &= \\sum_{i=1}^n (O^t W O)_{i,i} q_i^2 F(\\lambda_i), \\text{ and} \\\\\n    M_2 &= \\sum_{i > j}^n (O^t W O)_{i,j}q_iq_j H_0^{-}(\\lambda_i,\\lambda_j).\n  \\end{align*}\n  Moreover, this is the unique $M$ that vanishes in the first column.\n  The tridiagonal error is given by\n  \\begin{equation*}\n    \\begin{aligned}\n    [A,M]-W &= \n    \\sum_{i=1}^n (O^t W O)_{i,i} q_i^2 G(\\lambda_i)\n    -\\sum_{i > j}^n (O^t W O)_{i,j} q_iq_j\n    \\frac{B(\\lambda_i,\\lambda_i) - B(\\lambda_j,\\lambda_j)}{\\lambda_i-\\lambda_j} \\\\\n    &=\n    -\\frac{1}{2} \n    \\sum_{i,j=1}^n\n    (O^t W O)_{i,j} q_iq_j\n    \\Delta^{\\lambda_j}_{\\lambda_i} B(\\lambda_i,\\lambda_i).\n  \\end{aligned}\n  \\end{equation*}\n  All other solutions to the equation differ from this one by elements of\n  \\[\n    \\operatorname{Span}\\left\\{ E^{-}(\\lambda_i,\\lambda_i), i=1,2,\\dots,n \\right\\}.\n  \\]\n  \\label{thm:comm}\n\\end{theorem}\n\\begin{proof}\n  From \\eqref{eq:spec1} and \\eqref{eq:spec2} we can expand $W$ as\n  \\begin{align*}\n    W &= \\sum_{i=1}^n (O^t W O)_{i,i} \\cdot q_i^2 E^{+}(\\lambda_i,\\lambda_i)\\\\\n     &+ \\sum_{i > j}^n (O^t W O)_{i,j} \\cdot q_iq_j \\cdot\n     \\left\\{ \n       E^{+}(\\lambda_i,\\lambda_j) + E^{+}(\\lambda_j,\\lambda_i)\n     \\right\\}.\n  \\end{align*}\n  Applying \\eqref{eq:H-} and \\eqref{eq:F}, the theorem now follows.  Theorem~\\ref{thm:span} shows that any other solution to the equation differs from this one as desired.\n  The uniqueness of $M$ follows from the invertibility of $O,$ i.e.\\,the linear independence of $\\left\\{ (p_{k-1}(\\lambda_i))_k, i=1,\\dots,n \\right\\},$ which is what appears in the first column of $E^{-}(\\lambda_i,\\lambda_i)$ for $i=1,\\dots,n.$ \n\\end{proof}\n\nIf we restrict to the case that $W$ is tridiagonal itself, this leads to a family of orthogonality rules which have quartic dependence on the orthogonal polynomials. \n\\begin{corollary}\n  \\label{cor:orth}\n  For a symmetric tridiagonal matrix $W,$\n    \\[\n    W + \n    \\sum_{i=1}^n (O^t W O)_{i,i} q_i^2 G(\\lambda_i)\n    =\n    \\sum_{i > j}^n (O^t W O)_{i,j} q_iq_j\n    \\frac{B(\\lambda_i,\\lambda_i) - B(\\lambda_j,\\lambda_j)}{\\lambda_i-\\lambda_j}.\n  \\]\n\\end{corollary}\n\\begin{proof}\n  Suppose that $W$ is tridiagonal.  Then on the one hand, $M=M_1+M_2,$ given in the theorem, is one solution to\n  \\[\n    [A,M] - W \\in \\mathbb{T}.\n  \\]\n  A second solution is given by $M=0.$  As both vanish in the first column, we have that $M_1+M_2 = 0.$ \n\\end{proof}\n\nWe can also use Theorem~\\ref{thm:comm} to make a connection to the minor polynomials.  Suppose we wished to solve for a matrix $W$ that is supported in a single entry\n\\begin{corollary}\n  Let $W^{k,\\ell}$ be the matrix that is $1$ in the $(k,\\ell)$ and $(\\ell,k)$ entries, for $k > \\ell$, and $0$ elsewhere.  \n \n \n  The matrix $M^{k,\\ell}$ with $0$ in the first column so that $[A,M^{k,\\ell}] = W^{k,\\ell} + \\mathbb{T}$ is given by\n  \\[\n    \\begin{aligned}\n    M\n    &=\n    -\\frac{1}{2}\\sum_{i,j=1}^n\n    q_i^2\n    q_j^2\n    p_{k-1}(\\lambda_i)\n    p_{\\ell-1}(\\lambda_j)\n    \\Delta_{\\lambda_j}^{\\lambda_i} E^{-}(\\lambda_i,\\lambda_j), \\\\\n    M_{u,r}\n    &=\n    -\\frac{1}{2}\\sum_{i}\n    q_i^2\n    p_{u-1}(\\lambda_i)\n    p_{k-1}(\\lambda_i)p^{(\\ell-1)}_{r-1}(\\lambda_i),\n  \\end{aligned}\n  \\]\n  for any $u > r.$\n  When $k =\\ell+1,$ this matrix is identically $0.$ We also define $M = 0$ if $k = \\ell.$ \n  For other $k > \\ell+1,$ the entry $M_{u,r}$ vanishes when $u-r \\geq k-\\ell,$ and it vanishes when $u+r \\leq k+\\ell.$\n  The tridiagonal matrix $G^{k,\\ell} = [A,M^{k,\\ell}] - W^{k,\\ell}$ is given by \n \n  \\[\n    {G}^{k,\\ell}\n    =\n    -\n    \\frac{1}{2}\\sum_{i,j=1}^n\n    q_i^2\n    q_j^2\n    p_{k-1}(\\lambda_i)\n    p_{\\ell-1}(\\lambda_j)\n    \\Delta_{\\lambda_j}^{\\lambda_i} B(\\lambda_i,\\lambda_j)\n  \\]\n  When $k = \\ell+1,$ this matrix is equal to $W^{k,\\ell},$ and when $k=\\ell,$ this matrix is $0$ everywhere except in the $(k,k)$ entry where it is $1.$\n  \\label{cor:entry}\n\\end{corollary}\n\\begin{proof}\n  We expand the $(u,r),$ entry of $M=M_1+M_2$ for $u\\geq r$ from Theorem~\\ref{thm:comm}, which gives\n  \\[\n    M\n    =\n    -\\frac{1}{2}\\sum_{i,j}\n    p_{k-1}(\\lambda_i)\n    p_{\\ell-1}(\\lambda_j)\n    q_i^2 q_j^2\n    \\left( p_{u-1}(\\lambda_i) + p_{u-1}(\\lambda_j) \\right)\n    \\left\\{ \n      \\frac{p_{r-1}(\\lambda_i) - p_{r-1}(\\lambda_j)}{\\lambda_i-\\lambda_j}\n    \\right\\},\n  \\]\n  where the $i=j$ terms are defined by continuity.\n  In terms of the minor polynomials, this can be written as\n  \\[\n    \\begin{aligned}\n    M=\n    &-\\frac{1}{2}\\sum_{i}\n    q_i^2\n    p_{k-1}(\\lambda_i) \n    p_{u-1}(\\lambda_i)\n    p^{(\\ell-1)}_{r-1}(\\lambda_i) \\\\\n    &\n    -\\frac{1}{2}\\sum_{j}\n    q_j^2\n    p_{\\ell-1}(\\lambda_j) \n    p_{u-1}(\\lambda_j)\n    p^{(k-1)}_{r-1}(\\lambda_j).\n    \\end{aligned}\n  \\]\n  As $k > \\ell$ and $u > r,$ the second sum vanishes in all cases as $p_{\\ell-1} p^{(k-1)}_{r-1}$ has degree at most $r-1 < u - 1.$\n  To see the other vanishing conditions, if $u$ is too large, then $p_{u-1}$ is orthogonal to all polynomials of degree strictly smaller degree.  On the other hand, if $u+r$ is too small, then by the orthogonality of $p_{k-1}$ to lower degree polynomials, the first sum vanishes.  \n\\end{proof}\n\n\\begin{remark}\n  It is also possible to express the matrix $G^{k,\\ell}$ in terms on minor polynomials.\n  \\[\n    G^{k,\\ell}_{u,r}\n    =\n    \\frac{1}{2}\\sum_{i=1}^n\n    q_i^2\n    p_{k-1}(\\lambda_i)\n    \\begin{cases}\n    a_{r-1} p_{r-2}(\\lambda_i)p_{r-1}^{(\\ell-1)}(\\lambda_i) - a_r p_{r-1}(\\lambda_i) p_{r}^{(\\ell-1)}(\\lambda_i)&  \\\\\n    +a_{r-1} p_{r-1}(\\lambda_i)p_{r-2}^{(\\ell-1)}(\\lambda_i) - a_r p_{r}(\\lambda_i) p_{r-1}^{(\\ell-1)}(\\lambda_i),  & \\text{ if } u = r  < n, \\\\\n    a_r (p_{r-1}(\\lambda_i)p_{r-1}^{(\\ell-1)}(\\lambda_i) - p_r(\\lambda_i)p_r^{(\\ell-1)}(\\lambda_i)), & \\text{ if } u = r + 1 < n,\\\\\n    0, & \\text{ otherwise.}\n  \\end{cases}\n\\]\n\\label{rem:Gkl}\nThis characterization gives the result in Theorem \\ref{thm:lanczos}\n\\end{remark}\n\nThis leads us directly to a characterization of all differentials $dM_t$ that produce tridiagonal models.\n\n\n\\section{The Tridiagonal Models}\n\\label{sec:models}\n\nThe first characterization of the tridiagonal models associated to is a direct consequence of the work in the previous section. The following theorem gives the martingale structure of the model. \n\n\\begin{theorem}\n\\label{thm:martingales}\n  The rotation differentials $dM_t$ of Dyson Brownian Motion $(\\lambda_{i,t}, 1 \\leq i \\leq n)$ that produce \\emph{nondegenerate} tridiagonal models are all of the form\n  \\[\n    dM_t = \\sum_{i=1}^n \\left\\{q_{i,t}^2 d\\lambda_{i,t} F(\\lambda_{i,t}) + c_i E^{-}(\\lambda_{i,t},\\lambda_{i,t})\\right\\} + dM^{(2)}_t.\n  \\]\n  for some choice of coefficients $c_i$ and some finite variation process $dM^{(2)}_t.$  In particular, the martingale portion of $dA_t$ has the form\n  \\[\n    \\sum_{i=1}^n \n   \n    \\biggl\\{\n      -\n      dZ_{i,t}\n      \\cdot q_{i,t}^2 \\cdot G(\\lambda_{i,t})\n      +\n      dX_{i,t}\n      B(\\lambda_{i,t},\\lambda_{i,t})\n    \\biggr\\}\n  \\]\n  where $Z_{i,t}$ are the Brownian motions driving the Dyson Brownian motion and $X_{i,t}$ are some martingales. For the definitions of the matrices $F, E^{-}, G,$ and $B$ see the equations \\textup{(\\ref{eq:F}), (\\ref{eq:E}), (\\ref{eq:G})}, and \\textup{(\\ref{eq:B})} respectively\n  \\label{cor:trim}\n\\end{theorem}\nThere is a great deal of freedom in the general model. We specialize to two cases before writing down explicit finite variation processes, though this can be done for the general model. The two cases we study are the model where the spectral weights $q_i$ are frozen, and the second is where they evolve according to a finite variation process. \n\n\n\n\n\\subsection{The frozen spectral weight model}\n\\label{sec:tridiagonalfrozen}\n\nHaving the martingale portions of the SDE, we can then turn to solving for the finite variation portions of $dA_t.$  These second--order terms depend on the choices made in the martingale terms.  One possible choice is to make $dM_t$ have $0$ in the first column, which will lead to the frozen spectral weight model (where the spectral weights are constant in time).\n\n\n\n\\begin{theorem}\n\\label{thm:tridiagonalfrozen}\nLet the $q_i$ be fixed then the tridiagonal model associated to the $\\beta$-Dyson flow satisfies \n\\begin{equation}\n  \\begin{aligned}\n    dA_t &= \\sum_{i=1}^n -\\biggl\\{\\sqrt{\\frac{2}{\\beta}} dZ_{i,t} +  dt \\cdot \\biggl\\{-\\frac{V'(\\lambda_{i,t})}{2} + \\sum_{j \\neq i}\\frac{1}{\\lambda_{i,t} - \\lambda_{j,t}} \\biggr\\} \\biggr\\} \\cdot q_i^2 \\cdot G(\\lambda_{i,t})\n    + \\sum_{k \\geq \\ell} dP_{k,\\ell,t} \\cdot G^{k,\\ell} \\\\\n    dP_t &= \\sum_{i=1}^n -\\frac{q_{i}^4}{2} \\cdot [ -E^{+}(\\lambda_{i,t},\\lambda_{i,t}) + G(\\lambda_{i,t}), F(\\lambda_{i,t})] \\cdot dt  .\n  \\end{aligned}\n    \\label{eq:frozen}\n\\end{equation}\nMoreover if $\\Lambda$ is a stationary process then $A$ will be stationary in the frozen spectral weight model. The definitions of $G, E^+, F$ and $G^{k,\\ell}$ may be found in \\textup{(\\ref{eq:G}), (\\ref{eq:E}), (\\ref{eq:F})} and Remark \\ref{rem:Gkl} respectively\n\\end{theorem}\n\n\\begin{corollary}\nIf we take $A_0$ to have the same distribution as the tridiagonal model in (\\ref{eq:tridiagonal}) and $\\Lambda$ satisfies the $\\beta$-Dyson Brownian motion flow with $V(x)= 2x^2$ then $A_t$ will have the same distribution as (\\ref{eq:tridiagonal}).\n\\end{corollary}\n\nRecall that $dA_t$ satisfies\n\\[\n  dA_t = dW_t + [A_t, dM_t] + \\frac{1}{2}[2dW_t + [A_t,dM_t], dM_t]\n\\]\nwhere $dW_t = \\sum_{i=1}^n d\\lambda_{i,t} q_i^2 E^{+}(\\lambda_i,\\lambda_i).$  \nDefine\n\\begin{equation}\n  \\label{eq:frozenM}\n  \\begin{aligned}\n    dM^{(1)}_t &= \\sum_{i=1}^n -\\sqrt{\\frac{2}{\\beta}}dZ_{i,t} \\cdot q_i^2 \\cdot F(\\lambda_{i,t}), \\\\\n    dM^{(2)}_t &= \\sum_{i=1}^n -dt \\cdot q_i^2 \\cdot F(\\lambda_i)\\cdot \\biggl\\{-\\frac{V'(\\lambda_{i,t})}{2} + \\sum_{j \\neq i}\\frac{1}{\\lambda_{i,t} - \\lambda_{j,t}} \\biggr\\}, \\\\\n    dM^{(3)}_t &= \\sum_{k \\geq \\ell} -\\frac{1}{2}[2dW_t+[A_t,dM^{(1)}_t], dM^{(1)}_t]_{k,\\ell} \\cdot M^{k,\\ell},\n  \\end{aligned}\n\\end{equation}\nand let $dM_t$ be the sum of these three components.  Here $M^{k,\\ell}$ is the matrix defined in Corollary~\\ref{cor:entry}.  Observe that the first column of $dM_t$ is $0.$  Recalling the definitions of $F$ and $M^{k,\\ell},$ we have that\n\\[\n  \\begin{aligned}\n  &[A_t, dM^{(1)}_t+ dM^{(2)}_t] \\in \\sum_{i=1}^n -d\\lambda_{i,t} \\cdot q_i^2 \\cdot E^{+}(\\lambda_i,\\lambda_i) + \\mathbb{T} \\\\\n  &[A_t, dM^{(3)}_t] \\in -\\frac{1}{2}[ 2dW_t + [A_t,dM^{(1)}_t],dM^{(1)}_t] + \\mathbb{T} \\\\ \n  &[A_t, dM^{(3)}_t] \\in -\\frac{1}{2}[ 2dW_t + [A_t,dM_t],dM_t] + \\mathbb{T} \n  \\end{aligned}\n\\]\nThe equivalence of the last two terms comes from the fact that only products of two stochastic differentials survive.\nHence, this choice of $dM_t$ produces a tridiagonal diffusion, moreover, the produces the tridiagonal model in Theorem \\ref{thm:tridiagonalfrozen}. To arrive at the expression we have for $dP_t,$ we have used the independence of the Brownian motions $dZ_{i,t}.$\n\n\\subsubsection*{Computation of $dP_t$}\n\nThis computation reduces to that of the commutators $[G(\\lambda_i),F(\\lambda_i)]$ and $[E^{+}(\\lambda_i,\\lambda_i),F(\\lambda_i)]$ for an eigenvalue $\\lambda_i.$  We further identify the dominant terms in this expansion, at least for a sufficiently small principal submatrix.\nSpecifically we show that\n  \\[\n    \\sum_{k \\geq \\ell} dP_{k,\\ell,t} \\cdot G^{k,\\ell}\n    =\\sum_{i=1}^n \\frac{q_i^2}{4} \\partial_{xy} B(x,y) \\vert_{x=y=\\lambda_{i,t}}\n    + dR_{t},\n  \\]\n  where $dR_t$ is a lower order term, at least for sufficiently small windows of the upper left corner.  In the asymptotic regime considered in Section~\\ref{sec:asymptotics}, the magnitude of the displayed terms in principal submatrix of size $M$ will be $M^2\/n.$  In contrast, the $dR_t$ terms will be $M^4\/n^2.$\n\nWe begin with $[G(\\lambda_i), F(\\lambda_i)].$  For notational simplicity, we suppress $\\lambda_i$ in what follows.  All polynomials are evaluated at $\\lambda_i.$\nAs $G$ is tridiagonal, we recall for convenience the commutator equation \\eqref{eq:AM}\n\\begin{equation*}\n[A,M]_{k,\\ell} = a_{k-1} m_{k-1,\\ell} + b_k m_{k,\\ell} +a_k m_{k+1,\\ell} - (a_{\\ell-1} m_{k,\\ell-1} + b_\\ell m_{k,\\ell}+ a_\\ell m_{k,\\ell+1}).\n\\end{equation*}\nFor $k > \\ell +1,$ we get\n\\begin{align*}\n  [G,F]_{k,\\ell}\n  =\n  &-a_{k-1} (p_{k-2}p_{k-2}' - p_{k-1}p_{k-1}')\n  p_{k-2}p'_{\\ell-1} \\\\\n  &-\\biggl\\{a_{k-1} p_{k-2}p_{k-1}' - a_k p_{k-1} p_{k}'\n  +a_{k-1} p_{k-1}p_{k-2}' - a_k p_{k} p_{k-1}'\\biggr\\}\n  p_{k-1}p'_{\\ell-1} \\\\\n  &-a_k(p_{k-1}p_{k-1}' - p_k p_k')\n  p_{k}p'_{\\ell-1} \\\\\n  &+a_{\\ell-1} (p_{\\ell-2}p_{\\ell-2}' - p_{\\ell-1}p_{\\ell-1}')\n  p_{k-1}p'_{\\ell-2} \\\\\n  &+\\biggl\\{a_{\\ell-1} p_{\\ell-2}p_{\\ell-1}' - a_\\ell p_{\\ell-1} p_{\\ell}'\n  +a_{\\ell-1} p_{\\ell-1}p_{\\ell-2}' - a_\\ell p_{\\ell} p_{\\ell-1}'\\biggr\\}\n  p_{k-1}p'_{\\ell-1} \\\\\n  &+a_\\ell(p_{\\ell-1}p_{\\ell-1}' - p_\\ell p_\\ell')\n  p_{k-1}p'_{\\ell}. \n\\end{align*}\nThere is cancellation in this formula, which results in \n\\begin{equation}\n  \\label{eq:GF}\n  \\begin{aligned}\n  ~[G,F]_{k,\\ell}\n  = \n  \\mathcal{H}_{k,\\ell}\n  :=\n  &-p_{\\ell-1}'a_{k-1}p_{k-2}'(p_{k-2}^2 + p_{k-1}^2) \\\\\n  &+p_{\\ell-1}'a_{k}p_{k}'(p_{k-1}^2 + p_{k}^2) \\\\\n  &+p_{k-1}a_{\\ell-1}p_{\\ell-2}((p_{\\ell-2}')^2 + (p_{\\ell-1}')^2) \\\\\n  &-p_{k-1}a_{\\ell}p_{\\ell}((p_{\\ell-1}')^2 + (p_{\\ell}')^2). \n\\end{aligned}\n\\end{equation}\nHere, the formula for $[G,F]$ holds for $k > \\ell+1.$  We let $\\mathcal{H}_{k,\\ell}$ be given by this formula for all $k \\geq \\ell.$ \nFrom the symmetry of $G$ and the antisymmetry of $F,$ the commutator $[G,F]$ is symmetric, which allows its entries above the diagonal to be completed.  \nHence we extend the definition of $\\mathcal{H}$ to be $0$ on the diagonal and symmetric.\nOn the tridiagonal, there are correction terms coming from the behavior of $F$ near the diagonal.  \n\nThe correction to \\eqref{eq:GF} when $k = \\ell+1$ stems from the $0$ diagonal of $F.$  This $0$-diagonal implies that \\[\n  G_{\\ell+1,\\ell} F_{\\ell,\\ell} - F_{\\ell+1,\\ell+1} G_{\\ell+1,\\ell} = 0.\n\\]\nHence, the commutator for $k=\\ell+1$ is\n\\begin{equation}\n  \\label{eq:GFe1}\n  \\begin{aligned}\n  ~[G,F]_{\\ell+1,\\ell}\n  -\\mathcal{H}_{\\ell+1,\\ell}\n  &=\n  -G_{\\ell+1,\\ell} p_{\\ell-1} p_{\\ell-1}' + p_{\\ell}p_{\\ell}' G_{\\ell+1,\\ell} \\\\\n  &=\n  -a_\\ell( p_{\\ell-1} p_{\\ell-1}' - p_{\\ell}p_{\\ell}')^2. \n  \\end{aligned}\n\\end{equation}\nAs for the diagonal, we get\n\\begin{equation}\n  \\label{eq:GFe2}\n  \\begin{aligned}\n  ~[G,F]_{\\ell,\\ell}\n  =&\n  2G_{\\ell,\\ell+1} F_{\\ell+1,\\ell}\n  -2G_{\\ell-1,\\ell} F_{\\ell,\\ell-1} \\\\\n  =&-p_{\\ell-1}'a_{\\ell-1}p_{\\ell-2}'(0 + 2p_{\\ell-1}^2) \\\\\n  &+p_{\\ell-1}'a_{\\ell}p_{\\ell}'(0 + 2p_{\\ell}^2) \\\\\n  &+p_{\\ell-1}a_{\\ell-1}p_{\\ell-2}(2(p_{\\ell-2}')^2 + 0) \\\\\n  &-p_{\\ell-1}a_{\\ell}p_{\\ell}(2(p_{\\ell-1}')^2 + 0).  \\\\\n  =&\\mathcal{H}_{\\ell,\\ell} \\\\\n  &-p_{\\ell-1}'a_{\\ell-1}p_{\\ell-2}'(-p_{\\ell-2}^2 + p_{\\ell-1}^2) \\\\\n  &+p_{\\ell-1}'a_{\\ell}p_{\\ell}'(-p_{\\ell-1}^2 + p_{\\ell}^2) \\\\\n  &+p_{\\ell-1}a_{\\ell-1}p_{\\ell-2}((p_{\\ell-2}')^2 - (p_{\\ell-1}')^2) \\\\\n  &-p_{\\ell-1}a_{\\ell}p_{\\ell}((p_{\\ell-1}')^2 - (p_{\\ell}')^2).  \\\\\n  \\end{aligned}\n\\end{equation}\n\nTurning to $[E^{+},F],$ once again evaluated at an eigenvalue $\\lambda_i$ which we suppress, for any $k \\geq \\ell$\n\\begin{align*}\n  [E^{+},F]_{k,\\ell}\n  =&\n  \\sum_{m=1}^n E^{+}_{k,m} F_{m,\\ell} - F_{k,m} E^{+}_{m,\\ell} \\\\\n  =&\n  \\sum_{m=\\ell+1}^n -p_{k-1} p_{m-1} p_{m-1} p_{\\ell-1}' \n  -\\sum_{m=k+1}^n p_{k-1}' p_{m-1} p_{m-1} p_{\\ell-1} \\\\\n  +&\\sum_{m=1}^{\\ell-1} p_{k-1} p_{m-1} p_{m-1}' p_{\\ell-1}\n  +\\sum_{m=1}^{k-1} p_{k-1} p_{m-1}' p_{m-1} p_{\\ell-1}. \\\\\n\\end{align*}\nApplying the Christoffel--Darboux identities to these sums, we arrive at\n\\begin{align*}\n  [E^{+},F]_{k,\\ell}\n  =\n  &-\\{p_{k-1}p_{\\ell-1}'+p_{k-1}'p_{\\ell-1}\\}q_i^{-2} \\\\\n  &-p_{k-1}p_{\\ell-1}'\\left\\{ - a_\\ell p_\\ell'p_{\\ell-1}+a_\\ell p_\\ell p_{\\ell-1}' \\right\\} \\\\\n  &-p_{k-1}'p_{\\ell-1}\\left\\{ - a_k p_k'p_{k-1}+ a_k p_k p_{k-1}' \\right\\} \\\\\n  &+p_{k-1}p_{\\ell-1}a_{\\ell-1} \\left\\{ p_{\\ell-1}''p_{\\ell-2} - p_{\\ell-1}p_{\\ell-2}'' \\right\\} \\\\\n  &+p_{k-1}p_{\\ell-1}a_{k-1} \\left\\{ p_{k-1}''p_{k-2} - p_{k-1}p_{k-2}'' \\right\\}.\n\\end{align*}\nWe set $\\mathcal{E}_{k,\\ell}$ to be the sum of the final four lines from this equation.\n\n\\subsubsection*{Some simplifications of the FV terms}\n\nThere is some cancellation in the finite variation expressions which assists in estimating its true magnitude when passing to asymptotics. Much of the simplification is possible on account of the observation that for $k < \\ell,$\n\\[\n    \\frac{1}{2}\\sum_{i,j=1}^n\n    q_i^2\n    q_j^2\n    p_{k-1}(\\lambda_i)\n    p_{\\ell-1}(\\lambda_j)\n    \\Delta_{\\lambda_j}^{\\lambda_i} B(\\lambda_i,\\lambda_j)\n    =0.\n\\]\nThe left hand side of this expression gives $-G^{k,\\ell}$ for $k \\geq \\ell.$  Hence if we extend the definition of $G^{k,\\ell}$ to be $0$ for $k < \\ell,$ we can express the finite variation terms as four sums\n\\begin{equation}\n  \\sum_{k \\geq \\ell} dP_{k,\\ell,t} \\cdot G^{k,\\ell}\n  = \n  \\begin{aligned}\n  &\\sum_{i,k,\\ell=1}^n -\\frac{q_i^2}{2} \\cdot (p_{k-1}(\\lambda_i)p_{\\ell-1}'(\\lambda_i)+p_{k-1}(\\lambda_i)'p_{\\ell-1}(\\lambda_i) )\\cdot G^{k,\\ell} \\cdot dt  &\\bigg\\} =: dS_t \\\\\n  +&\\sum_{i,k,\\ell=1}^n -\\frac{q_i^4}{2} \\cdot \\mathcal{H}_{k,\\ell}(\\lambda_i)\\cdot G^{k,\\ell} \\cdot dt \\qquad &\\bigg\\} =: dR^1_t \\\\\n  +&\\sum_{i,k,\\ell=1}^n -\\frac{q_i^4}{2} \\cdot ( [G(\\lambda_i),F(\\lambda_i)]_{k,\\ell} - \\mathcal{H}_{k,\\ell}(\\lambda_i))\\cdot G^{k,\\ell} \\cdot dt \\qquad &\\bigg\\} =: dR^2_t \\\\\n  +&\\sum_{i,k,\\ell=1}^n \\frac{q_i^4}{2} \\cdot \\mathcal{E}_{k,\\ell}(\\lambda_i)\\cdot G^{k,\\ell} \\cdot dt.  &\\bigg\\} =: dR^3_t \\\\\n\\end{aligned}\n  \\label{eq:FVterms}\n\\end{equation}\n\nDefine for any eigenvalues $\\lambda_i$ and $\\lambda_u$\n\\[\n  \\mathfrak{D}(\\lambda_u,\\lambda_i)\n  =\\sum_{\\ell=1}^n p_{\\ell-1}(\\lambda_u) p_{\\ell-1}'(\\lambda_i).\n\\]\nObserve that for any $1 \\leq r \\leq n,$\n\\[\n  \\sum_{u=1}^n \n  \\mathfrak{D}(\\lambda_u,\\lambda_i)\n  p_{r-1}(\\lambda_u) q_u^2 = p_{r-1}'(\\lambda_i),\n\\]\nand hence we have that for any polynomial $p$ of degree less than $n,$\n\\begin{equation}\n  \\label{eq:Did}\n  \\sum_{u=1}^n \n  \\mathfrak{D}(\\lambda_u,\\lambda_i)\n  p(\\lambda_u) q_u^2 = p'(\\lambda_i).\n\\end{equation}\n\n\\subsubsection*{Simplifying $dS_t:$}\n\nWe begin this case by observing\nthat since $G^{k,\\ell} = 0$ for $k < \\ell,$\n\\[\n\\sum_{i,k,\\ell=1}^n -\\frac{q_i^2}{2} \\cdot (p_{k-1}(\\lambda_i)p_{\\ell-1}'(\\lambda_i))\\cdot G^{k,\\ell} = 0,\n\\]\nfrom the orthogonality of $p_{k-1}$ to lower degree polynomials.  Hence we only need consider the other term.  Expanding the definition of $G^{k,\\ell},$\n\\begin{align*}\n  dS_t\n  &=\n  \\sum_{i,k,\\ell=1}^n -\\frac{q_i^2}{2} \\cdot (p_{k-1}(\\lambda_i)'p_{\\ell-1}(\\lambda_i) )\\cdot G^{k,\\ell} \\cdot dt \\\\\n  &=\n  \\sum_{i,k,\\ell,u,j=1}^n \n  \\frac{q_i^2q_u^2q_j^2}{4} \\cdot \n  p_{k-1}(\\lambda_i)'p_{\\ell-1}(\\lambda_i) \\cdot \n  p_{k-1}(\\lambda_u)\n  p_{\\ell-1}(\\lambda_j)\n  \\Delta_{\\lambda_j}^{\\lambda_u} B(\\lambda_u,\\lambda_j)\n  \\cdot dt \\\\\n  &=\n  \\sum_{i,k,u=1}^n \n  \\frac{q_i^2q_u^2}{4} \\cdot \n  p_{k-1}(\\lambda_i)' \\cdot \n  p_{k-1}(\\lambda_u)\n  \\Delta_{\\lambda_i}^{\\lambda_u} B(\\lambda_u,\\lambda_i)\\cdot dt \\\\\n  &=\n  \\sum_{i,u=1}^n \n  \\frac{q_i^2q_u^2}{4} \\cdot \n  \\mathfrak{D}(\\lambda_u,\\lambda_i)\n  \\Delta_{\\lambda_i}^{\\lambda_u} B(\\lambda_u,\\lambda_i)\\cdot dt. \n\\end{align*}\nIn the third equality, we have used the orthogonality of the polynomials in $\\ell.$ In the fourth, we have used the definition of $\\mathfrak{D}.$ Hence, we conclude\n\\begin{lemma}\n  For $k,\\ell$ satisfying $2\\max(k,\\ell) \\leq n+1,$ we have\n  \\[\n    dQ_{k,\\ell,t} = \n    \\sum_{i=1}^n \n    \\frac{q_i^2}{4} \\partial_{xy}B_{k,\\ell}(x,y) \\vert_{x=y=\\lambda_{i,t}}\n    .\n  \\]\n  \\label{lem:R3}\n\\end{lemma}\n\\begin{proof}\n  For $k,\\ell$ as stated,\n  \\(\n    \\Delta_{y}^{x}B_{k,\\ell}(x,y)\n  \\)\n  are polynomials in $x$ of degree at most $n-1.$\n  Applying \\eqref{eq:Did} to $dS_t,$ it follows that \n  \\[\n    dQ_{k,\\ell,t} = \n    \\sum_{i=1}^n \n    \\frac{q_i^2}{4} (\\partial_{y} \\Delta_{\\lambda_i}^{y}B_{k,\\ell}(y,\\lambda_i))\\vert_{y=\\lambda_i}\n  \\]\n  Expanding,\n  \\[\n    \\partial_{y} \\Delta_{\\lambda_i}^{y}B(y,\\lambda_i)\n    =\n    \\partial_y\n    \\frac{B(y,y) - B(y,\\lambda_i)}{y-\\lambda_i}.\n  \\]\n  Hence, on setting $y=\\lambda_i,$ we get\n  \\[\n    (\\partial_{y} \\Delta_{\\lambda_i}^{y}B(y,\\lambda_i))\\vert_{y=\\lambda_i}\n    =(\\partial_{xy} + \\frac{1}{2}\\partial_{yy})B(x,y) \\vert_{x=y=\\lambda_i}.\n  \\]\n  By orthogonality, and recalling \\eqref{eq:B}, we have\n  \\[\n    \\sum_{i=1}^n \n    \\frac{q_i^2}{4}\n    (\\partial_{yy})B(x,y) \\vert_{x=y=\\lambda_i} = 0,\n  \\]\n  which completes the proof.\n\\end{proof}\n\n\\subsubsection*{Simplifications for quadratic constraining potential}\nFor $V'(t) = ct,$ some additional simplification is possible.  Specifically, we may take advantage of the identity\n\\begin{equation}\n  \\sum_{i=1}^n q_i^2 \\lambda_i G(\\lambda_i) =  -A\n  \\label{eq:AG}\n\\end{equation}\nWe prove this like follows.  By \\eqref{eq:F}, for an eigenvalue $\\lambda$ of $A,$\n\\[\n    [A,F(\\lambda)] = E^{+}(\\lambda,\\lambda) + G(\\lambda).\n\\]\nRecall that an entry of $F_{k,\\ell}(\\lambda)=p_{k-1}(\\lambda)p_{\\ell-1}'(\\lambda)$ for $k > \\ell.$ Hence, by orthogonality of $p_{k-1}$ to lower degree polynomials.\n\\[\n  \\sum_{i=1}^n q_i^2 \\lambda_i F(\\lambda_i) = 0.\n\\]\nOn the other hand, recalling \\eqref{eq:spec2},\n\\[\n  \\sum_{i=1}^n E^{+}(\\lambda_i,\\lambda_i)q_i^2 \\lambda_i = A,\n\\]\nwhich completes the proof.\n\n\\subsection{Smooth spectral weight model}\n\\label{sec:tridiagonalfv}\n\nA relatively tame perturbation of the frozen--spectral weight model is one in which the spectral weights are finite variation processes, which may or may not depend on the eigenvalues.  \n\\begin{theorem}\n\\label{thm:tridiagonalFV}\nSuppose that $dR_{i,t},$ for $1 \\leq i \\leq n$ are any finite variation processes.  Define\n\\[\n  dM^R_t = \n  dM^{F}_t\n  + \\sum_{i=1}^n dR_{i,t} q_{i,t} E^{-}(\\lambda_{i,t},\\lambda_{i,t}),\n\\]\nwhere $dM^{F}_t$ is the rotation differential from \\eqref{eq:frozenM}.  The associated tridiagonal model is given by \n\\begin{equation}\n  \\begin{aligned}\n    dA_t &= dA^{F}_t + \\sum_{i=1}^n dR_{i,t} q_i B(\\lambda_{i,t},\\lambda_{i,t}).\n  \\end{aligned}\n  \\label{eq:smooth}\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nThe eigenvector matrix $O_t$ of $A_t$ evolves (by Theorem~\\ref{thm:tri}) as\n\\begin{equation*}\n  dO_t^t = O_t^t (dM_t +\\frac{1}{2} dM_t^2).\n\\end{equation*}\nWhen $dM_t = dM^R_t,$ the first column of $dM_t^2$ vanishes, as $dM^F_t$ is identically $0$ in this column and $dM^R_t-dM^F_t$ is finite variation.  Hence, the first row of $dO_t$ evolves according to\n\\begin{align*}\n  dq_{i,t} = dO_{1,i,t} = \n  \\sum_{\\ell=1}^n O_{\\ell,i} dM^R_{\\ell,1,t} \n  &=\n  \\sum_{\\ell=2}^n q_{i,t} p_{\\ell-1}(\\lambda_{i,t}) \n  \\biggl\\{ \n    \\sum_{j=1}^n dR_{j,t} q_{j,t} p_{\\ell-1}(\\lambda_{j,t})\n  \\biggr\\} \n  \\\\\n  &=\n  -q_i \\sum_{j=1}^n dR_{j,t} q_{j,t} \n  +\n  \\sum_{j=1}^n \n  \\sum_{\\ell=1}^n q_{i,t} p_{\\ell-1}(\\lambda_{i,t}) \n    dR_{j,t} q_{j,t} p_{\\ell-1}(\\lambda_{j,t}) \\\\\n  &=\n  -q_{i,t} \\sum_{j=1}^n dR_{j,t} q_{j,t}\n  + dR_{i,t}.\n\\end{align*}\nIn other words, the spectral weights evolve exactly according to the projection of $( dR_{i,t} )_{i=1}^n$ in the direction orthogonal to $(q_{i,t})_{i=1}^n.$  Note that\n\\[\n  \\sum_{i=1}^n q_{i,t}^2 E^{-}(\\lambda_{i,t},\\lambda_{i,t}) = 0,\n\\]\nand hence one may as well assume that $dR_{i,t}$ already had this orthogonality.\n\\end{proof}\n\n\\subsection{The half finite variation model}\n\nThe frozen spectral weight and smooth spectral weight models featured in some sense the tamest choices for the evolution of the spectral weights.  However, it is also possible to choose evolutions of the spectral weights that lead to substantially different tridiagonal evolutions.  For example, one might hope to choose a spectral weight evolution that cancels some of the rough parts of the tridiagonal.  Here, we show it is possible to choose these weights such that the first half of the tridiagonal model will be finite variation.\n\n\nIn effect, it suffices to show the following lemma.\n\\begin{lemma}\nThere is a choice of coefficients $\\{c^j_i\\}$ so that\n\\[\n  F(\\lambda_i) = \\sum_{j=1}^n c^j_i E^{-}(\\lambda_j,\\lambda_j) + T,\n\\]\nfor some $T$ that vanishes for entries $(k,\\ell)$ with $k+\\ell \\leq n.$\n  \\label{lem:Fsolve}\n\\end{lemma}\n\\begin{proof}\n  Suppose that $p$ is a polynomial of degree at most $n-1.$  Then\n  \\[\n    \\sum_{i=1}^n \\Delta_{\\lambda_i}^{\\lambda_j} p(\\lambda_i) p_{n-1}(\\lambda_i) q_i^2 = 0\n  \\]\n  by orthogonality.\n  It follows that on rearranging the sum\n  \\[\n    \\sum_{i \\neq j} \\Delta_{\\lambda_i}^{\\lambda_j} p(\\lambda_i) p_{n-1}(\\lambda_i) q_i^2\n    =\n    p'(\\lambda_j) p_{n-1}(\\lambda_j) q_j^2.\n  \\]\n  In particular, we have for $k+\\ell \\leq n,$ when $E^{-}_{k,\\ell}(x,x)$ has degree at most $n-1,$ \n  \\[\n    F_{k,\\ell}(\\lambda_i) \n    = -\\frac{1}{2}\\partial_{\\lambda_i} E^{-}(\\lambda_i,\\lambda_i)\n    = \\frac{-1}{2p_{n-1}(\\lambda_i)q_i^2}\\sum_{j \\neq i} \\Delta_{\\lambda_j}^{\\lambda_i}  E^{-}_{k,\\ell}(\\lambda_j,\\lambda_j) p_{n-1}(\\lambda_j) q_j^2.\n  \\]\n  Hence choosing $c^j_i$ to be these coefficients, the lemma follows.\n\\end{proof}\n\n\\begin{corollary}\n  There is a tridiagonal model $A_t$ for Dyson Brownian motion so that its entries $(k,\\ell)$ with $k+\\ell \\leq n$ are finite variation.\n\\end{corollary}\n\n\n\\section{Asymptotics}\n\\label{sec:asymptotics}\n\nIn this section we will prove the large--$n$ asymptotics given in Theorem \\ref{thm:asymptotics} of the bounded--order principal submatrices of the frozen spectral weight model with external potential $V(x)=2x^2$ run from the Dumitriu--Edelman distribution.   \nLet $\\MP{A}$ be this tridiagonal matrix process.\nLet $M$ be a fixed natural number, which will be the order of the principal submatrix that we consider.  Recall that the model is given by \\eqref{eq:frozen}, which after applying \\eqref{eq:AG} is given by\n\\begin{equation}\n  \\begin{aligned}\n    dA_t &= -2A_t\\,dt + \\sum_{i=1}^n -dt \\cdot \\biggl\\{\\sum_{j \\neq i}\\frac{1}{\\lambda_{i,t} - \\lambda_{j,t}} \\biggr\\} \\cdot q_i^2 \\cdot G(\\lambda_{i,t}) + dA_t' \\\\\n    dA_t' &= \\sum_{i=1}^n -\\sqrt{\\frac{2}{\\beta}} dZ_{i,t}\\cdot q_i^2 \\cdot G(\\lambda_{i,t})\n    + \\sum_{k \\geq \\ell} dP_{k,\\ell,t} \\cdot G^{k,\\ell} \\\\\n    dP_t &= \\sum_{i=1}^n -\\frac{q_i^4}{2} \\cdot [ -E^{+}(\\lambda_{i,t},\\lambda_{i,t}) + G(\\lambda_{i,t}), F(\\lambda_{i,t})] \\cdot dt\n  \\end{aligned}\n  \\label{eq:frozen2}\n\\end{equation}\nTo show the convergence statement we will show that the contribution from $A'$ will be negligible and the contributions from $dA_t-dA'_t$ will be derived using an observation about the orthogonal polynomials being approximately Chebyshev and convergence of Stieltjes transforms.\n\nThis section is organized in the following way: we first show the observation about the polynomials being approximately Chebyshev holds in a precise way. Second we show that the contribution of we prove convergence of the appropriate Stieltjes transforms. Third we show the contributions from the $dA'_t$ are negligible. Fourth we give the asypmtotics for the main terms in $dA_t- dA'_t$. Fifth we show the cancellation of the leading order terms. Lastly we collect everything to get the appropriate convergence statement.\n\n\n\n\\subsection{Convergence to Chebyshev polynomials}\n\nThe asymptotics are based off the observation that the orthogonal polynomials in this upper corner are nearly Chebyshev polynomials.\nDefine the semi--infinite tridiagonal matrix\n\\[\n  D\n  =\n  \\begin{bmatrix}\n    0 & \\frac{1}{2} &   &  &  & \\\\\n    \\frac{1}{2} & 0 & \\frac{1}{2} &  &  & \\\\\n      & \\frac{1}{2} & 0 & \\frac{1}{2}&  & \\\\\n      &   & \\frac{1}{2} &  &  &  & \\\\\n      &   &   &  &  \\ddots &  & \\\\\n  \\end{bmatrix}.\n\\]\nThe orthogonal polynomials with this Jacobi matrix are the Chebyshev polynomials of the second kind, $\\left\\{ U_k \\right\\}_{k=0}^{\\infty}.$\n\nDefine stopping times, for any $R > 0,$\n\\[\n  \\tau_R = \\inf\\left\\{ t \\geq 0 ~:~ \\max_{1 \\leq i,j \\leq M} |\\MP{A}{t}[i,j]-\\sqrt{n}D_{i,j}| > R \\right\\}.\n\\]\nFor time before $\\tau_R,$ we have quantitative control on the proximity of $\\MP{A}$ to $D,$ and this implies the closeness of the scaled orthogonal polynomials associated to $\\MP{A}$ to the Chebyshev polynomials of the second kind.\n\\begin{lemma}\n  Uniformly in $0 \\leq k \\leq M,$ $0 \\leq j < k,$ $t \\leq \\tau_R,$ and locally uniformly in $x\\in{\\mathbb C},$\n  \\begin{align*}\n    &\\lim_{n\\to\\infty} p_{k,t}(\\sqrt{n} x) = U_k(x), \\\\\n    &\\lim_{n\\to\\infty} a_{j+1,t}p^{(j)}_{k,t}(\\sqrt{n} x) = U_{k-j-1}(x), \\\\\n    &\\lim_{n\\to\\infty} \\sqrt{n} p_{k,t}'(\\sqrt{n} x) = U_k'(x).\n  \\end{align*}\n  \\label{lem:cheb}\n\\end{lemma}\n\\begin{proof}\n  The three-term recurrence for $q_{k,t}(x) = p_{k,t}(\\sqrt{n}x)$ is given by\n  \\begin{align*}\n    x q_{k,t}(x) \n    =\n    x p_{k,t}(\\sqrt{n} x)\n    &=\n    \\frac{a_{k+1,t}}{\\sqrt{n}} p_{k+1,t}(\\sqrt{n} x)\n    +\\frac{b_{k+1,t}}{\\sqrt{n}} p_{k,t}(\\sqrt{n} x)\n    +\\frac{a_{k,t}}{\\sqrt{n}} p_{k-1,t}(\\sqrt{n} x) \\\\\n    &=\n    \\frac{a_{k+1,t}}{\\sqrt{n}} q_{k+1,t}(x)\n    +\\frac{b_{k+1,t}}{\\sqrt{n}} q_{k,t}(x)\n    +\\frac{a_{k,t}}{\\sqrt{n}} q_{k-1,t}(x).\n  \\end{align*}\n  From the uniform convergence of $n^{-1\/2} A_t,$ the $3$-term recurrence converges to that of the Chebyshev polynomials.  Hence by induction on $k,$ the result follows. \n\\end{proof}\n\nWe will need to find a uniform left--tail bound for these stopping times in $n$ which improves as $R \\to \\infty.$ To do so, define stopping times, for $K >0,$ \n\\[\n  \\sigma_K = \\inf\\left\\{ t \\geq 0 ~:~ \\max_{1 \\leq i \\leq N} |\\lambda_{i,t}| > K\\sqrt{n} \\right\\}.\n\\]\n\\begin{lemma}\n  There is a $K >0$ so that for all $T >0,$\n  \\[\n    \\lim_{n \\to \\infty} \\Pr\\left[ \\sigma_K \\leq T \\right] = 0.\n  \\]\n  \\label{lem:sigmaK}\n\\end{lemma}\n\\begin{proof}\n  See \\cite[Theorem 5.1]{Unterberger}.\n\\end{proof}\n\n\\subsection{Stieltjes transforms and their limits}\n\nThe Stieltjes transform of a measure $\\mu$ on the real line is given by \n\\[\ns^\\mu(z) =\\int_\\mathbb{R} \\frac{1}{x-z}d\\mu.\n\\]\nWe can also define the Stieltjes transform of a matrix by taking the measure to be the associated spectral measure. We make use of both the unweighted and weighted spectral measures and so define the Stieltjes transforms of $A_t$ by\n\\[\n  s_t(z) = \\frac{1}{n}\\sum_{i=1}^n \\frac{1}{\\lambda_{i,t}\/\\sqrt{n}-z}\n  \\quad\n  \\text{and}\n  \\quad\n  s_t^A(z) = \\sum_{i=1}^n \\frac{q_i^2}{\\lambda_{i,t}\/\\sqrt{n}-z},\n\\]\nfor $z$ in the complement of the spectrum with respect to the complex plane.\nRecall that $\\mathfrak{s}$ denotes the semicircle density on $[-1,1],$\n\\[\n  \\mathfrak{s}(x) = \\frac{2}{\\pi}\\sqrt{1-x^2}\\,\\one[|x| \\leq 1],\n\\]\nand that we define for $z \\in {\\mathbb C} \\setminus [-1,1]$\n\\[\n  s^{\\mathfrak{s}}(z) =\n  \\int_{-1}^1 \\frac{\\mathfrak{s}(x)\\,dx}{x-z}\n  =\n  2(-z+\\sqrt{z^2-1})\n  .\n\\]\n\\begin{proposition}\n  \\label{prop:stieltjes}\n  Let $K,T > 0$ and let $F \\subset {\\mathbb C} \\setminus [-K,K]$ be compact.  Then \n  \\[\n    \\max_{0 \\leq t \\leq \\sigma_K \\wedge T} \\max_{z \\in F} |s_t(z) - s^{\\mathfrak{s}}(z)| \\to 0\n  \\]\n  pointwise. This is essentially a version of uniform convergence for $0 \\le t \\le T$ of $\\mu_{n,t} = \\frac{1}{n} \\sum_{i=1}^n \\delta_{\\lambda_i\/\\sqrt{n}}$ weakly to $\\mathfrak{s}$.\n\\end{proposition}\n\n\\begin{proof}\nThe first statement is a consequence of Theorem 1 in Rogers and Shi \\cite{RogersShi}. To get that this implies the uniform weak convergence first observe that the measures are compactly supported and so $\\int x^k d\\mu_{n,t} \\to \\int x^k d\\mathfrak{s}(x)$ by contour integration, which then implies the weak convergence.\n\\end{proof}\n\nLet $Q_t$ be a standard Brownian motion defined for all positive and negative times, and set\n\\[\n  s^{Q}(z) = \n  s^{\\mathfrak{s}}(z)c_\\beta \\int_{-1}^{1}1\\ dQ_x+ c_\\beta \\int_{-1}^1 \\frac{\\sqrt{\\mathfrak{s}(x)}dQ_x}{x-z},\n\\]\nwith $c_\\beta = \\sqrt{\\beta}.$\n\\begin{proposition}\n  \\label{prop:stieltjes2}\n  Let $K,T > 0$ and let $F \\subset {\\mathbb C} \\setminus [-K,K]$ be compact.  Then there is a probability space so that\n  \\[\n    \\max_{0 \\leq t \\leq \\sigma_K \\wedge T} \\max_{z \\in F} |\\sqrt{n}(s_t^A(z)-s_t(z)) - s^{Q}(z)| \\overset{\\Pr}{\\to} 0.\n  \\]\n\\end{proposition}\n\\begin{proof}\nWe make use of the martingale central limit theorem to show convergence directly. Recall that the the vector $(q_1^2,...,q_n^2)$ has Dirichlet$(\\frac{\\beta}{2},...,\\frac{\\beta}{2})$ distribution. We couple the $q_i$'s to an independent family of i.i.d. random variables $Y_1,Y_2,...$ in the following way. Let $Y_i \\sim$ Gamma$(\\frac{\\beta}{2},\\theta)$ then for $V_n = \\sum_{i=1}^n Y_i$ we define \n\\[\n\\left( \\frac{Y_1}{V_n},...,\\frac{Y_n}{V_n}\\right) = (q_1^2,...,q_n^2).\n\\]\nUsing this we can use the decomposition  $\\sqrt {n} (s_t^A(z)-s_t(z)) = X_n^{(t)}(z)+M_n^{(t)}(1,z)$ where \n \\[\n X^{(t)}_n(z) = \\sqrt n \\sum_{i=1}^{n} \\left( \\frac{Y_i}{\\theta\\frac{\\beta}{2}n} - q_i^2\\right) \\frac{1}{\\lambda_{i,t}\/\\sqrt n-z} = \\frac{V_n- \\theta\\frac{\\beta}{2}n}{\\theta \\frac{\\beta}{2}\\sqrt n} s_t^A. \n\\]\nand $M_n^{(t)}$ is a family of continuous time martingales defined by \n\\[\nM_n^{(t)}(s,z) = \\sqrt{n} \\sum_{i=1}^{\\lfloor sn\\rfloor} \\left( \\frac{1}{n} - \\frac{Y_i}{\\theta \\frac{\\beta}{2} n}\\right) \\frac{1}{\\lambda_{i,t}\/\\sqrt n-z}.\n\\]\n\nTo show convergence we begin by defining\n\\[\nS_n(s) =\\sqrt{\\frac{2}{\\beta}}\\sqrt{n} \\sum_{k=1}^{\\lfloor sn\\rfloor} \\left(\\frac{Y_i}{\\theta \\frac{\\beta}{2} n}- \\frac{1}{n}\\right).\n\\]\nA strong approximation theorem by Koml\\'os, Major, and Tusn\\'ady \\cite{KMT} gives us that there exists a Brownian motion $B(s)$ so that  $\\sup_{s\\in[0,T] }|S_n(s) - B(s)| \\overset{\\Pr}{\\to} 0.$ Let $Q(x) = B((x+1)\/2)$. Notice that $X_n^{(t)}$ is bounded an so $|s_t(z)-s_t^A(z)|\\to 0$ for all $t$. In particular we get that $\\sup_{0\\le t \\le T} \\sup_{z\\in F}| s_t^A(z)- s^{\\mathfrak{s}}(z)|\\to 0$ in probability. From this we get \n\\begin{equation}\n \\max_{0 \\leq t \\leq \\sigma_K \\wedge T} \\max_{z \\in F}\\left| X^{(t)}_n(z) - s^{\\mathfrak{s}}(z)c_\\beta \\int_{-1}^{1}1\\ dQ_x \\right| \\to 0 \\quad \\text{ in probability.}\n \\label{eq:Xlimit}\n\\end{equation}\nWe now turn to the martingale term. In order to prove convergence we will separate the $\\lambda_{i,t}$ from the main term. Let $\\gamma_i$ be the $i\/n$th quantile of $\\mathfrak{s}$. That is $\\gamma_i$ satisfies\n\\[\n\\int_{-1}^{\\gamma_i} \\mathfrak{s}(x)dx = \\frac{i}{n}.\n\\]\nNow observe that $M_n^{(t)}(s)$ may be rewritten as\n\\[\nM_n^{(t)}(s) = \\sqrt{n} \\sum_{i=1}^{\\lfloor sn\\rfloor} \\left( \\frac{1}{n} - \\frac{Y_i}{\\theta \\frac{\\beta}{2} n}\\right)  \\frac{1}{\\gamma_i-z}\n+\\sqrt{n} \\sum_{i=1}^{\\lfloor sn\\rfloor} \\left( \\frac{1}{n} - \\frac{Y_i}{\\theta \\frac{\\beta}{2} n}\\right)\\left(\\frac{1}{\\lambda_{i,t}\/\\sqrt n-z}-  \\frac{1}{\\gamma_i-z}\\right).\n\\]\nThe second moment of the second term, conditional on the eigenvalue evolution, then may be bounded by \n\\[\nn  \\max_{1\\le i \\le n}\\left\\{ \\left(\\frac{1}{\\lambda_{i,t}\/\\sqrt n-z}-  \\frac{1}{\\gamma_i-z}\\right)^2 \\right\\} \\sum_{i=1}^{\\lfloor sn \\rfloor} \\mathbb{E}\\left( \\frac{1}{n} - \\frac{Y_i}{\\theta \\frac{\\beta}{2} n}\\right)^2.\n\\]\nIf we can show that the maximum converges to $0$ in probability that will show that the second term converges to 0 in probability. Recall that $\\sup_{0\\le t \\le T} \\max_{1\\le i \\le n}|\\frac{1}{\\lambda_{i,t}\/\\sqrt n-z}-  \\frac{1}{\\gamma_i-z}| \\to 0$ in probability. This follows from convergence of the quantile functions which can be obtained from the weak convergence in Proposition \\ref{prop:stieltjes}. \nWe now show the first sum appearing in $M_n^{(t)}$ converges to a stochastic integral.\nWe make use of the semicircle convergence again to get that \n\\[\n  \\frac{1}{n}\n\\sum_{i=1}^{\\lfloor sn\\rfloor}  \\frac{1}{\\gamma_i-z} \\to \\int_{-1}^{2s -1} \\frac{ \\mathfrak{s}(x)}{x-z} dx.\n\\]\nThis convergence is uniform in $s$.  By theorem 4.6 in Kurtz and Protter \\cite{KP} we get that \n\\[ \n\\max_{0 \\leq t \\leq \\sigma_K \\wedge T} \\max_{z \\in F} \\left|M_n^{(t)} - c_\\beta \\int_{-1}^1 \\frac{\\sqrt{\\mathfrak{s}(x)}dQ_x}{x-z}\\right| \\overset{\\Pr}{\\to} 0.\n\\]\nThis together with (\\ref{eq:Xlimit}) completes the proof.\n\\end{proof}\n\nHaving shown that the Stieltjes transforms converge, we can show that stopping times $\\tau_R$ advance to infinity for large $n$ and large $R.$\n\\begin{proposition}\n  Let $T > 0,$ then \n  \\[\n    \\limsup_{R \\to \\infty} \\limsup_{n \\to \\infty} \\Pr\\left[ \\tau_R \\leq T \\right] = 0.\n  \\]\n  \\label{prop:tightness}\n\\end{proposition}\n\\begin{proof}\n  Let $K > 1$ be a constant as in Lemma~\\ref{lem:sigmaK}.  Let $\\gamma$ be a smooth contour winding once in the positive orientation around $[-K,K]$ in the complex plane. \n  Define the $k$-th moment\n  \\[\n    c_{k,t} \n    = \\sum_{i=1}^n q_i^2 (\\lambda_{i,t}\/n)^2 \n    = \\frac{1}{2\\pi i} \\oint_\\gamma s^A_t(z) z^k\\,dz.\n  \\]\n  From \\cite[(2.2.6)]{Szego}, it is possible to express for all $1 \\leq k \\leq n-1$\n  \\[\n    p_k(\\sqrt{n} x) = \\frac{1}{\\sqrt{D_kD_{k-1}}}\n    \\left|\n    \\begin{matrix}\n      c_0 & c_1 & c_2 & \\dots & c_k          \\\\\n      c_1 & c_2 & c_3 & \\dots & c_{k+1}      \\\\\n      \\dots & \\dots & \\dots & \\dots & \\dots  \\\\\n      c_{k-1} & c_k & c_{k+1} & \\dots & c_{2k-1} \\\\\n      1 & x & x^2 & \\dots & x^{k} \\\\\n    \\end{matrix}\n    \\right|,\n  \\]\n  where $D_k$ is the Hankel determinant $\\det[c_{i+j}]_{i,j=0,\\dots, k}.$\n  By Proposition~\\ref{prop:stieltjes}, each coefficient of $p_k(\\sqrt{n}x)$ converges with rate $1\/\\sqrt{n}$ to the corresponding coefficient of $U_k(x),$ the orthogonal polynomials associated to the semicircle, uniformly for time up to $T \\wedge \\sigma_K.$  That is to say, for any natural numbers $i,j,$\n  \\[\n    \\max_{0 \\leq t \\leq T \\wedge \\sigma_K} |\\MP{A}{t}[i,j]\/\\sqrt{n} - D_{i,j}|\/\\sqrt{n} \n  \\]\nis tight as a family of variables in $n.$  Hence \n\\[\n    \\limsup_{R \\to \\infty} \\limsup_{n \\to \\infty} \\Pr\\left[ \\tau_R \\leq T \\right] \n    \\leq \n    \\limsup_{n \\to \\infty}\n    \\Pr\\left[ \\sigma_K \\leq T \\right],\n\\]\nwhich is equal to $0$ by Lemma~\\ref{lem:sigmaK}.\n\\end{proof}\n\n\\subsection{Negligible terms}\nRecall the decomposition given in (\\ref{eq:frozen2}). In this section we show that the $A'_t$ terms will be negligible. Recall\n\\begin{equation*}\n  \\begin{aligned}\n    dA_t' &= \\sum_{i=1}^n -\\sqrt{\\frac{2}{\\beta}} dZ_{i,t}\\cdot q_i^2 \\cdot G(\\lambda_i)\n    + \\sum_{k \\geq \\ell} dP_{k,\\ell,t} \\cdot G^{k,\\ell} \\\\\n    dP_t &= \\sum_{i=1}^n -\\frac{q_i^4}{2} \\cdot [ -E^{+}(\\lambda_i,\\lambda_i) + G(\\lambda_i), F(\\lambda_i)] \\cdot dt\n  \\end{aligned}\n\\end{equation*}\nHere we will also need that the spectral weights are relatively flat.  \n\\begin{lemma}\n  There is a constant $C_\\beta > 0$ so that with probability going to $1$\n  \\[\n    \\max_{1 \\leq i \\leq n} q_i^2 \\leq \\frac{C_\\beta \\log n}{n},\n  \\]\n  as $n \\to \\infty.$\n  \\label{lem:qibound}\n\\end{lemma}\n\n\\begin{proposition}\n  For any $T,R,K > 0$ as $n\\to\\infty,$\n  \\[\n    \\max_{1\\leq i,j \\leq M}\n    \\biggl|\n    \\int_0^{T \\wedge \\tau_R \\wedge \\sigma_K}\n    dA_{i,j,t}'\n    \\biggr|\n    \\overset{\\Pr}{\\to}\n    0.\n  \\]\n  \\label{prop:dAt}\n\\end{proposition}\n\\begin{proof}\n  Let $\\zeta$ be the stopping time $T \\wedge \\tau_R \\wedge \\sigma_K.$\n  We consider the local martingale and finite variation portions separately.  Recall that entries of $G$ are given by\n  \\begin{equation}\n    G_{k,\\ell}(\\lambda)\n  =\\begin{cases}\n    a_{\\ell-1} p_{\\ell-2}(\\lambda)p_{\\ell-1}'(\\lambda) - a_\\ell p_{\\ell-1}(\\lambda) p_{\\ell}'(\\lambda)&  \\\\\n    +a_{\\ell-1} p_{\\ell-1}(\\lambda)p_{\\ell-2}'(\\lambda) - a_\\ell p_{\\ell}(\\lambda) p_{\\ell-1}'(\\lambda),  & \\text{ if } k = \\ell , \\\\\n    a_\\ell (p_{\\ell-1}(\\lambda)p_{\\ell-1}'(\\lambda) - p_\\ell(\\lambda)p_\\ell'(\\lambda)), & \\text{ if } k = \\ell + 1 ,\\\\\n    0, & \\text{ otherwise.}\n  \\end{cases}\n  \\label{eq:GG}\n\\end{equation}\nHence in light of Lemma~\\ref{lem:cheb}, we have that\n\\[\n  G^* = \\max_{1 \\leq i \\leq n} \\max_{0 \\leq t \\leq \\zeta} \\max_{1\\leq k,\\ell \\leq M}\n  |G_{k,\\ell}(\\lambda_{i,t})|\n\\]\nis tight as a family of random variables in $n.$ Employing this bound, the quadratic variation of the local martingale is dominated by \n\\[\n  \\biggl\\langle\n  \\sum_{i=1}^n -\\sqrt{\\frac{2}{\\beta}} dZ_{i,t}\\cdot q_i^2 \\cdot G_{k,\\ell}(\\lambda_i)\n\\biggr\\rangle\n\\leq \n\\sum_{i=1}^n \\frac{2}{\\beta}q_i^4 (G^*)^2\\,dt.\n\\]\nBy Lemma~\\ref{lem:qibound}, it follows that\n\\[\n    \\max_{1\\leq k,\\ell \\leq M}\n    \\biggl|\n    \\int_0^{\\zeta}\n    \\sum_{i=1}^n dZ_{i,t}\\cdot q_i^2 \\cdot G_{k,\\ell}(\\lambda_i)\n    \\biggr|\n    \\overset{\\Pr}{\\to}\n    0.\n\\]\n\nAs for the finite variation terms, we consider each of the four classes of terms broken out in \\eqref{eq:FVterms}.  The estimates for $dS_t,$ $dR^1_t, dR^2_t,$ and $dR^3_t$ are similar\nand so we explain the estimates for $dR^1_t$ and do not discuss the others.\n\nRecall that\n\\(\ndR^1_t=\\sum_{i,k,\\ell=1}^n -\\frac{q_i^4}{2} \\mathcal{H}_{k,\\ell} \\cdot G^{k,\\ell} \\cdot dt,\n\\)\nwhere $\\mathcal{H}_{k,\\ell}$ is given in \\eqref{eq:GF} and $G^{k,\\ell}$ is given in Corollary~\\ref{cor:entry}.  The entries $G^{k,\\ell}_{u,r}$ vanish when $u+r < k+\\ell.$  In particular, this implies that for considering $dR^{1}_{u,r,t}$ for $1 \\leq u,r \\leq M$ it suffices to estimate $k,\\ell$ with $1 \\leq k,\\ell \\leq 2M.$  Using Remark~\\ref{rem:Gkl} and Lemma~\\ref{lem:cheb}, we have that\n\\[\n  G^{**}=\n  \\max_{0 \\leq t \\leq \\zeta} \n  \\max_{1 \\leq k,\\ell \\leq 2M} \n  \\max_{1 \\leq u,r \\leq M} \n  |G^{k,\\ell}_{u,r,t}|\n\\]\nis tight as a family of random variables in $n$, where we have used that $\\sum_{i=1}^n q_i^2=1.$ In a similar way,\n\\[\n  H^*\n  =\n  \\max_{1 \\leq i \\leq n}\n  \\max_{0 \\leq t \\leq \\zeta} \n  \\max_{1 \\leq k,\\ell \\leq 2M} \n  |\\mathcal{H}_{k,\\ell,t}(\\lambda_{i,t})|\\sqrt{n}\n\\]\nis tight.  Hence, \n\\[\n  \\biggl| \\int_0^\\zeta dR^1_t \\biggr| \\leq \n  \\zeta \\cdot \n  (\\max_{1 \\leq i \\leq n} q_i^4) \\cdot\n  \\sqrt{n} \\cdot H^* \\cdot G^{**} \\overset{\\Pr}{\\to} 0,\n\\]\nby Lemma~\\ref{lem:qibound}.\n\n\n\\end{proof}\n\n\\subsection{Principal terms}\nThe remaining terms will produce a nonvanishing contribution to the limit in the regime we consider.  Recalling \\eqref{eq:frozen2}, these remaining terms are given by\n\\[\n  dA_t - dA_t' = -2A_t\\,dt + \\sum_{i=1}^n -dt \\cdot \\biggl\\{\\sum_{j \\neq i}\\frac{1}{\\lambda_i - \\lambda_j} \\biggr\\} \\cdot q_i^2 \\cdot G(\\lambda_i).\n\\]\nThere are large cancellations between the $2A_t$ term and the sum.  Both are order $\\sqrt{n},$ but their leading order behavior cancels.  \n\nTo identify an exact leading term, we make a connection to the minor polynomials.\nRecall from \\eqref{eq:G} that $G(\\lambda) = -\\partial_\\lambda B(x,\\lambda) \\vert_{x=\\lambda}.$ By orthogonality, we have that \n\\[\n  \\sum_{i=1}^n q_i^2\\biggl\\{\\partial_{\\lambda \\lambda} B(\\lambda_i,\\lambda)\\vert_{\\lambda=\\lambda_i}  \\biggr\\}=0,\n\\]\non account of the orthogonality of $p_{k}$ to lower degree polynomials.  Hence, we have\n\\[\n  \\sum_{i=1}^n q_i^2\\biggl\\{\\sum_{j \\neq i} \\Delta_{\\lambda_j}^{\\lambda_i} \\partial_{\\lambda_j} B(\\lambda_i,\\lambda_j)  \\biggr\\}\n  =\n  \\sum_{j=1}^n \n  \\sum_{i=1}^n \n  q_i^2\\biggl\\{\\Delta_{\\lambda_j}^{\\lambda_i} \\partial_{\\lambda_j} B(\\lambda_i,\\lambda_j)  \\biggr\\}\n  =0,\n\\]\nwhere we have extended the formula to $i = j$ by continuity, and the second equality follows as the inner sum is identically $0,$ again by degree considerations.  This formula allows us to write\n\\[\n\\sum_{i=1}^n \\biggl\\{\\sum_{j \\neq i}\\frac{1}{\\lambda_i - \\lambda_j} \\biggr\\} \\cdot q_i^2 \\cdot G(\\lambda_i) \n =\\frac{-1}{2}\n\\sum_{i=1}^n \\biggl\\{\\sum_{j \\neq i}\\frac{ \n  \\partial_\\lambda B(\\lambda_i,\\lambda) \\vert_{\\lambda=\\lambda_i}\n  +\\partial_\\lambda B(\\lambda_i,\\lambda) \\vert_{\\lambda=\\lambda_j}\n}{\\lambda_i - \\lambda_j} \\biggr\\} \\cdot q_i^2 \n\\]\nSo, adding and subtracting $\\partial_\\lambda B(\\lambda_j,\\lambda) \\vert_{\\lambda=\\lambda_j},$ we can write\n\\begin{equation}\n \\sum_{i=1}^n \\biggl\\{\\sum_{j \\neq i}\\frac{1}{\\lambda_i - \\lambda_j} \\biggr\\} \\cdot q_i^2 \\cdot G(\\lambda_i) \n =\\frac{-1}{2}\n \\sum_{i,j:i\\neq j}^n\n \\Delta_{\\lambda_i}^{\\lambda_j} \\partial_{\\lambda_j} B(\\lambda_i,\\lambda_j) \\cdot q_i^2\n +\\frac{1}{2}\n \\sum_{i,j:i\\neq j}^n  \\frac{G(\\lambda_j)+G(\\lambda_i)}{\\lambda_i-\\lambda_j} \\cdot q_i^2\n \\label{eq:S2}\n\\end{equation}\nRewriting the model, we have\n\\begin{equation}\n  \\begin{aligned}\n  dA_t - dA_t' &= -2A_t\\,dt + \\frac{n}{2}\n \\sum_{i,j=1}^n\n \\Delta_{\\lambda_i}^{\\lambda_j} \\partial_{\\lambda_j} B(\\lambda_i,\\lambda_j) \\cdot q_i^2 \\cdot q_j^2 \\\\\n &- \\frac{n}{2}\n \\sum_{i,j=1}^n\n \\Delta_{\\lambda_i}^{\\lambda_j} \\partial_{\\lambda_j} B(\\lambda_i,\\lambda_j) \\cdot q_i^2 \\cdot \\biggl\\{ q_j^2 - \\frac{1}{n} \\biggr\\} \\\\\n &+\\frac{1}{2}\n \\sum_{i,j:i\\neq j}^n  \\frac{G(\\lambda_j)+G(\\lambda_i)}{\\lambda_i-\\lambda_j} \\cdot q_i^2 \\\\\n &-\n \\frac{1}{2}\n \\sum_{i=1}^n\n \\partial_{\\lambda_i\\lambda_j} B(\\lambda_i,\\lambda_j) \\vert_{\\lambda_j=\\lambda_i} \\cdot q_i^2 \\\\\n \\label{eq:S3}\n \\end{aligned}\n\\end{equation}\nThe first line can be expressed in terms of the minor polynomials.  Specifically, we define \n\\(\nG^{(0)}(\\lambda)\n=\n-\n \\sum_{i=1}^n\n \\Delta_{\\lambda_i}^{\\lambda} \\partial_{\\lambda} B(\\lambda_i,\\lambda) \\cdot q_i^2,\n\\)\nin terms of which\n\\begin{equation}\n  G_{k,\\ell}^{(0)}(\\lambda)\n  =-\\begin{cases}\n    -a_{\\ell-1} p^{(0)}_{\\ell-2}(\\lambda)p_{\\ell-1}'(\\lambda) + a_\\ell p^{(0)}_{\\ell-1}(\\lambda) p_{\\ell}'(\\lambda)&  \\\\\n    -a_{\\ell-1} p^{(0)}_{\\ell-1}(\\lambda)p_{\\ell-2}'(\\lambda) + a_\\ell p^{(0)}_{\\ell}(\\lambda) p_{\\ell-1}'(\\lambda),  & \\text{ if } k = \\ell , \\\\\n    -a_\\ell (p^{(0)}_{\\ell-1}(\\lambda)p_{\\ell-1}'(\\lambda) - p^{(0)}_\\ell(\\lambda)p_\\ell'(\\lambda)), & \\text{ if } k = \\ell + 1 ,\\\\\n    0, & \\text{ otherwise.}\n  \\end{cases}\n \\label{eq:BB}\n\\end{equation}\nWe will make a comparison between the minor polynomials and the original polynomials to expose the leading order behavior (c.f.\\,\\eqref{eq:MinorPerturb}).  The last line is an absolute constant multiple of $dR^3,$ which is negligible in the limit (c.f.\\,Lemma~\\ref{lem:R3}).  \n\nAs for the middle two lines, they can be expressed in terms of the fluctuations of the spectral weights:  \n\\begin{lemma}\n  Let $\\gamma$ be a smooth contour enclosing all the eigenvalues once in the positive orientation.\n  Then\n   \\[\n    \\begin{aligned}\n -\\sum_{i,j=1}^n\n \\Delta_{\\lambda_i}^{\\lambda_j} \\partial_{\\lambda_j} B(\\lambda_i,\\lambda_j) \\cdot q_i^2 \\cdot \\biggl\\{ q_j^2 - \\frac{1}{n} \\biggr\\} \n &= \\frac{1}{2\\pi i}\\oint (s^A_t(z) - s_t(z))G^{(0)}(\\sqrt{n} z)\\,dz \\\\ \n    \\end{aligned}\n  \\]\n  and\n  \\[\n    \\begin{aligned}\n    \\sum_{i,j:i\\neq j}^n  \\frac{G(\\lambda_j)+G(\\lambda_i)}{\\lambda_i-\\lambda_j} \\cdot \n    q_i^2\n   \n    &=\\frac{-\\sqrt{n}}{(2\\pi i)^2}\\oint\\oint  (s^A_t(z) - s_t(z))s_t(y) \\frac{G(\\sqrt{n} z)-G(\\sqrt{n} y)}{z-y}\\,dzdy \\\\\n    &+\\frac{2\\sqrt{n}}{2\\pi i}\\oint (s^A_t(z) - s_t(z))s_t(z) G(\\sqrt{n} z)\\,dz,\n    \\end{aligned}\n  \\]\n  where the contour integrals are over $\\gamma.$\n  \\label{lem:second}\n\\end{lemma}\n\\begin{proof}\n  By antisymmetry, we have that\n  \\[\n    \\sum_{i,j:i\\neq j}^n  \\frac{G(\\lambda_j)+G(\\lambda_i)}{\\lambda_i-\\lambda_j} \\cdot q_i^2\n    =\n    \\sum_{i,j:i\\neq j}^n  \\frac{G(\\lambda_j)+G(\\lambda_i)}{\\lambda_i-\\lambda_j} \\cdot \n    \\biggl(q_i^2-\\frac{1}{n}\\biggr).\n  \\]\n  This we now break into two parts\n  \\[\n    \\sum_{i,j:i\\neq j}^n  \\frac{G(\\lambda_j)+G(\\lambda_i)}{\\lambda_i-\\lambda_j} \\cdot \n    \\biggl(q_i^2-\\frac{1}{n}\\biggr)\n    =\n    -\\sum_{i,j:i\\neq j}^n  \\frac{G(\\lambda_i)-G(\\lambda_j)}{\\lambda_i-\\lambda_j} \\cdot \n    \\biggl(q_i^2-\\frac{1}{n}\\biggr)\n    +\\sum_{i,j:i\\neq j}^n  \\frac{2G(\\lambda_i)}{\\lambda_i-\\lambda_j} \\cdot \n    \\biggl(q_i^2-\\frac{1}{n}\\biggr)\n  \\]\n  For entire functions $f,$ by a residue computation\n  \\begin{equation}\n    \\label{eq:residues}\n    \\begin{aligned}\n      \\frac{1}{2\\pi i} \\oint (s^A_t(z) - s_t(z))f(z)\\,dz\n      =\n      &\\sum_{i=1}^n \\biggl\\{q_i^2-\\frac{1}{n}\\biggr\\}f(\\lambda_{i,t}\/\\sqrt{n}), \\text{ and } \\\\\n      \\frac{1}{2\\pi i} \\oint (s^A_t(z) - s_t(z)) s_t(z) f(z)\\,dz\n      =\n      &\\frac{1}{\\sqrt{n}}\n      \\sum_{i=1}^n \\biggl\\{q_i^2-\\frac{1}{n}\\biggr\\}\\cdot \\biggl\\{\\sum_{j \\neq i}\\frac{f(\\lambda_{i,t}\/\\sqrt{n})}{\\lambda_{i,t}-\\lambda_{j,t}}\\biggr\\}\n      \\\\\n      +&\\frac{1}{2n}\n      \\sum_{i=1}^n \\biggl\\{q_i^2-\\frac{1}{n}\\biggr\\}f'(\\lambda_{i,t}\/\\sqrt{n}),\n    \\end{aligned}\n  \\end{equation}\n  for a contour enclosing all the eigenvalues once in the positive orientation. Hence, appropriately accounting for $G'$ terms, we get the desired formula.\n\\end{proof}\n\n\\subsection{Cancellation at leading order}\n\nWe consider the leading order behavior of the sum\n\\[\n \\sum_{i,j=1}^n\n \\Delta_{\\lambda_i}^{\\lambda_j} \\partial_{\\lambda_j} B(\\lambda_i,\\lambda_j) \\cdot q_i^2\\cdot q_j^2.\n\\]\nOn account of \\eqref{eq:BB}, we are led to consider the following asymptotics. \n\\begin{lemma}\n\n  \\begin{align*}\n    &\\sum_{j=1}^n p_k^{(0)}(\\lambda_j) p_{k}'(\\lambda_j)\\cdot q_j^2 = \n    \\frac{k}{a_1 a_k} \n    + n^{-3\/2}\\sum_{j=1}^{k-1} 16 (a_j-\\tfrac{\\sqrt{n}}{2}) (k-2j) \n    + n^{-3\/2} 8k(a_1-a_k)\n    + o(n^{-3\/2}),\n    \\\\\n  &\\sum_{j=1}^n p_k^{(0)}(\\lambda_j) p_{k+1}'(\\lambda_j)\\cdot q_j^2 \n  =  \\frac{(b_1+\\cdots + b_k-kb_{k+1}) }{a_1a_k a_{k+1}}\n    + n^{-3\/2}\\sum_{j=1}^{k} 8b_j(k+1-2j)\n  +o(n^{-3\/2}),\\\\\n    &\\sum_{j=1}^n p_k^{(0)}(\\lambda_j) p_{k-1}'(\\lambda_j)\\cdot q_j^2\n    = n^{-3\/2}\\sum_{j=1}^{k} 8b_j(k+1-2j)\n    + o(n^{-3\/2}),\n  \\end{align*}\n  with the errors uniform in $0 \\leq t \\leq T \\wedge \\sigma_K \\wedge \\tau_R.$ \n  \\label{lem:mpeval}\n\\end{lemma}\n\\begin{proof}\n  We begin with the first asymptotic.\n  Equation \\eqref{eq:MinorPerturb} states that to leading order,\n  \\[\n    p_{k}^{(0)}(x) = \\frac{p_{k-1}{(x)}}{a_1}.\n  \\]\n  Hence, for the first identity, we have\n  \\begin{align*}\n    \\sum_{j=1}^n \n    \\frac{p_{k-1}{(\\lambda_j)}}{a_1}\n    p_k'(\\lambda_j)\n    \\cdot q_j^2\n    &=\n    \\frac{k }{a_1a_k}. \n  \\end{align*}\n\n  On the other hand, we also have the subleading behavior of $p_{k}^{(0)}$ given by\n  \\begin{equation*}\n  \\begin{aligned}\n  p^{(0)}_{k}(x) - \\frac{1}{a_1}p_{k-1}(x)\n  &=\n  \\frac{a_{k-1}-a_k}{a_{k}a_1}p_{k-1}(x) \\\\\n  &+\n  \\frac{a_{k-1}}{a_{k}}\\sum_{j=1}^{k-1} (b_{j} - b_{j+1})p^{(j-1)}_{k-1}(x)p_{j}^{(0)}(x) \n  \\\\\n  &+\n  \\frac{a_{k-1}}{a_{k}}\\sum_{j=1}^{k-2} (a_{j} - a_{j+1})\n  (\n  p^{(j)}_{k-1}(x)p_{j}^{(0)}(x) \n  +p^{(j-1)}_{k-1}(x)p_{j+1}^{(0)}(x) \n  ). \n\\end{aligned}\n\\end{equation*}\nHence, on scaling up by $n,$ we get locally uniformly\n  \\begin{equation}\n  \\label{eq:perturbationlimit}\n  \\begin{aligned}\n    \\frac{n}{4}(p^{(0)}_{k}(x) - \\frac{1}{a_1}p_{k-1}(x))\n  &\\to\n  (a_{k-1}-a_k)U_{k-1}(x) \\\\\n  &+\n  \\sum_{j=1}^{k-1} (b_{j} - b_{j+1})U_{k-j-1}(x)U_{j-1}(x)\n  \\\\\n  &+\n  \\sum_{j=1}^{k-2} (a_{j} - a_{j+1})\n  (\n  U_{k-j-2}(x)U_{j-1}(x)\n  +U_{k-j-1}(x)U_{j}(x)\n  ). \n\\end{aligned}\n\\end{equation}\nWe then need to evaluate the integral of this asymptotic against $U_k'(x)\\mathfrak{s}(x)\\,dx,$ for which purpose, we will need the following Chebyshev polynomial identities\n\\begin{align}\n  U_k'(x) &= \\sum_{j=0}^{k\/2} 2(k-2j) U_{k-2j-1}(x) \\label{eq:dU} \\\\\n  U_k(x)U_\\ell(x) &= \\sum_{j=0}^{\\ell} U_{k-\\ell+2j}(x), \\text{ if } k \\geq \\ell. \\label{eq:UU}\n\\end{align}\nBy parity considerations, some terms cancel.  For any $1 \\leq j \\leq k-1,$ $U_{k-j-1}(x)U_{j-1}(x)$ is a sum over $U_{p}(x)$ where $p$ has the same parity as $k.$  In particular, it follows that\n\\[\n  \\int_{-1}^{1} U_k'(x)\n  \\left\\{\n  \\sum_{j=1}^{k-1} (b_{j} - b_{j+1})U_{k-j-1}(x)U_{j-1}(x)\n\\right\\}\n  \\mathfrak{s}(x)\\,dx\n  =0.\n\\]\n\n\nOn the other hand, for $1 \\leq j \\leq k-1$\n\\begin{align*}\n  &\\int_{-1}^{1} U_k'(x)\n  (\n  U_{k-j-2}(x)U_{j-1}(x)\n  +U_{k-j-1}(x)U_{j}(x)\n  )\n  \\mathfrak{s}(x)\\,dx \\\\\n  &= \n  \\sum_{\\ell=1}^{ j \\wedge (k-j-1) } 2(k-2\\ell)\n  +\\sum_{\\ell=0}^{ j \\wedge (k-j-1) } 2(k-2\\ell).\n\\end{align*}\nDefine the array, for $1 \\leq j < k-1$\n  \\[\n    H_{k,j} = \\sum_{\\ell=1}^{j \\wedge (k-j-1)} 2(k-2\\ell),\n  \\]\n  and $H_{k,j} = 0$ for $j \\geq k-1.$\n  Then we can express\n\\[\n  n^{3\/2}\\sum_{j=1}^n p_k'(\\lambda_j)(p^{(0)}_{k}(\\lambda_j) - \\frac{1}{a_1}p_{k-1}(\\lambda_j))q_j^2 \\to \n  \\sum_{j=1}^{k-1} 4(a_j - a_{j+1})(2H_{k,j}+2k).\n\\]\n\nFor the second term we begin with the observation that for any $k \\in {\\mathbb N}$\n\\begin{equation}\n\\label{eq:pkcoeffiecients}\np_k(x) = \\frac{1}{a_1\\cdots a_k} x^k - \\frac{b_1+\\cdots +b_k}{a_1\\cdots a_k}x^{k-1} + ...\n\\end{equation}\nTherefore,\n\\[\n  p_{k+1}'(x) - \\frac{(k+1)p_k(x)}{a_{k+1}}\n  = \n  -k\\frac{b_1+\\cdots +b_{k+1}}{a_1\\cdots a_{k+1}}x^{k-1}\n  +(k+1)\\frac{b_1+\\cdots +b_{k}}{a_1\\cdots a_{k+1}}x^{k-1}\n  +O(x^{k-2})\n\\]\nHence,\n\\begin{align*}\n  n^{3\/2}\n\\sum_{j=1}^n \n    \\frac{p_{k-1}{(\\lambda_j)}}{a_1}\n    p_{k+1}'(\\lambda_j)\\cdot q_j^2\n    =\n    \\frac{n^{3\/2}(b_1+\\cdots + b_k-kb_{k+1}) }{a_1a_k a_{k+1}}.\n  \\end{align*}\nWe again use equation \\eqref{eq:perturbationlimit} but observe that the parity is the opposite of the previous case which leaves us with the terms of the form\n  \\[\n    \\begin{aligned}\n      \\frac{n^{3\/2}}{4}\n    &\\sum_{j=1}^n \n    \\biggl(p^{(0)}_{k}(\\lambda_i) - \\frac{p_{k-1}(\\lambda_i)}{a_1}\\biggr)\n    p_{k+1}'(\\lambda_j)\\cdot q_j^2 \\\\\n    &\\to\n      \\int_{-1}^{1} U_{k+1}'(x)\n  \\left\\{\n  \\sum_{j=1}^{k-1} (b_{j} - b_{j+1})U_{k-j-1}(x)U_{j-1}(x)\n\\right\\}\n  \\mathfrak{s}(x)\\,dx \\\\\n  &=\\sum_{j=1}^{k-1}(b_j-b_{j+1})\\sum_{m=1}^{(k-j)\\wedge(j)}2(k+1-2m).\n    \\end{aligned}\n  \\]\n\nFor the final term, we have by degree considerations that\n\\[\n  n^{3\/2}\n\\sum_{j=1}^n \n    \\frac{p_{k-1}{(\\lambda_j)}}{a_1}\n    p_{k-1}'(\\lambda_j)\\cdot q_j^2\n    =\n    0.\n\\]\nAnd we have that\n  \\[\n    \\begin{aligned}\n      \\frac{n^{3\/2}}{4}\n    &\\sum_{j=1}^n \n    \\biggl(p^{(0)}_{k}(\\lambda_i) - \\frac{p_{k-1}(\\lambda_i)}{a_1}\\biggr)\n    p_{k-1}'(\\lambda_j)\\cdot q_j^2 \\\\\n    &\\to\n      \\int_{-1}^{1} U_{k-1}'(x)\n  \\left\\{\n  \\sum_{j=1}^{k-1} (b_{j} - b_{j+1})U_{k-j-1}(x)U_{j-1}(x)\n\\right\\}\n  \\mathfrak{s}(x)\\,dx \\\\\n  &=\\sum_{j=1}^{k-1}(b_j-b_{j+1})\\sum_{m=1}^{(k-j)\\wedge(j)}2(k+1-2m),\n    \\end{aligned}\n  \\]\n  where we observe the limit agrees with that of the previous case.\n  \n  Lastly we use the identity\n \\begin{equation}\n \\label{eq:Hdiff}\n H_{k,j}-H_{k,j-1} = 2(k-2j)\n \\end{equation}\n for $j<k$ to simplify the solution.\n\\end{proof}\n\n\\begin{corollary}\n  Define the operator $\\mathcal{F}$ on Jacobi matrices by\n  \\[\n    \\mathcal{F}(A)_{k,\\ell}\n    =\n    \\begin{cases}\n      -b_\\ell(4\\ell-2)+\n       4\\sum_{j=1}^{\\ell-1} b_j & \\text{ if } k = \\ell , \\\\\n      -2 (\\ell+1)a_\\ell- 2\\ell a_{\\ell-1} \n      + 4\\sum_{j=1}^{\\ell-2} a_j \n      & \\text{ if } k = \\ell + 1 ,\\\\\n    0, & \\text{ otherwise.}\n    \\end{cases}\n  \\]\n  Uniformly in $1 \\leq k,\\ell \\leq M$ and $0 \\leq t \\leq T \\wedge \\sigma_K \\wedge \\tau_R.$\n  \\begin{equation}\n    -2A_t +\\frac{n}{2}\n    \\sum_{i,j=1}^n\n    \\Delta_{\\lambda_i}^{\\lambda_j} \\partial_{\\lambda_j} B_{k,\\ell}(\\lambda_i,\\lambda_j) \\cdot q_i^2 \\cdot q_j^2 \n    -\\mathcal{F}(A_t - D\\sqrt{n})_{k,\\ell} \n     \\to 0\n  \\end{equation}\n  \\label{cor:lead}\n\\end{corollary}\n\\begin{proof}\n  We arrive at these asymptotics substituting the asymptotics of Lemma~\\ref{lem:mpeval} into \\eqref{eq:BB}. \n\\end{proof}\n\n\\subsection{Convergence of the model}\n\nRecalling \\eqref{eq:S3} for convenience\n\\begin{equation*}\n  \\begin{aligned}\n  dA_t - dA_t' &= -2A_t\\,dt + \\frac{n}{2}\n \\sum_{i,j=1}^n\n \\Delta_{\\lambda_i}^{\\lambda_j} \\partial_{\\lambda_j} B(\\lambda_i,\\lambda_j) \\cdot q_i^2 \\cdot q_j^2 \\\\\n &- \\frac{n}{2}\n \\sum_{i,j=1}^n\n \\Delta_{\\lambda_i}^{\\lambda_j} \\partial_{\\lambda_j} B(\\lambda_i,\\lambda_j) \\cdot q_i^2 \\cdot \\biggl\\{ q_j^2 - \\frac{1}{n} \\biggr\\} \\\\\n &+\\frac{1}{2}\n \\sum_{i,j:i\\neq j}^n  \\frac{G(\\lambda_j)+G(\\lambda_i)}{\\lambda_i-\\lambda_j} \\cdot q_i^2 \\\\\n &-\n \\frac{1}{2}\n \\sum_{i=1}^n\n \\partial_{\\lambda_i\\lambda_j} B(\\lambda_i,\\lambda_j) \\vert_{\\lambda_j=\\lambda_i} \\cdot q_i^2 \\\\\n \\end{aligned}\n\\end{equation*}\nwe have shown $dA'_t$ is negligible in the limit (Proposition~\\ref{prop:dAt}).  The final line of this equaltity is negligible by the same argument, after identifying it as a multiple of $dR^3_t$ for the entries we consider.  In Corollary \\ref{cor:lead} we have identified the asymptotics of the first line, showing it is essentially a linear function of the entries of $A_t-D\\sqrt{n}.$  Finally, in Lemma~\\ref{lem:second} we have represented the middle two lines as contour integrals against $s_t^A(z) -s_t(z)$ and $s_t(z).$  By the convergenece of these Stieltjes transforms (Proposition~\\ref{prop:stieltjes}) and the various polynomials (Lemma~\\ref{lem:cheb}), the contour integrals in Lemma~\\ref{lem:second} converge to time--independent contour integrals.  All that remains to complete the proof is identify these limiting contour integrals.\nIn summary, we have that\n\\begin{equation}\n  \\begin{aligned}\n    dA_t &= \\mathcal{F}(A_t - D\\sqrt{n}) \\\\\n  &+\\frac{n}{4\\pi i}\\oint (s^A_t(z) - s_t(z))G^{(0)}(\\sqrt{n} z)\\,dz \\\\ \n  &+\\frac{-\\sqrt{n}}{2(2\\pi i)^2}\\oint\\oint  (s^A_t(z) - s_t(z))s_t(y) \\frac{G(\\sqrt{n} z)-G(\\sqrt{n} y)}{z-y}\\,dzdy \\\\\n  &+\\frac{\\sqrt{n}}{2\\pi i}\\oint (s^A_t(z) - s_t(z))s_t(z) G(\\sqrt{n} z)\\,dz + o(1)\n  \\label{eq:S4}\n\\end{aligned}\n\\end{equation}\nwith the error vanishing uniformly over bounded order principal minors and uniformly in $0 \\leq t \\leq T \\wedge \\tau_R \\wedge \\sigma_K.$ \n\nDefine the semi--infinite Jacobi matrix\n\\begin{equation}\n    \\mathfrak{U}(x,y)_{k,\\ell}\n    =\n    \\frac{1}{2}\n    \\begin{cases}\n      - U_{\\ell-2}(x)U_{\\ell-1}(y) + U_{\\ell-1}(x) U_{\\ell}(y)&  \\\\\n      -U_{\\ell-1}(x)U_{\\ell-2}(y) +U_{\\ell}(x) U_{\\ell-1}(y),  & \\text{ if } k = \\ell, \\\\\n      -U_{\\ell-1}(x)U_{\\ell-1}(y) + U_\\ell(x)U_\\ell(y), & \\text{ if } k = \\ell + 1,\\\\\n      0, & \\text{ otherwise,}\n    \\end{cases}\n    \\label{eq:UM}\n\\end{equation}\nAll expressions involving $G$ and $B$ scale to expressions involving $\\mathfrak{U}.$  For example, we have that uniformly in bounded order principal minors and uniformly in $0 \\leq t \\leq T \\wedge \\tau_R \\wedge \\sigma_K$ \n\\[\n  \\frac{\\sqrt{n}}{2\\pi i}\\oint (s^A_t(z) - s_t(z))s_t(z) G(\\sqrt{n} z)\\,dz\n  \\to\n  \\frac{-1}{4\\pi i}\\oint s^{Q}(z)s^{\\mathfrak{s}}(z) \\partial_z \\mathfrak{U}(z,z)\\,dz.\n\\]\nA similar limit can be given for the second contour integral of \\eqref{eq:S4}.  \nSpecifically,\n\\[\n  \\begin{aligned}\n  \\frac{-\\sqrt{n}}{2(2\\pi i)^2}\\oint\\oint  (s^A_t(z) - s_t(z))s_t(y) \n  &\\frac{G(\\sqrt{n} z)-G(\\sqrt{n} y)}{z-y}\\,dzdy \\\\\n  &\\to\n  \\frac{1}{2(2\\pi i)^2}\\oint\\oint \n  s^{Q}(z)s^{\\mathfrak{s}}(y)\n  (\\Delta_z^y + \\Delta_y^z)\\partial_z\\mathfrak{U}(y,z)\n  \\,dydz.\\\\\n\\end{aligned}\n\\]\nBy degree considerations,\n\\[\n  \\frac{1}{2\\pi i}\\oint\n  s^{\\mathfrak{s}}(y)\n  \\Delta_z^y\\partial_z\\mathfrak{U}(y,z)\\,dy\n  =\\int_{-1}^{1}\\Delta_z^y\\partial_z\\mathfrak{U}(y,z)\\mathfrak{s}(y)\\,dy=0.\n\\]\nAs for the first contour integral of \\eqref{eq:S4}, it converges to an expression involving the minor polynomials of the Chebyshev polynomials of the second kind, which we can express using contour integrals as\n\\[\n  \\frac{n}{4\\pi i}\\oint (s^A_t(z) - s_t(z))G^{(0)}(\\sqrt{n} z)\\,dz \n  \\to\n  \\frac{-1}{2(2\\pi i)^2}\\oint\\oint \n  s^{Q}(z)s^{\\mathfrak{s}}(y)\n  \\Delta_y^z \\partial_z\\mathfrak{U}(y,z)\n  \\,dydz.\n\\]\nIn summary we have $A_t - \\sqrt{n}D$ converges to the solution of \n\\begin{equation}\n  \\begin{aligned}\n  dA_t &= \\mathcal{F}(A_t) \n  -\\frac{1}{4\\pi i}\\oint s^{Q}(z)s^{\\mathfrak{s}}(z) \\partial_z \\mathfrak{U}(z,z)\\,dz.\n\\end{aligned}\n\\end{equation}\n\n\n\n\n\n\n{\\small\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{INTRODUCTION}\n\nNeutrino physics is one of the most rapidly developing areas of\nhigh-energy physics. The fundamental experimental result was\nobtained in this field in recent years --- neutrino oscillations\nwere discovered \\cite{osc.exp.}. From the theory of this phenomenon\n(see, for example, \\cite{books}), which is based on the ideas of\nPontecorvo \\cite{Pontecorvo} and Maki, Nakagawa and Sakata\n\\cite{MNS}, it follows that oscillations are possible only when the\nneutrino mass is nonzero. As a consequence of this circumstance, the\npossibility exists of neutrino spin rotation, i.e. of transitions\nbetween the active left-handed neutrino polarization state and the\nsterile right-handed state. As it is understood today, in contrast\nto flavor oscillations, this process can never be observed if the\nparticle moves in vacuum. For the effect of spin rotation to exist\nsome external influence leading to effective breaking of the\nLorentz-symmetry of the theory is needed. For instance, dense matter\nor electromagnetic fields can serve as the physical reason for this\nphenomenon.\nIn the latter case, nontrivial electromagnetic\nproperties of the neutrino, in particular, the anomalous magnetic\nmoment of a Dirac neutrino \\cite{LeeShr77}, can lead to rotation of\nits spin.\n\nThe conventional approach \\cite{theor}, \\cite{nun} to the theory of\nneutrino spin rotation, based on solving the Cauchy problem for\nthe Schr\\\"{o}dinger-type equation with an effective Hamiltonian,\nwas developed for the description of ultrarelativistic\nneutrinos (the particles that are observed in experiments now\navailable). This approach is in fact quasiclassical and is equivalent\nto solving the Bargmann--Michel--Telegdi (BMT)\nequation \\cite{BMT} in the spinor representation (see\n\\cite{16}).\n\nThe aim of this work is to develop a consistent quantum theory of\nneutrino spin rotation to describe\nneutrinos with nonzero mass, including the low-energy region that may play a significant role in\nastrophysics \\cite{Fargion}.\nFollowing papers \\cite{theor}, \\cite{nun},\nwe use the method of effective potential \\cite{Wolf}\nthat allows us to take into account\ncollective influence of background particles on the propagation of\nneutrino.\n\nTo this end, we find a complete system of solutions of the\nDirac-type equation\nthat describe the action of both electromagnetic field and dense\nmatter on neutrino dynamics. These solutions have the form of\nnonspreading wave packets. They enable one to consider the states\nof neutrino with rotating spin as pure quantum states and to\nevaluate probabilities of various processes with  participation of\na neutrino in the framework of the Furry picture \\cite{F51}.\n\n\nThe paper is organized as follows. In the first Section we present\nthe model we are working with and briefly describe how the effective\npotentials are constructed. In Section \\ref{II} we summarize the main\nideas of the method used for solving the Dirac equation. In Section\n\\ref{III} we introduce a kinetic momentum operator and justify the\nadvantages of using its eigenvalues as quantum numbers, instead of\nthe eigenvalues of the canonical momentum operator. Then we find\nboth stationary and nonstationary wave functions. Here we also derive\nthe dispersion law and calculate the group velocity for a neutrino\nin the dense magnetized matter. We demonstrate that the group velocity\nof a neutrino is independent of its spin orientation.\nIn Section \\ref{IV} we find the\nexplicit expressions for the kinetic momentum and spin projection\noperators. In Section \\ref{VV} we use the obtained results for\nquasiclassical interpretation of neutrino spin behavior in dense\nmagnetized matter. In the conclusive Section \\ref{VI} we compare our\nresults with those obtained by other authors.\n\nThe units $\\hbar =c=1$ are used throughout the paper; metric tensor, $g^{\\mu\\nu}={\\mathrm{diag}(1,-1,-1,-1)}$; totally antisymmetric Levi-Civit\\`{a} tensor,\n$e^{\\mu\\nu\\rho\\lambda}= +1$ if $\\{\\mu\\nu\\rho\\lambda\\}$ is an even permutation of $\\{0,1,2, 3\\}$, $e^{\\mu\\nu\\rho\\lambda}= -1$\nif it is an odd permutation, $e^{\\mu\\nu\\rho\\lambda}= 0$ otherwise.\n\n\n\\section{Wave equation for neutrino in dense matter and in electromagnetic\nfield}\\label{I}\n\nLet us first discuss the model we use for the\ndescription of the neutrino behavior in the matter and in the\nelectromagnetic field.\nAt present, there are no experimental\ndata that might help to make the choice between\nthe  Dirac or the Majorana nature of a neutrino \\cite{pdg}. Therefore, in this paper we discuss spin rotation of  a massive Dirac neutrino with anomalous magnetic moment\npropagating through dense magnetized matter.\n\nAs a first approximation, we can consider spin rotation\nindependently of variations of the particle flavor, i.e. we can deal\nwith the mass eigenstates of neutrino. It is quite obvious, due to the fact that\nthe physical origins of neutrino flavor\noscillations and its spin rotation are different. The Dirac--Pauli equation \\cite{Pauli}\nserves as the basis of this approach in electromagnetic fields.\nJust in this way, the behavior of the neutrino spin in a constant\nhomogeneous magnetic field was investigated in \\cite{Fu}.\nIn order to consider the influence of matter on neutrino dynamics through weak interaction,\nthe Dirac--Pauli equation should be modified.\n\n\n\n\nAs it was proposed in \\cite{Wolf}, \\cite{Mik}, if matter\ndensity is high enough for considering weak interaction of neutrino\nwith the background fermions as coherent, it is possible to describe\nneutrino interaction with matter by an effective potential.\nTo get covariant realization of this idea, it\nis necessary to take into account that the term that describes\ndirect interaction of neutrino with matter must depend on the\ncurrents\n\\begin{equation}\\label{3}\nj_{f}^\\mu=\\{n_f u^{0}_f,n_f{\\bf{u}}_f\\},\n\\end{equation}\nand on the polarizations\n\\begin{equation}\\label{4}\\lambda^{\\mu}_f =\\left\\{n_f\n({\\bb{\\zeta}}_f\\cdot{\\bf{u}}_f), n_f\\left({\\bb{\\zeta}}_f + \\frac{\n{\\bf{u}}_f ({\\bb{\\zeta}}_f\\cdot{\\bf{u}}_f)}{1+u^{0}}\\right)\\right\\}\n\\end{equation}\n\\noindent of the background fermions $f$ \\cite{RT,StT,SpinLight}. It\nis significant that only $j_{f}^\\mu$ and $\\lambda_{f}^\\mu$ characterize matter\nintegrally.\nIn formulas (\\ref{3}) and (\\ref{4}), $n_f$ and\n${\\bb{\\zeta}}_f \\; (0\\leqslant |{\\bb{\\zeta}}_f |^2 \\leqslant 1)$ are\nthe number density and the mean value of the polarization vector of\nthe background fermions $f$ in the center-of-mass system of matter\n($u^{\\mu}_f =\\{u^{0}_{f},{\\bf{u}}_{f}\\}$ denotes the four-velocity\nof the reference frame), where the mean momentum of the fermions $f$\nis equal to zero.\n\nIn the framework of the standard model (for a more general case, see\nAppendix \\ref{uuur}), the interaction term can be derived directly\nfrom the model Lagrangian. To this end, it is necessary to write\nthis Lagrangian in the low-energy effective form  using the\nfour-fermion interaction and replace the vector\n${\\bar{\\psi}}_{f}(x)\\gamma^{\\mu}{\\psi}_{f}(x)$ and axial vector\n${\\bar{\\psi}}_{f}(x)\\gamma^{5}\\gamma^{\\mu}{\\psi}_{f}(x)$ currents of\nthe background fermions by $j^{\\mu}_{f}$ and $\\lambda^{\\mu}_{f}$\nrespectively. The explicit form of $j^{\\mu}_{f}$ and\n$\\lambda^{\\mu}_{f}$ is obtained as a result of averaging of currents\nover the fermion statistical distribution function and depends on\nthe type of the  matter considered. In particular, the expression\nfor the polarization $\\lambda^{\\mu}_{\\mathrm{e}}$ of electron gas in\na magnetic field at finite temperature can be found in \\cite{nun}.\nNote that, since the origin of the effective potential is forward\nelastic scattering of neutrino, it can also be calculated using\nfield-theoretic methods \\cite{160}.\n\nAs a result we come to the equation\n\\begin{equation}\\label{6}\n\\left(i{\\gamma^{\\mu}\\partial_{\\mu}}  - \\frac{1}{2}\n\\gamma^{\\mu}{f_{\\mu}}(1 + \\gamma ^5) - \\frac{i}{2}\\mu_{0}F^{\\mu \\nu\n}\\sigma _{\\mu \\nu } - m\\right)\\varPsi(x) = 0,\n\\end{equation}\n\\noindent where $F^{\\mu\\nu}$ is\nthe electromagnetic field tensor. The $\\gamma$-matrices satisfy\n$\\gamma^{\\mu}\\gamma^{\\nu}+\\gamma^{\\nu}\\gamma^{\\mu}=2g^{\\mu\\nu}$\nwith $\\gamma^{0}$ Hermitian, ${\\bb{\\gamma}}$  anti-Hermitian, and are related to the $\\beta$ and ${\\bb{\\alpha}}$ matrices through\n$\\gamma^{0}=\\beta, {\\bb{\\gamma}}=\\beta{\\bb{\\alpha}}$;\n$\\sigma ^{\\mu \\nu }\n=\\frac{1}{2}(\\gamma^{\\mu}\\gamma^{\\nu}-\\gamma^{\\nu}\\gamma^{\\mu})$,\n$\\gamma^{5}=-i\\gamma^{0}\\gamma^{1}\\gamma^{2}\\gamma^{3}$. The effective\nfour-potential $f^\\mu$ is a linear combination of currents and\npolarizations of the background fermions\n\\begin{equation}\\label{2}\nf^\\mu =\n{\\sum\\limits_f\\big({\\rho_f^{(1)}j_f^\\mu+\\rho_f^{(2)}\\lambda_f^\\mu}\\big)}.\n\\end{equation}\n\\noindent Summation in (\\ref{2}) is carried out over all fermions $f$ of\nmatter. The expressions for the coefficients $\\rho_{f}^{(1,2)}$\nare determined as\n\\begin{equation}\\label{rho}\n\\rho_f^{(1)}\\!=\\sqrt 2 {G}_{{\\mathrm F}} \\left\\{\nI_{\\mathrm{e\\nu}}+T_3^{(f)}-2Q^{(f)}\\sin^2\\theta_{\\mathrm W} \\right\\}\\!,\\;\\;\n\\rho_f^{(2)}\\!=-\\sqrt 2 {G}_{{\\mathrm F}} \\left\\{ I_{\\mathrm{e\\nu}}+T_3^{(f)}\n\\right\\}\\!.\n\\end{equation}\n\n\\noindent Here $Q^{(f)}$ is the electric charge of the fermion $f$;\n$T_{3}^{(f)}$ is the third component of the weak isospin; ${G}_{{\\mathrm\nF}}$ and $\\theta_{\\mathrm W}$ are the Fermi constant and the\nWeinberg angle respectively; \\mbox{$I_{\\mathrm{e\\nu}}=1$} for the electron\nneutrino interaction with electrons, \\mbox{$I_{\\mathrm{e\\nu}}=-1$} for\ninteraction with positrons, otherwise $I_{\\mathrm{e\\nu}}= 0$.\n\nWhile equation (\\ref{6}) describes mass states of neutrino,\npotential (\\ref{2}) is flavor dependent, and in the general case, this\nleads to correlations between spin rotation and flavor oscillations.\nIn order to construct a mathematically consistent approach to\na description of neutrino spin rotation, we have to avoid the\ncorrelations. This is possible if we assume that the effective potential\nwhich describes the influence of dense matter on neutrino is the\nsame for different flavors. In this case we can construct flavor\nstates of the neutrino as the linear combination of its mass states with\ncoefficients that are elements of the mixing matrix of the neutrino in\nvacuum.\n\nThe physical justification of this model can be made in the\nfollowing way. In order to consider effective potentials as\nflavor independent, it is necessary to assume\nthat the fraction of electrons in the matter is small, i.e. the\nelectron number density ${n_{e}}$ is approximately equal to zero.\nCalculations which are in a satisfactory agreement with the\nexperimental data show that in the center of a neutron star the\nfraction of electrons does not exceed a few per cent \\cite{stars}.\nTherefore the model approximation $ {n_{e}= 0}$ is quite\nappropriate.\n\nWe restrict ourselves to considering equation (\\ref{6}) in the case\nof a constant homogeneous electromagnetic field and constant\ncurrents and polarizations of matter,\n\\begin{equation}\\label{5.0}\n    F^{\\mu\\nu}={\\mathrm{const}}, \\quad j_f^{\\mu}={\\mathrm{const}}, \\quad\n    \\lambda_f^{\\mu}={\\mathrm{const}},\n\\end{equation}\n\\noindent as the first order approximation of the realistic\nbackground.\nBecause of (\\ref{5.0}) additional conditions for\n$F^{\\mu\\nu}$ and $f^{\\mu}$ may be obtained. The strengths of the\nelectric and magnetic fields and average currents and polarizations\nof the background particles should obey the self-consistent system\nof equations including the Maxwell equations, the Lorentz equation\n\\begin{equation}\\label{01}\n    {\\dot{j}}^\\mu_{f}=\\frac{e_{f}}{m_{f}}\n    F^{{\\mu}{\\phantom{\\nu}}}_{{\\phantom{\\mu}}{\\nu}}j^\\nu_{f},\n\\end{equation}\n\\noindent and the the BMT quasiclassical\nspin evolution equation\n\\begin{equation}\\label{02}\n   {\\dot{\\lambda}}^\\mu_{f}=\\left[\\frac{e_{f}}{m_{f}}\n    F^{{\\mu}{\\phantom{\\nu}}}_{{\\phantom{\\mu}}{\\nu}}\n    +2\\mu_{f}\n    \\left( g^{{\\mu}{\\alpha}}\n    -u^{\\mu}_{f}u^{\\alpha}_{f}\\right)F_{\\alpha\\nu}\n    \\right]\\lambda^\\nu_{f}.\n\\end{equation}\n\\noindent Here the dot denotes differentiation with respect to the\nproper times $\\tau_{f}$ of particles. From equations (\\ref{01}) and\n(\\ref{02}) we find that for  charged particles the conditions\n$\\dot{j}^\\mu_{f}=0,\\; \\dot{\\lambda}^\\mu _{f}=0$ are equivalent to\n$F^{{\\mu}{\\phantom{\\nu}}}_{{\\phantom{\\mu}}{\\nu}}j^\\nu_{f}=0,$\n$F^{{\\mu}{\\phantom{\\nu}}}_{{\\phantom{\\mu}}{\\nu}}\\lambda^\\nu_{f}=0$.\nHowever, it seems reasonable to assume that in dense matter\nvelocities of\nthe center-of-mass systems\nfor all components are\nequal, thus for the neutral particles similar conditions should\nhold as well. In this way we\nobtain the restriction\n\\begin{equation}\\label{03}\n    F^{\\mu\\nu}f_\\nu=0.\n\\end{equation}\n\\noindent It should be emphasized\nthat condition (\\ref{03}) is the result of the\nfact that average currents and polarizations of particles of\nmatter in an external field should satisfy the classical equations\nof motion.  The physical\nmeaning of this condition is discussed in more\ndetail in Section \\ref{VV}.\n\nNote that equation (\\ref{6}) with constant coefficients appears in\nthe standard model extension \\cite{2}, \\cite{nambu} as well. In this case,\n$F^{{\\mu}{\\nu}}$ and $f^{{\\mu}}$ describe vacuum condensates that break the Lorentz-symmetry of the theory.\n\n\n\\section{Formulation of the problem}\\label{II}\n\nThe Dirac equation, in particular equation (\\ref{6}), is a\npartial differential equation. Therefore, as it is known, its\ngeneral solution is defined up to an arbitrary function.\nHowever, in quantum-mechanical\napplications we deal with the so-called complete integral which depends\non a set of constants, i.e. quantum numbers. Those are the eigenvalues\nof some self-adjoint operators. For the classification of\nparticle states it is necessary to introduce the complete set of the\noperators --- integrals of motion. Note that in nonrelativistic\nquantum mechanics any self-adjoint operator may serve as an\noperator of observable, which is not the case in the relativistic mechanics,\nwhere only integrals of motion, commuting with the operator of the\nequation, can be treated as operators of observables \\cite{LanP}.\nThe choice of the complete set of them\nis different in each particular case and should be adequate to the\nproblem being solved.\n\nIn the case of the problem of spin evolution, operators of kinetic momentum components should be included in the complete set. It becomes obvious if we start with the\nfollowing argument. The direction of particle polarization is well defined in\nits rest frame, and then one finds the polarization in the\nlaboratory frame upon carrying out an appropriate Lorentz transformation. This transformation is\ndefined by the group velocity of the particle, in other words,\nby its kinetic momentum. This implies that it is not the canonical momentum but the kinetic momentum operator that defines the direction of particle propagation.\n\nNow, there is only one problem that remains to be solved. The form of the\nkinetic momentum operator for a\nparticle with spin propagating under the influence of external fields is not known beforehand. So we\nhave to find a self-adjoint operator ${\\mathfrak{p}}^{\\mu}$ with\nthe eigenvalues $q^{\\mu}$, which satisfies the condition $q^{2}=m^{2}$\nand may be interpreted as the components of the particle kinetic momentum.\n\nLet us discuss this issue in more detail.\nIn the mathematical apparatus of quantum field theory, a particle is\nusually identified with an irreducible unitary representation of the\nPoincar\\'{e} group \\cite{BoLoTO}. The irreducible representations are\ncharacterized by two invariants of the group\n\\begin{equation}\\label{a4}\nP^2 \\equiv{P^{\\mu}}{P_{\\mu}}=m^2,\n\\end{equation}\n\\begin{equation}\\label{a5}\nW^2 \\equiv {W^{\\mu}}{W_{\\mu}} = -{m^2}s(s+1).\n\\end{equation}\n\\noindent The translation generators $P^\\mu$ are identified with the\ncomponents of the particle momentum, and the Pauli--Lubanski--Bargmann vector\n\\begin{equation}\\label{a6} W^{\\mu}\n  =-\\,{\\frac{1}{2}}\\,{e^{{\\mu}{\\nu}{\\rho}{\\lambda}}}{M_{{\\nu}{\\rho}}}\n  {P_{\\lambda}},\n\\end{equation}\n\\noindent where $M^{{\\mu}{\\nu}}$ are the Lorentz generators,\ncharacterizes the particle spin. The invariant $m^2$ is the particle\nmass squared and $s$ is the value of its spin.\n\n\nA space of unitary representation is defined by the condition called\n``the wave equation for a particle with mass $m$ and spin $s$.'' The\nwave equation for particles with spin $s ={1}\/{2}$ is the Dirac\nequation\n\\begin{equation}\\label{a9}\n\\left(i\\gamma^{\\mu}\\partial_{\\mu} - m\\right){\\varPsi}_{0}(x) = 0.\n\\end{equation}\n\\noindent In this case the realization of generators of the Poincar\\'{e}\ngroup and the Pauli--Lubanski--Bargmann vector in the coordinate\nrepresentation is\n\\begin{equation}\\label{a10}\n  p^{\\mu} = i{\\partial^{\\,\\mu}},\\quad m^{\\mu\\nu}=\n  i(x^{\\mu}\\partial^{\\nu}-x^{\\nu}\\partial^{\\mu})+\\frac{i}{2}\\sigma^{\\mu\\nu},\\quad\n  w^\\mu =\n  \\frac{i}{4}\\,{\\gamma^{5}}\\sigma^{\\mu\\nu}\\partial_{\\nu}.\n\\end{equation}\n\nThese operators commute with the operator of the Dirac equation and\ncan be identified with observables.  They have a\nself-adjoint extension on the subsets of solutions of equation\n(\\ref{a9}) with a fixed sign of the energy with regard to the standard scalar product,\n\\begin{equation}\\label{a010}\n({\\varPhi}, \\varPsi)= \\int\\! d{{\\bf\nx}}\\,{\\varPhi}^{\\dag}({\\bf x},t)\\,\\varPsi({\\bf x},t).\n\\end{equation}\n\\noindent Three-dimensional spin projection operator\n$\\bb{{s}}(p)$ is a set of coefficients\n${{s}}_{i}(p)$ of the expansion of the vector $w^\\mu$ in\nspacelike unit vectors ${{S}}_{i}^{\\mu}(p)\\; (i=1,2,3)$:\n\\begin{equation}\\label{a11}\n  s_{i}(p) = -\\,\\frac{1}{m}w_{\\mu}S_{i}^{\\mu}(p),\n\\end{equation}\n\\noindent where\n\\begin{equation}\\label{a110}\np_{\\mu}S^{\\mu}_{i}(p) =0,\\quad S^{\\mu}_{i}(p)S_{\\mu\nj}(p)=-\\delta_{ij}.\n\\end{equation}\n\\noindent Obviously,\n\\begin{equation}\\label{a12}\n\\big[s_{i}(p),s_{j}(p)\\big]\n=i{e_{ijk}}s_{k}(p).\n\\end{equation}\n\\noindent The choice of these unit vectors is not unique, and it is\npossible to construct operators that determine the spin projection\non any direction in an arbitrary Lorentz frame.\n\nThe above description of the particle characteristics cannot be\ndirectly used in the presence of external fields, where operators (\\ref{a10}) are not necessarily integrals of\nmotion. In this case the classification of particle states is usually realized  by linear combinations of operators\n$p^{\\mu}$ and $w^{\\mu}$ with coefficients depending on coordinates\n\\cite{BG90}.\nUnfortunately, the physical meaning of these operators  is usually not quite clear.\nTherefore, one should formulate a description of the particle motion in an external field in the same clear and detailed way as for a free particle.\n\nThis can be achieved basing on the following ideas.\nSince an irreducible representation of group is defined accurately\nup to an equivalence transformation, it is reasonable to state the\nproblem of finding such realization of the Lie algebra of the\nPoincar\\'{e} group for which the condition of\nirreducibility of the representation leads to wave equation describing a particle in a given external background. To solve this problem it is necessary to\nfind a unitary operator $U(x,x_{0})$ which converts solutions of the\nwave equation for a free particle (\\ref{a9}) to solutions\n$\\varPsi(x)$ of (\\ref{6}):\n\\begin{equation}\\label{a018}\nU(x,x_{0})\\varPsi_{0}(x)=\\varPsi(x).\n\\end{equation}\n\\noindent  Thus, $U(x,x_{0})$ is an intertwining operator in the\nsense of Darboux \\cite{Darboux}.  This operator in our case should\nsatisfy the equation\n\\begin{equation}\\label{a19}\n \\left(i\\gamma^{\\mu}{\\partial}_{\\mu} - \\frac{1}{2}\n\\gamma^{\\mu}{f_{\\mu}}(1 + \\gamma ^5) - \\frac{i}{2}\\mu_{0}F^{\\mu \\nu\n}\\sigma _{\\mu \\nu }\\right)U(x,x_{0}) -\nU(x,x_{0})\\left(i{\\gamma^{\\mu}\\partial_{\\mu}}\\right) =0.\n\\end{equation}\n\\noindent Therefore, operators\n\\begin{equation}\\label{a21}\n  {\\mathfrak{p}}^{\\mu} =U(x,x_{0})p^{\\mu}U^{-1}(x,x_{0}),\\quad {\\mathfrak m}^{\\mu\\nu}\n  =U(x,x_{0})m^{\\mu\\nu}U^{-1}(x,x_{0})\n\\end{equation}\n\\noindent commute with the operator of the wave equation. As a\nconsequence, the Pauli--Lubanski--Bargmann vector ${\\mathfrak{W}}^{\\mu}$ and the components\nof the three-dimensional spin projection operator ${\\mathfrak{S}}_{i}$\ncan be constructed in the same way as in the case of a free particle:\n\\begin{equation}\\label{aa21}\n  {\\mathfrak{W}}^{\\mu}\n  =-\\,{\\frac{1}{2}}\\,{e^{{\\mu}{\\nu}{\\rho}{\\lambda}}}{{\\mathfrak{m}}_{{\\nu}{\\rho}}}\n  {{\\mathfrak{p}}_{\\lambda}}, \\quad {\\mathfrak{S}}_{i}=\n  -\\,\\frac{1}{m}{\\mathfrak{W}}_{\\mu}S_{i}^{\\mu}({\\mathfrak{p}}).\n\\end{equation}\n\n\nThe above statement may be reduced to the following: the wave\nfunction of a neutrino in dense matter can be derived with the help\nof a solution of the Dirac equation for a free particle and of some\nunitary evolution operator. A complete set of integrals of motion\nmay be constructed with the help of operators (\\ref{a21}). The\nphysical meaning of eigenvalues of observables, i.e. quantum\nnumbers, is clear enough then. However,\nin the general case, $U(x,x_{0})$ is\nan integral operator and, as a consequence, operators (\\ref{a21})\nare also integral ones, so it is difficult to find\n$U(x,x_{0})$ by direct calculations.\nThat is why we are trying to find the evolution operator for equation\n(\\ref{6}) using the correspondence principle.\n\n\n\\section{Wave functions}\\label{III}\n\n\nWhen motion of a massive particle with spin in electromagnetic fields is\ndescribed in the framework of the quasiclassical approach, we\nuse two four-vectors, namely the four-velocity $u^{\\mu}$ and the spin\nvector $S^{\\mu}$ obeying the conditions\n\\begin{equation}\\label{x1}\nu^{2}=1,\\qquad S^{2}=-1,\\qquad (uS)\\equiv u^{\\mu}S_{\\mu}=0.\n\\end{equation}\n\\noindent  These vectors are solutions of the Lorentz\nand BMT equations, respectively. As it was shown in \\cite{Zw},\nboth four-vectors can be constructed with the help of one and the same evolution operator that acts on different initial values of these four-vectors, satisfying\nrelations (\\ref{x1}).\nIn other words, the evolution operator for the BMT equation completely describes\nquasiclassical behavior of the particle.\n\nAs far as the neutrino is a neutral particle,  the BMT equation for the spin\nvector $S^{\\mu}$ of a neutrino moving with the four-velocity $u^{\\mu}$\nhas the form\n\\begin{equation}\\label{BMT1}\n    \\dot{S}^{\\mu}=2\\mu_{0}\\left(g^{\\mu\\alpha}- u^{\\mu}u^{\\alpha}\\right)F_{\\alpha\\nu} S^{\\nu} .\n\\end{equation}\n\\noindent\nWe can extend the BMT equation to include weak interaction\nbetween the neutrino and dense matter. It was shown phenomenologically\nin \\cite{L43} that if effects of neutrino weak interactions are\ntaken into account, the Lorentz-invariant generalization of the BMT\nequation is\n\\begin{equation}\\label{x4}\n \\dot{S}^{\\mu}=2 \\left(g^{\\mu\\alpha}- u^{\\mu}u^{\\alpha}\\right)\\left(\\mu_{0} F_{\\alpha\\nu}+G_{\\alpha\\nu} \\right) S^{\\nu},\n\\end{equation}\n\\noindent where\n\\begin{equation}\\label{5}\nG^{\\mu \\nu}= \\frac{1}{2}e^{\\mu \\nu \\rho \\lambda}\nf_{\\rho}u_{\\lambda}.\n\\end{equation}\n\\noindent We can get an analogous result by averaging the equations of\nmotion for the Heisenberg operators (see \\cite{FRG}).\n\nA remarkable feature of the BMT equation is its universality --- at least in the zeroth order of\nthe Planck constant, particles with an arbitrary spin may be\ndescribed by the same BMT equation. However, a part of the information\nabout the particle behavior gets lost due to distinctions in\ntransformation properties of the states of the particles with\ndifferent spins. This information may be restored in the following\nway. We can describe a particle using quasiclassical spin\nwave functions $\\varPsi(\\tau)$ constructed in such a way that the\nvector current\n\\begin{equation}\\label{V}\nj _{V}^{\\mu}(\\tau)=\\bar{\\varPsi}(\\tau)\\gamma^{\\mu}\\varPsi(\\tau),\n\\end{equation}\n\\noindent as well as the axial current\n\\begin{equation}\\label{A}\nj_{A}^{\\mu}(\\tau)=\\bar{\\varPsi}(\\tau)\\gamma^{5}\\gamma^{\\mu}\\varPsi(\\tau)\n\\end{equation}\n\\noindent built on their base, obey  the BMT equation, and, as a\nconsequence, may be interpreted as four-velocity $u^{\\mu}$ and spin\nvector $S^{\\mu}$ respectively.\n\n\nWe can introduce quasiclassical spin wave functions as follows\n\\cite{LP}, \\cite{L}. Let the Lorentz equation be solved, i.e. the\ndependence of the particle coordinates on proper time is found.\nThen the BMT equation transforms to an ordinary differential equation,\nwhose resolvent determines a one-parameter subgroup of the\nLorentz group. The quasiclassical spin wave function is a\nspin tensor, whose evolution is determined by the same\none-parameter subgroup. It is easy to verify (see Appendix \\ref{rr}) that for a neutral particle, represented by a\nDirac bispinor, the equation that defines the evolution operator for\nthe wave function $\\varPsi(\\tau)$ takes the form\n\\begin{equation}\\label{103}\n\\dot{U}(\\tau,\\tau_{0}) = \\frac{i}{m}\\gamma^{5}\\sigma_{\\mu\\nu}\n{\\varphi}^{\\mu}{q}^{\\nu}\\,\n{U}(\\tau,\\tau_{0}).\n\\end{equation}\n\\noindent Here $\\varphi ^\\mu = f^\\mu\/2 + H^{\\mu\\nu}q_{\\nu}\/m$, $H^{\\mu\\nu}=\\mu_{0}{}^{\\star\\!}F^{\\mu\\nu}$, where ${}^{\\star\\!}F^{\\mu\\nu} =\n-\\frac{1}{2}e^{\\mu\\nu\\rho\\lambda}F_{\\rho\\lambda}$ is the dual\nelectromagnetic field tensor and $q^{\\mu}=mu^{\\mu}$.\n\nLet us look for a solution of equation (\\ref{6}) in the form\n\\begin{equation}\\label{102}\n\\varPsi (x) = e^{-iK(x)}U\\left(\\tau (x),0\\right)\\varPsi_{0}\n(x),\n\\end{equation}\n\\noindent where $U(\\tau,\\tau_{0})$ is an evolution operator for the\nquasiclassical spin wave function,  $e^{-iK(x)}$ is a phase factor, and\n$\\varPsi _0 (x)$ is a solution of the Dirac equation for a free\nparticle.\nSince we assume that $F^{\\mu\\nu}$ and\n$f^{\\mu}$ are constants, the evolution operator can be expressed\nas a matrix exponential,\n\\begin{equation}\\label{U}\n    U(\\tau,\\tau_{0}) = \\exp \\left\\{ \\frac{i}{m}{\n    \\gamma^{5}\\sigma_{\\mu\\nu}\n{\\varphi}^{\\mu}{q}^{\\nu}\n    (\\tau-\\tau_{0})} \\right\\},\n\\end{equation}\n\\noindent that depends on constant four-vector $q^\\mu$ satisfying\nthe condition $q^2=m^2$.\n\nUsing ansatz (\\ref {102}) we should choose such a basis in the space of\nsolutions of the Dirac equation for the free particle that the action\nof the evolution operator  on each element of this basis be reduced\nto multiplication by one and the same matrix function depending on quantum\nnumbers of the basis. It is obvious that in our case we must choose\nthe plane waves\n\\begin{equation}\\label{a3}\n\\varPsi _0 (x) = e^{ - i(qx)}(1 - \\zeta_{0}\\gamma\n^5\\gamma_{\\mu}{S}_0^{\\mu}(q) )(\\gamma^{\\mu}q_{\\mu} + m)\\psi _0.\n\\end{equation}\n\\noindent Here ${S}_0^{\\mu}(q)$ determines the direction of\npolarization of the particle; $\\zeta_{0} = \\pm 1 $ is the spin\nprojection on $ {S}_0^{\\mu}(q)$; $\\psi _{0}$ is a constant\nfour-component spinor. The wave function is normalized by the condition\n\\begin{equation}\\label{aaa3}\n {\\bar{\\varPsi}}_{0}(x)\\varPsi_{0}(x) =m\/q_{0},\n\\end{equation}\n\\noindent and four-vector $q^\\mu$ is a kinetic momentum of the particle. Though the explicit form of a kinetic momentum\noperator for a particle with spin interacting with dense matter and\nelectromagnetic field is not known beforehand, the correspondence\nprinciple allows us to construct solutions characterized by its eigenvalues.\n\nNaturally, as far as equation (\\ref {6}) is invariant under\ntranslations the canonical momentum operator\n$p^{\\mu}=i{{\\partial}^{\\mu}}$ is an integral of motion for this\nequation, too. However, the commonly adopted choice of eigenvalues\nof this operator as quantum numbers is not satisfactory if we prefer\nspin projection operators with clear physical meaning. The\ndirections of canonical and kinetic momenta are different in the general\ncase (see \\cite {nambu}) and, as it\nwas already mentioned in Section \\ref{II}, projection of the spin is well defined in the rest frame of the particle where its kinetic momentum is equal to zero: ${\\bf q}=0$.\nThat is why in the construction of spin projection operators (see\n(\\ref {a11})) it is necessary to select unit vectors orthogonal to\nfour-vector $q^{\\mu}$.\n\n\nLet us find the proper time $\\tau(x)$ and\nthe phase factor $K(x)$. Substitution of (\\ref{102}) into\n(\\ref{6}) gives\n\\begin{equation*}\\label{aa6}\n\\begin{array}{c}\n\\displaystyle\\left\\{\\gamma^{\\mu}{q}_{\\mu}\n+\\gamma^{\\mu}{\\partial}_{\\mu}K(x) -\n\\frac{1}{2}\\gamma^{\\mu}{f}_{\\mu}(1+\\gamma^{5}) + \\frac{1}{m}\\gamma^{5}\\sigma_{\\mu\\nu}\n{\\varphi}^{\\mu}{q}^{\\nu}\\gamma^{\\alpha}{N}_{\\alpha}\n\\right.\\\\[8pt] - \\displaystyle\\left.\\frac{i}{2}\\mu_{0}F^{\\mu \\nu }\\sigma\n_{\\mu \\nu }-m\\right\\}e^{-iK(x)}U(\\tau(x) )\\varPsi_{0} (x)\n = 0,\n\\end{array}\n\\end{equation*}\n\\noindent where $N^{\\mu} = \\partial ^{\\mu} \\tau $. Since the matrix\n$U(\\tau(x))$ is nondegenerate and the commutator $[\\gamma^{\\mu}q_{\\mu},U(\\tau(x))]$ is zero, we get\n\\begin{equation*}\\label{a7}\n\\begin{array}{l}\n\\displaystyle\\gamma^{\\mu}{\\partial}_{\\mu}K(x) -\n\\frac{1}{2}\\gamma^{\\mu}{f}_{\\mu}(1+\\gamma^{5}) + \\frac{1}{m}\\gamma^{5}\\sigma_{\\mu\\nu}\n{\\varphi}^{\\mu}{q}^{\\nu}\\gamma^{\\alpha}{N}_{\\alpha} -\n\\frac{i}{2}\\mu_{0}F^{\\mu \\nu }\\sigma _{\\mu \\nu }\n= 0.\n\\end{array}\n\\end{equation*}\n\nTo solve this equation for $N^\\mu$ and $K(x)$, we should set the\ncoefficients at the linearly independent elements of the algebra of the Dirac\nmatrices equal to zero. Thus\n\\begin{equation}\\label{a8}\n    {\\partial}^{\\mu}K(x) = f^{\\mu}\/2,\n\\end{equation}\n\\noindent and the system that defines the vector $N^{\\mu}$ is\n\\begin{equation}\\label{106}\n\\varphi ^\\mu (m - (Nq)) + (N\\varphi )q^\\mu = 0, \\quad\n     \\mu_{0}F^{\\mu\\alpha}q_{\\alpha} = - e^{\\mu \\nu \\rho \\lambda }\n    N_\\nu \\varphi _\\rho q_\\lambda,\n\\end{equation}\n\n\nThe system is consistent provided that\n\\begin{equation}\\label{dop1}\nq_{\\mu}F^{\\mu\\nu}\\varphi_{\\nu}=0.\n\\end{equation}\n\\noindent Equation (\\ref{dop1}) must be held for an\narbitrary $q^{\\mu}$, so we have\n\\begin{equation}\\label{dop11}\n  {}^{\\star\\!}F^{\\mu\\alpha}F_{\\alpha\\nu}\\equiv -\\frac{1}{4}\\delta^{\\mu}_{\\nu}\n  {}^{\\star\\!}F^{\\alpha\\beta}F_{\\alpha\\beta}=0,\n\\end{equation}\n\\begin{equation}\\label{dop12}\nF^{\\mu\\nu}f_{\\nu}=0.\n\\end{equation}\n\\noindent Any  antisymmetric tensor has an eigenvector corresponding\nto zero eigenvalue if and only if its second invariant\n$I_{2}=\\frac{1}{4}{}^{\\star\\!}F^{\\mu\\nu}F_{\\mu\\nu}$ is equal to zero. That is\nwhy conditions (\\ref{dop11}) and (\\ref{dop12}) are not independent\nand (\\ref{dop11}) is a consequence of (\\ref{dop12}). Comparing it with\n(\\ref{03}), we see that the condition obtained basing on the physical\nreasons alone and condition (\\ref{dop12}) totally coincide. This ensures quasiclassical behavior of the particle  in the\nbackground medium and allows obtaining the solution of\nequation (\\ref{6}) in  the form of (\\ref{102}).\n\nUsing an orthogonal basis in the Minkowski space,\n\\begin{equation}\\label{Basis}\n    \\displaystyle n_0^\\mu=q^\\mu\/m, \\;\\;\n    \\displaystyle n_1^\\mu=\\frac{H^{\\mu\\nu}q_\\nu}{\\sqrt{{\\cal N}}},\n    \\;\\;\n    \\displaystyle n_2^\\mu=\\frac{F^{\\mu\\nu}q_\\nu}{\\sqrt{\\tilde{{\\cal N}}}},\n    \\;\\;\n    \\displaystyle n_3^\\mu=\\frac{m^{2}H^{\\mu\\nu}H_{\\nu\\alpha}\n    q^\\alpha  -q^\\mu {\\cal N}}\n    {m\\sqrt{{\\cal N}\\tilde{{\\cal N}}}},\n\\end{equation}\n\\noindent where ${\\cal N}=q_{\\mu}\nH^{\\mu\\rho}H_{\\rho\\nu}q^{\\nu}$, $\\tilde{\\cal N}=\\mu_{0}^{2}q_{\\mu}\nF^{\\mu\\rho}F_{\\rho\\nu}q^{\\nu}$, and\nrelations (\\ref{A2.1}), (\\ref{A2.3}) from Appendix \\ref{ur}, we find\n\\begin{equation}\\label{N}\n\\displaystyle N^\\mu = - q^\\mu\\frac{m(f\\varphi)}\n    {{2((\\varphi q)^2-m^{2}\\varphi^2)}}\n     + f^\\mu\\frac{m}{2(\\varphi q)}  + \\varphi^\\mu\\frac{m^3(f \\varphi)}\n    {2(\\varphi q)((\\varphi q)^2-m^{2}\\varphi^2)}.\n\\end{equation}\n\\noindent According to the fact that $f^\\mu$ and\n$N^\\mu$ are constant values  we obtain the proper time\n\\begin{equation}\\label{xx} \\tau = (Nx),\n\\end{equation}\n\\noindent and the phase factor\n\\begin{equation}\\label{xx0}\nK(x) =(fx)\/2,\n\\end{equation}\nwhich determines an energy shift of the neutrino in matter.\n\n\nNow we can derive the expression for the wave function:\n\\begin{equation}\\label{Wf1}\n\\displaystyle \\varPsi_{q\\zeta_{0}}(x) =\n\\frac{1}{2}\\sum\\limits_{\\zeta = \\pm 1} e^{-i(P_{\\zeta} x)}{( 1 -\n\\zeta \\gamma^5 \\gamma_{\\mu}{S}_{\\mathrm{tp}}^{\\mu}(q))(1-\\zeta_{0}\\gamma^5\n\\gamma_{\\mu}{S}_{0}^{\\mu}(q))(\\gamma^{\\mu}{q}_{\\mu}+m)} \\psi_{0},\n\\end{equation}\nwhere\n\\begin{equation}\\label{P2}\n\\begin{array}{c}\n\\displaystyle P^\\mu_{\\zeta} = q^\\mu + f^{\\mu}\/2-\\zeta N^{\\mu}\\sqrt\n    {(\\varphi q)^2 - \\varphi^2 m^2}\/m\n\\displaystyle = q^\\mu \\bigg(1+\\zeta\\frac{(f\\varphi)}\n    {2\\sqrt{(\\varphi q)^2-m^2\\varphi^2}}\n    \\bigg)\\\\[12pt]\\displaystyle+ \\frac{1}{2}f^{\\mu}\\bigg(1  - \\frac{\\zeta\n\\sqrt{(\\varphi q)^2-m^2\\varphi^2}}{(\\varphi q)}\\bigg)\\,\n    \\displaystyle -\\,\\varphi^\\mu \\,\\frac{\\zeta (f \\varphi) m^2}\n    {2(\\varphi q)\\sqrt{(\\varphi q)^2-m^2\\varphi^2}},\n\\end{array}\n\\end{equation}\n\\begin{equation}\\label{P3}\n\\displaystyle {S}_{{\\mathrm{tp}}}^\\mu(q) = \\frac{q^\\mu (\\varphi q) \/ m -\n    \\varphi ^\\mu m }{\\sqrt {(\\varphi q)^2 -\n    \\varphi ^2m ^2}}.\n\\end{equation}\n\\noindent System (\\ref{Wf1}) represents the complete system of solutions of equation\n(\\ref{6}) characterized by kinetic momentum of the particle\n$q^{\\mu}$ and the quantum number $\\zeta_{0}= \\pm 1$ which can be\ninterpreted as the neutrino spin projection on the direction\n${S}_{0}^\\mu(q)$ at $\\tau = (Nx)=0$.\n\nSystem (\\ref{Wf1}) is nonstationary in the general case. The\nsolutions are stationary only when the initial polarization vector\n$S_{0}^{\\mu}(q)$ is equal to the vector of the total polarization\n$S_{{\\mathrm{tp}}}^{\\mu}(q)$ \\cite{x}, $S_{0}^{\\mu}(q)=S_{{\\mathrm{tp}}}^{\\mu}(q)$. In this case the wave functions are eigenfunctions of the spin\nprojection operator ${\\mathfrak{S}}_{\\mathrm{tp}}=-\\gamma^5 \\gamma_{\\mu}{S}_{\\mathrm{tp}}^{\\mu}(q)$ with eigenvalues $\\zeta = \\pm 1$, and of the canonical momentum operator $p^{\\mu}=i\\partial^{\\mu}$ with eigenvalues $P^{\\mu}_{\\zeta}$. The orthonormal system of the stationary solutions, the basis of\nsolutions of equation (\\ref{6}), can be written as (see Appendix \\ref{ru})\n\\begin{equation}\\label{Wf}\n    \\varPsi_{q\\zeta}(x) = e^{ - i(P_{\\zeta}x)}\\sqrt{|J_{\\zeta}(q)|}\n    (1 - \\zeta\\gamma^5\\gamma_{\\mu}{S}_{\\mathrm{tp}}^{\\mu}(q))(\\gamma^{\\mu}{q}_{\\mu} + m)\\psi_0,\n\\end{equation}\n\\noindent where $J_{\\zeta}(q)$ is the transition Jacobian between the variables\n$q^\\mu$ and $P^\\mu_{\\zeta}$\n\\begin{equation}\\label{nor3}\n  J_{\\zeta}(q)=  {\\mathrm{det}}(M_{ij}) =\n    {\\mathrm{det}} \\left[ \\frac{\\partial P^i_{\\zeta}}{\\partial\n    q^j}+\\frac{\\partial P^{i}_{\\zeta}}{\\partial q^{0}} \\frac{\\partial q^{0}}\n    {\\partial q^j} \\right].\n\\end{equation}\n\\noindent With the help of relations from Appendix \\ref{ur}, the explicit form of the matrix\n$M_{ij}$ may be written as\n\\begin{equation*}\n\\begin{array}{c}\n    \\displaystyle M_{ij}=\\delta_{ij}\n    \\left( 1+\\zeta\\frac{(f\\varphi)}{2\\sqrt{(\\varphi q)^2-m^2 \\varphi^2}}\n    \\right)\\\\[20pt]\n    \\displaystyle+\\zeta\\left( q_i-\\varphi_i\\frac{m^2}{(\\varphi q)} \\right)\n    \\left( \\varphi_j-q_j\\frac{\\varphi^0}{q^0} \\right)\\frac{m^2(f \\varphi)^2+f^2\n    (\\varphi q)^2-m^2 f^2\\varphi^2}{ 4(\\varphi q)((\\varphi\n    q)^2-m^2\\varphi^2)^{3\/2}}.\n\\end{array}\n\\end{equation*}\n\\noindent One can easily derive the following equality for arbitrary\nvectors ${\\bf g}$ and ${\\bf h}$\n$$\\det{(\\delta_{ij}+g_{i}h_{j})}=1+({\\bf{g}}\\cdot{\\bf{h}}).$$ So we have\n\\begin{equation}\\label{Det}\n    J_{\\zeta}(q)={\\left({1+\\zeta\\frac{(f \\varphi)}\n    {2\\sqrt{(\\varphi q)^2-m^2 \\varphi^2}}}\\right)}^2\n    {\\left({1+\\zeta\\frac{f_{\\mu}H^{\\mu\\nu}q_{\\nu}\/(2m)-2\\mu_{0}^{2}\n    I_1}\n    {\\sqrt{(\\varphi q)^2-m^2 \\varphi^2}}}\\right)}.\n\\end{equation}\n\\noindent Here $I_{1}=\\frac{1}{4}F^{\\mu\\nu} F_{\\mu\\nu}$ is\nthe first invariant of the tensor $F^{\\mu\\nu}$. Note that to obtain\na complete system of solutions for the antineutrino, it is necessary to\nchange the sign of the kinetic momentum $q^{\\mu}$.\n\nThe dispersion law for the neutrino in dense magnetized matter is\ndifferent from the one for the free particle and can be written as (see\nrelations (\\ref{vvv}), (\\ref{vv}) and (\\ref{z5}) in Appendix \\ref{ur})\n\\begin{equation}\\label{PP3}\n    {\\Tilde{P}}^2=m^2-f^{2}\/4-2\\mu_{0}^{2} I_1-2\\zeta \\Delta\\sqrt{{(\\Tilde{P}}\n    {\\Tilde{\\Phi}})^2-{\\Tilde{\\Phi}}^{2}m^2},\n\\end{equation}\nwhere\n\\begin{equation}\\label{bbb}\n\\begin{array}{c}\n \\displaystyle   \\tilde{P}^\\mu=P^{\\mu}_{\\zeta}-f^{\\mu}\/2, \\quad\n    \\tilde{\\Phi}^\\mu=f^\\mu\/2+H^{\\mu\\nu}\\tilde{P}_\\nu\/{m},\\\\[8pt]\n    \\displaystyle\\Delta =\n{\\mathrm{sign}}{\\left({1+\\zeta\\frac{f_{\\mu}H^{\\mu\\nu}q_{\\nu}\/(2m)-2\\mu_{0}^{2}\n    I_1}\n    {\\sqrt{(\\varphi q)^2-m^2 \\varphi^2}}}\\right)}=\n    {\\mathrm{sign}}\\left(1+\\frac{f_{\\mu}H^{\\mu\\nu}\\tilde{P}_{\\nu}\/m-4\\mu_{0}^{2}\n    I_1}{\\tilde{P}^2 - m^2 +\nf^{2}\/4+2\\mu_{0}^{2}\n    I_1-(\\tilde{\\Phi}f)}\\right).\n\\end{array}\n\\end{equation}\n\\noindent The appearance of the factor $\\Delta$ in equation\n(\\ref{PP3}) is a consequence of the fact that $\\zeta$ is projection of the\nparticle spin on the direction defined by the kinetic\nmomentum instead of the canonical one.\n\nIn spite of the modification of the dispersion law described above, we\nsee that the neutrino moving through dense matter and\nelectromagnetic field may still behave as a free particle, i.e.\nits group velocity\n\\begin{equation}\\label{gr}\n{\\bf v}_{{\\mathrm{gr}}} = \\frac{\\partial {P}_{\\zeta}^{0}}{\\partial {\\bf\n    P}_{\\zeta}}= \\frac{{\\bf\n    q}}{{q}^{0}}\n\\end{equation}\n\\noindent is the same for both polarization states of the\nparticle. However, in interactions with other particles some\nchannels of reactions which are closed for a free neutrino can be\nopened due to the modification of the dispersion law (see, for example, \\cite{SpinLight}, \\cite{zlm}, \\cite{pairs}).\n\nLet us discuss now properties of nonstationary solutions in more\ndetail. Solution (\\ref{Wf1}) is a plane-wave solution of equation (\\ref{6}), describing a pure quantum-mechanical state of a neutral particle with a nonconserved spin projection on the fixed space axis. Solutions (\\ref{Wf1}) do not form an orthogonal basis. However, the\nconsidered system is not overcomplete, since the spectrum of the spin projection operator is finite. So the system can be easily orthogonalized.\nGeneralization of the basis (\\ref{Wf}) is\n\\begin{equation}\\label{Wf2}\n\\displaystyle \\tilde{\\varPsi}_{q\\zeta_{0}}(x) =\n\\frac{1}{2}\\sum\\limits_{\\zeta = \\pm 1} e^{-i(P_{\\zeta}\nx)}\\sqrt{|J_{\\zeta}(q)|}{( 1 - \\zeta \\gamma^5\n\\gamma_{\\mu}{S}_{{\\mathrm{tp}}}^{\\mu}(q))(1-\\zeta_{0}\\gamma^5\n\\gamma_{\\mu}{S}_{0}^{\\mu}(q))(\\gamma^{\\mu}{q}_{\\mu}+m)} \\psi_{0}.\n\\end{equation}\n\nThus we have just established that the unitary intertwining operator\n(\\ref{a018}) in our case is the Fourier integral operator \\cite{FIO} and it acts on elements of the plane-wave basis of solutions of the free particle Dirac equation (\\ref{a3}) in the following way:\n\\begin{equation}\\label{Wf3}\n\\displaystyle\n\\tilde{{\\varPsi}}_{q\\zeta_{0}}(x)=U(x,x_{0}){\\varPsi}_{{0}}(x) =\n\\frac{1}{2}\\sum\\limits_{\\zeta = \\pm 1} e^{-i((P_{\\zeta}-q)\nx)}\\sqrt{|J_{\\zeta}(q)|}{( 1 - \\zeta \\gamma^5\n\\gamma_{\\mu}{S}_{{\\mathrm{tp}}}^{\\mu}(q))}{\\varPsi}_{{0}}(x).\n\\end{equation}\n\\noindent Action of the inverse operator is defined by the formula\n\\begin{equation}\\label{Wf4}\n\\displaystyle {\\varPsi}_{{0}}(x)=\nU^{-1}(x,x_{0})\\tilde{{\\varPsi}}_{q\\zeta_{0}}(x) =\n\\frac{1}{2}\\sum\\limits_{\\zeta = \\pm 1} e^{i((P_{\\zeta}-q)\nx)}\\frac{1}{\\sqrt{|J_{\\zeta}(q)|}}{( 1 - \\zeta \\gamma^5\n\\gamma_{\\mu}{S}_{{\\mathrm{tp}}}^{\\mu}(q))}\\tilde{{\\varPsi}}_{q\\zeta_{0}}(x).\n\\end{equation}\n\\noindent Since the intertwining operator is defined on the elements of the basis,\nits action on an arbitrary solution is defined as well. Hence, the explicit\nform of this operator as a function of coordinates and\ndifferential operators can be easily obtained.\n\n\n\nNote that an attempt to construct intertwining operator for equation\n(\\ref{6}) was undertaken in \\cite{RL} for the case where only\nparameter $f^{\\mu}$ is nontrivial. The result of the action of the operator suggested in \\cite{RL} on plane-wave solutions of the Dirac equation for a free\nparticle coincides with solutions obtained in our\nprevious work \\cite{SpinLight} for this particular case, and which\ncan be derived from (\\ref{Wf1}), if one sets $F^{\\mu\\nu}=0$. We should emphasize that solutions (\\ref{Wf1}) do not form an orthogonal basis. As a\nconsequence of this fact, the intertwining operator suggested in\n\\cite{RL} is not unitary with regard to the standard scalar product\n(\\ref{a010}).\n\nThe case of a massless neutrino is quite special. Equation (\\ref{103})\nobviously does not hold\nin the limit $m \\rightarrow 0$\nbecause massless particle helicity is conserved, while the BMT equation\ndescribes spin rotation.\nTo find wave functions of massless neutrino, one  must take into account that from the mathematical point of view, the small\ngroup of representation of the\nPoincar\\'{e} group for a massive particle differ from that for a\nmassless particle  (see, for example, \\cite{BoLoTO}).\nThe small group for a\nmassive particle is the rotation group of three-dimensional Euclidian\nspace $O_{+}(3)$, and nontrivial magnetic and electric\nmoments are allowed. In contrast to that, the small\ngroup for a massless particle is the movement group of Euclidian plane\n$E(2)$, which outlaws nonzero magnetic moment as\nwell as electric one or other nontrivial coefficients $\\mu_{n}$ and\n${\\varepsilon '}_{\\!n}$ in equation (\\ref{1.0}).  Therefore, one has to put\n$\\mu_{0}=0$ in equation (\\ref{6}).\nThen instead of (\\ref{Wf}), we obtain\n\\begin{equation}\\label{Wfx}\n    \\varPsi_{q\\zeta}(x)|_{m=0} = e^{- i(qx) (1+\\zeta \\eta)}e^{-i(fx)(1-\\zeta)\/2}|1+\\zeta \\eta|\n    (1 - \\zeta\\gamma^{5})\\gamma^{\\mu}{q}_{\\mu}\\psi_0,\n\\end{equation}\n\\noindent for the neutrino, and\n\\begin{equation}\\label{Wfxx}\n    \\varPsi_{q\\zeta}(x)|_{m=0} = e^{i(qx) (1+\\zeta \\eta)}e^{-i(fx)(1+\\zeta)\/2}|1+\\zeta \\eta|\n    (1 + \\zeta\\gamma^{5})\\gamma^{\\mu}{q}_{\\mu}\\psi_0,\n\\end{equation}\n\\noindent for the antineutrino. In these equations $\\eta=\\frac{f^{2}}{2(fq)}$ and $q^{2}=0$.\n\n\\section{Operators of observables}\\label{IV}\n\nLet us find the explicit forms of the\nkinetic momentum operators ${\\mathfrak{p}}^{\\mu}$ and the spin projection operator ${\\mathfrak{S}}_{{\\mathrm{tp}}}$ in the coordinate representation. For this purpose we might exploit formulas (\\ref{a21});\nhowever, we follow a simpler way.\n\nThe obtained solutions (\\ref{Wf}) are classified by eigenvalues of the operators\n${\\mathfrak{p}}^{\\mu}$ and ${\\mathfrak{S}}_{{\\mathrm{tp}}}$, so\n\\begin{equation}\\label{operact} {\\mathfrak{p}}^\\mu\n\\varPsi_{q\\zeta}(x)=q^\\mu \\varPsi_{q\\zeta}(x), \\quad\n{\\mathfrak{S}}_{{\\mathrm{tp}}} \\varPsi_{q\\zeta}(x)=\\zeta \\varPsi_{q\\zeta}(x).\n\\end{equation}\n\\noindent Since solutions (\\ref{Wf}) are also eigenfunctions of the canonical\nmomentum operator $p^{\\mu}=i\\partial^{\\mu}$ with eigenvalues\n$P^{\\mu}_{\\zeta}$, we have\n\\begin{equation}\\label{operP}\n    {{p}}^{\\mu}\\varPsi_{q\\zeta}(x)={{P}}^{\\mu}_{\\zeta}\\varPsi_{q\\zeta}(x).\n\\end{equation}\n\\noindent Now we should express eigenvalues of the kinetic momentum\noperator ${q^{\\mu}}$ in terms of eigenvalues of the canonical\nmomentum operator ${P}^{\\mu}_{\\zeta}$. From (\\ref{P2}) we have\n\\begin{equation}\\label{z4}\n\\begin{array}{c}\n\\displaystyle  q^\\mu =\\Bigg[{\\tilde{P}}^{\\mu}+ f^{\\mu} \\frac{\\zeta\n\\sqrt{(\\varphi q)^2-m^2\\varphi^2}}{2(\\varphi q)}\n    +\\varphi^\\mu \\,\\frac{\\zeta (f \\varphi) m^2}\n    {2(\\varphi q)\\sqrt{(\\varphi\n    q)^2-m^2\\varphi^2}}\\Bigg]\\\\[12pt]\n\\times\\displaystyle \\Bigg[1+\\zeta\\frac{(f\\varphi)}\n    {2\\sqrt{(\\varphi q)^2-m^2\\varphi^2}}\n    \\Bigg]^{-1}.\n\\end{array}\n\\end{equation}\n\\noindent With the help of relations from Appendix \\ref{ur} this can be\nrewritten as\n\\begin{equation}\\label{z13.x}\n\\begin{array}{l}\nq^{\\mu}=\\displaystyle {\\tilde{P}}^{\\mu}+\\frac{\n    {\\tilde{P}}^{\\mu}(\\tilde{\\Phi}f)\n    - f^{\\mu}\n(f\\tilde{P})\/2\n    -2mH^{\\mu\\nu}\\tilde{\\Phi}_{\\nu}\n    }\n{\\tilde{P}^2 - m^2 + f^{2}\/4+2\\mu_{0}^{2}\n    I_1-(\\tilde{\\Phi}f)}.\n\\end{array}\n\\end{equation}\n\\noindent The vector of total polarization in terms of the new variable\nis\n\\begin{equation}\\label{z15}\n\\begin{array}{c}\n\\displaystyle {S}_{{\\mathrm{tp}}}^\\mu(q) = \\Delta\\frac{q^\\mu\n(\\tilde{\\Phi}\\tilde{P})\/m -\\tilde{\\Phi}^{\\mu} m }{\\sqrt\n{(\\tilde{\\Phi}\\tilde{P})^2-m^2\\tilde{\\Phi}^2}}.\n\\end{array}\n\\end{equation}\nHere $\\tilde{P}, \\tilde{\\Phi}$ and $\\Delta$ are given by\n(\\ref{bbb}).\n\nBecause of (\\ref{operP}) we can interpret ${{P}}^{\\mu}_{\\zeta}$  as a\nresult of action of operator ${p}^\\mu=i\\partial^{\\mu}$ on the wave\nfunction. So by changing ${\\tilde{P}}^\\mu\\Rightarrow {{p}}^\\mu-f^\\mu\/2$\nand $\\tilde{\\Phi}^\\mu\\Rightarrow\nf^\\mu\/2+H^{\\mu\\nu}(p_\\nu-f_\\nu\/2)\/{m}$ in formulas (\\ref{z13.x}),\n(\\ref{z15}), we obtain  kinetic momentum operator ${\\mathfrak\np}^\\mu$ and spin projection operator ${\\mathfrak\nS}_{{\\mathrm{tp}}}=-\\gamma^{5}\\gamma_{\\mu}S^{\\mu}_{{\\mathrm{tp}}}(q)$ in the explicit form. These operators are pseudodifferential ones \\cite{FIO} and are\ndetermined on the solutions of equation (\\ref{6}) with fixed mass\n$m$.\n\n\nTo extend the domain of the definition of constructed operators, we need\nto replace mass $m$ in (\\ref{z13.x}) and (\\ref{z15}) by the matrix\noperator from equation (\\ref{6}):\n\\begin{equation}\\label{z14}\n\\begin{array}{c}\n\\displaystyle m= \\gamma^{\\mu}{{{p}}}_{\\mu}  -\n\\frac{1}{2}\\gamma^{\\mu}{f}_{\\mu}(1 + \\gamma ^5) - \\frac{i}{2}\\mu_{0}F^{\\mu\n\\nu }\\sigma _{\\mu \\nu },\n\\\\[8pt]\\displaystyle m^{2}=\n\\displaystyle {(p-f\/2)}^{2}-f^{2}\/4 +2\\mu_{0}^{2}I_{1}\n+\\gamma^{5}\\sigma^{\\mu\\nu}{f}_{\\mu}{p}_{\\nu}\n+H^{\\mu\\nu}\\gamma_{\\mu}f_{\\nu}(1+\\gamma^{5})+\n2\\gamma^{5}H^{\\mu\\nu}\\gamma_{\\mu}{p}_{\\nu}.\n\\end{array}\n\\end{equation}\n\\noindent In this way we obtain the covariant form for ${\\mathfrak\np}^\\mu$ and ${\\mathfrak S}_{{\\mathrm{tp}}}$.\n\nUnfortunately, the result of this substitution cannot be written\nas a compact formula, so we do not present it here. However, even\nif we do not know a covariant form of the operator ${\\mathfrak p}^\\mu$,\nwe may conclude that on the solutions of equation (\\ref{6}) the relations,\n\\begin{equation}\\label{z1}\n{\\mathfrak p}^{2}=m^{2},\\quad \\gamma^\\mu{\\mathfrak p}_\\mu=m,\n\\end{equation}\n\\noindent should hold. The first equality in (\\ref{z1}) is obvious; to proof the\nsecond one, recall that according to (\\ref{a21})\n\\begin{equation}\\label{z1.0}\nm=U(x,x_{0})\\gamma^{\\mu}p_{\\mu}U^{-1}(x,x_{0})=\n  U(x,x_{0})\\gamma^{\\mu}U^{-1}(x,x_{0}){\\mathfrak{p}}_{\\mu}=\n  (\\gamma^{\\mu}+\\varGamma^{\\mu}(q))\\,{\\mathfrak{p}}_{\\mu}.\n\\end{equation}\n\\noindent From formulas (\\ref{Wf3}) and (\\ref{Wf4}) it follows that\n$\\varGamma^{\\mu}(q) \\sim S_{{\\mathrm{tp}}}^{\\mu}(q)$, and as a consequence\n$\\varGamma^{\\mu}(q){\\mathfrak{p}}_{\\mu}=0 $. So equation (\\ref{6})\nmay be represented as\n\\begin{equation}\\label{z3}\n(\\gamma^{\\mu}{{\\mathfrak{p}}_{\\mu}} -m)\\varPsi(x) = 0.\n\\end{equation}\n\nConsider now special cases where the presented technique looks quite\nclear. Discuss the influence on the\nneutrino dynamics of the electromagnetic field alone, i.e. assume that  $f^{\\mu}=0$. In this case\neigenvalues of the kinetic momentum operator ${q^{\\mu}}$ would be\nexpressible in terms of eigenvalues of the canonical momentum\noperator ${P}^{\\mu}_{\\zeta}$ in the following way:\n\\begin{equation}\\label{z24.0}\nq^{\\mu}\n=P^{\\mu}_{\\zeta}-\\frac{2H^{\\mu\\alpha}H_{\\alpha\\nu}P^{\\nu}_{\\zeta}}\n{P^{2}_{\\zeta}- m^{2}+2\\mu_{0}^{2}I_{1}}.\n\\end{equation}\n\\noindent Then the covariant form of the kinetic momentum operator is\n\\begin{equation}\\label{z25}\n{\\mathfrak{p}}^{\\mu}=p^{\\mu}+\n\\frac{H^{\\mu\\alpha}H_{\\alpha\\nu}p^{\\nu}}\n{\\sqrt{p^{\\beta}H_{\\beta\\alpha}H^{\\alpha\\rho}p_{\\rho}}}\\,\\tilde{{\\mathfrak{S}}}_{{\\mathrm{tp}}},\n\\end{equation}\n\\noindent and the spin projection operator ${\\mathfrak{S}}_{{\\mathrm{tp}}}$ is\ndefined by the formula\n\\begin{equation}\\label{z22}\n{\\mathfrak{S}}_{{\\mathrm{tp}}} ={\\mathrm{sign}}\n\\left(1+\\frac{2\\mu_{0}^{2}I_{1}}\n{\\sqrt{p^{\\beta}H_{\\beta\\alpha}H^{\\alpha\\rho}p_{\\rho}}}\n\\,\\tilde{{\\mathfrak{S}}}_{{\\mathrm{tp}}}\\!\\right)\n\\tilde{{\\mathfrak{S}}}_{{\\mathrm{tp}}}.\n\\end{equation}\n\\noindent Here\n\\begin{equation}\\label{z23}\n\\tilde{{\\mathfrak{S}}}_{{\\mathrm{tp}}}=\\frac{\\gamma^{5}\\gamma_{\\mu}H^{\\mu\\nu}p_{\\nu}}\n{\\sqrt{p^{\\beta}H_{\\beta\\alpha}H^{\\alpha\\rho}p_{\\rho}}}.\n\\end{equation}\n\\noindent Thus operator ${\\mathfrak{S}}_{{\\mathrm{tp}}} $ has a simple physical\nmeaning. It characterizes a particle spin projection on the\ndirection of the magnetic field in the rest frame of the particle.\n\nVice versa, when the electromagnetic field is absent, but $f^\\mu$ is\nnontrivial, we have\n\\begin{equation}\\label{z13.xx}\n\\begin{array}{l}\nq^{\\mu}=\\displaystyle {P}_{\\zeta}^{\\mu}-\\frac{1}{2}f^{\\mu}+\\frac{\n    {P}_{\\zeta}^{\\mu}f^{2}\n    - f^{\\mu}\n(f{P}_{\\zeta})} {2({P}_{\\zeta}^2 -({P}_{\\zeta}f)- m^2)},\n\\end{array}\n\\end{equation}\n\\noindent and the covariant forms of the kinetic momentum and spin\nprojection operators are\n\\begin{equation}\\label{z18.0}\n{\\mathfrak{p}}^{\\mu}=p^{\\mu}-\n\\frac{f^{\\mu}}{2}-\\gamma^{5}\\frac{p^{\\mu}f^{2}-f^{\\mu}(fp)\n}{2((pf)^{2}-p^{2}f^{2})}\\sigma^{\\alpha\\nu}{f}_{\\alpha}{p}_{\\nu},\n\\end{equation}\n\\begin{equation}\\label{z23.0}\n{\\mathfrak{S}}_{{\\mathrm{tp}}}=\\frac{\\gamma^{5}\\sigma^{\\mu\\nu}{f}_{\\mu}{p}_{\\nu}\n}{\\sqrt{(pf)^{2}-p^{2}f^{2}}}.\n\\end{equation}\n\\noindent Note that if the matter is at rest and nonpolarized\n $({\\bf f}=0)$, then\n\\begin{equation}\\label{z23.00}\n{\\mathfrak{S}}_{{\\mathrm{tp}}}={\\mathrm{sign}}(f^{0})\\frac{({\\bb \\Sigma}\\cdot{\\bf p})}\n{|{\\bf p}|},\n\\end{equation}\n\\noindent in other words ${\\mathfrak{S}}_{{\\mathrm{tp}}}$ is equal to the\nstandard helicity operator up to the sign.\n\nWe can find now spin projection operators for nonstationary wave\nfunctions (\\ref{Wf1}) and (\\ref{Wf2}). For this purpose, we introduce operators ${\\mathfrak{S}}_{\\pm}$ that act on the elements of system (\\ref{Wf}) as follows:\n\\begin{equation}\\label{sa}\n{\\mathfrak{S}}_{+} {\\varPsi}_{q\\zeta}(x) = \\frac{(1-\\zeta)}{2}{\\varPsi}_{q(-\\zeta)}(x),\\quad {\\mathfrak{S}}_{-} {\\varPsi}_{q\\zeta}(x) = \\frac{(1+\\zeta)}{2}{\\varPsi}_{q(-\\zeta)}(x).\n\\end{equation}\n\\noindent Then operators ${\\mathfrak{S}}_{1}=\\frac{1}{2}({\\mathfrak{S}}_{+}+{\\mathfrak{S}}_{-})$,\n${\\mathfrak{S}}_{2}=\\frac{1}{2i}({\\mathfrak{S}}_{+}-{\\mathfrak{S}}_{-})$ and\n${\\mathfrak{S}}_{3}=\\frac{1}{2}{\\mathfrak{S}}_{{\\mathrm{tp}}}$ correspond to elements of the Lie algebra of the $SU\\!(2)$ group. Commutation relations for these operators are\n\\begin{equation}\\label{W}\n    [{\\mathfrak{S}}_{i},{\\mathfrak{S}}_{j}]=ie_{ijk}{\\mathfrak{S}}_{k}.\n\\end{equation}\n\\noindent To determine the explicit realization of operators\n${\\mathfrak{S}}_{\\pm}$ on eigenfunctions of operator ${\\mathfrak{p}}^{\\mu}$ let us choose the basis $S_{i}^{\\mu}(q)$ (see (\\ref{a110})) in the form\n${S}^{\\mu}_{{\\mathrm{tp}}}(q), {S}^{\\mu}_{1}(q), S^{\\mu}_{2}(q)$.\nHere ${S}^{\\mu}_{{\\mathrm{tp}}}(q)$ is defined by relation (\\ref{P3}), the\nother spacelike unit vectors are\n\\begin{equation}\\label{z18}\n{S}^{\\mu}_{1\n}(q)\\!=\\!\\frac{S_{0}^{\\mu}(q)+S_{{\\mathrm{tp}}}^{\\mu}(q)(S_{0}(q)\nS_{{\\mathrm{tp}}}(q))}\n{\\sqrt{1-(S_{0}(q)S_{{\\mathrm{tp}}}(q))^{2}}},\\quad {S}^{\\mu}_{2\n}(q)\\!=\\!\\frac{e^{\\mu\\nu\\rho\\lambda}q_{\\nu}S_{0\\rho}(q)S_{{{\\mathrm{tp}}}\\lambda}(q)}\n{m\\sqrt{1-(S_{0}(q)S_{{\\mathrm{tp}}}(q))^{2}}}.\n\\end{equation}\n\\noindent As a result we have\n\\begin{equation}\\label{zz18}\n{\\mathfrak{S}}_{\\pm}=-\\frac{1}{2}\\frac{\\sqrt{|J_{\\zeta =\\pm 1}(q)|}}{\\sqrt{|J_{\\zeta =\\mp 1}(q)|}}e^{\\pm 2i\\theta}\\gamma^{5}\\gamma_{\\mu}\\left({S}^{\\mu}_{1}(q) \\pm i{S}^{\\mu}_{2}(q)\\right),\n\\end{equation}\n\\noindent where\n\\begin{equation}\\label{g3}\n\\theta = (Nx)\\sqrt {(\\varphi q)^2 -\n    \\varphi^2 m^2}\/m.\n\\end{equation}\n\nOperators\n${\\mathfrak{S}}_{{\\mathrm{tp}}}$ and ${\\mathfrak{S}}_{\\pm}$ are integrals of motion. So the spin projection operator $\\tilde{\\mathfrak{S}}_{0}$ that has eigenfunctions (\\ref{Wf2}) and eigenvalues $\\zeta_{0}= \\pm 1$ is a linear combination of these operators.\nWe can rewrite wave functions (\\ref{Wf2}) in the form\n\\begin{equation}\\label{Wf20}\n\\displaystyle \\tilde{\\varPsi}_{q\\zeta_{0}}(x) =\\sqrt{1-\\zeta_{0}\n(S_{0}(q)S_{{\\mathrm{tp}}}(q))}\\;\\frac{{\\varPsi}_{q\\zeta=1}(x)}{\\sqrt{2}}+\n\\zeta_{0}\\sqrt{{1+\\zeta_{0}(S_{0}(q)S_{{\\mathrm{tp}}}(q))}}\n\\;\\frac{{\\varPsi}_{q\\zeta=-1}(x)}{\\sqrt{2}}.\n\\end{equation}\n\\noindent So it is obvious that\n\\begin{equation}\\label{z19.1}\n\\begin{array}{c}\n\\displaystyle{\\tilde{\\mathfrak{S}}}_{0}=-(S_{0}(q)S_{{\\mathrm{tp}}}(q))\n{\\mathfrak{S}}_{{\\mathrm{tp}}}\n+\\displaystyle \\sqrt{1-(S_{0}(q)S_{{\\mathrm{tp}}}(q))^{2}}\\,\\left[\\,{\\mathfrak{S}}_{+}+\n{\\mathfrak{S}}_{-}\\right].\n\\end{array}\n\\end{equation}\n\nSimilarly to (\\ref{z19.1}), one can construct the integral of motion ${\\mathfrak{S}}_{0}$\nwith eigenfunctions (\\ref{Wf1}) and eigenvalues $\\zeta_{0}= \\pm 1$. Since\n\\begin{equation}\\label{Wf10}\n\\displaystyle {\\varPsi}_{q\\zeta_{0}}(x) =\\sqrt{1-\\zeta_{0}\n(S_{0}(q)S_{{\\mathrm{tp}}}(q))}\\;\\frac{{\\varPsi}_{q\\zeta=1}(x)}\n{\\sqrt{2|J_{\\zeta =+ 1}(q)|}}+\n\\zeta_{0}\\sqrt{{1+\\zeta_{0}(S_{0}(q)S_{{\\mathrm{tp}}}(q))}}\n\\;\\frac{{\\varPsi}_{q\\zeta=-1}(x)}{\\sqrt{2|J_{\\zeta =- 1}(q)|}},\n\\end{equation}\n\\noindent we have\n\\begin{equation}\\label{z19}\n\\begin{array}{c}\n\\displaystyle{\\mathfrak{S}}_{0}=-(S_{0}(q)S_{{\\mathrm{tp}}}(q))\n{\\mathfrak{S}}_{{\\mathrm{tp}}}\n+\\displaystyle \\sqrt{1-(S_{0}(q)S_{{\\mathrm{tp}}}(q))^{2}}\\left[\\frac{\\sqrt{|J_{\\zeta =- 1}(q)|}}{\\sqrt{|J_{\\zeta =+ 1}(q)|}}\\,{\\mathfrak{S}}_{+}+\n\\frac{\\sqrt{|J_{\\zeta =+ 1}(q)|}}{\\sqrt{|J_{\\zeta =- 1}(q)|}}\\,{\\mathfrak{S}}_{-}\\right].\n\\end{array}\n\\end{equation}\n\\noindent Or, in the other form,\n\\begin{equation}\\label{z19.2}\n\\begin{array}{c}\n\\displaystyle{\\mathfrak{S}}_{0}=\n\\gamma^{5}\\gamma_{\\mu}\\left\\{(S_{0}(q)S_{\\mathrm{tp}}(q)){{S}}_{\\mathrm{tp}}^{\\mu}(q)\n-\\displaystyle \\sqrt{1-(S_{0}(q)S_{\\mathrm{tp}}(q))^{2}}\n\\left[\\cos2\\theta\\,{S}^{\\mu}_{1}(q)\n-\\sin2\\theta\\,{S}^{\\mu}_{2}(q)\\right]\\right\\}.\n\\end{array}\n\\end{equation}\n\\noindent Note that operator (\\ref{z19}) is not a self-adjoint operator with respect\nto the standard scalar product (\\ref{a010}). It seems quite natural,\nsince wave functions (\\ref{Wf1}) do not form an orthogonal system. However, the\nsystem of wave functions is orthonormalized to the condition ``one particle in the unit volume.'' In this sense wave functions (\\ref{Wf1}) minimize the uncertainty relation which is due to (\\ref{W})\n\\begin{equation}\\label{uuu}\n    \\langle({\\mathfrak{S}}_{1}-\n    \\langle{\\mathfrak{S}}_{1}\\rangle)^{2}\\rangle\n    \\langle({\\mathfrak{S}}_{2}-\n    \\langle{\\mathfrak{S}}_{2}\\rangle)^{2}\\rangle=\\frac{1}{4}\n    \\langle{\\mathfrak{S}}_{3}\\rangle^{2}.\n\\end{equation}\n\\noindent Therefore, these wave functions describe spin-coherent states of the neutrino\n(on properties of coherent states see, for example, \\cite{MM}). The given system of spin-coherent states is parametrized by four-vector $S^{\\mu}_{0}(q)$.\n\n\n\n\\section{Quasiclassical interpretation}\\label{VV}\n\nLet us discuss in more detail the physical meaning of spin-coherent\nstates of the neutrino. For this purpose we construct vector and axial\ncurrents with  the help of (\\ref{Wf1}). The vector current is\n\\begin{equation}\\label{g1}\nj^{\\mu}_{V}=\\bar{\\varPsi}(x)\\gamma^{\\mu}{\\varPsi}(x)=q^{\\mu}\/q^{0},\n\\end{equation}\n\\noindent and we see that (\\ref{Wf1}) describes neutrino propagation\nwith the constant velocity ${\\bf{v}}_{{\\mathrm{gr}}} ={\\bf{q}}\/q^{0}$. The axial\ncurrent is\n\\begin{equation}\\label{ggg1}\nj^{\\mu}_{A}=\\bar{\\varPsi}(x)\\gamma^{5}\\gamma^{\\mu}\n{\\varPsi}(x)=\\zeta_{0}\\frac{m}{q^{0}}S^{\\mu}.\n\\end{equation}\n\\noindent Here\n\\begin{equation}\\label{g2}\nS^{\\mu}=-S_{{\\mathrm{tp}}}^{\\mu}(q)(S_{0}(q)S_{{\\mathrm{tp}}}(q))\n+\\left[S_{0}^{\\mu}(q)+S_{{\\mathrm{tp}}}^{\\mu}(q)\n(S_{0}(q)S_{{\\mathrm{tp}}}(q))\n\\right]\\cos2\\theta\n-\\frac{1}{m}e^{\\mu\\nu\\rho\\lambda}q_{\\nu}S_{0\\rho}(q)\nS_{{{\\mathrm{tp}}}\\lambda}(q)\\sin2\\theta,\n\\end{equation}\n\\noindent where $\\theta$ is determined by (\\ref{g3}). As expected\n(see (\\ref{V}) and (\\ref{A})), vector and axial currents coincide\nwith the solutions of the BMT equation, if the proper time is defined as\nin (\\ref{xx}).\n\nThe three-vector of spin $\\bb{\\zeta}$ can be expressed in terms of the\nfour-vector $S^{\\mu}$ components as\n\\begin{equation}\\label{ad201}\n   \\bb{\\zeta} ={\\bf{S}}-\\frac{{\\bf{q}}S^0}{q^{0}+m}.\n\\end{equation}\nThen for $\\bb{\\zeta}$ we have\n\\begin{equation}\\label{ad201x}\n   \\bb{\\zeta} ={\\bb \\zeta}_{{\\mathrm{tp}}}({\\bb \\zeta}_{0}\\cdot{\\bb \\zeta}_{{\\mathrm{tp}}})+\n   [{\\bb \\zeta}_{0}- {\\bb \\zeta}_{{\\mathrm{tp}}}({\\bb \\zeta}_{0}\\cdot{\\bb \\zeta}_{{\\mathrm{tp}}})]\n   \\cos2\\theta\n   -[{\\bb \\zeta}_{{\\mathrm{tp}}}\\times\n   {\\bb \\zeta}_{0}]\\sin2\\theta.\n\\end{equation}\n\\noindent Expression (\\ref{ad201x})\nhas a simple quasiclassical\ninterpretation.\n\nThe antisymmetric tensor $G^{\\mu \\nu}$ (see equation (\\ref{5})) can\nbe written in the standard form\n\\begin{equation}\nG^{\\mu \\nu}= (\\mu_{0}{\\bf P}, \\mu_{0}{\\bf M}\\,), \\label{g6}\n\\end{equation}\nwhere\n\\begin{equation}\\label{g7}\n{\\bf M}= (f^{0} {\\bf q}-q^{0}{\\bf f})\/(2\\mu_{0}m), \\quad {\\bf P}=-\n\\big[{\\bf q \\times {\\bf f}}\\big]\/(2\\mu_{0}m).\n\\end{equation}\n\\noindent Vectors ${\\bf P}$ and ${\\bf M}$ are analogous to the\npolarization and the magnetization vectors of medium. Note that the\nsubstitution $F^{\\mu\\nu}\\Rightarrow F^{\\mu\\nu}+G^{\\mu \\nu}\/\\mu_{0}$ implies\nthat the magnetic ${\\bf H}$ and electric ${\\bf D}$ fields are\nshifted by the vectors ${\\bf M}$ and ${\\bf P}$, respectively (we use\nhere notation common in electrodynamics of continuous media\n\\cite{LL})\n\\begin{equation}\\label{g8}\n{\\bf H} \\Rightarrow {\\bf B} ={\\bf H} +{\\bf M}, \\quad {\\bf D}\n\\Rightarrow {\\bf E}= {\\bf D} -{\\bf P}.\n\\end{equation}\n\\noindent  Thus restriction (\\ref{03}) in the explicit form is\n\\begin{equation}\\label{fields}\n({\\bf{E}}\\cdot{\\bf{f}})=0, \\quad\n  {\\bf{E}}f^0-[{\\bf{B}}\\times{\\bf{f}}]=0.\n\\end{equation}\n\\noindent This means that the Lorentz force and moment of force\nacting on matter are equal to zero, i.e. matter is at equilibrium\nstate within the accuracy of our consideration. In particular,\nthe vector of polarization of matter is parallel to the magnetic field if\nthe matter is at rest.\n\nIn the rest frame of the particle, equation (\\ref{x4}) can\nbe written in the form\n\\begin{equation}\\label{g11}\n\\dot{{\\bb \\zeta}}={2}\\mu_{0} \\big[ {{\\bb\\zeta} \\times {\\bf B}_0} \\big],\n\\end{equation}\n\\noindent where the spin vector $\\bb\\zeta$ is related to four-vector\n$S^{\\mu}$ by equation (\\ref{ad201}) and the value ${\\bf B}_0$  is\nthe effective magnetic field in the neutrino rest frame.  This field\ncan be expressed in terms of quantities determined in the laboratory\nframe,\n\\begin{equation}\\label{g12}\n\\begin{array}{l}\n{\\bf B}_0= \\displaystyle\\frac{1}{m}\\left[q^{0}{\\bf B} -\\big[{\\bf\nq}\\times{\\bf E}\\big]-\\frac{{\\bf q}({\\bf q}\\cdot{\\bf\nB})}{q^{0}+m}\\right]\\\\[12pt]\\displaystyle =\\frac{1}{m}\\left[q^{0}{\\bf H}\n-\\big[{\\bf q}\\times{\\bf D}\\big]-\\frac{{\\bf q}({\\bf q}\\cdot{\\bf\nH})}{q^{0}+m}+\\frac{\\bf q}{2\\mu_{0}} \\Big(f^{0}-\\frac{({\\bf q}\\cdot{\\bf\nf})}{q^{0}+m}\\Big)\\right] -\\frac{\\bf f}{2\\mu_{0}}.\n\\end{array}\n\\end{equation}\n\\noindent We see that the neutrino spin precesses around the direction ${\\bf\nB}_{0}$ with the frequency $\\omega = 2m|{\\bf B}_{0}|\/q^{0}$,\nthe angle between ${\\bf B}_{0}$ and the vector of spin being\n$\\vartheta = \\arccos({\\bb{\\zeta}}_{0}\\cdot{\\bb{\\zeta}}_{{\\mathrm{tp}}})$. The spin vector direction corresponding to stationary\nstates $\\bb{\\zeta}_{{\\mathrm{tp}}}$ is connected to the effective magnetic\nfield as follows:\n\\begin{equation}\\label{g13}\n{\\bb{\\zeta}}_{{\\mathrm{tp}}}=\\frac{{\\bf B}_0}{|{\\bf B}_0|}.\n\\end{equation}\n\\noindent This fact explains in a simple way the stationarity of states with\n$S^{\\mu}_{0}(q)=S^{\\mu}_{{\\mathrm{tp}}}(q)$.\n\nLet us introduce a flight length $L$ and a spin oscillation length\n$L_{{\\mathrm{osc}}}$ of the particle, remember that these are related by\n$\\theta =\\pi L\/L_{{\\mathrm{osc}}}$ and that the scalar product $(Nx)= \\tau$\nmay be interpreted as the proper time of a particle. The spin oscillation\nlength is\n\\begin{equation}\\label{g4}\nL_{{\\mathrm{osc}}} = \\frac{2\\pi{|\\bf{q}|}}{\\sqrt {(fq)^2 - f^{2}\nm^2 - 4m\nH^{\\mu\\nu}f_{\\mu}q_{\\nu}+4H^{\\mu\\alpha}\nH_{\\alpha\\nu}q_{\\mu}q^{\\nu}}}.\n\\end{equation}\n\n\nWe can now write the probability for the neutrino which was arisen\nwith polarization ${\\bb \\zeta}_{0}$ to change the polarization to\n${-\\bb \\zeta}_{0}$ after traveling some distance $L$. This is\n\\begin{equation}\\label{g14}\n    W= \\sin^{2}\\vartheta\n    \\sin^{2}(\\pi L\/L_{{osc}}).\n\\end{equation}\n\\noindent It is clear that when $({\\bb{\\zeta}}_{0}\\cdot{\\bb\n{\\zeta}}_{{\\mathrm{tp}}})=0$ or $\\vartheta =\\pi\/2$, the probability can be\nequal to unity and the resonance takes place.\n\nIf the neutrino has the fixed helicity in the initial state\n\\begin{equation}\\label{g003}\nS_{0}^{\\mu}(q) =\\frac{1}{m}\\left\\{{|\\bf q|},\nq^{0}{\\bf q}\/{|\\bf q|}\\right\\},\\quad\n{\\bb{\\zeta}}_{0}= \\frac{{\\bf q}}{|{\\bf q}|},\n\\end{equation}\n\\noindent then $\\sin^{2}\\vartheta = 1-({\\bf{B}}_{0}\\cdot{\\bf\nq})^{2}\/|{\\bf{B}}_{0}|^{2}|{\\bf q}|^{2}$ and formula (\\ref{g14})\nsimplifies to the result widely discussed in the literature.\nThis fact is not surprising. As was mentioned in Introduction, the Schr\\\"{o}dinger-type equation with an effective Hamiltonian,\nwhich was used in papers \\cite{theor} for obtaining this\nresult, is merely the BMT equation in the spinor representation.\n\n\n\\section{Discussion and conclusions}\\label{VI}\n\nLet us compare our results with those of the standard\nquantum-mechanical approach to this problem. Since the neutrino\nbehavior in dense matter under the influence of magnetic field, to\nthe best of our knowledge, was not investigated before, we consider\nthe neutrino propagation in a constant homogeneous magnetic field\nalone.\n\nIn the studies of the influence of a stationary pure magnetic field on the\nneutrino spin rotation in the pioneer paper \\cite{Fu}, as well as in\nother papers, the stationary solutions\n$\\varPsi_{p{\\zeta}}(x)$ first found in \\cite{1} were used as\nthe wave functions of a particle. These solutions are the\neigenfunctions of the canonical momentum operator $p^{\\mu}$ and of\nthe spin projection operator ${{\\mathfrak{S}}}_{{\\mathrm{tp}}}$ (see (\\ref{z22})).\nThe description of the neutrino spin rotation there is based on\nsolving the Cauchy problem where the initial condition is chosen in\nsuch a way that the mean value of neutrino helicity is equal to $\\pm\n1$. It was taken for granted that the solution of the Cauchy problem can be\nexpressed as a linear combination of the above-mentioned wave\nfunctions:\n\\begin{equation}\\label{zx}\n\\varPsi(x)=\\sum_{{\\zeta} = \\pm\n1}c_{{\\zeta}}(p)\\varPsi_{p{\\zeta}}(x).\n\\end{equation}\n\\noindent However, such an assumption is incorrect.\nThe point is that, once in a pure state the mean value of some\nspin operator is equal to  $\\pm 1$, then this state is described by\nan eigenfunction of this operator. In the general case, the construction of the eigenfunction of the spin projection\noperator as a superposition of only positive-energy\nsolutions of  equation (\\ref{6}) is possible only when\nthis spin projection operator commutes with the operator of the sign of the energy.\nThe standard helicity operator $({\\bb \\Sigma}\\cdot{\\bf p})\/|{\\bf p}|$ does\nnot feature it.\n\nThe given phenomenon is a sort of the famous Klein paradox\n\\cite{Klein}. To avoid the indicated difficulties, in relativistic\nquantum mechanics only self-adjoint\noperators in the subspace of wave functions with a fixed energy sign can\nbe treated as operators of observables.\nThe choice of integrals of motion as operators of observables is the\nnecessary condition to satisfy this requirement \\cite{LanP}.\n\nIn the case considered the canonical momentum operator is an integral of motion. However, the conserved operator of the spin projection which should set\ninitial conditions to the Cauchy problem is uniquely --- up to the sign --- determined by the form of the Dirac--Pauli equation. This operator is ${{\\mathfrak{S}}}_{{\\mathrm{tp}}}$.\nTherefore, it is impossible to\nconstruct a wave function describing a neutrino with rotating spin\nin the form of an eigenfunction of the canonical momentum operator\nfor its arbitrary eigenvalues.\nThe solutions similar to (\\ref{zx}) can exist\nwhen the special values of the canonical\nmomentum are chosen. So, if a neutrino moves parallel or\nperpendicular to a constant homogeneous magnetic field,\neigenfunctions of the helicity\noperator are the superpositions of positive-energy solutions alone \\cite{BTZ}.\n\n\n\nTo solve the problem we abandon\nthe view that eigenvalues of the canonical momentum operator always impose\na direction of the particle propagation.\nWe found a self-adjoint operator ${\\mathfrak{p}}^{\\mu}$ which\ncan be interpreted as kinetic momentum operator of the particle and\nobtained the complete orthonormal system of the solutions of\nequation (\\ref{6}) with elements which are eigenfunctions of the given\noperator. On the base of this system we constructed solutions\ndescribing the neutrino with rotating spin. So our results enable one to\ntreat a possible effect of the neutrino polarization change as a\nreal precession of the particle spin.\n\nConsequently, the problem of\nneutrino spin rotation in dense matter and in strong electromagnetic\nfields is solved in full agreement with the basic principles of\nquantum mechanics. Using the wave functions of orthonormal\nbasis (\\ref{Wf}) or spin-coherent wave functions (\\ref{Wf1}),\nit is possible to calculate probabilities of various processes with\nthe neutrino in the framework of the Furry picture. When choosing\none or another type of the basis, it is necessary to take into\naccount that, due to the time-energy uncertainty, stationary\nstates of the neutrino can be generated only when the linear size of\nthe area occupied by the electromagnetic field and the matter is\ncomparable in the order of magnitude with the formation length of the process\n--- the spin oscillation length in our case.\n\n\\acknowledgments\n\nThe authors are grateful to V. G. Bagrov, A. V. Borisov, Ya. N. Istomin, O. S.\nPavlova, A. E. Shabad, and V.~Ch.~Zhukovsky  for helpful discussions.\n\n\\bigskip\n\nThe work was supported in part by grants of the President of the\nRussian Federation for leading scientific schools (Grant No. SS ---\n65255.2010.2 and Grant No. SS ---\n4142.2010.2).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe discovery of the D-branes, as an inevitable part\nof the string theory \\cite{1}, induced to study the\nproperties and interactions of the branes. One of the\nmost applicable methods for this purpose, is the\nboundary state formalism. Boundary state is a BRST\ninvariant state which describes the creation of closed\nstring from vacuum.\n\nAmong achievements in this formalism is extending it to the\nsuperstring theory and considering the contribution of the conformal\nand super conformal ghosts to the boundary state \\cite{2}. There are\nseparate studies which add background fields such as the Kalb-Ramond\nfield $B_{\\mu\\nu}$, $U(1)$ gauge field in the compact spacetime\n\\cite{3} and the tachyon field \\cite{4, 5, 6} to the subject of\nboundary state. These background fields give a more general feature\nto the subject. Apart from the longitudinal fluctuations of the\nbrane (for instance the $U(1)$ gauge field and tachyon field),\ntransverse fluctuations of the brane \\cite{7} should also be\nconsidered. This enables us to interpret it as a dynamical object.\nThis can be performed by considering velocity for the brane \\cite{8,\n9}. These motivated us to take into account all background fields\nand also compactification of some directions of the spacetime to\nstudy moving branes in a general framework of superstring theory.\nThis general set-up can not be found in the literatures of the\nboundary state and branes' interaction.\n\nSince open strings are quantum excitations of brane \\cite{10},\npresence of the open string tachyon reveals the instability of the\nbrane. In the bosonic string theory this is a natural property,\nwhile in the superstring theories this occurs in special cases. For\ninstance there are D$p$-branes with wrong dimensions in the type IIA\nand type IIB superstring theories; That is, there are D$p$-branes\nwith odd dimensions in the type IIA theory and even dimensions in\nthe type IIB theory! \\cite{11} which are unstable. Actually this\ninstability can be removed by rolling of the tachyon toward its\nminimum potential \\cite{12}. During this process tachyon energy\ndissipates to the bulk modes and an unstable system reach to a\nstable state which consists of lower dimensional branes or just the\nclosed string vacuum without any D-brane \\cite{10}. Usually in the\nliterature the tachyon field has been considered in just one\ndimension and its effects have been studied on a space-filling\nbrane, while in the present paper we consider a D$p$-brane with\narbitrary dimension, and hence the tachyon field possesses\ncomponents along all directions of the brane worldvolume.\n\nIn this manuscript we calculate the boundary state\ncorresponding to a moving D$p$-brane in the presence\nof the background fields $B_{\\mu\\nu}$, $U(1)$ gauge\nfield and tachyon. Then we use this boundary state to\ndetect the interaction between two moving D-branes.\nThere is no restriction on the branes' dimensions and\nthey can be parallel or perpendicular to each other.\nTo keep the generality, we let some\nof the spacetime directions be compact. We will\nobserve that presence of the tachyon prevents the\nclosed string from wrapping around the compact directions.\nUsing the boundary state, we calculate the interaction\namplitude between two branes in the NS-NS and the R-R\nsectors. Due to the presence of the velocities and\nthe background tachyon fields there is no cancellation\nbetween these amplitudes. This occurs\neven for the similar and parallel D$p$-branes with the\nsame background fields. We shall observe that the\ninteraction amplitude vanishes after a long time (or\nequivalently for large distances of branes). The origin of\nthis effect is the rolling of the background tachyon\nfield and decaying of the D-branes in this limit.\n\nSo putting all these together allows us to study\na system in the most general feature to obtain\nconsiderable results in spite of some mathematical\ndifficulties because of considering longitudinal\nand transverse fluctuations simultaneously.\n\\section{The boundary state associated with a D$p$-brane}\n\nTo obtain the boundary state corresponding to a\nmoving brane in the presence of the antisymmetric\nfield $B_{\\mu\\nu}$ in bulk, and tachyon and $U(1)$\ngauge fields on the boundary we consider the following\nsigma-model action for closed string\n\\begin{eqnarray}\nS =&~& -\\frac{1}{4\\pi\\alpha'} {\\int}_\\Sigma\nd^{2}\\sigma(\\sqrt{-g}g^{ab}G_{\\mu\\nu}\\partial_a\nX^{\\mu}\\partial_b X^{\\nu}+\\varepsilon^{ab}\nB_{\\mu\\nu}\\partial_a X^{\\mu}\\partial_b X^{\\nu})\n\\nonumber\\\\\n&~&\n+\\frac{1}{2\\pi\\alpha'} {\\int}_{\\partial\\Sigma} d\\sigma\n\\bigg{(} A_\\alpha\n\\partial_{\\sigma}X^{\\alpha}+V^{i}X^{0}\\partial_{\\tau}X^{i}\n+\\frac{1}{2}U_{\\alpha\\beta}X^{\\alpha}X^{\\beta}\\bigg{)},\n\\end{eqnarray}\nwhere the first integral is on the worldsheet of the closed string,\nexchanged between the branes, and the second one is on the boundary\nof this worldsheet which can be at $\\tau=0$ or $\\tau=\\tau_{0}$. The\n$U(1)$ gauge field $A_{\\alpha}$, lives on the D$p$-brane worldvolume\nand $V^{i}$ is the brane velocity component along $X^{i}$-direction.\nThe set $\\{X^{\\alpha}\\}$ and $\\{X^{i}\\}$ specify the directions\nalong and perpendicular to the D$p$-brane worldvolume, respectively.\nThe term $\\frac{1}{2}U_{\\alpha\\beta}X^{\\alpha}X^{\\beta}$ with\nconstant symmetric matrix $U_{\\alpha\\beta}$ specifies the tachyon\nprofile. According to \\cite{13} the tachyon field appears in a\nsquare form in the action to produce a Gaussian integral. We take\nthe tachyon field to have components along the D$p$-brane\nworldvolume. Here we consider $G_{\\mu\\nu}$ as the flat spacetime\nmetric with the signature $\\eta_{\\mu\\nu}={\\rm diag}\n(-1,1,\\cdot\\cdot\\cdot,1)$ and the Kalb-Ramond field $B_{\\mu\\nu}$ to\nbe a constant field.\n\nVanishing the variation of the action (1) with respect to\n$X^{\\mu}(\\sigma,\\tau)$, gives us the equations of motion as well as\nthe boundary equations of the emitted (absorbed) closed string.\n\\subsection{Bosonic part of the boundary state}\n\nBoundary equations resulted from the action (1) at\n$\\tau=0$ are as in the following\n\\[\n[\\partial_{\\tau} (X^{0}-V^{i}X^{i}) + {\\cal {F}}^{0}_{\\;\\;\\;\n\\alpha}\\partial_{\\sigma}X^{\\alpha}-U^{0}_{\\;\\;\\;\\alpha}\nX^{\\alpha}]|B_{x},\\tau=0\\rangle =0,\n\\]\n\\[\n(\\partial_{\\tau} X^{\\bar{\\alpha}} + {\\cal\n{F}}^{\\bar{\\alpha}}_{\\;\\;\\;\\beta}\\partial_{\\sigma}X^{\\beta}\n-U^{\\bar{\\alpha}}_{\\;\\;\\;\\beta}X^{\\beta}) |B_{x}, \\tau=0\\rangle\n=0,\\]\n\\begin{equation}\n(X^{i}-V^{i}X^{0}-y^{i})|B_{x}, \\tau=0\\rangle =0.\n\\end{equation}\nIn above boundary conditions $X^{\\overline{\\alpha}}$ shows the\nspatial directions of the brane worldvolume (i.e.\n$\\overline{\\alpha}\\neq0$) and $\\cal {F}$ is the total field\nstrength,\n${\\cal{F}}_{\\alpha\\beta}=\\partial_{\\alpha}A_{\\beta}-\\partial_{\\beta}\nA_{\\alpha}-B_{\\alpha\\beta}$, which contains $B$ field as well as the\n$U(1)$ gauge field. Note that we have assumed the mixed elements of the\nKalb-Ramond field to be zero, i.e, $B^{\\alpha}_{\\;\\;\\;i}=0$.\n\nThe solution of the closed string equation of motion is\n\\begin{equation}\nX^{\\mu}(\\sigma,\\tau)=x^{\\mu}+2\\alpha^{'}p^{\\mu}\\tau+\n2L^{\\mu}\\sigma+\\frac{i}{2}\n\\sqrt{2\\alpha^{'}}\\sum_{m\\neq0}\\frac{1}{m}(\\alpha^{\\mu}_{m}\ne^{-2im(\\tau-\\sigma)}\n+\\tilde{\\alpha}^{\\mu}_{m}e^{-2im(\\tau+\\sigma)}).\n\\end{equation}\n$L^{\\mu}$ is zero for non-compact directions, and\n$L^{\\mu}=N^{\\mu}R^{\\mu}$ for the compact direction $X^\\mu$ with\nthe compactification radius $R^{\\mu}$ and the closed string winding\nnumber $N^{\\mu}$. The closed string center of mass momentum is\n$p^{\\mu}=\\frac{M^{\\mu}}{R^{\\mu}}$, where $M^{\\mu}$ is the momentum\nnumber of it. Substituting this solution into the boundary equations\n(2), gives them in terms of oscillators and zero modes.\nDuring this process an interesting condition on the closed string\nwinding is obtained\n\\[\nU^{\\alpha}_{\\;\\;\\;\\overline{\\beta}}L_{\\rm op}^{\\overline{\\beta}}\n|B_{x}, \\tau=0 \\rangle=0 .\n\\]\nWe assumed that there is no compactification along time\ndirection and and hence $L^0=0$.\nIn the case of invertiblity of the matrix\n$U^{\\overline{\\alpha}}_{\\;\\;\\;\\overline{\\beta}}$, this\nequation reduces to\n$L_{\\rm op}^{\\overline{\\alpha}}|B_{x}, \\tau=0 \\rangle=0$.\nTherefore, the presence of\nthe background tachyon field prevents closed string from wrapping\naround compact directions which are parallel to the\nbrane worldvolume.\n\nUtilizing the coherent state method \\cite{14} to solve the boundary\nequations (2) for oscillating modes leads to the following state\n\\begin{equation}\n|B_{\\rm osc},\\tau=0\\rangle=\\prod^{\\infty}_{n=1}[\\det\nM_{(n)}]^{-1}\\exp\\bigg {[}-\\sum_{m=1}^{\\infty}\\bigg\n{(}\\frac{1}{m}\\alpha^{\\mu}_{-m}{\\cal{S}}_{(m)\\mu\\nu}\n\\widetilde{\\alpha}^{\\nu}_{-m}\\bigg {)}\\bigg{]}|0\\rangle,\n\\end{equation}\nwhere the matrix ${\\cal{S}}_{(m)}$ is defined by\n\\begin{eqnarray}\n&~&\n{\\cal{S}}_{(m)}=S_{(m)}+{(S_{(-m)}^{-1})}^{T},\n\\nonumber\\\\\n&~& S_{(m)}=M_{(m)}^{-1}N_{(m)}.\n\\end{eqnarray}\nThe matrices\n$M_{(m)}$ and $N_{(m)}$ which are functions of background fields\nare defined by\n\\begin{equation}\nM^{\\mu}_{(m)\\;\\nu}=\\Omega^{\\mu}_{\\;\\;\\;\\nu}-\\frac{i}{2m}\nU^{\\alpha}_{\\;\\;\\;\\beta}\\delta^{\\mu}_{\\;\\;\\;\\alpha}\n\\delta^{\\beta}_{\\;\\;\\;\\nu}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{rcl}\n&~&\\Omega^{0}_{\\;\\;\\;\\mu}=\n\\delta^{0}_{\\;\\;\\mu}-V^{i}{\\delta^{i}}_{ \\mu}-{{\\cal\n{F}}^{0}}_{\\alpha}{\\delta^{\\alpha}}_{\\mu}, \\\\\n&~&\\Omega^{\\bar{\\alpha}}_{\\;\\;\\;\\mu}=\\delta^{\\bar{\\alpha}}_{\\;\\;\\;\\mu}\n-{{\\cal{F}}^{\\bar{\\alpha}}}_{\\beta}{\\delta^{\\beta}}_{\n\\mu}, \\\\\n&~& \\Omega^{i}_{\\;\\;\\;\\mu}=\\delta^{i}_{\\;\\;\\;\\mu}\n-V^{i}{\\delta^{0}}_{\\mu},\n\\end{array}\\right.\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\left\\{ \\begin{array}{rcl}\n&~& N^{0}_{(m)\\mu}={\\delta^{0}}_{ \\mu}-V^{i}{\\delta^{i}}_{ \\mu}+\n{{\\cal {F}}^{0}}_{\\alpha}{\\delta^{\\alpha}}_{\\mu}\n+\\frac{i}{2m}U^{0}_{\\;\\;\\alpha}\\delta^{\\alpha}_{\\;\\;\\mu},\\\\\n&~& N^{\\bar{\\alpha}}_{(m)\\mu}={\\delta^{\\bar{\\alpha}}}_{\\mu}\n+{{\\cal {F}}^{\\bar{\\alpha}}}_{\\beta}{\\delta^{\\beta}}_{\\mu}\n+\\frac{i}{2m}U^{\\bar{\\alpha}}_{\\;\\;\\beta}\n\\delta^{\\beta}_{\\;\\;\\mu}, \\\\\n&~& N^{i}_{(m)\\mu}=-{\\delta^{i}}_{ \\mu}+V^{i}{\\delta^{0}}_{ \\mu}.\n\\end{array}\\right.\n\\end{eqnarray}\nWhen we solve the boundary equations, the matrix\n${(S^{-1}_{(-m)})}^{T}$ also appears in the Eq. (5). This is due to\nthe fact that the matrix $S_{(m)}$ is mode-dependent and generally\nis not orthogonal. In the absence of the tachyon field, $S$ becomes\nmode-independent and orthogonal, so ${\\cal{S}}=S$ \\cite{3}. The\ninfinite product in the Eq. (4), which comes from path integral, can\nbe regularized \\cite{15} as\n\\begin{equation}\n\\prod^{\\infty}_{n=1}[\\det M_{(n)}]^{-1}=\\sqrt{\\det\n\\Omega}\\;\\det\\bigg{[}\\Gamma \\bigg{(}\\frac{U}{1+2i\\Omega}\n\\bigg{)}\\bigg{]}.\n\\end{equation}\n\nFrom now on we consider a selected direction $X^{i_{0}}$ for the\nmotion of the D$p$-brane and hence the other components of the\nvelocity are zero. We also define $V^{i_{0}}=V$. By this assumption,\nthe zero mode part of the boundary state becomes\n\\begin{eqnarray}\n{|B_{x},\\tau=0\\rangle}^{(0)}=&~&\\frac{T_{p}}{2}\n{\\int^{\\infty}_{-\\infty}}\n\\prod_{\\alpha}dp^{\\alpha}\\;\\bigg{\\{} \\exp\\bigg{[}-4i\\alpha'\n(U^{-1})_{\\alpha\\beta}\\bigg{(}(1-\\frac{1}{2}\n\\delta_{\\alpha\\beta})\np^{\\alpha}p^{\\beta}+Vp^{i_{0}}p^{\\beta}\\delta^{\\alpha}_{\\;0}\n\\bigg{)}\\bigg{]}\n\\nonumber\\\\\n&~&\\times\\delta(x^{i_{0}}-Vx^{0}-y^{i_{0}}) \\prod_{i'\\neq\ni_{0}}\\delta(x^{i'}-y^{i'})\n\\nonumber\\\\\n&~&\\times\\prod_{\\alpha}|p^{\\alpha}_{L} =p^{\\alpha}_{\\rm R}\\rangle\n\\prod_{i'\\neq\ni_{0}}|p^{i'}_{L}=p^{i'}_{\\rm R}=0\\rangle\n|p^{i_{0}}_{L}=p^{i_{0}}_{\\rm R}=\\frac{1}{2}Vp^{0}\\rangle\n\\bigg{\\}}.\n\\end{eqnarray}\nTwo delta functions indicate the position of the\nbrane along the perpendicular directions. The integration\nover the momenta indicates that the effects of all values of\nthe momentum components have been taken into account.\nIn addition, the equality $p_L^\\alpha = p_R^\\alpha$ originates\nfrom the un-wrapping of the closed string around the brane\ndirections and non-compactness of the time direction.\n\nThere are two special limiting cases for $U$. In the limit\n$U_{\\alpha\\beta}\\rightarrow0$, the oscillating part of the boundary\nstate, i.e. the Eq. (4), reduces to a boundary state corresponding\nto a moving D$p$-brane in the absence of tachyon field, \\cite{9}.\n\nWhen we send some of the elements of $U$ to infinity, somehow\nwe are looking at the\nboundary state in the concept of tachyon condensation. This\ncondensation can be performed on some or all elements of the\ntachyon matrix $U$. Without loss of generality consider $U$ as a\ndiagonal matrix. By sending to infinity a spatial element\n$U_{\\overline{\\alpha}\\overline{\\alpha}}$, the boundary state\ntransforms to the one related to a moving D$(p-1)$-brane which has\nlost its dimension along the $X^{\\bar \\alpha}$-direction, and is\nin the presence of a new tachyon field $U'_{(p-1)\\times(p-1)}$\nwhich does not include the component\n$U_{\\overline{\\alpha}\\overline{\\alpha}}$.\n\nNotable point here is that although in the process of condensation\nalong the $X^{\\overline{\\alpha}}$-direction the matrices $M$ and\n$\\cal{S}$ in boundary state (4) change to lower dimensional ones, as\nexpected, the effect of condensated component remains as a\n$\\sqrt{U_{\\overline{\\alpha}\\overline{\\alpha}}}$ factor after\nregularization of infinite product $\\prod^{\\infty}_{n=1}[\\det\nM_{(n)}]^{-1}$. This result is different from the conventional case\nin which this factor is canceled by the factor\n$\\frac{1}{\\sqrt{U_{\\overline{\\alpha}\\overline{\\alpha}}}}$ from zero\nmode part, that is absent here.\n\nWhen condensation occurs along the time component of the tachyon\nmatrix, $U_{00}\\rightarrow\\infty$, beside the decreasing of the\nbrane worldvolume dimension in the $X^{0}$-direction, the brane loses its\nvelocity, too. In other words, the tachyon condensation\nalong the temporal direction fixes the D$p$-brane in time and\nspace, i.e. makes an instantonic D$p$-brane which has no\nvelocity.\n\n\\subsection{Fermionic part of the boundary state}\n\nTo find boundary equations for the fermionic degrees of freedom\nthere are two ways: 1) supersymmetrizing the action (1) and putting\nthe variation of the fermionic part of the action equal to zero; 2)\nSince the supersymmetrized action is invariant under the global\nworldsheet supersymmetry transformations, we can perform the\nworldsheet supersymmetry on the bosonic boundary Eqs. (2) and\ntransform them to fermionic ones. Here we choose the second\napproach. So the fermionic boundary equations are\n\\begin{eqnarray}\n&~& [-i\\eta(\\psi^{0}_{+}-V^{i}\\psi^{i}_{+})\n+(\\psi^{0}_{-}-V^{i}\\psi^{i}_{-})+{\\cal{F}}^{0}_{\\;\\;\\;\\alpha}\n(-i\\eta\\psi^{\\alpha}_{+}-\\psi^{\\alpha}_{-})\n\\nonumber\\\\\n&~& -U^{0}_{\\;\\;\\;\\nu}(-i\\eta\\psi^{\\nu}_{+}+\\psi^{\\nu}_{-})]|B_{\\psi},\n\\eta, \\tau=0\\rangle=0,\n\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n[-i\\eta\\psi^{\\overline{\\alpha}}_{+}\n-\\psi^{\\overline{\\alpha}}_{-}\n+{\\cal{F}}^{\\overline{\\alpha}}_{\\;\\;\\;\\beta}\n(-i\\eta\\psi^{\\beta}_{+}+\\psi^{\\beta}_{-})\n-U^{\\overline{\\alpha}}_{\\;\\;\\;\\nu}(-i\\eta\\psi^{\\nu}_{+}-\n\\psi^{\\nu}_{-})]|B_{\\psi}, \\eta, \\tau=0\\rangle=0,\n\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n[-i\\eta (\\psi^{i}_{+}-V^{i}\\psi^{0}_{+})\n- (\\psi^{i}_{-}-V^{i}\\psi^{0}_{-})]|B_{\\psi},\n\\eta,\\tau=0\\rangle=0.\n\\end{eqnarray}\nWith respect to the solution of the equations of motion for the\nfermions,\n\\begin{equation}\n\\psi^{\\mu}_{-}=\\sum_{k}\n\\psi^{\\mu}_{k}e^{-2ik(\\tau-\\sigma)}\\;\\;\\;\\;,\\;\\;\\;\\;\n\\psi^{\\mu}_{+}=\\sum_{k}\n\\widetilde{\\psi}^{\\mu}_{k}e^{-2ik(\\tau+\\sigma)},\n\\end{equation}\nthe boundary state Eqs. (11) can be represented as\n\\begin{equation}\n(\\psi^{\\mu}_{k}-i\\eta\n{S^{\\mu}}_{(k)\\nu}\\tilde{\\psi}^{\\nu}_{-k})|B_{\\psi},\\eta,\n\\tau=0\\rangle =0.\n\\end{equation}\nNote that in the Eqs. (12) and (13), $k$ is an integer number $m$\nfor the R-R sector,\n$\\psi^{\\mu}_{m}=d^{\\mu}_{m}$ and\n$\\widetilde{\\psi}^{\\mu}_{m}=\\widetilde{d}^{\\mu}_{m}$\nwhile in the NS-NS sector, $k$ is a half-integer number $r$ with\n$\\psi^{\\mu}_{r}=b^{\\mu}_{r}$ and\n$\\widetilde{\\psi}^{\\mu}_{r}=\\widetilde{b}^{\\mu}_{r}$. The constant\nnumber $\\eta$ can be $+1$ or $-1$. No matter we choose $+1$ or $-1$,\nbecause for gaining the interaction of the branes, we need to use\nthe boundary state which has been affected by the GSO projector. As\nwill be seen, this projection operator causes the both states with\n$\\eta=+1$ and $\\eta=-1$ to contribute to the interaction.\n\nSimilar to the bosonic part we should also consider the portion of\nthe super conformal ghosts in the fermionic boundary state. The\nsuper ghosts include the commuting fields $\\beta$, $\\gamma$,\n$\\widetilde{\\beta}$ and $\\widetilde{\\gamma}$.\n\\subsubsection{The NS-NS Sector}\n\nAccording to the Eq. (13), the resultant NS-NS sector boundary state of\nthe fermions is given by\n\\begin{equation}\n|B_{\\psi},\\eta,\\tau=0\\rangle_{\\rm NS}=\\prod^{\\infty}_{r=1\/2}[\\det\nM_{(r)}]\\exp\\bigg {[}i\\eta\\sum_{r=1\/2}^{\\infty}\n(b^{\\mu}_{-r}{\\cal{S}}_{(r)\\mu\\nu}\\widetilde{b}^{\\nu}_{-r})\n\\bigg{]}|0\\rangle_{\\rm NS}.\n\\end{equation}\nWhen the path integral is computed the determinant will be reversed\nin comparing to the bosonic case, the Eq. (4).\nThis is due to the Grassmann nature of\nthe integration variables \\cite{2}. As $r$ is half integer,\nregularization of this infinite product is\n\\begin{equation}\n\\prod^{\\infty}_{r=1\/2}[\\det M_{(r)}]=\\det\\bigg{(}\\frac{\\sqrt{\\pi}}{\n\\Gamma[\\frac{U}{2i\\Omega}+\\frac{1}{2}]}\\bigg{)}.\n\\end{equation}\n\n\\subsubsection{The R-R Sector}\n\nFor acquiring the boundary state in the\nR-R sector, we have to follow the same procedure of the NS-NS sector\nwith a bit difference which needs a careful notice. Since $k=m$ in\nthe Eq. (13) runs over integers in the R-R sector, there is a zero mode\nwhich affects the boundary state. Solving the Eq. (13)\nin the R-R sector, yields the following boundary state\n\\begin{equation}\n|B_{\\psi},\\eta,\\tau=0\\rangle_{\\rm R}=\\prod^{\\infty}_{m=1}[\\det\nM_{(m)}]\\exp\\bigg {[}i\\eta\\sum_{m=1}^{\\infty}\n(d^{\\mu}_{-m}{\\cal{S}}_{(m)\\mu\\nu}\\widetilde{d}\n^{\\nu}_{-m})\\bigg{]}|B_{\\psi},\\eta\\rangle^{(0)}_{\\rm R}.\n\\end{equation}\nSince $m$ is an integer number, regularization of the infinite\nproduct is exactly similar to the bosonic case (of course here\nthe determinant is inverse of the bosonic case)\n\\begin{equation}\n\\prod^{\\infty}_{m=1}[\\det M_{(m)}]=\\bigg{\\{}\\sqrt{\\det\n\\Omega}\\;\\det\\bigg{(}\\Gamma \\bigg{[} 1+\\frac{U}{2i\\Omega}\n\\bigg{]}\\bigg{)}\\bigg{\\}}^{-1}.\n\\end{equation}\n\nThe state $|B_{\\psi},\\eta \\rangle^{(0)}_{\\rm R}$ in the Eq. (16)\nis the zero mode boundary state\n\\begin{equation}\n|B_{\\psi},\\eta \\rangle^{(0)}_{\\rm\nR}=\\bigg{[}C\\Gamma_{11}\\bigg{(}\\frac{1+i\\eta\\Gamma_{11}}\n{1+i\\eta}\\bigg{)}\\exp\n(\\frac{1}{2}\\Phi_{\\mu\\nu}\\Gamma^{\\mu}\\Gamma^{\\nu})\\bigg{]}^{AB}|A\\rangle\n|\\widetilde{B}\\rangle,\n\\end{equation}\nwhere $|A\\rangle |\\widetilde{B}\\rangle$ is the vacuum of the zero\nmodes $d^{\\mu}_{0}$ and $\\widetilde{d}^{\\mu}_{0}$. $C$ is the\ncharge conjugate matrix, and the antisymmetric matrix $\\Phi$ is\ndefined in terms of the matrix ${\\cal {S}}$,\n\\begin{equation}\n{\\cal {S}}=(1-\\Phi)^{-1}(1+\\Phi).\n\\end{equation}\nDetails of obtaining the Eqs. (18) and (19) is shown in the appendix A. Since\nthe matrix ${\\cal {S}}$ should be orthogonal ${\\cal {S}}^{-1}={\\cal\n{S}}^{T}$, its definition\n${\\cal{S}}_{(m)}=S_{(m)}+[(S_{(-m)})^{-1}]^{T}$ implies that the\nmatrix $S$ should satisfy the following relation\n\\begin{equation}\nS_{(m)}^{T}-S_{(m)}^{-1}=S_{(-m)}^{T}+S_{(-m)}^{-1}.\n\\end{equation}\nAccording to the Eqs. (5)-(8) $S$ is defined in terms of the\nbackground fields. Thus, the Eq. (20) imposes a relation between\nthese background fields. When in the action (1) the tachyon and\nvelocity are put to zero, we receive $\\Phi=\\cal{F}$ and hence the\nterm $\\exp(\\frac{1}{2}{\\Phi}_{\\alpha\\beta}\n\\Gamma^{\\alpha}\\Gamma^{\\beta})$ reduces to the known\n$\\exp(\\frac{1}{2}{\\cal{F}}_{\\alpha\\beta}\n\\Gamma^{\\alpha}\\Gamma^{\\beta})$ \\cite{3}.\n\n\\section{Interaction of the branes}\n\nInteraction amplitude between D$p_1$ and D$p_2$-branes in each\nsector is defined by ${\\cal{A}}_{\\rm\nNS-NS,R-R}=2\\alpha'\\int^{\\infty}_{0}dt\\;_{\\rm NS,R}\\langle\nB^1,\\tau=0|e^{-tH_{\\rm NS,R}}|B^{2},\\tau=0\\rangle_{\\rm NS,R}$. Total\nHamiltonian $H_{\\rm NS,R}$ is sum of the Hamiltonians of\n$X^{\\mu}$'s, $\\psi^{\\mu}$'s, ghosts and superghosts in each sector.\nFor calculation of the interaction amplitude we need the total\nprojected boundary state. The total boundary state of each sector is\n\\[|B,\\eta,\\tau=0\\rangle_{\\rm NS,R}=|B_{X},\\tau=0\\rangle|B_{\\rm gh},\n\\tau=0\\rangle|B_{\\psi},\\eta,\\tau=0\\rangle_{\\rm NS,R}|B_{\\rm sgh},\n\\eta,\\tau=0\\rangle_{\\rm NS,R}.\\] In the appendix B the projection\nprocess is discussed. Thus, the total projected boundary states\nfind the feature of the Eqs. (40) and (41).\n\n\\subsection{Interaction amplitude in the NS-NS sector}\n\nUsing the boundary state (40) for NS-NS sector, after a long\ncalculation the total interaction amplitude in this sector is\nacquired as\n\\begin{eqnarray}\n{\\cal {A}}_{\\rm NS-NS} = &~&\n\\frac{\\alpha'V_{\\overline{u}}}{8(2\\pi)^{d_{\\overline{i}}}}\\;\n\\frac{T_{p_1}T_{p_2}}{|V_1-V_2|}\n\\prod^{\\infty}_{m=1}\\frac{\\det[\nM_{(m-1\/2)1}M_{(m-1\/2)2}]}{\\det[M_{(m)1}M_{(m)2}]}\n\\nonumber\\\\\n&~&\\times{\\int_{0}}^{\\infty}dt \\;\n\\bigg{\\{}\\prod_{\\bar{i}_{c}}\\Theta_{3}\\bigg{(}\\frac{y_{1}\n^{\\bar{i}_{c}}-y_{2}^{\\bar{i}_{c}}}{2\\pi\nR_{\\bar{i}_{c}}}\\mid\\frac{i\\alpha't}{\\pi\n(R_{\\bar{i}_{c}})^{2}}\\bigg{)}\n\\nonumber\\\\\n&~& \\times\n\\bigg{(}\\sqrt{\\frac{\\pi}{\\alpha't}}\\bigg{)}^{d_{\\overline{i}_{n}}}\n\\exp \\bigg{(}-\\frac{1}{4\\alpha't}\\sum_{\\overline{i}_{n}}\n({y_{1}}^{\\overline{i}_{n}}-{y_{2}}^{\\overline{i}_{n}})^{2}\n\\bigg{)}\n\\nonumber\\\\\n&~& \\times\\frac{1}{q}\\bigg{(}\\prod^{\\infty}_{m=1}\\bigg{[}\n{\\bigg{(}\\frac{1-q^{2m}}{1+q^{2m-1}}\n\\bigg{)}}^{2}\\frac{\\det(1+{\\cal{S}}_{(m-1\/2)1}\n{\\cal{S}}^{T}_{(m-1\/2)2}q^{2m-1})}\n{\\det(1-{\\cal{S}}_{(m)1}{\\cal{S}}^{T}_{(m)2}q^{2m})}\\bigg{]}\n\\nonumber\\\\\n&~& -\\prod^{\\infty}_{m=1}\\bigg{[}{\\bigg{(}\\frac{1-q^{2m}}\n{1-q^{2m-1}}\n\\bigg{)}}^{2}\\frac{\\det(1-{\\cal{S}}_{(m-1\/2)1}\n{\\cal{S}}^{T}_{(m-1\/2)2}q^{2m-1})}\n{\\det(1-{\\cal{S}}_{(m)1}{\\cal{S}}^{T}_{(m)2}q^{2m})}\\bigg{]}\n\\bigg{)}\\nonumber\\\\\n&~& \\times \\frac{1}{\\sqrt{\\det Q \\;\\det K_{1}\\;\\det\nK_{2}}}\\nonumber\\\\\n&~& \\times\\exp\\bigg{[}-\\frac{1}{4}\\bigg{(}E^{T}Q^{-1}E+\n\\sum_{\\alpha'_{1},\\beta'_{1}}\ny_{2}^{\\alpha'_{1}}y_{2}^{\\beta'_{1}}(K_{1}^{-1})_\n{\\alpha'_{1}\\beta'_{1}}\n+\\sum_{\\alpha'_{2},\\beta'_{2}}\ny_{1}^{\\alpha'_{2}}y_{1}^{\\beta'_{2}}(K_{2}^{-1})_\n{\\alpha'_{2}\\beta'_{2}}\\bigg{)}\\bigg{]}\\bigg{\\}},\n\\end{eqnarray}\nwhere $q=e^{-2t}$, and $V_{\\bar u}$ is common volume of the branes.\nThe set $\\{\\overline{i}\\}$ shows directions perpendicular to both\nbranes except $i_{0}$, $\\{\\overline{u}\\}$ is for the directions\nalong both branes except $0$, $\\{\\alpha'_{1}\\}$ is used for the\ndirections along the D$p_1$-brane and perpendicular to the\nD$p_2$-brane and $\\{\\alpha'_{2}\\}$ indicates the directions along\nthe D$p_2$-brane and perpendicular to the D$p_1$-brane.\n$\\overline{i}_{c}$ and $\\overline{i}_{n}$ are related to the compact\nand non-compact parts of $\\overline{i}$, respectively. The matrices\n$Q$, $K_{1}$, $K_{2}$ and the doublet $E$ are defined through their\nelements as in the following\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{rcl}\n&~& Q_{11}=\\frac{\\alpha't}{2(V_{2}-V_{1})^{2}}(1+{V_{1}}^{2})\n(1-{V_{2}}^{2})+2i\\alpha'(U^{-1}_{1})^{00}(1-V^2_2)^2, \\\\\n&~& Q_{22}=\\frac{\\alpha't}{2(V_{2}-V_{1})^{2}}(1+{V_{2}}^{2})\n(1-{V_{1}}^{2})-2i\\alpha'(U^{-1}_{2})^{00}(1-V_1^2)^2, \\\\\n&~& Q_{12}=Q_{21}=\\frac{\\alpha't}{(V_{2}-V_{1})^{2}}(1+{V_{1}}^{2})\n(1+{V_{2}}^{2})(1-V_{1}V_{2}),\n\\end{array}\\right.\n\\end{eqnarray}\n\\begin{eqnarray}\n\\left\\{ \\begin{array}{rcl} &~&\nE_{1}=\\frac{i}{2(V_{2}-V_{1})}\\bigg{[}{y_{2}}^{i_{0}}\n(1+{V_{1}}^{2})^{2}\n-{y_{1}}^{i_{0}}(1+V_{1}V_{2})\\bigg{]},\n\\\\ &~& E_{2}=\\frac{i}{2(V_{2}-V_{1})}\\bigg{[}{y_{1}}^{i_{0}}\n(1+{V_{2}}^{2})^{2}-{y_{2}}^{i_{0}}(1+V_{1}V_{2})\\bigg{]},\n\\end{array}\\right.\n\\end{eqnarray}\n\\begin{eqnarray}\n\\left\\{ \\begin{array}{rcl}\n&~& K^{\\alpha'_{1}\\beta'_{1}}_{1}\n=4i\\alpha'(1-\\frac{1}{2}\\delta_{\\alpha'_{1}\n\\beta'_{1}})(U^{-1}_{1})^{\\alpha'_{1}\\beta'_{1}}\n-\\alpha't\\delta^{\\alpha'_{1}\\beta'_{1}}, \\\\\n&~&\nK^{\\overline{u}\\overline{v}}_{1}\n=4i\\alpha'(1-\\frac{1}{2}\\delta_{\\overline{u}\\overline{v}})\n(U^{-1}_{1})^{\\overline{u}\\overline{v}}\n-\\frac{1}{2}\\alpha't\\delta^{\\overline{u}\\overline{v}}, \\\\\n&~& K^{\\alpha'_{1}\\overline{u}}_{1}=K^{\\overline{u}\\alpha'_{1}}_{1}=\n4i\\alpha'(U^{-1}_{1})^{\\alpha'_{1}\\overline{u}}.\n\\end{array}\\right.\n\\end{eqnarray}\nUnder the exchanges $1\\rightarrow2$ and $i\\rightarrow-i$ in\nthe elements of the matrix $K_{1}$ we receive the elements\nof the matrix $K_{2}$.  As is obvious $Q$, $E$ and $K$'s are\ncompletely velocity and tachyon dependent.\n\nIn the amplitude (21), the theta function comes from the compact\npart of the set $\\{X^{\\bar i}\\}$, while the exponential and its\npre-factor in the third line originate from the non-compact part of\nit. In fact, the exponential is a damping factor with respect to the\ndistance of the branes. If all directions $\\{X^{\\bar i}\\}$ are\ncompact the exponential and its pre-factor disappear. In this case\n${\\bar i}_c$ takes all values of ${\\bar i}$. In the case that all\ndirections $\\{X^{\\bar i}\\}$ are non-compact the $\\Theta_3$-factor is\nremoved, hence ${\\bar i}_n$ takes all values of ${\\bar i}$. The next\ntwo lines which contain the $\\cal{S}$ matrix reflect the portion of\nthe oscillators, conformal ghosts and super conformal ghosts. The\nremaining part, which is obtained by integration over the momenta,\nthe Eq. (10), is due to the presence of the velocities and the\nbackground tachyon fields. In absence of the velocities and tachyon\nfields, this factor is absent too and hence the interaction\namplitude resembles to the one in \\cite{3}.\n\\subsection{Interaction amplitude in the R-R sector}\n\nFor interaction amplitude in the R-R sector we use the total GSO\nprojected boundary state for the R-R sector, the Eq. (41), and\nfollow the same procedure in the NS-NS sector, so\n\\begin{eqnarray}\n{\\cal\n{A}}_{\\rm R-R} =&~&\n\\frac{\\alpha'V_{\\overline{u}}}{8(2\\pi)^{d_{\\overline{i}}}}\\;\n\\frac{T_{p_1}T_{p_2}}{|V_1-V_2|}{\\int_{0}}^{\\infty}dt\n\\;\n\\bigg{\\{}\\prod_{\\bar{i}_{c}}\\Theta_{3}\\bigg{(}\\frac{y_{1}\n^{\\bar{i}_{c}}-y_{2}^{\\bar{i}_{c}}}{2\\pi\nR_{\\bar{i}_{c}}}\\mid\\frac{i\\alpha't}{\\pi\n(R_{\\bar{i}_{c}})^{2}}\\bigg{)}\n\\nonumber\\\\\n&~&\\times \\bigg{(}\n\\sqrt{\\frac{\\pi}{\\alpha't}}\\bigg{)}^{d_{\\overline{i}_{n}}}\n\\exp \\bigg{(}-\\frac{1}{4\\alpha't}\\sum_{\\overline{i}_{n}}\n({y_{1}}^{\\overline{i}_{n}}-{y_{2}}^{\\overline{i}_{n}})^{2}\n\\bigg{)}\n\\nonumber\\\\\n&~&\\times\\bigg{(}\\zeta\\prod^{\\infty}_{m=1}\\bigg{[}\n{\\bigg{(}\\frac{1-q^{2m}}{1+q^{2m}}\n\\bigg{)}}^{2}\\frac{\\det(1+{\\cal{S}}_{(m)1}\n{\\cal{S}}^{T}_{(m)2}q^{2m})}\n{\\det(1-{\\cal{S}}_{(m)1}{\\cal{S}}^{T}_{(m)2}q^{2m})}\n\\bigg{]}+\\zeta'\\bigg{)}\n\\nonumber\\\\\n&~&\\times \\frac{1}{\\sqrt{\\det Q \\;\\det K_{1}\\;\\det\nK_{2}}}\n\\nonumber\\\\\n&~& \\times\\exp\\bigg{[}-\\frac{1}{4}\\bigg{(}E^{T}Q^{-1}E+\n\\sum_{\\alpha'_{1},\\beta'_{1}}\ny_{2}^{\\alpha'_{1}}y_{2}^{\\beta'_{1}}(K_{1}^{-1})_\n{\\alpha'_{1}\\beta'_{1}}\n+\\sum_{\\alpha'_{2},\\beta'_{2}}\ny_{1}^{\\alpha'_{2}}y_{1}^{\\beta'_{2}}(K_{2}^{-1})_\n{\\alpha'_{2}\\beta'_{2}}\\bigg{)}\\bigg{]}\\bigg{\\}},\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\zeta\\equiv-\\frac{1}{2} {\\rm Tr}[G_{1}C^{-1}G^{T}_{2}C],\n\\end{equation}\n\\begin{equation}\n\\zeta'\\equiv -i\n{\\rm Tr}[G_{1}C^{-1}G^{T}_{2}C\\Gamma_{11}],\n\\end{equation}\nand $G_{1,2}=\\exp[\\frac{1}{2}(\\Phi_{(1,2)})_\n{\\mu\\nu}\\Gamma^{\\mu}\\Gamma^{\\nu}]$. Note that the variables $\\zeta$\nand $\\zeta'$ implicitly depend on the branes' dimensions through\n$\\Phi_1$ and $\\Phi_2$ in $G_1$ and $G_2$.\n\nNow we are eager to study the total amplitude, i.e. the\ncombination of the amplitudes in the NS-NS and R-R sectors. Consider\nthe following special case: there is no compactification, the two D$p$-branes\nare parallel with the same dimensions, and the same fields living\non them. Thus, as in the literature, this interaction amplitude becomes zero\ndue to the cancellation of attractive and repulsive forces in the NS-NS\nand R-R sectors, respectively.\n\nIn the case at hand, beside the living fields on the branes, the\nvelocities which are transverse fluctuations of the branes are\npresent, too. In the amplitude (21) and (25) the relative speed\nappeared in the denominators. This puts a constraint on the system\nthat the velocities of the branes should be different, otherwise the\ntotal amplitude becomes infinite. In this case, we can not check the\nvanishing of the interaction amplitude for identical parallel branes\nwith the same fields. Therefore, even if all the fields are\nidentical, the velocities should be different. This causes the\nbranes to have different $\\Phi$'s and consequently different\n$\\cal{S}$'s. So the NS-NS and R-R amplitudes cannot cancel the\neffect of each other.\n\\section{Long distance behavior of the amplitude}\n\nNow we find the interaction between the branes when they are far\nfrom each other. That is, we find the behavior of the interaction\namplitudes (21) and (25) when time goes to infinity. Conventionally,\nin the large distance only the massless states of the closed string\ncontribute to the branes interaction.\n\nThe large distance amplitude is equivalent to the long time behavior\nof the branes. It can be acquired by sending $q$ to zero in the Eqs.\n(21) and (25). So the interaction amplitudes due to massless states\nin the NS-NS and R-R sectors are\n\\begin{eqnarray}\n\\lim_{q\\rightarrow0}{\\cal{A}}_{\\rm NS-NS}= &~&\n\\frac{V_{\\overline{u}}T_{p_1}T_{p_2}}{4(2\\pi)^{d_{\\overline{i}}}}\n\\;\\frac{i(-1)^{(p_1+p_2)\/2}\\;2^{d_{\\overline{u}}+1\/2}}\n{\\alpha'^{(p_1+p_2)\/2}(1+{V_1}^{2})(1+{V_2}^{2})}\\;\n\\prod^{\\infty}_{m=1}\\frac{\\det[\nM_{(m-1\/2)1}M_{(m-1\/2)2}]}{\\det[M_{(m)1}M_{(m)2}]}\n\\nonumber\\\\\n&~& \\times \\int^\\infty\ndt\\;\\bigg{\\{}\\bigg{(}\\sqrt{\\frac{\\pi}{\\alpha't}}\\bigg{)}^{d_{\\overline{i}_{n}}}\n\\exp \\bigg{(}-\\frac{1}{4\\alpha't}\\sum_{\\overline{i}_{n}}\n({y_{1}}^{\\overline{i}_{n}}-{y_{2}}^{\\overline{i}_{n}})^{2}\\bigg{)}\n\\nonumber\\\\\n&~& \\times\\lim_{t\\rightarrow\\infty}\n\\frac{2[{\\rm Tr}{({\\cal{S}}_{(1)1}{\\cal{S}}^{T}_{(1)2})-2]}}\n{t^{1+(p_1+p_2)\/2}}\\bigg{\\}},\n\\end{eqnarray}\n and\n\\begin{eqnarray}\n\\lim_{q\\rightarrow0}{\\cal{A}}_{\\rm R-R}= &~&\n\\frac{V_{\\overline{u}}\\;T_{p_1}T_{p_2}}{8(2\\pi)\n^{d_{\\overline{i}}}}\\;\n\\frac{i(-1)^{(p_1+p_2)\/2}\\;2^{d_{\\overline{u}}+1\/2}}\n{\\alpha'^{(p_1+p_2)\/2}(1+{V_1}^{2})(1+{V_2}^{2})}\n\\nonumber\\\\\n&~& \\times \\int^\\infty\ndt\\;\\bigg{\\{}\\bigg{(}\\sqrt{\\frac{\\pi}{\\alpha't}}\\bigg{)}^{d_{\\overline{i}_{n}}}\n\\exp \\bigg{(}-\\frac{1}{4\\alpha't}\\sum_{\\overline{i}_{n}}\n({y_{1}}^{\\overline{i}_{n}}-{y_{2}}^{\\overline{i}_{n}})^{2}\\bigg{)}\n\\nonumber\\\\\n&~& \\times\\lim_{t\\rightarrow\\infty}\n\\frac{1}{t^{1+(p_1+p_2)\/2}}\\bigg{\\}}.\n\\end{eqnarray}\nWe do not put the limit on the exponential part and its pre-factor\nin the Eqs. (28) and (29) because these factors are related to the\npositions of the branes, and closed string emission is independent\nof the locations of the branes. When there is no tachyonic\nbackground \\cite{3}, last factors in the Eqs. (28) and (29) do not\nhave the factor $1\/t^{1+(p_1+p_2)\/2}$. Thus, due to the presence of\ntachyon fields, the interaction amplitude decreases in time. In\nfact, the statement that for large distances of the branes the\nmassless closed string states dominate in the interaction is valid\nuntil there is no tachyon backgrounds in the system.\n\nThere is an interpretation for this unusual behavior. In fact, the\nopen string tachyon background causes an instability in the system.\nTherefore, after long enough time, by rolling of the tachyon\n\\cite{12} toward its minimum potential, unstable D-branes decay to\nthe bulk modes and their dimensions decrease to reach a stable\nsystem. Final products of this process are branes with lower\ndimensions or vacuum of the closed string \\cite{10}. The latter\nimplies that there are no physical perturbative open string states\naround the minimum of the potential. This is due to the fact that\nthe open string states live only on the branes. Thus, in the concept\nof interactive branes, by passing the time which leads to tachyon\nrolling and decreasing of their dimensions, the branes'\nconfiguration distorts and prevents them from the interaction.\n\nThe amplitude ${\\cal{A}}_{\\rm NS-NS}$ in the Eq. (28) depends on the\nbackground fields through the factor ${\\rm\nTr}({\\cal{S}}_{(1)1}{\\cal{S}}^{T}_{(1)2})$ and the determinants of\nthe matrices $\\{M_{(m-1\/2)}|m=1,2,3, \\cdot\\cdot\\cdot\\}$, while such\na dependence is absent in the amplitude ${\\cal{A}}_{\\rm R-R}$, the\nEq. (29). In other words, when the branes are far from each other,\nthe R-R amplitude becomes background independent.\n\nAnother interesting feature of the long time amplitude is\nits time-dependent behavior on the branes' dimensions.\nAn exception here is the D-instanton. When two D-instantons,\nwhich have the dimension $p_1=p_2=-1$, interact the\nfactor $1\/t^{1+(p_1+p_2)\/2}$ is removed and the amplitude\nbehavior in long time is resembled to a system without\ntachyon. For this system the presence of the\ntachyon does not affect the conventional behavior of the\nlarge distance interaction.\n\\section{Conclusions}\n\nThe boundary state of a closed superstring traveling between two\nmoving branes in the presence of $B_{\\mu\\nu}$, tachyon and $U(1)$\ngauge field was calculated. Notable feature in the boundary state\nequations is the prevention of the closed string wrapping around the\ncompact directions of spacetime, which is due to the presence of the\ntachyon field. As well, the boundary state includes a momentum\ndependent exponential factor which is absent in the conventional\nboundary states. This factor originates from the zero mode parts of\nthe velocity and tachyon terms in the boundary action.\n\nThe interaction amplitude of the branes via exchanging\nof closed string was calculated for the NS-NS and R-R\nsectors. It is shown that even for co-dimension parallel\nbranes with similar external fields, the total amplitude\nis not zero. This is due to the presence of the velocities\nand tachyon fields in the system.\n\nThe long distance behavior of the interaction amplitude was studied.\nIn this domain the instability of the branes, due to the background\ntachyon fields, weakens the interaction. This decreasing behavior\ncan be understood by dissipation of the branes to the bulk modes\nbecause of the rolling of the tachyon to its minimum potential in\nlong time regime. The interaction for two D-instantons obviates this\ndecreasing behavior. Thus, the long time amplitude in this case\nbehaves like the conventional case in which the massless states\ndominate.\n\\begin{center}\n\\textbf{Appendix A}\n\\end{center}\n\\begin{center}\n\\textbf{Zero mode boundary state in the R-R sector}\n\\end{center}\n\nThe state $|B_{\\psi},\\eta \\rangle^{(0)}_{\\rm R}$ in the Eq. (16)\nis the zero mode boundary state that obeys the following equation\n\\begin{equation}\n|B_{\\psi},\\eta \\rangle^{(0)}_{\\rm R}={\\cal{M}}^{(\\eta)_{AB}}\n|A\\rangle |\\widetilde{B}\\rangle,\n\\end{equation}\nwhere $|A\\rangle |\\widetilde{B}\\rangle$ is the vacuum of the zero\nmodes $d^{\\mu}_{0}$ and $\\widetilde{d}^{\\mu}_{0}$. The matrix\n${\\cal{M}}^{(\\eta)}$ has to satisfy the equation\n\\begin{equation}\n(\\Gamma^{\\mu})^{T}{\\cal {M}}^{(\\eta)}-i\\eta\n{\\cal{S}}^{\\mu}_{(m)\\nu}\\Gamma_{11}{\\cal\n{M}}^{(\\eta)}\\Gamma^{\\nu}=0.\n\\end{equation}\nConsider a solution in the form\n\\begin{equation}\n{\\cal {M}}^{(\\eta)}=C\\Gamma_{11}\\bigg{(}\\frac{1+i\\eta\\Gamma_{11}}\n{1+i\\eta}\\bigg{)}G,\n\\end{equation}\nin which $C$ is the charge conjugate matrix. Substitution of the Eq.\n(32) into the Eq. (31) leads to the following equation for the\nmatrix $G$,\n\\begin{equation}\n\\Gamma^{\\mu}G={\\cal {S}}^{\\mu}_{\\;\\;\\;\\nu}G\\Gamma^{\\nu}.\n\\end{equation}\nThere is a conventional solution for $G$ as\n\\begin{equation}\nG= \\exp (\\frac{1}{2}\\Phi_{\\mu\\nu}\\Gamma^{\\mu}\\Gamma^{\\nu}).\n\\end{equation}\nIndeed one must expand the exponential with the convention that\nall gamma matrices anticommute, therefore there are a finite\nnumber of terms. The antisymmetric matrix $\\Phi$ is defined in\nterms of the matrix ${\\cal {S}}$,\n\\begin{equation}\n{\\cal {S}}=(1-\\Phi)^{-1}(1+\\Phi).\n\\end{equation}\n\n\\begin{center}\n\\textbf{Appendix B}\n\\end{center}\n\\begin{center}\n\\textbf{GSO projected and ghosts boundary states}\n\\end{center}\n\nThe GSO projected boundary states are given by\n\\begin{equation}\n|B,\\tau=0\\rangle_{\\rm NS}=\\frac{1-(-1)^{F+G}}{2}\\frac{1-(-1)\n^{\\widetilde{F}+\\widetilde{G}}}{2}|B,\\eta=+1,\\tau=0\\rangle_{\\rm\nNS},\n\\end{equation}\n\\begin{equation}\n|B,\\tau=0\\rangle_{\\rm R}=\\frac{1+(-1)^{n}(-1)^{F+G}}{2}\n\\frac{1-(-1)^{\\widetilde{F}+\\widetilde{G}}}{2}\n|B,\\eta=+1,\\tau=0\\rangle_{\\rm R},\n\\end{equation}\nwhere $n$ is an even\nnumber for the type IIA superstring theory and is odd for the IIB\nsuperstring theory. The definitions of $F$ and $G$,  are\n\\begin{equation}\nF=\\sum^\\infty_{r=1\/2} b^\\mu_{-r} b_{r\\mu}\\;\\;\\;\\;,\\;\\;\\;\\;\nG=-\\sum^\\infty_{r=1\/2}(\\gamma_{-r}\\beta_r+\\beta_{-r}\\gamma_r),\n\\end{equation}\nfor the NS-NS sector, and\n\\begin{equation}\n(-1)^F=\\Gamma_{11}(-1)^{\\sum^\\infty_{m=1} d^\\mu_{-m}\nd_{m\\mu}}\\;\\;\\;\\;,\\;\\;\\;\\; G=-\\gamma_0\n\\beta_0-\\sum^\\infty_{m=1}(\\gamma_{-m}\\beta_m+\\beta_{-m}\\gamma_m),\n\\end{equation}\nfor the R-R sector. Similar definitions also hold for ${\\tilde F}$\nand ${\\tilde G}$. Thus, the total projected boundary states are\n\\begin{equation}\n|B,\\tau=0\\rangle_{\\rm NS}=\\frac{1}{2}\\bigg{(}|B,+,\n\\tau=0\\rangle_{\\rm NS}-|B,-,\\tau=0\\rangle_{\\rm NS}\\bigg{)},\n\\end{equation}\n\\begin{equation}\n|B,\\tau=0\\rangle_{\\rm R}=\\frac{1}{2}\\bigg{(}|B,+,\n\\tau=0\\rangle_{\\rm R}+|B,-,\\tau=0\\rangle_{\\rm R}\\bigg{)}.\n\\end{equation}\n\nSince the bulk action in the Eq. (1) preserves conformal symmetry,\nworking in covariant formalism necessitates including conformal\nghosts \\cite{2, 16}. In fact, what we need is the portion of ghosts\n(i.e. anti-commuting fields $b$, $c$, $\\widetilde{b}$ and\n$\\widetilde{c}$) in the bosonic boundary state. This part is\nindependent of the background fields and is expressed as\n\\begin{equation}\n|B_{\\rm gh},\\tau=0\\rangle=\\exp\\bigg{[}\\sum^{\\infty}_{m=1}\ne^{4im\\tau_{0}}\n(c_{-m}\\widetilde{b}_{-m}-b_{-m}\\widetilde{c}_{-m})\\bigg{]}\n\\frac{c_{0}\n+\\widetilde{c_{0}}}{2}|q=1\\rangle|\\widetilde{q}=1\\rangle.\n\\end{equation}\nIn the superstring theory, in addition to the conformal ghosts,\nwe should also consider the super conformal ghosts. Thus, the\nboundary state, corresponding to the super conformal\nghosts in the NS-NS and R-R sectors, are as in the following\n\\begin{equation}\n|B_{\\rm sgh},\\eta,\\tau=0\\rangle_{\\rm NS}=\\exp\\bigg{[}\ni\\eta\\sum^{\\infty}_{r=1\/2}\n(\\gamma_{-r}\\widetilde{\\beta}_{-r}-\\beta_{-r}\n\\widetilde{\\gamma}_{-r})\\bigg{]}\n|P=-1\\rangle|\\widetilde{P}=-1\\rangle.\n\\end{equation}\n\\begin{equation}\n|B_{\\rm sgh},\\eta,\\tau=0\\rangle_{\\rm R}=\\exp\\bigg{[}i\\eta\n\\sum^{\\infty}_{m=1}\n(\\gamma_{-m}\\widetilde{\\beta}_{-m}-\\beta_{-m}\n\\widetilde{\\gamma}_{-m})\\bigg{]}|P=-\\frac{1}{2}\n\\rangle|\\widetilde{P}\n=-\\frac{3}{2}\\rangle.\n\\end{equation}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe antiparticle to particle ratios of hadrons are important \nobservables of heavy ion collisions, and have been extensively studied\nsince the very beginning of RHIC operations\n\\cite{phobosrat:aa130,starrat:aa130}. They are more precisely \nmeasurable than invariant cross sections, and provide quantitative input \nto models concentrating on the hadro-chemistry of the collisions, such as \nstatistical thermal models. Antiproton to proton ratios and net \nproton yields provide \ninformation on baryon transport and baryon production in these \nultra-relativistic collisions, and are relevant quantities to estimate the \nfraction of the total collision energy that is available for particle \nproduction \\cite{Bearden:2003hx}.\n\nThe presented identified hadron spectra at \n$\\sqrt{s_{_{NN}}}$ = 62.4 GeV extend the \nenergy range for which the contributions of different particle species to \nthe inclusive charged hadron spectra are known, bridging a gap between \n17.2 and 130 GeV. Invariant cross sections at very low $p_T$, net \nproton yields and baryon enhancement in the intermediate $p_T$ range \n(reported earlier in \\cite{Adler:2003cb,Adcox:2001mf,Adcox:2003nr}) are \nstudied in Au+Au collisions at 62.4 GeV. \n\n\nThe sensors in the two arms of the PHOBOS silicon spectrometer \nare arranged into layers \\cite{phobos:si}. \nThe particles with $p_T$ measured to be less than 200 MeV\/c range \nout in these layers. Particle ratios are measured in the $0.2<p_T<0.8$ \nGeV\/c range in the spectrometer, and the identified spectra analysis uses\nthe time-of-flight detector as well, extending the $p_T$ reach to 3 \nGeV\/c.\n\nThe acceptance corrections of the particle ratios are cancelled by \nchanging the polarity of the magnetic field: the ratios are formed from \nthe yields of the positive and negative particles with similar \ntrajectories corresponding to the same bending direction, but to opposite \nmagnetic polarity.\nRatios in each bending direction and spectrometer arm are evaluated \nseparately and averaged, at the same time providing a handle on the \nsystematic errors. The ratios are corrected for absorption in the beam-pipe \nand in the detector materials, secondary particle production and feed-down \nfrom weak decays, based on detailed Monte Carlo simulations.\nThe size of the latter correction \nis only 1-2\\%, since the first silicon layer is positioned within 10~cm \nfrom the interaction point, thus allowing good rejection of tracks from \ndisplaced vertices. \n\n\\section{Results}\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=54.5mm]{Fig_1_a.eps}\n\\includegraphics[width=54.5mm]{Fig_1_b.eps}}\n\\caption{\\label{ratios}\nPreliminary $\\pi^-\/\\pi^+$ (top), $K^-\/K^+$ (center) and $\\overline{p}\/p$ \n(bottom) ratios as a function of the number of participant nucleons. \n(a) Cu+Cu collisions at $\\sqrt{s_{_{NN}}}$ = 62.4 GeV (open points)   \nand 200 GeV (closed symbols). (b) Comparison of p+p (triangles),\nCu+Cu (dots) and Au+Au (open squares) collisions at $\\sqrt{s_{_{NN}}}$ =\n200 GeV. The gray bands correspond to systematic errors at 90\\% C.L.}\n\\end{figure}\n\nThe preliminary $\\pi^-\/\\pi^+$, $K^-\/K^+$ and \n$\\overline{p}\/p$ ratios in p+p, Cu+Cu and Au+Au collisions are presented \nin figure \\ref{ratios} as a function of the number of participants in the \ncollision \\cite{Phobos:pid62}. \nThe comparison of different collision energies (left panel) \nshows that the $\\overline{p}\/p$ ratios are strongly energy dependent in \nCu+Cu collisions (similarly to collisions of heavier ions). \nThere is only an insignificantly small, if any, decrease of the \n$\\overline{p}\/p$ ratio observed with increasing collision centrality\nat both energies. On the right panel, the same ratios in p+p, Cu+Cu and \nAu+Au collisions are compared at 200 GeV, where again, almost no \nvariation of the $\\overline{p}\/p$ ratio with centrality is visible. This \nobservation challenges theoretical expectations about \nbaryon stopping in collisions with varying impact parameter.\nThese comparisons are being extended to 62.4 GeV where the ratios are \nfurther away from unity, possibly leaving more room for \nmeasurable variations.\n\nIn order to be able to study the ratios of different particle yields or\n$p_T$ integrated $dN\/dy$ rapidity densities, one has to measure invariant \nspectra of identified particles as well. Figure \n\\ref{spectra}a summarizes the measured $p_T$ spectra of charged hadrons\nin Au+Au collisions at $\\sqrt{s_{_{NN}}}$ = 62.4 GeV. \nDifferent rows correspond to three classes of centrality \n(expressed as the percentile of the total inelastic Au+Au cross section, \nstarting with the 15\\% most central collisions in the top row). \nA smooth evolution of the spectra with centrality can be observed, \nwith the proton spectrum being harder than the meson spectra, similar to \nresults at higher collision energies \\cite{Adler:2003cb,Adams:2003xp}.\nFigure \\ref{spectra}b compares the $\\pi^++\\pi^-$, $K^++K^-$ and\n$\\overline{p}+p$ spectra to the cross sections measured at very low $p_T$.\nA blast wave fit is applied to the higher $p_T$ data points (solid lines),\nand extrapolated to low $p_T$ (dashed lines). The extrapolation is \nin good agreement with the kaon and (anti)proton yields (the fit does not \ntake into account resonance production which populates the low $p_T$ \nregion with pions).\n\n\\begin{figure}[t] \n\\centerline{\\includegraphics[width=71mm]{Fig_2_a.eps}\n\\includegraphics[width=60mm]{Fig_2_bc.eps}} \n\\caption{\\label{spectra}\n(a) Invariant cross sections of identified hadrons in Au+Au collisions at\n$\\sqrt{s_{_{NN}}}$ = 62.4 GeV. Positives are\nplotted in the left, negatives in the right~column.\n(b) Invariant cross sections of $\\pi^++\\pi^-$, $K^++K^-$ and\n$\\overline{p}+p$, including~the very low $p_T$ data in the most central class.\n(c) Net proton yield close to mid-rapidity as a function of $N_{part}$ in\nAu+Au collisions at $\\sqrt{s_{_{NN}}}$ = 62.4 and 200 GeV.}\n\\end{figure}\n   \nThe proton and antiproton spectra can be integrated over $p_T$, with only \na small extrapolation of the yield to the unmeasured low and high-$p_T$ \nregion. Figure \\ref{spectra}c shows the integrated \nyield, $dN\/dy$, of the net protons ($\\overline{p}-p$) close to \nmid-rapidity at 62.4 \\cite{Phobos:pid62} and 200 GeV \\cite{Adler:2003cb} \ncollision energies. In both cases, the net \nproton yield is approximately proportional to $N_{part}$, which does not \nmeet expectations of increasing amount of baryon stopping with increasing \ncentrality. According to figure~\\ref{ratios}, the $\\overline{p}\/p$ ratio \ndoes not depend strongly on centrality, thus the $\\overline{p}$ and $p$ \nyields are also proportional to $N_{part}$ to a good approximation.\n\nThe less steeply falling proton $p_T$ spectrum dominates over the mesons \nat higher $p_T$ values, and at 200 GeV collision energy, even \nthe antiproton yield can exceed the pion yield at high $p_T$ \n\\cite{Adcox:2001mf}. The phenomenon may be important to identify the relevant \ndegrees of freedom and to study the energy loss of partons\nin the medium created in a heavy ion \ncollision. To illustrate the baryon dominance at high \n$p_T$, the fraction of protons among all positive hadrons ($p\/h^+$)\nand the fraction of antiprotons among all negative hadrons \n($\\overline{p}\/h^-$) are plotted in figure \n\\ref{fractions}a, for central Au+Au collisions at 62.4 GeV.\nThe $p\/h^+$ ratio approaches 1\/2 around 2.5--3 GeV\/c (dashed line), which \nindicates that the proton yield becomes dominant over both the $\\pi^+$ \nand $K^+$ yields.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=65mm]{Fig_3_a.eps}\n\\includegraphics[width=63mm]{Fig_3_b.eps}} \n\\caption{\\label{fractions}\n(a) $p\/h^+$ (closed symbols) and $\\overline{p}\/h^-$ (open\nsymbols) ratios as a function of $p_T$ in Au+Au collisions\nat $\\sqrt{s_{_{NN}}}$ = 62.4 GeV. (b) The $p_T$ value where the\nproton and $\\pi^+$ invariant yields become equal in central Au+Au (Pb+Pb)\ncollisions, as a function of $\\sqrt{s_{_{NN}}}$. All data have been\ncorrected for feed-down from weak decays (see text for references).}\n\\end{figure}\n\nFigure~\\ref{fractions}b illustrates the evolution of \nthe `baryon dominance' with collision energy.\nFor central heavy-ion (Au+Au or Pb+Pb) collisions at the AGS, SPS and RHIC \naccelerators, the approximate $p_T$ value at which the invariant cross \nsections of protons and positive pions at mid-rapidity approach each \nother is plotted.\nData (with the percentile of most central events) are taken \nfrom the \nE802~\\cite{Ahle:1998jc}~(4\\%),\nNA44~\\cite{Bearden:2002ib}~(3.7\\%),\nNA49~\\cite{Anticic:2004yj}~(5\\%),\nPHENIX~\\cite{Adler:2003cb,Adcox:2003nr}~(5\\%)\nand PHOBOS~(15\\%) experiments.\nA remarkably smooth collision energy dependence of the `crossing' $p_T$\nvalue is observed. At low energies, the abundance of produced pions is\nnaturally low compared to high collision energies, while baryon number\nconservation ensures that a significant fraction of the large number of\ninitial state protons are found in the final state, thus the invariant\nyields of protons and positive pions become comparable already at low\n$p_T$. With increasing energy, this $p_T$ value grows, mainly due to the\napproximately logarithmically increasing number of produced pions.\nOur new result at 62.4 GeV fits smoothly into the trend established by \nearlier experiments.\n\n\\ack\nThis work was partially supported by U.S. DOE grants \nDE-AC02-98CH10886,\nDE-FG02-93ER40802, \nDE-FG02-94ER40818, \nDE-FG02-94ER40865, \nDE-FG02-99ER41099, and\nDE-AC02-06CH11357, by U.S. \nNSF grants 9603486,\n0072204,           \nand 0245011,       \nby Polish KBN grant 1-P03B-062-27(2004-2007),\nby NSC of Taiwan Contract NSC 89-2112-M-008-024, and\nby Hungarian OTKA grant (F 049823).\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRecommender systems play a significant role in various applications, such as e-commerce and movie recommendation. Matrix Factorization (MF) ~\\cite{koren2009matrix}, as a typical Collaborative Filter (CF) method, has positioned itself as one of the effective means of generating recommendations and is widely adopted in real-world applications. \nTraditionally, for one company, it is essential to accumulate sufficient personal rating data to build a performant MF model. However, due to the sparse nature of user-item interactions, it can be hard for a single company to collect sufficient data to build an MF model. Moreover, recently enacted stringent laws and regulations such as General Data Protection Regulation (GDPR) ~\\cite{Albrecht2016HowTG} and California Consumer Privacy Act (CCPA) ~\\cite{ghosh2018you} stipulate rules on data sharing among companies and organizations, making collaboration between companies by sharing personal rating data illegal and impractical. \n\nTo tackle the challenge of protecting individual privacy and remitting the data shortage issue, federated learning (FL) ~\\cite{konevcny2016federated,mcmahan2016communication} provides a promising way that enables different parties collaboratively build a machine learning model without exposing private data in each party. It addresses data silos and privacy problems together. \nIn FL, data can be partitioned horizontally (example-partitioned) or vertically (feature-partitioned) into different parties. When records are not aligned between parties and the feature spaces among parties are heterogeneous, federated transfer learning can be adopted.\nThe use of FL in recommender systems has been studied over different data distributions. For example, ~\\cite{chai2019secure} considers horizontally partitioned rating data among clients, which hold ratings of the same user-item interaction matrix. Federated multi-view MF is studied where participants hold item interaction data, item features, or user features~\\cite{flanagan2020federated}. Each participant holds a part of model parameters, while some common parameters are shared among participants. \nExisting studies generally categorize horizontal and vertical federated recommender systems regarding on whether user alignment is required before FL ~\\cite{yang2019flbook,chai2019secure}. For example, participants sharing different users and the same set of items implies \\textit{horizontal federated recommender systems}. In our paper, we categorize different settings based on the partition of feature space, as shown in Fig. \\ref{Fig.sparse_data_partition}, which is consistent with other FL systems.\n\n\nAlthough most existing studies of federated recommender systems adopt privacy preserving techniques, including homomorphic encryption (HE)~\\cite{Paillier99public-keycryptosystems}, secure multiparty computation (MPC)~\\cite{DBLP:journals\/iacr\/MohasselZ17} and differential privacy (DP)~\\cite{Dwork:2008:DPS:1791834.1791836} to protect data privacy. \nThere is little exploration of how data privacy can be breached in federated MF. \nIt is shown that the gradient present in the global model is the potential to breach data privacy in horizontal FL for deep learning ~\\cite{Melis_2019} or logistic regression ~\\cite{li2019quantification}. \nHowever, a comprehensive study of the privacy threats against the plaintext federated matrix factorization in different data partitions is still required. \n\nInspired by this research gap, we investigate the potential privacy risks in federated matrix factorization. Specifically, we classify the federated MF into horizontal, vertical federated MF as well as federated transfer MF based on the data partition approaches. We then demonstrate how private user preference data can be breached during FL training to honest and curious participants. Finally, we discuss how cryptographic techniques can be adopted to protect privacy. \nThe contribution of this paper can be summarised as follows:\n\\begin{itemize}\n    \\item We identify and formulate three types of federated MF problems based on the way of data partition. \n    \\item We demonstrate the privacy threats in each type of federated MF, by showing how the private preference data can be leaked to honest-but-curious participants.\n    \\item We investigate privacy-preserving approaches to protect privacy in federated MF. \n\\end{itemize}\nIn the following parts of this paper, section \\ref{relatedwork} gives related works of privacy issues of MF method and federated MF. We give backgrounds of MF and security model in section \\ref{background}. Then, section \\ref{fedmf_privacy_threats} gives privacy attacks to each type of federated MF. In section \\ref{priv_prev}, we discuss privacy-preserving approaches. Finally, conclusions are drawn in section \\ref{conclusion}.\n\n\n\\section{Related Work}\\label{relatedwork}\nPrivacy risks in general recommender systems are studied in ~\\cite{Jeckmans2013PrivacyIR,Lam2006DoYT}, which analysis the privacy concerns that happened in different phases and caused by different entities. However, the federated learning framework is not considered, which introduces parameter exchanges between participants and thus enlarges the attack surface. Though ~\\cite{chai2019secure} investigates privacy risks in horizontal federated MF and adopts HE to harness the privacy risks, it assumes honest-and-curious participants, and the proposed approach only defenses against an honest-but-curious server. We demonstrate that other curious clients can easily infer private training data. \n\nMany works are exploring privacy-preserving techniques for federated recommender systems. ~\\cite{qi2020fedrec} adopts local differential privacy to train a neural network in horizontal federated news recommendation.  ~\\cite{flanagan2020federated} studies federated multi-view MF over heterogeneous data held by different participants. ~\\cite{chen2020practical} explores a two-tiered notion of privacy by introducing a set of public users.  ~\\cite{canny2002collaborative} proposes federated CF based on partial Singular Value Decomposition and adopts HE for model aggregation. Fully HE ~\\cite{kim2016efficient} as well as garbled circuit~\\cite{nikolaenko2013privacy} is also investigated for privacy-preserving MF, where secure MF is conducted between the server and a crypt-service provider. \n~\\cite{gao2019hhhfl,ju2020federated} are the first to investigate the feasibility of the FL framework to enable a distributed training of deep models from multiple heterogeneous datasets for brain-computer interface. \nAlthough these works propose various privacy-preserving approaches, they fail to investigate the potential privacy loss the intermediate transferred parameters can breach during FL training. \n\n\n\\section{Background}\\label{background}\nIn this section, we introduce the matrix factorization method based on stochastic gradient descent, as well as the security model considered in this paper. \n\n\\subsection{Matrix Factorization}\nWe consider $n$ users rate a subset of $m$ items. For $[n] := {1,\\dots,n}$ the set of $n$ users, and $[m] := {1, \\dots, m}$ the set of $m$ items, the user\/item pairs that generate the ratings are denoted by $\\mathcal{M} = \\subset [n] \\times [m]$. The total number of ratings is $M = |\\mathcal{M}|$. Finally, for $(i, j) \\in \\mathcal{M}$, we denote by $r_{i,j} \\in \\mathbb{R}$ the rating generated by user $i$ for item $j$. Matrix factorization uses a $d \\in \\mathbb{N}$ dimensional vector to represent a user as $u$ and an item as $v$, referred as a profile, and models the relevance of an item to a user as the inner product of their profiles. MF computes the user profiles and item profiles $u_i, v_j \\in \\mathbb{R}^d$ following the \\textit{regularized mean squared error} as follows:\n\\begin{equation}\\label{mf_optimization}\n    \\min_{U, V} \\frac{1}{M} \\sum_{(i,j) \\in \\mathcal{M}} (r_{ij} - \\langle {u_i}, v_j\\rangle)^2 + \\lambda_u \\sum_{i\\in [u]} ||u_i||_2^2 + \\lambda_v \\sum_{j\\in [m]} ||v_j||_2^2\n\\end{equation}\nfor positive constants $\\lambda_u, \\lambda_v$. the inner product $\\langle {u_i}, v_j \\rangle$ is the predicted unobserved ratings $r_{ij}$.\n\nStochastic gradient descent (SGD) is widely applied to optimize the profiles $u_i$ and $v_j$ as follows:\n\\begin{align}\n    u_i^t = u_i^{t-1} - \\gamma \\cdot (\\nabla_{u_i} F(U^{t-1},V^{t-1}) + 2 \\lambda_u u_i^{t-1}) \\\\\n    v_j^t = v_j^{t-1} - \\gamma \\cdot (\\nabla_{v_j} F(U^{t-1}, V^{t-1}) + 2 \\lambda_v v_j^{t-1})\n\\end{align}\nwhere $\\gamma>0$ is the learning rate. $U$ and $V$ are the user profile matrix and item profile matrix with each row as a profile, and $\\nabla_{u_i}F(U,V)$ and $\\nabla_{v_j}F(U,V)$ can be computed as follows:\n\\begin{align}\n    \\nabla_{u_i}F(U,V) = - \\frac{2}{M_{u_i}} \\sum_{j:(i,j)\\in \\mathcal{M}} v_j (r_{ij} - \\langle{u_i}, v_j\\rangle) \\label{gradient_U}\\\\\n    \\nabla_{v_j}F(U,V) = - \\frac{2}{M_{v_j}} \\sum_{i:(i,j)\\in \\mathcal{M}} u_i (r_{ij} - \\langle{u_i}, v_j\\rangle) \\label{gradient_V}\n\\end{align}\n\n\\subsection{Security Model}\nWe assume all participants as well as the server if there is any are \\textit{honest-but-curious} (a.k.a. semi-honest). An honest-but-curious participant follows the protocol honestly but tries to infer private information from the intermediate information it knows. \n\n\\section{Federated MF and Privacy Threats} \\label{fedmf_privacy_threats}\n\n\\begin{figure*}[!htb]\n\\centering \n\\subfigure[Horizontal Federated MF]{\n\\label{Fig.1.1}\n\\includegraphics[width=0.55\\linewidth]{fig\/horizontal_federated_MF_data.pdf}}\n\\subfigure[Vertical Federated MF]{\n\\label{Fig.1.2}\n\\includegraphics[width=0.34\\linewidth]{fig\/vertical_FedMF.pdf}}\n\\subfigure[Federated Transfer MF (Common item set)]{\n\\label{Fig.1.3}\n\\includegraphics[width=0.55\\linewidth]{fig\/federated_transfer_MF_data.pdf}}\n\\caption{The way of data partition in federated MF for horizontal federated MF, vertical federated MF and federated transfer MF (assuming participants shares common item set). Orange, blue, and green rectangles denote the profile matrices of the item, user, and auxiliary data, respectively. The small rectangles in the user-item interaction matrix represent user ratings.}\n\\label{Fig.data_partition}\n\\end{figure*}\n\n\\begin{figure*}[htb]\n\\centering \n\\subfigure[Horizontal Federated MF]{\n\\label{Fig.2.1}\n\\includegraphics[width=0.49\\linewidth]{fig\/feature_space_horizontal_FedMF_2.pdf}}\n\\subfigure[Federated Transfer MF (Common item set)]{\n\\label{Fig.2.2}\n\\includegraphics[width=0.49\\linewidth]{fig\/feature_space_Fed_Trans_MF_2.pdf}}\n\\subfigure[Vertical Federated MF]{\n\\label{Fig.2.3}\n\\includegraphics[width=0.48\\linewidth]{fig\/feature_space_vertical_FedMF_2.pdf}}\n\\caption{The sparse representation of partitioned data and feature space in federated MF for horizontal federated MF, vertical federated MF, and federated transfer MF (assuming participants share common item set). Orange, blue, and green rectangles denote the profile matrices of the item, user, and auxiliary data, respectively. The small rectangles represent non-zero values.}\n\\label{Fig.sparse_data_partition}\n\\end{figure*}\n\n\n\nIn this section, we discuss federated MF in three settings that differ in data partition. Then we investigate how privacy can be breached towards adversarial participants in each setting. For simplicity and without loss of generality, we consider FL systems consisting of two participants, $P^{A}$ and ${P^B}$. \nFig. \\ref{Fig.data_partition} compares the data partition in each setting of federated MF discussed in this paper. \nWe adopt a sparse representation of partitioned data in Fig.\\ref{Fig.sparse_data_partition}, to demonstrate the nature of horizontal, vertical, and transfer federated learning. In the horizontal FL setting Fig. \\ref{Fig.2.1}, participants share the same feature space. In the vertical FL setting Fig.\\ref{Fig.2.3}, participants hold heterogeneous feature space, and only $P^A$ holds the ratings. Whereas, in federated transfer MF Fig.\\ref{Fig.2.2}, participants share partial models (e.g., item profiles) for knowledge transfer.\n\n\\subsection{Horizontal Federated MF}\nIn horizontal federated MF, $P^A$ and $P^B$ share the same user-item interaction matrix (i.e., the same user and item feature space), as shown in Fig \\ref{Fig.1.1} and Fig. \\ref{Fig.2.1}. Therefore, each participant holds the profiles of all users and items, and can locally compute gradient of the whole MF model. Only model aggregation requires communication between A and B~\\cite{mcmahan2016communication}. In model aggregation, the global user profiles matrix is computed by $U_{Global} = \\frac{1}{2} (U_{P^A} + U_{P^B})$, and item profiles matrix $V_{Global} = \\frac{1}{2} (V_{P^A} + V_{P^B})$. $P^A$ can compute the gradient $\\nabla_{U}F_B(U,V)$ of $P^B$ following\n\\begin{align*}\n    & \\nabla_{U}F_B(U^{T-1},V^{T-1}) \\\\\n    & = \\frac{1}{\\gamma} (2 \\cdot U_{Global}^{T} - U_{P^A}^{T} - U^{T-1}) - 2 \\lambda_u U^{T-1}\n\\end{align*}\nwhere $T$ is the index of round. Since the user-item interaction matrix is sparse, that is, $M = \\Theta(n+m)$, which is much smaller than the number of potential ratings $n \\cdot m$ ~\\cite{nikolaenko2013privacy}. For one update in SGD, it is very likely to have one rating record for each item or user. Therefore, according to Equation \\ref{gradient_U}, $P^A$ can easily find the $(i,j)$ pair and the corresponding $v_j$, by checking the gradient and comparing $\\nabla_{u_i}F_B(U,V)$ to each of $\\{u_i\\}_{i=1}^{n}$ as well as comparing $\\nabla_{v_j}F_B(U,V)$ to each of $\\{u_j\\}_{j=1}^m$. Then, $P^A$ further infer the private rating score by\n\\begin{equation*}\n    \\hat{r}_{ij} = - \\frac{\\nabla_{u_i}F_B(U^{T-1},V^{T-1})}{2\\cdot v_j^{T-1}} + \\langle {u_i^{T-1}}, v_j^{T-1} \\rangle.\n\\end{equation*}\nThis way, $P^A$ may complete inference attack and extracts raw private user preference data $(i, j, \\hat{r}_{ij})$ of $P^B$ from the plaintext global model in horizontal federated MF. \n\n\\subsection{Vertical Federated MF}\nIn vertical federated MF as shown in Fig. \\ref{Fig.1.2} and Fig. \\ref{Fig.2.3}, $P^A$ holds the user-item interaction matrix, and $P^B$ holds some auxiliary data of users (or items). We adopt the model presented in ~\\cite{5197422} to leverage auxiliary data provided by $P^B$ in vertical federated MF. \nFor each user $i$, $P^B$ hold distinct factor vectors $y_a \\in \\mathbb{R}^d$ corresponds to each attribute. The user $i$ can thus be described through the set of user-associated attributes $A(u)$ as $\\sum_{a\\in A(u)} y_a$.\nFor vertical federated MF model, Equation \\ref{mf_optimization} can be modified as follows:\n\\begin{align*}\n        & \\min_{U, V} \\frac{1}{M} \\sum_{(i,j) \\in \\mathcal{M}} (r_{ij} - \\langle (u_i + k_i), {v_j}\\rangle)^2 + \\\\\n        & \\lambda_u \\sum_{i\\in [u]} ||u_i||_2^2 + \\lambda_v \\sum_{j\\in [m]} ||v_j||_2^2,\n\\end{align*}\nwhere $k_i = |N(u)|^{-0.5}\\sum_{l\\in N(u)} x_l \\sum_{a\\in A(u)}$. $y_a$ is the auxiliary information of user $i$. $x$ is the implicitly preferred item set, $y$ is $i$'s attributes (e.g., demographic info).\n\nTo conduct federated vertical MF, $P^B$ locally computes and sends $k_i$ to $P^A$, and $P^A$ sends nothing to $P^B$. Therefore, $P^A$ has no privacy leakage to $P^B$, while $P^B$ leaks $k_u$ to $P^A$. In such a setting, user ID leakage during the user alignment stage causes a major privacy threat.\n\n\\subsection{Federated Transfer MF}\nWithout loss of generality, in federated transfer MF, we assume $P^A$ and $P^B$ holds ratings given by different users on the same set of items, and $P^A$ tries to infer private data of $P^B$. That is, $P^A$ and $P^B$ holds the item profiles matrix $V$. $P^A$ holds $U^A$, and $P^B$ holds $U^A$ for their users, respectively. To train the model, each participant locally conducts SGD to update local models. For model aggregation, only $V$ is aggregated by the participants. \n$P^A$ can learn the gradient $\\nabla_V F_B(U^B,V)$ of $P^B$ as follows:\n\\begin{align*}\n    & \\nabla_{V}F_B({U^B}^{T-1},V^{T-1}) \\\\\n    & = \\frac{1}{\\gamma} (2 \\cdot V_{Global}^{T} - V_{P^A}^{T} - V^{T-1}) - 2 \\lambda_v V^{T-1}\n\\end{align*}\nAs the interaction matrix is sparse for each round, it is reasonable to assume $\\nabla_{V}F_B({U^B}^{T-1},V^{T-1}) = -2 {u^B_i}^{T-1} (r_{ij} - \\langle {u^B_i}^{T-1}, {v_j}^{T-1}\\rangle)$ for a user $i$. \nBy collecting the gradients of several steps and assuming the $u_i^B$ does not change, which is reasonable, then the model is nearly converged, we can use some iterative methods such as Newton's method to approximate the numeric value of $u^B_i$, as shown in ~\\cite{chai2019secure}. After computing $u^B_i$, the reconstructed rating score $\\hat{r^B}_{ij}$ of $P^B$ can easily be computed as follows:\n\\begin{equation*}\n    \\hat{r^B}_{ij} = - \\frac{\\nabla_{v_j}F_B({U^B}^{T-1},V^{T-1})}{2 \\cdot {u^B}_i^{T-1}} + \\langle {{u^B_i}^{T-1}}, v_j^{T-1} \\rangle.\n\\end{equation*}\nThus, $P^A$ completes the inference attack and extracts user profiles and the corresponding ratings of $P^B$ from the plaintext global model in horizontal federated MF. However, as the users are not aligned between participants, the user ID information can not be inferred by $P^A$. \n\nIt is worth noting that, although some works denote participants sharing the same set of items and a different set of users as \\textit{horizontal federated recommender systems}, and denote participants sharing the same set of users and a different set of items as \\textit{vertical federated recommender systems}, based on whether the users should be aligned before FL training. Such setting demonstrates the nature of federated transfer learning, where neither the global model is shared among participants, nor the feature space is fully partitioned without intersection. Privacy attacks on both settings are the same during FL training.\nTherefore, we denote both settings as federated transfer MF in this paper. \n\n\\subsection{Comparison}\\label{Comparison}\n\\begin{table*}[!htb]\n\\centering\n\\footnotesize\n    \\begin{tabular}{| c || c | c | c |}\n    \\hline \n    Problem setting & Parameter partition in each party & Gradient computation & Resilience against inference attack \\\\\n    \\hline \n    \\hline\n    Horizontal FedMF & The whole model params & Locally & Weak \\\\\n    \\hline\n    Vertical FedMF & Partial params without shared params & Collaboratively & P$^A$ strong, P$^B$ weak \\\\\n    \\hline\n    Federated Transfer MF & Partial params with shared params & Locally & Medium \\\\\n    \\hline\n\\end{tabular}\n\\caption{Comparison of different problem settings including horizontal and vertical federated MF as well as federated transfer MF. FedMF denotes federated MF.}\\label{table_comparison}\n\\end{table*}\n\nTab. \\ref{table_comparison} demonstrates the comparison of three settings based on the way to update the model, the partition of model parameters, and the resilience of the FL system against inference attack. \nIn horizontal federated MF, all participants share ratings from the same set of users and items. Therefore, each participant locally holds the whole user profiles matrix and item profiles matrix for local SGD. For federated transfer MF, participants only share the same set of users (or items), each participant locally holds its user (or item) profiles sub-matrix and the global item (or user) profiles matrix for local SGD. For vertical federated MF, one party holds rating data; the other holds auxiliary data, each party holds partial parameters with shared parameters such as user profiles matrix.\nFor both horizontal federated MF and Federated transfer MF, clients can locally conduct SGD optimization without the need for communication. Participants only need to exchange parameters during the model aggregation process. For vertical federated MF, two participants need to collaboratively compute the estimated rating for each update, which dramatically increases the communication cost. The resilience of each setting against the inference attack is also shown. Horizontal federated MF breaches most private information, including user ID and user preference data. For vertical federated MF, recommender $P^A$ leaks no information to data provider $P^B$, and data provider sends the intermediate data to the recommender. The user ID is breached for both participants. For federated transfer MF, only private rating data and user profiles are leaked, and no user ID is breached.\n\n\n\\section{Privacy Preservation in Federated MF}\\label{priv_prev}\nAccording to the privacy threats investigated in section \\ref{fedmf_privacy_threats}, we give some advises for privacy preservation in federated MF.\nFor horizontal federated MF, the global user and item profile matrices computed by aggregation should be protected against each participant. For vertical federated MF, the auxiliary data provider should keeping its computed feature sent to the recommender secret. For federated transfer MF, the shared user or item profile matrix should be kept secret to any honest-but-curious participant throughout the FL training, as the rating score and private profile can be potentially implied.\n\nTo keep intermediate parameters private, there are mainly three types of approaches \\textit{cryptography-based}, \\textit{obfuscation-based} and \\textit{hardware-based} approaches. Cryptography-based approaches generally use HE and MPC to keep intermediate transactions private. Obfuscation-based approaches such as DP obfuscate private data by randomization, generalization or repression. Hardware-based approaches rely on trusted execution environment (TEE) to conduct FL learning in a trusted enclave.\nBy using cryptography-based approaches, fully HE can be introduced to prevent decryption during training~\\cite{kim2016efficient}. Secret sharing schemes can also be introduced following a two-server architecture~\\cite{cryptoeprint:2011:535}. \nSince the user-item interaction matrix is sparse, applying DP may introduce too much noise and make the model unavailable. TEE can also be applied by encrypting private data and conducting private training inside TEE~\\cite{CHEN202069}.\n\n\\section{Conclusion}\\label{conclusion}\nWe identify and formulate three types of federated MF problems based on the partition of feature space. Then, we demonstrate the privacy threats against each type of federated MF. We show how the private user preference data, private user\/item profiles matrix, and user ID can be potentially leaked to honest-but-curious participants. Finally, We discuss privacy-preserving approaches to protect privacy in federated MF. \nFor future work, we will experimentally study the power of the proposed privacy attacks by measuring the portion and accuracy of the inferred private data. Privacy threats against alternating least squares-based MF and other recommender systems also require further comprehensive study.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{named}\n\n\\section{Introduction}\nFederated learning (FL) enables multiple parties to collaboratively train machine learning models without exposing ones' own data. It addresses the data silos and privacy problem together. \nIn this paper, we consider a typical federated learning scenario in the recommender systems. \nTaking the collaboration between a movie recommender and a data provider as an example, the intersection of their user space is large. The recommender records the user's rating, browsing and watching history of the recommended movies, while the data provider owns the demographic information. Their feature spaces are very different. We propose a method to integrate the features of these two parties in a secure and privacy-preserving way and improve the recommendation performance.\n\n\n\nThe feature interaction is a widely used method to integrate features and increase the models' expressiveness. For example, the movie recommender has features about the movie genres, and the data provider has user gender information. For instance, through modeling the interaction between ``gender-male'',  ``gender-female'', ``genres-romance'' and ``genres-adventure'' features, we can capture the preferences of different user groups, such as ``female on  romance'' and ``male on adventure''. Learning interactions among features is significant in improving the models' performance. In the federated learning setting, these features are stored in the movie recommender and data provider, respectively. They could not leave their own data repository and never be transmitted due to privacy concerns. Previous learning algorithms, which build a model on centrally collected data, are not able to model the interactions between features distributed on these two parties. Besides, although we can model the feature interaction through the weight parameters of feature pairs and learning the model by exchanging weight parameters as previous FL algorithms did, the communication cost could be extremely high. Taking the second-order feature interaction as an example, the cost is $O(d_A\\times d_B)$, where $d_A$ and $d_B$ are the feature dimensions. It is unacceptable with iterative optimization algorithms, which is widely adopted in FL. Furthermore, to avoid information leakage, the weight parameters are usually encrypted by the encryption algorithm. The bit lengths of the parameters increase in the encrypted space. Therefore, the communication cost further increases. \n\nIn this paper, we propose the \\textbf{Federated Collective Matrix Factorization} (FedCMF) algorithm. As shown in Figure ~\\ref{fig_intro_overview}, the FedCMF explicitly captures the feature interactions inside each party as well as between two parties in a privacy-preserving way. Only the model parameters and intermediate results are transmitted between parties instead of the raw data. As we prove in the following sections, the communication cost between two parties is at most $O(d\\times d')$, where $d'$ is a hyper-parameter and much smaller than the original dimensions $d\\in \\{d_A, d_B\\}$. The FedCMF is based on traditional collective matrix factorization (CMF), which is proposed in ~\\cite{rendle2010factorization}. Traditional CCMF parameterizes the weight of the feature interaction as the inner product of the embedding vectors of the constituent features. By learning an embedding vector for each feature, traditional CCMF can estimate the weight for any interacted features. However, traditional CCMF is not designed for decentralized data. In other words, all the interacted features should be collected together on a central server, then the traditional CMF can start to learn the model. For the federated learning setting, we propose a federated collective matrix factorization by reformulating the objective function of the traditional CCMF. The FedCMF exchanges embedding vectors and their partial gradients between parties to capture the feature interactions, without collecting raw data from isolated parties. \n\nNonetheless, exchanging the clear-text intermediate results in the FedCMF results in high privacy risks and is vulnerable to model inversion attack and data reconstruction attack. Theoretically, we prove that the model and raw data could be fully reconstructed by inserting a few known instances to the other party. To enhance the model security, we propose a secure computation protocol, where the intermediate results are encrypted by secret sharing and exchanged with MPC protocol~\\cite{7958569}. With the secure protocol, no bit of information will be leaked between parties. For the tasks that have high-dimensional sparse features, to protect the positional information of the non-null features, the secure computation protocol has to be conducted over all features, including the zero-value (and blank) features. It is not practical. Therefore, we further propose a secure sparse inner product protocol based on homomorphic encryption (HE). The secure computation is only conducted on non-null features, and the intermediate results that updated by the non-null features are encrypted by homomorphic encryption. The communication cost further decreases to $O(\\hat{d}\\times d')$, where $\\hat{d}\\in \\{\\hat{d}_A, \\hat{d}_B\\}$ is the number of non-null features in the instance. In most high-dimensional sparse cases, the $\\hat{d}$ is much smaller than the original dimension $d$ in each instance. \n\n\nIn summary, our contributions are three folds:\n\\begin{itemize}\n    \\item We propose a new FedCMF algorithm that models feature interactions between parties with small communication cost and is secure against model inversion attack and data reconstruction attack.\n    \\item We also propose a secure sparse inner product protocol to handle high-dimensional sparse features, which commonly exists in recommender systems and has not been studied in federated learning so far.\n\t\\item We evaluate the FedCMF over three real-world datasets, showing fast convergence and promising performance improvement.\n\\end{itemize}\n\nIn the following parts of this article, section~\\ref{problemdefinition} gives a formal definition of federated learning with feature interactions. In section~\\ref{vfCMF}, we introduce the traditional factorization machine and the federated factorization machine. In section~\\ref{securevfCMF}, first of all, we theoretically prove the information leakage when FedCMF transmits clear-text intermediate results. Then, we introduce secure computation protocols. In section~\\ref{exp}, we conduct experiments to evaluate the proposed FedCMF. We summary the related work in section~\\ref{relatedwork} and conclude the paper in section~\\ref{conclusion}.\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":"\\section{Introduction}\nGraph Neural Networks (GNNs) have emerged as a powerful tool for extracting information from graph-structured data and have achieved state-of-the-art performance on a variety of machine learning tasks. However, compared to research on constructing GNNs for unsigned and undirected graphs, and graphs with multiple types of edges, GNNs for graphs where the edges have a natural notion of sign, direction, or both, have received relatively little attention. \n\nThere is a demand for such tools because many important and interesting phenomena are naturally modeled as signed and\/or directed graphs, i.e., graphs in which objects may have either positive or negative relationships, and\/or in which such relationships are not necessarily symmetric~\\cite{he2022pytorch}. For example, in the analysis of social networks, positive and negative edges could model friendship or enmity, and directional information could model the influence of one person on another~\\cite{konect:kumar2016wsn,huang2021sdgnn}. Signed\/directed networks also arise when analyzing time-series data with lead-lag relationships~\\cite{bennett2021detection}, detecting influential groups in social networks~\\cite{he2021digrac}, and computing rankings from pairwise comparisons~\\cite{he2022gnnrank}. Additionally, signed and directed networks are a natural model for group conflict analysis~\\cite{zheng2015signed}, modeling the interaction network of the agents during a rumor spreading process~\\cite{hu2013bipartite}, and maximizing positive influence while formulating opinions~\\cite{ju2020new}. \nBroadly speaking, most GNNs may be classified as either spectral or spatial. Spatial methods typically define convolution on graphs as a localized aggregation operator whereas spectral methods rely on the eigen-decomposition of a suitable graph Laplacian. The goal of this paper is to introduce a novel Laplacian and an associated spectral GNN for signed  directed graphs. While  spatial GNNs exist, such as  SDGNN \\cite{huang2021sdgnn}, SiGAT \\cite{huang2019signed}, SNEA \\cite{li2020learning},  and SSSNET \\cite{he2022sssnet} proposed for signed (and possibly directed) networks, this is one of the first works to propose a spectral GNN for such networks.\n \nA principal challenge in extending traditional spectral GNNs to this setting is to define a proper notion of the signed, directed graph Laplacian. Such a Laplacian should be positive semidefinite, have a bounded spectrum when properly normalized, and encode information about both the sign and direction of each edge. Here, we unify the magnetic Laplacian, which has been used in \\cite{zhang2021magnet} to construct a GNN on an (unsigned) directed graph, with a signed Laplacian which has been used for a variety of data science tasks on (undirected) signed graphs~\\cite{kunegis2010spectral,zheng2015spectral, mercado2016clustering,Cucuringu2020Reg}. Importantly, our proposed matrix, which we refer to as the \\emph{magnetic signed \nLaplacian}, reduces to either the magnetic Laplacian or the signed Laplacian when the graph is directed, but not signed, or signed, but not directed.\n \nAlthough this magnetic signed Laplacian is fairly straightforward to obtain, it is novel and surprisingly powerful: We show that our proposed \\emph{\\textbf{M}agnetic \\textbf{S}igned \\textbf{GNN} ({MSGNN})} is effective for a variety of node clustering and link prediction tasks. Specifically, we consider several variations of the link prediction task, some of which prioritize signed information over directional information, some of which prioritize directional information over signed information, while others emphasize the method's ability to extract both signed and directional information simultaneously. \n\n\\looseness=-1 In addition to testing {MSGNN} on established data sets, we also devise a novel synthetic model which we call the \\emph{Signed Directed Stochastic Block Model (SDSBM)}, which generalizes both the (undirected) Signed Stochastic Block Model from \\cite{he2022sssnet} and the (unsigned) Directed Stochastic Block Model from \\cite{he2021digrac}. Analogous to these previous models, our SDSBM can be defined by a meta-graph structure and additional parameters describing density and noise levels.  We also introduce a number of signed directed networks for link prediction tasks using lead-lag relationships in real-world financial time series. \n\n\\textbf{Main Contributions.} The main contributions of our work are:  (1) We devise a novel matrix called the magnetic signed Laplacian, which can naturally be applied to signed and directed networks. The magnetic signed Laplacian is Hermitian, positive semidefinite, and the eigenvalues of its normalized counterpart lie in $[0,2]$. They reduce to existing Laplacians when the network is unsigned and\/or undirected. \n(2) We propose an efficient spectral graph neural network architecture, {MSGNN}, based on this magnetic signed Laplacian, which attains leading performance on extensive node clustering and link prediction tasks, including novel tasks that consider edge sign and directionality jointly. To the best of our knowledge, this is the first work to evaluate GNNs\\footnote{Some previous work, such as \\cite{huang2021sdgnn}, evaluates GNNs on signed and directed graphs. However, they focus on tasks where either only signed information is important, or where only directional information is important.} on tasks that are related to both edge sign and directionality. \n(3) We introduce a novel synthetic model for signed and directed networks, called Signed Directed Stochastic Block Model (SDSBM), and also contribute a number of new real-world data sets constructed from lead-lag relationships of financial time series data. \n\\section{Related Work} \\label{sec:related} \nIn this section, we review related work constructing neural networks for directed graphs and signed graphs. We refer the reader to \\cite{he2022pytorch} for more background information.\n\nSeveral works have aimed to define neural networks on directed graphs by constructing various directed graph Laplacians and defining convolution as multiplication in the associated eigenbasis. \\cite{ma:spectralDGCN2019} defines a directed graph Laplacian by generalizing identities involving the undirected graph Laplacian  and the stationary distribution of a random walk. \\cite{Tong2020DigraphIC} uses a similar idea, but with PageRank in place of a random walk. \\cite{tong:directedGCN2020} constructs three different first- and second-order symmetric adjacency matrices and uses these adjacency matrices to define associated Laplacians. Similarly, \\cite{monti:MotifNet2018} uses several different graph Laplacians based on various graph motifs. \n\n\nQuite closely related to our work, \\cite{zhang2021magnet} constructs a graph neural network using the magnetic Laplacian. Indeed, in the case where all links are positive, our GNN exactly reduces to the one proposed in \\cite{zhang2021magnet}. Importantly, unlike the other directed graph Laplacians mentioned here, the magnetic Laplacian is a complex, Hermitian matrix rather than a real, symmetric matrix. We also note\n\\cite{he2021digrac}, which constructs a GNN for node clustering on directed graphs based on flow imbalance. \n\nAll of the above works are restricted to unsigned graphs,  i.e., graphs with positive edge weights. However, there are also a number of neural networks introduced for signed (and possibly also directed) graphs, mostly focusing on the task of link sign prediction, i.e., predicting whether a link between two nodes will be positive  or negative. SGCN by  \\cite{derr2018signed} is one of the first graph neural network methods to be applicable to signed networks, using an approach based on balance theory~\\cite{harary1953notion}. However, its design is mainly aimed at undirected graphs. SiGAT~\\cite{huang2019signed} utilizes a graph attention mechanism based on \\cite{velivckovic2017graph} to learn node embeddings for signed, directed graphs, using a novel motif-based GNN architecture based on balance theory and status theory~\\cite{leskovec2010signed}. Subsequently, SDGNN by \\cite{huang2021sdgnn} builds upon this work by increasing its efficiency and proposing a new objective function. In a similar vein, SNEA  \\cite{li2020learning} proposes a signed graph neural network for link sign prediction based on a novel objective function. In a different line of work,\n\\cite{he2022sssnet} proposes SSSNET, a GNN not based on balance theory designed for semi-supervised node clustering in signed (and possibly directed) graphs. \n\nAdditionally, we note that several GNNs~\\cite{schlichtkrull2018modeling, vashishth2019composition, zhang2020relational} have been introduced for multi-relational graphs, i.e., graphs with different types of edges. In such networks, the number of learnable parameters typically increases linearly with the number of edge types. Signed graphs, at least if the graph is unweighted or the weighting function $w$ only takes finitely many values, can be thought of as special cases of multi-relational graphs. However, in the context of (possibly weighted) signed graphs, there is an implicit relationship between the different edge-types, namely that a negative edge is interpreted as the opposite of a positive edge and that edges with large weights are deemed more important than edges with small weights. These relationships will allow us to construct a network with significantly fewer trainable parameters than if we were considering an arbitrary multi-relational graph.\n\nWe are aware of a concurrent preprint~\\cite{fiorini2022sigmanet} which also constructs a GNN, SigMaNet, based on a signed magnetic Laplacian. Notably, the signed magnetic Laplacian considered in \\cite{fiorini2022sigmanet} is different from the magnetic signed Laplacian proposed here. The claimed advantage of SigMaNet is that it does not require the tuning of a charge parameter $q$ and is invariant to, e.g., doubling the weight of every edge. In our work, for the sake of simplicity, we usually set $q=0.25$, except for when the graph is undirected (in which case we set $q=0$). However, a user may choose to also tune $q$ through a standard cross-validation procedure as in \\cite{zhang2021magnet}. Moreover, one can readily address the latter issue by normalizing the adjacency matrix via a preprocessing step (e.g., \\cite{CLONINGER2017370}).\nIn contrast to our magnetic signed Laplacian, in the case where the graph is not signed but is weighted and directed, the matrix proposed in \\cite{fiorini2022sigmanet} does not reduce to the magnetic Laplacian considered in \\cite{zhang2021magnet}. For example, denoting the graph adjacency matrix by $\\mathbf{A}$, consider the case where $0< \\mathbf{A}_{j,i}< \\mathbf{A}_{i,j}$. Let $m=\\frac{1}{2}(\\mathbf{A}_{i,j}+\\mathbf{A}_{j,i})$, $\\delta=\\mathbf{A}_{i,j}-\\mathbf{A}_{j,i}$, and let $\\mathbbm{i}$ denote the imaginary unit. Then the $(i,j)$-th entry of the matrix $\\mathbf{L}^\\sigma$ proposed in \\cite{fiorini2022sigmanet} is given by \n$ \\mathbf{L}^\\sigma_{i,j}=m\\mathbbm{i}$, whereas the corresponding entry of the unnormalized magnetic Laplacian is given by\n$(\\mathbf{L}^{(q)}_U)_{i,j}=m\\exp(2\\pi \\mathbbm{i} q \\delta).$ Moreover, while SigMaNet is in principle well-defined on signed and directed graphs, the experiments in \\cite{fiorini2022sigmanet} are restricted to tasks where only signed or directional information is important (but not both). \nIn our experiments, we find that our proposed method outperforms SigMaNet on a variety of tasks on signed and\/or directed networks. Moreover, we observe that the signed magnetic Laplacian $\\mathbf{L}^\\sigma$ proposed in \\cite{fiorini2022sigmanet} has an undesirable property when the graph is unweighted --- a node is assigned to have degree zero if it has an equal number of positive and negative connections. Our proposed Laplacian does not suffer from this issue.\n\\section{Proposed Method}\n\\subsection{Problem Formulation}\n\\label{sec:formulation}\n\\looseness=-1 Let $\\mathcal{G}=(\\mathcal{V}, \\mathcal{E}, w, \\mathbf{X}_\\mathcal{V})$ denote\na signed, and possibly directed, weighted graph with node attributes, \nwhere $\\mathcal{V}$ is the set of nodes (or vertices), $\\mathcal{E}$ is the set of (directed) edges (or links), and $w: \n\\mathcal{E}\\rightarrow (-\\infty, \\infty)\\setminus\\{0\\}$ is the weighting function. Let $\\mathcal{E}^+=\\{e\\in\\mathcal{E}: w(e)>0\\}$ denote the set of positive edges and let $\\mathcal{E}^-=\\{e\\in\\mathcal{E}: w(e)<0\\}$ denote the set of negative edges so that $\\mathcal{E}=\\mathcal{E}^+\\cup\\mathcal{E}^-.$\nHere, we do allow self loops but not multiple edges; \nif $v_i, v_j\\in \\mathcal{V},$ there is at most one edge $e\\in\\mathcal{E}$ from $v_i$ to $v_j$.  \nLet $n=|\\mathcal{V}|$, and let $d_{\\text{in}}$ be the number of attributes at each node,  so that $\\mathbf{X}_\\mathcal{V}$ is an $n\\times d_\\text{in}$ matrix whose rows are the attributes of each node.  \nWe let $\\mathbf{A} = (A_{ij})_{i,j \\in \\mathcal{V}} $ denote the weighted, signed adjacency matrix where $\\mathbf{A}_{i,j}=w_{i,j}$ if $(v_i,v_j)\\in\\mathcal{E}$, and $\\mathbf{A}_{i,j}=0$ otherwise. \n\\subsection{Magnetic Signed Laplacian}\n\\label{sec:mag}\nIn this section, we define Hermitian matrices $\\mathbf{L}_U^{(q)}$ and $\\mathbf{L}_N^{(q)}$ which we refer to as the unnormalized and normalized magnetic signed Laplacian matrices, respectively. We first define a symmetrized adjacency matrix and an absolute degree matrix by\n$$\\Tilde{\\mathbf{A}}_{i,j} \\coloneqq \\frac{1}{2} ( \\mathbf{A}_{i,j} + \\mathbf{A}_{j,i} ), \\,\\:\\:1\\leq i,j\\leq n ,\\:\\:\n    \\Tilde{\\mathbf{D}}_{i,i} \\coloneqq \\frac{1}{2}\\sum_{j=1}^n (|\\mathbf{A}_{i,j}|+|\\mathbf{A}_{j,i}|),  \\:\\:1\\leq i\\leq n, $$\nwith $\\Tilde{\\mathbf{D}}_{i,j}=0$ for $i\\neq j$. Importantly, the use of absolute values ensures that the entries of $\\Tilde{\\mathbf{D}}$ are non-negative. Furthermore, it ensures that all $\\Tilde{\\mathbf{D}}_{i,i}$ will be strictly positive if the graph is connected. This is in contrast to the construction in \\cite{fiorini2022sigmanet} which will give a node degree zero if it has an equal number of positive and negative neighbors (for unweighted networks). To capture directional information, we next define a phase matrix $\\mathbf{\\Theta}^{(q)}$ by\n$\\mathbf{\\Theta}^{(q)}_{i,j} \\coloneqq 2\\pi q (\\mathbf{A}_{i,j} - \\mathbf{A}_{j,i}),$ where $q\\in\\mathbb{R}$ is the so-called ``charge parameter.\"\nIn our experiments, for simplicity, we set $q=0$ when the task at hand is unrelated to directionality, or when the underlying graph is undirected, and we set $q=0.25$ for all the other tasks (except in an ablation study on the role of $q$). \nWith  $\\odot$ denoting elementwise multiplication, and $\\mathbbm{i}$ denoting the imaginary unit, we now construct a complex Hermitian matrix $\\mathbf{H}^{(q)}$ by\n$$\n    \\mathbf{H}^{(q)} \\coloneqq \\Tilde{\\mathbf{A}} \\odot \\exp (\\mathbbm{i} \\mathbf{\\Theta}^{(q)}) \\, \n$$\nwhere \n$\\exp (\\mathbbm{i} \\mathbf{\\Theta}^{(q)})$ is defined elementwise by\n  $  \\exp (\\mathbbm{i} \\mathbf{\\Theta}^{(q)})_{i,j} \\coloneqq \\exp(\\mathbbm{i} \\mathbf{\\Theta}^{(q)}_{i,j})$.  \n  \nNote that $\\mathbf{H}^{(q)}$ is Hermitian, as  $\\Tilde{\\mathbf{A}}$ is symmetric and $\\mathbf{\\Theta}^{(q)}$ is skew-symmetric.  \nIn particular, when $q=0$, we have  $\\mathbf{H}^{(0)}=\\Tilde{\\mathbf{A}}$. Therefore, setting $q=0$ is  equivalent to making the input graph symmetric and discarding directional information. In general, however, $\\mathbf{H}^{(q)}$ captures information about a link's sign, through $\\tilde{\\mathbf{A}}$, and about its direction, through $\\boldsymbol{\\Theta}^{(q)}$. \n\nWe observe that flipping the direction of an edge, i.e., replacing a positive or negative link from $v_i$ to $v_j$ with a link of the same sign from $v_j$ to $v_i$ corresponds to complex conjugation of $\\mathbf{H}^{(q)}_{i,j}$ (assuming either that there is not already a link from $v_j$ to $v_i$ or that we also flip the direction of that link if there is one). \nWe also note that if $q=0.25,$ $\\mathbf{A}_{i,j}=\\pm1$, and $\\mathbf{A}_{j,i}=0$, we have \n\\begin{equation*}\n    \\mathbf{H}^{(0.25)}_{i,j}=\\pm\\frac{i}{2}=-\\mathbf{H}^{(0.25)}_{j,i}.\n\\end{equation*}\nThus, a unit-weight edge from $v_i$ to $v_j$ is treated as the opposite of a unit-weight edge from $v_j$ to $v_i$.  \n\nGiven $\\mathbf{H}^{(q)}$, we next define the unnormalized magnetic signed Laplacian by \n\\begin{equation}\n    \\mathbf{L}_{U}^{(q)} \\coloneqq \\Tilde{\\mathbf{D}} - \\mathbf{H}^{(q)} = \\Tilde{\\mathbf{D}} - \\Tilde{\\mathbf{A}} \\odot \\exp (\\mathbbm{i} \\mathbf{\\Theta}^{(q)}),\n\\end{equation}\nand also define the normalized magnetic signed Laplacian by\n\\begin{equation} \\mathbf{L}_{N}^{(q)} \\coloneqq  \\mathbf{I} - \\left(\\Tilde{\\mathbf{D}}^{-1\/2}\\Tilde{\\mathbf{A}}\\Tilde{\\mathbf{D}}^{-1\/2}\\right) \\odot \\exp (\\mathbbm{i} \\mathbf{\\Theta}^{(q)}) \\, .\n\\end{equation}\n\nWhen the graph $\\mathcal{G}$ is directed, but not signed, $\\mathbf{L}_U^{(q)}$ and $\\mathbf{L}_N^{(q)}$ reduce to the magnetic Laplacians utilized in works such as \\cite{zhang2021magnet,fanuel2017magnetic, fanuel:MagneticEigenmaps2018} and \\cite{f2020characterization}. Similarly, when $\\mathcal{G}$ is signed, but not directed,  $\\mathbf{L}_U^{(q)}$ and $\\mathbf{L}_N^{(q)}$ reduce to the signed Laplacian matrices considered in e.g., \\cite{kunegis2010spectral, Cucuringu2020Reg} and \\cite{ATAY2014165}. Additionally, when the graph is neither signed nor directed, they reduce to the standard normalized and unnormalized graph Laplacians~\\cite{defferrard2016convolutional}. The following theorems show that $\\mathbf{L}_U^{(q)}$ and $\\mathbf{L}_N^{(q)}$ satisfy properties analogous to the traditional graph Laplacians. The proofs are in Appendix \\ref{app:proofs}.\n\n\n\\begin{theorem}\\label{thm: posdef} \nFor any signed directed graph $\\mathcal{G}$ defined in Sec.~\\ref{sec:formulation}, $\\forall q\\in \\mathbb{R},$ both the unnormalized magnetic signed Laplacian $\\mathbf{L}_{U}^{(q)}$ and its normalized counterpart \n$\\mathbf{L}_{N}^{(q)}$ are positive semidefinite.\n\\end{theorem}\n\n\n\\begin{theorem}\\label{thm: normal02} For any signed directed graph $\\mathcal{G}$ defined in Sec.~\\ref{sec:formulation}, $\\forall q\\in \\mathbb{R},$ the eigenvalues of the normalized magnetic signed Laplacian $\\mathbf{L}_N^{(q)}$ are contained in the interval $[0,2]$.\n\\end{theorem}\n\nBy construction, $\\mathbf{L}_U^{(q)}$ and $\\mathbf{L}_N^{(q)}$ are Hermitian, and Theorem~\\ref{thm: posdef} shows they are positive semidefinite. In particular they are diagonalizable by an orthonormal basis of complex eigenvectors $\\mathbf{u}_1, \\ldots, \\mathbf{u}_n$ associated to real, nonnegative eigenvalues $\\lambda_1\\leq \\ldots\\leq \\lambda_n\\hspace{-.03in}=\\hspace{-.03in}\\lambda_{\\max}$. Thus,  \nsimilar to the traditional normalized Laplacian, we may factor \n    $\\mathbf{L}_N^{(q)}= \\mathbf{U} \\bm{\\Lambda} \\mathbf{U}^\\dagger$, \nwhere $\\mathbf{U}$ is an $n\\times n$ matrix whose $k$-th column is $\\mathbf{u}_k$, for $1\\leq k\\leq n$, $\\bm{\\Lambda}$ is a diagonal matrix with $\\bm{\\Lambda}_{k,k}=\\lambda_k$, and $\\mathbf{U}^{\\dagger}$ is the conjugate transpose of $\\mathbf{U}$. A similar formula holds for $\\mathbf{L}^{(q)}_U$.\n\n\\subsection{Spectral Convolution via the Magnetic Signed Laplacian}\n\nIn this section, we show how to use a Hermitian, positive semidefinite matrix $\\mathbf{L}$ such as the normalized or unnormalized magnetic signed Laplacian introduced in Sec.~\\ref{sec:mag}, to define convolution on a signed directed graph. This method is similar to the ones proposed for unsigned (possibly directed) graphs in, e.g., \\cite{kipf2016semi} and \\cite{zhang2021magnet}, but we provide details in order to keep our work reasonably self-contained.\n\nGiven $\\mathbf{L}$, let  $\\mathbf{u}_1\\ldots,\\mathbf{u}_n$ be an orthonormal basis of eigenvectors such that $\\mathbf{L}\\mathbf{u}_k=\\lambda_k\\mathbf{u}_k$, and let $\\mathbf{U}$ be an $n\\times n$ matrix whose $k$-th column is $\\mathbf{u}_k$, for $1\\leq k \\leq n$. For a signal $\\mathbf{x}:\\mathcal{V}\\rightarrow\\mathbb{C},$ we define its Fourier transform $\\widehat{\\mathbf{x}}\\in\\mathbb{C}^n$ by $ \\widehat{\\mathbf{x}}(k)= \\langle \\mathbf{x},\\mathbf{u}_k\\rangle\\coloneqq\\mathbf{u}_k^\\dagger\\mathbf{x},$ and equivalently, $\\widehat{\\mathbf{x}}=\\mathbf{U}^\\dagger \\mathbf{x}$. \nSince $\\mathbf{U}$ is unitary, we readily obtain the Fourier inversion formula\n\\begin{equation}\\label{eqn: inversion}\n    \\mathbf{x}=\\mathbf{U}\\widehat{\\mathbf{x}} = \\sum_{k=1}^n\\widehat{\\mathbf{x}}(k)\\mathbf{u}_k \\, .\n\\end{equation}\nAnalogous to the well-known convolution theorem in Euclidean domains, we define the \nconvolution of $\\mathbf{x}$ with a filter $\\mathbf{y}$ as multiplication in the Fourier domain, i.e., $\\widehat{\\mathbf{y}\\ast\\mathbf{x}}(k)=\\widehat{\\mathbf{y}}(k)\\widehat{\\mathbf{x}}(k)$. \nBy \\eqref{eqn: inversion}, this implies\n    $\\mathbf{y}\\ast\\mathbf{x} = \\mathbf{U}\\text{Diag}(\\widehat{\\mathbf{y}})\\widehat{\\mathbf{x}} = (\\mathbf{U}\\text{Diag}(\\widehat{\\mathbf{y}})\\mathbf{U}^\\dagger)\\mathbf{x}$, where $\\text{Diag}(\\mathbf{z})$ denotes a diagonal matrix with the vector $\\mathbf{z}$ on its diagonal.\nTherefore, we declare that $\\mathbf{Y}$ is a \\emph{generalized convolution matrix} if \n\\begin{equation}\\label{eqn: conv}\n    \\mathbf{Y} = \\mathbf{U} \\bm{\\Sigma} \\mathbf{U}^\\dagger \\, ,\n\\end{equation}\nfor a diagonal matrix $\\bm{\\Sigma}.$ This is a natural generalization of the class of  convolutions used in \\cite{bruna:spectralNN2014}. \n\nSome potential drawbacks exist when defining a convolution via \\eqref{eqn: conv}. First, it requires one to compute the eigen-decomposition of $\\mathbf{L}$ which is expensive for large graphs. Second, the number of trainable parameters equals the size of the graph (the number of nodes), rendering  GNNs constructed via \\eqref{eqn: conv} prone to overfitting. To remedy these issues, we follow \\cite{Defferrard2018}  (see also \\cite{hammond2011wavelets}) and observe that spectral convolution may also be implemented in the spatial domain via polynomials of  $\\mathbf{L}$ by setting   $\\bm{\\Sigma}$ equal to a polynomial of $\\bm{\\Lambda}$. This reduces the number of trainable parameters from the size of the graph to the degree of the polynomial and also enhances robustness \nto perturbations \\cite{levie2019transferability}.\nAs in \\cite{Defferrard2018}, we let  $\\widetilde{\\bm{\\Lambda}}=\\frac{2}{\\lambda_{\\max}} \\bm{\\Lambda}-\\mathbf{I}$ denote the  normalized eigenvalue matrix (with entries in $[-1,1]$) and choose  \n$\\bm{\\Sigma} = \\sum_{k=0}^K\\theta_k T_k(\\widetilde{\\bm{\\Lambda}}),$\nfor some $\\theta_1,\\ldots,\\theta_k\\in\\mathbb{R}$ where for $0\\leq k \\leq K$, \n$T_k$ is  the Chebyshev polynomials defined by $T_0(x)=1, T_1(x)=x,$ and $T_{k}(x)=2xT_{k-1}(x)+T_{k-2}(x)$ for $k\\geq 2.$ \nSince $\\mathbf{U}$ is unitary, we have  $(\\mathbf{U} \\widetilde{\\bm{\\Lambda}} \\mathbf{U}^\\dagger)^k = \\mathbf{U}\\widetilde{\\bm{\\Lambda}}^k\\mathbf{U}^\\dagger$, and thus, letting $\\widetilde{\\mathbf{L}} \\coloneqq \\frac{2}{\\lambda_{\\text{max}}}\\mathbf{L}-\\mathbf{I}$, we have\n\\begin{equation}\\label{eqn: ChebNet}\n    \\mathbf{Y}\\mathbf{x}=\\mathbf{U}\\sum_{k=0}^K\\theta_k T_k(\\widetilde{\\bm{\\Lambda}})\\mathbf{U}^\\dagger\\mathbf{x} = \\sum_{k=0}^K\\theta_k T_k(\\widetilde{\\mathbf{L}}) \\mathbf{x} \\, .\n\\end{equation}\nThis is the class of convolutional filters we will use in our experiments. However, one could also imitate Sec.~3.1 on \\cite{zhang2021magnet} to produce a class of filters based on \\cite{kipf2016semi} rather than \\cite{Defferrard2018}.\n\nIt is important to note that  $\\widetilde{\\mathbf{L}}$ is constructed so that, in \\eqref{eqn: ChebNet}, $(\\mathbf{Y}\\mathbf{x})_i$ depends on all nodes within $K$-hops from $v_i$ on the undirected, unsigned counterpart of $\\mathcal{G}$, i.e. the graph whose adjacency matrix is given by $\\mathbf{A}'_{i,j}=\\frac{1}{2}{(|\\mathbf{A}_{i,j}|+|\\mathbf{A}_{j,i}|})$. Therefore, this notion of convolution does not favor ``outgoing neighbors\"  $\\{v_j\\in \\mathcal{V}: (v_i,v_j)\\in\\mathcal{E}\\}$ over ``incoming neighbors\"  $\\{v_j\\in V: (v_j,v_i)\\in\\mathcal{E}\\}$ (or vice versa). This is important since for a given node $v_i$, both sets may contain different, useful information. Furthermore, since the phase matrix $\\mathbf{\\Theta}^{(q)}$ encodes an outgoing edge and an incoming edge differently, the filter matrix $\\mathbf{Y}$ is also able to aggregate information from these two sets in different ways. \nFor computational complexity, we note that while the matrix $\\exp (\\mathbbm{i} \\mathbf{\\Theta}^{(q)})$ is dense in theory,  in practice, one only needs to compute a small fraction of its entries corresponding to the nonzero entries of $\\Tilde{\\mathbf{A}}$ (which is sparse for most real-world data sets). Thus, the computational complexity of the convolution proposed here is equivalent to that of its undirected, unsigned counterparts.\n\n\\subsection{The {MSGNN} architecture}\n\nWe now define our network, {MSGNN}.  Let $\\mathbf{X}^{(0)}$ be an $n\\times F_0$ input matrix with columns $\\mathbf{x}_1^{(0)},\\ldots\\mathbf{x}_{F_0}^{(0)}$, and\n$L$ denote the number of convolution layers.\nAs in \\cite{zhang2021magnet}, we use a complex version of the Rectified Linear Unit \ndefined by $\\sigma(z)=z$, if $-\\pi\/2\\leq \\arg(z)<\\pi\/2$, and $\\sigma(z)=0$ otherwise, where $\\arg(\\cdot)$ is the complex argument of $z\\in\\mathbb{C}$. Let $F_\\ell$ be the number of channels in the $\\ell$-th layer. For $1\\leq\\ell\\leq L$, $1 \\leq i \\leq F_{\\ell}$, and $1\\leq j\\leq F_{\\ell-1},$ let $\\mathbf{Y}_{ij}^{(\\ell)}$ be a convolution matrix defined by \\eqref{eqn: conv} or \\eqref{eqn: ChebNet}. Given the $(\\ell-1)$-st layer hidden representation matrix $\\mathbf{X}^{(\\ell-1)}$, we define $\\mathbf{X}^{(\\ell)}$ columnwise by\n\\begin{equation}\\label{eqn: single hidden layer}\n\\vspace{-1mm}\n    \\mathbf{x}_j^{(\\ell)} = \\sigma \\left(\\sum_{i=1}^{F_{\\ell-1}} \\mathbf{Y}_{ij}^{(\\ell)} \\mathbf{x}_i^{(\\ell-1)} + \\mathbf{b}_j^{(\\ell)}\\right), \n\\end{equation}\nwhere $\\mathbf{b}_j^{(\\ell)}$ is a bias vector with equal real and imaginary parts,  $\\text{Real}(\\mathbf{b}_j^{(\\ell)}) = \\text{Imag}(\\mathbf{b}_j^{(\\ell)})$.\nIn matrix form we write $\\mathbf{X}^{(\\ell)} = \\mathbf{Z}^{(\\ell)} \\left(\\mathbf{X}^{(\\ell-1)}\\right) $, where $\\mathbf{Z}^{(\\ell)}$ is a hidden layer of the form \\eqref{eqn: single hidden layer}. In our experiments, we utilize convolutions of the form \\eqref{eqn: ChebNet} with $\\mathbf{L} = \\mathbf{L}_N^{(q)}$ and set $K=1$, \nin which case we obtain\n\\begin{equation*}\n    \\mathbf{X}^{(\\ell)} = \\sigma\\left( \\mathbf{X}^{(\\ell-1)} \\mathbf{W}_{\\text{self}}^{(\\ell)} + \\widetilde{\\mathbf{L}}_N^{(q)} \\mathbf{X}^{(\\ell-1)} \\mathbf{W}_{\\text{neigh}}^{(\\ell)} + \\mathbf{B}^{(\\ell)}\\right) \\, ,\n\\end{equation*}\nwhere $\\mathbf{W}_{\\text{self}}^{(\\ell)}$ and $\\mathbf{W}_{\\text{neigh}}^{(\\ell)}$ are learned weight matrices corresponding to the filter weights of different channels and $\\mathbf{B}^{(\\ell)} = (\\mathbf{b}_1^{(\\ell)}, \\ldots, \\mathbf{b}_{F_{\\ell}}^{(\\ell)})$. After the convolutional layers, we unwind the complex matrix $\\mathbf{X}^{(L)}$ into a real-valued $n\\times 2F_L$ matrix. For node clustering, we then apply a fully connected layer followed by the softmax function. \nBy default, we set $L=2$, in which case, our network is given by \n\\begin{equation*}\n    \\text{softmax}(\\text{unwind}(\\mathbf{Z}^{(2)}(\\mathbf{Z}^{(1)}(\\mathbf{X}^{(0)}))) \\mathbf{W}^{\\mathrm{(3)}}) \\, .\n\\end{equation*}\nFor link prediction, we apply the same method, except we concatenate rows corresponding to pairs of nodes after the unwind layer before applying the linear layer and softmax.\n\\section{Experiments}\n\\subsection{Tasks and Evaluation Metrics}\\label{subsec:tasks} \n\\paragraph{Node Clustering}\nIn the node clustering task, one aims to partition the nodes of the graph into the disjoint union of $C$ sets $\\mathcal{C}_0,\\ldots,\\mathcal{C}_{C-1}$. \nTypically in an unsigned, undirected network, one aims to choose the $\\mathcal{C}_i$'s so that there are many links within each cluster and comparably few links between clusters, in which case nodes within each cluster are \\emph{similar} due to dense connections. In general, however, similarity could be defined differently \\cite{he2022gnns}. \nIn a signed graph, clusters can be formed by grouping  together nodes with positive links and separating nodes with negative links (see \\cite{he2022sssnet}). \nIn a directed graph, clusters can  be determined by a directed flow on the network (see \\cite{he2021digrac}). More generally, we can define clusters based on an underlying meta-graph,  where meta-nodes, each of which corresponds to a cluster in the network, can be distinguished based on either signed or directional information (e.g., flow imbalance~\\cite{he2021digrac}). This general meta-graph idea motivates our introduction of a novel synthetic network model, which we will define in Sec.~\\ref{sec: novel sdsbm}, driven by both link sign and directionality. \n\nAll of our node clustering experiments are done in the semi-supervised setting, where one selects a fraction of the nodes in each cluster as seed nodes, with known cluster membership labels. In all of our node clustering tasks, we measure our performance using the Adjusted Rand Index (ARI) \\cite{hubert1985comparing}.\n\\paragraph{Link Prediction}\n\\label{subsec:link_pred}\n\\looseness=-1 On undirected, unsigned graphs, link prediction is simply the task of predicting whether or not there is a link between a pair of nodes. Here, we consider five different variations of the link prediction task for \\emph{signed and\/or directed} networks. In our first task, link sign prediction (SP), one assumes that there is a link from $v_i$ to $v_j$ and aims to predict whether that link is positive or negative, i.e., whether $(v_i, v_j) \\in \\mathcal{E}^+$ or $(v_i, v_j) \\in \\mathcal{E}^-$. Our second task, direction prediction (DP), one aims to predict whether  $(v_i, v_j)\\in\\mathcal{E}$ or $(v_j, v_i)\\in\\mathcal{E}$ under the assumption that exactly one of these two conditions holds. We also consider three-, four-, and five-class prediction problems. In the three-class problem (3C), the possibilities are $(v_i, v_j)\\in\\mathcal{E},$ $(v_j, v_i)\\in\\mathcal{E},$ or that neither $(v_i, v_j)$ nor $(v_j, v_i)$ are in $\\mathcal{E}$. For the four-class problem (4C), the possibilities are $(v_i, v_j)\\in\\mathcal{E}^+$, $(v_i, v_j)\\in\\mathcal{E}^-$, $(v_j, v_i)\\in\\mathcal{E}^+$, and $(v_j, v_i)\\in\\mathcal{E}^-$. For the five-class problem (5C), we also add in the possibility that neither $(v_i, v_j)$ nor $(v_j, v_i)$ are in $\\mathcal{E}$. For all tasks, we evaluate the performance with classification accuracy.  \nNotably, while (SP), (DP), and (3C) only require a method to be able to extract signed \\emph{or} directed information, the tasks (4C) and (5C) require it to be able to effectively process both sign \\emph{and} directional information. Also, we discard those edges that satisfy more than one condition in the possibilities for training and evaluation, but these edges are kept in the input network observed during training.\n\\subsection{Synthetic Data for Node Clustering}\n\\paragraph{Established Synthetic Models}\nWe conduct experiments on the Signed Stochastic Block Models (SSBMs) and polarized SSBMs (\\textsc{POL-SSBM}s) introduced in \\cite{he2022sssnet}. Notably, both of these models are signed, but undirected. In the SSBM($n, C, p, \\rho, \\eta$) model, $n$ represents the number of nodes, $C$ is the number of clusters, $p$ is the probability that there is a link (of either sign) between two nodes, $\\rho$ is the approximate ratio between the largest cluster size and the smallest cluster size, and $\\eta$ is the probability that an edge will have the ``wrong\" sign, i.e., that an intra-cluster edge will be negative or an inter-cluster edge will be positive. {\\textsc{Pol-SSBM}} ($n,r,p,\\rho,\\eta,N$) is a hierarchical variation of the SSBM model consisting of $r$ communities, each of which is itself an SSBM. We refer the reader to \\cite{he2022sssnet} for details of both models.  \n\\paragraph{A novel Synthetic Model: Signed Directed Stochastic Block Model (SDSBM)}\\label{sec: novel sdsbm}\nGiven a meta-graph adjacency matrix $\\mathbf{F} = (\\mathbf{F}_{k,l})_{k,l = 0, \\ldots, C-1}\n$, an edge sparsity level $p$, a number of nodes $n$, and a \nsign flip noise level parameter $0 \\leq \\eta \\leq 0.5$, we defined a  \n SDSBM model, denoted by \nSDSBM ($\\mathbf{F}, n, p, \\rho, \\eta$),  \n as follows: \n 1) Assign block sizes $n_0 \\le n_1 \\le \\cdots \\le n_{C-1}$ based on a parameter $\\rho \\geq 1$, which approximately represents the ratio between the size of largest block and the size of the smallest block, using the same method as in \\cite{he2022sssnet}. \n 2) \nAssign each node to one of the $C$ blocks, so that each block $C_i$ has size $n_i$. \n 3) For nodes $v_i\\in\\mathcal{C}_k$, and $v_j\\in\\mathcal{C}_l$, independently sample an edge from $v_i$ to $v_j$ with probability $ p \\cdot |\\mathbf{F}_{k,l}|$. Give this edge weight $1$ if $F_{k,l}\\geq 0$ and weight $-1$ if $F_{k,l}<0$. \n 4) Flip the sign of all the edges in the generated graph with sign-flip probability $\\eta$.\n\n\\looseness=-1 In our experiments, we use two sets of specific meta-graph structures $\\{\\mathbf{F}_1(\\gamma)\\},\\{\\mathbf{F}_2(\\gamma)\\}$, with three and four clusters, respectively, where  $0 \\leq \\gamma \\leq 0.5$ is the directional noise level. Specifically, we are interested in SDSBM ($\\mathbf{F}_1(\\gamma), n, p, \\rho, \\eta$) and SDSBM ($\\mathbf{F}_2(\\gamma), n, p, \\rho, \\eta$) models with varying $\\gamma$ where \n$$\\mathbf{F}_1(\\gamma)=\\begin{bmatrix}\n0.5 & \\gamma & -\\gamma\\\\\n1-\\gamma & 0.5 & -0.5\\\\\n-1+\\gamma&-0.5&0.5\n\\end{bmatrix},\\mathbf{F}_2(\\gamma)=\\begin{bmatrix}\n0.5 & \\gamma & -\\gamma&-\\gamma\\\\\n1-\\gamma & 0.5 &-0.5& -\\gamma\\\\\n-1+\\gamma&-0.5&0.5&-\\gamma\\\\\n-1+\\gamma&-1+\\gamma&-1+\\gamma&0.5\n\\end{bmatrix}.$$ \n\nTo better understand the above SDSBM models, toy examples are provided in Appendix~\\ref{appendix:examples}. We also note that the SDSBM model proposed here is a generalization of both the SSBM model from \\cite{he2022sssnet} and the Directed Stochastic Block Model from \\cite{he2021digrac} when we have suitable meta-graph structures.\n\n\\subsection{Real-World Data for Link Prediction}\n\\paragraph{Standard Real-World Data Sets}\nWe consider four standard real-world signed and directed data sets. \\textit{BitCoin-Alpha} and \\textit{BitCoin-OTC}~\\cite{konect:kumar2016wsn} describe bitcoin trading. \\textit{Slashdot}~\\cite{slashdot} is related to a technology news website, and \\textit{Epinions}~\\cite{epinions} describes consumer reviews. These networks range in size from 3783 to 131580 nodes. Only \\textit{Slashdot} and \\textit{Epinions} are unweighted. \n\\paragraph{Novel Financial Data Sets from Stock Returns}\n\\looseness=-1 Using financial time series data, we build signed directed networks where the weighted edges encode lead-lag relationships inherent in the financial market, for each year in the interval 2000-2020. The lead-lag matrices are built from time series of daily price returns\\footnote{Raw CRSP data accessed through \\url{https:\/\/wrds-www.wharton.upenn.edu\/}.}. We refer to these networks as our \\textbf{Fi}ancial \\textbf{L}ead-\\textbf{L}ag (FiLL) data sets. For each year in the data set, we build a signed directed graph (\\textit{FiLL-pvCLCL}) based on the price return of 444 stocks at market close times on consecutive days. We also build another graph (\\textit{FiLL-OPCL}), based on the price return of 430 stocks from market open to close. The difference between 444 versus 430 stems from the non-availability of certain open and close prices on some days for certain stocks. The lead-lag metric that is captured by the entry $\\mathbf{A}_{i,j}$ in each network encodes a measure that quantifies the extent to which stock $v_i$ leads stock $v_j$, and is obtained by computing the linear regression coefficient when regressing the time series (of length 245) of daily returns of stock $v_i$ against the lag-one  version of the time series (of length 245) of the daily returns of stock $v_j$. Specifically, we use the beta coefficient of the corresponding simple linear regression, to serve as the one-day lead-lag metric. The resulting  matrix is asymmetric and signed, rendering it amenable to a signed, directed network interpretation. The initial matrix is dense, with nonzero entries outside the main diagonal, since we do not consider the own auto-correlation of each stock. Note that an alternative approach to building the directed network could be based on  Granger causality, or other measures that quantify the lead-lag between a pair of time series, potentially while accounting for nonlinearity, such as second-order log signatures from rough paths theory as in \\cite{bennett2021detection}.\n\nNext, we sparsify each network, \nkeeping only 20\\% of the edges with the largest magnitudes. We also report the average results across the all the yearly data sets (a total of 42 networks) where the data set is denoted by \\textit{FiLL (avg.)}. To facilitate future research using these data sets as benchmarks, both the dense lead-lag matrices and their sparsified counterparts will be made publicly available.  \n\\subsection{Experimental Results}\nWe compare {MSGNN} against representative GNNs which are described in Section \\ref{sec:related}.\nThe six methods we consider \nare\n1)~SGCN \\cite{derr2018signed}, \n2)~SDGNN~\\cite{huang2021sdgnn}, \n3)~SiGAT~\\cite{huang2019signed}, \n4)~SNEA~\\cite{li2020learning}, \n5)~SSSNET~\\cite{he2022sssnet}, \nand 6)~SigMaNet~\\cite{fiorini2022sigmanet}. For all link prediction tasks, comparisons are carried out on all baselines; for the node clustering tasks, we only compare {MSGNN} against SSSNET and SigMaNet as adapting the other methods to this task is nontrivial. \nImplementation details are provided in Appendix~\\ref{appendix:implementation}, along with a runtime comparison which shows that {MSGNN} is the fastest method, see Table \\ref{tab:runtime} in Appendix \\ref{appendix:implementation}. Extended results are in Appendix~\\ref{appendix_sec:ablation} and \\ref{appendix:full_results}. Anonymous codes and some preprocessed data are available at \\url{https:\/\/anonymous.4open.science\/r\/MSGNN}.\n\\subsubsection{Node Clustering}\n\\begin{figure*}[hbt]\n    \\centering\n  \\begin{subfigure}[ht]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth,trim=0cm 0cm 0cm 1.8cm,clip]{figures\/signed_clustering\/SSBM\/1000p10K5100eta4010rho15N10000ARI.pdf}\n    \\vspace{-5mm}\n    \\caption{SSBM($n=10000,$\\\\$ C=5, p=0.01, \\rho=1.5$)}\n  \\end{subfigure}\n  \\begin{subfigure}[ht]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth,trim=0cm 0cm 0cm 1.8cm,clip]{figures\/signed_clustering\/polarized_SSBM\/1000p100K2100eta4010rho15N500total_n5000num_com5ARI.pdf}\n    \\vspace{-5mm}\n    \\caption{\\textsc{Pol-SSBM}($n=5000,$\\\\$ r=5, p=0.1, \\rho=1.5$) }\n  \\end{subfigure}\n  \\begin{subfigure}[ht]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth,trim=0cm 0cm 0cm 1.8cm,clip]{figures\/signed_clustering\/SDSBM\/3c\/1000p100K3100eta25100gamma010rho15N1000ARI.pdf}\n    \\vspace{-5mm}\n    \\caption{SDSBM($\\mathbf{F}_1(\\gamma=0), n=1000,$\\\\$ p=0.1, \\rho=1.5$)}\n  \\end{subfigure}\n  \\begin{subfigure}[ht]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth,trim=0cm 0cm 0cm 1.8cm,clip]{figures\/signed_clustering\/SDSBM\/3c\/1000p100K3100eta0100gamma2510rho15N1000_change_gammaARI.pdf}\n    \\vspace{-5mm}\n    \\caption{SDSBM($\\mathbf{F}_1(\\gamma),n=1000,$\\\\$ p=0.1, \\rho=1.5, \\eta=0$)}\n  \\end{subfigure}\n  \\begin{subfigure}[ht]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth,trim=0cm 0cm 0cm 1.8cm,clip]{figures\/signed_clustering\/SDSBM\/3c\/1000p1000K3100eta0100gamma2510rho15N1000_change_gammaARI.pdf}\n    \\vspace{-5mm}\n    \\caption{SDSBM($\\mathbf{F}_1(\\gamma),n=1000,$\\\\$ p=1, \\rho=1.5, \\eta=0$)}\n  \\end{subfigure}\n  \\begin{subfigure}[ht]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth,trim=0cm 0cm 0cm 1.8cm,clip]{figures\/signed_clustering\/SDSBM\/4c\/1000p100K4100eta25100gamma010rho15N1000ARI.pdf}\n    \\vspace{-5mm}\n    \\caption{SDSBM($\\mathbf{F}_2(\\gamma=0), n=1000,$\\\\$ p=0.1, \\rho=1.5$)}\n  \\end{subfigure}\n  \\begin{subfigure}[ht]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth,trim=0cm 0cm 0cm 1.8cm,clip]{figures\/signed_clustering\/SDSBM\/4c\/1000p100K4100eta0100gamma2510rho15N1000_change_gammaARI.pdf}\n    \\vspace{-5mm}\n    \\caption{SDSBM($\\mathbf{F}_2(\\gamma),n=1000,$\\\\$ p=0.1, \\rho=1.5, \\eta=0$)}\n  \\end{subfigure}\n  \\begin{subfigure}[ht]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth,trim=0cm 0cm 0cm 1.8cm,clip]{figures\/signed_clustering\/SDSBM\/4c\/1000p1000K4100eta0100gamma2510rho15N1000_change_gammaARI.pdf}\n    \\vspace{-5mm}\n    \\caption{SDSBM($\\mathbf{F}_2(\\gamma),n=1000,$\\\\$ p=1, \\rho=1.5, \\eta=0$)}\n  \\end{subfigure}\n  \\begin{subfigure}[ht]{\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth,trim=0cm 0cm 0cm 0cm,clip]{figures\/signed_clustering\/legend_SDSBM.jpg}\n  \\end{subfigure}\n    \\caption{Node clustering test ARI comparison on synthetic data. Dashed lines highlight {MSGNN}'s performance with $q=.25$. \n    Error bars indicate one standard error. Results are averaged over ten runs --- five different networks, each with two distinct data splits.\n    }\n    \\label{fig:signed_clustering_synthetic_compare}\n\\end{figure*}\nFigure~\\ref{fig:signed_clustering_synthetic_compare} \ncompares the node clustering performance of {MSGNN} with two other signed GNNs on synthetic data, and against variants of {MSGNN} on SDSBMs that take different $q$ values. For signed, undirected networks $q$ does not has an effect, and hence we only report one {MSGNN} variant. Error bars are given by one standard error. We conclude that {MSGNN} outperforms SigMaNet on all data sets and is competitive with SSSNET. On the majority of data sets,  {MSGNN} achieves leading performance, whereas on the others it is slightly outperformed by \nSSSNET, depending on the network density, noise level, network size, and the underlying meta-graph structure. On these relatively small  data sets, MSGNN and SSSNET have comparable runtime and are faster than SigMaNet. Comparing the {MSGNN} variants, we conclude that the directional information in these SDSBM models plays a vital role since {MSGNN} with $q=0$ is usually outperformed by {MSGNN} with nonzero $q$.\n\\subsubsection{Link Prediction}\n\\begin{table*}\n\\vspace{-1mm}\n\\caption{Test accuracy (\\%) comparison the signed and directed link prediction tasks introduced in\nSec.~\\ref{subsec:tasks}.\nThe best is marked in \\red{bold red} and the second best is marked in \\blue{underline blue}. }\n\\vspace{-2mm}\n  \\label{tab:link_sign_direction_prediction_performance}\n  \\centering\n  \\small\n  \\resizebox{0.8\\linewidth}{!}{\\begin{tabular}{c c c c  c  cccccc}\n    \\toprule\n    Data Set& Link Task&SGCN&SDGNN&SiGAT&SNEA&SSSNET&SigMaNet&{MSGNN} \\\\\n    \\midrule\n\\multirow{4}{*}{\\textit{BitCoin-Alpha}}&SP&64.7$\\pm$0.9&64.5$\\pm$1.1&62.9$\\pm$0.9&64.1$\\pm$1.3&\\blue{67.4$\\pm$1.1}&47.8$\\pm$3.9&\\red{69.8$\\pm$2.5}\\\\\n&DP&60.4$\\pm$1.7&61.5$\\pm$1.0&61.9$\\pm$1.9&60.9$\\pm$1.7&\\blue{68.1$\\pm$2.3}&49.4$\\pm$3.1&\\red{68.3$\\pm$3.7}\\\\\n&3C&81.4$\\pm$0.5&79.2$\\pm$0.9&77.1$\\pm$0.7&\\red{83.2$\\pm$0.5}&78.3$\\pm$4.7&37.4$\\pm$16.7&\\blue{82.7$\\pm$1.2}\\\\\n&4C&51.1$\\pm$0.8&52.5$\\pm$1.1&49.3$\\pm$0.7&52.4$\\pm$1.8&\\blue{54.3$\\pm$2.9}&20.6$\\pm$6.3&\\red{55.9$\\pm$4.0}\\\\\n&5C&79.5$\\pm$0.3&78.2$\\pm$0.5&76.5$\\pm$0.3&\\blue{81.1$\\pm$0.3}&77.9$\\pm$0.3&34.2$\\pm$6.5&\\red{81.4$\\pm$0.6}\\\\\n\\midrule\n\\multirow{4}{*}{\\textit{BitCoin-OTC}}&SP&65.6$\\pm$0.9&65.3$\\pm$1.2&62.8$\\pm$1.3&\\blue{67.7$\\pm$0.5}&\\red{70.1$\\pm$1.2}&50.0$\\pm$2.3&65.5$\\pm$10.1\\\\\n&DP&63.8$\\pm$1.2&63.2$\\pm$1.5&64.0$\\pm$2.0&65.3$\\pm$1.2&\\blue{69.6$\\pm$1.0}&48.4$\\pm$4.9&\\red{73.1$\\pm$1.2}\\\\\n&3C&79.0$\\pm$0.7&77.3$\\pm$0.7&73.6$\\pm$0.7&\\red{82.2$\\pm$0.4}&76.9$\\pm$1.1&26.8$\\pm$10.9&\\blue{81.3$\\pm$1.6}\\\\\n&4C&51.5$\\pm$0.4&55.3$\\pm$0.8&51.2$\\pm$1.8&\\blue{56.9$\\pm$0.7}&\\red{57.0$\\pm$2.0}&23.3$\\pm$7.4&55.5$\\pm$3.3\\\\\n&5C&77.4$\\pm$0.7&77.3$\\pm$0.8&74.1$\\pm$0.5&\\red{80.5$\\pm$0.5}&74.0$\\pm$1.6&25.9$\\pm$6.2&\\blue{79.0$\\pm$2.1}\\\\\n\\midrule\n\\multirow{4}{*}{\\textit{Slashdot}}&SP&74.7$\\pm$0.5&74.1$\\pm$0.7&64.0$\\pm$1.3&70.6$\\pm$1.0&\\blue{86.6$\\pm$2.2}&57.9$\\pm$5.3&\\red{92.4$\\pm$0.6}\\\\\n&DP&74.8$\\pm$0.9&74.2$\\pm$1.4&62.8$\\pm$0.9&71.1$\\pm$1.1&\\red{87.8$\\pm$1.0}&53.0$\\pm$4.0&\\blue{84.3$\\pm$17.2}\\\\\n&3C&69.7$\\pm$0.3&66.3$\\pm$1.8&49.1$\\pm$1.2&72.5$\\pm$0.7&\\blue{79.3$\\pm$1.2}&42.0$\\pm$7.9&\\red{84.7$\\pm$0.6}\\\\\n&4C&63.2$\\pm$0.3&64.0$\\pm$0.7&53.4$\\pm$0.2&60.5$\\pm$0.6&\\blue{72.7$\\pm$0.6}&25.7$\\pm$8.9&\\red{73.5$\\pm$3.7}\\\\\n&5C&64.4$\\pm$0.3&62.6$\\pm$2.0&44.4$\\pm$1.4&66.4$\\pm$0.5&\\blue{70.4$\\pm$0.7}&19.3$\\pm$8.6&\\red{74.2$\\pm$0.9}\\\\\n\\midrule\n\\multirow{4}{*}{\\textit{Epinions}}&SP&62.9$\\pm$0.5&67.7$\\pm$0.8&63.6$\\pm$0.5&66.5$\\pm$1.0&\\red{78.5$\\pm$2.1}&53.3$\\pm$10.6&\\blue{69.7$\\pm$16.2}\\\\\n&DP&61.7$\\pm$0.5&67.9$\\pm$0.6&63.6$\\pm$0.8&66.4$\\pm$1.2&\\blue{73.9$\\pm$6.2}&49.0$\\pm$3.2&\\red{81.0$\\pm$1.5}\\\\\n&3C&70.3$\\pm$0.8&\\blue{73.2$\\pm$0.8}&52.3$\\pm$1.3&72.8$\\pm$0.2&72.7$\\pm$2.0&30.5$\\pm$8.3&\\red{81.5$\\pm$1.3}\\\\\n&4C&66.7$\\pm$1.2&\\blue{71.0$\\pm$0.6}&62.3$\\pm$0.5&69.5$\\pm$0.7&70.2$\\pm$5.2&29.9$\\pm$6.4&\\red{74.9$\\pm$3.6}\\\\\n&5C&73.5$\\pm$0.8&\\blue{76.6$\\pm$0.7}&52.9$\\pm$0.7&74.2$\\pm$0.1&70.3$\\pm$4.6&22.1$\\pm$6.1&\\red{77.9$\\pm$1.4}\\\\\n\\midrule\n\\multirow{4}{*}{\\textit{FiLL (avg.)}}&SP&88.4$\\pm$0.0&82.0$\\pm$0.3&76.9$\\pm$0.1&\\red{90.0$\\pm$0.0}&\\blue{88.7$\\pm$0.3}&50.4$\\pm$1.8&88.4$\\pm$1.3\\\\\n&DP&88.5$\\pm$0.1&82.0$\\pm$0.2&76.9$\\pm$0.1&\\red{90.0$\\pm$0.0}&\\blue{88.8$\\pm$0.3}&48.0$\\pm$2.7&88.6$\\pm$2.1\\\\\n&3C&63.0$\\pm$0.1&59.3$\\pm$0.0&55.3$\\pm$0.1&\\red{64.3$\\pm$0.1}&62.2$\\pm$0.3&33.7$\\pm$1.3&\\blue{64.2$\\pm$0.8}\\\\\n&4C&\\blue{81.7$\\pm$0.0}&78.8$\\pm$0.1&70.5$\\pm$0.1&\\red{83.2$\\pm$0.1}&80.0$\\pm$0.3&24.9$\\pm$0.9&78.8$\\pm$1.2\\\\\n&5C&\\blue{63.8$\\pm$0.0}&61.1$\\pm$0.1&55.5$\\pm$0.1&\\red{64.8$\\pm$0.1}&60.4$\\pm$0.4&19.8$\\pm$1.1&62.1$\\pm$0.8\\\\\n\\bottomrule\n\\vspace{-4mm}  \n\\end{tabular}}\n\\vspace{-2mm}\n\\end{table*}\nOur results for link prediction in Table~\\ref{tab:link_sign_direction_prediction_performance} indicate that {MSGNN} is the top performing method, achieving the highest level of accuracy in 13 out of 25 total cases and being among the leading two in 19 out of 25 total cases. SNEA is the second best performing method, but is the least efficient in speed due to its use of graph attention, see runtime comparison in Appendix~\\ref{appendix:implementation}.\n\nSpecifically, the ``avg.\" results for the novel financial data sets first average the accuracy values across all individual networks (a total of 42 networks), then report the mean and standard deviation over the five runs. Results for individual \\textit{FiLL} networks are reported in Appendix~\\ref{appendix:full_results}. Note that \\textit{$\\pm0.0$} in the result tables indicates that the standard deviation is less than 0.05\\%.\n\n\\vspace{-2mm}\n\\subsubsection{Ablation Study and Discussion}\n\\looseness=-1 We now discuss the influence of the charge parameter $q$ in the Laplacian matrix, and how input features affect the performance. \nTable~\\ref{tab:link_prediction_ablation} in Appendix~\\ref{appendix_sec:ablation} compares different variants of {MSGNN} on the link prediction tasks, with respect to (1) whether we use a traditional signed Laplacian that is initially designed for undirected networks (for which $q=0$) and a magnetic signed Laplacian that strongly emphasizes directionality (for which $q=0.25$); (2) whether to include sign in input node features (if False, then only in- and out-degrees are computed like in \\cite{zhang2021magnet} regardless of edge signs, otherwise features are constructed based on the positive and negative subgraphs separately); and (3) whether we take edge weights into account (if False, we view all edge weights as having magnitude one). Taking the standard errors into account, we find that incorporating directionality into the Laplacian matrix (i.e., having nonzero $q$) typically leads to slightly better performance in the directionality-related tasks (DP, 3C, 4C, 5C). Drilling deeper, a comparison using different $q$ values in the magnetic part shown in Table~\\ref{tab:link_prediction_q_ablation} provides further evidence that nonzero $q$ values usually boost performance compared with no emphasis on directionality at all ($q=0$), in addition to the evidence from\nFigure~\\ref{fig:signed_clustering_synthetic_compare} for node clustering.\n\nMoreover, signed features are in general helpful for tasks involving sign prediction. \nFor constructing weighted features we see no significant difference in simply summing up entries in the adjacency matrix compared to summing the absolute values of the entries. Besides, it seems that calculating degrees based on unit-magnitude weights is generally not best, as even when it leads performance it is within one standard error of another method. Treating negative edge weights as the negation of positive ones is also not helpful (by not having separate degree features for the positive and negative subgraphs), which may explain why SigMaNet performs poorly in most scenarios due to its undesirable property. Surprisingly often, including only signed information but not weighted features does well. To conclude, constructing features based on the positive and negative subgraphs separately is helpful, and including directional information is generally beneficial.\n\n\\vspace{-2mm}\n\\section{Conclusion and Outlook}\nIn this paper, we propose a spectral graph neural network based on a novel magnetic signed Laplacian matrix, introduce a  novel synthetic network model and new real-world data sets, and conduct experiments on node clustering and link prediction tasks that are not restricted to considering either link sign or directionality alone. MSGNN performs as well or better than leading GNNs, while being considerably faster on real-world data sets. Future plans include investigating an extension to temporal\/dynamic graphs, where node features and\/or edge information could evolve over time~\\cite{dang2018link,rozemberczki2021pytorch}. We are also interested in extending our work to being able to encode nontrivial edge features, to develop objectives which explicitly handle heterogeneous edge densities throughout the graph, and to extend our approach to hypergraphs and other complex network structures.\n\n\\clearpage\n\\bibliographystyle{unsrtnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe fundamentals of fast magnetisation switching constitute one of the important problems in magnetism, related to technological applications \\cite{Hillebrands}. Experimentally  microwave-assisted switching has been reported as requiring a smaller applied field thus opening new possibilities for applications in magnetic recording and communication devices \\cite{Thirion, Nembach, Woltersdorf,Moriyama, Pimentel}. The microwave-assisted (mw) switching process with linearly-polarized excitation can be realized based, for example, on a fast magneto-optic Kerr effect set-up, applying simultaneously static and microwave fields \\cite{Nembach,Woltersdorf, Pimentel, Adam}. Another possibility to observe  mw-assisted switching is given by measuring the magnetoresistance as has been done, for example, in Co stripes \\cite{Grollier} or in NiFe magnetic tunnel junctions \\cite{Moriyama}. These experiments use relatively large (up to tens of microns) lithographically prepared magnetic elements whose dimensions do not allow the occurrence of homogeneous magnetisation processes.\n\n Differently from this, the micro-SQUID experiment \\cite{Thirion}, performed on Co nanoparticles with 20 nm diameter gives an example of a macrospin magnetisation behavior. In this system nonlinear effects are clearly observed experimentally in agreement with a  simple macrospin dynamical model.\n  The behavior found in the macrospin magnetisation dynamics has stimulated the technologically important proposal of micro-wave assisted (or ferromagnetic resonance (FMR) assisted) magnetic recording \\cite{Zhu, Rivkin, Scholtz}.\n\n The process of understanding the underlying physics is still going on and several ideas are found in the literature. From the theoretical point of view, probably the most comprehensive understanding has been achieved in the case of one single magnetic moment \\cite{Thirion, Rivkin, Scholtz, Sun, Zhang, Lee, Horley}.\n In what follows we would like to mention some of the proposed mechanisms, since most of them can also be partially encountered in the complex situation of non-homogeneous magnetisation reversal processes discussed in the present article. The most frequently used one is based on a simple idea of the efficient energy transfer during excitation with frequencies close to the FMR one \\cite{Woltersdorf, Sun}. It has been also argued that excitation with a circular polarised field would be much more efficient due to the possibility to synchronize with the rotation sense of the magnetic moment \\cite{Sun}. Although it seems to be less efficient,  synchronization is also possible with a linearly polarised field \\cite{Horley}. This idea remains certainly valid in a general case, however it should be noted that the notion of the FMR frequency, as corresponding to excitation of mode with zero wave vector, is not exact for micron-sized magnetic elements. Due to the minimization of the magnetostatic charges at the surfaces, even small angle magnetisation vibrations are not homogeneous \\cite{Adam} and the frequency spectrum depends strongly on the size and shape of the magnetic nanoelement \\cite{Kruglyak1, Kruglyak}.\n Moreover, in magnetic nanoelements the frequency spectrum becomes quantized \\cite{Bayer,Guslienko}. The second idea found in the literature is also simple:\n The microwave field provides a constant energy input which rises the magnetic energy of the particle up to a level higher than the saddle point of the magnetic energy landscape \\cite{Thirion, Sun, Moriyama, Horley}. Note that here also the thermal-activated process becomes possible \\cite{Lee,Nozaki}. In this sense, it can be said that the microwave field can act as an energy source which effectively decreases the energy barrier. It should be noticed here that this idea is relied on a simple picture of a two-level system which is generally not valid for a non-homogeneous magnetisation reversal scenario.\n\nThe fast mw-assisted magnetisation switching process in magnetic elements is closely related to the phenomenon of precessional switching. Indeed, a small perpendicular field helps the switching to take place via the excitation of the precessional motion \\cite{Back, BackScience, Hiebert} due to the torque acting on the magnetisation. The main problem is that with, for example, a perpendicularly applied field the magnetisation continues to precess (\"ringing\" phenomenon) and could switch back \\cite{Schumacker}. Furthermore, it has been suggested to use a pulsed field and to tune the pulse duration to avoid multiple switching. Even the pulse shape could be optimized \\cite{Bauer}, together with the use of two pulses with an optimised delay time \\cite{Gerrits, Schumacker}. This case again is mostly studied in the case of one magnetic moment, although micromagnetic simulations have been also widely performed in nanoelements with different shapes \\cite{Viena}.\n\nThe final idea which we would like to mention is that one magnetic moment under a microwave field represents a classical example of a parametrically excited nonlinear oscillator. Thus the problem is similar to the classical parametrically excited pendulum, where  parametric instabilities occur varying the strength and the frequency of external oscillation. Such instabilities may be considered as precursors of the switching process. Next, in these systems nonlinear phenomena such as bifurcations and chaos \\cite{Pla, Serpico, Horley} occur. A classical example of this is the appearance of an additional large-amplitude stationary orbit via the folding  bifurcation, again studied in the case of one macrospin only \\cite{Pla, Serpico}.  The nonlinear phenomena lead to a complex structure of the trajectories  near the separatrix, a  special trajectory separating the basis of attractions of the two minima. Under the influence of external periodic force, the separatrix may become fractal so that closely situated initial conditions may lead either to switching or not \\cite{Horley}. This produced, for example, fractal regions of switching beyond the Stoner-Wolfarth astroid observed experimentally in micro-SQUID experiments \\cite{Thirion} and numerically in Refs.\\onlinecite{Thirion, Zhang, Horley, Scholtz}.  However, since these phenomena are complex even in the case of one magnetic moment, in a large system  nonlinear phenomena are more difficult to interpret and may not lead to a clear pattern.\n\t\n\nThe reversal under oscillating fields of large-sized magnetic nanoelements was mostly studied in  cases with simple magnetisation reversal modes, such as domain wall\\cite{Grollier,Martinez}, $C$ or $S$ magnetisation states \\cite{Fidler}, the vortex to onion state in magnetic rings \\cite{Podbielski} or vortex core reversal \\cite{Wayenberge, LeeK}. The enhanced mobility of domain walls with some frequencies was numerically found \\cite{Martinez}. Experimentally, using the magnetic circular dichroism technique in a magnetic element having a Landau pattern, the asymmetry of the domain wall motion under a linearly-polarised mw field with the drift in one direction was reported \\cite{Krasyuk}. This has been explained by the entropy maximization involving the precessional excitation (the excitation of a so-called \"self-trapping spin-wave mode\").\n\n\nAs we mentioned above, most of the previous theoretical studies involve either fast switching in the macrospin case (or effectively conditions of coherent rotation) or the switching of individual objects as vortices or domain wall structures. However, the spatial resolution of the fast Kerr experiments \\cite{Nembach, Pimentel} requires quite large magnetic elements with not so simple and unique magnetisation reversal modes. Namely, the mw-assisted switching in large systems produces a sequence of nucleation - propagation - relaxation phenomena which is the subject of the present study. We show that the overall dynamics is a complex phenomenon where some of the above mentioned  processes could occur simultaneously and could play a role on different stages of the magnetisation reversal.\n\n\n\n\n\\section{Model}\nIn the present work and with the aim to understand the mw-assisted switching processes in large magnetic elements \\cite{Nembach,Pimentel}, we use a micromagnetic model to simulate the magnetisation dynamics in a permalloy ellipsoid. Due to the limitation of the simulational size we used  reduced dimensions of the ellipsoid:  the total magnetic volume is $V= 4 \\mu m$ x $2 \\mu m$ by $25 nm$. It has the same aspect ratio as in the experiment. We used the publicly available code Magpar \\cite{Magpar} with a finite element discretisation and in which we implemented the possibility to simultaneously apply dc and ac fields.\nThe permalloy easy axis was directed parallel to the large ellipsoid axis (y-axis, see Fig.~\\ref{geometry}). Initially the ellipsoid was magnetised along this axis. The dc-applied field is placed parallel to it in the opposite direction. The linearly polarized ac field with large amplitude is directed perpendicular to it (x-axis). The following micromagnetic parameters were used in the simulations:  anisotropy value $K=3.31\\times 10^3 erg\/cm^3$, saturation magnetisation value $M_{s}=860 \\;emu\/cm^3$ (the anisotropy field $H_{K}=7.7 Oe$),  exchange parameter $A=1.05 \\times 10^{-6} erg\/cm$,  damping parameter $\\alpha=0.012$.\n\n\\begin{figure}[h!]\n\\includegraphics[height=6cm]{sistema.eps}\n\\caption{Geometry of the simulated system.}\n\\label{geometry}\n\\end{figure}\n\nIn Fig. \\ref{Hyst} we present a hysteresis cycle for the permalloy ellipsoidal element with the field applied parallel to the easy axis direction. The coercive field is much larger than the anisotropy field value due to the large magnetostatic shape anisotropy.\n\n\n\\begin{figure}[h!]\n\\includegraphics[height=6cm]{Hyst1.EPS}\n\\caption{Simulated descending branch of the hysteresis cycle for the permalloy ellipsoid with static applied field only (full circles) and with simultaneously applied static and mw field with $H_{ac}=25.1 Oe$, and two frequencies (open symbols). The dc-field is applied along the long ellipsoid axis $Y$. The average ellipsoid magnetisation $<M_y>$  is normalized to the total saturation value.}\n\\label{Hyst}\n\\end{figure}\n\n\n\\section{Results}\nFirst, we model the hysteresis cycle with simultaneously applied dc and ac fields (see Fig.~\\ref{Hyst}) and note the reduction of coercivity  in the case of the mw-assisted switching process. The hysteresis cycle is square-like in the former case and rounded in the latter one. To understand the coercivity reduction we model the minimum strength of the applied field required to produce the magnetisation reversal in our system for different mw-field frequencies.\n\nWe notice that one of the large contributions to the coercivity reduction comes from the deviation of the magnetisation direction from the y-axis, following the ac-field. Thus, the coercivity is reduced  simply due to the fact that most of time the resulting field is applied at some angle to the easy axis. Consequently, we compare the mw-assisted switching process with the situation when a static field is applied at an angle equal to that formed during the mw-assisted case with the maximum amplitude $H_{ac}$ of the ac field. The maximum amplitude of the critical field $H_{cr}=\\sqrt{H_{ac}^2+H_{dc}^2}$, necessary to switch the magnetisation, is presented in Fig.~\\ref{DC_AC} as a function of the maximum applied field angle. We use the following condition for the magnetisation switching $<M_y> < 0$, where $<M_y>$ is the average magnetisation along the $y$ direction. Our results show that even with this more fair comparison,  mw-assisted switching always requires less field, except for the case of very large deviation angles and some frequencies. Therefore, the first important contribution to decrease the coercivity during the mw-assisted switching process is the deviation of field from the easy axis which leads to precessional switching. Moreover, for small field angles the results are almost independent of the ac-field frequency which role is just to induce the precession. The results are different when strong deviations of the magnetisation occur which stresses the importance of the nonlinear effects.\n\n\n\n\n\n\n\\begin{figure}[h!]\n\\includegraphics[height=6cm]{Hcr_vs_teta2.EPS}\n\\caption{Amplitude of the critical field necessary for magnetisation switching as a function of the maximum field angle with the anisotropy axis for static field and microwave-assisted switching. }\n\\label{DC_AC}\n\\end{figure}\n\n\nTo illustrate the differences between the static and mw-assisted switching processes we present the dynamical configurations during the magnetisation reversal in Figs.~\\ref{Static} and~\\ref{MW}. For the static process (Fig.~\\ref{Static}) the magnetisation nucleation starts with two vortices in the opposite upper and lower sides of the ellipsoid. During the irreversible jump these vortices propagate in  opposite directions creating two domain walls. The propagation of these domain walls toward the left and the right sides of the ellipsoid completes the magnetisation reversal.\n\n\n\\begin{figure}[h!]\n\\includegraphics[height=5cm]{Fig4.eps}\n\\caption{Magnetisation configurations during the static field hysteresis process at different time moments  (a) initial state at the coercive field, (b) intermediate state, showing the vortices propagation and\n(c)the final state at the coercive field. The field is applied at $45^o$ with the $Y$ axis.}\n\n\\label{Static}\n\\end{figure}\n\n\nFig.~\\ref{MW} shows the magnetisation configurations during the mw-assisted switching process. The complimentary video material (see Ref.~\\onlinecite{Videos}) illustrates the temporary dynamical evolution of the demagnetisation reversal. We observe that the ellipsoid is dynamically divided into domain structures. The number of domains depends strongly on the mw frequency and may be related to the length of the spinwave mode, excited in the system. In Fig.~\\ref{ripple} we represent an approximate number of magnetisation domains nucleated during the first two nanoseconds as a function of the ac-field frequency and in the magnetisation range $0.6 <\\; <M_y> \\; < 0.8$ (normalized to the total saturation magnetisation). Note that the domain size is larger in the central part of the ellipsoid than in the upper and lower sides, due to the ellipsoidal form, and initially grows with time (see Fig.~\\ref{size}).\nThe division of the ellipsoid into domains during mw-assisted switching is experimentally confirmed by Kerr images in Ref.~\\onlinecite{Pimentel}. Therefore, the mw-assisted case is characterized by a reversal mode completely different from the one appearing during the static hysteresis.\n\n\\begin{figure}[h!]\n\\includegraphics[height=6cm]{Fig5.eps}\n\\includegraphics[height=6cm]{Fig5_cd.eps}\n\\caption{Dynamical configurations: $M_x$ (upper) and $M_z$ (lower) components (grey scale) during the mw-assisted magnetisation reversal for applied fields $H_{dc}=-31.41 Oe$, $H_{ac}=18.84 Oe$ and for two different frequencies (left) $\\nu=1 GHz$ and (right) $\\nu=6 GHz$. For the dynamics of the reversal process, see the supplemental material \\cite{Videos}.}\n\\label{MW}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\includegraphics[height=6cm]{ripple1.eps}\n\\caption{ Number of domains in the ellipsoid nucleated during the first two nanoseconds as a function of mw frequency for applied field $H_{dc}=-31.41 Oe$, $H_{ac}=25.13 Oe$.}\n\\label{ripple}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\includegraphics[height=6cm]{ripples_vs_time1.EPS}\n\\caption{Domain size in the ellipsoid along the $y$ coordinate as a function of the position of its center for two time snapshots, corresponding to the first nucleation-expansion stage of the mw-assisted reversal processes and for $H_{dc}=-31.41 Oe$, $H_{ac}=25.13Oe$. Note that after approximately $t> 2.5 ns$ the domain size expansion is suppressed.}\n\\label{size}\n\\end{figure}\n\nGenerally speaking, the ac-driven process is complicated and involves different objects. The domains are separated by domain walls (of N\\'eel or cross-tie types) and the magnetic moments in the center of these domain walls are constantly precessing (see Fig.~\\ref{MW} (c) and (d)). This precession is important for the domain wall mobility. In the junctions between the domains vortices are created (see Fig.~\\ref{Vortex}).\nNext, during the process we observe constant spinwave generation and their reflection from the ellipsoid boundaries and domain walls. Therefore, the overall process contains an interplay of several important effects, discussed earlier in the literature in more simple situations (see the Introduction). Each of them has been reported previously to be influenced by the mw-field.\n\n\\begin{figure}[h!]\n\\includegraphics[height=5cm]{Fig7.eps}\n\\caption{More detailed magnetisation configuration in the upper part of the ellipsoid showing the occurrence of domain walls and vortices. $H_{dc}=-31.41Oe$, $H_{ac}= 25.13Oe$, $\\nu=0.1 GHz$.}\n\\label{Vortex}\n\\end{figure}\n\nFig.~\\ref{time} shows the dependence of the magnetisation switching time in the ellipsoid on the mw frequency for several amplitudes of the ac field. We clearly observe  oscillations in the switching time. Moreover there exists a region of the parameters (see Fig.~\\ref{contur}) where the switching required more than $16 \\;ns$. To understand the phenomenon we first plot in Fig. \\ref{My} the average $M_y$ magnetisation component as a function of time. The small-period magnetisation oscillations are related to the ac-field oscillations. During the switching process we can distinguish two main processes: The first one (occurring in the first 2 ns) is related to the efficiency of the magnetisation nucleation in the system while the second one is related to the magnetisation relaxation. Fig.~\\ref{My} shows that at several frequencies the magnetisation reversal process started faster in its nucleation part but proceeded slower in the relaxation part.\n\n\n\n The nucleation process is associated to efficient spin wave generation and then to  spinwave instability phenomena which is a precursor of the magnetisation reversal \\cite{Chubykalo, Dobin, Kashuba}. The role of the ac-field is to excite a spinwave with the external frequency.  Because of this, the data of Fig.~\\ref{ripple} resembles the spinwave dispersion relation and the ripple configurations - the spinwave modes reported in Ref.~\\onlinecite{Gubbiotti}. However, the excited mode may be unstable for a given condition and, thus, produce instabilities and magnetisation reversal. For example, we have seen that at these values of the applied field the main FMR mode is completely unstable. When we tried to excite the homogeneous mode, with the magnetisation anti-parallel to the applied field direction, the energy was immediately redistributed into inhomogeneous spinwaves provoking later the magnetisation reversal. The excitation of these spinwaves was acting as an additional damping source leading to the magnetisation reversal similar to the case of Ref.~\\onlinecite{Safonov}. In this case, the excitation with the frequencies close to FMR  (around $3 GHz$ in Fig.~\\ref{time}) is the most efficient one. The first stage (several ns) is also characterized by initial expansion of the domain size, especially in the center, as seen in Fig.~\\ref{size}.\n\n\n\n\n\\begin{figure}[h!]\n\\includegraphics[height=6cm]{tsw1.EPS}\n\\caption{Switching time of the magnetisation as a function of the mw frequency for various values of the mw-field amplitude $H_{ac}$ and  $H_{dc}= -31.41 Oe$ }\n\\label{time}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\\includegraphics[height=6.5cm]{contour1.eps}\n\\caption{Contour plot for the switching time of the ellipsoid for applied field $H_{dc}=- 31.41 Oe$. }\n\\label{contur}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\\includegraphics[height=6cm]{My_time1.EPS}\n\\caption{Temporal evolution for the average $<M_y>$ magnetisation component (normalized to the saturation value) during the microwave-assisted switching process at $H_{dc}=- 31.41 \\; Oe$ and $H_{ac}= 25.13\\; Oe$. }\n\\label{My}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\includegraphics[height=6cm]{desv1.EPS}\n\\caption{Temporal evolution for the average $<M_z^2>$ magnetisation component (normalized to the saturation value) during the microwave-assisted switching process at $H_{dc}=- 31.41 \\; Oe$ and $H_{ac}= 25.13 \\; Oe$.}\n\\label{desv}\n\\end{figure}\n\n\n\nTo illustrate the role of precession, we present in Fig.~\\ref{desv} the temporal evolution of the average value of the $<M_z^2>$ component (this value is proportional to the energy put into precession by the microwave). The first part of these curves characterizes the initial nucleation-expansion process (first 2 ns). It is clear that the energy is put efficiently into the precession with the frequency close to the main FMR mode. This is confirmed by the fast magnetisation decay of the $<M_y>$ component in Fig.~\\ref{My} and the fast growth of the $<M_z^2>$ value in Fig.~\\ref{desv}. This fact is consistent with the hypothesis introduced in many papers stating that for efficient switching the mw frequency should coincide with the FMR one \\cite{Woltersdorf, Sun}. In the case of magnetic elements behaving as one macrospin magnetic moment this would determine the overall switching. However, a fast nucleation is insufficient for  fast magnetisation switching of larger elements. Fig.~\\ref{desv} demonstrates that for frequencies close to $6GHz$ (double FMR frequency in the opposite well) the precessional energy remains constant. We note that only the magnetic moments in the centers of domain walls, separating domains with opposite $M_x$ signs (and in the vortex centers) are precessing (see Fig.~\\ref{MW}). Thus the expansion of domain walls necessary for magnetisation relaxation is dependent on the relaxation of these magnetic moments. For frequencies and timescales at which the $<M_z^2>$ relaxation is very slow, the energy is transferred efficiently into the precessional motion and not to the relaxation process. Consequently, the further domain expansion is extremely slow. We would like to note that this idea is in the spirit of the hypothesis introduced in Ref.~\\onlinecite{Krasyuk} (self-trapping of magnetic oscillations). At the long-time scale the magnetisation reversal at this stage proceeds again via additional spin-wave generation and reflection destabilizing the domain structure. Note also that the nucleation process occurs  when the magnetisation is anti-parallel to the field direction while in the relaxation part it is parallel. Thus, the relevant frequencies are different in both cases.\n\n\nFig.~\\ref{Twofields} shows the magnetisation switching time as a function of the mw frequency for two dc applied field values. As the applied dc field increases, the angle of the magnetisation precession increases in the nucleation part of the process but decreases in the relaxation part. Larger precessional angles lead to nonlinear phenomena. Associated with larger precessional angles there is a nonlinear shift of the frequencies to smaller values \\cite{Usatenko}. This is in agreement with the results presented in Fig.~\\ref{Twofields}, provided that the relevant frequencies are determined by the relaxation process. However, in Fig.~\\ref{time} the frequencies are shifted to larger values with larger ac-field amplitude. This shows that the overall process is much more complicated and cannot be analysed in terms of the FMR frequencies relevant to the switching of one magnetic moment only. In fact, different ac-frequencies excite modes with different spin wave vector and the instabilities of them occur at different threshold amplitudes.\n\n\\begin{figure}[h!]\n\\includegraphics[height=6cm]{Twofields2.EPS}\n\\caption{Switching time of the magnetisation as a function of the mw frequency for two values of the stationary field $H_{dc}$ and  the mw-field amplitude $H_{ac}= 25.13 \\; Oe$.}\n\\label{Twofields}\n\\end{figure}\n\n\n\\section{Conclusions}\n\nUsing the micromagnetic model, we have investigated the mechanism of  fast magnetisation switching assisted by a linearly polarised microwave field in micron-sized magnetic elements. Magnetisation dynamics in these magnetic patterns is governed by nucleation-propagation-relaxation processes and is different to those of  nanoscale elements behaving as one macrospin. Our simulations confirm that the microwave-assisted magnetisation reversal requires a smaller field than in the process under a static applied field. The first contribution to this is the deviation of the field from the easy-axis during the mw-assisted switching process. However, even when we compared the situations when the maximum applied field angle is the same in both cases,  mw-assisted switching  appeared to require less field. Moreover, our results show that at small  intensity\nof the mw-field its action is not described by the effective field tilt.\n This happens due to the fact that the static and the mw-assisted switching processes involve different reversal modes. In the first case this mode consists of the nucleation of two vortices in  opposite corners, their subsequent merging and creation of two domain walls which later span the whole ellipsoid. In the second case, the reversal mode consists of a ripple structure, the external frequency being responsible for the excitation of a spinwave mode with the corresponding ripple size. For fields below the static coercivity field, the magnetisation configuration corresponds to almost homogeneous background plus  excited spin wave mode and is metastable. The amplitude of the  excited spin wave mode grows with applied field  and becomes unstable, leading to  magnetisation reversal. Thus, the spin-wave instability mechanism \\cite{Chubykalo, Dobin,Kashuba} plays a major role for the fast magnetisation switching process in nano-size magnetic elements.\n\nWe have shown that the most efficient nucleation is at the FMR frequency (and its multiples). However, the magnetisation reversal process after the nucleation requires also an efficient magnetisation relaxation. For example, the domain growth is stimulated by the ac field. This happens due to the relaxation of precessing magnetic moments in the domain wall center. In the spirit of the mechanism suggested in Ref.~\\onlinecite{Krasyuk}, for some frequencies, the relaxation process is not efficient, this happens when the mw field is coupled to the precessional motion. In this case the microwave field efficiently puts the energy into precession and not into the propagation-relaxation processes which are slowed down.\nAs a consequence of the interplay of several mechanisms with different relevant frequencies, the switching time of magnetic elements is a complicated function of the external frequency. The above results show that the magnetisation dynamics in micron-sized magnetic elements could not be analysed as a simple FMR-related phenomenon and neither follows a direct correspondence with a spinwave spectrum.\n\n\nFinally, we would like to discuss our results in the context of available experimental data.\n Out results are in qualitative agreement with experimental observations. The Kerr images in Ref.~\\onlinecite{Pimentel} show the creation of  ripple structures during the mw-assisted switching process in a permalloy ellipsoid. Currently there is no data available for the switching time of the magnetisation in ellipsoids which could confirm the existence of the oscillations presented in Fig.~\\ref{time}. However, in several experimental papers on different magnetic elements we have found qualitative similarities with our results. For example, in Co bars the relaxation time observed by the fast Kerr technique showed a strong nonmonotonic behavior as a function of applied field \\cite{Adam}. Also in Co bars measured by the anisotropic magnetoresistance effect \\cite{Grollier}, the authors have observed the occurrence of several resonance peaks sweeping in frequency. The microwave power input necessary for the maximum coercivity reduction was reported to be frequency-dependent in the measurements of  mw-assisted switching in NiFe magnetic tunneling junctions \\cite{Moriyama}. The critical field necessary for the microwave assistance was shown to be a complicated function of the mw frequency with several minima for the mw- assisted process going from vortex to onion state in magnetic nanorings \\cite{Podbielski}.\n\n\n\\section*{Acknowledgements}\nThe results of the German group were obtained with the research funding from the European Commission under EU-RTN ULTRASWITCH (HPRN-CT-2002-00318) project. The Spanish authors acknowledge  financial support from the European COST-P19 Action and  Spanish National projects MAT2007-66719-C03-01 and\n Consolider CS2008-023. They are also grateful to J.M. Rodriguez Puerta for his help with computer facilities.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{APPROACH}\n\\label{sec::approach}\n\nIn this section, we introduce \\textsc{apple}, which has two novel features: (1) compared to learning from demonstration or interventions, which require human driving expertise in order to take full control of the moving robot, \\textsc{apple} requires instead just evaluative feedback that can be provided even by non-expert users; (2) in contrast to previous work \\cite{xiao2020appld, wang2021appli} that selects the planner parameter set based on how similar the deployment environment is to the demonstrated environment, \\textsc{apple} is based on the expected evaluative feedback, i.e., the actual navigation performance.  \n\\textsc{apple}'s performance-based parameter policy has the potential to outperform previous approaches that are based on similarity.\n\n\\subsection{Problem Definition}\n \nWe denote a classical parameterized navigation system as $G: \\mathcal{X} \\times \\Theta \\rightarrow \\mathcal{A}$, where $\\mathcal{X}$ is the state space of the robot (e.g., goal, sensor observations), $\\Theta$ is the parameter space for $G$ (e.g., max speed, sampling rate, inflation radius), and $\\mathcal{A}$ is the action space (e.g., linear and angular velocities). During deployment, the navigation system repeatedly estimates state $x$ and takes action $a$ calculated as $a = G(x; \\theta)$. Typically, a default parameter set $\\bar{\\theta}$ is tuned by a human designer trying to achieve good performance in most environments. However, being good at everything often means being great at nothing: $\\bar{\\theta}$ usually exhibits suboptimal performance in some situations and may even fail (is unable to find feasible motions, or crashes into obstacles) in particularly challenging ones~\\cite{xiao2020appld}. \n\nTo mitigate this problem, \\textsc{apple} learns a parameter policy from human evaluative feedback with the goal of selecting the appropriate parameter set $\\theta$ (from either a discrete parameter set library or from a continuous full parameter space) for the current deployment environment.\nIn detail, a human can supervise the navigation system's performance at state $x$ by observing its action $a$ and giving corresponding evaluative feedback $e$.\nHere, the evaluative feedback can be either discrete (e.g., ``good\/bad job\") or continuous (e.g., a score ranging in $[0, 1]$).\nDuring feedback collection, \\textsc{apple} finds (1) a parameterized predictor $F_\\phi: \\mathcal{X} \\times \\Theta \\rightarrow \\mathcal{E}$ that predicts human evaluative feedback for each state-parameter pair $(x, \\theta)$, and (2) a parameterized parameter policy $\\pi_\\psi: \\mathcal{X} \\rightarrow \\Theta$ that determines the appropriate planner parameter set for the current state.\n\nBased on whether \\textsc{apple} chooses the parameter set from a library or the parameter space, we introduce the discrete and continuous parameter policies in the following two sections, respectively.\n\n\\subsection{Discrete Parameter Policy}\n\nIn some situations, the user may already have $K$ candidate parameter sets (e.g., the default set or sets tuned for special environments like narrow corridors, open spaces, etc.) which together make up a parameter library $\\mathcal{L} = \\{\\theta^i\\}_{i=1}^K$ (superscript $i$ denotes the index in the library). In this case, \\textsc{apple} uses the provided evaluative feedback $e$ in order to learn a policy that selects the most appropriate of these parameters given the state observation $x$.\n\nTo do so, we parameterize the feedback predictor $F_\\phi$ in a way similar to the value network in DQN \\cite{mnih2015humanlevel}, where the input is the observation $x$ and the output is $K$ predicted feedback values $\\{\\hat{e}^i\\}_{i=1}^K$, one for each parameter set $\\theta^i$ in the library $\\mathcal{L}$, as a prediction of the evaluative feedback a human user would give if the planner were using the respective parameter set at state $x$. We form a dataset for supervised learning, $\\mathcal{D} := \\{x_j, \\theta_j, e_j\\}_{j=1}^N$ ($\\theta_j \\in \\mathcal{L}$, subscript $j$ denotes the time step) \nusing the evaluative feedback collected so far, and $F_\\phi$ is learned via supervised learning to minimize the difference between predicted feedback and the label,\n\\begin{equation}\n    \\phi^* = \\argmin_\\phi \\mathop{\\mathbb{E}}_{(x_j, \\theta_j, e_j) \\sim \\mathcal{D}} \\ell(F_\\phi(x_j, \\theta_j), e_j)\n    \\label{eq:critic_loss}\n\\end{equation}\nwhere $\\ell(\\cdot, \\cdot)$ is the categorical cross entropy loss if the feedback $e_j$ is discrete (e.g., ``good job\/bad job'' or an integer score between 1 to 5),  or mean squared error given continuous feedback.\n\nTo achieve the best possible performance, the parameter policy $\\pi(\\cdot|x)$ chooses the parameter set that maximizes the expected human feedback (the discrete parameter policy doesn't require any additional parameters beyond $\\phi$ for $F$, so the $\\psi$ is omitted here for simplicity). More specifically, \n\\begin{equation}\n     \\pi(\\cdot|x) = \\argmax_{\\theta \\in \\mathcal{L}} F_{\\phi^*}(x, \\theta).\n     \\label{eqn::discrete}\n\\end{equation}\n\nCompared to RL, especially DQN, discrete \\textsc{apple} has a similar architecture and training objective. However, an important difference is that while RL optimizes future (discounted) cumulative reward, \\textsc{apple} greedily maximizes the current feedback. The reason is that we assume, while supervising the robot's actions, the human will not only consider the current results but also future consequences and give the feedback accordingly. This assumption is consistent with past systems such as \\textsc{tamer}~\\cite{knox2009interactively}. Under this interpretation of feedback, \\textsc{apple} can also be though of as trying to maximize some notion of future performance.\n\n\\subsection{Continuous Parameter Policy}\n\nIf a discrete parameter library is not available or desired, \\textsc{apple} can also be used over continuous parameter spaces (e.g., deciding the max speed from $[0.1, 2]\\ \\mathrm{m\/s}$). \nIn this scenario, \\textsc{apple} can still learn from either discrete or continuous feedback. However, learning from discrete feedback of finer resolutions or even continuous feedback should lead to better performance.\n\nIn this setting, we parameterize the parameter policy $\\pi_\\psi$ and the feedback predictor $F_\\phi$ in the actor-critic style.\nWith the collected evaluative feedback $ \\mathcal{D} := \\{x_j, \\theta_j, e_j\\}_{j=1}^N$, the training objective of $F_\\phi$ is still to minimize the difference between predicted and collected feedback, as specified by Eqn. (\\ref{eq:critic_loss}). For the parameter policy $\\pi_\\psi$, beyond choosing the action that maximizes expected feedback, its training objective is augmented by maximizing the entropy of policy $\\mathcal{H}(\\pi_\\psi(\\cdot|x))$ at state $x$. Using the same entropy regularization as Soft Actor Critic (SAC)~\\cite{haarnoja2018soft}, $\\pi_\\psi$ favors more stochastic policies, leading to better exploration during training:\n\\begin{equation}\n     \\psi^* = \\argmin_\\psi \\mathop{\\mathbb{E}}_{\\substack{x_j \\in \\mathcal{D} \\\\ \\tilde{\\theta}_j \\sim \\pi_\\psi(\\cdot | x_j)}} \\left[-F_\\phi(x_j, \\tilde{\\theta}_j) + \\alpha \\log \\pi_\\psi(\\tilde{\\theta}_j | x_j)  \\right],\n     \\label{eqn::continuous}\n\\end{equation}\nwhere $\\alpha$ is the temperature controlling the importance of the entropy bonus and is automatically tuned as in SAC~\\cite{haarnoja2018soft}.\n\n\n\\subsection{Deployment}\nDuring deployment, we measure the state $x_t$, use the parameter policy to obtain a set of parameters $\\theta_t \\sim \\pi_{\\psi^*}(\\cdot | x_t)$ at each time step, and apply that parameter set to the navigation planner $G$.\n\n\\section{CONCLUSIONS}\n\\label{sec::conclusions}\n\nIn this work, we introduce \\textsc{apple}, \\emph{Adaptive Planner Parameter Learning from Evaluative Feedback}. In contrast to most existing end-to-end machine learning for navigation approaches, \\textsc{apple} utilizes existing classical navigation systems and inherits all their benefits, such as safety and explainability. Furthermore, instead of requiring a full expert demonstration or a few corrective interventions that need the user to take full control of the robot, \\textsc{apple} just needs evaluative feedback as simple as ``good job\" or ``bad job\" that can be easily collected from non-expert users. Moreover, comparing with \\textsc{appli} which selects the parameter set based on the similarity with demonstrated environments, \\textsc{apple} achieves better generalization by selecting the parameter set with a performance-based criterion, i.e., the expected evaluative feedback. We show \\textsc{apple}'s performance improvement with simulated and real human feedback, as well as its generalizability in both 50 unseen simulated environments and an unseen physical environment.\nIn this paper, we use relatively dense feedback signals from the human user in the physical experiments (and different resolutions of simulated feedback signals in the simulated experiments) to reduce the amount of time needed to train a good \\textsc{apple} policy. These dense feedback signals may not always be practical, for example, the user may not always be paying attention. Therefore an important direction for future investigation is to study how little feedback is needed to yield good performance. Another important direction is to evaluate APPLE's generality with human subjects with different expertise levels and feedback criteria using an extensive user study.\n\n\n\n\n\\section{EXPERIMENTS}\n\\label{sec::experiments}\nIn our experiments, we aim to show that \\textsc{apple} can improve navigation performance by learning from evaluative feedback, in contrast to a teleoperated demonstration or a few corrective interventions, both of which require the non-expert user to take control of the moving robot. We also show \\textsc{apple}'s generalizability to unseen environments. We implement \\textsc{apple} on a ClearPath Jackal ground robot in \\textsc{barn}~\\cite{perille2020benchmarking} with 300 navigation environments randomly generated using Cellular Automata, and in two physical obstacle courses. \n\n\\subsection{Implementation}\n\nThe Jackal is a differential-drive robot equipped with a Velodyne LiDAR that we use to obtain a 720-dimensional planar laser scan with a 270$^\\circ$ field of view, denoted as $l_t$.\nThe robot uses the Robot Operating System \\texttt{move\\textunderscore base} navigation stack with Dijkstra's global planner and the default \\textsc{dwa} local planner~\\cite{fox1997dynamic}.\nFrom the global planner, we query the relative local goal direction $g_t$ (in angle) as the averaged tangential direction of the first 0.5m global path. The state space of the robot is the combination of the laser scan and local goal $x_t = (l_t, g_t)$. The parameter space consists of the 8 parameters of the \\textsc{dwa} local planner as described in Tab. \\ref{tab::jackal_parameters}, and the action space is the linear and angular velocity of the robot, $a_t = (v_t, \\omega_t)$.\n\nFor discrete \\textsc{apple}, we construct the parameter library shown in Tab. \\ref{tab::jackal_parameters} with the default \\textsc{dwa} parameter set $\\theta_1$ and parameter sets $\\theta_{2 \\sim 7}$ learned in the \\textsc{appli} work~\\cite{wang2021appli}. Here, we use parameter sets learned in previous work, and we have found that this is important\u2014without a reasonably good set to start with, \\textsc{apple} is typically unable to learn, e.g., smaller \\emph{inflation\\_radius} and larger \\emph{vtheta\\_samples} in $\\theta_2$ achieve good navigation in tight spaces, while larger \\emph{max\\_vel\\_x} and \\emph{pdist\\_scale} in $\\theta_4$ perform well in open ones. Without prelearned parameter sets, one can obtain a library via coarse tuning to create various driving modes (e.g. increase \\emph{max\\_vel\\_x} to create an aggressive mode). For continuous \\textsc{apple}, the parameter ranges for the parameter policy $\\pi_\\psi$ to select from are listed in the same table.\n\n\\begin{table}[h]\n  \\caption{Parameter Library and Range: \\\\ \\emph{max\\_vel\\_x} \\textnormal{(v)}, \\emph{max\\_vel\\_theta} \\textnormal{(w)}, \\emph{vx\\_samples} \\textnormal{(s)}, \\emph{vtheta\\_samples} \\textnormal{(t)}, \\emph{occdist\\_scale} \\textnormal{(o)}, \\emph{pdist\\_scale} \\textnormal{(p)}, \\emph{gdist\\_scale \\textnormal{(g)}}, \\emph{inflation\\_radius \\textnormal{(i)}}}\n  \\label{tab::jackal_parameters}\n  \\centering\n  \\small\n  \\begin{tabular}{lrrrrrrrr}\n    \\toprule\n                & v & w & s & t & o & p & g & i \\\\\n    \\midrule\n    $\\theta_1$ & 0.50 & 1.57 &  6 & 20 & 0.10 & 0.75 & 1.00 & 0.30 \\\\\n    $\\theta_2$ & 0.26 & 2.00 & 13 & 44 & 0.57 & 0.76 & 0.94 & 0.02 \\\\\n    $\\theta_3$ & 0.22 & 0.87 & 13 & 31 & 0.30 & 0.36 & 0.71 & 0.30 \\\\\n    $\\theta_4$ & 1.91 & 1.70 & 10 & 47 & 0.08 & 0.71 & 0.35 & 0.23 \\\\\n    $\\theta_5$ & 0.72 & 0.73 & 19 & 59 & 0.62 & 1.00 & 0.32 & 0.24 \\\\\n    $\\theta_6$ & 0.37 & 1.33 &  9 &  6 & 0.95 & 0.83 & 0.93 & 0.01 \\\\\n    $\\theta_7$ & 0.31 & 1.05 & 17 & 20 & 0.45 & 0.61 & 0.22 & 0.23 \\\\\n    \\midrule\n    $\\min$ & 0.2 & 0.31 & 4  & 8  & 0.10 & 0.10 & 0.01 & 0.10 \\\\\n    $\\max$ & 2.0 & 3.14 & 20 & 40 & 1.50 & 2.00 & 1.00 & 0.60 \\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\nImplementation-wise, for discrete \\textsc{apple}, $F_\\phi(x, \\theta)$ is a fully-connected neural network with 2 hidden layers of 128 neurons, taking the 721 dimensional $x_t$ as input and outputting the 7 predicted feedback signals $\\hat{e}_{t, 1 \\sim 7}$ for $\\theta_{1 \\sim 7}$ respectively. Parameter policy $\\pi_\\psi$ uses $\\epsilon$-greedy exploration with $\\epsilon$ decreasing from $0.3$ to $0.02$ during the first half of the training. For continuous \\textsc{apple}, $F_\\phi(x, \\theta)$ shares the same architecture as discrete \\textsc{apple}, except for different input (concatenation of $x_t$ and $\\theta_t$) and output (scalar $\\hat{e}_t$). The parameter policy $\\pi_\\psi$ also uses the same architecture, mapping $x_t$ to $\\theta_t$.\n\nTo evaluate the performance of \\textsc{apple}, we use \\textsc{appli} with the same parameter library in Tab. \\ref{tab::jackal_parameters} upper part, \\textsc{applr} with the same parameter ranges in Tab. \\ref{tab::jackal_parameters} lower part, and the Default \\textsc{dwa} planner as three baselines. Since \\textsc{appld} does not generalize well without a confidence-based context predictor~\\cite{wang2021appli} and \\textsc{appli} can therefore outperform \\textsc{appld}, we do not include \\textsc{appld} as one of the baselines. Despite using the same library, \\textsc{appli} chooses the parameter set based on the similarity between the current observation and the demonstrated environments, while discrete \\textsc{apple} uses the expected feedback.\n\n\\subsection{Simulated Experiments}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{contents\/figures\/gazebo.jpg}\n  \\vspace{-15pt}\n  \\caption{Simulated Environments in \\textsc{barn} Dataset with Low, Medium, and High Difficulty Levels}\n  \\vspace{-15pt}\n  \\label{fig::simulated_envs}\n\\end{figure}\n\nWe begin by testing \\textsc{apple} on the \\textsc{barn} dataset, with simulated evaluative feedback generated by an oracle (proxy human). Note that although the proxy human for the simulated experiments appears to be similar to the \\textsc{applr} work \\cite{xu2020applr}, the simulated experiments aim to validate different \\textsc{apple} setups with easily accessible feedback before physical experiments. The intended use case for \\textsc{apple} is still during \\emph{physical} deployments with \\emph{real} humans. The benchmark dataset consists of 300 simulated navigation environments ranging from easy ones with a lot of open spaces to challenging ones where the robot needs to get through dense obstacles. Navigation trials in three example environments with low, medium, and high difficulty levels are shown in Fig. \\ref{fig::simulated_envs}. We randomly select 250 environments for training \\textsc{apple} and hold the remained 50 environments as the test set.\n\nFor the simulated feedback, we use the projection of the robot's linear velocity along the local goal direction, i.e., $e_t = v_t \\cdot \\cos(g_t)$, and it greedily encourages the robot to move along the global path as fast as possible. Then we discretize it to different number of levels ($\\infty$ levels mean using continuous feedback) to study the effect of feedback resolutions. Notice, this simulated evaluative feedback is suboptimal as it doesn't consider future states, and we expect actual human evaluative feedback would be more accurate. For example, when the robot is leaving an open space and is about to enter a narrow exit, a human would expect the robot to slow down to get through the exit smoothly, but the simulated oracle still encourages the robot to drive fast. The oracle provides its evaluative feedback at 1Hz, while \\textsc{apple}, \\textsc{appli} and  \\textsc{applr} dynamically adjust the parameter set for the \\textsc{dwa} planner at the same frequency.\n\nAfter training in 250 environments with a total of 2.5M feedback signals collected, we evaluate \\textsc{apple} with discrete and continuous parameter policies (denoted as \\textsc{apple} (disc.) and \\textsc{apple} (cont.)), as well as three baselines, on the 50 test environments by measuring the traversal time for 20 runs per environment. The proxy human aims at improving navigation efficiency and thus reducing traversal time.\nWe then conduct t-tests to compute the percentage of environments in which \\textsc{apple} with different feedback resolutions is significantly better\/worse ($p<0.05$) than baselines, as shown in Fig. \\ref{fig::apple_vs_baselines}.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{contents\/figures\/APPLE_different_feedback_levels.png}\n  \\vspace{-15pt}\n  \\caption{\\textsc{apple} Learning from Different Feedback Resolutions.}\n  \\label{fig::apple_vs_baselines}\n  \\vspace{-10pt}\n\\end{figure}\n\n\n\nDespite learning from suboptimal evaluative feedback, \\textsc{apple} (disc.) and \\textsc{apple} (cont.) still outperform the Default \\textsc{dwa} and \\textsc{appli} at all feedback resolutions.\nThese results demonstrate the advantage of \\textsc{apple} over \\textsc{appli}, which selects the parameter set with a performance-based predictor (Eqns. \\ref{eqn::discrete} and \\ref{eqn::continuous}) rather than a similarity-based predictor.\nAmong all feedback resolutions, there is significant improvement of 3 feedback levels over 2, for both discrete and continuous policies. Further increasing resolutions doesn't improve performance much (the slight decreases are likely due to stochasticity in learning), except for using continuous feedback (i.e., $\\infty$ levels) for continuous policy. These results suggest that as few as three discrete feedback levels are needed to improve navigation performance. \nMost of the cases where \\textsc{apple} achieves worse performance are due to a corner case for the global planner where it keeps switching between two global paths and thus confuses the local planner. \nSurprisingly, despite that in theory \\textsc{applr} is expected to perform the best, \\textsc{apple} (disc.) performs only slightly worse than \\textsc{applr}, and \\textsc{apple} (cont.) outperforms \\textsc{applr} in $29\\%$ of test environments with continuous feedback. The worse performance of \\textsc{applr} may come from challenging optimization of cumulative rewards or reward design. Lastly, comparing the left and right parts of Fig. \\ref{fig::apple_vs_baselines}, \\textsc{apple} (cont.) is comparable with \\textsc{apple} (disc.) with low feedback resolutions and performs much better with continuous feedback, because of its larger model capacity in the parameter space. \n\n\\subsection{Physical Experiments}\nWe also apply \\textsc{apple} on a physical Jackal robot. In a highly constrained obstacle course (Fig. \\ref{fig::apple}), we first apply \\textsc{appli} which provides 5 sets of parameters (1 default $\\theta_1$ and 4 learned $\\theta_{2\\sim5}$). Then to train a discrete \\textsc{apple} policy which uses the same parameter library, one of the authors follows the robot autonomously navigating and uses an Xbox joystick to give binary evaluative feedback (instead of 3 feedback levels for easier collection) at 2Hz. The author aims at teaching \\textsc{apple} to reduce traversal time. \nTo reduce the burden of giving a large amount of feedback, the user is only requested to give negative feedback by pressing a button on the joystick when he thinks the robot's navigation performance is bad, while for other instances, positive feedback is automatically given.\nIn other words, we interpret the absence of human feedback to be the same as if the human had provided positive feedback.\nWhile this interpretation is not standard in the literature (and even undesirable at times \\cite{faulkner2018policy}), we found that it yielded good results for the application studied here. Because \\textsc{applr} \\cite{xu2020applr} requires an infeasible amount of trial and error in the real world, it is not included as a baseline in the physical experiments.\n\nThe entire \\textsc{apple} training session lasts roughly 30 minutes, in which the robot navigates 10 trials in the environment shown in Fig. \\ref{fig::apple}. \\textsc{apple} learns in an online fashion with $30\\%$ probability of random exploration. \nAfter the training, the learned \\textsc{apple} model is deployed in the same training environment. We compare \\textsc{apple} to \\textsc{appli} with the same sets of parameters and the confidence measure of context prediction~\\cite{wang2021appli}, and the DWA planner with static default parameters. Each experiment is repeated five times. The results are shown in Tab \\ref{tab::physical}. \\textsc{apple} achieves the fastest average traversal time with the smallest variance in the training environment.\n\nTo test \\textsc{apple}'s generalizability, we also test \\textsc{apple} in an unseen environment (Fig. \\ref{fig::unseen}) and show the results in Tab. \\ref{tab::physical}. In the unseen environment, \\textsc{apple} has slightly increased variance, but still has the fastest average traversal time compared to the other two baselines.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{contents\/figures\/unseen.jpg}\n  \\vspace{-10pt}\n  \\caption{\\textsc{apple} Running in an Unseen Physical Environment}\n  \\vspace{-10pt}\n  \\label{fig::unseen}\n\\end{figure}\n\n\\begin{table}\n\\centering\n\\caption{Traversal Time in Training and Unseen Environment}\n\\begin{tabular}{cccc}\n\\toprule\n & \\textbf{Default} & \\textbf{\\textsc{appli}} & \\textbf{\\textsc{apple} (disc.)} \\\\ \n\\midrule\n\\textbf{Training} & 143.1$\\pm$20.0s & 79.8$\\pm$8.1s & 75.2$\\pm$4.1s \\\\\n\\textbf{Unseen} & 150.5$\\pm$24.0s & 86.4$\\pm$1.1s & 83.9$\\pm$4.6s \\\\\n\\bottomrule\n\\end{tabular}\n\\vspace{-15pt}\n\\label{tab::physical}\n\\end{table}\n\n\\section{INTRODUCTION}\n\\label{sec::intro}\n\n\\IEEEPARstart{M}{obile}\nrobot navigation is a well-studied problem in the robotics community. Many classical approaches have been developed over the last several decades and several of them have been robustly deployed on physical robot platforms moving in the real world~\\cite{quinlan1993elastic, fox1997dynamic}, with verifiable guarantees of safety and explainability.\n\nHowever, prior to deployment in a new environment, these approaches typically require parameter re-tuning in order to achieve robust navigation performance. For example, in cluttered environments, a low velocity and high sampling rate are necessary in order for the system to be able to generate safe and smooth motions, whereas in relatively open spaces, a large maximum velocity and relatively low sampling rate are needed in order to achieve optimal navigation performance. This parameter re-tuning requires robotics knowledge from experts who are familiar with the inner workings of the underlying navigation system, and may not be intuitive for non-expert users~\\cite{zheng2021ros}. Furthermore, using a single set of parameters assumes the same set will work well on average in different regions of a complex environment, which is often not the case. \n\nTo address these problems, Adaptive Planner Parameter Learning (\\textsc{appl}) is a recently-proposed paradigm that opens up the possibility of dynamically adjusting parameters to adapt to different regions, and enables non-expert users to fine-tune navigation systems through modalities such as teleoperated demonstration~\\cite{xiao2020appld} or corrective interventions~\\cite{wang2021appli}.\nThese interaction modalities require non-expert users to take full control of the moving robot during the entire navigation task, or, at least, when the robot suffers from poor performance. However, non-expert users who are inexperienced at controlling the robot may be unwilling or unable to take such responsibility due to perceived risk of human error and causing collisions.\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{contents\/figures\/apple.jpg}\n  \\vspace{-15pt}\n  \\caption{For non-expert users who are unable or unwilling to take control of the robot, evaluative feedback, e.g. {\\em good job} (green thumbs up) or {\\em bad job} (red thumbs down), is a more accessible human interaction modality, but still valuable for improving navigation systems during deployment. }\n  \\label{fig::apple}\n  \\vspace{-15pt}\n\\end{figure}\n\nFortunately, even non-expert users who are not willing to take control of the robot are typically still able to observe the robot navigating and provide real-time positive or negative assessments of the observed navigation behavior through {\\em evaluative feedback}. For example, even non-expert users can know to provide negative feedback when a robot is performing poorly, e.g., getting stuck in highly constrained spaces~\\cite{xiao2020toward, liu2020lifelong} or driving unnecessarily slowly in open spaces~\\cite{xiao2020agile}. This more-accessible modality provides an interaction channel for a larger community of non-expert users with mobile robots (Fig. \\ref{fig::apple}).\n\nIn this work, we introduce a machine learning method that can leverage evaluative feedback in the context of autonomous navigation called \\emph{Adaptive Planner Parameter Learning from Evaluative Feedback} (\\textsc{apple}). Based on a parameter library~\\cite{xiao2020appld, wang2021appli} or a parameter policy~\\cite{xu2020applr}, \\textsc{apple} learns how to choose appropriate navigation planner parameters at each time step in order to adapt to different parts of the deployment environment. \nSpecifically, \\textsc{apple} treats the scalar human feedback as the value for the state-action pair in the Reinforcement Learning framework during training, in which the action is the parameter set to be used by the underlying navigation system. During deployment, \\textsc{apple} selects the parameters to maximize the expected human feedback value. \nWe implement \\textsc{apple} both in the Benchmarking Autonomous Robot Navigation (\\textsc{barn}) ~\\cite{perille2020benchmarking} environments and also in real-world, highly constrained obstacle courses. In both training and unseen environments, \\textsc{apple} is able to outperform the planner with default parameters, and even to improve over \\textsc{appl} variants learned from richer interaction modalities, such as teleoperated interventions. Our experimental results indicate that evaluative feedback is a particularly valuable form of human interaction modality for improving navigation systems during deployment.\n\n\n\\section{RELATED WORK}\n\\label{sec::related}\n\nIn this section, we review existing work on machine learning for mobile robot navigation, adaptive planner parameters, and learning from human evaluative feedback. \n\n\\subsection{Learning for Navigation}\nWhile autonomous navigation has been studied by the robotics community for decades, machine learning approaches have recently been extensively applied to this problem as well. Xiao, et al.~\\cite{xiao2020motion} presented a survey on using machine learning for motion control in mobile robot navigation: while the majority of learning approaches tackle navigation in an end-to-end manner~\\cite{bojarski2016end, pfeiffer2017perception}, it was found that approaches using learning in conjunction with other classical navigation components were more likely to have achieved better navigation performance.\nThese methods included those that learned sub-goals~\\cite{stein2018learning}, local planners~\\cite{gao2017intention, chiang2019learning, xiao2020toward, xiao2020agile, liu2020lifelong}, or planner parameters~\\cite{teso2019predictive, bhardwaj2020differentiable, binch2020context, xiao2020appld, wang2021appli, xu2020applr}. Learning methods have also enabled navigation capabilities that complement those provided in the classical navigation literature, including terrain-aware~\\cite{wigness2018robot, siva2019robot, kahn2021badgr} and social~\\cite{hart2020using, liang2020crowd, everett2018motion} navigation. \n\n\\textsc{apple} leverages the aforementioned hybrid learning and classical architecture, where the learning component only learns to select appropriate set of planner parameters, and interacts with the underlying classical navigation system. \n\n\\subsection{Adaptive Parameters for Classical Navigation} \nConsidering classical navigation systems' verifiable safety, explainability, and stable generalization to new environments, and the difficulty in fine-tuning those systems, learning adaptive planner parameters is an emerging paradigm of combining learning and planning. Examples include finding trajectory optimization coefficients using Artificial Neural Fuzzy Inference Improvement~\\cite{teso2019predictive}, optimizing two different sets of parameters for straight-line and U-turn scenarios with genetic algorithms~\\cite{binch2020context}, or designing novel systems that can leverage gradient descent to match expert demonstrations~\\cite{bhardwaj2020differentiable}. Recently, the \\textsc{appl} paradigm~\\cite{xiao2020appld, wang2021appli, xu2020applr} has been proposed, which further allows parameters to be appropriately adjusted during deployment ``on-the-fly'', in order to adapt to different regions of a complex environment. \\textsc{appl} also learns from non-expert users using teleoperated demonstration~\\cite{xiao2020appld}, corrective interventions~\\cite{wang2021appli}, or trial-and-error in simulation~\\cite{xu2020applr}. \n\n\\textsc{apple} utilizes an accessible but sparse modality of human interaction in evaluative feedback, which is suitable for non-expert users who are not able to take control of the robot. It is also suitable for scenarios where extensive trial-and-error is not feasible and a handcrafted reward function is not available, e.g., using Reinforcement Learning (RL) \\cite{xu2020applr}. Similar, or even better, navigation performance, compared to that learned from richer interaction modalities, can be achieved using \\textsc{apple}.\n\n\\subsection{Learning from Human Feedback}\nThe method we propose in this paper uses evaluative feedback from a human to drive a machine learning process that seeks to increase the performance of an autonomous navigation system.\nBecause evaluative feedback is a relatively easy signal for humans to provide, several methods have been proposed to allow machines to learn from such signals over the past several decades.\nBroadly speaking, most of these methods can be understood as trying to interpret the feedback signal in the context of the classical RL framework.\nFor example, the \\textsc{coach} framework \\cite{macglashan2017interactive} interprets evaluative feedback as the policy-dependent {\\em advantage}, i.e., it is assumed that the feedback indicates how much better or worse the agent's current behavior is compared to what the human currently expects the agent to do.\nThe \\textsc{tamer} framework \\cite{knox2009interactively}, on the other hand, can be thought of as interpreting evaluative feedback to be the {\\em value}, or expected payoff, of the current behavior if the agent were to act in the future in the way the human desires.\nYet other approaches interpret evaluative feedback directly as reward or some related statistic \\cite{isbell2001social, thomaz2006reinforcement, pilarski2011online}.\n\n\\textsc{apple} adopts a similar learning from feedback paradigm, but instead of taking actions as raw motor commands, \\textsc{apple}'s action space is the parameters used by the underlying navigation system. During training, \\textsc{apple} learns the value of state-action pairs based on a scalar human feedback. During deployment, \\textsc{apple} selects the parameters to maximize the expected human feedback. \n\n\n\\section{INTRODUCTION}\n\\label{sec::intro}\n\nMobile robot navigation is a well-studied problem in the robotics community. Many classical approaches have been developed over the last \\DIFaddbegin \\DIFadd{several }\\DIFaddend decades and many of them have been robustly deployed on physical robot platforms moving in the real world~\\cite{quinlan1993elastic, fox1997dynamic}, with verifiable guarantees of safety and explainability.\n\nHowever, prior to deployment in a new environment, \\DIFaddbegin \\DIFadd{these approaches typically require }\\DIFaddend parameter re-tuning \\DIFdelbegin \\DIFdel{is most likely necessary }\\DIFdelend \\DIFaddbegin \\DIFadd{in order }\\DIFaddend to achieve robust navigation performance. For example, in cluttered \\DIFdelbegin \\DIFdel{environment, low maximum }\\DIFdelend \\DIFaddbegin \\DIFadd{environments, a low }\\DIFaddend velocity and high sampling rate are \\DIFdelbegin \\DIFdel{appropriate for generating }\\DIFdelend \\DIFaddbegin \\DIFadd{necessary in order for the system to be able to generate }\\DIFaddend safe and smooth motions, \\DIFdelbegin \\DIFdel{in contrast to high speed and }\\DIFdelend \\DIFaddbegin \\DIFadd{whereas in relatively open spaces, a large maximum velocity and relatively }\\DIFaddend low sampling rate \\DIFdelbegin \\DIFdel{for a relatively open space. Parameter }\\DIFdelend \\DIFaddbegin \\DIFadd{are needed in order to achieve optimal navigation performance. This parameter }\\DIFaddend re-tuning requires robotics knowledge from experts who are familiar with the inner workings of the underlying navigation system, and may not be intuitive for non-expert users~\\cite{zheng2017ros}. Furthermore, \\DIFdelbegin \\DIFdel{finding }\\DIFdelend \\DIFaddbegin \\DIFadd{using }\\DIFaddend a single set of parameters assumes the same set will work well on average in different regions of a complex environment, which is often not the case. \n\nTo address these problems, Adaptive Planner Parameter Learning (\\textsc{appl}) is a recently proposed paradigm that opens up the possibility of dynamically adjusting parameters to adapt to different regions, and enables non-expert users to fine-tune navigation systems with a teleoperated demonstration~\\cite{xiao2020appld} or a few corrective interventions~\\cite{wang2020appli}. \n\n\\DIFdelbegin \\DIFdel{However, interaction modalities such as teleoperated demonstration or corrective interventions }\\DIFdelend \\DIFaddbegin \\DIFadd{These interaction modalities (i.e., demonstration and corrective intervention) }\\DIFaddend require non-expert users to take full control of the moving robot during the entire navigation task\\DIFdelbegin \\DIFdel{or at least }\\DIFdelend \\DIFaddbegin \\DIFadd{, or, at least, }\\DIFaddend when the robot suffers from poor performance. \\DIFdelbegin \\DIFdel{Therefore}\\DIFdelend \\DIFaddbegin \\DIFadd{However}\\DIFaddend , non-expert users who are \\DIFdelbegin \\DIFdel{unversed in }\\DIFdelend \\DIFaddbegin \\DIFadd{inexperienced at }\\DIFaddend controlling the robot may be unwilling to take such responsibility \\DIFdelbegin \\DIFdel{, running the }\\DIFdelend \\DIFaddbegin \\DIFadd{due to perceived }\\DIFaddend risk of human \\DIFdelbegin \\DIFdel{errors }\\DIFdelend \\DIFaddbegin \\DIFadd{error }\\DIFaddend and causing collisions.\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{contents\/figures\/apple.jpg}\n  \\caption{For non-expert users who are \\DIFdelbeginFL \\DIFdelFL{also }\\DIFdelendFL not willing to take control of the robot, evaluative feedback, e.g. \\DIFaddbeginFL {\\em \\DIFaddendFL good job\\DIFaddbeginFL } \\DIFaddendFL (green thumbs up) or \\DIFaddbeginFL {\\em \\DIFaddendFL bad job\\DIFaddbeginFL } \\DIFaddendFL (red thumbs down), is a \\DIFdelbeginFL \\DIFdelFL{less engaging }\\DIFdelendFL \\DIFaddbeginFL \\DIFaddFL{more accessible }\\DIFaddendFL human interaction modality, but still valuable for improving navigation systems during deployment. }\n  \\label{fig::apple}\n\\end{figure}\n\n\\DIFdelbegin \\DIFdel{On the other hand}\\DIFdelend \\DIFaddbegin \\DIFadd{Fortunately}\\DIFaddend , even non-expert users who are not willing to take control of the robot \\DIFdelbegin \\DIFdel{can simply oversee }\\DIFdelend \\DIFaddbegin \\DIFadd{are typically still able to observe }\\DIFaddend the robot navigating and provide real-time positive or negative assessments of the observed navigation behavior through evaluative feedback. This \\DIFdelbegin \\DIFdel{less engaging }\\DIFdelend \\DIFaddbegin \\DIFadd{more-accessible }\\DIFaddend modality opens up a new interaction channel to a larger community of non-expert users with mobile robots (Fig. \\ref{fig::apple}). For example, it is apparent even to non-expert users when the robot gets stuck in highly-constrained spaces~\\cite{xiao2020toward, liu2020lifelong} or drives unnecessarily slowly in open spaces~\\cite{xiao2020agile}. \n\nIn this work, we introduce \\emph{Adaptive Planner Parameter Learning from Evaluative Feedback} (\\textsc{apple}). Based on a parameter library~\\cite{xiao2020appld, wang2020appli} or a parameter policy~\\cite{xu2020applr}, \\textsc{apple} learns how to choose appropriate planner parameters at each time step \\DIFaddbegin \\DIFadd{in order }\\DIFaddend to adapt to different parts of the deployment environment. \n\\DIFaddbegin \\DIFadd{Specifically, }\\textsc{\\DIFadd{apple}} \\DIFadd{treats the scalar human feedback as the value for the state-action pair during training, in which the action is the parameter set to be used by the underlying navigation system. During deployment, }\\textsc{\\DIFadd{apple}} \\DIFadd{selects the best parameters to maximize the expected human feedback value. \n}\\DIFaddend We implement \\textsc{apple} \\DIFaddbegin \\DIFadd{both }\\DIFaddend in the Benchmarking Autonomous Robot Navigation (BARN) ~\\cite{perille2020benchmarking} \\DIFdelbegin \\DIFdel{dataset and }\\DIFdelend \\DIFaddbegin \\DIFadd{environments and also in }\\DIFaddend real-world\\DIFaddbegin \\DIFadd{, }\\DIFaddend highly-constrained obstacle courses. In both training and unseen environments, \\textsc{apple} is able to outperform the planner with default parameters\\DIFdelbegin \\DIFdel{and even }\\DIFdelend \\DIFaddbegin \\DIFadd{, and even improve over }\\DIFaddend \\textsc{appl} variants learned from richer interaction modalities, such as teleoperated interventions. Our experimental results indicate that \\DIFdelbegin \\DIFdel{despite being sparse and less engaging, }\\DIFdelend evaluative feedback is a \\DIFdelbegin \\DIFdel{uniquely }\\DIFdelend \\DIFaddbegin \\DIFadd{particularly }\\DIFaddend valuable form of human interaction modality for improving navigation systems during deployment.\n\n\\section{RELATED WORK}\n\\label{sec::related}\n\nIn this section, we review existing work on machine learning for mobile robot navigation, adaptive planner parameters, and learning from human evaluative feedback. \n\n\\subsection{Learning for Navigation}\nWhile \\DIFdelbegin \\DIFdel{classical }\\DIFdelend \\DIFaddbegin \\DIFadd{autonomous }\\DIFaddend navigation has been studied by the robotics community for decades, \\DIFdelbegin \\DIFdel{recently }\\DIFdelend machine learning approaches have \\DIFaddbegin \\DIFadd{recently }\\DIFaddend been extensively applied to this problem as well. Xiao, et al.~\\cite{xiao2020motion} presented a survey on using machine learning for motion control in mobile robot navigation: while the majority of learning approaches tackle navigation in an end-to-end manner~\\cite{bojarski2016end, pfeiffer2017perception}, it \\DIFdelbegin \\DIFdel{is shown that }\\DIFdelend \\DIFaddbegin \\DIFadd{was found that approaches }\\DIFaddend using learning in conjunction with other classical navigation components \\DIFdelbegin \\DIFdel{is likely to achieve }\\DIFdelend \\DIFaddbegin \\DIFadd{were more likely to have achieved }\\DIFaddend better navigation performance\\DIFdelbegin \\DIFdel{, such as learning }\\DIFdelend \\DIFaddbegin \\DIFadd{.\nThese methods included those that learned }\\DIFaddend sub-goals~\\cite{stein2018learning}, \\DIFdelbegin \\DIFdel{learning }\\DIFdelend local planners~\\cite{gao2017intention, chiang2019learning, xiao2020toward, xiao2020agile, liu2020lifelong}, or \\DIFdelbegin \\DIFdel{learning }\\DIFdelend planner parameters~\\cite{teso2019predictive, bhardwaj2019differentiable, binch2020context, xiao2020appld, wang2020appli, xu2020applr}. Learning methods have also enabled navigation capabilities \\DIFdelbegin \\DIFdel{orthogonal to }\\DIFdelend \\DIFaddbegin \\DIFadd{that complement those provided in }\\DIFaddend the classical navigation \\DIFdelbegin \\DIFdel{problem, such as }\\DIFdelend \\DIFaddbegin \\DIFadd{literature, including }\\DIFaddend terrain-aware~\\cite{wigness2018robot, siva2019robot, kahn2020badgr} \\DIFdelbegin \\DIFdel{or }\\DIFdelend \\DIFaddbegin \\DIFadd{and }\\DIFaddend social~\\cite{hart2020using, liang2020crowdsteer, everett2018motion} navigation. \n\n\\textsc{apple} leverages the aforementioned hybrid learning and classical architecture, where the learning component only learns to select appropriate set of planner parameters, and interacts with the underlying classical navigation system. \n\n\\subsection{Adaptive Parameters for Classical Navigation} \nConsidering classical navigation systems' verifiable safety, explainability, and stable generalization to new environments, and the difficulty in fine-tuning those systems, learning adaptive planner parameters is an emerging paradigm of combining learning and planning. Examples include finding trajectory optimization coefficients using Artificial Neural Fuzzy Inference Improvement~\\cite{teso2019predictive}, optimizing two different sets of parameters for straight-line and U-turn scenarios with genetic \\DIFdelbegin \\DIFdel{algorithm}\\DIFdelend \\DIFaddbegin \\DIFadd{algorithms}\\DIFaddend ~\\cite{binch2020context}, or designing novel systems that can leverage gradient descent to match expert demonstrations~\\cite{bhardwaj2019differentiable}. Recently, the \\textsc{appl} paradigm~\\cite{xiao2020appld, wang2020appli, xu2020applr} has been proposed\\DIFaddbegin \\DIFadd{, }\\DIFaddend which further allows parameters to be appropriately adjusted during deployment ``on-the-fly'', in order to adapt to different regions of a complex environment. \\textsc{appl} also learns from non-expert users using teleoperated demonstration~\\cite{xiao2020appld}, corrective interventions~\\cite{wang2020appli}, or trial-and-error in simulation~\\cite{xu2020applr}. \n\n\\textsc{apple} utilizes \\DIFdelbegin \\DIFdel{another sparse and less engaging human interaction modality, i.e. }\\DIFdelend \\DIFaddbegin \\DIFadd{an accessible but sparse modality of human interaction in }\\DIFaddend evaluative feedback, which is suitable for non-expert users who are not able to take control of the robot. Similar, or even better, navigation performance, compared to that learned from richer interaction modalities, can be achieved using \\textsc{apple}.\n\n\\subsection{Learning from Human Feedback}\nThe method we propose in this paper uses evaluative feedback from a human to drive a machine learning process that seeks to increase the performance of an autonomous navigation system.\nBecause evaluative feedback is a relatively easy signal for humans to provide, several methods have been proposed to allow machines to learn from such signals over the past several decades.\nBroadly speaking, most of these methods can be understood as trying to interpret the feedback signal in the context of the classical \\DIFdelbegin \\DIFdel{reinforcement learning }\\DIFdelend \\DIFaddbegin \\DIFadd{Reinforcement Learning (RL) }\\DIFaddend framework.\nFor example, the \\textsc{coach} framework \\cite{macglashan2017interactive} interprets evaluative feedback as the policy-dependent {\\em advantage}, i.e., it is assumed that the feedback indicates how much better or worse the agent's current behavior is compared to what the human currently expects the agent to do.\nThe \\textsc{tamer} framework \\cite{knox2009interactively}, on the other hand, can be thought of as interpreting evaluative feedback to be the {\\em value}, or expected payoff, of the current behavior if the agent were to act in the future in the way the human desires.\nYet other approaches interpret evaluative feedback directly as reward \\DIFdelbegin \\DIFdel{, or something very close to it }\\DIFdelend \\DIFaddbegin \\DIFadd{or some related statistic }\\DIFaddend \\cite{isbell2001social, thomaz2006reinforcement, pilarski2011online}.\n\n\\DIFaddbegin \\textsc{\\DIFadd{apple}} \\DIFadd{adopts a similar learning from feedback paradigm, but instead of taking actions as raw motor commands, }\\textsc{\\DIFadd{apple}}\\DIFadd{'s action space is the parameters used by the underlying navigation system. During training, }\\textsc{\\DIFadd{apple}} \\DIFadd{learns the value of state-action pairs based on a scalar human feedback. During deployment, }\\textsc{\\DIFadd{apple}} \\DIFadd{selects the best parameters to maximize the expected human feedback. \n}\n\n\\DIFaddend %\n\\section{APPROACH}\n\\label{sec::approach}\n\nIn this section, we introduce \\textsc{apple}, which has two novel features: (1) compared \\DIFdelbegin \\DIFdel{with Learning from Demonstration or Interventions that requires expertise }\\DIFdelend \\DIFaddbegin \\DIFadd{to learning from demonstration or interventions, which require human driving expertise in order }\\DIFaddend to take full control of the moving robot, \\textsc{apple} \\DIFdelbegin \\DIFdel{only needs }\\DIFdelend \\DIFaddbegin \\DIFadd{requires instead just }\\DIFaddend evaluative feedback that can be provided even by non-expert users; (2) in contrast to previous work \\cite{xiao2020appld, wang2020appli} that selects the planner parameter set based on \\DIFdelbegin \\DIFdel{the similarity with demonstrated environments}\\DIFdelend \\DIFaddbegin \\DIFadd{how similar the deployment environment is to the demonstrated environment}\\DIFaddend , \\textsc{apple} is based on the expected evaluative feedback\\DIFdelbegin \\DIFdel{. }\\DIFdelend \\DIFaddbegin \\DIFadd{, i.e., the actual navigation performance.  \n}\\DIFaddend \\textsc{apple}'s performance-based parameter policy has the potential to outperform previous approaches \\DIFdelbegin \\DIFdel{only }\\DIFdelend \\DIFaddbegin \\DIFadd{that are }\\DIFaddend based on similarity.\n\n\\subsection{Problem Definition}\n\nWe denote a classical parameterized navigation system as $G: \\mathcal{X} \\times \\Theta \\rightarrow \\mathcal{A}$, where $\\mathcal{X}$ is the state space of the robot (e.g., goal, sensor observations), $\\Theta$ is the parameter space for $G$ (e.g., max speed, sampling rate, inflation radius), and $\\mathcal{A}$ is the action space (e.g., linear and angular velocities). During deployment, the navigation system repeatedly estimates state $x$ and takes action $a$ calculated as \\DIFdelbegin \\DIFdel{$a = G(x; \\bar{\\theta})$}\\DIFdelend \\DIFaddbegin \\DIFadd{$a = G(x; \\theta)$}\\DIFaddend . Typically, \\DIFdelbegin \\DIFdel{the }\\DIFdelend \\DIFaddbegin \\DIFadd{a }\\DIFaddend default parameter set $\\bar{\\theta}$ is tuned by a human designer trying to achieve good performance in most environments. However, being good at everything often means being great at nothing: $\\bar{\\theta}$ usually exhibits suboptimal performance in some situations and may even fail (is unable to find feasible motions, or crashes into obstacles) in particularly challenging ones\\DIFaddbegin \\DIFadd{~\\mbox{%\n\\cite{xiao2020appld}}\\hspace{0pt}%\n}\\DIFaddend . \n\nTo mitigate this problem, \\textsc{apple} learns a parameter policy from human evaluative feedback \\DIFdelbegin \\DIFdel{, to select }\\DIFdelend \\DIFaddbegin \\DIFadd{with the goal of selecting }\\DIFaddend the appropriate parameter set $\\theta$ (from \\DIFdelbegin \\DIFdel{a parameter }\\DIFdelend \\DIFaddbegin \\DIFadd{either a discrete parameter set }\\DIFaddend library or from \\DIFdelbegin \\DIFdel{the }\\DIFdelend \\DIFaddbegin \\DIFadd{a continuous full }\\DIFaddend parameter space) for \\DIFaddbegin \\DIFadd{the }\\DIFaddend current deployment environment.\nIn detail, a human can supervise the navigation system's performance at state $x$ by observing its action $a$ and giving corresponding evaluative feedback $e$.\nHere, the evaluative feedback can be either discrete (e.g., ``good job\"\/``bad job\") or continuous (e.g., a score ranging in $[0, 1]$).\nDuring \\DIFdelbegin \\DIFdel{the }\\DIFdelend feedback collection, \\textsc{apple} finds (1) a parameterized predictor $F_\\phi: \\mathcal{X} \\times \\Theta \\rightarrow \\mathcal{E}$ that predicts human evaluative feedback for each state-parameter pair $(x, \\theta)$, and (2) a parameterized parameter policy $\\pi_\\psi: \\mathcal{X} \\rightarrow \\Theta$ that determines the appropriate planner parameter set \\DIFaddbegin \\DIFadd{for the current state}\\DIFaddend .\n\nBased on whether \\textsc{apple} chooses the parameter set from a library or the parameter space, we introduce the discrete and continuous parameter policies in the following two sections, respectively.\n\n\\subsection{Discrete Parameter Policy}\n\n\\DIFdelbegin \\DIFdel{Including the default parameter set}\\DIFdelend \\DIFaddbegin \\DIFadd{In some situations}\\DIFaddend , the user may \\DIFdelbegin \\DIFdel{also has }\\DIFdelend \\DIFaddbegin \\DIFadd{already have }\\DIFaddend $K$ candidate parameter sets (e.g., \\DIFdelbegin \\DIFdel{ones }\\DIFdelend \\DIFaddbegin \\DIFadd{the default set or sets }\\DIFaddend tuned for special environments like narrow corridors, open spaces\\DIFdelbegin \\DIFdel{) which }\\DIFdelend \\DIFaddbegin \\DIFadd{, etc.) which together }\\DIFaddend make up a parameter library \\DIFdelbegin \\DIFdel{$\\{\\theta_i\\}_{i=1}^K$}\\DIFdelend \\DIFaddbegin \\DIFadd{$\\mathcal{L} = \\{\\theta^i\\}_{i=1}^K$ (superscript $i$ denotes the index in the library)}\\DIFaddend . In this case, \\textsc{apple} \\DIFdelbegin \\DIFdel{learns to select }\\DIFdelend \\DIFaddbegin \\DIFadd{uses the provided evaluative feedback $e$ in order to learn a policy that selects }\\DIFaddend the most appropriate \\DIFdelbegin \\DIFdel{one given the }\\DIFdelend \\DIFaddbegin \\DIFadd{of these parameters given the state }\\DIFaddend observation $x$\\DIFdelbegin \\DIFdel{and the provided evaluative feedback $e$}\\DIFdelend .\n\nTo do so, we parameterize the feedback predictor $F_\\phi$ in a way similar to the value network in DQN \\cite{mnih2015humanlevel}, where \\DIFaddbegin \\DIFadd{the }\\DIFaddend input is the observation $x$ and the \\DIFdelbegin \\DIFdel{outputs are the }\\DIFdelend \\DIFaddbegin \\DIFadd{output is }\\DIFaddend $K$ predicted feedback \\DIFdelbegin \\DIFdel{$\\{\\hat{e}_i\\}_{i=1}^K$}\\DIFdelend \\DIFaddbegin \\DIFadd{values $\\{\\hat{e}^i\\}_{i=1}^K$}\\DIFaddend , one for each parameter set \\DIFdelbegin \\DIFdel{$\\theta_i$ respectively. Then we form a supervised dataset }\\DIFdelend \\DIFaddbegin \\DIFadd{$\\theta^i$ in the library $\\mathcal{L}$, as a prediction of the evaluative feedback a human user would give if the planner were using the respective parameter set at state $x$. We form a dataset for supervised learning, }\\DIFaddend $\\mathcal{D} := \\{x_j, \\theta_j, e_j\\}_{j=1}^N$ \\DIFdelbegin \\DIFdel{from }\\DIFdelend \\DIFaddbegin \\DIFadd{($\\theta_j \\in \\mathcal{L}$, subscript $j$ denotes the time step) \nusing }\\DIFaddend the evaluative feedback collected so far, and $F_\\phi$ is learned via supervised learning to minimize the difference between predicted feedback and the label,\n\\begin{equation}\n    \\phi^* = \\argmin_\\phi \\mathop{\\mathbb{E}}\\DIFdelbegin \\DIFdel{_{(x_j, e_j) \\in \\mathcal{D}} \\| }\\DIFdelend \\DIFaddbegin \\DIFadd{_{(x_j, \\theta_j, e_j) \\sim \\mathcal{D}} \\ell(}\\DIFaddend F_\\phi(x_j, \\theta_j)\\DIFdelbegin \\DIFdel{- }\\DIFdelend \\DIFaddbegin \\DIFadd{, }\\DIFaddend e_j\\DIFdelbegin \\DIFdel{\\|\n    }\\DIFdelend \\DIFaddbegin \\DIFadd{)\n    }\\DIFaddend \\label{eq:critic_loss}\n\\DIFdelbegin %\n\\DIFdel{where $\\| \\cdot \\|$ }\\DIFdelend \\DIFaddbegin }\n\\DIFadd{where $\\ell(\\cdot, \\cdot)$ }\\DIFaddend is the binary cross entropy loss if the feedback $e_j$ is discrete (e.g., $e_j = 1$ for ``good job'', and $e_j = 0$ for ``bad job''),  or mean square error given continuous feedback.\n\n\\DIFdelbegin \\DIFdel{For the parameter policy $\\pi(\\cdot|x)$, to achieve }\\DIFdelend \\DIFaddbegin \\DIFadd{To achieve the }\\DIFaddend best possible performance, \\DIFdelbegin \\DIFdel{it }\\DIFdelend \\DIFaddbegin \\DIFadd{the parameter policy $\\pi(\\cdot|x)$ }\\DIFaddend chooses the parameter set that maximizes the expected human feedback (the discrete parameter policy doesn't \\DIFdelbegin \\DIFdel{need its own parameter, so }\\DIFdelend \\DIFaddbegin \\DIFadd{require any additional parameters beyond $\\phi$ for $F$, so the }\\DIFaddend $\\psi$ is omitted \\DIFaddbegin \\DIFadd{here }\\DIFaddend for simplicity). More specifically, \n\\begin{equation}\n     \\pi(\\cdot|x) = \\argmax\\DIFdelbegin \\DIFdel{_\\theta }\\DIFdelend \\DIFaddbegin \\DIFadd{_{\\theta \\in \\mathcal{L}} }\\DIFaddend F_{\\phi^*}(x, \\theta).\n     \\DIFaddbegin \\label{eqn::discrete}\n\\DIFaddend \\end{equation}\n\nWhen comparing with RL, especially DQN, discrete \\textsc{apple} has \\DIFaddbegin \\DIFadd{a }\\DIFaddend similar architecture and training objective. However, an \\DIFdelbegin \\DIFdel{importance }\\DIFdelend \\DIFaddbegin \\DIFadd{important }\\DIFaddend difference is that while RL optimizes \\DIFdelbegin \\DIFdel{for future cumulative rewards with a discount factor $\\gamma \\in [0, 1]$}\\DIFdelend \\DIFaddbegin \\DIFadd{future (discounted) cumulative reward}\\DIFaddend , \\textsc{apple} greedily maximizes the current feedback. The reason is that we assume, \\DIFdelbegin \\DIFdel{during }\\DIFdelend \\DIFaddbegin \\DIFadd{while }\\DIFaddend supervising the robot's actions, \\DIFaddbegin \\DIFadd{the }\\DIFaddend human will not only consider \\DIFaddbegin \\DIFadd{the }\\DIFaddend current results but also future consequences and give the feedback accordingly. \\DIFaddbegin \\DIFadd{This assumption is consistent with past systems such as }\\textsc{\\DIFadd{tamer}}\\DIFadd{~\\mbox{%\n\\cite{knox2009interactively}}\\hspace{0pt}%\n. }\\DIFaddend Hence, \\textsc{apple}'s greedy objective for human feedback still considers the goal of maximizing current and future performance.\n\n\\subsection{Continuous Parameter Policy}\n\n\\DIFdelbegin \\DIFdel{When not having a parameter library to select from}\\DIFdelend \\DIFaddbegin \\DIFadd{If a discrete parameter library is not available or desired}\\DIFaddend , \\textsc{apple} \\DIFdelbegin \\DIFdel{needs to select the optimal parameter set from the continuous parameter space }\\DIFdelend \\DIFaddbegin \\DIFadd{can also be used over continuous parameter spaces }\\DIFaddend (e.g., deciding the max speed from $[0.1, 2]\\ \\mathrm{m\/s}$). In this scenario, we assume the human will provide continuous evaluative feedback, as discrete feedback is not informative enough to learn a continuous policy efficiently.\n\nIn \\DIFdelbegin \\DIFdel{detail}\\DIFdelend \\DIFaddbegin \\DIFadd{this setting}\\DIFaddend , we parameterize the parameter policy $\\pi_\\psi$ and the feedback predictor $F_\\phi$ in the actor-critic style.\nWith the collected evaluative feedback $ \\mathcal{D} := \\{x_j, \\theta_j, e_j\\}_{j=1}^N$, the training objective of $F_\\phi$ is still to minimize the difference between predicted and collected feedback, \\DIFdelbegin \\DIFdel{the same as Eq}\\DIFdelend \\DIFaddbegin \\DIFadd{as specified by Eqn}\\DIFaddend . (\\ref{eq:critic_loss}). For the parameter policy $\\pi_\\psi$, beyond choosing the action that maximizes expected feedback, its training objective is augmented by maximizing the entropy of policy $\\mathcal{H}(\\pi_\\psi(\\cdot|x))$ at state $x$. \\DIFdelbegin \\DIFdel{With }\\DIFdelend \\DIFaddbegin \\DIFadd{Using }\\DIFaddend the same entropy regularization \\DIFdelbegin \\DIFdel{in }\\DIFdelend \\DIFaddbegin \\DIFadd{as }\\DIFaddend Soft Actor Critic \\DIFaddbegin \\DIFadd{(SAC)~}\\DIFaddend \\cite{haarnoja2018soft}, $\\pi_\\psi$ favors more stochastic policies\\DIFdelbegin \\DIFdel{and has }\\DIFdelend \\DIFaddbegin \\DIFadd{, leading to }\\DIFaddend better exploration during training:\n\\begin{equation}\n     \\psi^* = \\argmin_\\psi \\mathop{\\mathbb{E}}_{\\substack{x_j \\in \\mathcal{D} \\\\ \\tilde{\\theta}_j \\sim \\pi_\\psi(\\cdot | x_j)}} \\left[-F_\\phi(x_j, \\tilde{\\theta}_j) + \\alpha \\log \\pi_\\psi(\\tilde{\\theta}_j | x_j)  \\right],\n     \\DIFaddbegin \\label{eqn::continuous}\n\\DIFaddend \\end{equation}\nwhere $\\alpha$ is the temperature controlling the importance of \\DIFaddbegin \\DIFadd{the }\\DIFaddend entropy bonus and is automatically tuned as \\DIFdelbegin \\DIFdel{described in }\\DIFdelend \\DIFaddbegin \\DIFadd{in SAC~}\\DIFaddend \\cite{haarnoja2018soft_application}.\n\n\n\\subsection{Deployment}\nDuring deployment, we measure the state $x_t$, use the parameter policy to obtain a set of parameter $\\theta_t \\sim \\pi_\\psi^*(\\cdot | x_t)$ \\DIFaddbegin \\DIFadd{at each time step }\\DIFaddend and apply it to the navigation planner $G$.\n\n\\section{EXPERIMENTS}\n\\label{sec::experiments}\nIn our experiments, we aim to show that \\textsc{apple} can improve navigation performance by learning from evaluative feedback, in contrast to a teleoperated demonstration or a few corrective interventions, both of which require the non-expert user to take control of the moving robot. We also show \\textsc{apple}'s generalizability to unseen environments. We implement \\textsc{apple} on a ClearPath Jackal ground robot in \\DIFdelbegin \\DIFdel{the BARNdataset}\\DIFdelend \\DIFaddbegin \\DIFadd{BARN}\\DIFaddend ~\\cite{perille2020benchmarking} with 300 navigation environments randomly generated using Cellular Automata\\DIFdelbegin \\DIFdel{and }\\DIFdelend \\DIFaddbegin \\DIFadd{, and in }\\DIFaddend two physical obstacle courses. \n\n\\subsection{Implementation}\n\n\\DIFdelbegin \\DIFdel{Our }\\DIFdelend \\DIFaddbegin \\DIFadd{The }\\DIFaddend Jackal is a differential-drive robot equipped with a Velodyne LiDAR that we use to obtain a 720-dimensional planar laser scan with a 270$^\\circ$ field of view, denoted as $l_t$.\nThe robot uses the Robot Operating System \\texttt{move\\textunderscore base} navigation stack with Dijkstra's global planner and the default \\textsc{dwa} local planner\\DIFaddbegin \\DIFadd{~\\mbox{%\n\\cite{fox1997dynamic}}\\hspace{0pt}%\n}\\DIFaddend .\nFrom the global planner, we query the relative local goal direction $g_t$ (in angle) as \\DIFdelbegin \\DIFdel{a linear fitting }\\DIFdelend \\DIFaddbegin \\DIFadd{the averaged tangential direction }\\DIFaddend of the first 0.5m global path. The state space of the robot is the combination of the laser scan and local goal $x_t = (l_t, g_t)$. The parameter space \\DIFdelbegin \\DIFdel{are }\\DIFdelend \\DIFaddbegin \\DIFadd{consists of the }\\DIFaddend 8 parameters of the \\textsc{dwa} local planner as described in \\DIFdelbegin \\DIFdel{Table }\\DIFdelend \\DIFaddbegin \\DIFadd{Tab. }\\DIFaddend \\ref{tab::jackal_parameters}, and the action space \\DIFdelbegin \\DIFdel{are }\\DIFdelend \\DIFaddbegin \\DIFadd{is }\\DIFaddend the linear and angular velocity of the robot, $a_t = (v_t, \\omega_t)$.\n\nFor discrete \\textsc{apple}, we construct the parameter library shown in \\DIFdelbegin \\DIFdel{Table }\\DIFdelend \\DIFaddbegin \\DIFadd{Tab. }\\DIFaddend \\ref{tab::jackal_parameters} with the default \\textsc{dwa} parameter set $\\theta_1$ and parameter sets $\\theta_{2 \\sim 7}$ learned in \\DIFaddbegin \\DIFadd{the }\\textsc{\\DIFadd{appli}} \\DIFadd{work~}\\DIFaddend \\cite{wang2020appli}. Here we use parameter sets learned in previous work for simplicity. Without prelearned parameter sets, one can \\DIFdelbegin \\DIFdel{still }\\DIFdelend obtain a library \\DIFdelbegin \\DIFdel{by a }\\DIFdelend \\DIFaddbegin \\DIFadd{via }\\DIFaddend coarse tuning to create various driving modes (e.g. increase \\emph{max\\_vel\\_x} to create an aggressive mode). For continuous \\textsc{apple}, the parameter ranges for the parameter policy $\\pi_\\psi$ to select from are listed in the same table.\n\n\\begin{table}[h]\n  \\caption{Parameter Library and Range: \\\\ \\emph{max\\_vel\\_x} \\textnormal{(v)}, \\emph{max\\_vel\\_theta} \\textnormal{(w)}, \\emph{vx\\_samples} \\textnormal{(s)}, \\emph{vtheta\\_samples} \\textnormal{(t)}, \\emph{occdist\\_scale} \\textnormal{(o)}, \\emph{pdist\\_scale} \\textnormal{(p)}, \\emph{gdist\\_scale \\textnormal{(g)}}, \\emph{inflation\\_radius \\textnormal{(i)}}}\n  \\label{tab::jackal_parameters}\n  \\centering\n  \\small\n  \\begin{tabular}{lrrrrrrrr}\n    \\toprule\n                & v & w & s & t & o & p & g & i \\\\\n    \\midrule\n    $\\theta_1$ & 0.50 & 1.57 &  6 & 20 & 0.10 & 0.75 & 1.00 & 0.30 \\\\\n    $\\theta_2$ & 0.26 & 2.00 & 13 & 44 & 0.57 & 0.76 & 0.94 & 0.02 \\\\\n    $\\theta_3$ & 0.22 & 0.87 & 13 & 31 & 0.30 & 0.36 & 0.71 & 0.30 \\\\\n    $\\theta_4$ & 1.91 & 1.70 & 10 & 47 & 0.08 & 0.71 & 0.35 & 0.23 \\\\\n    $\\theta_5$ & 0.72 & 0.73 & 19 & 59 & 0.62 & 1.00 & 0.32 & 0.24 \\\\\n    $\\theta_6$ & 0.37 & 1.33 &  9 &  6 & 0.95 & 0.83 & 0.93 & 0.01 \\\\\n    $\\theta_7$ & 0.31 & 1.05 & 17 & 20 & 0.45 & 0.61 & 0.22 & 0.23 \\\\\n    \\midrule\n    $\\min$ & 0.2 & 0.31 & 4  & 8  & 0.10 & 0.10 & 0.01 & 0.10 \\\\\n    $\\max$ & 2.0 & 3.14 & 20 & 40 & 1.50 & 2.00 & 1.00 & 0.60 \\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\nImplementation-wise, for discrete \\textsc{apple}, $F_\\phi(x, \\theta)$ is a fully-connected neural network with 2 hidden layers of 128 neurons, taking the 721 dimensional $x_t$ as input and \\DIFdelbegin \\DIFdel{outing }\\DIFdelend \\DIFaddbegin \\DIFadd{outputting }\\DIFaddend the 7 predicted feedback \\DIFaddbegin \\DIFadd{signals }\\DIFaddend $\\hat{e}_{t, 1 \\sim 7}$ for $\\theta_{1 \\sim 7}$ respectively. \\DIFdelbegin \\DIFdel{For parameter }\\DIFdelend \\DIFaddbegin \\DIFadd{Parameter }\\DIFaddend policy $\\pi_\\psi$ \\DIFdelbegin \\DIFdel{, it }\\DIFdelend uses $\\epsilon$-greedy exploration with $\\epsilon$ decreasing from $0.3$ to $0.02$ during the first half of the training. For continuous \\textsc{apple}, $F_\\phi(x, \\theta)$ shares the same architecture as discrete \\textsc{apple}, except for different input (concatenation of $x_t$ and $\\theta_t$) and output (scalar $\\hat{e}_t$). The parameter policy $\\pi_\\psi$ also uses the same architecture, mapping $x_t$ to $\\theta_t$.\n\nTo evaluate the performance of \\textsc{apple}, we use \\DIFdelbegin \\textbf{\\textsc{\\DIFdel{appli}}%\n} %\n\\DIFdelend \\DIFaddbegin \\textsc{\\DIFadd{appli}} \\DIFaddend with the same parameter library in \\DIFdelbegin \\DIFdel{Table }\\DIFdelend \\DIFaddbegin \\DIFadd{Tab. }\\DIFaddend \\ref{tab::jackal_parameters} and the \\DIFdelbegin \\textbf{\\DIFdel{Default}} %\n\\DIFdelend \\DIFaddbegin \\DIFadd{Default }\\DIFaddend \\textsc{dwa} planner as two baselines. Since \\DIFdelbegin \\DIFdel{it is pointed out that }\\DIFdelend \\textsc{appld} does not generalize well without a confidence-based context predictor~\\cite{wang2020appli} and \\textsc{appli} can therefore outperform \\textsc{\\DIFdelbegin \\DIFdel{appli}\\DIFdelend \\DIFaddbegin \\DIFadd{appld}\\DIFaddend }, we do not include \\textsc{appld} as one of the baselines. Despite \\DIFaddbegin \\DIFadd{using }\\DIFaddend the same library, \\textsc{appli} chooses the parameter set based on the similarity between \\DIFaddbegin \\DIFadd{the }\\DIFaddend current observation and \\DIFaddbegin \\DIFadd{the }\\DIFaddend demonstrated environments, while discrete \\textsc{apple} uses the expected feedback.\n\n\\subsection{Simulated Experiments}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{contents\/figures\/gazebo.jpg}\n  \\caption{Simulated Environments in BARN Dataset with Low, Medium, and High Difficulty Levels\n  }\n  \\label{fig::simulated_envs}\n\\end{figure}\n\nWe begin \\DIFdelbegin \\DIFdel{with }\\DIFdelend \\DIFaddbegin \\DIFadd{by }\\DIFaddend testing \\textsc{apple} on the BARN dataset, with simulated evaluative feedback generated by an oracle (proxy human). The benchmark dataset consists of 300 simulated navigation environments ranging from easy ones with a lot of open spaces to challenging ones where the robot needs to get through dense obstacles. Navigation trials in three example environments with low, medium, and high difficulty levels are shown in Fig. \\ref{fig::simulated_envs}. We randomly select 250 environments for training \\textsc{apple} and hold the remained 50 environments as the test set.\n\nFor the simulated feedback, we use the projection of the robot's linear velocity along the local goal direction, i.e., $e_t = v_t \\cdot \\cos(g_t)$, and it greedily encourages the robot to move along the global path as fast as possible. Notice, this simulated evaluative feedback is suboptimal as it doesn't consider future states, and we expect actual human evaluative feedback would be more accurate. For example, when the robot is leaving an open space and is about to enter a narrow exit, \\DIFaddbegin \\DIFadd{a }\\DIFaddend human would expect the robot to slow down to get through the exit smoothly, but the simulated oracle still encourages the robot to drive fast. The oracle provides its evaluative feedback at 4Hz, while \\textsc{apple} and \\textsc{appli} dynamically adjust the parameter set for the \\textsc{dwa} planner at the same frequency.\n\nAfter training in 250 environments with a total of 12.5M feedback \\DIFaddbegin \\DIFadd{signals }\\DIFaddend collected, we evaluate \\textsc{apple} with discrete and continuous parameter \\DIFdelbegin \\DIFdel{policy }\\DIFdelend \\DIFaddbegin \\DIFadd{policies }\\DIFaddend (denoted as \\DIFdelbegin \\textbf{\\textsc{\\DIFdel{apple}} %\n\\DIFdel{(disc.)}} %\n\\DIFdel{and }\\textbf{\\textsc{\\DIFdel{apple}} %\n\\DIFdel{(cont.)}}%\n\\DIFdelend \\DIFaddbegin \\textsc{\\DIFadd{apple}} \\DIFadd{(disc.) and }\\textsc{\\DIFadd{apple}} \\DIFadd{(cont.)}\\DIFaddend ), as well as two baselines, on the 50 test environments by measuring the traversal time for 20 runs per environment. We then conduct a pair-wise t-test for all methods in order to compute the percentage of environments in which one method (denoted as Method 1) is significantly worse ($p<0.05$) than another (denoted as Method 2). For better illustration, we order the method by their performance and show the pairwise comparison in Tab. \\ref{tab::similation_results}.\n\n\\begin{table}\n\\vspace{0.25cm}\n\\centering\n\\caption{Percentage of Test Simulation Environments that Method 1 is Significantly Worse than Method 2 in Terms of Traversal Time\\\\\n\\scriptsize (Methods are listed in order of increasing performances. Best method when comparing with Method 1 are bold)}\n\\begin{tabular}{c | c c c c}\n\\hline\n \\diagbox{Method 1}{Method 2} & \\textbf{Default} & \\textbf{\\textsc{appli}} & \\thead{\\textbf{\\textsc{apple}} \\\\ \\textbf{(disc.)}}  & \\thead{\\textbf{\\textsc{apple}} \\\\ \\textbf{(cont.)}}\\\\\n\\hline\n\\textbf{Default}                & 0   & 23   & \\textbf{33}    & \\textbf{33} \\\\\n\\textbf{\\textsc{appli}}         & 6   & 0    & \\textbf{29}    & \\textbf{29} \\\\\n\\textbf{\\textsc{apple} (disc.)}   & 2   & 4    & 0              & \\textbf{27} \\\\\n\\textbf{\\textsc{apple} (cont.)}  & 4   & 4    & 10             & 0 \\\\\n\\hline\n\\end{tabular}\n\\label{tab::similation_results}\n\\end{table}\n\nDespite learning from suboptimal evaluative feedback, \\textsc{apple} (disc.) and \\textsc{apple} (cont.) still outperform the Default \\textsc{dwa} and \\textsc{appli} in $33\\%$ and $29\\%$ of test environments respectively, and they only perform significantly worse in a few cases. \\DIFdelbegin \\DIFdel{This proves }\\DIFdelend \\DIFaddbegin \\DIFadd{These results demonstrate }\\DIFaddend the advantage of \\textsc{apple} over \\textsc{appli}, which \\DIFdelbegin \\DIFdel{is selecting }\\DIFdelend \\DIFaddbegin \\DIFadd{selects }\\DIFaddend the parameter set with a performance-based predictor \\DIFaddbegin \\DIFadd{(Eqns. \\ref{eqn::discrete} and \\ref{eqn::continuous}) }\\DIFaddend rather than a similarity-based predictor. \\DIFdelbegin \\DIFdel{For self-comparison, }\\DIFdelend \\textsc{apple} (cont.) is much better than \\textsc{apple} (disc.), because of its larger model capacity in the parameter space.\n\n\n\\subsection{Physical Experiments}\nWe also apply \\textsc{apple} on a physical Jackal robot. \\DIFdelbegin \\DIFdel{One of the authors follows the robot autonomously navigating }\\DIFdelend \\DIFaddbegin \\DIFadd{In }\\DIFaddend a highly-constrained obstacle course (Fig. \\ref{fig::apple})\\DIFaddbegin \\DIFadd{, we first apply }\\textsc{\\DIFadd{appli}} \\DIFadd{which provides 5 sets of parameters (1 default and 4 learned). Then to train }\\textsc{\\DIFadd{apple}} \\DIFadd{which uses the same parameter library, one of the authors follows the robot autonomously navigating }\\DIFaddend and uses an Xbox joystick to give evaluative feedback at 2Hz. To reduce the burden of giving a large amount of feedback, the user is only requested to give negative feedback by pressing a button on the joystick when he thinks the robot's navigation performance is bad, while for other instances, positive feedback is automatically given.\nIn other words, we interpret the absence of human feedback to be the same as if the human had provided positive feedback.\nWhile this interpretation is not standard in the literature (and even undesirable at times \\cite{faulkner2018policy}), we found that it yielded good results for the application studied here.\n\nThe entire \\DIFaddbegin \\textsc{\\DIFadd{apple}} \\DIFaddend training session lasts roughly 30 minutes, in which the robot navigates 10 trials in the environment shown in Fig. \\ref{fig::apple}. \\textsc{apple} learns in an online fashion with $30\\%$ probability of random exploration. \n\nAfter the training, the learned \\textsc{apple} model is deployed in the same training environment. We compare \\textsc{apple} to \\textsc{appli} with the same \\DIFdelbegin \\DIFdel{four sets of planner }\\DIFdelend \\DIFaddbegin \\DIFadd{sets of }\\DIFaddend parameters and the confidence measure \\DIFaddbegin \\DIFadd{of context prediction}\\DIFaddend ~\\cite{wang2020appli}, and the DWA planner with static default parameters. Each experiment is repeated five times. The results are shown in Tab \\ref{tab::physical}. \\textsc{apple} achieves the fastest average traversal time with \\DIFaddbegin \\DIFadd{the }\\DIFaddend smallest variance in the training environment.\n\nTo test \\textsc{apple}'s generalizability, we also test \\textsc{apple} in an unseen environment (Fig. \\ref{fig::unseen}) and show the results in \\DIFdelbegin \\DIFdel{Table }\\DIFdelend \\DIFaddbegin \\DIFadd{Tab. }\\DIFaddend \\ref{tab::physical}. In the unseen environment, \\textsc{apple} has slightly increased variance, but still has the fastest average traversal time compared to the other two baselines.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{contents\/figures\/unseen.jpg}\n  \\caption{\\textsc{apple} Running in an Unseen Physical Environment}\n  \\label{fig::unseen}\n\\end{figure}\n\n\\begin{table}\n\\centering\n\\caption{Traversal Time in Training and Unseen Environment}\n\\begin{tabular}{cccc}\n\\toprule\n & \\textbf{Default} & \\textbf{\\textsc{appli}} & \\textbf{\\textsc{apple} (disc.)} \\\\ \n\\midrule\n\\textbf{Training} & 143.1$\\pm$20.0s & 79.8$\\pm$8.1s & 75.2$\\pm$4.1s \\\\\n\\textbf{Unseen} & 150.5$\\pm$24.0s & 86.4$\\pm$1.1s & 83.9$\\pm$4.6s \\\\\n\\bottomrule\n\\end{tabular}\n\\label{tab::physical}\n\\end{table}\n\n\\section{CONCLUSIONS}\n\\label{sec::conclusions}\n\nIn this work, we introduce \\textsc{apple}, \\emph{Adaptive Planner Parameter Learning from Evaluative Feedback}. In contrast to most existing end-to-end machine learning for navigation approaches, \\textsc{apple} utilizes existing classical navigation systems and inherits all their benefits, such as safety and explainability. Furthermore, instead of requiring a full expert demonstration or a few corrective interventions that \\DIFdelbegin \\DIFdel{needs the users }\\DIFdelend \\DIFaddbegin \\DIFadd{need the user }\\DIFaddend to take full control of the robot, \\textsc{apple} just needs evaluative feedback as simple as ``good job\" or ``bad job\" that can be easily collected from non-expert users. Moreover, comparing with \\textsc{appli} which \\DIFdelbegin \\DIFdel{select }\\DIFdelend \\DIFaddbegin \\DIFadd{selects }\\DIFaddend the parameter set based on the similarity with demonstrated environments, \\textsc{apple} \\DIFdelbegin \\DIFdel{achieve }\\DIFdelend \\DIFaddbegin \\DIFadd{achieves }\\DIFaddend better generalization by selecting the parameter set with a performance-based criterion, i.e., the expected evaluative feedback. We show \\textsc{apple}'s performance improvement with simulated and real human feedback, as well as its generalizability in both 50 unseen simulated environments and an unseen physical environment. While \\DIFdelbegin \\DIFdel{we can }\\DIFdelend \\DIFaddbegin \\DIFadd{in this paper we }\\DIFaddend only learn a continuous parameter policy from continuous evaluative feedback\\DIFdelbegin \\DIFdel{now}\\DIFdelend , an interesting direction for future work is to learn \\DIFdelbegin \\DIFdel{it }\\DIFdelend \\DIFaddbegin \\DIFadd{one }\\DIFaddend from discrete feedback, which can be augmented with instructive guidance (e.g. ``bad job\" augmented with ``should increase the velocity\"). \\DIFaddbegin \\DIFadd{In this paper, we use relatively dense feedback signals from the human user in the physical experiments and much denser simulated feedback signals in the simulated experiments to reduce the amount of time needed to train a good }\\textsc{\\DIFadd{apple}} \\DIFadd{policy. These dense feedback signals may not always be practical, for example, the user may not always be paying attention. Therefore another important direction for future investigation is to study how little feedback is needed to yield good performance.\n}\n\n\\DIFaddend \n\n\n\n\\section*{\\DIFdelbegin \\DIFdel{ACKNOWLEDGMENT}\\DIFdelend \\DIFaddbegin \\DIFadd{ACKNOWLEDGMENTS}\\DIFaddend }\nThis work has taken place in the Learning Agents Research\nGroup (LARG) at the Artificial Intelligence Laboratory, The University\nof Texas at Austin.  LARG research is supported in part by grants from\nthe National Science Foundation (CPS-1739964, IIS-1724157,\nNRI-1925082), the Office of Naval Research (N00014-18-2243), Future of\nLife Institute (RFP2-000), Army Research Office (W911NF-19-2-0333),\nDARPA, Lockheed Martin, General Motors, and Bosch.  The views and\nconclusions contained in this document are those of the authors alone.\nPeter Stone serves as the Executive Director of Sony AI America and\nreceives financial compensation for this work.  The terms of this\narrangement have been reviewed and approved by the University of Texas\nat Austin in accordance with its policy on objectivity in research.\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA central task in NLP is the generation of word embeddings to encode the meaning of words and their relationships in a vector space. This task is usually performed by a self-supervised Machine Learning (ML) model such as Word2Vec~\\cite{w2v}, ELMo~\\cite{elmo} or BERT~\\cite{bert}, with access to a large corpus of documents as input. These representations can then be used to perform advanced analytics on textual data. The larger and more complete the corpus is, the more accurate the representations will be.\n\nAs such, it can be useful for multiple organizations to collaborate with each other, each providing access to their corpora, in order to obtain the best results. However, different organizations typically cannot easily share their data, as they have to protect the privacy of their users and the details of their internal operations, or might be bound by external laws preventing the sharing of the data. One way organizations could overcome these issues is by employing Federated Learning protocol~\\cite{fl} to generate a global model without sharing any data. It is therefore fundamental to assess the performance, quality and privacy preservation characteristics of such approach.\n\n\\subsection{Distributed vs data-private massively-distributed approaches}\n\nDatacenter-scale ML algorithms work with vast amounts of data allowing the training of large models, exploiting multiple machines and multiple GPUs per machine. Currently, GPUs offer enough power to satisfy the needs of state of the art models. However, this exposes another important aspect for consideration - Data Privacy. In recent years, users and governments have started to be aware of this issue, publishing new and stricter regulations, such as the GDPR~\\cite{EUdataregulations2018}. Companies started making efforts to shield themselves from any security leak that could happen in a centralised system. This created a need to move the research towards distributed architectures where the data is not gathered by a central entity. \n\nThe fast development of smart devices, the growth of their computational power and of fast Internet connections, like 4G and 5G, enable new approaches that exploit them to train distributed models. This solution is currently not at the same scale of resources that a datacenter can offer yet, but the research and development of edge devices is making it feasible.\n\nFor these reasons, researchers are exploring the possibilities of different massively-distributed training designs. These new designs should offer scalability, ensure data privacy and reduce large traffic of data over the network. The main massively-distributed approach to large-scale training that fulfills all  these requirements is Federated Learning \\cite{fl}.\n\n\\subsection{Federated Learning in a small collaborative NLP scenario}\n\nFederated learning is frequently applied in a commercial, global-scale setting as an on-device learning framework. In this common scenario, many small devices, like smartphones, collaborate and grant access to their data, typically small due to storage limitations, to train a higher-quality model. These models are usually NLP applications to make suggestions \\cite{fl_1} or predict a user behaviour \\cite{fl_2}. \n\nHowever, Federated Learning could be applied in different contexts. This paper shifts the focus to address a new scenario where the users are a small number of large organizations with access to large corpora, which cannot be shared or centralised. These organizations are willing to cooperate in order to have access to a larger corpus with diverse topics and to overcome the very strict data privacy policies. A practical example could be a group of government agencies, each of which has only access to sensitive documents in a specific domain (e.g. taxes), which alone would not be sufficient for high-quality training.\n\n\\section{Federated Word2Vec}\n\nFederated Learning addresses the privacy concerns as the data is not shared between the organizations. It stays in the node of the owner and the information transferred through the network is only the gradient of the model. It also avoids the expensive transfer of the training data, replacing it with the repeated transfer of gradients, the total size of which is, generally, less than that of the dataset. And even if repeated gradient exchanges were to surpass the size of the dataset, their transfer would be spread on a long time and divided in smaller batches, so it would not delay the training as much as having to send a huge dataset before starting. The only common point that all the nodes share in this architecture is the existence of a central node which oversees the training process, directing the data transfers and merging the contributions of each node. Having the central node can facilitate the inclusion of additional safety measures to shield the training process from malicious attacks \\cite{secure_fl}. \n\nIn Federated Word2Vec, each organization owns a private dataset, which means that words that appear in the corpus of one organization may not be present in the one of another. This an issue as the input vocabulary must be common to all local models, so that the gradients can be aggregated. Preserving the privacy of the content of the text is very important, so this paper will overcome the aforementioned issue through a strategy that consists of a common agreement of all the participants in a global vocabulary. All must agree on a fixed vocabulary size $N$ and a minimum threshold of occurrences $T$. Each participant must provide a list of their top $N$ words that appear in their respective texts surpassing $T$ occurrences. The privacy is preserved as the organizations only share a set of isolated, unsorted words. However, the question arises of how to merge these sets. There are two operations that can be applied to produce the final vocabulary: intersection and union. We use, and recommend to use, the union operation. The final vocabulary is larger than the initial size $N$, but all organizations keep their relevant words. Although this approach requires more time to converge due to many words appearing only in certain datasets, the words meaning and knowledge return to the participants is enriched.\n\nOnce the participants receive the common agreed vocabulary, the training process can start following the FederatedSGD algorithm from \\cite{fl}. The gradient is transferred from all the external nodes to the main node in each iteration. The average gradient is calculated and transferred back to perform the updating process.\n\n\\section{Experiments}\n\n\\subsection{Data collection}\n\nWe generated topic-related datasets collected from Wikipedia articles and organised in different sizes. Two Wikipedia dumps were downloaded to satisfy the aforementioned datasets characteristics: a partial dump with a compressed size of 180 MB to simulate small organizations with short corpora; and a 16 GB compressed file with all the text content published in Wikipedia. From the whole Wikipedia dump, the Wikipedia Extractor script \\cite{extractor} is used with a tag to filter categories and prepare 5 different datasets divided by topic. The chosen topics are: biology, history, finance, geography, and sports. Although the themes are quite specific, some articles can appear in more than one dataset because of the distribution of the Wikipedia tree of categories. So, if an article is tagged with the biology category, it is included in the dataset of biological content. In order to simulate a larger number of organizations, every topic is split between two organizations.\n\n\\subsection{Setup}\n\nThe simulation is performed sequentially on a single machine and with a single GPU, a Nvidia Quadro RTX 5000. This limits the possibility to study the influence of the network, that is thus not covered in this work. The hyperparameters were set to sensible values based on existing literature and should provide a good compromise of training speed and quality. We use a \\textbf{fixed batch size} of 2,048 samples. It is important to notice that one iteration of Federated Word2Vec processes as much data as $N$ iterations of centralized Word2Vec, because each of the $N$ nodes processes one batch in parallel during each iteration, in our study $N$ is equal to 10 nodes. The \\textbf{embeddings size} is fixed to 200. The number of \\textbf{negative samples} per batch is 64, a small amount of negative samples, compared to the total batch size, but sufficient to achieve good results \\cite{w2v}. The \\textbf{vocabulary size} is 200,000 unique words, with a minimum threshold of 10 occurrences.\n\n\\section{Results}\n\n\\subsection{Proving convergence of the model with small datasets}\n\nFigure \\ref{fig:val_base} compares the validation loss of Federated Word2Vec with that of a baseline, centralized implementation. In order to compare the two models when both have processed the same amount of data, Federated Word2Vec is stopped at epoch 70, which is iteration 500,000.\n\n\\begin{figure}[!ht]\n  \\begin{center}\n    \\includegraphics[width=1\\linewidth]{base_together_scale.png}\n  \\end{center}\n  \\caption{On the left, validation loss per epoch in a full execution of Word2Vec. On the right, validation loss per epoch of the first 500.000 iterations of Federated Word2Vec. The red lanes represent the average of the validation loss calculated by aggregating all previous values from each epoch. Y-axis is in logarithmic scale.}\n  \\label{fig:val_base}\n\\end{figure}\n\n\\begin{figure}[!ht]\n  \\begin{center}\n    \\includegraphics[width=1\\linewidth]{bse_iterations.JPG}\n  \\end{center}\n  \\caption{On the left, validation loss per iteration in a full execution of Word2Vec. On the right, validation loss per iterarions in a full execution of Federated Word2Vec. The red lanes represent the average of the validation loss calculated by aggregating all previous values from each epoch. Y-axis is in logarithmic scale.}\n  \\label{fig:val_base_1}\n\\end{figure}\n\nThe loss of Word2Vec presents a value of ${\\small\\sim} 10^{4}$. It is stable, but with a small descending trend. On the other hand, although Federated Word2Vec does not reach the same loss (it is 10 times greater) its trend is clearly decreasing. To check if the trend continues, Figure \\ref{fig:val_base_1} illustrates the validation loss in terms of iteration for the full execution, 2 million iterations. The loss keeps going down until 1 million iterations when it stabilises. Overall, the two models provide very similar results. Centralized Word2Vec has a faster initial convergence; however, this might be overcome by adapting the hyperparameters to the distributed setting, for example with learning rate scaling \\cite{learning_rate_scaling}.\n\n\\subsection{Proving convergence of the model with large datasets}\n\nThe training of Federated Word2Vec with a large dataset presents improved results compared to the previous graphs. Figure \\ref{fig:large_loss} freezes the training in the iteration 500,000 as it was done in Figure \\ref{fig:val_base}. The number of epochs is fewer than in the former experiment but the loss presents a clear downward trend with a steeper slope. In Figure \\ref{fig:last}, where the execution continues until iteration 2 millions, the loss keeps decreasing reaching values of $10^{3}$, something that did not happen in Figure \\ref{fig:val_base_1}.\n\n\\begin{figure}[!h]\n  \\begin{center}\n    \\includegraphics[scale=0.49]{validation_loss_balanced_vs_baseline.JPG}\n  \\end{center}\n  \\caption{Validation loss of the first 500.000 iterations of Federated Word2Vec with a larger dataset, divided by epoch. The red lanes represent the average of the validation loss calculated by aggregating all previous values from each epoch. Y-axis is in logarithmic scale.}\n  \\label{fig:large_loss}\n\\end{figure}\n\n\n\\begin{figure}[!h]\n  \\begin{center}\n    \\includegraphics[scale=0.49]{validation_loss_balanced_all_iterations.JPG}\n  \\end{center}\n  \\caption{Validation loss of a full execution of Federated Word2Vec with a larger dataset, represented in blue. The red lanes represent the average of the validation loss calculated by aggregating all previous values from each epoch. Y-axis is in logarithmic scale.}\n  \\label{fig:last}\n\\end{figure}\n\nConsequently, Federated Word2Vec seems to work better with larger datasets as it benefits from learning from multiple sources at the same time. The results show that Federated Word2Vec is not better, and might perform slightly worse, than Word2Vec under the same settings. However, it is proven that Federated Word2Vec has a similar convergence pattern to Word2Vec and easily scales to a large dataset. \n\n\\subsection{How categorised data influence the results}\n\nWe then compare collaborative training with Federated Word2Vec to local training by a single organization, which only has access to the \\textit{finance} dataset. We analyse the organization of the words in the embedding space, using their cosine distance, by identifying the top-5 closest neighbours for a number of target words, as shown in Table \\ref{font-table}.\n\nThe most striking finding in this analysis is that clusters are populated with more meaningful words in Federated Word2Vec. This behaviour was expected for the target word \\textit{bacteria}, as it does not appear frequently in the \\textit{finance} dataset. However, the same situation happens with \\textit{market}, presenting meaningless words as the closest neighbours in its community, while the execution of Federated Word2Vec shows more specific context words. \n\nMoreover, \\textit{market} is not an outlier. Most words that are relevant to the \\textit{finance} dataset present similar results. The resultant neighbourhood of the word \\textit{money} trained with baseline Word2Vec on the financial dataset still presents generic words such as \\textit{\\{stated, said, there\\}}. In contrast, the community generated during the federated training clearly gathers meaning from the finance topic.\n\nThese results show the importance of having a full picture of the language to produce high-quality embeddings, even for domain-specific tasks. This, in turn, underscores the need for collaboration among organizations.\n\n\\begin{table}\n\\centering\n\\small\n\\begin{tabular}{l l l l l}\n\n\\hline \\textbf{Word} & \\multicolumn{2}{c}{\\textbf{W2V}} & \\multicolumn{2}{c}{\\textbf{Fed W2V}}\\\\ \\hline\n\\hline\n\n\\textbf{Market} & \\textbf{Top-5} & \\textbf{Dist} & \\textbf{Top-5} & \\textbf{Dist} \\\\ \\hline\n& this & 0.023 & markets & 0.029 \\\\\n& proposed & 0.024 & company & 0.035 \\\\\n& some & 0.024 & share & 0.042 \\\\\n& all & 0.025 & trading & 0.048 \\\\\n& other & 0.025 & assets & 0.049 \\\\ \\hline \\hline\n\n\\textbf{Bacteria} & \\textbf{Top-5} & \\textbf{Dist} & \\textbf{Top-5} & \\textbf{Dist} \\\\ \\hline\n\n& rare & 0.026 & organism & 0.070 \\\\\n& animals & 0.026 & toxic & 0.075 \\\\\n& applied & 0.026 & tissue & 0.077 \\\\\n& result & 0.027 & cells & 0.081 \\\\\n& plants & 0.027 & humans & 0.083  \\\\ \\hline \\hline\n\n\\textbf{Money} & \\textbf{Top-5} & \\textbf{Dist} & \\textbf{Top-5} & \\textbf{Dist} \\\\ \\hline\n& stated & 0.028 & paid & 0.045 \\\\\n& said & 0.028 & offer & 0.053 \\\\\n& there & 0.028 & sell & 0.062 \\\\\n& take & 0.029 & cash &  0.071 \\\\\n& help & 0.031 & interest & 0.073 \\\\\n\n\\end{tabular}\n\\caption{\\label{font-table} Top-5 nearest neighbours of each central word, using the cosine distance in the training of W2V with finance dataset and Fed W2V with all 5 datasets.}\n\\end{table}\n\n\\section{Conclusions}\n\nThe purpose of this paper was to implement and test the viability of a distributed, efficient, data-private approach that allows a small number of organizations, each owning a large private text corpus, to train global word representations. The results indicate the potential for applicability to real scenarios of collaborative training. The main contributions of this work are \n\\begin{enumerate*}\n\\item the viability of training NLP models like Word2Vec under the Federated Learning protocol with convergence times, at least, at the same level of the widely tested Word2Vec;\n\\item the importance for organizations to cooperate, as cooperation provides models that are not only globally good, but also locally better than locally-trained models; and\n\\item the quality of vector representations is not affected by the size of the corpora.\n\\end{enumerate*}\n\n\\noindent\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\uppercase{Introduction}}\n\\label{sec:introduction}\n\t\\IEEEPARstart{P}{oint} cloud registration is the problem of estimating the rigid relative pose transformation that aligns a pair of point clouds into the same coordinate system.\n\tThis is a key problem in many downstream applications including 3D scene reconstruction \\cite{li20183d}, localisation \\cite{Lu_2019_CVPR} and SLAM \\cite{ramezani2020online}.\n\tRecent applications such as Augmented Reality (AR) \\cite{gao2017stable}, cooperative (multi-agent) perception for autonomous vehicles \\cite{arnold2020cooperative} and multi-agent SLAM \\cite{dube2017online} introduce new challenges to this problem.\n\tSpecifically, these applications require registration methods that are robust to point clouds with low overlap, \\textit{e.g.} when sensors are far apart, and capable of operating in real-time.\n\t\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\input{images\/qualitative\/qualitative.pgf}\n\t\t\\caption{Qualitative results of the proposed method. Each row represents a different sample from the CODD test set and the vertical label indicates the relative translation between point clouds in meters, where $t^g$ indicates the ground-truth relative translation vector.}\n\t\t\\label{fig:qualitative}\n\t\\end{figure}\n\t\n\tExisting registration methods are often designed and evaluated assuming a significant overlap between the input point clouds.\n\tThis assumption is valid for applications such as SLAM \\cite{ramezani2020online} and lidar odometry \\cite{graeter2018limo}, where pairs of point clouds are obtained sequentially in adjacent time steps by a single vehicle navigating in a driving environment.\n\tOn the other hand, applications such as cooperative perception \\cite{arnold2020cooperative} and multi-agent SLAM \\cite{dube2017online} require registering point clouds obtained simultaneously from a pair of sensors on two different vehicles that are potentially far apart, and thus, may have low field-of-view overlap, \\textit{e.g.} Figure \\ref{fig:qualitative}.\n\tAs the relative translation between the sensors increases, the number of identifiable correspondences decreases, which poses challenges in registering the point clouds accurately.\n\t\n\tThe majority of existing point cloud registration methods cannot guarantee real-time execution.\n\tTraditional local registration methods such as Iterative Closest Point (ICP) \\cite{arun1987least} solve the problem iteratively assuming an initial relative pose.\n\tHowever, the iterative nature of such methods renders them unfeasible for real-time applications, particularly considering large scale point clouds.\n\tThese methods are also prone to non-optimal solutions when the initial pose estimate is poor, which may be addressed with global-optimisation variants \\cite{olsson2008branch,yang2013go} at the cost of higher computational complexity.\n\tAnother category of methods identify correspondences between point clouds using a distance metric between hand-engineered features \\cite{rusu2009fpfh} or learned point-wise features \\cite{choy2019fully}.\n\tThese correspondences are often contaminated by a large number outliers and must be filtered using RANSAC \\cite{fischler1981random,bustos2017guaranteed} or learned models \\cite{choy2020deep}, which further increases the registration execution time.\n\tFurthermore, state-of-the-art learning-based models \\cite{choy2019fully,choy2020deep} require computationally demanding 3D convolutions and generate numerous putative correspondences, introducing a bottleneck on the RANSAC loop and rendering real-time execution unfeasible.\n\n\tTo mitigate the aforementioned limitations, we propose a novel point cloud registration method capable of operating in real-time and robust to low-overlapping point clouds.\n\tThe proposed method identifies correspondences between the source and target point clouds by learning point-wise features.\n\tA novel encoder hierarchically subsamples the point clouds to reduce the number of key points and improve the run-time performance.\n\tThe resulting features are refined using self- and cross-attention based on a graph neural network.\n\tThe attention network leverages geometrical relationships between key points and their features across to improve the correspondence accuracy, particularly in regions of low overlap.\n\tThe relative pose parameters are obtained by fitting the learned correspondences using Random Sample Consensus (RANSAC) to robustly reject outliers.\t\n\tDuring inference, the RANSAC fitting is done efficiently considering a small number of correspondences, which allows end-to-end inference times below 410ms.\n\tThe model is trained and evaluated separately on the KITTI odometry dataset and a novel Cooperative Driving Dataset (CODD).\n\tThe relative translation between sensors in CODD ranges up to 30m, introducing challenging pairs of point clouds with low overlap, which we hope will create a new research benchmark.\n\tOur contributions are summarised as:\n\t\\begin{itemize}\n\t\t\\item A computationally efficient point-wise feature encoder that allows identifying correspondences between point clouds;\n\t\t\\item A graph neural network that provides self- and cross-attention between point clouds and improves the quality of correspondences;\n\t\t\\item A novel registration method for point clouds that is robust to partially-overlapping point clouds and capable of operating in real-time;\n\t\t\\item A new synthetic lidar dataset containing low overlapping point clouds in a wide range of driving scenarios;\n\t\\end{itemize}\n\n\n\\section{\\uppercase{Related Works}}\n\\label{sec:relatedwork}\n\tThis section reviews existing point cloud registration methods in the literature and highlights how the method proposed in this paper differs from these existing works.\n\tWe divide existing methods in the literature into two categories: traditional registration methods and learning-based methods.\n\n\t\\subsection{Traditional Registration Methods}\n\t\tIterative Closest Point (ICP) \\cite{arun1987least} is a local registration method that assumes an initial relative pose and iteratively computes the transformation parameters that minimise the distance between each point in the source point cloud and its closest neighbour in the target point cloud.\n\t\tThis method is highly sensitive to the initial pose estimate, and converges to non-optimal local-minima results when the initial pose estimate is poor.\n\t\tTo mitigate this, \\cite{olsson2008branch,yang2013go} estimate global optimum solutions for ICP considering branch-and-bound search over the transformation space.\n\t\tHowever, such global methods have significantly higher execution times, which prevents their usage in real-time applications.\n\t\t\t\t\n\t\tOther traditional approaches use handcrafted features to find correspondences between the point clouds.\n\t\tFast Point Feature Histogram (FPFH) \\cite{rusu2009fpfh} encodes the local geometry of 3D points using multi-dimensional feature vectors.\n\t\tBut the correspondences obtained by comparing FPFH features are often contaminated by a large number of outliers, which prevents accurate registration.\n\t\tFor this reason, RANSAC methods are used to filter out the outlier correspondences.\n\t\tMore recently, TEASER \\cite{Yang20tro-teaser} reformulates the registration problem using a truncated least-squares cost, which results in improved registration accuracy compared to RANSAC when considering a high number of outliers.\n\n\t\\subsection{Learning-based Methods}\n\t\tOne group of learning-based registration methods focus on learning accurate correspondences between the point clouds.\n\t\tGenerally, these methods learn a mapping from the original Euclidean space to a latent feature space and optimise the mapping such that corresponding points have a small distance in the latent space.\n\t\tDeng \\textit{et al.} \\cite{deng2018ppfnet} uses a PointNet \\cite{qi2017pointnet} model to learn point-wise features and trains the model using an $N$-tuple loss.\n\t\tVCR-Net \\cite{wei2020endtoend} learns point-wise feature vectors using multi-layer-perceptrons (MLPs) to extract local features, which are refined using global attention and used to identify correspondences between point clouds.\n\t\tIn contrast, \\cite{choy2019fully} uses sparse fully convolutional networks to obtain voxel-wise features and trains the model using variations of triplet loss with hard negative mining.\n\t\tThe resulting correspondences are often contaminated with outliers and need to be pruned using RANSAC \\cite{fischler1981random} or further learning-based filtering \\cite{choy2020deep} methods before estimating the pose transformation parameters.\n\t\t\n\t\tAnother group of methods solve the problem end-to-end by learning the relative pose transformation directly.\n\t\tFor example, PCRNet \\cite{vsarode2019pcrnet} uses a PointNet \\cite{qi2017pointnet} model to encode a global feature vector for both source and target point clouds and directly regress the transformation parameters.\n\t\tDeepVCP \\cite{lu2019deepvcp} uses PointNet++ \\cite{qi2017pointnetpp} to create point-wise features, then selects top K salient points using a learned weighting network to generate a deep feature embedding.\n\t\tThe embeddings are fed to a 3D CNN to obtain soft-correspondences which are finally used to compute the transformation parameters in closed form.\n\t\t\n\t\tThe proposed method differs from previous works in the following ways.\n\t\tFirst, our method considers a novel and computationally efficient point-wise feature encoder based on Set Abstraction (SA) and Feature Propagation (FP) layers \\cite{qi2017pointnetpp}.\n\t\tWhile previous works \\cite{lu2019deepvcp} have used PointNet++ feature encoders, we distinguish our encoder by adopting an architecture that hierarchically subsamples points at each layer, resulting in improved computational performance.\n\t\tSecondly, we improve the quality of the correspondences using a novel graph-based attention network that allows to efficiently combine self- and cross- information across point clouds.\n\t\tDifferently from the feature-based attention in \\cite{wei2020endtoend}, the proposed graph attention leverages both spatial and feature dimensions of local neighbourhoods to refine point-wise features.\n\t\tFinally, we extend our analysis beyond existing datasets and evaluate our model performance in challenging low overlapping point clouds using a novel dataset where the translation between sensor poses vary uniformly up to 30 meters.\n\n\\section{\\uppercase{Problem Formulation}}\n\\label{sec:problem}\n\tGiven two input point clouds $P_X  \\subset \\R^3$ and $P_Y \\subset \\R^3$, the registration problem is to estimate the rigid relative pose transformation that aligns $P_X$ into the coordinate system of $P_Y$.\n\tThis transformation is parametrised by a rotation matrix $R \\in SO(3)$ and a translation vector $t \\in \\R^3$.\n\tThe problem can be solved by identifying pairs of correspondences between $P_X$ and $P_Y$.\n\tGiven a set of correspondences, $X = \\{x_1,\\dots,x_N \\} \\subset P_X, Y = \\{y_1,\\dots,y_N \\} \\subset P_Y$, where $(x_i, y_i), i=1,\\dots,N$ are correspondence pairs, the transformation parameters are obtained by the minimisation of the least-squares error:\n\t\\begin{equation}\n\t\t\\label{eq:ls-error}\n\t\tE(R,t) = \\frac{1}{N} \\sum_{i=1}^N \\left\\| Rx_i + t - y_i \\right\\|^2.\n\t\\end{equation}\n\tThis error is a form of the Orthogonal Procrustes problem \\cite{1996weightedProcrustes} and admits the closed-form solution described below.\n\tFirst, the centroids are computed as\n\t\\begin{equation}\n\t\t\\bar{x} = \\frac{1}{N}\\sum_{i=1}^{N} x_i,\\quad\\quad  \\bar{y} = \\frac{1}{N}\\sum_{i=1}^{N} y_i,\n\t\\end{equation}\n\tand the covariance matrix, denoted by $H$, is obtained using\n\t\\begin{equation}\n\t\tH = \\sum_{i=1}^{N} (x_i - \\bar{x})(y_i - \\bar{y})^\\T.\n\t\\end{equation}\n\tFinally, the rotation matrix and translation vector $R,t$ that minimise Eq. \\ref{eq:ls-error} are computed in closed-form as\n\t\\begin{align}\n\t\t\\label{eq:procrustes-closed-form}\n\t\tR &= V \\left[\\begin{smallmatrix} 1 & & \\\\ & 1 &  \\\\ & & \\det(V^\\T U) \\end{smallmatrix}\\right] U^\\T, \\\\\n\t\tt &= -R\\bar{x} + \\bar{y}, \\nonumber\n\t\\end{align}\n\tconsidering the Singular Value Decomposition (SVD) $H = U S V^T$.\n\tThe next section proposes a novel method to efficiently obtain correspondences between pairs of point clouds.\n\t\n\\section{\\uppercase{Proposed Method}}\n\\label{sec:method}\n\tThis section presents a novel method for robust point cloud registration using learned correspondences targetting efficient, real-time inference.\n\tFigure \\ref{fig:diagram} describes the components and data flow of the proposed method.\n\tThe proposed method can be summarised as follows:\n\t\\begin{enumerate}[label=(\\Alph*)]\n\t\t\\item Both point clouds are fed to an encoder to obtain a subset of key points and associated point-wise features.\n\t\t\\item A graph neural network refines the point-wise features considering self- and cross-attention.\n\t\t\\item The resulting features are used to identify correspondences between the source and target key points.\n\t\t\\item The relative transformation parameters $R,t$ are robustly estimated using a RANSAC formulation of the problem defined in Section \\ref{sec:problem}. \n\t\\end{enumerate}\n\tThe aforementioned components and the training process are described in the following subsections.\n\t\n\t\\begin{figure*}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{images\/diagram.pdf}\n\t\t\\caption{Pipeline and data flow of the proposed point cloud registration method.}\n\t\t\\label{fig:diagram}\n\t\\end{figure*}\n\n\t\\subsection{Point-wise Feature Encoder}\n\t\tThe encoder is a core component of the pipeline, as it computes point-wise features that will be used to identify correspondences.\n\t\tWe propose a novel and computationally efficient encoder network based on Set Abstraction (SA) and Feature Propagation (FP) layers \\cite{qi2017pointnetpp}.\n\t\tThe proposed encoder architecture, including the hyper-parameters of each layer, is depicted in Figure \\ref{fig:encoder}.\n\t\tThe encoder outputs subset of sampled coordinates (key points) from the source and target point clouds, denoted by $X$ and $Y$, and their respective feature vectors, $f_X$ and $f_Y$.\n\t\tThe input to the encoder consists of 3D point coordinates and corresponding features, \\textit{e.g.} lidar return intensity.\n\t\tNote that the input features are optional, but in this work they consist of a single scalar per point representing the lidar intensity.\n\t\tThe source and target point clouds are fed to the encoder independently.\n\t\t\n\t\tThe first four encoder layers are SA layers.\n\t\tA SA layer consists of four operations:\n\t\t\\begin{enumerate}\n\t\t\t\\item $n$ coordinates are sampled from the previous layer using Farthest Point Sampling (FPS) \\cite{moenning2003fps}.\n\t\t\t\\item A local neighbourhood of each sampled coordinate is established by selecting all points within radius $r$ of the respective coordinate.\n\t\t\t\\item The features of the points in each neighbourhood are fed to a shared Multi Layer Perceptron (MLP), denoted as a list $L$ containing the number of intermediate nodes per layer.\n\t\t\t\\item The resulting feature vectors of the $n$ sampled coordinates is computed using an aggregation function (\\textit{max-pooling}) over the MLP output of the points in the respective neighbourhoods.\n\t\t\\end{enumerate}\n\t\tEach SA layer hierarchically subsamples and aggregates information from the previous layer with progressively larger receptive volumes, which is a fundamental step in reducing the computational cost of our pipeline.\n\t\tAt the same time, it is also important not to discard valuable information, \\textit{i.e.} prioritising that points from one layer are within the neighbourhood of sampled points in the next layer.\n\t\tThis trade-off is achieved by tuning the layers' hyper-parameters, namely $n,r,L$, such that the sampled points' neighbourhood include most points from the previous layer.\n\t\tThese hyper-parameters were optimised for large outdoor driving environments and would require fine-tuning for indoor scenarios.\n\t\t\n\t\tThe last encoder layer is an FP layer.\n\t\tIt propagates high level information from SA4 to the points in the previous layer (SA3) as illustrated in Figure \\ref{fig:encoder}.\n\t\tThis is achieved by interpolating the feature vectors in SA3 layer using the features from the three nearest-neighbour coordinates in SA4.\n\t\tThe final features are obtained fusing the original SA3 features with the interpolated SA4 features using a shared MLP, represented by a list $L$ of intermediate nodes.\n\t\tMore details about the interpolation can be found in \\cite{qi2017pointnetpp}.\n\t\t\t\n\t\t\\begin{figure}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\linewidth]{images\/encoder.pdf}\n\t\t\t\\caption{Point-wise feature encoder model architecture. The model consists of four Set-Abstraction (SA) layers and one Feature Propagation (FP) layer. Point coordinates and features are represented in yellow and green, respectively. The brackets indicate the dimensionality of each matrix. The number of input points, denoted by $M$, is arbitrary, and the model consistently outputs $N=512$ point coordinates and their respective feature vectors with $D=128$ dimensions.}\n\t\t\t\\label{fig:encoder}\n\t\t\\end{figure}\n\t\t\n\t\\subsection{Graph-based Attention}\n\t\tThe feature vectors obtained with the encoder network represent local point cloud information.\n\t\tHowever, these features are agnostic to the global context of the point cloud.\n\t\tFor example, if a point cloud contains multiple objects, \\textit{e.g.} trees, it would be difficult to distinguish between individual trees.\n\t\tAnother problem, most critical for low overlapping point clouds, is that regions of overlap generally have different point densities in each point cloud, challenging the correct identification of correspondences since a point's features change with the density of points in its neighbourhood.\n\t\tTo mitigate both problems, we propose a graph-based attention module which transforms points' feature vectors considering the wider point cloud context (self-attention) and the context of both source and target point-clouds (cross-attention).\n\t\tSelf-attention increases the distinctiveness of key points by attending to their surrounding context.\n\t\tThe cross-attention layer learns to refine point-wise features by attending to the most similar features across point clouds. \n\t\tBoth layers, illustrated in Figure \\ref{fig:graphatt}, increase the likelihood of finding accurate correspondences, even in cases of low overlap.\n\t\t\n\t\tThe self-attention layer introduces attention between points within the same point cloud.\n\t\tA graph connecting the points (nodes) is created for each point cloud using the $k$-Nearest-Neighbours of the points' spatial coordinates.\n\t\tLet $f_i$ be a feature vector from either $f_X$ or $f_Y$. \n\t\tThe self-attention layer computes a residual term for $f_i$ using the Crystal Graph Convolution Operation \\cite{xie2018cgconv}:\n\t\t\\begin{equation}\n\t\t\t\\label{eq:cgconv}\n\t\t\t\\hat{f}_i = f_i + \\max_{j \\in \\mathcal{N}(i)} \\sigma(z_{i,j} W_f) \\odot \\text{Softplus}(z_{i,j} W_s),\n\t\t\\end{equation}\n\t\twhere $\\mathcal{N}(i)$ indicates the $k$ neighbour nodes of $i$, $z_{i,j} = [f_i, f_j] $ is the aggregated features of nodes $i,j$.\n\t\t$\\sigma(\\cdot)$ indicates the sigmoid function, and the \\textit{Softplus} function is defined as $\\text{Softplus}(z) = \\log(1+e^z)$.\n\t\tThe attention matrices $W_f, W_s$ are parameters to be learned and the operation $\\odot$ represents element-wise multiplication.\n\t\tWe adopt the number of nearest neighbours $k=32$, which provides a good trade-off between accuracy and computational efficiency.\n\t\tThis process is performed independently with shared parameters for both source and target point clouds.\n\t\t\n\t\tFollowing the self-attention layer, the cross-attention layer allows interaction between the source and target point cloud features.\n\t\tThis layer creates a bi-partite graph between source and target points.\n\t\tEach source point is connected to the $k$-Nearest-Neighbours nodes in the target point cloud, where the distance metric is the dot product between the feature vectors of the respective points.\n\t\tThis layer uses the same residual update rule from Eq. \\ref{eq:cgconv}, considering the different underlying graph and independent attention matrices $W_f', W_s'$.\n\t\tThe graph-attention network output is given by the updated feature vectors from source and target point clouds, denoted by $\\hat{f}_X$ and $\\hat{f}_Y$, respectively.\n\t\t\n\t\t\\begin{figure}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\linewidth]{images\/graphatt.pdf}\n\t\t\t\\caption{Graph-based attention representation. Points are represented as graph nodes. The nodes are defined by their position, \\textit{i.e.} 3D coordinates, and their feature vector (not represented in the image). The graphs connecting the points are created based on the $k$-NN ($k=32$) between the points. In the self-attention layer, the $k$-NN distance is the Euclidean distance between points' spatial coordinates. In the cross-Attention graph, the $k$-NN distance is the dot product between points' feature vectors.}\n\t\t\t\\label{fig:graphatt}\n\t\t\\end{figure}\n\t\n\t\\subsection{Identifying Correspondences}\n\t\\label{sec:soft-corr}\n\t\tThe correspondences between the point clouds can be obtained by comparing the point-wise features of the source and target key points, denoted respectively as $\\hat{f}_X, \\hat{f}_Y \\in \\R^{N \\times D}$.\n\t\tThese feature vectors are normalised to unity $D$-dimensional vectors using the Euclidean norm.\n\t\tA matching probability map indicating the probability of correspondences between $X$ and $Y$ is computed as\n\t\t\\begin{equation}\n\t\t\t\\label{eq:softmap}\n\t\t\t\\phi = \\text{Softmax}\\left( \\frac{\\hat{f}_X \\cdot \\hat{f}_Y^\\T}{T} \\right) \\in \\R^{N \\times N},\n\t\t\\end{equation}\n\t\twhere $T$ is a temperature hyper-parameter and the \\textit{Softmax} function is applied row-wise.\n\t\tEach element $\\phi_{ij}$ represents the probability that the $i$-th key point in $X$ matches the $j$-th key point in $Y$.\n\t\tThe \\textit{Softmax} function scales the coefficients of each row $\\phi_i$, ensuring a probability distribution over the points in $Y$.\n\t\tThe temperature parameter, denoted by $T$, controls the entropy of distribution across points in $Y$.\n\t\tIn the limit, when $T \\to 0^+$, the coefficients become the one-hot encoding of the point in $Y$ with the highest similarity (\\textit{i.e.} dot product).\n\t\tFinally, each key point $x_i \\in X$ is matched to the key point in $Y$ with highest correspondence probability, resulting in the set of correspondence pairs $\\{ (x_i, y_{\\argmax_j \\phi_{ij}}), i=1,\\dots,N \\}$.\n\t\tThe ordered set of correspondences in $Y$ is denoted as $\\hat{Y} = \\{ y_{\\argmax_j \\phi_{1j}}, \\dots, y_{\\argmax_j \\phi_{Nj}} \\}$.\t\n\t\t\n\t\\subsection{Estimating Transformation Parameters}\n\t\tThe previous step computes a correspondence for every point in $X$.\n\t\tIn practice, only a fraction of points in $X$ will have correspondences in $Y$, particularly in the case of partially overlapping point clouds.\n\t\tSensor noise and varying point densities can also lead to encoding errors and erroneous correspondences.\n\t\tTo mitigate the effect of correspondence outliers, a common practice is to use sample consensus algorithms such as RANSAC \\cite{fischler1981random,rusu2009fpfh}.\n\t\tA general version of this algorithm applied to this problem consists of three steps:\n\t\t\\begin{enumerate}\n\t\t\t\\item Create a hypothesis: Sample a minimal set of three correspondences from the set of correspondences and compute the transformation parameters using Eq. \\ref{eq:procrustes-closed-form}.\n\t\t\t\\item Score the hypothesis based on consensus: Compute the number of inlier correspondences, where a correspondence $(x_i,\\hat{y}_i), x_i \\in X, \\hat{y}_i \\in \\hat{Y}$ is an inlier if $\\left\\| Rx_i + t - \\hat{y}_i \\right\\| \\leq \\kappa$, where $\\kappa$ is the inlier threshold, $R, t$ are the hypothesis parameters computed in the first step.\n\t\t\t\\item Repeat the previous steps for $L$ times and select the hypothesis with the highest number of inliers.\n\t\t\\end{enumerate}\n\t\tThe number of tested hypotheses, denoted by $H$, offers a trade-off between computational performance and robustness to outliers.\n\t\t$H$ can also be derived to achieve a desired confidence of the selected hypothesis \\cite{fischler1981random}, \\textit{i.e.} sampling an outlier-free set of correspondences.\n\t\tWhile previous works used RANSAC with a large set of putative correspondences \\cite{rusu2009fpfh}, this may be unfeasible for real-time systems.\n\t\tIn this work, we achieve negligible RANSAC computational cost by learning a small set of correspondences ($N=512$), which allows to reduce the number of hypotheses being tested.\n\t\t\n\t\\subsection{Training Process}\n\t\tThe training process consists of optimising the encoder and attention networks to find accurate correspondences where they exist.\n\t\tWe maximise the matching probabilities of ground-truth correspondences and minimise the matching probability of non-corresponding points.\n\t\tThis is achieved by directly minimising the following loss function:\n\t\t\\begin{equation}\n\t\t\t\\label{eq:loss}\n\t\t\t\\mathcal{L} = \\frac{1}{N_c} \\sum_{i=1}^N \\delta_i \\left[ -\\phi_{i\\hat{j}} + \\frac{\\lambda}{N-1} \\sum_{j=1,j \\neq \\hat{j}}^N \\phi_{ij} \\right],\n\t\t\\end{equation}\n\t\twhere the binary variable $\\delta_i$ indicates whether the source point $x_i$ has a correspondence in $Y$ and $\\hat{j}$ represents the index of the corresponding point in $Y$. \n\t\tAdditionally, $N_c=\\sum_{i=1}^N \\delta_i$ is the number of ground-truth correspondences and $\\lambda$ is a hyper-parameter scaling the contributions of incorrect matches into the loss function.\n\t\tA point in $X$ is considered to have a correspondence in $Y$ if, under the ground-truth transformation, it is within a distance smaller or equal to $1.6$ meters from a point in $Y$.\n\t\tThis inlier distance is arbitrary and was chosen based on the smallest encoder radius.\n\t\tData augmentation is employed by applying random rotation transformations to both input point clouds and adjusting the ground-truth rotation matrix accordingly.\n\t\tThe optimisation details are described in Section \\ref{sec:exp:implementation}.\n\t\t\n\\section{\\uppercase{Performance Evaluation}}\n\\label{sec:exp}\n\tIn this section, we first describe the datasets and the evaluation metrics, followed by the implementation details.\n\tWe then compare the performance of the proposed method with traditional baselines, including ICP \\cite{arun1987least}, FPFH RANSAC \\cite{rusu2009fpfh} and TEASER \\cite{Yang20tro-teaser}; and two state-of-the-art learning-based methods: FCGF \\cite{choy2019fully} and DGR \\cite{choy2020deep}.\n\tWe also provide an ablation study identifying the impact of the proposed attention network into the registration performance.\n\n\t\\subsection{Dataset}\n\t\\label{sec:exp:dataset}\n\t\tThe KITTI Odometry dataset \\cite{Geiger2012CVPR} is traditionally used to evaluate point cloud registration methods in outdoor environments.\n\t\tWe follow the evaluation protocol of recent methods \\cite{FGR2016,choy2019fully,choy2020deep,Bai_2020_CVPR}, which adopt sequences 0 to 5 for training, 6 to 8 for validation and 9 to 10 for testing.\n\t\tIn each sequence, the samples are created by selecting pairs of points clouds obtained sequentially by a single vehicle such that the translation between the poses is less than 10m.\n\t\tThe ground-truth pose is provided by GPS and refined using ICP to reduce misalignment.\n\t\t\n\t\tThe distribution of poses in the KITTI dataset is limited to the trajectory of a single vehicle as it navigates the environment.\n\t\tIn practice, registration methods must be resilient to point clouds with arbitrary relative pose, where the overlap between point clouds may vary significantly across samples.\n\t\tTo this end, we introduce Cooperative Driving Dataset (CODD) \\cite{codd}, an open-source synthetic dataset containing lidar point clouds collected simultaneously from multiple vehicles.\n\t\tThis dataset is created using CARLA \\cite{Dosovitskiy17} and features a diverse range of driving environments, including rural areas, suburbs, and dense urban centres.\n\t\tThe dataset consists of 108 sequences, which are split into three independent subsets for training, validation and testing, as detailed in Table \\ref{tab:dataset}.\n\t\tThe samples are created by selecting all pair-wise combinations of point clouds obtained from vehicles driving simultaneously within a vicinity considering a maximum distance of 30m.\n\t\tFigure \\ref{fig:plot:DatasetStats} presents the cumulative density plots of the relative distance (translation vector norm), rotation angle and overlap ratio of the pairs of point clouds in each dataset.\n\t\tThe overlap ratio measures the overlap between point clouds as the percentage of points in the source point cloud that, when aligned, are within a distance smaller than $\\gamma$ to any point in the target point cloud.\n\t\tOur CODD dataset has a significantly broader distribution of relative distance, rotation angles and overlap ratio between the point cloud pairs, which provides representative scenarios for cooperative perception and multi-agent SLAM.\n\n\t\t\\begin{table}[]\t\n\t\t\t\\centering\n\t\t\t\\caption{Dataset Details}\n\t\t\t\\label{tab:dataset}\t\t\t\n\t\t\t\\begin{tabular}{@{}llll@{}}\n\t\t\t\t\\toprule\n\t\t\t\t\\textbf{KITTI Odometry}            & \\textbf{Train} & \\textbf{Validation} & \\textbf{Test} \\\\ \\midrule\n\t\t\t\t\\# sequences                       & 5              & 2                   & 2    \\\\\n\t\t\t\t\\# samples (pairs of point clouds) & 1358           & 180                 & 555 \\\\ \t\\midrule\t\t\n\t\t\t\t\\textbf{CODD}                      & \\textbf{Train} & \\textbf{Validation} & \\textbf{Test} \\\\ \\midrule\n\t\t\t\t\\# exclusive maps                  & 6              & 1                   & 1    \\\\\n\t\t\t\t\\# sequences                       & 78             & 14                  & 16   \\\\\n\t\t\t\t\\# samples (pairs of point clouds) & 6129           & 1339                & 1315 \\\\ \\bottomrule\t\n\t\t\t\\end{tabular}\n\t\t\\end{table}\n\t\n\t\t\\begin{figure*}\n\t\t\t\\centering\n\t\t\t\\input{images\/datasetStats.pgf}\n\t\t\t\\caption{Cumulative density of the relative distance and rotation angle between coordinate systems of the pairs of point clouds in each subset.}\n\t\t\t\\label{fig:plot:DatasetStats}\n\t\t\\end{figure*}\n\n\t\\subsection{Evaluation Metrics}\n\t\\label{sec:exp:metrics}\n\t\tFollowing previous studies \\cite{choy2019fully,choy2020deep}, the registration performance is evaluated in terms of the translation and rotation errors given by\n\t\t\\begin{align}\n\t\t\t\\label{eq:metrics}\n\t\t\t\\text{TE} &= \\left\\|\\hat{t} - t^g\\right\\|_2, \\\\\n\t\t\t\\text{RE} &= \\arccos \\frac{\\Tr(\\hat{R}^\\T R^g) - 1}{2},\n\t\t\\end{align}\n\t\twhere $R^g,t^g$ denotes the ground-truth rotation matrix and translation vector, respectively.\n\t\tThese metrics are reported considering their mean value over all dataset samples, denoted as Mean Translation Error (MTE) and Mean Rotation Error (MRE), respectively.\n\t\tWe also consider the recall rate, measured as the ratio of successful registrations to the total number of samples, where the success criteria is $\\text{TE} < 0.6$m and $\\text{RE} < 5$deg following \\cite{choy2020deep}.\n\t\tThe runtime performance is evaluated as the average inference time for the registration of a pair of point clouds disregarding the data loading time.\n\t\n\t\\subsection{Implementation Details}\n\t\\label{sec:exp:implementation}\n\t\tOur proposed method is implemented using PyTorch \\cite{paszke2019pytorch}, PyTorch Geometric \\cite{feyPytorchGeometric}, the CUDA implementation of SA and FP layers from \\cite{qi2017pointnetpp} and the Open3D \\cite{Zhou2018} Procrustes RANSAC implementation.\n\t\tThe model is trained independently for each dataset using the Adam \\cite{kingma2014adam} optimiser with a learning rate of $0.1$, $\\epsilon=10^{-4}$, $\\beta_1=0.9, \\beta_2=0.999$ and a batch size of 6 (pairs of point clouds).\n\t\tThe model is trained for twenty epochs and the learning rate is reduced in half after every five epochs.\n\t\tFor evaluation, the model with the lowest validation loss is selected.\n\t\tThe temperature hyper-parameter, described in Section \\ref{sec:soft-corr}, is set to $T=10^{-2}$, and the loss scaling hyper-parameter is set to $\\lambda=10$.\n\t\tDuring inference, the RANSAC inlier threshold, $\\kappa$, is set to 0.5m, and the maximum number of RANSAC iterations, denoted by $H$, is computed to achieve $0.999$ confidence in the selected hypothesis within a limit of $10^5$ iterations.\n\t\tThe point clouds from both datasets are downsampled using voxel sizes of 0.3m following previous methods \\cite{choy2019fully,choy2020deep}.\n\t\tThe overlap ratio distance threshold, $\\gamma$, is set to 0.3m, following the down-sampling voxel size. \n\t\tThe experiments are carried out on a Xeon ES-1630 CPU and Quadro M4000 GPU with 8 GB of memory.\n\t\tFor fair comparison, all baselines are also evaluated on the same hardware.\n\t\tThe official implementation and pre-trained models (30cm voxel) are used for the evaluation of \\cite{choy2019fully,choy2020deep} in the KITTI dataset; likewise, the official TEASER \\cite{Yang20tro-teaser} implementation is adopted; and the Open3D \\cite{Zhou2018} implementation of FPFH, RANSAC and ICP is used for the evaluation of the respective methods.\n\t\t\n\t\\subsection{Performance on the KITTI dataset}\n\t\\label{sec:exp:results-kitti}\n\t\tThe evaluation results, presented in Table \\ref{tab:results-kitti}, show that our proposed method achieves competitive registration errors compared to other methods at a significantly lower inference time -- more than five times faster than the fastest baseline.\n\t\tWhile the proposed method has a marginal increase in mean translation error compared to baseline methods, it achieves on-par recall rate relative to baseline methods.\n\t\tThe mean translation and rotation errors of our method can be further reduced using ICP for refinement (Ours + ICP), at the cost of increased inference time.\n\t\t\n\t\t\\begin{table}[]\n\t\t\t\\centering\n\t\t\t\\caption{Evaluation Results on KITTI test set}\n\t\t\t\\label{tab:results-kitti}\n\t\t\t\\resizebox{\\linewidth}{!}{%\n\t\t\t\\begin{tabular}{@{}lllll@{}}\n\t\t\t\t\\toprule\n\t\t\t\t\\textbf{Model}                      & \\textbf{MTE [cm]} & \\textbf{MRE [deg]} & \\textbf{Recall} & \\textbf{Time\/sample [s]} \\\\ \\midrule\n\t\t\t\tFPFH TEASER \\cite{Yang20tro-teaser} & 18.8              &  0.76              &  0.984                &  2.26                  \\\\\t\t\t\t\n\t\t\t\tFCGF RANSAC \\cite{choy2019fully}    & 23.7              &  0.034             &  0.975                &  26.29                 \\\\\n\t\t\t\tDGR  \\cite{choy2020deep}            & 15.6              &  1.43              &  0.982                &  7.60                  \\\\ \\midrule\n\t\t\t\tOurs                                & 26.1              &  0.74              &  0.949                &  0.41                  \\\\\n\t\t\t\tOurs + ICP                          & 8.2               &  0.23              &  0.985                &  3.68                  \\\\ \\bottomrule\n\t\t\t\\end{tabular}\n\t\t\t}\n\t\t\\end{table}\n\t\n\t\\subsection{Performance on the CODD dataset}\n\t\\label{sec:exp:results-codd}\n\t\tWe aim to evaluate the performance of the proposed method and baselines on challenging low-overlapping pairs of point clouds.\n\t\tFor a fair comparison, learning-based methods \\cite{choy2019fully,choy2020deep} are also trained on the CODD dataset.\n\t\tTable \\ref{tab:results} presents the results on the CODD test set, aggregated by the overlap ratio between point clouds in four progressively larger intervals -- the last interval contains all samples.\n\t\tTraditional methods are not resilient to low overlapping points clouds, as the registration error increases significantly when considering lower overlap ratios, as shown in the first three rows of Table \\ref{tab:results}.\n\t\tIn contrast, the learning-based baselines are reasonably robust to low overlapping point clouds and achieve high recall rates on all intervals.\n\t\tHowever, the latter methods demand substantial running times due to their complex encoders and the filtering of a high number of putative correspondences.\n\t\tIn contrast, our proposed method achieves similar or better recall rates to the learning-based baselines with more than 35 times faster inference times.\n\t\tThis is achieved by our efficient encoder design which outputs a small number of correspondences, which in turn reduces the RANSAC inference time.\n\t\tOur efficient encoder strategy comes at the cost of a slight increase of the MTE and MRE metrics, as compared to DGR \\cite{choy2020deep}.\n\t\tTo mitigate this, we apply ICP refinement to our model's output (Ours + ICP), which allows achieving similar MTE and MRE for highly overlapping point clouds and outperforming all baselines on low-overlapping point clouds.\n\t\tAlthough the ICP refinement comes with an additional computational cost, we still achieve a nine-fold speed-up compared to learning-based baselines.\n\t\tQualitative results are presented in Figure \\ref{fig:qualitative}.\n\n\t\tFigure \\ref{fig:plot:results-ecdf} shows the Empirical Cumulative Density Function (ECDF) of the translation error, rotation errors, and inference time for different methods.\n\t\tThe distributions indicate that the proposed method with ICP refinement has the best translation error across samples, closely matched by DGR \\cite{choy2020deep}, however with one order of magnitude smaller inference time.\n\t\tThe inference time distributions show that the proposed method is the fastest among baselines, with an inference time of 320ms on average, with negligible standard deviation (17ms).\n\t\tThe proposed method can operate in real-time considering data input frequencies up to 3Hz.\n\t\t\n\t\t\\begin{table*}[]\n\t\t\t\\centering\n\t\t\t\\caption{Evaluation Results on CODD test set}\n\t\t\t\\label{tab:results}\n\t\t\t\\resizebox{\\linewidth}{!}{%\n\t\t\t\\begin{tabular}{@{}lllllllllllllll@{}}\n\t\t\t\\cmidrule(l){2-15}\n\t\t\t\\textbf{} &\n\t\t\t\\multicolumn{3}{c}{$\\text{Overlap Ratio} > 0.6$} &\n\t\t\t\\multicolumn{3}{c}{$\\text{Overlap Ratio} > 0.5$} &\n\t\t\t\\multicolumn{3}{c}{$\\text{Overlap Ratio} > 0.4$} &\n\t\t\t\\multicolumn{3}{c}{$\\text{Overlap Ratio} > 0$} &\n\t\t\t\\multicolumn{2}{c}{\\textbf{Time\/sample [s]}} \\\\ \\cmidrule(r){1-1} \\cmidrule(r){2-4} \\cmidrule(r){5-7} \\cmidrule(r){8-10} \\cmidrule(r){11-13} \\cmidrule(r){14-15}\n\t\t\t\\textbf{Method} &\n\t\t\t\\multicolumn{1}{c}{\\textbf{MTE [m]}} &\n\t\t\t\\multicolumn{1}{c}{\\textbf{MRE [deg]}} &\n\t\t\t\\multicolumn{1}{c}{\\textbf{Recall}} &\n\t\t\t\\multicolumn{1}{c}{\\textbf{MTE [m]}} &\n\t\t\t\\multicolumn{1}{c}{\\textbf{MRE [deg]}} &\n\t\t\t\\multicolumn{1}{c}{\\textbf{Recall}} &\n\t\t\t\\multicolumn{1}{c}{\\textbf{MTE [m]}} &\n\t\t\t\\multicolumn{1}{c}{\\textbf{MRE [deg]}} &\n\t\t\t\\multicolumn{1}{c}{\\textbf{Recall}} &\t\t\t\n\t\t\t\\multicolumn{1}{c}{\\textbf{MTE [m]}} &\n\t\t\t\\multicolumn{1}{c}{\\textbf{MRE [deg]}} &\n\t\t\t\\multicolumn{1}{c}{\\textbf{Recall}} &\t\t\t\n\t\t\t\\multicolumn{1}{c}{\\textbf{Mean}} &\n\t\t\t\\multicolumn{1}{c}{\\textbf{Std}} \\\\ \\cmidrule(r){1-1} \\cmidrule(r){2-4} \\cmidrule(r){5-7} \\cmidrule(r){8-10} \\cmidrule(r){11-13} \\cmidrule(r){14-15}\n\t\t\tICP \\cite{arun1987least}           & 1.69 & 36.11 & 0.67 & 4.81 & 68.67 & 0.38 & 9.11  & 66.42 & 0.17 & 16.14 & 74.84 & 0.07 & 0.38  & 0.048 \\\\\n\t\t\tRANSAC FPFH \\cite{rusu2009fpfh}    & 1.59 & 1.42  & 0.47 & 1.25 & 1.58  & 0.28 & 2.64  & 2.42  & 0.18 & 9.55  & 9.49  & 0.09 & 71.69 & 15.37 \\\\\n\t\t\tTEASER FPFH \\cite{Yang20tro-teaser}& 0.04 & 0.10  & 1.00 & 1.61 & 23.33 & 0.86 & 4.29  & 36.42 & 0.69 & 12.87 & 69.74 & 0.39 & 1.13  & 0.24  \\\\ \\midrule\n\t\t\tRANSAC FCGF \\cite{choy2019fully}   & 0.09 & 0.01  & 1.00 & 0.10 & 0.01  & 1.00 & 0.12  & 0.01  & 1.00 & 1.70  & 0.11  & 0.91 & 16.5  & 22.4  \\\\\n\t\t\tDGR \\cite{choy2020deep}            & 0.02 & 0.07  & 1.00 & 0.02 & 0.06  & 1.00 & 0.02  & 0.05  & 1.00 & 0.39  & 1.52  & 0.94 & 11.89 & 3.92  \\\\ \\midrule\n\t\t\tOurs                               & 0.14 & 0.21  & 1.00 & 0.19 & 0.25  & 0.99 & 0.22  & 0.29  & 0.98 & 0.28  & 0.41  & 0.94 & 0.32  & 0.017 \\\\\n\t\t\tOurs + ICP                         & 0.03 & 0.09  & 1.00 & 0.03 & 0.09  & 0.99 & 0.04  & 0.09  & 0.99 & 0.09  & 0.13  & 0.97 & 1.28  & 0.031 \\\\ \n\t\t\tOurs - Att                         & 0.29 & 0.48  & 0.87 & 0.39 & 0.56  & 0.86 & 0.59  & 0.86  & 0.76 & 2.22  & 5.69  & 0.57 & 0.30  & 0.003 \\\\ \\bottomrule\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\end{table*}\n\n\t\t\\begin{figure*}\n\t\t\t\\centering\n\t\t\t\\input{images\/error-time-ecdf.pgf}\n\t\t\t\\caption{ECDF of the Translation Error, Rotation Error and Sample Execution Time for different methods on the CODD test set.}\n\t\t\t\\label{fig:plot:results-ecdf}\n\t\t\\end{figure*}\n\n\n\t\\subsection{Ablation Study}\n\t\\label{sec:exp:results-ablation}\n\t\tWe assess the impact of the proposed attention network into the registration performance, measured in terms of translation and rotation errors.\n\t\tThis is achieved by removing the graph attention module, retraining the model and evaluating its performance on the CODD test set.\n\t\tThe results, indicated in Table \\ref{tab:results} ``Ours - Att'', show that the graph-attention network plays a key role in improving the accuracy of the correspondences, resulting in lower translation and rotation errors.\n\t\tThe benefits of the graph attention network is most significant for low-overlapping point clouds, as indicated by the last range group in Table \\ref{tab:results}, where the removal of the attention results in a 40\\% reduction of the registration recall and a significant increase in the mean translation and rotation errors.\n\n\n\\section{\\uppercase{Conclusion}}\n\\label{sec:conclusion}\n\tWe proposed a novel point cloud registration method focusing on fast inference of partially overlapping lidar point clouds. \n\tThe performance evaluation on the KITTI and CODD datasets indicates that the proposed model can operate with a latency lower than 410ms and 320ms, respectively.\n\tThe results show that the proposed model outperform baseline methods in terms of rotation and translation errors for pairs of point clouds with low overlap.\n\tFurthermore, we show that the proposed graph attention module plays a key role in improving the quality of the correspondences in low overlapping point clouds, which results in higher registration performance.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n{\\it Magnetic resonance imaging} (MRI) is a widely used imaging method in clinical applications and research. It is based on measuring the magnetic signal resulting from {\\it nuclear magnetic resonance} (NMR) of $\\rm ^1_1H$ nuclei (protons). In NMR, the magnetization rotates around an applied magnetic field $\\vec{B}$ at the proton Larmor frequency $f_{\\rm L}$, which is proportional to $B$ \\cite{Abragam}. This behavior of the magnetization is often referred to as {\\it precession} due to the direct connection to the quantum mechanical precession of nuclear spin angular momentum. \n\nConventionally, the magnetic precession signal has been detected using induction coils. The voltage induced in a coil by an oscillating magnetic field is proportional to the frequency of the oscillation, leading to vanishing signal amplitudes as $f_{\\rm L}$ approaches zero. Today, clinical MRI scanners indeed use a high main static field $\\vec B_0$; typically $B_0 = 3$\\unit{T}, corresponding to a frequency $f_0 = 128$\\unit{MHz}. However, when the signal is detected using magnetic field (or flux) sensors with a frequency-independent response, this need for high frequencies disappears. Combined with the so-called prepolarization technique for signal enhancement, highly sensitive magnetic field detectors, typically those based on {\\it superconducting quantum-interference devices} (SQUIDs), provide an NMR signal-to-noise ratio (SNR) that is independent of $B_0$ \\cite{Clarke2007}. In recent years, there has been growing interest in ultra-low-field (ULF) MRI, usually measured in a field on the order of Earth's magnetic field ($B_0 \\sim 10$--$100$\\unit{\\textmu T}). \n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=.98\\columnwidth]{array_schem_3d.pdf}\n        \\vspace{-2mm}\n\t\\caption{Helmet-type sensor array geometries consisting of (a) triple-sensor modules at 102 positions similar to standard Elekta\/Neuromag MEG configurations and (b) an array with larger overlapping pickup coils for increased perfomance. Magnetometers are marked in green and gradiometers in red or blue; see Sec.~\\ref{ssPickups} for descriptions of pickup coils. (The sample head shape is from MNE-Python \\cite{MNESoftware}.)}\n\t\\label{figGradArrays}\n\\end{figure}\n\nA number of ULF-MRI-specific imaging techniques have emerged, including rotary-scanning acquisition (RSA) \\cite{Hsu2016}, temperature mapping \\cite{VesanenTemperature2013}, signal-enhancing dynamic nuclear polarization \\cite{KRISSLee2010, Buckenmaier2018}, imaging of electric current density (CDI) \\cite{Vesanen2014, Nieminen2014, Hommen2019}, and making use of significant differences in NMR relaxation mechanisms at ULF compared to tesla-range fields \\cite{Lee2005, Hartwig2011, Vesanen2013Temperature}. Several groups have also investigated possibilities to directly detect changes in the NMR signal due to neural currents in the brain \\cite{KrausJr2008, Korber2013, Xue2006, KRISSKim2014} and electrical activation of the heart \\cite{KRISSKim2012}. A further notable field of research now focuses on combining ULF MRI with magnetoencephalography (MEG). In MEG, an array of typically $\\sim 100$ sensors \\cite{Lounasmaa2004, Vrba2002, Pizzella2001} is arranged in a helmet-shaped configuration around the head (see Fig.~\\ref{figGradArrays}) to measure the weak magnetic fields produced by electrical activity in the brain \\cite{Hamalainen1993, DelGratta2001}. SQUID sensors tailored for ULF MRI can typically also be used for MEG, and performing MEG and MRI with the same device can significantly improve the precision of localizing brain activity \\cite{MEGMRI2013,Magnelind2011co, Luomahaara2018, Roadmap2016, Makinen2019}. \n\nIn typical early ULF-MRI setups \\cite{Clarke2007}, the signal was detected by a single dc SQUID coupled to a superconducting pickup coil wound in a gradiometric configuration that rejects noise from distant sources. In this case, the maximum size of the imaging field of view (FOV) is roughly given by the diameter of the pickup coil. With large diameters such as 60\\unit{mm}, field sensitivities better than 1\\unit{fT$\/\\sqrt{\\rm Hz}$} have been achieved with a reasonable FOV. A large coil size, however, does have its drawbacks, including issues such as high inductance and increased requirements in dynamic range. Therefore, the most straightforward way to increase the available FOV and the SNR is to use an array of sensors. In addition, as is well known in the context of MEG \\cite{Uusitalo1997ssp, Vrba2002, Taulu2005}, a multi-channel measurement allows forming so-called software gradiometers and more advanced signal processing techniques to reduce noise that can be optimized separately for different noise environments. In ULF-MRI, this can even be done individually for each voxel (volume element) position within the imaging target, as will be shown later. While single-channel systems are still common, several groups have already been using arrays of sensors.\n\n\nAlso in conventional MRI, so-called parallel MRI is performed using an array of tens of induction coils, allowing full reconstruction of images from a reduced number of data acquisitions \\cite{Pruessmann1999, Larkman2007}. There are studies on designing arrays of induction coils for parallel MRI \\cite{Ohliger2006} with an emphasis on minimizing artefacts caused by the reduced number of acquisitions. At the kHz frequencies of ULF MRI, the dominant noise mechanisms are significantly different, and one needs to consider, for instance, electromagnetic interference from power lines and electrical equipment, thermal noise from the radiation shield of the cryostat required for operating the superconducting sensors, as well as noise and transients from other parts of the ULF MRI system structure and electronics \\cite{Zevenhoven2014amp}. Studies on the design of arrays for MEG \\cite{Vrba2002,Ahonen1993,Nurminen2014thesis},\nwhich mainly focus on the accuracy of localizing brain activity, are also not applicable to ULF MRI.  In terms of single-sensor ULF-MRI signals, there are existing studies of the depth sensitivity \\cite{Burmistrov2013} and SNR as a function of frequency with different detector types \\cite{Myers2007}. \n\nPreviously, in Ref.~\\cite{Zevenhoven2011}, we presented approaches for quantitative comparison of sensor arrays in terms of the combined performance of the sensors, the results indicating that the optimum sensor for ULF MRI of the brain would be somewhat larger than typical MEG sensors.\nExtending and refining those studies, we aim to provide a fairly general study of the optimization of ULF-MRI array performance, with special attention to SNR and imaging the human head.\n\nWe begin by defining relevant quantities and reviewing basic principles of ULF MRI in Sec.~\\ref{sBasics}. Then, we analyze the effects of sensor geometry and size with different noise mechanisms (Sec.~\\ref{sSingleSensor}), advancing to sensor arrays (Sec.~\\ref{sArrays}). Finally, we show computed estimations of array SNR as functions of pickup size and number, and provide more detailed comparison of spatial SNR profiles with different array designs (Secs.~\\ref{sMethods} and \\ref{sResults}).\n\n\\section{SQUID-detected MRI} \\label{sBasics}\n\n\\subsection{Signal model and single-channel SNR} \\label{ssULFMRI}\n\nIn contrast to conventional MRI, where the tesla-range main field is static and accounts for both polarizing the sample and for the main readout field, ULF MRI employs switchable fields. Dedicated electronics \\cite{Zevenhoven2014amp} are able to ramp on and off even the main field $\\vec B_0$ with an ultra-high effective dynamic range. An additional pulsed prepolarizing field $\\vec{B}_{\\rm p}$ magnetizes the target before signal acquisition. Typically, a dedicated coil is used to generate $\\vec{B}_{\\rm p}$  ($B_{\\rm p} \\sim 10$--$100$\\unit{mT}) in some direction to cause the proton bulk magnetization $\\vec{M}(\\vec r\\,)$ to relax with a longitudinal relaxation time constant $T_1$ towards its equilibrium value corresponding to $\\vec B_{\\rm p}$. After a polarizing time on the order of seconds or less, $\\vec B_{\\rm p}$ is switched off---adiabatically, in terms of spin dynamics---so that $\\vec M$ turns to the direction of the remaining magnetic field, typically $\\vec B_0$, while keeping most of its magnitude. \n\nNext, say at time $t=0$, a short excitation pulse $\\vec B_1$ is applied which flips $\\vec M$ away from $\\vec B_0$, typically by 90$^\\circ$, bringing $\\vec M$ into precession around the magnetic field at positions $\\vec r$ throughout the sample. While rotating, $\\vec M(\\vec r\\,)$ decays towards its equilibrium value corresponding to the applied magnetic field in which the magnetization precesses. This field, $\\vec B_\\mathrm L$, may sometimes simply be a uniform $\\vec B_0$, but for spatial encoding and other purposes, different non-uniform magnetic fields $\\mathrm\\Delta \\vec B(\\vec r, t)$ are additionally applied to affect the precession before or during acquisitions. The encoding is taken into account in the subsequent image reconstruction. \n\n\nThe ULF MRI signal can be modeled to a high accuracy given the absence of unstable distortions common at high frequencies and high field strengths. To obtain a model for image formation, we begin by examining $\\vec M$ at a single point. If the $z$ axis is set parallel to the total precession field $\\vec B_\\mathrm{L}$, then the $xy$ (transverse) components of $\\vec M$ account for the precession. Assuming, for now, a static $\\vec B_\\mathrm L$, and omitting the decay for simplicity, the transverse magnetization $\\vec M_{xy} = \\vec M_{xy}(t)$ can be written as \n\\begin{align}\n\\vec M_{xy}(t) = M_{xy}&\\left[\\widehat{e}_x\\cos(\\omega t+\\phi_0)\n - \\widehat{e}_y\\sin(\\omega t+\\phi_0)\\right]\\,,\n\\end{align}\nwhere $\\omega = 2\\pi f_\\mathrm{L}$ is the precession angular frequency, $\\widehat{e}_\\heartsuit$ is the unit vector along the $\\heartsuit$ axis ($\\heartsuit = x, y, z$), and $\\phi_0$ is the initial phase, which sometimes contains useful information.\n\nIn an infinitesimal volume $dV$ at position $\\vec r$ in the sample, the magnetic dipole moment of protons in the volume is $\\vec M(\\vec r\\,)\\, dV$. It is straightforward to show that the rotating components of this magnetic dipole are seen by any magnetic field or flux sensor as a sinusoidal signal $d\\psi_{\\rm s} = |\\beta|\\cos(\\omega t+\\phi_0 + \\phi_\\mathrm s) M_{xy}\\,dV$. Here $|\\beta| = |\\beta(\\vec r\\,)|$ is the peak sensitivity of the sensor to a unit dipole at $\\vec r$ that precesses in the $xy$ plane, and $\\phi_{\\mathrm s} = \\phi_{\\mathrm s}(\\vec r\\,)$ is a phase shift depending on the relative positioning of the sensor and the dipole. To obtain the total sensor signal $\\psi_{\\rm s}$, $d\\psi_{\\rm s}$ is integrated over all space:\n\\begin{align}\\label{eqRSignal}\n&\\psi_{\\rm s}(t) = \\int |\\beta(\\vec r\\,)|M_{xy}(\\vec r\\,)\\cos \\phi(\\vec r, t)\\, d^3\\vec r\\,,\\\\\\nonumber\n&\\text{where }~\\phi(\\vec r, t) = \\int_0^t\\omega(\\vec r, t^\\prime) \\,dt^\\prime +\\phi_0(\\vec r\\,)+\\phi_{\\mathrm s}(\\vec r\\,)\\,.\n\\end{align}\nHere, we have noted that the magnetic field can vary in both space and time and therefore $\\omega = \\omega(\\vec r, t) = \\gamma B(\\vec r, t)$, where $\\gamma$ is the gyromagnetic ratio; $\\gamma \/2\\pi = 42.58$\\unit{MHz\/T} for a proton.\n\nFor convenience, the signal given by Eq.~\\eqref{eqRSignal} can be demodulated at the angular Larmor frequency $\\omega_0=2\\pi f_0$ corresponding to $B_0$; using the quadrature component of the phase sensitive detection as the imaginary part, one obtains a complex-valued signal\n\\begin{align}\\nonumber\n\\Psi(t) &= \\int |\\beta(\\vec r\\,)|M_{xy}(\\vec r\\,)e^{-i[\\phi(\\vec r, t)-\\omega_0 t]}\\,d^3\\vec r\\\\\\label{eqSignal}\n&= \\int \\beta^*(\\vec r\\,) m(\\vec r\\,)e^{-i\\int_0^t\\mathrm\\Delta\\omega(\\vec r,t^\\prime)\\, dt^\\prime}\\,d^3\\vec r\\,,\n\\end{align}\nwhere $^*$ denotes the complex conjugate, $m(\\vec r\\,) = M_{xy}(\\vec r\\,)e^{-i\\phi_0(\\vec r\\,)}$ is the \\emph{uniform-sensitivity image}, $\\mathrm\\Delta\\omega = \\omega-\\omega_0$, and we define\n\\begin{equation}\n\\beta(\\vec r\\,) = |\\beta(\\vec r\\,)|e^{i\\phi_{\\mathrm s}(\\vec r\\,)}\n\\end{equation} \nas the single-channel \\emph{complex sensitivity profile}. Besides geometry, $\\beta$ generally also depends on the direction of the precession field; $\\beta = \\beta_{\\vec B_{\\mathrm L}}(\\vec r\\,)$. \n\nAfter acquiring enough data of the form of Eq.~\\eqref{eqSignal}, the image can be reconstructed---in the simplest case using only one sensor, or using multiple sensors, each having its own sensitivity profile $\\beta$. As a simplified model for understanding image formation, ideal Fourier encoding turns Eq.~\\eqref{eqSignal} into the 3-D Fourier transform of the sensitivity-weighted complex image $\\beta^*m = (\\beta^*m)(\\vec r\\,)$. In reality, however, the inverse Fourier transform only provides an approximate reconstruction, and more sophisticated techniques should be used instead \\cite{Hsu2014}. \n\nHere, we do not assume a specific spatial encoding scheme. Notably, however, the sensitivity profile is indistinguishable from $m$ based on the signal [Eq.~\\eqref{eqSignal}]. In other words, the spatial variation of $\\beta^*$ affects the acquired data in the same way as a similar variation of the actual image would, regardless of the spatial encoding sequence in $\\mathrm\\Delta \\omega$.\n\nConsider a small voxel of centered at $\\vec r$. The contribution of the  voxel to the signal in Eq.~\\eqref{eqSignal} is proportional to an effective voxel volume $V$. Due to measurement noise, the voxel value becomes $V\\beta^* m + \\xi$, where $\\xi$ is a random complex noise term. If $\\beta$ is known, the intensity-corrected voxel of a real-valued image from a single sensor is given by\n\\begin{equation} \\label{eqVoxelIntensity}\n{\\rm Re}\\left(m(\\vec r\\,) + \\frac{\\xi}{V\\beta^*(\\vec r\\,)}\\right) = \n m(\\vec r\\,) + \\frac{{\\rm Re}\\left(\\xi e^{i\\phi_{\\mathrm s}}\\right)}{|s(\\vec r\\,)|}\\, ,\n \\end{equation}\nwhere $s(\\vec r\\,)=V\\beta^*(\\vec r\\,)$ is the sensitivity of the sensor to $m$ in the given voxel.\nAssuming that the distribution of $\\xi=|\\xi|e^{i\\phi_\\xi}$ is independent of the phase $\\phi_\\xi$, the standard deviation $\\sigma$ of ${\\rm Re}\\left(\\xi e^{i\\phi_{\\mathrm s}}\\right)$ is independent of $\\phi_{\\mathrm s}$ and proportional to $\\sigma_{\\rm s}$, the standard deviation of the noise in the relevant frequency band of the original sensor signal. \n\nThe precision of a voxel value can be described by the (amplitude) SNR  of the voxel value. The voxel SNR is defined as the correct voxel value $m(\\vec r\\,)$ divided by the standard deviation of the random error and can be written as\n\\begin{equation} \\label{eqSNR0}\n{\\rm SNR} = \\frac{m(\\vec r\\,)V|\\beta(\\vec r\\,)|}{\\sigma}\n\\propto\\frac{B_{\\rm p}V|\\beta(\\vec r\\,)|\\sqrt{T_{\\rm tot}}}{\\sigma_{\\rm s}}\\, ,\n\\end{equation}\nwhere the last expression incorporates that $m \\propto B_{\\rm p}$, and that $\\sigma$ is inversely proportional to the square root of the total signal acquisition time, which is proportional to the total MRI scanning time $T_{\\rm tot}$. It should be recognized, however, that $\\sigma$ also depends heavily on factors not visible in Eq.~\\eqref{eqSNR0}, such as the imaging sequence.\n\nUltimately, the ability to distinguish between different types of tissue depends on the {\\it contrast-to-noise ratio} (CNR), which can be defined as the SNR of the difference between image values corresponding to two tissues. A better CNR can be achieved by improving either the SNR or the contrast, which both strongly depend also on the imaging sequence.\n\n\\subsection{SQUIDs, pickup coils and detection} \\label{ssPickups}\n\n\n\nSQUIDs are based on {\\it superconductivity}, the phenomenon where the electrical resistivity of a material completely vanishes below a critical temperature $T_{\\rm c}$ \\cite{SQUID-HB}. A commonly used material is niobium (Nb), which has $T_{\\rm c}=9.2\\,$K. It is usually cooled by immersion in a liquid helium bath that boils at $4.2\\,$K in atmospheric pressure. \n\nSQUIDs can be divided into two categories, rf and dc SQUIDs, of which the latter is typically used for biomagnetic signals as well as for ULF MRI \\cite{Lounasmaa2004, Roadmap2016}. The dc SQUID is a superconducting loop interrupted by two weak links, or Josephson junctions; see Fig.~\\ref{figSQUID}(a). With suitable shunting and biasing to set the electrical operating point, the current or voltage across the SQUID can be configured to exhibit an oscillatory dependence on the magnetic flux going through the loop---analogously to the well known double-slit interference of waves.\n\nA linear response to magnetic flux is obtained by operating the SQUID in a flux-locked loop (FLL), where an electronic control circuit aims to keep the flux constant by applying negative flux feedback via an additional feedback coil.\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=0.95\\columnwidth]{squid.pdf}\n\t\\caption{Schematic (a) of a simple SQUID sensor and the flux-locked loop (more detail in Secs.~\\ref{ssIntrinsic} and \\ref{ssCorrEffect}), and (b--f) of different types of pickup coils. Pickup coil types are (b) magnetometer (M0), (c) planar first-order gradiometer (PG1), (d) axial first-order gradiometer (AG1), (e) axial second-order gradiometer (AG2), (f) planar gradiometer with a long baseline, and (g) a magnetometer and two planar gradiometers in a triple-sensor unit (M0, PG1$x$, PG1$y$).}\n\t\\label{figSQUID}\n\\end{figure}\n\nTo avoid harmful resonances and to achieve low noise, the SQUID loop itself is usually made small. The signal is coupled to it using a larger pickup coil connected to the SQUID via an input circuit to achieve high sensitivity. An input circuit may simply consist of a {\\it pickup coil} and an {\\it input coil} in series, forming a continuous superconducting path which, by physical nature, conserves the flux through itself, and feeds the SQUID according to the signal received by the pickup coil, as explained in Sec.~\\ref{ssIntrinsic} along with more sophisticated input circuits.\n\nDifferent types of responses to magnetic fields can be achieved by varying the pickup coil geometry. Fig.~\\ref{figSQUID}(b--g) schematically depicts some  popular types. The simplest case is just a single loop, a {\\it magnetometer}, which in a homogeneous field responds linearly to the field component perpendicular to the plane of the loop (b). Two loops of the same size and orientation, but wound in opposite directions, can be used to form a {\\it gradiometer}. The resulting signal is that of one loop subtracted from that of the other. It can be used to approximate a derivative of the field component with respect to the direction in which the loops are displaced (by distance $b$, called the baseline). Typical examples are the planar gradiometer (c) and the axial gradiometer (d). By using more loops, one can measure higher-order derivatives. Some ULF-MRI implementations \\cite{Clarke2007,Zotev2007} use second-order axial gradiometers (e). If a source is close to one loop of a long-baseline gradiometer, that `pickup loop' can be thought of as a magnetometer, while the additional loops suppress noise from MRI coils or distant sources. However, adding loops also increases the inductance $L_\\mathrm p$. Before a more detailed theoretical discussion regarding $L_\\mathrm p$ and SQUID noise scaling, we study the detection of the MRI signal by the pickup coils. \n\n\\subsection{Sensitivity patterns and signal scaling}\n\nThe magnetic flux $\\Phi$ picked up by a coil made of a thin superconductor is given by the integral of the magnetic field $\\vec{B}$ over a surface $S$ bound by the coil path $\\partial S$,\n\\begin{equation} \\label{eqFlux}\n\\Phi = \\int_S \\vec{B}\\cdot d_{\\mathrm n}^2\\vec r = \\oint_{\\partial S} \\vec{A}\\cdot d\\vec r\\,.\n\\end{equation}\nHere, the line integral form was obtained by writing $\\vec{B}$ in terms of the vector potential $\\vec{A}$ as $\\vec{B} = \\nabla \\times \\vec{A}$, and applying Stokes's theorem.\n\nAs explained in Sec.~\\ref{ssULFMRI}, the signal in MRI arises from spinning magnetic dipoles. The quasi-static approximation holds well at signal frequencies, providing a vector potential for a dipole $\\vec{m}$ positioned at $\\vec r\\,'$ as $\\vec{A}(\\vec r\\,) = \\frac{\\mu}{4\\pi}\\frac{\\vec{m}\\times(\\vec r-\\vec r\\,^\\prime)}{|\\vec r-\\vec r\\,^\\prime|^3},$\nwhere $\\mu$ is the permeability of the medium, assumed to be that of vacuum; $\\mu = \\mu_0$. Substituting this into Eq.~\\eqref{eqFlux} and rearranging the resulting scalar triple product leads to\n\\begin{equation} \\label{eqLeadField}\n\\Phi = \\vec{m}\\cdot \\vec{B}_{\\rm s}(\\vec r\\,')\\,, \\;\\; \\vec{B}_{\\rm s}(\\vec r\\,') = \\frac{\\mu}{4\\pi}\\oint_{\\partial S} \\frac{d\\vec r \\times (\\vec r\\,'-\\vec r\\,)}{|\\vec r\\,'-\\vec r\\,|^3}\\,,\n\\end{equation}\nwhere the expression for the \\emph{sensor field} $\\vec{B}_{\\rm s}$ is the Biot--Savart formula for the magnetic field at $\\vec r\\,'$ caused by a hypothetical unit current in the pickup coil, as required by reciprocity.\n\nThe sensor field $\\vec B_\\mathrm s$ is closely related to the complex sensitivity pattern $\\beta$ introduced in Sec.~\\ref{ssULFMRI}. In an applied field $\\vec{B}_\\mathrm L = B_\\mathrm L\\widehat e_z$, the magnetization precesses in the $xy$ plane, and $\\beta$ can in fact be written as\n\\begin{equation} \\label{eqBeta0}\n\t\\beta(\\vec r\\,) = \\vec B_\\mathrm s(\\vec r\\,) \\cdot \\left(\\widehat e_x  + i \\,\\widehat e_y\\right)\\,.\n\\end{equation}\nFor arbitrary $\\vec B = B_\\mathrm L\\widehat e_\\mathrm L$, we have\n\\begin{equation} \\label{eqBetaNorm}\n |\\beta_{\\vec B} (\\vec r\\,)| = \\sqrt{|\\vec B_\\mathrm s(\\vec r\\,)|^2 - [\\vec B_\\mathrm s (\\vec r\\,) \\cdot\\widehat e_\\mathrm L]^2}\\,.\n\\end{equation}\n\n\nWe choose to define the measured signal as the \\emph{flux} through the pickup coil---a convention that appears throughout this paper. The measurement noise is considered accordingly, as flux noise. This contrasts looking at magnetic-field signals and noise, as is often seen in the literature. Working with magnetic flux signals allows for direct comparison of different pickup coil types. Moreover, the approximation that magnetometer and gradiometer pickups respond to the field and its derivatives, respectively, is not always valid.\n\nThe signal often scales as simple power laws $R^\\alpha$ with the pickup coil size $R$ (or radius, for circular coils). When the distance $l$ from the coil to the signal source is large compared to $R$, a magnetometer sees a flux $\\Phi\\propto BR^2$, giving an \\emph{amplitude scaling exponent} $\\alpha=2$. When scaling a gradiometer, however, also the baseline $b$ is proportional to $R$. This leads to $\\alpha=3$ for a first-order gradiometer, or $\\alpha=2+k$ for one of $k^{\\rm th}$ order. Conversely, the signal scales with the distance as $l^{-\\alpha-1}$, as is verified by writing the explicit forms of the field and its derivatives. The additional $-1$ in the exponent reflects the dipolar nature of the measured field ($-2$ for quadrupoles etc.).\n\nFor some cases, the detected flux can be calculated analytically using Eq.~\\eqref{eqLeadField}. First, as a simple example, consider a dipole at the origin, and a circular magnetometer pickup loop of radius $R$ parallel to the $xy$ plane at $z=l$, centered on the $z$ axis. The integral in Eq.~\\eqref{eqLeadField} is easily integrated in cylindrical coordinates to give\n\\begin{equation}\\label{eqCircleLead}\n\\vec{B}_{\\rm s} = B_{\\rm s}\\widehat{e}_z \n= \\frac{\\mu R^2}{2(R^2+l^2)^\\frac{3}{2}}\\widehat{e}_z\\,.\n\\end{equation}\nIf the dipole precesses in, for instance, the $xz$ plane, the corresponding sensitivity is $|\\beta| = B_{\\rm s}$. Instead, if precession takes place in the $xy$ plane, the sensitivity vanishes; $|\\beta|=0$, and no signal is received. In this case, moving the pickup loop away from the $z$ axis would cause a signal to appear. These extreme cases show that even the absolute value of a single-channel sensitivity is strongly dependent on the sensor orientation with respect to the source and the magnetic field, as is also seen in Fig.~\\ref{figSensContours}.\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=0.90\\columnwidth]{contours3d2.png}\n\t\\caption{Isosurfaces of sensitivity patterns $|\\beta(\\vec r\\,)|$ inside a helmet array for two of the magnetometer loops marked in red. The arrow depicts the direction of the precession field $\\vec B_\\mathrm L$ during readout ({\\it e.g.}\\ $\\vec{B}_0$). Note that, because of the precession plane, there are insensitive directions (``blind angles'') in the profiles, depending on the relative orientation of $\\vec B_\\mathrm L$.}\n\t\\label{figSensContours}\n\\end{figure}\n\n\nAnother notable property of the sensitivity $|\\beta|=B_{\\rm s}$ from Eq.~\\eqref{eqCircleLead} is that if $l$ is fixed, there is a value of $R$ above which the sensitivity starts to decrease, {\\it i.e.}, part of the flux going through the loop comes back at the edges canceling a portion of the signal. By requiring $\\partial B_{\\rm s}\/\\partial R$ to vanish, one obtains $R=l\\sqrt{2}$, the loop radius that gives the maximum signal. Interestingly, however, if instead of the perpendicular ($z$) distance, $l$ is taken as the closest distance to the pickup-coil winding, then the coil is on a spherical surface of radius $R_\\mathrm a = l$. Now, based on Pythagoras's theorem, $R^2 + l^2$ in Eq.~\\eqref{eqCircleLead} is replaced with $l^2$. In other words, the sensor field is simply $\\vec B_\\mathrm{s} = \\widehat e_z \\,\\mu R^2\/2l^3$, so the scaling of $\\alpha = 2$ happens to be the same as for distant sources in this simple case. \n\nImportantly, however, the \\emph{noise} mechanisms also depend on $R$, and moreover, the situation is complicated by the presence of multiple sensors. These matters are discussed in Secs.~\\ref{sSingleSensor}--\\ref{sArrays}.\n\n\n\n\n\\section{Noise mechanisms and scaling} \\label{sSingleSensor}\n\nThe signal from each measurement channel, corresponding to a pickup coil in the sensor array, contains flux noise that can originate from various sources. Examples of noise sources are the sensor itself, noise in electronics that drives MRI coils, cryostat noise, magnetic noise due to thermal motion of particles in other parts of the measurement device and in the sample, noise from other sensors, as well as environmental noise. This section is devoted to examining the various noise mechanisms and how the noise can be dealt with. Unless stated otherwise, noise is considered a random signal with zero average. We use amplitude scaling exponents $\\alpha$ to characterize the dependence of noise on pickup-coil size and type. \n\n\\subsection{Flux coupling and SQUID noise} \\label{ssIntrinsic}\n\nFor estimates of SQUID sensor noise as a function of pickup coil size, a model for the sensor is needed. As explained in Sec.~\\ref{ssPickups}, the signal is coupled into the SQUID loop via an input circuit. \nIn general, the input circuit may consist of a sequence of one or more all-superconductor closed circuits connected by intermediate transformers. Via inductance matching and coupling optimization, these circuits are designed to efficiently couple the flux signal into the SQUID loop. \n\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=\\columnwidth]{inputcircuit.pdf}\n\t\\caption{Simplified schematic of a superconducting SQUID input circuit. Zero or more intermediate transformers (dashed box) may be present.}\n\t\\label{figInputCircuit}\n\\end{figure}\nIntermediate transformers can be useful for optimal coupling of a large pickup coil to a SQUID-coupled input coil, as analyzed {\\it e.g.} in Ref.~\\cite{Mates2014}. To further understand the concept, consider a two-stage input circuit where a pickup coil ($L_\\mathrm p$) is connected to a transmitting inductor $L_1$ to form a closed superconducting path; see Fig.~\\ref{figInputCircuit}. Ideally, the distance between the two coils is fairly small in order to avoid signal loss due to parasitic inductances of the connecting traces or wiring. The total inductance of this flux-coupling circuit by itself is $L_\\mathrm p + L_1$. The primary is coupled to a secondary inductor $L_2$ with mutual inductance $M_{12}$. As the magnetic flux picked up in $L_\\mathrm p$ changes by $\\mathrm\\Delta\\Phi_{\\mathrm p}$, there is a corresponding change $\\mathrm\\Delta J_1$ in the supercurrent flowing in the circuit such that the flux through the closed path remains constant. This passes the flux signal onwards to $L_2$ which forms another flux-transfer circuit together with the input coil $L_\\mathrm i$, which couples inductively into the SQUID.\n\nSuperconductivity has two important effects on the transmission of flux into the next circuit. First, the presence of superconducting material close to a coil tends to reduce the coil inductance because of the Meissner effect: the magnetic flux is expelled and the material acts as a perfect diamagnet. This effect is included in the given inductances $L_\\mathrm{p}$ and $L_1$. The other effect emerges when the flux is transmitted into another closed superconducting circuit, such as via $M_{12}$. This is because the transmitting coil is subject to the counteracting flux $M_{12}^2 \\mathrm\\Delta J_1\/(L_2 + L_\\mathrm{i})$ from the receiving coil of the other circuit. Now current $\\mathrm\\Delta J_1$ only generates a flux $[L_1 - M_{12}^2\/(L_2 + L_\\mathrm{i})]\\mathrm\\Delta J_1$ in $L_1$. Closing the secondary circuit thus changes the inductance from $L_1$ to \n\\begin{equation}\nL_1^\\prime = L_1 - \\frac{M_{12}^2}{L_2 + L_\\mathrm{i}} = L_1\\left(1 - \\frac{k_{12}^2}{1 + L_\\mathrm{i}\/L_2}\\right)\\,, \n\\end{equation}\nwhere the last form is obtained by expressing the mutual inductance in terms of the coupling constant $k_{12}$ ($|k_{12}|<1$) as $M_{12} = k_{12}\\sqrt{L_1L_2}$. Note that we do not include a counteracting flux from the SQUID inductance $L_\\mathrm{S}$ back into $L_\\mathrm{i}$, {\\it i.e.}, no screening from the biased SQUID loop. However, like other inductances, $L_\\mathrm{i}$ does include the effect of the presence of the nearby superconductors through the Meissner effect.\n\n\nThe change of flux though the dc SQUID loop is now obtained as\n\\begin{align}\n\\mathrm\\Delta \\Phi_\\mathrm{S} &= M_\\mathrm{iS}\\mathrm\\Delta J_2 = \\frac{M_\\mathrm{iS}M_{12}}{L_2 + L_\\mathrm{i}}\\mathrm\\Delta J_1 \\\\&= \\frac{M_\\mathrm{iS}M_{12}}{(L_2 + L_\\mathrm{i})(L_\\mathrm{p} + L_1) - M_{12}^2} \\mathrm\\Delta\\Phi_\\mathrm{p} \\, ,\n\\end{align} \nor, with $M_\\mathrm{iS} = k_\\mathrm{iS}\\sqrt{L_\\mathrm{i}L_\\mathrm{S}}$ and defining $\\chi_1$ and $\\chi_2$ such that $L_1 = \\chi_1 L_\\mathrm{p}$ and $L_2 = \\chi_2 L_\\mathrm{i}$, we have\n\\begin{equation}\\label{eqSQUIDFluxChi}\n\\frac{\\mathrm\\Delta \\Phi_\\mathrm{S}}{\\mathrm\\Delta \\Phi_\\mathrm{p}} =  \\frac{k_\\mathrm{iS}\\sqrt{L_\\mathrm{S}}}{\\sqrt{L_\\mathrm{p}}}\\times \\frac{k_{12}\\sqrt{\\chi_1\\chi_2} }{\\chi_1\\chi_2(1 - k_{12}^2) + \\chi_1 + \\chi_2 + 1} \\, .\n\\end{equation}\n\nFor a given pickup coil, $\\chi_1$ and $\\chi_2$ can usually be chosen to maximize the flux seen by the SQUID. While the function in Eq.~\\eqref{eqSQUIDFluxChi} is monotonous in $k_{12}$, there is a single maximum with respect to parameters $\\chi_1,\\chi_2 > 0$. Noting the symmetry, we must have $\\chi_1 = \\chi_2 =: \\chi$, and the factor in Eq.~\\eqref{eqSQUIDFluxChi} becomes $k_{12}\\chi\/[\\chi^2(1-k_{12}^2) + 2\\chi + 1]$, which is maximized at $\\chi = 1\/\\sqrt{1-k_{12}^2}$. At the optimum, the coupled flux is given by\n\\begin{equation}\n\\frac{\\mathrm\\Delta \\Phi_\\mathrm{S}}{\\mathrm\\Delta \\Phi_\\mathrm{p}} =  \n\\frac{k_\\mathrm{iS}k_{12}\\sqrt{L_\\mathrm{S}}}{2\\sqrt{L_\\mathrm{p}}\\left(1 + \\sqrt{1-k_{12}^2}\\right)}  \\underset{k_{12} \\rightarrow 1^-}{\\longrightarrow}\n\\frac{k_\\mathrm{iS}}{2}\\sqrt\\frac{L_\\mathrm{S}}{L_\\mathrm{p}}\n\\,.\n\\end{equation}\nNotably, with a $k_{12} \\approx 1$, the coupling corresponds to a perfectly matched single flux-coupling circuit \\cite{SQUID-HB}. Already at $k_{12} = 0.8$, 50\\% of the theoretical maximum is achieved, while matching without an intermediate transformer may cause practical difficulties or parasitic resonances.\n\nWhen referred to SQUID flux $\\Phi_\\mathrm{S}$, the noise in the measured SQUID voltage in the flux-locked loop corresponds to a noise spectral density $S_{\\Phi_{\\rm S}}(f)$ at frequency $f$. As the signal transfer from the pickup coil to the SQUID is given by Eqs.~\\eqref{eqSQUIDFluxChi}, the equivalent flux resolution referred to the signal through the pickup coil can be written as\n\\begin{equation} \\label{eqNoiseSDens1}\nS_{\\Phi_{\\rm p}}^{1\/2}(f) = \\frac{2\\sqrt{L_{\\rm p}}\\left(1 + \\sqrt{1-k_{12}^2}\\right)}{k_\\mathrm{iS}k_{12}\\sqrt{L_\\mathrm{S}}}S_{\\Phi_{\\rm S}}^{1\/2}(f)\\,.\n\\end{equation}\nDue to resonance effects and thermal flux jumps, $L_\\mathrm{S}$ needs to be kept small \\cite{SQUID-HB}. The flexibility of intermediate transformers allows the same model to estimate noise levels with a wide range of pickup coil inductances $L_\\mathrm{p}$.\n\nIn general, the inductance of a coil with a given shape scales as the linear dimensions, or radius $R$, of the coil. If the wire thickness is not scaled accordingly, there will be an extra logarithmic term \\cite{Grover1973}. Even then, within a range small enough, the dependence is roughly $S_{\\Phi_{\\rm p}}^{1\/2} \\propto R^\\alpha$ with $\\alpha = 1\/2$. The case of a magnetometer loop in a homogeneous field then still has a field resolution $S_B^{1\/2}(f)$ proportional to $R^{-3\/2}$.\n\n\n\\subsection{Thermal magnetic noise from conductors} \\label{ssThermalNoise}\n\nElectric noise due to the thermal motion of charge carriers in a conducting medium is called Johnson--Nyquist noise \\cite{Nyquist1928, Johnson1928}. According to Amp$\\grave{\\rm e}$re's law $\\nabla \\times \\vec B = \\mu_0 \\vec J$, the noise currents in the current density $\\vec J$ also produce a magnetic field which may interfere with the measurement. In this view, devices should be designed in such a way that the amount of conducting materials in the vicinity of the sensors is small. However, there is a lower limit set by the conducting sample---the head. Estimations of the sample noise \\cite{Myers2007} have given noise levels below $0.1\\,{\\rm fT}\/\\sqrt{\\rm Hz}$, consistent with a recent experimental result of $55\\,{\\rm aT}\/\\sqrt{\\rm Hz}$ \\cite{Storm2019}. Other noise sources still exceed those values by more than an order of magnitude. More restrictingly, it is difficult to avoid metals in most applications. \n\n\nTo keep the SQUID sensors in the superconducting state, the array is kept in a helmet-bottom cryostat filled with liquid helium at $4.2\\,$K. The thermal superinsulation of a cryostat usually involves a vacuum as well as layers of aluminized film to suppress heat transfer by radiation \\cite{SQUID-HB}. The magnetic noise from the superinsulation can be reduced by breaking the conducting materials into small isolated patches. Seton {\\it et al.} \\cite{Seton2005} used aluminium-coated polyester textile, which efficiently breaks up current paths in all directions. By using very small patches, one can decrease the field noise at the sensors by orders of magnitude, although with increased He boil-off \\cite{Tervo2016MSc}.\n\n\nTo look at the thermal noise from the insulation layers in some more detail, consider first a thin slab with conductivity $\\sigma$ on the $xy$ plane at temperature $T$. Johnson--Nyquist currents in the conductor produce a magnetic field $\\vec{B}(x,y,z,t)$ outside the film. For an infinite (large) slab, the magnitude of the resulting field noise depends, besides the frequency, only on $z$, the distance from the slab (assume $z>0$). At low frequencies, the spectral densities $S_{B_\\alpha}$ ($\\alpha = x,y,z$) corresponding to Cartesian field noise components are then given by \\cite{Varpula1984}\n\\begin{equation} \\label{eqBz1}\nS_{B_z}^{1\/2} = \\sqrt{2} S_{B_x}^{1\/2} = \\sqrt{2} S_{B_y}^{1\/2} =\\frac{\\mu}{2}\\sqrt{ \\frac{k_{\\rm B}T}{2\\pi} \\frac{\\sigma d}{z(z+d)}}\\,, \n\\end{equation}\nwhere $d$ \nis the thickness of the slab and $k_{\\rm B}$ the Boltzmann constant.\n\nThe infinite slab is a good approximation when using a flat-bottom cryostat or when the radius of curvature of the cryostat wall is large compared to individual pickup loops. Consider a magnetometer pickup loop with area $A$ placed parallel to the conducting films in the insulation---to measure the $z$ component of the magnetic field, $B_z$. The coupled noise flux is the integral of $B_z$ over the loop area. If the loop is small, the noise couples to the pickup circuit as $S_{\\Phi}^{1\/2} = S_{B_z}^{1\/2}A$. A coil of size $R$ then sees a flux noise proportional to $S_{B_z}^{1\/2}R^2$, that is, $\\alpha = 2$. \n\nInstead, if the pickup coil is large, the situation is quite different. The instantaneous magnetic field depends on all coordinates and varies significantly over the large coil area. Consider the noise field at two points in the plane of the coil. The fields at the two points are nearly equal if the points are close to each other. However, if the points are separated by a distance larger than a correlation length $\\lambda_{\\rm c}(z)$, the fields are uncorrelated. Therefore, if $R \\gg \\lambda_c$, the coupled flux is roughly a sum of $A\/\\lambda_{\\rm c}^2$ uncorrelated terms from regions in which the field is correlated. Each term has a standard deviation of order $S_{B_z}^{1\/2}\\lambda_{\\rm c}^2$. The spectral density of the cryostat noise is then\n\\begin{equation} \\label{eqSPhiSlab}\nS_{\\Phi,\\rm c}(f) \\approx A S_{B_z}(\\vec r, f)\\lambda_{\\rm c}^2(\\vec r\\,)\\,.\n\\end{equation}\nMost importantly, the flux noise amplitude $S_{\\Phi,\\rm c}^{1\/2}$ is directly proportional to the coil size $R$, and we now have $\\alpha=1$. Still, the noise increases to a higher power of $R$ than the sensor noise, which according to section \\ref{ssIntrinsic} scales as $\\sqrt{R}$ and hence dominates in small pickup coils.\n\nFor a continuous film, the correlation length $\\lambda_{\\rm c}$ can be estimated from data in Ref.~\\cite{Nenonen96} to be around several times $z$. The correlation at distances smaller than $\\lambda_c$ is due to two reasons. First, the magnetic field due to a small current element in the conductor is spread in space according to the Biot--Savart law. Second, the noise currents in elements close to each other are themselves correlated. The latter effect is broken down when the film is divided into small patches; only very small current loops can occur, and the noise field starts to resemble that of Gaussian uncorrelated magnetic point dipoles throughout the surface. In this case, Eq.~\\eqref{eqBz1} is no longer valid, but the approximate relation of Eq.~\\eqref{eqSPhiSlab} still holds---now with a smaller $\\lambda_\\mathrm{c}$. \n\nThe magnetometer case is easily extended to first-order planar gradiometers parallel to the superinsulation layers [Fig.~\\ref{figSQUID}($\\mathrm b, \\mathrm f$)]. For a very small baseline, $b\\ll\\lambda_c$, the field noise is effectively homogeneous and thus cancels out. However, when $b\\gg\\lambda_c$, the spectral density of the noise power is twice that of a single loop. \n\n\n\n\n\\subsection{MRI electronics, coils and other noise sources}\n\nAs explained in Sec.~\\ref{ssULFMRI}, MRI makes heavy use of applied magnetic fields. The fields are generated with dedicated current sources, or amplifiers, to feed currents into coils wound in different geometries. As opposed to applying static fields, a major challenge arises from the need for oscillating pulses and the desire to quickly switch on and off all fields, including not only readout gradients but also the main field $\\vec B_0$, which requires an ultra-high dynamic range to avoid excess noise. Switching of $\\vec B_0$ enables full 3-D field mapping for imaging of small electric currents in volume \\cite{Zevenhoven2014amp}. Noise in the coil currents can be a major concern in the instrumentation. The contribution from $\\vec B_0$ ideally scales with pickup coil size as $R^\\alpha$, $\\alpha=2$ for a magnetometer, and noise in linear gradients essentially scales as $\\alpha=2$ in magnetometers as well as fixed-baseline gradiometers. With $b \\propto R$, first-order gradiometers experience noise from linear gradient coils according to $\\alpha=3$.\n\nMRI coils themselves also produce Johnson--Nyquist noise. In particular, the polarizing coil is often close to the sensors and made of thick wires as it should be able to produce relatively high fields. This allows thermal electrons to form current loops that generate field noise with complicated spatial characteristics, which is detrimental to image quality and should be eliminated. Another approach is to use litz wire, which is composed of thin wires individually coated with an insulating layer. This prevents significant noise currents perpendicular to the wire and eliminates large current loops. However, efficient uniform cooling of litz wire is problematic, leading to larger coil diameters. Increasing the coil size, however, significantly increases harmful transients in the system as well as the power and cooling requirements \\cite{Zevenhoven2011MSc}. Instead, we have had promising results with thin custom-made superconducting filament wire and DynaCAN (Dynamical Coupling for Additional dimeNsions) in-sequence degaussing waveforms to solve the problem of trapped flux \\cite{Zevenhoven2011MSc, Zevenhoven2013degauss}; optimized oscillations at the end of a pulse can expel the flux from the superconductor. Such coils contain much less metal, and significantly reduce the size of current loops that can generate magnetic noise.\n\nA significant amount of noise also originates from more distant locations. Power lines and electric devices, for instance, are sources that often can not be removed. Indeed, magnetically shielded rooms (MSRs) effectively attenuate such magnetic interference. However, pulsed magnetic fields inside the shielded room induce eddy currents exceeding $1\\,$kA in conductive MSR walls \\cite{Zevenhoven2014eddy}, leading to strong magnetic field transients that not only saturate the SQUID readout, but also seriously interfere with the nuclear spin dynamics in the imaging field of view. Even a serious eddy current problem can again be solved with a DynaCAN approach where optimized current waveforms are applied in additional coil windings to couple to the complexity of the transient \\cite{Zevenhoven2015}.\n\nNoise from distant sources typically scales with the pickup coil size with an exponent at least as large as the signal from far-away sources: $\\alpha=2+k$ for a $k^{\\rm th}$-order gradiometer (see Sec.~\\ref{ssPickups}). Although the noise detected by gradiometers scales to a higher power than with magnetometers ($k=0$), gradiometers have the advantage that they, in principle, do not respond to a uniform field. For a higher-order gradiometer that is not too large, the environmental noise is nearly uniform in space, and therefore effectively suppressed by the pickup coil geometry. Gradiometers with relatively long baselines can also be seen as magnetometers when the source is close to one of the loops. Still, they function as gradiometers from the perspective of distant noise sources. A similar result applies for so-called software gradiometers, which can, for example, be formed by afterwards taking the difference of the signals of two parallel magnetometers. However, in Sec.~\\ref{ssArrayNoise}, a more sophisticated technique is described for minimizing noise in the combination of multiple channels.\n\nAt very low system noise levels, other significant noise mechanisms include noise due to dielectric losses. Electrical activity in the brain can also be seen as a source of noise. This noise, however, is strongest at frequencies well below $1\\,$kHz. Using Larmor frequencies in the kHz range may therefore be sufficient for spectral separation of brain noise from MRI.\n\nThe amplitude scaling exponents $\\alpha$ for signal and noise are summarized in Table \\ref{tabExponents}. The notation in later sections refers to the scaling of flux signal and noise in terms of $\\alpha_\\mathrm s$ and $\\alpha_\\mathrm n$, respectively. For a single sensor, the SNR scaling $R^\\delta$ is given by $\\delta = \\alpha_\\mathrm s - \\alpha_\\mathrm n$.\n\n\n\n\n\n\\begin{table}\n\\caption{Amplitude scaling exponents $\\alpha$ for the flux noise standard deviation $\\sigma \\propto R^\\alpha$ as well as the signal, given different pickup-coil geometries and noise mechanisms.}\\label{tabExponents}\\vspace{5mm}\n\t\\centering\n\t\\begin{tabular}{l@{$\\;\\;$}c@{$\\;\\;$}c@{$\\;\\;$}c}\n\tPickup type (see Fig. 1) $\\rightarrow$ & M0 & AG$k$ & PG$k$\\\\\n\t\\hline\n\tSensor noise (optimally matched) & 1\/2 & 1\/2 & 1\/2 \\\\\n    Sensor noise (unmatched, large $L_\\mathrm p$) & 1 & 1 & 1 \\\\\n\tDistant source, $b \\propto R$ & 2 & $2 + k$ & $2 + k$ \\\\\n\tDistant source, $b$ fixed & 2 & 2 & -- \\\\\n        $\\vec B_0$ amplifier & 2 & $0^*$ & $0^*$ \\\\\n        Gradient amplifiers, $b \\propto R$, $k \\le 1$ & 2 & 3 & 3 \\\\\n        Gradient amplifiers, $b$ fixed & 2 & 2 & -- \\\\\n\tCryostat noise, small $R$ & 2  &  2  &  $2 + k$   \\\\\n\tCryostat noise, large $R$ & 1  &  1  &  1       \\\\\n\t\\hline\n\t\\end{tabular}\n    \\\\\\vspace{1mm}$^*$ Larger in practice, because of gradiometer \\\\imbalance and field inhomogeneities.\n\\end{table}\n\n\n\n\\section{Sensor arrays} \\label{sArrays}\n\n\n\n\n\n\n\\subsection{Combining data from multiple channels} \\label{ssArrayNoise}\n\nIt is common to work with absolute values of the complex images to eliminate phase shifts. Images from multiple channels can then be combined by summing the squares and taking the square root. This procedure, however, causes asymmetry in the noise distribution and loses information that can be used for improved combination of the data. If the sensor array and the correlations of noise between different sensors are known, the multi-channel data can be combined more effectively. \n\nIn the following, we show that, where multiple sensors can form a software gradiometer, an array of $N$ sensors can form an $N^\\text{th}$-order combination optimized to give the best SNR for each voxel.\n\nTo follow the derivation in Ref.~\\cite{Zevenhoven2011}, consider a voxel centered at $\\vec r$, and $N$ sensors indexed by $j = 1,2, ...,N$. Based on Sec.~\\ref{ssULFMRI}, each sensor has a unit magnetization image $s_j(\\vec r\\,) = \\beta_j^*(\\vec r\\,)V$, where $\\beta_j$ and $V$ are the sensitivity profile and voxel volume, respectively. The absolute value $|s_j|$ gives the sensed signal amplitude caused by a unit magnetization in the voxel, precessing perpendicular to $\\vec B_\\mathrm L$. The complex phase represents the phase shift in the signal due to the geometry. To study the performance of the array only, we set $V$ to unity.\n\nFor a voxel centered at $\\vec r$, we have a vector of reconstructed image values ${\\bf v} = [v_1,v_2, ...,v_N]^\\top  $ corresponding to the $N$ sensors. At this point, the values $v_j$ have not been corrected according to the sensitivity. The linear combination that determines the final voxel value $u$ can be written in the form\n\\begin{equation}\\label{eqVoxelLinComb}\nu = \\sum_{j=1}^{N} a_j^*v_j = {\\bf a}^\\dagger {\\bf v}\\,,\n\\end{equation}\nwhere $^\\dagger $ denotes the conjugate transpose. Requiring that the outcome is sensitivity-corrected sets a condition on the coefficient vector ${\\bf a} = [ a_1, ...,a_N]^\\top $. In the absence of noise, a unit source magnetization gives $v_j = s_j(\\vec r\\,)$. The final voxel value $u$ should represent the source, which leads to the condition\n\\begin{equation} \\label{eqConstr}\n{\\bf a}^\\dagger {\\bf s} = 1\\,.\n\\end{equation}\nBelow, we show how ${\\bf a} = [ a_1, ..., a_N]^\\top $ should be chosen in order to maximize the SNR in the final image given the sensor array and noise properties.\n\nThe single-sensor image values $v_i$ can be written in the form $v_j = w_j + \\xi_j$ where $w_j$ is the `pure' signal and $\\xi_j$ is the noise. The noise terms $\\xi_j$ can be modeled as random variables, which, for unbiased data, have zero expectation: ${\\rm E}(\\xi_j)=0$. If there is a bias, it can be measured and subtracted from the signals before this step. The expectation of the final value of this voxel is then\n\\begin{equation}\\label{eqExpectU}\n{\\rm E}(u) = {\\rm E}\\left[{\\bf a}^\\dagger  ({\\bf w + \\boldsymbol \\xi})\\right] = {\\bf a}^\\dagger  {\\bf w}\\,.\n\\end{equation}\nThe noise in the voxel is quantified by the variance of $u$. Eqs. \\eqref{eqVoxelLinComb} and \\eqref{eqExpectU} yield $u = {\\rm E}(u) + {\\bf a}^\\dagger  \\boldsymbol\\xi$, leading to\n\\begin{equation} \\label{eqVaru}\n{\\rm Var}(u) = {\\rm E}\\left[|u-{\\rm E}(u)|^2\\right] = {\\rm E}\\left[{\\bf a}^\\dagger  \\boldsymbol{\\xi}\\boldsymbol{\\xi}^\\dagger  {\\bf a}\\right] = {\\bf a}^\\dagger  {\\mathbf\\Sigma}{\\bf a}\\,,\n\\end{equation}\nwhere ${\\mathbf\\Sigma} = {\\rm E}(\\boldsymbol \\xi \\boldsymbol\\xi^\\dagger  )$ identifies as the noise covariance matrix.\nFor simple cases, ${\\mathbf\\Sigma}$ is the same for all voxels. However, it may vary between voxels if, for instance, the voxels are of different sizes.\n\nNow, the task is to minimize the noise ${\\bf a}^\\dagger  {\\mathbf\\Sigma}{\\bf a}$ subject to the constraint in  Eq.~\\eqref{eqConstr}. The Lagrange multiplier method turns the problem into finding the minimum of\n\\begin{equation} \\label{eqLagrange}\nL = {\\bf a}^\\dagger  {\\mathbf\\Sigma}{\\bf a} - \\lambda(1-{\\bf a}^\\dagger  {\\bf s})\n\\end{equation}\nwith respect to ${\\bf a}$, while still requiring that Eq.~\\eqref{eqConstr} holds. From the constraint it follows that ${\\bf a}^\\dagger  {\\bf s}$ is real, so it may be replaced by $({\\bf a}^\\dagger  {\\bf s}+{\\bf s}^\\dagger  {\\bf a})\/2$ in Eq.~\\eqref{eqLagrange}. By `completing the square' in Eq.~\\eqref{eqLagrange}, one obtains\n\\begin{equation}\nL = {(\\bf a - {\\bf \\tilde a})}^\\dagger  {\\mathbf\\Sigma}{(\\bf a- {\\bf\\tilde a})} - \\lambda + {\\rm constant}\\,,\n\\end{equation}\nwhere ${\\bf \\tilde a}$ satisfies \n\\begin{equation}\\label{eqLagrange2}\n2{\\mathbf\\Sigma}{\\bf\\tilde a} = -\\lambda {\\bf s}\\,.\n\\end{equation}\nSince $\\mathbf\\Sigma$, being a covariance matrix, is positive (semi)definite, the minimum of $L$ is found at ${\\bf a} = {\\bf\\tilde a}$. \n\nFurther, ${\\mathbf\\Sigma}$ is always invertible, as the contrary would imply that some non-trivial linear combination of the signals would contain zero noise. Multiplying Eq.~\\eqref{eqLagrange2} by ${\\bf s}^\\dagger  {\\mathbf\\Sigma}^{\\text{-1}}$ from the left and using Eq.~\\eqref{eqConstr} leads to $\\lambda = -2\/{\\bf s}^\\dagger  {\\mathbf\\Sigma}^{\\text{-1}}{\\bf s}$. When this expression for $\\lambda$ is put back into Eq.~\\eqref{eqLagrange2}, the optimal choice for the coefficient vector ${\\bf a} = \\tilde{\\bf a}$ is obtained as\n\\begin{equation} \\label{eqOptimalCoeff}\n{\\bf a} = \\frac{{\\mathbf\\Sigma}^{\\text{-1}}{\\bf s}}{{\\bf s}^\\dagger  {\\mathbf\\Sigma}^{\\text{-1}}{\\bf s}}\\,.\n\\end{equation}\nSimilar to Eq.~(7) of Ref.~\\cite{Capon1970}, Eqs. \\eqref{eqVaru} and \\eqref{eqOptimalCoeff} reveal the final noise variance $\\sigma_{\\rm fin}^2$ for the given voxel position,\n\\begin{equation}\\label{eqNoiseVar}\n\\sigma_{\\rm fin}^2 = {\\bf a}^\\dagger  {\\mathbf\\Sigma}{\\bf a} = \\frac{1}{{\\bf s}^\\dagger  \n{\\mathbf\\Sigma^{\\text{-1}}}{\\bf s}}\\,.\n\\end{equation}\n\nIn the above derivation, we assumed little about how the individual single-sensor data were acquired. In fact, the only significant requirement was that the sensitivities $s_i$ are well defined and accessible. As discussed previously, the signal can be modeled to high accuracy at ULF (see Sec.~\\ref{ssULFMRI}).\n\n\\subsection{Figures of merit and scaling for arrays} \\label{ssFigures}\n\nGiven the $N^\\mathrm{th}$-order combination from Eqs.~\\eqref{eqVoxelLinComb} and \\eqref{eqOptimalCoeff}, the contribution of the sensor array to the voxel-wise image SNR is given by Eq.~\\eqref{eqNoiseVar}. We define the \\emph{array-sensitivity-to-noise ratio} aSNR as\n\\begin{equation} \\label{eqaSNR}\n\\text{aSNR} = \\sqrt{\\mathbf s^\\dagger  \\mathbf \\Sigma^{\\text{-1}} \\mathbf s}\\,.\n\\end{equation}\nWhen each sensor in the array sees an equal flux noise level $\\sigma$, the aSNR$^{1\/2}$ takes the form\n\\begin{equation}\n\\text{aSNR} = \\frac{\\sqrt{\\mathbf s^\\dagger  \\mathbf X^{\\text{-1}} \\mathbf s}}{\\sigma} = \\frac{\\text{array sensitivity}}{\\text{noise level}}\\,,\n\\end{equation}\nwhere $\\mathbf X = \\mathbf\\Sigma\/\\sigma^2$ is the dimensionless noise \\emph{correlation} matrix. We refer to the quantity $\\sqrt{\\mathbf s^\\dagger  \\mathbf X^{\\text{-1}} \\mathbf s}$ as the \\emph{array sensitivity}, which for weak correlation is given approximately as $||\\mathbf s||_2$. Scaling law exponents for the array sensitivity are denoted by $\\alpha_\\mathrm a$, and for the aSNR by $\\delta = \\alpha_\\mathrm a - \\alpha_\\mathrm n$.\n\n\n\\subsection{Correlation of noise between sensors} \\label{ssCorrEffect}\n\nAs already seen in Secs.~\\ref{ssArrayNoise} and \\ref{ssFigures}, the aSNR is affected by the correlation of random noise between different single-sensor channels. There are two main reasons for such correlations. First, a noise source that is not an intrinsic part of a sensor can directly couple to many sensors. For instance, thermal noise in conductors close to the sensors may result in such correlated noise (see Sec.~\\ref{ssThermalNoise}). Second, the pickups of the sensors themselves are coupled to each other through their mutual inductances. This cross-coupling increases noise correlation and may also affect the sensitivity profiles via signal cross-talk.\n\nTo see the effect of noise correlation on the image SNR, consider a noise covariance matrix of the form\n\\begin{equation} \\label{eqCovSimplified}\n{\\mathbf\\Sigma} = \\sigma^2({\\bf I} + {\\bf C})\\,,\n\\end{equation}\nwhere ${\\bf I}$ is the identity matrix and $\\bf C$ contains the correlations between channels (the off-diagonal elements of $\\mathbf X$). In words, each channel has a noise variance of $\\sigma^2$ and channels $p$ and $q$ have correlation $C_{pq}={\\rm E}(\\xi_p\\xi_q^*)\/\\sigma^2$. Assume further that absolute values of the correlations $C_{pq}$ are substantially smaller than one. \n\nTo first order in $\\mathbf C$, the inverse of $\\mathbf\\Sigma$ is obtained as $\\mathbf\\Sigma^{-1} \\approx \\sigma^{-2}({\\bf I} - {\\bf C})$. The SNR in the final image, according to Eq.~\\eqref{eqNoiseVar}, is then proportional to $\\sigma_{\\rm fin}^{-1}$, with\n\\begin{align}\n \\sigma_{\\rm fin}^{-2} &\\approx \\sigma^{-2}\\left({\\bf s}^\\dagger  {\\bf s} - {\\bf s}^\\dagger  {\\bf C}{\\bf s}\\right)\\nonumber\\\\\n &= \\sigma^{-2}\\left\\|{\\bf s}\\right\\|_2^2 - 2\\sigma^{-2}\\sum_{p<q}{\\rm Re}\\left(s_p^*s_qC_{pq}\\right)\\,.\\label{eqCorrEffect}\n\\end{align}\nClearly, the effect of correlations on the image SNR is governed by the sum in Eq.~\\eqref{eqCorrEffect}. Assume that the dominant terms in the sum correspond to adjacent sensors $p$ and $q$. For voxels not too close to the sensors, the sensitivities $s_p$ and $s_q$ are similar, and therefore $s_p^*s_q \\approx |s_p|^2 \\approx |s_q|^2$. Further, if the noise correlation between the adjacent sensors is positive, one has ${\\rm Re}\\,(s_p^*s_q C_{pq}) > 0$. This leads to the conclusion that the noise correlation tends to decrease the image SNR.\n\nWhile the assumptions made in the above discussion may not always be exactly correct, the result is an indication that the correlation of noise between adjacent sensors is usually harmful---even if it is taken into account in reconstruction. Moreover, the actions taken in order to reduce noise correlation are often such that the noise variances decrease as well. For instance, eliminating a noise source from the vicinity of the sensor array does exactly that. \n\nCorrelation can also be reduced by minimizing the inter-sensor cross-talk, for instance by designing a sensor array with low mutual inductances between pickup coils. If the mutual inductances are non-zero, the cross-talk can be dramatically reduced by coupling the feedback of the SQUID flux-locked loop to the pickup circuit instead of more directly into the SQUID loop \\cite{SQUID-HB}. This way, the supercurrent in the pickup coil stays close to zero at all times. In theory, the cross-talk of the \\emph{flux signals} can be completely eliminated by this method.\n\nCorrelated noise originating from sources far from the subject's head and the sensor array can also be attenuated by signal processing methods prior to image reconstruction. The {\\it signal space separation} method (SSS) was developed at Elekta Neuromag Oy \\cite{Taulu2005} (now MEGIN) for use with `whole-head' MEG sensor arrays. The SSS method can distinguish between signals from inside the sensor helmet and those produced by distant sources. Now, the strong noise correlation is in fact exploited to significantly improve the SNR. Similar methods may be applicable to ULF MRI as well. To help such methods, additional sensors can be placed outside the helmet arrangement to provide an improved noise reference.\n\nFor sensor array comparisons, we assume that all measures have been taken to reduce correlated noise before image reconstruction. The details of the remaining noise correlation depend on many, generally unknown aspects. Therefore, we set ${\\bf C} = 0$ in Eq.~\\eqref{eqCovSimplified} for a slightly optimistic estimate, {\\it i.e.}, sensor noises are uncorrelated, each having variance $\\sigma^2$.\n\n\\subsection{Filling the array} \\label{ssSizeInfluence}\n\nIn this section, we use general scaling arguments to provide estimations of how the whole sensor array performs as a function of the pickup coil size. Consider a surface, for instance, of the shape of a helmet, and a voxel at a distance $l$ from the surface.\nThe surface is filled with $N$ pickup coils of radius $R$ to measure the field perpendicular to the surface. We assume the pickup coils are positioned either next to each other or in such a way that their areas overlap by a given fraction (see Fig.~\\ref{figGradArrays}). The number of sensors that fit the surface is then proportional to $R^{-2}$.\n\nTake, at first, a voxel far from the sensors; $l \\gg R$. Now, the signal from the voxel is spread over many sensors. For $\\mathbf\\Sigma = \\sigma^2{\\bf I}$, the aSNR is proportional to $\\|{\\bf s}\\|_2\/\\sigma$. Assume that $s_j\\propto R^{\\alpha_\\mathrm s}$ and $\\sigma \\propto R^{\\alpha_\\mathrm n}$, which leads to $\\|{\\bf s}\\|_2 \\propto \\sqrt{N}R^{\\alpha_\\mathrm s}\\propto R^{\\alpha_\\mathrm s-1}$, and finally, \n\\begin{equation}\n{\\rm aSNR} \\propto R^\\delta,\\quad \\delta = \\alpha_\\mathrm a - \\alpha_\\mathrm n = \\alpha_\\mathrm s - \\alpha_\\mathrm n-1\\, .\n\\end{equation}\nHere we thus have array sensitivity scaling according to $\\alpha_\\mathrm a = \\alpha_\\mathrm s - 1$, as opposed to $\\alpha_\\mathrm a = \\alpha_\\mathrm s$ when $N$ is fixed. \nRecall from Sec.~\\ref{ssPickups} that the flux sensitivities scale as $R^{\\alpha_\\mathrm s}$ with $\\alpha_\\mathrm s=2$ for magnetometers and $\\alpha_\\mathrm s=3$ for first-order planar gradiometers, given that $l\\gg R$. \nAssuming, for instance, optimally matched input circuits, the intrinsic flux noise of the sensor in both cases has a power law behavior with exponent $\\alpha_\\mathrm n=1\/2$ (see Sec.~\\ref{ssThermalNoise}), which yields $\\delta=0.5$ and $\\delta=1.5$. This is clearly in favor of using larger pickup coils. Especially for larger $R$, however, the cryostat noise may become dominant, and one has $\\alpha_\\mathrm n\\approx 1$. Now, magnetometer arrays have $\\delta\\approx 0$, {\\it i.e.}, the coils size does not affect the SNR. Still, gradiometer arrays perform better with larger $R$ ($\\alpha_\\mathrm a\\approx 1$).\n\nIn the perhaps unfortunate case that noise sources far from the sensors are dominant, the noise behaves like the signal, that is, $\\alpha_\\mathrm s=\\alpha_\\mathrm n$ and $\\delta=-1$. Unlike in the other cases, a higher SNR would be reached by decreasing the pickup coil size. However, such noise conditions are not realistic in the low-correlation limit. Instead, one should aim to suppress the external noise by improving the system design or by signal processing.\n\nThe breakdown of the assumption of $l \\gg R$ needs some attention. If the voxel of interest is close to the sensor array, the image value is formed almost exclusively by the closest pickup-loop. Now, for non-overlapping pickups, the results for single sensors ($\\alpha_\\mathrm a = \\alpha_\\mathrm s$) are applicable, and the optimum magnetometer size is $R\\approx l$. But then, if the voxel is far from the array (deep in the head), and $R$ is increased to the order of $l$, it is more difficult to draw conclusions. We therefore extend this discussion in Secs.~\\ref{sMethods} and \\ref{sResults} by a computational study.\n\n\n\n\n\n\n\n\n\n\\section{Methods for numerical study} \\label{sMethods}\n\nIn order to be able to compare the performance of different sensor configurations, we used 3-D computer models of sensor arrays and calculated their sensitivities to signals from different locations in the sample.\n\nThe sensitivities of single pickup coils were calculated using $\\vec B_\\textrm s$ from Eq.~\\eqref{eqLeadField}. Evaluating the line integral required the coil path $\\partial S$ to be discretized. The number of discretization points could be kept small by analytically integrating Eq.~\\eqref{eqLeadMethod0} over $n$ straight line segments between consecutive discretization points $\\vec r_j$ and $\\vec r_{j+1}$ (the end point $\\vec r_n = \\vec r_0$):\n\\begin{equation}\\label{eqLeadMethod0}\n \\vec B_\\textrm s(\\vec r\\,) = \\frac{\\mu}{4\\pi}\\sum_{k=0}^{n - 1} \\int_{\\vec r\\,^\\prime = \\vec r_j}^{\\vec r_{j+1}} \\frac{d\\vec r\\,^\\prime \\times (\\vec r-\\vec r\\,^\\prime)}{|\\vec r-\\vec r\\,^\\prime|^3}\\,.\n\\end{equation}\n As shown in Appendix A, this integrates exactly to\n\\begin{equation}\\label{eqLeadMethod1}\n \\vec B_\\mathrm s(\\vec r\\,) = \\frac{\\mu}{4\\pi}\\sum_{j=0}^{n - 1} \\frac{ a_j+a_{j+1}}{a_ja_{j+1}}\\,\\frac{\\vec a_j\\times \\vec a_{j+1}}{a_j a_{j+1} + \\vec a_j\\cdot\\vec a_{j+1}}\\,,\n\\end{equation}\nwhere $\\vec a_j = \\vec r_j-\\vec r$. Besides reducing computational complexity and increasing accuracy, this result allowed exact computation for polygonal coils.\n\nFor a precession field $\\vec B_\\mathrm L = B_\\mathrm L\\widehat e_\\mathrm L$, the single-sensor sensitivities were obtained from Eq.~\\eqref{eqBetaNorm} and the array-sensitivity and aSNR maps were computed according to Sec.~\\ref{ssFigures}. The normalization of the values computed here is somewhat arbitrary; the real image SNR depends on a host of details that are not known at this point (see Sec.~\\ref{ssULFMRI}). However, the results can be used for studying array sensitivity patterns and---with noise levels scaled according to estimated coil inductances---for comparing different possible array setups. \n\n\\section{Results} \\label{sResults}\n\nNumerical calculations were performed for simple spherical sensor arrays (Sec.~\\ref{ssSphereResults}) as well as for realistic configurations (Sec.~\\ref{ssHelmetResults}), {\\it e.g.}, of the shape of a helmet. The former were used for studying scaling behavior of array sensitivities with sensor size and number, extending the discussion in Sec.~\\ref{ssSizeInfluence}. The latter were used for comparing array sensitivity patterns of different potential designs. \n\n\\subsection{Effects of size and number} \\label{ssSphereResults}\n\nA sensor array model was built by filling the surface of a sphere of radius $10\\,$cm (see Fig.~\\ref{figSphere}) with $N$ magnetometers or $N\/2$ planar units of two orthogonal planar first-order gradiometers. Combining one of the magnetometers with one of the gradiometer units would thus give a sensing unit similar to those of the Elekta\/Neuromag MEG system, though circular (radius $R$). All sensors were oriented to measure the radial component of the field. A spherical surface of radius $6\\,$cm was chosen to represent the cerebral cortex. The cortex surface was thus at distance $4\\,$cm from the sensor shell. In addition, the center of the sphere was considered to represent deep parts of the brain.\n\\begin{figure}\n\t\\centering\n        \\includegraphics{sphere.pdf}\\\\\n\t\\caption{Geometry used in numerical analysis of the dependence of array sensitivity as functions of sensor size $R$ and number $N$ at different points inside the imaging volume. Sensors are on a spherical surface of radius $10\\,$cm. A shell with radius $6\\,$cm is representative of points on the cerebral cortex.}\n\t\\label{figSphere}\n\\end{figure}\n\nThe data in Fig.~\\ref{figRDep} show the dependence of the array sensitivity on $R$. Note that the number of sensors is approximately proportional to $R^{-2}$. The largest coil size $R=10\\,$cm corresponds to one magnetometer or gradiometer unit on each of the six faces of a cube. The solid lines correspond to the scaling of the sensitivity as $R^{\\alpha_\\mathrm a}$, $\\alpha_\\mathrm a = \\alpha_\\mathrm s - 1$. For smaller $R$, the scaling laws from Sec.~\\ref{ssSizeInfluence} hold in all cases, and particularly well for gradiometers and deep sources. The scaling law fails most notably with the magnetometer array at the cortex. Indeed, the sensitivity starts to \\emph{decrease} with $R$ when $R$ is very large, as was shown for a special case in Sec.~\\ref{ssPickups}. \n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1.00\\columnwidth]{array_snr_rdep.pdf}\n\t\\caption{Scaling of array sensitivity at the center and on the cortex as depicted in Fig.~\\ref{figSphere}: sphere filled with magnetometer loops and with planar units of two orthogonal gradiometers arranged side by side. Error bars correspond to the minimum and maximum values. Noise scaling with size is included in the figure, illustrating a potential cross-over from sensor noise with $\\alpha_\\mathrm n = 1\/2$ to cryostat noise or suboptimal input circuit matching with $\\alpha_\\mathrm n = 1$. With fixed $N$, array sensitivity scaling is steeper and given by $\\alpha_\\mathrm a = 2, 3$ for planar magnetometers and gradiometers.}\n\t\\label{figRDep}\n\\end{figure}\n\nThe error bars in Fig.~\\ref{figRDep} correspond to the minimum and maximum value of the sensitivity at the cortex while the data symbols correspond to the average value. Despite the strong orientational dependence of single sensors (see Sec.~\\ref{ssPickups}), the array sensitivities are fairly uniform at the cortex. Only at large $R$ do the orientational effects emerge. \n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=\\columnwidth]{array_snr_ndep.pdf}\n\t\\caption{Scaling of array sensitivity as $\\sqrt{N}$ at the center and on the cortex as depicted in Fig.~\\ref{figSphere}, when the pickup coil radius is fixed at $R=1.44\\,$cm: (left) $N$ magnetometers, (right) $N\/2$ planar units of two orthogonal gradiometers. Error bars correspond to the minimum and maximum values.}\n\t\\label{figSparse}\n\\end{figure}\n\nFigure \\ref{figSparse} shows a different dataset on how the array sensitivity changes with how densely the sensors are packed into the array. In this case, a varying number of magnetometer coils or gradiometer units with fixed radius $R=1.44\\,$cm was distributed on the spherical shell. The aSNR of voxels at the center scales as $\\sqrt{N}$ to an excellent accuracy. While the average sensitivity at points on the cortex also obeys $\\sqrt{N}$ scaling remarkably well, the uniformity drops dramatically when $N$ is lowered below roughly 30 sensors. Closer to the sensors, {\\it e.g.}\\ on the scalp, this effect is even more pronounced.\n\n\\subsection{Realistic sensor configurations} \\label{ssHelmetResults}\n\nFigure~\\ref{fig:snrfigures} presents several possible sensor configurations and provide maps of $\\log_{10}(\\text{aSNR})$ for their comparison for their comparison. The data shown are sagittal slices of the 3-D maps, {\\it i.e.}, on the symmetry plane of the sensor array. Other slices, however, displayed similar performance at the cortex. Also changing the direction of the precession field $\\vec B_\\mathrm L$ had only a minor effect on the SNR in the region of interest. In all cases shown here, $\\vec B_\\mathrm L$ was parallel to the $y$ axis, which is perpendicular to the visualization plane. Note that this contrasts MRI convention, where the $\\vec B_\\mathrm L$ direction is considered fixed and always along the $z$ axis.\n\nIn most cases, the sensors are arranged on a helmet surface at 102 positions as in the Elekta\/Neuromag system. Again, magnetometers and planar double-gradiometer units are considered separately (here, $R=1.25\\,$cm, resembling conventional MEG sensors). The same flux noise level was assumed for magnetometers and planar gradiometers of the same size. In addition, we consider arrays with axial gradiometers as well as radially oriented planar gradiometers, both cases having $k=1$, $b=4\\,$cm and $R=1.25\\,$cm. Configurations with 102 overlapping units with $R=2.5\\,$cm are also considered, as well as the existing Los Alamos 7-channel coil geometry \\cite{Zotev2007} and the single large second-order gradiometer at UC Berkeley \\cite{Clarke2007} (see figure caption). For long-baseline gradiometers with $k=1$, $L_\\mathrm p$ was estimated to be twice that of a single loop, and six times for $k=2$.\n\nWith planar sensor units of $R=1.25\\,$cm [Fig.~\\ref{fig:snrfigures}(a--b)], the aSNR for 102 magnetometers is three times that of 204 gradiometers at the cerebral cortex. At the center of the head, the difference is almost a whole order of magnitude in favor of the magnetometers. Therefore, the small gradiometers bring little improvement to the image SNR if the magnetometers are in use. However, as shown previously, especially gradiometer performance improves steeply with coil size. Allowing the coils to overlap with $R = 2.5\\,$cm [Fig.~\\ref{fig:snrfigures}(g--h)] leads to a vastly improved aSNR, especially with gradiometers, but also with magnetometers.\n\n\nGradiometers with long baselines provide somewhat magnetometer-like sensitivity patterns while rejecting external noise. However, their aSNR performance is inferior to magnetetometers because of their larger inductance, yielding higher flux noise when the sensor noise dominates; see Sec.~\\ref{ssIntrinsic}. Helmet arrays of magnetometers can provide a similar aSNR in the deepest parts of the brain as the Berkeley gradiometer provides at a small area on the scalp. \n\n\\begin{figure*}\n\t\\centering\n\t\t\\includegraphics[width=0.95\\textwidth]{aSNR_maps.pdf}\n\t\\caption{Base-10 logarithms of aSNR for different sensor-array geometries. To allow comparison of different arrays, we assumed SQUID noise scaling according to optimally matched input circuits. (a) Magnetometers: $R=1.25\\,$cm, (b) double-gradiometer units: $R=1.25\\,$cm, (c) axial gradiometers: $b=4\\,$cm, $R=1.25\\,$cm, (d) 7 Los Alamos second-order axial gradiometers: $b = 6\\,$cm, $R=1.85\\,$cm,   \n (e) Berkeley single second-order axial gradiometer: $b=7.5\\,$cm, $R=3.15\\,$cm, (f) radially oriented planar gradiometers [Fig.~\\ref{figSQUID}(f)]: $b=4\\,$cm, $R=1.25\\,$cm, (g) overlapping double-gradiometer units: $R=2.5\\,$cm, (h) overlapping magnetometers: $R=2.5\\,$cm. The data rate of the acquisition is proportional to the square  of the of aSNR.}\n\t\\label{fig:snrfigures}\n\\end{figure*}\n\n\\section{Conclusions and outlook}\n\nExtending Ref.~\\cite{Zevenhoven2011}, we analyzed a variety of factors that affect the noise and sensitivity of a SQUID-based sensor array for ULF MRI of the brain. Many of the principles, however, apply to non-SQUID arrays as well. We also derived numerical means for studying and comparing the SNR performances of any given sensor array designs.\n\nSignal- and noise-scaling arguments and calculations showed that filling a sensor array with a huge number of tiny sensors is usually not advantageous. Larger pickup coil sizes give a better image SNR at the center of the head and, up to some point, also at closer sources such as the cerebral cortex. This is true even if the number of sensors needs to be decreased due to the limited area available for the array. However, the average voxel SNR is proportional to the square root of the number of sensors.\n\n\n\\sloppy Several possible array designs were compared, including existing arrays designed for MEG and ULF MRI. The results are mostly in favor of magnetometers and large first-order gradiometers. While typically having inferior SNR, gradiometers do have the advantage of rejecting external fields, reducing also transient issues due to pulsed fields \\cite{Zevenhoven2011MSc}. An especially dramatic difference was found when comparing a magnetometer-filled helmet with a single larger gradiometer.\n\nIn general, using an array of sensors relaxes the dynamic range requirements for sensor readout. Splitting a large loop into smaller ones further allows interference rejection based on correlation, while also increasing the SNR close to the center of the loop. An array of many sensors also solves the single-sensor problem of `blind angles'.\n\nOur initial analysis of \\emph{overlapping} magnetometer and gradiometer coils gave promising results. Implementing such arrays, however, poses challenges. Practical considerations include how to fabricate such an array and what materials to use. For instance, wire-wound Type-I superconducting pickup coils have shown some favorable properties \\cite{Luomahaara2011,Hwang2014} in pulsed systems, and exploiting the dynamics of superconductor-penetrating flux \\cite{Zevenhoven2011MSc,Zevenhoven2013degauss,Al-Dabbagh2018} has been promising. However, existing techniques are not suitable for helmet configurations with overlapping coils. In addition, careful design work should be conducted to minimize mutual inductances and other coupling issues. Further significant improvements could be achieved by placing the sensors closer to the scalp, but that would require dramatic advancements in cryostat technology, and was not studied here.\n\nHere, we only considered the contribution of the sensor array to the imaging performance. Other things to consider are the polarizing technique as well as the ability of the instrumentation to apply more sophisticated sequences and reconstruction techniques, while preserving low system noise. A class of techniques enabled by multichannel magnetometers is accelerated parallel MRI \\cite{Larkman2007}. However, the so-called geometry factor should be taken into account \\cite{Lin2013} if large parallel acceleration factors are pursued.\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}\nAs cloud services and distributed data storage become increasingly prevalent, growing concerns about users' privacy have sparked much recent interest  in the problem of Private Information Retrieval (PIR). Originally introduced in  \\cite{PIRfirst,PIRfirstjournal}, the goal of PIR is to allow a user to efficiently retrieve a desired message from a server or a set of servers where multiple messages are stored, without revealing any information about which message is desired. In the information theoretic framework, which requires perfect privacy and assumes  long messages, the capacity of PIR is the maximum number of bits of desired information that can be retrieved per bit of download from the server(s) \\cite{Sun_Jafar_PIR}. Capacity characterizations have recently been obtained for various forms of PIR, especially for the multi-server setting  \\cite{Shah_Rashmi_Kannan, Sun_Jafar_PIR, Tajeddine_Gnilke_Karpuk_Hollanti, Sun_Jafar_TPIR, Sun_Jafar_SPIR, \nBanawan_Ulukus_BPIR, \nWang_Skoglund_PIRSPIRAd, \nWang_Skoglund_TSPIR,  \nFREIJ_HOLLANTI, Sun_Jafar_MDSTPIR, Wang_Skoglund_MDS, Jia_Sun_Jafar_XSTPIR, Jia_Jafar_MDSXSTPIR, Yang_Shin_Lee,\nJia_Jafar_GXSTPIR, Tandon_CachePIR,  Wei_Banawan_Ulukus,  Tajeddine_Gnilke_Karpuk, Yao_Liu_Kang_Collusion_Pattern, PIR_survey}. \n\nPIR in the basic single server setting would be most valuable if it could be made efficient. However, it was already shown in the earliest works on PIR \\cite{PIRfirst, PIRfirstjournal} that in the single server case there is no better alternative to the trivial solution of downloading everything, which is prohibitively expensive. Since the optimal solution turns out to be trivial, single server PIR generally received less attention from the information theoretic perspective, until recently. \nInterest in the capacity of single-server PIR was revived  by the seminal contribution of Kadhe et al. in \\cite{Kadhe_Garcia_Heidarzadeh_ElRouayheb_Sprintson_PIR_SI} which showed that the presence of \\emph{side information} at the user can significantly improve the efficiency of PIR, and that capacity characterizations under side information  are far from trivial. This crucial observation inspired much work on understanding the role of side-information in PIR \\cite{Chen_Wang_Jafar_Side, heidarzadeh2018oncapacity, li2018single, li2020single, kazemi2019single, heidarzadeh2019single, heidarzadeh2018capacity, heidarzadeh2019capacity, Wei_Banawan_Ulukus_Side, PIR_PCSI}, which remains an active topic of research. Among the  recent advances in this area is the study of single-server PIR with private coded side information (PIR-PCSI) that was initiated by Heidarzadeh, Kazemi and Sprintson in \\cite{PIR_PCSI}.  Heidarzadeh et al. obtain sharp capacity characterizations for PIR-PCSI in many cases, and also note an open problem, along with an intriguing conjecture that motivates our work in this paper.\n\nIn the PIR-PCSI problem, a single server stores $K$ independent messages $\\bm{W}_1, \\cdots, \\bm{W}_K$, each represented by $L$ i.i.d. uniform symbols from a finite field $\\mathbb{F}_q$. A user wishes to efficiently retrieve a desired message $\\bm{W}_{\\bm{\\theta}}$, while utilizing private side information $(\\bm{\\mathcal{S}}, \\bm{\\Lambda}, \\bm{Y}^{[\\bm{\\mathcal{S}},\\bm{\\Lambda}]})$ that is unknown to the server, comprised of a linear combination $ \\bm{Y}^{[\\bm{\\mathcal{S}},\\bm{\\Lambda}]}=\\sum_{m=1}^M\\bm{\\lambda}_m\\bm{W}_{i_m}$ of a uniformly chosen size-$M$ subset of messages, $\\bm{\\mathcal{S}}=\\{\\bm{i}_1,\\bm{i}_2,\\cdots,\\bm{i}_M\\}\\subset[K], \\bm{i}_1<\\bm{i}_2<\\cdots<\\bm{i}_M$, with the coefficient vector $\\bm{\\Lambda}=(\\bm{\\lambda}_1, \\cdots, \\bm{\\lambda}_M)$ whose elements are chosen i.i.d. uniform from $\\mathbb{F}_q^\\times$.  Depending on whether $\\bm{\\theta}$ is drawn uniformly from $[K]\\setminus\\bm{\\mathcal{S}}$ or uniformly from $\\bm{\\mathcal{S}}$, there are two settings,  known as PIR-PCSI-I and PIR-PCSI-II, respectively. In each case, $(\\bm{\\theta}, \\bm{\\mathcal{S}})$ must be kept private. Capacity of PIR is typically defined as the maximum number of bits of desired message that can be retrieved per bit of download from the server(s), and includes a supremum over message size $L$. Since the side-information formulation    specifies a finite field $\\mathbb{F}_q$, the capacity of PIR-PCSI can potentially depend on the field. A field-independent notion of capacity  is introduced in  \\cite{PIR_PCSI} by allowing a supremum over all finite fields. For PIR-PCSI-I, where $\\bm{\\theta}\\notin \\bm{\\mathcal{S}}$, Heidarzadeh et al. fully characterize the  capacity as $(K-M)^{-1}$ for $1 \\leq M \\leq K-1$. For PIR-PCSI-II, the   capacity is characterized as $(K-M+1)^{-1}$ for $\\frac{K+1}{2} < M \\leq K$. Capacity characterization for the remaining case of $2 \\leq M \\leq \\frac{K+1}{2}$ is noted as an open problem in \\cite{PIR_PCSI}, and it is conjectured that the capacity in this case is also $(K-M+1)^{-1}$. \n\nThe main motivation of our work is to settle this conjecture and obtain the capacity characterization for PIR-PCSI-II when $2 \\leq M \\leq \\frac{K+1}{2}$. Given the importance of better understanding the role of side information for single-server PIR, additional motivation comes from the following questions:  What is the infimum capacity (infimum over all finite fields instead of supremum)? What if the coefficient vector $\\bm{\\Lambda}$ (whose privacy is not required in \\cite{PIR_PCSI}) is also required to be private? Can the  side-information be reduced, e.g., to save storage, without reducing capacity? \n\n\n\\begin{table*}[!t]\n    \\caption{Capacity results for PIR-PCSI-I, PIR-PCSI-II and PIR-PCSI}\n    \\label{tab:capacity}\n    \\centering\n    \\scalebox{0.76}{\n    \\begin{tabular}{|c|c|c|}\n    \\hline\n    PIR-PCSI-I ($1 \\leq M \\leq K-1$) & PIR-PCSI-II ($2 \\leq M \\leq K$) & PIR-PCSI ($1 \\leq M \\leq K$)\\\\ \\hline\n    $C_{\\mbox{\\tiny PCSI-I}}^{\\sup} = \\frac{1}{K-M}$, \\cite{PIR_PCSI} \n    & $C_{\\mbox{\\tiny PCSI-II}}^{\\sup} =\n        \\begin{cases}\n            \\frac{2}{K}, & 2 \\leq M \\leq \\frac{K+1}{2}, \\text{Thm. \\ref{thm:cap_PCSI2_sup}}\\\\\t\n            \\frac{1}{K-M+1}, & \\frac{K+1}{2} < M \\leq K,\\text{\\cite{PIR_PCSI}}\n        \\end{cases}$ \n    & $C_{\\mbox{\\tiny PCSI}}^{\\sup} = \n    \\begin{cases}\n        \\frac{1}{K-1}, & M=1,\\\\\n        \\frac{1}{K-M+1}, & 2 \\leq M \\leq K,\n    \\end{cases}$, Thm. \\ref{thm:cap_PCSI_sup}\\\\ \\hline\n    $C_{\\mbox{\\tiny PCSI-I}}^{\\inf} =\n        \\begin{cases}\n            \\frac{1}{K-1}, & 1 \\leq M \\leq \\frac{K}{2},\\\\\n            \\big(K - \\frac{M}{K-M}\\big)^{-1}, & \\frac{K}{2} < M \\leq K-1,\n        \\end{cases}$, Thm. \\ref{thm:cap_PCSI1_inf}\n    & $C_{\\mbox{\\tiny PCSI-II}}^{\\inf} = \\frac{M}{(M-1)K}$, Thm. \\ref{thm:cap_PCSI2_inf} & $C_{\\mbox{\\tiny PCSI}}^{\\inf} = \\frac{1}{K-1}$, Thm. \\ref{thm:cap_PCSI_inf} \\\\ \\hline\n    \n    \\begin{tabular}{c}$ C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}, \\sup} = C_{\\mbox{\\tiny PCSI-I}}^{\\inf}$\\\\\n    $\\frac{1}{K-1} \\leq C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}, \\inf} \\leq \\min\\bigg(C_{\\mbox{\\tiny PCSI-I}}^{\\inf}, \\frac{1}{K-2}\\bigg)$\\end{tabular}, Thm. \\ref{thm:pcsi1_pub_pri} & $ C_{\\mbox{\\tiny PCSI-II}}^{\\mbox{\\tiny pri}}(q) = C_{\\mbox{\\tiny PCSI-II}}^{\\inf}$, Thm. \\ref{thm:pcsi2_pub_pri} & $ C_{\\mbox{\\tiny PCSI}}^{\\mbox{\\tiny pri}}(q) = C_{\\mbox{\\tiny PCSI}}^{\\inf}$, Thm. \\ref{thm:pcsi_pub_pri} \\\\ \\hline\n\n    \n    \\end{tabular}\n    }\n\\end{table*}\n\n\nThe contributions of this work are summarized in Table \\ref{tab:capacity}, along with prior results from \\cite{PIR_PCSI}. As our main contribution we show that the  capacity of PIR-PCSI-II for $2 \\leq M \\leq \\frac{K+1}{2}$ is equal to $2\/K$, which is strictly higher than the conjectured value in this parameter regime. The result reveals two surprising aspects  of this parameter regime. First, whereas previously known capacity characterizations of PIR-PCSI-II (and PIR-PCSI-I) in \\cite{PIR_PCSI} are all strictly increasing with $M$ (the size of the support set of side information), here the capacity does not depend on $M$. Second, in this parameter regime (and also when $M=\\left\\lfloor (K+1)\/2\\right\\rfloor +1$), half of the side information turns out to be redundant, i.e., the supremum capacity remains the same even if the user discards half of the side information. Half the side information is also the minimal necessary, because we show that the same capacity is not achievable with less than half of the side information. By contrast, in other regimes no redundancy exists in the side information, i.e., any reduction in side information would lead to a  loss in supremum capacity.\n\nThe optimal rate $2\/K$ is shown to be achievable for any finite field $\\mathbb{F}_q$ where $q$ is an \\emph{even} power of a prime. The achievable scheme requires downloads that are ostensibly non-linear in $\\mathbb{F}_q$, but in its essence the scheme is linear, as can be seen by interpreting $\\mathbb{F}_q$ as a $2$ dimensional vector space over the base field $\\mathbb{F}_{\\sqrt{q}}$, over which the downloads are indeed linear. Intuitively, the scheme may be understood as follows. A rate of $2\/K$ means a download of $K\/2$, which is achieved by downloading \\emph{half} of every message (one of the two dimensions in the $2$ dimensional vector space over $\\mathbb{F}_{\\sqrt{q}}$). The key idea is \\emph{interference alignment} -- for the undesired messages that appear in the side information, the halves that are downloaded are perfectly \\emph{aligned} with each other, whereas for the desired message, the half that is downloaded is not aligned with the downloaded halves of the undesired messages. For messages that are not included in the side information, any random half can be downloaded to preserve privacy. \n\nWith a bit of oversimplification for the sake of intuition, suppose  there are $K=4$ messages, that can be represented as $2$-dimensional vectors $\\bm{A}=[\\bm{a}_1~~ \\bm{a}_2], \\bm{B}=[\\bm{b}_1~~ \\bm{b}_2], \\bm{C}=[\\bm{c}_2~~ \\bm{c}_2], \\bm{D}=[\\bm{d}_1~~ \\bm{d}_2]$, the side information is comprised of  $M=3$ messages, say at first $\\bm{A}+\\bm{B}+\\bm{C}=[\\bm{a}_1+\\bm{b}_1+\\bm{c}_1~~ \\bm{a}_2+\\bm{b}_2+\\bm{c}_2]$, and the desired message is $\\bm{A}$. Then the user could recover $\\bm{A}$ by downloading $\\bm{a}_1, \\bm{b}_2, \\bm{c}_2$ and either $\\bm{d}_1$ or $\\bm{d}_2$, i.e., half of each message for a total download of $K\/2=2$ (normalized by message size). We may also note that half of the side information is redundant, i.e., the user only needs $\\bm{a}_2+\\bm{b}_2+\\bm{c}_2$, and can discard the rest. But there is a problem with this oversimplification -- this toy example  seemingly loses privacy because the matching indices reveal that $\\bm{b}_2$ aligns with $\\bm{c}_2$ but not $\\bm{a}_1$. This issue is resolved by noting that the side information is in fact $\\bm{\\lambda}_1\\bm{A}+\\bm{\\lambda}_2\\bm{B}+\\bm{\\lambda}_3\\bm{C}=\\bm{A}'+\\bm{B}'+\\bm{C}'$. Suppose $\\bm{\\lambda}_1, \\bm{\\lambda}_2, \\bm{\\lambda}_3$ are random (unknown to the server) independent linear transformations (\\emph{matrices}) that  independently `\\emph{rotate}' $\\bm{A}, \\bm{B}, \\bm{C}$ vectors into $\\bm{A}',\\bm{B}',\\bm{C}'$ vectors, respectively, such that the projections (combining coefficients) of each along any particular dimension become independent of each other. In other words, $\\bm{a}_i', \\bm{b}_i', \\bm{c}_i'$ are independent projections of $\\bm{A}, \\bm{B}, \\bm{C}$, and downloading, say $(\\bm{a}_1', \\bm{b}_2', \\bm{c}_2', \\bm{d}_2')$ reveals to the server no information about their relative alignments in the side information. From the server's perspective, each downloaded symbol is simply an independent random linear combination of the two components of each message. Intuitively, since the random rotation is needed to maintain privacy, it is important that $\\bm{\\lambda}_i$ are matrices, not scalars (because scalars only scale, they do not rotate vectors). This is not directly  the case in $\\mathbb{F}_q$ because $\\bm{\\lambda}_i$ are scalars in $\\mathbb{F}_q$. However, viewed as a $2$ dimensional vector space over $\\mathbb{F}_{\\sqrt{q}}$, the $\\bm{\\lambda}_i$ indeed act as invertible $2\\times 2$ matrices that act on the vectors $\\bm{A}, \\bm{B}, \\bm{C}, \\bm{D}$, rotating each vector randomly and independently, thus ensuring privacy. \n\nIn order for $\\mathbb{F}_{\\sqrt{q}}$ to be a valid finite field we need $q$ to be an \\emph{even} power of a prime. This suffices to characterize the  capacity because the capacity definition in \\cite{PIR_PCSI} allows a supremum over all fields. However, the question remains about whether the rate $2\/K$ is  achievable over every finite field. To understand this better, we explore an alternative definition of capacity (called infimum capacity in this work) which considers the infimum (instead of supremum) over all $\\mathbb{F}_q$. We find that the infimum capacity of PIR-PCSI-II is always equal to $M\/((M-1)K)$. Evidently, for $M=2$ the capacity is  field independent because the infimum and supremum over fields produce the same capacity result. In general however, the infimum capacity can be strictly smaller, thus confirming field-dependence. The worst case corresponds to the binary field $\\mathbb{F}_2$.  Intuitively, the reason that the infimum capacity corresponds to the binary field is that over $\\mathbb{F}_2$ the non-zero coefficients $\\bm{\\lambda}_m$ must all be equal to one, and thus the coefficients are essentially known to the server.  On the other hand, we also present an example with $q=3$ (and $M=3, K=4$) where $2\/K$ is achievable (and optimal), to show that the achievability of $2\/K$ for $M>2$ is not limited to just field sizes that are even powers of a prime number. We also show that for PIR-PCSI-II, the the infimum capacity with  private $(\\bm{\\theta},\\bm{\\mathcal{S}})$ is the same as the (supremum or infimum) capacity with private $(\\bm{\\theta},\\bm{\\mathcal{S}},\\bm{\\Lambda})$, i.e., when the coefficients $\\bm{\\Lambda}$  must also be kept private from the server. \n\nNext we consider PIR-PCSI-I where $\\bm{\\theta}$ is drawn from $[K]\\setminus\\bm{\\mathcal{S}}$. The supremum capacity of PIR-PCSI-I is fully characterized in \\cite{PIR_PCSI}. In this case, we show that there is no redundancy in the CSI. As in PIR-PCSI-II, we find that the infimum capacity of PIR-PCSI-I is strictly smaller than the supremum capacity in general, and the binary field $\\mathbb{F}_2$ yields the worst case. Unlike PIR-PCSI-II, however, the infimum capacity of PIR-PCSI-I with private $(\\bm{\\theta},\\bm{\\mathcal{S}})$ does not always match the infimum capacity with  private $(\\bm{\\theta},\\bm{\\mathcal{S}},\\bm{\\Lambda})$. For example, if $M=K-1$, then both the supremum and infimum capacities of PIR-PCSI-I are equal to $1$ for private $(\\bm{\\theta},\\bm{\\mathcal{S}})$, but if the  coefficient vector $\\bm{\\Lambda}$ must also be kept private then the infimum capacity is no more than $1\/(K-2)$. Thus, the loss in capacity from requiring privacy of coefficients can be quite significant.\n\nTo complete the picture, we finally consider the capacity of PIR-PCSI where $\\bm\\theta$ is drawn uniformly from $[K]$.  In PIR-PCSI the server is not allowed to learn anything about whether or not $\\bm{\\theta}\\in\\bm{\\mathcal{S}}$.  The supremum capacity of PIR-PCSI is found to be $(K-M+1)^{-1}$ for $2 \\leq M \\leq K$. Remarkably, this is not just the smaller  of the two capacities of PIR-PCSI-I and PIR-PCSI-II, so there is an additional cost to be paid for hiding from the server whether $\\bm{\\theta} \\in \\bm{\\mathcal{S}}$ or $\\bm{\\theta} \\notin \\bm{\\mathcal{S}}$. Depending on the relative values of $M$ and $K$, in this case we find that the redundancy in CSI can be as high as $1\/2$ or as low as $0$.  The infimum capacity of PIR-PCSI is smaller than the supremum capacity, the binary field $\\mathbb{F}_2$ yields the worst case, and as in PIR-PCSI-II, the infimum capacity with private $(\\bm{\\theta},\\bm{\\mathcal{S}})$ is the same as the (supremum or infimum) capacity with private $(\\bm{\\theta},\\bm{\\mathcal{S}},\\bm{\\Lambda})$.\n\nThis paper is organized as follows: Section \\ref{sec:state} states PIR-PCSI, PIR-PCSI-I, PIR-PCSI-II problems in \\cite{PIR_PCSI}. Section \\ref{sec:main} states our capacity and redundancy (in the CSI) results for PIR-PCSI-II, PIR-PCSI-I, PIR-PCSI with fourteen theorems which are proved in Section \\ref{sec:cap_PCSI2_sup} to Section \\ref{proof:pcsi_pub_pri}. Section \\ref{sec:con} concludes this paper and gives possible future directions.\n\n\\emph{Notation}: For a positive integer $a$, let $[a]$ denote the set $\\{1,2,\\cdots,a\\}$. For two integers $a, b$ where $a < b$, $[a:b]$ denotes the set $\\{a, a+1, \\cdots, b\\}$.  For a set $\\mathcal{S} = \\{i_1, i_2, \\cdots, i_n\\}$, $|\\mathcal{S}|$ denotes the cardinality of $\\mathcal{S}$. $\\mathbf{I}_{M}$ denotes the $M \\times M$ identity matrix, and $\\mathbf{0}_{M}$ denotes the $M \\times M$ all-zero matrix. For a matrix $\\mathbf{A}$, let $\\mathbf{A}(i,:)$ be the $i^{th}$ row of $\\mathbf{A}$. For a set $\\mathcal{A}$ whose elements are integers, let $\\mathcal{A}(i)$ denote the $i^{th}$ element of $\\mathcal{A}$ in ascending order. Let $\\mathbb{F}_{q}$ denote the finite field of order $q$ and $\\mathbb{F}_{q}^{\\times}$ contain all the non-zero elements of $\\mathbb{F}_{q}$. The notation $\\mathbb{F}_q^{a\\times b}$ represents the set of all $a\\times b$ matrices with elements in $\\mathbb{F}_q$. Let $\\mathfrak{S}$ be the set of all the subsets with cardinality $M$ of $[K]$, i.e., $|\\mathfrak{S}| = \\tbinom{K}{M}$, and let $\\mathfrak{C}$ be the set of all length $M$ sequences with elements in $\\mathbb{F}_{q}^{\\times}$, i.e., $|\\mathfrak{C}| = (q-1)^M$. For an index set $S\\subset[K]$, define the subscript notation $X_S=\\{X_s\\mid s\\in S\\}$. All entropies are in $q$-ary units.\n\n\\section{Problem Statement}\\label{sec:state}\n\\subsection{Capacity of PIR-PCSI-I, PIR-PCSI-II, PIR-PCSI}\nA single server stores $K$ independent messages $\\bm{W}_1, \\bm{W}_2, \\cdots, \\bm{W}_{K}\\in\\mathbb{F}_q^L$, each comprised of $L$ i.i.d. uniform symbols  from $\\mathbb{F}_{q}$, where we refer to $\\mathbb{F}_{q}$ as the \\emph{base field}. In terms of entropies,\n\\begin{align}\n    &H(\\bm{W}_{1}) = H(\\bm{W}_{2}) = \\cdots = H(\\bm{W}_{K}) = L,\\\\\n    &H(\\bm{W}_{[K]}) = \\sum_{k \\in [K]}H(\\bm{W}_{k}) = KL.\n\\end{align}\n\n A user who wishes to retrieve a message $\\bm{W}_{\\bm{\\theta}}$ for a privately generated index $\\bm{\\theta}$. The user has a linear combination of $M$ messages available as coded side information (CSI). $M$ is globally known. The CSI is comprised of $(\\bm{\\mathcal{S}}, \\bm{\\Lambda},  \\bm{Y}^{[\\bm{\\mathcal{S}},\\bm{\\Lambda}]})$,  defined as follows. The \\emph{support index set} $\\bm{\\mathcal{S}}$, drawn uniformly from $\\mathfrak{S}$, is a subset of $[K]$, of cardinality $M$. The vector of coefficients $\\bm{\\Lambda}=(\\bm{\\lambda}_1,\\bm{\\lambda}_2,\\cdots,\\bm{\\lambda}_M)$ is drawn uniformly from $\\mathfrak{C}$. \nThe linear combination available to the user is\n\\begin{align}\n    \\bm{Y}^{[\\bm{\\mathcal{S}},\\bm{\\Lambda}]}\\triangleq \\bm{\\lambda}_1\\bm{W}_{\\bm{\\mathcal{S}}(1)} + \\bm{\\lambda}_2\\bm{W}_{\\bm{\\mathcal{S}}(2)} + \\cdots + \\bm{\\lambda}_M\\bm{W}_{\\bm{\\mathcal{S}}(M)},\\label{eq:sideinfo_CSI}\n\\end{align}\nwhere we recall the notation that $\\bm{\\mathcal{S}}(m)$ denotes the $m^{th}$ element of $\\bm{\\mathcal{S}}$, in ascending order, i.e., $\\bm{\\mathcal{S}}(1)<\\bm{\\mathcal{S}}(2)<\\cdots<\\bm{\\mathcal{S}}(M)$. \nWe assume that $(\\bm{\\theta}, \\bm{\\mathcal{S}})$, $\\bm{\\Lambda}$, $\\bm{W}_{[K]}$ are  independent.\n\\begin{align}\n    H(\\bm{\\theta}, \\bm{\\mathcal{S}}, \\bm{\\Lambda}, \\bm{W}_{[K]}) = H(\\bm{\\theta}, \\bm{\\mathcal{S}}) + H(\\bm{\\Lambda}) + H(\\bm{W}_{[K]}).\n\\end{align}\n\nThere are three formulations of the problem depending on how $\\bm{\\theta}$ is chosen by the user.\n\\begin{enumerate}\n\\item{\\bf PIR-PCSI-I}: $\\bm{\\theta}$ is chosen uniformly from $[K]\\setminus\\bm{\\mathcal{S}}$.\n\\item{\\bf PIR-PCSI-II}: $\\bm{\\theta}$ is chosen uniformly from $\\bm{\\mathcal{S}}$.\n\\item{\\bf PIR-PCSI}: $\\bm{\\theta}$ is chosen uniformly from $[K]$.\n\\end{enumerate}\nWhen referring to all three formulations, we will  refer to the problem as {\\bf PIR-PCSI*} for brevity. In such statements, PCSI* can be replaced with PCSI-I, PCSI-II, or PCSI  to obtain corresponding statements for each of the three formulations.\n\nThe server knows the distributions but not the realizations of $\\bm\\theta, \\bm{\\mathcal{S}}, \\bm{\\Lambda},  \\bm{Y}^{[\\bm{\\mathcal{S}},\\bm{\\Lambda}]}$.\nIt is required that $(\\bm{\\theta},\\bm{\\mathcal{S}})$ be kept jointly private from the server. Note that the privacy of $\\bm{Y}^{[\\bm{\\mathcal{S}},\\bm{\\Lambda}]}$ or the coefficient vector $\\bm{\\Lambda}$ is not required. While the server initially knows nothing about the realization of $\\bm{\\Lambda}$, a PIR-PCSI* scheme may reveal some information about the coefficients, especially if it allows for efficient retrieval without leaking any information about $(\\bm{\\theta},\\bm{\\mathcal{S}})$. Leaking information about $\\bm{\\Lambda}$ has  implications for reusability of side-information, an issue that is explored recently in  \\cite{Anoosheh_reusable}.\n\nIn order to retrieve $\\bm{W_\\theta}$, we assume as in \\cite{PIR_PCSI} that the user generates a random query $\\bm{Q}$ that is independent of the messages. Specifically,\n\\begin{align}\n    I(\\bm{W}_{[K]}; \\bm{Q}, \\bm{\\theta}, \\bm{\\mathcal{S}}, \\bm{\\Lambda}) = 0.\\label{eq:indQ}\n\\end{align}\nLet $\\mathcal{Q}$ denote the alphabet of $\\bm{Q}$. \n\nBecause the messages are i.i.d. uniform, and the coefficients are non-zero, according to the construction of $\\bm{Y}^{[\\bm{\\mathcal{S}}, \\bm{\\Lambda}]}$, it follows that\n\\begin{align}\nL&=H(\\bm{Y}^{[\\bm{\\mathcal{S}}, \\bm{\\Lambda}]}),\\\\\n& = H(\\bm{Y}^{[\\bm{\\mathcal{S}}, \\bm{\\Lambda}]} \\mid \\bm{Q}, \\bm{\\mathcal{S}}, \\bm{\\Lambda},  \\bm{W}_{{[K]}\\setminus\\{\\bm{\\mathcal{S}}(m)\\}}), \\forall m \\in [M].\\label{eq:indY} \n\\end{align}\n\nThe user uploads $\\bm{Q}$ to the server. Mathematically, the privacy constraint is expressed as,\n\\begin{align}\n    &\\text{[$(\\bm{\\theta}, \\bm{\\mathcal{S}})$ Privacy]} &&I\\left(\\bm{\\theta}, \\bm{\\mathcal{S}}; \\bm{Q}, \\bm{W}_{[K]}\\right) = 0.\\label{eq:tsprivacy}\n\\end{align}\nThe server returns an answer $\\bm{\\Delta}$ as a function of $\\bm{Q}$ and the messages, i.e.,\n\\begin{align}\n    H\\left(\\bm{\\Delta} \\mid \\bm{Q}, \\bm{W}_{[K]}\\right) = 0.\n\\end{align}\nUpon receiving the answer, the user must be able to decode the desired message $\\bm{W}_{\\bm\\theta}$. \n\\begin{align}\n    &\\text{[Correctness]} &&H(\\bm{W}_{\\bm{\\theta}} \\mid \\bm{\\Delta}, \\bm{Q}, \\bm{Y}^{[\\bm{\\mathcal{S}}, \\bm{\\Lambda}]},  \\bm{\\mathcal{S}}, \\bm{\\Lambda},\\bm{\\theta}) = 0.\n\\end{align}\nThe rate of an achievable scheme is the ratio of the number of bits of the desired message to the total number of bits downloaded on average. If the average download is $D$ $q$-ary symbols, from which the $L$ $q$-ary symbols of the desired message are retrieved, then the rate achieved is,\n\\begin{align}\n    R = \\frac{L}{D}\n\\end{align}\nThe capacity is the supremum of achievable rates over all message sizes $L$,\n\\begin{align}\n    C_{\\mbox{\\tiny PCSI*}}(q) = \\sup_{L, \\mbox{\\tiny achievable $R$}}R.\n\\end{align}\nThe capacity can depend on the field $\\mathbb{F}_q$ which affects the nature of side information. Field-independent measures of capacity may be obtained by taking a supremum (as in \\cite{PIR_PCSI}) or infimum over all finite fields. These are called supremum and infimum capacity, respectively.\n\\begin{align}\n    C_{\\mbox{\\tiny PCSI*}}^{\\sup} &= \\sup_{q}C_{\\mbox{\\tiny PCSI*}}(q),\\\\\n    C_{\\mbox{\\tiny PCSI*}}^{\\inf} &= \\inf_{q}C_{\\mbox{\\tiny PCSI*}}(q).\n\\end{align}\n\n\\subsection{Capacity of PIR-PCSI* with Private Coefficients}\nRecall that in the formulation of PIR-PCSI* as presented above, while $(\\bm{\\theta},\\bm{\\mathcal{S}})$ must be kept private, the privacy of the coefficient vector $\\bm{\\Lambda}$ is not required. As an important benchmark, we  consider the setting where the privacy of coefficients must also be preserved. In this setting, the privacy constraint is modified so that instead of \\eqref{eq:tsprivacy} we require the following.\n\\begin{align}\n    &\\text{[$(\\bm{\\theta}, \\bm{\\mathcal{S}}, \\bm{\\Lambda})$ Privacy]} &&I\\left(\\bm{\\theta}, \\bm{\\mathcal{S}}, \\bm{\\Lambda}; \\bm{Q}, \\bm{W}_{[K]}\\right) = 0.\\label{eq:tscprivacy}\n\\end{align}\nThe capacity under this privacy constraint is referred to as the capacity with private coefficients and is denoted as $C_{\\mbox{\\tiny PCSI*}}^{\\mbox{\\tiny pri}}(q)$, which is potentially a function of the field size $q$. The supremum and infimum (over  $q$) of $C_{\\mbox{\\tiny PCSI*}}^{\\mbox{\\tiny pri}}(q)$ are denoted as $C_{\\mbox{\\tiny PCSI*}}^{\\mbox{\\tiny pri},\\sup}, C_{\\mbox{\\tiny PCSI*}}^{\\mbox{\\tiny pri},\\inf}$, respectively.\n\n\\subsection{Redundancy of CSI}\nIn addition to the capacity of PIR-PCSI*, we also wish to determine how much (if any) of the side information  is redundant, i.e., can be discarded without any loss in the \\emph{supremum capacity}. \n\nFor all $\\mathcal{S}\\in\\mathfrak{S}, \\Lambda\\in\\mathfrak{C}$, let $f_{\\mathcal{S},\\Lambda}$ be functions that produce\n\\begin{align}\n\\overline{\\bm{Y}}^{[{\\mathcal{S}}, {\\Lambda}]} = f_{\\mathcal{S}, \\Lambda}(\\bm{Y}^{[{\\mathcal{S}}, {\\Lambda}]}).\n\\end{align}\nLet us refer to all these functions collectively as $\\mathcal{F}=(f_{\\mathcal{S}, \\Lambda})_{\\mathcal{S}\\in\\mathfrak{S}, \\Lambda\\in\\mathfrak{C}}$. \nDefine, $\\overline{C}_{\\mbox{\\tiny PCSI*}}(q,\\mathcal{F})$ as the capacity (supremum of achievable rates) if the decoding must be based on $\\overline{\\bm{Y}}^{[{\\mathcal{S}}, {\\Lambda}]}$ instead of $\\bm{Y}^{[{\\mathcal{S}}, {\\Lambda}]}$, i.e., the correctness condition is modified to\n\\begin{align}\nH(\\bm{W}_{\\bm{\\theta}} \\mid \\bm{\\Delta}, \\bm{Q}, \\overline{\\bm{Y}}^{[\\bm{\\mathcal{S}}, \\bm{\\Lambda}]}, \\bm{\\mathcal{S}}, \\bm{\\Lambda},\\bm{\\theta}) = 0.\n\\end{align}\nWe say that $\\mathcal{F}$ uses $\\alpha$-CSI, where\n\\begin{align}\n\\alpha=\\max_{\\mathcal{S}\\in\\mathfrak{S}, \\Lambda\\in\\mathfrak{C}} H(\\overline{\\bm{Y}}^{[{\\mathcal{S}}, {\\Lambda}]})\/L\n\\end{align}\nWhereas storing $\\bm{Y}^{[{\\mathcal{S}}, {\\Lambda}]}$ requires $L$ $q$-ary symbols, note that storing $\\overline{\\bm{Y}}^{[{\\mathcal{S}}, {\\Lambda}]}$ requires only $\\alpha L$ storage, i.e., storage is reduced by a factor $\\alpha$. Define the $\\alpha$-CSI constrained capacity as\n\\begin{align}\n\\overline{C}_{\\mbox{\\tiny PCSI*}}(q,\\alpha)&=\\sup_{\n\\mathcal{F}: ~\\mbox{\\footnotesize uses no more than $\\alpha$-CSI}} \\overline{C}_{\\mbox{\\tiny PCSI*}}(q,\\mathcal{F})\n\\end{align}\nIn other words, $\\overline{C}_{\\mbox{\\tiny PCSI*}}(q,\\alpha)$ is the capacity when the user is allowed to retain no more than a fraction $\\alpha$ of the CSI $\\bm{Y}^{[{\\mathcal{S}}, {\\Lambda}]}$.\nThe notion of $\\alpha$-CSI constrained capacity is of broader interest on its own. However, in this work we will explore only the redundancy of CSI with regard to the supremum capacity. We say that `$\\alpha$-CSI is sufficient' if \n\\begin{align}\n\\sup_q\\overline{C}_{\\mbox{\\tiny PCSI*}}(q,\\alpha)&={C}_{\\mbox{\\tiny PCSI*}}^{\\sup}\n\\end{align}\nDefine $\\alpha^*$ as the smallest value of $\\alpha$ such that $\\alpha$-CSI is sufficient.\nThe redundancy of PCSI is defined as $\\rho_{\\mbox{\\tiny PCSI*}}=1-\\alpha^*$.\nNote that the opposite extremes of $\\rho_{\\mbox{\\tiny PCSI*}}=1$ and $\\rho_{\\mbox{\\tiny PCSI*}}=0$ correspond to situations where all of the side information is redundant, and where none of the side information is redundant, respectively.\n\nFor later use, it is worthwhile to note that for any scheme that uses no more than $\\alpha$-CSI,  because $\\overline{\\bm{Y}}^{[{\\mathcal{S}}, {\\Lambda}]}$ is a function of ${\\bm{Y}}^{[{\\mathcal{S}}, {\\Lambda}]}$, it follows from \\eqref{eq:indY} that for all\\footnote{We say $(Q,\\mathcal{S},\\Lambda)$ is feasible if $\\Pr((\\bm{Q}, \\bm{\\mathcal{S}}, \\bm{\\Lambda}) = (Q,\\mathcal{S},\\Lambda))>0$.} feasible $(Q,\\mathcal{S},\\Lambda)$,\n{\\small\n\\begin{align}\nH\\bigg(\\overline{\\bm{Y}}^{[{\\mathcal{S}}, {\\Lambda}]} \\mid (\\bm{Q},\\bm{\\mathcal{S}},\\bm{\\Lambda})=(Q,\\mathcal{S},\\Lambda)\\bigg)=H(\\overline{\\bm{Y}}^{[{\\mathcal{S}}, {\\Lambda}]} )\\leq\\alpha L.\\label{eq:invaYR}\n\\end{align}\n}\nThis is because of the property that if $A$ is independent of $B$, then any function of $A$ is also independent of $B$. In this case,  \\eqref{eq:indY} tells us that ${\\bm{Y}}^{[{\\mathcal{S}}, {\\Lambda}]}$ is independent of ${\\bf Q}$, therefore so is $\\overline{{\\bm{Y}}}^{[{\\mathcal{S}}, {\\Lambda}]}$.\n\n\\section{Main Results}\\label{sec:main}\nWe start with the setting of PIR-PCSI-II (where $\\bm{\\theta}\\in\\bm{\\mathcal{S}}$), which is the main motivation for this work. Note that the case $M=1$ is trivial, because in that case the user already has the desired message. Therefore, for PIR-PCSI-II we will always assume that $M>1$.\n\\subsection{PIR-PCSI-II (where $\\bm{\\theta}$ is drawn uniformly from $\\bm{\\mathcal{S}}$)}\n\\begin{theorem}\\label{thm:cap_PCSI2_sup}\n    The supremum capacity of PIR-PCSI-II  is\n    \\begin{align}\n        C_{\\mbox{\\tiny PCSI-II}}^{\\sup} &=\\max\\left(\\frac{2}{K},\\frac{1}{K-M+1}\\right)\\\\\n        &=\n        \\begin{cases}\n            \\frac{2}{K}, & 1 < M \\leq \\frac{K+1}{2},\\\\\t\n            \\frac{1}{K-M+1}, & \\frac{K+1}{2} < M \\leq K,\\text{\\cite{PIR_PCSI}}\n        \\end{cases}\n    \\end{align}\n\\end{theorem}\n\nThe case $(K+1)\/2<M\\leq K$ was already  settled by Heidarzadeh et al. in \\cite{PIR_PCSI}, and is included in Theorem \\ref{thm:cap_PCSI2_sup} primarily for the sake of completeness. Our contribution to Theorem \\ref{thm:cap_PCSI2_sup} is for the case $1< M\\leq (K+1)\/2$ which was noted as an open problem in \\cite{PIR_PCSI} along with a conjecture that the supremum capacity for this case may also be equal to $1\/(K-M+1)$. Theorem \\ref{thm:cap_PCSI2_sup} settles this open problem and resolves the conjecture by establishing that the supremum capacity in this case is $2\/K$. The proof of Theorem \\ref{thm:cap_PCSI2_sup} for the case $2\\leq M\\leq (K+1)\/2$ appears in Section \\ref{sec:cap_PCSI2_sup}.\n\nNote that when $2\\leq M\\leq (K+1)\/2$, the supremum capacity value $2\/K$  is strictly higher than the conjectured value $1\/(K-M+1)$, and does not depend on the support size $M$ of the coded side information. Achievability of $2\/K$ is shown in Section \\ref{sec:cap_PCSI2_sup} for any field $\\mathbb{F}_q$ where $q$ is an even power of a prime, by viewing $\\mathbb{F}_q$ as a $2$ dimensional vector space over $\\mathbb{F}_{\\sqrt{q}}$. Note that $q$ needs to be an even power of a prime, in order for $\\mathbb{F}_{\\sqrt{q}}$ to be a valid finite field. Over $\\mathbb{F}_{\\sqrt{q}}$, the  coefficients $\\bm{\\lambda}_m, m \\in [M]$  take the role of $2\\times 2$ matrices  that rotate $2\\times 1$ vector (in $\\mathbb{F}_{\\sqrt{q}}$) representation of the messages $\\bm{W}_{\\bm{\\mathcal{S}}_m}$ involved in the CSI, thus randomizing their relative alignments. Half of the desired message $\\bm{W}_{\\bm{\\theta}}$ is recovered by downloading  the corresponding halves of undesired messages that align (interfere) with that half of $\\bm{W}_{\\bm{\\theta}}$ (so that the interference can be subtracted),  while the other half of $\\bm{W}_{\\bm{\\theta}}$ is downloaded directly. The private rotations due to $\\bm{\\Lambda}$ in the CSI hide the alignments from the server. For the messages that are not involved in the CSI any random half can be downloaded. Since a random half of every message is downloaded, no information is leaked about $(\\bm{\\theta},\\bm{\\mathcal{S}})$ and the scheme is private.\n\nIntuitively, since one half of the desired message is directly downloaded, it stands to reason that the corresponding half of the CSI may be redundant and could be discarded by the user, thus saving storage cost. Indeed, this intuition turns out to be correct, as encapsulated in the next theorem which characterizes precisely how much of the side information in each parameter regime is redundant, i.e., can be discarded without any loss in the supremum capacity specified in Theorem \\ref{thm:cap_PCSI2_sup}.\n\n\\begin{theorem}\\label{thm:red}\nFor the supremum capacity of PIR-PCSI-II, the redundancy in coded side information is characterized as,\n\\begin{align}\n\\rho_{\\mbox{\\tiny PCSI-II}}&=\n\\begin{cases}\n          1\/2, & 1< M\\leq (K+2)\/2,\\\\\n           0, & (K+2)\/2<M\\leq K.\n        \\end{cases}\n\\end{align}\n\\end{theorem}\nIn particular, $\\rho_{\\mbox{\\tiny PCSI-II}}=1\/2$ implies that exactly half of the side information is redundant, and $\\rho_{\\mbox{\\tiny PCSI-II}}=0$ implies that there is no redundancy in the side information. The proof of Theorem \\ref{thm:red} appears in Section \\ref{proof:red}. Thus, for all $(M,K)$ parameters where the supremum capacity is equal to $2\/K$, half of the coded side information is redundant. Note that in the boundary case where $M=(K+2)\/2$, we have $2\/K=1\/(K-M+1)$, i.e., this boundary case could be included in either of the two cases in Theorem \\ref{thm:cap_PCSI2_sup}. Remarkably, these are the only cases where we have any redundancy in coded side information. According to Theorem \\ref{thm:red}, there is no redundancy when $(K+2)\/2<M\\leq K$. \n\nAs our next result for PIR-PCSI-II, we characterize the infimum capacity $C_{\\mbox{\\tiny PCSI-II}}^{\\inf}$ in the following theorem.\n\n\\begin{theorem}\\label{thm:cap_PCSI2_inf}\n    The infimum capacity of PIR-PCSI-II,\n    \\begin{align}\n        C_{\\mbox{\\tiny PCSI-II}}^{\\inf} = C_{\\mbox{\\tiny PCSI-II}}(q=2) = \\frac{M}{(M-1)K}.\n    \\end{align}\n\\end{theorem}\nThe proof of Theorem \\ref{thm:cap_PCSI2_inf} appears in Section \\ref{sec:cap_PCSI2_inf}. Evidently, the infimum capacity of PIR-PCSI-II matches its capacity over the binary field. Intuitively, one might expect that the binary field would represent the worst case because over $\\mathbb{F}_2$, the coefficients $\\bm{\\lambda}_m$, which must be non-zero, can only take the value $1$. Thus, the coefficients are known to the server. It is also worth noting that constant $\\bm{\\Lambda}$ trivially satisfy $(\\bm{\\theta}, \\bm{\\mathcal{S}}, \\bm{\\Lambda})$ privacy whenever $(\\bm{\\theta}, \\bm{\\mathcal{S}})$ privacy is satisfied.\n\nNote that for $M=2$, the infimum capacity of PIR-PCSI-II matches the supremum capacity, therefore for any field $\\mathbb{F}_q$, we have the exact capacity characterization, $C_{\\mbox{\\tiny PCSI-II}}(q)=C_{\\mbox{\\tiny PCSI-II}}^{\\inf}=C_{\\mbox{\\tiny PCSI-II}}^{\\sup}$. However, in general the infimum capacity is strictly smaller. The gap can be significant, for example when $M=K$  the supremum capacity is $1$ while the infimum capacity is $1\/(K-1)$. In general the capacity for arbitrary fields remains open. Intuitively, we expect that the capacity for most fields should be either equal to or close to the supremum capacity, whereas fields where the capacity is closer to the infimum capacity should be relatively rare. Indeed,  among the cases of PIR-PCSI-II where the capacity is known for a given field, other than $\\mathbb{F}_2$ which yields infimum capacity, the known capacity results for all other fields match the supremum capacity. This is further underscored by the next two capacity results, starting with the setting $M=K$ that yields the largest gap between the infimum and supremum capacities.\n\n\\begin{theorem}\\label{thm:MK}\nFor PIR-PCSI-II with $M=K$,\n\\begin{align}\nC_{\\mbox{\\tiny PCSI-II}}(q)&=\\begin{cases}\n1\/(K-1)=C_{\\mbox{\\tiny PCSI-II}}^{\\inf}, &q=2,\\\\\n1=C_{\\mbox{\\tiny PCSI-II}}^{\\sup},&q\\neq 2.\n\\end{cases}\n\\end{align}\n\\end{theorem}\nThe proof of Theorem \\ref{thm:MK} appears in Section \\ref{proof:MK}.\n\n\\begin{theorem}\\label{thm:M3K4}\nFor PIR-PCSI-II with $M=3, K=4$,\n\\begin{align}\nC_{\\mbox{\\tiny PCSI-II}}(q)&=\\begin{cases}\n3\/8=C_{\\mbox{\\tiny PCSI-II}}^{\\inf}, &q=2,\\\\\n1\/2=C_{\\mbox{\\tiny PCSI-II}}^{\\sup},&q\\neq 2.\n\\end{cases}\n\\end{align}\n\\end{theorem}\nNote that $M=3,K=4$ is a boundary case for which $1\/(K-M+1)=2\/K$, therefore the supremum capacity is achievable by both the Modified Specialized GRS Codes scheme presented in \\cite{PIR_PCSI} and by the interference alignment scheme that appears in the proof of Theorem \\ref{thm:cap_PCSI2_sup}. However, the former requires field size $q\\geq K=4$, and the latter requires $q$ to be an even power of a prime. Aside from $q=2$ which corresponds to the infimum capacity, this leaves only $q=3$, which is neither greater than or equal to $4$ nor an even power of a prime, as the only new result in Theorem \\ref{thm:M3K4}. \nThe proof for $q=3$ appears in Section \\ref{proof:M3K4}. \n\nBuilding on the observation that the infimum capacity corresponds to the binary field where the coefficients are essentially constants such that the $(\\bm{\\theta}, \\bm{\\mathcal{S}}, \\bm{\\Lambda})$ privacy is automatically satisfied, we next explore the capacity of PIR-PCSI-II for the case of private coefficients. The result appears as the next theorem.\n\\begin{theorem}\\label{thm:pcsi2_pub_pri}\n    The capacity of PIR-PCSI-II, for the setting with private coefficients, is given by\n    \\begin{align}\n        C_{\\mbox{\\tiny PCSI-II}}^{\\mbox{\\tiny pri}}(q) = C_{\\mbox{\\tiny PCSI-II}}^{\\inf}.\n    \\end{align}\n\\end{theorem}\nThe proof of Theorem \\ref{thm:pcsi2_pub_pri} appears in Section \\ref{proof:pcsi2_pub_pri}. Note that the capacity with private coefficients does not depend on the field (infimum and supremum are the same). The loss of supremum capacity due to privacy of coefficients is also notable. Evidently, partially private side information can be much more useful than fully private side information.\n\n\\subsection{PIR-PCSI-I (where $\\bm{\\theta}$ is drawn uniformly from $[K]\\setminus \\bm{\\mathcal{S}}$)}\nIn this section we consider the setting of PIR-PCSI-I (where $\\bm{\\theta} \\in [K]\\setminus \\bm{\\mathcal{S}}$). Note that the case $M=K$ is not valid, because in that case the desired message is also contained in the support set. Therefore, for PIR-PCSI-I we will always restrict $1 \\leq M \\leq K-1$.\n\nThe supremum capacity of PIR-PCSI-I is already found in \\cite{PIR_PCSI} as $C_{\\mbox{\\tiny PCSI}}^{\\sup} = (K-M)^{-1}$ and is achievable by Specialized GRS Codes. We start by characterizing the redundancy in the side information  in the following theorem, whose proof appears  in Section \\ref{proof:redundancy1}.\n\n\\begin{theorem}\\label{thm:redundancy1}\n    For the supremum capacity of PIR-PCSI-I, there is no redundancy in coded side information i.e., $ \\rho_{\\mbox{\\tiny PCSI-I}}= 0$.\n\\end{theorem}\n\nNext we characterize the infimum capacity of PIR-PCSI-I.\n\\begin{theorem}\\label{thm:cap_PCSI1_inf}\n    The infimum capacity of PIR-PCSI-I,\n    \\begin{align}\n        C_{\\mbox{\\tiny PCSI-I}}^{\\inf} &= C_{\\mbox{\\tiny PCSI-I}}(q=2) \\notag\\\\\n        &= \\max\\left(\\frac{1}{K-1}, \\left(K-\\frac{M}{K-M}\\right)^{-1}\\right)\\notag\\\\\n        &= \n        \\begin{cases}\n            \\frac{1}{K-1}, & 1 \\leq M \\leq \\frac{K}{2},\\\\\n            \\left(K - \\frac{M}{K-M}\\right)^{-1}, & \\frac{K}{2} < M \\leq K-1.\n        \\end{cases}\n    \\end{align}\n\\end{theorem}\nThe proof of Theorem \\ref{thm:cap_PCSI1_inf} appears in Section \\ref{sec:cap_PCSI1_inf}. The infimum capacity of PIR-PCSI-I also matches its capacity over binary field. The intuition why $\\mathbb{F}_{2}$ represents the worst case, is similar to the PIR-PCSI-II setting.\n\nNote that for $M=K-1$, the infimum capacity of PIR-PCSI-I matches the supremum capacity, therefore for any field $\\mathbb{F}_{q}$, we have the exact capacity characterization, $C_{\\mbox{\\tiny PCSI-I}}(q) = C_{\\mbox{\\tiny PCSI-I}}^{\\inf} = C_{\\mbox{\\tiny PCSI-I}}^{\\sup}$. However, in general the infimum capacity is strictly smaller and the gap can be significant. For example, when $M=K\/2$ the supremum capacity is $2\/K$ while the infimum capacity is $1\/(K-1)$, i.e., for large $K$, the infimum capacity is nearly half of the supremum capacity.\n\nWe next explore the capacity of PIR-PCSI-I for the case of private coefficients. \n\\begin{theorem}\\label{thm:pcsi1_pub_pri}\n    The supremum capacity of PIR-PCSI-I, for the setting with private coefficients, is given by\n    \\begin{align}\n        C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}, \\sup} = C_{\\mbox{\\tiny PCSI-I}}^{\\inf},\n    \\end{align}\n    while the infimum capacity of the private coefficients setting can be bounded as\n    \\begin{align}\n        \\frac{1}{K-1} \\leq C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}, \\inf} \\leq \\min\\bigg(C_{\\mbox{\\tiny PCSI-I}}^{\\inf}, \\frac{1}{K-2}\\bigg).\n    \\end{align}\n\\end{theorem}\nThe proof of Theorem \\ref{thm:pcsi1_pub_pri} appears in Section \\ref{proof:pcsi1_pub_pri}. Unlike PIR-PCSI-II, for PIR-PCSI-I the capacity with private coefficients may depend on the field, and may be strictly smaller than the infimum capacity. For example, if $M=K-1$, then the infimum capacity is $1$, but the infimum capacity with private coefficients is no more than $1\/(K-2)$. Remarkably, infimum capacity with private coefficients does not correspond to the binary field $\\mathbb{F}_2$, i.e., there exist other fields that yield strictly lower capacities than $\\mathbb{F}_2$ for PIR-PCSI-I when the coefficients are fully private. \n\n\n\\subsection{PIR-PCSI (where $\\bm{\\theta}$ is drawn uniformly  from $[K]$)}\nTo complete the picture, in this section we characterize the capacity of PIR-PCSI which was not studied in \\cite{PIR_PCSI}. Since $\\bm{\\theta} \\in [K]$, any $1\\leq M \\leq K$ is valid. We start with the supremum capacity.\n\n\\begin{theorem}\\label{thm:cap_PCSI_sup}\n    The supremum capacity of PIR-PCSI is\n    \\begin{align}\n        C_{\\mbox{\\tiny PCSI}}^{\\sup} &= \\max\\bigg(\\frac{1}{K-1}, \\frac{1}{K-M+1}\\bigg)\\notag\\\\\n        &=\\begin{cases}\n            \\frac{1}{K-1}, & M=1,\\\\\n            \\frac{1}{K-M+1}, & 2 \\leq M \\leq K,\n        \\end{cases}\n    \\end{align}\n\\end{theorem}\nThe proof of Theorem \\ref{thm:cap_PCSI_sup} appears in Section \\ref{sec:cap_PCSI_sup}.  For $M=1$, this problem is dominated by the PIR-PCSI-I setting, and the capacity is $(K-1)^{-1}$.\n\n\n\nThe redundancy of  CSI to achieve the supremum capacity of PIR-PCSI is bounded in the following theorem.\n\\begin{theorem}\\label{thm:redundancy}\n    The redundancy of the CSI to achieve the supremum capacity of PIR-PCSI is bounded as\n    \\begin{align}\n        \\rho_{\\mbox{\\tiny PCSI}} &= \\frac{1}{2}, && M = 2\\notag\\\\\n        \\rho_{\\mbox{\\tiny PCSI}} &\\leq \\frac{1}{M}, && 3 \\leq M \\leq \\frac{K+2}{2},\\notag\\\\\n        \\rho_{\\mbox{\\tiny PCSI}} &= 0, &&\\text{otherwise}.\n    \\end{align}\n\\end{theorem}\nThe proof of Theorem \\ref{thm:redundancy} appears in Section \\ref{proof:redundancy}. Evidently, for different values of $M$ the redundancy can be as high as $1\/2$ and as low as $0$.\n\nThe infimum capacity of PIR-PCSI is found next.\n\\begin{theorem}\\label{thm:cap_PCSI_inf}\n    The infimum capacity of PIR-PCSI corresponds to $q = 2$, and,\n    \\begin{align}\n        C_{\\mbox{\\tiny PCSI}}^{\\inf} = C_{\\mbox{\\tiny PCSI}}(q=2) = \\frac{1}{K-1}.\n    \\end{align}\n\\end{theorem}\nThe proof of Theorem \\ref{thm:cap_PCSI_inf} appears in Section \\ref{sec:cap_PCSI_inf}. \n\nNote that for $M=1$, the infimum capacity of PIR-PCSI matches the supremum capacity, therefore for any field $\\mathbb{F}_{q}$, we have the exact capacity characterization, $C_{\\mbox{\\tiny PCSI}}(q) = C_{\\mbox{\\tiny PCSI}}^{\\inf} = C_{\\mbox{\\tiny PCSI}}^{\\sup}$. However, in general the infimum capacity is strictly smaller and the gap can be significant. For example, when $M=K$ the supremum capacity is $1$ while the infimum capacity is $1\/(K-1)$.\n\nFinally, the capacity of PIR-PCSI for the case of private coefficients is characterized.\n\\begin{theorem}\\label{thm:pcsi_pub_pri}\n    The capacity of PIR-PCSI, for the setting with private coefficients, is given by\n    \\begin{align}\n        C_{\\mbox{\\tiny PCSI}}^{\\mbox{\\tiny pri}}(q) = C_{\\mbox{\\tiny PCSI}}^{\\inf}.\n    \\end{align}\n\\end{theorem}\nThe proof of Theorem \\ref{thm:pcsi_pub_pri} appears in Section \\ref{proof:pcsi_pub_pri}. Similar to PIR-PCSI-II, and unlike PIR-PCSI-I,  for PIR-PCSI the capacity  with private coefficients does not depend on the field, and is always equal to the infimum capacity. \n\n\n\n\n\\section{Proof of Theorem \\ref{thm:cap_PCSI2_sup}}\\label{sec:cap_PCSI2_sup}\n\n\\subsection{Converse}\nThe following lemma, which is essentially Lemma 1 of \\cite{PIR_PCSI}, states that for PIR-PCSI*, for every feasible $(Q, \\mathcal{S}, \\theta)$, there must exist at least one coefficient vector that allows successful decoding.\n\n\\begin{lemma}\\label{lem:privacy}\n\\begin{align}\n&\\mbox{PIR-PCSI: } \\forall (Q,\\mathcal{S},\\theta)\\in\\mathcal{Q}\\times \\mathfrak{S}\\times[K], ~~\\exists \\Lambda\\in\\mathfrak{C}, \\mbox{ s. t. }\\notag\\\\\n&\\hspace{0.2cm} H(\\bm{W}_{\\theta} \\mid \\bm{\\Delta}, \\bm{Y}^{[\\mathcal{S},\\Lambda]}, \\bm{Q}=Q) = 0.\\label{eq:lemma1pcsi}\\\\\n&\\mbox{PIR-PCSI-I: }\\forall (Q,\\mathcal{S},\\theta)\\in\\mathcal{Q}\\times \\mathfrak{S}\\times([K]\\setminus\\mathcal{S}),~~\\exists \\Lambda\\in\\mathfrak{C}, \\mbox{ s. t. }\\notag\\\\\n&\\hspace{0.2cm} H(\\bm{W}_{\\theta} \\mid \\bm{\\Delta}, \\bm{Y}^{[\\mathcal{S},\\Lambda]}, \\bm{Q}=Q) = 0.\\label{eq:lemma1pcsi1}\\\\\n&\\mbox{PIR-PCSI-II: }\\forall (Q,\\mathcal{S},\\theta)\\in\\mathcal{Q}\\times \\mathfrak{S}\\times\\mathcal{S},~~\\exists \\Lambda\\in\\mathfrak{C}, \\mbox{ s. t. }\\notag\\\\\n&\\hspace{0.2cm} H(\\bm{W}_{\\theta} \\mid \\bm{\\Delta}, \\bm{Y}^{[\\mathcal{S},\\Lambda]}, \\bm{Q}=Q) = 0.\\label{eq:lemma1pcsi2}\n\\end{align}\n\\end{lemma}\n\n\\proof Since the server  knows $\\bm{\\Delta}, \\bm{Q}$ and can test all possible realizations of $\\bm{\\theta}, \\bm{\\mathcal{S}}, \\bm{\\Lambda}$ for decodability, if no  coefficient vector exists for a particular $(\\bm{\\theta} = \\theta,\\bm{\\mathcal{S}} = \\mathcal{S})$ then that $(\\theta,\\mathcal{S})$ can be ruled out by the server. This contradicts the privacy constraint.$\\hfill\\square$\n\nLet us prove the converse for $2 \\leq M \\leq \\frac{K+1}{2}$.\n\n\nConsider any particular realization $Q \\in \\mathcal{Q}$ of $\\bm{Q}$. For all $i\\in[M-1]$, consider $\\mathcal{S}_i=[i:i+M-1]$ and $\\theta=i$, and let $\\Lambda_i$ be a coefficient vector that satisfies \\eqref{eq:lemma1pcsi2} according to Lemma \\ref{lem:privacy}, so that \n\\begin{align}\n    H(\\bm{W}_i \\mid \\bm{\\Delta}, \\bm{Y}^{[\\mathcal{S}_{i},\\Lambda_i]}, \\bm{Q}=Q) = 0.\\label{eq:decodability1}\n\\end{align}\nWriting $\\bm{Y}^{[\\mathcal{S}_{i},\\Lambda_i]}$ as $\\bm{Y}_i$ for compact notation, we have \n\\begin{align}\n    &H(\\bm{W}_{[M-1]}, \\bm{Y}_{[M-1]} \\mid \\bm{\\Delta}, \\bm{Q}=Q)\\\\\n    &\\leq \\sum_{i \\in [M-1]}H(\\bm{W}_i, \\bm{Y}_{i} \\mid \\bm{\\Delta}, \\bm{Q}=Q)\\\\\n    &= \\sum_{i \\in [M-1]}H(\\bm{Y}_i \\mid \\bm{\\Delta}, \\bm{Q}=Q)\\notag\\\\\n    &\\quad + \\sum_{i \\in [M-1]}H(\\bm{W}_i \\mid \\bm{\\Delta}, \\bm{Y}_i, \\bm{Q}=Q)\\\\\n    &= \\sum_{i \\in [M-1]}H(\\bm{Y}_i \\mid \\bm{\\Delta}, \\bm{Q}=Q)\\label{eq:decodability2}\\\\\n    &\\leq (M-1)L \n    \\label{eq:bound_Y}\n\\end{align}\nwhere \\eqref{eq:decodability2} holds since \\eqref{eq:decodability1} holds, and \\eqref{eq:bound_Y} is true since $\\forall i \\in [M-1], \\bm{Y}_i\\in\\mathbb{F}_{q}$.\n\nNext we note\\footnote{$2M-2 \\leq K$ since we consider the case where $2 \\leq M \\leq \\frac{K+1}{2}$.} that $\\bm{W}_{[2M-2]}$  can be obtained from ($\\bm{W}_{[M-1]}, \\bm{Y}_{[M-1]})$, as follows:  $\\bm{W}_{M}$ is obtained by subtracting $\\bm{W}_{[M-1]}$ terms from $\\bm{Y}_1$ which is a linear combination of $\\bm{W}_{[M]}$; $\\bm{W}_{M+1}$ by subtracting $\\bm{W}_{[2:M]}$ terms from $\\bm{Y}_2$ which is a linear combination of $\\bm{W}_{[2:M+1]}$; $\\cdots$; and finally $\\bm{W}_{2M-2}$ by subtracting $\\bm{W}_{[M-1:2M-3]}$ terms from $\\bm{Y}_{M-1}$ which is a linear combination of $\\bm{W}_{[M-1:2M-2]}$. Thus, \n\\begin{align}\n    &H(\\bm{W}_{[2M-2]} \\mid \\bm{\\Delta}, \\bm{Q}=Q) \\\\\n    &\\leq H(\\bm{W}_{[M-1]}, \\bm{Y}_{[M-1]}\\mid \\bm{\\Delta}, \\bm{Q}=Q)\\\\\n    &\\leq  (M-1)L,  \\hspace{1cm}\\forall Q \\in \\mathcal{Q}.\n\\end{align}\nAveraging over $\\bm{Q}$, we have\n\\begin{align}\n    H(\\bm{W}_{[2M-2]} \\mid \\bm{\\Delta}, \\bm{Q}) \\leq (M-1)L.\\label{eq:decodability3}\n\\end{align}\nWe can follow the same argument for any $2M-2$ out of the $K$ messages, thus \\eqref{eq:decodability3} must be true for any $2M-2$ of $K$ messages. Thus, by submodularity,\n\\begin{align}\n    H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q}) &\\leq \\frac{K(M-1)}{2M-2}L \\notag\\\\\n    &= \\frac{K}{2}L.\n\\end{align}\nSince\n\\begin{align}\n    &H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q}) \\notag\\\\\n    &= H(\\bm{W}_{[K]}, \\bm{\\Delta} \\mid \\bm{Q}) - H(\\bm{\\Delta} \\mid \\bm{Q})\\\\\n    &\\geq H(\\bm{W}_{[K]}) - H(\\bm{\\Delta} \\mid \\bm{Q})\\\\\n    & = KL - H(\\bm{\\Delta} \\mid \\bm{Q}),\n\\end{align}\nwe have\n\\begin{align}\n    D&\\geq H(\\bm{\\Delta} \\mid \\bm{Q})\\notag\\\\\n    &\\geq KL - H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q})\\\\\n    &\\geq \\frac{K}{2}L.\\label{eq:delta_bound}\n\\end{align}\nThus, the rate achieved must be bounded as $R=L\/D\\leq 2\/K$. Since this is true for every achievable scheme, $C_{\\mbox{\\tiny PCSI-II}}(q) \\leq 2\/K$ for $2 \\leq M \\leq \\frac{K+1}{2}$. The remaining case, $C_{\\mbox{\\tiny PCSI-II}}(q) \\leq (K-M+1)^{-1}$ for $\\frac{K+1}{2} < M \\leq K$ is already shown in \\cite{PIR_PCSI}.\n\n\\subsection{Achievability}\nWe present an interference alignment based scheme that works for arbitrary $2 \\leq M \\leq K$ and is capacity achieving for $2 \\leq M \\leq \\frac{K+1}{2}$. The capacity-achieving scheme for the remaining case is already shown in \\cite{PIR_PCSI}. The scheme requires that $q$ should be an even power of a prime number, so that $\\mathbb{F}_{\\sqrt{q}}$ is also a finite field. Recall that according to polynomial representations of finite fields, $\\mathbb{F}_q=\\mathbb{F}_{\\sqrt{q}}[x]\/g(x)$ for some degree $2$ irreducible polynomial $g(x)=x^2+a_1x+a_0\\in\\mathbb{F}_{\\sqrt{q}}[x]$, and  $\\mathbb{F}_q$ can be repesented as $\\mathbb{F}_q=\\{\\mu x+\\gamma \\mid \\forall\\mu,\\gamma\\in\\mathbb{F}_{\\sqrt{q}}\\}$. Alternatively, $\\mathbb{F}_q$ can be seen as a $2$ dimensional vector space over $\\mathbb{F}_{\\sqrt{q}}$. Any element ${c}= {\\mu_c} x + {\\gamma_c} \\in \\mathbb{F}_{q}$, where ${\\mu_c}, {\\gamma_c} \\in \\mathbb{F}_{\\sqrt{q}}$, has a corresponding $2\\times 1$ vector representation ${V_{c}}\\in\\mathbb{F}_{\\sqrt{q}}^{2\\times 1}$  and a $2\\times 2$ matrix  representation ${M_{c}}\\in\\mathbb{F}_{\\sqrt{q}}^{2\\times 2}$ as follows (see p. 65 of  \\cite{Finite_Fields}).\n\\begin{align}\n{V}_c&=\\begin{bmatrix}\\gamma_c\\\\ \\mu_c\\end{bmatrix},&&{M}_c=\\begin{bmatrix}\n        {\\gamma}_{{c}} & -{\\mu}_{{c}} a_{0}\\\\\n        {\\mu}_{\\bm{c}} & {\\gamma}_{{c}} - {\\mu}_{{c}} a_{1}\n    \\end{bmatrix}\n\\end{align}\nsuch that for any $a,b,c\\in\\mathbb{F}_q$ we have $a=bc$, if and only if\n\\begin{align}\nV_a&=M_bV_c.\n\\end{align}\nLet us start with the following lemma.\n\\begin{lemma}\\label{lem:uniform12}\nIf $\\bm{c}$ is chosen uniformly randomly over $\\mathbb{F}_q^\\times$, then each row of $M_{\\bm{c}}$ is uniformly distributed over $\\mathbb{F}_{\\sqrt{q}}^{1\\times 2}\\setminus\\{[0,0]\\}$.\n\\end{lemma}\n\\proof Since $a_0,a_1$ are given constants, the second row, $M_{\\bm{c}}(2,:)=[\\mu_{\\bm{c}}, \\gamma_{\\bm{c}}-\\mu_{\\bm{c}}a_1]$ is an invertible function of $V_{\\bm{c}}$. Next, note that $a_0\\neq 0$ because otherwise $g(x)=x(x+a_1)$ would not be irreducible. Therefore, the first row, $M_{\\bm{c}}(1,:)=[\\gamma_{\\bm{c}}, -\\mu_{\\bm{c}}a_0]$ is also an invertible function of $V_{\\bm{c}}$. Finally, since $\\bm{c}$ is uniform over $\\mathbb{F}_q^\\times$, it follows that $V_{\\bm{c}}$ is uniform over $\\mathbb{F}_{\\sqrt{q}}^{1\\times 2}\\setminus\\{[0,0]\\}$, and as an invertible function of $V_{\\bm{c}}$ that maps  non-zero vectors to  non-zero vectors, so is each row of $M_{\\bm{c}}$. \n $\\hfill\\square$\n\nThe scheme proposed in this section needs only $L=1$, so let us say $L=1$. \nRecall that the coded side information (CSI) $\\bm{Y}^{[\\bm{\\mathcal{S}}, \\bm{\\Lambda}]} \\triangleq \\bm{\\lambda}_{1}\\bm{W}_{\\bm{i}_1} + \\cdots + \\bm{\\lambda}_{M}\\bm{W}_{\\bm{i}_M}$ where $\\bm{\\mathcal{S}} = \\{\\bm{i}_1, \\cdots, \\bm{i}_M\\}$ and $\\bm{i}_1<\\bm{i}_2<\\cdots<\\bm{i}_M$.\n\nSince $L=1$, each message is a symbol in $\\mathbb{F}_{q}$. Thus each message $\\bm{W}_{k}, k \\in [K]$ has  vector representation ${V}_{\\bm{W}_k} \\in \\mathbb{F}_{\\sqrt{q}}^{2\\times 1}$. The first and second entry of ${V}_{\\bm{W}_k}$, namely ${V}_{\\bm{W}_k}(1)$ and ${V}_{\\bm{W}_k}(2)$ respectively, are both elements in $\\mathbb{F}_{\\sqrt{q}}$ and $\\bm{W}_k = {V}_{\\bm{W}_k}(2)x + {V}_{\\bm{W}_k}(1)$. \n\nEach coefficient $\\bm{\\lambda}_m, m \\in [M]$ is drawn from $\\mathbb{F}_{q}^{\\times}$, and can be represented as  ${M}_{\\bm{\\lambda}_m} \\in \\mathbb{F}_{\\sqrt{q}}^{2\\times 2}$ such that ${M}_{\\bm{\\lambda}_m}{V}_{\\bm{W}_{\\bm{i}_m}} \\in \\mathbb{F}_{\\sqrt{q}}^{2\\times 1}$ is the vector representation of $\\bm{\\lambda}_{m}\\bm{W}_{\\bm{i}_m} \\in \\mathbb{F}_{q}$.\n\nThus,\n\\begin{align}\n    {V}_{\\bm{Y}} = {M}_{\\bm{\\lambda}_{1}}{V}_{\\bm{W}_{\\bm{i}_1}} + \\cdots + {M}_{\\bm{\\lambda}_{M}}{V}_{\\bm{W}_{\\bm{i}_M}} \\in \\mathbb{F}_{\\sqrt{q}}^{2\\times 1},\\label{eq:vec_rep_CSI}\n\\end{align}\nis the vector representation of  $\\bm{Y}^{[\\bm{\\mathcal{S}}, \\bm{\\Lambda}]} \\in \\mathbb{F}_{q}$.\n\nLet $M_{\\bm{\\lambda}_{m}}(1,:), M_{\\bm{\\lambda}_{m}}(2,:)$ denote the first and second row of $M_{\\bm{\\lambda}_{m}}$ respectively, and $M_{\\bm{\\lambda}_{m}}(r,:)V_{\\bm{W}_{\\bm{i}_m}}$  the dot product of the $r^{th}$ row of $M_{\\bm{\\lambda}_{m}}$ with $V_{\\bm{W}_{\\bm{i}_m}}$ . Then the first and second entry of $V_{\\bm{Y}}$ are\n\\begin{align}\n    V_{\\bm{Y}}(1) = M_{\\bm{\\lambda}_{1}}(1,:)V_{\\bm{W}_{\\bm{i}_1}} + \\cdots + M_{\\bm{\\lambda}_{M}}(1,:)V_{\\bm{W}_{\\bm{i}_M}},\\label{eq:vec_rep_CSI_1}\\\\\n    V_{\\bm{Y}}(2) = M_{\\bm{\\lambda}_{1}}(2,:)V_{\\bm{W}_{\\bm{i}_1}} + \\cdots + M_{\\bm{\\lambda}_{M}}(2,:)V_{\\bm{W}_{\\bm{i}_M}},\n\\end{align}\nrespectively.\n\nTo privately  retrieve $\\bm{W}_{\\bm{\\theta}}$ for some $\\bm{\\theta} \\in \\bm{\\mathcal{S}}$, the user's download $\\bm{\\Delta}$ is \n\\begin{align}\n    \\bm{\\Delta} = (\\mathbf{L}_{k}V_{\\bm{W}_{k}})_{k \\in [K]},\n\\end{align}\nwhere $\\mathbf{L}_{\\bm{i}_m}=M_{\\bm{\\lambda}_{m}}(1,:)$ for $\\bm{i}_m \\in \\bm{\\mathcal{S}} \\setminus \\{\\bm{\\theta}\\}$,  and $\\mathbf{L}_{\\bm{i}_m}=M_{\\bm{\\lambda}_{m}}(2,:)$ for $\\bm{i}_m=\\bm{\\theta}$. For $k \\in [K] \\setminus \\bm{\\mathcal{S}}$, $\\mathbf{L}_{k}$ is uniformly drawn from $\\mathbb{F}_{\\sqrt{q}}^{1\\times 2}\\setminus\\{[0~~0]\\}$.\n\nUpon receiving $\\bm{\\Delta}$, by subtracting the $\\{M_{\\bm{\\lambda}_{m}}(1,:)V_{\\bm{W}_{\\bm{i}_m}}\\}_{\\bm{i}_m \\in \\bm{\\mathcal{S}} \\setminus \\{\\bm{\\theta}\\}}$ terms from $V_{\\bm{Y}}(1)$, the user is able to obtain $M_{\\bm{\\lambda}_{\\bm{t}}}(1,:)V_{\\bm{W}_{\\bm{\\theta}}}$, where $\\bm{i}_t=\\bm{\\theta}$. Together with $M_{\\bm{\\lambda}_{\\bm{t}}}(2,:)V_{\\bm{W}_{\\bm{\\theta}}}$, which is directly downloaded, the user is able to recover $M_{\\bm{\\lambda}_{\\bm{t}}}V_{\\bm{W}_{\\bm{\\theta}}}$, i.e., $\\bm{\\lambda}_{\\bm{t}}\\bm{W}_{\\bm{\\theta}}$, and since $\\bm{\\lambda}_{\\bm{t}}$ is a non-zero value in $\\mathbb{F}_q$ that is known to the user, the user is able to retrieve the desired message $\\bm{W}_{\\bm{\\theta}}$.\n\nSince $\\bm{\\lambda}_[M]$ are  i.i.d. uniform over $\\mathbb{F}_q^\\times$, it follows from Lemma \\ref{lem:uniform12} that all $\\mathbf{L}_k, k\\in[K]$ are i.i.d. uniform over $\\mathbb{F}_{\\sqrt{q}}^{1\\times 2}\\setminus\\{[0~~0]\\}$. Thus, the queries are independent of $(\\bm{\\theta},\\bm{\\mathcal{S}})$, and the privacy constraint is satisfied.\n\n\\begin{remark}\\label{rmk:PCSI2_margin}\n    The scheme is also capacity achieving for the boundary case $\\frac{K+1}{2} < M \\leq \\frac{K+2}{2}$ (i.e., $2M = K+2$) because in this case, $2\/K = (K-M+1)^{-1}$.\n\\end{remark}\n\n\\begin{remark}\\label{rmk:half_CSI}\n    The scheme only uses $V_{\\bm{Y}}(1)$ specified in \\eqref{eq:vec_rep_CSI}, i.e., $V_{\\bm{Y}}(2)$ is never used so it can be discarded by the user. Thus, at least half of the side-information is redundant.\n\\end{remark}\n\nLet us consider an example for illustration.\n\\begin{example}\\label{exp:F4}\nSuppose $q = 4$, $L=1$. There are $K=3$ messages $\\bm{A},\\bm{B},\\bm{C}\\in\\mathbb{F}_4$.   We have $M=2$. Say the CSI is the linear combination  $\\bm{Y}=\\bm{\\lambda}_1\\bm{A}+\\bm{\\lambda}_2\\bm{B}$, with $\\bm{\\lambda}_1,\\bm{\\lambda}_2$ i.i.d. uniform in $\\mathbb{F}_4^\\times$, and the desired message is $\\bm{A}$.\n\nWe note that $\\mathbb{F}_4=\\mathbb{F}_2[x]\/(x^2+x+1)$ has the $4$ elements: $0, 1, x, 1+x$, which have matrix representations:\n{\\small \n\\begin{align*}\nM_0= \\begin{bmatrix}\n        0&0\\\\\n        0&0\\\\\n    \\end{bmatrix}, M_1= \\begin{bmatrix}\n        1&0\\\\\n        0&1\\\\\n    \\end{bmatrix}, M_{x}= \\begin{bmatrix}\n        0&1\\\\\n        1&1\\\\\n        \\end{bmatrix},M_{1+x}= \\begin{bmatrix}\n        1&1\\\\\n        1&0\\\\\n        \\end{bmatrix}.\n\\end{align*}\n}\nNote that if ${\\bf c}$ is uniform over $\\mathbb{F}_4^\\times=\\{1,x,1+x\\}$ then $M_{\\bm{c}}(1,:)$ is uniform over $\\{[1~~0],[0~~1],[1~~1]\\}=\\mathbb{F}_{2}^{1\\times 2}\\setminus\\{[0~~0]\\}$, and so is $M_{\\bm{c}}(2,:)$, as claimed by Lemma \\ref{lem:uniform12}.\nDefine $\\bm{A}=\\bm{A}_1+\\bm{A}_2x$, where $\\bm{A}_1,\\bm{A}_2\\in\\mathbb{F}_2$, so that $V_{\\bm{A}}=[\\bm{A}_1~~\\bm{A}_2]^T$, and use similar definitions for $\\bm{B},\\bm{C}$ as well. Note that $\\bm{\\lambda}_1\\bm{A}$ is uniform over $\\{\\bm{A}_1,\\bm{A}_2,\\bm{A}_1+\\bm{A}_2\\}$ and $\\bm{\\lambda}_2\\bm{B}$ is uniform over $\\{\\bm{B}_1,\\bm{B}_2,\\bm{B}_1+\\bm{B}_2\\}$. For example, let's say $\\bm{\\lambda}_1=1+x, \\bm{\\lambda}_2=x$, then $V_{\\bm{Y}}(1)= \\bm{A}_1+\\bm{A}_2+\\bm{B}_2$. The user downloads, say $\\bm{\\Delta}=(\\bm{A}_1,\\bm{B}_2,\\bm{C}_1+\\bm{C}_2)$ which allows $\\bm{A}$ to be retrieved with the help of the side information $V_{\\bm{Y}}(1)$. The download is always uniform over $\\{\\bm{A}_1,\\bm{A}_2,\\bm{A}_1+\\bm{A}_2\\}\\times\\{\\bm{B}_1,\\bm{B}_2,\\bm{B}_1+\\bm{B}_2\\}\\times \\{\\bm{C}_1,\\bm{C}_2,\\bm{C}_1+\\bm{C}_2\\}$, regardless of the realization of $(\\bm{\\theta},\\bm{\\mathcal{S}})$, which guarantees privacy.\n\\end{example}\n\n\n\\section{Proof of Theorem \\ref{thm:red}}\\label{proof:red}\nWe need Lemma \\ref{lem:alpha_min_1} and \\ref{lem:alpha_min_2} to bound the redundancy $\\rho_{\\mbox{\\tiny PCSI-II}}$ from above, (equivalently, lower-bound $\\alpha^*$) for $2 \\leq M \\leq \\frac{K+2}{2}$ and $\\frac{K+2}{2} < M \\leq K$, respectively. \n\n\\begin{lemma}\\label{lem:alpha_min_1}\n    For $2 \\leq M \\leq \\frac{K+2}{2}$, the redundancy $\\rho_{\\mbox{\\tiny PCSI-II}}\\leq 1\/2$.\n\\end{lemma}\n\nIntuitively, the entropy of the download is $H(\\bm{\\Delta}) = \\frac{K}{2}L$. On average, at most $L\/2$ symbols of each message are contained in the download. In order to fully recover the desired message, the user must have at least another $L\/2$ $q$-ary symbols as the side information. \n\n\n\\proof Recall that the capacity for this case is $2\/K$, i.e., the optimal average download cost is $D\/L=K\/2$. Since this is the infimum across all achievable schemes, there must exist achievable schemes that achieve $D\/L\\leq K\/2+\\epsilon$ for any $\\epsilon>0$. So consider an achievable scheme such that $\\alpha$ PCSI is sufficient and the average download cost  $D\/L\\leq K\/2+\\epsilon$ for some $L$. Since $D\/L\\leq K\/2+\\epsilon$, we have\n\\begin{align}\n&LK\/2+\\epsilon L\\notag\\\\\n&\\geq D\\\\\n    &\\geq H(\\bm{\\Delta} \\mid \\bm{Q}) \\\\\n    &\\geq I(\\bm{\\Delta}; \\bm{W}_{[K]} \\mid \\bm{Q}) \\notag\\\\\n    &= \\sum_{k \\in [K]}I(\\bm{\\Delta}; \\bm{W}_k \\mid \\bm{Q}, \\bm{W}_{[k-1]})\\\\\n    &= \\sum_{k \\in [K]}\\bigg(H(\\bm{W}_k \\mid \\bm{Q}, \\bm{W}_{[k-1]})-H(\\bm{W}_k \\mid \\bm{\\Delta}, \\bm{Q}, \\bm{W}_{[k-1]})\\bigg)\\\\\n    &= \\sum_{k \\in [K]}\\bigg(H(\\bm{W}_k)-H(\\bm{W}_k \\mid \\bm{\\Delta}, \\bm{Q}, \\bm{W}_{[k-1]})\\bigg)\\label{eq:independent}\\\\\n    &\\geq \\sum_{k \\in [K]}\\bigg(H(\\bm{W}_k)-H(\\bm{W}_k \\mid \\bm{\\Delta}, \\bm{Q})\\bigg)\\\\\n    &= \\sum_{k \\in [K]}I(\\bm{W}_k; \\bm{\\Delta}, \\bm{Q}),\\\\\n    &\\geq K I(\\bm{W}_{k^*}; \\bm{\\Delta}, \\bm{Q})\n    \\label{eq:alpha_half}\n\\end{align}\nwhere \\eqref{eq:independent} holds since all the messages and the query are mutually independent, and\n\\begin{align}\nk^*=\\arg\\min_{k\\in[K]}I(\\bm{W}_k; \\bm{\\Delta}, \\bm{Q})\n\\end{align}\nFrom \\eqref{eq:alpha_half} we have,\n\\begin{align}\n    H(\\bm{W}_{k^*} \\mid \\bm{\\Delta}, \\bm{Q}) \\geq L\/2 -\\epsilon L\/K.\n\\end{align}\nThus, there must exist a feasible query $Q$ such that \n\\begin{align}\n   H(\\bm{W}_{k^*} \\mid \\bm{\\Delta}, \\bm{Q}=Q) \\geq L\/2-\\epsilon L\/K. \\label{eq:smallest}\n\\end{align}\nLet $\\mathcal{S} =\\{i_1, \\cdots, i_{M-1}, k^{*}\\}\\subset[K]$, such that $|\\mathcal{S}|=M$. Then according to Lemma \\ref{lem:privacy} and \\eqref{eq:invaYR}, there must exist $\\Lambda \\in \\mathfrak{C}$ such that \n\\begin{align}\n    &H(\\bm{W}_{k^*} \\mid \\bm{\\Delta}, \\overline{\\bm{Y}}^{[\\mathcal{S},\\Lambda]}, \\bm{Q}=Q) = 0,\\label{eq:dec_smallest}\\\\\n    &H(\\overline{\\bm{Y}}^{[\\mathcal{S},\\Lambda]} \\mid \\bm{Q} = Q) = H(\\overline{\\bm{Y}}^{[\\bm{\\mathcal{S}},\\bm{\\Lambda}]}) \\leq \\alpha L.\n\\end{align}\nCombining \\eqref{eq:smallest} and \\eqref{eq:dec_smallest}, we have\n\\begin{align}\n    I(\\overline{\\bm{Y}}^{[\\mathcal{S},\\Lambda]}; \\bm{W}_{k^*} \\mid \\bm{\\Delta}, \\bm{Q}=Q) \\geq L\/2-\\epsilon L\/K.\n\\end{align}\nThus \n\\begin{align}\n    \\alpha L &\\geq H(\\overline{\\bm{Y}}^{[\\mathcal{S},\\Lambda]} \\mid \\bm{Q} = Q)\\notag\\label{eq:indYQR}\\\\\n    &\\geq I(\\overline{\\bm{Y}}^{[\\mathcal{S},\\Lambda]}; \\bm{W}_{k^*} \\mid \\bm{\\Delta}, \\bm{Q}=Q) \\geq \\frac{L}{2}-\\epsilon L\/K\n\\end{align}\nwhich implies that $\\alpha \\geq 1\/2 - \\epsilon\/K$. In order to approach capacity, we must have $\\epsilon\\rightarrow 0$, therefore we need $\\alpha\\geq 1\/2$. Since this is true for any $\\alpha$ such that $\\alpha$ PCSI is sufficient, it is also true for $\\alpha^*$, and therefore the redundancy is $\\rho_{\\mbox{\\tiny PCSI-II}}\\leq 1\/2$.\n\n\\begin{lemma}\\label{lem:alpha_min_2}\n    For $\\frac{K+2}{2} < M \\leq K$, the redundancy $\\rho_{\\mbox{\\tiny PCSI-II}}\\leq 0$.\n    \\end{lemma}\n\n\\proof Recall that the capacity for this case is $(K-M+1)^{-1}$, i.e., the optimal average download cost is $D\/L=K-M+1$. Consider an achievable scheme such that $\\alpha$ PCSI is sufficient and the average download cost $D\/L\\leq K-M+1+\\epsilon$ for some $L$. Since $D\/L\\leq K-M+1+\\epsilon$, we have $L(K-M+1)+\\epsilon L\\geq D\\geq H(\\bm{\\Delta} \\mid \\bm{Q})$. Thus, there exists a feasible $Q$ such that \n\\begin{align}\n   H(\\bm{\\Delta} \\mid \\bm{Q} = Q) \\leq (K-M+1)L+\\epsilon L.\n\\end{align}\nFor all $i \\in [K-M+1]$, let $\\mathcal{S}_{i} = [i:i+M-1]$. Also, let $\\mathcal{S}_{K-M+2} = \\{1\\} \\cup [K-M+2:K]$. For all $i \\in [K-M+2]$, let $\\Lambda_i \\in \\mathfrak{C}$ satisfy\n\\begin{align}\n    H(\\bm{W}_i \\mid \\bm{\\Delta}, \\overline{\\bm{Y}}^{[\\mathcal{S}_i,\\Lambda_i]}, \\bm{Q} = Q) = 0.\n\\end{align}\nSuch $\\Lambda_i$'s must exist according to Lemma \\ref{lem:privacy}. \n\nWriting $\\overline{\\bm{Y}}^{[\\mathcal{S}_i,\\Lambda_i]}$ as $\\overline{\\bm{Y}}_{i}$ for compact notation, we have \n\\begin{align}\n    H(\\bm{W}_{[K-M+2]} \\mid \\bm{\\Delta}, \\overline{\\bm{Y}}_{[K-M+2]}, \\bm{Q}=Q) = 0. \\label{eq:redundancy2_1}\n\\end{align}\nAccording to \\eqref{eq:invaYR}, \n\\begin{align}\n    H(\\overline{\\bm{Y}}_i \\mid \\bm{Q} = Q)  \\leq \\alpha L.\n\\end{align}\nso we have\n\\begin{align}\n    &(K-M+1)L+\\epsilon L + H(\\overline{\\bm{Y}}_{[K-M+1]} \\mid \\bm{Q} = Q)+\\alpha L\\notag\\\\\n    &\\geq H(\\bm{\\Delta}, \\overline{\\bm{Y}}_{[K-M+2]} \\mid \\bm{Q} = Q)\\\\\n    &\\geq I(\\bm{\\Delta}, \\overline{\\bm{Y}}_{[K-M+2]}; \\bm{W}_{[K-M+2]},\\overline{\\bm{Y}}_{[K-M+2]}\\mid \\bm{Q}=Q)\\\\\n    &= H(\\bm{W}_{[K-M+2]},\\overline{\\bm{Y}}_{[K-M+2]} \\mid \\bm{Q}=Q)\\label{eq:redundancy2_2}\\\\\n    &\\geq H(\\bm{W}_{[K-M+2]},\\overline{\\bm{Y}}_{[K-M+1]} \\mid \\bm{Q}=Q)\\\\\n    &= H(\\bm{W}_{[K-M+2]} \\mid \\bm{Q}=Q)\\notag\\\\\n    &\\quad\\quad + H(\\overline{\\bm{Y}}_{[K-M+1]} \\mid \\bm{W}_{[K-M+2]}, \\bm{Q}=Q)\\\\\n    &\\geq (K-M+2)L\\notag\\\\\n    &\\quad\\quad + H(\\overline{\\bm{Y}}_{[K-M+1]} \\mid \\bm{W}_{[M-1]}, \\bm{Q}=Q),\\label{eq:region}\n\\end{align}\nwhere \\eqref{eq:redundancy2_2} follows from \\eqref{eq:redundancy2_1}. Step \\eqref{eq:region} uses the independence of messages and queries according to \\eqref{eq:indQ} and the fact that  $M-1 \\geq K-M+2$, because we require $M>(K+2)\/2$. We further bound\n\\begin{align}\n    &H(\\overline{\\bm{Y}}_{[K-M+1]} \\mid \\bm{W}_{[M-1]}, \\bm{Q}=Q)\\notag\\\\\n    &=H(\\overline{\\bm{Y}}_{1} \\mid\\bm{W}_{[M-1]}, \\bm{Q}=Q) + \\cdots \\notag\\\\\n    &\\quad\\quad + H(\\overline{\\bm{Y}}_{K-M+1} \\mid \\bm{W}_{[M-1]}, \\overline{\\bm{Y}}_{[K-M]}, \\bm{Q}=Q)\\\\\n    &\\geq \\sum_{i=1}^{K-M+1}H(\\overline{\\bm{Y}}_i\\mid \\bm{W}_{[i+M-2]}, \\bm{Q}=Q)\\label{eq:linearfunc}\\\\\n        &= \\sum_{i=1}^{K-M+1}H(\\overline{\\bm{Y}}_i\\mid  \\bm{Q}=Q)\\label{eq:pcsi2_red_indYO}\\\\\n        &\\geq H(\\overline{\\bm{Y}}_{[K-M+1]}\\mid  \\bm{Q}=Q)\\label{eq:plugin}\n\\end{align}\n \\eqref{eq:linearfunc} holds because $\\overline{\\bm{Y}}_{[i-1]}$ is a function of $\\bm{W}_{[i+M-2]}$ for all $i \\in [2:K-M+1]$. Step \\eqref{eq:pcsi2_red_indYO} follows from \\eqref{eq:invaYR}. Substituting from \\eqref{eq:plugin} into \\eqref{eq:region}, we have \n\\begin{align}\n    &(K-M+1)L + \\epsilon L+ \\alpha L \\geq (K-M+2)L ,\n\\end{align}\nwhich gives $\\alpha \\geq 1-\\epsilon$. In order to approach capacity, we must have $\\epsilon\\rightarrow 0$, so we need $\\alpha\\geq 1$, and since this is true for any $\\alpha$ such that $\\alpha$ PCSI is sufficient, it is also true for $\\alpha^*$. Thus, the redundancy is bounded as $\\rho_{\\mbox{\\tiny PCSI-II}}\\leq 0$. $\\hfill\\square$\n\nAccording to Remark \\ref{rmk:PCSI2_margin} and \\ref{rmk:half_CSI}, $\\alpha = 1\/2$ is sufficient for $2 \\leq M \\leq \\frac{K+2}{2}$ and by the construction of CSI (a linear combination of messages), $\\alpha \\leq 1$. Theorem \\ref{thm:red} is thus proved.\n\n\\section{Proof of Theorem \\ref{thm:cap_PCSI2_inf}}\\label{sec:cap_PCSI2_inf}\nWe prove Theorem \\ref{thm:cap_PCSI2_inf} by first showing that  $C_{\\mbox{\\tiny PCSI-II}}(q=2)\\leq M\/((M-1)K)$ and then presenting a PIR-PCSI-II scheme with rate $M\/((M-1)K)$ that works for any $\\mathbb{F}_{q}$.\n\n\\subsection{Converse for $C_{\\mbox{\\tiny PCSI-II}}(q=2)$}\nNote that Lemma \\ref{lem:privacy} is true for arbitrary $\\mathbb{F}_q$. In $\\mathbb{F}_{2}$, we can only have $\\bm{\\Lambda}=(1,1,\\cdots,1)=1_M$, i.e., the length $M$ vector whose elements are all equal to $1$. As a direct result of Lemma \\ref{lem:privacy}, for PIR-PCSI-II in $\\mathbb{F}_2$,\n\\begin{align}\n    H(\\bm{W}_{\\mathcal{S}} \\mid \\bm{\\Delta}, \\bm{Y}^{[\\mathcal{S},1_{M}]}, \\bm{Q}=Q) = 0, ~\\forall(Q,\\mathcal{S}) \\in \\mathcal{Q}\\times\\mathfrak{S}.\\label{eq:pcsi2_inf_dec}\n\\end{align}\nThus, $\\forall (Q,\\mathcal{S}) \\in \\mathcal{Q}\\times\\mathfrak{S}$,\n\\begin{align}\n    &H(\\bm{W}_{\\mathcal{S}} \\mid \\bm{\\Delta}, \\bm{Q}=Q)\\notag\\\\\n    &= H(\\bm{W}_{\\mathcal{S}}, \\bm{Y}^{[\\mathcal{S},1_M]} \\mid \\bm{\\Delta}, \\bm{Q}=Q)\\label{eq:F2CSI}\\\\\n    &= H(\\bm{Y}^{[\\mathcal{S},1_M]} \\mid \\bm{\\Delta}, \\bm{Q}=Q)\\notag\\\\\n    &\\quad + H(\\bm{W}_{\\mathcal{S}} \\mid \\bm{\\Delta}, \\bm{Y}^{[\\mathcal{S},1_M]}, \\bm{Q}=Q)\\\\\n    &\\leq L.\n\\end{align}\n\\eqref{eq:F2CSI} holds because $\\bm{Y}^{[\\mathcal{S},1_M]}$ is simply the summation of $\\bm{W}_{\\mathcal{S}}$. Averaging over $\\bm{Q}$, we have $H(\\bm{W}_{\\mathcal{S}} \\mid \\bm{\\Delta}, \\bm{Q}) \\leq L, \\forall \\mathcal{S} \\in \\mathfrak{S}$. By submodularity,\n\\begin{align}\n    H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q}) \\leq KL\/M.\n\\end{align}\nThe download cost can now be lower bounded as,\n\\begin{align}\n D\\geq   H(\\bm{\\Delta} \\mid \\bm{Q}) \\geq KL - H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q})) \\geq \\frac{(M-1)KL}{M}.\n\\end{align}\nThus, we have shown that $C_{\\mbox{\\tiny PCSI-II}}(q=2) \\leq \\frac{M}{(M-1)K}$.\n\n\\subsection{A PIR-PCSI-II Scheme  for Arbitrary $q$}\\label{sec:PCSI2_inf_ach}\nIn this section, we prove $C_{\\mbox{\\tiny PCSI-II}}(q) \\geq \\frac{M}{(M-1)K}$ for all $q$ by proposing a scheme, namely \\emph{Generic Linear Combination Based Scheme}, that can achieve the rate $\\frac{M}{(M-1)K}$ for any $\\mathbb{F}_{q}$.\n\nLet us choose $L = Ml$ where $M$ is the size of the support index set and $l$ is a positive integer which can be arbitrarily large. Thus, any message $\\bm{W}_k, k \\in [K]$ can be represented as a length-$M$ column vector $V_{\\bm{W}_k} \\in \\mathbb{F}_{q^l}^{M\\times 1}$. Let \n\\begin{align}\n    V_{\\bm{W}_{\\bm{\\mathcal{S}}}} = \n    \\begin{bmatrix}\n        V_{\\bm{W}_{\\bm{i}_1}}^{\\mathrm{T}} &  \\cdots & V_{\\bm{W}_{\\bm{i}_M}}^{\\mathrm{T}}\n    \\end{bmatrix}^{\\mathrm{T}} \\in \\mathbb{F}_{q^l}^{M^2\\times 1}\n\\end{align}\nwhere $\\bm{\\mathcal{S}} = \\{\\bm{i}_1, \\cdots, \\bm{i}_M\\}$ is the support index set. The CSI $\\bm{Y}$ can be represented as  $V_{\\bm{Y}} \\in \\mathbb{F}_{q^l}^{M\\times 1}$ such that,\n\\begin{align}\n    V_{\\bm{Y}} = \\underbrace{\n    \\begin{bmatrix}\n        \\bm{\\lambda}_{1}\\mathbf{I}_{M} & \\bm{\\lambda}_{2}\\mathbf{I}_{M} & \\cdots & \\bm{\\lambda}_{M}\\mathbf{I}_{M}\n    \\end{bmatrix}}_{M}V_{\\bm{W}_{\\bm{\\mathcal{S}}}},\n\\end{align}\nwhere $\\mathbf{I}_{M} \\in \\mathbb{F}_{q^l}^{M\\times M}$ is the $M \\times M$ identity matrix. \n\nThe download is specified as,\n\\begin{align}\n    \\bm{\\Delta} = \\{&\\mathbf{L}_1^{(1)}V_{\\bm{W}_{1}}, \\cdots, \\mathbf{L}_1^{(M-1)}V_{\\bm{W}_{1}}, \\notag\\\\ \n    &\\cdots, \\mathbf{L}_K^{(1)}V_{\\bm{W}_{K}}, \\cdots, \\mathbf{L}_K^{(M-1)}V_{\\bm{W}_{K}}\\},\n\\end{align}\nwhere $\\forall k \\in [K], m \\in [M-1], \\mathbf{L}_k^{(m)} \\in \\mathbb{F}_{q^l}^{1\\times M}$ is a length-$M$ row vector, i.e., for any message vector $V_{\\bm{W}_{k}} \\in \\mathbb{F}_{q^l}^{M\\times 1}$, $\\bm{\\Delta}$ contains $M-1$ linear combinations of that message vector.\n\nSuppose the vectors $\\mathbf{L}_k^{(m)}$ are chosen such that  $\\forall \\mathcal{S}=\\{j_1, \\cdots, j_{M}\\} \\in \\mathfrak{S}$ the following $M^2\\times M^2$ square matrix has full rank.\n\\begin{align}\n    \\mathbf{G}_{\\mathcal{S}} = \n    \\begin{bmatrix}\n        \\lambda_{1}\\mathbf{I}_{M} & \\cdots & \\lambda_{M}\\mathbf{I}_{M}\\\\\n         & \\mathbf{e}_{1}\\otimes\\mathbf{L}_{j_1}^{(1)} &\\\\\n         & \\cdots &\\\\\n         & \\mathbf{e}_{1}\\otimes\\mathbf{L}_{j_1}^{(M-1)} &\\\\\n         & \\cdots &\\\\\n         & \\mathbf{e}_{M}\\otimes\\mathbf{L}_{j_M}^{(1)} &\\\\\n         & \\cdots &\\\\\n         & \\mathbf{e}_{M}\\otimes\\mathbf{L}_{j_M}^{(M-1)} &\n    \\end{bmatrix},\\label{eq:F2_inv}\n\\end{align}\n$\\mathcal{S} = \\{j_1, \\cdots, j_{M}\\} \\in \\mathfrak{S}$. Note that $(\\lambda_1, \\cdots, \\lambda_M) \\in \\mathfrak{C}$ is the realization of $\\bm{\\Lambda}$, $\\mathbf{e}_{m}, m\\in [M]$ is the $m^{th}$ row of the $M\\times M$ identity matrix and ``$\\otimes$'' is the Kronecker product.\n\nThe correctness constraint is satisfied because the side-information and the downloads allow the user to obtain  $\\mathbf{G}_{\\bm{\\mathcal{S}}}V_{\\bm{W}_{\\bm{\\mathcal{S}}}}$, which can then be multiplied by the inverse of $\\mathbf{G}_{\\bm{\\mathcal{S}}}$ to obtain $V_{\\bm{W}_{\\bm{\\mathcal{S}}}}$, i.e., $\\bm{W}_{\\bm{\\mathcal{S}}}$ which contains $\\bm{W}_{\\bm{\\theta}}$. Specifically the side-information corresponds to the first $M$ rows of $\\mathbf{G}_{\\bm{\\mathcal{S}}}V_{\\bm{W}_{\\bm{\\mathcal{S}}}}$, the downloads $\\mathbf{L}_{\\bm{j}_1}^{(1)}V_{\\bm{W}_{\\bm{j}_1}},\\cdots,\\mathbf{L}_{\\bm{j}_1}^{(M-1)}V_{\\bm{W}_{\\bm{j}_1}}$ correspond to the next $M-1$ rows of $\\mathbf{G}_{\\bm{\\mathcal{S}}}V_{\\bm{W}_{\\bm{\\mathcal{S}}}}$, and so on.\n\nOn the other hand, the privacy constraint is satisfied because the construction is such that for every feasible $\\mathcal{S}$, the user is able to decode all $M$ messages $\\bm{W}_{\\mathcal{S}}$. \n\nFinally let us evaluate the rate achieved by this scheme. Since the user downloads $\\frac{M-1}{M}$ portion of every message, the download cost is $D=LK(M-1)\/M$, and the rate achieved is $M\/((M-1)K))$. Since this rate is achieved for any $\\mathbb{F}_q$, we have the lower bound $C_{\\mbox{\\tiny PCSI-II}}(q) \\geq M\/((M-1)K))$.\n\nIt remains to show the existence of such $\\mathbf{L}_k^{(m)}$, for which we need the following lemma.\n\\begin{lemma}\\label{lem:existence}\n    There exist $\\{\\mathbf{L}_{k}^{(m)}\\}_{k \\in [K], m \\in [M-1]}$ such that  for every $\\mathcal{S} = \\{j_1, \\cdots, j_{M}\\} \\in \\mathfrak{S}$, the matrix $ \\mathbf{G}_{\\mathcal{S}} $  in \\eqref{eq:F2_inv} has full rank, provided \n    \\begin{align}\n        q^l > \\tbinom{K}{M}M(M-1).\n    \\end{align}\n\\end{lemma}\n\\proof The proof is in Appendix \\ref{app:existence}. $\\hfill\\square$\n\nWith the help of Lemma  \\ref{lem:existence}, Theorem \\ref{thm:cap_PCSI2_inf} is  proved. Let us illustrate the scheme with an example.\n\n\\begin{example}\nConsider $M=2, K=4, L=2l, q = 2$. The $4$ messages are $\\bm{A},\\bm{B},\\bm{C},\\bm{D}$. The user has $\\bm{A}+\\bm{B}$ as the side information and wants to retrieve $\\bm{A}$.\n\n$\\bm{A}$ can be represented as a $2\\times 1$ vector $V_{\\bm{A}} = [V_{\\bm{A}}(1) \\quad V_{\\bm{A}}(2)]^{\\mathrm{T}}$ where $V_{\\bm{A}}(1), V_{\\bm{A}}(2) \\in \\mathbb{F}_{2^l}$. Similarly, $\\bm{B}, \\bm{C}, \\bm{D}$ can be represented as $V_{\\bm{B}}$, $V_{\\bm{C}}$, $V_{\\bm{D}}$, respectively. Thus, \n\\begin{align}\n    V_{\\bm{W}_{\\bm{\\mathcal{S}}}} = [V_{\\bm{A}}(1) \\quad V_{\\bm{A}}(2) \\quad V_{\\bm{B}}(1) \\quad V_{\\bm{B}}(2)]^{\\mathrm{T}},\n\\end{align}\n\\begin{align}\n    V_{\\bm{Y}} = \n    \\begin{bmatrix}\n        1&0&1&0\\\\\n        0&1&0&1\n    \\end{bmatrix}V_{\\bm{W}_{\\bm{\\mathcal{S}}}} =\n    \\begin{bmatrix}\n        V_{\\bm{A}}(1)+V_{\\bm{B}}(1)\\\\\n        V_{\\bm{A}}(2)+V_{\\bm{B}}(2)\n    \\end{bmatrix}.\n\\end{align}\n\nThe download from the server is \n\\begin{align}\n\\bm{\\Delta} = \\{&V_{\\bm{A}}(1)+\\alpha_1 V_{\\bm{A}}(2), V_{\\bm{B}}(1)+\\alpha_2 V_{\\bm{B}}(2),\\notag\\\\\n&V_{\\bm{C}}(1)+\\alpha_3 V_{\\bm{C}}(2), V_{\\bm{D}}(1)+\\alpha_4 V_{\\bm{D}}(2)\\}.\n\\end{align}\nwhere $\\alpha_1, \\cdots, \\alpha_4$ are elements of $\\mathbb{F}_{2^l}$ \nsuch that the $4\\times 4$ matrix over $\\mathbb{F}_{2^l}$, \n\\begin{align}\n\\begin{bmatrix}\n    1&0&1&0\\\\\n    0&1&0&1\\\\\n    1&\\alpha_i&0&0\\\\\n    0&0&1&\\alpha_j\n\\end{bmatrix}\n\\end{align}\nhas full rank for any $i < j, \\{i,j\\} \\subset [4]$, which is true if and only if $\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4,1,0$ are distinct. Thus any $2^l\\geq 6$ works, i.e., it suffices to choose $l=3$.\n\\end{example}\n\n\n\\section{Proof of Theorem \\ref{thm:MK}}\\label{proof:MK}\nFor the case $q=2$, it suffices to download any $K-1$ messages out of the $K$ messages to achieve the capacity $\\frac{1}{K-1}$, since the desired message is either directly downloaded or can be recovered by subtracting the $K-1$ downloaded messages from the CSI.\n\nFor $q \\neq 2$, to achieve the capacity $1$, it suffices to download a linear combination of all $K$ messages with non-zero coefficients. Specifically, \n\\begin{align}\n    \\bm{\\Delta} = \\bm{Y} + \\bm{\\lambda}^{\\prime}\\bm{W}_{\\bm{\\theta}},\n\\end{align}\nwhere $\\bm{Y}$ is the CSI and $\\bm{\\lambda}^{\\prime} \\in \\mathbb{F}_{q}^{\\times}$ is a non-zero element in $\\mathbb{F}_{q}$ such that $\\bm{\\lambda}_{\\bm{t}} + \\bm{\\lambda}^{\\prime} \\neq 0$ (let $\\bm{\\lambda}_{\\bm{t}}$ denote the coefficient in front of $\\bm{W}_{\\bm{\\theta}}$ in the CSI $\\bm{Y}$). Such $\\bm{\\lambda}^{\\prime}$ always exists for $q \\neq 2$. From the server's perspective, the user is downloading a random linear combination of $K$ messages so the privacy constraint is satisfied. The user is able to decode $\\bm{W}_{\\bm{\\theta}}$ by subtracting $\\bm{Y}$ from $\\bm{\\Delta}$ so the correctness constraint is satisfied. \n\n\\section{Proof of Theorem \\ref{thm:M3K4}}\\label{proof:M3K4}\nLet us denote the $K=4$ messages as $\\bm{W}_1=\\bm{A},\\bm{W}_2=\\bm{B},\\bm{W}_3=\\bm{C},\\bm{W}_4=\\bm{D}$ for simpler notation. We have $M = 3$, the base field is $\\mathbb{F}_{3}$ and the length of each message is $L=1$. Our goal is to prove the achievability of rate $1\/2$, i.e., download cost $D=2$ for $L=1$. The user downloads, \n\\begin{align}\n    \\bm{\\Delta} = \\{&\\bm{\\Delta}_1 = \\bm{A} + \\bm{\\eta}_{b}\\bm{B} + \\bm{\\eta}_{c}\\bm{C}, \\notag\\\\\n    &\\bm{\\Delta}_2 = 2\\bm{\\eta}_{b}\\bm{B} + \\bm{\\eta}_{c}\\bm{C} + \\bm{\\eta}_{d}\\bm{D}\\}.\\label{eq:queryfixed}\n\\end{align}\nFrom $\\bm{\\Delta}$, the user is able to also compute \n\\begin{align}\n    \\bm{L}_1 = \\bm{\\Delta}_1 + \\bm{\\Delta}_2 &= \\bm{A} + 2\\bm{\\eta}_{c}\\bm{C} + \\bm{\\eta}_{d}\\bm{D},\\\\\n    \\bm{L}_2 = \\bm{\\Delta}_1 + 2\\bm{\\Delta}_2 &= \\bm{A} + 2\\bm{\\eta}_{b}\\bm{B} + 2\\bm{\\eta}_{d}\\bm{D}.\n\\end{align}\nLet $\\bm{W}_{\\bm{\\theta}}$ denote the desired message. Let us normalize $\\bm{\\lambda_1}=1$ without loss of generality.\nThe $\\bm{\\eta}_b, \\bm{\\eta}_c, \\bm{\\eta}_d$ values are specified as follows. \n\\begin{enumerate}\n    \\item When $\\bm{\\mathcal{S}}=\\{1,2,3\\}$ and $\\bm{Y} = \\bm{A} + \\bm{\\lambda}_2\\bm{B} + \\bm{\\lambda}_3\\bm{C}$, then $\\bm{\\eta}_d$ is randomly chosen from $\\mathbb{F}_{3}^{\\times}=\\{1,2\\}$ and $\\bm{\\eta}_b,\\bm{\\eta}_c$ are chosen so that the desired message $\\bm{W}_{\\bm\\theta}$ can be recovered from $\\bm{Y}$ and $\\bm{\\Delta}_1$ as follows.\n     {\\small\n        \\begin{align}\n           \\bm{W}_{\\bm{\\theta}} = \\bm{A}:& ~(\\bm{\\eta}_b,\\bm{\\eta}_c) =( 2 \\bm{\\lambda}_2, 2 \\bm{\\lambda}_3),  2\\bm{A}=\\bm{Y}+\\bm{\\Delta}_1  \\notag\\\\\n         \\bm{W}_{\\bm{\\theta}} = \\bm{B}:&~(\\bm{\\eta}_b,\\bm{\\eta}_c) =( 2 \\bm{\\lambda}_2,  \\bm{\\lambda}_3), \\bm{\\lambda}_2\\bm{B}=2\\bm{Y}+\\bm{\\Delta}_1 \\notag\\\\\n  \\bm{W}_{\\bm{\\theta}} = \\bm{C}:&~ (\\bm{\\eta}_b,\\bm{\\eta}_c) =( \\bm{\\lambda}_2,  2\\bm{\\lambda}_3),\\bm{\\lambda}_3\\bm{C}=2\\bm{Y}+\\bm{\\Delta}_1 \\notag\n         \\end{align} \n        }\n    \n    \\item When $\\bm{\\mathcal{S}}=\\{2,3,4\\}$ and $\\bm{Y} = \\bm{B} + \\bm{\\lambda}_2\\bm{C} + \\bm{\\lambda}_3\\bm{D}$, then $\\bm{\\eta}_b$ is randomly chosen from $\\mathbb{F}_{q}^{\\times}=\\{1,2\\}$ and  $\\bm{\\eta}_c,\\bm{\\eta}_d$ are chosen so that the desired message $\\bm{W}_{\\bm\\theta}$ can be recovered from $\\bm{Y}$ and $\\bm{\\Delta}_2$ as follows.\n         {\\small\n        \\begin{align}\n           \\bm{W}_{\\bm{\\theta}} = \\bm{B}:& ~(\\bm{\\eta}_c,\\bm{\\eta}_d) =( \\bm{\\eta}_b \\bm{\\lambda}_2, \\bm{\\eta}_b \\bm{\\lambda}_3),  \\bm{B}=2\\bm{Y}+\\bm{\\Delta}_2\/\\bm{\\eta}_b \\notag \\\\\n         \\bm{W}_{\\bm{\\theta}} = \\bm{C}:&~(\\bm{\\eta}_c,\\bm{\\eta}_d) =( \\bm{\\eta}_b \\bm{\\lambda}_2, 2\\bm{\\eta}_b \\bm{\\lambda}_3), 2\\bm{\\lambda}_2 \\bm{C}=\\bm{Y}+\\bm{\\Delta}_2\/\\bm{\\eta}_b \\notag \\\\\n                  \\bm{W}_{\\bm{\\theta}} = \\bm{D}:&~(\\bm{\\eta}_c,\\bm{\\eta}_d) =(2 \\bm{\\eta}_b \\bm{\\lambda}_2, \\bm{\\eta}_b \\bm{\\lambda}_3), 2\\bm{\\lambda}_3 \\bm{D}=\\bm{Y}+\\bm{\\Delta}_2\/\\bm{\\eta}_b \\notag \n         \\end{align} \n        }\n    \\item When $\\bm{\\mathcal{S}}=\\{1,3,4\\}$ and $\\bm{Y} = \\bm{A} + \\bm{\\lambda}_2\\bm{C} + \\bm{\\lambda}_3\\bm{D}$, then $\\bm{\\eta}_b$ is randomly chosen from $\\mathbb{F}_{q}^{\\times}$ and   $\\bm{\\eta}_c,\\bm{\\eta}_d$ are chosen so that the desired message $\\bm{W}_{\\bm\\theta}$ can be recovered from $\\bm{Y}$ and $\\bm{L}_1$ as follows.\n             {\\small\n        \\begin{align}\n           \\bm{W}_{\\bm{\\theta}} = \\bm{A}:& ~(\\bm{\\eta}_c,\\bm{\\eta}_d) =(  \\bm{\\lambda}_2,  2\\bm{\\lambda}_3),  2\\bm{A}=\\bm{Y}+\\bm{L}_1 \\notag \\\\\n         \\bm{W}_{\\bm{\\theta}} = \\bm{C}:&~(\\bm{\\eta}_c,\\bm{\\eta}_d) =(  \\bm{\\lambda}_2,  \\bm{\\lambda}_3),  \\bm{\\lambda}_2\\bm{C}=2\\bm{Y}+\\bm{L}_1 \\notag \\\\\n                  \\bm{W}_{\\bm{\\theta}} = \\bm{D}:&~(\\bm{\\eta}_c,\\bm{\\eta}_d) =( 2\\bm{\\lambda}_2,  2\\bm{\\lambda}_3), \\bm{\\lambda}_3 \\bm{D}=2\\bm{Y}+\\bm{L}_1 \\notag \n         \\end{align} \n        }\n\n    \\item When $\\bm{\\mathcal{S}}=\\{1,2,4\\}$ and $\\bm{Y} = \\bm{A} + \\bm{\\lambda}_2\\bm{B} + \\bm{\\lambda}_3\\bm{D}$, then $\\bm{\\eta}_c$ is randomly chosen from $\\mathbb{F}_{q}^{\\times}$ and   $\\bm{\\eta}_b,\\bm{\\eta}_d$ are chosen so that the desired message $\\bm{W}_{\\bm\\theta}$ can be recovered from $\\bm{Y}$ and $\\bm{L}_2$ as follows. \n             {\\small\n        \\begin{align}\n           \\bm{W}_{\\bm{\\theta}} = \\bm{A}:& ~(\\bm{\\eta}_b,\\bm{\\eta}_d) =(  \\bm{\\lambda}_2,  \\bm{\\lambda}_3),  2\\bm{A}=\\bm{Y}+\\bm{L}_2 \\notag \\\\\n         \\bm{W}_{\\bm{\\theta}} = \\bm{B}:&~(\\bm{\\eta}_b,\\bm{\\eta}_d) =(  \\bm{\\lambda}_2,  2\\bm{\\lambda}_3),  \\bm{\\lambda}_2\\bm{B}=2\\bm{Y}+\\bm{L}_2 \\notag \\\\\n                  \\bm{W}_{\\bm{\\theta}} = \\bm{D}:&~(\\bm{\\eta}_b,\\bm{\\eta}_d) =( 2\\bm{\\lambda}_2,  \\bm{\\lambda}_3), \\bm{\\lambda}_3 \\bm{D}=2\\bm{Y}+\\bm{L}_2 \\notag \n         \\end{align} \n        }\n\\end{enumerate}\nCorrectness is already shown. For privacy, note that the form of the query is fixed as in \\eqref{eq:queryfixed} so the user only needs to specify $\\bm{\\eta}_b,\\bm{\\eta}_c,\\bm{\\eta}_d$, and  those are i.i.d. uniform over $\\mathbb{F}_3^{\\times}=\\{1,2\\}$, regardless of $(\\bm{\\mathcal{S}},\\bm{\\theta})$. Thus, the scheme is private, and the rate achieved is $1\/2$, which completes the proof of Theorem \\ref{thm:M3K4}.\n\n\\section{Proof of Theorem \\ref{thm:pcsi2_pub_pri}}\\label{proof:pcsi2_pub_pri}\n\\subsection{Converse}\nHere we prove that \n\\begin{align}\n    C_{\\mbox{\\tiny PCSI-II}}^{\\mbox{\\tiny pri}}(q) \\leq C_{\\mbox{\\tiny PCSI-II}}(q=2) = C_{\\mbox{\\tiny PCSI-II}}^{\\inf}.\n\\end{align}\n\nThe following lemma states that for PIR-PCSI*, for every feasible $Q$ and $(\\theta, \\mathcal{S})$ value, all  possible coefficient vectors must allow successful decoding.\n\\begin{lemma}\\label{lem:fullypri} Under the constraint of $(\\bm{\\theta}, \\bm{\\mathcal{S}, \\bm{\\Lambda}})$ privacy, \n    \\begin{align}\n    &\\mbox{PIR-PCSI: } \\forall (Q,\\mathcal{S},\\theta,\\Lambda)\\in\\mathcal{Q}\\times \\mathfrak{S}\\times[K]\\times\\mathfrak{C},\\notag\\\\\n    &\\hspace{0.2cm} H(\\bm{W}_{\\theta} \\mid \\bm{\\Delta}, \\bm{Y}^{[\\mathcal{S},\\Lambda]}, \\bm{Q}=Q) = 0.\\label{eq:pcsi_pri}\\\\\n    &\\mbox{PIR-PCSI-I: }\\forall (Q,\\mathcal{S},\\theta,\\Lambda)\\in\\mathcal{Q}\\times \\mathfrak{S}\\times([K]\\setminus\\mathcal{S})\\times\\mathfrak{C},\\notag\\\\\n    &\\hspace{0.2cm} H(\\bm{W}_{\\theta} \\mid \\bm{\\Delta}, \\bm{Y}^{[\\mathcal{S},\\Lambda]}, \\bm{Q}=Q) = 0.\\label{eq:pcsi1_pri}\\\\\n    &\\mbox{PIR-PCSI-II: }\\forall (Q,\\mathcal{S},\\theta,\\Lambda)\\in\\mathcal{Q}\\times \\mathfrak{S}\\times\\mathcal{S}\\times\\mathfrak{C},\\notag\\\\\n    &\\hspace{0.2cm} H(\\bm{W}_{\\theta} \\mid \\bm{\\Delta}, \\bm{Y}^{[\\mathcal{S},\\Lambda]}, \\bm{Q}=Q) = 0.\\label{eq:pcsi2_pri}\n    \\end{align}\n\\end{lemma}\n\\proof Since the server  knows $\\bm{\\Delta}, \\bm{Q}$ and can test all possible realizations of $\\bm{\\theta}, \\bm{\\mathcal{S}}, \\bm{\\Lambda}$ for decodability. If there exists $(\\theta, \\mathcal{S}, \\Lambda)$ such that $\\bm{W}_{\\theta}$ cannot be decoded, then that $(\\theta, \\mathcal{S}, \\Lambda)$ can be ruled out by the server. This contradicts the joint $(\\bm{\\theta}, \\bm{\\mathcal{S}, \\bm{\\Lambda}})$ privacy constraint.$\\hfill\\square$\n\nAs a direct result of \\eqref{eq:pcsi2_pri}, for any PIR-PCSI-II scheme that preserves joint $(\\bm{\\theta}, \\bm{\\mathcal{S}}, \\bm{\\Lambda})$ privacy, \n\\begin{align}\n    H(\\bm{W}_{\\mathcal{S}} \\mid \\bm{\\Delta}, \\bm{Y}^{[\\mathcal{S},\\Lambda]}, \\bm{Q}=Q) = 0,\\notag\\\\\n    \\forall (\\mathcal{S}, \\Lambda, Q) \\in \\mathfrak{S}\\times\\mathfrak{C}\\times\\mathcal{Q}.\\label{eq:pcsi2_pri_dec}\n\\end{align}\nNote that \\eqref{eq:pcsi2_pri_dec} is a \\emph{stronger} version of \\eqref{eq:pcsi2_inf_dec} which is sufficient to bound $C_{\\mbox{\\tiny PCSI-II}}(q=2)$. Thus, we have $C_{\\mbox{\\tiny PCSI-II}}^{\\mbox{\\tiny pri}}(q) \\leq C_{\\mbox{\\tiny PCSI-II}}(q=2) = C_{\\mbox{\\tiny PCSI-II}}^{\\inf}$.\n\n\\subsection{Achievability}\nThe \\emph{Generic Linear Combination Based Scheme} in Section \\ref{sec:PCSI2_inf_ach} where $M-1$ linear combinations of each messages (represented in the extended field $\\mathbb{F}_{q^l}$ where $L = Ml$) are downloaded, also works under $(\\bm{\\theta}, \\bm{\\mathcal{S}}, \\bm{\\Lambda})$ privacy, but with a slight modification. The only difference between the modified scheme and the infimum capacity achieving scheme of PIR-PCSI-II in Section \\ref{sec:PCSI2_inf_ach} is that, instead of the matrix in \\eqref{eq:F2_inv}, the following matrix \n\\begin{align}\n    \\mathbf{G}_{\\mathcal{S}}^{(\\gamma_1, \\gamma_2, \\cdots,\\gamma_M)} = \n    \\begin{bmatrix}\n        \\gamma_{1}\\mathbf{I}_{M} & \\cdots & \\gamma_{M}\\mathbf{I}_{M}\\\\\n        & \\mathbf{e}_{1}\\otimes\\mathbf{L}_{j_1}^{(1)} &\\\\\n        & \\cdots &\\\\\n        & \\mathbf{e}_{1}\\otimes\\mathbf{L}_{j_1}^{(M-1)} &\\\\\n        & \\cdots &\\\\\n        & \\mathbf{e}_{M}\\otimes\\mathbf{L}_{j_M}^{(1)} &\\\\\n        & \\cdots &\\\\\n        & \\mathbf{e}_{M}\\otimes\\mathbf{L}_{j_M}^{(M-1)} &\n    \\end{bmatrix},\\label{eq:Fq_inv_arb}\n\\end{align}\nmust have full rank for every $\\mathcal{S} = \\{j_1, \\cdots, j_{M}\\} \\in \\mathfrak{S}$ and every realization of $(\\gamma_1, \\gamma_2, \\cdots,\\gamma_M) \\in \\mathfrak{C}$. Let us prove that the scheme is correct, jointly private and such  $\\mathbf{L}_{\\cdot}^{(\\cdot)}$ vectors  exist when $l$ is large enough that,\n\\begin{align}\n    q^l > (q-1)^{M}\\tbinom{K}{M}M(M-1).\n\\end{align}\n\n\n\\proof For a particular realization of $(\\gamma_1, \\gamma_2, \\cdots, \\gamma_M)$, e.g., $(\\gamma_1, \\gamma_2, \\cdots, \\gamma_M) = (1, 1, \\cdots, 1)$, \\eqref{eq:Fq_inv_arb} yields a set of $\\tbinom{K}{M}$ matrices \n\\begin{align}\n    \\mathcal{G}^{(1,1,\\cdots,1)} = \\{\\mathbf{G}_{\\mathcal{S}_{1}}^{(1,1,\\cdots,1)}, \\mathbf{G}_{\\mathcal{S}_{2}}^{(1,1,\\cdots,1)}, \\cdots, \\mathbf{G}_{\\mathcal{S}_{\\tbinom{K}{M}}}^{(1,1,\\cdots,1)}\\}\\notag\n\\end{align}\ncorresponding to all possible $\\{j_1, j_2, \\cdots, j_M\\} \\in \\mathfrak{S}$. If all the $\\tbinom{K}{M}$ matrices in $\\mathcal{G}^{(1,1,\\cdots,1)}$ are invertible, this scheme preserves the joint privacy of $(\\bm{\\theta}, \\bm{\\mathcal{S}})$ and enables the user to decode all the $M$ messages in the support set, when all the coefficients in CSI are $1$, according to Appendix \\ref{app:existence}. \n\nGoing over all the possible realizations of $(\\gamma_1, \\cdots, \\gamma_M) \\in \\mathfrak{C}$ and $\\{j_1, j_2, \\cdots, j_M\\} \\in \\mathfrak{S}$, \\eqref{eq:Fq_inv_arb} yields $(q-1)^{M}$ sets of matrices \n\\begin{align}\n    \\mathcal{G}^{(1,\\cdots,1)}, \\mathcal{G}^{(1,\\cdots,1,2)}, \\cdots, \\mathcal{G}^{(q-1,\\cdots,q-1)},\n\\end{align}\neach of which contains $\\tbinom{K}{M}$ matrices, i.e., there are in total $(q-1)^{M}\\tbinom{K}{M}$ matrices. If all the $(q-1)^{M}\\tbinom{K}{M}$ matrices are invertible, then for arbitrary realization of $(\\gamma_1, \\gamma_2, \\cdots, \\gamma_M)$, i.e., arbitrary $M$ coefficients in the CSI, this scheme enables the user to decode all the $M$ messages in the support set and preserves the joint $(\\bm{\\theta}, \\bm{\\mathcal{S}})$ privacy. Since this scheme works for arbitrary coefficients, from the server's perspective, all the realizations of $M$ coefficients are equally likely. Thus, the joint privacy of coefficients $\\bm{\\Lambda}$, index $\\bm{\\theta}$, and support set $\\bm{\\mathcal{S}}$, is preserved.\n\nTo prove the existence of such linear combinations, note that the determinant of each one of the $(q-1)^{M}\\tbinom{K}{M}$ matrices yields a degree $M(M-1)$ multi-variate polynomial as proved in Appendix \\ref{app:existence}. Thus, the product of the determinants of all the matrices $F$ is a multi-variate polynomial of degree $(q-1)^{M}\\tbinom{K}{M}M(M-1)$. Again, as in Appendix \\ref{app:existence}, according to the Schwartz-Zippel Lemma, when $q^l > (q-1)^{M}\\tbinom{K}{M}$, there exists elements in $\\mathbb{F}_{q^l}$ such that the polynomial $F$ does not evaluate to $0$, i.e., all the $(q-1)^{M}\\tbinom{K}{M}M(M-1)$ matrices are invertible. \n\n\n\\section{Proof of Theorem \\ref{thm:redundancy1}}\\label{proof:redundancy1}\nHere we bound the redundancy $\\rho_{\\mbox{\\tiny PCSI-I}} $ from above (equivalently, lower-bound $\\alpha^{*}$) for $1 \\leq M \\leq K-1$.\n\nRecall that the supremum capacity for PIR-PCSI-I is $(K-M)^{-1}$, i.e., the optimal average download cost is $D\/L = K-M$. Consider an achievable scheme such that $\\alpha$ PCSI is sufficient and the average download cost $D\/L \\leq K-M+\\epsilon$ for some $L$. Since $D\/L \\leq K-M+\\epsilon$, we have $L(K-M) + \\epsilon L \\geq D \\geq H(\\bm{\\Delta} \\mid \\bm{Q})$. Thus, there exists a feasible $Q$ such that \n\\begin{align}\n    H(\\bm{\\Delta} \\mid \\bm{Q}=Q) \\leq (K-M)L + \\epsilon L.\n\\end{align}\nFor all $i \\in [M]$, let $\\mathcal{S}_{i} = [M+1] \\setminus \\{i\\}$. ALso, for all $i \\in [M+1:K]$, let $\\mathcal{S}_{i} = [M]$. For all $i \\in [K]$, let $\\Lambda_i \\in \\mathfrak{C}$ satisfy  \n\\begin{align}\n    H(\\bm{W}_{i} \\mid \\bm{\\Delta}, \\overline{\\bm{Y}}^{[\\mathcal{S}_i, \\Lambda_i]}, \\bm{Q}=Q) = 0.\\label{eq:redundancy1_1}\n\\end{align}\nSuch $\\Lambda_i$'s must exist according to \\eqref{eq:lemma1pcsi1} in Lemma \\ref{lem:privacy}.\n\nWriting $\\overline{\\bm{Y}}^{[\\mathcal{S}_i, \\Lambda_i]}$ as $\\overline{\\bm{Y}}_{i}$ for compact notation, we have \n\\begin{align}\n    &H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\overline{\\bm{Y}}_{[M]}, \\bm{Q}=Q)\\\\\n    &= H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\overline{\\bm{Y}}_{[M]}, \\bm{W}_{[M]}, \\bm{Q}=Q)\\label{eq:redundancy1_2}\\\\\n    &= H(\\bm{W}_{[M+1:K]} \\mid \\bm{\\Delta}, \\overline{\\bm{Y}}_{[K]}, \\bm{W}_{[M]}, \\bm{Q}=Q)\\label{eq:redundancy1_3}\\\\\n    &= 0\\label{eq:redundancy1_4},\n\\end{align}\nwhere \\eqref{eq:redundancy1_2} follows from \\eqref{eq:redundancy1_1}. \\eqref{eq:redundancy1_3} is correct since $\\overline{\\bm{Y}}_{[M+1:K]}$ are functions of $\\bm{W}_{[M]}$. \\eqref{eq:redundancy1_4} follows from \\eqref{eq:redundancy1_1}. Since we are considering the case where the supremum capacity is achieved, we have \n\\begin{align}\n    &(K-M)L + \\epsilon L + M\\alpha L\\notag\\\\\n    &\\geq H(\\bm{\\Delta}, \\overline{\\bm{Y}}_{[M]} \\mid \\bm{Q}=Q)\\label{eq:redundancy1_5}\\\\\n    &\\geq I(\\bm{\\Delta}, \\overline{\\bm{Y}}_{[M]}; \\bm{W}_{[K]} \\mid \\bm{Q}=Q)\\notag\\\\\n    &= H(\\bm{W}_{[K]} \\mid \\bm{Q}=Q) = KL.\\label{eq:redundancy1_6}\n\\end{align}\n\\eqref{eq:redundancy1_5} follows from \\eqref{eq:invaYR}. Step \\eqref{eq:redundancy1_6} follows from \\eqref{eq:redundancy1_4} and the fact that the query and the messages are mutually independent according to \\eqref{eq:indQ}. Thus we have $\\alpha \\geq 1 - \\frac{\\epsilon}{M}$. In order to approach capacity, we must have $\\epsilon\\rightarrow 0$, so we need $\\alpha\\geq 1$, and since this is true for any $\\alpha$ such that $\\alpha$ PCSI is sufficient, it is also true for $\\alpha^*$. Thus, the redundancy is bounded as $\\rho_{\\mbox{\\tiny PCSI-I}}\\leq 0$.\n\n\\section{Proof of Theorem \\ref{thm:cap_PCSI1_inf}}\\label{sec:cap_PCSI1_inf}\n\\subsection{Converse for $C_{\\mbox{\\tiny PCSI-I}}(q=2)$}\nAgain, \\eqref{eq:lemma1pcsi1} is true for arbitrary $\\mathbb{F}_{q}$. The only thing different in $\\mathbb{F}_{2}$ is that $\\bm{\\Lambda}$ must be the vector of all ones. As a direct result of \\eqref{eq:lemma1pcsi1}, for PIR-PCSI-I in $\\mathbb{F}_{2}$,\n\\begin{align}\n    H(\\bm{W}_{[K]\\setminus\\mathcal{S}} \\mid \\bm{\\Delta}, {\\bm{Y}}^{[\\mathcal{S}, 1_{M}]}, \\bm{Q} = Q) = 0, \\forall (Q,\\mathcal{S}) \\in \\mathcal{Q} \\times \\mathfrak{S}\\label{eq:dec_inf_PCSI1_1}\n\\end{align}\nand thus \n\\begin{align}\n    &H(\\bm{W}_{[K]\\setminus\\mathcal{S}} \\mid \\bm{\\Delta}, \\bm{Q} = Q) \\\\\n    &=I(\\bm{W}_{[K]\\setminus\\mathcal{S}}; {\\bm{Y}}^{[\\mathcal{S}, 1_{M}]}\\mid \\bm{\\Delta}, \\bm{Q} = Q)\\\\\n    &\\leq H({\\bm{Y}}^{[\\mathcal{S}, 1_{M}]}\\mid \\bm{\\Delta}, \\bm{Q} = Q)\\\\\n    &\\leq L, ~~\\forall (Q,\\mathcal{S}) \\in \\mathcal{Q} \\times \\mathfrak{S}.\n\\end{align}\nAveraging over $\\bm{Q}$ gives \n\\begin{align}\n    H(\\bm{W}_{[K]\\setminus\\mathcal{S}} \\mid \\bm{\\Delta}, \\bm{Q}) \\leq L, \\forall \\mathcal{S} \\in\\mathfrak{S}.\\label{eq:dec_inf_PCSI1_2}\n\\end{align}\nAlso, for all $\\mathcal{S} \\in \\mathfrak{S}$ and $Q \\in \\mathcal{Q}$, \n\\begin{align}\n    &H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q}=Q)\\\\ \n    &= H(\\bm{W}_{\\mathcal{S}}\\mid \\bm{\\Delta}, \\bm{Q}=Q)\\notag\\\\\n    &~~ + H(\\bm{W}_{[K]\\setminus\\mathcal{S}}\\mid \\bm{\\Delta}, \\bm{W}_{\\mathcal{S}}, \\bm{Q}=Q)\\\\\n    &= H(\\bm{W}_{\\mathcal{S}}\\mid \\bm{\\Delta}, \\bm{Q}=Q)\\notag\\\\ \n    &~~ + H(\\bm{W}_{[K]\\setminus\\mathcal{S}}\\mid \\bm{\\Delta}, \\bm{W}_{\\mathcal{S}}, {\\bm{Y}}^{[\\mathcal{S}, 1_{M}]}, \\bm{Q}=Q)\\label{eq:pcsi1_inf_Ysum}\\\\\n    &= H(\\bm{W}_{\\mathcal{S}}\\mid \\bm{\\Delta}, \\bm{Q}=Q),\n\\end{align}\nwhere \\eqref{eq:pcsi1_inf_Ysum} results from the fact that $\\overline{\\bm{Y}}^{[\\mathcal{S}, 1_{M}]} = \\sum_{s \\in \\mathcal{S}}\\bm{W}_{s}$, and the last step follows from \\eqref{eq:dec_inf_PCSI1_1}. Averaging over $\\bm{Q}$, it follows that \n\\begin{align}\n    H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q}) = H(\\bm{W}_{\\mathcal{S}}\\mid \\bm{\\Delta}, \\bm{Q}), &&\\forall \\mathcal{S} \\in \\mathfrak{S}.\\label{eq:pcsi1_inf_equ}\n\\end{align}\n\nLet us first prove $C_{\\mbox{\\tiny PCSI-I}}(q=2) \\leq (K-1)^{-1}$ in the regime where $1 \\leq M \\leq \\frac{K}{2}$. \n\\begin{align}\n    &H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q}) \\notag\\\\\n    &=H(\\bm{W}_{[M]} \\mid \\bm{\\Delta}, \\bm{Q})\\label{eq:dec_inf_PCSI1_r1_1}\\\\\n    &\\leq H(\\bm{W}_{[K-M]} \\mid \\bm{\\Delta}, \\bm{Q})\\label{eq:MtoK-M}\\\\\n    & \\leq L \\label{eq:K-MtoL},\n\\end{align}\nwhere \\eqref{eq:dec_inf_PCSI1_r1_1} is true according to \\eqref{eq:pcsi1_inf_equ}, \\eqref{eq:MtoK-M} follows from $(K-M \\geq M)$ and \\eqref{eq:dec_inf_PCSI1_1}, and \\eqref{eq:K-MtoL} follows from \\eqref{eq:dec_inf_PCSI1_2}. Thus\n\\begin{align}\n    H(\\bm{\\Delta} \\mid \\bm{Q}) &\\geq I(\\bm{\\Delta}; \\bm{W}_{[K]} \\mid \\bm{Q})\\\\\n    &= H(\\bm{W}_{[K]} \\mid \\bm{Q}) - H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q})\\\\\n    &\\geq KL - L.\n\\end{align} \nThus $D \\geq H(\\bm{\\Delta} \\mid \\bm{Q}) \\geq KL-L$ and since the rate $L\/D\\leq (K-1)^{-1}$ for every achievable scheme, we have shown that $C_{\\mbox{\\tiny PCSI-I}}(q=2) \\leq (K-1)^{-1}$ when $K-M\\geq M\\geq 1$, i.e., $1\\leq M\\leq K\/2$.\n\nNext let us prove that $C_{\\mbox{\\tiny PCSI-I}}(q=2) \\leq \\big(K - \\frac{M}{K-M}\\big)^{-1}$ for the regime $\\frac{K}{2} < M \\leq K-1$. It suffices to prove $H(\\bm{\\Delta} \\mid \\bm{Q}) \\geq KL - \\frac{ML}{K-M}$. Define,\n\\begin{align}\n    H_{m}^{K} = \\frac{1}{\\tbinom{K}{m}}\\sum_{\\mathcal{M}:\\mathcal{M}\\subset[K], |\\mathcal{M}|=m}\\frac{H(\\bm{W}_{\\mathcal{M}} \\mid \\bm{\\Delta}, \\bm{Q})}{m},\n\\end{align} \nwe have\n\\begin{align}\n    H_{K-M}^{K}&\\geq H_{M}^{K}\\label{eq:dec_inf_PCSI1_r2_1}\\\\\n    &= \\frac{H(\\bm{W}_{[K]}\\mid \\bm{\\Delta}, \\bm{Q})}{M},\\label{eq:dec_inf_PCSI1_r2_2}\n\\end{align}\nwhere \\eqref{eq:dec_inf_PCSI1_r2_1} follows from Han's inequality \\cite{Cover_Thomas}, and \\eqref{eq:dec_inf_PCSI1_r2_2} follows from \\eqref{eq:pcsi1_inf_equ}. Note that according to \\eqref{eq:dec_inf_PCSI1_2},\n\\begin{align}\n    \\frac{L}{K-M} \\geq H_{K-M}^{K},\n\\end{align}\nand therefore,\n\\begin{align}\n    H(\\bm{W}_{[K]}\\mid \\bm{\\Delta}, \\bm{Q}) \\leq \\frac{ML}{K-M}.\n\\end{align}\nThus, $H(\\bm{\\Delta} \\mid \\bm{Q}) \\geq KL - \\frac{ML}{K-M}$, which completes the converse proof for Theorem \\ref{thm:cap_PCSI1_inf}. We next prove  achievability.\n\n\\subsection{Two PIR-PCSI-I Schemes  for Arbitrary $q$}\\label{sec:PCSI1_inf_ach}\n\\subsubsection{Achieving rate $\\frac{1}{K-1}$ when $1 \\leq M \\leq \\frac{K}{2}$}\\label{sec:PCSI1_inf_ach1}\nThe goal here is to download $K-1$ generic linear combinations so that along with the one linear combination already available as side-information, the user has enough information to retrieve all $K$ messages. Let $L$ be large enough that  $q^L > \\tbinom{K}{M}(K-1)$. For all $k \\in [K]$, message $\\bm{W}_k \\in \\mathbb{F}_{q}^{L\\times 1}$ can be represented as a scalar $\\bm{w}_k \\in \\mathbb{F}_{q^L}$. Let \n\\begin{align}\n    \\bm{w}_{[K]} = \n    \\begin{bmatrix}\n        \\bm{w}_1 & \\bm{w}_2 & \\cdots & \\bm{w}_K\n    \\end{bmatrix}^{\\mathrm{T}} \\in \\mathbb{F}_{q^L}^{K\\times 1},\n\\end{align}\nbe the length $K$ column vector whose entries are the messages represented in $\\mathbb{F}_{q^{L}}$. Let $\\Psi \\in\\mathbb{F}_{q^L}^{K\\times (K-1)}$ be a  $K\\times (K-1)$ matrix whose elements are the variables $\\psi_{ij}$. The user downloads \n\\begin{align}\n    \\bm{\\Delta} =\\Psi^T \\bm{w}_{[K]} \\in\\mathbb{F}_{q^L}^{(K-1)\\times 1}.\n\\end{align}\nSuppose the realization of the coefficient vector is $\\bm{\\Lambda}=\\Lambda$. The linear combination available to the user can be expressed as $\\bm{Y}^{[{\\Lambda},\\bm{\\mathcal{S}}]}=U_{{{\\Lambda}},\\bm{\\mathcal{S}}}^T\\bm{w}_{[K]}$ for some $K\\times 1$ vector $U_{{\\Lambda},\\bm{\\mathcal{S}}}$ that depends on $({\\Lambda},\\bm{\\mathcal{S}})$. Combined with the download, the user has \n\\begin{align}\n[U_{{\\Lambda},\\bm{\\mathcal{S}}}, \\Psi]^T\\bm{w}_{[K]},\n\\end{align}\nso if the $K\\times K$ matrix $G_{{\\Lambda},\\bm{\\mathcal{S}}}=[U_{{\\Lambda},\\bm{\\mathcal{S}}}, \\Psi]$ is invertible (full rank) then the user can decode all $K$ messages. For all $\\mathcal{S}\\in\\mathfrak{S}$, let $f_{\\Lambda,\\mathcal{S}}(\\cdot)$ be the multi-variate polynomial of degree $K-1$ in variables $\\psi_{ij}$, representing the determinant of $G_{{\\Lambda},{\\mathcal{S}}}$. This is not the zero polynomial because the $K-1$ columns of $\\Psi$  can always be chosen to be linearly independent of the vector $U_{{\\Lambda},{\\mathcal{S}}}$ in a $K$ dimensional vector space. The product of all such polynomials, $f_\\Lambda=\\prod_{\\mathcal{S}\\in\\mathfrak{S}}f_{\\Lambda,\\mathcal{S}}$ is itself a multi-variate non-zero polynomial of degree $(K-1)\\binom{K}{M}$ in the variables $\\psi_{ij}$. By Schwartz-Zippel Lemma, if the $\\psi_{ij}$ are chosen randomly from $\\mathbb{F}_{q^L}$ then the probability that the corresponding evaluation of $f_\\Lambda$ is zero, is no more than $(K-1)\\binom{K}{M}\/q^L<1$, so there exists a choice of $\\psi_{ij}$ for which all $f_{\\Lambda,\\mathcal{S}}$ evaluate to non-zero values, i.e.,  $G_{\\Lambda, \\mathcal{S}}$ is invertible for every $\\mathcal{S}\\in\\mathfrak{S}$. Thus, with this choice of $\\Psi$, we have a scheme with rate $1\/(K-1)$ that is correct and private and allows the user to retrieve all $K$ messages. To verify privacy, note that the user constructs the query based on the realization of $\\bm\\Lambda$ alone, and does not need to know $(\\bm{\\mathcal{S}},\\bm{\\theta})$ before it sends the query, so the query is independent of $(\\bm{\\mathcal{S}},\\bm{\\theta})$. \n\n\\begin{remark}\\label{rmk:pcsi1_inf_pcsi_inf}\nSince the scheme allows the user to decode all messages, the scheme also works if $\\bm{\\theta}$ is uniformly drawn from $[K]$, i.e., in the PIR-PCSI setting.\n\\end{remark}\n\n\\subsubsection{Achieving rate $(K-\\frac{M}{K-M})^{-1}$ when $K\/2 < M \\leq K-1$}\nNow let us present a scheme with rate $(K-\\frac{M}{K-M})^{-1}$ which is optimal for the regime $\\frac{K}{2} < M \\leq K-1$. The scheme is comprised of two steps.\n\n\\emph{Step 1}: The user converts the $(M,K)$ PIR-PCSI-I problem to $(K-M,K)$ PIR-PCSI-II problem as follows.\n\nThe user first downloads\n\\begin{align}\n    \\bm{\\Delta}_{1} = \\sum_{k \\in [K]}\\bm{a}_{k}\\bm{W}_{k},\n\\end{align}\nwhere $\\bm{a}_{\\bm{i}_m} = \\bm{\\lambda}_m$ for $\\bm{i}_{m} \\in \\bm{\\mathcal{S}}$ while for $k \\notin \\bm{\\mathcal{S}}$, $\\bm{a}_k$'s are independently and uniformly drawn from $\\mathbb{F}_{q}^{\\times}$.\nThe user then computes\n\\begin{align}\n    \\bm{Y}^{\\prime} = \\bm{\\Delta}_1 - \\bm{Y}^{[\\bm{\\mathcal{S}}, \\bm{\\Lambda}]} = \\sum_{k \\in [K]\\setminus \\bm{\\mathcal{S}}}\\bm{a}_{k}\\bm{W}_{k}.\n\\end{align} \nIn this step, from the server's perspective, $\\bm{a}_1, \\cdots, \\bm{a}_K$ are i.i.d. uniform over $\\mathbb{F}_{q}^{\\times}$, thus there is no loss of privacy. The download cost of this step is $H(\\bm{\\Delta}_1) = L$.\n\n\\emph{Step 2}: The user has $\\bm{Y}^{\\prime}$ as  coded side information and applies the fully private PIR-PCSI-II scheme described in Section \\ref{proof:pcsi2_pub_pri} that  protects the privacy of all the coefficients. \n\nThe reason to apply the PIR-PCSI-II scheme that maintains the privacy of coefficients is that in \\emph{Step 1}, server knows $\\bm{a}_1, \\cdots, \\bm{a}_K$. If in the second step, the Query is not independent of $\\bm{a}_i, i \\in [K]\\setminus \\bm{\\mathcal{S}}$, then the server may be able to rule out some  realizations of $\\bm{\\mathcal{S}}$. The download cost of this step is $\\frac{K(K-M-1)L}{K-M}$. Thus, the total download cost of this scheme is $KL - \\frac{ML}{K-M}$ and the rate is $\\big(K - \\frac{M}{K-M}\\big)^{-1}$.\n\n\\section{Proof of Theorem \\ref{thm:pcsi1_pub_pri}}\\label{proof:pcsi1_pub_pri}\n\\subsection{Proof of $C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}, \\sup}=C_{\\mbox{\\tiny PCSI-I}}^{\\inf}$}\nFirst let us prove the converse. As a direct result of \\eqref{eq:pcsi1_pri} in Lemma \\ref{lem:fullypri}, for any PIR-PCSI-I scheme that preserves joint $(\\bm{\\theta}, \\bm{\\mathcal{S}}, \\bm{\\Lambda})$ privacy, \n\\begin{align}\n    H(\\bm{W}_{[K]\\setminus\\mathcal{S}} \\mid \\bm{\\Delta}, \\bm{Y}^{[\\mathcal{S},\\Lambda]}, \\bm{Q}=Q) = 0, \\notag\\\\\n    \\forall (\\mathcal{S}, \\Lambda, Q) \\in \\mathfrak{S}\\times\\mathfrak{C}\\times\\mathcal{Q}.\\label{eq:pcsi1_pri_dec}\n\\end{align}\nNote that \\eqref{eq:pcsi1_pri_dec} is a stronger version of \\eqref{eq:dec_inf_PCSI1_1} which is sufficient to bound $C_{\\mbox{\\tiny PCSI-I}}(q=2)$. Thus, we have $C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}}(q) \\leq C_{\\mbox{\\tiny PCSI-I}}(q=2) = C_{\\mbox{\\tiny PCSI-I}}^{\\inf}$, which completes the proof of converse.\n\nFor achievability, let us note that $C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}, \\sup}\\geq C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}}(q=2)=C_{\\mbox{\\tiny PCSI-I}}(q=2)=C_{\\mbox{\\tiny PCSI-I}}^{\\inf}$, because over $\\mathbb{F}_2$, the $\\bm{\\Lambda}$ vector is constant (all ones) and therefore trivially private.\n\n\\subsection{Proof of the bound: $C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}, \\inf} \\leq \\min\\bigg(C_{\\mbox{\\tiny PCSI-I}}^{\\inf}, \\frac{1}{K-2}\\bigg)$}\nSince privacy of $\\bm\\Lambda$ only further constrains PIR-PCSI, it is trivial that $C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}, \\inf} \\leq C_{\\mbox{\\tiny PCSI-I}}^{\\inf}$. For the remaining bound,  $C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}, \\inf}\\leq \\frac{1}{K-2}$, it suffices to show that $C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}}(q\\geq M) \\leq \\frac{1}{K-2}$, because $C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}, \\inf}\\leq C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}}(q\\geq M)$. Note that by $C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}}(q\\geq M)$ we mean $C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}}(q)$ for all $q\\geq M$.\n\nLet  \n\\begin{align}\n    \\bm{Y}_1 &= \\bm{W}_2 + \\alpha_3 \\bm{W}_3 + \\cdots \\alpha_{M+1} \\bm{W}_{M+1},\\\\\n    \\bm{Y}_2 &= \\bm{W}_1 + \\bm{W}_3 + \\bm{W}_4 + \\cdots \\bm{W}_{M+1},\n\\end{align}\nwhere $\\alpha_3, \\alpha_4, \\cdots, \\alpha_{M+1}$ are $M-1$ distinct elements in $\\mathbb{F}_{q}^\\times$.\n\nLet $\\beta_3, \\beta_4, \\dots \\beta_{M+1}$ be $M-1$ distinct  elements in $\\mathbb{F}_{q}^\\times$ such that $\\forall{m \\in [3:M+1]}, \\beta_{m}\\alpha_{m} + 1 = 0$ in $\\mathbb{F}_{q}$.\n\nNote that such $\\alpha$'s and $\\beta$'s exist since $q \\geq M$.\n\nThen let \n\\begin{align}\n    \\bm{Y}_{m} &= \\beta_m \\bm{Y}_1 + \\bm{Y}_2 \\notag\\\\\n    &= \\bm{W}_1 + \\beta_m \\bm{W}_2 + (\\beta_m \\alpha_3 + 1)\\bm{W}_3 + \\cdots \\notag\\\\\n    &\\quad +(\\beta_m \\alpha_i + 1) \\bm{W}_i + \\cdots + (\\beta_m \\alpha_{M+1} + 1)\\bm{W}_{M+1}, \\notag\\\\\n    &\\forall m \\in [3:M+1],\n\\end{align}\nbe $M-1$ linear combinations of the first $M+1$ messages $\\bm{W}_{[M+1]}$. Note that for any $m \\in [3:M+1]$, the coefficient for $\\bm{W}_m$ in $\\bm{Y}_m$ (i.e., $\\beta_{m}\\alpha_{m} + 1$) is $0$ while the coefficient for any $\\bm{W}_i, i\\in[M+1], i\\neq m$ (i.e., $\\beta_{m}\\alpha_{i} + 1$) is non-zero\\footnote{Since $\\beta_{m}\\alpha_{m}+1=0$, $\\beta_{m}\\alpha_{i}+1\\neq 0$ for $i \\neq m$.}. For example, \n\\begin{align}\n    \\bm{Y}_3 &= \\bm{W}_1 + \\beta_3 \\bm{W}_2 + 0\\bm{W}_3 + (\\beta_3\\alpha_4 + 1)\\bm{W}_4\\notag\\\\\n    &\\quad + \\cdots + (\\beta_3\\alpha_{M+1} + 1)\\bm{W}_{M+1}.\n\\end{align}\nThus, for any $m \\in [M+1]$, $\\bm{Y}_m$ is a linear combination of $M$ messages $\\bm{W}_{[M+1]\\setminus\\{m\\}}$ with non-zero coefficients. For $\\mathcal{S}_m=[M+1]\/\\{m\\}$ and $\\Lambda_m$ as the vector of coefficients that appear in $\\bm{Y}_m$, we $\\bm{Y}^{[\\mathcal{S}_m,\\Lambda_m]}=\\bm{Y}_m$.\n\nAccording to \\eqref{eq:pcsi1_pri_dec}, \n\\begin{align}\n    H(\\bm{W}_m, \\bm{W}_{[M+2:K]} \\mid \\bm{\\Delta}, \\bm{Y}_m, \\bm{Q} = Q) = 0,\\notag\\\\\n     \\forall m \\in [M+1], Q \\in \\mathcal{Q}. \\label{eq:PCSI1_pri_dec1}\n\\end{align}\nThus, for all $Q\\in\\mathcal{Q}$,\n\\begin{align}\n    &H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q} = Q) \\notag\\\\\n    &\\leq H(\\bm{W}_{[K]}, \\bm{Y}_{[M+1]} \\mid \\bm{\\Delta}, \\bm{Q}=Q) \\\\\n    &= H(\\bm{Y}_{[M+1]} \\mid \\bm{\\Delta}, \\bm{Q}=Q) + H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Y}_{[M+1]}, \\bm{Q}=Q)\\\\\n    &= H(\\bm{Y}_1, \\bm{Y}_2 \\mid \\bm{\\Delta}, \\bm{Q}=Q)\\label{eq:PCSI1_pri_dec2}\\\\\n    &\\leq 2L,\n\\end{align}\nwhere \\eqref{eq:PCSI1_pri_dec2} follows from \\eqref{eq:PCSI1_pri_dec1} and the fact that $\\bm{Y}_{[3:M+1]}$ are functions of $\\bm{Y}_{1}, \\bm{Y}_{2}$. Averaging over $\\bm{Q}$ we have \n\\begin{align}\n    H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q}) \\leq 2L.\n\\end{align}\n\n\\noindent Therefore, the average download cost is bounded as,\n\\begin{align}\n  D&\\geq  H(\\bm{\\Delta} \\mid \\bm{Q}) \\geq H(\\bm{W}_{[K]}\\mid\\bm{Q}) - H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q}) \\\\\n  & \\geq (K-2)L.\n\\end{align}\nThus, for $q\\geq M$, we have $C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}}(q) \\leq \\frac{1}{K-2}$.\n\n\\subsection{Proof of $C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}, \\inf} \\geq \\frac{1}{K-1}$}\\label{sec:pcsi1_pri_ach}\nWe need to show that $C_{\\mbox{\\tiny PCSI-I}}^{\\mbox{\\tiny pri}}(q)\\geq \\frac{1}{K-1}$ for all $\\mathbb{F}_q$. The scheme is identical to the scheme with rate $(K-1)^{-1}$ in Section \\ref{sec:PCSI1_inf_ach1} with a slight modification. Instead of fixing a realization $\\bm{\\Lambda}=\\Lambda$, we will consider all possible realizations $\\Lambda\\in\\mathfrak{C}$, and consider the product polynomial $f=\\prod_{\\Lambda\\in\\mathfrak{C}}f_{\\Lambda}$ which is a multi-variate polynomial of degree $(K-1)\\binom{K}{M}(q-1)^M$ in variables $\\psi_{ij}$. Following the same argument based on the Schwartz-Zippel Lemma, we find that there exists a $\\Psi$ for which all $G_{\\Lambda,\\mathcal{S}}$ are invertible matrices, provided that $L$ is large enough that $q^L>(q-1)^M(K-1)\\binom{K}{M}$. Thus, with this choice of $\\Psi$ we have a scheme that is allows the user to retrieve all $K$ messages. The scheme is also $(\\bm{\\mathcal{S}},\\bm{\\theta},\\bm{\\Lambda})$ private because we note that the user does not need to know the realization of $(\\bm{\\mathcal{S}},\\bm{\\theta},\\bm{\\Lambda})$ before it sends the query, so the query is independent of $(\\bm{\\mathcal{S}},\\bm{\\theta},\\bm{\\Lambda})$. \n\n\\begin{remark}\\label{rmk:pcsi1_pri_pcsi_pri}\nSince the scheme allows the user to decode all messages, and the query does not depend on $(\\bm{\\theta}, \\bm{\\mathcal{S}}, \\bm{\\Lambda})$, the scheme also works if $\\bm{\\theta}$ is uniformly drawn from $[K]$, i.e., in the PIR-PCSI setting.\n\\end{remark}\n\n\\section{Proof of Theorem \\ref{thm:cap_PCSI_sup}}\\label{sec:cap_PCSI_sup}\n\\subsection{Converse}\nThe converse is  divided into two regimes.\n\n\\textbf{Regime 1}: $2 \\leq M \\leq K$. The proof relies on \\eqref{eq:lemma1pcsi} in Lemma \\ref{lem:privacy}.\nConsider any particular realization $Q \\in \\mathcal{Q}$ of $\\bm{Q}$. For all $i \\in [K]$, consider $\\mathcal{S} = [M], \\theta = i$, and let $\\Lambda_i$ be a coefficient vector that satisfies \\eqref{eq:lemma1pcsi} according to Lemma \\ref{lem:privacy}, so that \n\\begin{align}\n    H(\\bm{W}_i \\mid \\bm{\\Delta}, \\bm{Y}^{[[M],\\Lambda_i]}, \\bm{Q} = Q) = 0.\\label{eq:con_PCSI_0}\n\\end{align}\nWriting $\\bm{Y}^{[[M],\\Lambda_i]}$ as $\\bm{Y}_{i}$ for compact notation, we have\n\\begin{align}\n    &H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Y}_{[M-1]}, \\bm{Q} = Q)\\notag\\\\\n    &= H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Y}_{[M-1]}, \\bm{W}_{[M-1]}, \\bm{Q} = Q)\\label{eq:con_PCSI_1}\\\\\n    &= H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{W}_{[M]}, \\bm{Q} = Q)\\label{eq:con_PCSI_2}\\\\\n    &= H(\\bm{W}_{[M+1:K]} \\mid \\bm{\\Delta}, \\bm{W}_{[M]}, \\bm{Y}_{[M+1:K]}, \\bm{Q} = Q)\\label{eq:con_PCSI_3}\\\\\n    &= 0,\\label{eq:con_PCSI_3a}\n\\end{align}\nwhere \\eqref{eq:con_PCSI_1} holds according to \\eqref{eq:con_PCSI_0}, and \\eqref{eq:con_PCSI_2} follows from the fact that $\\bm{W}_M$ is decodable by subtracting $\\bm{W}_{[M-1]}$ terms from $\\bm{Y}_1$. Then, \\eqref{eq:con_PCSI_3} uses the fact that $\\bm{Y}_{[M+1:K]}$ are functions of $\\bm{W}_{[M]}$. Finally, \\eqref{eq:con_PCSI_3a} follows from \\eqref{eq:con_PCSI_0}. \n\nAveraging over $\\bm{Q}$, \n\\begin{align}\n    H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Y}_{[M-1]}, \\bm{Q}) = 0.\\label{eq:con_PCSI_4}\n\\end{align}\nThen we have \n\\begin{align}\n    &H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q})\\\\\n    &= H(\\bm{W}_{[K]}, \\bm{Y}_{[M-1]} \\mid \\bm{\\Delta}, \\bm{Q})\\label{eq:con_PCSI_5}\\\\\n    &= H(\\bm{Y}_{[M-1]} \\mid \\bm{\\Delta}, \\bm{Q}) + H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q}, \\bm{Y}_{[M-1]})\\\\\n    &\\leq H(\\bm{Y}_{[M-1]})\\label{eq:con_PCSI_6}\\\\\n    &\\leq (M-1)L,\n\\end{align}\nwhere \\eqref{eq:con_PCSI_5} follows from the fact that $\\bm{Y}_{[M-1]}$ are linear combinations of $\\bm{W}_{[M]}$. Step \\eqref{eq:con_PCSI_6} holds because of  \\eqref{eq:con_PCSI_4}, and because conditioning reduces entropy.\n\nThus $D \\geq H(\\bm{\\Delta} \\mid \\bm{Q}) \\geq H(\\bm{W}_{[K]}) - H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q}) \\geq (K-M+1)L$, which implies that $C_{\\mbox{\\tiny PCSI}}^{\\sup}  \\leq (K-M+1)^{-1}$ for $2 \\leq M \\leq K$.\n\n\\textbf{Regime 2}: $M=1$.\n\nConsider any particular realization $Q \\in \\mathcal{Q}$ of $\\bm{Q}$. Since $M=1$, $\\bm\\Lambda$ is irrelevant, e.g., we may assume $\\bm{\\Lambda}=\\Lambda=1$ without loss of generality. For all $j \\in [2:K]$, consider $\\mathcal{S} = \\{1\\}, \\theta = j$, and apply \\eqref{eq:lemma1pcsi} according to Lemma \\ref{lem:privacy} so that \n\\begin{align}\n    H(\\bm{W}_j \\mid \\bm{\\Delta}, \\bm{Y}^{[\\{1\\},1]}, \\bm{Q}=Q) = 0\\\\\n\\implies        H(\\bm{W}_{[2:K]} \\mid \\bm{\\Delta}, \\bm{Y}^{[\\{1\\},1]}, \\bm{Q}=Q) = 0\\label{eq:dec_PCSI_corner}\n\\end{align}\n\\begin{align}\n    &H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q}=Q)\\\\\n    &\\leq H(\\bm{W}_1, \\bm{Y}^{[\\{1\\},1]} \\mid \\bm{\\Delta}, \\bm{Q} = Q)\\label{eq:corner_PCSI_1}\\\\\n    &= H(\\bm{W}_1 \\mid \\bm{\\Delta}, \\bm{Q}=Q)\\label{eq:corner_PCSI_2}\\\\\n    &\\leq L,\n\\end{align}\nwhere \\eqref{eq:corner_PCSI_1} holds since \\eqref{eq:dec_PCSI_corner} holds. Averaging over $\\bm{Q}$, $H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q}) \\leq L$. Thus $D \\geq H(\\bm{\\Delta} \\mid \\bm{Q}) \\geq H(\\bm{W}_{[K]}) - H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q}) \\geq KL-L$, which implies that $C_{\\mbox{\\tiny PCSI}}(q) \\leq (K-1)^{-1}$ for $M=1$.\n\n\\subsection{Achievability}\nFor $2 \\leq M \\leq K$, the achievable scheme will be a combination of \\emph{Specialized GRS Codes} and \\emph{Modified Specialized GRS Codes} which are schemes in \\cite{PIR_PCSI} for PIR-PCSI-I and PIR-PCSI-II setting, respectively.\n\nThe rate $(K-M)^{-1}$ is achievable by \\emph{Specialized GRS Codes} for PIR-PCSI-I setting and the rate $(K-M+1)^{-1}$ is achievable by \\emph{Modified Specialized GRS Codes} for the PIR-PCSI-II setting. Both  schemes work for $L=1$, so let us say $L=1$ here. Intuitively, these two achievable schemes have the same structures as explained below. \n\nFor the PIR-PCSI-I setting,  the desired message is not contained in the support set. The download will be $K-M$ linear equations of $K$ unknowns ($K$ messages). These $K-M$ linear equations are independent by design, so they allow the user to eliminate any $K-M-1$ unknowns and get an equation in the remaining $K-(K-M-1) = M+1$ unknowns (messages). Let these $M+1$ unknowns be the $M$ messages in the support set and the desired message. With careful design, the equation will be equal to $\\bm{Y}^{[\\bm{\\mathcal{S}},\\bm{\\Lambda}]} + \\bm{\\lambda}^{\\prime}\\bm{W}_{\\bm{\\theta}}$ for some non-zero $\\bm{\\lambda}^\\prime$. Thus by subtracting CSI from the equation the user is able to recover $\\bm{W}_{\\bm{\\theta}}$.\n\nFor the PIR-PCSI-II setting the desired message is contained in the support set. The download will be $K-M+1$ linear equations in $K$ unknowns (messages). These $K-M+1$ linear equations are independent by design, so they allow the user to eliminate any $K-M$ unknowns and get an equation in the remaining $K-(K-M) = M$ unknowns (messages). Let these $M$ unknowns be the $M$ messages in the support set. With careful design, the equation will be equal to $\\bm{Y}^{[\\bm{\\mathcal{S}},\\bm{\\Lambda}]} + \\bm{\\lambda}^{\\prime}\\bm{W}_{\\bm{\\theta}}$ for some $\\bm{\\lambda}^{\\prime} \\neq 0$. Thus by subtracting CSI from the equation the user is able to recover $\\bm{W}_{\\bm{\\theta}}$.\n\nConsider a scheme where the user applies \\emph{Specialized GRS Codes} when $\\bm{\\theta} \\notin \\bm{\\mathcal{S}}$ and applies \\emph{Modified Specialized GRS Codes} when $\\bm{\\theta} \\in \\bm{\\mathcal{S}}$. This scheme is obviously correct but not private because the server can tell if $\\bm{\\theta} \\in \\bm{\\mathcal{S}}$ or not from the download cost since the download cost of the two schemes are different. However, if the user always downloads one more redundant equation when applying \\emph{Specialized GRS Codes}, then there is no difference in the download cost. This is essentially the idea for the achievable scheme.\n\nLet us first present the \\emph{Specialized GRS Codes} in \\cite{PIR_PCSI} here for ease of understanding. There are $K$ distinct evaluation points in $\\mathbb{F}_{q}$, namely $\\omega_{1}, \\cdots, \\omega_{K}$. A polynomial $\\bm{p}(x)$ is constructed as \n\\begin{align}\n    \\bm{p}(x) &\\triangleq \\prod_{k \\in [K]\\setminus(\\bm{\\mathcal{S}} \\cup \\{\\bm{\\theta}\\})}(x - \\omega_{k})\\\\\n    & = \\sum_{i=1}^{K-M}\\bm{p}_i x^{i-1}.\\label{eq:polyGRS}\n\\end{align}\nThe query $\\bm{Q}$ is comprised of $K-M$ row vectors, each $1\\times K$, namely $\\bm{Q}_{1}, \\cdots, \\bm{Q}_{K-M}$ such that \n\\begin{align}\n    \\bm{Q}_i = [\\bm{v}_1\\omega_{1}^{i-1}~~ \\cdots~~ \\bm{v}_K\\omega_{K}^{i-1}], \\forall i \\in [K-M],\n\\end{align}\nwhere for $\\bm{i}_m \\in \\bm{\\mathcal{S}}, m \\in [M]$, $\\bm{v}_{\\bm{i}_m} = \\frac{\\bm{\\lambda}_m}{p(\\omega_{\\bm{i}_m})}$ ($\\bm{\\lambda}_m$ is the $m^{th}$ coefficient in the CSI), while for $k \\notin \\bm{\\mathcal{S}}$, $\\bm{v}_{k}$ is randomly drawn from $\\mathbb{F}_{q}^{\\times}$. Upon receiving $\\bm{Q}$, the server sends  \n\\begin{align}\n    \\bm{\\Delta} = \n    \\begin{bmatrix}\n        \\bm{\\Delta}_1\\\\\n        \\vdots\\\\\n        \\bm{\\Delta}_{K-M}\n    \\end{bmatrix}\n    =\n    \\begin{bmatrix}\n        \\bm{Q}_1\\\\\n        \\vdots\\\\\n        \\bm{Q}_{K-M}\n    \\end{bmatrix}\n    \\begin{bmatrix}\n        \\bm{W}_1\\\\\n        \\bm{W}_2\\\\\n        \\vdots\\\\\n        \\bm{W}_K\n    \\end{bmatrix}\n\\end{align}\nto the user. Let us call $[\\bm{Q}_1^{\\mathrm{T}} ~ \\cdots ~ \\bm{Q}_{K-M}^{\\mathrm{T}}]^{\\mathrm{T}}$ the \\emph{Specialized GRS Matrix} and $[\\bm{\\Delta}_1 ~ \\cdots ~ \\bm{\\Delta}_{K-M}]^{\\mathrm{T}}$  \\emph{Specialized GRS Codes} of $\\bm{W}_{[K]}$ for ease of reference. Note that the \\emph{Specialized GRS Matrix} is uniquely defined by $\\bm{v}_{1}, \\cdots, \\bm{v}_{K}$ as $\\omega$'s are constants.\n\nThe user gets $\\bm{W}_{\\bm{\\theta}}$ by subtracting $\\bm{Y}^{[\\bm{\\mathcal{S}}, \\bm{\\Lambda}]}$ from \n\\begin{align}\n    \\sum_{i=1}^{K-M}\\bm{p}_i\\bm{\\Delta}_{i} = \\bm{Y}^{[\\bm{\\mathcal{S}}, \\bm{\\Lambda}]} + \\bm{v}_{\\bm{\\theta}}\\bm{p}(\\omega_{\\bm{\\theta}})\\bm{W}_{\\bm{\\theta}}.\n\\end{align}\n\nOur PIR-PCSI scheme is as follows.\nFor any realization $(\\theta, \\mathcal{S})$ of $(\\bm{\\theta}, \\bm{\\mathcal{S}})$, \n\\emph{1)} When $\\theta \\in [K]\\setminus\\mathcal{S}$, first apply the Specialized GRS Codes in \\cite{PIR_PCSI}. Besides $Q_1, Q_2, \\cdots, Q_{K-M}$ as specified in the \\emph{Specialized GRS Codes} of \\cite{PIR_PCSI}, the user also has \n\\begin{align}\n    Q_{K-M+1} = [v_1\\omega_{1}^{K-M}, \\cdots, v_K\\omega_{K}^{K-M}]\n\\end{align}\nas part of the query. And the answer $\\bm{\\Delta}_{K-M+1} = \\sum_{j=1}^{K}v_j \\omega_j^{K-M} \\bm{W}_j$ will be generated for $Q_{K-M+1}$ and downloaded by the user as a redundant equation. Note that the matrix $[Q_1^{\\mathrm{T}}, Q_2^{\\mathrm{T}}, \\cdots, Q_{K-M+1}^{\\mathrm{T}}]^{\\mathrm{T}}$ is the generator matrix of a $(K, K-M+1)$ GRS code \\cite{Coding_Theory}.\n\n\\emph{2)} When $\\theta \\in \\mathcal{S}$, the user will directly apply \\emph{Modified Specialized GRS Codes} where the queries also form a generator matrix of a $(K, K-M+1)$ GRS code as specified in \\cite{PIR_PCSI}.\n\nSuch a scheme is private since the queries in both cases form a generator matrix of a $(K,K-M+1)$ GRS code, and the $v_1, \\cdots, v_{K}$ in both cases are identically uniform over $\\mathbb{F}_{q}^{\\times}$ for any realization of $\\bm{\\theta}, \\bm{\\mathcal{S}}$.\n\nFor the corner case $M=1$, it suffices to download $K-1$ generic linear combinations of all the $K$ messages such that from the $K-1$ downloaded linear combinations and the CSI, all the $K$ messages are decodable as noted in Remark \\ref{rmk:pcsi1_inf_pcsi_inf}.\n\n\\section{Proof of Theorem \\ref{thm:redundancy}}\\label{proof:redundancy}\nHere we bound the redundancy $\\rho_{\\mbox{\\tiny PCSI}}$ from above (equivalently, lower-bound $\\alpha^{*}$) for $1 \\leq M \\leq K$. For $\\frac{K+2}{2} < M \\leq K$, the proof for $\\rho_{\\mbox{\\tiny PCSI}} = 0$ is the same as in Section \\ref{proof:red} show that  so it will not be repeated. \n\n\nConsider an achievable scheme such that $\\alpha$ PCSI is sufficient and the average download cost, $D \\leq \\frac{1}{C_{\\mbox{\\tiny PCSI}}^{\\sup}}L+\\epsilon L$ for some $L$. Note that $D\\geq H(\\bm{\\Delta\\mid \\bm{Q}})$, therefore,\n\\begin{align}\nH(\\bm{\\Delta\\mid \\bm{Q}})\\leq \\frac{1}{C_{\\mbox{\\tiny PCSI}}^{\\sup}}L+\\epsilon L\\label{eq:deltabound}\n\\end{align}\n\nIt follows from \\eqref{eq:deltabound} that there exists a feasible $Q \\in \\mathcal{Q}$ such that \n\\begin{align}\n    H(\\bm{\\Delta} \\mid \\bm{Q}=Q) \\leq \\frac{1}{C_{\\mbox{\\tiny PCSI}}^{\\sup}}L+\\epsilon L.\n\\end{align}\nFor all $i \\in [K]$, let $\\Lambda_{i} \\in \\mathfrak{C}$ satisfy \n\\begin{align}\n    H(\\bm{W}_{i} \\mid \\bm{\\Delta}, \\overline{\\bm{Y}}^{[[M], \\Lambda_i]}, \\bm{Q} = Q) = 0.\\label{eq:red_M1_dec}\n\\end{align}\nThe argument that such $\\Lambda_i$'s must exist is identical to the proof of Lemma \\ref{lem:privacy}. \nWriting $\\overline{\\bm{Y}}^{[[M], \\Lambda_i]}$ as $\\overline{\\bm{Y}}_{i}$ for compact notation,\n\\begin{align}\n    &H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\overline{\\bm{Y}}_{[M]}, \\bm{Q} = Q)\\\\\n    &= H(\\bm{W}_{[M]} \\mid \\bm{\\Delta}, \\overline{\\bm{Y}}_{M}, \\bm{Q} = Q)\\notag\\\\\n    &~~~+ H(\\bm{W}_{[M+1:K]} \\mid \\bm{\\Delta}, \\overline{\\bm{Y}}_{[M]}, \\bm{W}_{[M]}, \\bm{Q} = Q)\\\\\n    &= 0 + H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{W}_{[M]}, \\overline{\\bm{Y}}_{[K]}, \\bm{Q} = Q)\\label{eq:red_M1_funcW}\\\\\n    &=0.\n\\end{align}\nwhere \\eqref{eq:red_M1_funcW} follows from \\eqref{eq:red_M1_dec} and the fact that $\\overline{\\bm{Y}}_{[K]}$ are functions of $\\bm{W}_{[M]}$. The last step also follows from \\eqref{eq:red_M1_dec}. Thus,\n\\begin{align}\n    &\\frac{1}{C_{\\mbox{\\tiny PCSI}}^{\\sup}}L + \\epsilon L + M\\alpha L\\notag\\\\\n    &\\geq H(\\bm{\\Delta}, \\overline{\\bm{Y}}_{[M]} \\mid \\bm{Q}=Q)\\label{eq:red_M1_indY}\\\\\n    &\\geq I(\\bm{\\Delta}, \\overline{\\bm{Y}}_{[M]}; \\bm{W}_{[K]} \\mid \\bm{Q}=Q)\\\\\n    &=H(\\bm{W}_{[K]} \\mid \\bm{Q}=Q) = KL.\\label{eq:red_M1_indW}\n\\end{align}\n\\eqref{eq:red_M1_indY} is true because \\eqref{eq:indQ}, \\eqref{eq:invaYR} hold. Step \\eqref{eq:red_M1_indW} follows from \\eqref{eq:red_M1_dec} and the fact that the query and messages are mutually independent according to \\eqref{eq:indQ}. Thus, $\\alpha \\geq (K-\\frac{1}{C_{\\mbox{\\tiny PCSI}}^{\\sup}})\/M - \\epsilon\/M$. In order to achieve capacity, we must have $\\epsilon \\rightarrow 0$, so we must have $\\alpha \\geq (K-\\frac{1}{C_{\\mbox{\\tiny PCSI}}^{\\sup}})\/M$, for all $1\\leq M\\leq K$.\n\nNow note that for $M=1$, since $C_{\\mbox{\\tiny PCSI}}^{\\sup} = (K-1)^{-1}$, we have shown that $\\alpha \\geq 1$, which implies $\\rho_{\\mbox{\\tiny PCSI}} = 0$ in this case. \n\nFor $2 \\leq M \\leq \\frac{K+2}{2}$, since $C_{\\mbox{\\tiny PCSI}}^{\\sup} = (K-M+1)^{-1}$, we have shown that $\\alpha \\geq \\frac{M-1}{M}$, which implies $\\rho_{\\mbox{\\tiny PCSI}} \\leq \\frac{1}{M}$ in this case.\n\nIt only remains to show that for $M=2$,  $\\rho_{\\mbox{\\tiny PCSI}} = \\frac{1}{2}$ is achievable, or equivalently, $\\alpha^{*} = \\frac{1}{2}$. For this case, let us present a PIR-PCSI scheme that achieves the rate $(K-M\/2)^{-1}$ for arbitrary $1 \\leq M \\leq K$. Note that $K-M\/2 = K-M+1$ when $M=2$, which is the only case where the supremum capacity is achieved by this scheme. The rate of this scheme is strictly smaller than $C_{\\mbox{\\tiny PCSI}}^{\\sup}$ for other $M \\neq 2$.\n\n\nLet the size of the base field $q$ be an even power of a prime number such that $\\sqrt{q}$ is a prime power and $\\sqrt{q} \\geq K$. For arbitrary realization $(\\theta, \\mathcal{S}) \\in [K]\\times\\mathfrak{S}$ of $(\\bm{\\theta},\\bm{\\mathcal{S}})$, if $\\theta \\in \\mathcal{S}$, the user can apply the \\emph{Interference Alignment} based PIR-PCSI-II scheme where half of each message is downloaded. If $\\theta \\in [K]\\setminus\\mathcal{S}$, then user can apply the \\emph{Specialized GRS Codes} based scheme for the halves of the messages corresponding to the CSI  dimension that is retained (while the other half of the CSI dimensions is discarded as redundant) and download the other half dimension of all the messages directly. Note that in both cases, a half-dimension of each of the $K$ messages is directly downloaded. The other halves are involved in the download corresponding to the \\emph{Specialized GRS Codes} which is not needed for decodability\/correctness if $\\theta \\in \\mathcal{S}$, but is still included for privacy, i.e., to hide whether or not $\\bm\\theta\\in\\bm{\\mathcal{S}}$. The download cost required is $K\\left(\\frac{L}{2}\\right)$ for the direct downloads of half of every message, plus  $(K-M)\\frac{L}{2}$ for the \\emph{Specialized GRS Codes} based scheme that usually requires $K-M$ downloads per message symbol, but is applied here to only half the symbols from each message, for a total download cost of $(K-M\/2)L$ which achieves the supremum capacity of PIR-PCSI for $M=2$. The details of the scheme are presented next.\n\nFor all $k \\in [K]$, let $V_{\\bm{W}_k} \\in \\mathbb{F}_{\\sqrt{q}}^{2\\times 1}$ be the length $2$ vector representation of $\\bm{W}_k \\in \\mathbb{F}_{q}$. For all $m \\in [M]$, let $M_{\\bm{\\lambda}_m} \\in \\mathbb{F}_{\\sqrt{q}}^{2\\times 2}$ be the matrix representation of $\\bm{\\lambda}_m \\in \\mathbb{F}_{q}^{\\times}$ where $\\bm{\\lambda}_m$ is the $m^{th}$ entry of the coefficient vector $\\bm{\\Lambda}$. Let \n\\begin{align}\n    \\overline{\\bm{Y}}^{[\\bm{\\mathcal{S}}, \\bm{\\Lambda}]} = M_{\\bm{\\lambda}_1}(1,:)V_{\\bm{W}_{\\bm{i}_1}} +\\cdots+ M_{\\bm{\\lambda}_M}(1,:)V_{\\bm{W}_{\\bm{i}_M}},\n\\end{align}\nwhere $\\bm{\\mathcal{S}} = \\{\\bm{i}_1, \\bm{i}_2, \\cdots, \\bm{i}_{M}\\}$ is the support index set, be the processed CSI where $H(\\overline{\\bm{Y}}^{[\\bm{\\mathcal{S}}, \\bm{\\Lambda}]}) = \\frac{1}{2}H(\\bm{W}_k)$. Note that $\\forall m \\in [M], M_{\\bm{\\lambda}_m}(1,:)$ is uniform over $\\mathbb{F}_{\\sqrt{q}}^{1\\times 2} \\setminus \\{[0~~0]\\}$ according to Lemma \\ref{lem:uniform12}.\n\nThe query $\\bm{Q} = \\{\\bm{Q}_1, \\bm{Q}_2, \\bm{Q}_3\\}$,\n\\begin{align}\n    \\bm{Q}_1 &= \\{\\mathbf{L}_1, \\mathbf{L}_2, \\cdots, \\mathbf{L}_{K}\\},\\\\\n    \\bm{Q}_2 &= \\{\\mathbf{L}_1^{\\prime}, \\mathbf{L}_2^{\\prime}, \\cdots, \\mathbf{L}_{K}^{\\prime}\\},\\\\\n    \\bm{Q}_3 &= \\{\\bm{v}_1, \\bm{v}_2, \\cdots, \\bm{v}_{K}\\}.\n\\end{align}\nwhere $\\mathbf{L}_{k}, \\mathbf{L}_{k}^{\\prime} \\in \\mathbb{F}_{\\sqrt{q}}^{1\\times 2} \\setminus \\{[0~~0]\\}$. $\\mathbf{L}_{k}, \\mathbf{L}_{k}^{\\prime}$ serve as two linearly independent projections that ask the server to split $\\bm{W}_k$ into two halves \n\\begin{align}\n    \\bm{w}_{k}(1) = \\mathbf{L}_{k}V_{\\bm{W}_k} \\in \\mathbb{F}_{\\sqrt{q}},\\\\\n    \\bm{w}_{k}(2) = \\mathbf{L}_{k}^{\\prime}V_{\\bm{W}_k} \\in \\mathbb{F}_{\\sqrt{q}}.\n\\end{align}\n$\\bm{Q}_3$ uniquely defines a \\emph{Specialized GRS Matrix} whose elements are in $\\mathbb{F}_{\\sqrt{q}}$.\n\nThe user will download the first halves of all the $K$ messages after projection, i.e., $\\bm{w}_{[K]}(1)$ and apply the \\emph{Specialized GRS Matrix} to download a \\emph{Specialized GRS Codes} of the second halves of all the $K$ messages after projection, i.e., $\\bm{w}_{[K]}(2)$.  \n\nLet us specify $\\mathbf{L}_{k}, \\mathbf{L}_{k}^{\\prime}, \\bm{v}_{k}$. Consider any realization $(\\theta, \\mathcal{S}) \\in [K]\\times\\mathfrak{S}$ of $(\\bm{\\theta},\\bm{\\mathcal{S}})$. Let us say $\\mathcal{S} = \\{i_1, i_2, \\cdots, i_M\\}$. For the messages not involved in the CSI, they are randomly projected to two linearly independent directions, i.e., for any $k \\in [K] \\setminus \\mathcal{S}$, $\\mathbf{L}_{k}, \\mathbf{L}_{k}^{\\prime}$ are linearly independent and are randomly drawn from $\\mathbb{F}_{\\sqrt{q}}^{1 \\times 2} \\setminus \\{[0~~0]\\}$. Also, for any $k \\in [K] \\setminus \\mathcal{S}$, $\\bm{v}_{k}$ is uniformly distributed in $\\mathbb{F}_{\\sqrt{q}}^{\\times}$. \n\nFor messages involved in the CSI, the construction of projections and $\\bm{v}$'s depends on whether $\\theta$ is in $\\mathcal{S}$ or not.\n\\begin{enumerate}\n    \\item When $\\theta \\in \\mathcal{S}$, for any $m \\in [M]$,\n    \\begin{align}\n        \\mathbf{L}_{i_m} = \n        \\begin{cases}\n            M_{\\bm{\\lambda}_m}(2,:), i_m = \\theta,\\\\\n            M_{\\bm{\\lambda}_m}(1,:), i_m \\neq \\theta.\n        \\end{cases}\n    \\end{align}\n    $\\mathbf{L}_{i_m}^{\\prime}$ is then chosen randomly from $\\mathbb{F}_{\\sqrt{q}}^{1 \\times 2} \\setminus \\{[0~~0]\\}$ such that it is linearly independent with $\\mathbf{L}_{i_m}$. Meanwhile, $\\bm{v}_{i_m}$ is randomly drawn from $\\mathbb{F}_{\\sqrt{q}}^{\\times}$. Under this case, the user has \n    \\begin{align}\n        \\overline{\\bm{Y}}^{[\\mathcal{S}, \\bm{\\Lambda}]} = \\sum_{i_m \\in \\mathcal{S}\\setminus\\{\\theta\\}}\\bm{w}_{i_m}(1) + \\bm{w}_{\\theta}(2)\n    \\end{align} \n    according to the construction of $\\mathbf{L}_{i_m}$. $\\bm{w}_{\\theta}(1)$ is directly downloaded and $\\bm{w}_{\\theta}(2)$ can be recovered by subtracting $\\{\\bm{w}_{i_m}(1)\\}_{i_m \\neq \\theta}$ from $\\overline{\\bm{Y}}^{[\\mathcal{S}, \\bm{\\Lambda}]}$. The user is then able to recover $\\bm{W}_{\\theta}$ as the two projections are linearly independent. $\\bm{Q}_3$ uniquely defines a \\emph{Specialized GRS Matrix} and applying $\\bm{Q}_3$ to download a \\emph{Specialized GRS Codes} of $\\bm{w}_{[K]}(2)$ is just for privacy.\n\n    \\item When $\\theta \\in [K]\\setminus\\mathcal{S}$, for any $m \\in [M]$, \n    \\begin{align}\n        \\mathbf{L}_{i_m}^{\\prime} = \\frac{1}{\\bm{a}_m}M_{\\bm{\\lambda}_m}(1,:),\n    \\end{align}\n    where $\\bm{a}_m$ is randomly drawn from $\\mathbb{F}_{\\sqrt{q}}^{\\times}$. $\\mathbf{L}_{i_m}$ is then chosen randomly from $\\mathbb{F}_{\\sqrt{q}}^{1 \\times 2} \\setminus \\{[0~~0]\\}$ such that they are linearly independent with $\\mathbf{L}_{i_m}^{\\prime}$. Under this case, the user has \n    \\begin{align}\n        \\sum_{m\\in[M]}\\bm{a}_{m}\\bm{w}_{i_m}(2) = \\overline{\\bm{Y}}^{[\\mathcal{S}, \\bm{\\Lambda}]},\n    \\end{align}\n    and sets \n    \\begin{align}\n        \\bm{v}_{i_m} = \\frac{\\bm{a}_m}{p(\\omega_{i_m})}, \\forall m \\in [M],\n    \\end{align}\n    where $p(\\omega_{i_m})$ is the evaluation of the polynomial specified in \\eqref{eq:polyGRS} (when $(\\bm{\\theta},\\bm{\\mathcal{S}}) = (\\theta$, $\\mathcal{S})$) at $\\omega_{i_m}$, which is a non-zero constant given $(\\theta, \\mathcal{S})$. Thus, given $(\\theta, \\mathcal{S})$, $\\bm{v}_{i_m}$ is still uniform over $\\mathbb{F}_{\\sqrt{q}}^{\\times}$. $\\bm{Q}_{3}$ uniquely defines a \\emph{Specialized GRS Matrix}. Applying $\\bm{Q}_3$ to download a \\emph{Specialized GRS Codes} of $\\bm{w}_{[K]}(2)$, together with $\\sum_{m\\in[M]}\\bm{a}_{m}\\bm{w}_{i_m}(2)$ as the side information, enable the user to recover $\\bm{w}_{\\theta}(2)$. Since the first halves of all the projected messages are also downloaded, the user also has $\\bm{w}_{\\theta}(1)$, thus, is able to decode $\\bm{W}_{\\theta}$.\n\\end{enumerate}\n\nNote that for arbitrary realization $(\\theta, \\mathcal{S})$ of $(\\bm{\\theta}, \\bm{\\mathcal{S}})$, no matter $\\theta \\in \\mathcal{S}$ or not, $\\mathbf{L}_1, \\cdots, \\mathbf{L}_{K}$, $\\mathbf{L}_1^{\\prime}, \\cdots, \\mathbf{L}_{K}^{\\prime}$, $\\bm{v}_1, \\cdots, \\bm{v}_{K}$ are independent, and for any $k \\in [K]$, the matrix whose first row is $\\mathbf{L}_{k}$ and second row is $\\mathbf{L}_{k}^{\\prime}$ is uniform over the set that contains all the full-rank matrix in $\\mathbb{F}_{\\sqrt{q}}^{2\\times 2}$, $\\bm{v}_{k}$ is uniform over $\\mathbb{F}_{\\sqrt{q}}^{\\times}$. Thus, the scheme is private.\n\n\n\\section{Proof of Theorem \\ref{thm:cap_PCSI_inf}}\\label{sec:cap_PCSI_inf}\nThe rate $\\frac{1}{K-1}$ PIR-PCSI-I scheme in Section \\ref{sec:PCSI1_inf_ach} is also the infimum capacity achieving PIR-PCSI scheme as noted in Remark \\ref{rmk:pcsi1_inf_pcsi_inf}, so we just prove the converse here.\n\nAs a result of \\eqref{eq:lemma1pcsi} and the fact that in $\\mathbb{F}_{2}$, we can only have $\\bm{\\Lambda} = 1_M$, i.e., the length-$M$ vector all of whose elements are equal to $1$, we have\n\\begin{align}\n    H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Y}^{[\\mathcal{S},1_M]}, \\bm{Q}=Q) = 0, \\notag\\\\\n    \\forall (Q,\\mathcal{S}) \\in \\mathcal{Q}\\times\\mathfrak{S}. \\label{eq:pcsi_inf_dec}\n\\end{align}\nWriting $\\bm{Y}^{[[M],1_M]}$ as $\\bm{Y}$ for compact notation, for any $Q \\in \\mathcal{Q}$, we have\n\\begin{align}\n    &H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q}=Q)\\notag\\\\\n    &= H(\\bm{W}_{[K]}, \\bm{Y} \\mid \\bm{\\Delta}, \\bm{Q}=Q) \\label{eq:dec_K_1}\\\\\n    &= H(\\bm{Y} \\mid \\bm{\\Delta}, \\bm{Q}=Q) + H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Y}, \\bm{Q} = Q)\\label{eq:dec_K_2}\\\\\n    &\\leq H(\\bm{Y}) = L.\n\\end{align}\n\\eqref{eq:dec_K_1} is true since $\\bm{Y}$ is a summation of the first $M$ messages, and \\eqref{eq:dec_K_2} follows from \\eqref{eq:pcsi_inf_dec}. Averaging over $\\bm{Q}$ we have,\n\\begin{align}\n    H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q}) \\leq L.\n\\end{align}\nThus, $D \\geq H(\\bm{\\Delta} \\mid \\bm{Q}) \\geq H(\\bm{W}_{[K]}) - H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Q}) \\geq KL-L$ which implies that $C_{\\mbox{\\tiny PCSI}}^{\\inf}(q = 2) \\leq (K-1)^{-1}$.\n\n\n\\section{Proof of Theorem \\ref{thm:pcsi_pub_pri}}\\label{proof:pcsi_pub_pri}\nThe rate $\\frac{1}{K-1}$ PIR-PCSI-I scheme which preserves $(\\bm{\\theta}, \\bm{\\mathcal{S}}, \\bm{\\Lambda})$ in Section \\ref{sec:pcsi1_pri_ach} is also the  capacity achieving PIR-PCSI scheme with private coefficients as noted in Remark \\ref{rmk:pcsi1_pri_pcsi_pri}, so we just prove the converse here. Specifically, we prove that $C_{\\mbox{\\tiny PCSI}}^{\\mbox{\\tiny pri}}(q)\\leq C_{\\mbox{\\tiny PCSI}}(q=2) = C_{\\mbox{\\tiny PCSI}}^{\\inf}$.\n\nAccording to \\eqref{eq:pcsi_pri} in Lemma \\ref{lem:fullypri}, for a fully private PIR-PCSI scheme,\n\\begin{align}\n    H(\\bm{W}_{[K]} \\mid \\bm{\\Delta}, \\bm{Y}^{[\\mathcal{S}, \\Lambda]}, \\bm{Q} = Q) = 0, \\notag\\\\\n    \\forall (Q,\\mathcal{S},\\Lambda) \\in \\mathcal{Q} \\times \\mathfrak{S} \\times \\mathfrak{C}.\\label{eq:pcsi_pri_dec}\n\\end{align}\nNote that \\eqref{eq:pcsi_pri_dec} is a \\emph{stronger} version of \\eqref{eq:pcsi_inf_dec} which is sufficient to bound $C_{\\mbox{\\tiny PCSI}}(q=2) = C_{\\mbox{\\tiny PCSI}}^{\\inf}$. Thus, $C_{\\mbox{\\tiny PCSI}}^{\\mbox{\\tiny pri}}(q) \\leq C_{\\mbox{\\tiny PCSI}}^{\\inf}$.\n\n\\section{Conclusion} \\label{sec:con}\nSide-information is a highly valuable resource for PIR in general, and for single-server PIR in particular. Building on the foundation laid by Heidarzadeh et al. in \\cite{PIR_PCSI}, this work  presents a more complete picture, as encapsulated in Table \\ref{tab:capacity}, revealing new insights that are described in the introduction. The redundancy of side-information is particularly noteworthy, because it allows the user to save storage cost, which may be used to store additional non-redundant side-information, e.g., multiple linear combinations instead of just one, as assumed in this work and in \\cite{PIR_PCSI}. An interesting direction for future work is to understand the trade-off between the size of side information and the efficiency of single-server PIR, e.g., by characterizing the $\\alpha$-CSI constrained capacity of PIR-PCSI-I, PIR-PCSI-II, PIR-PCSI. Other questions that remain open include  issues that are field-specific. For example, is the supremum capacity of PIR-PCSI-II for $M>2$ achievable for all fields except $\\mathbb{F}_2$?  Are there other fields besides $\\mathbb{F}_2$ over which the capacity is equal to the infimum capacity? Can the capacity over certain fields take values other than the supremum and infimum capacities? Progress on these issues may require field-dependent constructions of interference alignment schemes for achievability, and combinatorial arguments for converse bounds, both of which may be of broader interest.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nQuestion answering (QA) relates to the building of systems capable of automatically answering questions posed by humans in natural language. Various frameworks have been proposed for question answering, ranging from simple information-retrieval techniques for finding relevant knowledge articles or webpages, through methods for identifying the most relevant sentence in a text regarding a posed question, to methods for querying structured knowledge-bases or databases to produce an answer~\\cite{burke1997question,voorhees1999trec,kwok2001scaling,hirschman2001natural,ravichandran2002learning}\n\nA popular QA task is {\\em answer selection}, where, given a question, the system must pick correct answers from a pool of candidate answers~\\cite{xu2002trec,jijkoun2004answer,ko2007probabilistic,lee2009model,severyn2013automatic}.\n\nAnswer selection has many commercial applications. Virtual assistants such as Amazon Alexa and Google Assistant are designed to respond to natural language questions posed by users. In some cases such systems simply use a search engine to find relevant webpages; however, for many kinds of queries, such systems are capable of providing a concise specific answer to the posed question. \n\nSimilarly, various AI companies are attempting to improve customer service by automatically replying to customer queries. One way to design such a system is to curate a dataset of historical questions posed by customers and the responses given to these queries by human customer service agents. Given a previously unobserved query, the system can then locate the best matching answer in the curated dataset. \n\nAnswer selection is a difficult task, as typically there is a large number of possible answers which need to be examined. Furthermore, although in many cases the correct answer is lexically similar to the question, in other cases semantic similarities between words must be learned in order to find the correct answer~\\cite{kolomiyets2011survey,allam2012question}. Additionally, many of the words in the answer may not be relevant to the question. \n\nConsider, for example, the following question answer pair:\n\n\\begin{displayquote}\n\\textbf{How do I freeze my account?}\n\nHello, hope you are having a great day. You can freeze your account by logging into our site and pressing the freeze account button. Let me know if you have any further questions regarding the management of your account with us. \n\\end{displayquote}\n\n\\noindent Intuitively, the key section which identifies the above answer as correct is ``[...] you can freeze your account by [...]'', which represents a small fraction of the entire answer.\n\nEarlier work on answer selection used various techniques, ranging from information retrieval methods~\\cite{clarke2001exploiting} and machine learning methods relying on hand-crafted features~\\cite{parsetreeManning,wang2007jeopardy}. Deep learning methods, which have recently shown great success in many domains including image classification and annotation~\\cite{krizhevsky2012imagenet,zhou2014learning,lewenberg2016predicting}, multi-annotator data fusion~\\cite{albarqouni2016aggnet,gaunt2016training}, NLP and conversational models~\\cite{graves2013speech,bahdanau2014ntm,li2015diversity,kandasamy2017batch,shao2017generating} and speech recognition~\\cite{graves2013speech,albarqouni2016aggnet}, have also been successfully applied to question answering~\\cite{fengCNN}. Current state-of-the-art methods use recurrent neural network (RNN) architectures which incorporate attention mechanisms~\\cite{tan2016}.  These allow such models to better focus on relevant sections of the input~\\cite{bahdanau2014ntm}.\n\n{\\bf Our contribution: } We propose a new architecture for question answering. Our high-level approach is similar to recently proposed QA systems~\\cite{fengCNN,tan2016}, but we augment this design with a more sophisticated attention mechanism, combining the {\\em local} information in a specific part of the answer with a {\\em global} representation of the entire question and answer. \n\nWe evaluate the performance of our model using the recently released {\\em InsuranceQA dataset}~\\cite{fengCNN}, a large open dataset for answer selection comprised of insurance related questions such as: ``what can you claim on Medicare?''. \\footnote{As opposed to other QA tasks such as answers extraction or machine text comprehension and reasoning~\\cite{weston2015towards,rajpurkar2016squad}, the InsuranceQA dataset questions do not generally require logical reasoning.}\n\nWe beat state-of-the-art approaches ~\\cite{fengCNN,tan2016}, and achieve good performance even when using a relatively small network. \n\n\n\n\\section{Previous Work}\n\nAnswer selection systems can be evaluated using various datasets consisting of questions and answers. Early answer selection models were commonly evaluated against the QASent dataset \\cite{wang2007jeopardy}; however, this dataset is very small and thus less similar to real-world applications. Further, its candidate answer pools are created by finding sentences with at least one similar (non-stopword) word as compared to the question, which may create a bias in the dataset. \n\nWiki-QA~\\cite{yang2015wikiqa} is a dataset that contains several orders of magnitude more examples than QASent, where the candidate answer pools were created from the sentences in the relevant Wikipedia page for a question, reducing the amount of keyword bias in the dataset compared to QASent. \n\nOur analysis is based on the InsuranceQA~\\cite{fengCNN} dataset, which is much larger, and similar to real-world QA applications. The answers in InsuranceQA are relatively long (see details in Section~\\ref{sec:setup}), so the candidate answers are likely to contain content that does not relate directly to the question; thus, a good QA model for InsuranceQA must be capable of identifying the most important words in a candidate answer. \n\n\nEarly work on answer selection was based on finding the semantic similarity between question and answer parse trees using hand-crafted features \\cite{parsetreeManning, wang2007jeopardy}. Often, lexical databases such as WordNet were used to augment such models \\cite{ChangWordnet}. Not only did these models suffer from using hand-crafted features, those using lexical databases were also often language-dependent. \n\nRecent attempts at answer selection aim to map questions and candidate answers into n-dimensional vectors, and use a vector similarity measure such as cosine similarity to judge a candidate answer's affinity to a question. In other words, the similarity between a question and a candidate is high if the candidate answers the question well, low if the candidate is not a good match for the question. \n\nSuch models are similar to Siamese models, a good review of which can be found in Muller et al's paper~\\cite{mueller2016siamese}. Feng et al.~\\cite{fengCNN} propose using convolutional neural networks to vectorize both questions and answers before comparing them using cosine similarity. Similarly, Tan et al.~\\cite{tan2016} use a recurrent neural network \nto vectorize questions and answers. \nAttention mechanisms have proven to greatly improve the performance of recurrent networks in many tasks \\cite{bahdanau2014ntm, tan2016, rocktaschel2015entailment,rush2015neural,luong2015effective}, and indeed Tan et al.~\\cite{tan2016} incorporate a simple attention mechanism in their system.\n\n\n\\section{Preliminaries}\n\\label{l_sect_prelim}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.65\\textwidth]{attention0}\n\\caption{Model architecture using answer-localized attention \\cite{tan2016}. The left hand side used for the question. The right side of the architecture is used for both the answer and distractor.}\n\\label{fig:oldModel}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.8\\textwidth]{attention3}\n\\caption{Our proposed architecture with augmented attention. As in Figure~\\ref{fig:oldModel}, the right side of the model is used to embed answers and distractors.}\n\\label{fig:newModel}\n\\end{figure*}\n\nOur approach is similar to the {\\em Answer Selection Framework} of Tan et al.~\\cite{tan2016}, but we propose a different network architecture and a new attention mechanism. We first provide a high level description of this framework (see the original paper for a more detailed discussion), then discuss our proposed attention mechanism. \n\n  The framework is based on a neural network with parameters $\\theta$ which can embed either a question $q$ or a candidate answer $a$ into low dimensional vectors $r \\in \\!R^k$. The network can embed a question with no attention, which we denote as $f_{\\theta}(q)$, and embed a candidate answer with attention to the question, denoted as $g_{\\theta}(a, q)$. We denote the similarity function used as $s(x,y)$ ($s$ may be the dot product function, the cosine similarity function or some other similarity function).\n\nGiven a trained network, we compute the similarity between question and answer embeddings:\n\n$$s_i = s(f_{\\theta}(q), g_{\\theta}(A_i, q))$$\n\\noindent for any $i \\in {1,2,\\ldots,k}$ with $A_i$ being the $i$th candidate answer in the pool. We then select the answer yielding the highest similarity $\\arg \\max_i s_i$. \n\nThe embedding functions, $f_{\\theta}$ and $g_{\\theta}$, depend on the architecture used and the parameters $\\theta$. The network is trained by choosing a loss function $\\mathcal{L}$, and using stochastic gradient descent to tune the parameters given the training data. Each training item consists of a question $q$, the correct answer $a^*$ and a distractor $d$ (an incorrect answer). A prominent choice is using a shifted hinge loss, designating that the correct answer must have a higher score than the distractor by at least a certain margin $M$, where the score is based on the similarity to the question. \n\n$$ \\mathcal{L} =\\max \\Big\\{ 0, M - \\sigma_{a^*} + \\sigma_{d} \\Big\\} $$ \n\nwhere:\n$$\n\\sigma_{a^*} = s \\Big(f_{\\theta}(q), g_{\\theta}(a^*, q) \\Big) \n$$ \n$$\n\\sigma_{d} = s\\Big( f_{\\theta}(q), g_{\\theta}(d, q) \\Big)\n$$\n\nThe above expression has a zero loss if the correct answer has a score higher than the distractor by at least a margin $M$, and the loss linearly increases in the score difference between the correct answer and the distractor. \n\nAny reasonable neural network design for $f_{\\theta}$ can be used to build a working answer-selection systems using the above approach; however, the network design can have a big impact on the system's accuracy. \n\n\\subsection{Embedding Questions and Answers}\n\nEarlier work examined multiple approaches for embedding questions and answers, including convolutional neural networks, recurrent neural networks (RNNs) (sometimes augmented with an attention mechanism) and hybrid designs \\cite{fengCNN,tan2016}. \n\nAn RNN design ``digests'' the input sequence, one element at a time, changing its internal state at every timestep. The RNN is based on a cell, a parametrized function mapping a current state and an input element to the new state~\\cite{werbos1990backpropagation}. A popular choice for the RNN's cell is the Long Short Term Memory (LSTM) cell~\\cite{hochreiter1997long}.\n\nGiven a question comprised of words $q=(x_1,x_2,\\ldots,x_m)$, we denote the $i$'th output of an LSTM RNN digesting the question as $q_i$; similarly given an answer $a=(y_1,y_2,\\ldots,y_n)$ we denote the $j$'th output of an LSTM RNN digesting the question as $a_j$. \n\nOne simple approach is to have the embeddings of the question and answer be the last LSTM output, i.e. $f_{\\theta}(q) = q_m$ and $f_{\\theta}(a) = a_n$. Note that $q_i,a_i$ are vectors whose dimensionality depends on the dimensionality of the LSTM cell; we denote by $q_{i,j}$ the $j$'th coordinate of the LSTM output at timestep $i$.\n\nAnother alternative is to aggregate the LSTM outputs across the different timesteps by taking their coordinate-wise mean (mean-pooling):\n$$f_{\\theta}(q)_r = \\frac{1}{m} \\sum_{i=1}^m q_{i,r}$$\n\\noindent Alternatively, one may aggregate by taking the or coordinate-wise max (max-pooling):\n$$f_{\\theta}(q)_r = max_{i=1}^m q_{i,r}$$\n\nWe use another simple way of embedding the question and answer, which is based on term-frequency (TF) features. Given a vocabulary of words $V=(w_1,\\ldots,w_v)$, and a text $p$ we denote the TF representation of $p$ as $p^{\\text{tf}} = (d_1,\\ldots,d_v)$ where $d_j=1$ if the word $w_j$ occurs in $p$ and otherwise $d_j=0$. \\footnote{Another alternative is setting $d_j$ to the {\\em number} of times the word $w_j$ appears in $p$. A slightly more complex option is using TF-IDF features~\\cite{ramos2003using} or an alternative hand-crafted feature scheme; however we opt for the simpler TF representation, letting the neural network learn how to use the raw information.}\n\nA simple overall embedding of a text $p$ is $p' = W t(p)$ where $W$ is an $v \\times d$ matrix, and where $d$ determines the final embedding's dimensionality; the weights of $W$ are typically part of the neural network parameters, to be learned during the training of the network. Instead of a single matrix multiplication, one may use the slightly more elaborate alternative of applying a feedforward network, in order to allow for non-linear embeddings.\n\nWe note that a TF representation loses information regarding the {\\em order} of the words in the text, but can provide a good global view of key topics discussed in the text.  \n\nOur main contribution is a new design for the neural network that ranks candidate answers for a given question. Our design uses a TF-based representation of the question and answer, and includes a new attention mechanism which uses this global representation when computing the attention weights (in addition to the local information used in existing approaches). We describe existing attention designs (based on local information) in Section~\\ref{l_sect_loc_attn}, before proceeding to describe our approach in Section~\\ref{l_sect_glob_loc_attn}. \n\n\\subsection{Local Attention}\n\\label{l_sect_loc_attn}\n\nEarly RNN designs were based on applying a deep feedforward network at every timestep, but struggled to cope with longer sequences due to exploding and diminishing gradients \\cite{lstm}. Other recurrent cells such as the LSTM and GRU cells \\cite{lstm,gru} have been proposed as they alleviate this issue; however, even with such cells, tackling large sequences remains hard~\\cite{lstmsSUCK}. Consider using an LSTM to digest a sequence, and taking the final LSTM state to represent the entire sequence; such a design forces the system to represent the entire sequence using a single LSTM state, which is a very narrow channel, making it difficult for the network to represent all the intricacies of a long sequence~\\cite{bahdanau2014ntm}. \n\nAttention mechanisms allow placing varying amounts of emphasis across the entire sequence~\\cite{bahdanau2014ntm}, making it easier to process long sequences; in QA, we can give different weights to different parts of the answer while aggregating the LSTM outputs along the different timesteps: \n$$f_{\\theta}(a) =  \\sum_{i=1}^m \\alpha_i a_{i,r}$$\n\\noindent where $\\alpha_i$ denotes the weight (importance) placed on timestep $i$ and $a_{i,r}$ is the $r$th value of the $i$th embedding vector. \n\nTan et al.~\\cite{tan2016} proposed a very simple attention mechanism for QA, shown in Figure~\\ref{fig:oldModel}:\n$$ m_{a,q}(i) = W_{ad} a_i + W_{qd} f_{\\theta}(q) $$\n$$ \\alpha_i \\propto exp (w_{ms}^T \\tanh(m_{a,q}(i))) $$\n$$ \\hat{a} = \\sum_{i=1}^m \\alpha_i a_i $$ \n\\noindent where $\\alpha_i a(i)$ is the weighted hidden layer, $W_{ad}$ and $W_{qd}$ are matrix parameters to be learned, and $w_{ms}$ is a vector parameter to be learned.\n\n\n\\section{Global-Local Attention}\n\\label{l_sect_glob_loc_attn}\n\nA limitation of the attention mechanism of Tan et al.~\\cite{tan2016} is that it only looks at the the embedded question vector and one candidate answer word embedding at a time. Our proposed attention mechanism adds a {\\em global} view of the candidate, incorporating information from {\\em all} words in the answer. \n\n\\subsection{Creating Global Representations}\n\nOne possibility for constructing a global embedding is an RNN design. However, RNN cells tend to focus on the more recent parts of an examined sequence~\\cite{lstmsSUCK}. We thus opted for using a term-frequency vector representing the entire answer, as shown in Figure~\\ref{fig:newModel}. We denote this representation as:\n$$a^{\\text{tf}} = (d_1,d_2,\\ldots,d_v) $$ \n\\noindent where $d_i$ relates to the i'th word in our chosen vocabulary, and $d_i = 1$ if this word appears in the candidate answer, and $d_i = 0$ otherwise. \n\nConsider a candidate answer $a = (y_1,\\ldots,y_n)$, and let $(a_1,\\ldots,a_n)$ denote its sequence of RNN LSTM outputs, i.e. $a_i$ denotes the $i$'th output of a RNN LSTM processing this sequence (so $a_i$ is a vector whose dimensionality is as the hidden size of the LSTM cell). We refer to $a_i$ as the local-embedding at time $i$. \\footnote{Note that although we call $a_i$ a local embedding, the $i$'th LSTM state does of course take into account other words in the sequence (and not only the $i$'th word). By referring to it as ``local'' we simply mean to say that it is more heavily influenced by the $i$'th word or words close to it in the sequence.}\n\n\\subsection{Combining Local and Global Representations to Determine Attention Weights}\n\nThe goal of an attention mechanism is to construct an overall representation of the candidate answer $a$, which is later compared to the question representation to determine how well the candidate answers the question; this is achieved by obtaining a set of weights $w_1,\\ldots,w_n$ (where $w_i \\in \\mathbb{R}^+$), and constructing the final answer representation as a weighted average of the LSTM outputs, with these weights. \n\nGiven a candidate answer $a$, we compute the attention coefficient $w_i$ for timestep $i$ as follows.  \n\nFirst, we combine the local view (the LSTM output, more heavily influenced by the words around timestep $t$) with the global view (based on TF features of all the words in the answer). We begin by taking linear combinations of the TF features then passing them through a $\\tanh$ nonlinearity (so that the range of each dimension is bounded in $[-1,1]$):\n$$ b^{\\text{tf}} = \\tanh (W_{1} a^{\\text{tf}}) $$\n\\noindent The weights of the matrix $W_{1}$ are model parameters to be learned, and its dimensions are set so as to map the sparse TF vector $a^{\\text{tf}}$ to a dense low dimensional vector (in our implementation $b^{\\text{tf}}$ is a 50 dimensional vector). \n\nSimilarly, we take a linear combination of the different dimensions of the local representation $a_i$ (in this case there is no need for the $tanh$ operation, as the LSTM output is already bounded):\n$$ b_i^{\\text{loc}} = W_{2} a_i $$\n\\noindent where the weights of the $W_{2}$ are model parameters to be learned (and with dimensions set so that $b_i^{\\text{loc}}$ would be a 140 dimensional vector).  \n\nGiven a TF representation of a text $x^{\\text{tf}}$, whose dimensionality is the size of the vocabulary, and an RNN representation of the text $x^{\\text{rnn}}$, with a certain dimentionality $h$, we may wish construct a normalized representation of the text. As the norms of these two parts may differ, simply concatenating these parts may result in a vector dominated by one side. We thus define a joint representation \n$h(x^{\\text{tf}}, x^{\\text{rnn}})$ as follows. \n\nWe normalize each part so as to have a desired ratio of norms $\\frac{\\alpha}{\\beta}$ between the RNN and TF representations; this ratio reflects the relative importance of the RNN and TF embeddings in the combined representation (for instance when settings both $\\alpha, \\beta$ to $1$ both parts would have a unit norm, giving them equal importance): \n$$ c^{\\text{tf}}    = \\frac{\\alpha}{||x^{\\text{tf}}||} \\cdot x^{\\text{tf}} $$\n$$ c^{\\text{rnn}}   = \\frac{\\beta} {||x^{\\text{rnn}}||} \\cdot x^{\\text{rnn}} $$ \n\\noindent We then concatenate the normalized TF and RNN representations to generate the joint representation:\n$$h(x^{\\text{tf}}, x^{\\text{rnn}}) = c^{\\text{tf}} \\| c^{\\text{rnn}} $$\n\\noindent where $\\|$ represents vector concatenation. \n\nWe construct the local attention representation at the $i$'th word of the answer as:\n$$ a_i^{\\text{glob-loc}} = h(b^{\\text{tf}}, b_i^{\\text{loc}}) $$ \n\\\\ using values of $\\alpha=0.5, \\beta=1$.\n\nThe raw attention coefficient of the $i$'th word in the answer is computed by measuring the similarity of a vector representing the question, and a local-global representation of the answer at word $i$. We build these representations, of matching dimensions, by taking the same number of linear combinations from $a_i^{\\text{glob-loc}}$ (the raw global-local representation of the answer at word $i$). Thus the attention weight for the $i$'th word is:\n\n$$\n\\alpha'_i = sim\\Big(  W_{3} a_i^{\\text{glob-loc}}, W_{4} f_{\\theta}(q)  \\Big)\n$$\n\\noindent where $W_2$, $W_3$ are matrices whose weights are parameters to be learned (and whose dimensions are set so that $ W_{3} a_i^{\\text{glob-loc}} $ and $W_{4} f_{\\theta}(q)$ would be vectors of identical dimensionality, 140 in our implementation), and where $sim$ denotes the cosine similarity between vectors:\n$$ sim(u,v) = \\frac{u \\cdot v}{||u|| \\cdot ||v|| } $$ \n\\noindent with the $\\cdot$ symbol in the nominator denoting the dot product between two vectors.\n\n\nFinally, we normalize the attention coefficients with respect to their exponent to obtain the final attention weights, by applying the softmax operator on the raw attention coefficients. We take the raw attention coefficients, $\\alpha' = (\\alpha'_1, \\alpha'_2, \\ldots, \\alpha'_m)$ and define the final attention weights $\\alpha = (\\alpha_1, \\alpha_2, \\ldots, \\alpha_m)$ where $\\alpha_i \\propto exp(\\alpha'_i)$ and \n$\\alpha$ is the result of the softmax operator applied on $\\alpha$:\n$$ \\alpha_i = \\frac{\\exp{(\\alpha'_i)}}{\\sum_{j=1}^{m} \\exp{(\\alpha'_j)}} $$\n\n\\subsection{Building the Final Attention Based Representation}\n\nThe role of the attention weights is building a final representation of a candidate answer; different answers are ranked based on the similarity of their final representation and a final question representation. \nSimilarly to the TF representation of the answer, we denote the TF representation of the question as: $q^{\\text{tf}} = (r_1,r_2,\\ldots,r_v) $, where $r_i$ relates to the i'th word in our chosen vocabulary, and $r_i = 1$ if this word appears in the question, and $r_i = 0$ otherwise. Our final representation of the question is a joining of the TF representation of the question and the mean pooled RNN question representation (somewhat similarly to how we join the TF and RNN representation when determining the attention weights):\n$$ f'_{\\theta}(q) = h(q^{\\text{tf}}, f_{\\theta}(q)) $$ \n\nOur final representation of the answer is also a joining two parts, a TF part $a^{\\text{tf}}$ (as defined earlier) and an attention weighted RNN part $\\hat{a}$. We construct $\\hat{a}$ as the weighted average of the LSTM outputs, where the weights are the attention weights defined above:\n$$ \\hat{a} = \\sum_{i=1}^m \\alpha_i a_i $$ \n\nThe final representation of the answer is thus:\n$$ f'_{\\theta}(a) = h(a^{\\text{tf}}, \\hat{a}) $$ \n\nFigure \\ref{fig:newModel} describes the final architecture of our model, showing how we use a TF-based global embedding both in determining the attention weights and in the overall representation of the questions and answers. The dotted lines in the figures indicate that our model's attention weights depend not only on the local embedding but also on the global embedding. \n\n\\subsection{Tuning Parameters to Minimize the Loss}\n\nThe loss function $\\mathcal{L}$ we use is the shifted hinge loss defined in Section~\\ref{l_sect_prelim}. We compute the score of an answer candidate $a$ as the similarity between its final representation $f'_{\\theta}(a)$ and the final representation of the question $f'_{\\theta}(q)$ \n\\footnote{We use the cosine similarity as our similarity function for the loss, though other similarity functions can also be used.} : \n$$sim(f'_{\\theta}(q),f'_{\\theta}(a))$$ \n\\noindent Given the score of the correct answer candidate $\\sigma_{a^*} = sim(f'_{\\theta}(q),f'_{\\theta}(a))$ and the score of a distractor (incorrect) candidate $d$, $\\sigma_d = sim(f'_{\\theta}(q),f'_{\\theta}(d))$, our loss is \n$\\mathcal{L} =\\max \\Big\\{ 0, M - \\sigma_{a^*} + \\sigma_{d} \\Big\\}$. \n\nThe above loss relates to a single training item (consisting of a single question, its correct answer and an incorrect candidate answer). Training the neural network parameters involves iteratively examining items in a dataset consisting of many training items (each containing a question, its correct answer and a distractor) and modifying the current network parameters. We train our system using variant of stochastic gradient descent (SGD) with the Adam optimization~\\cite{kingma2014adam}.\n\n\\section{Empirical Evaluation}\n\nWe evaluate our proposed neural network design in a similar manner to earlier evaluations of Siamese neural network designs~\\cite{yang2015wikiqa,severyn2015learning}, where a neural network is trained to embed both questions and candidate answers as low dimensional vectors. \n\n\\subsection{Experiment Setup} \\label{sec:setup}\n\n\\begin{figure*}[h!t]\n\\includegraphics[width=\\textwidth]{att3_example5}\n\\includegraphics[width=0.8\\textwidth]{att3_example92}\n\\caption{A visualization of the attention weights for each word in a correct answer to a question. These examples show how the attention mechanism is focusing on relevant parts of the correct answer (although the attention is still quite noisy).}\n\\label{fig-example-attn-weights}\n\\end{figure*}\n\n\\begin{figure}[h!t]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{ModelPerf}\n\\caption{Performance of our system on InsuranceQA for various model sizes $h$ (both the LSTM hidden layer size and embedding size)}\n\\label{fig-size-to-perf}\n\\end{figure}\n\nWe use the InsuranceQA dataset and its  evaluation framework~\\cite{fengCNN}. \nThe InsuranceQA dataset contains question and answer pairs from the insurance domain, with roughly 25,000 unique answers, and is already partitioned into a training set and two test sets, called test 1 and test 2. \n\nThe InsuranceQA dataset has relatively short questions (mean length of 7). However, the answers are typically very long (mean length of 94). \n\nAt test time the system takes as input a question $q$ and a pool of candidate answers $P=(a_1,a_2,\\ldots,a_k)$ and is asked to select the best matching answer $a^*$ to the question from the pool. The InsuranceQA comes with answer pools of size $k=500$, consisting of the correct answers and random distractors chosen from the set of answers to other questions. \n\nState-of-the-art results for InsuranceQA were achieved by Tan et al~\\cite{tan2016}, which also provide a comparison with several baselines: Bag-of-words (with IDF weighted sum of word vectors and cosine similarity based ranking), the Metzler-Bendersky IR model~\\cite{bendersky2010}, and ~\\cite{fengCNN} - the CNN based Architecture-II and Architecture-II with Geometricmean of Euclidean and Sigmoid Dot product (GESD).\n\nWe implemented our model in TensorFlow~\\cite{abadi2016tensorflow} and conducted experiments on our GPU cluster. \n\nWe use the same hidden layer sizes and embedding size as Tan et al.~\\cite{tan2016}: $h=141$ for the bidirectional LSTM size and an embedding size of $e=100$; this allows us to investigate the impact of our proposed attention mechanism. \\footnote{As is the case with many neural networks, increasing the hidden layer size or embedding size can improve the performance on our InsuranceQA models; we compare our performance to the work of Tan et al.~\\cite{tan2016} with the same hidden and embedding sizes; similarly to them we use embeddings pre-trained using Word2Vec~\\cite{mikolov2013} and avoid overfitting by applying early stopping (we also apply Dropout~\\cite{dropout,zaremba2014recurrent}). } \n\n\\begin{table}[h!t]\n\\small\n\\centering\n  \\begin{tabular}{ | l | c | c | }\n    \\hline\n    Model & Test1 & Test2 \\\\\n    \\hline\n    \\hline\n    Bag-of-words & 32.1 & 32.2 \\\\\n    \\hline\n    Metzler-Bendersky & 55.1 & 50.8 \\\\\n    \\hline\n    Arch-II~\\cite{fengCNN} & 62.8 & 59.2 \\\\\n    \\hline\n    Arch-II GSED~\\cite{fengCNN} & 65.3 &   61.0 \\\\\n    \\hline \n    Attention LSTM~\\cite{tan2016} & 69.0 & 64.8 \\\\ \n    \\hline\n    \\hline\n    TF-LSTM Concatenation & 62.1 & {61.5} \\\\\n   \n    \\hline    \n    Local-Global Attention & {\\bf 70.1} & {\\bf 67.4} \\\\\n   \n    \\hline    \n  \\end{tabular}\n  \\vspace{0.2cm}\n\\caption{Performance of various models on InsuranceQA}\n\\label{table:perf_insqa}\n\\end{table}\n\n\\subsection{Results}\n\nTable~\\ref{table:perf_insqa} presents the results of our model and the various baselines for InsuranceQA. The performance metric used here is P@1, the proportion of instances where a correct answer was ranked higher than all other distractors in the pool. The table shows that our model outperforms the previous baselines.  \n\nWe have also examined the performance of our model as a function of its size (determining the system's runtime and memory consumption). We used different values $h \\in \\{10,20,30,40,50\\}$ for both the size of the LSTM's hidden layer size and embedding size, and examined the performance of the resulting QA system on InsuranceQA. Our results are given in Figure~\\ref{fig-size-to-perf}, which shows both the P@1 metric and the mean reciprocal rank (MRR)~\\cite{craswell2009mean,chapelle2009expected} \\footnote{The MRR metric assigns the model partial credit even in cases where the highest ranking candidate is an incorrect answer, with the score depending on the highest rank of a correct answer. }\n\nFigure~\\ref{fig-size-to-perf} shows that performance improves as the model gets larger, but the returns on extending the model size quickly diminish. Interestingly, even relatively small models achieve a reasonable question answering performance. \n\nTo show our attention mechanism is necessary to achieve good performance, we also construct a model that simply concatenates the output of the feedforward network (on TF features) and the output of the bidirectional LSTM, called TF-LSTM concatenation. While this model does make use of TF-based features in addition to the LSTM state of the RNN, it does not use an attention mechanism to allow it to focus on the more relevant parts of the text. As the table shows, the performance of the TF-LSTM model is significantly lower than that of our model with the global-local attention mechanism. This indicates that the improved performance stems from the model's improved ability to focus on the relevant parts of the answer (and not simply from having a larger capacity and including TF-features).\n\nFinally, we examine the the attention model's weights to evaluate it qualitatively. Figure~\\ref{fig-example-attn-weights} visualizes the weights for two question-answer pairs, where the color intensity reflects the relative weight placed on the word (the $\\alpha_i$ coefficients discussed earlier). The figure shows that our attention model can focus on the parts of the candidate answer that are most relevant for the given question. \n\n\n\n\\section{Conclusion}\n\nWe proposed a new neural design for answer selection, using\nan augmented attention mechanism, which combines both local and global information when determining the attention weight to place at a given timestep. Our analysis shows that our design outperforms earlier designs based on a simpler attention mechanism which only considers the local view. \n\nSeveral questions remain open for future research. First, the TF-based global view of our design was extremely simple; could a more elaborate design, possibly using convolutional neural networks, achieve better performance? \n\nSecond, our attention mechanism joins the local and global information in a very simple manner, by normalizing each vector and concatenating the normalized vectors. Could a more sophisticated joining of this information, perhaps allowing for more interaction between the parts, help further improve the performance of our mechanism?\n\nFinally, can the underlying principles of our global-local attention design improve the performance of other systems, such as machine translation or image processing systems? \n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe duality between symplectic and orthogonal groups has a long standing history,\nand has been noted in physics literature in various settings, see e.g.\n \\cite{Jungling-Oppermann-80,Mkrtchyan-81,Wegner-83,Witten-98}. Informally, the duality asserts that\n averages such as moments or partition functions\nfor the symplectic case of ``dimension'' $N$, can be derived from the respective\nformulas  for the orthogonal case of dimension $N$ by inserting $-N$\n into these expressions and by simple scaling.\nThe detailed study of  the moments of one-matrix Wishart ensembles, with duality explicitly noted,\nappears in \\cite{Hanlon-Stanley-Stembridge-92}, see \\cite[Corollary 4.2]{Hanlon-Stanley-Stembridge-92}.\nThe duality for one matrix Gaussian Symplectic Ensemble was noted\nby Mulase and Waldron \\cite{Mulase-Waldron-03} who introduced M\\\"obius graphs to write the\nexpansion for traces of powers of\nGOE\/GUE\/GSE expansions in a unified way. The duality appears also in\n \\cite[Theorem 6]{Ledoux-07} as a by-product of differential equations\n for the generating functions of moments.\nRef. \\cite{Goulden-Jackson-96,Goulden-Jackson-96b,Goulden-Jackson-97}\nanalyze the related ``genus series\" over locally orientable surfaces.\n\n\n\nThe purpose of this paper is to prove that the duality between moments of the Gaussian Symplectic Ensemble\nand the Gaussian Orthogonal Ensemble, and between real Wishart and quaternionic Wishart ensembles extends\nto several independent\nmatrices. Our technique consists of  elementary combinatorics; our proofs\n differ from \\cite{Mulase-Waldron-03} in the one matrix case,\n and provide a more geometric interpretation for the duality; in the one-matrix Wishart case,\n our proof completes the combinatorial approach initiated in\n  \\cite[Section 6]{Hanlon-Stanley-Stembridge-92}.\nThe technique limits the scope of our results to moments, but the\nrelations between moments suggest similar relations between other analytic objects,\nsuch as partition functions, see \\cite{Mulase-Waldron-03}, \\cite{Kodama-Pierce}.   The asymptotic expansion of the partition function and analytic description of\n the coefficients of this expansion\n  for $\\beta=2$ case appear in \\cite{Ercolani-McLaughlin-03,Guionnet-Maurel-Segala-0503064,Maurel-Segala-0608192}.\n\nThe paper is organized as follows. In Section \\ref{Sect1} we review basic properties of quaternionic\nGaussian random variables. In Section \\ref{Sect2} we introduce M\\\"obius graphs;  Theorems \\ref{T quaternion moments} and \\ref{thm2.1}\n give formulae\nfor the expected values of products of quaternionic Gaussian random\nvariables in terms of the Euler characteristics of sub-families of\nM\\\"obius graphs or of bipartite M\\\"obius graphs.\nIn Section \\ref{duality} we apply the formulae to the quaternionic Wigner and  Wishart families.\n\nIn this paper, we do not address the question of whether the duality\n can be extended to more general functions, or to more general\n $\\beta$-Hermite and $\\beta$-Laguerre ensembles introduced in \\cite{Dumitru-Edelman-06}.\n\n\n\\section{ Moments of quaternion-valued Gaussian random variables}\\label{Sect1}\n\n\\subsection{Quaternion Gaussian law}\nRecall that a quaternion $q\\in \\mathbb{H}$ can be represented as $q=x_0+i x_1+j x_2+ k x_3$\nwith\n$i^2=j^2=k^2=ijk=-1$ and with real coefficients $x_0,\\dots,x_3$.\nThe conjugate quaternion is $\\overline{q}=x_0-i x_1-j x_2- k x_3$,\nso $|q|^2:=q\\overline{q}\\geq 0$. Quaternions with $x_1=x_2=x_3=0$ are usually identified with real numbers;\nthe real part of a quaternion is  $\\Re(q)=(q+\\bar{q})\/2$.\n\nIt is well known that quaternions can be identified with the set of certain $2\\times2$ complex matrices:\n\\begin{equation}\n  \\label{H2C}\n \\mathbb{H}\\ni x_0+ix_1+jx_2+kx_3\\sim \\left[\\begin{matrix}\n  x_0+ix_1 & x_2+i x_3 \\\\\n  -x_2+ix_3&x_0-ix_1\n\\end{matrix}\\right]\\in \\mathcal{M}_{2\\times 2}(\\mathbb{C}),\n\\end{equation}\nwhere on the right hand side $i$ is the usual imaginary unit of $\\mathbb{C}$.\nNote that since $\\Re(q)$ is twice the trace of the matrix\nrepresentation in \\eqref{H2C}, this implies the cyclic property\n\\begin{equation}\n  \\label{tmp**}\\Re(q_1q_2\\dots q_n)=\\Re(q_2q_3\\dots q_nq_1).\n\\end{equation}\n\n\nThe (standard) quaternion Gaussian random variable is an $\\mathbb{H}$-valued random variable\nwhich can be represented as\n\\begin{equation}\n  \\label{HH}\n  Z=\\xi_0+i \\xi_1+ j\\xi_2+k\\xi_3\n\\end{equation} with independent real\nnormal $N(0,1)$ random variables $\\xi_0,\\xi_1,\\xi_2,\\xi_3$.\nDue to symmetry of the centered normal laws on $\\mathbb{R}$,\nthe law of $(Z,\\overline{Z})$ is the same as the law of $(\\overline{Z},Z)$.\nA  calculation shows  that if $Z$ is quaternion Gaussian then\nfor fixed $q_1, q_2 \\in\\mathbb{H}$,\n$$\n\\mathbb{E}(Z q_1 Zq_2)=\\mathbb{E}(Z^2)\\bar{q}_1q_2,\\;\n\\mathbb{E}(Z q_1 \\overline{Z}q_2)=\\mathbb{E}(Z\\bar{Z})\\Re(q_1)q_2\\,.\n$$\nFor future reference, we insert explicitly the moments:\n\\begin{eqnarray}\n \\label{q1}\n\\mathbb{E}(Z q_1 Zq_2)&=&-2\\bar{q}_1q_2,\\\\\n\\label{q2}\n\\mathbb{E}(Z q_1 \\overline{Z}q_2)&=& 2 (q_1+\\bar{q}_1)q_2.\n\\end{eqnarray}\nBy linearity, these formulas imply\n\\begin{eqnarray}\n\\label{q3}\n\\mathbb{E}( \\Re(Z q_1) \\Re( \\overline{Z} q_2) ) &=& \\Re( q_1 q_2 ) ,\\\\\n\\label{q4}\n\\mathbb{E}( \\Re(Z q_1) \\Re(Z q_2 ) ) &=& \\Re( \\bar{q}_1 q_2 ).\n\\end{eqnarray}\n\n\\subsection{Moments}\nThe following is  known as Wick's theorem  \\cite{Wick-50}.\n\\begin{theoremA}[Isserlis \\cite{Isserlis-1918}\n\\label{WickTHMR} If $(X_1,\\dots,X_{2n})$ is\na $\\mathbb{R}^{2n}$-valued Gaussian random vector with mean zero, then\n\\begin{equation}\\label{R-Wick}\n \n  E(X_1X_2\\dots X_{2n})=\\sum_{V}\\prod_{\\{j,k\\}\\in V} E(X_jX_k),\n\\end{equation}\nwhere the sum is taken over all pair partitions  $V$ of\n$\\{1,2,\\dots,2n\\}$, i.e.,\npartitions into two-element sets, so each $V$ has the form\n$$V=\\left\\{\\{j_1,k_1\\},\\{j_2,k_2\\},\\dots,\\{j_n, k_n\\}\\right\\}.$$\n\\end{theoremA}\n\n\n Theorem \\ref{WickTHMR} is a consequence of the moments-cumulants relation \\cite{Leonov-Shirjaev-59};\n  the connection is best visible in the partition formulation of \\cite{Speed-83}. For another proof,\n  see \\cite[page 12]{Janson-97}.\n\n  Our first goal is to extend this formula to certain quaternion Gaussian random variables.\nThe  general multivariate quaternion Gaussian law is discussed in\n\\cite{Vakhania-99}. Here we will only consider a special setting of\nsequences that are drawn with repetition from a sequence of\nindependent standard Gaussian quaternion random variables. In\nsection \\ref{duality}\n we apply this result to a multi-matrix version of the\nduality between GOE and GSE ensembles of random matrices.\n\n\n\nIn view of the Wick formula \\eqref{R-Wick} for real-valued jointly Gaussian random\nvariables, formulas \\eqref{q1} and \\eqref{q2} allow us to compute\nmoments of certain products of quaternion Gaussian random variables.\nSuppose  the $n$-tuple $(X_1,X_2,\\dots,X_{n})$\nconsists of random variables taken, possibly with repetition, from\nthe set\n $$\\{Z_1,\\bar{Z}_1,Z_2,\\bar{Z_2},\\dots\\},$$\n where\n$Z_1,Z_2,\\dots$ are independent quaternion Gaussian random variables.\n Consider an auxiliary family of independent pairs $\\{(Y_{j}^{(r)},Y_{k}^{(r)}): r=1,2,\\dots\\}$ which have the same laws as\n $(X_j,X_k)$, $1\\leq j,k\\leq n$ and are independent for different $r$.\n  Then the Wick formula for real-valued Gaussian variables implies $\\mathbb{E}(X_1X_2\\dots X_{n})=0$ for odd $n$, and\n   \\begin{equation}\\label{Wick0}\n\\mathbb{E}(X_1X_2\\dots X_{n})=\\sum_{f}\\mathbb{E}(Y_1^{(f(1))}Y_2^{(f(2))}\\dots Y_{n}^{(f(n))}),\n \\end{equation}\n where the sum is over the pair partitions $V$  that appear under the sum in Theorem \\ref{WickTHMR}, each represented by the level sets of a two-to-one valued function\n $f:\\{1,\\dots,n\\}\\to\\{1,\\dots,m\\}$ for $n=2m$.  (Thus the sum is over classes of equivalence of $f$, each of $m!$ representatives contributing the same value.)\n\n\n For example, if $Z$ is quaternion Gaussian then\napplying \\eqref{Wick0} with $f_1$ that is constant, say $1$,  on $\\{1,2\\}$,\n  $f_2$ that is constant  on $\\{1,3\\}$, and $f_3$ that is constant   on $\\{1,4\\}$\nwe get\n$$\n\\mathbb{E}(Z^4)=\\mathbb{E}(Z^2) \\left(\\mathbb{E}(Z^2) + \\mathbb{E}(\\bar{Z}Z)+ \\mathbb{E}(\\bar{Z}^2)\\right)=0.\n$$\n\n\nFormulas \\eqref{q1} and \\eqref{q2} then show that the Wick reduction step takes the\nfollowing form.\n\\begin{equation}\\label{Wick1}\n  \\mathbb{E}(X_1X_2\\dots X_n)=\\sum_{j=2}^n \\mathbb{E}(X_1X_j)\\mathbb{E}(U_j X_{j+1}\\dots X_n),\n\\end{equation}\nwhere\n$$\nU_j=\\begin{cases}\n  \\Re(X_2\\dots X_{j-1}) & \\mbox{ if $X_j=\\bar{X}_1$ }\\\\\n  \\bar{X}_{j-1}\\dots \\bar{X}_2 &\\mbox{ if $X_j={X}_1$}\\\\\n  0 &\\mbox{ otherwise} \\,.\n\\end{cases}\n$$\nThis implies that one can do inductively the calculations, but due to noncommutativity\narriving at explicit answers may still require significant work.\n\n\n\nFormula \\eqref{Wick1} implies that $\\mathbb{E}(X_1X_2\\dots X_n)$ is real, so on the left hand side of \\eqref{Wick1} we can write\n$\\mathbb{E}(\\Re(X_1X_2\\dots X_n))$; this form of the formula will be associated with one-vertex M\\\"obius graphs.\n\nFurthermore, we have a Wick reduction which will correspond to the multiple vertex M\\\"obius graphs:\n\\begin{multline}\\label{Wick2}\n\\mathbb{E}( \\Re( X_1 ) \\Re( X_2 X_3 \\dots X_n ) ) \\\\=\n\\sum_{j=2}^n \\mathbb{E}(\\Re(X_1)\\Re(X_j))\\mathbb{E}( \\Re(X_2 \\dots X_{j-1} X_{j+1} \\dots X_n  )).\n\\end{multline}\n(This is just a consequence of Theorem \\ref{WickTHMR}).\n\n\n\n\nIn the next section we will show that formulae (\\ref{Wick1}) and\n(\\ref{Wick2}) give a method of computing the expected values of\nquaternionic Gaussian random variables by the enumeration of M\\\"obius\ngraphs partitioned by their Euler characteristic.  This is analogous\nto similar results for complex Gaussian random variables and ribbon\ngraphs, and for real Gaussian random variables and M\\\"obius graphs.\nIn Section \\ref{duality} we will show  that this\nresult implies the duality of the GOE and GSE\nensembles of Wigner random matrices and the duality of real and\nquaternionic Wishart random matrices.\n\n\n\n\\section{M\\\"obius graphs and quaternionic Gaussian moments}\\label{Sect2}\n\nIn this section we introduce M\\\"obius graphs and then give formulae\nfor the expected values of products of quaternionic Gaussian random\nvariables in terms of the Euler characteristics of sub-families of\nM\\\"obius graphs.  This is an analogue of the method of t'Hooft\n\\cite{Bessis-Itzykson-Zuber-80,Goulden-Harer-Jackson-01, Goulden-Jackson-97,harer-zagier-86, Jackson-94, tHooft}.  M\\\"obius graphs have also been used to give combinatoric\ninterpretations of the expected values of traces of Gaussian\northogonal ensemble of random matrices and of Gaussian symplectic ensembles, see the articles\n\\cite{Goulden-Jackson-97} and \\cite{Mulase-Waldron-03}.  The  connection between M\\\"obius graphs and\nquaternionic Gaussian random variables is at the center of the work of Mulase\nand Waldron \\cite{Mulase-Waldron-03}.\n\n\n\n\n\\subsection{M\\\"obius graphs}\nM\\\"obius graphs are ribbon graphs where the edges (ribbons) are\nallowed to twist, that is they either preserve or reverse the local\norientations of the vertices.  As the convention is that the ribbons\nin \\textit{ribbon graphs} are not twisted we follow \\cite{Mulase-Waldron-03} and call the unoriented\nvariety M\\\"obius graphs. The vertices of a M\\\"obius graph are\nrepresented as disks together with a local orientation; the edges are represented as\nribbons, which preserve or reverse the local orientations of the\nvertices connected by that edge. Next we identify the collection of\ndisjoint cycles of the sides of the ribbons found by following the\nsides of the ribbon and obeying the local orientations at each\nvertex.  We then attach disks to each of these cycles by gluing the\nboundaries of the disks to the sides of the ribbons in the cycle.\nThese disks are called the faces of the M\\\"obius graph, and the\nresulting surface we find is the surface of maximal\nEuler\ncharacteristic on which the M\\\"obius graph may be drawn so that edges do not cross.\n\n\nDenote by $v(\\Gamma)$, $e(\\Gamma)$, and $f(\\Gamma)$ the number of vertices, edges, and faces of $\\Gamma$.\nWe say that the Euler characteristic of $\\Gamma$ is\n$$\n\\chi(\\Gamma)=v(\\Gamma)-e(\\Gamma)+f(\\Gamma),\n$$\nfor connected $\\Gamma$, this is also the maximal Euler characteristic of a connected surface into which\n$\\Gamma$ is embedded.\nFor example, in Fig. \\ref{F1}, the Euler characteristics are\n$\\chi_1=1-1+2=2$ and $\\chi_2=1-1+1=1$.  The two graphs may be embedded\ninto the sphere or projective sphere respectively.\n\n\nIf  $\\Gamma$ decomposes into connected components $\\Gamma_1$, $\\Gamma_2$, then\n$\\chi(\\Gamma)=\\chi(\\Gamma_1)+\\chi(\\Gamma_2)$.\n\nThroughout the paper our M\\\"obius graphs will have the following\nlabels attached to them:  the vertices are labeled to make them\ndistinct, in addition the edges emanating from each vertex are also\nlabeled so that rotating any vertex produces a distinct graph. These\nlabels may be removed if one wishes by rescaling all of our\nquantities by the number of automorphisms the unlabeled graph would\nhave.\n\n\\subsection{Quaternion version of Wick's theorem}\n\n\\tolerance=2000\nSuppose  the $2n$-tuple $$(X_1,X_2,\\dots,X_{2n})$$\nconsists of random variables taken, possibly with repetition, from\nthe set $\\{Z_1,\\bar{Z_1},Z_2,\\bar{Z_2},\\dots\\}$ where\n$Z_1,Z_2,\\dots$ are independent quaternionic Gaussian.\nFix a sequence $j_1,j_2,\\dots,j_m$ of natural numbers such that $j_1+\\dots+j_m=2n$.\n\nConsider the family $\\mathcal{M}=\\mathcal{M}_{j_1,\\dots,j_m}(X_1,X_2,\\dots,X_{2n})$, possibly empty, of M\\\"obius graphs with $m$ vertices of degrees $j_1,j_2,\\dots,j_m$ with edges labeled by $X_1,X_2,\\dots,X_{2n}$,\nwhose regular  edges  correspond to pairs $X_i=\\bar{X}_j$ and flipped edges  correspond to pairs $X_i=X_j$. No edges of  $\\Gamma\\in \\mathcal{M}$ can join random variables $X_i,X_j$  that are independent.\n\\tolerance=1000\n\n\\begin{theorem}\n  \\label{T quaternion moments}\nLet $\\left\\{ X_1, X_2, \\dots, X_{2n} \\right\\}$ be chosen, possibly\nwith repetition, from the set $\\{ Z_1, \\bar{Z_1}, Z_2, \\bar{Z_2},\n\\dots \\}$ where $Z_j$ are independent quaternionic Gaussian random\nvariables, then\n\\begin{multline}\n\\mathbb{E}\\big(\\Re(X_1X_2 \\dots X_{j_1})\\Re(X_{j_1+1}^{}\\dots X_{j_1+j_2}^{})\\times\\dots \\\\\\times\\Re(X_{j_1+j_2+\\ldots+j_{m-1}+1}^{}\\dots\n  X_{2n}^{})\\big)=4^{n-m}\\sum_{\\Gamma\\in \\mathcal{M}} (-2)^{\\chi(\\Gamma)}.\n\\end{multline}\n(The right hand side is interpreted as $0$ when $\\mathcal{M}=\\emptyset$.)\n\\end{theorem}\n\n\\begin{remark}\n  We would like to emphasize that in computing the Euler characteristic one must first break the graph into\n  connected components. For example, if $j_1=\\dots=j_m=1$ so that $m=2n$ is even,\nand $X_1, X_2, \\dots, X_{2n}$ are $n$ independent pairs, as the real\nparts are commutative we may assume that $X_{2k} = X_{2k-1}$, and\nthe moment is\n$$ 1=\\mathbb{E}( \\Re (X_1) \\Re (X_2) \\dots \\Re (X_{2n}) )  = 4^{-n}  (-2)^{\\chi(\\Gamma)}.  $$\nWe see that graphically $\\Gamma$\n is a collection of $2n$ degree one vertices connected together forming $n$\n dipoles (an edge with a vertex on either end).\n  Hence there are $n$ connected components each of Euler characteristic $2$,\n  therefore the total Euler characteristic is $\\chi = 2 n $  giving\n$$ 4^{-n}  (-2)^\\chi= 4^{-n} 4^{n} = 1. $$\n\\end{remark}\n\n\\begin{proof}[Proof of Theorem \\ref{T quaternion moments}]\nIn view of \\eqref{Wick0} and \\eqref{Wick1}, it suffices to show that  if $X_1,\\dots,X_{2n}$ consists of $n$ independent pairs, and each pair is either of the form $(X,X)$ or $(X,\\bar{X})$, then\n\\begin{multline}\n\\label{star2}\n  \\mathbb{E}\\big(\\Re(X_1X_2 \\dots X_{j_1})\\Re(X_{j_1+1}^{}\\dots X_{j_1+j_2}^{})\\times\\dots \\\\\\times\\Re(X_{j_1+j_2+ \\dots+\n  j_{m-1}+1}^{}\\dots\n  X_{2n}^{})\\big)=  4^{n-m}(-2)^{\\chi(\\Gamma)}\\;,\n\\end{multline}\nwhere $\\Gamma$ is the M\\\"obius graph that describes the pairings of the sequence.\n\nFirst we check the two M\\\"obius graphs for\n$n=1$, $m=1$:\n\\[   \\mathbb{E}( \\Re( X \\bar{X}) ) = (-2)^2  \\,, \\quad \\mbox{and}\\quad \\mathbb{E}( \\Re(X X))\n= (-2)^1   \\,.\\]\nOne checks that these correspond to the M\\\"obius graphs in Figure \\ref{F1},\nwhich gives a sphere ($\\chi=2$)  and projective sphere ($\\chi=1$) respectively.\n\\begin{figure}[htb]\n \\includegraphics[width=4in]{fig1}\n \\caption{The two possible M\\\"obius graphs with a single degree 2 vertex. The left hand one is a ribbon which is untwisted and the graph embeds into a copy of the Riemann sphere, while the\n right hand one is a ribbon which is twisted and the graph embeds into a copy of the projective sphere. \n \\label{F1}}\n\\end{figure}\n\n\nWe now proceed with the induction step.\n One notes that by independence of the pairs at different edges, the\nleft hand side of \\eqref{star2} factors into the product corresponding to\nconnected components of $\\Gamma$. It is therefore enough to consider\nconnected $\\Gamma$. \\label{tmp1}\n\nIf $\\Gamma$ has two vertices that are joined by an edge, we can use cyclicity\nof $\\Re$  to move the variables that label the edge to the first positions in\ntheir cycles, say $X_1$ and $X_{j_1+1}$ and use \\eqref{q3} or \\eqref{q4} to\neliminate this pair from the product. The use of relation (\\ref{q3})\nis just that of gluing the two vertices together removing the edge $x$\nwhich is labeled by the two appearances of $Z$.  Relation (\\ref{q4}) glues\ntogether the two vertices, removing\nthe edge $x$, and the reversal of orientation across the edge is\ngiven by the conjugate (see Figure \\ref{F4}).\nThese geometric operations reduce $n$ and $m$ by one without changing the\nEuler characteristic:\nthe number of edges and the number of vertices are reduced by 1; the faces are\npreserved -- in the case of edge flip\nin Fig. \\ref{F4}, the edges of the face from which we remove the edge, after\nreduction follow the same order.\n\n\n\n Therefore we will only need to prove the\nresult for the single vertex case of the induction step.\n\n\\begin{figure}[htb]\n\\includegraphics[height=4in]{fig2\n  \\caption{A M\\\"obius graph with two vertices connected by a ribbon may be reduced to a M\\\"obius graph with one less vertex and one less edge.\nIn these graphs the ``$\\dots$\" are to mean that there are arbitrary\nnumbers of other edges at the vertex, and the edges drawn are\nconnected to other vertices.\n The top graph is an example of this reduction when the connecting ribbon is untwisted, in this case the two vertices are glued together with no other changes in the ribbons.  The bottom graph is an example of this reduction when the connecting ribbon is twisted, in this case the two vertices are glued together and the order and orientation (twisted or untwisted) of the ribbons on one side are reversed.\n\\label{F4}}\n\\end{figure}\n\n\nWe wish to show that\n\\begin{equation} \\label{star3}\n \\mathbb{E}( X_1 X_2 X_3 X_4 \\dots X_{2n}) =\n(-2)^{\\chi({\\Gamma})}  4^{n-1} \\,,\n\\end{equation}\nwhere $\\Gamma$ is a one vertex M\\\"obius graph  with arrows (half edges)\n labeled by\n$X_k$.\nWe will do this by induction, there are two cases:\n\\begin{enumerate}\n\n\\item[\\bf Case 1:] $X_1 = \\bar{X}_j$ for $1< j \\leq 2n $,\n\\begin{align} \\nonumber\n\\mathbb{E}( X_1 X_2 \\dots X_{j-1} \\bar{X}_1 X_{j+1} \\dots X_{2n}) &=\n\\mathbb{E}( X_1 \\bar{X}_1 ) \\mathbb{E}( \\Re( X_2 \\dots X_{j-1} ) \\Re( X_{j+1} \\dots X_{2n}) ) \\\\\n\\label{oriented_reduction}\n&= 4 \\mathbb{E}( \\Re( X_2 \\dots X_{j-1}) \\Re( X_{j+1} \\dots X_{2n}) ) \\,.\n\\end{align}\nThis corresponds to the reduction of the M\\\"obius graph pictured\nin Figure \\ref{orient-reduct}, which splits the single vertex\ninto two vertices.  The Euler characteristic becomes $\\chi_2 = 2 - (n-1) + f_1\n= \\chi_1 +2 $.\nBy the induction assumption we find\n\\begin{equation*}\n\\mathbb{E}( X_1 \\dots X_{2n}) = 4 \\left[ 4^{(n-1) - 2} (-2)^{\\chi_2} \\right]\n= 4^{n-2} (-2)^{\\chi_1 +2} = 4^{n-1} (-2)^{\\chi_1} \\,.\n\\end{equation*}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{orient-reduct}\n\\end{center}\n\\caption{ Here we have an untwisted ribbon of the M\\\"obius graph\nreturning to the same vertex, this edge is removed in our reduction\nprocedure giving us a M\\\"obius graph with one more vertex and one\nless edge. \\label{orient-reduct}}\n\\end{figure}\n\n\\item[\\bf Case 2:] $X_1 = X_j$ for $1 < j \\leq 2n$,\n\\begin{align}\\nonumber\n\\mathbb{E}( X_1 X_2 \\dots X_{j-1} X_1 X_{j+1} \\dots X_{2n}) &=\n\\mathbb{E}( X_1 X_1 ) \\mathbb{E}(  \\bar{X}_{j-1} \\dots \\bar{X}_2  X_{j+1} \\dots X_{2n} )\n\\\\ \\label{unoriented-reduction}\n&= (-2) \\mathbb{E}( \\bar{X}_{j-1} \\dots \\bar{X}_2 X_{j+1} \\dots X_{2n} ) \\,.\n\\end{align}\nThis corresponds to the reduction of the M\\\"obius graph pictured\nin Figure \\ref{unorient-reduct}, which keeps the single vertex\nand flips the order and orientation of the edges between $X_1$ and $X_j$.\nThe Euler characteristic becomes $\\chi_2 = 1 - (n-1) + f_1 = \\chi_1 + 1$.\nBy the induction assumption we find\n\\begin{equation*}\n\\mathbb{E}( X_1 \\dots X_{2n}) = (-2) \\left[ 4^{(n-1) -1} (-2)^{\\chi_2} \\right]\n= (-2)^{-1} 4^{n-1} (-2)^{\\chi_1 + 1} = 4^{n-1} (-2)^{\\chi_1}\\,.\n\\end{equation*}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{unorient-reduct}\n\\end{center}\n\\caption{ Here we have a twisted ribbon of the M\\\"obius graph\nreturning to the same vertex, this edge is removed in our reduction\nprocedure giving us a M\\\"obius graph with one less edge and giving a\nreverse in both the order and orientation (twisted or untwisted) of\nthe ribbons on one side of the removed ribbon.\n\\label{unorient-reduct}}\n\\end{figure}\n\n\n\n\\end{enumerate}\n\n\nNote that taking away an oriented ribbon creates a new vertex. The\nremaining graph might still be connected, or it may split into two\ncomponents. If taking away a loop makes the graph disconnected, then\nthe counts of changes to edges and vertices are still the same. But\nthe faces need to be counted as follows: the inner face of the\nremoved edge becomes the outside face of one component, and the\nouter face at the removed edge becomes the outer face of the other\ncomponent. Thus the counting of faces is not affected by\nwhether the graph is connected.\n\nWith these two cases checked, by the induction hypothesis, the\nproof is completed.\n\n\\end{proof}\n\n\n\n\\subsection{Bipartite M\\\"obius graphs and quaternionic Gaussian moments} \\label{Sect bipartite}\nTo deal with quaternionic Wishart random matrices, we need to consider a special subclass of quaternionic\nGaussian variables from Theorem \\ref{T quaternion moments}.\nSuppose  the $2n$-tuple $(X_{\\pm 1},X_{\\pm 2},\\dots,X_{\\pm n})$\nconsists of $n$ pairs of random variables taken with repetition, from\nthe set $\\{Z_1,Z_2,\\dots,Z_n\\}$ of independent quaternionic Gaussian random variables. Note that in contrast to the setup for Theorem \\ref{T quaternion moments}, here all $Z$'s are without conjugation.\nFix a sequence $j_1,j_2,\\dots,j_m$ of natural numbers such that $j_1+\\dots+j_m=n$.\nTheorem \\ref{T quaternion moments} then says that\n\\begin{align*}\n \\mathbb{E}\\big( \\Re( \\bar{X}_{-1} X_1 \\cdots \\bar{X}_{-j_1} X_{j_1})\n\\times \\Re( \\bar{X}_{-j_1-1} X_{j_1 +1} \\cdots \\bar{X}_{-j_1-j_2} X_{j_1+j_2} )\n\\times \\cdots\n\\\\ \\dots \\times\n\\Re( \\bar{X}_{-j_1-j_2-\\dots -j_{m-1}-1} \\cdots \\bar{X}_{-n} X_{n} )\\big)\n\\\\ = 4^{n-m} (-2)^{\\chi(\\Gamma)},\n\\end{align*}\nwhere $\\Gamma$ is the  M\\\"obius graph with $m$ vertices and with edges labeled by $X_{\\pm 1},\\dots,X_{\\pm n}$ that describes the\npairings between the variables under the expectation. Our goal is to show that the same formula holds true for another graph, a bipartite M\\\"obius graph whose edges are labeled by $n$ pairs $(X_{-j},X_j)$, $1\\leq j \\leq n$.\n\nThe bipartite M\\\"obius  graph has two types of vertices: black vertices and white (or later, colored) vertices with ribbons that can only connect a black vertex to a white vertex. (As previously, the ribbons may carry a ``flip\" of orientation which we represent graphically as a twist.) To define this graph, we need to introduce three pair partitions on the set $\\{\\pm 1,\\dots,\\pm n\\}$.\nThe first partition, $\\delta$, pairs $j$ with $-j$. The second partition, $\\sigma$, describes the placement of $\\Re$: its pairs are\n\\begin{align*}\n\\{1,-2\\},\\{2,-3\\},\\dots,\\{j_1,-1\\},\\\\\n\\{j_1+1, -(j_1+2)\\},\\dots, \\{j_1+j_2,-(j_1+1)\\},\\\\\n\\vdots\\\\\n\\{1+\\sum_{k=1}^{m-1}j_k,-(2+\\sum_{k=1}^{m-1}j_k)\\},\\dots, \\{n,-(1+\\sum_{k=1}^{m-1}j_k)\\}.\n\\end{align*}\nThe third partition, $\\gamma$, describes the choices of pairs from $Z_1,\\dots,Z_n$. Thus\n$\n\\{j,k\\}\\in\\gamma\n$\nif $X_j=X_k$ when $jk>0$ or $X_j=\\bar{X}_k$ if $jk<0$.\n\nWe will also represent these pair partitions as graphs with vertices arranged in\ntwo rows, and\nwith the edges drawn between the vertices in each pair of  a partition.\nThus\n\\begin{equation}\n  \\label{eq:delta}\n  \\delta=\\begin{matrix}\n  \\xymatrix  @-1pc{\n {^1_ \\bullet} \\ar@{-}[d]& {^2_\\bullet}\n  \\ar@{-}[d]& \\dots & {^n_\\bullet} \\ar@{-}[d]\\\\\n {^{\\hspace{1.5mm}\\bullet}_{-1}} &{^{\\hspace{1.5mm}\\bullet}_{-2}} &\\dots &\n{^{\\hspace{1.5mm}\\bullet}_{-n}} \\\\\n\n}\n\\end{matrix}\n\\end{equation}\nand\n{\\small\n$$\n\\sigma=\\begin{matrix}\n \\xymatrix  @-1pc{\n {^1_ \\bullet} \\ar@{-}[dr]& {^2_\\bullet} \\ar@{-}[dr]& {^3_\\bullet}\n & \\dots & {^{j_1}_\\bullet}\\ar@{-}[dllll] & {^{j_1+1}_{\\hspace{2.5mm}\\bullet}}\\ar@{-}[dr] & {^{j_1+2}_{\\hspace{3mm}\\bullet}}&\\dots & {^{j_1+j_2}_{\\hspace{3.7mm}\\bullet}}\\ar@{-}[dlll]&\n  {^{j_1+j_2+1}_{\\hspace{5mm}\\bullet}}&\\dots \\\\\n {^{\\hspace{1.5mm}\\bullet}_{-1}} &{^{\\hspace{1.5mm}\\bullet}_{-2}} &{^{\\hspace{1.5mm}\\bullet}_{-3}}&\\dots &\n{^{}_\\bullet}& {^{}_\\bullet} & {^{}_\\bullet}&\\dots&{^{}_\\bullet}&{^{}_\\bullet}&\\dots\n}\n\\end{matrix}\n$$\n}\n\nConsider the 2-regular graphs $\\delta\\cup \\gamma$ and $\\delta\\cup \\sigma$. We orient the cycles of these graphs by ordering $(-j,j)$ on the left-most vertical edge of the cycle.\nFor example,\n\n{\\small $$\n\\sigma\\cup\\delta=\\begin{matrix}\n \\xymatrix  @-1pc{\n {^1_ \\bullet} \\ar@{-}[dr]& {^2_\\bullet} \\ar@{-}[dr]& {^3_\\bullet}\n & \\dots & {^{j_1}_\\bullet}\\ar@{-}[dllll] & {^{j_1+1}_{\\hspace{2.5mm}\\bullet}}\\ar@{-}[dr] & {^{j_1+2}_{\\hspace{3mm}\\bullet}}&\\dots & {^{j_1+j_2}_{\\hspace{3.7mm}\\bullet}}\\ar@{-}[dlll]& {^{j_1+j_2+1}_{\\hspace{5mm}\\bullet}}&\\dots \\\\\n {^{\\hspace{1.5mm}\\bullet}_{-1}}\\ar@{->}[u] &{^{\\hspace{1.5mm}\\bullet}_{-2}}\\ar@{-}[u] &{^{\\hspace{1.5mm}\\bullet}_{-3}}\\ar@{-}[u]&\\dots &\n{^{}_\\bullet}\\ar@{-}[u]& {^{}_\\bullet} \\ar@{->}[u]& {^{}_\\bullet}\\ar@{-}[u]&\\dots&{^{}_\\bullet}\\ar@{-}[u]& {^{}_\\bullet} \\ar@{->}[u]&\\dots\n}\n\\end{matrix}\n$$\n}\nWe now define the bipartite M\\\"obius graph by assigning black vertices to the $m$ cycles of  $\\delta\\cup \\sigma$, and white vertices to the cycles of  $\\delta\\cup \\gamma$.\n\nEach black vertex is oriented counter-clockwise. For each black vertex we follow the cycle of $\\delta\\cup \\sigma$, drawing a labeled line for each element of the partition. The lines corresponding to $-j,j$ are adjacent, and will eventually become two edges of a ribbon.\n\nEach white vertex is oriented, say, clockwise; we identify the graphs that differ only by\na choice of orientation at some of the white vertices.\nFor each white vertex, we follow the corresponding cycle of $\\delta\\cup \\gamma$, drawing a labeled line for each element of the partition. The lines corresponding to $-j,j$ are adjacent, but may appear in two different orders depending on the orientation of the corresponding edge of $\\delta$ on the cycle.\n\nThe final step is to connect pairs $(j,-j)$ on the black vertices with the same pairs on the white vertices. This creates the ribbons, which carry a flip if the orientation of the two lines\non the black vertex does not match the orientation of the same edges on the white vertex.\n\n\n\\begin{figure}\n\\begin{center}\\includegraphics[width=6cm]{black-white}\n\\end{center}\n\\caption{\\label{black_white} Representation of a bipartite M\\\"obius\ngraph, the edges drawn are ribbons that are either twisted or\nuntwisted.  }\n\\end{figure}\nThe individual edges pictured in Figure \\ref{black_white} are ribbons and are\nlabeled as in Figure \\ref{black-white-2}.\n\\begin{figure}\n\\begin{center}\\includegraphics[width=5cm]{black-white-2}\\end{center}\n\\caption{\\label{black-white-2} Example of the labeling we use for\nthe edges emanating from a black vertex in a bipartite M\\\"obius\ngraph.  }\n\\end{figure}\n  We allow twists of ribbons to\npropagate through a white vertex calling the two bipartite M\\\"obius graphs\nequivalent.\n\n\n\nSuppose  the $2n$-tuple $(X_{\\pm 1},X_{\\pm 2},\\dots,X_{\\pm n})$\nconsists of random variables taken with repetition, from\nthe set $\\{Z_1,Z_2,\\dots,Z_n\\}$ of independent quaternionic Gaussian random variables.\nLet $\\mathcal{M}=\\mathcal{M}(X_{\\pm 1},X_{\\pm 2},\\dots,X_{\\pm n})$ denote the set of all bipartite M\\\"obius graphs $\\Gamma$\nthat correspond to various ways of pairing all repeated $Z$'s in the sequence\n$(X_{\\pm 1},X_{\\pm 2},\\dots,X_{\\pm n})$; the pairs are given by adjacent half edges at each white\nvertex. (See the preceding construction.) $\\mathcal{M}=\\emptyset$ if there is a $Z_j$ that is repeated\nan odd number of times.\n\\begin{theorem} \\label{thm2.1}\n\\begin{align*}\n \\mathbb{E}\\big( \\Re( \\bar{X}_{-1} X_1 \\cdots \\bar{X}_{-j_1} X_{j_1})\n\\times \\Re( \\bar{X}_{-j_1-1} X_{j_1 +1} \\cdots \\bar{X}_{-j_1-j_2} X_{j_1+j_2} )\n\\times \\cdots\n\\\\ \\dots \\times\n\\Re( \\bar{X}_{-j_1-j_2-\\dots -j_{m-1}-1} \\cdots \\bar{X}_{-n} X_{n} )\\big)\n\\\\ = 4^{n-m}\\sum_{\\Gamma\\in\\mathcal{M}} (-2)^{\\chi(\\Gamma)}\\;,\n\\end{align*}\n(The right hand side is interpreted as $0$ when $\\mathcal{M}=\\emptyset$.)\n\\end{theorem}\n\n\n\n\\begin{proof}\nThe proof is fundamentally the same as that of the Wigner version of this\ntheorem.\nIn view of \\eqref{Wick0} and \\eqref{Wick1}, it suffices\n to consider  $\\{ X_{\\pm 1}, \\dots X_{\\pm n} \\}$  that  form $n$ independent pairs, and show that\n\\begin{multline}\\label{WWW314}\n \\mathbb{E}\\big( \\Re( \\bar{X}_{-1} X_1 \\cdots \\bar{X}_{-j_1} X_{j_1})\n\\times \\Re( \\bar{X}_{-j_1-1} X_{j_1 +1} \\cdots \\bar{X}_{-j_1-j_2} X_{j_1+j_2} )\n\\times \\cdots\n\\\\  \\dots \\times\n\\Re( \\bar{X}_{-j_1-j_2-\\dots -j_{m-1}-1} \\cdots \\bar{X}_{-n} X_{n} )\\big)\n = 4^{n-m} (-2)^{\\chi(\\Gamma)}\\;,\n\\end{multline}\nwhere $\\Gamma$ is the bipartite M\\\"obius graph that describes the\npairings.\n\nWe will prove \\eqref{WWW314} by induction; to that end we first check\nthat with $n=1, m=1$\nwe have $\\mathbb{E}(\\bar{X}X)=(-2)^2$ in agreement with \\eqref{WWW314}.\n\nIf $\\Gamma$ has two black vertices connected together by edges adjacent at a white vertex,\nwe can use the cyclicity of $\\Re$ to move the variables that label the\nrespective edges and share the same face to\nthe first position in their cycles, so that we may call them $\\bar{X}_{-1}$\nand either $X_j$ or $\\bar{X}_{-j}$.\nWe now use relations (\\ref{q3}) and (\\ref{q4}) to eliminate the pair from the\nproduct:\n\\begin{equation*}\n\\mathbb{E}\\left( \\Re( \\bar{X}_{-1} \\cdots X_{j_1}) \\Re( X_j \\dots X_{j_1+j_2} \\bar{X}_{-j} )\\right)\n=  \\mathbb{E}( X_1 \\cdots X_{j_1+j_2} \\bar{X}_{-j})\\,,\n\\end{equation*}\nor\n\\begin{equation*}\n\\mathbb{E}\\left( \\Re( \\bar{X}_{-1} \\cdots X_{j_1}) \\Re( \\bar{X}_j \\dots X_{j_1+j_2} )\\right)\n= \\mathbb{E}( \\bar{X}_{j_1} X_{-j_1} \\cdots \\bar{X}_1 X_j   \\cdots X_{j_1+j_2})\\,.\n\\end{equation*}\nThe use of relation (\\ref{q3}) corresponds to that of gluing together the two\nribbons along the halves adjacent at the white vertex, and gluing together the\ncorresponding black vertices (see Figure \\ref{two-gluing-orient}).\nThe use of relation (\\ref{q4}) corresponds to the same gluing, but in this\ncase one of the ribbons has an orientation reversal in it, resulting in an\norientation reversal for the remaining sides (see Figure\n\\ref{two-gluing-unorient}).\nThese geometric operations reduce $n$ and $m$ by one without changing the\nEuler characteristic: both the number of edges and the number of vertices are\nreduced by one while the number of faces is preserved.\n\n\\begin{figure}\n\\includegraphics[width=12cm]{two-gluing-orient}\n\\caption{ Here two black vertices are connected together through\nuntwisted edges adjacent at a white vertex.  This bipartite M\\\"obius\ngraph reduces to one with one less vertex and one less edge.  The\nreduction is found by gluing the two black vertices together and\ngluing the two ribbons together along their adjacent sides, here\nlabeled by $\\bar{X}_{-1}$ and $X_j$.  The same reduction would apply\nif the two edges were twisted as we could pass this twist through\nthe white vertex. \\label{two-gluing-orient}}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=12cm]{two-unorient}\n\\caption{ Here two black vertices are connected together through one\ntwisted edge and one untwisted edge adjacent at a white vertex.\nThis bipartite M\\\"obius graph reduces to one with one less vertex\nand one less edge.  The reduction is found by gluing the two black\nvertices together, gluing the two ribbons together along the sides\nadjacent at the white vertex, and reversing both the order and\norientations of remaining ribbons on the second black vertex.\n\\label{two-gluing-unorient}}\n\\end{figure}\n\nTherefore we will only need to prove the result for the single black vertex case of\nthe induction step.\nWe wish to show that\n\\begin{equation}\n\\mathbb{E}( \\bar{X}_{-1} X_1 \\bar{X}_{-2} X_2 \\cdots \\bar{X}_{-n} X_n ) = 4^{n-1}\n(-2)^{\\chi(\\Gamma)}\\;,\n\\end{equation}\nwhere $\\Gamma$ is a bipartite M\\\"obius graph with a single black vertex and\nhalf ribbons labeled by $X_{\\pm k}$.  We will\ndo this by induction, there are two cases:\n\\begin{enumerate}\n\n\\item[\\bf Case 1:] $X_{-1} = X_j$ for $1 \\leq j \\leq n$,\n\\begin{align*}\n\\mathbb{E}( \\bar{X}_{-1} X_1 \\cdots \\bar{X}_{-j} X_j \\cdots X_n ) &=\n\\mathbb{E}( \\bar{X}_{-1} X_j ) \\mathbb{E}( \\Re( X_1 \\cdots \\bar{X}_{-j} ) \\Re( \\bar{X}_{-j-1}\nX_{j+1} \\cdots X_n)) \\\\\n&= 4 \\mathbb{E}(\\Re( \\bar{X}_{-j} X_1 \\cdots X_{j-1} ) \\Re( \\bar{X}_{-j-1} X_{j+1}\n\\cdots X_n) )  \\,.\n\\end{align*}\nThis corresponds to the reduction of the bipartite M\\\"obius graph pictured in Figure\n\\ref{reduce-1}, which for $j>1$ splits the single black vertex into two black vertices,\nand glues the two edges labeled as $\\bar{X}_{-1}$ and $X_j$ together.\n\nThe edges $(1,-1,j,-j)$ are adjacent at the white vertex and appear either in this order, or in the reverse order. Thus after the removal of $\\{-1,j\\}$ we get an ordered pair of labels $(1,-j)$ or $(-j,1)$ to glue back into a ribbon.\nOn the black vertex, due to our conventions the edges of the ribbons appear in the following order\n $$((-1,1), (-2, 2),\\dots, (-j,j), (-k_1,k_1),\\dots,(-k_r,k_r)).$$\n Once we split the black vertex into two vertices with the edges of ribbons given by\n$((-1,1), (-2, 2),\\dots, (-j,j))$ and $((-k_1,k_1),\\dots,(-k_r,k_r))$, the removal of $\\{-1,j\\}$  creates a new pair $(1,-j)$ which we use to create the ribbon to the white vertex.\n[After this step, we relabel all edges to use again $\\pm 1,\\pm 2,\\dots$  consecutively.]\n\n\nWe note that the number of faces of the new graph is the same as the previous graph -- the face with the sequence of edges $$(\\dots, X_{k_r},\\bar{X}_{-1},X_j,\\bar{X}_{-k_1},\\dots)$$ becomes the face with edges $(\\dots, X_{k_r},\\bar{X}_{-k_1},\\dots)$ on the new graph.\n The\nEuler characteristic becomes $\\chi_2 = (v_1 + 1) - (e_1 - 1) + f_1 = \\chi_1 +\n2$ where $v_1, e_1$ and $f_1$ are the number of vertices, edges, and faces of\n$\\Gamma$.\nBy the induction assumption we then find\n\\begin{equation*}\n\\mathbb{E}( \\bar{X}_{-1} X_1 \\cdots X_n) = 4 \\left[ 4^{(n-1) - 2} (-2)^{\\chi_2}\n\\right] = 4^{n-2} (-2)^{\\chi_1 + 2} = 4^{n-1} (-2)^{\\chi_1} \\,.\n\\end{equation*}\n\n\n\\begin{figure}\n\\includegraphics[width=9cm]{reduce-1}\n\\caption{ Here we have a black vertex with two ribbons, both twisted\nor both untwisted, adjacent at the same white vertex.  The reduction\nglues these two ribbons together along their common side.  The\nresult is a bipartite M\\\"obius graph with one more vertex and one\nless edge.  The resulting graph may or may not be disconnected at\nthis point. \\label{reduce-1}}\n\\end{figure}\n\n\n\\item[\\bf Case 2:] $X_{-1} = X_{-j}$ for $1 < j \\leq n$,\n\\begin{multline*}\n\\mathbb{E}( \\bar{X}_{-1} X_1 \\cdots \\bar{X}_{-j} X_j \\cdots X_n ) \\\\=\n\\mathbb{E}( \\bar{X}_{-1} \\bar{X}_{-j} ) \\mathbb{E}( \\bar{X}_{j-1} X_{-j+1} \\cdots \\bar{X}_1\nX_j \\bar{X}_{-j-1} X_{j+1} \\cdots X_n) \\\\\n= (-2)  \\mathbb{E}( \\bar{X}_{j-1} X_{-j+1} \\cdots \\bar{X}_1\nX_j \\bar{X}_{-j-1} X_{j+1} \\cdots X_n)  \\,.\n\\end{multline*}\nThis corresponds to the reduction of the M\\\"obius graph pictured in Figure\n\\ref{reduce-2}, which switches the order and the orientations of the\nribbons on one side of the\nblack vertex from the $\\pm 1$ and $\\pm j$ ribbons and glues the two\nedges adjacent at the white vertex together as shown.\nAs previously, the removed edges are adjacent at the white vertex. At the black vertex, the labeled lines for the construction of the bipartite graph change from the sequence\n$$\n(-1,1),(-2,2),\\dots,(-j+1,j-1),(-j,j),(-k_1,k_1),\\dots,(-k_r,k_r)\n$$\nto the sequence\n$$\n(j-1,-j+1),\\dots,(2,-2),(1,j),(-k_1,k_1),\\dots,(-k_r,k_r)\n$$\nwhich then needs to be relabeled to use $\\pm 1,\\dots,\\pm n$. Again the number of faces on the bipartite graphs is preserved:\nthe face with edges\n$$(\\dots, 2, -1, -j,k_1,\\dots)$$\nbecomes the face\n$$\n(\\dots,2,1,j,k_1,\\dots).\n$$\nThe Euler\ncharacteristic becomes $\\chi_2 = v_1 - (e_1 -1) + f_1 = \\chi_1 + 1$.  By the\ninduction assumption we then find\n\\begin{equation*}\n\\mathbb{E}( \\bar{X}_{-1} X_1 \\cdots X_n) = (-2) \\left[ 4^{(n-1) - 1} (-2)^{\\chi_2}\n\\right] = (-2) 4^{n-2} (-2)^{\\chi_1 + 1} = 4^{n-1} (-2)^{\\chi_1} \\,.\n\\end{equation*}\n\n\n\\begin{figure}\n\\includegraphics[width=9cm]{reduce-2}\n\\caption{ Here we have a black vertex with two ribbons, one twisted\nand the other untwisted, adjacent at the same white vertex.  The\nreduction glues these two ribbons together along their common side.\nThe result is a bipartite M\\\"obius graph with one less edge, and\nwith the ribbons on one side of the removed ribbon now with reversed\norder and orientations. \\label{reduce-2}}\n\\end{figure}\n\n\\end{enumerate}\n\nWith these two cases checked, by the induction hypothesis, the proof is\ncompleted.\n\n\\end{proof}\n\nOne should note that this is fundamentally the same proof as in the Wigner\ncase, however in this case the geometric reduction is given by gluing together\ntwo ribbons, while in the Wigner case the geometric reduction is the\nelimination of one ribbon at a time.  The inductive steps remain\nthe same.\n\n\n\n\n\n\n\n\n\n\\section{\\label{duality} Duality between real and symplectic ensembles}\nBy $\\mathcal{M}_{M\\times N}(\\mathbb{H} )$ we denote the set of all $M\\times N$ matrices with entries from $\\mathbb{H}$.\nFor $\\mathbf A\\in \\mathcal{M}_{M\\times N}(\\mathbb{H})$, the adjoint matrix is $A^*_{i,j}:=\\overline{A}_{j,i}$. The trace is\n${\\rm tr}(\\mathbf A)=\n    \\sum_{j=1}^NA_{jj}\n$.\nSince the traces ${\\rm tr}(\\mathbf A)$ may fail to commute, in the formulas we will use $\\Re({\\rm tr}(\\mathbf A))$, compare to\n \\cite{Hanlon-Stanley-Stembridge-92}.\n\n\\subsection{Duality between GOE and GSE ensembles}\n\nThe Gaussian orthogonal ensembles consist of square symmetric matrices, $\\mathbf Z$,\nwhose independent entries are independent (real) Gaussian random\nvariables; the off diagonal entries have variance $1$ while the\ndiagonal entries have variance\n$2$.\nOne may show  that\n\\begin{theoremA} \\label{thm3.1} \\tolerance=2000\nFor $\\mathbf Z$ from the $N\\times N$ Gaussian orthogonal ensemble:\n\\begin{equation*}\n\\frac{1}{N^{n-m}} \\mathbb{E}( {\\rm tr}(\\mathbf Z^{j_1}) {\\rm tr}(\\mathbf Z^{j_2}) \\dots\n{\\rm tr}(\\mathbf Z^{j_m}) ) = \\sum_{\\Gamma} N^{\\chi(\\Gamma)}  \\,,\n\\end{equation*}\nwhere the sum is over labeled M\\\"obius graphs $\\Gamma$ with $m$\nvertices of degree $j_1, j_2, \\dots, j_m$, $\\chi(\\Gamma)$ is the Euler\ncharacteristic and $j_1 + j_2 + \\dots + j_m = 2n$.\nMore generally, if $\\mathbf Z_1, \\dots, \\mathbf Z_s$ are independent $N\\times N$ GOE\nensembles and $t: \\{ 1, 2, \\dots, 2n\\} \\to \\{ 1, \\dots, s\\}$ is fixed,\nthen\n\\begin{multline*}\n\\frac{1}{N^{n-m}} \\mathbb{E}( {\\rm tr}( \\mathbf Z_{t(1)} \\dots \\mathbf Z_{t(\\beta_1)} )\n{\\rm tr}( \\mathbf Z_{t(\\alpha_2)} \\dots \\mathbf Z_{t(\\beta_2)} ) \\times \\dots \\\\ \\dots\\times\n{\\rm tr}( \\mathbf Z_{t(\\alpha_m)} \\dots \\mathbf Z_{t(\\beta_m)}) )\n= \\sum_{\\Gamma} N^{\\chi(\\Gamma)} \\,,\n\\end{multline*}\nwhere $\\alpha_1 = 1$, $\\alpha_k = j_1 + j_2 +  \\dots +\n j_{k-1} + 1 $, and\n$\\beta_k = j_1 + j_2 + \\dots + j_k $ denote the ranges under the\ntraces, and where the sum is over labeled color-preserving M\\\"obius\ngraphs $\\Gamma$ with vertices of degree $j_1, j_2,\\dots, j_m$ whose edges are\ncolored by the mapping $t$.\nIf there are no $\\Gamma$ that are consistent with the coloring we\ninterpret the sum as being $0$.\n\\end{theoremA}\nThe single color version of this Theorem was given in Ref.\n\\cite{Goulden-Jackson-97}.\n\\tolerance=1000\n\nThe  Gaussian symplectic ensembles (GSE)\nconsist of square self-adjoint matrices\n\\begin{equation}\n  \\label{GUE}  \\mathbf Z=\\left[Z_{i,j}\\right],\n\\end{equation}\nwhere $\\{Z_{i,j}:i<j\\}$ is a family of independent $\\mathbb{H}$-Gaussian random variables \\eqref{HH}, $\\{Z_{i,i}\\}$ is a family\nof independent real Gaussian random variables of variance $2$,\nand $Z_{i,j}=\\bar{Z}_{j,i}$. The law of such a matrix has a density\n$C\\exp(-\\frac14 \\Re{\\rm tr}(\\mathbf x^2))$,\nwhich is supported on the self-adjoint subset of\n$\\mathcal{M}_{N\\times N}(\\mathbb{H})$. ($C=C(N)$ is a normalizing constant.)\n\n\n\n\nIn this section we will demonstrate a multi-matrix version of the theorem of Mulase and\nWaldron \\cite{Mulase-Waldron-03} from our Wick formula.\nThis represents an improvement to the argument\nof Mulase and Waldron as we will not need to rely on labelings of\nthe vertices by the quaternions $1, i, j,$ and $k$.  This part of\ntheir argument is encoded in relations (\\ref{q1}-\\ref{q4}).\n\nWe will compute here the expected values of the real traces of\npowers of a quaternionic self-dual matrix in the  Gaussian\nsymplectic ensemble \\eqref{GUE} where the off-diagonal matrix entries are\n\\eqref{HH}.\n\n\nThe basic theorem we will prove is\n\\begin{theorem}\\label{T-MW}\nFor $\\mathbf Z$ from the $N\\times N$ Gaussian symplectic ensemble:\n\\begin{equation}\n  \\label{MW1}\n  \\frac{1}{(4N)^{n-m}} \\mathbb{E}( \\Re( {\\rm tr}( \\mathbf Z^{j_1}))\n\\Re({\\rm tr}(\\mathbf Z^{j_2})) \\dots \\Re({\\rm tr}(\\mathbf Z^{j_m})) ) =\n\\sum_{\\Gamma} (-2N )^{\\chi(\\Gamma)}\\,,\n\\end{equation}\\tolerance=2000\nwhere the sum is over labeled M\\\"obius graphs $\\Gamma$ with $m$ vertices\nof degree\n$j_1, j_2, \\dots, j_m$,  $\\chi(\\Gamma)$ is the Euler\ncharacteristic and $j_1 + j_2 + \\dots + j_m = 2 n$.\n\nMore generally, suppose $\\mathbf Z_1,\\dots,\\mathbf Z_s$ are independent $N\\times N$  $GSE$ ensembles and $t:\\{1,\\dots,2n\\}\\to \\{1,\\dots, s\\}$ is fixed.\nLet $\\alpha_1=1$, $\\alpha_k=j_1 + j_2 + \\dots + j_{k-1}+1$, $\\beta_k=j_1 + j_2 + \\dots + j_{k}$ denote the ranges under traces.\nThen\n\\begin{multline}\n  \\label{MW expansion}\n  \\frac{1}{(4N)^{n-m}} \\mathbb{E}\\big( \\Re( {\\rm tr}( \\mathbf Z_{t(1)}\\dots \\mathbf Z_{t(\\beta_1)}))\n\\Re({\\rm tr}(\\mathbf Z_{t(\\alpha_2)}\\dots \\mathbf Z_{t(\\beta_2)}))\\times \\dots \\\\\\dots\\times \\Re({\\rm tr}(\\mathbf Z_{t(\\alpha_m)}\\dots \\mathbf Z_{t(\\beta_m)} ))\\big) =\n\\sum_{\\Gamma} (-2N )^{\\chi(\\Gamma)}\\,,\n\\end{multline}\nwhere the sum is over labeled color-preserving M\\\"obius graphs $\\Gamma$ with vertices\nof degree\n$j_1$, $j_2$, $\\dots$, $j_m$ that are colored with $s$ colors by the mapping $t$. As previously,\n  $\\chi(\\Gamma)$ is the Euler\ncharacteristic and $j_1 + j_2 + \\dots + j_m = 2 n$. (If there are no $\\Gamma$ that are consistent with the\n coloring $t$, we interpret the sum as $0$.)\n\\end{theorem}\n\\tolerance=1000\n\n\nIn view of Theorem \\ref{thm3.1}, Theorem \\ref{T-MW} gives the\nduality between the GOE and GSE ensembles of random matrices as $N\n\\to -2N$.\n\n\n\\begin{example}\n  To illustrate the multi-matrix aspect of the theorem, suppose $\\mathbf Z_1,\\mathbf Z_2$ are independent\n  $N\\times N$ GSE  ensembles and $\\mathbf X_1,\\mathbf X_2$\nare  independent\n  $N\\times N$  GOE ensembles. For $p,r,k\\geq 0$\n   let $b_{p,r,k}(N):=\\mathbb{E}(({\\rm tr}(\\mathbf X_1^{2p}\\mathbf X_2^{2r}))^k)$.\n  By Theorem \\ref{thm3.1}, $b_{p,r,k}(N)=N^{(p +r-1)k}\\sum_{\\Gamma} N^{\\chi(\\Gamma)}$.\n  Therefore Theorem \\ref{T-MW} implies that the moments for the independent\n  GSE ensembles are determined from the corresponding moments of independent GOE ensembles\n  by the ``dual formula''\n$$\\mathbb{E}(({\\rm tr}(\\mathbf Z_1^{2p}\\mathbf Z_2^{2r}))^k)=(-2)^{(p +r-1)k}b_{p,r,k}(-2N).$$\nIn particular,\n$$\\mathbb{E}({\\rm tr}(\\mathbf Z_1^{2p}))=(-2)^{p-1} b_{p,0,1}(N).$$\nThis is equivalent to \\cite[Theorem 6]{Ledoux-07} with $c_p(N)=\\mathbb{E}({\\rm tr}(\\mathbf Z^{2p}))\/2^p$;\nan extra factor of $2^p$ in our formula is due to the fact that\nthe symplectic ensemble in \\cite{Ledoux-07}\nhas the variance of $1\/2$ instead of our choice of $1$.\n\\end{example}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{T-MW}]\n\nWe begin by expanding out the traces in terms of the matrix entries\n\\begin{align}\\label{star1}\n &\\frac{N^m}{4^{n-m}} \\mathbb{E}(\\Re({\\rm tr}(\\mathbf Z^{j_1}))\n \\Re({\\rm tr}(\\mathbf Z^{j_2})) \\dots \\Re({\\rm tr}(\\mathbf Z^{j_m})) ) =\\\\\n =& \\nonumber\n \\sum_{\\begin{matrix}\n 1\\leq a_1, a_2, \\dots, a_{j_1} \\leq N \\\\\n 1\\leq b_1, b_2, \\dots, b_{j_2} \\leq N \\\\ \\vdots \\\\\n 1 \\leq c_1, c_2, \\dots, c_{j_m} \\leq N\n \\end{matrix} }\n \\begin{matrix}\n  \\frac{1}{(4n)^{n-m}} \\mathbb{E}(\\Re( Z_{a_1, a_2} Z_{a_2, a_3}\n \\dots Z_{a_{j_1}, a_1}) \\Re( Z_{ b_{1}, b_{2} } \\dots\n Z_{b_{j_2}, b_1} ) \\times\\dots\n\\\\\n\\hfill \\dots \\times \\Re( Z_{ c_{1}, c_{2} } \\dots\n Z_{ c_{j_m}, c_1} ) )\\,.\n \\end{matrix}\n \\end{align}\nNote that essentially the same expansion applies to \\eqref{MW expansion}, except that the consecutive\nentries $Z_{i,j}^{(t)}$ must now be labeled\nalso by the ``color\" $t$.\n It is then better to index the products\nby the cycles of a permutation. Put\n$$\\sigma=(1,\\dots,\\beta_1)(\\alpha_2,\\dots,\\beta_2)\\ldots(\\alpha_m,\\dots ,2n)\\,.$$\nIn this notation, the expansion of the  left hand side of \\eqref{MW expansion} takes the form\n $$\\sum_{a:\\{1\\dots2n\\}\\to\\{1\\dots N\\}}\\mathbb{E}\\left(\\prod_{c\\in\\sigma}\n \\Re(\\prod_{j\\in c}[\\mathbf Z_{t(j)}]_{a(j),a(\\sigma(j))})\\right).$$\n(Since  $\\Re$   has the cyclic property\n\\ref{tmp**}, this expression is well defined.)\n\n\nFrom \\eqref{Wick1} it follows that the right hand side of\n\\eqref{star1} can be expanded as a sum over all pairings, and we can\nassume that the pairs are independent.\n\nOf course, pairings that match two independent random variables do not contribute to the sum.\nThe pairings that contribute to the sum are of three different types: pairs that match $Z$ with another $Z$ at a different position in the product,\npairs that match $Z$ with $\\bar{Z}$, and pairings that match the diagonal (real) entries.\n\nWe first dispense with the pairings that match a diagonal entry $Z_{i,i}=\\xi$ with another diagonal entry\n$Z_{j,j}=\\xi$. Since real numbers commute with quaternions,\n$$\\Re(\\mathbb{E}(q_0\\xi q_1\\xi q_2))=\\mathbb{E}(\\xi^2) \\Re(q_0q_1q_2)=2 \\Re(q_0q_1q_2).$$\nOn the other hand,  using the cyclic property of $\\Re(\\cdot)$ and adding formulas \\eqref{q1} and \\eqref{q2}, we see that the same answer arises from\n$$\\mathbb{E}\\Re(q_0Z q_1Z q_2)+\\mathbb{E}\\Re(q_0Z q_1\\bar{Z} q_2)=\\mathbb{E}\\Re(Z q_1Z q_2q_0)+\\mathbb{E}\\Re(Z q_1\\bar{Z} q_2q_0)=2\\Re(q_1q_2q_0).$$\nSo the contribution of each such diagonal pairing is the same as that of two matches of quaternionic entries $(Z,Z)$ and $(Z,\\bar{Z})$.\n\nThus, once we replace all the real entries that came from the diagonal entries by the corresponding quaternion-Gaussian pairs,\nwe get the sum over all possible pairs of matches of the first two types only.\nWe label all such pairings by M\\\"obius graphs, with the interpretation that\npairings or random variables $(Z,Z)$ correspond to twisted ribbons, while pairings $(Z,\\bar{Z})$\ncorrespond to ribbons without a twist. The pairs of variables at different ribbons can now be assumed independent.\nIn the multi-matrix case, the edges of the graph at each vertex are colored according to the function $t$,\nwhich restricts the number of available pairings.\n\nWe now relabel the $Z_{i,j}$ by $Z_{a_k, a_{k+1}} = X_k$, $Z_{b_k, b_{k+1}}\n= X_{j_1 + k} $, $\\dots$,\n$Z_{c_k, c_{k+1}} = X_{j_1 + j_2 + \\dots\nj_{m-1} + k} $. Our claim is then that given a M\\\"obius graph $\\Gamma$\nwith vertices of degree $j_1$, $j_2, \\dots,$ and $j_m$ with edges\nlabeled by $X_k$, satisfies\n\\begin{multline}\n\\frac{1}{(4N)^{n-m}} \\mathbb{E}( \\Re(X_1 \\dots X_{j_1}) \\Re( X_{j_1+1}\n\\dots X_{j_1+j_2} ) \\dots \\Re( X_{j_1 + \\dots +j_{m-1}+1} \\dots\nX_{2n}) )\\\\= (-2N)^{\\chi(\\Gamma)} N^{-f(\\Gamma)} \\,,\n\\end{multline}\nwhere $f(\\Gamma)$ is the number of faces of $\\Gamma$.\nThe $N^{-f(\\Gamma)}$ terms are removed by the summations in\n(\\ref{star1}), the relations given by the edges of $\\Gamma$ reduce\nthe number of summations leaving us with $f(\\Gamma)$ sums from $1$\nto $N$.  Collecting the powers of $N$ we find\nthat this is the same as \\eqref{star2}.\n\\end{proof}\n\n\\subsection{Duality between the Laguerre Ensembles}\n\n\nWe will now consider the analogous problem for quaternionic Wishart random\nmatrices. These are $N\\times N$ matrices of the form\n\\begin{equation}\n  \\label{H-Wishart}\n  \\mathbf W = \\mathbf X^* \\mathbf X\\,,\n\\end{equation}\nwhere $\\mathbf X$ is an $M\\times N$ matrix with independent $\\mathbb{H}$-Gaussian entries \\eqref{HH}.\nThe Wick formula (\\ref{Wick1}) in this case is expressed in terms of\nbipartite M\\\"obius graphs defined in section \\ref{Sect bipartite};\nalthough in this case it is helpful to view our labeling of the ribbons around\neach black vertex in a slightly different way (see Figure\n\\ref{black-white-3}).\n\\begin{figure}\n\\includegraphics[width=8cm]{black-white-3-b}\n\\caption{ For the Wishart applications we will label the sides of\nthe ribbons emanating from a black vertex.  One side receives the\nlabel $X_{n_k m_k}$ while the other receives the label $\\bar{X}_{m_k\nn_{k+1} }$.  Note that the interior index, $m_k$, of the two labels\non each ribbon match; while the exterior index, $n_k$, of the labels\non ribbons adjacent at a black vertex match. \\label{black-white-3}}\n\\end{figure}\nOur heuristic is the following: when edges are glued to a white\nvertex we identify the labels on the interior of each ribbon as all\nequal, while the labels on the outside of each ribbon are equal to\nthe others which border the same face.  The reason for this\ndichotomy is that $\\mathbf X$ is not a square matrix and therefore the\n$n_j$'s and $m_j$'s are to be treated as separate families of\nvariables.\n\nIn the multi-matrix case, we color the ribbons by the number of a matrix, and color the white vertices so that only the ribbons of matching color can be attached.\n\nFor example,\n\\begin{equation*}\n\\mathbb{E}( {\\rm tr}(\\mathbf W) ) = \\sum_{m=1}^M \\sum_{n=1}^N \\mathbb{E}( X^*_{nm} X_{mn} )\n= 4 M N\n\\end{equation*}\ncorresponds to the bipartite M\\\"obius graph in Figure \\ref{black-white-ex1}.\n\\begin{figure}\n\\begin{center}\\includegraphics[width=5cm]{black-white-ex1}\n\\end{center}\n\\caption{ The only possible bipartite M\\\"obius graph with a single\nblack vertex of degree one.  In this case the graph is embedded on a\ncopy of the Riemann sphere. \\label{black-white-ex1}}\n\\end{figure}\nAs another example\n\\begin{align*}\n\\mathbb{E}({\\rm tr}(\\mathbf W^2)) &= \\sum_{m_1, m_2 =1}^M \\sum_{n_1, n_2 =1}^N\n\\mathbb{E}( X^*_{n_1 m_1} X_{m_1 n_2} X^*_{n_2 m_2} X_{m_2 n_1} )\n\\\\&= \\sum_{m_1, m_2=1}^M \\sum_{n_1, n_2=1}^N\\Big[\n\\mathbb{E}( X^*_{n_1 m_1} X_{m_1 n_2} ) \\mathbb{E}( X^*_{n_2 m_2} X_{m_2 n_1})\n\\\\&\\phantom{\\sum_{m_1, m_2=1}^M \\sum_{n_1, n_2=1}^N }\n+ \\mathbb{E}( X^*_{n_1 m_1} X^*_{n_2 m_2} ) \\mathbb{E}( \\bar{X}_{m_1 n_2} X_{m_2 n_1})\n\\\\&\\phantom{\\sum_{m_1, m_2=1}^M \\sum_{n_1, n_2=1}^N }\n+ \\mathbb{E}( X^*_{n_1 m_1} X_{m_2 n_1}) \\mathbb{E}( X_{m_1 n_2} X^*_{n_2 m_2} )\\Big]\n\\\\&= 16 N M^2 - 8 N M + 16 N^2 M\\,,\n\\end{align*}\ncorresponds to the bipartite M\\\"obius graphs in Figure \\ref{black-white-ex2}.\n\\begin{figure}\n\\begin{center}\\includegraphics[width=10cm]{black-white-ex2}\\end{center}\n\\caption{ The three possible bipartite M\\\"obius graphs with a single\nblack vertex of degree two.  In the top two cases the graph is\nembedded on a copy of the Riemann sphere, while in the bottom case\nthe graph is embedded on a copy of the projective sphere.\n\\label{black-white-ex2}}\n\\end{figure}\n\n\nIn this case our theorem takes the form.\n\\begin{theorem}\nWith $M = \\lambda N$ we find\n\\[ \\mathbb{E}( {\\rm tr}( \\mathbf W^{j_1} ) {\\rm tr}( \\mathbf W^{j_2}) \\cdots {\\rm tr}( \\mathbf W^{j_m}) )\n= N^{n-m}   4^{n-m} \\sum_{\\Gamma} \\lambda^{w(\\Gamma)} (-2N )^{\\chi(\\Gamma)}\\;,\n\\]\nwhere the sum is over the bipartite M\\\"obius graphs with black\nvertices having degrees $j_1, j_2, \\dots, j_m$, $n = j_1 + j_2 +\n\\dots + j_m $ is the number of edges, $w(\\Gamma)$ is the number of\nwhite vertices, and $\\chi(\\Gamma)$ is the Euler characteristic.\n\nMore generally, suppose that $\\mathbf W_1,\\mathbf W_2,\\dots,\\mathbf W_s$ are $N\\times N$ independent quaternionic Wishart with parameters $M_1,\\dots,M_s$. Denote $\\lambda_j=M_j\/N$.\nFix  $t:\\{1,\\dots,n\\}\\to \\{1,\\dots, s\\}$, and\nlet $\\alpha_1=1$, $\\alpha_k=j_1 + j_2 + \\dots + j_{k-1}+1$, $\\beta_k=j_1 + j_2 + \\dots + j_{k}$ denote the ranges under traces.\nThen\n\\begin{multline}\n  \\label{Wishart expansion}\n  \\frac{1}{(4N)^{n-m}} \\mathbb{E}\\big( \\Re( {\\rm tr}( \\mathbf W_{t(1)}\\dots \\mathbf W_{t(\\beta_1)}))\n\\Re({\\rm tr}(\\mathbf W_{t(\\alpha_2)}\\dots \\mathbf W_{t(\\beta_2)})) \\times\\dots\n\\\\\\dots\\times  \\Re({\\rm tr}(\\mathbf W_{t(\\alpha_m)}\\dots \\mathbf W_{t(\\beta_m)} ))\\big)=\n\\sum_{\\Gamma} \\lambda_1^{w_1(\\Gamma)}\\lambda_2^{w_2(\\Gamma)}\\dots \\lambda_s^{w_s(\\Gamma)} (-2N )^{\\chi(\\Gamma)},\n\\end{multline}\nwhere the sum is over the bipartite M\\\"obius graphs with $m$ black vertices of degrees $j_1,\\dots,j_m$, whose ribbons are colored according to $t$ by one of the\ncolors $1,\\dots,s$,\n and with white vertices that  are colored to match the ribbons. Here $w_j(\\Gamma)$ is the number of white vertices of color $j$,   $\\chi(\\Gamma)$ is the Euler\ncharacteristic and $j_1 + j_2 + \\dots + j_m = n$ is the number of edges\/ribbons.\n\\end{theorem}\n\n\n\n\\begin{proof}\nWe begin by expanding out the traces in terms of the matrix entries of\n$\\mathbf X$ where $\\mathbf W = \\mathbf X^* \\mathbf X $ and $\\mathbf X$ is an $M \\times N$ matrix of\nquaternionic Gaussian random variables\n\\begin{align} \\label{Wishart-expansion}\n& \\frac{N^{m-n}}{4^{n-m}} \\mathbb{E}( {\\rm tr}( \\mathbf W^{j_1}) {\\rm tr}( \\mathbf W^{j_2})\n\\cdots {\\rm tr}( \\mathbf W^{j_m}) )  = \\\\ \\nonumber\n& \\sum_{\n\\begin{matrix}\n 1 \\leq a_1, a_2, \\dots, a_{j_1} \\leq N \\\\\n1 \\leq A_1, A_2, \\dots, A_{j_1} \\leq M \\\\\n1 \\leq b_1, b_2, \\dots, b_{j_2} \\leq N \\\\\n1 \\leq B_1, B_2, \\dots, B_{j_2} \\leq M \\\\\n\\vdots\\\\\n1 \\leq C_1, C_2, \\dots, C_{j_m} \\leq M \\end{matrix}}\n\\begin{matrix}\n\\frac{N^{m-n}}{4^{n-m}} \\mathbb{E}( \\Re( [\\mathbf X^*]_{a_1 A_1} [\\mathbf X]_{A_1 a_2} [\\mathbf X^*]_{a_2\n  A_2} [\\mathbf X]_{A_2 a_3} \\dots [\\mathbf X^*]_{a_{j_1} A_{j_1}} [\\mathbf X]_{A_{j_1} a_1} )\n\\\\\n\\hfill \\Re( [\\mathbf X^*]_{b_1 B_1}  \\dots [\\mathbf X]_{B_{j_2} b_1} ) \\dots\n\\Re( [\\mathbf X^*]_{c_1 C_1} \\dots [\\mathbf X]_{C_{j_m} c_1} ) )\\,.\n\\end{matrix}\n\\end{align}\nNote that a similar expansion would be found for a multi matrix\nexpression, see below.\n\n\nFrom (\\ref{Wick1}) it follows that\n(\\ref{Wishart-expansion}) can be expanded as a sum over all pairings,\nand we can assume that the pairs are independent.\n\nWe label all such pairings by bipartite M\\\"obius graphs, that is\nM\\\"obius graphs with labeled black vertices of degree $j_1, j_2, \\dots,\nj_m$, and unlabeled white vertices of any degree.  The half edges of\neach graph are labeled by $\\bar{X}_{A_1 a_1}= \\bar{X}_{-1}$, $X_{A_1\n  a_2} = X_1 $ and so on.\nThe relations given by this M\\\"obius graph reduce the number of sums\nover capital indices (which go from 1 to $M$) to $w(\\Gamma)$ the\nnumber of white vertices, and reduce the number of sums over lower\ncase indices (which go from 1 to $N$) to $f(\\Gamma)$ the number of\nfaces of $\\Gamma$.  Therefore from Theorem \\ref{thm2.1},\nwith $M = \\lambda N$, the total\ncontribution from the Wishart M\\\"obius graph $\\Gamma$ to the sum is\n\\[ \\frac{N^{m-n}}{4^{n-m}}\n\\left[ 4^{n-m}  \\lambda^{w(\\Gamma)} (-2)^{\\chi(\\Gamma)} \\right]\nN^{w(\\Gamma) + f(\\Gamma)}\n= \\lambda^{w(\\Gamma)}  (-2N)^{\\chi(\\Gamma)}\\,, \\]\nas $\\chi(\\Gamma) = m + w(\\Gamma) - n + f(\\Gamma)$.\n\nThe multimatrix version of this proof requires no conceptual changes,  but  the notation becomes more cumbersome.\n In this case it is more convenient to index the products\nby the cycles of a permutation. Put\n$$\\sigma=(1,\\dots,\\beta_1)(\\alpha_2,\\dots,\\beta_2)\\ldots(\\alpha_m,\\dots, n).$$ \nIn this notation, the expansion on the right hand side of \\eqref{Wishart-expansion} is replaced by\n\\begin{multline*}\n\\sum_{a:\\{1\\dots n\\}\\to\\{1\\dots N\\}}\\mathbb{E}\\left(\\prod_{c\\in\\sigma}\\Re(\\prod_{j\\in c}\n[\\mathbf W_{t(j)}]_{a(j),a(\\sigma(j))\n)\n\\right)\\\\\n=\\sum_{a:\\{1\\dots n\\}\\to\\{1\\dots N\\}}\\sum_{b\\in\\mathcal{B}(t)} \\mathbb{E}\\left(\\prod_{c\\in\\sigma}\\Re(\\prod_{j\\in c}\n\\overline{[{\\mathbf X}_{t(j)}]}_{b(j)a(j)}[\\mathbf X_{t(j)}]_{b(j),a(\\sigma(j))})\\right),\n\\end{multline*}\nwhere $\\mathcal{B}(t\n=\\{1,\\dots,M_{t(1)}\\}\\times\\{1,\\dots,M_{t(2)}\\}\\times\\dots\\times \\{1,\\dots,M_{t(n)}\\}$.\n\nFrom (\\ref{Wick1}) the expected value can again be expanded as a sum over the pairings,\nwhere we can assume that the pairs are independent random variables and the pairings connect only the random variables of the same color $t(j)$. For each bipartite M\\\"obius graph $\\Gamma$,\n the sum over $b\\in\\mathcal{B}(t)$ contributes a factor of $M_j^{w_j(\\Gamma)}$ per color, where $w_j(\\Gamma)$ is the number of vertices of color $j$ in $\\Gamma$, while the sum over $a$ contributes $N^{f(\\Gamma)}$, as in the one-matrix case.\n Therefore from Theorem \\ref{thm2.1}, the left hand side of \\eqref{Wishart expansion} becomes\n\\begin{multline*}\n(4N)^{m-n}\\sum_\\Gamma   M_1^{w_1(\\Gamma)}\\dots M_s^{w_s(\\Gamma)}N^{f(\\Gamma)} 4^{n-m}(-2)^{\\chi(\\Gamma)}\\\\\n =\n N^{m-n}\\sum_\\Gamma  N^{w(\\Gamma)+f(\\Gamma)}(-2)^{\\chi(\\Gamma)}\\prod_{j=1}^s \\lambda_j^{w_j(\\Gamma)}\\\\\n =\n\\sum_\\Gamma  N^{w(\\Gamma)+m+f(\\Gamma)-n} (-2)^{\\chi(\\Gamma)}\\prod_{j=1}^s \\lambda_j^{w_j(\\Gamma)}\\\\=\n\\sum_\\Gamma  N^{v(\\Gamma)+f(\\Gamma)\n-e(\\Gamma)} (-2)^{\\chi(\\Gamma)}\\prod_{j=1}^s \\lambda_j^{w_j(\\Gamma)}=\\sum_\\Gamma  (-2N)^{\\chi(\\Gamma)}\\prod_{j=1}^s \\lambda_j^{w_j(\\Gamma)} .\n\\end{multline*}\n\\end{proof}\nWe remark that the result is again in duality to the\nformula for real Wishart matrices, where the analogue of \\eqref{Wishart expansion} has the\nfollowing form:\n\\begin{multline}\n  \\label{Real Wishart expansion}\n  \\frac{1}{N^{n-m}} \\mathbb{E}\\big(  {\\rm tr}( \\mathbf W_{t(1)}\\dots \\mathbf W_{t(\\beta_1)})\n{\\rm tr}(\\mathbf W_{t(\\alpha_2)}\\dots \\mathbf W_{t(\\beta_2)})\\times\\dots \\\\ \\dots \\times {\\rm tr}(\\mathbf W_{t(\\alpha_m)}\\dots \\mathbf W_{t(\\beta_m)} )\\big)\n=\n\\sum_{\\Gamma} \\lambda_1^{w_1(\\Gamma)}\\lambda_2^{w_2(\\Gamma)}\\dots \\lambda_s^{w_s(\\Gamma)} N ^{\\chi(\\Gamma)}.\n\\end{multline}\nFor the one-matrix case this can be read out from \\cite{Hanlon-Stanley-Stembridge-92}.\nFor the multivariate case this is a reinterpretation of \\cite[Theorem 2.10]{Bryc-07}.\n\nThe following illustrates a one-matrix version of this duality.\n\\begin{example}\nSuppose $\\mathbf W_\\mathbb{R}$ is $N\\times N$  real Wishart matrix with parameter $M$, and let\n$$Q_{j_1,\\dots,j_m}^{\\mathbb{R}}(M,N)=\\mathbb{E}( {\\rm tr}( \\mathbf W_\\mathbb{R}^{j_1}) {\\rm tr}( \\mathbf W_\\mathbb{R}^{j_2})\n\\cdots {\\rm tr}( \\mathbf W_\\mathbb{R}^{j_m}) ).$$\nIf  $\\mathbf W_\\mathbb{H}$ is $N\\times N$  quaternion Wishart matrix with parameter $M$, and $n=j_1+\\dots +j_m$,\nthen \\eqref{Real Wishart expansion} and \\eqref{Wishart expansion} give\n$$\\mathbb{E}( {\\rm tr}( \\mathbf W_\\mathbb{H}^{j_1}) {\\rm tr}( \\mathbf W_\\mathbb{H}^{j_2})\n\\cdots {\\rm tr}( \\mathbf W_\\mathbb{H}^{j_m}) )=(-2)^{n-m}Q_{j_1,\\dots,j_m}^{\\mathbb{R}}(-2M,-2N).$$\nThis is equivalent to \\cite[Corollary 4.2]{Hanlon-Stanley-Stembridge-92}, as with $\\lambda=(j_1,j_2,\\dots,j_m)$ in their notation\n\\begin{multline*}\n Q_{j_1,\\dots,j_m}^{\\mathbb{H}}(M,N)=2^{-n}\\mathbb{E}( {\\rm tr}( \\mathbf W_\\mathbb{H}^{j_1}) {\\rm tr}( \\mathbf W_\\mathbb{H}^{j_2})\n\\cdots {\\rm tr}( \\mathbf W_\\mathbb{H}^{j_m}) )\\\\=(-1)^{n-m} 2^{-m}Q_{j_1,\\dots,j_m}^{\\mathbb{R}}(-2M,-2N);\n\\end{multline*}\nan extra factor  of $2^n$ in our formula is due to the fact that the quaternion Gaussian law in \\cite{Hanlon-Stanley-Stembridge-92}\nhas the variance of $1\/2$ instead of our choice of $1$.\n\\end{example}\n\n\\subsection*{Acknowledgements}\nThe authors thank G\\'erard Letac for citation \\cite{Isserlis-1918}. We would like to thank the referee for several substantial corrections.\n\n\\bibliographystyle{acm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe advances achieved by deep learning models for facial landmark detection, segmentation, editing and enhancement tasks have made many new applications. \nHowever, the process of devising various deep learning applications is always accompanied by the challenges from data. \nOn the one hand, training models on a large-scale public dataset with fixed annotation formats cannot fulfill the highly customized and fast-changing needs. \nFor example, a landmark detection network trained on the 300W \\cite{300W} dataset generates the standard 68-point landmarks, which are unable to model the forehead and face skin for advance editing. \nA segmentation model trained on Helen \\cite{helen} dataset cannot parse out chin or cheeks for face analysis at a finer granularity. \nOn the other hand, though it is feasible to annotate new datasets at the cost of much time and labor, the conventional fully supervised learning paradigm is inapt in practice when facing more diverse, flexible and even customized demands, hindering researchers from coming up with more inspiring and novel applications for fast development iterations. \nA better solution is enabling a model to obtain a satisfying performance with only a minimal number of customized annotation, namely the few-shot model adaptation.\n\nSeveral recent works have attempted the few-shot model adaptation for facial images. \nThey leverage either unsupervised learning or meta-learning techniques to handle the facial landmark detection with limited data \\cite{3FacRec}, the video face reenactment with unseen identity \\cite{intro_review_reenactment} and the face anti-spoofing for unknown attacks \\cite{intro_review_antispoofing}. Nevertheless, these models are specifically designed for a single task. \nTo our best knowledge, there is no a versatile few-shot adaptation method that can be easily extended to face analysis, editing and enhancement tasks within a unified framework.\n\nIn this paper, we proposed the Few-Shot Model Adaptation (FSMA) framework which aims at data-efficient model adaptation as well as the versatility across a wide range of applications. The training pipeline and the model's structure are shown in Figure \\ref{fig:framework}.\nThe framework first exploits the compact feature embedding via large-scale unsupervised learning, in which an adversarial auto-encoder \\cite{aae} is trained on the image reconstruction task.\nThe high-fidelity reconstruction ensures the auto-encoder to capture the high-level `face knowledge' \\cite{face_knowledge}.\nThen the auto-encoder is further trained on supervised tasks with a small number of annotated images.\nSo far, the similar idea \\cite{3FacRec} has achieved progress in predicting the mid-level semantic information, e.g., facial landmarks.\nHowever, due to the bottleneck-like structure and lack of internal feature resolution, the vanilla auto-encoder hinders itself from adapting to more subtle analysis, editing and enhancement tasks, like face semantic segmentation, stylization and shadow removal. \nTo this end, we introduce additional skip layers between the encoder and decoder during the supervised adaptation stage.\nThe skip layers not only prevent the auto-encoder from cheating during the unsupervised learning but also realize the multi-scale feature fusion as well as the recovery of feature details.\nAs verified in the experiments (Table \\ref{tab:skip_layer_ablation_lm}\\ref{tab:skip_layer_ablation_seg} and Figure \\ref{fig:res_skip_layer}), the introduced skip layers are critical components to FSMA's versatility on various applications. \n\nExtensive experiments are conducted on both the small customized datasets as well as the standard datasets under few-shot setting. \nTo our best knowledge, the FSMA is the first solution to handle few-shot face semantic segmentation, few-shot facial image stylization and few-shot facial image shadow removal problems.\nBoth qualitative and quantitative results demonstrate the FSMA's superiority over existing task-specific models and versatility across different tasks.\nOn few-shot facial landmark detection task, the FSMA achieves state-of-the-art performance. \n\n\n\t\n\n\n\n\\section{Related Works}\nRepresentative works in few-shot methodology and each application area are reviewed in this section.\n\n\\subsection{Semi-Supervised Learning}\nThe FSMA framework is essentially a semi-supervised learning pipeline that follows the `unsupervised pretrain, supervised fine-tune' paradigm.\nOne series of related literature focus on the unsupervised pretraining stage, including the pseudo labeling methods \\cite{related_2_pseudolabel,related_2_pseudolabel_1} and the self-supervised methods \\cite{related_2_selfsup_1,related_2_selfsup_2,related_2_selfsup_3}.\nAnother series of works emphasized more on the supervised adaptation stage.\nTask-specific efforts for a single applications include \\cite{related_1_image_translation} for image translation, \\cite{3FacRec} for facial landmark detection, \\cite{related_1_segmentation} for image segmentation, etc.\nBesides, \\cite{simclrv2} proposed an extra distillation stage using unlabeled data after the supervised fine-tuning for further improvement.\nInstead of working on a particular task, the FSMA framework is designed for a wide range of facial image applications and has superior versatility than other task-specific models.\n\n\\subsection{Applications on Facial Images}\nThe validation of FSMA's capacity is conducted on facial image analysis, editing and enhancement applications.\nIn detail, the few-shot problems in landmark detection, segmentation, stylization and shadow removal are discussed.\n\n\\vspace{1mm}\n\\noindent\\textbf{Facial landmark detection.}\nIn this paper, we focus on the models that work with limited data.\nSemi-supervised models such as \\cite{Kingma2014} adopted partially and weakly annotated data. \nSolutions to generate pseudo label include landmark perturbation \\cite{Lv2016}, multi-views captured from 3D face model \\cite{Zhu2016}, image style translation \\cite{sa} and GAN generation \\cite{GAN_ref}.\nThe other attempts include multi-task learning \\cite{Ranjian2016, Zhang2014}, `teacher-student' framework for training quality improvement \\cite{TS3} and large-scale unsupervised learning \\cite{3FacRec}.\nSimilar to \\cite{3FacRec}, the FSMA framework also leverages large-scale unlabeled data, but is more expandable to various tasks.\n\n\\vspace{1mm}\n\\noindent\\textbf{Face segmentation.}\nMost previous works are studied under the fully supervised setting.\nStudies on network architecture worked on hierarchical model \\cite{Luo2012,aaai20_face_parsing}, image pyramids \\cite{icnn}, multi-task learning \\cite{cvpr15liusifei}, cascade model, CNN-RNN \\cite{cnn_rnn_seg} hybrid network and receptive field \\cite{adaptiveRF,tip_seg}.\n\\cite{tip_seg} also devised a new loss function and semi-supervised distillation.\n\\cite{tanh_warp} proposed a warping method to better segment hairs.\nDifferent from these works, the FSMA is the first attempt to solve the few-shot face segmentation problem.\n\n\\vspace{1mm}\n\\noindent\\textbf{Facial image stylization.}\nOn general images, \\cite{Gatys2016} firstly introduced deep learning models. \n\\cite{CycleGAN2017} developed a framework with unpaired images. \n\\cite{PerceptualLosses} devised the Perceptual Loss for better generation quality. \nOther works have contributed to improve the result's visual quality \\cite{Li2019, Xin2017, Artsiom2018, Yulun2019}, inference speed \\cite{Li2016, Dmitry2016, CQF2017}, diversities \\cite{Vincent2017, Dongdong2017, Yijun2017, Xun2017, Yijun20172} and controllablity \\cite{Jing2017, Pierre2017, Gatys2017}.\nAs the application on facial images, several works addressed specific tasks, such as makeup\/de-makeup \\cite{Qiao2019,Wai2007,Ying2017}, photo-sketch translation \\cite{Mingrui2017,Peng2020}, face aging \\cite{face_aging_1,face_aging_2}, etc.\nComparing to previous works, the FSMA framework focuses on the few-shot model adaptation problem on the artistic stylization.\n\n\\vspace{1mm}\n\\noindent\\textbf{Facial shadow removal.}\nMost recent related works concentrated on the network designs, including using novel GAN \\cite{Wang_2018,Cun_2020} and RNN \\cite{Hu_2019,Ding_2019} structures. \nBesides, \\cite{face_shadow_removal} manually selected a few thousand images without shadow and trained its model on synthesized shadowed images.\nThe FSMA framework aims at training with minimal data and alleviate the labor cost when collecting images for training.\n\n\\section{The FSMA Framework }\nThe goal of the FSMA framework is to reduce the manpower spent on data annotation and collection during the development of a customized application for facial images.\nBy leveraging large-scale unlabeled images and then a tiny number of annotated data, the FSMA enables a deep learning model to achieve satisfying performance on a new task.\n\nThe FSMA's training pipeline has two major stages: the unsupervised learning and the supervised adaptation stage.\nFigure \\ref{fig:framework} illustrates the pipeline and the model's structure.\n\n\\subsection{Exploiting Feature via Unsupervised Learning}\nIn the unsupervised learning stage, an adversarial auto-encoder is trained.\nFollowing the framework proposed in \\cite{aae,3FacRec}, the overall optimization objective comprises four loss functions.\nAn $L_1$ pixel-wise image reconstruction loss $L_{rec}$ and an adversarial loss $L_{adv}$ that penalize the difference between input and reconstructed image are used to guarantee the high-fidelity reconstruction ability.\nAn encoding feature loss $L_{enc}$ is applied to the latent vector $z$ to ensure its smoothness and continuousness in the feature space.\nWe also maximize the SSIM \\cite{ssim} between input and output image as a regularization term $L_{ssim}$ to maintain the face structural information.\nThe final objective for this stage is a weighted sum of the loss functions in mention, which is\n\\[L_{unsup} = \\lambda_1 L_{rec} + \\lambda_2 L_{adv} + \\lambda_3 L_{enc} + \\lambda_4 L_{ssim}\\,.\\]\nThe loss weights are selected empirically where $\\lambda_1=\\lambda_2=\\lambda_3=1.0$, $\\lambda_4=60$.\n\n\\subsection{Supervised Few-Shot Adaptation}\n\\subsubsection{Modifications on Network's Structure}\nThe supervised few-shot adaptation learning aims to extract task-specific features from the acquired robust face representations, and facilitate the expected functionality with the extracted information.\nTherefore, the parameters of the encoder and decoder are fixed.\nThis operation also prevents the model from overfiting on the small-scale annotated data.\nTo adapt the model to a particular application, interleaved transfer layers \\cite{3FacRec} are added into the decoder.\nEach interleaved transfer layer is a $3\\times3$ convolution layer inserted before every upsampling operation.\nBesides, the original output layer is replaced with a newly initialized one with different output channel number to fit the targeted task.\n\n\\subsubsection{Adding Skip Layers}\nIn practice, we find that the model is not capable enough to transfer to a wide range of tasks if only equipped with the interleaved transfer layers.\nDue to its high compactness, the latent code $z$ only contains high-level information.\nThe decoder is not able to recover the detailed features that authentically represent the original facial image.\nThis factor always leads to inferior performance when the targeted task requires detailed features, e.g. for face segmentation, stylization and shadow removal.\n\nTo solve this problem, we introduce feature connections between the corresponding feature scales in encoder and decoder and add newly initialized adaptation layers in the connections.\nWe name these additional layers as the `skip layers'.\nA skip layer consists of two $3\\times3$ convolutional layers with nonlinear activations and feature normalizations.\nIt takes the feature from the encoder as input and its output is added to the input of the interleaved transfer layer.\nAn illustration of its detailed structure is shown in Figure \\ref{fig:skiplayer}.\n\n\\begin{figure}[t]\n    \\vspace{-2mm}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.95\\linewidth]{skiplayer.pdf}\n\t\\end{center}\n\t\\vspace{-5mm}\n\t\\caption{\n\t\tThe detailed structure of skip layers.\n\t\tThis figure takes the feature with 32 downsampling rate (32s) as an example.\n\t\tThe skip layer processes feature from the encoder and sum its output to the input for interleaved transfer layer.\n\t}\n\t\\label{fig:skiplayer}\n\t\\vspace{-4mm}\n\\end{figure}\n\nFor conventional auto-encoders, the skip layers are intrinsically prohibited during its training.\nIt is because the short-circuited feature connections break the `compression and de-compression' model structure and degrade the learned feature quality.\nHowever, the FSMA framework adopts skip layers during its supervised adaptation stage with its backbone fixed, which avoids the failure of unsupervised learning while offering multi-scale features as needed in the targeted task.\n\nThe skip layers are the critical components in achieving the FSMA's versatility over different types of facial image applications.\nThe absence of some or all skip layers not only leads to less robustness to data scarcity, but also directly causes malfunctioning in editing and enhancement tasks.\n\nTo give a concise representation, we use binary digits to denote the existence of skip layers.\nFor example, `00000' stands for a model without skip layer while `11111' means all the five feature scales have a skip layer. \n`11000' shows the presences for features with 32 and 16 times down-sampling.\n\\label{sec:digit}\nNote that the `0000'\/`00000' model is identical to \\cite{3FacRec}, which serves as a baseline in the ablation studies.\n\n\n\\subsubsection{Optimization Objective}\nThe optimization objective $L_{task}$ in the supervised model adaptation stage is task-specific and is commonly used in fully supervised methods.\nIts details are elaborated in Section \\ref{sec:app} for each application.\nMore detailed experiment settings are also contained in the supplementary file.\n\n\\section{Applications}\n\\label{sec:app}\nWithout loss of generality, we validate FSMA's adaptation performance and task versatility on face analysis, editing and enhancement applications.\nSpecific tasks include facial landmark detection and segmentation for face analysis, facial image stylization for face editing and facial shadow removal for face enhancement.\n\nBesides, we also introduce customized data annotation formats for these tasks.\nThe customized data not only serves for FSMA's validation but also complements existing common datasets and remedies their failures.\n\n\\begin{figure}[t]\n    \\vspace{-2mm}\n\t\\begin{center}\n\t\t\\includegraphics[width=1\\linewidth]{lm_annotations.pdf}\n\t\\end{center}\n\t\\vspace{-8mm}\n\t\\caption{\n\t\tA demonstration of (a) standard 68-point landmark format from 300W dataset and (b) the customized 44-point complementary annotation.\n\t}\n\t\\label{fig:lm_annotations}\n\t\\vspace{-5mm}\n\\end{figure}\n\n\n\\subsection{Facial Landmarks}\nFacial landmark detection locates semantic key points and describes the basic facial structure.\nTherefore, it is a fundamental component in many applications.\n\n\\textbf{Customized data.}\nThe public datasets adopt predefined standard landmark formats to ensure a model can match the same local area in different faces.\nFor example, the\n300W dataset uses a 68-point format where the keypoints distribute along the face edge and facial components. However, the drawbacks are also obvious. \nThe foreheads are empty, making the landmarks unable to bound the whole facial area.\nExisting formats focus much more on facial components than the face skin where it is less textured but still varies on geometry.\n \nTo overcome these issues, we define additional 44 landmarks as complementary to the 68-point format.\nAs shown in Figure \\ref{fig:lm_annotations}, the new keypoints mainly focus on the face skin, among which 13 points cover the forehead and hairline, ten points on the lower eye sockets to circle around the eyes' with original points on eyebrows, six points locate nostrils and alae while others define chin and cheeks.\nWe annotate only 20 images collected from the 300W dataset.\nThe annotation workload costs less than an hour for an engineer.\n\n\\textbf{Learning objective.}\nDuring the adaptation, the model generates a heatmap for each landmark with half of the input image's resolution.\nTherefore, there are four feature scales in the decoder and the binary skip layer indicator  has only four digits.\nThe $L_{task}$ is an $L_2$ distance between the ground truth and predicted heatmaps.\nExperiments are conducted on both the 300W and customized data.\n\n\n\n\\subsection{Face Semantic Segmentation}\nA face segmentation model is to assign each pixel with a class, which is a combination of high-level semantic analysis and low-level detailed structure prediction.\n\n\\textbf{Customized data.}\nSince this task requires pixel-wise ground truth, the annotation is highly time-consuming.\nAcademic researches have been based on small-scale datasets (e.g. LFW \\cite{LFW}, Helen \\cite{helen}) over a long time.\nEven the latest dataset \\cite{aaai20_face_parsing} contains much less images than those for other facial image tasks.\nExcept for the shortage of labeled images, the annotation format is also over-simplified.\nCurrent datasets divide the whole face into the face skin and facial components.\nSimilar to facial landmarks, the public segmentation datasets emphasize more on facial components than the variances inside face skin.\n\nHere we propose a format that complements the Helen's annotation.\nSpecifically, the original face skin is divided into the forehead, left and right cheeks, chin, under nose area and eye sockets.\nThe eye sockets are further partitioned into left\/right and upper\/lower sockets, which are four parts in total.\nThe masks for common accessories like glasses, beard and mustache are also labeled.\nFew examples are given in Figure \\ref{fig:seg_annotations}.\nThe more fine-grained definition enables a model to distinguish various facial textures and geometries, and improve the performance of applications such as 3D face reconstruction and animation.\n\n\n\\begin{figure}[t]\n\\vspace{-4mm}\n\t\\begin{center}\n\t\t\\includegraphics[width=1.01\\linewidth]{seg_annotations.pdf}\n\t\\end{center}\n\t\\vspace{-8mm}\n\t\\caption{\n\t\tA demonstration of (b) Helen dataset's annotation and (c) the customized complementary annotation, which correspond to the original images (a).\n\t}\n\t\\label{fig:seg_annotations}\n\t\\vspace{-4mm}\n\\end{figure}\n\n\\textbf{Learning objective.}\nDuring adaptation training, the model output predicted masks that have the same size as input images.\nThe cross-entropy loss is used as $L_{task}$. \n\n\n\\subsection{Facial Image Stylization}\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=1\\linewidth]{style_annotations.pdf}\n\t\\end{center}\n\t\\vspace{-8mm}\n\t\\caption{\n\t\tExamples of the customized data for few-shot stylization training.\n\t\tEach column contains an original image and its stylized counterpart.\n\t\tThere are two different visual styles (a) and (b).\n\t}\n\t\\label{fig:style_annotations}\n\t\\vspace{-5mm}\n\\end{figure}\n\nImage stylization translates an image to a different visual domain and brings global visual changes.\nThis technique has been widely applied in many applications, such as face cartoonization and artistic stylization.\n\n\\textbf{Customized Data.}\nBeing an image editing task, training a stylization model always lacks enough direct supervision.\nOn the one hand, it costs a long time for an artist to process or paint the paired illustrations on a large scale.\nOn the other hand, the industrial demand for high-quality stylization emerges and changes rapidly.\nIn this case, building a conventional fully supervised pipeline to develop a stylization model is not feasible.\n\nWe verify FSMA's capability for the few-shot style adaptation by training the model with a few pairwise data.\nTo this end, we randomly collect a few images from the CelebA-HQ \\cite{celeba,celebahq} dataset and generate two different art styles with a commercial application.\nThis process is equivalent to the artist's manual image processing.\nSeveral pairwise data are shown in Figure \\ref{fig:style_annotations}.\nThe chosen styles contain significant color and texture changes.\n\n\\textbf{Learning objective.}\nDuring the supervised adaptation training, the $L_{rec}$ and $L_{adv}$ used in the unsupervised learning stage are kept as the task-specific loss $L_{task}$.\n\n\\subsection{Facial Shadow Removal}\n\\begin{figure}[t]\n    \\vspace{-3mm}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.9\\linewidth]{shadow_annotations.pdf}\n\t\\end{center}\n\t\\vspace{-6mm}\n\t\\caption{\n\t\tExamples of the synthesized facial shadow.\n\t\t(a): the face areas.\n\t\t(b): images with random shadows.\n\t}\n\t\\label{fig:shadow_annotations}\n\t\\vspace{-5mm}\n\\end{figure}\nUnlike facial image stylization that performs global transformation over an image, facial image enhancement usually makes local visual changes.\nFacial shadow removal is one of the enhancement applications which aims to compensate for the brightness within the shadow on the face caused by foreign objects.\nThis technique improves an image's aesthetic quality, especially when the photographer is limited by the environmental lighting conditions.\n\n\\textbf{Few-shot data collection.}\nA recent solution \\cite{face_shadow_removal} is to generate synthetic shadows on facial images.\nIn this case, the human labor is involved in choosing the images without foreign shadows, or the `clean images'.\nTo synthesize training data, binary masks are first randomly generated using the Perlin noise or certain object silhouettes.\nThen the pixel intensities inside the masks are decayed to imitate illumination changes.\nSeveral synthesized images are shown in Figure \\ref{fig:shadow_annotations}.\nThe \\cite{face_shadow_removal} used 5000 clean images in total.\nThe data collection in this amount is demanding and it would be more difficult for other customized needs.\nAs for the FSMA framework, only a minimal number of clean images are collected (less to 1\\%)\n\n\\textbf{Learning objective.}\nThe $L_{task}$ used here is the same as that for the stylization task.\nTo focus on FSMA's adaptation ability, other task-specific techniques in \\cite{face_shadow_removal} are not used.\n\n\n\\begin{figure*}[t]\n    \\vspace{-3mm}\n\t\\begin{center}\n\t\t\\includegraphics[width=1\\linewidth]{res_skip_layer.png}\n\t\\end{center}\n\t\\vspace{-6mm}\n\t\\caption{\n\t\tQualitative comparisons of the models for (A),(B) face segmentation, (C) image stylization and (D) shadow removal  with skip layer setting of (b) 11111, (c) 11110, (d) 11100, (e) 11000, (f) 10000 and (g) 00000.\n\t\t(a) are input images from (A)(B) Helen validation set, (C) CelebA-HQ dataset and (D) validation images with synthesized shadows.\n\t}\n\t\\label{fig:res_skip_layer}\n\t\\vspace{-4mm}\n\\end{figure*}\n\n\\begin{table}[htbp]\n    \\vspace{-2mm}\n\t\\centering\n\t\\caption{The ablation study on the skip layer settings w.r.t. the model's landmark detection accuracy on 300W validation set.\n\tThe skip layer setting is defined in \\ref{sec:digit}.\n    Each result contains the inside\/outlier\/all landmark accuracies.\n}\n\\resizebox{.35\\textwidth}{!}{\n\t\\begin{tabular}{c|ccc}\n\t\t\\hline\n\t\t\\textbf{skip layer settings} & \\multicolumn{3}{c}{\\textbf{NME(\\%)}} \\\\\n\t\t\\hline\n\t\t0000  & 3.65  & 6.40  & 4.31  \\\\\n\t\t1000  & 3.64  & 6.31  & 4.29 \\\\\n\t\t1100  & 3.61  & 6.24  & 4.15 \\\\\n\t\t1110  & 3.42  & 6.17  & 4.13 \\\\\n\t\t1111  & \\textbf{3.12}  & \\textbf{6.14}  & \\textbf{3.88}  \\\\\n\t\t\\hline\n\t\\end{tabular}%\n}\n\t\\label{tab:skip_layer_ablation_lm}%\n\\end{table}%\n\n\\begin{table}[t]\n\t\\centering\n\t\\caption{\n\t    The ablation study on the skip layer     settings w.r.t. the model's segmentation   accuracy on Helen validation set.\n\t    The skip layer setting is defined in \\ref{sec:digit}.\n\t\tSkip layers and feature fusions are critical to model's performances.\n\t}\n\\resizebox{.46\\textwidth}{!}{\n\t\\begin{tabular}{c|c|c|c|c|c|c}\n\t\t\\hline\n\t\t\\textbf{skip layer} & \\multicolumn{6}{c}{\\textbf{F1 scores (\\%)}}  \\\\\n\t\t\\cline{2-7}    \\textbf{settings} & \\textbf{face} & \\textbf{eyebrows} & \\textbf{eyes} & \\textbf{nose} & \\textbf{mouth} & \\textbf{overall} \\\\\n\t\t\\hline\n\t\t00000 & 92.10  & 73.55  & 79.11  & 90.71  & 87.62  & 85.23  \\\\\n\t\t10000 & 92.83  & 75.15  & 80.00  & 92.36  & 88.25  & 86.60  \\\\\n\t\t11000 & 93.69  & 76.51  & 83.44  & 92.35  & 89.89  & 87.78  \\\\\n\t\t11100 & 94.23  & 78.33  & 85.81  & 93.22  & 91.28  & 89.21  \\\\\n\t\t11110 & 94.70  & \\textbf{81.31}  & \\textbf{87.91}  & 93.28  & 92.11  & 90.09  \\\\\n\t\t11111 & \\textbf{94.72}  & 80.74  & 87.81  & \\textbf{93.67}  & \\textbf{92.47}  & \\textbf{90.32}  \\\\\n\t\t\\hline\n\t\\end{tabular}%\n}\n\t\\label{tab:skip_layer_ablation_seg}%\n\\end{table}%\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=1\\linewidth]{res_lm_skip_layer-min.png}\n\t\\end{center}\n\t\\vspace{-6mm}\n\t\\caption{\n\t\tQualitative comparisons between the models with skip layer setting (a) 1111 and (b) 0000.\n\t\tMajor differences are highlighted.\n\t\n\t\tBest viewed with zoom in.\n\t}\n\t\\label{fig:res_lm_skip_layer}\n\t\\vspace{-4mm}\n\\end{figure}\n\n\\section{Ablation Studies \\& Results}\nDetailed experimental settings including dataset description, hyper-parameter selection and evaluation metrics for the unsupervised learning and adaptation are elaborated in the supplementary files.\n\n\\subsection{The Skip Layers and FSMA's Versatility.}\nThe skip layers are the main factor that contributes to FSMA's task versatility.\nThey improve the model's prediction accuracy on face analysis tasks while, on face editing and enhancement tasks, the skip layers determine the model's success and failure.\nFrom a quantitative view, Table \\ref{tab:skip_layer_ablation_lm}, \\ref{tab:skip_layer_ablation_seg} show that more skip layers promote landmark detection and segmentation consistently in all columns, which indicates that the improvement for feature quality is comprehensive.\nAs for qualitative comparisons, the predicted landmarks around the mouth and eyes shown in Figure \\ref{fig:res_lm_skip_layer} locate closer to the ground truth.\nFigure \\ref{fig:res_skip_layer}(A)(B) show that the skip layers not only help to parse more detailed structures, like the glass frame, but also can identify whether the person's eyes are closed or not.\nThe results of stylization models in Figure \\ref{fig:res_skip_layer}(C) indicate that the model where skip layers are absent fails to produce stylized detailed texture and facial structures.\nThe output images are degraded noticeably in columns (f) and (g).\nFurthermore, the models shown in Figure \\ref{fig:res_skip_layer}(D) with fewer and no skip layers fail to capture the shadow areas, not to mention to remove the shadow.\n\n\\begin{table*}[htbp]\n\t\\centering\n\t\\caption{The quantitative comparison with state-of-the-art facial landmark detection models under the few-shot training setting on 300W validation set.\n\tResults are reported under the NME(\\%) metric.\n\tThe three values in a single result stand for the performances of inside\/outlier\/all landmarks.\n\tThe training set size is proportional to 300W's training set.\n\tFSMA's binary digits indicate the presence of skip layers as defined in \\ref{sec:digit}.\n}\n    \\vspace{-4mm}\n    \\resizebox{.80\\textwidth}{!}{\n\t\\begin{tabular}{l|ccc|ccc|ccc|ccc}\n\t\t\\hline\n\t\t\\multirow{2}[4]{*}{\\textbf{models}} & \\multicolumn{12}{c}{\\textbf{training set size}} \\\\\n\t\t\\cline{2-13}          & \\multicolumn{3}{c|}{100\\%} & \\multicolumn{3}{c|}{20\\%} & \\multicolumn{3}{c|}{10\\%} & \\multicolumn{3}{c}{5\\%} \\\\\n\t\t\\hline\n\t\tRCN+\\cite{rcn}  & 4.20  & 7.78  & 4.90  & -     & 9.56  & 5.88  & -     & 10.35  & 6.32  & -     & 15.54  & 7.22  \\\\\n\t\tSA\\cite{sa}    & 3.21  & \\textbf{4.98}  & \\textbf{3.46}  & 3.85  & -     & -     & 4.27  & -     & -     & 6.32  & -     & - \\\\\n\t\tTS$^3$\\cite{TS3} & \\textbf{2.91}  & 5.90  & 3.49  & 4.31  & 7.97  & 5.03  & 4.67  & 9.26  & 5.64  & -     & -     & - \\\\\n\t\t3FabRec\\cite{3FacRec} & 3.36  & 5.74  & 3.82  & 3.76  & \\textbf{6.53}  & 4.31  & 3.88  & \\textbf{6.88}  & 4.47  & 4.22  & \\textbf{6.95}  & 4.75  \\\\\n\t\t\\hline\n\t\tFSMA  &       &       &       &       &       &       &       &       &       &       &       &  \\\\\n\t\t$\\,\\,$-0000 & 3.65  & 6.40  & 4.31  & 3.63  & 7.08  & 4.49  & 4.05  & 7.36  & 4.87  & 4.13  & 7.78  & 5.04 \\\\\n\t\t$\\,\\,$-1111 & 3.12  & 6.14  & 3.88  & \\textbf{3.23}  & 6.87  & \\textbf{4.14}  & \\textbf{3.59}  & 7.01  & \\textbf{4.45}  & \\textbf{3.85}  & 7.30  & \\textbf{4.71}  \\\\\n\t\t\\hline\n\t\\end{tabular}%\n\t}\n\t\\vspace{-2mm}\n\t\\label{tab:sota_lm}%\n\\end{table*}%\n\n\\begin{table*}[t]\n\t\\centering\n\t\\caption{The quantitative comparison with state-of-the-art face segmentation models on Helen validation set.\n\t* means the model utilizes extra annotations.}\n\t\\vspace{-2mm}\n    \\resizebox{.82\\textwidth}{!}{\n\t\\begin{tabular}{l|c|c|c|c|c|c|c|c}\n\t\t\\hline\n\t\t\\multirow{2}{*}{\\textbf{models}} & \\textbf{image} & \\multirow{2}{4em}{\\textbf{backbone}}  & \\multicolumn{6}{c}{\\textbf{F1 scores (\\%)}} \\\\\n\t\t\\cline{4-9}          & \\textbf{resolution} &       & \\textbf{face}  & \\textbf{eyebrows} & \\textbf{eyes}  & \\textbf{nose}  & \\textbf{mouth} & \\textbf{overall} \\\\\n\t\t\\hline\n\t\tMO-GC\\cite{cvpr15liusifei} & 250$\\times$250   & other & 91.0  & 71.3  & 76.8  & 90.9  & 84.1  & 84.7 \\\\\n\t\tiCNN\\cite{icnn}  & 256$\\times$256   & other & -     & 81.3  & 87.4  & \\textbf{95.0}  & \\textbf{92.6}  & 87.3  \\\\\n\t\tCNN-RNN\\cite{cnn_rnn_seg} & 256$\\times$256   & other  & 92.1  & 77.0  & 86.8  & 93.0  & 89.1  & 88.6  \\\\\n\t\tAdaptive RF\\cite{adaptiveRF} & 256$\\times$256   & VGG16 & 91.48  & 78.61  & 84.66  & 93.65  & 91.48  & 90.21  \\\\\n\t\tFSMA  & 256$\\times$256   & Res18  & \\textbf{94.72}  & \\textbf{80.74}  & \\textbf{87.81}  & 93.67  & 92.47  & \\textbf{90.32} \\\\\n\t\t\\hline\n\t\tBSPNet+LaPa*\\cite{aaai20_face_parsing} & 473$\\times$473   & Res101     & 95.1  & 81.9  & 87.8  & 94.7  & 93.8  & 91.4 \\\\\n\t\tAFSRTS*\\cite{tip_seg} & 512$\\times$512   & VGG16     & 95.62  & 83.88 & 89.92  & 94.73  & 94.31  & 92.04  \\\\\n\t\tTanh-Warp*\\cite{tanh_warp} & 512$\\times$512   & Res18     & 95.3  & 85.9  & 89.7  & 95.6  & 95.2  & 93.1  \\\\\n\t\t\\hline\n\t\\end{tabular}%\n}\n\t\\label{tab:sota_seg}%\n\t\\vspace{-4mm}\n\\end{table*}%\n\n\\begin{figure}[t]\n    \\vspace{-2mm}\n\t\\begin{center}\n\t\t\\includegraphics[width=1\\linewidth]{res_customized_lm_shot-min.png}\n\t\\end{center}\n\t\\vspace{-6mm}\n\t\\caption{\n\t\tQualitative comparisons of landmark detection models trained on (b) \\textbf{\\textit{20}}, (c) \\textbf{\\textit{10}}, (d) \\textbf{\\textit{5}} and (e) \\textbf{\\textit{1}} images.\n\t}\n\t\\label{fig:res_customized_lm_shot}\n\t\\vspace{-2mm}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=1\\linewidth]{res_customized_seg_shot-min.png}\n\t\\end{center}\n\t\\vspace{-6mm}\n\t\\caption{\n\t\tQualitative comparisons of face segmentation models trained on (b) \\textbf{\\textit{100}}, (c) \\textbf{\\textit{50}}, (d) \\textbf{\\textit{25}} and (e) \\textbf{\\textit{10}} images.\n\t}\n\t\\label{fig:res_customized_seg_shot}\n\t\\vspace{-2mm}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=1\\linewidth]{res_style_shot.png}\n\t\\end{center}\n\t\\vspace{-6mm}\n\t\\caption{\n\t\tQualitative comparisons of the FSMA stylization models with (b) \\textbf{\\textit{50}}, (c) \\textbf{\\textit{25}}, (d) \\textbf{\\textit{15}} and (e) \\textbf{\\textit{10}} `clean images'.\n\t}\n\t\\label{fig:res_style_shot}\n\t\\vspace{-4mm}\n\\end{figure}\n\n\\subsection{Effect of the Amount of Training Data}\nFigure \\ref{fig:res_customized_lm_shot}-\\ref{fig:res_shadow_shot} demonstrate the qualitative comparisons among the FSMA models given various numbers of training images.\nFor all the tasks except stylization, the models generate visually satisfying results with less to 1\\% training supervision compared to previous fully supervised models.\nMeanwhile, as presented in Figure \\ref{fig:res_style_shot}, the models correspond to column (e) uses only 10 pairwise training data while maintaining most expected visual details in its outputs.\nFrom the quantitative aspect, Table \\ref{tab:sota_lm} shows that the FSMA model has the least performance drop compared with state-of-the-art few-shot models.\nIn table \\ref{tab:image_number_ablation_seg}, the FSMA model is much more robust to data scarcity than the ResNet18 \\cite{resnet} based UNet \\cite{unet} model, which is identical to the `11111' model's structure but is initialized with ImageNet \\cite{imagenet} pretrained weights.\nThe FSMA segmentation model also outperforms many fully supervised state-of-the-art methods with only 5\\% training data on Helen dataset.\n\n\\subsection{Comparison with State-of-the-Arts}\nWe compare the FSMA facial landmark detection with recent models under the few-shot setting and face segmentation model with state-of-the-art methods under the fully supervised setting.\nTable \\ref{tab:sota_lm} demonstrates the FSMA's superiority for landmarks on facial components and the overall landmarks given less training images.\nNote that the `0000' FSMA model is theoretically identical to \\cite{3FacRec}.\nFor the segmentation model compared in in Table \\ref{tab:sota_seg}, the FSMA model with the same input image resolution and supervision outperforms other models even with a less powerful backbone.\n\n\\begin{table}[t]\n\t\\centering\n\t\\caption{\n\t\tThe ablation study on the number of training images w.r.t. the model's F1 score on Helen val set.\n\t\tThe training set size in percentage is proportional to Helen's training set.\n\t    The skip layer setting is defined in \\ref{sec:digit}.\n\t    The UNet model has the same structure as FSMA's `11111' model but is initialized with the ResNet-18 pretrained on ImageNet.\n\t}\n\n\\resizebox{.48\\textwidth}{!}{\n\t\\begin{tabular}{c|c|c|c|c|c|c}\n\t\t\\hline\n\t\t\\multirow{2}{3em}{\\textbf{models}} & \\multicolumn{6}{c}{\\textbf{training set size}} \\\\\n\t\t\\cline{2-7}          & 100\\% & 25\\%  & 5\\%   & 2.5\\% & 0.5\\% & 5 \\\\\n\t\t\\hline\n\t\tUNet\\cite{unet} & 89.53  & 88.66  & 87.09  & 82.71  & 75.49  & 71.19   \\\\\n\t\t\\hline\n\t\tFSMA  &       &       &       &       &       &  \\\\\n\t\t$\\,\\,\\,$-00000 & 85.23  & 84.99  & 84.28  & 82.77  & 78.72  & 75.21  \\\\\n\t\t$\\,\\,\\,$-11111 & \\textbf{90.32}  & \\textbf{90.08}  & \\textbf{89.10}  & \\textbf{87.01}  & \\textbf{84.53}  & \\textbf{81.87}  \\\\\n\t\t\\hline\n\t\\end{tabular}%\n}\n\t\\label{tab:image_number_ablation_seg}%\n\t\\vspace{-4mm}\n\\end{table}%\n\n\\begin{figure}[t]\n\\begin{center}\n\t\\includegraphics[width=1\\linewidth]{res_shadow_shot-min.png}\n\\end{center}\n\\vspace{-4mm}\n\\caption{\n\tQualitative comparisons of facial shadow removal models with (b) \\textbf{\\textit{2500}}, (c) \\textbf{\\textit{500}}, (d) \\textbf{\\textit{100}} and (e) \\textbf{\\textit{50}} `clean images' manually selected for training.\n\t(a) are images with natural shadow.\n\tFor a comparison, \\cite{face_shadow_removal} used \\textbf{\\textit{5000}} `clean images' under the same training protocol.\n}\n\\label{fig:res_shadow_shot}\n\\end{figure}\n\n\\begin{table*}[tbp]\n\t\\centering\n\t\\caption{Empirical comparisons on annotation workloads between the conventional fully supervised pipeline and the few-shot learning pipeline with FSMA.}\n\t\\vspace{-2mm}\n\t\\resizebox{.75\\textwidth}{!}{\n\t\t\t\\begin{tabular}{l|llll|cl}\n\t\t\t\t\\hline\n\t\t\t\t& \\multicolumn{4}{l|}{\\textbf{man-hour\/day for labeling 2k images}} & \\multicolumn{2}{l}{\\textbf{man-hour\/day w\/ FSMA}}\\\\\n\t\t\t\t\\cline{2-7} \\textbf{tasks}     & \\multicolumn{1}{c}{first round } & \\multicolumn{1}{c}{\\multirow{2}[2]{*}{check}} & \\multicolumn{1}{c}{\\multirow{2}[2]{*}{correction}} & \\multicolumn{1}{c|}{\\multirow{2}[2]{*}{overall}} & \\multicolumn{1}{c}{annotated} & overall\\\\\n\t\t\t\t& \\multicolumn{1}{c}{annotation} &            &            &            & \\multicolumn{1}{c}{images} & \\\\\n\t\t\t\t\\hline\n\t\t\t\tlm.        & 100h\/12.5d   & 2h$\\;\\;\\,$\/0.3d     & 2.8h$\\;\\;$\/0.4d     & 108h\/13.3d   & 100        & 6h$\\;\\;$\/0.8d\\\\\n\t\t\t\tseg.       & 500h\/42.5d & 3.2h\/0.4d     & 16.8h\/2.1d    & 522h\/45.2d & 50         & 12h\/1.5d \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}%\n\t}\n\t\\label{tab:annotation_worklad}\n\\end{table*}%\n\n\\begin{figure}[t]\n\\begin{center}\n\t\\includegraphics[width=1\\linewidth]{res_cat_dog.pdf}\n\\end{center}\n\\vspace{-2mm}\n\\caption{\n\t\tThe customized landmark detection results of the FSMA-1111 model applied on the non-human facial images.\n\t\t15\\% of StanfordExtra \\cite{dog_dataset} and Cat-Head \\cite{cat_dataset} training images are used for the few-shot adaptations.\n}\n\\label{fig:cat_dog}\n\\end{figure}\n\n\n\\subsection{Reduction of Annotation Workloads}\nIn this paper, we emphasize on extending the few-shot model's functionalities to a wider range of customized facial image applications and producing practical values.\nTo demonstrate the significant advantages of FSMA in reducing the amount of annotation workloads, we conduct quantitative comparisons based on our empirical observations before and after deploying the FSMA.\nWe choose the landmark detection and segmentation tasks in this study since they have more predictable overall workloads.\nWe compare the man-hour\/day costs of labelling 2k images following a conventional fully-supervised training pipeline as well as labelling a small number of data with the few-shot adaptation pipeline FSMA.\nThe workloads in both settings are minimum in order to achieve adequate model performances in developing customized applications for facial images.\nAs shown in Table \\ref{tab:annotation_worklad}, the overall labor costs have drastically decreased with the usage of FSMA.\n\n\\subsection{Extendibility on Non-Human Facial Images}\nFSMA's efficacy on non-human faces is also investigated.\nWe verify the detection ability for customized landmarks on cats' and dogs' faces using the StanfordExtra \\cite{dog_dataset} and Cat-Head \\cite{cat_dataset} dataset.\nTo process the data, the head regions are detected and cropped out, contributing to 12000 and 7008 images during the unsupervised pretraining for dogs and cats, respectively.\nWith more than 100 times less data for pretraining comparing to human facial images, the FSMA manages to obtain descent performances with 15\\% of the original training sets for adaptation.\nFigure \\ref{fig:cat_dog} shows the examples.\n\n\n\n\\section{Conclusion}\nIn this paper, we have proposed a novel Few-Shot Model Adaptation (FSMA) framework and demonstrate its performance and versatility on various important computer vision tasks: facial landmark detection, face semantic segmentation, facial image stylization and facial shadow removal. These tasks requires both high-level semantic understanding and low-level detail reconstruction. In the unsupervised training stage, our adversarial auto-encoder is trained with millions of face images to capture face knowledge through reconstruction. In the model adaption stage, our design of skip layers, interleaved layers and output layers have provided a pathway to reconstruct the output of different tasks in high quality. Thus, enabling us to achieve the state-of-the-art performance in few-shot landmark detection, and satisfying good results for other tasks even with very limited data. We hope our FSMA framework can inspire and spark future research in few-shot learning applications.\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \\label{Ss.Intro}\n\nWe consider {\\it combinatorial games} of two players;\nthey take turns alternating and one who makes the last move wins.\nBoth players have perfect information and there are no moves of chance.\nA game is called {\\it impartial} if both players\nhave the same possible moves in each position and\n\\emph{acyclic} if it is impossible to revisit the same position.\nIn this paper we consider only impartial acyclic combinatorial games\nand call them simply games.\nA more detailed introduction to combinatorial games can be found in \\cite{BCG01-04, Con76}.\n\nIf there is a move from position $x$ to $y$, we write $x \\to y$.\nFor a set $S$ of nonnegative integers the {\\it minimum excluded value} of $S$\nis defined as the smallest nonnegative integer that is not in $S$ and\nis denoted by  $\\mex (S)$. In particular, $\\mex(\\emptyset) = 0$.\nThe {\\it Sprague-Grundy} (SG) value of a position $x$\nin a game $G$ is defined recursively by  $$\\G(x) = \\mex \\{g(y) \\mid x \\to y\\}.$$\nA position of SG value $t$ is called a {\\it $t$-position};\n$0$-positions are also known as $\\P$-positions.\nA player who moves into a $\\P$-position can win the game.\nThe SG function is instrumental in the theory of disjunctive sums of games;\nsee \\cite{BCG01-04, Con76, Gru39, Spr35, Spr37}.\n\n\nA classical example is {\\sc Nim} studied by Bouton \\cite{Bou901}.\nA position in {\\sc Nim} consists of $n$ piles of tokens.\nTwo players alternately choose one of the piles and remove\nan arbitrary (positive) number of tokens from that pile.\nBouton characterized the $\\P$-positions and in fact described the SG function of {\\sc Nim}.\n\nMoore \\cite{Moo910} introduced a generalization in which\na player can remove tokens from at least one and at most  $k$  of the piles, for some fixed  $k<n$.\nWe call this game Moore's {\\sc Nim} and denote it by {\\sc Nim}$^\\leq_{n, k}$.\nWhen $k=1$ {\\sc Nim}$^\\leq_{n, 1}$ is the traditional {\\sc Nim}.\n\nMoore generalized Bouton's results and\ncharacterized the $\\P$-positions of {\\sc Nim}$^\\leq_{n, k}$ as follows.\nWe denote by $\\Z_\\geq$ the set of nonnegative integers and\nby $x \\in \\Z^n_\\geq$ a position of the game,\nwhere $x_i$ is the number of tokens in pile $i$.\nLet us represent the components of $x$ in binary form\n$x_i=\\sum_{j}x_{ij}2^j$, $i=1,...,n$, define $y_j=\\sum_{i=1}^n x_{ij} \\mod (k+1)$, and set\n\\[\nM(x)=\\sum_{j} y_j (k+1)^j.\n\\]\nFor instance, if  $n=3$, $k=2$, and $x=(2,3,6)$, then using $3$ binary digits we can write:\n$2=010$, $3=011$, and $6=110$ yielding\n$y_0=1\\mod 3=1$, $y_1=3\\mod 3=0$, and $y_2=1\\mod 3=1$,\nfrom which we get $M(x)=1\\cdot 3^0+0\\cdot 3^1+1\\cdot 3^2=10$.\n\nMoore proved that $x$ is a $\\P$-position of {\\sc Nim}$^\\leq_{n, k}$ if and only if  $M(x)=0$.\nBerge \\cite[Theorem 3, page~55]{Ber62} claimed that  $\\G(x)$ is simply equal to  $M(x)$.\nHowever, Jenkyns and Mayberry \\cite{JM80} pointed out that this is an overstatement and\nthe equality holds only when  $M(x) \\leq 1$ or $\\G(x)\\leq 1$.\nFor example, direct calculations show that\n$2 = \\G(x) < M(x) = 3$   for  $k = 2, n = 3, x = (0,0,2)$ and\n$8 = \\G(x) > M(x) = 2$   for  $k = 2, n = 3, x = (2,3,3)$.\n\nFor the case of $n = k+1$ Jenkyns and Mayberry \\cite{JM80}\nprovided a formula for the SG function of {\\sc Nim}$^\\leq_{k+1, k}$.\nAn alternative proof for a slightly more general game\nwas given recently in \\cite{BGHM15}.\n\nIn this paper we introduce another generalization of {\\sc Nim}.\nGiven positive integers $n$  and  $k$  such that $1 \\leq k \\leq n$,\nwe define {\\sc Exact $k$-Nim}, denoted by {\\sc Nim$^=_{n,k}$},  as follows.\nGiven $n$ piles of tokens, by one move a player chooses exactly $k$ piles and\nremoves arbitrary positive number of tokens from each of them.\nThe game terminates when there are less than  $k$  nonempty piles.\n{\\sc Nim}$^=_{n, k}$  turns into the standard {\\sc Nim} when  $k=1$ and\nit is the trivial one-pile {\\sc Nim} when $k=n$.\n\n\\subsection*{Main results}\n\nGiven a position $x\\in \\Z_\\geq^n$ of {\\sc Nim$^=_{n,k}$},\nwe denote by $T_{n,k}(x)$ the maximum number of consecutive moves\none can make starting with  $x$.\nWe call $T_{n,k}$ the \\emph{Tetris} function of the game.\n\nThe following two theorems characterize the SG function of {\\sc Nim$^=_{n,k}$} for $2k\\geq n$.\n\\begin{theorem} \\label{thm.n<2k}\nIf $n < 2k$ then the SG function of {\\sc Nim$^=_{n,k}$} is\nequal to its Tetris function,  $\\G(x) = T_{n,k}(x)$.\n\\end{theorem}\n\n\\begin{theorem} \\label{thm.n=2k}\nLet $k \\geq 2$, $n = 2k$, and\nlet $x = (x_1, \\ldots, x_n)$ be a position of {\\sc Nim}$^=_{2k, k}$.\nSet\n\\begin{align}\nu(x) &= T_{2k,k}(x),  \\label{eq:u}\\\\\nm(x) &= \\min_{1\\leq i\\leq 2k} x_i,\\label{eq:m}\\\\\ny(x) &= T_{2k,k}\\big(x_1 - m(x), \\ldots, x_{2k} - m(x)\\big),\\label{eq:y}\\\\\nz(x) &= 1+\\binom{y(x)+1}{2}, \\text{ and}\\label{eq:z}\\\\\nv(x) &= \\big(z(x)-1\\big) +\\big[\\big(m(x)-z(x)\\big)\\mod \\big(y(x)+1\\big)\\big].\\label{eq:v}\n\\end{align}\nThen the SG function of {\\sc Nim}$^=_{2k,k}$ is given by formula\n\\begin{equation}\\label{eq-binomial;n=2k-Formula}\n\\G(x) ~=~ \\begin{cases}\nu(x), & \\text{if~~}  m(x) < z(x);\\\\\nv(x), & \\text{if~~}  m(x) \\geq z(x).\n\\end{cases}\n\\end{equation}\n\\end{theorem}\n\nNote that this formula fails for  $k = 1$.\nIn this case {\\sc Nim}$^=_{2, 1}$ is the standard $2$-pile {\\sc Nim} and\nits SG function is the modulo $2$ sum of the two coordinates of a position,\nas described by Bouton.\nThis function is different from the one described by the above formula.\n\nWe also would like to remark that the above formula is surprisingly\nsimilar to the one given by Jenkyns and Mayberry\nin \\cite{JM80} for {\\sc Nim}$^\\leq_{k+1, k}$.\n\n\\bigskip\n\nThe above result implies a simple characterization of\nthe $0$- and $1$-positions of {\\sc Nim}$^=_{2k, k}$.\nA position $x = (x_1, \\ldots, x_n)$ is said to be\n\\emph{nondecreasing} if $x_1 \\leq x_2 \\leq \\cdots \\leq x_n$.\n\n\n\\begin{corollary}   \\label{cor-0,1-exact:n=2k}\nGiven a nondecreasing position $x = (x_1, \\dots, x_{2k})$ of the game {\\sc Nim}$^=_{2k, k}$,\n\\begin{enumerate} \\itemsep0em\n\\item [\\rm{(i)}] $x$  is a $\\P$-position if and only if\nmore than half of its smallest coordinates are equal, that is,  $x_1 = \\cdots = x_k = x_{k+1}$.\n\\item [\\rm{(ii)}] $x$  is a $1$-position if and only if $x_1 = \\cdots = x_{k - \\ell} = 2c$  and\n$x_{k-\\ell+1} = \\cdots = x_{k+ \\ell + 1} = 2c+1$\nfor some integer  $c \\in \\ZZ_\\geq$  and  $\\ell \\in \\{0,1, \\ldots, k-1\\}$.\n\\end{enumerate}\n\\end{corollary}\n\nBoth statements \\rm{(i)} and  \\rm{(ii)} follow from Theorem \\ref{thm.n=2k},\nbut can also be derived much simpler, directly from the definitions.\n\n\\medskip\n\nThe case of $2k < n$ looks much more difficult and it is still open.\nMoreover, we have not even been able to characterize\nthe $\\P$-positions of {\\sc Exact $k$-Nim} for $1 < k < n\/2$,\ne.g., for {\\sc Nim}$^=_{5,2}$.\n\n\\bigskip\n\nThe rest of the paper is organized as follows.\nIn Section \\ref{Ss.n<2k} we characterize the SG function of {\\sc Nim$^=_{n,k}$} for the case $2k>n$.\nIn Section \\ref{Ss.n=2k} we characterize the SG function of {\\sc Nim$^=_{n,k}$} for the case $2k=n$.\nIn Section \\ref{Ss.Moore01} we provide an alternative proof for the above stated result of Jenkyns and Mayberry \\cite{JM80}.\nFinally in Section \\ref{Ss.Tetris} we show that for a given position we can compute efficiently the corresponding SG value.\n\n\n\n\n\\section{SG function in the case of $n < 2k$} \\label{Ss.n<2k}\n\n\nFor our proof we need the following basic properties of the Tetris function.\nGiven positions $x,x'\\in\\Z^n_\\geq$ we write $x\\leq x'$ if $x_i\\leq x'_i$ holds for $i=1,...,n$.\n\n\\begin{lemma}\\label{tbasic}\nConsider two positions $x, x'\\in\\ZZ^n_\\geq$.\n\\begin{itemize}\n\\item [\\rm{(i)}] If $x'\\leq x$ then $T_{n,k}(x')\\leq T_{n,k}(x)$.\n\\item [\\rm{(ii)}] If in addition we have $\\sum_{i=1}^n(x_i-x_i')=1$, then $T_{n,k}(x)-1\\leq T_{n,k}(x')\\leq T_{n,k}(x)$.\n\\end{itemize}\n\\end{lemma}\n\\proof\nIt is immediate by the definition.\n\\qed\n\nA move in {\\sc Nim}$^=_{n, k}$ is called {\\it slow} if exactly one token is taken from each of the $k$ chosen piles.\n\n\n\n\\begin{lemma}\\label{Tetrisnotdecrease}\nConsider a position $x=(x_1, \\ldots, x_n) \\in \\Z^n_\\geq$ with some indices $i, j$ such that\n$x_i < x_j$. Let $x'=(x'_1, \\ldots, x'_n)$ be defined by\n\\begin{equation}\\label{eq-l2}\nx_l'=\n\\begin{cases}\nx_i+1, & \\text{ if } l=i, \\\\\nx_j-1, & \\text{ if } l=j, \\\\\nx_l,   & \\text{ otherwise. }\\\\\n\\end{cases}\n\\end{equation}\nThen we have $T_{n,k}(x) \\leq T_{n,k}(x')$.\n\\end{lemma}\nIn other words, the Tetris function is nondecreasing when\nwe move a token from a larger pile to a smaller one.\n\n\n\\proof\nConsider any sequence of slow moves from $x \\to \\cdots \\to x''$. If $x''_j>0$  then the same sequence of slow moves can be made from $x'$ since $x'_l \\geq x_l$ for $l \\neq j$.\n\nIf $x''_j=0$ then since $x_j > x_i$, this sequence contains a slow move reducing $x_j$ but not $x_i$.\nLet us modify this move reducing $x_i$ rather than $x_j$ and keeping all other moves of the sequence unchanged.\nThe obtained sequence has the same length and consists of slow moves from $x'$. \\qed\n\nNotice that we can generalize  Lemma \\ref{Tetrisnotdecrease}\nreplacing $\\pm 1$ in \\raf{eq-l2} by $\\pm \\Delta$ for any integral $\\Delta \\in [0,x_j-x_i]$.\n\n\n\\begin{lemma}\\label{bestslowmove}\nThe slow move that reduces the $k$ largest piles of $x$ reduces the Tetris value $T_{n,k}(x)$ by exactly one.\n\\end{lemma}\n\n\\proof\nLet $x'$ be the position obtained from $x$ by reducing the $k$ largest piles of $x$ by exactly one each. Let $x''$ be another position obtained by some slow move.\nBy applying \\raf{eq-l2} repeatedly, we can obtain $x'$ from $x''$ with $T_{n,k}(x'') \\leq T_{n,k}(x')$ by Lemma \\ref{Tetrisnotdecrease}.\nThis implies that $x'$ has the  highest Tetris value among all positions each reachable from x by a slow move. By Lemma \\ref{tbasic},\neach slow move reduces the Tetris value by at least one and there exists a slow move reducing it by exactly one.\nHence, $T_{n,k}(x')=T_{n,k}(x)-1$. \\qed\n\n\\bigskip\n\\noindent\\textbf{Proof of Theorem }\\ref{thm.n<2k}:\nThe Tetris value is the largest number of moves one can take from a position $x$,\nimplying that $\\G(x)$ is at most $T_{n,k}(x)$.\nTherefore, it is enough to show that for all integral $g$ such that\n$0 \\leq g < T_{n,k}(x)$ there exists a move $x\\to x'$ such that $T_{n,k}(x')= g$.\n\nConsider the move $x \\to x'$ that reduces the largest $k$ piles to $0$.\nFor the resulting position we have $T_{n,k}(x')=0$ because $2k>n$.\nLet us also consider the move $x \\to x''$ that reduces the $k$ largest piles each by only $1$.\nThen we have $T_{n,k}(x'')=T_{n,k}(x)-1$ according to Lemma \\ref{bestslowmove}.\nAny position between $x'$ and $x''$ is reachable form $x$. Thus, by Lemma \\ref{tbasic} the claim follows.  \\qed\n\n\\section{SG function in case of $n=2k$} \\label{Ss.n=2k}\n\nLet us consider the game {\\sc Nim}$^=_{2k, k}$, where $k\\geq 2$, and let $x$ be a position of this game.\nRecall that to $x$ we associated several parameters in \\eqref{eq:u}--\\eqref{eq:v}.\nBased on these parameters, we classify the positions into the following two types:\n\\begin{itemize} \\itemsep0em\n\\item [\\rm{(i)}]   \\emph{type I}, if $m(x)<z(x)$, and\n\\item [\\rm{(ii)}]  \\emph{type II} otherwise.\n\\end{itemize}\n\n\nWe shall need some technical lemmas for our proof. In this section, all positions belong to the game {\\sc Nim}$^=_{2k, k}$.\n\n\\begin{lemma} \\label{continuity}\nConsider a position $x$ and two moves $x\\to x'$ and $x\\to x''$ such that $x'\\geq x''$ $($componentwise$)$ and $y(x')\\geq y(x'')$.\nThen, for every integer $g$ with $y(x')\\geq g \\geq y(x'')$ there exists a move $x\\to x'''$ such that $y(x''')= g$ and $x'\\geq x'''\\geq x''$.\n\\end{lemma}\n\\proof\nNote that $x'\\geq x''$ implies that in the two moves $x\\to x'$ and $x\\to x''$ the same $k$ components are decreased.\nLet us denote by $K\\subseteq \\{1,2,...,2k\\}$ these components.\nLet us now start decreasing the components $x'_j$, $j\\in K$ one by one,\nkeeping their values always greater than the corresponding $x''_j$ values.\nAfter $\\sum_{j\\in K} x'_j-x''_j$ steps we reach $x''$.\nIn each of these steps the corresponding value $y$ can only decrease, and by at most $1$.\nHence there will be at least one such position $x'\\geq x'''\\geq x''$ with $y(x''') = g$.\nIt also follows that $x\\to x'''$ is a move, completing the proof.\n\\qed\n\n\nLet us also note that in fact $z$ defined in \\eqref{eq:z} depends uniquely on $y(x)$ and hence can be considered as a function of $y$.\nWe shall also need the following easy arithmetical facts.\n\\begin{lemma}\\label{interval}\nEvery nonnegative integer $g$ belongs to exactly one of the intervals\n$$[z(y)-1, z(y)+y-1] = \\left[\\binom{y+1}{2},\\binom{y+2}{2}-1\\right] \\textrm{ for }y\\in \\{0, 1, 2, ...\\}.$$\n\\qed\n\\end{lemma}\n\\begin{corollary}\\label{delta-epsilon}\nFor every nonnegative integer $g$, there exist unique $\\nu(g)$ and $\\eps(g)$ integer values such that\n\\[\ng ~=~ \\binom{\\nu(g)+1}{2} +\\eps(g) ~~~\\textbf{ and }~~~ 0\\leq \\eps(g)\\leq \\nu(g).\n\\]\n\\end{corollary}\n\n\\smallskip\n\nGiven a nondecreasing position $x$, let us construct another position $\\bar x$ from $x$ by emptying the first $n-k$ piles and adding these $\\sum_{i=1}^{n-k}x_i$ tokens, one by one, to the last $k$ piles as follows:\nIn each step we add one token to the smallest of these $k$ piles.\nIf there are several such piles, we break the tie by adding this token to the pile of the largest index.\n\nIt is easy to see that we have $T_{n,k}(\\bar x)=\\min(\\bar x_i  |  n-k+1 \\leq i \\leq n)=\\bar x_{n-k+1}$.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[height=6cm]{Doc1}\n\\caption{$k=4,n=7$, $x=(1,2,2,3,4,4,7)$, and $\\bar x=(0,0,0,5,5,6,7)$.} \\label{ADD(1)}\n\\end{figure}\n\n\n\n\\begin{lemma}\\label{txequalstxbar}\nThe above construction of $\\bar x$ from $x$ keeps the Tetris value unchanged: $T_{n,k}(\\bar x)=T_{n,k}(x)$.\n\\end{lemma}\n\\proof\nLet us note that $T_{n,k}(\\bar x) \\leq T_{n,k}(x)$ by Lemma \\ref{Tetrisnotdecrease}.\nBy the above definition of $\\bar x$, none of the tokens from the smallest $n-k$ piles of $x$ are moved to any pile of size larger than $T_{n,k}(\\bar x)+1$ and hence we have\n\\begin{equation}\\label{volumeargument}\nV(x):=\\sum_{i=1}^n \\min(x_i,T_{n,k}(\\bar x)+1) =\\sum_{i=n-k+1}^n \\min(\\bar x_i,T_{n,k}(\\bar x)+1).\n\\end{equation}\nSince $\\bar x_{n-k+1}=T_{n,k}(\\bar x)$, we get by \\eqref{volumeargument} that\n\\begin{equation*}\nV(x)  \\leq k-1+ k T_{n,k}(\\bar x) < k(T_{n,k}(\\bar x)+1).\n\\end{equation*}\nAssume now indirectly that $T_{n,k}(\\bar x) < T_{n,k}(x)$.\nThen it is possible to construct a  sequence of $T_{n,k}(\\bar x)+1$ slow moves from $x$.\nBy such sequence any pile would be reduced at most $T_{n,k}(\\bar x)+1$ times,\nand therefore the total number of the removed tokens is at most $V(x)$,\nimplying $k(T_{n,k}(\\bar x)+1)\\leq V(x)$, contradicting the above inequality.\nThe obtained contradiction implies that $T_{n,k}(x)=T_{n,k}(\\bar x)$. \\qed\n\n\\begin{lemma}\\label{getmeTetris}\nLet $x = (x_1, \\ldots, x_n)$ be a position.\nIf there exists a nonnegative integer $g$ such that\n\\begin{equation}\\label{eq-getmeTetris}\nk g \\leq \\sum_{i=1}^{n} \\min(x_i,g) \\quad \\text{ and } \\quad \\sum_{i=1}^{n} \\min(x_i,g+1) < k(g+1),\n\\end{equation} \\label{E.delta}\nthen we have $T_{n,k}(x)=g$.\n\\end{lemma}\n\\proof\nThe sequence $s(g)=\\frac{1}{g}\\sum_{i=1}^n \\min(x_i,g)$ is monotone non-increasing for $g \\in \\ZZ_\\geq$,\nand hence, the inequalities \\raf{eq-getmeTetris} can hold for at most one $g$.\nWithout any loss of generality we can assume that $x$ is nondecreasing, and thus we can define $\\bar x$, as above.\nLemma \\ref{txequalstxbar} then implies that \\raf{eq-getmeTetris} holds for $g=T_{n,k}(x)=T_{n,k}(\\bar x)$. \\qed\n\n\\bigskip\n\n\\noindent{\\bf Proof of Theorem \\ref{thm.n=2k}.}\nOur main result claims that the SG function $\\G(x)$ of\n{\\sc Nim}$^=_{2k, k}$ for $k\\geq 2$ is equal to the function\n\n$$\ng(x)=\\begin{cases}\nu(x), & \\text{if }m(x) < z(x); \\\\\nv(x), & \\text{if }m(x) \\geq z(x);\n\\end{cases}\n$$\nwhere $ u,m,y,v,$ and $z$ are defined in \\eqref{eq:u}--\\eqref{eq:v}.\nTo prove this theorem, it is enough to show the following two properties of $g(x)$:\n\\begin{itemize}\n\\item[(I)] for any move $x \\rightarrow x'$ we have $g(x) \\neq g(x')$ and\n\\item[(II)] for every value $\\delta$ such that $0 \\leq \\delta < g(x)$ there exists a move $x \\to x'$ with $g(x')=\\delta$.\n\\end{itemize}\n\nWe assume that $x$ is nondecreasing.\n\n\\subsection{Proof of (I)}\n\nWe now prove property (I). Let us first consider the case $z(x) > m(x)$.\nObviously, any move $x \\rightarrow x'$ reduces the Tetris value by at least 1, implying $u(x) > u(x')$.\nUsing this and the definitions, we get $g(x)=u(x) > u(x')$. If $g(x') = u(x')$ then by the above inequality, we get $g(x) \\neq g(x')$.\nOn the other hand, if $g(x') = v(x')$ then we  must have $m(x') \\geq z(x')$ and thus $v(x')=z(x')-1 + \\left((m(x')-z(x')) \\mod (y(x')+1)\\right)\\leq z(x')-1 + (m(x')-z(x'))=m(x')-1$.\nThus, we get $g(x) = u(x) > u(x') > m(x')-1 \\geq  v(x') = g(x')$.\n\nIt remains to consider the case $z(x) \\leq m(x)$, in which case $v(x)\\leq m(x)-1$ follows by the definitions. \n\t\\begin{enumerate} \\itemsep0em\n\t\\item Suppose $z(x') > m(x')$.\n    We can estimate $u(x') \\geq m(x)+m(x')$ since $2k = n$.\n    Furthermore, $g(x)=v(x) \\leq m(x)-1$ and thus $g(x')=u(x') \\geq m(x)+m(x') > m(x)-1 \\geq g(x)$.\n    \\item Suppose $z(x') \\leq m(x')$. Then $g(x')=v(x')$.  Note that $z(x) \\leq m(x)$ and thus $g(x)=v(x)$.\n    Also note that $m(x') \\leq m(x)$. We examine the last inequality.\n    \t\\begin{enumerate} \\itemsep0em\n\t    \\item Suppose $m(x')=m(x)$.\n               \n                Then $x-m(x)\\to x'-m(x')$ is a move, and hence decreases the Tetris value implying $y(x)>y(x')$.\n\t           By Lemma \\ref{interval}, we have\n                $z(x')+y(x')-1 < z(x)-1$\n                and so\n                $z(x)  > z(x') + y(x')$\n                implying\n                $v(x) \\geq z(x)-1> z(x')-1+y(x') \\geq v(x')$,\n                since\n                $y(x') \\geq \\bigg(\\big(m(x') - z(x)\\big) \\mod\\big(y(x) + 1\\big)\\bigg)$,\n                regardless  of the value of $m(x') - z(x)$.\n\t    \\item Suppose $m(x') < m(x)$. We compare $y(x)$ with $y(x')$.\n            \\begin{enumerate} \\itemsep0em\n\t        \\item If $y(x)= y(x')$ then $z(x)=z(x')$. By the definition of a legal move, $x'$ has at least $k$ piles not smaller than $m(x)$. Therefore, $1 \\leq\n            m(x)-m(x') \\leq y(x')=y(x)$, which implies that $m(x) \\mod (y(x)+1) \\neq m(x') \\mod (y(x)+1)$ and thus $v(x) \\neq v(x')$.\n\t        \\item If $y(x) \\neq y(x')$, since the $y$ values are different, by Lemma \\ref{interval} we have that $v(x)$ and $v(x')$ are in different intervals,\n            therefore $v(x)\\neq v(x')$.\n            \\end{enumerate}\n\t   \\end{enumerate}\n\t\\end{enumerate}\n\n\\subsection{Proof of (II)}\n\nWe prove property (II) by considering type I and type II positions, separately.\n\n\\subsubsection{Type I positions: $m(x)<z(x)$}\n\nFirst, let us consider the case $m(x)=0$. Then there are at most $2k-1$ nonempty piles.\nSo we can reduce the Tetris value $T_{2k,k}(x)$ to $0$ by emptying the $k$ largest piles of $x$.\nTherefore, by Lemmas \\ref{achieveTetrisvalue} and \\ref{continuity}, for any $0 \\leq \\delta < T_{2k,k}(x)$, there exists a move $x \\to x'$ with $T_{2k,k}(x')=\\delta$.\nAll such moves also have $m(x')=0$, therefore $m(x')<z(x')$.\n\nFrom now on we can assume that $m(x) \\geq 1$. Let us consider the following four subcases, depending on the value $\\delta$.\n\n\\begin{enumerate}\n\\item $0\\leq \\delta <  m(x)$.\nTo simplify our proof, let us use simply $m$ instead of $m(x)$ in this section.\nLet us observe first that since $x$ is type I we have $m<z(x)$ implying by \\eqref{eq:z} that\n\\begin{equation}\\label{sqrt-m1}\ny(x)>\\nu(m-1)\\geq 0.\n\\end{equation}\nwith $\\nu$ being defined as in Corollary \\ref{delta-epsilon}.\n\nLet us next define a set $Q=Q(x)$ of pairs of integers by setting\n\\[\nQ^1(x)=\\left\\{(\\mu,\\ge)\\left| \\begin{array}{c}m-\\ge~\\leq~ \\mu~\\leq~ m\\\\\n\\ge ~\\leq \\nu(m-1)-1\n\\end{array}\\right.\\right\\}\n\\]\n\\[\nQ^2(x)=\\left\\{(\\mu,\\ge) \\left| \\begin{array}{c}m-\\eps(m-1)~\\leq~ \\mu~\\leq~ m\\\\\n\\ge ~= \\nu(m-1)\n\\end{array}\\right. \\right\\} ,\n\\]\nand defining $Q=Q^1(x)\\cup Q^2(x)$.\n\nWe show that if for a position $x^*$ we have $(m(x^*),y(x^*))\\in Q$, then $x^*$ is of type II.\nTo see this consider a pair $(\\mu,\\ge)\\in Q^1(x)$.\nThen we have by the definition of $Q^1(x)$ that $\\mu\\geq m-\\ge$ and that $\\nu(m-1)-1\\geq \\ge$ from which $z(\\nu(m-1)-1)\\geq z(\\ge)$ follows by the definition of $z$ in \\eqref{eq:z}.\nWe also have the inequality $m-\\nu(m-1)+1\\geq z(\\nu(m-1)-1)$ since $m\\geq \\binom{\\nu(m-1)+1}{2}$ by the definition of $\\nu$ in Corollary \\ref{delta-epsilon}.\nPutting these together, we obtain $\\mu\\geq z(\\ge)$ as stated.\nFor $(\\mu,\\ge)\\in Q^2(x)$ we have $\\mu\\geq m-\\epsilon(m-1)$ and $\\ge=\\nu(m-1)$ by the definition of $Q^2(x)$.\nSince $m-1=\\binom{\\nu(m-1)+1}{2}+\\epsilon(m-1)$ by Corollary \\ref{delta-epsilon},\nthe inequality $\\mu\\geq m-\\epsilon(m-1)=z(\\ge)$ follows again.\n\nWe show next that for all pairs $(\\mu,\\ge)\\in Q$ there exists a move $x\\to x^*$ such that $\\mu=m(x^*)$ and $\\ge=y(x^*)$ (and $x^*$ is of type II, as we argued in the previous paragraph.)\nFor this let us consider first $\\mu\\leq m-1$ and note that if $(\\mu,\\ge)\\in Q$ then $m-\\mu\\leq \\ge\\leq \\nu(m-1)$.\nOur plan is to use Lemma \\ref{continuity} and to cover this range of $\\ge$ values by two constructions.\n\nLet us define a pair of positions $x'\\geq x''$ by\n\\begin{align*}\nx'_i &=\n\\begin{cases}\nx_i,   &\\text{ for } i=1 \\text{ and } i\\geq k+2,\\\\\n\\mu,   &\\text{ for } i=2,                    \\\\\nx_i-1, &\\text{ for } i=3,...,k+1\n\\end{cases} \\\\\n\\intertext{and}\nx''_i &=\n\\begin{cases}\nx_i,  &\\text{ for } i=1 \\text{ and } i\\geq k+2,\\\\\n\\mu,  &\\text{ for } i=2,...,k+1.\n\\end{cases}\n\\end{align*}\nNote that since $\\mu<m=x_1\\leq x_2\\leq x_i$ for $i\\geq 3$, both $x'$ and $x''$ are reachable from $x$. We claim next that $y(x'')=m-\\mu$ and $y(x')\\geq \\nu(m-1)$.\nThe first claim follows easily, since in $x''$ we have exactly $k$ positions larger than $\\mu$, and $x_1=m=\\mu +(m-\\mu)$.\nFor the second let us note that since $x_1-\\mu=m-\\mu\\geq 1$ we have $y(x')\\geq x_{k+1}-\\mu$. We can now apply Lemma \\ref{continuity} for $x'$ and $x''$,\nand conclude that for all values $m-\\mu \\leq \\ge \\leq x_{k+1}-\\mu\\leq y(x')$ there exists a move $x\\to x^*$ such that $m(x^*)=\\mu$ and $y(x^*)=\\ge$.\nFor larger values of $\\ge$ we need a modified construction:\n\\begin{align*}\nx'_i & =\n\\begin{cases}\n\\mu&\\text{ for } i=1,\\\\\nx_i-1, &\\text{ for } i=2,...,k,\\\\\nx_i,   &\\text{ for } i=k+1,...,2k,\n\\end{cases}\\\\\n\\intertext{and}\nx''_i & = \\begin{cases}\n\\mu,   &\\text{ for } i=1,...,k,\\\\\nx_i,   &\\text{ for } i=k+1,...,2k.\n\\end{cases}\n\\end{align*}\nAssuming $\\mu < m$, we have $y(x')\\geq y(x)\\geq \\nu(m)\\geq \\nu(m-1)$,\nwhile $y(x'')=x_{k+1}-\\mu$. We have again $m(x')=m(x'')=\\mu$. Thus by applying\nLemma \\ref{continuity} for $x'$ and $x''$ we can conclude that for all values $x_{k+1}-\\mu\\leq \\ge \\leq \\nu(m-1)$ there exists a move $x\\to x^*$ such that $m(x^*)=\\mu$ and $y(x^*)=\\ge$.\n\n\\medskip\n\nFinally, for $\\mu=m$ we proceed analogously, but with a third construction. Note first that $(m,\\ge)\\in Q$ if and only if $0\\leq \\ge\\leq \\nu(m-1)$.\nLet us now proceed with constructing two positions reachable form $x$:\n\\begin{align*}\nx'_i &=\n\\begin{cases}\nx_i,  &\\text{ for } i=1,...,k,\\\\\nx_i-1, &\\text{ for } i=k+1,...,2k\n\\end{cases} \\\\\n\\intertext{and}\nx''_i &=\n\\begin{cases}\nx_i,   &\\text{ for } i=1,...,k,\\\\\nm,     &\\text{ for } i=k+1,...,2k.\n\\end{cases}\n\\end{align*}\nIt is easy to see that both are reachable from $x$, and that $m(x')=m(x'')=m$, $y(x')\\geq y(x)-1\\geq \\nu(m-1)$ by \\eqref{sqrt-m1} and $y(x'')=0$.\nThus, the existence of an $x^*$ reachable from $x$ with $m(x^*)=m$, $y(x^*)=\\ge$ follows by Lemma \\ref{continuity} for all $0\\leq \\ge\\leq \\nu(m-1)$.\n\nBy the above arguments we have a move $x\\to x^*$ to a type II position $x^*$ with $m(x^*)=\\mu$ and $y(x^*)=\\ge$ for all $(\\mu,\\ge)\\in Q$.\nTo conclude the proof for this case we claim that the corresponding $v(x^*)$ values include all integers in the interval $[0,m-1]$.\nNote that $v(x^*)$ depends only on $m(x^*)$ and $y(x^*)$ for type II positions by \\eqref{eq:v},\nand that for a fixed value of $\\ge\\leq \\nu(m-1)-1$ we have exactly $\\ge+1$ consecutive integer values for $\\mu$ such that $(\\mu,\\ge)\\in Q$, implying that the corresponding values\n$v(\\mu,\\ge)=\\big(z(\\ge)-1\\big) + \\big[\\big(\\mu-z(\\ge)\\big)\\mod (\\ge+1)\\big]$\nis exactly the set of integers in the interval\n$[z(\\ge)-1, z(\\ge)+ \\ge-1]$.\nThus our claim follows by the construction of $Q$ and by Lemma \\ref{interval}.\n\n\\bigskip\n\n\\item $m(x) \\leq \\delta < x_2$ (Figure \\ref{F1}).\nSet $x'_i=0$ for $i=k+1, \\ldots ,n-1$ and $x'_n=\\delta-x_1$.\n\nBy definition $x_n\\geq x_2>\\delta\\geq x_1$ which implies $x'_n=\\delta-x_1<x_n-x_1\\leq x_n$.\nFurthermore, we have $x_i\\geq x_2>0$ for $i=k+1,...,n-1$,\nand therefore we indeed decrease exactly $k$ piles of $x$ to obtain $x'$. Thus $x'$ is reachable from $x$.\n\nFor $x'$ we have\n\\begin{align*}\n\\sum_{i=1}^{2k}\\min(x_i',\\delta)   &\\geq x_1' + (k-1)\\delta + (\\delta-x_1') =k\\delta \\\\\n\\intertext{and}\n\\sum_{i=1}^{2k}\\min(x_i',\\delta+1) &\\leq x_1' + (k-1)(\\delta+1) + (\\delta - x_1)\\\\\n                                   &= \\delta + (k-1)(\\delta +1) < k(\\delta +1).\n\\end{align*}\nTherefore, by Lemma \\ref{getmeTetris}, we have $T_{2k,k}(x')=\\delta$.\n\nSince $k\\geq 2$, $x'_{k+1}=0$ and therefore $m(x')=0$. Thus, $m(x')=0<1\\leq z(x')$ implying\nthat $x'$ is of type I, from which $g(x')=u(x')=\\delta$ follows by the above.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[height=7cm]{Doc2}\n\\caption{$x'$ is obtained by removing the gray area.} \\label{F1}\n\\end{figure}\n\n\\item $x_2 \\leq \\delta  < u(x)-m(x)$ (Figure \\ref{F2}).  Let $A=\\sum_{i=2}^{k+1} \\max(0,\\delta-x_i)$.\nFor $i=k+2, \\ldots ,n$ choose $a_i$ such that $0 \\leq a_i \\leq \\min(x_i-1,\\delta)$ and $\\sum_{i=k+2}^n a_i=A$.\nFirst let us prove that this is possible, or equivalently that $A \\leq \\sum_{i=k+2}^n \\min(\\delta,x_i-1)$.\nTo see this let us consider two cases.\nIf $x_{k+1}>\\delta$ then $\\sum_{i=k+2}^n \\min(\\delta,x_i-1)\\geq (k-1)\\delta \\geq A$.\nIf $x_{k+1}\\leq \\delta$ then let us define\n\\begin{equation} \\label{B.def}\nB=\\sum_{i=2}^{k+1}x_i = k\\delta - A,\n\\end{equation}\nand observe that for any integer $t\\geq\\delta >x_{k+1}$ we have\n\\[\n\\sum_{i=1}^n\\min(t,x_i) = m + B +\\sum_{i=k+2}^n\\min(t,x_i).\n\\]\nConsequently, since we have $T_{2k,k}(x)=u(x)$, by Lemma \\ref{getmeTetris} we can write for $t=u(x)$ that\n\\[\nkt \\leq \\sum_{i=1}^n\\min(t,x_i) = m(x)+B+\\sum_{i=k+2}^n\\min(t,x_i).\n\\]\nNote that if we decrease $t=u(x)$ by $1$,\nthen the left hand side decreases by $k$ while the right hand side decreases by at most $k-1$,\nhence the inequality remains valid. Let us repeat this $m(x)$ times, obtaining the inequality\n\\[\nk(u(x)-m(x))+km(x) \\leq m(x)+B+\\sum_{i=k+2}^n\\min(u(x)-m(x),x_i)+(k-1)m(x)\n\\]\nfrom which\n\\[\nk(u(x)-m(x))\\leq B+\\sum_{i=k+2}^n\\min(u(x)-m(x),x_i)\n\\]\nfollows. Let us now decrease $t=u(x)-m(x)$ further by $1$, as well as replace $x_i$ by $x_i-1$.\nThen the left hand side decreases by exactly $k$, while the right hand side decreases by at most $k$, yielding the valid inequality\n\\[\nk(u(x)-m(x)-1)\\leq B+\\sum_{i=k+2}^n\\min(u(x)-m(x)-1,x_i-1).\n\\]\nFinally, we can decrease $t=u(x)-m(x)-1$ further on both sides to $t=\\delta$ and similarly to the above argument obtain\n\\[\nk\\delta\\leq B+\\sum_{i=k+2}^n\\min(\\delta,x_i-1).\n\\]\nBy \\eqref{B.def} we obtain the claimed inequality, and hence the proof for the existence of the $a_i$ values for $k=k+2,...,n$\nthat satisfy the desired inequalities.\n\nLet us now consider the position $x'$ defined by\n\\[\nx'_i =\n\\begin{cases}\n0,     & \\textrm{ for } i=1; \\\\\nx_i,   &\\textrm{ for }i=2, \\ldots, k+1; \\\\\na_i,   &\\textrm{ for }i=k+2, \\ldots ,n.\n\\end{cases}\n\\]\nBy the above arguments $x\\to x'$ is a move in the game.\nThe equality $g(x')=u(x')=\\delta$ now follows by the above analysis and Lemma \\ref{getmeTetris}, completing our proof in this case.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[height=9cm]{Doc3}\n\\caption{$x'$ is obtained by removing the gray area.} \\label{F2}\n\\end{figure}\n\n\n\\item $u(x)-m(x) \\leq \\delta < u(x)$ (Figure \\ref{F3}).\nLet us define position $x'$ as follow\n$$\nx_i' =\n\\begin{cases}\nx_i-u(x)+\\delta,  &\\textrm{ for } i \\in I_1=\\{i \\mid i\\leq k, x_{i+k} \\leq \\delta\\};   \\\\\nx_i-u(x)+\\delta,  &\\textrm{ for } i \\in I_2=\\{i \\mid i > k,   \\delta < x_i \\leq u(x)\\}; \\\\\n\\delta ,           &\\textrm{ for }i \\in I_3=\\{i \\mid  i > k,   x_i > u(x) \\}; \\\\\nx_i,              &\\textrm{ otherwise.}\n\\end{cases}\n$$\nNote that $x_i \\geq m(x)\\geq u(x)-\\delta$, therefore $x_i'$ are all nonnegative.\nIt is easy to see that $I_1+k,I_2,I_3$ form a partition of $\\{ k+1, \\ldots , n \\}$.\nWe have reduced $x_i$ for all $i \\in I_1 \\cup I_2 \\cup I_3$, therefore $x\\to x'$ is a move.\n\nNext we note that the above construction implies that\n\\[\n\\sum_{i=1}^n \\min(x_i',\\delta) \\geq \\sum_{i=1}^n\\min(x_i,u(x)) - k (u(x)-\\delta)\\geq k \\delta\n\\]\nwhere the last inequality follows by the fact that $u(x)=T_{2k,k}(x)$, and hence $\\sum_{i=1}^n\\min(x_i,u(x))\\geq ku(x)$ by Lemma \\ref{getmeTetris}.\nSimilarly, we get\n\\[\n\\sum_{i=1}^n \\min(x_i',\\delta+1) \\leq \\sum_{i=1}^n\\min(x_i,u(x)+1) - k (u(x)-\\delta)< k (\\delta+1)\n\\]\nsince $\\sum_{i=1}^n\\min(x_i,u(x)+1)< k(u(x)+1)$.\nTherefore $T_{n,k}(x')=u(x')= \\delta$ follows by Lemma \\ref{getmeTetris}.\n\nNote that $x_{k+1}\\leq \\delta$, since otherwise $u(x)\\geq \\delta+m(x)$ would follow, contradicting our choice of $\\delta$.\nIt follows that $x_1 \\in I_1$ and thus $m(x')=m(x)-u(x)+\\delta<m(x)$.\n\nLet us define $\\hat{x}_i=\\min(x_i,u(x))$ for $i=1,...,n$. Then, by Lemma \\ref{Tetrisnotdecrease}, we have $y(x)=y(\\hat{x})$ and by construction of $x'$\nwe have $\\hat{x}_i-m(x)\\leq x'_i-m(x')$ for all indices $i$, implying $y(x)=y(\\hat{x}) \\leq y(x')$ by Lemma \\ref{tbasic}.\n\nFinally, $m(x')=m(x)-u(x)+\\delta<m(x)<z(x)\\leq z(x')$, which implies $g(x')=u(x')=\\delta$.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[height=10.5cm]{Doc4}\n\\caption{$x'$ is obtained by removing the gray area.} \\label{F3}\n\\end{figure}\n\n\n\\end{enumerate}\n\n\\subsubsection{Type II positions: $m(x)\\geq z(x)$}\n\n\nFor a position $x$ let\n\\begin{align}\nD_1(x) &=\\{(m,y)\\mid 0\\leq m\\leq m(x) , m(x)-m\\leq y< y(x) \\},\\\\\nD_2(x) &=\\{(m,y)\\mid m(x)-v(x)+z(x)-1\\leq m <m(x) , y=y(x) \\},\\\\\nD(x) &=D_1(x)\\cup D_2(x).\n\\end{align}\nNote that $D_2(x)=\\emptyset$ whenever $(m(x)-z(x))\\mod (y(x)+1)$ is zero.\n\n\\begin{lemma}\\label{existsmoveinD}\nFor every $x$ and for every $(m,y) \\in D(x)$ there is a position $x'$ reachable from $x$ such that $m(x')=m$ and $y(x')=y$.\n\\end{lemma}\n\n\\proof\nFor $(m,y) \\in D(x)$ let us consider the following three cases.\n\nCase 1:\n$0 \\leq m< m(x)$ and $x_2-m \\leq y \\leq y(x)$.\nWe consider two positions $\\bar{x}$ and $\\hat{x}$ reachable from $x$ defined as follows\n$$\\bar{x}_i =\n\\begin{cases}\nm,    & \\textrm{ if either } i=1 \\textrm{ or } i\\geq k+2;\\\\\nx_i,  & \\textrm{ otherwise};\n\\end{cases}\n$$\n$$\n\\hat{x}_i =\n\\begin{cases}\nx_i,                    & \\textrm{ if } i \\leq k+1; \\\\\n\\min(x_i,m(x)+y(x))-1,  & \\textrm{ if } i \\geq k+2;\n\\end{cases}\n$$\nSince we have $y(\\bar{x})=x_2-m$ and $y(\\hat{x})\\geq y(x)-1+m(x)-m \\geq y(x)$,\nwe can apply Lemma \\ref{continuity} and obtain that\nfor all values $x_2-m\\leq y\\leq y(x)$ there exists a position $x'$ such that\n$\\bar{x}\\leq x'\\leq \\hat{x}$ and $y(x')=y$. All these positions have $m(x')=m$.\n\nLet us also note that all these positions $x'$\nare reachable from $x$, because exactly $k$ components of $x$ are decreased.\n\n\\medskip\n\nCase 2: $0\\leq m < m(x)$ and $m(x)-m \\leq y < x_2-m$.\n\nWe consider two positions $\\bar{x}$ and $\\hat{x}$ reachable from $x$ defined as\n$$\n\\bar{x}_i =\n\\begin{cases}\nm,   & \\textrm{ if } i=2 \\textrm{ or } i\\geq k+2;\\\\\nx_i, & \\textrm{ otherwise;}\n\\end{cases}\n$$\n$$\\hat{x}_i =\n\\begin{cases}\nm,   & \\textrm{ if } i=2;\\\\\nx_i, & \\textrm{ if } i=1 \\textrm{ or } i=3,4, \\ldots, k+1; \\\\\n\\min(x_i,m(x)+y(x))-1, & \\textrm{ if } i \\geq k+2.\n\\end{cases}\n$$\n\nSince we have $y(\\bar{x})=m(x)-m$ and\n$y( \\hat{x}) \\geq x_{k+1}-m-1 \\geq x_2-m-1$, we can apply Lemma \\ref{continuity}, and obtain\nthat for all $y \\in[m(x)-m,x_2-m-1]$ there exists a position $x' \\in [\\bar{x},\\hat{x}]$ such that $y=y(x')$.\n\n\\medskip\n\nCase 3: $m=m(x)$ and $0 \\leq y < y(x)$.\nLet us note that the last inequaliti implies $y(x) \\geq 1$ and therefore $x_{k+1}>m(x)$.\nWe consider two positions $\\bar{x}$ and $\\hat{x}$ reachable from $x$ defined as\n$$\\bar{x}_i =\n\\begin{cases}\nx_i,   & \\textrm{ if } i=1, \\ldots ,k;\\\\\nm(x),  & \\textrm{ if } i\\geq k+1; \\\\\n\\end{cases}\n~~~~\\hat{x}_i =\n\\begin{cases}\nx_i,   & \\textrm{ if } i=1, \\ldots ,k; \\\\\nx_i-1, & \\textrm{ if } i\\geq k+1. \\\\\n\\end{cases}\n$$\nSimilarly to the previous cases, we have $y(\\bar{x})=0$ and\n$y(\\hat{x}) = y(x)-1$, and thus by Lemma \\ref{continuity} it follows that\nfor all $y \\in[0,y(x)-1]$ there exists an $x' \\in [\\bar{x},\\hat{x}]$ such that $y=y(x')$.\n\nIf we put all three cases together we cover all values $(m,y) \\in D$.\n\\qed\n\n\\medskip\n\nLet us set\n\\begin{equation} \\label{v(m,y)}\nv(m,y)= {y+1 \\choose 2}+ \\Big[ \\Big(m-1-{y+1 \\choose 2}\\Big)\\mod (y+1)\\Big].\n\\end{equation}\nNote that if $m=m(x)$ and $y=y(x)$ then $v(m,y)=v(x)$.\nFurthermore, we have\n$$V(y):=\\{v(m,y) | m \\in \\ZZ_\\geq\\}=\\left[ {y+1 \\choose 2},{y+2 \\choose 2} \\right).$$\nTherefore the sets $V(y)$, $y \\in \\ZZ_\\geq$ form a partition of $\\ZZ_\\geq$, as shown in Lemma \\ref{interval}.\n\n\\begin{lemma}\\label{Dstar}\nIf $m(x) \\geq z(x)$, then every $(m,y) \\in D(x)$ satisfies the following relations\n$$m\\geq {y+1 \\choose 2}+1~~\\textrm{ and }~~\\{v(m,y) \\mid (m,y) \\in D(x) \\} = [0,v(x)).$$\n\\end{lemma}\n\n\\proof\nLet us first consider $(m,y) \\in D_1(x)$.\nBy the definition of $D_1$ and the assumption of  $m(x) \\geq z(x)={y(x)+1 \\choose 2}+1$ we get\n$$m\\geq m(x)-y \\geq {y(x)+1 \\choose 2}+1-y \\geq {y+2 \\choose 2}+1-y \\geq {y+1 \\choose 2}+1.$$\nBy the definition of $D_1(x)$, for all $y \\in [0,y(x))$ we have $(m,y) \\in D_1(x)$ for all $m \\in [m(x)-y,m(x)]$. Hence, by (\\ref{v(m,y)}) we have $\\{v(m,y) \\mid m \\in [m(x)-y,m(x)]\\}=V(y)$.\nThus, by Lemma~\\ref{interval} we get $$\\{v(m,y) \\mid (m,y) \\in D_1(x) \\}=\\bigcup_{y \\in [0,y(x))}V(y)=[0,z(x)-1).$$\n\n\\smallskip\n\nLet us next consider $(m,y) \\in D_2(x)$.\nBy the definition of $ v(x)$ we can write $v(x)=z(x)-1+r $, where $r=m(x)-z(x)-\\lambda(y(x)+1)$ for some $\\lambda \\in \\ZZ_\\geq$. This implies $m(x)-r\\geq z(x)$. By the definition of $D_2(x)$ we have $m\\geq m(x)-r$.\nThese two inequalities imply $m\\geq {y+1 \\choose 2} +1$.\nSince $m \\in [m(x)-r, m(x))$ takes $r$ consecutive values, we have\n$\\{v(m,y) \\mid (m,y) \\in D_2(x) \\}=[z(x)-1,v(x))$.\n\\qed\n\nThe above lemma implies that any position $x'$ with $m(x')=m$, $y(x')=y$ for some $(m,y)\\in D$\nis a type II position. Hence $g(x')=v(x')=v(m,y)$.\nThus the second claim in the above lemma together with Lemma \\ref{existsmoveinD} implies that for any $0\\leq \\delta <v(x)$ there exists a move $x\\to x'$ such that $g(x')=\\delta$.\n\nSince we proved this for both type I and type II positions we concluded the proof of (II).\n\nProperties (I) and (II) together now imply that $\\G=g$.\nThis concludes the proof of Theorem \\ref{thm.n=2k}.\n\\qed\n\n\\smallskip\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[height=7cm]{Doc5}\n\\caption{Case $n=6,k=3$. Both above positions have $m=1,u=7,y=5$; but the position obtained by removing the gray area has $m=1,u=2,y=0$ and such a position cannot be obtained by the picture on the right since $m=1$ implies that $u \\geq 4$.} \\label{n=6,k=3}\n\\end{figure}\n\n\\section{Characterization of the $0$- and $1$-positions in Moore's game} \\label{Ss.Moore01}\n\nThe following result was shown in \\cite{JM80}. We provide here a different proof for the convenience of the reader.\n\n\\begin{theorem} $($see \\cite{JM80}$)$. \\label{t-Moore-m=0,1}\nFor any $k \\in \\{1, \\ldots, n\\}$  and  $m \\in \\{0,1\\}$, a position\n$x$  is an $m$-position of the Moore game  {\\sc Nim}$^\\leq_{n, k}$\nif and only if  $M(x) = m$.\n\\end{theorem}\n\nLet us represent the components of $x$ as binary sequences\n\\[\nx_i=\\sum_{j=0}^\\infty x_{ij}2^j \\text{ for } i=1,...,n,\n\\]\nand define\n\\begin{equation}\\label{e0001}\ny_j=\\sum_{i=1}^n x_{ij} \\mod (k+1) \\text{ for } j=0,1,....\n\\end{equation}\nThen we have $M(x)=\\sum_{j=0}^\\infty y_j(k+1)^j$.\n\n\\begin{lemma}\\label{moore1}\nLet  $x\\to x'$ be a move, and let $j$ be the highest index such that $x_{ij}\\neq x'_{ij}$ for some $i$.\nThen we must have $x'_{ij}\\leq x_{ij}$ for all $i=1,...,n$.\n\\end{lemma}\n\\proof\nIn a move we can only decrease the components of $x$.\nTherefore, if $x'_{ij}>x_{ij}$, then we must have a $j'>j$ such that $x'_{ij'}<x_{ij'}$.\n\\qed\n\n\n\\subsection{Proof of Theorem \\ref{t-Moore-m=0,1} for $m=0$}   \\label{ss-p-t1-0}\n\nFirst, let us prove Moore's result:\n$\\cG(x)=0$ if and only if  $M(x)=0$.\n\n\\smallskip\n\nBy the properties of $\\P$-positions it is enough to show that\n\\begin{itemize}\n\\item [\\rm{(i0)}] for any position $x$ with $M(x)=0$, there exists no  move $x \\to x'$ such that $M(x') = 0$;\n\\item [\\rm{(a0)}] for any position $x$ with $M(x)> 0$,  there exists a move $x \\to x'$ such that  $M(x') = 0$.\n\\end{itemize}\n\nTo show \\rm{(i0)}, let us consider a move $x \\to x'$\nfrom a position  $x$  with  $M(x)=0$.\n\nLet  $j$  be the highest binary bit such that\n$x_{ij}$ and $x'_{ij}$  differ for some $i$. Such a $j$ must exist since in a move we must change at least one components.\nBy Lemma \\ref{moore1} we have $x'_{ij}\\leq x_{ij}$ for all $i=1,...,n$,\nimplying $1\\leq \\sum_{i=1}^n(x_{ij}-x'_{ij}) \\leq k$ because  in a move we can change at most $k$ components. Therefore,\n$(\\sum_{i}x'_{ij} \\mod (k+1)) \\not= 0$ and, thus,  $M(x') > 0$.\n\n\\medskip\n\nTo show \\rm{(a0)}, let us consider a position  $x$  with  $M(x)>0$. We will construct a move $x \\to x'$ such that $M(x') = 0$.\n\n\\begin{notation} \\label{T.N}\nLet  $t_1, \\dots , t_p$ denote the bits $j$ such that $y_j\\not=0$, assuming  $t_1 > \\dots > t_p$.\nSet $N = \\{1,2,\\ldots,n\\}$.\n\\end{notation}\n\n\nThe following algorithm defines index sets $\\emptyset=I_0\\subseteq I_1\\subseteq \\cdots I_p \\subseteq N$\nsuch that we can compute a move $x \\to x'$ with $M(x')=0$, by decreasing components $i\\in I_p$. Define $O_j=\\{i\\in I_j\\mid x_{it_{j+1}}=1\\}$, and set $\\ga_j = |I_j|$  and  $\\gb_j=|O_j|$.\n\n\n\n\\medskip\n\\noindent\n{\\bf Step 0}. Initialize $I_0=\\emptyset$\nand hence, $\\ga_0 = \\gb_0 = 0$, and set $x'_i:=x_i$ for all $i \\in N$.\n\n\\smallskip\n\\noindent\n{\\bf Step 1}. For $j=1, \\dots , p$, construct $I_j$ and update $x'$ as follows.\n\n\\smallskip\n\n{\\bf Case 1.} If $y_{t_j} \\leq \\beta_{j-1}$, then let  $I_j:=I_{j-1}$,\nchoose  $y_{t_j}$ many indices $i\\in O_{j-1}$,\nand update $x'_{it_j}:=0$.\n\n\\smallskip\n\n{\\bf Case 2.} If $y_{t_j} > \\beta_{j-1}$ and $(k+1) -y_{t_j} \\leq \\ga_{j-1}-\\gb_{j-1}$,  then\nlet  $I_j:=I_{j-1}$, choose  $(k+1)-y_{t_j}$ many indices $i\\in I_{j-1}\\setminus O_{j-1}$, and update $x'_{i{t_j}}:=1$.\n\n\\smallskip\n\n{\\bf Case 3.}  If $y_{t_j} > \\beta_{j-1}$ and $y_{t_j}-\\beta_{j-1} \\leq k-\\ga_{j-1}$, then\nlet $I_j$ be the index set obtained from $I_{j-1}$ by adding\n$y_{t_j} - \\beta_{j-1}$ many indices $i$ from $N \\setminus I_{j-1}$ such that\n$x_{i{t_j}}=1$.  Update $x'_{i{t_j}}:=0$ for $i\\in (I_j\\setminus I_{j-1})\\cup O_{j-1}$.\n\n\n\\medskip\n\\noindent\nWe first note that the three cases above are exclusive and\ncover all possible $y_{t_j}$, $\\ga_{j-1}$ and $\\gb_{j-1}$ values.\nMoreover, it is easily seen that\na position $x'$ after the execution of the algorithm satisfies  $M(x')=0$, and\n$x'_i=x_i$  holds for $i \\not\\in I_p$.\n\nNote that we increase the set $I_j$ only in Case 3, in which case we have\n\\[\n|I_j|=\\alpha_j=(y_{t_j}-\\beta_{j-1})+\\alpha_{j-1} \\leq k,\n\\]\nimplying $|I_p|\\leq k$.\n\nNote also that in Case 3 we must have at least $y_{t_j} - \\beta_{j-1}$ many indices $i\\in N\\setminus I_{j-1}$ with $x_{it_j}=1$ by the definition of $y_{t_j}$ in \\eqref{e0001}.\n\nIt remains to show that $x' < x$.\nAssume that $i$ is an index such that $x'_i \\not= x_i$.\nThen some $j$ satisfies $i \\not\\in I_{j-1}$ and $i \\in I_j$.\nThis implies that $x'_i$ was first updated during the $j$th iteration of Step 1.\nNamely, the $t_j$th bit of $x'_i$ is modified from $1$ to $0$.\nSince $x'_{it}=x_{it}$ holds for all $t$ with $t > t_j$,\nwe have $x'_i < x_i$, which completes the proof.\n\\qed\n\n\\subsection{Proof of Theorem \\ref{t-Moore-m=0,1} for $m=1$}\n\\label{ss-p-t1-1}\n\nNow, let us prove that\n$\\cG(x)=1$ if and only if  $M(x)=1$.\n\n\\smallskip\nThe proof in the previous subsection implies that for a position $x$ with $M(x)=1$ there exists a move $x\\to x'$ such that $M(x')=0$.\nBy the properties of the SG function, it remains to show that\n\\begin{itemize}\n\\item [\\rm{(i1)}] for any position $x$  with $M(x) = 1$, there exists\nno move $x \\to x'$ such that $M(x') = 1$;\n\\item [\\rm{(a1)}] for any position $x$ with $M(x) > 1$, there\nexists a move $x \\to x'$ such that  $M(x')=1$;\n\\end{itemize}\n\nWe prove \\rm{(i1)} similarly to (i0).\nLet us assume that $M(x)=1$ holds for a position $x$\nand consider a move $x \\to x'$.\nLet $j$ be the highest binary bit such that\n$x_{ij}$ and $x'_{ij}$ differ for some $i$.\nThen by Lemma \\ref{moore1} we have $x'_{ij} \\leq x_{ij}$ for all $i$,  and\n$1 \\leq \\sum_{i}(x_{ij}-x'_{ij}) \\leq k$.\nHence, $\\sum_{i}x'_{ij} \\not= \\sum_{i}x_{ij}  (\\mod (k+1))$ and, thus, $M(x') \\not= 1$.\n\n\nTo show \\rm{(a1)}, let us consider a position $x$  with $M(x) > 1$.\nSimilarly to  \\rm{(a0)}, we will algorithmically construct a move $x \\to x'$ such that $M(x')=1$.\n\nLet again $t_1, \\dots , t_{p-1}$ denote the bits $j>0$ such that $y_j>0$,\nwhere we assume that  $t_1 > \\dots > t_{p-1}$, and add $t_p=0$.\n\nThe algorithm remains the same, as for \\rm{(a0)},\nexcept for $j=p$, when $t_p=0$. We detail below the computation of $I_p$ from $I_{p-1}$:\n\n\n\\medskip\n\\noindent\n\n{\\bf Case 1.} If $y_0> 1$ and $y_0-1 \\leq \\gb_{p-1}$,\nthen let  $I_p:=I_{p-1}$,\nchoose  $y_0-1$ many indices ${i}$ from $I_p$  such that\n$x'_{i0}=1$, and update $x'_{i0}:=0$ for such indices.\n\n\\smallskip\n\n{\\bf Case 2.} If $y_0> 1$ and $(k+2)-y_0 \\leq \\ga_{p-1}-\\beta_{p-1}$,\nlet  $I_p:=I_{p-1}$, choose  $(k+2)-y_{0}$ many indices ${i}$ from $I_p$  such\nthat $x'_{i0}=0$ and update $x'_{i0}:= 1$ for such indices.\n\n\\smallskip\n\n{\\bf Case 3.} If $y_0> 1$ and $0< (y_0-1) -\\gb_{p-1} \\leq k-\\ga_{p-1}$, then\nlet $I_p$ be an index set obtained from $I_{p-1}$ by adding $(y_{0}-1)-\\beta_{p-1}$ many\nindices $i$ from $N \\setminus I_{p-1}$ such that $x_{i{0}}=1$, and\nupdate $x'_{i0}:=0$ for all $i \\in I_p$ with $x'_{i0}=1$.\n\n\\smallskip\n\n{\\bf Case 4.} If $y_0=0$ and $\\ga_{p-1} > \\gb_{p-1}$,\nthen let  $I_p:=I_{p-1}$,\nchoose  an index ${i}$ from $I_p$  such that $x'_{i0}=0$, and update $x'_{i0}:=1$.\n\n\\smallskip\n\n{\\bf Case 5.} If $y_0=0$ and $\\ga_{p-1} = \\gb_{p-1}=k$,\nthen let  $I_p:=I_{p-1}$, and update $x'_{i0}:=0$ for all $i \\in I_p$.\n\n\\smallskip\n\n{\\bf Case 6.} If $y_0=0$ and $\\ga_{p-1} = \\gb_{p-1}<k$,\nthen let $I_p$ be an index set obtained from $I_{p-1}$ by adding $k-\\ga_{p-1}$ many\nindices $i$ from $N \\setminus I_{p-1}$ such\nthat $x_{i{0}}=1$, and update $x'_{i0}:=0$ for all $i \\in I_p$.\n\n\\smallskip\n{\\bf Case 7.} If $y_0=1$, then we set $I_p=I_{p-1}$.\n\n\\medskip\n\\noindent\nNote that the above seven cases are exclusive and cover all possible $y_{0}$, $\\ga_{p-1}$, and $\\gb_{p-1}$ values.\nNote also that\nin Case 6,  $\\ga_{p-1} > 0$ since otherwise  $M(x)=0$, giving a contradiction.\nThus, in Case 6, we can choose $k - \\ga_{p-1}$ many\nindices $i$ from $N \\setminus I_{p-1}$ such that $x_{i{0}}=1$.\nLet $x'$ be a position obtained by the algorithm.\nThen, clearly  $M(x')=1$  and  $x \\to x'$ is a move. This completes the proof of (a1).\n\\qed\n\n\\section{More on the Tetris function} \\label{Ss.Tetris}\n\nIn this section we show that the SG function described in Theorems \\ref{thm.n<2k}\nand \\ref{thm.n=2k} can in fact be computed efficiently, in polynomial time. For this we need to show that the Tetris function can be computed in polynomial time for these games.\nWe also prove that for a given position $x$ and integer $0\\leq g < T_{n,k}(x)$\nwe can compute in polynomial time a move $x\\to x'$ such that $T_{n,k}(x')=g$.\nFinally, in subsection \\ref{tetris-degree} we recall\nsome relations to degree sequences of graphs and hypergraphs.\n\n\\subsection{Computing the Tetris function in polynomial time} \\label{comp-tet-in-p}\n\nLet us recall that to a position $x$ we associated a shifted position $\\bar x$ after Corollary \\ref{delta-epsilon} with the property that $T_{n,k}(x)=\\bar x_{n-k+1}$.\nThe procedure described there is a non-polynomial algorithm.\nHowever $\\bar x$ and consequently $T_{n,k}(x)=\\bar x_{n-k+1}$ can be computed in a more efficient way.\n\n\\begin{theorem}\\label{Tetrispolynomial}\nGiven a position $x$ we can compute $T_{n,k}(x)$ in linear time in $n$.\n\\end{theorem}\n\n\\proof\nWe can assume without loss of generality that $x$ is a nondecreasing position. We show that the corresponding $\\bar x$ can be constructed in linear time in $n$, and thus the claim follows by the equality $T_{n,k}(x)=\\bar x_{n-k+1}$.\n\nRecall that the input size is $\\log(\\prod_{i = 1}^n x_i)$. Let $s=\\sum_{i=1}^{n-k} x_i$ be the number of tokens we shift on top of the largest $k$ piles; see Figure \\ref{ADD(1)}.\nWe know that for some $\\ell <k$ the first $\\ell +1$ columns of $\\bar x$ have almost the same number of tokens (at most one difference.)\nTo determine this index $\\ell$ and the height of the resulting piles, we use simple volume based arguments.\nWe need to compute first the following parameters.\n\nFor each $i=1, \\ldots ,k-1$, we denote by $y_i=x_{n-k+i+1}-x_{n-k+i}$ the difference of the sizes of consecutive piles.\nSet $s_0 =0$, $s_{k}=\\infty$, and\nfor $i=1,...,k-1$, set $s_i=s_{i-1}+i \\cdot y_i$ (i.e., the number of tokens we need to shift on top of the first $i$ piles $(n-k+1), \\ldots, (n-k+i)$ to make them all equal to $x_{n-k+1+i}$.)\nWe define a unique $\\ell$ by $s_\\ell \\leq s < s_{\\ell+1}$.\nWe define $a = s - s_\\ell$, $\\alpha = \\lfloor\\frac{a}{\\ell+1}\\rfloor$, and $\\beta =a \\mod (\\ell+1)$.\nWe fill up the first $\\ell+1$ columns to level $x_{n-k+\\ell+1}$ using $s_\\ell$ tokens. Then,\nwe place the remaining $a$ tokens by increasing each of the first\n$\\ell+1$ columns (indexed $n-k+1,...,n-k+\\ell +1$) by $\\alpha$ and the last $\\beta$ of these by one more, as in the following expression.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[height=8cm]{Doc6}\n\\caption{An example of calculating $\\bar x$ for $x=(1,2,2,3,4,4,7)$ with $k = 4$.}\n\\end{figure}\n\n\n\\begin{align*}\n\\bar x_i=\n\\begin{cases}\n0,                        & \\text{if } i=1, \\ldots ,n-k; \\\\\nx_{n-k+\\ell+1}+\\alpha,    & \\text{if } i=n-k+1, \\ldots ,n-k+1+\\ell -\\beta; \\\\\nx_{n-k+\\ell+1}+\\alpha +1, & \\text{if } i=n-k+2+\\ell -\\beta, \\ldots, n-k+1+\\ell; \\\\\nx_i,                      & \\text{if } i=n-k+2+\\ell, \\ldots ,n.\n\\end{cases}\n\\end{align*}\nIt is easy to see that this defines $\\bar x$ correctly,\nand that all these parameters can be computed in $O(n)$ time, if $x$ is a nondecreasing vector.\n\\qed\n\n\\begin{remark}\nTechnically, only the computation of $s=\\sum_{i=1}^{n-k}x_i$ depends on $n$.\nAll other computations in the previous proof can be done in $O(k)$ time.\n\\end{remark}\n\\medskip\n\n\n\n\n\\subsection{Polynomial computation of a move to a given Tetris value} \\label{}\n\nLet $x = (x_1, \\ldots, x_n)$ be a nondecreasing position and $k > n\/2$.\nWe examine the question of how to move to some position of given Tetris value $g$.\n\nWe denote by\n\\begin{enumerate} \\itemsep0em\n\\item [\\rm{(i)}] $x^\\ell$ the position obtained from $x$ by removing all tokens from the largest $k$ piles of $x$, and by\n\\item [\\rm{(ii)}] $x^u$ the position obtained from $x$ by decreasing the largest $k$ piles by one unit each.\n\\end{enumerate}\n\nConsider the set\n$W = \\{T_{n,k}(x') \\mid \\forall x': x\\to x'\\}$.\nBy Lemma \\ref{tbasic} (i) we have $T_{n,k}(x^\\ell) = \\min(W)$.\nBy Lemma \\ref{bestslowmove} we have $T_{n,k}(x^u)=T_{n,k}(x)-1$. As in the proof of\nTheorem \\ref{thm.n<2k}, we can argue that for\nevery value $g$ such that $T_{n,k}(x^\\ell)\\leq g \\leq T_{n,k}(x)-1$ there exists a move $x\\to x'$ such that  $x^\\ell \\leq x' \\leq x$ and $T_{n,k}(x')=g$.\n\n\n\\begin{theorem}\\label{achieveTetrisvalue}  \nGiven $g \\in W$, computing a position $x'$ such that $x^\\ell \\leq x' \\leq x$ and $T_{n,k}(x')=g$ can be done in $O\\big(n\\log(\\sum_{i=n-k+1}^nx_i)\\big)$ time.\n\\end{theorem}\n\\proof\nWe have $T_{n,k}(x^\\ell)\\leq g \\leq T_{n,k}(x^u)$.\nUsing the monotonicity of the Tetris function we perform a binary search in the space of positions between $x^\\ell$ and $x^u$.\nIn a general step we compute $L=\\sum_{i=n-k+1}^nx^\\ell_i$ and $U=\\sum_{i=n-k+1}^nx^u_i$, set $M=\\lfloor\\frac{L+U}{2}\\rfloor$ and compute $y_i=int(\\frac{x^\\ell_i+x^u_i}{2})$ for $i=n-k+1$,\nwhere $int(\\cdot)$ is a rounding to a nearest integer value in such a way that $\\sum_{i=n-k+1}^ny_i=M$.\nFinally, we set $y_i=x_i$ for $i<n-k+1$.\nIf $T_{n,k}(y)<g$ then we replace $x^\\ell$ by $y$, otherwise we replace $x^u$ by $y$.\n\nClearly these computations can be done in each step in $O(n)$ time,\nand computing the Tetris value of $y$ can also be done in $O(n)$ time by Theorem~\\ref{Tetrispolynomial}.\n\\qed\n\n\\smallskip\n\n\\begin{remark}\nSimilarly to the proof of Proposition \\ref{Tetrispolynomial}\nwe could improve the complexity of the above algorithm to $O(n)$.\n\\end{remark}\n\n\\subsection{Tetris function and degree sequences of graphs and hypergraphs}\\label{tetris-degree}\n\nA related problem is the hypergraph realization of a given degree sequence.\nLet us fix $V=\\{1,2,...,n\\}$ as the set of vertices.\nA multi-hypergraph $\\cH=\\{H_1,...,H_m\\}$ is a family of subsets (called hyperedges) of $V$,\ni.e., $H_j\\subseteq V$ for all $j=1,...,m$. It is called $k$-uniform if $|H_j|=k$ for all $j=1,...,m$.\nThe degree $d_{\\cH}(i)$ of a vertex $i\\in V$ is the number of hyperedges $H_j$ of $\\cH$ that contain $i$.\nWe allow the same subset to appear multiple times in $\\cH$.\n\nGiven an integer vector $x\\in\\ZZ_{\\geq}^n$, one can ask if there exists a $k$-uniform multi-hypergraph $\\cH$ on the vertex set $V$ such that $d_{\\cH}(i)=x_i$ for all $i\\in V$.\nEquivalently, we examine the existence of a bipartite graph $G=(X,Y,E)$ such that $|X|=n$, $|Y|=m$, $d_G(i)=x_i$ for all $i\\in X$, and $d_G(j)=k$ for all $j\\in Y$.\nFor the latter we can apply the classical Gale-Ryser theorem claiming that the answer is yes if and only if\n\n\\begin{itemize} \\itemsep0em\n\\item [\\rm{(i)}]  $\\sum_{i=1}^nx_i=km$\n\\item [\\rm{(ii)}] $\\sum_{i=1}^n\\min(x_i,g) \\geq kg$ for all $g=1,...,m$.\n\\end{itemize}\nLet us note that checking these conditions may not be polynomial in $x$ and $k$, since $m=\\sum_{i=1}^nx_i\/k$ according to property \\rm{(i)}.\nLet us also note that following a sequence of slow moves starting from position $x$, each time the set of columns that we decrease by $1$ can be considered as a hyperedge of $\\cH$.\nThus a maximal sequence of slow moves will construct $\\cH$ if the Tetris function achieves its trivial bound $kT_{n,k}(x)=\\sum_{i=1}^nx_i$.\nThis equality is in fact equivalent with property \\rm{(i)}, since we must have $m=T_{n,k}(x) $ in this case.\nOur results in this section thus prove that for the above degree sequence realization problems the most efficient answer is to compute the Tetris function value in linear time, and then compare it to its trivial upper bound.\nIf these are the same then the answer is yes.\n\nHavel (1955) and Hakimi (1962) provided a simple greedy algorithm based on a characterization for the recognition of degree sequences of bipartite graphs.\nFor the above case their criterion states that $x$ is a degree sequence of a $k$-uniform multi-hypergraph if and only if the position $x'$ is also a degree sequence of a $k$-uniform multi-hypergraph, where $x'$ is obtained from $x$ by decreasing the $k$ largest components of $x$ by $1$.\nNote that this implies a recursive process that is one of the definitions we used for the Tetris function.\n\nLet us remark finally that in general $x$ is not the degree sequence of a $k$-uniform multi-hypergraph.\nIn this case however a move $x\\to x'$ such that $T_{n,k}(x')=0$ provides us with a minimal modification such that $x''=x-x'$ becomes the degree sequence of such a hypergraph.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe energy spectrum of primary cosmic rays\nextends over more than $12$ decades in energy.\nIt follows, over a large range, a simple power law\nindicating its non-thermal nature.\nThe highest energy cosmic rays (which means particles with energies above $10$~EeV $\\equiv 10^{19}$~eV)\nmust be produced by the most extreme non-thermal process in the Universe.\nTheir energy spectrum, arrival directions, and composition can be inferred from air shower observations.\nHowever, the origin of these particles is still unknown and there is not a consensus whether there is \na steepening in the energy spectrum at around $60$~EeV. \nDetermining the shape of the spectrum in this energy range in sufficient detail \nrequires an enormous detector exposure.\nThe Pierre Auger Observatory was designed to study these ultra-high energy cosmic rays.\nAlthough under construction, the Auger Southern site is the largest cosmic ray detector in the world.\n\nA fundamental characteristics of the Pierre Auger Observatory is its capability of \\textit{hybrid} reconstruction of cosmic ray \nshowers~\\cite{Mostafa:2005kd}. \nTwo independent detectors, the Surface Detector (SD), which samples the shower particles at ground, and the Fluorescence Detector (FD), \nwhich collects the fluorescence light emitted by the shower particles along their path in the atmosphere, \nare able to measure the energy and direction of the same cosmic ray shower.\nThe enhanced capabilities of the Auger hybrid detector are examined in this paper.\n\nThe Pierre Auger Observatory\nwas designed to observe, in coincidence, the shower particles at ground\nand the associated fluorescence light generated in the atmosphere.\nThis is\nachieved with a large array of water Cherenkov detectors coupled with\nair-fluorescence detectors that overlook the surface array.\nIt is not simply a dual experiment.\nApart from important cross-checks and measurement redundancy,\nthe two techniques see air showers in complementary ways.\nA single air shower is detected 3-dimensionally.\nThe ground array measures the 2-dimensional lateral structure of the shower at ground level,\nwith some ability to separate the electromagnetic and muon components.\nThe fluorescence detector records the longitudinal pro{f}ile of the shower during\nits development through the atmosphere.\n\nA \\textit{hybrid} event is an air shower that is simultaneously detected\nby the fluorescence detector and the ground array.\nThe Observatory was originally designed and is currently being built with a \\textit{cross--triggering}\ncapability.\nData are recovered from both detectors whenever either system is triggered.\nIf an air shower independently triggers both detectors the event is tagged\naccordingly.\nThere are also cases where the fluorescence detector,\nhaving a lower energy threshold,\npromotes a sub--threshold array trigger.\nSurface stations are then matched by timing and location.\nThis is an important capability because these sub--threshold hybrid events\nwould not have triggered the array otherwise.\nThe geometrical reconstruction of the air shower's axis is accomplished by minimizing a\n$\\chi^{2}$ function involving data from all triggered elements in the eye and on the ground.\nThe reconstruction accuracy is far better than the ground array counters and the single eye\ncould achieve independently~\\cite{Bonifazi:2005ns}.\n\nThe combination of the air fluorescence measurement and the particle detection on the ground provides\nan absolute energy calibration.\nThe FD measurements determine the longitudinal development profile,\nwhose integral is proportional to the total energy of the electromagnetic particle cascade.\nThe SD independently estimates the shower energy by evaluating the particle density at $1000$~m from the core.\nThis ground parameter, called $S(1000)$, is estimated from the surface stations by fitting the measured densities to a lateral\ndistribution function~\\cite{Ghia:2005bn}. \nA conversion factor that relates $S(1000)$ to the shower energy (based on the FD information) is extracted from hybrid events.\nThis reduces significantly the dependence on air shower models and on assumptions of the primary composition.\n\nThe Pierre Auger Collaboration started the construction of the Southern site in $2002$ in the city of \nMalarg\\\"{u}e, which is located at an elevation of $1400$~m a.s.l. in the province of Mendoza, Argentina.\nAfter a successful operation of a prototype experiment~\\cite{Abraham:2004dt},\nthe Southern Observatory started operation in \\textit{hybrid} production mode in January, 2004.\nSurface stations have a $100$\\% duty cycle,\nwhile fluorescence eyes can only operate on clear moonless nights.\nBoth surface and fluorescence detectors have been running simultaneously\n$14\\%$ of the time.\nThe number of hybrid events represents $10\\%$ the statistics of the surface array data.\nThe Southern site will be completed in 2007.\nAnother important objective is to obtain a uniform exposure over the full sky.\nThis will be achieved by constructing a second instrument in the Northern hemisphere.\nEach site is conceived to cover an area of $3000$~km$^{2}$ in order to collect about $1$ event\nper week and site above $60$~EeV.\nThe Northern Observatory is planned to be sited in the U.S.\n\n\n\n\\section{Hybrid Geometrical Reconstruction}\n\nA hybrid detector achieves the best geometrical accuracy by using timing information from all the detector components,\nboth FD pixels and SD counters.\nEach element records a pulse of light from which one can determine the central time of the pulse and its uncertainty.\nEach trial geometry for the shower axis yields a prediction for the times at each detector component.\nDifferences between actual and predicted times are weighted using their corresponding uncertainties,\nsquared, and summed to construct a $\\chi^{2}$ value.\nThe hypothesis with the minimum value of $\\chi^{2}$ is the reconstructed shower axis.\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.45\\textwidth]{golden_camera_bw.eps}\n\\caption{\nLight track of a hybrid event as seen by two adjacent fluorescence telescopes.\nThe full line is the fitted shower-detector plane.\nThe squares represent the surface detectors that also triggered in this event.%\n}\n\\label{fig:Gcamera} \n\\end{figure}\n\nIn the FD, cosmic ray showers are detected as a sequence of triggered pixels in the camera.\nAn example of an event propagating through two adjacent FD telescopes is presented in Fig.~\\ref{fig:Gcamera}.\nThe first step in the analysis is the determination of the shower-detector plane (SDP).\nThe SDP is the plane that includes the location of the eye and the line of the shower axis.\n(See the sketch in Fig.~\\ref{fig:sketch}.)\nExperimentally, it is the plane through the eye which most nearly contains the pointing directions of the FD pixels\ncentered on the shower axis.\n(See fitted line in Fig.~\\ref{fig:Gcamera}.)\nUsing a known axis provided from the Central Laser Facility (CLF), described in Ref.~\\cite{Arqueros:2005yn},\nthe SDP reconstruction error can be evaluated by comparing the space angle between\nthe normal vector to the experimentally determined SDP and the known true normal vector.\nThis uncertainty in the SDP is not greater than about $0.1^{\\circ}$.\n\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.45\\textwidth]{fd-geom_bw.eps}\n\\caption{Illustration of the geometrical shower reconstruction from the observables of the fluorescence detector.\n}\n\\label{fig:sketch} \n\\end{figure}\n\nNext, the timing information of the pixels is used for reconstructing the shower axis within the SDP.\nAs illustrated in Fig.~\\ref{fig:sketch}, the shower axis can be characterized by two parameters:\nthe perpendicular distance $R_{p}$ from the eye to the track,\nand the angle $\\psi$ that the track makes with the horizontal line in the SDP.\nEach pixel which observes the track has a pointing direction which makes an angle\n$\\chi_{i}$ with the horizontal line.\nLet $t_{0}$ be the time when the shower front on the axis passes the point of closest approach $R_{p}$ to the eye.\nThe light arrives at the $i$th pixel at the time \n\\begin{equation}\n\tt_i = t_0 + \\frac{R_p}{c} \\cot{[(\\psi+\\chi_i)\/2]}.\n\t\\label{eq:timefit}\n\\end{equation}\n\nThe shower parameters are then determined by fitting the data points to this functional form.\nUsing the FADC electronics, such a monocular reconstruction may achieve excellent accuracy.\nHowever, the accuracy of the monocular reconstruction is limited when the measured angular speed\n$d\\chi\/dt$ does not change much over the observed track length.\nAn example is shown in Fig.~\\ref{fig:timefit}.\nFor these events (usually short tracks) there is a small curvature in the functional form of Eq.~(\\ref{eq:timefit})\nsuch that there is a family of possible $(R_p,\\psi)$ axis solutions.\nThis is also illustrated in Fig.~\\ref{fig:correlation}.\nNot only the uncertainties in $R_p$ and $\\psi$ are large,\nbut also they are strongly correlated.\nThis translates directly into an uncertainty in the other shower parameters,\nespecially in the reconstructed shower energy.\nThis asymmetric uncertainty in the energy and angular resolution are important drawbacks of the\nmonocular reconstruction.\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.45\\textwidth]{timefit_bw.eps}\n\\caption{Functional form that correlates the time of arrival of the light at each pixel with the \nangle between the pointing direction of that particular pixel and the horizontal line within the shower-detector plane.\n}\n\\label{fig:timefit} \n\\end{figure}\n\nThe aforementioned degeneracy can be broken by combining\nthe timing information from the SD stations with that of the FD telescopes.\nThis is called the \\textit{hybrid} reconstruction.\nThe hybrid solution \nfor the example shown in Fig.~\\ref{fig:timefit} \nis shown in Fig.~\\ref{fig:correlation}\nas a white star and its uncertainty as the smaller (full) ellipse.\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.45\\textwidth]{correlation_bw.eps}\n\\caption{Solutions (and $1\\sigma$ regions) for the axis fit shown in Fig.~\\ref{fig:timefit}.\nThe large uncertainty (and strong correlation) of the monocular reconstruction is broken using the \ntiming information from the surface detectors (Hybrid).\nThe stars mark the solutions that minimize the $\\chi^2$ for the axis reconstruction.\n(Following the notation of this paper,\n$\\chi_0 = 180^{\\circ} - \\psi$.)\n}\n\\label{fig:correlation} \n\\end{figure}\n\nSince the SD operates with a $100\\%$ duty cycle,\nmost of the events observed by the FD are in fact hybrid events.\nThere are also cases where the fluorescence detector,\nhaving a lower energy threshold,\npromotes a sub--threshold array trigger.\nSurface stations are then matched by timing and location.\nThis is an important capability because these sub--threshold hybrid events\nwould not have triggered the array otherwise.\nIn fact, the time of arrival at a single counter at ground can suffice for the\nhybrid reconstruction. \n\nThe reconstruction uncertainties are evaluated\nusing events with \\textit{known} geometries, \\textit{i.e.} laser beams.\nSince the location of the CLF \n(approximately equidistant from the first three fluorescence sites)\nand the direction of the laser beam are\nknown to an accuracy better than the expected angular resolution of the\nfluorescence detector, laser shots from the CLF can be used to measure the accuracy of\nthe geometrical reconstruction.\nFurthermore, the laser beam is split and part of the laser light is sent\nthrough an optical fiber to a nearby ground array station.\nThus, the axis of the laser light can be reconstructed both in monocular mode and in the\n\\textit{single-tank} hybrid mode.\nThe resolution of the monocular and hybrid reconstructions are compared in\nFig.~\\ref{RESOLVRP} for the distance between the eye and the CLF, and\nin Fig.~\\ref{RESOLVCHI0} for the angle of the axis.\nThe results are very encouraging.\nWith the monocular reconstruction, the location of the CLF can be determined\nwith a resolution of $\\sim500$~m.\nAfter including the timing information of the single water tank,\nthe resolution improves by one order of magnitude with no systematic shift.\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.45\\textwidth]{resolRp_bw.eps}\n\\caption{\\label{RESOLVRP} Difference between the reconstructed and true distance\nfrom the eye to the vertical laser beam using the monocular and hybrid techniques.\nThe location of the laser is known to $5$~m.\n}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.45\\textwidth]{resolChi0_bw.eps}\n\\caption{\\label{RESOLVCHI0} Angular difference between reconstructed and true direction\nof the laser beam using the monocular and hybrid techniques.\nThe laser beam is vertical within $0.01^{\\circ}$.\n}\n\\end{figure}\n\nAs mentioned before, the laser light from the CLF produces simultaneous triggers in both the surface and\n(three) fluorescence detectors. \nThe recorded event times are used to measure and monitor the\nrelative timing between the two detectors.\nThe time offset between the first fluorescence eye and the surface detector is shown in\nFig.~\\ref{OFFSET}.\nThis time offset is crucial for the accuracy of the hybrid reconstruction, \nand it has been measured to\nbetter than $50$~ns~\\cite{Allison:2005vj}.\nThe contribution to the systematic uncertainty in the core location due\nto the uncertainty in the time synchronization is $20$~m.\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.45\\textwidth]{offset.eps}\n\\caption{\\label{OFFSET} Time offset between the surface detector and one of the fluorescence detectors.\nThe variation in time is within the uncertainty of $50$~ns.\n}\n\\end{figure}\n\nUsing the timing information from the eye's pixels together with the surface stations\nto reconstruct real air showers,\na core location resolution of $50$~m is achieved.\nThe resolution for the arrival direction of cosmic rays is\n$0.6^{\\circ}$~\\cite{Bonifazi:2005ns}.\nThese results for the \\textit{hybrid} accuracy are in good agreement with\nestimations using analytic arguments~\\cite{Sommers:1995dm},\nmeasurements on real data using a bootstrap method~\\cite{Fick:2003qp},\nand previous simulation studies~\\cite{Dawson:1996ci}.\n\n\\section{Energy Determination}\n\nAfter successful reconstruction of the event geometry,\nthe FADC traces of the FD pixels are analyzed in order to obtain the light emitted along the shower axis.\nAn atmospheric scattering model is used in this step to transform the light received at the FD back to the light emitted\nfrom the shower axis.\nThe geometrical height, as observed by the telescopes, is converted to grammage of atmosphere.\nThe amount of fluorescence light emitted from a volume of air is proportional to the energy dissipated by the shower\nparticles in that volume.\nThe observed longitudinal light profile represents the energy loss in the atmosphere, which in turn is proportional\nto the number of charged particles in a given volume.\nThe result for a hybrid shower of $\\theta\\sim60^{\\circ}$ \nis shown in Fig.~\\ref{fig:profile} (bottom).\nThe line represents the best fit to a Gaisser-Hillas function~\\cite{GH},\nyielding a primary energy of $E=(23\\pm6)$~EeV, in good agreement with the $S(1000)$ determination. \n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.45\\textwidth]{profile_bw.eps}\n\\caption{\\label{fig:profile} An air-shower event detected both in water tanks (\\textit{top:} array view), \nand in a fluorescence detector (\\textit{bottom:} longitudinal profile).\n}\n\\end{figure}\n\n\n\n\\section{Observatory Status}\n\n\\begin{figure}[!ht]\n\\centerline{\\includegraphics[width=0.45\\textwidth]{status_bw.eps}}\n\\caption{\\label{STATUS} Status of the Pierre Auger Southern Observatory on May 10, 2006.\nThere were 1138 deployed tanks\nand 953 of them were fully operational.\nAlso shown are the locations of \nthe fluorescence detectors with their fields of view \n(for the three fully operational detectors).\nThe fourth fluorescence site (Loma Amarilla) is under construction.\n}\n\\end{figure}\n\nThe status of the Southern Observatory on May 10, 2006 is summarized in Fig.~\\ref{STATUS}.\nThere were 953 fully operational tanks at that time.\nThe {f}irst two fluorescence sites (Los Leones and Coihueco) were fully operational,\n\\textit{i.e.} running six telescopes each, in June, 2004.\nThe third site (Los Morados) started operation on March 18, 2005.\nThe fourth and last site (Loma Amarilla) is currently under construction and is scheduled\nto start operation by the end of 2006.\nThe present average rate is $50$ hybrid events per night per eye,\nfor a total of $\\sim40000$ events up to June, 2006.\nAt this rate, $4000$ hybrid events per month are expected\nwhen the Southern site is completed.\n\n\\section{Hybrid Measurements}\n\nIn the Pierre Auger Observatory two complementary experimental approaches\nare combined on a shower-by-shower basis within one single experiment.\nSuch redundancy allows cross-correlations between experimental techniques,\nthereby controlling the systematic uncertainties.\nFurthermore, there is an improvement in the resolution of the energy, mass,\nand arrival direction of the reconstructed primary particles.\nData are being used to verify both the performance of the individual components~\\cite{Bellido:2005aw,Bertou:2005bx},\nas well as to produce the highest quality reconstructed air-shower events~\\cite{Matthews:2005ve}.\nResults are very promising~\\cite{Mantsch:2005uq}\nand underline the advantages of the hybrid approach.\n\n\\subsection{Energy Spectrum}\n\nHybrid air shower measurements \nare utilized in the spectrum analysis to avoid dependence on\nspecific numerical simulations of air showers and detector\nresponses to them.  The analysis is also free of assumptions\nabout the primary nuclear masses.  The FD\nprovides a nearly calorimetric, model-independent energy\nmeasurement: fluorescence light is produced in proportion to\nenergy dissipation by a shower in the atmosphere.\nHybrid data establish the relation of shower energy to a ground parameter. \nMoreover,\nhybrid data determine the trigger probability for individual\ntanks as a function of core distance and energy, from which it\nis found that the SD event trigger is fully efficient above $3$~EeV\nfor zenith angles less than $60^{\\circ}$.  The SD exposure is\nthen calculated simply by integrating the geometric aperture over\ntime.\nIt is the continuously operating surface array which provides the high\nstatistics with unambiguous exposure.\nAssigning energies to the SD event set is a two-step process.\nThe first step is to assign an energy parameter, $S_{38}$, to each\nevent.  Then the hybrid events are used to establish\nthe rule for converting $S_{38}$ to energy.\n\nThe energy parameter $S_{38}$ for each shower comes from its\nexperimentally measured $S(1000)$.\nIt \nmay be regarded as the $S(1000)$\nmeasurement the shower would have produced if it had arrived\n$38^{\\circ}$ from the zenith.\nAs it can be seen in Fig.~\\ref{fig:cic}, \n$S_{38}$ is well correlated with the FD energy measurements of \nhigh quality hybrid events.\nThe fitted line gives an empirical rule for\nassigning energies (in EeV) based on $S_{38}$ (in VEM):\n\\begin{equation}E = 0.16 \\times S_{38}^{1.06}.\\end{equation}\nThe SD acceptance is not saturated below $3$~EeV. \nBut the hybrid events used in Fig.~\\ref{fig:cic},  \nwhich start at $\\sim1$~EeV,  \nare those\nwith core locations and arrival directions such that they have\na probability greater than 0.9 for satisfying the SD trigger and\nquality conditions.  \nThe distribution over energy produced by this two-step procedure\nbecomes the cosmic ray spectrum presented in Ref.~\\cite{Sommers:2005vs}.\n\n\\begin{figure}[!ht]\n\\includegraphics*[width=0.45\\textwidth]{calibrate.eps}\n\\caption{\\label{fig:cic} Shower energy (measured from the longitudinal profile) as a function of the ground parameter\n  $S_{38}$.  Each point is a hybrid event recorded when\n  there were contemporaneous aerosol measurements, whose \n  longitudinal profile included shower maximum in a measured\n  range of at least 350~g\\ cm$^{-2}$, and in which there was less than 10\\%\n  Cherenkov contamination.  The fitted line is\n  $\\log(E)=-0.79+1.06\\ \\log(S_{38})$.}\n\\end{figure}\n\n\\subsection{Anisotropy}\n\nDue to the much improved angular accuracy,\nthe \\textit{hybrid} data sample is ideal for anisotropy studies\nand, in particular, for point source searches.\nResults on a search for a point--like source in the direction of the Galactic\nCenter (GC) using these hybrid events were presented in Ref.~\\cite{Aglietta:2006ur}.\nThese events have a better angular resolution\n($ 0.7^\\circ$ at 68\\% c.l. in the EeV energy range).\nConsidering the hybrid events with an energy between $10^{17.9}$~eV and $10^{18.5}$~eV,\nno significant excess is seen in the GC direction.\nFor instance, in an optimal top-hat window of\n$1.59\\sigma\\simeq 0.75^\\circ$ radius,\n 0.3 events are expected while\nno single event direction falls within that circle.\nThis leads to a source flux upper-bound at 95\\% c.l. of\n\\begin{equation}\n\\Phi_s^{95}=\\xi\\; 0.15 \\ {\\rm km^{-2}\\ yr^{-1}}.\n\\end{equation}\n(The factor $\\xi$ is close to unity and parametrises the\nuncertainties in the flux normalization.)\n\nNote that\nthe energy assignments of the FD apply regardless of the assumed primary\ncomposition (except for a small correction to account for the missing energy),\nbe they protons or heavy nuclei. However,\nthe acceptance has  a dependence on composition because different primaries\ndevelop at different depths in the atmosphere. Since a quality\nrequirement for hybrid events is to have the maximum of the shower\ndevelopment inside the field of view of the telescopes, this affects\nthe sensitivity to different primaries.\nThe bound obtained is indeed conservative even if the bulk of the cosmic rays are\nheavy nuclei.\n\n\\subsection{Composition}\n\nAn upper limit for the photon fraction in cosmic rays with energies above $10^{19}$~eV\nwas derived using the hybrid data set~\\cite{Abraham:2006ar}.\nThe importance of the photon fraction lies in the fact that \\textit{top-down}\nmodels predict a considerable proportion of photons among the generated particles.\nThe measured photon flux is thus a valuable indicator of these non-accelerator models.\n\nThe method used in Ref.~\\cite{Abraham:2006ar} to distinguish between hadrons and photons\nin the hybrid data exploits the information on the longitudinal profile of the air shower.\nIn fact, this is the first such limit on photons obtained by observing the fluorescence light\nprofile of air showers.\nThe atmospheric depth at the shower maximum, $X_{max}$, is commonly used as a discriminant observable\nfor the cosmic ray composition because\nlighter nuclei penetrate, on average, more deeply into the atmosphere.\nAbove $10^{19}$~eV, showers initiated by photons develop significantly deeper in the atmosphere than hadronic\nshowers.\n\nNo candidate for a primary photon was found in the hybrid data taken between Jan, 2004 and Feb, 2006.\nBy comparing the observed $X_{max}$ of each hybrid event to predictions from hadronic simulations,\nan upper limit of $16\\%$ (at $95\\%$ c.l.) was derived.\nThis results confirms and improves the existing limits above $10$~EeV.\n\nThis analysis is currently limited mainly by the the small number of events.\nThe number of hybrid events will considerably increase in the next few years,\nand much lower primary photon fractions can be tested.\nAn upper limit of $\\sim5\\%$ could be achieved, for example, with two more years of data taking.\nMoreover, the larger statistics will allow to increase the threshold energy\nabove $10$~EeV where even larger photon fractions are predicted by the models.\nA similar limit ($\\sim15\\%$) but at higher energy (e.g. above $40$~EeV), would be well below existing limits\nand severely constrain non-acceleration models.\n\n\\section{Conclusions}\n\nThe Pierre Auger Observatory is a hybrid detector with excellent capabilities\nfor studying the highest energy cosmic rays.\nMuch of its capability stems from the accurate geometric reconstruction it achieves.\nThe shower geometry is reconstructed combining information from the eyes and the ground detectors.\nArrival directions are determined to a small fraction of a degree and the shower core is located to an\naccuracy of about $60$~m~\\cite{Bellido:2005aw,Bonifazi:2005ns}.\n\nThe construction of the Southern Observatory is well under way.\nEighteen FD telescopes and more than $60\\%$ of the surface array are in operation\ntaking data routinely.\nAt the present rate of deployment,\nconstruction will be finish in mid 2007.\nDetectors are performing very well and the first results are very encouraging.\n\nEmphasis is placed on hybrid analysis that provide unprecedented quality\nin geometry, energy, and mass reconstruction.\nOf utmost importance for the near future will be the determination of the energy spectrum\nto study the GZK feature, and the search for anisotropies in arrival directions.\nIt is important to note that\nboth SD and FD techniques have different systematics,\nand results are preliminary at this stage while the Observatory is under construction.\nThe possibility of studying\nthe same set of air showers with two independent methods is\nvaluable in understanding the strengths and limitations of each technique.\nThe hybrid analysis bene{f}its from the calorimetry of the fluorescence technique\nand the uniformity of the surface detector aperture.\n\nIn parallel to the completion of the Southern Observatory and to the analysis of data towards\nthe first scientific results, R\\&D has started for the development of the Northern site in the US.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sintro}\n\nSince the nineteenth century, $j$-invariants associated with elliptic curves having complex multiplication (CM), the so-called singular moduli, have been an object of study in number theory. A theorem of Weber \\cite[Theorem~11.1]{Cox} states that every singular modulus is an algebraic integer. Under certain technical restrictions, Gross and Zagier \\cite{Gross1985} stated explicit formulas for the absolute norm of the difference between two singular moduli.\n\nMotivated by effective results of Andr\\'e-Oort type \\cite{BMZ13,\n  Ku12}, David Masser raised in 2011 the question whether only\nfinitely many singular moduli are algebraic units, that is, units of the\nring of all algebraic integers. Throughout this article, we call such\nhypothetical algebraic numbers \\textsl{singular units}. Since there\nis no example of a singular unit in the literature, it seems\nlegitimate to ask whether there are any singular units at all.\n\nIn~\\cite{hab:junit}, the second-named author answered Masser's original  question in the affirmative: There exist at most finitely many singular units. However, his proof is non-effective as it invokes Siegel's lower bounds on the class number of imaginary quadratic fields \\cite{Siegel1935} through Duke's Equidistribution Theorem \\cite{Duke:hyperbolic}.\n\nHere, we can give the following definite answer to Masser's question as our main theorem.\n\n\\begin{theorem}\n\\label{thmain}\nThere are no singular units.\n\\end{theorem}\n\n\n\n\nTheorem~\\ref{thmain} is a formal consequence of our Theorems~\\ref{thhighrange},~\\ref{thmidrange},~\\ref{thlowrange} and~\\ref{thm:habeggerspari}. Let us briefly \nsketch its proof.\nWe say that a\nsingular modulus is of discriminant~$\\Delta$ if it is the\n$j$-invariant of a CM-elliptic curve whose endomorphism ring is the\nimaginary quadratic order of discriminant~$\\Delta$. We also write\n${\\Delta=Df^2}$ where~$D$ is the discriminant of the CM-field\n$\\Q(\\sqrt\\Delta)$, the fundamental discriminant, and $f$ is the\nconductor of the endomorphism order. The singular moduli of a given\ndiscriminant $\\Delta$ form a full Galois orbit over~$\\Q$ of\ncardinality equal to the class number\\footnote{We do not use the more traditional notation $h(\\Delta)$ because of the risk of confusing it with the height ${\\mathrm{h}}(\\cdot)$.} ${\\mathcal{C}}(\\Delta)$. \n\nWrite $\\zeta_3$ (resp.\\ $\\zeta_6$) for the third (resp.\\ sixth) root\nof unity $e^{2\\pi i\/3}$ (resp.\\ $e^{\\pi i\/3}$). Note that $\\zeta_3$\n(resp.\\ $\\zeta_6$) is the left (resp.\\ right) vertex of the geodesic\ntriangle enclosing the standard fundamental domain $\\mathcal{F}$ in\nthe Poincar\u00e9 upper half-plane. Given ${\\varepsilon\\in (0,1\/3]}$, denote by\n  ${\\mathcal{C}}_\\varepsilon(\\Delta)$ the number of singular moduli of\n  discriminant~$\\Delta$ which can be written ${j(\\tau)}$ where~$\\tau\n  \\in \\mathcal{F}$ satisfies ${|\\tau-\\zeta_3|< \\varepsilon}$ or\n  ${|\\tau-\\zeta_6|< \\varepsilon}$ and ${j(\\cdot)}$ denotes Klein's $j$-function. Since $\\zeta_3$ and $\\zeta_6$ are the only\n  zeros  of the $j$-function contained in the closure of\n  $\\mathcal{F}$, a pivotal ingredient in the proof of Theorem\n  \\ref{thmain} is an upper bound on ${\\mathcal{C}}_\\varepsilon(\\Delta)$. Indeed, a main\n  point of the argument in \\cite{hab:junit} is the\n  estimate\n  $${{\\mathcal{C}}_\\varepsilon(\\Delta)\\ll {\\mathcal{C}}(\\Delta)\\varepsilon^2}$$\nwhich holds when~$|\\Delta|$ is sufficiently large (in terms of~$\\varepsilon$). Unfortunately, ``sufficiently large'' here\n   is not effective; in fact, this is the place  where \n Duke's Equidistribution Theorem~\\cite{Duke:hyperbolic}, generalized by Clozel and Ullmo~\\cite{CU:Equidistribution} to arbitrary discriminants, is used.\n\nOur main novelty is the following effective estimate (see Theorem~\\ref{thceps}):\n\\begin{equation}\n\\label{ecepsrough}\n{\\mathcal{C}}_\\varepsilon(\\Delta)  \\ll F\\left(\\frac{\\sigma_1(f)}{f}|\\Delta|^{1\/2}\\varepsilon^2+|\\Delta|^{1\/2}\\varepsilon+ \\sigma_0(f)  |\n\\Delta|^{1\/4}\\varepsilon+1\\right),\n\\end{equation}\nwhere\n$$\nF=F(\\Delta) = \\max \\bigl\\{2^{\\omega(a)}: a\\le {|\\Delta|^{1\/2}}\\bigr\\}.\n$$\nHere  and in the sequel all implicit constants are effective,  and we use the standard notation\n\\begin{align}\n  \\label{eq:standardnotation}\n\\omega(n)=\\sum_{p\\mid n}1,\\quad \\sigma_0(n)=\\sum_{d\\mid n}1,\\quad \\sigma_1(n)=\\sum_{d\\mid n}d. \n\\end{align}\nUsing that\n\\begin{equation}\n\\label{eolog}\n\\omega(n)=o(\\log n),\\quad \\log\\sigma_0(n)=o(\\log n), \\quad \\sigma_1(n)\\ll n\\log\\log n,\n\\end{equation}\nwe deduce from~\\eqref{ecepsrough} tha\n\\begin{equation}\n\\label{ecepseta}\n{\\mathcal{C}}_\\varepsilon(\\Delta)  \\le |\\Delta|^{1\/2+o(1)}\\varepsilon+  |\n\\Delta|^{o(1)}\n\\end{equation}\nas ${|\\Delta|\\to\\infty}$. \n\nThe height of a singular unit~$\\alpha$ of discriminant~$\\Delta$ can be easily estimated in terms of ${\\mathcal{C}}_\\varepsilon(\\Delta)$:\n\\begin{equation}\n\\label{eupperheight}\n{\\mathrm{h}}(\\alpha)\\ll\\frac{{\\mathcal{C}}_\\varepsilon(\\Delta)}{{\\mathcal{C}}(\\Delta)}\\log|\\Delta|+\\log(\\varepsilon^{-1}),\n\\end{equation}\nsee Theorem~\\ref{theceps}.\n(By the \\textit{height} we mean here the usual \\textit{absolute logarithmic height} of an algebraic number; its definition is recalled in the beginning of Section~\\ref{supper}.) Substituting \\eqref{ecepseta}, we obtain the following upper estimate:\n\\begin{equation*}\n{\\mathrm{h}}(\\alpha)\\le \\frac{|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}A\\varepsilon+  \\frac{|\n\\Delta|^{o(1)}}{{\\mathcal{C}}(\\Delta)}+O(\\log(\\varepsilon^{-1}))\n\\end{equation*}\nwhere ${A=A(\\Delta)=|\\Delta|^{o(1)}}$ \nas ${|\\Delta|\\to\\infty}$. Specifying \n$$\n\\varepsilon= \\frac{{\\mathcal{C}}(\\Delta)}{A|\\Delta|^{1\/2}}\n$$\n(which is a nearly optimal choice), we obtain the estimate \n$$\n{\\mathrm{h}}(\\alpha)\\le   \\frac{|\n\\Delta|^{o(1)}}{{\\mathcal{C}}(\\Delta)}+o(\\log|\\Delta|) +O\\left(\\log^+\\frac{|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}\\right)\n$$\nwhere ${\\log^+(x)=\\max\\{\\log x,0\\}}$. \n\nTo obtain an upper bound on $|\\Delta|$, we combine this bound with the following two lower estimates on ${\\mathrm{h}}(\\alpha)$ (see Section \\ref{slower})\n\\begin{align}\n\\label{ecolm}\n{\\mathrm{h}}(\\alpha)+1&\\gg \\log|\\Delta|, \\\\\n\\label{etriv}\n{\\mathrm{h}}(\\alpha)&\\gg \\frac{|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}.\n\\end{align}\nThe bound~\\eqref{ecolm} is rather deep and relies on work of Colmez~\\cite{Colmez} and Nakkajima-Taguchi \\cite{NT}. On\nthe contrary,~\\eqref{etriv}  follows easily from the fact that one of\nthe conjugates of our singular unit~$\\alpha$ is\n${j\\bigl((\\Delta+\\sqrt\\Delta)\/2\\bigr)}$. Nevertheless, \\eqref{etriv} plays a\ncrucial role when the class number is pathologically small so that it\nwould contradict the Generalized Riemann Hypothesis (GRH). In fact, \\eqref{etriv} becomes much stronger than \\eqref{ecolm} in these hypothetical cases.\n\nComparing upper and lower estimates, we obtain for large $|\\Delta|$ that\n$$\n\\max\\left\\{\\frac{|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}, \\log|\\Delta|\\right\\}\\le \\frac{|\n\\Delta|^{o(1)}}{{\\mathcal{C}}(\\Delta)}+o(\\log|\\Delta|) +O\\left(\\log^+\\frac{|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}\\right),\n$$\nwhich is clearly impossible.\n\nTo get an explicit bound on $|\\Delta|$, we need to replace all implicit constants above with explicit ones. This relies in particular on a numerically sharp estimate for the arithmetic function $\\omega(n)$ due to Robin~\\cite{Ro83}. In Section \\ref{stenfourteen}, we see that this leads to a bound $|\\Delta|< 10^{15}$. While already  effective, it is still not feasible to check directly by a computer-assisted proof that none of the singular moduli of discriminant ${\\Delta \\in (- 10^{15}, -3]}$ is an algebraic unit.\n\nA refinement of our original arguments comes to our rescue. When  ${|\\Delta|<10^{15}}$, we improve on the estimate~(\\ref{ecepsrough}) by bounding sums of the form $\\sum_{n \\in [a,b] \\cap \\Z} 2^{\\omega(n)}$ in a more refined way. A natural idea is to use the Selberg-Delange method, which yields the asymptotic expansion\n\\begin{equation}\n\\label{equation::selbergdelange}\nS(x)= \\sum_{n\\le x} 2^{\\omega(n)} = x \\log x\\left(\\lambda_0 + \\frac{\\lambda_1}{\\log x} + O\\bigl(e^{-c\\sqrt{\\log x}}\\bigr)\\right)\n\\end{equation} \nwith explicit constants $\\lambda_0,\\lambda_1 \\in \\mathbb{R}$ and some\n constant ${c>0}$ (see, for instance,  \\cite[Theorem\n  II.6.1]{Te15}). There are two downsides of this method. First, the\nerror term is suboptimal under assumption of the GRH. Second, it would need some effort to make the constant $c$ actually explicit. \n\nHowever, as $|\\Delta|<10^{15}$, we are only interested in the case\nwhere $[a,b]$ is a subinterval of  ${[1, 2 \\cdot 10^7]}$. In this\nrange, a simple \\textsf{SAGE} script using the \\textsf{MPFI} library\n\\cite{MPFI,sagemath} can be used to improve on\n\\eqref{equation::selbergdelange}  computationally (see Proposition\n\\ref{proposition::selbergdelange}). As a consequence, we obtain\n$|\\Delta|<10^{10}$ for any singular unit of discriminant $\\Delta$ in Theorem \\ref{thmidrange}.\n\nThis is still not sufficient to check all remaining cases, at least\nwith modest computational means. The range is nevertheless small\nenough to use a counting algorithm in order to bound\n${\\mathcal{C}}_{10^{-3}}(\\Delta)$ for all discriminants $\\Delta$ satisfying\n$|\\Delta|< 10^{10}$, see Lemma \\ref{lem:lowrangeCebound}. {This still\nneeds an appropriate counting strategy, as determining\n${\\mathcal{C}}_{10^{-3}}(\\Delta)$ for each discriminant is rather slow,\ncomparable to computing separately each class number ${\\mathcal{C}}(\\Delta)$ in the\nsame range. Our trick is to bound all ${\\mathcal{C}}_{10^{-3}}(\\Delta)$\nsimultaneously by running through a set containing all imaginary quadratic\n$\\tau \\in \\mathcal{F}$ satisfying ${|\\tau-\\zeta_3|< \\varepsilon}$ or\n${|\\tau-\\zeta_6|< \\varepsilon}$ and such that $j(\\tau)$ is of discriminant\n$\\Delta$ with $|\\Delta|<10^{10}$. For each $\\tau$ encountered, we\ncompute its discriminant $\\Delta(\\tau)$ after the fact and increment\nour counter for ${\\mathcal{C}}_{10^{-3}}(\\Delta(\\tau))$. The thus obtained\nbounds for ${\\mathcal{C}}_{10^{-3}}(\\Delta)$ refine once again our previous\ninequalities, and allow us to conclude that $|\\Delta|<10^7$. Repeating this\nprocedure once again, with a slightly changed $\\varepsilon$, we\nachieve even $|\\Delta|<3 \\cdot 10^5$ in Theorem \\ref{thlowrange}.\nThese remaining cases can now be dealt with directly, for which we use\na \\textsf{PARI}~\\cite{PARI} program to prove Theorem  \\ref{thm:habeggerspari}, completing thereby the proof of Theorem~\\ref{thmain}. \n\n\\bigskip\n\nIt is very probable that our argument can be adapted to solve a more general problem: given an algebraic integer~$\\beta$,  determine the singular moduli~$\\alpha$ such that ${\\alpha-\\beta}$ is a unit; or at least bound effectively the discriminants of such~$\\alpha$. For instance, one may ask whether~$0$ is the only singular modulus~$\\alpha$ such that ${\\alpha-1}$ is a unit. In the general case, as explained in~\\cite{hab:junit}, this would require lower bounds for elliptic logarithmic forms, but when~$\\beta$ itself is a singular modulus, our argument extends almost without changes. One may go further and obtain an effective version of Theorem~2 from~\\cite{hab:junit}, which is an analogue of Siegel's Finiteness  Theorem for\nspecial points.\n\n\nThe famous work of Gross-Zagier and Dorman \\cite{Dorman1988,Gross1985} inspires the following problem: determine all couples $(\\alpha,\\beta)$ of singular moduli such that ${\\alpha-\\beta}$ is a unit; presumably, there is none. As indicated above, when~$\\beta$ is fixed and~$\\alpha$ varying, a version of our argument does the job, but if we let both~$\\alpha$ and~$\\beta$ vary, the problem seems more intricate. Very recently Yingkun Li~\\cite{Li18} made important progress: he proved that ${\\alpha-\\beta}$ is not a unit if the discriminants of~$\\alpha$ and~$\\beta$ are  fundamental and coprime. In particular, his result implies the following partial version of our Theorem~\\ref{thmain}: the discriminant of a singular unit must be either non-fundamental or divisible by~$3$. \n\nAnother natural problem is extending our work to $S$-units. Recall that, given a finite set~$S$ of prime numbers, a non-zero algebraic number is called an $S$-unit if both its denominator and numerator are composed of prime ideals dividing primes from~$S$.  Recently\nHerrero,\nMenares and\nRivera-Letelier announced the proof of finiteness of the set of singular $S$-units (that is, singular moduli that are $S$-units) for any finite set of primes~$S$. However, to the best of our knowledge, their argument is not effective as of now. \n\n\n\\bigskip\n\nFinally, let us discuss an application of Theorem~\\ref{thmain} to\neffective results of Andr\\'e-Oort type. A point\n$(\\alpha_1,\\ldots,\\alpha_n) \\in \\mathbb{C}^n$ is called special if\neach $\\alpha_i$, $i \\in \\{ 1, \\dots, n\\}$, is a  singular modulus. Since singular moduli are algebraic integers, the following statement is an immediate consequence of our main result. \n\n\n\n\n\\begin{corollary} For each polynomial ${P}$ in unknowns $X_2,\\ldots,\n  X_n$ and  coefficients that are algebraic integers in~$\\C$,\n \n  the hypersurface defined by \n\\begin{equation*}\nX_1 P(X_1, \\dots, X_n) = 1\n\\end{equation*}\n contains no special points.\n\\end{corollary}\n\nIn particular, $\\alpha_1^{a_1}\\cdots \\alpha_n^{a_n}\\not=1$\nfor all special points $(\\alpha_1,\\ldots,\\alpha_n)$\nand all integers $a_1\\ge 1,\\ldots,a_n \\ge 1$. \nThis corollary exhibits a rather general class of algebraic varieties\nof arbitrary dimension and degree for which  the celebrated theorem of\nPila~\\cite{Pila2011a} can be proved effectively and even explicitly.\nIt is complementary to other recent effective results of Andr\\'e-Oort type~\\cite{Bilu2017,Bi18}.\n\n\\paragraph{Plan of the article}\nIn Section~\\ref{sceps} we obtain an explicit version of the\nestimate~\\eqref{ecepsrough}. In Section~\\ref{supper} we obtain an\nupper estimate  for the height of a singular unit. In\nSection~\\ref{slower} we obtain explicit versions of  the lower estimates~\\eqref{ecolm} and~\\eqref{etriv}. In Section~\\ref{stenfourteen} we use all  previous results to bound the discriminant of a singular unit as  ${|\\Delta|<10^{15}}$. This bound is reduced to $10^{10}$ in Section~\\ref{sec:midrange} and to $3\\cdot10^5$ in Section~\\ref{slowrange}.  Finally, in Section~\\ref{sfinal} we show that the discriminant of a singular unit satisfies ${|\\Delta|>3\\cdot10^5}$. \n\n\\paragraph{Convention}\nIn this article we fix, once and for all, an embedding ${\\bar\\Q\\hookrightarrow \\C}$; this means that all algebraic numbers in this article are viewed as elements of~$\\C$.\n\n\\paragraph{Acknowledgments}\nYuri Bilu was partially supported by the University of Basel, the Fields Institute (Toronto), and the Xiamen University. Lars K\u00fchne was supported by the Max-Planck Institute for Mathematics, the Fields Institute, and the Swiss National Science Foundation through an Ambizione grant. We thank Ricardo Menares and Amalia Pizarro for many useful conversations, Florian Luca and Aleksandar Ivic for helpful suggestions, Bill Allombert and Karim Belabas for a \\textsf{PARI} tutorial, and Jean-Louis Nicolas and Cyril Mauvillain for helping to access Robin's thesis~\\cite{Ro83a}.  Finally, we thank both anonymous referees for encouraging reports and helpful suggestions. \n\n\n\\section{An estimate for \\texorpdfstring{${\\mathcal{C}}_\\varepsilon(\\Delta)$}{Ceps(Delta)}}\n\\label{sceps}\nLet~$\\Delta$ be a negative integer satisfying ${\\Delta\\equiv 0,1\\bmod 4}$ and  \n$$\n{\\mathcal{O}}_\\Delta=\\Z[(\\Delta+\\sqrt\\Delta)\/2] \n$$\nthe  imaginary quadratic order of discriminant~$\\Delta$. Then ${\\Delta=Df^2}$, where~$D$ is the discriminant of the imaginary quadratic field ${\\Q(\\sqrt\\Delta)}$  (the ``fundamental discriminant'') and ${f=[{\\mathcal{O}}_D:{\\mathcal{O}}_\\Delta]}$ is the conductor. We denote by ${\\mathcal{C}}(\\Delta)$ the class number  of the order~${\\mathcal{O}}_\\Delta$. \n\n\n\nUp to $\\C$-isomorphism there exist ${\\mathcal{C}}(\\Delta)$ elliptic curves with CM by~${\\mathcal{O}}_\\Delta$. The $j$-invariants of these curves are called \\textsl{singular moduli} of discriminant~$\\Delta$.\nThe singular moduli  of discriminant~$\\Delta$ form a full Galois orbit over~$\\Q$\nof cardinality ${\\mathcal{C}}(\\Delta)$, see \\cite[Proposition~13.2]{Cox}. \n\n\nLet~${\\mathcal{F}}$ be the standard fundamental domain in the Poincar\u00e9  plane,  that is, the open hyperbolic triangle with vertices ${\\zeta_3,\\zeta_6,i\\infty}$, together with the geodesics $[i,\\zeta_6]$ and ${[\\zeta_6,i\\infty)}$; here \n$$\n\\zeta_3=e^{2\\pi i\/3}=\\frac{-1+\\sqrt{-3}}2, \\quad \\zeta_6=e^{\\pi i\/3}=\\frac{1+\\sqrt{-3}}2. \n$$\nEvery singular modulus can be uniquely presented as  ${j(\\tau)}$, where ${\\tau\\in {\\mathcal{F}}}$. \n\n\nNow fix ${\\varepsilon\\in (0,1\/3]}$ and denote by ${\\mathcal{C}}_\\varepsilon(\\Delta)$ the number of singular moduli of discriminant~$\\Delta$ that can be presented as ${j(\\tau)}$ where ${\\tau\\in {\\mathcal{F}}}$ satisfies\n\\begin{equation}\n\\label{etauzeze}\n\\min \\{|\\tau-\\zeta_3|,|\\tau-\\zeta_6|\\}< \\varepsilon. \n\\end{equation}\nIn this section we bound this quantity.\n\n\nDefine the \\textsl{modified conductor}~${\\tilde f}$ by\n\\begin{equation}\n\\label{etilf}\n{\\tilde f}=\n\\begin{cases}\nf,& D\\equiv 1\\bmod 4,\\\\\n2f, &D\\equiv 0\\bmod 4. \n\\end{cases}\n\\end{equation}\nThen ${\\Delta\/{\\tilde f}^2}$ is a square-free integer. \n\n\n\n\n\\begin{theorem}\n\\label{thceps}\nFor ${\\varepsilon\\in (0,1\/3]}$ we have\n\\begin{equation}\n\\label{enewbound}\n{\\mathcal{C}}_\\varepsilon(\\Delta)  \\le  F\\left( \\frac{16}3\\frac{\\sigma_1({\\tilde f})}{{\\tilde f}}|\\Delta|^{1\/2}\\varepsilon^2+\\frac83|\\Delta|^{1\/2}\\varepsilon+ 8|\\Delta\/3|^{1\/4}\\sigma_0({\\tilde f})\\varepsilon+4\\right),\n\\end{equation}\nwhere  \n\\begin{equation}\n\\label{ecapitalf}\nF=F(\\Delta) = \\max \\bigl\\{2^{\\omega(a)}: a\\le |\\Delta|^{1\/2}\\bigr\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{corollary}\n\\label{cseps}\nIn the set-up of Theorem~\\ref{thceps} assume that ${|\\Delta|\\ge 10^{14}}$.  Then  \n\\begin{equation}\n\\label{enewboundsimple}\n{\\mathcal{C}}_\\varepsilon(\\Delta)  \\le  F\\left( 9.83|\\Delta|^{1\/2} \\varepsilon^2\\log\\log(|\\Delta|^{1\/2})+3.605|\\Delta|^{1\/2}\\varepsilon+ 4\\right).\n\\end{equation}\n\\end{corollary}\n\n\n\\subsection{Some lemmas}\nWe need some lemmas. For a prime number~$\\ell$ and a non-zero integer~$n$ we denote by ${\\mathrm{ord}}_\\ell(n)$ the $\\ell$-adic order of~$n$; that is, ${\\ell^{{\\mathrm{ord}}_\\ell(n)}\\,\\|\\,n}$. \n\n\\begin{lemma}\n\\label{lrootsmodle}\nLet~$\\ell$ be a prime number, ${e \\ge 1}$ an integer, and~$\\Delta$ a\nnon-zero integer with ${\\nu={\\mathrm{ord}}_\\ell\\Delta}$.\nThen the set of ${b\\in \\Z}$ satisfying \n${b^2\\equiv \\Delta\\bmod \\ell^e}$ \nis a union of at most~$2$ residue classes modulo ${\\ell^{e-\\lfloor\n    \\min\\{e,\\nu\\}\/2\\rfloor}}$ in all cases except when ${\\ell=2}$ and\n${e\\ge 3}$; in this latter case it is a union of most~$4$ such\nclasses. Finally, the set of $b$ equals a single residue class modulo ${\\ell^{e-\\lfloor\n    \\min\\{e,\\nu\\}\/2\\rfloor}}$\nif $\\nu\\ge e$. \n\\end{lemma}\n\\begin{proof}\nWe suppose first that ${\\nu=0}$, that is, ${\\ell\\nmid \\Delta}$.  In this case we have to count the number of elements in the multiplicative group $(\\Z\/\\ell^e\\Z)^\\times$ whose\nsquare is represented by~$\\Delta$. If ${\\ell\\ge 3}$ or ${\\ell^e\\in \\{2,4\\}}$, then $(\\Z\/\\ell^e\\Z)^\\times$ is a cyclic group.\nThen there are at most~$2$ square roots and this implies our claim. If ${\\ell=2}$ and  ${e \\ge 3}$, then ${(\\Z\/2^e\\Z)^\\times\\cong \\Z\/2\\Z\\times \\Z\/2^{e-2}\\Z}$, and there are at most~$4$ square roots, as desired.\n\nNow assume that  ${\\nu <e}$. Then ${{\\mathrm{ord}}_\\ell(b^2) = \\nu}$. So~$\\nu$ is even and we can write ${b = \\ell^{\\nu\/2}b'}$, where ${b' \\in \\Z}$ is coprime to~$\\ell$. Now ${\\Delta=\\ell^\\nu\\Delta'}$ \nwith ${\\Delta'\\in \\Z}$ coprime to~$\\ell$,  and ${(b')^2 \\equiv \\Delta'\\bmod \\ell^{e-\\nu}}$.  Above we already determined\nthat, depending on the value of $\\ell^{e-\\nu}$, the set of\npossible~$b'$ consists of either at most~$2$ or at most~$4$ classes\nmodulo ${\\ell^{e-\\nu}}$. Hence the set of possible ${b =\n  \\ell^{\\nu\/2}b'}$ consists of the same number of classes modulo\n${\\ell^{e-\\nu\/2}}$, as desired.\n\nTo prove the final claim  assume that ${\\nu \\ge e}$. In this case\n${b^2\\equiv \\Delta\\bmod \\ell^e}$ \nis equivalent to ${b\\equiv0\\bmod \\ell^{\\lceil e\/2\\rceil}}$. This means\nthat the set of suitable~$b$ consists of exactly one class  modulo\n${\\ell^{\\lceil e\/2\\rceil}=\\ell^{e-\\lfloor e\/2\\rfloor}}$.\n\\end{proof}\n\n\n\nWe say that ${d\\in \\Z}$ is a \\textsl{quadratic divisor} of ${n\\in \\Z}$ if ${d^2\\mid n}$. We denote by ${\\gcd_2(m,n)}$ the greatest common quadratic divisor of~$m$ and~$n$. \n\n\\begin{lemma}\n\\label{lrootsmoda}\nLet~$a$ be a positive integer and~$\\Delta$ a non-zero integer. Then the set of \n${b\\in \\Z}$ satisfying \n${b^2 \\equiv \\Delta \\bmod a}$\nconsists of at most   ${2^{\\omega(a\/\\gcd(a,\\Delta))+1}}$ residue classes modulo \n${a\/\\gcd_2(a,\\Delta)}$\n\\end{lemma}\n\\begin{proof}\n  For a prime power $\\ell^e$ \n  we only need the following simple consequence of \n  Lemma~\\ref{lrootsmodle} on the number of residue classes counted there.\nThis number is at\nmost $2^{\\omega(\\ell^e\/\\gcd(\\ell^e,\\Delta))}$ if $\\ell\\ge 3$\nand at most $2^{\\omega(\\ell^e\/\\gcd(\\ell^e,\\Delta))+1}$ for $\\ell=2$.\n  The current lemma follows from the Chinese\nRemainder Theorem.\n\\end{proof}\n\nThe  following lemma is trivial, but we state it here because it is our principal counting tool.  \n\n\\begin{lemma}\n\\label{ltrivi}\nLet~$\\alpha$ and~$\\beta$ be real numbers, ${\\alpha<\\beta}$, and~$m$ a positive integer. Then every residue class modulo~$m$ has at most ${(\\beta-\\alpha)\/m+1}$ elements in the interval ${[\\alpha,\\beta]}$.\n\\end{lemma}\n\n\n\nGiven a negative integer ${\\Delta\\equiv0,1\\bmod 4}$, denote by ${T=T_\\Delta}$ the set of triples of integers $(a,b,c)$ such that \n\\begin{equation}\n\\label{ekuh}\n\\begin{aligned}\n&\\gcd(a,b,c)=1, \\quad \\Delta=b^2-4ac,\\\\\n&\\text{either\\quad $-a < b \\le a < c$\\quad or\\quad $0 \\le b \\le a = c$.}\n\\end{aligned}\n\\end{equation}\nFor  ${(a,b,c)\\in T_\\Delta}$ we set\n$$\n\\tau(a,b,c)=\\frac{b+\\sqrt{\\Delta}}{2a}.\n$$\n\\begin{lemma}\n\\label{lgauss}\n\\begin{enumerate}[label={(\\roman*)}]\n\\item\n\\label{ifund}\nFor every ${(a,b,c)\\in T_\\Delta}$ the number \n$\\tau(a,b,c)$\nbelongs to the standard fundamental domain. \n\n\\item\n\\label{iadelta}\nFor ${(a,b,c)\\in T_\\Delta}$ we have ${0<a\\le |\\Delta\/3|^{1\/2}}$, the equality being possible only if ${\\Delta=-3}$ (and ${a=b=c=1}$). We also have ${c\\ge |\\Delta|^{1\/2}\/2}$. \n\n\\item\n\\label{icl}\nThe map\n${(a,b,c)\\mapsto j(\\tau(a,b,c))}$\ndefines a bijection from $T_\\Delta$ onto the set of $\\Q$-conjugates\nof $j(\\tau)$. \nIn particular, ${{\\mathcal{C}}(\\Delta)=|T_\\Delta|}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nFor item~\\ref{ifund} just note that~\\eqref{ekuh} implies the inequalities \n$$\n-\\frac12 <\\frac{b}{2a}\\le \\frac12, \\qquad \\frac{b^2+|\\Delta|}{4a^2}\\ge 1\n$$\nand that the second one becomes equality only when ${a=c}$, in which case  ${b\\ge 0}$. \nFor item~\\ref{iadelta}, \nsince ${|b|\\le a\\le c}$,  we have \n$$\n4c^2\\ge |\\Delta|=4ac-b^2\\ge 4a^2-a^2=3a^2. \n$$\nwith equality on the right only when ${a=|b|=c}$. Since ${\\gcd(a,b,c)=1}$, this is only possible when ${a=b=c=1}$ and ${\\Delta=-3}$. \n\nItem~\\ref{icl} is a combination of several classical results that can be found, for instance, in~\\cite{Cox}. See \\cite[Proposition~2.5]{BLP16} for more details. \n\\end{proof}\n\n\n\\begin{lemma}\n\\label{lshort}\nLet ${\\varepsilon\\in (0,1\/3]}$ and let ${(a,b,c)\\in T_\\Delta}$ satisfy~\\eqref{etauzeze}. Then \n\\begin{align}\n\\label{eboundinga}\n\\frac{|\\Delta|^{1\/2}}{\\sqrt3+2\\varepsilon}&<\\hphantom{|}a\\hphantom{|}\\le \\frac{|\\Delta|^{1\/2}}{\\sqrt3},\\\\\n\\label{eboundingb}\na(1-2\\varepsilon)&<|b|\\le a,\\\\\n\\label{eboundingc}\na&\\le \\hphantom{|}c\\hphantom{|}<a(1+{\\sqrt3}\\varepsilon+\\varepsilon^2)\n\\end{align}\n\\end{lemma}\nNote that~\\eqref{eboundinga} and~\\eqref{eboundingb} will be used already in Subsection~\\ref{ssproofthceps}, while~\\eqref{eboundingc} will be used only in Section~\\ref{slowrange}.\n\n\\begin{proof}\nFor ${\\tau\\in {\\mathcal{F}}}$ condition~\\eqref{etauzeze} implies that \n$$\n\\frac{\\sqrt3}2\\le\\Im\\tau<\\frac{\\sqrt3}2+\\varepsilon, \\qquad \n\\frac12-\\varepsilon<|\\Re\\tau|\\le \\frac12.\n$$\nApplying this for ${\\tau=\\tau(a,b,c)}$, we obtain~\\eqref{eboundinga} and~\\eqref{eboundingb}. To prove~\\eqref{eboundingc}, write\n$$\n4ac= |\\Delta|+b^2 < a^2(\\sqrt3+2\\varepsilon)^2+a^2=4a^2(1+\\sqrt3\\varepsilon+\\varepsilon^2), \n$$ \nand~\\eqref{eboundingc} follows. \n\\end{proof}\n\n\n\\subsection{Proof of Theorem~\\ref{thceps}}\n\\label{ssproofthceps}\n\nNote that, by definition, \n$$\n{\\mathcal{C}}_\\varepsilon(\\Delta) = \\#\\{\\text{$(a,b,c)\\in T_\\Delta$~: ${\\tau=\\tau(a,b,c)}$ satisfies~\\eqref{etauzeze}}\\}. \n$$\nSetting \n$$\nI=\\left(\\frac{|\\Delta|^{1\/2}}{\\sqrt3+2\\varepsilon}, \\frac{|\\Delta|^{1\/2}}{\\sqrt3}\\right], \n$$\nfor ${\\tau=\\tau(a,b,c)}$ with ${(a,b,c)\\in T_\\Delta}$ we may re-write~\\eqref{eboundinga} and~\\eqref{eboundingb} as   \n\\begin{align}\n\\label{eainibin}\na\\in I,\\qquad b\\in [-a,-a(1-2\\varepsilon))\\cup (a(1-2\\varepsilon),a]. \n\\end{align}\nSince~$c$ is uniquely determined for given~$a$,~$b$ and~$\\Delta$, it suffices to bound the number of  pairs $(a,b)$ of integers satisfying ${b^2\\equiv \\Delta\\bmod a}$ and~\\eqref{eainibin}. \n\nFor every fixed~$a$ there are at most\n${(4\\varepsilon\\gcd_2(a,\\Delta)+2) 2^{\\omega(a)+1}}$ suitable~$b$, as follows from Lemmas~\\ref{lrootsmoda} and~\\ref{ltrivi};\nindeed, ${\\omega(a\/\\gcd(a,\\Delta))\\le \\omega(a)}$. Hence\n\\begin{align}\n\\label{estim0}\n{\\mathcal{C}}_\\varepsilon(\\Delta) &\\le 8\\varepsilon\\sum_{a\\in I\\cap\\Z} {\\gcd}_2(a,\\Delta) 2^{\\omega(a)} + 4\\sum_{a\\in I\\cap\\Z}2^{\\omega(a)}\\\\ \n\\label{estim1}\n&\\le  8\\varepsilon F\\sum_{a\\in I\\cap\\Z} {\\gcd}_2(a,\\Delta) +4F\\#(I\\cap\\Z).\n\\end{align}\nTo estimate the sum, note that \n\\begin{equation}\n\\label{estim2}\n\\sum_{a\\in I\\cap\\Z} {\\gcd}_2(a,\\Delta)\\le \\sum_{d^2\\mid \\Delta}d\\cdot \\#(I\\cap d^2\\Z). \n\\end{equation}\nRecall that we defined in~\\eqref{etilf} the modified conductor~${\\tilde f}$. \nSince $\\Delta\/{\\tilde f}^2$  is a square-free integer, we have ${d^2\\mid \\Delta}$ if and only if ${d\\mid {\\tilde f}}$. Also, since~$I$ is of length \n$$\n|\\Delta|^{1\/2}\\left(\\frac1{\\sqrt3}-\\frac1{\\sqrt3+2\\varepsilon}\\right)<\\frac23|\\Delta|^{1\/2}\\varepsilon,\n$$ \nwe have, by Lemma~\\ref{ltrivi},\n$$\n\\#(I\\cap d^2\\Z) \\le \n\\begin{cases}\n\\frac23\\frac{|\\Delta|^{1\/2}}{d^2}\\varepsilon +1, & d \\le {|\\Delta\/3|^{1\/4}}, \\\\\n0,& d >{|\\Delta\/3|^{1\/4}}. \n\\end{cases}\n$$\nHence\n\\begin{align}\n\\sum_{d^2\\mid \\Delta}d\\cdot \\#(I\\cap d^2\\Z) &\\le \\sum_{\\genfrac{}{}{0pt}{}{d\\mid{\\tilde f}}{d\\le|\\Delta\/3|^{1\/4}}}d\\left(\\frac23\\frac{|\\Delta|^{1\/2}}{d^2}\\varepsilon+1\\right)\\nonumber\\\\\n&\\le \\frac23|\\Delta|^{1\/2}\\varepsilon\\sum_{d\\mid{\\tilde f}}d^{-1}+ \\sum_{\\genfrac{}{}{0pt}{}{d\\mid{\\tilde f}}{d\\le |\\Delta\/3|^{1\/4}}}d\\nonumber\\\\\n\\label{estim3}\n&\\le \\frac23\\frac{\\sigma_1({\\tilde f})}{{\\tilde f}}|\\Delta|^{1\/2}\\varepsilon+ |\\Delta\/3|^{1\/4}\\sigma_0({\\tilde f}). \n\\end{align}\nFinally, Lemma~\\ref{ltrivi} implies that\n\\begin{equation}\n\\label{eintini}\n\\#(I\\cap\\Z)\\le \\frac23|\\Delta|^{1\/2}\\varepsilon+1. \n\\end{equation}\nPutting  the estimates~\\eqref{estim1},~\\eqref{estim2},~\\eqref{estim3} and~\\eqref{eintini} together, we obtain~\\eqref{enewbound}. \\qed\n\n\\subsection{Proof of Corollary~\\ref{cseps}}\n\nWe need to estimate  $\\sigma_0({\\tilde f})$ and $\\sigma_1({\\tilde f})$ in terms of~$|\\Delta|$.  The following lemma uses a simple estimate for $\\sigma_0(n)$ due to Nicolas and Robin~\\cite{NR83}. Much sharper  estimates can be found in Robin's thesis~\\cite{Ro83a}. \n\n\\begin{lemma}\n\\label{lsigzersimple}\nFor ${|\\Delta|\\ge 10^{14}}$ we have \n\\begin{align}\n\\label{euppersigzer}\n\\sigma_0({\\tilde f})&\\le |\\Delta|^{0.192},\\\\\n\\label{euppersigone}\n{\\sigma_1({\\tilde f})}\/{\\tilde f} &\\le 1.842\\log\\log(|\\Delta|^{1\/2}). \n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nFor proving~\\eqref{euppersigzer}  may assume that ${{\\tilde f}\\ge 16}$, otherwise there is nothing to prove. In~\\cite{NR83} it is proved that for ${n\\ge 3}$ we have\n$$\n\\frac{\\log\\sigma_0(n)}{\\log2} \\le 1.538\\frac{\\log n}{\\log\\log n}.\n$$\nThe function ${x\\mapsto (\\log x)\/(\\log\\log x)}$ is increasing for ${x\\ge 16}$. Since \n$$\n|\\Delta|\\ge 10^{14}, \\qquad 16\\le {\\tilde f}\\le |\\Delta|^{1\/2},\n$$ \nthis gives\n\\begin{align*}\n\\log\\sigma_0({\\tilde f}) \n&\\le 1.538\\log 2\\frac{\\log(|\\Delta|^{1\/2})}{\\log\\log(|\\Delta|^{1\/2})}\\\\\n&\\le \\frac{1.538}2\\log 2\\frac{\\log|\\Delta|}{\\log\\log(10^{7})}\\\\\n&<0.192\\log|\\Delta|,\n\\end{align*}\nas wanted. \n\nFor proving~\\eqref{euppersigone} we use the estimate \n${\\sigma_1(n)\\le 1.842n\\log\\log n}$\nwhich holds for ${n\\ge 121}$, see \\cite[Theorem~1.3]{AFJ07}. This\nproves~\\eqref{euppersigone} for ${{\\tilde f}\\ge 121}$. For ${{\\tilde f}\\le 120}$\none can check directly that  ${\\sigma_1({\\tilde f})\/{\\tilde f}\\le 3}$ so that inequality~\\eqref{euppersigone} is also\ntrue in this case.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Corollary~\\ref{cseps}]\nIf ${|\\Delta|\\ge 10^{14}}$ then Lemma~\\ref{lsigzersimple} implies that \n\\begin{align*}\n&8\\left|\\frac\\Delta3\\right|^{1\/4}\\sigma_0({\\tilde f}) \\le \\frac8{3^{1\/4}}|\\Delta|^{0.442}\\le \\frac8{3^{1\/4}\\cdot10^{0.812}}|\\Delta|^{1\/2}\\le 0.938|\\Delta|^{1\/2},\\\\\n&\\frac{16}3\\frac{\\sigma_1({\\tilde f})}{{\\tilde f}} \\le \\frac{16}3\\cdot 1.842 \\log\\log(|\\Delta|^{1\/2})\\le9.83 \\log\\log(|\\Delta|^{1\/2}).\n\\end{align*}\nSubstituting all this to~\\eqref{enewbound}, we obtain~\\eqref{enewboundsimple}. \n\\end{proof}\n\n\n\n\\section{An upper bound for the height of a singular unit}\n\\label{supper}\n\nIn this section we obtain a fully explicit version of estimate~\\eqref{eupperheight}.\nWe use the notation ${\\mathcal{C}}(\\Delta)$,  ${\\mathcal{C}}_\\varepsilon(\\Delta)$, ${\\mathcal{F}}$, $\\zeta_3$, $\\zeta_6$ introduced in Section~\\ref{sceps}. \n\nLet~$\\alpha$ be a complex algebraic number of degree~$m$ whose minimal polynomial over~$\\Z$ is \n$$\nP(x)=a_mx^m+\\cdots+a_0=a_m(x-\\alpha_1)\\cdots (x-\\alpha_m) \\in \\Z[x].\n$$\nHere ${\\gcd(a_0,a_1, \\ldots, a_m)=1}$ and ${\\alpha_1, \\ldots, \\alpha_m\\in \\C}$ are the conjugates of~$\\alpha$ over~$\\Q$. Then the height of~$\\alpha$ is defined by\n$$\n{\\mathrm{h}}(\\alpha) =\\frac1m\\left(\\log|a_m|+\\sum_{k=1}^m \\log^+|\\alpha_k|\\right), \n$$ \nwhere ${\\log^+(\\cdot)=\\log\\max\\{1,\\cdot\\}}$. \nIf~$\\alpha$ is an algebraic integer then \n$$\n{\\mathrm{h}}(\\alpha) =\\frac1m\\sum_{k=1}^m \\log^+|\\alpha_k|. \n$$ \nIt is known that ${{\\mathrm{h}}(\\alpha)={\\mathrm{h}}(\\alpha^{-1})}$ when ${\\alpha\\ne 0}$. \n\n\\begin{theorem}\n\\label{theceps}\nLet~$\\alpha$ be a singular unit of discriminant~$\\Delta$, and~$\\varepsilon$ a real number satisfying ${0<\\varepsilon \\le 4\\cdot10^{-3}}$. \nThen \n\\begin{equation}\n\\label{eupperheightbis}\n{\\mathrm{h}}(\\alpha)\\le 3\\frac{{\\mathcal{C}}_\\varepsilon(\\Delta)}{{\\mathcal{C}}(\\Delta)}\\log|\\Delta|+3\\log(\\varepsilon^{-1})-10.66. \n\\end{equation}\n\\end{theorem}\n\nCombining this with Corollary~\\ref{cseps} and optimizing~$\\varepsilon$, we obtain the following consequence. \n\n\\begin{corollary}\n\\label{cupper}\nIn the set-up of Theorem~\\ref{theceps} assume that ${|\\Delta|\\ge 10^{14}}$. Then\n\\begin{align}\n\\label{eupdel}\n{\\mathrm{h}}(\\alpha)&\\le \\frac{12A}{{\\mathcal{C}}(\\Delta)}+3\\log\\frac{A|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}-3.77,\n\\end{align}\nwhere ${A= F\\log|\\Delta|}$ and~$F$ is defined in~\\eqref{ecapitalf}.  \n\\end{corollary}\n\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{theceps}}\n\nWe start from some simple lemmas.\n\n\n\\begin{lemma}\n\\label{lanal}\nFor ${z\\in {\\mathcal{F}}}$ we have \n$$\n|j(z)|\\ge 42700\\bigl(\\min\\{|z-\\zeta_3|,|z-\\zeta_6|,4\\cdot10^{-3}\\}\\bigr)^3. \n$$\n\\end{lemma}\n\n\\begin{proof}\nThis is an easy modification  of Proposition~2.2 from~\\cite{BLP16}; just replace therein $10^{-3}$ by ${4\\cdot10^{-3}}$.   \n\\end{proof}\n\nIn the next lemma we use the notation $T_\\Delta$  and ${\\tau(a,b,c)}$ introduced before Lemma~\\ref{lgauss}. \n\n\\begin{lemma}\n\\label{lliouv}\nAssume that ${\\Delta\\ne -3}$. Let ${\\tau=\\tau(a,b,c)}$, where  ${(a,b,c)\\in T_\\Delta}$. Let~$\\zeta$ be one of the numbers~$\\zeta_3$ or~$\\zeta_6$. \nThen\n$$\n|\\tau-\\zeta|\\ge \\frac{\\sqrt3}{4|\\Delta|}. \n$$\n\\end{lemma}\n\n\\begin{proof}\nWe have\n$$\n|\\tau-\\zeta|\\ge|\\Im\\tau-\\Im\\zeta|= \\left|\\frac{\\sqrt{|\\Delta|}}{2a}-\\frac{\\sqrt3}{2}\\right|= \\frac{\\bigl||\\Delta|-3a^2\\bigr|}{2a(\\sqrt{|\\Delta|}+a\\sqrt3)}.\n$$\nSince ${\\Delta\\ne -3}$ we have ${\\Delta\\ne -3a^2}$, see item~\\ref{iadelta} of Lemma~\\ref{lgauss}.  Hence\n$$\n|\\tau-\\zeta|\\ge \\frac{1}{2a(\\sqrt{|\\Delta|}+a\\sqrt3)}\\ge \\frac{\\sqrt3}{4|\\Delta|}, \n$$\nthe last inequality being again by item~\\ref{iadelta} of Lemma~\\ref{lgauss}. \n\\end{proof}\n\nNow we are ready to prove Theorem~\\ref{theceps}. \n\n\\begin{proof}[Proof of Theorem~\\ref{theceps}]\nLet ${\\alpha=\\alpha_1, \\alpha_2, \\ldots, \\alpha_m\\in \\C}$ be the conjugates of~$\\alpha$ over~$\\Q$. Then ${m={\\mathcal{C}}(\\Delta)}$ and ${\\alpha_1, \\ldots, \\alpha_m}$ is the full list of singular moduli of discriminant~$\\Delta$. Write them as ${j(\\tau_1), \\ldots, j(\\tau_m)}$, where ${\\tau_1, \\ldots, \\tau_m\\in {\\mathcal{F}}}$. \n\nSince~$\\alpha$ is a unit, we have \n$$\n{\\mathrm{h}}(\\alpha)={\\mathrm{h}}(\\alpha^{-1})  = \\frac1m \\sum_{k=1}^m\\log^+|\\alpha_k^{-1}|,\n$$\nHence \n{\\footnotesize\n\\begin{align}\n{\\mathrm{h}}(\\alpha) &= \\frac{1}{{\\mathcal{C}}(\\Delta)}\\sum_{k=1}^m\\log^+|j(\\tau_k)^{-1}|\\nonumber\\\\\n\\label{etwosums}\n&= \\frac{1}{{\\mathcal{C}}(\\Delta)}\\left(\\sum_{\\genfrac{}{}{0pt}{}{1\\le k\\le m}{\\min\\{|\\tau_k-\\zeta_3|, |\\tau_k-\\zeta_6|\\}< \\varepsilon}}+\\sum_{\\genfrac{}{}{0pt}{}{1\\le k\\le m}{\\min\\{|\\tau_k-\\zeta_3|, |\\tau_k-\\zeta_6|\\}\\ge \\varepsilon}}\\right)\\log^+|j(\\tau_k)^{-1}|.\n\\end{align}\n}%\nWe estimate each of the two sums separately.\n\nSince ${\\varepsilon\\le 4\\cdot10^{-3}}$, Lemma~\\ref{lanal} implies that each term in the second sum satisfies \n$$\n\\log^+|j(\\tau_k)^{-1}| \\le 3\\log(\\varepsilon^{-1})-\\log42700 \\le 3\\log(\\varepsilon^{-1})-10.66. \n$$\nHence\n$$\n\\sum_{\\genfrac{}{}{0pt}{}{1\\le k\\le m}{\\min\\{|\\tau_k-\\zeta_3|, |\\tau_k-\\zeta_6|\\}\\ge \\varepsilon}}\\log^+|j(\\tau_k)^{-1}|\\le ({\\mathcal{C}}(\\Delta)-{\\mathcal{C}}_\\varepsilon(\\Delta))\\bigl(3\\log(\\varepsilon^{-1})-10.66\\bigr). \n$$\nSince ${\\varepsilon\\le 4\\cdot10^{-3}}$ we have\n ${3\\log(\\varepsilon^{-1})>10.66}$, which implies that \n\\begin{equation}\n\\label{esecondsum}\n\\sum_{\\genfrac{}{}{0pt}{}{1\\le k\\le m}{\\min\\{|\\tau_k-\\zeta_3|, |\\tau_k-\\zeta_6|\\}\\ge \\varepsilon}}\\log^+|j(\\tau_k)^{-1}|\\le {\\mathcal{C}}(\\Delta)\\bigl(3\\log(\\varepsilon^{-1})-10.66\\bigr). \n\\end{equation}\nAs for the first sum, Lemmas~\\ref{lanal} and~\\ref{lliouv} imply that each term in this sum satisfies\n$$\n\\log^+|j(\\tau_k)^{-1}| \\le \\max\\left\\{0, 3\\log\\frac{4|\\Delta|}{\\sqrt3}-\\log42700\\right\\} \\le 3\\log|\\Delta|.  \n$$ \nNote that we may use here Lemma~3.4 because the only singular modulus of discriminant $-3$ is~$0$, which is not a unit. \n\nSince the first sum has ${\\mathcal{C}}_\\varepsilon(\\Delta)$ terms, this implies the estimate \n\\begin{equation}\n\\label{efirstsum}\n\\sum_{\\genfrac{}{}{0pt}{}{1\\le k\\le m}{\\min\\{|\\tau_k-\\zeta_3|, |\\tau_k-\\zeta_6|\\}< \\varepsilon}}\\log^+|j(\\tau_k)^{-1}|\\le 3{\\mathcal{C}}_\\varepsilon(\\Delta)\\log|\\Delta|. \n\\end{equation}\nSubstituting~\\eqref{esecondsum} and~\\eqref{efirstsum} into~\\eqref{etwosums}, we obtain~\\eqref{eupperheightbis}.\n\\end{proof}\n\n\n\\subsection{Proof of Corollary~\\ref{cupper}}\n\nTo prove the corollary we need a lower bound for the quantity~$F$ defined in Theorem~\\ref{thceps} and an upper bound for the class number ${\\mathcal{C}}(\\Delta)$.  \n\n\\begin{lemma}\n\\label{lf}\nAssume that ${|\\Delta|\\ge 10^{14}}$. Then   ${F\\ge |\\Delta|^{0.34\/\\log\\log(|\\Delta|^{1\/2})}}$ and ${F\\ge 18.54\\log\\log(|\\Delta|^{1\/2})}$.  \n\\end{lemma}\n\n\\begin{proof}\nDefine, as usual \n\\begin{equation}\n\\label{ethetapi}\n\\vartheta(x)=\\sum_{p\\le x}\\log p, \\qquad \\pi(x)=\\sum_{p\\le x}1. \n\\end{equation}\nThen \n\\begin{align}\n\\vartheta(x) & \\le 1.017 x &&(x>0), \\nonumber\\\\\n\\label{elowerpi}\n\\pi(x) &\\ge \\frac{x}{\\log x} && (x\\ge 17),\n\\end{align} \nsee \\cite{RS62}, Theorem~9 on page~71 and Corollary~1 after Theorem~2 on page~69. Estimate~\\eqref{elowerpi} implies that\n\\begin{equation}\n\\label{elowerpibis}\n\\pi(x) \\ge 0.99995\\frac{x}{\\log x}  \\qquad (x\\ge 13).\n\\end{equation}\nSetting here \n$$\nx=\\frac{\\log(|\\Delta|^{1\/2})}{1.017} , \\qquad N =\\prod_{p\\le x}p,\n$$\nwe obtain ${N\\le |\\Delta|^{1\/2}}$ and \n$$\n\\omega(N) =\\pi(x) \\ge \\frac{0.99995\\log(|\\Delta|^{1\/2})}{1.017\\log\\log(|\\Delta|^{1\/2})}. \n$$\nNote that ${x\\ge \\bigl(\\log(10^{7})\\bigr)\/1.017>15}$, so we are allowed to use~\\eqref{elowerpibis}. \nWe obtain \n$$\nF\\ge 2^{\\omega(N)} \\ge |\\Delta|^{\\frac{0.99995\\log2}{2\\cdot1.017\\log\\log(|\\Delta|^{1\/2})}}\\ge |\\Delta|^{0.34\/\\log\\log(|\\Delta|^{1\/2})},\n$$\nproving the first estimate.\n\n\nTo prove the second estimate, we deduce from the first estimate that \n\\begin{equation}\n\\label{elog-log}\n\\log F-\\log\\log\\log(|\\Delta|^{1\/2})\\ge 0.68 \\frac{u}{\\log u}-\\log\\log u,\n\\end{equation} \nwhere we set ${u=\\log(|\\Delta|^{1\/2})}$. The right-hand side of~\\eqref{elog-log}, viewed as a function in~$u$, is increasing for ${u\\ge \\log(10^{7})}$. Hence \n$$\n\\log F-\\log\\log\\log(|\\Delta|^{1\/2})\\ge 0.68 \\frac{\\log(10^{7})}{\\log \\log(10^{7})}-\\log\\log \\log(10^{7}) \\ge 2.92,\n$$\nand ${F\\ge e^{2.92}\\log\\log(|\\Delta|^{1\/2}) \\ge 18.54 \\log\\log(|\\Delta|^{1\/2})}$. \n\\end{proof}\n\n\\begin{lemma}\n\\label{lcd}\nFor ${\\Delta\\ne -3,-4}$ we have \n$$\n{\\mathcal{C}}(\\Delta) \\le \\pi^{-1}|\\Delta|^{1\/2}(2+\\log|\\Delta|).\n$$ \n\\end{lemma}\n\n\\begin{proof}\nThis follows from Theorems~10.1 and~14.3 in \\cite[Chapter~12]{Hu82}. Note that in~\\cite{Hu82} the right-hand side has an extra factor $\\omega\/2$, where~$\\omega$ is the number of roots of unity in the imaginary quadratic order of discriminant~$\\Delta$. Since we assume that ${\\Delta \\ne -3,-4 }$, we have ${\\omega=2}$, so we may omit this factor. \n\\end{proof}\n\n\n\\begin{proof}[Proof of Corollary~\\ref{cupper}]\nSubstituting the estimate for ${\\mathcal{C}}_\\varepsilon(\\Delta)$ from~\\eqref{enewboundsimple} into~\\eqref{eupperheightbis}, we obtain the estimate\n\\begin{align*}\n{\\mathrm{h}}(\\alpha)&\\le 3A\\frac{9.83|\\Delta|^{1\/2} \\varepsilon^2\\log\\log(|\\Delta|^{1\/2})+3.605|\\Delta|^{1\/2}\\varepsilon+ 4}{{\\mathcal{C}}(\\Delta)}\n\\\\&\\hphantom{\\le}\n+3\\log(\\varepsilon^{-1})-10.66\n\\end{align*}\nwith ${A=F\\log|\\Delta|}$. Specifying \n$$\n\\varepsilon=0.27\\frac{{\\mathcal{C}}(\\Delta)}{A|\\Delta|^{1\/2}}\n$$\n(this is a nearly optimal value, and it satisfies ${\\varepsilon\\le 4\\cdot10^{-3}}$ as verified below), we obtain, using Lemmas~\\ref{lf} and~\\ref{lcd},\n\\begin{align*}\n{\\mathrm{h}}(\\alpha)&\\le  3\\cdot 9.83 \\cdot (0.27)^2\\frac{\\log\\log(|\\Delta|^{1\/2})}F\\frac{{\\mathcal{C}}(\\Delta)}{|\\Delta|^{1\/2}\\log|\\Delta|}+3\\cdot3.605\\cdot0.27\\\\ \n&\\hphantom{\\le}+\\frac{12A}{{\\mathcal{C}}(\\Delta)} +3\\log\\frac{A|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}-3\\log0.27-10.66\\\\\n&\\le \\frac{12A}{{\\mathcal{C}}(\\Delta)} +3\\log\\frac{A|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}+ \\frac{3\\cdot 9.83 \\cdot (0.27)^2\\cdot0.34}{18.54}+3\\cdot3.605\\cdot0.27\\\\ \n&\\hphantom{\\le}-3\\log0.27-10.66\\\\\n&\\le \\frac{12A}{{\\mathcal{C}}(\\Delta)} +3\\log\\frac{A|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}-3.77,\n\\end{align*}\nas wanted. \n\n \nWe only have to verify that ${\\varepsilon\\le 4\\cdot10^{-3}}$. We have\n${F\\ge 256}$ when ${|\\Delta|\\ge 10^{14}}$. Using Lemma~\\ref{lcd}, we obtain \n\\begin{align*}\n\\varepsilon=\\frac{0.27{\\mathcal{C}}(\\Delta)}{|\\Delta|^{1\/2}\\log |\\Delta|}\\frac1F\n\\le 0.27\\pi^{-1}\\frac{2+\\log(10^{14})}{\\log(10^{14})}\\cdot \\frac1{256}\n<4\\cdot 10^{-4}.\n\\end{align*}\nThe proof is complete.\n\\end{proof}\n\n\n\n\n\\section{Lower bounds for the height of a singular modulus}\n\\label{slower}\n\nNow we establish explicit lower bounds of the form~\\eqref{ecolm} and~\\eqref{etriv}. \n\n\\subsection{The ``easy'' bound}\n\nWe start by proving a bound of the form~\\eqref{etriv}. \n\n\\begin{proposition}\n\\label{ptriv}\nLet~$\\alpha$ be a singular modulus of discriminant~$\\Delta$. \nAssume that ${|\\Delta|\\ge 16}$. Then \n\\begin{equation}\n\\label{etrivbis}\n{\\mathrm{h}}(\\alpha) \\ge  \\frac{\\pi|\\Delta|^{1\/2}-0.01}{{\\mathcal{C}}(\\Delta)}.\n\\end{equation}\n\\end{proposition}\n\n\nWe need a simple lemma. \n\n\\begin{lemma}\n\\label{lttsn}\nFor ${z\\in {\\mathcal{F}}}$ with imaginary part~$y$ we have \n$$\n\\bigl||j(z)|-e^{2\\pi y}\\bigr|\\le 2079.\n$$ \nIf ${y\\ge 2}$ then we also have ${|j(z)|\\ge 0.992 e^{2\\pi y}}$. \n\\end{lemma}\n\n\\begin{proof}\nThe first statement is Lemma~1 of~\\cite{BMZ13}, and the second one is an immediate consequence. \n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Proposition~\\ref{ptriv}]\nOne of the conjugates of~$\\alpha$ over~$\\Q$ is equal to ${j((b+\\sqrt\\Delta)\/2)}$, with ${b=1}$ for~$\\Delta$ odd, and ${b=0}$ for~$\\Delta$ even;  it corresponds to the element ${(1,b,(-\\Delta+b^2)\/4)}$ of the set $T_\\Delta$. \nHence\n$$\n{\\mathrm{h}}(\\alpha) \\ge  \\frac{\\log|j((b+\\sqrt\\Delta)\/2)|}{{\\mathcal{C}}(\\Delta)}.\n$$\nUsing Lemma~\\ref{lttsn}, we obtain\n$$\n\\log|j((b+\\sqrt\\Delta)\/2)|\\ge \\pi|\\Delta|^{1\/2}+\\log 0.992 \\ge \\pi|\\Delta|^{1\/2}-0.01.\n$$\nWhence the result. \n\\end{proof}\n\n\\subsection{The ``hard'' bound}\n\n\\label{sshard}\n\nWe are left with  bound~\\eqref{ecolm}. We are going to prove the following. \n\n\\begin{proposition}\n  \\label{phlb2}\nLet~$\\alpha$ be a singular modulus of discriminant~$\\Delta$.\nThen\n\\begin{equation}\n\\label{ecolmbis}\n{\\mathrm{h}}(\\alpha) \\ge \\frac{3}{\\sqrt{5}}\\log|\\Delta| -9.79. \n\\end{equation}\n\\end{proposition}\n\n\nThe proof of Proposition~\\ref{phlb2} relies on the\nfact that it is possible to evaluate the Faltings height of an elliptic curve\nwith complex multiplication precisely, due to the work of Colmez~\\cite{Colmez} and Nakkajima-Taguchi~\\cite{NT}; for an exact statement see  \\cite[Lemma 4.1]{Ha10}. \n\nLet~$E$ be an elliptic curve with CM by an order of discriminant~$\\Delta$.  \nWe let ${\\mathrm{h}}_F(E)$ denote the stable Faltings height of~$E$ (using Deligne's normalization~\\cite{De85}). \nThe above-mentioned explicit formula for ${\\mathrm{h}}_F(E)$ is used in~\\cite{HJM:6} to obtain the lower bound  \n\\begin{equation}\n\\label{efaltlowerold}\n{\\mathrm{h}}_F(E) \\ge \\frac{1}{4\\sqrt5} \\log|\\Delta| - 5.93,\n\\end{equation}\nsee Lemma 14(ii) therein. Unfortunately, this bound is numerically too weak for our purposes.\n\nProposition~\\ref{phlb2} will be deduced from the following numerical refinement of~\\eqref{efaltlowerold}.\n\n\\begin{proposition}\n\\label{pfaltlower}\nLet~$E$ be an elliptic curve with CM by an order of discriminant~$\\Delta$.\nThen \n\\begin{equation}\n\\label{efaltlowernew}\n{\\mathrm{h}}_F(E) \\ge \\frac{1}{4\\sqrt5} \\log|\\Delta| -\\gamma -\\frac{\\log(2\\pi)}{2}-\\left(\\frac1{2\\sqrt5}-\\frac16\\right)\\log2,\n\\end{equation}\nwhere ${\\gamma=0.57721\\ldots}$ is the Euler constant. \n\\end{proposition}\n\n\nLet us first show how Proposition~\\ref{pfaltlower} implies Proposition~\\ref{phlb2}. \n\n\n\\begin{proof}[Proof of Proposition~\\ref{phlb2} (assuming Proposition~\\ref{pfaltlower})]\nLet~$E$ be an elliptic curve with ${j(E)=\\alpha}$. \nWe only need to relate ${\\mathrm{h}}_F(E)$ to ${\\mathrm{h}}(j(E))$. For this purpose we use Lemma~7.9 of Gaudron and R\\'emond~\\cite{GaudronRemond:periodes}\\footnote{The reader should be warned that our ${\\mathrm{h}}_F(E)$ is denoted ${\\mathrm{h}}(E)$ in~\\cite{GaudronRemond:periodes}.}.\nIn our\nnotation  they\nshow that \n\\begin{equation}\n\\label{egr}\n{\\mathrm{h}}_F(E)\\le {\\mathrm{h}}(j(E))\/12 - 0.72 \n\\end{equation}\nA quick calculation yields  our\nclaim. \n\\end{proof}\n\n\n\nTo prove Proposition~\\ref{pfaltlower} we need a technical lemma.\nSet \n\\begin{equation}\n\\label{elambda}\n\\lambda=\\frac12-\\frac1{2\\sqrt5},\n\\end{equation}\nand define the additive arithmetical functions $\\beta(n)$ and $\\delta(n)$ by  \n\\begin{equation}\n\\label{ebetadelta}\n\\beta(p^k) =\\frac{\\log p}{p+1}\\frac{1-p^{-k}}{1-p^{-1}}, \\quad \\beta(n)=\\sum_{p^k\\|n}\\beta(p^k), \\quad \\delta(n)= \\lambda\\log n -\\beta(n).\n\\end{equation}\n\n\\begin{lemma}\n\\label{ltechn} \nFor every positive integer~$n$ we have \n$$\n\\delta(n) \\ge\\delta(2)= \\left(\\frac16-\\frac1{2\\sqrt5}\\right)\\log2. \n$$\n\\end{lemma}\n\n\\begin{proof}\nSince ${1\/3>\\lambda>1\/4}$, we have ${\\delta(2)<0}$ and  ${\\delta(p)>0}$ for all primes ${p\\ge 3}$. Also, for ${k\\ge 1}$ and any prime~$p$ we have \n$$\n\\delta(p^{k+1})-\\delta(p^k)= \\left(\\lambda-\\frac1{p^k(p+1)}\\right)\\log p >0.\n$$\nSince ${\\delta(4)>0}$, this proves that ${\\delta(p^k)>0}$ for every prime power ${p^k\\ne 2}$, whence the result. \n\\end{proof}\n\nProposition~\\ref{pfaltlower} is an immediate consequence of Lemma~\\ref{ltechn} and the following statement.  \n\n\\begin{proposition}\n\\label{pfaltlowerf}\nIn the set-up of Proposition~\\ref{pfaltlower} we have\n\\begin{equation}\n\\label{efaltlowernewf}\n{\\mathrm{h}}_F(E) \\ge \\frac{1}{4\\sqrt5} \\log|\\Delta| + \\lambda \\log f- \\beta(f) -\\gamma -\\frac{\\log(2\\pi)}{2}.\n\\end{equation}\n\\end{proposition}\nSince \n$$\n\\lambda \\log f- \\beta(f)\\ge-\\left(\\frac1{2\\sqrt5}-\\frac16\\right)\\log2\n$$\nby Lemma~\\ref{ltechn}, this implies Proposition~\\ref{pfaltlower}. \n\n\\begin{proof}[Proof of Proposition~\\ref{pfaltlowerf}]\nWrite ${\\Delta=Df^2}$ with~$D$ the fundamental discriminant and~$f$ the conductor. Define \n$$\ne_f(p) = \\frac{1-\\chi(p)}{p-\\chi(p)}\\frac{1-p^{-{\\mathrm{ord}}_p(f)}}{1-p^{-1}}, \\qquad c(f) = \\frac 12 \\left(\\sum_{p\\mid f} e_f(p)\\log p\\right),\n$$\nwhere ${\\chi(\\cdot)=(D\/\\cdot)}$ is Kronecker's symbol.\n\n\nIn the proof of Lemma~14 of~\\cite{HJM:6}\\footnote{Note that our~$D$ is written~$\\Delta$ in~\\cite{HJM:6}.},\nthe stable Faltings height of~$E$  is estimated as \n\\begin{align*}\n{\\mathrm{h}}_F(E) &\\ge \\frac{1}{4\\sqrt5}\\log|D| + \\frac 12 \\log f\n-c(f)\n-\\gamma -\\frac{\\log(2\\pi)}{2},\\\\\n&=\\frac{1}{4\\sqrt5}\\log|\\Delta| + \\lambda \\log f\n-c(f)\n-\\gamma -\\frac{\\log(2\\pi)}{2}.\n\\end{align*}\nThus, to establish~\\eqref{efaltlowernewf}, we only have to prove that ${c(f)\\le \\beta(f)}$. \nWe have \n$$\n\\frac{1-\\chi(p)}{p-\\chi(p)} = \n\\begin{cases}\n0, &\\chi(p)=1,\\\\\n1\/p, &\\chi(p)=0,\\\\\n 2\/(p+1), &\\chi(p)=-1. \n\\end{cases}\n$$\nHence\n$$\n\\frac{1-\\chi(p)}{p-\\chi(p)} \\le \\frac2{p+1}\n$$\nin any case. This implies that ${c(f)\\le \\beta(f)}$.\nThe proposition is proved. \n\\end{proof}\n\n\n\\section{The estimate \\texorpdfstring{${|\\Delta|< 10^{15}}$}{|Delta|<10to15}}\n\\label{stenfourteen}\n\n\n\nIn this section we obtain the first explicit upper bound for the\ndiscriminant of a singular unit. \n\n\n\\begin{theorem}\n  \\label{thhighrange}\nLet~$\\Delta$  be the discriminant of a singular unit. Then\n  ${|\\Delta|<10^{15}}$. \n\\end{theorem}\n\n{\\sloppy\n\nThroughout this section~$\\Delta$ is the discriminant of a singular unit~$\\alpha$, and  we assume that  ${X=|\\Delta|\\ge 10^{15}}$, as otherwise there is nothing to prove. \nOur principal tools will be the upper estimate~\\eqref{eupdel}\nand  the lower estimates~\\eqref{etrivbis},~\\eqref{ecolmbis}. We reproduce them here for convenience:\n\\begin{align}\n\\label{eupx}\n{\\mathrm{h}}(\\alpha)&\\le \\frac{12A}{{\\mathcal{C}}(\\Delta)}+3\\log\\frac{AX^{1\/2}}{{\\mathcal{C}}(\\Delta)}-3.77,\\\\\n\\label{etrivx}\n{\\mathrm{h}}(\\alpha)&\\ge  \\frac{\\pi X^{1\/2}-0.01}{{\\mathcal{C}}(\\Delta)},\\\\\n\\label{ecolmx}\n{\\mathrm{h}}(\\alpha)&\\ge \\frac{3}{\\sqrt5}\\log X-9.79.\n\\end{align}\nNote that our assumption ${X\\ge 10^{15}}$ implies that the right-hand side of~\\eqref{ecolmx} is positive.\n\n}\n\n\n\\subsection{The main inequality}\n\\label{ssinequality}\n\nRecall that ${A= F\\log X}$. \nMinding~$0.01$ in~\\eqref{etrivx} we deduce from~\\eqref{eupx},~\\eqref{etrivx} and~\\eqref{ecolmx} the inequality\n$$\n\\frac{12A}{{\\mathcal{C}}(\\Delta)}+3\\log\\frac{AX^{1\/2}}{{\\mathcal{C}}(\\Delta)}-3.76 \\ge \\max\\left\\{\\frac{\\pi X^{1\/2}}{{\\mathcal{C}}(\\Delta)}, \\frac{3}{\\sqrt5}\\log X-9.78\\right\\}. \n$$\nDenoting\n\\begin{equation}\n\\label{ey}\nY=\\max\\left\\{\\frac{\\pi X^{1\/2}}{{\\mathcal{C}}(\\Delta)}, \\frac{3}{\\sqrt5}\\log X-9.78\\right\\}, \n\\end{equation}\nwe re-write this as \n\\begin{equation}\n\\label{enotyet}\n\\frac{12A\/{\\mathcal{C}}(\\Delta)}{Y}+\\frac{3\\log A-3.76}Y+ \\frac{\\log(X^{1\/2}\/{\\mathcal{C}}(\\Delta))}{Y} \\ge 1. \n\\end{equation}\nNote that ${3\\log A-3.76>0}$, because ${A\\ge  \\log X\\ge \\log(10^{15}) >30}$. Hence we may replace~$Y$ by ${\\frac{3}{\\sqrt5}\\log X-9.78}$ in the middle term of the left-hand side in~\\eqref{enotyet}. Similarly, in the first term we may replace~$Y$ by ${\\pi X^{1\/2}\/{\\mathcal{C}}(\\Delta)}$, and in the third term we may replace ${X^{1\/2}\/{\\mathcal{C}}(\\Delta)}$ by ${\\pi^{-1}Y}$. We obtain \n\\begin{equation}\n\\label{einequality}\n12\\pi^{-1} AX^{-1\/2}+ \\frac{3\\log A-3.76}{\\frac{3}{\\sqrt5}\\log X-9.78} + 3\\frac{\\log(\\pi^{-1}Y)}{Y}\\ge 1. \n\\end{equation}\n\n\nTo show that~\\eqref{einequality} is not possible for ${X\\ge 10^{15}}$, we will bound from above each of the three terms in its left-hand side.  To begin with, we bound~$A$. \n\n\n\\subsection{Bounding~$F$ and~$A$}\n\nRecall that ${F=\\max\\{2^{\\omega(a)}: a\\le X^{1\/2}\\}}$ and ${A=F\\log X}$. \n\nLet  \n${N_1 = 2\\cdot3\\cdot5\\cdots 1129}$  be the product of the first $189$ prime numbers. Define the real number~$c_1$ from \n\\begin{align*}\n\\omega(N_1)&=\\frac{\\log N_1}{\\log\\log N_1 -c_1}.\n\\end{align*}\nA calculation shows that ${c_1 < 1.1713142}$. \nRobin  \\cite[Th\u00e9or\u00e8me~13]{Ro83} proved that\n\\begin{equation*}\n\\omega(n) \\le \\frac{\\log n}{\\log\\log n-c_1} \n\\end{equation*}\nfor ${n\\ge 26}$. \nThis implies that \n\\begin{align}\n\\label{eboundf}\n\\frac{\\log F}{\\log 2} &\\le \\frac12\\frac{\\log X}{\\log\\log X-c_1-\\log2} ,\\\\\n\\label{ebounda}\n\\log A &\\le \\frac{\\log2}2\\frac{\\log X}{\\log\\log X-c_1-\\log2}+\\log\\log X. \n\\end{align}\nIndeed, the function \n$$\ng(x)= \\frac{\\log x}{\\log\\log x-c_1}\n$$\nis strictly increasing for ${x\\ge 6500}$ and ${g(6500)>8}$. \nIf ${a\\le X^{1\/2}}$ then either ${a\\le 6500}$ in which case \n${\\omega(a) \\le 5<g(6500)<g(X^{1\/2})}$\n(recall that ${X\\ge 10^{15}}$), or ${6500<a\\le X^{1\/2}}$, in which case \n${\\omega(a) \\le g(a)\\le g(X^{1\/2})}$.\nThus, in any case we have \n$$\n\\omega(a) \\le g(X^{1\/2})= \\frac12\\frac{\\log X}{\\log\\log X-c_1-\\log2}, \n$$\nwhich proves~\\eqref{eboundf}. The estimate~\\eqref{ebounda} is an immediate consequence of~\\eqref{eboundf}.  \n\n\n\\subsection{Bounding the first term in~\\eqref{einequality}}\nUsing~\\eqref{ebounda}, we estimate\n$$\n\\frac{\\log (AX^{-1\/2})}{\\log X}\\le u_0(X), \n$$\nwhere\n\\begin{align*}\nu_0(x)&= \\frac{\\log2}2\\frac{1 }{\\log\\log x-c_1-\\log2}+\\frac{\\log\\log x}{\\log x}-\\frac12. \n\\end{align*}\nThe function $u_0(x)$ is decreasing for ${x\\ge 10^{10}}$. Hence \nfor ${X\\ge 10^{15}}$ we have \n$$\n\\frac{\\log (AX^{-1\/2})}{\\log X}\\le u_0(10^{15}) <-0.1908. \n$$\nThis proves the estimate \n\\begin{equation*}\nAX^{-1\/2}<  10^{15\\cdot(-0.1908)}<0.0014\n\\end{equation*}\nfor ${X\\ge 10^{15}}$. \n\n\n\\subsection{Bounding the second term in~\\eqref{einequality}}\nUsing~\\eqref{ebounda}, we estimate \n$$\n\\frac{3\\log A-3.76}{\\frac{3}{\\sqrt5}\\log X-9.78}\\le u_1(X)u_2(X), \n$$\nwhere\n\\begin{align*}\nu_1(x) &= \\frac{3\\log2}2\\frac{1}{\\log\\log x -c_1-\\log2}+\\frac{3\\log\\log x-3.76}{\\log x},\\\\\nu_2(x)& = \\left(\\frac{3}{\\sqrt5}-\\frac{9.78}{\\log x}\\right)^{-1}. \n\\end{align*}\nBoth functions $u_1(x)$ and $u_2(x)$ are decreasing for ${x\\ge 10^{10}}$. Hence, for ${X\\ge 10^{15}}$ we have\n\\begin{equation*}\n\\frac{3\\log A-3.76}{\\frac{3}{\\sqrt5}\\log X-9.78}\\le u_1(10^{15})u_2(10^{15}) <0.7734. \n\\end{equation*}\n\n\n\\subsection{Bounding the third term in~\\eqref{einequality}}\n\nThe function ${x\\mapsto (\\log x)\/x}$ is decreasing for ${x\\ge e}$. Since for ${X\\ge 10^{15}}$ we have \n$$\n\\pi^{-1}Y\\ge \\pi^{-1}\\left(\\frac{3}{\\sqrt5}\\log X-9.78\\right)\\ge e,\n$$\nwe have, for ${X\\ge 10^{15}}$, the estimate \n$$\n\\frac{\\log(\\pi^{-1}Y)}{Y} \\le u_3(X), \n$$\nwhere \n$$\nu_3(x) = \\frac{\\log\\left(\\pi^{-1}\\bigl(\\frac{3}{\\sqrt5}\\log x-9.78\\bigr)\\right)}{\\frac{3}{\\sqrt5}\\log x-9.78}. \n$$\nMoreover, the function $u_3(x)$ is decreasing for ${x\\ge 10^{15}}$, which implies that \n\\begin{equation*}\n\\frac{\\log(\\pi^{-1}Y)}{Y}\\le u_3(X) \\le u_3(10^{15}) < 0.0672\n \\end{equation*}\nfor ${X\\ge 10^{15}}$. \n\n\\subsection{Summing up}\n\nNow, when ${X\\ge 10^{15}}$, we can combine the above estimates and bound the left-hand side of~\\eqref{einequality} by\n$$\n12\\pi^{-1}\\cdot0.0014 + 0.7734 + 3 \\cdot 0.0672 <0.981.\n$$\nHence, for ${X\\ge 10^{15}}$ we cannot have~\\eqref{einequality}. This contradiction completes the proof of Theorem~\\ref{thhighrange}. \n\n\n\n\n\n\\section{Handling the mid-range \\texorpdfstring{$10^{10}\\le |\\Delta|< 10^{15}$}{10to10<Delta<10to15}}\n\\label{sec:midrange}\n\n\nIn this section we rule out the existence of singular units with discriminants in the\nmid-range ${[10^{10}, 10^{15})}$, improving thereby the bound from the previous section.\n\n\n\\begin{theorem}\n  \\label{thmidrange}\nLet~$\\Delta$  be the discriminant of a singular unit. Then\n  ${|\\Delta|\\notin [10^{10},10^{15})}$. \n\\end{theorem}\n\n\n\nIn Section~\\ref{sceps} we estimate trivially ${2^{\\omega(a)}\\le F}$. One might expect to do better by estimating the average order rather than the maximal order of the arithmetical function ${2^{\\omega(n)}}$. This is accomplished in Subsection~\\ref{ssdelange} and  allows us to obtain, in Subsection~\\ref{ssnewceps}, a new bound for ${{\\mathcal{C}}_\\varepsilon(\\Delta)}$ in the range ${10^{10}\\le |\\Delta|< 10^{15}}$. Using this, Theorem~\\ref{thmidrange} is proved in Subsection~\\ref{ssproofmidrange} by  an argument similar to the proof of Theorem~\\ref{thhighrange}. \n\nThroughout this section~$n$ denotes a positive integer. \n\n\\subsection{Average order of the function $2^{\\omega(n)}$ on subintervals of ${[0,2\\cdot10^7]}$}\n\\label{ssdelange}\nFor a positive  real number~$x$ set \n$$\nS(x)= \\sum_{n\\le x} 2^{\\omega(n)}.\n$$\nWe define $S(0)=0$.\nAs Theorem II.6.1 from~\\cite{Te15} suggests, the function $S(x)$ can be well approximated by the function\n$$\ng(x)= \\lambda_0x\\log x+\\lambda_1 x,\n$$\nwhere \n\\begin{align*}\n\\lambda_0&= \\zeta(2)^{-1}=0.607927101854026\\dots,\\\\%62866327677925837\\dots,\\\\ \n\\lambda_1 &= -2\\frac{\\zeta^{\\prime}(2)}{\\zeta(2)^{2}}+\\frac{2\\gamma-1}{\\zeta(2)}=0.786872460166245\\dots\n\\end{align*}\nand~$\\gamma$ is the Euler constant. The function~$g$ is increasing on\n$[1,\\infty)$. \n\n\nAs already mentioned in the introduction, the error term ${|S(x)-g(x)|}$ can be estimated by the Selberg-Delange method~\\cite[Chapter~II.5]{Te15}, but on our\nlimited range it is more advantageous to obtain an optimal error term by a computer-assisted\ncalculation.\n\n\\begin{proposition}\n\\label{proposition::selbergdelange}\nFor ${2\\le x \\le 2\\cdot 10^7}$ we have \n\\begin{equation}\n\\label{esqrt}\ng(x)- 1.010x^{1\/2} \\le S(x) \\le g(x)+ 0.712x^{1\/2}, \n\\end{equation}\nand for ${4\\cdot10^4\\le x\\le 2\\cdot 10^7}$ we have \n\\begin{equation}\n\\label{esqrtlog}\ng(x)- 2.267\\frac{x^{1\/2}}{\\log x} \\le S(x) \\le g(x)+ 2.598\\frac{x^{1\/2}}{\\log x}. \n\\end{equation}\n\\end{proposition}\n\n\n\\begin{proof}\nSet\n\\begin{align*}\nc_1&=\\max_{2\\le n\\le 2\\cdot 10^7}\\frac{S(n)-g(n)}{\\sqrt n}, \\\\\nc_2&=\\max_{2\\le n\\le 2\\cdot 10^7}\\frac{g(n+1)-S(n)}{\\sqrt n},\\\\\nc_3&=\\max_{4\\cdot10^4\\le n\\le 2\\cdot 10^7}\\frac{(S(n)-g(n))\\log n}{\\sqrt n}, \\\\\nc_4&=\\max_{4\\cdot10^4\\le n\\le 2\\cdot 10^7}\\frac{(g(n+1)-S(n))\\log n}{\\sqrt n}.\n\\end{align*}\nThen for  ${2\\le n\\le 2\\cdot 10^7}$ we have\n$$\nS(n) \\le g(n)+c_1\\sqrt n,   \\qquad S(n) \\ge g(n+1)-c_2\\sqrt {n}.\n$$\nHence for ${2\\le x\\le 2\\cdot 10^7}$ we have \n\\begin{align*}\nS(x)&= S(\\lfloor x \\rfloor) \\le g(\\lfloor x \\rfloor)+c_1\\sqrt{\\lfloor x\\rfloor} \\le g(x)+c_1\\sqrt x,\\\\\nS(x)&= S(\\lfloor x \\rfloor) \\ge g(\\lfloor x \\rfloor+1)-c_2\\sqrt{\\lfloor x\\rfloor} \\ge g(x)-c_2\\sqrt x.\n\\end{align*}\nIn a similar way \nwe show that \nfor ${4\\cdot10^4\\le x\\le 2\\cdot 10^7}$ \n$$\ng(x)-c_4\\frac{\\sqrt x}{\\log x}\\le S(x) \\le g(x)+c_3\\frac{\\sqrt x}{\\log x}\n$$\nhaving used that $x\\mapsto \\sqrt{x}\/\\log x$ is increasing on\n$(e^2,\\infty)$.\nA computer-assisted calculation shows that \n$$\nc_1\\le 0.712, \\quad c_2\\le 1.010, \\quad\nc_3\\le 2.598, \\quad c_4\\le 2.267.\n$$\nWe verify this by means of a\n\\textsf{SAGE}~\\cite{sagemath} script\\footnote{A link to the  script\n  \\textsf{prop6\\_2.sage} is on  the second named\nauthor's homepage. The running time is roughly 30 minutes on a regular\ndesktop (Intel Xeon CPU E5-1620 v3, 3.50GHz, 32GB RAM).} using the interval arithmetic \\textsf{MPFI} package~\\cite{MPFI}.\n\\end{proof}\n\n\\begin{corollary}\nLet~$A$ and~$B$ be positive real numbers satisfying \n$$\n0<A<B\\le 2\\cdot 10^7, \\qquad B\\ge 1.\n$$\nThen \n\\begin{equation}\n\\label{esqrtab}\n\\sum_{A<n\\le B}2^{\\omega(n)}\\le \\lambda_0 (B-A)(1+\\log B)+\\lambda_1(B-A)+1.722B^{1\/2}\n\\end{equation}\nIf, in addition to this, \n$\n{A\\ge 4\\cdot10^4}, \n$\nthen \n\\begin{equation}\n\\label{esqrtlogab}\n\\sum_{A<n\\le B}2^{\\omega(n)}\\le \\lambda_0 (B-A)(1+\\log B)+\\lambda_1(B-A)+4.865\\frac{B^{1\/2}}{\\log B}. \n\\end{equation}\n\\end{corollary}\n\n\n\n\\begin{proof}\nIn general we have\n\\begin{equation*}\nS(B)-S(A)=  \\sum_{A<n\\le B} 2^{\\omega(n)} = \\sum_{n =\n    \\lfloor{A}\\rfloor+1}^{\\lfloor{B}\\rfloor}2^{\\omega(n)} =\n  S(\\lfloor B\\rfloor) - S(\\lfloor A\\rfloor).\n\\end{equation*}\nNote that since $B>A>0$ we remark that\n\\begin{equation}\n\\label{eremark}\nB\\log B - A \\log A \\le (B-A)(1+\\log B).\n\\end{equation}\nWhen ${A\\ge 2}$ estimate~\\eqref{esqrtab} follows immediately\nfrom~\\eqref{esqrt} and~\\eqref{eremark}.\n\nLet us assume  that ${A<2}$, hence $\\lfloor A\\rfloor$ is $0$ or $1$\nand $S(\\lfloor A\\rfloor)=0$ or $1$, respectively. For $B\\ge 2$ we find\n$$\n\\sum_{A<n\\le B}2^{\\omega(n)} \\le \\lambda_0 B\\log\nB+\\lambda_1B+0.712B^{1\/2} -S(\\lfloor A\\rfloor),\n$$\nand one easily verifies that \n$$\n-S(\\lfloor A\\rfloor)+\\lambda_0A(1+\\log B) +\\lambda_1A - \\lambda_0\n\\le 1.010B^{1\/2}\n$$\nby considering the cases $A\\in (0,1)$ and $A\\in [1,2)$ separately.\n  This implies (\\ref{esqrtab}). \nAnd if ${B<2}$, then \n${\\sum_{A<n\\le B} 2^{\\omega(n)}\\le 1\\le 1.722\nB^{1\/2}}$.  \nAs $B\\ge 1$, again we obtain (\\ref{esqrtab}). \n\nFinally, for ${4\\cdot10^4\\le A<B\\le 2\\cdot 10^7}$ estimate~\\eqref{esqrtlogab}\nfollows\nfrom~\\eqref{esqrtlog} and~\\eqref{eremark}.  \n\\end{proof}\n\n\n\n\\subsection{Bounding ${\\mathcal{C}}_\\varepsilon(\\Delta)$ for ${10^{10}\\le|\\Delta|< 10^{15}}$}\n\\label{ssnewceps}\n\nNow we can obtain a cardinal refinement of Theorem~\\ref{thceps}  for discriminants in the range ${10^{10}\\le |\\Delta|< 10^{15}}$.  We need a technical lemma using our\nnotation~\\eqref{eq:standardnotation}. \n\n\\begin{lemma}\n  \\label{lem:varsigma01}\n  Let $n$ be an integer with $1\\le n\\le 3.2\\cdot 10^7$.\n  \\begin{enumerate}[label={(\\roman*)}]\n  \\item \n  \\label{isigone} We have\n    $\\sigma_1(n)\/n \\le \\sigma_1(21621600)\/21621600= 3472\/715$. \n  \\item \n  \\label{isigzero} We have\n    $\\sigma_0(n)\\le 8.5 n^{1\/4}$. \n  \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nThis can be proved in at least two ways. In our relatively small range, we can\nperform a quick computer-assisted calculation\\footnote{A link to the \\textsf{PARI} script \\textsf{lemma6\\_4.gp} is on the second-named\nauthor's homepage. The running time is under 2 minutes on a regular\ndesktop (Intel Xeon CPU E5-1620 v3, 3.50GHz, 32GB RAM).}.  Alternatively, one can use the\n\\textit{On-Line Encyclopedia of Integer Sequences}~\\cite{oeis}. Inspecting T. D. Noe and D. Kilminster's\ntable in entry \\textsf{A004394}, one can deduce~\\ref{isigone}. Similarly,~\\ref{isigzero} follows from inspection of T. D. Noe's table in \\textsf{A002182}.\n\\end{proof}\n\n\n\\begin{proposition}\n  \\label{prop:Cepsmidrange}\nIn the set-up of Theorem~\\ref{thceps} assume that \n\\begin{equation}\n\\label{emidrange}\n{10^{10} \\le |\\Delta|< 10^{15}}.\n\\end{equation}\nThen\n\\begin{align*}\n{\\mathcal{C}}_\\varepsilon(\\Delta) &\\le (8\\varepsilon^2 + 0.811\\varepsilon)|\\Delta|^{1\/2} \\log|\\Delta|\n+(28\\varepsilon^2 +2.829\\varepsilon)|\\Delta|^{1\/2}  \\\\\n&\\hphantom{\\le}+89\\varepsilon {|\\Delta|^{3\/8}}+\n31.06\\frac{|\\Delta|^{1\/4}}{\\log|\\Delta|}\n\\end{align*}\n\\end{proposition}\n\n\\begin{proof}\nAs in the proof of Theorem~\\ref{thceps}, we\nset\n$$\nI=\\left(\\frac{|\\Delta|^{1\/2}}{\\sqrt3+2\\varepsilon}, \\frac{|\\Delta|^{1\/2}}{\\sqrt3}\\right]. \n$$\nWe again want to count pairs of integers $(a,b)$ such that \n\\begin{align*}\na\\in I,\\qquad b\\in [-a,-a(1-2\\varepsilon))\\cup (a(1-2\\varepsilon),a], \\qquad b^2\\equiv \\Delta\\bmod a. \n\\end{align*}\nLemmas~\\ref{lrootsmoda} and~\\ref{ltrivi} \nimply that,  for\nevery fixed~$a$, the number of  suitable~$b$ \ndoes not exceed ${(4\\varepsilon\\gcd_2(a,\\Delta)+2) 2^{\\omega(a\/\\gcd(a,\\Delta))+1}}$.\nHence\n\\begin{equation}\n\\label{ethreesums}\n{\\mathcal{C}}_\\varepsilon(\\Delta) \\le 8\\varepsilon\\sum_{a\\in I\\cap\\Z} {\\gcd}_2(a,\\Delta)\n2^{\\omega(a\/\\gcd_2(a,\\Delta)^2)}\n+ 4\\sum_{a\\in I\\cap\\Z}2^{\\omega(a)\n\\end{equation}\nwhere we used that $\\gcd_2(a,\\Delta)^2$ divides $\\gcd(a,\\Delta)$. \nWe estimate each of the terms separately.\n\n\n\\subsubsection*{Estimating the first term in~\\eqref{ethreesums}}\nRecall that $\\Delta \/ {\\tilde f}^2$ is a square-free integer. We have \n\\begin{equation}\n\\label{elefthand}\n\\sum_{a\\in I\\cap\\Z} {\\gcd}_2(a,\\Delta) 2^{\\omega(a\/\\gcd_2(a,\\Delta)^2)} \\le \\sum_{d\\mid {\\tilde f}}d\\sum_{a\\in I\\cap d^2\\Z}  2^{\\omega(a\/d^2)}.\n\\end{equation}\nTo estimate the inner sum, write ${a\\in I\\cap d^2\\Z}$ as ${a=d^2a'}$ with \n$$\na'\\in d^{-2}I = \\left(d^{-2}\\frac{|\\Delta|^{1\/2}}{\\sqrt3+2\\varepsilon}, d^{-2}\\frac{|\\Delta|^{1\/2}}{\\sqrt3}\\right].\n$$\nWe estimate $\\sum_{a'\\in d^{-2}I\\cap\\Z} 2^{\\omega(a')}$  using~\\eqref{esqrtab} with\n$$\nA=d^{-2}\\frac{|\\Delta|^{1\/2}}{\\sqrt3+2\\varepsilon}, \\quad B=d^{-2}\\frac{|\\Delta|^{1\/2}}{\\sqrt3}. \n$$ \nSince ${|\\Delta|\\le  10^{15}}$, we have ${B\\le 2\\cdot 10^7}$. From \n$$\nB-A\\le \\frac23 d^{-2}|\\Delta|^{1\/2}\\varepsilon, \\quad \\log B \\le \\frac12\\log |\\Delta|-\\frac12\\log 3,\n$$\nwe obtain \n$$\n\\sum_{a'\\in d^{-2}I\\cap \\Z}  2^{\\omega(a')}\\le \\frac{|\\Delta|^{1\/2}}{3d^2}(\\lambda_0 \\log |\\Delta| +2\\lambda_0+2\\lambda_1-\\lambda_0\\log3)\\varepsilon+\\frac{1.722|\\Delta|^{1\/4}}{\\sqrt[4]{3}d}  \n$$\nas long as $B\\ge 1$. If $B<1$, then the sum on the left is $0$ and the\ninequality remains valid as the right side is clearly positive. \nHence the left-hand side of~\\eqref{elefthand} is bounded by \n\\begin{align*}\n\\frac{|\\Delta|^{1\/2}}{3}(\\lambda_0 \\log |\\Delta|+ 2\\lambda_0+2\\lambda_1-\\lambda_0\\log3)\\varepsilon\\frac{\\sigma_1({\\tilde f})}{{\\tilde f}\n+\\frac{1.722|\\Delta|^{1\/4}}{\\sqrt[4]{3}}\\sigma_0({\\tilde f})\n\\end{align*}\nwhere we use  notation~\\eqref{eq:standardnotation} and the identity\n$\\sigma_1({\\tilde f})\/{\\tilde f} = \\sum_{d \\mid {\\tilde f}} d^{-1}$.\n\nRecall that ${\\tilde f} \\le |\\Delta|^{1\/2}\\le 3.2\\cdot 10^7$.\nHence Lemma \\ref{lem:varsigma01} implies that\n$$\n\\sigma_1({\\tilde f})\/{\\tilde f} \\le\n3472\/715, \\qquad \\sigma_0({\\tilde f})\\le 8.5 {\\tilde f}^{1\/4} \\le 8.5|\\Delta|^{1\/8}. \n$$\nTaking into account the factor $8\\varepsilon$, the first term in~\\eqref{ethreesums} is thus at most\n\\begin{alignat*}1\n   |\\Delta|^{1\/2} &(8\\log|\\Delta| + 28)\\varepsilon^2 +\n   89{|\\Delta|^{3\/8}}\\varepsilon.\n\\end{alignat*}\n\n\\subsubsection*{Estimating the second  term in~\\eqref{ethreesums}}\nSet\n$$\nA=\\frac{|\\Delta|^{1\/2}}{\\sqrt3+2\\varepsilon}, \\qquad B=\\frac{|\\Delta|^{1\/2}}{\\sqrt3}.\n$$ \nFrom~\\eqref{emidrange} and ${0<\\varepsilon \\le 1\/3}$ we deduce ${A\\ge 4\\cdot10^4}$  and ${B\\le\n2\\cdot 10^7}$. This allows us to apply~\\eqref{esqrtlogab}, and we obtain\n\\begin{align*}\n\\sum_{a\\in I\\cap\\Z}2^{\\omega(a)} &\\le \\frac{|\\Delta|^{1\/2}}{3}(\\lambda_0 \\log |\\Delta|+2\\lambda_0+2\\lambda_1 -\\lambda_0\\log3)\\varepsilon\\\\\n&\\hphantom{\\le}+\\frac{4.865|\\Delta|^{1\/4}}{\\sqrt[4]{3}\\log(|\\Delta\/3|^{1\/2})}\\\\\n &\\le \\frac{|\\Delta|^{1\/2}}{3}(\\lambda_0 \\log\n|\\Delta|+2\\lambda_0+2\\lambda_1 -\\lambda_0\\log3)\\varepsilon\\\\\n&\\hphantom{\\le}+\\frac{2\\cdot\n  4.865|\\Delta|^{1\/4}}{\\sqrt[4]{3}\\log |\\Delta|}\n\\frac{\\log10^{10}}{\\log(10^{10}\/3)}.\n\\end{align*}\nwhere  for the last estimate we used the assumption\n ${|\\Delta|\\ge\n  10^{10}}$. \nMinding the factor~$4$ we find that \nthe second term in~\\eqref{ethreesums} is at most\n\\begin{equation*}\n  |\\Delta|^{1\/2} (0.811 \\log|\\Delta| + 2.829 )\\varepsilon  + 31.06\n   \\frac{|\\Delta|^{1\/4}}{\\log |\\Delta|}. \n\\end{equation*}\nThis concludes our proof of Proposition \\ref{prop:Cepsmidrange}.\n\\end{proof}\n\n\\subsection{Proof of Theorem~\\ref{thmidrange}}\n\\label{ssproofmidrange}\nSuppose that~$\\alpha$ is a singular unit of discriminant~$\\Delta$\nand set ${X=|\\Delta|}$. Assuming  that ${10^{10}\\le X< 10^{15}}$,\nwe arrive at a contradiction. \nAs in   Section~\\ref{stenfourteen}  we use the estimates~\\eqref{etrivx} and~\\eqref{ecolmx} which follow from Propositions~\\ref{ptriv}\nand~\\ref{phlb2}, respectively. \n\nWe may no longer use Corollary~\\ref{cupper}, because its hypothesis ${X\\ge 10^{14}}$ is not valid in our current range. Instead, we will apply Theorem~\\ref{theceps}   directly now. For this, let\n${\\varepsilon \\in (0,4\\cdot 10^{-3}]}$. \n\n{\\sloppy\n\nWe define~$Y$ as in (\\ref{ey}) and recall that ${Y\\le 0.01 + {\\mathrm{h}}(\\alpha)}$. We find\n\\begin{equation}\n  \\label{eq:midrangeYbound}\n  Y \\le 3 \\frac{{\\mathcal{C}}_\\varepsilon(\\Delta)}{{\\mathcal{C}}(\\Delta)}\n  \\log X + 3 \\log(\\varepsilon^{-1}) - 10.65. \n\\end{equation}\nUsing Proposition \\ref{prop:Cepsmidrange} we find\n\\begin{alignat*}1\n  1&\\le 3 (8\\varepsilon^2 + 0.811\\varepsilon)\n  \\frac{X^{1\/2}(\\log X)^2}{Y{\\mathcal{C}}(\\Delta)} + 3 {(28\\varepsilon^2 + 2.829\n    \\varepsilon)}\\frac{X^{1\/2}\\log X}{Y{\\mathcal{C}}(\\Delta)}  \\\\&\n\\hphantom{\\le}    +\n3 \\frac{89 \\varepsilon X^{3\/8}\\log X}{Y{\\mathcal{C}}(\\Delta)}  \n + 3\\cdot 31.06\n  \\frac{X^{1\/4}}{Y{\\mathcal{C}}(\\Delta)} \n+ \\frac{3\\log(\\varepsilon^{-1}) - 10.65}{Y}.\n\\end{alignat*}\nFor all but the final term on the right we use ${Y{\\mathcal{C}}(\\Delta)\\ge \\pi X^{1\/2}}$\nand for the remaining term we use\n${Y\\ge \\frac{3}{\\sqrt{5}} \\log X - 9.78 > 0}$, as ${X\\ge  10^{5}}$,  \nto get\n\\begin{equation}\n  \\label{eq:1lowerbound}\\begin{aligned}\n  1&\\le \\frac{3}{\\pi}(8\\varepsilon^2 + 0.811\\varepsilon) (\\log X)^2\n  +\\frac{3}{\\pi}(28\\varepsilon^2 + 2.829\\varepsilon) \\log X\\\\\n  &\\hphantom{\\le}+  \\frac{267}{\\pi}\\varepsilon\\frac{\\log X}{X^{1\/8}} \n  +\\frac{93.18}{\\pi X^{1\/4}}\n   +\n    \\frac{3\\log(\\varepsilon^{-1})-10.65}{\\frac{3}{\\sqrt 5}\\log X - 9.78}.\n\\end{aligned}\n\\end{equation}\nOur choice is ${\\varepsilon = \n10^{-4}}$.\nThe first two terms in the right-hand side of~\\eqref{eq:1lowerbound} are \nmonotonously increasing, and the remaining three terms are  decreasing for ${X\\in[10^{10}, 10^{15})}$; note that ${x\\mapsto(\\log x)\/x^{1\/8}}$ is\ndecreasing for ${x \\ge 3000 > e^8}$. Using ${X < 10^{15}}$  for the first two terms and ${X\\ge  2\\cdot 10^{10}}$ for the\n remaining three terms, we see that the right-hand side of~\\eqref{eq:1lowerbound} is strictly smaller than $0.962$ if\n${X \\in [2\\cdot10^{10}, 10^{15})}$. Similarly, we infer that it is strictly smaller than $0.960$ if ${X \\in [10^{10}, 2\\cdot10^{10})}$. This completes the proof of Theorem~\\ref{thmidrange}. \n\\qed\n\n}\n\n\\section{Handling the low-range \\texorpdfstring{$3\\cdot 10^{5}\\le |\\Delta|< 10^{10}$}{3.10to5<|Delta|<10to10}}\n\\label{slowrange}\nWe now deal with the low-range ${|\\Delta|\\in [3\\cdot10^5, 10^{10})}$. For this range the upper\nbound on ${\\mathcal{C}}_\\varepsilon(\\Delta)$ arises from a computer-assisted search algorithm.\n\n \n\nWe prove the following. \n\\begin{theorem}\n  \\label{thlowrange}\n  Let~$\\Delta$  be the discriminant of a singular unit. Then\n ${|\\Delta|\\notin [3\\cdot10^5, 10^{10})}$. \n\\end{theorem}\n\nThe proof relies on the following lemma. \n\n\\begin{lemma}\n  \\label{lem:lowrangeCebound}\n  Let $\\Delta$ be the discriminant of a singular modulus.\n  \\begin{enumerate}[label={(\\roman*)}]\n  \\item \n  \\label{itenten} If ${10^7\\le|\\Delta|<  10^{10}}$ and ${\\varepsilon = 10^{-3}}$, then\n    ${{\\mathcal{C}}_\\varepsilon(\\Delta)\\le 16}$.\n      \\item \n      \\label{itenseven}\n      If ${3\\cdot10^5\\le|\\Delta|< 10^7}$ and ${\\varepsilon = 4\\cdot 10^{-3}}$, then\n    ${{\\mathcal{C}}_\\varepsilon(\\Delta)\\le 6}$. \n  \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nLet $X_{\\min}$ and $X_{\\max}$ be positive integers satisfying ${ X_{\\min}<X_{\\max}}$, and let ${\\varepsilon \\in (0, 1\/3]}$. We want to bound ${{\\mathcal{C}}_\\varepsilon(\\Delta)}$ for all~$\\Delta$ in the interval ${[-X_{\\max},-X_{\\min}]}$. \n \nRecall that ${\\mathcal{C}}_\\varepsilon(\\Delta)$ counts the triples    $(a,b,c)$ satisfying~\\eqref{ekuh} such that \n$$\n\\tau=\\tau(a,b,c)= \\frac{b + \\sqrt{\\Delta}}{2a}\n$$\nsatisfies\n  \\begin{equation}\n    \\label{eq:zrestriction}\n    \\min\\{|\\tau-\\zeta_3|,|\\tau-\\zeta_6|\\} < \\varepsilon. \n  \\end{equation}\nLemmas~\\ref{lgauss}\\ref{iadelta} and~\\ref{lshort} imply that such triples satisfy \n\\begin{align*}\n\\frac{|\\Delta|^{1\/2}}2 & \\le \\hphantom{|}c \\hphantom{|}\\le \\frac{|\\Delta|^{1\/2}}{\\sqrt3}(1+\\sqrt3\\varepsilon+\\varepsilon^2),\\\\\nc\/(1+\\sqrt3\\varepsilon+\\varepsilon^2)&<\\hphantom{|}a\\hphantom{|}\\le c,\\\\\na(1-2\\varepsilon)&<|b|\\le a. \n\\end{align*}\nNote that, since ${\\varepsilon \\in (0, 1\/3]}$, we have ${b\\ne 0}$ and \n${(1+\\sqrt3\\varepsilon+\\varepsilon^2)\/\\sqrt3<1}$. \n\nHence, to bound \n${\\mathcal{C}}_\\varepsilon(\\Delta)$ on the interval ${[-X_{\\max},-X_{\\min}]}$,\nit suffices, for every~$\\Delta$ in this interval, to count the triples $(a,b,c)$ satisfying \n\\begin{align*}\n{X_{\\min}^{1\/2}}\/2 & \\le c \\le X_{\\max}^{1\/2},\\\\\nc\/(1+\\sqrt3\\varepsilon+\\varepsilon^2)&<a\\le c,\\\\\na(1-2\\varepsilon)&<b\\le a \n\\end{align*}\nand ${b^2-4ac=\\Delta}$, and multiply the maximal count by~$2$ (because we counted only triples with positive~$b$). \nWe phrase this counting procedure formally as   Algorithm~\\ref{algo:CCDelta}. \n\n\nFor a correct implementation, we have to avoid floating point arithmetic in determining\nthe upper bounds on~$a$ and~$b$ used in the inner two for-loops. For this, we note that ${\\varepsilon = 10^{-3}}$\nimplies that \n$$\n0.998 c \\le c\/(1+\\sqrt3\\varepsilon+\\varepsilon^2), \\qquad 0.998a=a(1-2\\varepsilon). \n$$\nSimilarly, ${\\varepsilon = 4\\cdot 10^{-3}}$\nimplies that \n$$\n0.993 c \\le c\/(1+\\sqrt3\\varepsilon+\\varepsilon^2), \\qquad 0.992a=a(1-2\\varepsilon). \n$$\nAs we are only interested in an upper bound on ${\\mathcal{C}}_\\varepsilon(\\Delta)$ for these two specific values of~$\\varepsilon$, we use these\nweaker rational bounds in our implementation of\nAlgorithm~\\ref{algo:CCDelta}  by means of a\n\\textsf{C}-program\\footnote{A link to our program\n  \\textsf{algorithm1.c} is on the\nsecond-named author's homepage. The running time on a regular desktop\n(Intel Xeon CPU E5-1620 v3, 3.50GHz, 32GB RAM) was under a minute for item~\\ref{itenten} and a few milliseconds for item~\\ref{itenseven}. Its memory\nusage for ${[X_{\\min},X_{\\max}] = [1, 10^{10}]}$ is significant (5 GB) but this can be overcome by splitting\n${[1, 10^{10}]}$ into subintervals and running the program separately for each interval. This feature\nis also implemented in our program through the macro \\textsf{DISC\\underline{\\ }BLOCK\\underline{\\ }SIZE}.}. It\nverifies directly the assertions of the lemma.\n\\end{proof}\n\n\n\n    \\begin{algorithm}[t]{\\footnotesize\n      \\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n      \n    \\Input{Two positive integers $X_{\\mathrm{min}} <\n      X_{\\mathrm{max}}$ and $\\varepsilon > 0$}\n    \n    \n    \\Output{an upper bound for ${\\mathcal{C}}_\\varepsilon(\\Delta)$ for all\n      discriminants $\\Delta\\in [-X_{\\mathrm{max}},-X_{\\mathrm{min}}]$}\n    \\BlankLine\n\n    $counter \\leftarrow$ pointer to array of length\n    $X_{\\mathrm{max}}-X_{\\mathrm{min}}+1$ initialized to $0$\\;\n\n    $bound \\leftarrow 0$\\;\n    \n    \\For{$c\\leftarrow \\lfloor X_{\\mathrm{min}}^{1\/2}\/2\\rfloor$ \\KwTo\n      $\\lfloor X_{\\mathrm{max}}^{1\/2}\\rfloor$}{\n      \\For{$a\\leftarrow \\lfloor c\/(1+\\sqrt 3 \\varepsilon+\\varepsilon^2) \\rfloor$ to $c$}{\n        \\For{$b\\leftarrow \\lfloor (1-2\\varepsilon)a\\rfloor$ to $a$}{\n          $X\\leftarrow 4ac-b^2$\\;\n          \\If{$X\\ge X_{\\mathrm{min}}$  and $X \\le X_{\\mathrm{max}}$\n          }{$pos \\leftarrow X-X_{\\mathrm{min}}$\\;\n            $counter[pos] \\leftarrow counter[pos]+2$\\;\n            \\lIf{$counter[pos] > bound$}{$bound\\leftarrow counter[pos]$}\n            }\n        }\n    }}\n    \\Return{$bound$}\\;\n    \\caption{Compute an upper bound for ${\\mathcal{C}}_{\\epsilon}(\\Delta)$ in the\n      range ${\\Delta\\in [-X_{\\mathrm{max}},-X_{\\mathrm{min}}]}$}\\label{algo:CCDelta}\n      }%\n    \\end{algorithm}\n    \n\n\n\\begin{proof}[Proof of Theorem~\\ref{thlowrange}]\n  Assume that~$\\alpha$ is a singular unit of discriminant ${\\Delta\\in (-10^{-10}, -3\\cdot10^5]}$.\nLet ${0<\\varepsilon\\le 4\\cdot 10^{-3}}$. We set again~$Y$  as in (\\ref{ey}).\n  As in the proof of Theorem~\\ref{thmidrange} we find (\\ref{eq:midrangeYbound}).  \n \n \nWe infer that\n  \\begin{alignat*}1\n    1 &\\le 3\\frac{{\\mathcal{C}}_\\varepsilon(\\Delta)}{{\\mathcal{C}}(\\Delta) Y} \\log X +\n    \\frac{3\\log(\\varepsilon^{-1}) - 10.65}{Y}\\\\\n    & \\le \\frac {3{\\mathcal{C}}_\\varepsilon(\\Delta)}{\\pi} \\frac{\\log X}{X^{1\/2}} +\n    \\frac{3\\log(\\varepsilon^{-1}) - 10.65}{\\frac{3}{\\sqrt{5}}\\log X - 9.78}\n  \\end{alignat*}\nwhere we use ${{\\mathcal{C}}(\\Delta)Y \\ge \\pi X^{1\/2}}$\n  and ${Y\\ge \\frac{3}{\\sqrt{5}} \\log X - 9.78}$. \n\n  If ${X\\in [10^7,10^{10})}$ then we set ${\\varepsilon = 10^{-3}}$ and use the estimate\n  ${{\\mathcal{C}}_{\\varepsilon}(\\Delta) \\le 16}$ from Lemma~\\ref{lem:lowrangeCebound}\\ref{itenten}. \n  Recall that ${x\\mapsto (\\log x)\/x^{1\/2}}$\n  is decreasing for ${x\\ge e^2}$. So we find\n  \\begin{equation*}\n    1 \\le \\frac{3\\cdot 16}{\\pi}\\frac{\\log(10^7)}{10^{7\/2}} +\n    \\frac{3\\log(1000) - 10.65}{\\frac{3}{\\sqrt{5}} \\log(10^7)- 9.78}< 0.929,\n  \\end{equation*}\n  a contradiction.\n\nWhen ${X\\in [3\\cdot10^5,10^7)}$  we set ${\\varepsilon = 4\\cdot10^3}$.  Then\n  ${{\\mathcal{C}}_{\\varepsilon}(\\Delta)\\le 6}$ by  Lem\\-ma~\\ref{lem:lowrangeCebound}\\ref{itenseven}.\nUsing \n   ${X\\ge 3\\cdot 10^5}$ we find as before \n  \\begin{equation*}\n    1 \\le \\frac{3\\cdot 6}{\\pi}\\frac{\\log(3\\cdot 10^5)}{(3\\cdot 10^{5})^{1\/2}} +\n    \\frac{3\\log(250) - 10.65}{\\frac{3}{\\sqrt{5}} \\log(3\\cdot 10^{5})- 9.78}< 0.961,\n  \\end{equation*}\n  another contradiction which completes this proof. \n\\end{proof}\n\n\n\\section{The extra low-range}\n\n\\label{sfinal}\n\nThe results of the three previous sections reduce the proof of Theorem~\\ref{thmain}  to the\nfollowing assertion.\n\n\n\\begin{theorem}\n\\label{thm:habeggerspari}\n    Let~$\\Delta$ be the discriminant of a singular unit. Then \n  ${|\\Delta|\\ge 3\\cdot 10^5}$. \n\\end{theorem}\n\\begin{proof}\n  Let $\\alpha$ be a singular unit of discriminant $\\Delta$.  We write\n  ${X=|\\Delta|}$. \nWe may assume that ${X\\ge 4}$ because the only singular modulus of discriminant $-3$ is\n${j(\\zeta_3)=0}$, \nwhich is  not an algebraic unit.\n\n\nRecall from Section~\\ref{sceps} that the Galois conjugates of~$\\alpha$ are precisely the singular moduli\n$j(\\tau)$, where ${\\tau = \\tau(a,b,c)}$  with $(a,b,c)$ as in~\\eqref{ekuh}. The imaginary part of  such~$\\tau$ is\n$X^{1\/2}\/(2a)$ and ${a\\le (X\/3)^{1\/2}}$\nby Lemma~\\ref{lgauss}\\ref{iadelta}. \n  Lemma~\\ref{lttsn}  implies that\n$$\n|j(\\tau)|\\ge e^{2\\pi X^{1\/2}\/(2a)} -2079\n  =e^{\\pi X^{1\/2}\/a}-2079 > 23^{X^{1\/2}\/a}-2079\n$$\n  as ${e^\\pi > 23}$. Using Lemmas~\\ref{lanal} and~\\ref{lliouv}, we find that\n$$\n|j(\\tau)| \\ge 42700 \\min \\left\\{ \\frac{\\sqrt3}{4X},4\\cdot 10^{-3}\\right\\}^3.\n$$\nThese\n  bounds together show that\n  \\begin{equation}\n\\label{eq:finaljlb}\n    |j(\\tau)|\\ge \\max\\left\\{ 23^{\\lfloor X^{1\/2}\/a\\rfloor}-2079,\n    42700\\min\\Bigl\\{\\frac2{5X},\\frac1{250}\\Bigr\\}^3\\right\\}. \n  \\end{equation}\nBased on this observation,   Algorithm~\\ref{algo:exclude} prints a list of discriminants of potential singular\nunits in the range ${[-X_{\\max},-4]}$. For this purpose, it computes a \nrational lower bound~$P$ for the absolute value of the\n$\\Q(\\alpha)\/\\Q$-norm  of each singular moduli in this range. Those singular\nmoduli where ${P \\le 1}$ are then flagged as potential singular units.\n\n We have implemented this algorithm as a \\textsf{PARI}\n script\\footnote{A link to our \\textsf{PARI} script  \\textsf{algorithm2.gp}\nis on  the second-named author's homepage. The running time is about 23 minutes  on a regular desktop computer (Intel Xeon CPU E5-1620 v3, 3.50GHz, 32GB RAM). The only floating point operation used  approximates $X^{1\/2}$\n  which leads to  ${n=\\lfloor X^{1\/2}\/a\\rfloor}$.\nTo rule out a rounding error in the floating point arithmetic we\ncompare $(an)^2$  with~$X$ in our implementation.}. \nThe script flags only $-4$, $-7$ and $-8$ as discriminants of potential singular units. The\nsingular moduli of these discriminants are well-known \\cite[(12.20)]{Cox}: they are $12^3$, $-15^3$ and $20^3$, respectively. None of them is a unit, which concludes the proof. \n\\end{proof} \n  \n  \\begin{algorithm}[t]{\\footnotesize\n          \\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n          \\Input{An integer $X_{\\mathrm{max}}\\ge 1$}\n          \\Output{Print a list containing all discriminants\n            in $[-X_{\\mathrm{max}},-4]$  that are attached to a potential\n            singular unit.}\n\n          \\For{$X\\leftarrow 4$ \\KwTo $X_{\\mathrm{max}}$}{\n              $\\Delta \\leftarrow -X$\\;\n            \n            \\lIf{$\\Delta\\equiv 2 \\text{ or }3 \\mod 4$}{\n              next $X$\n             }\n            \n\n\n$P\\leftarrow 1$\\;\n            \\For{$a\\leftarrow 1$ \\KwTo $\\lfloor \\sqrt{X\/3}\\rfloor$} {\n\n              $n\\leftarrow \\lfloor X^{1\/2}\/a\\rfloor$\\;\n              \\For{$b\\leftarrow -a+1$ \\KwTo $a$}{\n                \\lIf{$b^2  \\not\\equiv \\Delta \\mod 4a$}{\n                  next $b$\n                }\n                $c\\leftarrow (b^2-\\Delta)\/(4a)$\\;\n                \\lIf{$a>c$}{next $b$}\n                \\lIf{$a=c \\text{ and } b<0$}{next $b$}\n                \\lIf{$\\mathrm{gcd}(a,b,c)\\not=1$}{next $b$}\n               \n               \n               \n               \n  $P\\leftarrow P \\cdot \\max\\{23^{n\n        }-2079,42700\\min\\{2\/(5X),1\/250\\}^3\\}$\\;\n}}\n\n\\lIf{$P\\le 1$}{print $\\Delta$}}\n\\caption{Exclude singular units}          \\label{algo:exclude}\n}%\n  \\end{algorithm}\n \n\n \n\n\nAs indicated in the introduction, Theorem~\\ref{thmain} is the combination of Theorems~\\ref{thhighrange},~\\ref{thmidrange},~\\ref{thlowrange} and~\\ref{thm:habeggerspari}. \n\n\n{\\footnotesize\n\n\\def$'${$'$}\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\n\\ \\ \\ \\ The time evolution of coherent states (CS) has attracted a great\ndeal of attention since the introduction of Glauber's CS of the harmonic\noscillator {\\normalsize \\cite{Glauber63}}. Of\nparticular interest has been the determination of the Hamiltonian operator\nfor which an initial coherent state remains coherent under time evolution.\nIt is established that this Hamiltonian has the form of the nonstationary\nbosonic forced oscillator Hamiltonian {\\normalsize \\cite{Glauber66, Mehta66,\n Stoler75, Kano76}:  \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\n\\begin{equation}\nH_{\\mathrm{cs}}=\\omega (t)a^{\\dagger }a+f(t)a^{\\dagger }+f^{\\ast }(t)a+\\beta\n(t),  \\label{Hcs}\n\\end{equation}%\nwhere $\\omega (t)$ and $\\beta (t)$ are arbitrary real functions of time $t$,\nand $f(t)$ is arbitrary complex function. \n$H_{\\mathrm{cs}}$ is a particular case of the nonstationary forced oscillator\nHamiltonian for which the exact time evolution of CS has been obtained in\n\\cite{MMT70,Holz70} by first constructing boson ladder operator dynamical %\ninvariants according to the Lewis and Riesenfeld scheme of time\ndependent invariants \\cite{Lewis69}.\n\nOur purpose in the present article is to study the dynamical invariants and\ntime evolution of CS for the {\\it fermionic} forced oscillator (FFO), which %\nin fact is the general (one mode) Hamiltonian.\n\nThe organization of the article is as follows. In Sec. 2 we construct fermionic %\nladder operator dynamical invariants and the corresponding Lewis-Riesenfeld\nHermitian invariant {\\normalsize \\cite{Lewis69}}, following the scheme related %\nto the boson system {\\normalsize \\cite{MMT70}}. Using these invariants, we\nconstruct in Sec. 3 fermionic CS and Fock states of FFO system as eigenstates %\nof the constructed invariant fermionic annihilation operator $B(t)$ and %\n$B^\\dagger(t)B(t)$ correspondingly. These CS can\nrepresent (under appropriate initial conditions) the exact time-evolution of\ninitial canonical fermionic CS. Finally the relation of the invariant ladder %\noperators method {\\normalsize \\cite{MMT70, Holz70}} to the Lewis-Riesenfeld method\n{\\normalsize \\cite{Lewis69}} is briefly described on the example of FFO. The\npaper ends with concluding remarks.\\\n\n\\medskip\n\n\\section{FFO and invariant ladder operators}\n\nWe consider the single nonstationary fermionic forced oscillator (FFO)\ndescribed by the following Hamiltonian,%\n\\begin{equation}\nH_{\\!f}=\\omega (t)b^{\\dagger }b+f(t)b^{\\dagger }+f^{\\ast }(t)b+g(t),\n\\label{Hf}\n\\end{equation}%\nwhere $\\omega (t)$ and $g(t)$ are arbitrary real functions of time, $f(t)$\nis arbitrary complex function, $b$ and $b^{\\dagger }$ are fermion\nannihilation and creation operators respectively, which obey to the fermion\nalgebra:\n\\begin{equation}\n\\{b,b^{\\dagger }\\} = 1,\\text{ \\ }%\nb^{2}=b^{\\dagger }{}^{2}=0,  \\label{013}\n\\end{equation}%\nwhere $\\{b,b^{\\dagger }\\}\\equiv bb^{\\dagger }+b^{\\dagger }b$.\nDue to the nilpotency of the fermionic operators $b$, $b^\\dagger$\nthe operator $H_{\\!f}$ represents the most general (one mode) fermionic %\nHamiltonian.\n\nThe Hilbert space $\\mathcal{H}$ of the single-fermion system is spanned by\nthe two eigenstates $\\left\\{ \\left\\vert 0\\right\\rangle ,\\left\\vert\n1\\right\\rangle \\right\\} $ of  number operator $b^{\\dagger }b$:\\, %\n$b^{\\dagger }b\\left\\vert n\\right\\rangle =n\\left\\vert n\\right\\rangle ,\\text{ \\ }n=0,1$.\nThe operators $b$ and $b^{\\dagger }$\\ allow transitions between number states,\n\\begin{equation}\nb\\left\\vert 0\\right\\rangle =0,\\text{\\ }b\\left\\vert 1\\right\\rangle\n=\\left\\vert 0\\right\\rangle \\,,\\text{ \\ }b^{\\dagger }\\left\\vert\n1\\right\\rangle =0,\\text{ }b^{\\dagger }|0\\rangle =|1\\rangle .\\,\n\\end{equation}%\nThe form of the Hamiltonian ({\\normalsize \\ref{Hf}}) is a Hermitian linear\ncombination of $b$, $b^{\\dagger }$ and $N = b^{\\dagger }b$. The fermion number\noperator $N$ obey the relation $N^{2}=N$ and the three operators $b$, $%\nb^{\\dagger }$and $N$ close under commutation the algebra:\n\\begin{equation}\n\\left[ b,N\\right] =b,\\text{ }\\left[ b^{\\dagger },N\\right] =-b^{\\dagger },%\n\\text{ }\\left[ b,b^{\\dagger }\\right] =1-2N,\\text{ \\ \\ \\ }\n\\end{equation}%\nLet us note that linear combinations of $b^{\\dagger }$, $b$ and $N$ produce\nthe half-spin operators $J_{i}$,\n\\begin{equation}\nJ_{1}=\\tfrac{1}{2}(b^{\\dagger }+b),\\quad J_{2}=\\tfrac{1}{2\\ri}(b^{\\dagger\n}-b),\\quad J_{3}=b^{\\dagger }b-\\tfrac{1}{2},  \\label{30a}\n\\end{equation}%\nclosing the \\textit{su}(2) algebra: $\\left[ J_{k},J_{l}\\right] =i\\epsilon\n_{klm}J_{m}.$\n\nIt is convenient to use raising and lowering operators $J_{\\pm }=J_{1}\\pm\niJ_{2}$ which satisfy the following commutation relation: $\\left[ J_{+},J_{-}%\n\\right] =2J_{3},\\text{ }\\left[ J_{3},J_{\\pm }\\right] =\\pm J_{\\pm }\\text{\\ }$%\n, where $J_{+}=b^{\\dagger },$ $J_{-}=b.$ So that in terms of these half spin\noperators the Hamiltonian ({\\normalsize \\ref{Hf}}) takes the form\n\\begin{equation}\nH_{\\!f}=\\omega (t)J_{3}+f(t)J_{+}+f^{\\ast }(t)J_{-}+g(t)+\\tfrac{\\omega (t)}{2%\n}.  \\label{Hf 2}\n\\end{equation}%\n\nOur task is the construction of the time-dependent invariants for the system\n{\\normalsize (\\ref{Hf})}, {\\normalsize (\\ref{Hf 2})}. The defining equation\nof the invariant operator $B(t)$ for a quantum system with Hamiltonian $H(t)$\nis%\n\\begin{equation}\n\\frac{\\partial }{\\partial t}B(t)-i[B(t),H]=0  \\label{19}\n\\end{equation}\nFormal solutions to Eq. {\\normalsize (\\ref{19})} are operators $%\nB(t)=U(t)B(0)U^{\\dagger }(t)$, where $U(t)$ is the evolution operator of the\nsystem, $U =T\\exp [-\\ri\\tint_{0}^{t}H(t^{^{\\prime }}){\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e} t^{^{\\prime }}]$. \nIn our case of FFO {\\normalsize (\\ref{Hf})}, {\\normalsize (\\ref{Hf 2})} we look\nfor the non-Hermitian invariants $B(t)$, $B^\\dagger(t)$ of the form of linear %\ncombination of the $SU(2)$ generators {\\normalsize (\\ref{30a})}, \\\n\\begin{equation}\n\\begin{aligned} B = \\nu_{-}(t)J_{-} + \\nu_{+}(t) J_{+} + \\nu_3 (t) J_{3}\n,\\\\ B^\\dagger = \\nu_{-}^* (t)J_{+} + \\nu_{+}^*(t) J_{-} + \\nu_3^* (t) J_{3},\n\\end{aligned}  \\label{B}\n\\end{equation}%\nwhere $\\nu _{\\pm }(t),\\,\\,\\nu _{3}(t)$ may be complex functions of the time.\nHermitian invariants then can be easily built up as Hermitian combinations\nof $B$ and $B^{\\dagger }$. In particular if $B$ is a non-Hermitian invariant\nthe operator $I=B^{\\dagger }B- 1\/2$ \nis a Hermitian invariant, the fermion analog of the Lewis-Riesenfeld quadratic\ninvariant {\\normalsize \\cite{Lewis69}}.\n\nLet us note at this point that we look for FFO invariants as elements of the\nsame algebra $su(2)$ to which the Hamiltonian belongs. Similar is the approach\nused in \\cite{Chumakov86} in construction of invariants for\nthe nonstationary singular oscillator, where the related algebra is $su(1,1)$.\nThis is to be compared with the case of nonsingular oscillator, for which invariant\nladder operators have been built up as elements of the Heisenberg-Weyl algebra %\n$h_w$ (i.e. as linear combinations of coordinate and momentum operators $x$ and %\n$p$ \\cite{MMT70,Holz70}), while the related nonstationary Hamiltonian belongs to \n$su(1,1)$. %\nFor {\\it forced} boson oscillator the Hamiltonian belongs to the large algebra of \nsemi-direct sum $su(1,1)\\dot{+}h_w$ but the ladder operator invariants %\n are again elements of the invariant subalgebra $h_w$  %\n\\cite{Prants86,Dattoli86,Trif93}.\n\n\nTo proceed with construction of FFO invariants we substitute {\\normalsize (\\ref{B}) and \n(\\ref{Hf 2}) } into {\\normalsize (\\ref{19})}, and find the following system of\ndifferential equations for the parameter functions $\\nu_\\pm,\\,\\nu_3$: \\\n\\begin{eqnarray}\n\\dot{\\nu}_{3} &=&2\\ri(\\nu _{+}f^{\\ast }-\\nu _{-}f),  \\label{(a)} \\\\\n\\dot{\\nu}_{+} &=&\\ri(\\nu _{3}f-\\nu _{+}\\omega ),  \\label{(b)} \\\\\n\\dot{\\nu}_{-} &=&\\ri(\\nu _{-}\\omega -\\nu _{3}f^{\\ast }).  \\label{(c)}\n\\end{eqnarray}%\nSolutions to the above linear system of first order equations are\nuniquely determined by the initial conditions $\\nu _{\\pm }(0)=\\nu _{0,\\pm }$%\n, $\\nu _{3}(0)=\\nu _{0,3}$. If we want the invariants $B(t)$ and $B^{\\dagger\n}(t)$ be again fermion ladder operators, i.e. to obey the conditions\n\\begin{equation}\nB^{2}=0, \\quad \\{B,B^{\\dagger}=1,  \\label{constr}\n\\end{equation}%\nwe have to take $\\nu _{0,\\pm }$ and $\\nu _{0,3}$ satisfying\n\\begin{equation}\n \\nu _{0,3}^{2} = -4\\nu _{0,+}\\nu _{0,-},\\quad\n |\\nu _{0,-}|+|\\nu _{0,+}| = 1\\,. \\label{0}\n\\end{equation}%\nIndeed, for $B^{2}(t)$ and $\\{B(t),B^{\\dagger }(t)\\}$ we find\n\\begin{equation}\n\\begin{tabular}{l}\n$\\displaystyle B^{2}=\\nu _{+}\\nu _{-}+\\tfrac{1}{4}\\nu _{3}^{2}\\equiv\n\\lambda _{1}$, \\\\[2mm]\n$\\displaystyle\\{B,B^{\\dagger}\\}=|\\nu _{-}|^{2}+|\\nu _{+}|^{2}+\\tfrac{1%\n}{2}|\\nu _{3}|^{2}\\equiv \\lambda _{2}$.%\n\\end{tabular}\n\\label{lam_i}\n\\end{equation}%\nThe quantities $\\lambda _{1}(\\nu _{\\pm },\\nu _{3})$, $\\lambda _{2}(\\nu _{\\pm\n},\\nu _{3})$ turned out to be two different '\\textit{constants of motion}'\nfor the system {\\normalsize (\\ref{(a)})}-{\\normalsize (\\ref{(c)})}, their\ntime derivatives being vanishing:\n\\begin{equation}\n\\begin{tabular}{l}\n$\\displaystyle \n\\frac{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}}{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e} t}\\lambda _{1} = \\frac{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}}{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e} t}\\left( \\nu _{+}\\nu _{-}+\\tfrac{1%\n}{4}\\nu _{3}^{2}\\right) =0$, \\\\[3mm]\n$\\displaystyle\n\\frac{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}}{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e} t}\\lambda _{2} = \\frac{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}}{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e} t}\\left( |\\nu _{-}|^{2}+|\\nu\n_{+}|^{2}+\\tfrac{1}{2}|\\nu _{3}|^{2}\\right) =0$.%\n\\end{tabular}\n\\label{dotlam_i}\n\\end{equation}%\nTherefore we can fix the values of these constants as $\\lambda _{1}=0$, $%\n\\lambda _{2}=1$, i.e.\n\\begin{equation}\n\\begin{tabular}{l}\n$ \\displaystyle \\nu _{+}\\nu _{-}+\\tfrac{1}{4}\\nu _{3}^{2}=0,$\\\\[3mm]\n$\\displaystyle |\\nu _{-}|^{2}+|\\nu_{+}|^{2}+\\tfrac{1}{2}|\\nu _{3}|^{2}=1$, \n\\end{tabular}\n\\label{lam 0}\n\\end{equation}%\nand satisfy the conditions {\\normalsize (\\ref{constr})}. If furthermore the initial\nconditions are taken as\n\\begin{equation}\n\\nu _{-}(0)=1,\\,\\,\\,\\nu _{+}(0)=0=\\nu _{3}(0),  \\label{nu_i0}\n\\end{equation}%\nthen $B(0)=b$. Later on we work with these fermionic ladder operator invariants, %\ni.e. we consider conditions (\\ref{lam 0}) satisfied.\n\nLet us now recall that in the case boson nonstationary oscillator the ladder %\noperator invariants, constructed first in \\cite{MMT70, Holz70} (see also %\n\\cite{Chumakov86, Prants86}), are expressed in terms of one only parameter function %\n$\\epsilon(t)$, which obeys a simple second order equation, namely that of the %\nclassical oscillator with varying frequency. It turned out that this can be done %\nin the case of  fermionic oscillator as well.\nIn this aim we first  express all the three parameter functions $\\nu _{\\pm }(t)$,%\n $\\nu _{3}(t)$ in terms of one of them, which has to obey a second order %\ndifferential equation.\nLet for example, express $\\nu _{3}(t)$ and $\\nu _{-}(t)$ in terms of $\\nu _{+}(t)$%\n and its derivatives.  We have %\n\\begin{equation} \\label{nu3}\n\\nu _{3} =  - \\tfrac{\\ri}{f}(\\dot{\\nu}_{+} + \\ri \\nu _{+}\\omega ),\n\\end{equation}%\n\\begin{equation}\n\\nu _{-}=\\tfrac{1}{2f^{2}}\\left[ \\ddot{\\nu}_{+}+\\left( \\ri \\omega -\\tfrac{\\dot{f%\n}}{f}\\right) \\dot{\\nu}_{+}+\\left( 2ff^{\\ast }+\\ri\\dot{\\omega}-\\ri \\frac{\\omega }{f%\n}\\dot{f}\\right) \\nu _{+}\\right] .  \\label{nu-}\n\\end{equation}%\nSubstituting these expressions into the expression of $\\lambda_1$ in terms of $\\nu_\\pm,\n\\,\\nu_3$ and taking into account that $\\lambda_1$ is fixed to $0$ we find that $\\nu_+$ %\nshould satisfy the following second order equation,\n\n\\begin{equation}\n2\\nu _{+}\\ddot{\\nu}_{+}-\\dot{\\nu}_{+}^{2} - 2\\nu_{+}\\dot{\\nu}_{+}\\tfrac{\\dot{f}}{f} +\n4\\nu _{+}^{2}\\left(\n|f|^{2}+\\tfrac{\\omega ^{2}}{4}+ \\ri\\tfrac{\\dot{\\omega}}{2}-\\ri \\tfrac{\\omega %\n\\dot{f }}{2f}\\right)  =  0 .  \\label{lam}\n\\end{equation}%\n Using this, and supposing that $\\nu_{+}\\neq 0$, we obtain for $\\nu _{-}$ a more %\ncompact expression in terms of $\\nu _{+}$ and $\\dot{\\nu}_{+}$, %\n\\begin{equation}\n\\nu _{-}= -\\nu _{3}^{2}\/4\\nu _{+},  \\label{nu- 2}\n\\end{equation}%\nwhere $\\nu_3$ is given again by eq. (\\ref{nu3}).\n\nThus the operators $B(t),\\,B^{\\dagger}(t)$, eq. (\\ref{B}), are fermionic %\nladder operator invariants for the forced oscillator (\\ref{Hf}), (\\ref{Hf 2})%\nif  $\\nu _{3}$ and $\\nu _{-}$ are given by eqs. (\\ref{nu3}) and (\\ref{nu- 2}),%\nand $\\nu _{+}(t)$ is a nonvanishing solution of the second order equation (\\ref{lam}).\n\nNext we try to linearize the auxiliary eq. (\\ref{lam}). In this purpose we put\n\\begin{equation}\n\\nu _{+}(t) = \\tfrac{1}{2}{\\epsilon'} ^{2}(t)  \\label{nu+ eps'}\n\\end{equation}%\nand obtain that  $\\epsilon' (t)$ satisfies the linear equation\n\\begin{equation} \\label{eps' eq}\n\\ddot{\\epsilon}' - \\tfrac{\\dot{f}}{f}\\dot{\\epsilon}'+\\Omega' (t)\\epsilon' =0,\n\\end{equation}%\nwhere\n\\begin{equation}\n\\Omega' (t)=|f(t)|^{2}+\\tfrac{1}{4}\\omega ^{2}(t)+\\tfrac{\\ri}{2}\\dot{\\omega}-%\n\\tfrac{\\ri}{2}\\omega \\tfrac{\\dot{f}}{f}.\n\\end{equation}\n\nIn terms of $\\epsilon'$ the formulas (\\ref{nu-}) and (\\ref{nu3} for $\\nu_-,\\, \\nu_3$ read\n\\begin{equation}\n\\begin{tabular}{l}\n$\\displaystyle\\nu _{-}=-\\frac{1}{2\\epsilon'}\\nu_3^{2}$, \\\\[2mm]\n$\\displaystyle\\nu _{3}=\\frac{1}{f}\\left( \\tfrac{\\omega }{2}{\\epsilon'}\n^{2}-\\ri\\epsilon' \\dot{\\epsilon}'\\right) $.%\n\\end{tabular}%\n\\end{equation}%\nThe term in (\\ref{eps' eq}) proportional to the first derivative can be eliminated %\nby the substitution %\n\\begin{equation}\\label{eps}\n\\epsilon'  = \\epsilon \\exp \\left( \\tfrac{1}{2}\\tint_{0}^{t}{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}\\tau \\,\\dot{f}%\n(\\tau )\/f(\\tau )\\right) .\n\\end{equation}%\nThis leads to the desired simple equation for $\\epsilon$,\n\\begin{equation}\\label{eps eq}\n\\ddot{\\epsilon}+\\Omega (t)\\epsilon = 0,\n\\end{equation}%\nwhere $ \\Omega(t)= \\Omega^{\\prime }(t)  + \\ddot{f}\/2f - 3\\dot{f}\\,^{2}\/4f^{2}$. %\nEquation (\\ref{eps eq}) is of the same type, as the auxiliary equation used in the %\ncase of nonstationary boson oscillator \\cite{MMT70, Holz70}. Here the %\n'squared frequency' $\\Omega$ however is complex and depends in a different manner %\non the corresponding Hamiltonian parameters. And the solutions are subject to %\ndifferent constraints, stemming from the different commutation relations: \nin terms of our $\\epsilon$, eq. (\\ref{eps eq}), the constraint $\\lambda_2=1$ reads %\n( $\\epsilon'$ is related to $\\epsilon$ according to (\\ref{eps})),\n\\begin{equation}\n\\tfrac{|\\epsilon' |^{4}}{4}\\left( 1+\\tfrac{2}{|f|^{2}}\\left\\vert\n\\tfrac{\\omega }{2}\\epsilon' -\\ri\\dot{\\epsilon}'\\right\\vert ^{2}+\\tfrac{1}{|f|^{4}%\n}\\left\\vert \\tfrac{\\omega }{2}\\epsilon' -\\ri\\dot{\\epsilon}'\\right\\vert^{4}\\right) =1 ,\n\\end{equation}%\n while in  the boson case the constraint is  ${\\rm Im}\\left({\\epsilon}^*\\dot{\\epsilon}\\right) = 1$ \n\\cite{MMT70, Holz70}.\n\nTo finalize this section let note that in the particular case of the %\n\\textit{free} fermion oscillator, $f(t)\\equiv 0$, the explicit solutions of the problem can %\n be easily found in the form %\n\\begin{equation}\n\\begin{tabular}{l}\n$\\displaystyle \n\\nu _{\\pm }(t)=\\nu _{0,\\pm }\\re^{\\pm \\ri\\tint^{t}\\omega (\\tau ){\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}\\tau },$\\\\\n$ \\displaystyle \n\\nu_{3}=\\nu _{0,3},$\n\\end{tabular}\n\\label{solutions1}\n\\end{equation}%\nwhere $\\nu _{0,\\pm }$, $\\nu _{0,3}$ are constants. To ensure the fermionic %\ncommutation relations of $B(t),\\,B^\\dagger(t)$ they have to obey the relations\n$\\nu _{0,-}\\,\\nu _{0,+} + \\nu _{0,3}^{2}\/4 =0$ \\,\\, and \\,\\, $ |\\nu\n_{0,-}|^{2} + |\\nu _{0,+}|^{2} + |\\nu _{0,3}|^{2}\/2  = 1.$\n\\medskip\n\n\\section{CS for the fermion forced oscillator}\n\nWe define coherent states (CS) for a given fermion system as eigenstates of\nthe corresponding invariant fermion annihilation (or creation) operator $B(t)\n$. Since the most general fermion one mode Hamiltonian operator is of the\nform of (nonstationary) forced oscillator (\\ref{Hf 2}), the one-mode fermion\nCS are defined as eigenstates of the invariant ladder operator $B(t)$\n(eqs. (\\ref{B}), (\\ref{constr})):\n\\begin{equation}\nB(t)|\\zeta ;t\\rangle =\\zeta |\\zeta ;t\\rangle .  \\label{|z;t> 1}\n\\end{equation}%\nSince $B(t)$ is invariant operator,  the eigenvalue $\\zeta $ does\nnot depend on time $t$. In terms of the $\\zeta $, $B(t)$, $B^{\\dagger }(t)$\nand the $B(t)$-vacuum $|0;t\\rangle $ we have for $|\\zeta ;t\\rangle $ the\nsame formulas as for the canonical fermion CS $|\\zeta \\rangle $ which are\ndefined {\\normalsize \\cite{Klauder, Abe89, Maam92, Junker98, Cahill99}} as\n\\begin{equation}\n\\left\\vert \\zeta \\right\\rangle =e^{-\\frac{1}{2}\\zeta ^{\\ast }\\zeta }\\left(\n\\left\\vert 0\\right\\rangle -\\text{ }\\zeta \\left\\vert 1\\right\\rangle \\right)\n\\,.  \\label{|z>}\n\\end{equation}%\nwhere the eigenvalue $\\zeta $ is a Grassmannian variable: $\\zeta ^{2}=0,\\\n\\zeta \\zeta ^{\\ast }+\\zeta ^{\\ast }\\zeta =0$, $\\left\\vert 0\\right\\rangle $\nis the fermionic vacuum, $b\\left\\vert 0\\right\\rangle =0$, and $\\left\\vert\n1\\right\\rangle $ is the one-fermion state, $\\left\\vert 1\\right\\rangle\n=b^{\\dagger }\\left\\vert 0\\right\\rangle $. In particular\n\\begin{equation}\n|\\zeta ;t\\rangle = \\re^{-\\tfrac{1}{2}\\zeta ^{\\ast }\\zeta }\\left( |0;t\\rangle\n-\\zeta B^{\\dagger }(t)|0;t\\rangle \\right) .  \\label{|z;t> 2}\n\\end{equation}%\nIt remains therefore to construct the (normalized) new ground state $%\n|0;t\\rangle $ according to its defining equations\n\\begin{equation}\n\\begin{aligned} B(t) |0;t\\rangle = 0 ,\\quad \n\\ri\\frac{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}}{dt} |0;t\\rangle = H_{f}|0;t\\rangle. \\end{aligned}  \\label{|0;t> 1}\n\\end{equation}%\nWe put\n\\begin{equation}\n|0;t\\rangle =\\alpha _{0}(t)|0\\rangle +\\alpha _{1}(t)|1\\rangle ,\n\\label{|0;t> 2}\n\\end{equation}%\nsubstitute this into (\\ref{|0;t> 1}) and after some tedious calculations find%\n\\begin{eqnarray}\n\\alpha _{1}(t) &=&\\alpha _{0}(t)\\frac{\\nu _{3}^{\\ast }(t)}{2\\nu _{+}^{\\ast\n}(t)}, \\\\\n\\alpha _{0}(t) &=&\\sqrt{|\\nu _{+}(t)|}\\exp \\left[ -\\tfrac{\\ri}{2}\\left(\n\\varphi _{\\nu _{+}}(t)+\\tint_{0}^{t}(2g(\\tau )+\\omega (\\tau )){\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}\\tau \\right) %\n\\right] ,\n\\end{eqnarray}%\nwhere $\\varphi _{\\nu _{+}}$ is the phase of $\\nu _{+}(t)$. The state $|\\zeta\n;t\\rangle $ will represent the exact time evolution of an initial canonical\nCS $|\\zeta \\rangle $ if the initial conditions (\\ref{nu_i0}) are imposed: %\n$|\\zeta ;0\\rangle = |\\zeta \\rangle $. In this case, the time evolved state %\n$ |\\zeta ;t\\rangle $ could be again an eigenstate of $b$ if the oscillator %\nis not 'forced', i.e. if $f(t)=0$.\nLet us note that the time-dependence of the constructed states is obtained %\nin terms of solutions to the system of auxiliary equations (\\ref{(a)})-(\\ref{(c)}),%\nor equivalently to the 'classical oscillator' equation (\\ref{eps eq}).\n\nOur method of construction of dynamical invariants differs slightly from the\nLewis-Riesenfeld method \\cite{Lewis69} (developed for bosonic oscillators).\nLewis and Riesenfeld used to first construct Hermitian invariant, which then\nis represented as a product of normally ordered ladder operators. To make\nconnection to their approach let us suppose that we first succeeded to\nconstruct the Hermitian invariant $N(t)$ and to find some ladder operators $%\n\\tilde{B}(t)$, $\\tilde{B}^{\\dagger }(t)$ that factorize it: $N(t)=\\tilde{B}%\n^{\\dagger }(t)\\tilde{B}(t)$. It is clear that $\\tilde{B}(t)$ may differ from\nour non-Hermitian invariant $B(t)$ in a phase factor:\\thinspace\\ $\\tilde{B}%\n(t) = {\\rm e}^{\\ri\\varphi (t)}B(t)$.\\newline\nWe can then in a standard way construct normalized eigenstates of $N(t)$,\n\\begin{equation}\nN(t)\\widetilde{|0;t\\rangle }=0,\\quad N(t)\\widetilde{%\n|1;t\\rangle }=\\widetilde{|1;t\\rangle },  \\label{tld 2}\n\\end{equation}%\nand of $\\tilde{B}(t)$, \\\n\\begin{equation}\n\\tilde{B}(t)\\widetilde{|\\zeta ;t\\rangle }=\\zeta \\widetilde{|\\zeta ;t\\rangle },\n\\end{equation}%\n\\begin{equation}\n\\widetilde{|\\zeta ;t\\rangle }=\\left( 1-\\tfrac{1}{2}\\zeta ^{\\ast }\\zeta\n\\right) \\left[ \\widetilde{|0;t\\rangle }-\\zeta \\widetilde{|1;t\\rangle }\\right]\n\\label{tld 3}\n\\end{equation}%\nwhich however do not obey the Schr\\\"odinger equation since, in general $ \\tilde{B}(t)$\nmay not be invariant. To obtain solutions $|n;t\\rangle $ and $ |\\zeta ;t\\rangle $ the\nabove eigenstates $\\widetilde{|n;t\\rangle }$, $n=0,1$, should also be multiplied by\nphase factors,\n\\begin{equation}\n|n;t\\rangle = {\\rm e}^{\\ri\\phi _{n}(t)}\\widetilde{|n;t\\rangle },\\quad n=0,1,\n\\label{tld 4}\n\\end{equation}%\n\\begin{equation}\n|\\zeta ;t\\rangle =\\left( 1-\\tfrac{1}{2}\\zeta ^{\\ast }\\zeta \\right) \\left[\n{\\rm e}^{\\ri\\phi _{0}(t)}\\widetilde{|0;t\\rangle }-\\zeta {\\rm e}^{\\ri\\phi _{1}(t)}%\n\\widetilde{|1;t\\rangle }\\right]  \\label{tld 5}\n\\end{equation}%\nwhich should obey the equations \\\n\\begin{equation}\n\\tfrac{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}}{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e} t}\\phi _{n}=\\widetilde{\\langle n;t|}\\ri\\tfrac{\\partial }{\\partial t}%\n-H\\widetilde{|n;t\\rangle }.  \\label{tld 6}\n\\end{equation}%\nEvidently the state (\\ref{tld 5}) is an eigenstate of $\\tilde{B}(t)$ with time\ndependent eigenvalue $\\zeta (t)=\\zeta \\exp (\\ri\\varphi (t))$, $\\varphi\n(t)=\\phi _{1}(t)-\\phi _{0}(t)$.\\newline\nThe phase $\\varphi (t)=\\phi _{1}(t)-\\phi _{0}(t)$ consists of two parts -\ngeometrical one $\\varphi ^{G}$, and dynamical one $\\varphi ^{D}=\\varphi\n-\\varphi ^{G}$ \\cite{Maam99},\n\\begin{eqnarray}\n\\varphi ^{G}(t)  &=&\\varphi (t)+\\int_{0}^{t}\\left( \\widetilde{\\langle 1;t^{\\prime }|}H%\n\\widetilde{|1;t^{\\prime }\\rangle }-\\widetilde{\\langle 0;t^{\\prime }|}H%\n\\widetilde{|0;t^{\\prime }\\rangle }\\right) {\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e} t^{\\prime }.\n\\end{eqnarray}\n\n\\subsection*{Concluding Remarks}\n\n\\ \\ \\ In this article, we have studied fermionic system of nonstationary\nforced oscillator and we have constructed invariant ladder operators and the\nrelated Fock and coherent states. We succeeded to express these invariants\nand the time evolution of the corresponding states in terms of the same\nclassical equation, that describes the evolution of coherent states of the\nboson nonstationary (forced) oscillator \\cite{MMT70, Holz70}. The relation of\nthe invariant ladder operators method to the Lewis-Riesenfeld method \\cite{Lewis69}\nwas briefly described on the example of nonstationary fermion systems. \\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:level1}INTRODUCTION}\n\nIt is well recognized that the $\\alpha$ particle interaction with atomic nuclei is important in astrophysics \\cite{1}. Even if astrophysical reactions involving helium do not proceed through the strong $\\alpha$-cluster states (because of their high excitation energy), these states can provide $\\alpha$ width to the states that are closer to the region of astrophysical interest through configuration mixing.\n\nFor a long time, the surprising alpha cluster structure has been a stimulus for the development of classical shell model approaches (see \\cite{2} for new results). Additionally, work by the authors of Ref. \\cite{3} recently ``strengthened the theoretical motivation for experimental searches of alpha cluster states in alpha-like nuclei\" \\cite{3}. The authors of Ref. \\cite{3} related the nuclear structure in light systems of even and equal numbers of protons and neutrons with the first-order transition at zero temperature from a Bose-condensed gas of alpha particles (\\textsuperscript{4}He nuclei) to a nuclear liquid.\n\n\\textsuperscript{20}Ne nucleus presents a famous example of the manifestation of the alpha-cluster structure, and therefore it makes this nucleus a touchstone for \\textit{ab initio} approaches \\cite{4}. The nucleus \\textsuperscript{20}Ne is a benchmark case for the traditional shell model and its extension into algebraic and clustering domains. The well-established effective interaction Hamiltonians such as \\cite{5} not only shows an outstanding agreement with experimental data for \\textit{sd}-shell nuclei but also generates configuration mixing that shows transition to deformation and clustering.\n\nThe remarkable feature of \\textsuperscript{20}Ne nucleus shows up in the fact that almost all the observed states below 10 MeV can be classified into several overlapping rotational-like bands with the first one based on the ground  0$^+_1$ state. There are three other bands based on 0$^+$ levels: on 0$^+_2$ at 6.725 MeV, on a very narrow 0$^+_3$ at 7.191 MeV, on a very broad 0$^+_4$ at $\\sim$ 8.7 MeV \\cite{6} which are of evident cluster structure. The 0$^+_2$ and 0$^+_4$ bands have $\\alpha$+\\textsuperscript{16}O core structure as can be seen from their reduced $\\alpha$ particles widths, and probably the 0$^+_3$ band has predominant \\textsuperscript{12}C+\\textsuperscript{8}Be structure which manifests used itself in the selectivity of the \\textsuperscript{8}Be transfer reactions \\cite{7}. As the ground state band and the 0$^+_2$ band in \\textsuperscript{20}Ne can be related with similar structures in \\textsuperscript{16}O and \\textsuperscript{12}C, the ``additional\" structure of 0$^+_4$ states is not understood \\cite{4}. The cluster approaches \\cite{8} related the 0$^+_4$ band with large $\\alpha$-widths, starting with the so-called ``\\textsuperscript{16}O+$\\alpha$\" higher nodal band, which has one more nodal point in ``\\textsuperscript{16}O+$\\alpha$\" relative wave function than the lower bands have. However it appeared that there are too many bands with a similar structure.\n\nThe $\\alpha$ particle decay threshold in \\textsuperscript{20}Ne is 4.73 MeV, while the threshold for proton decay is at 12.8 MeV (neutron decay threshold is even higher). Therefore resonance $\\alpha$ particle scattering should be considered as an evident way to obtain data on the natural parity levels in \\textsuperscript{20}Ne up to 13 MeV excitation energy. Indeed the majority of adopted data \\cite{6} on level properties in the region in question based on a resonance work and an analysis made in 1960 year. More recently the $\\alpha$+\\textsuperscript{16}O resonance scattering experiment were developed further to backward angles and the data were reanalyzed using \\textit{R} matrix code Multi 6 \\cite{9}. The authors \\cite{9} obtained quite different results from those used in for many levels (see Table 1); in particular the broad 0$^+_4$ and 2$^+_4$ levels appeared even much more broader. The authors \\cite{9} noted difficulties of the fit in the region of 6-8 MeV excitation energies, in a region mainly free from narrow resonances. Evidently strong states of over 1 MeV width should influence a very broad excitation region (see for instance \\cite{10}).\n\nT. Fortune et al. \\cite{11,12} were the first who recognized the importance of the fact that single particle structure of the broad states is in drastic contradiction with the shell model predictions. They \\cite{11,12} also proposed the idea of mixing different configurations to explain the effect. The same idea was used in \\cite{13}. However, the authors of Refs. \\cite{11,12} used old data for the 0$^+_2$ state and used some estimates for the properties of the broad states proposed in Ref. \\cite{14} to support the idea. Later measurements \\cite{6} gave the width of 19 keV for the 6.72 MeV state (which is about 25\\% larger than that used in work \\cite{11}).\n\n\\begin{figure}[!t]\n    \\begin{center}\n    \\include{fig1}\n    \\end{center}\n    \\caption{E-T spectrum for the zero degrees detector. Alpha particles dominate; one can see also a weaker proton locus below the alpha particles.}\n\\end{figure}\n\nThe experimental aim of this work is to obtain new information on the structure of \\textsuperscript{20}Ne states, especially the broad 0$^+$, 2$^+$ states. Unlike other experimentalists, we used the Thick Target Inverse Kinematics method (TTIK) (see \\cite{15,16,17,18,19} and references therein) to study the excitation functions for the \\textsuperscript{16}O($\\alpha$, $\\alpha$)\\textsuperscript{16}O elastic scattering in the \\textsuperscript{20}Ne excitation region of 5.5-9.6 MeV and in a broad angular interval. On the theoretical side, we also used multi configuration shell model calculation to understand the limits of this approach in a description of the cluster states.\n\n\\section{\\label{sec:level2}EXPERIMENT}\n\n\\begin{figure}[!t]\n    \\begin{center}\n    \\include{fig2}\n    \\end{center}\n    \\caption{The \\textsuperscript{16}O($\\alpha$, $\\alpha$)\\textsuperscript{16}O elastic scattering excitation function  at 180$^\\circ$ cm. The excitation energies in \\textsuperscript{20}Ne, E$_x$ in Table 1,  are related with cm energy, E$_{cm}$, by expression,  E$_x$=E$_{cm}$+4.73 MeV.  The bold (red) line is the \\textit{R} matrix fit with the parameters of the present work. The dot (cyan) line is a fit with the 0$^+_4$ excitation energy of 8.3 MeV \\cite{11}, and dot (black) line is a fit with the 0$^+_4$ energy excitation energy of 8.62 MeV and the width of 1.472MeV \\cite{9}.}\n\\end{figure}\n\nThe experiment was performed at the DC-60 cyclotron (Astana) \\cite{17} which can accelerate heavy ions up to the 1.9 MeV\/A energy. While the TTIK method can't compete with a classical approach in terms of energy resolution, the possibility of observing excitation functions at and close to 180 degrees, where the resonance scattering dominates over the potential scattering, enables one to obtain more reliable information on the broad states.  In the TTIK technique the inverse kinematics is used; and the incoming ions are slowed in a helium target gas. The light recoils, $\\alpha$ particles, are detected from a scattering event. These recoils emerge from the interaction with the beam ions and hit a Si detector array located at forward angles while the beam ions are stopped in the gas, as $\\alpha$ particles have smaller energy losses than the scattered ions. The TTIK approach provides a continuous excitation function as a result of the slowing down of the beam.\n\nFor the present experiment, the scattering chamber was filled with helium of 99.99\\% purity. The 30 MeV \\textsuperscript{16}O beam entered the scattering chamber through a thin entrance window made of 2.0 $\\mu$m Ti foil. Eight monitor Si detectors were placed in the chamber to detect \\textsuperscript{16}O ions elastically scattered from the Ti foil at 21$^\\circ$ angle. This array monitors the intensity of the beam with precision better than 4\\%. Fifteen 10x10 mm$^2$ Si detectors were placed at a distance of $\\sim$ 500 mm from the entrance window in the forward hemisphere at different laboratory angles starting from zero degrees. The gas pressure was chosen to stop the beam at distance of 40 mm from the zero degrees detector. The detector energy calibration and resolution ($\\sim$ 30 keV) were tested with a \\textsuperscript{226}Ra, \\textsuperscript{222}Rn, \\textsuperscript{218}Po and \\textsuperscript{214}Po $\\alpha$-source. The experimental set up was similar to that used before \\cite{19}, and more details can be found in Ref. \\cite{17,19}. The main errors in the present experimental approach are related to the uncertainties of the beam energy loss in the gas. To test the energy loss, we placed a thin Ti foil (2.0 $\\mu$m) at different distances from the entrance window. This can be used during the experiment without cycling vacuum. We found that the data \\cite{20} for energy loss of \\textsuperscript{16}O in helium are correct. The details of these tests will be published elsewhere. As a result, we estimated that the uncertainties in the absolute cross section are less than 6\\%. This conclusion was tested by comparison with the Rutherford cross sections at low energies. The agreement with the Rutherford scattering is within 5\\% error bars at all angles (see Fig.3).\n\nTogether with the amplitude signal, the Si detectors provided for a fast signal. This signal together with a ``start\" signal from RF of the cyclotron was used for the Time-of-Flight measurements. This E-T combination is used for particle identification in the TTIK approach \\cite{16,17,18}. Of course, only $\\alpha$ particles should be detected as a result of the interaction of \\textsuperscript{16}O with helium at the chosen conditions. However, protons can be created in the Ti window, and protons can appear due to hydrogen admixtures in the gas. Indeed, we have observed a weak proton banana, likely as a result of  reactions in the window. These protons were easily identified by TF and separated from the $\\alpha$ particles, as  seen in Fig.1.\n\n\\section{\\label{sec:level3}Experimental Results and Discussion}\n\n\\begin{figure}[!t]\n    \\begin{center}\n    \\include{fig3}\n    \\caption{\\textit{R} matrix fit (bold red curve) of the excitation functions for the $\\alpha$+\\textsuperscript{16}O elastic scattering. (a) The dashed (cyan) curve presents the data of  the 0$^+$ level at the 8.3 MeV excitation energy \\cite{11} and (e) dot (black) line is a fit with 2$^+$ level at the excitation energy of 9.0 MeV \\cite{6}.}\n    \\end{center}\n\\end{figure}   \n\n\\begin{table*}[!t]\n\\label{tab:1}\n\\caption{\\textsuperscript{20}Ne levels}\n\\begin{center}\n\\begin{tabular}{lccccccccccclccl}\n\\hline\n\\hline\n\\multirow{1}{*}{N} & \\multicolumn{3}{c}{TUNL data \\cite{6}} & \\multirow{1}{*}{} & %\n    \\multicolumn{2}{c}{H. Shen et al. \\cite{9}} & \\multirow{1}{*}{} & %\n    \\multicolumn{3}{c}{This work} & \\multirow{1}{*}{} & \\multicolumn{4}{c}{CNCIM} \\\\\n\\cline{2-4}\\cline{6-7}\\cline{9-11}\\cline{13-16}\n & E$_x$ & J$^\\pi$ & $\\Gamma_\\alpha$ & & E$_x$ & $\\Gamma_\\alpha$ & & E$_x$ &\n $\\Gamma_\\alpha$ & $\\gamma_\\alpha$ & & E$_x$ & J$^\\pi$ & SF$_p$ & SF$_\\alpha$\\\\\n & (MeV) & & (keV) & & (MeV) & (keV) & & (MeV) & (keV) & & & (MeV) &  &  & \\\\\n\\hline\n\n1 &\t0 & 0$^+_1$ &  & & - & - & & 0 & & Large & & 0 & 0$^+$ & 0.36 & 0.73\\\\\n2 &\t1.63 & 2$^+_1$ & &  & - & - & & 1.63 & & Large & & 2.242 & 2$^+$ & 0.41 & 0.67\\\\\n3 &\t4.25 & 4$^+_1$ & &  & - & - & & 4.25 & & Large & & 4.58 & 4$^+$ & & 0.62\\\\\n\n4 &\t5.78 & 1$^-$ & (28$\\pm$3)x10$^{-3}$ &  & - & - &  & 4.45 & 0.03 & 1.4\\\\\t\t\t\t\n5 &\t6.73 & 0$^+_2$ & 19$\\pm$0.9 & & 6.72 & 11 & & 6.78 & 20.6 & 0.47 & & 6.94 & 0$^+_3$ & 0.55 & 0.46\\\\\n6 &\t7.16 & 3$^-$ & 8.2$\\pm$0.3 & & 7.16 & 10 & & 7.18 & 8.3 & 1.37\\\\\t\t\t\t\n7 &\t7.19 & 0$^+_3$ & 3.4$\\pm$0.2 & & 7.19 & 5 & & 7.20 & 3\t& 0.019 & & 6.27** & 0$^+_2$ & 0.055 & 0.44**\\\\\n8 &\t7.42 & 2$^+_2$ & 15.1$\\pm$0.7 & & 7.43 & 7 & & 7.44 & 14.3 & 0.19 & & 7.39 & 2$^+_3$ & 0.01 & 0.12\\\\\n9 &\t7.83 & 2$^+_3$ & 2 & & 7.83 & 1 & & 7.85 & 3.68 & 0.01 & & 7.15** & 2$^+_2$ & 0.12 & 0.18**\\\\\n\n10 & 8.45 & 5$^-$ & 0.013$\\pm$0.004 & & 8.45 & 0.02 & & 8.45 & 0.013\\\\\t\t\t\t\t\n11 & 8.71 & 1$^-$ & 2.1$\\pm$0.8 & & & & & 8.71 & 3.5\\\\\t\t\t\t\t\n12 & $\\approx$8.7 & 0$^+_4$ & $>$800 & & 8.62 & 1470 & & 8.77$\\pm$0.15 & 750$\\pm$220 & $\\sim$0.25 & & 9.66** & 0$^+_4$ & 0.002 & 0.18**\\\\\n13 & 8.78 & 6$^+_1$ & 0.11$\\pm$0.02 & & - & - & & 8.78 & 0.14 & 0.5 & & 9.49 & 6$^+$ & & 0.51\\\\\n14 & 8.85 & 1$^-$ & 19 & & 8.84 & 27 & & 8.85 & 18.0\\\\\t\t\t\t\t\n15 & 9.00 & 2$^+_4$ & $\\approx$800 & & 8.87 & 1250 & & 8.79$\\pm$0.10 & 695$\\pm$120 & 0.86 & & 8.36** & 2$^+_4$ & 0.02 & 0.02**\\\\\n16 & 9.03 & 4$^+_3$ & 3 & & 9.02 & & & 9.03 & 1.9 & 0.03 & & 9.0 & 4$^+$ & & 0.09\\\\\n\n17 & 9.12 & 3$^-$ & 3.2 & & 9.09 & 4 & & 9.13 & 4.1\\\\\n18 & 9.19 & 2$^+$ &  &  & - & - & & (9.29) & $\\leq$10\t\\\\\t\t\t\t\n19 & 9.48 & 2$^+$ & 29$\\pm$15 & & 9.48 & 46 & & 9.48 & 65$\\pm$20 & 0.02?\\\\\t\t\t\t\n20 & 9.99 & 4$^+_4$ & 155$\\pm$30 & & 10.02 & 150 & & 9.97 & 157 & 0.38 & & 9.5 & 4$^+$ & & 0.009\\\\\n21* & 10.26 & 5$^-$ & 145$\\pm$40 & & 10.26 & 190 & & 10.26 &\t& 1.9\t\\\\\t\t\t\n22 & 10.41 & 3$^-$ & 80 & & 10.40 & 101 & & 10.41 & &\t\\\\\n\n23 & 10.58 & 2$^+$ & 24 & & 10.56 & 15 & & 10.58 & & & & 10.2 & 2+ & 0.005 & 0.04\\\\\n24 & 10.80 & 4$^+_4$ & 350 & & 10.75 & 400 & & 10.80 & & & & 10.7 & 4$^+$ & & 0.04\\\\\n25 & 10.97 & 0$^+_5$ & 580 & & 10.99 & 700 & & 10.97 & & & & 11.9 & 0$^+$\t\\\\\t\n26 & 11.24 & 1$^-$ & 175 & & 11.19 & 85 & & 11.24 & &\\\\\t\t\t\t\t\n27 & 11.95 & 8$^+$ &  (3.5$\\pm$1.0)x10$^{-2}$  & & & & & 11.95 & & 0.35 & & 11.50 & 8$^+$ & & 0.40\\\\\n\\hline\n\\hline\n\\multicolumn{16}{l}{ * For the levels with numbers 21-27 the parameters of the present fit were fixed as in \\cite{6} } \\\\\n\\multicolumn{16}{l}{ ** Calculated in \\textit{psd} space. SF is to the first excited state in \\textsuperscript{16}O; SFs for the ground state in \\textsuperscript{16}O are $\\leq$ 0.1 } \\\\\n\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\nThe experimental excitation functions were analyzed using multilevel multichannel \\textit{R} matrix code \\cite{21}. The calculated curves were convoluted with the experimental energy resolution. The experimental energy resolution was $\\sim$ 30 keV at zero degrees and deteriorated up to $\\sim$100 keV with angles estranging from zero degrees. We did not notice a deterioration of the energy resolution with the energy loss of the beam in the chamber. As it seen in Table 1 the excitation energies of the resonances of the present work agree with the adopted ones \\cite{6} within 10-15 keV. This agreement is an evidence of our overall good energy calibrations and the correct account of the ion energy loss in helium in the present work. Fig.2 and Fig.3 give the experimental excitation functions together with the present \\textit{R} matrix fit. Fig.2 demonstrates the data at 180$^\\circ$ and illustrates the differences in the fits due to different parameters of the broad 0$^+_4$ resonance. The data on the resonances used in the present \\textit{R}-matrix fit are summarized in the Table 1 together with the adopted data \\cite{6}. Data of the last \\textit{R} matrix analysis \\cite{9} are also given in Table 1. The analysis \\cite{9} resulted in level parameters which are often different from the adopted.\n\nOur analysis (Table 1) resulted in  small discrepancies with the data \\cite{6} in the detail description of the narrow states (widths less than 10 keV). The \\textit{R}-matrix code \\cite{21} is tuned for the TTIK measurements and for the analyses of states with a width of over 10 keV to accelerate automatic fit calculations. Therefore the small disagreements for the narrow states are not significant. We focused on the broader states and on the part of the excitation function changing slowly with energy and angle.\n\nOur analysis indicates that all strong alpha cluster states at 5$\\sim$6 MeV below or up the investigated excitation energy region can influence the \\textit{R}-matrix fit. Therefore we included in the fit the \\textsuperscript{20}Ne ground state, the first 2$^+$ and 4$^+$ states and 1$^-$ (5.67 MeV) state in the fit, even though  below the investigated region (see Table 1). Among these states, only the 1$^-$ state is above the $\\alpha$ particle decay threshold in \\textsuperscript{20}Ne; the reduced width of this state is known, and it is large. Shell model calculations also give very large spectroscopic factors for all members of the ground state band. Indeed, a good agreement needs large values of the corresponding amplitudes (over 0.7). Above the investigated excitation energy region, high spin $\\alpha$-cluster resonances are mainly known. Each of these resonances (see Table 1) considered separately influences the fit, especially at 180$^\\circ$. However, their joint influence is much weaker. This cancellation is due to different parities of the spins. Only the influence of the closest to the investigated region, the 4$^+$ (9.99 MeV) resonance, can be noticed. A somewhat better fit needs the width of this resonance to be slightly larger $\\sim$160 keV (well within the quoted uncertainties, see Table 1). Parameters of all other resonances above the investigated region were fixed according to the data of Ref. \\cite{6}. A good general fit ($\\chi^2$=1.1) was reached in this way without any backward resonance inclusion.\n\n\\begin{table*}[!t]\n    \\caption{$\\alpha$+\\textsuperscript{12}C levels in \\textsuperscript{16}O}\n    \\centering\n    \\begin{tabular}{ccccccccc}\n    \\hline\n    \\hline    \n    \\multicolumn{1}{c}{\\textsuperscript{16}O level} & \\multirow{1}{*}{} & \\multicolumn{1}{c}{$\\Gamma_\\alpha$ $_{exp}$ keV \\cite{6} } & \\multirow{1}{*}{} & \\multicolumn{1}{c}{$\\gamma_\\alpha$(1); -V$_0$ MeV} & \\multirow{1}{*}{} & \\multicolumn{1}{c}{$\\gamma_\\alpha$(2); -V$_0$ MeV} & \\multirow{1}{*}{} & \\multicolumn{1}{c}{$\\gamma_\\alpha$(3); -V$_0$ MeV}\\\\ \n    \\cline{1-1}\\cline{3-3}\\cline{5-5}\\cline{7-7}\\cline{9-9}\n    1$^-$; 9.58 MeV & & 420$\\pm$20 & & 0.70; 138.2 & & 0.72; 150.0 & & 0.84; 158.5 \\\\\n    4$^+$; 10.36 MeV & & 26$\\pm$3 & & 0.68; 125.3 & & 0.88; 139,6 & & 1.21; 143.2 \\\\\n    \\hline\n    \\hline    \n    \\end{tabular}\n    \\label{tab:2}\n\\end{table*}\n\nAll resonances are at the maximum at 180$^{\\circ}$. The broad hump at this angle at cm energy of 4 MeV (Fig. 2) is a clear indication for the presence of low spin states, 0$^+$ and 2$^+$. Higher spin states are narrower. A single level (2$^+$) cannot produce the strong peak at different angles, and levels of different parity, such as 2$^+$ and 1$^-$, interfere destructively at 180$^{\\circ}$. The present analysis resulted in two practically degenerate states at $\\sim$ 8.8 MeV with the same width (see Table 1). Our results for the broad 2$^+$ are rather close to the adopted values. However, if this level is moved to 9.0 MeV excitation energy (as in \\cite{6}) then the fit becomes worse, especially in the vicinity of the dip die to the presence of another 2$^+$ level at 9.48 MeV (Fig. 3(e)).  The fit becomes even worse with a very broad 2$^+$ level resulting from the parameters of Ref. \\cite{9}.\n\nT. Fortune at al., \\cite{11} observed a broad distribution with a center at 8.3 MeV excitation energy in \\textsuperscript{20}Ne and related it with the 0$^+$ level. Fig. 2 presents \\textit{R} matrix calculations with the 0$^+$ level to be moved to 8.3 MeV excitation energy. This move destroyed the good fit. Fig.2 shows also that a very broad 0$^+$ level of the fit \\cite{9} destroys the agreement.\n\nThe 18$^{th}$ (2$^+$) level of \\cite{6} is in the energy region of the present investigation. It was observed in a single work in a study of \\textsuperscript{20}Na $\\beta^+$ decay. We have not found any reliable evidence for the presence of this state. If it exists, its width should be less than 10 keV. We observe a fluctuation of points which might be associated with a narrow 2$^+$ level at excitation energy of 9.29 MeV.\n\nWe noted that the 19$^{th}$ level, 2$^+$, has the adopted \\cite{6} excitation energy in our fit but a different width of 65$\\pm$20 keV. While the two times difference with Ref. \\cite{6} in the widths is marginally exceeds the error bars, the influence on the better fit is evident. The adopted data \\cite{6} for this level are based on the results of a single older work \\cite{22}. The authors \\cite{22} observed a weak $\\gamma$ decay of this state in the presence of a large background. A broader level than in Ref. \\cite{22} was also found in work \\cite{9} as shown in Table 1.\n\nWe characterized the alpha-cluster properties of the states above the alpha particle decay threshold by SF=$\\gamma_\\alpha$=$\\Gamma_\\alpha$ $_{exp}$\/$\\Gamma_\\alpha$ $_{calc}$, where $\\Gamma_\\alpha$ $_{calc}$  is the single alpha particle width calculated in the $\\alpha$-core potential. To normalize SFs we calculated these values for the well-known alpha-cluster states in \\textsuperscript{16}O.\n\nThe Woods-Saxon potential was used to calculate the limit ($\\Gamma_\\alpha$ $_{cal}$) as the widths of single particle states in the potential. First we tried to fit the widths of known alpha-cluster states, 1$^-$ and 4$^+$, in \\textsuperscript{16}O so that $\\gamma_\\alpha$ = $\\Gamma_\\alpha$ $_{exp}$\/$\\Gamma_\\alpha$ $_{cal}$ $\\sim$ 1.0. The real part of the potential was changed to fit the binding energy of the states. The radius of the potential was chosen to be R = r$_0$ $\\times$12$^{1\/3}$; the Coulomb potential was taken in to account as a charge sphere potential with R$_{coul}$ = R.  We made first calculations (1) with r$_0$ = 1.31 fm and the diffuseness a = 0.65 fm, then we set r$_0$ = 1.23 fm (2), and we finally performed the third calculations (3)  with r$_0$ = 1.23 fm and a = 0.6 fm. The results are summarized in Table 2. The $\\gamma_\\alpha$ calculations for the \\textsuperscript{20}Ne states were made with the final (3) parameters  should be compared with SF given by a theory.\n\n\\section{\\label{sec:level4}Theoretical description of the $\\alpha$ cluster states in \\textsuperscript{20}$\\textbf{Ne}$}\n\nCNCIM \\cite{2} is among the latest developments of the classical shell model approaches towards clustering. This model targets a combination of classical configuration interaction techniques with algebraic methods that emerge in the description of clustering.  The ability to construct a fully normalized set of orthogonal cluster channels is at the core of this approach; the overlaps of the shell model states with these channels are associated with spectroscopic factors and compared in Table 1 with the reduced widths obtained earlier. The CNCIM allows us to study clustering features that emerge in models with well-established traditional shell model Hamiltonians. These effective model Hamiltonians are built from fundamental nucleon-nucleon interactions followed by phenomenological adjustments to select observables; thus, they generally describe of a broad scope of experimental data with high accuracy. Apart from using these phenomenological shell model Hamiltonians, our study does not involve any adjustable parameters. In order to fully explore the problem, we considered several different model spaces and corresponding Hamiltonians: the \\textit{sd} model space with USDB interaction \\cite{5}; unrestricted \\textit{p-sd} shell model Hamiltonian \\cite{23}, the same Hamiltonian has been used in Refs. \\cite{2};  WBP Hamiltonian \\cite{24} allowing 0$\\hbar\\omega$, 1$\\hbar\\omega$  and 2$\\hbar\\omega$ excitations in \\textit{p-sd-pf} valence space; and the \\textit{sd-pf} Hamiltonian \\cite{25}. This sequence of Hamiltonians represents and expansions of the valence space from \\textit{sd} to \\textit{p-sd}, to \\textit{p-sd-pf}. All models are in good agreement for the \\textit{sd}-states; the low-lying negative parity states as well as positive 2ph excitations are dominated by the \\textit{p-sd} configurations. Thus, in Table 1 we only include the results from the \\textit{p-sd} Hamiltonian which turned out to be most representative although the following discussion and conclusions are largely based on comparisons. The lowest states associated with significant \\textit{fp} shell component appear at excitation energies above 15 MeV.\n\nShell model calculations for \\textsuperscript{20}Ne with open \\textit{2s-1d (sd)} shells predict well the ground state band. The structure of this band is based on the dominating SU(3) configuration (about 75\\%) with quantum numbers (8,0). The model predicts (Table 1) large SF for all members of the band based on the ground state in \\textsuperscript{20}Ne. The 0$^+$, 2$^+$, and 4$^+$ members of the band are below the $\\alpha$ particle decay threshold and do not have observable $\\alpha$ particle widths. However, large $\\alpha$ cluster SFs for these states provide for a better \\textit{R} matrix fit. While uncertainties for the \\textit{R} matrix amplitudes for these states are large, the fit (Fig. 2 and 3) requires these amplitudes to be close to those of the negative parity states with the known extreme $\\alpha$ cluster structure. The $\\alpha$ particle widths of the highest 6$^+$ and 8$^+$ members of this band are known. There is a long history of attempts and ideas to describe these widths using shell model approaches (see, for instance \\cite{5}). The most calculations predicted large clustering for the band but could not explain the decrease of the reduced width for the 6$^+$ and 8$^+$ members. As one sees in Table 1, the CNCIM calculations are in fine agreement with the experimental data for these states. All members of this band have significant clustering that diminishes at higher energies due to configuration mixing.  The second 0$^+$ state within \\textit{sd} space appears at around 6.7 MeV of excitation (in \\textit{psd} model in Table 1, this is a third 0$^+$ state at 6.9 MeV). This level also has a substantial clustering component and absorbs nearly all 15\\% of the remaining strength of the SU(3) (8,0) component. The following 2$^+_2$ (in experiment and in \\textit{sd} model, but third in \\textit{p-sd} model as discussed in what follows) at about 7.4 MeV of excitation energy being a member of the 0$^+_2$ band can be described in a similar way.\n\nA defrost of the \\textit{1p} shell (which is filled in \\textsuperscript{16}O) results in the doubling of the levels, as it is observed and is shown in Table 1 new levels 0$^+$ and 2$^+$, marked with asterisks, appear. In our calculations, the ordering in energy is reversed for both doublets. Structurally the members of each doublet are very different which allows them to be so close in energy and inhibits configuration mixing and Wigner repulsion. One of the doublet levels (the lower 0$^+$ and 2$^+$ levels) has a large $\\alpha$ cluster SF relative to the first excited state in \\textsuperscript{16}O and much smaller SF relative to the ground state in \\textsuperscript{16}O (pay attention that the theory gives the wrong order for the levels in question, see Table 1). Indeed the predicted difference in the structure is supported by population selectivity in different nuclear reactions. The 6.72 MeV 0$^+$ and 7.42 MeV 2$^+$ are populated much stronger than neighboring 7.20 and 7.83 MeV levels in the \\textsuperscript{16}O(\\textsuperscript{6}Li, d) reaction \\cite{26}. The opposite is the case in the \\textsuperscript{12}C(\\textsuperscript{9}Be, n) or \\textsuperscript{12}C(\\textsuperscript{12}C,$\\alpha$) \\cite{27} reactions. \n\nThe theory gives large single particle spectroscopic factors for the \\textit{sd} states and smaller for the 7.20 and 7.83 MeV states. Indeed, one expects that states in \\textsuperscript{20}Ne with a hole in the \\textit{p1\/2} shell will be weakly excited in the single nucleon transfer, \\textsuperscript{19}F(\\textsuperscript{3}He, d) reaction in accordance with the experimental data \\cite{28}. The experiment \\cite{11,12,28} supports also detailed single particle SF calculations giving SF for the ground state smaller than for the 0$^+$ 6.73 MeV state and much higher SF for the first excited 2$^+$ than for 2$^+$ member of the band based on the 6.73 MeV state.\n\nIn the \\textit{sd} valence space (USDB) the third 0$^+$ state appears only at 11.9 MeV, and it has a relatively small alpha spectroscopic factor; opening of the \\textit{p}-shall in addition to 0$^+$ at 6.27 MeV, leads to 0$^+$ state at 9.7 MeV. However, the predicted state has a low proton spectroscopic factor which is not as well supported by observations. A similar serious discrepancy is observed with a broad 4$^{th}$ 2$^+$ state at around 9 MeV, both \\textit{sd} and \\textit{p-sd} shell models produce candidates but with very low alpha SF.  As evident from our studies the only strong coupling to alpha channels could come from \\textit{fp} shell and higher shells. The lowest two bands saturate the alpha strength within \\textit{sd} configurations, holes in the \\textit{p}-shell do not lead to a significant contribution due to low level of core excitation in the ground state of \\textsuperscript{16}O. While our models predict high excitation energies of states with significant \\textit{fp} components, we can speculate that strong configuration mixing, collective effects such as deformation, and coupling to continuum via super radiance mechanism \\cite{29,30} can enhance admixture needed to reproduce the broad resonances observed. There is a similar problem with $\\alpha$ cluster negative parity states 1$^-$ and 3$^-$, the \\textit{p-sd} Hamiltonian produces an acceptable spectrum but the alpha spectroscopic factors are low (see also \\cite{31}).\n\n\\section{\\label{sec:level5}Conclusions}\n\nIn this work we study $\\alpha$-clustering in \\textsuperscript{20}Ne. This nucleus is a benchmark example of many theoretical techniques targeting clustering in light nuclei. Our \\textit{R}-matrix analysis of TTIK experimental data confirms previously known results and establishes new constraints for the positions and widths of the resonances. We compared our findings with those obtained theoretically using cluster-nucleon configuration interaction approach developed in several previous works \\cite{2} and references therein. There is good overall agreement between theoretically predicted and observed spectra. Our theoretical approach describes very well the ground state band and the band built on the first 0$^+$ state. Allowing cross shell excitations from the \\textit{p}-shell, it was possible to reproduce the band built on the second 0$^+$ state. For these states, all spectroscopic factors for alpha transitions to the ground state of \\textsuperscript{16}O and to the first excited state in \\textsuperscript{16}O as well as proton spectroscopic factors to the ground state of \\textsuperscript{19}F are well reproduced. The situation is not as good when it comes to resonances 1$^-$ 3$^-$ and 4$^{th}$ 0$^+$ and 4$^{th}$ 2$^+$, all of these states are broad and have exceptionally large alpha spectroscopic factors. \n\nIn order to describe strong clustering features these states must include configurations from \\textit{fp} shell and from higher oscillator shells, however Hamiltonians that we explored predict these contributions to be negligible below 15 MeV of excitation. Thus, inability of theoretical models to describe broad states exclusively while working well elsewhere suggests an additional coupling mechanism unaccounted for in the traditional shell model Hamiltonians. The super radiance suggested in Refs. \\cite{29,30} could provide this mechanism. Alternatively, the problem could be associated with relatively unknown cross shell interactions. Therefore, work shows that the experimental study of alpha clustering represents an outstanding tool for exploring cross shell excitations especially those of multi-particle multi-hole nature.\n\n\\begin{acknowledgments}\nThis work was supported by Ministry of Education and Science of the Republic of Kazakhstan (grant number \\#0115\u0420\u041a03029 ``NU-Berkeley\", 2014-2018; grant number \\#0115\u0420\u041a022465, 2015-2017). This material is also based upon work supported by the U.S. Department of Energy Office of Science, Office of Nuclear Physics under Grants No DE-FG02-93ER40773 and No. DE-SC0009883.  G.V.R is also grateful to the Welch Foundation (Grant No. A-1853).\n\\end{acknowledgments}\n\n\\nocite{*}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \\label{Introduction}\n\nLiDAR point clouds, compared against other sensors such as camera and radar in the autonomous driving perception, have advantages of both accurate distance measurements and fine semantic descriptions. Studies on point clouds have gained increasing popularity in the computer vision area. Typical research topics include 3D shape recognition, part segmentation, indoor scenario parsing, and outdoor large-scale scene understanding. Several benchmark datasets such as ModelNet40 \\cite{wu20153d}, ShapeNet \\cite{chang2015shapenet}, S3DIS \\cite{armeni20163d}, and Semantic3D \\cite{hackel2017semantic3d} have been established for these topics. However, there exists spatial property disparity between these datasets and outdoor 360$^\\circ$ sweep scans, which are typically produced by the on-board vehicle LiDAR. Per swept LiDAR point clouds are much sparser, and their sparsity is generally increased with the reflection distance. Examples in Figure \\ref{fig2} demonstrate the difference between dense and sparse point clouds in outdoor areas.\n\nPoint clouds processing approaches could be generally classified into three categories. First, projection into 2D representations \\cite{lawin2017deep, wu2018squeezeseg, caltagirone2017fast, meyer2019lasernet}. The basic idea of this approach is to transform unstructured point clouds into 2D images, which could be directly plugged into image-based Convolutional Neural Networks (CNN) \\cite{krizhevsky2012imagenet}. Consequently, the 2D projection benefits to an easier fusion strategy with the image \\cite{chen2017multi, meyer2019sensor} or multi-views \\cite{su2015multi, qi2016volumetric}. However, the drawback is inevitably occluding a large number of points from the projection perspective, which suffers massive information loss. Second, voxelization into 3D volumetric grid cells \\cite{maturana2015voxnet, huang2016point, li20173d, tchapmi2017segcloud, jiang2018pointsift} or their structural variations (e.g., Octree \\cite{riegler2017octnet} or Spherical \\cite{rao2019spherical}). Even though this approach is more effective to maintain points in the 3D space, the data loss problem is not eliminated due to the grid quantization. 3D convolutions are also computationally expensive. Third, directly process raw point clouds. A fundamental model PointNet \\cite{qi2017pointnet} has been specifically designed to process raw unstructured point clouds, which inspires many other studies \\cite{qi2017pointnet++, wang2018dynamic, wu2019pointconv, wang2019graph, wang2019associatively} to follow this idea. Many existing methods have reported their performance on several dense data classification and segmentation benchmarks, however, their effectiveness for sparse data is unknown.\n\nTransfer learning \\cite{pan2009survey} and domain adaptation \\cite{tzeng2017adversarial} techniques have been recently discussed to bridge the gap between the source data and target data. They either extend the training procedure with new labeled data \\cite{rist2019cross}, or re-design the network architecture by adopting an adversarial generator-discriminator \\cite{lai2010object, wu2019squeezesegv2} for semi-supervised or unsupervised learning. In this particular study, however, we focus on the specific problem of semantic segmentation for sparse point clouds. We retain the cross-domain adaptation in our continued work. \n\nTo the best of our knowledge and to the time of our work, VirtualKITTI \\cite{3dsemseg_ICCVW17}, SemanticKITTI \\cite{behley2019dataset}, and DeepenaiKITTI\\footnote{https:\/\/www.deepen.ai\/kitti-labels-download\/} are currently available public datasets that provide semantic labels for sparse point clouds. We investigate a decent comparison of more than 10 state-of-the-art methods that directly process raw point clouds. With the evaluation of their effectiveness on the sparse data, we reveal that network architecture, neighborhood selection, and local sparsity are essential factors that affect the segmentation performance. This investigation has shown us advantages\/shortcomings of previous methods, and therefore we are motivated to propose our method, Multi-domain Neighborhood Embedding and Weighting (MNEW). The key idea is illustrated in Figure \\ref{fig1}. In MNEW, we collect multi-scale neighborhood points in both the static geometry domain and dynamic feature domain. Given a query point, its geometry neighbors are based on the Euclidean distance, and its feature neighbors are based on the similarity that are dynamically varied across different network layers. For each neighborhood point, we first assign attentions according to their location distance and feature similarity. We also compute the geometry\/feature sparsity at each neighbor point, which are then transformed as adaptive weighting factors. The embedded neighborhood feature is a combination of weighted convolution outputs in the two domains. The overall network structure inherits PointNet, which is able to capture both pointwise details and global semantics. In addition, MNEW also extends its capability to embody the local contextual information. Experiments in Section \\ref{Experiments} manifest the effectiveness of our method for the sparse data.\n\nThe major contributions of this paper are summarized as follows:\n\\begin{itemize}\n\t\\itemsep 0em\n\t\\item We introduce MNEW, a novel semantic segmentation model that is effective for per sweep LiDAR data, which is crucial for the application of autonomous driving perception.\n\t\\item We design a neighborhood embedding method in both static geometry domain and dynamic feature space, which embodies attention mechanism by location distance and feature similarity, as well as sparsity-adapted weighting mechanism.\n\t\\item We investigate a thorough comparison for a number of recent methods, evaluating their effectiveness on sparse point clouds. We claim that network architecture, neighborhood selection, and weighting mechanism are essential factors.\n\t\\item We achieve state-of-the-art performance on sparse point clouds, and we observe that performance is varied by distance and local sparsity.\n\\end{itemize}\n\n\n\n\n\n\t\n\n\n\n\\begin{table*}[!htb]\n\t\\begin{center}\n\t\t\\footnotesize\n\t\n\t\t\\begin{tabular}{c c c *3c c c}\n\t\t\t\\toprule\n\t\t\t\\multirow{2}{*}{Method}\t\t\t\t& \\multirow{2}{*}{Architecture} \t& \\multirow{2}{*}{\\makecell{Feature \\\\Extractor}}\t& \\multicolumn{3}{c}{Neighborhood}\t\t\t\t\t& \\multirow{2}{*}{Weighting}\t& \\multirow{2}{*}{Loss}\t\t\t\t\t\t\t\\\\ \\cline{4-6}\n\t\t\t&\t\t\t\t\t\t\t\t\t&\t\t\t\t\t\t\t\t\t\t\t\t\t& Domain\t\t\t& Selection\t\t& Embedding \t&\t\t\t\t\t\t\t\t&\t\t\t\t\t\t\t\t\t\t\t\t\\\\ \n\t\t\t\\midrule\n\t\t\tPointNet \\cite{qi2017pointnet} \t\t& Dilated \t\t\t\t\t\t\t& MLP\t\t\t\t\t\t\t\t\t\t\t\t& -\t\t\t\t\t& - \t\t\t& Points\t\t& -\t\t\t\t\t\t\t\t& $\\mathcal{L}_{CE}$\t\t\t\t\t\t\t\\\\\n\t\t\tPointNet++ \\cite{qi2017pointnet++}\t& Encoder-Decoder \t\t\t\t\t& MLP\t\t\t\t\t\t\t\t\t\t\t\t& Geometry\t\t\t& Multi-radius\t& Points \t\t& -\t\t\t\t\t\t\t\t& $\\mathcal{L}_{CE}$\t\t\t\t\t\t\t\\\\\n\t\t\tA-CNN \\cite{komarichev2019cnn}\t\t& Encoder-Decoder \t\t\t\t\t& MLP\t\t\t\t\t\t\t\t\t\t\t\t& Geometry\t\t\t& Ring-shaped \t& Points\t\t& -\t\t\t\t\t\t\t\t& $\\mathcal{L}_{CE}$\t\t\t\t\t\t\t\\\\ \n\t\t\tKP-FCNN \\cite{thomas2019kpconv}\t\t& Encoder-Decoder\t\t\t\t\t& KP-Conv\t\t\t\t\t\t\t\t\t\t\t& Geometry\t\t\t& kNN in Radius\t& Points\t\t& Geometry Distance\t\t\t\t& $\\mathcal{L}_{CE} + \\mathcal{L}_{Reg}$ \t\t\\\\\n\t\t\tDGCNN \\cite{wang2018dynamic}\t\t& Dilated\t\t\t\t\t\t\t& MLP\t\t\t\t\t\t\t\t\t\t\t\t& Feature\t\t\t& kNN\t\t\t& Query-Edges\t& -\t\t\t\t\t\t\t\t& $\\mathcal{L}_{CE}$ \t\t\t\t\t\t\t\\\\\n\t\t\tRS-CNN \\cite{liu2019relation}\t\t& Encoder-Decoder\t\t\t\t\t& RS-Conv\t\t\t\t\t\t\t\t\t\t\t& Geometry\t\t\t& Random-pick\t& Query-Edges\t& -\t\t\t\t\t\t\t\t& $\\mathcal{L}_{CE}$ \t\t\t\t\t\t\t\\\\\n\t\t\tPointWeb \\cite{zhao2019pointweb}\t& Encoder-Decoder\t\t\t\t\t& MLP\t\t\t\t\t\t\t\t\t\t\t\t& Geometry\t\t\t& kNN\t\t\t& Pairwise-Edges& -\t\t\t\t\t\t\t\t& $\\mathcal{L}_{CE}$ \t\t\t\t\t\t\t\\\\\n\t\t\tGACNet \\cite{wang2019graph}\t\t\t& Encoder-Decoder\t\t\t\t\t& MLP\t\t\t\t\t\t\t\t\t\t\t\t& Geometry\t\t\t& Radius\t\t& Points\t\t& Feature Similarity\t\t\t& $\\mathcal{L}_{CE}$ \t\t\t\t\t\t\t\\\\\n\t\t\tPointConv \\cite{wu2019pointconv}\t& Encoder-Decoder\t\t\t\t\t& MLP\t\t\t\t\t\t\t\t\t\t\t\t& Geometry\t\t\t& Radius\t\t& Points\t\t& Local Density\t\t\t\t\t& $\\mathcal{L}_{CE}$ \t\t\t\t\t\t\t\\\\\n\t\t\tASIS \\cite{wang2019associatively}\t& Encoder-Decoder\t\t\t\t\t& MLP\t\t\t\t\t\t\t\t\t\t\t\t& Geometry\t\t\t& Radius\t\t& Points\t\t& -\t\t\t\t\t\t\t\t& $\\mathcal{L}_{CE} + \\mathcal{L}_{Disc}$\t\t\\\\ \n\t\t\t\\hline\n\t\t\t\\multirow{3}{*}{Ours (MNEW)}\t& \\multirow{3}{*}{\\makecell{Dilated \\\\ (improved)}}\t& \\multirow{3}{*}{MLP}\t& \\multirow{3}{*}{\\makecell{Geometry \\\\+ Feature}}\t& \\multirow{3}{*}{\\makecell{Multi-radius \\\\+ Multi-kNN}}\t& \\multirow{3}{*}{Query-Edges}\t& \\multirow{3}{*}{\\makecell{Geometry Distance \\\\+ Feature Similarity \\\\+ Neighbor Sparsity}}\t& \\multirow{3}{*}{$\\mathcal{L}_{CE} + \\mathcal{L}_{Reg}$} \\\\ \n\t\t\n\t\t\t&\t\t\t\t\t\t\t\t\t&\t\t\t\t\t\t\t\t\t\t\t\t\t&\t\t\t\t\t&\t\t\t\t&\t\t\t\t&\t\t\t\t\t\t\t\t&\t\t\t\t\t\t\t\t\t\t\t\t\\\\\n\t\t\t&\t\t\t\t\t\t\t\t\t&\t\t\t\t\t\t\t\t\t\t\t\t\t&\t\t\t\t\t&\t\t\t\t&\t\t\t\t&\t\t\t\t\t\t\t\t&\t\t\t\t\t\t\t\t\t\t\t\t\\\\ \n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\vspace{-0.5cm}\n\t\\caption{Comparison of methods that directly process raw point clouds.}\n\n\t\\label{tab1}\n\\end{table*}\n\n\n\t\n\\section{Related Work} \\label{Related Work}\n\n\\subsection{Methods} \\label{sec2.1}\nSince conversion-based approaches like 2D projection or 3D voxelization inevitably suffer the problem of losing points, in this section we focus on the related semantic segmentation methods that directly process raw point clouds, which are well-suited to explore the 3D data capability and close to our work.\n\nPointNet \\cite{qi2017pointnet} is considered a milestone method that inputs raw point clouds without any format transformation. This method adopts shared Multi-Layer Perceptrons (MLP) \\cite{haykin1994neural} as the key component to learn pointwise features, and a pooling operation is followed to obtain global features representing all-points maximal response. The limit of PointNet is that it does not consider the local spatial relationship with neighborhood points. To address this issue, PointNet++ \\cite{qi2017pointnet++} is proposed with a hierarchical encoder-decoder structure. Analogous to the Fully Convolutionally Networks (FCN) \\cite{long2015fully} used in image segmentation, PointNet++ extracts local features by grouping and subsampling points in increasing contextual scales, and propagates subsampled features to their original points by interpolation.\n\nSeveral subsequent studies improve PointNet and PointNet++ by their upgraded network designs. A-CNN \\cite{komarichev2019cnn} introduces annular convolution with ring-shape neighborhoods to reduce the duplicated computation that exists in PointNet++ multi-scale grouping. Inspired from kernel pixels in the image-based convolution, KP-FCNN \\cite{thomas2019kpconv} creates local 3D spatial filters using a set of kernel points. A function KP-Conv between kernel points and input points is defined, which is used to replace the MLP operation in PointNet\/PointNet++. Alternative to the process of independent points, DGCNN \\cite{wang2018dynamic} and RS-CNN \\cite{liu2019relation} employ the idea of graph which embeds edges between a query point and its neighborhood points. The difference is that DGCNN follows the PointNet pipeline whereas RS-CNN follows the encoder-decoder structure of PointNet++. PointWeb \\cite{zhao2019pointweb} extends this idea and proposes a pairwise edge embedding between every two points within the selected local region. Instead of the typical approach which collects neighborhood points based on their location distance, DGCNN collects neighbors in the dynamic feature space based on their similarity. Likewise, GACNet \\cite{wang2019graph} assigns proper attention weights to different geometry neighbor points according to their feature attributes, which is to focus on the most relevant part of the neighbors. Weighting mechanism is also utilized in PointConv \\cite{wu2019pointconv}, which estimates kernel density to re-weight the continuous function learned by MLP. SPG \\cite{landrieu2018large} and its continued work \\cite{landrieu2019point} partition point clouds into superpoint graphs and perform Edge-Conditioned Convolution (ECC) \\cite{simonovsky2017dynamic} to assign a label on each superpoint. The difference lies in the graph construction approach, which is solved as an unsupervised minimal geometric partition problem in \\cite{landrieu2018large} and a leaning-based method that minimizes the graph contrastive loss in \\cite{landrieu2019point}. However, neither of these two methods is end-to-end. The purpose of graph contrastive loss is to detect the borders between adjacent objects. It pulls points belonging to the same object towards their centroid, while repelling those belonging to different objects. This idea is intuitively derived as the discriminative loss ($\\mathcal{L}_{Disc}$) \\cite{de2017semantic} in ASIS \\cite{wang2019associatively}, which is added with the cross-entropy loss ($\\mathcal{L}_{CE}$) and regularization loss ($\\mathcal{L}_{Reg}$) for a joint semantic and instance segmentation.\n\nTable \\ref{tab1} summarizes an extensive comparison of selected methods, which are varied by the network architecture, feature extractor, neighborhood selection\/embedding, weighting mechanism, and loss function. Our proposed method is also listed to overview the relation and distinction. More details are discussed in Section \\ref{Method}.\n\n\n\n\n\\begin{table*}[!htp]\n\t\\begin{center}\n\t\t\\footnotesize\n\t\n\t\t\\begin{tabular}{c c c c c c c c}\n\t\t\t\\toprule\n\t\t\tDataset\t\t\t\t\t\t\t\t\t& Type \t\t\t\t& Attributes\t& Size (Train + Test)\t\t\t\t\t& Classes\t\t\t\t& Instance\t& Sequential\t& Train\/Valid \t\t\t\\\\\n\t\t\t\\midrule\n\t\t\tS3DIS \\cite{armeni20163d}\t\t\t\t& indoor dense\t\t& XYZ + RGB\t\t& 6 indoor area, 273M points\t\t\t& 13\t\t\t\t\t& Yes\t\t& No\t\t\t& -\t\t\t\t\t\t\\\\\n\t\t\tScanNet \\cite{dai2017scannet}\t\t\t& indoor dense\t\t& XYZ + RGB\t\t& 1.5K scans, 2.5M frames\t\t\t\t& 21\t\t\t\t\t& Yes\t\t& Yes\t\t\t& -\t\t\t\t\t\t\\\\\n\t\t\tSemantic3D \\cite{hackel2017semantic3d}\t& outdoor dense\t\t& XYZ + RGB\t\t& 30 scenarios, 4009M points\t\t\t& 9\t\t\t\t\t\t& No\t\t& No\t\t\t& -\t\t\t\t\t\t\\\\\n\t\t\tNPM3D \\cite{roynard2018paris}\t\t\t& outdoor dense\t\t& XYZ\t\t\t& 6 scenarios, 143M points\t\t\t\t& 50 (10)\t\t\t\t& Yes\t\t& No\t\t\t& -\t\t\t\t\t\t\\\\\n\t\t\t\\midrule\n\t\t\tVirtualKITTI \\cite{3dsemseg_ICCVW17}\t& outdoor sparse\t& XYZ + RGB\t\t& 4 simulated scenes, 90 frames\t\t\t& 14\t\t\t\t\t& No\t\t& No\t\t\t& 80\\%\/20\\%\trandom\t\t\\\\\n\t\t\tDeepenaiKITTI\t\t\t\t\t\t\t& outdoor sparse\t& XYZ\t\t\t& 1 sequence, 100 frames\t\t\t\t& 17\t\t\t\t\t& No\t\t& Yes\t\t\t& 80\\%\/20\\% random\t\t\\\\\n\t\t\tSemanticKITTI \\cite{behley2019dataset}\t& outdoor sparse\t& XYZ\t\t\t& 22 sequences, 43.5K frames\t\t\t& 28 (20)\t\t\t\t& Yes\t\t& Yes\t\t \t& 10\/1 sequence\t\t\t\\\\\n\t\t\n\t\t\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\vspace{-0.5cm}\n\t\\caption[Caption for LOF]{Comparison of selected datasets with dense and sparse point clouds.}\n\n\t\\label{tab2}\n\\end{table*}\n\t\n\t\n\\begin{table*}[!htb]\n\t\\begin{center}\n\t\t\\footnotesize\n\t\n\t\t\\begin{tabular}{c| *2c *2c *2c| *2c *2c *2c}\n\t\t\t\\toprule\n\t\t\t\\multirow{2}{*}{Method}\t\t\t\t& \\multicolumn{2}{c}{S3DIS} \t& \\multicolumn{2}{c}{ScanNet}\t& \\multicolumn{2}{c|}{Semantic3D}\t\t& \\multicolumn{2}{c}{VirtualKITTI}\t& \\multicolumn{2}{c}{DeepenaiKITTI}\t& \\multicolumn{2}{c}{SemanticKITTI}\t\t\\\\ \\cline{2-13}\n\t\t\t\t\t\t\t\t\t\t\t\t& OA\t\t\t& mIoU\t\t\t& OA\t\t\t& mIoU\t\t\t& OA\t\t\t& mIoU\t\t\t\t\t& OA\t\t\t& mIoU\t\t\t\t& OA\t\t\t& mIoU\t\t\t\t& OA\t\t\t& mIoU\t\t\t\t\t\\\\\n\t\t\t\\midrule\n\t\t\tPointNet \\cite{qi2017pointnet}\t\t& 78.62\t\t\t& 47.71\t\t\t& 73.9\t\t\t& -\t\t\t\t& -\t\t\t\t& -\t\t\t\t\t\t& 88.07\t\t\t& 50.36\t\t\t\t& 98.41\t\t\t& 64.85\t\t\t\t& 66.12\t\t\t& 19.74\t\t\t\t\t\\\\\n\t\t\tPointNet++ \\cite{qi2017pointnet++}\t& -\t\t\t\t& -\t\t\t\t& 84.5\t\t\t& 33.9\t\t\t& 82.5\t\t\t& 52.1\t\t\t\t\t& 81.99\t\t\t& 44.74\t\t\t\t& 96.66\t\t\t& 54.40\t\t\t\t& 72.35\t\t\t& 22.90\t\t\t\t\t\\\\\n\t\t\tA-CNN \\cite{komarichev2019cnn}\t\t& 87.3\t\t\t& 62.9\t\t\t& \\textbf{85.4}\t& -\t\t\t\t& -\t\t\t\t& -\t\t\t\t\t\t& 42.80\t\t\t& 18.75\t\t\t\t& 43.15\t\t\t& 7.31\t\t\t\t& 33.35\t\t\t& 7.85\t\t\t\t\t\\\\\n\t\t\tKP-FCNN \\cite{thomas2019kpconv}\t\t& -\t\t\t\t&\\textbf{65.4}\t& -\t\t\t\t& \\textbf{68.6}\t& \\textbf{92.9}\t& \\textbf{74.6}\t\t\t& 75.02\t\t\t& 30.49\t\t\t\t& 36.75\t\t\t& 4.54\t\t\t\t& 78.05\t\t\t& 26.71\t\t\t\t\t\\\\\n\t\t\tDGCNN \\cite{wang2018dynamic}\t\t& 84.1\t\t\t& 56.1\t\t\t& -\t\t\t\t& -\t\t\t\t& -\t\t\t\t& -\t\t\t\t\t\t& 92.04\t\t\t& 60.19\t\t\t\t& 98.28\t\t\t& 64.54\t\t\t\t&\\textbf{80.64}\t&\\textbf{30.51}\t\t\t\\\\\n\t\t\tPointWeb \\cite{zhao2019pointweb}\t& 86.97\t\t\t& 60.28\t\t\t& 85.9\t\t\t& -\t\t\t\t& -\t\t\t\t& -\t\t\t\t\t\t& 57.06\t\t\t& 18.94\t\t\t\t& 67.98\t\t\t& 16.67\t\t\t\t& 32.17\t\t\t& 6.84\t\t\t\t\t\\\\\n\t\t\tGACNet \\cite{wang2019graph}\t\t\t&\\textbf{87.79}\t& 62.85\t\t\t& -\t\t\t\t& -\t\t\t\t& 91.9\t\t\t& 70.8\t\t\t\t\t&\\textbf{92.57}\t&\\textbf{60.58}\t\t& 95.56\t\t\t& 51.38\t\t\t\t& 76.51\t\t\t& 26.06\t\t\t\t\t\\\\\n\t\t\tPointConv \\cite{wu2019pointconv}\t& -\t\t\t\t& -\t\t\t\t& -\t\t\t\t& 55.6\t\t\t& -\t\t\t\t& -\t\t\t\t\t\t& 85.26\t\t\t& 47.60\t\t\t\t&\\textbf{98.50}\t&\\textbf{65.74}\t\t& 72.51\t\t\t& 23.24\t\t\t\t\t\\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\vspace{-0.5cm}\n\t\\caption{Comparison of existing methods on dense and sparse point clouds.}\n\n\t\\label{tab3}\n\\end{table*}\n\t\n\t\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\begin{center}\n\t\t\\includegraphics[width=0.9\\linewidth]{img\/Fig3_MNEWdesign.png}\n\t\\end{center}\n\t\\caption{Design of our proposed network. The key module MNEW is zoomed in for a detailed illustration. In MNEW, the upper branch embeds static geometry neighbors based on their location distance (by multi-radius), and the lower branch embeds dynamic feature neighbors based on their similarity (by multi-kNN). Local sparsity is computed in both geometry and feature domain, and transformed to weight the convolution output. After concatenation, a pooling operation aggregates neighborhood features for each query point.}\n\t\\label{fig3}\n\\end{figure*}\n\n\n\n\\subsection{Datasets} \\label{sec2.2}\n\nFor the task of 3D scene semantic segmentation, publicly available datasets such as S3DIS \\cite{armeni20163d} and ScanNet \\cite{dai2017scannet} are indoor dense data, whereas Semantic3D \\cite{hackel2017semantic3d} and NPM3D \\cite{roynard2018paris} are outdoor. For all of the four benchmark datasets, point clouds are collected by accumulating multiple scans in stationary environments to obtain fine detailed measurements. However, sparse point clouds for the application of autonomous driving perception, like the example shown in Figure \\ref{fig2b}, are much different. A sweeping LiDAR sensor is mounted on a moving vehicle, correspondingly the scanning environment is also changing. In a single frame, point clouds are generally denser in close areas, and much sparser far away from the sensor. This is determined by hardware characteristics of the rotation LiDAR sensor. To the time of our work, we found three public datasets with sparse point clouds. VirtualKITTI \\cite{3dsemseg_ICCVW17} is simulated virtual data, while DeepenaiKITTI and SemanticKITTI \\cite{behley2019dataset} provide small\/large-sized semantic labels on the real world KITTI \\cite{geiger2012we} sequential data.\n\nTable \\ref{tab2} summarizes the selected datasets. Other popular benchmarks such as ModelNet40 \\cite{wu20153d} or ShapeNet \\cite{chang2015shapenet} are focused on small 3D CAD objects, which are beyond our interest scope. Note for the number of classes, NPM3D has 50-class fine annotations and 10-class coarse annotations; SemanticKITTI has 28-class labels to separate moving\/stationary objects, and mapped into 20-class for closest equivalent. Since labels for their test subsets are invisible before submission, we split the annotated training data into training\/validation in our experiments for verification.\n\n\n\n\\subsection{Analysis} \\label{sec2.3}\n\nTable \\ref{tab3} compares the performance of selected methods on selected datasets. For dense point clouds, results are borrowed from their original reports. For sparse data, we use the train\/valid splits in Table \\ref{tab2}, and re-produce experiments with their official implementations. Evaluation metrics are the overall accuracy (OA) for every point and the mean intersection-over-union (mIoU) for averaged IoU across all classes. From Table \\ref{tab3}, we summarize our findings as follows.\n\n{\\bf Architecture.} \nPointNet and DGCNN are the networks retaining the number of points unchanged, while other methods are hierarchical encoder-decoders where points are down-sampled and then up-sampled. Despite the fact that encoder-decoder networks performs better in dense data benchmarks, their effectiveness is depreciated for sparse point clouds. One possible explanation is that, 3D interpolations for up-sampling might be suitable for the near-uniformly distributed dense point clouds, but not for the irregular sparse data. Since no down-sample\/up-sampling exists in PointNet and DGCNN, they are similar to the dilated convolution \\cite{yu2015multi} in the image segmentation, which maintains resolution unchanged end-to-end for element-wise feature learning. \n\n{\\bf Neighborhood.} \nDGCNN selects neighboring points in the dynamic feature domain, which differs from other methods whose neighbors are selected by the static geometry distance. GACNet collects geometry neighbors but assigns attention weights based on their feature similarity. Since the performances of DGCNN and GACNet seem promising for SemanticKITTI and VirtualKITTI, we infer that dynamic feature-based neighborhoods are critical. This is interpretable, for example, sparse points on the road in far distances are isolated in geometry location, but they are similar in their feature representations. In contrast, traffic signs hidden in trees are close to leaves, but they should be distinguished from the surrounding vegetation.\n\n{\\bf Weighting.} \nSimilar to the attention schema in GACNet, PointConv compromises the density estimation as a weighting function, which is learned to adapt with different local densities. This method directly considers the data density and therefore obtaining encouraging performance on DeepenaiKITTI. We infer that weighting mechanisms in GACNet and PointConv could compensate for the effectiveness depreciation of their encoder-decoder architecture. \n\n{\\bf Dataset Discrepancy.} \nAccording to the experimental results in Table \\ref{tab3}, DGCNN, GACNet and PointConv are the preferred methods on sparse point clouds. However, the performance is inconsistent across the three selected sparse datasets. The major reason is essentially the data intrinsic discrepancy. VirtualKITTI is generated by the simulator, and it is the only sparse data with RGB colors. SemanticKITTI is a large-scale sequential dataset, which suggests 10 sequences for training and 1 other sequence for validation. DeepenaiKITTI is small-sized probe data, and all its currently available frames are extracted from the same sequence. Due to the small size and high correlation of DeepenaiKITTI, we neglect it in Section \\ref{Experiments} and evaluate our method on VirtualKITTI and SemanticKITTI.\n\t\n\t\n\n\n\n\\begin{table*}[!htb]\n\n\t\\scriptsize\n\t\\centering\n\t\\newcolumntype{M}{ >{\\arraybackslash} p{0.27cm} }\n\n\t\\begin{subtable}[!htb]{\\textwidth}\n\t\t\\centering\n\t\t\\begin{tabular}{c| M M M M M M M M M M M M M M| M M}\n\t\t\t\\toprule\n\t\t\tMethod\t&\\rotatebox[origin=c]{90}{Terrian}\t&\\rotatebox[origin=c]{90}{Tree}\t&\\rotatebox[origin=c]{90}{Vegetation}\t&\\rotatebox[origin=c]{90}{Building}\t&\\rotatebox[origin=c]{90}{Road}\t&\\rotatebox[origin=c]{90}{Guardrail}\t&\\rotatebox[origin=c]{90}{Traffic Sign}\t&\\rotatebox[origin=c]{90}{Traffic Light}\t&\\rotatebox[origin=c]{90}{Pole}\t&\\rotatebox[origin=c]{90}{Misc}\t&\\rotatebox[origin=c]{90}{Truck}\t&\\rotatebox[origin=c]{90}{Car}\t&\\rotatebox[origin=c]{90}{Van}\t&\\rotatebox[origin=c]{90}{Unlabeled} \t& OA\t& mIoU\t\\\\\n\t\t\n\t\t\t\\midrule\n\t\t\n\t\t\n\t\t\tDGCNN \\cite{wang2018dynamic}\t\t& 86.7\t\t\t& 92.5\t\t\t& 70.8\t\t\t& 81.2\t\t\t& 94.8\t\t\t& 93.9\t\t\t& 38.0\t\t\t& 78.0\t\t\t& 65.5\t\t\t& 27.3\t\t\t& 29.2\t\t\t& 76.3\t\t\t& 8.6\t\t\t& 0.0\t& 92.0\t\t\t& 60.2\t\\\\\n\t\t\tGACNet \\cite{wang2019graph}\t\t\t& 82.0\t\t\t& 95.2\t\t\t& 72.5\t\t\t& 86.6\t\t\t& 92.1\t\t\t& 90.6\t\t\t& 51.4\t\t\t& 48.1\t\t\t& 42.6\t\t\t& 31.1\t\t\t& 46.6\t\t\t& 81.6\t\t\t& \\textbf{27.8}\t& 0.0\t& 92.6\t\t\t& 60.6\t\\\\\n\t\t\tPointConv \\cite{wu2019pointconv}\t& 58.1\t\t\t& 89.4\t\t\t& 57.0\t\t\t& 76.0\t\t\t& 80.6\t\t\t& 66.9\t\t\t& 25.2\t\t\t& 59.3\t\t\t& 25.1\t\t\t& 35.6\t\t\t& 9.14\t\t\t& 72.4\t\t\t& 12.0\t\t\t& 0.0\t& 85.3\t\t\t& 47.6\t\\\\\n\t\t\t\\midrule\n\t\t\tMNEW-4096\t\t\t\t\t\t\t& 92.6\t\t\t& \\textbf{97.7}\t& 84.5\t\t\t& 90.7\t\t\t& 97.6\t\t\t& 97.3 \t\t\t& 68.8\t\t\t& 71.9\t\t\t& 62.6\t\t\t& 52.9\t\t\t& 11.0\t\t\t& 85.9\t\t\t& 23.5\t\t\t& 0.0\t& 95.9\t\t\t& 67.0\t\\\\\n\t\t\tMNEW-2048\t\t\t\t\t\t\t& \\textbf{97.0}\t& \\textbf{97.7} & \\textbf{91.2} & \\textbf{92.4} & \\textbf{98.8} & \\textbf{98.2} & \\textbf{70.6} & \\textbf{83.8} & \\textbf{72.8} & \\textbf{64.9} & \\textbf{58.4} & \\textbf{88.3} & 12.7\t\t\t& 0.0\t& \\textbf{97.1} & \\textbf{73.3}\t\\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\t\\vspace{-0.1cm}\n\t\t\\caption{Validation results on VirtualKITTI dataset}\n\t\t\\label{tab4a}\n\t\t\\vspace{0.1cm}\n\t\\end{subtable}\n\t\t\n\t\t\n\t\\begin{subtable}[!htb]{\\textwidth}\n\t\t\\centering\n\t\t\\begin{tabular}{c| M M M M M M M M M M M M M M M M M M M M| M M}\n\t\t\t\\toprule\n\t\t\tMethod\t&\\rotatebox[origin=c]{90}{Car}\t&\\rotatebox[origin=c]{90}{Bicycle}\t&\\rotatebox[origin=c]{90}{Motorcyclist}\t&\\rotatebox[origin=c]{90}{Truck}\t&\\rotatebox[origin=c]{90}{Other Vehicle}\t&\\rotatebox[origin=c]{90}{Person}\t&\\rotatebox[origin=c]{90}{Bicyclist}\t&\\rotatebox[origin=c]{90}{Motorcyclist}\t&\\rotatebox[origin=c]{90}{Road}\t&\\rotatebox[origin=c]{90}{Parking}\t&\\rotatebox[origin=c]{90}{Sidewalk}\t&\\rotatebox[origin=c]{90}{Other Ground}\t&\\rotatebox[origin=c]{90}{Building}\t&\\rotatebox[origin=c]{90}{Fence}\t&\\rotatebox[origin=c]{90}{Vegetation}\t&\\rotatebox[origin=c]{90}{Trunk}\t&\\rotatebox[origin=c]{90}{Terrain}\t&\\rotatebox[origin=c]{90}{Pole}\t&\\rotatebox[origin=c]{90}{Traffic Sign}\t&\\rotatebox[origin=c]{90}{Unlabeled}\t& OA\t& mIoU \\\\\n\t\t\n\t\t\t\\midrule\n\t\t\n\t\t\n\t\t\tDGCNN \\cite{wang2018dynamic}\t\t& 78.3\t\t\t& 0.0\t\t\t& 1.1\t\t\t& \\textbf{17.3}\t& 1.4\t\t\t& 1.9\t\t\t& 3.9\t\t\t& 0.0\t& 88.7\t\t\t& 10.2\t\t\t& 65.1\t\t\t& 0.1\t\t\t& 74.1\t\t\t& 18.8\t\t\t& 71.6\t\t\t& 25.0\t\t\t& 62.1\t\t\t& 28.5\t\t\t& 8.8\t\t\t& 47.8\t\t\t& 80.8\t\t\t& 30.2\t\t\t\\\\\n\t\t\tGACNet \\cite{wang2019graph}\t\t\t& 71.5\t\t\t& 0.0\t\t\t& 0.0 \t\t\t& 12.2\t\t\t& 1.4\t\t\t& 0.0\t\t\t& 0.0\t\t\t& 0.0\t& 80.3\t\t\t& 13.4\t\t\t& 55.3\t\t\t& 0.2\t\t\t& 63.1\t\t\t& 16.7\t\t\t& 67.8\t\t\t& 15.7\t\t\t& 56.4\t\t\t& 12.1\t\t\t& \\textbf{22.9}\t& 38.8\t\t\t& 76.0 \t\t\t& 26.4 \t\t\t\\\\\n\t\t\tPointConv \\cite{wu2019pointconv}\t& 60.5\t\t\t& 0.1\t\t\t& 0.2\t\t\t& 0.6\t\t\t& 3.3\t\t\t& 1.0\t\t\t& 0.9\t\t\t& 0.0\t& 82.1\t\t\t& 3.8\t\t\t& 55.4\t\t\t& \\textbf{0.4}\t& 63.6\t\t\t& 10.8\t\t\t& 59.6\t\t\t& 14.2 \t\t\t& 52.1\t\t\t& 14.1\t\t\t& 8.0\t\t\t& 34.4\t\t\t& 72.5\t\t\t& 23.2\t\t\t\\\\\n\t\t\tRangeNet \\cite{milioto2019rangenet++}\t& 74.1\t\t& \\textbf{14.3}\t& 2.6\t\t\t& 9.4\t\t\t& 10.5\t\t\t& \\textbf{7.2}\t& 21.9\t\t\t& 0.0\t& \\textbf{90.7}\t& \\textbf{36.2}\t& \\textbf{74.2}\t& 0.2\t\t\t& 67.8\t\t\t& \\textbf{33.4}\t& 71.9\t\t\t& 30.7\t\t\t& \\textbf{68.5}\t& 23.0\t\t\t& 22.2\t\t\t& 34.1\t\t\t& 81.4\t\t\t& 34.6 \t\t\t\\\\\n\t\t\t\\midrule\n\t\t\tMNEW-4096\t\t\t\t\t\t\t& \\textbf{81.3}\t& 0.0\t\t\t& \\textbf{13.3}\t& 8.8\t\t\t& \\textbf{12.9}\t& 6.2\t\t\t& \\textbf{31.7}\t& 0.0\t& 88.7\t\t\t& 22.3\t\t\t& 70.4\t\t\t& 0.1\t\t\t& \\textbf{79.3}\t& 30.0\t\t\t& \\textbf{76.9}\t& \\textbf{34.2}\t& 66.4\t\t\t& \\textbf{33.1}\t& 1.1\t\t\t& \\textbf{49.3}\t& \\textbf{84.1}\t& \\textbf{35.3} \\\\\n\t\t\tMNEW-2048\t\t\t\t\t\t\t& 79.8\t\t\t& 0.0\t\t\t& 10.5\t\t\t& 6.5\t\t\t& 7.8\t\t\t& 5.5\t\t\t& 25.5\t\t\t& 0.0\t& 88.8\t\t\t& 22.7\t\t\t& 67.4\t\t\t& 0.0\t\t\t& 77.2\t\t\t& 29.1\t\t\t& 75.0\t\t\t& 29.6\t\t\t& 61.9\t\t\t& 27.3\t\t\t& 1.4 \t\t\t& 47.5\t\t\t& 82.5\t\t\t& 33.2 \t\t\t\\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\t\\vspace{-0.1cm}\n\t\t\\caption{Validation results on SemanticKITTI dataset}\n\t\t\\label{tab4c}\n\t\t\\vspace{0.1cm}\n\t\\end{subtable}\n\t\n\t\\caption{Semantic segmentation results on sparse point clouds. Metrics are OA(\\%), mIoU(\\%), and per class IoU(\\%).}\n\t\\label{tab4}\n\t\\end{table*}\n\t\n\t\n\t\n\n\\section{Methodology} \\label{Method}\n\n\\subsection{Network Architecture} \\label{sec3.1}\n\nInspired from the findings in Section \\ref{sec2.3}, we propose our network design illustrated in Figure \\ref{fig3}. The overall architecture inherits a dilated structure like PointNet \\cite{qi2017pointnet} or DGCNN \\cite{wang2018dynamic}, which eliminates re-sampling operations end-to-end. The model takes batches of input $N_p$ points, and passes through a sequence of three MNEW modules to extract pointwise features $L_1$, $L_2$, and $L_3$ in a hierarchical order. Note that local neighborhood information is carried inside MNEW, which yields the upgrade from PointNet. We increase the number of neighbors in $L_1$, $L_2$, and $L_3$, which correspond to hierarchical-scale feature encoders but keep the number of query points fixed. A global 2D convolution, max pooling, and two fully convolution layers are followed to aggregate the global feature $G_3$. The hierarchical pointwise features and tiled global feature are concatenated as a descriptor for each point, and passed through three regular 1D convolutions to get the segmentation score, i.e., category-level probability. \n\n\\subsection{Multi-domain Neighborhood Embedding and Weighting} \\label{sec3.2}\n\nThe key component in our network design is the multi-domain neighborhood embedding and weighting (MNEW) module. As shown in Figure \\ref{fig3}, the input of MNEW is batches of points with their xyz coordinates and original features, shaped as $[B, N_p, D_{xyz+fea}]$. We compute pairwise distances in both geometry and feature domain, resulting in two matrices with shape $[B, N_p, N_p]$ representing the geometry distance and feature similarity between every two points. \n\nAs discussed in KP-Conv \\cite{thomas2019kpconv} and RS-Conv \\cite{liu2019relation}, radius-based geometry neighborhood selection is more robust than k-NN \\cite{weinberger2006distance} in the non-uniform sampling settings. However, feature neighborhoods are dynamically shifting and hard to be encircled. Therefore, we use multiple radius in the geometry domain and multiple k-NN in the feature domain to collection multi-scale neighbor indices. We gather their original features to compose two initial neighborhood embedding matrices $\\mathbf{X}_g^0$ and $\\mathbf{X}_f^0$ with shape $[B, N_p, N_{ng}, D_{embed}]$ and $[B, N_p, N_{nf}, D_{embed}]$ respectively, where $ng=\\sum r_i$ represents the number of accumulated neighbors in multi-radius geometry space, and $nf=\\sum k_i$ represents the number of accumulated neighbors in multi-kNN feature space. Similar to the graph embedding $G(V,E)$ in DGCNN and RS-Conv which includes vertices and edges, we embed each element in $\\mathbf{X}_g^0$ and $\\mathbf{X}_f^0$ as,\n\\begin{equation} \\label{eq1}\n\\mathbf{f}(x_{i,n_j}) = \\mathbf{f}(x_{n_j}, x_{n_j}-x_i), \\quad x_{i,n_j} \\in (\\mathbf{X}_g^0 \\cup \\mathbf{X}_f^0)\n\\end{equation}\nwhere $x_i$ denotes the $i$-th query point, and $x_{n_j}$ denotes the $j$-th neighbor point. Given indices of selected neighbors, we also gather their geometry distance matrix $\\mathbf{D}_g$ (shape $[B, N_p, N_{ng}, 1]$) and feature similarity matrix $\\mathbf{D}_f$ (shape $[B, N_p, N_{nf}, 1]$). Next, a transformation function $\\mathbf{T}(d_{i,n_j})=\\mathbf{w}_{i,n_j}\\cdot\\mathbf{f}(d_{i,n_j})$ is utilized to obtain adaptive attention weights. The attended embedding $\\mathbf{X}_g^a$ and $\\mathbf{X}_f^a$ are computed as, \n\\begin{equation} \\label{eq2}\n\\begin{split}\n\\mathbf{X}_g^a = \\mathbf{T}(\\mathbf{D}_g) \\cdot \\mathbf{X}_g^0 \\\\\n\\mathbf{X}_f^a = \\mathbf{T}(\\mathbf{D}_f) \\cdot \\mathbf{X}_f^0\n\\end{split}\n\\end{equation}\n\nMotivated by the density estimation in PointConv \\cite{wu2019pointconv}, we calculate the neighborhood sparsity using,\n\\begin{equation} \\label{eq3}\n\\mathbb{P}(x_{i,n_j}|\\mu, \\sigma^2) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}} \\exp{[-\\frac{(x_{n_j}-x_i)^2}{2\\sigma^2}]}\n\\end{equation}\n\\begin{equation} \\label{eq4}\n\\mathbb{S}(x_i|\\mu, \\sigma^2) = (\\frac{1}{N_n}\\log[\\sum_{n_j \\in N_n} \\mathbb{P}(x_{i,n_j}|\\mu, \\sigma^2)])^{-1}\n\\end{equation}\n$\\mathbb{P}(x_{i,n_j}|\\mu, \\sigma^2)$ in Equation (\\ref{eq3}) is equivalent to the Gaussian probability density function computed for every neighbor $x_{n_j}$ with respect to the query point $x_{i}$. $\\mathbb{S}(x_{i,n_j})$ in Equation (\\ref{eq4}) is the estimated sparsity which inverses the averaged density. We also take the log-scale value to obtain a better sparsity distribution (see Figure \\ref{fig4b}). Geometry sparsity $\\mathbb{S}_g$ and feature sparsity $\\mathbb{S}_f$ are computed individually, which are transformed as the weighting factor for the 2D convolution activation $\\mathbf{h}(x)$. The weighted outputs $\\mathbf{X}_g^w$ (shape $[B, N_p, N_{ng}, D_{conv}]$) and $\\mathbf{X}_f^w$ (shape $[B, N_p, N_{nf}, D_{conv}]$) are computed as,\n\\begin{equation} \\label{eq5}\n\\begin{split}\n\\mathbf{X}_g^w = \\mathbf{T}(\\mathbb{S}_g) \\cdot \\mathbf{h}(\\mathbf{X}_g^a) \\\\\n\\mathbf{X}_f^w = \\mathbf{T}(\\mathbb{S}_f) \\cdot \\mathbf{h}(\\mathbf{X}_f^a)\n\\end{split}\n\\end{equation}\nAfter concatenating the neighborhood information from geometry and feature domain, an average pooling operation is followed to aggregate a feature vector for each query point, yielding the output of MNEW module $\\mathbf{X}_{mnew}^{out}$ with shape $[B, N_p, D_{conv}]$.\n\\begin{equation} \\label{eq6}\n\\mathbf{X}_{mnew}^{out} = \\frac{1}{N_p} \\sum_{i \\in N_p} (\\mathbf{X}_g^w \\oplus \\mathbf{X}_f^w)\n\\end{equation}\n\n\n\\subsection{Loss Function} \\label{sec3.3}\n\nThe loss function is a combination of softmax cross-entropy loss $\\mathcal{L}_{CE}$ and regularization loss $\\mathcal{L}_{Reg}$ adjusted by $\\lambda$. Since the task is semantic segmentation only (i.e., no instance-level labels), the discriminitive loss suggested by ASIS \\cite{wang2019associatively} is not applicable.\n\\begin{equation} \\label{eq7}\n\\mathcal{L}_{Total} = \\mathcal{L}_{CE} + \\lambda \\mathcal{L}_{Reg}\n\\end{equation}\n\n\n\n\n\\subsection{Comparison to Existing Methods} \\label{sec3.4}\n\nReferring to Table \\ref{tab1}, we compare existing methods and summarize the essential distinctions of MNEW as follows.\n\nThe dilated network architecture in our work excludes downsample grouping and upsample interpolation, which differs from all recent works that are based on hierarchical encoder-decoder structures. Compared with PointNet \\cite{qi2017pointnet} whose feature contains pointwise and global information, we also include local neighborhood features. Compared with DGCNN \\cite{wang2018dynamic} which collects neighbors in the feature space only, we embed neighbor points in both geometry and feature domain.\n\nIn terms of the neighborhood embedding, our method adopts multi-scaling in multi-domain. This differs from all existing methods where neighbors are collected in only one single domain (i.e., either geometry or feature). Hierarchical PointNet++ \\cite{qi2017pointnet++} and A-CNN \\cite{komarichev2019cnn} use multi-radius ball-shaped or ring-shaped scales in the geometry domain, while DGCNN using single-scale kNN in the feature domain. In our method, there may exist overlapping points selected in geometry and feature neighborhoods. However, since we compute adaptive attention \\& weighting factors in each domain separately, their impact are learned individually.\n\nFor the attention\/weighting mechanism, KP-FCNN \\cite{thomas2019kpconv} and GACNet \\cite{wang2019graph} compute the geometry distance or feature similarity as fixed weighting factors, while PointConv \\cite{wu2019pointconv} transforms the local sparsity as a learning-based flexible variable. In our method, all these factors are trainable.\n\n\n\n\n\n\n\\section{Experiments} \\label{Experiments}\n\n\\subsection{Sparse Point Cloud Segmentation} \\label{sec4.1}\n\nExperimental results on VirtualKITTI and SemanticKITTI are shown in Table \\ref{tab4}, using the train\/valid splits from Table \\ref{tab2}. Evaluation metrics include OA, mIoU, and per class IoU. We select DGCNN, GACNet, and PointConv as baselines since their accuracies are higher than other related methods (see Table \\ref{tab3}). Our proposed method, MNEW, achieves outstanding performances on both VirtualKITTI and SemanticKITTI. We experimentally set the number of points (i.e., $N_p$) 4096 or 2048, since it affects the global feature extraction and neighborhood selection. Table \\ref{tab4} indicates that MNEW-2048 performs better on VirtualKITTI, while MNEW-4096 is superior on SemanticKITTI. Comparing against the top-performed baseline method, MNEW facilitates 4.5\\% OA and 12.7\\% mIoU increments on VirtualKITTI, a greater margin than those on SemanticKITTI (3.3\\% OA and 5.1\\% mIoU higher than DGCNN). This is because VirtualKITTI provides RGB information as its original feature, which is independent from the XYZ geometry coordinates. For SemanticKITTI, we use intensity, and compute 2D\\_range and 3D\\_distance to fill the input feature slots. Therefore, the multi-domain neighborhood combination is more effective for VirtualKITTI. \n\nSemanticKITTI contains more categorical labels, but the percentage of unlabeled points (class-0) is also higher than VirtualKITTI (4.49\\% vs. $<$0.01\\%). For SemanticKITTI, our experimental results slightly differ from those reported in \\cite{behley2019dataset}. One of the critical reason is that unlabeled points, despite occupying considerable large percentage, are ignored in \\cite{behley2019dataset}. With the class-0 included, we reproduced the RangeNet \\cite{milioto2019rangenet++}, an upgraded DarkNet in \\cite{behley2019dataset}. We provide a consistent comparison as listed in Table \\ref{tab4c}. In addition to DarkNet\/RangeNet, \\cite{behley2019dataset} and \\cite{milioto2019rangenet++} also claim that projection-based methods such as SqueezeSeg \\cite{wu2018squeezeseg} or TangentConv \\cite{tatarchenko2018tangent} are more effective than those directly processing raw data. Since per sweep point clouds are collected by a rotational LiDAR, a cylinder-view projection exactly looks at those points from the perspective of LiDAR sensor. This could ensure its effectiveness for small objects, such as bicycles and traffic-signs which are better viewed from the sensor's perspective. Although reasonable, MNEW still obtains higher mIoU (35.3\\% vs. 34.6\\%) and OA (84.1\\% vs. 81.4\\%) than RangeNet.\n\n\n\n\n\n\n\\begin{table}[t]\n\t\\begin{center}\n\t\n\t\t\\scriptsize\n\t\t\\begin{tabular}{c c c c| c c}\n\t\t\t\\toprule\n\t\t\tGeometry\t\t& Feature\t\t\t& Distance\/Similarity\t& Sparsity\t\t& OA\t\t& mIoU\t\t\\\\\n\t\t\t\\midrule\n\t\t\t\\checkmark\t\t&\t\t\t\t\t&\t\t\t\t\t\t&\t\t\t\t& 92.1\t\t& 57.0\t\t\\\\\n\t\t\t\t\t\t\t& \\checkmark\t\t&\t\t\t \t\t\t&\t\t\t\t& 95.8\t\t& 65.9\t\t\\\\\n\t\t\t\\checkmark\t\t& \\checkmark\t\t&\t\t\t \t\t\t&\t\t\t\t& 95.9\t\t& 69.7\t\t\\\\\n\t\t\t\\midrule\n\t\t\t\\checkmark\t\t& \\checkmark\t\t& \\checkmark \t\t\t&\t\t\t\t& 96.3\t\t& 72.6\t\t\\\\\n\t\t\t\\checkmark\t\t& \\checkmark\t\t&  \t\t\t\t\t\t& \\checkmark\t& 96.9\t\t& 70.6\t\t\\\\\n\t\t\t\\checkmark\t\t& \\checkmark\t\t& \\checkmark \t\t\t& \\checkmark\t& 97.1\t\t& 73.3\t\t\\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\vspace*{-1mm}\n\t\\caption{Effect of neighborhood selection, distance\/similarity attention, and sparsity weighting. Experimented MNEW-2048 on VirtualKITTI}\n\t\\label{tab5}\n\\end{table}\n\n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\begin{subfigure}[t]{\\textwidth}\n\t\t\\centering\n\t\t\\begin{subfigure}[t]{0.4\\textwidth}\n\t\t\t\\includegraphics[width=\\textwidth, height=3.8cm]{img\/Fig4-1_DistanceDistribution.png}\n\t\t\\end{subfigure}\n\t\n\t\n\t\n\t\t\\begin{subfigure}[t]{0.5\\textwidth}\n\t\t\t\\includegraphics[width=\\textwidth, height=3.8cm]{img\/Fig4-2_DistanceEffects.png}\n\t\t\\end{subfigure}\n\t\t\\vspace*{-1mm}\n\t\t\\caption{Distance (m) Distribution and OA (\\%) by Distance Effects}\n\t\t\\label{fig4a}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{\\textwidth}\n\t\t\\centering\n\t\t\\begin{subfigure}[t]{0.4\\textwidth}\n\t\t\t\\includegraphics[width=\\textwidth, height=3.8cm]{img\/Fig4-3_SparsityDistribution.png}\n\t\t\\end{subfigure}\n\t\n\t\n\t\n\t\t\\begin{subfigure}[t]{0.5\\textwidth}\n\t\t\t\\includegraphics[width=\\textwidth, height=3.8cm]{img\/Fig4-4_SparsityEffects.png}\n\t\t\\end{subfigure}\n\t\t\\vspace*{-1mm}\n\t\t\\caption{Sparsity (normalized 0-1) Distribution and OA (\\%) by Sparsity Effects}\n\t\t\\label{fig4b}\n\t\\end{subfigure}\n\t\\caption{Points distribution and performance variation by distance and sparsity. Experimented MNEW-4096 on SemanticKITTI.}\n\t\\label{fig4}\n\\end{figure*}\n\n\n\n\n\\subsection{Ablation Studies} \\label{sec4.2}\n\n{\\bf Neighborhood Embedding and Weighting.}\nIn Table \\ref{tab5}, we compare several variations of the model design. For the neighborhood selection, we compare the neighbors collected by geometry location, feature similarity, and both. The experiment reveals that the accuracy significantly increased with the supplement of feature domain neighborhood. Next, we optionally enable the neighbor attention of geometry distance and feature similarity ($\\mathbf{T}(\\mathbf{D}_g)$+$\\mathbf{T}(\\mathbf{D}_f)$), as well as the weighting of local sparsity ($\\mathbf{T}(\\mathbb{S}_g)$+$\\mathbf{T}(\\mathbb{S}_f)$). Both attention\/weighting settings contribute to the performance improvement, which validate the effectiveness of our MNEW design.\n\n\n{\\bf Performance varied by Distance and Sparsity.}\nSince we target sparse point clouds segmentation towards the application of autonomous driving perception, it is interesting to know the performance for points with different distances (with respect to the LiDAR sensor) or sparsity (with respect to nearby points). Euclidean distances are computed in the geometry space, and sparsity is calculated using the normalized value of Equation (\\ref{eq4}). Investigated on SemanticKITTI, we demonstrate distribution histograms of distance and sparsity, as well as result variations in Figure \\ref{fig4}. We observe in Figure \\ref{fig4a}, the performance occurs an abrupt elevation at $\\approx$50 meters distance. We explicit that points in far away areas (e.g., $>$50m) are limited in quantity (refer to the distance distribution), but generally involved in well-leaned categories (e.g., road, building, terrain) which contain relatively large percentage of samples. In Figure \\ref{fig4b}, as the sparsity increases, the performance starts to decrease until 0.7$\\sim$0.8 and then increase. This is corresponded with the sparsity distribution, implying that the amount of samples affect the performance. Since sparsity distributions for dense datasets are relatively uniform, we infer it an essential reason for the effectiveness disparity of existing methods. It is also observed from Figure \\ref{fig4} that MNEW achieves winning performance against other methods across the distance and sparsity distribution.\n\n\n\n\\section{Conclusion} \\label{Conclusion}\n\nIn this work, we propose MNEW for sparse point clouds segmentation. MNEW inherits a dilation architecture to capture pointwise and global features, and involves multi-scale local semantics adopted from the hierarchical encoder-decoder structure. Neighborhood information is embedded in both static geometry and dynamic feature domain. The geometry distance, feature similarity, and local sparsity are computed and transformed as adaptive weighting factors. We obtain outstanding performances on both sparse point clouds. We believe this study will contribute to the application of LiDAR-based autonomous driving perception. \n\nIn the continued work, one direction is to extend the model for joint semantic and instance segmentation, i.e., panoptic segmentation \\cite{kirillov2019panoptic} for point clouds. The second ongoing direction is domain adaption to resolve the issue of cross-sensor disparity. The individual model could also be light-weighted and exported version as a LiDAR feature extractor, which is useful to fuse with other sensors such as radar and camera \\cite{zheng2019gfd}.\n\n\n\n\n{\\small\n\t\\bibliographystyle{ieee_fullname}\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:Introduction}\nIn the field of illumination optics, optical engineers design optical elements to transport the light from a source, which can be an LED, laser, or incandescent lamp, to obtain a desired irradiance (spatial density of the luminous flux) or intensity (angular density of the luminous flux) \\citep{Grant2011}.\nTo transport the light from the source to the target, the optical engineer can construct a system consisting of various optical elements such as lenses, mirrors, diffusers, and light guides \\citep{John2013}.\nOne particular type of optic used in automotive and road lighting applications is the freeform lens, a lens without any form of symmetry \\citep{Falaggis2022, Mohedano2016}.\nThe design of these lenses is a complex problem. It is currently solved by numerically solving system-specific differential equations or through optimization, with every step validated using a (non-differentiable) non-sequential ray tracer \\citep{Wu2018}.\nGreat effort is involved in generalizing these methods to account for varying amounts of optical surfaces \\citep{Anthonissen2021}, their optical surface and volume properties \\citep{Kronberg2022, lippman_prescribed_2020}, or the source model~\\citep{Muschaweck2022, tukker_efficient_2007, sorgato_design_2019}. \n\nThe performance of an optical system is evaluated using ray tracing, which is the process of calculating the path of a ray originating from a source through the optical system. Sequential ray tracers such as Zemax~\\citep{zemax} and Code V~\\citep{codev}, primarily used in the design of imaging optics, trace a small number of rays to determine the quality of the image. \nNon-sequential ray tracers such as LightTools~\\citep{LightTools} and Photopia~\\citep{photopia} use many rays to simulate the optical flux through the system and share similarities with the rendering procedures in computer graphics, with the main difference being that the rays are traced from source to camera.\n\nAlgorithmically differentiable ray tracing, a generalization of differential ray tracing \\citep{feder_differentiation_1968, stone_differential_1997, oertmann_differential_1989, chen_second-order_2012}, is a tool that is being developed for both sequential~\\citep{Sun2021, Volatier2017}  and non-sequential \\citep{Mitsuba2} ray tracing.\n\\emph{Differential ray tracing} obtains system parameter gradients using numerical or algebraic differentiation. The gradient can be calculated numerically using numerical differentiation or the adjoint method \\citep{givoli_tutorial_2021}, requiring the system to be ray traced twice, once for its current state and once with perturbed system parameters. Analytic expressions for the gradient can be obtained by tracing the rays analytically through the system, calculating where the ray intersects the system's surfaces and how the ray's trajectory is altered. However, these expressions can become long and complicated depending on the system. In addition, the method is limited to optics described by conics as finding analytic ray surface intersection with surfaces of higher degrees becomes complicated or even impossible. Algorithmic differentiable ray tracing can handle these issues by obtaining the gradients with one single forward simulation for an almost arbitrary system. In addition, it can be seamlessly integrated into gradient-descent-based optimization pipelines. A modern framework for this is \\emph{Physics Informed Machine Learning} \\citep{Karniadakis2021}, where a neural network is trained to approximate the solution to a physics problem formulated using data, a set of differential equations, or an implemented physics simulation (or a combination of these). \n\nWe investigate the reliability of designing freeform lenses with B-spline surfaces \\citep{Piegl1995} using algorithmically differentiable non-sequential ray tracing and gradient-based optimization to redirect the light of a light source into a prescribed irradiance distribution. The source models will be the collimated light source, point source, and finally, sources with a finite extent. The results are validated using the commercial ray trace program LightTools \\citep{LightTools}. In addition, we investigate the effectiveness of optimizing a network to determine the optimal B-spline control points as proposed in \\citep{Moller2021PIML} and \\citep{GASICK2023115839}, and compare it to optimizing the control points directly and seeing the possible speed-up.\n\n\\section{Gradient-based freeform design}\nThe overall structure of our pipeline is depicted in Fig.~\\ref{fig:optimizationloop}. \nA freeform surface is defined by the parameters $P \\in \\mathscr{P}$, where $\\mathscr{P}$ is the set of permissible parameter values. This surface is combined with a flat surface to create a lens, and an irradiance distribution $\\mathcal{I}$ is produced by tracing rays through the lens onto a screen. The irradiance distribution is compared to a target $\\mathcal{I}_\\text{ref}$ yielding a loss $\\mathscr{L}(\\mathbf{P};\\mathcal{I}_\\text{ref})$. The optimization problem we are trying to solve can then be formulated as\n\\begin{equation}\n    \\min_\\mathbf{P \\in \\mathscr{P}} \\; \\mathscr{L}(\\mathbf{P};\\mathcal{I}_\\text{ref}),\n\\end{equation}\nwhich we solve by using gradient descent.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = \\textwidth]{figures\/Caustic_design_optimization.png}\n    \\caption{Overview of our learning-based freeform design pipeline.}\n    \\label{fig:optimizationloop}\n\\end{figure}\n\nThe freeform surface of the lens is defined in terms of a B-spline surface. From a manufacturing standpoint, this is convenient since B-spline surfaces can be chosen to be $C^1$ smooth (in fact, B-spline surfaces can be $C^n$ smooth for arbitrarily large $n$). From an optimization perspective, B-spline surfaces have the property that the control points that govern the shape of the surface and which will be optimized have a local influence on the surface geometry, which in turn has a local influence on the resulting irradiance distribution.\n\n\\subsection{The lens model using a B-spline surface}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.2\\textwidth]{figures\/lens_schematic.png}\n    \\caption{The used lens type: a volume enclosed between a flat surface and a freeform surface with a uniform refractive index.}\n    \\label{fig:lenschematic}\n\\end{figure}\n\nWe define a lens as in Fig.~\\ref{fig:lenschematic} as the volume between a flat surface and a B-spline surface, with a uniform refractive index.\n\nA B-spline surface $\\mathbf{S}$ in $\\mathbb{R}^3$ is a parametric surface, see Fig.~\\ref{fig:Bsplinesurface}. It has rectangular support $[a,b]\\times[c,d]$ where $a<b$ and $c<d$. It is defined as a linear combination of an $(n_1+1) \\times (n_2+1)$ grid of control points $\\mathbf{P}_{i,j} \\in \\mathbb{R}^3$ where $n_1$ and $n_2$ are positive integers which define the size of the control net $\\{\\mathbf{P}_{i,j} :  0 \\leq i \\leq n_1, \\; 0 \\leq j \\leq n_2\\}$. This linear combination is defined in terms of univariate B-spline basis functions $N_{i,p}$ and $N_{j,q}$, which are recursively defined as follows by the Cox-de Boor formula \\cite[eq. 2.5]{Piegl1995}:\n\\begin{align}\n    & N_{i,0}(u) =\n    \\begin{cases}\n        1 \\text{ if } u_i \\le u < u_{i+1} \\\\\n        0 \\text{ otherwise}\n    \\end{cases} \\nonumber \\\\\n    & N_{i,p}(u) = \\frac{u-u_i}{u_{i+p}-u_i}N_{i,p-1}(u) + \\frac{u_{i+p+1}-u}{u_{i+p+1}}N_{i+1,p-1}(u),\n\\end{align}\nfor $i=0,\\ldots,n_1$ and similarly for $N_{j,q}$ with $j=0,\\ldots,n_2$. The basis functions are $p$-degree piece-wise polynomials. The knots $u_i$ are non-decreasing real numbers in $[a,b]$ collected in a knot vector\n\\cite[eq. 2.13]{Piegl1995}:\n\\begin{equation}\n    \\mathcal{V} = (\\underbrace{0,\\ldots,0}_{p+1},u_{p+1},\\ldots,u_{r-p-1},\\underbrace{1,\\ldots,1}_{p+1}) \\in \\mathbb{R}^{r+1}.\n\\end{equation}\n Here we assume $a=0$ and $b=1$. For the number of knots $r+1$, the number of control points $n_1+1$ and the degree $p$, there is the relationship $n_1 = r-p-1$. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.5\\textwidth]{figures\/BsplineSurface.png}\n    \\caption{A B-spline surface of degrees $(p,q) = (3,3)$ with indicated directions of the $u$ and $v$ parameters. The control points are shown in black.}\n    \\label{fig:Bsplinesurface}\n\\end{figure}\n\nAn interval $[u_i, u_{i+1})$ of consecutive knots is called a knot span, on which the basis functions are analytic. At the knots, the basis functions are $p-k$ times continuously differentiable, with $k$ being the multiplicity of the knot, that is, how many times the same knot is repeated in the knot vector. The specific type of knot vector here with the multiplicity of $p+1$ at the first and last knots is called clamped or open, which yields the property\n\\begin{equation}\n    N_{0,p}(0) = N_{n_1,p}(1) = 1.\n\\end{equation}\nAll other basis functions are $0$ at $u=0,1$, which means that the boundary control points determine the boundary of the surface.  \n\n\\noindent The interior knots are chosen to be equispaced, \\emph{i.e.}\n\\begin{equation}\n    u_i = \\frac{i-p}{n_1-p+1}, \\qquad i = p,\\ldots, n_1+1.\n\\end{equation}\nThus there are no knots with multiplicity larger than $1$ apart from those at the boundary. So if $p,q \\ge 2$, then the surface is at least $C^1$, and the gradient of the B-spline surface is defined everywhere.\n\nCompletely analogously, there is the knot vector $\\mathcal{W}$ with the knots $v_j$ for the basis functions $N_{j,q}$  of degree $q$, for $j=0,\\ldots,n_2$. The definition of the surface is then \\cite[eq. 3.11]{Piegl1995}\n\\begin{equation}\n    \\mathbf{S}(u,v) = \\sum_{i=0}^{n_1}\\sum_{j=0}^{n_2} N_{i,p}(u)N_{j,q}(v)\\mathbf{P}_{i,j}, \\quad (u,v) \\in [0,1]^2. \\label{eq:Bsplinesurfacedef}\n\\end{equation}\n\n\\subsubsection{Linearizing the B-spline parametrizations}\nThe volume $V \\subset \\mathbb{R}^3$ of the modeled lens has a rectangular extent  $[-r_x,r_x] \\times [-r_y,r_y]$ in the ($x,y$)-plane with $2r_x$ and $2r_y$ being the width and height of the lens, respectively. The lens volume is enclosed on one side by a B-spline surface\n\\begin{equation}\n    \\mathbf{S}(u,v) = (X(u,v), Y(u,v), Z(u,v)), \\label{eq:coordfuncs}\n\\end{equation}\nwhere $X,Y,Z$ are the individual coordinate parameterizations, for instance\n\\begin{subequations}\n    \\begin{align}\n        X(u,v) = \\sum_{i=0}^{n_1}\\sum_{j=0}^{n_2} N_{i,p}(u)N_{j,q}(v)P^x_{i,j}, \\\\\n        Y(u,v) = \\sum_{i=0}^{n_1}\\sum_{j=0}^{n_2} N_{i,p}(u)N_{j,q}(v)P^x_{i,j}.\n    \\end{align}\n\\end{subequations}\nFor simplicity of calculations on ray-sampling and ray-intersection (Section \\ref{subsec:raytracer}), it is helpful to define the mapping $(u,v) \\mapsto (X(u,v),Y(u,v))$ in a way that it is analytically invertible. Therefore the coordinates of the control points are chosen such that the parametrizations $X$ and $Y$ are linear:\n\\begin{subequations}\n    \\begin{align}\n        X &: u \\mapsto (2u-1)r_x \\in [-r_x,r_x], \\label{eq:linearX}\\\\\n        Y &: v \\mapsto (2v-1)r_y \\in [-r_y,r_y]. \\label{eq:linearY}\n    \\end{align}\n\\end{subequations}\nIn general, $X$ and $Y$ are degree $p$ and $q$ piece-wise polynomials, respectively, and thus not linear. Linearity can be achieved by making use of the nodal representation of the B-spline \n basis functions \\cite[eq. 23]{Cohen2010Iso}:\n\\begin{equation}\n    u = \\sum_{i=0}^{n_1} u_{i,p}^* N_{i,p}(u), \\quad u \\in [0,1], \\quad u_{i,p}^* = \\frac{u_{i+1}+\\ldots+u_{i+p}}{p}, \\label{eq:nodalrepresentation}\n\\end{equation}\nwhich provides a specific knot vector-dependent linear combination of the basis functions that yields the identity function on the domain $[0,1]$. The values $u_{i,p}^*$ are called the Greville abscissae \\cite[sec. 8.6]{Farin2002}.\n\nWe assume that the $P^x_{i,j}$ are independent of $j$, and choose $j=0$ as a representative.  Then we obtain by the definition of $X$:\n\\begin{subequations}\n    \\begin{align}\n        X(u,v) &= \\sum_{i=0}^{n_1}\\sum_{j=0}^{n_2} P^x_{i,j}N_{i,p}(u)N_{j,q}(v) \\\\\n        &= \\sum_{i=0}^{n_1} P^x_{i,0}N_{i,p}(u) \\underbrace{\\sum_{j=0}^{n_2} N_{j,q}(v)}_{=1},\n    \\end{align}\n\\end{subequations}\nwhere $\\sum_{i=0}^n N_{i,p}(u) = 1$ by the partition of unity property of the basis functions \\cite[P2.4]{Piegl1995}. Now we see that if we define $P^x_{i,j} := u^*_{i,p}$ then $X(u) = u$. Thus if we apply the mapping $u \\mapsto (2u-1)r_x$ to both sides of eq. ~\\ref{eq:nodalrepresentation}, we obtain\n\\begin{equation}\n    (2u-1)r_x = \n    \\sum_{i=0}^{n_1} (2u^*_{i,p}-1)r_x N_{i,p}(u). \\label{eq:lineartaucontrolpoints}\n\\end{equation}\nThis equality can be understood by expanding the $1$ into the sum over all $N_{i,p}(u)$ by again exploiting the partition of unity property. Thus if we define $P^x_{i,j} := (2u^*_{i,p}-1)r_x$ and equivalently $P^y_{i,j} := (2v^*_{j,q}-1)r_y$, then Eqs.~\\ref{eq:linearX} and \\ref{eq:linearY} and are satisfied.\n\nThe lens is then defined as the volume in $\\mathbb{R}^3$ enclosed by the B-spline surface $\\mathbf{S}$ and the flat surface given by $z=z_\\text{in}$ on $[-r_x,r_x] \\times [-r_y,r_y]$:\n\\begin{equation}\n    V = \\left\\{ (x,y,z) \\in \\mathbb{R}^3 \\; \\mid \\;\n    z_\\text{in} \\le z \\le \n    Z\\left(X^{-1}(x), Y^{-1}(y)\\right), \\vert x\\vert\\le r_x, \\vert y\\vert\\le r_y\\right\\}.\n\\end{equation}\nFor the arguments of $Z(u,v)$ the inverses of $X$ and $Y$ are used:\n\\begin{equation}\n    X^{-1}(x) = \\frac{1}{2}\\left(\\frac{x}{r_x}+1\\right), \\quad\n    Y^{-1}(y) = \\frac{1}{2}\\left(\\frac{y}{r_y}+1\\right).\n\\end{equation}\n\n\\subsubsection{Lens constraints} \\label{subsubsec:lensconstraints}\nTo let the lens be well-defined the surfaces of the lens should not intersect:\n\\begin{equation}\n    z_\\text{in} < Z(u,v), \\quad(u,v) \\in [0,1]^2.\n\\end{equation}\nBy the convex hull property of B-spline surfaces \\cite[P3.22]{Piegl1995} it suffices that\n\\begin{equation}\n    P^z_{i,j} > z_\\text{in} \\quad \\forall (i,j). \\label{eq:nosurfintersect}\n\\end{equation}\nManufacturing can require that the lens has some minimal thickness $\\delta$, so that the constraint is stronger:\n\\begin{equation}\n    P^z_{i,j} \\ge \\delta + z_\\text{in} \\quad \\forall (i,j).\n\\end{equation}\n\n\n\\subsection{Differentiable ray tracer} \\label{subsec:raytracer}\nOur implementation traces rays from a source through the flat lens surface and the freeform lens surface to the detector screen as depicted in Figs.~\\ref{fig:planetraceschematic} and \\ref{fig:pointtraceschematic}. Other ray paths, e.g., total internal reflection at lens surfaces, are not considered since it is assumed that the contribution of these to the resulting irradiance distribution is negligible.\n\n\\subsubsection{Sources and ray-sampling}\nNon-sequential ray tracing is a Monte-Carlo approximation method of the solution to the continuous integration formulation of light transport through an optical system. For a detailed discussion of this topic, see \\cite[ch. 14]{pharr2016physically}. Thus to perform ray tracing, the light emitted by a source must be discretized into a finite set of rays\n\\begin{equation}\n    l: t \\rightarrow \\mathbf{o} + \\hat{\\mathbf{d}}t,\n\\end{equation}\nwhere $\\mathbf{o}$ is the origin of the ray and $\\hat{\\mathbf{d}}$ its normalized direction vector. Both collimated ray bundle and point sources will be considered, see Figs.~\\ref{fig:planetraceschematic} and \\ref{fig:pointtraceschematic}, respectively. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.6\\textwidth]{figures\/plane_trace_schematic.png}\n    \\caption{Schematic of the ray tracing with a collimated ray bundle source.}\n    \\label{fig:planetraceschematic}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.6\\textwidth]{figures\/point_trace_schematic.png}\n    \\caption{Schematic of the ray tracing with a point source.}\n    \\label{fig:pointtraceschematic}\n\\end{figure}\n\nTracing rays from a collimated ray bundle can be understood from Fig.~\\ref{fig:planetraceschematic}. The path of all rays from the source plane to the B-spline surface is a line segment parallel to the $z$-axis. Therefore, we can sample the incoming rays directly on the B-spline surface, with $\\hat{\\mathbf{d}} = (0,0,1)^\\top$. By the linearity of $X$ and $Y$ sampling on the B-spline domain $[0,1]^2$ is analogous to sampling on the lens extent $[-r_x,r_x]\\times [-r_y,r_y]$ in terms of distribution. Rays are sampled in a (deterministic) square grid on $[0,1]^2$. \n\nFor a point source, each ray starts at the location of the source, and the direction vector $\\hat{\\mathbf{d}}$ is sampled over the unit sphere $\\mathbb{S}^2$. More precisely, $\\hat{\\mathbf{d}}$ is given by\n\\begin{equation}\n    \\hat{\\mathbf{d}} = \\left(\\cos\\theta\\sin\\phi,\\sin\\theta\\sin\\phi,\\cos\\phi\\right)^\\top,\n\\end{equation}\nwith $\\theta \\in [0,2\\pi)$ and $\\phi \\in [0,\\phi_\\text{max}]$ for some $0\\le \\phi_\\text{max} < \\frac{\\pi}{2}$, see Fig.~\\ref{fig:pointtraceschematic}. $\\phi_\\text{max}$ is chosen as small enough to minimize the number of rays that miss the lens entrance surface but large enough such that the whole surface is illuminated. For instance, if the source is on the $z$-axis, then $\\phi_\\text{max} = \\arctan\\left(\\frac{\\sqrt{r_x^2 + r_y^2}}{z_\\text{in}-z_s}\\right)$ where $z_\\text{in}$ is the $z$-coordinate location of the entrance surface and $z_s$ the $z$-coordinate of the source. To uniformly sample points on this sphere segment, $\\theta$ is sampled (non-deterministically) uniformly in $[0,2\\pi)$ and $\\phi$ is given by\n\\begin{equation}\n    \\phi = \\arccos\\left(1-(1-\\cos\\phi_\\text{max})a\\right)\n\\end{equation}\nwhere $a$ is sampled (non-deterministically) uniformly in $[0,1]$. This sampling is used to produce the results in Section~\\ref{sec:results}.\n\nFor the point source, the calculation of the intersection of a ray with the B-spline surface is non-trivial. This calculation comes down to finding the smallest positive root of the $p+q$ degree piece-wise polynomial function\n\\begin{equation}\n    f(t) = \n    Z\\left(\\begin{pmatrix}o_u \\\\ o_v\\end{pmatrix} + \n    \\begin{pmatrix}d_u \\\\ d_v\\end{pmatrix}t\n    \\right)\n     - d_zt - o_z, \\label{eq:surfaceintersect}\n\\end{equation}\nif such a root exists and yields a point in the domain of $Z$.  Here the subscripts $u$ and $v$ denote that the ray is considered in $(u,v,z)$ space instead of $(x,y,z)$ space, so for instance\n\\begin{equation}\n    o_u = X^{-1}(o_x) = \\frac{1}{2}\\left(\\frac{o_x}{r_y}+1\\right), \\quad d_v = \\frac{d_y}{2r_y}.\n\\end{equation}\nThe roots of eq. \\ref{eq:surfaceintersect} cannot generally be found analytically for $p+q>4$, and thus an intersection algorithm is implemented, which is explained in the next section.\n\n\\subsubsection{B-spline surface intersection algorithm}\nThe intersection algorithm is based on constructing a triangle mesh approximation of the B-spline surface and computing intersections with that mesh.\n\n\\paragraph{Triangle mesh intersection phase 1: bounding boxes}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{figures\/bb_per_knotspanproduct.png}\n    \\caption{Triangles and corresponding bounding box for a few knot span products of a spherical surface.}\n    \\label{fig:bb_per_knotspanproduct}\n\\end{figure}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{figures\/uv_triangle_check.png}\n    \\caption{Example of which triangles are candidates for a ray-surface intersection with the ray plotted in red, based on their $u,v$-domain.}\n    \\label{fig:uvtriangleint}\n\\end{figure}\nChecking every ray against every triangle for intersection is computationally expensive, so it is helpful to have bounding box tests that provide rough information about whether the ray is even near some section of the B-spline surface. B-spline theory provides a tool for this: the strong convex hull property, which yields the bounding box \n\n\\begin{equation}\n    B_{i_0,j_0} = \\left[u_{i_0},u_{i_0 + 1}\\right) \\times \\left[v_{j_0},v_{j_0 + 1}\\right) \\times \\left[z^{\\min}_{i_0,j_0}, z^{\\max}_{i_0,j_0}\\right]\n\\end{equation}\nwhere $z^{\\min}_{i,j}$ and $z^{\\max}_{i,j}$ are the minimum and maximum $z$-values of the control points that affect the B-spline surface on the knot span product $\\left[u_{i_0},u_{i_0 + 1}\\right) \\times \\left[v_{j_0},u_{j_0 + 1}\\right)$, hence those with indices $i_0-p\\le i\\le i_0, j_0-q \\le j\\le j_0$. Formulated in terms of $Z(u,v)$ this yields\n\\begin{equation}\n    z^{\\min}_{i_0,j_0} \\le Z(u,v) \\le z^{\\max}_{i_0,j_0}, \\quad\n    (u,v) \\in \\left[u_{i_0},u_{i_0 + 1}\\right) \\times \\left[v_{j_0},v_{j_0 + 1}\\right).\n\\end{equation}\nExamples of such bounding boxes are shown in Fig.~\\ref{fig:bb_per_knotspanproduct}.\n\nThere are two steps in applying the bounding boxes in the intersection algorithm. First, a test for the entire surface (in $(u,v,z)$-space):\n\\begin{equation}\n    [0,1]^2 \\times \\left[\\min_{i,j}P^z_{i,j},\\max_{i,j}P^z_{i,j}\\right].\n\\end{equation}\nSecond, a recursive method where, starting with all knot span products, each rectangle of knot span products is divided into at most 4 sub-rectangles for a new bounding box test until individual knot span products are reached.\n\n\n\\paragraph{Triangle mesh intersection phase 2: $(u,v)$-space triangle intersection}\nEach non-trivial knot span product $[u_{i_0},u_{i_0+1}) \\times [v_{j_0},v_{j_0+1})$ is divided into a grid of $n_u$ by $n_v$ rectangles. Thus we can define the boundary points\n\\begin{subequations}\n    \\begin{align}\n         u_{i_0,k} =& u_{i_0} + k\\Delta u_{i_0},\n         \\quad \\Delta u_{i_0} = \\frac{u_{i_0+1}-u_{i_0}}{n_u}, \n         \\quad k = 0, \\ldots, n_u, \\\\\n         v_{i_0,\\ell} =&  v_{j_0}+ \\ell \\Delta v_{j_0}, \\quad \\Delta v_{j_0} = \\frac{v_{j_0+1}-v_{j_0}}{n_v},\n         \\quad \\ell = 0,\\ldots, n_v.\n     \\end{align}\n\\end{subequations}\n\nEach rectangle is divided into a lower left and an upper right triangle, as demonstrated in Fig.~\\ref{fig:uvtriangleint}. In this figure it is shown for a ray projected onto the $(u,v)$-plane in some knot span which triangles are candidates for an intersection in $(u,v,z)$-space. This is determined by the following rules:\n\\begin{itemize}\n    \\item A lower left triangle is intersected in the $(u,v)$-plane if either its left or lower boundary is intersected by the ray;\n    \\item an upper right triangle is intersected in the $(u,v)$-plane if either its right or upper boundary is intersected by the ray.\n\\end{itemize}\n\nThe intersection of these boundaries can be determined by finding the indices of the horizontal lines at which the vertical lines are intersected:\n\\begin{equation}\n    \\ell_k = \\left\\lfloor\\frac{ o_v+(u_{i_0,k}-o_u)\\frac{d_v}{d_u} - v_{j_0}}{\\Delta v_{j_0}}\\right\\rfloor,\n\\end{equation}\nand analogously $k_\\ell$.\n\n\\paragraph{Triangle mesh intersection phase 3: $u,v,z$-space triangle intersection}\nA lower left triangle can be expressed by a plane\n\\begin{equation}\n    T(u,v) = Au + Bv + C\n\\end{equation}\n    defined by the following linear system:\n\\begin{equation}\n    \\begin{pmatrix}\n    u_{i_0,k} & v_{j_0,\\ell} & 1 \\\\\n    u_{i_0,k+1} & v_{j_0,\\ell} & 1 \\\\\n    u_{i_0,k} & v_{j_0,\\ell+1} & 1\n    \\end{pmatrix}\n    \\begin{pmatrix}\n    A \\\\ B \\\\ C\n    \\end{pmatrix}\n    =\n    \\begin{pmatrix}[1.75]\n    z_{i_0,k}^{j_0,\\ell} \\\\ z_{i_0,k+1}^{j_0,\\ell} \\\\ z_{i_0,k}^{j_0,\\ell+1}\n    \\end{pmatrix}.\n\\end{equation}\nHere we use the following definition:\n\\begin{equation}\n    z_{i_0,k}^{j_0,\\ell} = Z(u_{i_0,k},v_{j_0,\\ell}).\n\\end{equation}\nThis yields the plane\n\\begin{align}\n    T(u,v) =&& z_{i_0,k}^{j_0,\\ell} + n_u\\left(z_{i_0,k+1}^{j_0,\\ell}-z_{i_0,k}^{j_0,\\ell}\\right)\\frac{u-u_{i_0,k}}{u_{i_0+1}-u_{i_0}} \\\\ \n    && +n_v\\left(z_{i_0,k}^{j_0,\\ell+1}-z_{i_0,k}^{j_0,\\ell}\\right)\\frac{v-v_{j_0,\\ell}}{v_{j_0+1}-v_{j_0}}. \\label{eq:trianglefuncdetermined}\n\\end{align}\nNote that to define this triangle, the B-spline basis functions are evaluated at fixed points in $[0,1]^2$ independent of the rays or the $P^z_{i,j}$. This means that for a lens that will be optimized these basis function values can be evaluated and stored only once rather than in every iteration, for computational efficiency.\n\nComputing the intersection with the ray $\\tilde{\\mathbf{r}}(t) = \\tilde{\\mathbf{o}} + \\tilde{\\hat{\\mathbf{d}}}t$ is now straight-forward, and yields\n\\begin{equation}\n    t_\\text{int} = - \\frac{C+\\langle\\tilde{\\mathbf{o}}, \\mathbf{n}\\rangle}{\\langle\\tilde{\\hat{\\mathbf{d}}},\\mathbf{n}\\rangle}, \\quad \\mathbf{n} = \n    \\begin{pmatrix} 0 \\\\ 1 \\\\ \\partial_u T\\end{pmatrix} \\times\n    \\begin{pmatrix} 1 \\\\ 0 \\\\ \\partial_v T\\end{pmatrix} = \n    \\begin{pmatrix}A \\\\ B \\\\ -1\\end{pmatrix},\n\\end{equation}\nwhere $\\mathbf{n}$ is a normal vector to the triangle, computed using the cross product. This also explains why $\\langle\\tilde{\\hat{\\mathbf{d}}},\\mathbf{n}\\rangle=0$ does not yield a well-defined result: in this situation the ray is parallel to the triangle.\n\nThe last thing to check is whether $\\tilde{l}(t_\\text{int})$ lies in the $(u,v)$-domain of the triangle, which can be checked by three inequalities for the three boundaries of the triangle:\n\n\\begin{subequations}\n    \\begin{align}\n        o_u + d_u t_\\text{int} \\ge u_{i_0,k} \\\\\n        0 \\leq o_v + d_v t_\\text{int} - v_{j_0,\\ell}< \\frac{n_u}{n_v}\\frac{v_{j_0+1}-v_{j_0}}{u_{i_0+1}-u_{i_0}}(u_{i_0,k+1}-(o_u + d_u t_\\text{int})).\n    \\end{align}\n\\end{subequations}\n\nThe computation for an upper right triangle is completely analogous. The upper triangle has a closed boundary, whereas the lower triangle has an open one and vice versa, which means that the $(u,v)$ domains of the triangles form an exact partition of $[0,1]^2$. Thus the triangle mesh is `water-tight', meaning that no ray intersection should be lost by rays passing in between triangles.\n\n\\subsection{Image reconstruction}\nThe ray tracing produces an irradiance distribution in the form of an image matrix $\\mathcal{I} \\in \\mathbb{R}^{n_x \\times n_y}_{\\ge 0}$, where the elements correspond to a grid of rectangles called pixels that partition the detector screen positioned at $z=z_\\text{screen} > \\max_{i,j} P_{i,j}^z$. The screen resolution $(n_x,n_y)$ and the screen radii $(R_x,R_y)$ together yield the pixel size\n\\begin{equation}\n    (w_x,w_y) = \\left(\\frac{2R_x}{n_x},\\frac{2R_y}{n_y}\\right).\n\\end{equation}\nFor reasons explained later in this section, sometimes a few `ghost pixels' are added, so the effective screen radii are\n\\begin{equation}\n    R_x^* := R_x + \\frac{\\nu_x - 1}{2}w_x, \\quad\n    R_y^* := R_y + \\frac{\\nu_y - 1}{2}w_y,\n\\end{equation}\nand the effective screen resolution is $(n_x + \\nu_x -1, n_y + \\nu_y - 1)$ where $\\nu_x$ and $\\nu_y$ are odd positive integers whose meaning will become clear later in this section.\n\n\nProducing the irradiance distribution from the rays that intersect the detector screen is called image reconstruction \\cite[sec. 7.8]{pharr2016physically}. The way that a ray contributes to a pixel with indices $i,j$ is governed by a reconstruction filter\n\\begin{equation}\n    F_{i,j} : [-R_x,R_x] \\times [-R_y,R_y] \\rightarrow \\mathbb{R}_{\\ge 0},\n\\end{equation}\nyielding for the irradiance distribution\n\\begin{equation}\n    \\mathcal{I}_{i,j} = \\sum_{k=1}^N \\omega_k F_{i,j}(\\mathbf{x}_k),\n\\end{equation}\nfor a set of ray intersections $\\{\\mathbf{x}_k\\}_{k=1}^N$ with corresponding final ray weights $\\{\\omega_k\\}_{k=1}^N$. The ray weights are initialized at the sampling of the ray at the source. They are slightly modified by the lens boundary interactions as a small portion of the light is reflected rather than refracted. The amount by which the ray weights are modified is governed by the Fresnel equations \\cite[sec. 2.7.1]{Fowles1975}. In our implementation, the Fresnel equations are approximated by Schlick's approximation \\cite[eq. 24]{Schlick1994}. In the current implementation, all ray weights are initialized equally. The precise value does not matter since the relationship between the initial and final weights is linear. The loss function (section \\ref{lossfunc}) compares scaled versions of the produced and target irradiance distribution.\n\nIn the simplest reconstruction case, the value of a pixel is given by the sum of the weights of the rays that intersect the detector screen at that pixel (called box reconstruction in \\cite[sec. 7.8.1]{pharr2016physically}). In this case the reconstruction filter of pixel $i,j$ is simply the indicator function of the pixel $\\left[(i-1)w_x,iw_x\\right) \\times \\left[(j-1)w_y,jw_y\\right)$.\n\nTo obtain a ray tracing implementation where the irradiance $\\mathcal{I}$ is differentiable with respect to geometry parameters of the lens, say, the parameter $\\theta$, the irradiance distribution must vary smoothly with this parameter. The dependency on this parameter is carried from the lens to the screen by the rays through the screen intersections $\\mathbf{x}_k = \\mathbf{x}_k(\\theta)$. Thus to obtain a useful gradient $\\frac{\\partial \\mathcal{I}}{\\partial \\theta}$ the filter function $F_{i,j}$ should be at least $C^1$, see Fig.~\\ref{fig:reconstructiondiffb} which is achieved by introducing a filter function that spreads out the contribution of a ray over a kernel of pixels of size $(\\nu_x,\\nu_y)$ centered at the intersection location. For the conservation of light, we require that $\\sum_{i,j}F_{i,j}(\\mathbf{x}) \\equiv 1$.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.7\\textwidth]{figures\/reconstruction_diffb.png}\n    \\caption{$\\mathbf{x}(\\theta)$ in the left plot shows the intersection location of a ray with the screen, dependent on a lens geometry parameter $\\theta$. The right plot then shows the reconstruction filter value for the green pixel in the left plot dependent on $\\theta$. In order to obtain a useful gradient of the pixel value with respect to $\\theta$, a smooth reconstruction filter is needed.}\n    \\label{fig:reconstructiondiffb}\n\\end{figure}\n\nTherefore, the Gaussian reconstruction function is introduced, based on the identically named one described in \\cite[sec. 7.8.1]{pharr2016physically}. This filter function is based on the product \n\\begin{equation}\n    \\tilde{F}_{i,j}(x,y;\\alpha,\\nu_x,\\nu_y) := f_{i}^x(x;\\alpha,\\nu_x)f_{j}^y(y;\\alpha,\\nu_y),\n\\end{equation}\nwhere\n\\begin{equation}\n    f_{i_0}^x(x;\\alpha,\\nu_x) = \n    \\begin{cases}\n        e^{-\\alpha\\left(x-c^x_{i_0}\\right)^2} - e^{-\\alpha\\left(\\frac{\\nu_x w_x}{2}\\right)^2} &\\text{ if } \\lvert x-c^x_i\\rvert < \\frac{\\nu_x w_x}{2},\\\\\n        0 & \\text{otherwise.}\n    \\end{cases} \\label{eq:filter1dim}\n\\end{equation}\nThe centers of the pixels are given by\n\\begin{equation}\n    (c_i^x,c_j^y) := \\left(\\left(i + \\textstyle\\frac{1}{2}\\right)w_x - R_x, \\left(j + \\textstyle\\frac{1}{2}\\right)w_y - R_y\\right).\n\\end{equation}\nNote that the support of $\\tilde{F}_{i,j}$ is of size $\\nu_xw_x$ by $\\nu_yw_y$, the size of the kernel on the detector screen. The normalized reconstruction filter is then given by\n\\begin{equation}\n    F_{i,j}(x,y;\\alpha,\\nu_x,\\nu_y) = \\frac{\\tilde{F}_{i,j}(x,y;\\alpha,\\nu_x,\\nu_y)}{\\sum_{i',j'}\\tilde{F}_{i',j'}(x,y;\\alpha,\\nu_x,\\nu_y)}.\n\\end{equation}\nThe function $F_{i,j}$ is plotted in Fig.~\\ref{fig:recfilter3d}. Note that the function is not differentiable at the boundary of its support, but this yields no problems in the optimization.\n\\begin{figure}\n    \\centering\n    \\includegraphics{figures\/Gaussianfilter.png}\n    \\caption{Gaussian reconstruction filter $F_{i_0,j_0}$ for $\\alpha = 1$ and $(\\nu_x,\\nu_y)=(3,3)$.}\n    \\label{fig:recfilter3d}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = \\textwidth]{figures\/image_reconstr_examples.png}\n    \\caption{Image reconstruction based on a small set of ray-screen intersections, for bincount and various reconstruction filter sizes and $\\alpha = 1$.}\n    \\label{fig:imagereconstr}\n\\end{figure}\n\nGaussian image reconstruction is shown in Fig.~\\ref{fig:imagereconstr} for various values of $\\nu_x = \\nu_y$. There is a trade-off here since the larger $\\nu_x$, and $\\nu_y$ are the blurrier the resulting image is, and the larger the computational graph becomes, but also the larger the section of the image is that is aware of a particular ray which yields more informative gradients. \n\nUp to this point, this section has discussed the ray tracing part of the pipeline, the next subsections will discuss the role of the neural network and the optimization.\n\n\\subsection{Multi-layer perceptron as optimization accelerator} \\label{subsec:nnarchitectures}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.6\\textwidth]{figures\/nn_architecture_dense.png}\n    \\caption{The dense multi-layer perceptron architecture based on the size of the control net $(n_1+1)\\times(n_2+1)$.}\n    \\label{fig:densenn}\n\\end{figure}\n\nSeveral neural network architectures are considered, all with a trivial input of 1, meaning that the neural networks will not, strictly speaking, be used to approximate a function since the considered domain is trivial. Non-trivial network inputs of system parameters like the source location will probably be part of follow-up research.\n\nIn this configuration, the neural network can be considered a transformation of the space over which is optimized: from the space of trainable neural network parameters to the space of control point $z$-coordinate values.\nThe goal of choosing the network architecture is that optimizing the trainable neural network parameters of this architecture yields better training behavior than optimizing the control point z-coordinate values directly. The used networks are multi-layer perceptions (MLPs), feed-forward networks consisting of several layers of neurons, as depicted in Fig.~\\ref{fig:densenn}. The considered architectures are:\n\\begin{enumerate}\n    \\item No network at all. \\label{item:nonetcase}\n    \\item A sparse MLP where the sparsity structure is informed by the overlap of the B-spline basis function supports on the knot spans. In other words: this architecture aims to precisely let those control points 'communicate' within the network that share influence on some knot span product on the B-spline surface, yielding a layer with the same connectivity as a convolutional layer with kernel size $(2p+1,2q+1)$. However, each connection has its own weight and each kernel its own bias, instead of only having a weight per element of the convolution kernel and one single bias for all kernels.\n    \\item Larger fully connected architectures are also considered, with 3 layers of control net size. Note that two consecutive such layers yield many weight parameters: $n^4$ for a square control net with `side length' $n$.\n\\end{enumerate}\nThe activation function used for all neurons is the hyperbolic tangent, which is motivated below.\n\n\\subsubsection{Control point freedom} \\label{subsec:controlpointfreedom}\nControl over the range of values that can be assumed by the control point $z$-coordinates is essential to make sure that the systems stays physical (as mentioned in Section~\\ref{subsubsec:lensconstraints}), but also to be able to take into account restrictions imposed on the lens as part of mechanical construction in a real-world application. Note that the restriction $P_{i,j}^z > z_\\text{in}$ for the control points being above the lens entrance surface is not critical for a collimated ray bundle simulation since, the entrance surface can be moved arbitrarily to the $-z$ direction without affecting the ray tracing. \n\nSince the final activation function $\\tanh$ has finite range $(-1,1)$, this can easily be mapped to a desired interval $(z_{\\min},z_{\\max})$:\n\\begin{equation}\n    y_{i,j} \\mapsto z_{\\min} + \\textstyle\\frac{1}{2} (y_{i,j} + 1)(z_{\\max} - z_{\\min}), \\label{eq:outputcorrection}\n\\end{equation}\nwhich can even vary per control point if desired. Here $y_{i,j}$ denotes an element of the total output $Y$ of the network.\nThe above can also be used as an offset from certain fixed values:\n\\begin{equation}\n    y_{i,j} \\mapsto f\\left(P^x_{i,j},P^y_{i,j}\\right) + z_{\\min} + \\textstyle\\frac{1}{2} (y_{i,j} + 1)(z_{\\max} - z_{\\min}). \\label{eq:outputcorrectionwfunc}\n\\end{equation}\nThe resulting B-spline surface approximates the surface given by $f(x,y) + \\textstyle\\frac{1}{2}(z_{\\max}+z_{\\min})$ if $Y \\approx 0$ can be used to optimize a lens that is globally at least approximately convex\/concave. The choice of the hyperbolic tangent activation function accommodates this: since this activation function is smooth around its fixed point $0$ when initializing the weights and biases of the network close to $0$, there is no cumulative value-increasing effect in a forward pass through the network so that indeed $Y\\approx 0$ in this case.\n\nFor comparability, in the case without a network, the optimization is not performed directly on the control point $z$-coordinates. Instead, for each control point, a new variable for optimization is created, which is passed through the activation function and the correction as in Eq.~\\ref{eq:outputcorrection} or \\ref{eq:outputcorrectionwfunc} before being assigned to the control point.\n\n\\subsection{The optimization}\n\n\\label{lossfunc}\nThe lens is optimized such that the irradiance distribution $\\mathcal{I}$ projected by the lens approximates a reference image $\\mathcal{I}_\\text{ref}$, where $\\mathcal{I} ,\\mathcal{I}_\\text{ref}\\in \\mathbb{R}^{n_x\\times n_y}_{\\ge 0 }$. The loss function used to calculate the difference between the two uses the normalized matrices: \n\\begin{equation}\n    \\widehat{\\mathcal{I}} = \\frac{\\mathcal{I}}{\\sum_{i,j}^{n_x,n_y} \\mathcal{I}_{i,j}} \n    \\quad\n    \\mathrm{and}\n    \\quad\n    \\widehat{\\mathcal{I}}_\\text{ref} = \\frac{\\mathcal{I}_\\text{ref}}{\\sum_{i,j}^{n_x,n_y} \\mathcal{I}_{\\mathrm{ref},i,j}}.\n\\end{equation}\n\n\\noindent\nThe loss function is given by\n\\begin{equation}\n    \\mathcal{L}(\\mathcal{I};\\mathcal{I}_\\text{ref}) = \\frac{1}{\\sqrt{n_x n_y}}\n   \\left\\| \\widehat{\\mathcal{I}}-\\widehat{\\mathcal{I}}_\\text{ref} \\right\\|_F \n    \\label{eq:pipelineLoss},\n\\end{equation}\nwhere $\\| \\cdot \\|_F$ is the Frobenius or matrix norm, which is calculated as follows:\n\\begin{equation}\n    \\| \\mathcal{A} \\|_F = \\sqrt{\\sum_{i}^{n_x}\\sum_{j}^{n_y} \\lvert a_{i,j} \\rvert ^2}.\n\\end{equation}\nFig.~\\ref{fig:optimizationloop} shows the conventional stopping criterion of the loss value being smaller than some $\\varepsilon > 0$, but in our experiments, we use a fixed number of iterations.\n\nThe neural network parameters (weights and biases) are updated using the Adam optimizer \\citep{Kingma2014}  by back-propagation of the loss to these parameters.\n\n\\section{Results} \\label{sec:results}\n\nSeveral results produced with the optimization pipeline discussed in the previous sections are displayed and discussed in this section. The implementation mainly uses PyTorch, a Python wrapper of Torch \\citep{Collobert2002Torch}.\n\nNone of the optimizations performed for this section took more than a few hours to complete, on a \\verb|HP ZBook Power G7 Mobile Workstation| with a \\verb|NVIDIA Quadro T1000 with Max-Q Design| GPU.\n\nMost of the results have been validated with \\emph{LightTools} \\citep{LightTools}, an established ray tracing software package in the optics community. Lens designs were imported to LightTools as a point cloud, then interpolated to obtain a continuous surface, and all simulations were conducted using $10^6$ rays.\n\nUnits of length are mostly unspecified since the obtained irradiance distributions are invariant under uniform scaling of the optical system. This invariance to scaling is reasonable as long as the lens details are orders of magnitude larger than the wavelength of the incident light such that diffraction effects do not play a role. Furthermore, the irradiance distributions are directly proportional to the scaling of all ray weights and thus the source power, so the source and screen power also need no unit specification. Note that relative changes have a non-trivial effect, like changes to the power proportion between sources or the distance proportions of the optical system.\n\n\\newpage\n\\subsection{Irradiance derivatives with respect to a control point}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.75\\textwidth]{figures\/results\/controlpoint_derivatives.png}\n    \\caption{Gradients of an irradiance distribution of a collimated ray bundle through a flat lens (parallel sides), with respect to the $z$-coordinate of one control point. The zeros are masked with white to show the extend of the influence of the control point. These irradiation distributions differ by: (a): degrees $(3,3)$, reconstruction filter size $(3,3)$, (b): degrees $(3,3)$, reconstruction filter size $(11,11)$, (c): degrees $(5,3)$, reconstruction filter size $(3,3)$.}\n    \\label{fig:controlpointderivs}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figures\/results\/renderderivsexplanation.png}\n    \\caption{Demonstration of how one control point influences the irradiance distribution in the case of a flat lens with B-spline degrees $(3,3)$ and a collimated ray bundle source.}\n    \\label{fig:renderderivsexplanation}\n\\end{figure}\nThis section gives a simple first look at the capabilities of the implemented differentiable ray tracer: computing the derivative of an irradiance distribution with respect to a single control point. Obtaining this data is inefficient in the current PyTorch implementation as a forward mode automatic differentiation pass is required, which is not currently (entirely) supported by PyTorch. Therefore these derivatives are computed with pixel-wise back-propagation.\n\nFig.~\\ref{fig:controlpointderivs} shows the derivative of an irradiance distribution produced by a collimated ray bundel through a flat lens for various B-spline degrees and reconstruction filter sizes, and Fig.~\\ref{fig:renderderivsexplanation} shows what one of these systems looks like. The overall `mountain with a surrounding valley' structure can be understood as follows: as one of the control points rises, it creates a local convexity in the otherwise flat surface. This convexity has a focusing effect, redirecting light from the negative valley region toward the positive mountain region.\n\nNoteworthy of these irradiance derivatives is also their total sum: (a) $\\SI{-1.8161e-08}{}$, (b) $\\SI{3.4459e-08}{}$, (c) $\\SI{9.7095e-05}{}$. These small numbers with respect to the total irradiance of about $93$ and therefore indicate conservation of light; as the control point moves out of the flat configuration, at first, the total amount of power received by the screen will not change much. This is expected from cases (a) and (b), where the control point does not affect rays that reach the screen on the boundary pixels. However, in all cases, all rays intersect the lens at right angles. Around $\\theta = 0$, the slope of Schlick's approximation is very shallow, indicating a small decrease in refraction in favor of reflection.\n\n\\subsection{Sensitivity of the optimization to initial state and neural network architecture} \\label{subsec:results_nnsensitivity}\nAs with almost any iterative optimization procedure, choosing a reasonable initial guess of the solution is crucial for reaching a good local\/global minimum. For training neural networks, this comes down to how the network weights and biases are initiated. In this section, we look at three target illuminations: the circular top hat distribution (Fig.~\\ref{fig:circtophat}), the TU Delft logo (Fig.~\\ref{fig:TUDflameinv}), and an image of a faceted ball (Fig.~\\ref{fig:facball}). For some experiments, black padding or Gaussian blurring is applied to these images. We design lenses to produce these distributions from a collimated ray bundle, given various neural network architectures (introduced in section \\ref{subsec:nnarchitectures}) and parameter initializations.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{figures\/results\/circular_tophat.png}\n    \\caption{The circular tophat target illumination.}\n    \\label{fig:circtophat}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{figures\/TUD_400_inverse.png}\n    \\caption{The TU Delft flame target illumination.}\n    \\label{fig:TUDflameinv}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.4\\textwidth]{figures\/results\/shaded_faceted_ball.png}\n    \\caption{The faceted ball target illumination.}\n    \\label{fig:facball}\n\\end{figure}\n\n\\paragraph{Circular top hat distribution from collimated ray bundle} \\label{subsec:circtophat}\nFig.~\\ref{fig:tophatloss} shows the progress of the loss over 1000 iterations, with each iteration taking $2.5$ seconds, for various neural network architectures and parameter initialization combinations. For the other parameters in these simulations, see the supplementary information. For a few moments during the training, the resulting freeform surfaces and irradiance distributions are shown in Figs.~\\ref{fig:lensshapes}, \\ref{fig:tophatrandomsparse}, \\ref{fig:tophatunifsparse}, \\ref{fig:tophatunifnonet} and \\ref{fig:tophatunifdense}. Uniform here means that the initial trainable parameter values are sampled from a small interval: $U\\left(\\left[-10^{-4},10^{-4}\\right]\\right)$, except for the no-network case; this is initialized with all zeros.\n\nThe first notable difference is between the random and uniformly initialized sparse neural networks. The uniformly initialized neural network performs much better, and no network performs better. This is probably because the uniformly initialized cases converge to a better (local) minimum than the randomly initialized case. Of course, it could happen that the random initialization lands in a very favorable spot in the design landscape, but intuitively this seems very unlikely.\n\nAnother property of the uniformly initialized cases is their preservation of symmetry in these setups. As Fig.~\\ref{fig:lensshapes} shows, this leads to much simpler lenses, which are probably much less sensitive to manufacturing errors due to their relative lack of small detail. What is interesting to note here is that if the sparse network is initialized with all parameters set to $0$, then its optimization is identical to the no-network case, as only the biases in the last layer achieve non-zero gradients.\n\nNo rigorous investigation has been conducted to the extent that this behavior of increased convergence speed carries over to other target distributions and system configurations and what the optimal hyper-parameters are. A thorough investigation of the hyper-parameter space that defines a family of network architectures could reveal where in the increase of the architecture complexity, diminishing returns for optimizing these lenses arises. However, based on these initial findings the fully connected network is used for all the following optimizations in the results.\n\n\n\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{figures\/results\/tophat_loss_progress.png}\n    \\caption{Loss progress over the iterations for various pipeline-setups for forming a tophat distribution from a collimated ray bundle.}\n    \\label{fig:tophatloss}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figures\/results\/tophat_lens.png}\n    \\caption{The lens height field after initialization ($n=0$), and $n=50,100$ and $1000$ iterations respectively, for different network architectures (Section~\\ref{subsec:nnarchitectures}) and network parameter initializations (Section~\\ref{subsec:circtophat}).}\n    \\label{fig:lensshapes}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figures\/results\/tophat_randomsparse.png}\n    \\caption{Irradiance distributions and pixel-wise errors in the optimization progress of a random lens with a sparse network towards a circular tophat illumination.}\n    \\label{fig:tophatrandomsparse}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figures\/results\/tophat_unifsparse.png}\n    \\caption{Irradiance distributions and pixel-wise errors in the optimization progress of a flat lens with a sparse network towards a circular tophat illumination.}\n    \\label{fig:tophatunifsparse}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figures\/results\/tophat_unifnonet.png}\n    \\caption{Irradiance distributions and pixel-wise errors in the optimization progress of a flat lens without a network towards a circular tophat illumination.}\n    \\label{fig:tophatunifnonet}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figures\/results\/tophat_unifdense.png}\n    \\caption{Irradiance distributions and pixel-wise errors in the optimization progress of a flat lens with a dense network towards a circular tophat illumination.}\n    \\label{fig:tophatunifdense}\n\\end{figure}\n\n\\clearpage\n\\paragraph{TU flame and faceted ball from collimated ray bundle}\nIn what follows, we consider complex target distributions: the TU Delft flame (for a complex shape) and a faceted ball (for a target with various brightness levels). Here we still use the collimated ray bundle illumination, but lenses are now optimized for various magnifications; see Table.~\\ref{tab:magndata}. These magnifications are defined as the scaling of the screen size with respect to the smallest screen size $(0.64,0.64)$.  The other parameters of these optimizations are shown in the supplementary information. All these iterations took about $4$ seconds each.\n\nThe final irradiance distributions and corresponding LightTools results are shown in Figs.~\\ref{fig:flamerenders} and \\ref{fig:facetedballrenders}, respectively. These figures show that the optimization pipeline can handle these more complex target illuminations well. The LightTools results predict some artifacts within the irradiance distribution, which the implemented ray tracer does not, especially in the TU flame magnification 1 case. By visual inspection, based on the LightTools results, one would probably rate these results in the exact opposite order than as indicated by the losses shown in Fig.~\\ref{fig:ManufacturingLoss}.\n\nA potential explanation of the increase in loss with the magnification factor in Fig.~\\ref{fig:ManufacturingLoss} is that the bigger the screen is: the rays require higher angles to reach the edges of the screen, which is apparent in the cases of magnification 3 and 5 Fig.~\\ref{fig:ManufacturingRays}. This results in a larger sensitivity of the irradiance to the angle with which a ray leaves the screen. This in turn gives larger gradients of the irradiance with respect to the control points. Therefore the optimization takes larger steps in the neural network parameter space, possibly overshooting points that result in a lower loss.\n\nFor the magnification, $3$ and $5$, the irradiance distributions from LightTools show artifacts at the screen boundaries. A possible explanation for this is that the way the B-spline surfaces are transferred to LightTools is inaccurate at the surface boundaries.\\footnote{Assuming only rays from the B-spline surface boundaries reach the screen boundary area.} This is because surface normals are inferred from fewer points on the B-spline surface at the boundary than in the middle of the surface by LightTools.\n\nFurthermore, a significant amount of rays are lost during optimization because the target illuminations are black at the borders, so rays near the screen boundary will be forced off the screen by the optimization. Once rays are off the screen, they no longer contribute to the loss function. Once a ray misses the screen, the patch on the B-spline surface these rays originate from does not influence the irradiance and, thus, the loss function. \nHowever, this does not mean that this patch is idle for the rest of the optimization, as this patch can be in support of a basis function that corresponds to a control point that still affects rays that hit the screen. Therefore, the probability of getting idle lens patches with this setup decreases with the B-spline degrees since these determine the size of the support of the B-spline basis functions but might, in some cases, lead to oscillatory behavior, with rays alternating between hitting and missing the screen. \n\nFig. \\ref{fig:ManufacturingSurfaces} shows the optimized B-spline lens surface height field. A densely varying color map is chosen since the deviations from a flat or smooth concave shape are quite subtle, which is due to the large lens exit angle sensitivity of the ray-screen intersections since the ratio lens size to screen size is large with respect to the ratio lens size to screen distance.\n\n\\begin{table}[h]\n    \\centering\n    {\n    \\def1.5{1.5}\n        \\begin{tabular}{c|c|c|c}\n        \\textbf{Magnification} & \\textbf{screen size} & $f(x,y)$ & \\textbf{starting shape type}\\\\\n        \\hline\n        $1$ & $(0.64,0.64)$ & $\\textstyle\\frac{1}{2}$ & flat\\\\\n        $3$ & $(1.92,1.92)$ & $\\textstyle\\frac{1}{2} + 8 - \\sqrt{8^2-x^2-y^2}$ & concave\\\\\n        $5$ & $(3.20,3.20)$ & $\\textstyle\\frac{1}{2} + 4 - \\sqrt{4^2 - x^2 - y^2}$ & concave\n        \\end{tabular}\n    }\n    \\caption{The screen size and control point offset function $f$ used per magnification in the TU flame and faceted ball optimizations (distances in centimeters).}\n    \\label{tab:magndata}\n\\end{table}\n\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{figures\/results\/Manufacturing_Loss.png}\n    \\caption{Loss progress for the various magnifications and target distributions.}\n    \\label{fig:ManufacturingLoss}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/results\/Manufacturing_flames.png}\n    \\caption{Implementation and LightTools irradiance distributions of the TU flame target from the final lens design of the optimization.}\n    \\label{fig:flamerenders}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/results\/Manufacturing_balls.png}\n    \\caption{Implementation and LightTools irradiance distributions of the faceted ball target from the final lens design of the optimization.}\n    \\label{fig:facetedballrenders}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/results\/Manufacturing_surfaces.png}\n    \\caption{The lens designs for the different magnifications and two target distributions.}\n    \\label{fig:ManufacturingSurfaces}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/results\/Manufacturing_rays.png}\n    \\caption{$25\\times25$ traced rays through the final lens designs for the different magnifications and two target distributions.}\n    \\label{fig:ManufacturingRays}\n\\end{figure}\n\n\n\n\n\\clearpage\n\\subsection{Optimization with a point source and a grid of point sources}\nWe now consider an optimization that uses the B-spline intersection algorithm. First, we design a lens with one point source at $(0,0,-5)$ with $5\\times 10^5$ rays to again form the TU flame. Then after $\\sim 200$ iterations, we change the source to an equispaced grid of $25\\times 25$ point sources with $10^3$ rays each on $[-1,1]\\times[-1,1]\\times\\{-5\\}$, approximating a source of non-negligible size. The other (hyper-)~parameters of this optimization are shown in the supplementary information. Due to the additional B-spline intersection procedures, each iteration takes approximately $50$ seconds.\nThe resulting final irradiance distribution and LightTools verifications can be seen in Fig.~\\ref{fig:point_source_renders}. The final irradiance distribution similar to the that obtained by LightTools, indicating that ray tracing with the implemented B-spline intersection algorithm works correctly. The irradiance are blurred due to the reconstruction filter.\nThe single-source point optimization performs well, although the illumination is less uniform than in the collimated ray bundle case (Figs.~\\ref{fig:flamerenders} and \\ref{fig:facetedballrenders}). \nThe non-uniformity can be attributed to the gaussian reconstruction filter used during optimization, as it smoothes out the small uniformities. \n\n\nAs seen in Fig.~\\ref{fig:point_source_renders} the irradiance distribution obtained with a grid of point sources accurately approximates the extended source illumination distribution quite well for the unoptimized case. \nFinding a lens design that redirects light from a source of non-negligible size into a desired irradiance distribution is a complex problem for which it is hard to indicate how good the optimal irradiance distribution can become.\nThe progress of the loss, as seen in Fig.~\\ref{fig:point_source_loss}, shows that the optimization can still improve the loss, even after the transition to the grid of point sources. Interestingly, looking at Fig.~\\ref{fig:point_source_renders} again, the optimization seems to adopt the coarse strategy of filling up the target distribution with images of the source square, as shown in Fig.~\\ref{fig:sourceimages}. This strategy does hinder the possible quality of the final irradiance distribution as the image of the source on the target is larger than the fine details in the desired irradiance. Optimizing both the front and back surfaces of the freeform could resolve this issue, as this will cause the image of the source to change shape depending on where it ends up on the target screen.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{figures\/results\/source_images_indication.png}\n    \\caption{Indication of images of the source square in the irradiance distribution obtained by LightTools using the point source grid.}\n    \\label{fig:sourceimages}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/results\/point_source_renders.png}\n    \\caption{The final irradiation distribution of the lens optimizations with point sources and the corresponding LightTools verifications. The extended source is not implemented in our ray tracer, but is approximated by the point source grid.}\n    \\label{fig:point_source_renders}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=0.75\\textwidth]{figures\/results\/point_source_surfaces.png}\n    \\caption{Height fields of the lenses optimized for the TU flame with point sources.}\n    \\label{fig:point_source_surfaces}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width= 0.6\\textwidth]{figures\/results\/point_source_loss.png}\n    \\caption{Loss over the iterations optimizing for the TU flame. The system is initiated with a point source, and after $\\sim 200$ iterations the point source is replaced by an equispaced grid of $25\\times 25$ point sources.}\n    \\label{fig:point_source_loss}\n\\end{figure}\n\n\\section{Conclusion} \\label{sec:conclusion}\n\nWe demonstrated that non-sequential differentiable ray tracing is a viable tool for designing freeform lenses for collimated ray bundles, points, and extended sources.\nUsing a B-spline allows for the design of a continuous surface, which is desirable for manufacturing, and its control point allows for locally altering the irradiance distribution. For both cases, collimated and point source lens designs were found that could accurately project the desired irradiance distribution in both the differentiable ray tracer and in commercial software LightTools. Some artifacts still exist and resolving this issue will be a part of further research.\n\nFor the source with a finite extent, the optimizer improved upon the design obtained for a point source. However, the final irradiance distribution was made up of images of the source, which hinders the minimum that can be obtained as the source image is larger than the details in the desired irradiance distribution. This issue can be resolved by optimizing multiple surfaces simultaneously, as the image of the source on the target plane can then be optimized to vary with location.\n\nUsing a neural network to remap the optimization space provides an interesting way to increase the convergence speed of the optimization. However, further investigation is required to see whether this generally holds and what the effect is on other network architectures.\n\nThe developed ray tracing implementation is currently a proof of concept and needs to be optimized for speed. The B-spline intersection algorithm, in particular, adds roughly a factor of $10$ to the computation time. A significant speedup can be achieved here by leveraging efficient lower-level GPU programming languages, such as CUDA. \n\n\n\\section{Acknowledgements}\nWe acknowledge support by NWO-TTW Perspectief program (P15-36) ``Free-form scattering optics\".\n\n\n\\section{Introduction}\\label{sec:Introduction}\nIn the field of illumination optics, optical engineers design optical elements to transport the light from a source, which can be an LED, laser, or incandescent lamp, to obtain a desired irradiance (spatial density of the luminous flux) or intensity (angular density of the luminous flux) \\citep{Grant2011}.\nTo transport the light from the source to the target, the optical engineer can construct a system consisting of various optical elements such as lenses, mirrors, diffusers, and light guides \\citep{John2013}.\nOne particular type of optic used in automotive and road lighting applications is the freeform lens, a lens without any form of symmetry \\citep{Falaggis2022, Mohedano2016}.\nThe design of these lenses is a complex problem. It is currently solved by numerically solving system-specific differential equations or through optimization, with every step validated using a (non-differentiable) non-sequential ray tracer \\citep{Wu2018}.\nGreat effort is involved in generalizing these methods to account for varying amounts of optical surfaces \\citep{Anthonissen2021}, their optical surface and volume properties \\citep{Kronberg2022, lippman_prescribed_2020}, or the source model~\\citep{Muschaweck2022, tukker_efficient_2007, sorgato_design_2019}. \n\nThe performance of an optical system is evaluated using ray tracing, which is the process of calculating the path of a ray originating from a source through the optical system. Sequential ray tracers such as Zemax~\\citep{zemax} and Code V~\\citep{codev}, primarily used in the design of imaging optics, trace a small number of rays to determine the quality of the image. \nNon-sequential ray tracers such as LightTools~\\citep{LightTools} and Photopia~\\citep{photopia} use many rays to simulate the optical flux through the system and share similarities with the rendering procedures in computer graphics, with the main difference being that the rays are traced from source to camera.\n\nAlgorithmically differentiable ray tracing, a generalization of differential ray tracing \\citep{feder_differentiation_1968, stone_differential_1997, oertmann_differential_1989, chen_second-order_2012}, is a tool that is being developed for both sequential~\\citep{Sun2021, Volatier2017}  and non-sequential \\citep{Mitsuba2} ray tracing.\n\\emph{Differential ray tracing} obtains system parameter gradients using numerical or algebraic differentiation. The gradient can be calculated numerically using numerical differentiation or the adjoint method \\citep{givoli_tutorial_2021}, requiring the system to be ray traced twice, once for its current state and once with perturbed system parameters. Analytic expressions for the gradient can be obtained by tracing the rays analytically through the system, calculating where the ray intersects the system's surfaces and how the ray's trajectory is altered. However, these expressions can become long and complicated depending on the system. In addition, the method is limited to optics described by conics as finding analytic ray surface intersection with surfaces of higher degrees becomes complicated or even impossible. Algorithmic differentiable ray tracing can handle these issues by obtaining the gradients with one single forward simulation for an almost arbitrary system. In addition, it can be seamlessly integrated into gradient-descent-based optimization pipelines. A modern framework for this is \\emph{Physics Informed Machine Learning} \\citep{Karniadakis2021}, where a neural network is trained to approximate the solution to a physics problem formulated using data, a set of differential equations, or an implemented physics simulation (or a combination of these). \n\nWe investigate the reliability of designing freeform lenses with B-spline surfaces \\citep{Piegl1995} using algorithmically differentiable non-sequential ray tracing and gradient-based optimization to redirect the light of a light source into a prescribed irradiance distribution. The source models will be the collimated light source, point source, and finally, sources with a finite extent. The results are validated using the commercial ray trace program LightTools \\citep{LightTools}. In addition, we investigate the effectiveness of optimizing a network to determine the optimal B-spline control points as proposed in \\citep{Moller2021PIML} and \\citep{GASICK2023115839}, and compare it to optimizing the control points directly and seeing the possible speed-up.\n\n\\section{Gradient-based freeform design}\nThe overall structure of our pipeline is depicted in Fig.~\\ref{fig:optimizationloop}. \nA freeform surface is defined by the parameters $P \\in \\mathscr{P}$, where $\\mathscr{P}$ is the set of permissible parameter values. This surface is combined with a flat surface to create a lens, and an irradiance distribution $\\mathcal{I}$ is produced by tracing rays through the lens onto a screen. The irradiance distribution is compared to a target $\\mathcal{I}_\\text{ref}$ yielding a loss $\\mathscr{L}(\\mathbf{P};\\mathcal{I}_\\text{ref})$. The optimization problem we are trying to solve can then be formulated as\n\\begin{equation}\n    \\min_\\mathbf{P \\in \\mathscr{P}} \\; \\mathscr{L}(\\mathbf{P};\\mathcal{I}_\\text{ref}),\n\\end{equation}\nwhich we solve by using gradient descent.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = \\textwidth]{figures\/Caustic_design_optimization.png}\n    \\caption{Overview of our learning-based freeform design pipeline.}\n    \\label{fig:optimizationloop}\n\\end{figure}\n\nThe freeform surface of the lens is defined in terms of a B-spline surface. From a manufacturing standpoint, this is convenient since B-spline surfaces can be chosen to be $C^1$ smooth (in fact, B-spline surfaces can be $C^n$ smooth for arbitrarily large $n$). From an optimization perspective, B-spline surfaces have the property that the control points that govern the shape of the surface and which will be optimized have a local influence on the surface geometry, which in turn has a local influence on the resulting irradiance distribution.\n\n\\subsection{The lens model using a B-spline surface}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.2\\textwidth]{figures\/lens_schematic.png}\n    \\caption{The used lens type: a volume enclosed between a flat surface and a freeform surface with a uniform refractive index.}\n    \\label{fig:lenschematic}\n\\end{figure}\n\nWe define a lens as in Fig.~\\ref{fig:lenschematic} as the volume between a flat surface and a B-spline surface, with a uniform refractive index.\n\nA B-spline surface $\\mathbf{S}$ in $\\mathbb{R}^3$ is a parametric surface, see Fig.~\\ref{fig:Bsplinesurface}. It has rectangular support $[a,b]\\times[c,d]$ where $a<b$ and $c<d$. It is defined as a linear combination of an $(n_1+1) \\times (n_2+1)$ grid of control points $\\mathbf{P}_{i,j} \\in \\mathbb{R}^3$ where $n_1$ and $n_2$ are positive integers which define the size of the control net $\\{\\mathbf{P}_{i,j} :  0 \\leq i \\leq n_1, \\; 0 \\leq j \\leq n_2\\}$. This linear combination is defined in terms of univariate B-spline basis functions $N_{i,p}$ and $N_{j,q}$, which are recursively defined as follows by the Cox-de Boor formula \\cite[eq. 2.5]{Piegl1995}:\n\\begin{align}\n    & N_{i,0}(u) =\n    \\begin{cases}\n        1 \\text{ if } u_i \\le u < u_{i+1} \\\\\n        0 \\text{ otherwise}\n    \\end{cases} \\nonumber \\\\\n    & N_{i,p}(u) = \\frac{u-u_i}{u_{i+p}-u_i}N_{i,p-1}(u) + \\frac{u_{i+p+1}-u}{u_{i+p+1}}N_{i+1,p-1}(u),\n\\end{align}\nfor $i=0,\\ldots,n_1$ and similarly for $N_{j,q}$ with $j=0,\\ldots,n_2$. The basis functions are $p$-degree piece-wise polynomials. The knots $u_i$ are non-decreasing real numbers in $[a,b]$ collected in a knot vector\n\\cite[eq. 2.13]{Piegl1995}:\n\\begin{equation}\n    \\mathcal{V} = (\\underbrace{0,\\ldots,0}_{p+1},u_{p+1},\\ldots,u_{r-p-1},\\underbrace{1,\\ldots,1}_{p+1}) \\in \\mathbb{R}^{r+1}.\n\\end{equation}\n Here we assume $a=0$ and $b=1$. For the number of knots $r+1$, the number of control points $n_1+1$ and the degree $p$, there is the relationship $n_1 = r-p-1$. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.5\\textwidth]{figures\/BsplineSurface.png}\n    \\caption{A B-spline surface of degrees $(p,q) = (3,3)$ with indicated directions of the $u$ and $v$ parameters. The control points are shown in black.}\n    \\label{fig:Bsplinesurface}\n\\end{figure}\n\nAn interval $[u_i, u_{i+1})$ of consecutive knots is called a knot span, on which the basis functions are analytic. At the knots, the basis functions are $p-k$ times continuously differentiable, with $k$ being the multiplicity of the knot, that is, how many times the same knot is repeated in the knot vector. The specific type of knot vector here with the multiplicity of $p+1$ at the first and last knots is called clamped or open, which yields the property\n\\begin{equation}\n    N_{0,p}(0) = N_{n_1,p}(1) = 1.\n\\end{equation}\nAll other basis functions are $0$ at $u=0,1$, which means that the boundary control points determine the boundary of the surface.  \n\n\\noindent The interior knots are chosen to be equispaced, \\emph{i.e.}\n\\begin{equation}\n    u_i = \\frac{i-p}{n_1-p+1}, \\qquad i = p,\\ldots, n_1+1.\n\\end{equation}\nThus there are no knots with multiplicity larger than $1$ apart from those at the boundary. So if $p,q \\ge 2$, then the surface is at least $C^1$, and the gradient of the B-spline surface is defined everywhere.\n\nCompletely analogously, there is the knot vector $\\mathcal{W}$ with the knots $v_j$ for the basis functions $N_{j,q}$  of degree $q$, for $j=0,\\ldots,n_2$. The definition of the surface is then \\cite[eq. 3.11]{Piegl1995}\n\\begin{equation}\n    \\mathbf{S}(u,v) = \\sum_{i=0}^{n_1}\\sum_{j=0}^{n_2} N_{i,p}(u)N_{j,q}(v)\\mathbf{P}_{i,j}, \\quad (u,v) \\in [0,1]^2. \\label{eq:Bsplinesurfacedef}\n\\end{equation}\n\n\\subsubsection{Linearizing the B-spline parametrizations}\nThe volume $V \\subset \\mathbb{R}^3$ of the modeled lens has a rectangular extent  $[-r_x,r_x] \\times [-r_y,r_y]$ in the ($x,y$)-plane with $2r_x$ and $2r_y$ being the width and height of the lens, respectively. The lens volume is enclosed on one side by a B-spline surface\n\\begin{equation}\n    \\mathbf{S}(u,v) = (X(u,v), Y(u,v), Z(u,v)), \\label{eq:coordfuncs}\n\\end{equation}\nwhere $X,Y,Z$ are the individual coordinate parameterizations, for instance\n\\begin{subequations}\n    \\begin{align}\n        X(u,v) = \\sum_{i=0}^{n_1}\\sum_{j=0}^{n_2} N_{i,p}(u)N_{j,q}(v)P^x_{i,j}, \\\\\n        Y(u,v) = \\sum_{i=0}^{n_1}\\sum_{j=0}^{n_2} N_{i,p}(u)N_{j,q}(v)P^x_{i,j}.\n    \\end{align}\n\\end{subequations}\nFor simplicity of calculations on ray-sampling and ray-intersection (Section \\ref{subsec:raytracer}), it is helpful to define the mapping $(u,v) \\mapsto (X(u,v),Y(u,v))$ in a way that it is analytically invertible. Therefore the coordinates of the control points are chosen such that the parametrizations $X$ and $Y$ are linear:\n\\begin{subequations}\n    \\begin{align}\n        X &: u \\mapsto (2u-1)r_x \\in [-r_x,r_x], \\label{eq:linearX}\\\\\n        Y &: v \\mapsto (2v-1)r_y \\in [-r_y,r_y]. \\label{eq:linearY}\n    \\end{align}\n\\end{subequations}\nIn general, $X$ and $Y$ are degree $p$ and $q$ piece-wise polynomials, respectively, and thus not linear. Linearity can be achieved by making use of the nodal representation of the B-spline \n basis functions \\cite[eq. 23]{Cohen2010Iso}:\n\\begin{equation}\n    u = \\sum_{i=0}^{n_1} u_{i,p}^* N_{i,p}(u), \\quad u \\in [0,1], \\quad u_{i,p}^* = \\frac{u_{i+1}+\\ldots+u_{i+p}}{p}, \\label{eq:nodalrepresentation}\n\\end{equation}\nwhich provides a specific knot vector-dependent linear combination of the basis functions that yields the identity function on the domain $[0,1]$. The values $u_{i,p}^*$ are called the Greville abscissae \\cite[sec. 8.6]{Farin2002}.\n\nWe assume that the $P^x_{i,j}$ are independent of $j$, and choose $j=0$ as a representative.  Then we obtain by the definition of $X$:\n\\begin{subequations}\n    \\begin{align}\n        X(u,v) &= \\sum_{i=0}^{n_1}\\sum_{j=0}^{n_2} P^x_{i,j}N_{i,p}(u)N_{j,q}(v) \\\\\n        &= \\sum_{i=0}^{n_1} P^x_{i,0}N_{i,p}(u) \\underbrace{\\sum_{j=0}^{n_2} N_{j,q}(v)}_{=1},\n    \\end{align}\n\\end{subequations}\nwhere $\\sum_{i=0}^n N_{i,p}(u) = 1$ by the partition of unity property of the basis functions \\cite[P2.4]{Piegl1995}. Now we see that if we define $P^x_{i,j} := u^*_{i,p}$ then $X(u) = u$. Thus if we apply the mapping $u \\mapsto (2u-1)r_x$ to both sides of eq. ~\\ref{eq:nodalrepresentation}, we obtain\n\\begin{equation}\n    (2u-1)r_x = \n    \\sum_{i=0}^{n_1} (2u^*_{i,p}-1)r_x N_{i,p}(u). \\label{eq:lineartaucontrolpoints}\n\\end{equation}\nThis equality can be understood by expanding the $1$ into the sum over all $N_{i,p}(u)$ by again exploiting the partition of unity property. Thus if we define $P^x_{i,j} := (2u^*_{i,p}-1)r_x$ and equivalently $P^y_{i,j} := (2v^*_{j,q}-1)r_y$, then Eqs.~\\ref{eq:linearX} and \\ref{eq:linearY} and are satisfied.\n\nThe lens is then defined as the volume in $\\mathbb{R}^3$ enclosed by the B-spline surface $\\mathbf{S}$ and the flat surface given by $z=z_\\text{in}$ on $[-r_x,r_x] \\times [-r_y,r_y]$:\n\\begin{equation}\n    V = \\left\\{ (x,y,z) \\in \\mathbb{R}^3 \\; \\mid \\;\n    z_\\text{in} \\le z \\le \n    Z\\left(X^{-1}(x), Y^{-1}(y)\\right), \\vert x\\vert\\le r_x, \\vert y\\vert\\le r_y\\right\\}.\n\\end{equation}\nFor the arguments of $Z(u,v)$ the inverses of $X$ and $Y$ are used:\n\\begin{equation}\n    X^{-1}(x) = \\frac{1}{2}\\left(\\frac{x}{r_x}+1\\right), \\quad\n    Y^{-1}(y) = \\frac{1}{2}\\left(\\frac{y}{r_y}+1\\right).\n\\end{equation}\n\n\\subsubsection{Lens constraints} \\label{subsubsec:lensconstraints}\nTo let the lens be well-defined the surfaces of the lens should not intersect:\n\\begin{equation}\n    z_\\text{in} < Z(u,v), \\quad(u,v) \\in [0,1]^2.\n\\end{equation}\nBy the convex hull property of B-spline surfaces \\cite[P3.22]{Piegl1995} it suffices that\n\\begin{equation}\n    P^z_{i,j} > z_\\text{in} \\quad \\forall (i,j). \\label{eq:nosurfintersect}\n\\end{equation}\nManufacturing can require that the lens has some minimal thickness $\\delta$, so that the constraint is stronger:\n\\begin{equation}\n    P^z_{i,j} \\ge \\delta + z_\\text{in} \\quad \\forall (i,j).\n\\end{equation}\n\n\n\\subsection{Differentiable ray tracer} \\label{subsec:raytracer}\nOur implementation traces rays from a source through the flat lens surface and the freeform lens surface to the detector screen as depicted in Figs.~\\ref{fig:planetraceschematic} and \\ref{fig:pointtraceschematic}. Other ray paths, e.g., total internal reflection at lens surfaces, are not considered since it is assumed that the contribution of these to the resulting irradiance distribution is negligible.\n\n\\subsubsection{Sources and ray-sampling}\nNon-sequential ray tracing is a Monte-Carlo approximation method of the solution to the continuous integration formulation of light transport through an optical system. For a detailed discussion of this topic, see \\cite[ch. 14]{pharr2016physically}. Thus to perform ray tracing, the light emitted by a source must be discretized into a finite set of rays\n\\begin{equation}\n    l: t \\rightarrow \\mathbf{o} + \\hat{\\mathbf{d}}t,\n\\end{equation}\nwhere $\\mathbf{o}$ is the origin of the ray and $\\hat{\\mathbf{d}}$ its normalized direction vector. Both collimated ray bundle and point sources will be considered, see Figs.~\\ref{fig:planetraceschematic} and \\ref{fig:pointtraceschematic}, respectively. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.6\\textwidth]{figures\/plane_trace_schematic.png}\n    \\caption{Schematic of the ray tracing with a collimated ray bundle source.}\n    \\label{fig:planetraceschematic}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.6\\textwidth]{figures\/point_trace_schematic.png}\n    \\caption{Schematic of the ray tracing with a point source.}\n    \\label{fig:pointtraceschematic}\n\\end{figure}\n\nTracing rays from a collimated ray bundle can be understood from Fig.~\\ref{fig:planetraceschematic}. The path of all rays from the source plane to the B-spline surface is a line segment parallel to the $z$-axis. Therefore, we can sample the incoming rays directly on the B-spline surface, with $\\hat{\\mathbf{d}} = (0,0,1)^\\top$. By the linearity of $X$ and $Y$ sampling on the B-spline domain $[0,1]^2$ is analogous to sampling on the lens extent $[-r_x,r_x]\\times [-r_y,r_y]$ in terms of distribution. Rays are sampled in a (deterministic) square grid on $[0,1]^2$. \n\nFor a point source, each ray starts at the location of the source, and the direction vector $\\hat{\\mathbf{d}}$ is sampled over the unit sphere $\\mathbb{S}^2$. More precisely, $\\hat{\\mathbf{d}}$ is given by\n\\begin{equation}\n    \\hat{\\mathbf{d}} = \\left(\\cos\\theta\\sin\\phi,\\sin\\theta\\sin\\phi,\\cos\\phi\\right)^\\top,\n\\end{equation}\nwith $\\theta \\in [0,2\\pi)$ and $\\phi \\in [0,\\phi_\\text{max}]$ for some $0\\le \\phi_\\text{max} < \\frac{\\pi}{2}$, see Fig.~\\ref{fig:pointtraceschematic}. $\\phi_\\text{max}$ is chosen as small enough to minimize the number of rays that miss the lens entrance surface but large enough such that the whole surface is illuminated. For instance, if the source is on the $z$-axis, then $\\phi_\\text{max} = \\arctan\\left(\\frac{\\sqrt{r_x^2 + r_y^2}}{z_\\text{in}-z_s}\\right)$ where $z_\\text{in}$ is the $z$-coordinate location of the entrance surface and $z_s$ the $z$-coordinate of the source. To uniformly sample points on this sphere segment, $\\theta$ is sampled (non-deterministically) uniformly in $[0,2\\pi)$ and $\\phi$ is given by\n\\begin{equation}\n    \\phi = \\arccos\\left(1-(1-\\cos\\phi_\\text{max})a\\right)\n\\end{equation}\nwhere $a$ is sampled (non-deterministically) uniformly in $[0,1]$. This sampling is used to produce the results in Section~\\ref{sec:results}.\n\nFor the point source, the calculation of the intersection of a ray with the B-spline surface is non-trivial. This calculation comes down to finding the smallest positive root of the $p+q$ degree piece-wise polynomial function\n\\begin{equation}\n    f(t) = \n    Z\\left(\\begin{pmatrix}o_u \\\\ o_v\\end{pmatrix} + \n    \\begin{pmatrix}d_u \\\\ d_v\\end{pmatrix}t\n    \\right)\n     - d_zt - o_z, \\label{eq:surfaceintersect}\n\\end{equation}\nif such a root exists and yields a point in the domain of $Z$.  Here the subscripts $u$ and $v$ denote that the ray is considered in $(u,v,z)$ space instead of $(x,y,z)$ space, so for instance\n\\begin{equation}\n    o_u = X^{-1}(o_x) = \\frac{1}{2}\\left(\\frac{o_x}{r_y}+1\\right), \\quad d_v = \\frac{d_y}{2r_y}.\n\\end{equation}\nThe roots of eq. \\ref{eq:surfaceintersect} cannot generally be found analytically for $p+q>4$, and thus an intersection algorithm is implemented, which is explained in the next section.\n\n\\subsubsection{B-spline surface intersection algorithm}\nThe intersection algorithm is based on constructing a triangle mesh approximation of the B-spline surface and computing intersections with that mesh.\n\n\\paragraph{Triangle mesh intersection phase 1: bounding boxes}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{figures\/bb_per_knotspanproduct.png}\n    \\caption{Triangles and corresponding bounding box for a few knot span products of a spherical surface.}\n    \\label{fig:bb_per_knotspanproduct}\n\\end{figure}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{figures\/uv_triangle_check.png}\n    \\caption{Example of which triangles are candidates for a ray-surface intersection with the ray plotted in red, based on their $u,v$-domain.}\n    \\label{fig:uvtriangleint}\n\\end{figure}\nChecking every ray against every triangle for intersection is computationally expensive, so it is helpful to have bounding box tests that provide rough information about whether the ray is even near some section of the B-spline surface. B-spline theory provides a tool for this: the strong convex hull property, which yields the bounding box \n\n\\begin{equation}\n    B_{i_0,j_0} = \\left[u_{i_0},u_{i_0 + 1}\\right) \\times \\left[v_{j_0},v_{j_0 + 1}\\right) \\times \\left[z^{\\min}_{i_0,j_0}, z^{\\max}_{i_0,j_0}\\right]\n\\end{equation}\nwhere $z^{\\min}_{i,j}$ and $z^{\\max}_{i,j}$ are the minimum and maximum $z$-values of the control points that affect the B-spline surface on the knot span product $\\left[u_{i_0},u_{i_0 + 1}\\right) \\times \\left[v_{j_0},u_{j_0 + 1}\\right)$, hence those with indices $i_0-p\\le i\\le i_0, j_0-q \\le j\\le j_0$. Formulated in terms of $Z(u,v)$ this yields\n\\begin{equation}\n    z^{\\min}_{i_0,j_0} \\le Z(u,v) \\le z^{\\max}_{i_0,j_0}, \\quad\n    (u,v) \\in \\left[u_{i_0},u_{i_0 + 1}\\right) \\times \\left[v_{j_0},v_{j_0 + 1}\\right).\n\\end{equation}\nExamples of such bounding boxes are shown in Fig.~\\ref{fig:bb_per_knotspanproduct}.\n\nThere are two steps in applying the bounding boxes in the intersection algorithm. First, a test for the entire surface (in $(u,v,z)$-space):\n\\begin{equation}\n    [0,1]^2 \\times \\left[\\min_{i,j}P^z_{i,j},\\max_{i,j}P^z_{i,j}\\right].\n\\end{equation}\nSecond, a recursive method where, starting with all knot span products, each rectangle of knot span products is divided into at most 4 sub-rectangles for a new bounding box test until individual knot span products are reached.\n\n\n\\paragraph{Triangle mesh intersection phase 2: $(u,v)$-space triangle intersection}\nEach non-trivial knot span product $[u_{i_0},u_{i_0+1}) \\times [v_{j_0},v_{j_0+1})$ is divided into a grid of $n_u$ by $n_v$ rectangles. Thus we can define the boundary points\n\\begin{subequations}\n    \\begin{align}\n         u_{i_0,k} =& u_{i_0} + k\\Delta u_{i_0},\n         \\quad \\Delta u_{i_0} = \\frac{u_{i_0+1}-u_{i_0}}{n_u}, \n         \\quad k = 0, \\ldots, n_u, \\\\\n         v_{i_0,\\ell} =&  v_{j_0}+ \\ell \\Delta v_{j_0}, \\quad \\Delta v_{j_0} = \\frac{v_{j_0+1}-v_{j_0}}{n_v},\n         \\quad \\ell = 0,\\ldots, n_v.\n     \\end{align}\n\\end{subequations}\n\nEach rectangle is divided into a lower left and an upper right triangle, as demonstrated in Fig.~\\ref{fig:uvtriangleint}. In this figure it is shown for a ray projected onto the $(u,v)$-plane in some knot span which triangles are candidates for an intersection in $(u,v,z)$-space. This is determined by the following rules:\n\\begin{itemize}\n    \\item A lower left triangle is intersected in the $(u,v)$-plane if either its left or lower boundary is intersected by the ray;\n    \\item an upper right triangle is intersected in the $(u,v)$-plane if either its right or upper boundary is intersected by the ray.\n\\end{itemize}\n\nThe intersection of these boundaries can be determined by finding the indices of the horizontal lines at which the vertical lines are intersected:\n\\begin{equation}\n    \\ell_k = \\left\\lfloor\\frac{ o_v+(u_{i_0,k}-o_u)\\frac{d_v}{d_u} - v_{j_0}}{\\Delta v_{j_0}}\\right\\rfloor,\n\\end{equation}\nand analogously $k_\\ell$.\n\n\\paragraph{Triangle mesh intersection phase 3: $u,v,z$-space triangle intersection}\nA lower left triangle can be expressed by a plane\n\\begin{equation}\n    T(u,v) = Au + Bv + C\n\\end{equation}\n    defined by the following linear system:\n\\begin{equation}\n    \\begin{pmatrix}\n    u_{i_0,k} & v_{j_0,\\ell} & 1 \\\\\n    u_{i_0,k+1} & v_{j_0,\\ell} & 1 \\\\\n    u_{i_0,k} & v_{j_0,\\ell+1} & 1\n    \\end{pmatrix}\n    \\begin{pmatrix}\n    A \\\\ B \\\\ C\n    \\end{pmatrix}\n    =\n    \\begin{pmatrix}[1.75]\n    z_{i_0,k}^{j_0,\\ell} \\\\ z_{i_0,k+1}^{j_0,\\ell} \\\\ z_{i_0,k}^{j_0,\\ell+1}\n    \\end{pmatrix}.\n\\end{equation}\nHere we use the following definition:\n\\begin{equation}\n    z_{i_0,k}^{j_0,\\ell} = Z(u_{i_0,k},v_{j_0,\\ell}).\n\\end{equation}\nThis yields the plane\n\\begin{align}\n    T(u,v) =&& z_{i_0,k}^{j_0,\\ell} + n_u\\left(z_{i_0,k+1}^{j_0,\\ell}-z_{i_0,k}^{j_0,\\ell}\\right)\\frac{u-u_{i_0,k}}{u_{i_0+1}-u_{i_0}} \\\\ \n    && +n_v\\left(z_{i_0,k}^{j_0,\\ell+1}-z_{i_0,k}^{j_0,\\ell}\\right)\\frac{v-v_{j_0,\\ell}}{v_{j_0+1}-v_{j_0}}. \\label{eq:trianglefuncdetermined}\n\\end{align}\nNote that to define this triangle, the B-spline basis functions are evaluated at fixed points in $[0,1]^2$ independent of the rays or the $P^z_{i,j}$. This means that for a lens that will be optimized these basis function values can be evaluated and stored only once rather than in every iteration, for computational efficiency.\n\nComputing the intersection with the ray $\\tilde{\\mathbf{r}}(t) = \\tilde{\\mathbf{o}} + \\tilde{\\hat{\\mathbf{d}}}t$ is now straight-forward, and yields\n\\begin{equation}\n    t_\\text{int} = - \\frac{C+\\langle\\tilde{\\mathbf{o}}, \\mathbf{n}\\rangle}{\\langle\\tilde{\\hat{\\mathbf{d}}},\\mathbf{n}\\rangle}, \\quad \\mathbf{n} = \n    \\begin{pmatrix} 0 \\\\ 1 \\\\ \\partial_u T\\end{pmatrix} \\times\n    \\begin{pmatrix} 1 \\\\ 0 \\\\ \\partial_v T\\end{pmatrix} = \n    \\begin{pmatrix}A \\\\ B \\\\ -1\\end{pmatrix},\n\\end{equation}\nwhere $\\mathbf{n}$ is a normal vector to the triangle, computed using the cross product. This also explains why $\\langle\\tilde{\\hat{\\mathbf{d}}},\\mathbf{n}\\rangle=0$ does not yield a well-defined result: in this situation the ray is parallel to the triangle.\n\nThe last thing to check is whether $\\tilde{l}(t_\\text{int})$ lies in the $(u,v)$-domain of the triangle, which can be checked by three inequalities for the three boundaries of the triangle:\n\n\\begin{subequations}\n    \\begin{align}\n        o_u + d_u t_\\text{int} \\ge u_{i_0,k} \\\\\n        0 \\leq o_v + d_v t_\\text{int} - v_{j_0,\\ell}< \\frac{n_u}{n_v}\\frac{v_{j_0+1}-v_{j_0}}{u_{i_0+1}-u_{i_0}}(u_{i_0,k+1}-(o_u + d_u t_\\text{int})).\n    \\end{align}\n\\end{subequations}\n\nThe computation for an upper right triangle is completely analogous. The upper triangle has a closed boundary, whereas the lower triangle has an open one and vice versa, which means that the $(u,v)$ domains of the triangles form an exact partition of $[0,1]^2$. Thus the triangle mesh is `water-tight', meaning that no ray intersection should be lost by rays passing in between triangles.\n\n\\subsection{Image reconstruction}\nThe ray tracing produces an irradiance distribution in the form of an image matrix $\\mathcal{I} \\in \\mathbb{R}^{n_x \\times n_y}_{\\ge 0}$, where the elements correspond to a grid of rectangles called pixels that partition the detector screen positioned at $z=z_\\text{screen} > \\max_{i,j} P_{i,j}^z$. The screen resolution $(n_x,n_y)$ and the screen radii $(R_x,R_y)$ together yield the pixel size\n\\begin{equation}\n    (w_x,w_y) = \\left(\\frac{2R_x}{n_x},\\frac{2R_y}{n_y}\\right).\n\\end{equation}\nFor reasons explained later in this section, sometimes a few `ghost pixels' are added, so the effective screen radii are\n\\begin{equation}\n    R_x^* := R_x + \\frac{\\nu_x - 1}{2}w_x, \\quad\n    R_y^* := R_y + \\frac{\\nu_y - 1}{2}w_y,\n\\end{equation}\nand the effective screen resolution is $(n_x + \\nu_x -1, n_y + \\nu_y - 1)$ where $\\nu_x$ and $\\nu_y$ are odd positive integers whose meaning will become clear later in this section.\n\n\nProducing the irradiance distribution from the rays that intersect the detector screen is called image reconstruction \\cite[sec. 7.8]{pharr2016physically}. The way that a ray contributes to a pixel with indices $i,j$ is governed by a reconstruction filter\n\\begin{equation}\n    F_{i,j} : [-R_x,R_x] \\times [-R_y,R_y] \\rightarrow \\mathbb{R}_{\\ge 0},\n\\end{equation}\nyielding for the irradiance distribution\n\\begin{equation}\n    \\mathcal{I}_{i,j} = \\sum_{k=1}^N \\omega_k F_{i,j}(\\mathbf{x}_k),\n\\end{equation}\nfor a set of ray intersections $\\{\\mathbf{x}_k\\}_{k=1}^N$ with corresponding final ray weights $\\{\\omega_k\\}_{k=1}^N$. The ray weights are initialized at the sampling of the ray at the source. They are slightly modified by the lens boundary interactions as a small portion of the light is reflected rather than refracted. The amount by which the ray weights are modified is governed by the Fresnel equations \\cite[sec. 2.7.1]{Fowles1975}. In our implementation, the Fresnel equations are approximated by Schlick's approximation \\cite[eq. 24]{Schlick1994}. In the current implementation, all ray weights are initialized equally. The precise value does not matter since the relationship between the initial and final weights is linear. The loss function (section \\ref{lossfunc}) compares scaled versions of the produced and target irradiance distribution.\n\nIn the simplest reconstruction case, the value of a pixel is given by the sum of the weights of the rays that intersect the detector screen at that pixel (called box reconstruction in \\cite[sec. 7.8.1]{pharr2016physically}). In this case the reconstruction filter of pixel $i,j$ is simply the indicator function of the pixel $\\left[(i-1)w_x,iw_x\\right) \\times \\left[(j-1)w_y,jw_y\\right)$.\n\nTo obtain a ray tracing implementation where the irradiance $\\mathcal{I}$ is differentiable with respect to geometry parameters of the lens, say, the parameter $\\theta$, the irradiance distribution must vary smoothly with this parameter. The dependency on this parameter is carried from the lens to the screen by the rays through the screen intersections $\\mathbf{x}_k = \\mathbf{x}_k(\\theta)$. Thus to obtain a useful gradient $\\frac{\\partial \\mathcal{I}}{\\partial \\theta}$ the filter function $F_{i,j}$ should be at least $C^1$, see Fig.~\\ref{fig:reconstructiondiffb} which is achieved by introducing a filter function that spreads out the contribution of a ray over a kernel of pixels of size $(\\nu_x,\\nu_y)$ centered at the intersection location. For the conservation of light, we require that $\\sum_{i,j}F_{i,j}(\\mathbf{x}) \\equiv 1$.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.7\\textwidth]{figures\/reconstruction_diffb.png}\n    \\caption{$\\mathbf{x}(\\theta)$ in the left plot shows the intersection location of a ray with the screen, dependent on a lens geometry parameter $\\theta$. The right plot then shows the reconstruction filter value for the green pixel in the left plot dependent on $\\theta$. In order to obtain a useful gradient of the pixel value with respect to $\\theta$, a smooth reconstruction filter is needed.}\n    \\label{fig:reconstructiondiffb}\n\\end{figure}\n\nTherefore, the Gaussian reconstruction function is introduced, based on the identically named one described in \\cite[sec. 7.8.1]{pharr2016physically}. This filter function is based on the product \n\\begin{equation}\n    \\tilde{F}_{i,j}(x,y;\\alpha,\\nu_x,\\nu_y) := f_{i}^x(x;\\alpha,\\nu_x)f_{j}^y(y;\\alpha,\\nu_y),\n\\end{equation}\nwhere\n\\begin{equation}\n    f_{i_0}^x(x;\\alpha,\\nu_x) = \n    \\begin{cases}\n        e^{-\\alpha\\left(x-c^x_{i_0}\\right)^2} - e^{-\\alpha\\left(\\frac{\\nu_x w_x}{2}\\right)^2} &\\text{ if } \\lvert x-c^x_i\\rvert < \\frac{\\nu_x w_x}{2},\\\\\n        0 & \\text{otherwise.}\n    \\end{cases} \\label{eq:filter1dim}\n\\end{equation}\nThe centers of the pixels are given by\n\\begin{equation}\n    (c_i^x,c_j^y) := \\left(\\left(i + \\textstyle\\frac{1}{2}\\right)w_x - R_x, \\left(j + \\textstyle\\frac{1}{2}\\right)w_y - R_y\\right).\n\\end{equation}\nNote that the support of $\\tilde{F}_{i,j}$ is of size $\\nu_xw_x$ by $\\nu_yw_y$, the size of the kernel on the detector screen. The normalized reconstruction filter is then given by\n\\begin{equation}\n    F_{i,j}(x,y;\\alpha,\\nu_x,\\nu_y) = \\frac{\\tilde{F}_{i,j}(x,y;\\alpha,\\nu_x,\\nu_y)}{\\sum_{i',j'}\\tilde{F}_{i',j'}(x,y;\\alpha,\\nu_x,\\nu_y)}.\n\\end{equation}\nThe function $F_{i,j}$ is plotted in Fig.~\\ref{fig:recfilter3d}. Note that the function is not differentiable at the boundary of its support, but this yields no problems in the optimization.\n\\begin{figure}\n    \\centering\n    \\includegraphics{figures\/Gaussianfilter.png}\n    \\caption{Gaussian reconstruction filter $F_{i_0,j_0}$ for $\\alpha = 1$ and $(\\nu_x,\\nu_y)=(3,3)$.}\n    \\label{fig:recfilter3d}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = \\textwidth]{figures\/image_reconstr_examples.png}\n    \\caption{Image reconstruction based on a small set of ray-screen intersections, for bincount and various reconstruction filter sizes and $\\alpha = 1$.}\n    \\label{fig:imagereconstr}\n\\end{figure}\n\nGaussian image reconstruction is shown in Fig.~\\ref{fig:imagereconstr} for various values of $\\nu_x = \\nu_y$. There is a trade-off here since the larger $\\nu_x$, and $\\nu_y$ are the blurrier the resulting image is, and the larger the computational graph becomes, but also the larger the section of the image is that is aware of a particular ray which yields more informative gradients. \n\nUp to this point, this section has discussed the ray tracing part of the pipeline, the next subsections will discuss the role of the neural network and the optimization.\n\n\\subsection{Multi-layer perceptron as optimization accelerator} \\label{subsec:nnarchitectures}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.6\\textwidth]{figures\/nn_architecture_dense.png}\n    \\caption{The dense multi-layer perceptron architecture based on the size of the control net $(n_1+1)\\times(n_2+1)$.}\n    \\label{fig:densenn}\n\\end{figure}\n\nSeveral neural network architectures are considered, all with a trivial input of 1, meaning that the neural networks will not, strictly speaking, be used to approximate a function since the considered domain is trivial. Non-trivial network inputs of system parameters like the source location will probably be part of follow-up research.\n\nIn this configuration, the neural network can be considered a transformation of the space over which is optimized: from the space of trainable neural network parameters to the space of control point $z$-coordinate values.\nThe goal of choosing the network architecture is that optimizing the trainable neural network parameters of this architecture yields better training behavior than optimizing the control point z-coordinate values directly. The used networks are multi-layer perceptions (MLPs), feed-forward networks consisting of several layers of neurons, as depicted in Fig.~\\ref{fig:densenn}. The considered architectures are:\n\\begin{enumerate}\n    \\item No network at all. \\label{item:nonetcase}\n    \\item A sparse MLP where the sparsity structure is informed by the overlap of the B-spline basis function supports on the knot spans. In other words: this architecture aims to precisely let those control points 'communicate' within the network that share influence on some knot span product on the B-spline surface, yielding a layer with the same connectivity as a convolutional layer with kernel size $(2p+1,2q+1)$. However, each connection has its own weight and each kernel its own bias, instead of only having a weight per element of the convolution kernel and one single bias for all kernels.\n    \\item Larger fully connected architectures are also considered, with 3 layers of control net size. Note that two consecutive such layers yield many weight parameters: $n^4$ for a square control net with `side length' $n$.\n\\end{enumerate}\nThe activation function used for all neurons is the hyperbolic tangent, which is motivated below.\n\n\\subsubsection{Control point freedom} \\label{subsec:controlpointfreedom}\nControl over the range of values that can be assumed by the control point $z$-coordinates is essential to make sure that the systems stays physical (as mentioned in Section~\\ref{subsubsec:lensconstraints}), but also to be able to take into account restrictions imposed on the lens as part of mechanical construction in a real-world application. Note that the restriction $P_{i,j}^z > z_\\text{in}$ for the control points being above the lens entrance surface is not critical for a collimated ray bundle simulation since, the entrance surface can be moved arbitrarily to the $-z$ direction without affecting the ray tracing. \n\nSince the final activation function $\\tanh$ has finite range $(-1,1)$, this can easily be mapped to a desired interval $(z_{\\min},z_{\\max})$:\n\\begin{equation}\n    y_{i,j} \\mapsto z_{\\min} + \\textstyle\\frac{1}{2} (y_{i,j} + 1)(z_{\\max} - z_{\\min}), \\label{eq:outputcorrection}\n\\end{equation}\nwhich can even vary per control point if desired. Here $y_{i,j}$ denotes an element of the total output $Y$ of the network.\nThe above can also be used as an offset from certain fixed values:\n\\begin{equation}\n    y_{i,j} \\mapsto f\\left(P^x_{i,j},P^y_{i,j}\\right) + z_{\\min} + \\textstyle\\frac{1}{2} (y_{i,j} + 1)(z_{\\max} - z_{\\min}). \\label{eq:outputcorrectionwfunc}\n\\end{equation}\nThe resulting B-spline surface approximates the surface given by $f(x,y) + \\textstyle\\frac{1}{2}(z_{\\max}+z_{\\min})$ if $Y \\approx 0$ can be used to optimize a lens that is globally at least approximately convex\/concave. The choice of the hyperbolic tangent activation function accommodates this: since this activation function is smooth around its fixed point $0$ when initializing the weights and biases of the network close to $0$, there is no cumulative value-increasing effect in a forward pass through the network so that indeed $Y\\approx 0$ in this case.\n\nFor comparability, in the case without a network, the optimization is not performed directly on the control point $z$-coordinates. Instead, for each control point, a new variable for optimization is created, which is passed through the activation function and the correction as in Eq.~\\ref{eq:outputcorrection} or \\ref{eq:outputcorrectionwfunc} before being assigned to the control point.\n\n\\subsection{The optimization}\n\n\\label{lossfunc}\nThe lens is optimized such that the irradiance distribution $\\mathcal{I}$ projected by the lens approximates a reference image $\\mathcal{I}_\\text{ref}$, where $\\mathcal{I} ,\\mathcal{I}_\\text{ref}\\in \\mathbb{R}^{n_x\\times n_y}_{\\ge 0 }$. The loss function used to calculate the difference between the two uses the normalized matrices: \n\\begin{equation}\n    \\widehat{\\mathcal{I}} = \\frac{\\mathcal{I}}{\\sum_{i,j}^{n_x,n_y} \\mathcal{I}_{i,j}} \n    \\quad\n    \\mathrm{and}\n    \\quad\n    \\widehat{\\mathcal{I}}_\\text{ref} = \\frac{\\mathcal{I}_\\text{ref}}{\\sum_{i,j}^{n_x,n_y} \\mathcal{I}_{\\mathrm{ref},i,j}}.\n\\end{equation}\n\n\\noindent\nThe loss function is given by\n\\begin{equation}\n    \\mathcal{L}(\\mathcal{I};\\mathcal{I}_\\text{ref}) = \\frac{1}{\\sqrt{n_x n_y}}\n   \\left\\| \\widehat{\\mathcal{I}}-\\widehat{\\mathcal{I}}_\\text{ref} \\right\\|_F \n    \\label{eq:pipelineLoss},\n\\end{equation}\nwhere $\\| \\cdot \\|_F$ is the Frobenius or matrix norm, which is calculated as follows:\n\\begin{equation}\n    \\| \\mathcal{A} \\|_F = \\sqrt{\\sum_{i}^{n_x}\\sum_{j}^{n_y} \\lvert a_{i,j} \\rvert ^2}.\n\\end{equation}\nFig.~\\ref{fig:optimizationloop} shows the conventional stopping criterion of the loss value being smaller than some $\\varepsilon > 0$, but in our experiments, we use a fixed number of iterations.\n\nThe neural network parameters (weights and biases) are updated using the Adam optimizer \\citep{Kingma2014}  by back-propagation of the loss to these parameters.\n\n\\section{Results} \\label{sec:results}\n\nSeveral results produced with the optimization pipeline discussed in the previous sections are displayed and discussed in this section. The implementation mainly uses PyTorch, a Python wrapper of Torch \\citep{Collobert2002Torch}.\n\nNone of the optimizations performed for this section took more than a few hours to complete, on a \\verb|HP ZBook Power G7 Mobile Workstation| with a \\verb|NVIDIA Quadro T1000 with Max-Q Design| GPU.\n\nMost of the results have been validated with \\emph{LightTools} \\citep{LightTools}, an established ray tracing software package in the optics community. Lens designs were imported to LightTools as a point cloud, then interpolated to obtain a continuous surface, and all simulations were conducted using $10^6$ rays.\n\nUnits of length are mostly unspecified since the obtained irradiance distributions are invariant under uniform scaling of the optical system. This invariance to scaling is reasonable as long as the lens details are orders of magnitude larger than the wavelength of the incident light such that diffraction effects do not play a role. Furthermore, the irradiance distributions are directly proportional to the scaling of all ray weights and thus the source power, so the source and screen power also need no unit specification. Note that relative changes have a non-trivial effect, like changes to the power proportion between sources or the distance proportions of the optical system.\n\n\\newpage\n\\subsection{Irradiance derivatives with respect to a control point}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.75\\textwidth]{figures\/results\/controlpoint_derivatives.png}\n    \\caption{Gradients of an irradiance distribution of a collimated ray bundle through a flat lens (parallel sides), with respect to the $z$-coordinate of one control point. The zeros are masked with white to show the extend of the influence of the control point. These irradiation distributions differ by: (a): degrees $(3,3)$, reconstruction filter size $(3,3)$, (b): degrees $(3,3)$, reconstruction filter size $(11,11)$, (c): degrees $(5,3)$, reconstruction filter size $(3,3)$.}\n    \\label{fig:controlpointderivs}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figures\/results\/renderderivsexplanation.png}\n    \\caption{Demonstration of how one control point influences the irradiance distribution in the case of a flat lens with B-spline degrees $(3,3)$ and a collimated ray bundle source.}\n    \\label{fig:renderderivsexplanation}\n\\end{figure}\nThis section gives a simple first look at the capabilities of the implemented differentiable ray tracer: computing the derivative of an irradiance distribution with respect to a single control point. Obtaining this data is inefficient in the current PyTorch implementation as a forward mode automatic differentiation pass is required, which is not currently (entirely) supported by PyTorch. Therefore these derivatives are computed with pixel-wise back-propagation.\n\nFig.~\\ref{fig:controlpointderivs} shows the derivative of an irradiance distribution produced by a collimated ray bundel through a flat lens for various B-spline degrees and reconstruction filter sizes, and Fig.~\\ref{fig:renderderivsexplanation} shows what one of these systems looks like. The overall `mountain with a surrounding valley' structure can be understood as follows: as one of the control points rises, it creates a local convexity in the otherwise flat surface. This convexity has a focusing effect, redirecting light from the negative valley region toward the positive mountain region.\n\nNoteworthy of these irradiance derivatives is also their total sum: (a) $\\SI{-1.8161e-08}{}$, (b) $\\SI{3.4459e-08}{}$, (c) $\\SI{9.7095e-05}{}$. These small numbers with respect to the total irradiance of about $93$ and therefore indicate conservation of light; as the control point moves out of the flat configuration, at first, the total amount of power received by the screen will not change much. This is expected from cases (a) and (b), where the control point does not affect rays that reach the screen on the boundary pixels. However, in all cases, all rays intersect the lens at right angles. Around $\\theta = 0$, the slope of Schlick's approximation is very shallow, indicating a small decrease in refraction in favor of reflection.\n\n\\subsection{Sensitivity of the optimization to initial state and neural network architecture} \\label{subsec:results_nnsensitivity}\nAs with almost any iterative optimization procedure, choosing a reasonable initial guess of the solution is crucial for reaching a good local\/global minimum. For training neural networks, this comes down to how the network weights and biases are initiated. In this section, we look at three target illuminations: the circular top hat distribution (Fig.~\\ref{fig:circtophat}), the TU Delft logo (Fig.~\\ref{fig:TUDflameinv}), and an image of a faceted ball (Fig.~\\ref{fig:facball}). For some experiments, black padding or Gaussian blurring is applied to these images. We design lenses to produce these distributions from a collimated ray bundle, given various neural network architectures (introduced in section \\ref{subsec:nnarchitectures}) and parameter initializations.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{figures\/results\/circular_tophat.png}\n    \\caption{The circular tophat target illumination.}\n    \\label{fig:circtophat}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{figures\/TUD_400_inverse.png}\n    \\caption{The TU Delft flame target illumination.}\n    \\label{fig:TUDflameinv}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.4\\textwidth]{figures\/results\/shaded_faceted_ball.png}\n    \\caption{The faceted ball target illumination.}\n    \\label{fig:facball}\n\\end{figure}\n\n\\paragraph{Circular top hat distribution from collimated ray bundle} \\label{subsec:circtophat}\nFig.~\\ref{fig:tophatloss} shows the progress of the loss over 1000 iterations, with each iteration taking $2.5$ seconds, for various neural network architectures and parameter initialization combinations. For the other parameters in these simulations, see the supplementary information. For a few moments during the training, the resulting freeform surfaces and irradiance distributions are shown in Figs.~\\ref{fig:lensshapes}, \\ref{fig:tophatrandomsparse}, \\ref{fig:tophatunifsparse}, \\ref{fig:tophatunifnonet} and \\ref{fig:tophatunifdense}. Uniform here means that the initial trainable parameter values are sampled from a small interval: $U\\left(\\left[-10^{-4},10^{-4}\\right]\\right)$, except for the no-network case; this is initialized with all zeros.\n\nThe first notable difference is between the random and uniformly initialized sparse neural networks. The uniformly initialized neural network performs much better, and no network performs better. This is probably because the uniformly initialized cases converge to a better (local) minimum than the randomly initialized case. Of course, it could happen that the random initialization lands in a very favorable spot in the design landscape, but intuitively this seems very unlikely.\n\nAnother property of the uniformly initialized cases is their preservation of symmetry in these setups. As Fig.~\\ref{fig:lensshapes} shows, this leads to much simpler lenses, which are probably much less sensitive to manufacturing errors due to their relative lack of small detail. What is interesting to note here is that if the sparse network is initialized with all parameters set to $0$, then its optimization is identical to the no-network case, as only the biases in the last layer achieve non-zero gradients.\n\nNo rigorous investigation has been conducted to the extent that this behavior of increased convergence speed carries over to other target distributions and system configurations and what the optimal hyper-parameters are. A thorough investigation of the hyper-parameter space that defines a family of network architectures could reveal where in the increase of the architecture complexity, diminishing returns for optimizing these lenses arises. However, based on these initial findings the fully connected network is used for all the following optimizations in the results.\n\n\n\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{figures\/results\/tophat_loss_progress.png}\n    \\caption{Loss progress over the iterations for various pipeline-setups for forming a tophat distribution from a collimated ray bundle.}\n    \\label{fig:tophatloss}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figures\/results\/tophat_lens.png}\n    \\caption{The lens height field after initialization ($n=0$), and $n=50,100$ and $1000$ iterations respectively, for different network architectures (Section~\\ref{subsec:nnarchitectures}) and network parameter initializations (Section~\\ref{subsec:circtophat}).}\n    \\label{fig:lensshapes}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figures\/results\/tophat_randomsparse.png}\n    \\caption{Irradiance distributions and pixel-wise errors in the optimization progress of a random lens with a sparse network towards a circular tophat illumination.}\n    \\label{fig:tophatrandomsparse}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figures\/results\/tophat_unifsparse.png}\n    \\caption{Irradiance distributions and pixel-wise errors in the optimization progress of a flat lens with a sparse network towards a circular tophat illumination.}\n    \\label{fig:tophatunifsparse}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figures\/results\/tophat_unifnonet.png}\n    \\caption{Irradiance distributions and pixel-wise errors in the optimization progress of a flat lens without a network towards a circular tophat illumination.}\n    \\label{fig:tophatunifnonet}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figures\/results\/tophat_unifdense.png}\n    \\caption{Irradiance distributions and pixel-wise errors in the optimization progress of a flat lens with a dense network towards a circular tophat illumination.}\n    \\label{fig:tophatunifdense}\n\\end{figure}\n\n\\clearpage\n\\paragraph{TU flame and faceted ball from collimated ray bundle}\nIn what follows, we consider complex target distributions: the TU Delft flame (for a complex shape) and a faceted ball (for a target with various brightness levels). Here we still use the collimated ray bundle illumination, but lenses are now optimized for various magnifications; see Table.~\\ref{tab:magndata}. These magnifications are defined as the scaling of the screen size with respect to the smallest screen size $(0.64,0.64)$.  The other parameters of these optimizations are shown in the supplementary information. All these iterations took about $4$ seconds each.\n\nThe final irradiance distributions and corresponding LightTools results are shown in Figs.~\\ref{fig:flamerenders} and \\ref{fig:facetedballrenders}, respectively. These figures show that the optimization pipeline can handle these more complex target illuminations well. The LightTools results predict some artifacts within the irradiance distribution, which the implemented ray tracer does not, especially in the TU flame magnification 1 case. By visual inspection, based on the LightTools results, one would probably rate these results in the exact opposite order than as indicated by the losses shown in Fig.~\\ref{fig:ManufacturingLoss}.\n\nA potential explanation of the increase in loss with the magnification factor in Fig.~\\ref{fig:ManufacturingLoss} is that the bigger the screen is: the rays require higher angles to reach the edges of the screen, which is apparent in the cases of magnification 3 and 5 Fig.~\\ref{fig:ManufacturingRays}. This results in a larger sensitivity of the irradiance to the angle with which a ray leaves the screen. This in turn gives larger gradients of the irradiance with respect to the control points. Therefore the optimization takes larger steps in the neural network parameter space, possibly overshooting points that result in a lower loss.\n\nFor the magnification, $3$ and $5$, the irradiance distributions from LightTools show artifacts at the screen boundaries. A possible explanation for this is that the way the B-spline surfaces are transferred to LightTools is inaccurate at the surface boundaries.\\footnote{Assuming only rays from the B-spline surface boundaries reach the screen boundary area.} This is because surface normals are inferred from fewer points on the B-spline surface at the boundary than in the middle of the surface by LightTools.\n\nFurthermore, a significant amount of rays are lost during optimization because the target illuminations are black at the borders, so rays near the screen boundary will be forced off the screen by the optimization. Once rays are off the screen, they no longer contribute to the loss function. Once a ray misses the screen, the patch on the B-spline surface these rays originate from does not influence the irradiance and, thus, the loss function. \nHowever, this does not mean that this patch is idle for the rest of the optimization, as this patch can be in support of a basis function that corresponds to a control point that still affects rays that hit the screen. Therefore, the probability of getting idle lens patches with this setup decreases with the B-spline degrees since these determine the size of the support of the B-spline basis functions but might, in some cases, lead to oscillatory behavior, with rays alternating between hitting and missing the screen. \n\nFig. \\ref{fig:ManufacturingSurfaces} shows the optimized B-spline lens surface height field. A densely varying color map is chosen since the deviations from a flat or smooth concave shape are quite subtle, which is due to the large lens exit angle sensitivity of the ray-screen intersections since the ratio lens size to screen size is large with respect to the ratio lens size to screen distance.\n\n\\begin{table}[h]\n    \\centering\n    {\n    \\def1.5{1.5}\n        \\begin{tabular}{c|c|c|c}\n        \\textbf{Magnification} & \\textbf{screen size} & $f(x,y)$ & \\textbf{starting shape type}\\\\\n        \\hline\n        $1$ & $(0.64,0.64)$ & $\\textstyle\\frac{1}{2}$ & flat\\\\\n        $3$ & $(1.92,1.92)$ & $\\textstyle\\frac{1}{2} + 8 - \\sqrt{8^2-x^2-y^2}$ & concave\\\\\n        $5$ & $(3.20,3.20)$ & $\\textstyle\\frac{1}{2} + 4 - \\sqrt{4^2 - x^2 - y^2}$ & concave\n        \\end{tabular}\n    }\n    \\caption{The screen size and control point offset function $f$ used per magnification in the TU flame and faceted ball optimizations (distances in centimeters).}\n    \\label{tab:magndata}\n\\end{table}\n\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{figures\/results\/Manufacturing_Loss.png}\n    \\caption{Loss progress for the various magnifications and target distributions.}\n    \\label{fig:ManufacturingLoss}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/results\/Manufacturing_flames.png}\n    \\caption{Implementation and LightTools irradiance distributions of the TU flame target from the final lens design of the optimization.}\n    \\label{fig:flamerenders}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/results\/Manufacturing_balls.png}\n    \\caption{Implementation and LightTools irradiance distributions of the faceted ball target from the final lens design of the optimization.}\n    \\label{fig:facetedballrenders}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/results\/Manufacturing_surfaces.png}\n    \\caption{The lens designs for the different magnifications and two target distributions.}\n    \\label{fig:ManufacturingSurfaces}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/results\/Manufacturing_rays.png}\n    \\caption{$25\\times25$ traced rays through the final lens designs for the different magnifications and two target distributions.}\n    \\label{fig:ManufacturingRays}\n\\end{figure}\n\n\n\n\n\\clearpage\n\\subsection{Optimization with a point source and a grid of point sources}\nWe now consider an optimization that uses the B-spline intersection algorithm. First, we design a lens with one point source at $(0,0,-5)$ with $5\\times 10^5$ rays to again form the TU flame. Then after $\\sim 200$ iterations, we change the source to an equispaced grid of $25\\times 25$ point sources with $10^3$ rays each on $[-1,1]\\times[-1,1]\\times\\{-5\\}$, approximating a source of non-negligible size. The other (hyper-)~parameters of this optimization are shown in the supplementary information. Due to the additional B-spline intersection procedures, each iteration takes approximately $50$ seconds.\nThe resulting final irradiance distribution and LightTools verifications can be seen in Fig.~\\ref{fig:point_source_renders}. The final irradiance distribution similar to the that obtained by LightTools, indicating that ray tracing with the implemented B-spline intersection algorithm works correctly. The irradiance are blurred due to the reconstruction filter.\nThe single-source point optimization performs well, although the illumination is less uniform than in the collimated ray bundle case (Figs.~\\ref{fig:flamerenders} and \\ref{fig:facetedballrenders}). \nThe non-uniformity can be attributed to the gaussian reconstruction filter used during optimization, as it smoothes out the small uniformities. \n\n\nAs seen in Fig.~\\ref{fig:point_source_renders} the irradiance distribution obtained with a grid of point sources accurately approximates the extended source illumination distribution quite well for the unoptimized case. \nFinding a lens design that redirects light from a source of non-negligible size into a desired irradiance distribution is a complex problem for which it is hard to indicate how good the optimal irradiance distribution can become.\nThe progress of the loss, as seen in Fig.~\\ref{fig:point_source_loss}, shows that the optimization can still improve the loss, even after the transition to the grid of point sources. Interestingly, looking at Fig.~\\ref{fig:point_source_renders} again, the optimization seems to adopt the coarse strategy of filling up the target distribution with images of the source square, as shown in Fig.~\\ref{fig:sourceimages}. This strategy does hinder the possible quality of the final irradiance distribution as the image of the source on the target is larger than the fine details in the desired irradiance. Optimizing both the front and back surfaces of the freeform could resolve this issue, as this will cause the image of the source to change shape depending on where it ends up on the target screen.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{figures\/results\/source_images_indication.png}\n    \\caption{Indication of images of the source square in the irradiance distribution obtained by LightTools using the point source grid.}\n    \\label{fig:sourceimages}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/results\/point_source_renders.png}\n    \\caption{The final irradiation distribution of the lens optimizations with point sources and the corresponding LightTools verifications. The extended source is not implemented in our ray tracer, but is approximated by the point source grid.}\n    \\label{fig:point_source_renders}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width=0.75\\textwidth]{figures\/results\/point_source_surfaces.png}\n    \\caption{Height fields of the lenses optimized for the TU flame with point sources.}\n    \\label{fig:point_source_surfaces}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\centering\n    \\includegraphics[width= 0.6\\textwidth]{figures\/results\/point_source_loss.png}\n    \\caption{Loss over the iterations optimizing for the TU flame. The system is initiated with a point source, and after $\\sim 200$ iterations the point source is replaced by an equispaced grid of $25\\times 25$ point sources.}\n    \\label{fig:point_source_loss}\n\\end{figure}\n\n\\section{Conclusion} \\label{sec:conclusion}\n\nWe demonstrated that non-sequential differentiable ray tracing is a viable tool for designing freeform lenses for collimated ray bundles, points, and extended sources.\nUsing a B-spline allows for the design of a continuous surface, which is desirable for manufacturing, and its control point allows for locally altering the irradiance distribution. For both cases, collimated and point source lens designs were found that could accurately project the desired irradiance distribution in both the differentiable ray tracer and in commercial software LightTools. Some artifacts still exist and resolving this issue will be a part of further research.\n\nFor the source with a finite extent, the optimizer improved upon the design obtained for a point source. However, the final irradiance distribution was made up of images of the source, which hinders the minimum that can be obtained as the source image is larger than the details in the desired irradiance distribution. This issue can be resolved by optimizing multiple surfaces simultaneously, as the image of the source on the target plane can then be optimized to vary with location.\n\nUsing a neural network to remap the optimization space provides an interesting way to increase the convergence speed of the optimization. However, further investigation is required to see whether this generally holds and what the effect is on other network architectures.\n\nThe developed ray tracing implementation is currently a proof of concept and needs to be optimized for speed. The B-spline intersection algorithm, in particular, adds roughly a factor of $10$ to the computation time. A significant speedup can be achieved here by leveraging efficient lower-level GPU programming languages, such as CUDA. \n\n\n\\section{Acknowledgements}\nWe acknowledge support by NWO-TTW Perspectief program (P15-36) ``Free-form scattering optics\".\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nHigh-redshift star-forming galaxies are becoming an important probe of galaxy\nformation, reionization and cosmology (Robertson et al. 2010; Shapley 2011).  A\npopular method for finding high redshift star forming galaxies is to target\ntheir often bright Ly$\\alpha$ emission (Partridge \\& Peebles 1967). This\nemission can be easily detected in narrow-band imaging surveys, and can be\nfurther confirmed by spectroscopic observations (Hu et al. 1998; Ouchi et al.\n2008; Yamada et al.  2012a,b). In addition to discovering numerous Ly$\\alpha$\nemitters (LAEs), a particular class of objects, also known as \"Ly$\\alpha$\nblobs\" (LABs), has been most commonly found in the dense environment of\nstar-forming galaxies at high redshift, and they are very extended (30 to 200\nkpc) and Ly$\\alpha$-luminous (10$^{43}$ to 10$^{44}$ erg~s$^{-1}$) (see,\ne.g., Francis et al. 1996; Steidel et al. 2000; Palunas et al. 2004; Matsuda et\nal. 2004, 2009, 2011; Dey et al. 2005; Saito et al. 2006; Yang et al. 2009,\n2010; Erb et al.  2011; Prescott et al. 2012a, 2013; Bridge et al. 2013). In\ncontrast to the large Ly$\\alpha$ nebulae surrounding some high-redshift radio\ngalaxies (e.g., Reuland et al. 2003; Venemans et al. 2007), these objects do\nnot always have obvious sources for energy responsible for their strong\nemission.\n\nWhile the LABs' preferential location in overdense environments indicates an\nassociation with massive galaxy formation, the origin of Ly$\\alpha$ emission in\nthe LABs is still unclear and under debate (Faucher-Giguere et al. 2010; Cen \\&\nZheng 2013; Yajima et al. 2013). Proposed sources have generally fallen into\ntwo categories: cooling radiation from cold streams of gas accreting onto\ngalaxies (e.g., Haiman et al. 2000; Dijkstra \\& Loeb 2009; Goerdt et al. 2010)\nand photoionization\/recombination from starbursts or active\ngalactic nuclei (AGNs) (e.g., Taniguchi \\& Shioya 2000; Furlanetto et al. 2005;\nMori \\& Umemura 2006; Zheng et al. 2011). Supporting evidence for the cooling\nflow scenario comes from those LABs lacking any visible power source (e.g.,\nNilsson et al. 2006; Smith \\& Jarvis 2007). Ionizing photons from young stars\nin star-forming galaxies and\/or AGNs can ionize neutral hydrogen atoms and the\nsubsequent recombination gives off Ly$\\alpha$\\, emission. The resonant scattering of\nLy$\\alpha$\\, photons in the circumgalactic medium makes the emission extended (Geach\net al.  2005, 2009; Colbert et al. 2006, 2011; Beelen et al. 2008; Webb et al.\n2009; Zheng et al. 2011; Cen \\& Zheng 2013; Overzier et al. 2013). \n\nExcept for cooling flows and photoionization from star-forming galaxies and\/or\nAGNs, other possible mechanisms, such as galactic super-winds and obscured AGNs,\nare also proposed to explain the nature of the LABs (e.g., Ohyama et al. 2003;\nWilman et al. 2005; Colbert et al. 2006; Matsuda et al. 2007). All these\nsources of energy may be activated in an environment where violent interactions are\nfrequent between gas rich galaxies as expected in over-dense regions at high\nredshift (Matsuda et al. 2009, 2011; Prescott et al. 2012b; Kubo et al. 2013).\n\nThe 110 Mpc filament with 37 LAEs related to the protocluster J2143-4423 at\n$z$=2.38 (Francis et al. 1996, 2004; Palunas et al. 2004) is one of the largest\nknown structures at high redshift, and this field also includes four large\nextended LABs with extensions of $\\sim$ 50 kpc and above, named {B1}, {B5}, {B6}\\,\nand {B7}. In this paper, we present our deep radio observations and {\\it\nHerschel} released far-infrared (FIR) data in J2143-4423\\, to study the powering\nsource of these LABs. Throughout this paper, we use a $\\Lambda$ cosmology with\n$\\rm H_{\\rm0}$ = 67.3~$\\rm \\ifmmode{{\\rm \\ts km\\ts s}^{-1}}\\else{\\ts km\\ts s$^{-1}$}\\fi\\ Mpc^{-1}$, $\\rm \\Omega_\\Lambda$ = 0.685 and\n$\\rm \\Omega_{\\rm m}$ = 0.315 (Planck Collaboration XVI 2013), and 1$\\arcsec$\ncorresponds to 8.37~kpc at $z$=2.38.\n\n\n\\section{Observations}\n\\subsection{ATCA observations}\nWe observed J2143-4423\\, with the Australia Telescope Compact Array\n(ATCA)\\footnote{The Australia Telescope Compact Array is part of the Australia\nTelescope, which is funded by the Commonwealth of Australia for operation as a\nNational Facility managed by CSIRO.} in its extended configuration 6A. During\nthe observations from 2009 June 14 to 17, only five out of six antennas were\navailable. The observations were performed at a central frequency of 1.75 GHz.\nWe used the Compact Array Broadband Backend (Wilson et al. 2011) in a\nwide-band mode, with a total bandwidth of 2~GHz and a channel width of 1~MHz.\nThe nearby source PKS~2134-470 served as a gain calibrator. Absolute fluxes\nwere calibrated with the ATCA standard PKS~1934-638. The total observing time\nwas about 70 hours. \n\nThe data were reduced with the MIRIAD software package. Although the observations\nwere carried out with a total bandwidth of 2~GHz, the effective bandwidth was\nabout 489 MHz with a central frequency of 1.51~GHz. We carefully flagged the\nchannels affected by radio frequency interference (RFI) by checking the\nvisibility data sorted by time, channels and baselines.\nThe image was deconvolved with MIRIAD task MFCLEAN, and\ntask SELFCAL was used to reduce the noise from strong radio continuum sources.\nWe first created cleaned images in a normal procedure and made model images for\nthe strong sources. The models were used as inputs for task SELFCAL to perform\nself-calibration of visibility data. We ran this cycle for three times, and\nthen obtained the model images to create the visibility data with\nself-calibration, which were used to make the final images. The noise of the\nimages after applying self-calibration was about one order of magnitude lower\nthan that without self-calibration. The field of view was about 31 arcmins and\nthe synthesized beam size was 7.8$\\arcsec$$\\times$4.8$\\arcsec$. The noise was\nabout 15~$\\mu$Jy\/beam before applying primary beam correction.\n\n\\begin{figure*}[t]\n\\vspace{-0.0cm}\n\\centering\n\\includegraphics[angle=0,width=1.0\\textwidth]{fig1.pdf}\n\\vspace{-0.0cm}\n\\caption{ATCA 20~cm, {\\it Spitzer} MIPS 24$\\mu$m and {\\it Herschel}\\, PACS and SPIRE data for the four Ly$\\alpha$\\, blobs\n(LABs) in J2143-4423.  {\\bf a) }Contours and gray scale maps of ATCA radio\nemission.  The contours are -2, 2, 3, 4, 5 and 6~$\\times$ 15~$\\mu$Jy\n(1~$\\sigma$), with a synthesized beam of 7.8$\\arcsec$$\\times$4.8$\\arcsec$,\nwhich is shown in the lower left corner of each panel.  {\\bf b) }Gray maps of\n{\\it Spitzer} MIPS 24~$\\mu$m emission (Colbert et al. 2006).  {\\bf c-g)\n}Contours and gray scale maps of {\\it Herschel}\\, FIR emission.  The contours are\n-2$\\sigma$, 2$\\sigma$, 3$\\sigma$, 4$\\sigma$, 5$\\sigma$ and 6$\\sigma$ (see\n$\\S$~\\ref{obsher} for the noise level of each band). A circle with a diameter\nof 40$\\arcsec$ is shown in each panel. The circles in\n{B7}\\, are on an off-center position (5$\\arcsec$, 0$\\arcsec$) to cover most FIR\nemission. All sources are centered on the positions of the four LABs (see\nColbert et al.  2006) as shown with plus signs in each panel. All offsets are\nrelative to the positions of the LABs.}\n\\label{map} \n\\end{figure*}\n\n\\subsection{Archival {\\it Herschel}\\, observations}\\label{obsher}\n{\\it Herschel}\\, observations towards J2143-4423\\, were carried out with PACS (Poglitsch et\nal. 2010) at 100 and 160~$\\mu$m and SPIRE (Griffin et al. 2010) at 250, 350 and\n500~$\\mu$m in 2010 to 2011. J2143-4423\\, was imaged in a field size of\n15$^\\prime$$\\times$15$^\\prime$ for each band, and the observing time was\n$\\sim$2.9 hours for PACS ({\\it Herschel}\\, OD: 686) and $\\sim$0.6 hours for SPIRE ({\\it Herschel}\\,\nOD: 558). The level 2.5 product for PACS and the level 2 product for SPIRE from\nthe pipeline procedures are used for our data analysis. Source photometry\nis carried out using DAOphot algorithm in the Herschel Interactive Processing\nEnvironment (HIPE). We apply beam correction, colour correction, aperture\ncorrection for a spectral index of $-$2 and adopt a flux calibration error of 5\\%\nat PACS bands and 7\\% at SPIRE bands as recommended in the PACS and SPIRE\nObserver's Manual. The full width at half power (FWHP) beam sizes are\n6.8$\\arcsec$ at 100~$\\mu$m, 11.4$\\arcsec$ at 160~$\\mu$m, 17.6$\\arcsec$ at\n250~$\\mu$m, 23.9$\\arcsec$ at 350~$\\mu$m and 35.2$\\arcsec$ at 500~$\\mu$m,\nrespectively.\n\n\n\\section{Results}\n\\subsection{Radio emission from ATCA observations}\nIn Fig.~\\ref{map}(a) we present the radio continuum emission images at 20~cm\nfrom the ATCA. Among the four LABs, {B6}\\, and {B7}\\, are detected with fluxes\nof 67\\ppm17~$\\mu$Jy and 77\\ppm16~$\\mu$Jy, respectively, and {B5}\\, is marginally\ndetected at 3~$\\sigma$ (51\\ppm16~$\\mu$Jy). For all detected sources, their\npositions are consistent with the central positions of the LABs.  Only {B1}\\, is\nnot detected by the observations. \n\n\\subsection{FIR emission from {\\it Herschel}\\, observations}\nAll four LABs are observed with {\\it Herschel}\\, PACS at 100 and 160~$\\mu$m and SPIRE at\n250, 350 and 500~$\\mu$m, and the images are shown in Fig.~\\ref{map}(c-g). The\nobserved flux densities are calculated for the areas within the blue circles as\nshown in Fig.~\\ref{map} and are listed in Table~\\ref{tab1}. {B1}\\, is not\ndetected but contaminated by a nearby strong source about 20$\\arcsec$ in the\nnorth-west,  which is the background QSO LBQS2138-4427 at $z$\\,=\\,3.2 (Francis \\&\nHewett 1993), and its emission features at different FIR bands appear to reach\nout to {B1}\\, from this location. There is no FIR counterpart for {B5}\\, in any\n{\\it Herschel}\\, band.\n\n\\begin{center}\n\\begin{table*}[t]\n\\centering\n\\caption{Observational and derived parameters towards the four LABs$^a$}\\label{tab1}\n\\begin{tabular}{ccccccccc}\n\\hline\nSource & 20~cm$^{b}$ & 100~$\\mu$m & 160~$\\mu$m & 250~$\\mu$m & 350~$\\mu$m & 500~$\\mu$m  \n& \\ifmmode{L_{\\rm FIR}}\\else{$L_{\\rm FIR}$}\\fi$^c$  & M$\\rm_{dust}$\\\\ \n       & [$\\mu$Jy]   & [mJy]  & [mJy]  &  [mJy]   &  [mJy] & [mJy]     &  [10$^{12}$\\ifmmode{L_{\\odot}}\\else{$L_{\\odot}$}\\fi]  & [$10^8$\\ifmmode{{\\rm M}_{\\odot} }\\else{M$_{\\odot}$}\\fi] \\\\\n\\hline\nB1  & $<$51    &   $<$4.2   &  $<$9.0  &  $<$17.9  &  $<$19.6  &   $<$22.5    &  $<$2.8 & \\\\\nB5  & 51\\ppm16 &   $<$2.1   &   $<$11.1  & $<$17.5  & $<$18.7  &   $<$19.8    &  $<$2.5 & \\\\\nB6  & 67\\ppm17 &  13.2\\ppm3.2   &  53.9\\ppm8.0  & 49.7\\ppm9.0  & 53.7\\ppm10.7  &  36.7\\ppm10.3    & 10.0\\ppm1.9  & 3.2\\ppm0.6 \\\\\nB7  & 77\\ppm16 &  12.9\\ppm4.0   &  33.5\\ppm10.0  & 41.6\\ppm7.8  & 48.0\\ppm10.6  &  39.2\\ppm8.6    & 8.6\\ppm2.3  & 5.0\\ppm1.0\\\\\n\\hline\n\\end{tabular}\n\\begin{list}{}{}\n\\item{$^{\\mathrm{a}}$ The wavelengths shown in this table are the redshifted values.}\n\\item{$^{\\mathrm{b}}$ Measured fluxes have been modified by a primary beam correction (less than 15\\%).}\n\\item{$^{\\mathrm{c}}$ The total luminosities are calculated between rest frame\nwavelengths of 40~$\\mu$m to 200~$\\mu$m from the dust models (see\n$\\S$~\\ref{dust} for details). The 3~$\\sigma$ upper limits are given for undetected sources.} \n\\end{list}\n\\end{table*}\n\\end{center}\n\n\\subsection{Redshifts of the FIR sources}\\label{red}\nTo estimate the redshift of the FIR sources associated with {B6}\\, and {B7}, we\ntry to fit the data with the SEDs of different templates (Polletta et al.\n2007) at different redshifts and find that the starburst templates can well\nreproduce the data.  With the observational data and the SEDs of the templates,\nthe minimum reduced $\\chi^2$ value for each redshift can be calculated and the\ncorresponding probability can be estimated. In this analysis, we include\nfive {\\it Herschel}\\, band, APEX 870~$\\mu$m data (Beelen et al. 2008), and {\\it Spitzer}\nMIPS 24~$\\mu$m data (Colbert et al.  2006).\n\nAmong four typical templates, Arp~220, M~82, Mrk~231 and NGC~6240, we find\nthat the spectral energy distribution of starburst galaxies NGC 6240 and Arp 220\nfit the data best, and Mrk 231 doesn't fit well because it has warm IR emission \nfrom its AGN which is not really consistent with the data.\nFig.~\\ref{redshift} shows the probability distribution\nagainst redshift for both LABs. The estimated redshifts are\n2.20$^{+0.30}_{-0.35}$ for {B6}\\, and 2.20$^{+0.45}_{-0.30}$ for {B7}\\,\nrespectively. Considering the uncertainty of this method to determine the\nredshifts, both values are consistent with the Ly$\\alpha$\\, redshift of 2.38 of the\nLABs. Adopting the number count study of {\\it Herschel}\\, sources in Clements et al.\n(2010), the probability of finding a 350 $\\mu$m source with a flux greater than 40 mJy\nwithin 20 arcsec is 2\\%. According for\nsuch a low number density of strong FIR sources and the positional coincidence\nof the LABs with strong FIR sources, the FIR sources are very likely associated\nwith the LABs. Nevertheless, future spectroscopic observations from molecular\nlines at millimeter or from forbidden lines at near-infrared will be quite\nimportant to confirm it. In the following sections, the Ly$\\alpha$\\, redshift of 2.38\nwill be adopted for the LABs.\n\n\n\\begin{figure*}[t]\n\\vspace{-0.0cm}\n\\centering\n\\includegraphics[angle=0,width=1.0\\textwidth]{fig2.pdf}\n\\vspace{-0.0cm}\n\\caption{Probability as a function of redshift for {B6}\\, and {B7}. NGC~6240 and\nArp~220 are adopted as the most appropriate starburst templates for B6 and B7, respectively.\nA red vertical line denotes a redshift of 2.38 for\nLy$\\alpha$\\, emission.}\n\\label{redshift} \n\\end{figure*}\n\n\\subsection{Dust properties}\\label{dust}\nFor {B6}\\,and {B7}, we have included the measurements from the five {\\it Herschel}\\, bands\nas well as the 870~$\\mu$m data taken from Beelen et al. (2008)\nin the dust continuum analysis using a single-component dust model as described in\nWei\\ss\\ et al. (2007). \n{\\it Spitzer} MIPS 24~$\\mu$m data (Colbert et al. 2006) are not used in the\nmodel fitting because they are strongly affected by PAH features, but are shown\nin Fig.~\\ref{sed} to allow for a better comparison with overlaid templates.\nWe find a dust temperature, $T\\rm_{dust}$, of\n70\\ppm5 K and a dust mass, M$\\rm_{dust}$, of (3.2\\ppm0.8)$\\times$10$^8$ M$\\rm\n_\\odot$ for {B6}, and $T\\rm_{dust}$\\,=\\,70\\ppm5 K and\nM$\\rm_{dust}$\\,=\\,(5.0\\ppm1.0)$\\times$10$^8$ M$\\rm _\\odot$ for {B7}, respectively. \nThe implied FIR luminosities are \\ifmmode{L_{\\rm FIR}}\\else{$L_{\\rm FIR}$}\\fi\\, = (10.0\\ppm1.9)$\\times$10$^{12}$\\,\\ifmmode{L_{\\odot}}\\else{$L_{\\odot}$}\\fi\\,\nfor {B6}, and \\ifmmode{L_{\\rm FIR}}\\else{$L_{\\rm FIR}$}\\fi\\, = (8.6\\ppm2.3)$\\times$10$^{12}$\\,\\ifmmode{L_{\\odot}}\\else{$L_{\\odot}$}\\fi\\, for {B7},\nrespectively, where \\ifmmode{L_{\\rm FIR}}\\else{$L_{\\rm FIR}$}\\fi\\, is integrated from 40~$\\mu$m to 200~$\\mu$m in the\nrest frame. The upper \\ifmmode{L_{\\rm FIR}}\\else{$L_{\\rm FIR}$}\\fi\\,limits for both {B1}\\, and {B5}\\, are\n$\\sim$2.5$-$2.8$\\times$10$^{12}$\\,\\ifmmode{L_{\\odot}}\\else{$L_{\\odot}$}\\fi.\n\n\n\\begin{figure*}[t]\n\\vspace{-0.0cm}\n\\centering\n\\includegraphics[angle=0,width=1.0\\textwidth]{fig3.pdf}\n\\vspace{-0.0cm}\n\\caption{Single-component dust models for B6 and B7 (a redshift of 2.38 is\nadopted). The black solid lines show the thermal dust continuum emission\nof the 70~K dust components for both {B6} and {B7}. The open circles represent\nthe measurements at five {\\it Herschel}\\, bands in this paper and the filled circles\nindicate the flux densities at 24~$\\mu$m (Colbert et al.  2006).  The filled\nsquare denotes the flux density (or its upper limit) at 870~$\\mu$m, taken from\nBeelen et al. (2008). The wavelengths at the rest frame are labelled on the\ntop. For the single-component dust models adopted in the figure (see\n$\\S~\\ref{dust}$ for details of the dust models.), the $\\chi^2$ values are 1.1\nfor {B6}\\, and 1.0 for {B7}, respectively. In $\\S~\\ref{red}$, four typical\nstarburst templates, NGC~6240, M~82, Mrk~231 and Arp~220 (Polletta et al.\n2007), are adopted to estimate the redshifts for {B6}\\, and {B7}, and their best\nfits are overlaid in colored lines.\n}\n\\label{sed} \n\\end{figure*}\n\n\\subsection{Star formation rates}\nHere we derive the star formation rates from the Ly$\\alpha$, far-infrared and\nradio luminosities.  To estimate the star formation rate (SFR) from the Ly$\\alpha$\\,\nluminosity, we first assume that star formation (SF) powers the observed\nLy$\\alpha$\\, flux.  We use an unreddened Ly$\\alpha$\/H$\\alpha$ ratio of 8:1 and the\nconversion factor between H$\\alpha$ luminosity and SFR (Kennicutt 1998),\nyielding SFR($\\rm{Ly\\alpha}$)\/(\\ifmmode{{\\rm M}_{\\odot} }\\else{M$_{\\odot}$}\\fi\/yr)\\,=\\,$L_{\\rm Ly\\alpha}$\/(10$^{42}$ erg\ns$^{-1}$). This provides a lower limit because the extinction of Ly$\\alpha$\\, emission\ncaused by dust will largely reduce the observed Ly$\\alpha$\\, luminosity. With the FIR\nluminosity derived from {\\it Herschel}\\,data, we can estimate the SFR by using the\nrelation SFR($L_{\\rm FIR}$)\/(\\ifmmode{{\\rm M}_{\\odot} }\\else{M$_{\\odot}$}\\fi\/yr)\\,=\\,$1.7\\times$$L_{\\rm FIR}$\/(10$^{10}$\n\\ifmmode{L_{\\odot}}\\else{$L_{\\odot}$}\\fi) (Kennicutt 1998). If the observed radio emission, with a rest wavelength\nof 6~cm, is dominated by free-free emission in H{\\small II} regions, one can\nalso relate the SFR by the relation SFR($L_{\\rm\n1.4~GHz}$)\/(\\ifmmode{{\\rm M}_{\\odot} }\\else{M$_{\\odot}$}\\fi\/yr)\\,=\\,5.52$\\times$10$^{-22}$~$L_{\\rm 1.4~GHz}$\/(W Hz$^{-1}$)\n(Bell 2003). The radio luminosity at 1.4 GHz at the rest frame can be estimated\nfrom the observed flux at 1.51 GHz by assuming a relation\n$S \\propto \\nu^\\alpha$, where S is the flux density and the typical spectral\nindex $\\alpha$ of $-$0.8 is commonly adopted for the SMGs (e.g., Ivison et al.\n2010). These values are listed in Table~\\ref{tab2}.\n\n\n\\begin{center}\n\\begin{table}[h]\n\\centering\n\\caption{Derived star formation rates towards the four LABs}\\label{tab2}\n\\begin{tabular}{ccccc}\n\\hline\nSource &  SFR(${L_{\\rm FIR}}$) & SFR($L_{\\rm 1.4GHz}$) & log $L_{\\rm Ly\\alpha}$$^a$  &  SFR($\\rm{Ly\\alpha}$) \\\\ \n       &  [\\ifmmode{{\\rm M}_{\\odot} }\\else{M$_{\\odot}$}\\fi\/yr] & [\\ifmmode{{\\rm M}_{\\odot} }\\else{M$_{\\odot}$}\\fi\/yr] &  [ergs~s$^{-1}$] & [\\ifmmode{{\\rm M}_{\\odot} }\\else{M$_{\\odot}$}\\fi\/yr]  \\\\\n\\hline\nB1  &  $<$480 & $<$1090 & 43.9 & 79 \\\\\nB5  &  $<$430 & 1090\\ppm340 & 43.8 & 63 \\\\\nB6  &  1700\\ppm320    & 1430\\ppm360 & 43.8 & 63 \\\\\nB7  &  1460\\ppm390    & 1650\\ppm340 & 43.5 & 32  \\\\\n\\hline\n\\end{tabular}\n\\begin{list}{}{}\n\\item{$^{\\mathrm{a}}$ The Ly$\\alpha$\\,luminosities are adopted from Colbert et al. (2006).}\n\\end{list}\n\\end{table}\n\\end{center}\n\n\n\n\\section{Discussion and Conclusions}\n\nA high detection rate of radio emission (three out of four) around LABs\nsuggests that most LABs do not originate from cooling radiation. Instead,\nphotoionization from starbursts and\/or AGNs may power the LABs in most cases.\nThe high rate of FIR detections (two out of four) points to a star-formation\norigin of the LABs. \nThe SEDs of {B6}\\, and {B7}\\, can be well described by starburst dominated\ntemplates, as shown in Fig.~\\ref{sed}, further supporting Ly$\\alpha$\\, emission\nrelated to the SF in the LABs. In {B6}\\, and {B7}, the SFRs derived from Ly$\\alpha$\\,\nfluxes are far below those estimated from FIR luminosities (Table~\\ref{tab2}).\nThis suggests that the dust indeed greatly reduces the measured Ly$\\alpha$\\, flux.\nComparing the different SFRs, the dust absorption optical depth of the Ly$\\alpha$\\,\nemission becomes $\\sim$3.1$-$3.6. The SFRs estimated from the FIR and radio\nluminosities are comparable, indicating that the radio emission is dominated by\nSF, not by AGNs. The energetic starbursts can provide enough ionizing photons\nto ionize neutral hydrogen atoms in the interstellar medium (ISM), and each\nsubsequent recombination has a probability of $\\sim$ 2\/3 of ending up as a\nLy$\\alpha$ photon (Partridge \\& Peebles 1967).  After escaping the galaxy's\nISM, these Ly$\\alpha$ photons can be resonantly scattered by neutral hydrogen\natoms in the intergalactic medium (IGM), which tends to make the Ly$\\alpha$\nemission extended (Zheng et al.  2011). \n\nCen \\& Zheng (2013) propose an SF-based model and predict that LABs at high\nredshift correspond to protoclusters containing the most massive galaxies and\ncluster halos in the early universe as well as ubiquitous strong infrared\nsources undergoing extreme starbursts. \nThis may be supported by the multiple Spitzer\/MIPS sources detected in both\nLABs (see Fig~\\ref{map}(b), Colbert et al.  2006, 2011).  Indeed, Prescott et\nal. (2012b) suggest that LABs may be the seeds of galaxy clusters by resolving\nthe galaxies within a LAB at $z$\\,=\\,2.7. The strong FIR emission and the\ninferred high SFRs support the presence of a strong starburst in both {B6}\\, and\n{B7}. However, AGN-dominated templates like Mrk~231 can not well reproduce\nthe data (see $\\S$~\\ref{red}), suggesting that the SF instead of AGN may power\nthe Ly$\\alpha$\\, emission in both LABs. The model also predicts that the most\nluminous FIR source in each LAB is likely representing the gravitational center\nof the protocluster. Fig.~\\ref{map}(c-g) shows that the FIR emission indeed\npeaks in the centers of {B6}\\, and {B7}. The radio continuum emission is detected\nexclusively in the centers, which suggests that the source with most luminous\nFIR emission (therefore highest SFR) is in the gravitational center of each\nLAB. Another very important prediction of this model is that the Ly$\\alpha$\\, emission\nfrom photons that escape the galaxy are expected to be significantly polarized,\nwhich has been for the first time confirmed towards LAB1 in the SSA22 field by\nHayes et al. (2011), supporting models with central power sources. Adopting a\ngas-to-dust mass ratio of 150 and the SFRs estimated above, the timescales of\n{B6}\\, and {B7}\\, are relatively short ($\\sim$100 Myr), which is much shorter\nthan the galaxy building timescale. Note that this timescale is a lower limit\nbecause (1) the LABs may have been alive for a while now, and (2) additional\ngas may be continuously accreted. In any case, the LABs are visible only during\na short time interval during the lifetime of their parent clusters.\n\nNote that the so-called ``SF-based model'' proposed by Cen \\& Zheng (2013) also\nincludes AGN powering or any central powering. The morphologies of the\nLy$\\alpha$\\,emission of the four LABs are quite different (Palunas et al. 2004):\n{B1}\\,and {B5}\\, have core-like structures, while {B6}\\,and {B7}\\, are\ncharacterized by diffuse and extended emission with physical sizes of\n$\\sim$60-70~kpc. The latter may be driven by multiple sources as suggested\nby the MIPS data and are consistent with the SF-based model.  There is no\nclear FIR emission detected around {B1}\\,and {B5}. Therefore, the Ly$\\alpha$\\, emission\nin both LABs is unlikely predominantly triggered by SF. Overzier et al.  (2013)\nconclude that in {B1}\\, the photoionization from an AGN is the main driver of\nLy$\\alpha$\\, emission. However, Francis et al. (2013) shows that the observed Ly$\\alpha$\\,\nemission in {B1}\\, is of complex origin, dominated by the sum of the emission\nfrom the sub-haloes where the cold gas is being lit up most likely by a\ncombination of tidally triggered star formation, bow shocks, resonant\nscattering of Ly$\\alpha$\\, from the filament collisions and tidal stripping of the\ngas. In {B5}\\, radio emission is tentatively detected and therefore the AGN may\nalso power the Ly$\\alpha$\\, emission. Among the four LABs in J2143-4423, two of them, {B6}\\,\nand {B7}, are mainly driven by SF. However, the other two LABs, {B1}\\, and {B5},\nwithout clear FIR detection, are predominantly driven by the AGNs or other\nsources of energy still to be specified, but not mainly by star formation. We\nthus conclude that LABs must be powered by quite diverse sources of energy.\n\nWith its high angular resolution and superb sensitivity, future observations\nwith the Large Atacama Millimeter Array (ALMA) will reveal more details about\nthe nature of LABs such as testing the predictions of models where the\nionization is provided by intense star formation and confirming the\nsignificantly polarized dust emission at mm\/submm wavelength.\n\n\\begin{acknowledgements}\nWe thank the anonymous referee for valuable comments that improved this manuscript. \nY.A. acknowledges partial support by NSFC grant 11373007 and Youth Innovation Promotion Association CAS.\nR.C. is supported in part by NASA grant NNX11AI23G.\nY.M. acknowledges support from JSPS KAKENHI Grant Number 20647268. \nZZ was partially supported by NSF grant AST-1208891 and NASA grant NNX14AC89G.\nThis research has made use of NASA's Astrophysical Data System (ADS).\n\nPACS has been developed by a consortium of institutes led by MPE (Germany) and\nincluding UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France);\nMPIA (Germany); INAF-IFSI\/OAA\/OAP\/OAT, LENS, SISSA (Italy); IAC (Spain). This\ndevelopment has been supported by the funding agencies BMVIT (Austria),\nESA-PRODEX (Belgium), CEA\/CNES (France), DLR (Germany), ASI\/INAF (Italy), and\nCICYT\/MCYT (Spain).\n\nSPIRE has been developed by a consortium of institutes led by Cardiff\nUniversity (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA, LAM\n(France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory\n(Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and\nCaltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported\nby national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS\n(France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA\n(USA).\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{PACS and its view}\nThe PACS spectrometer (\\citeauthor{pog2010}~\\citeyear{pog2010}) on-board the Herschel Space Telescope (\\citeauthor{pil2010}~\\citeyear{pil2010}), covers the wavelength range between $50\\ \\mu$m and $200\\ \\mu$m at a resolution of typically 1000. In comparison to the ISO LWS instrument, PACS offers a higher resolution and a better sensitivity. With PACS, we can observe a wide range of molecular emission lines in the innermost regions of circumstellar envelopes (CSE) of AGB stars, including CO and H$_2$O, which are major coolants and thus important for determining the thermodynamical structure of these environments.\n\n\\section{Methodology and preliminary results}\nKinematical, thermodynamical and chemical information about the circumstellar shell is provided by molecular emission lines and dust features. This information is derived through the use of two radiative transfer codes. The non-LTE line radiative transfer code, \\emph{GASTRoNOoM} (\\citeauthor{dec2006}~\\citeyear{dec2006}), calculates the velocity, temperature and density profiles of the gas envelope, the level populations of the molecules accounted for and the emergent line profiles for the different transitions of each molecule. The continuum radiative transfer code, \\emph{MCMax} (\\citeauthor{min2009a}~\\citeyear{min2009a}), calculates the temperature structure of the dust envelope and the final SED. In order to get a full understanding of the entire envelope around an AGB source, both modelling approaches need be used while maintaining consistency between dust and gas (see Lombaert et al., in prep).\n\nThe need for a consistent treatment of both the gas and dust components is important because of the high sensitivity of the water emission models to the dust-to-gas ratio. Modelling has shown that CO emission lines do not share this sensitivity. This behaviour is shown in Figure~1, which gives an excerpt of the PACS data of V669 Cas (full black), overlayed with two models differing only in the dust-to-gas ratio. This indicates that CO can be safely used to determine the gas temperature profile and the mass loss rate and that the water abundance profile can then be derived if the dust-to-gas ratio is constrained. Therefore, we suggest to determine the dust-to-gas ratio empirically from both gas emission (i.e. CO lines) and SED (i.e. dust continuum) modelling, with a consistent iterative treatment of both gas and dust. Consequently, one can improve the constraints on the water abundance profile.\n\n\\begin{figure}\\centering\n\\includegraphics[height=5.7cm]{lombaert_fig1.ps}\n\\caption{An excerpt of the PACS spectrum of V669 Cas (full black). Two models are overplotted, as well as the molecular transitions for CO, $^{13}$CO, o-H$_2$O, p-H$_2$O, o-H$_2^{18}$O, p-H$_2^{18}$O and SiO at their expected frequencies. Model 1 (full gray) has a dust-to-gas ratio $\\psi = 0.001$, whereas Model 2 (dashed) has $\\psi = 0.005$.}\n\\end{figure}\n\\acknowledgements\nRL acknowledges support from the KULeuven under grant number GOA-B6995, BdV and LD from the Fund for Scientific Research of Flanders (FWO), EDB from FWO under grant number G.0470.07 and JB and PR from the Belgian Federal Science Policy Office via the PRODEX Programme of ESA.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\vspace{-0.3em}\n\n\n   Ultra-reliable and low-latency communications (URLLC) have attracted a lot of attention in 5G and the upcoming 6G for mission-critical services\\cite{3GPPRelease16,Mahyar2019ShortCode,popovski2019wireless}. URLLC is renowned by its requirements for significantly lower latency compared to 4G long term evolution (4G-LTE) and high transmission reliability requiring the block error rate (BLER) lower than $10^{-4}$. These stringent requirements necessitate the use of low-complexity decoders and short block-length codes ($\\leq 150$ bits) in coding scheme design \\cite{R1-1608770}. Additionally, URLLC requires bit-level granularity in the block lengths and coding rates, to accommodate scenarios with varying latency and bandwidth constraints \\cite{Mahyar2019ShortCode}, which complicates the coding system further.\n   \n    Main candidates of short block-length codes for URLLC have been thoroughly reviewed in \\cite{Mahyar2019ShortCode,liva2016codeSurvey}. It has been identified that short Bose-Chaudhuri-Hocquenghem (BCH) codes and cyclic-redundancy-check-aided Polar (CRC-Polar) codes have superior BLER performance that is close to the normal approximation (NA) bound\\cite{erseghe2016coding}. However, it is challenging for BCH and CRC-Polar to provide bit-level granularity with the optimal error-correction capability, as they are originally available at certain block-lengths and rates \\cite{lin2004ECC,Designofpolar}, while the best known linear codes of different lengths and rates have different structures \\cite{Grassl:codetables}. Recently, \\cite{papadopoulou2021short} demonstrated that when using universal decoders, high-performance rate-compatible short codes can also be conveniently constructed by selecting good random codes. Therefore, universal decoders are favored in URLLC, as they can decode any linear block codes, including BCH and CRC-Polar codes, as well as codes that do not adhere to specific code structures.\n    \n    Ordered-statistics decoding (OSD) \\cite{Fossorier1995OSD} is a near-maximum-likelihood (near-ML) universal decoder that rekindles interests recently. It can decode any linear block code with near-ML BLER performance. OSD has two main phases, namely, \\textit{preprocessing} and \\textit{reprocessing}. In preprocessing, it sorts the received codeword bits according to their reliabilities (by which high reliable bits are distinguished), and permutes the columns of the code generator matrix accordingly. The permuted matrix is then transformed into the systematic form by Gaussian elimination (GE), where the information set is associated with high reliable bits. In reprocessing, a number of test error patterns (TEPs) are attempted to flip the high reliable bits. The remaining low reliable bits are recovered by \\textit{re-encoding} the flipped high reliable bits with the systematic permuted generator matrix.\n    \n    Let $C$ be the notation of complexity. In general, the average complexity of OSD is roughly characterized as\n    \\begin{equation} \\label{equ::CmpOSD}\n        C_{\\mathrm{OSD}} =  C_{\\mathrm{Preprocessing}} + N_a  C_{\\mathrm{Re-encoding}},\n    \\end{equation}\n    where $N_a$ is the average number of TEPs attempted in single decoding. Recent years have seen many works towards reducing $C_{\\mathrm{OSD}}$ by reducing the number $N_a$ \\cite{yue2021probability,Chentao2019SDD,NewOSD-5GNR,yue2021linear,Wu2007OSDMRB,FossorierBoxandMatch,improvedTEPwithMean,choi2019fast, jin2006probabilisticConditions,WJin2007MultipleBiases}. For example, \\cite{yue2021probability,Chentao2019SDD,Wu2007OSDMRB,choi2019fast} proposed techniques to identify unpromising TEPs that can be discarded without processing, and \\cite{yue2021probability,Wu2007OSDMRB, jin2006probabilisticConditions} designed approaches to terminate OSD early rather than processing all possible TEPs. These approaches can usually reduce $N_a$ to a very low level at high signal-to-noise ratios (SNRs). In this situation, $C_{\\mathrm{OSD}}$ will be dominated by preprocessing rather than reprocessing. Specifically, GE has the complexity as high as $O(nk^2)$ for a $k\\times n$ generator matrix \\cite{Fossorier1995OSD}, where $O(\\cdot)$ is the big-O operand. As a result, when $N_a$ is not large, GE introduces a \\textit{complexity floor} to OSD. That is, a complexity component hardly reducible. This complexity floor hinders the application of OSD in URLLC, especially for scenarios operating at high SNRs. Recently, \\cite{choi2021fast} proposed using multiple offline produced generator matrices to replace GE in OSD. Nevertheless, this approach could introduce extra overheads at low-to-moderate SNRs, as it performs several reprocessings over multiple generator matrices.\n    \n    In this paper, we design an OSD decoder with adaptive GE reduction. Specifically, the decoder will skip GE if doing so is still very likely to produce the correct decoding result. Two conditions are introduced in the proposed decoder. The first condition decides whether to skip GE based on evaluating BLER performance with and without GE. If GE is skipped, the re-encoding process will be performed with the original generator matrix of the code. The second condition determines whether the correct decoding result has been found in the decoding process without GE. If so, decoding is terminated early to reduce the complexity, and if not, the standard OSD is performed as a supplement to prevent BLER from degrading. Our verification shows that at high SNRs, the proposed decoder can avoid GE in almost all decoding attemps with nearly the same BLER performance as the standard OSD. Owing to the effective GE reduction, the proposed decoder can achieve a significantly lowered complexity at high SNRs compared to the latest OSD approaches from the literature \\cite{choi2021fast,yue2021probability}.\n    \n    The rest of this paper is organized as follows. Section \\ref{sec::Preliminaries} presents the preliminaries. Section \\ref{sec::Algorithm} describes the proposed decoding algorithm. Section \\ref{sec::Performance and Complexity} discusses the decoding error performance and complexity. Section \\ref{Sec::Simulation} verifies the BLER and complexity of the proposed decoder via simulations. Finally, Section \\ref{sec::Conclusion} concludes the paper.\n    \n    \\emph{Notation}: We use $[a]_u^v = [a_u,\\ldots,a_v]$ to denote a row vector containing element $a_{\\ell}$ for $u\\le \\ell\\le v$. For simplicity, we do not distinguish random variables and their samples throughout the paper, while possible abuse of notations will be specified.\n\n\\vspace{-0.1em} \n\\section{Preliminaries} \\label{sec::Preliminaries}\n\\vspace{-0.1em} \nLet $\\mathcal{C}(n,k)$ denote a binary linear block code, where $n$ and $k$ are the lengths of the codeword and the information block, respectively. $\\mathcal{C}(n,k)$ is defined by its generator matrix $\\mathbf{G}$; that is, an information sequence $\\mathbf{b} =  [b]_1^k$ is uniquely encoded to a codeword $\\mathbf{c} = [c]_1^n$ by $\\mathbf{c} = \\mathbf{b}\\mathbf{G}$. In this paper, we assume that $\\mathbf{G}$ is systematic, i.e.,  $\\mathbf{G} =  [\\mathbf{I}_k \\  \\mathbf{P}]$, where $\\mathbf{I}_k$ is a $k\\times k$ identity matrix and $\\mathbf{P}$ is the parity sub-matrix.\n\nWe consider an additive white Gaussian Noise (AWGN) channel and binary phase shift keying (BPSK) modulation. Let $\\mathbf{s} = [s]_1^n$ denote the modulated signals, where $s_{i} = (-1)^{c_{i}}\\in \\{\\pm 1\\}$. At the channel output, the received signal is given by $\\mathbf{r} = \\mathbf{s} + \\mathbf{w}$, where $\\mathbf{w}$ is the AWGN vector with zero mean and variance $N_{0}\/2$, for $N_0$ being the single-band noise power spectrum density. SNR is accordingly defined as $\\mathrm{SNR} = 2\/N_0$. Without loss of generality, approaches provided in this paper can be extended to other modulation schemes.\n\nAt the receiver, the bitwise hard-decision estimate $\\mathbf{y}= [y]_{1}^n$ of codeword $\\mathbf{c}$ is obtained according to: $y_{i} = 1 $ for $r_{i}<0$ and $y_{i} = 0$ for $r_{i}\\geq 0$. If codewords in $\\mathcal{C}(n,k)$ have equal transmission probabilities, the log-likelihood ratio (LLR) of the $i$-th received symbol is defined as ${\\ell}_{i} \\triangleq \\ln \\frac{\\mathrm{Pr}(c_{i}=0|r_{i})}{\\mathrm{Pr}(c_{i}=1|r_{i})}$, which is further simplified to ${\\ell}_{i} = \\frac{4r_{i}}{N_{0}}$ if employing BPSK \\cite{lin2004ECC}. Thus, we define $\\alpha_{i} \\triangleq |r_{i}|$ (the scaled magnitude of LLR) as the reliability of $y_i$, where $|\\cdot|$ is the absolute operation.\n\nOSD preprocesses received signals according to reliabilities. First, a permutation $\\pi_{1}$ is performed  to sort the reliabilities $\\bm{\\alpha} = [\\alpha]_1^n$ in descending order, and the ordered reliabilities are obtained as $\\pi_1(\\bm{\\alpha})$. Then, the generator matrix are accordingly permuted (in terms of columns) to $\\pi_1(\\mathbf{G})$. Next, OSD obtains the systematic form of $\\pi_1(\\mathbf{G})$ as $\\widetilde{\\mathbf{G}} =  [\\mathbf{I}_k \\  \\widetilde{\\mathbf{P}}]$ by performing GE. We represent the GE operation as $\\widetilde{\\mathbf{G}} = \\mathbf{E}(\\pi_2(\\pi_1(\\mathbf{G})))$, where $\\mathbf{E}$ (dimension $k\\times k$) represents row operations, and $\\pi_{2}$ represents an additional column permutation. $\\pi_{2}$ occurs to ensure that the first $k$ columns of $\\pi_1(\\mathbf{G})$ are linearly independent. Accordingly, $\\mathbf{r}$, $\\mathbf{y}$, and $\\bm{\\alpha}$, are permuted into $\\widetilde{\\mathbf{r}} = \\pi_2(\\pi_1(\\mathbf{r}))$, $\\widetilde{\\mathbf{y}} = \\pi_2(\\pi_1(\\mathbf{y}))$, $\\widetilde{\\bm{\\alpha}} = \\pi_2(\\pi_1(\\bm{\\alpha}))$, respectively.\n\nSince $\\pi_2$ only marginally disrupts the descending order of $\\widetilde{\\bm{\\alpha}}$ \\cite[Eq. (59)]{Fossorier1995OSD}, the first $k$ positions of $\\widetilde{\\mathbf{y}}$, denoted by $\\widetilde{\\mathbf{y}}_{\\mathrm{B}} =[\\widetilde{y}]_1^k$, are referred to as the most reliable basis (MRB) \\cite{Fossorier1995OSD}. To eliminate errors in MRB, a length-$k$ TEP $\\mathbf{e} = [e]_1^k$ is added to $\\widetilde{\\mathbf{y}}_{\\mathrm{B}}$ to obtain a codeword estimate by re-encoding according to $\\widetilde{\\mathbf{c}}_{\\mathbf{e}} = \\left(\\widetilde{\\mathbf{y}}_{\\mathrm{B}}\\oplus \\mathbf{e}\\right)\\widetilde{\\mathbf{G}}$, where $\\widetilde{\\mathbf{c}}_{\\mathbf{e}}$ is the ordered codeword estimate with respect to the TEP $\\mathbf{e}$. In reprocessing, a list of TEPs are re-encoded to generate multiple codeword candidates. The maximum Hamming weight of TEPs attempted is limited by a parameter $m$, namely, decoding order. Thus, the number of TEPs processed in an order-$m$ OSD can be up to $\\sum_{i=0}^{m}\\binom{k}{i}$. For a code with the minimum Hamming weight $d_{\\mathrm{H}}$, OSD with order $m = \\lceil d_{\\mathrm{H}}\/4-1\\rceil$ can achieve the ML decoding performance \\cite{Fossorier1995OSD}.\n\nWith BPSK, the best ordered codeword estimate $\\widetilde{\\mathbf{c}}_{\\mathrm{best}}$ is found by minimizing the weighted Hamming distance between codeword estimate $\\widetilde{\\mathbf{c}}_{\\mathbf{e}}$ and $\\widetilde{\\mathbf{y}}$, which is defined as \\cite{valembois2002comparison}\n    \t\\begin{equation} \\small \\label{equ::Prelim::WHD_define}\n    \t\t \\mathcal{D}(\\widetilde{\\mathbf{c}}_{\\mathbf{e}},\\widetilde{\\mathbf{y}}) \\triangleq \\sum_{1 \\leq i \\leq n } (\\widetilde{c}_{\\mathbf{e},i} \\oplus \\widetilde{y}_{i}) \\widetilde{\\alpha}_{i}.\n    \t\\end{equation}\nFinally, the decoding result $\\hat{\\mathbf{c}}_{\\mathrm{best}}$ is output by performing inverse permutations over $\\widetilde{\\mathbf{c}}_{\\mathrm{best}}$, i.e., $\\hat{\\mathbf{c}}_{\\mathrm{best}} = \\pi_1^{-1}(\\pi_2^{-1}(\\widetilde{\\mathbf{c}}_{\\mathrm{best}}))$.\n\n\\vspace{-0.25em} \n\\section{OSD with Adaptive GE Reduction}   \\label{sec::Algorithm}\n\\vspace{-0.25em} \n\\subsection{Overall Description}\n\\vspace{-0.25em} \n     \\begin{figure} \n     \\centering\n        \\definecolor{mycolor1}{rgb}{0.00000,0.44706,0.74118}%\n        \\definecolor{mycolor2}{rgb}{0.00000,0.44700,0.74100}%\n        \\tikzstyle{terminator} = [rectangle, draw, text centered, rounded corners, minimum height=2em]\n        \\tikzstyle{process} = [rectangle, draw, text centered, minimum height=2em]\n        \\tikzstyle{decision} = [diamond, draw, text centered, minimum height=1em,aspect=2.5]\n        \\tikzstyle{connector} = [draw, -latex']\n        %\n        \\begin{tikzpicture}[node distance=2cm]\n        \\node at (-2,0) [terminator] (start) {\\footnotesize Start Decoding};\n        \\node [decision] at (-2,-1.3) (con1) {\\footnotesize Condition 1};\n        \\node [process] at (2,-1.3) (NonGE) {\\footnotesize Non-GE OSD (order $m\\!-\\!1$)};\n        \\node [decision] at (2,-2.6) (con2) {\\footnotesize Condition 2};\n        \\node [process] at (-2,-2.6) (GE) {\\footnotesize Standard OSD (order $m$)};\n        \\node at (0,-3.9) [terminator] (end) {\\footnotesize Finish Decoding};\n        \\path [connector] (start) -- (con1);\n        \\path [connector] (con1) -- (NonGE);\n        \\path [connector] (con1) -- (GE);\n        \\path [connector] (NonGE) -- (con2);\n        \\path [connector] (con2) -- (GE);\n        \\path [connector] (con2) |- (end);\n        \\path [connector] (GE) |- (end);\n        \n        \\node[draw=none] at (-1.6, -2.0) (No) {\\footnotesize No};\n        \\node[draw=none] at (-0.4, -1.1) (Yes) {\\footnotesize Yes};\n        \\node[draw=none] at (1.6, -3.3) (yes) {\\footnotesize Yes};\n        \\node[draw=none] at (0.4, -2.4) (No) {\\footnotesize No};\n        \n        \\end{tikzpicture}\n    \t\\vspace{-0em}\n        \\caption{The structure of the proposed decoder.}\n    \t\\vspace{-0em}\n    \t\\label{Fig::structure}\n        \n\t\\end{figure}\n\nWe proposed an OSD algorithm that can adaptively skip its reprocessing stage (including GE). When reprocessing is skipped, the original hard-decision estimate $\\mathbf{y}$ and generator matrix $\\mathbf{G}$ will be used for re-encoding (instead of using $\\widetilde{\\mathbf{y}}$ and $\\widetilde{{\\mathbf{G}}}$). Specifically, let $\\mathbf{y}_{\\mathrm{B}}$ denote the first $k$ positions of $\\mathbf{y}$, i.e., $\\mathbf{y}_{\\mathrm{B}} = [y]_1^k$. Then, a codeword estimate is directly recovered by  $\\mathbf{c}_{\\mathbf{e}} = \\left(\\mathbf{y}_{\\mathrm{B}}\\oplus \\mathbf{e}\\right)\\mathbf{G}$ with respect to a TEP $\\mathbf{e}$. Similar to the standard OSD, a list of TEPs will be used in re-encoding, and the maximum allowed Hamming weight of TEPs is limited by $m'$. Finally, if $\\mathbf{c}_{\\mathbf{e}}$ is identified as the best codeword estimate, it will be directly output as the decoding result with no inverse permutation required. We referred to such a decoding process without GE as the Non-GE OSD, and $m'$ is its decoding order. \n\nThe structure of the proposed decoder is illustrated in Fig \\ref{Fig::structure}. At the start of decoding, the decoder decides whether to conduct the Non-GE OSD or the standard OSD according to ``Condition 1''. If the Non-GE OSD is performed, ``Condition 2\" will determine whether the Non-GE OSD has found the correct decoding result. If not, the standard OSD will be conducted following the Non-GE OSD to avoid degraded decoding performance. We set $m'=\\max(m-1,0)$ in the proposed decoder, i.e., the Non-GE OSD is one-order lower than the standard OSD. The reasons are that 1) if $m'\\geq m$, the Non-GE OSD has a higher complexity than the standard OSD, which negates the need for GE reduction and worsens the worst-case decoding complexity, and 2) if $m'$ is too small, the Non-GE OSD may easily fail to find the correct result.\n\nAs seen, the design of ``Condition 1'' and ``Condition 2'' is of importance for the structure illustrated in Fig. \\ref{Fig::structure}. We note that the standard OSD  in Fig. \\ref{Fig::structure} can be implemented by any improved variants of OSD; for example, the efficient probability-based OSD (PB-OSD) proposed recently \\cite{yue2021probability}.\n\n\\vspace{-0.25em} \n\\subsection{The First Condition}\n\\vspace{-0.25em} \nTo derive the first condition, let us first consider the BLER performance of OSD, which is represented as \\cite{Fossorier1995OSD}\n\\begin{equation}\n    \\mathrm{P_e} \\leq (1 - \\mathrm{P_{list}}) + \\mathrm{P_{ML}},\n\\end{equation}\nwhere $\\mathrm{P_{ML}}$ is the ML BLER of $\\mathcal{C}(n,k)$. $\\mathrm{P_{ML}}$ is mainly characterized by the structure of $\\mathcal{C}(n,k)$ and in particular, the minimum Hamming weight $d_{\\mathrm{H}}$. $\\mathrm{P_{list}}$ is the probability that some TEP can eliminate the the errors over MRB, whose value depends on the decoding order $m$ \\cite[Eq. (24)]{dhakal2016error}.\n\n\n\nLet $\\mathrm{P'_{list}}$ denote the probability that the Non-GE OSD can eliminate the errors over $\\mathbf{y}_{\\mathrm{B}}$ with some TEP. Thus, BLER of Non-GE OSD is upper bounded by $\\mathrm{P_e} \\leq (1 - \\mathrm{P'_{list}}) + \\mathrm{P_{ML}}$. Therefore, if $\\mathrm{P'_{list}} =\n \\mathrm{P_{list}}$, the Non-GE OSD will deliver the same BLER as OSD. In other words, the Non-GE OSD is sufficient to find the correct decoding result and thus GE is not necessary. $\\mathrm{P'_{list}} =\n \\mathrm{P_{list}}$ can be satisfied when SNR is asymptotically large (i.e., $N_0\\to 0$). To see this, consider $\\mathrm{P_{list}}$ derived from \\cite[Lemma 1]{yue2021revisit}), i.e.,\n \\begin{equation} \\label{equ::Ana::Plist}\n     \\mathrm{P_{list}} = \\sum_{i=0}^{m}\\int_{x = 0}^{\\infty} \\binom{k}{i} (p(x))^{i}(1-p(x))^{k-i} f_{\\widetilde{\\alpha}_{k+1}}(x) dx,\n \\end{equation}\n where $f_{\\widetilde{\\alpha}_{k+1}}(x)$ is the probability density function ($\\mathrm{pdf}$) of the $(k+1)$-th ordered reliability, $\\widetilde{\\alpha}_{k+1}$ (as a random variable). $p(x)$ is the average bitwise error probability of $\\widetilde{\\mathbf{y}}_{\\mathrm{B}}$ conditioning on $\\{\\widetilde{\\alpha}_{k+1} = x\\}$, which is given by \\cite[Eq. (13)]{yue2021revisit}\n    \\begin{equation}   \\label{equ::Ana::Pe::px}\n         p(x) = \\frac{Q(\\frac{2x+2}{\\sqrt{2N_0}}) }{ Q(\\frac{2x+2}{\\sqrt{2N_0}}) +  Q(\\frac{2x-2}{\\sqrt{2N_0}}) } .\n     \\end{equation}\nThen, when $N_0\\to \\infty$, we have \\cite[Eq. (28)]{yue2021linear}\n     \\begin{equation}   \n         \\lim_{N_0 \\to 0}p(x) = \\frac{1}{1 + \\lim\\limits_{N_0 \\to 0}\\exp\\left(\\frac{4x}{N_0}\\right)}.\n     \\end{equation}\nwhen $N_0 \\to 0$, there are $\\widetilde{\\alpha}_{k+1} \\to 1$ and $f_{\\widetilde{\\alpha}_{k+1}}(x) \\to \\delta(x-1)$, where $\\delta(x)$ is the Dirac delta function. We then obtain \n\\begin{equation} \\label{equ::Plist::N0to0}\n     \\lim_{N_0 \\to 0}\\mathrm{P_{list}} =  \\lim_{N_0 \\to 0}\\sum_{i=0}^{m} \\binom{k}{i} \\left(\\frac{1}{1 + e^{4\/N_0}}\\right)^{\\!i\\!}\\left(\\frac{e^{4\/N_0}}{1 + e^{4\/N_0}}\\right)^{\\!\\!k-i}\\!\\!\\!.\n\\end{equation}\n\nOn the other hand, we can derive $\\mathrm{P'_{list}}$ as\n \\begin{equation}  \\label{equ::Ana::Plist'}\n     \\mathrm{P'_{list}} = \\sum_{i=0}^{m'} \\binom{k}{i} (p')^{i}(1-p')^{k-i},\n \\end{equation}\nwhere $p'$ is the bitwise error probability of $\\mathbf{y}_{\\mathrm{B}}$. Under AWGN and BPSK, $p'$ is readily given by $p' = Q(\\sqrt{2\/N_0})$, which also has the following asymptotic property,\n\\begin{equation}   \\label{equ::Ana::P'::app}\n    \\lim_{N_0 \\to 0}p' = \\frac{1}{1 + \\lim\\limits_{N_0 \\to 0}\\exp\\left(\\frac{4}{N_0}\\right)}.\n\\end{equation}\nTherefore, substituting (\\ref{equ::Ana::P'::app}) into (\\ref{equ::Ana::Plist'}) and taking $m'=m-1$, we observe that \n\\begin{equation} \\label{equ::ana::same}\n\\begin{split}\n   \\lim_{N_0 \\to 0}&\\mathrm{P_{list}}- \\mathrm{P'_{list}} \\\\\n   &= \\lim_{N_0 \\to 0}\\binom{k}{m} \\left(\\frac{1}{1 + e^{4\/N_0}}\\right)^{\\!m\\!}\\left(\\frac{e^{4\/N_0}}{1 + e^{4\/N_0}}\\right)^{\\!\\!k-m} \\\\\n   &\\to 0,   \n\\end{split}\n\\end{equation}\nwhich indicates that when $N_0\\to 0$, the Non-GE OSD could completely replace the standard OSD. However, when $N_0$ is not negligible, there is $\\mathrm{P'_{list}} < \\mathrm{P_{list}}$ for $m'<m$, since the Non-GE OSD applies TEPs over $\\mathbf{y}_{\\mathrm{B}}$ rather than MRB $\\widetilde{\\mathbf{y}}_{\\mathrm{B}}$, which has a higher reliability.\n\nIn this regard, we define a parameter $\\lambda \\in (0,1]$ for relaxation and introduce the ``Condition 1'' as follows. If \n $$\\mathrm{P'_{list}} \\geq (1-\\lambda)\\mathrm{P_{list}} $$\nis satisfied, we regard that the Non-GE OSD is highly possible to find the correct decoding result, and it should be performed before the standard OSD. According to (\\ref{equ::ana::same}), For an arbitrarily small $\\lambda$, $\\mathrm{P'_{list}} \\geq (1-\\lambda) \\mathrm{P_{list}}$ can always be satisfied at a sufficient large SNR.\n\nIn terms of the implementation, $\\mathrm{P'_{list}}$ and $\\mathrm{P_{list}}$ could be computed prior according to (\\ref{equ::Ana::Plist}) and (\\ref{equ::Ana::Plist'}) for a given $N_0$. Alternatively, the decoder can perform an adaptive GE reduction based on the quality of received signals. That is, to evaluate $\\mathrm{P'_{list}}$ and $\\mathrm{P_{list}}$ based on the current received signal on the fly. Given the ordered reliability $\\widetilde{\\alpha}_i$ (i.e., the random variable is realized to $\\widetilde{\\alpha}_i$), the bit-wise error probability of $\\widetilde{y}_i$ is given by \n\\begin{equation}   \\label{equ::Ana::Pi}\n    \\mathrm{P}(\\widetilde{i}) =\\frac{1}{1+\\exp\\left(4\\widetilde{\\alpha}_i\/N_0\\right)}.\n\\end{equation}\nThen, the average error probability of MRB, $p$, is evaluated as $p = \\frac{1}{k}\\sum_{i=1}^{k} \\mathrm{P}(\\widetilde{i})$,\nand $\\mathrm{P_{list}}$ is computed as\n\\begin{equation}   \\label{equ::Ana::Plist::eva}\n    \\mathrm{P_{list}} = \\sum_{i=0}^{m} \\binom{k}{i} (p)^{i}(1-p)^{k-i}\n\\end{equation}\nIt is important to distinguish between (\\ref{equ::Ana::Plist::eva}) and (\\ref{equ::Ana::Plist}); (\\ref{equ::Ana::Plist::eva}) is instantaneous and can vary from block to block in receiving, while (\\ref{equ::Ana::Plist}) is theoretically derived based on $N_0$. Similarly, the bit-wise error probability of $y_i$ is given by $\\mathrm{P}(i) =\\frac{1}{1+\\exp\\left(\\alpha_i\/N_0\\right)}$. Then, $\\mathrm{P'_{list}}$ is evaluated by substituting $p' = \\frac{1}{k}\\sum_{i=1}^{k} \\mathrm{P}(i)$ into (\\ref{equ::Ana::Plist'}).\n\nWe note that $\\mathrm{P'_{list}}$ and $\\mathrm{P_{list}}$ can be evaluated on the fly very efficiently. For example, the exponential in (\\ref{equ::Ana::Pi}) can be implemented by its polynomial approximation or by table lookup. Moreover, $p$ is tiny at high SNRs and (\\ref{equ::Ana::Plist::eva}) can be approximated to $\\mathrm{P_{list}}\\approx \\sum_{i=0}^{m} \\binom{k}{i} p^{i}$.\n\n\\vspace{-0.25em} \n\\subsection{The Second Condition}\n\\vspace{-0.25em} \nAs depicted in Fig. \\ref{Fig::structure}, ``Condition 2'' identifies if the Non-GE OSD has found the correct decoding result. ``Condition 2'' should be checked once a better codeword is generated during the decoding process, so that the decoding can be terminated early to reduce the complexity.\n\nLeveraging the idea of PB-OSD \\cite{yue2021probability}, the decoder can compute the probability that a generated codeword estimate $\\mathbf{c}_{\\mathbf{e}}$ is the correct decoding result, conditioning on the difference pattern $\\mathbf{d}_{\\mathbf{e}} = \\mathbf{c}_{\\mathbf{e}} \\oplus \\mathbf{y}$. We denote this probability as $\\mathrm{Pr}\\left(\\mathbf{c}_{\\mathbf{e}} | \\mathbf{d}_{\\mathbf{e}} \\right)$. Thus, if $\\mathrm{Pr}\\left(\\mathbf{c}_{\\mathbf{e}} | \\mathbf{d}_{\\mathbf{e}} \\right)$ is close enough to 1, $\\mathbf{c}_{\\mathbf{e}}$ is very likely to be the correct decoding result. This leads to the ``Condition 2'' as follows. Given a parameter $\\tau \\in [0,1]$, if \n$$\\mathrm{Pr}\\left(\\mathbf{c}_{\\mathbf{e}} | \\mathbf{d}_{\\mathbf{e}} \\right) \\geq \\tau,$$\n$\\mathbf{c}_{\\mathbf{e}}$ will be claimed as the decoding result and the decoding is terminated.\n\nProbability $\\mathrm{Pr}\\left(\\mathbf{c}_{\\mathbf{e}} | \\mathbf{d}_{\\mathbf{e}} \\right)$ can be approximately computed similar to \\cite[Eq. (2)]{yue2021probability}. Specifically,\n    \\begin{equation} \\label{equ::Psuc::itis}\n        \\mathrm{Pr}\\left(\\mathbf{c}_{\\mathbf{e}} | \\mathbf{d}_{\\mathbf{e}} \\right) \\approx\n          \t\t\t\\Bigg(1  +  \\frac{(1-\\mathrm{P}(\\mathbf{e}))2^{k-n}}{\\exp\\Big(-\\frac{4}{N_0}\\sum\\limits_{d_{\\mathbf{e},i} \\neq 0}\\alpha_i\\Big) \\prod\\limits_{i=1}^{n} (1-\\mathrm{P}(i))} \\Bigg)^{-1}  ,\n    \\end{equation}\nwhere $\\mathrm{P}(\\mathbf{e})$ is the probability that TEP $\\mathbf{e}$ can eliminate the errors over $\\mathbf{y}_{\\mathrm{B}}$, which is derived as\n\\begin{equation} \\label{Equ::Psuc1}\n\\begin{split}\n  \\mathrm{P}(\\mathbf{e}) & = \\prod_{e_i\\neq 0} \\mathrm{P}(i) \\prod_{e_i = 0} (1-\\mathrm{P}(i))  \\\\\n  & \\overset{(a)}{=} \\exp\\Big(-\\frac{4}{N_0}\\sum_{\\substack{1 < i \\leq k\\\\d_{\\mathbf{e},i} \\neq 0}}\\alpha_i\\Big)\\prod_{i=1}^{k} (1-\\mathrm{P}(i)) ,\n\\end{split}\n\\end{equation}\nwhere step (a) applies the fact $[d_{\\mathbf{e}}]_1^k = \\mathbf{e}$. We omit the detailed derivation of (\\ref{equ::Psuc::itis}), as it can be obtained by applying unordered reliabilities to  \\cite[Eq. (2)]{yue2021probability}.\n\nThe computation of (\\ref{equ::Psuc::itis}) and (\\ref{Equ::Psuc1}) is fast. First, the terms $\\prod_{i=1}^{n} (1-\\mathrm{P}(i))$ and $\\prod_{i=1}^{k} (1-\\mathrm{P}(i))$ can be reused for different codeword estimates. Then, $\\sum_{d_{\\mathbf{e},i} \\neq 0}\\alpha_i$ is exactly the WHD $\\mathcal{D}(\\mathbf{c}_{\\mathbf{e}},\\mathbf{y})$, while $\\sum_{\\substack{1 < i \\leq k\\\\d_{\\mathbf{e},i} \\neq 0}}\\alpha_i$ is a partial WHD $\\mathcal{D}([c_{\\mathbf{e}}]_1^k,[y]_1^k)$ with respect to $\\mathcal{D}(\\mathbf{c}_{\\mathbf{e}},\\mathbf{y})$. Therefore, considering that $\\mathcal{D}(\\mathbf{c}_{\\mathbf{e}},\\mathbf{y})$ is readily computed when generating each codeword estimate (in order to find the best estimate), (\\ref{equ::Psuc::itis}) and (\\ref{Equ::Psuc1}) are efficiently implemented.\n\\vspace{-0.25em} \n\\subsection{Algorithm}\n\\vspace{-0.25em} \nWe summarize the proposed decoding algorithm in Algorithm \\ref{ago::OSDGEreduction}. We employ the PB-OSD algorithm \\cite{yue2021probability} to place the standard OSD component. As the latest OSD improvement, PB-OSD shows a significantly reduced complexity compared to the original standard OSD \\cite{Fossorier1995OSD}. To further reduce the possible overhead of the Non-GE OSD, we also integrate the TEPs discarding rule of \\cite[Eq. (9)]{yue2021probability} into the Non-GE OSD. Specifically, a probability $\\mathrm{P}_p(\\mathbf{e})$ \\cite[Eq. (8)]{yue2021probability} is computed for TEP $\\mathbf{e}$. Then, if $\\mathrm{P}_p(\\mathbf{e})$ is less than a threshold $\\tau_{p}$, the TEP $\\mathbf{e}$ is skipped without re-encoding. This discarding rule can be implemented efficiently to reduce the complexity of OSD \\footnote{By using a \nmonotonicity trick $\\mathrm{P}_p(\\mathbf{e})$ only needs to be computed for a small amount of TEPs, resulting in a small self-overhead. We refer interested readers to \\cite{yue2021probability}.}. Following the approach in \\cite{yue2021probability}, $\\tau_p$ is set to $\\tau_p = 0.002\\sqrt{p'\/N_{m'}}$ in Algorithm \\ref{ago::OSDGEreduction}, where $p'=Q(\\sqrt{2\/N_0})$ and $N_{m'} = \\sum_{i=1}^{m'}\\binom{k}{i}$.\n\n\nIf ``Condition 2'' is not satisfied when finishing the Non-GE OSD, the so-far found best estimate $\\mathbf{c}_{\\mathrm{best}}$ and the recorded minimum WHD will be passed into PB-OSD, which will be executed to find a better codeword estimate than $\\mathbf{c}_{\\mathrm{best}}$.\n\n\\begin{spacing}{1.25}\n    \\begin{algorithm}\n    \\small\n\t\\caption{OSD with Adaptive GE Reduction}\n\t\\label{ago::OSDGEreduction}\n\t\\begin{algorithmic} [1]\n\t\t\\REQUIRE Received signal $\\mathbf{r}$, Parameters $\\lambda$, $\\tau$, and $\\tau_p$\n\t\t\\ENSURE ~Optimal codeword estimate $\\hat{\\mathbf{c}}_{\\mathrm{best}}$\n\t\t\n\t\t\n\t\t~\\\\\n\t\t\\STATE Initialize $\\mathcal{D}_{\\min} = +\\infty$.\n\t\t\\STATE Compute $\\mathrm{P_{list}}$ and $\\mathrm{P'_{list}}$ according to (\\ref{equ::Ana::Plist::eva}) and (\\ref{equ::Ana::Plist'}), respectively.\n\t\t$\/\/$ \\textbf{Perform the Non-GE OSD if Condition 1 is true}\n\t\t\n\t\t\\IF{$\\mathrm{P'_{list}} \\geq (1-\\lambda)\\mathrm{P_{list}}$} \n\t\t\n                \\FOR{$i=1:\\sum\\limits_{j = 0}^{m}\\binom{k}{j}$}\n        \t\t\\STATE Select an unprocessed TEP $\\mathbf{e}_i$ with the lowest Hamming weight\n        \t\t\\STATE Compute $\\mathrm{P}_{p}(\\mathbf{e})$ according to \\cite[Eq. (8)]{yue2021probability}\n        \t\t\\STATE \\textbf{if} $\\mathrm{P}_{p}(\\mathbf{e})\\leq \\tau_p$ \\textbf{then} \\textbf{Continue}\n        \t\t\\STATE Generate codeword estimate $\\mathbf{c}_{\\mathbf{e}_i} = \\left(\\mathbf{y}_{\\mathrm{B}}\\oplus \\mathbf{e}\\right)\\mathbf{\\widetilde G}$\n        \t\t\\STATE Compute the WHD $\\mathcal{D}(\\mathbf{c}_{\\mathbf{e}_i} , \\mathbf{y})$\n\n    \t\t     \\IF{$\\mathcal{D}(\\mathbf{c}_{\\mathbf{e}_i} , \\mathbf{y}) \\leq \\mathcal{D}_{\\min}$}\n        \t\t\\STATE Claim $\\mathbf{c}_{\\mathbf{e}_i}$ as $\\mathbf{c}_{\\mathrm{best}}$, and  $\\mathcal{D}_{\\min} \\leftarrow \\mathcal{D}(\\mathbf{c}_{\\mathbf{e}_i} , \\mathbf{y})$\n        \t\t\\STATE Calculate $\\mathrm{Pr}\\left(\\mathbf{c}_{\\mathbf{e}} | \\mathbf{d}_{\\mathbf{e}} \\right)$ according to  (\\ref{equ::Psuc::itis})\n        \t\t\\STATE \\textbf{if} {$\\mathrm{Pr}\\left(\\mathbf{c}_{\\mathbf{e}} | \\mathbf{d}_{\\mathbf{e}} \\right) \\geq \\tau$}\n        \t\t    \\textbf{then} \\textbf{return} $\\mathbf{c}_{\\mathrm{best}}$ \\hspace*{\\fill}  $\/\/$\\textbf{Condition 2}\n        \t\t\\ENDIF\n        \t\t\n    \t\t    \\ENDFOR\n\t\t\\ENDIF\n\t\t\\STATE Perform PB-OSD \\cite{yue2021probability}, with $\\mathcal{D}_{\\min}$ and $\\mathbf{c}_{\\mathrm{best}}$ passed in.\n\t\\end{algorithmic}\n\\end{algorithm} \n\\end{spacing}\n\\vspace{-0.25em} \n\\section{Performance and Complexity Consideration} \\label{sec::Performance and Complexity}\n\\vspace{-0.25em} \n\\subsubsection{Performance} In terms of the BLER performance, the proposed decoder will have the same BLER performance as the standard OSD if ``Condition 2'' is effective. Precisely, there will be a decoding error that does not occur in the standard OSD but does occur in the proposed decoder, if and only if \"Condition 2\" identifies an incorrect estimate as the decoding result. To avoid such decoding errors, the parameter $\\tau$ should be chosen with care. Recalling $\\mathrm{Pr}\\left(\\mathbf{c}_{\\mathbf{e}} | \\mathbf{d}_{\\mathbf{e}} \\right)$, a large $\\tau$ (close to 1) can ensure that $\\mathbf{c}_{\\mathbf{e}}$ is the correct decoding result with high confidence. In contrast, a small $\\tau$ (far from 1) can identify the decoding output earlier while compromising the accuracy. According to our verification, $\\tau = 0.95$ can provide a good performance of ``Condition 2'', i.e., identifying correct decoding result accurately while keeping the complexity at a low degree.\n\n\\subsubsection{Average Complexity} The average decoding complexity of the proposed decoder can be roughly represented as\n\\begin{equation} \\label{equ::Complexity1}\n\\begin{split}\n    C_{\\mathrm{Prop.}} &=  p_1 (C_{\\mathrm{NonGE}} + (1-p_2)C_{\\mathrm{OSD}}) + (1-p_1)C_{\\mathrm{OSD}}\\\\\n    &=  p_1 C_{\\mathrm{NonGE}} + (1- p_1p_2)C_{\\mathrm{OSD}},\n\\end{split}\n\\end{equation}\nwhere $p_1$ and $p_2$ are the probability that ``Condition 1'' and ``Condition 2'' are satisfied, respectively. The complexity $C_{\\mathrm{NonGE}}$ is given by $C_{\\mathrm{NonGE}} = N_{\\mathrm{NonGE}} C_{\\mathrm{Re-encoding}}$, where $N_{\\mathrm{NonGE}}$ is the average number of TEPs re-encoded in the Non-GE OSD. Therefore, (\\ref{equ::Complexity1}) is re-written as\n\\begin{equation} \\label{equ::Complexity2}\n\\begin{split}\n     C_{\\mathrm{Prop.}} = & (1- p_1p_2)C_{\\mathrm{Preprocessing}} \\\\\n     & + (p_1N_{\\mathrm{NonGE}}+(1- p_1p_2)N_a)C_{\\mathrm{Re-encoding}}\\\\\n     =& (1- p_1p_2)C_{\\mathrm{Preprocessing}} + N_a'C_{\\mathrm{Re-encoding}},\n\\end{split}\n\\end{equation}\nwhere $N_a' = p_1N_{\\mathrm{NonGE}}+(1- p_1p_2)N_a$ is the average number of TEPs in the entire procedure of the proposed decoder. Comparing (\\ref{equ::Complexity2}) and (\\ref{equ::CmpOSD}), the proposed decoder can have a lower complexity than the standard OSD if $ (1- p_1p_2)<1$ and $N_a' \\approx N_a$, i.e., reducing the overhead of preprocessing while maintaining the overhead of reprocessing. \n\nRigorously determining $p_1$, $p_2$, and $C_{\\mathrm{Prop. }}$ is out of the scope of this paper \\footnote{We note that a rigorous $C_{\\mathrm{Prop.}}$ shall be determined by considering the correlations 1) between $p_1$ and $p_2$ and 2) between $N_{\\mathrm{NonGE}}$ and $p_2$. For example, when $p_1$ is small (a small $\\lambda$), $\\mathrm{P'_{list}}$ will be large, which implies that the Non-GE OSD is likely to find the correct decoding results. As a result, $p_2$ will accordingly be large, }; nevertheless, (\\ref{equ::Complexity2}) still provides insights into the parameter selection. A large $\\tau$ will decrease $p_2$, and therefore increases $ C_{\\mathrm{Prop.}}$. If $\\tau$ is assumed to be fixed, a larger $\\lambda$ will raise $p_1$, which lowers the preprocessing overhead. Nonetheless, this may increase $N_a'$, hence increasing the cost of reprocessing. We will further demonstrate the impacts of $\\lambda$ over $N_a'$ and $(1- p_1p_2)$ via simulations in Section \\ref{Sec::Simulation},\n\n\\subsubsection{Worst-Case Complexity} The proposed decoder is probable to have a higher worst-case complexity than the standard OSD. However, the increase in the worst-case complexity could be minor. Given the decoding order $m$, the worst-case complexity of OSD is given by $ C_{\\mathrm{OSD}}^{\\mathrm{(worst)}} = C_{\\mathrm{Preprocessing}} + N_mC_{\\mathrm{Re-encoding}}$, where $N_m$ is the maximum possible number of TEPs, i.e., $N_m = \\sum_{i=0}^{m}\\binom{k}{m}$. On the other hand, the worst-case decoding of the proposed decoder is derived as\n\\begin{equation} \\label{equ::Complexityworst}\n\\begin{split}\n     C_{\\mathrm{Prop.}}^{\\mathrm{(worst)}} = & C_{\\mathrm{Preprocessing}} + (N_{m'}+N_m)C_{\\mathrm{Re-encoding}},\\\\\n\\end{split}\n\\end{equation}\nwhere $N_m'$ is the maximum possible number of TEPs in Non-GE OSD, i.e.,  $N_m' = \\sum_{i=0}^{m'}\\binom{k}{m}$. Eq. (\\ref{equ::Complexityworst}) is obtained by assuming that the Non-GE OSD is always conducted and ``Condition2'' fails for all estimates. Then, by noticing $m' = m-1$ and $k\\gg m$, we have $N_m \\gg N_m'$, indicating that \n\\begin{equation}\n    C_{\\mathrm{Prop.}}^{\\mathrm{(worst)}}\\approx C_{\\mathrm{OSD}}^{\\mathrm{(worst)}}.\n\\end{equation}\n\\vspace{-0.25em} \n\\section{Simulation Results} \\label{Sec::Simulation}\n\\vspace{-0.25em} \nIn this section, we evaluate the performance and complexity of the proposed decoder via simulations. We take extended BCH (eBCH) codes as examples. Simulations in this paper were executed on MATLAB 2021a with an i7-10700 CPU. All results were obtained by simulating decoding until 500 decoding errors were collected.\n\\vspace{-0.25em} \n\\subsection{BLER performance}\n\\vspace{-0.25em} \nWe demonstrate the BLER performance of the proposed decoder in decoding various length-64 and length-128 eBCH codes in Fig \\ref{Fig::Performance}. Parameters are set to $\\tau = 0.95$ and $\\lambda = 0.05$. The proposed decoder is compared with the original OSD \\cite{Fossorier1995OSD}. The decoder orders of both decoders are set to $m = \\lceil d_{\\mathrm{H}}\/4-1\\rceil$. The Non-GE part in the proposed decoder accordingly has the order of $m-1$. It can be seen that the proposed decoder has nearly the same BLER as the original OSD. In other words, the proposed decoder can also approximately approach the ML performance (Recall that OSD at the order $m = \\lceil d_{\\mathrm{H}}\/4-1\\rceil$ is near ML \\cite{Fossorier1995OSD}).\n\n\n    \\begin{figure}  [t]\n     \\centering\n    \\begin{tikzpicture}\n    \\definecolor{applebrown}{rgb}{0.55, 0.71, 0.0}\n    \\definecolor{english}{rgb}{0.0, 0.5, 0.0}\n    \\begin{axis}[%\n    width=2.8in,\n    height=1.4in,\n    at={(0.822in,0.529in)},\n    scale only axis,\n    xmin=-1,\n    xmax=8,\n    xlabel style={at={(0.5,1ex)},font=\\color{white!15!black},font = \\small},\n    xlabel={SNR [dB]},\n    ymin=1e-6,\n    ymax=1,\n    ymode=log,\n    ylabel style={at={(1.5ex,0.5)},font=\\color{white!15!black},font = \\small},\n    ylabel={BLER},\n    ytick = {1,1e-1,1e-2,1e-3,1e-4,1e-5},\n    yminorticks=true,\n    axis background\/.style={fill=white},\n    tick label style={font=\\footnotesize},\n    xmajorgrids,\n    ymajorgrids,\n    yminorgrids,\n    major grid style={dotted,black},\n    minor grid style={dotted},\n    legend style={at={(0,0)}, anchor=south west, legend cell align=left, align=left, draw=white!15!black,font = \\tiny\t,row sep=-1pt, legend columns=1}\n    ]\n    \n    \\addplot [thick, color=black]\n          table[row sep=crcr]{%\n        10 10\\\\\n        };\n    \\addlegendentry{The original OSD \\cite{Fossorier1995OSD}}\n    \n    \\addplot [ color=red, dashed]\n          table[row sep=crcr]{%\n        10 10\\\\\n        };\n    \\addlegendentry{The proposed decoder}\n    \n    \n    \\addplot [only marks, color=black, mark=square ,mark size = 1.5pt, mark options={solid, black}]\n          table[row sep=crcr]{%\n        10 10\\\\\n        };\n    \\addlegendentry{$(64,24)$ eBCH}\n    \n    \\addplot [only marks, color=black, mark=triangle ,mark size = 2.0pt, mark options={solid, black}]\n          table[row sep=crcr]{%\n        10 10\\\\\n        };\n    \\addlegendentry{$(64,36)$ eBCH}\n    \n    \\addplot [only marks, color=black, mark=diamond ,mark size = 2.0pt, mark options={solid, black}]\n          table[row sep=crcr]{%\n        10 10\\\\\n        };\n    \\addlegendentry{$(64,45)$ eBCH}\n    \n    \\addplot [only marks, color=black, mark=o ,mark size = 2.0pt, mark options={solid, black}]\n          table[row sep=crcr]{%\n        10 10\\\\\n        };\n    \\addlegendentry{$(128,106)$ eBCH}\n\n    \n    \\addplot [ color=black, mark=square, mark size = 1.54 pt, mark options={solid, black}, forget plot]\n    table[row sep=crcr, x expr={\\thisrow{SNR}}]{%\n                SNR  Cpx\\\\\n    -1.24938736608300\t0.228519195612431\\\\\n    -0.849387366083000\t0.170706725844998\\\\\n    -0.449387366083000\t0.122010736944851\\\\\n    -0.0493873660829993\t0.0656512605042017\\\\\n    0.350612633917001\t0.0386966953022212\\\\\n    0.750612633917001\t0.0195152414035362\\\\\n    1.15061263391700\t0.00750187546886722\\\\\n    1.55061263391700\t0.00322514061613086\\\\\n    1.95061263391700\t0.000990073522859808\\\\\n    2.35061263391700\t0.000297017587005362\\\\\n    2.75061263391700\t8.12398368964043e-05\\\\\n    3.15061263391700\t1.66992524746622e-05\\\\\n    };\n    \\addlegendentry{$(64,24)$ eBCH}\n    \n\n    \\addplot [dashed,color=red, mark=square, mark size = 1.54 pt, mark options={solid, red}, forget plot]\n    table[row sep=crcr, x expr={\\thisrow{SNR}}]{%\n                SNR  Cpx\\\\\n    -0.249387366083000\t0.0963081861958267\\\\\n    0.250612633917001\t0.0440076279888514\\\\\n    0.750612633917001\t0.0182937984023416\\\\\n    1.25061263391700\t0.00640396191777313\\\\\n    1.75061263391700\t0.00177455990914253\\\\\n    2.25061263391700\t0.000451175311686945\\\\\n    2.75061263391700\t9.81526585379287e-05\\\\\n    3.25061263391700\t1.4035743395750e-05\\\\\n    };\n    \\addlegendentry{$(64,24)$ eBCH}\n    \n\n    \n    \\addplot [color=black, mark=triangle, mark size = 2 pt, mark options={solid, black}, forget plot]\n    table[row sep=crcr, x expr={\\thisrow{SNR}}]{%\n                SNR  Cpx\\\\\n      \n        1.01\t0.299043062200957\\\\\n        1.51\t0.175438596491228\\\\\n        2.01\t0.0968616815187912\\\\\n        2.51\t0.0428595919766844\\\\\n        3.01\t0.0134220981423816\\\\\n        3.51\t0.00394343536315096\\\\\n        4.01\t0.000758223694187154\\\\\n        4.51\t0.000127519888639432\\\\\n        5.01\t1.47450469323048e-05\\\\\n        5.51    1.68e-06\\\\\n       \n    };\n    \\addlegendentry{$(64,36)$ eBCH}\n    \n    \n    \\addplot [dashed,color=red, mark=triangle, mark size = 2 pt, mark options={solid, red}, forget plot]\n    table[row sep=crcr, x expr={\\thisrow{eBN0}+0.5115}]{%\n                eBN0  Cpx\\\\\n    1\t0.208333333333333\\\\\n    1.50000000000000\t0.106685633001422\\\\\n    2\t0.0457526307762696\\\\\n    2.50000000000000\t0.0149476831091181\\\\\n    3\t0.00406157345355591\\\\\n    3.50000000000000\t0.000825509339262325\\\\\n    4\t0.000131736702427636\\\\\n    4.50000000000000\t1.66550191360077e-05\\\\\n    5\t1.78000000000000e-06\\\\\n    };\n    \\addlegendentry{$(64,36)$ eBCH}\n    \n    \n\n    \n    \n    \\addplot [color=black, mark=diamond, mark size = 2 , mark options={solid, black}, forget plot]\n    table[row sep=crcr, x expr={\\thisrow{eBN0}+ 1.4806}]{%\n                eBN0  Cpx\\\\\n   \n    0.5\t0.549450549450549\\\\\n    1\t0.373134328358209\\\\\n    1.5\t0.259336099585062\\\\\n    2\t0.126198889449773\\\\\n    2.5\t0.0598229241445322\\\\\n    3\t0.0229568411386593\\\\\n    3.5\t0.00712271004871934\\\\\n    4\t0.00139593849599449\\\\\n    4.5\t0.000290677394600377\\\\\n    5\t3.259780940113e-05\\\\\n    5.5\t3.89843567403314e-06\\\\\n    };\n    \\addlegendentry{$(64,45)$ eBCH}\n    \n    \n    \\addplot [dashed,color=red, mark=diamond, mark size = 2 , mark options={solid, red}, forget plot]\n    table[row sep=crcr, x expr={\\thisrow{eBN0}+ 1.4806}]{%\n                eBN0  Cpx\\\\\n    1\t0.390625000000000\\\\\n    1.5\t0.257289879931389\\\\\n    2\t0.131694468832309\\\\\n    2.5\t0.0594059405940594\\\\\n    3\t0.0242502627111794\\\\\n    3.5\t0.00674930819590992\\\\\n    4\t0.00159797164132994\\\\\n    4.5\t0.000275109975212591\\\\\n    5\t4.41087409850923e-05\\\\\n    5.5\t4.69277717458210e-06\\\\\n    };\n    \\addlegendentry{$(64,45)$ eBCH}\n    \n    \n    \n    \\addplot [color=black, mark=o ,mark size = 2pt, mark options={solid, black}, forget plot]\n      table[row sep=crcr]{%\n    2.19125891280883\t0.97799511002445\\\\\n    2.69125891280883\t0.934579439252336\\\\\n    3.19125891280883\t0.849256900212314\\\\\n    3.69125891280883\t0.683760683760684\\\\\n    4.19125891280883\t0.471142520612485\\\\\n    4.69125891280883\t0.236127508854782\\\\\n    5.19125891280883\t0.0888296691094826\\\\\n    5.69125891280883\t0.0281729821101564\\\\\n    6.19125891280883\t0.00505439795802323\\\\\n    6.69125891280883\t0.000686511423550088\\\\\n    7.19125891280883\t6.36064849577776e-05\\\\\n    7.49125891280883\t1.211e-05\\\\\n    };\n    \\addlegendentry{$(128,106)$ eBCH}\n    \n    \\addplot [dashed,color=red, mark=o ,mark size = 2pt, mark options={solid, red}, forget plot]\n      table[row sep=crcr]{%\n    3.19125891280883\t0.87977 \\\\\n    3.69125891280883\t0.66815\\\\\n    4.19125891280883\t0.42433\\\\\n    4.69125891280883\t0.24773\\\\\n    5.19125891280883\t0.09446\\\\\n    5.69125891280883\t0.03010\\\\\n    6.19125891280883\t0.00530\\\\\n    6.69125891280883    0.00077\\\\\n    7.19125891280883\t0.00007\\\\\n    7.69125891280883\t4.811e-06\\\\\n    };\n    \\addlegendentry{$(128,106)$ eBCH}\n    \n    \n    \\end{axis}\n    \\end{tikzpicture}%\n\t\\vspace{-0.5em}\n    \\caption{BLER performance of the proposed decoder and the original OSD \\cite{Fossorier1995OSD}. The decoding order is optimal, i.e., $m = \\lceil d_{\\mathrm{H}}\/4-1\\rceil$. }\n    \\vspace{-0.25em}\n\t\\label{Fig::Performance}\n        \n\t\\end{figure}\n\n\\vspace{-0.25em} \n\\subsection{Computational Complexity}\n\\vspace{-0.25em} \nWe evaluate the complexity from two aspects; 1) the average decoding time, and 2) the number of average TEPs in decoding a single codeword. The average decoding time reflects the decoding speed in our MATLAB implementation. On the other hand, according to (\\ref{equ::CmpOSD}), the number of average TEPs can reflect the overall computational complexity of OSD and its variants when TEPs are many (i.e., reprocessing dominates the decoding complexity rather than preprocessing). \n\nWe compare the proposed decoder with the original OSD \\cite{Fossorier1995OSD}, PB-OSD \\cite{yue2021probability}, as well as the recent approach proposed by Choi \\emph{et. al.} \\cite{choi2021fast}. We implemented Choi's approach with three offline permuted generator matrices as per the implementation in \\cite{choi2021fast}. Decoding the $(64,36)$ eBCH code, we compare the decoding time and numbers of TEPs of various decoders in Fig. \\ref{Fig::64-36-Time} and Fig. \\ref{Fig::64-36-Na}, respectively. Parameters of the proposed decoder are set to $\\tau = 0.95$ and $\\lambda = 0.05$. As shown, PB-OSD has a significantly reduced complexity compared to the original OSD; however, it shows a complexity floor at high SNRs due to the overhead of GE. We highlight that this complexity floor is broken by the proposed decoder, which significantly reduces the decoding complexity of PB-OSD at high SNRs. We note that the proposed algorithm slightly increases the number of TEPs at high SNRs compared to PB-OSD; nevertheless, this is acceptable as the reduction of GE results in a lower overall decoding time. Choi's approach can also break the complexity floor. However, it may have a significantly increased number of TEPs at low SNRs compared to PB-OSD or even the original OSD. This is because it performs additional reprocessing over multiple generator matrices\\footnote{We note that although the decoding time of Choi's approach is high at low SNRs in Fig. \\ref{Fig::64-36-Time}, \\cite{choi2021fast} indicated that the decoding efficiency can be improved by employing parallelism; nevertheless, decoders in this paper were not implemented with parallel structures. We refer interested readers to \\cite{choi2021fast} for details.}. In contrast, the proposed algorithm introduces virtually no additional TEPs and decoding complexity at low SNRs compared to PB-OSD.\n\nWe further depict the average decoding time and number of TEPs in decoding the $(128,106)$ eBCH code in Fig. \\ref{Fig::128-106}. Breaking the complexity floor is more critical for such high-rate codes, because 1) OSD usually requires low orders to decode high-rate codes and therefore the number of required TEPs is not large, and 2) the overhead of GE is higher for generator matrices with more rows (a larger $k$). As shown by Fig. \\ref{Fig::128-106}, the proposed decoder has a significantly low complexity in decoding $(128,106)$ eBCH code at high SNRs. For example, at the SNR of 9 dB, the proposed decoder requires 0.09 ms to decode a codeword, compared to 1.9 ms of PB-OSD suffering from the complexity floor. \n\n\n           \\begin{figure}\n             \\centering\n             \\vspace{-0.25em}\n             \\hspace{-0.81em}\n             \\begin{subfigure}[b]{0.49\\columnwidth}\n                 \\centering\n                \\begin{tikzpicture}\n                \\begin{axis}[%\n                width=1.35in,\n                height=1.5in,\n                at={(0.785in,0.587in)},\n                scale only axis,\n                xmin=2.5,\n                xmax=8.5,\n                xlabel style={at={(0.5,2ex)},font=\\color{white!15!black},font=\\scriptsize},\n                xlabel={SNR (dB)},\n                ymode=log,\n                ymin=1e-2,\n                ymax=100,\n                yminorticks=true,\n                ylabel style={at={(3ex,0.5)},font=\\color{white!15!black},font=\\scriptsize},\n                ylabel={Average Decoding Time (ms)},\n                axis background\/.style={fill=white},\n                tick label style={font=\\tiny},\n                xmajorgrids,\n                ymajorgrids,\n                yminorgrids,\n                minor grid style={dotted},\n                major grid style={dotted,black},\n                legend style={at={(0,0)}, anchor=south west, legend cell align=left, align=left, draw=white!15!black,font = \\tiny,row sep=-3pt}\n                ]\n                \n                \\addplot [color=black, mark=square ,mark size = 1.5pt, mark options={solid, black}]\n                  table[row sep=crcr]{%\n                2.5  17.207   \\\\\n                3\t17.207  \\\\\n                3.5\t17.207  \\\\\n                4\t17.207  \\\\\n                4.5\t17.207  \\\\\n                5\t17.207  \\\\\n                5.5\t17.207  \\\\\n                6\t17.207  \\\\\n                6.5\t17.207  \\\\\n                7\t17.207  \\\\\n                7.5\t17.207  \\\\\n                8\t17.207  \\\\\n                8.5\t17.207  \\\\\n                };\n              \n\n                \n                \\addplot [color=black, mark=o ,mark size = 2pt, mark options={solid, black}]\n                  table[row sep=crcr]{%\n                2.5000    0.4939\\\\\n                3.0000    0.3067\\\\\n                3.5000    0.1933\\\\\n                4.0000    0.1283\\\\\n                4.5000    0.1030\\\\\n                5.0000    0.0926\\\\\n                5.5000    0.0903\\\\\n                6.0000    0.0895\\\\\n                6.5000    0.0895\\\\\n                7.0000    0.0904\\\\\n                7.5000    0.0912\\\\\n                8.0000    0.0929\\\\\n                8.5000    0.0906\\\\\n                };\n               \n                \n                \\addplot [color=black, mark=x ,mark size = 2pt, mark options={solid, black}]\n                  table[row sep=crcr]{%\n                2.5  0.529  \\\\\n                3\t0.299472683393288\\\\\n                3.5\t0.193465876192601\\\\\n                4\t0.131855751000000\\\\\n                4.5\t0.104433565000000\\\\\n                5\t0.0891703520000000\\\\\n                5.5\t0.0761599070000000\\\\\n                6\t0.0625690520000000\\\\\n                6.5\t0.0477919320000000\\\\\n                7\t0.0373623350000000\\\\\n                7.5\t0.0300554940000000\\\\\n                8\t0.0261969560000000\\\\\n                8.5\t0.0244859210000000\\\\\n                };\n               \n                \n                \\addplot [color=black, mark=diamond ,mark size = 2pt, mark options={solid, black}]\n                  table[row sep=crcr]{%\n                2.5000   23.3453\\\\\n                3.0000   19.1801\\\\\n                3.5000   15.8998\\\\\n                4.0000   11.8227\\\\\n                4.5000    7.1446\\\\\n                5.0000    4.4865\\\\\n                5.5000    2.3678\\\\\n                6.0000    1.2179\\\\\n                6.5000    0.3403\\\\\n                7.0000    0.1016\\\\\n                7.5000    0.0540\\\\\n                8.0000    0.0376\\\\\n                8.5000    0.0347\\\\\n                };\n               \n                \n               \n                \n          \n                \\end{axis}\n                \\end{tikzpicture}%\n                \\vspace{-0.3em}\n                \\caption{Average Decoding Time}  \n                \\vspace{-0.3em}\n                \\label{Fig::64-36-Time}\n\n             \\end{subfigure}\n            \n             \\hspace{-0.4em}\n             \\begin{subfigure}[b]{0.49\\columnwidth}\n                \\centering\n                \\begin{tikzpicture}\n                \\begin{axis}[%\n                width=1.35in,\n                height=1.5in,\n                at={(0.642in,0.505in)},\n                scale only axis,\n                xmin=2.5,\n                xmax=8.5,\n                xlabel style={at={(0.5,2ex)},font=\\color{white!15!black},font=\\scriptsize},\n                xlabel={SNR (dB)},\n                ymode=log,\n                ymin=0,\n                ymax=1e5,\n                ylabel style={at={(4ex,0.5)},font=\\color{white!15!black},font=\\scriptsize},\n                ylabel={Average number of TEPs},\n                axis background\/.style={fill=white},\n                tick label style={font=\\tiny},\n                xmajorgrids,\n                ymajorgrids,\n                yminorgrids,\n                minor grid style={dotted},\n                major grid style={dotted,black},\n                legend style={at={(1,1.05)}, anchor=south east, legend cell align=left, align=left, draw=white!15!black,font = \\tiny,row sep=-3pt, legend columns=2}\n                ]\n                \n    \n                \\addplot [color=black, mark=square ,mark size = 1.5pt, mark options={solid, black}]\n                      table[row sep=crcr]{%\n                    10 10\\\\\n                    };\n                \\addlegendentry{Original OSD \\cite{Fossorier1995OSD}}\n                \n                \\addplot  [color=black, mark=o ,mark size = 2pt, mark options={solid, black}]\n                      table[row sep=crcr]{%\n                    10 10\\\\\n                    };\n                \\addlegendentry{PB-OSD \\cite{yue2021probability}}\n                \n                \\addplot  [color=black, mark=diamond ,mark size = 2pt, mark options={solid, black}]\n                      table[row sep=crcr]{%\n                    10 10\\\\\n                    };\n                \\addlegendentry{Choi's \\cite{choi2021fast}}\n                \n                \\addplot [color=black, mark=x ,mark size = 2pt, mark options={solid, black}]\n                      table[row sep=crcr]{%\n                    10 10\\\\\n                    };\n                \\addlegendentry{Proposed}\n                \n                \\addplot [color=black, mark=x ,mark size = 2pt, mark options={solid, black}, forget plot]\n                  table[row sep=crcr]{%\n                2.5000  237.7881\\\\\n                3.0000  126.0819\\\\\n                3.5000   60.1544\\\\\n                4.0000   21.7310\\\\\n                4.5000    8.1435\\\\\n                5.0000    5.6081\\\\\n                5.5000    6.1709\\\\\n                6.0000    6.4867\\\\\n                6.5000    6.2388\\\\\n                7.0000    5.1092\\\\\n                7.5000    3.3950\\\\\n                8.0000    2.2321\\\\\n                8.5000    1.5305\\\\\n                };\n                \\addlegendentry{The proposed}\n                \n                \\addplot [color=black, mark=o ,mark size = 2pt, mark options={solid, black}, forget plot]\n                  table[row sep=crcr]{%\n                    2.5000  227.3613\\\\\n                    3.0000  127.2286\\\\\n                    3.5000   60.0300\\\\\n                    4.0000   21.1256\\\\\n                    4.5000    7.2184\\\\\n                    5.0000    2.4126\\\\\n                    5.5000    1.3807\\\\\n                    6.0000    1.0722\\\\\n                    6.5000    1.0136\\\\\n                    7.0000    1.0094\\\\\n                    7.5000    1.0019\\\\\n                    8.0000    1.0003\\\\\n                    8.5000    1.0000\\\\\n                };\n                \\addlegendentry{PB-OSD \\cite{yue2021probability}}\n                \n                \\addplot [color=black, mark=diamond ,mark size = 2pt, mark options={solid, black}, forget plot]\n                  table[row sep=crcr]{%\n                2.5\t18560.8650000000\\\\\n                3\t15360.3655000000\\\\\n                3.5\t12126.3660000000\\\\\n                4\t9369.74250000000\\\\\n                4.5\t5870.60000000000\\\\\n                5\t3666.04900000000\\\\\n                5.5\t1858.25800000000\\\\\n                6\t852.211500000000\\\\\n                6.5\t250.959000000000\\\\\n                7\t55.7085000000000\\\\\n                7.5\t17.3245000000000\\\\\n                8\t4.28850000000000\\\\\n                8.5\t2.01350000000000\\\\\n                };\n               \n                \n                \\addplot [color=black, mark=square ,mark size = 1.5pt, mark options={solid, black}, forget plot]\n                  table[row sep=crcr]{%\n                2.5  7807\\\\\n                3\t  7807\\\\\n                3.5\t  7807\\\\\n                4\t  7807\\\\\n                4.5\t  7807\\\\\n                5\t  7807\\\\\n                5.5\t  7807\\\\\n                6\t  7807\\\\\n                6.5\t  7807\\\\\n                7\t  7807\\\\\n                7.5\t  7807\\\\\n                8     7807\\\\\n                8.5\t  7807\\\\\n                };\n               \n                \n\n\n                \\end{axis}\n                \\end{tikzpicture}%\n                \\vspace{-0.3em}\n                 \\caption{Average Number of TEPs}\n                 \\vspace{-0.3em}\n                 \\label{Fig::64-36-Na}\n             \\end{subfigure}\n             \n              \\vspace{-0.11em}\n             \\caption{The average decoding time and number of TEPs of various decoders in decoding the$(64,36)$ eBCH code with order 3.}\n            \\vspace{-0.61em}\n             \\label{Fig::64-36}\n        \\end{figure}\n\n\n\n           \\begin{figure}\n             \\centering\n             \\vspace{-0.25em}\n             \\hspace{-0.81em}\n             \\begin{subfigure}[b]{0.49\\columnwidth}\n                 \\centering\n                \\begin{tikzpicture}\n                \\begin{axis}[%\n                width=1.35in,\n                height=1.5in,\n                at={(0.785in,0.587in)},\n                scale only axis,\n                xmin= 4,\n                xmax= 10,\n                xlabel style={at={(0.5,2ex)},font=\\color{white!15!black},font=\\scriptsize},\n                xlabel={SNR (dB)},\n                ymode=log,\n                ymin=1e-2,\n                ymax=100,\n                yminorticks=true,\n                ylabel style={at={(3ex,0.5)},font=\\color{white!15!black},font=\\scriptsize},\n                ylabel={Average Decoding Time (ms)},\n                axis background\/.style={fill=white},\n                tick label style={font=\\tiny},\n                xmajorgrids,\n                ymajorgrids,\n                yminorgrids,\n                minor grid style={dotted},\n                major grid style={dotted,black},\n                legend style={at={(0,0)}, anchor=south west, legend cell align=left, align=left, draw=white!15!black,font = \\tiny,row sep=-3pt}\n                ]\n                \n                \\addplot [color=black, mark=square ,mark size = 1.5pt, mark options={solid, black}]\n                  table[row sep=crcr]{%\n                4\t27.268 \\\\\n                4.5\t27.268 \\\\\n                5\t27.268 \\\\\n                5.5\t27.268 \\\\\n                6\t27.268 \\\\\n                6.5\t27.268 \\\\\n                7\t27.268 \\\\\n                7.5\t27.268 \\\\\n                8\t27.268 \\\\\n                8.5\t27.268 \\\\\n                9\t27.268 \\\\\n                9.5\t27.268 \\\\\n                10\t27.268 \\\\\n                };\n              \n\n                \n                \\addplot [color=black, mark=o ,mark size = 2pt, mark options={solid, black}]\n                  table[row sep=crcr]{%\n                4.0000    3.7135\\\\\n                4.5000    3.2725\\\\\n                5.0000    2.7513\\\\\n                5.5000    2.1758\\\\\n                6.0000    1.9852\\\\\n                6.5000    1.8191\\\\\n                7.0000    1.8289\\\\\n                7.5000    1.8615\\\\\n                8.0000    1.7834\\\\\n                8.5000    1.8478\\\\\n                9.0000    1.8499\\\\\n                9.5000    1.8798\\\\\n               10.0000    1.9112\\\\\n                };\n               \n                \n                \\addplot [color=black, mark=x ,mark size = 2pt, mark options={solid, black}]\n                  table[row sep=crcr]{%\n                4.0000    3.5236\\\\\n                4.5000    3.3048\\\\\n                5.0000    2.7370\\\\\n                5.5000    2.3815\\\\\n                6.0000    2.0316\\\\\n                6.5000    1.8902\\\\\n                7.0000    1.7499\\\\\n                7.5000    1.3460\\\\\n                8.0000    0.8306\\\\\n                8.5000    0.4576\\\\\n                9.0000    0.1997\\\\\n                9.5000    0.0908\\\\\n               10.0000    0.0599\\\\\n                };\n               \n                \n                \\addplot [color=black, mark=diamond ,mark size = 2pt, mark options={solid, black}]\n                  table[row sep=crcr]{%\n                4.0000   75.0353\\\\\n                4.5000   70.8606\\\\\n                5.0000   74.0493\\\\\n                5.5000   61.5666\\\\\n                6.0000   54.5038\\\\\n                6.5000   39.9972\\\\\n                7.0000   25.6771\\\\\n                7.5000   14.4564\\\\\n                8.0000    6.9522\\\\\n                8.5000    3.4117\\\\\n                9.0000    1.3649\\\\\n                9.5000    0.6544\\\\\n               10.0000    0.2916\\\\\n                };\n               \n                \n            \n                \n          \n                \\end{axis}\n                \\end{tikzpicture}%\n                \\vspace{-0.3em}\n                \\caption{Average Decoding Time}  \n                \\vspace{-0.3em}\n                \\label{Fig::128-106-Time}\n\n             \\end{subfigure}\n            \n             \\hspace{-0.4em}\n             \\begin{subfigure}[b]{0.49\\columnwidth}\n                \\centering\n                \\begin{tikzpicture}\n                \\begin{axis}[%\n                width=1.35in,\n                height=1.5in,\n                at={(0.642in,0.505in)},\n                scale only axis,\n                xmin=4,\n                xmax=10,\n                xlabel style={at={(0.5,2ex)},font=\\color{white!15!black},font=\\scriptsize},\n                xlabel={SNR (dB)},\n                ymode=log,\n                ymin=0,\n                ymax=1e5,\n                ylabel style={at={(4ex,0.5)},font=\\color{white!15!black},font=\\scriptsize},\n                ylabel={Average number of TEPs},\n                axis background\/.style={fill=white},\n                tick label style={font=\\tiny},\n                xmajorgrids,\n                ymajorgrids,\n                yminorgrids,\n                minor grid style={dotted},\n                major grid style={dotted,black},\n                legend style={at={(1,1.05)}, anchor=south east, legend cell align=left, align=left, draw=white!15!black,font = \\tiny,row sep=-3pt, legend columns=2}\n                ]\n                \n    \n                \\addplot [color=black, mark=square ,mark size = 1.5pt, mark options={solid, black}]\n                      table[row sep=crcr]{%\n                    100 100\\\\\n                    };\n                \\addlegendentry{Original OSD \\cite{Fossorier1995OSD}}\n                \n                \\addplot  [color=black, mark=o ,mark size = 2pt, mark options={solid, black}]\n                      table[row sep=crcr]{%\n                    100 100\\\\\n                    };\n                \\addlegendentry{PB-OSD \\cite{yue2021probability}}\n                \n                \\addplot  [color=black, mark=diamond ,mark size = 2pt, mark options={solid, black}]\n                      table[row sep=crcr]{%\n                    100 100\\\\\n                    };\n                \\addlegendentry{Choi's \\cite{choi2021fast}}\n                \n                \\addplot [color=black, mark=x ,mark size = 2pt, mark options={solid, black}]\n                      table[row sep=crcr]{%\n                    100 100\\\\\n                    };\n                \\addlegendentry{Proposed}\n                \n                \\addplot [color=black, mark=x ,mark size = 2pt, mark options={solid, black}, forget plot]\n                  table[row sep=crcr]{%\n                     4.0000  506.3795\\\\\n                    4.5000  434.6542\\\\\n                    5.0000  310.6941\\\\\n                    5.5000  148.9069\\\\\n                    6.0000   49.2371\\\\\n                    6.5000   13.7714\\\\\n                    7.0000    5.2818\\\\\n                    7.5000    4.4104\\\\\n                    8.0000    4.5196\\\\\n                    8.5000    4.3223\\\\\n                    9.0000    2.8542\\\\\n                    9.5000    1.8470\\\\\n                   10.0000    1.3805\\\\\n                };\n                \\addlegendentry{The proposed}\n                \n                \\addplot [color=black, mark=o ,mark size = 2pt, mark options={solid, black}, forget plot]\n                  table[row sep=crcr]{%\n                    4.0000  560.9407\\\\\n                    4.5000  447.0570\\\\\n                    5.0000  307.8416\\\\\n                    5.5000  145.6763\\\\\n                    6.0000   50.1076\\\\\n                    6.5000   14.0725\\\\\n                    7.0000    3.7646\\\\\n                    7.5000    1.3143\\\\\n                    8.0000    1.0450\\\\\n                    8.5000    1.0067\\\\\n                    9.0000    1.0049\\\\\n                    9.5000    1.0000\\\\\n                   10.0000    1.0000\\\\\n                };\n                \\addlegendentry{PB-OSD \\cite{yue2021probability}}\n                \n                \\addplot [color=black, mark=diamond ,mark size = 2pt, mark options={solid, black}, forget plot]\n                  table[row sep=crcr]{%\n                    4\t20683.2699228792\\\\\n                    4.5\t19378.4144144144\\\\\n                    5\t16969.3082489146\\\\\n                    5.5\t14220.3410000000\\\\\n                    6\t11730.3995000000\\\\\n                    6.5\t8581.93050000000\\\\\n                    7\t5441.44050000000\\\\\n                    7.5\t3374.11000000000\\\\\n                    8\t1697.45850000000\\\\\n                    8.5\t819.381000000000\\\\\n                    9\t298.622000000000\\\\\n                    9.5\t145.275500000000\\\\\n                    10\t59.0740000000000\\\\\n                };\n               \n                \n                \\addplot [color=black, mark=square ,mark size = 1.5pt, mark options={solid, black}, forget plot]\n                  table[row sep=crcr]{%\n                4\t  5672\\\\\n                4.5\t  5672\\\\\n                5\t  5672\\\\\n                5.5\t  5672\\\\\n                6\t  5672\\\\\n                6.5\t  5672\\\\\n                7\t  5672\\\\\n                7.5\t  5672\\\\\n                8     5672\\\\\n                8.5\t  5672\\\\\n                9\t  5672\\\\\n                9.5\t  5672\\\\\n                10\t  5672\\\\\n                };\n               \n                \n\n\n                \\end{axis}\n                \\end{tikzpicture}%\n                \\vspace{-0.3em}\n                 \\caption{Number of decoding iterations}\n                 \\vspace{-0.3em}\n                 \\label{Fig::128-106-Na}\n             \\end{subfigure}\n             \n              \\vspace{-0.11em}\n             \\caption{The average decoding time and number of TEPs of various decoders in decoding the $(128,106)$ eBCH code with order 2.}\n            \\vspace{-0.61em}\n             \\label{Fig::128-106}\n        \\end{figure}\n\n\n\n\\vspace{-0.25em} \n\\subsection{Impacts of Parameter}\n\\vspace{-0.25em} \nIn this section, we investigate the impact of $\\lambda$ on the number of TEPs, i.e., $N_a'$, and the probability of GE reduction, i.e., $p_1p_2$ in (\\ref{equ::Complexity2}). In Fig \\ref{Fig::64-36-PARA}. we illustrate $N_a'$ and $p_1p_2$ at different selections of $\\lambda$ in decoding the $(64,36)$ eBCH code. As shown, the proposed approach can almost reduce all GEs from OSD at high SNRs. Furthermore, it can be seen that there is a trade-off in selecting $\\lambda$. Specifically, if $\\lambda$ is large, the decoder can reduce more GEs at low SNRs at the expense of increasing the number of TEPs. On the other hand, a smaller $\\lambda$ results in a smaller number of TEPs, but also comprises the performance of GE reduction. We identify that the optimal selection of $\\lambda$ should be found by further research. Nevertheless, from our simulations, $\\lambda = 0.05$ could provide a good trade-off between $N_a'$ and $p_1p_2$, and result in a low decoding complexity.\n\n\n\n\n    \\begin{figure}\n             \\centering\n             \\vspace{-0.25em}\n             \\hspace{-0.81em}\n             \\begin{subfigure}[b]{0.49\\columnwidth}\n                 \\centering\n                \\begin{tikzpicture}\n                \\begin{axis}[%\n                width=1.35in,\n                height=1.5in,\n                at={(0.785in,0.587in)},\n                scale only axis,\n                xmin=2.5,\n                xmax=8.5,\n                xlabel style={at={(0.5,2ex)},font=\\color{white!15!black},font=\\scriptsize},\n                xlabel={SNR (dB)},\n                ymin=0,\n                ymax=1,\n                yminorticks=true,\n                ylabel style={at={(3ex,0.5)},font=\\color{white!15!black},font=\\scriptsize},\n                ylabel={Probability of GE Reduction, $p_1p_2$},\n                axis background\/.style={fill=white},\n                tick label style={font=\\tiny},\n                xmajorgrids,\n                ymajorgrids,\n                yminorgrids,\n                minor grid style={dotted},\n                major grid style={dotted,black},\n                legend style={at={(0,0)}, anchor=south west, legend cell align=left, align=left, draw=white!15!black,font = \\tiny,row sep=-3pt}\n                ]\n                \n                \\addplot [color=black, mark=square ,mark size = 1.5pt, mark options={solid, black}]\n                  table[row sep=crcr]{%\n                    2.5000    0.1242\\\\\n                    3.0000    0.2599\\\\\n                    3.5000    0.4318\\\\\n                    4.0000    0.5938\\\\\n                    4.5000    0.7425\\\\\n                    5.0000    0.8433\\\\\n                    5.5000    0.9101\\\\\n                    6.0000    0.9496\\\\\n                    6.5000    0.9767\\\\\n                    7.0000    0.9899\\\\\n                    7.5000    0.9960\\\\\n                    8.0000    0.9986\\\\\n                    8.5000    0.9996\\\\\n                };\n              \n\n                \n                \\addplot [color=black, mark=o ,mark size = 2pt, mark options={solid, black}]\n                  table[row sep=crcr]{%\n                2.5000    0.0055\\\\\n                3.0000    0.0225\\\\\n                3.5000    0.0814\\\\\n                4.0000    0.2026\\\\\n                4.5000    0.3931\\\\\n                5.0000    0.6114\\\\\n                5.5000    0.7881\\\\\n                6.0000    0.9033\\\\\n                6.5000    0.9618\\\\\n                7.0000    0.9860\\\\\n                7.5000    0.9956\\\\\n                8.0000    0.9983\\\\\n                8.5000    0.9993\\\\\n                };\n               \n                \n                \\addplot [color=black, mark=x ,mark size = 2pt, mark options={solid, black}]\n                  table[row sep=crcr]{%\n                2.5000         0\\\\\n                3.0000    0.0028\\\\\n                3.5000    0.0155\\\\\n                4.0000    0.0588\\\\\n                4.5000    0.1636\\\\\n                5.0000    0.3452\\\\\n                5.5000    0.5561\\\\\n                6.0000    0.7493\\\\\n                6.5000    0.8853\\\\\n                7.0000    0.9572\\\\\n                7.5000    0.9853\\\\\n                8.0000    0.9947\\\\\n                8.5000    0.9991\\\\\n                };\n               \n                \n                \\addplot [color=black, mark=diamond ,mark size = 2pt, mark options={solid, black}]\n                  table[row sep=crcr]{%\n                2.5000         0\\\\\n                3.0000    0.0004\\\\\n                3.5000    0.0032\\\\\n                4.0000    0.0154\\\\\n                4.5000    0.0605\\\\\n                5.0000    0.1613\\\\\n                5.5000    0.3378\\\\\n                6.0000    0.5538\\\\\n                6.5000    0.7407\\\\\n                7.0000    0.8719\\\\\n                7.5000    0.9430\\\\\n                8.0000    0.9792\\\\\n                8.5000    0.9933\\\\\n                };\n               \n\n                \\end{axis}\n                \\end{tikzpicture}%\n                \\vspace{-0.3em}\n                \\caption{Probability of GE reduction, $p_1p_2$}  \n                \\vspace{-0.3em}\n                \\label{Fig::64-36-p1p2}\n\n             \\end{subfigure}\n            \n             \\hspace{-0.4em}\n             \\begin{subfigure}[b]{0.49\\columnwidth}\n                \\centering\n                \\begin{tikzpicture}\n                \\begin{axis}[%\n                width=1.35in,\n                height=1.5in,\n                at={(0.642in,0.505in)},\n                scale only axis,\n                xmin=2.5,\n                xmax=8.5,\n                xlabel style={at={(0.5,2ex)},font=\\color{white!15!black},font=\\scriptsize},\n                xlabel={SNR (dB)},\n                ymode=log,\n                ymin=0,\n                ymax=1e3,\n                ylabel style={at={(4ex,0.5)},font=\\color{white!15!black},font=\\scriptsize},\n                ylabel={Average number of TEPs, $N_a'$},\n                axis background\/.style={fill=white},\n                tick label style={font=\\tiny},\n                xmajorgrids,\n                ymajorgrids,\n                yminorgrids,\n                minor grid style={dotted},\n                major grid style={dotted,black},\n                legend style={at={(1,1.05)}, anchor=south east, legend cell align=left, align=left, draw=white!15!black,font = \\tiny,row sep=-3pt, legend columns=2}\n                ]\n                \n    \n                \\addplot [only marks , color=black, mark=square ,mark size = 1.5pt, mark options={solid, black}]\n                      table[row sep=crcr]{%\n                    10 10\\\\\n                    };\n                \\addlegendentry{$\\lambda = 0.5$}\n                \n                \\addplot  [only marks , color=black, mark=o ,mark size = 2pt, mark options={solid, black}]\n                      table[row sep=crcr]{%\n                    10 10\\\\\n                    };\n                \\addlegendentry{$\\lambda = 0.2$}\n                \n                \\addplot  [only marks ,  color=black, mark=x ,mark size = 2pt, mark options={solid, black}]\n                      table[row sep=crcr]{%\n                    10 10\\\\\n                    };\n                \\addlegendentry{$\\lambda = 0.1$}\n                \n                \\addplot [only marks , color=black, mark=diamond ,mark size = 2pt, mark options={solid, black}]\n                      table[row sep=crcr]{%\n                    10 10\\\\\n                    };\n                \\addlegendentry{$\\lambda = 0.05$}\n                \n                \\addplot [color=black, mark=square ,mark size = 1.5pt, mark options={solid, black}, forget plot]\n                  table[row sep=crcr]{%\n                2.5000  286.2038\\\\\n                3.0000  234.2912\\\\\n                3.5000  210.4110\\\\\n                4.0000  194.6784\\\\\n                4.5000  153.7456\\\\\n                5.0000  110.8260\\\\\n                5.5000   71.8238\\\\\n                6.0000   43.2638\\\\\n                6.5000   22.5609\\\\\n                7.0000   11.2657\\\\\n                7.5000    5.6918\\\\\n                8.0000    2.9838\\\\\n                8.5000    1.7554\\\\\n                };\n                \\addlegendentry{$\\lambda = 0.5$}\n                \n                \\addplot [dashed, color=black, forget plot]\n                  table[row sep=crcr]{%\n                    2.5000  227.3613\\\\\n                    3.0000  127.2286\\\\\n                    3.5000   60.0300\\\\\n                    4.0000   21.1256\\\\\n                    4.5000    7.2184\\\\\n                    5.0000    2.4126\\\\\n                    5.5000    1.3807\\\\\n                    6.0000    1.0722\\\\\n                    6.5000    1.0136\\\\\n                    7.0000    1.0094\\\\\n                    7.5000    1.0019\\\\\n                    8.0000    1.0003\\\\\n                    8.5000    1.0000\\\\\n                };\n                \\addlegendentry{PB-OSD \\cite{yue2021probability}}\n                \n                \\addplot [color=black, mark=o ,mark size = 2pt, mark options={solid, black}, forget plot]\n                  table[row sep=crcr]{%\n                    2.5000  231.0426\\\\\n                    3.0000  133.1721\\\\\n                    3.5000   67.0858\\\\\n                    4.0000   45.4232\\\\\n                    4.5000   44.7203\\\\\n                    5.0000   46.5570\\\\\n                    5.5000   43.1711\\\\\n                    6.0000   31.1091\\\\\n                    6.5000   20.3303\\\\\n                    7.0000   10.4390\\\\\n                    7.5000    5.3479\\\\\n                    8.0000    3.1065\\\\\n                    8.5000    1.9397\\\\\n                };\n               \n                \n                \\addplot [color=black, mark=x ,mark size = 2pt, mark options={solid, black}, forget plot]\n                  table[row sep=crcr]{%\n                    2.5000  236.2947\\\\\n                    3.0000  126.9369\\\\\n                    3.5000   64.0263\\\\\n                    4.0000   25.0845\\\\\n                    4.5000   16.4156\\\\\n                    5.0000   15.9237\\\\\n                    5.5000   16.9914\\\\\n                    6.0000   16.6750\\\\\n                    6.5000   12.7273\\\\\n                    7.0000    7.9841\\\\\n                    7.5000    4.5692\\\\\n                    8.0000    2.8716\\\\\n                    8.5000    1.6094\\\\\n                };\n               \n                \n                \n                  \\addplot [color=black, mark=diamond ,mark size = 2pt, mark options={solid, black}, forget plot]\n                  table[row sep=crcr]{%\n                    2.5000  228.1837\\\\\n                    3.0000  127.0546\\\\\n                    3.5000   59.8820\\\\\n                    4.0000   22.7882\\\\\n                    4.5000    8.0843\\\\\n                    5.0000    5.7473\\\\\n                    5.5000    6.4787\\\\\n                    6.0000    6.9854\\\\\n                    6.5000    6.2725\\\\\n                    7.0000    5.0019\\\\\n                    7.5000    3.5111\\\\\n                    8.0000    2.2742\\\\\n                    8.5000    1.6317\\\\\n                };\n               \n\n                \\draw[->,black] (axis cs: 4.8,1.5) -- (axis cs: 5.25,1.5);\n                \\node[anchor=south] at (axis cs: 4,1){\\scriptsize PB-OSD};\n\n\n                \\end{axis}\n                \\end{tikzpicture}%\n                \\vspace{-0.3em}\n                 \\caption{Average number of TEPs, $N_a'$}\n                 \\vspace{-0.3em}\n                 \\label{Fig::64-36-N'}\n             \\end{subfigure}\n             \n              \\vspace{-0.11em}\n             \\caption{The impacts of values of $\\lambda$ in decoding the $(64,36)$ eBCH code with order 3.}\n            \\vspace{-0.61em}\n             \\label{Fig::64-36-PARA}\n        \\end{figure}\n\n\n\n\n\\vspace{-0.3em}\n\\section{Conclusion} \\label{sec::Conclusion}\n\\vspace{-0.3em}\n\nIn this paper, we designed an efficient ordered-statistics decoding (OSD) algorithm with adaptive GE reduction. The proposed decoder employs two conditions. The first condition decides whether to conduct the OSD decoding without Gaussian elimination (GE). If the OSD without GE is performed, the second condition identifies if the correct decoding result has been found in the Non-GE decoding process. If so, the standard OSD with GE can be avoided. The proposed decoding algorithm is an effective solution to the ``complexity floor'' owning to the overhead of GE in OSD decoders. Simulation results indicated that compared to the approaches from the literature, the proposed decoder can significantly reduce the decoding complexity at high SNRs while maintaining the error performance of the original OSD.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\\vspace{-0.3em}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzittr b/data_all_eng_slimpj/shuffled/split2/finalzzittr
new file mode 100644
index 0000000000000000000000000000000000000000..3d7baa09b80deba241a7de75d53f43b9a5834821
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzittr
@@ -0,0 +1,5 @@
+{"text":"\\section{Resonances}\nResonances are unstable particles usually observed as bell-shaped structures in scattering cross sections of their decay products. For a simple narrow resonance, its fundamental properties correspond to the visible cross-section features: mass $M$ is at the peak position, and decay width $\\Gamma$ is the width of the bell shape. These parameters, along with the branching fraction $x$, are known as the Breit-Wigner parameters \\cite{BW}. In reality, resonance peaks may be very broad, and the shape so deformed that it is not at all clear where exactly is the mass, or what would be the width of that resonance. In such cases resonance parameters are treated as energy dependent functions. These functions are often defined differently for different resonances. For example, in the case of $\\rho(770)$ resonance in the $\\pi \\pi$ channel, modern analyses include the pion-pion P-wave potential barrier (momentum to three halves) in the energy dependent width \\cite{GS}, while in the case of $Z$ boson the width function is \nproportional to the energy squared \\cite{Sir91}. \n\nWith such model dependent parameterizations, the simple connection between physical properties of a resonance and its model parameters is lost, and the choice of the \"proper\" resonance parameters becomes the matter of preference. There are many definitions for Breit-Wigner mass, which is assumed by some to be the proper resonance physical property. Other will prefer the real part of pole position in the complex energy plane. Some will even define the resonance mass to be something unrelated to these two most common definitions, as we will soon see, or assume that there is no difference between poles and Breit-Wigner parameters whatsoever. All that makes the comparison between cited resonance parameters quite confusing and potentially hinders the direct comparison between microscopic theoretical predictions (such is \\cite{Dur08}) and experimentally obtained resonance properties \\cite{PDG}. \n\nTo clarify this situation we try to devise a simple model-independent formula for resonant scattering, with well defined resonance physical properties, which will be capable of successfully fitting the realistic data for broad resonances.\n\nIn this letter we show how to dramatically improve the simple Breit-Wigner formula by incorporating in it just one additional (phase) parameter. This new formula has two equivalent forms that can be used to estimate either pole or Breit-Wigner parameters in a model independent way. \n\nWe begin our analysis by noting that resonant cross section is commonly parameterized by a simple Breit-Wigner formula \\cite{BW}\n\\begin{equation}\n\\sigma =\\frac{4\\pi}{q^2}\\frac{2J+1}{(2s_1+1)(2s_2+1)}\\, |A|^2,\n\\end{equation}\nwhere $q$ is a c.m.~momentum, $J$ is the spin of the resonance, while $s_1$ and $s_2$ are spins of the two incoming particles. Resonant scattering amplitude $A$ is given by\n\\begin{equation}\\label{eq:PDGparameterization}\nA=\\frac{x\\, \\Gamma\/2}{M-W-i\\,\\Gamma\/2},\n\\end{equation}\nwhere $M$ is the resonant mass, $\\Gamma$ is the total decay width, $x$ is the branching fraction to a particular channel (for inelastic scattering it is $\\sqrt{x_{\\mathrm{in}}\\,x_{\\mathrm{out}}}$), and W is c.m.~energy. \n\nThis simple parameterization cannot describe most of the realistic cross sections since the resonance shapes are seldom symmetric. To fix this, a background contribution is usually added. Unfortunately, there is no standard way to add background, but polynomials in $W^2$ (i.e., Mandelstam $s$) are commonly used in the literature (see e.g.~\\cite{BES06}) \n\\begin{equation}\\label{eq:polybg}\n|A|^2\\rightarrow|A|^2+\\sum_{k=0}^n B_k\\,W^{2k}.\n\\end{equation}\n\nTo extract the resonance parameters, we do local fits (in energy) of this parameterization to a broad range of data points in the vicinity of the resonance peak. To estimate the proper order $n$ of polynomial background, we vary endpoints of the data range and check the convergence of the physical fit parameters: $M$, $\\Gamma$, and $x$. Goodness of the convergence is estimated by calculating $c_{n,l}$ parameters for each data range and for all polynomial orders $n$ and $l$ \n\\begin{equation}\\label{eq:crit1}\nc_{n,l}=\\sum_{y=m,\\Gamma,x}(y_l-y_n)^2\/y_n^2.\n\\end{equation}\nSmaller $c_{n,l}$ means better convergence. \n\nTo avoid false positive convergence signals as much as possible, we demand good convergence not just for two, but for three consecutive polynomial orders by using \n\\begin{equation}\\label{eq:crit2}\nc_{n}=c_{n,n+1}+c_{n,n+2}.\n\\end{equation}\nFinal result is the one having the smallest reduced $\\chi^2_R$ among several fits (we use ten) with lowest convergence parameters $c_n$. When statistical errors turn out to be unrealistically small due to the dataset issues, the spread in extracted pole parameter values is used to estimate parameters errors. \n\nTo test this extraction approach, we analyze five broad resonances with well known properties, and masses ranging from less then 1 GeV, to almost 100 GeV. For $\\Delta(1232)$ and $N(1440)$, we analyze GWU \\cite{GWU} $\\pi N$ elastic partial-wave amplitudes squared. For $\\rho(770)$ and $Z$ boson we analyze $e^+e^-$ scattering ratio R (between hadronic an muonic channels) from PDG compilation \\cite{PDG}, and for $\\Upsilon(11020)$  the new {\\sc BaBar} data \\cite{Aub09}.\n\n\nUsing the Breit-Wigner parameterization (\\ref{eq:PDGparameterization}) on broad resonances does not produce very good results. Therefore, in the advanced approaches, resonance width $\\Gamma$ (and other parameters) are considered to be energy dependent, which drastically improves the fit. However, parameterization then becomes model dependent, obfuscating the connection between the model parameters and physical properties of the resonance. We want to find a simple model independent form, as close to the original Breit-Wigner parameterization as possible, that will be capable of successfully fitting the realistic data for broad resonances. To do so, we assume that the numerator and the denominator in Rel.~(\\ref{eq:PDGparameterization}) are functions of energy, expand them and keep only the linear terms. The amplitude $A$ becomes\n\\begin{equation}\\label{eq:res.amplitude}\nA = \\frac{x_p\\,\\Gamma_p\/2\\, \\,e^{i\\theta_p}}{M_p-W-i\\,\\Gamma_p\/2}+|A_{B}|\\,e^{i\\theta_{B}},\n\\end{equation}\nwhich turns out to be the lowest order Laurent expansion of amplitude A about its pole position at \\mbox{$W= M_p-i\\,\\Gamma_p\/2$}. Therefore, $M_p$ and $\\Gamma_p$ are pole mass and width, while $x_p\\,\\Gamma_p\/2$ and $\\theta_p$ are the complex residue magnitude and phase, respectively. (Note that we use the standard convention for residue phase $\\theta_p$, as used in PDG \\cite{PDG}, which differs from the mathematical residue phase by $\\pm\\pi$.) Three additional fit parameters are residue phase $\\theta_p$, (coherently added) background magnitude $|A_B|$, and the background phase $\\theta_B$. We can extract only the relative phase $\\delta_{p}=\\theta_{p}-\\theta_{B}$, since absolute square of this amplitude will be compared to the data. In order to ease the numerical analysis, we rewrite the new parameterization in a compact form\n\\begin{equation}\\label{eq:betterparameterization}\n|A|^2 = |A_B|^2\\frac{(\\mu-W)^2+\\lambda^2}{(M_p-W)^2+\\Gamma_p^2\/4},\n\\end{equation}\nwhere $\\mu$ and $\\lambda$ are simple fit parameters related to the pole parameters through \n\\begin{align}\nx_p\\,\\sin\\delta_p&=|A_B|\\,\\frac{\\Gamma_p\/2-|\\lambda|}{\\Gamma_p\/2},\\\\\nx_p\\,\\cos\\delta_p&=-|A_B|\\,\\frac{M_p-\\mu}{\\Gamma_p\/2}.\n\\end{align}\n\nUsing parameterization (\\ref{eq:betterparameterization}), we should be able to extract pole mass, width, branching fraction, magnitude of the background amplitude, and the relative phase from the data. We again use the same polynomial background from relation (\\ref{eq:polybg}) and convergence criteria from  relations (\\ref{eq:crit1}) and (\\ref{eq:crit2}). However, at the very beginning of this analysis we stumbled upon a problem with our fits. When we fitted $\\Delta(1232)$ resonance, parameter $\\lambda$ was rather unstable, ranging from zero to several thousands MeV. In addition, fits often did not converge, even for a carefully chosen initial values. \n\nWe looked into it more closely and realised that since $\\Delta(1232)$ is almost elastic resonance (decaying by more than 99 percent to $\\pi N$ channel), $\\lambda$ should be zero due to the elastic two-body unitarity condition \\mbox{$A^\\dag A=\\mathrm{Im}\\,A$}. When $\\lambda$ was set to zero, everything worked almost perfectly. It is important to note  that $x_p$ should be 1 for elastic resonances, again due to the unitarity, but setting $\\lambda$ to zero does not imply that $x_p$ is 1.\n\nThings became really interesting when we tried to extract Z boson parameters from $e^+e^-$ scattering data. The unstable $\\lambda$ behavior seen in the case of $\\Delta(1232)$ was observed again, even though Z boson is definitely not an elastic resonance. Fit could not be stabilized, and eventually we tried $\\lambda=0$ again ($x_p$ still can take care of inelasticity). This choice smoothed the fitting procedure, and the extracted resonance parameters were in excellent agreement with the PDG (pole) estimates \\cite{PDG}. Assuming that $\\lambda=0$ for other processes, we rewrite amplitude defined in Eq.~(\\ref{eq:res.amplitude}) as:\n \\begin{equation}\\label{eq:newamplitude}\nA = x_p\\,e^{i\\eta}\\left(\\frac{\\Gamma_p\/2 \\,\\, e^{2i\\delta_p}} {M_p-W-i\\,\\Gamma_p\/2} + e^{i\\delta_p}\\sin\\delta_p\\right),\n\\end{equation}\nwith unmeasurable overall phase $\\eta$ equal to $2\\theta_{B}-\\theta_p$. Square of this amplitude is then\n \\begin{equation}\\label{eq:pole}\n |A|^2 = x_p^2\\,\\frac{\\left[(M_p-W)\\,\\sin\\delta_p+\\Gamma_p\/2\\,\\cos\\delta_p\\right]^2}{(M_p-W)^2+\\Gamma_p^2\/4}.\n\\end{equation}\n\nWe know that for $\\Delta(1232)$, the overall phase $\\eta$ iz zero due to unitarity, which means that $\\theta_{B} = \\delta_p$ and $\\theta_p=2\\delta_p$. We  compare our results for $2\\delta_p$ to published results of $\\theta_p$ for other analyzed resonances to check whether the same relation is valid for them as well.  Roper resonance N(1440) is $\\pi N$ resonance with $\\pi N$ branching fraction $x$ estimated to 65\\%, and the $2\\delta_p$ value of $-81^\\circ$ is a surprisingly close to the newest residue phase estimate  $-85^\\circ$ from \\cite{PDG}. For the Z boson, these two values are even closer: $2\\delta_p$ is $-2.2^\\circ$, while $\\theta_p$ is $-2.35^\\circ$. Extracted masses and widths are much closer to the pole parameters listed in the literature, than to the Breit-Wigner ones, as can be seen in Table \\ref{tab:poles}. The best fits for all analyzed resonances are shown in Figures \\ref{fig:results5}-\\ref{fig:results11}.\n\n\n\n\\begin{table}[h!]\n\\caption{\\label{tab:poles} Resonance pole parameters extracted by using pole formula (\\ref{eq:pole}). Our $2\\delta_p$ is compared to the residue phase $\\theta_p$ from the literature. PDG pole estimates are from Ref.~\\cite{PDG}. The $\\rho$ meson pole and the Z boson residue phase are estimated by analytic continuation of Gounaris-Sakurai \\cite{GS} and Breit-Wigner \\cite{PDG} parameterization, respectively. }\n\\begin{ruledtabular}\n\\begin{tabular}{lllll}\nResonance & $M_p$ \/ MeV & $\\Gamma_p$ \/ MeV & $x_p$ \/ \\% & $2\\delta_p$ \/ $^o$\\\\\\hline\n$\\rho(770)$                &  762   $\\pm$  1   &   138  $\\pm$  2   &   0.71    &   1  $\\pm$  1   \\\\\n\\scriptsize POLE                               &  763   &   144    & \\scriptsize  N\/A   & \\scriptsize N\/A   \\\\\\hline \n$\\Delta(1232)$               &  1211   $\\pm$ 1    &  102  $\\pm$  1   &  103   $\\pm$  1   &  $-$47   $\\pm$ 1   \\\\\n\\scriptsize PDG\/POLE                       &  1210   $\\pm$  1   &  100   $\\pm$  2   &  104 $\\pm$ 2 &   $-$47  $\\pm$  1   \\\\\\hline\n$N(1440) $           &  1362   $\\pm$  5   &   191  $\\pm$  10   &  61   $\\pm$  4   &  $-$81  $\\pm$  10   \\\\\n\\scriptsize PDG\/POLE               &  1365   $\\pm$  15   &  190   $\\pm$  30   &  65   $\\pm$  10   &  $-$85   $\\pm$  $^{15}_{10}$   \\\\\\hlin\n$\\Upsilon(11020)$         &   11000  $\\pm$  2   &  43   $\\pm$  6   &  0.10   &  $-$52   $\\pm$  8   \\\\\n {\\scriptsize\\sc BaBar}  \\cite{Aub09}                   &   10996  $\\pm$  2   &  37   $\\pm$  3     & \\scriptsize N\/A  & \\scriptsize N\/A  \\\\\\hline\n\n$Z(91188)$                    &  91167   $\\pm$  6   &  2493   $\\pm$  5   &  15.4    &   $-$2.2  $\\pm$   0.2  \\\\\n\\scriptsize PDG\/POLE                               &  91162   $\\pm$  2   &  2494   $\\pm$  2   &  \\scriptsize N\/A   &     $-2.35$    \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n \nThe new pole parameterization (\\ref{eq:pole}) may be used instead of (\\ref{eq:PDGparameterization}), since it works much better and adds only one quite important parameter $\\delta_p$. This parameter is the main ingredient of the shape of the resonance contribution to the cross section. When $\\delta_p$ is equal to zero, the new pole and the old simple Breit-Wigner parameterization are exactly the same. \n\n\nWe began this study in the first place to find an improved Breit-Wigner parameterization, but ended up with pole parameterization instead. Following the original notion of Breit and Wigner, that the resonance mass is at the peak position \\cite{BW}, and by noting the convenient form of our pole parameterization (\\ref{eq:newamplitude}) that looks very similar to a single channel elastic amplitude (apart from $x_p\\neq1$ and $\\eta\\neq0$). We now define the new Breit-Wigner parameters as single-channel K-matrix pole $M_b$, residue $\\Gamma_b$, branching fraction $x_b$, and background phase $\\delta_b$\n\\begin{align}\nK&= \\frac{\\Gamma_{b}\/2}{M_{b}-W} + \\tan\\delta_{b},\\\\\nA&=x_{b}\\,\\frac{K}{1-iK},\\\\\n|A|^2 &=x_{b}^2\\frac{\\left[\\Gamma_{b}\/2+(M_{b}-W)\\tan{\\delta_{b}}\\right]^2}{(M_{b}-W)^2+\\left[\\Gamma_{b}\/2+(M_{b}-W)\\tan{\\delta_{b}}\\right]^2}. \\label{eq:BW}\n\\end{align}\n\nIn this form, $x_{b}$ and $\\delta_{b}$ will be mathematically equal to $x_p$ and $\\delta_p$, respectively. When we fit parameterization (\\ref{eq:BW}) to the data, extracted fit parameters $M_{b}$, $\\Gamma_{b}$, and $x_{b}$ are consistent with Breit-Wigner parameters in PDG \\cite{PDG}, as is clearly visible from Table \\ref{tab:breitwigner}. As expected,  $x_{b}$ and $\\delta_{b}$ have almost exactly equal values as their pole counterparts $x_p$ and $\\delta_p$ in Table \\ref{tab:poles}. \nFurthermore, the extracted pole and Breit-Wigner parameters are interrelated through Manley relations \\cite{Man95}\n\\begin{align}\nM_{b}&=M_p-\\Gamma_p\/2\\,\\,\\tan\\delta_p,\\\\\n\\Gamma_{b}&=\\Gamma_p\/\\cos^2\\delta_p. \n\\end{align}\n\n\n\\begin{table}[h!]\n\\caption{\\label{tab:breitwigner}  Resonance parameters extracted by using new Breit-Wigner formula (\\ref{eq:BW}). PDG estimates are from Ref.~\\cite{PDG}.}\n\\begin{ruledtabular}\n\\begin{tabular}{lllll}\nResonance                    & $M_{b}$ \/ MeV         & $\\Gamma_{b}$ \/ MeV & $x_{b}$ \/ \\% & $2\\delta_{b}$ \/ $^o$ \\\\\\hline\n$\\rho(770)$                &  761   $\\pm$  1  &    139  $\\pm$  2   &   0.71    &   0  $\\pm$ 1   \\\\\n\\scriptsize  PDG                               &   \\scriptsize 775.5   $\\pm$  0.3  &   \\scriptsize 146.2   $\\pm$  0.7   & 0.69   & \\scriptsize N\/A   \\\\\n\\hline\n$\\Delta(1232)$               &   1233  $\\pm$    1         &   120  $\\pm$   1  &  102   $\\pm$   1  &  $-$46  $\\pm$ 1    \\\\\n\\scriptsize  PDG\/BW                         &  1232   $\\pm$  2   &  117   $\\pm$  3   &  100   &  \\scriptsize N\/A   \\\\\\hline\n$N(1440) $                    &  1443 $\\pm$ 2   &  325 $\\pm$ 11  &  61 $\\pm$ 4   &   $-$80 $\\pm$ 2  \\\\\n\\scriptsize  PDG\/BW                        &  1440   $\\pm$  $^{30}_{20}$   &  300   $\\pm$  $^{150}_{100}$   &  65   $\\pm$  10   & \\scriptsize N\/A   \\\\\\hline\n$\\Upsilon(11020)$           &       11010        $\\pm$    2    &      53        $\\pm$  8   &    0.10  &  $-$52 $\\pm$ 8     \\\\\n\\scriptsize  PDG                                &   11019  $\\pm$  8    &  79  $\\pm$  16   &   \\scriptsize N\/A   &  \\scriptsize  N\/A      \\\\\\hline\n$Z(91188)$                     & 91191 $\\pm$  5   &   2494           $\\pm$ 5    &   15.4  &  $-$2.2 $\\pm$ 0.2     \\\\\n\\scriptsize  PDG\/BW                         &  91188  $\\pm$  2   &  2495   $\\pm$  2   & 15.3   &  \\scriptsize   N\/A       \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\begin{figure}[h!]\n\\includegraphics[width=8.5cm]{fig5.eps}\n\\caption{Fitting pole parameterization (\\ref{eq:pole}) to the data \\cite{PDG}. The pole and BW masses have almost the same value, quite different from PDG estimate (dotted line). (We removed the data from the peak to eliminate the influence of the $\\omega(782)$ resonance.) \\label{fig:results5}}\n\\end{figure}\n\n \\begin{figure}[h!]\n\\includegraphics[width=8.5cm]{fig1.eps}\n\\caption{Fitting pole parameterization (\\ref{eq:pole}) to the SAID data \\cite{GWU}. Pole (solid) and BW mass (dashed) are clearly distinct, while the PDG estimates (dotted lines) are indistinguishable from them. All the data in this figure are analyzed, but only the black data points are used in the best fit. \\label{fig:results1}}\n\\end{figure}\n \\begin{figure}[h!]\n\\includegraphics[width=8.5cm]{fig2.eps}\n\\caption{Fitting pole parameterization (\\ref{eq:pole}) to the SAID data \\cite{GWU}. This resonance has the largest difference between pole and BW mass. PDG estimates (dotted liness) are  consistent with pole (solid) and BW (dashed) parameters. \\label{fig:results2}}\n\n\\end{figure}\n \\begin{figure}[h!]\n\\includegraphics[width=8.5cm]{fig14.eps}\n\\caption{Fitting pole parameterization (\\ref{eq:pole}) to the {\\sc BaBar}  data \\cite{Aub09}. Pole and BW masses are clearly distinct, but PDG estimate coincides with neither of them. \\label{fig:results14}}\n\\end{figure}\n \\begin{figure}[h!]\n\\includegraphics[width=8.5cm]{fig11.eps}\n\\caption{Fitting pole parameterization (\\ref{eq:pole}) to the data \\cite{PDG}. The pole and BW parameters are consistent with their PDG estimates (dotted lines).  \\label{fig:results11}}\n\\end{figure}\n\nThese Breit-Wigner parameters are uniquely defined and model independent, with directly observable mass as the peak of the squared amplitude $|A|^2$. However, they strongly depend on phase $\\delta_p$, which may change from reaction to reaction. That means that for the same pole position, there will be different Breit-Wigner masses and widths in different channels. Therefore, it is more practical to have one (pole) mass and width for each resonance in particle data tables \\cite{PDG} than stockpiling different (Breit-Wigner) masses and widths for each process in which the resonance contributes.\n\nIn our study, there are two resonances that show systematic discrepancy between the PDG mass estimates \\cite{PDG} and our results presented here: the $\\rho(770)$ and the $\\Upsilon(11020)$. For $\\rho$ meson an alternative parameterization by Gounaris and Sakurai  \\cite{GS} is used, where mass and width are defined somewhat unconventionally to take into account its mixing with $\\omega(782)$ and $\\rho(1450)$. $\\Upsilon(11020)$ mass and width were fit parameters to a Gaussian with a relativistic tail \\cite{CUSBCLEO}. For both resonances cited values in the PDG tables are neither masses consistent with the original Breit-Wigner idea, being at the peak of the resonance, nor the pole positions. Since the resonance parameters are collected in PDG tables to be used as an input for various models and for comparison between theory and experiment, placing all these resonance parameters in a single table may generally create considerable confusion. This confusion is evident in the $\\rho$ meson case where in the table \nwith predominantly Gounaris-Sakurai masses (about 775 MeV) one can find pole masses (roughly 760 MeV).  \n\nIn conclusion, we have shown here that the original Breit-Wigner formula may be drastically improved by including a single additional (phase) parameter $\\delta_p$. Our results suggest that parameter $\\delta_p$ seems to be equal to the half of the resonance residue phase $\\theta_p$, regardless of the resonance inelasticity. This new formula has two equivalent forms that can be used to estimate either the pole, or the Breit-Wigner parameters in a model independent way. Having both forms enabled us to learn that in the PDG tables \\cite{PDG} there are values that do not correspond neither to pole, nor to Breit-Wigner parameters. Such an outcome undermines the proper matching between microscopic theories (e.g., lattice QCD \\cite{Dur08}) and experiment.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFrom the time of Connes's 1995 paper~\\cite{Connes95}, spectral triples with finite-di{\\-}men{\\-}sion{\\-}al \\ensuremath{\\ast}-algebra and Hilbert space, or \\term{finite spectral triples}, have been central to the noncommutative-geometric (NCG) approach to the Standard Model of elementary particle physics, where they are used to encode the fermionic physics. As a result, they have been the focus of considerable research activity.\n\nThe study of finite spectral triples began in earnest with papers by Paschke and Sitarz~\\cite{PS98} and by Krajewski~\\cite{Kraj98}, first released nearly simultaneously in late 1996 and early 1997, respectively, which gave detailed accounts of the structure of finite spin geometries, {\\it i.e.\\\/}\\  of finite real spectral triples of $KO$-dimension $0 \\bmod 8$ satisfying orientability and Poincar{\\'e} duality. In their approach, the study of finite spectral triples is reduced, for the most part, to the study of \\term{multiplicity matrices}, integer-valued matrices that explicitly encode the underlying representation-theoretic structure. Krajewski, in particular, defined what are now called \\term{Krajewski diagrams} to facilitate the classification of such spectral triples. Iochum, Jureit, Sch{\\\"u}cker, and Stephan have since undertaken a programme of classifying Krajewski diagrams for finite spectral triples satisfying certain additional physically desirable  assumptions~\\cites{ACG1,ACG2,ACG3,Sch05} using combinatorial computations~\\cite{JS08}, with the aim of fixing the finite spectral triple of the Standard Model amongst all other such triples.\n\nHowever, there were certain issues with the then-current version of the NCG Standard Model, including difficulty with accomodating massive neutrinos and the so-called fermion doubling problem, that were only to be resolved in the 2006 papers by Connes~\\cite{Connes06} and by Chamseddine, Connes and Marcolli~\\cite{CCM07}, which use the Euclidean signature of earlier papers, and by Barrett~\\cite{Bar07}, which instead uses Lorentzian signature; we restrict our attention to the Euclidean signature approach of~\\cite{Connes06} and~\\cite{CCM07}, which has more recently been set forth in the monograph~\\cite{CM08} of Connes and Marcolli. The finite spectral triple of the current version has $KO$-dimension $6 \\bmod 8$ instead of $0 \\bmod 8$, fails to be orientable, and only satisfies a certain modified version of Poincar{\\'e} duality. It also no longer satisfies $S^0$-reality, another condition that holds for the earlier finite geometry of~\\cite{Connes95}, though only because of the Dirac operator. Jureit, and Stephan~\\cites{ACG4,ACG5} have since adopted the new value for the $KO$-dimension, but further assume orientability and Poincar{\\'e} duality. As well, Stephan~\\cite{St06} has proposed an alternative finite spectral triple for the current NCG Standard Model with the same physical content but satisfying Poincar{\\'e} duality; it also just fails to be $S^0$-real in the same manner as the finite geometry of~\\cite{CCM07}; in the same paper, Stephan also discusses non-orientable finite spectral triples.\n\nMore recently, Chamseddine and Connes~\\cites{CC08a,CC08b} have sought a purely algebraic method of isolating the finite spectral triple of the NCG Standard Model, by which they have obtained the correct \\ensuremath{\\ast}-algebra, Hilbert space, grading and real structure using a small number of fairly elementary assumptions. In light of these successes, it would seem reasonable to try to view this new approach of Chamseddine and Connes through the lens of the structure theory of Krajewski and Paschke--Sitarz, at least in order to understand better their method and the assumptions involved. This, however, would require adapting that structure theory to handle the failure of orientability and Poincar{\\'e} duality, yielding the initial motivation of this work.\n\nTo that end, we provide, for the first time, a comprehensive account of the structure theory of Krajewski and Paschke--Sitarz for finite real spectral triples of arbitrary $KO$-dimension, without the assumptions of orientability or Poincar{\\'e} duality; this consists primarily of straightforward generalisations of the results and techniques of~\\cite{PS98} and~\\cite{Kraj98}. In this light, the main features of the approach presented here are the following:\n\\begin{enumerate}\n \\item A finite real spectral triple with algebra $\\alg{A}$ is to be viewed as an $\\alg{A}$-bimodule with some additional structure, together with a choice of Dirac operator compatible with that structure.\n \\item For fixed algebra $\\alg{A}$, an $\\alg{A}$-bimodule is entirely characterised by its multiplicity matrix (in the ungraded case) or matrices (in the graded case), which also completely determine(s) what sort of additional structure the bimodule can admit; this additional structure is then unique up to unitary equivalence.\n \\item The form of suitable Dirac operators for an $\\alg{A}$-bimodule with real structure is likewise determined completely by the multiplicity matrix or matrices of the bimodule and the choice of additional structure.\n\\end{enumerate}\nHowever, we do not discuss Krajewski diagrams, though suitable generalisation thereof should follow readily from the generalised structure theory for Dirac operators.\n\nOnce we view a real spectral triple as a certain type of bimodule together with a \\emph{choice} of suitable Dirac operator, it then becomes natural to consider moduli spaces of suitable Dirac operators, up to unitary equivalence, for a bimodule with fixed additional structure, yielding finite real spectral triples of the appropriate $KO$-dimension. The construction and study of such moduli spaces of Dirac operators first appear in~\\cite{CCM07}, though the focus there is on the sub-moduli space of Dirac operators commuting with a certain fixed subalgebra of the relevant \\ensuremath{\\ast}-algebra. Our last point above almost immediately leads us to relatively concrete expressions for general moduli spaces of Dirac operators, which also appear here for the first time. Multiplicity matrices and moduli spaces of Dirac operators are then worked out for the bimodules appearing in the Chamseddine--Connes--Marcolli formulation of the NCG Standard Model~\\cites{CCM07,CM08} as examples.\n\nFinally, we apply these methods to the work of Chamseddine and Connes~\\cites{CC08a,CC08b}, offering concrete proofs and some generalisations of their results. In particular, the choices determining the finite geometry of the current NCG Standard Model within their framework are made explicit.\n\nThis work, a revision of the author's qualifying year project (master's thesis equivalent) at the Bonn International Graduate School in Mathematics (BIGS) at the University of Bonn, is intended as a first step towards a larger project of investigating in generality the underlying noncommutative-geometric formalism for field theories found in the NCG Standard Model, with the aim of both better understanding current versions of the NCG Standard Model and facilitating the further development of the formalism itself.\n\nThe author would like to thank his supervisor, Matilde Marcolli, for her extensive comments and for her advice, support, and patience, Tobias Fritz for useful comments and corrections, and George Elliott for helpful conversations. The author also gratefully acknowledges the financial and administrative support of BIGS and of the Max Planck Institute for Mathematics, as well as the hospitality and support of the Department of Mathematics at the California Institute of Technology and of the Fields Institute.\n\n\\section{Preliminaries and Definitions}\n\n\\subsection{Real {\\ensuremath{C^*}}-algebras}\n\nIn light of their relative unfamiliarity compared to their complex counterparts, we begin with some basic facts concerning real {\\ensuremath{C^*}}-algebras.\n\nFirst, recall that a \\term{real \\ensuremath{\\ast}-algebra} is a real associative algebra $\\alg{A}$ together with an \\term{involution} on $\\alg{A}$, namely an antihomomorphism $\\ast$ satisfying $\\ast^2 = \\Id$, and that the \\term{unitalisation} of a real \\ensuremath{\\ast}-algebra $\\alg{A}$ is the unital real \\ensuremath{\\ast}-algebra $\\tilde{\\alg{A}}$ defined to be $\\alg{A} \\oplus \\field{R}$ as a real vector space, together with the multiplication $(a,\\alpha)(b,\\beta) := (ab + \\alpha b + \\beta a, \\alpha\\beta)$ for $a$, $b \\in \\alg{A}$, $\\alpha$, $\\beta \\in \\field{R}$ and the involution $\\star \\oplus \\Id_{\\field{R}}$. Note that if $\\alg{A}$ is already unital, then $\\unit{\\alg{A}}$ is simply $\\alg{A}\\oplus\\field{R}$.\n\n\\begin{definition}\n A \\term{real {\\ensuremath{C^*}}-algebra} is a real \\ensuremath{\\ast}-algebra $\\alg{A}$ endowed with a norm $\\norm{\\cdot}$ making $\\alg{A}$ a real Banach algebra, such that the following two conditions hold:\n\\begin{enumerate}\n \\item $\\forall a \\in \\alg{A}$, $\\norm{a^* a} = \\norm{a}^2$ (\\term{{\\ensuremath{C^*}}-identity});\n \\item $\\forall a \\in \\unit{\\alg{A}}$, $1 + a^* a$ is invertible in $\\unit{\\alg{A}}$ (\\term{symmetry}).\n\\end{enumerate}\n\\end{definition}\n\nThe symmetry condition is redundant for complex {\\ensuremath{C^*}}-algebras, but not for real {\\ensuremath{C^*}}-algebras. Indeed, consider $\\field{C}$ as a real algebra together with the trivial involution $\\ensuremath{\\ast} = \\Id$ and the usual norm $\\norm{\\zeta} = |\\zeta|$, $\\zeta \\in \\field{C}$. Then $\\field{C}$ with this choice of involution and norm yields a real Banach \\ensuremath{\\ast}-algebra satisfying the {\\ensuremath{C^*}}-identity but not symmetry, for $1 + i^* i = 0$ is certainly not invertible in $\\unit{\\field{C}} = \\field{C}\\oplus\\field{R}$.\n\nNow, in the finite-dimensional case, one can give a complete description of real {\\ensuremath{C^*}}-algebras, which we shall use extensively in what follows:\n\n\\begin{theorem}[Wedderburn's theorem for real {\\ensuremath{C^*}}-algebras \\cite{Fare}]\n Let $\\alg{A}$ be a finite-dimensional real {\\ensuremath{C^*}}-algebra. Then\n\\begin{equation}\n \\alg{A} \\cong \\bigoplus_{i=1}^N M_{n_i}(\\field{K}_i),\n\\end{equation}\nwhere $\\field{K}_i = \\field{R}$, $\\field{C}$, or $\\field{H}$, and $n_i \\in \\semiring{N}$. Moreover, this decomposition is unique up to permutation of the direct summands.\n\\end{theorem}\n\nNote, in particular, that a finite-dimensional real {\\ensuremath{C^*}}-algebra is necessarily unital.\n\nGiven a finite-dimensional real {\\ensuremath{C^*}}-algebra $\\alg{A}$ with fixed \\term{Wedderburn decomposition} $\\oplus_{i=1}^N M_{n_i}(\\field{K}_i)$ we can associate to $\\alg{A}$ a finite dimensional complex {\\ensuremath{C^*}}-algebra $\\alg{A}_{\\field{C}}$, the \\term{complex form} of $\\alg{A}$, by setting\n\\begin{equation}\n \\alg{A}_{\\field{C}} := \\bigoplus_{i=1}^N M_{m_i}(\\field{C}),\n\\end{equation}\nwhere $m_i = 2n_i$ if $\\field{K}_i = \\field{H}$, and $m_i = n_i$ otherwise. Then $\\alg{A}$ can be viewed as a real \\ensuremath{\\ast}-subalgebra of $\\alg{A}_\\field{C}$ such that $\\alg{A}_{\\field{C}} = \\alg{A} + i \\alg{A}$, that is, as a \\term{real form} of $\\alg{A}_{\\field{C}}$. Here, $\\field{H}$ is considered as embedded in $M_2(\\field{C})$ by\n\\[\n \\zeta_1 + j \\zeta_2 \\mapsto\n \\begin{pmatrix}\n  \\zeta_1 & \\zeta_2 \\\\\n  -\\overline{\\zeta_2} & \\overline{\\zeta_1}\n \\end{pmatrix},\n\\]\nfor $\\zeta_1$, $\\zeta_2 \\in \\field{C}$.\n\nIn what follows, we will consider only finite-dimensional real {\\ensuremath{C^*}}-algebras with fixed Wedderburn decomposition.\n\n\n\\subsection{Representation theory}\n\nIn keeping with the conventions of noncommutative differential geometry, we shall consider \\ensuremath{\\ast}-representations of real {\\ensuremath{C^*}}-algebras on complex Hilbert spaces. Recall that such a (left) representation of a real {\\ensuremath{C^*}}-algebra $\\alg{A}$ consists of a complex Hilbert space $\\hs{H}$ together with a \\ensuremath{\\ast}-homomorphism $\\lambda : \\alg{A} \\to \\mathcal{L}(\\hs{H})$ between real {\\ensuremath{C^*}}-algebras. Similarly, a \\term{right representation} of $\\alg{A}$ is defined to be a complex Hilbert space $\\hs{H}$ together with a \\ensuremath{\\ast}-\\emph{antihomomorphism} $\\rho : \\alg{A} \\to \\mathcal{L}(\\hs{H})$ between real {\\ensuremath{C^*}}-algebras. For our purposes, then, an \\term{$\\alg{A}$-bimodule} consists of a complex Hilbert space $\\hs{H}$ together with a left \\ensuremath{\\ast}-representation $\\lambda$ and a right \\ensuremath{\\ast}-representation $\\rho$ that commute, {\\it i.e.\\\/}\\  such that $[\\lambda(a),\\rho(b)]=0$ for all $a$, $b \\in \\alg{A}$. In what follows, we will consider only finite-dimensional representations and hence only finite-dimensional bimodules; since finite-dimensional {\\ensuremath{C^*}}-algebras are always unital, we shall require all representations to be unital as well.\n\nNow, given a left [right] representation $\\alpha = (\\hs{H},\\pi)$ of an algebra $\\alg{A}$, one can define its \\term{transpose} to be the right [left] representation $\\alpha^T = (\\hs{H}^*,\\pi^T)$ , where $\\pi^T(a) := \\pi(a)^T$ for all $a \\in \\alg{A}$. Note that for any left or right representation $\\alpha$, $(\\alpha^T)^T$ can naturally be identified with $\\alpha$ itself. In the case that $\\hs{H} = \\field{C}^N$, we shall identify $\\hs{H}^*$ with $\\hs{H}$ by identifying the standard ordered basis on $\\hs{H}$ with the corresponding dual basis on $\\hs{H}^*$. The notion of the transpose of a representation allows us to reduce discussion of right representations to that of left representations.\n\nSince real {\\ensuremath{C^*}}-algebras are semisimple, any left representation can be written as a direct sum of irreducible representations, unique up to permutation of the direct summands, and hence any right representation can be written as a direct sum of transposes of irreducible representations, again unique up to permutation of the direct summands.\n\n\\begin{definition}\n The \\term{spectrum} $\\spec{\\alg{A}}$ of a real {\\ensuremath{C^*}}-algebra $\\alg{A}$ is the set of unitary equivalence classes of irreducible representations of $\\alg{A}$.\n\\end{definition}\n\nNow, let $\\alg{A}$ be a real {\\ensuremath{C^*}}-algebra with Wedderburn decomposition $\\oplus_{i=1}^N M_{k_i}(\\field{K}_i)$. Then\n\\begin{equation}\n \\spec{\\alg{A}} = \\bigsqcup_{i=1}^N \\spec{M_{k_i}(\\field{K}_i)},\n\\end{equation}\nwhere the embedding of $\\spec{M_{k_i}(\\field{K}_i)}$ in $\\spec{\\alg{A}}$ is given by composing the representation maps with the projection of $\\alg{A}$ onto the direct summand $M_{k_i}(\\field{K}_i)$. The building blocks for $\\spec{\\alg{A}}$ are as follows:\n\\begin{enumerate}\n \\item $\\spec{M_{n}(\\field{R})} = \\{[(\\field{C}^n,\\lambda)]\\}$,\n \\item $\\spec{M_{n}(\\field{C})} = \\{[(\\field{C}^n,\\lambda)],[(\\field{C}^N,\\overline{\\lambda})]\\}$,\n \\item $\\spec{M_{n}(\\field{H})} = \\{[(\\field{C}^{2n},\\lambda)]\\}$,\n\\end{enumerate}\nwhere $\\lambda(a)$ denotes left multiplication by $a$ and $\\overline{\\lambda}(a)$ denotes left multiplication by $\\overline{a}$.\n\n\\begin{definition}\n Let $\\alg{A}$ be a real {\\ensuremath{C^*}}-algebra, and let $\\alpha \\in \\spec{\\alg{A}}$. We shall call $\\alpha$ \\term{conjugate-linear} if it arises from the conjugate-linear irreducible representation $(a \\mapsto \\overline{a},\\field{C}^{n_i})$ of a direct summand of $\\alg{A}$ of the form $M_{n_i}(\\field{C})$; otherwise we shall call it \\term{complex-linear}.\n\\end{definition}\n\nThus, a representation $\\alpha$ of the real {\\ensuremath{C^*}}-algebra $\\alg{A}$ extends to a $\\field{C}$-linear \\ensuremath{\\ast}-representation of $\\alg{A}_{\\field{C}}$ if and only if $\\alpha$ is the sum of complex-linear irreducible representations of $\\alg{A}$.\n\nFinally, for an individual direct summand $M_{k_i}(\\field{K}_i)$ of $\\alg{A}$, let $e_i$ denote its unit, $n_i$ the dimension of its irreducible representations (which is therefore equal to $2 k_i$ if $\\field{K}_i = \\field{H}$, and to $k_i$ itself otherwise), $\\rep{n}_i$ its complex-linear irreducible representation, and, if $\\field{K}_i = \\field{C}$, $\\crep{n}_i$ its conjugate-linear irreducible representation. We define a strict ordering $<$ on $\\spec{\\alg{A}}$ by setting $\\alpha < \\beta$ whenever $\\alpha \\in \\spec{M_{n_i}(\\field{K}_i)}$, $\\beta \\in \\spec{M_{n_j}(\\field{K}_j)}$ for $i < j$, and by setting $\\rep{n}_i < \\crep{n}_i$ in the case that $\\field{K}_i = \\field{C}$. Note that the ordering depends on the choice of Wedderburn decomposition, {\\it i.e.\\\/}\\  on the choice of ordering of the direct summands. Let $S$ denote the cardinality of $\\spec{\\alg{A}}$. We shall identify $M_S(\\field{R})$ with the real algebra of functions $\\spec{\\alg{A}}^2 \\to \\ring{R}$, and hence index the standard basis $\\{E_{\\alpha\\beta}\\}$ of $M_S(\\field{R})$ by $\\spec{\\alg{A}}^2$.\n\n\\subsection{Bimodules and spectral triples}\n\nLet us now turn to spectral triples. Recall that we are considering only finite-dimensional algebras and representations ({\\it i.e.\\\/}\\  Hilbert spaces), so that we are dealing only with what are termed \\term{finite} or \\term{discrete} spectral triples.\n\nLet $\\hs{H}$ and $\\hs{H}^\\prime$ be $\\alg{A}$-bimodules. We shall denote by $\\bdd^{\\textup{L}}_\\alg{A}(\\hs{H},\\hs{H}^\\prime)$, $\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H},\\hs{H}^\\prime)$, and $\\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H},\\hs{H}^\\prime)$ the subspaces of $\\mathcal{L}(\\hs{H},\\hs{H}^\\prime)$ consisting of left $\\alg{A}$-linear, right $\\alg{A}$-linear, and left and right $\\alg{A}$-linear operators, respectively. In the case that $\\hs{H}^\\prime = \\hs{H}$, we shall write simply $\\bdd^{\\textup{L}}_\\alg{A}(\\hs{H})$, $\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H})$ and $\\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H})$. If $N$ is a subalgebra or linear subspace of a real or complex {\\ensuremath{C^*}}-algebra, we shall denote by $N_\\textup{sa}$ the real linear subspace of $N$ consisting of the self-adjoint elements of $N$, and we shall denote by $\\U(N)$ set of unitary elements of $N$. Finally, for operators $A$ and $B$ on a Hilbert space, we shall denote their anticommutator $AB + BA$ by $\\{ A,B \\}$.\n\n\\subsubsection{Conventional definitions}\n\nWe begin by recalling the standard definitions for spectral triples of various forms. Since we are working with the finite case, all analytical requirements become redundant, leaving behind only the algebraic aspects of the definitions.\n\nThe following definition first appeared in a 1995 paper~\\cite{Connes95a} by Connes:\n\n\\begin{definition}\n A \\term{spectral triple} is a triple $(\\alg{A},\\hs{H},D)$, where:\n\\begin{itemize}\n \\item $\\alg{A}$ is a unital real or complex \\ensuremath{\\ast}-algebra;\n \\item $\\hs{H}$ is a complex Hilbert space on which $\\alg{A}$ has a left representation  $\\lambda : \\alg{A} \\to \\mathcal{L}(\\hs{H})$;\n \\item $D$, the \\term{Dirac operator}, is a self-adjoint operator on $\\hs{H}$.\n\\end{itemize}\n\nMoreover, if there exists  a $\\ring{Z}\/2\\ring{Z}$-grading $\\gamma$ on $\\hs{H}$ ({\\it i.e.\\\/}\\  a self-adjoint unitary on $\\hs{H}$) such that:\n\\begin{enumerate}\n \\item $[\\gamma, \\lambda(a)]=0$ for all $a \\in \\alg{A}$,\n \\item $\\{\\gamma,D\\}=0$;\n\\end{enumerate}\nthen the spectral triple is said to be \\term{even}. Otherwise, it is said to be \\term{odd}.\n\\end{definition}\n\nIn the context of the general definition for spectral triples, a finite spectral triple necessarily has metric dimension $0$.\n\nIn a slightly later paper~\\cite{Connes95}, Connes defines the additional structure on spectral triples necessary for defining the noncommutative spacetime of the NCG Standard Model; indeed, the same paper also contains the first version of the NCG Standard Model to use the language of spectral triples, in the form of a reformulation of the so-called Connes-Lott model.\n\n\\begin{definition}\\label{realdef}\n A spectral triple $(\\alg{A},\\hs{H},D)$ is called a \\term{real spectral triple of $KO$-dimension $n \\bmod 8$} if, in the case of $n$ even, it is an even spectral triple, and if there exists an antiunitary $J: \\hs{H} \\to \\hs{H}$ such that:\n\\begin{enumerate}\n \\item $J$ satisfies $J^2 = \\varepsilon$, $JD = \\varepsilon^\\prime DJ$ and $J\\gamma = \\varepsilon^{\\prime\\prime} \\gamma J$ (in the case of even $n$), where $\\varepsilon$, $\\varepsilon^\\prime$, $\\varepsilon^{\\prime\\prime} \\in \\{-1,1\\}$ depend on $n \\bmod 8$ as follows:\n \\begin{center}\n  \\begin{tabular}{crrrrrrrr}\n   \\toprule\n   $n$ & $0$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ \\\\\n   \\midrule\n   $\\varepsilon$ & $1$ & $1$ & $-1$ & $-1$ & $-1$ & $-1$ & $1$ & $1$ \\\\\n   $\\varepsilon^\\prime$ & $1$ & $-1$ & $1$ & $1$ & $1$ & $-1$ & $1$ & $1$ \\\\\n   $\\varepsilon^{\\prime\\prime}$ & $1$ & & $-1$ & & $1$ & & $-1$ & \\\\\n   \\bottomrule\n  \\end{tabular}\n \\end{center}\n \\item The \\term{order zero condition} is satisfied, namely $[\\lambda(a),J\\lambda(b)J^*]=0$ for all $a$, $b \\in \\alg{A}$;\n \\item The \\term{order one condition} is satisfied, namely $[[D,\\lambda(a)],J\\lambda(b)J^*]=0$ for all $a$, $b \\in \\alg{A}$.\n\\end{enumerate}\n\nMoreover, if there exists a self-adjoint unitary $\\epsilon$ on $\\hs{H}$ such that:\n\\begin{enumerate}\n \\item $[\\epsilon,\\lambda(a)]=0$ for all $a \\in \\alg{A}$;\n \\item $[\\epsilon, D]=0$;\n \\item $\\{\\epsilon, J\\}=0$;\n \\item $[\\epsilon, \\gamma]=0$ (even case);\n\\end{enumerate}\nthen the real spectral triple is said to be \\term{$S^0$-real}.\n\\end{definition}\n\n\\begin{remark}[Krajewski~\\cite{Kraj98}*{\\S 2.2}, Paschke--Sitarz~\\cite{PS98}*{Obs.~1}]\n If $(\\alg{A},\\hs{H},D)$ is a real spectral triple, then the order zero condition is equivalent to the statement that $\\hs{H}$ is an $\\alg{A}$-bimodule for the usual left action $\\lambda$ and the right action $\\rho: a \\mapsto J\\lambda(a^*)J^*$.\n\\end{remark}\n\nIt was commonly assumed until fairly recently that the finite geometry of the NCG Standard Model should be $S^0$-real. Though the current version of the NCG Standard Model no longer makes such an assumption~\\cites{Connes06,CCM07}, we shall later see that its finite geometry can still be seen as satisfying a weaker version of $S^0$-reality.\n\n\\subsubsection{Structures on bimodules}\n\nIn light of the above remark, the order one condition, the strongest algebraic condition placed on Dirac operators for real spectral triples, should be viewed more generally as a condition applicable to operators on bimodules~\\cite{Kraj98}*{\\S 2.4}. This then motivates our point of view that a finite real spectral triple $(\\alg{A},\\hs{H},D)$ should be viewed rather as an $\\alg{A}$-bimodule with additional structure, together with a Dirac operator satisfying the order one condition that is compatible with that additional structure. We therefore begin by defining a suitable notion of ``additional structure'' for bimodules.\n\n\\begin{definition}\n A \\term{bimodule structure} $P$ consists of the following data:\n\\begin{itemize}\n \\item A set $\\alg{P} = \\alg{P}_\\gamma \\sqcup \\alg{P}_J \\sqcup \\alg{P}_\\epsilon$, where each set $\\alg{P}_X$ is either empty or the singleton $\\{X\\}$, and where $\\alg{P}_\\epsilon$ is non-empty only if $\\alg{P}_J$ is non-empty;\n \\item If $\\alg{P}_J$ is non-empty, a choice of \\term{$KO$-dimension} $n \\bmod 8$, where $n$ is even if and only if $\\alg{P}_\\gamma$ is non-empty.\n\\end{itemize}\n\nIn particular, we call a structure $P$:\n\\begin{itemize}\n \\item \\term{odd} if $\\alg{P}$ is empty;\n \\item \\term{even} if $\\alg{P} = \\alg{P}_\\gamma = \\{\\gamma\\}$;\n \\item \\term{real} if $\\alg{P}_J$ is non-empty and $\\alg{P}_\\epsilon$ is empty\n \\item \\term{$S^0$-real} if $\\alg{P}_\\epsilon$ is non-empty.\n\\end{itemize}\n\nFinally, if $P$ is a graded structure, we call $\\gamma$ the \\term{grading}, and if $P$ is real or $S^0$-real, we call $J$ the \\term{charge conjugation}.\n\\end{definition}\n\nSince this notion of $KO$-dimension is meant to correspond with the usual $KO$-dimension of a real spectral triple, we assign to each real or $S^0$-real structure $P$ of $KO$-dimension $n \\bmod 8$ constants $\\varepsilon$, $\\varepsilon^\\prime$ and, in the case of even $n$, $\\varepsilon^{\\prime\\prime}$, according to the table in Definition~\\ref{realdef}.\n\nWe now define the \\term{structure algebra} of a structure $P$ to be the real associative algebra with generators $\\alg{P}$ and relations, as applicable,\n\\[\n \\gamma^2 = 1, \\quad J^2 = \\varepsilon, \\quad \\epsilon^2 = 1; \\gamma J = \\varepsilon^{\\prime\\prime} J \\gamma, \\quad [\\gamma,\\epsilon] = 0, \\quad \\{\\epsilon,J\\} = 0.\n\\]\n\n\\begin{definition}\n An $\\alg{A}$-bimodule $\\hs{H}$ is said to have structure $P$ whenever it admits a faithful representation of the structure algebra of $P$ such that, when applicable, $\\gamma$ and $\\epsilon$ are represented by self-adjoint unitaries in $\\bdd^{\\textup{LR}}_{\\alg{A}}(\\hs{H})$, and $J$ is represented by an antiunitary on $\\hs{H}$ such that\n\\begin{equation}\\label{realintertwine}\n \\forall a \\in \\alg{A}, \\quad \\rho(a) = J \\lambda(a^*) J.\n\\end{equation}\n\\end{definition}\n\nNote that a $S^0$-real bimodule can always be considered as a real bimodule, and a real bimodule of even [odd] $KO$-dimension can always be considered as an even [odd] bimodule. Note also that an even bimodule is simply a graded bimodule such that the algebra acts from both left and right by degree $0$ operators, and the grading itself respects the Hilbert space structure; an odd bimodule is then simply an ungraded bimodule. We use the terms ``even'' and ``odd'' so as to keep the terminology consistent with that for spectral triples.\n\nNote also that for a real or $S^0$-real structure $P$, the structure algebra of $P$ is independent of the value of $\\varepsilon^\\prime$. Thus the notions of real [$S^0$-real] $\\alg{A}$-bimodule with $KO$-dimension $1 \\bmod 8$ and $7 \\bmod 8$ are identical, as are the notions of [$S^0$-real] $\\alg{A}$-bimodule with $KO$-dimension $3 \\bmod 8$ and $5 \\bmod 8$; again, we make the distinction with an eye to the discussion of Dirac operators (and hence of spectral triples) later on.\n\nNow, a \\term{unitary equivalence} of $\\alg{A}$-bimodules $\\hs{H}$ and $\\hs{H}^\\prime$ with structure $P$ is a unitary equivalence of $\\alg{A}$-bimodules ({\\it i.e.\\\/}\\  a unitary element of $\\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H},\\hs{H}^\\prime)$) that effects unitary equivalence of the representations of the structure algebra of $P$. We denote the set of all such unitary equivalences $\\hs{H} \\to \\hs{H}^\\prime$ by $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},\\hs{H}^\\prime;\\alg{P})$. In particular, $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},\\hs{H};\\alg{P})$, which we denote by $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H};\\alg{P})$, is a subgroup of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}) := \\U(\\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}))$. In all such notation, we suppress the argument $\\alg{P}$ whenever $\\alg{P}$ is empty.\n\n\\begin{definition}\n Let $\\alg{A}$ be a real {\\ensuremath{C^*}}-algebra, and let $P$ be a bimodule structure. The abelian monoid $(\\Bimod(\\alg{A},P),+)$ of $\\alg{A}$-bimodules with structure $P$ is defined as follows:\n\\begin{itemize}\n \\item $\\Bimod(\\alg{A},P)$ is the set of unitary equivalence classes of $\\alg{A}$-bimodules with structure $P$;\n \\item For $[\\hs{H}]$, $[\\hs{H}^\\prime] \\in \\Bimod(\\alg{A},P)$, $[\\hs{H}] + [\\hs{H}^\\prime] := [\\hs{H} \\oplus \\hs{H}^\\prime]$.\n\\end{itemize}\n\\end{definition}\n\nFor convenience, we shall denote $\\Bimod(\\alg{A},P)$ by:\n\\begin{itemize}\n \\item $\\Bimod(\\alg{A})$ if $P$ is the odd structure;\n \\item $\\Bimod^\\textup{even}(\\alg{A})$ if $P$ is the even structure;\n \\item $\\Bimod(\\alg{A},n)$ if $P$ is the real structure of $KO$-dimension $n \\bmod 8$;\n \\item $\\Bimod^0(\\alg{A},n)$ if $P$ is the $S^0$-real structure of $KO$-dimension $n \\bmod 8$.\n\\end{itemize}\nThese monoids will be studied in depth in the next section. In light of our earlier comment, we therefore have that\n\\[\n \\Bimod(\\alg{A},1) = \\Bimod(\\alg{A},7), \\quad \\Bimod(\\alg{A},3) = \\Bimod(\\alg{A},5).\n\\]\nand\n\\[\n \\Bimod^0(\\alg{A},1) = \\Bimod^0(\\alg{A},7), \\quad \\Bimod^0(\\alg{A},3) = \\Bimod^0(\\alg{A},5).\n\\]\n\nFinally, for the sake of completeness, we now define the notions of orientabilty and Poincar{\\'e} duality in this more general context; in the case of a real spectral triple $(\\alg{A},\\hs{H},D,\\gamma, J)$ of even $KO$-dimension, where the right action is given by $\\rho(a) := J\\lambda(a^*)J^*$, these definitions yield precisely the usual ones ({\\it cf.\\\/}\\ ~\\cite{Kraj98}*{\\S\\S 2.2, 2.3}).\n\n\\begin{definition}\n We call an even $\\alg{A}$-bimodule $(\\hs{H},\\gamma)$ \\term{orientable} if there exist $a_1, \\dotsc, a_k$, $b_1, \\dotsc, b_k \\in \\alg{A}$ such that\n\\begin{equation}\n \\gamma = \\sum_{i=1}^k \\lambda(a_i)\\rho(b_i).\n\\end{equation}\n\\end{definition}\n\n\\begin{definition}\n Let $\\alg{A}$ be a real ${\\ensuremath{C^*}}$-algebra, and let $(\\hs{H},\\gamma)$ be an even $\\alg{A}$-bimodule. Then the \\term{intersection form} $\\left\\langle \\cdot, \\cdot \\right\\rangle : KO_0(\\alg{A}) \\times KO_0(\\alg{A}) \\to \\ring{Z}$ associated with $(\\hs{H},\\gamma)$ is defined by setting\n\\begin{equation}\n \\left\\langle \\left[e\\right], \\left[f\\right] \\right\\rangle := \\tr(\\gamma\\lambda(e)\\rho(f))\n\\end{equation}\nfor projections $e$, $f \\in \\alg{A}$.\n\nIn the case that the intersection form is non-degenerate, we shall say that $(\\hs{H},\\gamma)$ satisfies \\term{Poincar{\\'e} duality}.\n\\end{definition}\n\nThe orientability assumption was used extensively in \\cite{PS98} and \\cite{Kraj98}, as it leads to considerable algebraic simplifactions; we shall later define a weakened version of orientability that will yield precisely those simplifications.\n\n\\subsubsection{Bilateral spectral triples}\n\nWe now turn to Dirac operators on bimodules satisfying a generalised order one condition, and define the appropriate notion of compatibility with additional structure on the bimodule.\n\n\\begin{definition}\n A \\term{Dirac operator} for an $\\alg{A}$-bimodule $\\hs{H}$ with structure $P$ is a self-adjoint operator $D$ on $\\hs{H}$ satisfying the \\term{order one condition}:\n\\begin{equation}\n \\forall a, b \\in \\alg{A}, \\quad [[D,\\lambda(a)],\\rho(b)] = 0,\n\\end{equation}\ntogether with the following relations, as applicable:\n\\[\n \\{D,\\gamma\\} = 0, \\quad DJ = \\varepsilon^\\prime JD, \\quad [D,\\epsilon]=0.\n\\]\n\\end{definition}\n\nWe denote the finite-dimensional real vector space of Dirac operators for an an $\\alg{A}$-bimodule $\\hs{H}$ with structure $P$ by $\\ms{D}_0(\\alg{A},\\hs{H},\\alg{P})$.\n\n\\begin{definition}\n A \\term{bilateral spectral triple} with structure $P$ is a triple of the form $(\\alg{A},\\hs{H},D)$, where $\\alg{A}$ is a real {\\ensuremath{C^*}}-algebra, $\\hs{H}$ is an $\\alg{A}$-bimodule with structure $P$, and $D$ is a Dirac operator for $(\\hs{H},P)$.\n\\end{definition}\n\nWe shall generally denote such a spectral triple by $(\\alg{A},\\hs{H},D;\\alg{P})$, where $\\alg{P}$ is the set of generators of the structure algebra; in cases where the presence or absence of a grading $\\gamma$ is immaterial, we will suppress the generator $\\gamma$ in this notation.\n\n\\begin{remark}\n In the case that $P$ is a real [$S^0$-real] structure of $KO$-dimension $n \\bmod 8$, a bilateral spectral triple with structure $P$ is precisely a real [$S^0$-real] spectral triple of $KO$-dimension $n \\bmod 8$.\n\nMore generally, an odd [even] bilateral spectral triple $(\\alg{A},\\hs{H},D)$ is equivalent to an odd [even] spectral triple $(\\alg{A} \\otimes \\alg{A}^\\textnormal{op},\\hs{H},D)$ such that $[[D,\\alg{A}\\otimes 1],1\\otimes\\alg{A}^\\textnormal{op}]=\\{0\\}$, an object that first appears in connection with $S^0$-real spectral triples~\\cite{Connes95}\n\\end{remark}\n\nA \\term{unitary equivalence} of spectral triples $(\\alg{A},\\hs{H},D)$ and $(\\alg{A},\\hs{H}^\\prime, D^\\prime)$ is then a unitary $U \\in \\U^{\\textup{LR}}_\\alg{A}(\\hs{H},\\hs{H}^\\prime)$ such that $D^\\prime = U D U^*$. This concept leads us to the following definition:\n\n\\begin{definition}\n Let $\\alg{A}$ be a real {\\ensuremath{C^*}}-algebra, and let $\\hs{H}$ be an $\\alg{A}$-bimodule with structure $P$. The \\term{moduli space of Dirac operators} for $\\hs{H}$ is defined by\n\\begin{equation}\n \\ms{D}(\\alg{A},\\hs{H},\\alg{P}) := \\ms{D}_0(\\alg{A},\\hs{H},\\alg{P})\/\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},\\alg{P}),\n\\end{equation}\nwhere $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},\\alg{P})$ acts on $\\ms{D}_0(\\alg{A},\\hs{H},\\alg{P})$ by conjugation.\n\\end{definition}\n\nIf $\\alg{C}$ is a central subalgebra of $\\alg{A}$, we can form the subspace\n\\begin{equation}\n \\ms{D}_0(\\alg{A},\\hs{H},\\alg{P};\\alg{C}) := \\{D \\in \\ms{D}_0(\\alg{A},\\hs{H},\\alg{P}) \\mid [D,\\lambda(\\alg{C})] = [D,\\rho(\\alg{C})] = \\{0\\}\\}.\n\\end{equation}\nand hence the sub-moduli space\n\\begin{equation}\n \\ms{D}(\\alg{A},\\hs{H},\\alg{P};\\alg{C}) := \\ms{D}_0(\\alg{A},\\hs{H},\\alg{P};\\alg{C})\/\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},\\alg{P}),\n\\end{equation}\nof $\\ms{D}_0(\\alg{A},\\hs{H},\\alg{P})$; the moduli space of Dirac operators studied by Chamseddine, Connes and Marcolli~\\cite{CCM07}*{\\S 2.7},\\cite{CM08}*{\\S 13.4} is in fact a sub-moduli space of this form.\n\nSince $\\ms{D}(\\alg{A},\\hs{H},\\alg{P})$ [$\\ms{D}(\\alg{A},\\hs{H},\\alg{P};\\alg{C})$] is the orbit space of a smooth finite-di\\-men\\-sion\\-al representation of a compact Lie group, it is \\latin{a priori} locally compact Hausdorff, and is thus homeomorphic to a semialgebraic subset of $\\field{R}^d$ for some $d$~\\cite{Schwarz75}. The dimension of $\\ms{D}(\\alg{A},\\hs{H},\\alg{P})$ [$\\ms{D}(\\alg{A},\\hs{H},\\alg{P};\\alg{C})$] can then be defined as the dimension of this semialgebraic set. Such moduli spaces will be discussed in some detail.\n\n\\subsubsection{$S^0$-reality}\n\nFollowing Connes~\\cite{Connes95}, we now describe how to reduce the study of $S^0$-real bimodules of even [odd] $KO$-dimension to the study of even [odd] bimodules.\n\nLet $(\\hs{H},J,\\epsilon)$ be an $S^0$-real $\\alg{A}$-bimodule of even [odd] $KO$-dimension. Define mutually orthogonal projections $P_i$, $P_{-i}$ in $\\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H})$ by $P_{\\pm i} = \\frac{1}{2}(1 \\pm \\epsilon)$. Then, at the level of even [odd] bimodules, $\\hs{H} = \\hs{H}_i \\oplus \\hs{H}_{-i}$ for $\\hs{H}_{\\pm i} := P_{\\pm i} \\hs{H}$, where the left and right actions on $\\hs{H}_{\\pm i}$ are given by\n\\[\n \\lambda_{\\pm i}(a) := P_{\\pm i}\\lambda(a)P_{\\pm i}, \\quad \\rho_{\\pm i}(a) := P_{\\pm i}\\rho(a)P_{\\pm i},\n\\]\nfor $a \\in \\alg{A}$, and, in the case of even $KO$-dimension, the grading on $\\hs{H}_{\\pm i}$ is given by $\\gamma_{\\pm i} := P_{\\pm i} \\gamma P_{\\pm i}$. Moreoever,\n\\[\n J =\n\\begin{pmatrix}\n 0 & \\varepsilon \\tilde{J}^*\\\\\n \\tilde{J} & 0\n\\end{pmatrix},\n\\]\nwhere $\\tilde{J} := P_{-i} J P_i$ is an antiunitary $\\hs{H}_i \\to \\hs{H}_{-i}$, so that for $a \\in \\alg{A}$,\n\\[\n \\lambda_{-i}(a) = \\tilde{J} \\rho_i(a^*) \\tilde{J}^*, \\quad \\rho_{-i}(a) = \\tilde{J} \\lambda_i(a^*) \\tilde{J}^*,\n\\]\nand in the case of even $KO$-dimension, $\\gamma_{-i} = \\varepsilon^{\\prime\\prime} \\tilde{J} \\gamma \\tilde{J}^*$. Finally, note that $\\tilde{J}$ can also be viewed as a unitary $\\overline{\\hs{H}_i} \\to \\hs{H}_{-i}$, where $\\overline{\\hs{H}_i}$ denotes the conjugate space of $\\hs{H}$. Hence, for fixed $KO$-dimension, an $S^0$-real $\\alg{A}$-bimodule $\\hs{H}$ is determined, up to unitary equivalence, by the bimodule $\\hs{H}_i$.\n\nOn the other hand, if $\\hs{V}$ is an even [odd] $\\alg{A}$-bimodule, we can construct an $S^0$-real $\\alg{A}$-bimodule $\\hs{H}$ for any even [odd] $KO$-dimension $n \\bmod 8$ such that $\\hs{H}_i = \\hs{V}$, by setting $\\hs{H} := \\hs{H}_i \\oplus \\hs{H}_{-i}$ for $\\hs{H}_i := \\hs{V}$, $\\hs{H}_{-i} := \\overline{\\hs{V}}$, defining $\\tilde{J} : \\hs{H}_i \\to \\hs{H}_{-i}$ as the identity map on $\\hs{V}$ viewed as an antiunitary $\\hs{V} \\to \\overline{\\hs{V}}$, then using the above formulas to define $J$, $\\gamma$ (as necessary), $\\lambda$, $\\rho$, and finally setting $\\epsilon = 1_\\hs{V} \\oplus (-1_{\\overline{\\hs{V}}})$. In the case that $\\hs{V}$ is already $\\hs{H}_i$ for some $S^0$-real bimodule $\\hs{H}$, this procedure reproduces $\\hs{H}$ up to unitary equivalence. We have therefore proved the following:\n\n\\begin{proposition}\\label{s0reduction}\n Let $\\alg{A}$ be a real {\\ensuremath{C^*}}-algebra, and let $n \\in \\ring{Z}_8$. Then the map\n\\[\n \\Bimod^0(\\alg{A},n) \\to\n \\begin{cases}\n  \\Bimod(\\alg{A}), &\\text{if $n$ is odd,}\\\\\n  \\Bimod^\\textup{even}(\\alg{A}), &\\text{if $n$ is even,}\n \\end{cases}\n\\]\ndefined by $[\\hs{H}] \\mapsto [\\hs{H}_i]$ is an isomorphism of monoids.\n\\end{proposition}\n\nNow, let $\\hs{H}$ is an $S^0$-real $\\alg{A}$-bimodule, and suppose that $D$ is a Dirac operator for $\\hs{H}$. We can define Dirac operators $D_i$ and $D_{-i}$ on $\\hs{H}_i$ and $\\hs{H}_{-i}$, respectively, by $D_{\\pm i} := P_{\\pm i} D P_{\\pm i}$; then $D = D_i \\oplus D_{-i}$ and, in fact, $D_{-i} = \\varepsilon^\\prime \\tilde{J} D_i \\tilde{J}^*$. Thus, a Dirac operator $D$ on $\\hs{H}$ is completely determined by $D_i$; indeed, the map $D \\mapsto D_i$ defines an isomorphism $\\ms{D}_0(\\alg{A},\\hs{H},J,\\epsilon) \\cong \\ms{D}_0(\\alg{A},\\hs{H})$. \n\nAlong similar lines, one can show that $\\U^{\\textup{LR}}_{\\alg{A}}(\\hs{H},J) \\cong \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}_i)$ by means of the map $U \\mapsto U_i := P_i U P_i$; this isomorphism is compatible with the isomorphism $\\ms{D}_0(\\alg{A},\\hs{H},J,\\epsilon) \\cong \\ms{D}_0(\\alg{A},\\hs{H})$. Hence, the functional equivalence between $\\hs{H}$ and $\\hs{H}_i$ holds at the level of moduli spaces of Dirac operators:\n\n\\begin{proposition}\\label{diracs0reduction}\n Let $\\hs{H}$ be an $S^0$-real $\\alg{A}$-bimodule. Then\n\\begin{equation}\n \\ms{D}(\\alg{A},\\hs{H},J,\\epsilon) \\cong \\ms{D}(\\alg{A},\\hs{H}_i).\n\\end{equation}\n\\end{proposition}\n\nOne can similarly show that for a central subalgebra $\\alg{C}$ of $\\alg{A}$,\n\\[\n \\ms{D}(\\alg{A},\\hs{H},J,\\epsilon;\\alg{C}) \\cong \\ms{D}(\\alg{A},\\hs{H}_i;\\alg{C}).\n\\]\n\nLet us conclude by considering the relation between orientability and Poincar{\\'e} duality for an $S^0$-real bimodule $\\hs{H}$ of even $KO$-dimension and orientability and Poincar{\\'e} duality, respectively, for the associated even bimodule $\\hs{H}_i$.\n\n\\begin{proposition}\\label{s0orient}\n Let $\\hs{H}$ be an $S^0$-real $\\alg{A}$-bimodule of even $KO$-dimension. Then $\\hs{H}$ is orientable if and only if there exist $a_1, \\dotsc, a_k, b_1, \\dotsc, b_k \\in \\alg{A}$ such that\n\\begin{equation}\n \\gamma_i = \\sum_{j=1}^k \\lambda_i(a_j)\\rho_i(b_j) = \\varepsilon^{\\prime\\prime} \\sum_{j=1}^k \\lambda_i(b_j^*)\\rho_i(a_j^*).\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\n Let $a_1, \\dotsc, a_k$, $b_1, \\dotsc, b_k \\in \\alg{A}$, and set $T = \\sum_{j=1}^k \\lambda(a_j)\\rho(b_j)$. Then\n\\[\n T_i := P_i T P_i = \\sum_{j=1}^k \\lambda_i(a_j)\\rho_i(b_j),\n\\]\nwhile\n\\[\n \\quad T_{-i} := P_{-i} T P_{-i} = \\sum_{j=1}^k \\lambda_{-i}(a_j)\\rho_{-i}(b_j) = \\tilde{J} \\biggl( \\sum_{j=1}^k \\lambda_i(b_j^*)\\rho_i(a_j^*) \\biggr) \\tilde{J}^*.\n\\]\nHence, $T_{-i} = \\varepsilon^{\\prime\\prime} \\tilde{J} T_i \\tilde{J}^*$ if and only if\n\\[\n \\varepsilon^{\\prime\\prime} \\sum_{j=1}^k \\lambda_i(b_j^*)\\rho_i(a_j^*) = T_i = \\sum_{j=1}^k \\lambda_i(a_j)\\rho_i(b_j).\n\\]\nApplying this intermediate result to $a_j$ and $b_j$ such that $\\gamma = \\sum_{j=1}^k \\lambda(a_j)\\rho(b_j)$, in the case that $\\hs{H}$ is orientable, and then to $a_j$ and $b_j$ such that $\\gamma_i = \\sum_{j=1}^k \\lambda_i(a_j)\\rho_i(b_j)$, in the case that $\\hs{H}_i$ is orientable, yields the desired result.\n\\end{proof}\n\nThus, orientability of an $S^0$-real bimodule $\\hs{H}$ is equivalent to a stronger version of orientability on the bimodule $\\hs{H}_i$.\n\nTurning to Poincar{\\'e} duality, we can obtain the following result:\n\n\\begin{proposition}\\label{s0poincare}\n Let $\\hs{H}$ be an $S^0$-real $\\alg{A}$-bimodule of even $KO$-dimension with intersection form $\\form{\\cdot,\\cdot}$, and let $\\form{\\cdot,\\cdot}_i$ be the intersection form for $\\hs{H}_i$. Then for any $p$, $q \\in KO_0(\\alg{A})$,\n\\[\n \\form{p,q} = \\form{p,q}_i + \\varepsilon^{\\prime\\prime} \\form{q,p}_i.\n\\]\n\\end{proposition}\n\n\\begin{proof}\n Let $e$, $f \\in \\alg{A}$ be projections. Then\n\\begin{align*}\n \\form{[e],[f]} &= \\tr(\\gamma\\lambda(e)\\rho(f))\\\\\n &= \\tr(\\gamma_i \\lambda_i(e) \\rho_i(f)) + \\tr(\\gamma_{-i}\\lambda_{-i}(e)\\rho_{-i}(f))\\\\\n &= \\tr(\\gamma_i \\lambda_i(e) \\rho_i(f)) + \\varepsilon^{\\prime\\prime} \\tr(\\tilde{J}\\gamma_i \\lambda_i(f) \\rho_i(e) \\tilde{J}^*)\\\\\n &= \\tr(\\gamma_i \\lambda_i(e) \\rho_i(f)) + \\varepsilon^{\\prime\\prime} \\overline{\\tr(\\gamma_i \\lambda_i(f) \\rho_i(e))}\\\\\n &= \\form{[e],[f]}_i + \\varepsilon^{\\prime\\prime} \\form{[f],[e]}_i,\n\\end{align*}\nwhere we have used the fact that the intersection forms are integer-valued.\n\\end{proof}\n\nThus, Poincar{\\'e} duality on an $S^0$-real bimodule $\\hs{H}$ is equivalent to nondegeneracy of either the symmetrisation or antisymmetrisation of the intersection form on $\\hs{H}_i$, as the case may be.\n\n\\section{Bimodules and Multiplicity Matrices}\n\nWe now turn to the study of bimodules, and in particular, to their characterisation by multiplicity matrices. We shall find that a bimodule admits, up to unitary equivalence, at most one real structure of any given $KO$-dimension, and that the multiplicity matrix or matrices of a bimodule will determine entirely which real structures, if any, it does admit.\n\nIn what follows, $\\alg{A}$ will be a fixed real {\\ensuremath{C^*}}-algebra.\n\n\\subsection{Odd bimodules}\n\nLet us begin with the study of odd bimodules.\n\nFor $m \\in M_S(\\semiring{Z}_{\\geq 0})$, we define an $\\alg{A}$-bimodule $\\hs{H}_m$ by setting\n\\begin{align*}\n \\hs{H}_m &:= \\bigoplus_{\\alpha,\\beta \\in \\spec{\\alg{A}}} \\field{C}^{n_\\alpha} \\otimes \\field{C}^{m_{\\alpha\\beta}} \\otimes \\field{C}^{n_\\beta},\\\\\n \\lambda_m(a) &:= \\bigoplus_{\\alpha,\\beta \\in \\spec{\\alg{A}}} \\lambda_\\alpha(a) \\otimes 1_{m_{\\alpha\\beta}} \\otimes 1_{n_\\beta}, \\quad a \\in \\alg{A},\\\\\n \\rho_m(a) &:= \\bigoplus_{\\alpha, \\beta \\in \\spec{\\alg{A}}} 1_{n_\\alpha} \\otimes 1_{m_{\\alpha\\beta}} \\otimes \\lambda_\\beta(a)^T, \\quad a \\in \\alg{A}.\n\\end{align*}\nHere we use the convention that $1_n$ is the identity on $\\field{C}^n$, with $\\field{C}^0 := \\{0\\}$ and hence $1_0 := 0$. \n\n\\begin{proposition}[Krajewski~\\cite{Kraj98}*{\\S 3.1}, Paschke--Sitarz~\\cite{PS98}*{Lemmas 1, 2}] \\label{oddmult}\nThe map $\\bimod : M_S(\\semiring{Z}_{\\geq 0}) \\to \\Bimod(\\alg{A})$ given by $m \\mapsto [\\hs{H}_m]$ is an isomorphism of monoids.\n\\end{proposition}\n\n\\begin{proof}\nBy construction, $\\bimod$ is an injective morphism of monoids. It therefore suffices to show that $\\bimod^\\textup{odd}$ is also surjective.\n\nNow, let $\\hs{H}$ be an $\\alg{A}$-bimodule. For $\\alpha \\in \\spec{\\alg{A}}$ define projections $P^L_\\alpha$ and $P^R_\\alpha$ by\n\\[\n P^L_\\alpha :=\n \\begin{cases}\n  \\lambda(e_i) &\\text{if $\\alpha = \\rep{n}_i$ for $\\field{K}_i \\neq \\field{C}$,}\\\\\n  \\frac{1}{2}\\left(\\lambda(e_i) - i \\lambda(i e_i) \\right) &\\text{if $\\alpha = \\rep{n}_i$ for $\\field{K}_i = \\field{C}$,}\\\\\n  \\frac{1}{2}\\left(\\lambda(e_i) + i \\lambda(i e_i) \\right) &\\text{if $\\alpha = \\crep{n}_i$ for $\\field{K}_i = \\field{C}$,}\n \\end{cases}\n\\]\nand\n\\[\n P^R_\\alpha :=\n \\begin{cases}\n  \\rho(e_i) &\\text{if $\\alpha = \\rep{n}_i$ for $\\field{K}_i \\neq \\field{C}$,}\\\\\n  \\frac{1}{2}\\left(\\rho(e_i) - i \\rho(i e_i) \\right) &\\text{if $\\alpha = \\rep{n}_i$ for $\\field{K}_i = \\field{C}$,}\\\\\n  \\frac{1}{2}\\left(\\rho(e_i) + i \\rho(i e_i) \\right) &\\text{if $\\alpha = \\crep{n}_i$ for $\\field{K}_i = \\field{C}$,}\n \\end{cases}\n\\]\nrespectively; by construction, $P^L_\\alpha \\in \\lambda(\\alg{A}) + i \\lambda(\\alg{A})$ and $P^R_\\alpha \\in \\rho(\\alg{A}) + i \\rho(\\alg{A})$, so that for $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$, $P^L_\\alpha$ and $P^R_\\beta$ commute. We can therefore define projections $P_{\\alpha\\beta} := P^L_\\alpha P^R_\\beta$ for each $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$; it is then easy to see that each $\\hs{H}_{\\alpha\\beta} := P_{\\alpha\\beta}\\hs{H}$ is a sub-$\\alg{A}$-bimodule of $\\hs{H}$, and that $\\hs{H} = \\oplus_{\\alpha,\\beta \\in \\spec{\\alg{A}}} \\hs{H}_{\\alpha\\beta}$.\n\nLet $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$. As noted before, the left action of $\\alg{A}$ on $\\hs{H}_{\\alpha\\beta}$ must decompose as a direct sum of irreducible representations, but by construction of $\\hs{H}_{\\alpha\\beta}$, those irreducible representations must all be $\\alpha$. Similarly, the right action on $\\hs{H}_{\\alpha\\beta}$ must be a direct sum of copies of $\\beta$. Since the left action and right action commute, we must therefore have that $\\hs{H}_{\\alpha\\beta} \\cong \\hs{H}_{m_{\\alpha\\beta} E_{\\alpha\\beta}}$ for some $m_{\\alpha\\beta} \\in \\semiring{Z}_{\\geq 0}$. Taking the direct sum of the $\\hs{H}_{\\alpha\\beta}$, we therefore see that $\\hs{H}$ is unitarily equivalent to $\\hs{H}_m$ for $m = (m_{\\alpha\\beta}) \\in M_S(\\semiring{Z}_{\\geq 0})$, that is, $[\\hs{H}] = \\bimod(m)$.\n\\end{proof}\n\nWe denote the inverse map $\\bimod^{-1} : \\Bimod(\\alg{A}) \\to M_S(\\semiring{Z}_{\\geq 0})$ by $\\mult$.\n\n\\begin{definition}\n Let $\\hs{H}$ be an $\\alg{A}$-bimodule. Then the \\term{multiplicity matrix} of $\\alg{A}$ is the matrix $\\mult[\\hs{H}] \\in M_S(\\semiring{Z}_{\\geq 0})$.\n\\end{definition}\n\nFrom now on, without any loss of generality, we shall assume that an $\\alg{A}$-bimodule $\\hs{H}$ with multiplicity matrix $m$ is $\\hs{H}_m$ itself.\n\n\\begin{remark}\n Multiplicity matrices readily admit a $K$-theoretic interpretation~\\cite{Ell08}. For simplicity, suppose that $\\alg{A}$ is a complex {\\ensuremath{C^*}}-algebra and consider only complex-linear representations. Then for $\\hs{H}$  an $\\alg{A}$-bimodule, $\\mult[\\hs{H}]$ is essentially the \\term{Bratteli matrix} of the inclusion $\\lambda(\\alg{A}) \\hookrightarrow \\rho(\\alg{A})^\\prime \\subset \\mathcal{L}(\\hs{H})$ (cf.~\\cite{Ell07}*{\\S 2}), and can thus be interpreted as representing the induced map $K_0(\\lambda(\\alg{A})) \\to K_0(\\rho(\\alg{A})^\\prime)$ in complex $K$-theory. Likewise, $\\mult[\\hs{H}]^T$ can be interpreted as representing the map $K_0(\\rho(\\alg{A})) \\to K_0(\\lambda(\\alg{A})^\\prime)$ induced by the inclusion $\\rho(\\alg{A}) \\hookrightarrow \\lambda(\\alg{A})^\\prime \\subset \\mathcal{L}(\\hs{H})$. Similar interpretations can be made in the more general context of real {\\ensuremath{C^*}}-algebras and $KO$-theory.\n\\end{remark}\n\nWe shall now characterise left, right, and left and right $\\alg{A}$-linear maps between $\\alg{A}$-bimodules. Let $\\hs{H}$ and $\\hs{H}^\\prime$ be $\\alg{A}$-bimodules with multiplicity matrices $m$ and $m^\\prime$, respectively, let $P_{\\alpha\\beta}$ be the projections on $\\hs{H}$ defined as in the proof of Proposition~\\ref{oddmult}, and let $P_{\\alpha\\beta}^\\prime$ be the analogous projections on $\\hs{H}^\\prime$. Then any linear map $T : \\hs{H} \\to \\hs{H}^\\prime$ is characterised by the components\n\\begin{equation}\n T_{\\alpha\\beta}^{\\gamma\\delta} := P^\\prime_{\\gamma\\delta} T P_{\\alpha\\beta},\n\\end{equation}\nwhich we view as maps $T_{\\alpha\\beta}^{\\gamma\\delta} : \\field{C}^{n_\\alpha} \\otimes \\field{C}^{m_{\\alpha\\beta}} \\otimes \\field{C}^{n_\\beta} \\to \\field{C}^{n_\\gamma} \\otimes \\field{C}^{m_{\\gamma\\delta}^\\prime} \\otimes \\field{C}^{n_\\delta}$, or equivalently, as elements $\n T_{\\alpha\\beta}^{\\gamma\\delta} \\in M_{n_\\gamma \\times n_\\alpha}(\\field{C}) \\otimes M_{m_{\\gamma\\delta}^\\prime \\times m_{\\alpha\\beta}} \\otimes M_{n_\\delta \\times n_\\beta}(\\field{C})$. Thus we have an isomorphism\n\\[\n \\comp : \\mathcal{L}(\\hs{H},\\hs{H}^\\prime) \\to \\bigoplus_{\\alpha,\\beta,\\gamma,\\delta \\in \\spec{\\alg{A}}} M_{n_\\gamma \\times n_\\alpha}(\\field{C}) \\otimes M_{m_{\\gamma\\delta}^\\prime \\times m_{\\alpha\\beta}} \\otimes M_{n_\\delta \\times n_\\beta}(\\field{C})\n\\]\ngiven by $\\comp(T) := (T_{\\alpha\\beta}^{\\gamma\\delta})_{\\alpha,\\beta,\\gamma,\\delta \\in \\spec{\\alg{A}}}$. Note that when $\\hs{H} = \\hs{H}^\\prime$, $T$ is self-adjoint if and only if $T_{\\gamma\\delta}^{\\alpha\\beta} = (T_{\\alpha\\beta}^{\\gamma\\delta})^*$ for all $\\alpha$, $\\beta$, $\\gamma$, $\\delta \\in \\spec{\\alg{A}}$.\n\n\\begin{proposition}[Krajewski~\\cite{Kraj98}*{\\S3.4}]\\label{linear}\n Let $\\hs{H}$ and $\\hs{H}^\\prime$ be $\\alg{A}$-bimodules with multiplicity matrices $m$ and $m^\\prime$, respectively. Then\n\\begin{align}\n \\comp(\\bdd^{\\textup{L}}_{\\alg{A}}(\\hs{H},\\hs{H}^\\prime)) &= \\bigoplus_{\\alpha,\\beta,\\delta \\in \\spec{\\alg{A}}} 1_{n_\\alpha} \\otimes M_{m_{\\alpha\\delta}^\\prime \\times m_{\\alpha\\beta}}(\\field{C}) \\otimes M_{n_\\delta \\times n_\\beta}(\\field{C}),\\\\\n \\comp(\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H},\\hs{H}^\\prime)) &= \\bigoplus_{\\alpha,\\beta,\\gamma \\in \\spec{\\alg{A}}} M_{n_\\gamma \\times n_\\alpha}(\\field{C}) \\otimes M_{m_{\\gamma\\beta}^\\prime \\times m_{\\alpha\\beta}}(\\field{C}) \\otimes 1_{n_\\beta},\\\\\n \\comp(\\bdd^{\\textup{LR}}_{\\alg{A}}(\\hs{H},\\hs{H}^\\prime)) &= \\bigoplus_{\\alpha,\\beta \\in \\spec{\\alg{A}}} 1_{n_\\alpha} \\otimes M_{m_{\\alpha\\beta}^\\prime \\times m_{\\alpha\\beta}}(\\field{C}) \\otimes 1_{n_\\beta}.\n\\end{align} \n\\end{proposition}\n\n\\begin{proof}\nObserve that $T \\in \\mathcal{L}(\\hs{H},\\hs{H}^\\prime)$ is left, right, or left and right $\\alg{A}$-linear if and only if each $T_{\\alpha\\beta}^{\\gamma\\delta}$ is left, right, or left and right $\\alg{A}$. Thus, let $\\alpha$, $\\beta$, $\\gamma$ and $\\delta \\in \\spec{\\alg{A}}$ be fixed, and let $T \\in M_{n_\\gamma \\times n_\\alpha}(\\field{C}) \\otimes M_{m_{\\gamma\\delta}^\\prime \\times m_{\\alpha\\beta}} \\otimes M_{n_\\delta \\times n_\\beta}(\\field{C})$.\n\nFirst, write $T = \\sum_{i=1}^k A_i \\otimes B_i$ for $A_i \\in M_{n_\\gamma \\times n_\\alpha}(\\field{C})$ and for linearly independent $B_i \\in M_{m_{\\gamma\\delta}^\\prime \\times m_{\\alpha\\beta}} \\otimes M_{n_\\delta \\times n_\\beta}(\\field{C})$. Then, for $a \\in \\alg{A}$, \n\\[\n (\\lambda_\\gamma(a) \\otimes 1_{m_{\\gamma\\delta}^\\prime} \\otimes 1_{n_\\delta}) T - T (\\lambda_\\alpha(a) \\otimes 1_{m_{\\alpha\\beta}} \\otimes 1_{n_\\beta}) = \\sum_{i=1}^k (\\lambda_\\gamma(a) A_i - A_i \\lambda_\\alpha(a)) \\otimes B_i, \n\\]\nso that by linear independence of the $B_i$, $T$ is left $\\alg{A}$-linear if and only if each $A_i$ intertwines the irreducible representations $\\alpha$ and $\\gamma$, and hence, by Schur's lemma, if and only if $\\alpha = \\gamma$ and each $A_i$ is a constant multiple of $1_{n_\\alpha}$ or each $A_i = 0$. Thus,\n\\begin{multline*}\n \\bdd^{\\textup{L}}_{\\alg{A}}(\\field{C}^{n_\\alpha} \\otimes \\field{C}^{m_{\\alpha\\beta}} \\otimes \\field{C}^{n_\\beta}, \\field{C}^{n_\\gamma} \\otimes \\field{C}^{m_{\\gamma\\delta}^\\prime} \\otimes \\field{C}^{n_\\delta})\\\\ =\n \\begin{cases}\n  1_{n_\\alpha} \\otimes M_{m_{\\alpha\\delta}^\\prime \\times m_{\\alpha\\beta}}(\\field{C}) \\otimes M_{n_\\delta \\times n_\\beta}(\\field{C}) &\\text{if $\\alpha = \\gamma$,}\\\\\n  \\{0\\} &\\text{otherwise.}\n \\end{cases}\n\\end{multline*}\n\nAnalogously, one can show that\n\\begin{multline*}\n \\bdd^{\\textup{R}}_{\\alg{A}}(\\field{C}^{n_\\alpha} \\otimes \\field{C}^{m_{\\alpha\\beta}} \\otimes \\field{C}^{n_\\beta}, \\field{C}^{n_\\gamma} \\otimes \\field{C}^{m_{\\gamma\\delta}^\\prime} \\otimes \\field{C}^{n_\\delta})\\\\ = \n \\begin{cases}\n  M_{n_\\gamma \\times n_\\alpha}(\\field{C}) \\otimes M_{m_{\\gamma\\beta}^\\prime \\times m_{\\alpha\\beta}}(\\field{C}) \\otimes 1_{n_\\beta} &\\text{if $\\beta = \\delta$,}\\\\\n  \\{0\\} &\\text{otherwise},\n \\end{cases}\n\\end{multline*}\nand then these first two results together imply that\n\\begin{multline*}\n \\bdd^{\\textup{LR}}_{\\alg{A}}(\\field{C}^{n_\\alpha} \\otimes \\field{C}^{m_{\\alpha\\beta}} \\otimes \\field{C}^{n_\\beta}, \\field{C}^{n_\\gamma} \\otimes \\field{C}^{m_{\\gamma\\delta}^\\prime} \\otimes \\field{C}^{n_\\delta})\\\\ = \n \\begin{cases}\n  1_{n_\\alpha} \\otimes M_{m_{\\alpha\\beta}^\\prime \\times m_{\\alpha\\beta}}(\\field{C}) \\otimes 1_{n_\\beta} &\\text{if $(\\alpha,\\beta)=(\\gamma,\\delta)$,}\\\\\n  \\{0\\} &\\text{otherwise,}\n \\end{cases}\n\\end{multline*}\nas was claimed.\n\\end{proof}\n\nAn immediate consequence is the following description of the group $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H})$:\n\n\\begin{corollary}\\label{oddunitary}\n Let $\\hs{H}$ be an $\\alg{A}$-bimodule. Then\n\\[\n \\comp(\\U^{\\textup{LR}}_\\alg{A}(\\hs{H})) = \\bigoplus_{\\alpha,\\beta \\in \\spec{\\alg{A}}} 1_{n_\\alpha} \\otimes \\U(m_{\\alpha\\beta}) \\otimes 1_{n_\\beta} \\cong \\prod_{\\alpha,\\beta \\in \\spec{\\alg{A}}} \\U(m_{\\alpha\\beta}),\n\\]\nwith the convention that $\\U(0) = \\{0\\}$ is the trivial group.\n\\end{corollary}\n\n\\subsection{Even bimodules}\n\nWe now turn to the study of even bimodules; let us begin by considering the decomposition of an even bimodule into its even and odd sub-bimodules.\n\nLet $(\\hs{H},\\gamma)$ be an even $\\alg{A}$-bimodule. Define mutually orthogonal projections $P^\\textup{even}$ and $P^\\textup{odd}$ by \n\\[\n P^\\textup{even} = \\frac{1}{2}(1+\\gamma), \\quad P^\\textup{odd} = \\frac{1}{2}(1-\\gamma).\n\\]\nWe can then define sub-bimodules $\\hs{H}^\\textup{even}$ and $\\hs{H}^\\textup{odd}$ of $\\hs{H}$ by $\\hs{H}^\\textup{even} = P^\\textup{even} \\hs{H}$, $\\hs{H}^\\textup{odd} = P^\\textup{odd} \\hs{H}$; one has that $\\hs{H} = \\hs{H}^\\textup{even} \\oplus \\hs{H}^\\textup{odd}$ at the level of bimodules.\n\nOn the other hand, given $\\alg{A}$-bimodules $\\hs{H}_1$ and $\\hs{H}_2$, we can construct an even $\\alg{A}$-bimodule $(\\hs{H},\\gamma)$ such that $\\hs{H}^\\textup{even} = \\hs{H}_1$ and $\\hs{H}^\\textup{odd} = \\hs{H}_2$ by setting $\\hs{H} = \\hs{H}_1 \\oplus \\hs{H}_2$ and $\\gamma = 1_{\\hs{H}_1} \\oplus (-1_{\\hs{H}_2})$. If $\\hs{H}_1$ and $\\hs{H}_2$ are already $\\hs{H}^\\textup{even}$ and $\\hs{H}^\\textup{odd}$ for some $(\\hs{H},\\gamma)$, then this procedure precisely reconstructs $(\\hs{H},\\gamma)$. Since this procedure manifestly respects direct summation and unitary equivalence at either end, we have therefore proved the following:\n\n\\begin{proposition}\\label{evensplit}\n Let $\\alg{A}$ be a real {\\ensuremath{C^*}}-algebra. The map \n\\[\n C : \\Bimod^\\textup{even}(\\alg{A}) \\to \\Bimod(\\alg{A}) \\times \\Bimod(\\alg{A})\n\\]\ngiven by\n\\[\n C([\\hs{H}]) := ([\\hs{H}^\\textup{even}],[\\hs{H}^\\textup{odd}])\n\\]\nis an isomorphism of monoids.\n\\end{proposition}\n\nOne readily obtains a similar decomposition at the level of unitary groups:\n\n\\begin{corollary}\nLet $(\\hs{H},\\gamma)$ be an even $\\alg{A}$-bimodule. Then\n\\[\n \\U^{\\textup{LR}}_\\alg{A}(\\hs{H},\\gamma) = \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even}) \\oplus \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd}).\n\\]\n\\end{corollary}\n\nAnother immediate consequence is the following analogue of Proposition~\\ref{oddmult}:\n\n\\begin{proposition}\\label{evenmult}\n Let $\\alg{A}$ be a real {\\ensuremath{C^*}}-algebra. The map \n\\[\n \\bimod^\\textup{even} : M_S(\\semiring{Z}_{\\geq 0}) \\times M_S(\\semiring{Z}_{\\geq 0}) \\to \\Bimod^\\textup{even}(\\alg{A})\n\\]\ndefined by $bimod^\\textup{even} := C^{-1} \\circ (\\bimod \\times \\bimod)$ is an isomorphism of monoids.\n\\end{proposition}\nJust as in the odd case, we will find it convenient to denote $(\\bimod^\\textup{even})^{-1} : \\Bimod^\\textup{even}(\\alg{A}) \\to M_S(\\semiring{Z}_{\\geq 0}) \\times M_S(\\semiring{Z}_{\\geq 0})$ by $\\mult^\\textup{even}$. It then follows that $\\mult^\\textup{even} = (\\mult \\times \\mult) \\circ C$.\n\n\\begin{definition}\n Let $(\\hs{H},\\gamma)$ be an even $\\alg{A}$-bimodule. Then the \\term{multiplicity matrices} of $(\\hs{H},\\gamma)$ are the pair of matrices\n\\[\n (\\mult[\\hs{H}^\\textup{even}],\\mult[\\hs{H}^\\textup{odd}]) = \\mult^\\textup{even}[(\\hs{H},\\gamma)] \\in M_S(\\semiring{Z}_{\\geq 0}) \\times M_S(\\semiring{Z}_{\\geq 0}).\n\\]\n\\end{definition}\n\nLet us now consider orientability of even bimodules.\n\n\\begin{lemma}[Krajewski~\\cite{Kraj98}*{\\S3.4}]\\label{orientable}\n Let $(\\hs{H},\\gamma)$ be an even $\\alg{A}$-bimodule. Then $(\\hs{H},\\gamma)$ is orientable only if $\\bdd^{\\textup{LR}}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})=\\{0\\}$.\n\\end{lemma}\n\n\\begin{proof}\n Suppose that $(\\hs{H},\\gamma)$ is orientable, so that $\\gamma = \\sum_{i=1}^k \\lambda(a_i)\\rho(b_i)$ for some $a_1,\\dotsc,a_k$, $b_1,\\dotsc,b_k \\in \\alg{A}$. Now, let $T \\in \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$, and define $\\tilde{T} \\in \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H})$ by\n\\[\n \\tilde{T} =\n \\begin{pmatrix}\n  0 & T^*\\\\\n  T & 0\n \\end{pmatrix}.\n\\]\nThen, on the one hand, since $\\gamma = 1_{\\hs{H}^\\textup{even}} \\oplus (-1_{\\hs{H}^\\textup{odd}})$, $\\tilde{T}$ anticommutes with $\\gamma$, and on the other, since $\\gamma = \\sum_{i=1}^k \\lambda(a_i)\\rho(b_i)$, $\\tilde{T}$ commutes with $\\gamma$, so that $\\tilde{T} = 0$. Hence, $T = 0$.\n\\end{proof}\n\nThis last result motivates the following weaker notion of orientability:\n\n\\begin{definition}\n An even $\\alg{A}$-bimodule $(\\hs{H},\\gamma)$ shall be called \\term{quasi-orientable} whenever $\\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd}) = \\{0\\}$.\n\\end{definition}\n\nThe subset of $\\Bimod^\\textup{even}(\\alg{A})$ consisting of the unitary equivalence classes of the quasi-orientable even $\\alg{A}$-bimodules will be denoted by $\\Bimod^\\textup{even}_q(\\alg{A})$. \n\nWe define the \\term{support} of a real $p \\times q$ matrix $A$ to be the set\n\\[\n \\supp(A) := \\{(i,j) \\in \\{1,\\dotsc,p\\} \\times \\{1,\\dotsc,q\\} \\mid A_{ij} \\neq 0\\}.\n\\]\nFor $A \\in M_S(\\field{R})$, we shall view $\\supp(A)$ as a subset of $\\spec{\\alg{A}}^2$ by means of the identification of $\\{1,\\dotsc,S\\}$ with $\\spec{\\alg{A}}$ as ordered sets. We shall also find it convenient to associate to each matrix $m \\in M_S(\\ring{Z})$ a matrix $\\widehat{m} \\in M_N(\\ring{Z})$ by\n\\begin{equation}\n \\widehat{m}_{ij} := \\sum_{\\alpha \\in \\spec{M_{n_i}(\\field{K}_i)}} \\sum_{\\beta \\in \\spec{M_{n_j}(\\field{K}_j)}} m_{\\alpha\\beta}.\n\\end{equation}\nOne can check the map $M_S(\\ring{Z}) \\to M_N(\\ring{Z})$ defined by $m \\mapsto \\widehat{m}$ is linear and respects transposes.\n\nWe can now offer the following characterisation of quasi-orientable bimodules:\n\n\\begin{proposition}[Krajewski~\\cite{Kraj98}*{\\S3.3}, Paschke--Sitarz~\\cite{PS98}*{Lemma 3}]\n Let $\\alg{A}$ be a real {\\ensuremath{C^*}}-algebra. Then\n\\begin{multline}\n \\mult^\\textup{even}(\\Bimod^\\textup{even}_q(\\alg{A})) \\\\ = \\{(m^\\textup{even},m^\\textup{odd}) \\in M_S(\\semiring{Z}_{\\geq 0})^2 \\mid \\supp(m^\\textup{even}) \\cap \\supp(m^\\textup{odd}) = \\emptyset\\}.\n\\end{multline}\n\\end{proposition}\n\n\\begin{proof}\nLet $(\\hs{H},\\gamma)$ be an even $\\alg{A}$-bimodule and let $(m^\\textup{even},m^\\textup{odd})$ be its multiplicity matrices. Then by Proposition~\\ref{linear},\n\\[\n \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd}) \\cong \\bigoplus_{\\alpha,\\beta \\in \\spec{\\alg{A}}} M_{m_{\\alpha\\beta}^\\textup{odd} \\times m_{\\alpha\\beta}^\\textup{even}}(\\field{C}),\n\\]\nwhence the result follows immediately.\n\\end{proof}\n\nWe therefore define the \\term{signed multiplicity matrix} of a quasi-orientable even $\\alg{A}$-bimodule $(\\hs{H},\\gamma)$, or rather, the unitary equivalence class thereof, to be the matrix\n\\[\n \\mult_q[(\\hs{H},\\gamma)] := \\mult[\\hs{H}^\\textup{even}] - \\mult[\\hs{H}^\\textup{odd}] \\in M_S(\\ring{Z}).\n\\]\nThe map $\\Bimod^\\textup{even}_q(\\alg{A}) \\to M_S(\\ring{Z})$ defined by \n\\[\n [(\\hs{H},\\gamma)] \\mapsto \\mult_q[(\\hs{H},\\gamma)]\n\\]\nis then bijective, and $\\mult^\\textup{even}[(\\hs{H},\\gamma)]$ is readily recovered from $\\mult_q[(\\hs{H},\\gamma)]$. Indeed, if $(\\hs{H},\\gamma)$ is a quasi-orientable even $\\alg{A}$-bimodule with signed multiplicity matrix $\\mu$, then (cf.~\\cite{PS98}*{Lemma 3},\\cite{Kraj98}*{3.3})\n\\begin{equation}\n \\gamma = \\bigoplus_{\\alpha,\\beta \\in \\spec{\\alg{A}}} \\mu_{\\alpha\\beta} 1_{\\hs{H}_{\\alpha\\beta}}.\n\\end{equation}\nThese algebraic consequences of quasi-orientability, which were derived from the stronger condition of orientability in the original papers \\cite{PS98} and \\cite{Kraj98}, are key to the formalism developed by Krajewski and Paschke--Sitarz, and hence to the later work by Iochum, Jureit, Sch{\\\"u}cker, and Stephan~\\cites{ACG1,ACG2,ACG3,Sch05}.\n\nWe can now characterise orientable bimodules amongst quasi-orientable bimodules:\n\n\\begin{proposition}[Krajewski~\\cite{Kraj98}*{\\S 3.3}]\\label{converseorientable}\n Let $(\\hs{H},\\gamma)$ be a quasi-orientable $\\alg{A}$-bimodule with signed multiplicity matrix $\\mu$. Then $(\\hs{H},\\gamma)$ is orientable if and only if the following conditions all hold:\n\\begin{enumerate}\n \\item For each $i \\in \\{1,\\dotsc,N\\}$ such that $\\field{K}_i = \\field{C}$ and all $\\beta \\in \\spec{\\alg{A}}$,\n\\[\n \\mu_{\\rep{n}_i \\beta}\\mu_{\\crep{n}_i \\beta} \\geq 0;\n\\]\n \\item For all $\\alpha \\in \\spec{\\alg{A}}$ and each $j \\in \\{1,\\dotsc,N\\}$ such that $\\field{K}_j = \\field{C}$, \n\\[\n \\mu_{\\alpha \\rep{n}_j}\\mu_{\\alpha \\crep{n}_j} \\geq 0;\n\\]\n \\item For all $i$, $j \\in \\{1,\\dotsc,N\\}$ such that $\\field{K}_i = \\field{K}_j = \\field{C}$,\n\\[\n \\mu_{\\rep{n}_i \\crep{n}_j}\\mu_{\\crep{n}_i \\rep{n}_j} \\geq 0.\n\\]\n\\end{enumerate}\nIn particular, if $(\\hs{H},\\gamma)$ is orientable, then\n\\begin{equation}\n \\gamma = \\sum_{i,j=1}^N \\lambda(\\sgn(\\widehat{\\mu}_{ij})e_i)\\rho(e_j).\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\n First, suppose that $(\\hs{H},\\gamma)$ is indeed orientable, so that there exist $a_1,\\dotsc,a_n$, $b_1,\\dotsc,b_n \\in \\alg{A}$ such that $\\gamma = \\sum_{l=1}^n \\lambda(a_l)\\rho(b_l)$; in particular, then, for each $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$,\n\\[\n \\sgn(\\mu_{\\alpha\\beta}) 1_{n_\\alpha} \\otimes 1_{|\\mu_{\\alpha\\beta}|} \\otimes 1_{n_\\beta} = \\gamma_{\\alpha\\beta}^{\\alpha\\beta} = \\sum_{l=1}^n \\lambda_\\alpha(a_l) \\otimes 1_{|\\mu_{\\alpha\\beta}|} \\otimes \\lambda_\\beta(b_l)^T.\n\\]\n\nNow, let $i \\in \\{1,\\dotsc,N\\}$ be such that $\\field{K}_i = \\field{C}$, and let $\\beta \\in \\spec{\\alg{A}}$, and suppose that $\\mu_{\\rep{n}_i \\beta}$ and $\\mu_{\\crep{n}_i \\beta}$ are both non-zero. It then follows that\n\\[\n \\sgn(\\mu_{\\rep{n}_i \\beta}) 1_{n_i} \\otimes 1_{n_\\beta} = \\sum_{l=1}^n (a_l)_i \\otimes \\lambda_\\beta(b_l)^T, \\quad\n \\sgn(\\mu_{\\crep{n}_i \\beta}) 1_{n_i} \\otimes 1_{n_\\beta} = \\sum_{l=1}^n \\overline{(a_l)_i} \\otimes \\lambda_\\beta(b_l)^T,\n\\]\nwhere $(a_l)_i$ denotes the component of $a_l$ in the direct summand $M_{k_i}(\\field{C})$ of $\\alg{A}$. If $X$ denotes complex conjugation on $\\field{C}^{n_i}$, it then follows from this that\n\\[\n \\sgn(\\mu_{\\rep{n}_i \\beta}) 1_{n_i} \\otimes 1_{n_\\beta} = (X \\otimes 1_{n_\\beta}) (\\sgn(\\mu_{\\rep{n}_i \\beta}) 1_{n_i} \\otimes 1_{n_\\beta}) (X \\otimes 1_{n_\\beta}) = \\sgn(\\mu_{\\crep{n}_i \\beta}) 1_{n_i} \\otimes 1_{n_\\beta},\n\\]\nso that $\\sgn(\\mu_{\\rep{n}_i \\beta}) = \\sgn(\\mu_{\\crep{n}_i \\beta})$, or equivalently $\\mu_{\\rep{n}_i \\beta}\\mu_{\\crep{n}_i \\beta} > 0$. One can similarly show that the other two conditions hold.\n\nNow, suppose instead that the three conditions on $\\mu$ hold. Then for all $i$, $j \\in \\{1,\\dotsc,N\\}$, all non-zero entries $\\mu_{\\alpha\\beta}$ for $\\alpha \\in \\spec{M_{k_i}(\\field{K}_i)}$, $\\beta \\in \\spec{M_{k_j}(\\field{K}_j)}$, have the same sign, so set $\\gamma_{ij}$ equal to this common value of non-zero $\\sgn(\\mu_{\\alpha\\beta})$ if at least one such $\\mu_{\\alpha\\beta}$ is non-zero, and set $\\gamma_{ij} = 0$ otherwise. One can then easily check that $\\gamma = \\sum_{i,j=1}^{N} \\lambda(\\gamma_{ij}e_i)\\rho(e_j)$, so that $(\\hs{H},\\gamma)$ is indeed orientable. Moreover, using the same three conditions, one can readily check that $\\gamma_{ij} = \\sgn(\\widehat{\\mu}_{ij})$, which yields the last part of the claim.\n\\end{proof}\n\nLet us now turn to intersection forms and Poincar{\\'e} duality. In particular, we are now able to provide explicit expressions for intersection forms in terms of multiplicity matrices.\n\nRecall that for $\\field{K} = \\field{R}$, $\\field{C}$ or $\\field{H}$, $KO_0(M_k(\\field{K}))$ is the infinite cyclic group generated by $[p]$ for $p \\in M_k(\\field{K})$ a minimal projection, so that for $\\alg{A}$ a real {\\ensuremath{C^*}}-algebra with Wedderburn decomposition $\\oplus_{i=1}^N M_{n_i}(\\field{K}_i)$,\n\\[\n KO_0(\\alg{A}) \\cong \\prod_{i=1}^N KO_0(M_{n_i}(\\field{K}_i)) \\cong \\ring{Z}^N,\n\\]\nwhich can be viewed as the infinite abelian group generated by $\\{[p_i]\\}_{i=1}^N$ for $p_i$ a minimal projection in $M_{n_i}(\\field{K}_i)$. Since\n\\[\n \\tau_i := \\tr(p_i) =\n \\begin{cases}\n  2 &\\text{if $\\field{K}_i = \\field{H}$,}\\\\\n  1 &\\text{otherwise,}\n \\end{cases}\n\\]\nit follows that for $\\alpha \\in \\spec{\\alg{A}}$,\n\\begin{equation}\n \\tr(\\lambda_\\alpha(p_i)) =\n \\begin{cases}\n  \\tau_i &\\text{if $\\alpha \\in \\spec{M_{n_i}(\\field{K}_i)}$,}\\\\\n  0 &\\text{otherwise.}\\\\\n \\end{cases}\n\\end{equation}\n\nNow, if $(\\hs{H},\\gamma)$ is an even $\\alg{A}$-bimodule with intersection form $\\form{\\cdot,\\cdot}$, we can define a matrix $\\cap \\in M_N(\\ring{Z})$ by\n\\begin{equation}\n \\cap_{ij} := \\form{[p_i],[p_j]}.\n\\end{equation}\nThe intersection form $\\form{\\cdot,\\cdot}$ is completely determined by the matrix $\\cap$, and in particular, $\\form{\\cdot,\\cdot}$ is non-degenerate ({\\it i.e.\\\/}\\  $(\\hs{H},\\gamma)$ satisfies Poincar{\\'e} duality) if and only if $\\cap$ is non-degenerate.\n\n\\begin{proposition}[Krajewski~\\cite{Kraj98}*{\\S 3.3}, Paschke--Sitarz~\\cite{PS98}*{\\S 2.4}]\\label{intform}\n Let $(\\hs{H},\\gamma)$ be an even $\\alg{A}$-bimodule with pair of multiplicity matrices $(m^\\textup{even},m^\\textup{odd})$. Then\n\\begin{equation}\n \\cap_{ij} = \\tau_i \\tau_j \\left(\\widehat{m^\\textup{even}}_{ij} - \\widehat{m^\\textup{odd}}_{ij}\\right),\n\\end{equation}\nso that $(\\hs{H},\\gamma)$ satisfies Poincar{\\'e} duality if and only if the matrix $\\widehat{m^\\textup{even}} - \\widehat{m^\\textup{odd}}$ is non-degenerate.\n\\end{proposition}\n\n\\begin{proof}\nFirst, since $\\hs{H} = \\hs{H}^\\textup{even} \\oplus \\hs{H}^\\textup{odd}$, we can write\n\\[\n \\gamma = \\bigoplus_{\\alpha,\\beta\\in\\spec{\\alg{A}}} 1_{n_\\alpha} \\otimes \\gamma_{\\alpha\\beta} \\otimes 1_{n_\\beta},\n\\]\nwhere $\\gamma_{\\alpha\\beta} = 1_{m_{\\alpha\\beta}^\\textup{even}} \\oplus (-1_{m_{\\alpha\\beta}^\\textup{odd}})$. Then,\n\\begin{align*}\n\\cap_{ij} &= \\form{[p_i],[p_j]}\\\\\n&= \\tr(\\gamma\\lambda(p_i)\\rho(p_j))\\\\ \n&= \\tr\\left(\\bigoplus_{\\alpha,\\beta\\in\\spec{\\alg{A}}} \\lambda_\\alpha(p_i) \\otimes \\gamma_{\\alpha\\beta} \\otimes \\lambda_\\beta(p_j)\\right)\\\\\n&= \\sum_{\\alpha,\\beta \\in \\spec{\\alg{A}}} \\tr(\\lambda_\\alpha(p_i))\\tr(\\lambda_\\beta(p_j))(m_{\\alpha\\beta}^\\textup{even} - m_{\\alpha\\beta}^\\textup{odd})\\\\\n&= \\sum_{i,j=1}^N \\tau_i \\tau_j (\\widehat{m^\\textup{even}}_{ij} - \\widehat{m^\\textup{odd}}_{ij}).\n\\end{align*}\nThis calculation implies, in particular, that $\\cap$ can be obtained from $\\widehat{m^\\textup{even}} - \\widehat{m^\\textup{odd}}$ by a finite sequence of elementary row or column operations, so that $\\cap$ is indeed non-degenerate if and only if $\\widehat{m^\\textup{even}} - \\widehat{m^\\textup{odd}}$ is.\n\\end{proof}\n\n\\begin{corollary}\n Let $(\\hs{H},\\gamma)$ be a quasi-orientable $\\alg{A}$-bimodule with signed multiplicity matrix $\\mu$. Then $(\\hs{H},\\gamma)$ satisfies Poincar{\\'e} duality if and only if $\\widehat{\\mu}$ is non-degenerate.\n\\end{corollary}\n\nIn particular, if we restrict ourselves to complex {\\ensuremath{C^*}}-algebras and complex-linear representations, a quasi-orientable bimodule is completely characterised by the $K$-theoretic datum of its intersection form.\n\n\\subsection{Real bimodules of odd $KO$-dimension}\n\nLet us now consider real bimodules of odd $KO$-dimension. Before continuing, recall that\n\\[\n \\Bimod(\\alg{A},1) = \\Bimod(\\alg{A},7), \\quad \\Bimod(\\alg{A},3) = \\Bimod(\\alg{A},5).\n\\]\n\nFor $m \\in \\Sym_S(\\semiring{Z}_{\\geq 0})$, we define an antilinear operator $X_m$ on $\\hs{H}_m$ by defining $(X_m)_{\\alpha\\beta}^{\\gamma\\delta} : \\field{C}^{n_\\alpha} \\otimes \\field{C}^{m_{\\alpha\\beta}} \\otimes \\field{C}^{n_\\beta} \\to \\field{C}^{n_\\gamma} \\otimes \\field{C}^{m_{\\gamma\\delta}} \\otimes \\field{C}^{n_\\delta}$ by\n\\begin{equation}\n (X_m)_{\\alpha\\beta}^{\\beta\\alpha}: \\xi_1 \\otimes \\xi_2 \\otimes \\xi_3 \\mapsto \\overline{\\xi_3} \\otimes \\overline{\\xi_2} \\otimes \\overline{\\xi_1},\n\\end{equation}\nand by setting $(X_m)_{\\alpha\\beta}^{\\gamma\\delta} = 0$ whenever $(\\gamma,\\delta) \\neq (\\beta,\\alpha)$.\n\n\\subsubsection{$KO$-dimension $1$ or $7 \\bmod 8$}\n\nWe begin by determining the form of the multiplicity matrix for a real bimodule of $KO$-dimension $1$ or $7 \\bmod 8$.\n\n\\begin{lemma}[Krajewski~\\cite{Kraj98}*{\\S3.2}, Paschke--Sitarz~\\cite{PS98}*{Lemma 4}]\\label{real17a}\n Let $(\\hs{H},J)$ be a real $\\alg{A}$-bimodule of $KO$-dimension $1$ or $7 \\bmod 8$ with multiplicity matrix $m$. Then $m$ is symmetric, and the only non-zero components of $J$ are of the form $J_{\\alpha\\beta}^{\\beta\\alpha}$ for $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$, which are anti-unitaries $\\hs{H}_{\\alpha\\beta} \\to \\hs{H}_{\\beta\\alpha}$ satisfing the relations $J_{\\beta\\alpha}^{\\alpha\\beta} = (J_{\\alpha\\beta}^{\\beta\\alpha})^*$.\n\\end{lemma}\n\n\\begin{proof}\nLet the projections $P_{\\alpha}^L$, $P_{\\beta}^R$ and $P_{\\alpha\\beta}$ be defined as in the proof of Proposition~\\ref{oddmult}, and recall that $P_{\\alpha\\beta} = P_{\\alpha}^L P_{\\beta}^R$. By Equation~\\ref{realintertwine}, it follows that for all $\\alpha \\in \\spec{\\alg{A}}$,$J P_{\\alpha}^L = P_{\\alpha}^R J$ and $J P_{\\alpha}^R = P_{\\alpha}^L J$, and hence that for all $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$, $J P_{\\alpha\\beta} = J P_\\alpha^L P_\\beta^R = P_\\alpha^R P_\\beta^L J = P_{\\beta\\alpha} J$. Thus, the only non-zero components of $J$ are the anti-unitaries $J_{\\alpha\\beta}^{\\beta\\alpha} : \\hs{H}_{\\alpha\\beta} \\to \\hs{H}_{\\beta\\alpha}$ which satisfy $J_{\\beta\\alpha}^{\\alpha\\beta} = (J_{\\alpha\\beta}^{\\beta\\alpha})^*$; this, in turn, implies that $m$ is indeed symmetric.\n\\end{proof}\n\nNext, we show that for every $m \\in \\Sym_S(\\semiring{Z}_{\\geq 0})$, not only does $\\hs{H}_m$ admit a real structure of $KO$-dimension $1$ or $7 \\bmod 8$, but it is also unique up to unitary equivalence.\n\n\\begin{lemma}[Krajewski~\\cite{Kraj98}*{\\S3.2}, Paschke--Sitarz~\\cite{PS98}*{Lemma 5}]\\label{real17b}\n Let $m \\in \\Sym_S(\\semiring{Z}_{\\geq 0})$. Then, up to unitary equivalence, $J_m := X_m$ is the unique real structure on $\\hs{H}_m$ of $KO$-dimension $1$ or $7 \\bmod 8$.\n\\end{lemma}\n\n\\begin{proof}\n First, $X_m$ is indeed by construction a real structure on $\\hs{H}_m$ of $KO$-dimension $1$ or $7 \\bmod 8$.\n \n Now, let $J$ be another real structure on $\\hs{H}_m$ of $KO$-dimension $1$ or $7 \\bmod 8$. Define a unitary $K$ on $\\hs{H}$ by $K = J X_m$; thus, $J = K X_m$. Since the intertwining condition of Equation~\\ref{realintertwine} applies to both $J$ and $X_m$, we have, in fact, that $K \\in \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}_m)$, and hence\n\\[\n K = \\bigoplus_{\\alpha,\\beta \\in \\spec{\\alg{A}}} 1_{n_\\alpha} \\otimes K_{\\alpha\\beta} \\otimes 1_{n_\\beta},\n\\]\nfor $K_{\\alpha\\beta} \\in \\U(m_{\\alpha\\beta})$. In particular, since $K^* = X_m J = X_m K X_m$, we have that $K_{\\beta\\alpha} = K_{\\alpha\\beta}^T$.\n\nLet $(\\alpha,\\beta) \\in \\supp(m)$, and suppose that $\\alpha < \\beta$. Let $K_{\\alpha\\beta} = V_{\\alpha\\beta} \\tilde{K}_{\\alpha\\beta} V_{\\alpha\\beta}^*$ be a unitary diagonalisation of $K_{\\alpha\\beta}$, and let $L_{\\alpha\\beta}$ be a diagonal square root of $\\tilde{K}_{\\alpha\\beta}$. Then$ K_{\\alpha\\beta} = V_{\\alpha\\beta} L_{\\alpha\\beta} L_{\\alpha\\beta} V_{\\alpha\\beta}^* = (V_{\\alpha\\beta}L_{\\alpha\\beta})(\\overline{V_{\\alpha\\beta}} L_{\\alpha\\beta})^T$, and hence $K_{\\beta\\alpha} = (\\overline{V_{\\alpha\\beta}} L_{\\alpha\\beta})(V_{\\alpha\\beta}L_{\\alpha\\beta})^T$. If, instead, $\\alpha = \\beta$, then $K_{\\alpha\\alpha}$ is unitary and complex symmetric, so that there exists a unitary $W_{\\alpha\\alpha}$ such that $K_{\\alpha\\alpha} = W_{\\alpha\\alpha}W_{\\alpha\\alpha}^T$. We can now define a unitary $U \\in \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}_m)$ by\n\\[\n U = \\bigoplus_{\\alpha,\\beta \\in \\spec{\\alg{A}}} 1_{n_\\alpha} \\otimes U_{\\alpha\\beta} \\otimes 1_{n_\\beta},\n\\]\nwhere $U_{\\alpha\\beta} = 0$ if $m_{\\alpha\\beta} = 0$, and for $(\\alpha,\\beta) \\in \\supp(m)$,\n\\[\n U_{\\alpha\\beta} = \n\\begin{cases}\n V_{\\alpha\\beta}L_{\\alpha\\beta}, &\\text{if $\\alpha < \\beta$,}\\\\\n \\overline{V_{\\beta\\alpha}}L_{\\beta\\alpha}, &\\text{if $\\alpha > \\beta$,}\\\\\n W_{\\alpha\\alpha}, &\\text{if $\\alpha = \\beta$.}\n\\end{cases}\n\\]\nThen, by construction, $K = U  X_m U^* X_m$, and hence, $J = U X_m U^*$, so that $U$ is the required unitary equivalence between $(\\hs{H}_m,X_m)$ and $(\\hs{H}_m,J)$.\n\\end{proof}\n\nWe can now give our characterisation of real bimodules of $KO$-dimension $1$ or $7 \\bmod 8$:\n\n\\begin{proposition}[Krajewski~\\cite{Kraj98}*{\\S3.2}]\\label{real17mult}\nLet $n = 1$ or $7 \\bmod 8$. Then the map $\\iota_n : \\Bimod(\\alg{A},n) \\to \\Bimod(\\alg{A})$ defined by $\\iota_n : [(\\hs{H},J)] \\mapsto [\\hs{H}]$ is injective, and\n\\begin{equation}\n (\\mult \\circ \\iota_n)(\\Bimod(\\alg{A},n)) = \\Sym_S(\\semiring{Z}_{\\geq 0}).\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nFirst, since a unitary equivalence of real $\\alg{A}$-bimodules of $KO$-di\\-men\\-sion $n \\bmod 8$ is, in particular, a unitary equivalence of odd $\\alg{A}$-bimodules, the map $\\iota_n$ is well defined.\n\nNext, let $(\\hs{H},J)$ and $(\\hs{H}^\\prime,J^\\prime)$ be real $\\alg{A}$-bimodules of $KO$-dimension $n \\bmod 8$, and suppose that $\\hs{H}$ and $\\hs{H}^\\prime$ are unitarily equivalent as bimodules; let $U \\in \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\prime,\\hs{H})$. Now, if $m$ is the multiplicity matrix of $\\hs{H}$, then $\\hs{H}$ and $\\hs{H}_m$ are unitarily equivalent, so let $V \\in \\U^{\\textup{LR}}_\\alg{A}(\\hs{H},\\hs{H}_m)$. Then $V J V^*$ and $V U J^\\prime U^* V^*$ are both real structures of $KO$-dimension $n \\bmod 8$, so by Lemma~\\ref{real17b}, they are both unitarily equivalent to $J_m$. This implies that $J$ and $U J^\\prime U^*$ are unitarily equivalent as real structures on $\\hs{H}$, and hence that $(\\hs{H},J)$ and $(\\hs{H}^\\prime,J^\\prime)$ are unitarily equivalent. Thus, $\\iota_n$ is injective.\n\nFinally, Lemma~\\ref{real17a} implies that $(\\mult \\circ \\iota_n)(\\Bimod(\\alg{A},n)) \\subseteq \\Sym_S(\\semiring{Z}_{\\geq 0})$, while Lemma~\\ref{real17b} implies the reverse inclusion.\n\\end{proof}\n\nThus, without any loss of generality, a real bimodule $\\hs{H}$ of $KO$-dimension $1$ or $7 \\bmod 8$ with multiplicity matrix $m$ can be assumed to be simply $(\\hs{H}_m,J_m)$.\n\nOne following characterisation of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},J)$ now follows by direct calculation:\n\n\\begin{proposition}\\label{real17unitary}\nLet $(\\hs{H},J)$ be a real $\\alg{A}$-bimodule of $KO$-dimension $1$ or $7 \\bmod 8$ with multiplicity matrix $m$. Then\n\\begin{equation}\\begin{split}\n \\comp(\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},J)) &= \\{(1_{n_\\alpha} \\otimes U_{\\alpha\\beta} \\otimes 1_{n_\\beta})_{\\alpha,\\beta \\in \\spec{\\alg{A}}} \\in \\comp(\\U^{\\textup{LR}}_\\alg{A}(\\hs{H})) \\mid U_{\\beta\\alpha} = \\overline{U_{\\alpha\\beta}}\\}\\\\ &\\cong \\prod_{\\alpha \\in \\spec{\\alg{A}}} \\biggl( \\Orth(m_{\\alpha\\alpha}) \\times \\prod_{\\substack{\\beta \\in \\spec{\\alg{A}}\\\\ \\beta > \\alpha}} \\U(m_{\\alpha\\beta}) \\biggr).\n\\end{split}\\end{equation}\n\\end{proposition}\n\n\\subsubsection{$KO$-dimension $3$ or $5 \\bmod 8$}\n\nLet us now turn to real bimodules of $KO$-dimension $3$ or $5 \\bmod 8$. We begin with the relevant analogue of Lemma~\\ref{real17a}.\n\n\\begin{lemma}\\label{real35a}\nLet $(\\hs{H},J)$ be a real $\\alg{A}$-bimodule of $KO$-dimension $3$ or $5 \\bmod 8$ with multiplicity matrix $m$. Then $m$ is symmetric with even diagonal entries, and the only non-zero components of $J$ are of the form $J_{\\alpha\\beta}^{\\beta\\alpha}$ for $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$, which are anti-unitaries $\\hs{H}_{\\alpha\\beta} \\to \\hs{H}_{\\beta\\alpha}$ satisfying the relations $J_{\\beta\\alpha}^{\\alpha\\beta} = -(J_{\\alpha\\beta}^{\\beta\\alpha})^*$.\n\\end{lemma}\n\n\\begin{proof}\nThe proof follows just as for Lemma~\\ref{real17a}, except that the equation $J^2 = -1$ forces the relations $J_{\\beta\\alpha}^{\\alpha\\beta} = -(J_{\\alpha\\beta}^{\\beta\\alpha})^*$, which imply, in particular, that for each $\\alpha \\in \\spec{\\alg{A}}$, $(J_{\\alpha\\alpha}^{\\alpha\\alpha})^2 = -1$, so that $m_{\\alpha\\alpha}$ must be even.\n\\end{proof}\n\nLet us denote by $\\Sym_S^0(\\semiring{Z}_{\\geq 0})$ the set of all matrices in $\\Sym_S(\\semiring{Z}_{\\geq 0})$ with even diagonal entries. For $n = 2k$, let\n\\[\n \\Omega_n =\n \\begin{pmatrix}\n 0 & -1_k\\\\\n 1_k & 0\n \\end{pmatrix}.\n\\]\n\n\\begin{lemma}~\\label{real35b}\nLet $m \\in \\Sym_S^0(\\semiring{Z}_{\\geq 0})$. Define an antiunitary $J_m$ on $\\hs{H}_m$ by\n\\[\n (J_m)_{\\alpha\\beta}^{\\gamma\\delta} =\n \\begin{cases}\n  (X_m)_{\\alpha\\beta}^{\\beta\\alpha} &\\text{if $(\\gamma,\\delta)=(\\beta,\\alpha)$ and $\\alpha < \\beta$,}\\\\\n  -(X_m)_{\\alpha\\beta}^{\\beta\\alpha} &\\text{if $(\\gamma,\\delta)=(\\beta,\\alpha)$ and $\\alpha > \\beta$,}\\\\\n  \\Omega_{m_{\\alpha\\alpha}}(X_m)_{\\alpha\\alpha}^{\\alpha\\alpha} &\\text{if $\\alpha = \\beta = \\gamma = \\delta$,}\\\\\n  0 &\\text{otherwise.}\n \\end{cases}\n\\]\nThen, up to unitary equivalence, $J_m$ is the unique real structure on $\\hs{H}_m$ of $KO$-dimension $3$ or $5 \\bmod 8$.\n\\end{lemma}\n\n\\begin{proof}\nThe proof follows that of Lemma~\\ref{real17b}, except we now have that\n$K_{\\alpha\\alpha}^T = \\Omega_{m_{\\alpha\\alpha}} K_{\\alpha\\alpha} \\Omega_{m_{\\alpha\\alpha}}^T$ instead of $K_{\\alpha\\alpha}^T = K_{\\alpha\\alpha}$; each $K_{\\alpha\\alpha} \\Omega_{m_{\\alpha\\alpha}}$ is therefore unitary and complex skew-symmetric, so that we choose $W_{\\alpha\\alpha}$ unitary such that\n\\[\n K_{\\alpha\\alpha}\\Omega_{m_{\\alpha\\alpha}} = W_{\\alpha\\alpha}\\Omega_{m_{\\alpha\\alpha}}W_{\\alpha\\alpha}^T,\n\\]\nor equivalently, $K_{\\alpha\\alpha} = W_{\\alpha\\alpha}\\Omega_{m_{\\alpha\\alpha}}W_{\\alpha\\alpha}^T\\Omega_{m_{\\alpha\\alpha}}^T$. One can then construct the unitary equivalence $U$ between $(\\hs{H}_m,J)$ and $(\\hs{H},J_m)$ as before.\n\\end{proof}\n\nMuch as in the analogous case of $KO$-dimension $1$ or $7 \\bmod 8$, Lemmas~\\ref{real35a} and~\\ref{real35b} together imply the following characterisation of real bimodules of $KO$-dimension $3$ or $5 \\bmod 8$:\n\n\\begin{proposition}\\label{real35mult}\nLet $n=3$ or $5 \\bmod 8$. Then the map $\\iota_n : \\Bimod(\\alg{A},n) \\to \\Bimod(\\alg{A})$ defined by $\\iota_n : [(\\hs{H},J)] \\mapsto [\\hs{H}]$ is injective, and\n\\begin{equation}\n (\\mult \\circ \\iota_n)(\\Bimod(\\alg{A},n)) = \\Sym_S^0(\\semiring{Z}_{\\geq 0}).\n\\end{equation}\n\\end{proposition}\n\nFinally, these results immediately imply the following description of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},J)$:\n\n\\begin{proposition}\\label{real35unitary}\nLet $(\\hs{H},J)$ be a real $\\alg{A}$-bimodule of $KO$-dimension $3$ or $5 \\bmod 8$ with multiplicity matrix $m$. Then\n\\begin{equation}\\begin{split}\n \\comp(\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},J)) &= \\left\\{(1_{n_\\alpha} \\otimes U_{\\alpha\\beta} \\otimes 1_{n_\\beta})_{\\alpha,\\beta \\in \\spec{\\alg{A}}} \\in \\comp(\\U^{\\textup{LR}}_\\alg{A}(\\hs{H})) \\mid \\substack{U_{\\alpha\\alpha} \\in \\Sp(m_{\\alpha\\alpha}), \\\\ U_{\\beta\\alpha} = \\overline{U_{\\alpha\\beta}}, \\: \\alpha \\neq \\beta}\\right\\}\\\\\n &\\cong \\prod_{\\alpha \\in \\spec{\\alg{A}}} \\biggl(\\Sp(m_{\\alpha\\alpha}) \\times \\prod_{\\substack{\\beta \\in \\spec{\\alg{A}}\\\\ \\beta > \\alpha}} \\U(m_{\\alpha\\beta})\\biggr).\n\\end{split}\\end{equation}\n\\end{proposition}\n\n\\subsection{Real bimodules of even $KO$-dimension}\n\nWe now come to the case of even $KO$-dimension. Before continuing, note that for $(\\hs{H},\\gamma,J)$ a real bimodule of even $KO$-dimension,\n\\[\n \\forall p,q \\in KO_0(\\alg{A}), \\: \\form{q,p} = \\varepsilon^{\\prime\\prime} \\form{p,q},\n\\]\nas a direct result of the relation $J \\gamma = \\varepsilon^{\\prime\\prime} \\gamma J$; this is then equivalent to the condition\n\\begin{equation}\n \\cap = \\varepsilon^{\\prime\\prime} \\cap^T,\n\\end{equation}\nwhere $\\cap$ is the matrix of the intersection form. Thus, for $KO$-dimension $0$ or $4 \\bmod 8$, the intersection form is symmetric, whilst for $KO$-dimension $2$ or $6 \\bmod 8$, it is anti-symmetric. It then follows, in particular, that a real $\\alg{A}$-bimodule of $KO$-dimension $2$ or $6 \\bmod 8$ satisfies Poincar{\\'e} duality only if $\\alg{A}$ has an even number of direct summands in its Wedderburn decomposition, as an anti-symmetric $k \\times k$ matrix for $k$ odd is necessarily degenerate.\n\n\\subsubsection{$KO$-dimension $0$ or $4 \\bmod 8$}\n\nWe begin with the case where $\\varepsilon^{\\prime\\prime} = 1$ and hence $[\\gamma,J]=0$, {\\it i.e.\\\/}\\  of $KO$-dimension $0$ or $4 \\bmod 8$.\n\nLet $(\\hs{H},\\gamma,J)$ be a real $\\alg{A}$-bimodule of $KO$-dimension $n \\bmod 8$, for $n = 0$ or $4$; let the mutually orthogonal projections $P^\\textup{even}$ and $P^\\textup{odd}$ on $\\hs{H}$ be defined as before. Then, since $[J,\\gamma]=0$, we have that $J = J^\\textup{even} \\oplus J^\\textup{odd}$, where $J^\\textup{even} = P^\\textup{even} J P^\\textup{even}$ and $J^\\textup{odd} = P^\\textup{odd} J P^\\textup{odd}$. One can then check that $(\\hs{H}^\\textup{even},J^\\textup{even})$ and $(\\hs{H}^\\textup{odd},J^\\textup{odd})$ are real $\\alg{A}$-bimodules of $KO$-dimension $1$ or $7 \\bmod 8$ if $n = 0$, and $3$ or $5 \\bmod 8$ if $n=4$. On the other hand, given $(\\hs{H}^\\textup{even},J^\\textup{even})$ and $(\\hs{H}^\\textup{odd},J^\\textup{odd})$, one can immediately reconstruct $(\\hs{H},\\gamma,J)$ by setting $\\gamma = 1_{\\hs{H}^\\textup{even}} \\oplus (-1_{\\hs{H}^\\textup{odd}})$ and $J = J^\\textup{even} \\oplus J^\\textup{odd}$. Thus we have proved the following analogue of Proposition~\\ref{evensplit}:\n\n\\begin{proposition}~\\label{real04split}\n Let $\\alg{A}$ be a real {\\ensuremath{C^*}}-algebra. Let $k_0$ denote $1$ or $7 \\bmod 8$, and let $k_4$ denote $3$ or $5 \\bmod 8$. Then for $n = 0, 4 \\bmod 8$, the map\n \\[\n  C_n : \\Bimod(\\alg{A},n) \\to \\Bimod(\\alg{A},k_n) \\times \\Bimod(\\alg{A},k_n)\n \\]\n given by $C_n([(\\hs{H},\\gamma,J)]) := ([(\\hs{H}^\\textup{even},J^\\textup{even})],[(\\hs{H}^\\textup{odd},J^\\textup{odd})])$ is an isomorphism of monoids.\n\\end{proposition}\n\nOne can then apply this decomposition to the group $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},\\gamma,J)$ to find:\n\n\\begin{corollary}\\label{real04unitary}\n Let $(\\hs{H},\\gamma,J)$ be a real $\\alg{A}$-bimodule of $KO$-dimension $0$ or $4 \\bmod 8$. Then\n \\begin{equation}\n \\U^{\\textup{LR}}_\\alg{A}(\\hs{H},\\gamma,J) = \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},J^\\textup{even}) \\oplus \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd},J^\\textup{odd}).\n \\end{equation}\n\\end{corollary}\n\nCombining Proposition~\\ref{real04split} with our earlier characterisations of real bimodules of odd $KO$-dimension, we immediately obtain the following:\n\n\\begin{proposition}\\label{real04mult}\n Let $n=0$ or $4 \\bmod 8$. Then the map $\\iota_n : \\Bimod(\\alg{A},n) \\to \\Bimod^\\textup{even}(\\alg{A})$ defined by\n $[(\\hs{H},\\gamma,J)] \\mapsto ([(\\hs{H},\\gamma)])$ is injective, and\n \\[\n  (\\mult^\\textup{even} \\circ \\iota_n)(\\Bimod(\\alg{A},n)) =\n  \\begin{cases}\n   \\Sym_S(\\semiring{Z}_{\\geq 0}) \\times \\Sym_S(\\semiring{Z}_{\\geq 0}) &\\text{if $n = 0 \\bmod 8$,}\\\\\n   \\Sym_S^0(\\semiring{Z}_{\\geq 0}) \\times \\Sym_S^0(\\semiring{Z}_{\\geq 0}) &\\text{if $n = 4 \\bmod 8$.}\n  \\end{cases}\n \\]\n\\end{proposition}\n\nIn particular,\n\\[\n \\Bimod_q(\\alg{A},n) := \\iota_n^{-1}(\\Bimod^\\textup{even}_q(\\alg{A}))\n\\]\nis thus the set of all equivalence classes of quasi-orientable real $\\alg{A}$-bimodules of $KO$-dimension $n \\bmod 8$; the last Proposition then implies the following:\n\n\\begin{corollary}\n Let $n=0$ or $4 \\bmod 8$. Then\n\\begin{equation}\n (\\mult_q \\circ \\iota_n)(\\Bimod_q(\\alg{A},n)) = \\Sym_S(\\ring{Z}).\n\\end{equation}\n\\end{corollary}\n\n\\subsubsection{$KO$-dimension $2$ or $6 \\bmod 8$}\n\nFinally, let us consider the remaining case where $\\varepsilon^{\\prime\\prime} = -1$ and hence $\\{\\gamma,J\\}=0$, {\\it i.e.\\\/}\\  of $KO$-dimensions $2$ and $6 \\bmod 8$.\n\nLet $(\\hs{H},\\gamma,J)$ be a real $\\alg{A}$-bimodule of $KO$-dimension $n \\bmod 8$ for $n = 2$ or $6$. Since $\\{J,\\gamma\\}$, we have that\n\\[\n J =\n \\begin{pmatrix}\n  0 & \\varepsilon \\tilde{J}^*\\\\\n  \\tilde{J} & 0\n \\end{pmatrix},\n\\]\nwhere $\\tilde{J} := P^\\textup{odd} J P^\\textup{even}$ is an antiunitary $\\hs{H}^\\textup{even} \\to \\hs{H}^\\textup{odd}$, so that for $a \\in \\alg{A}$,\n\\[\n \\lambda^\\textup{odd}(a) = \\tilde{J} \\rho^\\textup{even}(a^*) \\tilde{J}^*, \\quad \\rho^\\textup{odd}(a) = \\tilde{J} \\lambda^\\textup{even}(a^*) \\tilde{J}^*.\n\\]\nIt then follows, in particular, that $\\mult[\\hs{H}^\\textup{odd}] = \\mult[\\hs{H}^\\textup{even}]^T$. \n\nNow, let $J^\\prime$ be another real structure on $(\\hs{H},\\gamma)$ of $KO$-dimension $n \\bmod 8$, and let $\\tilde{J^\\prime} = P^\\textup{odd} J^\\prime P^\\textup{even}$. Define $K \\in \\U^{\\textup{LR}}_\\alg{A}(\\hs{H},\\gamma)$ by $K = 1_{\\hs{H}^\\textup{even}} \\oplus (\\tilde{J^\\prime} \\tilde{J}^*)$. Then, by construction, $J^\\prime = KJK^*$, {\\it i.e.\\\/}\\  $K$ is a unitary equivalence of real structures between $J$ and $J^\\prime$. Thus, real structures of $KO$-dimension $2$ or $6 \\bmod 8$ are unique. As a result, we have proved the following analogue of Proposition~\\ref{s0reduction}:\n\n\\begin{proposition}~\\label{real26reduction}\nLet $\\alg{A}$ be a real {\\ensuremath{C^*}}-algebra, and let $n = 2$ or $6 \\bmod 8$. Then the map\n\\[\n C_n : \\Bimod(\\alg{A},n) \\to \\Bimod(\\alg{A})\n\\]\ngiven by $C_n([(\\hs{H},\\gamma,J)]) := ([\\hs{H}^\\textup{even}])$ is an isomorphism of monoids.\n\\end{proposition}\n\nAgain, as an immediate consequence, we obtain the following characterisation of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},\\gamma,J)$:\n\n\\begin{corollary}\\label{real26unitary}\nLet $(\\hs{H},\\gamma,J)$ be a real $\\alg{A}$-bimodule of $KO$-dimension $2$ or $6 \\bmod 8$. Then\n\\begin{equation}\\begin{split}\n \\U^{\\textup{LR}}_\\alg{A}(\\hs{H},\\gamma,J) &= \\{ U^\\textup{even} \\oplus U^\\textup{odd} \\in \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even}) \\oplus \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd}) \\mid U^\\textup{odd} = \\tilde{J}U^\\textup{even}\\tilde{J}^*\\}\\\\ &\\cong \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even}).\n\\end{split}\\end{equation}\n\\end{corollary}\n\nFinally, one can combine Proposition~\\ref{real26reduction} with our observation concerning the uniqueness up to unitary equivalence of real structures of $KO$-dimension $2$ or $6 \\bmod 8$ and earlier results on multiplicity matrices to obtain the following characterisation:\n\n\\begin{proposition}\n Let $n=2$ or $6 \\bmod 8$. Then the map $\\iota_n : \\Bimod(\\alg{A},n) \\to \\Bimod^\\textup{even}(\\alg{A})$ defined by $[(\\hs{H},\\gamma,J)] \\mapsto ([\\hs{H},\\gamma])$ is injective, and\n\\begin{equation}\\begin{split}\n (\\mult^\\textup{even} \\circ \\iota_n)(\\Bimod(\\alg{A},n)) &= \\{(m^\\textup{even},m^\\textup{odd}) \\in M_S(\\semiring{Z}_{\\geq 0})^2 \\mid m^\\textup{odd} = (m^\\textup{even})^T\\}\\\\ &\\cong M_S(\\semiring{Z}_{\\geq 0}).\n\\end{split}\\end{equation}\n\\end{proposition}\n\nOnce more, it follows that\n\\[\n \\Bimod_q(\\alg{A},n) := \\iota_n^{-1}(\\Bimod^\\textup{even}_q(\\alg{A})),\n\\]\nis the set of all equivalence classes of quasi-orientable real $\\alg{A}$-bimodules of $KO$-dimension $n \\bmod 8$, for which we can again obtain a characterisation in terms of signed multiplicity matrices:\n\n\\begin{corollary}\n Let $n=2$ or $6 \\bmod 8$. Then\n\\begin{equation}\n (\\mult_q \\circ \\iota_n)(\\Bimod_q(\\alg{A},n)) = \\{m \\in M_S(\\ring{Z}) \\mid m^T = -m\\}.\n\\end{equation}\n\\end{corollary}\n\n\\subsubsection{$S^0$-real bimodules of even $KO$-dimension}\n\nLet us now characterise quasi-orientability, orientability and Poincar{\\'e} duality for an even $KO$-dimensional $S^0$-real $\\alg{A}$-bimodule $(\\hs{H},\\gamma,J,\\epsilon)$ by means of suitable conditions on $(\\hs{H}_i,\\gamma_i)$ expressible entirely in terms of the pair of multiplicity matrices of $(\\hs{H}_i,\\gamma_i)$\n\nWe begin by considering quasi-orientability:\n\n\\begin{proposition}\\label{s0quasiorient}\n  Let $(\\hs{H},\\gamma,J,\\epsilon)$ be an $S^0$-real $\\alg{A}$-bimodule of even $KO$-dimension $n \\bmod 8$. Then $(\\hs{H},\\gamma)$ is quasi-orientable if and only if $(\\hs{H}_i,\\gamma_i)$ is quasi-orientable and\n\\[\n \\begin{cases}\n  \\supp(m^\\textup{even}_i) \\cap \\supp((m^\\textup{odd}_i)^T) = \\emptyset &\\text{if $n=0$, $4$,}\\\\\n  \\supp(m^\\textup{even}_i) \\cap \\supp((m^\\textup{even}_i)^T) = \\supp(m^\\textup{odd}_i) \\cap \\supp((m^\\textup{odd}_i)^T) = \\emptyset &\\text{if $n=2$, $6$,}\n \\end{cases}\n\\]\nfor $(m^\\textup{even}_i,m^\\textup{odd}_i)$ the multiplicity matrices of $(\\hs{H}_i,\\gamma_i)$, in which case, if $\\mu$ and $\\mu_i = m^\\textup{even}_i - m^\\textup{odd}_i$ are the signed multiplicity matrices of $(\\hs{H},\\gamma)$ and $(\\hs{H}_i,\\gamma_i)$, respectively, then\n\\begin{equation}\n \\mu = \\mu_i + \\varepsilon^{\\prime\\prime} \\mu_i^T.\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\n First, let $(m^\\textup{even},m^\\textup{odd})$ and $(m^\\textup{even}_i,m^\\textup{odd}_i)$ denote the pairs of multipicity matrices of $(\\hs{H},\\gamma)$ and $(\\hs{H}_i,\\gamma_i)$, respectively. It then follows that \n\\begin{align*}\n m^\\textup{even} &=\n \\begin{cases}\n  m^\\textup{even}_i + (m^\\textup{even}_i)^T &\\text{if $n = 0$, $4$,}\\\\\n  m^\\textup{even}_i + (m^\\textup{odd}_i)^T &\\text{if $n = 2$, $6$;}\n \\end{cases}\\\\\n m^\\textup{odd} &=\n \\begin{cases}\n  m^\\textup{odd}_i + (m^\\textup{odd}_i)^T &\\text{if $n = 0$, $4$,}\\\\\n  m^\\textup{odd}_i + (m^\\textup{even}_i)^T &\\text{if $n=2$, $6$.}\n \\end{cases}\n\\end{align*}\n\nThus $\\supp(m^\\textup{even}) = \\supp(m^\\textup{even}_i) \\cup S^\\textup{even}, \\quad \\supp(m^\\textup{odd}) = \\supp(m^\\textup{odd}_i) \\cup S^\\textup{odd}$, where\n\\[\n S^\\textup{even} =\n\\begin{cases}\n \\supp((m^\\textup{even}_i)^T) &\\text{if $n = 0$, $4$,}\\\\\n \\supp((m^\\textup{odd}_i)^T) &\\text{if $n = 2$, $6$;}\n\\end{cases} \\quad\n S^\\textup{odd} =\n\\begin{cases}\n \\supp((m^\\textup{odd}_i)^T) &\\text{if $n = 0$, $4$,}\\\\\n \\supp((m^\\textup{even}_i)^T) &\\text{if $n = 2$, $6$.}\n\\end{cases}\n\\]\n\nThen,\n\\begin{multline*}\n \\supp(m^\\textup{even}) \\cap \\supp(m^\\textup{odd}) = (\\supp(m^\\textup{even}_i) \\cap \\supp(m^\\textup{odd}_i)) \\cup (S^\\textup{even} \\cap \\supp(m^\\textup{odd}_i)) \\\\ \\cup (\\supp(m^\\textup{even}_i) \\cap S^\\textup{odd}) \\cup (S^\\textup{even} \\cap S^\\textup{odd}),\n\\end{multline*}\nso that $(\\hs{H},\\gamma)$ is quasi-orientable if and only if $(\\hs{H}_i,\\gamma_i)$ is quasi-orientable and\n\\[\n (S^\\textup{even} \\cap \\supp(m^\\textup{odd}_i)) \\cup (\\supp(m^\\textup{even}_i) \\cap S^\\textup{odd}) = \\emptyset,\n\\]\nas required. \n\nFinally, if $\\mu = m^\\textup{even} - m^\\textup{odd}$ and $\\mu_i = m^\\textup{even}_i - m^\\textup{odd}_i$ are the signed multiplicity matrices of $(\\hs{H},\\gamma)$ and $(\\hs{H}_i,\\gamma_i)$, respectively, then the relations amongst $m^\\textup{even}$, $m^\\textup{odd}$, $m^\\textup{even}_i$, and $m^\\textup{odd}_i$ given at the beginning immediately yield the equation $\\mu = \\mu_i + \\varepsilon^{\\prime\\prime} \\mu_i^T$.\n\\end{proof}\n\nLet us now turn to orientability:\n\n\\begin{proposition}\n Let $(\\hs{H},\\gamma,J,\\epsilon)$ be a quasi-orientable $S^0$-real $\\alg{A}$-bimodule of even $KO$-dimension $n \\bmod 8$. Then $(\\hs{H},\\gamma)$ is orientable if and only if $(\\hs{H}_i,\\gamma_i)$ is orientable and, if $n = 2$ or $6 \\bmod 8$, for all $j \\in \\{1,\\dotsc,N\\}$ such that $\\field{K}_j = \\field{C}$, \n\\begin{equation}\n (\\mu_i)_{\\rep{n}_j \\crep{n}_j} = (\\mu_i)_{\\crep{n}_j \\rep{n}_j},\n\\end{equation}\nwhere $\\mu_i$ is the signed multiplicity matrix of $(\\hs{H}_i,\\gamma_i)$.\n\\end{proposition}\n\n\\begin{proof}\n Let $\\mu$ be the signed multiplicity matrix of $(\\hs{H},\\gamma)$. Propositions~\\ref{s0orient} and~\\ref{converseorientable} together imply that $(\\hs{H},\\gamma,J,\\epsilon)$ is orientable if and only if\n\\[\n \\gamma_i = \\sum_{k,l=1}^N \\lambda_i(\\sgn(\\widehat{\\mu}_{kl})e_k)\\rho_i(e_l) = \\varepsilon^{\\prime\\prime} \\sum_{k,l=1}^N \\lambda_i(e_l)\\rho_i(\\sgn(\\widehat{\\mu}_{kl})e_k),\n\\]\nand by considering individual components $(\\gamma_i)_{\\alpha\\beta}$, one can easily check that this in turn holds if and only if $(\\hs{H}_i,\\gamma_i)$ is orientable and for all $k \\in \\{1,\\dotsc,N\\}$,\n\\[\n \\sgn(\\widehat{\\mu}_{kk}) = \\varepsilon^{\\prime\\prime} \\sgn(\\widehat{\\mu}_{kk}).\n\\]\n\nThis last condition is trivial when $\\varepsilon^{\\prime\\prime} = 1$, {\\it i.e.\\\/}\\  when $n = 0$ or $4 \\bmod 8$, so let us suppose instead that $n = 2$ or $6 \\bmod 8$, so that $\\varepsilon^{\\prime\\prime} = -1$. If $(\\hs{H},\\gamma)$ is orientable, then, by the above discussion, $(\\hs{H}_i,\\gamma_i)$ is orientable and the diagonal entries of $\\widehat{\\mu}$ vanish, which in turn implies by Proposition~\\ref{converseorientable} that for each $l \\in \\{1,\\dotsc,N\\}$ and all $\\alpha$, $\\beta \\in \\spec{M_{k_l}(\\field{K}_l)}$, $\\mu_{\\alpha\\beta} = 0$. By antisymmetry of $\\mu$, this is equivalent to having, for all $l \\in \\{1,\\dotsc,N\\}$ such that $\\field{K}_l = \\field{C}$, $\\mu_{\\rep{n}_l \\crep{n}_l} = 0$, or equivalently,\n\\[\n (\\mu_i)_{\\rep{n}_j \\crep{n}_j} = (\\mu_i)_{\\crep{n}_j \\rep{n}_j},\n\\]\nwhere $\\mu_i$ is the signed multiplicity matrix of $(\\hs{H}_i,\\gamma_i)$. On the other hand, if $(\\hs{H}_i,\\gamma_i)$ is orientable and this condition on $\\mu_i$ holds, then $\\mu$ certainly satisfies the above condition, so that $(\\hs{H},\\gamma)$ is indeed orientable.\n\\end{proof}\n\nFinally, let us consider Poincar{\\'e} duality.\n\n\\begin{proposition}\n Let $(\\hs{H},\\gamma,J,\\epsilon)$ be an $S^0$-real $\\alg{A}$-bimodule of even $KO$-dimension $n \\bmod 8$, let $(m^\\textup{even}_i,m^\\textup{odd}_i)$ denote the multiplicity matrices of $(\\hs{H}_i,\\gamma_i)$, and let $\\cap$ denote the matrix of the intersection form of $(\\hs{H},\\gamma)$. Finally, let $\\mu_i = m^\\textup{even}_i - m^\\textup{odd}_i$. Then\n\\begin{equation}\n \\cap_{kl} = \\tau_k \\tau_l (\\widehat{\\mu_i} + \\varepsilon^{\\prime\\prime} \\widehat{\\mu_i}^T)_{kl},\n\\end{equation}\nso that $(\\hs{H},\\gamma)$ satisfies Poincar{\\'e} duality if and only if $\\widehat{\\mu_i} + \\varepsilon^{\\prime\\prime} \\widehat{\\mu_i}^T$ is non-degenerate.\n\\end{proposition}\n\n\\begin{proof}\n By Proposition~\\ref{s0poincare}, $\\cap = \\cap_i + \\varepsilon^{\\prime\\prime} \\cap_i^T$ for $\\cap_i$ the matrix of the intersection form of $(\\hs{H}_i,\\gamma_i)$, which, together with Proposition~\\ref{intform}, yields the desired result.\n\\end{proof}\n\n\\subsection{Bimodules in the Chamseddine--Connes--Marcolli model}\n\nTo illustrate the structure theory outlined thus far, let us apply it to the construction of the finite spectral triple of the NCG Standard Model given by Chamseddine, Connes and Marcolli~\\cite{CCM07}*{\\S\\S 2.1, 2.2, 2.4} (cf.~also~\\cite{CM08}*{\\S 1.13}).\n\nLet $\\alg{A}_{LR} = \\field{C} \\oplus \\field{H}_L \\oplus \\field{H}_R \\oplus M_3(\\field{C})$, where the labels $L$ and $R$ serve to distinguish the two copies of $\\hs{H}$; we can therefore write $\\spec{\\alg{A}_{LR}} = \\{\\rep{1},\\crep{1},\\rep{2}_L,\\rep{2}_R,\\rep{3},\\crep{3}\\}$ without ambiguity. Now, let $(\\hs{M}_F,\\gamma_F,J_F)$ be the orientable real $\\alg{A}_{LR}$-bimodule of $KO$-dimension $6 \\bmod 8$ with signed multiplicity matrix\n\\[\n \\mu =\n\\begin{pmatrix}\n 0 & 0 & -1 & 1 & 0 & 0\\\\\n 0 & 0 & 0 & 0 & 0 & 0\\\\\n 1 & 0 & 0 & 0 & 1 & 0\\\\\n -1 & 0 & 0 & 0 & -1 & 0\\\\\n 0 & 0 & -1 & 1 & 0 & 0\\\\\n 0 & 0 & 0 & 0 & 0 & 0\n\\end{pmatrix}.\n\\]\nThis bimodule is, in fact, an $S^0$-real bimodule for $\\epsilon_F = \\lambda(-1,1,1,-1)$; $\\hs{E} = (\\hs{M}_F)_i$ is then the orientable even $\\alg{A}_{LR}$-bimodule with signed multiplicity matrix\n\\[\n \\mu_\\hs{E} = \n\\begin{pmatrix}\n 0 & 0 & 0 & 0 & 0 & 0\\\\\n 0 & 0 & 0 & 0 & 0 & 0\\\\\n 1 & 0 & 0 & 0 & 1 & 0\\\\\n -1 & 0 & 0 & 0 & -1 & 0\\\\\n 0 & 0 & 0 & 0 & 0 & 0\\\\\n 0 & 0 & 0 & 0 & 0 & 0\n\\end{pmatrix}.\n\\]\nNote, however, that neither $\\hs{M}_F$ nor $\\hs{E}$ satisfies Poincar{\\'e} duality, as\n\\[\n \\hat{\\mu} =\n\\begin{pmatrix}\n 0 & -1 & 1 & 0\\\\\n 1 & 0 & 0 & 1\\\\\n -1 & 0 & 0 & -1\\\\\n 0 & -1 & 1 & 0\n\\end{pmatrix}, \\quad\n \\widehat{\\mu_\\hs{E}} =\n\\begin{pmatrix}\n 0 & 0 & 0 & 0\\\\\n 1 & 0 & 0 & 1\\\\\n -1 & 0 & 0 & -1\\\\\n 0 & 0 & 0 & 0\n\\end{pmatrix}\n\\]\nare both clearly degenerate; the intersection forms of $\\hs{M}_F$ and $\\hs{E}$ are given by the matrices $\\cap = 2 \\hat{\\mu}$ and $\\cap_\\hs{E} = 2 \\widehat{\\mu_\\hs{E}}$, respectively. \n\nIn order to introduce $N$ generations of fermions and anti-fermions, one now considers the real $\\alg{A}$-bimodule $\\hs{H}_F := (\\hs{M}_F)^{\\oplus N}$; by abuse of notation, $\\gamma_F$, $J_F$ and $\\epsilon_F$ now also denote the relevant structure operators on $\\hs{H}_F$. In terms of multiplicity matrices and intersection forms, the sole difference from our discussion of $\\hs{M}_F$ is that all matrices are now multiplied by $N$.\n\nNow, let $\\alg{A}_F = \\field{C} \\oplus \\hs{H} \\oplus M_3(\\field{C})$, which we consider as a subalgebra of $\\alg{A}_{LR}$ by means of the embedding\n\\[\n (\\zeta,q,m) \\mapsto \\left(\\zeta,q,\\begin{pmatrix}\\lambda & 0\\\\0 & \\overline{\\lambda}\\end{pmatrix},m\\right);\n\\]\njust as we could for $\\alg{A}_{LR}$, we can write $\\spec{\\alg{A}_F} = \\{\\rep{1},\\crep{1},\\rep{2},\\rep{3},\\crep{3}\\}$ without ambiguity. We can therefore view $\\hs{H}_F$ as a real $\\alg{A}_F$-bimodule of $KO$-dimension $6 \\bmod 8$, whose pair of multiplicity matrices $(m^\\textup{even},m^\\textup{odd})$ is then given by\n\\[\n m^\\textup{even} = N\n\\begin{pmatrix}\n1 & 1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0 & 0\\\\\n1 & 0 & 0 & 1 & 0\\\\\n1 & 1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0 & 0\n\\end{pmatrix}, \\quad\nm^\\textup{odd} = N\n\\begin{pmatrix}\n1 & 0 & 1 & 1 & 0\\\\\n1 & 0 & 0 & 1 & 0\\\\\n0 & 0 & 0 & 0 & 0\\\\\n0 & 0 & 1 & 0 & 0\\\\\n0 & 0 & 0 & 0 & 0\n\\end{pmatrix};\n\\]\nthe essential observation is that the irreducible representation $\\rep{2}_R$ of $\\alg{A}_{LR}$ corresponds to the representation $\\rep{1}\\oplus\\crep{1}$ of $\\alg{A}_F$, whilst $\\rep{2}_L$, $\\rep{3}$ and $\\crep{3}$ correspond to $\\rep{2}$, $\\rep{3}$ and $\\crep{3}$, respectively. \n\nNote that $\\hs{H}_F$ now fails even to be quasi-orientable let alone orientable, with the sub-bimodule $(\\hs{H}_F)_{\\rep{1}\\rep{1}}$ providing the obstruction, and even if we were to restore quasi-orientability by setting $(\\hs{H}_F)_{\\rep{1}\\rep{1}} = 0$, $(\\hs{H}_F)_{\\rep{1}\\crep{1}}$ and $(\\hs{H}_F)_{\\crep{1}\\rep{1}}$ would still present an obstruction to orientability by Proposition~\\ref{converseorientable}. Note also that $\\hs{H}_F$ must necessarily fail to satisfy Poincar{\\'e} duality, as the matrix $\\cap_F$ of its intersection form is a $3 \\times 3$ anti-symmetric matrix, and thus \\latin{a priori} degenerate. Let us nonetheless compute $\\cap_F$:\n\\[\n \\widehat{m^\\textup{even}} - \\widehat{m^\\textup{odd}} = \n N \\begin{pmatrix}\n 2 & 0 & 0\\\\\n 1 & 0 & 1\\\\\n 2 & 0 & 0\n\\end{pmatrix} -\nN \\begin{pmatrix}\n 2 & 1 & 2\\\\\n 0 & 0 & 0\\\\\n 0 & 1 & 0   \n\\end{pmatrix} =\nN \\begin{pmatrix}\n 0 & -1 & -2\\\\\n 1 & 0 & 1\\\\\n 2 & -1 & 0\n\\end{pmatrix},\n\\]\nand hence, by Proposition~\\ref{intform},\n\\[\n \\cap_F =\n2N \\begin{pmatrix}\n 0 & -1 & -1\\\\\n 1 & 0 & 1\\\\\n 1 & -1 & 0\n\\end{pmatrix}.\n\\]\n\nFinally, let us consider the $S^0$-real structure on $\\hs{H}_F$ the $\\alg{A}_F$-bimodule, inherited from $\\hs{H}_F$ as an $\\alg{A}_{LR}$-bimodule; we now denote $(\\hs{H}_F)_i$ by $\\hs{H}_f$. One still has that $\\hs{H}_f = \\hs{E}^{\\oplus N}$, which is still orientable and thus specified by the signed multiplicity matrix\n\\[\n \\mu_{f} = N\n\\begin{pmatrix}\n -1 & 0 & 0 & -1 & 0\\\\\n -1 & 0 & 0 & -1 & 0\\\\\n 1 & 0 & 0 & 1 & 0\\\\\n 0 & 0 & 0 & 0 & 0\\\\\n 0 & 0 & 0 & 0 & 0\n\\end{pmatrix};\n\\]\nthe intersection form is then given by the matrix\n\\[\n \\cap_{f} = 2N\n\\begin{pmatrix}\n -1 & 0 & -1\\\\\n 1 & 0 & 1\\\\\n 0 & 0 & 0\n\\end{pmatrix},\n\\]\nso that $\\hs{H}_f$ fails to satisfy Poincar{\\'e} duality as an $\\alg{A}_F$-bimodule.\n\n\n\\section{Dirac Operators and their Structure}\n\n\\subsection{The order one condition}\n\nWe now examine the structure of Dirac operators in detail. We will find it useful to begin with the study of operators between $\\alg{A}$-bimodules (for fixed $\\alg{A}$) satisfying a further generalisation of the order one condition. Thus, let $\\alg{A}$ be a fixed real {\\ensuremath{C^*}}-algebra, and let $\\hs{H}_1$ and $\\hs{H}_2$ be fixed $\\alg{A}$-bimodules with multiplicity matrices $m_1$ and $m_2$, respectively. \n\n\\begin{definition}\n We shall say that a map $T \\in \\mathcal{L}(\\hs{H}_1,\\hs{H}_2)$ satisfies the \\term{generalised order one condition} if\n\\begin{equation}\\label{genorderone}\n  \\forall a, b \\in \\alg{A}, \\: (\\lambda_2(a)T - T\\lambda_1(a))\\rho_1(b) = \\rho_2(b)(\\lambda_2(a)T - T\\lambda_1(a)).\n\\end{equation}\n\\end{definition}\n\nNote that if $\\hs{H}_1 = \\hs{H}_2$, then the generalised order one condition reduces to the usual order one condition on Dirac operators.\n\nIt is easy to check that the generalised order one condition is, in fact, equivalent to the following alternative condition:\n\\begin{equation}\n \\forall a, b \\in \\alg{A}, \\: (\\rho_2(a)T - T\\rho_1(a))\\lambda_1(b) = \\lambda_2(b)(\\rho_2(a)T - T\\rho_1(a)).\n\\end{equation}\nThus, the following are equivalent for $T \\in \\mathcal{L}(\\hs{H}_1,\\hs{H}_2)$:\n\\begin{enumerate}\n \\item $T$ satisfies the generalised order one condition;\n \\item For all $a \\in \\alg{A}$, $\\lambda_2(a)T - T\\lambda_1(a)$ is right $\\alg{A}$-linear;\n \\item For all $a \\in \\alg{A}$, $\\rho_2(a)T - T\\rho_1(a)$ is left $\\alg{A}$-linear.\n\\end{enumerate}\n\nNow, since the unitary group $\\U(\\alg{A})$ of $\\alg{A}$ is a compact Lie group, let $\\mu$ be the normalised bi-invariant Haar measure on $\\U(\\alg{A})$. \n\n\\begin{lemma}~\\label{decompproj}\nLet $\\hs{H}_1$ and $\\hs{H}_2$ be $\\alg{A}$-bimodules. Define operators $E_\\lambda$ and $E_\\rho$ on $\\mathcal{L}^1_\\alg{A}(\\hs{H}_1,\\hs{H}_2)$ by\n\\begin{equation}\n E_\\lambda(T) := \\int_{\\U(\\alg{A})} \\mathop{}\\!\\mathrm{d}\\mu(u) \\lambda_2(u)T\\lambda_1(u^{-1}), \\quad E_\\rho(T) := \\int_{\\U(\\alg{A})} \\mathop{}\\!\\mathrm{d}\\mu(u) \\rho_2(u^{-1})T\\rho_1(u).\n\\end{equation}\nThen $E_\\lambda$ and $E_\\rho$ are commuting idempotents such that\n\\[\n \\im(E_\\lambda) = \\bdd^{\\textup{L}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2), \\quad \\im(E_\\rho) = \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2),\n\\]\nand\n\\[\n \\ker(E_\\lambda) = \\im(\\Id - E_\\lambda) \\subseteq \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2), \\quad \\ker(E_\\rho) = \\im(\\Id - E_\\rho) \\subseteq \\bdd^{\\textup{L}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2),\n\\]\nwhile\n\\[\n \\im(E_\\lambda E_\\rho) = \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2).\n\\]\n\\end{lemma}\n\n\\begin{proof}\nFirst, the fact that $E_\\lambda$ and $E_\\rho$ are idempotents follows immediately from the Fubini-Tonelli theorem together with translation invariance of the Haar measure $\\mu$, whilst commutation of $E_\\lambda$ and $E_\\rho$ follows from the Fubini-Tonelli theorem together with the commutation of left and right actions on $\\hs{H}_1$ and on $\\hs{H}_2$. Moreover, by construction, $E_\\lambda$ and $E_\\rho$ act as the identity on $\\bdd^{\\textup{L}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)$ and $\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)$, respectively, so that\n\\[\n \\im(E_\\lambda) \\supseteq \\bdd^{\\textup{L}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2), \\quad \\im(E_\\rho) \\supseteq \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2).\n\\]\n\nNow, let $T \\in \\mathcal{L}^1_\\alg{A}(\\hs{H}_1,\\hs{H}_2)$. Then, by translation invariance of the Haar measure, it follows that for any $u \\in \\U(\\alg{A})$,\n\\[\n E_\\lambda(T) = \\lambda_2(u) E_\\lambda(T) \\lambda_1(u)^*, \\quad E_\\rho(T) = \\rho_2(u) E_\\rho(T) \\rho_1(u)^*,\n\\]\nor equivalently,\n\\[\n \\lambda_2(u) E_\\lambda(T) = E_\\lambda(T) \\lambda_1(u), \\quad \\rho_2(u)E_\\rho(T) = E_\\rho(T) \\rho_1(u).\n\\]\nBy the real analogue of the Russo-Dye theorem~\\cite{Li}*{Lemma 2.15.16}, the convex hull of $\\U(\\alg{A})$ is weakly dense in the unit ball of $\\alg{A}$, so that\n\\[\n \\lambda_2(a) E_\\lambda(T) = E_\\lambda(T) \\lambda_1(a), \\quad \\rho_2(a)E_\\rho(T) = E_\\rho(T) \\rho_1(a)\n\\]\nfor all $a \\in \\alg{A}$, {\\it i.e.\\\/}\\  $E_\\lambda(T) \\in \\bdd^{\\textup{L}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)$ and $E_\\rho(T) \\in \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)$.\n\nOn the other hand,\n\\begin{align*}\n (\\Id - E_\\lambda)(T) &= \\int_{\\U(\\alg{A})} \\mathop{}\\!\\mathrm{d}\\mu(u) (T\\lambda_1(u) - \\lambda_2(u)T)\\lambda_1(u^{-1}),\\\\ (\\Id - E_\\rho)(T) &= \\int_{\\U(\\alg{A})} \\mathop{}\\!\\mathrm{d}\\mu(u) (T\\rho_1(u^{-1}) - \\rho_2(u^{-1})T)\\rho_1(u),\n\\end{align*}\nso that by the generalised order one condition, $(\\Id - E_\\lambda)(T) \\in \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)$ and $(\\Id - E_\\rho)(T) \\in \\bdd^{\\textup{L}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)$.\n\nFinally, the commutation of $E_\\lambda$ and $E_\\rho$ together with our identification of $\\im(E_\\lambda)$ and of $\\im(E_\\rho)$ imply the desired result about $\\im(E_\\lambda E_\\rho)$.\n\\end{proof}\n\nNow, since\n\\[\n \\im(\\Id - E_\\lambda) \\subseteq \\im(E_\\rho), \\quad \\im(\\Id - E_\\rho) \\subseteq \\im(E_\\lambda),\n\\]\none has that\n\\[\n (\\Id - E_\\lambda)E_\\rho = \\Id - E_\\lambda, \\quad (\\Id - E_\\rho)E_\\lambda = \\Id - E_\\rho,\n\\]\nwhich implies in turn that $\\Id - E_\\rho$, $E_\\lambda E_\\rho$ and $\\Id - E_\\lambda$ are mutually orthogonal idempotents such that\n\\[\n (\\Id - E_\\rho) + E_\\lambda E_\\rho + (\\Id - E_\\lambda) = \\Id.\n\\]\nWe have therefore proved the following:\n\n\\begin{proposition}[Krajewski~\\cite{Kraj98}*{\\S 3.4}]\\label{order1decomp}\nLet $\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)^0$ denote $\\ker(E_\\lambda)$, and let $\\bdd^{\\textup{L}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)^0$ denote $\\ker(E_\\rho)$. Then\n\\begin{equation}\n \\mathcal{L}^1_\\alg{A}(\\hs{H}_1,\\hs{H}_2) = \\bdd^{\\textup{L}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)^0 \\oplus \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2) \\oplus \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)^0,\n\\end{equation}\nwhere\n\\begin{equation}\n \\bdd^{\\textup{L}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)^0 \\oplus \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2) = \\bdd^{\\textup{L}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)\n\\end{equation}\nand\n\\begin{equation}\n \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2) \\oplus \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)^0 = \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2).\n\\end{equation}\n\\end{proposition}\n\nThus, elements of $\\bdd^{\\textup{L}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)^0$ can be interpreted as the ``purely'' left $\\alg{A}$-linear maps $\\hs{H}_1 \\to \\hs{H}_2$, whilst elements of $\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)^0$ can be interpreted as the ``purely'' right $\\alg{A}$-linear maps $\\hs{H}_1 \\to \\hs{H}_2$.\n\nOne can readily check that the decomposition of Proposition~\\ref{order1decomp} is respected by left multiplication by elements of $\\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}_2)$ and right multiplication by elements of $\\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}_1)$:\n\n\\begin{proposition}~\\label{lrorder1decomp}\nFor any $T \\in \\mathcal{L}^1_\\alg{A}(\\hs{H}_1,\\hs{H}_2)$, $A \\in \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}_1)$, $B \\in \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}_2)$,\n\\[\n E_\\lambda(A T) = A E_\\lambda(T), \\quad E_\\rho(T B) = E_\\rho(T) B.\n\\]\n\\end{proposition}\n\nNow, if $T \\in \\mathcal{L}(\\hs{H}_1,\\hs{H}_2)$, it is easy to see that $T$ satisfies the generalised order one condition if and only if each $T_{\\alpha\\beta}^{\\gamma\\delta}$ satisfies the generalised order one condition within $\\mathcal{L}((\\hs{H}_1)_{\\alpha\\beta},(\\hs{H}_2)_{\\gamma\\delta})$; by abuse of notation, we will also denote by $E_\\lambda$ and $E_\\rho$ the appropriate idempotents on each $\\mathcal{L}((\\hs{H}_1)_{\\alpha\\beta},(\\hs{H}_2)_{\\gamma\\delta})$. It then follows that\n\\[\n E_\\lambda(T)_{\\alpha\\beta}^{\\gamma\\delta} = E_\\lambda(T_{\\alpha\\beta}^{\\gamma\\delta}), \\quad E_\\rho(T)_{\\alpha\\beta}^{\\gamma\\delta} = E_\\rho(T_{\\alpha\\beta}^{\\gamma\\delta}).\n\\]\n\nFinally, let us turn to characterising $\\ker(E_\\lambda)$ and $\\ker(E_\\rho)$; before proceeding, we first need a technical lemma:\n\n\\begin{lemma}\\label{schur}\n Let $G$ be a compact Lie group, and let $\\mu$ be the bi-invariant Haar measure on $G$. Let $(\\hs{H},\\pi)$ and $(\\hs{H}^\\prime,\\pi^\\prime)$ be finite-dimensional irreducible unitary matrix representations of $G$. Then for any $T \\in \\mathcal{L}(\\hs{H}^\\prime,\\hs{H})$, if $\\pi \\ncong \\pi^\\prime$ then\n\\begin{equation}\n \\int_G \\mathop{}\\!\\mathrm{d}\\mu(g) \\pi(g)T\\pi^\\prime(g^{-1}) = 0,\n\\end{equation}\nand if $\\pi \\cong \\pi^\\prime$, then for any unitary $G$-isomorphism $U: \\hs{H}^\\prime \\to \\hs{H}$,\n\\begin{equation}\n \\int_G \\mathop{}\\!\\mathrm{d}\\mu(g) \\pi(g)T\\pi^\\prime(g^{-1}) = \\frac{1}{\\dim \\hs{H}} \\tr(T U^*)U.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n  Let\n\\[\n \\tilde{T} = \\int_G \\mathop{}\\!\\mathrm{d}\\mu(g) \\pi(g)T\\pi^\\prime(g^{-1}).\n\\]\nwhich, by translation invariance of the Haar measure $\\mu$, is a $G$-invariant map. If $\\pi \\ncong \\pi^\\prime$, then Schur's Lemma forces $\\tilde{T}$ to vanish. If instead $\\pi \\cong \\pi^\\prime$, let $U: \\hs{H} \\to \\hs{H}^\\prime$ be a unitary $G$-isomorphism. Then by Schur's Lemma there exists some $\\alpha \\in \\field{C}$ such that $\\tilde{T} = \\alpha U$; in fact,\n\\[\n \\alpha = \\alpha \\frac{1}{\\dim \\hs{H}}\\tr(U U^*) = \\frac{1}{\\dim \\hs{H}} \\tr(\\tilde{T} U^*).\n\\]\nOne can then show that $\\tr(\\tilde{T} U^*) = \\tr(T U^*)$ by introducing an orthonormal basis of $\\hs{H}$ and then calculating directly.\n\\end{proof}\n\nWe now arrive at the desired characterisation:\n\n\\begin{proposition}~\\label{zeromean}\nIf $T \\in \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)$, then $E_\\lambda(T) = 0$ if and only if for all $\\alpha$, $\\beta \\in \\supp(m_1) \\cap \\supp(m_2)$,\n\\[\n T_{\\alpha\\beta}^{\\alpha\\beta} \\in \\mathfrak{sl}(n_\\alpha) \\otimes M_{(m_2)_{\\alpha\\beta} \\times (m_1)_{\\alpha\\beta}}(\\field{C}) \\otimes 1_{n_\\beta},\n\\]\nand if $T \\in \\bdd^{\\textup{L}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)$, then $E_\\rho(T) = 0$ if and only if for all $\\alpha$, $\\beta \\in \\supp(m_1) \\cap \\supp(m_2)$,\n\\[\n T_{\\alpha\\beta}^{\\alpha\\beta} \\in 1_{n_\\alpha} \\otimes M_{(m_2)_{\\alpha\\beta} \\times (m_1)_{\\alpha\\beta}}(\\field{C}) \\otimes \\mathfrak{sl}(n_\\beta).\n\\]\n\\end{proposition}\n\n\\begin{proof}\nLet $T \\in \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}_1,\\hs{H}_2)$. Then, by Proposition~\\ref{linear}, it suffices to consider components $T_{\\alpha\\beta}^{\\gamma\\beta}$ for $\\alpha$, $\\beta$, $\\gamma \\in \\spec{\\alg{A}}$, which take the form\n\\[\n T_{\\alpha\\beta}^{\\gamma\\beta} = M_{\\alpha\\beta}^{\\gamma} \\otimes 1_{n_\\beta}\n\\]\nfor $M_{\\alpha\\beta}^\\gamma \\in M_{n_\\gamma \\times n_\\alpha}(\\field{C}) \\otimes M_{(m_2)_{\\gamma\\beta} \\times (m_1)_{\\alpha\\beta}}(\\field{C})$.\n\nNow fix $\\alpha$, $\\beta$, $\\gamma \\in \\spec{\\alg{A}}$, and write\n\\[\n M_{\\alpha\\beta}^\\gamma = \\sum_{i=1}^k A_i \\otimes B_i\n\\]\nfor $A_i \\in M_{n_\\gamma \\times n_\\alpha}(\\field{C})$ and for $B_i \\in M_{(m_2)_{\\gamma\\beta} \\times (m_1)_{\\alpha\\beta}}(\\field{C})$ linearly independent. It then follows by direct computation together with Lemma~\\ref{schur} that\n\\[\n E_\\lambda(T_{\\alpha\\beta}^{\\gamma\\beta}) = \n \\begin{cases}\n  \\frac{1}{n_\\alpha} \\left(\\sum_{i=1}^k \\tr(A_i) 1_{n_\\alpha} \\otimes B_i \\right) \\otimes 1_{n_\\beta} &\\text{if $\\alpha = \\gamma$,}\\\\\n  0 &\\text{otherwise},\n \\end{cases} \n\\]\nso that by linear independence of the $B_i$,  $E_\\lambda(T_{\\alpha\\beta}^{\\gamma\\beta})$ vanishes if and only if either $\\alpha \\neq \\gamma$ or, $\\alpha = \\beta$ and each $A_i$ is traceless, and hence, if and only $\\alpha \\neq \\gamma$ or, $\\alpha = \\beta$ and $M_{\\alpha\\beta}^\\alpha \\in \\mathfrak{sl}(n_\\alpha) \\otimes M_{(m_2)_{\\gamma\\beta} \\times (m_1)_{\\alpha\\beta}}(\\field{C})$, as required.\n\n\\latin{Mutatis mutandis}, this argument also establishes the desired characterisation of $\\ker(E_\\rho)$.\n\\end{proof}\n\n\\subsection{Odd bilateral spectral triples}\n\nLet us now take $\\hs{H}_1 = \\hs{H}_2 = \\hs{H}$. By construction of $E_\\lambda$ and $E_\\rho$, the following conditions are readily seen to be equivalent for $T \\in \\mathcal{L}_\\alg{A}^1(\\hs{H})$:\n\\begin{enumerate}\n \\item $T$ is self-adjoint;\n \\item $E_\\lambda(T)$ and $(\\Id - E_\\lambda)(T)$ are self-adjoint;\n \\item $(\\Id - E_\\rho)(T)$ and $E_\\rho(T)$ are self-adjoint;\n \\item $(\\Id - E_\\rho)(T)$, $(E_\\lambda E_\\rho)(T)$ and $(\\Id - E_\\lambda)(T)$ are self-adjoint.\n\\end{enumerate}\nThus, in particular,\n\\begin{equation}\n \\ms{D}_0(\\alg{A},\\hs{H}) = \\bdd^{\\textup{L}}_\\alg{A}(\\hs{H})^0_\\textup{sa} \\oplus \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H})_\\textup{sa} \\oplus \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H})^0_\\textup{sa}.\n\\end{equation}\nIn light of Proposition~\\ref{lrorder1decomp}, we therefore have the following description of $\\ms{D}(\\alg{A},\\hs{H})$:\n\n\\begin{proposition}\n Let $\\hs{H}$ be an $\\alg{A}$-bimodule. Then\n\\begin{equation}\n \\ms{D}(\\alg{A},\\hs{H}) = \\left( \\bdd^{\\textup{L}}_\\alg{A}(\\hs{H})^0_\\textup{sa} \\times \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H})_\\textup{sa} \\times \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H})^0_\\textup{sa} \\right) \/ \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}),\n\\end{equation}\nwhere $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H})$ acts diagonally by conjugation. \n\\end{proposition}\n\nNow, in light of Propositions~\\ref{linear},~\\ref{order1decomp} and~\\ref{zeromean}, we can describe how to construct an arbitrary Dirac operator on an odd $\\alg{A}$-bimodule $\\hs{H}$ with multiplicity matrix $m$:\n\\begin{enumerate}\n \\item For $\\alpha$, $\\beta$, $\\gamma \\in \\spec{\\alg{A}}$ such that $\\alpha < \\gamma$, choose $M_{\\alpha\\beta}^\\gamma \\in M_{n_\\gamma m_{\\gamma\\beta} \\times n_\\alpha m_{\\alpha\\beta}}(\\field{C})$;\n \\item For $\\alpha$, $\\beta$, $\\delta \\in \\spec{\\alg{A}}$ such that $\\beta < \\delta$, choose $N_{\\alpha\\beta}^\\delta \\in M_{m_{\\alpha\\delta} n_\\delta \\times m_{\\alpha\\beta} n_\\beta}(\\field{C})$;\n \\item For $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$, choose $M_{\\alpha\\beta}^\\alpha \\in M_{n_\\alpha m_{\\alpha\\beta}}(\\field{C})_\\textup{sa}$ and $N_{\\alpha\\beta}^\\beta \\in M_{m_{\\alpha\\beta} n_\\beta}(\\field{C})_\\textup{sa}$;\n \\item Finally, for $\\alpha$, $\\beta$, $\\gamma$, $\\delta \\in \\spec{\\alg{A}}$, set\n \\begin{equation}\n  D_{\\alpha\\beta}^{\\gamma\\delta} = \n  \\begin{cases}\n   M_{\\alpha\\beta}^\\gamma \\otimes 1_{n_\\beta} &\\text{if $\\alpha < \\gamma$ and $\\beta = \\delta$,}\\\\\n   (M_{\\gamma\\beta}^\\alpha)^* \\otimes 1_{n_\\beta} &\\text{if $\\alpha > \\gamma$ and $\\beta = \\delta$,}\\\\\n   1_{n_\\alpha} \\otimes N_{\\alpha\\beta}^\\delta &\\text{if $\\alpha = \\gamma$ and $\\beta < \\delta$,}\\\\\n   1_{n_\\alpha} \\otimes (N_{\\alpha\\delta}^\\beta)^* &\\text{if $\\alpha = \\gamma$ and $\\beta > \\delta$,}\\\\\n   M_{\\alpha\\beta}^\\alpha \\otimes 1_{n_\\beta} + 1_{n_\\alpha} \\otimes N_{\\alpha\\beta}^\\beta &\\text{if $(\\alpha,\\beta) = (\\gamma,\\delta)$,}\\\\\n   0 &\\text{otherwise}.\n  \\end{cases}\n \\end{equation}\n\\end{enumerate}\n\nNote that for any $K = (1_{n_\\alpha} \\otimes K_{\\alpha\\beta} \\otimes 1_{n_\\beta})_{\\alpha,\\beta\\in\\spec{\\alg{A}}} \\in \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H})_\\textup{sa}$ (so that each $K_{\\alpha\\beta}$ is self-adjoint), we can make the replacements\n\\[\n M_{\\alpha\\beta}^\\alpha \\mapsto M_{\\alpha\\beta}^\\alpha + 1_{n_\\alpha} \\otimes K_{\\alpha\\beta}, \\quad N_{\\alpha\\beta}^\\beta \\mapsto N_{\\alpha\\beta}^\\beta - K_{\\alpha\\beta} \\otimes 1_{n_\\beta},\n\\]\nand still obtain the same Dirac operator $D$; by Proposition~\\ref{order1decomp}, this freedom is removed by requiring either that $M_{\\alpha\\beta}^\\alpha \\in \\mathfrak{sl}(n_\\alpha) \\otimes M_{m_{\\alpha\\beta}}(\\field{C})$ or that $N_{\\alpha\\beta}^\\beta \\in M_{m_{\\alpha\\beta}}(\\field{C}) \\otimes \\mathfrak{sl}(n_\\beta)$.\n\nWe now turn to the moduli space $\\ms{D}(\\alg{A},\\hs{H})$ itself. By the above discussion and Corollary~\\ref{oddunitary}, we can identify the space $\\ms{D}_0(\\alg{A},\\hs{H})$ with\n\\begin{multline}\n \\ms{D}_0(\\alg{A},m) := \\prod_{\\alpha,\\beta \\in \\spec{\\alg{A}}} \\prod_{\\substack{\\gamma \\in \\spec{\\alg{A}} \\\\ \\gamma > \\alpha}} M_{n_\\gamma m_{\\gamma\\beta} \\times n_\\alpha m_{\\alpha\\beta}}(\\field{C}) \\times \\left(\\mathfrak{sl}(n_\\alpha) \\otimes M_{m_{\\alpha\\beta}}(\\field{C})\\right)_\\textup{sa} \\\\ \\times \\prod_{\\substack{\\delta \\in \\spec{\\alg{A}} \\\\ \\delta \\geq \\alpha}} M_{m_{\\alpha\\delta} n_\\delta \\times m_{\\alpha\\beta} n_\\beta}(\\field{C}) \\times M_{m_{\\alpha\\beta} n_\\beta}(\\field{C})_\\textup{sa},\n\\end{multline}\nand identify $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H})$ with\n\\begin{equation}\\label{modunit}\n \\U(\\alg{A},m) := \\prod_{\\alpha,\\beta \\in \\spec{\\alg{A}}} \\U(m_{\\alpha\\beta}).\n\\end{equation}\nBy checking at the level of components, one sees that the action of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H})$ on the space $\\ms{D}_0(\\alg{A},\\hs{H})$ corresponds under these identifications to the action of $\\U(\\alg{A},m)$ on $\\ms{D}_0(\\alg{A},m)$ defined by having $(U_{\\alpha\\beta}) \\in \\U(\\alg{A},m)$ act on\n\\[\n (M_{\\alpha\\beta}^\\gamma;M_{\\alpha\\beta}^\\alpha;N_{\\alpha\\beta}^\\delta;N_{\\alpha\\beta}^\\beta) \\in \\ms{D}_0(\\alg{A},m)\n\\]\nby\n\\[\n M_{\\alpha\\beta}^\\gamma \\mapsto (1_{n_\\gamma} \\otimes U_{\\gamma\\beta}) M_{\\alpha\\beta}^\\gamma (1_{n_\\alpha\\beta} \\otimes U_{\\alpha\\beta}^*), \\quad N_{\\alpha\\beta}^\\delta \\mapsto (U_{\\alpha\\delta} \\otimes 1_{n_\\delta}) N_{\\alpha\\beta}^\\delta (U_{\\alpha\\beta}^* \\otimes 1_{n_\\beta}).\n\\]\nWe have therefore proved the following:\n\n\\begin{proposition}\n Let $\\hs{H}$ be an odd $\\alg{A}$-bimodule with multiplicity matrix $m$. Then\n \\begin{equation}\n  \\ms{D}(\\alg{A},\\hs{H}) \\cong \\ms{D}_0(\\alg{A},m) \/ \\U(\\alg{A},m).\n \\end{equation}\n\\end{proposition}\n\n\\subsection{Even bilateral spectral triples}\\label{evenbilateral}\n\nFor this section, let $(\\hs{H},\\gamma)$ be a fixed even $\\alg{A}$-bimodule with pair of multiplicity matrices $(m^\\textup{even},m^\\textup{odd})$.\n\nNow, let $D$ be a self-adjoint operator on $\\hs{H}$ anticommuting with $\\gamma$. Then, with respect to the decomposition $\\hs{H} = \\hs{H}^\\textup{even} \\oplus \\hs{H}^\\textup{odd}$ we can write\n\\[\n D =\n \\begin{pmatrix}\n 0 & \\Delta^*\\\\\n \\Delta & 0\\\\\n \\end{pmatrix},\n\\]\nwhere $\\Delta = P^\\textup{odd} D P^\\textup{even}$, viewed as a map $\\hs{H}^\\textup{even} \\to \\hs{H}^\\textup{odd}$. Thus, $D$ is uniquely determined by $\\Delta$ and \\latin{vice versa}. Moreover, one can  check that $D$ satisfies the order one condition if and only if $\\Delta$ satisfies the generalised order one condition as a map $\\hs{H}^\\textup{even} \\to \\hs{H}^\\textup{odd}$. We therefore have the following:\n\n\\begin{lemma}\nLet $(\\hs{H},\\gamma)$ be an even $\\alg{A}$-bimodule. Then the map $\\ms{D}_0(\\alg{A},\\hs{H},\\gamma) \\to \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$ defined by $D \\mapsto P^\\textup{odd} D P^\\textup{even}$ is an isomorphism.\n\\end{lemma}\n\nWe now apply this Lemma to obtain our first result regarding the form of $\\ms{D}(\\alg{A},\\hs{H},\\gamma)$:\n\n\\begin{proposition}\\label{evendirac}\nThe map \n\\[\n \\ms{D}(\\alg{A},\\hs{H},\\gamma) \\to \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd}) \\backslash \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd}) \/ \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even})\n\\] \ndefined by $[D] \\mapsto [P^\\textup{odd} D P^\\textup{even}]$ is a homeomorphism.\n\\end{proposition}\n\n\\begin{proof}\nRecall that $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},\\gamma) = \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$. We therefore have for $D \\in \\ms{D}_0(\\alg{A},\\hs{H},\\gamma)$ and $U = U^\\textup{even} \\oplus U^\\textup{odd} \\in \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$ that\n\\[\n  P^\\textup{odd} U D U^* P^\\textup{even} = U^\\textup{odd} P^\\textup{odd} D P^\\textup{even} (U^\\textup{even})^*.\n\\]\nThus, under the correspondence $\\ms{D}_0(\\alg{A},\\hs{H},\\gamma) \\cong \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$, the action of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},\\gamma)$ decouples into an action of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd})$ by multiplication on the left and an action of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even})$ by multiplication by the inverse on the right. Thus, the map $[D] \\to [P^\\textup{odd} D P^\\textup{even}]$ is not only well-defined but manifestly homeomorphic.\n\\end{proof}\n\nCombining this last Proposition with Proposition~\\ref{order1decomp}, we immediately obtain the following:\n\n\\begin{corollary}\n Let $(\\hs{H},\\gamma)$ be an even $\\alg{A}$-bimodule. Then\n \\begin{multline}\n  \\ms{D}(\\alg{A},\\hs{H},\\gamma) \\cong \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd}) \\backslash ( \\bdd^{\\textup{L}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})^0 \\\\ \\times \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd}) \\times \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})^0 ) \/ \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even}),\n \\end{multline}\n where $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd})$ acts diagonally by multiplication on the left, and $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even})$ acts diagonally by multiplication on the right by the inverse.\n\\end{corollary}\n\nNow, just as we did in the odd case, let us describe the construction of an arbitary Dirac operator $D$ on $(\\hs{H},\\gamma)$:\n\\begin{enumerate}\n \\item For $\\alpha$, $\\beta$, $\\gamma \\in \\spec{\\alg{A}}$, choose $M_{\\alpha\\beta}^\\gamma \\in M_{n_\\gamma m^\\textup{odd}_{\\gamma\\beta} \\times n_\\alpha m^\\textup{even}_{\\alpha\\beta}}(\\field{C})$;\n \\item For $\\alpha$, $\\beta$, $\\delta \\in \\spec{\\alg{A}}$, choose $N_{\\alpha\\beta}^\\delta \\in M_{m^\\textup{odd}_{\\alpha\\delta} n_\\delta \\times m^\\textup{even}_{\\alpha\\beta} n_\\beta}(\\field{C})$;\n \\item Construct $\\Delta \\in \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$ by setting, for $\\alpha$, $\\beta$, $\\gamma$, $\\delta \\in \\spec{\\alg{A}}$,\n \\begin{equation}\n  \\Delta_{\\alpha\\beta}^{\\gamma\\delta} =\n  \\begin{cases}\n   M_{\\alpha\\beta}^\\gamma \\otimes 1_{n_\\beta} &\\text{if $\\alpha \\neq \\gamma$ and $\\beta = \\delta$,}\\\\\n   1_{n_\\alpha} \\otimes N_{\\alpha\\beta}^\\delta &\\text{if $\\alpha = \\gamma$ and $\\beta \\neq \\delta$,}\\\\\n   M_{\\alpha\\beta}^\\alpha \\otimes 1_{n_\\beta} + 1_{n_\\alpha} \\otimes N_{\\alpha\\beta}^\\beta &\\text{if $(\\alpha,\\beta) = (\\gamma,\\delta)$,}\\\\\n   0 &\\text{otherwise;}\\\\\n  \\end{cases}\n \\end{equation}\n \\item Finally, set $D = \\bigl( \\begin{smallmatrix} 0 & \\Delta^* \\\\ \\Delta & 0 \\end{smallmatrix} \\bigr)$.\n\\end{enumerate}\n\nAgain, note that for any $K = (1_{n_\\alpha} \\otimes K_{\\alpha\\beta} \\otimes 1_{n_\\beta})_{\\alpha,\\beta\\in\\spec{\\alg{A}}} \\in \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$ , we can make the replacements\n\\[\n M_{\\alpha\\beta}^\\alpha \\mapsto M_{\\alpha\\beta}^\\alpha + 1_{n_\\alpha} \\otimes K_{\\alpha\\beta}, \\quad N_{\\alpha\\beta}^\\beta \\mapsto N_{\\alpha\\beta}^\\beta - K_{\\alpha\\beta} \\otimes 1_{n_\\beta},\n\\]\nand still obtain the same Dirac operator $D$; by Proposition~\\ref{order1decomp}, this freedom is removed by requiring either that\n\\[\n M_{\\alpha\\beta}^\\alpha \\in \\mathfrak{sl}(n_\\alpha) \\otimes M_{m^\\textup{odd}_{\\alpha\\beta} \\times m^\\textup{even}_{\\alpha\\beta}}(\\field{C})\n\\]\nor that\n\\[\nN_{\\alpha\\beta}^\\beta \\in M_{m^\\textup{odd}_{\\alpha\\beta} \\times m^\\textup{even}_{\\alpha\\beta}}(\\field{C}) \\otimes \\mathfrak{sl}(n_\\beta).\n\\]\n\nJust as in the odd case, the above discussion and Corollary~\\ref{oddunitary} imply that we can identify $\\ms{D}_0(\\alg{A},\\hs{H},\\gamma)$ with\n\\begin{multline}\n \\ms{D}_0(\\alg{A},m^\\textup{even},m^\\textup{odd}) := \\prod_{\\alpha,\\beta \\in \\spec{\\alg{A}}} \\prod_{\\substack{\\gamma \\in \\spec{\\alg{A}} \\\\ \\gamma \\neq \\alpha}} M_{n_\\gamma m^\\textup{odd}_{\\gamma\\beta} \\times n_\\alpha m^\\textup{even}_{\\alpha\\beta}}(\\field{C}) \\\\ \\times \\left(\\mathfrak{sl}(n_\\alpha) \\otimes M_{m^\\textup{odd}_{\\alpha\\beta} \\times  m^\\textup{even}_{\\alpha\\beta}}(\\field{C})\\right) \\times \\prod_{\\delta \\in \\spec{\\alg{A}}} M_{m^\\textup{odd}_{\\alpha\\delta} n_\\delta \\times m^\\textup{even}_{\\alpha\\beta} n_\\beta}(\\field{C}),\n\\end{multline}\nand identify $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even})$ and $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd})$ with $\\U(\\alg{A},m^\\textup{even})$ and $\\U(\\alg{A},m^\\textup{odd})$, respectively, which are defined according to Equation~\\ref{modunit}. The actions of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even})$ and $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd})$ on $\\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$ therefore correspond under these identifications to the actions of $\\U(\\alg{A},m^\\textup{even})$ and $\\U(\\alg{A},m^\\textup{odd})$, respectively, on $\\ms{D}_0(\\alg{A},m^\\textup{even},m^\\textup{odd})$ defined by having $(U_{\\alpha\\beta}^\\textup{odd}) \\in \\U(\\alg{A},m^\\textup{odd})$ and $(U_{\\alpha\\beta}^\\textup{even}) \\in \\U(\\alg{A},m^\\textup{even})$ act on\n\\[\n (M_{\\alpha\\beta}^\\gamma;M_{\\alpha\\beta}^\\alpha;N_{\\alpha\\beta}^\\delta) \\in \\ms{D}_0(\\alg{A},m^\\textup{even},m^\\textup{odd})\n\\]\nby\n\\[\n M_{\\alpha\\beta}^\\gamma \\mapsto (1_{n_\\gamma} \\otimes U_{\\gamma\\beta}^\\textup{odd})M_{\\alpha\\beta}^\\gamma, \\quad N_{\\alpha\\beta}^\\delta \\mapsto (U_{\\alpha\\delta}^\\textup{odd} \\otimes 1_{n_\\delta}) N_{\\alpha\\beta}^\\delta,\n\\]\nand\n\\[\n M_{\\alpha\\beta}^\\gamma \\mapsto M_{\\alpha\\beta}^\\gamma (1_{n_\\alpha} \\otimes (U_{\\alpha\\beta}^\\textup{even})^*), \\quad N_{\\alpha\\beta}^\\delta \\mapsto  N_{\\alpha\\beta}^\\delta ((U_{\\alpha\\beta}^\\textup{even})^* \\otimes 1_{n_\\beta}),\n\\]\nrespectively. Thus we have proved the following:\n\n\\begin{proposition}\nLet $(\\hs{H},\\gamma)$ be an even $\\alg{A}$-bimodule with multiplicity matrices $(m^\\textup{even},m^\\textup{odd})$. Then\n\\begin{equation}\n \\ms{D}(\\alg{A},\\hs{H},\\gamma) \\cong \\U(\\alg{A},m^\\textup{odd}) \\backslash \\ms{D}_0(\\alg{A},m^\\textup{even},m^\\textup{odd}) \/ \\U(\\alg{A},m^\\textup{even}).\n\\end{equation}\n\\end{proposition}\n\nIn the quasi-orientable case, the picture simplifies considerably, as all components $\\Delta_{\\alpha\\beta}^{\\alpha\\beta}$ necessarily vanish. One is then left, essentially, with the situation described by Krajewski~\\cite{Kraj98}*{\\S 3.4} and Paschke--Sitarz~\\cite{PS98}*{\\S 2.II} ; as mentioned before, one can find in the former the original definition of what are now called \\term{Krajewski diagrams}. These diagrams, used extensively by Iochum, Jureit, Sch{\\\"u}cker and Stephan~\\cites{ACG1,ACG2,ACG3,ACG4,ACG5,Sch05}, offer a concise, diagrammatic approach to the study of quasi-orientable even bilateral spectral triples that strongly emphasizes the underlying combinatorics. Though they do admit ready generalisation to the non-quasi-orientable case, we will not discuss them here.\n\nWe conclude our discussion of even bilateral spectral triples by recalling a result of Paschke and Sitarz of particular interest in relation to the NCG Standard Model.\n\n\\begin{proposition}[Paschke--Sitarz~\\cite{PS98}*{Lemma 7}]\nLet $(\\hs{H},\\gamma)$ be an orientable $\\alg{A}$-bimodule. Then for all $D \\in \\ms{D}_0(\\alg{A},\\hs{H},\\gamma)$,\n\\begin{equation}\n D = \\sum_{\\substack{i,j = 1\\\\ i \\neq j}}^N \\lambda(e_i)[D,\\lambda(e_j)] + \\sum_{\\substack{k,l = 1\\\\ k \\neq l}}^N \\rho(e_k)[D,\\rho(e_l)].\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\n Fix $D \\in \\ms{D}_0(\\alg{A},\\hs{H},\\gamma)$, and let\n\\begin{align*}\n T &:= D - \\sum_{\\substack{i,j = 1\\\\ i \\neq j}}^N \\lambda(e_i)[D,\\lambda(e_j)] - \\sum_{\\substack{k,l = 1\\\\ k \\neq l}}^N \\rho(e_k)[D,\\rho(e_l)]\\\\\n &= D - \\sum_{\\substack{i,j = 1\\\\ i \\neq j}}^N \\lambda(e_i)D\\lambda(e_j) - \\sum_{\\substack{k,l = 1\\\\ k \\neq l}}^N \\rho(e_k)D\\rho(e_l).\n\\end{align*}\nThen for all $\\alpha$, $\\beta$, $\\gamma$, $\\delta \\in \\spec{\\alg{A}}$,\n\\[\n T_{\\alpha\\beta}^{\\gamma\\delta} = \n\\begin{cases}\n D_{\\alpha\\beta}^{\\gamma\\delta} &\\text{if $r(\\alpha) = r(\\gamma)$, $r(\\beta) = r(\\delta)$,}\\\\\n -D_{\\alpha\\beta}^{\\gamma\\delta} &\\text{if $r(\\alpha) \\neq r(\\gamma)$, $r(\\beta) \\neq r(\\delta)$,}\\\\\n 0 &\\text{otherwise,}\n\\end{cases}\n\\]\nwhere for $\\alpha \\in \\spec{\\alg{A}}$, $r(\\alpha)$ is the value of $j \\in \\{1,\\dotsc,N\\}$ such that $\\alpha \\in \\spec{M_{k_j}(\\field{K}_j)}$. However, by Proposition~\\ref{order1decomp}, $D_{\\alpha\\beta}^{\\gamma\\delta}$ must vanish in the second case, whilst by Proposition~\\ref{converseorientable}, $D_{\\alpha\\beta}^{\\gamma\\delta}$ must vanish in the first, so that $T = 0$.\n\\end{proof}\n\nNow, let $(\\alg{A},\\hs{H},D,J,\\gamma)$ be a real spectral triple of even $KO$-dimension. A \\term{gauge potential} for the triple is then a self-adjoint operator on $\\hs{H}$ of the form\n\\[\n \\sum_{k=1}^n \\lambda(a_k) [D,\\lambda(b_k)],\n\\]\nwhere $a_1,\\dotsc,a_n$, $b_1,\\dotsc,b_n \\in \\alg{A}$, and an \\term{inner fluctuation of the metric} is a Dirac operator $D_A \\in \\ms{D}_0(\\alg{A},\\hs{H},J,\\gamma)$ of the form\n\\[\n D_A := D + A + \\varepsilon^\\prime J A J^* = D + A + J A J^*,\n\\]\nwhere $A$ is a gauge potential. One then has that for any gauge potential $A$, $(\\alg{A},\\hs{H},D,J,\\gamma)$ and $(\\alg{A},\\hs{H},D_A,J,\\gamma)$ are Morita equivalent. In this light, the last Proposition admits the following interpretation:\n\n\\begin{corollary}\\label{orientmorita}\n Let $(\\hs{H},J,\\gamma)$ be an orientable real $\\alg{A}$-bimodule of even $KO$-dimen\\-sion. Then for all $D \\in \\ms{D}_0(\\alg{A},\\hs{H},\\gamma,J)$, \n\\begin{equation}\n A = - \\sum_{\\substack{i,j = 1\\\\ i \\neq j}}^N \\lambda(e_i)[D,\\lambda(e_j)]\n\\end{equation}\nis a gauge potential for the real spectral triple $(\\alg{A},\\hs{H},D,J,\\gamma)$ such that $D_A = 0$.\n\\end{corollary}\n\nThus, every finite orientable real spectral triple $(\\alg{A},\\hs{H},D,J,\\gamma)$ of even $KO$-dimension is Morita equivalent to the dynamically trivial triple $(\\alg{A},\\hs{H},0,J,\\gamma)$.\n\n\\subsection{Real spectral triples of odd $KO$-dimension}\n\nFor this section, let $(\\hs{H},J)$ be a real $\\alg{A}$-bimodule of odd $KO$-dimension $n \\bmod 8$ with multiplicity matrix $m$. We begin by reducing the study of Dirac operators on $(\\hs{H},J)$ to that of self-adjoint right $\\alg{A}$-linear operators on $\\hs{H}$.\n\n\\begin{proposition}[Krajewski~\\cite{Kraj98}*{\\S 3.4}]\\label{oddrealdirac}\nLet $(\\hs{H},J)$ be a real $\\alg{A}$-bimodule of odd $KO$-dimension $n \\bmod 8$. Then the map $R_n : \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H})_\\textup{sa} \\to \\ms{D}_0(\\alg{A},\\hs{H},J)$ defined by $R_n(M) := M + \\varepsilon^\\prime J M J^*$ is a surjection interwining the action of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},J)$ on $\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H})_\\textup{sa}$ by conjugation with the action on $\\ms{D}_0(\\alg{A},\\hs{H},J)$ by conjugation, and $\\ker(R_n) \\subseteq \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H})_\\textup{sa}$.\n\\end{proposition}\n\n\\begin{proof}\nFirst, note that $R_n$ is indeed well-defined, since by Equation~\\ref{realintertwine}, for any $M \\in \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H})_\\textup{sa}$, $J M J^* \\in \\bdd^{\\textup{L}}_\\alg{A}(\\hs{H})_\\textup{sa}$, and hence $R_n(M) \\in \\ms{D}_0(\\alg{A},\\hs{H},J)$.\n\nNow, let $E_\\lambda$ and $E_\\rho$ be defined as in Lemma~\\ref{decompproj}, and let $E_\\lambda^\\prime = \\Id - E_\\lambda$, $E_\\rho^\\prime = \\Id - E_\\rho$. Then, by construction of $E_\\lambda$ and $E_\\rho$ and Equation~\\ref{realintertwine}, for any $T \\in \\mathcal{L}^1_\\alg{A}(\\hs{H})$,\n\\[\n E_\\lambda(J T J^*) = J E_\\rho(T) J^*,\\quad E_\\rho(J T J^*) = J E_\\lambda(T) J^*.\n\\]\nHence, in particular, for $D \\in \\ms{D}_0(\\alg{A},\\hs{H},J)$, since $J D J^* = \\varepsilon^\\prime D$,\n\\begin{align*}\n D &= \\frac{1}{2}(E_\\lambda^\\prime + E_\\rho)(T) + \\frac{1}{2}(E_\\lambda + E_\\rho^\\prime)(T) \\\\ &= \\frac{1}{2}(E_\\lambda^\\prime + E_\\rho)(T) + \\varepsilon^\\prime J \\frac{1}{2}(E_\\lambda^\\prime + E_\\rho)(T) J^*\\\\ &= R_n\\left(\\frac{1}{2}(E_\\lambda^\\prime + E_\\rho)(T)\\right),\n\\end{align*}\nwhere $\\frac{1}{2}(E_\\lambda^\\prime + E_\\rho)(T) \\in \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H})_\\textup{sa}$.\n\nFinally, that $R_n$ interwtines the actions of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},J)$ follows from Proposition~\\ref{lrorder1decomp} together with the fact that elements of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},J)$, by definition, commute with $J$, whilst the fact that $R_n(M) = 0$ if and only if $M = -\\varepsilon^\\prime J M J^*$ implies that $\\ker(R_n) \\subseteq \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H})_\\textup{sa}$.\n\\end{proof}\n\nIt follows, in particular, that $\\ker(R_n)$ is invariant under the action of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},J)$ by conjugation, so that the action of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},J)$ on $\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H})_\\textup{sa}$ induces an action on the quotient $\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H})_\\textup{sa} \/ \\ker(R_n)$, and hence $R_n$ induces an isomorphism\n\\begin{equation}\n \\ms{D}_0(\\alg{A},\\hs{H},J) \\cong \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H})_\\textup{sa} \/ \\ker(R_n)\n\\end{equation}\nof $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},J)$-representations. Thus we have proved the following:\n\n\\begin{corollary}\nLet $(\\hs{H},J)$ be a real $\\alg{A}$-bimodule of odd $KO$-dimension $n \\bmod 8$. Then\n\\begin{equation}\n \\ms{D}(\\alg{A},\\hs{H},J) \\cong \\left(\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H})_\\textup{sa} \/ \\ker(R_n)\\right) \/ \\U^{\\textup{LR}}_\\alg{A}(\\hs{H},J).\n\\end{equation}\n\\end{corollary}\n\nDiscussion of $\\ms{D}(\\alg{A},\\hs{H},J)$ thus requires discussion first of $\\ker(R_n)$:\n\n\\begin{lemma}\\label{oddker}\nIf $K = (1_{n_\\alpha} \\otimes K_{\\alpha\\beta} \\otimes 1_{n_\\beta})_{\\alpha,\\beta\\in\\spec{\\alg{A}}} \\in \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H})_\\textup{sa}$, then $K \\in \\ker(R_n)$ if and only if for each $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$ such that $\\alpha \\neq \\beta$,\n\\begin{equation}\n K_{\\beta\\alpha} = -\\varepsilon^\\prime K_{\\alpha\\beta}^T,\n\\end{equation}\nand for each $\\alpha \\in \\spec{\\alg{A}}$,\n\\begin{equation}\n K_{\\alpha\\alpha} \\in \\hs{R}_\\alpha(n) = \n \\begin{cases}\n  \\Sym_{m_{\\alpha\\alpha}}(\\field{R}) &\\text{if $n=1$,}\\\\\n  i \\mathfrak{sp}(m_{\\alpha\\alpha}) &\\text{if $n=3$,}\\\\\n  M_{m_{\\alpha\\alpha}\/2}(\\field{H})_\\textup{sa} &\\text{if $n=5$,}\\\\\n  i \\mathfrak{so}(m_{\\alpha\\alpha}) &\\text{if $n=7$.}\n \\end{cases}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nBy definition of $R_n$, $K \\in \\ker(R_n)$ if and only if $K = -\\varepsilon^\\prime J K J^* = -\\varepsilon\\varepsilon^\\prime J K J$, and this in turn holds if and only if, for $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$ such that $\\alpha \\neq \\beta$, \n\\[\n K_{\\alpha\\beta} = -\\varepsilon^\\prime K_{\\beta\\alpha}^T,\n\\]\nwhile for $\\alpha \\in \\spec{\\alg{A}}$,\n\\[\n K_{\\alpha\\alpha} =\n \\begin{cases}\n  -\\varepsilon^\\prime \\overline{K_{\\alpha\\alpha}}, &\\text{if $n=1$ or $7$,}\\\\\n  \\varepsilon^\\prime I_{\\alpha} K_{\\alpha\\alpha} I_{\\alpha}^* &\\text{if $n=3$ or $5$,}\n \\end{cases}\n\\]\nwhere $I_\\alpha = \\Omega_{m_{\\alpha\\alpha}} \\circ \\text{complex conjugation}$. In the case that $n =3$ or $5$, however, by construction, $M_{m_{\\alpha\\alpha}\/2}(\\field{H})$, viewed in the usual way as a real form of $M_{m_{\\alpha\\alpha}}(\\field{C})$, is precisely the set of matrices in $M_{m_{\\alpha\\alpha}}(\\field{C})$ commuting with $I_\\alpha$. This, together with the hypothesis that $K$ is self-adjoint, so that each $K_{\\alpha\\beta}$ is self-adjoint, yields the desired result.\n\\end{proof}\n\nWe can now describe the the construction of an arbitrary Dirac operator $D$ on $(\\hs{H},J)$:\n\\begin{enumerate}\n \\item For $\\alpha$, $\\beta$, $\\gamma \\in \\spec{\\alg{A}}$ such that $\\alpha < \\gamma$, choose $M_{\\alpha\\beta}^\\gamma \\in M_{n_\\gamma m_{\\gamma\\beta} \\times n_\\alpha m_{\\alpha\\beta}}(\\field{C})$;\n \\item For $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$, choose $M_{\\alpha\\beta}^\\alpha \\in M_{n_\\alpha m_{\\alpha\\beta}}(\\field{C})_\\textup{sa}$;\n \\item For $\\alpha$, $\\beta$, $\\gamma$, $\\delta \\in \\spec{\\alg{A}}$, set\n \\begin{equation}\n  M_{\\alpha\\beta}^{\\gamma\\delta} = \n  \\begin{cases}\n   M_{\\alpha\\beta}^\\gamma \\otimes 1_{n_\\beta} &\\text{if $\\alpha < \\gamma$ and $\\beta = \\delta$,}\\\\\n   (M_{\\gamma\\beta}^\\alpha)^* \\otimes 1_{n_\\beta} &\\text{if $\\alpha > \\gamma$ and $\\beta = \\delta$,}\\\\\n   M_{\\alpha\\beta}^\\alpha \\otimes 1_{n_\\beta} &\\text{if $(\\alpha,\\beta) = (\\gamma,\\delta)$,}\\\\\n   0 &\\text{otherwise}.\n  \\end{cases}\n \\end{equation}\n \\item Finally, set $D = R_n(M)$.\n\\end{enumerate}\n\nNow, let $K = (1_{n_\\alpha} \\otimes K_{\\alpha\\beta} \\otimes 1_{n_\\beta})_{\\alpha,\\beta\\in\\spec{\\alg{A}}} \\in \\ker(R_n)$, so that each $K_{\\alpha\\beta}$ is self-adjoint, and for $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$ such that $\\alpha \\neq \\beta$, $K_{\\beta\\alpha} = -\\varepsilon^\\prime K_{\\alpha\\beta}^T$ and $K_{\\alpha\\alpha} \\in \\hs{R}_\\alpha(n)$. Thus, $K$ is uniquely specified by the matrices $K_{\\alpha\\beta} \\in M_{m_{\\alpha\\beta}}(\\field{C})_\\textup{sa}$ for $\\alpha < \\beta$ and by the $K_{\\alpha\\alpha} \\in \\hs{R}_\\alpha(n)$. Then, we can replace $M$ by $M + K$, {\\it i.e.\\\/}\\  make the replacements, for $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$ such that $\\alpha < \\beta$,\n\\begin{gather*}\n M_{\\alpha\\beta}^\\alpha \\mapsto M_{\\alpha\\beta}^\\alpha + 1_{n_\\alpha} \\otimes K_{\\alpha\\beta}, \\quad M_{\\beta\\alpha}^\\beta \\mapsto M_{\\beta\\alpha}^\\beta + 1_{n_\\beta} \\otimes (-\\varepsilon^\\prime K_{\\alpha\\beta}^T), \\\\ M_{\\alpha\\alpha}^\\alpha \\mapsto M_{\\alpha\\alpha}^\\alpha + 1_{n_\\alpha} \\otimes K_{\\alpha\\alpha}\n\\end{gather*}\nand obtain the same Dirac operator $D$. However, this is a freedom cannot generally be removed as we did in earlier cases, as it reflects precisely the non-injectivity of $R_n$. \n\nBy the above discussion and Propositions~\\ref{real17unitary} and~\\ref{real35unitary}, we can identify the space $\\ms{D}_0(\\alg{A},\\hs{H},J)$ with\n\\begin{multline}\n \\ms{D}_0(\\alg{A},m,n) := \\prod_{\\alpha \\in \\spec{\\alg{A}}} \\biggl[ M_{n_\\alpha m_{\\alpha\\alpha}}(\\field{C})_\\textup{sa} \/ (1_{n_\\alpha} \\otimes \\hs{R}_\\alpha(n)) \\\\ \\times \\prod_{\\substack{\\beta \\in \\spec{\\alg{A}} \\\\ \\beta > \\alpha}} (M_{n_\\alpha m_{\\alpha\\beta}}(\\field{C})_\\textup{sa} \\oplus M_{n_\\beta m_{\\alpha\\beta}}(\\field{C})_\\textup{sa})\/M_{m_\\alpha\\beta}(\\field{C})_\\textup{sa} \\times \\prod_{\\substack{\\beta,\\gamma \\in \\spec{\\alg{A}}\\\\ \\gamma > \\alpha}} M_{n_\\gamma m_{\\gamma\\beta} \\times n_\\alpha m_{\\alpha\\beta}}(\\field{C}) \\biggr],\n\\end{multline}\nwhere $M_{m_{\\alpha\\beta}}(\\field{C})_\\textup{sa}$ is viewed as embedded in $M_{n_\\alpha m_{\\alpha\\beta}}(\\field{C})_\\textup{sa} \\oplus M_{n_\\beta m_{\\alpha\\beta}}(\\field{C})_\\textup{sa}$ via the map\n\\[\n K \\mapsto (1_{n_\\alpha} \\otimes K) \\oplus (-\\varepsilon^\\prime 1_{n_\\beta} \\otimes K^T),\n\\]\nand $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H},J)$ with\n\\begin{equation}\n \\U(\\alg{A},m,n) := \\prod_{\\alpha \\in \\spec{\\alg{A}}} \\biggl( \\hs{U}_\\alpha(n) \\times \\prod_{\\substack{\\beta \\in \\spec{\\alg{A}}\\\\ \\beta > \\alpha}} \\U(m_{\\alpha\\beta}) \\biggr),\n\\end{equation}\nwhere\n\\[\n \\hs{U}_\\alpha(n) :=\n\\begin{cases}\n \\Orth(m_{\\alpha\\alpha}) &\\text{if $n = 1$ or $7$,}\\\\\n \\Sp(m_{\\alpha\\alpha}) &\\text{if $n=3$ or $5$.}\n\\end{cases}\n\\]\nThen the action of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H};J)$ on $\\ms{D}_0(\\alg{A},\\hs{H};J)$ corresponds under these identifications to the action of $\\U(\\alg{A},m,n)$ on $\\ms{D}_0(\\alg{A},m,n)$ defined by having the element $(U_{\\alpha\\alpha};U_{\\alpha\\beta}) \\in \\U(\\alg{A},m,n)$ act on $\\bigl([M_{\\alpha\\alpha}^\\alpha];[(M_{\\alpha\\beta}^\\alpha,M_{\\beta\\alpha}^\\beta)];M_{\\alpha\\beta}^\\gamma\\bigr) \\in \\ms{D}_0(\\alg{A},m,n)$ by\n\\begin{align*}\n [M_{\\alpha\\alpha}^\\alpha] &\\mapsto \\bigl[(1_{n_\\alpha} \\otimes U_{\\alpha\\alpha}) M_{\\alpha\\alpha} (1_{n_\\alpha} \\otimes U_{\\alpha\\alpha}^*)\\bigr];\\\\\n [(M_{\\alpha\\beta}^\\alpha,M_{\\beta\\alpha}^\\beta)] &\\mapsto \\biggl[\\bigl((1_{n_\\alpha} \\otimes U_{\\alpha\\beta}) M_{\\alpha\\beta}^\\alpha (1_{n_\\alpha} \\otimes U_{\\alpha\\beta}^*),(1_{n_\\beta} \\otimes \\overline{U_{\\alpha\\beta}})M_{\\beta\\alpha}^\\beta(1_{n_\\beta} \\otimes U_{\\alpha\\beta}^T)\\bigr)\\biggr];\\\\\n M_{\\alpha\\beta}^\\gamma &\\mapsto\n \\begin{cases}\n  (1_{n_\\gamma} \\otimes U_{\\gamma\\beta}) M_{\\alpha\\beta}^\\gamma (1_{n_\\alpha} \\otimes U_{\\alpha\\beta}^*) &\\text{if $\\alpha < \\beta$, $\\gamma < \\delta$,}\\\\\n  (1_{n_\\gamma} \\otimes U_{\\gamma\\beta}) M_{\\alpha\\beta}^\\gamma (1_{n_\\alpha} \\otimes U_{\\beta\\alpha}^T) &\\text{if $\\alpha > \\beta$, $\\gamma < \\delta$,}\\\\\n  (1_{n_\\gamma} \\otimes \\overline{U_{\\beta\\gamma}}) M_{\\alpha\\beta}^\\gamma (1_{n_\\alpha} \\otimes U_{\\alpha\\beta}^*) &\\text{if $\\alpha < \\beta$, $\\gamma > \\delta$,}\\\\\n  (1_{n_\\gamma} \\otimes \\overline{U_{\\beta\\gamma}}) M_{\\alpha\\beta}^\\gamma (1_{n_\\alpha} \\otimes U_{\\beta\\alpha}^T) &\\text{if $\\alpha > \\beta$, $\\gamma > \\delta$.}\n \\end{cases}\n\\end{align*}\nWe have therefore proved the following:\n\n\\begin{proposition}\n Let $(\\hs{H},J)$ be a real $\\alg{A}$-bimodule of odd $KO$-dimension $n \\bmod 8$ with multiplicity matrix $m$. Then\n \\begin{equation}\n  \\ms{D}(\\alg{A},\\hs{H},J) \\cong \\ms{D}_0(\\alg{A},m,n) \/ \\U(\\alg{A},m,n).\n \\end{equation}\n\\end{proposition}\n\n\\subsection{Real spectral triples of even $KO$-dimension}\n\nWe now turn to real spectral triples of even $KO$-dimension. Because of the considerable qualitative differences between the two cases, we consider separately the case of $KO$-dimension $0$ or $4 \\bmod 8$ and $KO$-dimension $2$ or $6 \\bmod 8$. \n\nIn what follows, $(\\hs{H},\\gamma,J)$ is a fixed real $\\alg{A}$-bimodule of even $KO$-dimension $n \\bmod 8$ with multiplicity matrices $(m^\\textup{even},m^\\textup{odd})$; we  denote by $\\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd};J)$ the subspace of $\\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$ consisting of $\\delta$ such that \n\\[\n \\begin{pmatrix}\n 0 & \\Delta^*\\\\\n\\Delta & 0\n\\end{pmatrix} \\in \\ms{D}_0(\\alg{A},\\hs{H};\\gamma,J).\n\\]\nIt then follows that\n\\begin{equation}\n \\ms{D}_0(\\alg{A},\\hs{H},\\gamma,J) \\cong \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd};J)\n\\end{equation}\nvia the map $D \\mapsto P^\\textup{odd} D P^\\textup{even}$.\n\n\\subsubsection{$KO$-dimension $0$ or $4 \\bmod 8$}\n\nLet us first consider the case where $n = 0$ or $4 \\bmod 8$, {\\it i.e.\\\/}\\  where $\\varepsilon^\\prime = 1$. Then $J = J^\\textup{even} \\oplus J^\\textup{odd}$ for anti-unitaries $J^\\textup{even}$ and $J^\\textup{odd}$ on $\\hs{H}^\\textup{even}$ and $\\hs{H}^\\textup{odd}$, respectively, such that $(\\hs{H}^\\textup{even},J^\\textup{even})$ and $(\\hs{H}^\\textup{odd},J^\\textup{odd})$ are real $\\alg{A}$-bimodules of $KO$-dimension $n^\\prime \\bmod 8$, where $n^\\prime = 1$ or $7$ if $n=0$, $3$ or $5$ if $n=4$. In light of Corollary~\\ref{real04unitary}, one can readily check the following analogue of Proposition~\\ref{evendirac}:\n\n\\begin{proposition}\\label{real04dirac}\n The map \n\\[\n \\ms{D}(\\alg{A},\\hs{H},\\gamma,J) \\to \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd},J^\\textup{odd}) \\backslash \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd};J) \/ \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},J^\\textup{even})\n\\] \ndefined by $[D] \\mapsto [P^\\textup{odd} D P^\\textup{even}]$ is a homeomorphism.\n\\end{proposition}\n\nHere, as before, $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd},J^\\textup{odd})$ acts by multiplication on the left, whilst the group $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},J^\\textup{even})$ acts by multiplication on the right by the inverse.\n\nWe now prove the relevant analogue of Proposition~\\ref{oddrealdirac}:\n\n\\begin{proposition}\\label{even04dirac}\n The map $R_n : \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd}) \\to \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd},J)$ defined by $R_n(M) := M + J^\\textup{odd} M (J^\\textup{even})^*$ is a surjection interwining the actions of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd},J^\\textup{odd})$ by multiplication on the left and of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},J^\\textup{even})$ by multiplication on the right by the inverse on $\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$ and $\\mathcal{L}^1_\\alg{A}(\\alg{A},\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd},J)$, and $\\ker(R_n) \\subseteq \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$.\n\\end{proposition}\n\n\\begin{proof}\n First note that\n\\[\n \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd},J) = \\{\\Delta \\in \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd}) \\mid \\Delta = J^\\textup{odd} \\Delta (J^\\textup{even})^*\\},\n\\]\nso that $R_n$ is indeed well-defined by construction. Moreover, since $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},J^\\textup{even})$ and $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd},J^\\textup{odd})$ commute by definition with $J^\\textup{even}$ and $J^\\textup{odd}$, respectively, it then follows by construction of $R_n$ that $R_n$ does indeed have the desired intertwining properties.\n\nNext, for $M \\in \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$, we have that $R_n(M) = 0$ if and only if $M = -J^\\textup{odd} M (J^\\textup{even})^*$, but $M$ is right $\\alg{A}$-linear if and only if $J^\\textup{odd} M (J^\\textup{even})^* = \\varepsilon J^\\textup{odd} M J^\\textup{even}$ is left $\\alg{A}$-linear, so that $M \\in \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$ as claimed.\n\nFinally, it is easy to check, just as in the proof of Proposition~\\ref{oddrealdirac}, that for $\\Delta \\in \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd},J)$,\n\\[\n \\Delta = R_n\\left(\\frac{1}{2}(E_\\lambda^\\prime + E_\\rho)(\\Delta)\\right),\n\\]\nwhere $\\frac{1}{2}(E_\\lambda^\\prime + E_\\rho)(\\Delta) \\in \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$.\n\\end{proof}\n\nAgain, just as in the case of odd $KO$-dimension, this last result not only implies that the actions of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},J^\\textup{even})$ and $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd},J^\\textup{odd})$ on $\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$ descend to actions on $\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd}) \/ \\ker(R_n)$, but that $R_n$ descends to an isomorphism $\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd}) \/ \\ker(R_n) \\cong \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd};J)$ intertwining the actions of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},J^\\textup{even})$ and $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd},J^\\textup{odd})$, thereby yielding the following\n\n\\begin{corollary}\n Let $(\\hs{H},\\gamma,J)$ be a real $\\alg{A}$-bimodule of $KO$-dimension $n \\bmod 8$ for $n = 0$ or $4$. Then \n\\begin{equation}\n \\ms{D}(\\alg{A},\\hs{H},\\gamma,J) \\cong \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd},J^\\textup{odd}) \\backslash \\left(\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd}) \/ \\ker(R_n)\\right) \/ \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},J^\\textup{even}) \n\\end{equation}\n\\end{corollary}\n\n\\latin{Mutatis mutandis}, the proof of Lemma~\\ref{oddker} yields the following characterisation of $\\ker(R_n)$:\n\n\\begin{lemma}\\label{even04ker}\nIf $K = (1_{n_\\alpha} \\otimes K_{\\alpha\\beta} \\otimes 1_{n_\\beta})_{\\alpha,\\beta\\in\\spec{\\alg{A}}} \\in \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$, then $K \\in \\ker(R_n)$ if and only if for each $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$ such that $\\alpha \\neq \\beta$,\n\\begin{equation}\n K_{\\beta\\alpha} = -\\overline{K_{\\alpha\\beta}},\n\\end{equation}\nand for each $\\alpha \\in \\spec{\\alg{A}}$,\n\\begin{equation}\n K_{\\alpha\\alpha} \\in \\hs{R}_\\alpha(n) = \n \\begin{cases}\n  i M_{m^\\textup{odd}_{\\alpha\\alpha} \\times m^\\textup{even}_{\\alpha\\alpha}}(\\field{R}) &\\text{if $n=0$,}\\\\\n  i M_{m^\\textup{odd}_{\\alpha\\alpha}\/2 \\times m^\\textup{even}_{\\alpha\\alpha}\/2}(\\field{H}) &\\text{if $n=4$.}\\\\\n \\end{cases}\n\\end{equation}\n\\end{lemma}\n\nNote that such a map $K \\in \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$ is therefore entirely specified by the $K_{\\alpha\\beta} \\in M_{m^\\textup{odd}_{\\alpha\\beta} \\times m^\\textup{even}_{\\alpha\\beta}}(\\field{C})$ for $\\alpha < \\beta$ and by the $K_{\\alpha\\alpha} \\in \\hs{R}_\\alpha(n)$.\n\nLet us now describe the construction of an arbitrary Dirac operator $D$ on the real $\\alg{A}$-bimodule $(\\hs{H},\\gamma,J)$ of $KO$-dimension $n = 0$ or $4 \\bmod 8$:\n\\begin{enumerate}\n \\item For $\\alpha$, $\\beta$, $\\gamma \\in \\spec{\\alg{A}}$, choose $M_{\\alpha\\beta}^\\gamma \\in M_{n_\\gamma m^\\textup{odd}_{\\gamma\\beta} \\times n_\\alpha m^\\textup{even}_{\\alpha\\beta}}(\\field{C})$;\n \\item Construct $M \\in \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$ by setting for $\\alpha$, $\\beta$, $\\gamma$, $\\delta \\in \\spec{\\alg{A}}$,\n \\begin{equation}\n  M_{\\alpha\\beta}^{\\gamma\\delta} =\n  \\begin{cases}\n   M_{\\alpha\\beta}^\\gamma \\otimes 1_{n_\\beta} &\\text{if $\\beta = \\delta$,}\\\\\n   0 &\\text{otherwise;}\\\\\n  \\end{cases}\n \\end{equation}\n \\item Finally, set \n \\begin{equation}\n  D =\n  \\begin{pmatrix}\n   0 & R_n(M)^*\\\\\n   R_n(M) & 0\n  \\end{pmatrix}.\n \\end{equation}\n\\end{enumerate}\n\nJust as before, if $R_n$ is non-injective, we can make the substitution $M \\mapsto M + K$ for any $K \\in \\ker(R_n)$ and obtain the same Dirac operator D; at the level of components, we have for $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$ such that $\\alpha < \\beta$,\n\\begin{gather*}\n M_{\\alpha\\beta}^\\alpha \\mapsto M_{\\alpha\\beta}^\\alpha + 1_{n_\\alpha} \\otimes K_{\\alpha\\beta}, \\quad M_{\\beta\\alpha}^\\beta \\mapsto M_{\\beta\\alpha}^\\beta + 1_{n_\\alpha} \\otimes (-\\overline{K_{\\alpha\\beta}})\\\\\n M_{\\alpha\\alpha}^\\alpha \\mapsto M_{\\alpha\\alpha}^\\alpha + 1_{n_\\alpha} \\otimes K_{\\alpha\\alpha}.\n\\end{gather*}\nWith these observations in hand, we can revisit the moduli space $\\ms{D}(\\alg{A},\\hs{H},\\gamma,J)$.\n\nBy the discussion above and Corollaries~\\ref{real17unitary} and~\\ref{real35unitary}, we can identify the space $\\ms{D}_0(\\alg{A},\\hs{H},\\gamma,J)$ with\n\\begin{multline}\n \\ms{D}_0(\\alg{A},m^\\textup{even},m^\\textup{odd},n) := \\prod_{\\alpha \\in \\spec{\\alg{A}}} \\biggl[ M_{n_\\alpha m^\\textup{odd}_{\\alpha\\alpha} \\times n_\\alpha m^\\textup{even}_{\\alpha\\alpha}}(\\field{C}) \/ (1_{n_\\alpha} \\otimes \\hs{R}_\\alpha(n)) \\\\ \\times \\prod_{\\substack{\\beta \\in \\spec{\\alg{A}} \\\\ \\beta > \\alpha}} (M_{n_\\alpha m^\\textup{odd}_{\\alpha\\beta} \\times n_\\alpha m^\\textup{even}_{\\alpha\\beta}}(\\field{C}) \\oplus M_{n_\\beta m^\\textup{odd}_{\\alpha\\beta} \\times n_\\beta m^\\textup{even}_{\\alpha\\beta}}(\\field{C}))\/M_{m^\\textup{odd}_{\\alpha\\beta} \\times m^\\textup{even}_{\\alpha\\beta}}(\\field{C}) \\\\ \\times \\prod_{\\substack{\\beta,\\gamma\\in\\spec{\\alg{A}} \\\\ \\gamma \\neq \\alpha}} M_{n_\\gamma m^\\textup{odd}_{\\gamma\\beta} \\times n_{\\alpha} m^\\textup{even}_{\\alpha\\beta}}(\\field{C})\\biggr],\n\\end{multline}\nwhere $M_{m^\\textup{odd}_{\\alpha\\beta} \\times m^\\textup{even}_{\\alpha\\beta}}(\\field{C})$ is viewed as embedded in the space\n\\[\n M_{n_\\alpha m^\\textup{odd}_{\\alpha\\beta} \\times n_\\alpha m^\\textup{even}_{\\alpha\\beta}}(\\field{C}) \\oplus M_{n_\\beta m^\\textup{odd}_{\\alpha\\beta} \\times n_\\beta m^\\textup{even}_{\\alpha\\beta}}(\\field{C})\n\\]\nvia the map $K \\mapsto (1_{n_\\alpha} \\otimes K) \\oplus (-1_{n_\\beta} \\otimes \\overline{K})$, and identify the groups $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even};J^\\textup{even})$ and $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd};J^\\textup{odd})$ with $\\U(\\alg{A},m^\\textup{even},n^\\prime)$ and $\\U(\\alg{A},m^\\textup{odd},n^\\prime)$, respectively. Then the actions of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even};J^\\textup{even})$ and $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{odd};J^\\textup{odd})$ on $\\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd};J)$ corresponds under these identifications to the actions of the groups $\\U(\\alg{A},m^\\textup{even},n^\\prime)$ and $\\U(\\alg{A},m^\\textup{odd},n^\\prime)$, respectively, on $\\ms{D}_0(\\alg{A},m^\\textup{even},m^\\textup{odd},n)$ defined by having \n\\[\n (U_{\\alpha\\alpha}^\\textup{odd};U_{\\alpha\\beta}^\\textup{odd}) \\in \\U(\\alg{A},m^\\textup{odd};n^\\prime), \\quad (U_{\\alpha\\alpha}^\\textup{even};U_{\\alpha\\beta}^\\textup{odd}) \\in \\U(\\alg{A},m^\\textup{even};n^\\prime)\n\\]\nact on $\\bigl([M_{\\alpha\\alpha}^\\alpha];[(M_{\\alpha\\beta}^\\alpha,M_{\\beta\\alpha}^\\beta)];M_{\\alpha\\beta}^\\gamma\\bigr) \\in \\ms{D}_0(\\alg{A},m,n)$ by\n\\begin{align*}\n [M_{\\alpha\\alpha}^\\alpha] &\\mapsto \\bigl[(1_{n_\\alpha} \\otimes U_{\\alpha\\alpha}^\\textup{odd}) M_{\\alpha\\alpha}\\bigr];\\\\\n [(M_{\\alpha\\beta}^\\alpha,M_{\\beta\\alpha}^\\beta)] &\\mapsto \\biggl[\\bigl((1_{n_\\alpha} \\otimes U_{\\alpha\\beta}^\\textup{odd}) M_{\\alpha\\beta}^\\alpha,(1_{n_\\beta} \\otimes \\overline{U_{\\alpha\\beta}^\\textup{odd}})M_{\\beta\\alpha}^\\beta\\bigr)\\biggr];\\\\\n M_{\\alpha\\beta}^\\gamma &\\mapsto\n \\begin{cases}\n  (1_{n_\\gamma} \\otimes U_{\\gamma\\beta}^\\textup{odd}) M_{\\alpha\\beta}^\\gamma  &\\text{if $\\gamma < \\delta$,}\\\\\n  (1_{n_\\gamma} \\otimes \\overline{U_{\\beta\\gamma}^\\textup{odd}}) M_{\\alpha\\beta}^\\gamma &\\text{if $\\gamma > \\delta$;}\n \\end{cases}\n\\end{align*}\nand\n\\begin{align*}\n [M_{\\alpha\\alpha}^\\alpha] &\\mapsto \\bigl[M_{\\alpha\\alpha} (1_{n_\\alpha} \\otimes (U_{\\alpha\\alpha}^\\textup{even})^*)\\bigr];\\\\\n [(M_{\\alpha\\beta}^\\alpha,M_{\\beta\\alpha}^\\beta)] &\\mapsto \\biggl[\\bigl(M_{\\alpha\\beta}^\\alpha (1_{n_\\alpha} \\otimes (U_{\\alpha\\beta}^\\textup{even})^*), M_{\\beta\\alpha}^\\beta(1_{n_\\beta} \\otimes (U_{\\alpha\\beta}^\\textup{even})^T)\\bigr)\\biggr];\\\\\n M_{\\alpha\\beta}^\\gamma &\\mapsto\n \\begin{cases}\n  M_{\\alpha\\beta}^\\gamma (1_{n_\\alpha} \\otimes (U_{\\alpha\\beta}^\\textup{even})^*) &\\text{if $\\alpha < \\beta$,}\\\\\n  M_{\\alpha\\beta}^\\gamma (1_{n_\\alpha} \\otimes (U_{\\beta\\alpha}^\\textup{even})^T) &\\text{if $\\alpha > \\beta$;}\n \\end{cases}\n\\end{align*}\nrespectively. We have therefore proved the following:\n\n\\begin{proposition}\n Let $(\\hs{H},\\gamma,J)$ be a real $\\alg{A}$-bimodule of even $KO$-dimension $n \\bmod 8$ for $n = 0$ or $4$, with multiplicity matrices $(m^\\textup{even},m^\\textup{odd})$. Then\n \\begin{equation}\n  \\ms{D}(\\alg{A},\\hs{H},\\gamma,J) \\cong \\U(\\alg{A},m^\\textup{odd},n^\\prime) \\backslash \\ms{D}_0(\\alg{A},m^\\textup{even},m^\\textup{odd},n) \/ \\U(\\alg{A},m^\\textup{even},n^\\prime).\n \\end{equation}\n\\end{proposition}\n\nIt is worth noting that considerable simplifications are obtained in the quasi-orientable case, as all components of the form $M_{\\alpha\\beta}^\\alpha \\otimes 1_{n_\\beta}$ of $M \\in \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$ must necessarily vanish, as must $\\ker(R_n)$ itself. In particular, then, one is left with\n\\[\n \\ms{D}_0(\\alg{A},m^\\textup{even},m^\\textup{odd},n) = \\prod_{\\alpha,\\beta,\\gamma\\in\\spec{\\alg{A}}} M_{n_\\gamma m^\\textup{odd}_{\\gamma\\beta} \\times n_{\\alpha} m^\\textup{even}_{\\alpha\\beta}}(\\field{C}).\n\\]\n\n\\subsubsection{$KO$-dimension $2$ or $6 \\bmod 8$}\\label{ko6}\n\nLet us now consider the case where $n = 2$ or $n=6 \\bmod 8$, {\\it i.e.\\\/}\\  where $\\varepsilon^\\prime = -1$. Then\n\\[\n J =\n\\begin{pmatrix}\n 0 & \\varepsilon \\tilde{J}^*\\\\\n \\tilde{J} & 0\n\\end{pmatrix}\n\\]\nfor $\\tilde{J} : \\hs{H}^\\textup{even} \\to \\hs{H}^\\textup{odd}$ anti-unitary, and $m^\\textup{odd} = (m^\\textup{even})^T$. In light of Corollary~\\ref{real26unitary}, one can easily establish, along the lines of Propositions~\\ref{evendirac} and~\\ref{even04dirac}, the following result:\n\n\\begin{proposition}\n Let $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even})$ act on $\\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd};J)$ by\n\\[\n (U,\\Delta) \\mapsto \\tilde{J}U\\tilde{J}^* \\Delta U^*\n\\]\nfor $U \\in \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even})$ and $\\Delta \\in \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd};J)$. Then the map\n\\[\n \\ms{D}(\\alg{A},\\hs{H},\\gamma,J) \\to \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd};J)\/\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even})\n\\]\ndefined by $[D] \\mapsto [P^\\textup{odd} D P^\\textup{even}]$ is a homeomorphism.\n\\end{proposition}\n\nIn the same way, we can define an action of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even})$ on $\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$. We now give the relevant analogue of Propositions~\\ref{oddrealdirac} and~\\ref{even04dirac}:\n\n\\begin{proposition}\\label{even26dirac}\n The map $R_n : \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd}) \\to \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd};J)$ defined by $R_n(M) := M + \\varepsilon \\tilde{J} M^* \\tilde{J}$ is a surjection intertwining the actions of the group $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even})$ on $\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$ and $\\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd};J)$, and $\\ker(R_n) \\subset \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$.\n\\end{proposition}\n\n\\begin{proof}\n First note that\n\\[\n \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd};J) = \\{\\Delta \\in \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd}) \\mid \\Delta = \\varepsilon \\tilde{J} \\Delta^* \\tilde{J}\\},\n\\]\nas can be checked by direct calculation, so that $R_n$ is indeed well-defined. It also readily follows by construction of $R_n$ and the definition of the actions of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even})$ that $R_n$ has the desired intertwining properties.\n\nNow, for $M \\in \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$, one has that $R_n(M) = 0$ if and only if $M = -\\varepsilon \\tilde{J} M^* \\tilde{J}$, but $\\tilde{J} M^* \\tilde{J}$ is manifestly left $\\alg{A}$-linear, so that $M \\in \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$, as claimed.\n\nFinally, just as in the proof of Propositions~\\ref{oddrealdirac} and~\\ref{even04dirac}, one can easily check that for $\\Delta \\in \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd};J)$,\n\\[\n \\Delta = R_n\\bigl(\\frac{1}{2}(E^\\prime_\\lambda + E_\\rho)(\\Delta)\\bigr),\n\\]\nwhere $\\frac{1}{2}(E^\\prime_\\lambda + E_\\rho)(\\Delta)$ is right $\\alg{A}$-linear.\n\\end{proof}\n\nJust as in the earlier cases, the action of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even})$ on $\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$ descends to an action on the quotient $\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd}) \/ \\ker(R_n)$, so that $R_n$ descends to an $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$-isomorphism\n\\begin{equation}\n \\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd}) \/ \\ker(R_n) \\cong \\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd};J),\n\\end{equation}\nthereby yielding the following:\n\n\\begin{corollary}\n Let $(\\hs{H},\\gamma,J)$ be a real $\\alg{A}$-bimodule of $KO$-dimension $n \\bmod 8$ for $n = 2$ or $6$. Then\n\\begin{equation}\n \\ms{D}(\\alg{A},\\hs{H},\\gamma,J) \\cong (\\bdd^{\\textup{R}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})\/\\ker(R_n)) \/ \\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even}).\n\\end{equation}\n\\end{corollary}\n\nAgain, \\latin{mutatis mutandis}, the proof of Lemma~\\ref{oddker} yields the following characterisation of $\\ker(R_n)$:\n\n\\begin{lemma}\n If $K = (1_{n_\\alpha} \\otimes K_{\\alpha\\beta} \\otimes 1_{n_\\beta})_{\\alpha,\\beta\\in\\spec{\\alg{A}}} \\in \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd})$, then $K \\in \\ker(R_n)$ if and only if for each $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$ such that $\\alpha \\neq \\beta$,\n\\begin{equation}\n K_{\\beta\\alpha} = -\\varepsilon K_{\\alpha\\beta}^T,\n\\end{equation}\nand for each $\\alpha \\in \\spec{\\alg{A}}$,\n\\begin{equation}\n K_{\\alpha\\alpha} \\in \\hs{R}_\\alpha(n) =\n\\begin{cases}\n \\Sym_{m^\\textup{even}_{\\alpha\\alpha}}(\\field{C}) &\\text{if $n = 2$,}\\\\\n \\mathfrak{so}(m^\\textup{even}_{\\alpha\\alpha},\\field{C}) &\\text{if $n = 6$.}\n\\end{cases}\n\\end{equation}\n\\end{lemma}\n\nThus, such a map $K \\in \\ker(R_n)$ is entirely specified by the components $K_{\\alpha\\beta} \\in M_{m^\\textup{even}_{\\beta\\alpha} \\times m^\\textup{even}_{\\alpha\\beta}}(\\field{C})$ for $\\alpha < \\beta$ and by the $K_{\\alpha\\alpha} \\in \\hs{R}_\\alpha(n)$.\n\nNote that the discussion of the construction of Dirac operators and of the freedom in the construction provided by $\\ker(R_n)$ in the case of $KO$-dimension $0$ or $4 \\bmod 8$ holds also in this case. Thus we can identify $\\ms{D}(\\alg{A},\\hs{H},\\gamma,J)$ with\n\\begin{multline}\n \\ms{D}_0(\\alg{A},m^\\textup{even},n) := \\prod_{\\alpha \\in \\spec{\\alg{A}}} \\biggl[ M_{n_\\alpha m^\\textup{even}_{\\alpha\\alpha}}(\\field{C}) \/ \\bigl(1_{n_\\alpha} \\otimes \\hs{R}_\\alpha(n)\\bigr) \\\\\n\\times \\prod_{\\substack{\\beta \\in \\spec{\\alg{A}} \\\\ \\beta > \\alpha}} \\bigl(M_{n_\\alpha m^\\textup{even}_{\\beta\\alpha} \\times n_\\alpha m^\\textup{even}_{\\alpha\\beta}}(\\field{C}) \\oplus M_{n_\\beta m^\\textup{even}_{\\alpha\\beta} \\times n_\\beta m^\\textup{even}_{\\beta\\alpha}}(\\field{C})\\bigr) \/ M_{m^\\textup{even}_{\\beta\\alpha} \\times m^\\textup{even}_{\\alpha\\beta}}(\\field{C}) \\\\\n\\times \\prod_{\\substack{\\beta,\\gamma\\in\\spec{\\alg{A}}\\\\\\gamma \\neq \\alpha}} M_{n_\\gamma m^\\textup{even}_{\\beta\\gamma} \\times n_\\alpha m^\\textup{even}_{\\alpha\\beta}}(\\field{C})\\biggr],\n\\end{multline}\nwhere $M_{m^\\textup{even}_{\\beta\\alpha} \\times m^\\textup{even}_{\\alpha\\beta}}(\\field{C})$ is viewed as embedded in the space\n\\[\n M_{n_\\alpha m^\\textup{even}_{\\beta\\alpha} \\times n_\\alpha m^\\textup{even}_{\\alpha\\beta}}(\\field{C}) \\oplus M_{n_\\beta m^\\textup{even}_{\\alpha\\beta} \\times n_\\beta m^\\textup{even}_{\\beta\\alpha}}(\\field{C})\n\\]\nvia the map $K \\mapsto (1_{n_\\alpha} \\otimes K) \\oplus (-\\varepsilon 1_{n_\\beta} \\otimes K^T)$, and identify $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even})$ with $\\U(\\alg{A},m^\\textup{even})$. Then the action of $\\U^{\\textup{LR}}_\\alg{A}(\\hs{H}^\\textup{even})$ on $\\mathcal{L}^1_\\alg{A}(\\hs{H}^\\textup{even},\\hs{H}^\\textup{odd};J)$ corresponds under these identifications with the action of $\\U(\\alg{A},m^\\textup{even})$ on $\\ms{D}_0(\\alg{A},m^\\textup{even},n)$ defined by having $(U_{\\alpha\\beta}) \\in \\U(\\alg{A},m^\\textup{even})$ act on $\\bigl([M_{\\alpha\\alpha}^\\alpha];[(M_{\\alpha\\beta}^\\alpha,M_{\\beta\\alpha}^\\beta)];M_{\\alpha\\beta}^\\gamma\\bigr) \\in \\ms{D}_0(\\alg{A},m,n)$ by\n\\begin{align*}\n [M_{\\alpha\\alpha}^\\alpha] &\\mapsto \\bigl[(1_{n_\\alpha} \\otimes \\overline{U_{\\alpha\\alpha}}) M_{\\alpha\\alpha}^\\alpha (1_{n_\\alpha} \\otimes U_{\\alpha\\alpha}^*)\\bigr];\\\\\n \\bigl[(M_{\\alpha\\beta}^\\alpha,M_{\\beta\\alpha}^\\beta)\\bigr] &\\mapsto \\biggl[\\bigl((1_{n_\\alpha} \\otimes \\overline{U_{\\beta\\alpha}}) M_{\\alpha\\beta}^\\alpha (1_{n_\\alpha} \\otimes U_{\\alpha\\beta}^*), (1_{n_\\beta} \\otimes \\overline{U_{\\alpha\\beta}}) M_{\\beta\\alpha}^\\alpha (1_{n_\\beta} \\otimes U_{\\beta\\alpha}^*)\\bigr)\\biggr];\\\\\n M_{\\alpha\\beta}^\\gamma &\\mapsto (1_{n_\\gamma} \\otimes \\overline{U_{\\beta\\gamma}}) M_{\\alpha\\beta}^\\gamma (1_{n_\\alpha} \\otimes U_{\\alpha\\beta}^*).\n\\end{align*}\nThis, then, proves the following:\n\n\\begin{proposition}\\label{moduli26}\n Let $(\\hs{H},\\gamma,J)$ be a real $\\alg{A}$-bimdoule of even $KO$-dimension $n \\bmod 8$ for $n = 2$ or $6$, with multiplicity matrices $(m^\\textup{even},(m^\\textup{even})^T)$. Then\n\\begin{equation}\n \\ms{D}(\\alg{A},\\hs{H},\\gamma,J) \\cong \\ms{D}_0(\\alg{A},m^\\textup{even},n) \/ \\U(\\alg{A},m^\\textup{even}).\n\\end{equation}\n\\end{proposition}\n\nAgain, considerable simplifications are obtained in the quasi-orientable case, just as for $KO$-dimension $0$ or $4 \\bmod 8$.\n\n\n\\subsection{Dirac operators in the Chamseddine--Connes--Marcolli model}\n\nLet us now apply the above results on Dirac operators and moduli spaces thereof to the bimodules appearing in the Chamseddine--Connes--Marcolli model. \n\nWe begin with $(\\hs{H}_F,\\gamma_F,J_F,\\epsilon_F)$ as an $S^0$-real $\\alg{A}_{LR}$-bimodule of $KO$-dimension $6 \\bmod 8$, which, as we shall now see, is essentially $S^0$-real in structure:\n\n\\begin{proposition}\n For the $S^0$-real $\\alg{A}_{LR}$-bimodule $(\\hs{H}_F,\\gamma_F,J_F,\\epsilon_F)$ of $KO$-dimen\\-sion $6 \\bmod 8$,\n\\[\n \\ms{D}_0(\\alg{A}_{LR},\\hs{H}_F,\\gamma_F,J_F) = \\ms{D}_0(\\alg{A}_{LR},\\hs{H}_F,\\gamma_F,J_F,\\epsilon_F),\n\\]\nand\n\\[\n \\U^{\\textup{LR}}_{\\alg{A}_{LR}}(\\hs{H},\\gamma_F,J_F) = \\U^{\\textup{LR}}_{\\alg{A}_{LR}}(\\hs{H},\\gamma_F,J_F,\\epsilon_F),\n\\]\nso that\n\\[\n \\ms{D}(\\alg{A}_{LR},\\hs{H}_F,\\gamma_F,J_F) = \\ms{D}(\\alg{A}_{LR},\\hs{H}_F,\\gamma_F,J_F,\\epsilon_F).\n\\]\n\n\\end{proposition}\n\n\\begin{proof}\n To prove the first part of the claim, by Proposition~\\ref{even26dirac}, it suffices to show that any right $\\alg{A}_{LR}$-linear operator $\\hs{H}_F^\\textup{even} \\to \\hs{H}_F^\\textup{odd}$ commutes with $\\epsilon_F$. Thus, let $T \\in \\bdd^{\\textup{R}}_{\\alg{A}_{LR}}(\\hs{H}_F^\\textup{even},\\hs{H}_F^\\textup{odd})$. Then, since the signed multiplicity matrix $\\mu$ of $(\\hs{H}_F,\\gamma_F)$ as an orientable even $\\alg{A}_{LR}$-bimodule is given by\n\\[\n \\mu =\n N \\begin{pmatrix}\n 0 & 0 & -1 & +1 & 0 & 0\\\\\n 0 & 0 & 0 & 0 & 0 & 0\\\\\n +1 & 0 & 0 & 0 & +1 & 0\\\\\n -1 & 0 & 0 & 0 & -1 & 0\\\\\n 0 & 0 & -1 & +1 & 0 & 0\\\\\n 0 & 0 & 0 & 0 & 0 & 0\n\\end{pmatrix},\n\\]\nit follows from Proposition~\\ref{linear} that the only non-zero components of $T$ are $T_{\\rep{2}_R \\rep{1}}^{\\rep{2}_L \\rep{1}}$ and $T_{\\rep{2}_R \\rep{3}}^{\\rep{2}_L \\rep{3}}$, which both have domain and range within $\\hs{H}_f = (\\hs{H}_F)_i$, where $\\epsilon$ acts as the identity. Thus, $T$ commutes with $\\epsilon_F$.\n\nTo prove the next part of the claim, it suffices to show that any left and right $\\alg{A}_{LR}$-linear operator on $\\hs{H}_F$ commutes with $\\epsilon_F$. But again, if $K \\in \\bdd^{\\textup{LR}}_{\\alg{A}_{LR}}(\\hs{H}_F)$, then the only non-zero components of $K$ are of the form $K_{\\alpha\\beta}^{\\alpha\\beta}$, each of which therefore has both domain and range either within $\\hs{H}_f$ or $\\hs{H}_{\\overline{f}} = J_F \\hs{H}_f$, so that $K$ commutes with $\\epsilon_F$. The last part of the claim is then an immediate consequence of the first two parts.\n\\end{proof}\n\nThus, by Proposition~\\ref{diracs0reduction}, we have that\n\\begin{equation}\n \\ms{D}_0(\\alg{A}_{LR},\\hs{H}_F,\\gamma_F,J_F) = \\ms{D}_0(\\alg{A}_{LR},\\hs{H}_F,\\gamma_F,J_F,\\epsilon_F) \\cong \\ms{D}_0(\\alg{A}_{LR},\\hs{H}_f,\\gamma_f)\n\\end{equation}\nand\n\\begin{equation}\n \\ms{D}(\\alg{A}_{LR},\\hs{H}_F,\\gamma_F,J_F) = \\ms{D}(\\alg{A}_{LR},\\hs{H}_F,\\gamma_F,J_F,\\epsilon_F) \\cong \\ms{D}(\\alg{A}_{LR},\\hs{H}_f,\\gamma_f),\n\\end{equation}\nwhere $(\\hs{H}_f,\\gamma_f) = ((\\hs{H}_F)_i,(\\gamma_F)_i)$ is the orientable even $\\alg{A}_{LR}$-bimodule with signed multiplicity matrix\n\\[\n \\mu_f =\n N \\begin{pmatrix}\n 0 & 0 & 0 & 0 & 0 & 0\\\\\n 0 & 0 & 0 & 0 & 0 & 0\\\\\n +1 & 0 & 0 & 0 & +1 & 0\\\\\n -1 & 0 & 0 & 0 & -1 & 0\\\\\n 0 & 0 & 0 & 0 & 0 & 0\\\\\n 0 & 0 & 0 & 0 & 0 & 0\n\\end{pmatrix}.\n\\]\nIn particular, then, $(\\hs{H}_F,\\gamma_F,J_F)$ as a real $\\alg{A}_{LR}$-bimodule admits no \\term{off-diagonal} Dirac operators, that is, Dirac operators with non-zero $P_{-i} D P_i : \\hs{H}_f \\to \\hs{H}_{\\overline{f}}$, or equivalently, that have non-vanishing commutator with $\\epsilon_F$. Let us now examine $\\ms{D}_0(\\alg{A}_{LR},\\hs{H}_F,\\gamma_F,J_F)$ and $\\ms{D}(\\alg{A}_{LR},\\hs{H}_F,\\gamma_F,J_F)$, or rather, $\\ms{D}_0(\\alg{A}_{LR},\\hs{H}_f,\\gamma_f)$ and $\\ms{D}(\\alg{A}_{LR},\\hs{H}_f,\\gamma_f)$, in more detail.\n\nFirst, it follows from the form of $\\mu_f$ and Proposition~\\ref{linear} that $\\bdd^{\\textup{L}}_{\\alg{A}_{LR}}(\\hs{H}_f^\\textup{even},\\hs{H}_f^\\textup{odd})$ vanishes, whilst\n\\[\n \\bdd^{\\textup{R}}_{\\alg{A}_{LR}}(\\hs{H}_f^\\textup{even},\\hs{H}_f^\\textup{odd}) = M_{2N}(\\field{C}) \\oplus (M_{2N}(\\field{C}) \\otimes 1_3) \\cong M_{2N}(\\field{C}) \\oplus M_{2N}(\\field{C}).\n\\]\nso that any Dirac operator on $\\hs{H}_f$ (and hence on $\\hs{H}_F$) is completely specified by a choice of $M_{\\rep{2}_L \\rep{1}}^{\\rep{2}_R}$, $M_{\\rep{2}_L \\rep{3}}^{\\rep{2}_R} \\in M_{2N}(\\field{C})$. Indeed, if $(m^\\textup{even},m^\\textup{odd})$ denotes the pair of multiplicity matrices of $(\\hs{H}_f,\\gamma_f)$, then, in the notation of subsection~\\ref{ko6},\n\\[\n \\ms{D}_0(\\alg{A}_{LR},m^\\textup{even},6) = M_{2N}(\\field{C}) \\oplus M_{2N}(\\field{C}).\n\\]\nAt the same time,\n\\[\n \\U^{\\textup{LR}}_{\\alg{A}_{LR}}(\\hs{H}_f^\\textup{even}) = (1_2 \\otimes \\U(N)) \\oplus (1_2 \\otimes \\U(N) \\otimes 1_3) \\cong \\U(N) \\times \\U(N) =: \\U(\\alg{A}_{LR},m^\\textup{even})\n\\]\nand\n\\[\n \\U^{\\textup{LR}}_{\\alg{A}_{LR}}(\\hs{H}_f^\\textup{odd}) = (1_2 \\otimes \\U(N)) \\oplus (1_2 \\otimes \\U(N) \\otimes 1_3) \\cong \\U(N) \\times \\U(N) =: \\U(\\alg{A}_{LR},m^\\textup{odd}).\n\\]\nIt then follows that\n\\begin{align}\n \\ms{D}(\\alg{A}_{LR},\\hs{H}_f,\\gamma_f) &\\cong \\U(\\alg{A}_{LR},m^\\textup{odd}) \\backslash \\ms{D}_0(\\alg{A}_{LR},m^\\textup{even},6) \/ \\U(\\alg{A}_{LR},m^\\textup{even}) \\\\\n &= \\bigl(\\U(N) \\backslash M_{2N}(\\field{C}) \/ \\U(N)\\bigr)^2,\n\\end{align}\nwhere $\\U(N)$ acts on the left by multiplication and on the right by multiplication by the inverse as $1_2 \\otimes U(N)$. The two factors of the form $\\U(N) \\backslash M_{2N}(\\field{C}) \/ \\U(N)$ can thus be viewed as the parameter spaces of the components $M_{\\rep{2}_L \\rep{1}}^{\\rep{2}_R}$ and $M_{\\rep{2}_L \\rep{3}}^{\\rep{2}_R}$, respectively. \n\nLet us now consider $(\\hs{H}_F,\\gamma_F,J_F,\\epsilon_F)$ as an $S^0$-real $\\alg{A}_F$-bimodule, so that the multiplicity matrices $(m^\\textup{even},m^\\textup{odd})$ of $(\\hs{H}_F,\\gamma_F)$ are given by\n\\[\n m^\\textup{even} = N\n\\begin{pmatrix}\n1 & 1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0 & 0\\\\\n1 & 0 & 0 & 1 & 0\\\\\n1 & 1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0 & 0\n\\end{pmatrix}, \\quad\nm^\\textup{odd} = N\n\\begin{pmatrix}\n1 & 0 & 1 & 1 & 0\\\\\n1 & 0 & 0 & 1 & 0\\\\\n0 & 0 & 0 & 0 & 0\\\\\n0 & 0 & 1 & 0 & 0\\\\\n0 & 0 & 0 & 0 & 0\n\\end{pmatrix} = (m^\\textup{even})^T.\n\\]\n\nNow it follows from the form of $(m^\\textup{even},m^\\textup{odd})$ that\n\\begin{multline*}\n \\bdd^{\\textup{R}}_{\\alg{A}_F}(\\hs{H}_F^\\textup{even},\\hs{H}_F^\\textup{odd}) = M_N(\\field{C})^{\\oplus 2} \\oplus M_{N \\times 2N}(\\field{C})^{\\oplus 2} \\oplus M_{N \\times 3N}(\\field{C})^{\\oplus 2} \\\\ \\oplus (M_{N \\times 2N}(\\field{C}) \\otimes 1_3)^{\\oplus 2},\n\\end{multline*}\nwhilst\n\\[\n \\ker(R_6) = \\mathfrak{sl}(N,\\field{C}) \\subseteq M_N(\\field{C})\n\\]\nfor the copy of $M_N(\\field{C})$ corresponding to $\\bdd^{\\textup{R}}_\\alg{A}((\\hs{H}_F^\\textup{even})_{\\rep{1}\\rep{1}},(\\hs{H}_F^\\textup{odd})_{\\rep{1}\\rep{1}})$. Since\n$M_N(\\field{C}) = \\Sym_N(\\field{C}) \\oplus \\mathfrak{sl}(N,\\field{C})$, $M_N(\\field{C})\/\\mathfrak{sl}(N,\\field{C})$ can be identified with $\\Sym_N(\\field{C})$, so that\n\\begin{align*}\n &\\ms{D}_0(\\alg{A}_F,\\hs{H}_F,\\gamma_F,J_F)\\\\  \\cong &\\bdd^{\\textup{R}}_{\\alg{A}_F}(\\hs{H}_F^\\textup{even},\\hs{H}_F^\\textup{odd}) \/ \\ker(R_6)\\\\\n = &\\Sym_N(\\field{C}) \\oplus M_N(\\field{C}) \\oplus M_{N \\times 2N}(\\field{C})^{\\oplus 2} \\oplus M_{N \\times 3N}(\\field{C})^{\\oplus 2} \\oplus (M_{N \\times 2N}(\\field{C}) \\otimes 1_3)^{\\oplus 2}.\n\\end{align*}\nThus, a Dirac operator $D$, which is specified by a choice of class \n\\[\n [M] \\in \\bdd^{\\textup{R}}_{\\alg{A}_F}(\\hs{H}_F^\\textup{even},\\hs{H}_F^\\textup{odd}) \/ \\ker(R_6),\n\\]\nis therefore specified in turn by the choice of the following matrices:\n\\begin{itemize}\n \\item $M_{\\rep{1}\\rep{1}}^{\\rep{1}} \\in \\Sym_N(\\field{C})$, $M_{\\rep{1}\\rep{1}}^{\\crep{1}} \\in M_N(\\field{C})$;\n \\item $M_{\\rep{2}\\rep{1}}^{\\rep{1}}$, $M_{\\rep{2}\\rep{1}}^{\\crep{1}} \\in M_{N \\times 2N}(\\field{C})$;\n \\item $M_{\\rep{3}\\rep{1}}^{\\rep{1}}$, $M_{\\rep{3}\\rep{1}}^{\\crep{1}} \\in M_{N \\times 3N}(\\field{C})$;\n \\item $M_{\\rep{2}\\rep{3}}^{\\rep{1}}$, $M_{\\rep{2}\\rep{3}}^{\\crep{1}} \\in M_{N \\times 2N}(\\field{C})$.\n\\end{itemize}\nIndeed, it follows that\n\\begin{multline}\n \\ms{D}_0(\\alg{A}_F,m^\\textup{even},6) = \\Sym_N(\\field{C}) \\oplus M_N(\\field{C}) \\oplus M_{N \\times 2N}(\\field{C})^{\\oplus 2} \\oplus M_{N \\times 3N}(\\field{C})^{\\oplus 2} \\\\ \\oplus M_{N \\times 2N}(\\field{C})^{\\oplus 2}.\n\\end{multline}\n\nNext, we have that $\\U(\\alg{A}_F,m^\\textup{even}) = \\U(N)^6$, with a copy of $\\U(N)$ corresponding to each of $(\\hs{H}_F^\\textup{even})_{\\rep{1}\\rep{1}}$, $(\\hs{H}_F^\\textup{even})_{\\rep{1}\\crep{1}}$, $(\\hs{H}_F^\\textup{even})_{\\rep{2}\\rep{1}}$, $(\\hs{H}_F^\\textup{even})_{\\rep{2}\\rep{3}}$, $(\\hs{H}_F^\\textup{even})_{\\rep{3}\\rep{1}}$, and $(\\hs{H}_F^\\textup{even})_{\\rep{3}\\crep{1}}$. Then, by Proposition~\\ref{moduli26},\n\\begin{equation}\n \\ms{D}(\\alg{A}_F,\\hs{H}_F,\\gamma_F,J_F) \\cong \\ms{D}_0(\\alg{A}_F,m^\\textup{even},6) \/ \\U(\\alg{A}_F,m^\\textup{even})\n\\end{equation}\nfor the action of $\\U(\\alg{A}_F,m^\\textup{even})$ on $\\ms{D}_0(\\alg{A}_F,m^\\textup{even},6)$ given by having the element $(U_{\\alpha\\beta}) \\in \\U(\\alg{A}_F,m^\\textup{even})$ act on $(M_{\\alpha\\beta}^\\gamma) \\in \\ms{D}_0(\\alg{A}_F,m^\\textup{even},6)$ by\n\\[\n M_{\\alpha\\beta}^\\gamma \\mapsto (1_{n_\\gamma} \\otimes \\overline{U_{\\beta\\gamma}}) M_{\\alpha\\beta}^\\gamma (1_{n_\\alpha} \\otimes U_{\\alpha\\beta}^*).\n\\]\nNote that in the notation of~\\cite{CM08}*{\\S\\S 13.4, 13.5}, for $(M_{\\alpha\\beta}^\\gamma) \\in \\ms{D}_0(\\alg{A}_F,m^\\textup{even},6)$,\n\\[\n M_{\\rep{1}\\rep{1}}^{\\rep{1}} = \\frac{1}{2}\\Upsilon_R,\n\\]\nso that the so-called Majorana mass term is already present in its final form, whilst for $U \\in \\U(\\alg{A}_F,m^\\textup{even})$,\n\\[\n U = (U_{\\rep{1}\\rep{1}},U_{\\rep{1}\\crep{1}},U_{\\rep{2}\\rep{1}},U_{\\rep{2}\\rep{3}},U_{\\rep{3}\\rep{1}},U_{\\rep{3}\\crep{1}}) = (\\overline{V_2},\\overline{V_1},V_3,W_3,\\overline{W_2},\\overline{W_1}).\n\\]\n\nFinally, let us compute the sub-moduli space $\\ms{D}(\\alg{A}_F,\\hs{H}_F,\\gamma_F,J_F;\\field{C}_F)$ for \n\\[\n \\field{C}_F = \\{(\\zeta,\\diag(\\zeta,\\overline{\\zeta}),0) \\in \\alg{A}_F \\mid \\lambda \\in \\field{C}\\} \\cong \\field{C}.\n\\]\nIt is easy to see that $[M] \\in \\bdd^{\\textup{R}}_{\\alg{A}_F}(\\hs{H}_F^\\textup{even},\\hs{H}_F^\\textup{odd}) \/ \\ker(R_6)$ yields an element of the subspace $\\ms{D}_0(\\alg{A}_F,\\hs{H}_F,\\gamma_F,J_F;\\field{C}_F)$ if and only if $M$ commutes with $\\lambda(\\field{C}_F)$, but this holds if and only if for all $\\zeta \\in \\field{C}$ and $\\beta \\in \\spec{\\alg{A}_F}$,\n\\begin{align*}\n \\zeta M_{\\rep{1}\\beta}^{\\rep{1}} &= M_{\\rep{1}\\beta}^{\\rep{1}} \\zeta, \n &\\overline{\\zeta} M_{\\rep{1}\\beta}^{\\crep{1}} &= M_{\\rep{1}\\beta}^{\\crep{1}} \\zeta,\\\\\n \\zeta M_{\\rep{2}\\beta}^{\\rep{1}} &= M_{\\rep{2}\\beta}^{\\rep{1}} (\\diag(\\zeta,\\overline{\\zeta}) \\otimes 1_N),\n &\\overline{\\zeta} M_{\\rep{2}\\beta}^{\\crep{1}} &= M_{\\rep{2}\\beta}^{\\crep{1}} (\\diag(\\zeta,\\overline{\\zeta}) \\otimes 1_N),\\\\\n 0 M_{\\rep{3}\\beta}^{\\rep{1}} &= M_{\\rep{3}\\beta}^{\\rep{1}} \\zeta,\n &0 M_{\\rep{3}\\beta}^{\\crep{1}} &= M_{\\rep{3}\\beta}^{\\crep{1}} \\zeta,\n\\end{align*}\nwhich is in turn equivalent to having $M_{\\rep{1}\\rep{1}}^{\\crep{1}}$, $M_{\\rep{3}\\rep{1}}^{\\rep{1}}$ and $M_{\\rep{3}\\rep{1}}^{\\crep{1}}$ all vanish, and\n\\[\n M_{\\rep{2}\\rep{1}}^{\\rep{1}} = \\begin{pmatrix} \\Upsilon_\\nu & 0 \\end{pmatrix}, \\quad M_{\\rep{2}\\rep{1}}^{\\crep{1}} = \\begin{pmatrix} 0 & \\Upsilon_e \\end{pmatrix}, \\quad M_{\\rep{2}\\rep{3}}^{\\rep{1}} = \\begin{pmatrix} \\Upsilon_u & 0 \\end{pmatrix}, \\quad M_{\\rep{2}\\rep{3}}^{\\crep{1}} = \\begin{pmatrix} 0 & \\Upsilon_d \\end{pmatrix},\n\\]\nfor $\\Upsilon_\\nu$, $\\Upsilon_e$, $\\Upsilon_u$, $\\Upsilon_d \\in M_N(\\field{C})$. One can check that our notation is consistent with that of~\\cite{CM08}*{\\S\\S 13.4, 13.5}. Indeed, if $\\ms{D}_0(\\alg{A}_F,m^\\textup{even},6;\\field{C}_F)$ denotes the subspace of $\\ms{D}_0(\\alg{A}_F,m^\\textup{even},6)$ corresponding to $\\ms{D}_0(\\alg{A}_F,\\hs{H}_F,\\gamma_F,J_F;\\field{C}_F)$, then\n\\begin{equation}\n \\ms{D}(\\alg{A}_F,\\hs{H}_F,\\gamma_F,J_F;\\field{C}_F) \\cong \\ms{D}_0(\\alg{A}_F,m^\\textup{even},6;\\field{C}_F) \/ \\U(\\alg{A}_F,m^\\textup{even}) \\cong \\ms{C}_q \\times \\ms{C}_l\n\\end{equation}\nfor\n\\begin{equation}\n \\ms{C}_q := \\bigl(\\U(N) \\times \\U(N)\\bigr) \\backslash \\bigl(M_N(\\field{C}) \\times M_N(\\field{C})\\bigr) \/ \\U(N),\n\\end{equation}\nwhere $\\U(N)$ acts diagonally by multiplication on the right, and\n\\[\n \\ms{C}_l := \\bigl(\\U(N) \\times \\U(N)\\bigr) \\backslash \\bigl(M_N(\\field{C}) \\times M_N(\\field{C}) \\times \\Sym_N(\\field{C})\\bigr) \/ \\U(N),\n\\]\nwhere $\\U(N) \\times \\U(N)$ acts trivially on $\\Sym_N(\\field{C})$ and $\\U(N)$ acts on $\\Sym_N(\\field{C})$ by\n\\[\n (V_2,\\Upsilon_R) \\mapsto V_2 \\Upsilon_R V_2^T;\n\\]\nnote that $\\ms{C}_q$ is the parameter space for the matrices $(\\Upsilon_u,\\Upsilon_d)$, whilst $\\ms{C}_l$ is the parameter space for the matrices $(\\Upsilon_\\nu,\\Upsilon_e,\\Upsilon_R)$. Thus we have recovered the sub-moduli space of Dirac operators considered by Chamseddine--Connes--Marcolli~\\cite{CCM07}*{\\S\\S 2.6, 2.7} ({\\it cf.\\\/}\\  also~\\cite{CM08}*{\\S\\S 13.4, 13.5}).\n\n\\section{Applications to the Recent Work of Chamseddine and Connes}\n\nIn this section, we reformulate the results of Chamseddine and Connes in~\\cites{CC08a,CC08b} and give new proofs thereof using the theory of bimodules and bilateral triples developed above.\n\nBefore continuing, recall that, up to automorphisms, the only real forms of $M_n(\\field{C})$ are $M_n(\\field{C})$, $M_n(\\field{R})$, and, if $n$ is even, $M_{n\/2}(\\field{H})$.\n\n\\subsection{Admissible real bimodules}\n\nWe begin by studying what Chamseddine and Connes call \\term{irreducible triplets}, namely, real $\\alg{A}$-bimodules satisfying certain representation-theoretic conditions, along the lines of~\\cite{CC08b}*{\\S 2}. However, we shall progress by adding Chamseddine and Connes's various requirements for irreducible triplets one by one, bringing us gradually to their classification of irreducible triplets. \n\nIn what follows, $\\alg{A}$ will once more denote a fixed real {\\ensuremath{C^*}}-algebra, and for $(\\hs{H},J)$ a real $\\alg{A}$-bimodule of odd $KO$-dimension, $\\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H};J)$ will denote the real \\ensuremath{\\ast}-subalgebra of $\\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H})$ consisting of elements commuting with $J$.\n\nLet us now introduce the first explict requirement for irreducible triplets.\n\n\\begin{definition}\n Let $(\\hs{H},J)$ be a real $\\alg{A}$-bimodule of odd $KO$-dimension. We shall say that $(\\hs{H},J)$ is \\term{irreducible} if $0$ and $1$ are the only projections in $\\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H};J)$.\n\\end{definition}\n\nTo proceed, we shall need the following:\n\n\\begin{lemma}\\label{irreduciblelemma}\n Let $(\\hs{H},J)$ be a real $\\alg{A}$-bimodule of odd $KO$-dimension $n \\bmod 8$ with multiplicity matrix $m$. Then\n\\begin{equation}\n \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H};J) \\cong\n\\begin{cases}\n  \\biggl(\\bigoplus_{\\alpha \\in \\spec{\\alg{A}}} M_{m_{\\alpha\\alpha}}(\\field{R}) \\biggr) \\oplus \\bigoplus_{\\substack{\\alpha,\\beta \\in \\spec{\\alg{A}}\\\\ \\alpha < \\beta}} M_{m_{\\alpha\\beta}}(\\field{C}), &\\text{if $n = 1$ or $7 \\bmod 8$,}\\\\\n \\biggl(\\bigoplus_{\\alpha \\in \\spec{\\alg{A}}} M_{m_{\\alpha\\alpha}\/2}(\\field{H}) \\biggr) \\oplus \\bigoplus_{\\substack{\\alpha,\\beta \\in \\spec{\\alg{A}}\\\\ \\alpha < \\beta}} M_{m_{\\alpha\\beta}}(\\field{C}), &\\text{if $n = 3$ or $5 \\bmod 8$.}\n\\end{cases}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n Let $T = (1_{n_\\alpha} \\otimes T_{\\alpha\\beta} \\otimes 1_{n_\\beta}) \\in \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H})$. Just as for Propositions~\\ref{real17unitary} and~\\ref{real35unitary}, one can show that $[T,J] = 0$ if and only if for all $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$, $T_{\\beta\\alpha} = \\overline{T_{\\alpha\\beta}}$ if $\\alpha \\neq \\beta$ and\n\\[\n T_{\\alpha\\alpha} \\in\n\\begin{cases}\n M_{m_{\\alpha\\alpha}}(\\field{R}), &\\text{if $n = 1$ or $7 \\bmod 8$,}\\\\\n M_{m_{\\alpha\\alpha}\/2}(\\field{H}), &\\text{if $n = 3$ or $5 \\bmod 8$.}\n\\end{cases}\n\\]\nThus, $T \\in \\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H};J)$ is completely specified by the matrices $T_{\\alpha\\alpha}$ and $T_{\\alpha\\beta}$ for $\\alpha > \\beta$, giving rise to the isomorphisms of the claim.\n\\end{proof}\n\nWe can now formulate the part of the results of~\\cite{CC08b}*{\\S 2} that depends only on this notion of irreducibility.\n\n\\begin{proposition}\n Let $(\\hs{H},J)$ be a real $\\alg{A}$-bimodule of odd $KO$-dimension $n \\bmod 8$ with multiplicity matrix $m$. Then $(\\hs{H},J)$ is irreducible if and only if one of the following holds:\n\\begin{enumerate}\n \\item There exists $\\alpha \\in \\spec{\\alg{A}}$ such that $m = 2^{(1-\\varepsilon)\/2} E_{\\alpha\\alpha}$;\n\\item There exist $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$, $\\alpha \\neq \\beta$, such that $m = E_{\\alpha\\beta} + E_{\\beta\\alpha}$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nBy definition, $(\\hs{H},J)$ is irreducible if and only if the only projections in the real {\\ensuremath{C^*}}-algebra $\\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H},J)$ are $0$ and $1$, but by Lemma~\\ref{irreduciblelemma}, this in turn holds if and only if one of the following holds:\n\\begin{enumerate}\n \\item $\\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H};J) \\cong \\field{R}$, so that $n = 1$ or $7 \\bmod 8$, and $m = E_{\\alpha\\alpha}$ for some $\\alpha \\in \\spec{\\alg{A}}$,\n \\item $\\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H};J) \\cong \\field{H}$, so that $n = 3$ or $5 \\bmod 8$, and $m = 2 E_{\\alpha\\alpha}$ for some $\\alpha \\in \\spec{\\alg{A}}$,\n \\item $\\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H};J) \\cong \\field{C}$, so that $m = E_{\\alpha\\beta} + E_{\\beta\\alpha}$ for some $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$, $\\alpha \\neq \\beta$,\n\\end{enumerate}\nwhich yields in turn the desired result.\n\\end{proof}\n\nWe shall call an irreducible odd $KO$-dimensional real $\\alg{A}$-bimod\\-ule $(\\hs{H},J)$ \\term{type A} if the first case holds, and \\term{type B} if the second case holds; Chamseddine and Connes's first and second case for irreducible triplets~\\cite{CC08b}*{Lemma 2.2} correspond to the type A and type B case, respectively. We shall also find it convenient to define the \\term{skeleton} $\\skel(\\hs{H},J)$ of such a bimodule as follows:\n\\begin{enumerate}\n \\item if $(\\hs{H},J)$ is type A, then $\\skel(\\hs{H},J) := \\{\\alpha\\}$, where $\\alpha \\in \\spec{\\alg{A}}$ is such that $\\mult[\\hs{H}] = 2^{\\frac{1-\\varepsilon}{2}} E_{\\alpha\\alpha}$;\n \\item if $(\\hs{H},J)$ is type B, then $\\skel(\\hs{H},J) := \\{\\alpha,\\beta\\}$, where $\\alpha$, $\\beta \\in \\spec{\\alg{A}}$, $\\alpha \\neq \\beta$, are such that $\\mult[\\hs{H}] = E_{\\alpha\\beta} + E_{\\beta\\alpha}$.\n\\end{enumerate}\n\nLet us now introduce the second explicit requirement for irreducible triplets.\n\n\\begin{definition}\n An $\\alg{A}$-bimodule $\\hs{H}$ is \\term{(left) separating} if there exists some $\\xi \\in \\hs{H}$ such that $\\lambda(\\alg{A})^\\prime \\xi = \\hs{H}$. Such a vector $\\xi$ is then called a \\term{separating vector} for $\\alg{A}$.\n\\end{definition}\n\nRecall that for a representation $\\hs{X}$ of a complex {\\ensuremath{C^*}}-algebra $\\alg{C}$, $\\xi \\in \\hs{X}$ is a separating vector if and only if the map $\\alg{C} \\to \\hs{X}$ given by $c \\mapsto c \\xi$ is injective.\n\n\\begin{lemma}\\label{separatinglemma}\n Let $p$, $q \\in \\semiring{N}$. There exists a separating vector $\\xi$ for the usual action of $M_p(\\field{C})$ on $\\field{C}^p \\otimes \\field{C}^q$ as $M_p(\\field{C}) \\otimes 1_q$ if and only if $p \\leq q$.\n\\end{lemma}\n\n\\begin{proof}\n Let $\\{e_i\\}_{i=1}^p$ be a basis for $\\field{C}^p$, and let $\\{f_j\\}_{j=1}^q$ be a basis for $\\field{C}^q$. \n\nFirst suppose that $p \\leq q$. Let $\\xi \\in \\field{C}^p \\otimes \\field{C}^q$ be given by $\\xi = \\sum_{i=1}^p e_i \\otimes f_i$. Then for any $a$, $b \\in M_p(\\field{C})$,\n\\[\n \\left(a \\otimes 1_q\\right) \\xi - \\left(b \\otimes 1_q\\right) \\xi = \\sum_{i=1}^p \\left(\\sum_{l=1}^p (a_i^l - b_i^l) e_l \\right) \\otimes f_i\n\\]\nso that by linear independence of the $e_i$ and $f_j$, the left-hand side vanishes if and only if for each $i$ and $l$, $a_i^l - b_i^l = 0$, {\\it i.e.\\\/}\\  $a = b$. Hence, $\\xi$ is indeed a separating vector.\n\nNow suppose that $p > q$. Then $\\dim_{\\field{C}} M_p(\\field{C}) - \\dim_{\\field{C}} \\field{C}^p \\otimes \\field{C}^q = p(p-q) > 0$, so that for any $\\xi \\in \\field{C}^p \\otimes \\field{C}^q$, the map $M_p(\\field{C}) \\mapsto \\field{C}^p \\otimes \\field{C}^q$ given by $a \\mapsto \\left(a \\otimes 1_q \\right) \\xi$ cannot possibly be injective, and hence $\\xi$ cannot possibly be separating.\n\\end{proof}\n\nWe can now reformulate that part of the results in~\\cite{CC08b}*{\\S 2} that depends only on irreducibility and the existence of a separating vector.\n\n\\begin{proposition}\n Let $(\\hs{H},J)$ be an irreducible real $\\alg{A}$-bimodule of odd $KO$-dimen\\-sion $n \\bmod 8$.\n\\begin{enumerate}\n \\item If $(\\hs{H},J)$ is type A, then it is separating;\n \\item If $(\\hs{H},J)$ is type B with skeleton $(\\alpha,\\beta)$, then $(\\hs{H},J)$ is separating if and only if $n_{\\alpha} = n_{\\beta}$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nFirst suppose that $(\\hs{H},J)$ is type A. Let $\\{\\alpha\\} = \\skel(\\hs{H},J)$, and let $m_n = 2^{(1-\\varepsilon)\/2}$. Then $\\hs{H} = \\field{C}^{n_\\alpha} \\otimes \\field{C}^{m_n} \\otimes \\field{C}^{n_\\alpha} = \\field{C}^{n_\\alpha} \\otimes \\field{C}^{m_n n_\\alpha}$, and the left action $\\lambda$ of $\\alg{A}$ on $\\hs{H}$ is thus given by $\\lambda_\\alpha \\otimes 1_{m_n n_{\\alpha}}$. Now\n\\[\n \\lambda(\\alg{A})^\\prime = \\left(\\lambda_{\\alpha}(\\alg{A}) \\otimes 1_{m_n n_{\\alpha}}\\right)^\\prime = \\left(M_{n_\\alpha}(\\field{C}) \\otimes 1_{m_n n_\\alpha}\\right)^\\prime,\n\\]\nso that the action $\\lambda$ of $\\alg{A}$ admits a separating vector if and only if the action of $M_{n_\\alpha}(\\field{C})$ as $M_{n_\\alpha}(\\field{C}) \\otimes 1_{m_n n_\\alpha}$ admits a separating vector, but by Lemma~\\ref{separatinglemma} this is indeed the case, as $n_\\alpha \\leq m_n n_\\alpha$.\n\nNow, suppose that $(\\hs{H},J)$ is type B. Let $\\{\\alpha,\\beta\\} = \\skel(\\hs{H},J)$. Then\n\\[\n \\hs{H} = (\\field{C}^{n_\\alpha} \\otimes \\field{C}^{n_\\beta}) \\oplus (\\field{C}^{n_\\beta} \\otimes \\field{C}^{n_\\alpha}),\n\\]\nand the left action $\\lambda$ of $\\alg{A}$ on $\\hs{H}$ is given by $\\lambda = (\\lambda_{\\alpha} \\otimes 1_{n_\\beta}) \\oplus (\\lambda_{\\beta} \\otimes 1_{n_\\alpha})$. Since $\\alpha \\neq \\beta$,\n\\begin{align*}\n \\lambda(\\alg{A})^\\prime &= \\left((\\lambda_{\\alpha}(\\alg{A}) \\otimes 1_{n_\\beta}) \\oplus (\\lambda_\\beta(\\alg{A}) \\otimes 1_{n_\\alpha})\\right)^\\prime\\\\ &= \\left((M_{n_\\alpha}(\\field{C}) \\otimes 1_{n_\\beta}) \\oplus (M_{n_\\beta}(\\field{C}) \\otimes 1_{n_\\alpha})\\right)^\\prime,\n\\end{align*}\nso that the action $\\lambda$ of $\\alg{A}$ admits a separating vector if and only if the action of $M_{n_\\alpha}(\\field{C}) \\oplus M_{n_\\beta}(\\field{C})$ as $(M_{n_\\alpha}(\\field{C}) \\otimes 1_{n_\\beta}) \\oplus (M_{n_\\beta}(\\field{C}) \\otimes 1_{n_\\alpha})$ admits a separating vector. Since $\\dim_{\\field{C}}M_{n_\\alpha}(\\field{C}) \\oplus M_{n_\\beta}(\\field{C}) - \\dim_{\\field{C}}\\hs{H} = (n_\\alpha - n_\\beta)^2$, if $n_\\alpha \\neq n_\\beta$ then no injective linear maps $M_{n_\\alpha}(\\field{C}) \\oplus M_{n_\\beta}(\\field{C}) \\to \\hs{H}$ can exist, and in particular, there exist no separating vectors for the action of $M_{n_\\alpha}(\\field{C}) \\oplus M_{n_\\beta}(\\field{C})$, and hence for $\\lambda$. Suppose instead that $n_\\alpha = n_\\beta = n$. Then\n\\[\n \\hs{H} = (\\field{C}^n \\otimes \\field{C}^n) \\oplus (\\field{C}^n \\otimes \\field{C}^n)\n\\]\nso that, since $\\alpha \\neq \\beta$, $\\lambda(\\alg{A})^\\prime = (M_n(\\field{C}) \\otimes 1_n)^\\prime \\oplus (M_n(\\field{C}) \\otimes 1_n)^\\prime$. Thus, if $\\xi$ is the separating vector for the action of $M_n(\\field{C})$ on $\\field{C}^n \\otimes \\field{C}^n$ given by the proof of Lemma~\\ref{separatinglemma}, then $\\xi\\oplus\\xi$ is also a separating vector for the action $\\lambda$ of $\\alg{A}$, and hence $(\\hs{H},J)$ is indeed separating.\n\\end{proof}\n\nLet us now introduce the final requirement for irreducible triplets; recall that the complex form of a real {\\ensuremath{C^*}}-algebra $\\alg{A}$ a real {\\ensuremath{C^*}}-algebra is denoted by $\\alg{A}_\\field{C}$.\n\n\\begin{definition}\n We shall call an $\\alg{A}$-bimodule $\\hs{H}$ \\term{complex-linear} if both left and right actions of $\\alg{A}$ on $\\hs{H}$ extend to $\\field{C}$-linear actions of $\\alg{A}_{\\field{C}}$, making $\\hs{H}$ into a complex $\\alg{A}_{\\field{C}}$-bimodule.\n\\end{definition}\n\nIt follows immediately that a $\\alg{A}$-bimodule $\\hs{H}$ is complex-linear if and only if for $m = \\mult[\\hs{H}]$, $m_{\\alpha\\beta} = 0$ whenever $\\alpha$ or $\\beta$ is conjugate-linear. In particular, by Proposition~\\ref{converseorientable}, it follows that a complex-linear quasi-orientable graded bimodule is always orientable.\n\nWe can now reformulate Chamseddine and Connes's definition for irreducible triplets:\n\n\\begin{definition}\n An \\term{irreducible triplet} is a triplet $(\\alg{A},\\hs{H},J)$, where $\\alg{A}$ is a finite-dimensional real {\\ensuremath{C^*}}-algebra and $(\\hs{H},J)$ is a complex-linear, separating, irreducible real $\\alg{A}$-bimodule of odd $KO$-dimension such that the left action of $\\alg{A}$ on $\\hs{H}$ is faithful.\n\\end{definition}\n\nNote that for $\\hs{H}$ a real $\\alg{A}$-bimodule, the left action of $\\alg{A}$ is faithful if and only if the right action is faithful.\n\nBy combining the above results, we immediately obtain Cham\\-sed\\-dine and Connes's classification of irreducible triplets:\n\n\\begin{proposition}[Chamseddine--Connes~\\cite{CC08b}*{Propositions 2.5, 2.8}]\n Let $\\alg{A}$ be a finite-dimensional real {\\ensuremath{C^*}}-algebra, and let $(\\hs{H},J)$ be a real $\\alg{A}$-bimodule of odd $KO$-dimension $n \\bmod 8$. Then $(\\alg{A},\\hs{H},J)$ is an irreducible triplet if and only if one of the following cases holds:\n\\begin{enumerate}\n \\item There exists $n \\in \\semiring{N}$ such that $\\alg{A} = M_{k}(\\field{K})$ for a real form $M_{k}(\\field{K})$ of $M_n(\\field{C})$, and\n\\begin{equation}\n \\mult[\\hs{H}] = 2^{(1-\\varepsilon)\/2} E_{\\rep{n}\\rep{n}};\n\\end{equation}\n \\item There exists $n \\in \\semiring{N}$ such that $\\alg{A} = M_{k_1}(\\field{K}_1) \\oplus M_{k_2}(\\field{K}_2)$ for real forms $M_{k_1}(\\field{K}_1)$ and $M_{k_2}(\\field{K}_2)$ of $M_n(\\field{C})$, and \n\\begin{equation}\n \\mult[\\hs{H}] = E_{\\rep{n}_1 \\rep{n}_2} + E_{\\rep{n}_2 \\rep{n_1}}.\n\\end{equation}\n\\end{enumerate}\n\\end{proposition}\n\n\\subsection{Gradings}\n\nWe now seek a classification of gradings inducing even $KO$-dimen\\-sional real bimodules from irreducible triplets.\n\n\\begin{definition}\n Let $(\\alg{A},\\hs{H},J)$ be an irreducible triplet. We shall call a $\\ring{Z}_2$-grading $\\gamma$ on $\\hs{H}$ as a Hilbert space \\term{compatible} with $(\\alg{A},\\hs{H},J)$ if and only if the following conditions all hold:\n\\begin{enumerate}\n \\item For every $a \\in \\alg{A}$, $\\gamma \\lambda(a) \\gamma \\in \\lambda(\\alg{A})$;\n \\item The operator $\\gamma$ either commutes or anticommutes with $J$.\n\\end{enumerate}\n\\end{definition}\n\nGiven a compatible grading $\\gamma$ for an irreducible triplet $(\\alg{A},\\hs{H},J)$, one can view $(\\hs{H},\\gamma,J)$ as a real $\\alg{A}^\\textup{even}$-bimodule of even $KO$-dimension, for $\\alg{A}^\\textup{even} = \\{a \\in \\alg{A} \\mid [\\lambda(a),\\gamma] = 0\\}$, with $KO$-dimension specified by the values of $\\varepsilon$ and $\\varepsilon^{\\prime\\prime}$ such that $J^2 = \\varepsilon$, $\\gamma J = \\varepsilon^{\\prime\\prime} J \\gamma$.\n\nNow, recall that a $\\ring{Z}_2$-grading on a real {\\ensuremath{C^*}}-algebra $\\alg{A}$ is simply an automorphism $\\Gamma$ on $\\alg{A}$ satisfying $\\Gamma^2 = \\Id$; we call such a grading \\term{admissible} if and only if $\\Gamma$ extends to a $\\field{C}$-linear grading on $\\alg{A}_\\field{C}$. Thus, if $(\\alg{A},\\hs{H},J)$ is an irreducible triplet and $\\gamma$ is a grading on $\\hs{H}$, then $\\gamma$ satisfies the first condition for compatibility if and only if there exists some admissible grading $\\Gamma$ on $\\alg{A}$ such that $\\Ad_\\gamma \\circ \\lambda = \\lambda \\circ \\Gamma$, where $\\Ad_x$ denotes conjugation by $x$.\n\n\\begin{lemma}\\label{admissiblegrading}\n Let $M_{k}(\\field{K})$ be a real form of $M_n(\\field{C})$, and let $\\alpha \\in \\Aut(M_n(\\field{C}))$. Then $\\alpha$ is an admissible grading on $M_{k}(\\field{K})$ if and only if there exists a self-adjoint unitary $\\gamma$ in $M_{k}(\\field{K})$ or $i M_{k}(\\field{K})$, such that $\\alpha = \\Ad_\\gamma$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $\\alpha$ is an admissible grading. Let $\\field{K}_0$ be $\\field{C}$ if $\\field{K}=\\field{C}$, and $\\field{R}$ otherwise. Then $M_k(\\field{K})$ is central simple over $\\field{K}_0$, so that there exists some invertible element $S$ of $M_k(\\field{K})$ such that $\\alpha = \\Ad_S$. Since $\\alpha$ respects the involution, for any $A \\in M_k(\\field{K})$ we must have\n\\[\n (S^{-1})^*A^*S^* = (SAS^{-1})^* = \\alpha(A)^* = \\alpha(A^*) = SA^*S^{-1},\n\\]\n{\\it i.e.\\\/}\\  $[A,S^*S]=0$, so that $S^*S$ is a positive central element of $M_k(\\field{K})$, and hence $S^*S = c1$ for some $c > 0$. Thus, $U = c^{-1\/2}S$ is a unitary element of $M_k(\\field{K})$ such that $\\alpha = \\Ad_U$. Now, recall that $\\alpha^2 = \\Id$, so that $\\Ad_{U^2} = \\Id$, and hence $U^2 = \\zeta 1$ for some $\\zeta \\in \\mathbb{T} \\cap \\field{K}_0$. If $\\field{K} = \\field{C}$, then one can simply set $\\gamma = \\overline{\\lambda}U$ for $\\lambda$ is a square root of $\\zeta$. Otherwise, $U^2 = \\pm 1$, so that if $U^2 = 1$, set $\\gamma = U \\in M_k(\\field{K})$, and if $U^2 = -1$, set $\\gamma = iU \\in i M_k(\\field{K})$.\n\nOn the other hand, if $\\gamma$ is a self-adjoint unitary in either $M_k(\\field{K})$ or $i M_k(\\field{K})$, then $\\Ad_\\gamma$ is readily seen to be an admissible grading on $M_k(\\field{K})$. \n\\end{proof}\n\nLet us now give the classification of compatible gradings for a type A irreducible triplet; it is essentially a generalisation of \\cite{CC08b}*{Lemma 3.1}.\n\n\\begin{proposition}\n Let $(\\alg{A},\\hs{H},J)$ be a type A irreducible triplet of odd $KO$-dimen\\-sion $n \\bmod 8$, so that $\\alg{A}$ is a real form $M_k(\\field{K})$ of $M_n(\\field{C})$ for some $n$, and let $\\gamma$ be a grading on $\\hs{H}$ as a Hilbert space.  Then $\\gamma$ is compatible if and only if there exists a self-adjoint unitary $g$ in $M_k(\\field{K})$ or $i M_k(\\field{K})$ such that\n\\begin{equation}\n \\gamma = \\pm g \\otimes 1_{m_k} \\otimes g^T,\n\\end{equation}\nin which case $\\gamma$ necessarily commutes with $J$.\n\\end{proposition}\n\n\\begin{proof}\nLet $m_n = 2^{(1-\\varepsilon)\/2}$. Then $\\hs{H} = \\field{C}^n \\otimes \\field{C}^{m_n} \\otimes \\field{C}^n$, and for all $a \\in \\alg{A}$,\n\\[\n \\lambda(a) = \\lambda_\\alpha(a) \\otimes 1_{m_n} \\otimes 1_n = a \\otimes 1_{m_nk} \\otimes 1_n, \\quad \\rho(a) = 1_n \\otimes 1_{m_n} \\otimes \\lambda_\\alpha(a)^T = 1_n \\otimes 1_{m_n} \\otimes a^T.\n\\]\n\nSuppose that $\\gamma$ is compatible. Then by Lemma~\\ref{admissiblegrading} there exists some self-adjoint unitary $g$ in either $M_k(\\field{K})$ or $i M_k(\\field{K})$ such that for all $a \\in \\alg{A}$,\n\\[\n \\gamma(a \\otimes 1_{m_n} \\otimes 1_n)\\gamma = (gag) \\otimes 1_{m_k} \\otimes 1_n.\n\\]\nNow, let $\\gamma_0 = g \\otimes 1_{m_n} \\otimes g^T$. Then, by construction, $\\gamma_0$ is a compatible grading for $(\\alg{A},\\hs{H},J)$ that induces the same admissible grading on $\\alg{A}$ as $\\gamma$, and moreover commutes with $J$. Then $\\nu := \\gamma \\gamma_0 \\in \\U^{\\textup{LR}}_\\alg{A}(\\hs{H};J)$, so that $\\nu = 1_n \\otimes \\nu_{\\rep{n}\\rep{n}} \\otimes 1_n$ for some\n\\[\n \\nu_{\\rep{n}\\rep{n}} \\in\n\\begin{cases}\n \\{\\pm1\\}, &\\text{if $n = 1$ or $7 \\bmod 8$,}\\\\\n \\SU(2), &\\text{if $k = 3$ or $5 \\bmod 8$.}\n\\end{cases}\n\\]\nThus $\\gamma = g \\otimes \\nu_{\\rep{n}\\rep{n}} \\otimes g^T$, and hence, since $\\gamma$ is self-adjoint, $\\nu_{\\rep{n}\\rep{n}}$ must also be self-adjoint. Therefore $\\nu_{\\alpha\\alpha} = \\pm 1_{m_k}$, or equivalently, $\\gamma^\\prime = \\pm \\gamma$.\n\nOn the other hand, if $g$ is a self-adjoint unitary in either $M_k(\\field{K})$ or $i M_k(\\field{K})$, then $\\gamma = g \\otimes 1_{m_k} \\otimes g^T$ is certainly a compatible grading that commutes with $J$.\n\\end{proof}\n\nThus, irreducible triplets can only give rise to real $\\alg{A}^\\textup{even}$-bimodules of $KO$-dimension $0$ or $4 \\bmod 8$.\n\nLet us now turn to the type B case.\n\n\\begin{proposition}\n Let $(\\alg{A},\\hs{H},J)$ be a type B irreducible triplet of odd $KO$-dimen\\-sion $n \\bmod 8$, so that for some $n \\in \\semiring{N}$, $\\alg{A} = M_{k_1}(\\field{K}_1) \\oplus M_{k_2}(\\field{K}_2)$ for real forms $M_{k_1}(\\field{K}_1)$ and $M_{k_2}(\\field{K}_2)$ of $M_n(\\field{C})$, and let $\\gamma$ be a grading on $\\hs{H}$ as a Hilbert space. Then $\\gamma$ is compatible if and only if one of the following holds:\n\\begin{enumerate}\n \\item There exist gradings $\\gamma_1$ and $\\gamma_2$ on $\\field{C}^n$, with $\\gamma_j \\in M_{k_j}(\\field{K}_j)$ or $i M_{k_j}(\\field{K}_j)$, such that\n \\begin{equation}\n  \\gamma =\n  \\begin{pmatrix}\n   \\gamma_1 \\otimes \\gamma_2^T & 0\\\\\n   0 & \\varepsilon^{\\prime\\prime} \\gamma_2 \\otimes \\gamma_1^T\n  \\end{pmatrix},\n \\end{equation}\n in which case $\\gamma J = \\varepsilon^{\\prime\\prime} J \\gamma$, and if $\\gamma^\\prime$ is any other compatible grading, $\\Ad_{\\gamma^\\prime} = \\Ad_\\gamma$ if and only if $\\gamma^\\prime = \\pm \\gamma$.\n \\item One has that $\\field{K}_1 = \\field{K}_2 = \\field{K}$ and $k_1 = k_2 = k$, and there exist a unitary $u \\in M_k(\\field{K})$ and $\\eta \\in \\mathbb{T}$ such that\n \\begin{equation}\n  \\gamma =\n  \\begin{pmatrix}\n   0 & \\overline{\\eta} u^* \\otimes \\overline{u}\\\\\n   \\eta u \\otimes u^T & 0\n  \\end{pmatrix},\n \\end{equation}\n in which case $\\gamma$ necessarily commutes with $J$, and if $\\gamma^\\prime$ is any other compatible grading, $\\Ad_{\\gamma^\\prime} = \\Ad_\\gamma$ if and only if $\\gamma^\\prime = (\\zeta 1_{n^2} \\oplus \\overline{\\zeta} 1_{n^2})\\gamma$ for some $\\zeta \\in \\mathbb{T}$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nLet $\\gamma$ be a compatible grading. Then, with respect to the decomposition $\\hs{H} = (\\field{C}^n \\otimes \\field{C}^n) \\oplus (\\field{C}^n \\oplus \\field{C}^n)$, let us write\n\\[\n \\gamma = \n \\begin{pmatrix}\n  A & B\\\\\n  C & D\n \\end{pmatrix}\n\\]\nfor $A$, $B$, $C$ and $D \\in M_n(\\field{C})\\otimes M_n(\\field{C})$. Applying self-adjointness of $\\gamma$, we find that $A$ and $D$ must be self-adjoint, and that $B = C^*$, and then applying the fact that $\\gamma^2 = 1$, we find that\n\\[\n A^2 + C^*C = 1, \\qquad CA + DC = 0, \\qquad CC^* + D^2 = 1.\n\\]\nFinally, applying the condition that $\\gamma$ commutes or anticommutes with $J$, {\\it i.e.\\\/}\\  that $\\gamma J = \\varepsilon^{\\prime\\prime} J \\gamma$ for $\\varepsilon^{\\prime\\prime} = \\pm 1$, we find that\n\\[\n D = \\varepsilon^{\\prime\\prime} X A X, \\qquad C^* = \\varepsilon^{\\prime\\prime} X C X,\n\\]\nwhere $X$ is the antiunitary on $\\field{C}^n \\otimes \\field{C}^n$ given by $X : \\xi_1 \\otimes \\xi_2 \\mapsto \\overline{\\xi_2} \\otimes \\overline{\\xi_1}$.\n\nNow, since $\\gamma$ is compatible, and since $(1,0)$ and $(0,1)$ are projections in $\\alg{A}$ satisfying $(1,0) + (0,1) = 1$, there exist projections $P$ and $Q$ in $\\alg{A}$ such that\n\\[\n \\Ad_\\gamma \\lambda(1,0) = \\lambda(P,1-Q), \\qquad \\Ad_\\gamma \\lambda(0,1) = \\lambda(1-P,Q),\n\\]\nthat is,\n\\[\n \\begin{pmatrix}\n  P \\otimes 1_n & 0\\\\\n  0 & (1-Q) \\otimes 1_n\n \\end{pmatrix} =\n \\gamma\n \\begin{pmatrix}\n  1 & 0\\\\\n  0 & 0\n \\end{pmatrix}\n \\gamma =\n \\begin{pmatrix}\n  A^2 & AC^*\\\\\n  CA & CC^*\n \\end{pmatrix}\n\\]\nand\n\\[\n \\begin{pmatrix}\n  (1-P) \\otimes 1_n & 0\\\\\n  0 & Q \\otimes 1_n\n \\end{pmatrix} =\n \\gamma\n \\begin{pmatrix}\n  0 & 0\\\\\n  0 & 1\n \\end{pmatrix}\n \\gamma =\n \\begin{pmatrix}\n  C^*C & C^*D\\\\\n  DC & D^2\n \\end{pmatrix}.\n\\]\nThus, $A$ is a self-adjoint partial isometry with support and range projection $P \\otimes 1_n$, $D$ is a self-adjoint partial isometry with support and range projection $Q \\otimes 1_n$, and $C$ is a partial isometry with support projection $(1-P)\\otimes 1_n$ and range projection $(1-Q)\\otimes 1_n$.\n\nNow, recalling that $D = \\varepsilon^{\\prime\\prime} X A X$, we see that\n\\[\n Q \\otimes 1_n = D^2 = X A^2 X = X P \\otimes 1_n X = 1_n \\otimes \\overline{P}.\n\\]\nIf $Q = 0$, then certainly $P = 0$. Suppose instead that $Q \\neq 0$, and let $\\xi \\in Q\\field{C}^n \\otimes \\field{C}^n$ be non-zero. Then\n\\[\n \\Id_{\\xi \\otimes \\field{C}^n} = (Q \\otimes 1_n)|_{\\xi \\otimes \\field{C}^n} = (1 \\otimes \\overline{P})|_{\\xi \\otimes \\field{C}^n},\n\\]\nso that $P = 1$ and hence $Q = 1$ also. We therefore have two possible cases:\n\\begin{enumerate}\n \\item We have\n \\[\n  \\gamma =\n  \\begin{pmatrix}\n   A & 0\\\\\n   0 & \\varepsilon^{\\prime\\prime} X A X\n  \\end{pmatrix}\n \\]\n for $A$ a grading on $\\field{C}^n \\otimes \\field{C}^n$;\n \\item We have\n \\[\n  \\gamma =\n  \\begin{pmatrix}\n   0 & C^*\\\\\n   C & 0\n  \\end{pmatrix}\n \\]\n for $C$ a unitary on $\\field{C}^n \\otimes \\field{C}^n$ such that $C^* = (-1)^m X C X$.\n\\end{enumerate}\n \n First suppose that the first case holds. Then, on the one hand, $\\Ad_A|_{M_n(\\field{C}) \\otimes 1_n}$ induces an admissible grading for $M_{k_1}(\\field{K}_1)$, so that there exists a self-adjoint unitary $\\gamma_1$ in either $M_{k_1}(\\field{K}_1)$ or $i M_{k_1}(\\field{K}_1)$ such that $\\Ad_A|_{M_n(\\field{C}) \\otimes 1_n} = \\Ad_{\\gamma_1 \\otimes 1_n}$, and on the other hand, $\\Ad_{\\varepsilon^{\\prime\\prime} X A X}|_{M_n(\\field{C}) \\otimes 1_n}$ induces an admissible grading for $M_{k_2}(\\field{K}_2)$, so that there exists a self-adjoint unitary $\\gamma_2$ in $M_{k_2}(\\field{K}_1)$ or $i M_{k_2}(\\field{K}_1)$ such that $\\Ad_{\\varepsilon^{\\prime\\prime} X A X}|_{M_n(\\field{C}) \\otimes 1_n} = \\Ad_{\\gamma_2 \\otimes 1_n}$. Since for $a \\otimes b \\in M_n(\\field{C}) \\otimes M_n(\\field{C})$ we can write\n\\[\n a \\otimes b = (a \\otimes 1_n) X (\\overline{b} \\otimes 1_n) X,\n\\]\nit therefore follows that $\\Ad_A = \\Ad_{\\gamma_1 \\otimes \\gamma_2^T}$ on the central simple algebra $M_n(\\field{C}) \\otimes M_n(\\field{C}) \\cong M_{n^2}(\\field{C})$ over $\\field{C}$. Hence, there exists some non-zero $\\eta \\in \\field{C}$ such that $A = \\eta \\gamma_1 \\otimes \\gamma_2^T$, and since both $A$ and $\\gamma_1 \\otimes \\gamma_2^T$ are self-adjoint and unitary, it follows that $\\eta = \\pm 1$. Absorbing $\\pm 1$ into $\\gamma_1$ or $\\gamma_2$, we therefore find that\n\\[\n \\gamma =\n \\begin{pmatrix}\n  \\gamma_1 \\otimes \\gamma_2^T & 0\\\\\n  0 & \\varepsilon^{\\prime\\prime} \\gamma_2 \\otimes \\gamma_1^T\n \\end{pmatrix}.\n\\]\nOn the other hand, $\\gamma$ so constructed is readily seen to be a compatible grading satisfying $\\gamma J = \\varepsilon^{\\prime\\prime} J \\gamma$.\n\nNow suppose that the second case holds. Then, since $\\gamma$ is compatible, it is clear that the automorphisms $\\alpha$, $\\beta$ of $M_n(\\field{C})$ specified by\n\\[\n \\alpha(a) \\otimes 1_n = C (a \\otimes 1_n) C^*, \\qquad \\beta(a) \\otimes 1_n = C^*(a \\otimes 1_n) C,\n\\]\nare inverses of each other, and that $\\alpha$, in particular, induces an isomorphism $M_{k_1}(\\field{K}_1) \\to M_{k_2}(\\field{K}_2)$, so that $\\field{K}_1 = \\field{K}_2 = \\field{K}$ and $k_1 = k_2 = k$. Next, by the proof of Lemma~\\ref{admissiblegrading}, there exists some unitary $u$ in $M_n(\\field{C})_I$ such that $\\alpha = \\Ad_u$, from which it follows that $\\beta = \\Ad_{u^*}$. By the same trick as above, we then find that $\\Ad_C = \\Ad_{u \\otimes u^T}$ on the central simple algebra $M_n(\\field{C}) \\otimes M_n(\\field{C}) \\cong M_{n^2}(\\field{C})$ over $\\field{C}$. Hence, there exists some non-zero $\\eta \\in \\field{C}$ such that $C = \\eta u \\otimes u^T$, and since both $C$ and $u \\otimes u^T$ are unitary, it follows that $\\eta \\in \\mathbb{T}$. Thus,\n\\[\n \\gamma =\n \\begin{pmatrix}\n  0 & \\overline{\\eta} u^* \\otimes \\overline{u}\\\\\n  \\eta u \\otimes u^T & 0\n \\end{pmatrix}.\n\\]\nOn the other hand, $\\gamma$ so constructed is readily seen to be a compatible grading satisfying $[\\gamma,J]=0$.\n\nFinally, let $\\gamma$ and $\\gamma^\\prime$ be two compatible gradings. Suppose that $\\Ad_\\gamma = \\Ad_{\\gamma^\\prime}$, and set $U = \\gamma^\\prime \\gamma$. Then, by construction, $U$ is a unitary element of $\\bdd^{\\textup{LR}}_\\alg{A}(\\hs{H};J)$, so that there exists some $\\zeta \\in \\mathbb{T}$ such that\n\\[\n U = \\zeta 1_{n^2} \\oplus \\overline{\\zeta} 1_{n^2}.\n\\]\nIf the second case holds, then nothing more can be said, but if the first case holds, so that\n\\[\n \\gamma =\n \\begin{pmatrix}\n  \\gamma_1 \\otimes \\gamma_2^T & 0\\\\\n  0 & \\varepsilon^{\\prime\\prime} \\gamma_2 \\otimes \\gamma_1^T\n \\end{pmatrix}\n\\]\nfor suitable $\\gamma_1$ and $\\gamma_2$, then\n\\[\n \\gamma^\\prime =\n \\begin{pmatrix}\n  \\zeta \\gamma_1 \\otimes \\gamma_2^T & 0\\\\\n  0 & \\varepsilon^{\\prime\\prime} \\overline{\\zeta} \\gamma_2 \\otimes \\gamma_1^T\n \\end{pmatrix},\n\\]\nso that by self-adjointness of $\\gamma^\\prime$, $\\gamma_1$ and $\\gamma_2$, we must have $\\zeta = \\pm 1$, as required.\n\\end{proof}\n\nThus, we can obtain a real bimodule of $KO$-dimension $6 \\bmod 8$ only from a type B irreducible triplet together with a compatible grading satisfying the first case of the last result.\n\n\\subsection{Even subalgebras and even $KO$-dimensional bimodules}\n\nWe now consider real bimodules of $KO$-di\\-men\\-sion $6 \\bmod 8$ obtained from irreducible triplets. Thus, let $(\\alg{A},\\hs{H},J)$ be a fixed type B irreducible triplet of $KO$-dimension $1$ or $7 \\bmod 8$, and let $\\gamma$ be a fixed compatible grading for $(\\alg{A},\\hs{H},J)$ anticommuting with $J$, so that for some $n \\in \\semiring{N}$,\n\\begin{itemize}\n \\item $\\alg{A} = M_{k_1}(\\field{K}_1) \\oplus M_{k_2}(\\field{K}_2)$ for real forms $M_{k_j}(\\field{K}_j)$ of $M_n(\\field{C})$;\n \\item $\\mult[\\hs{H}] = E_{\\rep{n}_1 \\rep{n}_2} + E_{\\rep{n}_2 \\rep{n}_1}$;\n \\item There exist self-adjoint unitaries $\\gamma_j \\in M_{k_j}(\\field{K}_j)$ or $i M_{k_j}(\\field{K}_j)$ with signature $(r_j,n - r_j)$ such that\n\\[\n \\gamma = \n\\begin{pmatrix}\n \\gamma_1 \\otimes \\gamma_2^T & 0\\\\\n 0 & -\\gamma_2 \\otimes \\gamma_1^T\n\\end{pmatrix}.\n\\]\n\\end{itemize}\nIt is worth noting that $(\\hs{H},J)$ admits, up to sign, a unique $S^0$-real structure, given by $\\epsilon = 1_{n^2} \\oplus -1_{n^2}$, which certainly commutes with $\\gamma$. We can exploit the symmetries present to simplify our discussion by taking, without loss of generality, $r_j > 0$, and requiring that $\\gamma_1 \\in i M_{k_1}(\\field{K}_1)$ only if $\\gamma_2 \\in i M_{k_2}(\\field{K}_2)$, and that $\\gamma_1 = 1_n$ only if $\\gamma_2 = 1_n$.\n\nOur main goal in this section is to give an explicit description of $\\alg{A}^\\textup{even}$ and of $(\\hs{H},\\gamma,J)$ as a real $\\alg{A}^\\textup{even}$-bimodule. To do so, however, we first need the following:\n\n\\begin{lemma}\n Let $M_k(\\field{K})$ be a real form of $M_n(\\field{C})$, let $g$ be a self-adjoint unitary in $M_k(\\field{K})$ or $i M_k(\\field{K})$, and let $r = \\nullity(g-1)$. Set $M_k(\\field{K})^g := \\{a \\in M_k(\\field{K}) \\mid [a,g] = 0\\}$.\n\\begin{itemize}\n \\item If $g \\in M_k(\\field{K})$, then $M_k(\\field{K})^g \\cong M_{kr\/n}(\\field{K}) \\oplus M_{k(n-r)\/n}(\\field{K})$;\n \\item If $g \\in i M_k(\\field{K})$, then $r = n\/2$ and\n \\[\n \tM_k(\\field{K})^g \\cong \\{(a,b) \\in M_{k\/2}(\\field{C})^2 \\mid b = \\overline{a}\\} \\cong M_{k\/2}(\\field{C}).\n \\]\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nLet $P^+ := \\frac{1}{2}(1 + g)$ and $P^- := \\frac{1}{2}(1-g)$, which are thus projections in $M_n(\\field{C})$ of rank $r$ and $n-r$, respectively. Define an injection $\\phi : M_k(\\field{K})^g \\mapsto M_{r}(\\field{C}) \\oplus M_{n-r}(\\field{C})$ by $\\phi(A) := (P^\\textup{even} A P^\\textup{even}, P^\\textup{odd}  A P^\\textup{odd})$.\n\nFirst, suppose that $g \\in M_k(\\field{K})$. Then $P^+$ and $P^-$ are also in $M_k(\\field{K})$, from which it immediately follows that $\\phi(M_k(\\field{K})^g) = M_{kr\/n}(\\field{K}) \\oplus M_{k(n-r)\/n}(\\field{K})$.\n\nSuppose instead that $g \\in i M_k(\\field{K})$ and  $\\field{K} \\neq \\field{C}$. Then $M_k(\\field{K}) = \\{A \\in M_n(\\field{C}) \\mid [A,I]=0\\}$ for a suitable antiunitary $I$ on $\\field{C}^n$ satisfying $I^2 = \\alpha 1$, where $\\alpha = 1$ if $\\field{K} = \\field{R}$ and $\\alpha = -1$ if $\\field{K} = \\field{H}$. Then $\\{g,I\\} = 0$, and hence, with respect to the decomposition $\\field{C}^n = P^+ \\field{C}^n \\oplus P^- \\field{C}^n \\cong \\field{C}^r \\oplus \\field{C}^{n-r}$,\n\\[\n I =\n\\begin{pmatrix}\n 0 & \\alpha \\tilde{I}^*\\\\\n \\tilde{I} & 0\n\\end{pmatrix},\n\\]\nwhere $\\tilde{I} = P^\\textup{odd} I P^\\textup{even}$ is an antiunitary $\\field{C}^r \\mapsto \\field{C}^{n-r}$. Thus, $n$ is even and $r = n\/2$, and taking $\\tilde{I}$, without loss of generality, to be complex conjugation on $\\field{C}^r$, for all $A \\in M_n(\\field{C})$ commuting with $g$, $[A,I] = 0$ if and only if $P^- A P^- = \\overline{P^+ A P^+}$, and hence $\\phi(M_k(\\field{K})^g) = \\{(a,\\overline{a}) \\mid a \\in M_{n\/2}(\\field{C})\\} \\cong M_{n\/2}(\\field{C})$.\n\\end{proof}\n\nIn light of the form of $\\gamma$, this last Lemma immediately implies the aforementioned explicit description of $\\alg{A}^\\textup{even}$ and $(\\hs{H},\\gamma,J)$:\n\n\\begin{proposition}\n Let $(m^\\textup{even},m^\\textup{odd}) = (m^\\textup{even},(m^\\textup{even})^T)$ be the pair of multiplicity matrices of $(\\hs{H},\\gamma,J)$ as an even $KO$-dimensional real $\\alg{A}^\\textup{even}$-bimodule. Let $r_i^\\prime = n - r_i$, and, when $n$ is even, let $c = n\/2$. Then:\n\\begin{enumerate}\n \\item If $\\gamma_1 \\in i M_{k_1}(\\field{K}_1)$, $\\gamma_2 \\in i M_{k_2}(\\field{K})$, then\n\\begin{equation}\n \\alg{A}^\\textup{even} = M_{c}(\\field{C}) \\oplus M_{c}(\\field{C}),\n\\end{equation}\nand\n\\begin{equation}\n m^\\textup{even} = E_{\\rep{c}_1 \\rep{c}_2} + E_{\\crep{c}_1 \\crep{c}_2} + E_{\\rep{c}_2 \\crep{c}_1} + E_{\\crep{c}_2 \\rep{c}_1};\n\\end{equation}\n \\item If $\\gamma_1 \\in i M_{k_1}(\\field{K}_1)$, $\\gamma_2 \\in M_{k_2}(\\field{K}) \\setminus \\{1_n\\}$, then\n\\begin{equation}\n \\alg{A}^\\textup{even} = M_{c}(\\field{C}) \\oplus M_{k_2 r_2\/n}(\\field{K}_2) \\oplus M_{k_2 r_2^\\prime \/n}(\\field{K}_2).\n\\end{equation}\nand\n\\begin{equation}\n m^\\textup{even} = E_{\\rep{c} \\rep{r}_2} + E_{\\crep{c} \\rep{r}_2^\\prime} + E_{\\rep{r}_2 \\crep{c}} + E_{\\rep{r}_2^\\prime \\rep{c}};\n\\end{equation}\n \\item If $\\gamma_1 \\in i M_{k_1}(\\field{K}_1)$, $\\gamma_2 = 1$, then\n\\begin{equation}\n \\alg{A}^\\textup{even} = M_{c}(\\field{C}) \\oplus M_{k_2}(\\field{K}_2),\n\\end{equation}\nand\n\\begin{equation}\n m^\\textup{even} = E_{\\rep{c} \\rep{n}} + E_{\\rep{n} \\crep{c}};\n\\end{equation}\n \\item If $\\gamma_1 \\in M_{k_1}(\\field{K}_1) \\setminus \\{1_n\\}$, $\\gamma_2 \\in M_{k_2}(\\field{K}_2) \\setminus \\{1_n\\}$, then\n\\begin{equation}\n \\alg{A}^\\textup{even} = M_{k_1 r_1 \/n}(\\field{K}_1) \\oplus M_{k_1 r_1^\\prime \/n}(\\field{K}_1) \\oplus M_{k_2 r_2 \/ n}(\\field{K}_2) \\oplus M_{k_2 r_2^\\prime \/n}(\\field{K}_2),\n\\end{equation}\nand\n\\begin{equation}\n m^\\textup{even} = E_{\\rep{r}_1 \\rep{r}_2} + E_{\\rep{r}_1^\\prime \\rep{r}_2^\\prime} + E_{\\rep{r}_2 \\rep{r}_1^\\prime} + E_{\\rep{r}_2^\\prime \\rep{r}_1};\n\\end{equation}\n \\item If $\\gamma_1 \\in M_{k_1}(\\field{K}_1) \\setminus \\{1_n\\}$, $\\gamma_2 = 1_n$, then\n\\begin{equation}\n \\alg{A}^\\textup{even} = M_{k_1 r_1 \/n}(\\field{K}_1) \\oplus M_{k_1 r_1^\\prime \/n}(\\field{K}_1) \\oplus M_{k_2}(\\field{K}_2),\n\\end{equation}\nand\n\\begin{equation}\n m^\\textup{even} = E_{\\rep{r}_1 \\rep{n}} + E_{\\rep{n} \\rep{r}_1^\\prime};\n\\end{equation}\n item If $\\gamma_1 = \\gamma_2 = 1_n$, then\n\\begin{equation}\n \\alg{A}^\\textup{even} = M_{k_1}(\\field{K}_1) \\oplus M_{k_2}(\\field{K}_2),\n\\end{equation}\nand\n\\begin{equation}\n m^\\textup{even} = E_{\\rep{n}_1 \\rep{n}_2}.\n\\end{equation}\n\\end{enumerate}\n\\end{proposition}\n\nOne can check in each case that $(\\hs{H},\\gamma)$ is quasi-orientable as an even $\\alg{A}^\\textup{even}$-bimodule. However, Propositions~\\ref{converseorientable} and~\\ref{intform} immediately imply the following:\n\n\\begin{corollary}\n The following are equivalent for $(\\hs{H},\\gamma)$ as an even $\\alg{A}^\\textup{even}$-bimodule:\n\\begin{enumerate}\n \\item $\\gamma_1 \\in M_{k_1}(\\field{K}_1)$ and $\\gamma_2 \\in M_{k_2}(\\field{K}_2)$;\n \\item $(\\hs{H},\\gamma)$ is orientable;\n \\item $(\\hs{H},\\gamma)$ has non-vanishing intersection form;\n \\item $(\\hs{H},\\gamma)$ is complex-linear.\n\\end{enumerate}\n\\end{corollary}\n\nThis then motivates us to restrict ourselves to the case where $\\gamma_1 \\in M_{k_1}(\\field{K}_1)$ and $\\gamma_2 \\in M_{k_2}(\\field{K}_2)$. Note, however, that in no case is Poincar{\\'e} duality possible.\n\n\\subsection{Off-diagonal Dirac operators}\n\nLet us now consider the slightly more general $S^0$-real $\\alg{A}^\\textup{even}$-bimodule $(\\hs{H}_F,\\gamma_F,J_F,\\epsilon_F)$ of $KO$-dimension $6 \\bmod 8$ given by taking the direct sum of $N$ copies of $(\\hs{H},\\gamma,J,\\epsilon)$, where $N \\in \\semiring{N}$. If we modify our earlier conventions slightly to allow for the summand $0$ in Wedderburn decompositions, we can therefore write\n\\begin{equation}\n \\alg{A}^\\textup{even} = M_{k_1 r_1 \/n}(\\field{K}_1) \\oplus M_{k_1 r_1^\\prime \/n}(\\field{K}_1) \\oplus M_{k_2 r_2 \/ n}(\\field{K}_2) \\oplus M_{k_2 r_2^\\prime}(\\field{K}_2),\n\\end{equation}\nso that $(\\hs{H}_F,\\gamma_F,J_F)$ is the real $\\alg{A}^\\textup{even}$-bimodule of $KO$-dimension $6 \\bmod 8$ with signed multiplicity matrix\n\\begin{equation}\n \\mu_F = N(E_{\\rep{r}_1 \\rep{r}_2} - E_{\\rep{r}_1 \\rep{r}_2^\\prime} - E_{\\rep{r}_1^\\prime \\rep{r}_2} + E_{\\rep{r}_1^\\prime \\rep{r}_2^\\prime} - E_{\\rep{r}_2 \\rep{r}_1} + E_{\\rep{r}_2 \\rep{r}_1^\\prime} + E_{\\rep{r}_2^\\prime \\rep{r}_1} - E_{\\rep{r}_2^\\prime \\rep{r}_1^\\prime}),\n\\end{equation}\nwhilst $(\\hs{H}_f,\\gamma_f) := ((\\hs{H}_F)_i,(\\gamma_F)_i)$ is the even $\\alg{A}^\\textup{even}$-bimodule with signed multiplicity matrix\n\\begin{equation}\n \\mu_f = N(E_{\\rep{r}_1 \\rep{r}_2} - E_{\\rep{r}_1 \\rep{r}_2^\\prime} - E_{\\rep{r}_1^\\prime \\rep{r}_2} + E_{\\rep{r}_1^\\prime \\rep{r}_2^\\prime}).\n\\end{equation}\nIt then follows also that $(\\hs{H}_{\\overline{f}},\\gamma_{\\overline{f}}) := (J_F \\hs{H}_f, - (J_F \\gamma_f J_F)|_{J_F \\hs{H}_f})$ is the even $\\alg{A}^\\textup{even}$-bimodule with signed multiplicty matrix\n\\[\n \\mu_{\\overline{f}} = -\\mu_f^T = N(- E_{\\rep{r}_2 \\rep{r}_1} + E_{\\rep{r}_2 \\rep{r}_1^\\prime} + E_{\\rep{r}_2^\\prime \\rep{r}_1} - E_{\\rep{r}_2^\\prime \\rep{r}_1^\\prime}).\n\\]\n\nNow, for $\\alg{C}$ a unital \\ensuremath{\\ast}-subalgebra of $\\alg{A}^\\textup{even}$, let us call a Dirac operator $D \\in \\ms{D}_0(\\alg{C},\\hs{H}_F,\\gamma_F,J_F)$ \\term{off-diagonal} if it does not commute with $\\epsilon_F$, or equivalently~\\cite{CC08b}*{\\S 4} if $[D,\\mathcal{Z}(\\alg{A})] \\neq \\{0\\}$. If $\\ms{D}_1(\\alg{C},\\hs{H}_F,\\gamma_F,J_F,\\epsilon_F) \\subseteq \\ms{D}_0(\\alg{C},\\hs{H}_F,\\gamma_F,J_F)$ is the subspace consisting of Dirac operators anti-commuting with $\\epsilon_F$, then, in fact,\n\\begin{equation}\n \\ms{D}_0(\\alg{C},\\hs{H}_F,\\gamma_F,J_F) = \\ms{D}_0(\\alg{C},\\hs{H}_F,\\gamma_F,J_F,\\epsilon_F) \\oplus \\ms{D}_1(\\alg{C},\\hs{H}_F,\\gamma_F,J_F,\\epsilon_F),\n\\end{equation}\nas can be seen from writing \n\\[\n D = \\frac{1}{2}\\{D,\\epsilon_F\\}\\epsilon_F + \\frac{1}{2}[D,\\epsilon_F]\\epsilon_F\n\\]\nfor $D \\in \\ms{D}_0(\\alg{C},\\hs{H}_F,\\gamma_F,J_F)$. Thus, non-zero off-diagonal Dirac operators exist for $(\\hs{H}_F,\\gamma_F,J_F,\\epsilon_F)$ as an $S^0$-real $\\alg{C}$-bimodule if and only if \n\\[\n \\ms{D}_1(\\alg{C},\\hs{H}_F,\\gamma_F,J_F,\\epsilon_F) \\neq \\{0\\}.\n\\]\nOur goal is to generalise Theorem 4.1 in~\\cite{CC08b}*{\\S 4} and characterise subalgebras of $\\alg{A}^\\textup{even}$ of maximal dimension admitting off-diagonal Dirac operators.\n\nThe following result is the first step in this direction:\n\n\\begin{proposition}[\\cite{CC08b}*{Lemma 4.2}]\n A unital \\ensuremath{\\ast}-subalgebra $\\alg{C} \\subseteq \\alg{A}^\\textup{even}$ admits off-diagonal Dirac operators if and only if there exists some partial unitary $T \\in \\mathcal{L}(\\field{C}^{r_1} \\oplus \\field{C}^{r_1^\\prime} \\oplus \\field{C}^{r_2} \\oplus \\field{C}^{r_2^\\prime})$ with support contained in one of $\\field{C}^{r_1}$ or $\\field{C}^{r_1^\\prime}$ and range contained in one of $\\field{C}^{r_2}$ or $\\field{C}^{r_2^\\prime}$, such that\n\\[\n \\alg{C} \\subseteq \\alg{A}(T) := \\{a \\in \\alg{A}^\\textup{even} \\mid [a,T] = [a^*,T] = 0\\}.\n\\]\n\\end{proposition}\n\n\\begin{proof}\n First note that the map $\\ms{D}_1(\\alg{C},\\hs{H}_F,\\gamma_F,J_F,\\epsilon_F) \\to \\mathcal{L}_\\alg{C}^1(\\hs{H}_f,\\hs{H}_{\\overline{f}})$ given by $D \\mapsto P_{-i} D P_i$ is an isomorphism, so that $\\alg{C}$ admits off-diagonal Dirac operators if and only if $\\mathcal{L}_\\alg{C}^1(\\hs{H}_f,\\hs{H}_{\\overline{f}}) \\neq \\{0\\}$. Since a map $S \\in \\mathcal{L}(\\hs{H}_f,\\hs{H}_{\\overline{f}})$ satisfies the generalised order one condition for $\\alg{C}$ if and only if $\\rho_{\\overline{f}}(c) S - S \\rho_f(C)$ is left $\\alg{C}$-linear for all $c \\in \\alg{C}$, $\\alg{C}$ admits off-diagonal Dirac operators only if\n\\[\n \\{S \\in \\bdd^{\\textup{L}}_\\alg{C}(\\hs{H}_f,\\hs{H}_{\\overline{f}}) \\mid -\\gamma_{\\overline{f}}S = S\\gamma_f\\} \\neq \\{0\\},\n\\]\nor equivalently,\n\\[\n \\alg{C} \\subseteq \\alg{A}_S := \\{a \\in \\alg{C}^\\textup{even} \\mid \\lambda_{\\overline{f}}(a)S = S\\lambda(a), \\; \\lambda_{\\overline{f}}(a^*)S = S\\lambda(a^*)\\}\n\\]\nfor some non-zero $S \\in \\mathcal{L}(\\hs{H}_f,\\hs{H}_{\\overline{f}})$ such that $-\\gamma_{\\overline{f}}S = S\\gamma_f$. \n\nNow, let $S \\in \\mathcal{L}(\\hs{H}_f,\\hs{H}_{\\overline{f}})$ be non-zero and such that $-\\gamma_{\\overline{f}}S = S\\gamma_f$. Then, the support of $S$ must have non-zero intersection with one of $(\\hs{H}_F)_{\\rep{r}_1\\rep{r}_2}$ or $(\\hs{H}_F)_{\\rep{r}_1^\\prime \\rep{r}_2^\\prime}$, and the range of $S$ must have non-zero intersection with one of $(\\hs{H}_F)_{\\rep{r}_2\\rep{r}_1}$ or $(\\hs{H}_F)_{\\rep{r}_2^\\prime \\rep{r}_1^\\prime}$. Thus, $S_{\\alpha\\beta}^{\\gamma\\delta} \\neq 0$ for some $(\\alpha,\\beta) \\in \\{(\\rep{r}_1,\\rep{r}_2),(\\rep{r}_1^\\prime,\\rep{r}_2^\\prime)\\}$ and $(\\gamma,\\delta) \\in \\{(\\rep{r}_2,\\rep{r}_1),(\\rep{r}_2^\\prime, \\rep{r}_1^\\prime)\\}$, so that $\\alg{A}_S \\subseteq \\alg{A}_{S_{\\alpha\\beta}^{\\gamma\\delta}}$. Let us now write\n\\[\n S_{\\alpha\\beta}^{\\gamma\\delta} = \\sum_{i} A_i \\otimes B_i\n\\]\nfor non-zero $A_i \\in M_{n_\\gamma \\times n_\\alpha}(\\field{C})$ and for linearly independent $B_i \\in M_{N n_\\delta \\times N n_\\beta}(\\field{C})$. Then for all $a \\in \\alg{A}^\\textup{even}$,\n\\[\n \\lambda_{\\overline{f}}(a)S - S\\lambda_f(a) = \\sum_i \\bigl(\\lambda_\\gamma(a) A_i - A_i \\lambda_\\alpha(a)\\bigr) \\otimes B_i,\n\\]\nso by linear independence of the $B_i$, $a \\in \\alg{A}_{S_{\\alpha\\beta}^{\\gamma\\delta}}$ if and only if for each $i$,\n\\[\n \\lambda_\\gamma(a) A_i = A_i \\lambda_\\alpha(a), \\quad \\lambda_\\gamma(a^*) A_i = A_i \\lambda_\\alpha(a),\n\\]\nand hence\n\\[\n \\alg{A}(S) \\subseteq \\alg{A}_{S_{\\alpha\\beta}^{\\gamma\\delta}} \\subseteq \\alg{A}(T_0) := \\{a \\in \\alg{A}^\\textup{even} \\mid [a,T_0] = 0, [a^*,T_1]=0\\}\n\\]\nfor $T_0 = A_1$, say, viewing $T_0$ and the elements of $\\alg{A}^\\textup{even}$ as operators on $\\field{C}^{r_1} \\oplus \\field{C}^{r_1^\\prime} \\oplus \\field{C}^{r_2} \\oplus \\field{C}^{r_2^\\prime}$. However, if $T_0 = P T$ is the polar decomposition of $T_0$ into a positive operator $P$ on $\\field{C}^{n_\\gamma}$ and a partial isometry $T : \\field{C}^{n_\\alpha} \\to \\field{C}^{n_\\gamma}$, it follows that $a \\in \\alg{A}^\\textup{even}$ commutes with $T_0$ only if it commutes with $T$, and hence $\\alg{A}_0 \\subseteq \\alg{A}(T_0) \\subseteq \\alg{A}(T)$, proving the one direction of the claim.\n\nNow suppose that $\\alg{C} = \\alg{A}(T)$ for a suitable partial isometry $T$, which we view as a partial isometry $\\field{C}^{n_{\\alpha_0}} \\to \\field{C}^{n_{\\gamma_0}}$ for some $\\alpha_0 \\in \\{\\rep{r}_1,\\rep{r}_1^\\prime\\}$, $\\gamma_0 \\in \\{\\rep{r}_2,\\rep{r}_2^\\prime\\}$. Then for any non-zero $\\Upsilon \\in M_N(\\field{C})$, we can define an element $S(\\Upsilon) \\in \\bdd^{\\textup{LR}}_\\alg{C}(\\hs{H}_f,\\hs{H}_{\\overline{f}})$ by setting\n\\[\n S(\\Upsilon)_{\\alpha\\beta}^{\\gamma\\delta} =\n\\begin{cases}\n T \\otimes \\Upsilon \\otimes T^* &\\text{if $\\alpha = \\delta = \\alpha_0$, $\\beta = \\gamma = \\gamma_0$,}\\\\\n 0 &\\text{otherwise,}\n\\end{cases}\n\\]\nwhich, as noted above, corresponds to a unique non-zero element of the space $\\ms{D}_1(\\alg{C},\\hs{H}_F,\\gamma_F,J_F,\\epsilon_F)$, so that $\\alg{C}$ does indeed admit off-diagonal Dirac operators.\n\\end{proof}\n\nIn light of the above characterisation, it suffices to consider subalgebras $\\alg{A}(T)$ for partial isometries $T : \\field{C}^{r_1} \\to \\field{C}^{r_2}$, so that\n\\begin{align}\n \\alg{A}(T) &= \\{(a_1,a_2,b_1,b_2) \\in \\alg{A}^\\textup{even} \\mid b_1 T = T a_1, \\; b_1^* T = T a_1^*\\}\\\\\n &\\cong \\alg{A}_0(T) \\oplus M_{k_1 r_1^\\prime \/ n}(\\field{K}_1) \\oplus M_{k_2 r_2^\\prime \/n}(\\field{K}_k),\n\\end{align}\nwhere\n\\begin{equation}\n \\alg{A}_0(T) := \\{(a,b) \\in M_{k_1 r_1\/n}(\\field{K}_1) \\oplus M_{k_2 r_2 \/n}(\\field{K}_2) \\mid b T = T a, \\; b^* T = T a^*\\},\n\\end{equation}\nso that our problem is reduced to that of maximising the dimension of $\\alg{A}_0(T)$.\n\nIt is reasonable to assume that $T$ is, in some sense, compatible with the algebraic structures of $M_{k_1 r_1\/n}(\\field{K}_1)$ and $M_{k_2 r_2\/n}(\\field{K}_2)$, so as to minimise the restrictiveness of the defining condition on $\\alg{A}_0(T)$, and hence maximise the dimension of $\\alg{A}_0(T)$. It turns out that this notion of compatibility takes the form of the following conditions on $T$:\n\\begin{enumerate}\n \\item The subspace $\\supp(T)$ of $\\field{C}^{r_1}$ is either a $\\field{K}_1$-linear subspace of $\\field{C}^{r_1} = \\field{K}_1^{k_1 r_1\/n}$ or, if $\\field{K}_1 = \\field{H}$, $\\supp(T) = E \\oplus \\field{C}$ for $E$ an $\\field{H}$-linear subspace of $\\field{C}^{r_1} = \\field{H}^{r_1\/2}$;\n \\item The subspace $\\im(T)$ of $\\field{C}^{r_2}$ is either a $\\field{K}_2$-linear subspace of $\\field{C}^{r_2} = \\field{K}_1^{k_2 r_2\/n}$ or, if $\\field{K}_2 = \\field{H}$, $\\im(T) = E \\oplus \\field{C}$ for $E$ an $\\field{H}$-linear subspace of $\\field{C}^{r_2} = \\field{H}^{r_2\/2}$.\n\\end{enumerate}\nNow, let $r = \\rank(T)$, let $d(r) = \\dim_{\\field{R}}(\\alg{A}_0(T))$, and let\n\\[\n d_i = \n\\begin{cases}\n 1 &\\text{if $\\field{K}_i = \\field{R}$,}\\\\\n 2 &\\text{if $\\field{K}_i = \\field{C}$,}\\\\\n \\frac{1}{2} &\\text{if $\\field{K}_i = \\field{H}$.}\n\\end{cases}\n\\]\nUnder these assumptions, then, one can show that\n\\begin{enumerate}\n \\item If $\\field{K}_1 = \\field{K}_2$ or $\\field{K}_2 = \\field{C}$, and, if $\\field{K}_1 = \\field{H}$, $r$ is even, then\n\\begin{equation}\n \\alg{A}_0(T) \\cong M_{k_1 r \/n}(\\field{K}_1) \\oplus M_{k_1 (r_1 - r)\/n}(\\field{K}_1) \\oplus M_{k_2 (r_2 - r)\/n}(\\field{K}_2),\n\\end{equation}\nand hence\n\\[\n d(r) = d_1 r^2 + d_1 (r - r_1)^2 + d_2 (r - r_2)^2;\n\\]\n \\item If $(\\field{K}_1,\\field{K}_2) = (\\field{H},\\field{R})$ and $r$ is odd, then\n\\begin{equation}\n \\alg{A}_0(T) \\cong \\bigl(M_{(r-1)\/2}(\\field{H}) \\cap M_{r-1}(\\field{R})\\bigr) \\oplus \\field{R} \\oplus M_{(r_2-r-1)\/2}(\\field{H}) \\oplus M_{r_1 - r}(\\field{R}),\n\\end{equation}\nand hence\n\\[\n d(r) = (r-1)^2 + 1 + \\frac{1}{2}(r - r_2 + 1)^2 + (r - r_1)^2;\n\\]\n \\item If $(\\field{K}_1,\\field{K}_2) = (\\field{H},\\field{C})$ and $r$ is odd, then\n\\begin{equation}\n \\alg{A}_0(T) \\cong M_{(r-1)\/2}(\\field{H}) \\oplus \\field{C} \\oplus M_{(r_2 - r - 1)\/2}(\\field{H}) \\oplus M_{r_1 - r}(\\field{C}),\n\\end{equation}\nand hence\n\\[\n d(r) = \\frac{1}{2}(r-1)^2 + 2 + \\frac{1}{2}(r-r_2+1)^2 + 2(r-r_1)^2;\n\\]\n \\item If $\\field{K}_1 = \\field{K}_2 = \\field{H}$ and $r$ is odd, then\n\\begin{equation}\n \\alg{A}_0(T) \\cong M_{(r-1)\/2}(\\field{H}) \\oplus \\field{C} \\oplus M_{(r_1 - r - 1)\/2}(\\field{H}) \\oplus M_{(r_2-r-1)\/2}(\\field{H}),\n\\end{equation}\nand hence\n\\begin{equation}\n d(r) = \\frac{1}{2}(r-1)^2 + 2 + \\frac{1}{2}(r-r_1+1)^2 + \\frac{1}{2}(r-r_2+1)^2.\n\\end{equation}\n\\end{enumerate}\nThe other cases are obtained easily, by symmetry, from the ones listed above.\n\nNow, let $R_{\\textrm max}$ be the set of all $r \\in \\{1, \\dotsc, \\min(r_1,r_2)\\}$ maximising the value of $d(r)$. By checking case by case, one can arrive at the following generalisation of Theorem 4.1 in~\\cite{CC08b}:\n\n\\begin{proposition}\n Let $T : \\field{C}^{r_1} \\to \\field{C}^{r_2}$ be a partial isometry. Then $\\alg{A}(T)$ attains maximal dimension only if $\\rank(T) \\in R_{\\textrm max}$, where $R_{\\textrm max} = \\{1\\}$ except in the following cases:\n\\begin{enumerate}\n \\item $(\\field{K}_1,\\field{K}_2)=(\\field{C},\\field{C})$ and $(r_1,r_2)=(2,2)$, in which case $R_{\\textrm max} = \\{2\\}$;\n \\item $(\\field{K}_1,\\field{K}_2)=(\\field{C},\\field{C})$ and $(r_1,r_2)=(3,3)$, in which case $R_{\\textrm max} = \\{1,2\\}$;\n \\item $(\\field{K}_1,\\field{K}_2)=(\\field{C},\\field{R})$ and $(r_1,r_2)=(2,2)$, in which case $R_{\\textrm max} = \\{1,2\\}$;\n \\item $(\\field{K}_1,\\field{K}_2)=(\\field{C},\\field{H})$ and $(r_1,r_2)=(2,2)$, in which case $R_{\\textrm max} = \\{1,2\\}$;\n \\item $(\\field{K}_1,\\field{K}_2)=(\\field{R},\\field{C})$ and $(r_1,r_2)=(2,2)$, in which case $R_{\\textrm max} = \\{1,2\\}$;\n \\item $(\\field{K}_1,\\field{K}_2)=(\\field{R},\\field{R})$ and $(r_1,r_2)=(2,2)$, in which case $R_{\\textrm max} = \\{2\\}$;\n \\item $(\\field{K}_1,\\field{K}_2)=(\\field{R},\\field{R})$ and $(r_1,r_2)=(3,3)$, in which case $R_{\\textrm max} = \\{1,2\\}$;\n \\item $(\\field{K}_1,\\field{K}_2)=(\\field{R},\\field{H})$ and $r_1=2$, in which case $R_{\\textrm max} = \\{1,2\\}$;\n \\item $(\\field{K}_1,\\field{K}_2)=(\\field{H},\\field{C})$ and $(r_1,r_2)=(2,2)$, in which case $R_{\\textrm max} = \\{1,2\\}$;\n \\item $(\\field{K}_1,\\field{K}_2)=(\\field{H},\\field{R})$ and $r_2=2$, in which case $R_{\\textrm max} = \\{1,2\\}$;\n \\item $(\\field{K}_1,\\field{K}_2)=(\\field{H},\\field{H})$ and $(r_1,r_2)=(4,4)$, in which case $R_{\\textrm max} = \\{4\\}$;\n \\item $(\\field{K}_1,\\field{K}_2)=(\\field{H},\\field{H})$ and $(r_1,r_2)\\neq(4,4)$, in which case $R_{\\textrm max} = \\{2\\}$.\n\\end{enumerate}\nMoreover, if $T$ satisfies the aforementioned compatibility conditions, then $\\alg{A}(T)$ does indeed attain maximal dimension whenever $\\rank(T) \\in R_{\\textrm max}$.\n\\end{proposition}\n\nOne must carry out the same calculations for the other possibilities for the domain and range of $T$, but this can be done simply by replacing $(r_1,r_2)$ in the above equations and claims with $(r_1,r_2^\\prime)$, $(r_1^\\prime,r_2)$ and $(r_1^\\prime,r_2^\\prime)$. Thus, one can determine the maximal dimension of a subalgebra of $\\alg{A}^\\textup{even}$ admitting off-diagonal operators by comparing the maximal values of $\\dim_{\\field{R}}(\\alg{A}(T))$ for $T : \\field{C}^{r_1} \\to \\field{C}^{r_2}$, $T : \\field{C}^{r_1} \\to \\field{C}^{r_2^\\prime}$, $T : \\field{C}^{r_1^\\prime} \\to \\field{C}^{r_2}$, and $T : \\field{C}^{r_1^\\prime} \\to \\field{C}^{r_2^\\prime}$.\n\nFinally, by means of the discussion above and the fact that $\\Sp(n)$ acts transitively on $1$-dimensional subspaces of $\\field{C}^n$, one can readily check that the real {\\ensuremath{C^*}}-algebra $\\alg{A}_F$ and the $S^0$-real $\\alg{A}_F$-bimodule $(\\hs{H}_F,\\gamma_F,J_F,\\epsilon_F)$ of $KO$-dimension $6 \\bmod 8$ of the NCG Standard Model are uniquely determined, up to inner automorphisms of $\\alg{A}^\\textup{even}$ and unitary equivalence, by the following choice of inputs:\n\\begin{itemize}\n \\item $n = 4$;\n \\item $(\\field{K}_1,\\field{K}_2) = (\\field{H},\\field{C})$;\n \\item $g_1 \\in M_2(\\field{H})$, $g_2 \\in M_4(\\field{C})$;\n \\item $(r_1,r_2) = (2,4)$;\n \\item $N = 3$.\n\\end{itemize}\nThe value of $N$, by construction, corresponds to the number of generations of fermions, whilst the values of $n$, $r_1$ and $r_2$ give rise to the number of species of fermion of each chirality per generation. The significance of the other inputs remains to be seen.\n\n\\section{Conclusion}\n\nAs we have seen, the structure theory first developed by Paschke and Sitarz~\\cite{PS98} and by Krajewski~\\cite{Kraj98} for finite real spectral triples of $KO$-dimension $0 \\bmod 8$ and satisfying orientability and Poincar{\\'e} duality can be extended quite fully to the case of arbitrary $KO$-dimension and without the assumptions of orientability and Poincar{\\'e} duality. In particular, once a suitable ordering is fixed on the spectrum of a finite-dimensional real {\\ensuremath{C^*}}-algebra $\\alg{A}$, the study of finite real spectral triples with algebra $\\alg{A}$ reduces completely to the study of the appropriate multiplicity matrices and of certain moduli spaces constructed using those matrices. This reduction is what has allowed for the success of Krajewski's diagrammatic approach~\\cite{Kraj98}*{\\S 4} in the cases dealt with by Iochum, Jureit, Sch{\\\"u}cker, and Stephan~\\cites{ACG1,ACG2,ACG3,ACG4,ACG5,JS08,Sch05}. We have also seen how to apply this theory both to the ``finite geometries'' of the current version of the NCG Standard Model~\\cites{Connes06,CCM07,CM08} and to Chamseddine and Connes's framework~\\cites{CC08a,CC08b} for deriving the same finite geometries.\n\nDropping the orientability requirement comes at a fairly steep cost, as even bimodules of various sorts generally have fairly intricate moduli spaces of Dirac operators. It would therefore be useful to characterise the precise nature of the failure of orientability (and of Poincar{\\'e} duality) for the finite spectral triple of the current noncommutative-geometric Standard Model. It would also be useful to generalise and study the physically-desirable conditions identified in the extant literature on finite spectral triples, such as dynamical non-degeneracy~\\cite{Sch05} and anomaly cancellation~\\cite{Kraj98}. Indeed, it would be natural to generalise Krajewski diagrams~\\cite{Kraj98} and the combinatorial analysis they facilitate~\\cite{JS08} to bilateral spectral triples of all types. The paper by Paschke and Sitarz~\\cite{PS98} also contains further material for generalisation, namely discussion of the noncommutative differential calculus of a finite spectral triple and of quantum group symmetries. In particular, one might hope to characterise finite spectral triples equivariant under the action or coaction of a suitable Hopf algebra~\\cites{PS00,Sit03}.\n\nFinally, as was mentioned earlier, the finite geometry of the current NCG Standard Model fails to be $S^0$-real. However, this failure is specifically the failure of the Dirac operator $D$ to commute with the $S^0$-real structure $\\epsilon$. The ``off-diagonal'' part of $D$ does, however, take a very special form; we hope to provide in future work a more geometrical interpretation of this term, which provides for Majorana fermions and for the so-called see-saw mechanism~\\cite{CCM07}.\n\n\n\\begin{bibdiv}\n\\begin{biblist}\n\\bib{Bar07}{article}{\n\ttitle={A Lorentzian version of the non-commutative geometry of the standard model of particle physics},\n\tauthor={Barrett, John W.},\n\tjournal={J. Math. Phys.},\n\tvolume={48},\n\tdate={2007},\n\tnumber={012303},\n\n}\n\\bib{CC08a}{article}{\n\ttitle={Conceptual explanation for the algebra in the noncommutative approach to the Standard Model},\n\tauthor={Chamseddine, Ali H.},\n\tauthor={Connes, Alain},\n\tjournal={Phys. Rev. Lett.},\n\tvolume={99},\n\tdate={2007},\n\tnumber={191601},\n\n}\n\\bib{CC08b}{article}{\n\ttitle={Why the standard model},\n\tauthor={Chamseddine, Ali H.},\n\tauthor={Connes, Alain},\n\tjournal={J. Geom. Phys.},\n\tvolume={58},\n\tdate={2008},\n\tpages={38--47},\n\n}\n\\bib{CCM07}{article}{\n\ttitle={Gravity and the Standard Model with neutrino mixing},\n\tauthor={Chamseddine, Ali H.},\n\tauthor={Connes, Alain},\n\tauthor={Marcolli, Matilde},\n\tjournal={Adv. Theor. Math. Phys.},\n\tvolume={11},\n\tdate={2007},\n\tpages={991--1089},\n\n}\n\\bib{Connes95a}{article}{\n\ttitle={Geometry from the spectral point of view},\n\tauthor={Connes, Alain},\n\tjournal={Lett. Math. Phys.},\n\tvolume={34},\n\tdate={1995},\n\tnumber={3},\n\tpages={203--238},\n}\n\\bib{Connes95}{article}{\n\ttitle={Noncommutative geometry and reality},\n\tauthor={Connes, Alain},\n\tjournal={J. Math. Phys.},\n\tvolume={6},\n\tdate={1995},\n\tpages={6194--6231},\n}\n\\bib{Connes06}{article}{\n\ttitle={Noncommutative geometry and the Standard Model with neutrino mixing},\n\tauthor={Connes, Alain},\n\tjournal={JHEP},\n\tvolume={11},\n\tdate={2006},\n\tnumber={81},\n\n}\n\\bib{CM08}{book}{\n\ttitle={Noncommutative Geometry, Quantum Fields and Motives},\n\tauthor={Connes, Alain},\n\tauthor={Marcolli, Matilde},\n\tseries={Colloquium Publications},\n\tvolume={55},\n\tpublisher={American Mathematical Society},\n\taddress={Providence, RI},\n\tdate={2007},\n}\n\\bib{Ell07}{article}{\n\ttitle={Towards a theory of classification},\n\tauthor={Elliott, George A.},\n\tjournal={Adv. in Math.},\n\tvolume={223},\n\tdate={2010},\n\tnumber={1},\n\tpages={30--48},\n\n}\n\\bib{Ell08}{misc}{\n\ttitle={private conversation},\n\tauthor={Elliott, George A.},\n\tdate={2008},\n}\n\\bib{Fare}{book}{\n\ttitle={Algebras of Linear Transformations},\n\tauthor={Farenick, Douglas R.},\n\tpublisher={Springer},\n\taddress={New York},\n\tdate={2000},\n}\n\\bib{ACG1}{article}{\n\ttitle={On a classification of irreducible almost commutative geometries},\n\tauthor={Iochum, Bruno},\n\tauthor={Sch{\\\"u}cker, Thomas},\n\tauthor={Stephan, Christoph},\n\tjournal={J. Math. Phys.},\n\tvolume={45},\n\tdate={2004},\n\tpages={5003--5041},\n\n}\n\\bib{ACG2}{article}{\n\ttitle={On a classification of irreducible almost commutative geometries, a second helping},\n\tauthor={Jureit, Jan-H.},\n\tauthor={Stephan, Christoph A.},\n\tjournal={J. Math. Phys.},\n\tvolume={46},\n\tdate={2005},\n\tnumber={043512},\n\n}\n\\bib{ACG3}{article}{\n\ttitle={On a classification of irreducible almost commutative geometries III},\n\tauthor={Jureit, Jan-Hendrik},\n\tauthor={Sch{\\\"u}cker, Thomas},\n\tauthor={Stephan, Christoph},\n\tjournal={J. Math. Phys.},\n\tvolume={46},\n\tdate={2005},\n\tnumber={072303},\n\n}\n\\bib{ACG4}{article}{\n\ttitle={On a classification of irreducible almost commutative geometries IV},\n\tauthor={Jureit, Jan-Hendrik},\n\tauthor={Stephan, Christoph A.},\n\tjournal={J. Math. Phys.},\n\tvolume={49},\n\tdate={2008},\n\tpages={033502},\n\n}\n\\bib{ACG5}{article}{\n\ttitle={On a classification of irreducible almost commutative geometries, V},\n\tauthor = {Jureit, Jan-Hendrik},\n\tauthor={Stephan, Christoph A.},\n\tdate={2009},\n\n}\n\\bib{JS08}{article}{\n\ttitle={Finding the standard model of particle physics, a combinatorial problem},\n\tauthor={Jureit, Jan-H.},\n\tauthor={Stephan, Christoph A.},\n\tjournal={Comp. Phys. Comm.},\n\tvolume={178},\n\tdate={2008},\n\tpages={230--247},\n\n}\n\\bib{Kraj98}{article}{\n\ttitle={Classification of finite spectral triples},\n\tauthor={Krajewski, Thomas},\n\tjournal={J. Geom. Phys.},\n\tvolume={28},\n\tdate={1998},\n\tpages={1--30},\n\n}\n\\bib{Li}{book}{\n\ttitle={Introduction to Operator Algebras},\n\tauthor={Li, Bing-Ren},\n\tpublisher={World Scientific},\n\taddress={Singapore},\n\tdate={1992},\n}\n\\bib{PS98}{article}{\n\ttitle={Discrete spectral triples and their symmetries},\n\tauthor={Paschke, Mario},\n\tauthor={Sitarz, Andrzej},\n\tjournal={J. Math. Phys.},\n\tvolume={39},\n\tdate={1998},\n\tpages={6191--6205},\n\n}\n\\bib{PS00}{article}{\n\ttitle={The geometry of noncommutative symmetries},\n\tauthor={Paschke, Mario},\n\tauthor={Sitarz, Andrzej},\n\tjournal={Acta Physica Polonica B},\n\tvolume={31},\n\tdate={2000},\n\tpages={1897--1911},\n}\n\\bib{Sch05}{article}{\n\ttitle={Krajewski diagrams and spin lifts},\n\tauthor={Sch{\\\"u}cker, Thomas},\n\tdate={2005},\n\teprint={arXiv:hep-th\/0501181v2},\n}\n\\bib{Sit03}{article}{\n\ttitle={Equivariant spectral triples},\n\tauthor={Sitarz, Andrzej},\n\tbook={\n\t\ttitle={Noncommutative Geometry and Quantum Groups},\n\t\teditor={Hajac, Piotr M.},\n\t\teditor={Pusz, Wies{\\l}aw},\n\t\tseries={Banach Center Publ.},\n\t\tvolume={61},\t\n\t\tpublisher={Polish Acad. Sci.},\n\t\taddress={Warsaw},\n\t\tdate={2003},\n\t},\n\tpages={231--268},\n}\n\\bib{Schwarz75}{article}{\n\ttitle={Smooth functions invariant under the action of a compact Lie group},\n\tauthor={Schwarz, Gerald W.},\n\tjournal={Topology},\n\tvolume={14},\n\tdate={1975},\n\tpages={63--68},\n}\n\\bib{St06}{article}{\n\ttitle={Almost-commutative geometry, massive neutrinos and the orientability axiom in $KO$-dimension $6$},\n\tauthor={Stephan, Christoph A.},\n\tdate={2006},\n\teprint={arXiv:hep-th\/0610097v1},\n}\n\\end{biblist}\n\\end{bibdiv}\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe variety of cuprate compounds that exhibit superconductivity is remarkable; however, the set of cuprate families for which large superconducting crystals can readily be grown is much more limited.  Such large crystals are needed for neutron scattering studies of the magnetic excitations, where it is of interest to establish universal behaviors across multiple families.  While suitable crystals of La$_{2-x}$Sr$_x$CaCu$_2$O$_6$ (La-Sr-2126) \\cite{ulri02} as well as La$_{2-x}$Ca$_{1+x}$Cu$_2$O$_6$ (La-Ca-2126)  \\cite{wang03} have been grown previously and studied by neutron diffraction \\cite{ulri02,huck05}, those samples exhibited little to no superconductivity.  Obtaining crystals with a large superconducting volume fraction has been a challenge, because synthesis of superconducting samples requires annealing in high-pressure oxygen \\cite{cava90b}.  We have finally been able to achieve this last step, and in this paper we describe the synthesis conditions and magnetic and structural characterizations of the resulting crystals.\n\nThe La-Sr\/Ca-2126 system is certainly not new.   The first report of La$_{2-x}$$A_{1+x}$Cu$_2$O$_{6-x\/2}$ ($A = $Ca, Sr) was made by Raveau and coworkers \\cite{nguy80} almost four decades ago.  Following the discovery of superconductivity in La$_{2-x}$Ba$_x$CuO$_4$\\ \\cite{bedn86}, Torrance {\\it et al.}\\ \\cite{torr88} reported on the metallic, but non-superconducting, character of La$_2$SrCu$_2$O$_{6.2}$, followed by crystallographic studies of La-Ca-2126 \\cite{izum89} and La$_2$SrCu$_2$O$_6$ \\cite{caig90}.  It was not long before Cava {\\it et al.}\\ \\cite{cava90b} announced the discovery of superconductivity with a transition temperature, $T_c$, of $\\sim60$~K in La-Sr-2126; the key here was to anneal in high-pressure oxygen.  Numerous studies of synthesis conditions and superconductivity in La-Sr-2126 \\cite{saku91,liu91} and La-Ca-2126 \\cite{fuer90,kino90,kino92a,kino92b} quickly followed.\n\nMillimeter-size crystals of La-Ca-2126 were initially grown from a CuO flux; these were rendered superconducting, with $T_c$ as high as 40~K by annealing in an O$_2$ partial pressure of 300 atm at 1080$^\\circ$C \\cite{ishi91}.  Larger crystals (4 mm $\\phi \\times 30$ mm) were grown by the travelling-solvent floating-zone (TSFZ) method, where annealing in 400 atm of O$_2$ at 1080$^\\circ$C yielded $T_c\\approx45$~K \\cite{okuy94}.   One of us (GDG) was able to grow large crystals of La-Sr-2126 in 11 bar O$_2$, which showed onset $T_c$'s over 40~K, but with a very small superconducting volume fraction ($<7$\\%\\ as measured by magnetic shielding) \\cite{gu06b}.\n\nHere we focus on crystals of La$_{2-x}$Ca$_{1+x}$Cu$_2$O$_6$ with $x=0.10$ and 0.15.  Annealing in $\\gtrsim0.11$~GPa (1100 atm) partial pressure of O$_2$ yields sharp superconducting transitions with $T_c$ up to 60~K (after post-annealing).  The magnetic shielding fraction is essentially 100\\%; however, there are substantial volume fractions of two other phases present, as we will demonstrate.\n\nThe rest of the paper is organized as follows.  The next section describes the crystal growth, annealing treatments, and the characterization methods [magnetization, neutron diffraction, and muon-spin-rotation ($\\mu$SR)].  In Sec.~III, we present and analyze the diffraction and $\\mu$SR data, followed by a description of the post-annealing study in Sec.~IV.  The results are discussed in Sec.~V and summarized in Sec.~VI.  We will present studies of the spin fluctuations by inelastic neutron scattering and anisotropic resistivity as a function of magnetic field in separate papers.\n\n\n\\section{Experimental Methods}\n\nSingle crystals of La-Ca-2126 were grown by the TSFZ method \\cite{gu06a}. High purity powders of La$_{2}$O$_{3}$, CaCO$_{3}$, and CuO (99.99\\%) were mixed in their metal ratio, ground well, and then sintered in a crucible. This grind-sinter procedure was repeated three times in order to achieve a homogeneous mixture, before the final feed-rod sintering, performed with the rod hung vertically from a Pt wire in a box furnace at 1100$^\\circ$C\\ for 72 hours in air. The Cu-rich solvent material with lower melting point was prepared through the same procedures, and then sintered at 950 $^\\circ$C\\ for 48 hours. Single-crystal growth was performed in flowing oxygen gas ($P_{\\rm O_{2}}=1$~atm) in the floating-zone furnace. During the crystal growth, the feed and seed rods rotated in opposite directions at 30 rpm, to stir the liquid zone, and simultaneously translated through the heating zone at a velocity of 0.4 mm\/h. After growth, the resulting rod was cut into sections, with polished sections checked with an optical polarization microscope to identify regions of single-crystal domain, allowing large single crystals to be selected for the high-pressure annealing experiments. \n\nThe as-grown crystals of La-Ca-2126 are non-superconducting. We induced superconductivity in large crystals with both $x=0.10$ and 0.15 by annealing under a high-pressure mixture of 20\\% oxygen and 80\\% argon in a hot isostatic press (HIP).  The annealing was performed at temperatures within 1130--1180$^\\circ$C\\ and pressures within 0.55--0.69 GPa in two separate runs, with run A lasting 31 hours and run B lasting 8 days {\\color{black} \\footnote{{\\color{black} Note that the annealing temperature was selected to test the performance of the HIP.  Typical annealing temperatures in previous studies have been somewhat lower, in the range of 970$^\\circ$C to 1080$^\\circ$C \\cite{cava90b,kino90,ishi91}.}}}.\n\nMagnetization measurements were performed using a commercial SQUID (superconducting quantum interference device) magnetometer. The magnetic susceptibility $\\chi$ was calculated assuming a density of 6.244 g\/cm$^{3}$, which was calculated from the lattice constants listed in Ref.\\ \\cite{huck05} assuming a composition of La$_{1.9}$Ca$_{1.1}$Cu$_{2}$O$_{6}$.   The susceptibility of two of the annealed crystals measured in a field of 1 mT (with uncertain crystal orientation) is shown in Fig.~\\ref{fg:susc1}.  While the Meissner fraction, measured while field cooling, is small, the shielding fraction, obtained after zero-field cooling, is large.  \n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=1.0\\columnwidth]{fig_diamagnetism.pdf}}\n\\caption{Volume magnetic susceptibility for samples SC45 (blue circles) and SC55 (red squares).  Open symbols: Meissner fraction (measured on field cooling); filled symbols: shielding fraction (zero-field cooling).}\n\\label{fg:susc1} \n\\end{figure}\n\nInitial neutron scattering experiments were performed on the SEQUOIA time-of-flight spectrometer at the Spallation Neutron Source, Oak Ridge National Laboratory \\cite{tof_sns14}. Three crystals were used for neutron scattering, which we label NSC, SC45, and SC55. NSC is an as-grown, non-superconducting crystal with $x=0.10$ and mass 7.5g. SC45 is a 6.3~g crystal of $x=0.10$ annealed in run A.  SC55 is a 7.4-g crystal of $x=0.15$ annealed in run B.    As one can see from Fig.~\\ref{fg:susc1}, SC45 has an onset of diamagnetism at $T_c=45$~K, while SC55 has $T_c=55$~K.  \n\nWhile the inelastic scattering results will be reported separately, the data in the elastic channel provided interesting clues as to the structural phases in the sample, motivating further measurements.\nNeutron diffraction measurements were then performed on triple-axis spectrometers HB-1A and HB-1 at the High Flux Isotope Reactor, Oak Ridge National Laboratory.  Sample SC55 was studied at HB-1A, where the incident energy was $E_i=14.6$~meV and horizontal collimations of $40'$-$40'$-S-$40'$-$80'$ were used.  The elastic scans of SC45 were done on a 1.2-g piece at HB-1 with $E_i=13.5$~meV and horizontal collimations $48'$-$40'$-S-$40'$-$120'$.  \n\n\\begin{figure}[b]\n\\centerline{\\includegraphics[width=0.9\\columnwidth]{fig_chi_LCCO10.pdf}}\n\\caption{Normalized susceptibility data for pieces of the SC55$\\mu$ sample.  Red line: zero-field-cooled bulk susceptibility; blue circles: diamagnetic shift of the internal field observed by muons in a field of 30~mT.}\n\\label{fg:susc2} \n\\end{figure}\n\n\\begin{figure*}[t]\n\\centerline{\\includegraphics[width=1.8\\columnwidth]{fig_fitted00LPlots.pdf}}\n\\caption{Neutron diffraction intensities measured along ${\\bf Q}=(0,0,L)$ for the SC55 (red circles, measured on HB-1A) and SC45 (blue circles, measured on HB-1) crystals at $T=4$~K.  Intensity scale is in arbitrary units; data for the two samples have been normalized at the (004) peak.  Counting time was $\\sim5$~s\/pt, with an attenuator in the beam (to avoid detector saturation).  Top (bottom) panel uses a linear (logarithmic) intensity scale.  The solid lines are model calcuations as described in the text.}\n\\label{fg:00L} \n\\end{figure*}\n\nComplementary $\\mu$SR measurements were performed at the $\\pi$M3 beam line of the Paul Scherrer Institut (Switzerland), using the general purpose instrument (GPS), on an NSC crystal and a piece of $x=0.10$ annealed in run B, which we will label SC55$\\mu$.   Experiments were performed both with the sample in zero field (ZF), to test for a finite magnetic hyperfine field, and in a transverse field (TF) of 3 mT, to probe the paramagnetic fraction in the normal state, or 30 mT, to probe the superconducting state.  The $\\mu$SR spectra were analyzed in the time domain  using least-squares optimization routines from the {\\tt musrfit} software suite \\cite{sute12}.\n\nFigure~\\ref{fg:susc2} shows measurements on the SC55$\\mu$ sample, comparing the normalized diamagnetic response obtained by weak-TF $\\mu$SR and from a bulk susceptibility measurement.  Both measurements are quite consistent with $T_c\\approx55$~K.\n\n\\section{Results and Analysis}\n\\label{sc:results}\n\nA number of studies, largely on polycrystalline samples of La-Sr\/Ca-2126, have demonstrated that high-pressure annealing can lead to secondary phases \\cite{liu91,saku93,hu14b}.  This behavior can depend on the concentration of Ca or Sr.  We have chosen to focus on $x=0.10$ and 0.15 because previous work \\cite{kino90} has indicated that these concentrations span the composition range for which homogeneous, single-phase samples can be prepared with the available oxygen partial pressure during crystal growth.  The high-pressure annealing of the crystals, essential for achieving bulk superconductivity, can result in some inhomogeneity. To identify specific phases and relative volumes of secondary phases in our superconducting crystals, we have combined neutron diffraction and $\\mu$SR measurements.\n\n\\subsection{La-Ca-2126 and La-214}\n\nIn an earlier study of a high-pressure annealed La-Ca-2126 with $x=0.10$, imaging with transmission electron microscopy demonstrated the presence of intergrowth-like thin layers of La$_{2-x}$Ca$_x$CuO$_4$ (La-214) within the La-Ca-2126 matrix \\cite{hu14b}.  (Note that such layers were not observed in an as-grown crystal.)  From neutron diffraction measurements, such as those shown in Fig.~\\ref{fg:00L}, we again find evidence for both phases.  The sharp Bragg peaks at $(00L)$ with $L$ an even integer come from the La-Ca-2126 domains, while the La-214 phase shows up as broad diffuse scattering  with prominent  peaks at $L\\sim3.2$ and 8.9.  (Other peaks, such as those at $L\\approx2.5$, 5.1, 7.7, 10.2, and 12.8, come from a third phase that will be discussed in the following subsection.)\n\nTo model these data, we created a random stacking along the $c$ axis of two structural units: 0.5 unit cell of La$_2$CaCu$_2$O$_6$ and 1.5 unit cells of La$_2$CuO$_4$.  These units are terminated by LaO layers, and there is a shift of $(0.5,0.5,0)$ applied between neighboring units.  To calculate the structure factor, structural parameters for La-Ca-2126 were taken from \\cite{ulri02} and for La-214 from \\cite{jorg88}, where we take the space group to be $I4\/mmm$ in both cases.  The lattice parameters used for La-Ca-2126 are $a=3.83$~\\AA\\ and $c=19.37$~\\AA.  For La-214, we started with $c=13.1$~\\AA, corresponding to the undoped bulk compound; however, we found that the fit was considerably improved by using $c_{214}=\\frac23 c_{2126} = 12.91$~\\AA.  \n\nIf we allow that the La-214 layers have their $a$ lattice parameter epitaxially-constrained to match that of La-Ca-2126, an expansion of 1.007, then the reduction in $c$ corresponds to a uniaxial strain of the unit cell that does not change the unit cell volume.  Given that Ca solubility in La$_2$CuO$_4$ is limited and its presence tends to change the lattice parameters in directions opposite to the apparent strain \\cite{mood92}, it seems likely that the La-214 phase contains minimal Ca ($x\\lesssim0.1$).\n\nThe model calculations shown in Fig.~\\ref{fg:00L} correspond to the square of the structure factor calculated with 29,500 La-Ca-2126 units and 3,500 La-214 units.  These parameters are a compromise that is close to the best fit for both samples.  The intensity has been multiplied by an $L$-dependent correction for spectrometer resolution volume; the only differences between the two calculated curves are due to the differences in spectrometer configurations for the measurements of the two samples.  We find that this model gives a very good description of the measured scattering.  Taking into account the relative thicknesses of the structural units, the calculation shown corresponds to an 80.8\\%\\ volume fraction of La-Ca-2126 and 19.2\\%\\ of La-214.  The actual best fits to each data set correspond to a La-214 volume fraction of 17.7\\%\\ for SC45 and 19.9\\%\\ for SC55.  The difference between these values is comparable to the uncertainty, but it suggests that the volume of La-214 increases with annealing time and $T_c$.\n\n\\subsection{La-8-8-20}\n\n\\begin{figure}[b]\n\\centerline{\\includegraphics[width=1.0\\columnwidth]{fig_4.png}}\n\\caption{(a) Elastic scattering obtained at SEQUOIA on the SC55 sample at $T=4$~K.  Rings of scattering correspond to powder diffraction from the Al sample holder and from the sample itself.  (b) Calculated diffraction pattern for the La-8-8-20 phase using the structural parameters from \\cite{erra88}.}\n\\label{fg:tof} \n\\end{figure}\n\nAnother phase that has been identified in La-Sr-2126 samples is La$_{8-x}$Sr$_x$Cu$_8$O$_{20}$ (La-8-8-20) \\cite{liu91,saku93,erra88}.  This phase was first reported in 1987 as La$_5$SrCu$_6$O$_{15}$ \\cite{toku87,torr88}; the proper formula per unit cell was later determined in a neutron powder diffraction study \\cite{erra88}.    Given the absence of Sr in our samples and evidence (discussed in the next subsection) that the relevant phase is an antiferromagnetic insulator, we believe that our case corresponds to $x=0$.  The phase is essentially a version of the perovskite LaCuO$_{3-\\delta}$ with an ordered arrangement of oxygen vacancies.  Taking $a_0$ as the average Cu-O-Cu distance, the unit cell is tetragonal with $a'\\approx2\\sqrt{2}a_0$ and $c'\\approx a_0$.  Extrapolating reported lattice parameters for finite $x$ to $x=0$ gives $a'\\approx10.89$~\\AA\\ and $c'\\approx3.85$~\\AA, with $a_0\\approx3.85$~\\AA.\n\nWe can fully explain the extra $(00L)$ peaks in Fig.~\\ref{fg:00L} if the La-8-8-20 phase is oriented coherently with the La-Ca-2126 phase, such that a [110] axis of the former is parallel to the [001] axis of the latter.   This identification also allows us to explain an array of superlattice peaks observed in the elastic channel of time-of-flight measurements performed at SEQUOIA.  An example of peaks in the $(H0L)$ zone for the SC55 sample is shown in Fig.~\\ref{fg:tof}, together with peak positions and intensities calculated from the reported structural parameters \\cite{erra88}.  Based on the analysis of Bragg peak intensities, we estimate that the La-8-8-20 phase corresponds to $\\sim15\\%$ of the sample volume.   The presence of this third phase then renormalizes the fractions of the other phases to 69\\%\\ La-Ca-2126 and 16\\%\\ La-214.\n\n\n\n\\subsection{Antiferromagnetism}\n\n\\begin{figure}[b]\n\\centerline{\\includegraphics[width=0.9\\columnwidth]{fig_mag_vol_frac.pdf}}\n\\caption{Magnetic volume fraction in the NSC and SC55$\\mu$ samples as determined from $\\mu$SR measured in a weak transverse field of 3 mT. }\n\\label{fg:mvol} \n\\end{figure}\n\nZero-field $\\mu$SR data provide clear evidence for long-range antiferromagnetic ordering in the NSC sample, with two distinct precession frequencies in the $\\mu$SR signal, corresponding (at low temperature) to the local magnetic fields 37.7(1)~mT (70\\%\\ of the signal) and 102.6(3)~mT (30\\%\\ of the signal).  The hyperfine fields (sampled at only a few temperatures) grow substantially on cooling, especially between 100~K and 5~K.  Similar measurements on the SC55$\\mu$ sample provide evidence for ordering, but without any coherent precession signal; instead, a fast decaying ${\\mu}$SR signal is found, which could be due to a wide distribution of static fields.  Figure~\\ref{fg:mvol} shows the temperature-dependent magnetic volume fractions for the two samples determined from weak TF $\\mu$SR.  Both samples show a significant enhancement of the magnetic volume fraction at low temperature.\n\nThe rise in magnetic volume and hyperfine fields at low $T$ in the NSC sample is reminiscent of related behavior in lightly-doped La$_{2-x}$Sr$_x$CuO$_4$, with $0<x<0.02$.  There, the Sr doping caused a reduction of the N\\'eel temperature and of the hyperfine field observed by $\\mu$SR \\cite{bors95} and La nuclear quadrupole resonance \\cite{chou93}, but the full hyperfine field of the $x=0$ system was recovered on cooling to low temperature, with the recovery beginning below 30~K.  Neutron scattering studies later showed that the low-temperature recovery corresponds to phase separation of the doped holes into patches of diagonal spin stripes \\cite{mats02}.  The segregation of the holes allows the hyperfine field in the commensurate antiferromagnetic regions to achieve its full strength.\n\nReturning to the NSC sample, previous neutron scattering measurements on a similar as-grown La-Ca-2126 crystal with $x=0.1$ found a slight downturn in the magnetic Bragg peak intensity below 25~K; however, there was a concomitant increase in the elastic diffuse magnetic scattering from CuO$_2$ bilayers \\cite{huck05}.  It seems likely that these changes reflect some type of segregation of a low density of doped holes, which is associated with the rise in the ordered magnetic volume fraction below 50~K, as indicated in Fig.~\\ref{fg:mvol}.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=1.0\\columnwidth]{fig_6.pdf}}\n\\caption{Neutron diffraction scans along ${\\bf Q}=(\\frac12,\\frac12,L)$ measured at $T=4$~K (blue points) and 150~K (violet points) on HB-1A.  Background data (black points) measured at ${\\bf Q}=(0.45,0.45,L)$ and $T=4$.  Three of the $L$ values at which the temperature dependence was followed in detail are indicated by arrows.}\n\\label{fg:magL} \n\\end{figure}\n\nFor the SC55$\\mu$ sample, a somewhat different interpretation is required.  Here we believe that none of the magnetic order is in the La-Ca-2126 phase.  The neutron diffraction data discussed below indicate that the magnetic order that onsets at $\\sim130$~K is associated with the La-8-8-20 phase.  The upturn below 10~K is reminiscent of the spin-glass behavior observed in lightly-doped La$_{2-x}$Sr$_x$CuO$_4$\\ \\cite{nied98}, and we propose that it is associated with the thin La-214 layers.  {\\color{black} Based on the upturn temperature, we infer doping by Ca to a hole concentration of less than 0.1.  Assuming that the behavior is similar to that of the bulk, oxygen doping could not explain this behavior since phase separation tends to yield an antiferromagnetic phase with a N\\'eel temperature of $\\sim260$~K plus a superconducting phase \\cite{well97}.}\n\nTo characterize antiferromagnetic order in SC55, we performed diffraction scans along \n$(\\frac12,\\frac12,L)$, as shown in Fig.~\\ref{fg:magL}.  Much of the scattering is nuclear, such as the broad diffuse scattering peaked at $L\\sim3.5$, which is partially reduced on warming from 4~K to 150~K.  In contrast, the relatively sharp peaks at $L=0$ and $L=2.53$ have largely disappeared by 150~K.  In Fig.~\\ref{fg:magT}, we compare the temperature dependences of these peaks with diffuse scattering at $L=3.2$ and a strong La2126 structural superlattice peak at $L=6$.  The intensities of the former peaks drop to zero at $\\sim130$~K, consistent with the magnetic ordering temperature from the $\\mu$SR measurements shown in Fig.~\\ref{fg:mvol}, while the diffuse and structural superlattice intensities only approach zero near 300~K.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=1.0\\columnwidth]{fig_7.pdf}}\n\\caption{Temperature dependence of intensities for ${\\bf Q}=(0.5,0.5,L)$ at several values of $L$, after appropriate background subtraction.  Red squares: $L=0$; black circles: $L=2.53$; violet diamonds: $L=3.2$; green triangles: $L=6$.}\n\\label{fg:magT} \n\\end{figure}\n\nThe fact that the $L=0$ and 2.53 peak intensities show the same temperature dependence suggests that they come from the same phase.  The $L$ value of 2.53 is uniquely consistent with the La-8-8-20 phase, corresponding to a commensurate reflection.  [In terms of the cubic perovskite cell with $a=3.85$~\\AA, it corresponds to $(\\frac12,\\frac12,\\frac12)$.] Hence, it appears that the La-8-8-20 phase is magnetic, and the corresponding magnetic volume fraction for $T\\gtrsim50$~K is consistent with our diffraction estimate of about 15\\%\\ sample volume.  The upturn below 10~K is compatible with the net 17\\%\\ volume of La-214 layers estimated from the analysis of the $(00L)$ neutron scattering data.\n\n\\section{Post-annealing in air}\n\nAs a test, a 67-mg piece of the SC55 sample was annealed in air at several different temperatures.   As a reference on mass change, a 92-mg as-grown, non-superconducting crystal with $x=0.15$ was included in the annealing.  Sample masses were measured before and after each annealing, each annealing was for 24h, and the temperature dependence of the magnetic susceptibility was measured (zero-field cooling) after each anneal.\n\nThe results are summarized in Fig.~\\ref{fg:anneal}.  We make the assumption that any mass change corresponds to a variation in the oxygen content, and Fig.~\\ref{fg:anneal}(a) shows the cumulative change in oxygen content, normalized to the formula unit.  The first meaningful change in oxygen content occurs from annealing at 900$^\\circ$C, and it grows substantially at 1000$^\\circ$C.  The superconducting $T_c$ actually rises to a maximum {\\color{black} (60~K)} after annealing at 700$^\\circ$C, before decreasing slightly at higher annealing temperatures, as shown in Fig.~\\ref{fg:anneal}(b).  The relative superconducting volume (shielding) fraction seems to show small changes that roughly correlate with $T_c$, but it plummets when the oxygen content drops, as shown in Fig.~\\ref{fg:anneal}(c).\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=0.65\\columnwidth]{fig_anneal.pdf}}\n\\caption{Results obtained from a series of 24h annealings in air at steadily increasing temperatures.  (a) Cumulative change in oxygen content, normalized to the La-Ca-2126 formula unit, estimated from mass changes for a piece of SC55 and an as-grown, non-superconducting piece of La-Ca-2126 with $x=0.15$.  (b) Resulting $T_c$ after each annealing, determined from ZFC susceptibility at $\\chi=-0.1$.  (c)  Superconducting (shielding) volume at 40~K, relative to that of the 700$^\\circ$C sample at 5~K, from the ZFC susceptibility data.}\n\\label{fg:anneal} \n\\end{figure}\n\n\\section{Discussion}\n\\label{sc:discussion}\n\n{\\color{black} We have assumed that the superconductivity in our sample comes from the La-Ca-2126 phase, but we discovered that the annealed crystals also contain two other phases.  Could one of the latter phases be responsible for the superconductivity?  We believe that the combination of the neutron and $\\mu$SR data make a convincing case that the La-8-80-20 phase must be antiferromagnetic with a N\\'eel temperature of $\\sim130$~K; such order is inconsistent with superconductivity.  La-214 can be superconducting; however, the highest reported $T_c$ due to Ca doping is just 34~K \\cite{dabr93}.  Oxygen doping of La-214 can raise $T_c$ as high as 45~K \\cite{blak98}; however, oxygen intercalation increases the $c$ lattice parameter \\cite{rada93}, which is inconsistent with the decreased $c$ parameter inferred for our La-214 intergrowths.  Hence, we believe that the main source of the superconductivity in our crystals must be the La-Ca-2126 phase, especially given the observed $T_c$ of 55~K. }\n\nAn unresolved issue for La-Ca-2126 concerns the mechanism for doping sufficient holes into the CuO$_2$ planes to achieve the observed superconducting transition temperatures.  Suppose we write the chemical formula as La$_{2-x}$Ca$_{1+x}$Cu$_2$O$_{6+\\delta}$, to allow for oxygen interstitials or vacancies.  Then the effective hole concentration $p$ per Cu would be $p = x\/2 + \\delta$, and the maximum $T_c$ in a given cuprate family, corresponding to ``optimal'' doping, typically occurs for $p_{\\rm opt}\\sim0.16$.  In principle, we could achieve $p_{\\rm opt}$ by using sufficient Ca, adding excess oxygen, or using a combination of these components.   If we consider the composition originally studied by Cava {\\it et al.} \\cite{cava90b,cava90c}, La$_{1.6}$Sr$_{0.4}$CaCu$_2$O$_{6+\\delta}$ with $\\delta\\sim-0.07$, application of our formula gives $p = 0.13$, which is in the vicinity of $p_{\\rm opt}$.  There, the doping by excess alkaline-earth ions was partially offset by oxygen vacancies.\n\nNext, consider our samples and assume, for the moment, that $\\delta=0$ and that we have a single phase in the superconducting samples.  For $x=0.10$, we would get $p=0.05$, which would generally be too low to realize superconductivity; increasing $x$ to 0.15 only just crosses the anticipated threshold into the superconducting regime.  One might imagine that annealing in high-pressure oxygen would increase the oxygen content of the sample, raising $p$ towards $p_{\\rm opt}$.   A few studies have used neutron powder diffraction to analyze the change in oxygen occupancy due to such annealing, including consideration of vacancies at regular lattice sites and  additional atoms at an extra oxygen site within the CuO$_2$ bilayers \\cite{saku91,kino92a,shak93}; however, the associated change in hole concentration would only appear to be $\\Delta p\\sim0.03$ (with a large spread in results), and this would not be sufficient to shift the $T_c$ of our samples from zero to 60~K.\n\nOf course, our crystals have extra phases in them, and we believe that the formation of these secondary phases is essential to the doping process.  The key point is that the La-214 and La-8-8-20 phases contain very little Ca compared to the La-Ca-2126 phase.  We start with a nominally uniform composition of La-Ca-2126 with an excess Ca concentration of $x$.  Removing quantities of the elements La, Cu, and O to form La-214 and La-8-8-20 phases will increase the Ca concentration to $x'$ in the remaining La-Ca-2126 phase.  (A small amount of Ca in the La-214 phase has only a small impact on $x'$.)  Using the estimated phase fractions, a simple analysis yields $x' \\sim 0.6$, which, in the absence of oxygen vacancies, would predict $p\\gg p_{\\rm opt}$ {\\color{black} \\footnote{{\\color{black} In principle, one could estimate $x'$ from a refinement of neutron diffraction data; however, we did not collect data suitable for such a refinement.  The triple-axis measurements covered only limited directions in reciprocal space.  The time-of-flight measurements covered a bigger range of reciprocal space; however, those measurements were aimed at inelastic scattering and the data were not collected in a fashion that would ensure reliable Bragg peak intensities.}}}.  {\\color{black} Based on the fact that the $T_c$'s of our samples are comparable to the maximum values obtained in previous studies, we assume that the actual hole concentration cannot be more than slightly beyond optimal doping.}\n\nIn equilibrium, the relative volumes of the various phases should be controlled by the chemical potential.  Kinetics may limit whether we reach equilibrium.  The hole concentration in the La-Ca-2126 phase is also associated with a chemical potential.  As the volume fraction of the secondary phases increases, a constant chemical potential could cause oxygen vacancies to develop in the La-Ca-2126 phase to compensate for the growing Ca concentration.  This may keep our superconducting samples from reaching the over-doped regime.\n\nThis surprising doping mechanism also seems to be compatible with the observation that the final $T_c$ of a crystal does not seem to depend on the initial value of $x$ (for our narrow range of $x$), but only the length of time spent at high pressure.  We have observed that the volume fraction of La-214 is increased by a longer annealing time. Note that we have attempted to maintain the same temperature during annealing; we have not yet studied the impact of varying the annealing temperature in a controlled fashion.   If doping were controlled by introducing excess oxygen into La-Ca-2126, then we would expect that the oxygen content should be determined by the annealing temperature and pressure; changing the annealing time at a fixed temperature and pressure might impact the transition width, but it should not affect the onset $T_c$.  Instead, we see sharp transitions with $T_c$ values that depend on time alone.  Given the large inferred value of $x'$ in the La-Ca-2126 phase, it may be possible to attain a more modest value of $x'$ and achieve superconductivity, with a smaller volume fraction of secondary phases, by annealing at a somewhat lower temperature.  This will be tested in the future.\n\nThe impact of the post-annealing on $T_c$ will require further work to explain.  Does it involve a tuning of the doping through a change in concentration of the secondary phases, or possibly a relaxation of strain between epitaxial phases {\\color{black} or ordering of oxygen interstitials\/vacancies}?  Resolving this will require a careful {\\color{black} structural analysis}\nof samples at various stages of post annealing.  \n\n\\section{Summary}\n\\label{sc:summary}\n\nWe have grown large crystals of La$_{2-x}$Ca$_{1+x}$Cu$_2$O$_6$ with $x=0.10$ and 0.15.  The crystals are non-superconducting as grown.  Superconductivity has been induced by annealing in $\\sim0.12$~GPa partial pressure of O$_2$ at $T\\sim 1150^\\circ$C.  We have obtained crystals with sharp transitions at either $T_c=45$~K or 55~K, with $T_c$ growing with annealing time.\n\nWe have used neutron scattering and $\\mu$SR to characterize the structural and magnetic phases in these crystals.  The as-grown crystals are essentially single phase, with significant antiferromagnetic order.  The superconducting crystals, in contrast, contain two secondary epitaxial phases, La$_{2-x}$Ca$_x$CuO$_4$ and La$_8$Cu$_8$O$_{20}$.  The La-8-8-20 phase exhibits antiferromagnetic order below 130~K, while the La-214 may develop spin-glass order below 10~K.   It appears that the formation of the secondary phases enhances the Ca concentration of the La-Ca-2126 phase, and it is the resulting change in hole-doping that induces the superconductivity.\n\n\n\\acknowledgments\n \nWork at Brookhaven was supported by the Office of Basic Energy Sciences (BES), Division of Materials Sciences and Engineering, U.S. Department of Energy (DOE), through Contract No.\\ DE-SC0012704.     Z.G. gratefully acknowledges financial support by the Swiss National Science Foundation (Early Postdoc Mobility SNF fellowship P2ZHP2-161980), and sincerely thanks Hubertus Luetkens and Alex Amato for support on the $\\mu$SR experiments.\nRZ was supported by the Center for Emergent Superconductivity, an Energy Frontier Research Center funded by BES.  The experiments at ORNL's SNS and HFIR were sponsored by the Scientific User Facilities Division, BES, U.S. DOE.  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA Lipschitz game is a multi-player game in which there is an additive limit $\\lambda$ (called the Lipschitz constant of the game) on how much any player's payoffs can change due to a deviation by any other player. Thus, any player's payoff function is $\\lambda$-Lipschitz continuous as a function of the other players' mixed strategies. Lipschitz games were introduced about ten years ago by Azrieli and Shmaya \\cite{AS13}. A key feature of Lipschitz games is that they are guaranteed to have approximate Nash equilibria \\emph{in pure strategies}, where the quality of the approximation depends on the number of players $n$, the number of actions $m$, and the Lipschitz constant $\\lambda$. In particular, \\cite{AS13} showed that this guarantee holds (keeping the number of actions constant) for Lipschitz constants of size $o(1\/\\sqrt{n \\log n})$ (existence of pure approximate equilibria is trivial for Lipschitz constants of size $o(1\/n)$ since then, players have such low effect on each others' payoffs that they can best-respond independently to get a pure approximate equilibrium). The general idea of the existence proof is to take a mixed Nash equilibrium (guaranteed to exist by Nash's theorem \\cite{N51}), and prove that there is a positive probability that a pure profile sampled from it will constitute an approximate equilibrium.\n\nAs noted in \\cite{AS13} (and elsewhere), solutions in pure-strategy profiles are a more plausible and satisfying model of a game's outcome than solutions in mixed-strategy profiles. On the other hand, the existence guarantee raises the question of how to \\emph{compute} an approximate equilibrium. In contrast with potential games, in which pure-strategy equilibria can often be found via best- and better-response dynamics, there is no obvious natural approach in the context of Lipschitz games, despite the existence guarantee. The general algorithmic question (of interest in the present paper) is:\n\\begin{quote}\n\\emph{Given a Lipschitz game, how hard is it to find a pure-strategy profile that constitutes an approximate equilibrium?}\n\\end{quote}\nRecent work \\cite{DFS20,GCW19} has identified algorithms achieving additive constant approximation guarantees, but as noted by Babichenko \\cite{Bab19}, the extent to which we can achieve the pure approximate equilibria that are guaranteed by \\cite{AS13} (or alternatively, potential lower bounds on query or computational complexity) is unknown.\n\nVariants and special cases of this question include classes of Lipschitz games having a concise representation, as opposed to unrestricted Lipschitz games for which an algorithm has query access to the payoff function (as we consider in this paper). In the latter case, the question subdivides into what we can say about the query complexity, and about the computational complexity (for concisely-represented games the query complexity is low, by Theorem 3.3 of \\cite{GR16}). Moreover, if equilibria can be easily computed, does that remain the case if we ask for this to be achievable via some kind of natural-looking decentralized process? Positive results for these questions help us to believe in ``approximate pure Nash equilibrium'' as a solution concept for Lipschitz games. Alternatively, it is of interest to identify computational obstacles to the search for a Nash equilibrium.\n\n\\subsection{Prior Work}\n\nIn this paper we apply various important lower bounds on the query complexity of computing approximate Nash equilibria of \\emph{unrestricted} $n$-player games. In general, lower bounds on the query complexity are known that are exponential in $n$ (which motivates a focus on subclasses of games, such as Lipchitz games, and others).  Hart and Mansour \\cite{HM10} showed that the communication (and thus query) complexity of computing an exact Nash equilibrium (pure or mixed) in a game with $n$ players is $2^{\\Omega(n)}$.  Subsequent results have iteratively strengthened this lower bound.  First, Babichenko \\cite{Bab16} showed an exponential lower bound on the \\emph{randomized} query complexity of computing an $\\epsilon$-well supported Nash equilibrium (an approximate equilibrium in which every action in the support of a given player's mixed strategy is an $\\epsilon$-best response) for a constant value of $\\epsilon$, even when considering $\\delta$-distributional query complexity, as defined in Definition \\ref{def:query}.  Shortly after, Chen, Cheng, and Tang \\cite{CCT17} showed a $2^{\\Omega\\left(n\/\\log n\\right)}$ lower bound on the randomized query complexity of computing an $\\epsilon$-approximate Nash equilibrium for a constant value of $\\epsilon$, which Rubinstein \\cite{Rub16} improved to a $2^{\\Omega(n)}$ lower bound, even allowing a constant fraction of players to have regret greater than $\\epsilon$ (taking regret as defined in Definition \\ref{def:reg}). These intractability results motivate us to consider a restricted class of games (Lipschitz, or large, games) which contain significantly more structure than do general games.\n\nLipschitz games were initially considered by Azrieli and Shmaya \\cite{AS13}, who showed that any $\\lambda$-Lipschitz game (as defined in Section \\ref{subsec:game}) with $n$ players and $m$ actions admits an $\\epsilon$-approximate \\emph{pure} Nash equilibrium for any $\\epsilon\\geq\\lambda\\sqrt{8n\\log2mn}$ \\footnote{General Lipschitz games cannot be written down concisely, so we assume black-box access to the payoff function of a Lipschitz game. This emphasizes the importance of considering \\emph{query complexity} in this context. Note that a pure approximate equilibrium can still be checked using $mn$ queries.}.  In Section \\ref{sec:results} we provide a lower bound on the query complexity of finding such an equilibrium.\n\nPositive algorithmic results have been found for classes of games that combine the Lipschitz property with others, such as \\emph{anonymous} games \\cite{DP15} and \\emph{aggregative} games \\cite{Bab18}. For anonymous games (in which each player's payoffs depend only on the \\emph{number} of other players playing each action, and not \\emph{which} players), Daskalakis and Papadimitriou \\cite{DP15} improved upon the upper bound of \\cite{AS13} to guarantee the existence of $\\epsilon$-approximate pure Nash equilibria for $\\epsilon=\\Omega(\\lambda)$ (the only dependence on $n$ coming from $\\lambda$ itself). Peretz et al.\\ \\cite{PSS20} analyze Lipschitz values that result from $\\delta$-perturbing anonymous games, in the sense that every player is assumed to randomize uniformly with probability $\\delta$. Goldberg and Turchetta \\cite{GT17} showed that a $3\\lambda$-approximate pure Nash equilibrium of a $\\lambda$-Lipschitz anonymous game can be found querying $O(n\\log n)$ individual payoffs.\n\nGoldberg et al.\\ \\cite{GCW19} showed a logarithmic upper bound on the randomized query complexity of computing $\\frac{1}{8}$-approximate Nash equilibria in binary-action $\\frac{1}{n}$-Lipschitz games.  They also presented a randomized algorithm finding a $(\\frac{3}{4}+\\alpha)$-approximate Nash equilibrium when the number of actions is unbounded.\n\n\\subsection{Our Contributions}\n\nThe primary contribution of this work is the development and application of a query-efficient version of a reduction technique used in \\cite{AS13,Bab13a} in which an algorithm finds an equilibrium in one game by reducing it to a population game with a smaller Lipschitz parameter.  As the former is a known hard problem, we prove hardness for the latter.\n\nIn Section \\ref{sec:prelim} we introduce notation and relevant concepts, and describe the query model assumed for our results.  Section \\ref{sec:results} contains our main contributions.  In particular, Theorem \\ref{thm:pure} utilizes a query-efficient reduction to a population game with a small Lipschitz parameter while preserving the equilibrium.  Hence, selecting the parameters appropriately, the hardness of finding well-supported equilibria in general games proven in \\cite{Bab16} translates to finding approximate pure equilibria in Lipschitz games.  Whilst several papers have discussed both this problem and this technique, none has put forward this observation.\n\nIn Section \\ref{subsec:multi} we introduce ``Multi-Lipschitz'' games, a generalization of Lipschitz games that allows player-specific Lipschitz values (the amount of influence the player has on others). We show that certain results of Lipschitz games extend to these, and the measure of interest is the sum of individual Lipschitz values (in a standard Lipschitz game, they are all equal). Theorem \\ref{thm:multi} provides a query-efficient reduction from finding equilibria in Multi-Lipschitz games to finding equilibria in Lipschitz games. In particular, if there is a query-efficient approximation algorithm for the latter, there is one for the former as well.\n\nFinally, Section \\ref{subsec:det} provides a simpler proof of the result of \\cite{HN18} showing exponential query lower-bounds on finding correlated equilibria with approximation constants better than $\\frac{1}{2}$.  Theorem \\ref{thm:det} provides a more general result for games with more than $2$ actions, and Corollary \\ref{cor:det} extends this idea futher to apply to Lipschitz games.  While \\cite{HN18} relies on a reduction from the $\\texttt{ApproximateSink}$ problem, we explicitly describe a class of games with vastly different equilibria between which no algorithm making a subexponential number of queries can distinguish.  To any weak deterministic algorithm, these games look like pairs of players playing Matching Pennies against each other - however the equilibria are far from those of the Matching Pennies game.\n\nFor the sake of brevity, some technical details are omitted from this work, and can be found in the appendix.\n\n\\section{Preliminaries}\\label{sec:prelim}\n\nThroughout, we use the following notation.\n\\begin{itemize}\n\\item Boldface capital letters denote matrices, and boldface lowercase letters denote vectors.\n\\item The symbol $\\textbf{\\textup{a}}$ is used to denote a pure action profile, and $\\textbf{\\textup{p}}$ is used when the strategy profiles may be mixed.  Furthermore, $\\textbf{\\textup{X}}$ is used to denote \\emph{correlated} strategies.\n\\item $[n]$ and $[m]$ denote the sets $\\{1,\\ldots,n\\}$ of players and $\\{1,\\ldots,m\\}$ of actions, respectively.  Furthermore, $i\\in[n]$ will always refer to a player, and $j\\in[m]$ will always refer to an action.\n\\item Whenever a query returns an approximate answer, the payoff vector $\\tilde{\\textbf{\\textup{u}}}$ will be used to represent the approximation and $\\textbf{\\textup{u}}$ will represent the true value.\n\\end{itemize}\n\n\n\\subsection{The Game Model}\\label{subsec:game}\n\nWe introduce standard concepts of strategy profiles, payoffs, regret, and equilibria for pure, mixed, and correlated strategies.\n\n\\begin{paragraph}{Types of strategy profile; notation:}\n    \\begin{itemize}\n        \\item A \\emph{pure} action profile $\\textbf{\\textup{a}}=(a_1,\\ldots,a_n)\\in[m]^n$ is an assignment of one action to each player. We use $\\textbf{\\textup{a}}_{-i}=(a_1,\\ldots,a_{i-1},a_{i+1},\\ldots,a_n)\\in[m]^{n-1}$ to denote the set of actions played by players in $[n]\\setminus\\{i\\}$.\n        \\item A (possibly \\emph{mixed}) strategy profile $\\textbf{\\textup{p}}=(p_1,\\ldots,p_n)\\in(\\Delta[m])^n$ (where $\\Delta(S)$ is the probability simplex over $S$) is a collection of $n$ independent probability distributions, each taken over the action set of a player, where $p_{ij}$ is the probability with which player $i$ plays action $j$.  The set of distributions for players in $[n]\\setminus\\{i\\}$ is denoted $\\textbf{\\textup{p}}_{-i}=(p_1,\\ldots,p_{i-1},p_{i+1},\\ldots,p_n)$.  When $\\textbf{\\textup{p}}$ contains just $0$-$1$ values, $\\textbf{\\textup{p}}$ is equivalent to some action profile $\\textbf{\\textup{a}}\\in[m]^n$.\n        \n        Furthermore, when considering binary-action games with action set $\\{1,2\\}$, we instead describe strategy profiles by $\\textbf{\\textup{p}}=\\left(p_1,\\ldots,p_n\\right)$, where $p_i$ is the probability that player $i$ plays action $1$.\n        \\item A \\emph{correlated} strategy profile $\\textbf{\\textup{X}}\\in\\Delta([m]^n)$ is a single joint probability distribution taken over the space of all pure action profiles $\\textbf{\\textup{a}}$.\n    \\end{itemize}\n\\end{paragraph}\n\n\\begin{paragraph}{Notation for payoffs:}\n    Given player $i$, action $j$, and pure action profile $\\textbf{\\textup{a}}$,\n    \\begin{itemize}\n        \\item $u_i(j,\\textbf{\\textup{a}}_{-i})$ is the payoff that player $i$ obtains for playing action $j$ when all other players play the actions given in $\\textbf{\\textup{a}}_{-i}$.\n        \\item $u_i(\\textbf{\\textup{a}})=u_i(a_i,\\textbf{\\textup{a}}_{-i})$ is the payoff that player $i$ obtains when all players play the actions given in $\\textbf{\\textup{a}}$.\n        \\item Similarly for mixed-strategy profiles:\\newline$u_i(j,\\textbf{\\textup{p}}_{-i})=\\mathbb{E}_{\\textbf{\\textup{a}}_{-i}\\sim\\textbf{\\textup{p}}_{-i}}[u_i(j,\\textbf{\\textup{a}}_{-i})]$ and $u_i(\\textbf{\\textup{p}})=\\mathbb{E}_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{p}}}[u_i(\\textbf{\\textup{a}})]$.\n        \\item For a given player $i\\in[n]$, consider a deviation function $\\phi:[m]\\rightarrow[m]$.  Then, similarly, $u_i^{(\\phi)}(\\textbf{\\textup{X}})=\\mathbb{E}_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}}[u_i(\\phi(a_i),\\textbf{\\textup{a}}_{-i})]$ and $u_i(\\textbf{\\textup{X}})=\\mathbb{E}_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}}[u_i(\\textbf{\\textup{a}})]$.  Furthermore, given an event $E$, $u_i(\\textbf{\\textup{X}}\\mid E)=\\mathbb{E}_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}}[u_i(\\textbf{\\textup{a}})\\mid E]$.\n    \\end{itemize}\n\\end{paragraph}\n\n\\begin{definition}[Regret]\\label{def:reg}\n    \\begin{itemize}\n        \\item Given a player $i$ and a strategy profile $\\textbf{\\textup{p}}$, define the regret\n        \\[\\textnormal{reg}_i(\\textbf{\\textup{p}})=\\max_{j\\in[m]}u_i(j,\\textbf{\\textup{p}}_{-i})-u_i(\\textbf{\\textup{p}})\\]\n        to be the difference between the payoffs of player $i$'s best response to $\\textbf{\\textup{p}}_{-i}$ and $i$'s strategy $p_i$.\n        \\item Given a player $i$ and a correlated strategy profile $\\textbf{\\textup{X}}$, define\n        \\[\\textnormal{reg}_i^{(\\phi)}(\\textbf{\\textup{X}})=u_i^{(\\phi)}(\\textbf{\\textup{X}})-u_i(\\textbf{\\textup{X}}),\\qquad\\qquad\\textnormal{reg}_i(\\textbf{\\textup{X}})=\\max_{\\phi:[m]\\rightarrow[m]}\\textnormal{reg}_i^{(\\phi)}(\\textbf{\\textup{X}}),\\]\n        the regret $\\textnormal{reg}_i(\\textbf{\\textup{X}})$ being the difference between the payoffs of player $i$'s best deviation from $\\textbf{\\textup{X}}$, and $\\textbf{\\textup{X}}$.\n    \\end{itemize}\n\\end{definition}\n\n\\begin{definition}[Equilibria]\n    \\begin{itemize}\n        \\item An {\\em $\\epsilon$-approximate Nash equilibrium} ($\\epsilon$-ANE) is a strategy profile $\\textbf{\\textup{p}}^*$ such that, for every player $i\\in[n]$, $\\textnormal{reg}_i(\\textbf{\\textup{p}}^*)\\leq\\epsilon$.\n        \\item An {\\em $\\epsilon$-well supported Nash equilibrium} ($\\epsilon$-WSNE) is an $\\epsilon$-ANE $\\textbf{\\textup{p}}^*$ for which every action $j$ in the support of $p^*_i$ is an $\\epsilon$-best response to $\\textbf{\\textup{p}}^*_{-i}$.  \n        \\item An {\\em $\\epsilon$-approximate pure Nash equilibrium} ($\\epsilon$-PNE) is a pure action profile $\\textbf{\\textup{a}}$ such that, for every player $i\\in[n]$, $\\textnormal{reg}_i(\\textbf{\\textup{a}})\\leq\\epsilon$.\n        \\item An {\\em $\\epsilon$-approximate correlated equilibrium} ($\\epsilon$-ACE) is a correlated strategy profile $\\textbf{\\textup{X}}^*$ such that, for every player $i\\in[n]$, $\\textnormal{reg}_i(\\textbf{\\textup{X}}^*)\\leq\\epsilon$.\n    \\end{itemize}\n\\end{definition}\n\nNote that any $\\epsilon$-PNE is an $\\epsilon$-WSNE, and any $\\epsilon$-ANE constitutes an $\\epsilon$-ACE.  Consequently, any algorithmic lower bounds on correlated equilibria also apply to Nash equilibria.\n\nFinally, to end this section, we introduce the class of games that is the focus of this work.\n\n\\begin{definition}[Lipschitz Games]\n    For any value $\\lambda\\in(0,1]$, a {\\em $\\lambda$-Lipschitz game} is a game in which a change in strategy of any given player can affect the payoffs of any other player by at most an additive $\\lambda$, or for every player $i$ and pair of action profiles $\\textbf{\\textup{a}},\\textbf{\\textup{a}}'$, $|u_i(\\textbf{\\textup{a}})-u_i(\\textbf{\\textup{a}}')|\\leq\\lambda||\\textbf{\\textup{a}}_{-i}-\\textbf{\\textup{a}}'_{-i}||_1$.  Here we consider games in which all payoffs are in the range $[0,1]$ (in particular, note that any general game is, by definition, $1$-Lipschitz).\n\\end{definition}\nThe set of $n$-player, $m$-action, $\\lambda$-Lipschitz games will be denoted $\\G nm\\lambda$.\n\n\\subsection{The Query Model}\n\nThis section introduces the model of queries we consider.\n\n\\begin{definition}[Queries]\\label{def:query}\n\\begin{itemize}\n    \\item A {\\em profile query} of a pure action profile $\\textbf{\\textup{a}}$ of a game $G$, denoted $\\mathcal{Q}^G(\\textbf{\\textup{a}})$, returns a vector $\\textbf{\\textup{u}}$ of payoffs $u_i(\\textbf{\\textup{a}})$ for each player $i\\in[n]$.\n    \\item A {\\em $\\delta$-distribution query} of a strategy profile $\\textbf{\\textup{p}}$ of a game $G$, denoted $\\mathcal{Q}^G_{\\delta}(\\textbf{\\textup{p}})$, returns a vector $\\tilde{\\textbf{\\textup{u}}}$ of $n$ values such that $\\left|\\left|\\tilde{\\textbf{\\textup{u}}}-\\textbf{\\textup{u}}\\right|\\right|_{\\infty}\\leq\\delta$, where $\\textbf{\\textup{u}}$ is the players' expected utilities from $\\textbf{\\textup{p}}$.  We also define a {\\em $(\\delta,\\gamma)$-distribution query} to be a $\\delta$-distribution query of a strategy profile $\\textbf{\\textup{p}}$ in which every action $j$ in the support of $p_i$ is allocated probability at least $\\gamma$ for every player $i\\in[n]$.\n    \\item The {\\em (profile) query complexity} of an algorithm $A$ on input game $G$ is the number of calls $A$ makes to $\\mathcal{Q}^G$.  The {\\em $\\delta$-distribution query complexity} of $A$ is the number of calls $A$ makes to $\\mathcal{Q}^G_\\delta$.\n\\end{itemize}\n\\end{definition}\n\nBabichenko \\cite{Bab16} points out that it is uninteresting to consider $0$-distribution queries, as any game in which every payoff is a multiple of $\\frac{1}{M}$ for some $M\\in\\mathbb{N}$ can be completely learned by a single $0$-distribution query. On the other hand, additive approximations to the expected payoffs can be computed via sampling from $\\textbf{\\textup{p}}$. Indeed, for general binary-action games we have from \\cite{GR16}:\n\n\\begin{theorem}[\\cite{GR16}]\\label{thm:GR16}\n    Take $G\\in\\G n21,\\eta>0$.  Any $(\\delta,\\gamma)$-distribution query of $G$ can be simulated with probability at least $1-\\eta$ by\n    \\[\\max\\left\\{\\frac{1}{\\gamma\\delta^2}\\log\\left(\\frac{8n}{\\eta}\\right),\\frac{8}{\\gamma}\\log\\left(\\frac{4n}{\\eta}\\right)\\right\\}\\]\n    profile queries.\n\\end{theorem}\n\n\\begin{corollary}\\label{cor:GR16}\n    Take $G\\in\\G n21,\\eta>0$.  Any $(\\delta,\\gamma)$-distribution query of $G$ can be simulated with probability at least $1-\\eta$ by\n    \\[\\frac{8}{\\gamma^2\\delta^2}\\log^2\\left(\\frac{8n}{\\eta}\\right)\\]\n    profile queries.  Furthermore, any algorithm making $q$ $(\\delta,\\gamma)$-distribution queries of $G$ can be simulated with probability at least $1-\\eta$ by\n    \\[\\frac{8q}{\\gamma^2\\delta^2}\\log^2\\left(\\frac{8nq}{\\eta}\\right)=\\textnormal{poly}\\left(n,\\frac{1}{\\gamma},\\frac{1}{\\delta},\\log\\frac{1}{\\eta}\\right)\\cdot q\\log q\\]\n    profile queries.\n\\end{corollary}\n\n\\begin{proof}\nThe first claim is a weaker but simpler version of the upper bound of Theorem \\ref{thm:GR16}. The second claim follows from the first by a union bound.\\qed\n\\end{proof}\n\n\n\\subsection{The Induced Population Game}\n\nFinally, this section introduces a reduction utilized by \\cite{AS13} in an alternative proof of Nash's Theorem, and by \\cite{Bab13a} to upper bound the support size of $\\epsilon$-ANEs.\n\n\\begin{definition}\\label{def:gG}\n    Given a game $G$ with payoff function $\\textbf{\\textup{u}}$, we define the {\\em population game} induced by $G$, $G'=g_G(L)$ with payoff function $\\textbf{\\textup{u}}'$ as follows.  Every player $i$ is replaced by a population of $L$ players ($v^i_\\ell$ for $\\ell\\in[L]$), each playing $G$ against the aggregate behavior of the other $n-1$ populations. More precisely,\\\\ $u'_{v^i_\\ell}(\\textbf{\\textup{p}}')=u_i\\left(p'_{v^i_\\ell},\\textbf{\\textup{p}}_{-i}\\right)$ where $p_{i'}=\\frac{1}{L}\\sum_{\\ell=1}^Lp'_{v^{i'}_{\\ell}}$ for all $i'\\neq i$.\n\\end{definition}\n\nPopulation games date back even to Nash's thesis \\cite{N50}, in which he uses them to justify the consideration of mixed equilibria.  To date, the reduction to the induced population game has been focused on proofs of existence.  We show that the reduction can be made query-efficient: an equilibrium of $g_G(L)$ induces an equilibrium on $G$ \\emph{which can be found with few additional queries}.  This technique is the foundation for the main results of this work.\n\n\\begin{lemma}\\label{lem:Nfactor}[Appendix \\ref{app:Nfactor}]\n    Given an $n$-player, $m$-action game $G$ and a population game $G'=g_G(L)$ induced by $G$, if an $\\epsilon$-PNE of $G'$ can be found by an algorithm making $q$ $(\\delta,\\gamma)$-distribution queries of $G'$, then an $\\epsilon$-WSNE of $G$ can be found by an algorithm making $n\\cdot m\\cdot q$ $(\\delta,\\gamma\/L)$-distribution queries of $G$.\n\\end{lemma}\n\n\\section{Results}\\label{sec:results}\n\nIn this section, we present our three main results:\n\n\\begin{itemize}\n    \\item In Section \\ref{subsec:pure}, Theorem \\ref{thm:pure} shows a lower bound exponential in $\\frac{n\\lambda}{\\epsilon}$ on the randomized query complexity of finding $\\epsilon$-approximate \\emph{pure} Nash equilibria of games in $\\G n2\\lambda$.\n    \\item In Section \\ref{subsec:multi}, we generalize the concept of Lipschitz games.  Theorem \\ref{thm:multi} provides a reduction from finding approximate equilibria in our new class of ``Multi-Lipschitz'' games to finding approximate equilibria of Lipschitz games.\n    \\item In Section \\ref{subsec:det}, Theorem \\ref{thm:det} and Proposition \\ref{prop:det} provide a complete dichotomy of the query complexity of deterministic algorithms finding $\\epsilon$-approximate correlated equilibria of $n$-player, $m$-action games.  Corollary \\ref{cor:det} scales the lower bound to apply to Lipschitz games, and motivates the consideration of explicitly randomized algorithms for the above results.\n\\end{itemize}\n\nThese results also use the following simple lemma (which holds for all types of queries and equilibria mentioned in Section \\ref{sec:prelim}).\n\n\\begin{lemma}\\label{lem:scale}\n    For any constants $\\lambda'<\\lambda\\leq1,\\epsilon>0$, there is a query-free reduction from finding $\\epsilon$-approximate equilibria of games in $\\G nm\\lambda$ to finding $\\frac{\\lambda'}{\\lambda}\\epsilon$-approximate equilibria of games in $\\G nm{\\lambda'}$.\n\\end{lemma}\n\nIn other words, query complexity upper bounds hold as $\\lambda$ and $\\epsilon$ are scaled up together, and query complexity lower bounds hold as they are scaled down.  The proof is very simple - the reduction multiplies every payoff by $\\frac{\\lambda'}{\\lambda}$ (making no additional queries) and outputs the result.  Note that the lemma does not hold for $\\lambda'>\\lambda$, as the reduction could introduce payoffs that are larger than $1$.\n\n\\subsection{Hardness of Approximate Pure Equilibria}\\label{subsec:pure}\n\nIn this section we will rely heavily on the following result of Babichenko.\n\n\\begin{theorem}[\\cite{Bab16}]\\label{thm:Bab16}\n    There is a constant $\\epsilon_0>0$ such that, for any $\\beta=2^{-o(n)}$, the randomized $\\delta$-distribution query complexity of finding an $\\epsilon_0$-WSNE of $n$-player binary-action games with probability at least $\\beta$ is $\\delta^22^{\\Omega(n)}$.\n\\end{theorem}\n\nFor the remainder of this work, the symbol $\\epsilon_0$ refers to this specific constant.  A simple application of Lemma \\ref{lem:scale} yields\n\n\\begin{corollary}\\label{cor:Bab16}\n    There is a constant $\\epsilon_0>0$ such that, for any $\\beta=2^{-o(n)}$, the randomized $\\delta$-distribution query complexity of finding an $\\epsilon_0\\lambda$-WSNE of games in $\\G n2\\lambda$ with probability at least $\\beta$ is $\\delta^22^{\\Omega(n)}$.\n\\end{corollary}\n\nWe are now ready to state our main result -- an exponential lower bound on the randomized query complexity of finding $\\epsilon$-PNEs of $\\lambda$-Lipschitz games.\n\n\\begin{theorem}[Main Result]\\label{thm:pure}\n    There exists some constant $\\epsilon_0$ such that, for any $n\\in\\mathbb{N},\\epsilon<\\epsilon_0,\\lambda\\leq\\frac{\\epsilon}{\\sqrt{8n\\log4n}}$, while every game in $\\G n2\\lambda$ has an $\\epsilon$-PNE, any randomized algorithm finding such equilibria with probability at least $\\beta=1\/\\textnormal{poly}(n)$ must make $\\lambda^22^{\\Omega(n\\lambda\/\\epsilon)}$ profile queries.\n\\end{theorem}\n\nThe proof follows by contradiction.  Assume such an algorithm $A$ exists making $\\lambda^22^{o(n\\lambda\/\\epsilon)}$ profile queries, convert it to an algorithm $B$ making $\\lambda^22^{o(n\\lambda\/\\epsilon)}$ $\\delta$-distribution queries, then use Lemma \\ref{lem:Nfactor} to derive an algorithm $C$ finding $\\epsilon_0\\lambda$-WSNE in $\\lambda$-Lipschitz games contradicting the lower bound of Corollary \\ref{cor:Bab16}.\n\n\\begin{proof}\nAssume that some such algorithm $A$ exists finding $\\epsilon$-PNEs of games in $\\G n2\\lambda$ making at most $\\lambda^22^{o(n\\lambda\/\\epsilon)}$ profile queries.  Consider any $\\epsilon<\\epsilon_0,\\lambda'<\\frac{\\epsilon}{\\sqrt{8n\\log4n}}$, and define $\\lambda=\\frac{\\epsilon}{\\epsilon_0},L=\\frac{\\lambda}{\\lambda'},N=Ln$.  We derive an algorithm $C$ (with an intermediate algorithm $B$) that contradicts Corollary \\ref{cor:Bab16}.\n\n\\begin{enumerate}[$A$]\n    \\item Note that $A$ finds $\\frac{\\epsilon}{2}$-PNEs of games in $\\G N2{\\frac{3\\lambda'}{2}}$ with probability at least $\\beta$ making at most $\\lambda^{\\prime2}2^{o\\left(N\\lambda'\/\\epsilon\\right)}$ profile queries ($\\beta$ can be amplified to constant).\n    \\item Let $\\delta=\\frac{\\epsilon_0\\lambda'}{4}$.  For any game $G'\\in\\G N2{\\lambda'}$, consider an algorithm making $\\delta$-distribution queries of \\emph{pure action profiles} of $G'$ (introducing the uncertainty without querying mixed strategies).\n\n\\begin{claim}[Appendix \\ref{app:G''}]\n\tThere is a game $G''\\in\\G N2{\\frac{3\\lambda'}{2}}$ that is consistent with all $\\delta$-distribution queries (i.e. $\\textbf{\\textup{u}}''(\\textbf{\\textup{a}})=\\tilde{\\textbf{\\textup{u}}}'(\\textbf{\\textup{a}})$ for all queried $\\textbf{\\textup{a}}$) in which no payoff differs from $G'$ by more than an additive $\\delta$.  Futhermore, any $\\frac{\\epsilon}{2}$-PNE of $G''$ is an $\\epsilon$-PNE of $G'$.  Figure \\ref{fig:dist} visually depicts this observation.\n\\end{claim}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=0.3]{deltadistvexact.jpeg}\n    \\caption{Taking $G'$ to be the Coordination Game for fixed values of $\\epsilon$ and $\\delta$, the blue region shows the set of $\\epsilon$-approximate equilibria of $G'$ (the acceptable outputs of algorithm $B$) while the orange region shows the set of all $\\frac{\\epsilon}{2}$-approximate equilibria of any possible game $G''$ in which each payoff may be perturbed by at most $\\delta$ (the possible outputs of algorithm $B$).}\n    \\label{fig:dist}\n\\end{figure}\n\nSo define the algorithm $B$ that takes input $G'$ and proceeds as though it is algorithm $A$ (but makes $\\delta$-distribution queries instead).  By the claim above, after at most $\\lambda^{\\prime2}2^{o\\left(N\\lambda'\/\\epsilon\\right)}$ queries, it has found an $\\frac{\\epsilon}{2}$-PNE of some $G''\\in\\G N2{\\frac{3\\lambda'}{2}}$ that it believes it has learned, which is also an $\\epsilon$-PNE of $G'$.\n\n    \\item Consider any game $G\\in\\G n2\\lambda$, and let $G'=g_G(L)$ be the population game induced by $G$.  There is an algorithm $C$ described by Lemma \\ref{lem:Nfactor} that takes input $G$ and simulates $B$ on $G'$ (making $2n\\cdot\\lambda^{\\prime2}2^{o\\left(N\\lambda'\/\\epsilon\\right)}=\\delta^22^{o\\left(n\\lambda\/\\epsilon\\right)}$ $\\delta$-distribution queries) and correctly outputs an $\\epsilon$-WSNE (i.e. an $\\epsilon_0\\lambda$-WSNE) of $G$ with probability constant probability (so certainly $2^{-o(n)}$).\n\\end{enumerate}\n\nThe existence of algorithm $C$ directly contradicts the result of Corollary \\ref{cor:Bab16}, proving that algorithm $A$ cannot exist.\\qed\n\\end{proof}\n\n\\begin{remark}\n    Note that, if we instead start the proof with the assumption of such an algorithm $B$, we can also show a $\\delta^22^{o(n\\lambda\/\\epsilon)}$ lower bound for the $\\delta$-distribution query complexity of finding $\\epsilon$-PNEs of $\\lambda$-Lipschitz games.\n\\end{remark}\n\n\\subsection{Multi-Lipschitz Games}\\label{subsec:multi}\n\nIn this section, we consider a generalization of Lipschitz games in which each player $i\\in[n]$ has a ``player-specific'' Lipschitz value $\\lambda_i$ in the sense that, if player $i$ changes actions, the payoffs of all other players are changed by at most $\\lambda_i$.\n\n\\begin{definition}\n    A $\\Lambda$-Multi-Lipschitz game $G$ is an $n$-player, $m$-action game $G$ in which each player $i\\in[n]$ is associated with a constant $\\lambda_i\\leq1$ such that $\\sum_{i'=1}^n\\lambda_{i'}=\\Lambda$\n    and, for any player $i'\\neq i$ and action profiles $\\textbf{\\textup{a}}^{(1)},\\textbf{\\textup{a}}^{(2)}$ with $\\textbf{\\textup{a}}^{(1)}_{-i}=\\textbf{\\textup{a}}^{(2)}_{-i}$,\n    $\\left|u_{i'}\\left(\\textbf{\\textup{a}}^{(1)}\\right)-u_{i'}\\left(\\textbf{\\textup{a}}^{(2)}\\right)\\right|\\leq\\lambda_i$.\n    The class of such games is denoted $\\GL\\Lambda nm$, and for simplicity it is assumed that $\\lambda_1\\leq\\ldots\\leq\\lambda_n$.\n\\end{definition}\n\nThe consideration of this generalized type of game allows real-world situations to be more accurately modeled.  Geopolitical circumstances, for example, naturally take the form of Multi-Lipschitz games, since individual countries have different limits on how much their actions can affect the rest of the world.  Financial markets present another instance of such games; they not only consist of individual traders who have little impact on each other, but also include a number of institutions that might each have a much greater impact on the market as a whole.  This consideration is further motivated by the recent GameStop frenzy; the institutions still wield immense power, but so do the aggregate actions of millions of individuals \\cite{PL21}.\n\nNotice that a $\\lambda$-Lipschitz game is a $\\Lambda$-Multi-Lipschitz game, for $\\Lambda=n\\lambda$. Any algorithm that finds $\\epsilon$-ANEs of $\\Lambda$-Multi-Lipschitz games is also applicable to $\\Lambda\/n$-Lipschitz games. Theorem \\ref{thm:multi} shows a kind of converse for query complexity, reducing from finding $\\epsilon$-ANE of $\\Lambda$-Multi-Lipschitz games to finding $\\epsilon$-ANE of $\\lambda$-Lipschitz games, for $\\lambda$ a constant multiple of $\\Lambda\/n$.\n\n\\begin{theorem}\\label{thm:multi}\n    There is a reduction from computing $\\epsilon$-ANEs of games in $\\GL\\Lambda n2$ with probability at least $1-\\eta$ to computing $\\frac{\\epsilon}{2}$-ANEs of games in $\\G{2n}2{\\frac{3\\Lambda}{2n}}$ with probability at least $1-\\frac{\\eta}{2}$ introducing at most a multiplicative $\\textnormal{poly}(n,\\frac{1}{\\epsilon},\\log\\frac{1}{\\eta})$ query blowup.\n\\end{theorem}\n\nAs we now consider $\\epsilon$-ANEs, existence is no longer a question: such equilibria are \\emph{always} guaranteed to exist by Nash's Theorem \\cite{N51}.  This proof will also utilize a more general population game $G'=g_G(L_1,\\ldots,L_n)$ in which player $i$ is replaced by a population of size $L_i$ (where the $L_i$ may differ from each other), and the queries in Lemma \\ref{lem:Nfactor} become $(\\delta,\\min_{i\\in[n]}\\{\\gamma\/L_i\\})$-distribution queries (this will now be relevant, as we need to apply Corollary \\ref{cor:GR16}).  Otherwise, the proof follows along the same lines as that of Theorem \\ref{thm:pure}.\n\n\\begin{proof}\n    Consider some $\\epsilon>0$ and a game $G\\in\\GL\\Lambda n2$ (WLOG take $\\lambda_1\\leq\\ldots\\leq\\lambda_n)$.  First, if $\\Lambda<\\frac{\\epsilon}{n}$, finding an $\\epsilon$-ANE is trivial (each player can play their best-response to the uniform mixed strategy, found in $2n$ queries).  So assume $\\Lambda\\geq\\frac{\\epsilon}{n}$.  Define $L_i=\\max\\{\\frac{n\\lambda_i}{\\Lambda},1\\}$ and, taking $i'=\\max_{i\\in[n]}\\{i:L_i=1\\}$, note that\n    \\[\\sum_{i=1}^nL_i=\\sum_{i=1}^{i'}1+\\sum_{i=i'+1}^n\\frac{n\\lambda_i}{\\Lambda}=\\sum_{i=1}^{i'}1+\\frac{n}{\\Lambda}\\sum_{i=i'+1}^n\\lambda_i\\leq\\sum_{i=1}^{i'}1+\\frac{n}{\\Lambda}\\Lambda\\leq2n.\\]\n    Thus the population game $G'=g_G(L_1,\\ldots,L_n)\\in\\G{2n}2{\\frac{\\Lambda}{n}}$.\n    \\begin{enumerate}[$A$]\n        \\item Consider an algorithm $A$ that finds $\\frac{\\epsilon}{2}$-ANEs of games in $\\G{2n}2{\\frac{3\\Lambda}{2n}}$, with probability at least $1-\\frac{\\eta}{2}$ making $q$ profile queries.\n        \\item Taking $\\delta=\\frac{\\epsilon^2}{4n^2}<\\frac{\\epsilon\\Lambda}{4n}$, the algorithm $B$ from the proof of Theorem \\ref{thm:pure} that simulates $A$ but makes $(\\delta,1)$-distribution queries finds an $\\epsilon$-ANE of $G'$ (the proof in Appendix \\ref{app:G''} also holds for these parameters with this choice of $\\delta$).\n        \\item By Lemma \\ref{lem:Nfactor}, there is an algorithm $C$ on input $G\\in\\GL\\Lambda n2$ that simulates $B$ (replacing each $(\\delta,1)$-distribution query of $G'$ with $2n$ $(\\delta,\\frac{1}{n})$-distribution queries of $G$ since $\\frac{1}{L_n}\\geq\\frac{1}{n}$) finding an $\\epsilon$-ANE with probability at least $1-\\eta$.\n    \\end{enumerate}\n    Applying Corollary \\ref{cor:GR16} (using $\\delta=\\frac{\\epsilon^2}{4n^2},\\gamma=\\frac{1}{n}$) to create a profile-query algorithm from $C$ completes the proof.\\qed\n\\end{proof}\n\nAs an example application of Theorem \\ref{thm:multi}, an algorithm of \\cite{GCW19} finds $\\left(\\frac{1}{8}+\\alpha\\right)$-approximate Nash equilibria of games in $\\G n2{\\frac{1}{n}}$; Theorem \\ref{thm:GCW19} states that result in detail, and Corollary \\ref{cor:GCW} extends it to Multi-Lipschitz games.\n\n\\begin{theorem}[\\cite{GCW19}]\\label{thm:GCW19}\n  Given constants $\\alpha,\\eta>0$, there is a randomized algorithm that, with probability at least $1-\\eta$, finds $\\left(\\frac{1}{8}+\\alpha\\right)$-approximate Nash equilibria of games in $\\G n2{\\frac{1}{n}}$ making $O\\left(\\frac{1}{\\alpha^4}\\log\\left(\\frac{n}{\\alpha\\eta}\\right)\\right)$ profile queries.  \n\\end{theorem}\n\nWe now have some ability to apply this to Multi-Lipschitz games; if $1\\leq\\Lambda<4$ we can improve upon the trivial $\\frac{1}{2}$-approximate equilibrium of Proposition \\ref{prop:det}.\n\n\\begin{corollary}\\label{cor:GCW}\n    For $\\alpha,\\eta>0,\\Lambda\\geq1,\\epsilon\\geq\\frac{\\Lambda}{8}+\\alpha$, there is an algorithm finding $\\epsilon$-ANEs of games in $\\GL\\Lambda n2$ with probability at least $1-\\eta$ making at most $\\textnormal{poly}(n,\\frac{1}{\\alpha},\\log\\frac{1}{\\eta})$\n    profile queries.\n\\end{corollary}\n\n\\begin{remark}\n    This is actually a slight improvement over just combining Theorems \\ref{thm:multi} and \\ref{thm:GCW19}, since the choice of $\\delta$ can be made slightly smaller to shrink $\\alpha$ as necessary.\n\\end{remark}\n\n\\subsection{A Deterministic Lower Bound}\\label{subsec:det}\n\nWe complete this work by generalizing the following result of Hart and Nisan.\n\n\\begin{theorem}[\\cite{HN18}]\\label{thm:HN18}\n    For any $\\epsilon<\\frac{1}{2}$, the deterministic profile query complexity of finding $\\epsilon$-ACEs of $n$-player games is $2^{\\Omega(n)}$.\n\\end{theorem}\n\nWhile the proof of Theorem \\ref{thm:HN18} utilizes a reduction from $\\texttt{ApproximateSink}$, we employ a more streamlined approach, presenting an explicit family of ``hard'' games that allows us to uncover the optimal value of $\\epsilon$ as a function of the number of actions:\n\n\\begin{theorem}\\label{thm:det}\n    Given some $m\\in\\mathbb{N}$, for any $\\epsilon<\\frac{m-1}{m}$, the deterministic profile query complexity of finding $\\epsilon$-ACEs of $n$-player, $m$-action games is $2^{\\Omega(n)}$.\n\\end{theorem}\n\nFurthermore, this value of $\\epsilon$ cannot be improved:\n\n\\begin{proposition}\\label{prop:det}\n    Given some $n,m\\in\\mathbb{N}$, for any $\\epsilon\\geq\\frac{m-1}{m}$, an $\\epsilon$-ANE of an $n$-player, $m$-action game can be found making no profile queries.\n\\end{proposition}\n\nThe upper bound of Proposition \\ref{prop:det} can be met if every player plays the uniform mixed strategy over their actions.  Finally, we can apply Lemma \\ref{lem:scale} to scale Theorem \\ref{thm:det} and obtain our intended result:\n\n\\begin{corollary}\\label{cor:det}\n    Given some $m\\in\\mathbb{N},\\lambda\\in(0,1]$, for any $\\epsilon<\\frac{m-1}{m}\\lambda$, the deterministic profile query complexity of finding $\\epsilon$-ACEs of $n$-player, $m$-action, $\\lambda$-Lipschitz games is $2^{\\Omega(n)}$.\n\\end{corollary}\n\nIn order to prove these results, we introduce a family of games $\\{G_{k,m}\\}$.  For any $k,m\\in\\mathbb{N}$, $G_{k,m}$ is a $2k$-player, $m$-action generalization of $k$ Matching Pennies games in which every odd player $i$ wants to match the even player $i+1$ and every even player $i+1$ wants to mismatch with the odd player $i$.\n\n\\begin{definition}\\label{def:gk}\n    Define $G_{1,2}$ to be the generalized Matching Pennies game, as described in Figure \\ref{fig:MP}(a).  Define the generalization $G_{k,m}$ to be the $2k$-player $m$-action game such that, for any $i\\in[k]$, player $2i-1$ has a payoff $1$ for matching player $2i$ and $0$ otherwise (and vice versa for player $2i$) ignoring all other players.\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\begin{tabular}{cc|c|c|}\n            & \\multicolumn{1}{c}{} & \\multicolumn{2}{c}{Player $2$} \\\\\n            & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{$1$}  & \\multicolumn{1}{c}{$2$} \\\\\\cline{3-4}\n            \\multirow{2}*{Player $1$} & $1$ & $1,0$ & $0,1$ \\\\\\cline{3-4}\n            & $2$ & $0,1$ & $1,0$ \\\\\\cline{3-4}\n        \\end{tabular}\n        \\caption{The payoff matrix of $G_{1,2}$, the Matching Pennies game.}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.54\\textwidth}\n        \\centering\n        \\begin{tabular}{cc|c|c|c|}\n            & \\multicolumn{1}{c}{} & \\multicolumn{3}{c}{Player 2} \\\\\n            & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{$1$}  & \\multicolumn{1}{c}{$2$}  &\\multicolumn{1}{c}{$3$} \\\\\\cline{3-5}\n            \\multirow{3}*{Player 1} & $1$ & $1,0$ & $0,1$ & $0,1$ \\\\\\cline{3-5}\n            & $2$ & $0,1$ & $1,0$ & $0,1$ \\\\\\cline{3-5}\n            & $3$ & $0,1$ & $0,1$ & $1,0$ \\\\\\cline{3-5}\n        \\end{tabular}\n        \\caption{The payoff matrix of $G_{1,3}$, the generalized Matching Pennies game.}\n    \\end{subfigure}\n    \\caption{The payoff matrices of $G_{1,2}$ and $G_{1,3}$.}\n    \\label{fig:MP}\n\\end{figure}\n\\end{definition}\n\nThe critical property of the generalized Matching Pennies game is that we can bound the probability that any given action profile is played in any $\\epsilon$-ACE of $G_{k,m}$.  If too much probability is jointly placed on matching actions, player $2$ will have high regret.  Conversely, if too much probability is jointly placed on mismatched actions, player $1$ will have high regret.\n\n\\begin{lemma}\\label{lem:game}\n    For any $k,m\\in\\mathbb{N},\\alpha>0$, take $\\epsilon=\\frac{m-1}{m}-\\alpha$.  In any $\\epsilon$-ACE $\\textbf{\\textup{X}}^*$ of $G_{k,m}$, every action profile $\\textbf{\\textup{a}}'\\in[m]^n$ satisfies $\\Pr_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}^*}(\\textbf{\\textup{a}}=\\textbf{\\textup{a}}')<\\rho^{\\frac{n}{2}}$ where\n    \\[\\rho=\\frac{(2-\\alpha)m-1}{2m}.\\]\n\\end{lemma}\n\nThis phenomenon can be seen in Figure \\ref{fig:cregion}.\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[scale=0.3]{alphac13.jpeg}\n        \\caption{$\\alpha=\\frac{1}{3}$}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[scale=0.3]{alphac16.jpeg}\n        \\caption{$\\alpha=\\frac{1}{6}$}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[scale=0.3]{alphac0.jpeg}\n        \\caption{$\\alpha\\rightarrow0$}\n    \\end{subfigure}\n    \\caption{The region of possible values for $(\\Pr_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}^*}(a_1=1,a_2=1),\\Pr_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}^*}(a_1=2,a_2=2)$ in any $\\left(\\frac{1}{2}-\\alpha\\right)$-approximate correlated equilibrium of $G_{k,2}$.  The only exact correlated equilibrium is shown by the red point, and the corresponding values of $\\rho$ are displayed as the orange lines.}\n    \\label{fig:cregion}\n\\end{figure}\n\n\\begin{proof}\n    Define $n=2k$ to be the number of players in $G_{k,m}$, and consider some $\\epsilon$-ACE $\\textbf{\\textup{X}}^*$.  Now WLOG consider players $1$ and $2$ and assume, for the sake of contradiction, that there exist some actions $j_1,j_2$ such that $\\Pr_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}^*}(a_1=j_1,a_2=j_2)>\\rho$.  We will need to consider the two cases $j_1=j_2$ and $j_1\\neq j_2$.\n    \\item\\paragraph{Matching Actions} In this case, WLOG assume $j_1=j_2=1$.  We show that player $2$ can improve her payoff by more than $\\epsilon$.  Under $\\textbf{\\textup{X}}^*$, with probability $>\\rho$, any random realization $\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}^*$ will yield player $2$ a payoff of $0$.  In other words, $u_2(\\textbf{\\textup{X}}^*)<1-\\rho$.  Furthermore, considering the marginal distribution over player $1$'s action, we are guaranteed that\n    \\[\\sum_{j=2}^m\\Pr_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}^*}(a_1=j)<1-\\rho\\]\n    so there must exist some action (WLOG action $2$) for which $\\Pr_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}^*}(a_1=2)<\\frac{1-\\rho}{m-1}$.  As such, define $\\phi(j)=2$.  Then $u_2^{(\\phi)}(\\textbf{\\textup{X}}^*)>1-\\frac{1-\\rho}{m-1}$, so\n    \\[\\textnormal{reg}_2^{(\\phi)}(\\textbf{\\textup{X}}^*)>\\underbrace{\\left(1-\\frac{1-\\rho}{m-1}\\right)}_{u_2^{(\\phi)}(\\textbf{\\textup{X}}^*)}-\\underbrace{(1-\\rho)}_{u_2(\\textbf{\\textup{X}}^*)}=\\frac{\\rho m-1}{m-1}\\geq\\frac{m-1}{m}-\\alpha\\]\n    for all $m\\geq2$.  This contradicts our assumption of an $\\epsilon$-ACE.\n    \\item\\paragraph{Mismatched Actions}\n    In this case, WLOG assume $j_1=1,j_2=2$.  The situation is simpler, taking $\\phi(j)=2$, $u_1(\\textbf{\\textup{X}}^*)<1-\\rho$ and $u_1^{(\\phi)}(\\textbf{\\textup{X}}^*)>\\rho$, so $\\textnormal{reg}_1^{(\\phi)}(\\textbf{\\textup{X}}^*)>\\frac{m-1}{m}-\\alpha$.  This too contradicts our assumption of an $\\epsilon$-ACE, and thus completes the proof of Lemma \\ref{lem:game}.\\qed\n\\end{proof}\n\nWe can now prove Theorem \\ref{thm:det}.  The general idea is that, should an efficient algorithm exist, because any equilibrium of $G_{k,m}$ must have large support by Lemma \\ref{lem:game}, there is significant probability assigned to action profiles that are not queried by the algorithm.  We show there is a game that the algorithm cannot distinguish from $G_{k,m}$ that shares no approximate equillibria with $G_{k,m}$.\n\n\\begin{proof}[Theorem \\ref{thm:det}]\n    Consider any $\\alpha>0$ and let $\\epsilon=\\frac{m-1}{m}-\\alpha$.  Taking $\\rho$ as in the statement of Lemma \\ref{lem:game}, assume there exists some deterministic algorithm $A$ that takes an $n$-player, $m$-action game $G$ as input and finds an $\\epsilon$-ACE of $G$ querying the payoffs of $q<\\frac{\\alpha}{2}\\rho^{-\\frac{n}{2}}$ action profiles.  Fix some $k\\in\\mathbb{N}$ and consider input $G_{k,m}$ as defined in Definition \\ref{def:gk}.  Then $\\textbf{\\textup{X}}^*=A\\left(G_{k,m}\\right)$ is an $\\epsilon$-ACE of $G_{k,m}$.  Note that, for some $j$, $\\Pr_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}^*}(a_1=j)\\leq\\frac{1}{m}$ (WLOG assume $j=1$).\n    \n    Now define the perturbation $G'_{k,m}$ of $G_{k,m}$ with payoffs defined to be equal to $G_{k,m}$ for every action profile queried by $A$, $1$ for every remaining action profile in which player $1$ plays action $1$ (chosen because it is assigned low probability by $\\textbf{\\textup{X}}^*$ by assumption), and $0$ otherwise.  Note that, by definition, $A$ cannot distinguish between $G_{k,m}$ and $G'_{k,m}$, so $A(G'_{k,m})=\\textbf{\\textup{X}}^*$.\n    \n    Taking the function $\\phi(j)=1$, the quantity we need to bound is $\\textnormal{reg}_i^{\\prime(\\phi)}(\\textbf{\\textup{X}}^*)\\geq u_i^{\\prime(\\phi)}(\\textbf{\\textup{X}}^*)-u'_i(\\textbf{\\textup{X}}^*)$.\n    We must bound the components of this expression as follows:\n\\begin{claim}[Appendix \\ref{app:Gkm}]\n$u^{\\prime(\\phi)}_1(\\textbf{\\textup{X}}^*)>(1-q\\rho^{\\frac{n}{2}})$ and $u'_1(\\textbf{\\textup{X}}^*)<\\left(\\frac{1}{m}+q\\rho^{\\frac{n}{2}}\\right)$.\n\\end{claim}\n    Using the claim and once again recalling the assumption that $q<\\frac{\\alpha}{2}\\rho^{-\\frac{n}{2}}$, we see\n    \\[\\textnormal{reg}^{\\prime(\\phi)}_1(\\textbf{\\textup{X}}^*)>\\left(1-\\frac{1}{m}-2q\\rho^{\\frac{n}{2}}\\right)=\\frac{m-1}{m}-\\alpha=\\epsilon.\\]\n    So $\\textbf{\\textup{X}}^*$ cannot actually be an $\\epsilon$-ACE of $G_{k,m}$. This completes the proof of Theorem \\ref{thm:det}.\\qed\n\\end{proof}\n\n\\section{Further Directions}\n\nAn important additional question is the query complexity of finding $\\epsilon$-PNEs of $n$-player, $\\lambda$-Lipschitz games in which $\\epsilon=\\Omega(n\\lambda)$.  Theorem \\ref{thm:pure} says nothing in this parameter range, yet Theorem \\ref{thm:GCW19} provide a logarithmic upper bound in this regime.  The tightness of this bound is of continuing interest.  Furthermore, the query- and computationally-efficient reduction discussed in Lemma \\ref{lem:Nfactor} provides a hopeful avenue for further results bounding the query, and computational, complexities of finding equilibria in many other classes of games.\n\n\\paragraph{Acknowledgements}{We thank Francisco Marmalejo Coss\\'{i}o and Rahul Santhanam for their support in the development of this work, and the reviewers of an earlier version for helpful comments.  Matthew Katzman was supported by an Oxford-DeepMind Studentship for this work.}\n\n\\bibliographystyle{splncs04}\n\n\\section{Introduction}\n\nA Lipschitz game is a multi-player game in which there is an additive limit $\\lambda$ (called the Lipschitz constant of the game) on how much any player's payoffs can change due to a deviation by any other player. Thus, any player's payoff function is $\\lambda$-Lipschitz continuous as a function of the other players' mixed strategies. Lipschitz games were introduced about ten years ago by Azrieli and Shmaya \\cite{AS13}. A key feature of Lipschitz games is that they are guaranteed to have approximate Nash equilibria \\emph{in pure strategies}, where the quality of the approximation depends on the number of players $n$, the number of actions $m$, and the Lipschitz constant $\\lambda$. In particular, \\cite{AS13} showed that this guarantee holds (keeping the number of actions constant) for Lipschitz constants of size $o(1\/\\sqrt{n \\log n})$ (existence of pure approximate equilibria is trivial for Lipschitz constants of size $o(1\/n)$ since then, players have such low effect on each others' payoffs that they can best-respond independently to get a pure approximate equilibrium). The general idea of the existence proof is to take a mixed Nash equilibrium (guaranteed to exist by Nash's theorem \\cite{N51}), and prove that there is a positive probability that a pure profile sampled from it will constitute an approximate equilibrium.\n\nAs noted in \\cite{AS13} (and elsewhere), solutions in pure-strategy profiles are a more plausible and satisfying model of a game's outcome than solutions in mixed-strategy profiles. On the other hand, the existence guarantee raises the question of how to \\emph{compute} an approximate equilibrium. In contrast with potential games, in which pure-strategy equilibria can often be found via best- and better-response dynamics, there is no obvious natural approach in the context of Lipschitz games, despite the existence guarantee. The general algorithmic question (of interest in the present paper) is:\n\\begin{quote}\n\\emph{Given a Lipschitz game, how hard is it to find a pure-strategy profile that constitutes an approximate equilibrium?}\n\\end{quote}\nRecent work \\cite{DFS20,GCW19} has identified algorithms achieving additive constant approximation guarantees, but as noted by Babichenko \\cite{Bab19}, the extent to which we can achieve the pure approximate equilibria that are guaranteed by \\cite{AS13} (or alternatively, potential lower bounds on query or computational complexity) is unknown.\n\nVariants and special cases of this question include classes of Lipschitz games having a concise representation, as opposed to unrestricted Lipschitz games for which an algorithm has query access to the payoff function (as we consider in this paper). In the latter case, the question subdivides into what we can say about the query complexity, and about the computational complexity (for concisely-represented games the query complexity is low, by Theorem 3.3 of \\cite{GR16}). Moreover, if equilibria can be easily computed, does that remain the case if we ask for this to be achievable via some kind of natural-looking decentralized process? Positive results for these questions help us to believe in ``approximate pure Nash equilibrium'' as a solution concept for Lipschitz games. Alternatively, it is of interest to identify computational obstacles to the search for a Nash equilibrium.\n\n\\subsection{Prior Work}\n\nIn this paper we apply various important lower bounds on the query complexity of computing approximate Nash equilibria of \\emph{unrestricted} $n$-player games. In general, lower bounds on the query complexity are known that are exponential in $n$ (which motivates a focus on subclasses of games, such as Lipchitz games, and others).  Hart and Mansour \\cite{HM10} showed that the communication (and thus query) complexity of computing an exact Nash equilibrium (pure or mixed) in a game with $n$ players is $2^{\\Omega(n)}$.  Subsequent results have iteratively strengthened this lower bound.  First, Babichenko \\cite{Bab16} showed an exponential lower bound on the \\emph{randomized} query complexity of computing an $\\epsilon$-well supported Nash equilibrium (an approximate equilibrium in which every action in the support of a given player's mixed strategy is an $\\epsilon$-best response) for a constant value of $\\epsilon$, even when considering $\\delta$-distributional query complexity, as defined in Definition \\ref{def:query}.  Shortly after, Chen, Cheng, and Tang \\cite{CCT17} showed a $2^{\\Omega\\left(n\/\\log n\\right)}$ lower bound on the randomized query complexity of computing an $\\epsilon$-approximate Nash equilibrium for a constant value of $\\epsilon$, which Rubinstein \\cite{Rub16} improved to a $2^{\\Omega(n)}$ lower bound, even allowing a constant fraction of players to have regret greater than $\\epsilon$ (taking regret as defined in Definition \\ref{def:reg}). These intractability results motivate us to consider a restricted class of games (Lipschitz, or large, games) which contain significantly more structure than do general games.\n\nLipschitz games were initially considered by Azrieli and Shmaya \\cite{AS13}, who showed that any $\\lambda$-Lipschitz game (as defined in Section \\ref{subsec:game}) with $n$ players and $m$ actions admits an $\\epsilon$-approximate \\emph{pure} Nash equilibrium for any $\\epsilon\\geq\\lambda\\sqrt{8n\\log2mn}$ \\footnote{General Lipschitz games cannot be written down concisely, so we assume black-box access to the payoff function of a Lipschitz game. This emphasizes the importance of considering \\emph{query complexity} in this context. Note that a pure approximate equilibrium can still be checked using $mn$ queries.}.  In Section \\ref{sec:results} we provide a lower bound on the query complexity of finding such an equilibrium.\n\nPositive algorithmic results have been found for classes of games that combine the Lipschitz property with others, such as \\emph{anonymous} games \\cite{DP15} and \\emph{aggregative} games \\cite{Bab18}. For anonymous games (in which each player's payoffs depend only on the \\emph{number} of other players playing each action, and not \\emph{which} players), Daskalakis and Papadimitriou \\cite{DP15} improved upon the upper bound of \\cite{AS13} to guarantee the existence of $\\epsilon$-approximate pure Nash equilibria for $\\epsilon=\\Omega(\\lambda)$ (the only dependence on $n$ coming from $\\lambda$ itself). Peretz et al.\\ \\cite{PSS20} analyze Lipschitz values that result from $\\delta$-perturbing anonymous games, in the sense that every player is assumed to randomize uniformly with probability $\\delta$. Goldberg and Turchetta \\cite{GT17} showed that a $3\\lambda$-approximate pure Nash equilibrium of a $\\lambda$-Lipschitz anonymous game can be found querying $O(n\\log n)$ individual payoffs.\n\nGoldberg et al.\\ \\cite{GCW19} showed a logarithmic upper bound on the randomized query complexity of computing $\\frac{1}{8}$-approximate Nash equilibria in binary-action $\\frac{1}{n}$-Lipschitz games.  They also presented a randomized algorithm finding a $(\\frac{3}{4}+\\alpha)$-approximate Nash equilibrium when the number of actions is unbounded.\n\n\\subsection{Our Contributions}\n\nThe primary contribution of this work is the development and application of a query-efficient version of a reduction technique used in \\cite{AS13,Bab13a} in which an algorithm finds an equilibrium in one game by reducing it to a population game with a smaller Lipschitz parameter.  As the former is a known hard problem, we prove hardness for the latter.\n\nIn Section \\ref{sec:prelim} we introduce notation and relevant concepts, and describe the query model assumed for our results.  Section \\ref{sec:results} contains our main contributions.  In particular, Theorem \\ref{thm:pure} utilizes a query-efficient reduction to a population game with a small Lipschitz parameter while preserving the equilibrium.  Hence, selecting the parameters appropriately, the hardness of finding well-supported equilibria in general games proven in \\cite{Bab16} translates to finding approximate pure equilibria in Lipschitz games.  Whilst several papers have discussed both this problem and this technique, none has put forward this observation.\n\nIn Section \\ref{subsec:multi} we introduce ``Multi-Lipschitz'' games, a generalization of Lipschitz games that allows player-specific Lipschitz values (the amount of influence the player has on others). We show that certain results of Lipschitz games extend to these, and the measure of interest is the sum of individual Lipschitz values (in a standard Lipschitz game, they are all equal). Theorem \\ref{thm:multi} provides a query-efficient reduction from finding equilibria in Multi-Lipschitz games to finding equilibria in Lipschitz games. In particular, if there is a query-efficient approximation algorithm for the latter, there is one for the former as well.\n\nFinally, Section \\ref{subsec:det} provides a simpler proof of the result of \\cite{HN18} showing exponential query lower-bounds on finding correlated equilibria with approximation constants better than $\\frac{1}{2}$.  Theorem \\ref{thm:det} provides a more general result for games with more than $2$ actions, and Corollary \\ref{cor:det} extends this idea futher to apply to Lipschitz games.  While \\cite{HN18} relies on a reduction from the $\\texttt{ApproximateSink}$ problem, we explicitly describe a class of games with vastly different equilibria between which no algorithm making a subexponential number of queries can distinguish.  To any weak deterministic algorithm, these games look like pairs of players playing Matching Pennies against each other - however the equilibria are far from those of the Matching Pennies game.\n\nFor the sake of brevity, some technical details are omitted from this work, and can be found in the appendix.\n\n\\section{Preliminaries}\\label{sec:prelim}\n\nThroughout, we use the following notation.\n\\begin{itemize}\n\\item Boldface capital letters denote matrices, and boldface lowercase letters denote vectors.\n\\item The symbol $\\textbf{\\textup{a}}$ is used to denote a pure action profile, and $\\textbf{\\textup{p}}$ is used when the strategy profiles may be mixed.  Furthermore, $\\textbf{\\textup{X}}$ is used to denote \\emph{correlated} strategies.\n\\item $[n]$ and $[m]$ denote the sets $\\{1,\\ldots,n\\}$ of players and $\\{1,\\ldots,m\\}$ of actions, respectively.  Furthermore, $i\\in[n]$ will always refer to a player, and $j\\in[m]$ will always refer to an action.\n\\item Whenever a query returns an approximate answer, the payoff vector $\\tilde{\\textbf{\\textup{u}}}$ will be used to represent the approximation and $\\textbf{\\textup{u}}$ will represent the true value.\n\\end{itemize}\n\n\n\\subsection{The Game Model}\\label{subsec:game}\n\nWe introduce standard concepts of strategy profiles, payoffs, regret, and equilibria for pure, mixed, and correlated strategies.\n\n\\begin{paragraph}{Types of strategy profile; notation:}\n    \\begin{itemize}\n        \\item A \\emph{pure} action profile $\\textbf{\\textup{a}}=(a_1,\\ldots,a_n)\\in[m]^n$ is an assignment of one action to each player. We use $\\textbf{\\textup{a}}_{-i}=(a_1,\\ldots,a_{i-1},a_{i+1},\\ldots,a_n)\\in[m]^{n-1}$ to denote the set of actions played by players in $[n]\\setminus\\{i\\}$.\n        \\item A (possibly \\emph{mixed}) strategy profile $\\textbf{\\textup{p}}=(p_1,\\ldots,p_n)\\in(\\Delta[m])^n$ (where $\\Delta(S)$ is the probability simplex over $S$) is a collection of $n$ independent probability distributions, each taken over the action set of a player, where $p_{ij}$ is the probability with which player $i$ plays action $j$.  The set of distributions for players in $[n]\\setminus\\{i\\}$ is denoted $\\textbf{\\textup{p}}_{-i}=(p_1,\\ldots,p_{i-1},p_{i+1},\\ldots,p_n)$.  When $\\textbf{\\textup{p}}$ contains just $0$-$1$ values, $\\textbf{\\textup{p}}$ is equivalent to some action profile $\\textbf{\\textup{a}}\\in[m]^n$.\n        \n        Furthermore, when considering binary-action games with action set $\\{1,2\\}$, we instead describe strategy profiles by $\\textbf{\\textup{p}}=\\left(p_1,\\ldots,p_n\\right)$, where $p_i$ is the probability that player $i$ plays action $1$.\n        \\item A \\emph{correlated} strategy profile $\\textbf{\\textup{X}}\\in\\Delta([m]^n)$ is a single joint probability distribution taken over the space of all pure action profiles $\\textbf{\\textup{a}}$.\n    \\end{itemize}\n\\end{paragraph}\n\n\\begin{paragraph}{Notation for payoffs:}\n    Given player $i$, action $j$, and pure action profile $\\textbf{\\textup{a}}$,\n    \\begin{itemize}\n        \\item $u_i(j,\\textbf{\\textup{a}}_{-i})$ is the payoff that player $i$ obtains for playing action $j$ when all other players play the actions given in $\\textbf{\\textup{a}}_{-i}$.\n        \\item $u_i(\\textbf{\\textup{a}})=u_i(a_i,\\textbf{\\textup{a}}_{-i})$ is the payoff that player $i$ obtains when all players play the actions given in $\\textbf{\\textup{a}}$.\n        \\item Similarly for mixed-strategy profiles:\\newline$u_i(j,\\textbf{\\textup{p}}_{-i})=\\mathbb{E}_{\\textbf{\\textup{a}}_{-i}\\sim\\textbf{\\textup{p}}_{-i}}[u_i(j,\\textbf{\\textup{a}}_{-i})]$ and $u_i(\\textbf{\\textup{p}})=\\mathbb{E}_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{p}}}[u_i(\\textbf{\\textup{a}})]$.\n        \\item For a given player $i\\in[n]$, consider a deviation function $\\phi:[m]\\rightarrow[m]$.  Then, similarly, $u_i^{(\\phi)}(\\textbf{\\textup{X}})=\\mathbb{E}_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}}[u_i(\\phi(a_i),\\textbf{\\textup{a}}_{-i})]$ and $u_i(\\textbf{\\textup{X}})=\\mathbb{E}_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}}[u_i(\\textbf{\\textup{a}})]$.  Furthermore, given an event $E$, $u_i(\\textbf{\\textup{X}}\\mid E)=\\mathbb{E}_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}}[u_i(\\textbf{\\textup{a}})\\mid E]$.\n    \\end{itemize}\n\\end{paragraph}\n\n\\begin{definition}[Regret]\\label{def:reg}\n    \\begin{itemize}\n        \\item Given a player $i$ and a strategy profile $\\textbf{\\textup{p}}$, define the regret\n        \\[\\textnormal{reg}_i(\\textbf{\\textup{p}})=\\max_{j\\in[m]}u_i(j,\\textbf{\\textup{p}}_{-i})-u_i(\\textbf{\\textup{p}})\\]\n        to be the difference between the payoffs of player $i$'s best response to $\\textbf{\\textup{p}}_{-i}$ and $i$'s strategy $p_i$.\n        \\item Given a player $i$ and a correlated strategy profile $\\textbf{\\textup{X}}$, define\n        \\[\\textnormal{reg}_i^{(\\phi)}(\\textbf{\\textup{X}})=u_i^{(\\phi)}(\\textbf{\\textup{X}})-u_i(\\textbf{\\textup{X}}),\\qquad\\qquad\\textnormal{reg}_i(\\textbf{\\textup{X}})=\\max_{\\phi:[m]\\rightarrow[m]}\\textnormal{reg}_i^{(\\phi)}(\\textbf{\\textup{X}}),\\]\n        the regret $\\textnormal{reg}_i(\\textbf{\\textup{X}})$ being the difference between the payoffs of player $i$'s best deviation from $\\textbf{\\textup{X}}$, and $\\textbf{\\textup{X}}$.\n    \\end{itemize}\n\\end{definition}\n\n\\begin{definition}[Equilibria]\n    \\begin{itemize}\n        \\item An {\\em $\\epsilon$-approximate Nash equilibrium} ($\\epsilon$-ANE) is a strategy profile $\\textbf{\\textup{p}}^*$ such that, for every player $i\\in[n]$, $\\textnormal{reg}_i(\\textbf{\\textup{p}}^*)\\leq\\epsilon$.\n        \\item An {\\em $\\epsilon$-well supported Nash equilibrium} ($\\epsilon$-WSNE) is an $\\epsilon$-ANE $\\textbf{\\textup{p}}^*$ for which every action $j$ in the support of $p^*_i$ is an $\\epsilon$-best response to $\\textbf{\\textup{p}}^*_{-i}$.  \n        \\item An {\\em $\\epsilon$-approximate pure Nash equilibrium} ($\\epsilon$-PNE) is a pure action profile $\\textbf{\\textup{a}}$ such that, for every player $i\\in[n]$, $\\textnormal{reg}_i(\\textbf{\\textup{a}})\\leq\\epsilon$.\n        \\item An {\\em $\\epsilon$-approximate correlated equilibrium} ($\\epsilon$-ACE) is a correlated strategy profile $\\textbf{\\textup{X}}^*$ such that, for every player $i\\in[n]$, $\\textnormal{reg}_i(\\textbf{\\textup{X}}^*)\\leq\\epsilon$.\n    \\end{itemize}\n\\end{definition}\n\nNote that any $\\epsilon$-PNE is an $\\epsilon$-WSNE, and any $\\epsilon$-ANE constitutes an $\\epsilon$-ACE.  Consequently, any algorithmic lower bounds on correlated equilibria also apply to Nash equilibria.\n\nFinally, to end this section, we introduce the class of games that is the focus of this work.\n\n\\begin{definition}[Lipschitz Games]\n    For any value $\\lambda\\in(0,1]$, a {\\em $\\lambda$-Lipschitz game} is a game in which a change in strategy of any given player can affect the payoffs of any other player by at most an additive $\\lambda$, or for every player $i$ and pair of action profiles $\\textbf{\\textup{a}},\\textbf{\\textup{a}}'$, $|u_i(\\textbf{\\textup{a}})-u_i(\\textbf{\\textup{a}}')|\\leq\\lambda||\\textbf{\\textup{a}}_{-i}-\\textbf{\\textup{a}}'_{-i}||_1$.  Here we consider games in which all payoffs are in the range $[0,1]$ (in particular, note that any general game is, by definition, $1$-Lipschitz).\n\\end{definition}\nThe set of $n$-player, $m$-action, $\\lambda$-Lipschitz games will be denoted $\\G nm\\lambda$.\n\n\\subsection{The Query Model}\n\nThis section introduces the model of queries we consider.\n\n\\begin{definition}[Queries]\\label{def:query}\n\\begin{itemize}\n    \\item A {\\em profile query} of a pure action profile $\\textbf{\\textup{a}}$ of a game $G$, denoted $\\mathcal{Q}^G(\\textbf{\\textup{a}})$, returns a vector $\\textbf{\\textup{u}}$ of payoffs $u_i(\\textbf{\\textup{a}})$ for each player $i\\in[n]$.\n    \\item A {\\em $\\delta$-distribution query} of a strategy profile $\\textbf{\\textup{p}}$ of a game $G$, denoted $\\mathcal{Q}^G_{\\delta}(\\textbf{\\textup{p}})$, returns a vector $\\tilde{\\textbf{\\textup{u}}}$ of $n$ values such that $\\left|\\left|\\tilde{\\textbf{\\textup{u}}}-\\textbf{\\textup{u}}\\right|\\right|_{\\infty}\\leq\\delta$, where $\\textbf{\\textup{u}}$ is the players' expected utilities from $\\textbf{\\textup{p}}$.  We also define a {\\em $(\\delta,\\gamma)$-distribution query} to be a $\\delta$-distribution query of a strategy profile $\\textbf{\\textup{p}}$ in which every action $j$ in the support of $p_i$ is allocated probability at least $\\gamma$ for every player $i\\in[n]$.\n    \\item The {\\em (profile) query complexity} of an algorithm $A$ on input game $G$ is the number of calls $A$ makes to $\\mathcal{Q}^G$.  The {\\em $\\delta$-distribution query complexity} of $A$ is the number of calls $A$ makes to $\\mathcal{Q}^G_\\delta$.\n\\end{itemize}\n\\end{definition}\n\nBabichenko \\cite{Bab16} points out that it is uninteresting to consider $0$-distribution queries, as any game in which every payoff is a multiple of $\\frac{1}{M}$ for some $M\\in\\mathbb{N}$ can be completely learned by a single $0$-distribution query. On the other hand, additive approximations to the expected payoffs can be computed via sampling from $\\textbf{\\textup{p}}$. Indeed, for general binary-action games we have from \\cite{GR16}:\n\n\\begin{theorem}[\\cite{GR16}]\\label{thm:GR16}\n    Take $G\\in\\G n21,\\eta>0$.  Any $(\\delta,\\gamma)$-distribution query of $G$ can be simulated with probability at least $1-\\eta$ by\n    \\[\\max\\left\\{\\frac{1}{\\gamma\\delta^2}\\log\\left(\\frac{8n}{\\eta}\\right),\\frac{8}{\\gamma}\\log\\left(\\frac{4n}{\\eta}\\right)\\right\\}\\]\n    profile queries.\n\\end{theorem}\n\n\\begin{corollary}\\label{cor:GR16}\n    Take $G\\in\\G n21,\\eta>0$.  Any $(\\delta,\\gamma)$-distribution query of $G$ can be simulated with probability at least $1-\\eta$ by\n    \\[\\frac{8}{\\gamma^2\\delta^2}\\log^2\\left(\\frac{8n}{\\eta}\\right)\\]\n    profile queries.  Furthermore, any algorithm making $q$ $(\\delta,\\gamma)$-distribution queries of $G$ can be simulated with probability at least $1-\\eta$ by\n    \\[\\frac{8q}{\\gamma^2\\delta^2}\\log^2\\left(\\frac{8nq}{\\eta}\\right)=\\textnormal{poly}\\left(n,\\frac{1}{\\gamma},\\frac{1}{\\delta},\\log\\frac{1}{\\eta}\\right)\\cdot q\\log q\\]\n    profile queries.\n\\end{corollary}\n\n\\begin{proof}\nThe first claim is a weaker but simpler version of the upper bound of Theorem \\ref{thm:GR16}. The second claim follows from the first by a union bound.\\qed\n\\end{proof}\n\n\n\\subsection{The Induced Population Game}\n\nFinally, this section introduces a reduction utilized by \\cite{AS13} in an alternative proof of Nash's Theorem, and by \\cite{Bab13a} to upper bound the support size of $\\epsilon$-ANEs.\n\n\\begin{definition}\\label{def:gG}\n    Given a game $G$ with payoff function $\\textbf{\\textup{u}}$, we define the {\\em population game} induced by $G$, $G'=g_G(L)$ with payoff function $\\textbf{\\textup{u}}'$ as follows.  Every player $i$ is replaced by a population of $L$ players ($v^i_\\ell$ for $\\ell\\in[L]$), each playing $G$ against the aggregate behavior of the other $n-1$ populations. More precisely,\\\\ $u'_{v^i_\\ell}(\\textbf{\\textup{p}}')=u_i\\left(p'_{v^i_\\ell},\\textbf{\\textup{p}}_{-i}\\right)$ where $p_{i'}=\\frac{1}{L}\\sum_{\\ell=1}^Lp'_{v^{i'}_{\\ell}}$ for all $i'\\neq i$.\n\\end{definition}\n\nPopulation games date back even to Nash's thesis \\cite{N50}, in which he uses them to justify the consideration of mixed equilibria.  To date, the reduction to the induced population game has been focused on proofs of existence.  We show that the reduction can be made query-efficient: an equilibrium of $g_G(L)$ induces an equilibrium on $G$ \\emph{which can be found with few additional queries}.  This technique is the foundation for the main results of this work.\n\n\\begin{lemma}\\label{lem:Nfactor}[Appendix \\ref{app:Nfactor}]\n    Given an $n$-player, $m$-action game $G$ and a population game $G'=g_G(L)$ induced by $G$, if an $\\epsilon$-PNE of $G'$ can be found by an algorithm making $q$ $(\\delta,\\gamma)$-distribution queries of $G'$, then an $\\epsilon$-WSNE of $G$ can be found by an algorithm making $n\\cdot m\\cdot q$ $(\\delta,\\gamma\/L)$-distribution queries of $G$.\n\\end{lemma}\n\n\\section{Results}\\label{sec:results}\n\nIn this section, we present our three main results:\n\n\\begin{itemize}\n    \\item In Section \\ref{subsec:pure}, Theorem \\ref{thm:pure} shows a lower bound exponential in $\\frac{n\\lambda}{\\epsilon}$ on the randomized query complexity of finding $\\epsilon$-approximate \\emph{pure} Nash equilibria of games in $\\G n2\\lambda$.\n    \\item In Section \\ref{subsec:multi}, we generalize the concept of Lipschitz games.  Theorem \\ref{thm:multi} provides a reduction from finding approximate equilibria in our new class of ``Multi-Lipschitz'' games to finding approximate equilibria of Lipschitz games.\n    \\item In Section \\ref{subsec:det}, Theorem \\ref{thm:det} and Proposition \\ref{prop:det} provide a complete dichotomy of the query complexity of deterministic algorithms finding $\\epsilon$-approximate correlated equilibria of $n$-player, $m$-action games.  Corollary \\ref{cor:det} scales the lower bound to apply to Lipschitz games, and motivates the consideration of explicitly randomized algorithms for the above results.\n\\end{itemize}\n\nThese results also use the following simple lemma (which holds for all types of queries and equilibria mentioned in Section \\ref{sec:prelim}).\n\n\\begin{lemma}\\label{lem:scale}\n    For any constants $\\lambda'<\\lambda\\leq1,\\epsilon>0$, there is a query-free reduction from finding $\\epsilon$-approximate equilibria of games in $\\G nm\\lambda$ to finding $\\frac{\\lambda'}{\\lambda}\\epsilon$-approximate equilibria of games in $\\G nm{\\lambda'}$.\n\\end{lemma}\n\nIn other words, query complexity upper bounds hold as $\\lambda$ and $\\epsilon$ are scaled up together, and query complexity lower bounds hold as they are scaled down.  The proof is very simple - the reduction multiplies every payoff by $\\frac{\\lambda'}{\\lambda}$ (making no additional queries) and outputs the result.  Note that the lemma does not hold for $\\lambda'>\\lambda$, as the reduction could introduce payoffs that are larger than $1$.\n\n\\subsection{Hardness of Approximate Pure Equilibria}\\label{subsec:pure}\n\nIn this section we will rely heavily on the following result of Babichenko.\n\n\\begin{theorem}[\\cite{Bab16}]\\label{thm:Bab16}\n    There is a constant $\\epsilon_0>0$ such that, for any $\\beta=2^{-o(n)}$, the randomized $\\delta$-distribution query complexity of finding an $\\epsilon_0$-WSNE of $n$-player binary-action games with probability at least $\\beta$ is $\\delta^22^{\\Omega(n)}$.\n\\end{theorem}\n\nFor the remainder of this work, the symbol $\\epsilon_0$ refers to this specific constant.  A simple application of Lemma \\ref{lem:scale} yields\n\n\\begin{corollary}\\label{cor:Bab16}\n    There is a constant $\\epsilon_0>0$ such that, for any $\\beta=2^{-o(n)}$, the randomized $\\delta$-distribution query complexity of finding an $\\epsilon_0\\lambda$-WSNE of games in $\\G n2\\lambda$ with probability at least $\\beta$ is $\\delta^22^{\\Omega(n)}$.\n\\end{corollary}\n\nWe are now ready to state our main result -- an exponential lower bound on the randomized query complexity of finding $\\epsilon$-PNEs of $\\lambda$-Lipschitz games.\n\n\\begin{theorem}[Main Result]\\label{thm:pure}\n    There exists some constant $\\epsilon_0$ such that, for any $n\\in\\mathbb{N},\\epsilon<\\epsilon_0,\\lambda\\leq\\frac{\\epsilon}{\\sqrt{8n\\log4n}}$, while every game in $\\G n2\\lambda$ has an $\\epsilon$-PNE, any randomized algorithm finding such equilibria with probability at least $\\beta=1\/\\textnormal{poly}(n)$ must make $\\lambda^22^{\\Omega(n\\lambda\/\\epsilon)}$ profile queries.\n\\end{theorem}\n\nThe proof follows by contradiction.  Assume such an algorithm $A$ exists making $\\lambda^22^{o(n\\lambda\/\\epsilon)}$ profile queries, convert it to an algorithm $B$ making $\\lambda^22^{o(n\\lambda\/\\epsilon)}$ $\\delta$-distribution queries, then use Lemma \\ref{lem:Nfactor} to derive an algorithm $C$ finding $\\epsilon_0\\lambda$-WSNE in $\\lambda$-Lipschitz games contradicting the lower bound of Corollary \\ref{cor:Bab16}.\n\n\\begin{proof}\nAssume that some such algorithm $A$ exists finding $\\epsilon$-PNEs of games in $\\G n2\\lambda$ making at most $\\lambda^22^{o(n\\lambda\/\\epsilon)}$ profile queries.  Consider any $\\epsilon<\\epsilon_0,\\lambda'<\\frac{\\epsilon}{\\sqrt{8n\\log4n}}$, and define $\\lambda=\\frac{\\epsilon}{\\epsilon_0},L=\\frac{\\lambda}{\\lambda'},N=Ln$.  We derive an algorithm $C$ (with an intermediate algorithm $B$) that contradicts Corollary \\ref{cor:Bab16}.\n\n\\begin{enumerate}[$A$]\n    \\item Note that $A$ finds $\\frac{\\epsilon}{2}$-PNEs of games in $\\G N2{\\frac{3\\lambda'}{2}}$ with probability at least $\\beta$ making at most $\\lambda^{\\prime2}2^{o\\left(N\\lambda'\/\\epsilon\\right)}$ profile queries ($\\beta$ can be amplified to constant).\n    \\item Let $\\delta=\\frac{\\epsilon_0\\lambda'}{4}$.  For any game $G'\\in\\G N2{\\lambda'}$, consider an algorithm making $\\delta$-distribution queries of \\emph{pure action profiles} of $G'$ (introducing the uncertainty without querying mixed strategies).\n\n\\begin{claim}[Appendix \\ref{app:G''}]\n\tThere is a game $G''\\in\\G N2{\\frac{3\\lambda'}{2}}$ that is consistent with all $\\delta$-distribution queries (i.e. $\\textbf{\\textup{u}}''(\\textbf{\\textup{a}})=\\tilde{\\textbf{\\textup{u}}}'(\\textbf{\\textup{a}})$ for all queried $\\textbf{\\textup{a}}$) in which no payoff differs from $G'$ by more than an additive $\\delta$.  Futhermore, any $\\frac{\\epsilon}{2}$-PNE of $G''$ is an $\\epsilon$-PNE of $G'$.  Figure \\ref{fig:dist} visually depicts this observation.\n\\end{claim}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=0.3]{deltadistvexact.jpeg}\n    \\caption{Taking $G'$ to be the Coordination Game for fixed values of $\\epsilon$ and $\\delta$, the blue region shows the set of $\\epsilon$-approximate equilibria of $G'$ (the acceptable outputs of algorithm $B$) while the orange region shows the set of all $\\frac{\\epsilon}{2}$-approximate equilibria of any possible game $G''$ in which each payoff may be perturbed by at most $\\delta$ (the possible outputs of algorithm $B$).}\n    \\label{fig:dist}\n\\end{figure}\n\nSo define the algorithm $B$ that takes input $G'$ and proceeds as though it is algorithm $A$ (but makes $\\delta$-distribution queries instead).  By the claim above, after at most $\\lambda^{\\prime2}2^{o\\left(N\\lambda'\/\\epsilon\\right)}$ queries, it has found an $\\frac{\\epsilon}{2}$-PNE of some $G''\\in\\G N2{\\frac{3\\lambda'}{2}}$ that it believes it has learned, which is also an $\\epsilon$-PNE of $G'$.\n\n    \\item Consider any game $G\\in\\G n2\\lambda$, and let $G'=g_G(L)$ be the population game induced by $G$.  There is an algorithm $C$ described by Lemma \\ref{lem:Nfactor} that takes input $G$ and simulates $B$ on $G'$ (making $2n\\cdot\\lambda^{\\prime2}2^{o\\left(N\\lambda'\/\\epsilon\\right)}=\\delta^22^{o\\left(n\\lambda\/\\epsilon\\right)}$ $\\delta$-distribution queries) and correctly outputs an $\\epsilon$-WSNE (i.e. an $\\epsilon_0\\lambda$-WSNE) of $G$ with probability constant probability (so certainly $2^{-o(n)}$).\n\\end{enumerate}\n\nThe existence of algorithm $C$ directly contradicts the result of Corollary \\ref{cor:Bab16}, proving that algorithm $A$ cannot exist.\\qed\n\\end{proof}\n\n\\begin{remark}\n    Note that, if we instead start the proof with the assumption of such an algorithm $B$, we can also show a $\\delta^22^{o(n\\lambda\/\\epsilon)}$ lower bound for the $\\delta$-distribution query complexity of finding $\\epsilon$-PNEs of $\\lambda$-Lipschitz games.\n\\end{remark}\n\n\\subsection{Multi-Lipschitz Games}\\label{subsec:multi}\n\nIn this section, we consider a generalization of Lipschitz games in which each player $i\\in[n]$ has a ``player-specific'' Lipschitz value $\\lambda_i$ in the sense that, if player $i$ changes actions, the payoffs of all other players are changed by at most $\\lambda_i$.\n\n\\begin{definition}\n    A $\\Lambda$-Multi-Lipschitz game $G$ is an $n$-player, $m$-action game $G$ in which each player $i\\in[n]$ is associated with a constant $\\lambda_i\\leq1$ such that $\\sum_{i'=1}^n\\lambda_{i'}=\\Lambda$\n    and, for any player $i'\\neq i$ and action profiles $\\textbf{\\textup{a}}^{(1)},\\textbf{\\textup{a}}^{(2)}$ with $\\textbf{\\textup{a}}^{(1)}_{-i}=\\textbf{\\textup{a}}^{(2)}_{-i}$,\n    $\\left|u_{i'}\\left(\\textbf{\\textup{a}}^{(1)}\\right)-u_{i'}\\left(\\textbf{\\textup{a}}^{(2)}\\right)\\right|\\leq\\lambda_i$.\n    The class of such games is denoted $\\GL\\Lambda nm$, and for simplicity it is assumed that $\\lambda_1\\leq\\ldots\\leq\\lambda_n$.\n\\end{definition}\n\nThe consideration of this generalized type of game allows real-world situations to be more accurately modeled.  Geopolitical circumstances, for example, naturally take the form of Multi-Lipschitz games, since individual countries have different limits on how much their actions can affect the rest of the world.  Financial markets present another instance of such games; they not only consist of individual traders who have little impact on each other, but also include a number of institutions that might each have a much greater impact on the market as a whole.  This consideration is further motivated by the recent GameStop frenzy; the institutions still wield immense power, but so do the aggregate actions of millions of individuals \\cite{PL21}.\n\nNotice that a $\\lambda$-Lipschitz game is a $\\Lambda$-Multi-Lipschitz game, for $\\Lambda=n\\lambda$. Any algorithm that finds $\\epsilon$-ANEs of $\\Lambda$-Multi-Lipschitz games is also applicable to $\\Lambda\/n$-Lipschitz games. Theorem \\ref{thm:multi} shows a kind of converse for query complexity, reducing from finding $\\epsilon$-ANE of $\\Lambda$-Multi-Lipschitz games to finding $\\epsilon$-ANE of $\\lambda$-Lipschitz games, for $\\lambda$ a constant multiple of $\\Lambda\/n$.\n\n\\begin{theorem}\\label{thm:multi}\n    There is a reduction from computing $\\epsilon$-ANEs of games in $\\GL\\Lambda n2$ with probability at least $1-\\eta$ to computing $\\frac{\\epsilon}{2}$-ANEs of games in $\\G{2n}2{\\frac{3\\Lambda}{2n}}$ with probability at least $1-\\frac{\\eta}{2}$ introducing at most a multiplicative $\\textnormal{poly}(n,\\frac{1}{\\epsilon},\\log\\frac{1}{\\eta})$ query blowup.\n\\end{theorem}\n\nAs we now consider $\\epsilon$-ANEs, existence is no longer a question: such equilibria are \\emph{always} guaranteed to exist by Nash's Theorem \\cite{N51}.  This proof will also utilize a more general population game $G'=g_G(L_1,\\ldots,L_n)$ in which player $i$ is replaced by a population of size $L_i$ (where the $L_i$ may differ from each other), and the queries in Lemma \\ref{lem:Nfactor} become $(\\delta,\\min_{i\\in[n]}\\{\\gamma\/L_i\\})$-distribution queries (this will now be relevant, as we need to apply Corollary \\ref{cor:GR16}).  Otherwise, the proof follows along the same lines as that of Theorem \\ref{thm:pure}.\n\n\\begin{proof}\n    Consider some $\\epsilon>0$ and a game $G\\in\\GL\\Lambda n2$ (WLOG take $\\lambda_1\\leq\\ldots\\leq\\lambda_n)$.  First, if $\\Lambda<\\frac{\\epsilon}{n}$, finding an $\\epsilon$-ANE is trivial (each player can play their best-response to the uniform mixed strategy, found in $2n$ queries).  So assume $\\Lambda\\geq\\frac{\\epsilon}{n}$.  Define $L_i=\\max\\{\\frac{n\\lambda_i}{\\Lambda},1\\}$ and, taking $i'=\\max_{i\\in[n]}\\{i:L_i=1\\}$, note that\n    \\[\\sum_{i=1}^nL_i=\\sum_{i=1}^{i'}1+\\sum_{i=i'+1}^n\\frac{n\\lambda_i}{\\Lambda}=\\sum_{i=1}^{i'}1+\\frac{n}{\\Lambda}\\sum_{i=i'+1}^n\\lambda_i\\leq\\sum_{i=1}^{i'}1+\\frac{n}{\\Lambda}\\Lambda\\leq2n.\\]\n    Thus the population game $G'=g_G(L_1,\\ldots,L_n)\\in\\G{2n}2{\\frac{\\Lambda}{n}}$.\n    \\begin{enumerate}[$A$]\n        \\item Consider an algorithm $A$ that finds $\\frac{\\epsilon}{2}$-ANEs of games in $\\G{2n}2{\\frac{3\\Lambda}{2n}}$, with probability at least $1-\\frac{\\eta}{2}$ making $q$ profile queries.\n        \\item Taking $\\delta=\\frac{\\epsilon^2}{4n^2}<\\frac{\\epsilon\\Lambda}{4n}$, the algorithm $B$ from the proof of Theorem \\ref{thm:pure} that simulates $A$ but makes $(\\delta,1)$-distribution queries finds an $\\epsilon$-ANE of $G'$ (the proof in Appendix \\ref{app:G''} also holds for these parameters with this choice of $\\delta$).\n        \\item By Lemma \\ref{lem:Nfactor}, there is an algorithm $C$ on input $G\\in\\GL\\Lambda n2$ that simulates $B$ (replacing each $(\\delta,1)$-distribution query of $G'$ with $2n$ $(\\delta,\\frac{1}{n})$-distribution queries of $G$ since $\\frac{1}{L_n}\\geq\\frac{1}{n}$) finding an $\\epsilon$-ANE with probability at least $1-\\eta$.\n    \\end{enumerate}\n    Applying Corollary \\ref{cor:GR16} (using $\\delta=\\frac{\\epsilon^2}{4n^2},\\gamma=\\frac{1}{n}$) to create a profile-query algorithm from $C$ completes the proof.\\qed\n\\end{proof}\n\nAs an example application of Theorem \\ref{thm:multi}, an algorithm of \\cite{GCW19} finds $\\left(\\frac{1}{8}+\\alpha\\right)$-approximate Nash equilibria of games in $\\G n2{\\frac{1}{n}}$; Theorem \\ref{thm:GCW19} states that result in detail, and Corollary \\ref{cor:GCW} extends it to Multi-Lipschitz games.\n\n\\begin{theorem}[\\cite{GCW19}]\\label{thm:GCW19}\n  Given constants $\\alpha,\\eta>0$, there is a randomized algorithm that, with probability at least $1-\\eta$, finds $\\left(\\frac{1}{8}+\\alpha\\right)$-approximate Nash equilibria of games in $\\G n2{\\frac{1}{n}}$ making $O\\left(\\frac{1}{\\alpha^4}\\log\\left(\\frac{n}{\\alpha\\eta}\\right)\\right)$ profile queries.  \n\\end{theorem}\n\nWe now have some ability to apply this to Multi-Lipschitz games; if $1\\leq\\Lambda<4$ we can improve upon the trivial $\\frac{1}{2}$-approximate equilibrium of Proposition \\ref{prop:det}.\n\n\\begin{corollary}\\label{cor:GCW}\n    For $\\alpha,\\eta>0,\\Lambda\\geq1,\\epsilon\\geq\\frac{\\Lambda}{8}+\\alpha$, there is an algorithm finding $\\epsilon$-ANEs of games in $\\GL\\Lambda n2$ with probability at least $1-\\eta$ making at most $\\textnormal{poly}(n,\\frac{1}{\\alpha},\\log\\frac{1}{\\eta})$\n    profile queries.\n\\end{corollary}\n\n\\begin{remark}\n    This is actually a slight improvement over just combining Theorems \\ref{thm:multi} and \\ref{thm:GCW19}, since the choice of $\\delta$ can be made slightly smaller to shrink $\\alpha$ as necessary.\n\\end{remark}\n\n\\subsection{A Deterministic Lower Bound}\\label{subsec:det}\n\nWe complete this work by generalizing the following result of Hart and Nisan.\n\n\\begin{theorem}[\\cite{HN18}]\\label{thm:HN18}\n    For any $\\epsilon<\\frac{1}{2}$, the deterministic profile query complexity of finding $\\epsilon$-ACEs of $n$-player games is $2^{\\Omega(n)}$.\n\\end{theorem}\n\nWhile the proof of Theorem \\ref{thm:HN18} utilizes a reduction from $\\texttt{ApproximateSink}$, we employ a more streamlined approach, presenting an explicit family of ``hard'' games that allows us to uncover the optimal value of $\\epsilon$ as a function of the number of actions:\n\n\\begin{theorem}\\label{thm:det}\n    Given some $m\\in\\mathbb{N}$, for any $\\epsilon<\\frac{m-1}{m}$, the deterministic profile query complexity of finding $\\epsilon$-ACEs of $n$-player, $m$-action games is $2^{\\Omega(n)}$.\n\\end{theorem}\n\nFurthermore, this value of $\\epsilon$ cannot be improved:\n\n\\begin{proposition}\\label{prop:det}\n    Given some $n,m\\in\\mathbb{N}$, for any $\\epsilon\\geq\\frac{m-1}{m}$, an $\\epsilon$-ANE of an $n$-player, $m$-action game can be found making no profile queries.\n\\end{proposition}\n\nThe upper bound of Proposition \\ref{prop:det} can be met if every player plays the uniform mixed strategy over their actions.  Finally, we can apply Lemma \\ref{lem:scale} to scale Theorem \\ref{thm:det} and obtain our intended result:\n\n\\begin{corollary}\\label{cor:det}\n    Given some $m\\in\\mathbb{N},\\lambda\\in(0,1]$, for any $\\epsilon<\\frac{m-1}{m}\\lambda$, the deterministic profile query complexity of finding $\\epsilon$-ACEs of $n$-player, $m$-action, $\\lambda$-Lipschitz games is $2^{\\Omega(n)}$.\n\\end{corollary}\n\nIn order to prove these results, we introduce a family of games $\\{G_{k,m}\\}$.  For any $k,m\\in\\mathbb{N}$, $G_{k,m}$ is a $2k$-player, $m$-action generalization of $k$ Matching Pennies games in which every odd player $i$ wants to match the even player $i+1$ and every even player $i+1$ wants to mismatch with the odd player $i$.\n\n\\begin{definition}\\label{def:gk}\n    Define $G_{1,2}$ to be the generalized Matching Pennies game, as described in Figure \\ref{fig:MP}(a).  Define the generalization $G_{k,m}$ to be the $2k$-player $m$-action game such that, for any $i\\in[k]$, player $2i-1$ has a payoff $1$ for matching player $2i$ and $0$ otherwise (and vice versa for player $2i$) ignoring all other players.\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\begin{tabular}{cc|c|c|}\n            & \\multicolumn{1}{c}{} & \\multicolumn{2}{c}{Player $2$} \\\\\n            & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{$1$}  & \\multicolumn{1}{c}{$2$} \\\\\\cline{3-4}\n            \\multirow{2}*{Player $1$} & $1$ & $1,0$ & $0,1$ \\\\\\cline{3-4}\n            & $2$ & $0,1$ & $1,0$ \\\\\\cline{3-4}\n        \\end{tabular}\n        \\caption{The payoff matrix of $G_{1,2}$, the Matching Pennies game.}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.54\\textwidth}\n        \\centering\n        \\begin{tabular}{cc|c|c|c|}\n            & \\multicolumn{1}{c}{} & \\multicolumn{3}{c}{Player 2} \\\\\n            & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{$1$}  & \\multicolumn{1}{c}{$2$}  &\\multicolumn{1}{c}{$3$} \\\\\\cline{3-5}\n            \\multirow{3}*{Player 1} & $1$ & $1,0$ & $0,1$ & $0,1$ \\\\\\cline{3-5}\n            & $2$ & $0,1$ & $1,0$ & $0,1$ \\\\\\cline{3-5}\n            & $3$ & $0,1$ & $0,1$ & $1,0$ \\\\\\cline{3-5}\n        \\end{tabular}\n        \\caption{The payoff matrix of $G_{1,3}$, the generalized Matching Pennies game.}\n    \\end{subfigure}\n    \\caption{The payoff matrices of $G_{1,2}$ and $G_{1,3}$.}\n    \\label{fig:MP}\n\\end{figure}\n\\end{definition}\n\nThe critical property of the generalized Matching Pennies game is that we can bound the probability that any given action profile is played in any $\\epsilon$-ACE of $G_{k,m}$.  If too much probability is jointly placed on matching actions, player $2$ will have high regret.  Conversely, if too much probability is jointly placed on mismatched actions, player $1$ will have high regret.\n\n\\begin{lemma}\\label{lem:game}\n    For any $k,m\\in\\mathbb{N},\\alpha>0$, take $\\epsilon=\\frac{m-1}{m}-\\alpha$.  In any $\\epsilon$-ACE $\\textbf{\\textup{X}}^*$ of $G_{k,m}$, every action profile $\\textbf{\\textup{a}}'\\in[m]^n$ satisfies $\\Pr_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}^*}(\\textbf{\\textup{a}}=\\textbf{\\textup{a}}')<\\rho^{\\frac{n}{2}}$ where\n    \\[\\rho=\\frac{(2-\\alpha)m-1}{2m}.\\]\n\\end{lemma}\n\nThis phenomenon can be seen in Figure \\ref{fig:cregion}.\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[scale=0.3]{alphac13.jpeg}\n        \\caption{$\\alpha=\\frac{1}{3}$}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[scale=0.3]{alphac16.jpeg}\n        \\caption{$\\alpha=\\frac{1}{6}$}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[scale=0.3]{alphac0.jpeg}\n        \\caption{$\\alpha\\rightarrow0$}\n    \\end{subfigure}\n    \\caption{The region of possible values for $(\\Pr_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}^*}(a_1=1,a_2=1),\\Pr_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}^*}(a_1=2,a_2=2)$ in any $\\left(\\frac{1}{2}-\\alpha\\right)$-approximate correlated equilibrium of $G_{k,2}$.  The only exact correlated equilibrium is shown by the red point, and the corresponding values of $\\rho$ are displayed as the orange lines.}\n    \\label{fig:cregion}\n\\end{figure}\n\n\\begin{proof}\n    Define $n=2k$ to be the number of players in $G_{k,m}$, and consider some $\\epsilon$-ACE $\\textbf{\\textup{X}}^*$.  Now WLOG consider players $1$ and $2$ and assume, for the sake of contradiction, that there exist some actions $j_1,j_2$ such that $\\Pr_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}^*}(a_1=j_1,a_2=j_2)>\\rho$.  We will need to consider the two cases $j_1=j_2$ and $j_1\\neq j_2$.\n    \\item\\paragraph{Matching Actions} In this case, WLOG assume $j_1=j_2=1$.  We show that player $2$ can improve her payoff by more than $\\epsilon$.  Under $\\textbf{\\textup{X}}^*$, with probability $>\\rho$, any random realization $\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}^*$ will yield player $2$ a payoff of $0$.  In other words, $u_2(\\textbf{\\textup{X}}^*)<1-\\rho$.  Furthermore, considering the marginal distribution over player $1$'s action, we are guaranteed that\n    \\[\\sum_{j=2}^m\\Pr_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}^*}(a_1=j)<1-\\rho\\]\n    so there must exist some action (WLOG action $2$) for which $\\Pr_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}^*}(a_1=2)<\\frac{1-\\rho}{m-1}$.  As such, define $\\phi(j)=2$.  Then $u_2^{(\\phi)}(\\textbf{\\textup{X}}^*)>1-\\frac{1-\\rho}{m-1}$, so\n    \\[\\textnormal{reg}_2^{(\\phi)}(\\textbf{\\textup{X}}^*)>\\underbrace{\\left(1-\\frac{1-\\rho}{m-1}\\right)}_{u_2^{(\\phi)}(\\textbf{\\textup{X}}^*)}-\\underbrace{(1-\\rho)}_{u_2(\\textbf{\\textup{X}}^*)}=\\frac{\\rho m-1}{m-1}\\geq\\frac{m-1}{m}-\\alpha\\]\n    for all $m\\geq2$.  This contradicts our assumption of an $\\epsilon$-ACE.\n    \\item\\paragraph{Mismatched Actions}\n    In this case, WLOG assume $j_1=1,j_2=2$.  The situation is simpler, taking $\\phi(j)=2$, $u_1(\\textbf{\\textup{X}}^*)<1-\\rho$ and $u_1^{(\\phi)}(\\textbf{\\textup{X}}^*)>\\rho$, so $\\textnormal{reg}_1^{(\\phi)}(\\textbf{\\textup{X}}^*)>\\frac{m-1}{m}-\\alpha$.  This too contradicts our assumption of an $\\epsilon$-ACE, and thus completes the proof of Lemma \\ref{lem:game}.\\qed\n\\end{proof}\n\nWe can now prove Theorem \\ref{thm:det}.  The general idea is that, should an efficient algorithm exist, because any equilibrium of $G_{k,m}$ must have large support by Lemma \\ref{lem:game}, there is significant probability assigned to action profiles that are not queried by the algorithm.  We show there is a game that the algorithm cannot distinguish from $G_{k,m}$ that shares no approximate equillibria with $G_{k,m}$.\n\n\\begin{proof}[Theorem \\ref{thm:det}]\n    Consider any $\\alpha>0$ and let $\\epsilon=\\frac{m-1}{m}-\\alpha$.  Taking $\\rho$ as in the statement of Lemma \\ref{lem:game}, assume there exists some deterministic algorithm $A$ that takes an $n$-player, $m$-action game $G$ as input and finds an $\\epsilon$-ACE of $G$ querying the payoffs of $q<\\frac{\\alpha}{2}\\rho^{-\\frac{n}{2}}$ action profiles.  Fix some $k\\in\\mathbb{N}$ and consider input $G_{k,m}$ as defined in Definition \\ref{def:gk}.  Then $\\textbf{\\textup{X}}^*=A\\left(G_{k,m}\\right)$ is an $\\epsilon$-ACE of $G_{k,m}$.  Note that, for some $j$, $\\Pr_{\\textbf{\\textup{a}}\\sim\\textbf{\\textup{X}}^*}(a_1=j)\\leq\\frac{1}{m}$ (WLOG assume $j=1$).\n    \n    Now define the perturbation $G'_{k,m}$ of $G_{k,m}$ with payoffs defined to be equal to $G_{k,m}$ for every action profile queried by $A$, $1$ for every remaining action profile in which player $1$ plays action $1$ (chosen because it is assigned low probability by $\\textbf{\\textup{X}}^*$ by assumption), and $0$ otherwise.  Note that, by definition, $A$ cannot distinguish between $G_{k,m}$ and $G'_{k,m}$, so $A(G'_{k,m})=\\textbf{\\textup{X}}^*$.\n    \n    Taking the function $\\phi(j)=1$, the quantity we need to bound is $\\textnormal{reg}_i^{\\prime(\\phi)}(\\textbf{\\textup{X}}^*)\\geq u_i^{\\prime(\\phi)}(\\textbf{\\textup{X}}^*)-u'_i(\\textbf{\\textup{X}}^*)$.\n    We must bound the components of this expression as follows:\n\\begin{claim}[Appendix \\ref{app:Gkm}]\n$u^{\\prime(\\phi)}_1(\\textbf{\\textup{X}}^*)>(1-q\\rho^{\\frac{n}{2}})$ and $u'_1(\\textbf{\\textup{X}}^*)<\\left(\\frac{1}{m}+q\\rho^{\\frac{n}{2}}\\right)$.\n\\end{claim}\n    Using the claim and once again recalling the assumption that $q<\\frac{\\alpha}{2}\\rho^{-\\frac{n}{2}}$, we see\n    \\[\\textnormal{reg}^{\\prime(\\phi)}_1(\\textbf{\\textup{X}}^*)>\\left(1-\\frac{1}{m}-2q\\rho^{\\frac{n}{2}}\\right)=\\frac{m-1}{m}-\\alpha=\\epsilon.\\]\n    So $\\textbf{\\textup{X}}^*$ cannot actually be an $\\epsilon$-ACE of $G_{k,m}$. This completes the proof of Theorem \\ref{thm:det}.\\qed\n\\end{proof}\n\n\\section{Further Directions}\n\nAn important additional question is the query complexity of finding $\\epsilon$-PNEs of $n$-player, $\\lambda$-Lipschitz games in which $\\epsilon=\\Omega(n\\lambda)$.  Theorem \\ref{thm:pure} says nothing in this parameter range, yet Theorem \\ref{thm:GCW19} provide a logarithmic upper bound in this regime.  The tightness of this bound is of continuing interest.  Furthermore, the query- and computationally-efficient reduction discussed in Lemma \\ref{lem:Nfactor} provides a hopeful avenue for further results bounding the query, and computational, complexities of finding equilibria in many other classes of games.\n\n\\paragraph{Acknowledgements}{We thank Francisco Marmalejo Coss\\'{i}o and Rahul Santhanam for their support in the development of this work, and the reviewers of an earlier version for helpful comments.  Matthew Katzman was supported by an Oxford-DeepMind Studentship for this work.}\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn biomolecular simulations, electrostatic interactions are of paramount importance \ndue to their ubiquitous existence and significant contribution in the force fields, \nwhich governs the dynamics of molecular simulation.  \nHowever, computing non-bonded interactions is challenging \nsince these pairwise interactions are long-range with $O(N^2)$ computational cost, \nwhich could be prohibitively expensive for large systems. \nTo reduce the degree of freedom of the system in terms of electrostatic interactions,  \nan implicit solvent Poisson-Boltzmann (PB) model is used \\cite{Baker:2005}. \nIn this model, the explicit water molecules are treated as continuum \nand the dissolved electrolytes are approximated using the statistical Boltzmann distribution. \nThe PB model has broad applications in biomolecular simulations such as \nprotein structure \\cite{Cherezov:2007}, \nprotein-protein interaction \\cite{Huang:2012}, \nchromatin packing \\cite{Beard:2001}, \npKa \\cite{Alexov:2011,Hu:2018,Chen:2021a}, \nmembrane \\cite{Zhou:2010}, \nbinding energy \\cite{Nguyen:2017}, \nsolvation free energy\\cite{Wagoner:2006},  \nion channel profiling \\cite{Unwin:2005}, etc.\n\nThe PB model is an elliptic interface problem \nwith several numerical difficulties \nsuch as \ndiscontinuous dielectric coefficients, \nsingular sources, \na complex interface, \nand unbounded domains. \nGrid-based finite difference or finite volume discretization that discretize the entire\nvolumetric domain have been developed in, e.g.,\n\\cite{\nBaker:2004, \nIm:1998, \nHonig:1995,\nBaker:2001,\nLuo:2002,\nDeng:2015,\nYing:2015}.\nThe grid-based discretization is efficient and robust and is \ntherefore popular.  \nHowever, solvers that are based on discretizing the partial\ndifferential equation may suffer from accuracy reduction due to \ndiscontinuity of the coefficients, \nnon-smoothness of the solution, \nsingularity of the sources, \nand truncation of the domains, \nunless special interface \\cite{Qiao:2006, Yu:2007} \nand singularity \\cite{Geng:2007, Cai:2009,Geng:2017, Lee:2020} treatments are applied. \nThese treatments come at the price of more complicated discretization scheme \nand possibly reduced convergence speed of the iterative solver for the\nlinear system. \n\n\nAn alternative approach is to reformulate the PB equation as a\nboundary integral equation and use the boundary elements \nto discretize the molecular surface,\ne.g.~\\cite{\nJuffer:1991,\nBoschitsch:2002,\nLu:2007,\nGreengard:2009,\nBajaj:2011,\nZhang:2013,\nGeng:2013b,\nZhong:2018,\nQuan:2019}.\nBesides the reduction from\nthree dimensional space to the two dimensional molecular boundary,\nthis approach has the advantage that singular charges, interface\nconditions, and far-field condition are incorporated analytically in\nthe formulation, and hence do not impose additional approximation\nerrors.\n\nIn addition, due to the structures hidden in the linear algebraic system \nafter the discretization of the boundary integral and molecular surface, \nthe matrix-vector product in each iteration can be accelerated by fast methods \nsuch as fast multipole methods (FMM) \\cite{Greengard:2002,Tausch:2004,\nBoschitsch:2002,Lu:2007,Bajaj:2011}\nand\ntreecodes~\\cite{Barnes:1986, Li:2009, Wang:2020a}.\nOur recently developed treecode-accelerated boundary integral (TABI) Poisson-Boltzmann solver \n\\cite{Geng:2013b} is an example of a code that combines the advantages of \nboth boundary integral equation and multipole methods. \nThe TABI solver uses the well-posed derivative form of the Fredholm second kind integral equation \\cite{Juffer:1991}\nand the $O(N\\log N)$ treecode \\cite{Li:2009}. \nIt also has advantages in memory use and parallelization\n\\cite{Geng:2013b, Chen:2021}. The TABI solver has been\nused by many computational biophysics\/biochemistry groups and it has\nbeen disseminated as a standalone code \\cite{Geng:2013b} or as a contributing module of the popular APBS software package \\cite{Baker:2001a,Jurrus:2018}.\n\nRecently, based on feedback from TABI solver users \nand our gained experience in theoretical development and practical applications, \nwe realized that we could still improve the TABI solver in the following aspects.\nFirst, the $O(N \\log N)$ treecode can be replaced by the $O(N)$ FMM method \nwith manageable extra costs in memory usage and complexity of the algorithms. \nSecond, the singularity that occurs \nwhen the Poisson's or Yukawa's kernel is evaluated \nwas previously handled by simply removing the singular triangle \\cite{Geng:2013b, Lu:2007}\nin fact can be treated by using the Duffy transformation \\cite{Duffy:1982} analytically, \nachieving improved accuracy. \nThird,  the collocation scheme used in TABI solver \ncan be updated by using Galerkin discretization\nwith further advantage in maintaining desired accuracy. \nFourth, the treecode-based preconditioning scheme that was used\nin TABI solver \\cite{Chen:2018a} can be similarly developed and used under the FMM frame, \nreceiving significant improvement in convergence and robustness. \nBy combining all these new features, we developed a Cartesian fast multipole method (FMM) \naccelerated Galerkin boundary integral (FAGBI) Poisson--Boltzmann\nsolver. In the remainder of this article, we provide more detail\nabout the theoretical background of the numerical algorithms related\nto the FAGBI solver. We conclude with a discussion of the numerical\nresults obtained with our implementation.  \n\n\n\n\n\n \n\n \n\n\n\\section{Theory and Algorithms}\n\\label{theory}\n\nIn this section we briefly describe \nthe Poisson-Boltzmann (PB) implicit solvent model and review the\nboundary integral form of the PB equation and its\nGalerkin discretization. \nBased on this background information, \nwe then provide details of our recently developed FMM-accelerated Galerkin\nboundary integral (FAGBI)  Poisson-Boltzmann solver, which involves\nthe boundary integral form, multipole expansion scheme, and a  block diagonal preconditioning scheme. \n\n\n\\begin{figure}[htb]\n\\setlength{\\unitlength}{1cm}\n\\begin{picture}(5,1)\n\\put( 5.2, -3.2){\\Large $\\Omega_1$}\n\\put( 2.4, -0.9){\\Large $\\Omega_2$}\n\\put( 4.8, -5.2){\\Large $\\Gamma$}\n\\end{picture}\n\\begin{center}\n\\includegraphics[width=2.5in]{figures\/pbe1layer}~~~~~\n\\includegraphics[width=2.5in]{figures\/1b2s_barstar_d5}\\\\\n\\hskip -0.7in (a) \\hskip 3in (b)\n\\caption{\nSchematic models;\n(a) the PB implicit solvent model, in which\nthe molecular surface $\\Gamma$ separates space into the solute region $\\Omega_1$ \nand solvent region $\\Omega_2$; \n(b) the triangulation of molecular surface of protein Barstar at MSMS \\cite{Sanner:1996} density $d=5$ (\\# of vertices per \\AA$^2$). \n}\n\\label{fig_1}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{The Poisson-Boltzmann (PB) model for a solvated biomolecule}\n\nThe PB model for a solvated biomolecule is depicted in Fig.~\\ref{fig_1}(a) in which\nthe molecular surface $\\Gamma$ \nseparates the solute domain $\\Omega_1$ from the solvent domain $\\Omega_2$.\nFigure~\\ref{fig_1}(b) is an example of the molecular surface $\\Gamma$ \nas the triangulated surface of protein barstar \\cite{Dong:2003}.  \nIn domain $\\Omega_1$, \nthe solute is represented by $N_c$ partial charges $q_k$ located at atomic centers ${\\bf r}_k$ for $k=1,\\cdots,N_c$,\nwhile in domain $\\Omega_2$,\na distribution of ions is described by a Boltzmann distribution \nand we consider a linearized version in this study. \nThe solute domain has a low dielectric constant $\\epsilon_1$\nand\nthe solvent domain has a high dielectric constant $\\epsilon_2$.\nThe modified inverse Debye length $\\bar{\\kappa}$ is given as $\\bar{\\kappa}^2 = \\epsilon_2\\kappa^2$,\nwhere $\\kappa$ is the inverse Debye length measuring the ionic strength;\n$\\bar{\\kappa} = 0$ in $\\Omega_1$ and is nonzero only in $\\Omega_2$. \nThe electrostatic potential $\\phi({\\bf x})$ satisfies the linear PB equation,\n\\begin{equation}\n-\\nabla \\cdot \\epsilon({\\bf x}) \\nabla \\phi({\\bf x}) + \\bar{\\kappa}^2({\\bf x})\\phi({\\bf x}) =\n\\sum_{k=1}^{N_c} q_k \\delta({\\bf x}-{\\bf x}_k),\n\\label{eqNPBE}\n\\end{equation}\nsubject to continuity conditions for the potential and electric flux density on $\\Gamma$,\n\\begin{equation}\n[\\phi] = 0, \\quad [\\epsilon \\phi_\\nu] = 0,\n\\label{eqJump}\n\\end{equation}\nwhere \n$[f] = f_1 - f_2$ is the difference of the quantity $f$ across the interface,\nand\n$\\phi_\\nu = \\partial\\phi\/\\partial{\\nu}$ is the partial derivative in the outward normal direction $\\nu$.\nThe model also incorporates the far-field condition,\n\\begin{equation}\n\\label{eqfar-field}\n\\lim_{{\\bf x} \\rightarrow \\infty} \\phi({\\bf x }) = 0.\n\\end{equation}\nNote that Eqs.~(\\ref{eqNPBE})-(\\ref{eqfar-field}) \ndefine a boundary value problem for the potential $\\phi({\\bf x})$\nwhich in general must be solved numerically.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Boundary integral form of PB model}\n\nThis section summarizes the well-conditioned boundary integral form of the \nPB implicit solvent model we employ~\\cite{Juffer:1991,Geng:2013b}.\nApplying Green's second identity and properties of fundamental solutions \nto Eq.~(\\ref{eqNPBE}) yields the electrostatic potential in each domain,\n\\begin{subequations}\n\\begin{align}\n\\label{eqbim_1}\n\\phi({\\bf x}) = \n&\\int_\\Gamma \\left[G_0({\\bf x},\n{\\bf y})\\frac{\\partial\\phi({\\bf y})}{\\partial{\\nu}} -\\frac{\\partial G_0({\\bf x}, {\\bf y})}{\\partial{\\nu}_{{\\bf y}}} \\phi({\\bf y})\n\\right]dS_{{\\bf y}} + \\sum_{k=1}^{N_{c}}q_k G_0({\\bf x}, {\\bf y}_k), \\quad {\\bf x} \\in \\Omega_1, \\\\\n\\label{eqbim_2}\n\\phi({\\bf x}) = \n&\\int_\\Gamma\\left[-G_\\kappa({\\bf x},\n{\\bf y})\\frac{\\partial\\phi({\\bf y})}{\\partial{\\nu}} + \\frac{\\partial G_\\kappa({\\bf x}, {\\bf y})}{\\partial{\\nu}_{{\\bf y}}}\n\\phi({\\bf y})\\right]dS_{{\\bf y}}, \\quad {\\bf x} \\in \\Omega_2,\n\\end{align}\n\\end{subequations}\nwhere $G_0({\\bf x},{\\bf y})$ and $G_\\kappa({\\bf x}, {\\bf y})$ are the \nCoulomb and screened Coulomb potentials,\n\\begin{equation}\nG_0({\\bf x}, {\\bf y}) = \\frac{1}{4\\pi|{\\bf x}-{\\bf y}|},\n\\quad\n\\label{eq_potential}\nG_\\kappa({\\bf x}, {\\bf y}) = \\frac{e^{-\\kappa|{\\bf x}-{\\bf y}|}}{4\\pi|{\\bf x}-{\\bf y}|},\n\\end{equation}\nand ${\\bf y}_k \\in \\Omega_1$ are the location of the atomic centers.\n\nApplying the interface conditions in Eq.~(\\ref{eqJump}) \nwith the differentiation of electrostatic potential in each domain\nyield \na set of boundary integral equations relating the\nsurface potential $\\phi_1$ (the subscript 1 denotes the inside domain)\nand\nits normal derivative $\\partial\\phi_1\/\\partial{\\nu}$ on $\\Gamma$, \\cite{Juffer:1991, Geng:2013b},\n\\begin{subequations}\n\\begin{align}\n\\label{eqbim_3}\n\\frac{1}{2}\\left(1+\\varepsilon\\right)\\phi_1({\\bf x}) & =\n\\int_\\Gamma \\left[K_1({\\bf x}, {\\bf y})\\frac{\\partial\\phi_1({\\bf y})}{\\partial{\\nu}} +\nK_2({\\bf x}, {\\bf y})\\phi_1({\\bf y})\\right]dS_{{\\bf y}}+S_{1}({\\bf x}),\n\\quad {\\bf x}\\in\\Gamma, \\\\\n\\label{eqbim_4}\n\\frac{1}{2}\\left(1+\\frac{1}{\\varepsilon}\\right)\\frac{\\partial\\phi_1({\\bf x})}{\\partial{\\nu}} & =\n\\int_\\Gamma \\left[K_3({\\bf x}, {\\bf y})\\frac{\\partial\\phi_1({\\bf y})}{\\partial{\\nu}} +\nK_4({\\bf x}, {\\bf y})\\phi_1({\\bf y})\\right]dS_{{\\bf y}}\n+S_{2}({\\bf x}),\n\\quad {\\bf x} \\in \\Gamma,\n\\end{align}\n\\end{subequations}\nwhere $\\varepsilon = \\varepsilon_2\/\\varepsilon_1$, \nand\nthe kernels $K_{1,2,3,4}$ and source terms $S_{1,2}$ are\n\\begin{subequations}\n\\begin{align} \n\\label{Eq_K1}\nK_1({\\bf x}, {\\bf y}) = \n&\\,{G_{0}({\\bf x},{\\bf y})}-{G_{\\kappa}({\\bf x},{\\bf y})},\n\\quad\nK_2({\\bf x}, {\\bf y}) = \\varepsilon\\frac{\\partial G_{\\kappa}({\\bf x},{\\bf y})}{\\partial\\nu_{{\\bf y}}}-\\frac{\\partial\nG_{0}({\\bf x},{\\bf y})}{\\partial\\nu_{{\\bf y}}}, \\\\\n\\label{Eq_K2}\nK_3({\\bf x}, {\\bf y}) = \n&\\,\\frac{\\partial G_{0}({\\bf x},{\\bf y})}{\\partial\\nu_{{\\bf x}}}-\\frac{1}{\\varepsilon}\\frac{\\partial G_{\\kappa}({\\bf x},{\\bf y})}{\\partial\\nu_{{\\bf x}}},\n\\quad\nK_4({\\bf x}, {\\bf y}) =\n\\frac{\\partial^2 G_\\kappa({\\bf x},{\\bf y})}{\\partial\\nu_{{\\bf x}}\\partial\\nu_{{\\bf y}}}-\\frac{\\partial^2G_{0}({\\bf x},{\\bf y})}{\\partial\\nu_{{\\bf x}}\\partial\\nu_{\\bf y}},\n\\end{align}\n\\end{subequations}\nand \n\\begin{equation}\nS_{1}({\\bf x}) = \\frac{1}{\\varepsilon_1}\\sum_{k=1}^{N_{c}}q_kG_{0}({\\bf x}, {\\bf y}_k),\n\\quad\nS_{2}({\\bf x}) = \\frac{1}{\\varepsilon_1}\\sum_{k=1}^{N_{c}}q_k\n\\frac{\\partial G_{0}({\\bf x},{\\bf y}_k)}{\\partial\\nu_{{\\bf x}}}.\n\\label{source_terms}\n\\end{equation}\nAs given in Eqs.~(\\ref{Eq_K1}-\\ref{Eq_K2}) and (\\ref{source_terms}), the kernels $K_{1,2,3,4}$ and source terms $S_{1,2}$ are linear combinations of $G_0$, $G_k$, \nand their first and second order normal\nderivatives~\\cite{Juffer:1991,Geng:2013b}. Since the Coulomb potential\nis singular, the kernels have the following behavior\n\\begin{equation*}\n  K_1({\\bf x}, {\\bf y}) = O(1),\\;\n  K_{2,3,4}({\\bf x}, {\\bf y}) = O\\left(\\frac{1}{|{\\bf x}-{\\bf y}|}\\right),\\;\n \n \n\\end{equation*}\nas $ {\\bf y} \\to  {\\bf x}$. \n\n\n\n\n\n\nAfter the potentials $\\phi_1$ and $\\partial \\phi_1\/\\partial \\nu$ have\nbeen found by solving the boundary integral equation, the  electrostatic\nsolvation energy can be obtained by\n\\begin{equation}\nE_{\\rm sol} = \\frac{1}{2}\\sum_{k=1}^{N_c}q_k\\phi_{\\rm reac}({\\bf y}_k) =\n\\frac{1}{2}\\displaystyle \\sum\\limits_{k=1}^{N_c} q_k\n\\int_\\Gamma \\left[K_1({\\bf y}_k, {\\bf y})\\frac{\\partial\\phi_1({\\bf y})}\n{\\partial\\nu}+K_2({\\bf y}_k, {\\bf y})\\phi_1({\\bf y})\\right]dS_{{\\bf y}},\n\\label{solvation_energy}\n\\end{equation}\nwhere $\\phi_{\\rm reac}({\\bf x}) = \\phi_1({\\bf x})-S_1({\\bf x})$ is the\nreaction potential \\cite{Juffer:1991,Geng:2013b}.\n\n\n\\subsection{Galerkin Discretization}\n\nIn solving the boundary integral PB equation, \nboth the molecular surface and the solution function need to be discretized. \nThe molecular surface $\\Gamma$ is usually approximated by\na collection of triangles\n\\begin{equation}\n\\label{eqbemesh}\n\\Gamma_{N}=\\bigcup\\limits^{N}_{i=1} \\tau_{i},\n\\end{equation}\nwhere $N$ is number of elements \nand $\\tau_i$ for $i=1,\\dots,N$ is a planar triangular boundary element with mid-point $\\textbf{x}^c_i$.\nThis triangulation must be conforming, i.e., the intersection of two\ndifferent triangles is either empty, or a common vertex or edge.\nFortunately, surface generators are available, and\nour choice for our computations is MSMS~\\cite{Sanner:1996}, though\nother packages could also be used.\nHere, the resolution of the surface can be controlled by the parameter\n$d$ that controls the number of vertices per \\AA$^2$.\nFor example, Fig.~\\ref{fig_1} (b)\nshows the triangulated molecule surface\nof the protein barstar, which will bind another protein barnase to form a\nbiomolecular complex (PDB: 1b2s) \\cite{Dong:2003}.\n\n\nEach triangle $\\tau_i$ of $\\Gamma_N$ is the parametric image of the reference triangle $\\tau$\n\\begin{equation}\n\\label{eqreftri}\n\\tau = \\big\\{ \\eta=(\\eta_1, \\eta_2) \\in \\mathbb{R}^2 : 0 \\le \\eta_1 \\le 1 , 0 \\le \\eta_2 \\le \\eta_1 \\big\\}.\n\\end{equation}\nIf $\\textbf{u}_i$, $\\textbf{v}_i$ and $\\textbf{w}_i$ are the vertices\nthen the parameterization is given by\n\\begin{equation}\n\\label{eqparameter}\n\\textbf{x}(\\eta) = \\textbf{u}_i + \\eta_1(\\textbf{v}_i-\\textbf{u}_i) + \\eta_2(\\textbf{w}_i-\\textbf{u}_i) \\in \\tau_i \\quad \\text{for } \\eta=(\\eta_1,\\eta_2)\\in\\tau.\n\\end{equation}\nThe area of the element, the local mesh size, and the global mesh size of the boundary elements $\\tau_i$ are given as\n$A_i=\\frac{1}{2}|(\\textbf{v}_i-\\textbf{u}_i)\\times (\\textbf{w}_i-\\textbf{u}_i)|$, $h_i=\\sqrt{A_i}$, and \n$h=\\max\\limits_{1\\le i \\le N}h_i$.\n\nSince a function $f$ defined on $\\tau_i$ can be interpreted as a function $g(\\eta)$ with respect to the reference element $\\tau$,\n\\begin{equation}\n\\label{eqf2g}\nf(\\textbf{x}) = f(\\textbf{x}(\\eta)) = g(\\eta) \\quad \\text{for } \\eta\\in\\tau, \\quad {\\bf x}\\in\\tau_i.\n\\end{equation}\nwe can define a finite element space by functions on $\\Gamma_N$ whose\npullbacks to the reference triangle are polynomials in $\\eta$. The\nsimplest example is the space of piecewise constant functions, which\nare polynomials of order zero on each triangle, which will be denoted\nby $S^0_h(\\Gamma_N)$. Obviously, the dimension of this space is $N$\nand the basis is given by the box functions\n\\begin{equation}\n\\label{eqpiecewiseconstant}\n\\psi^0_i(\\bf{x}) =\\begin{cases}\n1 & \\text{if $\\bf{x}\\in\\tau_i$},\\\\\n0 & \\text{otherwise},\n\\end{cases}\n\\end{equation}\nwhere $i$ is an index of a triangle.\n\nThe next step up are piecewise linear\nfunctions. Since there are three independent linear functions on\n$\\tau$, namely,\n\\begin{equation}\n\\label{eqlinearfcn}\n\\psi^{1}_1(\\eta) = 1-\\eta_1, \\quad \\psi^{1}_2(\\eta) = \\eta_1-\\eta_2, \\quad \\psi^{1}_3(\\eta) = \\eta_2\n\\quad \\text{for }\\eta=(\\eta_1, \\eta_2)\\in\\tau.\n\\end{equation}\nthe dimension is $3N$. Usually, one works with the space of\ncontinuous linear functions, denoted by $S^1_h(\\Gamma_N)$. It is not hard to see that the dimension\nof this space is the number of vertices and that the basis is given by\n\\begin{equation}\n\\label{eqglinearcfcn}\n\\psi^c_{i}(\\textbf{x}) = \\begin{cases}\n1 & \\text{for $\\textbf{x}=\\textbf{v}_i$,}\\\\\n0 & \\text{for $\\textbf{x}=\\textbf{v}_j\\neq\\textbf{v}_i$,}\\\\\n\\text{piecewise linear}& \\text{elsewhere,}\n\\end{cases}\n\\end{equation}\nwhere ${v}_i$ is the $i$-th vertex. \n\nThe approximation powers of piecewise polynomial spaces are well\nknown, see, e.g., \\cite{Rjasanow:2006}. For a function\n$w\\in H^1_{pw}(\\Gamma_N)$ we denote by $w_h^0$ the $L_2$-orthogonal\nprojection of $w$ into the space of piecewise constant functions, then\n\\begin{equation}\n\\label{eqconstanterr}\n\\|w-w_h^0\\|_{L_2(\\Gamma_N)} \\le\nc\\left(\\sum_{i=1}^{N}h^2_i|w|^2_{H^1(\\tau_i)}\\right)^{\\frac{1}{2}}\n  \\le c h |w|_{H^1_{pw}(\\Gamma_N)},\n\\end{equation}\nwhere $c$ is the upper bound of the mesh ratio $h_{\\max}\/h_{\\min}$,\n$h$ is the maximal diameter of a triangle\nand\n\\begin{equation}\n\\label{eqHpw}\n|w|_{H^1_{pw}(\\Gamma_N)} = \\Bigg(\\sum_{i=1}^{N}|w|^2_{H^1(\\tau_i)}\\Bigg)^{1\/2}.\n\\end{equation}\nThus, the constant piecewise basis function can give a convergence rate of maximum $O(h)$.\n\nLikewise, the error for the $L_2$-orthogonal projection $w_h^1$ of $w\\in\nH^2_{pw}(\\Gamma_N)$ into $S^1_h(\\Gamma_N)$ is\n\\begin{equation}\n\\label{lineerr}\n\\|w-w_h^1\\|_{L_2(\\Gamma_N)} \\le\nc\\left(\\sum_{i=1}^{N}h^2_i|w|^2_{H^2(\\tau_i)}\\right)^{\\frac{1}{2}}\n  \\le c h^2 |w|_{H^2_{pw}(\\Gamma_N)},\n\\end{equation}\n\n\nFinite element spaces with higher order polynomials could also be\nconsidered, however their practical value for surfaces with low\nregularity and complicated geometries is limited.\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\nThe Galerkin discretization is based on a variational formulation of\nintegral equations (\\ref{eqbim_3}) and (\\ref{eqbim_4}). That is,\ninstead of understanding the equations pointwise for ${\\bf x} \\in \\Gamma_N$\nthe equations are multiplied by test functions $\\psi$ and\n$\\psi^\\nu$ and integrated again over $\\Gamma_N$. Solving the\nvariational form amounts to finding $\\phi_1$ and\n$\\partial\\phi_1\/\\partial \\nu$ such that \n{\\small\n\\begin{subequations}\n\\begin{align}\n\\label{eqbim_5}\n\\int_{\\Gamma_N} \\bigg\\{ \\frac{1}{2}\\left(1+\\varepsilon\\right)\\phi_1({\\bf x}) & -\n\\int_{\\Gamma_N} \\left[K_1({\\bf x}, {\\bf y})\\frac{\\partial\\phi_1({\\bf y})} {\\partial{\\nu}} +\nK_2({\\bf x}, {\\bf y})\\phi_1({\\bf y})\\right]dS_{{\\bf y}} \\bigg\\} \\psi({\\bf x}) dS_{\\bf{x}}= \\int_{\\Gamma_N} S_{1}({\\bf x}) \\psi({\\bf x}) dS_{\\bf{x}},\\\\\n\\label{eqbim_6}\n\\int_{\\Gamma_N} \\bigg\\{ \\frac{1}{2}\\left(1+\\frac{1}{\\varepsilon}\\right)\\frac{\\partial\\phi_1({\\bf x})}{\\partial{\\nu}} & -\n\\int_{\\Gamma_N} \\left[K_3({\\bf x}, {\\bf y})\\frac{\\partial\\phi_1({\\bf y})}{\\partial{\\nu}} +\nK_4({\\bf x}, {\\bf y})\\phi_1({\\bf y})\\right]dS_{{\\bf y}} \\bigg\\} \\psi^\\nu({\\bf x}) dS_{\\bf{x}}\n= \\int_{\\Gamma_N} S_{2}({\\bf x}) \\psi^\\nu({\\bf x}) dS_{\\bf{x}},\n\\end{align}\n\\end{subequations}\n}\nholds for all test functions $\\psi, \\psi^\\nu$. In the Galerkin method the solution and\nthe test functions are formally replaced by functions in the finite element\nspace. To that end, the unknowns are expanded by basis functions\n$\\psi_i$ (which\ncould be either box or hat functions)\n\\begin{equation}\n  \\label{eqprojectphi}\n \\phi \\approx \\sum\\limits_{i=1}^{N} \\phi_{i}\\psi_{i}, \\quad\\mbox{and}\\quad\n \\frac{\\partial \\phi}{\\partial \\nu} \\approx \\sum\\limits_{i=1}^{N} \\phi^\\nu_{i}\\psi_{i}\n\\end{equation}\nand integral equations are tested against the basis functions. This\nleads to the linear system $Ax=b$ where $x$ contains the coefficients\nin \\eqref{eqprojectphi} and\n\\begin{equation}\\label{def:linsys}\nA=\\left[\\begin{array}{cc}A_{11} & A_{12} \\\\A_{21} & A_{22}\\end{array}\\right]\n\\quad \\mbox{and} \\quad                                               \nb =   \\left[\\begin{array}{c} b_1 \\\\ b_2 \\end{array}\\right] .\n\\end{equation}\nThe entries of these block matrices are given as\n\\begin{eqnarray}\n\\label{eq_linsysEntry}\n\\nonumber\n  A_{11}(i,j) &=& \\int_{\\Gamma_N}\n  \\frac{1}{2}\\left(1+{\\varepsilon}\\right) \\psi_i({\\bf x}) \\psi_j({\\bf x}) \\text{d}S_{\\bf x}\n+ \\int_{\\Gamma_N} \\int_{\\Gamma_N} K_2({\\bf x}, {\\bf y}) \n   \\psi_i({\\bf x}) \\psi_j({\\bf y}) \\text{d} S_{\\bf y}  \\text{d}S_{\\bf x}\\\\\\nonumber\n  A_{12}(i,j) &=& \\int_{\\Gamma_N} \\int_{\\Gamma_N} K_1({\\bf x}, {\\bf y}) \n   \\psi_i({\\bf x}) \\psi_j({\\bf y}) \\text{d} S_{\\bf y} \\text{d}S_{\\bf x}\\\\\\nonumber\n  A_{21}(i,j) &=& \\int_{\\Gamma_N} \\int_{\\Gamma_N} K_4({\\bf x}, {\\bf y}) \n   \\psi_i({\\bf x}) \\psi_j({\\bf y}) \\text{d} S_{\\bf y}  \\text{d}S_{\\bf x}\\\\\n  A_{22}(i,j) &=& \\int_{\\Gamma_N}\n  \\frac{1}{2}\\left(1+\\frac{1}{\\varepsilon}\\right) \\psi_i({\\bf x}) \\psi_j({\\bf x}) \\text{d}S_{\\bf x}\n+ \\int_{\\Gamma_N} \\int_{\\Gamma_N} K_3({\\bf x}, {\\bf y}) \n   \\psi_i({\\bf x}) \\psi_j({\\bf y}) \\text{d} S_{\\bf y}  \\text{d}S_{\\bf x} \n\\end{eqnarray}\nand the right hand side is\n\\begin{equation}\n  b_1(i) = \\int_{\\Gamma_N} S_{1}({\\bf x}) \\psi_i({\\bf x}) dS_{\\bf{x}}\n  \\quad \\mbox{and} \\quad\n  b_2(i) = \\int_{\\Gamma_N} S_{2}({\\bf x}) \\psi_i({\\bf x}) dS_{\\bf{x}}\n\\end{equation}\nSince the basis functions vanish on most triangles, the integrations for\nthe coefficients are only local. For instance, for piecewise constant\nelements, the integral $\\int_{\\Gamma}\\dots\\psi_i  dS_{\\bf{x}}$ reduces\nto $\\int_{\\tau_i}\\dots dS_{\\bf{x}}$.\nSince the coefficients cannot be expressed in analytical form they\nhave to be calculated by a suitable choice of quadrature rule.\nHowever singularities will appear \nif triangles $\\tau_i$ and $\\tau_j$ are identical or sharing common edges and vertices.  \nTo overcome this issue,\nwe apply the singularity removing transformation of\n\\cite{Sauter:1996}. This results smooth integrals over a four\ndimensional cube. The latter integrals are then approximated by\ntensor product Gauss-Legendre quadrature.\n\n\n\n\n\nAfter the solution of the linear system has been obtained, \nthe electrostatic free solvation energy can be calculated using the\napproximations for the surface potentials and its normal derivative\n\\begin{equation}\n\\label{eqsolengdis}\nE_{sol} = \\frac{1}{2}\\sum_{n=1}^{N_c}q_n\\sum_{i=1}^{N}\n\\int_{\\tau_{i}}\\Big[K_1(\\textbf{x}_n,\\textbf{x})  \\phi_{1i}^{\\nu} + K_2(\\textbf{x}_n,\\textbf{x})\\phi_{1i} \\Big] dS_{\\textbf{x}}.\n\\end{equation}\n\n\nSince the matrix $A$ is a dense and non-symmetric our choice of solver\nis the GMRES method.\nIn each step of GMRES iteration, a matrix-vector product is calculated and a direct summation for this requires $O(N^2)$ complexity.\nBelow we will introduce the $O(N)$ Cartesian Fast Multipole Method (FMM) to accelerate the matrix-vector product. Calculating the electrostatic solvation free energy $E_{sol}$ in Eq.~(\\ref{eqsolengdis}) is $O(N_cN)$ and we use a Cartesian treecode to reduce the cost to $O(N_c\\log(N))$. Both FMM and treecode algorithm are described the next for comparison and for the reason that both are used to accelerate the N-body particle-particle interactions.\n\n\n\n\n\\subsection{Cartesian fast multipole method (FMM)}\nIn this section, \nwe introduce the Cartesian FMM to evaluate matrix vector products with\nthe matrix in (\\ref{def:linsys}) efficiently. This is a\nkernel-independent version of the FMM, where instead of the multipole\nexpansion, truncated Taylor series are used to approximate  Coulomb ($\\kappa=0$) the screened Coulomb ($\\kappa \\ne 0$) potential.\n\nThis considerably simplifies the translation operators for the kernels $K_{1-4}$ because\nthey involve different values of $\\kappa$. It was shown in\n\\cite{Tausch:2003} how the derivatives of the Coulomb kernel can be\ncomputed by simple recurrence formulas. \nFurthermore, the moment-to-moment (MtM) and local-to-local (LtL) translation\nare easily derived using the binomial formula. \n\nWhen a refined mesh is required for a larger $N$, \nincreasing the expansion order is essential to control the accuracy.\nThe FMM error analysis implies that the truncated Taylor expansion error \nhas the same magnitude as the discretization error if the expansion\norder is adjusted to the level according to the formula\n\\begin{equation}\\label{variable:order}\n  p_l = p_L + L - l\n\\end{equation}\nwhere $l = 0,1,\\cdots,L$ with $l=0$ the coarsest level and $l=L$ the finest level. That\nis, the finest level uses a low-order expansion $p_L$, and the order\nis incremented in each coarser level, see \\cite{Tausch:2004}.\n\nNote that the multipole series is more efficient as it contains $(p+1)^2$\nterms, while the Taylor series has $p(p+1)(p+2)\/6$ terms. This\ndifference becomes significant with larger values of $p$. However,\nwith the variable order scheme the advantage of the multipole series\nis much less because most translation operators are in the fine levels\nwhere the number of terms in both series are comparable.\n\n\nThe matrix vector product can be considered as generalized $N$-body\nproblem of the form\n\\begin{equation}\n\\label{eqgeneralKernel}\nV_i = \\int_{\\Gamma_N} \\int_{\\Gamma_N} \\psi_i({\\bf x})\\frac{\\partial^l}{\\partial\\nu_{\\bf x}^l} \\frac{\\partial^k}{\\partial\\nu_{\\bf y}^k} G({\\bf x},{\\bf y}) f_h({\\bf y}) dS_{\\bf y} dS_{\\bf x}.\n\\end{equation}\nwhere $k,l \\in \\{0,1\\}$ and $f_h$ is a linear combination of the basis\nfunctions $\\psi_j$.\n\n\n\nNext we show how the Cartesian FMM is used under the framework of boundary element method. \n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=3.3in]{figures\/precinterlist}\n\\includegraphics[width=3.3in]{figures\/particle_cluster}\n\\caption{FMM vs Treecode structure. \nLeft: FMM cluster-cluster interaction list; \nRight: treecode particle-cluster interaction ($R$ is the distance from the charge to the blue cluster's center;\n$r_c$ is the radius of the blue cluster $c$ which is the farthest particle inside $c$ to the center of $c$.}\n\\label{fig_2}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{fig_2}a is a 2-D illustration of a discretized molecular surface $\\Gamma_N$ \nembedded in a hierarchy of cubes (squares in the image). \nEach black solid dot represents a boundary triangle  $\\tau_{i}$ for $i=1,\\cdots,N$. \n\nA cluster $c$ in any level $l$ is defined as the union of triangles\nwhose centroid are located in a cube of that level. $C_l$ is the set\nof all clusters in level $l$.\n\n\n\n\nThe level-0 cube is the smallest axiparallel cube that contains\n$\\Gamma_N$ and \nthus $C_0 = \\Gamma_N$. The refinement of coarser cubes into finer\ncubes stops when clusters in the finest level contain at most a\npredetermined (small) number of triangles. For a cluster $c$ we denote\nby $B_c$ the smallest axiparallel rectangular box that contains $c$\nand write ${\\bf x}_c$ for its center and $\\rho_c$ for the half-length\nof its longest diagonal. Note that $B_c$ can be\nconsiderably smaller than its cube, this is why we call this process\nthe shrink scheme. For two clusters $c$ and $c'$ in the same level we\ndenote by\n\\begin{equation}\n  \\eta(c,c') = \\frac{ \\rho_c + \\rho_{c'}}{|{\\bf x}_c - {\\bf x}_{c'}|}\\label{def:eta}\n\\end{equation}\nthe separation ratio of the two clusters. This number determines the\nconvergence rate of the Taylor series\nexpansion, see~\\cite{Tausch:2004}. Two clusters in the same level are neighbors if\ntheir separation ratio is larger than a predetermined constant.  \n$\\mathcal{N}(c)$ denotes the set of its neighbors for a given cluster $c$.\nThe set of nonempty children of $c$ that are generated in the refinement\nprocess is denoted by $\\mathcal{K}(c)$.\nFinally, we use $\\mathcal{I}(c)$ to denote the interaction list for a cluster $c$,\nwhich are clusters at the same level such that \nfor any $c'\\in \\mathcal{I}(c)$, \nthe parent of $c'$ is a neighbor of the parent of $c$, but $c'$ itself\nis not a neighbor of $c$.\n\nUnder the FMM framework, \nthe evaluation of Eq.~(\\ref{eqgeneralKernel}) consists of the near field direct summation \nand the Taylor expansion approximation for well-separated far field.\nThe near field direct summation happens in between neighboring panels\nin the finest level.  The far field summation is done by multipole or\nTaylor expansions between interaction lists in all levels. This\nprocess is described in many papers, so we do not give details about\nthe derivation. \n\n\n\nTo emphasize the differences of the cartesian FMM, we \nconsider a cluster-cluster interaction between two clusters $c$ and\n$c' \\in \\mathcal{I}(c)$. Let $u$ be the potential due to sources in\n$c'$ which is evaluated in $c$, then  \nby Taylor expansion of the kernel with center ${\\bf x} = {\\bf x}_c$ and ${\\bf y} =\n{\\bf x}_{c'}$ one finds easily that\n\\begin{equation}\n\\label{eqapproximation}\nu_{c,c'}({\\bf x}) = \\int_{c'} \\frac{\\partial^l}{\\partial\\nu_{\\bf x}^l} \\frac{\\partial^k}{\\partial\\nu_{\\bf y}^k} G({\\bf x}, {\\bf y})f_h({\\bf y}) dS_{\\bf y} \\approx \\sum_{|\\alpha| \\leq p}\\lambda ^{\\alpha}_{c}  \\frac{\\partial^l}{\\partial\\nu_{\\bf x}^l} ({\\bf x}-{\\bf x}_{c})^{\\alpha} .\n\\end{equation}\nwhere \nand $\\alpha = (\\alpha_1, \\alpha_2, \\alpha_3) \\in \\mathbb{N}^3$\nis a multi-index. \nThe expansion coefficients are given by \n\\begin{equation}\n\\label{eqcoeff}\n\\lambda^{\\alpha}_{c} = \\sum\\limits^{p-|\\alpha|}_{|\\beta|=0}\\frac{D^{\\alpha+\\beta} G({\\bf x}_{c}, {\\bf x}_{c'})}{\\alpha!\\beta!}(-1)^{\\beta} m^{\\beta}_{c'}(f_h), \\quad |\\alpha| \\leq p.\n\\end{equation}\nwhere $\\alpha!=\\alpha_1!\\alpha_2!\\alpha_3!$ and $m^{\\beta}_{c'}(f)$ is the\nmoment of $f_h$, given by\n\\begin{equation}\n\\label{eqmoment}\nm^{\\beta}_{c'}(f) = \\int_{S_{c'}} \\frac{\\partial^k}{\\partial\\nu_{\\bf x}^k} ({\\bf x} - {\\bf x}_{c'})^{\\beta}f_h({\\bf x})dS_{\\bf x}, \\quad |\\beta| \\leq p.\n\\end{equation}\nEquation~(\\ref{eqcoeff}) translates the moment of $c'$ to the local expansion coefficients of the cluster $c$,\nand it is therefore called MtL translation. Since $f_h$ is a linear\ncombination of basis functions, we obtain from linearity that\n\\begin{equation}\\label{eqmomentL}\nm^{\\beta}_{c'}(f) = \\sum_{i\\in c'} m^{\\beta}_{c'}(\\psi_i) f_i\n\\end{equation}\nwhere $f_i$ are the coefficients of $f_h$ with respect to the\n$\\psi_i$-basis and the summation is taken over basis functions that\nwhose support overlaps with $c$.\nSince we consider a Galerkin discretization we have to integrate the\nfunction $u_{c,c'}({\\bf x})$ against the test functions, to obtain the\ncontribution $g_i$ of the two clusters to the matrix vector product. Thus we\nget from (\\ref{eqapproximation})\n\\begin{equation}\\label{LtP}\ng_i = \\int_{c} \\psi_i({\\bf x}) u_{c,c'}({\\bf x}) dS_{\\bf x} =\n\\sum_{|\\alpha| \\leq p}\\lambda ^{\\alpha}_{c}  \\int_{c}\n\\frac{\\partial^l}{\\partial\\nu_{\\bf x}^l} ({\\bf x}-{\\bf\n  x}_{c})^{\\alpha}  \\psi_i({\\bf x}) dS_{\\bf x} =\n\\sum_{|\\alpha| \\leq p}\\lambda^{\\alpha}_{c} m_c^{\\alpha}(\\psi_i)\n\\end{equation}\nThis operation converts expansion coefficients to potentials and is\ndenoted as LtP translation.\n\nTo move moments and local expansion coefficients between levels, we also\nneed the moment-to-moment (MtM) and local-to-local (LtL)\ntranslations. They can be derived easily from the multivariate\nbinomial formula. We obtain\n\\begin{equation}\\label{eqMtM}\nm^{\\alpha}_{c}(f)\n= \\sum_{c'\\in\\mathcal{K}(c)}\\sum_{\\beta \\leq \\alpha} \\binom {\\alpha}{\\beta} ({\\bf x}_{c'} - {\\bf x}_{c})^{\\alpha-\\beta} m^{\\beta}_{c'}(f),\n\\end{equation}\nand \n\\begin{equation}\\label{eqLtL}\n\\lambda ^{\\beta}_{c'} = \\sum_{\\substack{\\alpha\\le\\beta \\\\ |\\alpha|\\le p}} \\binom {\\alpha}{\\beta} ( {\\bf x}_{c'} - {\\bf x}_{c})^{\\alpha-\\beta} \\lambda ^{\\alpha}_{c},  \\quad |\\beta|\\le p.\n\\end{equation}\nwhere $c' \\in \\mathcal{K}(c)$.\n\n\n\n\n\n\nWe see that moments and expansion coefficients are computed by\nrecurrence from the previous level. In the finest level the moments of\nthe basisfunctions $m_c(\\psi_i)$ can be either computed by numerical\nquadrature, or even analytically, because we consider flat panels and\npolynomial ansatz functions. We skip the details, as these formulas are\nstraightforward application of the binomial formula. \n\nIn summary, the Cartesian FMM under the framework of boundary element method is described as the following.\n\n\n\\begin{enumerate}\n\\item \\emph{Nearfield Calculation.}\\\\\nfor $c \\in C_L$\\\\\n\\hspace*{1.5em} for $c' \\in \\mathcal{N}(c)$ \\\\\n\\hspace*{3em} multiply matrix block of $c$ and $c'$ directly.\n\n\\item \\emph{Moment Calculation.} \\\\\nfor $c \\in C_L$\\\\\n\\hspace*{1.5em} Compute the moments $m^{\\beta}_{c'}(f)$ in (\\ref{eqmomentL}).\n\n\\item \\emph{Upward Pass.} \\\\\nfor $l=L-1,\\dots,l_{min}$\\\\\n\\hspace*{1.5em}for $c \\in C_l$\\\\\n\\hspace*{3em}for $c' \\in \\mathcal{K}(c)$\\\\\n\\hspace*{4.5em} Compute the MtM translation (\\ref{eqMtM})\n\n\\item \\emph{Interaction Phase.} \\\\\nfor $l=L,\\dots,l_{min}$\\\\\n\\hspace*{1.5em} for $\\nu \\in C_l$\\\\\n\\hspace*{3em} for $\\nu' \\in \\mathcal{N}(c)$\\\\\n\\hspace*{4.5em} Compute the MtL translation (\\ref{eqcoeff})\n\n\\item \\emph{Downward Pass.} \\\\\nfor $l=l_{min},\\dots,L-1$\\\\\n\\hspace*{1.5em} for $c \\in C_l$\\\\\n\\hspace*{3em} for $c' \\in \\mathcal{K}(c)$\\\\\n\\hspace*{4.5em} Compute the LtL translation (\\ref{eqLtL})\n\n\\item \\emph{Evaluation Phase.}\\\\\nfor $c \\in C_L$\\\\\n\\hspace*{1.5em} Compute the LtP translation (\\ref{LtP})\n\\end{enumerate}\n\nIn this algorithm $l_{min}$ is the coarsest level that contains\nclusters with non-empty interaction lists.\n\nSince the finest level contain a fixed number of triangles the number\nof levels grows logarithmically with $N$ as the mesh is refined. \nWith a geometric series argument, one can show that the total number\nof interaction lists in all levels is $O(N)$. If the translations in\nall levels are computed with the same order $p$ then the complexity of\nall translations is  $O(N p^3)$. If the variable order method is used\nwhere the order is given by (\\ref{variable:order}), then the\ncomplexity reduces to $O(N)$. More details can be found in ~\\cite{Tausch:2004}.\n\n\n\n\n\n\n\\subsection{Cartesian treecode}\nThe Cartesian treecode can be considered as a fast multipole method\nwithout the downward pass.  \nThe computational cost of treecode is order of $O(N\\log N)$ as opposed to the $O(N)$ FMM.  \nHowever, the constants in this complexity estimate are smaller, and we\nfound it to be useful for the computation of the free solvation energy\n(\\ref{eqsolengdis}), where\nthe source and evaluation points are different and zero or limited near field\ncalculations are required.\nThe direct computation of the solvation energy \nas interactions between $N$ boundary elements and $N_c$ atomic centers\nhas $O(N_cN)$ complexity. \nThis is shown in Fig.~\\ref{fig_2}b, \nin which a charge located at ${\\bf x}_n$ will interact with induced charges ($\\phi_1$ or $\\frac{\\partial \\phi_1}{\\partial \\nu}$)\nlocated at the center of each panel. \nThese interactions consist of near field particle-particle interaction by direction summation \nand far field particle-cluster interaction controlled by maximum acceptance criterion (MAC) as specified below. \nFor simplicity, we write the involved calculations as \n\\begin{equation}\n\\label{eqsolengform}\nE = \\sum_{n=1}^{N_c} q_n V_n\n = \\sum_{n=1}^{N_c} q_n\\sum_{l=1}^{N}\n\\int_{\\tau_{l}} \\frac{\\partial^k}{\\partial\\nu_\\textbf{x}^k} G(\\textbf{x}_n,\\textbf{x})f(\\textbf{x})dS_\\textbf{x},\n\\end{equation}\nwhere $G$ is Coulomb or screened Coulomb potential kernel, $k \\in\n\\{0,1\\}$, \n$q_n$'s are partial charges,\nand $f$ is either $\\phi_1$ or $\\partial\\phi_1\/\\partial\\nu$.\n\n\n\n\\begin{figure}\n\\begin{subfigure}{.5\\textwidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/treecode}\n\\caption{}\n\\label{fig_3_1}\n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/pc_interaction}\n\\caption{}\n\\label{fig_3_2}\n\\end{subfigure}\n\\caption{The hierarchical tree structure for the treecode: Left: the form of the tree by recursively dividing a parent cluster into eight\/four children clusters using $N_0=8$; Right:  The adaptive shrink scheme improves the treecode efficiency.}\n\\label{fig_3}\n\\end{figure}\n\n\nIn our implementation, we use the same clustering scheme of\n$\\Gamma_N$ as in the FMM. Instead of using the interaction lists to calculate the far field\ninteraction, the treecode use the following multipole acceptance criterion (MAC) to determine if the particle and the cluster are well separated or thus a far-field particle-cluster interaction will be considered.  \nThis is similar to the separation ratio in the FMM. The MAC is given as \n\\begin{equation}\n\\label{eqMAC}\n\\frac{r_c}{R} \\le \\theta,\n\\end{equation}\nwhere $r_c=\\text{max}_{\\textbf{x}_j\\in c}|\\textbf{x}_j-\\textbf{x}_c|$ is the cluster radius,\n$R=|\\textbf{x}_n-\\textbf{x}_c|$ is the particle-cluster distance,\nand $\\theta<1$ is a user-specified parameter.\nIf the criterion is not satisfied,\nthe program checks the children of the cluster recursively\nuntil either the MAC is satisfied\nor the leaves (the finest level cluster) are reached\nat which direct summation is applied.\nOverall, the treecode evaluates the potentials~(\\ref{eqsolengform}) \nas a combination of particle-cluster interactions and direct summations.\nThus, when $\\textbf{x}_n$ and $c$ are well-separated,\nthe potential can be evaluated as\n\\begin{equation}\n\\label{eqsolengformTree}\n\\int_{c} \\frac{\\partial^{k}}{\\partial\\nu_{\\bf x}^{k}} G({\\bf x}_n,{\\bf x}) f({\\bf x})dS_{\\bf x}\n\\approx \n\\sum\\limits^{p}_{|\\beta|=0} D^{\\beta}G({\\bf x}_{n}, {\\bf x}_{c}) (-1)^{\\beta} {m'}^{\\beta}_{c}(f),\n\\end{equation}\nwhere the moment ${m'}^{\\beta}_{c}(f)$ is calculated by the same operator-MtM in FMM.\n\nThe treecode method therefore can be concluded as\n\n\\begin{enumerate}\n\\item \\emph{Moment Calculation.} \\\\\nfor $c \\in C_L$\\\\\n\\hspace*{1.5em} Compute the moments $m^{\\beta}_{c'}(f)$ in (\\ref{eqmomentL}).\n\n\\item \\emph{Upward Pass.} \\\\\nfor $l=L-1,\\dots,l_{min}$\\\\\n\\hspace*{1.5em}for $c \\in C_l$\\\\\n\\hspace*{3em}for $c' \\in \\mathcal{K}(c)$\\\\\n\\hspace*{4.5em} Compute the MtM translation (\\ref{eqMtM})\n\n\\item \\emph{Interaction Phase}\\\\\n  for $n=1,...,N_c$\\\\\n\\hspace*{1.5em}$E_n = 0$\\\\\n\\hspace*{1.5em}for $c \\in C_0$\\\\\n\\hspace*{3em} addCluster(c,$\\textbf{x}_n$,$E_n$)    \n\\end{enumerate}\nwhere addCluster(c,$\\textbf{x}_n$,$E_n$) as shown below  is a routine that\nrecurses from the coarse clusters to the finer clusters until the\nseparation is sufficient to use the Taylor series approximation\\\\\n\\begin{itemize}\n  \\item[]if $\\textbf{x}_n$ and $\\textbf{x}_c$ satisfy the MAC for $c$\\\\\n\t$~~~~~~~~~~~~~~E_n\\mathrel{+}=\\sum\\limits^{p}_{|\\beta|=0} D^{\\beta}\\Phi({\\bf x}_{n}, {\\bf x}_{c}) (-1)^{\\beta} {m'}^{\\beta}_{c}(f)$\n\t\\item[] else if $\\mathcal{K}(c) \\ne \\emptyset$\\\\\n          $~~~~~~$for $c'\\in \\mathcal{K}(c)$\\\\\n          $~~~~~~~~~~~~~~$ addCluster($c'$,$\\textbf{x}_n$,$E_n$)\n\t\\item[] else\\\\ \n\t$~~~~~~~~~~~~~~E_n\\mathrel{+}=\\displaystyle\\int_{c} \\frac{\\partial^{\\kappa}}{\\partial\\nu_{\\bf x}^{\\kappa}} \\Phi({\\bf x}_n,{\\bf x})f(x)dS_{\\bf x}$\n\\end{itemize}\n\n\n\n\n\n\nNote that steps $1$ and $2$ are analogous to the steps the in FMM,\nhence the addition of the treecode to the FMM code requires little\nextra work.\n\n\n\\subsection{Preconditioning}\nThe results in our previous work \\cite{Geng:2013b, Geng:2013a} shows\nthat the PB boundary integral formulation in Eqs.~(\\ref{eqbim_3}) and (\\ref{eqbim_4})\nis well-conditioned thus will only require a small number of GMRES iteration if\nthe triangulation quality is satisfied (e.g. nearly quasi-uniform). \nHowever, due to the complexity of the molecular surface, \nthe triangulation unavoidably has a few triangles with defects (e.g. narrow triangles and tiny triangles)\nwhich deteriorate the condition number of the linear algebraic matrix, resulting in      \nincreased GMRES iteration number required to reach the desired convergence accuracy. \n \nRecently, we designed a block-diagonal preconditioning scheme \nto improve the matrix condition for the treecode-accelerated boundary integral (TABI) Poisson-Boltzmann solver \\cite{Chen:2018a}.\nThe essential idea for this preconditioning scheme is \nto use the short range interactions within the leaves of the tree to form the preconditioning matrix $M$. \nThis preconditioning matrix $M$ can be permuted into a block diagonal form thus $Mx=y$\ncan solved by the efficient and accurate direct methods. \nIn the current study of FAGBI solver,  \nthe same conditioning issue rises and it can be resolved \nby a similar but FMM structure adjusted and controlled preconditioning scheme. \n\nThe key idea is to find an approximating matrix $M$ of $A$ such that\n$M$ is similar to $A$ and  the linear system $My=z$ is easy to solve.\nTo this end, our choice for $M$ is the matrix \ninvolving only direct sum interactions in cubes\/clusters at a designated level \n(an optimal choice considering both cost and efficiency) \nas opposed to $A$, \nwhich involves all interactions.\n\nThe definition of $M$ will be essentially similar to $A$ in\n(\\ref{def:linsys}) except that the entries of $M$ are zero \nif $\\tau_i$ and $\\tau_j$ are not on the same cube at a designated level of the tree, i.e. \n\\begin{equation}\nM_{mn}(i,j) = \\left\\{\\begin{array}{l}\nA_{mn}(i,j) \\text{~~~if~$\\tau_i, \\tau_j$ are on the same cube at a designated level of the tree}  \\\\\n0 \\text{~~~~~~~~~~~~~otherwise.}\\end{array}\\right. \n\\end{equation}\n\n\\begin{figure}[htb]\n\t\\begin{center}\n\t\t\\includegraphics[width=2.18in]{figures\/fig_A4}\n\t\t\\includegraphics[width=2.18in]{figures\/fig_M4}\n\t\t\\includegraphics[width=2.18in]{figures\/fig_M4d}\n\t\t\\hskip 0.8in (a) \\hskip 2.0in (b) \\hskip 2.0in (c)\n\t\n\t\t\\caption{A schematic illustration of the boundary element dense matrix $A$ and its preconditioning matrix $M$:  (a) matrix $A$ for the case of $N=20$ elements (the size of the matrix entry shows the strength of the interaction; the four different color-coded region relates to $K_{1-4}$ in Eqs.~(\\ref{eqbim_3})-(\\ref{eqbim_4})); (b) the ``block diagonal block\" preconditioning matrix $M$ (assuming the cube at the designated level contains at most 3 panels; (c)  the ``block diagonal\"  preconditioning matrix $M$, which is a permuted matrix from $M$ in (b) after switching the order of the unknowns. \n\t\t}\n\t\t\\label{fig_matrices}\n\t\\end{center}\n\\end{figure}\n\n\nHere we use Fig.~\\ref{fig_matrices} to illustrate \nhow we design our preconditioning scheme and its advantage.\nFigure~\\ref{fig_matrices}(a) is the illustration of the dense boundary element matrix $A$\nfor the discretized system~(\\ref{def:linsys}) with 20 boundary elements. \nThe four different colors represent the four kernels $K_{1-4}$ related entries of the linear algebraic matrix $A$ in Eq.~(\\ref{eq_linsysEntry}). \nNote the unknowns are ordered by the potentials $\\phi_1$ on all elements, \nfollowed by the normal derivative of the potential $\\frac{\\partial \\phi_1}{\\partial \\nu}$. \nThe size of the matrix entry in Fig.~\\ref{fig_matrices} indicates the magnitude of the interaction \nbetween a target element and a source element, which decays from the main diagonal to its two wings. \nBy only including the interactions between elements on the same cube at a designated level, \nwe obtain our designed preconditioning matrix $M$ as illustrated in \nFig.~\\ref{fig_matrices}(b). This preconditioning matrix $M$ has four blocks, and each block is \na diagonal block matrix.\nFollowing the procedure detailed in \\cite{Chen:2018a}, \nby rearranging the order of the unknowns, a block diagonal matrix $M$ \nis achieved as illustrated in Fig.~\\ref{fig_matrices}(c). \nSince $M=\\text{diag}\\{M_1, M_2, \\cdots, M_{N_l}\\}$ as shown in  Fig.~\\ref{fig_matrices}(c) is a block diagonal matrix \nsuch that $My=z$ can be solved using direct method e.g. LU factorization by solving each individual $M_iy_i=z_i$. Here each $M_i$ is a square nonsingular matrix, which represents the interaction between particles\/elements on the $i$th cube of the tree at a designated level.  As shown in \\cite{Chen:2018a}, the total cost of solving $My=z$ is essentially $O(N)$ thus is very efficient. Results for the preconditioning performance will be shown in the next section. \n\n\n\n\n  \n\n\\section{Results}\nOur numerical results are mostly produced \non a desktop with an i5 7500 CPU and 16G Memory, \nusing GNU Fortran 7.2.0 compiler with compiling option ``-O2''. \nA few results for the long elapsed direct summation are obtained from the SMU high performance computing cluster, ManeFrame II (M2), with Intel Xeon Phi 7230 Processors, using openmpi\/3.1.3 compiler with compiling option ``-O2''. \nNote these direct sum results are needed for the evaluation of accuracy only \nand low resolutions results are checked on different machines to ensure the consistency of the accuracy.\nAll protein structures are obtained from Protein Data Bank (\\url{https:\/\/www.wwpdb.org)} and partial charges are assigned by CHARMM22 force field \\cite{Brooks:1983} using PDB2PQR software \\cite{Dolinsky:2007}.\n\nThe physical quantity we computed in this manuscript is the electrostatic free energy of solvation with the unit kcal\/mol. The electrostatic potential $\\phi$ or $\\phi_1$ governed in Eq.~(\\ref{eqNPBE}) or Eqs.~(\\ref{eqbim_3}-\\ref{eqbim_4}) uses the unit of $e_c\/(4\\pi$\\AA), where $e_c$ is the elementary charge. By doing this, we can directly use the partial charge obtained from PDB2PQR \\cite{Dolinsky:2007} for solving the PB equation. \nAfter obtaining the potential, we can convert the unit $e_c\/(4\\pi$\\AA) to kcal\/mol\/$e_c$ by multiplying the constant $4\\pi332.0716$ at room temperature T=300K. From potential to energy, only a multiplication of $e_c$ is needed. \n\n\nWe solved the PB equation first on the Kirkwood sphere \\cite{Kirkwood:1934}, where the analytic solution is available to validate the accuracy and efficiency of FAGBI solver, then on a typical protein 1a63 to demonstrate the overall performance, and finally on a series of proteins to emphasize the preconditioning scheme and the broad usage of the FAGBI solver. \n\n\\subsection{The Kirkwood sphere}\nOur first test case is the Kirkwood sphere of radius 50\\AA~with an atomic charge $q=50e_c$ at the center of the sphere.\nThe dielectric constant is $\\varepsilon_1=1$ inside the sphere and $\\varepsilon_2=40$ outside the sphere. \nWe provide three parts of this test case on the Kirkwood sphere, \nwhich show first the discretization error, \nthen the impact of the quadrature orders toward the convergence of accuracy, \nand finally the comparison between using the Cartesian FMM and using direct sum \nin terms of error, CPU time, and memory usage. \\\\ \n\n\\subsubsection{Overall discretization error}\nWe first solve the boundary integral PB equation on the Kirkwood sphere \nusing the direct summation for matrix-vector product instead of using the FMM acceleration. \nThe linear algebraic system is solved using GMRES iterative solver with $L_2$ relative tolerance $\\tau = 10^{-6}$. \nThe Galerkin method is applied to form the matrix combined with a single point Gauss quadrature. \nCubature methods \\cite{Sauter:1996} are applied for treating the singularities \narising from Galerkin discretization of boundary integral equations.\n\n\n\\begin{table}[htp!]\n\\caption{\\small Discretization error from solving the PB equation on a Kirkwood sphere with a centered charge.  \nResults include electrostatic solvation free energy $E^{ds}_{sol}$ with error $e^{ds}_{sol}$ and convergence rate $r^{ds}_{sol}$, and discretization error in surface potential $e^{ds}_{\\phi}$, normal derivative $e^{ds}_{\\partial_n \\phi}$ with their convergence rates $r^{ds}_{\\phi}$ and $r^{ds}_{\\partial_n \\phi}$.}\n{\n\\small\n\\begin{center}\n\\begin{tabular}{lccccccccc}\n\\hline\n\\rule{0pt}{12pt}\n$N^1$ &$h$& $E^{ds}_{sol}$ (kcal\/mol) & $e^{ds}_{sol}$ (\\%) & $r^{ds}_{sol}$ & $e^{ds}_{\\phi}$ (\\%) & $r^{ds}_{\\phi}$ & $e^{ds}_{\\partial_n \\phi}$ (\\%) & $r^{ds}_{\\partial_n \\phi}$ & $\\text{Iters}^2$ \\\\\\hline\n320      &9.90&-8413.28&1.692&3.6&17.925&4.1&0.652&1.9&3\\\\\n1280     &4.95&-8328.18&0.663&2.6& 4.419&4.1&0.212&3.1&3\\\\\n5120     &2.48&-8293.42&0.243&2.7& 1.112&4.0&0.092&2.3&3\\\\\n20,480   &1.24&-8280.70&0.089&2.7& 0.285&3.9&0.046&2.0&3\\\\\n81,920   &0.62&-8276.33&0.036&2.5& 0.077&3.7&0.023&2.0&3\\\\\n327,680  &0.31&-8274.64&0.016&2.3& 0.022&3.5&0.011&2.0&3\\\\\n1,310,720&0.15&-8273.91&0.007&2.2& 0.007&3.2&0.006&2.0&4\\\\\n$\\infty^3$ &&-8273.31&&&&&&&\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n${}^1$ $N$ is number of triangles in triangulation; $h$ is the average of largest edge length of all triangles; $h\\approx O(N^{-2})$\\\\\n${}^2$ Number of GMRES iterations.\\\\\n${}^3$ This row displays the exact electrostatic solvation energy $E^{ex}_{sol}$,\nwhich is known analytically.\n}\n\\label{tb_discretization}\n\\end{table}%\n\nTable~\\ref{tb_discretization} shows the total discretization errors, \nwhich is related to triangulation, quadrature, and basis function. \nIn this table, Column 1 is the number of triangles $N$ for the sphere with the refinement of the mesh and \nColumn 2 is the average of largest edge length of all triangles $h$. Note we have $h \\approx O(N^{-2})$, which can be seen from the comparison of values in the two columns. Since using $N$ is more convenient  to specify mesh refinement in our numerical simulation, we use it to quantify the mesh refinement for the rest of the paper.\n\nColumns 3-4 show that the electrostatic solvation energy $E^{ds}_{sol}$ \nand its error $e^{ds}_{sol}$ compared with the true value in the last row of the table. \nThe convergence rate $r^{ds}_{sol}$ defined as the ratio of the error is shown in column 5 with an $O(N^{-1\/2})$ pattern.  \nThe relative $L_\\infty$ errors of surface potential $\\phi$, $e^{ds}_{\\phi}$ \nand normal derivative $\\partial_n \\phi$, $e^{ds}_{\\partial_n \\phi}$,  \nare shown in columns 6 and 8. \nThe surface potential converges with a pattern of $O(N^{-1})$ as shown in column 7, \nwhich is faster than its normal derivative with a pattern of $O(N^{-1\/2})$ as shown in column 9. \nWe believe that this is due to the continuity of surface potential \nand the discontinuity of the normal derivatives  across the interface. \nWe also can observe that  the GMRES iterations shown in column 10 in all the tests \nare less than or equal to four, \nwhich verifies that the boundary integral formulation is well-posed.\n\nNote a back-to-back comparison between Table~\\ref{tb_discretization} in this manuscript \nand Table 2 in our previous work \\cite{Geng:2013b} shows improvements in convergence of \n$E^{ds}_{sol}$, $\\phi$, and  $\\partial_n \\phi$ for the present work. \nThis is due to the Galerkin scheme with Duffy's trick and the Cubature method in treating the singularity\nas opposed to the collocation scheme with simply the removal of \nsingular integral whenever it occurs on an element \\cite{Geng:2013b}. \n\n\n\n\n\\subsubsection{Quadrature error}\nIn part 1, we noticed that the converge rate for $E^{ds}_{sol}$ is about $O(N^{-1\/2})$ \nas the rate of $\\partial_n \\phi$, \nbut is less than the $O(N^{-1})$ rate of $\\phi$. \nTo investigate the possible reason, \nwe study the influence of the  quadrature rule the next.\n\nWe increase the order of the tensor product Gauss-Legendre rule to $2$, $3$ and $4$ \nand test their effects on the discretization error of $E^{ds}_{sol}$ \nas shown in Table~\\ref{tb_quadrature}. \nComparing with results in Table~\\ref{tb_discretization}, \nusing higher order quadratures improves both the convergence rate of $E^{ds}_{sol}$\nand the required GMRES iterations. \nWhen the quadrature order is $4$, the electrostatic solvation energy $E^{ds}_{sol}$ \nconverges to the exact energy at the rate $O(N^{-1})$ approximately. \n\nIncreasing the quadrature order further\nwill not significantly improve the convergence rate of $E^{ds}_{sol}$,\nbecause then the discretization error will be greater than the\nquadrature error.  \nSince higher quadrature requires more computational cost,   \nin practice, due to the large size of the protein solvation problem, \nwe will use quadrature order 1 as it shows the optimal combination or accuracy and efficiency. \n\n\\begin{table}[htp]\n\\caption{\\small Discretization error of solvation free energy $E^{ds}_{sol}$ \nfor solving the same problem in case 1 using Gaussian quadrature orders of 2-4}\n{\n\\small\n\\begin{center}\n\\begin{tabular}{lccccccccc}\n\\hline\n & \\multicolumn{3}{c}{Quad. Order 2} & \\multicolumn{3}{c}{Quad. Order 3} & \\multicolumn{3}{c}{Quad. Order 4}\n\\\\\n\\rule{0pt}{12pt}\n$N$ & $E^{ds}_{sol}$ & $r^{ds}_{sol}$ & Iters & $E^{ds}_{sol}$ & $r^{ds}_{sol}$ & Iters & $E^{ds}_{sol}$ & $r^{ds}_{sol}$ & Iters\\\\\\hline\n320      &-8378.82&3.7&2 &-8369.13&3.9&2 &-8369.21&3.9&2\\\\\n1280     &-8305.24&3.3&3 &-8298.57&3.8&2 &-8297.67&4.0&2\\\\\n5120     &-8284.01&3.0&3 &-8280.12&3.7&3 &-8279.40&3.9&3\\\\\n20,480   &-8277.27&2.7&3 &-8275.21&3.6&3 &-8374.81&4.0&3\\\\\n81,920   &-8274.94&2.4&3 &-8273.89&3.3&3 &-8373.68&4.1&3\\\\\n327,680  &-8274.03&2.3&3 &-8273.51&3.0&3 &-8373.40&4.0&3\\\\\n1,310,720&-8273.64&2.2&3 &-8273.38&2.7&3 &-8273.33&4.7&3\\\\\n$\\infty$ &-8273.31&   &  &-8273.31&   &  &-8273.31&   & \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n}\n\\label{tb_quadrature}\n\\end{table}%\n\n\\subsubsection{FMM}\nThis part of the test studies the role of Cartesian FMM \nrelating to the accuracy and efficiency of the algorithm. \nWe applied the FMM to replace the direct-sum for accelerating the matrix-product calculation in GMRES. \nHere we use the first order quadrature rule for simplicity.\nWe set $\\eta = 0.8$, which is defined in (\\ref{def:eta}) and adjust\n{the number of levels $L$ in the FMM algorithm} for different $N$.\nFigure~\\ref{fig_sphere_err} shows (a) the error in electrostatic solvation energy, (b) the CPU time, and (c) the memory usage \nversus the number of triangles $N$.\nHere the error is computed as compared with the exact value $E_{sol}^{ex} = -8273.31$. \nWe provide results using fixed Taylor expansion order $p=1,3,5,7$ \nand adaptive order start from $p=1$, and $p=3$. \nHere the adaptive order represents the idea that \nexpansion order should be adjusted to the level (e.g. higher expansion order at higher level) \nin order to match the discretization error \\cite{Tausch:2003}. \nIn this figure, the solid blue line with square marks is results of direct summation with one point quadrature, \nwhich shows in (a) an $O(N^{-1\/2})$ order of convergence in accuracy as observed in Table~\\ref{tb_discretization},\nan $O(N^2)$ CPU time in (b), and an $O(N)$ memory usage in (c). \n\nAs seen in Fig.~~\\ref{fig_sphere_err}(a), \nthe use of FMM introduces truncation error in addition to the discretization error. \nTruncation errors are more significant than the discretization error when the order $p$ is small and are less significant when $p$ is large.   \n\n\nFurthermore, we observed that when expansion orders $p=5,7$ of the FMM are used, \nthe errors are even smaller than those obtained with the direct sum. \nThis is due to the fact that the error of the truncation error of\nTaylor approximation is smaller than the quadrature error of the far\nfield coefficients in the direct sum.\n\n\n\n\n\\begin{figure}[htb!]\n\t\\begin{center}\t\n\t\t\\includegraphics[width=2.188in]{figures\/FMM_sphere_err}\n\t\t\\includegraphics[width=2.188in]{figures\/FMM_sphere_time}\n\t\t\\includegraphics[width=2.188in]{figures\/FMM_sphere_mem}\\\\\n\t\t(a) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (b)  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (c)\n\t\t\\caption{(Compute electrostatic solvation energy on a Kirkwood sphere as number of triangles\/particles $N$ in creases:  (a) Error, (b) CPU time, and (c) Memory usage; discretization error $e^{ds}_{sol}$ (solid line), Cartesian FMM approximation error $e^{cf}_{sol}$ (dashed line); Taylor expansion order $p = 1, 3, 5, 7$ and adaptive Taylor order $p=1,3$}\n\t\t\\label{fig_sphere_err}\n\t\\end{center}\n\\end{figure}\n\nAs seen in Fig.~\\ref{fig_sphere_err}(b),  the use of FMM significantly reduce the CPU time, \nwhich shows a $O(N)$ pattern as opposed to the $O(N^2)$ pattern of the direct sum. \nFigures.~\\ref{fig_sphere_err}(a-b) combined also justify the use of adaptive order. \nAdaptive order 1 and 3 use about the same amount of CPU time as regular order 1 and 3 but achieved significant improvements in accuracy. \nMeanwhile, Figure~\\ref{fig_sphere_err}(c) shows that FMM use additional memory in trading of efficiency. \nHowever, the $O(N)$ pattern of memory usage \nare well preserved at different orders with only an adjustment in a factor. \n \nIn summary, from Tables~\\ref{tb_discretization} and \\ref{tb_quadrature}, \nwe observed that the Galerkin discretization with piecewise constant basis functions \ncan achieve $O(N^{-1\/2})$ convergence rate with low quadrature order (e.g. 1 or 2)\nand can achieve $O(N^{-1})$ convergence rate with high quadrature order (e.g. 4). \nApplying FMM algorithm for acceleration significantly reduced the $O(N^2)$ CPU time to $O(N)$ \nwhile maintains desired accuracy and $O(N)$ memory usage. \nFor later tests, we apply the adaptive FMM with starting order $1$ \nand $\\eta=0.8$ as an optimal choice at the consideration of both efficiency and accuracy.\n\n\\subsection{The protein 1a63}\nIn this section, we use the FAGBI solver to compute the solvation energy for protein 1A63, \nwhich has 2065 atoms. In computation involving proteins,\nthe molecular surface is triangulated by MSMS \\cite{Sanner:1996},\nwith atom locations from the Protein Data Bank \\cite{PDB}\nand partial charges from the CHARMM22 force field \\cite{Brooks:1983}.\nMSMS has a user-specified density parameter $d$\ncontrolling the number of vertices per $\\text{\\AA}^2$ in the triangulation.\nMSMS constructs an irregular triangulation\nwhich becomes smoother as $d$ increases.\nThe tree structure level is adjusted according to different number of particles.\nThe GMRES tolerance is $\\tau=10^{-4}$.\nThese are representative parameter values chosen \nto ensure that the FMM approximation error and GMRES iteration error are smaller\nthan the direct sum discretization error,\nand to keep efficient performance in CPU time and memory based on tests on spheres previously.\n\n\\begin{table}[htp]\n\\caption{(protein 1A63). FAGBI results; PB equations; showing electrostatic solvation energy $E_{sol}$, error, CPU time, memory usage; columns show MSMS density ($d$~in $\\text{\\AA}^{-2}$), number of triangles $N$, $E_{sol}$ values computed by direct sum (ds) and Cartesian FMM (cf), discretization error $e^{ds}_{sol}$, Cartesian FMM approximation error $e^{cf}_{sol}$ and their convergence rate $r^{ds}_{sol}$ and $r^{cf}_{sol}$; adaptive Taylor expansion order $p=1$, separate rate $\\eta = 0.8$.}\n{\n\\small\n\\begin{center}\n\\begin{tabular}{lrrrcrrcrrcrrcrr}\n\\hline\n$d$ & \\multicolumn{1}{l}{$N^a$} & \\multicolumn{2}{l}{$E_{sol}$ (kcal\/mol)} & & \\multicolumn{2}{l}{Error $(\\%)$} & & \\multicolumn{2}{l}{Rate} & & \\multicolumn{2}{l}{CPU (s)}  & & \\multicolumn{2}{l}{Mem. (MB)} \n\\\\\n\\cline{3-4}\\cline{6-7}\\cline{9-10}\\cline{12-13}\\cline{15-16}\n\\rule{0pt}{13pt}\n & & $ds$ & $cf$ & & \\multicolumn{1}{r}{$e^{ds}_{sol}$} & \\multicolumn{1}{r}{$e^{cf}_{sol}$} & & \\multicolumn{1}{r}{$r^{ds}_{sol}$} & \\multicolumn{1}{r}{$r^{cf}_{sol}$} & & \\multicolumn{1}{r}{$ds$} & \\multicolumn{1}{r}{$cf$} & & \\multicolumn{1}{r}{$ds$} & \\multicolumn{1}{r}{$cf$}\n\\\\\n\\hline\n1 & 20,227&-2755.05&-2756.82&&16.12&16.20&&   &   &&    632&  6&& 18& 30\\\\\n2 & 30,321&-2498.20&-2499.20&& 5.30& 5.34&&5.5&5.4&&   1135&  8&& 23& 40\\\\\n5 & 69,969&-2412.40&-2413.02&& 1.68& 1.71&&2.7&2.7&&   5912& 17&& 66&111\\\\\n10&132,133&-2383.09&-2382.50&& 0.45& 0.42&&4.1&4.4&& 36,530& 37&& 92&165\\\\\n20&264,927&-2375.21&-2376.52&& 0.11& 0.17&&3.9&2.6&&149,651& 69&&249&423\\\\\n40&536,781&-2371.52&-2372.77&& 0.04& 0.01&&2.9&7.5&&618,879&141&&359&654\\\\\n  &${\\infty}^b$&-2372.48&-2372.48&&&&&&&&&\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n${}^a$ Number of elements in triangulation.\\\\\n${}^b$ This row shows the estimates of exact energy $E^{ex}_{sol}$ obtained by the parallel computing on high order quadrature method.\n}\n\\label{tb_1a63}\n\\end{table}%\n\nIn Table~\\ref{tb_1a63}, the first two columns give the MSMS density ($d$) and number of faces $N$ in the triangulation .\nThe next two columns give the electrostatic solvation energy $E_{sol}$ computed by\ndirect sum $(ds)$ and Cartesian FMM $(cf)$.\nWe use a parallel version of direct sum\nto compute an estimate of the exact energy \nwith high order quadrature methods.\nWe computed the discretization errors $e^{ds}_{sol}$ and $e^{cf}_{sol}$\non the fifth and sixth columns, \nwhich shows convergence rate faster than $O(N^{-1})$ as\nobserved for the geodesic grid triangulation of the Kirkwood\nsphere in Case 1. \nThe faster convergence seen here is due to\nnon-uniform adaptive treatment of MSMS triangulation \\cite{Geng:2013b}.\n\nA back-to-back comparison of results from direct sum ($ds$) and Cartesian FMM results $cf$\nin Error, Rate, CPU time, and Memory in table~\\ref{tb_1a63} provides the following conclusions. \n(1) the adoption of FMM only slightly modify the error and its convergence rate in accuracy, not even necessarily in a negative way; \n(2) Cartesian FMM dramatically reduces the  $O(N^2)$ direct sum CPU time to $O(N)$. \nFor example, the simulation with $d = 10\\text{\\AA}^{-2}$\nand $N=132,133$ took $36,530 \\text{s} \\approx 10 \\text{h}$ by direct sum\nand $37 \\text{s} \\approx 1\/2 \\text{min}$ by FMM; \n(3) Moreover, the memory usage shows that\nboth the direct sum and FMM memory usage is $O(N)$.\nFor the FMM, more memory is used for the moment and local coefficient storage\nbut this only adds a pre-factor rather than increases the growth rate.\\\\\n\n\\subsection{27 proteins}\n\\begin{table}[t!]\n{\\small\n\\caption{Convergence comparison using diagonal preconditioning (d) and block diagonal preconditioning (bd) on a set of 27 proteins; MSMS density $d=10$.}\n\\begin{center}\n\\begin{tabular}{rrr|rrr|rr|rrr}\n\\hline\nInd. & PDB & \\# of ele. & \\multicolumn{3}{c|}{$E_\\text{sol}$ (kcal\/mol)} &\\multicolumn{2}{c|}{\\# of it.} & \\multicolumn{3}{c}{CPU time (s)}  \\\\\n&&& d & bd & \\text{diff.} (\\%) & d & bd & d & bd & ratio \\\\\\hline\n1&1ajj&40496  &-1141.17&-1141.15&0.00&22&14&12.5&9.7&1.28\\\\\n2&2erl&43214  & -953.43& -953.42&0.00&15&10&9.2&7.8&1.18\\\\\n3&1cbn&44367  & -305.94& -305.94&0.00&12&11&7.4&8.3&0.88\\\\\n4&1vii&47070  & -906.11& -906.11&0.00&16&14&10.6&11.5&0.92\\\\\n5&1fca&47461  &-1206.46&-1206.48&0.00&16&11&10.2&8.8&1.16\\\\\n6&1bbl&49071  & -991.21& -991.22&0.00&19&13&13.3&11.2&1.18\\\\\n7&2pde&50518  & -829.49& -829.46&0.00&{\\bf 75}&23&50.7&19.5&{\\bf 2.60}\\\\\n8&1sh1&51186  & -756.64& -756.63&0.00&{\\bf \\underline{100}}&21&70.7&18.2&{\\bf 3.89}\\\\\n9&1vjw&52536  &-1242.55&-1242.56&0.00&11&10&8.2&9.3&0.87\\\\\n10&1uxc&53602 &-1145.38&-1145.38&0.00&20&13&14.7&11.9&1.23\\\\\n11&1ptq&54256 & -877.83& -877.84&0.00&16&13&11.9&12.2&0.97\\\\\n12&1bor&54628 & -857.28& -857.27&0.00&14&13&10.9&12.5&0.87\\\\\n13&1fxd&54692 &-3318.18&-3318.14&0.00&10&10&7.8&9.9&0.79\\\\\n14&1r69&57646 &-1094.86&-1094.86&0.00&13&12&10.6&12.6&0.84\\\\\n15&1mbg&58473 &-1357.32&-1357.33&0.00&18&13&14.8&13.6&1.09\\\\\n16&1bpi&60600 &-1309.61&-1310.02&0.03&18&12&16.2&14.5&1.11\\\\\n17&1hpt&61164 & -816.47& -817.34&0.11&15&13&12.8&14.0&0.92\\\\\n18&451c&79202 &-1031.74&-1031.91&0.02&27&20&30.3&28.8&1.05\\\\\n19&1svr&88198 &-1718.97&-1718.97&0.00&15&12&21.4&21.3&1.01\\\\\n20&1frd&81792 &-2868.29&-2867.32&0.00&14&12&18.1&17.2&1.05\\\\\n21&1a2s&84527 &-1925.23&-1925.24&0.00&20&17&26.4&24.8&1.06\\\\\n22&1neq&89457 &-1740.50&-1740.49&0.00&19&15&26.7&22.8&1.17\\\\\n23&1a63&132133&-2382.50&-2382.50&0.00&21&16&41.3&36.8&1.12\\\\\n24&1a7m&147121&-2171.13&-2172.12&0.00&{\\bf 55}&21&111.2&51.4&{\\bf 2.16}\\\\\\hline\n25&2go0&111615&-1968.61&-1968.65&0.00&{\\bf 44}&24&67.6&43.0&1.57\\\\\n26&1uv0&128497&-2296.43&-2296.43&0.00&{\\bf 73}&25&130.7&52.6&{\\bf 2.48}\\\\\n27&4mth&123737&-2479.62&-2479.61&0.00&{\\bf 36}&18&64.3&37.0&1.74\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tb_proteinSets}\n}\n\\end{table}%\n\n\nWe finally provide testing results on a set of 27 proteins \nfor the purpose of demonstrating the general application of FAGBI solver to broader macromolecules  \nand the efficiency of the preconditioning scheme. \nTable~\\ref{tb_proteinSets} shows the convergence tests using diagonal preconditioning (d)\nand block diagonal preconditioning (bd) for a set of 27 proteins.\nAfter applying the block diagonal preconditioning scheme,\nthe cases with slow convergence using diagonal preconditioning \nhas been well resolved.\nIn this table, the first column is the protein index,\nfollowed by the PDB ID in the second column, \nand the number of elements in the third column generated by MSMS with density $d=10$.\nColumns 4 and 5 are the solvation energy of the proteins \napplying both preconditioning schemes,\nand column 6 is the relative difference between both methods, \nwhich shows no significant difference.\nA significant reduction of number of iterations using block diagonal preconditioning (bd)\nis shown in column 8 compared with results in column 7 using diagonal preconditioning (d).\nOne can see that the worse the diagonal preconditioning result is, \nthe larger improvements block diagonal preconditioning can achieve.\nFor example, proteins 2pde, 1sh1, 1a7m, 2go0, 1uv0 and 4mth \nfirst use 75, 100, 55, 44, 73, and 36 iterations for diagonal preconditioning as highlighted in column 7, \nbut only use 23, 21, 21, 24, 25, and 18 iterations for block diagonal preconditioning.\nThe CPU time comparison in columns 9 and 10, as well as their ratio in column 11,\nfurther confirms the results in columns 7 and 8 \nas CPU time is related to the number of iterations.\nThe ratio of CPU reduction for some proteins are more than 2 times as highlighted in the last column.\nWe plot the results of columns 7,8,9 and 10 in Fig.~\\ref{fig_proteinSets}\nwhich shows the improvements on both number of iterations\nand CPU time when block diagonal preconditioning is used to replace the diagonal preconditioning.\nIt shows that the block diagonal preconditioning does not impair\nthe originally well-conditioned cases\nbut significantly improve the slow convergence cases, \nwhich suggests that we can uniformly use block diagonal preconditioning \nin replace of the original diagonal preconditioning.\nFigures~\\ref{fig_proteinSets}(a) and~\\ref{fig_proteinSets}(b) shows a similar pattern\nas CPU time and the number of iterations are highly correlated. \n\n\n\n\n\\begin{figure}[htb]\n\t\\begin{center}\n\t\t\\includegraphics[width=3.3in]{figures\/fig_fmm27ite}~~~~~\n\t\t\\includegraphics[width=3.3in]{figures\/fig_fmm27cpu}\\\\\n\t\t\\hskip 0.3in (a) \\hskip 3.25in (b)\n\t\n\t\t\\caption{convergence comparison using diagonal preconditioning and block diagonal preconditioning.\n\t\t\t(a) number of iterations; \n\t\t\t(b) CPU time (s). \n\t\t}\n\t\t\\label{fig_proteinSets}\n\t\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\\section{Conclusion}\nIn this paper, we report recent work in developing \nan FMM accelerated Galerkin boundary integral (FAGBI) method \nfor solving the Poisson-Boltzmann equation. \nThe solver has combined advantages in accuracy, efficiency, and memory\nas it applies a well-posed boundary integral formulation \nto circumvent many numerical difficulties\nand \nuses an $O(N)$ Cartesian FMM  to accelerate the GMRES iterative solver. \nSpecial treatments such as\nadaptive FMM order,\nblock diagonal preconditioning, \nGalerkin discretization,  \nand Duffy's transformation are combined to improve the performance,\nwhich is validated on benchmark Kirkwood's sphere and a series of testing proteins.  \nWith its attractive  $O(N^{-1})$ convergence rate in accuracy, $O(N)$ CPU run time, and $O(N)$ memory usage, the FAGBI solver and its broad usage can contribute significantly to the greater computational biophysics\/biochemistry community as a powerful tool for the study of electrostatics of solvated biomolecules.  \n\n\n\n\\section*{Acknowledgments}\nThe work of W.G. and J.C. was supported by NSF grant DMS-1418957,\nDMS-1819193, SMU new faculty startup fund and SMU center for\nscientific computing. The work of J.T. is in part funded by NSF grant\nDMS-1720431. \n\n\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nSubmodular functions arise in a wide range of applications: graph theory, optimization, economics, game theory, to name a few. \nA function $f: 2^V\\to \\mathbb{R}$ on a ground set $V$ is {\\em submodular} if\n$f(X)+f(Y)\\geq f(X\\cap Y)+f(X\\cup Y)$ for all sets $X,Y\\subseteq\nV$. Submodularity can also be interpreted as a decreasing marginals property.\n\nThere has been significant interest in submodular optimization in the machine learning and computer vision communities. The {\\em submodular function minimization} (SFM) problem arises\nin problems in image segmentation or MAP inference tasks in Markov Random Fields. Landmark results in combinatorial optimization give polynomial-time exact algorithms for SFM. However, the high-degree polynomial dependence in the running time is prohibitive for large-scale problem instances. The main objective in this context is to develop fast and scalable SFM algorithms.\n\nInstead of minimizing arbitrary submodular functions, several recent papers aim to exploit special structural properties of submodular functions arising in practical applications. A popular model is {\\em decomposable submodular functions}: these can be written as sums of several ``simple'' submodular functions defined on small supports.\n\nSome definitions are in order. Let $f:2^V\\to\\mathbb{R}$ be a submodular function, and let $n:=|V|$. We can assume w.l.o.g. that $f(\\emptyset)=0$. We are interested in solving the {\\em submodular function minimization problem}\n\\begin{equation}\\label{prob:sfm}\n\\min_{S\\subseteq V} f(S).\\tag{SFM}\n\\end{equation}\nThe base polytope of a submodular function is defined as\n\\begin{align*}\n  B(f) := \\{ x \\in \\mathbb{R}^V \\colon x(S) \\leq f(S) \\; \\forall S \\subseteq V, x(V) = f(V) \\}.\n\\end{align*}\nOne can optimize linear functions over $B(f)$ using the greedy algorithm.\nThe problem \\eqref{prob:sfm} can be reduced to finding the minimum norm point of the base polytope $B(f)$ \\cite{fujishige80}.\n\\begin{equation}\\label{prob:fuji}\n\\min\\left\\{ \\frac12 \\|y\\|_2^2\\colon y \\in B(f)\\right\\}.\\tag{Min-Norm}\n\\end{equation}\nThis reduction is the starting point of convex optimization approaches for \\eqref{prob:sfm}.\nWe refer the reader to Sections 44--45 in \\cite{Schrijver03} for concepts and results in submodular optimization, and  to \\cite{Bach-monograph} on machine learning applications.\n\nWe assume that $f$ is given in the decomposition \n\\[f(S) = \\sum_{i = 1}^r f_i(S), \\]\nwhere each $f_i: 2^{V} \\rightarrow \\mathbb{R}$ is a submodular function. Such functions are called \\emph{decomposable} or\n\\emph{Sum-of-Submodular (SoS)} in the literature. In this paper, we will use the abbreviation DSFM.\n\nFor each $i\\in [r]$, the function $f_i$ has an effective support $C_i$ such that  $f_i(S)=f_i(S\\cap C_i)$ for every $S\\subseteq V$. For each $i\\in [r]$, we assume that two oracles are provided: {\\em (i)} a value oracle that returns $f_i(S)$ for any set $S\\subseteq V$ in time $\\mathrm{EO}_i$; and {\\em (ii)} a quadratic minimization oracle ${\\cal O}_i(w)$.\nFor any input vector $w \\in \\mathbb{R}^n$, this oracle returns an optimal solution to \\eqref{prob:fuji} for the function $f_i+w$, or equivalently, an optimal solution to  $\\min_{y \\in B(f_i)} \\|y + w\\|^2_2$. We let $\\Theta_i$ denote the running time of a single call to the oracle ${\\cal O}_i$,  $\\Theta_{\\max}:=\\max_{i\\in[r]}\\Theta_i$ denote the maximum time of an oracle call, $\\Theta_{\\mathrm{avg}} := {1 \\over r} \\sum_{i \\in [r]} \\Theta_i$ denote the average time of an oracle call.\\footnote{For flow-type algorithms for DSFM, a slightly weaker oracle assumption suffices, returning a minimizer of $\\min_{S \\subseteq C_i} f_i(S) + w(S)$ for any given $w \\in \\mathbb{R}^{C_i}$. This oracle and the quadratic minimization oracle are reducible to each other:  the former reduces to a single call to the latter, and one can implement the latter using $O(|C_i|)$ calls to the former (see e.g. \\cite{Bach-monograph}).} We let $F_{i,\\max} := \\max_{S \\subseteq V} |f_i(S)|$, $F_{\\max}:= \\max_{S \\subseteq V} |f(S)|$ denote the maximum function values.\n\nDecomposable SFM thus requires algorithms on two levels. The \\emph{level-0} algorithms\nare the subroutines used to evaluate the oracles  ${\\cal O}_i$ for every $i\\in [r]$. The \\emph{level-1} algorithm minimizes the function $f$ using the level-0 algorithms as black boxes.\n\n\\subsection{Prior work}\nSFM has had a long history in combinatorial optimization since the early 1970s, following the influential work of Edmonds \\cite{Edmonds70}. The first polynomial-time algorithm was obtained via the ellipsoid method \\cite{Grotschel1981}; recent work presented substantial improvements using this approach \\cite{LeeSW15}.\nSubstantial work focused on designing strongly polynomial combinatorial algorithms \\cite{Schrijver00,iwata2001,fleischer03,Iwata03,Orlin09,IwataO09}. \nStill, designing practical algorithms for SFM that can be applied to large-scale problem instances remains an open problem.\n\nLet us now turn to decomposable SFM.\nPrevious work mainly focused on level-1 algorithms. These can be classified as \\emph{discrete} and \\emph{continuous} optimization methods. The discrete approach builds on techniques of classical discrete algorithms for  network flows and for submodular flows. Kolmogorov \\cite{kolmogorov12} showed that the problem can be reduced to submodular flow maximization, and also presented a more efficient augmenting path algorithm. Subsequent discrete approaches were given in \\cite{Arora12,Fix13,Fix14}.\nContinuous approaches start with convex programming formulation \\eqref{prob:fuji}. Gradient methods were \napplied for the decomposable setting in \\cite{StobbeK10,NishiharaJJ14,EneN15}. \n\nLess attention has been given to the level-0 algorithms. Some papers mainly focus on theoretical guarantees on the running time of level-1 algorithms, and treat the level-0 subroutines as black-boxes (e.g. \\cite{kolmogorov12,NishiharaJJ14,EneN15}). In other papers (e.g. \\cite{StobbeK10,JegelkaBS13}), the model is restricted to functions $f_i$ of a simple specific type that are easy to minimize. An alternative assumption is that all $C_i$'s are small, of size at most $k$; and thus these oracles can be evaluated by exhaustive search, in $2^k$ value oracle calls (e.g. \\cite{Arora12,Fix13}). \n\nShanu {\\em et al.}\\xspace \\cite{Shanu16} use a block coordinate descent method for level-1, and allow arbitrary functions $f_i$. The oracles are evaluated via the Fujishige-Wolfe minimum norm point algorithm \\cite{Fujishige11, Wolfe76} for level-0. \n\n\\subsection{Our contributions}\nOur paper establishes connections between discrete and continuous methods for decomposable SFM, as well as provides a systematic experimental comparison of these approaches.\nOur main theoretical contribution improves the worst-case complexity bound of the most recent continuous optimization methods \\cite{NishiharaJJ14,EneN15} by a factor of $r$, the number of functions in the decomposition. This is achieved by improving the bounds on the relevant condition numbers. Our proof exploits ideas from the discrete optimization approach. This provides not only better, but also considerably simpler arguments than the algebraic proof in \\cite{NishiharaJJ14}. \n\nThe guiding principle of our experimental work is the clean conceptual distinction between the level-0 and level-1 algorithms. Previous experimental studies considered the level-0 and level-1 algorithms as a single ``package''. For example, Shanu {\\em et al.}\\xspace \\cite{Shanu16} compare the performance of their \\emph{SoS Min-Norm algorithm} to the continuous approach of Jegelka {\\em et al.}\\xspace \\cite{JegelkaBS13} and the combinatorial approach of Arora {\\em et al.}\\xspace \\cite{Arora12}. However, these implementations are difficult to compare since they  use three different level-0 algorithms: Fujishige-Wolfe in SoS Min-Norm, a general QP solver for the algorithm of \n\\cite{JegelkaBS13}, and exhaustive search for \\cite{Arora12}. For potentials of large support, Fujishige-Wolfe outperforms these other level-0 subroutines, hence  the algorithms in \\cite{JegelkaBS13,Arora12} could have compared more favorably using the same Fujishige-Wolfe subroutine.\n\nIn our experimental setup, we compare level-1 algorithms by using the same level-0 subroutines. We compare the state-of-the-art continuous and discrete  algorithms: RCDM and ACDM from \\cite{EneN15} with Submodular IBFS from \\cite{Fix13}. We consider multiple options for the level-0 subroutines. For certain potential types, we use tailored subroutines exploiting the specific form of the problem. We also consider a variant of the Fujishige-Wolfe algorithm as a subroutine applicable for arbitrary potentials. \nOur experimental results reveal the following tradeoff. Discrete algorithms on level-1 require more calls to the level-0 oracle, but less overhead computation. Hence using algorithms such as IBFS on level-1 can be significantly faster than gradient descent as long as the potentials have fairly small supports. However, as the size of the potentials grow, or we do need to work with a generic level-0 algorithm, the better choice is using gradient methods. Gradient methods can perform better for larger potentials also due to weaker requirements on the level-0 subroutines: approximate level-0 subroutines suffice for them, whereas discrete algorithms require exact optimal solutions on level-0. \n\n{\\bf Paper outline.} The rest of the paper is structured as follows.\nSection~\\ref{sec:flow} describes the level-1 algorithmic framework for DSFM that is based on network flows, and outlines the IBFS algorithm. Section~\\ref{sec:conv-opt} describes the level-1 algorithmic framework for DSFM that is based on convex optimization, and outlines the gradient descent algorithms. Section~\\ref{sec:kappa} gives improved convergence guarantees for the gradient descent algorithms outlined in Section~\\ref{sec:conv-opt}. Section~\\ref{sec:level-0} discusses the different types of level-0 algorithms and how they can be used together with the level-1 frameworks. Section~\\ref{sec:experiments} presents our experimental results.\n\n\\section{Level-1 algorithms based on network flow}\n\\label{sec:flow}\n\nIn this section, we outline a level-1 algorithmic framework for DSFM that is based on a combinatorial framework first studied in \\cite{FujishigeZhang92}.\\footnote{The framework was introduced in a slightly different context, for the submodular intersection problem. The dual of this problem is minimizing a submodular function of the form $f=f_1+f_2$, with access to oracles minimizing $f_1$ and $f_2$.} \n\nFor a decomposable function $f$, every $x\\in B(f)$ can be written as $x = \\sum_{i = 1}^r x_i$, where\n$\\text{supp}(x_i)\\subseteq C_i$ and $x_i\\in B(f_i)$ (see e.g. Theorem 44.6 in \\cite{Schrijver03}). A natural algorithmic approach is to maintain an $x\\in B(f)$ in such a representation, and iteratively update it using the combinatorial framework described below. \nDSFM can be casted as a maximum network flow instance in a network that is suitably defined based on the current point $x$. This  can be viewed as an analogue of the residual graph in the maxflow\/mincut setting, and it is precisely the residual graph if the DSFM instance was a mincut instance.\n\n{\\bf The auxiliary graph.} For an $x\\in B(f)$ of the form $x=\\sum_{i = 1}^r x_i$, we construct the following directed auxiliary graph $G = (V, E)$, with $E = \\bigcup_{i = 1}^r E_i$ and capacities $c:E\\to {\\mathbb{R}}_+$. The arc sets $E_i$ are complete directed graphs (cliques) on $C_i$, and for an arc $(u,v)\\in E_i$, we define $c(u,v):=\\min\\{ f_i(S)-x_i(S)\\colon S\\subseteq C_i, u\\in S, v\\notin S\\}$. This is the maximum value $\\varepsilon$ such that $x'_i\\in B(f_i)$, where $x'_i(u)=x_i(u)+\\varepsilon$, $x'_i(v)=x_i(v)-\\varepsilon$, $x'_i(z)=x_i(z)$ for $z\\notin\\{u,v\\}$. \n\nLet \n $N := \\{ v \\in V \\colon x(v) < 0 \\}$ and $P := \\{ v \\in V \\colon x(v) > 0\\}$. \nThe algorithm aims to improve the current $x$ by updating along shortest directed paths from $N$ to $P$ with positive capacity; there are several ways to update the solution, and we discuss specific approaches later in the section. If there exists no such directed path, then we let $S$ denote the set reachable from $N$ on directed paths\nwith positive capacity; thus, $S\\cap P=\\emptyset$. It is easy to show that $S$ is a minimizer of the function $f$. \n\nUpdating along a shortest path ${\\cal Q}$ from $N$ to $P$ amounts to the following. Let $\\varepsilon$ denote the minimum capacity of an arc on $\\cal Q$. If $(u,v)\\in {\\cal Q}\\cap E_i$, then we increase $x_i(u)$ by $\\varepsilon$ and decrease $x_i(v)$ by $\\varepsilon$. The crucial technical claim \\cite{FujishigeZhang92} is the following. Let $d(u)$ denote the shortest path distance of positive capacity arcs from $u$ to the set $P$. Then, an update along a shortest directed path from $N$ to $P$ results in a feasible $x\\in B(f)$, and further, all distance labels $d(u)$ are non-decreasing.\n\n{\\bf Level-1 algorithms based on the network flow approach.} Using this auxiliary graph, and updating on shortest augmenting paths, one can generalize several maximum flow algorithms to a level-1 algorithm of DSFM.\nThese algorithms include: the Edmonds-Karp-Dinitz maximum flow algorithm, the preflow-push algorithm \\cite{GoldbergTarjan88}, the incremental breadth first search algorithm (IBFS) \\cite{goldberg11}, and the excesses incremental breadth first search algorithm \\cite{goldberg15}. Our experiments will use an implementation of IBFS, following~\\cite{Fix13}. \n\n{\\bf Submodular incremental breadth first search (IBFS).} Fix {\\em et al.}\\xspace\n\\cite{Fix13} adapt the IBFS algorithm to the above\ndescribed submodular framework  using the above\nmentioned claims by Fujishige \\& Zhang \\cite{FujishigeZhang92}. IBFS is an augmenting path algorithm\nfor the maximum flow problem. It identifies a shortest path from\nthe source set $N$ to the sink set $P$ via \ngrowing shortest path trees simultaneously forwards from $N$ and\nbackwards from $P$. \n\nThe submodular IBFS algorithm provides us with a level-1 algorithm for DSFM. Each step of the algorithm involves determining the capacity of an arc in the auxiliary graph; as we explain in Section~\\ref{sec:level-0}, each of these capacities can be computed using a single call to a level-0 subroutine ${\\cal O}_i$.\n\nBy combining the level-1 IBFS algorithm with appropriate level-0 subroutines, we obtain an algorithm for DSFM whose running time can be upper bounded as follows. On a directed graph with $n$ nodes and $m$ arcs, IBFS runs in time $O(n^2m)$. In the DSFM setting, we have $m=O(\\sum_{i\\in [r]} |C_i|^2 )$. Every step  involves determining an auxiliary capacity, which can be implemented using a single call to a level-0 subroutine ${\\cal O}_i$ (see Section~\\ref{sec:level-0}); the maximum time of such an oracle call is $\\Theta_{\\max}$. Hence, the running time bound for submodular IBFS can be given as $O(n^2\\Theta_{\\max} \\sum_{i\\in [r]}|C_i|^2)$. If all $C_i$'s are small, $O(1)$, then this gives $O(n^2r\\Theta_{\\max})$.\n\n\\section{Level-1 algorithms based on convex optimization}\n\\label{sec:conv-opt}\nIn this section, we outline the level-1 algorithms for DSFM that are based on gradient descent. \nRecall the convex quadratic program (\\ref{prob:fuji}) from the Introduction. This program has a unique optimal solution $s^*$; the set $S=\\{v \\in V \\colon s^*(v) < 0\\}$ is the unique smallest minimizer to \\eqref{prob:sfm}. We will refer to this optimal solution $s^*$ throughout the section. \n\nIn the DSFM setting, one can write (\\ref{prob:fuji}) in multiple equivalent forms \\cite{JegelkaBS13}. For the first formulation, we let $\\mathcal{P} := \\prod_{i = 1}^r B(f_i)\\subseteq {\\mathbb{R}}^{rn},$ and let $A\\in {\\mathbb{R}}^{n\\times (rn)}$ denote the following matrix:\n\t$$A := \\underbrace{[I_n I_n \\dots I_n]}_\\text{$r$ times}.$$\nNote that, for every $y \\in \\mathcal{P}$, $Ay = \\sum_{i = 1}^r y_i$, where $y_i$ is the $i$-th block of $y$, and thus $Ay \\in B(f)$. The problem \\eqref{prob:fuji} can be reformulated for DSFM as follow.\n\\begin{equation}\n\\label{Prox-DSFM}\n\t\\min\\left\\{ {1 \\over 2} \\left\\|Ay \\right\\|^2_2 \\colon \\tag{Prox-DSFM} y\\in \\mathcal{P}\\right\\}.\n\\end{equation}\nThe second formulation is the following. Let us define the subspace $\\mathcal{A} := \\{ a \\in \\mathbb{R}^{nr} \\colon Aa= 0\\}$, and minimize its distance from $\\mathcal{P}$:\n\\begin{equation}\n\\label{Best-Approx}\n\t\\min\\left\\{ \\|a - y\\|^2_2 \\tag{Best-Approx} \\colon a \\in \\mathcal{A}, y \\in \\mathcal{P}\\right\\}.\n\\end{equation}\nThe set of optimal solutions for both formulations (\\ref{Prox-DSFM}) and (\\ref{Best-Approx}) is the set \n$\\mathcal{E} := \\{y \\in \\mathcal{P} \\colon Ay = s^*\\}$,\n where $s^*$ is the optimum of (\\ref{prob:fuji}). We note that, even though the set of solutions to (\\ref{Best-Approx}) are pairs of points $(a, y) \\in \\mathcal{A} \\times \\mathcal{P}$, the optimal solutions are uniquely determined by $y \\in \\mathcal{P}$, since the corresponding $a$ is the projection of $y$ to   $\\mathcal{A}$.\n\n\\begin{lemma}[{\\cite{JegelkaBS13}, Lemma 2}]\n\\label{lem:decomp-opt-set}\n The set $\\mathcal{E}$ is non-empty and it coincides with the set of optimal solutions of (\\ref{Prox-DSFM}) and (\\ref{Best-Approx}).\n\\end{lemma}\n\n{\\bf Gradient methods.} The gradient descent algorithms of \\cite{NishiharaJJ14,EneN15} provide level-1 algorithms for DSFM. In the following, we provide a brief overview of these algorithms and we refer the reader to the respective papers for more details. \n\n{\\bf The alternating projections algorithm.} Nishihara {\\em et al.}\\xspace minimize (\\ref{Best-Approx}) using \\emph{alternating projections} \\cite{NishiharaJJ14}. The algorithm starts with a point $a_0 \\in \\mathcal{A}$ and it iteratively constructs a sequence $\\set{(a^{(k)}, x^{(k)})}_{k \\geq 0}$ by projecting onto $\\mathcal{A}$ and $\\mathcal{P}$: $x^{(k)} = {\\operatorname{argmin}}_{x \\in \\mathcal{P}}\\|a^{(k)} - x\\|_2$, $a^{(k + 1)} = {\\operatorname{argmin}}_{a \\in \\mathcal{A}}\\|a - x^{(k)}\\|_2$.\n\n{\\bf Random coordinate descent algorithms.} Ene and Nguyen minimize (\\ref{Prox-DSFM}) using \\emph{random coordinate descent} \\cite{EneN15}. The RCDM algorithm adapts the random coordinate descent algorithm of Nesterov \\cite{Nesterov10} to (\\ref{Prox-DSFM}). In each iteration, the algorithm samples a block $i \\in [r]$ uniformly at random and it updates $x_i$ via a standard gradient descent step for smooth functions. ACDM, the accelerated version of the algorithm, presents a further enhancement using techniques from  \\cite{FR13}.\n\n{\\bf Rate of convergence.} The algorithms mentioned above enjoy a \\emph{linear convergence rate} despite the fact that the objective functions of (\\ref{Best-Approx}) and (\\ref{Prox-DSFM}) are not strongly convex. Instead, these works show that there are certain parameters that one can associate with the objective functions such that the convergence is at the rate $(1 - \\alpha)^k$, where $\\alpha \\in (0, 1)$ is a quantity that depends on the appropriate parameter. Let us now precisely define these parameters and state the convergence guarantees as a function of these parameters.\n\n Let $\\mathcal{A}'$ be the affine subspace \n$\\mathcal{A}':= \\{ a \\in \\mathbb{R}^{nr} \\colon Aa = s^*\\}.$ \nNote that $\\mathcal{E} = \\mathcal{P} \\cap \\mathcal{A}'$.\nFor $y\\in {\\mathbb{R}}^{nr}$ and a closed set $K\\subseteq {\\mathbb{R}}^{nr}$, we let $d(y,K)=\\min\\set{\\|y-z\\|_2 \\colon z\\in K}$ denote the distance between $y$ and $K$.\n The relevant parameter for the Alternating Projections algorithm is defined as follows.\n\\begin{definition}[\\cite{NishiharaJJ14}]\nFor every $y\\in (\\mathcal{P} \\cup \\mathcal{A}') \\setminus {\\cal E}$, let\n\t\\begin{align*}\n\t\t\\kappa(y) &\\coloneqq \\frac{d(y, {\\cal E})}{\\max\\set{d(y, \\mathcal{P}), d(y, \\mathcal{A}')}},\\quad \\mbox{ and }\\\\\n\t\t\\kappa_* &\\coloneqq \\sup\\left\\{ \\kappa(y) \\colon y \\in (\\mathcal{P} \\cup \\mathcal{A}') \\setminus {\\cal E} \\right\\}.\n\t\\end{align*}\n\\end{definition}\n\nThe relevant parameter for the random coordinate descent algorithms is the following.\n\n\\begin{definition}[\\cite{EneN15}]\n\tFor every $y\\in\\mathcal{P}$, let $y^* \\coloneqq {\\operatorname{argmin}}_{p} \\{\\|p-y\\|_2 \\colon Ap=s^*\\}$ be the optimal solution to (\\ref{Prox-DSFM}) that is closest to $y$. We say that the objective function ${1 \\over 2} \\|Ay\\|^2_2$ of (\\ref{Prox-DSFM}) is \\emph{restricted $\\ell$-strongly convex} if, for all $y \\in \\mathcal{P}$, we have\n\t\\begin{align*}\n\t\t&\\|A(y - y^*)\\|^2_2 \\geq \\ell \\|y - y^*\\|^2_2,\\quad \\mbox{ and }\\\\\n\t\t\\ell_* \\coloneqq \\sup&\\left\\{\\ell \\colon {1 \\over 2} \\|Ay\\|^2_2 \\text{ is restricted $\\ell$-strongly convex}\\right\\}.\n\t\\end{align*}\n\\end{definition}\n\nThe running time dependence of the algorithms on these parameters is given in the following theorems.\n\\begin{theorem}[\\cite{NishiharaJJ14}]\n\tLet $(a^{(0)}, x^{(0)} =  {\\operatorname{argmin}}_{x \\in \\mathcal{P}}\\|a^{(0)} - x\\|_2)$ be the initial solution and let  $(a^*, x^*)$ be an optimal solution to (\\ref{Best-Approx}). The alternating projection algorithm produces in\n\t\t\\[k = \\Theta\\left(\\kappa_*^2  \\ln\\left( {\\|x^{(0)} - x^*\\|_2 \\over \\epsilon} \\right) \\right) \\]\n\titerations a pair of points $a^{(k)} \\in \\mathcal{A}$ and $x^{(k)} \\in \\mathcal{P}$ that is $\\epsilon$-optimal, i.e.,\n\t\t\\[\\|a^{(k)} - x^{(k)}\\|_2^2 \\leq \\|a^* - x^*\\|_2^2 + \\varepsilon. \\]\n\\end{theorem}\n\n\\begin{theorem}[\\cite{EneN15}]\n\tLet $x^{(0)} \\in \\mathcal{P}$ be the initial solution and let $x^*$ be an optimal solution to (\\ref{Prox-DSFM}) that minimizes $\\|x^{(0)} - x^*\\|_2$. The random coordinate descent algorithm produces in\n\t\\[ k =  \\Theta\\left(  {r \\over \\ell_*} \\ln\\left( {\\|x^{(0)}  - x^* \\|_2 \\over \\epsilon} \\right)\\right)\\]\n\titerations a solution $x^{(k)}$ that is $\\epsilon$-optimal in expectation, i.e., $\\mathbb{E}\\left[{1 \\over 2} \\|A x^{(k)} \\|^2_2 \\right] \\leq {1 \\over 2} \\|Ax^*\\|^2_2 + \\epsilon$.\n\t\n\tThe accelerated coordinate descent algorithm produces in \n\t\\[k = \\Theta \\left (r \\sqrt{{1 \\over \\ell_*}} \\ln\\left( { \\|x^{(0)} - x^* \\|_2 \\over \\epsilon} \\right) \\right)  \\]\n\titerations (specifically, $\\Theta\\left( \\ln\\left( { \\|x^{(0)} - x^*\\|_2 \\over \\epsilon} \\right) \\right)$ epochs with $\\Theta\\left(r \\sqrt{{1 \\over \\ell_*}}\\right)$ iterations in each epoch) a solution $x^{(k)}$ that is $\\epsilon$-optimal in expectation, i.e., $\\mathbb{E}\\left[{1 \\over 2} \\|A x^{(k)} \\|^2_2 \\right] \\leq {1 \\over 2} \\|Ax^*\\|^2_2 + \\epsilon$.\n\\end{theorem}\n\nNishihara {\\em et al.}\\xspace show that $\\kappa_* \\leq nr$, and a family of instances (in fact, minimum cut instances) is given for which $\\kappa_* \\geq \\Omega(n \\sqrt{r})$. Ene and Nguyen show that $\\ell_* \\geq {r \/\\kappa^2_*}$. In Theorem~\\ref{thm:kappa-bound}, we close the remaining gap and show that $\\kappa_* = \\Theta(n \\sqrt{r})$ and $\\ell_* = \\Theta(1 \/ n^2)$, and thus we obtain tight analyses for the running times of the above mentioned algorithms.\n\nBy combining the level-1 gradient descent algorithms with appropriate level-0 subroutines, we obtain algorithms for DSFM whose running times can be upper bounded as follows. Using our improved convergence guarantees, it follows that RCDM obtains in time $O\\left(n^2 r\\Theta_{\\mathrm{avg}}\\ln\\left( { \\|x^{(0)} - x^* \\|_2 \\over \\epsilon}\\right)\\right)$ a solution that is $\\varepsilon$-approximate in expectation. For ACDM, the improved time bound is $O\\left(nr \\Theta_{\\mathrm{avg}} \\ln\\left( { \\|x^{(0)} - x^* \\|_2 \\over \\epsilon} \\right) \\right)$. We can upper bound the diameter of the base polytope by $O(\\sqrt{n} F_{\\max})$ \\cite{Jegelka11}. For integer-valued functions, a $\\varepsilon$-approximate solution can be converted to an exact optimum if $\\varepsilon=O(1\/n)$ \\cite{Bach-monograph}.\n\n\\section{Tight convergence bounds for the continuous algorithms}\n\\label{sec:kappa}\n\nIn this section, we show that the combinatorial approach introduced in Section~\\ref{sec:flow} can be applied to obtain better bounds on the parameters $\\kappa_*$ and $\\ell_*$ defined in Section~\\ref{sec:conv-opt}. Besides giving a stronger bound, our proof is considerably simpler than the algebraic one using Cheeger's inequality in \\cite{NishiharaJJ14}.\nThe key is the following lemma.\n\\begin{lemma}\n\\label{lem:decompose}\n\tLet $y \\in \\mathcal{P}$ and $s^* \\in B(f)$. Then there exists a point $x \\in \\mathcal{P}$ such that $Ax = s^*$ and $\\|x - y\\|_2 \\leq \\frac{\\sqrt{n}}{2} \\|Ay - s^*\\|_1$.\n\\end{lemma}\nBefore proving this lemma, we show how it can be used to derive the bounds.\n\\begin{theorem}\\label{thm:kappa-bound}\n\tWe have $\\kappa_* \\leq n \\sqrt{r}\/2 + 1$ and $\\ell_* \\geq {4 \/ n^2}$.\n\\end{theorem}\n\\begin{proof}\n\tWe start with the bound on $\\kappa_*$. In order to bound $\\kappa_*$, we need to upper bound $\\kappa(y)$ for any $y\\in (\\mathcal{P}\\cup\\mathcal{A}')\\setminus \\mathcal{E}$. We distinguish between two cases: $y \\in \\mathcal{P} \\setminus\\mathcal{E}$ and $y \\in \\mathcal{A}' \\setminus\\mathcal{E}$.\n\n\t\\noindent{\\bf Case I: $y \\in \\mathcal{P}\\setminus\\mathcal{E}$.} The denominator in the definition of $\\kappa(y)$ is equal to $d(y, \\mathcal{A}') = {\\|Ay - s^*\\|_2}\/{\\sqrt{r}}$. This follows since the closest point $a=(a_1,\\ldots,a_r)$ to $y$ in $\\mathcal{A}'$ is to set $a_i=y_i+(s^*-Ay)\/r$ for each $i\\in [r]$. \nLemma~\\ref{lem:decompose} gives an $x \\in \\mathcal{P}$ such that $Ax = s^*$ and $\\|x - y\\|_2 \\leq \\frac{\\sqrt{n}}{2}  \\|Ay - s^*\\|_1\\le \\frac{n}2 \\|Ay - s^*\\|_2$. Since $Ax = s^*$, we have $x \\in {\\cal E}$ and thus the numerator of $\\kappa(y)$ is at most $\\|x - y\\|_2$. Thus $\\kappa(y) \\leq {\\|x - y\\|_2 \/(\\|Ay - s^*\\|_2\/\\sqrt{r})} \\leq n\\sqrt{r}\/2$.\n\n\t\\noindent{\\bf Case II: $y \\in \\mathcal{A}'\\setminus\\mathcal{E}$.}\nThis means that $Ay = s^*$. The denominator of $\\kappa(y)$ is equal to $d(y, \\mathcal{P})$. For each $i \\in [r]$, let $q_i \\in B(f_i)$ be the point that minimizes $\\|y_i - q_i\\|_2$. Let $q = (q_1, \\dots, q_r)\\in {\\cal P}$. Then $d(y, \\mathcal{P}) = \\|y - q\\|_2$. Lemma~\\ref{lem:decompose} with $q$ in place of $y$ gives a point $x\\in {\\cal E}$ such that $\\|q - x\\|_2 \\leq \\frac{\\sqrt{n}}2 \\|Aq - s^*\\|_1$. We have\n\t $\\|Aq - s^*\\|_1 = \\|Aq - Ay\\|_1 \\leq \\sum_{i = 1}^r \\|q_i - y_i\\|_1 = \\|q - y\\|_1 \\leq {\\sqrt{nr}} \\|q - y\\|_2.$\n\tThus $\\|q - x\\|_2 \\leq \\frac{{n \\sqrt{r}} }2\\|q - y\\|_2$. \n\tSince $x \\in {\\cal E}$, we have \n\t$d(y, {\\cal E}) \\leq \\|x - y\\|_2 \\leq \\|x - q\\|_2 + \\|q - y\\|_2 \\leq \\left(1 + \\frac{n \\sqrt{r}}2\\right) \\|q - y\\|_2 = \\left(1 + \\frac{n \\sqrt{r}}2\\right) d(y, \\mathcal{P}).$\n\tTherefore $\\kappa(p) \\leq 1 + \\frac{n \\sqrt{r}}2$, as desired.\n\nLet us now prove the bound on $\\ell_*$. Let $y \\in \\mathcal{P}$ and let  $y^* \\coloneqq {\\operatorname{argmin}}_{p} \\{\\|p-y\\|_2\\colon Ap=s^*\\}$. We need to verify that $\\|A(y - y^*)\\|^2_2 \\geq {4 \\over n^2} \\|y - y^*\\|^2_2$.\nAgain, we apply Lemma~\\ref{lem:decompose} to obtain a point $x\\in \\mathcal{P}$ such that $Ax = s^*$ and  \n$\\|x - y\\|_2^2 \\leq \\frac{n}4\\|Ax-Ay\\|_1^2 \\le \\frac{n^2}4 \\|Ax - Ay\\|_2^2$.\nSince $Ax= s^*$, the definition of $y^*$ gives $\\|y - y^*\\|_2^2 \\leq \\|x - y \\|_2^2$. Using that $Ax = Ay^* = s^*$, we have $\\|Ax - Ay\\|_2 =\\|Ay-Ay^*\\|_2$. The same calculation as in Case II above implies the required $\\|y - y^*\\|_2^2 \\leq \\frac{n^2}4 \\|A(y - y^*)\\|_2^2$.\n\\end{proof}\n\n\\begin{proofof}{Lemma~\\ref{lem:decompose}}\nWe give an algorithm that transforms $y$ to a vector $x\\in {\\cal P}$ as in the statement through a sequence of path augmentations in the auxiliary graph defined in Section~\\ref{sec:flow}.\nWe initialize $x=y$ and maintain $x\\in {\\cal P}$ (and thus $Ax\\in B(f)$) throughout. We now define the set of source and sink nodes as \n$N := \\{ v \\in V \\colon (Ax)(v) < s^*(v) \\}$ and $P := \\{ v \\in V \\colon (Ax)(v) > s^*(v)\\}$. Once $N=P=\\emptyset$, we have $Ax=s^*$ and terminate. Note that since $Ax,s^*\\in B(f)$, we have $\\sum_v (Ax)(v)=\\sum_v s^*(v)=f(V)$, and therefore $N=\\emptyset$ is equivalent to $P=\\emptyset$.\nThe blocks of $x$ are denoted as $x=(x_1,x_2,\\ldots,x_r)$, with $x_i\\in B(f_i)$.\n\\begin{claim} If $N\\neq\\emptyset$, then there exists a directed path of positive capacity in the auxiliary graph between the sets $N$ and $P$.\n\\end{claim}\n\\begin{proof}\nWe say that a set $T$ is $i$-tight, if $x_i(T)=f_i(T)$. It is a simple consequence of submodularity that the intersection and union of two $i$-tight sets are also $i$-tight sets.\nFor every $i\\in [r]$ and every $u\\in V$, we define $T_i(u)$ as the unique minimal $i$-tight set containing $u$. It is easy to see that for an arc $(u,v)\\in E_i$, \n$c(u,v)>0$ if and only if $v\\in T_i(u)$. We note that if $u\\notin C_i$, then $x(u)=f_i(\\{u\\})=0$ and thus $T_i(u)=\\{u\\}$.\n\nLet $S$ be the set of vertices reachable from $N$ on a directed path of positive capacity in the auxiliary graph. For a contradiction, assume $S\\cap P=\\emptyset$. \nBy the definition of $S$, we must have $T_i(u)\\subseteq S$ for every $u\\in S$ and every $i\\in[r]$. Since the union of $i$-tight sets is also $i$-tight, we see that  $S$ is $i$-tight for every $i\\in[r]$, and consequently, $x(S)=f(S)$. On the other hand, since $N \\subseteq S$, $S \\cap P = \\emptyset$, and $N \\neq \\emptyset$, we have $x(S) < s^*(S)$. Since $s^* \\in B(f)$, we have $x(S) < s^*(S) \\leq f(S)$, which is a contradiction. We conclude that $S \\cap P \\neq \\emptyset$.\n\\end{proof}\n\nIn every step of the algorithm, we take a shortest directed path ${\\cal Q}$ of positive capacity from $N$ to $P$, and update $x$ along this path. That is, \nif $(u,v)\\in {\\cal Q}\\cap E_i$, then we increase $x_i(u)$ by\n$\\varepsilon$ and decrease $x_i(v)$ by $\\varepsilon$, where\n$\\varepsilon$ is the minimum capacity of an arc on\n$\\cal Q$. Note that this is the same as running the Edmonds-Karp-Dinitz algorithm in the submodular auxiliary graph. Using the analysis in \\cite{FujishigeZhang92}, one can show that this change maintains $x\\in {\\cal P}$, and that the algorithm terminates in finite (in fact, strongly polynomial) time.\n\nIt remains to bound $\\|x-y\\|_2$. At every path update, the change in $\\ell_{\\infty}$-norm of $x$ is at most $\\varepsilon$, and the change in $\\ell_1$-norm is at most $n\\varepsilon$, since the length of the path is $\\le n$. At the same time, $\\sum_{v\\in N} (s^*(v)-(Ax)(v))$ decreases by $\\varepsilon$.\nThus, $\\|x-y\\|_\\infty\\le \\|Ay-s^*\\|_1 \/2$ and $\\|x-y\\|_1\\le n\\|Ay-s^*\\|_1 \/2$. Using the inequality $\\|p\\|_2\\le \\sqrt{\\|p\\|_1\\|p\\|_{\\infty}}$, we obtain $\\|x-y\\|_2\\le \\frac{\\sqrt{n}}2 \\|Ay-s^*\\|_1$, completing the proof.\n\\end{proofof}\n\n\\section{The level-0 algorithms}\n\\label{sec:level-0}\n\nIn this section, we briefly discuss the level-0 algorithms and the interface between the level-1 and level-0 algorithms.\n\n{\\bf Two-level frameworks via quadratic minimization oracles.}\nRecall from the Introduction the assumption on the subroutines ${\\cal O}_i(w)$ that finds the minimum norm point in $B(f_i+w)$ for the input vector $w\\in{\\mathbb{R}}^n$. \nThe continuous methods in Section~\\ref{sec:conv-opt} directly use the  subroutines ${\\cal O}_i(w)$ for the alternating projection or coordinate descent steps. For the flow-based algorithms in Section~\\ref{sec:flow}, the main oracle query is to find the auxiliary graph capacity $c(u,v)$ of an arc $(u,v)\\in E_i$ for some $i\\in[r]$. This can be easily formulated as minimizing the function $f_i+w$ for an appropriate $w$ with $\\text{supp}(w)\\subseteq C_i$; the details are given in Lemma~\\ref{lem:exchange-cap-oracle}. As explained at the beginning of Section~\\ref{sec:conv-opt}, an optimal solution to \\eqref{prob:fuji} immediately gives an optimal solution to \\eqref{prob:sfm} for the same submodular function. Hence, the auxiliary graph capacity queries can be implemented via the subroutines ${\\cal O}_i(w)$. \nLet us also remark that, while the functions $f_i$ are formally defined on the entire ground set $V$, their effective support is $C_i$, and thus it suffices to solve the quadratic minimization problems on the ground set $C_i$.\n\n\\begin{lemma}\n\\label{lem:exchange-cap-oracle}\n  The capacity $c(u, v) := \\min\\{f_i(S) - x_i(S) \\colon S \\subseteq\n  C_i, u \\in S, v \\notin S\\}$ can be computed as the minimum value of\n  $\\min_{S\\subseteq C_i} f_i(S)+w(S)$ for an appropriately chosen vector $w\\in{\\mathbb{R}}^n$,\n  $\\text{supp}(w)\\subseteq C_i$.\n\\end{lemma}\n\\begin{proof}\n  We define a weight vector $w \\in \\mathbb{R}^n$ as follows: $w(u)\n  = - (f_i(\\{u\\}) + 1)$; $w(v) = - (f_i(C_i) - f_i(C_i \\setminus\n  \\{v\\}) - 1)$; $w(a) = - x(a)$ for all $a \\in C_i \\setminus \\{u,\n  v\\}$, and $w(a)=0$ for all $a\\notin C_i$.\nLet $A\\subseteq C_i$ be a minimizer of $\\min_{S \\subseteq C_i} f_i(S)\n+ w(S)$.\nIt suffices to show that $u \\in A$ and $v \\notin A$. Note that\n$f_i(\\{u\\}) = f_i(\\{u\\}) - f(\\emptyset)$ is the maximum marginal value\nof $u$, i.e., $\\max_{S} (f_i(S \\cup \\{u\\}) - f_i(S))$. Moreover,\n$f_i(C_i) - f_i(C_i \\setminus \\{v\\})$ is the minimum marginal value of\n$v$. To show $u\\in A$, let us assume for a contradiction that $u\\notin A$.\n  \\begin{align*}\n    f_i(A \\cup \\{u\\}) + w(A \\cup \\{u\\}) &= (f_i(A) + w(A)) + (f_i(A \\cup \\{u\\}) - f_i(A)) + w(u)\\\\\n    &= (f_i(A) + w(A)) + (f_i(A \\cup \\{u\\}) - f_i(A)) - f_i(\\{u\\}) + 1\\\\\n    &\\leq f_i(A) + w(A) - 1.\n  \\end{align*}\n  Similarly, to show that $v\\notin A$, suppose for a contradiction that $v \\in A$, and consider the set $A \\setminus \\{v\\}$. Since $f_i(C_i) - f_i(C_i \\setminus \\{v\\}) \\leq f_i(A) - f_i(A \\setminus \\{v\\})$, we have\n  \\begin{align*}\n    f_i(A \\setminus \\{v\\}) + w(A \\setminus \\{v\\}) &= (f_i(A) + w(A)) - (f_i(A) - f_i(A \\setminus \\{v\\})) - w(v)\\\\\n    &= (f_i(A) + w(A)) - (f_i(A) - f_i(A \\setminus \\{v\\})) + (f_i(C_i) - f_i(C_i \\setminus \\{v\\})) - 1\\\\\n    &\\leq f_i(A) + w(A) - 1.\n  \\end{align*}\n  Therefore $u \\in A$ and $v \\notin A$, and hence $A \\in {\\operatorname{argmin}}\\{f_i(S) - x_i(S) \\colon u \\in S, v \\notin S\\}$. \n\\end{proof}\n\n\n\nWhereas discrete and continuous algorithms require the same type of oracles, there is an important difference between the two algorithms in terms of exactness for the oracle solutions. The discrete algorithms require exact values of the auxiliary graph capacities $c(u,v)$, as they must maintain $x_i\\in B(f_i)$ throughout. Thus, the oracle  must always return an optimal solution. The continuous algorithms are more robust, and return a solution with the required accuracy even if the oracle only returns an approximate solution. As discussed in Section~\\ref{sec:experiments}, this difference leads to the continuous methods being applicable in settings where the combinatorial algorithms are prohibitively slow.\n\n{\\bf Level-0 algorithms.}\nWe now discuss specific algorithms for quadratic minimization over the base polytopes of the functions $f_i$. Several functions that arise in applications are ``simple'', meaning that there is a function-specific quadratic minimization subroutine that is very efficient. If a function-specific subroutine is not available, one can use a general-purpose submodular minimization algorithm. The works \\cite{Arora12,Fix13} use a {\\em brute force search} as the subroutine for each each $f_i$, whose running time is $2^{|C_i|} \\mathrm{EO}_i$. However, this is applicable only for small $C_i$'s and is not suitable for our experiments where the maximum clique size is quite large. \nAs a general-purpose algorithm, we used the {\\em Fujishige-Wolfe minimum norm point algorithm} \\cite{Fujishige11, Wolfe76}. This provides an $\\varepsilon$-approximate solution in $O(|C_i| F^2_{i,\\max}\/\\varepsilon)$ iterations, with overall running time bound $O((|C_i|^4 + |C_i|^2 \\mathrm{EO}_i) F^2_{i, \\max} \/ \\varepsilon)$. \\cite{Chakrabarty14}. \nThe experimental running time of the Fujishige-Wolfe algorithm can be prohibitively large \\cite{jegelka11fast}. As we discuss in Section~\\ref{sec:experiments}, by warm-starting the algorithm and performing only a small number of iterations, we were able to use the algorithm in conjunction with the gradient descent level-1 algorithms. \n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\n\\begin{table}\n\\caption{Instance sizes}\n\\label{tb:instance-sizes}\n\\centering\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n{\\bf image} & {\\bf \\# pixels} & {\\bf \\# edges} & {\\bf \\# squares}\\\\\n\\hline\nbee & 273280 & 1089921 & 68160 \\\\\n\\hline\noctopus & 273280 & 1089921 & 68160\\\\\n\\hline\npenguin & 154401 & 615200 & 38400\\\\\n\\hline\nplant & 273280 & 1089921 & 68160\\\\\n\\hline\nplane & 154401 & 615200 & 38400 \\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n{\\bf \\# regions} & \\multicolumn{3}{|c|}{{\\bf min, max, and average region size}}\\\\\n\\hline\n50 & 298 & 299 & 298.02\\\\\n\\hline\n49 & 7 & 299 & 237.306\\\\\n\\hline\n50 & 5 & 299 & 279.02\\\\\n\\hline\n50 & 8 & 298 & 275.22\\\\\n\\hline\n50 & 10 & 299 & 291.48\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Minimum cut experiments}\n\\label{tb:mincut}\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\n{\\bf image} & {\\bf \\# functions ($r$)} & {\\bf IBFS time (sec)}\\\\\n\\hline\nbee & 1363201 & 1.70942\\\\\n\\hline\noctopus & 1363201 & 1.09101 \\\\\n\\hline\npenguin & 769601 & 0.684413 \\\\\n\\hline\nplant & 1363201 & 1.30977 \\\\\n\\hline\nplane & 769601 & 0.745521 \\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{{\\bf UCDM time (sec)}}\\\\\n\\hline\n{\\bf \\# iter = $5r$} & {\\bf \\# iter = $10r$} & {\\bf \\# iter = $100r$} & {\\bf \\# iter = $1000r$}\\\\\n\\hline\n0.951421 & 1.6234 & 13.4594 & 134.719 \\\\\n\\hline\n0.937317 & 1.6279 & 13.9887 & 137.969 \\\\\n\\hline\n0.492372 & 0.836147 & 7.1069 & 70.1742\\\\\n\\hline\n0.943306 & 1.63492 & 13.9559 & 137.865 \\\\\n\\hline\n0.521685 & 0.850145 & 7.31664 & 71.8874 \\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{{\\bf ACDM time (sec)}}\\\\\n\\hline\n{\\bf \\# iter = $5r$} & {\\bf \\# iter = $10r$} & {\\bf \\# iter = $100r$} & {\\bf \\# iter = $1000r$}\\\\\n\\hline\n1.3769 & 2.2696 & 18.4351 & 182.069 \\\\\n\\hline\n1.40884 & 2.33431 & 19.0471 & 188.887 \\\\\n\\hline\n0.757929 & 1.24094 & 9.99443 & 98.5717 \\\\\n\\hline\n1.39893 & 2.29446 & 18.6846 & 185.274 \\\\\n\\hline\n0.766455 & 1.26081 & 10.1244 & 99.0298 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{table}\n\\caption{Small cliques experiments}\n\\label{tb:smallcliques}\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\n{\\bf image} & {\\bf \\# functions ($r$)} & {\\bf IBFS time (sec)}\\\\\n\\hline\nbee & 1431361 & 14.5125 \\\\\n\\hline\noctopus & 1431361 & 12.9877 \\\\\n\\hline\npenguin & 808001 & 7.58177 \\\\\n\\hline\nplant & 1431361 & 13.7403 \\\\\n\\hline\nplane & 808001 & 7.67518 \\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{{\\bf RCDM time (sec)}}\\\\\n\\hline\n{\\bf \\# iter = $5r$} & {\\bf \\# iter = $10r$} & {\\bf \\# iter = $100r$} & {\\bf \\# iter = $1000r$}\\\\\n\\hline\n4.14091 & 7.57959 & 66.0576 & 660.496 \\\\\n\\hline\n4.29358 & 7.80816 & 68.5862 & 675.23 \\\\\n\\hline\n2.16441 & 4.08777 & 37.8157 & 372.733 \\\\\n\\hline\n4.6404 & 8.21702 & 69.059 & 672.753 \\\\\n\\hline\n2.182 & 4.12521 & 37.8602 & 373.825 \\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{{\\bf ACDM time (sec)}}\\\\\n\\hline\n{\\bf \\# iter = $5r$} & {\\bf \\# iter = $10r$} & {\\bf \\# iter = $100r$} & {\\bf \\# iter = $1000r$}\\\\\n\\hline\n5.24474 & 10.0951 & 98.7737 & 932.954 \\\\\n\\hline\n5.5891 & 10.7124 & 99.4081 & 924.076 \\\\\n\\hline\n2.95226 & 5.71215 & 52.9766 & 512.665 \\\\\n\\hline\n5.8395 & 11.0806 & 102.023 & 900.979 \\\\\n\\hline\n2.95003 & 5.70771 & 53.7524 & 486.294 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{table}\n\\caption{Large cliques experiments with potential specific quadratic minimization for the region potentials. In order to be able to run IBFS, we used smaller regions: $50$ regions with an average size between $45$ and $50$.}\n\\label{tb:largecliques}\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\n{\\bf image} & {\\bf \\# functions ($r$)} & {\\bf IBFS time (sec)}\\\\\n\\hline\nbee & 1431411 & 14.7271\\\\\n\\hline\noctopus & 1431411 & 12.698 \\\\\n\\hline\npenguin & 808051 & 7.51067 \\\\\n\\hline\nplant & 1431411 & 13.6282 \\\\\n\\hline\nplane & 808051 & 7.64527 \\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{{\\bf RCDM time (sec)}}\\\\\n\\hline\n{\\bf \\# iter = $5r$} & {\\bf \\# iter = $10r$} & {\\bf \\# iter = $100r$} & {\\bf \\# iter = $1000r$}\\\\\n\\hline\n 4.29954 & 7.87555 & 67.8876 & 664.816\\\\\n\\hline\n 4.18879 & 7.61576 & 66.7 & 656.71\\\\\n\\hline\n 2.132 & 4.01926 & 36.9896 & 364.694\\\\\n\\hline\n 4.55894 & 8.06429 & 67.72 & 659.685\\\\\n\\hline\n 2.16248 & 4.0713 & 37.1917 & 366.272\\\\\n\\hline\n\\end{tabular}\n\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{{\\bf ACDM time (sec)}}\\\\\n\\hline\n{\\bf \\# iter = $5r$} & {\\bf \\# iter = $10r$} & {\\bf \\# iter = $100r$} & {\\bf \\# iter = $1000r$}\\\\\n\\hline\n 5.34726 & 10.3231 & 100.24 & 912.477\\\\\n\\hline\n 5.44726 & 10.4446 & 96.2384 & 898.579\\\\\n\\hline\n 2.90223 & 5.60117 & 51.9775 & 500.083\\\\\n\\hline\n 5.72946 & 10.8512 & 99.6597 & 879.872\\\\\n\\hline\n 2.89726 & 5.61102 & 52.5439 & 475.967\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\nWe evaluate the algorithms on energy minimization problems that arise\nin image segmentation problems.\n We follow the standard approach and model the image segmentation task of segmenting an object from the background as finding a minimum cost $0\/1$ labeling of the pixels. The total labeling cost is the sum of labeling costs corresponding to \\emph{cliques}, where a clique is a set of pixels. We refer to the labeling cost functions as clique potentials.\n\nThe main focus of our experimental analysis is to compare the running times of the decomposable submodular minimization algorithms. Therefore we have chosen to use the simple hand-tuned potentials that were used in previous work~\\cite{Shanu16,Arora12,StobbeK10}: the \\emph{edge-based costs} defined by \\cite{Arora12} and the \\emph{count-based costs} defined by \\cite{StobbeK10}. Specifically, we used the following clique potentials in our experiments, all of which are submodular:\n\\begin{compactitem}\n\\item {\\bf Unary potentials} for each pixel. The unary potentials are derived from Gaussian Mixture Models of color features \\cite{RotherKB04}.\n\\item {\\bf Pairwise potentials} for each edge of the $8$-neighbor grid graph. Each graph edge $(i, j)$ between pixels $i$ and $j$ is assigned a weight that is a function of $\\exp(- \\|v_i - v_j\\|^2)$, where $v_i$ is the RGB color vector of pixel $i$. The clique potential for the edge is the cut function of the edge: the cost of a labeling is equal to zero if the two pixels have the same label and it is equal to the weight of the edge otherwise.\n\\item {\\bf Square potentials} for each $2 \\times 2$ square of pixels. We view a $2 \\times 2$ square as a graph on $4$ nodes connected with $4$ edges (two horizontal and two vertical edges). The cost of a labeling is the square root of the number of edges of the square that have different labels. This is the basic edge-based potential defined by \\cite{Arora12}.\n\\item {\\bf Region potentials} for a set of regions of the image. We compute a set of regions of the image using the region growing algorithm suggested by \\cite{StobbeK10}. For each region $C_i$, we define a count-based clique potential as in \\cite{StobbeK10,Shanu16}: for each set $S \\subseteq C_i$ of pixels, $f_i(S) = |S| |C_i \\setminus S|$.\n\\end{compactitem}\nWe used five image segmentation instances to evaluate the algorithms\\footnote{The data is available at \\url{http:\/\/melodi.ee.washington.edu\/~jegelka\/cc\/index.html} and \\url{http:\/\/research.microsoft.com\/en-us\/um\/cambridge\/projects\/visionimagevideoediting\/segmentation\/grabcut.htm}}. Table~\\ref{tb:instance-sizes} provides the sizes of the resulting instances. The experiments were carried out on a single computer with a 3.3 GHz Intel Core i5 processor and 8 GB of memory. The reported times are averaged over 10 trials.\n\n\\begin{table}\n\\caption{Large cliques experiments with Fujishige-Wolfe quadratic minimization algorithm for the region potentials. The Fujishige-Wolfe algorithm was run for $10$ iterations starting from the current gradient descent solution. The region sizes are given in Table~\\ref{tb:instance-sizes}.}\n\\label{tb:largecliquesmnp}\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{{\\bf RCDM time (sec)}}\\\\\n\\hline\n{\\bf \\# iter = $5r$} & {\\bf \\# iter = $10r$} & {\\bf \\# iter = $100r$} & {\\bf \\# iter = $1000r$}\\\\\n\\hline\n4.4422 & 8.18077 & 69.0444 & 674.526 \\\\\n\\hline\n4.30835 & 7.86231 & 68.1428 & 665.57\\\\\n\\hline\n2.2724 & 4.28243 & 38.1329 & 366.549\\\\\n\\hline\n4.61008 & 8.20094 & 68.8351 & 660.469\\\\\n\\hline\n2.28484 & 4.30316 & 38.0435 & 366.825\\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{{\\bf ACDM time (sec)}}\\\\\n\\hline\n{\\bf \\# iter = $5r$} & {\\bf \\# iter = $10r$} & {\\bf \\# iter = $100r$} & {\\bf \\# iter = $1000r$}\\\\\n\\hline\n5.29305 & 10.2853 & 103.452 & 936.613\\\\\n\\hline\n5.55511 & 10.6411 & 97.955 & 901.875\\\\\n\\hline\n2.95909 & 5.74585 & 54.3808 & 505.977\\\\\n\\hline\n5.71402 & 10.8467 & 99.6515 & 873.694\\\\\n\\hline\n2.9556 & 5.73271 & 54.0599 & 482.496\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n{\\bf Number of iterations for the coordinate methods.} We have run the coordinate descent algorithms for $1000 r$ iterations, where $r$ is the number of functions in the decomposition. Our choice is based on the empirical results of Jegelka {\\em et al.}\\xspace \\cite{JegelkaBS13} that showed that this number of iterations suffices to obtain good results.\n\n{\\bf Minimum cut experiments.} We evaluated the algorithms on instances containing only the unary potentials and the pairwise potentials. Table~\\ref{tb:mincut} gives the running times in seconds.\n\n{\\bf Small cliques experiments.} We evaluated the algorithms on instances containing the unary potentials, the pairwise potentials, and the square potentials. Table~\\ref{tb:smallcliques} gives the running times in seconds.\n\n\\iffalse\nFor the coordinate descent methods, we report the running times after a given number of iterations between $5r$ and $1000r$, where $r$ is the number of functions. \n\\fi\n\n{\\bf Large cliques experiments.} We evaluated the algorithms on instances containing all of the potentials: the unary potentials, the pairwise potentials, the square potentials, and the region potentials. For the region potentials, we used a potential-specific level-$0$ algorithm that performs quadratic minimization over the base polytope in time $O(|C_i| \\log(|C_i|) + |C_i| \\mathrm{EO}_i)$. Additionally, due to the slow running time of IBFS, we used smaller regions: $50$ regions with an average size between $45$ and $50$.\n\n{\\bf Large cliques experiments with Fujishige-Wolfe algorithm.} We also ran a version of the large cliques experiments with the Fujishige-Wolfe algorithm as the level-$0$ algorithm for the region potentials. The Fujishige-Wolfe algorithm was significantly slower than the potential-specific quadratic minimization algorithm and in our experiments it was prohibitive to run the Fujishige-Wolfe algorithm to near-convergence. Since the IBFS algorithm requires almost exact quadratic minimization in order to compute exchange capacities, it was prohibitive to run the IBFS algorithm with the Fujishige-Wolfe algorithm. In contrast, the coordinate descent methods can potentially make progress even if the level-$0$ solution is far from being converged.\n\nIn order to empirically evaluate this hypothesis, we made a simple but crucial change to the Fujishige-Wolfe algorithm: we \\emph{warm-started} the algorithm with the current solution. Recall that the coordinate descent algorithms maintain a solution $x_i \\in B(f_i)$ for each function $f_i$ in the decomposition. We warm-started the Fujishige-Wolfe algorithm with the current solution $x_i$, and we ran the algorithm for a small number of iterations. In our experiments, we ran the Fujishige-Wolfe algorithm for $10$ iterations. These changes made the level-$0$ running time considerably smaller, which made it possible to run the level-$1$ coordinate descent algorithms for as many as $1000r$ iterations. At the same time, performing $10$ iterations starting from the current solution seemed enough to provide an improvement over the current solution. Table~\\ref{tb:largecliquesmnp} gives the running times.\n\n{\\bf Conclusions.} The combinatorial level-$1$ algorithms such as IBFS are exact and can be significantly faster than the gradient descent algorithms provided that the sizes of the cliques are fairly small. For instances with larger cliques, the combinatorial algorithms are no longer suitable if the only choice for the level-$0$ algorithms are generic methods such as the Fujishige-Wolfe algorithm. The experimental results suggest that in such cases, the coordinate descent methods together with a suitably modified Fujishige-Wolfe algorithm provides an approach for obtaining an approximate solution.\n\n\n\\newpage \\clearpage\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe largest globular cluster associated with our Galaxy, $\\omega$\nCentauri, shows substantial ranges in abundance of all elements that\nare studied (e.g. \\citealt{nd95a, scl95, smi00, pan02}).  The iron\nabundance ranges from \\mbox{[Fe\/H]$\\approx$--2.0} up to\n\\mbox{[Fe\/H]$\\approx$--0.4}.  A peak at [Fe\/H]=--1.7 exists which accounts for\napproximately 70\\% of the population, and there is a long tail to\nhigher metallicities \\citep{nfm96, sk96, lee99, pan00, sta06a}.\n\n\nStudies of individual elements have also shown a range in their\nabundances within the cluster members.  Carbon, nitrogen and oxygen\nshow large scatters in their abundance ratios for a given\n[Fe\/H]~\\citep{per80, bw93, nd95a}.  The $\\alpha$ elements (Mg, Si, Ca\nand Ti) show a constant value of \\mbox{[$\\alpha$\/Fe]{$\\approx$}0.3} below\n\\mbox{[Fe\/H]$<$--1.0} \\citep{bw93, scl95, nd95a, smi00} but decrease\nto \\mbox{[$\\alpha$\/Fe]=0.0} at higher metallicities\n\\mbox{([Fe\/H]$>$--1.0)} \\citep{pan02}. The constancy with iron at lower\nabundances indicates they are produced by the same source, most likely\nsupernovae Type II.  At the higher metallicities, the [$\\alpha$\/Fe]\ndecreases, consistent with supernovae Type Ia contributions.\n\n\nSodium and aluminium abundance ratios are correlated, and both are\nanticorrelated with [O\/Fe] \\citep{bw93, nd95a, nd95b, smi00}.\n\\citet{smi00} also report [Al\/Fe] being anticorrelated with [Mg\/Fe].\n[Cu\/Fe] has been found to be constant for \\mbox{[Fe\/H]$<$--0.8}\n\\citep{smi00, cun02}, but \\citet{pan02} reported a trend of increasing\n[Cu\/Fe] as the metallicity increased from \\mbox{[Fe\/H]=--1.2} to~--0.5.\nStudies of the iron peak elements (Cr, Ni, Ti) and metals (Sc, V) have\nshown that there is no trend with respect to iron (up to\n\\mbox{[Fe\/H]$\\approx$--0.8)}, consistent with primordial enrichment from Type\nII supernovae.  The heavy neutron capture elements have been shown by\n\\citet{nd95a} and \\citet{scl95} to increase sharply as a function of\niron abundance.  These results are in contrast to normal globular\nclusters, and suggest that the stellar winds from AGB stars (sources\nof s-process elements) were involved with the enrichment of {$\\omega$ Cen}.\n\n\nThe cluster also has several members that are of distinct classes,\nsuch as CH stars, Ba stars, S stars \\citep{le83} and stars with strong\nCO \\citep{per80} to name a few.  Here we present another object that\nhas unusual abundance ratios, the formation of which is difficult to\nexplain.\n\n\nWe have obtained observations of a sample of main sequence and turnoff\n(MSTO) stars within the cluster for several purposes.  Firstly, the\npossibility of an age-metallicity relation within the cluster's member\nstars was investigated \\citep{sta06a}.  We found an age range of\n\\mbox{2--4} Gyrs exists between the most metal-poor and metal-rich\npopulations within the cluster, similar to that found by other studies\n\\citep{hil04, rey04, sol05}. Positions, photometry, metallicities and\nages of the stars in the catalogue can be found in the electronic\nversion of \\citet{sta06a}.  The second goal was to determine abundance\nratios of several elements --- carbon, nitrogen, strontium and barium\nas functions of [Fe\/H] \\citep{sta06b}. In that analysis a metal-rich\nmain sequence (MS) member, 2015448, was found that exhibited unusual\ns-process abundance ratios. The purpose of this Letter is to detail\nthe abundance analysis for star 2015448, and to propose possible\nformation scenarios.  \\S \\ref{OR_sect} briefly describes the\nobservation and reduction process for our sample of main sequence and\nturnoff members. The stellar parameters and analysis are detailed in\n\\S \\ref{SPA_sect}, while \\S \\ref{D_sect} discusses the possible\nformation scenarios for this object.\n\n\n\\section{Observations} \\label{OR_sect}\n\nThe observations and reduction of our data are described in detail in\n\\citet{sta06a} to which we refer the reader.  Briefly, photometry for\nthe cluster was obtained in the V and B bands, and samples were chosen\nwithin an annulus \\mbox{15'---25'} from the cluster center.  Two\nregions were defined near the MSTO, shown in Figure \\ref{cmd}, and\nspectra were obtained for objects within these CMD regions using the\nTwo Degree Field Spectrograph (2dF) on the Anglo-Australian Telescope\n\\citep{lew02}.  Figure \\ref{cmd} shows the color-magnitude diagram of\nall objects with no membership information as small dots, the sample\nof 420 radial velocity members as large dots, and the object that is\nthe topic of this Letter as a large star.  This object, 2015448, has\n$V=18.22$, and $B-V=0.69$.  This star immediately stood out in a\nvisual inspection of the spectra, due to the anomalously strong Sr\nlines, seen in a Figure \\ref{spec}.  Its position is\n\\mbox{RA=13$^h$24$^m$49.10$^s$} and\n\\mbox{Dec=--47$\\degr$40$'$34.7$''$} (J2000).  With a radial velocity\nof \\mbox{228$\\pm$11~kms$^{-1}$}, star 2015448 is likely to be a member of\nthe cluster as {$\\omega$ Cen} has a radial velocity of\n\\mbox{232$\\pm$0.7~kms$^{-1}$} \\citep{din99}\n\n\nThe observations were carried out in half hour exposures, and, as\nseveral such exposures were needed to obtain the required\nsignal-to-noise of $\\sim$30 for each star, there are several\nindividual observations for star 2015448.  Also, this object was observed\nin observing sessions in 1998, 1999 and 2002, in different fibres and\nspectrographs. This gives an independent check on the relative\nstrengths of the Sr and Ba features.  It was found that the two Sr\nfeatures were present in all observations, while Ba was not clearly\ndetected in any of our spectra.\n\nA spectroscopic abundance analysis of the full sample is detailed in\n\\citet{sta06b}.  Abundances relative to iron of C, N and Sr were\ndetermined using spectrum synthesis techniques.  The CH feature at\n$\\sim$4300{\\AA} was used to determine the [C\/Fe] abundance for each\nstar.  This [C\/Fe] was then adopted when the CN feature at 3883{\\AA}\nwas analyzed to find [N\/Fe].  The Sr abundance ratio was determined\nusing the two Sr lines at 4077.71{\\AA} and 4215.52{\\AA}.  When\npossible, [Ba\/Fe] was also determined from the feature at\n4554.03{\\AA}.\n\n\n\\section{Stellar Parameters and Analysis} \\label{SPA_sect}\n\nThe process for calculating metallicities is described in detail in\n\\citet{sta06a}.  The Sr-rich star has a metallicity of\n\\mbox{[Fe\/H]=--0.74$\\pm$0.23~dex}, determined solely from the\nAuto-Correlation Function method, and an age of\n\\mbox{14.5$\\pm$2.2~Gyrs}.  This age is consistent with the mean age of\nthe bulk of the stars in the cluster.  The ages for the individual\nstars in the sample were calculated by fitting an isochrone to each\nobject using the magnitude, color, metallicity and [$\\alpha$\/Fe]\nabundance (Y$^2$, \\citet{yi01}, Green Table).  This also enabled an\nindividual temperature and gravity to be assigned to each star.  A\ntemperature of 5820 (K), and \\mbox{log$g$=4.2~(cgs)} were adopted for\nthis object.  A reddening \\mbox{$E(B-V)$=0.11} \\citep{lub02} and\ndistance modulus \\mbox{(m--M)$_{V}$=14.10} were assumed.  Although the\nstar's color is redder than the bulk of the population in the cluster,\nit is also more metal-rich than that population. The hydrogen line\nstrengths in the observed spectrum are consistent with those expected\non the basis of the star's color.\n\nSynthetic spectra were obtained using the stellar models of Kurucz\n(1993), atomic line lists of Bell (2000, private communication) and\nmolecular line lists of Kurucz. The spectrum synthesis code was\ndeveloped by \\citet{cn78}.  More details of this process are given in\n\\citet{sta06b}.\n\n\n\n\\subsection{Abundance Ratios} \\label{A_sect}\n\nFigure \\ref{spec1} shows the synthetic spectrum fits to the CH, CN, Sr\nand Ba features.  The G band at $\\sim$4300{\\AA} was analyzed first,\nand gave a C abundance of \\mbox{[C\/Fe]=--0.5$\\pm$0.3~dex}.  This\nfeature loses its sensitivity for abundances less than\n\\mbox{[C\/Fe]=--0.5}, shown by the close proximity of the\n\\mbox{[C\/Fe]=--0.8} and --0.5~dex synthetic lines.  As we have no\ninformation on O, which affects the C abundance obtained from the CH\nfeature, a value of \\mbox{[O\/Fe]=0.18~dex} was assumed.  This star was\nassumed to have a \\mbox{[C\/Fe]=--0.5}, which was then used when\ndetermining [N\/Fe]. Synthesis of the CN feature at $\\sim$3883{\\AA} led\nto a N abundance ratio of \\mbox{[N\/Fe]$<$0.5~dex}.  A range of N\nabundance ratios are shown in the synthetic spectra, enabling an upper\nlimit on to be placed on the N abundance ratio.  The CN feature in\nFigure \\ref{spec1} shows that abundance ratios of \\mbox{[N\/Fe]=1.5}\nand 1.0 are too high.  However, this feature loses its sensitivity as\nsmaller N abundance ratios are considered and only an upper limit can\nbe determined.\n\n\n\nThe Sr and Ba lines were analyzed together.  There were two Sr lines\nused, Sr {\\sc} II at 4077.71{\\AA} and Sr {\\sc II} at 4215.52{\\AA}.\nThe Ba {\\sc II} 4554.03{\\AA} line was used to constrain the Ba\nabundance ratio.  The two Sr lines are in agreement and yield\n\\mbox{[Sr\/Fe]=1.6$\\pm$0.1~dex}.  It can be seen clearly that a solar\nSr abundance ratio does not fit the observed spectrum.  The CN\nbandhead at $\\sim$4216{\\AA} can affect the abundance ratio obtained\nfor the Sr 4215{\\AA} feature. However with an assumed solar N\nabundance ratio, there is little effect to the Sr line.  While the N\nabundance ratio could be as high as \\mbox{[N\/Fe]=0.5~dex}, spectrum\nsynthesis calculations show this does not affect on the Sr 4215{\\AA}\nline significantly, and does not alter the abundance ratio obtained.\nUsing the same enhancement for Ba that was found for Sr, one finds the\npredicted strength to be too high to fit the observed Ba line.  The\nsensitivity of the Ba feature is low up until abundance ratios of\n\\mbox{[Ba\/Fe]=0.6~dex}, and an upper limit can be placed at this\nvalue.\n\n\n\\subsection{Errors} \\label{E_sect}\n\nThe stellar parameters temperature, gravity and metallicity, for\nstar 2015448 were varied individually by their uncertainties to give an\nestimate of the error in our determined abundance ratios.  These were\nthen added in quadrature.  The error in temperature comes from\n$\\Delta(B-V)$ and the reddening, equating to $\\pm$100K uncertainty in\ntemperature.  This propagated to errors of $\\pm$0.2~dex in C and\n$\\pm$0.1~dex in Sr.  Gravity was changed by $\\pm$0.2~dex, and did not\nlead to any significant errors in the final abundance ratios.  The\nuncertainty in metallicity propagated to errors in abundance ratios\nfor \\mbox{$\\Delta$C$\\pm$0.2~dex} and \\mbox{$\\Delta$Sr$\\pm$0.1~dex}.\nThe CN feature used to determine N, adopting the previously determined\nC abundance ratio for 2015448, did not show any change with the above\nchanges in temperature, gravity, metallicity, or the determined error\nin the C abundance ratio, due to the low sensitivity of this feature.\nTo determine the effect of the assumed oxygen abundance,\n\\mbox{[O\/Fe]=0.0} and +0.3 were used and the CH feature reanalyzed.\nIt was found to have no noticeable effect on the [C\/Fe] abundance\ndetermined.\n\n\n\\section{Discussion} \\label{D_sect}\n\nThis peculiar star in {$\\omega$ Cen} shows \\mbox{[C\/Fe]=--0.5},\n\\mbox{[N\/Fe]$<$0.5}, enhanced \\mbox{[Sr\/Fe]=1.6}, and\n\\mbox{[Ba\/Fe]$<$0.6}.  The formation process that created it was\nunusual as there are no similar objects yet found within the cluster.\nWe do find MS objects with high Sr, but these also have similar\nenhancements in Ba.  This star was one out of 420 stars studied on the\nmain sequence turnoff.  Given that it is a metal-rich star, and we\nonly found 25 such objects, a more extensive search of other areas of\nthe center of the cluster may prove fruitful in determining if there\nare more of these objects.  On the RGB, at a metallicity of\n\\mbox{[Fe\/H]=--0.8}, most stars have \\mbox{[Ba\/Fe]$\\approx$+0.6}\n\\citep{nd95a}, which is consistent with the upper limit found for\nstar 2015448.  \\citet{nd95a} did not observe Sr, and a direct comparison\ncannot be made.  However, they did investigate other light s-process\nelements such as Y and Zr.  Both of these elements have abundance\nratios equal to or less than 0.6, and no stars on the RGB exhibit\ns-process abundances as high as is found here for [Sr\/Fe].\n\nThere are at least two possibilities that lead to the formation of\nthis star. Firstly, it may have been in a binary system with a\ncompanion that underwent unusual s-process enrichment and transfered\nmass to the star we see today.  At present we have no evidence that\nthis star is in a binary system: the individual velocities from the\n1998, 1999 and 2002 observations are consistent to within the\nmeasurement errors of \\mbox{$\\sim$10kms$^{-1}$}.  A second possibility\nis that the Sr-rich star formed out of already enriched material.\n\nAGB stars of low and intermediate mass dredge up $^{12}$C to the\nsurface via helium shell thermal pulses.  In general, if the star is\nmore massive than $\\sim$3M$_{\\odot}$, the $^{12}$C is processed to\n$^{14}$N via the CN cycle \\citep{vdm02}.  The main s-process is\nthought to occur within the $^{13}$C pocket where the neutron source\nis the \\mbox{$^{13}$C($\\alpha$,n)$^{16}$O} reaction\n(e.g. \\citealt{lk01}).  To generate the abundance ratio seen here it\nwould seem necessary that the s-processing occurs only for the light\ns-process elements, perhaps up to the peak at Zr, rather than\nprogressing through to heavy s-process elements which include Ba.\nThis was may be due to the neutron density not being high enough, as\nhigher neutron densities, assuming all other parameters being equal,\nwill produce larger amounts of heavier nuclei such as Ba or La,\nrelative to the lighter ones \\citep{smi05}.\n\nAnother source of s-processing, known as the weak s-process, occurs in\nmassive stars \\citep{phn90}. The \\mbox{$^{22}$Ne($\\alpha$,n)$^{25}$Mg}\nreaction is the source of the neutrons.  Most of the neutrons produced\nfrom this reaction are captured by the light elements, and only a\nsmall fraction are captured by the $^{56}$Fe seed nucleus, a process\nknown as ``self-poisoning''. This is the reason for the limited\nefficiency of the \\mbox{$^{22}$Ne($\\alpha$,n)$^{25}$Mg} source for\ns-process. It allows for the production of only light s-process nuclei\nwith mass numbers \\mbox{65$<$A$<$90} and Sr, with a mass number of 87,\nfalls into this range. Ba, on the other hand with mass number 137,\ndoes not and little of this element is produced by this process\n\\citep{phn90}. It is unclear whether massive stars could also produce\nthe necessary C and N abundance pattern found here, or whether star\n2015448 contains other elemental abundance patterns that may have\noriginated from these objects.\n\n\nThe r-process occurs in an environment that is rich in neutrons, and\nthe mean time between successive neutron captures in very short\ncompared with the time to undergo $\\beta$-decay.  This scenario as the\nenrichment source also requires the presence of SNe products such as\ncalcium and iron, or to have the Sr enriched material transferred but\nnot any SNe products.\n\nUsing the abundance yields for the r and (main) s-process (occurring\nin low-to-intermediate mass AGB stars) from \\citet{cam82}, and\nnormalizing to [Sr\/Fe] one finds a difference greater than one dex\nbetween predicted and observed abundances for Ba.  For the main\ns-process, the predicted abundance is \\mbox{[Ba\/Fe]=1.68}, while the\nr-process predicts \\mbox{[Ba\/Fe]=1.84}.  Both of these values are\nobviously too large, confirming the unusual nature of the source of\nthe abundance patterns found in star 2015448.  However, it should be\nnoted that AGB star yields do not explain the abundance anomalies of\nNa, Mg, Al, and O found in other stars of {$\\omega$ Cen} or globular clusters\n(see e.g. \\citealt{fen04}).  This may be due to the inadequacy of the\nmodels used to calculate yields for these objects.\n\nFrom weak s-process yields in massive stars \\citep{phn90}, using an\ninitial metallicity of \\mbox{Z\/Z$_{\\odot}$=10$^{-1}$} and mass of\n16M$_{\\odot}$, \\mbox{[Sr\/Fe]\\eq1.8} and \\mbox{[Ba\/Fe]\\eq0.7}, in good\nagreement with the Sr abundance and upper limit of the Ba one obtained\nhere.  This indicates the weak s-process is the more favorable option\nfor the source of enrichment in star 2015448.\n\n\nThis star is also highly unusual when compared to observations of\nfield stars.  Studies of n-capture elements in samples of field stars\nfor a range of metallicities show a spread in abundance ratios of Sr\nand Ba at low metallicities \\mbox{([Fe\/H]$<$-2.0)}, with most stars\nhaving [Sr\/Fe] or [Ba\/Fe] $\\leq$0.0~\\citep{mcw98, bur00, hon04}.  At\nhigher metallicities, the abundances are within $\\sim$0.5 of the solar\nabundance.\n\nA cool field giant, U Aquarii, has enhanced [Sr\/Fe] and [Y\/Fe]\nabundances, and low [Ba\/Fe] \\citep{bln79}.  This star is a faint R CrB\ntype variable star and shows no CH features, but strong $^{12}$C$_{2}$\nbands.  \\citet{bln79} concluded that U Aqr is a hydrogen deficient\ncarbon star with enhanced abundances of the light s-process elements\nSr and Y (by a factor of $\\sim$100) and little or no Ba.  It is now a\nHe-C core of an evolved star of $\\sim$1M$\\odot$ that ejected its H\nrich envelope at the He core flash. \\citet{bln79} postulated that a\nsingle neutron exposure occurred at the flash resulting in a brief\nneutron irradiation producing only the light s-process elements. A\nsimilar giant, known as Sakurai's Object \\citep{asp97} shows similarly\nenhanced C and light s-process elements, and is H deficient.  These\ntypes of stars may be responsible for the abundance pattern found in\nstar 2015448.  However, a significant difference is the C-rich nature of\nthe giants compared with the carbon depleted nature of star 2015448.\n\nOur results provide a challenging puzzle to determine the source of\nthe abundance patterns found for this star.  That said, the resolution\nof our data is inadequate to address this question.  Higher resolution\nspectra with high S\/N are needed to analyze as many elements as\npossible, in particular the s-process ones, to be able to obtain an\naccurate history for the evolution of star 2015448.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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This has many technical and conceptual advantages: Complicated higher bundles become just bundles in the category of Q-manifolds in this approach (the many structural identities being encoded in the one operator $Q$ squaring to zero), gauge transformations are generated by internal vertical automorphisms in these bundles and even for a relatively intricate field content the gauge algebra can be determined in some lines only and is given by the so-called derived bracket construction. \n\nThis article aims equally at mathematicians and theoretical physicists; each more physical section is followed by a purely mathematical one. While the considerations are valid for arbitrary highest form-degree $p$, we pay particular attention to $p=2$, i.e.~1-~and 2-form gauge fields coupled non-linearly to scalar fields (0-form fields). The structural identities of the coupled system correspond to a Lie 2-algebroid in this case and we provide different axiomatic descriptions of those, inspired by the application, including e.g.~one as a particular kind of a vector-bundle twisted Courant algebroid. \n\\end{abstract}\n\\newpage\n\\tableofcontents\n\\section{Introduction and Results}\n\\subsection{Motivation}\nThese are basic and general considerations on higher gauge theories, i.e.~gauge theories where the standard connection 1-forms of Yang-Mills (YM) gauge theories are replaced by a whole tower of such gauge fields of different form degrees. Standard Yang-Mills theories are governed by some (finite-dimensional, quadratic) Lie algebra $\\g = \\Lie(G)$, where $G$ is the  structure group of the underlying principal bundle. On the other hand, when one attempts to construct an interacting theory of 1-form gauge fields, various consistency conditions would lead one to a constraint that has to be satisfied by the interaction constants $C^a_{bc}$, namely\\footnote{There is a second condition that one obtains: The constants $\\eta_{ab}$ entering the ``free'' kinetic term in a functional is also constrained; it has to correspond to an ad-invariant scalar product.}\n\\begin{equation} \\label{Jacobi}\nC^a_{eb} C^e_{cd} + {\\text{cycl.}}(bcd) = 0 \\, .\n\\end{equation}\nThis constraint is easily recognized as the Jacobi identity of a Lie algebra. Thus, also in the other direction, one is led by itself to a Lie algebra $\\g$ which, in a second step, then can be related to a principal bundle as mentioned before. In this paper we want to follow this strategy for the much more involved context of gauge fields of form degrees up to some $p \\in \\N$, including 0-forms for completeness. \n\nAt the place of \\eqref{Jacobi} one then obtains a much more involved set of equations for the unknown parameters introduced in an ansatz for the multiple interactions. Due to the fact that we permit also the 0-forms, these are even differential equations. It is one of the goals of this paper to identify this coupled system of structural equations with an underlying mathematical structure which it corresponds to, in generalization of the incomparably simpler relation of \\eqref{Jacobi} with a structural Lie algebra $\\g$, needed for the construction of an ordinary Yang-Mills theory. \n\nAnother goal is, however, to provide one mathematical formulation of what ``consistency conditions'', often imposed in more physically oriented constructions, can mean. In particular, we aim at isolating a basic or minimal set of such requirements, which we believe that the physics community would usually want to have and which we found to be fulfilled in all the examples we know or are aware of. \n\nTwo, three remarks are in place here in this context: First, there exists a very general and elegant approach to the \\emph{deformation} of free gauge theories to interacting ones, given by a BRST-BV-formalism, cf.~\\cite{BBH00} for a review. There one starts with just the kinetic terms of the fields that one wants to consider and the related, simple-to-find free gauge transformations. The result of the procedure provides the most general possible action that contains the original terms to lowest order and that preserves the ``number'' of the gauge symmetries, although not their form; e.g.~starting with $r$ 1-form fields $A^a$ and the free (abelian) action functional \n\\begin{equation}\\label{Skin}\nS[A] = \\int_\\Sigma \\eta_{ab} \\, \\ud A^a \\wedge *\\ud A^b \\, ,\n\\end{equation}\none is led to the most general Yang-Mills functional for a quadratic Lie algebra $\\g$ of dimension $r$ (cf.~the above discussion including the previous footnote), the abelian curvature or field strength $\\ud A^a$ being effectively replaced by its non-abelian generalization $F^a = \\ud A^a + \\frac{1}{2}C^a_{bc}A^b A^c$ in the above functional. There are several advantages of this procedure: except for possible problems of global nature or of convergence (one works in the setting of formal power series) the approach is completely systematic and extensive, providing the most general (formal) deformation moreover already up to trivial ones, that can be induced by changes of coordinates on the space of fields and that is captured by BRST-exact terms vanishing in the cohomological treatment; furthermore, the resulting theory already comes in its BV-form, thus ready for a subsequent quantization. There is, however, one decisive drawback of this formalism: The calculations are generically very technical and lengthy and the resulting constraints on the parameters of the deformation do not come with a mathematical interpretation. This does not pose a problem if they are of a comparatively simple form like in Equation \\eqref{Jacobi} (accompanied by a second condition expressing the ad-invariance of $\\eta$ in this case), where one retrieves a more mathematical interpretation rather easily. But it does so in a context like the present one, where the structural equations that one finds can easily fill a page (cf., e.g., results like in \\cite{Bizdadea09,Bizdadea09b,Bizdadea10}).\n\nSecond, what is also transparent from the above YM example already, in the case of blowing up the number of equations by adding another permitted form degree, it may be useful and will prove so to separate equations that replace the defining condition of the Lie algebra, Eq.~\\eqref{Jacobi}, from objects defined on it, like the ad-invariant tensor $\\eta$, that is needed for the functional  (cf.~the discussion above) and thus the dynamics of the physical fields, but not yet for the underlying geometry of a principal bundle. Note that a metric $h$ on $\\Sigma$ entered the functional (in terms of the Hodge duality operation $*$) and in more complicated theories the gauge transformations leaving invariant a functional can and will depend on $h$ also (cf.~\\cite{DSM} for an example), while some important part of the gauge transformations, to be specified below, will prove to be independent of this additional structure. Both these observations lead us to focus on generalizations of the curvature or field strength in a first step, where neither $\\eta$ nor $h$ enters. \n\nThis leads us also to postpone questions of the global bundle structure to a separate investigation: one may want to find the generalization of the structural Lie algebra, that governs the theory already on a local level, in a first instance. \nThis restricted focus permits one to work in a local chart on the spacetime or base manifold $\\Sigma$, to replace a connection by a set of 1-forms (in the ordinary YM case, but generalized to include higher form degrees here) and to deal with the question of how to represent an (adequately generalized) connection on a possibly non-trivial (higher) bundle in a second step only.\\footnote{In fact, this investigation appeared in the mean time already, at least on the arXiv, cf.~\\cite{KS07}. The present article thus can be seen, not solely but also, as a possible introduction or motivation from the perspective of physics to \\cite{KS07},\nwhich has an essentially mathematical focus. We will describe the resulting global picture in subsection \\ref{sec:phys} below, cf., e.g., Fig.~\\ref{fig1}.} \n\n\\subsection{Structure of the article}\nThe paper is organized as follows.  In the two subsections \\ref{sec:phys} and \\ref{sec:math} we summarize the main results of the present paper. \n\nIn Section~\\ref{s:Bian} we expand in detail on the first part of what is explained in subsections \\ref{sec:phys} below. In particular, we draw the necessary mathematical conclusions from the physical requirements, which we believe to be minimal or mandatory for any higher gauge theory. The main result of these considerations is summarized in Theorem~\\ref{thm:phys1} below. In Section~\\ref{s:examQ} we unravel the geometric structure of Q2-manifolds, as they appear as what replaces the structural Lie algebra of an ordinary YM gauge theory when considering a higher gauge theory with 0-form, 1-form, and 2-form gauge fields. They are in equivalence with Lie 2-algebroids or 2-term $L_\\infty$-algebroids. In Section~\\ref{s:examQ} we provide, in particular, a description generalizing the one of a Lie 2-algebra as a crossed module, in the Appendix~\\ref{s:Linf} we complement this by a descriptions in terms of multiple brackets as usual for $L_\\infty$-algebras (cf., also, subsection \\ref{sec:math} below).\n\nIn Section~\\ref{s:gauge} we take another step in the construction of a ``physically acceptable'' higher gauge theory, focusing on the gauge symmetries in this context. The main result of this investigation is summarized in Theorems~\\ref{thm:phys2} and \\ref{theo:Lie}) below. Having been led to regard derived brackets in this context, we take another view on Q2-manifolds or Lie 2-algebras from this perspective in the subsequent Section~\\ref{sec:derived}. This motivates the definition of what we call a V-twisted Courant algebroid. We derive its elementary properties in that section and provide sufficient conditions under which it is equivalent to a Lie 2-algebroid again. \n\nAppendix~\\ref{s:Linf} reviews briefly the relation between Q-structures and $L_\\infty$-algebras\/ algebroids in general.  \n\n\\subsection{Chronology of the results}\\label{s:hist\n\nSections \\ref{s:Bian}, \\ref{s:examQ}, and \\ref{s:gauge} were written already in 2005, Section  \\ref{sec:derived} between 2005 and 2008 (as visible also by several talks of TS and a partial distribution of preliminary versions). The Introduction and the Appendix, as well as some fine-tuning of the remainder, is from the end of 2013\/beginning of 2014, where we finally finished the article. For TS the results of this work as of 2005 were among the main motivations for starting his work with Alexei Kotov on characteristic classes and Q-bundles, the first results of which appeared as a preprint in 2007 on the arXiv and is now published right behind this article in the present journal \\cite{KS07}. We attempted to leave the Sections  \\ref{s:Bian}, \\ref{s:examQ}, \\ref{s:gauge}, and \\ref{sec:derived} as much as possible in their original form from the time of 2005 to 2008, including an updated perspective only in the two subsections to follow (to which we then also moved important formulas and statements from those older sections, like Equation \\eqref{ideal} and Theorem \\ref{thm:phys1}, respectively)\n\nFor what concerns our work summarized in section  \\ref{sec:derived}, we found out later that there were parallel developments arriving at a similar notion of a vector-bundle twisted Courant algebroid as the one given in the present paper. It is interesting to see that quite different motivations and starting points led to the same or similar objects of interest at about the same time.  In the following, we comment on the relation in more detail: Chen, Liu and Sheng have introduced $E$-Courant algebroids in \\cite{CLS08}.  An $E$-Courant algebroid as defined by them is a $V$-twisted Courant algebroid as defined in this paper after identifying their vector bundle $E$ with our $V$ iff in addition their inner product is surjective to this vector bundle.  Conversely a $V$-twisted Courant algebroid is a $V$-Courant algebroid in their sense iff the anchor $\\rho$ factors through $\\mathfrak{D}V$, the derived module of $V$.  Similarly, Li-Bland introduced  an $AV$-Courant algebroid in \\cite{LiB11}. This is a  $V$-twisted Courant algebroid in our sense and, conversely, every $V$-twisted Courant algebroid is an $AV$-Courant algebroid iff the anchor map factors through a Lie algebroid $A$ and the action of the twisted Courant algebroid on $V$ induces an action of $A$ on $V$.\n  \n\n\n\\subsection{Acknowledgments}\nWe thank the TPI in Jena and in particular A.~Wipf for the support of our work in 2005 and 2013, the ESI in Vienna for support in 2007, as well as the ICJ in Lyon for likewise support in 2008 and 2013. \nWe also thank the anonymous referee for helpful suggestions, such as a change of the subtitle. T.S.~thanks A.~Kotov for many valuable discussions during our stimulating parallel work. \n\n\\subsection{Mathematical framework for physical theories} \\label{sec:phys}\nLet us now describe our approach in some detail and summarize the main results of the paper. As mentioned above, we consider a tower of differential forms. Let us denote them collectively by $A^\\alpha$ and call them the ``gauge fields'' of the theory under investigation. So $A^\\alpha$ collects differential forms of various form degrees, in general at each level of a different number: $A^\\alpha = (X^i, A^a,B^D, \\ldots)$, where $(X^i)_{i=1}^n$ are $n$ 0-forms (or scalar fields), $(A^a)_{a=1}^r$ are $r$ 1-forms (called, in a somewhat misleading way, ``vector fields'' in the physics literature), $(B^C)_{C=1}^s$ are $s$ 2-forms etc., up to some highest form degree $p$. At this stage of the consideration we do not want to attach any geometrical interpretation to those differential forms yet, treating them in a most elementary way possible in the first step and believing in the power of ``physically oriented'' argumentation, supposed to lead us to a geometrically interesting picture more or less by itself in the end. \n\nNext we make a most general ansatz for the ``field strengths'' $F^\\alpha$ compatible with the form degrees and not using any additional structures (like a metric $h$ on $\\Sigma$), since we believe that these objects, which in the end should represent in one way or another some generalized curvature(s), should not depend on anything else but the data specified up to here. Most physicists would agree, moreover, that $F^\\alpha$ should start with the exterior derivative acting on the gauge fields, $F^\\alpha = \\ud A^\\alpha + \\ldots$, and that this should be complemented by terms, represented by  the dots, that are polynomial in the gauge fields of form degree 1 and higher, but, to stay as general as possible, with a priory unrestricted coefficient functions of the scalar fields $X^i$. This ansatz can be considered as still too restrictive, however. First, one may possibly want to replace the first term in $F^\\alpha$ by a coefficient matrix $M^\\alpha_\\beta$---depending on the $X^i$s---times $\\ud A^\\beta$; second, one may want to permit also $\\mathcal{A}^\\beta \\wedge \\ud A^\\gamma$-contributions to the remaining  terms indicated by the dots above. However, as we will argue in the main text, both of these apparent generalizations of our ansatz can be mapped in a first transformation to a situation of the simpler form above, provided only that the matrix $M^\\alpha_\\beta$ is invertible.\\footnote{Correspondingly, also theories that have field strengths containing Chern-Simons type of contributions can be tackled by the present formalism and, e.g., the Theorem \\ref{thm:phys1} below is applicable also in such situation. This is exemplified for instance in \\cite{SLS14}.} Thus, anticipating this argument, without loss of generality we can thus assume the field strengths to have the form\n\\begin{equation} \\label{curv}\nF^\\alpha = \\ud A^\\alpha + t^\\alpha_\\beta(X) A^\\beta + \\tfrac{1}{2} C^\\alpha_{\\beta \\gamma}(X) A^\\beta \\wedge A^\\gamma +\\tfrac{1}{6} H^\\alpha_{\\beta \\gamma\\delta}(X) A^\\beta \\wedge A^\\gamma \\wedge A^\\delta + \\ldots ,\n\\end{equation}\nwhere the coefficients or coefficient functions  $t^\\alpha_\\beta$,  $C^\\alpha_{\\beta \\gamma}$, $H^\\alpha_{\\beta \\gamma\\delta}$, $\\ldots$ are assumed to be fixed for defining the theory (on a local level and at this stage of constructing the theory). Their choice is now assumed to be constrained by some kind of ``physical consistency condition'', like one that would lead to \\eqref{Jacobi} in the case of only 1-form gauge fields $A^a$. It is thus now our task and declared goal to specify one or several possible consistency requirements, motivated by a more or less physical argument, to display the resulting constraints on the coefficients in some explicit form, and, in a final step, to give them some algebraic-geometrical meaning so as to unravel what replaces the structural Lie algebra $\\g$ of a standard YM theory in this much more general setting. \n\nBefore turning to these issues two more remarks on the nomenclature and the setting, however. We chose to include scalar fields (i.e.~0-forms) into that what we call gauge fields. In some sense this is unconventional, since they normally describe matter, while the conventional gauge fields (the 1-forms) represent the interaction forces. There are two reasons for this: First, it is suggested by the study of the Poisson sigma model (PSM) \\cite{Ikeda94, Str94}: As argued in  \\cite{Str04b} and \\cite{KS07}, the PSM can be seen as a ``Chern Simons (CS) theory''  of the Lie algebroid $T^*M$ associated to a Poisson manifold $M$; like the CS-theory corresponds to the Pontryagin class, also the PSM, and more generally all AKSZ sigma models \\cite{AKSZ}, correspond to characteristic classes (cf.~also \\cite{FRS13}). In this context, all the tower of differential forms correspond to a single map or section of an adequate higher bundle, and it in particular also includes the 0-forms. \n\nThe second reason for including the 0-forms is that even if one wants to consider the 0-forms $X^i$ as matter fields, it has advantages to include them right away.\\footnote{In this case the ``1-form field strength'' $F^i \\equiv \\ud  X^i - \\rho^i_a(X) A^a$ should be rather interpreted as a (generalized) covariant derivative. All the considerations concerning the ideal $\\I$ below will apply also to those.} Suppose one considers an ordinary YM-connection with scalar fields being a section of an associated (vector) bundle induced by a representation $\\rho$ of $\\g$\non a vector space or, more generally, by a $\\g$-action on a manifold $M$ that serves as a fiber of the associated bundle. In a local chart, the connection and the section correspond to a collection of 0-forms and 1-forms, $X^i$ and  $A^a$, respectively. On the other hand, $E:=\\g \\times M\\to M$ becomes a Lie algebroid in a canonical way and the data of the fields $X^i$ and $A^a$ are the same as a vector bundle morphism from $T\\Sigma \\to E$; moreover, the covariant derivative of the scalar fields and the curvature of the connection combine into a field strength map, cf.~\\cite{Str04b,Str09,KS07}. There are many more Lie algebroids\n(cf., e.g., \\cite{Mack87,CaWe99})\n than those coming from a Lie algebra action $\\rho$; if one intends to treat the 0-forms in a second step only, one misses all these additional possibilities. From a more physical perspective,  scalar fields are sometimes seen to deform the structural identities underlying the gauge fields of degrees at least one in a highly nontrivial way (cf, e.g., \\cite{Str04b} or some supergravity theories). Thus also from this perspective it seems reasonable to not exclude a geometrical explanation of such structures (in terms of some Lie algebroids for example) from the outset by the restriction to a two-step procedure.\n \n \n \n \nThe second remark concerns the nomenclature. We have chosen to call the $A^\\alpha$-fields \\emph{gauge fields} and the combinations $F^\\alpha$ as in \\eqref{curv} \\emph{field strengths}, following a ``physics oriented'' wording. We in particular avoid to call them generalized connections and curvatures, as it may seem more adequate from a (superficial) mathematical perspective. The reason for this is the following. As already obvious from the previously given example of the PSM, that what replaces the structural Lie algebra $\\g$ (as well as its integration $G$) of an ordinary principal bundle can and in general will be a bundle itself, in fact, even some kind of higher bundle. So, the fibers of what replace principal and associated bundles will be bundles themselves (and it may be useful  to choose auxiliary connections for technical reasons, that must be distinguished strictly from the ``dynamical'' gauge fields $A^\\alpha$). But even worse, a Poisson manifold for example does not always give rise to an integrating Lie groupoid, there can be obstructions in the integration process \\cite{CrF01}, while still the above mentioned characteristic classes as well as the PSM do exist as meaningful objects for any Poisson manifold $M$. To cover all these examples in a common mathematical framework it thus is indicated to work with bundles (over the base $\\Sigma$ or its tangent space $T\\Sigma$) where one does not need the analogue of the structural Lie group $G$, but it is sufficient to work with its Lie algebra $\\g$. In the ordinary setting this is the so-called Atiyah algebroid $A\\to T\\Sigma$ associated to a principal bundle $P\\to \\Sigma$, where $A$ can be identified with $TP\/G$ and, as a bundle over $T\\Sigma$, its fiber is isomorphic to $\\g$ only (cf.~\\cite{KS07} for further details). Now a connection in $P$, i.e.~an ordinary gauge field in physics language, corresponds to a \\emph{section} in $A$, to be distinguished from connections in $A$, that exist as well as on any bundle. (The generalization of $A$ will be what was called a $Q$-bundle in \\cite{KS07}.) These potential sources of a possible confusion with other natural connections lead us to avoid calling $A^\\alpha$ a generalized connection.\n\nWe now come back to our principal goal, finding conditions for the restriction of the coefficients appearing in \\eqref{curv}. The approach nearest at hand may seem to look at gauge transformations and to require some generalized form of invariance of the field strengths. This, however, has a decisive disadvantage: There is a lot of freedom in defining gauge transformations, and, as examples show (cf., e.g., the Dirac sigma model (DSM) \\cite{DSM}), they may depend on several additional structures not needed for the definition of the above field strengths. \n\n\nThe main idea for the departure is, therefore, to use a generalization of the Bianchi identities. Normally, they would contain a covariant derivative, that we have not yet defined as well, not yet knowing the adequate algebraic-geometric setting to use. Not wanting to use any additional new parameters\/coefficients, we thus will want to formulate a minimalistic version of Bianchi that at least most people in the field would agree with. Regarding the standard Bianchi identity for an ordinary YM theory, $\\ud F^a + C^a_{bc}A^b\\wedge F^c = 0$, we note that the application of the exterior derivative on the field strength becomes ``proportional'' to the field strength itself, albeit with a field dependent ``prefactor''. \n\nMathematically we express this as follows: We require that the exterior derivative $\\ud$ applied to any monomial containing a field strength $F^\\alpha$, i.e.~terms of the form $\\ldots \\wedge  A^\\alpha \\wedge F^\\beta \\wedge A^\\gamma \\wedge \\ldots$, yields expressions that again contain at least one field strength. In other words, we regard the ideal $\\I$ generated by expressions of the form \\eqref{curv} within the (graded commutative, associative) algebra $\\mathcal{A}$ generated by abstract elements $A^\\alpha$ and $\\ud A^\\alpha$. The condition we require is then simply\n\\beq \\boxed{\\ud \\I \\subset \\I } \\, ,\\label{ideal} \\eeq\ni.e.~$\\I$ has to be what is called a differential ideal. As innocent and weak as this condition may seem, it turns out to restrict the coefficient functions of \\eqref{curv} already very drastically, at least if the dimension $d$ of spacetime $\\Sigma$ exceeds the highest permitted form degree $p$ of $A^\\alpha$ by two, i.e.~$d \\geq p+2$. In that case, we will find that \\eqref{ideal} holds true iff the following vector field squares to zero:\n\\begin{equation} \\label{Q1}\nQ=  \\left(t^\\alpha_\\beta(x) q^\\beta + \\frac{1}{2} C^\\alpha_{\\beta \\gamma}(x) q^\\beta q^\\gamma +\\frac{1}{6} H^\\alpha_{\\beta \\gamma\\delta}(x) q^\\beta  q^\\gamma q^\\delta + \\ldots\\right) \\dfrac{\\partial}{\\partial q^\\alpha} .\n\\end{equation}\nHere we introduced for each gauge field $A^\\alpha$ an abstract coordinate $q^\\alpha$, carrying a degree that equals the form degree of  $A^\\alpha$ (for the degree zero variables we also use $q^i \\equiv x^i$ to distinguish them from the others, since they can appear non-polynomially).\nThese coordinates commute (anticommute) iff the respective differential forms commute (anticommute). Also, since by its definition, $F^\\alpha = \\ud A^\\alpha + \\ldots$ has a form degree that exceeds the one of $A^\\alpha$ by precisely one, the vector field $Q$ is an odd vector field of degree $+1$. In general, the condition\n\\beq\\label{Qsquare}\nQ^2 \\equiv \\tfrac{1}{2} [Q,Q] = 0\n\\eeq\nsubsummarizes a whole list of equations and in particular reduces precisely to \\eqref{Jacobi} in the case of coordinates of only degree one. \n\nThis observation turns out to be rather easy to prove, but so general that we consider it worth being called a fundamental theorem in the context of higher gauge theories:\n\\begin{thm} \\label{thm:phys1} If $\\dim \\Sigma \\geq p+2$ the generalized Bianchi identities \\eqref{ideal} hold true if and only if the vector field \\eqref{Q1} squares to zero, $Q^2=0$.\n\\end{thm}\nThus any higher gauge theory is associated to what is called a differential graded manifold or a Q-manifold \n$\\CM$. \n(In the case of an ordinary YM theory this corresponds to a reformulation of the Lie algebra $\\g$ in terms of its Chevalley-Eilenberg complex; in this case, simply $ (C^\\infty(\\CM), Q) \\cong (\\Lambda^\\cdot \\g^*, \\udCE)$). The most important part of the above theorem is not that a Q-manifold, as a generalization of a Lie algebra, gives rise to some generalized Bianchi identities, but the reverse direction: Even if Bianchi identities are formulated in as weak a version as (\\ref{bian}), one \\emph{necessarily}\ndeals with a Q-manifold on the target if one observes it or not. This is the strength of the above observation. \n\nTogether with \\cite{BKS,KS07} (and the considerations on the gauge symmetries to follow) this paves the way for an elegant, general, and simple picture of higher gauge theories. Their field content (and symmetries) is compactly described by a section $a$ in a Q-bundle (cf.~Fig.~\\ref{fig1} below):\n\\begin{figure}\n$$\\begin{xy}\/r0.15pc\/:\n  (-50,-70)*{}; (50,-70)*{} **\\crv{(0,-60)}+(0,-7)*\n  (T[1]\\Sigma,\\ud)};\n  (-50,0)*{}; (50,0)*{} **\\crv{(-20,20) & (20,-20)};  {\\ar_{\\textstyle a}(-10,-65)*{};(-10,2)*{}};\n  (-50,-50)*{}=\"A\"; (50,-50)*{} **\\crv{(0,-40)};    (72,30)*{(\\CM_{tot},Q_{tot})};\n  (50,-50)*{}; (50,50)*{} **\\crv{(40,-25)&(60,25)};\n  (50,50)*{}; (-50,50)*{} **\\crv{(0,60)};\n  \"A\"; (-50,50)*{} **\\crv{(-60,-25)&(-40,25)};\n  (-35,-47.5)*{};(-35,52.5)*{} **\\dir{-};\n  {\\ar(-60,10)*{(\\CM,Q)};(-35,20)*{}};\n\n\n\n  (-48,20)*{}; (52,20)*{} **\\crv{~*=<4pt>{.} (-20,30) & (20,10)};  {\\ar_{\\textstyle \\delta a}(0,0)*{};(0,20)*{}};\n\\end{xy}\n$$\n\\caption{\\label{fig1}  The resulting geometry governing all higher gauge theories: The generalization of a connection, called a \\emph{gauge field} in this work, corresponds to a section $a$ of a graded bundle $\\CM_{tot}$. The bundle is a Q-bundle and (the principal part of) gauge transformations correspond to vertical inner automorphisms. The section is a Q-morphism iff the generalization of curvature, called the \\emph{field strength} here, vanishes. In physical theories, however, it is almost exclusively those gauge fields which have a non-vanishing field strength that, up to gauge invariance, are of interest.}\n\\end{figure}\nThe tangent bundle $T\\Sigma$ of spacetime $\\Sigma$ can be viewed as a graded manifold when fiber-linear coordinates are regarded as carrying degree 1, functions on this graded manifold $T[1]\\Sigma$ correspond to differential forms, and the nilpotent, degree 1 de Rham differential $\\ud$ equips it with a Q-structure. The Q-manifold $(\\CM,Q)$ that we introduced above in the context of (\\ref{Q1}) can be viewed as a target if we interpret the tower of gauge fields $A^\\alpha$ as corresponding to a degree-preserving map from $T[1]\\Sigma$  to \n$\\CM$.\\footnote{We will explain all this in more detail in the main text, but we want to give an overview of the resulting picture already as this point as a guide for what we are going to obtain or derive as a final framework.} As a straightforward generalization of the considerations in \\cite{BKS} this can be reinterpreted as a section $a$ in a trivial Q-bundle, $(\\CM_{tot},Q_{tot}) := (\\CM \\times T[1]\\Sigma, Q+\\ud)$,  an observation that proves essential in the context of describing the gauge symmetries (in \\cite{BKS} for the PSM, here for general higher gauge theories). One of the observations fundamental to \\cite{KS07} is, on the other hand, that this picture generalizes also to the twisted case, i.e.~that a connection in an ordinary principal bundle corresponds just to a section of a (possibly non-trivial) Q-bundle. \n\nWe add a remark in parenthesis: The section $a \\colon T[1]\\Sigma \\to \\CM_{tot}$, corresponding to our gauge fields, is a section of the category of ($\\N_0$-)graded manifolds, and, not, in the sense of the category augmented by a Q-structure. It becomes an ``honest'' section of the category of Q-bundles, i.e.~a Q-morphism, iff all the field strengths $F^\\alpha$ vanish. As nice as this may seem from a purely mathematical perspective, as inadequate it is from the one of physics: It would exclude all the particles (like photons, gluons, etc) for which gauge fields were introduced in physics in the first place. In fact, even in topological models, such as the PSM, the DSM, or the AKSZ sigma models, the gauge fields become Q-morphisms only ``on-shell'', i.e.~on the level of the Euler-Lagrange equations of the corresponding action functionals, cf.~\\cite{BKS,DSM,KS08} for the proofs, respectively.\n\nLet us now address the gauge symmetries. Similarly to the general ansatz (\\ref{curv}), we make one for the gauge symmetries\n\\begin{equation}\\label{gaugedelta0}\n\\delta^{(0)}_{\\epsilon} A^\\alpha = \\ud \\epsilon^\\alpha +  \\left(s^\\alpha_\\beta(X)  + D^\\alpha_{\\gamma \\beta}(X) A^\\beta  +\\tfrac{1}{2} G^\\alpha_{ \\gamma\\delta\\beta}(X) A^\\gamma \\wedge A^\\delta + \\ldots \\right)\\wedge \\epsilon^\\beta,\n\\end{equation}\nfor some collection $\\epsilon^\\alpha$ of differential forms of degree up to  $p-1$, where we may allow the total gauge transformations to also \ncontain a part proportional to $F^\\alpha$ again:\n\\begin{equation} \\label{gaugedelta}\n\\delta_{\\epsilon} A^\\alpha = \\delta^{(0)}_\\epsilon A^\\alpha + O(F^\\alpha) \\, . \n\\end{equation} \nIt is important to remark that the terms $O(F^\\beta)$ have to be in the ideal $\\I$ for the following to be still true. This still includes higher gauge theories as they appear as a part of gauged supergravity for example, cf.~\\cite{SSW11,SLS14,SKS14}, and where they play an important role, but excludes terms containing e.g.~$*F^\\alpha$ (as they appear in the DSM \\cite{DSM}) or terms containing ``contractions'' of $F$ (with the inverse of the part of $A$ that may describe a vielbein) as they typically appear in the gravitational part of supergravity theories. In analogy to (\\ref{ideal}), we now require that also our gauge transformations do not leave the ideal \n$\\I$ generated by the field strengths,\n\\beq \\boxed{\\delta_\\varepsilon \\I \\subset \\I } \\, .\\label{deltaI} \\eeq\nThe result we obtain, illustrated and proven in detail in the main text, is the following (cf.~also Fig.~\\ref{fig1})\n\\begin{thm} \\label{thm:phys2}\nProvided $\\dim \\Sigma \\geq p+1$ and $Q^2=0$, infinitesimal gauge transformations of the form (\\ref{gaugedelta}) leave the field strength ideal $\\I$ invariant, Eq.~\\eqref{deltaI}, iff their principal part $\\delta^{(0)}_\\varepsilon$ corresponds to vertical inner automorphisms of the Q-bundle $(\\CM_{tot},Q_{tot})$. Moreover, if the infinitesimal gauge transformations are of the more restricted form (\\ref{gaugedelta0}), $\\delta_\\varepsilon \\equiv \\delta^{(0)}_\\varepsilon$, one does not need to require $Q^2=0$, but one can conclude it together with the generalized Bianchi identities (\\ref{bian}). \n\\end{thm}\nLet us briefly explain what inner automorphisms of a Q-manifold should be thought of to our mind. An automorphism should be degree preserving diffeomporphism that leaves invariant the vector field $Q$. Infinitesimally this corresponds to degree zero vector field annihilated by $\\ad_Q$. It is an immediate consequence of (\\ref{Qsquare}) that $\\ad_Q$ squares to zero as well. We call an inner automorphism of a Q-manifold one that is generated by the image of $\\ad_Q$, i.e.~by vector fields of the form  $\\ad_Q(\\epsilon)$, where $\\epsilon$ is a degree -1 on the Q-manifold. If the Q-manifold is an ordinary Lie algebra (shifted in degree by one) $\\g[1]$, this corresponds to the usual definition of inner automorphisms, i.e.~to  the adjoint action of the integrating group $G$ on $\\g=\\Lie(G)$ (generated infinitesimally by the left action on itself). For $(T[1]\\Sigma, \\ud)$ this definition of inner automorphism gives infinitesimally precisely the Lie derivatives of vector fields on $\\Sigma$, so they correspond to the canonical lift of a diffeomorphism of $\\Sigma$ to its tangent bundle. \n\nGauge transformations of an ordinary YM-theory are generated by vertical automorphisms of the underlying principal bundle $P$. They correspond precisely to inner vertical automorphisms in the above sense for the associated Q-bundle $T[1]P\/G$. The advantage of the present formulation is that like this it generalizes in a straightforward way to \\emph{all} higher gauge theories. \n\nSome further explanations are in place: First, additional structures entering a particular theory can certainly constrain the symmetries permitted in the respective case. For example, in the Poisson sigma model or, more generally, the AKSZ sigma models, the inner automorphisms also have to preserve the symplectic form defined on the fibers and that are compatible with the Q-structure on the fibers. For the PSM, for example, this implies that its gauge symmetries are generated infinitesimally by the Lie subalgebra $\\Omega^1_{closed}(M)$ of $\\Gamma(T^*M)$ ($T^*M$ viewed upon as the Lie algebroid of the Poisson manifold $M$ in this case). Second, there are cases where the gauge transformations including the additional terms of the form $O(F)$ in (\\ref{gaugedelta}) can be given a similarly good geometrical interpretation. For example, as will be shown in \\cite{SLS14}, for the higher gauge theory sector of the (1,0) superconformal model in six spacetime dimensions \\cite{SSW11}, the $O(F)$-part is essential for an off-shell closed algebra of gauge transformations on the one hand. On the other hand, in this case the gauge transformations (\\ref{gaugedelta}) correspond to nothing but inner automorphisms of another Q-bundle associated to the one in Fig.~\\ref{fig1}, where the typical fiber $(\\CM,Q)$ is replaced by $(T[1]\\CM, \\ud_{\\CM} \n+ \\L_Q)$ (cf.~also \\cite{KS07,SaS13}).\\footnote{Any Q-bundle $(\\CM_{tot},Q_{tot}) \\to (\\CM_1,Q_1)$ can be lifted to a tangent Q-bundle $(T[1]\\CM_{tot}, \\ud_{\\CM_{tot}} \n+ \\L_{Q_{tot}}) \\to (T[1]\\CM_1, \\ud_{\\CM_1} + \\L_{Q_1})$. Using the canonical 0-section $\\sigma \\colon \\CM_1\\to T[1]\\CM_1$, we can pull back $T[1]\\CM_{tot}$ to become a bundle over $\\CM_1$, which turns out to be compatible with the respective Q-structure\n\n}\n\n\n\nThird, it is important to be aware that $Q_{tot}$ projects to $\\ud$ on the base of the bundle, and it is like this that $\\ud \\varepsilon^\\alpha$ is generated in (\\ref{gaugedelta0}), (\\ref{gaugedelta}); this becomes possible only by going to the picture of a bundle, even in the case of a local treatment, where the bundle (at least when defined as in \\cite{KS07}) is trivial. \n\nIn the present description, the vector fields generating the gauge transformation always form a Lie algebra. They are vertical vector fields of degree zero of the form $\\ad_{Q_{tot}}(\\epsilon)$ and they close with respect to the usual commutator of vector fields. They are parameterized by vertical vector fields $\\epsilon$ of degree minus one \\emph{up to $\\ad_{Q_{tot}}$-closed} contributions. On the vector fields $\\epsilon$ one discovers that one has an induced bracket of the form \n\\begin{equation}\\label{derQtot}\n[\\epsilon,\\epsilon']_{Q_{tot}} \\equiv [\\ad_{Q_{tot}}(\\epsilon), \\epsilon'] \\, ,\n\\end{equation} called the ``derived bracket'' (following Cartan and \\cite{YKS03en}). For example, if one considers the Q-manifold $(\\g[1], \\udCE)$, the derived bracket just reproduces the Lie algebra bracket. While in general, however, this bracket is a Leibniz--Loday algebra only and not necessarily antisymmetric, it becomes so for the coexact elements, i.e.~for those $\\epsilon$ where one considers the quotient by closed contributions (which do not change the gauge transformation, as argued above). In this way one obtains the Theorem  \\ref{theo:Lie} in the main text.\\footnote{To simplify the notation in that section, we called the total Q-structure simply $Q$ and renamed the Q-structure on the fiber by $Q_2$.}\n\nIn a conventional treatment of gauge symmetries (i.e.~along the lines as described e.g.~in \\cite{HT}), they form a so-called \\emph{open} algebra for a generic higher gauge theory. This is not in contradiction to the present, to our mind more elegant formulation of its gauge symmetries. To resolve the apparent paradox, it is useful to consider the PSM---as a simpler toy-model, showing already all the essential features of this behavior. In \\cite{BKS} it is shown, in particular, that one can obtain the usual, only on-shell closed algebra of gauge symmetries of the PSM from a incomparably shorter calculation in the above-mentioned framework. This generalizes to the context of any higher gauge theory. For instance for a higher gauge theory containing 0-forms $X^i$, 1-forms $A^a$, and 2-forms $B^C$ one obtains in this way the result of Proposition \\ref{prop:onshellclosed} below. We also will comment further on this issue in \\cite{SLS14,GSS14}. \n \nWhile one can still discuss the usefulness or necessity  of using the exterior derivative $\\ud$ in vector analysis or $ \\udCE$ for Lie algebras, there will be no doubt in the case of $p \\ge 2$; already for $p=2$ there is, in general, a whole list of coupled differential equations satisfied by the structural quantities, cf.\\ Eqs.~\\eqref{B1}--\\eqref{B7} in the main text below, which are to be used in almost every calculation related to such a gauge theory and which can be replaced and subsummarized in a single operator $Q$ squaring to zero. The technical usefulness becomes particularly obvious in the above-mentioned context of the calculation of the commutator of gauge symmetries, but also already for the concrete form of the Bianchi identities (cf., e.g., the derivation of Eqs.~\\eqref{BFi}--\\eqref{BFB} in the main text below).\n\nLet us briefly come back to Fig.~\\ref{fig1} and the corresponding two theorems, Thm.~\\ref{thm:phys1} and \\ref{thm:phys2}. Not only the base of the Q-bundle is in fact a bundle in ordinary differential-geometric terms, but even worse so for the fibers: the Q-manifold $(\\CM,Q)$ isomorphic to the  typical fiber is generically a rather intricate (higher) bundle structure itself. We will come back to this issue in more detail in subsection \\ref{sec:math} below. What we consider important in this context is that the infinite-dimensional space of gauge fields is described as sections in \\emph{finite-dimensional} (graded) bundles. This gives additional mathematical control. The situation can be compared with the interest in Lie algebroids (for a definition cf., e.g., \\ref{def:Loid}): While for infinite-dimensional Lie algebras topological questions usually play an important role and are intricate to deal with, the situation simplifies greatly when this infinite-dimensional Lie algebra comes from the sections of a Lie algebroid. \n\nLikewise, the above operator $Q$ (or also $Q_{tot}$) must not be confused also with a BV- or BRST-operator. Although the BV-BRST operator $Q_{BV}$  also squares to zero, $Q_{BV}^2=0$,  and is of degree +1, it is defined over the \\emph{infinite-dimensional} graded manifold of (gauge) fields, ghosts, and their antifields, and, more importantly, it, in general, needs many more structures to be defined, namely those needed for an action functional that we have not yet addressed much up to here (and for which there are many options still open).\\footnote{Correspondingly, also the discovery of the role of the derived bracket in the context of the gauge symmetries, cf.~Thm.~\\ref{theo:Lie}, is not or at most vaguely related of the role of the derived bracket in the BV-context as observed in \\cite{YKS97}.} \n\n\nThere are very particular models of topological nature, like the Poisson sigma model or, more generally, the AKSZ-sigma models \\cite{AKSZ}, on the other hand, where there \\emph{is} a close relation of the two Q-structures $Q$ and $Q_{BV}$: $Q$ corresponds to a defining Q-structure on the target of the sigma model, which is also equipped with a compatible symplectic form $\\omega$, $\\ud$ is a Q-structure on its source $T[1]\\Sigma$, all very much like in Fig.~\\ref{fig1}. Then $a \\colon T[1]\\Sigma \\to \\CM$ is no more degree preserving (so as to capture ghosts and antifields), but a general supermap and $\\Omega_{BV}$ is, roughly speaking, the ``difference'' of $Q$ and $\\ud$ induced on this space of supermaps $\\CM_{BV}$. Using $\\omega$ and a canonical measure on $T[1]\\Sigma$, one then induces also a (weakly) symplectic form $\\omega_{BV}$ on $\\CM_{BV}$ and shows that $Q_{BV}$ is Hamiltonian, i.e.~$Q_{BV}=\\{ S_{BV}, \\cdot \\}_{BV}$, where $S_{BV}$ is the BV-action, extending the classical action of the PSM and AKSZ-sigma model, respectively (cf., e.g., \n\\cite{AKSZ,Catt01,Royt06}). \n\n\nThis changes quite drastically, however, in the case of non-topological, physically relevant theories; already for an ordinary YM gauge theory the BV operator is not constructed in such a simple way (unfortunately). Even for \\emph{relatively} simple generalizations like (for the PSM) the topological DSM \\cite{DSM} or (for YM) the non-topological Lie algebroid Yang-Mills theories \\cite{Str04b}, the BV operator is even not known at the present time; however, in both cases the target Q-structures are very well known, corresponding to Lie algebroids in both cases. And, again, the field content for any such a theory, and, more generally, for any higher gauge theory, its described compactly by sections in the finite-dimensional Q-bundle as sketched in Fig.~\\ref{fig1}. \n\nAlthough originally intended to do so, we will not address the construction of action functionals within this paper, leaving it for eventual later work (but cf.~also \\cite{Str04b,KS08,SLS14}). \n\n\n\\subsection{Further mathematical results} \\label{sec:math}\nHere we briefly summarize the ideas and results contained in the more mathematical sections \\ref{s:examQ} and \\ref{sec:derived}. It mainly consists in reformulating supergeometrically defined objects appearing in the Q-bundles as potential fibers in \nmore conventional algebraic-geometrical terms. \n\nOur graded or supermanifolds, on which the homological vector fields $Q$ are defined, always will be $\\N_0$ graded manifolds, being simultaneously supermanifolds by means of the induced $\\Z\/2\\Z$-grading. These are conventionally called simply\n N-manifolds and are defined as follows (for further details see also \\cite{Royt02,Gru09}): An N-manifold is a ringed second countable Hausdorff topological space $\\CM=(M,{C^\\infty})$ where the structure sheaf, which by abuse of notation we also denote as ${C^\\infty}$, is locally of the form smooth functions in tensor product with the exterior algebra in the odd coordinates tensor product with the polynomial algebra in the even coordinates not of degree 0. This implies that it \n consists of a tower of affine fibrations.  In degree 0 we just have a smooth manifold $M$.  In degree 1, also after truncation of a higher degree N-manifold, we have an odd vector bundle $E[1]$, i.e.\\ the fiber-linear coordinates of $E$ are declared as functions of degree 1.  Their parity is odd and thus the function algebra are the $E$-forms.  The topological structure of such an N-manifold comes from the smooth base only.  \n\nMorphisms of N-manifolds are morphisms of ringed spaces that preserve the $\\N_0$-grading of the structure sheaf.  Coordinate changes are particular local isomorphisms. Like this one finds e.g.~that the coordinates of degree 1 transform as the coordinates of a vector bundle so that any N-manifold $\\CM$  with degrees up to 1, i.e.~an N1-manifold,\\footnote{In general, we call an N$k$-manifold an N-manifold, where the highest coordinate degree is $k$. \\label{fn:Nk}} can be canonically identified with vector bundles $E\\to M$, $\\CM \\cong E[1]$.  \n\nIn degree 2, coordinate changes contain, however, also affine terms beside the terms one would expect for a vector bundle.  Globally an N2-manifold $\\CM$ is an affine bundle over the base $E[1]$ modeled after the pullback $\\pi_1^*V[2]$ of a vector bundle $V\\to M$ where $\\pi_1\\colon E[1]\\to M$.  Since the fibers $V_x$ for $x\\in M$ of this bundle are contractible, there exists a (non-canonical) global section (splitting) that identifies the affine bundle $\\CM$ with the graded vector bundle $E[1]\\oplus V[2]$ over the base $M$. In all of our theorems on Q2-manifolds, i.e.~N2-manifolds equipped with a degree +1 homological vector field, we will need such a splitting permitting us to identify the underlying graded manifold with the direct sum of two vector bundles $E\\to M$ and $V\\to M$, $\\CM \\cong E[1]\\oplus V[2]$. \n\nAs mentioned already in the previous subsection, the definition of an ordinary Lie algebra $\\g$ is equivalent to a Q1-manifold over a point, i.e.~with coordinates of degree 1 only. The corresponding N1-manifold gives the vector space (vector bundle $E$ over a point), $\\CM \\cong \\g[1]$ (as graded vector spaces), the functions $C^\\infty(\\CM)$ are identified with elements of $\\Lambda^\\cdot \\g^*$, and the degree +1 vector field $Q$ corresponds precisely to the Chevalley-Eilenberg differential, $Q\\cong \\udCE$, from which one certainly can also reconstruct the Lie bracket on the vector space. \n\nA general Q1-manifold can be seen to be equivalent to a Lie algebroid, as was first observed in \\cite{Vai97}. For completeness, we briefly provide a definition:\n\\begin{vdef} \\label{def:Loid} A \\emph{Lie algebroid} consists of a vector bundle $E\\to M$ together with a bundle map $\\rho \\colon E \\to TM$ (over the identity) and a Lie algebra $(\\Gamma(E),[\\cdot,\\cdot])$ on the sections of $E$ such that for all $\\psi_1$, $\\psi_2 \\in \\Gamma(E)$ and $f \\in C^\\infty(M)$ one has the following Leibniz rule\n\\begin{equation} \\label{Leib}\n[\\psi_1, f \\psi_2] = f [\\psi_1,  \\psi_2] + \\rho(\\psi_1)f \\,\\cdot  \\psi_2 \\;.\n\\end{equation}\n\\end{vdef}\nThese data can be recovered from a homological vector field of degree 1 on an N1-manifold. In local coordinates $x^i$ and $\\xi^a$ of degree 0 and 1, respectively, it always is of the form \n\\beq \\label{Q0} Q = \\rho^i_a(x) \\xi^a \\frac{\\partial}{\\partial x^i}\n- \\frac12 C^a_{bc}(x) \\xi^b \\xi^c \\frac{\\partial}{\\partial \\xi^a} \\, . \\eeq\nThe axioms can be reconstructed\nby using $Q^2=0$ \\emph{and} the behavior of the coefficient functions $ \\rho^i_a$ and \n$C^a_{bc}$ with respect to (degree preserving) coordinate changes on $\\CM \\cong E[1]$ (so as to recover also \\eqref{Leib}). The details of this process can be found in the main text. It is also true, moreover, that the Q-morphisms (morphisms of N-manifolds such that their pullback intertwine the two Q-structures) correspond precisely to Lie algebroid morphisms in the sense of \\cite{MaH90} (cf.~\\cite{BKS} for an explicit proof). \nWe summarize this in \n\\begin{thm}[Vaintrob 97] \\label{Vain}\nA Q1-manifold is in 1:1-correspondence with a Lie algebroid. This is an equivalence of categories.\n\\end{thm}\nIn the present paper we treat the analogue question for one degree higher, i.e.~we address the description of a Q2-manifold. Instead of two coefficient functions $\\rho$ and $C$ satisfying two equations resulting from $Q^2=0$, one ends up with five coefficient functions satisfying seven coupled differential equations, cf.~Eqs.~\\eqref{Q} and \\eqref{B1}--\\eqref{B7}, respectively, below. For an orientation we first collect related results, corresponding to special cases.\n\nIf the Q2 manifold is one over a point, it corresponds to a (semistrict) Lie 2-algebra or, equivalently, to a 2-term $L_\\infty$-algebra. A special case of this are strict Lie 2-algebras, which, according to \\cite{Baez03vi}, are equivalent to Lie algebra crossed modules. We thus here define a Lie 2-algebra by means of this description:\n\\begin{vdef} A \\emph{strict Lie 2-algebra} \\label{strictL2}\nis a pair $(\\g,\\h)$ of Lie algebras \ntogether with a homomorphism $t \\colon \\h\\to \\g$ and a  representation\n$\\alpha \\colon \\g\\to\\Der(\\h)$ such that  $\\, t(\\alpha (x)v) =[x,t(v)]$   \nand $ \\alpha(t(v))w =[v,w]$  hold true for all $x \\in \\g$, $v,w \\in \\h$.  \n\\end{vdef}\nWe remark in parenthesis that this definition can be shortened:\n\\begin{lemma} A strict Lie 2-algebra $(\\g,\\h)$  is the same as an intertwiner, $t \\colon \\h \\to \\g$, from a $\\g$-representation $\\h$ to the adjoint representation on $\\g$ such that $t(v)\\cdot v = 0$ for all $v\\in \\h$.\n\\end{lemma}\nThe data of Definition \\ref{strictL2} can be retrieved easily with all their properties; e.g.~the Lie algebra structure on $\\h$ can then be \\emph{defined} by means of $[v,w]_\\h := t(v) \\cdot w$.\n\nThe definition of a semistrict Lie 2-algebra or simply \\emph{a Lie 2-algebra} is analogous, \nbut contains an additional map $H \\colon \\Lambda^3 \\g \\to \\h$, satisfying a ``coherence-type'' consistency relation. $H$ turns out to control the Jacobiators of brackets on $\\g$ and $\\h$ and enters a modification of the definition above in several places. For $H=0$ one recovers a strict Lie 2-algebra. (In fact, one may extract its precise definition from specializing the Definition \\ref{def:Lie2d} below to the case of $M$ being a point.)\n\n\nAlternatively, one can describe a Lie 2-algebra also by means of a 2-term $L_\\infty$-algebra. We recapitulate its definition in detail in the Appendix \\ref{s:Linf1} (cf.~also \\cite{SHLA,SHLA2} for the original definitions). Essentially this consists of a 2-term complex $\\h[-1] \\xrightarrow{t} \\g$,\\footnote{Note that in the context of N-manifolds, $\\h[-1]$ means that the linear coordinates are of degree -1, which implies that the \\emph{elements} of this vector space are of degree +1. The shifting is slightly changed with respect to the previous occurrence of $\\g$ and $\\h$, but this is due to different but equivalent conventions, cf.~also Appendix \\ref{s:Linf1}.} a unary bracket $[\\cdot]_1=t$ of degree -1, a binary bracket $[\\cdot,\\cdot]_2$ of degree 0 and a ternary bracket $[\\cdot,\\cdot,\\cdot]_3$ of degree 1 \nthat have to fulfill several quadratic relations which can be understood as generalized Jacobi identities. One of those is, e.g., for $x,y,z\\in\\g$\n\\begin{equation}\n [x,[y,z]_2]_2+[z,[x,y]_2]_2 +[y,[z,x]_2]_2 = [\\,[x,y,z]_3]_1 \\, ,\n\\end{equation}\nshowing that the 2-bracket forms a Lie algebra iff the image of the 3-bracket lie in the kernel of $t$. It is thus not surprising that this 3-bracket is the above mentioned map $H$. \nIt is thus again a non-vanishing $H$ that impedes the appearance of ordinary Lie algebras and that controls their deviation from being so on the other hand. In particular, $H$ itself has to fulfill a Jacobi-type quadratic relation. Further details on this perspective of a Q2-manifold over a point and $L_\\infty$-algebras in general are deferred to Appendix \\ref{s:Linf1}.\n\nFinally, concerning Q2-manifolds with an honest base manifold $M$, there is the following result (see also \\cite{Royt02}). \n\\begin{thm}[Roytenberg]$\\quad$\\\\ \\label{thm:Roy}\nA QP2-manifold is in 1:1-correspondence with a Courant algebroid.\n\\end{thm}\nSome comments are in order: A QP-manifold is a Q-manifold (in the above sense) equipped with a compatible symplectic form $\\omega$. QP2 signifies that the underlying N-manifold is an N2-manifold in our sense (cf.~footnote \\ref{fn:Nk}) or, equivalently, that the 2-form $\\omega$ has total degree $4=2+2$.\nA Courant algebroid can be defined as follows\n\\begin{vdef}\n\\label{def:Cour} A \\emph{Courant algebroid} consists of a vector bundle $E\\to M$ together with a bundle map $\\rho \\colon E \\to TM$ (over the identity), a Leibniz--Loday algebra $(\\Gamma(E),[\\cdot,\\cdot])$ on the sections of $E$, and fiber metric   $\\<.,.\\>$ on $E$\nsuch that for all $\\psi_1$, $\\psi_2 \\in \\Gamma(E)$ one has the following rules\n\\begin{align} \\label{Coura}\n  \\< \\psi_1,[\\psi_2,\\psi_2]\\>&=  \\tfrac{1}{2}\\rho(\\psi_1)\\<\\psi_2,\\psi_2\\>  = \\<[\\psi_1,\\psi_2],\\psi_2\\> \\, .\n\\end{align}\n\\end{vdef}\nHere some further explanations may be in place: First, we call Leibniz--Loday algebra an algebra, where the left-multiplication is a derivation, i.e.\n\\begin{equation}\\label{JaCou}\n[\\psi_1,[\\psi_2,\\psi_3]] = [[\\psi_1,\\psi_2],\\psi_3] +[\\psi_2,[\\psi_1,\\psi_3]]\n\\end{equation}\n holds true $\\forall \\psi_k \\in \\Gamma(E)$. Moreover, one can deduce the Leibniz rule \\eqref{Leib} from the axioms. So, \\emph{if} the bracket $[\\cdot , \\cdot]$ were antisymmetric, this would be a Lie algebroid (with an additional fiber metric). However, the first equation of \\eqref{Coura} prohibits this: The inner product being non-degenerate, the symmetric part of the bracket can vanish only in the relatively uninteresting case of an identically vanishing map $\\rho$ \n(in which case one obtains a bundle of quadratic Lie algebras over $M$---ad-invariance following from the remainder of \\eqref{Coura} then). So, it is precisely the inner product (together with $\\rho$) that governs the symmetric part of the bracket. The second equation of \\eqref{Coura} expresses a (for non-vanishing $\\rho$ not just pointwise) ad-invariance of the fiber-metric.\n\nThis completes the explanations of the terms appearing in the above theorem. As an aside we mention that one may also rephrase the axioms of a Courant algebroid in terms of the anti-symmetrization of the above bracket, \n\\begin{equation}\\label{anti}\n[\\phi,\\psi]_2=\\tfrac12\\left([\\phi,\\psi] -[\\psi,\\phi]\\right) \\, .\n\\end{equation}\nIn that case, however, (\\ref{JaCou}) does not yield a Jacobi identity for this new bracket. It thus does not come as a surprise that this can be rephrased in terms of an (infinite-dimensional) 2-term $L_\\infty$-algebra $[.]_1\\:V_1\\to V_0$ \\cite{Royt98}: Here the vector spaces are $V_0=\\Gamma(E)$ and $V_1={C^\\infty}(M)$, respectively, the map $[.]_1$ is provided by means of the de Rham differential $\\ud$ followed by the transpose of the anchor map and a subsequent use of the inner product (to identify $E^*$ with $E$ and likewise so for the sections). Between two elements of $V_0$ the degree 0 binary bracket is given by \\eqref{anti}, while for $f\\in V_1$ and $\\phi \\in V_0$ one has $[\\phi,f]_2:=\\rho(\\phi)f$.\nThe degree 1 ternary bracket, finally, can be non-vanishing only on $[.,.,.]_3:\\Lambda^3V_0\\to V_1$, where it is given by $[\\psi_1,\\psi_2,\\psi_3]_3=\\tfrac13\\<[\\psi_1,\\psi_2]_2,\\psi_3\\>+{\\text{cycl.}}$  All higher brackets vanish for degree reasons.  \n\nWe now turn to describing our main result of section \\ref{s:examQ}. As in the previously mentioned results on Q1-manifolds, Lie 2-algebras, and QP2-manifolds, an essential ingredient of it is a definition (specifying to what precisely the Q2-manifold is equivalent). To obtain this definition, we proceeded as sketched above for Q1-manifolds, i.e.~analyzed the behavior of coefficient functions with respect to coordinate changes on the N2-manifold, which we identified with the sum of two vector bundles (in a non-canonical way, but different choices being related by an isomorphism). We first provide the generalization of Definition \\ref{strictL2} to the case of a general base $M$ of $\\CM$, trying to stay as close as possible to the known one for $M=\\mathrm{pt}$: \n\\begin{vdef}\\label{d:sLie2}\nA \\emph{strict Lie 2-algebroid} or a \\emph{crossed module of Lie algebroids} is a pair of \n $(E,V)$ of Lie algebroids over the same base $M$, where $\\rho_V \\colon V \\to TM$ vanishes,\ntogether with a homomorphism $t \\colon V\\to E$ and a representation ${{}^E{}\\nabla}$ of $E$ on $V$\nsuch that  $t\\left(\\Econn_\\psi v\\right) = [\\psi,t(v)]$\n and  $\\Econn_{t(v)}w = [v,w]$   hold true for all $\\phi,\\psi\\in\\Gamma(E)$, $v,w\\in\\Gamma(V)$.\n\\end{vdef}\nThe generalization of a representation of a Lie algebra $\\g$ on a vector space to a representation of a Lie algebroid $E$ on a vector bundle $V$ (over the same base) uses the notion of an $E$-covariant derivative: This is a map ${{}^E{}\\nabla}\\:\\Gamma(E)\\times\\Gamma(V)\\to\\Gamma(V)$ such that for all $\\psi\\in\\Gamma(E)$, $v\\in\\Gamma(V)$, and $f\\in{C^\\infty}(M)$\n\\[\\label{EConn}\n  {{}^E{}\\nabla}_{f\\psi} v = f {{}^E{}\\nabla}_\\psi v, \\quad\\quad \n  {{}^E{}\\nabla}_\\psi (f\\cdot v) = \\rho(\\psi)[f]\\cdot v +f{{}^E{}\\nabla}_\\psi v.\n\\]\nThis becomes a representation iff its $E$-curvature, defined by means of  ${{}^E\\!{}R}(\\phi,\\psi)v:=[\\Econn_\\phi,\\Econn_\\psi]v -\\Econn_{[\\phi,\\psi]}v$, vanishes,${{}^E\\!{}R}=0$.  This flatness condition implies also that $\\Econn_\\psi$ is a derivation of the (bundle of) Lie algebra(s) $(\\Gamma(V),[.,.])$.\n\n\n\\begin{vdef}\\label{def:Lie2d}  A (semi-strict) \\emph{Lie 2-algebroid} is defined as in Definition \\ref{d:sLie2} except that the (still antisymmetric) brackets on $\\Gamma(E)$ and $\\Gamma(V)$ are no more required to satisfy the Jacobi identity and\n ${{}^E{}\\nabla}$ is no more flat in general. Instead, all these deviations are governed by a single $E$-3-form $H\\in\\Omega^3(E,V)$ with values in $V$:  For all $\\psi_k\\in\\Gamma(E)$, $v,w\\in\\Gamma(V)$ and $f\\in{C^\\infty}(M)$ one has\n\\begin{align}\n  [\\psi_1,[\\psi_2,\\psi_3]] +{\\text{cycl.}} &= t\\left(H(\\psi_1,\\psi_2,\\psi_3)\\right), \\label{HJacobi} \\\\\n   {{}^E\\!{}R}(\\psi_1,\\psi_2)v &= H(\\psi_1,\\psi_2,t(v)), \\label{HRep}\\\\\n\n \n  {{}^E\\!{}\\mathrm{D}} H &= 0 \\, . \\label{DH}\n\\end{align}\n\\end{vdef}\nHere ${{}^E\\!{}\\mathrm{D}}$ is the exterior $E$-covariant derivative associated to ${{}^E{}\\nabla}$. We did not need to specify the Jacobiator of the bracket on $\\Gamma(V)$, since it follows at once from \\eqref{HRep} and $\\Econn_{t(v)}w = [v,w]$: \n\\begin{equation}\n[u,[v,w]] + {\\text{cycl.}} = H(t(u),t(v),t(w)) \\qquad \\forall u,v,w \\in \\Gamma(V) \\, .\n\\end{equation} \nIt is noteworthy that in this formulation, the coherence condition for $H$ takes the simple form \\eqref{DH}. \nIn this paper we will then prove the following:\n\\begin{thm}\\label{thm1.5} $\\quad$  \\\\ A Q2-manifold is in 1:1-correspondence with a (semi-strict) Lie 2-algebroid \\\\ up to isomorphism.\n\\end{thm}\nSimilarly to Theorem \\ref{Vain}, this statement can be turned also into an equivalence of categories.\n\n\nIn this context it is important to remark that the notion of a morphism of Lie 2-algebroids (as defined above in Definition \\ref{def:Lie2d}) is less restrictive than the one of  strict Lie 2-algebroids (Definition \\ref{d:sLie2}). This can be seen as follows: A definition of a morphism of a Lie 2-algebroid can be tailored from the corresponding Q-morphisms and the identification given in the above theorem  (such as one can do it for ordinary Lie algebroids and Q1-manifolds). On the one hand, the change of a splitting of the Q-manifold corresponds to a Q-morphism.\nOn the other hand, it corresponds to an element $B\\in\\Omega^2(E,V)$ and $H$ changes under this $B$ (in a slightly tricky way, cf.~Equation \\eqref{split3} below) and in particular this can be used to generate a non-zero $H$ from a vanishing one. \n\nA remark to the nomenclature seems important. One of the goals here is to give a definition of a mathematical structure equivalent to a Q2-manifold on the one hand and generalizing the definition of a Lie 2-algebra as given in \\cite{Baez03vi} on the other hand. In other words, we strive at a definition equivalent to the \\emph{super}-geometrical notion of a Q2-manifold in terms of notions used only in \\emph{ordinary} differential geometry. This can be viewed as the same philosophy underlying the Theorems \\ref{Vain} and \\ref{thm:Roy}---with the difference that in these two cases the definitions \\ref{def:Loid} and \\ref{def:Cour} were already established as a geometrical notion before the equivalence  (expressed in the two theorems mentioned above) to the (simpler) super- or graded-geometrical structure was observed. Some also call Q$k$-manifolds simply Lie $k$-algebroids \\emph{by definition}. To emphasize the non-tautological nature of Theorem \\ref{thm1.5}, it may thus be useful to add the specification ``semi-strict'' in the equivalence, with \\emph{semi-strict} Lie 2-algebroids being then \\emph{introduced} (to our knowledge for the first time) in \nDefinition \\ref{def:Lie2d} above. \n\nNote that as a simple corollary of the above theorem, we find that any Q2-manifold gives rise to an infinite-dimensional Lie 2-algebra or also 2-term $L_\\infty$-algebra, $\\Gamma(V) \\xrightarrow{t} \\Gamma(E)$. In fact, a Lie 2-algebroid provides the additional control over such an infinite-dimensional Lie 2-algebra in the same way as a Lie algebroid provides one for its infinite-dimensional Lie algebra of sections. \n\nCertainly, a QP2-manifold is also a Q2-manifold. The infinite-dimensional Lie 2-algebra found in the above way is, however, \\emph{not} the same one as the one constructed in \\cite{Royt98}: There the  2-term complex is $\\Omega^0(M) \\xrightarrow{t} \\Gamma(E)$ and, containing  the de Rham differential  $\\ud$, neither $t$ nor the Jacobiator are  $C^\\infty(M)$-linear, \nwhereas in our construction one has $\\Omega^1(M) \\xrightarrow{t} \\Gamma(E)$ rendering  the map $t$ and the Jacobiator $C^\\infty(M)$-linear.\n\nFurther perspectives on Q$2$-manifolds in terms of particular  $L_\\infty$-algebras or algebroids are deferred to Appendix \\ref{s:Linf}).\n\n\nThere is also another way of obtaining the axioms for a Lie algebroid from studying a Q1-manifold than the one we sketched above (by a study of components of $Q$ with respect to changes of coordinates). This is provided by regarding the Q-derived bracket on the space of degree $-1$ vector fields on the Q-manifold $\\CM$: Let $\\psi$, $\\psi'$ be two such vector fields, then\n\\begin{equation}\n[\\psi,\\psi']_Q := [[\\psi,Q],\\psi']] \\label{QderLie}\n\\end{equation}\nis another vector field of degree -1 on $\\CM$. While in the case of a Q1-manifold one easily identifies vector fields of degree -1 on $\\CM = E[1]$ with sections in the Lie algebroid $E$ and the bracket \\eqref{QderLie} with the Lie bracket between them, so that at the end one rediscovers also by this procedure the usual axioms of a Lie algebroid, for a Q2-manifold one obtains a \\emph{different} picture. First of all, the derived bracket does not need to be antisymmetric anymore, since $[\\psi,\\psi]$ can be non-zero now as, in contrast to a Q1-manifold, a Q2-manifold carries non-trivial degree -2 vector fields (for more details and explanations we refer to section \\ref{sec:derived}). The derived bracket, however, always gives a Leibniz--Loday algebra, satisfying a condition of the form \\eqref{JaCou}. These observations, which are motivated moreover by our study of the gauge transformations (cf., in particular, Equation \\eqref{derQtot})\\footnote{Note that $\\ad_Q(\\psi) \\equiv [Q,\\psi] = [\\psi,Q]$ since both vector fields are odd.}, lead to an attempt of finding another description of a Q2-manifold $\\CM$. In fact, since $[\\cdot,\\cdot]$, restricted to degree -1 (and thus odd) vector fields, yields a $C^\\infty(M)$-linear symmetric map from the degree -1 vector fields on $\\CM$ to the degree -2 vector fields, and  since vector fields of negative degrees on an N-manifold can always be identified with sections of vector bundles, one swiftly extracts from the above an axiomatic formulation very close to the one of a Courant algebroid, with the difference that now the inner product can take values in another bundle. \n\n\\begin{vdef}\n\\label{def:VCour} A \\emph{$V$-twisted Courant algebroid} consists of a vector bundle $W\\to M$, a bundle map $\\rho \\colon W \\to TM$, a Leibniz--Loday algebra $(\\Gamma(W),[\\cdot,\\cdot])$, a non-degenerate surjective symmetric product $\\<.,.\\>$ taking values in another vector bundle $V \\to M$, and  a $W$-connection  ${{}^W\\nabla}$ on $V$\nsuch that for all $\\psi_1$, $\\psi_2 \\in \\Gamma(W)$ one has the following axioms\n\\begin{align} \\label{VCoura}\n  \\< \\psi_1,[\\psi_2,\\psi_2]\\>&=  \\tfrac{1}{2}{{}^W\\nabla}_{\\psi_1}\\<\\psi_2,\\psi_2\\>  = \\<[\\psi_1,\\psi_2],\\psi_2\\> \\, .\n\\end{align}\n\\end{vdef}\nIt is easy to see that this reduces to an ordinary Courant algebroid if $V=M \\times \\Real$ and ${{}^W\\nabla}(1)=0$ (in this case of the trivialized bundle we can identify its sections with functions and if the unit section is parallel, ${{}^W\\nabla}$ reduces to $\\rho$ on these functions). More generally, we can say that a $\\rk V=1$, $V$-twisted Courant algebroid admitting a non-vanishing, $W$-constant section $v\\in\\Gamma(V)$ (i.e.~$v(x)\\neq 0 \\; \\forall x\\in M$ and $\\Wconn_\\psi v=0 \\; \\forall \\psi \\in \\Gamma(W)$) is in a canonical correspondence with a Courant algebroid. However, line-bundle twisted Courant algebroids are a strictly more general notion than the one of a Courant algebroid.\n\nA nontrivial example of a $V$-twisted Courant algebroid can be provided as follows: Let $A$ be a Lie algebroid together with a flat $A$-connection ${}^A\\nabla$ on a vector bundle $V$. Then the couple $W$, $V$, becomes a $V$-twisted Courant algebroid if one takes $W:=A\\oplus \\Hom(A,V) \\cong A\\oplus A^*\\otimes V$ and equips it with the bracket\n\\begin{equation}\\label{ACourant}\n[X\\oplus\\alpha,Y\\oplus\\beta] = [X,Y] \\oplus [\\imath_X,\\uD]\\beta -\\imath_Y\\uD\\alpha +H(X,Y,\\bullet)\n\\end{equation}\nwhere $\\uD\\:\\Omega^k(A,V)\\to\\Omega^{k+1}(A,V)$ is the exterior covariant derivative constructed from ${}^A\\nabla$, $[.,.]$ the Lie algebroid bracket on $A$, and $H\\in\\Omega^3(A,V)$ is any $A$-3-form with values in $V$ satisfying $\\uD H=0$.  The other maps are obvious, for example $\\Wconn_{X\\oplus\\alpha} v={}^A\\conn_X v$.  Certainly, this reduces to the standard Courant algebroid $W\\cong TM \\oplus T^*M$ twisted by a closed 3-form $H$ if $A$ is chosen be the standard Lie algebroid $TM$ (and $V$ and ${{}^W\\nabla}$ as mentioned above), in which case \\eqref{ACourant} reduces to the known, $H$-twisted Courant-Dorfman bracket \n\\begin{equation} \\label{standardtwistCour}\n[X\\oplus\\alpha,Y\\oplus\\beta] = [X,Y] \\oplus \\L_X \\beta -\\imath_Y\\ud\\alpha + \\imath_Y \\imath_X H \\, .\n\\end{equation}\nWhile an ordinary Courant algebroid reduces to a quadratic Lie algebra when $M$ is a point, the ``decoupling'' of $\\rho$ and ${{}^W\\nabla}$ has the advantage that now we can also host (non-Lie) Leibniz--Loday algebras in this framework. For $M=\\mathrm{pt}$, the definition reduces to a Leibniz--Loday algebra together with an invariant inner product with values in a vector space that has a representation.  Also, conversely, one can promote \\emph{any} Leibniz--Loday algebra $W$ to a $V$-twisted Courant algebra $W$ in a \\emph{canonical} way, cf.~Proposition \\ref{canonical} below.\n\n\n\nThe main result of section \\ref{sec:derived} is then the following (for further details, cf.~Proposition~\\ref{p:FWder} and Theorem~\\ref{thm:der})\n\\begin{thm}$\\,$ \\\\ Up to isomorphism and  provided that $\\rk V>1$, there is a 1:1 correspondence between Lie 2-algebroids $V \\xrightarrow{t} E$ and $V$-twisted Courant algebroids $W$ fitting into the following exact sequence \\begin{equation} \\label{VEW}\n0\\longrightarrow\\Hom(E,V) \\longrightarrow W\\xrightarrow{\\;\\;\\pi\\; \\;} E\\longrightarrow 0\n\\end{equation} with $\\rho_W = \\rho_E \\circ \\pi$. In particular, if $\\rk V >1$ and after the choice of a splitting of \\eqref{VEW},\nthe bracket on a $V$-twisted Courant algebroid $W$ as above has the form\n\\begin{equation}\\label{VCourant2}\n[X\\oplus a,Y\\oplus b]_W = [X,Y]_E - t(a(Y)) \\oplus [\\imath_X,{{}^E\\!{}\\mathrm{D}}]b -\\imath_Y{{}^E\\!{}\\mathrm{D}} a +\\imath_Y \\imath_X H+ b \\circ t \\circ a -  a\\circ t \\circ b  ,\n\\end{equation}\nwhere all the operations on the r.h.s.~are the induced ones on the corresponding Lie 2-algebroid.\\label{thm1.6}\n\\end{thm}\nIn the above formula, $X$, $Y \\in \\Gamma(E)$ and $a$, $b$ are either viewed as elements of $\\Omega^1(E,V)$ (in the first two terms following the $\\oplus$) or as homomorphisms from $E$ to $V$ so that they can be composed with the fixed map $t \\in \\Hom(V,E)$ of the Lie 2-algebroid. \n\nThe similarity of formula \\eqref{VCourant2} with Equations \\eqref{standardtwistCour} and \\eqref{ACourant} is striking, in particular in view of the fact that, in contrast to those examples, now the (anti-symmetric) Lie 2-algebroid bracket $[ \\cdot , \\cdot ]_E$ does no more satisfy a Jacobi condition in general, while still \\eqref{VCourant2} obeys the left-derivation property \\eqref{JaCou}. Similarly, ${{}^E\\!{}\\mathrm{D}}$ does no more square to zero in general (such as $\\uD$ and $\\ud$ in the analogous formulas \\eqref{ACourant} and \\eqref{standardtwistCour}, respectively),\nbut rather gives the $E$-curvature \\eqref{HRep}. It is here where the simple $t$-contributions to the new bracket become essential.\n\n\n\nTheorems \\ref{thm1.5} and \\ref{thm1.6} certainly also induce a 1:1 correspondence between the above mentioned ``exact V-twisted Courant algebroids'' (with $\\rk V >1$) and  Q2-manifolds $\\CM$ (with at least two independent generators of $C^\\infty(\\CM)$ of degree 2). Exact $V$-twisted Courant algebroids thus provide an alternative, ordinary differential geometric description of the graded-geometrical notion of a Q2-manifold---with respective restrictions provided in the parenthesis of the previous sentence.\n\nThe condition of $\\rk V \\neq 1$ seems necessary, since there exist examples of line bundle twisted Courant algebroids not corresponding to Q2-manifolds (in the above way). In fact, one to our mind interesting problem left open in the analysis of the present paper is a classification of line bundle twisted Courant algebroids, i.e.~$V$-twisted Courant algebroids $W$ with $\\rk V = 1$. \n\nOn the technical level this restriction on the rank can be traced back to the following simple fact of linear algebra (proven in Lemma \\ref{p:Jh=0}): Let $V_1$ and $V_2$ be two vector spaces. Then any linear operator $\\Delta \\colon \\Hom(V_1,V_2) \\to V_1$ satisfying $a(\\Delta(a))=0$ for any $a \\in \\Hom(V_1,V_2)$ has to necessarily vanish \\emph{iff} $\\dim V_2 >1$, while, e.g., for $V_2 = \\Real$ one may identify $\\Delta$ with an element of $V^1 \\otimes V^1$ and then any $\\Delta \\in \\Lambda^2 V_1$ obviously satisfies the condition for symmetry reasons. \n\nIn view of the different descriptions of QP2 manifolds as infinite dimensional Lie 2-algebras sketched before, it is at this point no more astonishing that if the QP2 manifold is regarded as a Q2 manifold (by the forgetful functor), one does not obtain the usual Courant algebroid in the above fashion. Instead, as before, $M \\times \\Real$ is replaced by $T^*M$, so one obtains a particular $T^*M$-twisted Courant algebroid in the above identification (when applied to a QP2 manifold viewed upon as a Q2 manifold). \n\nThe bracket \\eqref{standardtwistCour} is also well-defined for sections of $TM \\oplus \\Lambda^k T^*M$ if $H \\in \\Omega^{k+2}_{closed}(M)$, (without the $H$-twist, this bracket goes back to Vinogradov, cf.~\\cite{Vino90}). One obtains similar properties to those of the standard Courant algebroid, but, for example, the symmetrized contraction of its elements provides an inner product which now takes values in $V=\\Lambda^{k-1} T^*M$. In section \\ref{sec:derived} we propose an axiomatization of these properties:  Very much as \\cite{Xu97} does it for the case of $k=1$, leading  (together with its later reformulations\/simplifications) to the Definition \\ref{def:Cour} above, we propose the definition of a \\emph{Vinogradov algebroid} (cf.~Definition \\ref{def:Vino} below). We also show that $V$-twisted Courant algebroids are particular Vinogradov algebroids. \n\nSection \\ref{sec:derived} is a relatively long section and contains further potentially interesting subjects (like the axiomatization of the bracket \\eqref{standardtwistCour} when applied to higher form degrees into a Vinogradov algebroid and a simple non-standard Q-description of $H$-twisted standard Courant and Vinogradov algebroids). We do not want to summarize all this here, so as to keep some tension for the interested reader. \n\n\n\n\\vskip\\newpassage\n\n\nFinally, in the Appendix \\ref{s:Linf} we provide a further reformulation of Q$k$-manifolds in terms of $k$-term $L_\\infty$-algebroids, which we also define there so as to generalize the more standard relation between Q-manifolds over a point and $L_\\infty$-algebras \\cite{SHLA}.\n\n\\newpage\n\\section{From Bianchi identities to Q-structures} \\label{s:Bian}\nWe want to construct a gauge theory for some $p$-form gauge field. As examples\nshow, at least if we want to go beyond a trivial abelian version of such a theory with $p\\ge 1$, \nthis necessarily involves a tower of gauge fields, where the highest form\ndegree is $p$. For definiteness we take $p=2$, but the generalization to\narbitrary $p$ is immediate. We thus consider a collection of 2-forms $B^D$,\ntogether with 1-form fields $A^a$, and in general also 0-form fields\n$X^i$. The scalar fields are permitted to take values in some internal\nmanifold $M$, which is not necessarily a vector space---so in general the\ntheory is a sigma model. Denoting spacetime by $\\Sigma$, one has\n $X \\colon \\Sigma \\to M$, and the particular collection of scalar fields\n $X^i=X^i(\\x)$ ($\\x \\in \\Sigma$) results after a (local) choice of\n coordinates $x^i$ \non $M$, $X^i = X^*(x^i)$.\\footnote{It is\ntempting to exchange the notation of $M$ and\n$\\Sigma$; then $M$ would be spacetime, and if $\\Sigma$ is nontrivial, one has a\nsigma model. However, here we conform to previous publications.}  In\nmany examples there are no scalar fields and the tower of gauge fields starts\nwith 1-forms only. This just corresponds to the choice $M$ a point, in which\ncase one may drop the map $X$ as it contains no information. However, even in this case, scalar fields would then enter the theory in a second step representing matter; intending to stay as general as possible, we permit their presence thus right from the beginning. \n\nOne of the main ingredients in a gauge theory are the field strengths. We\nalso know how they should start, namely with the exterior derivative acting\non the respective gauge field, possibly then modified by adding wedge\nproducts of gauge fields to this expression. We thus make the most\ngeneral ansatz of this kind:\n\\begin{eqnarray}  F^i &:=& \\ud X^i - \\rho^i_a A^a \\, , \\label{Fi}\\\\\n        F^a&:=& \\ud A^a + \\frac12 C^a_{bc} A^b A^c - t^a_D B^D\\, , \\label{Fa}\\\\\n        F^D&:=& \\ud B^D + \\Gamma^D_{aC} A^a B^C - \\frac{1}{6} \n        H^D_{abc} A^a A^b A^c \\, ,\n\\label{FB}\n\\end{eqnarray}\nwhere wedge products between differential forms are understood \nand\nall the yet unrestricted coefficients can be functions of  $X$.\nFor some purposes it will prove convenient to use a more condensed\nnotation: We then denote the field strengths \ncollectively by $F^\\alpha$ and likewise \nthe gauge fields by $A^\\alpha$ (so $A^i \\equiv X^i$, $A^D \\equiv B^D$).\n\n\nWe now want to require a  very mild form of  Bianchi identities. In\nordinary YM-gauge theories Bianchi identities tell us that $\\ud\nF^a\\equiv -C^a_{bc} A^b F^c$. We want to relax\/generalize this\ncondition on the complete set of field strengths $F^\\alpha$ by requiring\nthat the exterior derivative of them produces terms that all \n\\emph{contain} field strengths again, $\\ud F^\\alpha = \\lambda^\\alpha_\\beta F^\\beta$,\nwhere $\\lambda^\\alpha_\\beta$ may be  arbitrary, field dependent factors.\nInterpreted in this sense, we require that one has:\n\\beq \\boxed{\\ud F^\\alpha |_{F^\\beta=0} \\equiv 0}\\, . \\label{bian}\\end{equation}\nThis equation has to be understood in such a way that for \\emph{every} choice of the gauge fields $A^\\alpha$ satisfying $F^\\alpha =0$ one has $\\ud F^\\alpha =0$.\nWe remark right away that this is satisfied also for generalizations\nsuch as for \nhigher YM theories as discussed in \\cite{Baez02jn}\nor the Algebroid YM theories of \\cite{Str04b}. Moreover, there are\noccasions where\nthe $X$-fields above, considered as part of the gauge field here, can\nbe identified with some Higgs type scalar fields  and $F^i$ then\ncorresponds to the covariant\nderivative of this scalar field $X^i$. In standard\nYM-theory e.g.~we may use the fact that the square of the\nexterior covariant derivative\nis proportional to the curvature to show\nthat also in this enlarged setting \n\\eqref{bian} is satisfied.\n\nLet us rephrase the\nrequirement \\eqref{bian}\nin more mathematical terms: Within\nthe differential graded commutative algebra $(\\mathcal{A},\\ud)$\ngenerated by abstract elements $A^\\alpha$ and $\\ud A^\\alpha$ there is an ideal $\\I$\ngenerated by $F^\\alpha$ as \ndefined in Eqs.~\\eqref{Fi}--\\eqref{FB}, i.e.~by the choice of\nstructural functions $\\rho^i_a(X)$, $\\ldots$, $H^D_{abc}(X)$. The\ngeneralized Bianchi identities hold iff (by definition) this ideal is\nan invariant subspace with respect to the exterior derivative, i.e.~iff Eq.~(\\ref{ideal}) holds true.\n\nThere are some further possible reformulations of the above\nconditions.\nOne may rephrase them as saying e.g.~that the\nstructural functions must be such that the quotient\nalgebra $\\mathcal{A}\/\\I$ forms a so-called free differential algebra\n(cf.~\\cite{Sull77b} or, for a physical\nexample in low dimensions, e.g.~\\cite{Izq99}). Since this quotient\ncorresponds to looking at vanishing field strengths only, which will\n\\emph{not} be our field equations when interested in constructing\na (non-topological) generalization of Yang-Mills theories, we avoid\nthis perspective, such as that we avoid the notation $\\ud F^\\alpha\n\\approx 0$ instead of \\eqref{bian} since for non-topological\ntheories the ``on-shell'' equality would not coincide with the on-shell\nnotion dictated by field equations. Finally, Eq.~\\eqref{bian} may\nalso be viewed as an integrability condition in the following sense:\nFor all fields $A^\\alpha$ which satisfy $F^\\alpha(\\sigma)=0$ for some point $\\sigma\n\\in \\Sigma$, it follows automatically (by the choice of structural\nfunctions) that also $\\left(\\ud F^\\alpha\\right)(\\sigma)=0$. \n\n \n\n\nThe main purpose of this section is twofold: First, we want to show\nthat the simple ``physical'' and rather general requirement \n\\eqref{bian}\n\nautomatically and naturally leads to\nQ-structures.\\footnote{By definition (but cf.~also the more detailed\n explanation below), Q-structures, introduced in the \n context of topological field theories in \\cite{AKSZ}\\label{ct:Scha}\n  are homological vector fields of degree one on a\nsupermanifold.}\nSecondly, to obtain the explicit form\nof the generalized Bianchi identities (cf., e.g.,\nEqs.~\\eqref{BFi}--\\eqref{BFB}\nbelow), the use of this formalism shortens calculations extremely. \n\n\nFor this purpose we first unify the gauge fields into a single map.\nThe scalar fields $X ^i(\\x)$ describe some map  $a_0 \\colon \\Sigma \\to\nM$ ($a_0 \\equiv X$ in previous notation); as already mentioned, they\nare the pullback\nof coordinate functions \non $M$ to functions on $\\Sigma$  by precisely this map (possibly\nrestricted to local charts). To imitate\nthis for the other components $A^a, \nB^D, \\ldots$ of the gauge \nfield, we introduce auxiliary spaces spanned by coordinates \n$\\xi^a, b^D, \\ldots$, taking these spaces to be linear and thus \ndefined by the respective linear coordinates: $W = \\langle \\xi_a   \\rangle$,\n$\\V = \\langle b_I   \\rangle$, where $\\xi_a$ is a basis \nof  vectors \ndual to $\\xi^a$ (linear coordinates on a vector space are a basis in the\ndual vector space) etc. \n\nSince every 1-form on $\\Sigma$ may be viewed as a (fiberwisely linear)\nfunction on $T\\Sigma$, we may thus\nregard $A^a$ as the pullback of the coordinate function $\\xi^a$ by a map from\n$T\\Sigma$ to $W$. A local coordinate system on $T\\Sigma$ can be chosen in the form\n$\\x^\\mu, \\theta^\\mu$, where $\\theta^\\mu=\\ud \\x^\\mu$. Since the natural\nmultiplication of 1-forms is the antisymmetric wedge product, it is near at\nhand to declare the fiber coordinates $\\theta^\\mu$ of $T\\Sigma$ as anti-commuting;\nthis is taken into account by the notation $T[1]\\Sigma$: the coordinates $\\x^\\mu$\nof the base are of degree zero (and thus even and commuting), while\n$\\theta^\\mu$ is of degree one (and in particular odd). In this way we may interpret $A^a$ as\nthe pullback of $\\xi^a$ to functions on $T[1]\\Sigma$ (i.e.~differential forms) by\na map $a_1 \\colon T[1]\\Sigma \\to W[1]$, $A^a = (a_1)^* \\xi^a$; here we declared\nthe coordinates $\\xi^a$ to be of degree one, too, so that the map $a_1$ becomes\ndegree preserving, and $(a_1)^*$ a morphism of algebras.  Likewise,\n $B^D = (a_2)^* b^D$, where $a_2 \\: T[1]\\Sigma \\to \\V[2]$, and $b^D$ are declared to\nbe of degree two, since $B^D$ is a 2-form (that is a function on $T[1]\\Sigma$ of\ndegree two) and we want $a_2$ to be degree preserving again.\\footnote{If one considers supersymmetric or supergravity theories, the form degree does not determine if a homogeneous element is commuting or anti-commuting. If one has bosonic and fermionic 0-forms, for example, then already $M$ will be a corresponding supermanifold. Likewise, there may be anticommuting and commuting 1-forms in a supergravity theory; to describe such objects one would introduce a bidegree on the target, so that the first degree corresponds to the form degree and the other one to the commutativity property. We will not go into further details about this possible generalization here; as a simple prototype of such a theory one may look at supergravity theories reformulated as Poisson sigma models with a super-Poisson structure on the target as given in \\cite{Strobl:1999zz}.}\n\nIf $X^i$ denote local coordinates on $U \\subset M$, we can assemble the above\nmaps into a single map $a$ from (open domains of) $T[1]\\Sigma$ to $U \\times W[1] \\times\n\\V[2]$. It is tempting to require this direct product form of the target\nmanifold only on local charts $U$. Globally we may permit a twist (nontrivial\ntransition functions) in changing the vector spaces when changing charts in\n$M$.  In fact, what we permit is a super-manifold $\\CM_2$ of degree 2: Locally\nit is described by degree 0 coordinates $x^i$, degree 1 coordinates $\\xi^a$,\nand degree 2 coordinates $b^D$.\\footnote{\\label{fn:pos} \nIn this and the following section we will \nconsider only non-negatively graded supermanifolds. (We will often call an integer graded manifold with its induced superalgebra of functions determined by the parity of the degrees simply a supermanifold, understanding implicitly from the context that we consider the full $\\Z$-grading (and not just modulo 2)). Speaking of the degree of it we will \nthen always mean the maximal degree of ``local coordinate functions'' (local generators). \nLater on also supermanifolds with (coordinate) functions of negative degrees will play \nsome role, cf in particular Section~\\ref{sec:derived}.} \nThe transition functions need to be degree\npreserving (by definition). This implies that forgetting about the degree two\ncoordinates one has a vector bundle $E \\to M$ with typical fiber\n$W$. Similarly, $x^i, b^D$ define a vector bundle $V \\to M$ with typical fiber\n$\\V$.  (Since the transition functions of $b^D$s may also contain a part\nquadratic in the degree one coordinates $\\xi^a$, the total space\n$\\CM_2$ does not correspond to the direct sum vector bundle $E \\oplus\nV$ canonically, but does so only after an additional choice of\nsection; we will come back to this subtlety in the subsequent section).\n\nThus in sum we obtain a degree preserving map $a \\colon \\CM_1 \\to \\CM_2$,\nwhere $\\CM_1 = T[1]\\Sigma$ is a degree one super-manifold.\\footnote{More generally,\nwe may also permit a nontrivial fibration over $\\CM_1$, such that $\\CM_2$ is\nonly a typical fiber, leading to the notion of ``$Q$-bundles''. This\nis dealt with in \\cite{KS07}, while in this paper we restrict to\ntrivial bundles (except for the target bundle, encoded in the\nsupermanifold $\\CM_2$). Still, the present \nformulas remain correct also in the more general setting if they are\nreinterpreted as local ones.}\nChoosing local coordinates $q^\\alpha : = (x^i, \\xi^a, b^D)$\non $\\CM_2$ this gives rise to \nfunctions on $\\CM_1$ by means of  the pullback $a^*$, which then \njust yields the set of  local 0-forms, 1-forms, and 2-forms,\nrespectively, with which we started this section, $A^\\alpha = a^*(q^\\alpha)$. \n\nThe de Rham differential $\\ud$ on $\\Sigma$ corresponds to a degree one\nvector field \non $T[1]\\Sigma \\equiv \\CM_1$, namely $\\Qo = \\theta^\\mu \n\\partial\/\\partial \\x^\\mu$. It squares to zero, $(\\Qo)^2 \\equiv 0$, and thus defines \nwhat is called a $Q$-structure on $\\CM_1$. We now define a likewise vector field on $\\CM_2$ \nusing the above data. It is sufficient to know its action on coordinates $q^\\alpha$: \n\\begin{eqnarray} {Q_2} x^i &:=& \\rho^i_a \\xi^a \\, , \\quad \\label{Qi}\\\\\n     {Q_2} \\xi^a& := &- \\frac12 C^a_{bc} \\xi^b \\xi^c +t^a_D b^D\\, , \n     \\label{Qa}\\\\\n        {Q_2} b^D&:= & -\\Gamma^D_{aC} \\xi^a b^C + \\frac{1}{6} \n        H^D_{abc} \\xi^a \\xi^b \\xi^c \\, .\\label{QB}\n\\end{eqnarray}\nThis vector field is obviously also of degree one, i.e.~it raises the degree\nof the respective coordinate by precisely one unit. Furthermore we\nintroduce (cf.~also \\cite{BKS})\n\\beq \\CF := \\Qo \\circ a^* - a^* \\circ {Q_2} \\qquad \\Rightarrow \\qquad \\CF (q^\\alpha) \n\\equiv F^\\alpha \\, . \\label{CF}\\end{equation}\n\nUp to here we only rewrote the initial data contained in\nEqs.~\\eqref{Fi} --  \\eqref{FB}. We now come to analyze\nCondition~\\eqref{bian}. Denoting by $\\ev_\\x$ evaluation of a\nfunction on $\\CM_1=T[1]\\Sigma$ at a point $\\x \\in \\Sigma$, in the present\nlanguage the generalized Bianchi identities (in their formulation as\nintegrability conditions for $F^\\alpha=0$ equations) can be translated into the \nrequirement that for every $\\x \\in \\Sigma$\nthe composed operator $\\ev_\\x \\circ \\Qo \\circ\n\\CF$ vanishes whenever \n$a\\colon\\CM_1 \\to \\CM_2$ is such that the operator $\\ev_\\x\n\\circ \\CF$ vanishes.\n\nNow, using  \n$(\\Qo)^2 \\equiv 0$, we find the identity\n\\begin{equation}\n\\ev_\\x \\circ \\Qo \\circ \\CF =- \\ev_\\x \\circ \\left( \\CF \\circ {Q_2}\n  +a^* \\circ ({Q_2})^2\\right). \\label{preBian}\n\\end{equation}\nThus, for \\eqref{bian} to hold true\nit is obviously sufficient that the vector field ${Q_2}$ \nsquares to zero. But also the converse is true, provided only that the\ndimension of spacetime $\\Sigma$ is at least $p+2$ ($p$ is the degree of\nthe supermanifold $\\CM_2$, in our discussion above $p=2$): Assume that\n$({Q_2})^2 \\neq 0$, that $a\\colon \\CM_1 \\to \\CM_2$ is such that\n$\\ev_\\x \\circ \\CF \\equiv 0$, and that the integrability\ncondition holds true. Then there exists at least one\nbasic coordinate function $q^{\\alpha_0} \\in C^\\infty(\\CM_2)$ for which\n$({Q_2})^2 q^{\\alpha_0}=: \\varphi$ is a non-vanishing function on $\\CM_2$.\nNote that this function has at most degree $p+2$, so $a^* \\varphi$ is a\ndifferential form on $\\Sigma$ of at most degree $p+2$. \nApplying \\eqref{preBian} to $q^{\\alpha_0}$ under the above assumptions,\nwe obtain $\\ev_\\x a^* \\varphi =0$, valid for every $\\sigma \\in \\Sigma$. \nIf we are able to show that\nthere always exists some $a$ fulfilling $(F^\\alpha)_\\x=0$ for which this\nequation is violated, we obtained a contradiction, proving that ${Q_2}$\nmust square to zero. For this to be the case, we need the condition\n$\\dim \\Sigma \\ge p+2$, because, otherwise, \\emph{any} violation\n$\\varphi$ of $({Q_2})^2=0$ in a degree higher than $\\dim \\Sigma$ will not contribute\nto $\\ev_\\x a^* \\varphi$. On the other hand, it is precisely the\nintegrability of \n the equations $F^\\alpha =0$ that  ensure that for \\emph{every} choice of\n $\\ev_\\x \\circ a$ (i.e.~for every choice of $(A^\\alpha)_\\x$)\nwe can \nfind a map $a$, defined at least in a neighborhood of $\\x \\in \\Sigma$, such\nthat also $(F^\\alpha)_\\x=0$. This concludes the proof. \n\n\n\n\nWe thus managed to translate the relatively weak form of Bianchi identities \n\\eqref{bian}, $\\ud F^\\alpha \\sim \\sum F^\\beta$ (where the sum contains\narbitrary field dependent coefficients), \ninto a $Q$-structure on a target super manifold $\\CM_2$. This gives the first part of Theorem \\ref{thm:phys1} mentioned in the introduction, or, in the\nparticular case $p=2$, it means in detail:\n\\begin{thm} \\label{theo:nilpot}\nFor a spacetime dimension of at least 4, the nilpotency of the\nsuper vector field ${Q_2}$\ndefined in Eqs.~\\eqref{Qi} -- \\eqref{QB} is necessary and\nsufficient for the validity of \\eqref{bian}. \n\\end{thm}\n\nThe relation $({Q_2})^2=0$ contains \\emph{all} the conditions we\nwant to place on the \nstructural functions in \\eqref{Fi} -- \\eqref{FB}. They contain the \ngeneralization of the Jacobi identity of the structure constants in ordinary \nYang-Mills gauge theories. In the subsequent section we will display all \nthe conditions (in the example $p=2$), cf.~Eqs.~\\eqref{B1} --\n\\eqref{B7} below, discuss\ntheir general structure and present examples of it\nsuch as Lie 2-algebras or Lie algebroids (an alternative perspective, containing Courant \nalgebroids and their generalizations will be discussed separately in Section~\\ref{sec:derived}).\n\nIn the present section we still want to display the explicit form of\nthe Bianchi identities that we obtain when \\emph{using} the nilpotency \nof ${Q_2}$. One finds\n\\begin{align}  \\ud F^i &\\equiv \\rho^i_{a,j}A^a F^j - \\rho^i_a F^a \\, ,\n        \\label{BFi}\\\\\n      \\ud   F^a&\\equiv \\frac12C^a_{bc,i}A^b A^c F^i -t^a_{D,i}B^D F^i\n         -C^a_{bc} A^b F^c - t^a_D F^D\\, ,  \\label{BFa}\\\\\n      \\ud   F^D&\\equiv -\\Gamma^D_{aC,i} A^aB^C F^i +\\Gamma^D_{aC} B^C F^a\n        +\\frac16H^D_{abc,i} A^aA^bA^c F^i  \\nonumber\\\\\n      &\\quad  -\\frac12H^D_{abc} A^aA^b F^c -\\Gamma^D_{aC} A^a F^C \\, .\n      \\label{BFB}\n\\end{align}\nHere the power of the employed super language becomes particularly transparent: \nWhereas in every explicit calculation one needs to make heavily use \nof the Eqs.~\\eqref{B1} -- \\eqref{B7} below, one may derive the above \nconditions within some lines by use of the following obvious two relations\n(which\nfollow directly from the defining Equation~\\eqref{CF} and the fact\nthat $\\Qo$, ${Q_2}$ are nilpotent vector fields, cf.~also \\cite{BKS,Str04b})\n\\begin{eqnarray}  \\Qo  \\circ \\CF &=& - \\CF \\circ  {Q_2} \\label{dF} \\\\\n   \\CF(\\phi) &=& F^\\alpha \\, a^*(\\partial_\\alpha \\phi) \\label{FLeibniz} \n\\end{eqnarray}\nwhere the second equation holds for all functions $\\phi$ on the target\nand $\\partial_\\alpha$ denotes a left derivation w.r.t.~$q^\\alpha$. \nWe illustrate this for $\\alpha=i$: \nApplying the left hand side of \\eqref{dF} to $x^i$, we get $\\ud F^i$. On the \nother hand $\\CF ({Q_2} x^i) = \\CF(\\rho^i_a\\xi^a) = \\CF(\\rho^i_a) a^*(\\xi^a)+\na^*(\\rho^i_a) \\CF(\\xi^a) = \\rho^i_{a,j} \\CF(x^j) A^a + \\rho^i_a F^a$.\n(Following \nusual physics notation,\nwe did not explicitly display $a^*$ when acting on \nfunctions over $M$ such as  $\\rho^i_{a,j}\\equiv \\partial_j \\rho^i_a$,\nwhich are then understood as\nfunctions over $\\Sigma$ via the ``scalar fields'' $X^i$).\n\n\n   \nThe result above is readily specialized to particular cases of interest. \nE.g., for $M$ a point, we merely drop all terms containing $F^i$. If there is \nno $B$-field (i.e.~there are no degree two generators on $\\CM_2$), we \nput $F^D$ and $B^D$ to zero wherever they occur. Combining both steps\nwe regain \nthe Bianchi identities of ordinary Yang-Mills theory. \n\n\\vskip\\newpassage\n\nBefore continuing we briefly comment on some further\ngeneralization. In the ansatz \\eqref{Fi}--\\eqref{FB} we could\nhave permitted also terms proportional to $\\ud A^\\alpha$, adding e.g.~a\nterm $\\nu^a_i \\ud X^i$ to the r.h.s.~of \\eqref{Fa} etc. Since\nsuch terms enter field strengths $F^\\beta$ only for\nvalues of $\\beta$ corresponding to a higher form degree than the one\nof $A^\\alpha$, it is always possible to replace $\\ud A^\\alpha$ by $F^\\alpha$ by\nmerely renaming coefficient functions in $F^\\beta$. Thus, in the above\nexample of Eq.~\\eqref{Fa}, we may instead equally well add $\\nu^a_i\nF^i$ to the ansatz, merely by\nreplacing $C^a_{bc}$ with new structure functions. Constructing\nout of these new structural functions---except for the additional\ncoefficients $\\nu^\\beta_\\alpha$, which we may just ignore at this point---the \nvector field $Q_2$ as before, Eqs.~\\eqref{Qi}--\\ref{QB}, \nTheorem~\\ref{theo:nilpot}\nstill holds true: While the ideal $\\I$ generated by field strengths changes with \nthe change of structural functions in the first step of the rewriting, it does not change by adding or dropping the terms with lower degree field strengths on the right hand sides in their definition and thus $\\ud \\I \\subset \\I$ gives the\nsame conditions.\n\nMore generally, the ideal $\\I$ remains unaltered by every\nredefinition of the form $\\widetilde F^\\alpha = M^\\alpha_\\beta F^\\beta$ if the field\ndependent matrix $M^\\alpha_\\beta$ is invertible. If the form degree\nof $A^\\alpha$ is denoted by $|\\alpha|$, then $M^\\alpha_\\beta$ has form\ndegree $|\\alpha| - |\\beta|\\ge 0$. It thus is a lower triangular matrix, where\nthe non-vanishing off-diagonal components correspond to $\\nu^\\alpha_\\beta$\nintroduced above. The (block) diagonal pieces, those where $|\\alpha| =\n|\\beta|$, correspond to $X$-dependent coefficient matrices in front of \n$\\ud A^\\alpha$ in the definition of the new field strength $\\widetilde\nF^\\alpha$---the matrix $M^\\alpha_\\beta$ is thus invertible, if and only if each of its \ndiagonal components is nonzero everywhere. For example, instead of \n\\eqref{Fi} we would consider\n$\\widetilde F^i = M^i_j \\ud X^j - \\widetilde \\rho^i_a A^a$. Now, if the\nmatrix $M^i_j(X)$ is invertible (for every point in the target $M$),\nwe may introduce $\\rho^i_a := (M^{-1})^i_j \\widetilde \\rho^j_a$, and\nthis is the first component of the vector field $Q_2$ of Theorem\n\\ref{theo:nilpot}, \nand so on.  Only\nredefinitions of the above form with a non maximal and possibly even\nvarying rank of the\ndiagonal pieces of $M^\\alpha_\\beta$ would require a new and in general more\nintricate analysis. \n\nAlso coordinate transformations on $M$, or, more\ngenerally, on the supermanifold $\\CM_2$, will induce\nredefinitions of field strengths. In an active interpretation of coordinate transformations, i.e.~viewing them as (local expressions of) diffeomorphisms instead of a \nchange of a chart (passive interpretation in physics terminology), the ideal $\\I$ changes in general, but in a covariant way. In this active interpretation $\\I$ remains unchanged only under very particular super-diffeomorphisms, namely those that leave the $Q$-structure invariant; we will come back to them in the context of gauge symmetries in section \\ref{s:gauge} below. \n\nIf $M$ is not a point, \nthe field strength components $F^\\alpha$ themselves do, however, \\emph{not} transform covariantly, except for the lowest ones with $|\\alpha| =0$. (Only the complete set\nof the whole ideal $\\I$ does). \n This can be corrected by means of the choice of connections in\nthe vector bundles $E$ and\n $V$.\\footnote{The bundles $E$ and $V$ and these connections are not to\nbe confused with the (total) bundle of the gauge theory and the gauge fields,\nrespectively. The first one, the total bundle, is assumed to be trivial\nwithin this paper---for a generalization cf.~\\cite{KS07}---and we\non purpose did not call $A^\\alpha$ a connection since this would require\nfurther geometrical justification, and in some way is even the wrong\nnotion. The choice of $M$, $E$, $V$, and possibly connections on the\nlatter two, are fixed background data; they replace the choice of a\nLie algebra in ordinary Yang-Mills theories. Only the gauge fields $A^\\alpha$ \nare dynamical.} One way of seeing this is to first determine the\ntransformation behavior of all the structural coefficient functions \nunder a change of coordinates on ${\\CM_2}$---we will do this within the\nsubsequent section---and then to correct for the overall behavior of\n$F^a$ and $F^D$ by appropriate additions. There is a much simpler route,\nhowever, using the fact\nthat $\\CF$ is a (graded) Leibniz-type operator\nover $a^*$, cf.~Eq.~\\eqref{FLeibniz}:\nA change of local frame in the vector bundle $E$, e.g.,\ncorresponds to a change of coordinates $\\xi^a \\mapsto\n\\widetilde{\\xi^a} = M^{\\tilde a}_b \\xi^b$ on ${\\CM_2}$, where $M^{\\tilde a}_b$ are\nfunctions on the base\\footnote{which is not to be confused with\nspacetime $\\Sigma$---both $M$ and $E$ are data of the target.}\n $M$ of $E$. With \\eqref{FLeibniz} this implies $F^{\\widetilde{a}}= M^{\\tilde\n a}_b(X) F^b +  M^{\\tilde a}_{b,i}(X) F^i  A^b$, \n where $M^{\\tilde a}_{b,i} \\equiv \\partial\n M^{\\tilde a}_{b}\/\\partial x^i$. Only the first term of the two\n corresponds to a tensorial behavior. From this it is obvious\n that if $\\Gamma^a_{ib}$ and\n $\\Gamma^D_{iC}$ denote connection coefficients in local frames in $E$\n and $V$, respectively, the redefined field strengths\n \\begin{eqnarray} F^a_{(\\Gamma)} &:=& F^a + \\Gamma^a_{ib} F^i  A^b \\label{FaG}\n \\\\\n     F^D_{(\\Gamma)} &:=& F^D + \\Gamma^D_{iC} F^i  B^C\n  \\label{FBG} \\end{eqnarray}\n(together with the unaltered $F^i$) are tensorial. In some gauge invariant \naction functionals $S$  there may be\na contribution of the form $\\Lambda_i F^i$, where $\\Lambda_i$ is a\nLagrange multiplier field;\\footnote{Cf, e.g., \\cite{Str04b,KS08}.} in such a case one may replace $F^a_{(\\Gamma)},\nF^D_{(\\Gamma)}$ by the simpler expressions\n$F^a, F^D$ in the remaining part of the action $S$---at\nthe mere expense that the new, redefined Lagrange multipliers loose\ntheir tensorial behavior. For other functionals these contributions\nmay be mandatory, however. \n\nFinally, an interesting alternative to the use of $\\CF$ or the auxiliary connections needed for a tensorial behavior of the field strengths like in Eqs.~\\eqref{FaG} and \\eqref{FBG} is to introduce a map $f \\colon \\CM_1 \\to T[1]\\CM_2$ defined by means of $f^*(q^\\alpha)=a^*(q^\\alpha)$ and $f^*(\\ud q^\\alpha)=\\CF (q^\\alpha)$;\\footnote{This was found later in the paper \\cite{KS07}.} \n equipping $T[1]\\CM_2$ with an appropriate canonical Q-structure, $f$ becomes a Q-morphism for \\emph{every} map $a \\colon \\CM_1 \\to \\CM_2$ and generalizes the Chern-Weil map in the case of non-trivial Q-bundles as is useful for constructing corresponding characteristic classes. In particular, $f^* \\colon {C^\\infty}(T[1]\\CM_2 \\to \\Omega^\\bullet(\\Sigma))$ is an algebra morphism then, $f^*(x^i\\xi^a \\ud \\xi^D) = X^i A^a \\wedge F^D$ etc. \n\n\\newpage\n\n\\section{Q-structures for $p=2$ --- Lie 2-algebroids}\n\\label{s:examQ}\nIn this section we study general vector fields of the form\n\\eqref{Qi}--\\eqref{QB}. Dropping the index 2 for notational\nconvenience within \\emph{this} section,\n\\beq \\label{Q} Q = \\rho^i_a \\xi^a \\frac{\\partial}{\\partial x^i}\n- \\frac12 C^a_{bc} \\xi^b \\xi^c \\frac{\\partial}{\\partial \\xi^a} +t^a_D b^D\n\\frac{\\partial}{\\partial \\xi^a}  -\\Gamma^D_{aC} \\xi^a b^C\n\\frac{\\partial}{\\partial b^D}\n+ \\frac{1}{6} \n        H^D_{abc} \\xi^a \\xi^b \\xi^c \\frac{\\partial}{\\partial b^D} \\eeq\nis the most general degree one vector field on a degree two\nsupermanifold (cf.~also footnote \\ref{fn:pos}). \nThe range of indices is $i = 1, \\ldots, n$, $a = 1,\n\\ldots r$, $D=1 \\ldots s$, respectively,\nwhere, in previous notation,  $n = \\dim M$, $r = \\dim V$, and $s  = \\dim W$.\nNote that $C^c_{ab}$ and $H^D_{abc}$ are completely\nantisymmetric in their lower indices by construction. \nRequiring $Q$ to be homological, that is to square to\nzero and thus to define a Q-structure, one obtains the following identities to\nhold true: \n\\begin{align}\n  \\rho^j_{[a}\\rho^i_{b],j} -\\frac12\\rho^i_c C^c_{ab}  &= 0  \\label{B1}\\\\\n  \\rho^i_a t^a_D &= 0  \\label{B2}\\\\\n  3C^e_{[ab} C^d_{c]e} +3\\rho^i_{[c} C^d_{ab],i} - t^d_D H^D_{abc} &= 0\n    \\label{B3}\\\\\n  \\rho^i_c t^a_{D,i} -\\Gamma^F_{cD} t^a_F -C^a_{bc}t^b_D &= 0  \\label{B4}\\\\\n  \\rho^i_{[b}\\Gamma^D_{a]F,i}  +\\frac12 \\Gamma^D_{eF}C^e_{ab}\n    -\\Gamma^D_{[aC}\\Gamma^C_{b]F} +\\frac12H^D_{abc}t^c_F &= 0  \\label{B5}\\\\\n  t^a_{(A} \\Gamma^D_{aC)} &= 0  \\label{B6}\\\\\n  \\Gamma^D_{[aF}H^F_{bcd]} +\\rho^i_{[a}H^D_{bcd],i} -\\frac32H^D_{e[ab}C^e_{cd]}  &= 0  \\label{B7}\n\\end{align}\nThe first two equations result upon twofold application of $Q$ to\ncoordinates $X^i$, the next three to $\\xi^a$, and the last two when\n$Q^2$ is applied to $b^D$ (the result being put to zero\nin all cases). Round\/square brackets indicate symmetrization\/antisymmetrization\nof enclosed\nindices of equal type (in Eq.~\\eqref{B5}, e.g., this concerns the indices $a$\nand $b$ in the respective two terms, and only them).\n\nWe now discuss the meaning of these equations in more standard,\nindex-free terms. We start with $M =\\mathrm{pt}$ and $s=0$, i.e.~there are no $X^i$\nand $b^D$ coordinates. Correspondingly,  in \\eqref{Q} only the term\ncontaining $C^c_{ab}$  survives. From the above set of equations, the\nonly nontrivial one is Eq.~\\eqref{B3}, reducing to $C^e_{[ab}\nC^d_{c]e}=0$, which is the Jacobi identity for the coefficients\n$C^a_{bc}$. Thus in this simple case, in which the supermanifold\n$\\CM$ has a\ntrivial base and is actually only of degree one,\none finds the Q-manifold\n$(\\CM,Q)= (\\g[1],\\ud_{CE})$, where $\\g$ is some Lie algebra and\n$\\ud_{CE}$ the corresponding Chevalley-Eilenberg\ndifferential. Super vector spaces of degree one\nwith Q-structure are equivalent to\nLie algebras. \n\nLet us in a next step drop the restriction that $M$ is a point,\ni.e.~we deal with a general degree one supermanifold (which,\ncertainly, may also be viewed as a particular degree two\nsupermanifold) with Q-structure. This leaves the first two\nterms in \\eqref{Q}. The corresponding equations, \\eqref{B1} and\n\\eqref{B3} with $t^d_D \\equiv 0$, are recognized as the\nstructural equations for a Lie algebroid: A Lie algebroid is a vector\nbundle $E$ over $M$ together with a bundle map $\\rho \\colon E \\to TM$,\ncalled the anchor map, \nand a Lie algebra defined on the sections of $E$ satisfying the\nLeibniz rule $[\\psi, f\n\\varphi] = f [\\psi, \\varphi] + \\rho(\\psi) f \\, \\varphi$.\\footnote{For\nlater use we mention that we call $E$ an \\emph{almost Lie algebroid} if it\nis defined as above with the mere difference that the bracket $[\n\\cdot, \\cdot ]$ needs to\nbe antisymmetric only, but not necessarily to satisfy the Jacobi\nidentity. If $\\rho \\equiv 0$ in an (almost) Lie algebroid, the bracket\ndefines a fiberwise\nproduct and $E$ becomes a bundle of (almost) Lie algebras.}\n{} From these\ndata one can infer that $\\rho$ is also a morphism of Lie algebras, $\\rho([\\psi, \n\\varphi]) =  [\\rho(\\psi), \\rho(\\varphi)]$. Choosing a local frame\n$\\xi_a$ in $E$ and local coordinates $x^i$ on $M$, the anchor gives\nrise to structural functions $\\rho^i_a=\\rho^i_a(x)$ \nvia $\\rho(\\xi_a) = \\rho^i_a \\partial_i$. Likewise, the bracket between\ntwo sections from the basis induces $C^c_{ab}=C^c_{ab}(x)$:\n$[\\xi_a,\\xi_b]= C^c_{ab} \\xi_c$. Now it is easy to see that the\nmorphism property of $\\rho$ implies Eq.~\\eqref{B1} and the Jacobi\nidentity for the Lie bracket on $E$, together with the Leibniz rule,\nyields (the respective remainder of) Eq.~\\eqref{B3}. \n\nFor the reverse direction, i.e.~to recover the index free geometrical\nstructure from the above formulas, one needs to address coordinate\nchanges on the Q-manifold $\\CM$. By definition, coordinate changes on\na graded manifold need to be degree preserving; therefore, \n$\\widetilde{x^i}= \\widetilde{x^i}(x)$,\n$\\widetilde{\\xi^a} = M^{\\tilde a}_b(x) \\xi^b$. Thus one deals with a\nvector bundle $E$ over the base $M$ of $\\CM$ and $\\CM = E[1]$ (by\ndefinition, for a vector bundle the shift in degree\nconcerns only the fiber coordinates). Implementing this coordinate\nchange on $Q$, we see that $\\rho^i_a$ transforms like a tensor, but\n$C_{ab}^c$ does not. Indeed,\n$\\partial_i=\\frac{\\partial \\widetilde{x^j}}{\\partial x^i}\n\\widetilde{\\partial_j} + \\frac{\\partial \\widetilde{\\xi^a}}{\\partial\nx^i} \\widetilde{\\partial_a}$, where \n$\\partial_i = \\partial\/\\partial x^i$, $\\partial_a = \\partial\/\\partial \\xi^a$\netc. So, there is a contribution proportional to $\\rho^i_a$ and the\n$x$-derivative of $M^{\\tilde a}_b$ to the transformation law of\n$C_{ab}^c$. Since $C_{ab}^c \\xi_c = [\\xi_a, \\xi_b]$ is to hold true\nin every frame---after all we do not want to have frame-dependent\ndefinitions of a bracket---this results in the Leibniz property of the\nbracket. The rest is then obvious.\n\nSummarizing, a degree one supermanifold with Q-structure, \ncalled Q1-manifold for simplicity,  is \\emph{tantamount}\nto a Lie algebroid, $(\\CM,Q)=(E[1],{}^E\\ud)$.\\footnote{This was observed first by Vaintrob \\cite{Vai97}, including also the compact formulation of Lie algebroid morphisms (a proof of equivalence of this definition with the standard, more complicated one of \\cite{MaH93} can be found in \\cite{BKS}). --- Recall \\label{f:8}  that we \nrestricted to non-negatively graded supermanifolds; Q-manifolds with \nthis restriction are also called NQ-manifolds in the literature, cf., e.g., \n\\cite{Sch93,AKSZ}.} Here ${}^E\\ud$ is the\ncanonical differential that generalizes $\\ud_{CE}$ from above for\n$M=\\mathrm{pt}$ and the de Rham differential for $E=TM$, $\\rho = \\id$,\nthe standard Lie algebroid (cf., e.g., \\cite{CaWe99} for\n${}^E\\ud$  or just\nuse the above formulas for $Q$, reinterpreted correspondingly, as a\npossible definition). \n \nIn order to reveal the differential geometry contained in a general\n``Q2-manifold'', i.e.~a degree two graded manifold with Q-structure, \nEq.~\\eqref{Q} together\nwith Eqs.~\\eqref{B1}--\\eqref{B7}, we follow the same strategy: We\nfirst look at degree preserving coordinate changes on $\\CM$,\n\\beq \\widetilde{x^i}= \\widetilde{x^i}(x), \\qquad \n\\widetilde{\\xi^a} = M^{\\tilde a}_b(x) \\xi^b , \\qquad\n \\widetilde{b^D}  = N^{\\tilde D}_C(x)  b^C +\\frac12 L^{\\tilde D}_\n{ab}(x)\\xi^a\\xi^b  \\, . \\label{trafo} \\eeq\n If there were no $\\xi^a$ coordinates,\ni.e.~$r=0$, similarly to before we would obtain $\\CM = V[2]$ for some\nvector bundle $V$ over $M$. The above $L$-terms imply that in general\n$\\CM$ is not just the direct sum of two vector bundles $E$ and $V$,\nbut instead one has the sequence of supermanifolds\n\\beq V[2] \\to \\CM \\to  E[1] \\, . \\label{sequence} \\eeq\n(The first map is an embedding, characterized \nby $\\xi^a =0$, and the second map a (surjective)\nprojection, dropping the $b^D$ coordinates). \nOnly after the choice of a\nsection $\\imath \\colon E[1] \\to \\CM$, characterized locally by $b^D = L^D_{ab}(x) \\xi^a \\xi^b$, \nthis gives rise to $\\CM = E[1] \\oplus V[2]$. The \\emph{difference} between two \nsections is a \nsection $B$ of $\\Lambda^2 E^* \\otimes V$---similarly to the fact that the\ndifference between two connections in a vector bundle\ncorresponds to a tensorial object.\nIn what follows the choice of a section \nshall be understood, so that in terms of ordinary differential geometry \nthere is a total space corresponding to the Whitney sum  \n$E \\oplus V$ of two vector bundles over $M$. However, let us mention that when the Q-manifold under discussion carries \nadditional structures, like a compatible symplectic form, this may not be the best adapted description.\n\n\n\n\nSo, under the above assumption, a (non-negatively) graded manifold of (maximal) degree two gives rise to\na vector bundle $E \\oplus V$ \nover $M$. Next we\ndetermine the transformation properties of the coefficients in\n\\eqref{Q}. For later purposes we keep also $L$-contributions,\ndisregarding them only afterwards within the rest of this\nsection. With \\eqref{trafo} one\nobtains:\n\\begin{eqnarray}\n  \\widetilde{\\rho^i_a} &=& \\frac{\\partial\\tilde x^\\imath}{\\partial\nx^j} \\rho^j_b (M^{-1}) ^b_{\\tilde a}  \\label{trafo1} \\\\\n  \\widetilde{C_{ab}^c} M^{\\tilde a}_d M^{\\tilde b}_e    \n   &=& M^{\\tilde c}_f C_{de}^f\n    -2M^{\\tilde c}_{d],i}\\rho^i_{[e}\n    +M^{\\tilde c}_f t^f_D (N^{-1})^D_{\\tilde E} L^{\\tilde E}_{de}\n   \\label{trafo2} \\\\\n  \\widetilde {t^a_D} &=& M^{\\tilde a}_b t^b_C  (N^{-1})^C_{\\tilde D}\n  \\label{trafo3}\n  \\\\\n  \\widetilde {\\Gamma^C_{aE}} M^{\\tilde a}_b N^{\\tilde E}_D \n   &=&  N^{\\tilde C}_E \\Gamma^E_{bD}\n    -N^{\\tilde C}_{D,i} \\rho^i_b\n    -L^{\\tilde C}_{db} t^d_D  \\label{trafo4} \\\\\n  \\widetilde {H^D_{def} } M^{\\tilde d}_a M^{\\tilde e}_b M^{\\tilde f}_c\n       \n   &=& N^{\\tilde D}_C H^C_{abc} \n    +3N^{\\tilde D}_F \\Gamma_{a]C}^F (N^{-1})^C_{\\tilde E} L^{\\tilde E}_{[bc}\n    +3\\rho^i_{[a} L^{\\tilde D}_{bc],i}\n    -3L^{\\tilde D}_{d[a}C^d_{bc]}\n  \\nonumber\\\\  &&\n    -3N^{\\tilde D}_{C,i}\\rho^i_{[a} \\big(N^{-1}\\big)^C_{\\tilde E}\n    L^{\\tilde E}_{bc]}\n    -3L^{\\tilde D}_{d[a} t^d_C (N^{-1})^C_{\\tilde E} L^{\\tilde E}_{bc]}\n  \\label{trafo5}\n\\end{eqnarray}\nNote that $C_{ab}^c$, $H^D_{abc}$ and $\\Gamma_{aD}^C$ are affected by the affine \n$L$-contributions. Only the two quantities $\\rho^i_a$ and $t_D^a$ are\nnot; they both correspond to bundle maps, namely $t \\colon V\n\\to E$ and $\\rho \\colon E \\to TM$ or, equivalently, to sections of\n$V^* \\otimes E$ and $E^* \\otimes TM$, respectively.\n If $L$-terms are suppressed, $H$ is likewise just a \nsection of $\\Lambda^3 E^* \\otimes V$. But the nature of all three\nobjects, $H$,\nthe bracket on $E$ induced by $C^c_{ab}$, and the object corresponding\nto $\\Gamma_{aD}^C$ discussed further below\ncan change if $L^D_{ab}$ is not forced to vanish. \nFor the next steps we will first restrict to vanishing $L$-contributions, i.e.~to\na fixed choice of a section of \\eqref{sequence}\n\nEven under this assumption, the nature of $\\Gamma_{aD}^C$ is more\nintricate than that of $H$. \nAs anticipated by the notation,  $\\Gamma_{aD}^C$ are\nthe coefficients of a kind of connection. More precisely, \nintroduce an\n$E$-connection ${{}^E{}\\nabla}$ in $V$, i.e.\\ for every $\\psi \\in \\Gamma(E)$ an\noperator $\\Econn_\\psi \\colon \\Gamma(V)\n\\to \\Gamma(V)$ satisfying the Leibniz rule \\mbox{\\( \\Econn_\\psi (fv) =\n\\rho(\\psi)f \\cdot v +f \\Econn_\\psi v \\)} such that the assignment $\\psi\n\\mapsto \\Econn_\\psi$ is $C^\\infty(M)$-linear in\n$\\psi$.\\footnote{More abstractly, $\\Econn_\\cdot$ corresponds to a map from $E$\n  to $A(V)$, the Atiyah algebroid of $V$. $A(V)$ is a vector bundle over $M$\n  itself, namely the Lie algebroid\n  corresponding to the canonical Lie groupoid of automorphisms of the\n  vector bundle $V$.}\n This definition reduces to the one of an ordinary covariant\nderivative in the case of $E=TM$. Then $\\Gamma_{aD}^C$ are just the\nE-connection coefficients in a local frame, $\\Econn_{\\xi_a} b_D =\n\\Gamma_{aD}^C b_C$; now Eq.~\\eqref{trafo4}  follows from\n\\eqref{trafo} (both without the respective\n$L$-contribution, \\eqref{trafo} interpreted as a change of frames in\nthe two vector bundles) by means of the properties of ${{}^E{}\\nabla}$ and vice versa.     \n\nFinally the coefficients $C_{ab}^c$ induce a\nbracket $[ \\cdot , \\cdot ]$ on sections of $E$  via $[\\xi_a,\\xi_b] :=\nC_{ab}^c \\xi_c$, which is antisymmetric by construction and, \n\\emph{according to} \\eqref{trafo2}\n without $L$-terms, satisfies a Leibniz\nrule. \n\nNow, using the above data, we are to reinterpret the conditions\nensuring $Q^2=0$. In this way we arrive at:\n\\begin{prop} \\label{prop:double}\nA Q-structure on a supermanifold of degree two (a Q2-manifold)  \nwith section of the sequence \\eqref{sequence} as described above \nis equivalent to a Lie 2-algebroid.\n\\end{prop}\n\\begin{vdef} \\label{Lie2def} A \\emph{Lie 2-algebroid} is a complex of vector bundles\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\beq   \\mbox\n\\setlength{\\unitlength}{3947sp}%\n\\begin{picture}(1297,885)(601,-361)\n\\thicklines\n{\\put(1351,314){\\vector( 1, 0){375}}\n}%\n{\\put(1876,164){\\vector(-2,-1){540}}\n}%\n{\\put(751,314){\\vector( 1, 0){375}}\n}%\n{\\put(1276,164){\\vector( 0,-1){300}}\n}%\n{\\put(676,164){\\vector( 2,-1){540}}\n}%\n\\put(601,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{V}%\n}}}\n\\put(1201,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{E}%\n}}}\n\\put(1801,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{TM}%\n}}}\n\\put(1201,-361){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{M}%\n}}}\n\\put(826,389){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{$t$}%\n}}}\n\\put(1426,389){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{$\\rho$}%\n}}}\n\\end{picture}\n\n \\label{compl} \n\\eeq \ntogether with an antisymmetric bracket $[ \\cdot , \\cdot ]$ on $E$, an $E$-connection ${{}^E{}\\nabla}$ \non $V$, and an ``anomaly'' $H \\in \\Gamma(\\Lambda^3 E^* \\otimes V)$ satisfying\nthe following compatibility conditions: \n\\begin{align} [\\psi_1,f\\psi_2] &= \\rho(\\psi_1)f \\, \\psi_2 + \nf[\\psi_1,\\psi_2] \\, ,  \\label{Leibn}\\\\\n  {}[\\psi_1,[\\psi_2,\\psi_3]\\,] +\\textup{cycl(123)} &= t \\big(H(\\psi_1,\n  \\psi_2, \\psi_3)\\big) \\, , \n    \\label{Jac}\\\\{}\n[\\Econn_{\\psi_1},\\Econn_{\\psi_2}]v - \\Econn_{[\\psi_1,\\psi_2]}v &= \nH\\big(\\psi_1,\\psi_2,t(v)\\big) \\, ,  \\label{repr1}\n\\end{align}\nas well as  $t(\\Econn_\\psi v) = [\\psi, t(v)]$ and $\\Econn_{t(v)} w = -\n\\Econn_{t(w)} v$. In addition, $H$ has to satisfy\n\\beq\n  \\Econn_{\\psi_1} H(\\psi_2,\n  \\psi_3, \\psi_4) -\\frac32 H([\\psi_1,\\psi_2],\\psi_3,\\psi_4)\n  +\\textup{Alt}(1234)= 0  \\label{strange} \\, .\n\\eeq\n\\end{vdef}\nWe will now make further remarks on the statement of the above\nproposition and some parts of the above definition. In this way we\nwill arrive also at another, somewhat simpler form of the description of Q2-manifolds (or of what we just called Lie 2-algebroids), provided by Theorem\n\\ref{theo1} below.\\footnote{In this part of the paper we provide a \nclassical differential geometry description of a Q2-manifold. The relatively simple notion of a Q2 manifold in graded geometry encodes relatively involved data in the classical geometry setting. These are, however, chosen so as to mimic the standard definition of a Lie algebroid (corresponding to Q1-manifolds) and, simultaneously,  generalize existing definitions of strict and semi-strict Lie 2-algebras. It may be reasonable to call the above or any equivalent data in \\emph{classical} differential geometry \\emph{semi-strict} Lie 2-algebroids, so as to clearly distinguish from definitions like the ones given in the Appendix or increasingly popular conventions where Q$k$-manifolds are considered as a defintion of Lie $k$-algebroids (cf also the corresponding discussion in the Introduction following Theorem \\ref{thm1.5}).}\n\n\nIn the above, $\\psi_i \\in \\Gamma(E)$, $f \\in\nC^\\infty(M)$, and $v,w \\in \\Gamma(V)$. \\eqref{compl} defining a\ncomplex just means that $ \\rho \\circ t = 0$, as corresponds to\nEq.~\\eqref{B2}. The three projections going to one and the same\ncopy of the base $M$ means that the bundle maps $t$ and $\\rho$ cover\nthe identity map on $M$. Eqs.~\\eqref{Jac}, \\eqref{repr1}, and\n\\eqref{strange} correspond to Eqs.~\\eqref{B3}, \\eqref{B5},\nand \\eqref{B7}, respectively; the two equations in the text are\n\\eqref{B4} and \\eqref{B6}. From Eqs.~\\eqref{Leibn} and\n\\eqref{Jac} together with $ \\rho \\circ t = 0$ one may show that\n$\\rho$ is a morphism of brackets:\n\\beq   \\rho\\big([\\psi_1,\\psi_2]\\big) = [\\rho (\\psi_1), \\rho (\\psi_2)] \\, , \n\\label{rhomor}  \\eeq\nwhere the bracket on the r.h.s.~denotes the commutator of \nvector fields (i.e.~the bracket in the standard Lie algebroid $TM$). \nThis equation, finally, reduces to \\eqref{B1} in components. \n\n\n\n\nCondition \\eqref{Leibn} is the Leibniz rule for the bracket on\n$E$, $t \\circ H$ defining its Jacobiator according to\n\\eqref{Jac}. So, for $H=0$, or even for $H$ living only in the\nkernel of $t$, $E$ becomes a Lie algebroid. As follows from\n\\eqref{repr1}, $H$ also governs the $E$-curvature ${{}^E\\!{}R}$ of ${{}^E{}\\nabla}$; indeed,\n$$ {{}^E\\!{}R}(\\psi_1,\\psi_2)  \\equiv [\\Econn_{\\psi_1},\\Econn_{\\psi_2}] -\n\\Econn_{[\\psi_1,\\psi_2]} \\quad , \\qquad \n{{}^E\\!{}R} \\in \\Gamma(\\Lambda^2E^* \\otimes \\End(V)) \\, , \n$$  where $[\\Econn_{\\psi_1},\\Econn_{\\psi_2}]$ denotes the ordinary commutator of derivations, \nis nothing but the generalization of\ncurvature to the context of $E$-connections in $V$ for the \nalmost Lie algebroid $E$. Denoting by ${{}^E{}\\Omega}^p(M,V)$ the sections of\n$\\Lambda^p E^* \\otimes V$, the $E$-$p$-forms with values in $V$, where $V$ is some\nvector bundle over the base $M$ of $E$, ${{}^E\\!{}R}$ becomes an $E$-2-form\nwith values in the endomorphisms of $V$. Note that we used\nEq.~\\eqref{rhomor} to show \nthat the operator ${{}^E\\!{}R}(\\psi_1,\\psi_2)$ is\n$C^\\infty(M)$-linear and thus indeed defines an \nendomorphism of $V$. $H=0$ implies a flat E-connection, $ {{}^E\\!{}R}=0$, which \nis a \\emph{representation} of $E$ on $V$.  The condition\n\\eqref{strange} implies that the Jacobi condition for $[ \\cdot ,\n\\cdot ]$  and the\nrepresentation property of ${{}^E{}\\nabla}$ are violated in a relatively mild,\ncontrolled way.\n\nEq.~\\eqref{strange} can be rewritten in a more compact form.\nIntroduce the generalization of an exterior covariant derivative\n${{}^E\\!{}\\mathrm{D}} \\colon {{}^E{}\\Omega}^\\cdot (M,V) \\to {{}^E{}\\Omega}^{\\cdot \\, +1} (M,V)$ \nby means of a generalized Cartan\nformula\n\\begin{align} \\label{genCar} {{}^E\\!{}\\mathrm{D}} \\omega (\\psi_0, \\ldots , \\psi_p) = \n  &  \\sum_i (-1)^i \\Econn_{\\psi_i}\n  \\omega(\\psi_0,\\ldots,\\widehat{\\psi_i},\\ldots,\\psi_p)  \\nonumber \\\\\n  & +\\sum_{i<j} (-1)^{i+j} \\omega\\big( [\\psi_i,\\psi_j], \\psi_0, \\ldots,\\widehat{\\psi_i},\\ldots\\widehat{\\psi_j},\\ldots,\\psi_p \\big)\n  \\, .  \n\\end{align}\nNote that this defines a map in the indicated manner only because\n${{}^E{}\\nabla}$ and the bracket satisfy a Leibniz rule---with the same anchor\n$\\rho$. \nThen the condition to be satisfied by $H$ is nothing but simply\n${{}^E\\!{}\\mathrm{D}} H=0$. \n\nAt this point it is worthwhile to return to the question of changes \nof the section of \\eqref{sequence}. As\nalready mentioned above it corresponds to an $E$-2-form $B$ with\nvalues in $V$. For this purpose we thus need to interpret\nEqs.~\\eqref{trafo1} to \\eqref{trafo5} for the case that\nboth $M^{\\tilde a}_b$ and\n$N^{\\tilde D}_C$ are the identity and $L$ equals $B$.\\footnote{Note\nthat this is\n\\emph{not} the only option one has. Indeed, $L$ may be a function of\n$M$ and $N$ and then changes of the frame in $E$ and $V$ induced by\n$M$ and $N$ result in a different nature of e.g.~the bracket, which\nthen will in general \\emph{not} satisfy a Leibniz rule as before. This\nis what happens in the case of the Courant algebroid, which we will\ndiscuss briefly in a later section; here additional structures, present\nonly in that case, give rise to a natural unique relation of $L$ to\nchanges of frames.} This induces the following changes of the bracket,\nthe $E$-connection, and the anomaly:\n\\begin{eqnarray} [\\psi_1,\\psi_2] &\\mapsto&\\widetilde{ [\\psi_1,\\psi_2]}\n\\equiv\n [\\psi_1,\\psi_2] + t(B(\\psi_1,\\psi_2)) \\label{split1}\n\\\\ \\Econn_\\psi v &\\mapsto& ^E{}\\widetilde{\\nabla}_{\\!\\psi\\,} v \\equiv  \\Econn_\\psi v +\nB(\\psi,t(v))\\label{split2} \n\\\\ H &\\mapsto& H +{{}^E\\!{}\\mathrm{D}} B -C \\, ,  \\label{split3}\n\\end{eqnarray}\nwhere $C(\\psi_1,\\psi_2,\\psi_3) \\equiv B(t(B(\\psi_1,\\psi_2)),\\psi_3) +{\\text{cycl.}}$\nNote in this context that ${{}^E\\!{}\\mathrm{D}}$ does not define a differential for two\nreasons: first, ${{}^E{}\\nabla}$ has a non-zero curvature, and, secondly, the\nbracket on $E$ does not satisfy the Jacobi condition---and these two\nfacts do \\emph{not} cancel against one another. Correspondingly, the\nanomaly $H$ up to changes of the section is not just an element of a\ncohomology. The new and the old $H$ are closed only w.r.t.~the\nnew and the old  $E$-derivatives, ${{}^E\\!{}\\mathrm{D}}$ and  \n$^E\\!\\!{}\\widetilde{D}$, respectively.\n\n\n\n\n\n\nFinally,\n${{}^E{}\\nabla}$ and $t$ may be used to define naturally a bracket on $V$ according to \n\\([v,w]_V:= \\Econn_{t (v)} w \\). According to Prop.~\\ref{prop:double}, \nthis bracket is antisymmetric. Moreover, $\\rho_V := \\rho \\circ t = 0$\ndefines a canonical \nanchor map from $V$ to $TM$ with respect to\nwhich $[ \\cdot , \\cdot ]_V$ may be seen to \nsatisfy a Leibniz rule, i.e.~just $[v, fw]_V = f[v,w]_V$. \nIn this way also $V$ becomes an \nalmost Lie algebroid, in fact a bundle of almost Lie algebras,\nwith Jacobiator determined by $H$ via\n \\beq [v_1,[v_2,v_3]_V]_V +\n{\\text{cycl.}}(123) = H(tv_1,tv_2, tv_3) \\, ,\\label{Jac2} \\eeq\nas can be derived from the equations in Prop.~\\ref{prop:double}.  \nAlso, $t \\colon V \\to E$ may be seen to be a morphism of brackets, \n$t\\big([v,w]_V\\big) = [t (v), t(w)]$. Finally, it remains to check in what sense $\\Econn_\\psi$ \nrespects the bracket on $V$; one obtains\n\\beq \\Econn_\\psi ([v,w]_V) = [ \\Econn_\\psi (v),w]_V + [v, \\Econn_\\psi (w)]_V + \nH(\\psi, t(v),t(w)) \\, . \\label{der} \\eeq\nIn summary, \n\\begin{thm} \\label{theo1}\nA Q2-manifold with section or a Lie 2-algebroid is equivalent to an\nalmost Lie algebroid $(E,[ \\cdot , \\cdot ])$ and a bundle of almost\nLie algebras $(V,[ \\cdot , \\cdot ]_V)$ together with a morphism $t\n\\colon V \\to E$, covering the identity on $M$, an $E$-connection on\n$V$ such that \\beq \\Econn_{t(v)} w= [v,w]_V \\quad , \\qquad t(\\Econn_{\n\\psi} v) = [\\psi, t(v)] \\, , \\label{repr2} \\eeq as well as an\nanomaly $H\\in {{}^E{}\\Omega}^3(M,V)$ such that the Jacobiators on $E$ and $V$ are\n$t \\circ H$ and $H \\circ t$, respectively, and the representation and\nderivation property of ${{}^E{}\\nabla}$ are violated according to the\n$E$-curvature ${{}^E\\!{}R} \\cdot = H( \\, , \\, , t(\\cdot))$ and\nEq.~\\eqref{der}, respectively.  $H$ fulfills the consistency\ncondition ${{}^E\\!{}\\mathrm{D}} H=0$.\n\nA change of section of \\eqref{sequence} corresponds to $L \\in\n{{}^E{}\\Omega}^2(M,V)$ and induces Eqs.~\\eqref{split1} to \\eqref{split3} as\nwell as $[v,w]_V \\mapsto [v,w]_V + L(t(v),t(w))$, with all remaining\nstructures unmodified. \n\\end{thm}\nInstead of the first three lines we could have said also \n``A Q2-manifold with section is equivalent to a complex of almost \nLie algebroids\n\\begingroup\\makeatletter\n\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\beq \\mbox\n\\setlength{\\unitlength}{3947sp}%\n\\begin{picture}(2120,885)(601,-361)\n\\thicklines\n{\\put(1276,314){\\vector( 1, 0){300}}\n}%\n\\put(1351,389){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{$t$}%\n}}}\n{\\put(2251,314){\\vector( 1, 0){300}}\n}%\n\\put(2251,389){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{$\\rho$}%\n}}}\n{\\put(1726,164){\\vector( 0,-1){300}}\n}%\n{\\put(906,172){\\vector( 2,-1){749.200}}\n}%\n{\\put(2699,169){\\vector(-2,-1){870}}\n}%\n\\put(601,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{(V,[.,.])}%\n}}}\n\\put(1651,-361){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{M}%\n}}}\n\\put(2626,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{(TM,[.,.])}%\n}}}\n\\put(1651,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{(E,[.,.])}%\n}}}\n\\end{picture}\n\n  \\label{compl2} \n\\eeq\ntogether with an $E$-connection on $V$ such that \\ldots''. \nIn this sequence $TM$ is regarded as \nthe standard Lie algebroid over \n$M$ and arrows are morphisms. Since $t$ and $\\rho$ cover the identity\nmap, the \nmorphism of almost Lie algebroids then implies that\n$\\rho_V = \\rho \\circ t$ and this has to  vanish due to the complex \nproperty. Note also that we did not try to formulate minimal \nconditions: e.g.~Eqs.\\ \\eqref{Jac2} and  \\eqref{der} \nmay be established from the rest as obvious from our derivation. \n\nOf particular interest is certainly the special case of vanishing \nanomaly, where the situation simplifies considerably: \n\\begin{prop} \n A Q2-manifold with a section such that $H =0$ is equivalent to a \ncomplex of Lie algebroids\n\\begingroup\\makeatletter\n\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\beq \\mbox\n\\setlength{\\unitlength}{3947sp}%\n\\begin{picture}(1297,885)(601,-361)\n\\thicklines\n{\\put(1351,314){\\vector( 1, 0){375}}\n}%\n{\\put(1876,164){\\vector(-2,-1){540}}\n}%\n{\\put(751,314){\\vector( 1, 0){375}}\n}%\n{\\put(1276,164){\\vector( 0,-1){300}}\n}%\n{\\put(676,164){\\vector( 2,-1){540}}\n}%\n\\put(601,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{V}%\n}}}\n\\put(1201,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{E}%\n}}}\n\\put(1801,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{TM}%\n}}}\n\\put(1201,-361){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{M}%\n}}}\n\\put(826,389){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{$t$}%\n}}}\n\\put(1426,389){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{$\\rho$}%\n}}}\n\\end{picture}\n} \n  \\label{compl1} \n\\eeq \ntogether with a representation ${{}^E{}\\nabla}$ of \n$E$ on $V$ satisfying the compatibility conditions \\eqref{repr2}. \n\\label{cor:Liecomp}\n\\end{prop}\nIn the notion of a representation of one Lie algebroid on another one we \nmay include $\\Econn_\\psi ([v,w]_V) = [ \\Econn_\\psi (v),w]_V + [v,\n\\Econn_\\psi (w)]_V$, or otherwise we may even derive \nthis condition from the other assumptions in the proposition.\n\nIt remains to specialize to the case of $M=\\mathrm{pt}$, \ni.e.~to ``Q-vector spaces''. E.g.~then Prop.~\\ref{cor:Liecomp} reduces to \n\\begin{prop} A Q2-vector space with section such that $H=0$ is \nequivalent to a Lie 2-algebra. \n\\end{prop}\n\\begin{vdef} A Lie-2-algebra (a differential crossed module) \nis a pair $(\\g,\\h)$ of Lie algebras \ntogether with a homomorphism $t \\colon \\h\\to \\g$ and a  representation\n$\\alpha \\colon \\g\\to\\Der(\\h)$ such that $\\, t((\\alpha x)v) =[x,t(v)]$   \nand $ \\alpha(t(v))w =[v,w]_\\h$  for all $x \\in \\g$, $v,w \\in \\h$ hold true.  \n\\end{vdef}\nHere we adapted to the notation of \\cite{Baez02jn}. The relation is \nestablished easily via $\\g =E$, $\\h = V$ and $\\alpha(x) \\equiv \\Econn_x$ for \n$x \\in \\g \\cong \\Gamma(\\g)$, $\\alpha_{aD}^C \\equiv \\Gamma_{aD}^C$.\\footnote{$\\alpha$ \nis not an arbitrary representation on $\\h$, \n$\\alpha \\in \\End(\\h)$ and $[\\alpha(x),\\alpha(y)] = \\alpha([x,y])$---the flatness property of ${{}^E{}\\nabla}$, \nbut the fact that $\\h$ is a Lie algebra itself is taken into account \nby restricting to $\\Der(\\h) \\subset  \\End(\\h)$: \n$\\alpha(x) \\in \\Der(\\h) \\Leftrightarrow \\alpha(x) ([v,w]) =  [\\alpha(x)(v),w]\n+[v,\\alpha(x)(w)]$---which follows from our Condition~\\eqref{der} with $H=0$.} \nThe case of a Q2-vector space with non-vanishing $H$ (and fixed\nsection $\\imath$) corresponds to a \nsemistrict Lie 2-algebra in the nomenclature of Baez, cf.~e.g.~\\cite{Baez03vi}. \n\n  \n\n\n\\newpage\n\n\\section{Gauge transformations} \\label{s:gauge}\nHere we introduce a notion of infinitesimal gauge\ntransformations. The present section is intended to\nparallel Section~\\ref{s:Bian}, where we introduced an as general as possible\nnotion of field strength of gauge fields, imposing conditions that a\nphysicist would generically consider mandatory---including certainly\nordinary Yang--Mills gauge theories as particular example, but also\nknown generalizations like those of \\cite{Str04b}, \\cite{SaWei:2005bp,SaWit:2005hv}, \\cite{BM05}, \\cite{ACJ05} or \\cite{BS05}. \nSuch as we required the field strengths $F^\\alpha$ to start with $\\ud\nA^\\alpha$, likewise now we want to require that the gauge\ntransformations\nfor $A^\\alpha$ start with $\\ud \\epsilon^\\alpha$ (except those for the zero\nforms $X^i$ certainly). Again this is satisfied in all the \nexamples we just mentioned---at least after choosing appropriate parameters. Correspondingly, infinitesimal gauge\ntransformations are\nparameterized by quantities $(\\epsilon^\\alpha) :\\equiv\n(\\epsilon^a,\\mu^D)$ which are differential\nforms on $\\Sigma$ of form degree one less\nthan the respective $A^\\alpha$ (so $\\mu^D$ are 1-forms); due to the index\n$\\alpha$, in general $\\epsilon^\\alpha$ should be considered as functions of\n$X^i(\\sigma)$ as well---otherwise formulas are at least frame\ndependent---but we postpone this issue to a more careful\ndiscussion below. We thus arrive at the following rather general ansatz: \n\\begin{align}  \\delta_\\varepsilon X^i &= \\bar\\rho^i_a  \\varepsilon^a  \\nonumber\\\\\n  \\delta_\\varepsilon A^a &= \\ud\\varepsilon^a +\\bar C^a_{bc}A^b \\varepsilon^c -\\bar t^a_D \\mu^D +O(F^i) \\nonumber\\\\\n  \\delta_\\varepsilon B^D &= \\ud\\mu^D -\\bar\\Gamma^D_{aC} \\varepsilon^a B^C +\\bar\\Gamma^{\\prime\\,D}_{aC}\n    A^a \\mu^C +\\frac12 \\bar H^D_{abc} A^a  A^b \\varepsilon^c\\,\n    +O(F^i,F^a) \\,.  \\label{gsymm0}\n\\end{align}\nHere $O(F^\\alpha)$ is supposed to denote contributions that vanish for\n(the respective)\n$F^\\alpha=0$. They are required already to obtain frame-independence of\nformulas, while there exist several options to achieve this \n(cf.~also the discussion in \\cite{BKS}). Moreover, there may be\nadditional, model dependent contributions necessary to render a\nparticular action functional $S$ invariant w.r.t.~gauge transformations.\nSuch contributions may use additional data present in $S$,\nincluding even a metric $h$ on $\\Sigma$; an illustration is provided by\nthe Dirac sigma model \\cite{DSM}, where the $A^a$ variations contain\ncontributions proportional to $\\ast F^i$, where $\\ast$ denotes the\nHodge duality operation induced by $h$ on $\\Sigma$, which is 2-dimensional\nin this case (moreover, such a contribution makes sense only in this\ndimension).\nIn this way one\ncould even consider completely general\n$F^a$ contributions to $\\delta A^a$, or likewise\ncontributions to $\\delta X^i$. We excluded them in the above (thus\nonly ``rather general'') ansatz, where we assume that the additions live in the ideal $\\I$ and are homogeneous of the correct form-degree (without any additional structures); that is why, e.g., only $F^i$ is permitted to enter $\\delta A^a$ in \\eqref{gsymm0}. \n\nThe coefficients in the ansatz~\\eqref{gsymm0} are arbitrary\nfunctions of $X(\\sigma)$ at this point. The notation chosen anticipates\nthe result one obtains, where one simply drops the bars\n(and also the prime in the third contribution to the last formula).\n\n\n\n\n\nIn analogy to the Bianchi identities \\eqref{bian}, \\eqref{ideal}\nwe require that gauge transformations of field strengths result into\ncombinations that \\emph{contain} field strengths again, i.e.~we\nrequire: $\\delta_\\varepsilon F^\\alpha = \\L_\\beta^\\alpha\nF^\\gamma$ for all possible choices of gauge parameters \n$\\varepsilon^\\alpha$ and field configurations $A^\\alpha$---where  the coefficients\n$\\L_\\beta^\\alpha$ depend \nlinearly on $\\varepsilon^\\alpha$ (and, in principle, its first derivatives) and are\npermitted to be also field dependent---or,  in other\nwords, we require\n\\beq  \\boxed{\\delta_\\varepsilon F^\\alpha|_{F^\\beta=0} \\equiv 0 \\quad \\textrm{or}  \\quad \\delta_\\varepsilon\n\\I \\subset \\I } \\; .\n\\label{gauge}\\eeq\nSo the ideal $\\I$, defined in Section~\\ref{s:Bian}, should not only\nbe invariant w.r.t.~the operator $\\ud$ but also w.r.t.~the operator\n$\\delta_\\varepsilon$ (for arbitrary choices of the gauge transformation\nparameters $\\varepsilon^a$ and $\\mu^D$). \n\nSuch relations are certainly always satisfied in all the examples\nmentioned above. In particular, even if some of the $X^i$ fields are\nconsidered as scalar fields (and not as part of the gauge field) and,\ncorrespondingly, the respective $F^i = \\ud X^i - \\rho^i_a A^a$ then\nhas the interpretation of a covariant derivative of $X^i$, again it is\na minimal requirement on what one would usually like to\ncall a covariant derivative.\n \nFollowing the same strategy as in the Section~\\ref{s:Bian}, the\nCondition~\\eqref{gauge} is now used to restrict the coefficient\nfunctions in the ansatz~\\eqref{gsymm0}. There we obtained \nTheorem \\ref{theo:nilpot}. Consequently, \nwe now assume that $Q_2$\nsquares to zero in the definition of field strengths.\\footnote{At the\nend of the following proof we will\nmake a brief remark on what still can be said if we drop this\nrequirement.}\nWe obtain\n\\begin{thm} \\label{theo:gauge1}  \nFor a spacetime dimension of at least 3, equality of the coefficients\nin \\eqref{gsymm0} with the respective unbared (and unprimed)\ncoefficient in the\nfield strengths \\eqref{Fi}--\\eqref{FB}---or, likewise, \nin the (nilpotent) super vector field ${Q_2}$\nin Eqs.~\\eqref{Qi}--\\eqref{QB}---is necessary and\nsufficient for the validity of \\eqref{gauge}.\n\\end{thm}\nAgain, a likewise statement holds true for every choice of $p$, provided\nonly the dimension of $\\Sigma$ exceeds $p$ by at least one (otherwise one\nstill has sufficiency, but necessity is maybe the more striking part). \nNote that we assumed the Bianchi\nidentities to be satisfied, so \\emph{any}\ncontribution of the form $O(F^\\alpha)$, i.e.~lying inside the ideal $\\I$, to \\eqref{gsymm0} does not\nmodify the result!\n\nTo prove the statement we will content ourselves with a halfway\nQ-language at this point, not yet\nintroducing any further abstractness beyond what we already did\nso far, while still avoiding any lengthy direct calculation in this manner.\nIn a condensed notation we may summarize the ansatz~\\eqref{gsymm0}\nas follows:\n\\beq \\delta^{(0)}_\\varepsilon A^\\alpha \\equiv\n\\ud \\varepsilon^\\alpha + \\varepsilon^\\beta a^*V^\\alpha_\\beta \\label{gauge0}\n\\, ,\n\\eeq\nwhere $V^\\alpha_\\beta$ is a function on $\\CM_2$ (of degree $|\\alpha| - |\\beta| +1\\ge\n0$, vanishing otherwise as well as for $|\\beta|=0$),\ncollecting all the yet undetermined\ncoefficients of the ansatz. The superscript $(0)$\nwas used, moreover, so as to indicate that we dropped the\nterms $O(F^\\alpha)$, which are irrelevant for the present\npurposes. We assume for simplicity, moreover,\nthat $\\varepsilon^\\alpha$\ndepends on $\\sigma$ only (and not also\non fields).\\footnote{Both steps, dropping $O(F^\\alpha)$ terms and having $\\varepsilon^\\alpha$\ndepend on $\\sigma$ only, imply, among others, that we fixed a frame or target space coordinates for\nthe calculation. But this is legitimate at this point.} We also kept\nthe slightly more suggestive notation $\\ud$ instead of $Q_1$ here.\n\nNow we are to calculate $ \\delta^{(0)}_\\varepsilon F^\\alpha$, where, according to\n\\eqref{CF}, $F^\\alpha= \\ud A^\\alpha -a^* Q_2^\\alpha$ with  $Q_2^\\alpha$ denoting \n the (left) components of the super vector field\n$Q_2$ in a basis $\\partial_\\alpha \\equiv \\partial \/\\partial q^\\alpha$. A\nrather simple calculation (using at this point only $\\ud^2=0$) yields\n\\begin{eqnarray}\\delta^{(0)}_\\varepsilon F^\\alpha  &= \n&  \\ud \\varepsilon^\\beta \\, a^* (V^\\alpha_\\beta - \\partial_\\beta\nQ_2^\\alpha) \\nonumber \\\\ \n  & & - \\varepsilon^\\beta \\, a^* \\left[ \\partial_\\beta (Q_2^\\gamma \\partial_\\gamma\n  Q_2^\\alpha) + O(V^\\alpha_\\beta - \\partial_\\beta\nQ_2^\\alpha) \\right] \n\\nonumber \\\\ \n  & &- (-1)^{|\\beta|} \\varepsilon^\\beta \\CF(V^\\alpha_\\beta) \\, , \\label{Fchangeold} \n\\end{eqnarray}\nwhere $ O(V^\\alpha_\\beta - \\partial_\\beta\nQ_2^\\alpha)$ denotes terms vanishing with $V^\\alpha_\\beta = \\partial_\\beta\nQ_2^\\alpha$ and, as before, $\\CF \\equiv \\ud a^* - a^* Q_2$.\nSo, obviously the sum of the first two lines has to vanish.\nFor every point $\\sigma_0 \\in \\Sigma$ we may choose $\\varepsilon^\\alpha$ such that $\\ud \\varepsilon^\\alpha(\\sigma_0)$\nis non-zero and $\\varepsilon^\\alpha(\\sigma_0)$ vanishes and vice versa. Also the map $a \\colon\nT[1]\\Sigma \\to \\CM_2$ is at our disposal. Correspondingly,\nif the dimension of $\\Sigma$ exceeds $p$, this requires\n\\beq V^\\alpha_\\beta = \\partial_\\beta\nQ_2^\\alpha \\qquad \\textrm{and} \\qquad  \\partial_\\beta (Q_2^\\gamma \\partial_\\gamma\n  Q_2^\\alpha) = 0 \\, \\qquad \\textrm{for} \\quad\n   |\\beta| \\ge 1 \\,\n   . \\label{consistency}\n   \\eeq\nThe second equation is fulfilled identically for $(Q_2)^2=0$, while\nthe first one gives the claimed necessity. \n\nOn the other hand, if we restrict ourselves to gauge transformations of the form \\eqref{gauge0} (and in general only then!), one may even\n\\emph{conclude} $(Q_2)^2=0$ from \\eqref{gauge}. In this case one needs\nonly $\\dim \\Sigma \\ge p+1$. The reason is that according to the second Equation~\\eqref{consistency} $Q_2^\\gamma \\partial_\\gamma\n  Q_2^\\alpha \\equiv (Q_2 \\circ Q_2)^\\alpha$ can only be a function of degree zero, which contradicts with the fact that it must have a strictly positive degree by knowing that $Q_2$ has degree plus one. This gives the result summarized in Theorem \\ref{thm:phys2}.\n\n\n\nFor convenience of the reader, we display the resulting\nformulas\nexplicitly: \n\\begin{align}  \\delta^{(0)}_\\varepsilon X^i &= \\rho^i_a  \\varepsilon^a \\, ,  \\label{g1a}\\\\\n  \\delta_\\varepsilon^{(0)} A^a &= \\ud\\varepsilon^a + C^a_{bc}A^b \\varepsilon^c +t^a_D \\mu^D \\, , \n    \\label{g2a}\\\\\n  \\delta_\\varepsilon^{(0)} B^D &= \\ud\\mu^D -\\Gamma^D_{aC} \\varepsilon^a B^C +\\Gamma^D_{aC}\n    A^a \\mu^C +\\frac12 H^D_{abc} A^a A^b \\varepsilon^c \\, ,  \\label{g3a}\n\\end{align}\nwhile, according to \\eqref{gsymm0} and the above result, the full\ntransformation then takes the form\n\\begin{align}\n  \\delta_\\varepsilon X^i &= \\delta^{(0)}_\\varepsilon X^i  \\label{g1b}\\\\\n  \\delta_\\varepsilon A^a &= \\delta^{(0)}_\\varepsilon A^a +F^i \\lambda^a_i  \\label{g2b} \\\\\n  \\delta_\\varepsilon B^D &= \\delta^{(0)}_\\varepsilon B^D +F^i \\lambda^D_i \n    +F^a \\lambda^D_a   \\,, \\label{g3b}\n\\end{align}\nfor some coefficients $\\lambda^\\alpha_\\beta$ depending on $\\varepsilon$ (and the gauge fields). We will\nspecify two covariant options for a choice of $\\lambda$s below. Again we\nmention that this is not yet the most general possible form; there may\nbe further model- and\/or dimension-dependent contributions of the more\ngeneral form that they just vanish with vanishing $F^\\alpha$\\footnote{If, for example, we add a term containing $*F^\\alpha$ linearly to, say, $\\delta A^\\beta$, $*$ denoting the Hodge dual with respect to some metric on $\\Sigma$, then the gauge transformation of $F^\\beta = \\ud A^\\beta + \\ldots$ produces a term linear in $\\ud *F^\\alpha$ from which, in general, we cannot say that it belongs to the ideal $\\I$ (while it still would vanish together with $F^\\alpha=0$).}---but in any case the above $\\delta_\\varepsilon$ (as\nwell as $\\delta^{(0)}_\\varepsilon$) is always an important part of such gauge\ntransformations as well.\n\n\n\\vskip\\newpassage\n\nIn the following we will now refine the Q-language so as to also be\nable to cover the gauge transformations of the form \\eqref{g1a}--\\eqref{g3a} or even \\eqref{g1b}--\\eqref{g3b}. We restrict ourselves\nto infinitesimal ones and we will not strive to present their\ngeometrical meaning here (this is rather subject of a more\nmathematical accompanying paper \\cite{KS07}); the main goal here\nis to provide a technical tool, which e.g.~permits one to calculate\nthe commutator of two gauge transformations in an efficient manner.\nAgain we want to be model independent here and thus we aim at\ncalculating the generic part of such a commutator,\n$[\\delta^{(0)}_\\varepsilon,\\delta^{(0)}_{\\bar \\varepsilon}] A^\\alpha$;\nevery explicit calculation, making use of \\eqref{B1}--\\eqref{B7}, \nwould be rather tedious and lengthy---the method presented\nbelow\\footnote{Cf.~also \\cite{BKS} for the same idea, illustrated\nthere at the example of the\nPoisson sigma model.} shortens the calculation considerably; also it\nprovides a relation to so-called derived brackets \\cite{YKS96},\nwhich were studied in the mathematical literature and which we will\nthen develop further for our particular case in a subsequent section. \n\nIn Section~\\ref{s:Bian} we found that $A^\\alpha = a^* (q^\\alpha)$, the\nmap $a \\colon {\\CM_1} \\to {\\CM_2}$ containing all the information about our\ngauge fields. Likewise, we now could attempt\nto cast gauge transformations into the form\n$\\delta_\\varepsilon A^\\alpha = (\\delta_\\varepsilon a)^* (q^\\alpha)$ with $(\\delta_\\varepsilon\na)^* = a^* \\circ  V_\\varepsilon$  for some degree preserving vector field $V_\\varepsilon$ on\n${\\CM_2}$ (parameterized somehow by $\\varepsilon$). Observing moreover that $\\varepsilon^\\alpha =\n(0,\\varepsilon^a,\\mu^D)$ naturally carries an upper index of ${\\CM_2}$, it is natural\nto regard these coefficients as components of a vector field\non ${\\CM_2}$, $\\varepsilon := \\varepsilon^\\alpha \\partial_\\alpha$. There are two obvious problems\nwith this ansatz: First, $\\varepsilon$ should depend on $\\sigma$ not just via the\nmap $X$ and, second, in this manner we would not be able to have $\\ud = Q_1$ enter respective formulas. \n\nBoth issues are cured simultaneously by the following construction\n\\cite{BKS,KS07}: we can trivially extend the map $a$ from before\nto a map \\beq a \\colon {\\CM_1} \\to\n\\CM := {\\CM_1} \\times {\\CM_2} \\, , \\label{aext}\n\\eeq\nwhich, by definition, is the identity on the\nfirst factor. For notational simplicity we do not introduce a new\nsymbol for this extended map, henceforth $a$ always corresponds to\nthis extended map; as before $A^\\alpha = a^* (q^\\alpha)$, since (local)\nfunctions $q^\\alpha$ on ${\\CM_2}$ can be regarded also as functions on $\\CM$\n(together with coordinate functions $\\sigma^\\mu$ on ${\\CM_1}$ they provide\ncoordinate functions on the product supermanifold $\\CM$). Obviously,\n$\\CM$ naturally becomes a Q-manifold itself; just add the two nilpotent\nvector fields to obtain\n\\beq Q = Q_1 + Q_2 \\, . \\eeq\nThis is again a nilpotent degree one vector field since by definition\n$Q_1$ and $Q_2$ supercommute (i.e.~they anticommute). In other words,\ngiven the two {complexes} $(C^\\infty({\\CM_1}), Q_1)$ and  $(C^\\infty({\\CM_2}),\nQ_2)$, one forms a double complex in the standard way,  \n$C^\\infty({\\CM_1}) \\otimes C^\\infty({\\CM_2}) \\cong C^\\infty(\\CM)$. In this\nextended setting, the operator $\\CF$ of\nEquation~\\eqref{CF} is replaced by\n\\beq \\CF := \\Qo \\circ a^* - a^* \\circ Q \\qquad \\Rightarrow \\qquad \\CF (q^\\alpha) \n\\equiv F^\\alpha \\, , \\label{CFext}\\end{equation}\nsince essentially all remains the same only that the target\n$({\\CM_2},{Q_2})$ is replaced by $(\\CM,Q)$---merely with an additional restriction on \n\\eqref{aext} that $a$ is the identity on the first factor.\\footnote{In other\nwords, $a$ is a section of the bundle $\\CM \\to {\\CM_1}$. Cf.~\\cite{KS07}\nfor the obvious generalization to nontrivial (total) bundles, which,\ndespite its simplicity,\nmay be seen to generalize the notion of an ordinary\nconnection in a nontrivial principal bundle (while certainly\nin this case a connection is not just\na globally defined Lie algebra valued 1-form on the base of that bundle).} Note that\nthis extended $\\CF$ vanishes identically on functions coming from\n${\\CM_1}$ (i.e.~constant along ${\\CM_2}$). \n\n\nThe space of vector fields $\\X_\\bullet(\\CM)= \\Gamma(T\\CM)$ of every Q-manifold $\\CM$ carries canonically the structure of a complex. If $\\CM$ is a supermanifold of degree $p$\n(cf.~Section~\\ref{s:examQ}), the space $\\X_\\bullet(\\CM)$ has also elements of\nnegative degree, bounded from below by $-p$. The differential is\nprovided by the graded commutator with $Q$, $\\ud_Q \\colon\nV \\mapsto [Q,V]$, since the bracket\n$[ \\cdot , \\cdot ]$ maps derivations of $C^\\infty(\\CM)$ into itself,\n$\\ud_Q \\equiv [Q,\\cdot]$ is of degree one since $Q$ is, and\n$(\\ud_Q)^2=0$ due to $Q^2 = \\frac{1}{2} [Q,Q]=0$ as one shows easily by means of\nthe (graded) Jacobi identity of the bracket.\n\nFor the purpose of gauge symmetries, we will be interested in a\nsubcomplex $({\\X_\\bullet^{\\mathrm{vert}}(\\CM)},\\ud_Q)\\subset ({\\X_\\bullet(\\CM)},\\ud_Q)$,  spanned by ``vertical''\nvector fields, i.e.~vector fields parallel to ${\\CM_2}$.\\footnote{Here we made\nuse of the fact that $\\ud_{Q_1}$ maps ${\\X_\\bullet^{\\mathrm{vert}}(\\CM)}$ into itself: $\\epsilon =\\epsilon^\\alpha\n\\partial_\\alpha \\in {\\X_\\bullet^{\\mathrm{vert}}(\\CM)}\n\\Rightarrow [Q_1,\\epsilon] = Q_1(\\epsilon^\\alpha)\n\\partial_\\alpha \\in {\\X_\\bullet^{\\mathrm{vert}}(\\CM)}$.}\n This complex is\nisomorphic to $(C^\\infty({\\CM_1}),Q_1) \\otimes (\\X_\\bullet({\\CM_2}),\\ud_{Q_2})$,\nas one verifies easily (cf.~also Eq.~\\eqref{complex}  below for\nmore details). In this form we recognize that both of the potential problems of the ansatz \nof symmetries generated by vector fields are cured at the same time: while still dealing essentially with vector fields on the target, the coefficient functions now depend also on ${\\CM_1}$ and there is $Q_1\\cong \\ud$ in the game as well. \n\nLet $V \\in \\X_0^{\\mathrm{vert}}(\\CM)$, a vertical vector field of degree zero,\nand set $\\delta A^\\alpha = (a^* \\circ V) (q^\\alpha)$. Together with\n\\eqref{CFext} this implies\n\\beq \\delta F^\\alpha = ( \\Qo \\circ a^*\\circ V - a^*\\circ V \\circ Q)(q^\\alpha)\n\\equiv \\CF ( V(q^\\alpha)) + (a^* \\circ \\ud_Q(V))(q^\\alpha) \\, . \\label{Ftrafo1}\n\\eeq\nThus if $\\dim \\Sigma \\ge p+1$, $\\delta \\I \\subset \\I$ is tantamount \nto requiring $V$ to be $\\ud_Q$-closed. The solution found before in\nTheorem \\ref{theo:gauge1} corresponds to an exact $V$, i.e.~$V=\\ud_Q\n\\epsilon$, where $\\epsilon$ is\na degree minus one vertical vector field,\n$\\epsilon \\in \\X_{-1}^{\\mathrm{vert}}(\\CM)$. We will now make this more explicit (and thereafter\nadd a remark about the $\\ud_Q$-cohomology). \n\n\nThe degree $k$ elements of\n$C^\\infty({\\CM_1})$ may be identified with $\\Omega^k(\\Sigma)$ and $Q_1$ is the\nde Rham differential $\\ud$ on $\\Sigma$. Correspondingly,\n\\begin{align}\n (\\X_{-1}^{\\mathrm{vert}}(\\CM),\\ud_Q)\n\\cong & \\; (\\Omega^0(\\Sigma),\\ud) \\otimes (\\X_{-1}({\\CM_2}),\\ud_{Q_2})\n  \\nonumber \\\\ &  \\oplus\n (\\Omega^1(\\Sigma),\\ud) \\otimes\n(\\X_{-2}({\\CM_2}),\\ud_{Q_2})  \\; \\oplus \\ldots \\nonumber \\\\\n  &  \\oplus  (\\Omega^{p-1}(\\Sigma),\\ud) \\otimes\n(\\X_{-p}({\\CM_2}),\\ud_{Q_2})\n  \\, .  \\label{complex}\n\\end{align}\nThus, for $p=2$, the most general $\\epsilon \\in \\X_{-1}^{\\mathrm{vert}}(\\CM)$ takes the form\n\\beq \\epsilon = \\epsilon^a \\frac{\\partial}{\\partial \\xi^a} + \\epsilon^D\n\\frac{\\partial}{\\partial b^D} \n\\eeq\nwith $\\epsilon^a \\in C^\\infty(\\Sigma) \\otimes C^\\infty(M)\\cong\nC^\\infty(\\Sigma\\times M)$ and  $\\epsilon^D = (\\epsilon^D)_{\\mathrm{main}} + \\xi^a\n\\epsilon^D_a$, where $(\\epsilon^D)_{\\mathrm{main}}\n\\in \\Omega^1(\\Sigma) \\otimes C^\\infty(M)$ and $\\epsilon^D_a \\in\nC^\\infty(\\Sigma\\times M)$. Here we used that the degree zero\npart of $C^\\infty({\\CM_2})$ is $C^\\infty(M)$ and that $\\xi^a\n\\frac{\\partial}{\\partial b^D} \\in \\X_{-1}({\\CM_2})$; \neverything is understood as\ndefined only locally over the target base $M$ certainly. We now\nintroduce\n\\beq \\varepsilon^a := a^* \\epsilon^a \\; , \\quad  \\mu^D \\equiv \\varepsilon^D:= a^* \\epsilon^D \\, ;\n\\eeq\nthus obviously, $\\varepsilon^a \\in  C^\\infty(\\Sigma)$ and $\\mu^D \\in\n\\Omega^1(\\Sigma)$.\\footnote{Note: $\\varepsilon^a(\\sigma) = \\epsilon^a(\\sigma,X(\\sigma))$---it was,\namong others, this extra \\emph{explicit}\n$\\sigma$-dependence that made it necessary to extend the target as\nin \\eqref{aext}: For a fixed choice of $X \\colon \\Sigma \\to M$ that is\nan embedding this would not have been necessary, but, on the other\nextreme side---which, however, is \\emph{always} realized in ordinary YM\ngauge theories---if the\nimage of $X$ is just a point in $M$, one now\nstill has the possibility for a nontrivial\n$\\sigma$-dependence of $\\varepsilon^a$. The situation with $\\mu^D$ is similar, but\na bit more involved: The coefficient $\\epsilon^D_a$ corresponds to a part\nof $\\mu^D$ that is not only $X$, but also $A^a$--dependent. We mostly\nwill consider only the case with $\\epsilon^D_a=0$.}\nNow, to prove the above claim, we just\nperform the computation:\n\\begin{align}\n\\left(a^* \\circ \\ud_Q(\\epsilon)\\right)(q^\\alpha)& =\n  a^* \\left(\\epsilon(Q_2^\\alpha)\\right)  + \\left(a^* \\circ Q\\right) (\\epsilon^\\alpha) \n  \\nonumber \\\\ \n& =  a^*(\\epsilon^\\beta\\partial_\\beta Q_2^\\alpha) - \\CF (\\epsilon^\\alpha) + \\left(Q_1 \\circ\na^*\\right)  (\\epsilon^\\alpha) \\nonumber \\\\\n& = \\ud \\varepsilon^\\alpha + \\varepsilon^\\beta \\, a^*(\\partial_\\beta Q_2^\\alpha) - F^\\beta \\, a^*\n( \\partial_\\beta \\epsilon^\\alpha ) \\, .\n  \\end{align}\nComparison with Eqs.~\\eqref{gauge0}, \\eqref{consistency}, and \n\\eqref{g1b}--\\eqref{g3b} shows that indeed\n\\beq \\boxed{\\delta_\\varepsilon A^\\alpha = (a^* \\circ [Q,\\epsilon] )\\, (q^\\alpha)}\n\\label{gaugebox} \\eeq\nwith particular coefficients $\\lambda^\\alpha_\\beta$ in\n\\eqref{g1b}--\\eqref{g3b}, namely\n\\beq \\lambda^\\alpha_\\beta = \n-a^*(\\partial_\\beta \\epsilon^\\alpha)\\, , \\label{lambdas1} \\eeq\ni.e., in\ndetail, $\\lambda^a_i = \n-a^*(\\partial_i \\epsilon^a)$, $\\lambda^D_i = \n-a^*(\\partial_i \\epsilon^D) \\equiv -\na^*\\left(\\partial_i (\\epsilon^D)_{\\mathrm{main}}\\right) -\n A^a a^*(\\partial_i \\epsilon^D_a)$, and $\\lambda^D_a = \n-a^*(\\epsilon^D_a)$. For the particular case of prime interest that\n$\\epsilon^D =(\\epsilon^D)_{\\mathrm{main}}$, this simplifies to\n\\beq \\lambda^a_i = \n-a^*(\\partial_i \\epsilon^a) \\, , \\quad  \\lambda^D_i = \n-a^*(\\partial_i \\epsilon^D) \\, , \\quad \\lambda^D_a =0 \\, ,   \\label{lambdas2}\n\\eeq\nwhere $\\lambda^D_i$ depends on the choice of $X$ but not also on\n$A^a$.\n\nNote that, given a fixed map $a$, i.e.~a\nfixed choice of $A^\\alpha$, we can always adapt for an arbitrary choice of\n$\\varepsilon^\\alpha$ \\emph{and}  $\\lambda^\\alpha_\\beta$ in \\eqref{g1b}--\\eqref{g3b}\nby an appropriate choice of\n$\\epsilon^\\alpha$. On the other hand, from the point of view of\nEq.~\\eqref{gaugebox}, gauge transformations are governed by\n``vertical vector fields'' on the target, which in turn determine, for\n\\emph{every} independent choice of $A^\\alpha$, both $\\varepsilon^\\alpha$ and\n$\\lambda^\\alpha_\\beta$. \n\nWe use the opportunity for a side remark about \\emph{one} alternative covariant\nchoice of $\\lambda$s in  \\eqref{g1b}--\\eqref{g3b}. At the\nexpense of introducing a connection in both  $E$ and $V$, say those\nused to define the field strengths \\eqref{FaG} and \\eqref{FBG},\none may choose\n\\beq \\lambda^a_i=\\varepsilon^b\\Gamma^a_{ib} \\qquad\n\\lambda^D_i=\\mu^C\\Gamma^D_{iC} \\label{lambdas3}\n\\eeq \nbesides $\\lambda^D_a =0$. This  has the advantage that now already $\\varepsilon^\\alpha \\equiv a^*(\\epsilon^\\alpha)$\ndetermines $\\lambda^\\alpha_\\beta$ uniquely (in \\eqref{lambdas1} also the first germ of $\\epsilon^\\alpha$ at the section $a$ is needed). To see that this choice is indeed\ncovariant, one can argue by comparison with Eqs.~\\eqref{lambdas1},\n\\eqref{lambdas2}: By construction, Eq.~\\eqref{gaugebox}\ngives a covariant gauge transformation---note that here covariance\nmeans that $\\delta_\\varepsilon$ is to act as a derivation, but the r.h.s.~of\n\\eqref{gaugebox} obviously does so since $[Q,\\varepsilon]$ is a\nvector field.\\footnote{Cf.~also \\cite{BKS} for a  more detailed\ndiscussion of this issue.} In detail this implies that when we change\nlocal frames in $E$ and $V$ according to \\(\\tilde{\\xi^a}=M^a_b \\xi^b\\),\n\\(\\tilde{b^D}=N^D_C b^C\\), every transformation of $\\lambda$s such as\nthose of Eqs.~\\eqref{lambdas1}, \\eqref{lambdas2}, namely\n\\begin{align}  \\widetilde{\\lambda^a_i} &=\n    M^{\\tilde a}_b \\lambda^b_i - M^{\\tilde a}_{b,i}\\varepsilon^b\n  \\label{lambdatrafo1} \\\\\n  \\widetilde{\\lambda^D_a} &=\n    N^{\\tilde D}_F \\lambda^F_c\n   (M^{-1})^c_{\\tilde a}  \\label{lambdatrafo2} \\\\\n  \\widetilde{\\lambda^D_i} &=\n    N^{\\tilde D}_C\\lambda^C_i - N^{\\tilde D}_{C,i}\\mu^C\n   \n    \\label{lambdatrafo3}\n  \\, ,\n\\end{align}\ngive a covariant gauge transformation in \n\\eqref{g1b}--\\eqref{g3b}.\n(We suppressed writing pullback by $X$ here: E.g., $M^{\\tilde a}_b$ in\n\\eqref{lambdatrafo1} is rather $X^*(M^{\\tilde a}_b) \\equiv M^{tilde a}_b(X)$ and likewise\n$M^{\\tilde a}_{b,i}$ denotes $X^*(\\partial_i M^{\\tilde a}_b)$.)\nThe above choice of $\\lambda$s using background connections\nobviously fulfills the transformation requirements\n\\eqref{lambdatrafo1}--\\eqref{lambdatrafo3}. Note that the\nmethod of obtaining these transformation properties is\nincomparably simpler than an elementary one where, in a first step, one would \ncompute all changes in \\eqref{g1a}--\\eqref{g3a} using \n\\eqref{trafo1}--\\eqref{trafo5}. \n\nIn the considerations around Eq.~\\eqref{Ftrafo1} we obtained only that\n$V$ needs to be $\\ud_Q$-closed, but not necessarily $\\ud_Q$-exact. This\nreconciles with the result of Theorem \\ref{theo:gauge1} since \nunderlying \\eqref{gauge} is a particular form of\ngauge transformations, namely the one specified in\nEqs.~\\eqref{gsymm0}---starting with exact forms on $\\Sigma$.\nThis already gives the restriction\nto trivial elements of the otherwise nontrivial cohomology of\n$\\ud_Q$. It may be interesting to relax this condition, however.\n\nNote that the $\\ud_Q$--cohomology on vertical vector fields\ncan be computed by means of an algebraic\nversion of the K\\\"unneth formula, in detail:  $H_{\\ud_Q}^0(\\CM) =\n\\bigoplus_{k=0}^2 \\left(H_{\\textrm{deRham}}^k(\\Sigma) \\otimes\nH_{\\ud_{Q_2}}^{-k}({\\CM_2})\\right)$. E.g.~a change of coordinates on the\nsupermanifold ${\\CM_2}$ induces a symmetry only if it leaves the Q-structure invariant, i.e.~if its (degree zero) generator $V=V^\\alpha \\partial_\\alpha$ commutes with $Q_2$. \nThus such a generator corresponds to an element with $k=0$\nin the previous sum. \n\n\n\\vskip\\newpassage\n\nWe now come to the announced application of the above framework for\ngauge transformations. As a warm-up exercise we return to\nEq.~\\eqref{Ftrafo1} and use it to\ncompute the explicit form of the change of field strengths $F^\\alpha$\nunder the generic part $\\delta^{(0)}_\\varepsilon$ of gauge transformations of\nour gauge fields. Needless to say that again every explicit\ncalculation would be very lengthy, making heavy use of\n\\eqref{B1}--\\eqref{B7}. We first observe that locally and for every\nfixed choice of $(\\varepsilon^a,\\mu^D)$ we can choose frames in $E$ and $V$\nsuch that $\\epsilon^a$ and $\\epsilon^D$ are constant along ${\\CM_2}$---correspondingly,\nin \\emph{this} particular frame, $\\delta_\\varepsilon A^\\alpha = \\delta^{(0)}_\\varepsilon\nA^\\alpha$ and we may make use of \\eqref{Ftrafo1} with $V=\\ud_{Q}\\epsilon$:\\footnote{Alternatively, here we may equally well\nuse \\eqref{Fchangeold} together with Eqs.~\\eqref{consistency}.}\n\\begin{eqnarray}\n  \\delta^{(0)}_\\varepsilon F^i &=& \\varepsilon^a \\, \\rho^i_{a,j} \\, F^j \\, , \\label{deltaFi} \\\\\n  \\delta^{(0)}_\\varepsilon F^a &=&-C^a_{bc}\\, \\varepsilon^b F^c + \\left(C^a_{cb,i}\\, \\varepsilon^bA^c\n  -t^a_{D,i}\\, \\mu^D \\right)F^i \\, , \\label{deltaFa} \\\\\n  \\delta^{(0)}_\\varepsilon F^D &=& -\\Gamma^D_{aC} \\, \\varepsilon^a F^C + \\left(\\Gamma^D_{aC}\n  \\, \\mu^C  - H^D_{abc} \\,\n  \\varepsilon^b A^c \\right)  F^a \\nonumber \\\\\n  && + \\left[\\tfrac{1}{2} H^D_{abc,i}\\, \\varepsilon^a A^bA^c -\\Gamma^D_{aC,i}\n  \\left(\\varepsilon^a B^C + \\mu^CA^a \\right) \\right] F^i \\, . \\label{deltaFB}\n\\end{eqnarray}\nFor example, the first line was obtained as follows:\n$\\delta_\\varepsilon F^i=\\CF([Q,\\epsilon]x^i)=\\CF(\\rho^i_a\\epsilon^a) =\nX^*\\left(\\partial_j(\\rho^i_a\\epsilon^a)\\right) F^j$,\nwhere in the last step we  used \\eqref{FLeibniz}. \n\n\n\n  \n\n\nWe now come to the commutator of gauge transformations. For this, we first make\nthe following simple but important observation\n\\begin{thm} \\label{theo:Lie}\nInfinitesimal gauge transformations, written in form of\n\\eqref{gaugebox},\nform a Lie algebra isomorphic to $(\\X_{-1}^{\\mathrm{vert}}(\\CM)\/\\ker\\ud_Q,[\\cdot ,\\cdot]_Q)$,\nwith\nthe derived bracket:\n\\beq [\\epsilon,\\bar \\epsilon]_Q = [\\ud_Q(\\epsilon),\\bar \\epsilon] \\equiv  [[\\epsilon,Q],\\bar\n\\epsilon] \\, . \\label{induced}\n\\eeq \n\\end{thm}\nTo prove this we observe that infinitesimal gauge transformations\n\\eqref{gaugebox} are generated by  degree minus one vertical\nvector fields \\emph{modulo} elements that are annihilated by\n$\\ud_Q \\equiv [Q, \\cdot]$. Such elements are called \\emph{coexact} in\n\\cite{YKS96}, in generalization of coexact differential forms. \nIt is now easy to convince oneself that the commutator of two gauge\ntransformations \\eqref{gaugebox} gives rise to a new bracket\nbetween vector fields:\n\\beq \n [\\delta_\\epsilon , \\delta_{\\bar \\epsilon} ] A^\\alpha = \\left(a^* \\circ\n [\\ud_Q(\\epsilon), \\ud_Q(\\bar \\epsilon)]\\right) (q^\\alpha) =\n \\left( a^* \\circ \\ud_Q ( [\\ud_Q(\\epsilon),\\bar \\epsilon])\\right) (q^\\alpha) \\, ,\n\\label{comm}\n \\eeq\nwhere we made use of the (graded) Leibniz property of $\\ud_Q$ \nw.r.t.~$[ \\cdot , \\cdot ]$---noting that\n$\\ud_Q(\\epsilon)$ is of degree zero---and of \n$(\\ud_Q)^2=0$. This proves \\eqref{induced}. The induced or\n\\emph{derived} bracket\n$[\\epsilon,\\bar \\epsilon]_Q = [\\ud_Q(\\epsilon),\\bar \\epsilon]$ on $\\X_{-1}^{\\mathrm{vert}}(\\CM)$ is \\emph{not}\na Lie bracket; however, as a result of a general construction,\nrecalled in the subsequent section, on the\nquotient by the kernel of the differential $\\ud_Q$ it is,\ncf.~Proposition \\ref{p:YKSder} below.\n\nIt is possible to characterize Q2-manifolds \nalso by means of its derived bracket---as an alternative to the\ndescription found in Section~\\ref{s:examQ}. Since\nwe were now naturally led to this bracket, and since this formulation\nis well-suited also for comparison with the usual formulation of what\nis a Courant algebroid, we will reanalyze\nQ2-manifolds from this perspective in the subsequent section.\n\n\nThe above theorem\nimplies in particular that if gauge transformations are\ninterpreted---in generalization\nof \\cite{BKS}---in the sense advocated in the present section, they close\noff-shell---and this also in theories which in the \\emph{usual}\nterminology have an\n\\emph{open algebra} of gauge transformations. In fact,\nas a kind of corollary of Theorem \\ref{theo:Lie}, we will now verify\n\\begin{prop}  For the\ngauge transformations \\eqref{g1a}--\\eqref{g3a} one finds\n\\begin{eqnarray} [\\delta^{(0)}_\\varepsilon,\\delta^{(0)}_{\\bar \\varepsilon}] X^i &=&\n\\delta^{(0)}_{\\widetilde \\varepsilon}\nX^i  \\label{gcom1}\\\\{}\n  [\\delta^{(0)}_\\varepsilon,\\delta^{(0)}_{\\bar \\varepsilon}] A^a &=&\n  \\delta^{(0)}_{\\widetilde \\varepsilon}\n \n  A^a -C^a_{bc,i} \\varepsilon^b\n    \\bar \\varepsilon^{c} F^i  \\label{gcom2}\\\\{}\n  [\\delta^{(0)}_\\varepsilon,\\delta^{(0)}_{\\bar \\varepsilon}] B^D &=&  \n  \\delta^{(0)}_{\\widetilde \\varepsilon}\n\n B^D \n     +H^D_{abc} \\varepsilon^b  \\bar \\varepsilon^{c} F^a \\nonumber \\\\ \n  & & +\\,\\Gamma^D_{aC,i}\n     (\\varepsilon^a\\bar \\mu^{C}-\\bar \\varepsilon^{a}\\mu^C) F^i\n  - H^D_{abc,i} \\varepsilon^{a}\\bar\\varepsilon^b A^c F^i  \\label{gcom3}\n\\end{eqnarray}\n   \nwhere $\\,{\\widetilde \\varepsilon}^a\n\\equiv C^a_{bc}   \\varepsilon^b \\bar \\varepsilon^c$\nand  \n \n \n \n   \n    $\\,{\\widetilde \\varepsilon}^D\n   \n    \\equiv  \\Gamma^D_{aC}(\n \\varepsilon^a \\bar \\varepsilon^{C}-\\bar \\varepsilon^{a}\\varepsilon^C ) \n -  H^D_{abc}\\varepsilon^a\\bar \\varepsilon^{ b}\n  A^c \\, .  $\n\\label{prop:onshellclosed}  \n\\end{prop}\nThe idea is simple: As when computing the variation of\nfield strengths w.r.t.~$\\delta^{(0)}_\\varepsilon$, for a given choice of \n$\\varepsilon^\\alpha \\in C^\\infty({\\CM_1})$ \nand given a particular coordinate system $q^\\alpha$ on ${\\CM_2}$, we may\ndefine $\\epsilon := \\varepsilon^\\alpha \\partial_\\alpha$, which now is a vertical vector\nfield. In that coordinate system, we then have\n$\\delta^{(0)}_\\varepsilon A^\\alpha = \\delta_\\epsilon A^\\alpha$ and likewise so for the\nbarred variations.\\footnote{Here we\nwrote  $\\delta_\\epsilon A^\\alpha$ instead of  $\\delta_\\varepsilon A^\\alpha$\nas in \\eqref{gaugebox} or \\eqref{g1b}--\\eqref{g3b}, so as to\nmake clear that the gauge transformation \\eqref{gaugebox} is\nparameterized not only by $\\varepsilon^\\alpha(\\sigma)=\\epsilon^\\alpha(\\sigma,A^\\beta(\\sigma))$, but also by the\nfirst derivative of $\\epsilon^\\alpha$\nw.r.t.~$q^\\beta$---cf.~Eq.~\\eqref{lambdas1}. This is in contrast to the\nsituation when choosing \\eqref{lambdas3} in\n\\eqref{g1b}--\\eqref{g3b}.} \nThus also for\nthe commutators we have $[\\delta^{(0)}_\\varepsilon,\\delta^{(0)}_{\\bar \\varepsilon}] A^\\alpha\n= [\\delta_\\epsilon,\\delta_{\\bar \\epsilon}] A^\\alpha$. Now we can apply\n\\eqref{comm} and Theorem \\ref{theo:Lie}. This contains the ``heavy\ncalculation'', so to say (which, in the present formulation, is almost a\ntriviality). We are left with splitting  $\\delta_{[\\epsilon,\\bar \\epsilon]_Q}\nA^\\alpha$ into the part $\\delta^{(0)}_{[\\epsilon,\\bar \\epsilon]_Q} A^\\alpha$ and, using\n\\eqref{lambdas1}, the remaining \nterms containing $F^\\alpha$s according to \\eqref{g2b} and \\eqref{g3b}.\nThe main point is that even if $\\epsilon$ and $\\bar \\epsilon$ are constant along\n${\\CM_2}$, their induced bracket is not! Let us be explicit about this:\n\\beq [ [\\varepsilon^\\alpha \\partial_\\alpha , Q_1 + Q_2 ], \\bar \\varepsilon^\\beta \\partial_\\beta ] =\n\\bar \\varepsilon^\\beta \\varepsilon^\\alpha  [[\\partial_\\alpha ,Q_2 ],\\partial_\\beta ]\n\\eeq\nsince $\\varepsilon^\\alpha$ and $\\bar \\varepsilon^\\beta$ are constant along ${\\CM_2}$ and\n$[Q_1(\\varepsilon^\\alpha) \\partial_\\alpha, \\bar \\varepsilon^\\beta \\partial_\\beta ] =0$. Thus we need to\ncompute $[\\partial_\\alpha,\\partial_\\beta ]_{{Q_2}}$, which is easily found to\nequal\n$(-1)^{|\\beta|+1}(\\partial_\\alpha \\partial_\\beta {Q_2}^\\gamma) \\partial_\\gamma$. So,\n\\begin{eqnarray}\n\\widetilde \\epsilon := [\\varepsilon^\\alpha \\partial_\\alpha,  \\bar \\varepsilon^\\beta \\partial_\\beta ]_Q \n  &=& -(-1)^{|\\beta|}\\bar \\varepsilon^\\beta \\varepsilon^\\alpha\n(\\partial_\\alpha \\partial_\\beta {Q_2}^\\gamma) \\, \\partial_\\gamma\n\\label{derivedsymm} \\\\\n&\\equiv&   \\varepsilon^a \\bar \\varepsilon^b C^c_{ab} \\partial_{c}\n\\,+ (\\varepsilon^a \\bar \\mu^D- \\bar \\varepsilon^a \\mu^D ) \\, \\Gamma^C_{aD}\n \\partial_{C} \n \\, - \\varepsilon^a\\bar \\varepsilon^{ b}H^D_{abc}\n  \\xi^c \\partial_{D}\n \\, , \\nonumber\n \\end{eqnarray}\nwhere we used the general form \\eqref{Q} for ${Q_2}$ and $\\varepsilon^D \\equiv\n\\mu^D$. Despite the fact that $\\epsilon$ and  $\\bar \\epsilon$\nwere vertical vector fields ``constant along ${\\CM_2}$'' (in the chosen\nframe $\\partial_\\alpha$), the vector field $\\widetilde \\epsilon = \\widetilde\n\\epsilon^\\alpha \\partial_\\alpha$ has coefficients which \\emph{do} depend\nnon-trivially on ${\\CM_2}$. By comparison of coefficients it is now easy to\nconclude the above statement from\n\\beq [\\delta^{(0)}_\\varepsilon,\\delta^{(0)}_{\\bar \\varepsilon}] A^\\alpha\n= [\\delta_\\epsilon,\\delta_{\\bar \\epsilon}] A^\\alpha = \\delta_{\\widetilde \\epsilon} A^\\alpha =\n\\delta^{(0)}_{\\widetilde \\varepsilon} A^\\alpha + F^\\beta \\widetilde \\lambda_\\beta^\\alpha\n\\, , \n\\eeq\nwith $ \\widetilde \\varepsilon^\\alpha = a^* \\widetilde \\epsilon^\\alpha$\nand $\\widetilde \\lambda_\\beta^\\alpha = - a^*(\\partial_\\beta \\widetilde\n\\epsilon^\\alpha)$.\nIt is seen also that the nontrivial\n$\\xi^a$--dependence of $\\widetilde \\epsilon^D$ is responsible for both, the\n$A^a$-dependence of $\\widetilde \\varepsilon^D$ and the $F^a$-contribution to \n\\eqref{gcom3}. \n\nWe conclude this section by illustrating how the previous considerations can be  used to also compute the commutator of symmetries of the form  \n\\begin{align}  \\delta^{}_\\varepsilon X^i &= \\rho^i_a  \\varepsilon^a \\, ,  \\label{g1an}\\\\\n  \\delta_\\varepsilon^{} A^a &= \\ud\\varepsilon^a + C^a_{bc}A^b \\varepsilon^c +t^a_D \\varepsilon^D +\\Gamma^a_{ib}\\varepsilon^bF^i  \\label{g2bn} \\\\\n  \\delta_\\varepsilon B^D  &= \\ud\\varepsilon^D -\\Gamma^D_{aC} \\varepsilon^a B^C +\\Gamma^D_{aC}\n    A^a \\wedge \\varepsilon^C +\\frac12 H^D_{abc} A^a \\wedge A^b \\varepsilon^c  -  \\Gamma^D_{iC}  \\epsilon^C \\wedge F^i \\,, \\label{g3bn}\n\\end{align}\ninduced by (ordinary, ``non-dynamical'') connections on the (target) bundles $E$ and $V$\nand which result from implementing \\eqref{lambdas3} (together with $\\lambda_a^D=0$) into the general symmetries \\eqref{g1b}--\\eqref{g3b}. The symmetries on the 0-forms $X^i$ coincide with the ones in \\eqref{g1a} and we thus can take recourse to \\eqref{gcom1} for the corresponding commutator. Changes occur when calculating the commutator on the 1-form fields $A^a$ and the 2-form fields $B^D$, where it is useful to split the infinitesimal symmetries according to $\\delta_\\varepsilon= \\delta^{(0)}_\\varepsilon+ \\delta^{(1)}_\\varepsilon$, where $\\delta^{(0)}_\\varepsilon$ is given by \\eqref{g2a} and \\eqref{g3a}. In the following we restrict ourselves to the 1-form fields, leaving the rest as an exercise to the reader. Evidently \\begin{equation}\n\\delta^{(1)}_{\\varepsilon} A^a = \\Gamma^a_{ib}\\varepsilon^b F^i \\, .\n\\end{equation}\nMoreover, \n\\begin{equation}\n[\\delta^{}_\\varepsilon,\\delta^{}_{\\bar \\varepsilon}] A^a \n= [\\delta^{(0)}_\\varepsilon,\\delta^{(0)}_{\\bar \\varepsilon}]A^a \n + [\\delta^{(1)}_\\varepsilon,\\delta^{(1)}_{\\bar \\varepsilon}]A^a \n + \\left(\\delta^{(0)}_\\varepsilon \\delta^{(1)}_{\\bar \\varepsilon}A^a \n    +\\delta^{(1)}_\\varepsilon \\delta^{(0)}_{\\bar \\varepsilon}A^a  \n\t- (\\varepsilon \\leftrightarrow \\bar{\\varepsilon})\\right)\\, .\n\\end{equation}\nThe first commutator is given by \\eqref{gcom2}. For the second commutator, one observes    $\\delta^{(1)}X^i=0$ such that  $\\delta_{\\varepsilon}^{(1)}\\left(\\Gamma^a_{ib}\\bar{\\varepsilon}^b F^i\\right)=\\Gamma^a_{ib}\\bar{\\varepsilon}^b\\delta_{\\varepsilon}^{(1)}\\!F^i= \n-\\Gamma^a_{ib}\\bar{\\varepsilon}^b\\rho^i_c \\Gamma^c_{jd}\\varepsilon^d F^j$. Since we also know how $F^i$ transforms with respect to $\\delta^{(0)}$, cf.~Eq.~\\eqref{deltaFi}, we are now left with merely collecting all terms so as to arrive at\\footnote{There were some later developments about such commutators in the mean time. First, in \\cite{Str09} the terms violating closeness were given a geometric interpretation in the case that $E$ is a Lie algebroid: An ordinary connection on $E$ induces an $E$-connection generalizing the adjoint representation of a Lie algebra \\cite{Str04b}; its $E$-curvature factorizes over the anchor map  $\\rho$,  ${{}^E\\!{}R}^a{}_{bcd}=\\rho_b^i \\, S^a{}_{icd}$. With this one obtains $ [\\delta_\\epsilon,\\delta_{\\bar{\\epsilon}}]A^a = \\delta_{\\widetilde{\\epsilon}}^{}A^a+S^a_{ibc}\\epsilon^b\\bar{\\epsilon}^{c}F^i$.\nSecond, the methods used here were extended so as to perform calculations about such commutators more directly, replacing $a^*$ by $f^*$---cf., e.g., \\cite{SaS13,SLS14}.}\n\\begin{align}\n [\\delta_\\epsilon,\\delta_{\\bar{\\epsilon}}]A^a =& \\delta_{\\widetilde{\\epsilon}}^{}A^a\n -(C^a_{bc,i}+\\Gamma_{id}^aC^d_{bc} ) \\epsilon^b\\bar{\\epsilon}^{c}F^i\n  \\\\\n & +\\left(C^a_{db}\\Gamma^d_{ic} -\\Gamma^a_{jb}\\rho^j_{c,i}-\\Gamma^a_{ib,j}\\rho^j_c\n +\\Gamma^a_{jb}\\rho^j_d\\Gamma^d_{ic} \\right)( \\epsilon^b\\bar{\\epsilon}^{c}-\\bar{\\epsilon}^{b}\\epsilon^c)F^i , \\nonumber\n\\end{align}\nwhere the new parameter $\\widetilde{\\epsilon}$ is given by \nProposition \\ref{prop:onshellclosed}. Note that the part of the symmetries on the $A^a$ fields parameterized by the 1-forms $\\varepsilon^D\\equiv \\mu^D$, cf.~Eq.~\\eqref{g2bn}, do commute\nevidently. Still, these symmetries do not commute with those parameterized by the 0-forms $\\epsilon^a$. This explains why on the r.h.s.~the parameter $\\widetilde{\\epsilon}^D$ is non-vanishing in general---in fact, as we learn from Proposition \\ref{prop:onshellclosed}, I 1-form part of the symmetries is even induced if the initial two symmetries $\\epsilon$ and $\\bar{\\epsilon}$ have vanishing 1-form parts (like in a central extension).  \n\n\n\n\\newpage\n\n\\section{Derived bracket and extended Courant algebroids} \\label{sec:derived}\nIn Section~\\ref{s:examQ}\nwe studied Q-manifolds $\\CM$ in terms of ordinary differential\ngeometric notions by analyzing the components of the vector field $Q$\nin some chosen (graded) coordinate system, taking into account their\ntransformation properties. There is, however, also an alternative\nprocedure using the space of vector fields on $\\CM$ with its graded\ncommutator bracket as well as its derived one, as is inspired by the\nprevious section on gauge symmetries. While in the case of\nQ1-manifolds or Lie algebroids this provides an alternative route\nleading to an \\emph{evidently} equivalent picture, for Q2-manifolds\none obtains an alternative description of Lie 2-algebroids, closer to\nthe description of Courant algebroids (they will be a particular case\nof what we will call vector bundle twisted Courant algebroids), with a\nnontrivial translation of one picture into the other one,\ncf.~definition \\ref{def:VCourant} and, e.g., Theorem \\ref{thm:der}\nbelow (or, equivalently, Theorem \\ref{thm1.6} in the summary).\n\n\nAs mentioned, the bracket between gauge symmetries \\eqref{induced} is\na so called derived bracket \\cite{YKS03en}. In Subsection~\\ref{sec:Der}\nwe first recall some generic features of derived brackets, starting\nfrom a general differential graded Lie algebra (DGLA). We extend the\nconsiderations of the literature to the context of algebroids by\nintroducing an anchor map; vector fields on a Q$p$-manifold equipped\nwith the differential $\\ud_Q$ are an example of this construction for\narbitrary $p \\in \\N$. We show that applying this perspective to\n$p=1$ reproduces the respective considerations of section\n\\ref{s:examQ} (i.e.~it provides the usual definition of a Lie \nalgebroid) in a rather straightforward way. We also provide the\ndefinition of Courant algebroids in this subsection as another example\nof the derived bracket construction. \n\nInspired by the definition of Courant algebroids and the derived\nbracket construction applied to vector fields on Q$p$-manifolds for\narbitrary $p$ we are led to define Vinogradov algebroids and vector\nbundle twisted Courant algebroids in Subsection~\\ref{subsec:Vino}. The\nformer ones are also inspired by the Vinogradov bracket (which\ngeneralizes the Courant or Dorfman bracket to higher form degrees). \nIn vector bundle twisted\nCourant algebroids, which\nare special Vinogradov algebroids, the inner product in the definition of an\nordinary Courant algebroid is replaced by a bilinear form taking\nvalues in a second vector bundle $V$ over the same base. Although in part\ninspired by Q-manifolds, it is not clear that all Vinogradov and V-twisted\nCourant algebroids have a description in terms of Q-manifolds. \n\nIn Subsection~\\ref{subsec:Lie2} we show that particular V-twisted\nCourant algebroids are in one-to-one correspondence to (split)\nQ2-manifolds, and, consequently (cf.~Theorem \\ref{theo1}) to Lie\n2-algebroids. This equivalence, cf.~Theorem \\ref{thm:der} below, was\nsuggested by the changed perspective on Q-manifolds as stemming from\nthe gauge transformations in our context and would not be that obvious\nto invent without such a motivation. The subsection also contains a\nclassification of these type of V-twisted Courant algebroids, which is\nsomewhat analogous to the one of exact Courant algebroids.\n\n\n\\subsection{Derived brackets and anchored DGLAs}\n\\label{sec:Der}\nThe prototype of a derived bracket goes back to E.~Cartan. Consider the space of differential \nforms $\\Omega^\\cdot(M)$. Besides the de Rham operator $\\ud$ we can also consider contractions \n$\\imath_v$ with a vector field $v$ as an operator. The Lie derivative  $\\L_v$ \nthen results as the graded commutator $\\L_v = [\\imath_v, \\ud]$. On the other hand, certainly, \n$[\\L_v, \\imath_w] = \\imath_{\\L_v(w)}$, by the Leibniz rule of the Lie derivative. Using \n$\\L_v(w)=[v,w]$ and the above formula for the Lie derivative, we see that the bracket $[v,w]$ \nof two vector fields can be viewed as arising from the left hand side of the following \nequation\n\\beq[[\\imath_v,\\ud], \\imath_w] = \\imath_{[v,w]}\\,.\\label{derbasic}\\eeq\nNote the similarity of the left hand side with Eq.~\\eqref{induced}. In fact, the latter \nequation can be almost literally  translated into that setting: Consider the\nsupermanifold $T[1]M$ with the canonical Q-structure $\\ud$ on it. Every vertical vector\nfield on $T[1]M$ corresponds to some $\\imath_v$ for $v \\in \\Gamma(TM)$. It is a nontrivial\nbut easy-to-check fact that the result of the l.h.s.~of \\eqref{derbasic} is a vertical\nvector field again; therefore it comes from a vector field on $M$, defining a product\n$[v,w]$. \n\nThe above construction can be easily extended to include also exterior multiplication \nwith differential forms. Let $\\alpha,\\beta \\in \\Omega^1(M)$ and compute\\footnote{A.~Alekseev,\nprivate communication 1999.} \n\\beq [[\\imath_v+\\alpha\\wedge,\\,\\ud],\\,\\imath_w+\\beta \\wedge]\n  =\\imath_{[v+\\alpha,w+\\beta ]_W} \\, ; \\label{Anton} \\eeq\nIn this way one obtains the  (non-skewsymmetric) Courant, i.e.~the Dorfman bracket \\cite{Cour90,Dor87,Dor93},\n\\beq [v\\oplus \\alpha,w\\oplus \\beta]_W = [v,w]\\oplus \\L_v\\beta -\\imath_w\\ud\\alpha \\, . \\label{Cour}\n\\eeq\nAlso note that the mere bracket,\n$[\\imath_v+\\alpha\\wedge,\\imath_w+\\beta \\wedge]$ induces a natural\nbilinear, symmetric and non-degenerate pairing: when acting on\n$\\Omega^\\cdot(M)$ this operator reduces to multiplication with $\\alpha(w)\n+ \\beta(v)$. Viewing $v \\oplus \\alpha$ as a section of $W=TM\\oplus T^*M$, one\ncan axiomatize \\cite{Xu97,SevLett} rules that can be derived for these\noperations  so as to end up with the definition\n \n \\newpage\n\\begin{vdef} \\label{def:Cour2} A {\\rm Courant algebroid} is a vector bundle $W \\to M$ together with a bundle map\n$\\rho \\colon W \\to TM$, a pairing of its sections $[ \\cdot , \\cdot ]_W \\colon \\Gamma(W)\n\\otimes \\Gamma(W) \\to \\Gamma(W)$ as well as a non-degenerate fiber metric $\\< \\cdot, \\cdot\\>$\nsuch that the following three identities hold true \n\\begin{align}\n  [\\psi,[\\phi_1,\\phi_2]_W]_W &= [[\\psi,\\phi_1]_W,\\phi_2]_W +[\\phi_1,[\\psi,\\phi_2]_W]_W\n    \\label{cJacobi}  \\\\\n  [\\psi,\\psi]_W &= \\frac12\\uD\\<\\psi,\\psi\\>  \\label{cnSkew}\\\\\n  \\rho(\\psi)\\<\\phi_1,\\phi_2\\> &=\n    \\<[\\psi,\\phi_1]_W,\\phi_2\\> + \\<\\phi_1,[\\psi,\\phi_2]_W\\> \n\\, ,  \\label{cInvar}\n\\end{align}\nwhere $\\uD$ is induced by means of $\\rho$ and the inner product via $\\<\\uD f , \\phi\\> =\n\\rho(\\phi) f$.  \n \\end{vdef}\nThe first axiom is the (left-)Leibniz rule of the bracket with respect\nto itself. From \\eqref{cInvar} one can also conclude \\cite{LiB11} a\nLeibniz rule w.r.t.~multiplication of sections by functions\n\\begin{equation} [\\psi,f\\phi]_W = \\rho(\\psi)f\\phi +f[\\psi,\\phi]_W\n\\label{cLeibniz} \\, . \\end{equation}\nThe properties \\eqref{cJacobi} and \\eqref{cLeibniz} render  \n$(W,\\rho,[ \\cdot , \\cdot ]_W)$\nwhat we want to call a Leibniz--Loday algebroid (cf.~also \\cite{KS08})\\footnote{In fact, in \\cite{KS08} we called it Loday algebroid. We finally decided for the present paper to call an algebra with the property \\eqref{cJacobi} a Leibniz-Loday algebra and, if it is an algebra of sections and the condition \\eqref{cLeibniz} is satisfied, a Leibniz-Loday algebroid.}. \nThe second line brings in the fiber metric so as\nto control the violation of the antisymmetry of the bracket, while the last line is the\nad-invariance of the inner product.\nEqs.~\\eqref{cJacobi} and\n\\eqref{cLeibniz} permit to conclude that $\\rho$ is a morphism of\nbrackets, cf.~e.g.~the corresponding generalization in\nProp.~\\ref{p:Vinogrelem} below; together with \\eqref{cLeibniz} this\ngives the five axioms often demanded in the literature for the\ndefinition of a Courant algebroid. \n\nFor a general Courant algebroid one always has the complex\n\\beq 0 \\to T^*M \\to W \\to TM \\to 0 \\, , \\label{seq}\\eeq\ninduced by $\\rho$ and its adjoint. Exact Courant algebroids, which, by\ndefinition, means that the above sequence is exact, are classified by\nan element $[H] \\in H^3_{{\\mathrm{dR}}}(M)$ \\cite{SevLett}; up to a\ncontribution $H(v,w,\\cdot)$  added to the r.h.s.~of\n\\eqref{Cour}, this reproduces the explicit formulas on $W=TM\\oplus\nT^*M$ introduced above (with $\\rho$ being projection to the first\nfactor and a change of splitting in \\eqref{seq} corresponding to a\nchange $H \\mapsto H + \\ud B$).\n\nThe construction in \\eqref{Anton} can be translated easily into\nsuper-language. As before we may use $\\CM=T[1]M$ and consider\n$\\imath_v$ as a vertical vector field and $\\alpha \\wedge$ as\nmultiplication with a function of degree one. If we want to have\n$\\alpha$'s enter as vector fields as well, getting a model for what\nhappens in the previous section, we can simply extend $T[1]M$ by one\n(graded) copy of $\\Real$. Denoting this coordinate by $b$, and\nchoosing local $x^i$ on $M$ with its induced odd coordinates $\\xi^i =\n\\ud x^i$ on $T[1]M$, we then associate to every $\\psi = v \\oplus \\alpha$\nthe vector field\n\\begin{equation} \\psi = v^i \\frac{\\partial}{\\partial \\xi^i} + \\alpha_i \\xi^i  \\frac{\\partial}{\\partial b} \\, ; \\end{equation}\ndeclaring $b$ to have degree two, this becomes a homogeneous vector\nfield on the graded manifold $\\CM = T[1]M \\times \\Real [2]$ of degree\nminus one (and, moreover, it has the form of the most general vector\nfield of this degree). The bracket \\eqref{Cour} now follows as a\nderived bracket of (graded) vector fields on $\\CM$, $[ v \\oplus \\alpha , w\n\\oplus \\beta]_W = [\\psi,\\psi']_Q$ where the latter is defined by \n\\begin{equation} [ \\psi, \\psi']_Q := [[ \\psi , Q ], \\psi'] , \\label{Cour2} \\end{equation}\nwith $Q$ being the\ncanonical vector field on $T[1]M$ corresponding to the de Rham\ndifferential on $M$ extended trivially to the product and the bracket\ndenoting the graded Lie bracket of vector fields.\n\nObviously, this\ncan be ``twisted'' by a closed 3-form $H$ by replacing the above $Q$\nby \\begin{equation} Q = \\xi^i  \\frac{\\partial}{\\partial x^i} + \\frac{1}{3!} H_{ijk}\n\\xi^i \\xi^j \\xi^k  \\frac{\\partial}{\\partial b} \\label{Qtwist1} \\end{equation}\nwhich again yields a Q-structure on $\\CM$. This reproduces the general\nsituation of an exact Courant algebroid mentioned above. Note also\nthat in this language the inner product on $W=TM \\oplus T^*M$ results\nfrom the (graded) bracket of the corresponding vector fields on\n$\\CM$: \n\\begin{equation} [\\psi,\\psi'] = \\<\\psi,\\psi'\\>   \\frac{\\partial}{\\partial b} \\, .\n\\label{inner2}\\end{equation} \n\n\nTo have vector fields $v$ and 1-forms $\\alpha$ enter more symmetrically,\none may alternatively lift the graded construction on $\\CM = T[1]M$\ncanonically to the (super)symplectic manifold $T^*[2]\\CM$: The local\ncoordinates $x^i$ and $\\xi^i$ of degree 0 and 1 on $\\CM$ are then\naccompanied by canonically conjugate momenta $p_i$ and $\\xi_i$ of\ndegree 2 and 1, respectively. $\\psi= v + \\alpha = v^i \\xi_i + \\alpha_i \\xi^i$\nthen describes the most general function of (total) degree\none\\footnote{A vector field on $\\CM$ is always a fiber linear function\non its cotangent bundle and a function can be pulled back canonically\nby the projection. The shift in grading by two has been chosen so as\nto have them both of the same degree.}, and the canonical Hamiltonian $\\tilde Q$ for\nlifting the vector field $Q=\\xi^i \\partial\/\\partial x^i$ to the\ncotangent bundle, $\\tilde Q = \\xi^i p_i$, then reproduces\n\\eqref{Anton} and the inner product by means of the (graded)\nPoisson brackets\n\\begin{align}\n\\{ \\, \\{ \\psi , \\tilde Q \\} , \\psi' \\} &= [ \\psi ,  \\psi' ]_W \n\\label{dBrack1}\\\\\n\\{  \\psi , \\psi' \\} &= \\< \\psi , \\psi' \\> \\, .\n\\label{dInner}\\end{align}\nIn fact, such a construction can be extended to the general setting of\na Courant algebroid as defined above. One then finds that Courant\nalgebroids are in {\\em one-to-one} correspondence to {\\it symplectic}\nQ2-manifolds \\cite{Royt02} (where, by definition, the symplectic form\nis required to be of degree two and to also be compatible with the\nQ-structure, i.e.\\ to be preserved by the vector field $Q$).\n\nOne observes the similarity of the formulas \\eqref{Cour2},\n\\eqref{inner2} with \\eqref{dBrack1}, \\eqref{dInner}. In both\ncases the original bracket is a (graded) Lie bracket, in the second\ncase of degree minus two, in the first case of degree\nzero.\\footnote{\\label{f:24} In fact, the second bracket is even a\n(graded) Poisson bracket, i.e.~there exists a second compatible graded\ncommutative multiplication $\\cdot$, usually taken to be of degree zero. (Here\ncompatibility means a graded Leibniz condition: $\\{F,G \\cdot H \\} = \\{F,G\n\\} \\cdot H + (-1)^{(|F|+d)|G|} G \\cdot \\{F, H \\}$, where $d$ is the degree of the bracket, cf.~also the subsequent footnote).  If one\nprefers, also the bracket of vector fields can be embedded into a\nPoisson algebra, by viewing vector fields as fiber linear functions on\nthe cotangent bundle and thus replacing $\\CM$ by the graded manifold\n$T^*\\CM$, if one prefers, shifted also by some degree (by degree one\nif the canonical Poisson bracket is to reproduce the\nSchouten--Nijenhuis  bracket of (graded)\nvector fields on $\\CM$). We will come back to this perspective below.}\nIn the previous section, moreover, we viewed the left-adjoint action\nof $Q$ as a (compatible) differential $\\ud_Q$. Thus, to cover all\nrelevant cases simultaneously, one is led to regard the derived\nbracket construction of a differential graded Lie algebra (DGLA),\ni.e.\\ a graded Lie algebra with a differential (nilpotent odd\noperator) compatible with the bracket:\\footnote{We call\n$(A^\\bullet,[.,.])$ a (graded) \\emph{Leibniz--Loday algebra} of degree $d$, if\n$[A^k,A^l]\\subset A^{k+l+d}$ and $[a,[b,c]]=[[a,b],c]\n+(-1)^{(|a|+d)(|b|+d)}[b,[a,c]]$. It is a \\emph{differential graded\nLeibniz--Loday algebra} (DGLoA) if in addition there is an operation $D \\colon\nA^\\bullet \\to A^{\\bullet+1}$ squaring to zero and satisfying\n$D([a,b])=[Da,b] + (-1)^{(|a|+d)} [a,Db]$. If the bracket $[.,.]$ is\nalso graded antisymmetric, i.e.~$[b,a]=-(-1)^{(|a|+d)(|b|+d)}[a,b]$, we\ncan replace ``Leibniz--Loday'' by ``Lie'' in these definitions.}\n\n\\begin{prop}[Kosmann-Schwarzbach \\cite{YKS96}]\\label{p:YKSder} Let\n$(A^\\bullet,[.,.],\\uD)$ be a DGLA (with bracket\nof degree $d$) and define its derived bracket as:\n\\begin{equation}\\label{Dder} [\\phi,\\psi]_D = (-1)^{|\\phi|+d+1}[\\uD\\phi,\\psi] \\;,\n\\end{equation}\nwhere $|\\phi|$ denotes the degree of $\\phi$ (assumed to be\nhomogeneous).  \n\n\\noindent Then the following facts are true: \n\\begin{enumerate}\n\\item This new bracket has degree $d+1$ and is a graded\nLeibniz-Loday bracket,\n\\begin{equation}\\label{Loday} [\\phi,[\\psi_1,\\psi_2]_D]_D = [[\\phi,\\psi_1]_D,\\psi_2]_D\n  +(-1)^{(|\\phi|+d+1)(|\\psi_1|+d+1)}[\\psi_1,[\\phi,\\psi_2]_D]_D \\;.\n\\end{equation}\n\\item If $|\\phi|$, $|\\psi_!|$, and $|\\psi_2|$ all have the opposite parity of $d$, one has \n\\begin{eqnarray}\\label{dSkew} [\\phi,\\phi]_D &=&\\frac12 \\uD[\\phi,\\phi] \\;, \\\\\n             {}[\\phi,[\\psi_1,\\psi_2]]_D &=& [[\\phi,\\psi_1]_D,\\psi_2]+ \n[[\\psi_1,[\\phi,\\psi_2]_D]\\label{Dcomp} \\; .  \\label{adInvar}\n\\end{eqnarray}\n\\item The quotient of  $(A^\\bullet,[.,.]_D)$ by $\\ker D$ is a graded Lie\nalgebra. \n\\end{enumerate} \n\\end{prop}\nThese facts are easy to verify and recommended to the reader as an\nexercise. (To see that $A^\\bullet\/\\ker D$ is well-defined one observes\nthat $[\\phi,\\psi]_D$ equals $[\\phi,D\\psi]$ modulo a $D$-exact term,\nwhich can also be used to establish the graded antisymmetry of the\ninduced bracket). \n\nIn fact, the Leibniz-Loday Property~\\eqref{Loday} also follows in the more\ngeneral context of $(A^\\bullet,[.,.],\\uD)$ being a DGLoA only, cf.~the\npreceding footnote and Lemma \\ref{p:der2} below. The choice of the\nsign in \\eqref{Dder} is essential and can be motivated also from\nexamples where $\\uD$ is an inner derivation, $\\uD=[Q,.]$ with\n$[Q,Q]=0$, and the brackets arise as in \\eqref{Cour2}. This\ndefinition also agrees with (the first part of) Eq.~\\eqref{induced}\nwith the identification $\\uD=\\ud_Q$, since there the bracket has\ndegree zero and the elements are odd.\n\n\\bigskip\n\nWe now turn to the alternative description of Q-manifolds when\nstudying the vector fields on them and their derived bracket. To\nexplain this procedure, we start with a Q1-manifold, $\\CM = E[1]$:\nSections $\\psi$ in $E$ can be identified with degree minus one vector\nfields on $\\CM$, $\\psi = \\psi^a\n\\partial\/\\partial \\xi^a$ in our notation.  Replacing $\\ud$ in Cartan's\nformula \\eqref{derbasic} with the differential of the Lie algebroid (acting\n on ${{}^E{}\\Omega}^\\bullet(M)$) every Lie algebroid bracket occurs naturally as a\nderived bracket. In super-language this means explicitly that the\nLie algebroid bracket on $E$, which for clarity we now denote by $[\n\\cdot , \\cdot ]_E$ is a derived bracket w.r.t.~$Q$,\n\\begin{equation} [\\phi,\\psi]_E = [[\\phi,Q],\\psi] \\, , \\label{Eder}\n\\end{equation}\nwhere, for notational simplicity, we did not introduce a symbol for\nthe map identifying sections in $E$ (as they appear on the l.h.s.~of\nthis equation) with the respective degree minus one vector field (as\nunderstood on the r.h.s.)---in \\eqref{derbasic} the symbol\n$\\imath_\\cdot$ could be interpreted as such an isomorphism. We will\nkeep such a notational convention also further on. From the previous\ngeneral considerations about DGLAs and their derived brackets, we can\nnow at once conclude about two of the three defining properties for\n$[\\cdot , \\cdot ]_E$ to be a Lie algebroid bracket: Since there are\n\\emph{no} degree minus two vector fields on $\\CM = E[1]$, the right\nhand side of \\eqref{dSkew} vanishes identically. With the fact that\n$\\phi$, $\\psi$ have odd degree and the natural bracket of super vector\nfields degree $d=0$, Eqs.~\\eqref{dSkew} and \\eqref{Loday} yield\n$[\\cdot , \\cdot ]_E$ to be (ungraded) antisymmetric and to satisfy an\n(ungraded) Jacobi identity, respectively. To retrieve the Leibniz \nproperty w.r.t.~an or the anchor map of $E$,\nwe also need to make use of the module structure of vector fields\nw.r.t.~functions on $\\CM$, where we may restrict to degree zero\nfunctions $C^\\infty_0(\\CM) \\cong C^\\infty(M)$ so as to preserve the\ndegree of our vector fields. This will be provided by\nEq.~\\eqref{eq:Leibniz} below, then establishing that we obtain a Lie\nalgebroid structure on $E$ by this procedure. \n\nSince we will need such type of anchor maps also in the general\ncontext of higher degree $Q$-manifolds and their compatibility with\nthe derived bracket construction, we will extend the\nconsiderations on DGL(o)As to this setting. A rather minimalistic way of\ndoing so is \n\\begin{vdef}\nWe call $(A^\\bullet,[.,.],\\uD, R, \\rho_0)$ a  \\emph{weakly anchored} DGL(o)A\nover the ring $R$ (commutative, with unity) if $(A^\\bullet,[.,.],\\uD)$\nis a DGL(o)A of degree $d$ and $\\cdot \\colon R \\times A^k\n\\to A^k$ and $\\rho_0 \\colon A^{-d} \\to \\End_{\\Real} (R)$\nare maps such that \n\\begin{equation} [\\phi, f \\cdot \\psi] = \\left(\\rho_0(\\phi)\\right)(f) \\cdot \\psi + f\n\\cdot [\\phi, \\psi] \\label{ring}\n\\end{equation}\nholds true for every $\\phi \\in  A^{-d}$, $\\psi \\in A$, and\n$f\\in R$.\n\nWe call it merely \\emph{anchored} iff in addition $\\rho_0$ is $R$-linear and\nso is its composition with $\\uD$, i.e.~for all $\\phi \\in  A^{-d-1}$\none has\n\\[ \\label{rhoBund} \\rho_0(\\uD(f\\phi))=f\\rho_0(\\uD\\phi) \\,. \\]\nWe call it \\emph{strongly anchored} if also ($\\forall f,g \\in R, \\psi \\in\nA$) \n\\[  \\uD(fg\\psi) +fg\\uD \\psi= f\\uD(g\\psi) +g\\uD(f\\psi) \\,. \\label{1stder} \\]\n\\end{vdef} \n\nIn fact, \\eqref{1stder} is \\emph{equivalent} to the\nexistence of an operator  $\\CD \\colon R\\to \\Hom_R(A^\\bullet,A^{\\bullet+1})$ such that\n\\begin{align}\n  \\uD(f\\phi) &= (\\CD f)(\\phi) +f\\uD \\phi  \\label{DLeib} \\,, \\\\\n  \\CD(fg) &= f\\CD g +g\\CD f \\label{D0Leib}\n\\end{align}  holds true. \\eqref{D0Leib} implies that  $\\CD$ is a first \norder differential operator and \\eqref{DLeib} that likewise is $\\uD$\nw.r.t.~$R$. In fact Eq.~\\eqref{DLeib} also shows constructively how to\ndefine $\\CD$ once \\eqref{1stder} is fulfilled. We have chosen the\nformulation as in \\eqref{1stder} since it is intrinsic and does not\nneed the introduction of new operations such as the above map $\\CD$. \n\nA graded Lie algebroid over an ungraded manifold $M$ gives an example\nof a strongly anchored DGLA with $\\uD=0$; here $A^\\bullet$ are the\nsections in the graded vector bundle and $R=C^\\infty(M)$, as an\n$R$-module $A^\\bullet$ is projective in this example. Obviously the\nabove definition can be easily extended to graded commutative rings\n$R$ in which case general graded Lie algebroids can be covered as\nwell; this is however not needed for the applications we are having in\nmind here (cf.~Proposition \\ref{p:tangentLie} below). An example with\nnontrivial $\\uD$ can be constructed from Lie bialgebroids satisfying\nEq.~\\eqref{rhoBund}; here $A^\\bullet$ consists of the sections of the\nexterior powers of the (ungraded) vector bundle. But more important\nfor our context is the following observation:\\footnote{Cf.~also\nfootnote\n\\ref{f:8} concerning our simplified nomenclature and footnote \\ref{f:24} for\na possible relation of the two parts of the proposition. A differential\ngraded Poisson algebra $(A^\\bullet, \\{ . , . \\}, \\cdot , \\uD)$ of degree $d$ is\na DGLA $(A^\\bullet, \\{ . , . \\}, \\uD)$ and a graded Poisson algebra\n$(A^\\bullet, \\{ . , . \\}, \\cdot)$, both of degree $d$, such that $\\uD(F \\cdot\nG) = \\uD F \\cdot G + (-1)^{|F|} F \\cdot \\uD G$ for all $F\n\\in A^\\bullet$, $G \\in A$.}\n\\begin{prop}\\label{p:tangentLie}\n$\\,$\n\\begin{enumerate}\n\\item $(A^\\bullet,[.,.])=\\X(\\CM)$ with $\\uD=\\ud_Q$\nof a Q-manifold is a strongly anchored DGLA with bracket of degree zero \nw.r.t.\\ $R=C^\\infty_0(\\CM)\\cong{C^\\infty}(M)$, where $\\rho_0(\\phi)[f]=\\phi(f)$.\n\\item \nA differential graded Poisson algebra (DGPA) on $C^\\infty(\\CM)$ is a weakly\nanchored DGLA w.r.t.\\ $R=C^\\infty_0(\\CM)$ and $\\rho_0(\\phi)f=\\{\\phi,f\\}$. It\nis strongly anchored iff $\\{f,g\\}=0=\\{\\phi \\cdot \\uD f,g\\}$ for all $f,g\\in\nR$ and $\\phi \\in C^\\infty_{-d-1}(\\CM)$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof} For the case of vector fields use $\\uD \\phi = [Q,\\phi]$ and find\n\\eqref{DLeib} with \n$(Df)(\\phi)$ being the multiplication of $\\phi$ by $Q(f)$. $Q$ being a vector\nfield, this proves also \\eqref{D0Leib} and thus \\eqref{1stder}. From its\ndefinition it is immediate that $\\rho_0$ is $R$-linear in $\\phi\n\\in \\X_0(\\CM)$. Moreover using \\eqref{DLeib} and observing that $\\rho_0\n\\left(Q(f) \\phi\\right)=0$, which follows since here $\\phi$ is degree minus\none and vanishes when applied to degree zero functions, also \\eqref{rhoBund}\nfollows. \n \n\nIn the Poisson case the two additional conditions are found by directly\nchecking the desired conditions using the definition of\n$\\rho_0$. E.g.~$\\{f,g\\}=0$ follows immediately from $R$-linearity of\n$\\rho_0$. \n\\end{proof}\n\n\\pagebreak[2]These additional structures are compatible with the\nderived bracket construction in the following sense:\\nopagebreak[3]\n\\begin{lemma}\\label{p:der2}\n  Let $(A^\\bullet,[.,.],\\uD,R,\\rho_0)$ be a (weakly\/strongly) anchored DGLoA, \n  then $(A^\\bullet,[.,.]_\\uD,\\uD,R,\\rho)$ is also a (weakly\/strongly)\n  anchored DGLoA with \n\\begin{equation} \\rho(\\phi) = \\rho_0(\\uD\\phi)  \\;. \\label{eq:rho1}\n\\end{equation}\n\\end{lemma}\nThis can be proven by simple straightforward calculations. The Lemma contains\nthree different statements. \n\nWe now apply these considerations first by returning to Q1-manifolds. As a\nconsequence of Proposition \\ref{p:tangentLie} and the preceding lemma,\nwe notice that the derived bracket \\eqref{Eder} satisfies a Leibniz\nrule \\begin{equation} [\\phi, f \\cdot \\psi]_E =\\rho(\\phi)(f) \\cdot \\psi + f\n\\cdot [\\phi, \\psi]_E \\label{eq:Leibniz} \n\\end{equation} with an $R$- or $C^\\infty(M)$-linear anchor map \n$\\rho$ given by $\\rho(\\phi)f = [Q,\\phi] f$. $C^\\infty(M)$-linearity\nimplies  that $\\rho$ indeed comes from a vector bundle map $E\\to TM$. \n\n\n\nThus, indeed, the bracket on $\\Gamma(E)$ defined via the derived\nbracket construction\n\\eqref{Eder} induces a Lie algebroid structure on $E$. Moreover, this is the \n\\emph{same} Lie algebroid structure as found before in Section~\\ref{s:examQ}, \nwhich one easily verifies on a basis: Using the above formulas as definitions \nfor the bracket and the anchor, one verifies easily that $[\\partial_a,\n\\partial_b]_E = C^c_{ab} \\partial_c$ and $\\rho(\\partial_a)=\\rho^i_a \n\\partial_i$ (with $\\partial_a \\equiv \\frac{\\partial}{\\partial \\xi^a}$ and \n$\\partial_i \\equiv \\frac{\\partial}{\\partial x^i}$ for the parameterization used \nin \\eqref{Q}).\n\nWe now want to imitate this second procedure, illustrated at the\nexample of Q1-manifolds and in spirit close to the considerations of\nthe previous section on gauge symmetries, so as to find an alternative\ncharacterization of degree two Q-manifolds. In fact, we want to\nperform some of the considerations common to all choices of degree $p$\nfirst and later specialize the discussion to $p=2$. Courant algebroids\nshould be particular examples of the latter case, moreover, and we\nwill find a description of Q2-manifolds or Lie-2-algebroids\ncomplementary to the one given in Section~\\ref{s:examQ} and much closer to\nthe usual formulation of Courant algebroids.\n\n\n\\subsection{Vinogradov and $V$-twisted Courant algebroids}\n\\label{subsec:Vino}\nTo obtain a prototype of a bracket for higher $p$ than two, we\nmay look at a generalization of the Courant-Dorfman bracket, the\nso-called Vinogradov\nbracket. It is in fact the bracket \\eqref{Cour} with $\\alpha,\\beta\n\\in \\Omega^k(M)$ for some arbitrary $k \\in \\N$, and it can be \nobtained precisely as for the case $k=1$. Accordingly, we again have\nat least two options translating this scenario into super-language. In\nthe $\\CM=T[1]M \\times \\Real[p]$ picture\\footnote{We have found this description in 2005. In the mean time it appeared also in \\cite{Uri13}}, sections $v\\oplus\\alpha$ correspond\nto \\begin{equation}  \\label{psi}\n\\psi = \\imath_v + \\alpha \\frac{\\partial}{\\partial b} \\, , \\end{equation} \nwhere $b$ obviously needs to have degree $p=k+1$ for $\\psi$ to be\nhomogeneous of degree minus one. The Vinogradov bracket then follows\nagain as a derived bracket \\eqref{Cour2} with $Q$ being the de Rham\ndifferential. In generalization of Eq.~\\eqref{Qtwist1} this may\ncertainly again be twisted, \\begin{equation} Q = \\ud + H \\frac{\\partial}{\\partial\nb} \\, , \\label{Qtwist2} \\end{equation} where now $H$ needs to be a closed\n($k+2$)-form obviously. Like in the Courant algebroid, also here the\noriginal (graded) Lie bracket plays an important role,\ncf.~Eq.~\\eqref{inner2} (or \\eqref{dInner}), generating a\n$C^\\infty(M)$ bilinear pairing; it comes from the contraction of the\nvector fields with the $k$-forms, as one easily verifies using\n\\eqref{psi}. Thus, this time the pairing maps two elements in\n$W=TM \\oplus \\Lambda^k T^*M$ to an element of another bundle, namely\nof $V=\\Lambda^{k-1} T^*M$. This, the appearance of a second bundle $V$\nof relevance, will be one feature we want to keep also for the more\ngeneral setting in what follows.\n\nBefore axiomatizing these data, in analogy to the step from\n\\eqref{Cour} to Definition \\ref{def:Cour}, into what one may want\nto call a Vinogradov algebroid, we briefly translate the above picture\nalso into a graded Poisson language. We therefore regard \n$\\CM=T^*[p]T[1]M$ with its canonical Poisson bracket $\\{ \\cdot , \\cdot\n\\}$ of degree $-p$. Local coordinates $x^i$ and $\\xi^i$ of degree 0\nand 1, respectively, are then accompanied by momenta $p_i$ and $\\xi_i$\nof degree $p$ and $p-1$, respectively. Sections of $W$ correspond to\nfunctions on this $\\CM$ of degree $k=p-1$, $\\psi= v^i \\xi_i +\n\\frac{1}{k!}\\alpha_{i_1\n\\ldots i_k} \\xi^{i_1} \\ldots \\xi^{i_k}$, and the bracket and pairing \non $W$ results from formulas \\eqref{dBrack1} and \\eqref{dInner},\nrespectively, where $\\tilde{Q}=\\xi^i p_i +H$. \n\n\nWe now return to the task of axiomatization. In particular, one may\nwant to check properties of the bracket and pairing, finding\nappropriate generalizations of those written in definition\n\\ref{def:Cour}. There are natural candidates for the analogues of the\nobjects appearing in \\eqref{cLeibniz} and \\eqref{cnSkew}, which will\nalso turn out to be realized for what one may call the Standard\nVinogradov algebroid ($W=TM \\oplus \\Lambda^k T^*M$, $V=\\Lambda^{k-1}\nT^*M$), possibly twisted by some $H\\in \\Omega_{cl}^{k+2}(M)$. The\nanchor map $\\rho$ will be projection to $TM$, for $\\uD$ one can choose\nthe de Rham differential (succeeded by the natural embedding of\n$k$-forms into sections of $W$). For the ad-invariance of the pairing\n(cf.~Eq.~\\eqref{cInvar}), however, we can no more use solely the\nanchor map, since the pairing lands in sections of $V$. The operator\non the l.h.s.~of\n\\eqref{cInvar} will then be found to be replaced by the Lie derivative\nw.r.t.~the anchor of the section $\\psi$. Note that this is for $k>1$\n(or, likewise, $p>2$) no more $C^\\infty(M)$-linear in $\\psi$. So, in\nthe axioms there will be besides $\\rho$ and $\\uD$ one more new object,\nwhich we will call $\\rho_D$, not necessarily coming from a bundle map,\nbut being restricted appropriately otherwise. Also, the relation\nbetween $\\rho$ and $\\uD$ being adjoint to one another does not make\nany sense anymore.\\footnote{Only the generalization that $\\rho_D$\nwould be adjoint of $\\uD$ w.r.t.~the pairing can be meaningful. In\nfact, although it is not satisfied in the Standard Vinogradov\nalgebroid for $k>1$, posing this additional condition will be studied\nin detail the context of $p=2$ below, cf., e.g., definition\n\\ref{def:VCourant} and Proposition \\ref{p:FW}.}  $\\uD$ needs to be\nrestricted then by something like Equation~\\eqref{1stder} so as to\nensure that it is a first order differential operator. Likewise for\n$\\rho_D$ we require it to take values in the module $\\Gamma(\\CDO)(V)\\equiv \\Gamma(\\CDO(V))$ of covariant differential operators  on $V$: Keeping in\nmind that the $\\Real$-linear first order differential operators on a\nvector bundle $V\\to M$ are themselves a projective module over $M$, we\ncall the generating bundle $\\CDO(V)$, the bundle of covariant\ndifferential operators on $V$; elements in  $\\Gamma(\\CDO)(V)$ are then sections of this bundle. There is a canonical map from $\\CDO(V)$\nto $TM$ which we may use to require $\\rho_D$ to cover the\nmap from $\\Gamma(W)$ to $\\Gamma(TM)$ induced by $\\rho$ (and\nconventionally denoted by the same letter), i.e.~in formulas we require \n\\begin{equation} \n  \\rho_D(\\phi)[fv] = \\rho(\\phi)[f]v +f\\rho_D(\\phi)[v] \\, .\n\\label{vLeibn3} \n\\end{equation} where $\\phi \\in \\Gamma(W)$, $f \\in C^\\infty(M)$, and $v \\in \\Gamma(V)$.\nWe thus are led to the following\n\n\\newpage\n\\begin{vdef} \\label{def:Vino} A \\emph{Vinogradov algebroid} are two vector bundles $W$ and $V$\ntogether with a bracket $[.,.]_W:\\Gamma(W)\\times\\Gamma(W)\\to \\Gamma(W)$,\na map $\\rho\\:W\\to TM$, a non-degenerate\\footnote{A (skew)-symmetric bilinear form $\\<.,.\\>$ is non-degenerate if for every $\\phi\\in E_x$, $\\phi\\ne0$ there is a $\\psi\\in E_x$ such that $\\<\\phi,\\psi\\>\\ne0$.} surjective inner product\n$\\<.,.\\>\\: W\\otimes_M W\\to V$, an $\\Real$-linear map\n$\\uD:\\Gamma(V)\\to \\Gamma(W)$,\nand a map $\\rho_D: \\Gamma(W)\\to \\Gamma(\\CDO)(V)$ covering the map\n$\\rho \\colon  \\Gamma(W)\\to \\Gamma(TM)$ subject to the\nfollowing axioms.\n\\begin{align}\n  [\\psi,[\\phi_1,\\phi_2]_W]_W &= [[\\psi,\\phi_1]_W,\\phi_2]_W +\n[\\phi_1,[\\psi,\\phi_2]_W]_W\n    \\label{vJacobi}  \\\\\n  [\\psi,\\psi]_W &= \\frac12\\uD\\<\\psi,\\psi\\>  \\label{vSkew}\\\\\n  \\rho_D(\\psi)\\<\\phi,\\phi\\> &=\n  2\\<[\\psi,\\phi]_W,\\phi\\>  \\label{vInvar}\\\\\n  \\uD(fgv) +fg\\uD(v) &= f\\uD(gv) +g\\uD(fv)  \\label{vLeibn2}\n\\end{align} where $\\phi_i,\\psi \\in \\Gamma(W)$, $f,g \\in {C^\\infty}(M)$,\nand $v \\in \\Gamma(V)$.\n\\end{vdef}\nIt is easy to see that \\eqref{vInvar} and \\eqref{vLeibn3} imply \n\\begin{equation}\n[\\psi,f\\phi]_W = \\rho(\\psi)f\\phi +f[\\psi,\\phi]_W \\label{vLeibniz} \\;\n\\end{equation} and, conversely,\none may also conclude \\eqref{vLeibn3} from \\eqref{vInvar}\nand \\eqref{vLeibniz}.  One may consider adding $[\\uD v,\\phi]_W =\n0\\label{brwExact}$ to the axioms as this rule is suggested from\nseveral perspectives (e.g.~it is satisfied for the Vinogradov\nbracket). For the case of\n $\\rho_D(\\phi)[v]=\\<\\phi,\\uD v\\>$, on the other hand,\nthis follows automatically, cf.~Proposition \\ref{p:Royt} below.\n\nTo require surjectivity of the pairing goes almost without loss of\ngenerality since $V$ enters the axioms only via the image of  $\\<\n\\cdot , \\cdot \\>$; only if this image changes its rank, one cannot\nconsistently restrict to it as a vector bundle. On the other side, it\nis suggested from some perspectives to drop the condition of\nnon-degeneracy,\nwhich, however, then would go with a partially\ndifferent formulation of the axioms and we\nwill not pursue this idea in the present article. Let us mention that\nin the application to Q-manifolds the inner product is automatically\nsurjective and non-degenerate, cf.~Proposition \\ref{p:genStdVinogr}\nbelow.\n\n\nNote that the Equation~\\eqref{vInvar} is nothing but the invariance of the\ninner product under sections of $W$, being equivalent to\n\\begin{align*}\n  \\rho_D(\\psi)\\<\\phi_1,\\phi_2\\> &=\n    \\<[\\psi,\\phi_1]_W,\\phi_2\\> + \\<\\phi_1,[\\psi,\\phi_2]_W\\> \\,\n\\end{align*}\nby a standard polarization argument. It is an instructive exercise to\ncompute the analog of the Leibniz rule (Eq.~\\eqref{vLeibniz}) for\nthe left hand side of the bracket, which shows that the bracket\nis also a first order linear partial differential operator in the l.h.s.\nAlso one has the following \n\\begin{prop}\\label{p:Vinogrelem}  The maps $\\rho$ and $\\rho_D$ are morphisms of\nbrackets.\\footnote{In the case of Courant algebroids it was apparently\nfirst K. Uchino \\cite{Uchi02} who observed that the morphism property\nof $\\rho$ can be deduced from the other axioms.} Furthermore, their\ncomposition with $\\uD$ vanishes ($\\rho\\circ\\uD =0=\\rho_D\\circ\\uD$).\n\\end{prop}\n\n\\begin{proof}\n \nThe morphism property of $\\rho$ can be concluded easily from\n\\eqref{vJacobi} and \\eqref{vLeibniz}, cf., e.g., \\cite{HS08}. \nFor the morphism property of $\\rho_D$ write a section of\n$V$ locally as scalar product of two sections of $W$.  Then use \\eqref{vInvar}\nand \\eqref{vJacobi} to proceed in a similar manner.\nThe last two properties follow from \\eqref{vSkew} and the first or second property, respectively.\n\\end{proof}\n\n\n\n\nLet us now check that the axioms are indeed all fulfilled for the\n(twisted standard) Vinogradov bracket described above.  Since we\nalready introduced some super language for the bracket, let us start\nfrom there:\n\\begin{prop}\\label{p:derVino}  Given a strongly anchored DGLA structure\n  $(A^\\bullet,[.,.],\\uD,\\rho_0)$ with bracket of degree $d$ over\n  $R={C^\\infty}(M)$ of a smooth manifold $M$, where $A^{-d-1}$ and\n  $A^{-d-2}$ are projective modules over $M$, then there exist vector\n  bundles $W\\to M$ and $V\\to M$ such that $A^{-d-1}$ is (canonically\n  isomorphic to) the space of sections of $W$ denoted\n  $\\phi,\\psi,\\ldots$ in what follows and $A^{-d-2}$ is the space of\n  sections of $V$ a typical element of which is denoted by $v$ below.\n  Thus $\\uD \\colon \\Gamma(V)\\to \\Gamma(W)$. Now the following operations\n  can be defined by their right hand sides:\n\\begin{align}\n  [.,.]_W &: \\Gamma(W)\\otimes\\Gamma(W)\\to \\Gamma(W),&  [\\phi,\\psi]_W&:=[\\uD\\phi,\\psi] \\label{dBrack2}\\\\\n  \\rho &: W\\to TM,&\n \\rho(\\phi)[f]&:= \\rho_0(\\uD\\phi)[f]  \\label{dAnchor}\\\\\n  \\<.,.\\> &: \\Gamma(W)\\otimes\\Gamma(W)\\to \\Gamma(V),&\n    \\<\\phi,\\psi\\>&:=[\\phi,\\psi] \\label{dInner2}\\\\\n  \\rho_D &: \\Gamma(W)\\to \\Gamma(\\CDO)(V), \\;& \\rho_D(\\phi)[v]&:= [\\uD\\phi,v] \\,. \\label{rhoQ}\n\\end{align}\n  $\\<.,.\\>$ is ${C^\\infty}(M)$-linear iff $[\\psi,f\\phi]=f[\\psi,\\phi]$ for\n  all functions $f$ on $M$. Requiring this and that the induced inner product\n  $\\<.,.\\>\\:W\\otimes_M W\\to V$ is non-degenerate and surjective, these\n  operations form a Vinogradov algebroid.\n\\end{prop}\n\\begin{proof}  Follows from Proposition \\ref{p:YKSder}\n  and Lemma \\ref{p:der2} with the interpretation that \\eqref{vLeibn3} is just\n  the Leibniz rule for the derived bracket restricted to $A^{-d-2}$ in the\n  right-hand argument.\n\\end{proof}\n\nDue to Proposition \\ref{p:tangentLie} the vector fields on our super manifold\n$\\CM=T[1]M\\times\\Real[k+1]$ together with the above $Q$-structure fulfill the\nfirst condition.  ${C^\\infty}(M)$-linearity of the inner product is\nautomatically fulfilled for Q-manifolds (since $d=0$ and thus $\\Gamma(W)$\nis identified with degree minus one vector fields in this case). \nNon-degeneracy follows by direct inspection, noting that  \n$W \\cong TM\\oplus \\Lambda^kT^*M$ for\n$1\\le k\\le\\dim M$. This shows that the (twisted) Vinogradov\nbracket and the canonical pairing indeed provide an example of a\nVinogradov algebroid; we will call this the ($H$-twisted) standard \nVinogradov algebroid.\n\nAn immediate generalization of this is to replace the tangent bundle\nin the above construction by a Lie algebroid $A$.  Starting with a Lie\nalgebroid $(A,[.,.]_A,\\rho_A)$ and a positive integer $k\\le\\rk A$ we\nconsider the bundle $W:=A\\oplus\\Lambda^kA^*$ with the canonical\nsymmetric (non-degenerate) pairing $\\<X\\oplus\\alpha,Y\\oplus\\beta\\>\n:=\\imath_{X}\\alpha+\\imath_{Y}\\beta$.  The choice for $V$ is therefore\n$\\Lambda^{k-1}A^*$.  The choice of the super manifold $\\CM$ is analog\nto the tangent case $\\CM=T^*[k+1]A[1]$.  $A[1]$ also has a canonical\n$Q$-structure, the differential $\\ud_A$ of the algebroid $A$.  Its\nHamiltonian lift to $\\CM$ is the $Q$-structure we use.  Since the\nHamiltonian is itself of degree $k+2$, we can add an $A$-$(k+2)$-form\n$H$, satisfying $\\ud_AH=0$ so as to render the total charge further on\nnilpotent.  This leads us to\n\\begin{prop}\\label{p:genStdVinogr}  Let $(A,[.,.]_A,\\rho_A)$ be a Lie algebroid over\n$M$. Let $W=A\\oplus\\Lambda^kA^*$ and $V= \\Lambda^{k-1}A^*$. Then  \n\\begin{align}\n  (X\\oplus\\alpha,Y\\oplus\\beta) &:= \\imath_X\\beta+\\imath_Y\\alpha \\\\\n  [X\\oplus\\alpha,Y\\oplus\\beta]_W &:= [X,Y]_A\\oplus \\L_X\\beta\n    -\\imath_Y\\ud_A\\alpha +\\imath_X\\imath_Y H  \\label{stdVbrack}\\\\\n  \\rho(X\\oplus\\alpha) &:= \\rho_A(X)  \\\\\n  \\uD(v) &:= 0\\oplus\\ud_Av  \\label{stdVder}\\\\\n  \\rho_D(X\\oplus\\alpha) &:= \\L_X  \\, ,  \\label{stdVrhoQ}\n\\end{align}  where $X,Y \\in \\Gamma(A)$, $\\alpha,\\beta \\in\n\\Gamma(\\Lambda^kA^*)$,  and $v \\in \\Gamma(V)$, equips $(W,V)$ with the\nstructure of a Vinogradov algebroi\n.\n\\end{prop}\n\\begin{proof}  Follows from Proposition \\ref{p:derVino} with the\nrealization as a Q-structure. \n\\end{proof}\nWe will call this example an ($H$-twisted) generalized standard\nVinogradov algebroid, reducing to an ($H$-twisted) standard Vinogradov\nalgebroid for the choice of $A$ being a standard Lie algebroid, $A\n=TM$. Note that for every generalized standard Vinogradov algebroid we\nhave two quite distinct super-geometric descriptions, one as described\nabove where $\\CM = T^*[k+1]A[1]$ and another one on the supermanifold\n$\\CM' = A[1] \\times\n\\Real[k+1]$  with $Q'=\\ud_A + H \\partial\/\\partial b$, $b$ being the\ncanonical coordinate on $\\Real[k+1]$.\\footnote{Also these two pictures\nare not just related by the Hamiltonian reformulation of $(\\CM',Q')$\naccording to footnote \\ref{f:24}, $T^*[\\bullet]\\CM'$ does not agree\nwith $\\CM$.}\n\nAn important class of Vinogradov algebroids is provided by those where\nthe map $\\rho_D$ is $C^\\infty(M)$-linear (note that for a generalized\nstandard Vinogradov algebroid this is the case only for $\\rk\nV=1$). A special case of this is provided by the following definition\n(cf.~Prop.~\\ref{p:FW} below):\n\\begin{vdef}[{$V$}-twisted Courant algebroid]\\label{def:VCourant}\nLet $V,W$ be vector bundles over $M$, $W$ equipped with an anchor map $\\rho\n\\colon W\\to TM$, a bracket $[ . , . ]_W$ on its sections, and a non-degenerate\nsurjective symmetric product $\\<.,.\\>$ taking values in $V$. Let\n ${{}^W\\nabla}$ be a $W$-connection on $V$. We call this a vector bundle twisted (or\nV-twisted) Courant algebroid if these data are subject to the\nfollowing axioms\n\\begin{align}\n  [\\psi,[\\phi_1,\\phi_2]_W]_W &= [[\\psi,\\phi_1]_W,\\phi_2]_W +[\\phi_1,[\\psi,\\phi_2]_W]_W\n    \\label{fwJacobi}  \\\\\n  \\<[\\psi,\\phi]_W,\\phi\\> &=\\frac12\\Wconn_\\psi \\<\\phi,\\phi\\> =  \\<\\psi,[\\phi,\\phi]_W\\>\n  \\label{fwInvar} \\;.\n\\end{align}\n\\end{vdef}\n\nWe remark that the usual Leibniz rule \\eqref{vLeibniz} follows from \\eqref{fwInvar} after polarization, $\\<[\\psi,\\phi_1],\\phi_2\\>+\\<\\phi_1,[\\psi,\\phi_2]\\> = \\Wconn_\\psi\\<\\phi_1,\\phi_2\\> $, from the Leibniz rule for ${{}^W\\nabla}$.\n\n\n\\begin{prop} \\label{p:FW} A V-twisted Courant algebroid\n  is a Vinogradov algebroid with   $\\rho_D(\\phi):=\\Wconn_\\phi$ and $\\uD$\nits adjoint, i.e.~$\\uD$ being defined via\n\\begin{align} \n  \\label{FW}  \\<\\phi,\\uD v\\> =\\Wconn_\\phi v \\;.\n\\end{align}\nConversely, a Vinogradov algebroid is a V-twisted Courant algebroid if\n\\begin{align} \n  \\label{FW'}  \\<\\phi,\\uD v\\> =\\rho_D(\\phi) v \\;.\n\\end{align}\n\\end{prop}\n\\begin{proof} Given \\eqref{FW} then \\eqref{fwInvar} is equivalent to \n  \\eqref{vSkew}--\\eqref{vInvar}.\n  It remains to check that $\\uD$ fulfills the Leibniz rule \\eqref{vLeibn2},\n  but this follows since \\eqref{vLeibn3} shows that the adjoint\n  ${{}^W\\nabla}$ of $\\uD$ is a first order linear PDO. For the other direction note that \\eqref{FW'} replaces \\eqref{FW} thus \\eqref{vSkew}--\\eqref{vInvar} are fulfilled.  Further ${{}^W\\nabla}=\\rho_D$ the adjoint of $\\uD$ implies that it is ${C^\\infty}$-linear in $\\phi$.\n\\end{proof}\n\nNote that the l.h.s.~ of \\eqref{FW} also permits the interpretation that\n$\\uD\\:\\Gamma(V)\\to{}^W{}\\Omega^1(M,V)$ and this 1-form is then applied to a vector\n$\\phi\\in W$, i.e.\\ $\\uD$ is an exterior $W$-covariant derivative. From\nProposition \\ref{p:Vinogrelem} we learn that ${{}^W\\nabla}$ is a\nmorphism of brackets,\ni.e.~$$\\Wconn_{[\\phi,\\psi]_W}=[\\Wconn_\\phi,\\Wconn_\\psi] \\, .$$  In other words,\nthe $W$-connection of a V-twisted Courant algebroid is always\nflat and thus also $\\uD^2=0$. Furthermore, one has \n\\begin{prop}\\label{p:Royt}  In a twisted generalized standard Vinogradov \n  algebroid as well as in a V-twisted Courant algebroid\n   brackets for $\\uD$-exact sections satisfy \\\\(for all $v,v' \\in \\Gamma(V)$ and\n$\\phi \\in \\Gamma(W)$):\n\\begin{align*} \\<\\uD v,\\uD v'\\>&=0 \\\\ [\\uD v,\\phi]_W &=0\\\\\n  [\\phi,\\uD v]_W &= \\uD\\<\\phi,\\uD v\\>  \n\\end{align*}\n\\end{prop}\n\\begin{proof} The first equality follows from $ \\<\\uD v,\\uD\nv'\\>=\\Wconn_{\\uD v}v' $ and Proposition \\ref{p:Vinogrelem}. For the\nremaining two equations one can use the explicit formula\n\\eqref{stdVbrack} for the derived bracket \nin the case of twisted generalized standard Vinogradov algebroids and\ngeneralize the proof in \\cite[p.20, lemma 2.6.2]{Royt99} to\n$V$-twisted Courant algebroids in a straightforward way, respectively.\n\\end{proof}\n\nAs the terminology suggests, ordinary Courant algebroids\n$(W,[.,.],\\rho,\\<.,.\\>)$ provide examples of $V$-twisted ones: one just\ntakes $V$ to be  the trivial $\\Real$ bundle over the base $M$ so that its\nsections can be identified with functions on $M$ and identify ${{}^W\\nabla}$\nwith $\\rho$ in this case. For the reverse direction we need an\nadditional condition: Consider a V-twisted Courant algebroid with $V$\nof rank one. Suppose we can find a section $v$ of $V$ which vanishes\nnowhere (consequently $V$ has to be trivial) and which is annihilated\nby $\\uD$, $\\uD v=0$. Then we can define a non-degenerate symmetric\nbilinear form $\\<.,.\\>_C$ on $W$ by means of\n$\\<\\phi,\\psi\\>=\\<\\phi,\\psi\\>_Cv$.  Moreover, $\\<\\uD (fv),\\phi\\>=\n\\left(\\rho(\\phi) f\\right) v$. It now is easy to see that $([.,.]_W,\n\\rho, \\<.,.\\>_C)$ defines a Courant algebroid.\n\nLet us remark, however, that a line-bundle twisted Courant algebroid,i.e.~a $V$-twisted Courant algebroid with $\\rk V = 1$, is a strictly more general notion than the one of an ordinary Courant algebroid.  We intend to come back to this elsewhere.\n\n\nOrdinary Courant algebroids over a point reduce to quadratic Lie\nalgebras. This is no more the case for the above $V$-twisted\ngeneralizations, where even the bracket need not be antisymmetric. In\nthis case $W$ is what one calls a Leibniz--Loday algebra and $V$ becomes a\n$W$-module by means of $\\Wconn_\\cdot \\colon W \\to \\End(V)$. Whereas the\nr.h.s.~of \\eqref{cnSkew} vanishes identically for $M$ being a point,\nthe analogous second equality in \\eqref{fwInvar} does \\emph{not} lead\nto $[\\phi,\\phi]_W=0$ for any representation $\\Wconn_\\cdot$. An explicit\nexample with a bracket having a symmetric part is given by the\nfollowing construction which was motivated by the example in \\cite[chap.\\ 1]{San07}\n\n\\begin{prop}  Let $W:=\\End(\\Real^2)$, $V:=\\Real^3$\n\\begin{align*}\n P\\: W&\\to W: (x_{ij})\\mapsto\\pmatrix{cc}{x_{11}&0\\\\0&0} \\\\\n [X,Y]_W &:= P(X)Y -YP(X) \\\\\n \\<X,Y\\> &:= (x_{11}y_{12}+x_{12}y_{11}, -x_{11}y_{21}-x_{21}y_{12},x_{22}y_{22}) \\\\\n \\Wconn_Y(p,q,r) &:= (2y_{11}p,2y_{11}q,0)\n\\end{align*}  Then these 5 structures form a V-twisted Courant algebroid over a point.\n\\end{prop}\n\\begin{proof}  Straightforward calculations show that\n $$P(P(X)Y)=P(X)P(Y)=P(XP(Y)) \\;. $$  Due to \\cite{San07} $(W,[.,.]_W)$ forms a Leibniz--Loday algebra.  By inspection the given inner product is non degenerate.  It is now a straightforward calculation that the given connection fulfills \\eqref{fwInvar}.  Thus the given structure is a V-twisted Courant algebroid.\n\\end{proof}\n\nIn fact, even the following statement is true: \n\\begin{prop} \\label{canonical}\n\\emph{Any} Leibniz--Loday algebra $(W,[ \\cdot , \\cdot ]_W)$ becomes a $V$-twisted Courant algebroid over a point in a \\emph{canonical} way. \n\\end{prop}\n\\begin{proof}  For this purpose one takes $V$ to be the two-sided ideal generated by quadratic elements, $V=\\langle [w,w]_W| w \\in W \\rangle$, and the inner product to be nothing but the symmetrization of the original bracket, $(w_1, w_2) := \\frac{1}{2} [w_1 , w_2 ]_W + \\frac{1}{2}[w_2 , w_1 ]_W$, mapping elements from $S^2W$ into $V\\subset W$. With the evident choice $\\Wconn_{w_1} (w_2) := [w_1,w_2]_W$, the conditions \\eqref{fwInvar} follow directly from \\eqref{fwJacobi}. \n\\end{proof}\n\nIn the present context the most important examples of $V$-twisted Courant algebroids are, however, provided by Q2-manifolds. This is the subject of the\nfollowing subsection.\n\n\n\\subsection{Lie-2-algebroids and  V-twisted Courant algebroids}\n\\label{subsec:Lie2}\nEvery Q2-manifold gives rise to a V-twisted Courant algebroid as follows:\n\\begin{prop}\\label{p:derVcour} For the strongly anchored DGLA $\\X_\\bullet (\\CM)$ of\na Q2-manifold (cf.~Prop.~\\ref{p:tangentLie}) the conditions in \nProp.~\\ref{p:derVino} are satisfied and the resulting Vinogradov\nalgebroid is even a V-twisted Courant algebroid.\n\\end{prop}\n\n\\begin{proof}  First we recall that $d=0$ and thus vector fields of\ndegree -2 are sections of $V\\to M$ while those of degree -1 are\nsections of the vector bundle $W\\to M$\n(cf.~Prop..~\\ref{p:tangentLie}). We have \n $[\\psi,f\\phi]=f[\\psi,\\phi]$ for all $\\psi, \\phi \\in \\Gamma(W) \\cong \\X_{-1}(\\CM)$ and\n $f \\in {C^\\infty}(M)\\cong C^\\infty_0(\\CM)$ since there are no negative\ndegree functions on $\\CM$. In every  (local) splitting of\n\\eqref{sequence} we find $W=E\\oplus E^*\\otimes V$ and  the bracket \nbetween tangent vector fields of degree -1 is just the canonical pairing\non this direct sum \\begin{equation} \\<X\\oplus a,Y\\oplus b\\> = a(Y)+b(X)\n\\label{pair} \\, ,  \\end{equation}\nwhere $X,Y \\in \\Gamma(E)$ and $a,b \\in \\Gamma(E^* \\otimes V)\\cong {}^E\\Omega^1(M,V)$.\nThis is\nobviously non-degenerate and surjective. Thus by Prop.~\\ref{p:derVino}\nwe conclude to have a Vinogradov algebroid. According to\nProp.~\\ref{p:FW} it now suffices to verify Eq.~\\eqref{FW'}: Indeed,\nthis follows from the compatibility of the Lie bracket with $\\uD$ and\nthe fact that $[\\psi,v]\\equiv 0$ $\\forall \\psi \\in \\Gamma(W) \\cong \\X_{-1}(\\CM)$ and $v \\in \n\\Gamma(V) \\cong \\X_{-2}(\\CM)$, since there are no vector fields of\ndegree -3.\\end{proof} In Section~\\ref{s:examQ} we found an\ninterpretation of the components of the vector field $Q$,\nEq.~\\eqref{Q}, on a (split) Q2-manifold in terms of a Lie\n2-algebroid. Using the same $Q$ to calculate explicitly the derived\nbracket given in the previous proposition, one arrives at the\nfollowing\n\\begin{prop}\\label{p:FWder} Let $(E,V, [.,.]_E,\\rho_E,t,{{}^E{}\\nabla},H)$ be a Lie\n2-algebroid as defined in Def.~\\ref{Lie2def\n. Then $W:=E\\oplus E^*\\otimes V$ with \n\\begin{align} \\rho_W(X\\oplus a):=&\\; \\rho_E(X) \\, , \\label{Rho}\\\\\n  \\Wconn_{X\\oplus a}v := &\\Econn_Xv -\\imath_{t(v)}a \\label{rhoQCont} \\\\\n  \\<X\\oplus a,Y\\oplus b\\> :=&\\; \\imath_Y a + \\imath_X b \\, ,  \\label{Paar}\\\\\n  [X\\oplus a, Y\\oplus b]_W :=&\\; \\left([X,Y]_E  \\oplus\n    [\\imath_X,{{}^E\\!{}\\mathrm{D}}]b -\\imath_Y{{}^E\\!{}\\mathrm{D}} a  + \\imath_X \\imath_Y H  \\right) \\nonumber\\\\\n  &-\\left(t(a(Y))\\oplus T(a,b)\\right)\\, , \\label{brCont}\n\\end{align} where $X,Y \\in \\Gamma(E)$, $a,b \\in \n{{}^E{}\\Omega}^1(M,V)$, and $v \\in \\Gamma(V)$, is a $V$-twisted Courant\nalgebroid. Here ${{}^E\\!{}\\mathrm{D}}$ has been defined in Eq.~\\eqref{genCar} and the\noperator $T\\colon (E^*\\otimes V)^{\\otimes2} \\to E^*\\otimes V$ is induced by $t$ according to the composition of maps \n\\begin{equation} \nT(a,b)\\equiv a\\circ t\\circ b - b\\circ t\\circ a \\, . \\label{Tdefn}\n\\end{equation}\n \n\\end{prop}\n\\begin{proof}  Instead of checking all axioms separately one shows that the given algebroid is the V-twisted Courant algebroid from Proposition \\ref{p:derVcour}.\n\nThe Lie 2-algebroid induces a Q-structure via Theorem \\ref{theo1}.  To prove the above formulas one can e.g.\\ use local coordinates on the Q-manifold.\n\\end{proof}\n\nIt turns out that under certain circumstances Proposition \\ref{p:FWder} can be reversed. For that purpose we now consider a generalization of exact Courant algebroids.\nLet $W$ be an ordinary Courant algebroid. Then the anchor map $\\rho\\colon W\\to TM$ together with the inner product induce a map $\\rho^* \\colon T^*M\\to W$, where we used the inner product to identify $W^*$ with $W$.  Due to Proposition \\ref{p:Vinogrelem} the composition $\\rho\\circ\\rho^*$ vanishes. Thus for \\emph{any} Courant algebroid $W$ one has the complex\n \\begin{equation}\n 0\\to T^*M\\xrightarrow{\\rho^*} W\\xrightarrow{\\rho} TM\\to 0 \\, ; \\label{Wexact}\n\\end{equation}   \n$W$ is said to be exact iff the above short sequence is exact.\nExamples of exact Courant algebroids occur on $W=TM\\oplus T^*M$ via the choice of a closed 3-form $H$ with the bracket \\eqref{Cour}.  Conversely given an exact Courant algebroid we can choose an isotropic splitting $j\\:TM\\to E$\\footnote{We will be more explicit on this in the more general context below.} and define a 3-form $H$ as\n $$ H(X,Y,Z):= \\<[j(X),j(Y)],j(Z)\\> \\;. $$  \\eqref{cnSkew} shows that $H$ is skew-symmetric in the arguments $X$ and $Y$, while \\eqref{cInvar} implies that it is invariant under cyclic permutations of $X$, $Y$ and $Z$. Thus $H$ is a 3-form. The remaining  identity on the bracket implies that this 3-form has to be closed under the de Rham differential.  Finally, a change of splitting, which is necessarily induced by a skew-symmetric 2-form $B$, changes $H$ by an exact term, $H \\mapsto H+\\ud B$.  Thus, exact Courant algebroids are in 1:1 correspondence to classes of the third real cohomology on $M$, an observation going back to \\v{S}evera \\cite{SevLett}.\n\nWe wish to generalize this result to certain V-twisted Courant algebroids $W$.  Since the inner product on $W$ has values in $V$, the map $\\rho \\equiv \\rho_W \\colon W \\to TM$ induces a map $\\rho^* \\colon T^*M \\otimes V\\to W$ by means of $( \\rho^*(\\alpha \\otimes v), \\psi ) = \\alpha(\\rho(\\psi))$ for all $\\psi \\in W$. Thus, \\emph{any} $V$-twisted Courant algebroid $W$ gives rise to the complex\n\\begin{equation}\n 0\\to T^*M\\otimes V \\xrightarrow{\\rho^*} W\\xrightarrow{\\rho} TM\\to 0 \\, . \\label{VWexact}\n\\end{equation}\nLet us call a $V$-twisted Courant algebroid exact if this complex is acyclic, in generalization of ordinary exact Courant algebroids (which fit into this definition since for $V=M\\times \\Real$ \\eqref{VWexact} reduces to \\eqref{Wexact}). More generally, consider a map $\\pi\\colon W\\to E$. Together with the V-valued inner product it induces a map $\\pi^*\\colon E^*\\otimes V\\to W$.  Here $E^*\\otimes V$ is the $V$-dual of E and $W$ the $V$-dual of itself. We are thus led to \n\\begin{vdef}\nSuppose that in a V-twisted \nCourant algebroid $(V,W)$ the anchor map $\\rho_W \\colon W \\to TM$ factors through a bundle map $\\pi\\colon W\\to E$ for some vector bundle $E$. If the then induced complex \n\\[  0 \\rightarrow E^*\\otimes V \\xrightarrow{\\pi^*} W \\xrightarrow{\\pi} E \\to 0\n   \\label{Vexact}\\]\nis acyclic, we call $(V,W)$ an \\emph{exact $V$-twisted (or $V$-exact) Courant algebroid over $E$} and for $E=TM$, $\\pi=\\rho_W$ simply a \\emph{$V$-exact Courant algebroid}.\n\\end{vdef}\nThere then is the following remarkable fact:\n\\begin{thm}\\label{thm:der} Let $(V,W)$ be a $V$-exact Courant algebroid over some $E$ with $\\rk V>1$ and let $j\\: E\\to W$ be an isotropic \nsplitting  of \\eqref{Vexact}.\n\nThen $E$ and $V$ carry canonically the structure of a semistrict Lie 2-algebroid, $(E,V,[\\cdot, \\cdot]_E,\\rho,t,{{}^E{}\\nabla},H)$, and the structural data of the $V$-twisted Courant algebroid $W$ are necessarily of the form  of Proposition \\ref{p:FWder} above. \n\nA change of splitting $j(X)\\mapsto j(X)-B^\\#(X)$ results into a change of the underlying data $[.,.]_E$, $H$, and ${{}^E{}\\nabla}$ according to \\eqref{split1}--\\eqref{split3} furthermore.\n\\end{thm}\n\nWe first remark that an isotropic splitting of \\eqref{Vexact} always exists. This is seen as follows: First, as for any exact sequence of vector bundles, we can choose \\emph{some} splitting $j_0\\:E\\to W$ of the short exact sequence \\eqref{Vexact}.  This permits us to identify $E\\oplus E^*\\otimes V\\xrightarrow\\sim W$ via $(X,a)\\mapsto j_0(X)+\\pi^*(a)$ for $X,Y\\in E_x$, $a,b\\in E^*\\otimes V_x$ and $x\\in M$. Computing the symmetric 2-form $L(X,Y):=\\<j_0(X),j_0(Y)\\>$, $L\\:E\\otimes E\\to V$, it may or may not vanish.  If it does, $j_0$ provides an isotropic splitting.  If it does not, it induces a map $L^\\#\\:E\\to E^*\\otimes V,\\, X\\mapsto L(X,\\bullet)$.  If we change $j_0$ by $-\\pi^*\\circ L^\\#$, we obtain an isotropic splitting $W\\cong E\\oplus E^*\\otimes V$ and the inner product on $W$ is identified with the standard product on $E\\oplus E^*\\otimes V$, $\\<j(X)+\\pi^*(a),j(Y)+\\pi^*(b)\\>=a(Y)+b(X)$.\n\nWe henceforth do no more write $j$ and $\\pi^*$ explicitly, but use the identification  $W\\cong E\\oplus E^*\\otimes V$ induced by the isotropic splitting to write elements of $W$ in the form $X\\oplus a$ (instead of $j(X)+\\pi^*(a)$) with $X \\in E$ and $a \\in \\Hom(E,V)$.\n\nWe also remark that the condition on the rank of $V$ is necessary. The 1:1 relation between $V$-exact Courant algebroids over $E$ and semi-strict Lie 2-algebroids $(E,V)$ breaks down explicitly in the case of $\\rk V=1$. While due to Proposition \\ref{p:FWder} also for $V$ being a line bundle, one obtains a $V$-exact Courant algebroid over $E$ from any semi-strict Lie 2-algebroid, not any such a generalized Courant algebroid is of this form, even if the line bundle is trivial, $V=M \\times \\Real$.\nThis will become clear within the proof below, which will also permit us to provide an explicit example of this fact right after the proof.\n\\begin{proof}\nOur task now is to retrieve the data\n$[.,.]_E$, $\\rho_E$, $t$, ${{}^E{}\\nabla}$, and $H$ from the data of the $V$-twisted exact Courant algebroid over $E$ and to show that \nthey satisfy all the identities they have to so as to form a Lie 2-algebroid.\n\nWe start with Equation \\eqref{rhoQCont} and \\emph{define} ${{}^E{}\\nabla}$ and $t$ implicitly by means of\n\\[\n  \\Wconn_{X\\oplus a} v =: \\Econn_Xv -a(t(v)) \\, , \\label{naTdefn}\n\\]\nwhich has to hold true for all $v \\in \\Gamma(V)$ and $X \\oplus a \\in \\Gamma(E) \\oplus \\Gamma(\\Hom(E,V)) \\cong \\Gamma(W)$. We also know by assumption that $\\rho_W = \\rho_E \\circ \\pi$ for some $\\rho_E \\colon E \\to TM$. Noting that after the choice of a splitting, $\\pi$ just becomes projection to the first factor, $\\pi(X \\oplus a)=X$, the Leibniz property of ${{}^W\\nabla}$ and Definition \\eqref{naTdefn} (for simplicity one may set $a=0$) ensure that ${{}^E{}\\nabla}$ is an $E$-covariant derivative on $V$ with the anchor map $\\rho_E$, as anticipated by the notation chosen.  Returning with this information to \\eqref{naTdefn}, it implies that the map $t$ is $C^\\infty(M)$-linear, and thus comes from a bundle map $t\\colon V\\to E$.\n\nNext, anticipating what we want to achieve in \\eqref{brCont}, we \\emph{define} $[.,.]_E$ and some  ${H}$ from $[X,Y]_W$ by means of\n$$  [X\\oplus0,Y\\oplus0]_W =: [X,Y]_E\\oplus -{H}(X,Y) \\, .$$\nFirst we can conclude from the known properties of the left-hand-side that $[X,Y]_E$ satisfies a Leibniz property with respect to the second argument for the anchor map $\\rho_E$. Likewise we also follow that $H(\\cdot, \\cdot)$ is $C^\\infty$-linear in the second argument. Moreover, as a consequence of the isotropy of the chosen splitting, $(X\\oplus0,Y\\oplus0)=0$, and the second equation of \\eqref{fwInvar} yields antisymmetry of the $E$-bracket and $H$ in $X$ and $Y$. By definition, $H(X,Y) \\in \\Gamma(E^* \\otimes V)$. The first equation of \\eqref{fwInvar} then implies that $H(X,Y,Z) := \\imath_Z H(X,Y)$ \nis also antisymmetric in $Y$ and $Z$. Thus it is antisymmetric in all three arguments and the $C^\\infty$-linearity extends to all these arguments. All together this implies that  $H\\in{{}^E{}\\Omega}^3(M,V)$.\n\nWe now have all of the ingredients $[.,.]_E$, $\\rho_E$, $t$, ${{}^E{}\\nabla}$, and $H$ necessary to define the bracket \\eqref{brCont}. Let us denote the r.h.s.~of this equation by $[X\\oplus a,Y\\oplus b]_W^0$ and parameterize the difference to this form of the bracket by several bilinear maps $\\delta_i$ as follows:\n\\begin{align}\\label{ansatzbr}\n  [X\\oplus a,Y\\oplus b] =: &\\;[X\\oplus a,Y\\oplus b]_W^0 +\\\\ &+\\delta_1(a,Y)+\\delta_2(b,X)+\\delta_3(a,b)  \n    \\oplus  \\delta_4(X,b)+\\delta_5 (Y,a) +\\delta_6(a,b) \\, . \\nonumber\n\\end{align}\nThese maps are certainly $\\Real$-bilinear, but due to the property of $[X\\oplus a,Y\\oplus b]_W^0$ they are even $C^\\infty(M)$-bilinear (due to the already established features of the quantities entering this bracket).\n\nWe now want to show that for $\\rk V\\ne1$, the Axiom~\\eqref{fwInvar} implies that $\\delta_1 = \\ldots = \\delta_6=0$, while the Leibniz--Loday property \\eqref{fwJacobi} implies the three Equations \\eqref{HJacobi}, \\eqref{HRep}, and \\eqref{DH} needed to endow $(E,V)$ with the structure of a semi-strict Lie 2-algebroid. \n\nThe term in the center of  \\eqref{fwInvar} does not depend on the bracket at all and these terms are precisely balanced by the terms coming from $[X\\oplus a,Y\\oplus b]_W^0$ in the right equation. This implies that for equal arguments, $X=Y$ and $a=b$, all the delta-contributions have to vanish in \\eqref{ansatzbr}. This is the case iff\n\\begin{equation}\n\\delta_1=-\\delta_2 \\; , \\quad \\delta_4 = -\\delta_5 \\; , \\label{delta4} \n\\end{equation}\nand  $\\delta_3$ and $\\delta_6$ are antisymmetric in their two arguments. We now turn to the l.h.s.~of \\eqref{fwInvar}. The left Equation \\eqref{fwInvar} then gives the following additional constraints:\n\\begin{eqnarray}\n\\delta_5 &=& 0 \\, ,\\label{delta5}\\\\\nb(\\delta_2(b,X)) &=& 0 \\, ,\\label{delta2}\\\\\nb(\\delta_3(a,b)) &=& 0\\, , \\label{delta3}\\\\\n\\iota_Y\\delta_6(a,b)+ b(\\delta_1(a,Y)) &=& 0 \\, ,\\label{delta6} \n\\end{eqnarray}\nto hold true for all choices of arguments. To exploit Equations \\eqref{delta2} and \\eqref{delta3}, we prove the following\n\\begin{lemma}\\label{p:Jh=0}\n Let $V_1$ and $V_2$ be two vector spaces and let $\\Delta \\in \\left(\\Hom(V_1, V_2)\\right)^* \\otimes V_1 \\cong (V_1)^{\\otimes 2} \\otimes V_2^*$ satisfying \\begin{equation} \\label{aa}\n  a(\\Delta(a))=0\n\\end{equation} for all $a \\in \\Hom(V_1, V_2)$. Then, for $\\dim V_2 >1$, necessarily $\\Delta =0$, while for $V_2 \\cong \\Real$ any $\\Delta \\in \\Lambda^2 V_1$ has this property.\n\\end{lemma}\n\\begin{proof}[{ (of Lemma \\ref{p:Jh=0})}] \nLet us first consider the case of a factorizing $a$: $a = \\alpha \\otimes v$ with $\\alpha \\in V_1^*$ and $v \\in V_2$. Then \\eqref{aa} holds true iff $\\Delta \\in \\Lambda^2(V_1) \\otimes V_2^*$. It remains to check \\eqref{aa} for $a = a_1 + a_2$ with both summands such products. This is satisfied iff the contraction of $\\Delta$ in the $V_2$-slot with the first factor of any term of the form $v_1 \\wedge v_2 \\in \\Lambda^2 V_2$ has to vanish. If $\\dim V_2 = 1$, this condition is empty and the statement follows from $\\Lambda^2(V_1) \\otimes \\Real^* \\cong \\Lambda^2(V_1)$. In the case that $V_2$ has at least two directions, on the other hand, evidently one is forced to a vanishing $\\Delta$.\n\\end{proof}\nBy $C^\\infty(M)$-linearity, the Equations \\eqref{delta2} and \\eqref{delta3} are pointwise in $M$ and for $\\rk V>1$ we can conclude the vanishing of $\\delta_2$ and $\\delta_3$ by setting $V_1 := E|_x$, $V_2 :=V|_x$ for any fixed $x \\in M$ and $\\delta(\\cdot)$ equal to \n$\\delta_2(\\cdot , X)|_x$ and $\\delta_3(a,\\cdot)|_x$, respectively (again for any fixed $X$ and $a$, respectively). \n\nThus, if the rank condition on $V$ is satisfied, Equations \\eqref{delta4}---\\eqref{delta6} imply the vanishing of $\\delta_i$, $i=1,\\ldots,6$. It now remains to exploit Equation \\eqref{fwJacobi}. \n\nIt is a straightforward calculation to check that  \\eqref{fwJacobi}---together with the already established form of the bracket $[ \\cdot , \\cdot ]_W = [ \\cdot , \\cdot]_W^0$, i.e.~\\eqref{brCont} holding true,---implies the remaining identities \\eqref{HJacobi}, \\eqref{HRep}, and \\eqref{DH}. For example, regarding the $\\Gamma(E)$-part of  \\eqref{fwJacobi} yields \n\\begin{gather}\n [[X,Y]_E,Z]_E -[t(\\imath_Y a),Z]_E-t\\left( \\imath_Z\\left([\\imath_X,{{}^E\\!{}\\mathrm{D}}]b -\\imath_Y{{}^E\\!{}\\mathrm{D}} a  + \\imath_X \\imath_Y H  -T(a,b)\\right)\\right)= \\nonumber \\\\\n  = [X,[Y,Z]_E]_E -[X,t(\\imath_Zb)]_E -t\\left( a\\left([Y,Z]_E-t(\\imath_Z b)\\right)\\right) - \\left(X \\leftrightarrow Y \\,\\,\\text{and}\\,\\, a \\leftrightarrow b\\right)\\, ,\n\\end{gather}\nwhich, after the dust clears and using the definitions \\eqref{genCar} and \\eqref{Tdefn} for ${{}^E\\!{}\\mathrm{D}}$ and $T$, respectively, yields the the first of the three equations \nand the right hand equation of \\eqref{repr2}. Establishing the three quadratic equations on the structural data for $E$ and $V$ shows that they form a semi-strict Lie 2-algebroid.\n\nAlternatively one may also proceed as follows in the last step: Given the above data, we can define a vector field of the form \\eqref{Q} on  $E[1]\\oplus V[2]$. Vector fields of degree -1 can then be identified with sections of $W$. Equation~\\eqref{vJacobi} then translates into $0=\\frac12[[[[Q,Q],\\psi],\\phi_1],\\phi_2]$. \n It remains to check that\nthis equation contains the three Equations \\eqref{B3}, \\eqref{B5}, and \\eqref{B7}. In fact, it contains even all seven components \\eqref{B1}--\\eqref{B7} of\n$Q^2=0$ as one may convince oneself.\n\nA change of splitting of the degree 2 Q-manifold used above corresponds precisely to a change of splitting of the sequence \\eqref{Vexact} if one regards the vector fields of degree -1 on that graded manifold.  This then proves the  theorem.\n\\end{proof}\n\nWe conclude with some related remarks. First, we saw that in the case $\\rk V =1$ the discussion of $V$-exact Courant algebroids over some $E$ is at least more involved. Let us restrict to a trivial bundle $V=M \\times \\Real$ so that we can identify $\\Hom(E,V)$ with $E^*$. Then, according to the above considerations, the data appearing in Proposition \\ref{p:FWder} can be extended by a modification of the bracket by means of two tensors: $\\delta_1 = - \\delta_2 \\sim \\delta_6 \\in \\Gamma(\\Lambda^2E \\otimes E^*)$ and $\\delta_3 \\in \\Gamma(\\Lambda^3 E)$. While the general analysis of this special case gets somewhat involved---one still needs to analyze the consequences of \\eqref{fwJacobi}---, we just make two remarks at this point here: First, \\emph{every} Lie 2-algebra gives rise to a $V$-exact Courant algebroid over $E$ according to Proposition \\ref{p:FWder}, also in the case of $\\rk V =1$. Second, one simple example of a $V$-exact Courant algebroid over $E$ with $\\rk V = 1$ \\emph{not} covered by this parameterization may be provided by the following construction: Choose the Lie 2-algebroid to have all structural data vanish and take, in the above parameterization also $\\delta_1=0$ but $\\delta_3\\equiv h \\in \\Gamma(\\Lambda^3 E)$ arbitrary.  \nThen for arbitrary sections in $E \\oplus E^*\\otimes V \\cong E \\oplus E^*$ one has \n\\begin{equation}\n[X \\oplus \\alpha , Y \\oplus \\beta ]_W := h(\\alpha, \\beta) \\oplus 0 \\, . \n\\end{equation} \nIt is evident that double brackets always vanish with this definition, and thus Equation \\eqref{fwJacobi} is satisfied trivially. But also each of the three terms in \\eqref{fwInvar} vanish by the antisymmetry of $h$ in all three arguments and the vanishing of ${{}^W\\nabla}$, so that also these two equations are satisfied. \n\nWe intend to come back to a more systematic investigation of line-bundle-twisted exact Courant algebroids over some $E$ elsewhere. \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\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOne of the main  aims of research in nuclear physics is to make reliable predictions for the ground state nuclear properties of all nuclei in the periodic table with one nuclear model. For this purpose, several approaches have been developed and they can be classified into three categories: The first one called\nthe macroscopic models such as the Bethe-Weizs\u00e4cker mass formula~\\cite{samanta2002}. The second one is the macroscopic-microscopic models such as the Finite Range Droplet Model\n(FRDM)~\\cite{moller1995}. The last one consists of the microscopic models such as the conventional Hartree Fock method~\\cite{skyrme1956,decharg1980,bassem2015,bassem2016,antonov2017} with effective density-dependent interactions and its relativistic analog, the relativistic mean field (RMF) theory~\\cite{walecka1974,reinhard1989}; the former is based on the nonrelativistic kinematics, in which the density-dependent and the spin-orbit interactions are important ingredients, while the latter is based on the relativistic kinematics, \nwhere the ingredients are the nucleons, the mesons and their interactions.\nTherefore, the spin appears automatically and the interplay of the scalar and the vector potentials leads naturally to a proper spin orbit interaction and appropriate shell structure. The RMF theory  has received wide attention due to its successful description of lots of properties of nuclei not only in the valley of stability but also far away from it.~\\cite{meng1998,zhou2003,meng1999,geng2004,ring1996,meng1996,ginocchio1997}.\n\n\nDensity functional theories (DFT's) are extremely useful in understanding nuclear many-body dynamics. Among different nuclear DFTs, the covariant density functional theory (CDFT)~\\cite{niki2014,lalazissis2005,rocamaza2011,niki2002} based on the energy density functionals (EDFs) is one of the most attractive and is very successful in  describing very well the ground and excited states throughout the\nchart of nuclei~\\cite{abusara2012,agbemava2015,agbemava2017} as well as in the nuclear structure analysis~\\cite{meng2015,matev2007,afanasjev2008}.\nIn Ref.~\\refcite{Agbemava2014}, the global performance of some covariant\nenergy density functionals on some nuclear observables was analyzed.\n\nIn this work, we are interested in calculating and analyzing some ground-state properties of even-even Pt isotopes, N=82-160, within the framework of the covariant density functional theory by using two of the state-of-the-art functionals which provide a complete and an accurate description of different ground states and excited states over the whole nucleic chart~\\cite{karim2015,afanasjev2010,elbassem2019}, namely : The density-dependent point-coupling DD-PC1~\\cite{niki2008} and the density-dependent meson-exchange DD-ME2~\\cite{DD-ME2}.\n\nThe paper is organized in the following way: The covariant density functional theory and details of the numerical calculations are presented in section~\\ref{Theoretical Framework}. \nSection~\\ref{Results and Discussion} is devoted to present our results and discussion.\nFinally, the conclusions of this study are presented in section~\\ref{Conclusion}.\n\n\\section{Covariant density functional theory}\n\\label{Theoretical Framework}\nTwo classes of covariant density functional models are used throughout this paper: the density-dependent point-coupling (DD-PC) model and the density-dependent meson-exchange (DD-ME) model. \nThe first has a finite interaction range and has been fitted \nto binding energies and radii of spherical nuclei; while the latter uses a zero-range interaction and has been fitted to nuclear matter data and for finite nuclei only to binding energies  of a large range of deformed nuclei.\\\\\n\n\nIn the meson-exchange model, the nucleus is considered as a system of Dirac nucleons which interact via the exchange of mesons with finite masses leading to finite-range interactions~\\cite{typel1999,lalazissis2009}.\nThe standard Lagrangian density with medium dependence vertices that defines the meson-exchange model~\\cite{gambhir1990} is given by:\n\\begin{eqnarray}\n\\mathcal{L}  &  =\\bar{\\psi}\\left[\n\\gamma(i\\partial-g_{\\omega}\\omega-g_{\\rho\n}\\vec{\\rho}\\,\\vec{\\tau}-eA)-m-g_{\\sigma}\\sigma\\right]  \\psi\\nonumber  +\\frac{1}{2}(\\partial\\sigma)^{2}-\\frac{1}{2}m_{\\sigma}^{2}\\sigma^{2}\\\\\n&\n-\\frac{1}{4}\\Omega_{\\mu\\nu}\\Omega^{\\mu\\nu}+\\frac{1}{2}m_{\\omega}^{2}\\omega\n^{2}\\label{lagrangian}  -\\frac{1}{4}{\\vec{R}}_{\\mu\\nu}{\\vec{R}}^{\\mu\\nu}+\\frac{1}{2}m_{\\rho}%\n^{2}\\vec{\\rho}^{\\,2}-\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}\n\\end{eqnarray}\nwhere $m$ is the bare nucleon mass and $\\psi$ denotes the Dirac spinors. $m_\\rho$, $m_\\sigma$ and $m_\\omega$ are the masses of  $\\rho$ meson,  $\\sigma$ meson and  $\\omega$ meson, \nwith the corresponding coupling constants for the mesons to the nucleons as $g_\\rho$, $g_\\sigma$ and $g_\\omega$ respectively, and $e$ is the charge of the proton.\\\\\n\nThe point-coupling model represents  an alternative formulation of the self-consistent RMF framework~\\cite{manakos1988,rusnak1997,brvenich2002,zhao2010}. The Lagrangian for the DD-PC model~\\cite{niki2008,nikolaus1992} is given by:\n\\begin{eqnarray}\n&\\mathcal{L}    =\\bar{\\psi}\\left(i\\gamma \\cdot \\partial-m\\right)  \\psi\n-\\frac{1}{2}\\alpha_S(\\hat\\rho)\\left(\\bar{\\psi}\\psi\\right)\\left(\\bar{\\psi}\\psi\\right)-\\frac{1}{2}\\alpha_V(\\hat\\rho)\\left(\\bar{\\psi}\\gamma^{\\mu}\\psi\\right)\\left(\\bar{\\psi}\\gamma_{\\mu}\\psi\\right)\\label{Lag-pc}\\\\\n&\n-\\frac{1}{2}\\alpha_{TV}(\\hat\\rho)\\left(\\bar{\\psi}\\vec\\tau\\gamma^{\\mu}\\psi\\right)\\left(\\bar{\\psi}\\vec\\tau\\gamma_{\\mu}\\psi\\right)-\\frac{1}{2}\\delta_S\\left(\\partial_v\\bar{\\psi}\\psi\\right)\\left(\\partial^v\\bar{\\psi}\\psi\\right) - e\\bar\\psi\\gamma \\cdot A\n\\frac{(1 - \\tau_3)}{2}\\psi .\\nonumber\n\\end{eqnarray}\nThe Eq.~(\\ref{Lag-pc}) contains the free-nucleon Lagrangian, the point coupling interaction terms, and the coupling of the proton to the electromagnetic field, the derivative terms account for the leading effects of finite-range interaction\nwhich are important in nuclei. \n\nIn the present work, we have used the recently developed density-dependent meson-exchange relativistic energy functional DD-ME2~\\cite{DD-ME2} and the very successful density-dependent point-coupling interaction DD-PC1~\\cite{niki2008} with the parameter sets listed in Tables~\\ref{tab1}~and~\\ref{tab2} respectively. \n\n\n\\begin{table}[!ht\n\t\\caption{ The parameters of the  DD-ME2 functional. \n\t\t\\label{tab1}}\n\t\\begin{center}%\n\t\t\\begin{tabular}\n\t\t\t[c]\t\t{@{\\hspace{0pt}}c@{\\hspace{24pt}}@{\\hspace{24pt}}c@{\\hspace{0pt}}}\n\t\t\t\\hline Parameter &  DD-ME2 \\\\\\hline\n\t\t\t$m$   & 939  \\\\\n\t\t\t$m_{\\sigma}$  & 550.1238 \\\\\n\t\t\t$m_{\\omega}$  & 783.000  \\\\\n\t\t\t$m_{\\rho}$    & 763.000 \\\\\n\t\t\t$g_{\\sigma}$  &  10.5396\\\\\n\t\t\t$g_{\\omega}$  & 13.0189\\\\\n\t\t\t$g_{\\rho}$    & 3.6836  \\\\\n\t\t\t$a_{\\sigma}$  & 1.3881 \\\\\n\t\t\t$b_{\\sigma}$  & 1.0943 \\\\\n\t\t\t$c_{\\sigma}$  & 1.7057 \\\\\n\t\t\t$d_{\\sigma}$  & 0.4421 \\\\\n\t\t\t$a_{\\omega}$  & 1.3892 \\\\\n\t\t\t$b_{\\omega}$  & 0.9240 \\\\\n\t\t\t$c_{\\omega}$  & 1.4620 \\\\\n\t\t\t$d_{\\omega}$  & 0.4775 \\\\\n\t\t\t$a_{\\rho}$    & 0.5647 \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table}\n\n\\begin{table}[!ht]\n\t\\caption{The parameters of the DD-PC1 functional.}%\n\t\\label{tab2}\n\t\\begin{center}\n\t\t\\begin{tabular}\n\t\t\t[c]\t\t{@{\\hspace{0pt}}c@{\\hspace{24pt}}@{\\hspace{24pt}}c@{\\hspace{0pt}}}\n\t\t\t\\hline Parameter &  DD-PC1 \\\\\\hline\n\t\t\t$m$ &   939\\\\\n\t\t\t$a_{\\sigma}$ & -10.04616\\\\\n\t\t\t$b_{\\sigma}$ & -9.15042\\\\\n\t\t\t$c_{\\sigma}$ & -6.42729\\\\\n\t\t\t$d_{\\sigma}$ &  1.37235\\\\\n\t\t\t$a_{\\omega}$ &  5.91946\\\\\n\t\t\t$b_{\\omega}$ &  8.86370\\\\\n\t\t\t$d_{\\omega}$ &  0.65835\\\\\n\t\t\t$b_{\\rho}$   &  1.83595\\\\\n\t\t\t$d_{\\rho}$   &  0.64025\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table}\n\n\n\n\\section{Results and Discussion}\n\\label{Results and Discussion}\n\nIn this section, we present the ground-state properties of $^{160-238}$Pt nuclei obtained in the framework of the CDFT by using the interactions DD-ME2~\\cite{DD-ME2} and DD-PC1~\\cite{niki2008}. \nOur results are compared with the available experimental data, the predictions of the RMF model with NL3~\\cite{lalazissis1999} functional and with the results of HFB theory with SLy4 Skyrme force calculated by using the computer code HFBTHO v2.00d~\\cite{Stoitsov2013} as in Ref~\\refcite{bassem2017}. \n\n\\subsection{Binding energy}\n\nBinding energy (BE) is directly related to the stability of nuclei and is a very important quantity in nuclear physics. \nThe binding energies per nucleon (BE\/A) of ground states for platinum isotopes, $^{160-238}$Pt, are presented in Fig.~\\ref{BEexp} as a function of the neutron number N. The available\nexperimental data~\\cite{wang2012} as well as the predictions of the RMF(NL3)~\\cite{lalazissis1999} and HFB(SLy4) theories are also shown for comparison.\n\n\\begin{figure}[ht]\n\t\\centering \\includegraphics[scale=0.5]{BE_Pt.eps}\n\t\\caption{(Color online) The binding energies per nucleon for even-even $^{166-238}$Pt isotopes.}\n\t\\label{BEexp}\n\\end{figure}\t\t\n\nIt can be clearly seen from Fig. \\ref{BEexp} that the theoretical predictions reproduce the experimental data accurately and, qualitatively, all curves show a similar behavior.\n\nIn order to provide a further check of the accuracy of our results, the differences between the experimental total BE for even-even platinum isotopes and the calculated results obtained in this work are presented in Fig.~\\ref{Delta_BE} as a function of the neutron number N. \n\n\\begin{figure}[ht]\n\t\\centering \\includegraphics[scale=0.5]{deltaBE_Pt.eps}\n\t\\caption{(Color online) Differences between theoretical total binding energies and experimental values for even-even Pt isotopes.}\n\t\\label{Delta_BE}\n\\end{figure}\n\nThe comparison of the available experimental total binding energies of the ground state for $^{166-238}$Pt with the present calculations and with the other theoretical models, is also done by the root mean square (rms) deviation tabulated in Table~\\ref{rms}. \n\n\n\\begin{table}[!htb]\n\t\\caption{The rms deviations of the total binding energies of platinum isotopes.\\label{rms}}\n\t\\centering\n\t\\begin{tabular}\n\t\t{@{\\hspace{0pt}}c@{\\hspace{24pt}}c@{\\hspace{24pt}}c@{\\hspace{0pt}}@{\\hspace{24pt}}c@{\\hspace{0pt}}}\n\t\t\\hline\\noalign{\\smallskip}\n\t\tDD-ME2 \t&   DD-PC1\t&  NL3\t\t & SLy4\\\\ \n\t\t\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\t\t1.864 \t&\t1.880\t&\t2.755\t & \t4.019\t\\\\  \n\t\t\\noalign{\\smallskip}\\hline\n\t\\end{tabular}\n\\end{table}\n\nAs we can see from Fig.~\\ref{Delta_BE} and Table~\\ref{rms}, DD-ME2 exceeds in accuracy the other nuclear models DD-PC1, NL3 and SLy4.\n\n\\subsection{Two neutron separation energy ($S_{2n}$) and shell gap ($\\delta_{2n}$)}\n\nIn the present work, we have calculated two-neutron separation energies, \n$S_{2n}(N,Z)$ = BE$(N,Z)$ - BE$(N - 2,Z)$, for Pt isotopes by using the density-dependent effective interactions DD-PC1 and DD-ME2. \n\nIn Fig.~\\ref{S2n}, we display the calculated S$_{2n}$ of  even-even platinum isotopes, as a function of the neutron number N, in comparison with the available experimental data~\\cite{wang2012} and the predictions of RMF(NL3)~\\cite{lalazissis1999} and HFB(SLy4).\n\n\\begin{figure}[ht]\n\t\\centering \\includegraphics[scale=0.52]{S2n_Pt.eps}\n\t\\caption{(Color online) The two-neutron separation energies, S$_{2n}$, for Pt\n\t\tisotopes.}\n\t\\label{S2n}\n\\end{figure}\n\nAs one can see from Fig.~\\ref{S2n}, the results of the two density-dependent models DD-ME2 and DD-PC1 as well as those of NL3 and SLy4 reproduce the experimental data quite well except some small discrepancies which are mainly due to the missing beyond mean field corrections~\\cite{Litvinova2006}. S$_{2n}$ gradually decreases with N, and a sharp drop is distinctly seen at N = 126 in both experimental and theoretical curves, except in SLy4 curve, which corresponds to the closed shell at this magic neutron number. \nThe discontinuity of S$_{2n}$ at this nucleus indicates that it corresponds to a critical point of first order phase transition to prolate shape.\n\nA more sensitive observable for locating the shell closure is the two-neutron shell gap $\\delta_{2n}=S_{2n}(N,Z)-S_{2n}(N+2,Z)$, also known as the S$_{2n}$~differential. \n$\\delta_{2n}$ is shown in Fig.~\\ref{ShellGap} as a function of the neutron number N. The strong peaking in the two-neutron shell gap ($\\delta_{2n}$) clearly seen at N~=~126 further supports the shell closure at this neutron magic number as shown in Fig.~\\ref{S2n} by two-neutron separation energy ($S_{2n}$). Here again, only the SLy4 curve does not show a strong peak at the neutron magic number N~=~126. \n\n\n\\begin{figure}[!ht]\n\t\\centering \\includegraphics[scale=0.55]{ShellGap_Pt.eps}\n\t\\caption{(Color online) The two-neutron shell gap $\\delta_{2n}$ for even-even $^{94-168}$Pt isotopes.}\n\t\\label{ShellGap}\n\\end{figure}\n\n\n\n\\subsection{Quadrupole deformation} \n\nThe quadrupole deformation is also an important property for describing the\nstructure and shape of the nucleus. \nIn Fig.~\\ref{23beta}, we show for every Pt isotope (covering the mass\ninterval $160 \\leqslant A \\leqslant 204$) the energy curves along\nthe axial symmetry axis, as a function of the deformation\nparameter, $\\beta$, obtained within CDFT framework by using the density-dependent effective interactions DD-ME2 and DD-PC1.\n\n\n\\begin{figure}[!htb]\n\t\\centering \\includegraphics[scale=0.6]{23beta_Pt.eps}\n\t\\caption{(Color online) The total energy curves for $^{160-204}$Pt as a function of the axial quadrupole deformation parameter $\\beta_2$.}\n\t\\label{23beta}\n\\end{figure}\n\nAs we can see from Fig.~\\ref{23beta}, the interaction DD-PC1 provides potential\nenergy curves which are extremely similar to the ones obtained with DD-ME2. The deformations of the oblate and prolate minima are practically independent of the force. \n\nThe lightest isotopes, $^{160-162}$Pt, exhibit spherical shape. The next isotope, $^{164}$Pt, starts to develop two shallow degenerate minima, oblate\nand prolate, that correspond to a small value of $\\beta$. The next isotope, $^{166}$Pt, starts to develop a more pronounced prolate minimum. The $^{168-186}$Pt isotopes show a similar structure, with a well-deformed prolate minimum, $\\beta \\approx 0.3$, and an oblate local minimum. \n\nA transition from prolate to oblate shapes occurs smoothly between $^{188}$Pt(prolate) and $^{190}$Pt(oblate). In $^{190-200}$Pt two minima appear, with the opposite situation\noccurring in $^{168-186}$Pt. As the mass number increases, the two well-deformed minima gradually disappear and we get a flat potential energy curve at A = 202. At A = 204, we get a sharp single minimum, which confirms the spherical shape at the magic neutron number N = 126. \n\nThese results are in good agreement with recent works \\cite{Rodriguez2010,Garcia2014,Nomura2011} in terms of predicting $^{188}$Pt as critical. However, other calculations have different results that are not in agreement with ours, such as: Ref.~\\refcite{Yao2013} which predicts that the shape transition in Pt isotopes within a beyond-mean-field approach with the Skyrme SLy6 occurs at A = 186 to 188 instead of A = 188 to 190 in our calculations. In the same line, \nconstrained Hartree-Fock+ BCS calculations with the Skyrme forces Sk3, SGII, and SLy4 suggest a prolate to oblate shape transition at $^{182}$Pt~\\cite{Boillos2015}. Furthermore, triaxial D1M-Gogny calculations predict a smooth shape transition at A = 184 to 186~\\cite{Nomura2013}.\n\nThese differences between theoretical methods in predicting the exact location of the shape transition are due, firstly, to the difference between the models used and, secondly, to the fact that the shape transition is very sensitive to small details of the calculation, since the shape transition occurs exactly around the region where the energy of the competing shapes are practically degenerate.\n\nIn Fig.~\\ref{PESs} we display the triaxial contour plots of $^{186-190}$Pt\nisotopes in the ($\\beta$, $\\gamma$) plane. To study the dependency on $\\gamma$,  systematic constrained triaxial calculations have been done for mapping the quadrupole deformation space defined by $\\beta_2$ and $\\gamma$ using the effective interaction DD-ME2. \nThe constrained calculations are performed by imposing constraints on both axial and triaxial mass quadrupole moments. The potential energy surface\n(PES) study as a function of the quadrupole deformation parameter is performed by the method of quadratic constraint~\\cite{Ring1980}  (see Ref.~\\refcite{Abusara2017} for more details).\nEnergies are normalized with respect to the binding energy of the global minimum such that the ground state has a zero MeV energy.\n\n\n\\begin{figure}[!htb]\n\t\\minipage{0.5\\textwidth}\n\t\\centering \\includegraphics[scale=0.4]{Pt186.eps}\n\t\\endminipage\\hfill\n\t\\minipage{0.5\\textwidth}\n\t\\centering \\includegraphics[scale=0.4]{Pt188.eps}\n\t\\endminipage\\hfill\n\t\\centering \\includegraphics[scale=0.4]{Pt190.eps}\n\t\\caption{(Color online) Potential energy surfaces for  $^{186-190}$Pt in the ($\\beta$, $\\gamma$) plane, obtained from a CDFT calculations with the DD-ME2 parameter set. The color scale shown at the right has the unit of MeV, and scaled such that the ground state has a zero MeV energy.}\n\t\\label{PESs}\n\\end{figure}\n\nFrom this figure, we can notice that the location of the ground state minimum moves from near prolate shape at $^{186}$Pt to near oblate shape at $^{190}$Pt. $^{188}$Pt is slightly triaxial with its global minimum at (0.25, 10\\si{\\degree}). Thus, the shape transition is smooth, and there are no sudden changes in the nuclear shape. These  results confirm those seen previously in Fig.~\\ref{23beta} and are in full agreement with the results shown in Fig. 5 of Ref.~\\cite{Garcia2014} obtained with Hartree-Fock-Bogoliubov based on Gogny-D1S interaction. \n\n\n\\subsection{Neutron, Proton and Charge radii}\n\nThe charge radii calculated within the framework of the covariant density functional theory by using the functionals DD-ME2 and DD-PC1 are compared with the available experimental data~\\cite{angeli2004} and with the predictions of RMF(NL3)~\\cite{lalazissis1999} and HFB(SLy4). \nGood agreement between theory and experiment can be clearly seen in Fig.~\\ref{Rc}, except in the region $100 \\le N \\le 108$ where a small difference between the models and experiment is seen. This discrepancy is mainly due to the deformation effect since the experimental $\\beta_2$ values are extracted from experimental $B(E2)$$\\uparrow$ values (the reduced probability of the transition from the ground state of the nucleus to the first excited $2^+$ state) by using the Bohr model, which is valid only in the case of well-deformed nuclei, as it has been explained in Ref.~\\refcite{elbassem2019}.\\\\\n\n\\begin{figure}[ht]\t\t\t\t\t\t\t\t   \n\t\\centering \\includegraphics[scale=0.55]{Rc_Pt.eps}\n\t\\caption{(Color online) The charge radii of Pt isotopes}\n\t\\label{Rc}\n\\end{figure}\t\t\t\t\t\t\t\t   \n\nWe display in Fig. \\ref{RnRp} the calculated neutron and proton radii, $R_n$ and $R_p$, of platinum isotopes obtained by using the two functionals DD-ME2 and DD-PC1. The results of HFB theory with SLy4 Skyrme force as well as those of RMF model with NL3 functional are also shown for comparison.  \n\n\n\\begin{figure}[!htb]\n\t\\minipage{0.48\\textwidth}\n\t\\centering \\includegraphics[scale=0.4]{R_Pt.eps}\n\t\\endminipage\\hfill\n\t\\minipage{0.48\\textwidth}\n\t\\centering \\includegraphics[scale=0.41]{Rn-Rp_Pt.eps}\n\t\\endminipage\\hfill\n\t\\caption{(Color online) The neutron and proton radii of Pt isotopes (left panel) and the neutron skin thicknesses ($\\Delta R$ = $R_n$ \u2212 $R_p$)(right panel).}\n\t\\label{RnRp}\n\\end{figure}\n\nIn the left panel of Fig.~\\ref{RnRp}, we see that the neutron rms radii obtained by using the density-dependent\neffective interactions DD-ME2 and DD-PC1 and by HFB(SLy4) are almost equal\nthroughout the isotopic chain, but the NL3 results are overestimated. The origin of this difference is the missing density-dependence in the iso-vector channel of NL3~\\cite{lalazissis2005}. The proton rms radii obtained by all the formalisms, CDFT(DD-ME2 and DD-PC1), HFB(SLy4) and RMF(NL3) are almost similar.\n\nWe also notice from Fig.~\\ref{RnRp} (right panel) that the difference between the rms radii of neutrons and protons ($\\Delta R = R_n - R_p$) increases by increasing the neutron number in favor of developing a neutron skin. \n$\\Delta R$ attains its maximum for $^{238}$Pt in which it reaches 0.41~fm in the case of DD-ME2 and DD-PC1 formalisms, 0.38~fm in HFB(SLy4) and 0.55~fm in RMF(NL3) calculations.\n\nThere are also some abnormalities in the charge and proton radii of Pt isotopes as we can see in Figs.~\\ref{Rc} and~\\ref{RnRp} (left panel), these little jumps in $R_c$ and $R_p$ in the two regions $98 \\leqslant N \\leqslant 108$ and $140 \\leqslant N \\leqslant 160$ are due to the deformation effect~\\cite{elbassem2019}.\\\\\n\n\\section{Conclusion}\n\\label{Conclusion}\nIn the present work, we have studied the ground state properties of even-even\nplatinum isotopes, $^{160-238}$Pt, from the proton-rich side up to the neutron-rich one within the framework of the covariant density functional theory, by using two of the most recent functionals: The density-dependent point-coupling DD-PC1 and the density-dependent meson-exchange DD-ME2.\nThe bulk ground state properties are quite well reproduced in our calculations and are in good agreement with the experimental data.\nA strong shell closure is clearly seen at N=126. The neutron skin in this study reaches 0.41 fm for $^{238}$Pt.  \nThe total energy curves for $^{160-204}$Pt obtained in this work suggest a smooth\nprolate to oblate shape transition at $^{188}$Pt.\\\\\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe sampling Kantorovich operators have been introduced to approximate and reconstruct not necessarily continuous signals. In \\cite{BABUSTVI}, the authors introduced this operators starting from the well-known generalized sampling operators (see e.g. \\cite{BUFIST,BUST2,BAVI2,VI1,BABUSTVI2}) and replacing, in their definition, the sample values $f(k\/w)$ with $w \\int_{k\/w}^{(k+1)\/w}f(u)\\,du$. Clearly, this is the most natural mathematical modification to obtain operators which can be well-defined also for general measurable, locally integrable functions, not necessarily continuous. Moreover, this situation very often occur in Signal Processing, when one cannot match exactly the sample at the point $k\/w$: this represents the so-called \"time-jitter'' error. The theory of sampling Kantorovich operators allow us to reduces the time-jitter error, calculating the information in a neighborhood of $k\/w$ rather that exactly in the node $k\/w$. These operators, as the generalized sampling operators, represent an approximate version of classical sampling series, based on the Whittaker-Kotelnikov-Shannon sampling theorem (see e.g. \\cite{ANVI}).\n\n   Subsequently, the sampling Kantorovich operators have been studied in various settings. In \\cite{COVI,COVI2} the multivariate version of these operators were introduced. Results concerning the order of approximation are shown in \\cite{COVI3}. Extensions to more general contexts are presented in \\cite{VIZA1,VIZA2,VIZA3,BAMA2}.\n\n   The multivariate sampling Kantorovich operators considered in this paper are of the form:\n$$\n(S_w f)(\\underline{x})\\ :=\\ \\sum_{\\underline{k} \\in \\Z^n} \\chi(w\\underline{x}-t_{\\underline{k}})\\left[\\frac{w^n}{A_{\\underline{k}}} \\int_{R_{\\underline{k}}^w}f(\\underline{u})\\ d\\underline{u}\\right], \\hskip0.7cm (\\underline{x} \\in \\R^n), \\hskip0.8cm \\mbox{(I)}\n$$\nwhere $f: \\R^n \\to \\R$ is a locally integrable function such that the above series is convergent for every $\\underline{x} \\in \\R^n$. The symbol $t_{\\underline{k}}=\\left(t_{k_1},...,t_{k_n}\\right)$ denotes vectors where each $(t_{k_i})_{k_i \\in \\Z}$, $i=1,...,n$ is a certain strictly increasing sequence of real numbers with $\\Delta_{k_i}=t_{k_{i+1}}-t_{k_i}>0$.\nNote that, the sequences $(t_{k_i})_{k_i \\in \\Z}$ are not necessary equally spaced. We denote by $R_{\\underline{k}}^w$ the sets:\n$$\nR_{\\underline{k}}^w\\ :=\\ \\left[\\frac{t_{k_1}}{w},\\frac{t_{k_1+1}}{w}\\right]\\times\\left[\\frac{t_{k_2}}{w},\\frac{t_{k_2+1}}{w}\\right]\\times...\\times\\left[\\frac{t_{k_n}}{w},\\frac{t_{k_n+1}}{w}\\right], \\hskip2.2cm \\mbox{(II)}\n$$\n$w>0$ and $A_{\\underline{k}} = \\Delta_{k_1} \\cdot \\Delta_{k_2} \\cdot...\\cdot \\Delta_{k_n}$, $\\underline{k} \\in \\Z^n$. Moreover, the function $\\chi:\\R^n \\to \\R$ is a kernel satisfying suitable assumptions. \n\n   For the operators in (I) we recall some convergence results. We have that the family $(S_wf)_{w>0}$ converges pointwise to $f$, when $f$ is continuous and bounded and $(S_w f)_{w>0}$ converges uniformly to $f$, when $f$ is uniformly continuous and bounded. Moreover, to cover the case of not necessarily continuous signal, we study our operators in the general setting of Orlicz spaces $L^{\\varphi}(\\R^n)$. For functions $f$ belonging to $L^{\\varphi}(\\R^n)$ and generated by the convex $\\varphi$-function $\\varphi$, the family of sampling Kantorovich operators is \"modularly'' convergent to $f$, being the latter the natural concept of convergence in this setting.  \n\n   The latter result, allow us to apply the theory of the sampling Kantorovich operators to approximate and reconstruct images. In fact, static gray scale images are characterized by jumps of gray levels mainly concentrated in their contours or edges and this can be translated, from a mathematical point of view, by discontinuities (see e.g. \\cite{GOWO}).\n\n  Here, we introduce and analyze in detail some practical applications of the sampling Kantorovich algorithm to thermographic images, very useful for the analysis of buildings in seismic engineering. The thermography is a remote sensing technique, performed by the image acquisition in the infrared. Thermographic images are widely used to make non-invasive investigations of structures, to analyze the story of the building wall, to make diagnosis and monitoring buildings, and to make structural measurements. A further important use, is the application of the texture algorithm for the separation between the bricks and the mortar in masonries images. Through this procedure the civil engineers becomes able to determine the mechanical parameters of the structure under investigation. Unfortunately, the direct application of the texture algorithm to the thermographic images, can produce errors, as an incorrect separation between the bricks and the mortar. \n\n  Then, we use the sampling Kantorovich operators to process the thermographic images before to apply the texture algorithm. In this way, the result produced by the texture becomes more refined and therefore we can apply structural analysis after the calculation of the various parameters involved. In order to show the feasibility of our applications, we present in detail a real-world case-study.\n\n\n\\section{Preliminaries}\n\nIn this section we recall some preliminaries, notations and definitions.\n\n  We denote by $C(\\R^n)$ (resp. $C^0(\\R^n)$) the space of all uniformly continuous and bounded (resp. continuous and bounded) functions $f:\\R^n\\to \\R$ endowed with the usual sup-norm $\\|f\\|_{\\infty} := \\sup_{\\uu \\in \\R^n}\\left|f(\\uu)\\right|$, $\\uu = (u_1,\\, ...,\\, u_n)$, and by $C_c(\\R^n) \\subset C(\\R^n)$ the subspace of the elements having compact support. Moreover, $M\\left(\\R^n\\right)$ will denote the linear space of all (Lebesgue) measurable real functions defined on $\\R^n$.\n\n   We now recall some basic fact concerning Orlicz spaces, see e.g. \\cite{MUORL,MU1,RAO1,BAVI_1,BAMUVI}.  \n\n\\noindent   The function $\\varphi: \\R^+_0 \\to \\R^+_0$ is said to be a $\\varphi$-function if it satisfies the following assumptions: $(i)$ $\\varphi \\left(0\\right)=0$, and $\\varphi \\left(u\\right)>0$ for every $u>0$; $(ii)$ $\\varphi$ is continuous and non decreasing on $\\R^+_0$; $(iii)$\n$\\lim_{u\\to \\infty}\\varphi(u)\\ =\\ + \\infty$.\n\n  The functional $I^{\\varphi} : M(\\R^n)\\to [0,+\\infty]$ (see e.g. \\cite{MU1,BAMUVI}) defined by\n$$\nI^{\\varphi} \\left[f\\right] := \\int_{\\R^n} \\varphi(\\left| f(\\underline{x}) \\right|)\\ d\\underline{x},\\ \\hskip0.5cm \\left(f \\in M(\\R^n)\\right),\n$$\nis a modular in $M(\\R^n)$ . The Orlicz space generated by $\\varphi$ is given by\n$$\nL^{\\varphi}(\\R^n) := \\left\\{f \\in M\\left(\\R^n\\right):\\ I^{\\varphi} [\\lambda f]<+\\infty,\\ \\mbox{for\\ some}\\ \\lambda>0\\right\\}.\n$$\nThe space $L^{\\varphi}(\\R^n)$ is a vector space and an important subspace is given by\n$$\nE^{\\varphi}(\\R^n)  := \\left\\{f \\in M\\left(\\R^n\\right):\\ I^{\\varphi} [\\lambda f]<+\\infty,\\ \\mbox{for\\ every}\\ \\lambda>0\\right\\}.\n$$\n$E^{\\varphi}(\\R^n)$ is called the space of all finite elements of $L^{\\varphi}(\\R^n)$. It is easy to see that the following inclusions hold:\n$\nC_c(\\R^n) \\subset E^{\\varphi}(\\R^n) \\subset L^{\\varphi}(\\R^n).\n$\nClearly, functions belonging to $E^{\\varphi}(\\R^n)$ and $L^{\\varphi}(\\R^n)$ are not necessarily continuous. A norm on $L^{\\varphi}(\\R^n)$, called Luxemburg norm, can be defined by\n$$\n\\left\\|f\\right\\|_{\\varphi}\\ :=\\ \\inf \\left\\{\\lambda>0:\\ I^{\\varphi}[f\/\\lambda] \\miu \\lambda\\right\\}, \\hskip0.7cm (f \\in L^{\\varphi}(\\R^n)).\n$$\n\n   We will say that a family of functions $(f_w)_{w>0} \\subset L^{\\varphi}(\\R^n)$ is norm convergent to a function $f \\in L^{\\varphi}(\\R^n)$, i.e., $\\left\\|f_w-f\\right\\|_{\\varphi}\\rightarrow 0$ for $w\\to +\\infty$, if and only if \n$\\lim_{w \\to +\\infty} I^{\\varphi}\\left[\\lambda(f_w-f)\\right]\\ =\\ 0$, for every $\\lambda>0$.\nMoreover, an additional concept of convergence can be studied in Orlicz spaces: the \"modular convergence\". The latter induces a topology (modular topology) on the space $L^{\\varphi}(\\R^n)$ (\\cite{MU1,BAMUVI}). \n\n  We will say that a family of functions $(f_w)_{w>0} \\subset L^{\\varphi}(\\R^n)$ is modularly convergent to a function $f \\in L^{\\varphi}(\\R^n)$ if\n$\n\\lim_{w \\to +\\infty} I^{\\varphi}\\left[\\lambda(f_w-f)\\right]\\ =\\ 0$,\nfor some $\\lambda>0$. Obviously, norm convergence implies modular convergence, while the converse implication does not hold in general. The modular and norm convergence are equivalent if and only if the $\\varphi$-function $\\varphi$ satisfies the $\\Delta_2$-condition, see e.g., \\cite{MU1,BAMUVI}. Finally, as last basic property of Orlicz spaces, we recall the following.\n\\begin{lemma}[\\cite{BAMA}] \\label{lemma1}\nThe space $C_c(\\R^n)$ is dense in $L^{\\varphi}(\\R^n)$ with respect to the modular topology, i.e., for every $f \\in L^{\\varphi}(\\R^n)$ and for every $\\ep>0$ there exists a constant $\\lambda>0$ and a function $g \\in C_c(\\R^n)$ such that $I^{\\varphi}[\\lambda(f-g)]<\\ep$.\n\\end{lemma}\n\n\n\\section{The sampling Kantorovich operators}\n\nIn this section we recall the definition of the operators with which we will work. We will denote by \n$t_{\\underline{k}}=\\left(t_{k_1},...,t_{k_n}\\right)$ a vector where each element $(t_{k_i})_{k_i \\in \\Z}$, $i=1,...,n$ is a sequence of real numbers with $-\\infty < t_{k_i}<t_{k_{i+1}}<+\\infty$, $\\lim_{k_i \\to \\pm \\infty}t_{k_i} = \\pm \t\\infty$, for every $i=1, ..., n$, and such that there exists $\\Delta$, $\\delta>0$ for which $\\delta \\leq \\Delta_{k_i}:= t_{k_{i+1}}-t_{k_{i}} \\leq  \\Delta$, for every $i=1, ... , n$.\nNote that, the elements of $(t_{k_i})_{k_i \\in \\Z}$ are not necessary equally spaced.  In what follows, we will identify with the symbol $\\Pi^n$ the sequence $(t_{\\underline{k}})_{\\kk \\in \\Z^n}$.\n\n  A function $\\chi: \\R^n \\to \\R$ will be called a kernel if it satisfies the following properties:\n\\begin{itemize}\n\t\\item[$(\\chi1)$] $\\chi \\in L^1(\\R^n)$ and is bounded in a neighborhood of $\\underline{0} \\in \\R^n$; \n\t\\item[$(\\chi 2)$] for every $\\underline{u} \\in \\R^n$, \\hskip0.2cm $\\displaystyle \\sum_{\\underline{k} \\in \\Z^n} \\chi(\\underline{u} - t_{\\underline{k}})\\ =\\ 1$;\n\t\\item[$(\\chi 3)$] for some $\\beta>0$,\n$$\nm_{\\beta, \\Pi^n }(\\chi)\\ =\\ \\sup_{\\underline{u} \\in \\R^n}\\sum_{\\underline{k} \\in \\Z^n}\\left|\\chi(\\underline{u}-t_{\\underline{k}})\\right|\\cdot \\left\\|\\underline{u}-t_{\\underline{k}}\\right\\|^{\\beta}_2\\ <\\ +\\infty,\n$$\nwhere $\\| \\cdot \\|_2$ denotes the usual Euclidean norm. \n\\end{itemize}\nWe now recall the definition of the linear multivariate sampling Kantorovich operators introduced in \\cite{COVI}. Define:\n\\begin{equation} \\label{KANTO}\n(S_w f)(\\underline{x})\\ :=\\ \\sum_{\\underline{k} \\in \\Z^n} \\chi(w\\underline{x}-t_{\\underline{k}})\\left[\\frac{w^n}{A_{\\underline{k}}} \\int_{R_{\\underline{k}}^w}f(\\underline{u})\\ d\\underline{u}\\right], \\hskip0.7cm (\\underline{x} \\in \\R^n),\n\\end{equation}\nwhere $f: \\R^n \\to \\R$ is a locally integrable function such that the above series is convergent for every $\\underline{x} \\in \\R^n$, where\n\\begin{displaymath}\nR_{\\underline{k}}^w\\ :=\\ \\left[\\frac{t_{k_1}}{w},\\frac{t_{k_1+1}}{w}\\right]\\times\\left[\\frac{t_{k_2}}{w},\\frac{t_{k_2+1}}{w}\\right]\\times...\\times\\left[\\frac{t_{k_n}}{w},\\frac{t_{k_n+1}}{w}\\right],\n\\end{displaymath}\n$w>0$ and $A_{\\underline{k}} = \\Delta_{k_1} \\cdot \\Delta_{k_2} \\cdot...\\cdot \\Delta_{k_n}$, $\\underline{k} \\in \\Z^n$. The operators in (\\ref{KANTO}) have been introduced in \\cite{BABUSTVI} in the univariate setting.\n\\begin{remark} \\label{remark1} \\rm\n(a) Under conditions $(\\chi 1)$ and $(\\chi 3)$, the following properties for the kernel $\\chi$ can be proved:\n$$\nm_{0, \\Pi^n}(\\chi)\\ :=\\ \\sup_{\\uu \\in \\R^n} \\sum_{\\kk \\in \\Z^n} \\left|\\chi(\\uu-\\tk)\\right|\\ <\\ +\\infty,\n$$\nand, for every $\\gamma >0$\n\\begin{equation} \\label{eee}\n\\lim_{w \\to +\\infty} \\sum_{\\left\\|w \\uu-\\tk\\right\\|_2 >\\gamma w}\\left|\\chi(w \\uu - \\tk)\\right|\\ =\\ 0,\n\\end{equation}\nuniformly with respect to $\\uu \\in \\R^n$, see \\cite{COVI}.\n\\vskip0.2cm\n\\noindent (b) By (a), we obtain that $S_w f$ with $f \\in L^{\\infty}(\\R^n)$ are well-defined. Indeed,\n\\begin{displaymath}\n\\left|(S_w f)(\\xx)\\right|\\ \\miu\\ m_{0, \\Pi^n}(\\chi)\\left\\|f\\right\\|_{\\infty}\\ < +\\infty,\n\\end{displaymath}\nfor every $\\xx \\in \\R^n$, i.e. $S_w : L^{\\infty}(\\R^n)\\rightarrow L^{\\infty}(\\R^n)$.\n\\end{remark}\n\\begin{remark} \\label{remark3} \\rm\nNote that, in the one-dimensional setting, choosing $t_{k}=k$, for every $k \\in \\Z$, condition $(\\chi 2)$ is equivalent to \n\\begin{displaymath}\n\\widehat{\\chi}(k) :=\\ \\left\\{\n\\begin{array}{l}\n0, \\hskip0.5cm k \\in \\Z\\setminus \\left\\{0\\right\\}, \\\\\n1, \\hskip0.5cm k=0,\n\\end{array}\n\\right.\n\\end{displaymath}\nwhere $\\widehat{\\chi}(v):=\\int_{\\R}\\chi(u)e^{-ivu}\\ du$, $v \\in \\R$, denotes the Fourier transform of $\\chi$; see \\cite{BUNE,BABUSTVI,COVI,COVI3}.\n\\end{remark}\n\n\n\n\n\\section{Convergence results}\n\nIn this section, we show the main approximation results for the multivariate sampling Kantorovich operators. In \\cite{COVI}, the following approximation theorem for our operators has been proved. \n\\begin{theorem} \\label{th1}\nLet $f \\in C^0(\\R^n)$. Then, for every $\\underline{x} \\in \\R^n$,\n\\begin{displaymath}\n\\lim_{w \\to +\\infty} (S_w f)(\\underline{x})\\ =\\ f(\\underline{x}).\n\\end{displaymath}\nIn particular, if $f \\in C(\\R)$, then\n\\begin{displaymath}\n\\lim_{w \\to +\\infty} \\left\\|S_w f -f\\right\\|_{\\infty}\\ =\\ 0.\n\\end{displaymath}\n\\end{theorem}\n\\begin{proof}\nHere we highlight the main points of the proof.\n\n\\noindent Let $f \\in C^0(\\R^n)$ and $\\xx \\in \\R^n$ be fixed. By the continuity of $f$ we have that for every fixed $\\ep>0$ there exists $\\gamma>0$ such that $|f(\\xx)-f(\\uu)|<\\ep$ for every $\\| \\xx -\\uu\\|_2 \\miu \\gamma$, $\\uu \\in \\R^n$. Then, by $(\\chi 2)$ we obtain:\n$$\n\\left|(S_w f)(\\xx)-f(\\xx)\\right|\\ \\miu\\ \\sum_{\\kk \\in \\Z^n} \\left|\\chi(w \\xx - \\tk)\\right| \\frac{w^n}{A_{\\kk}}\\int_{R^w_{\\kk}}\\left|f(\\uu)-f(\\xx)\\right|\\, d\\uu\\ \n$$\n\\vskip-0.5cm\n\\begin{eqnarray*}\n&=& (\\sum_{\\left\\|w \\xx-\\tk\\right\\|_2 \\miu w \\gamma\/2}+\\sum_{\\left\\|w \\xx-\\tk\\right\\|_2 > w \\gamma\/2})\\left|\\chi(w \\xx - \\tk)\\right|\\frac{w^n}{A_{\\kk}}\\int_{R^w_{\\kk}}\\left|f(\\uu)-f(\\xx)\\right|\\ d\\uu \\\\\n&=&\\ I_1+I_2.\n\\end{eqnarray*}\nFor $\\uu \\! \\in \\! R^w_{\\kk}$ and $\\left\\|w \\xx-\\tk\\right\\|_2\\! \\miu \\! w \\gamma\/2$ we have $\\|\\uu-\\xx\\|_2\\! \\miu \\! \\gamma$ for  $w\\! > \\!0$ sufficiently large, then by the continuity of $f$ we obtain $I_1 \\miu m_{0,\\Pi^n}(\\chi) \\ep$ (see Remark \\ref{remark1} (a)). Moreover, by the boundedness of $f$ and (\\ref{eee}) we obtain $I_2 \\miu 2\\|f\\|_{\\infty} \\ep$ for $w>0$ sufficiently large, then the first part of the theorem follows since $\\ep>0$ is arbitrary. The second part of the theorem follows similarly replacing $\\gamma>0$ with the parameter of the uniform continuity of $f$. \n\\end{proof}\n\n  In order to obtain a modular convergence result, the following norm-convergence theorem for the sampling Kantorovich operators (see \\cite{COVI}) can be formulated.\n\\begin{theorem} \\label{norm_conv} \nLet $\\varphi$ be a convex $\\varphi$-function. For every $f \\in C_c(\\R^n)$ we have\n$$\n\\lim_{w \\to +\\infty} \\left\\|S_w f - f \\right\\|_{\\varphi}\\ =\\ 0.\n$$\n\\end{theorem}\nNow, we recall the following modular continuity property for $S_w$, useful to prove the modular convergence for the above operators in Orlicz spaces.\n\\begin{theorem} \\label{mod_cont} \nLet $\\varphi$ be a convex $\\varphi$-function. For every $f \\in L^{\\varphi}(\\R^n)$ there holds\n$$\nI^{\\varphi}[\\lambda S_w f]\\ \\miu\\ \\frac{\\left\\|\\chi\\right\\|_1}{\\delta^n\\cdot m_{0,\\Pi^n}(\\chi)}I^{\\varphi}[\\lambda m_{0,\\Pi^n}(\\chi)f],\n$$\nfor some $\\lambda>0$. In particular, $S_w$ maps $L^{\\varphi}(\\R^n)$ in $L^{\\varphi}(\\R^n)$.\n\\end{theorem}\n\\noindent Now, the main result of this section follows (see \\cite{COVI}).\n\\begin{theorem} \\label{th2}\nLet $\\varphi$ be a convex $\\varphi$-function. For every $f \\in L^{\\varphi}(\\R^n)$, there exists $\\lambda>0$ such that\n$$\n\\lim_{w \\to +\\infty} I^{\\varphi}[\\lambda (S_w f-f)]\\ =\\ 0.\n$$\n\\end{theorem}\n\\begin{proof}\nLet $f \\in L^{\\varphi}(\\R^n)$ and $\\ep>0$ be fixed. By Lemma \\ref{lemma1}, there exists $\\overline{\\lambda}>0$ and $g \\in C_c(\\R^n)$ such that $I^{\\varphi}[\\overline{\\lambda} (f-g)]\\!<\\! \\ep$. Let now $\\lambda>0$ such that $3\\lambda(1 + m_{0,\\Pi^n}(\\chi)) \\miu \\overline{\\lambda}$. By the properties of $\\varphi$ and Theorem \\ref{mod_cont}, we have\n\\begin{eqnarray*}\n&& \\hskip-0.7cm I^{\\varphi}[\\lambda(S_w f-f)]\\ \\miu\\ I^{\\varphi}[3\\lambda(S_w f - S_w g)]+I^{\\varphi}[3\\lambda(S_w g - g)] + I^{\\varphi}[3\\lambda(f-g)]\\\\\n&\\miu&\\ \\frac{1}{m_{0,\\Pi^n}(\\chi)\\cdot \\delta^n}\\left\\|\\chi \\right\\|_1 I^{\\varphi}[\\overline{\\lambda} (f-g)] + I^{\\varphi}[3\\lambda(S_w g - g)] + I^{\\varphi}[\\overline{\\lambda} (f-g)] \\\\\n&<&\\ \\left(\\frac{1}{m_{0,\\Pi^n}(\\chi) \\cdot \\delta^n}\\left\\|\\chi \\right\\|_1+1\\right) \\ep + I^{\\varphi}[3\\lambda(S_w g - g)].\n\\end{eqnarray*}\nThe assertion follows from Theorem \\ref{norm_conv}.\n\\end{proof} \nThe setting of Orlicz spaces allows us to give a unitary approach for the reconstruction since we may obtain convergence results for particular cases of Orlicz spaces. For instance, choosing $\\varphi(u)=u^p$, $1 \\leq p < +\\infty$, we have that $L^{\\varphi}(\\R^n) = L^p(\\R^n)$ and $I^{\\varphi}[f]=\\|f\\|^p_{p}$, where $\\| \\cdot \\|_p$ is the usual $L^p$-norm. Then, from Theorem \\ref{mod_cont} and Theorem \\ref{th2} we obtain the following corollary.\n\\begin{corollary}\nFor every $f \\in L^p(\\R^n)$, $1 \\leq p < +\\infty$, \nthe following inequality holds:\n\\begin{displaymath}\n\\left\\|S_w f\\right\\|_p\\ \\miu\\ \\delta^{-n\/p}\\, [m_{0,\\Pi^n}(\\chi)]^{(p-1)\/p}\\, \\left\\|\\chi\\right\\|^{1\/p}_1\\, \\left\\|f\\right\\|_p.\n\\end{displaymath}\nMoreover, we have:\n$$\n\\lim_{w \\to +\\infty} \\|S_w f-f\\|_p\\ =\\ 0.\n$$\n\\end{corollary}\nThe corollary above, allows us to reconstruct $L^p$-signals (in $L^p$-sense), therefore signals\/images not necessarily continuous. Other examples of Orlicz spaces for which the above theory can be applied can be found e.g., in \\cite{MUORL,MU1,BAMUVI,BABUSTVI,COVI}. The theory of sampling Kantorovich operators in the general setting of Orlicz spaces allows us to obtain, by means of a unified treatment, several applications in many different contexts.\n\n\n\\section{Examples of special kernels} \\label{sec5}\n\nOne important fact in our theory is the choice of the kernels, which influence the order of approximation that can be achieved by our operators (see e.g. \\cite{COVI3} in one-dimensional setting).\n\n  For instance, one  can take into consideration  {\\em radial kernels}, i.e., functions for which the value depends on the Euclidean norm of the argument only. Example of such a kernel can be given, for example, by the Bochner-Riesz kernel, defined as follows $b^{\\alpha}(\\underline{x}):=2^{\\alpha}\\Gamma(\\alpha+1)\\|\\underline{x}\\|_2^{-(n\/2)+\\alpha}{\\cal B}_{(n\/2)+\\alpha}(\\|\\underline{x}\\|_2)$,\nfor $\\underline{x} \\in \\R^n$, where $\\alpha > (n-1)\/2$, ${\\cal B}_{\\lambda}$ is the Bessel function of order $\\lambda$ and $\\Gamma$ is the Euler function. For more  details about this matter,  see e.g. \\cite{BUFIST}. \n\n  To construct, in general, kernels satisfying all the assumptions $(\\chi_i)$, $i=1,2,3$ is not very easy.\n\n For this reason, here we show a procedure useful to construct examples using product of univariate kernels, see e.g. \\cite{BUFIST,COVI,COVI2}. In this case, we consider the case of uniform sampling scheme, i.e., $t_{\\underline{k}}=\\underline{k}$.\n\n Denote by $\\chi_1, ..., \\chi_n$, the univariate functions $\\chi_i : \\R \\to \\R$, $\\chi_i \\in L^1(\\R)$, satisfying\nthe following assumptions:\n\\begin{equation} \\label{uno-dim-}\nm_{\\beta,\\Pi^1}(\\chi_i)\\ :=\\ \\sup_{u \\in \\R}\\sum_{k \\in \\Z}\\left|\\chi_i(u-k)\\right| |u-k|^{\\beta}\\ <\\ +\\infty,\n\\end{equation}\nfor some $\\beta >0$, $\\chi_i$ is bounded in a neighborhood of the origin and \n\\begin{equation} \\label{sing}\n\\sum_{k \\in \\Z}\\chi_i(u-k) =1,\n\\end{equation}\nfor every $u \\in \\R$, for $i=1,...,n$. \nNow, setting $\\chi(\\underline{x}) := \\prod_{i=1}^n\\chi_i(x_i)$, $\\underline{x}=(x_1,...,x_n) \\in \\R^n$,\nit is easy to prove that $\\chi$ is a multivariate kernel for the operators $S_w$ satisfying all the assumptions of our theory, see e.g., \\cite{BUFIST,COVI}. \n\n  As a first example, consider the univariate Fej\\'{e}r's kernel defined by $F(x)\\ :=\\ \\frac{1}{2}\\mbox{sinc}^2\\left(\\frac{x}{2}\\right)$, $x \\in \\R$, where the sinc function is given by\n\\begin{displaymath}\n\\mbox{sinc}(x)\\ :=\\ \\left\\{\n\\begin{array}{l}\n\\disp \\frac{\\sin \\pi x}{\\pi x}, \\hskip1cm x \\in \\R\\setminus \\left\\{0\\right\\}, \\\\\n\\hskip0.5cm 1, \\hskip1.5cm x=0.\n\\end{array}\n\\right.\n\\end{displaymath}\nClearly, $F$ is bounded, belongs to $L^1(\\R)$ and satisfies the moment conditions $(\\ref{uno-dim-})$ for $\\beta = 1$, as shown in \\cite{BUNE,BABUSTVI,COVI}. \nFurthermore, taking into account that the Fourier transform of $F$ is given by (see \\cite{BUNE})\n\\begin{displaymath}\n\\widehat{F}(v) :=\\ \\left\\{\n\\begin{array}{l}\n1-|v\/\\pi|,\\ \\hskip0.5cm |v|\\miu \\pi, \\\\\n0,\\ \\hskip1.85cm |v|>\\pi,\n\\end{array}\n\\right.\n\\end{displaymath}\nwe obtain by Remark \\ref{remark3} that condition $(\\ref{sing})$ is fulfilled. Then, we can define $\\disp \\mathcal{F}_n(\\xx)= \\prod^n_{i=1}F(x_i)$, $\\underline{x}=(x_1,...,x_n) \\in \\R^n,$ the multivariate Fej\\'{e}r's kernel, satisfying the condition upon a multivariate kernel. The Fej\\'{e}r's kernel $F$ and the bivariate Fej\\'{e}r's kernel $F(x)\\cdot F(y)$ are plotted in Figure \\ref{fig1}.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.22]{fejer}\n\\hskip0.8cm\n\\includegraphics[scale=0.22]{multidimFejer}\n\\caption{\\small{The Fej\\'{e}r's kernel $F$ (left) and the bivariate Fej\\'{e}r's kernel ${\\cal F}(x,y)$ (right).}} \\label{fig1}\n\\end{figure}\n\n  The Fej\\'{e}r's kernel $\\mathcal{F}_n$ is an example of kernel with unbounded support, then to evaluate our sampling Kantorovich series at any given $\\xx \\in \\R^n$, we need of an infinite number of mean values $w^n \\int_{R^w_{\\kk}}f(\\uu)\\ d\\uu$. However, if the function $f$ has compact support, this problem does not arise. In case of function having unbounded support the infinite sampling series must be truncated to a finite one, which leads to the so-called truncation error. In order to avoid the truncation error, one can take kernels $\\chi$ with bounded support. Remarkable examples of kernels with compact support, can be constructed using the well-known central B-spline of order $k \\in \\N$, defined by\n$$\nM_k(x) :=\\ \\frac{1}{(k-1)!}\\sum^k_{i=0}(-1)^i \\left(\\begin{array}{l} \\!\\! \nk\\\\\n\\hskip-0.1cm i\n\\end{array} \\!\\! \\right)\n\\left(\\frac{k}{2}+x-i\\right)^{k-1}_+,\n$$\nwhere the function $(x)_+ := \\max\\left\\{x,0\\right\\}$ denotes the positive part of $x \\in \\R$ (see \\cite{BABUSTVI,VIZA1,COVI}). The central B-spline $M_3$ and the bivariate B-spline kernel $M_3(x)\\cdot M_3(y)$ are plotted in Figure \\ref{fig2}.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.22]{spline}\n\\hskip0.7cm\n\\includegraphics[scale=0.22]{bidimspline}\n\\caption{\\small{The B-spline $M_3$ (left) and the bivariate B-spline ${\\cal M}^2_3(x,y)$ (right).}} \\label{fig2}\n\\end{figure}\nWe have that, the Fourier transform of $M_k$ is given by\n$\\widehat{M_k}(v)\\ :=\\ \\mbox{sinc}^n\\left( \\frac{v}{2 \\pi} \\right)$, $v \\in \\R$, and then, if we consider the case of the uniform spaced sampling scheme, condition (\\ref{sing}) is satisfied by Remark \\ref{remark3}. Clearly, $M_k$ are bounded on $\\R$, with compact support $[-n\/2,n\/2]$, and hence $M_k \\in L^1(\\R)$, for all $k \\in \\N$. Moreover, it easy to deduce that condition $(\\ref{uno-dim-})$ is fulfilled for every $\\beta>0$, see \\cite{BABUSTVI}.\nHence $\\mathcal{M}^n_k(\\xx)\\ :=\\ \\prod^n_{i=1}M_k(x_i)$, $\\underline{x}=(x_1,...,x_n) \\in \\R^n$,\nis the multivariate B-spline kernel of order $k \\in \\N$.\n\n  Finally, other important examples of univariate kernels are given by the Jackson-type kernels, defined by\n$J_k(x)=c_k\\, \\mbox{sinc}^{2k}\\left(\\frac{x}{2k\\pi\\alpha}\\right)$, $x \\in  \\R$,\nwith $k \\in \\N$, $\\alpha \\geq 1$, where the normalization coefficients $c_k$ are given by\n$c_k\\ :=\\ \\left[ \\int_{\\R} \\mbox{sinc}^{2k}\\left(\\frac{u}{2 k \\pi \\alpha} \\right) \\, du \\right]^{-1}$.\nSince $J_k$ are band-limited to $[-1\/\\alpha,\\, 1\/\\alpha]$, i.e., their Fourier transform vanishes outside this interval, condition $(\\chi 2)$ is satisfied by Remark \\ref{remark3} and $(\\ref{uno-dim-})$ is satisfied since $J_k(x)=\\mathcal{O}(|x|^{-2k})$, as $x \\to \\pm \\infty$, $k \\in \\N$. In similar manner, by the previous procedure we can construct a multivariate version of Jackson kernels. For more details about Jackson-type kernels, and for others useful examples of kernels see e.g. \\cite{BUNE,BAMUVI,BABUSTVI0,BABUSTVI}.\n\n\n\\section{The sampling Kantorovich algorithm for image reconstruction}\n\nIn this section, we show applications of the multivariate sampling Kantorovich operators to image reconstruction. First of all, we recall that every bi-dimensional gray scale image $A$ is represented by a suitable matrix and can be modeled as a step function $I$, with compact support, belonging to $L^p(\\R^2)$, $1 \\leq p <+\\infty$. The definition of $I$ arise naturally as follows:\n$$\nI(x,y)\\ :=\\ \\sum^{m}_{i=1}\\sum^{m}_{j=1}a_{ij} \\cdot \\textbf{1}_{ij}(x,y), \\hskip0.3cm (x,y) \\in \\R^2,\n$$\nwhere ${\\bf 1}_{ij}(x,y)$, $i,j =1,2,...,m$, are the characteristics functions of the sets $(i-1,\\ i]\\times(j-1,\\ j]$ (i.e. ${\\bf 1}_{ij}(x,y)=1$, for $(x,y)\\ \\in\\ (i-1,\\ i]\\times(j-1,\\ j]$ and ${\\bf 1}_{ij}(x,y)=0$ otherwise). \n\n  The above function $I(x,y)$ is defined in such a way that, to every pixel $(i,j)$ it is associated the corresponding gray level $a_{ij}$. \n\n  We can now consider approximation of the original image by the bivariate sampling Kantorovich operators $(S_w I)_{w>0}$ based upon some kernel $\\chi$. \n\n  Then, in order to obtain a new image (matrix) that approximates in $L^p$-sense the original one, it is sufficient to sample $S_w I$ (for some $w>0$) with a fixed sampling rate. In particular, we can reconstruct the approximating images (matrices) taking into consideration different sampling rates and this is possible since we know the analytic expression of $S_w I$.\n\n If the sampling rate is chosen higher than the original sampling rate, one can get a new image that has a better resolution than the original one's. The above procedure has been implemented by using MATLAB and tools of matrix computation in order to obtain an optimized algorithm based on the multivariate sampling Kantorovich theory.\n\n  In the next sections, examples of thermographic images reconstructed by the sampling Kantorovich operators will be given to show the main applications of the theory to civil engineering. In particular, a real world case-study is analyzed in term of modal analysis to obtain a model useful to study the response of the building under seismic action.\n\n\n\\section{An application of thermography to civil engineering}\n\nIn the present application, thermographic images will be used to locate the resisting elements and to define their dimensions, and moreover to investigate the actual texture of the masonry wall, i.e., the arrangement of blocks (bricks and\/or stones) and mortar joints. \nIn general, thermographic images have a resolution too low to accurately analyze  the texture of the masonries,  therefore it has to be increased by means of suitable tools.\n\nIn order to obtain a consistent separation of the phases, that is a correct identification of the pixel which belong to the blocks and those who belong to mortar joints, the image is converted from gray-scale representation to black-and-white (binary) representation by means of an image texture algorithm, which employs techniques belonging to the field of digital image processing. \nThe image texture algorithm, described in details in \\cite{CACLGU}, leads to areas of white pixels identified as blocks and areas of black pixels identified as mortar joints.\nHowever, the direct application of the image texture algorithm to the thermographic images  (see Figure~\\ref{fig:original_ric}), can produce errors, as an incorrect separation between the bricks and the mortar (see Figure~\\ref{fig:original_BW}). \nTherefore, we can use the sampling Kantorovich operators to process the thermographic images. \nIn particular, here we used the operators $S_w$ based upon the bivariate Jackson-type kernel with $k=12$ (see Section \\ref{sec5}) and the parameter $w=40$ (see Figure~\\ref{fig:reconstructed_ric}). \nThe application of the image texture algorithm produces a consistent separation of the phases (see Figure~\\ref{fig:reconstructed_BW}).\n\n\\begin{figure}[!ht]\n\\centering\n\\subfigure[]{ \\includegraphics[width = 0.2\\textwidth]{sez_im3_color} \\label{fig:original_ric}}\n\\hskip0.15cm\n\\subfigure[]{ \\includegraphics[width = 0.2\\textwidth]{sez_im3_color_BW} \\label{fig:original_BW}}\n\\hskip0.15cm \n\\subfigure[]{ \\includegraphics[width = 0.2\\textwidth]{ric_sez_im3_color_JACK_N=12_450pix_w=40} \\label{fig:reconstructed_ric}}\n\\hskip0.15cm\n\\subfigure[]{ \\includegraphics[width = 0.2\\textwidth]{ric_sez_im3_color_JACK_N=12_450pix_w=40_BW} \\label{fig:reconstructed_BW}} \n\\caption{{\\small (a) Original thermographic image ($75\\times 75$ pixel resolution) and (b) its texture; (c) Reconstructed thermographic image ($450\\times 450$ pixel resolution) and (d) its texture.}}\n\\label{fig:original}\n\\end{figure}\n\nIn order to perform structural analysis, the mechanical characteristics of an homogeneous material equivalent to the original heterogeneous material are sought. \nThe equivalence is in the sense that, when subjected to the same boundary conditions (b.c.), the overall responses in terms of mean values of stresses and deformations are the same.\nThe equivalent elastic properties are estimated by means of the ``test-window'' method \\cite{CLGU}.\n\n\n\\section{Case-study}\n\nThe image texture algorithm described previously  has been used to analyze a real-world case-study: a building  consisting of two levels and an attic, with a very simple architectural plan, a rectangle with sides 11 m and 11.4 m (see Figure~\\ref{fig:pianta_prospetto}). \nThe vertical structural elements consist of masonry walls. \nAt first level the masonry walls have thickness of 40 cm with blocks made of  stones, while at second level the masonry is made of squared tuff block and it has thickness of about 35 cm. \nBoth surfaces of the walls are plastered.\nThe slab can be assumed to be rigid in the horizontal plane and \nthe building is placed in a medium risk seismic area. \nAccording to a preliminary visual survey, the structure does not have strong asymmetries between principal directions and the distribution of the masses are quite uniform.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width = 0.75\\textwidth]{pianta_prospetto} \n\\caption{{\\small Case-study building: plant (left) and main facade (right).}}\n\\label{fig:pianta_prospetto}\n\\end{figure}\n\n\nThe response of the building is evaluated by means of three different models.\nIn the first model, only information that can be gathered by the visual survey have been used, and the characteristics of material has been assume according to Italian Building Code. \nIn the second model information concerning the actual geometry of vertical structural elements acquired by means of thermographic survey have been used; \nfor example, the survey showed that the openings at ground level in the main facade were once larger and have been subsequently partially filled with non-structural masonry, and thus the dimensions of all the masonry walls in the facade have to be reduced.\nEventually, in the third model information about the actual texture of masonries of ground level (chaotic masonry, Fig.~\\ref{fig:chaotic}) and of second level (periodic masonry, Fig.~\\ref{fig:periodic}) have been used. \nIn particular, the actual textures where established using the reconstructed thermographic images. \n\n\\begin{figure}[!ht]\n\\centering\n\\subfigure[]{ \\includegraphics[width = 0.2\\textwidth]{ir5463_dx_ric_BW} \\label{fig:chaotic} }\n\\hskip0.8cm\n\\subfigure[]{ \\includegraphics[width = 0.2\\textwidth]{ir5462_dx_ric_BW} \\label{fig:periodic}} \n\\caption{{\\small Texture of (a) ground level and (b) second level masonries.}}\n\\label{fig:masonries}\n\\end{figure}\n\nThe main characteristics of the models are reported in Table~\\ref{tab:char_models}.\nFor the mechanical characteristic the Young's modulus $E$ and the shear modulus $G$ are shown.\nIn the third model, the following mechanical characteristics (Young's modulus $E$ and Poisson's ratio $\\nu$) of the constituent phases have been used: for the stones of the ground level, $E = 25000\\,\\mathrm{N\\,mm^{-2}}, \\nu = 0.2$, for the bricks of the second level $E = 1700\\,\\mathrm{N\\,mm^{-2}}, \\nu = 0.2$, for the mortar of both levels  $E = 2500\\,\\mathrm{N\\,mm^{-2}}, \\nu = 0.2$.\n\n\\begin{table}[!ht]\n\\center\n{\\small\n\\begin{tabular}{c|c|cc|cc}\n\\hline\n & & \\multicolumn{2}{|c}{ground level} & \\multicolumn{2}{|c}{second level}\\\\\n & geometry & $E$ & $G$ & $E$ & $G$ \\\\\n & & $\\mathrm{N\\,mm^{-2}}$ & $\\mathrm{N\\,mm^{-2}}$ & $\\mathrm{N\\,mm^{-2}}$ & $\\mathrm{N\\,mm^{-2}}$ \\\\\n\\hline\n Model \\#1 & visual survey & 3346 & 1115 & 1620 & 540 \\\\\n Model \\#2 & thermogr. survey & 3346 & 1115 & 1620 & 540 \\\\\n Model \\#3 & thermogr. survey & 7050 & 2957 & 1996 & 833 \\\\\n \\hline\\end{tabular}\n}\n\\caption{{\\small Geometry and masonries' mechanical characteristics in the models.}}\n\\label{tab:char_models}\n\\end{table}\n\n\nThe behavior under seismic actions is estimated by means of modal analysis, using a commercial code based on the Finite Element Method.\nThe periods of the first three modes and the corresponding mass participating ratio  for the two principal direction of seismic action are reported in Tab.~\\ref{tab:periodi_masse}.\nFor all the models, the first mode is along the $y$ axis, the second is along the $x$ axis, and the third is mainly torsional. \nAnyway, the second mode is not a pure translational one since the asymmetry in the walls distribution produces also a torsional component. \n\n\\begin{table}[h!]\n\\center\n{\\small\n\\begin{tabular}{c|p{0.7cm}p{0.7cm}p{0.7cm}|p{0.7cm}p{0.7cm}p{0.7cm}|p{0.7cm}p{0.7cm}p{0.7cm}}\n\\hline\n& \\multicolumn{3}{|c|}{Periods}\n & \\multicolumn{6}{|c}{Mass partecipating ratio} \\\\\n & \\multicolumn{3}{|c|}{ }\n  & \\multicolumn{3}{|c|}{ $x$ direction} &    \\multicolumn{3}{|c}{$y$ direction} \\\\\n & 1st mode & 2nd mode & 3rd mode  & 1st mode & 2nd mode & 3rd mode  & 1st mode & 2nd mode & 3rd mode\\\\\n\\hline\n Model \\#1 & 0.22 & 0.15 & 0.13 & 0.00 & 0.64 & 0.13 & 0.73 & 0.00 & 0.00 \\\\\n Model \\#2 & 0.22 & 0.14 & 0.13 & 0.00 & 0.51 & 0.27 & 0.73 & 0.01 & 0.00 \\\\\n Model \\#3 & 0.18 & 0.13 & 0.11 & 0.00 & 0.46 & 0.26 & 0.68 & 0.01 & 0.00 \\\\\n \\hline\\end{tabular}\n}\n\\caption{{\\small Periods and mass participating ratio for two directions of seismic action of the first three modes .}}\n\\label{tab:periodi_masse}\n\\end{table}\n\nAs can be noted, the first two models have the same period for the fundamental mode in $y$ direction; nevertheless, in $x$ direction there is a slight difference due to the reduced dimensions of masonry walls width discovered by means of thermographic survey. In fact, the reduction of dimensions leads to a decrease of global stiffness while the total mass is almost the same.  \nThe periods of third model show a reduction of about 20\\% due to  the greater value of equivalent elastic moduli estimated by means of homogenized texture.\nAs already noted, for seismic action in $x$ direction the structure response is dominated by second mode, with also a significant contribution from the third mode (which is torsional); for seismic action in $y$ the response is dominated by first mode.\nIt is also worth noting that Model \\#3 shows a reduced mass participating ratio for the fundamental mode in each direction, and therefore a greater number of modes should be considered in the evaluation of seismic response in order to achieve a suitable accuracy. \n\n\n\n\n\n\n\\section{Concluding remarks}\n\nThe sampling Kantorovich operators and the corresponding MATLAB algorithm for image reconstruction are very useful to enhance the quality of thermographic images of portions of masonry walls. In particular, after the processing by sampling Kantorovich algorithm, the thermographic image has higher definition with respect to the original one and therefore, it was possible to estimate the mechanical characteristics of homogeneous materials equivalent to actual masonries, taking into account the texture (i.e., the arrangement of blocks and mortar joints). These materials were used to model the behavior of a case study under seismic action. This model has been compared with others constructed by well-know methods, using ``naked eyes'' survey and the mechanical parameters for materials taken from the Italian Building Code. Our method based on the processing of thermographic images by sampling Kantorovich operators enhances the quality of the model with respect to that based on visual survey only.\n\n\\noindent In particular the proposed approach allows to overcome some difficulties that arise when dealing with the vulnerability analysis of existing structures, which are: i) the knowledge of the actual geometry of the walls (in particular the identification of hidden doors and windows); ii) the identification of the actual texture of the masonry and the distribution of inclusions and mortar joints, and from this iii) the estimation of the elastic characteristics of the masonry.\nIt is noteworthy that, for item i) the engineer has usually limited knowledge, due to the lack of documentation, while for items ii) and iii) he usually use tables proposed in technical manuals and standards which however give large bounds in order to encompass the generality of the real masonries. Instead, the use of reconstruction techniques on thermographic images coupled with homogenization permits to reduce these latter uncertainty on the estimation of the mechanical characteristics of the masonry.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{1}\n\nEdges play an important role in many vision tasks~\\cite{YangZYPML20,wu2012strong,ramamonjisoa2020predicting,wang2015designing}. While generic edge detection~\\cite{xie2015hed,liu2017rcf,he2019bdcn,wang2017ced} has been extensively studied for decades, specific-edge detection recently attracts increasing amount of efforts due to its practical applications concerning on different types of edges, such as occlusion contours~\\cite{wang2016doc,wang2018doobnet,lu2019ofnet} or semantic boundaries~\\cite{yu2017casenet,dff19}.\n\nIn his seminal work \\cite{marr1982vision}, David Marr summarized four basic ways edges can arise: (1) \\textit{surface-reflectance discontinuity}, (2) \\textit{illumination discontinuity}, (3) \\textit{surface-normal discontinuity}, and (4) \\textit{depth discontinuity}, as shown in Fig.~\\ref{fig:fig1}. Recent studies~\\cite{YangZYPML20,wu2012strong,ramamonjisoa2020predicting,wang2015designing} have shown that the above types of edges are beneficial for downstream tasks. For example, pavement crack detection (reflectance discontinuity) is a critical task for intelligent transportation \\cite{YangZYPML20}; shadow edge (illumination discontinuity) detection is a prerequisite for shadow removal and path detection \\cite{wu2012strong}; \\cite{ramamonjisoa2020predicting} and \\cite{wang2015designing} show that depth edge and normal edge representation prompt refined normal and sharp depth estimation, respectively. Besides, \\cite{KimTO15Joint} utilizes four types of cues simultaneously to improve the performance of depth refinement.\n\nDespite their importance, fine-grained edges are still under-explored, especially when compared with generic edges. Generic edge detectors usually treat edges indistinguishably; while existing studies for specific edges focus on individual edge type. By contrast, the four fundamental types of edges, to our best knowledge, have never been explored in an integrated edge detection framework.\n\n\n\nIn this paper, for the first time, we propose to detect simultaneously the four types of edges, namely \\textit{reflectance edge} (RE), \\textit{illumination edge} (IE), \\textit{normal edge} (NE) and \\textit{depth edge} (DE). Although edges share similar patterns in intensity variation in images, they have different physical bases. Specifically, REs and IEs are mainly related to photometric reasons -- REs are caused by changes in material appearance (\\emph{e.g.}} \\def\\Eg{\\emph{E.g.}, texture and color), while IEs are produced by changes in illumination (\\emph{e.g.}} \\def\\Eg{\\emph{E.g.}, shadows, light sources and highlights). By contrast, NEs and DEs reflect the geometry changes in object surfaces or depth discontinuity. Considering the correlations and distinctions among all types of edges, we develop a CNN-based solution, named \\textit{RINDNet}, for jointly detecting the above four types of edges.\n\nRINDNet works in three stages. In stage \\uppercase\\expandafter{\\romannumeral1}, it extracts general features and spatial cues from a backbone network for all edges. Then, in stage \\uppercase\\expandafter{\\romannumeral2}, it proceeds with four separate decoders. Specifically, low-level features are first integrated under the guidance of high-level hints by Weight Layer (WL), and then fed into RE-Decoder and IE-Decoder to produce features for REs and IEs respectively. At the same time, NE\/DE-Decoder take the high-level features as input and explore effective features. After that, these features and accurate spatial cues are forwarded to four decision heads in stage \\uppercase\\expandafter{\\romannumeral3} to predict the initial results. Finally, the attention maps obtained by the Attention Module (AM), which captures the underlying relations between all types, are aggregated with the initial results to generate the final predictions. All these components are differentiable, making RINDNet an end-to-end architecture to jointly optimize the detection of four types of edges.\n\n\nTraining and evaluating edge detectors for all four types of edges request images with all such edges annotated. In this paper, we create the first known such dataset, named \\textit{BSDS-RIND}, by carefully labeling images from the BSDS~\\cite{arbelaez2010bsds} benchmark (see Fig.~\\ref{fig:fig1}). BSDS-RIND allows the first thorough evaluation of edge detection of all four types. The proposed RINDNet shows clear advantages over previous edge detectors, both quantitatively and qualitatively. The source code, dataset, and benchmark are available at \\url{https:\/\/github.com\/MengyangPu\/RINDNet}.\n\nWith the above efforts, our study is expected to stimulate further research along the line, and benefit more downstream applications with rich edge cues. Our contributions are summarized as follows: (1) We develop a novel end-to-end edge detector, RINDNet, to jointly detect the four types of edges. RINDNet is designed to effectively investigate shared information among different edges (\\emph{e.g.}} \\def\\Eg{\\emph{E.g.}, through feature sharing) and meanwhile flexibly model the distinction between them (\\emph{e.g.}} \\def\\Eg{\\emph{E.g.}, through edge-aware attention). (2) We present the first public benchmark, BSDS-RIND, dedicated to studying simultaneously the four edge types, namely reflectance edge, illumination edge, normal edge and depth edge. (3) In our experiments, the proposed RINDNet shows clear advantages over state of the arts.\n\n\\section{Related Works}\n\\label{2}\n\n\n\\vspace{-2mm}\n\\paragraph{Edge Detection Algorithms.}\nEarly edge detectors~\\cite{kittler1983accuracy,canny1986computational,winnemoller2011xdog} obtain edges based directly on the analysis of image gradients. By contrast, learning-based methods \\cite{martin2004learning,dollar2006supervised,lim2013sketch} exploit different low-level features that respond to characteristic changes, then a classifier is trained to generate edges. CNN-based edge detectors~\\cite{kokkinos2015pushing,deng2020dscd,xu2017AMHNet,shen2015deepcontour,bertasius2015deepedge,bertasius2015hfl,liu2016rds,maninis2016cob,deng2018lpcb,kelm2019rcn,poma2020dexined} do not rely on hand-crafted features and achieve better performance.\nCombining multi-scale and multi-level features, \\cite{xie2015hed,liu2017rcf,poma2020dexined,he2019bdcn} yield outstanding progresses on generic edge detection. A novel refinement architecture is also proposed in \\cite{wang2017ced} using a top-down backward refinement pathway to generate crisp edges.\nRecent works~\\cite{zhen2020joint,acuna2019devil,yu2018simultaneous,yang2016object} pay more attention to special types of edges. In~\\cite{hariharan2011semantic} the generic object detector is combined with bottom-up contours to infer object contours. CASENet~\\cite{yu2017casenet} adopts a nested architecture to address semantic edge detection. For better prediction, DFF~\\cite{dff19} learns adaptive weights to generate specific features of each semantic category. For occlusion boundary detection, DOC~\\cite{wang2016doc} decomposes the task into occlusion edge classification and occlusion orientation regression, then two sub-networks are used to separately perform the above two tasks. DOOBNet~\\cite{wang2018doobnet} uses an encoder-decoder structure to obtain multi-scale and multi-level features, and shares the backbone features with two branches. OFNet~\\cite{lu2019ofnet} considers the relevance and distinction for the occlusion edge and orientation, thus it shares the occlusion cues between two sub-networks.\n\n\n\\vspace{-5mm}\n\\paragraph{Edge Datasets.} \nMany datasets have been proposed for studying edges.\nBSDS \\cite{arbelaez2010bsds} is a popular edge dataset for detecting generic edges containing $500$ RGB natural images. Although each image is annotated by multiple users, they usually pay attention to salient edges related to objects. BIPED~\\cite{mely2016multicue} is created to explore more comprehensive and dense edges, and contains $250$ outdoor images. NYUD~\\cite{silberman2012indoor} contains $1,449$ RGB-D indoor images, and lacks edge types pertaining to outdoor scenes. Significantly, Multicue~\\cite{mely2016multicue} considers the interaction between several visual cues (luminance, color, stereo, motion) during boundary detection.\n\nRecently, SBD \\cite{hariharan2011semantic} is presented for detecting semantic contours, using the images from the PASCAL VOC challenge \\cite{everingham2010pascal}. Cityscapes \\cite{cordts2016cityscapes} provides the object or semantic boundaries focusing on road scenes. For reasoning occlusion relationship between objects, the dataset in \\cite{ren2006figure} consists of $200$ images, where boundaries are assigned with figure\/ground labels. Moreover, PIOD~\\cite{wang2016doc} contains $10,000$ images, each with two annotations: a binary edge map denotes edge pixels and a continuous-valued occlusion orientation map. The recent dataset in~\\cite{ramamonjisoa2020predicting} annotates NYUD test set for evaluating the occlusion boundary reconstruction.\n\n\nOur work is inspired by the above pioneer studies, but makes novel contributions in two aspects: the proposed RINDNet, to the best of our knowledge, is the first edge detector to jointly detect all four types of edges, and the proposed BSDS-RIND is the first benchmark with all four types of edges annotated.\n\n\\section{Problem Formulation and Benchmark}\n\\label{3}\n\n\\subsection{Problem Formulation}\n\\label{3.1}\nLet $X \\in \\mathbb{R}^{3 \\times W \\times H}$ be an input image with ground-truth labels $\\mathcal{E}=\\{E^{r},E^{i},E^{n},E^{d}\\}$, where $E^{r}, E^{i}, E^{n}, E^{d} \\in\\{0,1\\}^{W \\times H}$ are binary edge maps indicating the reflectance edges (REs), illumination edges (IEs), surface-normal edges (NEs) and depth edges (DEs), respectively. Our goal is to generate the final predictions $\\mathcal{Y}=\\{Y^{r}, Y^{i}, Y^{n}, Y^{d}\\}$, where $Y^{r}, Y^{i}, Y^{n}, Y^{d}$ are the edge maps corresponding to REs, IEs, NEs and DEs, respectively. In our work, we aim to learn a CNN-based edge detector $\\psi$: \n$\\mathcal{Y} = \\psi (X)$.\n\nThe training of $\\psi$ can be done over training images by minimizing some loss functions between $\\mathcal{E}$ and $\\mathcal{Y}$. Therefore, a set of images with ground-truth labels are required to learn the mapping $\\psi$. We contribute such annotations in this work, and the detailed processes are shown in \\S\\ref{3.2}.\n\n\n\\subsection{Benchmark}\n\\label{3.2}\nOne aim of our work is to contribute a first public benchmark, named BSDS-RIND, over BSDS images~\\cite{arbelaez2010bsds}. The original images contain various complex scenes, which makes it challenging to jointly detect all four types of edges. Fig.~\\ref{fig:fig1} (Right) shows some examples of our annotations.\n\n\n\\vspace{-3mm}\n\\paragraph{Edge Definitions.}\nIt is critical to define four types of edges for the annotation task. Above all, we give the definition of each type and illustrate it with examples.\n\\begin{itemize}\n    \\vspace{-1.6mm}\\item \\textbf{\\textit{Reflectance Edges} (REs)} usually are caused by the changes in material appearance (\\emph{e.g.}} \\def\\Eg{\\emph{E.g.}, texture and color) of smooth surfaces. Notably, although the edges within paintings in images (see Fig.~\\ref{fig:fig1} (c)) could be classified to DEs by human visual system, these edges are assigned as REs since there is no geometric discontinuity. \n    \\vspace{-1.6mm}\\item \\textbf{\\textit{Illumination Edges} (IEs)} are produced by shadows, light sources, highlights, \\emph{etc.}} \\def\\vs{\\emph{vs.}\\ (as shown in Fig.~\\ref{fig:fig1}).\n    \\vspace{-1.6mm}\\item \\textbf{\\textit{Normal Edges} (NEs)} mark the locations of discontinuities in surface orientation and normally arise between parts. As shown in Fig.~\\ref{fig:fig1} (b), we take the edges between the building and the ground as an example, the change of depth across these two surfaces is continuous but not smooth, which is caused by the surface-normal discontinuity between them.\n    \\vspace{-1.6mm}\\item \\textbf{\\textit{Depth Edges} (DEs)} are resulted by depth discontinuity, and often coincide with object silhouettes. It is difficult to measure the depth difference (\\emph{e.g.}} \\def\\Eg{\\emph{E.g.}, Fig.~\\ref{fig:fig1} (a)), thus the relative depth difference is used to determine whether an edge belongs to DEs. Although there exists the depth changes between windows and walls in the building (Fig.~\\ref{fig:fig1} (b)), the small distance ratio is caused by the long distance between them and camera, so such edges are classified to REs rather than DEs.\n\\end{itemize}\n\n\\vspace{-6.5mm}\n\\paragraph{Annotation Process.}\nThe greatest effort for constructing a high-quality edge dataset is devoted to, not surprisingly, manual labeling, checking, and revision. For this task, we manually construct the annotations using ByLabel~\\cite{qin2018bylabel}.\nTwo annotators collaborate to label each image. One annotator first manually labels the edges, and another annotator checks the result and may supplement missing edges. Those edges with labels are added directly to the final dataset if both annotators agree with each other. Ambiguous edges will be revised by both annotators together: the consistent annotations are given after discussion. After iterating several times, we get the final annotations. Moreover, for some edges that are difficult to determine the main factors of their formation, multiple labels are assigned for them. It is only about 53k (2\\%) pixels with multi-labels in BSDS-RIND. In addition, we use the average Intersection-over-Union (IoU) score to measure agreement between two annotators, and get $0.97$, $0.92$, $0.93$ and $0.95$ for REs, IEs, NEs and DEs, respectively. The statistics show good consistency. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.86\\linewidth, height=.28\\linewidth]{edgeNum.pdf}\n\\caption{The distribution of pixels for each type of edges on BSDS-RIND training set and testing set.}\n\\label{fig:edge_num}\n\\vspace{-13pt}\n\\end{figure}\n\nWith all efforts, finally, a total of $500$ images are carefully annotated, leading to a densely annotated dataset, named BSDS-RIND. Then it is split into $300$ training images, and $200$ testing images, respectively. The total number of pixels for each type on the BSDS-RIND training and testing set are reported in Fig. \\ref{fig:edge_num}. Significantly, the number of edge pixels in BSDS-RIND are twice as in BSDS. Moreover, edge detection is a pixel-wise task, thus the number of samples provided by BSDS-RIND decently supports learning-based algorithms. More examples and details are given in the supplementary material.\n\n\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.95\\linewidth, height=.44\\linewidth]{figure2.pdf}\n\\caption{The three-stage architecture of RINDNet.  \\textbf{Stage \\uppercase\\expandafter{\\romannumeral1}}: the input image is fed into a backbone to extract features shared with all edge types. \\textbf{Stage \\uppercase\\expandafter{\\romannumeral2}}: features across different levels are fused via Weight Layers (WLs), and are forwarded to four decoders in two clusters: RE-Decoder\/IE-Decoder and NE-Decoder\/DE-Decoder. \\textbf{Stage \\uppercase\\expandafter{\\romannumeral3}}: four decision heads predict the four types of initial results. In addition, the attention maps learned by the attention module are integrated into the final prediction  ($A^b$ used only in training). (Best viewed in color)}\n\\label{fig:fig2}\n\\vspace{-13pt}\n\\end{figure*}\n\n\n\n\\section{RINDNet}\n\\label{4}\n\nIn this work, we design an end-to-end network (\\S\\ref{4.1}), named \\textit{RINDNet}, to learn distinctive features for optimizing the detection of four edge types jointly. Fig.~\\ref{fig:fig2} shows an overview of our proposed RINDNet, which includes three stages of initial results inference (\\emph{i.e.}} \\def\\Ie{\\emph{I.e.}, extracting common features, preparing distinctive features, and generating initial results) and final predictions integrated by an Attention Module. We also explain the loss functions and training details in \\S\\ref{4.2} and \\S\\ref{4.3}.\n\n\\subsection{Methodology}\n\\label{4.1}\n\n\\paragraph{Stage \\uppercase\\expandafter{\\romannumeral1}: Extracting Common Features for All Edges.}\nWe first use a backbone to extract common features for all edges because these edges share similar patterns in intensity variation in images.\nThe backbone follows the structure of ResNet-50 \\cite{he2016deepres} which is composed of five repetitive building blocks. Specifically, the feature maps from the above five blocks of ResNet-50 \\cite{he2016deepres} are denoted as $res_1$, $res_2$, $res_3$, $res_4$ and $res_5$, respectively.\n\nThen, we generate spatial cues from the above features. It is well known that different layers of CNN features encode different levels of appearance\/semantic information, and contribute differently to different edge types. Specifically, bottom-layer feature maps $res_{1-3}$ focus more on low-level cues (\\emph{e.g.}} \\def\\Eg{\\emph{E.g.}, color, texture and brightness), while top-layer maps $res_{4-5}$ are in favor of object-aware information. Thus it is beneficial to capture multi-level spatial responses from different layers of feature maps. Given multiple feature maps $res_{1-5}$, we obtain the spatial response maps:\n\\begin{equation}\n    f_{sp}^k = \\psi^{k}_{\\rm sp}(res_k), \\quad k \\in \\{1,2,3,4,5\\}\n\\end{equation}\nwhere the spatial responses $f_{sp}^k \\in \\mathbb{R}^{2 \\times W \\times H}$ are learned by Spatial Layer $\\psi^{k}_{\\rm sp}$ which is composed of one convolution layer and one deconvolution layer.\n\n\n\\vspace{-3mm}\n\\paragraph{Stage \\uppercase\\expandafter{\\romannumeral2}: Preparing Distinctive Features for REs\/IEs and NEs\/DEs.} Afterwards, RINDNet learns particular features for each edge type separately by the corresponding decoder in stage \\uppercase\\expandafter{\\romannumeral2}. Inspired by \\cite{lu2019ofnet}, we design the Decoder with two streams to recover fine location information, as shown in Fig.~\\ref{fig:fig6} (b). Two-stream decoder can work collaboratively and learn more powerful features from different views in the proposed architecture. Although four decoders have the same structure, some special designs are proposed for different types of edges, and we will give the detailed descriptions below. To distinguish each type of edges reasonably and better depict our work, we next cluster the four edge types into two groups, \\emph{i.e.}} \\def\\Ie{\\emph{I.e.}, REs\/IEs and NEs\/DEs, to prepare features for them respectively.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth, height=.7\\linewidth]{components.pdf}\n\\caption{The architectures of (a) Weight Layer, (b) Decoder and (c) Attention Module. $\\odot$ is the element-wise multiplication.}\n\\label{fig:fig6}\n\\vspace{-13pt}\n\\end{figure}\n\n\n\\textbf{\\textit{REs and IEs.}} In practice, the low-level features (\\emph{e.g.}} \\def\\Eg{\\emph{E.g.}, $res_{1-3}$) capture detailed intensity changes that are often reflected in REs and IEs. Besides, REs and IEs are related to the global context and surrounding objects provided by the high-level features (\\emph{e.g.}} \\def\\Eg{\\emph{E.g.}, $res_{5}$). Thus, it is desirable that semantic hints may give the felicitous guidance to aware the intensity changes, before forwarding to the Decoder. Moreover, it is notable that simply concatenating the low-level and high-level features may be too computationally expensive due to the increased number of parameters. We therefore propose the Weight Layer (WL) to adaptively fuse the low-level features and high-level hints in a learnable manner, without increasing the dimension of the features. \n\nAs shown in Fig.~\\ref{fig:fig6} (a), WL contains two paths: the first path receives high-level feature $res_{5}$ to recover high resolution through a deconvolution layer, and then two $3\\times3$ convolution layers with Batch Normalization (BN) and ReLU excavate adaptive semantic hints; another path is implemented as two convolution layers with BN and ReLU, which encodes low-level features $res_{1-3}$. Afterwards, they are fused by element-wise multiplication. Formally, given the low-level features $res_{1-3}$ and high-level hints $res_{5}$, we generate the fusion features for REs and IEs individually,\n\\begin{equation}\n\\begin{array}{ll}\n    g^{r}= \\psi^{r}_{\\rm wl} \\big(res_5, [res_1,res_2,\\rm up (res_3)]\\big), \\\\\n    g^{i}= \\psi^{i}_{\\rm wl} \\big(res_5, [res_1,res_2,\\rm up (res_3)]\\big),\n\\end{array}\n\\end{equation}\nwhere the WL of REs and IEs are indicated as $\\psi^{r}_{\\rm wl}$ and $\\psi^{i}_{\\rm wl}$ respectively, $g^{r}$\/$g^{i}$ are the fusion features for REs\/IEs, and $[\\cdot]$ is the concatenation. Note that, the resolution of $res_{3}$ is smaller than $res_{1}$ and $res_{2}$, so one up-sampling operation $\\rm up(\\cdot)$ is used on $res_{3}$ to increase resolution before feature concatenation.\nNext, the fusion features are fed into the corresponding Decoder to generate specific features with accurate location information separately for IEs and REs,\n\\begin{equation}\n\\begin{array}{ll}\n    f^{r} = \\psi^{r}_{\\rm deco}(g^{r}) , &\n    f^{i} = \\psi^{i}_{\\rm deco}(g^{i}) , \\\\\n\\end{array}\n\\end{equation}\nwhere $\\psi^{r}_{\\rm deco}$ and $\\psi^{i}_{\\rm deco}$ indicate Decode of REs and IEs respectively, and $f^{r}$\/$f^{i}$ are decoded feature maps for REs\/IEs.\n\n\n\\textbf{\\textit{NEs and DEs.}} Since the high-level features (\\emph{e.g.}} \\def\\Eg{\\emph{E.g.}, $res_5$) express strong semantic responses that are usually epitomized in NEs and DEs, we utilize $res_5$ to obtain the particular features for NEs and DEs,\n\\begin{equation}\n\\begin{array}{ll}\n    f^{n} = \\psi^{n}_{\\rm deco}(res_5) , &\n    f^{d} = \\psi^{d}_{\\rm deco}(res_5) ,\\\\\n\\end{array}\n\\end{equation}\nwhere NE-Decode and DE-Decoder are denoted as $\\psi^{n}_{\\rm deco}$ and $\\psi^{d}_{\\rm deco}$ respectively, and $f^{n}$\/$f^{d}$ are the decoded features of NEs\/DEs. Since DEs and NEs commonly share some relevant geometry cues, we share the weights of the second stream of NE-Decoder and DE-Decoder to learn the collaborative geometry cues. At the same time, the first stream of NE-Decoder and DE-Decoder is responsible for learning particular features for REs and DEs, respectively.\n\n\\vspace{-3mm}\n\\paragraph{Stage \\uppercase\\expandafter{\\romannumeral3}: Generating Initial Results.}\nWe predict the initial results for each type of edges by the respective decision head in final stage. The features from previous stages, containing rich location information of edges, can be used to predict edges. Specifically, we concatenate the decoded features $f^{r}$\/$f^{i}$ with spatial cues $f_{sp}^{1-3}$ to predict REs\/IEs,\n\\begin{equation}\n\\begin{array}{ll}\n    O^{r} = \\psi^{r}_{\\rm h}\\big([f^{r}, f_{sp}^{1-3}]\\big) , & \n    O^{i} = \\psi^{i}_{\\rm h}\\big([f^{i}, f_{sp}^{1-3}]\\big) , \n\\end{array}\n\\end{equation}\nwhere $O^{r}$\/$O^{i}$ are the initial predictions of REs\/IEs. The decision heads of REs and IEs, named $\\psi^{r}_{\\rm h}$ and $\\psi^{i}_{\\rm h}$ respectively, are modeled as a $3\\times3$ convolution layer and a $1\\times1$ convolution layer. Note that REs and IEs do not directly rely on the location cues provided by top-layer, thus spatial cues $f_{sp}^{4-5}$ are not used for them. By contrast, all spatial cues $f_{sp}^{1-5}$ are concatenated with the decoded features to generate initial results for NEs and DEs, respectively,\n\\begin{equation}\n\\begin{array}{ll}\n    O^{n} = \\psi^{n}_{\\rm h}\\big([f^{n}, f_{sp}^{1-5}]\\big) , &\n    O^{d} = \\psi^{d}_{\\rm h}\\big([f^{d}, f_{sp}^{1-5}]\\big) , \n\\end{array}\n\\end{equation}\nwhere $\\psi^{n}_{\\rm h}$ and $\\psi^{d}_{\\rm h}$ respectively indicate the decision heads of NEs and DEs, which are composed of three $1 \\times 1$ convolutional layers to integrate hints at each position. In summary, $\\mathcal{O}=\\{O^{r},O^{i},O^{n},O^{d}\\}$ denotes the initial result set.\n\n\n\\begin{table*}\n\\caption{Quantitative comparison for REs, IEs, NEs, DEs and Average (best viewed in color: ``\\textcolor{red}{\\bf{red}}'' for best, and ``\\textcolor{blue}{\\bf{blue}}'' for second best).}\n  \\centering\n  \\small\n  \\renewcommand\\tabcolsep{3.5pt}\n  \\renewcommand\\arraystretch{0.9}\n  \\begin{tabular}{|l|ccc|ccc|ccc|ccc|ccc|}\n    \\hline\n    \\multirow{2}{*}{Method}\n    &\\multicolumn{3}{c|}{Reflectance} & \\multicolumn{3}{c|}{Illumination} & \\multicolumn{3}{c|}{Normal} &\\multicolumn{3}{c|}{Depth} &\\multicolumn{3}{c|}{Average}\\\\  \n    \\cline{2-16}\n                    & ODS   & OIS   & AP       & ODS   & OIS   & AP        & ODS   & OIS   & AP        & ODS   & OIS   & AP       & ODS   & OIS   & AP\\\\\n    \\hline\n    HED~\\cite{xie2015hed}             & 0.412 & 0.466 & 0.343    & 0.256 & 0.290 & 0.167     & 0.457 & 0.505 & \\bf{\\textcolor{blue}{0.395}}     & 0.644 & 0.679  & 0.667    & 0.442 & 0.485 & \\bf{\\textcolor{blue}{0.393}}\\\\\n    CED~\\cite{wang2017ced}                      & 0.429 & 0.473 & 0.361\t & 0.228 & 0.286 & 0.118     & 0.463 & 0.501 & 0.372    & 0.626\t& 0.655\t& 0.620      & 0.437 & 0.479 & 0.368\\\\\n    RCF~\\cite{liu2017rcf}             & 0.429 & 0.448 & 0.351    & 0.257 & 0.283 & \\bf{\\textcolor{red}{0.173}}     & 0.444 & 0.503 & 0.362     & 0.648 & 0.679  & 0.659    & 0.445 & 0.478 & 0.386\\\\\n    BDCN~\\cite{he2019bdcn}            & 0.358 & 0.458 & 0.252    & 0.151 & 0.219 & 0.078     & 0.427 & 0.484 & 0.334     & 0.628 & 0.661  & 0.581    & 0.391 & 0.456 & 0.311\\\\\n    DexiNed~\\cite{poma2020dexined}                   & 0.402 & 0.454 & 0.315\t & 0.157 & 0.199 & 0.082     & 0.444 & 0.486 & 0.364     & 0.637 & 0.673  & 0.645   & 0.410 & 0.453 & 0.352\\\\\n    CASENet~\\cite{yu2017casenet}      & 0.384 & 0.439 & 0.275    & 0.230 & 0.273 & 0.119     & 0.434 & 0.477 & 0.327     & 0.621 & 0.651  & 0.574    & 0.417 & 0.460 & 0.324\\\\\n    DFF~\\cite{dff19}                  & \\bf{\\textcolor{blue}{0.447}} & 0.495 & 0.324    & \\bf{\\textcolor{red}{0.290}} & \\bf{\\textcolor{red}{0.337}} & 0.151     & \\bf{\\textcolor{blue}{0.479}} & \\bf{\\textcolor{blue}{0.512}} & 0.352     & \\bf{\\textcolor{blue}{0.674}} & \\bf{\\textcolor{blue}{0.699}}  & 0.626    & \\bf{\\textcolor{blue}{0.473}} & \\bf{\\textcolor{blue}{0.511}} & 0.363\\\\\n    *DeepLabV3+~\\cite{chen2018deeplabv3}    & 0.297 & 0.338 & 0.165    & 0.103 & 0.150 & 0.049     & 0.366 & 0.398 & 0.232     & 0.535 & 0.579  & 0.449    & 0.325 & 0.366 & 0.224\\\\\n    *DOOBNet~\\cite{wang2018doobnet}         & 0.431 & 0.489 & 0.370    & 0.143 & 0.210 & 0.069     & 0.442 & 0.490 & 0.339     & 0.658 & 0.689  & 0.662    & 0.419 & 0.470 & 0.360\\\\\n    *OFNet~\\cite{lu2019ofnet}         & 0.446 & 0.483 & \\bf{\\textcolor{blue}{0.375}}    & 0.147 & 0.207 & 0.071     & 0.439 & 0.478 & 0.325     & 0.656 & 0.683  & \\bf{\\textcolor{blue}{0.668}}    & 0.422 & 0.463 & 0.360\\\\\n    \\hline\n    DeepLabV3+~\\cite{chen2018deeplabv3}     & 0.444 & 0.487 & 0.356    & 0.241 & \\bf{\\textcolor{blue}{0.291}} & 0.148     & 0.456 & 0.495 & 0.368     & 0.644 &0.671  & 0.617    & 0.446 & 0.486 & 0.372\\\\\n    DOOBNet~\\cite{wang2018doobnet}    & 0.446 & \\bf{\\textcolor{blue}{0.503}} & 0.355   & 0.228 & 0.272 & 0.132     & 0.465 & 0.499 & 0.373     & 0.661 & 0.691 & 0.643     & 0.450 & 0.491 & 0.376\\\\\n    OFNet~\\cite{lu2019ofnet}          & 0.437 & 0.483 & 0.351   & 0.247 & 0.277 & 0.150     & 0.468 & 0.498 & 0.382     & 0.661 & 0.687 & 0.637     & 0.453 & 0.486 & 0.380\\\\\n    \n    \\hline\n    RINDNet (Ours) & \\bf{\\textcolor{red}{0.478}} & \\bf{\\textcolor{red}{0.521}} & \\bf{\\textcolor{red}{0.414}}    &\\bf{\\textcolor{blue}{0.280}} &\\bf{\\textcolor{red}{0.337}} &\\bf{\\textcolor{blue}{0.168}}      &\\bf{\\textcolor{red}{0.489}} &\\bf{\\textcolor{red}{0.522}} &\\bf{\\textcolor{red}{0.440}}     &\\bf{\\textcolor{red}{0.697}} &\\bf{\\textcolor{red}{0.724}} &\\bf{\\textcolor{red}{0.705}}      &\\bf{\\textcolor{red}{0.486}} &\\bf{\\textcolor{red}{0.526}} &\\bf{\\textcolor{red}{0.432}}\\\\\n    \\hline\n  \\end{tabular}\n  \\label{tab:tab1}\n\\vspace{-8pt}\n\\end{table*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.99\\linewidth, height=.20\\linewidth\n]{eval.pdf}\n\\caption{Evaluation results on BSDS-RIND for (a) REs, (b) IEs, (c) NEs, (d) DEs and (e) generic edges.}\n\\label{fig:fig3}\n\\vspace{-8pt}\n\\end{figure*}\n\n\\vspace{-3mm}\n\\paragraph{Attention Module.} Finally, RINDNet integrates initial results with attention maps obtained by Attention Module (AM) to generate the final results. Since different types of edges are reflected in different locations, when predicting each type of edges, it is necessary to pay more attention to the related locations. Fortunately, the edge annotations provide the label of each location. Accordingly, the proposed AM could infer the spatial relationships between multiple labels with pixel-wise supervision by the attention mechanism. The attention maps could be used to activate the responses of related locations. Formally, given the input image $X$, AM learns spatial attention maps,\n\\begin{equation} \\label{eq:eq4.4}\n    \\mathcal{A} =\\{A^{b},A^{r},A^{i},A^{n},A^{d}\\} ={\\rm softmax} \\big(\\psi_{\\rm att}(X) \\big),\\\\\n\\end{equation}\nwhere $\\mathcal{A}$ is the normalized attention maps by a softmax function, and $A^{b},A^{r},A^{i},A^{n},A^{d} \\in [0, 1]^{W \\times H}$ are the attention maps corresponding to background, REs, IEs, NEs and DEs respectively. Obviously, if a label is tagged to one pixel, the location of this pixel should be assigned with higher attention values. The AM $\\psi_{\\rm att}$ is achieved by the first building block of ResNet, four $3 \\times 3$ convolution layers (each layer is followed by ReLU and BN operations), and one $1 \\times 1$ convolution layers, as shown in Fig. \\ref{fig:fig6} (c). \n\nFinally, the initial results are integrated with the attention maps to generate the final results $\\mathcal{Y}$,\n\\begin{equation} \n    \\mathcal{Y} = {\\rm sigmoid}\\big(\\mathcal{O} \\odot (1 + A^{\\{r,i,n,d\\}})\\big),\n\\end{equation}\nwhere $\\odot$ is the element-wise multiplication.\n\n\n\n\\subsection{Loss Function}\n\\label{4.2}\n\\paragraph{Edge Loss.}\nWe use the loss function presented in \\cite{wang2018doobnet} to supervise the training of our edge predictions:\n\\begin{equation}\n\\mathcal{L}_{\\rm e}(\\mathcal{Y},\\mathcal{E})= \\sum_{k \\in {\\left\\{r,i,n,d\\right\\}}} \\ell_{\\rm e} \\left (Y^{k},E^{k} \\right),\n\\end{equation}\n\\begin{equation}\\label{eq10}\n\\begin{split}\\ell_{\\rm e} \\left (Y,E \\right)  & = - \\sum_{i,j} \\Big(   E_{i,j}\\alpha_{1}\\beta^{(1-Y_{i,j})^{\\gamma_{1}}}{\\rm log}(Y_{i,j}) \\\\\n    &  + (1-E_{i,j})(1-\\alpha_{1})\\beta^{Y^{\\gamma_{1}}}{\\rm log}(1-Y_{i,j}) \\Big), \n\\end{split}\n\\end{equation}\nwhere $\\mathcal{Y}=\\{Y^{r}, Y^{i}, Y^{n}, Y^{d}\\}$ is the final prediction, $\\mathcal{E} = \\{E^{r},E^{i},E^{n},E^{d}\\}$ is the corresponding ground-truth label, and $E_{i,j}$\/$Y_{i,j}$ are the $(i,j)^{th}$ element of matrix $E$\/$Y$ respectively.\nMoreover, $\\alpha_{1}=|E_{-}|\/|E|$ and $1-\\alpha_{1}=|E_{+}|\/|E|$, where $E_{-}$ and $E_{+}$ denote the non-edge and edge ground truth label sets, respectively. In addition, $\\gamma_{1}$ and $\\beta$ are the hyperparameters. We drop the superscript $k$ in Eq. \\ref{eq10} and Eq. \\ref{eq13} for simplicity.\n\n\n\\vspace{-5mm}\n\\paragraph{Attention Module Loss.}\nSince the pixel-wise edge annotations provide spatial labels, it is easy to obtain the ground truth of attention.\nLet $\\mathcal{T} = \\{T^{b},T^{r},T^{i},T^{n},T^{d}\\}$ be the ground-truth label of attention, where $T^{b}$ specifies the non-edge pixels. $T^{b}_{i,j} = 1$ if the $(i,j)^{th}$ pixel is located on non-edge\/background, otherwise $T^{b}_{i,j} = 0$.\n$T^{r},T^{i},T^{n},T^{d}$ indicate attention labels of REs, IEs, NEs and DEs respectively, which are obtained from $\\mathcal{E} = \\{E^{r},E^{i},E^{n},E^{d}\\}$,\n\\begin{equation}\n\\label{eq:att_gt}\nT_{i,j}^k = \\left\\{\\begin{array}{ll}\n    E_{i,j}^k, & {\\rm if} \\sum_{k} E_{i,j}^k =1, k\\in\\{r,i,n,d\\} \\\\\n    255, & {\\rm if} \\sum_{k} E_{i,j}^k > 1, k\\in\\{r,i,n,d\\} \\\\\n  \\end{array} \\right.,\n\\end{equation}\nwhere $k$ denotes the type of edges, $T_{i,j}^k$ and $E_{i,j}^k$ indicate the attention label and edge label of the $(i,j)^{th}$ pixel, respectively. The attention label equals the edge label if one pixel is only assigned one type of edge label, or it will be tagged 255 that will be ignored during training if one pixel has multiple types. It is should be noted that multi-labeled edges are used when training four decision heads for each type of edges, and only excluded when training the AM. The loss function $\\mathcal{L}_{\\rm att}$ of the AM is formulated as:\n\\begin{equation}\n\\mathcal{L}_{\\rm att}(\\mathcal{A},\\mathcal{T}) = \n\\sum_{k \\in {\\left\\{b,r,i,n,d\\right\\}}} \\ell_{\\rm foc} \\left (A^{k},T^{k} \\right),\n\\end{equation}\n\\begin{equation}\\label{eq13}\n\\begin{split}\n\\ell_{\\rm foc} &\\left (A,T \\right)  = - \\sum_{i,j} \\big( T_{i,j}\\alpha_{2}(1-A_{i,j})^{\\gamma_{2}} {\\rm log}(A_{i,j}) \\\\\n& + (1-T_{i,j})(1-\\alpha_{2})A_{i,j}^{\\gamma_{2}} {\\rm log}(1-A_{i,j})\\big),\n\\end{split}\n\\end{equation}\nwhere $\\ell_{\\rm foc}$ indicates the Focal Loss \\cite{lin2017focal} and $\\mathcal{A}$ is the output of Attention Module. Note that $\\alpha_{2}$ and $\\gamma_{2}$ are a balancing weight and a focusing parameter, respectively.\n\n\\textbf{Total Loss.} Finally, we optimize RINDNet by minimizing the total loss defined as:\n\\begin{equation}\n\\label{eq:eq4}\n\\mathcal{L} = \\lambda \\mathcal{L}_{\\rm e} + (1-\\lambda) \\mathcal{L}_{\\rm att} ,\n\\end{equation}\nwhere $\\lambda$ is the weight for balancing the two losses.\n\n\n\n\\begin{table*}\n  \\caption{Ablation study to verify the effectiveness of each component in our proposed RINDNet.}\n  \\label{tab:wlam}\n  \\centering\n  \\small\n  \\renewcommand\\tabcolsep{3.7pt}\n  \\renewcommand\\arraystretch{0.9}\n  \\begin{tabular}{|l|ccc|ccc|ccc|ccc|ccc|}\n    \\hline\n    \\multirow{2}{*}{Method}\n    &\\multicolumn{3}{c|}{Reflectance} & \\multicolumn{3}{c|}{Illumination} & \\multicolumn{3}{c|}{Normal} &\\multicolumn{3}{c|}{Depth} &\\multicolumn{3}{c|}{Average}\\\\  \n    \\cline{2-16}\n                        & ODS   & OIS   & AP       & ODS   & OIS   & AP         & ODS   & OIS   & AP        & ODS   & OIS   & AP    & ODS   & OIS   & AP\\\\\n    \\hline\n    Ours             & 0.478 & 0.521 & 0.414     & 0.280 & 0.337 & 0.168     & 0.489 & 0.522 & 0.440     & 0.697 & 0.724 & 0.705     & 0.486 & 0.526 & 0.432\\\\\n    Ours w\/o WL    & 0.422 & 0.468 & 0.357    & 0.280 & 0.321 & 0.180      & 0.476 & 0.515 & 0.425     & 0.693 & 0.713 & 0.700 & 0.468 & 0.504 & 0.416\\\\\n    Ours w\/o AM   & 0.443 & 0.494 & 0.338    & 0.268 & 0.327 & 0.139      & 0.473 & 0.506 & 0.378     & 0.670 & 0.699 & 0.649 & 0.464 & 0.507 & 0.376\\\\   \n    Ours w\/o AM\\&WL & 0.409 & 0.460 & 0.316  & 0.277 & 0.331 & 0.178      & 0.471 & 0.507 & 0.389     & 0.677 & 0.707 & 0.662 & 0.459 & 0.501 & 0.386\\\\\n    \\hline\n  \\end{tabular}\n  \\vspace{-13pt}\n\\end{table*}\n\n\n\\subsection{Training Details}\n\\label{4.3}\nOur network is implemented using PyTorch \\cite{paszke2017automatic} and finetuned from a ResNet-50 model pre-trained on ImageNet \\cite{DengDSLL009}. Specifically, we adopt the Stochastic Gradient Descent optimizer with momentum=$0.9$, initial learning rate=$10^{-5}$, and we decay it by the ``poly'' policy on every epoch. We train the model for $70$ epochs on one GPU with a batch size of $4$. Moreover, we set $\\beta=4$ and $\\gamma_{1}=0.5$ for $\\mathcal{L}_{\\rm e}$; $\\alpha_{2}=0.5$ and $\\gamma_{2}=2$ for $\\mathcal{L}_{\\rm att}$; and $\\lambda=0.1$ for total loss. In addition, following \\cite{wang2018doobnet}, we augment our dataset by rotating each image by four different angles of $\\{0,90,180,270\\}$ degrees. Each image is randomly cropped to $320\\times320$ during training while retaining the original size during testing.\n\n\n\\section{Experiment Evaluation}\n\\label{5}\nWe compare our model with $10$ state-of-the-art edge detectors. HED \\cite{xie2015hed}, RCF \\cite{liu2017rcf}, CED \\cite{wang2017ced}, DexiNed \\cite{poma2020dexined}, and BDCN \\cite{he2019bdcn} exhibit excellent performance in general edge detection; DeepLabV3+ \\cite{chen2018deeplabv3}, CASENet \\cite{yu2017casenet} and DFF \\cite{dff19} show outstanding accuracy on semantic edge detection; DOOBNet \\cite{wang2018doobnet} and OFNet \\cite{lu2019ofnet} yield competitive results for occlusion edge detection. \nAll models are trained on $300$ training images and evaluated on $200$ test images. In addition, more qualitative results are provided in the supplementary material. \nWe evaluate these models with three metrics introduced by \\cite{arbelaez2010bsds}: fixed contour threshold (ODS), per-image best threshold (OIS), and average precision (AP). Moreover, a non-maximum suppression \\cite{canny1986computational} is performed on the predicted edge maps before evaluation.\n\n\n\n\\subsection{Experiments on Four Types of Edges}\n\\vspace{-2mm}\n\n\\paragraph{Comparison with State of the Arts.} To adapt existing detectors for four edge types simultaneously, they are modified in two ways: (1) The output $Y \\!\\in\\! \\{0,1\\}^{W \\times H}$ is changed to $\\mathcal{Y} \\!\\in\\! \\{0,1\\}^{4 \\times W \\times H}$. In particular, for DeepLabV3+ focusing on segmentation, the output layer of DeepLabV3+ is replaced by an edge path (same as DOOBNet~\\cite{wang2018doobnet} and OFNet~\\cite{lu2019ofnet}, containing a sequence of four $3\\!\\times\\!3$ convolution blocks and one $1\\!\\times\\!1$ convolution layer) to predict edge maps. As shown in Table \\ref{tab:tab1}, ten compared models are symbolized as HED, RCF, CED, DexiNed, BDCN, CASENet, DFF, *DeepLabV3+, *DOOBNet and *OFNet, respectively. (2) DeepLabV3+, DOOBNet and OFNet only provide one edge prediction branch without structure suitable for multi-class predictions, thus we provide the second modification: the last edge prediction branch is expanded to four, and each branch predicts one type of edges. The modification is similar to the prediction approach of our model and aims to explore the capabilities of these models. They are symbolized as DeepLabV3+, DOOBNet, and OFNet, respectively.\n\n\nTable~\\ref{tab:tab1} and Fig.~\\ref{fig:fig3} present the F-measure of the four types of edges and their averages. We observe that the proposed RINDNet outperforms other detectors over most metrics across the dataset. Essentially, \\cite{xie2015hed,liu2017rcf,wang2017ced,he2019bdcn,poma2020dexined,wang2018doobnet,lu2019ofnet} are designed for generic edge detection, so the specific features of different edges are not fully explored. Even if we extend the prediction branch of OFNet~\\cite{lu2019ofnet}, DOOBNet~\\cite{wang2018doobnet} and DeepLabV3+~\\cite{chen2018deeplabv3} to four for learning specific features respectively, the results are still unsatisfactory. DFF~\\cite{dff19} could learn the specific cues in a certain extent by introducing the dynamic feature fusion strategy, but the performance is still limited. On the contrary, our proposed RINDNet achieves promising results by extracting the corresponding distinctive features based on different edge attributes.\n\n\n\\begin{table}\n\\caption{Ablation study on the choices of features or spatial cues from different layers for the proposed RINDNet.}\n\\centering\n\\small\n\\renewcommand\\tabcolsep{4.9pt}\n\\renewcommand\\arraystretch{1.0}\n\\begin{tabular}{|c|c|c|ccc|}\n    \\hline\n    \\multirow{2}{*}{Reference} & \\multirow{2}{*}{RE\\&IE} & \\multirow{2}{*}{NE\\&DE} &\\multicolumn{3}{c|}{Average} \\\\\n    \\cline{4-6}\n            &        &                 & ODS   & OIS   & AP\\\\\n    \\hline\n    \\hline\n    \\multirow{4}{*}{\\makecell[c]{ Different-Layer \\\\ Features \\\\}} \n    & $res_{1-3}$  & $res_{5}$  & 0.486 & 0.526 & 0.432\\\\\n    & $res_{1-3}$  & $res_{1-3}$  & 0.467 & 0.499 & 0.422\\\\\n    & $res_{5}$  & $res_{1-3}$  & 0.452 & 0.482 & 0.381\\\\   \n    & $res_{5}$  & $res_{5}$  & 0.464 & 0.489 & 0.396\\\\\n    \\hline\n    \\hline\n    \\multirow{4}{*}{Spatial Cues} \n    & $f_{sp}^{1-3}$  & $f_{sp}^{1-5}$  & 0.486 & 0.526 & 0.432\\\\\n    & $f_{sp}^{1-3}$  & $f_{sp}^{1-3}$  & 0.472 & 0.504 & 0.416\\\\\n    & $f_{sp}^{1-5}$  & $f_{sp}^{1-5}$  & 0.478 & 0.512 & 0.418\\\\\n    & w\/o $f_{sp}$     & w\/o $f_{sp}$     & 0.478 & 0.516 & 0.420\\\\\n    \\hline\n\\end{tabular}\n\\label{tab:sp}\n\\vspace{-8pt}\n\\end{table}\n\n\\begin{table}\n\\caption{Ablation study to verify the effectiveness of Decoder for the proposed RINDNet. SW refers to ``Share Weight'', $1^{st}$ and $2^{nd}$ refer to the first stream and the second stream in Decoder.}\n\\centering\n\\small\n\\renewcommand\\tabcolsep{5.0pt}\n\\renewcommand\\arraystretch{1.1}\n\\begin{tabular}{|cc|cc|ccc|}\n    \\hline\n    \\multicolumn{2}{|c|}{RE\\&IE-Decoder} &\\multicolumn{2}{c|}{NE\\&DE-Decoder} &\\multicolumn{3}{c|}{Average}\\\\\n    \\cline{1-7}\n    $1^{st}$ & $2^{nd}$ & $1^{st}$ & $2^{nd}$ & ODS   & OIS   & AP\\\\\n    \\hline\n    $\\surd$ & $\\surd$ & $\\surd$ & w SW  & \\bf{0.486} & \\bf{0.526} & \\bf{0.432}\\\\\n    $\\surd$ & $\\times$ & $\\surd$ & $\\times$  & 0.457 & 0.492 & 0.398\\\\\n    $\\surd$ & $\\times$ & $\\surd$ & w SW  & 0.476 & 0.514 & 0.415\\\\\n    $\\surd$ & $\\surd$ & $\\surd$ & w\/o SW     & 0.474 & 0.517 & 0.408\\\\\n    \\hline\n\\end{tabular}\n\\label{tab:decoder}\n\\vspace{-13pt}\n\\end{table}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.975\\linewidth]{test.pdf}\n\\caption{Qualitative comparison for (a) Reflectance Edges, (b) Illumination Edges, (c) Normal Edges, (d) Depth Edges and (e) Generic Edges (best viewed in color: ``\\textcolor{green2}{\\textbf{green}}'' for true positive, ``\\textcolor{red}{\\textbf{red}}'' for false negative (missing edges), and ``\\textcolor{yellow}{\\textbf{yellow}}'' for false positive). We provide visualization results of the top 6 scores in this figure, more results can be found in the \\textbf{supplementary material}.}\n\\label{fig:fig4}\n\\vspace{-13pt}\n\\end{figure*}\n\n\\vspace{-3mm}\n\\paragraph{Ablation Study.}\nWe first conduct the ablation study to verify the role of Weight Layer (WL) and Attention Module (AM) in RINDNet. In the experiments, each module is removed separately or together to construct multiple variants for evaluation, as shown in Table~\\ref{tab:wlam} (rows 2 -- 4). Intuitively, WL plays a significant role for REs and IEs. Especially for REs, in terms of ODS, OIS and AP, the WL improves the performance significantly from $42.2\\%$, $46.8\\%$ and $35.7\\%$ to $47.8\\%$, $52.1\\%$ and $41.4\\%$, respectively. Besides, with AM, RINDNet achieves noticeable improvements for all types of edges, as shown in row 1 and row 3 of Table~\\ref{tab:wlam}. This illustrates the effectiveness of the proposed AM for capturing the distinctions between different edges. Overall, the cooperation of WL and AM allows RINDNet to successfully capture the specific features of each type of edges and thus delivers remarkable performance gain.\n\nNext, we perform an experiment to verify the effectiveness of different-layer features for detecting REs\/IEs and NEs\/DEs. As shown in Table~\\ref{tab:sp} (rows 1 -- 4), the combination based on edge attributes (row 1) performs better than using other choices (rows 3 -- 4). Besides, we also explore the impact of choosing different spatial cues for REs\/IEs and NEs\/DEs, and report the quantitative results in Table~\\ref{tab:sp} (rows 5 -- 7). Similarly, there are average performance drops in different combinations of spatial cues (rows 6 -- 7). Basically, ours (row 5) is the reasonable combination that helps achieve the best performance.\n\nWe also design careful ablation experiments (Table \\ref{tab:decoder}) to study the effectiveness of each stream in Decoder. The two-stream design combined with share weight (referred to as SW) for NEs\/DEs performs better together (row 1) than using either of them separately (rows 3 -- 4). More detailed results are provided in the supplementary material.\n\n\n\\subsection{Experiments on Generic Edges}\n\nTo fully examine the performance of RINDNet, we modify and test it on generic edges setting (the ground truth of four types of edges are merged to one ground-truth edge map). The outputs of four decision heads are combined and fed into a final decision head $\\psi_{\\rm e}$ (one $1\\times1$ convolution layer) to predict generic edges: $P=\\psi_{\\rm e}\\big([Y^{r}, Y^{i}, Y^{n}, Y^{d}]\\big)$. Moreover, the original ground-truth labels of Attention Module (AM) are unavailable in this setting. Thus we take ground-truth labels of generic edges as the supervisions of AM, so that AM could capture the location of generic edges.\n\n\\begin{table}\n\\caption{Comparison of generic edge detection on \\textbf{BSDS-RIND}.\n}\n\\centering\n\\small\n\\renewcommand\\tabcolsep{14pt}\n\\renewcommand\\arraystretch{0.9}\n\\begin{tabular}{|l|ccc|}\n    \\hline\n    Method                              & ODS   & OIS   & AP       \\\\\n    \\hline\n    HED~\\cite{xie2015hed}               & 0.786 & 0.805 &\\bf{\\textcolor{red}{0.834}} \\\\\n    CED~\\cite{wang2017ced}              & \\bf{\\textcolor{red}{0.801}} & \\bf{\\textcolor{red}{0.814}} & \\bf{\\textcolor{blue}{0.824}} \\\\\n    RCF~\\cite{liu2017rcf}               & 0.771 & 0.791 & 0.800 \\\\\n    BDCN~\\cite{he2019bdcn}              & 0.789 & 0.803 & 0.757 \\\\\n    DexiNed~\\cite{poma2020dexined}      & 0.789\t& 0.805\t& 0.816 \\\\\n    CASENet~\\cite{yu2017casenet}        & 0.792 & 0.806 & 0.786 \\\\\n    DFF~\\cite{dff19}                    & 0.794 & 0.806 & 0.767 \\\\\n    DeepLabV3+~\\cite{chen2018deeplabv3} & 0.780 & 0.792 & 0.776 \\\\\n    DOOBNet~\\cite{wang2018doobnet}      & 0.790 & 0.805 & 0.809 \\\\\n    OFNet~\\cite{lu2019ofnet}            & 0.794 & 0.807 & 0.800 \\\\\n    \\hline\n    RINDNet (Ours)                      &\\bf{\\textcolor{blue}{0.800}} &\\bf{\\textcolor{blue}{0.811}} & 0.815 \\\\\n    \\hline\n\\end{tabular}\n\\label{tab:generic}\n\\vspace{-12pt}\n\\end{table}\n\nWe report the quantitative results over generic edges in Table~\\ref{tab:generic} and Fig.~\\ref{fig:fig3} (e). Note that CED is pre-trained on HED-BSDS. In contrast, our model is trained from scratch and still achieves competitive results. This confirms the integration capability of our model, especially considering that RINDNet is not specially designed for generic edges. Some qualitative results on BSDS-RIND are shown in the last row in Fig. \\ref{fig:fig4}.\n\n\\section{Conclusions}\n\\label{6}\n\nIn this work, we study edge detection on four types of edges including \\textit{Reflectance Edges}, \\textit{Illumination Edges}, \\textit{Normal Edges} and \\textit{Depth Edges}. We propose a novel edge detector RINDNet that, for the first time, simultaneously detects all four types of edges. In addition, we contribute the first public benchmark with four types of edges carefully annotated. Experimental results illustrate that RINDNet yields promising results in comparison with state-of-the-art edge detection algorithms.\n\n\\vspace{-4mm}\n\\paragraph{Acknowledgements.}\nThis work is supported by Fundamental Research Funds for the Central Universities (2019JBZ104) and National Natural Science Foundation of China (61906013, 51827813). The work is partially done while Mengyang Pu was with Stony Brook University.\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nOne of the important observations in the LHC era is that the self-normalised production of hadrons increases faster than linearly as a function of the self-normalised charged-particle multiplicity \\cite{aliceHFhadrons}. This enhancement in high-multiplicity events is not fully understood, but calculations including multiple parton interactions (MPI) and colour reconnection (CR) are able to reproduce the trend. The measurement of the multiplicity-dependent production of the \\W boson, which is colourless, with associated hadron in proton--proton (pp) collisions can then provide further understanding of the MPI and CR mechanisms.\n\nIn heavy-ion collisions (HIC), electroweak bosons are valuable probes of the initial phase of the collision, on which a precise knowledge is required to disentangle QGP-induced phenomena from other nuclear effects. Being hard processes, the production of the \\W and \\Z bosons is highly sensitive to the initial state, especially the modifications of the Parton Distribution Functions (PDF) in the nucleus with respect to that in the proton. The bosons decay before the typical time of creation of the QGP and can be measured via their leptonic decays, providing a final state insensitive to the strong force. The whole process is then medium-blind, carrying the information from the initial state to the detector where it can be recorded. \\\\\n\nThe measurements presented here were performed using data collected with the ALICE detector~\\cite{alice}. In pp collisions at \\thirteen, electrons from \\W-boson decays were measured at midrapidity, in the interval $|y| < 0.6$. Electrons coming from \\W-boson decays typically have a large transverse momentum (\\pt) and are well isolated from other particles. They were thus identified by considering electrons in the range $30 < \\pt^{\\rm e} < 60$~\\GeVc, then defining an isolation cone surrounding the electron in which the total energy is requested to be below a certain threshold. The associated hadrons, produced together with \\W bosons, were detected away-side in azimuth with respect to the electron coming from the decay of the boson, applying a lower minimum \\pt selection of 5~\\GeVc.\n\nMeasurements in HIC were performed in proton--lead (p--Pb) collisions at \\eightnn and lead--lead (Pb--Pb) collisions at \\fivenn. The data were collected from muon-triggered events in the muon forward spectrometer, covering the rapidity interval $2.5 < y < 4$. The \\Z-boson candidates were reconstructed by combining high-\\pt muons ($\\pt^\\mu > 20$ \\GeVc) in pairs of opposite charge, considering the pairs with an invariant mass in the range $60 < m_{\\mu^+\\mu^-} < 120$ \\GeVmass. The number of \\Wminus and \\Wplus bosons was extracted via a template fit to the single-muon \\pt distribution, accounting for the various contributions to the inclusive spectrum. As the low-\\pt region features a very low signal-to-background ratio, the measurements were performed on muons with \\pt > 10 \\GeVc.\n\nThe available measurements of electroweak bosons performed by the ALICE Collaboration at forward rapidity can be found in Refs.~\\cite{aliceWZ,aliceZ,aliceW}.\n\n\\section{Results}\n\\label{sec:results}\n\n\\subsection{Measurements in pp collisions}\n\\label{sec:pp}\n\nFigure~\\ref{fig:pp} shows the self-normalised multiplicity-dependent yield of electrons from \\W boson decays (combining \\Wminus and \\Wplus for increased precision), and that of hadrons associated with a \\W boson, measured in pp collisions at \\thirteen. The yield of electrons from \\W-boson decays is consistent with a linear increase as a function of the charged-particle multiplicity, while the yield of associated hadron shows a faster-than-linear increase. This measurement thus suggests a significant correlation of the associated hadron production with the event multiplicity, and the trend is well reproduced by PYTHIA~8~\\cite{pythia8} calculations including MPI and CR mechanisms. This analysis also constitutes the first measurement of electroweak bosons at midrapidity with ALICE, and will serve as reference for similar measurements in HIC.\n\n\\begin{figure}[h]\n    \\centering\n    \\sidecaption\n    \\includegraphics[width=0.5\\linewidth,clip]{figures\/pp13tev_multiplicity.pdf}\n    \\caption{Self-normalised yield of electron from \\W decays (red), and hadron associated with \\W bosons (blue), as a function of the charged-particle multiplicity. The coloured line correspond to calculations with PYTHIA~8~\\cite{pythia8} including MPI and CR.}\n    \\label{fig:pp}\n\\end{figure}\n\n\\subsection{Measurements in p--Pb collisions}\n\\label{sec:ppb}\n\nThe measurements of the \\Z and \\Wplus bosons as a function of rapidity in p--Pb collisions at \\eightnn are shown in the left and right panels of Fig.~\\ref{fig:ppb}, respectively. The larger number of events available for the \\Wplus boson allowed splitting the acceptance into several rapidity intervals. The \\Z-boson measurement is compared with predictions from the EPPS16~\\cite{epps16} and nCTEQ15~\\cite{ncteq15} nuclear PDFs (nPDFs), as well as predictions from the CT14~\\cite{ct14} free PDF accounting for the isospin effect but without nuclear modifications. The models are in good agreement with the measurement, although no strong conclusion can be derived on the nuclear modifications due to statistical limitations.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.43\\linewidth,clip]{figures\/pPb8tev_Z.pdf}\n    \\includegraphics[width=0.55\\linewidth,clip]{figures\/pPb8tev_W_xSecWplusDiff.pdf}\n    \\caption{\\textbf{Left}: cross section of dimuons from \\Z decays measured in p--Pb collisions at \\eightnn, compared with theoretical calculations with and without nuclear modifications of the PDF. \\textbf{Right}: cross section of muons from \\Wplus decays in the same collisions systems, compared with model predictions from various PDF and nPDF sets.}\n    \\label{fig:ppb}\n\\end{figure}\n\nThe \\Wplus cross section shown in the right panel of Fig.~\\ref{fig:ppb} is compared with the same models, as well as two recent nPDF sets: nCTEQ15WZ~\\cite{ncteq15wz}, an improvement of nCTEQ15 with the addition of electroweak-boson measurements from the LHC for the nPDF determination; and nNNPDF2.0~\\cite{nnnpdf2}, a new model obtained following a methodology based on machine learning. The models including nuclear modifications are in agreement with one another, and provide a good description of the data. Calculations without nuclear modifications, on the contrary, overestimate the production at large positive rapidity. The resulting 3.5$\\sigma$ deviation is the strongest observation of nuclear modification of the PDF measured by the ALICE Collaboration from electroweak-boson measurements. This measurement helps constraining the nPDF models in the very low Bjorken-$x$ region (down to about 10$^{-4}$), where the available constraints are scarce. \\\\\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.46\\linewidth,clip]{figures\/ZPbPb5tev_IntegratedYield.pdf}\n    \\includegraphics[width=0.52\\linewidth,clip]{figures\/PbPb5tev_W_yieldW_HGpythia.pdf}\n    \\caption{\\textbf{Left}: \\avTaa-scaled yield of dimuons from \\Z-boson decays in Pb--Pb collisions at \\fivenn, compared with theoretical calculations with and without nuclear modifications of the PDF. \\textbf{Right}: \\avTaa-scaled yield of muons from \\W-boson decays as a function of centrality, in Pb--Pb at \\fivenn. The measured distribution is compared with HG-PYTHIA~\\cite{hg-pythia} predictions of the nuclear modification factor of hard scatterings, scaled to the measured value in 0--90\\% centrality.}\n    \\label{fig:pbpb}\n\\end{figure}\n\n\\subsection{Measurements in Pb--Pb collisions}\n\\label{sec:pbpb}\n\nThe results obtained in Pb--Pb collisions at \\fivenn are illustrated in Fig.~\\ref{fig:pbpb}. In the left panel, the yield of dimuons from \\Z decays, scaled with the average nuclear overlap function \\avTaa, is compared with a previously published result obtained from the 2015 data only, as well as predictions with and without nuclear modifications. The combination of the 2015 and 2018 data periods for this new measurement improved the available integrated luminosity by a factor three, allowing for a significant reduction of the uncertainty. The various nPDFs provide a good description of the measured yields, while a deviation by 3.4$\\sigma$ from the CT14-only calculation is observed, constituting another strong sign of nuclear modifications.\n\nIn the right panel of Fig.~\\ref{fig:pbpb}, the \\avTaa-scaled yield of electrons and positrons from \\W decays are shown as a function of the collision centrality. They are compared with HG-PYTHIA~\\cite{hg-pythia} calculations of the nuclear modification factor of hard scatterings, scaled with the value measured in 0--90\\% centrality. The scaled calculations are in good agreement with the data. They predict a sizeable drop for the most peripheral collisions, but statistical limitations prevent to have enough granularity in this region in the measurement, such that no experimental conclusion can be drawn. Nonetheless, this measurement is the first measurement of the \\W-boson production in Pb--Pb collisions at forward rapidity, where low values of Bjorken-$x$ are attained.\n\n\\section{Conclusion}\n\nThe recent measurements of electroweak-boson production show their usefulness in the characterization of key features of QCD. The measurement at midrapidity in pp collisions is well described by calculations including MPI and CR. In heavy-ion collisions, the measurements are generally well described by nuclear models, while significant deviations from free-PDF calculations can be observed. These measurements can thus be used to help constraining the nPDF models.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $\\frg$ be a finite-dimensional simple Lie algebra over $\\bbC$. Fix a Cartan subalgebra $\\frh$ of $\\frg$.\nThe associated root system is $\\Delta=\\Delta(\\frg, \\frh)\\subseteq\\frh_{\\bbR}^*$. Recall that a decomposition\n\\begin{equation}\\label{grading}\n\\frg=\\bigoplus_{i\\in \\bbZ}\\frg(i)\n\\end{equation}\nis  a \\emph{$\\bbZ$-grading} of $\\frg$ if $[\\frg(i), \\frg(j)]\\subseteq \\frg(i+j)$ for any $i, j\\in\\bbZ$.\nIn particular, in such a case, $\\frg(0)$ is a Lie subalgebra of $\\frg$. Since each derivation of $\\frg$ is inner, there exists $h_0\\in\\frg(0)$ such that $\\frg(i)=\\{x\\in\\frg\\mid [h_0, x]=i x\\}$. The element $h_0$ is said to be \\emph{defining} for the grading \\eqref{grading}. Without loss of generality, one may assume that $h_0\\in\\frh$. Then $\\frh\\subseteq\\frg(0)$. Let $\\Delta(i)$ be the set of roots in $\\frg(i)$. Then we can\nchoose a set of positive roots $\\Delta(0)^+$ for $\\Delta(0)$ such that\n$$\n\\Delta^+ :=\\Delta(0)^+\\sqcup \\Delta(1)\\sqcup \\Delta(2)\\sqcup \\cdots\n$$\nis a set of positive roots of $\\Delta(\\frg, \\frh)$. Let $\\Pi$ be the\ncorresponding simple roots, and put $\\Pi(i)=\\Delta(i)\\cap \\Pi$. Note\nthat the grading \\eqref{grading} is fully determined by\n$\\Pi=\\bigsqcup_{i\\geq 0} \\Pi(i)$. We refer the reader to Ch.~3, \\S 3\nof \\cite{GOV} for generalities on gradings of Lie algebras. Each\n$\\Delta(i)$, $i\\geq 1$, inherits a poset structure from the usual\none of $\\Delta^+$. That is, let $\\alpha$ and $\\beta$ be two roots of\n$\\Delta(i)$, then $\\beta\\geq\\alpha$ if and only if $\\beta-\\alpha$ is\na nonnegative integer combination of simple roots.\n\n\n\n\n\n\nRecently, Panyushev initiated the study of the rich structure of\n$\\Delta(1)$ in \\cite{P}. In particular,  he raised five\nconjectures concerning the $\\mathcal{M}$-polynomial,\n$\\mathcal{N}$-polynomial and the reverse operator of $\\Delta(1)$.\nNote that Conjectures 5.1, 5.2 and 5.12 there have been solved by\nWeng and the author \\cite{DW}. The current paper aims\nto handle conjecture 5.11 of \\cite{P}. Let us prepare more notation.\n\nRecall that a subset $I$ of a finite poset $(P, \\leq)$ is a\n\\emph{lower} (resp., \\emph{upper}) \\emph{ideal} if $x\\leq y$ in $P$\nand $y\\in I$ (resp. $x\\in I$) implies that $x\\in I$ (resp. $y\\in\nI$). We collect the  lower ideals of $P$ as $J(P)$, which is\npartially ordered by inclusion. A subset $A$ of $(P, \\leq)$ is an\n\\emph{antichain} if any two elements in $A$ are non-comparable under\n$\\leq$. We collect the antichains of $P$ as $\\mathrm{An}(P)$. For\nany $x\\in P$, let $I_{\\leq x}=\\{y\\in P\\mid y\\leq x\\}$. Given an\nantichain $A$ of $P$, let $I(A)=\\bigcup_{a\\in A} I_{\\leq a}$. The\n\\emph{reverse operator} $\\mathfrak{X}$ is defined by\n$\\mathfrak{X}(A)=\\min (P\\setminus I(A))$. Since antichains of $P$\nare in bijection with lower (resp. upper) ideals of $P$, the reverse\noperator acts on lower (resp. upper) ideals of $P$ as well. Note\nthat the current $\\mathfrak{X}$ is inverse to the reverse operator\n$\\mathfrak{X}^{\\prime}$ in Definition 1 of \\cite{P}, see Lemma\n\\ref{lemma-inverse-reverse-operator}. Thus replacing $\\mathfrak{X}^{\\prime}$ by $\\mathfrak{X}$ does not affect our\nforthcoming discussion on orbits.\n\n\nWe say the $\\bbZ$-grading \\eqref{grading} is \\emph{extra-special} if\n\\begin{equation}\\label{extra-special}\n\\frg=\\frg(-2)\\oplus \\frg(-1) \\oplus \\frg(0) \\oplus \\frg(1)\n\\oplus \\frg(2) \\mbox{  and  }\\dim\\frg(2)=1,\n\\end{equation}\nUp to conjugation, any simple Lie algebra $\\frg$ has a unique extra-special $\\bbZ$-grading. Without loss of generality, we assume that $\\Delta(2)=\\{\\theta\\}$ , where $\\theta$ is the highest root of $\\Delta^+$.\nNamely, we may assume that the grading \\eqref{extra-special} is defined by the element $\\theta^{\\vee}$, the dual root of $\\theta$. In such a case, we have\n\\begin{equation}\\label{Delta-one}\n\\Delta(1)=\\{\\alpha\\in\\Delta^+\\mid (\\alpha, \\theta^{\\vee})=1\\}.\n\\end{equation}\nLet $\\mathrm{ht}$ be the height function. Recall that $h:=\\mathrm{ht}(\\theta)+1$ is the \\emph{Coxeter number}\nof $\\Delta$. Let $h^*$ be the \\emph{dual Coxeter number }of\n$\\Delta$. That is, $h^*$ is the height of\n$\\theta^{\\vee}$ in $\\Delta^{\\vee}$. As noted on p.~1203 of \\cite{P},\nwe have $|\\Delta(1)|=2h^*-4$. We call a lower (resp. upper) ideal\n$I$ of $\\Delta(1)$ \\emph{Lagrangian} if $|I|=h^*-2$. Write\n$\\Delta_l$ (resp. $\\Pi_l$) for the set of \\emph{all} (resp.\n\\emph{simple}) \\emph{long} roots. In the simply-laced cases, all\nroots are assumed to be both long and short. Note that $\\theta$ is\nalways long, while $\\theta^{\\vee}$ is always short.\n\n\n\n\n\n\n\n\nNow Conjecture 5.11 of \\cite{P} is stated as follows.\n\n\n\\medskip\n\\noindent \\textbf{Panyushev conjecture.}\\quad In any extra-special\n$\\bbZ$-grading of $\\frg$, the number of\n$\\mathfrak{X}_{\\Delta(1)}$-orbits equals $|\\Pi_l|$, and each orbit\nis of size $h-1$. Furthermore, if $h$ is even (which only excludes the case $A_{2k}$ where $h=2k+1$), then each\n$\\mathfrak{X}_{\\Delta(1)}$-orbit contains a unique Lagrangian lower\nideal.\n\n\\medskip\n\nOriginally, the conjecture is stated in terms of upper ideals and\nthe reverse operator $\\mathfrak{X}^{\\prime}$. One agrees that we can\nequivalently phrase it using lower ideals and $\\mathfrak{X}$. The\nmain result of the current paper is the following.\n\n\\begin{thm}\\label{thm-main}\nPanyushev conjecture is true.\n\\end{thm}\n\nAfter collecting necessary preliminaries in Section 2, the above\ntheorem will be proven in Section 3. Moreover, we note that by our\ncalculations in Section 3, one checks easily that for any\nextra-special $1$-standard $\\bbZ$-grading of $\\frg$, all the\nstatements of Conjecture 5.3 in \\cite{P} hold.\n\n\n\n\n\n\n\n\\medskip\n\n\\noindent\\textbf{Notation.} Let $\\bbN =\\{0, 1, 2, \\dots\\}$, and let\n$\\mathbb{P}=\\{1, 2, \\dots\\}$. For each $n\\in\\mathbb{P}$, $[n]$\ndenotes the poset $(\\{1, 2, \\dots, n\\}, \\leq)$.\n\n\\section{Preliminaries}\n\nLet us collect some preliminary results in this section. Firstly,\nlet us compare the two reverse operators. Let $(P, \\leq)$ be any\nfinite poset. For any $x\\in P$, let $I_{\\geq x}=\\{y\\in P\\mid y\\geq\nx\\}$. For any antichain $A$ of $P$, put $I_{+}(A)=\\bigcup_{a\\in A}\nI_{\\geq a}$. Recall that in Definition 1 of \\cite{P}, the reverse\noperator $\\mathfrak{X}^{\\prime}$ is given by\n$\\mathfrak{X}^{\\prime}(A)=\\max (P\\setminus I_{+}(A))$.\n\n\\begin{lemma}\\label{lemma-inverse-reverse-operator}\nThe operators $\\mathfrak{X}$ and  $\\mathfrak{X}^{\\prime}$ are\ninverse to each other.\n\\end{lemma}\n\\begin{proof}\nTake any antichain $A$ of $P$, note that\n$$I_{+}(\\min(P\\setminus\nI(A)))=P\\setminus I(A)\\mbox{ and } I(\\max(P\\setminus\nI_{+}(A)))=P\\setminus I_{+}(A).\n$$\nThen the lemma follows.\n\\end{proof}\n\n\nLet $(P_i,\\leq), i=1, 2$ be two finite posets. One can define a\nposet structure on $P_1\\times P_2$ by setting $(u_1, v_1)\\leq (u_2,\nv_2)$ if and only if $u_1\\leq u_2$ in $P_1$ and $v_1\\leq v_2$ in\n$P_2$. We simply denote the resulting poset by $P_1 \\times P_2$. The\nfollowing well-known lemma describes the lower ideals of\n$[m]\\times P$.\n\n\\begin{lemma}\\label{lemma-ideals-CnP}\nLet $P$ be a finite poset. Let $I$ be a subset of $[m]\\times P$. For\n$1\\leq i\\leq m$, denote $I_i=\\{a\\in P\\mid (i, a)\\in I\\}$. Then $I$\nis a lower ideal of $[m]\\times P$ if and only if each $I_i$ is a\nlower ideal of $P$, and $I_m\\subseteq I_{m-1}\\subseteq \\cdots\n\\subseteq I_{1}$.\n\\end{lemma}\n\n\nIn this section, by a \\emph{finite graded poset} we always mean a\nfinite poset $P$ with a rank function $r$ from  $P$ to the positive\nintegers $\\mathbb{P}$ such that all the minimal elements  have rank\n$1$, and $r(x)=r(y)+1$ if $x$ covers $y$. In such a case, let $P_i$\nbe the set of elements in $P$ with rank $i$. The sets $P_i$ are said\nto be the \\emph{rank levels} of $P$. Suppose that\n$P=\\bigsqcup_{j=1}^{d} P_j$. Let $P_0$ be the empty set $\\emptyset$.\nPut $L_i=\\bigsqcup_{j=1}^{i} P_j$ for $1\\leq j\\leq d$, and let $L_0$ be the empty set.\nWe call those $L_i$ \\emph{rank level lower ideals}.\n\n\nLet $\\mathfrak{X}$  be the reverse operator on $[m]\\times P$. In\nview of Lemma \\ref{lemma-ideals-CnP}, we denote by $(I_1, \\cdots,\nI_m)$ a general lower ideal of $[m]\\times P$, where each $I_i\\in\nJ(P)$ and $I_m\\subseteq  \\cdots \\subseteq I_{1}$. We say that the\nlower ideal $(I_1, \\cdots, I_m)$ is \\emph{full rank} if each $I_i$\nis a rank level lower ideal of $P$. Let $\\mathcal{O}(I_1, \\cdots,\nI_m)$ be the $\\mathfrak{X}_{[m]\\times P}$-orbit of $(I_1, \\cdots,\nI_m)$. The following lemma will be helpful in determining\n$\\mathfrak{X}_{[m]\\times P}$-orbits  consisting of rank level lower\nideals.\n\n\\begin{lemma}\\label{lemma-operator-ideals-CmP}\nKeep the notation as above.  Then\nfor any $n_0\\in \\bbN$, $n_i\\in\\mathbb{P}$ ($1\\leq i\\leq s$) such that $\\sum_{i=0}^{s} n_i =m$, we have\n\\begin{equation}\\label{rank-level}\n\\mathfrak{X}_{[m]\\times P}(L_d^{n_0}, L_{i_1}^{n_1}, \\cdots, L_{i_s}^{n_s})=\n(L_{i_1+1}^{n_0+1}, L_{i_2+1}^{n_1}, \\cdots, L_{i_s+1}^{n_{s-1}}, L_0^{n_s-1}),\n\\end{equation}\nwhere $0\\leq i_s<\\cdots <i_1<d$, $L_d^{n_0}$ denotes $n_0$ copies of $L_d$ and so on.\n\n\\end{lemma}\n\\begin{proof}\nNote that under the above assumptions, $(L_d^{n_0}, L_{i_1}^{n_1}, \\cdots, L_{i_s}^{n_s})$ is a lower ideal of $[m]\\times P$ in view of Lemma \\ref{lemma-ideals-CnP}. Then analyzing the minimal elements of $([m]\\times P)\\setminus (L_d^{n_0}, L_{i_1}^{n_1}, \\cdots, L_{i_s}^{n_s})$ leads one to \\eqref{rank-level}.\n\n\\end{proof}\n\n\\begin{lemma}\\label{lemma-operator-types}\nLet $(I_1, \\cdots, I_m)$ be an arbitrary lower ideal of $[m]\\times P$.\nThen $(I_1, \\cdots, I_m)$ is full rank if and only if each lower ideal\nin the orbit $\\mathcal{O}(I_1, \\cdots, I_m)$ is full rank.\n\\end{lemma}\n\\begin{proof}\nUse Lemma \\ref{lemma-operator-ideals-CmP}.\n\\end{proof}\n\nThe above lemma tells us that there are two types of $\\mathfrak{X}$-orbits: in the first type each lower ideal is full rank, while in the second type each lower ideal is not. We call them \\emph{type I} and \\emph{type II}, respectively.\n\nFor any $n\\geq 2$, let $K_{n-1}=[n-1]\\oplus([1]\\sqcup [1])\\oplus [n-1]$ (the ordinal sum, see\np.~246 of \\cite{St}).  We label the elements of $K_{n-1}$ by $1$, $2$, $\\cdots$,\n$n-1$, $n$, $n^{\\prime}$, $n+1$, $\\cdots$, $2n-2$, $2n-1$. Figure 1\nillustrates the labeling for the Hasse diagram of $K_3$. Note that $L_i$ ($0\\leq i\\leq 2n-1$) are all the full rank lower ideals. For instance, we have $L_{n}=\\{1, 2, \\cdots, n, n^{\\prime}\\}$.\nMoreover, we put $I_{n}=\\{1,  \\cdots, n-1,\nn\\}$ and $I_{n^{\\prime}}=\\{1,  \\cdots, n-1, n^{\\prime}\\}$. The following lemma will be helpful in analyzing the  $\\mathfrak{X}_{[m]\\times K_{n-1}}$-orbits of type II.\n\n\n\\begin{lemma}\\label{lemma-operator-ideals-CmK}\nFix $n_0\\in \\bbN$, $n_i\\in\\mathbb{P}$ ($1\\leq i\\leq s$),\n$m_j\\in\\mathbb{P}$ ($0\\leq j\\leq t$) such that $\\sum_{i=0}^{s} n_i +\n\\sum_{j=0}^{t} m_j=m$. Take any $0\\leq j_t< \\cdots<j_1<n\\leq\ni_s<\\cdots <i_1<2n-1$, we have\n\\begin{align*}\n\\mathfrak{X}_{[m]\\times K_{n-1}}&(L_{2n-1}^{n_0}, L_{i_1}^{n_1}, \\cdots, L_{i_s}^{n_s}, I_n^{m_0}, L_{j_1}^{m_1}, \\cdots, L_{j_t}^{m_t})=\\\\\n&\\begin{cases}\n( L_{i_1+1}^{n_0+1}, L_{i_2+1}^{n_1}, \\cdots, L_{i_s+1}^{n_{s-1}}, I_{n^{\\prime}}^{n_s}, L_{j_1+1}^{m_0},\nL_{j_2+1}^{m_1}, \\cdots, L_{j_t+1}^{m_{t-1}}, L_0^{m_t-1} ) & \\mbox { if } j_1 < n-1;\\\\\n( L_{i_1+1}^{n_0+1}, L_{i_2+1}^{n_1}, \\cdots, L_{i_s+1}^{n_{s-1}}, L_{n}^{n_s}, I_n^{m_0},\n\\, \\, \\, \\, L_{j_2+1}^{m_1}, \\cdots, L_{j_t+1}^{m_{t-1}}, L_0^{m_t-1} )& \\mbox { if } j_1 = n-1.\n\\end{cases}\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nAnalyzing the minimal elements of $$([m]\\times K_{n-1})\\setminus (L_{2n-1}^{n_0}, L_{i_1}^{n_1}, \\cdots, L_{i_s}^{n_s}, I_n^{m_0}, L_{j_1}^{m_1}, \\cdots, L_{j_t}^{m_t})$$ leads one to the desired expression.\n\\end{proof}\n\n\\begin{figure}[]\n\\centering \\scalebox{0.4}{\\includegraphics{K3_Labelled.eps}}\n\\caption{The labeled Hasse diagram of $K_3$}\n\\end{figure}\n\n\\section{Panyushev conjecture}\n\nThis section is devoted to proving Theorem \\ref{thm-main}.\n\n\n\n\\noindent \\emph{Proof of Theorem \\ref{thm-main}.} Note that when\n$\\frg$ is $A_n$,  the extra-special $\\Delta(1)\\cong [n-1]\\sqcup\n[n-1]$; when $\\frg$ is $C_n$, the extra-special $\\Delta(1)\\cong\n[2n-2]$. One can verify Theorem \\ref{thm-main} for these two cases\nwithout much effort. We omit the details.\n\n\n\nFor $\\frg=B_n$,  the extra-special $\\Delta(1)= [2]\\times [2n-3]$.\nNow $|\\Pi_{l}|=n-1$, $h-1=2n-1$, and $h^*-2=2n-3$. As in Section 2,\nlet $L_i$ ($0\\leq i\\leq 2n-3$) be the rank level lower ideals. For\nsimplicity, we simply denote $\\mathfrak{X}_{[2]\\times [2n-3]}$ by\n$\\mathfrak{X}$. For any $1\\leq i\\leq n-2$, let us analyze the type I\n$\\mathfrak{X}$-orbit $\\mathcal{O}(L_i, L_i)$ via the aid of Lemma\n\\ref{lemma-operator-ideals-CmP}:\n\\begin{align*}\n\\mathfrak{X}(L_i, L_i)&=(L_{i+1}, L_0),\\\\\n\\mathfrak{X}^{2n-4-i}(L_{i+1}, L_0)&=(L_{2n-3}, L_{2n-4-i}),\\\\\n\\mathfrak{X}(L_{2n-3}, L_{2n-4-i})&=(L_{2n-3-i}, L_{2n-3-i}),\\\\\n\\mathfrak{X}(L_{2n-3-i}, L_{2n-3-i})&=(L_{2n-2-i}, L_{0}),\\\\\n\\mathfrak{X}^{i-1}(L_{2n-2-i}, L_{0})&=(L_{2n-3}, L_{i-1}),\\\\\n\\mathfrak{X}(L_{2n-3}, L_{i-1})&=(L_{i}, L_{i}).\n\\end{align*}\nThus $\\mathcal{O}(L_i, L_i)$ consists of $2n-1$ elements. Moreover,\nin this orbit,  $(L_{2n-2-\\frac{i+1}{2}}, L_{\\frac{i-1}{2}})$ is the\nunique ideal with size $2n$ when $i$ is odd,  $(L_{n+\\frac{i}{2}-1},\nL_{n-\\frac{i}{2}-2})$ is the unique ideal with size $2n$ when $i$ is\neven. Similarly, the orbit $\\mathcal{O}(L_0, L_0)$ consists of\n$2n-1$ elements and contains a unique ideal with size $2n$:\n$(L_{n-1}, L_{n-2})$. Since there are $(n-1)(2n-1)$ lower ideals in\n$[2]\\times [2n-3]$ by Lemma \\ref{lemma-ideals-CnP}, one sees that\nall the $\\mathfrak{X}$-orbits have been exhausted, and Theorem\n\\ref{thm-main} holds for $B_{n}$.\n\n\n\n\nLet us consider $D_{n+2}$, where the extra-special $\\Delta(1)\\cong\n[2]\\times K_{n-1}$. We adopt the notation as in Section 2. For simplicity,\nwe write $\\mathfrak{X}_{[2]\\times K_{n-1}}$ by $\\mathfrak{X}$.  We propose\nthe following.\n\n\\textbf{Claim.} $\\mathcal{O}(L_i, L_i)$, $0\\leq i\\leq n-1$,\n $\\mathcal{O}(I_n, I_n)$, and $\\mathcal{O}(I_{n^{\\prime}},\nI_{n^{\\prime}})$ exhausts the orbits of $\\mathfrak{X}$ on $[2]\\times\nK_{n-1}$. Moreover, each orbit has size $2n+1$ and contains a unique\nlower ideal with size $2n$.\n\nIndeed, firstly, for any $0\\leq i\\leq n-1$, observe that by Lemma\n\\ref{lemma-operator-ideals-CmP}, we have\n\\begin{align*}\n\\mathfrak{X}(L_i, L_i)&=(L_{i+1}, I_0),\\\\\n\\mathfrak{X}^{2n-i-2}(L_{i+1}, L_0)&=(L_{2n-1}, L_{2n-i-2}),\\\\\n\\mathfrak{X}(L_{2n-1}, L_{2n-i-2})&=(L_{2n-i-1}, L_{2n-i-1}),\\\\\n\\mathfrak{X}(L_{2n-i-1}, L_{2n-i-1})&=(L_{2n-i}, L_{0}),\\\\\n\\mathfrak{X}^{i-1}(L_{2n-i}, L_{0})&=(L_{2n-1}, L_{i-1}),\\\\\n\\mathfrak{X}(L_{2n-1}, L_{i-1})&=(L_{i}, L_{i}).\n\\end{align*}\nThus the type I orbit $\\mathcal{O}(L_i, L_i)$ consists of $2n+1$ elements. Moreover,\nin this orbit,  $(L_{2n-i+\\frac{i-1}{2}}, L_{\\frac{i-1}{2}})$ is the\nunique ideal with size $2n$ when $i$ is odd,  $(L_{n+\\frac{i}{2}},\nL_{n-\\frac{i}{2}-1})$ is the unique ideal with size $2n$ when $i>0$\nis even, while $(L_{n}, L_{n-1})$ is the unique ideal\nwith size $2n$ when $i=0$.\n\n\nSecondly, assume that $n$ is even and let us analyze the orbit\n$\\mathcal{O}(I_n, I_n)$. Indeed, by Lemma \\ref{lemma-operator-ideals-CmK}, we have\n\\begin{align*}\n\\mathfrak{X}(I_n, I_n)&=(I_{n^{\\prime}}, L_0),\\\\\n\\mathfrak{X}^{n-1}(I_{n^{\\prime}}, L_0)&=(I_{n}, L_{n-1}),\\\\\n\\mathfrak{X}(I_{n}, L_{n-1})&=(L_{n}, I_{n}),\\\\\n\\mathfrak{X}^{n-1}(L_{n}, I_{n})&=(L_{2n-1}, I_{n^{\\prime}}),\\\\\n\\mathfrak{X}(L_{2n-1}, I_{n^{\\prime}})&=(I_{n}, I_{n}).\n\\end{align*}\nThus the type II orbit $\\mathcal{O}(I_n, I_n)$ consists of $2n+1$ elements. Moreover,\nin this orbit,  $(I_n, I_n)$ is the unique ideal with size $2n$. The\nanalysis of the orbit $\\mathcal{O}(I_{n^{\\prime}}, I_{n^{\\prime}})$\nis entirely similar.\n\n\nFinally, assume that $n$ is odd and let us analyze the orbit\n$\\mathcal{O}(I_n, I_n)$. Indeed,  by Lemma \\ref{lemma-operator-ideals-CmK},  we have\n\\begin{align*}\n\\mathfrak{X}(I_n, I_n)&=(I_{n^{\\prime}}, L_0),\\\\\n\\mathfrak{X}^{n-1}(I_{n^{\\prime}}, L_0)&=(I_{n^{\\prime}}, L_{n-1}),\\\\\n\\mathfrak{X}(I_{n^{\\prime}}, L_{n-1})&=(L_{n}, I_{n^{\\prime}}),\\\\\n\\mathfrak{X}^{n-1}(L_{n}, I_{n^{\\prime}})&=(L_{2n-1}, I_{n^{\\prime}}),\\\\\n\\mathfrak{X}(L_{2n-1}, I_{n^{\\prime}})&=(I_{n}, I_{n}).\n\\end{align*}\nThus the type II  orbit $\\mathcal{O}(I_n, I_n)$ consists of $2n+1$ elements. Moreover,\nin this orbit,  $(I_n, I_n)$ is the unique ideal with size $2n$. The\nanalysis of the orbit $\\mathcal{O}(I_{n^{\\prime}}, I_{n^{\\prime}})$\nis entirely similar.\n\nTo sum up, we have verified the claim since there are $(n+2)(2n+1)$\nlower ideals in $[2]\\times K_{n-1}$ by Lemma \\ref{lemma-ideals-CnP}.\nNote that $|\\Pi_{l}|=n+2$, $h=h^*=2n+2$ for $\\frg=D_{n+2}$, one sees that Theorem\n\\ref{thm-main} holds for $D_{n+2}$.\n\n\n\n\nTheorem \\ref{thm-main} has been verified for all exceptional Lie\nalgebras using \\texttt{Mathematica}. We only present the details for\n$E_6$, where $\\Delta(1)=[\\alpha_2]$, and the Dynkin diagram is as follows.\n\n\\begin{figure}[H]\n\\centering \\scalebox{0.5}{\\includegraphics{E6-Dynkin.eps}}\n\\end{figure}\n\nNote that $|\\Pi_l|=6$,\n$h-1=11$, $h^*-2=10$. On the other hand, $\\mathfrak{X}$ has six\norbits on $\\Delta(1)$, each has $11$ elements. Moreover, the size of\nthe lower ideals in each orbit is distributed as follows:\n\\begin{itemize}\n\\item[$\\bullet$] $0, 1, 2, 4, 7, \\textbf{10}, 13, 16, 18, 19, 20$;\n\n\\item[$\\bullet$] $3, 4, 5, 6, 9, \\textbf{10}, 11, 14, 15, 16, 17$;\n\n\\item[$\\bullet$] $3, 4, 5, 6, 9, \\textbf{10}, 11, 14, 15, 16, 17$;\n\n\\item[$\\bullet$] $7, 7, 8, 8, 9, \\textbf{10}, 11, 12, 12, 13, 13$;\n\n\\item[$\\bullet$] $5, 6, 6, 8, 9, \\textbf{10}, 11, 12, 14, 14, 15$;\n\n\\item[$\\bullet$] $7, 7, 8, 8, 9, \\textbf{10}, 11, 12, 12, 13, 13$.\n\\end{itemize}\nOne sees that each orbit has a unique Lagrangian lower ideal.\n\n\n\n\nThis finishes the proof of Theorem \\ref{thm-main}. \\hfill\\qed\n\n\n\\medskip\n\n\\centerline{\\scshape Acknowledgements} The research is supported by\nthe National Natural Science Foundation of China (grant no.\n11571097) and the Fundamental Research Funds for the Central\nUniversities.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nAccording to the International Diabetes Federation, the number of people affected by diabetes is expected to reach 700 million by 2045 \\cite{Saeedi2019}. Diabetic retinopathy (DR) affects over one-third of this population and is the leading cause of vision loss worldwide \\cite{ogurtsova2017idf}. This happens when the retinal blood vessels are damaged by high blood sugar levels, causing swelling and leakage. In fundus retina images, lesions appear as leaking blood and fluids. Red and bright lesions are the type of lesions  that can be commonly identified during DR screening. The blindness incidence can be reduced if the DR is detected at an early stage. In clinical routine, color fundus photographs (CFP) are employed to identify the morphological changes of the retina by examining the presence of retinal lesions such as microaneurysms, hemorrhages, and soft or hard exudates. The international clinical DR severity scale includes no apparent DR, mild non-proliferative diabetic retinopathy (NPDR), moderate NPDR, severe NPDR, and proliferative diabetic retinopathy (PDR), labeled as grades 0, 1, 2, 3, (illustrated in Fig.\\ref{fig:progresion_DR}) and 4. NPDR (grades 1, 2, 3) corresponds to the early-to-middle stage of DR and deals with a progressive microvascular disease characterized by small vessel damages and occlusions. PDR (grade 4) corresponds to the period of potential visual loss which is often due to a massive hemorrhage. Early identification and adequate treatment, particularly in the mild to moderate stage of NPDR, may slow the progression of DR, consequently preventing the establishment of diabetes-related visual impairments and blindness.\n\nIn the past years, deep learning has achieved great success in medical image analysis. Many supervised learning techniques based on convolutional neural networks have been proposed to tackle the automated DR grading task \\cite{PRATT2016200,QUELLEC2017178,gayathri2020lightweight}. Nevertheless, these approaches rarely take advantage\nof longitudinal information. In this direction, Yan et al. \\cite{Yan} proposed to exploit a Siamese network with different pre-training and fusion schemes to detect the early stage of RD using longitudinal pairs of CFP acquired from the same patient. Further, self-supervised learning (SSL) held great promise as it can learn robust high-level representations by training on pretext tasks \\cite{e24040551} before solving a supervised downstream task. Current self-supervised models are largely based on contrastive learning \\cite{liu2020self,https:\/\/doi.org\/10.48550\/arxiv.2002.05709}. However, the choice of the pretext task to learn a good representation is not straightforward, and the application of contrastive learning to medical images is relatively limited. To tackle this, a self-supervised framework using lesion-based contrastive learning was employed for automated diabetic retinopathy (DR) grading \\cite{Lesion-based}.\n\n\n\\begin{figure}[h!]\n\\includegraphics[width=\\textwidth]{image_paper\/longitudinal_progression_DR.png}\n\\caption{Evolution from no DR to severe NPDR for a patient in OPHDIAT \\cite{ophdiat} dataset.\n} \n\\label{fig:progresion_DR}\n\\end{figure}\n\n\n\nMore recently, a new pretext task has appeared for classification purposes in a longitudinal context. Rivail et al. \\cite{Rivail2019} proposed a longitudinal self-supervised learning Siamese model trained to predict the time interval between two consecutive longitudinal retinal optical coherence tomography (OCT) acquisitions and thus capturing the disease progression. Yang et al. \\cite{Zhao2021} proposed an auto-encoder named LSSL that takes two consecutive longitudinal scans as inputs. They added to the classic reconstruction term a time alignment term that forces the topology of the latent space to change in the direction of longitudinal changes. An extension of such principle was provided in \\cite{Ouyang}. To reach a smooth trajectory field, a dynamic graph in each training batch was computed to define a neighborhood in the latent space for each subject. The graph then connects nearby subjects and enforces their progression directions to be maximally aligned.\n\nIn this regard, we aim to use LSSL approaches to capture the disease progression to predict the change between no DR\/mild NPDR (grade 0 or 1) and more severe DR (grade $\\geq$2) through two consecutive follow-ups. To this end, we explore three methods incorporating current and prior examinations. Finally, a comprehensive evaluation is conducted by comparing these pipelines on the OPHDIAT dataset \\cite{ophdiat}. To the best of our knowledge, this work is the first to automatically assess the early DR severity changes between consecutive images using self-supervised learning applied in a longitudinal context.\n\n\\section{Methods}\n\nIn this work, we study the use of different longitudinal pretext tasks. We use the encoders trained with those pretext tasks as feature extractors embedded with longitudinal information. The aim is to predict the severity grade change from normal\/mild NPDR to more severe DR between a pair of follow-up CFP images. Let $\\mathcal{X}$ be the set of subject-specific image pairs for the collection of all CFP images. $\\mathcal{X}$ contains all $(x^t, x^s)$ that are from the same subject with $x^t$ scanned before $x^s$. These image pairs are then provided as inputs to an auto-encoder (AE) structure (Fig.\\ref{fig:overview}\\textit{c}). The latent representations generated by the encoder are denoted by $z^t=F(x^t)$ and $ z^s=F(x^s)$ where $F$ is the encoder. From this encoder, we can define the $\\Delta z = (z^s - z^t)$ trajectory vector and then formulate $\\Delta z^{(t,s)} = (z^s - z^t) \/ \\Delta t^{(t,s)}$ as the normalized trajectory vector where $\\Delta t^{(t,s)}$ is the time interval between the two acquisitions. The decoder $H$ uses the latent representation to reconstruct the input images such that $\\tilde{x}^t=H(z^t)$ and $\\tilde{x}^s=H(z^s)$. $\\mathbf{E}$ denotes the expected value. In what follows, three longitudinal self-supervised learning schemes are further described.\n\n\\begin{figure}[h!]\n\\includegraphics[width=\\textwidth]{image_paper\/image_paper_LSSL_norm_delta_z_final.png}\n\\caption{The figure a) illustrates to longitudinal Siamese and takes as inputs a pair of consecutive images and predict the time between the examinations. The figure b) represents the longitudinal self-supervised learning which is composed of two independent modules, an AE and dense layers. The AE takes as input the pair of consecutive images and reconstruct the image pairs while the dense layer maps a dummy vector to the direction vector $\\tau$. The figure c) corresponds to the LNE, and takes as input the consecutive pairs and build a dynamic graph to align in a neighborhood the subject-specific trajectory vector ($\\Delta z$) and the pooled trajectory vector ($\\Delta h$) that represents the local progression direction in latent space (green circle).}\n\\label{fig:overview}\n\\end{figure}\n\n\n\\subsection{Longitudinal Siamese} \n\nThe Siamese network takes the two image pairs $(x^t, x^s)$. These images are encoded into a compact representation ($z^t$, $z^s$) by the encoder network, $F$. A feed forward neural network (denoted $G$) then predicts $\\Delta t^{(t,s)}$, the time interval between the pair of CFP images (Fig.\\ref{fig:overview}\\textit{a}). The regression model is trained by minimizing the following L2 loss: $\\parallel G(z^t,z^s) - \\Delta t^{(t,s)} \\parallel_2^2$.\n\n\\subsection{Longitudinal self-supervised learning} \n\nThe longitudinal self-supervised learning (LSSL) exploits a standard AE. The AE is trained with a loss that forces the trajectory vector $\\Delta z$ to be aligned with a direction that could rely in the latent space of the AE called $\\tau$. This direction is learned  through a sub-network composed of single dense layers which map a dummy data into a vector $\\tau \\in \\Omega_{\\alpha}$, the dimension of the latent space. The high-level representation of the network is illustrated in Fig.\\ref{fig:overview}\\textit{b}. Enforcing the AE to respect this constraint is equivalent to encouraging $\\cos\\left(\\Delta z,\\boldsymbol{\\tau}\\right)$ to be close to 1, i.e. a zero-angle between $\\boldsymbol{\\tau}$ and the direction of progression in the representation space.\n\n\n\n\n\\noindent\\textbf{Objective Function.}\n\\begin{equation}\n\\mathbf{E}_{(x^t, x^s) \\sim \\mathcal{X}} \\left(\\lambda_{rec}\\cdot(\\parallel x^t - \\tilde{x}^t \\parallel_2^2 + \\parallel x^s - \\tilde{x}^s \\parallel_2^2)-\\lambda_{dir} \\cdot \\cos(\\Delta z,\\tau)\\right)\n\\label{eq:loss_1}\n\\end{equation}\n\n\n\n\n\\subsection{Longitudinal neighbourhood embedding}\n\n\nLongitudinal neighborhood embedding (LNE) is based on the LSSL framework. The main difference is that a directed graph $\\mathcal{G}$ is built in each training batch. A pair of sample ($x_{t},x_{s}$) serves as a node in the graph with node representation $\\Delta z$. For each node $i$, Euclidean distances to other nodes $j\\neq i$ are computed by $D_{i,j} = \\parallel z^t_i - z^t_j\\parallel_2$. The neighbour size ($N_{nb}$) is the closest nodes of node $i$ form its 1-hop neighbourhood $\\mathcal{N}_i$ with edges connected to $i$. The adjacency matrix $A$ for the directed graph ($\\mathcal{G}$) is then defined as:\n\n\\begin{align*} \n    A_{i,j} := \n   \\begin{cases}\n    exp(-\\frac{D_{i,j}^2}{2\\sigma_i^2})  & j \\in \\mathcal{N}_i\\\\\n    0, & j \\notin \\mathcal{N}_i\n    \\end{cases}~. \\\\\n    \\mbox{with } \\sigma_i := max(D_{i,j \\in \\mathcal{N}_i}) - min(D_{i,j \\in \\mathcal{N}_i})\n    \\label{eqn:adj}\n\\end{align*}\nThis matrix regularizes each node's representation by a longitudinal neighbourhood embedding $\\Delta h$ pooled from the neighbours' representations. The neighborhood embedding for a node $i$ is computed by:\n\\begin{equation*}\n\\Delta h_i := \\sum _{j \\in \\mathcal{N}_i} A_{i,j} O^{-1}_{i,j} \\Delta z_j,\n\\end{equation*} where $O$ is the out-degree matrix of graph $\\mathcal{G}$, a diagonal matrix that describes the sum of the weights for outgoing edges at each node. They define $\\theta_{\\langle \\Delta z,\\Delta h \\rangle}$ the angle between $\\Delta z$ and $\\Delta h$, and only incite $\\cos(\\theta_{\\langle \\Delta z,\\Delta h \\rangle}) = 1$, i.e., a zero-angle between the subject-specific trajectory vector and the pooled trajectory vector that represents the local progression direction in the latent space (Fig. \\ref{fig:overview}\\textit{c}). \n\n\\noindent\\textbf{Objective Function.} \n\\begin{equation}\n    \\mathbf{E}_{(x^t, x^s) \\sim \\mathcal{X}} \\left(\\lambda_{rec} \\cdot(\\parallel x^t - \\tilde{x}^t \\parallel_2^2 + \\parallel x^s - \\tilde{x}^s \\parallel_2^2)- \\lambda_{dir} \\cdot \\cos(\\theta_{\\langle \\Delta z,\\Delta h \\rangle})\\right) \\label{eq:loss_2}\n\\end{equation}\n\n\\section{Dataset}\n\nThe proposed models were trained and evaluated on OPHDIAT \\cite{ophdiat}, a large CFP database collected from the Ophthalmology Diabetes Telemedicine network consisting of examinations acquired from 101,383 patients between 2004 and 2017. Within 763,848 interpreted CFP images 673,017 are assigned with a DR severity grade, the others being non-gradable. Image sizes vary from 1440 $\\times$ 960 to 3504 $\\times$ 2336 pixels. Each examination has at least two images for each eye. Each subject had 2 to 16 scans with an average of 2.99 scans spanning an average time interval of 2.23 years. The age range of the patients is from 9 to 91.  \n\n\\noindent\\textbf{Image pair selection.} The majority of patients from the OPHDIAT database have multiple images with different fields of view for both eyes. To facilitate the pairing, we propose to select a single image per eye for each examination: we select the one that best characterizes the DR severity grade, as detailed hereafter. For this purpose, we train a standard classifier using the complete dataset that predicts the DR severity grade (5 grades). During the first epoch, we randomly select one image per eye and per examination for the full dataset. After the end of the first epoch with the learned weights of the model, for each image present in every examination, we select the image that gives the highest classification probability. We repeat this process until the selected images by the model converge to a fixed set of images per examination. From the set of selected images, we construct consecutive pairs for each patient and finally obtain 100,033 pairs of images from 26,483 patients. Only 6,690 (6.7\\%) pairs have severity grade changes from grade 0 or 1 to grade $\\geq 2$ against 93,343 (93.3\\%) pairs with severity  changes that lie between grades 0 and 1. The resulting dataset exhibits the following proportions in gender (Male 52\\%, Female 48\\%) and diabetes type (type 2 69\\%, type 1 31\\%). This dataset was further divided into training (60\\%), validation (20\\%), and test (20\\%) based on subjects, i.e., images of a single subject belonged to the same split and in a way that preserves the same proportions of examples in each class as observed in the original dataset.  \n\n\\noindent\\textbf{Image pre-processing.} Image registration is a fundamental pre-processing step for longitudinal analysis \\cite{Saha2019}. Therefore, using an affine transformation, we first conducted a registration step to align $x_{t}$ to $x_{s}$. Images are then adaptively cropped to the width of the field of view (i.e., the eye area in the CPF image) and then resized to 256$\\times$256. A Gaussian filter estimates the background in each color channel to attenuate the strong intensity variations among the dataset which is then subtracted from the image. Finally, the field of view is eroded by 5\\% to eliminate illumination artifacts around the edges. During the training, random resized crops ([0.96, 1.0] as scale range and [0.95, 1.05] as aspect ratio range) are applied for data augmentation purposes. \n\n\n\\section{Experiments and Results}\n\n\\noindent\\textbf{Implementation Details.} As it was conducted in \\cite{Zhao2021,Ouyang}, we constructed a standard AE for all the compared methods to focus only on the presented methodology and make a fair comparison between approaches, with the hope that using advanced AE structures could lead to better encoding and generalization. In our basic architecture, we employed a stack of $n$ pre-activated residual blocks, where $n$ determines the depth of that scale for the encoder. In each res-block, the residual feature map was calculated using a series of three 3$\\times$3 convolutions, the first of which always halves the number of the feature maps employed at the present scale, such that the residual representations live on a lower-dimensional manifold. Our encoder comprises five levels; the first four levels are composed of two residual blocks, and the latter only one residual block. This provides a latent representation of size $64\\times4\\times4$. The employed decoder is a reverse structure of the encoder. The different networks were trained for 100 epochs by the AdamW optimizer with a learning rate of $5 \\times 10^{-4}$, OneCycleLR as scheduler and a weight decay of $10^{-5}$, using an A6000 GPU with the PyTorch framework. The regularization weights were set to $\\lambda_{dir}=1.0$ and $\\lambda_{rec}=5.0$. A batch size of 64 was used for all models, and a neighbour size $N_{nb}=5$ and $\\Delta z^{(t,s)}$ were used for the LNE, as in the original paper \\cite{Ouyang}.\n\\subsection{Comparison of the approaches on the early change detection}\n\nWe evaluate the LSSL encoders on detecting the severity grade change from normal\/mild NPDR to more severe DR between a pair of follow-up CFP images. The classifier was constructed as the concatenation of the learned backbone (feature extractor) and a multi-layer perceptron (MLP). The MLP consists of two fully connected layers of dimensions 1024 and 64 with LeakyReLU activation followed by a last single perceptron. Receiving the flattened representation of the trajectory vector $\\Delta z$, the MLP predicts a score between 0 and 1.\nWe compared the area under the receiver operating characteristics curve (AUC) and the accuracy (Acc) in Tab.\\ref{table:table_comparaison_AUC} for different pre-training strategies (from scratch, trained on LSSL methods, encoder from a standard AE). We also pre-trained on the OPHDIAT dateset (classification of the DR severity grade) to compare the LSSL pre-training strategies with a conventional pre-training method. The statistical significance was estimated using DeLong's t-test \\cite{Robin2011} to analyze and compare ROC curves. The results in Tab.\\ref{table:table_comparaison_AUC} and Fig.\\ref{fig:ROC_curve} show the clear superiority of the LSSL encoder, with a statistical significance p-value < 2.2e-16. Due to class imbalance, the Longitudinal-siamese (L-siamese) have a high Acc while exhibiting a lower AUC than the baseline (trained from scratch). \n\n\n\n\n\n\n\\newfloatcommand{capbtabbox}{table}[][\\FBwidth]\n\n\n\n\\begin{figure}\n\\begin{floatrow}\n\\ffigbox{%\n \\includegraphics[width=6cm]{image_paper\/roc_curve.png}%\n}{%\n  \\caption{ROC Curve Analysis of the compared methods}\\label{fig:ROC_curve}%\n}\n\\capbtabbox{%\n\\begin{adjustbox}{width=\\columnwidth,center}%\n\\begin{tabular}{l c c}\n   \n   \n   \n    \\cline{2-3} \n      & AUC (95\\% CI) & Acc \\tabularnewline\n    \\cline{2-3} \n    Model & & \\tabularnewline\n    \\hline\n    No pretrain& 0.8758 (0.8688-0.883) & 0.8379 \\tabularnewline\n    Pre-train on OPHDIAT& 0.8994 (0.8921-0.9068) &\t0.8289 \\tabularnewline\n    AE& 0.7724 (0.7583-0.7866) & 0.5599 \\tabularnewline\n    L-siamese \\cite{Rivail2019}& 0.8253  (0.8127-0.838) & 0.9354 \\tabularnewline\n    LSSL \\cite{Zhao2021}& \\textbf{0.9624} (0.9593-0.9655) & 0.8871 \\tabularnewline\n    LNE  \\cite{Ouyang} & \\textbf{0.9448} (0.9412-0.9486)  & 0.8646 \\tabularnewline\n    \\hline\n    & & \\tabularnewline\n    & & \\tabularnewline\n    & & \\tabularnewline\n    & & \\tabularnewline\n    & & \\tabularnewline\n     \n  \n\\end{tabular}\n\\end{adjustbox}\n\n}{%\n\\caption{Comparison of the approach on the early change detection with the frozen encoder.}\\label{table:table_comparaison_AUC}%\n}%\n\\end{floatrow}\n\\end{figure}\n\n\n\n\n\n\n\n\n     \t \t\n\n\n\n\n\n\n\\subsection{Norm of trajectory vector analyze}\n\nWe constructed different histograms in Fig.\\ref{fig:norm_plot} representing the mean value of the norm of the trajectory vector with respect to both diabetes type and progression type. According to Fig.\\ref{fig:norm_plot}, only the models with the direction alignment loss term are able to capture the change detection in the DR relative to the longitudinal representation. Therefore, we observe in the histogram that the trajectory vector $(\\Delta z)$ is able to dissociate the two types of diabetes (t-test p-value < 0.01) and change detection (t-test p-value < 0.01). For the diabetes type, a specific study \\cite{Chamard2020} about the OPHDIAT dataset indicates that the DR progression is faster for \n\\begin{figure}[h!]\n\\includegraphics[width=\\textwidth]{image_paper\/norm_z_change.png}\n\\caption{Mean of the trajectory vector norm for the different self-supervised method used} \n\\label{fig:norm_plot}\n\\end{figure}\npatients with type 1 diabetes. Based on the fact that the $\\Delta z$ can be seen as a relative speed, this observation agrees with the histogram plot of the mean of the $\\Delta z$ norm represented in Fig. \\ref{fig:norm_plot}. We also observed that the norm of the $\\Delta z$ vector is lower for the normal stage of the DR than for mild NPDR to more severe. This was expected because only the methods with a direction alignment term in their objective explicitly modeled longitudinal effects, resulting in more informative $\\Delta z$. This also implies that simply computing the trajectory vector itself is not enough to force the representation to capture the longitudinal change. \n\n\\section{Discussion}\nWe applied different LSSL techniques to encode diabetic retinopathy (DR) progression. The accuracy boost, relative to a network trained from scratch or transferred from conventional tasks, demonstrates that longitudinal pre-trained self-supervised representation learns clinically meaningful information. Concerning the limitations of the LSSL methods, we first observe that the models with no time alignment loss perform poorly and provide no evidence of disease progression encoding. Also, we report for the LNE that the normalized trajectory vector for some pairs, that have a large time between examinations, is almost all zeros, which results in a non-informative representation. This could explain the difference between the LSSL and LNE prediction performances. Moreover, during the LSSL and the LNE training, we often faced a plateau with the direction loss alignment. Therefore, we also claim that intensive work should be done regarding the choice of the hyperparameters : constant weights for the losses, latent space size ($\\Omega_{\\alpha}$), neighbour size ($N_{nb}$). The results concerning quantifying the encoding of the disease progression from the models trained with a time direction alignment are encouraging but not totally clear. As it was mentioned in \\cite{Vernhet2021}, one limitation of the LSSL approach pretains to the cosine loss (direction alignment term from equations (\\ref{eq:loss_1},\\ref{eq:loss_2}) used to encode the longitudinal progression in a specific direction in the latent space and learned while training. The loss only focuses on the correlation with the disease progression timeline but not disentanglement of the disease progression itself. Therefore, a more in-depth analysis of the latent space is required to evaluate if the trajectory vector could be used to find a specific progression trajectory according to patient characteristics (diabetes types, age, DR severity). The pairing and the registration are critical steps in the longitudinal study. As it was previously mentioned, by using a better registration method and exploiting different fusion schemes and backbone architectures, we could get enriched latent representation and, thus, hopefully, better results. Also, the frozen encoders could be transferred to other types of longitudinal problems. In summary, LSSL techniques are quite promising: preliminary results are encouraging, and we expect further improvements. \\\\\n\n\n\n\\noindent\\textbf{Acknowledgements}\nThe work takes place in the framework of the ANR RHU project Evired. This work benefits\nfrom State aid managed by the French National Research Agency under \"Investissement\nd'Avenir\" program bearing the reference ANR-18-RHUS-0008\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLong period helical spin order resulting from\nDzyaloshinskii-Moriya (DM) spin-orbit\ncoupling\\cite{dzyaloshinskii58} in non-centrosymmetric\nitinerant magnets (e.g. MnSi, Fe$_x$Co$_{1-x}$Si, FeGe with crystal\nstructure B20) has been intensively studied in the\npast.\\cite{dzyaloshinskii64,ishikawa76, moriya76,nakanishi80,bak80,beille83,lebech89,ishimoto95,uchida06}\nOne material of this family, MnSi, has attracted recent interest\nbecause of its puzzling behavior under applied hydrostatic\npressure.\\cite{pfleiderer97,thessieu97} \n A state with non-Fermi liquid transport\nproperties\\cite{pfleiderer01} is obtained over a wide range of\npressures above a critical threshold $p_c$. \nBeginning at the same critical pressure, but over a smaller\npressure range,  magnetic 'partial order' is observed in neutron\nscattering experiments.\\cite{yu04,pfleiderer04} While usual helical\norder gives rise to sharp Bragg peaks in neutron scattering\n(corresponding to the periodicity of the helical spin-density wave),\n'partial order' is characterized by a neutron scattering signal which\nis smeared over a wavevector sphere rather than localized at\ndiscrete points in reciprocal space.\n\nRecent theoretical work on electronic properties, critical fluctuations and\ncollective modes of helimagnets \nare in Refs.~\\cite{fischer04,grigoriev05,belitz06}. Theoretical\nproposals for the high pressure state of MnSi have invoked proximity to a\nquantum multi-critical point \\cite{schmalian04} or magnetic liquid-gas transitions.\\cite{tewari05} Closest in spirit to our approach are the skyrmion-like magnetic patterns studied in Refs.~\\cite{bogdanov05,fischer06}.\n\nRecently, we have proposed a novel kind of magnetic order, the\nhelical spin crystal, as a promising starting point for a theory of\n'partial order'.\\cite{binz06} Helical spin crystals are magnetic\npatterns, which are obtained by superposition of several helical\nspin-density waves which propagate in different directions. There is a substabtial resemblance to multi-$k$ magnetic structures (also known as multiple-$q$ or multiple spin density wave states).\\cite{mukamel76,jo83,forgan89,longfield02,steward04} But in contrast to most other magnetic multi-$k$  systems,  in helical spin crystals the ordering wavevectors are selected  from an infinite number of degenerate modes lying on a sphere in reciprocal space - a process analogous to the crystallization of liquids.\n\nIn this\nwork, we present a detailed theory of such structures. The\nstability, structure and distinctive properties of such states are\ndescribed, and the consequences of \n coupling to non-magnetic disorder is discussed.\n\nThe paper is organized as follows. First, we review the standard\ntheory of helimagnetism in Section \\ref{GLsection}, finishing with a\nshort remark about more general helical magnetic states.\n Then, the theory of helical spin crystals is developed.\nThe requirements\nto energetically stabilize  helical spin crystal states are\ninvestigated in Section \\ref{energetics}. The analysis works in two directions. First, we establish a phase diagram in terms of natural parameters which tune the interaction between helical modes and second, we give simple rules to construct  model interactions which stabilize a large class of helical spin crystals. The remaining parts of the\npaper are dedicated to extracting testable consequences of these\nnovel magnetic states. In Section \\ref{structure},  we give a\ndescription of the most prominent spin crystals which emerge from\nour energetic analysis in terms of their symmetry. It is shown that the symmetry  of the magnetic state may stabilize topological textures like merons and anti-vortices which are otherwise not expected to be stable in the present context, given the order parameter and dimensionality of the system.\nSymmetry also determines, which higher-harmonics Bragg peaks these structures would produce.\nWe subsequently study the response of helical spin crystals with respect to\ndifferent perturbations in Section \\ref{response}. For example, sub-leading spin orbit coupling (crystal anisotropy) locks the magnetic crystal to the\n underlying atomic lattice and thus determines the location of magnetic Bragg peaks. \nWe also study the response to an\nexternal magnetic field which, apart from producing a uniform\nmagnetic moment, also leads to distinctive distortions of the helical\nmagnetic structure, which could be observable by neutron scattering. \nFinally, in Section \\ref{disorder} we investigate the implications of\nnon-magnetic impurities, which are expected to destroy long-range\nmagnetic order and produce diffuse scattering.\n\n\n\\section{Landau-Ginzburg theory of helimagnetism}\\label{GLsection}\n\nFor a cubic magnet without a center of inversion,  the Landau-Ginzburg free energy to quadratic order in the magnetization $\\bv M(\\bv r)$ is\n\\begin{equation}\nF_2=\\left\\langle r_0 \\bv M^2 + J (\\partial_\\alpha M_\\beta)(\\partial_\\alpha M_\\beta)+2D\\bv M\\cdot(\\bv \\nabla\\times\\bv M)\\right\\rangle,\\label{F2}\n\\ee\nwhere $\\left\\langle\\ldots\\right\\rangle$ indicates sample averaging,  $r_0,J,D$ are parameters ($J>0$)  and Einstein summation is understood. The last term of Eq.~\\eref{F2} is the DM interaction, which is odd under spatial inversion and originates in spin-orbit coupling.\\cite{dzyaloshinskii58} Fourier transformation, $\\bv M(\\bv r)=\\sum_{\\bv q} \\bv m_{\\bv q}e^{i\\bv q\\cdot\\bv r}$ with $\\bv m_{-\\bv q}=\\bv m^*_{\\bv q}$, leads to\n\\begin{equation}\nF_2=\\sum _{\\bv q} \\left[\\left(r_0+Jq^2\\right)|\\bv m_{\\bv q}|^2+2D \\bv m_{\\bv q}^*\\cdot\\left(i\\bv q\\times\\bv m_{\\bv q}\\right)\\right].\n\\ee\nClearly, the energy is minimal for circularly polarized spiral modes,\n where  $\\nabla\\times\\bv M$ points in the direction of $-D\\,\\bv M$. For such modes,\n\\begin{equation}\nF_2=\\sum _{\\bv q} r(q)|\\bv m_{\\bv q}|^2,\n\\ee\nwhere $r(q)=r_0-JQ^2+J(q-Q)^2$ with $Q=|D|\/J$.\nThe Gaussian theory thus determines both the chirality of low-energy helical modes and their wavelength $\\lambda=2\\pi\/Q$. The latter  is typically long (between $180\\mathring{A}$ in MnSi and $2300\\mathring{A}$ in Fe$_{0.3}$Co$_{0.7}$Si), reflecting the smallness of spin-orbit coupling effects compared to exchange.\n However, no preferred spiraling direction is selected by Eq.~\\eref{F2}, since $F_2$ is rotation-invariant. Cubic anisotropy terms which break this invariance are of higher order in the spin-orbit interaction and therefore small. We neglect them for the moment and reintroduce them later.\n\nThe isotropic Gaussian theory leaves us with an infinite number of modes which become soft as $r(Q)\\to0$. They consist of helical spin-density waves with given chirality (determined by the sign of $D$), whose wave-vectors lie on a sphere $|\\bv q|=Q$ in reciprocal space.\n Each of these helical modes is determined by an amplitude and a phase.  Hence, for each point  $\\bv q$ on the sphere, we define a complex order parameter $\\psi_{\\bv q}$ (with $\\psi_{-\\bv q}=\\psi^*_{\\bv q}$) through\n\\begin{equation}\n\\bv m_{\\bv q}=\\frac12\\psi_{\\bv q}(\\uv\\epsilon'_{\\bv q}+i \\uv\\epsilon''_{\\bv q}),\\label{psis}\n\\ee\nwhere $\\uv \\epsilon'_{\\bv q}$, $\\uv \\epsilon''_{\\bv q}$, and $\\uv q$, are mutually orthogonal unit vectors (with a defined handedness, given by the sign of $D$). Obviously, changing the phase of $\\psi_{\\bv q}$ is equivalent to rotating $\\uv \\epsilon'_{\\bv q}$ and $\\uv \\epsilon''_{\\bv q}$ around $\\uv q$. The phase of $\\psi_{\\bv q}$  is thus only defined relative to some initial choice of $\\epsilon'_{\\bv q}$. The neutron scattering intensity is proportional to $|\\uv q\\times \\bv m_{\\bv q}|^2=1\/2|\\psi_{\\bv q}|^2$, independent of the phase. Changing the phase of $\\psi_{\\bv q}$ is also equivalent to translating $\\bv M(\\bv r)$ along  $\\uv q$.\n\n\nIn the following, we  study minima of the free energy in the ordered phase [$r(Q)<0$].\nThese depend on the interactions between  degenerate modes (i.e., free energy contributions, which are quartic or higher order in $\\bv M$).  We only consider interactions which, as $F_2$, have full rotation symmetry and we will include the weak crystal anisotropy last. The most general quartic term which has full rotation symmetry (transforming space and spin together) is of the form\n\\begin{equation}\n F_4=\\!\\!\\sum_{\\bv q_1,\\bv q_2,\\bv q_3}\\!\\!U(\\bv\nq_1,\\bv q_2,\\bv q_3)\\left(\\bv m_{\\bv q_1}\\cdot\\bv m_{\\bv\nq_2}\\right)\\left(\\bv m_{\\bv q_3}\\cdot\\bv m_{\\bv\nq_4}\\right),\\label{F4}\n\\ee\n with $\\bv q_4=-(\\bv q_1+\\bv q_2+\\bv q_3)$.\n\n\\subsection{Single-spiral state}\n\nFor example, if $U(\\bv q_1,\\bv q_2,\\bv q_3)$ is a  constant, then $F_4\\propto \\left\\langle \\bv M^4\\right\\rangle$.\nIf the interaction depends only on the local magnetization amplitude, i.e., in general if $F=F_2+\\left\\langle f(\\bv M^2)\\right\\rangle$ for some function $f$, then the absolute minimum of $F$ is given by a single-spiral state (also known as helical spin density wave) $\\bv M(\\bv r)=\\bv m_{\\bv k}e^{i\\bv k\\cdot\\bv r}+\\bv m^*_{\\bv k}e^{-i\\bv k\\cdot\\bv r}$, where a single pair of opposite momenta $\\pm \\bv k$ is selected. \n To proof this, we write $F$ as\n\\begin{equation}\n\\sum_{\\bv q} [r(q)-r(Q)]\\,|\\bv m_{\\bv q}|^2 + \\left\\langle r(Q)\\bv M^2+f(\\bv M^2)\\right\\rangle.\\label{proof}\n\\ee\nIn the single-spiral state, $\\bv M^2$ is constant in space and it minimizes the first and the second term of Eq.~\\eref{proof} independently. Therefore, no other magnetic state can be lower in energy.\n\nBecause $Q$ is small,  the relevant wavevectors entering Eq.~\\eref{F4} are also small and  $U(\\bv q_1,\\bv q_2,\\bv q_3)$ is effectively\n close to a constant. Therefore, the single-spiral state, as observed in Fe$_x$Co$_{1-x}$Si, FeGe and in MnSi at ambient pressure, is the most natural helical magnetic order from the point of view of Landau theory.\n\n\n\n\n\\subsection{Linear superpositions of single-spiral states}\\label{continuous}\n\nMotivated by the phenomenology of  'partial order', we will now  extend the theory beyond this standard solution. We speculate that $U(\\bv q_1,\\bv q_2,\\bv q_3)$ is not constant, such that $F_4$  favors a  linear superpositions of multiple  spin-spirals with different wave-vectors on the sphere of degenerate modes $|\\bv q|=Q$.\n\n\nOne may first speculate about magnetic patterns whose Fourier transform\nis non-zero everywhere on the wave-vector sphere and\n peaked infinitely sharply perpendicular to the sphere, i.e.\n\\begin{equation}\n|\\psi_{\\bv q}|^2\\propto \\delta(|\\bv q|-Q)\\label{po-literally}\n\\ee\n[see Eq.~\\eref{psis}]. This idea turns out to be complicated for at least two reasons.\n\nThe first complication is that there is no continuous way of attributing a finite-amplitude spiral mode to each point on the wavevector sphere. This can be seen by noting that $\\uv \\epsilon'_{\\bv q}$ [Eq.~\\eref{psis}] is a tangent vector field on the sphere. Thus, it cannot be continuous (impossibility of combing a hedgehog).\\cite{mermin79} Thus there is no ``uniform''  superposition of helical modes on the sphere. The problem of singularities can be avoided if one assumes a $\\psi_{\\bv q}$ with point nodes on the sphere.\n\nThe second complication is that   higher harmonics would result in a broadening of the delta-function in Eq.~\\eref{po-literally}. This is seen as follows. Consider three momenta $\\bv q_1, \\bv q_2, \\bv q_3$ on the wavevector sphere and $\\bv q_4$ which is off the sphere. The non-vanishing modes $\\bv m_{\\bv q_1},\\bv m_{\\bv q_2}, \\bv m_{\\bv q_3}$  couple linearly to  $\\bv m_{\\bv q_4}$ via Eq.~\\eref{F4} and thus induce a higher harmonic ``off-shell'' mode $\\bv m_{\\bv q_4}\\neq0$. Since this happens for every point away from the sphere,\n the effect is an intrinsic broadening of the peak in $|\\psi_{\\bv q}|^2$, in contradiction with the initial  assumption of Eq.~\\eref{po-literally}.\n\n\n\\section{Energetics of helical spin crystals}\\label{energetics}\n\nIn the following, we study magnetic structures which are superpositions of a {\\em  finite} number of degenerate helical modes $\\psi_j$ with wavevectors $\\pm\\bv k_j$, $j=1,\\ldots,N$. We call the resulting states helical spin crystals, because of the analogy with weak crystallization theory of the solid-liquid transition.\\cite{brazovskii87}\n\n\\subsection{Structure of the quartic interaction}\\label{section_int}\n\nWe assume that $F_4$ is small, and that its main effect is to provide an\ninteraction between the modes which are degenerate under $F_2$.\nThus, the relevant terms of $F_4$ are those with $|\\bv q_1|= |\\bv q_2|=|\\bv\nq_3|=|\\bv q_4|=Q$. This phase-space constraint and  rotational symmetry implies that the coupling function $U$ depends only on two relative angles between the momenta\n\\begin{equation}\n\\left.U(\\bv q_1,\\bv q_2,\\bv q_3)\\right|_{|\\bv q_i|=Q}=U(\\theta,\\phi),\\label{effint}\n\\ee\n where we have chosen the following parameterization:\n\\begin{equation}\n\\begin{split}\n2\\,\\theta&=\\arccos(\\uv q_1\\cdot\\uv q_2)\\\\\n\\phi\/2&=\\arccos\\left[\\frac{(\\uv q_2-\\uv q_1)\\cdot \\uv q_3}{1-\\uv q_1\\cdot\\uv q_2}\\right].\n\\end{split}\\label{thph}\n\\ee\n Geometrically,\n$\\phi\/2$ is the angle between the two planes\nspanned by $(\\bv q_1, \\bv q_2)$ and  $(\\bv q_3,\\bv q_4)$ (Fig.~\\ref{angles}). In the special case $\\bv q_1+\\bv q_2=0$, it\n becomes the angle between $\\bv q_2$ and  $\\bv q_3$. This mapping allows $\\theta$ and $\\phi$ to be interpreted as the polar and azimuthal angles of a sphere and the coupling $U(\\theta,\\phi)$ is a function on that sphere.   Since it describes an effective coupling between modes on the wavevector sphere, the coupling  $U(\\theta,\\phi)$ has a status similar to that of Fermi liquid parameters in the theory of metals.\n\n\n\\begin{figure}\n\\includegraphics[scale=0.6]{fig1.eps}\n\\caption{The set of quartets $(\\bv q_1,\\ldots,\\bv q_4)$  satisfying  $|\\bv q_i|=Q$ and  $\\bv q_1+\\ldots+\\bv q_4=0$, modulo global rotations, may be parameterized by two angles $\\theta$ and $\\phi$ as shown in this figure.  \\label{angles}}\n\\end{figure}\n\n\nThe Landau free energy $F=F_2+F_4$ of helical spin crystal states is calculated as follows. Obviously,\n\\begin{equation}\nF_2=r(Q)\\sum_j|\\psi_j|^2.\\label{F2psi}\n\\ee\nThe quartic term, Eq.~\\eref{F4}, may be split into three distinct contributions $F_4=F_{s}+F_{p}+F_{\\rm nt}$. The first term, $F_{s}$, is the self-interaction of each spiral mode with itself:\n\\begin{equation}\nF_{s}=U_{s}\\sum_j|\\psi_j|^4,\\label{Fs}\n\\ee\nwhere $U_{s}=U\\!(\\theta\\!=\\!\\pi\/2,\\phi\\!=\\!0)$. A minimum requirement for stability of the  theory is $U_{s}>0$.\n\nThe next term, $F_{p}$, stems from pairwise interactions between modes. It is of the form\n\\begin{equation}\nF_{p}=2\\sum_{i<j} V_{p}(\\theta_{ij})\\, |\\psi_i|^2|\\psi_j|^2,\\label{Fp}\n\\ee\nwhere  $2\\theta_{ij}=\\arccos(\\uv k_i\\cdot\\uv k_j)$ and\n\\begin{equation}\nV_{p}(\\theta)=U\\!(\\frac\\pi2,4\\theta)+\\sin^4\\!\\theta\\,U\\!(\\theta,0)+\\cos^4\\!\\theta\\,U\\!(\\frac\\pi2\\!-\\!\\theta,0).\\label{V}\n\\ee\nSince $\\lim_{\\theta\\to0}V_{p}(\\theta)=2U_{s}$ is large and positive, $F_{p}$ provides an efficient repulsion of modes which are too close on the wavevector sphere. Energetically stable superpositions of spirals require substantially smaller values of\n $V_{p}(\\theta)$ and therefore big-enough angles between them (Section \\ref{exact}). This ``mode repulsion'' suggests that a continuous distribution of Fourier modes on the wavevector sphere, as discussed in the  Section \\ref{continuous}, is difficult to stabilize energetically by  slowly varying coupling functions $U(\\theta,\\phi)$. However, singular behavior of the coupling  function is conceivable when discussing effective interactions of low energy modes.\\cite{binz02}\n\n\nFinally, the non-trivial quartic term $F_{\\rm nt}$ stems from  quartets of modes, whose wavevectors sum up to zero. The geometry of four equal-length  wavevectors summing to zero is depicted in Fig.~\\ref{angles}.  A  quartet-contribution occurs for $0<2\\theta<\\pi$ and $0<\\phi\/2<\\pi$. In this case, the eight wavevectors $\\pm\\bv k_j$  form the vertices of a cuboid. Hence,\n\\begin{equation}\nF_{\\rm nt}={\\sum_{j_1<\\ldots<j_4}}^{\\!\\!\\!\\!\\!\\!\\prime}\\,\\, F_{j_1,j_2,j_3,j_4},\\label{Fq}\n\\ee\nwhere the summation is over such quartets. For example, if $\\bv k_{j_1}+\\bv k_{j_2}+\\bv k_{j_3}+\\bv k_{j_4}=0$, there is a term $ F_{j_1,j_2,j_3,j_4}\\propto\\psi_{j_1}\\psi_{j_2}\\psi_{j_3}\\psi_{j_4}$. \n The algebra of these terms, which depend on the relative phases of  modes, is rather lengthy and obviously  depending on the phase convention used to define the $\\psi$-variables.\n\n\nThe term $F_{\\rm nt}$ fixes the relative phases of $\\psi_j$  to minimize the energy.  \n If there is no frustration between minimizing each individual quartet-term, the phases arrange in such a way that all $F_{j_1,j_2,j_3,j_4}\\leq 0$. It follows that $F_{\\rm nt}\\leq0$, after minimization for the $\\psi$-phases.  A special case is the quartet of modes, whose wavevectors $\\pm k_j$ form the vertices of a cube (e.g. modes along  $\\langle111\\rangle$). \n  In this case,  $F_{j_1,j_2,j_3,j_4}=0$ independently of the coupling function $U(\\theta,\\phi)$ as a consequence of rotational symmetries.\\cite{ashvin06} This is the reason why no such quartet term appears in the theory of Bak and Jensen.\\cite{bak80} As a consequence, a superposition of four modes along all $\\langle111\\rangle$ directions is necessarily unstable towards  shifting the wavevectors away from the perfect cubic configuration in order to gain energy from $F_{\\rm nt}$ (see description of fcc$^*$  below).\n\n\n\n\\subsection{Phase diagram}\n\n\nIf $U(\\theta,\\phi)$ is only slowly varying, it is justified to expand it in spherical harmonics ($Y_{lm}$). That is,\n\\begin{equation}\n\\begin{split}\nU(\\theta,\\phi)&=U_0+U_{11}\\sin\\theta\\cos\\phi\\\\\n& \\quad +U_{20}(3\\cos^2\\theta-1)+U_{22}\\sin^2\\theta\\cos2\\phi,\n\\end{split}\\label{Ulm}\n\\ee\nwhere we retained all terms with $l\\leq2$, which satisfy the relation $U(\\theta,\\phi)=U(\\pi-\\theta,\\phi)=U(\\theta,2\\pi-\\phi)$.\n\nDifferent quartic expressions in terms of the real-space magnetization $\\bv M(\\bv r)$ may lead to the same projected coupling $U(\\theta,\\phi)$. For example, the  three  terms\n\\begin{equation}\n\\left\\langle W\\,\\bv M^4+W'\\,\\left[\\bv\\nabla\\left(\\bv M^2\\right)\\right]^2+W''\\, (\\partial_\\alpha M_\\beta)(\\partial_\\alpha M_\\beta)\\bv M^2\\right\\rangle \\label{W'}\n\\ee\nlead to $U(\\bv q_1,\\bv q_2,\\bv q_3)=W+W'(\\bv q_1+\\bv q_2)^2-W''\\bv q_1\\cdot\\bv q_2$. When projected onto the wavevector sphere, they generate only two spherical harmonics, namely\n\\begin{equation}\n\\begin{split}\nU_0&=W+\\frac13(4W'+W'')Q^2,\\\\\nU_{20}&=\\frac23(2W'-W'')Q^2.\n\\end{split}\\label{dict}\n\\ee\nThe terms $W$ and $W'$ have  intuitive interpretations: $W$ restricts  the magnetization amplitude,  whereas $W'$ favors or disfavors modulations of $\\bv M^2$, depending on the sign.\\footnote{Whenever $W'<0$, Eq.~\\eref{W'} becomes negative for a rapidly varying $\\bv M^2$ and the quartic theory is thus unstable. We consciously ignore this instability, since we only study local minima at small $\\bv M^2$, where $F_2$ is dominant and forbids rapid variations.} In the following, we set $W''=0$ and use the four parameters $W,W',U_{11}$ and $U_{22}$ to tune the interaction $U(\\theta,\\phi)$.\n\n\nIt is hard to find the exact global minimum of $F_2+F_4$ for a general coupling function $U(\\theta,\\phi)$. We therefore restrict ourselves to a certain variational  class of magnetic states and determine the minimum within this class. Recently,\\cite{binz06} we studied  only states which can be obtained by superposition of those six modes which  propagate along the  $\\langle110\\rangle$  directions (6-mode model).\nHere, we explore a much broader class of states. We include\n\\begin{enumerate}\n\\item any superposition of those 13 spin-spirals which propagate along the directions\\footnote{Only the relative orientation between these modes enters the calculation, not their orientation with respect to the atomic lattice, since crystal anisotropy is neglected.} $\\langle111\\rangle$,  $\\langle110\\rangle$ and  $\\langle100\\rangle$ (13 complex variables $\\psi_j$ as variational parameters),\n\\item  any superposition of up to four spirals with arbitrary propagation direction (four  variables $\\psi_j$ and five independent angles between wavevectors as  variational parameters).\n\\end{enumerate}\nWithin these constraints, we  have computed\\footnote{To obtain the phase diagram of Figs.~\\ref{pdiag0} and \\ref{pdiag-2}, we have used the approximation $(1+x)^{5\/2}+(1-x)^{5\/2}\\approx 2+(4\\sqrt{2}-2)x^2$ for $x=\\cos\\theta$ in the evaluation of $V_{p}(\\theta)$. The error is smaller than  0.7\\%.} a phase diagram as a function of the coupling parameters $W,W',U_{11}$ and $U_{22}$, shown in Figs.~\\ref{pdiag-2} and \\ref{pdiag0}.\n\n\n\\begin{figure}\n\\includegraphics[scale=0.6]{fig2.eps}\n\\caption{Phase diagram for  $W>0$ and $W'=0$.  In the grey region, $F4<0$ and the quartic theory is unstable. The various phases are explained in the text. In contrast to our earlier use of these symbols,\\cite{binz06}  ``$\\bigtriangleup$'' and ``$\\square$''  denote general states with 3 and 4 helical modes, respectively.\n\\label{pdiag0}}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[scale=0.6]{fig3.eps}\n\\caption{Same as Fig.~\\ref{pdiag-2} with $Q^2W'=-0.5W$. The point $U_{11}=U_{22}=0$ is now at the phase boundary between spiral order and simple cubic.\n\\label{pdiag-2}}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[scale=0.6]{fig4.eps}\n\\caption{a) and b) For the locations $A,\\ldots,E$ in the phase diagram of Fig.~\\ref{pdiag0},\nthe ratio between the pair interaction $V_{p}(\\theta)$ and the self-interaction $U_{s}$ is plotted as a function of  the angle $2\\theta$ (the angle between propagation directions of modes). A: single-spiral state; B: phase boundary between single-spiral and bcc1; C: bcc1; D: sc; E: $\\bigtriangleup$.\n\\label{Vpfig}}\n\\end{figure}\n\nAll magnetic ground states are equal-amplitude superpositions of 1,2,3,4 or 6 spiral modes.\n The body-centered cubic (bcc) states are superpositions of all  $\\langle110\\rangle$-modes.\n bcc1 and bcc2 differ by the relative phases of the six interfering helical modes (see Section \\ref{bcc1}).  The simple cubic (sc) crystal consists of three mutually orthogonal spirals (e.g. along all $\\langle100\\rangle$ directions). A face-centered cubic (fcc) helical spin crystal is obtained by superposing all four  $\\langle111\\rangle$-modes. However, the  ground state is not fcc but a small distortion of it: fcc$^*$. In fcc$^*$, the wavevectors are  shifted slightly  away from $\\langle111\\rangle$  in order to gain energy from the quartet-term $F_{\\rm nt}$.\\footnote{The shift\n to directions $[\\pm\\sqrt{1+\\delta}\\,\\pm\\!\\!\\sqrt{1+\\delta}\\,\\sqrt{1-2\\delta}\\,]$  is very small with $\\delta\\sim 0.02$ and even without the deformation, fcc would still beat the neighboring phases in the fcc$^*$-region.}  The symbols ``$\\bigtriangleup$'' and ``$\\square$'' are used differently here than in our previous paper.\\cite{binz06} Here, ``$\\square$''  stands for a superposition of four modes with wavevectors as shown in Fig.~\\ref{angles} with  $\\phi\/2=\\pi\/2$. Hence,  wavevectors $\\pm k_j$ form a square cuboid and allow for  one quartet-term $F_{\\rm nt}$. The angle $2\\theta$ changes as a function of interaction parameters within the range  $0.24\\pi<2\\theta<0.38\\pi$.\nFinally, the phase ``$\\bigtriangleup$'' consists of  three modes. The wavevectors $\\bv k_1,\\bv k_2,\\bv k_3$ point to the vertices of an equilateral triangle on the sphere, whose size is determined by the requirement that the mutual angle  between two wavevectors, $2\\theta$, minimizes Eq.~\\eref{V}. The angle $2\\theta$ is parameter-dependent  and lies in the  range $0.14\\pi<2\\theta<0.24\\pi$.\n\nAs expected, a negative $W'$ [Eq.~\\eref{W'}] favors multi-mode spin crystal states with varying magnetization amplitude relative to the spiral state with constant $\\bv M^2$ (compare Figs.~\\ref{pdiag0} and \\ref{pdiag-2}). Positive $W'$ has the opposite effect, and enhances the region of the spiral phase (not shown). The term $W'$ alone (i.e., with $U_{11}=U_{22}=0$) stabilizes sc in the regime $Q^2W'<-W\/2<0$. However, a small positive $U_{22}$ is sufficient to favors bcc1 over sc.\n\nIn conclusion, we observe that  two helical spin crystals, bcc1 and sc, appear adjacent to the single-spiral state and are stable at relatively small values of  $Q^2W'$, $U_{11}$ and $U_{22}$.\n In the following, we study the properties of the bcc1, and sc states since they are the most likely candidates of helical spin crystals from the point of view of energetics.\n\n\n\n\\subsection{Model interactions with exact ground states}\\label{exact}\n\nIn the preceding Section, we established a variational phase diagram for ``natural'', i.e., slowly varying coupling functions $U(\\theta,\\phi)$. Most phases in this phase diagram, (``spiral'', sc, bcc1, bcc2, fcc and ``$\\bigtriangleup$'') can  be shown to be the {\\em exact} global minima for some fine-tuned model interaction, that are constructed below.\n\nLet us consider the toy model $F_{{\\rm toy}}=F_2+F_{s}+F_{p}$ [Eqs.~(\\ref{F2psi}-\\ref{Fp})] where the quartet term $F_{\\rm nt}$ is dropped and we replace $V_{p}(\\theta)$ by a constant $V$. In this model,  local minima with $N$ non-vanishing modes ($N\\geq1$) must have equal amplitudes  $|\\psi_j|^2=|r|\/(U_{s}-V+N V)\/2$ and the minimum energy with $N$ modes is\n\\begin{equation}\nF_{{\\rm toy},N}=-\\frac14\\,\\frac{r^2}{V+\\frac{U_{s}-V}N}.\n\\ee\nThere are three regimes. If $0<U_{s}<V$, the ground state is the single-mode phase with $N=1$ (all but one $\\psi$ are zero). For $0<V<U_{s}$, the energy is monotonically decreasing with $N$. It means that the system includes as many modes as possible to lower its energy. Finally, for $V<0$ or $U_{s}<0$, $F_{{\\rm toy}}$ is not bounded from below and therefore unstable.\n\nTo make use of this toy model, we must tune the interaction such that $U(\\theta,\\phi)\\to0$ unless $\\theta=\\pi\/2$ or $\\phi=0$. This removes the quartet term $F_{\\rm nt}$ and we are left  with a model similar to $F_{{\\rm toy}}$ with the difference that $V_{p}(\\theta)$ is not constant.\n\nWe now tune the interaction such that $V_{p}(\\theta)$ is very big\\cite{verybig} everywhere except at  some angle $\\theta_{\\rm opt}$,\n where it has a  narrow minimum. For such a pair interaction, all arrangements of modes which involve angles other than $2\\theta_{\\rm opt}$ are  excluded. Within this constraint, we are left with our toy model with a constant pair interaction $V=V_{p}(\\theta_{\\rm opt})$, but the number of modes is restricted to $1\\leq N\\leq 3$, since no more than three modes can have equal mutual angles between them. It is clear that in the region $-3U_{s}\/2<V_{p}(\\theta_{\\rm opt})<U_{s}$, the ground state is a helical spin crystal with three modes, i.e., the state ``$\\bigtriangleup$'' or sc (in the case $2\\theta_{\\rm opt}=\\pi\/2$).\n\n\nA different class of exact ground states is obtained, if\nthe interaction is tuned such that\n\\begin{enumerate}\n\\item $V_{p}(\\theta)$ is very big\\cite{verybig} in the region $0<2\\theta<2\\theta_c$ and\n\\item  $V_{p}(\\theta)=V$ (constant) in the region  $2\\theta_c\\leq2\\theta\\leq\\pi\/2$\n\\end{enumerate}\nfor some critical angle $2\\theta_c$. In this way, modes whose wavevectors are too close are excluded. That is, it enforces a ``hard sphere'' constraint  $|\\uv k_i\\pm\\uv k_j|\\geq 2(1-\\cos{2\\theta_c})$ on the wavevectors.\nThe interaction between modes which satisfy this constraint is constant and  reduces to the toy model. Therefore, if $V<U_s$ the ground state will include as many modes as geometrically possible by the ``hard sphere'' constraint. In the case $2\\theta_c=\\arccos 1\/3$, we obtain fcc, which has  wavevectors at the vertices of a cube. In the case $2\\theta_c=\\pi\/3$, we obtain bcc whose wavevectors are the vertices of a cuboctahedron. This can be seen as follows.\nA real-space bcc lattice corresponds to a fcc reciprocal lattice and the wavevectors $\\pm k_j$ of the bcc spin crystals are the  12 nearest neighbors of the origin in the fcc reciprocal lattice. Because fcc is the cubic close packing of spheres, this represents the only arrangement of twelve vectors $|\\bv k_j|=Q$  which satisfies the constraint $|\\uv k_i-\\uv k_j|\\geq1$. (The hexagonal close packing is not acceptable because it does not consist of pairs $\\pm\\bv k_j$.) \n In principle, this construction can be used to create models whose ground state contains an arbitrarily high number of modes.\n\n\n\nWe can now use the insight from these constructed models to understand certain features of the phase diagram of Figs.~\\ref{pdiag0} and \\ref{pdiag-2}.\nIn Fig. \\ref{Vpfig}, we plot $V_{p}(\\theta)$ as obtained from the expansion Eq.~\\eref{Ulm} at different places in the phase diagram of Fig.~\\ref{pdiag0}. In the bcc1 region of the phase diagram (curve C),  $V_{p}(\\theta)$\nis similar to the one constructed above: close to constant and small for  $\\pi\/3\\leq 2\\theta\\leq\\pi\/2$, big for $2\\theta<\\pi\/3$. At the phase boundary between single-spiral and bcc1 (curve B), $V_{p}$ is still bigger than $U_{s}$, indicating that the quartet term $F_{\\rm nt}$ is essential to stabilize bcc1 close to the phase boundary. The phase boundaries between single-spiral and three-mode states (sc or $\\bigtriangleup$) are exactly determined by the crossing of the minimum of $V_{p}(\\theta)$ with $U_{s}$, as illustrated by curves D and E.\n\n\n\n\\section{Structure of helical spin crystals}\\label{structure}\n\n\\subsection{Symmetry properties and topology of helical spin crystals.}\\label{symmetry}\n\nTime-reversal symmetry  ($\\mathcal{T}$)  reverses the magnetization direction $\\bv M\\to-\\bv M$ and may be implemented in terms of the $\\psi$-variables as $\\psi_j\\to-\\psi_j$.\n\nUnder spatial translation by a vector $\\bv a$, the $\\psi$-variables transform as $\\psi_j'=\\psi_j\\exp(i\\bv k_j\\bv a)$, i.e., they experience a phase change.\n\nIf the set of vectors $\\bv k_j$ is linearly independent, every phase-change of modes simply amounts to a global translation. This is the case in helical spin crystals with up to three non-planar modes, e.g. in sc and $\\bigtriangleup$.  These states are periodic with Bravais vectors $\\bv a_1,\\bv a_2,\\bv a_3$, which are determined by $\\bv a_i\\cdot\\bv k_j=2\\pi\\delta_{ij}$. It follows that  time-reversal $\\mathcal{T}$ is  equivalent to a translation by $(\\bv a_1+\\bv a_2+\\bv a_3)\/2$ for these states, which leads to a 2-fold symmetry inside  the unit cell reminiscent of antiferromagnetism.\n\n\nIn contrast, helical spin crystals with more than three modes like bcc, fcc and $\\square$ depend essentially on the relative phases between the helical modes. For these states,  $\\mathcal{T}$ is {\\em not}  equivalent to any translation, since it is not possible that $\\exp(i\\bv k_j\\bv a)=-1$ for all $\\bv k_j$.  As a consequence, these states are doubly degenerate in addition to the translational degeneracy.\n\nTo obtain an understanding of the real-space picture of helical spin crystals, it is useful to study their symmetries under rotations (reflections change the chirality and therefore never appear in the symmetry group).   For example, if the structure $\\bv M(\\bv r)$ has a $n-$fold rotation axis with direction $\\uv u$, then $\\bv M\\parallel\\uv u$ along this axis.\nTherefore,  $\\bv M_\\perp$, which is the projection  to the plane orthogonal to $\\uv u$, has a node at the axis.\n The  winding number\\cite{mermin79} (vorticity) of the node is restricted by symmetry\n to the values $1,1\\pm n,1\\pm 2n$, etc.  \n The resulting pattern in the vicinity of the rotation axis, $\\bv M$ pointing along $\\uv u$ in the center and $\\bv M_\\perp$ winding around it, resembles that of a skyrmion or meron. Such patterns are currently being discussed  in the context of helimagnets.\\cite{bogdanov05,fischer06} Here, we observe that they naturally appear resulting from rotational symmetries.\nThe simplest cases are winding numbers of $+1$ (for any $n$-fold axis)  or $-1$ (only for $n=2$).\n\nApart from proper rotations, the point group may contain anti-rotations, i.e.,  symmetry operations which are composed of a rotation followed by $\\mathcal{T}$. $n$-fold anti-rotation axes are only possible for even numbers $n$, since they imply  $n\/2$-fold rotation symmetry. In the special case $n=2$, it merely follows that $\\bv M\\perp\\uv u$ along the axis, where $\\uv u$ is the  axis direction. Higher anti-rotation symmetries with $n=4,6,\\ldots$ imply $\\bv M=0$ along the axis, i.e., they create a line-node in the magnetization. In the vicinity of this line-node, $\\bv M$ is approximately orthogonal to $\\uv u$ (as can be seen by expanding $\\bv M(\\bv r)$ to linear order around a point on the axis). The winding number of $\\bv M_\\perp$ around the node, is restricted  by symmetry to the values $1\\pm n\/2,1\\pm3n\/2,1\\pm5n\/2$, etc.     \n The simplest case for a 4-fold anti-rotation axis is a winding number of $-1$ (anti-vortex line) and  the simplest case for $n=6$ is a winding number of $-2$.\n\nWe have thus demonstrated the emergence of topological objects like merons and antivortices, which are not expected to be stable in the present context of a vectorial order parameter in three dimensions but which are stabilized by symmetry. Thus, rotation axes are meron lines and 4-fold anti-rotation axes are anti-vortex node-lines.\n\n\n\\subsubsection{Symmetry and real-space picture of bcc1}\\label{bcc1}\n\nThe bcc helical spin crystals consist of six helical modes with wavevectors $\\uv k_1=[1\\bar10]$,  $\\uv k_2=[\\bar1\\bar10]$,  $\\uv k_3=[0\\bar11]$,  $\\uv k_4=[0\\bar1\\bar1]$,  $\\uv k_5=[101]$ and  $\\uv k_6=[10\\bar1]$, as shown in Fig.~\\ref{6modes}, and has the periodicity of a body centered cubic lattice.\n We chose the convention [see Eq.~\\eref{psis}]\n\\begin{equation}\n\\uv \\epsilon_j''=\\frac{\\uv z\\times\\uv k_j}{|\\uv z\\times\\uv k_j|},\\label{conv}\n\\ee\nfor $j=1,\\ldots,6$ with $\\uv z=[001]$, and we consider negative chirality [i.e., $\\bv M\\cdot(\\nabla\\times\\bv M)<0$], such that $\\uv \\epsilon_j'=\\uv k_j\\times\\uv \\epsilon_j''$.\nThe geometry of the wavevectors allows for three quartet terms in the free energy produced by $T_x=\\psi_1^*\\psi_2\\psi_5\\psi_6$,\n$T_y=\\psi_1^*\\psi_2^*\\psi_3\\psi_4$ and\n$T_z=-\\psi_3\\psi_4^*\\psi_5^*\\psi_6$.  The sign in the definition of $T_z$ has been introduced for convenience.\n\n\\begin{figure}\n\\includegraphics[scale=0.6]{fig5.eps}\n\\caption{Geometry of the wavevectors $\\bv k_1,\\ldots,\\bv k_6$, which constitute the bcc spin crystal states. The vectors $\\pm \\bv k_j$ form the vertices of a cuboctahedron.\n\\label{6modes}}\n\\end{figure}\n\nThe transformation properties of the $\\psi$-variables and the three quartic terms  under rotations are shown in Table \\ref{trans}. From this, it can be deduced that the (rotation-invariant) quartet contribution to the free energy is\n\\begin{equation}\nF_{\\rm nt}=2\\lambda_{\\rm nt}\\, \\Re(T_x+T_y+T_z).\n\\ee\nThe value of the parameter $\\lambda_{\\rm nt}$ is not determined by symmetry. Direct calculation yields\n\\begin{equation}\n\\lambda_{\\rm nt}=\\frac74U[\\frac\\pi3,2\\arccos(\\frac13)]-\\frac12U(\\frac\\pi4,\\pi).\n\\ee \n\nThere are thus two bcc states, depending on the sign of $\\lambda_{\\rm nt}$. In bcc1 (bcc2), which corresponds to $\\lambda_{\\rm nt}>0$ ($<0$), the phases of the $\\psi$'s are such that $T_x$, $T_y$, $T_z$ are all negative (positive). Three out of the six phases are arbitrary due to global translation symmetry. This means that the magnetic pattern of bcc1 and bcc2 is uniquely determined up to translational and time-reversal degeneracy.\n\n\\begin{table\n\\begin{ruledtabular}\n\\begin{tabular}{c|cccccc|ccc}\n & $\\psi_1'$ &  $\\psi_2'$ &  $\\psi_3'$ &  $\\psi_4'$ &  $\\psi_5'$ &  $\\psi_6'$ & $T_x'$ & $T_y'$ & $T_z'$ \\\\\n\\hline\n$R_z$ & $\\psi_2$ & $\\psi_1^*$  & $\\psi_6^*$ & $\\psi_5^*$ & $\\psi_3$ & $\\psi_4$ & $T_y$ & $T_x^*$ & $T_z^*$ \\\\\n$R_x$ & $i\\psi_5$ & $-i\\psi_6^*$  & $-\\psi_4^*$ & $\\psi_3$ & $i\\psi_2^*$ & $i\\psi_1$ &  $T_x^*$ & $T_z$ & $T_y^*$\n\\end{tabular}\n\\caption{Transformation properties of the $\\psi$-variables and three quartic terms (defined in Section \\ref{bcc1}) of the bcc spin crystals under rotations. $R_z$ and $R_x$, respectively, are $\\pi\/2$ rotations around the $z$- and $x$-axis. These two rotations generate the cubic point group $O$ and therefore, the behavior under any rotation which maps the 12 wavevectors onto each other may be obtained by combining these two operations.\n\\label{trans}}\n\\end{ruledtabular}\n\\end{table}\n\nThe solution for $\\lambda_{\\rm nt}>0$, bcc1, turns out to be the bcc structure with the highest point group symmetry.\nBy selecting  the coordinate origin conveniently, we obtain  \n $-\\psi_1=\\psi_2=-i\\psi_3=i\\psi_4=i\\psi_5=-i\\psi_6=SM_0$ for bcc1, where $S=\\pm1$ is the time-reversal symmetry label and $M_0>0$ is the amplitude. From Table \\ref{trans}, we deduce that $\\bv M(\\bv r)$ changes sign under a $\\pi\/2$ rotation about the  $x$, $y$ or $z$ axis. That is, the magnetic point group is $O(T)$ (international notation $\\underline{4}32$) with 4-fold anti-rotation axes  at $\\langle 100\\rangle$, 3-fold rotation axes at $\\langle111\\rangle$ and 2-fold anti-rotation axes at $\\langle110\\rangle$.\n\nThe real-space representation of the bcc1 state is\n\\begin{equation}\n\\bv M(\\bv r)=SM_0\\left(\\begin{array}{c}\\sqrt2\\,s_x(c_y-c_z)-2\\,s_ys_z\\\\ \\sqrt2\\,s_y(c_z-c_x)-2\\,s_zs_x\\\\ \\sqrt2\\,s_z(c_x-c_y)-2\\,s_xs_y\\end{array}\\right),\n\\ee\n where $s_x=\\sin(Qx\/\\sqrt{2})$,  $c_x=\\cos(Qx\/\\sqrt{2})$, etc.\n The resulting pattern was shown in Fig.~2 of our earlier paper.\\cite{binz06} In Fig.~\\ref{real-space}, we show the symmetry axes. As discussed above, the magnetization must vanish along the 4-fold anti-rotation axes, which are anti-vortices with winding number $-1$. The  $x$, $y$, $z$ axes, and their translations according to the bcc periodicity, form two interpenetrating cubic latices of such line-nodes. The  cubic space diagonals are 3-fold and the red arrowed lines of Fig.~\\ref{real-space} are 2-fold  rotation axes.\n In the vicinity of these lines, the magnetization field is skyrmion-like (i.e., $\\bv M_\\perp$ has  a winding number of $+1$).\n\nThe fact that the bcc1 state breaks $\\mathcal{T}$ in a non-trivial way and cannot be restored by any translation is manifest in the occurrence of the $\\mathcal{T}$-breaking order parameter $\\langle M_xM_yM_z\\rangle=SM_0^3\/2\\neq0$, which is a magnetic octupole. This curious property may lead to distinctive anomalous effects, e.g. in the magnetotransport.\\cite{binz06b} \nOctupolar magnetic ordering has recently been discussed in different contexts.\\cite{octupole}\n\n\n\\begin{figure}\n\\includegraphics[scale=0.6]{fig6.eps}\n\\caption{(Color online). Symmetry of the bcc1 state. The figure shows a cubic unit cell of bcc1.  Black lines are anti-vortex lines with 4-fold anti-rotation symmetry and vanishing magnetization. The red (dark gray) lines are 2-fold rotation axes and the arrows indicate the direction of $\\bv M$. The structure has 3-fold rotation symmetry about all cubic space diagonals.\n\\label{real-space}}\n\\end{figure}\n\n\\subsubsection{Symmetry and real-space picture of sc}\n\n\nThe simple cubic (sc) helical spin crystal consists of three modes with  $\\uv k_1=[100]$,  $\\uv k_2=[010]$ and  $\\uv k_3=[001]$. It forms a periodic structure with a cubic unit cell and the lattice constant is $\\lambda=2\\pi\/Q$.\n The convention for $ \\epsilon_j''$ is given by  Eq.~\\eref{conv}, where  the unit vector $\\uv z$ is replaced by $[111]$.\n\n\\begin{table\n\\begin{ruledtabular}\n\\begin{tabular}{c|ccccc}\n& $R_x$ & $R_y$ & $R_z$ & $R_{[111]}$ & $R_{[1\\bar 1 0]}$ \\\\\n\\hline\n $\\psi_1'$ & $i\\psi_1$ & $-i\\psi_3$ & $\\psi_2^*$ & $\\psi_3$ & $-\\psi_2^*$ \\\\\n $\\psi_2'$ & $\\psi_3^*$ & $i\\psi_2$ & $-i\\psi_1$ & $\\psi_1$ & $-\\psi_1^*$ \\\\\n $\\psi_3'$ & $-i\\psi_2$ & $\\psi_1^*$ & $i\\psi_3$ & $\\psi_2$ & $-\\psi_3^*$\n\\end{tabular}\n\\caption{Transformation for the $\\psi$-variables of sc under rotations. $R_z$ and $R_x$ are $\\pi\/2$-rotations, $R_{[111]}$ is a $2\\pi\/3$-rotation and $R_{[1\\bar 1 0]}$ a $\\pi$-rotation around the indicated axis.\n\\label{transsc}}\n\\end{ruledtabular}\n\\end{table}\n\n\n\n The transformation properties of the $\\psi$-variables under rotations are given in Table \\ref{transsc}. By choosing the center of coordinates corresponding to $\\psi_1=\\psi_2=\\psi_3=iM_0$, we obtain from Table \\ref{transsc} that the  point group symmetry is $D_3(D_3)$ (international notation 32). That is, the chosen origin has a 3-fold rotation axis along $[111]$ and three two-fold axes along $[1\\bar10]$, $[10\\bar1]$ and $[01\\bar1]$. Obviously, $\\bv M$ must vanish at a point of such high symmetry. Hence there is a point node at the origin.\n\nSymmetry operations consisting of a rotation followed by an appropriate translation yield similar point nodes at $\\frac 12\\frac {3}4\\frac {1}4$ (with 3-fold axis along $[\\bar111]$), $\\frac 14\\frac {1}2\\frac 34$ (3-fold axis $[1\\bar11]$) and   $\\frac {3}4\\frac {1}4\\frac {1}2$ (3-fold axis $[11\\bar1]$). Finally, each of these nodes is doubled inside one unit cell\n because a translation by $(\\frac \\lambda2,\\frac \\lambda2,\\frac \\lambda2)$ amounts to $\\bv M\\to-\\bv M$.\nThe 2- and 3-fold rotation axes form a  complex array of skyrmion-like lines, all with winding numbers of $+1$.\nThe real-space representation is\n\\begin{equation}\n\\bv M(\\bv r)=SM_0\\left(\\begin{array}{c}\n\\tilde c_y-\\tilde s_z\\\\\n\\tilde c_z-\\tilde s_x\\\\\n\\tilde c_x-\\tilde s_y\n\\end{array}\\right),\n\\ee\nwhere  $\\tilde s_x=\\sin[Q(x+\\lambda\/8)]$,   $\\tilde c_x=\\cos[Q (x+\\lambda\/8)]$, etc.\n\n\n\\subsection{Higher harmonics Fourier modes}\\label{higher-harmonics}\n\nAs briefly mentioned in Section \\ref{continuous}, magnetic ordering in wavevectors $\\pm\\bv k_j$ generally induces higher harmonics in the magnetic structure. In the presence of magnetic order $\\bv m_{\\bv k_j}\\neq0$,  the Landau free energy for the modes $\\bv m_{\\bv q}$, which do not belong to the set  $\\bv m_{\\bv k_j}$, is (to quartic order)\n\\begin{equation}\n\\Delta F=\\sum_{\\bv q} \\tilde r_\\psi(\\bv q) |\\bv m_{\\bv q}|^2-\\bv h_{\\psi}(\\bv q)\\cdot \\bv m^*_{\\bv q}-\\bv h^*_{\\psi}(\\bv q)\\cdot \\bv m_{\\bv q}, \\label{induce}\n\\ee\nwith $\\tilde r_\\psi(\\bv q)=r(q)+O\\left(|\\psi_j|^2\\right)$ and\n\\begin{equation}\n\\bv h_{\\psi}(\\bv q)=-4\n \\!\\!{\\sum_{\\bv q_1,\\bv q_2,\\bv q_3}}^{\\!\\!\\!\\!\\!\\!\\prime}\\,\\,U(\\bv\nq_1,\\bv q_2,\\bv q_3)\\left(\\bv m_{\\bv q_1}\\cdot\\bv m_{\\bv\nq_2}\\right)\\,\\bv m_{\\bv q_3},\n\\label{hpsi}\n\\ee\nwhere the sum  is restricted to $\\bv q_1,\\bv q_2,\\bv q_3\\in \\{\\pm\\bv k_{j}\\}$ such that $\\bv q_1+\\bv q_2+\\bv q_3=\\bv q$. The origin of the exchange field $\\bv h_{\\psi}$ is  the  coupling term  Eq.~\\eref{F4}. In the following, we assume that $\\tilde r(\\bv q)>0$.  Obviously, Eq.~\\eref{induce} then leads  to induced modes\n\\begin{equation}\n\\bv m_{\\bv q}=\\frac{\\bv h_{\\psi,\\bv q}}{\\tilde r(\\bv q)}\\label{harmonics}\n\\ee\nat momenta $\\bv q=\\pm\\bv k_{j_1}\\pm\\bv k_{j_2}\\pm\\bv k_{j_3}$.\\footnote{In non-magnetic crystals, the coupling term is cubic and therefore  higher harmonics are generated at all sums of {\\em two} momenta $\\bv k_{j_1}\\pm\\bv k_{j_2}$. In magnetic structures, \n higher-harmonics wavevectors are restricted to  sums of {\\em three} ordering momenta.} These modes modify the detailed magnetic structure, but they do not change its symmetry, since the field $\\bv h_\\psi$ respects all the symmetries of the spin crystal.\n\nWe now briefly discuss the consequences for the three helical magnetic structures under consideration.\n\n\n A single spin-density wave involving wavevectors $\\pm\\bv k$ might create  higher harmonics at $\\pm3\\bv k$ via  Eq.~\\eref{harmonics}. However, in the case of spin spirals, $\\bv m_{\\bv k}^2=0$ and therefore $\\bv h_{\\psi,3\\bv k}=0$.\n Thus, there are  {\\em no} higher harmonics created by a single spin spiral.\n\n\nThe sc spin structure with principal ordering wave-vectors along $\\langle001\\rangle$ with $|\\bv k_j|=Q$\n generates higher harmonics along $\\langle111\\rangle$  (with $|\\bv q|=\\sqrt{3}Q$) and\n along $\\langle012\\rangle$   (with $|\\bv q|=\\sqrt{5}Q$). Note that throughout the current and last sections, all crystal directions refer to the magnetic crystals.\nThe orientation of a magnetic crystal with respect to the atomic crystal depends on the  anisotropy term $F_{\\rm a}$, which will be considered in  Section \\ref{crystalanisotropy}.\n\nIn contrast to the former cases, bcc structures couple linearly to the $\\bv q=0$ mode (i.e., the uniform magnetization), since some triples of ordering vectors  add to zero. This coupling will be further investigated in Section \\ref{H}. Here, we only notice that for the  bcc1 and bcc2 states,  $\\bv h_{\\psi,\\bv q=0}=0$. This results can be understood in terms of the symmetry of these states. In the case of bcc1, the point group symmetry is too high to support a non-zero axial vector $\\bv h_{\\psi,\\bv q=0}$. Therefore, bcc1 and bcc2 do not create a spontaneous net magnetization.\nThe next set of wavevectors which can be reached by adding three ordering vectors are along $\\langle001\\rangle$ (with $|\\bv q|=\\sqrt{2}Q$).  However  for bcc1 and bcc2,  direct calculation shows  $\\bv h_{\\psi}=0$ for these modes. As before, this can be understood in terms of symmetry. Higher harmonics along  $\\langle001\\rangle$  would  have the structure of a  sc spin crystal. We have seen in Section \\ref{symmetry}, that the point group symmetry of sc is lower than that of bcc1. Therefore, bcc1 can not create such an exchange field $\\bv h_{\\psi}$. \n We conclude that bcc1 creates {\\em no} secondary Bragg peaks at  $(0,0,\\sqrt{2}Q)$, etc. The same is true for bcc2.\nThe shortest wavevectors which are created by bcc1 or bcc2 as higher harmonics are along $\\langle 112\\rangle$ (with $|\\bv q|=\\sqrt{3}Q$). Others are at  $\\langle110\\rangle$ ($|\\bv q|=2Q$),   $\\langle013\\rangle$ ($|\\bv q|=\\sqrt{5}Q$),  $\\langle111\\rangle$ ($|\\bv q|=\\sqrt{6}Q$) and  $\\langle123\\rangle$ ($|\\bv q|=\\sqrt{7}Q$).\n\n\\section{Response to crystal anisotropy, magnetic field and disorder.}\\label{response}\n\n\\subsection{Effect of crystal anisotropy}\n\nSo far, our  free energy has\nbeen completely rotation invariant. In the magnetically ordered states,  full rotation symmetry\nis spontaneously broken, but any global rotation of the spin\nstructure leaves the energy invariant.\n This degeneracy is lifted by an additional anisotropy term  $F_{\\rm a}$, which couples the  magnetic crystal to  the underlying atomic lattice. The crystal anisotropy energy is small and may be treated as a perturbation which merely selects the directional orientation, but does not otherwise affect the magnetic state.\n\n\n\\subsubsection{Single-spiral state}\\label{spiral-anisotropy}\n\nIn the case of a single-spiral state, crystal anisotropy is a function $F_{\\rm a}(\\uv k)$, where $\\uv k$ is the spiral direction.\n The function  $F_{\\rm a}(\\uv k)$ may depend on various parameters, it should be symmetric under the\n point group of the (atomic) crystal lattice and satisfy $F_{\\rm a}(\\uv k)=F_{\\rm a}(-\\uv k)$. For concreteness, we assume the cubic point group $T$, relevant for the $B20$ crystal structure. We further assume that $F_{\\rm a}(\\uv k)$ is a slowly varying function, since a singular or rapidly oscillating function in reciprocal space would translate into a (non-local) interaction between magnetic moments and the atomic crystal. Such a function $F_{\\rm a}(\\uv k)$ generally has its minimum at either $\\left\\langle100\\right\\rangle$ or $\\langle111\\rangle$, which can be shown in two different ways.\n\nThe first argument is based on combining symmetry with Morse's theory of critical points.\\cite{morse34}  Morse theory implies that\n\\begin{equation}\n\\mbox{maxima}-\\mbox{saddles}+\\mbox{minima}=2 \\label{morse}\n\\ee\nfor a function on the unit sphere. Symmetry requires that $F_{\\rm a}(\\uv k)$ has stationary points (points with vanishing first derivative, i.e., maxima, minima or saddles) at $\\left\\langle100\\right\\rangle$ (6 directions),  $\\left\\langle111\\right\\rangle$ (8 directions) and  $\\left\\langle110\\right\\rangle$ (12 directions).\nIf  $F_{\\rm a}(\\uv k)$ is slowly varying, we  suspect that these are the only stationary points, since adding more maxima, minima and saddles means that the function is more rapidly oscillating. Under this hypothesis, it follows from Eq.~\\eref{morse}, that the  $\\left\\langle110\\right\\rangle$ directions are saddle points and that the extrema are at  $\\left\\langle111\\right\\rangle$ and  $\\left\\langle100\\right\\rangle$. For   $\\left\\langle110\\right\\rangle$ to be minima,  $F_{\\rm a}(\\uv k)$ needs to have additional stationary points (e.g. saddles) at non-symmetric, parameter-dependend locations. We conclude that an anisotropy  which favors  $\\left\\langle110\\right\\rangle$ would need to be more rapidly oscillating than required by symmetry.\n\nThe second argument is based on an expansion of  $F_{\\rm a}(\\uv k)$ in powers of the directions cosines $\\hat k_x, \\hat k_y,\\hat k_z$:\n\\begin{equation}\n F_{\\rm a}(\\uv k)=\\alpha\\, (\\hat k_x^4+\\hat k_y^4+\\hat k_z^4) + \\alpha' \\, \\hat k_x^2\\hat k_y^2\\hat k_z^2+\\ldots,\\label{Fa}\n\\ee\nwhere we retained the first two terms allowed by cubic symmetry. Because of the smallness of the wave-vector sphere $Q$, one  typically expects $|\\alpha'|\\ll |\\alpha|$ and subsequent terms even smaller. It is easily checked that for most values of the parameters $\\alpha,\\alpha'$, Eq.~\\ref{Fa} has its global minima at either  $\\left\\langle100\\right\\rangle$ (for $\\alpha<\\min\\{0,\\alpha'\/18\\}$) or  $\\left\\langle100\\right\\rangle$ (for $a>\\max\\{2\\alpha'\/9,\\alpha'\/18\\}$). Only in the narrow parameter regime $0<\\alpha<2\\alpha'\/9$, the minima are indeed at $\\left\\langle110\\right\\rangle$.\\footnote{In this regime, all 26 high-symmetry points are extrema of $F_{\\rm a}$ and 24 saddle points are located at $\\left\\langle\\sqrt{2\\alpha}\\,\\sqrt{2\\alpha}\\,\\sqrt{\\alpha'-4\\alpha}\\right\\rangle$, in agreement with Eq.~\\eref{morse}.} We conclude that crystal anisotropy  which favors  $\\left\\langle110\\right\\rangle$ may only appear in a narrow regime between two phases which favor  $\\left\\langle111\\right\\rangle$ and   $\\left\\langle100\\right\\rangle$, respectively.\n\nAccordingly,   $\\left\\langle111\\right\\rangle$ or   $\\left\\langle100\\right\\rangle$ are the selected spiral directions in all cubic helimagnets known so far.\\cite{ishikawa76,beille83,lebech89}\nThe preferred direction in MnSi at low pressure is $\\left\\langle 111\\right\\rangle$ and in  Fe$_x$Co$_{1-x}$Si, it is  $\\left\\langle 100\\right\\rangle$.  In FeGe, there is a phase transition between these two directions,\\cite{lebech89} but no  intermediate phase with  $\\left\\langle110\\right\\rangle$ spiral orientation has been reported. However,  the neutron scattering data\\cite{pfleiderer04}  in the partially ordered phase of MnSi clearly show a maximum signal along the $\\left\\langle110\\right\\rangle$ crystal directions. While it is initially tempting to  interpret the partially ordered state of MnSi as a  single-spiral state that has lost it's orientational long-range order by some mechanism,  one would still expect a maximal scattering intensity in the energetically preferred lattice direction.\n Theories of the partially ordered state in terms of disordered helical spin-density waves\\cite{tewari05,grigoriev05} thus depend on a crystal anisotropy that prefers spiral directions along $\\left\\langle 110\\right\\rangle$. As we have shown, this seems very unlikely  .\n\n\n\n\n\\subsubsection{Helical spin crystals}\\label{crystalanisotropy}\n\n\n For multi-mode spin crystals, \n $F_{\\rm a}$  is no longer determined by a single direction $\\uv k$, so the arguments of Section \\ref{spiral-anisotropy} do not apply.  Rather, the anisotropy energy depends on  three Euler angles, which rotate the full three-dimensional magnetic structure relatively to the atomic crystal. In other words, $F_{\\rm a}$  is a function of the rotation group $SO(3)$. Relative to some standard orientation $\\bv k_j$ of the mode directions,  the  leading-order anisotropy term is\n\\begin{equation}\nF_{\\rm a}(R)=a\\sum_{j}g({R\\uv k_j})\\,|\\psi_j|^2\n\\label{crysFa}\n\\ee\nwhere \n $R$ is a rotation operator and   $g(\\uv k)=\\hat k_x^4+\\hat k_y^4+\\hat k_z^4$. As before, we have assumed a cubic point group symmetry.\n\nIn the case $a>0$, the modes of the bcc  spin crystals get locked to the  $\\langle110\\rangle$ directions. The  orientation of sc is four times degenerate if $a>0$. The four minima of $F_{\\rm a}$ are obtained from the standard orientation along $\\langle100\\rangle$ through a $\\pi\/3$-rotation around any of the four space diagonals, such that the three spiral modes point along  $\\langle122\\rangle$.\n\nIn the opposite case ($a<0$), sc is oriented along $\\langle100\\rangle$. This time, it is the bcc spin crystals that get rotated by  $\\pi\/3$  around any  $\\langle111\\rangle$ axis to reach one of four stable orientations.  Under such  $\\pi\/3$-rotations, three of the six modes remain along $\\langle110\\rangle$  and three  move to $\\langle114\\rangle$. Each individual  $\\langle114\\rangle$ direction appears only in one of the four solutions but each  $\\langle110\\rangle$ direction appears in two of four solutions.\n\n\nIf there is more than one degenerate\n orientation, the sample typically breaks up into domains such that  full cubic symmetry is restored in the neutron scattering signal.  Table \\ref{orienttable} lists the directions of magnetic Bragg peaks for the \n different cases.\n Out of the three prominent phases in our phase diagram,  the bcc1 spin crystal is the only one  that can explain the neuron-scattering peaks along $\\langle110\\rangle$ in the 'partial order' phase of MnSi. It does so most naturally for the case $a>0$, which is the known sign of the anisotropy in MnSi at low pressure.\n\n\nIn order to compare the energy scale of $F_{\\rm a}$ (i.e., the locking energy) for the different magnetic states, we note the following.\n At the phase boundary between two equal-amplitude spin-crystal phases (one phase with amplitudes $|\\psi_1|=\\ldots=|\\psi_N|$ and the second phase with $|\\tilde\\psi_1|=\\ldots=|\\tilde\\psi_{\\tilde N}|$)\nthe amplitudes of the two neighboring phases are related by\n\\begin{equation}\nN|\\psi_j|^2=\\tilde N|\\tilde\\psi_{j'}|^2.\n\\ee\nIn the vicinity of the phase boundary, the anisotropy term is therefore proportional to the mode-average  $1\/N\\sum_j g(R\\uv k_j)$. Using this result, we find  that the effective anisotropy energy is smaller for the bcc spin crystals than for the single-spiral state by a factor of 4 - 4.5.\\footnote{Using $\\max\\{F_{\\rm a}\\}-\\min\\{F_{\\rm a}\\}$ to estimate the locking energy scale leads to a reduction factor of 4.5. Expanding $F_{\\rm a}$ around its minimum for $a>0$ leads to a  factor of 4.}\nFor the sc state, the locking energy is  anisotropic. Certain rotations are equally costly in energy as for the single-spiral case while some small rotations about the minimum for $a>0$ are softer than for the single-spiral  by a factor of 4.5.\n\n\n\\begin{table\n\\begin{ruledtabular}\n\\begin{tabular}{c|cc|cc|cc}\n & spiral & No.  & bcc & No.  & sc & No.  \\\\\n\\hline\n $a>0$ &$\\langle111\\rangle$ & 4 & $\\langle110\\rangle$ & 1 &$\\langle122\\rangle$ & 4\\\\\n $a<0$ &$\\langle100\\rangle$ & 3 & $\\langle110\\rangle$,$\\langle114\\rangle$\\footnote{When averaged over domains, Bragg peaks along $\\langle110\\rangle$ are twice as intense as peaks along $\\langle114\\rangle$.} & 4 &$\\langle100\\rangle$ & 1\n\\end{tabular}\n\\caption{Crystal directions of magnetic Bragg peaks and number  of degenerate orientations for three magnetic structures, sc, bcc and single-spiral, with crystal anisotropy given by Eq.~\\eref{crysFa}.\n\\label{orienttable}}\n\\end{ruledtabular}\n\\end{table}\n\n\\subsection{Effect of magnetic field}\\label{H}\n\n\nA uniform external magnetic field $\\bv H$ couples to the $\\bv q=0$ mode of the\nmagnetization, $\\bv m=\\langle\\bv M\\rangle$, via Zeeman coupling.\nThe uniform magnetization, in turn, couples to the helical modes $\\psi_j$ through\n\\begin{equation}\n\\begin{split}\nF=&\\left.F\\right|_{\\bv m=0}+r_0\\,\\bv m^2+U(0,0,0)\\,\\bv m^4-\\bv h_{\\psi}(0)\\,\\bv m\\\\\n&+2\\sum_j\\left(U(\\bv k_j,-\\bv k_j,0)\\,\\bv m^2+U(\\bv k_j,0,0)\\,\\bv m^2_{\\perp,j}\\right)|\\psi_j|^2,\\label{Fm}\n\\end{split}\n\\ee\nwhere  $\\bv m_{\\perp,j}=\\bv m-(\\bv m\\cdot\\uv k_j)\\uv k_j$ and we have used Eqs.~\\eref{F4},\\eref{hpsi}.  Plumer and Walker\\cite{plumer81} argued that $U(0,0,0)\\approx U(\\bv k_j,-\\bv k_j,0)\\approx U_{s}$, which we will use in the following for simplicity.\n\n\\subsubsection{Response of the single-spiral state}\n\n\nThe behavior of the single-mode spiral state under a\nmagnetic field has been studied both experimentally\\cite{ishikawa76,lebech89,ishimoto95, thessieu97,uchida06} and theoretically.\\cite{kataoka81,plumer81}\n It is characterized by a strongly anisotropic susceptibility,\ninduced by the last term in Eq.~\\eref{Fm}. For a fixed spiral direction $\\bv k$, the  susceptibilities parallel and orthogonal to the spiral direction are given by\n\\begin{equation}\n\\begin{split}\n\\chi_\\parallel\\,\\approx&\\,\\Delta^{-1}\\\\\n\\chi_\\perp\\approx&\\left(\\Delta+2\\frac{U'}{U_{s}}|r(Q)|\\right)^{-1},\n\\end{split}\n\\ee\nwhere $\\Delta=2[r_0-r(Q)]=2JQ^2$ and $U'=U(\\bv k,0,0)$. Well below the critical ordering temperature, $\\Delta\\ll 2|r(Q)|$ and therefore $\\chi_\\parallel\\gg \\chi_\\perp$. This strong anisotropy leads to a spin reorientation transition at $H=H_{\\rm sr}$, where the spiral axis gets oriented along the field direction. The value of $H_{\\rm sr}$ depends on the anisotropy [Eq.~\\eref{Fa}] and the field direction. For $\\alpha>0$ and $\\bv H\\parallel \\langle 100\\rangle$, Plumer and Walker obtained\n\\begin{equation}\nH_{\\rm sr}^2=\\frac{4\\alpha}{\\chi_\\parallel-\\chi_\\perp}\\gtrsim 4 \\alpha \\Delta,\\label{Hsr}\n\\ee\nwhere we have used  $\\chi_\\parallel\\gg \\chi_\\perp>0$. Once the spiral is oriented, the susceptibility is large (equal to $\\chi_\\parallel$).\n\nThe spiral amplitude decreases as a function of the external field and vanishes at\n\\begin{equation}\n H_{\\rm c}=|\\psi_0|\\Delta,\\label{Hc}\n\\ee\nwhere $|\\psi_0|^2=|r(Q)|\/(2U_{s})$. Above $H_{\\rm c}$, the magnetization is uniform.\n\n\n\\subsubsection{Response of the bcc1 spin crystal}\\label{bcc-and-field}\n\n In the bcc spin crystal\nstates, the linear response is isotropic, because\ntheir symmetry group does not allow for an anisotropic\nsusceptibility tensor. As a consequence, there is no orientation of the bcc state towards the magnetic field at the level of linear response (i.e., from energies up to order $\\bv H^2$). However, there is a sub-leading contribution to the energy $\\propto\\langle M_xM_yM_z\\rangle H_xH_yH_z$. This contribution splits the degeneracy between the $S=1$ and $S=-1$ states and it may lead to a reorientation of the bcc crystal towards the field.\n\n\nIn terms of the six $\\psi$-variables of bcc, the exchange field $\\bv h_\\psi(0)$, which enters Eq.~\\eref{Fm}, amounts to\n \\begin{equation}\n\\bv h_\\psi(0)=-\\mu \\Re{\\left[(5-i\\sqrt{2})\\bv{\\tilde h}_\\psi\\right]},\n\\ee\nwhere $\\mu=U(\\bv k_{j_1},\\bv k_{j_2},\\bv k_{j_3})$ for $\\bv k_{j_1}+\\bv k_{j_2}+\\bv k_{j_3}=0$ and\n\\begin{equation}\n\\begin{split}\n\\bv{\\tilde h}_\\psi= & \\,\\psi_1\\psi_3^*\\psi_6^*\\left(\\begin{array}{c}1\\\\1\\\\1\\end{array}\\right)\n+\\psi_1^*\\psi_4\\psi_5\\left(\\begin{array}{c}-1\\\\-1\\\\1\\end{array}\\right) \\\\\n& -\\psi_2\\psi_4^*\\psi_6\\left(\\begin{array}{c}-1\\\\1\\\\-1\\end{array}\\right)\n-\\psi_2^*\\psi_3\\psi_5^*\\left(\\begin{array}{c}1\\\\-1\\\\-1\\end{array}\\right).\n\\end{split}\n\\ee\n\nThe (isotropic)  inverse spin susceptibility (see Appendix) in the bcc1 state is composed of three contributions\n\\begin{equation}\n\\chi^{-1}_{\\rm bcc1}=\\chi^{-1}_{\\rm bare}+\\chi_{\\rm phase}^{-1}+\\chi_{\\rm amp}^{-1}.\\label{chibcc1}\n\\ee\nThe first term\n\\begin{equation}\n\\chi^{-1}_{\\rm bare}=2\\left(r_0+\\frac{U_s+\\frac23 U'}{U_{\\rm bcc1}}|r(Q)|\\right),\n\\ee\nwhere $U_{\\rm bcc1}=1\/6[U_{s}+V_{p}(\\pi\/4)+4V_{p}(\\pi\/6)-\\lambda_{\\rm nt}]$,  can be derived in analogy to the single-spiral case. In fact, $\\chi_{\\rm bare}$ is a ``mixture'' of $\\chi_\\parallel$ and $\\chi_\\perp$, determined geometrically by the angles between the mode directions $\\bv k_j$ and the magnetic field. It follows that $\\chi_{bare}\\ll \\chi_\\parallel$, provided $U_{\\rm bcc1}\\sim U_{s}$ (the two couplings are equal at the phase boundary between single-spiral and bcc1).\nThe remaining terms in Eq.~\\eref{chibcc1}\n\\begin{equation}\n\\begin{split}\n\\chi_{\\rm phase}^{-1}&=-\\,\\frac{\\mu^2\\,|r(Q)|}{3\\,U_{\\rm bcc1}\\,\\lambda_{\\rm nt}}\\\\\n\\chi_{\\rm amp}^{-1}&=-\\,\\frac{25\\,\\mu^2\\,|r(Q)|}{12\\,U_{\\rm bcc1}\\,[U_{s}-V_{p}(\\pi\/4)+\\lambda_{\\rm nt}]},\n\\end{split}\\label{chi-phase-amp}\n\\ee\nstem from the response of the bcc magnetic structure to the field. That is, they originate from the adjustments of relative phases and amplitudes, respectively, of the helical modes as a result of the term $-\\bv h_\\psi(0)\\cdot\\bv m$ in  Eq.~\\eref{Fm}. The effect of $\\chi_{\\rm phase}$ and $\\chi_{\\rm amp}$, which are necessarily negative, is to increase the susceptibility of the bcc1 state.\n\nThe change in\nthe relative amplitudes and phases of the six interfering spirals as\na function of the magnetic field may  be calculated  (see Appendix). For example, the\nlinear response of the amplitudes of bcc1 is\n\\begin{equation}\n\\left(\\begin{array}{c}\n\\delta|\\psi_1|\\\\ \\delta|\\psi_2|\\\\ \\delta|\\psi_3|\\\\ \\delta|\\psi_4|\\\\ \\delta|\\psi_5|\\\\ \\delta|\\psi_6|\\\\\n\\end{array}\\right)=\\frac{5\\,\\mu\\, S}{4 [U_{s}-V_{p}(\\pi\/4)+\\lambda_{\\rm nt}]}\\left(\\begin{array}{r}\n-m_z\\\\ m_z\\\\ -m_x\\\\ m_x\\\\ m_y\\\\ -m_y\n\\end{array}\\right).\n\\ee\nThis response should be observable by neutron scattering, if it is\npossible to prepare the sample in a single-domain state (i.e.\nwithout mixture of the two time-reversal partners). For example, a field in $\\uv z$ direction affects $|\\psi_1|$ and $|\\psi_2|$, the amplitudes of the modes propagating orthogonally to $\\uv z$ (Fig. \\ref{6modes}), which get enhanced and suppressed by the magnetic field, respectively. \n\nThe expected effects of external magnetic field on the resistivity of bcc spin crystal are presented elsewhere.\\cite{binz06b}\n\n\n\\section{Effect of impurities: a possible route to 'partial order'.}\\label{disorder}\n\nWhile the helical spin crystal states are expected to show Bragg\nspots at particular wavevectors, a variety of effects such as\nthermal or quantum fluctuations or disorder can destroy the long\nrange order while preserving the helical spin crystal structure at\nshorter scales. Here, we investigate in more detail the effect of\nnon-magnetic disorder on helical spin crystal structures.\n\n Although\nthe experimentally studied helimagnets are very clean from the\nelectrical resistivity point of view, the helical magnetic\nstructures are sensitive to disorder at a much longer length scale.\nIn addition, the low energy scales required to distort them means\nthat one needs to consider disorder effects. An observation that can\nimmediately be made is that for the physically relevant case of\nnon-magnetic disorder ($V_{dis}(\\bv r)$), the single-spiral state\nand the spin crystal states respond very differently. By symmetry,\nthe coupling of disorder to the magnetic structure is given by\n$F_{dis}=\\langle V_{dis}(\\bv r) |\\bv M(\\bv r)|^2\\rangle$. Hence,\nsingle-spiral states which are unique in having a spatially uniform\nmagnitude of magnetization ($|\\bv M(\\bv r)|={\\rm constant}$) are\nunaffected by this coupling; in contrast the spin-crystal states\nnecessarily have a modulated magnitude\\cite{binz06} and hence are\naffected by non-magnetic disorder. Therefore the neutron scattering\nsignal of the spin-crystal state is expected to have more diffuse\nscattering than the single mode state. This is consistent with the\nexperimental observation that the high pressure phase has diffuse\nscattering peaked about $\\langle110\\rangle$ while the low pressure\nphase has sharper spots, consistent with identifying the two as\nspin-crystal and single-spiral states respectively.\n\n The effect of disorder on the\nspin-crystal state is closely related to the problem\nof the ordering of an XY model in the presence of a random external field.\nThe phase rotation symmetry of the XY model captures the\ntranslational invariance of the spin-crystal in the clean state.\nDisorder destroys this invariance and behaves like a random field\napplied to the XY system. Using the insights from the study of that\nproblem in three dimensions,\\cite{giamarchi95}\n one expects that for weak\ndisorder a Bragg glass will result, where although true long range\norder is destroyed, power law divergent peaks at the Bragg\nwavevectors remain, and the elastic constants remain finite. For\nstronger disorder one expects this algebraic phase to also be\ndestroyed, and recover a short range correlated phase without\nelasticity. Nevertheless, for the case of the bcc1 and bcc2\ncrystals, due to time reversal symmetry ($\\mathcal{T}$) breaking in\nthese states, the disordered states also spontaneously break time\nreversal symmetry, and hence a phase transition is expected on\ncooling despite the absence of long range order. It is difficult to\npredict which of these two scenarios (Bragg glass or only\n$\\mathcal{T}$ breaking) is more appropriate for MnSi. In the latter\ncase one may estimate the spreading of the Bragg spots due to\ndisorder by considering the energetic cost to deform the\nspin-crystal state in different ways.\n\nIgnoring elastic contributions,\n there are two\ndistinct types of deformations - ones that involve a change in the\nmagnitude of the ordering wavevectors $\\delta q_\\parallel$\n and others that do not change\nthe wavevector magnitude but rotate the structure from its preferred\norientation: $\\delta q_\\perp$. The second is expected to be low in energy because\nrotations of the structure are locked by the crystal anisotropy\nterm, which is weak.\n From Eq.~\\eref{Fa}, we obtain the energy cost to shift the ordering vector  by $\\delta q_\\perp$ along the sphere $|\\bv q|=Q$\n\\begin{equation}\n\\delta F_\\perp=\\frac{4\\alpha}{3\\kappa}\\left(\\frac{\\delta  q_\\perp}Q\\right)^2,\\label{Fperp}\n\\ee\nwhere $\\kappa=1$ for the single-spiral. For multi-mode spin crystals, the energy cost of rotation is reduced, as explained in Section \\ref{crystalanisotropy}. Thus, $\\kappa\\approx4$ for the bcc spin crystals.\nIn contrast, deformations that change the magnitude of the ordering\nwavevectors, must contend with the DM\ninteraction scale, and hence pay a higher energy penalty\n\\begin{equation}\n\\delta F_\\parallel=\\frac12\\Delta\\cdot|\\psi_0|^2\\cdot\\left(\\frac{\\delta  q_\\parallel}Q\\right)^2.\\label{Fpar}\n\\ee\nAssuming\nthe disorder couples to these deformations equally, we can estimate\nthe ratio of their amplitudes in the limit of weak deformations, by equating Eqs.~\\eref{Fperp} and \\eref{Fpar}. It follows\n\\begin{equation}\n\\left(\\frac{\\delta  q_\\perp}{\\delta  q_\\parallel}\\right)^2=\\frac{3\\kappa\\Delta|\\psi_0|^2}{8\\alpha}.\n\\ee\nUsing Eqs.~\\eref{Hsr} and \\eref{Hc}, we can relate this ratio to the experimentally known ratio between the critical magnetic fields for, respectively, reorienting and  polarizing the single-spiral state\n\\begin{equation}\n\\frac{\\delta  q_\\perp}{\\delta  q_\\parallel}\n\\gtrsim\\sqrt{\\frac{3\\kappa}2}\\frac{H_{\\rm c}}{H_{\\rm sr}}.\n\\ee\n\nWe can now apply these results to the case of MnSi and test the\nhypothesis that the 'partial order' state is in fact a\ndisordered bcc spin crystal. Setting in $\\kappa=4$ and the\nexperimentally measured\\cite{thessieu97} critical fields for MnSi,\n$H_{c}=0.6{\\rm Tesla}$ and $H_{c1}=0.1{\\rm Tesla}$, one obtains\n$\\delta q_{\\perp}\/\\delta q_\\parallel\\gtrsim 15$. Neutron scattering\nexperiments do indeed find that the transverse broadening is larger\nthan the longitudinal broadening, but since the latter is resolution\nlimited, this only gives us an lower bound that is consistent with\nthe estimate above: $[\\delta q_{\\perp}\/ \\delta\nq_{\\parallel}]_{\\rm{expt}}> 2.3$. Nevertheless, the trend that the\nwidth of the spot is greater along the equal magnitude sphere than\ntransverse to it is clearly seen in the experimental data.\n\nThus, weak non-magnetic disorder of the atomic crystal is expected to destroy magnetic long range order in multi-mode helical spin crystal states and lead to a neutron scattering signal compatible with the observations in the 'partial order' phase of MnSi. \n However in the case of bcc spin crystals, time-reversal symmetry breaking is expected to persist even in the presence of disorder. The scenario of interpreting 'partial order' in MnSi as a bcc1 state disordered by impurities thus predicts  quasi-static local magnetic moments \n  and implies  a finite temperature phase transition on cooling into this phase. \n \n\\section{Conclusion}\n\nWe have analyzed the magnetic properties of non-centrosymmetric weak ferromagnets subject to DM spin-orbit coupling. This problem falls into the general class of systems where the low energy excitations live on a surface in reciprocal space rather than on discrete points.\nThe addition of DM interactions to a ferromagnetic state produces a\nlarge degeneracy of magnetic states characterized by arbitrary\nsuperpositions of spin helices of a fixed helicity and fixed\nwavevector magnitude. This enormous degeneracy is broken by\ninteractions between modes, and the single-spiral state is realized\nfor slowly varying interactions, by virtue of its unique property of\nhaving a spatially uniform magnitude of magnetization. For more\ngeneral interactions, multi-mode helical spin crystal states are\nobtained. We show that for the model interactions considered, the\nphase diagram is largely determined just by considering the\ninteractions between pairs of modes. The phase that is eventually\nrealized may be readily deduced from the range of angles in which\nthis interaction drops below a critical value. In particular, the\nbcc structure is stabilized by virtue of the fact that its\nreciprocal lattice, fcc, is a close packed structure. These results\nmay also be relevant in other physical situations where\ncrystallization occurs, such as the Larkin-Ovchinnikov-Fulde-Ferrel\n instability in spin-imbalanced superconductors, which may\npotentially be realized in solid state systems,\\cite{CeCoIn5} cold\natomic gases\\cite{zwierlein} and dense nuclear matter.\\cite{Rajagopal}\n\nHelical spin crystals typically give rise to complicated real space\nmagnetic structures which we discussed in this paper. In particular,\ntopological textures like merons and anti-vortices can be seen about\nspecial axes in particular realizations, although these are not\nexpected to be stable given the order parameter and spatial\ndimensionality of the system. We show here that such topological\nstructures exist as a consequence of symmetry, which also dictates\nthe absence of certain higher Bragg reflections, which a naive\nanalysis would predict.\n\nThe response of helical spin crystals to crystalline anisotropy and\napplied magnetic field are considered with a special emphasis on the\nbcc structures which are contrasted against the response of the\nsingle helix state. An unusual transfer of spectral intensity in the\npresence of an applied magnetic field, which is strongly dependent\non the direction of applied field is noted for the bcc structures.\nThis is a consequence of broken time reversal symmetry in the\nabsence of a net magnetization (which is symmetry forbidden). The\nunusual magnetotransport in such a state, a linear in field\nmagnetoresistance and quadratic Hall effect, has been discussed\nbriefly in Ref.~\\cite{binz06} and was elaborated upon in Ref.~\\cite{binz06b}.\n\n Helical spin crystals exhibit Bragg peaks at specific\nwave-vectors, and hence are not directly consistent with the\nexperimental observation of 'partial order'. The point of view taken\nin our earlier work\\cite{binz06} is that the short distance and\nshort time properties are captured by the appropriate helical spin\ncrystal structure. Studying the properties of helical spin crystals\nwith long range order is a theoretically well defined task with\ndirect consequences for a proximate disordered phase with similar\ncorrelations up to some intermediate scale. The mechanism\nthat leads to the destruction of long range helical spin crystal\norder is uncear; in Ref.~\\cite{binz06}, this was assumed to be the coupling\nto non-magnetic disorder. Then, as elaborated in this paper,\nbeginning with a bcc helical spin crystal a neutron scattering\nsignature consistent with that of 'partial-order' may be obtained.\nHowever, within the simplest version of this scenario, one also\nexpects a finite temperature phase transition where time reversal\nsymmetry breaking develops, and static magnetic order which may be\nseen in nuclear magnetic resonance or muon spin rotation\nexperiments. Other mechanism for the destruction of long range order\nof the bcc spin crystal state, such as thermal or quantum\nfluctuations may also be considered, but are left for future work.\n\n\n\\acknowledgments We would like to thank L. Balents, P. Fazekas, I. Fischer, D.\nHuse, J. Moore, N. Nagaosa, M.P. Ong, C. Pfleiderer, D. Podolsky, A. Rosch, T.\nSenthil, H. Tsunetsugu and C. Varma for useful and stimulating discussions and V.\nAji for an earlier collaboration on related topics. This work was\nsupported by the Swiss National Science Foundation, the A.P. Sloan\nFoundation and Grant No. DE-AC02-05CH11231.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe paper is written for the special issue dedicated to the outstanding physicist Mark Azbel. It addresses the problems of quantum physics, to which Prof. Azbel made a number of seminal \ntheoretical contributions \\cite{AzbelP,AzbelN,Azbel}. \n\nThe angle variable $\\varphi$ is ubiquitous in classical  and  quantum physics. Among  examples of this variable are the rotation angle of the plane rotator  (mechanical pendulum  as a particular case) and the phase difference across  the single Josephson junction (JJ).  The  canonically  conjugate variable to the phase is the angular momentum $M$ (further called moment) in the first example and  the charge  in the second one. \n\nThe history of using the canonically conjugate  pair  ``angle  (phase)--moment'' in quantum mechanics is full of controversies and disputes. In particular, the commutation  relation \n\\begin{equation}\n[\\hat \\varphi,\\hat M]=i\\hbar\n    \\ee{ComS}\nintroduced by \\citet{dirac} was challenged \\cite{judge,PhasRev,Pegg}. Here  $\\hat \\varphi$ and $\\hat  M$ are operators of the angle  (phase) and  the moment, respectively. The problem with the commutation relation was connected with  the non-Hermitian character  of  the phase operator. It was suggested to repair this mostly mathematical flaw  by rewriting the commutation relation in the terms of Hermitian operators  $\\sin \\hat \\varphi$ and $\\cos \\hat \\varphi$ \\cite{suss}. The uncertainty relation \n\\begin{equation}\n\\Delta M \\Delta \\varphi > {\\hbar\\over 2} \n   \\ee{unc}\nwas also under scrutiny \\cite{judge,PhasRev,Pegg}.  Here  $\\Delta M$ and $ \\Delta \\varphi$ are uncertainties of the moment and the phase, respectively.\n \nAnother problem  (but connected with the first one) is that the phase  $\\varphi$ is defined modulo $2\\pi$. One can choose the phase defined in  an   interval from an arbitrarily chosen phase  $\\varphi_0$  to $\\varphi_0  +2\\pi$  (compact phase),  or  the phase ranging from $-\\infty$ to $\\infty$ (extended phase). If  the  phases differing by the integer number $s$ of  $2\\pi$ describe the same state, it does not matter  at all which phase, compact or extended,  one uses in  the theoretical analysis.  The analysis (if correct) must lead to the same results. However, in quantum mechanics there are some  subtleties, and there is no consensus  on the dilemma ``compact vs. extended phase''. \n\nThe proponents of  the suggestion that only the compact phase must be used in the quantum theory of the JJ argue that it is natural to expect that that the states with the phases $\\varphi$ and $\\varphi+2\\pi$ are the same state and  only states with wave functions periodic in  $\\varphi$ with the period $2\\pi$ are possible. This means that the variable canonically conjugated to the phase (moment in a quantum rotator, or charge in a JJ) is quantized. The proponents of the extended  phase argue, that the ``natural expectation'' of the identity of the states with the phases $\\varphi$ and $\\varphi+2\\pi$ is not so natural and is invalid in the case of the JJ because of its interaction with the environment.  Then different phases in the whole interval  $-\\infty<\\varphi <\\infty$ always correspond to different states. This was called ``decompactification of the phase''.  \n\nIt  is important to stress, however, that the compact phase is sufficient for description of states, but not for description of dynamical processes of transitions between states with different phases. In these processes it is necessary  to know  not only the variation of the compact phase but also an integer number $s$ of full $2\\pi$ winding during the process.  It is more convenient instead of two variables to deal only with one variable, which is an extended phase determined in the interval $(-\\infty,\\infty)$\n\\begin{equation}\n\\varphi(t)  =2\\pi s(t) +  \\varphi_c(t).\n      \\ee{comp}\nHere $\\varphi_c$ is the compact phase determined in any interval of the length $2\\pi$. The voltage drop over the JJ is determined by the time derivative of the extended but not the compact phase. The time derivative of the compact phase has unphysical jumps when the phase reaches borders of the $2\\pi$ interval chosen for the compact phase.\nThus, one should not interpret  the requirement of using only the compact phase (phase compactification) literally but interpret it as  the requirement of using only wave functions periodic in the extended phase $\\varphi$ with the period $2\\pi$. Under  the assumption of decompactification this requirement is abandoned.\nThus, the dilemma ``compact vs. extended  phase'' is in fact  the dilemma whether the  states with the phases $\\varphi$ and $\\varphi+2\\pi$  indistinguishable or distinguishable. Nevertheless, further in the paper the dilemma will be called  ``compact vs. extended  phase'' as widely accepted in the literature.\n\n\nActuality of this dilemma  reemerged in the recent dispute about the Dissipative Quantum Phase  Transition (DQPT) \\cite{Sacl,CommMurani,ReplMurani}  between the superconducting and insulating states of a single JJ predicted about 40 years ago \\cite{Schmid,Bulg}.  \\citet{Sacl} claimed that their  experiment and theory proved the absence of  the  DQPT, because the single JJ cannot be an insulator. \n\nThe estimation done in Ref.  \\onlinecite{CommMurani} demonstrated that the experiment of Murani {\\em et al.} was done at the conditions, in which the existing theory did not predict the DQPT. Therefore, the experiment could not provide  any evidence either pro or contra the DQPT. Their theoretical arguments were also rejected, but they deserve an  analysis more detailed than it was possible within  a short Comment \\cite{CommMurani}. In particular, Murani {\\em et al.} \\cite{Sacl,ReplMurani} argued that the conventional theory failed because it  used the extended phase while  only the compact phase must be used. This bring us to the topic of the present paper.\n\nBecause of generality of the aforementioned problems with the phase variable, the paper addresses three systems: quantum rotator, particle in a periodic potential, and single JJ.  In the quantum rotator a particle moves around a 1D ring. The quantum pendulum \\cite{Cond} and an electron moving around a 1D normal ring \\cite{silver,AzbelN,Buttiker,Gefen} are examples of the quantum rotator. The paper explores analogies between these systems, but looking for possible differences at the same time. \n\nIn order to resolve the dilemma ``compact vs. extended  phase'',  it is necessary to answer to three questions:\n\\begin{enumerate}\n\\item\nAre the states with the phases $\\varphi$ and $\\varphi+2\\pi$ indistinguishable in the JJ?\n\\item\nMust the wave function be periodic in $\\varphi$ if the states with the phases $\\varphi$ and $\\varphi+2\\pi$ are indistinguishable?\n\\item\nThe last but not the least: Is it important for the theory of DQPT whether the states with the phases $\\varphi$ and $\\varphi+2\\pi$ are  distinguishable, or not?\n\\end{enumerate}\nThe paper looks for answers to these three questions. \n\n\n\nOur analysis has fully confirmed the final conclusions of the about 40-years old conventional theory of the  DQPT in the small JJ. But it reassessed  justifications of  these conclusions and analogies of the single JJ with other systems  (quantum rotator and particle in a periodic potential).  While in the past it was widely (but not unanimously) believed that the single JJ is an analog of particle in a periodic potential, but not of a mechanical pendulum, we argue that the opposite is true. This means that the states with the phases $\\varphi$ and $\\varphi+2\\pi$ are distinguishable both in the JJ and the quantum rotator.  However, whatever analogy is more correct, the  DQPT must take place both in the JJ and  in the quantum rotator. \n\n\n\\section{The phase in the classical theory} \\label{cl}\n\n\\subsection{Plane rotator in the conjugate variable ``angle--moment''}            \\label{QRc}\n\nThe Hamiltonian of the classical plane rotator  is \n\\begin{equation}\nH={m^2\\over 2 J}+G (1-\\cos \\varphi )-\\varphi N,\n      \\ee{Ho}\nwhere $m$ is the moment of the particle moving in the rotator, $J$ is  the moment of  inertia, $N$ is the external torque, and the periodic in $\\varphi$ potential $\\propto G$\nemerges from a constant force acting on the rotator (the gravity force in the case of the pendulum or the constant  electric field in the case of a charged particle  in a 1D ring). The $\\varphi$-dependent part of the Hamiltonian is the well known washboard potential.\nThe Hamilton equations are\n\\begin{eqnarray}\n{dm\\over dt}= -{\\delta H\\over \\delta \\varphi }=- G\\sin\\varphi +N,\n \\nonumber \\\\\n {d\\varphi \\over dt}={\\delta H\\over \\delta m}={m\\over J}.\n   \\eem{phit}\n\n\nThe Hamiltonian \\eq{Ho} is not periodic in $\\varphi$. This looks as a flaw, since violates the principle that the states with the phases $\\varphi$  and $\\varphi+2\\pi $ are indistinguishable. \nAccording to this Hamiltonian, the energies of these states differ by the energy $2\\pi N$ pumped to the rotator from the environment after full $2\\pi$ winding of the rotator. \n\n\nThe flaw can be eliminated by using another Hamiltonian \n\\begin{equation}\nH={( M+M_0)^2\\over 2 J}+G (1-\\cos \\varphi ), \n            \\ee{Hm}\nwhich is periodic  in $\\varphi$. Here  the moment $M_0$ transferred to the rotator by  the external  torque,\n\\begin{equation}\n{dM_0\\over dt}=N,\n    \\ee{}\nwas introduced. Since $M_0$ emerges from the interaction with the environment, we shall call it {\\em external moment}. \nThe Hamilton equations for the Hamiltonian \\eq{Hm} are\n\n\\begin{eqnarray}\n{d M\\over dt}= -{\\delta H\\over \\delta \\varphi }=- G\\sin\\varphi ,\n\\nonumber \\\\\n{d\\varphi \\over dt}={\\delta H\\over   \\delta  M}={ M+M_0\\over J}.\n   \\eem{phgP}\nThe moment  \n\\begin{equation}\n M ={\\partial {\\cal L}\\over \\partial \\dot \\varphi}\n    \\ee{}\nis the canonical moment determined by the Lagrangian \n\\begin{equation}\n{\\cal L}={(J \\dot \\varphi -M_0)^2\\over 2 J}-G (1-\\cos \\varphi ).\n    \\ee{}\nHowever, the angular  velocity $\\omega=\\dot \\varphi$ of the phase rotation is determined not by the canonical  but by the kinetic moment $m=  M+M_0$. The equations of motion \\eq{phit}  in  the terms of the observables $m$ and $\\varphi$ do not depend on the choice of  the Hamiltonians \\eq{Ho}     or \\eq{Hm}, since they differ by the full time derivative from $\\varphi M_0$, which does not affect the equations of motion. Later on we shall call the gauges with the periodic Hamiltonian like in \\eq{Hm} and with the non-periodic Hamiltonian like in \\eq{Ho} {\\em periodic }and {\\em non-periodic gauge}, respectively.  \n\n\n\nThe terms  {\\em canonical moment} and {\\em kinetic moment} were introduced by the analogy with the {\\em canonical momentum} and {\\em kinetic momentum}  of a charged particle in the electromagnetic field.  Splitting of the kinetic moment $m$ onto the canonical and the external moment obtained  from the environment is purely formal in the classical theory. But this splitting is more important in the quantum theory. \n\n In the  absence of the torque   $ N$, which pumps the moment and the energy to the system, there are two types of motion: (i) an oscillation around the ground state $\\varphi=0$ with $m$ vanishing  in average, and (ii) a monotonic rotation with $\\langle m \\rangle \\neq 0$  $\\varphi(t)$ being a periodic function in the time interval from $-\\infty$ to $\\infty$. \nThe stationary state with constant \n\\begin{equation}\n\\varphi =\\arcsin(N\/G)+2\\pi s\n     \\ee{}\n  is possible if $N <G$.   In this state the kinetic moment $m$ vanishes.  The  states of the rotator at the phases $\\varphi$ and $\\varphi+2\\pi s$ are indistinguishable.  The stationary state with   constant $\\varphi $  at  $N <G$ is an analog of the superconducting state of the JJ, while the regime of monotonic rotation  is an analog of the insulating state of the JJ.  At  $N >G$  the torque  drives the  quantum rotator to rotate with acceleration, but the monotonic  rotation with the angular velocity $\\omega=d\\varphi \/dt$ periodically oscillating around the constant average angular velocity $\\bar \\omega$  is possible  in the presence of  friction.\n\n\n\n\\subsection{Plane rotator vs. particle in a periodic potential}\n\nLet us  compare the rotator dynamics  with the dynamics of a particle with charge $q$ moving in a periodic potential and a classical electromagnetic field. In the latter case the Hamiltonian is \n\n\\begin{equation}\nH={( P- {q\\over c} A)^2\\over 2 m_0 }+G\\left(1-\\cos {2\\pi x\\over l} \\right) +q\\Phi. \n            \\ee{Hg}\nHere $m_0$ is the particle mass, $c$ is the speed of light, $P$ and $A$ are  the $x$-components of the canonical momentum $\\bm P$ and  of the electromagnetic vector potential $\\bm A$ (we consider the 1D problem), and $\\Phi$ is the electromagnetic scalar potential. The gauge transformation,\n\\begin{equation}\nA=A' +\\nabla \\chi(x,t),~~\\Phi=\\Phi' -\\dot   \\chi(x,t),\n     \\ee{GT}\nwith $\\chi(x,t)$ being an arbitrary function of $x$ and $t$,\nyields the Hamiltonian \\eq{Hg} with $A'$ and $\\Phi'$ instead of $A$ and $\\Phi$ and with the full time derivative of $q\\dot   \\chi(x,t)$ added, which  does not affect the Hamilton equations of motion.\n\nThe torque on the plane rotator can be a result  of the magnetic field normal to the plane of the rotator. The Hamiltonian   \\eq{Hg} describes also the dynamics of the plane rotator with $x$ being the coordinate along the circumference of the 1D ring of the rotator and $l$ being the length of this circumference.  The relations between variables in two presentations are \n\\begin{equation}\n\\varphi={2\\pi x\\over l} ,~~M={Pl\\over 2\\pi},~~J={m_0l^2\\over 4\\pi^2} .\n    \\ee{cord}\nThe periodic gauge in Sec.~\\ref{QRc} corresponds to the gauge without the scalar potential $\\Phi$ and with the Hamiltonian \n\\begin{equation}\nH={( P- {q\\over c} A)^2\\over 2 m_0 }+G\\left(1-\\cos {2\\pi x\\over l} \\right), \n            \\ee{HmS}\nThe external moment and the external torque of Sec.~\\ref{QRc} are\n\\begin{equation}\nM_0=-{ql \\over 2\\pi c }A=-\\hbar{\\phi\\over \\phi_0},~~N=\\dot M_0={ ql  \\over 2\\pi}E,\n    \\ee{cor}\nwhere $E=-\\dot A\/c$ is the azimuthal component of the electric field and $\\phi_0=hc\/q$ is the magnetic flux quantum  for the particle of charge  $q$.\nThe external  moment $M_0$  is determined by the  magnetic flux $\\phi=Al$ through the area restricted by the 1D ring of the rotator. The magnetic field is supposed to be  axisymmetric.\n\nThe transformation with the gradient  $\\nabla \\chi = A(t)$ independent from $x$ yields the non-periodic gauge, in which the potential $A(t)$ is absent, but instead the linear in $x$ scalar potential $\\Phi(t)=-  E(t) x=\\dot A(t) x\/c$ appears: \n\\begin{equation}\nH={ P^2\\over 2 m_0 }+G \\left(1-\\cos {2\\pi x\\over l} \\right)- q E x. \n            \\ee{WS}\n \n\nThe scalar potential in the non-periodic gauge is multivalued. \nThis does not produce any problem, since only fields but not potentials are observable quantities.\n\n\n\n Whatever gauge one uses, there is no difference between  dynamics of the rotator and the charged particle in a periodic potential. The dynamics does not depend on whether  the positions of the particle with the coordinates $x$ and  $x+l$ (or angles $\\varphi$ and $\\varphi+ 2\\pi $) are distinguishable,  or not.  Thus, in the classical theory the dilemma ``compact vs. extended  phase'' does not exist and there is no difference between the dynamics in a 1D ring and in the infinite 1D space with a periodic potential.\n\n\n\\section{The phase in the quantum  theory}\n\n\n\\subsection{Axisymmetric quantum  rotator: commutation and uncertainty relations, and wave packets} \\label{WP}\n\n\nThe standard way to go from the classical to the quantum theory is to replace  in the Hamiltonian the canonical moment $ M$  by the operator\n\\begin{equation}\n\\hat { M} =-i\\hbar {\\partial  \\over \\partial \\varphi}.\n    \\ee{Mphi}\n\nGeneral problems  with the canonically conjugate  pair  ``angle  (phase)--moment'' can be discussed for\nthe simple case of  the axisymmetric rotator ($G=0$). This case has already been investigated in the works on persistent currents in 1D normal rings \\cite{Buttiker,Gefen}. For a while we also ignore \n the external moment $M_0$. This means that we ignore any interaction with the environment, either at the present moment, or in the past. \n \n The objection to commutation relation \\eq{ComS} was based on the following calculation of the matrix elements of the commutation relation between two eigenstates of the moment operators with eigenvalues $M_s$ and $M_{s'}$ \\cite{judge,PhasRev,Pegg}:\n\\begin{eqnarray}\n\\int\\limits_0^{2\\pi}\\psi^*_{s'} [\\hat \\varphi,\\hat M]\\psi_s\\,d\\varphi\n\\nonumber \\\\\n=-i\\hbar \\int\\limits_0^{2\\pi}\\psi^*_{s'}\\left[ \\varphi{\\partial \\psi_s\\over \\partial \\varphi}-{\\partial (\\varphi \\psi_s)\\over \\partial \\varphi} \\right]\\,d\\varphi\n\\nonumber \\\\\n=-i\\hbar\\int\\limits_0^{2\\pi}\\left(\\psi^*_{s'} \\varphi{\\partial \\psi_s \\over \\partial \\varphi}+{\\partial \\psi^*_{s'} \\over \\partial \\varphi}\\varphi \\psi_s\\right)\\,d\\varphi\n \\nonumber \\\\\n=\\left. i\\hbar \\psi^*_{s'} \\varphi\\psi_s  \\right|_0^{2\\pi}+(M_s-M_{s'})\\int\\limits_0^{2\\pi}\\psi^*_{s'} \\varphi\\psi_s \\,d\\varphi\n    \\eem{ComC}\nThe opponents of the commutation relation \\eq{ComS} neglected the first term emerging from the borders of the integration interval $(0,2\\pi)$.  Then the diagonal matrix elements ($M_s=M_{s'}$) of the commutator  vanish, while the diagonal matrix elements of the righthand side of the commutation relation \\eq{ComS} are not zero. The justification\nfor ignoring of the border contribution  was that it should not appear in a matrix element of a Hermitian  operator when the integrand is a periodic function of $\\varphi$. \nThe operator  $\\hat \\varphi$ in the commutation relation is not Hermitian and breaks periodicity. \n\nVarious modifications of the commutation relation were suggested (one of them is discussed below).  However, there is another resolution of the problem, which rehabilitates the commutation relation \\eq{ComS}. The border  term in \\eq{ComC}  appears after the integration by parts of only one from two terms in the original commutator. While any of two terms in the commutator separately is non-Hermitian and breaks periodicity, their sum is  Hermitian and does not break periodicity. The matrix element of the commutator can be calculated without integration by parts:\n\\begin{eqnarray}\n\\int\\limits_0^{2\\pi}\\psi^*_{s'} [\\hat \\varphi,\\hat M]\\psi_s\\,d\\varphi\n\\nonumber \\\\\n=-i\\hbar \\int\\limits_0^{2\\pi}\\psi^*_{s'}\\left( \\varphi{\\partial \\psi_s\\over \\partial \\varphi}-\\varphi {\\partial \\psi_s\\over \\partial \\varphi} -\\psi_s\\right)\\,d\\varphi\n\\nonumber \\\\\n=i\\hbar \\int\\limits_0^{2\\pi}\\psi^*_{s'}\\psi_s\\,d\\varphi.\n    \\eem{ComP}\nThis is equal to the matrix element of the righthand side of the commutation relation. Similar arguments rehabilitating the standard commutation relation \\eq{ComS} were presented by \\citet{LossM}.\n\nWe checked the commutation relation using the wave functions in the continuous space of the phase $\\varphi$. Another route is to do it in the discrete space of quantized moments $M$. Then one encounters the  problem that because of discreteness of $M$ the expression conjugate to \\eq{Mphi}\n\\begin{equation}\n\\hat \\varphi =i\\hbar {\\partial  \\over \\partial M}\n    \\ee{phiM}\nfor the operator $\\hat \\varphi$ in the moment space is invalid. Instead one can use the operator $e^{i\\hat \\varphi}$, which shifts from one quantized eigenvalue of the operator $\\hat M$   to the next  one.\nThe commutation relation with this operator is\n\\begin{equation}\n[e^{i\\hat \\varphi},\\hat M]=-\\hbar e^{i\\hat \\varphi}.\n    \\ee{eph}\n\nThe operator $e^{i\\hat \\varphi}$ is a superposition of two Hermitian operators $\\cos \\hat \\varphi$ and $\\sin \\hat \\varphi$. The commutation relations for these operators, which are equivalent  to \\eq{eph}, were suggested by \\citet{suss}. Although the commutation relation \\eq{eph} contains only Hermitian  operators, its expansion in $\\hat \\varphi$ consists of non-Hermitian  operators, which must be treated correspondingly. A failure of some operation valid only for Hermitian  operators, means the failure of the operation, but not of the commutation relation.\n\nThe problem with the canonical commutation relation naturally leads to the problem with the uncertainty relation \\eq{unc}. The uncertainty relation is derived from the analysis of semiclassical wave packets, which demonstrates the correspondence principle: the transition from the quantum mechanical to the classical description. The wave packet is formed by a superposition of the states with  moments $M$ in the interval of the width $\\Delta M$. In the continuous moment space the superposition is determined by an integral. In the discrete moment space the integral must be replaced by a sum over the quantized values of the moment.\n \n \n \n Let us assume that the phase uncertainty is essentially less than $2\\pi$.  Then the moment uncertainty\n\\begin{equation}\n\\Delta M \\gg {\\hbar\\over 4\\pi} \n   \\ee{}\nis much larger than the distance between quantized values of the moment. Then the quantization can be ignored, and the summation determining the wave packet can be replaced by the integration. This provides a sufficiently accurate description of the wave packet within the phase interval $2\\pi$. \n\nBut there is a hurdle in this picture. The original wave function of the wave packet with summation over quantized values of the moment is periodic in $\\varphi $ because any term of the sum is periodic. However, the replacing of summation by integration definitely breaks the periodicity. The flaw is easily healed. The wave packet is replaced by the periodic chain of packets. The procedure can be considered as a compactification of phase, which the proponents of the compact phase insist on. Namely, one calculates the phase only in one $2\\pi$ interval and then continues this wave function periodically on all other phase intervals.  At the same time this demonstrates that the compactification is necessary only due to inaccuracy  of the approximation and is not needed for a sufficiently exact analysis. \n\n\nAnother situation emerges if one tries to construct a wave packet with the phase uncertainty $ \\Delta \\varphi$ exceeding $2\\pi$.  In this case the moment uncertainty $\\Delta M <\\hbar\/4\\pi$ is less than the moment quantum, the summation reduces to only one term, and the picture of wave packets fails. Then the physical meaning of the uncertainty relation \\eq{unc} becomes unclear.\n\nThere were suggestions to modify the standard uncertainty relation \\eq{unc} as a reaction to the aforementioned problem \\cite{judge,Pegg}. They were based on the concept of the compact phase, which assumed that the phase uncertainty cannot exceed $2\\pi$. This concept ignores that a phase fluctuation cannot be described only by the fluctuation of the compact phase (Sec.~\\ref{cl}) . A number $s$ of full $2\\pi$ rotations [see \\eq{comp}] in the course of the fluctuation is also important. We do not dwell more on this issue, since our analysis of the slow dynamics is based on the adiabatic approximation and does not use the wave packet concept. \n\n\n\\subsection{Particle moving in a 1D ring vs. particle moving in an infinite 1D space} \\label{per}\n\nLet us now compare the quantum mechanical dynamics of  a particle moving  in a 1D ring of the rotator and a  particle moving in the infinite 1D space. As in the previous subsection,\nwe ignore the periodic potential $G\\left(1-\\cos {2\\pi x\\over l} \\right)$. This allows to deal with simple analytical solutions of the Schr\\\"odinger equation.\n\nIn the quantum mechanics the canonical momentum becomes an operator:\n\\begin{equation}\n\\hat P=-i\\hbar {\\partial \\over \\partial x}.\n     \\ee{}\nThe quantum mechanical version of the Hamiltonian  \\eq{Hg} and the Schr\\\"odinger equation at $G=0$ in the periodic gauge  are \n\\begin{equation}\nH={1\\over 2 m_0}\\left|-i \\hbar{\\partial \\psi \\over \\partial x} -{q\\over c}A(t)\\psi\\right|^2, \n            \\ee{Hcx}\n\\begin{equation}\ni\\hbar {\\partial \\psi \\over \\partial t}=-{1\\over 2m_0}\\left[ \\hbar{\\partial  \\over \\partial x} -i{q\\over c}A(t)\\right]^2\\psi  .\n            \\ee{Shcx}\nThe Schr\\\"odinger equation  has a solution for an arbitrary time dependence of  $A(t)$:\n\\begin{equation}\n\\psi(\\varphi,t) = e^{iP x \/\\hbar-i \\int^t{{\\cal E}(t')\\over \\hbar}dt'},\n    \\ee{wfc}\nwhich is an eigenstate on the canonical momentum with the  eigenvalue $P$.  Here the time dependent energy is\n\\begin{equation}\n{\\cal E}(t)={[P -{q\\over c}A(t)]^2\\over 2m_0}.\n     \\ee{}\nThe particle  velocity \n\\begin{equation}\nv ={dx\\over dt} ={d{\\cal E}\\over d P}={P -{q\\over c}A(t)\\over m_0}\n    \\ee{omega0}\ndepends on the kinetic momentum $p=P -{q\\over c}A(t)$ and is well-defined, while the coordinate $x$ itself is not defined at all. There is an equal probability for any value of $x$.  \nAn electric field  $E=-\\dot A\/c$ monotonically accelerates the particle, as in the classical theory.\n\nIn a constant electric field\n\\begin{equation}\n\\psi(\\varphi,t) = e^{iP x \/\\hbar- {i(P+qEt)^3\\over 6m_0 qE}}.\n    \\ee{}\n\n\n\nIn the quantum mechanics  the difference of the particle dynamics in the 1D infinite space  and in the 1D ring  becomes  important. In the former case any value of $P$ is allowed. In the latter case  \nthe canonical momentum $P$ is quantized and cannot differ from the values $sh\/l$ with integer $s$. Only at these quantized  values the wave function \\eq{wfc} is periodic with the period $l$.  \nIn the variables ``moment--angle'' the quantized values of the canonical moment $M=Pl\/2\\pi$ [see \\eq{cord}] are $s\\hbar$.\nThe plot the energy vs. the external moment $M_0$ for different quantized values of the canonical moment  is shown in Fig.~\\ref{f1}(a).\n\n \n\\begin{figure}[!b]\n\\includegraphics[width=0.4 \\textwidth]{Energ} \n \\caption{ Plot the energy vs. the external moment $M_0$ at various quantized values of the canonical moment $M=s\\hbar$. (a) Axisymmetric rotator.  (b) Quantum rotator in a constant field.      \\label{f1}}\n \\end{figure}\n\n\n\n\nIn the quantum theory the gauge transformation \\eq{GT} must be accompanied by the transformation of the wave function \\cite{LLqu}:\n\\begin{equation} \n\\psi =\\psi'e^{iq\\chi\/\\hbar c}.\n      \\ee{GTp}\nAfter the transformation with $\\chi =xA=-xc\\int^tE(t')dt'$,\n\\begin{equation} \n\\psi =\\psi'e^{i qAx\/\\hbar c},\n      \\ee{gauge}\n  from the periodic to the non-periodic gauge, the Hamiltonian and the  Schr\\\"odinger equation become\n\\begin{equation}\nH={\\hbar^2\\over 2 m_0}\\left|{\\partial \\psi' \\over \\partial x}\\right|^2- qE(t)x|\\psi'|^2,\n            \\ee{Hwb0}\n\\begin{equation}\ni\\hbar {\\partial   \\psi' \\over \\partial t}=-{\\hbar^2\\over 2m_0}{\\partial^2 \\psi '\\over \\partial x^2} - qE(t)x\\psi'.\n   \\ee{Sht0}\nThe gauge transformation  \\eq{gauge} yields an non-stationary state with the non-periodic wave function\n \\begin{eqnarray}\n\\psi'=\\psi e^{-qAx\/c} = e^{i\\left(P +qE t\\right) x \/\\hbar-i \\int^t{{\\cal E}(t')\\over \\hbar}dt'}.\n    \\eem{nonpf}\n In a constant electric field\n\\begin{equation}\n\\psi'(\\varphi,t) = e^{i(P+qEt) x \/\\hbar- {i(P+qEt)^3\\over 6m_0 qE}}.\n    \\ee{}\n\n\n\nAfter the gauge transformation the canonical and the  kinetic momentum do not differ and are determined by the operator\n\\begin{equation}\n\\hat p'=\\hat P'=-{\\partial \\over \\partial x}\n    \\ee{}   \nin the space of functions $\\psi'$. In the quantum rotator  the quantization of the canonical momentum must be done in the periodic but not in the non-periodic gauge. This means that not the momentum $p'$ but the canonical momentum $P=sh\/l$  is equal  to an integer number $s$ of the momentum quanta. \n\nIn the non-periodic gauge the wave function is non-periodic since the gauge transformation \\eq{GTp} and the Hamiltonian \\eq{Hwb0} are non-periodic. One can rewrite  \\eq{nonpf} as \n\\begin{eqnarray}\n\\psi'= e^{iPx \/\\hbar-i \\int^t{{\\cal E}'(t')\\over \\hbar}dt'},\n    \\eem{}\nwhere \n\\begin{equation}\n{\\cal E}'(t)={\\left(P +qEt\\right)^2\\over 2J}-qEx\n     \\ee{}\nis the energy after the  gauge transformation. It is evident that the  wave function is not periodic because of the non-periodic term $-qEx$ in the energy.  It is impossible to satisfy the requirement of the wave function periodicity in the non-periodic gauge. If the wave function  is periodic at some moment of time it will become non-periodic at the next moment of time because of the non-periodic Schr\\\"odinger equation.\n\n\n\nThe loss of periodicity of the wave function in the non-periodic gauge should not be a matter of concern, as well as not  a matter of concern is the non-periodic electric scalar potential.  The phase factor, which makes the wave functions $\\psi(x)$ and $\\psi(x+l)$ different, means that the state is described by a multivalued wave function. The property of the gauge transformation to make the wave  function multivalued was pointed out by \\citet{LLqu} in Sec. 111 of their book. Multivaluedness (non-periodicity) of the wave function of the quantum rotator compensates multivaluedness (non-periodicity) of the washboard potential  in the non-periodic gauge \\cite{david}.\n\n\nIn summary, the important and the only difference between the dynamics of the particle in the quantum rotator and the particle moving in the infinite 1D space is that the Hilbert space of wave functions in the former case is discrete and is a subspace of the continuous Hilbert state in the latter case.\n\n\\subsection{Quantum  rotator in an external constant field} \\label{rcf}\n\nWhile in Sec.~\\ref{per} the variables ``coordinate--momentum'' were more convenient for comparison of the rotator with the particle moving in the infinite 1D space, here we return back to the variables ``angle (phase)--moment'', which are more convenient for comparison with the JJ.\n\nAt the presence of the external constant field the quantum mechanical version of  the  Hamiltonian \\eq{Hm} in the periodic gauge is\n\\begin{equation}\nH={1\\over 2 J}\\left|-i \\hbar{\\partial \\psi \\over \\partial \\varphi} +M_0\\psi\\right|^2+G (1-\\cos \\varphi ) |\\psi|^2.\n            \\ee{Hms}\nThe Schr\\\"odinger equation for this Hamiltonian is\n\\begin{equation}\ni\\hbar {\\partial \\psi \\over \\partial t}=-{1\\over 2J}\\left(\\hbar{\\partial \\over \\partial \\varphi}+iM_0\\right)^2\\psi +G (1-\\cos \\varphi ) \\psi.\n   \\ee{Sh}\n\n\nAccording to  the Bloch theorem,  any stationary  solution of \\eq{Sh} is the Bloch function\n\\begin{equation}\n\\psi(\\varphi,t)=u(\\tilde M+M_0,\\varphi)e^{i (\\tilde M \\varphi- E_0 t)\/\\hbar},\n   \\ee{BF}\nwhere $u(\\tilde M+M_0,\\varphi)=u(\\tilde M+M_0,\\varphi+2\\pi)$ is a periodic in  $\\varphi$  function and   $\\tilde M$ is a canonical quasimoment (analog of the canonical quasimomentum in the solid body theory \\cite{LLstPh2}). The energy spectrum of Bloch states consists of bands  with forbidden  gaps between them. In the quantum rotator the  Bloch wave function must be periodic in $\\varphi$. It is possible only  for quantized values  of the canonical quasimoment  $\\tilde M=s\\hbar$. \n\n\n\nWe consider only the lowest band with the energy $E_0(\\tilde M  +M_0)$, which depends on  the kinetic quasimoment  $\\tilde m=\\tilde M  +M_0$. \nBy the analogy with the quasimomentum and the coordinate operators for a particle in a periodic potential, one can consider the energy $E_0(\\tilde M  +M_0)$ as a Hamiltonian   \\cite{LLstPh2}, \nwhich yields the Hamilton equations \n\\begin{equation}\n{d \\tilde M\\over dt}=- {\\partial E_0\\over \\partial \\varphi}=0,\n   \\ee{Mt}\n\\begin{equation}\n\\omega={d \\varphi\\over dt}={\\partial E_0(\\tilde M  +M_0)\\over \\partial \\tilde M}.\n            \\ee{omega}\nWhile  the canonical quasimoment $\\tilde M$ does not vary in time,  the kinetic quasimoment $\\tilde m=\\tilde M  +M_0$ depends on time:\n\\begin{equation} \n{d\\tilde m\\over dt}={d\\tilde M\\over dt}  +\\dot M_0=\\dot M_0.\n      \\ee{tlM}\nIn general, these equations must be operator equations for the conjugate operators of the canonical quasimoment   $\\tilde M$ and of the angle $\\varphi$  \\cite{LLstPh2}. But if the torque is weak, one can use the adiabatic approximation with $M_0$ being a slowly varying adiabatic parameter. This allows to assume that at any moment the state does not differ essentially from the eigenstate of  the canonical quasimoment with the eigenvalue $\\tilde M$  at fixed $M_0$. Then Eqs.~(\\ref{Mt}--\\ref{tlM}) can be treated as classical equations. \n\n \n In the solid body theory the classical treatment of Eqs.~(\\ref{Mt}) and  (\\ref{omega}) is sometimes justified by considering them as written for  semiclassical wave packets \\cite{ziman}. Since for the quantum rotator the concept of wave packets is problematic (see Sec.~\\ref{WP}), it is important that for this  justification we used the adiabatic principle, but not the concept of wave packets.\n\n\nThe function $E_0(\\tilde m)$ is determined by the solution of the Schr\\\"odinger equation in Mathieu functions. Close to the bottom of the band\n\\begin{equation}\n E_0(\\tilde M  +M_0)={(\\tilde M  +M_0)^2\\over 2 J^*},~~\\omega={\\tilde M  +M_0\\over J^*},\n     \\ee{}\nwhere \n\\begin{equation}\nJ^* = \\left[{\\partial ^2E_0(\\tilde M  +M_0)\\over \\partial \\tilde M^2}\\right]^{-1}\n    \\ee{}\nis  the effective moment of inertia, an analog of the effective mass in the Bloch theory for solids.\n\n\n\n\n\nIn the weak binding limit $G \\ll \\hbar^2\/J$ the energy in the Brillouin zone $-\\hbar\/2 < \\tilde m < \\hbar\/2$ is $E_0={\\tilde m^2\\over 2J}$ excepting the close vicinity to the zone borders.\nThe effective moment of inertia $J^*$ does not differ from the bare moment of inertia $J$. The dependence of the energy on the external moment $M_0$  in the weak binding limit is shown for two Bloch bands at quantized values of $\\tilde M=s\\hbar$ in Fig.~\\ref{f1}(b).\n\nIn the strong binding limit $G \\gg \\hbar^2\/J$ there is the narrow band\n \\begin{equation}\n   E_0=\\Delta \\left(1-\\cos {2\\pi \\tilde m\\over \\hbar}\\right), \n      \\ee{}\nwhere $\\Delta $ is the band half-width, which goes to zero at $GJ\/\\hbar^2\\to \\infty$. The effective  moment of inertia is\n\\begin{equation}\nJ^*={\\hbar^2 \\over 4\\pi^2\\Delta}.\n   \\ee{}\n\nThe band energy has extrema at $M_0=s\\hbar$, where the phase angular velocity $\\omega$ vanishes. Thus, at zero external moment $M_0$, i.e., in the absence of any connection with the environment, either at the present moment, or in the past, the monotonic phase rotation is impossible. This follows  from the analysis of the quantum pendulum  \n\\cite{Cond} and of the quantum rotator in a constant electric field \\cite{silver} ignoring the connection  with the environment. \n \n Impossibility of monotonic phase rotation without any connection with the environment can be explained by the following arguments. In  an axisymmetric rotator, i.e., without an external constant field, there are two degenerate eigenstates  of fixed energy, which are either two states $e^{\\pm i M\\varphi\/\\hbar}$ with rotating phase, or states $\\cos {M\\varphi\\over \\hbar}$ and  $\\sin {M\\varphi\\over \\hbar}$  with  vanishing average angular velocity $\\langle\\dot \\varphi\\rangle$. But in a coherent superposition of  degenerate states $\\cos {M\\varphi\\over \\hbar}$ and  $\\sin {M\\varphi\\over \\hbar}$ the  nonzero angular velocity $\\langle\\dot \\varphi\\rangle$ is possible. However, whatever weak phase dependent potential (even a single impurity)  breaks the axial symmetry  and lifts degeneracy of states $\\cos {M\\varphi\\over \\hbar}$ and  $\\sin {M\\varphi\\over \\hbar}$. Then the superposition of two states  is not an eigenstate of the energy operator. In any eigenstate of the energy phase rotation is impossible.\n \n\nLet us apply  a constant weak torque $N$ to the rotator.  The external moment $M_0=Nt$  is proportional to time, and the time  can be excluded from Eqs.~(\\ref{omega}) and (\\ref{tlM}). This yields the equation \n\\begin{equation}\nN{d \\varphi\\over d\\tilde m}={\\partial E_0(\\tilde m)\\over \\partial \\tilde m}.\n            \\ee{Tm}\nThe equation describes Bloch oscillations with the  time period\n \\begin{equation}\n T={\\hbar \\over N }. \n       \\ee{t}\nWhile in the absence of the external field ($G=0$), the torque produces rotation with acceleration as in the classical theory (Sec.~\\ref{per}), in the presence of the periodic external field\nthe group velocity  $\\partial E_0(\\tilde m)\/\\partial \\tilde m$ is a periodic function of $\\tilde m$, and  the phase  angular velocity $\\omega$ performs periodic Bloch oscillations.  The angular velocity   vanishes after averaging over the time  and the amplitude of the phase oscillation is  determined by the  width $\\Delta E_0=E_{0\\mbox{max}}-E_{0\\mbox{min}}$ of the Bloch band:  \n\\begin{equation}\n\\Delta \\varphi ={ \\Delta E_0\\over N }.\n     \\ee{}\nThus, any finite torque makes  a monotonic rotation impossible. However, in the limit $N \\to 0$ when the period  $ T$ becomes much longer than the time of observation, one cannot discern the Bloch oscillation from monotonic rotation.\n\nNext we  phenomenologically introduce dissipation. The environment not  only pumps the moment into the rotator, but also provides a friction torque proportional to the phase angular velocity:\n\\begin{equation}\n\\dot M_0= N -f  \\omega = N -f {\\partial E_0(\\tilde m)\\over \\partial \\tilde m}.\n        \\ee{dis}\nHere $f$ is the friction coefficient. This equation has a solution with constant   $M_0$ and  the phase angular velocity \n\\begin{equation}\n\\omega={N\\over f}.\n     \\ee{Tf}\nNote that  no moment is pumped to the rotator since the moment $M_0$ does not vary in time. The external torque $ N$ is balanced by the friction torque $f\\omega$, i.e., the pumped moment  is returned back to the environment.\n\nRotation of the particle of charge $q$ with the angular velocity $\\omega$ produces the current $j=q\\omega\/2\\pi$. At the same time,  the torque is connected with the electric field $E$ [see \\eq{cor}]. Thus, \\eq{Tf} is in fact the Ohm law $j=El \/R_r$, where \n\\begin{equation}\nR_r ={4\\pi^2 f\\over q^2}\n     \\ee{Rf}\nis the resistance to the circular current $j$ around the 1D ring of the quantum rotator.\n\n\\begin{figure}[!t]\n\\includegraphics[width=0.4 \\textwidth]{VIcurve} \n \\caption{Plot the average angular velocity $\\bar \\omega$ vs. torque $N$ for the quantum rotator and plot the average voltage $\\bar V$ vs. current $I$ for the JJ in dimensionless variables.  Curve 1:  Weak binding limit, $\\omega_0=2\\hbar\/J$, $N_0=\\hbar f\/2 J$, $V_0=e\/C$, $I_0=e\/RC$. Curve 2: Strong binding limit, $\\omega_0=\\hbar\/2\\pi J^*$, $N_0=\\hbar f\/2\\pi  J^*$, $V_0=e\/\\pi C^*$, $I_0=e\/\\pi RC$.       \\label{f2}}\n \\end{figure}\n\n\n\nThe derivative $\\partial E_0(\\tilde m)\/ \\partial \\tilde m$ of the periodic function has a maximum which determines the  maximum of the phase angular velocity $\\omega_m$. When the torque $N$ becomes larger than $f  \\omega_m$ the steady phase rotation is  impossible and the Bloch oscillation starts. In the presence of the dissipation Eqs.~(\\ref{Tm}) and (\\ref{t}) become\n\\begin{equation}\n{d \\varphi\\over d\\tilde m}=\\frac{{\\partial E_0(\\tilde m)\\over \\partial \\tilde m}}{N-f\\ {\\partial E_0(\\tilde m)\\over \\partial \\tilde m}},\n            \\ee{Tmf}\n \\begin{equation}\n T=\\int\\limits_{-\\hbar\/2}^{\\hbar\/2}{d\\tilde m \\over N-f {\\partial E_0(\\tilde m)\\over \\partial \\tilde m }}. \n       \\ee{tf}\nNow the phase not only oscillates but also rotates with the average angular velocity\n\\begin{equation}\n\\bar \\omega ={1\\over T} \\int\\limits_{-\\hbar\/2}^{\\hbar\/2}{{\\partial E_0(\\tilde m)\\over \\partial \\tilde m } \\over N-f {\\partial E_0(\\tilde m)\\over \\partial \\tilde m }}d\\tilde m={N\\over f}-{\\hbar \\over fT}.\n   \\ee{omf}\nAt large $N$ the average angular velocity decreases as $1\/N$. \n\nIn the weak binding limit $G \\ll \\hbar^2\/J$ \n\\begin{equation}\nT= {J\\over f} \\ln \\frac{1+{\\hbar f\\over 2N J}}{1-{\\hbar f\\over 2N J}} ,~~\\bar \\omega={N \\over  f}- {\\hbar  \\over J}\\ln^{-1} \\frac{1+{\\hbar f\\over 2N J}}{1-{\\hbar f\\over 2N J}} .\n     \\ee{WC}\n In the strong binding limit $G \\gg \\hbar^2\/J$\n\\begin{equation}\nT= {J^*\\over f}{1 \\over \\sqrt{{J^{*2}N^2\\over \\hbar^2 f^2 }- {1\\over 4\\pi^2}}},~~\\bar \\omega ={  N\\over f }-{\\hbar \\over J^*}\\sqrt{{J^{*2}N^2\\over \\hbar^2 f^2 }- {1\\over 4\\pi^2}} .\n     \\ee{WCs}\nThe dependences of the average angular velocity $\\bar\\omega$ on the torque $N$ in the weak and the strong limit are shown in the dimensionless variables  in Fig.~\\ref{f2}. \n\nThe gauge transformation \\eq{gauge} in the variables ``phase--moment'' is \n\\begin{equation} \n\\psi =\\psi'e^{-M_0\\varphi\/\\hbar }.\n      \\ee{gaug}\nIt transforms the Hamiltonian \\eq{Hms} and the  Schr\\\"odinger equation \\eq{Sh} to \n\\begin{equation}\nH={\\hbar^2\\over 2 J}\\left|{\\partial \\psi' \\over \\partial \\varphi}\\right|^2+[G (1-\\cos \\varphi )- \\dot M_0 \\varphi]|\\psi'|^2,\n            \\ee{Hwb}\n\\begin{equation}\ni\\hbar {\\partial \\psi' \\over \\partial t}=-{\\hbar^2\\over 2J}{\\partial^2 \\psi' \\over \\partial \\varphi^2}+[G (1-\\cos \\varphi )- \\dot M_0 \\varphi] \\psi',\n   \\ee{Sht}\nwhich are not periodic in $\\varphi$. The  Bloch  wave function after the transformation is also non-periodic:\n\\begin{eqnarray}\n \\psi'(\\varphi)=u(\\tilde M+M_0,\\varphi)e^{i \\left[(\\tilde M+M_0) \\varphi- \\int^{t}E_0( t')dt'\\right]\/\\hbar}\n \\nonumber \\\\\n=u(\\tilde M+M_0,\\varphi)e^{i\\left[\\tilde M\\varphi-\\int^{t}E'_0 (t')dt' \\right]\/\\hbar}\n     \\eem{M0}\nwhere\n\\begin{equation}\nE'_0=E_0(\\tilde M+M_0) -\\dot M_0 \\varphi=E_0(\\tilde M+M_0) -N\\varphi\n     \\ee{}\nis  the energy in the non-periodic gauge.\nFurther discussion of the non-periodic gauge does not differ essentially from the discussion of this gauge for the axisymmetric rotator (Sec.~\\ref{per}). The wave function is not periodic because in the non-periodic gauge the  energy with the washboard potential  is non-periodic.\n\n \n \\begin{figure}[!t]\n\\includegraphics[width=0.4 \\textwidth]{Energ0} \n \\caption{The band energy ${\\cal E}_0$ vs. the external moment $M_0$. (a) The canonical quasimoment $\\tilde M =s\\hbar$ is quantized. (b) The canonical quasimoment $\\tilde M $ is not quantized. There is a continuous manifold of curves for various values of $\\tilde M$.       \\label{f1.1}}\n \\end{figure}\n\n\n\n Let us address now the case of the non-quantized canonical quasimoment $\\tilde M$ (particle in a periodic potential in the infinite 1D space). Figure \\ref{f1.1} compares the dependence of the energy ${\\cal E}_0$ in  the lowest Bloch band on the external moment $M_0$ for the quantized [Fig.~\\ref{f1.1}(a)] and the non-quantized  [Fig.~\\ref{f1.1}(b)] canonical quasimoment $\\tilde M$.  Instead of a single curve for quantized  $\\tilde M$ one has a continuous manifold of curves. Figure~\\ref{f1.1}(b), however, shows only a discrete manifold of curves in order to demonstrate that the curves are obtained from the single curve in Fig.~\\ref{f1.1}(a) by a shift without deformation. \n\n \n Despite this difference between quantized and non-quantized $\\tilde M$, for the regimes discussed above (monotonous phase rotation and Bloch oscillation)  the effect of quantization is practically absent.  The dynamics of these regimes is governed by the kinetic quasimoment $\\tilde m =\\tilde M+M_0$. In the absence of quantization the division of  $\\tilde m$ into  quantized $\\tilde M$ and non-quantized $M_0$ is meaningless since the both are not quantized. If in Fig.~\\ref{f1.1}(b) one plots the energy as a function of $\\tilde m$ instead of $M_0$ this yields the same single curve as in  Fig.~\\ref{f1.1}(a).\n\nSummarizing, the slow dynamics of the particle moving in the 1D ring of the rotator with indistinguishable states with the phases $\\varphi$ and $\\varphi+2\\pi$ does not differ from the dynamics of the particle moving in the infinite 1D space with the periodic potential, when the phases $\\varphi$ and $\\varphi+2\\pi$ correspond to different states.\nThis is because only one Bloch state participates  in the adiabatic processes. During its tuning by the external torque there are neither  intraband,  nor interband\ntransitions between Bloch states.\n\n\n\\section{JJ and DQPT}\n\nThere is one to one correspondence (ideal mapping) between the quantum rotator in the constant external field and the single JJ. The correspondence between variables of two systems is shown in Table 1. \n\n\\begin{table}[ht\n\\begin{tabular}{|l|l|} \\hline  {\\bf Rotator}~~~~~~~~~~~~~~~~ & {\\bf JJ}\n \\\\\n  &   \\\\ \\hline\n Canonical moment $M$ & Canonical charge $Q \\to {2e \\over \\hbar} M$ \\\\ \\hline\n  External moment $M_0$ &  External charge $Q_0 \\to {2e \\over \\hbar}M_0$ \\\\ \\hline \n Torque $N$ &  Electrical current $I\\to {2e \\over \\hbar}N $ \\\\ \\hline\n Rotation angle $\\varphi$ & Quantum mechanical phase $\\varphi$ \\\\ \\hline\nAngular velocity $\\omega=\\dot  \\varphi$ & Voltage  $V= {\\hbar\\over 2e}\\dot \\varphi$ \\\\ \\hline\nMoment of inertia $J$ & Capacitance  $C\\to {4e ^2\\over \\hbar^2}J$ \\\\ \\hline\nFriction coefficient $f$ & Conductance $1\/R\\to {4e ^2\\over \\hbar^2 }f$  \\\\ \\hline\n \\end{tabular}\n\\caption{Correspondence of variables of the rotator and the JJ} \\end{table}\n\n\nAs well as in the case of quantum rotator, the theory of the JJ can use either  the periodic gauge, in which the Hamiltonian and the wave function are periodic in $\\varphi$, or  the non-periodic gauge, in which both the Hamiltonian and the wave function  are not periodic in $\\varphi$. \n\nTranslating the periodic  Hamiltonian \\eq{Hms} and the  Schr\\\"odinger equation \\eq{Sh} for the rotator to the JJ one obtains\n\\begin{equation}\nH={1\\over 2 C}\\left|-i 2e{\\partial \\psi \\over \\partial \\varphi} +Q_0\\psi\\right|^2+E_J (1-\\cos \\varphi ) |\\psi|^2,\n            \\ee{HmQ}\n\\begin{equation}\ni\\hbar {\\partial \\psi \\over \\partial t}=-{1\\over 2C}\\left(2e{\\partial \\over \\partial \\varphi}+iQ_0\\right)^2\\psi +E_J (1-\\cos \\varphi ) \\psi.\n   \\ee{ShQ}\nHere $Q_0$ is the external charge, while the canonical charge is an operator\n\\begin{equation}\n\\hat Q =-2ie{\\partial \\over \\partial \\varphi}.\n    \\ee{}\nThe gauge transformation analogous to \\eq{gauge},\n\\begin{equation}\n\\psi=\\psi' e^{-Q_0 \\varphi \/2e},\n     \\ee{}\nyields the   Hamiltonian and the  Schr\\\"odinger equation  in the non-periodic gauge. The wave function  is periodic in $\\varphi$ in the periodic gauge, but not periodic in the non-periodic gauge. \n\nSolutions of the  Schr\\\"odinger equation \\eq{ShQ} are Bloch functions \n\\begin{equation} \n\\psi(\\varphi,t) =u(\\tilde Q+Q_0)e^{i(\\tilde Q\\varphi\/2e -E_0t\/\\hbar)},\n      \\ee{}\nwhere  $\\tilde Q$ is the canonical quasicharge, which is quantized in the JJ, and $u(\\tilde Q+Q_0)$ is a periodic function of the kinetic quasicharge $\\tilde q= \\tilde Q+Q_0$ with the period $2e$.\n\nThe further analysis is similar to that for the quantum rotator (Sec.~\\ref{rcf}). The analog of  \\eq{dis} is Kirchhoff's law\n\\begin{equation}\n\\dot {\\tilde q}=\\dot  Q_0 =I - {V\\over R},\n   \\ee{}\nwhere \n \\begin{equation}\nV={\\hbar \\over 2e}{d\\varphi\\over dt} \n     \\ee{}\nis the voltage drop across the JJ. The ohmic resistance \n\\begin{equation}\nR= {R_sR_{qp}\\over R_s+R_{qp}},\n   \\ee{}\nis determined by the resistance $R_{qp}$ of the normal channel in the JJ and by the resistance $R_s$ of the external shunt parallel to the JJ.\n\nAt small current $I$ $\\dot Q_0=0$, and the phase rotates with the constant angular velocity, i.e., at the constant voltage $V=IR$. This means that the whole current goes through the ohmic channel, and the JJ is an insulator. The insulating state is possible as far as the voltage $V$ dies not exceeds the voltage $V_0$ equal to the maximum of the derivative ${\\partial E_0\/ \\partial Q_0}$ in the Bloch band. In the limits of weak and strong binding\n\\begin{equation}\nV_0=\\left\\{\\begin{array}{ccc}  {e\\over C} & &   E_J\\ll {e^2\\over C} \\\\ {e\\over\\pi C^*}&  & E_J\\gg {e^2\\over C}   \\end{array}\\right.  .\n   \\ee{}\nThe voltage $V_0$ is  the electric breakdown voltage of the insulator \\cite{CommMurani}. At $V>V_0$  the steady rotation of the phase is impossible. Instead the Bloch oscillation regime takes place accompanied by a slow drift of the phase. The JJ becomes a conductor.  The $VI$ curve of the JJ  is described by the same plot as the plot ``angular velocity vs. torque'' for the quantum rotator shown in Fig.~\\ref{f2}. The $VI$ curves in Fig.~\\ref{f2} were calculated for the JJ by \\citet{widom2} and  by Averin, Likharev, and Zorin \\cite{AverLikh,Likh}. \\citet{widom2} called the voltage maximum on this curve current-voltage anomaly. \n\\citet{Schon} called it Bloch nose. The corresponding maximum on the curve ``resistance--current'' was called the Coulomb blockade bump \\cite{SI,Penttila2001,CommMurani}.\n\nWhile \\citet{widom2}  calculated the $VI$ curve using the analogy with the pendulum, i.e., assuming that the states with the phases $\\varphi$ and $\\varphi+2\\pi$ are  identical, Averin, Likharev, and Zorin \\cite{AverLikh,Likh} assumed that they are not identical as in the case of  a particle in a periodic potential. Nevertheless, the both groups obtained the same $VI$ curves in agreement with our conclusion in the end of Sec.~\\ref{rcf}. \n\nThe possibility of monotonic phase rotation in the Bloch band theory is due to quantum tunneling between neighboring wells of the periodic potential. Dissipation can suppress quantum tunneling \\cite{CL}. Then the particle (virtual particle in the JJ case) becomes localized in one of the wells of the periodic potential. This is the superconducting state of the JJ. The transition from the superconducting to the insulating state is the DQPT. \n\nThe DQPT is a joint effect of Coulomb interaction, dissipation, and quantum mechanics. The Coulomb blockade of Cooper pairs makes the JJ an insulator at small bias. However, it is effective only if the Coulomb energy $E_C= e^2\/C$ exceeds the quantum-mechanical uncertainty $\\hbar \/\\tau$ \\cite{Tin}, where $\\tau=RC$ is the time of the charge relaxation in the circuit.  According to the condition $E_C \\sim \\hbar\/\\tau$, the DQPT is expected at the resistance $R$ of the order of the quantum resistance  $R_q=h\/4e^2$. This qualitative estimation \\cite{Tin,CommMurani} agrees with the more detailed and accurate theory predicting the DQPT  exactly at $R = R_q$  \\cite{Schmid,Bulg}.   \n \nA similar DQPT must exist in the quantum rotator. The analog of the Coulomb energy $e^2\/C$  is the energy \n$\\hbar^2\/ J$ necessary for a transfer of the moment quantum $\\hbar$ to the rotator. The time $\\tau=J\/f$ is the decay time for the moment in the rotator with friction. Thus, the critical friction coefficient is of the order of $f \\sim \\hbar$. \\Eq{Rf} yields the relation between the friction coefficient $f$ and the ohmic resistance $R_r$ for the current produced by the particle of charge $q$ rotating in the rotator. According to this relation, the condition  $f \\sim \\hbar$ for the DQPT in the rotator is identical to the condition $R_r\\sim R_q $. Now $R_q \\sim \\hbar \/q^2$ is the quantum resistance for the charge $q$.\n\n\nHowever, there is a difference in the role of the resistance $R$ in the JJ and of the resistance $R_r$ in the quantum rotator. In the quantum rotator the monotonic phase rotation is possible at small $R_r<R_q$, while in the JJ the phase rotates at large $R>R_q$. In the regime of the phase rotation the JJ is an insulator, while the quantum rotator is a conductor. In the regime of the localized phase the JJ is a superconductor, while the quantum rotator is an insulator. In the both cases the insulating state takes place at  resistances larger than the quantum resistance. \n\n\nOne can estimate the resistance of the quantum rotator using the Drude formula for the conductivity $l\/R_r$, which for the 1D system in the weak-binding limit yields\n\\begin{equation}\n{l\\over R_r}\\sim  {q^2\\tau \\over m_0}={q^2l_0 \\over \\hbar k l},\n    \\ee{}\nwhere $l_0$ is the mean-free path of the particle and $\\tau =l_0\/v=m_0l_0\/\\hbar k$ is the relaxation time at elastic scattering by impurities. Since the wave number $k$ of the particle is on the order of the inverse space period $l$, the phase transition condition \n\\begin{equation}\n{R_q\\over R_r}\\sim {l_0 \\over l^2 k} \\sim kl_0 \\sim 1\n   \\ee{}\nbecomes the Ioffe--Regel condition for the metal--insulator transition \\cite{Mott}.\n\nAlthough the pioneer theoretical investigations \\cite{Schmid,Bulg} predicted the DQPT at the line $R=R_q$ independently from the ratio $E_J\/ E_C$, later it became clear that in the real experiment it is impossible to detect the transition at this line at $E_J\\gg E_C$.  One reason is an inevitable non-zero temperature of the experiment  \\cite{Zaik}. But even at strictly zero temperature the insulator state is not observable at $E_J\\gg E_C$ because  either  the observation time is short compared with the average time interval between tunneling events, which are phase slips destroying superconductivity \\cite{Schon}, or the accuracy of the voltage measurement is not sufficient for detection of the phase rotation since the voltage error bar exceeds the electric breakdown voltage $V_0$ \\cite{SI,Penttila2001}. \n\nThe results of the experimental and theoretical investigation of the phase diagram by Penttil\\\"a {\\em et al.} \\cite{SI,Penttila2001}  are shown in Fig.~\\ref{PD}. Open and black circles show observations of the superconductorlike and the insulatorlike behavior, respectively. The insulating  (I) state differs from the superconducting (S) state by the presence of the Coulomb blockade bump at dependences of the resistance on the current.\nThe solid line was determined from the condition that the error bar of voltage measurements is equal to the voltage $V_0$ at which the electric breakdown of the insulating state takes place. It is impossible to detect the insulating state above the solid line.  The observation of  the Coulomb blockade bump below the solid line  is  a smoking gun of the DQPT.\n\nIf one increases measurement accuracy (or lowers temperature), the  solid line  moves closer to the Schmid--Bulgadaev line. Thus, the vertical  Schmid--Bulgadaev line is an idealized asymptotic limit, which remains experimentally unattainable in practice for large $E_J\/E_C$.  \n\n\\section{Discussion and conclusions}\n\n\n \\begin{figure}[t]\n\\includegraphics[width=.45\\textwidth]{PhD}\n \\caption{The phase diagram of the JJ \\cite{SI}. The dashed vertical line shows the DQPT of  \\citet{Schmid} and \\citet{Bulg}. Due to  voltage measurement error bar the experimental detection of the DQPT is expected at the solid line  \\cite{SI}.\n Black and open symbols show observations of the superconductorlike ($S$) and the insulatorlike ($I$) $RI$ curves (see insets), respectively. Squares shows  results of experimental observations of unshunted junctions when the resistance $R$ is equal to a very large quasiparticle resistance $R_{qp}$ of the junction itself. \\footnote{ In the caption to Fig.~3 in Ref.~\\onlinecite{SI}  it was stated that \"unshunted samples (squares) are collected at  $R_q\/R=0$\". However, as said in the text of the paper, because of a large shunting quasiparticle resistance $R_{qp}$, $R_q\/R$ was never truly equal to 0 in the experiments.}\n \\label{PD}}\n \\end{figure}\n\nAlthough the analogy of the JJ with the quantum rotator was pointed out in the past \\cite{AzbelP,Rogov,widom2,Loss}, the opinion that the JJ is an analog of  a particle in a periodic potential was more prevalent   \\cite{AverLikh,Likh,GefenJ,Zwerg,Apen,SI,Penttila2001,CommMurani,morel,Schon,golub}. Averin, Likharev, and Zorin \\cite{AverLikh,Likh} argued that   the states of the JJ with $\\varphi$ and $\\varphi+2\\pi$ are not identical because the states of the environment (electric circuit) for two values of the phase are different. As a result, they concluded that all states in  the Bloch band are possible and used the concept of wave packets at the derivation  of Bloch oscillations.  \\citet{Zwerg} and  \\citet{morel} explained the distinguishability of the states  with $\\varphi$ and $\\varphi+2\\pi$ (decompactification of the phase) by the effect of dissipation. \\citet{Apen} argued that the distinguishability assumption is justified by taking into account fluctuations of the external classical charge, which were not taken into account in the previous calculations of the $VI$ curve in Refs.~\\cite{widom2,AverLikh,Likh} and in the present paper. \n\n The assumption that the states of the JJ with $\\varphi$ and $\\varphi+2\\pi$ can be different was disputed by \\citet{Loss}. We also believe that the indistinguishability of the states with $\\varphi$ and $\\varphi+2\\pi$ is a just requirement for the quantum rotator and the JJ. Although the connection between the JJ and the environment is of utmost importance and must not be ignored, we consider  the wave function of the junction alone, but not the wave function of the junction+environment.  There is an important difference between two questions: (i) whether the states of the JJ with $\\varphi$ and $\\varphi+2\\pi$ can be different or not, and (ii) whether the states of the JJ and the environment are the same before and after a phase slip, which is  a jump $2\\pi$ of the phase difference across the JJ. \n \nAverin, Likharev, and Zorin \\cite{AverLikh,Likh}  looked for an answer to the second question. Let us illustrate this for a simple SQUID circuit, which  is a superconducting loop interrupted by a JJ. There is a phase $\\varphi$ across the JJ and a phase difference $\\Phi$ along the rest part of the loop. They must satisfy the boundary condition $\\varphi+\\Phi =2\\pi N$, where $N$ is an integer. Averin, Likharev, and Zorin compared the state with $\\varphi$, $\\Phi$,  and $N$,  with  the state with $\\varphi+2\\pi $, $\\Phi-2\\pi$,  and $N$. These are states before and after a phase slip, which are definitely different. While before the phase slip the state was stationary, after the slip the state is not stationary: Adding $2\\pi$ to the Josephson phase does not change the current across the JJ  but  does change the current in the rest part of the circuit and the magnetic flux because the phase difference $\\Phi$ is not the same before and after the slip.\n\nThe states of the JJ with the phases $\\varphi$ and $\\varphi+2\\pi $ must be compared at fixed state of the environment. One should compare  the state with $\\varphi$,  $\\Phi$, and $N$ with the state with $\\varphi+2\\pi $, $\\Phi$, and $N+1$. The environment (the rest part of the circuit) remains in the same state because the phase $\\Phi$  remains the same, and the current and the magnetic flux do not differ.\n \n Through the whole paper it was assumed that the external moment in the quantum rotator (charge in the JJ) is a well defined classical variable slowly varying in time.  As mentioned above,\n fluctuations of the external charge were considered as a reason for the phase decompactification in the JJ. Indeed, for any {\\em given} external charge only {\\em one} state in the Bloch band  is possible, but this state is different for different  external charges. Thus, for a broad ensemble of external charges any state in the Bloch band  is possible as in the case of a particle in the infinite 1D state. However, this does not eliminate the difference between the Hilbert space  of all possible Bloch states in the case of a particle in the infinite 1D state and its much smaller discrete subspace of the rotator states with quantized canonical moments (charges). If one has the ensemble of $n$ external moments, the number of states of the  system ``quantum rotator+environment'' is equal to $n$, while  the number of states for ``particle in an infinite 1D space+environment'' is $n\\times n_B$, where $n_B$ is the number of all Bloch states in a Bloch band.\n  \n This difference is not important for the conventional theory on the insulating state and the DQPT. This is true because the theory deals only with a slow adiabatic tuning of only {\\em one} state in the lowest Bloch band. The question whether this state is a single state or other Bloch states  are possible, is irrelevant. This is another example when  according to \\citet{Legg} ``it is largely unnecessary  to address the vexed question of whether or not states differing in $\\varphi$ value by $2\\pi$ should be identified.''\n \n This does not mean that difference between dynamics of a particle in a 1D ring and a particle in the infinite 1D space is not important in general, at any experimental conditions. \n \\citet{Loss} made calculations of the resonant tunneling between quantum levels in the neighboring potential wells of the washboard potential in the JJ.  In the Bloch theory this is the Zener interband tunneling. The results depended on the choice of the initial state. The latter  was either periodic in the extended phase as must be for a particle in a 1D ring, or was confined in the interval $2\\pi$ that is possible only for a particle in the infinite 1D space. This calculation has not yet resolved the dilemma ``compact vs. extended phase''. \\citet{Loss}  solved the Schr\\\"odinger equation \\eq{ShQ} at the constant current $I= \\dot Q_0$, while the proponents of the extended phase assumption for the JJ connected the decompactification with fluctuations of the external charge $Q_0$, which were not taken into account in their calculation. Thus,  the answer to the question whether and how the difference between motion of a particle in a 1D ring and motion of a particle in the infinite 1D space with a periodic potential can affect observable phenomena  is still lacking.   \n\n\nArguing that the conventional theory failed and the insulating state is impossible in the JJ, \\citet{Sacl}  referred to the existence of supercurrent in the Cooper pair box, which is identical to a JJ in the limit $R\\to \\infty$. Indeed, in Sec.~\\ref{rcf}  we demonstrated that without dissipation the phase (angle) of the quantum rotator is localized and  performs Bloch oscillations around the localization point. Thus, a supercurrent flows across the JJ. However, from the same subsection it is clear that {\\em any} dissipation whatever large $R$ be,  delocalizes the phase and the JJ becomes an insulator. One can see in Fig.~\\ref {f1.1} that at $R \\to \\infty $ the voltage vanishes at currents $I\\gg e\/RC^*$. Although the insulating state is possible at any $R$, the electric breakdown  of the insulator occurs at the current  $I\\sim e\/RC^*$, which vanishes at $R \\to \\infty $. This makes observation of the DQPT impossible in this limit, but it is not an argument against its existence. \n\n\nSummarizing, our answers to three questions formulated in the introduction are following:\n\\begin{enumerate}\n\\item\nThe states with the phases $\\varphi$ and $\\varphi+2\\pi$ are indistinguishable in the JJ for the fixed state of the environment. In this aspect the JJ is an analog of the quantum rotator, but not of a particle in a periodic potential.\n\\item\nThe assumption that in the quantum rotator and in the JJ the states  with the phases $\\varphi$ and $\\varphi+2\\pi$ are indistinguishable, does not mean that the wave functions must be also periodic in all gauges. The wave functions can be  periodic only in the periodic gauge where the Hamiltonian is periodic in $\\varphi$. \n\\item\nThe conventional  theory of the DQPT is valid independently from whether the states with the phases $\\varphi$ and $\\varphi+2\\pi$ are  distinguishable, or not.\n\\end{enumerate}\n\nThe present analysis and its conclusions referred only to the case of a single particle, which excludes  interaction between particles. The case of many interacting particles in a quantum rotator requires another approach to the dilemma ``compact vs. extended  phase'' \\cite{averin}.\n\n\nThe close analogy between the regime of phase rotation in the quantum  rotator and the phase rotation in the insulating state of  the JJ allows to  expect the DQPT also in the quantum rotator. This can be checked by experimental investigations of persistent currents in 1D normal rings put in a constant electric field.   \n\n\n\n\n\n\n\\begin{acknowledgments}\nI thank Dmitri Averin, Pertti  Hakonen, and Andrei Zaikin for discussions and useful comments.\n\\end{acknowledgments}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{XP algorithm parameterized by neighborhood diversity}\\label{sec:nd}\nIn this section, we deal with neighborhood diversity ${\\nu}$, which is a more general parameter than vertex cover number. We first give an $O^*((n\/{\\nu})^{{\\nu}})$-time algorithm for Arc Kayles. This is an XP algorithm parameterized by neighborhood diversity. On the other hand, we show that there is a graph having at least $O^*((n\/{\\nu})^{{\\rm \\Omega}({\\nu})})$ non-isomorphic induced subgraphs, which implies the analysis of the proposed algorithm is\nasymptotically tight. \n\n\nBy Proposition \\ref{prop:isomorphic}, if we list up all non-isomorphic induced subgraphs, the winner of Arc Kayles can be determined by using recursive formulas (\\ref{eq:recursion1}) and (\\ref{eq:recursion2}).\nLet $\\mathcal{M}=\\{M_1,M_2,\\ldots,M_{{\\nu}}\\}$ be a partition such that $\\bigcup_i M_i=V$ and vertices of $M_i$ are twins each other.  We call each $M_i$ a \\emph{module}.\nWe can see that non-isomorphic induced subgraphs of $G$ are identified by how many vertices are selected from which module.\n\n\\begin{lemma}\\label{lem:positions:nd}\nThe number of non-isomorphic induced subgraphs of a graph of neighborhood diversity ${\\nu}$ is at most $(n\/{\\nu}+1)^{{\\nu}}$.\n\\end{lemma}\n\\begin{proof}\nBy the definition of neighborhood diversity, vertices in a module are twins each other. Therefore, the number of non-isomorphic induced subgraphs of $G$ is at most \n    $\\prod_{i=1}^{{\\nu}}(\\card{M_i}+1) \\le (\\sum_{i=1}^{{\\nu}}(\\card{M_i}+1)\/{\\nu})^{{\\nu}}\n    \\le (n\/{\\nu}+1)^{{\\nu}}$.\n   \n\\end{proof}\n\nWithout loss of generality, we select an edge whose endpoints are the minimum indices of vertices in the corresponding module. \nBy Proposition \\ref{prop:isomorphic}, the algorithm in Section \\ref{sec:Exalgo} can be modified to run in time $O^*((n\/{\\nu}+1)^{{\\nu}})$.\n\n\n \n\\begin{theorem}\\label{thm:algo:nd}\nThere is an $O^*((n\/{\\nu}+1)^{{\\nu}})$-time algorithm for Arc Kayles.\n\\end{theorem}\n \nThe idea can be extended to Colored Arc Kayles and BW-Arc Kayles. \nIn $G=(V,E_{\\CG}\\cup E_{\\CB}\\cup E_{\\CW})$, two vertices $u,v\\in V$ are called \\emph{colored twins} if $c(\\{u,w\\})=c(\\{v,w\\})$ holds $\\forall w \\in V\\setminus \\{u,v\\}$. \nWe then define the notion of colored neighborhood diversity. \n\\begin{definition}\\label{def:cnd}\nThe \\emph{colored neighborhood diversity} of $G=(V,E)$ is defined as minimum ${\\nu'}$ such that $V$ can be partitioned into ${\\nu'}$ vertex sets of colored twins.\n\\end{definition} \nIn Colored Arc Kayles or BW-Arc Kayles, we can utilize a partition of $V$ into modules each of which consists of colored twins. \nIf we are given a partition of the vertices into colored modules, we can decide the winner of Colored Arc Kayles or BW-Arc Kayles like \nTheorem \\ref{thm:algo:nd}. Different from ordinary neighborhood diversity,  it might be hard to compute colored neighborhood diversity in polynomial time. \n\\begin{theorem}\\label{thm:algo:cnd}\nGiven a graph $G=(V,E_{\\CG}\\cup E_{\\CB}\\cup E_{\\CW})$ with a partition of $V$ into ${\\nu'}$ modules of colored twins, we can compute the winner of Colored Arc Kayles on $G$ in time $O^*((n\/{\\nu'}+1)^{{\\nu'}})$.   \n\\end{theorem}\n\n\n\n    \n\n  \n     \n \n \nIn the rest of this section, we give a bad instance for the proposed algorithm as shown in Figure \\ref{fig:lower:nd}.\nThe result implies that the analysis of Theorem \\ref{thm:algo:nd} is\nasymptotically tight. \n\n\\begin{theorem}\\label{thm:lower:nd}\nThere is a graph having at least $(n\/{\\nu}+1-o(1))^{{\\nu}(1 - o(1))}$ non-isomorphic positions of Arc Kayles. \n\\end{theorem}\n\n\\begin{proof}\nWe construct such a graph $G$. Assume that $k$ is a number forming power of two minus one, that is, $k=2^{k'}-1$. \nFirst, we prepare $k$ cliques of $s$ vertices, $C_1,\\ldots,C_k$, and vertex set $X=\\{x_1,x_2,\\ldots,x_{\\log_2 (k+1)}\\}$. The subscript $i$ of $x_i$ represents $i$-th bit used below. \nFor each $x_i$, we attach $i-1$ pendant vertices, which is used to distinguish from another $x_j$. For $j$, $\\mathrm{bin}(j)$ and $\\mathrm{bin}(j,i)$ denote the $j$'s binary representation and its $i$-th bit, respectively. For example, $\\mathrm{bin}(6)=110$, \n$\\mathrm{bin}(6,1)=0, \\mathrm{bin}(6,2)=1, \\mathrm{bin}(6,3)=1$, and $\\mathrm{bin}(6,i)=0$ for $i\\ge 4$. \nWe connect the vertices in $C_j$ and vertices in $X$ according to the binary representation of $j$; vertices in $C_j$ are connected with $x_i$ if and only if $\\mathrm{bin}(j,i)=1$. Finally, we connect all the vertices in $\\bigcup_i C_i$, which form a large clique with size $sk$. \nFigure \\ref{fig:lower:nd} shows the constructed graph $G$.\n\nThe number of vertices in $G$ is $n=sk + \\log_2 (k+1)(\\log_2 (k+1)+1)\/2$, that is, $s=(n-\\log_2 (k+1)(\\log_2 (k+1)+1)\/2)\/k$, \nand the neighborhood diversity of $G$ is ${\\nu}=k+2\\log_2 (k+1)$, \nbecause vertices in each clique are twins, and also pendant vertices connected to each $x_i$ are twins. \n\n\n\nWe estimate the number of non-isomorphic induced subgraphs of $G$. \nWe restrict vertices to delete only from $\\bigcup_i C_i$, \nthat is, edges to select are inside of $\\bigcup_i C_i$. \nSince the number of pendant vertices for each $x_i$ is different, \n$x_i$'s substantially have IDs, and thus vertices from two distinct cliques are distinguishable. Hence, the number of non-isomorphic induced subgraphs obtained by removing edges inside $\\bigcup_i C_i$ are decided by the numbers of remaining vertices in $C_i$'s, each of which varies from $0$ to $s$. \nTherefore, the number of non-isomorphic induced subgraphs is at least\n\\begin{align*}\n(s+1)^{k}\/2 &= \\frac{1}{2}\\left(\\frac{n-\\log_2 (k+1)(\\log_2 (k+1)+1)\/2}{k}+1\\right)^{k}\\\\\n& = \\frac{1}{2}\\left(\\frac{n-o(k)}{k}+1\\right)^{k} \\ge \n\\frac{1}{2}\\left(\\frac{n}{k}+1-o(1)\\right)^{k},\n\\end{align*}\nwhere the division of $2$ comes from the fact that the number of deleted vertices must be even. \nSince ${\\nu}=k+2\\log_2 (k+1)$, $k={\\nu} - 2\\log_2 (k+1)\\ge {\\nu} - 2\\log_2  {\\nu}$ holds. We thus have lower bound $\\left(n\/{\\nu}+1-o(1)\\right)^{{\\nu}(1 - o(1))}$. \n\\end{proof}\n\n\\begin{figure}[tbp]\n \\centering\n \\includegraphics[width=0.65\\linewidth]{nd_lowler.pdf}\n \\caption{The constructed graph $G$ with neighborhood diversity ${\\nu}=k+2\\log_2 (k+1)$.}\n \\label{fig:lower:nd}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section\nBasic Algorithm }\\label{sec:Exalgo}\nIn this section, we show that the winner of \\emph{Colored} Arc Kayles on $G$ can be determined in time $O^*(2^n)$. We first observe that  the following lemma holds by the definition of the game.\n\\begin{lemma}\\label{lem:winner:Colored_Arc_Kayles}\nSuppose that Colored Arc Kayles is played on $G=(V,E_{\\CG}\\cup E_{\\CW}\\cup E_{\\CB})$. \nThen, player $\\mathrm{B}$ (resp., $\\mathrm{W}$) wins on $G$ with player $\\mathrm{B}$'s (resp., $\\mathrm{W}$'s) turn if and only if there is an edge $\\{u,v\\}\\in E_{\\CG}\\cup E_{\\CB}$ (resp., $\\{u,v\\}\\in E_{\\CG}\\cup E_{\\CW}$) such that player $\\mathrm{W}$ (resp., $\\mathrm{B}$) loses on $G-u-v$ with player $\\mathrm{W}$'s (resp., $\\mathrm{B}$'s) turn. \n\\end{lemma}\nThis lemma is interpreted by the following two recursive formulas:\n\\begin{align}\\label{eq:recursion1}\n    f_{\\mathrm{B}}(G) & = \\bigvee_{\\{u,v\\}\\in{E_{\\CG}\\cup E_{\\CB}}}\\lnot \\left(f_{\\mathrm{W}}(G-u-v)\\right),\\\\ \\label{eq:recursion2} \n    f_{\\mathrm{W}}(G) & = \\bigvee_{\\{u,v\\}\\in{E_{\\CG}\\cup E_{\\CW}}}\\lnot \\left(f_{\\mathrm{B}}(G-u-v)\\right).      \n\\end{align}\nBy these formulas, we can determine the winner of $G$ with either first or second player's turn by computing $f_{\\mathrm{B}}(G)$ and $f_{\\mathrm{W}}(G)$ for all induced subgraphs of $G$. Since the number of all induced subgraphs of $G$ is $2^n$,  it can be done in time $O^*(2^n)$ by a standard dynamic programming algorithm. \n\n\\begin{comment}\nOur algorithm is based on dynamic programming with respect to induced subgraphs of an input graph $G$. \nAs a key observation, any position of Colored Arc Kayles corresponds to an induced subgraph of $G$.\n\nFurther, we have the following lemma.\n\nBased on Lemma \\ref{lem:winner:Colored_Arc_Kayles}, we present Algorithm \\ref{alg:Colored_Arc_Kayles}. The input of Algorithm \\ref{alg:Colored_Arc_Kayles} is a graph $G=(V,E_{\\CG}\\cup E_{\\CB}\\cup E_{\\CW})$ and three edge sets $E_{\\CG}$,$E_{\\CB}$, and $E_{\\CW}$. Algorithm \\ref{alg:Colored_Arc_Kayles} returns 1 if the first player, which can choose edges in $E_{\\CG}\\cup E_{\\CB}$, wins. Otherwise, it returns 0. In the algorithm, if $E_{\\CG}\\cup E_{\\CB}$ is empty, the first player $p_1$ cannot choose an edge, and hence he\/she loses. Otherwise, he\/she picks an edge from $E_{\\CG}\\cup E_{\\CB}$. By Lemma \\ref{lem:winner:Colored_Arc_Kayles}, the winner is determined on $G-u-v$. \nHere, the second player $p_2$ loses on $G-u-v$ if and only if there is no edge $\\{u',v'\\}\\in E_{\\CG}\\cup E_{\\CW}\\setminus\\Gamma(\\{u,v\\})$ such that $p_1$ loses on $G-u-v-u'-v'$. That is, the winner on $G$ can be found by considering a game such that the first player is $p_2$, the second player is $p_1$, and the input graph is $G'-u-v$. Thus, in Line 3 of Algorithm \\ref{alg:Colored_Arc_Kayles}, it returns $\\bigvee_{\\{u,v\\}\\in{E_{\\CG}\\cup E_{\\CB}}}\\lnot \\left({\\bf Colored\\_Arc\\_Kayles}(G-u-v)\\right)$. \n\nSince the number of all induced subgraphs is at most $2^n$, the running time of Algorithm \\ref{alg:Colored_Arc_Kayles} is $O^*(2^n)$.\n\n\\begin{algorithm*}[t]\n     \\caption{${\\bf Colored\\_Arc\\_Kayles}(G,E_{\\CG},E_{\\CW},E_{\\CB})$, {\\bf Input:} $G=(V,E_{\\CG}\\cup E_{\\CW}\\cup{E_{\\CB}})$}\n     \\begin{algorithmic}[1]\\label{alg:Colored_Arc_Kayles}\n\n     \\IF {$E_{\\CG}\\cup E_{\\CB}=\\emptyset$}\n        \\RETURN $0$\n\n     \\ELSE\n         \\RETURN \\\\\n         $\\bigvee_{\\{u,v\\}\\in{E_{\\CG}\\cup E_{\\CB}}}\\lnot \\left({\\bf Colored\\_Arc\\_Kayles}(G-u-v)\\right)$\n     \\ENDIF\n \\end{algorithmic}\n \\end{algorithm*}\n\\end{comment}\n\n\\begin{theorem}\\label{thm:Colored_Arc_Kayles:Exp}\nThe winner of Colored Arc Kayles can be determined in time $O^*(2^n)$.\n\\end{theorem}\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\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\subsection{Background and Motivation} \nCram, Domineering, and Arc Kayles are well-studied two-player mathematical games and \ninterpreted as combinatorial games on graphs. \nDomineering (also called Stop-Gate) was   \nintroduced by G\\\"oran Andersson around 1973 under the name of Crosscram~\\cite{conway2000numbers,gardner1974mathematical}.\nDomineering is usually played on a checkerboard. The two players are denoted by\nVertical and Horizontal. Vertical (resp., Horizontal) player is\nonly allowed to place its dominoes vertically (resp., horizontally) on the board.\nNote that placed dominoes are not allowed to overlap. \nIf no place is left to place a domino, the player in the turn loses the game. Domineering is a partisan game, where players use different pieces. The impartial version of the game is Cram, where two players can place dominoes both vertically and horizontally. \n\nAn analogous game played on an undirected graph $G$ is Arc Kayles.\nIn Arc Kayles, the action of a player in a turn is to select an edge of $G$, and then the selected edge and its neighboring edges are removed from $G$. If no edge remains in the resulting graph, the player in the turn loses the game.  \nFigure \\ref{ex:arcKayles} is a play example of Arc Kayles. In this example, the first player selects edge $e_1$, and then the second player selects edge $e_2$. By the first player selecting edge $e_3$, no edge is left; the second player loses.  \nNote that the edges selected throughout a play form a maximal matching on the graph.\n\nSimilarly, we can define BW-Arc Kayles, which is played on an undirected graph with black and white edges. The rule is the same as the ordinary Arc Kayles except that the black (resp., white) player can select only black (resp., white) edges. \nNote that Cram and Domineering are respectively interpreted as Arc Kayles and BW-Arc Kayles on a two-dimensional grid graph, which is the graph Cartesian product of two path graphs.  \n\nTo focus on the common nature of such games with matching structures, we newly define {Colored Arc Kayles}. {Colored Arc Kayles} is played on a graph whose edges are colored in black, white, or gray, and black (resp., white) edges can be selected only by the black (resp., white) player, though grey edges can be selected by both black and white players. \n{BW-Arc Kayles} and ordinary {Arc Kayles} are special cases of {Colored Arc Kayles}. In this paper, we investigate {Colored Arc Kayles} from the algorithmic point of view.\n\n\\begin{figure}[tbp]\n \\centering\n \\includegraphics[width=0.8\\linewidth]{arc_kayles.pdf}\n \\caption{A play example of Arc Kayles}\n \\label{ex:arcKayles}\n\\end{figure}\n\n\\subsection{Related work}\n\\subsubsection{Cram and Domineering}\nCram and Domineering are well studied in the field of combinatorial game theory. In \\cite{gardner1974mathematical}, Gardner gives winning strategies for some simple cases. For Cram on $a\\times b$ board, the second player can always win if both $a$ and $b$ are even, and the first player can always win if one of $a$ and $b$ is even and the other is odd. This can be easily shown by the so-called Tweedledum and Tweedledee strategy. \nFor specific sizes of boards, computational studies have been conducted~\\cite{uiterwijk2019solving}. \nIn \\cite{uiterwijk2018construction}, Cram's endgame databases for all board sizes with at most 30 squares are constructed. \nAs far as the authors know, the complexity to determine the winner for Cram on general boards still remains open. \n\n\nFinding the winning strategies of Domineering for specific sizes of boards by using computer programs is well studied. \nFor example, the cases of $8\\times 8$ and $10\\times 10$ are solved in 2000~\\cite{BREUKER2000195} and 2002~\\cite{bullock2002domineering}, respectively. The first player wins in both cases. Currently, the status of boards up to $11\\times 11$ is known~\\cite{uiterwijk201611}. \nIn \\cite{uiterwijk2015new}, endgame databases for all single-component positions up to 15 squares for Domineering are constructed.\nThe complexity of Domineering on general boards also remains open. Lachmann, Moore, and Rapaport show that the winner and a winning strategy Domineering on $m\\times n$ board can be computed in polynomial time for $m\\in \\{1, 2, 3, 4, 5, 7, 9, 11\\}$ and all $n$~\\cite{lachmann2000wins}.\n\n\n\\subsubsection{Kayles, Node Kayles, and Arc Kayles}\nKayles is a simple impartial game, introduced by Henry Dudeney in 1908~\\cite{dudeney2002canterbury}.  \nThe name ``Kayles'' derives from French word ``quilles'', meaning ``bowling''. The rule of Kayles is as follows.  \nGiven bowling pins equally spaced in a line, players take turns to knock out either one pin or two adjacent pins, until all the pins are gone.\nAs graph generalizations, Node Kayles and Arc Kayles are introduced by Schaefer~\\cite{SCHAEFER1978185}. \nNode Kayles is the vertex version of Arc Kayles. Namely, \nthe action of a player is to select a vertex instead of an edge, and then the selected vertex and its neighboring vertices are removed. \nNote that both generalizations can describe the original Kayles; Kayles is represented as Node Kayles on sequentially linked triangles or as Arc Kayles on a caterpillar graph.\n\nNode Kayles is known to be PSPACE-complete \\cite{SCHAEFER1978185}, whereas the winner determination is solvable in polynomial time on graphs of bounded asteroidal numbers such as cocomparability graphs and cographs by using Sprague-Grundy theory~\\cite{BODLAENDER2002}. \nFor general graphs, Bodlaender et al. propose an $O(1.6031^n)$-time algorithm~\\cite{BODLAENDER2015165}. Furthermore, they show that the winner of Node Kayles can be determined in time $O(1.4423^{n})$ on trees. In \\cite{Kobayashi2018}, Kobayashi sophisticates the analysis of the algorithm in \\cite{BODLAENDER2015165} from the perspective of the parameterized complexity and shows that it can be solved in time $O^*(1.6031^{{\\mathtt{\\mu}}})$, where ${\\mathtt{\\mu}}$ is the modular width of an input graph\\footnote[1]{The $O^*(\\cdot)$ notation suppresses polynomial factors in the input size.}. He also gives an $O^*(3^{{\\tau}})$-time algorithm, where ${\\tau}$ is the vertex cover number, \nand a linear kernel when parameterized by neighborhood diversity.\n\nDifferent from Node Kayles, the complexity of Arc Kayles has remained open for more than 30 years. Even for subclasses of trees, not much is known. For example, Huggans and Stevens study Arc-Kayles on subdivided stars with three paths~\\cite{huggan2016polynomial}. \nTo our best knowledge,\nno exponential-time algorithm for Arc Kayles is presented except for an $O^*(4^{{\\tau}^2})$-time algorithm proposed in \\cite{LM2014}.\n\n\\subsection{Our contribution}\n\nIn this paper, we address winner determination algorithms for Colored Arc Kayles. \nWe first propose an $O^*(2^n)$-time algorithm for Colored Arc Kayles.\nNote that this is generally faster than applying the Node Kayles algorithm  to the line graph of an instance of Arc Kayles;  \nit takes time $O(1.6031^m)$, where $m$ is the number of the original edges.\nWe then focus on algorithms based on graph parameters.\nWe present an $O^*(1.4143^{{\\tau}^2+3.17{\\tau}})$-time algorithm for Colored Arc Kayles, where ${\\tau}$ is the vertex cover number. \nThe algorithm runs in time $O^*(1.3161^{{\\tau}^2+4{\\tau}})$ and $O^*(1.1893^{{\\tau}^2+6.34{\\tau}})$ for BW-Arc Kayles, and Arc Kayles, respectively.\nThis is faster than the previously known time complexity $O^*(4^{{\\tau}^2})$ in  \\cite{LM2014}. \n\nOn the other hand, we give a bad instance for the proposed algorithm, which implies the running time analysis is asymptotically tight.\nFurthermore, we show that the winner of Arc Kayles can be determined in time $O^*((n\/{\\nu}+1)^{{\\nu}})$,  where ${\\nu}$ is the neighborhood diversity of an input graph. This analysis is also asymptotically tight, because there is an instance having $(n\/{\\nu}-o(1))^{{\\nu}(1-o(1))}$.\nWe finally show that the winner determination of Arc Kayles on trees can be solved in $O^*(2^{n\/2})=O(1.4143^n)$ time, which improves $O^*(3^{n\/3})(=O(1.4423^n))$ by a direct adjustment of the analysis of Bodlaender et al.'s $O^*(3^{n\/3})$-time algorithm for Node Kayles. \n\n\n\n\n\n\n\n\n\n\n\n\\section{Preliminaries}\\label{sec:preliminaries}\n\\subsection{Notations and terminology}\nLet $G=(V,E)$ be an undirected graph. We denote $n=\\card{V}$ and $m=\\card{E}$, respectively. \nFor an edge $e=\\{u,v\\}\\in E$, we define $\\Gamma(e)=\\{e'\\mid e\\cap e'\\neq \\emptyset\\}$.\nFor a graph $G=(V,E)$ and a vertex subset $V'\\subseteq V$, we denote by $G[V']$ the subgraph induced by $V'$. For simplicity, we denote $G-v$ instead of $G[V\\setminus \\{v\\}]$. For an edge subset $E'$, we also denote by $G-E'$ the subgraph obtained from $G$ by removing all edges in $E'$ from $G$. \nA vertex set $S$ is called a \\emph{vertex cover} if  $e\\cap S\\neq \\emptyset$ for every edge $e\\in E$. We denote by ${\\tau}$ the size of a minimum vertex cover of $G$.\nTwo vertices $u,v\\in V$ are called \\emph{twins} if $N(u)\\setminus \\{v\\}=N(v)\\setminus \\{u\\}$.\n\n\\begin{definition}\\label{def:nd}\nThe \\emph{neighborhood diversity} ${\\nu}(G)$ of $G=(V,E)$ is defined as the minimum number $w$ such that $V$ can be partitioned into $w$ vertex sets of twins. \n\\end{definition}\n\nIn the following, we simply write ${\\nu}$ instead of ${\\nu}(G)$ if no confusion arises. We can compute the neighborhood diversity of $G$ and the corresponding partition in polynomial time \\cite{Lampis2012}. For any graph $G$,  ${\\nu} \\le 2^{{\\tau}}+{\\tau}$ holds.\n\n\\subsection{Colored Arc Kayles}\nColored Arc Kayles is played on a graph $G=(V,E_{\\CG}\\cup E_{\\CB}\\cup E_{\\CW})$, where $E_{\\CG},E_{\\CB}, E_{\\CW}$ are mutually disjoint. The subscripts $\\mathrm{G}$, $\\mathrm{B}$, and $\\mathrm{W}$ of $E_{\\CG}, E_{\\CB}, E_{\\CW}$ respectively, stand for gray, black, and white. \nFor every edge $e\\in E_{\\CG} \\cup E_{\\CB} \\cup E_{\\CW}$, let $c(e)$ be the color of $e$, that is, $c(e)=\\mathrm{G}$ if $e\\in E_{\\CG}$, $\\mathrm{B}$ if $e\\in E_{\\CB}$, and $\\mathrm{W}$ if $e\\in E_{\\CW}$.\nIf $\\{u,v\\}\\not\\in E_{\\CG} \\cup E_{\\CB} \\cup E_{\\CW}$, we set $c(\\{u,v\\})=\\emptyset$ \nfor convenience. \nAs explained below, the first (black or $\\mathrm{B}$) player can choose only gray or black edges, and the second (white or $\\mathrm{W}$) player can choose only gray or white edges. \n\nTwo players alternatively choose an edge of $G$. Player B can choose an edge in $E_{\\CG}\\cup E_{\\CB}$ and player W can choose an edge in $E_{\\CG}\\cup E_{\\CW}$.\nThat is, there are three types of edges; $E_{\\CB}$ is the set of edges that only the first player can choose, $E_{\\CW}$ is the set of edges that only the second player can choose, and $E_{\\CG}$ is the set of edges that both the first and second players can choose. Once an edge $e$ is selected, the edge and its neighboring edges (i.e., $\\Gamma(e)$) are removed from the graph, and the next player chooses an edge of $G- \\Gamma(e)$.  \nThe player that can take no edge loses the game. \nSince (Colored) Arc Kayles is a two-person zero-sum perfect information game and ties are impossible, one of the players always has a winning strategy. \nWe call the player having a winning strategy the \\emph{definite winner}, or simply \\emph{winner}. \n\nThe problem that we consider in this paper is defined as follows: \n\\begin{description}\n\\item {\\bf Input}: $G=(V,E_{\\CG}\\cup E_{\\CB}\\cup E_{\\CW})$, active player in $\\{\\mathrm{B},\\mathrm{W}\\}$. \n\\item {\\bf Question}: Suppose that players $\\mathrm{B}$ and $\\mathrm{W}$ play Colored Arc Kayles on $G$ from the active player's turn. Which player is the winner?\n\\end{description}\nRemark that if $E_{\\CB}=E_{\\CW}=\\emptyset$, Colored Arc Kayles is equivalent to Arc Kayles and if $E_{\\CG}=\\emptyset$, it is equivalent to BW-Arc Kayles.\n\nTo simply represent the definite winner of Colored Arc Kayles, we introduce two Boolean functions $f_{\\mathrm{B}}$ and $f_{\\mathrm{W}}$. The $f_{\\mathrm{B}}(G)$ is defined such that $f_{\\mathrm{B}}(G)=1$ if and only if the winner of Colored Arc Kayles on $G$ from player B's turn is player B. Similarly, $f_{\\mathrm{W}}(G)$ is the function such that $f_{\\mathrm{W}}(G)=1$ if and only if the winner of Colored Arc Kayles on $G$ from player W's turn is the player W. If two graphs $G$ and $G'$ satisfy that \n$f_{\\mathrm{B}}(G)=f_{\\mathrm{B}}(G')$ and $f_{\\mathrm{W}}(G)=f_{\\mathrm{W}}(G')$, we say that \\emph{$G$ and $G'$ have the same game value on Colored Arc Kayles}.    \n\n\n\n\\section{Arc Kayles for Trees}\\label{sec:trees}\nIn \\cite{BODLAENDER2015165}, Bodlaender et al. show that the winner of Node Kayles on trees can be determined in time $O^*(3^{n\/3})=O(1.4423^n)$. \n It is easy to show by a similar argument that the winner of Arc Kayles can also be determined in time $O(1.4423^n)$. \n It is also mentioned that the analysis is sharp apart from a polynomial factor because there is a tree for which the algorithm takes $\\Omega(3^{n\/3})$ time. The example is also available for Arc Kayles; namely, as long as we use the same algorithm, the running time cannot be improved.  \n \n \\medskip\n \nIn this section, we present that the winners of Arc Kayles on trees can be determined in time $O^*(2^{n\/2})=O(1.4143^n)$, which is attained by considering a tree (so-called) unordered. Since a similar analysis can be applied to Node Kayles on trees, the winner of Node Kayles on trees can be determined in time $O^*(2^{n\/2})$.\n\n\\medskip \n\n \n\nLet us consider a tree $T=(V,E)$. \nBy Sprague\u2013Grundy theory, if all connected subtrees of $T$ are enumerated, one can determine the winner of Arc Kayles. \nFurthermore, by Proposition \\ref{prop:isomorphic}, once a connected subtree $T'$ is listed, we can ignore subtrees isomorphic to $T'$. Here we adopt \nisomorphism of rooted trees. \n\\begin{definition}\\label{def:isomorphic:tree}\nLet $T^{(1)}=(V^{(1)},E^{(1)},r^{(1)})$ and $T^{(2)}=(V^{(2)},E^{(2)},r^{(2)})$ be trees rooted at $r^{(1)}$ and $r^{(2)}$, respectively. Then,  \n$T^{(1)}$ and $T^{(2)}$ are called \\emph{isomorphic with respect to root} if for any pair of $u,v\\in V^{(1)}$ there is a bijection $f: V^{(1)}\\to V^{(2)}$ such that $\\{u,v\\}\\in E^{(1)}$ if and only if $\\{f(u),f(v)\\}\\in E^{(2)}$ and $f(r^{(1)})=f(r^{(2)})$. \n\\end{definition}\nFor a tree $T$ rooted at $r$, two subtrees $T'$ and $T''$ are simply said \\emph{non-isomorphic} \nif $T'$ with root $r$ and $T''$ with root $r$ are not isomorphic with respect to root. \nNow, we estimate the number of non-isomorphic connected subgraphs of $T$ based on isomorphism of rooted trees. \nFor $T=(V,E)$ rooted at $r$, a connected subtree $T'$ rooted at $r$ is called an \\emph{AK-rooted subtree} of $T$, if there exists a matching $M  \\subseteq E$ such that $T[V\\setminus \\bigcup M]$ consists of $T'$ and isolated vertices.\nNote that $M$ can be empty, AK-rooted subtree $T'$ must contain root $r$ of $T$, and the graph consisting of only vertex $r$ can be an AK-rooted subtree.\n\n\n\\begin{lemma}\\label{lem:K-set:tree}\nAny tree rooted at $r$ has $O^*(2^{n\/2})(=O(1.4143^n))$ non-isomorphic AK-rooted subtrees rooted at $r$.  \n\\end{lemma}\n\n\\begin{proof}\n Let $R(n)$ be the maximum number of non-isomorphic AK-rooted subtrees of any tree rooted at some $r$ with $n$ vertices. \nWe claim that $R(n)\\le 2^{n\/2}-1$ for all $n\\geq 4$, which proves the lemma. \n\nWe will prove the claim by induction. For $n\\le 4$, the values of $R(n)$'s are as follows: $R(1)=1, R(2)=1, R(3)=2$, and $R(4)=3$. These can be shown by concretely enumerating trees. For example, for $n=2$, a tree $T$ with $2$ vertices is unique, and an AK-rooted subtree of $T$ containing $r$ is also unique, which is $T$ itself. For $n=3$, the candidates of $T$ are shown in Figure \\ref{tree1}. For Type A in Figure \\ref{tree1}, AK-rooted subtrees are the tree itself and isolated $r$, and for Type B, an AK-rooted subtree is only the tree itself; thus we have $R(3)=2$. Similarly, we can show $R(4)=3$ as seen in Figure \\ref{tree2}.\nNote that $R(1)>2^{1\/2}-1$, $R(2)=1\\le 2^{2\/2}-1=1$, $R(3)=2 > 2^{3\/2}-1$, and $R(4)=3\\le 2^{4\/2}-1=3$. This $R(4)$ is used as the base case of induction. \n\n\\begin{figure}[htbp]\n\\begin{minipage}{0.48\\hsize}\n \\centering\n \\includegraphics[width=0.55\\linewidth]{tree3.pdf}\n \\caption{Trees with 3 vertices rooted at $r$}\n \\label{tree1}\n\\end{minipage}\n\\begin{minipage}{0.48\\hsize}\n \\centering\n \\includegraphics[width=1.05\\linewidth]{tree4.pdf}\n \\caption{Trees with 4 vertices rooted at $r$}\n \\label{tree2}\n\\end{minipage}\n\\end{figure}\nAs the induction hypothesis, let us assume that the claim is true \nfor all $n' < n$ except $1$ and $3$, and consider a tree $T$ rooted at $r$ with $n$ vertices. Let $u_1, u_2, \\ldots, u_p$ be the children of root $r$, and $T_i$ be the subtree of $T$ rooted at $u_i$ with $n_i$ vertices for $i=1,2,\\ldots,p$. \nNote that for an AK-rooted subtree $T'$ of $T$, the intersection of $T'$ and $T_i$ for each $i$ is either empty or an AK-rooted subtree of $T_i$ rooted at $u_i$. Based on this observation, we take a combination of the number of AK-rooted subtrees of $T_i$'s, which gives an upper bound on the number of AK-rooted subtrees of $T$. We consider two cases: (1) for any $i$, $n_i \\neq 3$, (2) otherwise. For case (1), the number of AK-rooted subtrees of $T$ is at most \n\\[\n\\prod_{i:n_i>1}(R(n_i)+1)\\cdot \\prod_{i:n_i=1}1 \\le \\prod_{i:n_i>1} 2^{n_i\/2}=2^{\\sum_{i:n_i>1} n_i\/2} \\le 2^{(n-1)\/2} \\le 2^{n\/2}-1.    \n\\]\nThat is, the claim holds in this case. Here, in the left hand of the first inequality, $R(n_i)+1$ represents the choice of AK-rooted subtree of $T_i$ rooted at $u_i$ or empty, \nand ``$1$'' for $i$ with $n_i=1$ represents that $u_i$ needs to be left as is, because otherwise edge $\\{r,u_i\\}$ must be removed, which violates the condition ``rooted at $r$''.  The first inequality holds since any $n_i$ is not $3$ and thus the induction hypothesis can be applied.  \nThe last inequality holds by $n\\ge 5$.  \n\nFor case (2), we further divide into two cases: (2.i), for every $i$ such that $n_i=3$, $T_i$ is Type B, and (2.ii) otherwise. For case (2.i), since an AK-rooted subgraph of $T_i$ of Type B in Figure \\ref{tree1} is only $T_i$ itself, the number is $1\\le 2^{3\/2}-1$. Thus, the similar analysis of Case (1) can be applied as follows: \n\\begin{align*}\n\\prod_{i:n_i\\neq 1, 3}(R(n_i)+1) \\cdot \\prod_{i: T_i \\mathrm{{\\ is\\ Type\\ B }}  }(2^{3\/2}-1+1) \\le \\prod_{i:n_i>1} 2^{n_i\/2}\\le 2^{n\/2}-1,     \n\\end{align*}\nthat is, the claim holds also in case (2.i). \n\nFinally, we consider case (2.ii). By the assumption, at least one $T_i$ is Type A in Figure \\ref{tree1}. Suppose that $T$ has $q$ children of $r$ forming Type A, which are renamed $T_1,\\ldots,T_q$ as canonicalization. Such renaming is allowed because we count non-isomorphic subtrees.  \nFurthermore, we can sort AK-rooted subtrees of $T_1,\\ldots, T_q$ as canonicalization. Since each Type A tree can form in $T'$ empty, a single vertex, or Type A tree itself, $T_1,\\ldots,T_q$ of $T$, the number of possible forms of subforests of $T_1,\\ldots,T_q$ of $T$ is \n\\[\n\\multiset{q}{3}=\\binom{q+2}{2}. \n\\]\nSince the number of subforests of $T_i$'s other than $T_1,\\ldots,T_q$ are similar evaluated as above, we can bound the number of AK-rooted subtrees by \\[\n\\binom{q+2}{2} \\prod_{i:i>q} 2^{n_i\/2} \\le \\frac{(q+2)(q+1)}{2} 2^{\\sum_{{i:i>q}} n_i\/2} \\le \\frac{(q+2)(q+1)}{2} 2^{(n-3q-1)\/2}.  \n\\]\nThus, to prove the claim, it is sufficient to show that $(q+2)(q+1)2^{(n-3q-3)\/2}\\le 2^{n\/2}-1$ for any pair of integers $n$ and $q$ satisfying $n\\ge 5$ and $1\\le q\\le (n-1)\/3$.  \nThis inequality is transformed to the following\n\\[\n\\frac{(q+1)(q+2)}{2^{\\frac{3(q+1)}{2}}} \\le 1-\\frac{1}{2^{\\frac{n}{2}}}.\n\\]\nSince the left hand and right hand of the inequality are monotonically decreasing with respect to $q$ and monotonically increasing with respect to $n$, respectively, the inequality always holds if it is true for $n=5$ and $q=1$. In fact, we have \n\\[\n\\frac{(1+1)(1+2)}{2^{\\frac{3(1+1)}{2}}}= \\frac{3}{4}=1 - \\frac{1}{2^2} \\le 1-\\frac{1}{2^{\\frac{5}{2}}}, \n\\]\nwhich completes the proof. \n\\end{proof}\n\n\\begin{theorem}\nThe winner of Arc Kayles on a tree with $n$ vertices can be determined in time $O^*(2^{n\/2})=O(1.4143^n)$.\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{comment}\nEach type A has three position, no edge is selected, upper edge is selected and lower edge is selected.\nSo, this tree has rooted K-set at most\n\\begin{align*}\nT(n)\n&\\leq \\binom{a+2}{2}(T(n_1)+1)(T(n_2)+1)\\cdots(T(n_k)+1)\\\\\n&\\leq\\frac{1}{2}(a+1)(a+2)\\cdot2^\\frac{n_1}{2}\\cdot2^\\frac{n_2}{2}\\cdots2^\\frac{n_k}{2}\\\\\n&=\\frac{1}{2}(a+1)(a+2)\\cdot2^\\frac{n_1+n_2+\\cdots+n_k}{2}\\\\\n&=\\frac{1}{2}(a+1)(a+2)\\cdot2^\\frac{n-3a-1}{2}.\n\\end{align*}\nThen,\n\\begin{align}\n2^\\frac{n}{2}-1-T(n)\n&\\geq2^\\frac{n}{2}-1-\\frac{1}{2}(a+1)(a+2)\\cdot2^\\frac{n-3a-1}{2}\\notag\\\\\n&=2^\\frac{n}{2}\\left\\{1-\\frac{(a+1)(a+2)}{2^\\frac{3a+3}{2}}\\right\\}-1. \\label{eq1}\n\\end{align}\n\n\nWe consider function $f(a)=(2^\\frac{5}{2}-1)2^\\frac{3a+3}{2}-2^\\frac{5}{2}(a+1)(a+2)$.\nWhen $a$ is non-negative integer, $f(a)>0$ and\n\\begin{align}\n(2^\\frac{5}{2}-1)2^\\frac{3a+3}{2}-2^\\frac{5}{2}(a+1)(a+2)&>0\\notag\\\\\n(2^\\frac{5}{2}-1)2^\\frac{3a+3}{2}&>2^\\frac{5}{2}(a+1)(a+2)\\notag\\\\\n1-\\frac{(a+1)(a+2)}{2^\\frac{3a+3}{2}}&>\\frac{1}{2^\\frac{5}{2}}\\label{eq2}.\n\\end{align}\nFrom the formula\\ref{eq1} and formula\\ref{eq2},\n\\begin{align*}\n2^\\frac{n}{2}-1-T(n)\n&\\geq2^\\frac{n}{2}\\left(1-\\frac{(a+1)(a+2)}{2^\\frac{3a+3}{2}}\\right)-1\\\\\n&>2^\\frac{n}{2}\\cdot\\frac{1}{2^\\frac{5}{2}}-1\\\\\n&\\geq2^\\frac{5}{2}\\cdot\\frac{1}{2^\\frac{5}{2}}-1\\\\\n&>0.\n\\end{align*}\nTherefore, $n\\geq4$, $T(n)\\leq2^\\frac{n}{2}-1=O(1.4143^n)$\n\\end{comment}\n\n\\bigskip\n\nIn the rest of this section, we mention that we can determine the winner of Node Kayles for a tree in the same running time as Arc Kayles. The outline of the proof is also almost the same as Arc Kayles. Only the difference is to utilize the notion of \\emph{NK-rooted subtree} instead of \nAK-rooted subtree for Arc Kayles. \nFor $T=(V,E)$ rooted at $r$, a connected subtree $T'$ rooted at $r$ is called an NK-rooted subtree of $T$, if there exists an independent set $U  \\subseteq V$ such that $T[V\\setminus N[U]]=T'$.\n\n\n\n\n\\begin{lemma}\nAny tree rooted at $r$ has $O^*(2^{n\/2})(=O(1.4143^n))$ non-isomorphic NK-rooted subtrees rooted at $r$.  \n\\end{lemma}\n\n\\begin{proof}\nLet $\\hat{R}(n)$ be the maximum number of non-isomorphic NK-rooted subtrees of any tree rooted at some $r$ with $n$ vertices. \nSimilarly to Arc Kayles, we can show that $\\hat{R}(n)\\le 2^{n\/2}-1$ for all $n\\geq 4$ by induction. For $n\\le 4$, it is easy to see that the values of $\\hat{R}(n)$'s coincide with those of $R(n)$'s: $\\hat{R}(1)=R(1)=1, \\hat{R}(2)=R(2)=1, \\hat{R}(3)=R(3)=2$ (see Figure \\ref{tree1}), and $\\hat{R}(4)=R(4)=3$ (see Figure \\ref{tree2}). That is, $\\hat{R}(n)\\le 2^{n\/2}-1$ does not hold for $n=1$ and $3$, whereas it holds for $n=2$ and $4$, which shows the base step of the induction. We then consider the induction step. \n\nIn the induction step, we again take the same strategy as Arc Kayles; \nwe take a combination of the number of NK-rooted subtrees of $T_i$'s, which gives an upper bound on the number of NK-rooted subtrees of $T$. Since all the arguments use the same induction hypothesis and the same values of $\\hat{R}(n)=R(n)$ for $n=1,2,3,4$, the derived bound is the same as Arc Kayles. \n\\end{proof}\n\n\n\n\n\n\n\n\n    \n       \n       \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{FPT algorithm parameterized by vertex cover}\\label{sec:vc}\nIn this section, we propose winner determination algorithms for Colored Arc Kayles parameterized by the vertex cover number. As mentioned in Introduction, the selected edges in a play of Colored Arc Kayles form a matching. This implies that the number of turns is bounded above by the maximum matching size of $G$ and thus by the vertex cover number.  \nFurthermore, the vertex cover number of the input graph is bounded by twice of the number of turns of Arc Kayles. \nIntuitively, we may consider that a game taking longer turns is harder to analyze than games taking shorter turns. In that sense, the parameterization by the vertex cover number is quite natural. \n\n\nIn this section, we propose an $O^*(1.4143^{{\\tau}^2+3.17{\\tau}})$-time algorithm for Colored Arc Kayles, where ${\\tau}$ is the vertex cover number of the input graph.\nIt utilizes similar recursive relations shown in the previous section, \nbut we avoid to enumerate all possible positions by utilizing equivalence classification.\n\nBefore explaining the equivalence classification, we give a simple observation based on isomorphism. The isomorphism on edge-colored graphs is defined as follows. \n\n\\begin{definition}\\label{def:isomorphic:colored}\nLet $G^{(1)}=(V^{(1)},E_{\\CG}^{(1)}\\cup E_{\\CB}^{(1)}\\cup E_{\\CW}^{(1)})$ and $G^{(2)}=(V^{(2)},E_{\\CG}^{(2)}\\cup E_{\\CB}^{(2)}\\cup E_{\\CW}^{(2)})$ be edge-colored graphs. \nThen\n$G^{(1)}$ and $G^{(2)}$ are called \\emph{isomorphic} if for any pair of $u,v\\in V$ there is a bijection $f: V^{(1)}\\to V^{(2)}$ such that (i) $\\{u,v\\}\\in E_{\\CG}^{(1)}$ if and only if $\\{f(u),f(v)\\}\\in E_{\\CG}^{(2)}$, (ii) $\\{u,v\\}\\in E_{\\CB}^{(1)}$ if and only if $\\{f(u),f(v)\\}\\in E_{\\CB}^{(2)}$, and (iii) $\\{u,v\\}\\in E_{\\CW}^{(1)}$ if and only if $\\{f(u),f(v)\\}\\in E_{\\CW}^{(2)}$. \n\\end{definition}\n\nThe following proposition is obvious. \n\n\\begin{proposition}\\label{prop:isomorphic}\nIf edge-colored graphs $G^{(1)}$ and $G^{(2)}$ are isomorphic, \n$G^{(1)}$ and $G^{(2)}$ have the same game value for Colored Arc Kayles.\n\\end{proposition}\n\n\\begin{comment}\n\\begin{definition}\\label{def:isomorphic}\nLet $G=(V,E)$ and $G'=(V',E')$ be graphs. Then $G$ is \\emph{isomorphic} to $G'$ if for ant pair of $u,v\\in V$ there is a bijection $f: V\\to V'$ such that $\\{u,v\\}\\in E$ if and only if $\\{f(u),f(v)\\}\\in E'$. \n\\end{definition}\n\nWe further define isomorphism for colored graphs.\n\n\\begin{definition}\\label{def:isomorphic:colored}\nLet $G=(V,E)$ and $G'=(V',E')$ be edge-colored graphs where $E=\\bigcup_{i=1}^r E_i$ and $E'=\\bigcup_{i=1}^r E'_i$. Then $G$ is isomorphic to $G'$ if for ant pair of $u,v\\in V$ there is a bijection $f: V\\to V'$ such that $\\{u,v\\}\\in E_i$ if and only if $\\{f(u),f(v)\\}\\in E'_i$. \n\\end{definition}\n\nOur algorithm is also based on dynamic programming. However, we do not compute the winner on $G$ if the winner of $G'$ isomorphic to $G$ has already been computed. Instead, we use the result of $G'$.\nTo do this, we utilize memoization and preserve the result of $G$ if the winner of a graph isomorphic to $G$ is not computed.\n\n\n\\end{comment}\n\nLet $S$ be a vertex cover of $G=(V,E_{\\CG}\\cup E_{\\CW}\\cup E_{\\CB})$, that is, any $e=\\{u,v\\}\\in E_{\\CG} \\cup E_{\\CW} \\cup E_{\\CB}$ satisfies that $\\{u,v\\}\\cap S\\neq \\emptyset$. \nNote that for $v\\in V\\setminus S$, $N(v)\\subseteq S$ holds. \nWe say that two vertices $v,v'\\in V\\setminus S$ are \\emph{equivalent with respect to $S$ in $G$} if $N(v)=N(v')$ and $c(\\{u,v\\})=c(\\{u,v'\\})$ holds for $\\forall u\\in N(v)$. If two vertices $v,v'\\in V\\setminus S$ are equivalent with respect to $S$ in $G$, $G-u-v$ and $G-u-v'$ are isomorphic because the bijective function swapping only $v$ and $v'$ satisfies the isomorphic condition. Thus, we have the following lemma.\n\\begin{comment}\n\\begin{lemma}\\label{lem:isomorphic:type}\nFor $v,v'\\in V$ of the same type, the winner of Kayles on $G-u-v$ is the same as the one on $G-u-v'$ where $u\\in N(v)$.\n\\end{lemma}\n\\end{comment}\n\\begin{lemma}\\label{lem:isomorphic:type}\nSuppose that two vertices $v,v'\\in V\\setminus S$ are equivalent with respect to $S$ in $G$. Then, for any $u\\in N(v)$, $G-u-v$ and $G-u-v'$ have the same game value.\n\\end{lemma}\n\nBy the equivalence with respect to $S$, we can split $V\\setminus S$ into equivalence classes. Note here that the number of equivalence classes is at most $4^{\\card{S}}$, because for each $u\\in S$ and $v \\in V\\setminus S$, edge $\\{u,v\\}$ does not exist, or it can be colored with one of three colors if exists; \nwe can identify an equivalent class with ${\\bm{x}}\\in \\{\\emptyset,\\mathrm{G},\\mathrm{B},\\mathrm{W}\\}^{S}$, a 4-ary vector with length $\\card{S}$. For $S'\\subseteq S$, let ${\\bm{x}}[S']$ denotes\nthe vector by dropping the components of ${\\bm{x}}$ except the ones corresponding to $S'$. Also for $u\\in S$, ${\\bm{x}}[u]$ denotes the component  corresponding to $u$ in ${\\bm{x}}$. \nThen, $V$ is partitioned into $V_S^{({\\bm{x}})}$'s, where $V_S^{({\\bm{x}})}=\\{v\\in V\\setminus S \\mid \\forall u\\in S: c(\\{v,u\\})={\\bm{x}}[u]\\}$. We arbitrarily define  the representative of non-empty $V_S^{({\\bm{x}})}$ (e.g., the vertex with the smallest ID), which is denoted by $\\rho(V_S^{({\\bm{x}})})$. By using $\\rho$, we also define the representative edge set by\n\\begin{align*}\n    E^R(S)=\\bigcup_{{\\bm{x}}\\in \\{\\emptyset,\\mathrm{G},\\mathrm{B},\\mathrm{W}\\}^{S}} \\{\\{u,\\rho(V_S^{({\\bm{x}})})\\} \\in E_{\\CG}\\cup E_{\\CB} \\cup E_{\\CW} \\mid u \\in S\\}.\n\\end{align*}\nBy Lemma \\ref{lem:isomorphic:type}, we can assume that both players choose an edge only in $E^R(S)$, which enables to modify the recursive equations (\\ref{eq:recursion1}) and (\\ref{eq:recursion2}) as follows: \n\\begin{comment}\nBy Lemma \\ref{lem:isomorphic:type}, if there are at least two vertices with the same type, we only choose edges incident to one of them.\nThus, we define the representative edge set $E^R$ by\n\\begin{align*}\n    E^R=\\left\\{\\{u,v_i\\}\\in E \\mid \\nexists v_j (j<i) \\mbox{ such that $v_i$ and $v_j$ are equivalent.} \\right\\}.\n\\end{align*}\nWithout loss of generality, each player selects an edge from $E^R$.\n\n\\begin{lemma}\\label{lem:compute:representative}\nThe representative edge set $E^R$ can be computed in polynomial time.\n\\end{lemma}\n\\begin{proof}\nFor each $v\\in V$, we only check the neighborhoods of $v$. Thus, $E^R$ can be computed in polynomial time.\n\\end{proof}\nIn the step of the algorithm in Section \\ref{sec:Exalgo}, we compute the representative edge set $E^R$ and only select edges from $E^R$. That is, the recurrence formulas are changed as follows.\n\\end{comment}\nFor a vertex cover $S$ of $G$, we have  \n\\begin{align}\\label{eq:recursion3}\n    f_B(G) & = \\bigvee_{\\{u,v\\}\\in{(E_{\\CG}\\cup E_{\\CB})\\cap (E^R(S) \\cup S\\times S)}}\\lnot \\left(f_W(G-u-v))\\right),\\\\ \\label{eq:recursion4} \n    f_W(G) & = \\bigvee_{\\{u,v\\}\\in{(E_{\\CG}\\cup E_{\\CW})\\cap (E^R(S) \\cup S\\times S)}}\\lnot \\left(f_B(G-u-v))\\right).      \n\\end{align}\nNote that this recursive formulas imply that the winner of Colored Arc Kayles can be determined in time $O^*((\\tau^2 +\\tau\\cdot 4^{\\tau})^{\\tau})=\nO^*((4^{\\tau+\\log_4 \\tau})^{\\tau})=O^*(4^{\\tau^2+\\tau\\log_4 \\tau})$ $=O^*(5.6569^{\\tau^2})$, because the recursions are called at most $\\card{S}$ times and $\\tau+\\log_4 \\tau \\le 1.25 \\tau$ for $\\tau\\ge 1$. \n\n\\medskip \n\nIn the following, we give a better estimation of the number of induced subgraphs appearing in the recursion. Once such subgraphs are listed up, we can apply a standard dynamic programming to decide the necessary function values, or we can compute $f_B$ and $f_W$ according to the recursive formulas with memorization, by which we can skip redundant recursive calls. \nIn order to estimate the number of induced subgraphs appearing in the recursion, we focus on the fact that the position of a play in progress corresponds to the subgraph induced by a matching. \n \\begin{lemma}\\label{lem:positions:vc}\nThe number of nodes in recursion trees of equations (\\ref{eq:recursion3}) and (\\ref{eq:recursion4}) for Colored Arc Kayles is $O((r+1)^{\\card{S}^2\/4}3^{\\card{S}}\\card{S}^2)$, where $r$ is the used colors.\n \\end{lemma}\n\\begin{proof}\nSuppose that $S$ is a vertex cover of $G$ and players play Colored Arc Kayles on $G$. At some point, some edges selected by players together with their neighboring edges are removed, and the left subgraph represents a game position. Note that at least one endpoint of such a selected edge is in $S$, and selected edges form a matching. \nTo define such a subgraph, let us imagine that some matching $M$ is the set of edges that have been selected until the point. \nAlthough we do not specify $M$, the $M$ defines a partition $(X,Y,Z)$ of $S$; \n$X=\\{u\\in S \\mid  \\exists \\{u,v\\}\\in M: v\\in S\\}$,\n$Y=\\{u\\in S \\mid  \\exists \\{u,v\\}\\in M: v\\not\\in S\\}$,\nand $Z=S\\setminus (X\\cup Y)$. \nWe now count the number of positions having a common $(X,Y,Z)$. \nSince $X$ and $Y$ are removed, the remaining vertices in $V\\setminus S$ are classified into $V_Z^{({\\bm{x}})}$'s \n${\\bm{x}} \\in \\{\\emptyset,\\mathrm{G},\\mathrm{B},\\mathrm{W}\\}^{\\card{Z}}$.   \nThis is the common structure defined by $(X,Y,Z)$, and the positions vary as $\\card{Y}$ vertices in $\\bigcup_{{\\bm{x}} \\in \\{\\emptyset,\\mathrm{G},\\mathrm{B},\\mathrm{W}\\}^{\\card{Z}}} V_Z^{({\\bm{x}})}$ \nare matched with $Y$. Thus, we estimate the number of positions by counting the number of choices of $\\card{Y}$ vertices in $\\bigcup_{{\\bm{x}} \\in \\{\\emptyset,\\mathrm{G},\\mathrm{B},\\mathrm{W}\\}^{\\card{Z}}} V_Z^{({\\bm{x}})}$. Here, let $\\gamma$ be the number of used colors of edges. For example, BW-Arc Kayles and ordinary Arc Kayles use $\\gamma=2$ and $\\gamma=1$ colors, respectively, and which may reduce the numbers of ${\\bm{x}}$'s for smaller $\\gamma$. Then, it is above bounded\nby the multiset coefficient of $(\\gamma+1)^{\\card{Z}}$ multichoose $\\card{Y}$, \n\\[ \n\\multiset{(\\gamma+1)^{\\card{Z}}}{\\card{Y}}=\\binom{(\\gamma+1)^{\\card{Z}}+\\card{Y}-1}{\\card{Y}}\\le (\\gamma+1)^{\\card{Z}\\card{Y}}. \n\\]\nThis is an upper bound of the number of subgraphs to consider with respect to $(X,Y,Z)$. By considering all possible $(X,Y,Z)$, the total number of subgraphs is bounded by \n\\begin{align*}\n\\sum_{X,Y,Z\\subseteq S}(\\gamma+1)^{\\card{Z}\\card{Y}}      \n& \\le 3^{\\card{S}}\\cdot {\\card{S}}\\sum_{y=0}^{\\card{S}}(\\gamma+1)^{y({\\card{S}}-y)}\\le 3^{\\card{S}} \\cdot {\\card{S}}^2 \\cdot {(\\gamma+1)^{\\frac{{\\card{S}}^2}{4}}}, \n\\end{align*}\nwhere the first inequality comes from the choices of $X$, $Y$, and $Z$ from $S$. \n\\end{proof}\n\nThe following theorem immediately holds by Lemma \\ref{lem:positions:vc} and the fact that a minimum vertex cover of $G$ can be found in time $O^*(1.2738^{\\tau})$, where ${\\tau}$ is the vertex cover number of $G$~\\cite{chen2010improved}. \n\\begin{theorem}\\label{thm:algo:vc}\nThe winners of Colored Arc Kayles, BW-Arc Kayles, and Arc Kayles can be determined in time $O^*(1.4143^{{\\tau}^2+3.17{\\tau}})$, \n$O^*(1.3161^{{\\tau}^2+4{\\tau}})$, and  $O^*(1.1893^{{{\\tau}}^2+6.34{{\\tau}}})$, respectively, where ${\\tau}$ is the vertex cover number of a graph.  \n\\end{theorem}\n\n\n\n\\begin{comment}\n\\begin{align*}\n\\sum_{X,Y,Z\\subseteq S}\\left( (r+1)^{\\card{Z}}\\right)^{\\card{Y}}\n&=O(3^{{\\tau}}\\cdot {{\\tau}}\\sum_{y=0}^{{\\tau}} (r+1)^{y({{\\tau}}-y)})\\\\\n&=O(3^{{\\tau}} \\cdot {{\\tau}}^2 \\cdot {(r+1)^{\\frac{{{\\tau}}^2}{4}}})\\\\\n&=O(1.1893^{{{\\tau}}^2+6.34{{\\tau}}}\\cdot {{\\tau}}^2)\n\\end{align*}\n\\end{comment}\n\n\\begin{comment}\n\\begin{theorem}\\label{thm:algo:vc}\nGiven a graph $G$ of vertex cover number ${{\\tau}}$, ${\\bf Arc\\_Kayles\\_vc}(G)$ runs in time $O^*(2^{{{\\tau}}^2\/4}3^{{\\tau}})$.\n\\end{theorem}\n\\begin{proof}\nToDo\n\\end{proof}\n\n\\begin{corollary}\\label{cor:fpt:arc_kayles:vc}\nThe winner of Arc Kayles can be determined in time $O^*(1.1893^{{{\\tau}}^2+6.34{{\\tau}}})$\n\\end{corollary}\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\\if0\n\\subsection{Arc Kayles}\nIn this section, we propose an $O^*(1.1893^{{\\tau}^2+6.340{\\tau}})$-time algorithm for Arc Kayles, where $k$ is the vertex cover number of the input graph.\nIt utilizes the recursive relation shown in the previous section,\nbut we avoid enumerating all possible positions based on isomorphism. \nThis achieves the improvement of the running time for Arc Kayles.\n\n\n\n\n\n\n\\begin{definition}\\label{def:isomorphic}\nLet $G=(V,E)$ and $G'=(V',E')$ be graphs. Then $G$ is isomorphic to $G'$ if for ant pair of $u,v\\in V$ there is a bijection $f: V\\to V'$ such that $\\{u,v\\}\\in E$ if and only if $\\{f(u),f(v)\\}\\in E'$. \n\\end{definition}\n\n\nThen we can observe that the following key lemma holds.\n\\begin{lemma}\\label{lem:winner:isomorphic}\nIf $G$ is isomorphic to $G'$, then the winner of Arc Kayles on $G$ is the same as the one on $G'$.\n\\end{lemma}\n\nOur algorithm is also based on dynamic programming. However, we do not compute the winner on $G$ if the winner of $G'$ isomorphic to $G$ has already been computed. Instead, we use the result of $G'$.\nTo do this, we preserve the result of $G$ if the winner of a graph isomorphic to $G$ is not computed.\n\n\n\nIt is easily seen that twins $v,v'\\in V$, $G-u-v$ and $G-u-v'$ are isomorphic. Thus, we have the following lemma.\n\\begin{lemma}\\label{lem:isomorphic:twins}\nFor twins $v$ and $v'$, the winner of Keyles on $G-u-v$ is the same as the one on $G-u-v'$ where $u\\neq v,v'$.\n\\end{lemma}\n\n\n\n\nBy Lemma \\ref{lem:isomorphic:twins}, if there are twins, we only choose edges incident to one of them.\nThus, we define the representative edge set for $G$ by\n\\begin{align*}\nE^R(G)=\\left\\{\\{u,v_i\\}\\in E \\mid \\nexists v_j (j<i)  \\mbox{ such that $v_i$ and $v_j$ are twins.} \\right\\}.\n\\end{align*}\nWithout loss of generality, each player selects an edge from $E^R(G)$.\n\n\n\\begin{lemma}\\label{lem:compute:representative}\nGiven a vertex cover $S$ of $G$, the representative edge set $E^R(G)$ for $S$ can be computed in time $O(\\card{S}n)$.\n\\end{lemma}\n\\begin{proof}\nFor each $v\\in V$, we only check the neighborhoods of $v$. Since $S$ is a vertex cover, the degree of $v\\in V\\setminus S$ is at most $\\card{S}$. Thus, $E^R(G)$ can be computed in time $O(\\card{S}n)$.\\qed\n\\end{proof}\n\n\n\n\\begin{algorithm}[t]\n     \\caption{${\\bf Arc\\_Kayles\\_vc}(G)$, {\\bf Input:} a graph $G=(V,E)$}\n     \\begin{algorithmic}[1]\\label{keyles:vc}\n    \n    \\STATE Compute a minimum vertex cover $S$ of $G$\n    \\RETURN ${\\bf Recursive\\_Arc\\_Kayles\\_vc}(G,S)$\n \\end{algorithmic} \n \\end{algorithm}\n\n\n \\begin{algorithm}[t]\n    \\caption{${\\bf Recursive\\_Arc\\_Kayles\\_vc}(G,S)$, {\\bf Input:} a graph $G=(V,E)$, a vertex cover $S$}\n    \\begin{algorithmic}[1]\\label{keyles:recursive:vc}\n  \n    \\IF {$E=\\emptyset$}\n        \\RETURN $0$\n     \n    \\ELSIF {there is $G'$ isomorphic to $G$ that has already been computed}\n        \\RETURN ${\\bf Recursive\\_Arc\\_Kayles\\_vc}(G',S)$\n    \\ELSE\n         \\STATE Compute the representative edge set $E^R(G)$\n         \\RETURN $\\bigvee_{\\{u,v\\}\\in E(S)\\cup E^R}\\lnot \\left({\\bf Recursive\\_Arc\\_Kayles\\_vc}(G-\\Gamma(\\{u,v\\}),S)\\right)$\n     \\ENDIF\n \\end{algorithmic} \n \\end{algorithm}\n \n \n \\begin{lemma}\\label{lem:positions:vc}\nThe number of nodes in the recursive tree of ${\\bf Recursive\\_Arc\\_Kayles\\_vc}(G,S)$ is $O(2^{\\card{S}^2\/4}3^{\\card{S}})$.\n \\end{lemma}\n \\begin{proof}\n ToDo.\\qed\n \\end{proof}\n \n\\begin{theorem}\\label{thm:algo:vc}\nGiven a graph $G$ of vertex cover number ${{\\tau}}$, ${\\bf Arc\\_Kayles\\_vc}(G)$ runs in time $O^*(2^{{{\\tau}}^2\/4}3^{{\\tau}})$.\n\\end{theorem}\n\\begin{proof}\nWe can compute a minimum vertex cover $S$ of $G$ in time $O^*(2^{\\tau})$ by a standard branching algorithm.\n\nThe running time of ${\\bf Recursive\\_Arc\\_Kayles\\_vc}(G,S)$ can be bounded by the number of nodes of the recursive tree multiplied by the running time of the procedure in each node.\nIn Line 3, we have to check whether there is $G'$ isomorphic to $G$ has already been computed. To do this, we use memoization that preserves the information how vertices in vertex cover $S$ are connected to each other and how vertices in $V\\setminus S$ are connected to $S$ as a key. We only need $O(kn)$ bits to preserve a key and can detect \nThis can be bounded by $O^*(2^{{\\tau}^2\/4}3^{{\\tau}})$ by Lemmas \\ref{lem:compute:representative} and \\ref{lem:positions:vc}.\n\n\nThus, the total running time of ${\\bf Arc\\_Kayles\\_vc}(G)$ is $O^*(2^{{\\tau}})+O^*(2^{{{\\tau}}^2\/4}3^{{\\tau}})=O^*(2^{{{\\tau}}^2\/4}3^{{\\tau}})$.\\qed\n\\end{proof}\n\\fi\n\n\n\n\n\nWe have shown that the winner of Arc Kayles can be determined in time $O^*(1.1893^{{{\\tau}}^2+6.34{{\\tau}}})$. \nThe following theorem shows that the analysis is asymptotically tight,  \nwhich implies that for further improvement, we need additional techniques apart from ignoring vertex-cover-based isomorphic positions. We here give such an example in Figure \\ref{fig:lower}. \n\n\\begin{theorem}\\label{thm:lower:vc}\nThere is a graph for which the algorithm requires $2^{{\\tau}^2\/2}$ recursive calls for Colored Arc Kayles.\n\\end{theorem}\n\n\\begin{proof}\nIn this proof, we assume that $k$ is a multiple of $4$. \nWe explain how we systematically construct such a graph $G$ (see Figure \\ref{fig:lower}). \nLet $k$ be an even number. \nWe first define $U=\\{u_1,\\ldots, u_{k\/2}\\}$ and $V= \\{v_1, \\ldots, v_{k\/2}\\}$ as vertex sets. The union $U\\cup V$ will form a vertex cover after the graph $G$ are constructed.\nFor every 4-ary vector ${\\bm{x}}\\in \\{\\emptyset, \\mathrm{G},\\mathrm{B},\\mathrm{W}\\}^{U}$ and every $i\\in \\{1,\\ldots,k\/2\\}$, \nwe define $x_{i,{\\bm{x}}}$ as a vertex, and let $X$ be the collections of $x_{i,{\\bm{x}}}$'s, i.e., $X=\\{x_{i,{\\bm{x}}} \\mid {\\bm{x}}\\in \\{\\emptyset, \\mathrm{G},\\mathrm{B},\\mathrm{W}\\}^{U}, i\\in \\{1,\\ldots,k\/2\\}\\}$. These are the vertices of $G$. \nWe next define the set of edges of $G$. \nWe connect $v_i$ and $x_{i,{\\bm{x}}}$'s by black edges and by white edges for each $i\\in \\{1,2,\\ldots,k\/4\\}$ and $\\{k\/4+1,k\/4+2,\\ldots,k\/2\\}$, respectively.  \nFurthermore, we connect $x_{i,{\\bm{x}}}$ \nand $u\\in U$ \nfor each ${\\bm{x}}$ so that $\\{u,x_{i,{\\bm{x}}}\\}$ has \ncolor ${\\bm{x}}[u]$; if ${\\bm{x}}[u]=\\emptyset$, no edge exists between $x_{i,{\\bm{x}}}$ and $u\\in U$. Figure~\\ref{fig:lower} shows an example how we connect $x_{1,{\\bm{x}}}$ and $u_i$'s, where \n${\\bm{x}}[u_1]=\\emptyset$, ${\\bm{x}}[u_2]=\\mathrm{G}$, ${\\bm{x}}[u_i]=\\mathrm{W}$, and ${\\bm{x}}[u_{k\/2}]=\\mathrm{B}$. Notice that in Figure \\ref{fig:lower}, $\\{x_{1,{\\bm{x}}},u_1\\}$ is connected with edge $\\emptyset$ for explanation, which means that there is no edge between $x_{1,{\\bm{x}}}$ and $u_1$. \nNote that the number of vertices in $G$ is $\\card{U}+\\card{V}+\\card{X}=k+4^{k\/2}k\/2$. \nMoreover,\n$U$, $V$, and $X$ form independent sets and $X$ separates $U$ and $V$.\nThus, $U\\cup V$ forms a vertex cover of size $k$ in $G$.\n\n\nWe are ready to explain that $G$ has different $2^{{\\tau}^2\/2}$ subgraphs called by the algorithm. \nStarting from $G$, we call the recursive formulas (\\ref{eq:recursion3}) and (\\ref{eq:recursion4}) $k\/2$ times by selecting edges incident to only $v_i$'s. \nThen, all the vertices in $V$ are removed from $G$, and the neighbors of remaining $x_{i,{\\bm{x}}}$'s are in $U$. That is, each $x_{i,{\\bm{x}}}$ has its inherent set of neighbors, and thus \n$x_{i,{\\bm{x}}}$'s are not equivalent each other for vertex cover $U\\cup V$.\nThis implies that if the set of removed edges are different, the resulting subgraphs are also different. \n\nIn a step before $k\/2+1$, an edge connecting some $v_i$ is removed, and such an edge is chosen from $\\{\\{v_i,x_{i,{\\bm{x}}}\\}\\mid {\\bm{x}}\\in \\{\\emptyset, \\mathrm{G},\\mathrm{B},\\mathrm{W}\\}^{U}\\}$, that is, the number of candidates is $4^{k\/2}$ for each $i$. \nThus, the total way to choose edges is $(4^{k\/2})^{k\/2}=4^{k^2\/4}=2^{k^2\/2}$; at least $2^{k^2\/2}$ recursion calls occur.  \n\\end{proof}\n\n\\begin{figure}[htbp]\n \\centering\n \\includegraphics[width=0.8\\linewidth]{vc_lower.pdf}\n \\caption{The constructed graph $G=(U\\cup V\\cup X, E)$.}\n \\label{fig:lower}\n\\end{figure}\n\nBy the similar construction, we can show the following theorem.\n\n\\begin{theorem}\\label{thm:lower:vc2}\nThere is a graph for which the algorithm requires $1.3161^{{\\tau}^2}$ and $1.1893^{{\\tau}^2}$ recursive calls for BW-Arc Kayles and Arc Kayles, respectively.\n\\end{theorem}\n\n\n\\begin{remark}\nAlthough Theorems \\ref{thm:lower:vc} and \\ref{thm:lower:vc2} give lower bounds on the running time of the vertex cover-based algorithms, the proof implies a stronger result. In the proof of Theorem \\ref{thm:lower:vc}, we use ID's of the vertices in $U$. By connecting $i$ pendant vertices to $u_i$, we can regard them as ID of $u_i$. Furthermore, we make $U$ a clique by adding edges. These make the graphs not automorphic, which implies that the time complexity of an algorithm utilizing only isomorphism is at least the value shown in Theorems \\ref{thm:lower:vc} or \\ref{thm:lower:vc2}.    \\end{remark}\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\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection*{Acknowledgments}\nThis work is partially supported by JSPS KAKENHI JP20H05967, JP21H05852, JP21K17707, JP21K19765, JP21K21283, JP22H00513.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWhen observing and characterising the Universe on large scales, there are two broadly different, yet intertwined, types of observations \\citep{1980lssu.book.....P}. In the first type of observation, one endeavours to constrain the expansion rate of the Universe at different times. This can be done by measuring the expansion rate itself or through a variety of cosmological distance measures: angular diameter distances, luminosity distances, standard sirens, etc. In particular, if $a(t)$ is the scale factor of the Universe at cosmic time $t$, the expansion rate at that time can be defined as $H[a(t)]={\\dot a}\/{a}$ (where the overdot is derivative with regards to cosmic time). Then, one can derive an expression for the radial comoving distance, $D_M$, by following a radial null-geodesic.\n\\begin{equation} \\label{eq:distance}\n    D_M \\equiv \\chi = \\int_t^{t_0}\\frac{dt'}{a(t')}=\\int_0^{z}\\frac{dz'}{H(z')}\n    \\, ,\n\\end{equation}\nwhere we use unit $c=1$ throughout, $t_0$ is cosmic time today and $z$ is the cosmological redshift, $1+z=1\/a$. Moreover, assuming zero spatial curvature, it is possible to derive expressions for the luminosity ($D_L$) and angular diameter distances ($D_A$) by multiplying and dividing Eqn. \\ref{eq:distance} by $(1+z)$, respectively:\n\\begin{equation} \\label{eq:distances}\n    D_L = (1+z) D_M \\quad \\text{and} \\quad\n    D_A = D_M\/(1+z).\n\\end{equation}\nThus, measurements of $D_L$ and $D_A$ at different redshifts can be used to reconstruct $H(z)$ over time.\n\nIn the second type of observation, one focuses on the rate with which structures undergo gravitational collapse on the expanding background. Under the assumptions that the bulk of matter in the recent past can be described by a presureless fluid and that neutrino effects can be neglected, the density contrast, $\\delta\\equiv\\rho\/{\\bar \\rho}-1$ (where ${\\bar \\rho}$ is the mean matter density of the Universe), obeys an evolution equation of the form:\n\\begin{eqnarray}\nf'+f^2+\\left(1+\\frac{{\\rm d}\\ln aH}{{\\rm d}\\ln a}\\right)f=\\frac{3}{2}\\Omega_m(z) \\, ,  \\label{eq:growth}\n\\end{eqnarray}\nwhere the prime denotes a derivative with regards to $\\ln a$, the growth rate is defined as $f\\equiv {\\rm d}\\ln \\delta\/{\\rm d}\\ln a$ and $\\Omega_m(z)$ is the fractional energy density in matter as a function of redshift. The latter quantity depends, through the Einstein field equations, on $H \\equiv H(z)$ so that\n\\begin{eqnarray} \\label{eq:jeans}    \n\\Omega_m(z)=\\frac {\\Omega_m H^2_0}{a^3H^2},\n\\end{eqnarray}\nwith $\\Omega_m \\equiv \\Omega_m(0)$, which we use for ease of notation.\nThus, a measurements of the growth rate of structure at different redshifts (or times) can also be used to reconstruct $H(z)$ over time as well as the fractional matter density today, $\\Omega_m$.\n\nThe challenge of modern cosmology has been to use a wide range of different cosmological observables to constrain $H(z)$ as a function of time or redshift (and, of course, $\\Omega_m$) and, crucially, to pin down the underlying cosmological model which describes $H(z)$ in terms of a greatly reduced set of cosmological parameters. The frontrunner is the $\\Lambda$ Cold Dark Matter ($\\Lambda$CDM) model, whose parameters are now constrained to an unprecedented precision \\citep{Planck}. However, interestingly, inconsistencies, or \"tensions\", have begun to emerge. Different probes are leading to different constraints on, for example, the Hubble constant, $H_0$ \\citep[e.g.][]{riess_42, TRGB_H0, Maser_H0} or the density-weighted amplitude of fluctuations, $S_8 \\equiv \\sigma_8 \\left(\\Omega_m\/0.3\\right)^{0.5}$ \\citep[e.g.][]{2007.15633, 2105.13549, 1607.00008, 2106.01208, K1K, 2105.09545, Garcia-Garcia21, 2021arXiv211109898W}, where $\\sigma_8$ is the variance of $\\delta$, in spheres of radius $8\\,h^{-1}$Mpc.\n\nIn this paper we ask if it is possible to obtain meaningful, or precise, constraints on cosmological parameters with minimal assumptions about the cosmological model. To be more specific, we step back and try to find model-independent constraints on $H(z)$ and $\\Omega_m$  from measurements of the expansion history itself, cosmological distances and the growth rate. This allows us to contrast the constraining power of these two very different sets  of observables, to explore how combining them improves constraints and, most importantly, how much constraints improve once one assumes a cosmological model.\nThe hope is that understanding this process will shed light on the analysis of theories that go beyond  $\\Lambda$CDM, but also may have bearing on the current inconsistencies in parameter constraints. We will implement our model-independent approach using Gaussian processes \\citet{gp_book, Seikel_12}. Gaussian processes allow us to  reconstruct a well-defined distribution of histories of $H(z)$. The data then allows us to constrain the parameters of this distribution and, in doing so, tells us at what redshifts $H(z)$ is well determined and at what redshifts it is determined poorly.\n\nThe literature already hosts a number of examples of the possible uses of Gaussian processes to test certain aspects of the present cosmological paradigm in model-agnostic ways. For example,  one can find tests for the model dependence of the $H_0$ \\citep{1802.01505, 1907.10813, Liao_H_gp, Bonilla_H0_gp} and $S_8$ \\citep{S8_gp} tensions, for the non-zero curvature of space-time \\citep{2011.11645, 2007.05714}, as well as tests for the density and the equation of state of dark energy \\citep{1902.0942, 1806.02981}.  More closely related to the topic of this work, in \\citet{2105.01613} Gaussian processes were used to study the statistical correlations between the expansion history, cosmological distances, and the linear growth rate without appealing to the physical relationships between the three functions. However, the clearest precedent of the methodology used here is \\citet{1911.12076} who already employ Gaussian processes to obtain model-independent constraints for $\\Omega_m$ and $\\sigma_{8}$ based on the relationship between the expansion history and the linear growth rate.  The methodology developed in the present paper expands and improves many aspects of their analysis. First, our Gaussian process extends all the way to recombination. This allows us to solve the Jeans equation (Eqn. \\ref{eq:growth}) without using fitting formulas. Moreover, we also employ cosmological distances to constrain the evolution of the expansion history. This, in combination with the extended range of the Gaussian process, allows us to use the position of the first acoustic peak of the CMB temperature power spectrum to constrain expansion history far into the past. Finally, and most importantly, we sample our Gaussian process simultaneously with the cosmological parameters, allowing us to observe potential correlations between the two.\n\nThe structure of this paper is as follows: in Section~\\ref{Sect: Methodology} we present the methodology of this work regarding the use of Gaussian Process to compute predictions for cosmological observables. In Section~\\ref{Sect: Observables and data sets} we describe the cosmological observables from which we employ data and motivate their use in the context of this work. We present our results in Section~\\ref{Sect: Results} and discuss the implications of our work in Section ~\\ref{Sect: Conclusions}.\n\n\\section{Methodology} \\label{Sect: Methodology}\n\nThroughout this paper we will be working with the Eqns. \\ref{eq:distance} and \\ref{eq:growth} to connect the time evolution of $H(z)$ with current and future observations. In the case of direct measurements of $H(z)$, the expansion history can be trivially matched to observations without the need to apply any transformation.  In the case of distances, we can see from Eqn. \\ref{eq:distance} that it is possible to generate predictions for the observables from any choice of $H(z)$ by performing the relevant integral. In the case of growth (Eqn. \\ref{eq:growth}), we are faced with the problem that most observations do not report $f$ but the combined quantity $ f\\!\\sigma_{8}$. Thus, it is convenient to rewrite Eqn. \\ref{eq:growth} in terms of the latter. This can be done as follows: assuming that perturbations grow in a self-similar manner, we can  define $\\delta(t,{\\bf x})\\equiv D(t)\\delta_0({\\bf x})$, where $\\delta_0$ is the density contrast today. Then, we can rewrite Eqn. \\ref{eq:growth} as\n\\begin{equation} \\label{eq:jeans_D}\n    \\frac{d}{da}\\left(a^2H\\frac{dD}{d\\log a}\\right)=\\frac{3}{2}\\Omega_m(a)\\,a\\,H\\,D \\, .\n\\end{equation}\nDefining $y\\equiv a^2E\\frac{dD}{d\\log a}$, with $E\\equiv H\/H_0$, and switching to the integration variable $s\\equiv\\log(1+z)$, this equation can be written in terms of a system of coupled  first-order equations:\n\n\\begin{equation} \\label{eq:Jeans_sys_D}\n    \\frac{dy}{ds}=-\\frac{3}{2}\\frac{\\Omega_m}{a E}D \\, ,\\hspace{12pt} \n    \\frac{dD}{ds}=-\\frac{y}{a^2E}.\n\\end{equation}\n\nThen, it is possible to transform the quantities $y(s)$ and $D(s)$ into $f\\!\\sigma_{8}(s)$ and $\\sigma_{8}(s)$ by applying the following transformation\n\\begin{equation} \\label{eq:D_to_s8}\n    \\sigma_{8}(s) = \\sigma_{8}(0) \\frac{D(s)}{D(0)}\\,,\\hspace{12pt}\n    f\\!\\sigma_{8}(s) = \\frac{y(s) \\, \\sigma_{8}(0)}{E(s)D(0)}e^{2s}\\, .\n\\end{equation}\nThe thrust of this paper is to keep the analysis as model-independent as possible. Yet, as we can see, it is still possible to extract information about some of the cosmological paramaters. For a start, combining the information from distances and growth allows us to constrain $\\Omega_m$. But we also have, automatically, $H_0\\equiv H(z=0)$ and, as we just saw, we can calculate $\\sigma_8\\equiv\\sigma_8(z=0)$ (or $S_8$ as a derived parameter).\n\nIn this paper we will model the time (or redshift) dependent Hubble rate, $H(z)$, as a Gaussian Process spanning over the redshift range $0<z<1100$. A Gaussian process (GP) is a collection of random variables, any finite number of which have a joint Gaussian distribution \\citep{gp_book}. A GP $g(\\textbf{x})$, where $\\boldsymbol{x}$ is a collection of random variables, is fully specified by a mean function $m(\\textbf{x}) \\equiv \\mathcal{E}[g(\\textbf{x})]$, where $\\mathcal{E}[\\cdots]$ is the expectation value over the ensemble,  and a covariance function $k(\\textbf{x}, \\textbf{x'}) \\equiv \\mathcal{E}[(g(\\textbf{x})-m(\\textbf{x}))(g(\\textbf{x'})-m(\\textbf{x'}))^T]$. In combination, the mean and covariance functions determine the statistical properties of the ensemble of random variables which defines the family of shapes that the GP can take. \n\nThus, GP's can be used as agnostic function-space priors which, in combination with a likelihood for the observed data $\\mathcal{L}(\\textbf{y}| g(\\textbf{x}), \\boldsymbol{\\sigma})$, where $\\textbf{y}$ is a set of data points with a set of errors $\\boldsymbol{\\sigma}$),  define a regression model. Then, observations can be used to inform the GP posterior (i.e. the statistical properties of the set of random variables) $\\mathcal{P}(g(\\textbf{x})| \\textbf{y},\\boldsymbol{\\sigma})$, which determines the subset of functions most consistent with the data.\n\n Since the Hubble rate is generally regarded as a monotonically increasing function, it is important to define a non-zero mean for the GP. This prevents the GP from simply fitting the long range upwards trend of $H(z)$ while washing out interesting local features \\citep{Shafieloo_gp}. In this paper, we will define the mean of the GP in terms of a fiducial $\\Lambda$CDM prediction:\n\\begin{equation} \\label{H_bar}\n    \\overline{H}(z) \\equiv  \\overline{H_0} \\sqrt{\\overline{\\Omega}_m(1+z)^3+\\overline{\\Omega}_r(1+z)^4+ \\overline{\\Omega}_{\\Lambda}} \\, ,\n\\end{equation}\nwhere $\\overline{\\Omega}_m$,  $\\overline{\\Omega}_r$ and $\\overline{ \\Omega}_{\\Lambda}$ stand for the present values of the cosmological densities of matter, radiation and dark energy , respectively,\nas given by the \\textit{Planck} 2018 TTTEEE+lowE cosmology (see Tab. \\ref{tab:fiducial}). In other words, they are completely fixed and are not variables.\n\nIn order to prevent this (ultimately arbitrary) choice of mean from biasing our constraints on the cosmological parameters, we define a free amplitude parameter $A_0$ that multiplies the mean of the GP such that\n\\begin{equation} \\label{H_mean}\n    H_m(z) \\equiv A_0 \\, \\overline{H}(z) \\, .\n\\end{equation}\nSimply put, if the data is somewhat away from what one might expect from the fiducial $\\Lambda$CDM background, the Gaussian process will have to soak up the large scale differences, or trends, in detriment to the small scale, local features. Moreover, in those regions where data is sparse, the GPs constraints are highly dominated by their prior; i.e. the chosen mean. If such mean systematically falls beneath or above the data, it can lead to spurious trends in the final results. In App. \\ref{app:validation}, we show these effects can substantially bias the constraints we obtain in the absence of $A_0$. For these reasons, we advise against common practices such as employing a constant as a mean for a GP modelling the recent expansion history or extrapolating GP results to regions where they become dominated by the choice of mean and covariance function. \n\n\\begin{table} \n    \\caption{ Values of the cosmological parameters used to define the fiducial cosmology. Values based of \\textit{Planck} 2018 TTTEEE+lowE cosmology \\citep{Planck}}\n    \\centering\n    \\begin{tabular}{|p{1.5 cm}|p{1.5cm}}\n    \\hline\n    Parameter & value   \\\\\n    \\hline\n    $\\overline{\\Omega}_{\\rm{m}}$   &  0.316  \\\\\n    $\\overline{\\Omega}_{\\Lambda}$ & 0.683 \\\\\n    $\\overline{\\Omega}_r$ &  $9.245 10^{-5}$ \\\\\n    $\\overline{H}_0$ & 67.27 \\\\\n\\end{tabular}\n\\centering\n\\label{tab:fiducial}\n\\end{table}\n\nThe Hubble rate must also be a continuous and smooth function. Therefore it is important that the covariance function of the GP reflect such properties \\citep{Dialektopoulos21}. Ultimately, we chose a a square exponential covariance function to model the correlations between the different nodes of the GP. This decision was made based on the fact that the square exponential is computationally inexpensive and infinitely differentiable kernel appropriate for modelling smooth fluctuations around the mean of the GP. Mathematically, the square exponential covariance function is defined as\n\\begin{eqnarray} \nk\\left[g(\\boldsymbol{x}),g(\\boldsymbol{x}')\\right]=\\eta^2\\exp\\left[\\frac{-|g(\\boldsymbol{x})-g(\\boldsymbol{x}')|^2}{2l^2} \\right] \\, ,  \\label{eq:QuadExp}\n\\end{eqnarray}\nwhere $\\eta$ is the amplitude of the oscillations around the mean and $l$ is the correlation length between the GP nodes. The parameters $\\eta$ and $l$ are known as hyperparameters since they constrain the possible values that the nodes; i.e. the parameters the GP, can take. In other words, the hyperparameters dictate the family of functions available to the GP.  In addition to this, a white noise term with amplitude $10^{-3}$ was added to the covariance function to ensure numerical stability.   \n\nSampling over GPs is extremely efficient when the model for the observed data is linearly related to the GP and the likelihood is Gaussian. In this case, it can be shown that the posterior distribution also takes the form of a GP for which the mean and covariance functions can be calculated analytically by marginalizing over the nodes of the original GP \\citep{gp_book}. However, when the observed data is not linearly related to the GP, the posterior distribution becomes non-Gaussian. Thus, each node of the GP; i.e. each dimension of the multivariate Gaussian distribution, must be treated as a newly added degree of freedom in the sampling process (i.e. a new parameter). \n \n\\begin{figure*} \n\\centering\n    \\includegraphics[width=.65\\linewidth]{{\"schematic\"}.pdf} \n    \\caption{Shows a schematic representation of the data analysis pipeline.}\n    \\label{fig:code}\n\\end{figure*} \n\nExploring such a vast parameter space is effectively unfeasible with traditional non-gradient-based sampling algorithms and requires more sophisticated samplers. In this work we employ the No U-turn Sampler (NUTS) \\citep{NUTS}, a self-tuning variant of Hamiltonian Monte Carlo (HMC) \\citep{HMC, Betancourt17} which simulates Hamiltonian trajectories through the parameter space to generate efficient proposal steps. This allows us to handle hundreds or thousands of parameters during the inference process efficiently. \nWe employ the NUTS implementation of the python package \\texttt{Pymc3} \\citep{Pymc3} which makes use of the tensor package \\texttt{Theano} \\citep{Theano} as a back-end to build a symbolic graph of the model used to fit the data. This graph is then used to perform automatic-differentiation \\citep{Autodiff18} to obtain the gradient of the regression model needed for HMC. Through out the likelihood we make use of simple integration schemes; e.g. Runge-Kutta, trapezoidal rule, on which \\textsc{Theano} can apply automatic-differentiation to make the likelihood differentiable in all parameters.\n\nIn the case of distances (see Eqn. \\ref{eq:distance}), the integral over $1\/H_{\\mathcal{GP}}$ was performed using the trapezoidal rule with $s\\equiv \\log(1+z)$ as the integration variable. In the case of the linear growth rate, the system of differential equations shown in Eqns. \\ref{eq:Jeans_sys_D} was solved for $y(s)$ and $D(s)$ employing a second order Runge-Kutta scheme, integrating over the redshift range $0 < z < 1100$ from the past into the present with initial conditions $D(z=1100)=a(z=1100)$ and $y(z=1100)=a(z=1100)^3E(z=1100)$. Note that these initial conditions are only strictly correct for a purely matter-dominated universe, which is not accurate at $z\\sim1000$. However, the impact of this assumption is negligible by comparison to current uncertainties on growth measurements. In combination, these methods allow us to obtain sub-percentage accuracy through out the entire redshift range of the GP with respect the output of cosmological CLASS \\citep{Class}, which performs the full numerical calculation.\n\nIn order to cover the redshift range $0 < z < 1100$ while keeping the numerical errors under control, we sampled the GP evenly in $s$ as opposed to redshift itself. This variable concentrates most of the GP samples at low redshift where most of our data lays while offering great numerical accuracy \\citep{1710.0423}. This allowed us to cover our desired redshift range with only $200$ nodes; i.e. with a 200-dimensional GP. \n\nIn addition to the $200$ parameters associated with the GP, we also sample over the cosmological parameters present in Eqn. \\ref{eq:growth}; i.e. $\\Omega_m$ and $\\sigma_{8}$, as well as the GP hyperparameters $\\eta$ and $l$ described in Eqn. \\ref{eq:QuadExp}. We define the GP not over the Hubble rate itself but over as relative deviations from the $\\Lambda$CDM background \n\\begin{eqnarray}\n  H_{\\mathcal{GP}} \\equiv H_m(s) [1 + \\delta H_{\\mathcal{GP}}(s)], \\nonumber\n\\end{eqnarray}\nwhere $\\delta H_{\\mathcal{GP}}$ is a GP centred at zero. This approach has the added benefit of normalizing the amplitude of the oscillations of the GP with respect the mean making the sampling of $\\eta$ far more efficient by virtue of reducing its potential range of values. Note that the correlation length scale $l$ is defined in units of $s$ as opposed to $z$. We also marginalize over the absolute magnitude of the supernovae $M$ (See Sect. \\ref{subsec: Supernovae} for details), the scale of the sound horizon $r_s$, and the amplitude of $H_m$, $A_0$.\n\nIt is important to emphasise that we perform a fully Bayesian inference over the GP, as opposed to what it is known as an empirical Bayesian analysis in which first, the marginal likelihood of the GP hyperparameters is maximized; and then, keeping their values fixed, the conditional posterior over the GP is inferred. While for large sets of data the output of the two approaches converges to the same results, when only sparse data is available the fact that the empirical Bayesian analysis does not account for the full posterior volume of the hyperparameters  can lead to an under-accounting of uncertainties.\n\nIn order to establish the reliability of the methodology developed in this work for cosmological inference, and the robustness of the fiducial results, we also considered a series of alternative analyses. In the first test, the hyperparameters of the GP ($\\eta$, $l$) were kept fixed. We will label results from this analysis as \"Fixed HP\". In the second test, instead of sampling over the $A_0$ parameter, we performed a two-step analysis. In the first step, a standard $\\Lambda$CDM model was fit to the data. Based on this best-fit $\\Lambda$CDM model, a expansion history was derived. Then, in the second step, the best-fit expansion history was used as the mean of the GP during the analysis while fixing $A_0 = 1$. We will label results from this analysis as \"Two-Steps\". \n\n\\begin{table} \n\\caption{Sampled parameters and their priors in the standard analysis (first column), the fixed hyperparameters analysis (second column) and the Two-Steps analysis (third column). }\n\\begin{tabular}{ |p{1cm}|p{1.75cm}|p{1.75cm}|p{1.75cm}|p{1.75cm}|p{1.75cm}}\n \\hline\n  Parameter & Fiducial & Fixed HP & Two-Steps  \\\\\n    \\hline\n    $\\eta$ & $\\mathcal{N}_{1\/2}(0,0.2)$  & 0.2 & $\\mathcal{N}_{1\/2}(0,0.2)$  \\\\\n    $l$ & $U(0.01, 6)$  & 1.0 & $U(0.01, 6)$   \\\\\n    \\hline\n    $\\Omega_m$ & $U(0,1)$ & $U(0,1)$ & $U(0,1)$ \\\\\n    $\\sigma_{8}$ & $\\mathcal{N}(0.8, 0.5)$ & $\\mathcal{N}(0.8, 0.5)$ & $\\mathcal{N}(0.8, 0.5)$   \\\\\n    \\hline\n    $A_0$ & $U(0.8, 1.2)$ & $U(0.8, 1.2)$ & 1 \\\\\n    $M$ & $\\mathcal{N}(-19.2, 1)$ & $\\mathcal{N}(-19.2, 1)$ & $\\mathcal{N}(-19.2, 1)$   \\\\\n    $r_s$ & $\\mathcal{N}(150, 5)$ & $\\mathcal{N}(150, 5)$ & $\\mathcal{N}(150, 5)$ \\\\\n\\end{tabular}\n\\label{tab:prior}\n\\end{table}\n\nA summary of the priors used in the fiducial and alternative analyses can be found in Tab. \\ref{tab:prior}. As a general rule, when performing Bayesian inference one should avoid broad, uniform priors \\citep{Gelman17}. This is mainly due to the fact that they do not accurately represent the prior knowledge, put a lot of posterior mass in unlikely values and introduce hard boundaries which can be difficult to motivate. This last consideration is particularly important when using \\textsc{HMC} since sharp edges in parameter space can lead to the sampling process becoming inefficient. Following this principle, we employed a half-Gaussian distribution with zero mean and standard deviation $0.2$ as prior distribution for the hyperparameter $\\eta$ to down-weight extreme deviations (i.e. $20\\%$ and above, well within the observational errors on $H(z)$) from the chosen GP mean without introducing any unnecessary hard boundaries. In the case of $l$ we used a uniform distribution between $0.01 < l < 6$ for the following reasons. First, there is no reason to down-weight long correlation modes against short ones or vice versa. Second, looking at Eqn. \\ref{eq:QuadExp}, it is possible to see that as $l \\rightarrow 0$ the value of the covariance function approaches zero regardless of the value of $\\eta$. This opens a vast a volume in the parameter space which can lead to an inefficient sampling. Third, the likelihood function becomes flat at such small scales since there is no information in the data to constrain those small scale modes. Forth, the expansion rate is expected to have some degree of temporal correlations, and therefore the limit $l=0$ must be excluded. The parameter $\\Omega_m$ was sampled from a uniform distribution between $0$ and $1$ to enforce the physical boundaries on the allowed values for the cosmological matter fraction. In the case of $\\sigma_8$ for which there is no physically-motivated upper limit, we employ a better behaved Gaussian prior centered at $0.8$ and with $0.5$ standard deviation. Given the degeneracy of $A_0$ with the GP parameters, the parameter was sampled from a uniform distribution between $0.8$ and $1.2$, which amply encompasses the current discrepancy in $H_0$ between CMB data and local measurements. The supernova absolute magnitude parameter $M$ was sampled from a Gaussian distribution centered at value found by the SH0ES collaboration for the parameter $M = -19.2$, with a standard deviation of $1$. While the mean value corresponds to a local expansion rate of $H_0 \\simeq 74.0$ km\/s\/Mpc, the standard deviation ensures that all values in the range $50 < H_0 < 100$ km\/s\/Mpc fall within the $1\\sigma$ region of the prior distribution. Finally, the sound horizon scale, $r_s$,  was sampled from a Gaussian distribution centered at $150$ Mpc with a standard deviation of $5$ Mpc. We do this instead of computing $r_s$ from the expansion history in combination with the BBN prior to allow for largest deviations from the fiducial cosmology. This was needed to recover the test cosmologies of App. \\ref{app:validation}.\n\n\n\\section{Observables and Data sets} \\label{Sect: Observables and data sets}\nIn order to make the most of the modelling freedom offered by GPs we consider as much data as possible. In this work we use a combination of several different probes that together account for 91 data points for a variety of cosmological observables. A summary of the data used in this work can be found in Tab. \\ref{tab:data}.\n\n\\begin{figure*} \n\\centering\n    \\includegraphics[width=\\linewidth]{{\"data\"}.pdf} \n    \\caption{Data points from the different surveys used in this work across redshift for the three cosmological functions of interest $H(z)$, $D_M$ and $f \\! \\sigma_8$.}\n    \\label{fig:data}\n\\end{figure*} \n\n\\begin{table*} \n\\centering\n\\caption{Data sets used in our analysis, listing the probe, the redshift range of the probe, the choice of observable and the size of the data vector. See Fig. \\ref{fig:data} for a pictorial representation. }\n\\begin{tabular}{|p{5.5cm}|p{2cm}|p{1.5cm}|p{0.7cm}|p{.7cm}|p{0.7cm}|p{1cm} |  }\n \\hline\nData set &  Probe & Redshifts &\\multicolumn{3}{|c|}{Observable} & Data Points\\\\\n&  & & $H(z)$ & $D_M(z)$ & $f \\! \\sigma_8 $ & \\\\\n \\hline\n CCs \\citep{2011.11645} &  Cosmic Chronometers & 0.07 - 2.36 & $\\checkmark$ & $\\times$ & $\\times$ & 35\\\\\n \nPantheon DS17\\citep{1710.00845} & SNe Ia & 0.38 - 0.61 & $\\times$ & $\\checkmark$ & $\\times$ & 40\\\\\n \n BOSS DR12 \\citep{1607.03155} & BAO+RSD & 0.38 - 0.61 & $\\checkmark$ & $\\checkmark$ & $\\checkmark$ & $3\\times3$\\\\\n\n eBOSS DR16 \\citep{2007.08991} & BAO+RSD & 1.48 & $\\checkmark$ & $\\times$ & $\\times$  & $1\\times3$ \\\\ \n \n Wigglez \\citep{1204.3674} & RSD & 0.44 - 0.73 & $\\times$ & $\\times$ & $\\checkmark$  & $1\\times3$ \\\\\n \n DSS \\citep{2105.05185} & RSD & 0 & $\\times$ & $\\times$ & $\\checkmark$  & 1 \\\\\n \n \\textit{Planck} 2018  \\citep{Planck} & CMB & 1090.30 & $\\times$ & $\\checkmark$ & $\\times$  & 1 \\\\\n \\hline\n DESI \\citep{Font-Ribera14} & BAO+RSD & 0.15 - 1.85 & $\\checkmark$ & $\\checkmark$ & $\\checkmark$ & $3\\times18$\\\\\n\n\\end{tabular}\n\\label{tab:data}\n\\end{table*}\n\n\\subsection{Cosmic Chronometers}\n\nCosmic Chronometers (CCs) are tracers of the evolution of the differential age of the Universe as a function of redshift. Since $H(z) \\equiv \\Dot{a}\/a = -(dt\/dz)\/(1+z)$ a measurement of dt\/dz directly yields the expansion rate \\citep{0106145}.  By measuring the age difference between two ensembles of passively evolving galaxies at different redshifts, one can determine the derivative of redshift with respect to cosmic time, $dz\/dt$. Massive, early, passively-evolving galaxies have been found to be very good tracers in this sense \\citep{Cimatti06, Thomas11, Moresco15, Moresco18, Moresco20} and have been used extensively over the past two decades to measure $H(z)$ up to $z \\approx 2$. In this work we make use the $H(z)$ measurements from CCs summarized in Table 1 of \\citet{2011.11645}.\n\n\n\\subsection{Supernovae}\n\\label{subsec: Supernovae}\nType Ia supernovae (SNe Ia) \u2014 titanic explosions of white dwarfs in multi-star systems  \\citep{Hoyle60, Colgate69}, are highly prized observations in cosmology due to their capacity to act as standard candles \\citep{9907052, 0012376}.\n\nHowever, SNe Ia by themselves can only inform their relative distance to one another and need to be calibrated with nearby SNe Ia of known redshift and luminosity distance $D_L$ to obtain absolute distances. Thus, SNe Ia inform the relationship\n\\begin{equation} \\label{eq: light_curve}\n    \\mu(z) = 5 \\log_{10} D_L(z) + 25 + M \\,, \n\\end{equation}\nknown as the luminosity distance modulus where $M$ is the calibrator known as the absolute magnitude of the SNe Ia. Therefore, once $M$ has been determined SNe Ia luminosity distance measurements can be used to inform the Hubble rate.\n\nIn this work, we fit the compressed data vector of the Pantheon sample, known as \"DS17\", composed of 40 measurements of the distance modulus (See Eqn. \\ref{eq: light_curve}) in the range $0.15\\leq z\\leq1.615$ \\citep{1710.00845}. The original Pantheon sample is composed of the optical light curves and redshifts for 365 spectroscopically confirmed Type Ia supernovae (SNe Ia) discovered by the Pan-STARRS1 (PS1) Medium Deep Survey combined with the subset of 279 PS1 SN Ia ($0.03 < z < 0.68$) with useful distance estimates of SN Ia from SDSS, SNLS, various low-z and HST samples to form the largest combined sample of SN Ia consisting of a total of 1048 SN Ia ranging from $0.01 < z < 2.3$ \\citep{1710.00845}. \n\nIn the light of recent works in the literature questioning the accuracy of the absolute calibration of important sectors of Pantheon sample \\citep{H0_ofer}, we marginalize over the absolute magnitude of the supernovae as opposed to fixing its value. Due to the degeneracy between $M$ and $H_0$, this is equivalent to fitting the expansion rate, $E(z)= H(z)\/H_0$, as opposed to the Hubble rate, $H(z)$.\n\n\\subsection{Baryon Acoustic Oscillations}\nBaryon acoustic oscillations (BAOs) enhance matter overdensities  at a characteristic physical separation scale which corresponds to the size of the sound horizon at the end of the drag epoch, $r_{\\rm{s}}(z_{\\rm d})$ \\citep{Peebles70, 0110414}. The sound horizon is defined as the distance a pressure wave can travel from its time of emission in the very early Universe up to a given redshift. This can be expressed as \n\\begin{equation} \\label{eq:rsound}\n    r_{\\rm{s}}(z)= \\int^{\\infty}_z \\frac{c_{\\rm{s}}  \\, {\\rm d}z'}{H(z')} \\;,\n\\end{equation}\nwhere $c_{\\rm s}$ denotes the speed of sound, and where $H(z)$ is the expansion rate at redshift $z$. The end of the drag epoch is defined as the time when photon pressure can no longer prevent gravitational instability in baryons around $z \\sim 1020$ \\citep{0803.0547}. \n\nThe BAO feature can be measured in the directions parallel and perpendicular to the line of sight. Perpendicular to the line of sight, the BAO feature informs the trigonometric relationship\n\\begin{equation}\\label{eq:acoustic}\n    \\theta = \\frac{r_{\\rm{s}}(z_{\\rm d})}{D_M(z)} \\;,\n\\end{equation}\nwhere $\\theta$ is the angle under which the scale of the sound horizon is observed.  Parallel to the line of sight, the BAO feature informs the relationship $ \\Delta z = H(z) r_{\\rm{s}}(z_{\\rm{d}})$ which can be used to constrain the expansion history of the Universe directly.\n\nIn this work, we make use of the twelfth data release of the galaxy clustering data set of the Baryon Oscillation Spectroscopic Survey (BOSS DR12) which forms part of the Sloan Digital Sky Survey (SDSS) III. BOSS DR12 comprises 1.2 million galaxies over an area of 9329 deg$^2$ and volume of 18.7 Gpc$^3$, divided into three partially overlapping redshift slices centred at effective redshifts 0.38, 0.51, and 0.61.\n\nWe fit the Alcock-Paczynski (AP) parameters  $\\alpha_\\parallel$ and $\\alpha_\\perp$ as reported by the BOSS DR12 data products\n \\begin{equation} \\label{eq:alphas_B}\n    \\frac{[H(z)]_{\\rm{fid}}}{\\alpha_\\parallel} = \\frac{H(z) [r_{\\rm{s}}(z_{\\rm{d}})]_{\\rm{fid}} }{ r_{\\rm{s}}(z_{\\rm{d}})}, \\quad\n    \\alpha_\\perp [\\chi(z)]_{\\rm{fid}} = \\frac{\\chi \/ r_{\\rm{s}}(z_{\\rm{d}})}{[r_{\\rm{s}}(z_{\\rm{d}})]_{\\rm{fid}}} ,\n\\end{equation}\nfrom the reconstruction of the BAO feature at the three different redshift bins where $[r_{\\rm{s}}(z_{\\rm{d}})]_{\\rm{fid}} = 147.78$ Mpc is the scale of the sound horizon at drag epoch as given by the fiducial cosmology used for the reconstruction. $[H(z)]_{\\rm{fid}}$ and $[\\chi(z)]_{\\rm{fid}}$ are the corresponding Hubble parameter and comoving radial distance for the fiducial cosmology, respectively. \n\nIn addition to this, we employ the  anisotropic clustering of quasars in the sixteenth data release of the extended Baryon Oscillation Spectroscopic Survey (eBOSS DR16 \\citealp{1703.00052}), which forms part of the Sloan Digital Sky Survey (SDSS) IV  \\citep{1508.04473}. The eBOSS DR16 catalog contains 343,708 quasars  between $0.8 < z < 2.2$, from which BAO and RSD meausurements are obtained at an effective redshift of $z_{\\rm eff} = 1.48$ \\citep{2007.08998}.  We use the results from the configuration space analysis performed by measuring the two-point correlation function and decomposing it using the Legendre polynomials.  Similarly to BOSS DR12, the BAO signal is measured both parallel and perpendicular to the line of sight. This allows for the measurement of the geometrical relationships $D_H(z_{\\rm eff})\/[r_{\\rm d}]_{\\rm fid}$ and $D_M(z_{\\rm eff})\/[r_{\\rm d}]_{\\rm fid}$ respectively, where $D_H(z) \\equiv c\/H(z)$ and $[r_{\\rm d}]_{\\rm fid} = 147.3$.\n\nFinally, we make use of the \\textit{Planck} 2018 measurement of the BAO angular scale $\\theta_* = \\frac{D_{\\rm{A}}(z_*)}{r_{\\rm{s}}(z_*)}$, where $z_* \\sim 1100$ is the redshift of the last scattering surface. We use the \\textit{Planck} measurement from the temperature and polarization maps denoted as TTTEEE + lowE.\n\n\\subsection{Redshift Space Distortions}\nRedshift space distortions (RSDs) are modifications to the observed redshift of a given object caused by its radial peculiar velocity \\citep{kaiser87}. RSDs are caused by deviations\nfrom the Hubble flow that are gravitationally induced by inhomogeneities in the gravitational potential of the surrounding matter distribution. On large, linear scales RSDs are dominated by the infall towards overdense structures, known as the Kaiser effect \\citep{Hamilton98}. Clustering two-point statistics as a function of transverse and line-of-sight separation are sensitive to the quantity $f \\! \\sigma_8$ via RSDs \\citep{1607.03150}.\nAlternatively, SNe Ia themselves can be used as a probe of the local velocity field \\citep{2105.05185, 9908237, 1605.01765}, which can also be used to measure this parameter.\n\nWe use the three measurements of  $f\\sigma_{\\rm{8}}(z)$ from RSD from the BOSS DR12 data, obtained using the anisotropic clustering of the pre-reconstruction density field \\citep{1607.03155}. We also include the value of $f \\! \\sigma_8 (z_{\\rm eff})$ measured from the BOSS DR16 quasar sample. The BAO and RSD measurements of both datasets are extracted from the same set of observations. As such, they are not statistically independent, and we account for their full covariance matrix in our analysis \\citep{1607.03155,2007.08998}. In addition to these, we use the $f \\! \\sigma_8$ measurements reported by the WiggleZ Dark Energy Survey at the Australian Astronomical Observatory \\citep{Drinkwater10} at redshifts 0.44, 0.60 and 0.73.\n\nFinally, we also use the value of $f \\! \\sigma_8 (z=0)$ derived from the measured peculiar velocities of the Democratic Samples of Supernovae (DSS), the largest  catalog used to study bulk flows in the nearby Universe, compiled of 775 low-redshift Type Ia and II supernovae (SNe Ia $\\&$ II). \\dam{You need to include a citation here!}\n\n\\subsection{Synthetic Stage-IV Data}\n\\label{sec: forecasts}\nIn addition to the previously discussed observables and data sets, we also produce forecasts for future experiments, focusing on the Dark Enegy Spectroscopic Instrument (DESI). DESI is a galaxy and quasar redshift survey currently taking data from the Mayall 4 meter telescope at Kitt Peak National Observatory. The baseline area is 14000 sq. deg. We use the forecast errors for the measurements of the Hubble rate, $H(z)$; the angular diameter distance, $D_A(z)$; and the growth rate measurement $f \\! \\sigma_8$ reported by \\citet{Font-Ribera14} to create a set synthetic measurements for $H(z)$, $D_A(z)$, and $f \\! \\sigma_8$ at an array of 18 redshifts from 0.15 to 1.85. The synthetic measurements were generated using the best-fit \\textit{Planck} 2018 cosmology, including measurement noise following the statistical uncertainties reported by \\citet{Font-Ribera14}.\n\n\\section{Results} \\label{Sect: Results}\n\n\\begin{table} \n\\caption{One dimensional constraints on the hyperparameter $\\eta$ (first column) and the mean variance of $\\delta H(z)$ (second column) for different analyses.}\n\\begin{tabular}{|p{3.2cm}|p{2cm}|p{2cm}|}\n \\hline\n  Analysis & $\\eta$ & $\\overline{\\sigma}(\\delta H(s))$ \\\\\n    \\hline\n    All data ($\\Lambda$CDM) & $0 \\pm 0$ & $0.011 \\pm 0.003$ \\\\\n    DESI+CMB ($\\Lambda$CDM) & $0 \\pm 0$ & $0.002 \\pm 0.001$ \\\\\n    \\hline\n    All data & $0.113 \\pm 0.075$ & $0.094 \\pm 0.004$ \\\\\n    All data (Fixed HP) & $0.2$ & $0.2$ \\\\\n    All data (Two-Steps) & $0.044 \\pm 0.039$ & $0.025 \\pm 0.01$ \\\\\n    No CMB & $0.142 \\pm 0.096$ & $0.125 \\pm 0.016$ \\\\\n    No DSS & $0.111 \\pm 0.074$ & $0.091 \\pm 0.005$ \\\\\n    Growth data & $0.147 \\pm 0.114$ & $0.183 \\pm 0.003$ \\\\\n    Geometry data & $0.122 \\pm 0.077$ & $0.098 \\pm 0.005$ \\\\\n    \\hline\n    DESI $+$ CMB & $0.085 \\pm 0.074$ & $0.08 \\pm 0.004$ \\\\\n\\end{tabular}\n\\label{tab:variance}\n\\end{table}\n\nIn this section we present the results of our analysis, including both the constraints on expansion history and cosmological parameters from current data, and forecasts for ongoing and future surveys. \n\n\\subsection{Validity of the methodology}\n\nBefore discussing the results of our analysis, we must first verify its reliability. Our methodology produces two types of results: constraints on the GP used to model the expansion history, and on the cosmological parameters (mainly $\\Omega_M$) involved in solving the Jeans equation (Eqn. \\ref{eq:growth}). We thus report results on both aspects. In App. \\ref{app: Alternative analyses} we present the results of the alternative analyses described at the end of Sect. \\ref{Sect: Methodology} and show that, despite the different treatments of the GP, they produce statistically compatible constraints for the cosmological functions as well as for the cosmological parameters, and similar uncertainties on both.  In App. \\ref{app:validation}, we considered five different cosmologies, each progressively more discrepant with our fiducial cosmology (\\textit{Planck} 2018), and generated mock data based on each them. We then showed that our methodology recovers the input cosmologies within statistical uncertainty. In addition to this, in App. \\ref{app: Using HMC}, we discuss the specific \\textsc{NUTS} set up used to produce our results. These settings allow us to achieve a Gelman-Rubin convergence test~\\citep{GelmanRubin} value of $R -1 < 0.01$ for all the combinations of data considered.\n\n\\subsection{Current constraints}\n\\subsubsection{Cosmological functions}\n\nBefore presenting the results for the cosmological functions of interest; i.e. the expansion history $H(s)$ and linear growth rate $f \\! \\sigma_8$, we must first discuss the results on the core of our analysis, the GP on $\\delta H(s)$. We found that our analysis of current data produces $\\delta H(s)$ constraints compatible with $0$ at all redshifts at less than $1\\sigma$ deviation. This means that the mean of the GP, $H_m(s) = A_0 \\overline{H}(s)$, is capable of capturing the long range correlations of the data allowing the GP to fit local features. The recovered bounds on $\\delta H$ are shown in Fig. \\ref{fig:dH_geo_vs_gro}.\n\nMoreover, we quantified how well different combinations of data can constrain $\\delta H(s)$. In other words, we measured how strong the agreement of $\\delta H(s)$ with zero is, and how it is affected by the data considered and the analysis choices. However, the fact that $\\delta H(s)$ has a multivariate distribution means that there is not a unique figure of merit for how well data constrains it. In this work we focused in two numbers. On the one hand, we looked at the hyperparameter $\\eta$ that constrains the prior distribution of values that $\\delta H(s)$ can take. On the other hand, we computed the mean variance of the $\\delta H(s)$ samples across redshift. In other words, for each parameter of $\\delta H(s)$ (i.e. $\\{\\delta H(s_1),\\,\\delta H(s_2),\\,...,\\,\\delta H(s_{200})\\}$), we computed the variance of the \\textsc{HMC} samples. Then, we took mean value of those variances, which is equivalent to averaging over redshift. We refer to this statistic as $\\overline{\\sigma}(\\delta H(s))$. The motivation behind $\\overline{\\sigma}(\\delta H(s))$ lies in the fact that it directly translates to average fractional constraints on $H(z)$ that can be readily interpreted. \n\nA summary of the impact of the different data sets and analysis choices on the distribution of $\\eta$ and $\\overline{\\sigma}(\\delta H(s))$ can be found in Tab. \\ref{tab:variance}. For the combination of data sets employed in our fiducial analysis, we find $\\eta = 0.113 \\pm 0.075$ and an $\\overline{\\sigma}(\\delta H(s)) = 0.094 \\pm 0.004$ corresponding to an average $9.4 \\pm 0.4\\%$ constraint on $H(z)$ across redshift. Removing the CMB data point significantly worsened both constraints finding $\\eta = 0.142 \\pm 0.096$ and an average $12.5 \\pm 1.6\\%$ constraint on $H(z)$ across redshift. In order to better understand this effect we can look at the first panel in Fig. \\ref{fig:dH_geo_vs_gro}. In this figure, we can observe how removing the CMB data from the analysis significantly widens the constraints of the GP, specially beyond $z>2.5$ (see Table \\ref{tab:variance}). This is to be expected as the CMB data is the only point we have above $z=2.5$. Thus, the constraints from this integrated effect are expected to become dominant in the redshift range between $2.5 < z < 1100$. However, it is important to note that the CMB is not the only contributor to the $\\delta H(s)$ constraints over this redshift range since $f \\! \\sigma_8$ data also constraints $\\delta H(s)$ over its whole domain through its role in solving the Jeans equation. The associated constraint is however very weak.\n\nFor the purpose of studying the effect of different data types, we split the data points within the data sets employed in the fiducial analysis in two groups: ``geometry'' -- exclusively containing measurements of the expansion history -- and ``growth'' -- solely containing $f \\! \\sigma_8$ measurements. \n\n\\begin{figure} \n\\centering\n    \\includegraphics[width=\\linewidth]{{\"dH_gp_geo_vs_gro\"}.png} \n    \\caption{$1\\sigma$-constraints on $\\delta H(z)$ broken down by type of data considered. Solid lines represent the mean of the GPs at each redshift. In red we display the constraints resulting from the analysis of all present data, in blue the effect of removing the CMB data set, in black the effect of fixing the $\\Omega_m$ and $\\sigma_8$ to their best-fit (BF) value; in magenta, the constraints resulting of only considering growth data; and in green, the constraints resulting of only considering geometry data.}\n    \\label{fig:dH_geo_vs_gro}\n\\end{figure}\n\nAs we can see in the second panel of Fig. \\ref{fig:dH_geo_vs_gro}, growth data only is much weaker  at constraining $\\delta H(z)$. From Table \\ref{tab:variance} we see that constraints from growth data alone on $\\overline{\\sigma}(\\delta H(s))$ are approximately twice as wide as those resulting from analysing the entire data set. These constraints are consistent with the prior on the hyperparameter $\\eta$. On the other hand, the constraining power of the geometry data is only slightly weaker than that of the entire data set. Hence the $\\delta H(z)$ constraints are mostly dominated by the geometry data sets as one would expect. Nonetheless, the addition of growth data increases constraining power. This is shown explicitly in the last panel of Fig. \\ref{fig:dH_geo_vs_gro}, which shows the results of using geometry data alone compared to those using the full data set. This recovers the expected behavior: more data increases the constraining power and the contours shrinks.\n\n\\begin{figure} \n\\centering\n    \\includegraphics[width=\\linewidth]{{\"data_cosmo_functions\"}.pdf} \n    \\caption{$1\\sigma$-constraints for the cosmological functions $H(z)$ and $f \\! \\sigma_{8}$ (top and bottom panel respectively) broken down by combination of data set. Solid lines represent the mean of the GPs at each redshift. The dashed black lines show the prediction for each cosmological function using our Planck 2018 fiducial cosmology (see Table~\\ref{tab:fiducial}). In red we show the constraints resulting of the analysis of all present data; in green, the impact of removing the CMB data set; and in black, the impact of removing the DSS data set.}\n    \\label{fig:comp_data}\n\\end{figure} \n\nWe now shift the focus of our discussion to the constraints we derive from $\\delta H(s)$ for the expansion history itself, $H(z)$, and the linear growth of matter anisotropies, $f\\!\\sigma_8$. Comparing the constraints for both cosmological functions from our analysis of current data with the \\textit{Planck} 2018 predictions, we find an overall good agreement, finding both functions to contain the \\textit{Planck} 2018 predictions within their $2\\sigma$ confidence contours. This can be seen in  Fig. \\ref{fig:comp_data}. Nonetheless, we observed a greater than 1$\\sigma$ preference for a lower $f\\!\\sigma_8$ between $0<z<0.75$, mostly driven by the DSS data point. However, the constraining power of current $f\\!\\sigma_8$ data is too weak to make a case for the presence of new physics. \n\n\\begin{figure*} \n\\hspace*{-0.7cm} \n\\centering\n    \\includegraphics[width=\\linewidth]{{\"1D_comp\"}.pdf} \n    \\caption{Comparison between the different $1 \\sigma$ constraints for the cosmological parameters $S_8$,  $\\sigma_{8}$ and $\\Omega_m$ (first, second and third panel respectively). In each panel the \\textit{Planck} 2018 constraint is displayed in the form of a blue band for comparison.}\n    \\label{fig:1D_comp}\n\\end{figure*} \n\n\\subsubsection{Cosmological parameters}\nWe now focus on the constraints on the specific cosmological parameters $\\Omega_M$ and $\\sigma_8$. We will also show constraints on the derived parameter $S_8 = \\sigma_8 \\sqrt{ \\Omega_m \/ 0.3}$ and on the local value of the expansion rate, given in our case by $H_{0}\\equiv A_0 \\, \\overline{H}_0 \\, (1+ \\delta H(s=0))$. A summary of the constraints obtained by the different analyses considered in this work can be found in Table \\ref{tab:constraints} with a graphical illustration in Fig \\ref{fig:1D_comp}. Full 2D contours for the respective sets of constraints, including hyperparameters, can be found in Appendix \\ref{app:full_results}. We primarily focus our discussion on the cosmological parameter $\\Omega_m$, the only remaining degree of freedom in Eqn. \\ref{eq:growth}. \n\n\\begin{table*} \n\\caption{Constraints on the cosmological parameters $\\Omega_m$, $\\sigma_8$, $S_8$,  and $H_0$ (first to fourth columns respectively) for each of the different analyses (rows), as well as the \\textit{Planck} 2018 constraints and $\\Lambda$CDM analyses of current and DESI+CMB for reference (first to third row respectively).}\n\\begin{tabular}{|p{3cm}|p{1.75cm}|p{1.75cm}|p{1.75cm}|p{3cm}| }\n \\hline\n  Analysis & $\\Omega_m$ & $\\sigma_{8}$ & $S_8$ & $H_0 = \\Bar{H_{0}} A_0 (1+\\delta H(0))$ \\\\ \n    \\hline\n    \n    \\textit{Planck} 2018 &   $0.317 \\pm 0.008$ &  $0.812 \\pm 0.007$  & $0.834 \\pm 0.016$ &  $67.27 \\pm 0.006 $\\\\\n    All data ($\\Lambda$CDM) & $0.283 \\pm 0.007$ & $0.8 \\pm 0.029$ & $0.777 \\pm 0.028$ & $68.601 \\pm 0.775$ \\\\\n    DESI+CMB ($\\Lambda$CDM) & $0.316 \\pm 0.006$ & $0.812 \\pm 0.003$ & $0.834 \\pm 0.008$ & $66.992 \\pm 0.311$ \\\\\n    \\hline\n    All data & $0.224 \\pm 0.066$ & $0.946 \\pm 0.158$ & $0.788 \\pm 0.029$ & $67.715 \\pm 0.93$ \\\\\n    All data (Fixed HP) & $0.228 \\pm 0.067$ & $0.932 \\pm 0.155$ & $0.785 \\pm 0.03$ & $68.113 \\pm 1.061$ \\\\\n    All data (Two-Steps) & $0.226 \\pm 0.066$ & $0.936 \\pm 0.151$ & $0.787 \\pm 0.029$ & $68.144 \\pm 0.718$ \\\\\n    No CMB & $0.227 \\pm 0.068$ & $0.936 \\pm 0.156$ & $0.786 \\pm 0.03$ & $67.94 \\pm 1.034$ \\\\\n    No DSS & $0.229 \\pm 0.106$ & $0.974 \\pm 0.218$ & $0.791 \\pm 0.047$ & $67.766 \\pm 0.96$ \\\\\n    DESI $+$ CMB & $0.293 \\pm 0.013$ & $0.839 \\pm 0.014$ & $0.828 \\pm 0.008$ & $66.788 \\pm 0.371$ \\\\\n\\end{tabular}\n\\label{tab:constraints}\n\\end{table*}\n\nTo establish a benchmark against which to compare our method, we start by examining the constraints obtained assuming a $\\Lambda$CDM model in which we vary $H_0$, $\\Omega_m$ and $\\sigma_8$. This allows us to quantify the impact of performing a model-independent analysis using GPs on the final constraining power. Looking at Fig. \\ref{fig:1D_comp} and Table \\ref{tab:constraints} we can observe that, assuming $\\Lambda$CDM,  $\\Omega_m = 0.283 \\pm 0.007$. In other words, for the setup used in this work, we can constrain $\\Omega_m $ to around $2\\%$ precision if we undertake a model-dependent analysis with  $\\Lambda$CDM. In this case, $\\Omega_m$ receives information from both background and perturbations.\n\nIn turn, our fiducial model-independent analysis yields $\\Omega_m = 0.224 \\pm 0.066$, inflating the uncertainty by a factor $\\sim9$. Comparing this result with the \\textit{Planck}  2018 $\\Omega_m = 0.317 \\pm  0.008$, our $\\Omega_m$ constraint is lower but statistically compatible with the \\textit{Planck} 2018 constraint and our $\\Lambda$CDM analysis both at $1.5 \\sigma$.\n\nLooking at the relevance of the different individual data sets on the constraints, we observe an excellent agreement between all the different combinations considered (see Fig. \\ref{fig:1D_comp} and Tab. \\ref{tab:constraints}).  Removing the DSS data, one of the most precise $f \\! \\sigma_8$ measurements, significantly worsens the $\\Omega_m$ constraints by nearly $60 \\%$. On the other hand, removing the CMB data set resulted in nearly identical constraints for the cosmological parameters. This is due to the fact that, in the presence of other geometry data to inform the expansion history, $\\Omega_m$ constraints become dominated by growth data through the relationship between both. Thus, it is fair to ask what the impact of completely removing any type of geometry data from the analysis is. Analytically, we can see from Eqn. \\ref{eq:jeans} that in the presence of measurements of the linear growth rate and an arbitrary value of $\\Omega_m$ one can always find a expansion history that solves the differential equation. This degeneracy between the expansion history and $\\Omega_m$ in Eqn. \\ref{eq:jeans}, prevents our methodology from obtaining any meaningful constraints on $\\Omega_m$ in the absence of Geometry data that is not completely dominated by our choice of GP mean and hyperparameter priors.\n\nLooking at the parameters $\\sigma_8$ and $S_8$ in more detail, we found compatible constraints with the \\textit{Planck} 2018 cosmology up to 1$\\sigma$. However, it is worth mentioning that our results show a mild tendency towards higher $\\sigma_8$ values which, combined with the tendency towards lower  $\\Omega_m$ values, results in lower $S_8$ values. This is consistent with the underprediction of $f \\! \\sigma_8$ between $0<z<0.75$ we discussed in the previous section. A lower value of $S_8$ would also be consistent with the most recent measurements by large scale structure experiments \\citep{K1K, DESY3, Garcia-Garcia21, 2021arXiv211109898W, 2021arXiv210503421K}, the origin of which could lie on a lower value of $\\Omega_m$ \\citep{2021MNRAS.501.1481H}.\n\n\\subsection{Forecasts}\nOur model-independent analysis leads to far weaker constraints than assuming the $\\Lambda$CDM model. Being data-driven, the performance of the method used here may improve significantly with the advent of next-generation surveys with significantly tighter uncertainties. To quantify this, we repeated our fiducial analysis pipeline on mock data generated based on the forecasted errors for the DESI mission \\citep{Font-Ribera14} in combination with the \\textit{Planck} measurement of $\\theta_*$. The results can be found in Fig. \\ref{fig:dH_forecast}. In this figure we can observe a $10\\%$ improvement between the expected $\\delta H(z)$ constraints from DESI with respect to those of current data. The corresponding $\\eta$ and $\\overline{\\sigma}(\\delta H(s))$ constraints (shown in Tab. \\ref{tab:variance}) improve by approximately $15\\%$: $\\eta = 0.085 \\pm 0.074$ and $\\overline{\\sigma}(\\delta H(s)) = 0.080 \\pm 0.004$. Looking at the cosmological parameters,  DESI \\citep{Font-Ribera14} in combination with the CMB data set results in nearly 5 times tighter $\\Omega_m$ constraints and 10 times better constraints on $\\sigma_8$.\n\n\\begin{figure} \n\\centering\n    \\includegraphics[width=\\linewidth]{{\"dH_gp_forecast\"}.png} \n    \\caption{Forecast $1\\sigma$-constraints on $\\delta H(z)$ from DESI (blue) and using current data (red). Solid lines represent the mean of the GPs at each redshift.}\n    \\label{fig:dH_forecast}\n\\end{figure}\n\nWhile these constraints are still significantly worse than the \\textit{Planck} 2018 results, they are comparable if not better than using the $\\Lambda$CDM model to analyze the currently available data used in this work. For example, while GP constraints on $\\Omega_m$ from DESI are two times wider than those currently found assuming a $\\Lambda$CDM model, the constraints on $\\sigma_8$ and $S_8$ are 2.3 and 3.6 times tighter respectively. \n\nIf we, instead, compare how a model-independent analysis of DESI data using this methodology will pitch against a $\\Lambda$CDM analysis, we find that the gap between the model-independent and the model-dependent constraints shrinks. This can be see in the fact that, with current data, the $\\Lambda$CDM constraint on $\\Omega_m$ is $9.2$ times tighter than the one obtained with our methodology. However, in our DESI forecast, it is only twice as good. Similarly, the improvement for the parameter $\\sigma_8$ reduces from five times tighter constraints to about four times. Our understanding is that, as better data allow us to better reconstruct $H(z)$, the difference between the reconstructed and model-dependent $H(z)$ shrinks, as long as the assumed model fits the data well. As a consequence, the constraints on the other parameters ($\\Omega_m$, $\\sigma_8$ and $S_8$) become more similar. Thus, we expect that in the future, as the quality of the data keeps improving, cosmological constraints from model-independent methodologies, such the one proposed in this work, will rival model-dependent ones.\n\n\\section{Conclusions} \n\\label{Sect: Conclusions}\n In this work we have developed a method to obtain  constraints on $H(z)$ and $\\Omega_m$ purely based on the relationship between the expansion history and the linear growth rate. In order to do so, we employ a Gaussian process to model the evolution of the expansion history from present time to recombination.\n\nFrom this expansion history, we have derived predictions for the comoving distance and the linear growth rate $f \\! \\sigma_8$ making use of the physical relationships between the three quantities. Constraints for $\\Lambda$CDM parameters were obtained by simultaneously fitting a suite of the latest measurements of these three cosmological functions. The data combination used for our fiducial analysis consisted of cosmic chronometers, the Pantheon supernova catalog, BAO and RSD data, peculiar velocity data from supernovae, and the position of the first acoustic peak in the CMB power spectrum. Moreover, we also obtained forecast constraints on these cosmological functions from the future DESI data.\n\nCurrent data can constrain the $H(z)$ Gaussian process at an average $9.4\\%$ throughout all redshifts. These constraints are compatible up to $1 \\sigma$ with the best-fit \\textit{Planck} 2018 expansion history across $0 < z < 1100$. Our constraints on the expansion rate $f \\! \\sigma_8$ lie below the corresponding \\textit{Planck} prediction by less than 2-$\\sigma$ in the range $0 < z < 0.75$ (a result driven by the Democratic Supernova Sample data).\n\nTranslating the Gaussian process constraints into constraints of cosmological parameters, we find a model-independent measurement of $\\Omega_m = 0.224 \\pm 0.066$. This result is lower than, but statically compatible with, the Planck 2018 \\citep{Planck} cosmology. We also find $S_8 = 0.788 \\pm 0.029$, an intermediate value, statistically compatible with both the Planck 2018 cosmology \\citep{Planck} and recent local measurements from weak lensing and galaxy  clustering\\citep{DESY3, K1K, Garcia-Garcia21, 2021arXiv210503421K, 2021arXiv211109898W}.\n\nThe forecast analysis performed using the methodology of this work predicts that combining the DESI measurements with the CMB BAO data used in this work will yield $15\\%$ tighter constraints on $H(z)$ across redshift as well as five times tighter constraints on $\\Omega_m$. While this constraints are still significantly looser than the model-dependent cosmological constraints of \\textit{Planck} 2018, we show that a model-independent analysis of DESI plus CMB BAO data would be as powerful as a $\\Lambda$CDM analysis of current data. Indeed, we find that model-independent constraints using this methodology would achieve four times tighter $S_8$ constraints. Thus, in the future it will be possible to weigh in on the ongoing $S_8$ tension making use of model-independent methods. Moreover, we also show that, as the quality of the data increases as we go into the future, the gap between the constraining power of model-independent and model-dependent constraints will significantly shrink. \n\nFuture implementations of this methodology could explore several possible extensions. On the one hand, the types of data used in this work are greatly limited by lack of differentiable tools to obtain theoretical predictions for observables with greater constraining power. The development of tools such as differentiable Boltzmann codes, differentiable emulators of the non-linear matter power spectrum, or differentiable Limber integrators would allow us to fit the power spectrum data directly. These developments would enable model-agnostic analyses similar to that of \\citet{Garcia-Garcia21} but with a greatly reduced number of assumptions and a more reliable measure of uncertainty in their results. On the other hand, it would also be possible to explore the use of Gaussian processes to constrain general forms of modified gravity, generalizing works such as \\citet{2017PhRvD..96h3509R, 2019PhRvD..99b3512E, 2021PhRvD.103j3530P, 2021arXiv210712990R, 2021arXiv210712992P}, and study how these theories can be informed by the relationship between background and perturbations. Gaussian processes are an exceptional tool to constrain modified gravity since they don't require assuming a particular parametrisation of such deviations. A comprehensive list of different departures from $\\Lambda$CDM that could be explored with a similar methodology to the one presented in this work can be found in \\citet{Baker14}.\n\n\\section*{Acknowledgements}\n\\textit{Author contributions}: All authors contributed to the development and writing of\nthis paper. We acknowledge support from the Beecroft Trust. CGG and PGF acknowledge funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 693024). DA acknowledges support from the Science and Technology Facilities Council through an Ernest Rutherford Fellowship, grant reference ST\/P004474. JRZ is supported by an STFC doctoral studentship.\n\nThe analysis made use of the software tools\n\\textsc{SciPy} \\citep{scipy}, \\textsc{NumPy} \\citep{np}, \\textsc{Matplotlib} \\citep{np},\n\\textsc{CLASS} \\citep{Class}, GetDist \\citep{getDist}.\n\nBased on observations obtained with \\textit{Planck} (\\url{http:\/\/www.esa.int\/Planck}), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada.\n\n\nFunding for the Sloan Digital Sky Survey IV has been provided by the  Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and  resources from the Center for High Performance Computing  at the University of Utah. The SDSS website can be found at \\url{www.sdss.org}.\nSDSS-IV is managed by the  Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including  the Brazilian Participation Group,  the Carnegie Institution for Science,  Carnegie Mellon University, Center for Astrophysics | Harvard \\& Smithsonian, the Chilean Participation, the French Participation Group,  Instituto de Astrof\\'isica de Canarias, The Johns Hopkins  University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) \/ University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut f\\\"ur Astrophysik Potsdam (AIP),  Max-Planck-Institut f\\\"ur Astronomie (MPIA Heidelberg), Max-Planck-Institut f\\\"ur  Astrophysik (MPA Garching), Max-Planck-Institut f\\\"ur Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observat\\'ario Nacional \/ MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Aut\\'onoma de M\\'exico, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.\n\n\\section*{Data Availability}\nThe code developed for this work as well as the data sets used are available upon request. \n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nStrong spin-orbit coupled unconventional superconductors, whose superconducting (SC) state cannot be described by electron-phonon coupling, provide a platform for experimental and theoretical studies of emergent quantum behavior \\cite{Pfleiderer2009,Smidman2017}. Time-reversal and parity are key symmetries to characterize these materials, and striking states of matter often emerge when one (or both) of these symmetries are broken. For instance, odd-parity superconductors have been identified as a promising route for topological superconductivity, which hosts edge modes or vortices with non-abelian statistics required for topological quantum computing~\\cite{SatoAndo2017}. A chiral superconductor further breaks time-reversal symmetry and lowers the energy of the SC condensate by removing nodes from the gap function~\\cite{Kallin2016}. Odd-parity chiral superconductors are remarkably rare, but their experimental manifestation has been observed in superfluid $^3$He and actinide superconductor UPt$_3$~\\cite{Avers2020Broken}. \n\nUTe$_2$ is a newly-discovered candidate for odd-parity chiral superconductivity~\\cite{Ran2019Nearly,Aoki2022Unconventional}. Nuclear magnetic resonance (NMR) Knight shift measurements strongly suggest that UTe$_2$ is an odd-parity superconductor with a dominant $B_{3u}$ order parameter~\\cite{Nakamine2019Sup,Nakamine2021Ani,Fujibayashi2022Super}. Point nodes in the SC gap structure are supported by transport measurements~\\cite{Ran2019Nearly,Metz2019Point,Kittaka2020Ori}, Knight shift measurements~\\cite{Fujibayashi2022Super}, and non-local superfluid density measurements~\\cite{Bae2021Ano}. \nThe position of the point nodes, however, is still controversial. Thermal conductivity, microwave surface impedance, and specific heat measurements suggest point nodes in the $ab$-plane or along $a$~\\cite{Metz2019Point,Bae2021Ano,Kittaka2020Ori}, whereas magnetic penetration depth measurements argue for a multicomponent SC state with multiple point nodes near the $b$- and $c$-axes~\\cite{Ishihara2021Chiral}.  \n\nEvidence for chiral superconductivity was found in UTe$_2$ by scanning tunneling spectroscopy on the step edges of a ($0\\bar{1}1$)-plane~\\cite{Jiao2020Chiral} and by polar Kerr rotation measurements~\\cite{Hayes2021Multi}. Multiple SC phase transitions were also reported in UTe$_2$ even at ambient pressure by specific heat measurements~\\cite{Hayes2021Multi, Thomas2020}. Subsequently, it was found that the observed ``double peak\" in the specific heat can arise from sample inhomogeneity~\\cite{Thomas2021Spat}. In addition, a single phase transition is reported in higher quality samples with higher  SC critical temperature  $T_{\\rm c}$, higher residual resistivity ratios, lower residual resistivities, and quantum oscillations~\\cite{Rosa2021Single,Aoki2022Unconventional}. \n\nTo microscopically investigate the SC state of UTe$_2$, here we report the temperature dependence of the local superfluid response using scanning SQUID (superconducting quantum interference device) susceptometry on a cleaved (011)-plane of UTe$_2$. We also image the pinned vortices induced by field cooling. Our results show no evidence for multiple phase transitions in the temperature dependence of the superfluid density and imply an anisotropic nodal gap structure in UTe$_2$.\n\n\\begin{figure*}[htb]\n\\begin{center}\n\\includegraphics*[width=16cm]{.\/Fig1.jpg}\n\\caption{ Local susceptibility is microscopically homogeneous on (011) surface of UTe$_2$ sample\\#1. (a) Optical image of the scanned area, which includes the cleaved (011) surface with small bumps creating no signals in our scans and the edge. (b-e) Temperature dependence of the susceptometry scan indicates the homogeneous superfluid density on UTe$_2$. Stripes along the scan directions are the scanning noise. (f) The temperature dependence of the local susceptibility at the points from A to A' in Fig. 1(e) has no kink below $T_{\\rm c}$. The susceptibilities are shifted by 0.2 $\\Phi_0$\/A for clarity except for the data at A.}\n\\end{center}\n\\end{figure*}\n\nBulk single crystals of UTe$_2$ were grown by chemical vapor transport. Samples \\#1 and \\#2 used in this paper were obtained from the same batch as sample s2 in Ref.~\\cite{Rosa2021Single}. Heat capacity measurements confirmed a single SC transition at $T_{\\rm c} =1.68$~K with a width of 50 mK on a single crystal which was subsequently cleaved into two samples used in this study. We used a scanning SQUID susceptometer to obtain the local ac susceptibility on a cleaved (011)-plane of UTe$_2$ at temperatures from 80 mK to 2 K in a Bluefors LD dilution refrigerator. Our scanning SQUID susceptometer has two pickup loop and field coil pairs configured with a gradiometric structure~\\cite{Kirtleyrsi2016}. The inner radius of the pickup loop is 0.4 $\\mu$m and the inner radius of the field coil is 1.5 $\\mu$m. The scan height is $\\sim$500 nm. The pickup loop provides the local dc magnetic flux $\\Phi$ in units of the flux quantum $\\Phi_0=h\/2e$, where $h$ is the Planck constant and $e$ is the elementary charge. The pickup loop also detects the ac magnetic flux $\\Phi^{ac}$ in response to an ac magnetic field $He^{i\\omega t}$, which was produced by an ac current of $|I^{ac}| =$ 1 mA at a frequency $\\omega\/2\\pi\\sim$ 1 kHz through the field coil, using an SR830 Lock-in-Amplifier. Here we report the local ac susceptibility as $\\chi=\\Phi^{ac}\/|I^{ac}|$ in units of $\\Phi_0$\/A.\n\n\nTo obtain the homogeneity of the superfluid density and its local temperature dependence, we measured the local susceptibility near the edge of the sample [Figs. 1(a)-1(e)]. The susceptibility far from the edge has a homogeneous temperature dependence on the micron-scale [Fig. 1(f)]. We note that our results do not rule out possible inhomogeneity either on the nanoscale or on scales larger than the scan area (e.g  sub-millimeter). Our data also cannot rule out fluctuations in time. There is no kink in the temperature dependence of the local susceptibility below $T_{\\rm c}$, wherein the temperature step of the scans is 25 mK. The susceptibility is positive above $T_{\\rm c}$ due to paramagnetism. The local susceptibility was negative (diamagnetic) near the edge but positive far from the edge at 1.69 K [Figs. 1(c),1(f)]. \n\n\n\nWe defined the local $T_{\\rm c}$ as the temperature that satisfies the relation of $\\chi(T>T_{\\rm c})>$-0.1 $\\Phi_0\/$A. The local $T_{\\rm c}$ mapping clearly shows that the local $T_{\\rm c}$ is weakly enhanced at the edge but is homogeneous 30~$\\mu$m away from the edge into the sample [Fig. 2, sample\\#1; Fig. S2, sample\\#2]. We note that the reported local $T_{\\rm c}$ near the edge is a lower bound relative to the actual value because the penetration depth near $T_{\\rm c}$ is longer than the pickup loop's scale and the susceptibility loses some signal at the edge. If two phase transitions do exist, they must be closer to each other than 25 mK, or the second kink below $T_{\\rm c}$ is much smaller than our experimental noise.\n\n\\begin{figure}[!hb]\n\\begin{center}\n\\includegraphics*[width=7cm]{.\/Fig2.jpg}\n\\caption{ Small enhancement of local $T_{\\rm c}$ near the edge of sample\\#1. (a) The local $T_{\\rm c}$ mapping is obtained from the local susceptometry scans. (b) Cross section of the local $T_{\\rm c}$ from A to A' shows the local $T_{\\rm c}$ enhancement of 30 mK at the edge. The plotted area and the cross section from A to A' are the same with Fig. 1.}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure*}[htb]\n\\begin{center}\n\\includegraphics*[width=13cm]{.\/Fig3.jpg}\n\\caption{The vortex density is homogeneous over many-micron distances. The existence of vortices and antivortices in low-field scans may indicate a local magnetic source in the sample\\#1. (a,b) Local magnetometry scan after field cooling shows the vortices pinned parallel to the sample edge, as denoted by the dashed lines. (c) There are a few vortices and antivortices pinned far from the edge after near zero field cooling. (d) Magnetometry scan near zero field above $T_{\\rm c}$ shows a small magnetic dipole at the sample's edge, but no other indication of magnetism. The \"tail\" of the vortices and dipoles are due to the asymmetric shielding structure of the scanning SQUID~\\cite{Kirtleyrsi2016}.}\n\\end{center}\n\\end{figure*}\n\nNow we turn to the pinned vortex density, which reflects the impurity density on the crystal surface for small applied magnetic fields. The distance between vortices is on the order of microns. Our magnetometry scan imaged the pinned vortices induced by cooling in an applied uniform magnetic field from 2 K to 100 mK [Figs. 3(a),3(b), sample\\#1; Figs. S3(a),S3(b), sample\\#2]. The number of vortices corresponds to the applied field, but the vortices are preferentially pinned along lines in one direction, which is parallel to the sample edge. This linear pinning indicates the existence of a line anomaly, such as nanometer-scale step edges along crystal axes. Near zero magnetic field, there are still a few vortices and antivortices pinned far from the edge [Fig. 3(c), sample\\#1; Fig. S3(c), sample\\#2]. Notably, these vortices and antivortices do not disappear after zero field cooling with slower cooling rates, which is expected to cancel the uniform background field normal to the sample surface by the application of an external field. These data are inconsistent with the argument from polar Kerr effect measurements that there are no vortices in UTe$_2$ within the beam size area ($\\sim11~\\mu$m radius)~\\cite{Wei2022}. Further, our results indicate the existence of a local magnetic source that induces vortices and antivortices, in spite of the absence of long-range order or strong magnetic sources on the scan plane above $T_{\\rm c}$ \\cite{Ran2019Nearly,Miyake2019Meta,Hutanu2020Low}. \nSmall dipole fields are observed at the edge of the sample, which may stem from U impurities; however, these impurities cannot induce pinned vortices and antivortices as they are too far away [Fig. 3(d), sample\\#1]. Muon spin resonance and NMR measurements have detected the presence of strong and slow magnetic fluctuations in UTe$_2$ at low temperatures~\\cite{Tokunaga2022Slow,Sundar2022Ubi}. Therefore, a sensible scenario is that these fluctuations are pinned by defects and become locally static.\n\n\n\\begin{figure*}[htb]\n\\begin{center}\n\\includegraphics*[width=16cm]{.\/Fig4.jpg}\n\\caption{The temperature dependence of the  superfluid density = best matches an anisotropic, rather than isotropic, gap structure. (a-c) Temperature dependence of the normalized superfluid density $n_{(011)}$ at the fixed position in sample\\#1,\\#2 that are indicated by the blue dot in Figs. 1(b), S1(b), respectively. The thick lines are simulation curves best fit for 4 gap symmetries with (a) Isotropic FS, (b) Ellipsoidal FS, and (c) Cylindrical FS~\\cite{supple}. (d-i) Best fit models of gap symmetries (d,e), (f,g) and (h,i) for (a), (b) and (c), respectively. FS is plotted by yellow color. The distance between larger surfaces and FS represents the angular dependence of the SC gap $\\Omega$ in (a,b) the spherical coordinate and (c) the cylindrical coordinate. All surfaces are cut for clarity.}\n\\end{center}\n\\end{figure*}\n\n\nTo estimate the local superfluid density, we measure the local susceptibility at different temperatures with the pickup loop position fixed. The local superfluid density is obtained using the numerical expression of the susceptibility assuming a homogeneous penetration depth, $\\lambda$, as described below. Kirtley {\\it et al}. developed the expression for the susceptibility as a function of the distance between the susceptometer and the sample surface~\\cite{Kirtley2012}. In this model, wherein the sample surface is at $z=0$, we consider three regions.  Above the sample ($z>0$), the pickup loop and field coil are at $z$ in vacuum and $\\mu_1 = \\mu_0$, where $\\mu_0$ is the permeability in vacuum. In the sample ($-t\\leq z \\leq 0$), the London penetration depth is $\\lambda=\\sqrt{m\/4\\pi ne^2}$, and the permeability is $\\mu_2$. Below the sample ($z<-t$), there is a nonsuperconducting substrate with a permeability $\\mu_3$. The radius of the field coil and the pickup loop are $a$ and $b$, respectively. By solving Maxwell's equations and the London equation for the three regions, the SQUID height dependence of the susceptibility $\\chi(z)$ is expressed as \n\\begin{widetext}\n\\begin{equation}\\label{phi-z}\n    \\chi(z)\/\\phi_s = \\int_0^\\infty{dx e^{-2x\\bar{z}}xJ_1(x)}\\left[\\frac{-(\\bar{q}+\\bar{\\mu}_2x)(\\bar{\\mu}_3\\bar{q}-\\bar{\\mu}_2x)+e^{2\\bar{q}\\bar{t}}(\\bar{q}-\\bar{\\mu}_2x)(\\bar{\\mu}_3q+\\bar{\\mu}_2x)}{-(\\bar{q}-\\bar{\\mu}_2x)(\\bar{\\mu}_3\\bar{q}-\\bar{\\mu}_2x)+e^{2\\bar{q}\\bar{t}}(\\bar{q}+\\bar{\\mu}_2x)(\\bar{\\mu}_3q+\\bar{\\mu}_2x)}\\right],\n\\end{equation}\n\\end{widetext}\nwhere $\\phi_s = A\\mu_0\/2\\Phi_0a$ is the self inductance between the field coil and the pickup loop, $A$ is the effective area of the pickup loop, $\\bar{z} = z\/a$, $J_1$ is the Bessel function of first order, $\\bar{t} = t\/a$, $\\bar{q} = \\sqrt{x^2 + \\bar{\\lambda}^{-2}}$, and $\\bar{\\lambda} = \\lambda\/a$. For the bulk sample on a copper substrate ($\\bar{t}>>1, \\mu_3 = 1$), the observed susceptibility only depends on $\\lambda$, $\\mu_2$, and the SQUID structure. \n\nThe penetration depth $\\lambda(T)$ was calculated using Eq.~\\eqref{phi-z} and the observed susceptibility [Fig. 4(a)]. The normalized superfluid density $n_{s} = \\lambda^2(0)\/\\lambda^2(T)$ was calculated from the obtained penetration depth's temperature dependence, where $\\lambda(0) = 1620\\pm150$~nm [sample\\#1], $1730\\pm300$~nm [sample\\#2] [Fig.~4(b)]. Here the error for $\\lambda$ and $n_s$ is roughly calculated from the pickup loop height uncertainty. We note that sample\\#2 had a dead layer of 700~nm on the surface, which we estimated by assuming that sample\\#2 has a similar penetration depth at zero temperature with sample\\#1, because sample\\#2 was accidentally exposed in air about one extra hour. The locally obtained superfluid density $n_s$ saturates below $T\/T_{\\rm c}=0.1$.\n\n\nWe examine the SC gap structure through the temperature dependence of the superfluid density. The superfluid density $n_{i}$ is sensitive to low-energy excitations along the $i$ axis, which is perpendicular to the applied field. In our case, $n_{i}$ is sensitive to excitations within the plane normal to [011], and the extracted $n_{(011)}$ is the average of $n_a$ and $n_{\\perp[011],a}$)~\\cite{Chandrasekhar1993}. The SC gap function of UTe$_2$, $\\Delta$, is most likely odd-parity within the orthorhombic $D_{2h}$ point group. In the presence of strong spin-orbit coupling, $\\Delta(T,\\Vec{k})=\\Psi(T)\\Omega(\\Vec{k})$, and the angle dependence of the gap function is expressed as $\\Omega(\\Vec{k})\\propto\\sqrt{\\Vec{d}\\cdot\\Vec{d}^*\\pm|\\Vec{d}\\times\\Vec{d}^*|}$. In this case, the possible irreducible representations are $A_{1u}$ [full gap, $\\Vec{d}=(c_1k_x,c_2k_y,c_3k_z)$)], $B_{1u}$ [point nodes along $c$, $\\Vec{d}=(c_1k_y,c_2k_x,c_3k_xk_yk_z)$], $B_{2u}$ [point nodes along $b$, $\\Vec{d}=(c_1k_z,c_2k_xk_yk_z,c_3k_x)$], and $B_{3u}$ [point nodes along $a$, $\\Vec{d}=(c_1k_xk_yk_z,c_2k_z,c_3k_y)$]~\\cite{Annett1990}. We note that coefficients $c_1, c_2$, and $c_3$ may differ by orders of magnitude~\\cite{IshizukaPRB2021}. \n\nFor the sake of completeness, here we assume three cases of Fermi surface structure to calculate the temperature-dependent superfluid density with fit parameters $c_1, c_2$, and $c_3$~\\cite{supple,Kogan2021}: (I) isotropic Fermi Surface (FS) based on the isotropic heavy 3D Fermi surface observed by angle-resolved photoemission spectroscopy (ARPES) measurements ~\\cite{FujimoriJPSJ2019,MiaoPRL2020}; (II) ellipsoidal FS based on the  upper critical field~\\cite{AokiJPSJ2019}; and (III) cylindrical FS. Case (III) is based on both ARPES measurements, which observed cylindrical light electron bands~\\cite{FujimoriJPSJ2019,MiaoPRL2020} and recent de Haas\u2013van Alphen measurements that reveal heavy cylindrical bands ~\\cite{Aoki2022First}. \n\nThe isotropic fully gapped model $A_{1u}$ saturates at $T\/T_{\\rm c}=0.2$ [Fig.~S6]. In contrast, the experimental data saturate at a lower temperature, which implies an anisotropic structure in the SC gap function. The calculated normalized superfluid density $n_{(011)}\\sim(n_a+n_{\\perp[011],a})\/2$ for highly anisotropic $A_{1u}$ and $B_{1u}$ have a similar temperature dependence compared to our experimental results, whereas $n_{(011)}$ for $B_{2u}$ and $B_{3u}$ do not agree with our data because of their point nodes near the (011) plane with isotropic or ellipsoidal 3D Fermi surfaces [Figs.~4(a),4(b)]. For highly anisotropic $A_{1u}$ and $B_{3u}$, $n_{(011)}$ agrees with our experimental results, whereas $n_{(011)}$ for $B_{2u}$ inconsistent with the data for a cylindrical Fermi surface [Fig.~4(c)]. We note that our calculations with point nodes do not completely explain our results near zero temperature, which may be caused by our assumptions of the simplest structures of Fermi surface and gap functions, the simple averaging of $n_a$ and $n_{\\perp[011],a}$, or by the assumption of a single band.\nOur results indicate the existence of point nodes along the $a$ axis for a cylindrical Fermi surface. A highly anisotropic fully-gapped component is also allowed.\n\n\nIn summary, we microscopically imaged the superfluid density and the vortex density in high quality samples of UTe$_2$. The superfluid density is homogeneous, and the temperature dependence below the SC transition $T_{\\rm c}$ does not show evidence for a second phase transition. The observed temperature dependence of the superfluid density can be explained by a $B_{1u}$ order parameter for a 3D ellipsoidal (or isotropic) Fermi surfaces or by a $B_{3u}$ order parameter for a quasi-2D cylindrical Fermi surface. A highly anisotropic $A_{1u}$ symmetry component is also allowed for any Fermi surface structures. Combining our results with previous studies about the gap symmetry of UTe$_2$~\\cite{Bae2021Ano,Metz2019Point,Kittaka2020Ori,Fujibayashi2022Super}, we conclude that the SC order parameter is most likely dominated by the $B_{3u}$ symmetry. In light of our results, evidence for time-reversal symmetry breaking and chiral superconductivity in UTe$_2$ could be understood either through the presence of vortices and antivortices even at zero applied field or by the presence of a finite anisotropic $A_{1u}$ symmetry in the SC order parameter. \n\n\n\n\n\\begin{acknowledgments}\nThe authors thank J. Ishizuka for fruitful discussions. This work was primarily supported by the Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract No. DE- AC02-76SF00515. Sample synthesis at LANL was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering \"Quantum Fluctuations in Narrow-Band Systems\" project, while heat capacity measurements were performed with support from the LANL LDRD program. Y.I. was supported by the Japan Society for the Promotion of Science (JSPS), Overseas Research Fellowship.\n\\end{acknowledgments}\n\n\n\n\t\n\t\n\\section*{Contributions}\nY.I. carried out the scanning SQUID microscopy, analyzed experimental data, simulated the superfluid density, and wrote the manuscript. H.M. carried out the scanning SQUID microscopy. S.M.T., F.R., and P.F.S.R. synthesized the crystals. K.A.M. supervised the project. All the authors discussed the results and implications and commented on the manuscript.\n\t\n\t\n\\section*{Additional information}\nCorrespondence and requests for materials should be addressed to Y. I. (yiguchi@stanford.edu)\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nConsider the problem of finding a local minimizer of the expectation\n$F(x)\\eqdef\\bE(f(x,\\xi))$ w.r.t. $x\\in \\bR^d$, where $f(\\,.\\,,\\xi)$ is\na possibly non-convex function depending on some random\nvariable~$\\xi$.\nThe distribution of $\\xi$ is assumed unknown, but revealed online by\nthe observation of iid copies $(\\xi_n:n\\geq 1)$ of the r.v.~$\\xi$.\nStochastic gradient descent (SGD) is the most classical algorithm\nto search for such a minimizer.\nVariants of SGD which include an inertial term have also become very popular.\nIn these methods, the update rule depends on a parameter called\nthe \\emph{learning rate}, which is generally assumed constant or vanishing.\nThese algorithms, although widely used, have at least two limitations.\nFirst, the choice of the learning rate is generally difficult; large learning rates result in\nlarge fluctuations of the estimate, whereas small learning rates\ninduce slow convergence.  Second, a common learning rate is used for\nevery coordinate despite the possible discrepancies in the values of\nthe gradient vector's coordinates.\n\nTo alleviate these limitations, the popular\n\\adam\\ algorithm \\cite{kingma2014adam} adjusts the learning rate\ncoordinate-wise, as a function of the past values of the squared\ngradient vectors' coordinates. The algorithm thus combines the assets\nof inertial methods with an adaptive per-coordinate learning rate\nselection. Finally, the algorithm includes a so-called\n\\emph{bias correction} step. Acting on the current estimate of the gradient vector,\nthis step is especially useful during the early iterations.\n\nDespite the growing popularity of the algorithm, only few works\ninvestigate its behavior from a theoretical point of\nview (see the discussion in Section~\\ref{sec:related-works}).\nThe present paper studies the convergence of \\adam\\ from a dynamical system viewpoint.\\\\\n\n\\noindent{\\bf Contributions}\n\\begin{itemize}[leftmargin=*]\n\\item We introduce a continuous-time version of the \\adam\\ algorithm under the form\n  of a non-autonomous ordinary differential equation (ODE).\n  Building on the existence of an explicit Lyapunov function for the ODE,\n  we show the existence of a unique global solution to the ODE. This first result\n  turns out to be non-trivial due to the irregularity of the vector field.\n  We then establish the convergence of the continuous-time \\adam\\\n  trajectory to the set of critical points of the objective function $F$.\nThe proposed continuous-time version of \\adam\\ provides useful\n  insights on the effect of the bias correction step.  It is shown\n  that, close to the origin, the objective function~$F$ is\n  non-increasing along the \\adam\\ trajectory,\n  suggesting that early iterations of \\adam\\ can only improve the initial guess.\n\n\\item Under a \\L{}ojasiewicz-type condition,\n  we prove that the solution to the ODE converges to a single critical point of the objective\n  function $F$. We provide convergence rates in this case.\n\n\\item\n  In discrete time, we first analyze the \\adam\\ iterates in the constant stepsize\n  regime as originally introduced in \\cite{kingma2014adam}.\n  In this work, it is shown that the discrete-time \\adam\\ iterates shadow the\n  behavior of the non-autonomous ODE in the asymptotic regime where\n  the stepsize parameter $\\gamma$ of \\adam\\ is small.  More\n  precisely, we consider the interpolated process $\\sz^\\gamma(t)$\n  which consists of a piecewise linear interpolation of the \\adam\\ iterates.\n  The random process $\\sz^\\gamma$ is indexed by the parameter~$\\gamma$, which is\n  assumed constant during the whole run of the algorithm.\n  In the space of continuous functions on $[0,+\\infty)$ equipped\n  with the topology of uniform convergence on compact sets,\n  we establish that  $\\sz^\\gamma$ converges in probability\n  to the solution to the non-autonomous ODE when $\\gamma$ tends to zero.\n\n\\item Under a stability condition, we prove the asymptotic ergodic convergence\n  of the probability of the discrete-time \\adam\\ iterates to approach the set of critical\n  points of the objective function in the doubly asymptotic regime\n  where $n\\to\\infty$ then $\\gamma\\to 0$.\n\\item\nBeyond the original constant stepsize \\adam\\,, we propose\na decreasing stepsize version of the algorithm.\nWe provide sufficient conditions ensuring the stability and the almost sure convergence of the iterates towards\nthe critical points of the objective function.\n\\item  We establish a convergence rate of the stochastic iterates of the decreasing stepsize algorithm under the\nform of a conditional central limit theorem.\n\n\\end{itemize}\nWe claim that our analysis can be easily extended to\nother adaptive algorithms such as e.g. \\textsc{RmsProp} or\n\\textsc{AdaGrad} \\cite{tieleman2012lecture,duchi2011adaptive}\nand \\textsc{AmsGrad} (see Section~\\ref{sec:related-works}).\\\\%\\medskip\n\nThe paper is organized as follows.\nIn Section~\\ref{sec:adam}, we present\nthe \\adam\\ algorithm and the main assumptions.\nOur main results are stated in Sections~\\ref{sec:continuous_time} to \\ref{sec:discrete_decreasing}.\nWe provide a review of related works in Section~\\ref{sec:related-works}.\nThe rest of the paper addresses the proofs of our results\n(Sections~\\ref{sec:proofs_cont_time} to \\ref{sec:proofs_sec_discrete_decreasing}).\n\\hfill\\\\\n\n\\noindent{\\bf Notations}. If $x$, $y$ are two vectors on $\\bR^d$ for some $d\\geq 1$, we denote by $x \\odot y$, $x^{\\odot 2}$, $x\/y$, $|x|$, $\\sqrt{|x|}$ the vectors\non $\\bR^d$ whose $i$-th coordinates are respectively given by $x_iy_i$, $x_i^2$, $x_i\/y_i$, $|x_i|$, $\\sqrt{|x_i|}$.\nInequalities of the form $x\\leq y$ are read componentwise.\nDenote by $\\|\\cdot\\|$ the standard Euclidean norm.\nFor any vector $v\\in (0,+\\infty)^d$, write $\\|x\\|^2_v = \\sum_i v_i x_i^2$.\nNotation $A^T$ represents the transpose of a matrix $A$.\nIf  $z\\in \\bR^d$ and $A$ is a non-empty subset of $\\bR^d$,\nwe use the notation $\\mathsf d(z,A) \\eqdef \\inf\\{ \\|z-z'\\| :z'\\in A\\}$.\nIf $A$ is a set, we denote  by $\\1_A$ the function equal to one on that set and to zero elsewhere.\n We denote by $C([0,+\\infty),\\bR^d)$ the space of continuous functions from $[0,+\\infty)$ to $\\bR^d$ endowed with the topology of\nuniform convergence on compact intervals.\n\n\\section{The \\adam\\ Algorithm}\n\\label{sec:adam}\n\n\\subsection{Algorithm and Assumptions}\n\nLet $(\\Omega,\\cF,\\bP)$ be a probability space, and let\n$(\\Xi,\\mathfrak{S})$ denote an other measurable space.  Consider a\nmeasurable map $f:\\bR^d\\times \\Xi\\to \\bR$, where $d$ is an integer.\nFor a fixed value of $\\xi$, the mapping $x\\mapsto f(x,\\xi)$ is\nsupposed to be differentiable, and its gradient w.r.t. $x$ is denoted\nby $\\nabla f(x,\\xi)$.  Define $\\cZ\\eqdef \\bR^d\\times\\bR^d\\times\n\\bR^d$, $\\cZ_{+}\\eqdef \\bR^d\\times\\bR^d\\times [0,+\\infty)^d$ and\n$\\cZ_{+}^*\\eqdef \\bR^d\\times\\bR^d\\times (0,+\\infty)^d$. \\adam\\\ngenerates a sequence $z_n\\eqdef (x_n,m_n,v_n)$ on\n$\\cZ_+$ given by Algorithm~\\ref{alg:adam}.\n\\begin{algorithm}[tb]\n   \\caption{\\bf \\adam$(\\gamma,\\alpha,\\beta,\\varepsilon)$.}\n   \\label{alg:adam}\n\\begin{algorithmic}\n   \\STATE {\\bfseries Initialization:} $x_0\\in \\bR^d, m_0=0$, $v_0=0$.\n   \\FOR{$n=1$ {\\bfseries to} $n_{\\text{iter}}$}\n   \\STATE $m_n = \\alpha m_{n-1} + (1-\\alpha)  \\nabla f(x_{n-1},\\xi_n)$\n   \\STATE $v_n = \\beta v_{n-1} + (1-\\beta) \\nabla f(x_{n-1},\\xi_n)^{\\odot 2}$\n   \\STATE $\\hat m_{n} = m_{n}\/(1-\\alpha^{n})$\n   \\COMMENT{bias correction step}\n   \\STATE $\\hat v_{n} = v_{n}\/(1-\\beta^{n})$\n   \\COMMENT{bias correction step}\n   \\STATE $x_{n} = x_{n-1} - \\gamma \\hat m_{n} \/ (\\varepsilon+\\sqrt{\\hat v_{n}}) \\,.$\n   \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\nIt satisfies:\n$\nz_n=T_{\\gamma,\\alpha,\\beta}(n,z_{n-1},\\xi_n)\\,,\n$\nfor every $n\\geq 1$, where for every $z=(x,m,v)$ in $\\cZ_+$, $\\xi\\in \\Xi$,\n\\begin{equation}\n  T_{\\gamma,\\alpha,\\beta}(n,z,\\xi) \\eqdef\n  \\begin{pmatrix}\n x -\\frac{\\gamma (1-\\alpha^{n})^{-1}(\\alpha m+(1-\\alpha)\\nabla f(x,\\xi))}{ \\varepsilon+{(1-\\beta^{n})^{-1\/2} (\\beta v+(1-\\beta)\\nabla f(x,\\xi)^{\\odot 2})^{1\/2}}} \\\\\n\\alpha m + (1-\\alpha) \\nabla f(x,\\xi) \\\\\n\\beta v + (1-\\beta) \\nabla f(x,\\xi)^{\\odot 2}\n\\end{pmatrix}\\,.\\label{eq:T}\n\\end{equation}\n\n\\begin{remark}\n\\label{rem:debiasing}\n\\quad\\quad The iterates $z_n$ form a non-homogeneous Markov chain, because $T_{\\gamma,\\alpha,\\beta}(n,z,\\xi)$ depends on $n$.\nThis is due to the so-called debiasing step, which consists of replacing $m_n,v_n$ in Algorithm~\\ref{alg:adam}\nby their ``debiased'' versions $\\hat m_n,\\hat v_n$. The motivation becomes clear when expanding the expression:\n\\begin{equation*}\n\\hat m_n = \\frac{m_n}{1-\\alpha^n} = \\frac {1-\\alpha}{1-\\alpha^n}\\sum_{k=0}^{n-1}\\alpha^k \\nabla f(x_k,\\xi_{k+1})\\,. \\label{eq:debiasing}\n\\end{equation*}\nFrom this equation, it is observed that,  $\\hat m_n$ forms a convex combination of the past gradients.\nThis is unlike $m_n$, which may be small during the first iterations.\n\n\\end{remark}\n\n\\begin{assumption} \\label{hyp:model}\nThe mapping $f:\\bR^d\\times\\Xi\\to \\bR$ satisfies the following.\n  \\begin{enumerate}[{\\sl i)}]\n  \\item For every $x\\in \\bR^d$, $f(x,\\,.\\,)$ is $\\mathfrak{S}$-measurable.\n  \\item For almost every $\\xi$, the map $f(\\,.\\,,\\xi)$ is continuously differentiable.  \\label{hyp:dif}\n  \\item There exists $x_*\\in \\bR^d$ s.t. $\\bE(|f(x_*,\\xi)|)<\\infty$ and  $\\bE(\\|\\nabla f(x_*,\\xi)\\|^2)<\\infty$.\n  \\item For every compact subset $K\\subset \\bR^d$, there exists $L_K>~0$ such that\n   for every $(x,y)\\in K^2$, $\\bE(\\|\\nabla f(x,\\xi)-\\nabla f(y,\\xi)\\|^2)\\leq L_K^2\\|x-y\\|^2$.\n  \\end{enumerate}\n\\end{assumption}\n\nUnder Assumption~\\ref{hyp:model}, it is an easy exercise to show that the mappings\n$F:\\bR^d\\to\\bR$ and $S:\\bR^d\\to\\bR^d$, given by:\n\\begin{equation}\n  F(x) \\eqdef \\bE(f(x,\\xi)) \\quad \\text{and} \\quad S(x) \\eqdef \\bE(\\nabla f(x,\\xi)^{\\odot 2})\\label{eq:F_and_S}\n\\end{equation}\nare well defined; $F$ is continuously differentiable and by\nLebesgue's dominated convergence theorem, $\\nabla F(x) = \\bE(\\nabla\nf(x,\\xi))$ for all $x$. Moreover, $\\nabla F$ and $S$ are locally Lipschitz continuous.\n\\begin{assumption}\n  \\label{hyp:coercive}\n$F$ is coercive.\n\\end{assumption}\n\\begin{assumption}\n  \\label{hyp:S>0}\nFor every $x\\in \\bR^d$, $S(x)>0$.\n\\end{assumption}\nIt follows from our assumptions that the set of critical points of\n$F$, denoted by $$\\cS \\eqdef \\nabla F^{-1}(\\{0\\}),$$ is non-empty.\nAssumption~\\ref{hyp:S>0} means that there is \\emph{no} point $x\\in\n\\bR^d$ satisfying $\\nabla f(x,\\xi) = 0$ with probability one (w.p.1). This is\na mild hypothesis in practice.\n\n\\subsection{Asymptotic Regime}\n\nWe address the constant stepsize regime, where\n$\\gamma$ is fixed along the iterations (the default value recommended\nin \\cite{kingma2014adam} is $\\gamma = 0.001$). As opposed to the decreasing stepsize context,\nthe sequence $z_n^\\gamma\\eqdef z_n$ \\emph{cannot} in general converge as $n$ tends to infinity, in an almost sure sense.\nInstead, we investigate the asymptotic behavior of the {family} of processes $(n\\mapsto z_n^\\gamma)_{\\gamma>0}$\nindexed by $\\gamma$, in the regime where $\\gamma\\to0$. We use the so-called ODE method (see e.g. \\cite{ben-(cours)99}).\nThe interpolated process $\\sz^{\\gamma}$ is the\npiecewise linear function defined on $[0,+\\infty)\\to \\cZ_+$\nfor all $t \\in [n\\gamma ,(n+1)\\gamma)$  by:\n\\begin{equation}\n\\sz^{ \\gamma}(t) \\eqdef z_{n}^{ \\gamma} + (z_{n+1}^{ \\gamma}-z_{n}^{ \\gamma})\\left(\\frac {t-n\\gamma}{\\gamma}\\right)\\,.\n\\label{eq:interpolated-process}\n\\end{equation}\nWe establish the convergence in probability of the\nfamily of random processes $(\\sz^{ \\gamma})_{\\gamma>0}$ as $\\gamma$ tends to zero, towards a\ndeterministic continuous-time system defined by an ODE.\nThe latter ODE, which we provide below at Eq.~(\\ref{eq:ode}), will be referred to\nas the continuous-time version of \\adam.\n\nBefore describing the ODE, we need to be more specific about our\nasymptotic regime. As opposed to SGD, \\adam\\ depends on\ntwo parameters $\\alpha$, $\\beta$, in addition to the stepsize $\\gamma$.\nThe  paper \\cite{kingma2014adam} recommends choosing the constants $\\alpha$ and $\\beta$ close to one\n(the default values $\\alpha=0.9$ and $\\beta=0.999$ are suggested).\nIt is thus legitimate to assume that\n$\\alpha$ and $\\beta$ tend to one, as $\\gamma$ tends to zero.\nWe set  $\\alpha \\eqdef \\bar \\alpha(\\gamma)$ and $\\beta\\eqdef\\bar\\beta(\\gamma)$, where\n$\\bar \\alpha(\\gamma)$ and $\\bar\\beta(\\gamma)$ converge to one as $\\gamma\\to 0$.\n\\begin{assumption}\n\\label{hyp:alpha-beta}\nThe functions $\\bar \\alpha:\\bR_+\\to [0,1)$ and $\\bar \\beta:\\bR_+\\to [0,1)$ are\ns.t. the following limits exist:\n  \\begin{equation}\n    a\\eqdef \\lim_{\\gamma_\\downarrow 0} \\frac{1-\\bar\\alpha(\\gamma)}\\gamma ,\\quad b\\eqdef  \\lim_{\\gamma_\\downarrow 0}\\frac{1-\\bar\\beta(\\gamma)}\\gamma\\,.\\label{eq:regime}\n  \\end{equation}\nMoreover, $a>0$, $b>0$, and the following condition holds: $b\\leq 4a\\,.$\n\\end{assumption}\nNote that the condition $b\\leq 4a$ is compatible with the default settings recommended\nby \\cite{kingma2014adam}.\nIn our model, we shall now replace the map $T_{\\gamma,\\alpha,\\beta}$ by $T_{\\gamma,\\bar \\alpha(\\gamma),\\bar \\beta(\\gamma)}$.\nLet $x_0\\in \\bR^d$ be fixed. For any fixed $\\gamma>0$, we define the sequence $(z_n^\\gamma)$ generated by\n\\adam\\ with a fixed stepsize~$\\gamma>0$:\n\\begin{equation}\nz_n^\\gamma \\eqdef T_{\\gamma,\\bar \\alpha(\\gamma),\\bar \\beta(\\gamma)}(n,z^\\gamma_{n-1},\\xi_n)\\,,\\label{eq:znT}\n\\end{equation}\nthe initialization being chosen as $z_0^\\gamma=(x_0,0,0)$.\n\n\n\\section{Continuous-Time System}\n\\label{sec:continuous_time}\n\n\\subsection{Ordinary Differential Equation}\n\nIn order to gain insight into the behavior of the sequence $(z_n^\\gamma)$\ndefined by (\\ref{eq:znT}),\nit is convenient to rewrite the \\adam\\ iterations under the following equivalent form, for every $n\\geq 1$:\n\\begin{equation}\nz_n^{\\gamma} = z^{\\gamma}_{n-1} + \\gamma h_{\\gamma}(n,z^{\\gamma}_{n-1}) + \\gamma \\Delta^\\gamma_{n}\\,,\\label{eq:RM}\n\\end{equation}\nwhere we define for every $\\gamma>0$, $z\\in \\cZ_+$,\n\\begin{equation}\nh_\\gamma(n,z) \\eqdef \\gamma^{-1}\\bE(T_{\\gamma,\\bar\\alpha(\\gamma),\\bar\\beta(\\gamma)}(n,z,\\xi)-z)\\,,\\label{eq:hgamma}\n\\end{equation}\nand where $\\Delta^\\gamma_{n}\\eqdef \\gamma^{-1}(z_n^{\\gamma} - z^{\\gamma}_{n-1}) - h_{\\gamma}(n,z^{\\gamma}_{n-1})$.\nNote that $(\\Delta^\\gamma_{n})$ is a martingale increment noise sequence in the sense that\n$\\bE(\\Delta^\\gamma_{n}|\\cF_{n-1}) = 0$ for all $n\\geq 1$, where $\\cF_n$ stands for the $\\sigma$-algebra generated\nby the r.v. $\\xi_1,\\dots,\\xi_n$.\nDefine the map $h:(0,+\\infty)\\times \\cZ_+\\to\\cZ$ for all $t>0$, all $z=(x,m,v)$ in $\\cZ_+$ by:\n\\begin{equation}\n  \\label{eq:h}\n  h(t,z)= \\begin{pmatrix}\n  -\\frac{(1-e^{-at})^{-1}m}{\\varepsilon+\\sqrt{(1-e^{-bt})^{-1} v}} \\\\\na (\\nabla F(x)-m) \\\\\nb (S(x)-v)\n\\end{pmatrix} \\,,\n\\end{equation}\nwhere $a,b$ are the constants defined in Assumption~\\ref{hyp:alpha-beta}.\nWe prove that, for any fixed $(t,z)$, the quantity $h(t,z)$ coincides with the limit\nof $h_\\gamma(\\lfloor t\/\\gamma\\rfloor,z)$ as $\\gamma\\downarrow 0$. This remark along with Eq.~(\\ref{eq:RM})\nsuggests that, as $\\gamma\\downarrow 0$, the interpolated process $\\sz^\\gamma$ shadows the non-autonomous differential equation\n\\begin{equation}\n  \\begin{array}[h]{l}\n\\dot z(t) = h(t, z(t))\\,.\n\\end{array}\n\\tag{ODE}\n\\label{eq:ode}\n\\end{equation}\n\n\\subsection{Existence, Uniqueness, Convergence}\n\\label{subsec:odeanalysis}\n\nSince $h(\\,.\\,,z)$ is non-continuous\nat point zero for a fixed $z\\in \\cZ_+$, and since $h(t,\\,.\\,)$\nis not locally Lipschitz continuous for a fixed~$t~>~0$,\nthe existence and uniqueness of the solution to (\\ref{eq:ode}) do not stem directly from off-the-shelf theorems.\n\nLet {\\color{black}$x_0$ be fixed}.\nA continuous map $z:[0,+\\infty)\\to\\cZ_+$ is said to be a global solution to (\\ref{eq:ode}) with initial condition {\\color{black}$(x_0,0,0)$}\nif $z$ is continuously differentiable on $(0,+\\infty)$, if Eq.~(\\ref{eq:ode})\nholds for all $t>0$, and if {\\color{black}$z(0)=(x_0,0,0)$}.\n\n\\begin{theorem}[Existence and uniqueness]\n  \\label{th:exist-unique}\n \n  Let Assumptions~\\ref{hyp:model} to \\ref{hyp:alpha-beta} hold true.\n  There exists a unique global solution $z:[0,+\\infty)\\to\\cZ_+$\n  to~(\\ref{eq:ode}) with\n  initial condition $(x_0,0,0)$.\n  Moreover, $z([0,+\\infty))$ is a bounded subset of $\\cZ_+$.\n\\end{theorem}\nOn the other hand, we note that a solution may not exist for\nan initial point$(x_0,m_0,v_0)$ with arbitrary (non-zero) values of $m_0, v_0$.\n\n\\begin{theorem}[Convergence]\n\\label{th:cv-adam}\nLet Assumptions~\\ref{hyp:model} to \\ref{hyp:alpha-beta} hold true.\nAssume that $F(\\cS)$ has an empty interior.\nLet $z:t\\mapsto (x(t),m(t),v(t))$ be\nthe global solution to~(\\ref{eq:ode}) with the initial condition\n$(x_0,0,0)$.\\,\nThen, the set $\\cS$ is non-empty and $\\lim_{t\\to\\infty} \\sd(x(t),\\cS) =0$,\n $\\lim_{t\\to\\infty}m(t)=0$, $\\lim_{t\\to\\infty}S(x(t))-v(t)=0$.\n\n\n\\end{theorem}\n\n\\noindent{\\bf Lyapunov function.} The proof of Th.~\\ref{th:exist-unique}\n  relies on the existence of a Lyapunov function for the non-autonomous\n  equation~(\\ref{eq:ode}).  Define  $V:(0,+\\infty)\\times \\cZ_+\\to \\bR$ by\n  \\begin{equation}\n    \\label{eq:V}\n    V(t,z)\\eqdef F(x)+\\frac 12 \\left\\|m\\right\\|^2_{U(t,v)^{-1}}\\,,\n  \\end{equation}\n  for every $t>0$ and every $z=(x,m,v)$ in $\\cZ_+$, where\n  $U:(0,+\\infty)\\times [0,+\\infty)^d\\to \\bR^d$ is the map given by:\n  \\begin{equation}\n    \\label{eq:U}\n    U(t,v) \\eqdef a(1-e^{-at})\\left(\\varepsilon+\\sqrt{\\frac{v}{1-e^{-bt}}}\\right)\\,.\n  \\end{equation}\nThen, $t\\mapsto V(t,z(t))$ is decreasing if $z(\\cdot)$ is the global solution to~(\\ref{eq:ode}).\n\n\\noindent{\\bf Cost decrease at the origin.} As $F$ itself is not a Lyapunov function for~(\\ref{eq:ode}),\nthere is no guarantee that $F(x(t))$ is decreasing w.r.t. $t$.\nNevertheless, the statement holds at the origin. Indeed, it can be shown that\n$\\lim_{t\\downarrow 0}V(t,z(t))=F(x_0)$ (see Prop.~\\ref{prop:adam-bounded}). As a consequence,\n\\begin{equation}\n  \\forall t\\geq 0,\\ F(x(t))\\leq F(x_0)\\,.\n\\label{eq:costdecrease}\n\\end{equation}\n\nIn other words, the (continuous-time) \\adam\\ procedure\n\\emph{can only improve} the initial guess $x_0$.\nThis is the consequence of the so-called bias correction steps in \\adam\\ (see Algorithm~\\ref{alg:adam})\\,.\nIf these debiasing steps were deleted in the \\adam\\ iterations,\nthe early stages of the algorithm could degrade the initial estimate $x_0$.\n\n\\noindent{\\bf Derivatives at the origin.}\nThe proof of Th.~\\ref{th:exist-unique} reveals that the initial derivative is given\nby $\\dot x(0) = -\\nabla F(x_0)\/(\\varepsilon+\\sqrt{S(x_0)})$ (see Lemma~\\ref{lem:m-v-derivables-en-zero}).\nIn the absence of debiasing steps, the initial derivative $\\dot x(0)$ would be a function of the initial\nparameters $m_0$, $v_0$, and the user would be required to tune these hyperparameters.\nNo such tuning is required thanks to the debiasing step.\nWhen $\\varepsilon$ is small and when the variance of $\\nabla f(x_0,\\xi)$ is small (\\emph{i.e.}, $S(x_0)\\simeq \\nabla F(x_0)^{\\odot 2}$),\nthe initial derivative $\\dot x(0)$ is approximately equal to $-\\nabla F(x_0)\/|\\nabla F(x_0)|$.\nThis suggests that in the early stages of the algorithm, the \\adam\\ iterations\nare comparable to the \\emph{sign} variant of the gradient descent, the properties of which were\ndiscussed in previous works, see \\cite{balles2018dissecting}\n\n\\subsection{Convergence rates}\n\nIn this paragraph, we establish the convergence to a single critical\npoint of $F$ and quantify the convergence rate, using the following\nassumption \\cite{lojasiewicz1963}.\n\\begin{assumption}[\\L{}ojasiewicz property]\n  \\label{hyp:lojasiewicz_prop}\nFor any  $x^* \\in {\\mathcal S}$, there exist\n$c >0\\,, \\sigma >0$ and $\\theta \\in (0,\\frac 12]$ s.t.\n\\begin{equation}\n  \\label{eq:lojasiewicz}\n\\forall x \\in \\bR^d \\,\\,\\text{s.t}\\,\\,\\|x-x^*\\|\\leq \\sigma\\,,\\quad  \\|\\nabla F(x)\\| \\geq c |F(x) - F(x^*)|^{1-\\theta}\\,.\n\\end{equation}\n\\end{assumption}\nAssumption~\\ref{hyp:lojasiewicz_prop} holds for real-analytic functions\nand semialgebraic functions.\nWe refer to\n\\cite{harauxjendoubi2015,attouch2009convergence,bolte2014proximal}\nfor a discussion and a review of applications.\nWe will call any $\\theta$ satisfying (\\ref{eq:lojasiewicz}) for some $c,\\sigma>0$,\nas a \\L{}ojasiewicz exponent of $f$ at $x^*$. The next result establishes the convergence\nof the function $x(t)$ generated by the ODE to a single critical point of $f$,\nand provides the convergence rate as a function of the \\L{}ojasiewicz exponent of $f$\nat this critical point. The proof is provided in subsection~\\ref{sec:cont_asymptotic_rates}.\n\\begin{theorem}\n  \\label{thm:asymptotic_rates}\n \n  Let Assumptions~\\ref{hyp:model} to \\ref{hyp:alpha-beta} and \\ref{hyp:lojasiewicz_prop} hold true.\n  Assume that $F(\\cS)$ has an empty interior. Let $x_0 \\in \\bR^d$ and let $z:t\\mapsto (x(t),m(t),v(t))$ be\n  the global solution to~(\\ref{eq:ode}) with initial condition $(x_0,0,0)$.\n  Then, there exists $x^* \\in \\cS$ such that $x(t)$ converges to $x^*$ as $t \\to +\\infty$.\n\n  Moreover, if $\\theta\\in (0,\\frac 12]$ is a \\L{}ojasiewicz exponent of $f$ at $x^*$,\n there exists a constant $C >0$ s.t. for all $t\\geq 0$,\n  \\begin{align*}\n    \\|x(t)-x^*\\| &\\leq C t^{-\\frac{\\theta}{1-2\\theta}}\\,,\\quad \\text{if}\\,\\,\\, 0<\\theta<\\frac{1}{2}\\,,\\\\\n    \\|x(t)-x^*\\| &\\leq C e^{- \\delta t} \\,,\\quad \\text{for some}\\,\\, \\delta >0\\,\\,\\text{if}\\,\\, \\theta = \\frac 12 \\,.\n  \\end{align*}\n\\end{theorem}\n\n\\section{Discrete-Time System: Convergence of \\adam}\n\\label{sec:discrete}\n\n\\begin{assumption}\n  \\label{hyp:iid}\nThe sequence $(\\xi_n:n\\geq 1)$ is iid, with the same distribution as~$\\xi$.\n\\end{assumption}\n\n\\begin{assumption}\n  \\label{hyp:moment-f}\nLet $p>0$. Assume either one of the following conditions.\n\\begin{enumerate}[i)]\n\\item \\label{momentegal} For every compact set $K\\subset \\bR^d$, $\\sup_{x\\in K} \\bE(\\|\\nabla f(x,\\xi)\\|^{p})<\\infty\\,.$\n\\item \\label{momentreinforce} For every compact set $K\\subset \\bR^d$,  $\\exists\\, p_K>p$, $\\sup_{x\\in K} \\bE(\\|\\nabla f(x,\\xi)\\|^{p_K})<\\infty.$\n\\end{enumerate}\n\\end{assumption}\nThe value of $p$ will be specified in the sequel, in the statement of\nthe results. Clearly, Assumption~\\ref{hyp:moment-f}~\\ref{momentreinforce} is stronger than Assumption~\\ref{hyp:moment-f}~\\ref{momentegal}.\nWe shall use either the latter or the former in our statements.\n\n\\begin{theorem}\n  \\label{th:weak-cv}\nLet Assumptions~\\ref{hyp:model} to \\ref{hyp:alpha-beta} and \\ref{hyp:iid} hold true. Let Assumption~\\ref{hyp:moment-f}~\\ref{momentreinforce}\nhold with $p=2$.\nConsider $x_0\\in \\bR^d$. For every $\\gamma>0$, let $(z_n^\\gamma:n\\in \\bN)$ be the random sequence defined by the \\adam\\ iterations~(\\ref{eq:znT})\nand $z_0^\\gamma = (x_0,0,0)$. Let $\\sz^\\gamma$ be the corresponding interpolated process defined by Eq.~(\\ref{eq:interpolated-process}).\nFinally, let $z$ denote the unique global solution to (\\ref{eq:ode}) issued from $(x_0,0,0)$.\nThen,\n$$\n\\forall\\, T>0,\\ \\forall\\, \\delta>0,\\ \\ \\lim_{\\gamma\\downarrow 0}\\bP\\left(\\sup_{t\\in[0,T]}\\|\\sz^\\gamma(t)-z(t)\\|>\\delta\\right)=0\\,.\n$$\n\\end{theorem}\nRecall that a family of r.v. $(X_\\alpha)_{\\alpha\\in I}$ is called \\emph{bounded in probability}, or \\emph{tight}, if for every\n$\\delta>0$, there exists a compact set $K$ s.t. $\\bP(X_\\alpha\\in K)\\geq 1-\\delta$ for every $\\alpha\\in I$.\n\\begin{assumption}\n\\label{hyp:tight}\n  There exists $\\bar \\gamma_0>0$ s.t. the family of r.v. $(z_n^\\gamma:n\\in \\bN,0<\\gamma<\\bar \\gamma_0)$ is bounded in probability.\n\\end{assumption}\n\n\\begin{theorem}\nConsider $x_0\\in \\bR^d$. For every $\\gamma>0$, let $(z_n^\\gamma:n\\in \\bN)$ be the random sequence defined by the \\adam\\ iterations~(\\ref{eq:znT})\nand $z_0^\\gamma = (x_0,0,0)$.\nLet Assumptions~\\ref{hyp:model} to \\ref{hyp:alpha-beta}, \\ref{hyp:iid} and \\ref{hyp:tight} hold.\nLet Assumption~\\ref{hyp:moment-f}~\\ref{momentreinforce} hold with $p=2$. Then,\nfor every $\\delta > 0$,\n  \\begin{equation}\n    \\label{eq:long-run}\n    \\lim_{\\gamma\\downarrow 0}\\limsup_{n\\to\\infty} \\frac 1{n} \\sum_{k=1}^n \\bP(\\sd(x_k^\\gamma, \\cS)>\\delta) = 0\\,.\n  \\end{equation}\n\\label{th:longrun}\n\\end{theorem}\n\n\n\\noindent{\\bf Convergence in the long run.}\nWhen the stepsize $\\gamma$ is constant, the sequence\n$(x_n^\\gamma)$ cannot converge in the almost sure sense as\n$n\\to\\infty$.\nConvergence may only hold in the doubly asymptotic regime where\n$n\\to\\infty$ then $\\gamma\\to 0$.\n\n\\noindent{\\bf Randomization.} For every $n$, consider a r.v. $N_n$  uniformly\ndistributed on  $\\{1,\\dots,n\\}$. Define $\\tilde x_n^\\gamma = x_{N_n}^\\gamma$.\nWe obtain from Th.~\\ref{th:longrun} that for every $\\delta>0$,\n$$\n\\limsup_{n\\to\\infty} \\ \\bP(\\sd(\\tilde x_n^\\gamma, \\cS)>\\delta) \\xrightarrow[\\gamma\\downarrow 0]{} 0\\,.\n$$\n\n\\noindent{\\bf Relationship between discrete and continuous time \\adam.}\nTh.~\\ref{th:weak-cv} means that the family of random processes $(\\sz^\\gamma:~\\gamma>0)$ converges\nin probability as $\\gamma\\downarrow 0$ towards the unique solution to~(\\ref{eq:ode}) issued from $(x_0,0,0)$.\nThis motivates the fact that the non-autonomous system~(\\ref{eq:ode})\nis a relevant approximation to the behavior of the iterates $(z_n^\\gamma:n\\in \\bN)$ for a small\nvalue of the stepsize~$\\gamma$.\n\n\\noindent{\\bf Stability.} Assumption~\\ref{hyp:tight} ensures\nthat the iterates $z_n^\\gamma$ do not explode in the long run.\nA sufficient condition is for instance that\n$\\sup_{n,\\gamma} \\bE \\|z_n^\\gamma\\|<\\infty\\,.$\n{In theory, this assumption can be difficult to verify.}\nNevertheless, in practice, a projection step on a compact set\ncan be introduced to ensure the boundedness of the estimates.\n\n\\section{A Decreasing Stepsize \\adam\\, Algorithm}\n\\label{sec:discrete_decreasing}\n\n\\subsection{Algorithm}\n\n\\adam\\, inherently uses constant stepsizes. Consequently, the\niterates~(\\ref{eq:znT}) do not converge in the almost sure sense.\nIn order to achieve convergence, we introduce in this section a decreasing\nstepsize version of \\adam. The iterations are given in Algorithm~\\ref{alg:adam-decreasing}.\n\\begin{algorithm}[tb]\n   \\caption{\\bf \\adam - decreasing stepsize $(((\\gamma_n,\\alpha_n,\\beta_n):n\\in \\bN^*), \\varepsilon)$.}\n   \\label{alg:adam-decreasing}\n\\begin{algorithmic}\n  \\STATE {\\bfseries Initialization:} $x_0\\in \\bR^d, m_0=0$, $v_0=0$, $r_0=\\bar r_0=0$.\n   \\FOR{$n=1$ {\\bfseries to} $n_{\\text{iter}}$}\n   \\STATE $m_n = \\alpha_n m_{n-1} + (1-\\alpha_n)  \\nabla f(x_{n-1},\\xi_n)$\n   \\STATE $v_n = \\beta_n v_{n-1} + (1-\\beta_n) \\nabla f(x_{n-1},\\xi_n)^{\\odot 2}$\n   \\STATE $r_n = \\alpha_n r_{n-1} + (1-\\alpha_n)$\n   \\STATE $\\bar r_n = \\beta_n \\bar r_{n-1} + (1-\\beta_n)$\n  \\STATE $\\hat m_{n} = m_{n}\/r_n$\n   \\COMMENT{bias correction step}\n   \\STATE $\\hat v_{n} = v_{n}\/\\bar r_n$\n   \\COMMENT{bias correction step}\n   \\STATE $x_{n} = x_{n-1} - \\gamma_n \\hat m_{n} \/ (\\varepsilon+\\sqrt{\\hat v_{n}}) \\,.$\n   \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\nThe algorithm generates a sequence $z_n=(x_n,m_n,v_n)$ with initial point $z_0=(x_0,0,0)$,\nwhere $x_0\\in\\bR^d$. Apart from the fact that the hyperparameters $(\\gamma_n,\\alpha_n,\\beta_n)$\nnow depend on $n$, the main difference w.r.t Algorithm~\\ref{alg:adam} lies in the expression of the\ndebiasing step. As noted in Remark~\\ref{rem:debiasing}, the aim is to rescale $m_n$ (resp. $v_n$)\nin such a way that the rescaled version $\\hat m_n$ (resp. $\\hat v_n$) is a convex combination of\npast stochastic gradients (resp. squared gradients). While in the constant step case the rescaling\ncoefficient is $(1-\\alpha^n)^{-1}$ (resp. $(1-\\beta^n)^{-1}$), the decreasing step case requires dividing\n$m_n$ by the coefficient $r_n=1-\\prod_{i=1}^n\\alpha_i$ (resp. $v_n$ by $\\bar r_n=1-\\prod_{i=1}^n\\beta_i$),\nwhich keeps track of the previous weights:\n$$\n\\hat m_n = \\frac{m_n}{r_n} = \\frac{\\sum_{k=1}^{n} \\rho_{n,k} \\nabla f(x_{k-1},\\xi_{k})}{\\sum_{k=1}^{n} \\rho_{n,k}}\\,,\n$$\nwhere for every $n,k$, $\\rho_{n,k} = \\alpha_n\\cdots\\alpha_{k+1}(1-\\alpha_k)$. A similar equation holds for $\\hat v_n$.\n\n\\subsection{Almost sure convergence\n  \\begin{assumption}[Stepsizes]\n    \\label{hyp:stepsizes}\n \n  The following holds.\n  \\begin{enumerate}[{\\sl i)}]\n  \\item For all $n \\in \\bN$, $\\gamma_n > 0$ and $\\gamma_{n+1}\/\\gamma_n\\to 1$,\n  \\item $\\sum_n \\gamma_n = +\\infty$ and $\\sum_n \\gamma_n^2 <+\\infty$,\n  \\item For all $n \\in \\bN$, $0 \\leq \\alpha_n \\leq 1$ and $0 \\leq \\beta_n \\leq 1$,\n  \\item There exist $a,b$ s.t. $0<b<4a$,\\,\n  $\\gamma_n^{-1}(1-\\alpha_n) \\to a$ and $\\gamma_n^{-1}(1-\\beta_n) \\to b$\\,.\n\n  \n  \n  \n \n  \\end{enumerate}\n  \\end{assumption}\n\n\n\n\n\n\n\n\n \\begin{theorem}\n \\label{thm:as_conv_under_stab}\nLet Assumptions~\\ref{hyp:model} to \\ref{hyp:S>0}, \\ref{hyp:iid} and~\\ref{hyp:stepsizes} hold.\nLet Assumption \\ref{hyp:moment-f}~\\ref{momentegal} hold with $p=4$.\nAssume that $F(\\cS)$ has an empty interior\nand that the random sequence $((x_n,m_n,v_n):n\\in \\bN)$\ngiven by Algorithm~\\ref{alg:adam-decreasing}\nis bounded, with probability one.\nThen, w.p.1, $\\lim_{n\\to\\infty}\n\\sd (x_n,\\cS)=0$, $lim_{n\\to\\infty}m_n = 0$\nand $lim_{n\\to\\infty} (S(x_n)-v_n)=0$.\nIf moreover $\\cS$ is finite or countable, then w.p.1, there exists $x^*\\in \\cS$\ns.t. $\\lim_{n\\to\\infty} (x_n,m_n,v_n) = (x^*,0,S(x^*))$.\n\n\n\n\n\n\n\n\n \\end{theorem}\nTh.~\\ref{thm:as_conv_under_stab} establishes the almost sure convergence of\n$x_n$ to the set of critical points of~$F$, under the assumption that the sequence\n$((x_n,m_n,v_n))$ is a.s. bounded. The next result provides a sufficient condition\nunder which almost sure boundedness holds.\n\\begin{assumption}\n\\label{hyp:stab}\n  The following holds.\n  \\begin{enumerate}[{\\sl i)}]\n\\item \\label{lipschitz} $\\nabla F$ is Lipschitz continuous.\n\\item \\label{momentgrowth} There exists $C>0$ s.t. for all $x \\in \\bR^d$, $\\bE[\\|\\nabla f(x,\\xi)\\|^2] \\leq C (1+F(x))$.\n\\item We assume the condition:\n\n\\lim\\sup_{n\\to\\infty}\n\\left(\\frac 1{\\gamma_n}-\\left(\\frac{1-\\alpha_{n+2}}{1-\\alpha_{n+1}}\\right)\\frac 1{\\gamma_{n+1}}\\right)\n< 2(a-\\frac b4)\\,,\n$\\\\%$$\nwhich is satisfied for instance if $b<4a$ and $1-\\alpha_{n+1} = a\\gamma_n$.\n  \\end{enumerate}\n\\end{assumption}\n \\begin{theorem}\n \\label{thm:stab}\n\nLet Assumptions~\\ref{hyp:model}, \\ref{hyp:coercive}, \\ref{hyp:iid}, \\ref{hyp:stepsizes} and \\ref{hyp:stab} hold.\nLet Assumption~\\ref{hyp:moment-f}~\\ref{momentegal} hold with $p=4$.\nThen, the sequence\n$((x_n,m_n,v_n):n\\in \\bN)$ given by Algorithm~\\ref{alg:adam-decreasing} is bounded with probability one.\n \\end{theorem}\n\n\\subsection{Central Limit Theorem}\n\n \\begin{assumption}\n   \\label{hyp:mean_field_tcl}\nLet $x^*\\in \\cS$. There exists a neighborhood $\\cV$ of $x^*$ s.t.\n\\begin{enumerate}[{\\sl i)}]\n\\item $F$ is twice continuously differentiable on $\\cV$,\n  and the Hessian $\\nabla^2 F(x^*)$ of $F$ at $x^*$ is positive\n  definite.\n\\item $S$ is continuously differentiable on $\\cV$.\n\\end{enumerate}\n \\end{assumption}\nDefine $\nD \\eqdef \\textrm{diag}\\left((\\varepsilon + \\sqrt{S_1(x^*)})^{-1},\\cdots,(\\varepsilon + \\sqrt{S_d(x^*)})^{-1}\\right)\\,.\n$\nLet $P$ be an orthogonal matrix s.t. the following spectral decomposition holds:\n$$\nD^{1\/2}\\nabla^2F(x^*)D^{1\/2} = P\\textrm{diag}(\\lambda_1,\\cdots,\\lambda_d)P^{-1}\\,,\n$$\nwhere $\\lambda_1, \\cdots,\\lambda_d$ are the (positive) eigenvalues of\n$D^{1\/2} \\nabla^2F(x^*)D^{1\/2}$.\nDefine\n\\begin{equation}\nH \\eqdef\n\\begin{pmatrix}\n  0 & -D & 0\\\\ a\\nabla^2F(x^*) & -a I_d & 0 \\\\ b \\nabla S(x^*) & 0 & -b I_d\n\\end{pmatrix}\\label{eq:H}\n\\end{equation}\nwhere $I_d$ represents the $d\\times d$ identity matrix and $\\nabla S(x^*)$ is the\nJacobian matrix of~$S$ at $x^*$.\nThe largest real part of the eigenvalues of $H$ coincides with $-L$, where\n\\begin{equation}\nL\\eqdef b\\wedge \\frac a2\\left( 1-\\sqrt{\\left(1-\\frac{4\\lambda_1}a\\right)\\vee 0}\\right) >0\\,.\\label{eq:L}\n\\end{equation}\nFinally, define the $3d\\times 3d$ matri\n\\begin{equation}\n  \\resizebox{.9\\hsize}{!}{$\nQ \\eqdef\n\\left( \\begin{array}[h]{cc}\n   0 &\n   \\begin{array}[h]{cc}\n     0 & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 0\n   \\end{array}\\\\\n   \\begin{array}[h]{c}\n     0 \\\\ 0\n   \\end{array}\n & \\bE\\left[\n  \\begin{pmatrix}\n    a\\nabla f(x^*,\\xi) \\\\ b(\\nabla f(x^*,\\xi)^{\\odot 2}-S(x^*))\n  \\end{pmatrix}\\begin{pmatrix}\n    a\\nabla f(x^*,\\xi) \\\\ b(\\nabla f(x^*,\\xi)^{\\odot 2}-S(x^*))\n  \\end{pmatrix}^T\\right]\n \\end{array}\\right)\\,.\\label{eq:Q}\n $}\n\\end{equation}\n\\begin{assumption}\n\\label{hyp:step-tcl}\nThe following holds.\n\\begin{enumerate}[{\\sl i)}]\n\\item \\label{step-tcl-i} There exist $\\kappa \\in (0,1]$, $\\gamma_0>0$, s.t. the sequence $(\\gamma_n)$\nsatisfies\n$\\gamma_n = {\\gamma_0}\/{(n+1)^\\kappa}$ for all $n$.\nIf $\\kappa = 1$, we assume moreover that $\\gamma_0 > \\frac{1}{2L}$.\n\\item The sequences $\\left(\\frac{1}{\\gamma_n}(\\frac{1-\\alpha_n}{\\gamma_n}-a)\\right)$ and $\\left(\\frac{1}{\\gamma_n}(\\frac{1-\\beta_n}{\\gamma_n}-b)\\right)$ are bounded.\n\\end{enumerate}\n\\end{assumption}\n\n\\noindent For an arbitrary sequence $(X_n)$ of random variables on some Euclidean space, a probability measure\n$\\mu$ on that space and an event $\\Gamma$ s.t. $\\bP(\\Gamma)>0$, we say that $X_n$ converges in distribution\nto $\\mu$ \\emph{given $\\Gamma$} if the measures $\\bP (X_n\\in \\cdot\\,|\\Gamma)$ converge weakly to $\\mu$.\n\\begin{theorem}\n \\label{thm:clt}\nLet Assumptions~\\ref{hyp:model}, \\ref{hyp:S>0}, \\ref{hyp:iid}, \\ref{hyp:mean_field_tcl} and \\ref{hyp:step-tcl} hold true.\nLet Assumption~\\ref{hyp:moment-f}~\\ref{momentreinforce} hold with $p=4$.\nConsider the iterates $z_n=(x_n,m_n,v_n)$ given by Algorithm~\\ref{alg:adam-decreasing}. Set $z^*=(x^*,0,S(x^*))$.\nSet $\\zeta \\eqdef 0$ if $0<\\kappa<1$ and $\\zeta \\eqdef \\frac{1}{2 \\gamma_0}$ if $\\kappa =1$.\nAssume $\\bP(z_n \\to z^*)>0$. Then, given the event $\\{z_n\\to z^*\\}$,\nthe rescaled vector $\\sqrt{\\gamma_n}^{-1}(z_n-z^*)$\nconverges in distribution to a zero mean Gaussian distribution on $\\bR^{3d}$ with a covariance matrix $\\Sigma$\nwhich is solution to the Lyapunov equation: $  \\left(H + \\zeta I_{3d} \\right) \\Sigma + \\Sigma \\left( H^T +  \\zeta I_{3d} \\right) = - Q$.\nIn particular, given $\\{z_n\\to z^*\\}$, the vector $\\sqrt{\\gamma_n}^{-1}(x_n-x^*)$\nconverges in distribution to a zero mean Gaussian distribution with a covariance matrix $\\Sigma_1$ given by:\n\\begin{equation}\n\\Sigma_1 = D^{1\/2} P\n\\left(\n\\frac{C_{k,\\ell}}{ (1 - \\frac{2\\zeta}{a})(\\lambda_k+\\lambda_\\ell-2\\zeta + \\frac 2a \\zeta^2) +\\frac 1{2(a-2\\zeta)}(\\lambda_k-\\lambda_\\ell)^2}\n\n\\right)_{k,\\ell=1\\dots d}\nP^{-1}D^{1\/2}\\label{eq:cov}\n\\end{equation}\nwhere $C\\eqdef P^{-1}D^{1\/2}\\bE\\left(\\nabla f(x^*,\\xi)\\nabla f(x^*,\\xi)^T\\right)D^{1\/2}P$.\n\\end{theorem}\n\nThe following remarks are useful.\n\\begin{itemize}[leftmargin=*]\n\\item The variable $v_n$ has an impact on the limiting covariance $\\Sigma_1$ through its limit $S(x^*)$ (used to define $D$),\nbut the fluctuations of $v_n$ and the parameter $b$ have no effect on $\\Sigma_1$.\nAs a matter of fact, $\\Sigma_1$ coincides with the limiting covariance matrix that would have been obtained by considering\niterates of the form\n\\begin{equation*}\n     \\begin{cases}\n       x_{n+1} &= x_n - \\gamma_{n+1} p_{n+1} \\\\\n       p_{n+1} &= p_n + a\\gamma_{n+1}(D\\nabla f(x_n,\\xi_{n+1})-p_n) \\,,\n     \\end{cases}\n\\end{equation*}\n\nwhich can be interpreted as a preconditioned version of the stochastic heavy ball algorithm~\\cite{gadat2018stochastic}.\nOf course, the above iterates are not implementable because the preconditioning matrix $D$ is unknown.\n\\item \nWhen $a$ is large, $\\Sigma_1$ is close to the matrix $\\Sigma_1^{(0)}$ obtained\nwhen letting $a \\to +\\infty$ in Eq.~(\\ref{eq:cov}).\nThe matrix $\\Sigma_1^{(0)}$ is the solution to the  Lyapunov equation\n$$\n(D \\nabla^2F(x^*) - \\zeta I_d)  \\Sigma_1^{(0)} + \\Sigma_1^{(0)} (\\nabla^2F(x^*) D - \\zeta I_d) = D \\bE\\left(\\nabla f(x^*,\\xi)\\nabla f(x^*,\\xi)^T\\right) D\\,.\n$$\nThe matrix $\\Sigma_1^{(0)}$ can be interpreted as the asymptotic covariance matrix of the $x$-variable\nin the absence of the inertial term (that is, when one considers \\textsc{RmsProp} instead of \\adam).\nThe matrix $\\Sigma_1^{(0)}$ approximates $\\Sigma_1$ in the sense that $\\Sigma_1 = \\Sigma_1^{(0)}+\\frac 1a\\Delta + O(\\frac 1{a^2})$\nfor some symmetric matrix $\\Delta$ which can be explicited. The matrix $\\Delta$ is neither positive nor negative definite\nin general.\nThis suggests that the question of the potential benefit of the presence of an inertial term\nis in general problem dependent.\n\\item In the statement of Th.~\\ref{thm:clt},\nthe conditioning event $\\{z_n\\to z^*\\}$ can be replaced by the event $\\{x_n\\to x^*\\}$\nunder the additional assumption that $\\sum_n \\gamma_n^2 < +\\infty$.\n\\end{itemize}\n\n \\section{Related Works}\n \\label{sec:related-works}\n\n Although the idea of adapting the\n (per-coordinate) learning rates as a function of past gradient values\n is not new (see \\emph{e.g.} variable metric methods such as the BFGS\n algorithms),\n \\textsc{AdaGrad} \\cite{duchi2011adaptive} led the way to a new class of algorithms\n that are sometimes referred to as adaptive gradient methods. \\textsc{AdaGrad}\n consists of dividing the learning rate by the square root of the sum\n of previous gradients squared\n componentwise. \n The idea was to give larger learning rates to highly informative but\n infrequent features instead of using a fixed predetermined schedule.\n However, in practice, the division by the cumulative sum of squared\n gradients may generate small learning rates, thus freezing the\n iterates too early.  Several works proposed\n heuristical ways to set the learning rates using a less aggressive\n policy.\n The work \\cite{tieleman2012lecture} introduced an unpublished, yet popular, algorithm\n referred to as \\textsc{RmsProp} where the cumulative sum\n used in \\textsc{AdaGrad} is replaced by a moving average of squared\n gradients.\n\n\n\n \\adam\\ combines the advantages of both \\textsc{AdaGrad},\n \\textsc{RmsProp} and inertial methods\n\n As opposed to \\textsc{AdaGrad}, for which theoretical convergence guarantees exist\n \\cite{duchi2011adaptive,chen2018convergence,zhou2018convergence,ward2018adagrad},\n \\adam\\ is comparatively less studied.\n The initial paper \\cite{kingma2014adam} suggests a $\\mathcal{O}(\\frac{1}{\\sqrt{T}})$ average regret bound in the convex setting,\n but \\cite{j.2018on} exhibits a counterexample in contradiction with this statement.\n The latter counterexample implies that the average regret bound of \\adam\\ does\n not converge to zero. A first way to overcome the problem is to modify the \\adam\\ iterations\n themselves in order to obtain a vanishing average regret. This led \\cite{j.2018on}\n to propose a variant called \\textsc{AmsGrad} with the aim to recover, at least in the convex case, the sought guarantees.\n\n\n\n\n\n\n\n\n\n The work \\cite{balles2018dissecting} interprets \\adam\\ as a variance-adapted sign descent combining an update direction given by the sign and\n a magnitude controlled by a variance adaptation principle. A ``noiseless'' version\n of \\adam\\ is considered in \\cite{basu2018convergence}. Under quite specific values of the \\adam-hyperparameters, it is shown that for every $\\delta>0$,\n there exists some time instant\n for which the norm of the gradient of the objective\n at the current iterate is no larger than~$\\delta$.\n\n\n The recent paper \\cite{chen2018convergence} provides a similar result\n for \\textsc{AmsGrad} and \\textsc{AdaGrad}, but the generalization to \\adam\\ is subject\n to conditions which are not easily verifiable.\n The paper \\cite{zaheer2018adaptive} provides a convergence result for \\textsc{RmsProp}\n using the objective function $F$ as a Lyapunov function.\n\n However, our work suggests that unlike \\textsc{RmsProp},\n \\adam\\ does not admit $F$ as a Lyapunov function.\n This makes the approach of\n\n \\cite{zaheer2018adaptive} hardly generalizable to \\adam.\n Moreover, \\cite{zaheer2018adaptive} considers biased gradient estimates instead of the debiased\n estimates used in \\adam.\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 In the present work, we study the behavior of an ODE, interpreted as the\n limit in probability of the (interpolated) \\adam\\ iterates as the stepsize tends to zero.\n Closely related continuous-time dynamical systems are also studied in \\cite{attouch2000heavy,cabot2009long}.\n We leverage the idea of approximating a discrete time stochastic system by a deterministic continuous one,\n often referred to as the ODE method.\n A recent work \\cite{gadat2018stochastic} fruitfully exploits this method to study\n a stochastic version of the celebrated heavy ball algorithm.\n We refer to \\cite{davis2018stochastic} for the reader interested in the non-differentiable setting\n with an analysis of the stochastic subgradient algorithm for non-smooth non-convex objective functions.\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\n\n Concomitant to the present paper, Da Silva and Gazeau \\cite{da2018general}\n (posted only four weeks after the first version of the present work)\n\n study the asymptotic behavior of a similar dynamical system as the one introduced here.\n They establish several results in continuous time, such as avoidance of traps\n as well as convergence rates in the convex case; such aspects are out of the scope of this paper.\n However, the question of the convergence of the (discrete-time) iterates is left open.\n In the current paper, we also exhibit a Lyapunov function which allows, amongst others, to draw useful conclusions on the effect of the\n debiasing step of \\adam\\,. Finally, \\cite{da2018general} studies a slightly modified version of \\adam\\, allowing to recover an\n ODE with a locally Lipschitz continuous vector field, whereas the original \\adam\\ algorithm \\cite{kingma2014adam} leads\n to an ODE with an irregular vector field. This technical issue is tackled in the present paper.\n\n\\section{Proofs of Section~\\ref{sec:continuous_time}}\n\\label{sec:proofs_cont_time}\n\n\\subsection{Preliminaries}\n\\label{subsec:setting}\nThe results in this section are not specific to the case where $F$ and $S$ are defined as in\nEq.~(\\ref{eq:F_and_S}): they are stated for\n\\emph{any} mappings $F$, $S$ satisfying the following hypotheses.\n\\begin{assumption}\n  \\label{hyp:F}\nThe function $F:\\bR^d\\to\\bR$ is s.t.: $F$ is continuously differentiable and\n$\\nabla F$ is locally Lipschitz continuous.\n\\end{assumption}\n\\begin{assumption}\n\\label{hyp:S}\nThe map $S:\\bR^d\\to [0,+\\infty)^d$ is locally Lipschitz continuous.\n\\end{assumption}\nIn the sequel, we consider the following generalization of Eq. (\\ref{eq:ode}) for any $\\eta >0$:\n\\begin{equation}\n  \\begin{array}[h]{l}\n\\dot z(t) = h(t+\\eta, z(t))\\,.\n\\end{array}\n\\tag{ODE\\mbox{$_\\eta$}}\n\\label{eq:odeeta}\\end{equation}\nWhen $\\eta=0$, Eq. (\\ref{eq:odeeta}) boils down to the equation of interest (\\ref{eq:ode}).\nThe choice $\\eta\\in (0,+\\infty)$ will be revealed useful to prove Th.~\\ref{th:exist-unique}.\nIndeed, for $\\eta>0$, a solution to Eq. (\\ref{eq:odeeta}) can be shown to exist (on some interval) due to the continuity of\nthe map $h(\\,.+\\eta,\\,.\\,)$. Considering a family of such solutions indexed by $\\eta\\in (0,1]$,\nthe idea is to prove the existence of a solution to (\\ref{eq:ode}) as a cluster point of the latter family when $\\eta\\downarrow 0$.\nIndeed, as the family is shown to be equicontinuous, such a cluster point does exist thanks to the Arzel\u00e0-Ascoli theorem.\nWhen $\\eta=+\\infty$,\nEq. (\\ref{eq:odeeta}) rewrites\n\\begin{equation}\n  \\label{eq:ode-a}\n  \\begin{array}[h]{l}\n\\dot z(t) = h_\\infty(z(t))\\,,\n\\end{array}\n\\tag{ODE\\mbox{$_\\infty$}}\n\\end{equation}\nwhere $h_\\infty(z)\\eqdef \\lim_{t\\to \\infty} h(t,z)$.\nIt is useful to note that for $(x,m,v)\\in \\cZ_+$,\n\\begin{equation}\n  \\label{eq:h_infty}\nh_{\\infty}((x,m,v)) = \\left(-m \/ (\\varepsilon+\\sqrt{v})\\,,\\, a (\\nabla F(x)-m) \\,,\\,b (S(x)-v) \\right)\\,.\n\\end{equation}\nContrary to Eq. (\\ref{eq:ode}), Eq.~(\\ref{eq:ode-a}) defines an  autonomous ODE.\nThe latter admits a unique global solution for any initial condition in $\\cZ_+$,\nand defines a dynamical system $\\cD$. We shall exhibit a strict Lyapunov function\nfor this dynamical system $\\cD$, and deduce that any solution to (\\ref{eq:ode-a}) converges\nto the set of equilibria of $\\cD$ as $t\\to\\infty$.\nOn the otherhand, we will prove that the solution to (\\ref{eq:ode}) with a proper initial condition is a so-called asymptotic pseudotrajectory (APT) of $\\cD$. Due to the\nexistence of a strict Lyapunov function, the APT shall inherit the convergence behavior of the autonomous system as $t\\to\\infty$,\nwhich will prove Th.~\\ref{th:cv-adam}.\n\nIt is convenient to extend the map $h:(0,+\\infty)\\times \\cZ_+\\to\\cZ$ on  $(0,+\\infty)\\times \\cZ\\to\\cZ$ by\nsetting $h(t,(x,m,v))\\eqdef h(t,(x,m,|v|))$ for every $t>0$, $(x,m,v)\\in \\cZ$.\nSimilarly, we extend $h_\\infty$ as $h_\\infty((x,m,v)) \\eqdef h_\\infty((x,m,|v|))$.\nFor any $T\\in (0,+\\infty]$ and any $\\eta\\in [0,+\\infty]$, we say that a map $z:[0,T)\\to \\cZ$ is a solution\nto (\\ref{eq:odeeta}) on $[0,T)$ with initial condition $z_0\\in \\cZ_+$,\nif $z$ is continuous on $[0,T)$, continuously differentiable\non $(0,T)$, and if (\\ref{eq:odeeta}) holds for all $t\\in (0,T)$.\nWhen $T=+\\infty$, we say that the solution is global.\nWe denote by $Z^\\eta_T(z_0)$ the subset of $C([0,T),\\cZ)$ formed by the solutions to (\\ref{eq:odeeta})\non $[0,T)$ with initial condition $z_0$.\nFor any $K\\subset\\cZ_+$, we define $Z^\\eta_T(K)\\eqdef \\bigcup_{z\\in K}Z^\\eta_T(z)$.\n\n\n\\begin{lemma}\n  \\label{lem:m-v-derivables-en-zero}\n  Let Assumptions~\\ref{hyp:F} and \\ref{hyp:S} hold.  Consider $x_0\\in \\bR^d$,\n  $T\\in (0,+\\infty]$ and let $z\\in Z_T^0((x_0,0,0))$,\n  which we write $z(t) = (x(t),m(t),v(t))$.  Then, $z$\n  is continuously differentiable on $[0,T)$,\n  $\\dot m(0)=a\\nabla F(x_0)$, $\\dot v(0)=bS(x_0)$ and\n$\n\\dot x(0) =  \\frac{-\\nabla F(x_0)}{\\varepsilon + \\sqrt{S(x_0)}}.\n$\n\\end{lemma}\n\\begin{proof}\nBy definition of $z(\\,.\\,)$, $m(t)=\\int_0^ta(\\nabla F(x(s))-m(s))ds$ for all $t\\in [0,T)$\n(and a similar relation holds for $v(t)$).\nThe integrand being continuous, it holds\nthat $m$ and $v$ are differentiable at  zero and $\\dot m(0)=a\\nabla F(x_0)$, $\\dot v(0)=bS(x_0)$.\nSimilarly, $x(t) = x_0+\\int_0^t h_x(s,z(s))ds$, where\n$h_x(s,z(s)) \\eqdef -(1-e^{-as})^{-1}m(s)\/(\\varepsilon+\\sqrt{(1-e^{-bs})^{-1}v(s)})\\,.$\nNote that $m(s)\/s \\to \\dot m(0) = a\\nabla F(x_0) $ as $s\\downarrow 0$.\nThus, $(1-e^{-as})^{-1}m(s)\\to \\nabla F(x_0)$ as $s\\to 0$. Similarly,\n$(1-e^{-bs})^{-1}v(s)\\to S(x_0)$. It follows that\n$h_x(s,z(s))\\to -(\\varepsilon+\\sqrt{S(x_0)})^{-1}\\nabla F(x_0)$.\nThus, $s\\mapsto h_x(s,z(s))$\ncan be extended\nto a continuous map on $[0,T)\\to\\bR^d$ and the differentiability of $x$ at zero\nfollows.\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{lem:v-positif}\nLet Assumptions~\\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\nFor every $\\eta\\in [0,+\\infty]$, $T\\in (0,+\\infty]$,  $z_0\\in \\cZ_+$,  $z\\in Z_T^\\eta(z_0)$,\nit holds that $z((0,T))\\subset \\cZ_+^*$.\n\\end{lemma}\n\\begin{proof}\nSet $z(t) = (x(t),m(t),v(t))$ for all $t$. Consider $i\\in \\{1,\\dots,d\\}$.\nAssume by contradiction that there exists $t_0\\in (0,T)$ s.t.\n$v_i(t_0)<0$. Set $\\tau\\eqdef\\sup\\{t\\in [0,t_0]:v_i(t)\\geq 0\\}$.\nClearly, $\\tau<t_0$ and $v_i(\\tau)=0$ by the continuity of $v_i$.\nSince $v_i(t)\\leq 0$ for all $t\\in (\\tau,t_0]$, it holds that $\\dot v_i(t) = b(S_i(x(t))-v_i(t))$\nis nonnegative for all  $t\\in (\\tau,t_0]$. This contradicts the fact that $v_i(\\tau)>v_i(t_0)$.\nThus, $v_i(t)\\geq 0$ for all $t\\in [0,T)$.\nNow assume by contradiction that there exists $t\\in (0,T)$  s.t.\n$v_i(t)=0$. Then, $\\dot v_i(t)=bS_i(x(t))>0$.\nThus,\n$\n\\lim_{\\delta\\downarrow 0} \\frac{v_i(t-\\delta)}{-\\delta} = bS_i(x(t))\\,.\n$\nIn particular, there exists $\\delta>0$ s.t.\n$v_i(t-\\delta)  \\leq  -\\frac{\\delta b}2S_i(x(t))\\,.$ This contradicts the first point.\n\\end{proof}\n\n Recall the definitions of $V$ and $U$ from Eqs.~(\\ref{eq:V}) and (\\ref{eq:U}).\n Clearly, $U_\\infty(v)\\eqdef\\lim_{t\\to \\infty} U(t,v)=a(\\varepsilon+\\sqrt{v})$ is well defined for every $v\\in [0,+\\infty)^d$.\n Hence, we can also define $V_\\infty(z)\\eqdef \\lim_{t\\to \\infty} V(t,z)$ for every $z\\in \\cZ_+$.\n\n\n\n\n\\begin{lemma}\n\\label{lem:V}\n \n  Let Assumptions~\\ref{hyp:F} and \\ref{hyp:S} hold.\n   Assume that $0< b\\leq 4a$.\nConsider $(t,z)\\in (0,+\\infty)\\times \\cZ_+^*$ and set $z=(x,m,v)$.\nThen, $V$ and $V_\\infty$ are  differentiable at points  $(t,z)$ and $z$ respectively. Moreover,\n$\\ps{\\nabla V_\\infty(z),h_{\\infty}(z)}\\leq -\\varepsilon \\left\\|\\frac {am}{U_\\infty(v)}\\right\\|^2\\,$ and\n\\begin{equation*}\n\\ps{\\nabla V(t,z), (1,h(t,z))} \\leq -\\frac{\\varepsilon }2\\left\\|\\frac{a\\,m}{U(t,v)}\\right\\|^2\\,.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nWe only prove the second point, the proof of the first point follows the same line.\n Consider $(t,z)\\in (0,+\\infty)\\times \\cZ_+^*$.\nWe decompose\n$\\ps{\\nabla V(t,z), (1,h(t,z))} = \\partial_tV(t,z)\n+\\ps{\\nabla_z V(t,z),h(t,z)}$. After tedious but straightforward derivations, we get:\n\\begin{equation}\n  \\label{eq:partial-t}\n  \\resizebox{0.99\\hsize}{!}{$\n  \\partial_t V(t,z) =- \\sum_{i=1}^d \\frac{a^2m_i^2}{U(t,v_i)^2}\\left(\\frac{e^{-at}\\varepsilon}2+\n\\left(\\frac{e^{-at}}2-\\frac{be^{-bt}(1-e^{-at})}{4a(1-e^{-bt})}\\right)\\sqrt{\\frac{v_i}{1-e^{-bt}}}\\right)\\,,\n$}\n\\end{equation}\nwhere $U(t,v_i)=a(1-e^{-at})\\left(\\varepsilon+\\sqrt{\\frac{v_i}{1-e^{-bt}}}\\right)$ and $\\ps{\\nabla_z V(t,z),h(t,z)}$ is equal to:\n\\begin{equation*}\n    \\sum_{i=1}^d \\frac{-a^2m_i^2(1-e^{-at})}{U(t,v_i)^2}\n  \\left(\\varepsilon\n+ (1-\\frac b{4a})\\sqrt{\\frac{v_i}{1-e^{-bt}}}\n+\\frac{bS_i(x)}{4a\\sqrt{v_i(1-e^{-bt})}}\n\\right)\\,.\n\\end{equation*}\nUsing that $S_i(x)\\geq 0$, we obtain:\n\\begin{equation}\n\\ps{\\nabla V(t,z), (1,h(t,z))} \\leq  -\\sum_{i=1}^d \\frac{a^2m_i^2}{U(t,v_i)^2}\\left(\n(1-\\frac{e^{-at}}2)\\varepsilon+c_{a,b}(t)\\sqrt{\\frac{v_i}{1-e^{-bt}}}\n\\right)\\,,\\label{eq:ineg-V}\n\\end{equation}\nwhere $c_{a,b}(t)\\eqdef 1-\\frac{e^{-at}}2-\\frac b{4a}\\frac{1-e^{-at}}{1-e^{-bt}}\\,.$\nUsing inequality $1-{e^{-at}}\/2\\geq 1\/2$ in (\\ref{eq:ineg-V}), the inequality~(\\ref{eq:ineg-V})\nproves the Lemma, provided that one is able to show that $c_{a,b}(t)\\geq 0$, for all $t>0$\nand all $a,b$ satisfying $0< b\\leq 4a$. We prove this last statement.\nIt can be shown that the function $b\\mapsto c_{a,b}(t)$ is decreasing on $[0,+\\infty)$.\nHence, $c_{a,b}(t)\\geq c_{a,4a}(t)$. Now, $c_{a,4a}(t) = q(e^{-at})$ where $q:[0,1)\\to\\bR$ is the\nfunction defined for all $y\\in [0,1)$ by $q(y) = y \\left(y^4-2y^3+1\\right)\/(2(1-y^4))\\,$.\nHence $q \\geq 0$. Thus, $c_{a,b}(t)\\geq q(e^{-at})\\geq 0$.\n\\end{proof}\n\n\\subsection{Proof of Th.~\\ref{th:exist-unique}}\n\n\\subsubsection{Boundedness}\nDefine $\\cZ_0 \\eqdef \\{(x,0,0):x\\in \\bR^d\\}$.\nLet $\\bar e:(0,+\\infty)\\times \\cZ_+\\to\\cZ_+$ be defined by\n$\\bar e(t,z)\\eqdef (x,m\/(1-e^{-at}),v\/(1-e^{-bt}))\\,$\nfor every $t>0$ and every $z=(x,m,v)$ in $\\cZ_+$.\n\n\\begin{proposition}\n\\label{prop:adam-bounded}\nLet Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\nAssume that $0< b\\leq 4a$.\nFor every $z_0\\in \\cZ_0$, there exists a compact set $K\\subset \\cZ_+$ s.t.\nfor all $\\eta\\in [0,+\\infty)$, all $T\\in (0,+\\infty]$ and all $z\\in Z_T^\\eta(z_0)$,\n$\\left\\{\\bar e(t+\\eta,z(t)) :t\\in (0,T)\\right\\} \\subset K\\,.$\nMoreover, choosing $z_0$ of the form $z_0=(x_0,0,0)$ and $z(t) = (x(t), m(t),v(t))$, it holds that $F(x(t))\\leq F(x_0)$ for all $t\\in [0,T)$.\n\\end{proposition}\n\n\\begin{proof}\nLet $\\eta\\in [0,+\\infty)$.\nConsider a solution $z_\\eta(t) = (x_\\eta(t),m_\\eta(t),v_\\eta(t))$  as in the statement, defined on some interval $[0,T)$.\nDefine\n$\\hat m_\\eta(t) = m_\\eta(t)\/(1-e^{-a(t+\\eta)})$,\n$\\hat v_\\eta(t) = v_\\eta(t)\/(1-e^{-b(t+\\eta)})$.\nBy Lemma~\\ref{lem:v-positif}, $t\\mapsto V(t+\\eta,z(t))$ is continuous on $[0,T)$, and\ncontinuously differentiable on $(0,T)$.\nBy Lemma~\\ref{lem:V}, $\\dot V(t+\\eta,z_\\eta(t)) \\leq 0$ for all $t>0$.\nAs a consequence, $t\\mapsto V(t+\\eta,z_\\eta(t))$ is non-increasing on $[0,T)$.\nThus, for all $t\\geq 0$, $F(x_{\\eta}(t))\\leq \\lim_{t'\\downarrow 0}V(t'+\\eta,z_\\eta(t'))$. Note that\n$\n  V(t+\\eta,z_\\eta(t)) \\leq F(x_\\eta(t))+\\frac 12 \\sum_{i=1}^d\n  \\frac{m_{\\eta,i}(t)^2}{a(1-e^{-a(t+\\eta)})\\varepsilon}\\,. \\label{eq:majV}\n$\nIf $\\eta>0$, every term in the sum in the righthand side\ntends to zero, upon noting that\n$m_{\\eta}(t)\\to 0$ as $t\\to 0$.\nThe statement still holds if $\\eta=0$. Indeed, by Lemma~\\ref{lem:m-v-derivables-en-zero},\nfor a given $i\\in \\{1,\\dots,d\\}$, there exists $\\delta>0$ s.t. for all $0<t<\\delta$,\n$m_{\\eta,i}(t)^2\\leq 2 a^2 (\\partial_i F(x_0))^2 t^2$\nand $1-e^{-at}\\geq (at)\/2$. As a consequence, each term of the sum\nis no larger than $4 (\\partial_i F(x_0))^2 t\/\\varepsilon$, which tends to zero as $t\\to 0$.\nWe conclude that for all $t\\geq 0$, $F(x_\\eta(t))\\leq F(x_0)$.\nIn particular, $\\{x_\\eta(t):t\\in [0,T)\\}\\subset \\{F\\leq F(x_0)\\}$, the latter set being bounded\nby Assumption~\\ref{hyp:coercive}.\n\n\nWe prove that $v_{i,\\eta}(t)$ is (upper)bounded.\nDefine $R_{i} \\eqdef \\sup  S_i(\\{F\\leq F(x_0)\\})$, which is finite by continuity of $S$.\nAssume by contradiction that the set $\\{t\\in [0,T):v_{\\eta,i}(t)\\geq R_{i}+1\\}$ is non-empty, and denote\nits infimum by $\\tau$. By continuity of $v_{\\eta,i}$, one has\n$v_{\\eta,i}(\\tau) =  R_{i}+1$. This by the way implies that $\\tau>0$. Hence,\n$\\dot v_{\\eta,i}(\\tau) = b(S_i(x_\\eta(\\tau))-v_{\\eta,i}(\\tau)) \\leq -b$.\nThis means that there exists $\\tau'<\\tau$ s.t. $v_{\\eta,i}(\\tau')>v_{\\eta,i}(\\tau)$, which contradicts the definition of $\\tau$.\nWe have shown that $v_{\\eta,i}(t)\\leq R_i+1$ for all $t\\in (0,T)$.\nIn particular, when $t\\geq 1$, $\\hat v_{\\eta,i}(t) = v_{\\eta,i}(t)\/(1-e^{-bt}) \\leq (R_i+1)\/(1-e^{-b})\\,.$\nConsider $t\\in (0,1\\wedge T)$.\nBy the mean value theorem, there exists $\\tilde t_\\eta\\in [0,t]$ s.t. $v_{\\eta,i}(t) = \\dot v_{\\eta,i}(\\tilde t_\\eta)t$.\nThus, $v_{\\eta,i}(t) \\leq b S_i(x(\\tilde t_\\eta)) t\\leq b R_i t$. Using that the map $y\\mapsto y\/(1-e^{-y})$ is increasing on $(0,+\\infty)$,\nit holds that for all $t\\in (0,1\\wedge T)$,\n$\\hat v_{\\eta,i}(t)\n\\leq bR_i \/(1-e^{-b})\\,.$\nWe have shown that, for all $t\\in (0,T)$ and all $i\\in \\{1,\\dots,d\\}$, $0\\leq \\hat v_{\\eta,i}(t)\\leq M$, where\n$M\\eqdef (1-e^{-b})^{-1}(1+ b)(1+\\max\\{R_\\ell:\\ell\\in \\{1,\\dots,d\\})$.\n\nAs $V(t+\\eta,z_\\eta(t))\\leq F(x_0)$, we obtain: $F(x_0) \\geq F(x_\\eta(t))+\\frac 12\n\\left\\|m_\\eta(t)\\right\\|^2_{U(t+\\eta,v_\\eta(t))^{-1}}$.\nThus, $F(x_0) \\geq \\inf F+\\frac 1{2a(\\varepsilon+\\sqrt{M})} \\|m_{\\eta}(t)\\|^2\\,$.\nTherefore, $m_\\eta(\\,.\\,)$ is bounded on $[0,T)$, uniformly in $\\eta$.\nThe same holds for $\\hat m_\\eta$ by using the mean value theorem\nin the same way as for $\\hat v_\\eta$. The proof is complete.\n\\end{proof}\n\n\n\\begin{proposition}\n\\label{prop:bounded}\nLet Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\nAssume that $0< b\\leq 4a$.\nLet $K$ be a compact subset of $\\cZ_+$.\nThen, there exists an other compact set $K'\\subset \\cZ_+$ s.t.\nfor every $T\\in (0,+\\infty]$ and every $z\\in Z_{T}^\\infty(K)$,\n$z([0,T))\\subset K'$.\n\\end{proposition}\n\\begin{proof}\nThe proof follows the same line as Prop.~\\ref{prop:adam-bounded} and is omitted.\n\\end{proof}\n\n\n\nFor any $K\\subset \\cZ_+$, define $v_{\\min}(K)\\eqdef\\inf\\{v_k: (x,m,v)\\in K,i\\in \\{1,\\dots,d\\}\\}$.\n\\begin{lemma}\n\\label{lem:v-lowerbound}\nUnder Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S},\nthe following holds true\n\\begin{enumerate}[{\\it i)},leftmargin=*]\n\\item For every compact set $K\\subset \\cZ_+$, there exists $c>0$, s.t. for every $z\\in Z^\\infty_{\\infty}(K)$, of the form\n$z(t)= (x(t),m(t),v(t))$, $v_i(t)\\geq c \\min\\left(1 ,\\frac{v_{\\min}(K)}{2c}+ t\\right)\\qquad(\\forall t\\geq 0, \\forall i\\in\\{1,\\dots,d\\})\\,.$\n\\item For every $z_0\\in \\cZ_0$, there exists $c>0$ s.t. for every $\\eta\\in [0,+\\infty)$ and every $z\\in Z_\\infty^\\eta(z_0)$,\n$v_i(t)\\geq c\\min(1,t)\\qquad(\\forall t\\geq 0, \\forall i\\in\\{1,\\dots,d\\})\\,.$\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nWe prove the first point. Consider a compact set $K\\subset \\cZ_+$.\nBy Prop.~\\ref{prop:bounded}, one can find a compact set $K'\\subset \\cZ_+$ s.t.\nfor every $z\\in Z^\\infty_{\\infty}(K)$, it holds that $\\{z(t):t\\geq 0\\}\\subset K'$.\nDenote by $L_S$ the Lipschitz constant of $S$ on the compact set $\\{x:(x,m,v)\\in K'\\}$.\nIntroduce the constants $M_1\\eqdef \\sup\\{\\|m\/(\\varepsilon + \\sqrt v)\\|_\\infty:(x,m,v)\\in K'\\}$,\n$M_2\\eqdef \\sup\\{\\|S(x)\\|_\\infty:(x,m,v)\\in K'\\}$.\nThe constants $L_S, M_1, M_2$ are finite.\nNow consider a global solution $z(t)=(x(t),m(t),v(t))$ in $Z^\\infty_{\\infty}(K)$.\nChoose $i\\in \\{1,\\dots,d\\}$ and consider $t\\geq 0$. By the mean value theorem,\nthere exists $t'\\in [0,t]$ s.t. $v_i(t) = v_i(0)+\\dot v_i(t')t$. Thus,\n$ v_i(t) = v_i(0) + \\dot v_i(0) t + b(S_i(x(t')) - v_i(t') - S_i(x(0))+ v_i(0)) t$,\nwhich in turn implies\n$ v_i(t)\\geq v_i(0) + \\dot v_i(0) t - b L_S\\|x(t')-x(0)\\|t - b  |v_i(t') - v_i(0)| t$.\nUsing again the mean value theorem, for every $\\ell\\in \\{1,\\dots,d\\}$, there exists $t''\\in [0,t']$ s.t.\n$\n|x_\\ell(t')-x_\\ell(0)| = t' |\\dot x_\\ell(t'')| \\leq t M_1\\,.\n$\nTherefore, $\\|x(t')-x(0)\\|\\leq \\sqrt d M_1 t$. Similarly, there exists $\\tilde t$ s.t.:\n$|v_i(t') - v_i(0)|=  t'|\\dot v_i(\\tilde t)|\\leq t'b S_i(x(\\tilde t)) \\leq t bM_2\\,.$\nPutting together the above inequalities, $v_i(t) \\geq  v_i(0) (1-bt) + bS_i(x(0)) t - bC t^2 \\,$,\nwhere $C\\eqdef (M_2+L_S\\sqrt d M_1)$.\nFor every $t\\leq 1\/(2b)$, $v_i(t) \\geq \\frac{v_{\\min}}{2} + tbC\\left(\\frac{S_{\\min}}C  - t\\right) \\,,$\nwhere we defined $S_{\\min}\\eqdef\\inf\\{S_i(x):i\\in \\{1,\\dots,d\\}, (x,m,v)\\in K\\}$.\nSetting $\\tau \\eqdef 0.5\\min(1\/b,S_{\\min}\/C)$,\n\\begin{equation}\n\\forall t\\in [0,\\tau],\\  v_i(t) \\geq \\frac{v_{\\min}}{2} + \\frac{bS_{\\min}t}{2}\\,.\\label{eq:vlin}\n\\end{equation}\nSet $\\kappa_1\\eqdef 0.5(v_{\\min} + bS_{\\min}\\tau)$. Note that $v_i(\\tau)\\geq \\kappa_1$.\nDefine $S_{\\min}'\\eqdef\\inf\\{S_i(x):i\\in \\{1,\\dots,d\\}, (x,m,v)\\in K'\\}\\,.$\nNote that $S_{\\min}'>0$ by Assumptions~\\ref{hyp:S} and \\ref{hyp:S>0}.\nFinally, define $\\kappa = 0.5\\min(\\kappa_1,S_{\\min}')$.\nBy contradiction, assume that the set $\\{t\\geq \\tau : v_i(t)<\\kappa\\}$ is non-empty, and\ndenote by $\\tau'$ its infimum. It is clear that $\\tau'>\\tau$ and\n$v_i(\\tau')=\\kappa$. Thus, $b^{-1}\\dot v_i(\\tau') =S_i(x(\\tau'))-\\kappa$.\nWe obtain that $b^{-1}\\dot v_i(\\tau') \\geq 0.5S_{\\min}'>0$.\nAs a consequence, there exists $t\\in (\\tau,\\tau')$ s.t. $v_i(t)<v_i(\\tau')$. This contradicts\nthe definition of $\\tau'$. We have shown that for all $t\\geq \\tau$, $v_i(t)\\geq \\kappa$. Putting this together\nwith Eq.~(\\ref{eq:vlin}) and using that $\\kappa\\leq v_{\\min} + bS_{\\min}\\tau$,\nwe conclude that:\n$\\forall t\\geq 0,\\ v_i(t)\\geq \\min\\left(\\kappa\\,, \\frac{v_{\\min}}{2} + \\frac{bS_{\\min}t}{2}\\right)\\,.$\nSetting $c\\eqdef \\min(\\kappa,bS_{\\min}\/2)$, the result follows.\n\nWe prove the second point.  By Prop.~\\ref{prop:adam-bounded}, there exists a compact set $K\\subset \\cZ_+$ s.t.\nfor every $\\eta\\geq 0$, every $z\\in Z_\\infty^\\eta(x_0)$ of the form $z(t) = (x(t),m(t),v(t))$\nsatisfies\n$\\{(x(t),\\hat m(t),\\hat v(t)):t\\geq 0\\}\\subset K$,\nwhere $\\hat m(t) = m(t)\/(1-e^{-a(t+h)})$ and $\\hat v(t) = v(t)\/(1-e^{-b(t+h)})$.\nDenote by $L_S$ the Lipschitz constant of $S$ on the set $\\{x:(x,m,v)\\in K\\}$.\nIntroduce the constants $M_1\\eqdef \\sup\\{\\|m\/(\\varepsilon + \\sqrt v)\\|_\\infty:(x,m,v)\\in K\\}$,\n$M_2\\eqdef \\sup\\{\\|S(x)\\|_\\infty:(x,m,v)\\in K'\\}$. These constants being introduced,\nthe rest of the proof follows the same line as the proof of the first point.\n\\end{proof}\n\n\\subsubsection{Existence}\n\\label{sec:existence-adam}\n\n\\begin{corollary}\n\\label{coro:existence}\nLet Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\nAssume that $0< b\\leq 4a$.\nFor every $z_0\\in \\cZ_+$,$Z_{\\infty}^\\infty(z_0)\\neq\\emptyset$.\nFor every $(z_0,\\eta)\\in \\cZ_0\\times (0,+\\infty)$,$Z_{\\infty}^\\eta(z_0)\\neq\\emptyset$.\n\\end{corollary}\n\\begin{proof}\nWe prove the first point (the proof of the second point follows the same line).\nUnder Assumptions~\\ref{hyp:F} and \\ref{hyp:S}, $h_\\infty$ is continuous.\nTherefore, Cauchy-Peano's theorem guarantees the existence of a solution to the (\\ref{eq:ode}) issued from $z_0$,\nwhich we can extend over a maximal interval of existence $[0,T_{\\max})$\nWe conclude that the solution is global ($T_{\\max} = +\\infty$) using the boundedness of the solution given by Prop.~\\ref{prop:bounded}.\n\\end{proof}\n\\begin{lemma}\n\\label{lem:equicont-eta}\n \n  Let Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\n  Assume that $0< b\\leq 4a$.\nConsider $z_0\\in \\cZ_0$. Denote by $(z_\\eta:\\eta\\in (0,+\\infty))$ a family of functions on $[0,+\\infty)\\to \\cZ_+$\ns.t. for every $\\eta>0$, $z_\\eta\\in Z_\\infty^\\eta(z_0)$.\nThen,  $(z_\\eta)_{\\eta>0}$ is equicontinuous.\n\\end{lemma}\n\\begin{proof}\nFor every such solution $z_\\eta$, we set $z_\\eta(t)=(x_\\eta(t),m_\\eta(t),v_\\eta(t))$ for all $t\\geq 0$,\nand define $\\hat m_\\eta$ and $\\hat v_\\eta$ as in Prop.~\\ref{prop:adam-bounded}.\nBy Prop.~\\ref{prop:adam-bounded}, there exists a constant $M_1$ s.t. for all $\\eta>0$ and all $t\\geq 0$,\n$\\max(\\|x_\\eta(t)\\|,\\|\\hat m_\\eta(t)\\|_\\infty,\\|\\hat v_\\eta(t)\\|)\\leq M_1$.\nUsing the continuity of $\\nabla F$ and $S$, there exists an other finite constant $M_2$ s.t.\n$M_2\\geq \\sup\\{\\|\\nabla F(x)\\|_\\infty:x\\in \\bR^d, \\|x\\|\\leq M_1\\}$ and\n$M_2\\geq \\sup\\{\\|S(x)\\|_\\infty:x\\in \\bR^d, \\|x\\|\\leq M_1\\}$.\nFor every $(s,t)\\in [0,+\\infty)^2$,  we have for all $i\\in \\{1,\\dots,d\\}$,\n$|x_{\\eta,i}(t)-x_{\\eta,i}(s)|\\leq \\int_s^t\\left|\\frac{\\hat m_{\\eta,i}(u)}{\\varepsilon + \\sqrt{\\hat v_{\\eta,i}(u)}}\\right|du\\,\\leq \\frac{M_1}\\varepsilon |t-s|$,\nand similarly $|m_{\\eta,i}(t)-m_{\\eta,i}(s)| \\leq a(M_1+M_2)|t-s|$,\n$|v_{\\eta,i}(t)-v_{\\eta,i}(s)|\\leq b(M_1+M_2)|t-s|$\\,.\nTherefore, there exists a constant $M_3$, independent from $\\eta$, s.t. for all $\\eta>0$ and all $(s,t)\\in [0,+\\infty)^2$,\n$\\|z_\\eta(t)-z_\\eta(s)\\|\\leq M_3 |t-s|$\n\\end{proof}\n\n\n\\begin{proposition}\n  \\label{prop:existence-adam}\n \n  Let Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\n  Assume that $0< b\\leq 4a$.\nFor every  $z_0\\in \\cZ_0$, $Z_\\infty^0(z_0)\\neq \\emptyset$ \\emph{i.e.},\n(\\ref{eq:ode}) admits a global solution issued from $z_0$.\n\\end{proposition}\n\\begin{proof}\n  By Cor.~\\ref{coro:existence}, there exists  a family $(z_\\eta)_{\\eta>0}$ of functions on $[0,+\\infty)\\to \\cZ$\ns.t. for every $\\eta>0$, $z_\\eta\\in Z^\\eta_\\infty(z_0)$.\nWe set as usual $z_\\eta(t)=(x_\\eta(t),m_\\eta(t),v_\\eta(t))$. By Lemma~\\ref{lem:equicont-eta},\nand the Arzel\u00e0-Ascoli theorem, there exists a map $z:[0,+\\infty)\\to \\cZ$ and a sequence $\\eta_n\\downarrow 0$ s.t.\n$z_{\\eta_n}$ converges to $z$ uniformly on compact sets, as $n\\to\\infty$. Considering some fixed scalars $t>s> 0$,\n$z(t) = z(s) + \\lim_{n\\to\\infty}\\int_s^t h(u+\\eta_n, z_{\\eta_n}(u))du\\,.$\nBy Prop.~\\ref{prop:adam-bounded}, there exists a compact set $K\\subset \\cZ_+$ s.t.\n$\\{z_{\\eta_n}(t):t\\geq 0\\}\\subset K$ for all $n$.\nMoreover, by Lemma~\\ref{lem:v-lowerbound}, there exists a constant $c>0$ s.t.\nfor all $n$ and all $u\\geq 0$, $v_{\\eta_n,k}(u)\\geq c \\min(1,u)$.\nDenote by $\\bar K\\eqdef K\\cap (\\bR^d\\times\\bR^d\\times [c\\min(1,s),+\\infty)^d)$.\nIt is clear that $\\bar K$ is a compact subset of $\\cZ_+^*$.\nSince $h$ is continuously differentiable on the set\n$[s,t]\\times \\bar K$, it is Lipschitz continuous on that set. Denote by $L_h$ the corresponding\nLipschitz constant. We obtain:\n$$\n\\int_s^t\\|h(u+\\eta_n, z_{\\eta_n}(u)) - h(u, z(u))\\|du \\leq L_h\\left(\\eta_n + \\sup_{u\\in [s,t]}\\|z_{\\eta_n}(u)-z(u)\\|\\right)(t-s)\\,,\n$$\nand the righthand side converges to zero. As a consequence, for all $t>s$,\n$\nz(t) = z(s) + \\int_s^t h(u, z(u))du\\,.\n$ Moreover, $z(0)=z_0$. This proves that $z\\in Z^0_\\infty(z_0)$.\n\\end{proof}\n\n\\subsubsection{Uniqueness}\n\n\\begin{proposition}\n  \\label{prop:unique-adam}\nLet Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\nAssume  $b\\leq 4a$.\nFor every $z_0\\in\\cZ_0$, $Z_\\infty^0(z_0)$ is a singleton\n\\emph{i.e.}, there exists a unique global solution to (\\ref{eq:ode})\nwith initial condition $z_0$.\n\\end{proposition}\n\\begin{proof}\n \n  Consider solutions $z$ and $z'$ in $Z^0_\\infty(z_0)$.  We denote by $(x(t),m(t),v(t))$ the blocks of $z(t)$,\n  and we define $(x'(t),m'(t),v'(t))$ similarly.  For all $t>0$, we\n  define $\\hat m(t)\\eqdef m(t)\/(1-e^{-at})$, $\\hat v(t)\\eqdef\n  v(t)\/(1-e^{-bt})$, and we define $\\hat m'(t)$ and $\\hat v'(t)$\n  similarly.  By Prop.~\\ref{prop:adam-bounded}, there exists a compact\n  set $K\\subset \\cZ_+$ s.t.  $ (x(t),\\hat m(t),\\hat v(t))$ and\n  $(x'(t),\\hat m'(t),\\hat v'(t))$ are both in $K$ for all $t> 0$.  We\n  denote by $L_S$ and $L_{\\nabla F}$ the Lipschitz constants of $S$\n  and $\\nabla F$ on the compact set $\\{x:(x,m,v)\\in K\\}$. These\n  constants are finite by Assumptions~\\ref{hyp:F}\n  and \\ref{hyp:S}.\nWe define $M\\eqdef \\sup\\{\\|m\\|_\\infty:(x,m,v)\\in K\\}$.\nDefine $u_x(t) \\eqdef \\|x(t)-x'(t)\\|^2$,\n$u_m(t)\\eqdef \\|\\hat m(t)-\\hat m'(t)\\|^2$ and $u_v(t)\\eqdef \\|\\hat v(t)-\\hat v'(t)\\|^2$.\nLet $\\delta>0$. Define: $u^{(\\delta)}(t) \\eqdef u_x(t)+\\delta u_m(t)+\\delta u_v(t)\\,.$\nBy the chain rule and the Cauchy-Schwarz inequality,\n$\\dot u_x(t)\\leq 2\\|x(t)-x'(t)\\|\\|\\frac{\\hat m(t)}{\\varepsilon +\\sqrt{\\hat v(t)}}-\\frac{\\hat m'(t)}{\\varepsilon +\\sqrt{\\hat v'(t)}}\\|$. Thus,\nusing Lemma~\\ref{lem:v-lowerbound}, there exists $c>0$ s.t.\n\\begin{equation*}\n  \\dot u_x(t)\\leq 2\\|x(t)-x'(t)\\|\\left(\\varepsilon^{-1}\\left\\|\\hat m(t)-\\hat m'(t)\\right\\|\n+\\frac M{2\\varepsilon^2\\sqrt{c\\min(1,t)}}\\left\\|{\\hat v(t)}-{\\hat v'(t)}\\right\\|\\right)\\,.\n\\end{equation*}\nFor any $\\delta>0$,\n$2\\|x(t)-x'(t)\\|\\,\\|\\hat m(t)-\\hat m'(t)\\|\\leq \\delta^{-1\/2}(u_x(t)+\\delta u_m(t))\\leq \\delta^{-1\/2}u^{(\\delta)}(t)$.\nSimilarly, $2\\|x(t)-x'(t)\\|\\,\\|\\hat v(t)-\\hat v'(t)\\|\\leq \\delta^{-1\/2}u^{(\\delta)}(t)$.\nThus, for any $\\delta>0$,\n\\begin{align}\n\\label{eq:ux}\n  \\dot u_x(t)&\\leq \\left(\\frac 1{\\varepsilon\\sqrt \\delta}+\\frac M{2\\varepsilon^2\\sqrt{\\delta c\\min(1,t)}}\\right) u^{(\\delta)}(t)\\,.\n\\end{align}\nWe now study $u_m(t)$. For all $t>0$, we obtain after some algebra:\n$\\frac {d}{dt}\\hat m(t) = a(\\nabla F(x(t)) - \\hat m(t))\/(1-e^{-at})\\,.$\nTherefore,\n$\n\\dot u_m(t) \\leq \\frac{2aL_{\\nabla F}}{1-e^{-at}}\\|\\hat m(t)-\\hat m'(t)\\|\\,\\|x(t) -x'(t)\\|\\,.\n$\nFor any $\\theta>0$, it holds that $2\\|\\hat m(t)-\\hat m'(t)\\|\\,\\|x(t) -x'(t)\\|\\leq \\theta u_x(t)+ \\theta^{-1}u_m(t)$.\nIn particular, letting $\\theta\\eqdef 2L_{\\nabla F}$, we obtain that for all $\\delta>0$,\n\\begin{equation}\n\\resizebox{\\hsize}{!}{$\n\\delta \\dot u_m(t)\\leq  \\frac{a }{2(1-e^{-at})}\\left(4\\delta L_{\\nabla F}^2 u_x(t)+ \\delta u_m(t)\\right)\n                    \\leq  \\left(\\frac a2+\\frac 1{2t}\\right)\\left(4\\delta L_{\\nabla F}^2 u_x(t)+ \\delta u_m(t)\\right)\\,,\n$}\n\\label{eq:um}\n\\end{equation}\nwhere the last inequality is due to the fact that $y\/(1-e^{-y})\\leq 1+y$ for all $y>0$.\nUsing the exact same arguments, we also obtain that\n\\begin{align}\n \\delta \\dot u_v(t)&\\leq  \\left(\\frac b2+\\frac 1{2t}\\right)\\left(4\\delta L_{S}^2 u_x(t)+ \\delta u_m(t)\\right)\\,.\n\\label{eq:uv}\n\\end{align}\nWe now choose any $\\delta$ s.t. $4\\delta \\leq  1\/\\max(L_S^2,L_{\\nabla F}^2)$.\nThen, Eq.~(\\ref{eq:um}) and~(\\ref{eq:uv}) respectively imply that\n$\\delta \\dot u_m(t)\\leq 0.5(a+t^{-1})u^{(\\delta)}(t)$ and\n$\\delta \\dot u_v(t)\\leq 0.5(b+t^{-1})u^{(\\delta)}(t)$.\nSumming these inequalities along with Eq.~(\\ref{eq:ux}), we obtain that for every $t>0$,\n$\\dot u^{(\\delta)}(t) \\leq \\psi(t) u^{(\\delta)}(t)\\,$,\nwhere: $\\psi(t) \\eqdef \\frac{a+b}2+\\frac 1{\\varepsilon\\sqrt \\delta}+\\frac M{2\\varepsilon^2\\sqrt{\\delta c\\min(1,t)}}\n+ \\frac 1t\\,.$\nFrom Gr\\\"onwall's inequality, it holds that for every $t>s>0$,\n$u^{(\\delta)}(t)\\leq u^{(\\delta)}(s)\\exp\\left(\\int_s^t \\psi(s')ds'\\right)\\,$.\nWe first consider the case where $t\\leq 1$. We set $c_1\\eqdef  (a+b)\/2+(\\varepsilon\\sqrt \\delta)^{-1}$\nand $c_2\\eqdef M\/(\\varepsilon^2\\sqrt{\\delta c})$. With these notations,\n$\\int_s^t \\psi(s')ds' \\leq c_1t+c_2\\sqrt t + \\ln \\frac ts\\,.$\nTherefore, $u^{(\\delta)}(t)\\leq \\frac{u^{(\\delta)}(s)}{s}\n\\exp\\left(c_1t+c_2\\sqrt t + \\ln t\\right)\\,$.\nBy Lemma~\\ref{lem:m-v-derivables-en-zero}, recall that $\\dot x(0)$ and\n$\\dot x'(0)$ are both well defined (and coincide). Thus,\n$$\nu_x(s) = \\|x(s)-x'(s)\\|^2\n\\leq  2\\|x(s)-x(0)-\\dot x(0)s\\|^2+2\\|x'(s)-x'(0)-\\dot x'(0)s\\|^2\\,.\n$$\nIt follows that $u_x(s)\/s^2$ converges to zero as $s\\downarrow 0$.\nWe now show the same kind of result for $u_m(s)$ and $u_v(s)$.\nConsider $i\\in \\{1,\\dots,d\\}$. By the mean value theorem, there exists $\\tilde s$ (resp. $\\tilde s'$) in\n$[0,t]$ s.t. $m_i(s)=\\dot m_i(\\tilde s)s$ (resp. $m_i'(s)=\\dot m_i'(\\tilde s')s$).\nThus, $\\hat m_i(s) = \\frac{as}{1-e^{-as}} \\left(\\partial_i F(x(\\tilde s))-m_i(\\tilde s)\\right)$,\nand a similar equality holds for $\\hat m_i'(s)$. Then,\ngiven that $\\|x(\\tilde s)-x'(\\tilde s')\\| \\vee \\|m(\\tilde s)-m'(\\tilde s')\\| \\leq \\|z(\\tilde s)-z'(\\tilde s')\\|$\n, $\\tilde s\\leq s$ and $\\tilde s'\\leq s$,\n$$\n\\frac{|\\hat m_i(s) -\\hat m_i'(s) |}s\n\\leq \\frac{2a(L_{\\nabla F}\\vee 1)s}{1-e^{-as}} \\left(\\frac{\\|z(\\tilde s)-z(0)\\|}{\\tilde s}+\\frac{\\|z'(\\tilde s')-z'(0)\\|}{\\tilde s'}\\right)\\,.\n$$\nBy Lemma~\\ref{lem:m-v-derivables-en-zero}, $z$ and $z'$ are differentiable at point zero.\nThen, the above inequality gives $\\limsup_{s\\downarrow 0}\\frac{|\\hat m_i(s) -\\hat m_i'(s) |}s \\leq 4(L_{\\nabla F}\\vee 1)\\|\\dot z(0)\\|$\nand\n\n\\limsup_{s\\downarrow 0}\\frac{u_m(s)}{s^2} \\leq 16d(L_{\\nabla F}^2\\vee 1)\\|\\dot z(0)\\|^2\\,.\n\nTherefore, $u_m(s)\/s$ converges to zero as $s\\downarrow 0$.\nBy similar arguments, it can be shown that\n$\\limsup_{s\\downarrow 0}{u_v(s)}\/{s^2} \\leq 16d(L_{S}^2\\vee 1)\\|\\dot z(0)\\|^2$,\nthus $\\lim u_v(s)\/s=0$.\nFinally, we obtain that\n${u^{(\\delta)}(s)}\/{s}$ converges to zero as $s\\downarrow 0$.\nLetting $s$ tend to zero, we obtain that for every\n$t\\leq 1$, $u^{(\\delta)}(t)=0$. Setting $s=1$ and $t>1$,\nand noting that $\\psi$ is integrable on $[1,t]$, it follows that $u^{(\\delta)}(t)=0$ for all $t>1$.\nThis proves that $z=z'$.\n\\end{proof}\n\nWe recall that a semiflow $\\Phi$ on the space $(E,\\sd)$ is a continuous map\n$\\Phi$ from $[0,+\\infty)\\times E$ to $E$ defined by $(t,x) \\mapsto \\Phi(t,x) = \\Phi_t(x)$\nsuch that $\\Phi_0$ is the identity and $\\Phi_{t+s} = \\Phi_t\\circ\\Phi_s$ for all $(t,s)\\in [0,+\\infty)^2$.\n\n\\begin{proposition}\n  \\label{prop:semiflow}\nLet Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\nAssume that $0< b\\leq 4a$.\nThe map $Z_{\\infty}^{\\infty}$ is single-valued on $\\cZ_+\\to C([0,+\\infty),\\cZ_+)$\n\\emph{i.e.}, there exists a unique global solution to~(\\ref{eq:ode-a})\nstarting from any given point in $\\cZ_+$.\nMoreover, the following map is a semiflow:\n\\begin{equation}\n  \\label{eq:flot}\n  \\begin{array}[h]{rcl}\n    \\Phi:[0,+\\infty)\\times \\cZ_+&\\to& \\cZ_+ \\\\\n(t,z) &\\mapsto& Z_{\\infty}^{\\infty}(z)(t)\n  \\end{array}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe result is a direct consequence of Lemma~\\ref{prop:unique-adam}.\n\\end{proof}\n\n\\subsection{Proof of Th.~\\ref{th:cv-adam}}\n\\label{sec:convergence}\n\\subsubsection{Convergence of the semiflow}\n\nWe first recall some useful definitions and results.\nLet $\\Psi$ represent any semiflow on an arbitrary metric space $(E,\\sd)$.\nA point $z\\in E$ is called an \\emph{equilibrium point} of the semiflow $\\Psi$ if $\\Psi_t(z)=z$ for all $t\\geq 0$.\nWe denote by $\\Lambda_\\Psi$ the set of equilibrium points of~$\\Psi$.\nA continuous function $\\sV:E\\to\\bR$ is called a \\emph{Lyapunov function} for the semiflow $\\Psi$\nif $\\sV(\\Psi_t(z))\\leq \\sV(z)$ for all $z\\in E$ and all $t\\geq 0$.\nIt is called a \\emph{strict Lyapunov function} if, moreover,\n$\n\\{ z\\in E\\,:\\, \\forall t\\geq 0,\\,\\sV(\\Psi_t(z))=\\sV(z) \\}= \\Lambda_\\Psi\n$.\nIf $\\sV$ is a strict Lyapunov function for $\\Psi$ and if $z\\in E$ is a point s.t. $\\{\\Psi_t(z):t\\geq 0\\}$ is relatively compact,\nthen it holds that $\\Lambda_\\Psi\\neq \\emptyset$ and $\\sd(\\Psi_t(z),\\Lambda_\\Psi)\\to 0$, see \\cite[Th.~2.1.7]{haraux1991systemes}.\nA continuous function $z:[0,+\\infty)\\to E$ is said to be an asymptotic pseudotrajectory (APT)\nfor the semiflow $\\Psi$ if for every $T\\in (0,+\\infty)$,\n$\n\\lim_{t\\to+\\infty} \\sup_{s\\in [0,T]} \\sd(z(t+s),\\Psi_s(z(t))) = 0\\,.\n$\nThe following result follows from \\cite[Th.~5.7]{ben-(cours)99} and \\cite[Prop.~6.4]{ben-(cours)99}.\n\\begin{proposition}[\\cite{ben-(cours)99}]\\hfill\\\\\n\\label{prop:benaim}\n  Consider a semiflow $\\Psi$ on $(E,d)$ and a map $z:[0,+\\infty)\\to E$. Assume the following:\n  \\begin{enumerate}[{\\it i)}]\n  \\item $\\Psi$ admits a strict Lyapunov function $\\sV$.\n  \\item The set $\\Lambda_\\Psi$ of equilibrium points of $\\Psi$ is compact.\n  \\item  $\\sV(\\Lambda_\\Psi)$ has an empty interior.\n  \\item $z$ is an APT of $\\Psi$.\n  \\item $z([0,\\infty))$ is relatively compact.\n  \\end{enumerate}\nThen, $\n\\bigcap_{t\\geq 0}\\overline{z([t,\\infty))}$ is a compact connected subset of $\\Lambda_\\Psi$\\,.\n\\end{proposition}\n\nFor every $\\delta>0$ and every $z = (x,m,v)\\in \\cZ_+$, define:\n\\begin{equation}\nW_\\delta(x,m,v) \\eqdef V_{\\infty}(x,m,v) - \\delta \\ps{\\nabla F(x),m} + \\delta \\|S(x)-v\\|^2\\,,\\label{eq:Wdelta}\n\\end{equation}\nwhere we recall that $V_\\infty(z)\\eqdef \\lim_{t\\to \\infty} V(t,z)$ for every $z\\in \\cZ_+$ and $V$ is defined by Eq.(\\ref{eq:V}).\nConsider the set $\\cE\\eqdef h_\\infty^{-1}(\\{0\\})$ of all equilibrium points of (\\ref{eq:ode-a}), namely:\n$\\cE = \\{(x,m,v)\\in \\cZ_+:\\nabla F(x)=0,m=0,v=S(x)\\}\\,$.\nThe set $\\cE$ is non-empty by Assumption~\\ref{hyp:coercive}.\n\n\\begin{proposition}\n  \\label{prop:Wstrict}\nLet Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\nAssume that $0< b\\leq 4a$.\nLet $K\\subset \\cZ_+$ be a compact set. Define $K'\\eqdef \\overline{\\{\\flot(t,z):t\\geq 0, z\\in K\\}}$.\nLet $\\bflot:[0,+\\infty)\\times K'\\to K'$ be the restriction of the semiflow $\\flot$ to $K'$ \\emph{i.e.},\n$\\bflot(t,z) = \\flot(t,z)$ for all  $t\\geq 0, z\\in K'$. Then,\n\\begin{enumerate}[{\\it i)}]\n\\item $K'$ is compact.\n\\item $\\bflot$ is well defined and is a semiflow on $K'$.\n\\item The set of equilibrium points of $\\bflot$ is equal to $\\cE\\cap K'$.\n\\item There exists $\\delta>0$ s.t. $W_\\delta$ is a strict Lyapunov function for the semiflow $\\bflot$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nThe first point is a consequence of Prop.~\\ref{prop:bounded}.\nThe second point stems from Prop.~\\ref{prop:semiflow}.\nThe third point is immediate from the definition of $\\cE$ and the fact that $\\bflot$ is valued in $K'$.\nWe now prove the last point.\nConsider $z\\in K'$ and write $\\bflot_t(z)$ under the form $\\bflot_t(z) = (x(t),m(t),v(t))$. For \\emph{any} map ${\\mathsf W}:\\cZ_+\\to\\bR$, define\nfor all $t>0$, $ \\cL_{\\mathsf W}(t) \\eqdef \\limsup_{s\\to 0} s^{-1}({\\mathsf W}(\\bflot_{t+s}(z)) -   {\\mathsf W}(\\bflot_{t}(z)))\\,.$\nIntroduce $G(z)\\eqdef -\\ps{\\nabla F(x),m}$ and $H(z)\\eqdef \\|S(x)-v\\|^2$ for every $z=(x,m,v)$.\nConsider $\\delta>0$ (to be specified later on).\nWe study $\\cL_{W_\\delta} = \\cL_V + \\delta \\cL_{G} + \\delta \\cL_{H}$.\nNote that $\\bflot_t(z)\\in K'\\cap \\cZ_+^*$ for all $t>0$ by Lemma~\\ref{lem:v-positif}. Thus, $t\\mapsto V_\\infty(\\bflot_t(z))$\nis differentiable at any point $t>0$ and the derivative coincides with $\\cL_V(t) = \\dot V_\\infty(\\bflot_t(z))$.\nDefine $C_1\\eqdef \\sup\\{\\|v\\|_\\infty:(x,m,v)\\in K'\\}$.\nThen, by Lemma~\\ref{lem:V}, $\\cL_V(t)\\leq -\\varepsilon(\\varepsilon+\\sqrt{C_1})^{-2} \\left\\|m(t)\\right\\|^2$.\nLet $L_{\\nabla F}$ be the Lipschitz constant of $\\nabla F$\non $\\{x:(x,m,v)\\in K'\\}$.\nFor every $t>0$,\n\\begin{align*}\n  \\cL_G(t\n&\\leq  \\limsup_{s\\to 0} s^{-1}\\|\\nabla F(x(t))- \\nabla F(x(t+s))\\| \\|m(t+s)\\| - \\ps{\\nabla F(x(t)),\\dot m(t)}\\\\\n&\\leq  L_{\\nabla F}\\varepsilon^{-1}\\|m(t)\\|^2 - a\\|\\nabla F(x(t))\\|^2  + a\\ps{\\nabla F(x(t)),m(t)}\\\\\n&\\leq - \\frac a2\\|\\nabla F(x(t))\\|^2  +  \\left(\\frac a2+\\frac{L_{\\nabla F}}{\\varepsilon}\\right)\\|m(t)\\|^2\\,.\n\\end{align*}\nDenote by $L_S$ the Lipschitz constant of $S$ on $\\{x:(x,m,v)\\in K'\\}$.\nFor every $t>0$,\n\\begin{eqnarray*}\n  \\cL_H(t\n&=&  \\limsup_{s\\to 0} s^{-1}(\\|S(x(t+s))-S(x(t)) + S(x(t))-v(t+s)\\|^2-\\|S(x(t))-v(t)\\|^2)\\\\\n&=& - 2\\ps{S(x(t))-v(t),\\dot v(t)\n+\\limsup_{s\\to 0} 2s^{-1}\\ps{S(x(t+s))-S(x(t)),S(x(t))-v(t+s)} \\\\\n&\\leq&   - 2b\\|S(x(t))-v(t)\\|^2+2L_S\\varepsilon^{-1} \\|m(t)\\| \\|S(x(t))-v(t)\\|\\,.\n\\end{eqnarray*}\nUsing that $2 \\|m(t)\\| \\|S(x(t))-v(t)\\|\\leq \\frac {L_S}{b\\varepsilon}\\|m(t)\\|^2 + \\frac {b\\varepsilon}{L_S}\\|S(x(t))-v(t)\\|^2$, we obtain\n$\n\\cL_H(t) \\leq - b\\|S(x(t))-v(t)\\|^2+\\frac {L_S^2}{b\\varepsilon^2}\\|m(t)\\|^2\\,.\n$\nHence,\nfor every $t>0$,\n$$\n  \\cL_{W_\\delta}(t) \\leq -M(\\delta)\n  \\|m(t)\\|^2  - \\frac {a\\delta}2\\|\\nabla\n  F(x(t))\\|^2 - \\delta b \\|S(x(t))-v(t)\\|^2\\,.\n$$\nwhere $M(\\delta)\\eqdef \\varepsilon(\\varepsilon+\\sqrt{C_1})^{-2} -\\frac {\\delta\n      L_S^2}{b\\varepsilon^2} - \\delta \\left(\\frac a2+\\frac{L_{\\nabla\n        F}}{\\varepsilon}\\right)\\,.$\nChoosing $\\delta$ s.t. $M(\\delta)>0$,\n\\begin{equation}\n\\forall t>0,\\ \\  \\cL_{W_\\delta}(t) \\leq -c\\left( \\|m(t)\\|^2 + \\|\\nabla  F(x(t))\\|^2 + \\|S(x(t))-v(t)\\|^2\\right)\\,,\\label{eq:ae-derivee}\n\\end{equation}\nwhere $c \\eqdef \\min\\{ M(\\delta), \\frac {a\\delta}2, \\delta b\\}$.\nIt can easily be seen that for every $z\\in K'$,  $t\\mapsto W_\\delta(\\bflot_t(z))$ is Lipschitz continuous, hence absolutely continuous.\nIts derivative almost everywhere coincides with $\\cL_{W_\\delta}$, which is non-positive.\nThus, $W_\\delta$ is a Lyapunov function for $\\bflot$.\nWe prove that the Lyapunov function is strict.\nConsider $z\\in K'$ s.t. $W_\\delta(\\bflot_t(z))=W_\\delta(z)$ for all $t>0$.\nThe derivative almost everywhere of  $t\\mapsto W_\\delta(\\bflot_t(z))$ is identically zero,\nand by Eq. (\\ref{eq:ae-derivee}), this implies that\n $-c\\left(\\|m_t\\|^2 + \\|\\nabla F(x_t)\\|^2 +\\|S(x_t)-v_t\\|^2\\right)$ is equal to zero\nfor every $t$ a.e. (hence, for every~$t$, by continuity of $\\bflot$).\nIn particular for $t=0$, $m=\\nabla F(x)=0$ and $S(x)-v=0$.\nHence, $z\\in h_\\infty^{-1}(\\{0\\})$.\n\\end{proof}\n\n\\begin{corollary}\n  \\label{coro:cv}\nLet Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\nAssume that $0< b\\leq 4a$.\nFor every $z\\in \\cZ_+$, $\\lim_{t\\to \\infty} \\sd(\\Phi(z,t),\\cE)=0\\,.$\n\\end{corollary}\n\\begin{proof}\nUse Prop.~\\ref{prop:Wstrict} with $K\\eqdef\\{z\\}$.\nand \\cite[Th.~2.1.7]{haraux1991systemes}.\n\\end{proof}\n\n\n\\subsubsection{Asymptotic Behavior of the Solution to (\\ref{eq:ode})}\n\\label{sec:cv-non-autonomous}\n\n\n\\begin{proposition}[APT]\n \n  Let Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold true.\n  Assume that $0< b\\leq 4a$.\nThen, for every $z_0\\in \\cZ_0$, $Z^0_\\infty(z_0)$ is an asymptotic pseudotrajectory\nof the semiflow  $\\Phi$ given by~(\\ref{eq:flot}).\n\\label{prop:apt}\n\\end{proposition}\n\\begin{proof}\n  Consider  $z_0\\in \\cZ_0$, $T\\in (0,+\\infty)$ and define $z\\eqdef Z_\\infty^0(z_0)$.\nConsider $t\\geq 1$. For every $s\\geq 0$, define $\\Delta_t(s)\\eqdef  \\|z(t+s)-\\flot(z(t))(s)\\|$.\nThe aim is to prove that $\\sup_{s\\in [0,T]}\\Delta_t(s)$ tends to zero as~$t\\to\\infty$.\nPutting together Prop.~\\ref{prop:adam-bounded} and Lemma~\\ref{lem:v-lowerbound},\nthe set $K\\eqdef \\overline{\\{z(t):t\\geq 1\\}}$\nis a compact subset of $\\cZ_+^*$.\nDefine $C(t)\\eqdef \\sup_{s\\geq 0}\\sup_{z'\\in K}\\|h(t+s,z')-h_\\infty(z')\\|$.\nIt can be shown that $\\lim_{t\\to\\infty} C(t)=0$.\nWe obtain that for every $s\\in [0,T]$, $\n\\Delta_t(s)\\leq T C(t)  + \\int_0^s\n  \\|h_\\infty(z(t+s'))-h_{\\infty}(\\flot(z(t))(s'))\\|ds'\\,.$\nBy Lemma~\\ref{lem:v-lowerbound}, $K'\\eqdef \\overline{\\bigcup_{z'\\in\\flot(K)}z'([0,\\infty))}$ is a compact subset of $\\cZ_+^*$.\nIt is immediately seen from the definition that $h_{\\infty}$ is Lipschitz continuous on every compact subset of $\\cZ_+^*$, hence on $K\\cup K'$. Therefore, there exists a constant $L$, independent from $t,s$, s.t.\n$\n\\Delta_t(s)\\leq T C(t)  + \\int_0^s L \\Delta_t(s')ds'\\qquad (\\forall t\\geq 1, \\forall s\\in[0,T])\\,.\n$\nUsing Gr\\\"onwall's lemma, it holds that for all $s\\in [0,T]$,\n$\n\\Delta_t(s)\\leq TC(t) e^{Ls}\\,.\n$\nAs a consequence,\n$\\sup_{s\\in [0,T]}\\Delta_t(s)\\leq TC(t) e^{LT}$ and the righthand side converges to zero\nas $t\\to\\infty$.\n\\end{proof}\n\n\\subsubsection*{End of the Proof of Th.~\\ref{th:cv-adam}}\n\nBy Prop.~\\ref{prop:adam-bounded}, the set $K\\eqdef \\overline{Z^0_\\infty(z_0)([0,\\infty))}$ is a compact subset of $\\cZ_+$.\nDefine $K'\\eqdef \\overline{\\{\\flot(t,z):t\\geq 0, z\\in K\\}}\\,,$ and let\n $\\bflot:[0,+\\infty)\\times K'\\to K'$ be the restriction $\\flot$ to $K'$.\nBy Prop.~\\ref{prop:Wstrict}, there exists $\\delta>0$ s.t.\n$W_\\delta$ is  a strict Lyapunov function for the semiflow $\\bflot$.\nMoreover, the set of equilibrium points coincides with $\\cE\\cap K'$.\nIn particular, the equilibrium points of $\\bflot$ form a compact set.\nBy Prop.~\\ref{prop:apt}, $Z^0_\\infty(z_0)$ is an APT of $\\bflot$.\nNote that every $z\\in\\cE$ can be written under the form\n$z=(x,0,S(x))$ for some $x\\in \\cS$.\nFrom the definition of $W_\\delta$ in (\\ref{eq:Wdelta}),\n$W_\\delta(z)=W_\\delta(x,0,S(x)) =V_{\\infty}(x,0,S(x)) = F(x)$.\nSince $F(\\cS)$ is assumed to have an empty interior, the same holds\nfor $W_\\delta(\\cE\\cap K')$. By Prop.~\\ref{prop:benaim},\n$\n\\bigcap_{t\\geq 0}\\overline{Z^0_\\infty(z_0)([t,\\infty))} \\subset \\cE\\cap K'\\,.\n$\nThe set in the righthand side coincides with the set of limits of convergent\nsequences of the form $Z^0_\\infty(z_0)(t_n)$ for $t_n\\to\\infty$.\nAs $Z^0_\\infty(z_0)([0,\\infty))$ is a bounded set, $\\sd(Z^0_\\infty(z_0)(t),\\cE)$ tends to zero.\n\n\n\\subsection{Proof of Th.~\\ref{thm:asymptotic_rates}}\n\\label{sec:cont_asymptotic_rates}\n\nThe proof follows the path of \\cite[Th.~10.1.6, Th.~10.2.3]{harauxjendoubi2015},\nbut requires specific adaptations to deal with the dynamical system at hand.\nDefine for all $\\delta>0$, $t>0$, and $z=(x,m,v)$,\n\\begin{equation}\n\\tilde W_\\delta(t,(x,m,v)) \\eqdef V(t,(x,m,v)) - \\delta \\ps{\\nabla F(x),m} + \\delta \\|S(x)-v\\|^2\\,.\\label{eq:Wdelta-t}\n\\end{equation}\nThe function $\\tilde W_\\delta$ is the non-autonomous version of the function (\\ref{eq:Wdelta}).\nConsider a fixed $x_0\\in \\bR^d$, and define $w_{\\delta}(t)\\eqdef \\tilde W_\\delta(t,z(t))$\nwhere $z(t)=(x(t),m(t),v(t))$ is the solution to~(\\ref{eq:ode}) with initial condition $(x_0,0,0)$.\nThe proof uses the following steps.\n\n\\begin{enumerate}[{\\sl i)},leftmargin=11pt]\n\\item \\textit{Upper-bound on  $w_\\delta(t)$.}\nFrom Eq.~(\\ref{eq:V}),\nwe obtain that for every $t\\geq 1$,\n\nV(t,z(t))\\leq |F(x(t))|+\\frac {\\|m(t)\\|^2}{2a\\varepsilon (1-e^{-a})}\\,.\n\nUsing $\\ps{\\nabla F(x),m}\\leq (\\|\\nabla F(x)\\|^2+\\|m\\|^2)\/2$, we obtain that there exists a constant\n$c_1$ (depending on $\\delta$) s.t. for every $t\\geq 1$,\n\\begin{equation}\n  \\label{eq:w-up}\n  w_\\delta(t) \\leq c_1\\left(|F(x(t))|+\\|m(t)\\|^2+\\|\\nabla F(x(t))\\|^2+ \\|S(x(t))-v(t)\\|^2\\right)\\,.\n\\end{equation}\n\n\\item \\textit{Upper-bound on  $\\frac d{dt} w_\\delta(t)$.}\nThe function $w_{\\delta}$ is absolutely continuous on $[1,+\\infty)$.\n   \n   \n   \n   \n   \n   \nMoreover, there exist $\\delta>0$, $c_2>0$ (both depending on\n$x_0$) s.t. for every $t\\geq 1$ a.e.,\n\\begin{equation}\n  \\label{eq:dw-up}\n  \\frac d{dt} w_\\delta (t) \\leq -c_2\\left(\\|m(t)\\|^2+\\|\\nabla F(x(t))\\|^2+\\|S(x(t))-v(t)\\|^2\\right)\\,.\n\\end{equation}\nThe proof of Eq.~(\\ref{eq:dw-up}) uses arguments that are similar to\nthe ones used in the proof of Prop.~\\ref{prop:Wstrict} (just\nuse Lemma~\\ref{lem:V} to bound the derivative of the first term in\nEq.~(\\ref{eq:Wdelta-t})).  For this reason, it is omitted.\n\\item \\textit{Positivity of $w_\\delta(t)$.} By Lemma~\\ref{lem:V},\nthe function $t\\mapsto V(t,z(t))$ is decreasing. As it is lower bounded,\n$\\ell\\eqdef \\lim_{t\\to\\infty}V(t,z(t))$ exists. By Th.~\\ref{th:cv-adam},\n$m(t)$ tends to zero, hence this limit coincides with $\\lim_{t\\to\\infty} F(x(t))$.\nReplacing $F$ with $F-\\ell$, one can assume without loss of generality that $\\ell=0$.\nBy Eq.~(\\ref{eq:dw-up}), $w_\\delta$ is non-increasing on $[1,+\\infty)$, hence converging\nto some limit. Using again Th.~\\ref{th:cv-adam}, $\\ps{\\nabla F(x(t)),m(t)}\\to 0$\nand $S(x(t))-v(t)\\to 0$. Thus, $\\lim_{t\\to\\infty} w_\\delta(t) = \\ell = 0$.\nAssume that there exists $t_0\\geq 1$ s.t. $w_\\delta(t_0)=0$. Then, $w_\\delta$\nis constant on $[t_0,+\\infty)$. By Eq.~(\\ref{eq:dw-up}), this implies that $m(t)=0$ on this interval.\nHence, $d x(t)\/dt = 0$. This means that $x(t) = x(t_0)$ for all $t\\geq t_0$. By Th.~\\ref{th:cv-adam},\nit follows that $x(t_0)\\in \\cS$. In that case, the final result is shown. Therefore, one\ncan assume that $w_\\delta(t)>0$ for all $t\\geq 1$.\n\n\\item \\textit{Putting together (\\ref{eq:w-up}) and (\\ref{eq:dw-up}) using the \\L{}ojasiewicz condition.}\nBy Prop. \\ref{prop:benaim} and \\ref{prop:apt}, the set\n$\nL\\eqdef \\overline{\\bigcup_{s\\geq 0}\\{z(t):t\\geq s\\}}\\,\n$\nis a compact connected subset of $\\cE = \\{(x,0,S(x)):\\nabla F(x)=0\\}$.\nThe set $\\mathcal{U}\\eqdef \\{x:(x,0,S(x))\\in L\\}$ is a compact and connected subset of $\\cS$.\nUsing Assumption~\\ref{hyp:lojasiewicz_prop} and \\cite[Lemma 2.1.6]{harauxjendoubi2015},\nthere exist constants $\\sigma,c>0$ and $\\theta\\in (0,\\frac 12]$, s.t.\n$\\|\\nabla F(x)\\| \\geq c|F(x)|^{1-\\theta}\\,$ for all $x$ s.t.~$\\sd(x,\\mathcal{U})<\\sigma\\,$.\nAs $\\sd(x(t), \\mathcal{U})\\to 0$, there exists $T\\geq 1$ s.t. for all $t\\geq T$,\n$\\|\\nabla F(x(t))\\|\\geq c |F(x(t))|^{1-\\theta}$. Thus, we may replace the term $\\|\\nabla F(x(t))\\|^2$\nin the righthand side of Eq.~(\\ref{eq:dw-up}) using\n$\\|\\nabla F(x(t))\\|^2\\geq \\frac 12\\|\\nabla F(x(t))\\|^2+ \\frac 12|F(x(t))|^{2(1-\\theta)}$.\nUpon noting that $2(1-\\theta)\\geq 1$,\nwe thus obtain that there exists a constant $c_3$ and some $T'\\geq 1$ s.t. for $t\\geq T'$ a.e.,\n$$\n\\frac d{dt} w_\\delta (t) \\leq -c_3\\left(\\|m(t)\\|^2+\\|\\nabla F(x(t))\\|^2+|F(x(t))|+\\|S(x(t))-v(t)\\|^2\\right)^{2(1-\\theta)}\\,.\n$$\nPutting this inequality together with Eq.~(\\ref{eq:w-up}), we obtain that for some constant $c_4>0$ and for all\n$t\\geq T'$ a.e.,\n$\n\\frac d{dt} w_\\delta (t) \\leq -c_4 w_\\delta(t)^{2(1-\\theta)}\\,.\n$\n\\item \\textit{End of the proof.} Following the arguments of \\cite[Th.~10.1.6]{harauxjendoubi2015},\nby integrating the preceding inequality,\nover $[T',t]$, we obtain\n$\nw_\\delta(t) \\leq c_5 t^{-\\frac 1{1-2\\theta}\n$\nfor $t\\geq T'$\nin the case where $\\theta<\\frac 12$, whereas $w_\\delta(t)$ decays exponentially if $\\theta=\\frac 12$.\nFrom now on, we focus on the case $\\theta<\\frac 12$.\nBy definition of~(\\ref{eq:ode}), $\\|\\dot{x}(t)\\|^2\\leq \\|m(t)\\|^2\/((1-e^{-a T'})^2\\varepsilon^2)$\nfor all $t\\geq T'$. Since Eq.~(\\ref{eq:dw-up}) implies $\\|m(t)\\|^2\\leq -\\dot w_\\delta(t)\/c_2$,\nwe deduce that there exists $c,c'>0$ s.t. for all $t\\geq T'$,\n\n \n  \\int_t^{2t} \\|\\dot{x}(s)\\|^2 ds \\leq c w_\\delta(t) \\leq c' t^{-\\frac 1{1-2\\theta}}\\,.\n\nApplying \\cite[Lemma 2.1.5]{harauxjendoubi2015}, it follows that\n$\\int_t^{\\infty} \\|\\dot{x}(s)\\|^2 ds  \\leq c  t^{-\\frac{\\theta}{1-2\\theta}}$\nfor some other constant $c$.\nTherefore $x^* \\eqdef \\lim_{t \\to +\\infty} x(t)$ exists by Cauchy's criterion and\nfor all $t\\geq T'$, $\\|x(t)-x^*\\| \\leq c  t^{-\\frac{\\theta}{1-2\\theta}}$\\,.\nFinally, since $x(t)\\to a$, we remark that, using the same arguments,\nthe global \\L{}ojasiewicz exponent $\\theta$ can be replaced by any \\L{}ojasiewicz exponent\nof $f$ at $x^*$.\nWhen $\\theta=\\frac 12$, the proof follows the same line.\n\\end{enumerate}\n\n\n\\section{Proofs of Section~\\ref{sec:discrete}}\n\\label{sec:proofs_sec_discrete}\n\\subsection{Proof of Th.~\\ref{th:weak-cv}}\n\nGiven an initial point $x_0\\in \\bR^d$ and a stepsize $\\gamma>0$,\nwe consider the iterates $z_n^{\\gamma}$ given by~(\\ref{eq:znT})\nand $z_0^\\gamma\\eqdef (x_0,0,0)$.\nFor every $n\\in \\bN^*$ and every $z\\in \\cZ_+$, we define\n\\begin{equation*}\nH_\\gamma(n,z,\\xi)\\eqdef \\gamma^{-1} (T_{\\gamma,\\bar\\alpha(\\gamma),\\bar\\beta(\\gamma)}(n,z,\\xi)-z)\\,.\n\\end{equation*}\nThus, $z_n^\\gamma = z_{n-1}^\\gamma+\\gamma H_\\gamma(n,z_{n-1}^\\gamma,\\xi_n)$ for every $n\\in\\bN^*$.\nFor every $n\\in\\bN^*$ and every $z\\in \\cZ$ of the form\n$z=(x,m,v)$, we define $e_\\gamma(n,z)\\eqdef (x,(1-\\bar\\alpha(\\gamma)^n)^{-1}m,(1-\\bar\\beta(\\gamma)^n)^{-1}v)$,\nand set $e_\\gamma(0,z)\\eqdef z$.\n\\begin{lemma}\n  \\label{lem:moment-H}\nLet Assumptions~\\ref{hyp:model}, \\ref{hyp:alpha-beta} and \\ref{hyp:moment-f} hold true.\nThere exists $\\bar \\gamma_0>0$ s.t. for every $R>0$, there exists $s>0$,\n\\begin{equation}\n\\sup\\left\\{\\bE\\left(\\left\\|H_\\gamma(n+1,z,\\xi)\\right\\|^{1+s}\\right):\\gamma\\in (0,\\bar \\gamma_0], n\\in \\bN, z\\in \\cZ_+\\,\\text{s.t.}\\,\\|e_\\gamma(n,z)\\|\\leq R\\right\\}<\\infty\\,.\n\\label{eq:UI}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\n\n\nLet $R>0$.\nWe denote by $(H_{\\gamma,\\sx},H_{\\gamma,\\sm},H_{\\gamma,\\sv})$ the block components of $H_\\gamma$.\nThere exists a constant $C_s$ depending only on $s$ s.t.\n$\\|H_\\gamma\\|^{1+s}\\leq C_s(\\|H_{\\gamma,\\sx}\\|^{1+s}+\\|H_{\\gamma,\\sm}\\|^{1+s}+\\|H_{\\gamma,\\sv}\\|^{1+s})$.\nHence, it is sufficient to prove that Eq.~(\\ref{eq:UI}) holds\nrespectively when replacing $H_\\gamma$ with each of $H_{\\gamma,\\sx},H_{\\gamma,\\sm},H_{\\gamma,\\sv}$.\nConsider $z=(x,m,v)$ in $\\cZ_+$. We write:\n$\\|H_{\\gamma,\\sx}(n+1,z,\\xi)\\| \\leq \\varepsilon^{-1}(\\|\\frac{m}{1-\\bar \\alpha(\\gamma)^n}\\|+\\|\\nabla f(x,\\xi)\\|)\\,.$\nThus, for every $z$ s.t. $\\|e_\\gamma(n,z)\\|\\leq R$, there exists a constant $C$ depending only on $\\varepsilon$, $R$ and $s$ s.t.\n$\\|H_{\\gamma,\\sx}(n+1,z,\\xi)\\|^{1+s} \\leq C(1+ \\|\\nabla f(x,\\xi)\\|^{1+s})$. By Assumption~\\ref{hyp:moment-f}, (\\ref{eq:UI}) holds for\n$H_{\\gamma,\\sx}$ instead of $H_\\gamma$. Similar arguments hold for $H_{\\gamma,\\sm}$ and $H_{\\gamma,\\sv}$\nupon noting that the functions $\\gamma\\mapsto(1-\\bar \\alpha(\\gamma))\/\\gamma$ and $\\gamma\\mapsto (1-\\bar \\beta(\\gamma))\/\\gamma$\nare bounded under Assumption~\\ref{hyp:alpha-beta}.\n\\end{proof}\n\nFor every $R>0$, and every arbitrary sequence $z=(z_n:n\\in\\bN)$ on $\\cZ_+$, we define\n$\\tau_R(z) \\eqdef \\inf\\{n\\in \\bN:\\|e_\\gamma(n,z_n)\\|>R\\}$ with the convention that $\\tau_R(z)=+\\infty$ when the set is empty.\nWe define the map $B_R:\\cZ_+^\\bN\\to\\cZ_+^\\bN$ given for any arbitrary sequence  $z=(z_n:n\\in\\bN)$ on $\\cZ_+$ by\n$B_R(z)(n) = z_n\\1_{n<\\tau_R(z)}+z_{\\tau_R(z)}\\1_{n\\geq\\tau_R(z)}$.\nWe define the random sequence $z^{\\gamma,R}\\eqdef B_R(z^\\gamma)$. Recall that a family $(X_i:i\\in I)$ of random variables on some Euclidean space\nis called \\emph{uniformly integrable} if $\\lim_{A\\to+\\infty} \\sup_{i\\in I}\\bE(\\|X_i\\|\\1_{\\|X_i\\|>A})=0$.\n\\begin{lemma}\n  \\label{lem:UI}\nLet Assumptions~\\ref{hyp:model}, \\ref{hyp:alpha-beta}, \\ref{hyp:moment-f} and \\ref{hyp:iid} hold true.\nThere exists $\\bar \\gamma_0>0$ s.t. for every $R>0$, the family of r.v.\n$(\\gamma^{-1}(z_{n+1}^{\\gamma,R} - z_n^{\\gamma,R}):n\\in \\bN,\\gamma\\in (0,\\bar \\gamma_0])$ is uniformly integrable.\n\\end{lemma}\n\\begin{proof}\nLet $R>0$.\nAs the event $\\{n<\\tau_R(z^\\gamma)\\}$ coincides with\n$\\bigcap_{k=0}^n\\{\\|e_\\gamma(k,z_k^\\gamma)\\|\\leq R\\}$, it holds that\nfor every $n\\in \\bN$,\n\\begin{equation*}\n  \\frac{z_{n+1}^{\\gamma,R} - z_n^{\\gamma,R}}\\gamma =\\frac{z_{n+1}^{\\gamma} - z_n^{\\gamma}}\\gamma \\1_{n<\\tau_R(z^\\gamma)}\n = H_\\gamma(n+1,z_n^\\gamma,\\xi_{n+1}) \\prod_{k=0}^n\\1_{\\|e_\\gamma(k, z_k^\\gamma)\\|\\leq R} \\,.\n\\end{equation*}\nChoose $\\bar \\gamma_0>0$ and $s>0$ as in Lemma~\\ref{lem:moment-H}.\nFor every $\\gamma\\leq \\bar \\gamma_0$,\n\\begin{equation*}\n  \\resizebox{\\hsize}{!}{$\n  \\bE\\left(\\left\\|\\gamma^{-1}(z_{n+1}^{\\gamma,R} - z_n^{\\gamma,R})\\right\\|^{1+s}\\right)\n \\leq \\sup\\left\\{ \\bE\\left(\\left\\|H_{\\gamma'}(\\ell+1,z,\\xi)\\right\\|^{1+s} \\right):\\gamma'\\in (0,\\bar \\gamma_0],\\ell\\in \\bN,z\\in \\cZ_+, \\|e_\\gamma(\\ell,z)\\|\\leq R\\right\\}\\,.\n $}\n\\end{equation*}\nBy Lemma~\\ref{lem:moment-H}, the righthand side is finite and does not depend on $(n,\\gamma)$.\n\\end{proof}\nFor a fixed $\\gamma>0$, we define the interpolation map $\\sX_\\gamma:\\cZ^\\bN\\to C([0,+\\infty),\\cZ)$ as follows for every\nsequence $z=(z_n:n\\in \\bN)$ on $\\cZ$:\n$$\n  \\sX_\\gamma(z)\\,:t \\mapsto z_{\\lfloor \\frac t\\gamma\\rfloor} + (t\/\\gamma-\\lfloor t\/\\gamma\\rfloor)(z_{\\lfloor \\frac t\\gamma\\rfloor+1}-z_{\\lfloor \\frac t\\gamma\\rfloor})\\,.\n$$\nFor every $\\gamma,R>0$, we define $\\sz^{\\gamma,R}\\eqdef \\sX_\\gamma(z^{\\gamma,R}) = \\sX_\\gamma\\circ B_R (z^\\gamma)$.\nNamely, $\\sz^{\\gamma,R}$ is the interpolated process associated with the sequence $(z^{\\gamma,R}_n)$.\nIt is a random variable on $C([0,+\\infty),\\cZ)$.\nWe recall that $\\cF_n$ is the $\\sigma$-algebra generated by the r.v. $(\\xi_k:1\\leq k\\leq n)$.\nFor every $\\gamma,n,R$, we use the notation: $\\Delta_0^{\\gamma,R}\\eqdef 0$ and\n$$\n\\Delta_{n+1}^{\\gamma,R} \\eqdef \\gamma^{-1}(z_{n+1}^{\\gamma,R} - z_n^{\\gamma,R}) - \\bE(\\gamma^{-1}(z_{n+1}^{\\gamma,R} - z_n^{\\gamma,R})|\\cF_n)\\,.\n$$\n\\begin{lemma}\n  \\label{lem:tightness-in-C}\nLet Assumptions~\\ref{hyp:model}, \\ref{hyp:alpha-beta}, \\ref{hyp:moment-f} and \\ref{hyp:iid} hold true.\nThere exists $\\bar \\gamma_0>0$ s.t. for every $R>0$, the family of r.v. $(\\sz^{\\gamma,R}:\\gamma\\in (0,\\bar \\gamma_0])$\nis tight. Moreover, for every $\\delta>0$,\n$ \n\\bP\\left(\\max_{0\\leq n\\leq \\lfloor\\frac T\\gamma\\rfloor}\\gamma\\left\\|\\sum_{k=0}^n\\Delta_{k+1}^{\\gamma,R}\\right\\|>\\delta\\right)\\xrightarrow[]{\\gamma\\to 0} 0\\,.\n\n\\end{lemma}\n\\begin{proof}\n  It is an immediate consequence of Lemma~\\ref{lem:UI} and \\cite[Lemma 6.2]{bianchi2019constant}\n\\end{proof}\n\\noindent The proof of the following lemma is omitted.\n\\begin{lemma}\n  \\label{lem:cv-h}\nLet Assumptions~\\ref{hyp:model} and \\ref{hyp:alpha-beta} hold true.\nConsider $t>0$ and $z\\in \\cZ_+$. Let $(\\varphi_n,z_n)$ be a sequence on $\\bN^*\\times \\cZ_+$ s.t.\n$\\lim_{n\\to\\infty}\\gamma_n\\varphi_n= t$ and $\\lim_{n\\to\\infty}z_n= z$. Then, $\\lim_{n\\to\\infty} h_{\\gamma_n}(\\varphi_n,z_n)= h(t,z)$\nand $\\lim_{n\\to\\infty} e_{\\gamma_n}(\\varphi_n,z_n)= \\bar e(t,z)$.\n\\end{lemma}\n\n\\noindent\\textbf{End of the Proof of Th.~\\ref{th:weak-cv}}\nConsider $x_0\\in \\bR^d$ and set $z_0=(x_0,0,0)$. Define\n$R_0\\eqdef \\sup\\left\\{\\|\\bar e(t,Z_\\infty^0(z_0)(t))\\|:t>0\\right\\}\\,.$\nBy Prop.~\\ref{prop:adam-bounded}, $R_0<+\\infty$. We select an arbitrary $R$ s.t. $R\\geq  R_0+1$.\nFor every $n\\geq 0$, $z\\in \\cZ_+$,\n$$\nz_{n+1}^{\\gamma,R} = z_{n}^{\\gamma,R}+\\gamma H_\\gamma(n+1,z_{n}^{\\gamma,R},\\xi_{n+1}) \\1_{\\|e_\\gamma(n,z_{n}^{\\gamma,R})\\|\\leq R}  \\,.\n$$\nDefine for every $n\\geq 1$, $z\\in \\cZ_+$,\n$h_{\\gamma,R}(n,z)\\eqdef h_\\gamma(n,z)\\1_{\\|e_\\gamma(n-1,z)\\|\\leq R}$. Then,\\\\\n$\n\\Delta_{n+1}^{\\gamma,R} = \\gamma^{-1}(z_{n+1}^{\\gamma,R} - z_n^{\\gamma,R}) - h_{\\gamma,R}(n+1,z_{n}^{\\gamma,R})\\,.\n$\nDefine also for every $n\\geq 0$, \\\\\n\n  M_n^{\\gamma,R} \\eqdef \\sum_{k=1}^n\\Delta_k^{\\gamma,R} =\\gamma^{-1}(z_n^{\\gamma,R} - z_0) - \\sum_{k=0}^{n-1} h_{\\gamma,R}(k+1,z_{k}^{\\gamma,R})\\,.\n$\nConsider $t\\geq 0$ and set $n\\eqdef \\lfloor t\/\\gamma\\rfloor$.\nFor any $T>0$, it holds that :\n\\begin{equation*}\n  \\sup_{t\\in [0,T]} \\left\\| \\sz^{\\gamma,R}(t)-z_0 - \\int_{0}^{t}h_{\\gamma,R}(\\lfloor s\/\\gamma\\rfloor+1,\\sz^{\\gamma,R}(\\gamma\\lfloor s\/\\gamma\\rfloor))ds\\right\\|\\\\\n \n  \\leq \\max_{0\\leq n\\leq \\lfloor T\/\\gamma\\rfloor+1} \\gamma \\|M_n^{\\gamma,R}\\|\\,.\n\\end{equation*}\nBy Lemma~\\ref{lem:tightness-in-C},\n\\begin{equation}\n  \\label{eq:Mn-cv-proba-zero}\n \\bP\\left( \\sup_{t\\in [0,T]} \\left\\| \\sz^{\\gamma,R}(t)-z_0 - \\int_{0}^{t}h_{\\gamma,R}\\left(\\lfloor s\/\\gamma\\rfloor+1,\\sz^{\\gamma,R}(\\gamma\\lfloor s\/\\gamma\\rfloor)\\right)ds\\right\\|\n>\\delta\\right) \\xrightarrow[]{\\gamma\\to 0} 0\\,.\n\\end{equation}\nAs a second consequence of Lemma~\\ref{lem:tightness-in-C}, the family of r.v. $(\\sz^{\\gamma,R}:0<\\gamma\\leq \\bar \\gamma_0)$ is tight, where $\\bar \\gamma_0$\nis chosen as in Lemma~\\ref{lem:tightness-in-C} (it does not depend on $R$).\nBy Prokhorov's theorem, there exists a sequence $(\\gamma_k:k\\in \\bN)$ s.t. $\\gamma_k\\to 0$ and\ns.t. $(\\sz^{\\gamma_k,R}:k\\in \\bN)$ converges in distribution to some probability measure $\\nu$ on $C([0,+\\infty),\\cZ_+)$.\nBy Skorohod's representation theorem, there exists a r.v. $\\sz$ on some probability space $(\\Omega',\\cF',\\bP')$, with distribution $\\nu$,\nand a sequence of r.v. $(\\sz_{(k)}:k\\in \\bN)$ on that same probability space where for each $k \\in \\bN$, the r.v. $\\sz_{(k)}$ has the same distribution\nas the r.v. $\\sz^{\\gamma_k,R}$,\nand s.t. for every $\\omega\\in\\Omega'$, $\\sz_{(k)}(\\omega)$ converges\nto $\\sz(\\omega)$ uniformly on compact sets. Now select a fixed $T>0$. According to Eq.~(\\ref{eq:Mn-cv-proba-zero}), the sequence\n$$\n\\sup_{t\\in [0,T]} \\left\\| \\sz_{(k)}(t)-z_0 - \\int_{0}^{t}h_{\\gamma_{k},R}\\left(\\lfloor s\/\\gamma_{k}\\rfloor+1,\\sz_{(k)}(\\gamma_{k}\\lfloor s\/\\gamma_{k}\\rfloor)\\right)ds\\right\\|\\,,\n$$\nindexed by $k\\in \\bN$, converges in probability to zero as $k\\to\\infty$. One can therefore extract a further subsequence $\\sz_{(\\varphi_k)}$,\ns.t. the above sequence converges to zero almost surely. In particular, since $\\sz_{(k)}(t)\\to \\sz(t)$ for every $t$, we obtain that\n\\begin{equation}\n\\sz(t) = z_0 + \\lim_{k\\to\\infty} \\int_{0}^{t}h_{\\gamma_{\\varphi_k},R}\\left(\\lfloor s\/\\gamma_{\\varphi_k}\\rfloor+1,\\sz_{(\\varphi_k)}(\\gamma_{\\varphi_k}\\lfloor s\/\\gamma_{\\varphi_k}\\rfloor)\\right)ds\\quad (\\forall t\\in [0,T])\\,.\\label{eq:cv_z}\n\\end{equation}\nConsider $\\omega\\in \\Omega'$ s.t. the r.v. $\\sz$ satisfies (\\ref{eq:cv_z}) at point $\\omega$.\nFrom now on, we consider that $\\omega$ is fixed, and we handle $\\sz$ as an element of $C([0,+\\infty),\\cZ_+)$,\nand no longer as a random variable.\nDefine $\\tau\\eqdef \\inf\\{t\\in [0,T]: \\|\\bar e(t,\\sz(t))\\|>R_0+\\frac 12\\}$ if the latter set is non-empty,\nand $\\tau\\eqdef T$ otherwise.\nSince $\\sz(0)=z_0$ and $\\|z_0\\|<R_0$, it holds that $\\tau>0$ using the continuity of $\\sz$.\nChoose any $(s,t)$ s.t. $0<s<t<\\tau$.\nNote that $\\sz_{(k)}(\\gamma_{k}\\lfloor s\/\\gamma_{k}\\rfloor)\\to \\sz(s)$\nand $\\gamma_k(\\lfloor s\/\\gamma_{k}\\rfloor+1)\\to s$. Thus, by Lemma~\\ref{lem:cv-h},\n$h_{\\gamma_{k}}\\left(\\lfloor s\/\\gamma_{k}\\rfloor+1,\\sz_{(k)}(\\gamma_{k}\\lfloor s\/\\gamma_{k}\\rfloor)\\right)$\nconverges to $h(s,\\sz(s))$ and\n$e_{\\gamma_k}(\\lfloor s\/\\gamma_{k}\\rfloor,\\sz_{(k)}(\\gamma_{k}\\lfloor s\/\\gamma_{k}\\rfloor))$\nconverges to $\\bar e(s,\\sz(z))\\,.$\nSince $s<\\tau$,  $\\bar e(s,\\sz(z))\\leq R_0+\\frac 12$. As $R\\geq R_0+1$, there exists a certain\n$K(s)$ s.t. for every $k\\geq K(s)$,\n$\n\\1_{\\|e_{\\gamma_k}(\\lfloor s\/\\gamma_{k}\\rfloor,\\sz_{(k)}(\\gamma_{k}\\lfloor s\/\\gamma_{k}\\rfloor))\\|\\leq R} = 1\\,.\n$\nAs a consequence, $h_{\\gamma_{k},R}(\\lfloor s\/\\gamma_{k}\\rfloor+1,\\sz_{(k)}(\\gamma_{k}\\lfloor s\/\\gamma_{k}\\rfloor))$\nconverges to $h(s,\\sz(s))$ as $k\\to\\infty$.\nUsing Lebesgue's dominated convergence theorem,  we obtain, for all $t\\in [0,\\tau]$:\n$\n\\sz(t) = z_0 +  \\int_{0}^{t}h\\left(s,\\sz(s))\\right)ds\\,\n$\nTherefore $\\sz(t) = Z_\\infty^0(x_0)(t)$ for every $t\\in [0,\\tau]$. In particular,\n$\\|\\sz(\\tau)\\|\\leq R_0$\nand this means that\n$\\tau = T$. Thus, $\\sz(t)=Z_\\infty^0(x_0)(t)$ for every $t\\in [0,T]$\n(and consequently for every $t\\geq 0$). We have shown that for every $R\\geq R_0+1$, the sequence of r.v.\n$(\\sz^{\\gamma,R}:\\gamma \\in (0,\\bar \\gamma_0])$ is tight and converges in probability to $Z_\\infty^0(z_0)$ as $\\gamma\\to 0$.\nTherefore, for every $T>0$,\n\\begin{equation}\n\\forall \\delta>0,\\ \\lim_{\\gamma\\to 0} \\bP\\left(\\sup_{t\\in [0,T]}\\left\\| \\sz^{\\gamma,R}(t) - Z_\\infty^0(x_0)(t)\\right\\|>\\delta \\right) =0\\,.\n\\label{eq:cv-proba-R}\n\\end{equation}\nIn order to complete the proof, we show that\n$\n\\bP\\left(\\sup_{t\\in [0,T]}\\left\\| \\sz^{\\gamma,R}(t) - \\sz^{\\gamma}(t)\\right\\|>\\delta \\right) \\to 0\n$\nas $\\gamma \\to 0$, for all $\\delta >0$.\nwhere we recall that $\\sz^\\gamma=\\sX_\\gamma(z^\\gamma)$.\nNote that $\\left\\| \\sz^{\\gamma,R}(t)\\right\\|\\leq\n\\left\\| \\sz^{\\gamma,R}(t)-Z_\\infty^0(z_0)(t)\\right\\| + R_0$ by the triangular inequality.\nTherefore, for every $T,\\delta>0$,\n\\begin{align*\n  \\bP\\left(\\sup_{t\\in [0,T]}\\left\\| \\sz^{\\gamma,R}(t) - \\sz^{\\gamma}(t)\\right\\|>\\delta \\right)\n  &\\leq \\bP\\left(\\sup_{t\\in [0,T]}\\left\\| \\sz^{\\gamma,R}(t)\\right\\|\\geq R \\right)\\\\\n  &\\leq \\bP\\left(\\sup_{t\\in [0,T]}\\left\\| \\sz^{\\gamma,R}(t)-Z_\\infty^0(z_0)(t)\\right\\| \\geq R-R_0\\right)\\,.\n\\end{align*\nBy Eq.~(\\ref{eq:cv-proba-R}), the RHS of the above inequality tends to zero as $\\gamma\\to 0$.\nThe proof is complete.\n\n\\subsection{Proof of Th.~\\ref{th:longrun}}\n\\label{sec:longrun}\n\n\nWe start by stating a general result. Consider a Euclidean space $\\sX$ equipped with its Borel $\\sigma$-field $\\cal X$.\nLet $\\bar \\gamma_0>0$, and consider two families\n$(P_{\\gamma,n}:0<\\gamma<\\bar \\gamma_0, n\\in \\bN^*)$ and $(\\bar P_{\\gamma}:0<\\gamma<\\bar \\gamma_0)$\nof Markov transition kernels on $\\sX$.\nDenote by $\\cP(\\sX)$ the set of probability measures on $\\sX$.\nLet $X=(X_n:n\\in\\bN)$ be the canonical process on $\\sX$.\nLet $(\\bP^{\\gamma,\\nu}:0<\\gamma<\\bar \\gamma_0,\\nu\\in \\cP(\\sX))$ and\n$(\\bar \\bP^{\\gamma,\\nu}:0<\\gamma<\\bar \\gamma_0,\\nu\\in \\cP(\\sX))$ be two families of measures on the canonical space\n$(X^\\bN,\\cal X^{\\otimes\\bN})$ such that the following holds:\n\\begin{itemize}[leftmargin=*]\n\\item Under $\\bP^{\\gamma,\\nu}$, $X$ is a non-homogeneous Markov chain with transition kernels $(P_{\\gamma,n}:n\\in \\bN^*)$\nand initial distribution $\\nu$, that is, for each $n\\in\\bN^*$,\n$\n\\bP^{\\gamma,\\nu}(X_{n}\\in dx|X_{n-1}) = P_{\\gamma,n}(X_{n-1},dx)\\,.\n$\n\\item Under $\\bar\\bP^{\\gamma,\\nu}$, $X$ is an homogeneous Markov chain with transition kernel $\\bar P_{\\gamma}$\nand initial distribution $\\nu$.\n\\end{itemize}\nIn the sequel, we will use the notation $\\bar P^{\\gamma,x}$ as a shorthand notation for\n$\\bar P^{\\gamma,\\delta_x}$ where $\\delta_x$ is the Dirac measure at some point $x\\in \\sX$.\nFinally, let $\\Psi$ be a semiflow on $\\sX$. A Markov kernel $P$ is \\emph{Feller}\nif $Pf$ is continuous for every bounded continuous  $f$.\n\\begin{assumption} Let $\\nu\\in \\cP(\\sX)$.\n  \\begin{enumerate}[{\\sl i)}]\n  \\item For every $\\gamma$, $\\bar P_{\\gamma}$ is Feller.\n  \\item $(\\bP^{\\gamma,\\nu}X_n^{-1}:n\\in \\bN,0<\\gamma<\\bar \\gamma_0)$ is a tight family of measures.\n  \\item For every $\\gamma\\in (0,\\bar \\gamma_0)$ and every bounded Lipschitz continuous function $f:\\sX\\to\\bR$,\n$P_{\\gamma,n}f$ converges to $\\bar P_\\gamma f$ as $n\\to\\infty$, uniformly on compact sets.\n\\item For every $\\delta>0$, for every compact set $K\\subset \\sX$, for every $t>0$,\\\\\n\n\\lim_{\\gamma\\to 0}\\sup_{x\\in K}\\bar P^{\\gamma,x}\\left(\\|X_{\\lfloor t\/\\gamma\\rfloor} - \\Psi_t(x)\\|>\\delta\\right)=0\\,.\n\n  \\end{enumerate}\n\\label{hyp:general}\n\\end{assumption}\nLet $BC_\\Psi$ be the Birkhof center of $\\Psi$ \\emph{i.e.}, the closure of the set of recurrent points.\n\n\\begin{theorem}\n\\label{longrun}\nConsider $\\nu\\in \\cP(\\sX)$ s.t. Assumption~\\ref{hyp:general} holds true. Then, for every $\\delta>0$,\n$\n\\lim_{\\gamma\\to 0}\\limsup_{n\\to\\infty} \\frac 1{n+1}\\sum_{k=0}^n \\bP^{\\gamma,\\nu}\\left(d(X_k,BC_\\Psi)>\\delta\\right)=0\\,.\n$\n\\end{theorem}\nWe omit the proof of this result which follows a similar reasoning to \\cite[Th.~5.5 and Proof in section 8.4]{bianchi2019constant} and makes use of results from \\cite{for-pag-99}.\n\n\\noindent\\textbf{End of the Proof of Th.~\\ref{th:longrun}.}\nWe apply Th.~\\ref{longrun} in the case where $P_{\\gamma,n}$ is the kernel\nof the non-homogeneous Markov chain $(z_n^\\gamma)$ defined by~(\\ref{eq:znT}) and\n$\\bar P_\\gamma$ is the kernel of the homogeneous Markov chain $(\\bar z_n^\\gamma)$\ngiven by\n$\\bar z_n^\\gamma = \\bar z_{n-1}^\\gamma+\\gamma H_\\gamma(\\infty,\\bar z_{n-1}^\\gamma,\\xi_n)$\nfor every $n\\in\\bN^*$ and $\\bar z_0 \\in \\cZ_+$ where $H_\\gamma(\\infty,\\bar z_{n-1}^\\gamma,\\xi_n) \\eqdef \\lim_{k \\to \\infty} H_\\gamma(k,\\bar z_{n-1}^\\gamma,\\xi_n)$. The task is merely to verify Assumption~\\ref{hyp:general}{\\sl iii)}, the other assumptions being easily verifiable using Th.~\\ref{th:weak-cv},\nConsider $\\gamma\\in (0,\\bar \\gamma_0)$. Let $f: \\cZ \\to \\mathbb{R}$ be a bounded $M$-Lipschitz continuous function and $K$ a compact.\nFor all $z=(x,m,v) \\in K$:\n\\begin{align*}\n&|P_{\\gamma,n}(f)(z) - \\bar P_{\\gamma}(f)(z)| \\leq M \\gamma \\bE \\left \\| \\frac{(1-\\alpha^n)^{-1}\\tilde{m}_\\xi}{ \\varepsilon+(1-\\beta^n)^{-\\frac{1}{2}}{\\tilde{v}_\\xi^{1\/2}}} -\\frac{\\tilde{m}_\\xi}{ \\varepsilon+{\\tilde{v}_\\xi^{1\/2}}}\\right \\| \\\\\n&\\resizebox{.99\\hsize}{!}{$\\leq \\frac{M \\gamma\\alpha^n}{\\varepsilon(1- \\alpha^n)} \\sup_{x,m}\\left(\\alpha ||m|| + (1-\\alpha)\\bE||\\nabla f(x,\\xi)|| \\right)\n + \\frac{M \\gamma \\bE ||\\tilde{m}_\\xi \\odot \\tilde{v}_\\xi^{1\/2}||}{\\varepsilon^2}\\left(1- \\frac{1}{(1-\\beta^n)^{1\/2}}\\right)$}\\,\n\\end{align*}\nwhere we write $\\alpha=\\bar \\alpha(\\gamma)$, $\\beta=\\bar \\beta(\\gamma)$,\n$\\tilde{m}_\\xi \\eqdef \\alpha m+(1-\\alpha)\\nabla f(x,\\xi)$ and\n$\\tilde{v}_\\xi \\eqdef  \\beta v+(1- \\beta)\\nabla f(x,\\xi)^{\\odot 2}$.\nThus, condition~\\ref{hyp:general}{\\sl iii)} follows.\nFinally, Cor.~\\ref{coro:cv} implies $BC_\\Phi=\\cE$.\n\n\\section{Proofs of Section~\\ref{sec:discrete_decreasing}}\n\\label{sec:proofs_sec_discrete_decreasing}\nIn this section, we denote by $\\bE_n = \\bE(\\cdot|\\cF_n)$ the conditional expectation w.r.t. $\\cF_n$.\nWe also use the notation $\\nabla f_{n+1} \\eqdef \\nabla f(x_n,\\xi_{n+1})$.\n\nThe following lemma will be useful in the proofs.\n\\begin{lemma}\n\\label{lemma:r_n}\nLet the sequence $(r_n)$ be defined as in Algorithm~\\ref{alg:adam-decreasing}.\nAssume that $0 \\leq \\alpha_n \\leq 1$ for all $n$ and that $(1-\\alpha_n)\/\\gamma_n \\to a > 0$ as $n \\to +\\infty$.\nThen,\n\\begin{enumerate}[{\\sl i)}]\n\\item $\\forall n \\in \\bN, r_n = 1 - \\prod_{i=1}^n \\alpha_i$,\n\\item The sequence $(r_n)$ is nondecreasing and converges to $1$.\n\\item Under Assumption~\\ref{hyp:step-tcl}~\\ref{step-tcl-i}, for every $\\epsilon>0$, for sufficiently large $n$, we have\n$r_n-1 \\leq e^{-\\frac{a\\gamma_0}{2(1-\\kappa)}n^{1-\\kappa}}$ if $\\kappa \\in (0,1)$ and\n$r_n-1 \\leq n^{-a \\gamma_0\/(1+\\epsilon)}$ if $\\kappa = 1$.\n\\end{enumerate}\n\\end{lemma}\nA similar lemma holds for the sequence $(\\bar{r}_n)$.\n\\begin{proof}\ni) stems from observing that $r_{n+1} -1 = \\alpha_{n+1}(r_n -1)$ for every $n \\in \\bN$ and\niterating this relation ($r_0=0$). As a consequence, the sequence $(r_n)$ is nondecreasing.\nWe can write :\n$\n0 \\leq 1-r_n \\leq \\exp(- \\sum_{i=1}^n (1-\\alpha_i))\\,.\n$\niii) As $\\sum_{n\\geq 1} \\gamma_n = + \\infty$\nand $(1-\\alpha_n) \\sim a\\gamma_n$, we deduce that $\\sum_{i=1}^n (1-\\alpha_i) \\sim \\sum_{i=1}^n a\\gamma_i$.\nThe results follow from the fact that $\\sum_{i=1}^n \\gamma_i \\sim \\frac{\\gamma_0}{1-\\kappa} n^{1-\\kappa}$\nwhen $\\kappa \\in (0,1)$ and $\\sum_{i=1}^n \\gamma_i \\sim \\gamma_0 \\ln n$ for $\\kappa=1$.\n\\end{proof}\n\n\\subsection{Proof of Th.~\\ref{thm:as_conv_under_stab}}\n\nWe define\n$\\bar z_n = (x_{n-1},m_n,v_n)$ (note the shift in the index of the variable $x$).\nWe have\n$$\n\\bar z_{n+1} = \\bar z_n + \\gamma_{n+1} h_\\infty(\\bar z_n) + \\gamma_{n+1}\\chi_{n+1} + \\gamma_{n+1} \\varsigma_{n+1}\\,,\n$$\nwhere $h_\\infty$ is defined in Eq.~(\\ref{eq:h_infty}) and where we set\n$$\n\\chi_{n+1} = \\left(0,\\gamma_{n+1}^{-1}(1-\\alpha_{n+1})(\\nabla f_{n+1}-\\nabla F(x_n)),\\gamma_{n+1}^{-1}(1-\\beta_{n+1})(\\nabla f_{n+1}^{\\odot 2}-S(x_n))\\right)\n$$\nand $\\varsigma_{n+1} = (\\varsigma_{n+1}^x,\\varsigma_{n+1}^m,\\varsigma_{n+1}^v)$ with the components defined by:\n$\\varsigma_{n+1}^x = \\frac{m_n}{\\varepsilon + \\sqrt{v_n}} - \\frac{\\gamma_n}{\\gamma_{n+1}}\\frac{\\hat m_n}{\\varepsilon + \\sqrt{\\hat v_n}}$,\n$\\varsigma_{n+1}^m = \\left(\\frac{1-\\alpha_{n+1}}{\\gamma_{n+1}}-a\\right)(\\nabla F(x_n)-m_n) + a(\\nabla F(x_n) - \\nabla F(x_{n-1}))$ and\n$\\varsigma_{n+1}^v = \\left(\\frac{1-\\beta_{n+1}}{\\gamma_{n+1}}-b\\right)(S(x_n)-v_n) + b(S(x_n) - S(x_{n-1}))$ \\,.\nWe prove that $\\varsigma_n\\to 0$ a.s.\nUsing the triangular inequality,\n\\begin{align*}\n  \\|\\varsigma_n^x\\|\n&\\leq \\left\\| \\frac{m_n}{\\varepsilon + \\sqrt{v_n}} -\\frac{m_n}{\\bar r_n^{1\/2}\\varepsilon + \\sqrt{v_n}}\\right\\|\n+ \\left|1-\\frac{\\gamma_nr_n^{-1}}{\\gamma_{n+1}\\bar r_n^{-1\/2}}\\right|\\left\\| \\frac{m_n}{\\bar r_n^{1\/2}\\varepsilon + \\sqrt{v_n}}\\right\\|\\\\\n&\\leq \\varepsilon^{-1}|1-\\bar r_n^{-1\/2}|\\|m_n\\| + \\varepsilon^{-1}\\left|\\bar r_n^{-1\/2}-\\frac{\\gamma_nr_n^{-1}}{\\gamma_{n+1}}\\right|\\|m_n\\|\\,,\n\\end{align*}\nwhich converges a.s. to zero because of the boundedness of $(z_n)$ combined with Assumption~\\ref{hyp:stepsizes} and Lemma~\\ref{lemma:r_n} for $(\\bar r_n)$.\nThe components $\\varsigma_{n+1}^m$ and $\\varsigma_{n+1}^v$ converge a.s. to zero,\nas products of a bounded term and a term converging to zero.\nIndeed, note that $\\nabla F$ and $S$ are locally Lipschitz continuous under Assumption~\\ref{hyp:model}.\nHence, there exists a constant $C$ s.t. $\\|\\nabla F(x_n) - \\nabla F(x_{n-1})\\| \\leq C \\|x_n-x_{n-1}\\| \\leq \\frac{C}{\\varepsilon}\\gamma_n\\|m_n\\|$.\nThe same inequality holds when replacing $\\nabla F$ by $S$.\nNow consider the martingale increment sequence $(\\chi_n)$, adapted to $\\cF_n$.\nEstimating the second order moments, it is easy to show using Assumption~\\ref{hyp:moment-f}~\\ref{momentegal} that\nthere exists a constant $C'$ s.t.\n$\\bE_n(\\|\\chi_{n+1}\\|^2)\\leq C'$.\nUsing that $\\sum_k\\gamma_k^2<\\infty$, it follows that $\\sum_n\\bE_n(\\|\\gamma_{n+1}\\chi_{n+1}\\|^2)<\\infty$ a.s.\nBy Doob's convergence theorem, $\\lim_{n\\to\\infty} \\sum_{k\\leq n} \\gamma_k\\chi_k$ exists almost surely.\nUsing this result along with the fact that $\\varsigma_n$ converges a.s. to zero, it follows from\nusual stochastic approximation arguments \\cite{ben-(cours)99} that the interpolated process\n$\\bar\\sz: [0,+\\infty)\\to \\cZ_+$ given by\n\\[\n\\bar \\sz(t) = \\bar z_n + (t-\\tau_n) \\frac{\\bar z_{n+1}-\\bar z_n}{\\gamma_{n+1}} \\qquad \\left(\\forall n \\in \\bN\\,,\\, \\forall t \\in [\\tau_n,\\tau_{n+1})\\right)\n\\]\n(where $\\tau_n = \\sum_{k=0}^n\\gamma_k$), is almost surely a bounded APT of the semiflow $\\bar\\Phi$ defined by~(\\ref{eq:ode-a}).\nThe proof is concluded by applying Prop.~\\ref{prop:benaim} and Prop.~\\ref{prop:Wstrict}.\n\n\\subsection{Proof of Prop.~\\ref{thm:stab}}\n\\label{sec:stability}\n\nAs $\\inf F>-\\infty$, one can assume without loss of generality that $F\\geq 0$.\nIn the sequel,\n$C$ denotes some positive constant which may change from line to line.\nWe define $a_n \\eqdef (1-\\alpha_{n+1})\/\\gamma_n$ and\n$P_n\\eqdef \\frac 1{2a_nr_n}\\ps{m_n^{\\odot 2},\\frac 1{\\varepsilon+\\sqrt{\\hat v_n}}}$.\nWe have $a_n\\to a$ and $r_n\\to 1$.\nBy Assumption~\\ref{hyp:stab}-\\ref{lipschitz},\n\\begin{align}\nF(x_{n}) \n&\\leq F(x_{n-1}) - \\gamma_{n} \\ps{\\nabla F(x_{n}),\\frac{\\hat m_n}{\\varepsilon + \\sqrt{\\hat v_n}}} + C\\gamma_n^2 P_n\\label{eq:lip}.\n\\end{align}\nWe set $u_n \\eqdef 1-\\frac{a_{n+1}}{a_{n}}$ and\n$D_n \\eqdef \\frac {r_n^{-1}}{\\varepsilon+\\sqrt{\\hat v_{n}}}$, so that $P_n = \\frac 1{2a_n}\\ps{D_n,m_n^{\\odot 2}}$.\nWe can write:\n\\begin{equation}\nP_{n+1}-P_n= u_nP_{n+1} +\\ps{\\frac{D_{n+1}-D_n}{2a_n},m_{n+1}^{\\odot 2}}+\\ps{\\frac{D_n}{2a_n},m_{n+1}^{\\odot 2}-m_n^{\\odot 2}}.\\label{eq:P-P}\n\\end{equation}\nWe estimate the vector $D_{n+1}-D_n$.\nUsing that $(r_n^{-1})$ is non-increasing,\n$$\nD_{n+1}-D_n \\leq r_{n}^{-1} \\frac{\\sqrt{\\hat v_n} - \\sqrt{\\hat v_{n+1}}}{(\\varepsilon + \\sqrt{\\hat v_{n+1}})\\odot(\\varepsilon + \\sqrt{\\hat v_n})}\\,.\n$$\nRemarking that $v_{n+1} \\geq \\beta_{n+1} v_n$, recalling that\n$(\\bar{r}_n)$ is nondecreasing and using the update rules of $v_n$ and\n$\\bar{r}_n$, we obtain after some algebra\n\\begin{align}\n  \\label{eq:subsubterm2}\n\\sqrt{\\hat v_n} - \\sqrt{\\hat v_{n+1}} &=  \\bar{r}_{n+1}^{-\\frac 12}(1-\\beta_{n+1})\\frac{v_n-\\nabla f_{n+1}^{\\odot 2}}{\\sqrt{v_n}+\\sqrt{v_{n+1}}\n                                          + \\frac{\\bar{r}_{n+1}-\\bar{r}_n}{\\sqrt{\\bar{r}_n}(\\sqrt{\\bar{r}_n}+\\sqrt{\\bar{r}_{n+1}})} \\sqrt{\\frac{v_n}{\\bar{r}_{n+1}}} \\nonumber\\\\\n                                         \n                                          &\\leq c_{n+1} \\sqrt{\\hat v_{n+1}}  \\,\\, \\text{where} \\,\\,\n                                          c_{n+1} \\eqdef \\frac{1-\\beta_{n+1}}{\\sqrt{\\beta_{n+1} }}\\left( \\frac{1}{1+\\sqrt{\\beta_{n+1}}} + \\frac{1- \\bar{r}_n}{2\\bar{r}_n} \\right)\\,.\n\\end{align}\nIt is easy to see that $c_n\/\\gamma_n\\to b\/2$. Thus, for any $\\delta>0$, $c_{n+1}\\leq (b+2\\delta)\\gamma_{n}\/2$ for all $n$ large enough.\nUsing also that $\\sqrt{\\hat v_{n+1}}\/(\\varepsilon +\\sqrt{\\hat v_{n+1}})\\leq 1$, we obtain that\n\nD_{n+1}-D_n \\leq\n\\frac{b+2\\delta}2 \\gamma_n D_n\\,.\n$\nSubstituting this inequality in Eq.~(\\ref{eq:P-P}), we get\n\\begin{align*}\n  P_{n+1}-P_n&\\leq u_nP_{n+1} +\\gamma_n\\ps{\\frac{b+2\\delta}{4a_n} D_n,m_{n+1}^{\\odot 2}}+\\ps{\\frac{D_n}{2a_n},m_{n+1}^{\\odot 2}-m_n^{\\odot 2}}\\,.\n\\end{align*}\nUsing $m_{n+1}^{\\odot 2} - m_n^{\\odot 2} =  2 m_n\n\\odot (m_{n+1}-m_n)+(m_{n+1}-m_n)^{\\odot 2} $, and noting that\\\\ $\\bE_n(m_{n+1}-m_n) = a_n\\gamma_n(\\nabla F(x_n) -m_n)$,\n\\begin{equation*}\n  \\bE_n\\ps{\\frac {D_n}{2a_n},m_{n+1}^{\\odot 2}-m_n^{\\odot 2}}\n = \\gamma_n\\ps{\\nabla F(x_n),\\frac{\\hat m_n}{\\varepsilon+\\sqrt{\\hat v_n}}}-2a_n\\gamma_nP_n+\\ps{\\frac {D_n}{2a_n}, \\bE_n[(m_{n+1}-m_n)^{\\odot 2}] }\n\\end{equation*}\nAs $a_n\\to a$, we have $a_n-\\frac{b+2\\delta}{4}\\geq a-\\frac{b+\\delta}{4}$ for all $n$ large enough. Hence,\n\\begin{align*}\n \\bE_n P_{n+1}-P_n&\\leq\nu_nP_{n+1} -2(a-\\frac{b+\\delta}{4})\\gamma_n P_n + \\gamma_n\\ps{\\nabla F(x_n),\\frac{\\hat m_n}{\\varepsilon+\\sqrt{\\hat v_n}}}\n\\\\\n&+\n\\gamma_n^2\\frac{b+2\\delta}2\\ps{\\nabla F(x_n),\\frac{\\hat m_n}{\\varepsilon+\\sqrt{\\hat v_n}}}\n+C\\ps{\\frac {D_n}{2a_n}, \\bE_n[(m_{n+1}-m_n)^{\\odot 2}] }\\,.\n\\end{align*}\nUsing the Cauchy-Schwartz inequality and Assumption~\\ref{hyp:stab}~\\ref{momentgrowth},\nit is easy to show the inequality\n$\\ps{\\nabla F(x_n),\\frac{\\hat m_n}{\\varepsilon+\\sqrt{\\hat v_n}}}\\leq C(1+F(x_n)+P_n)$.\nMoreover, using the componentwise inequality $(\\nabla f_{n+1}-m_n)^{\\odot 2} \\leq 2 \\nabla f_{n+1}^{\\odot 2} + 2 m_n^{\\odot 2}$\nalong with Assumption~\\ref{hyp:stab}~\\ref{momentgrowth}, we obtain\n\\begin{equation*}\n  \\resizebox{\\hsize}{!}{$\n  \\ps{\\frac {D_n}{2a_n}, \\bE_n[(m_{n+1}-m_n)^{\\odot 2}] } \\leq\n2(1-\\alpha_{n+1})^2\\ps{\\frac {D_n}{2a_n},\n\\bE_n[ \\nabla f_{n+1}^{\\odot 2}]+ m_n^{\\odot 2}\n}\n\\leq C\\gamma_n^2(1+F(x_n)+P_n)\\,.\n$}\n\\end{equation*}\nPutting all pieces together with Eq.~(\\ref{eq:lip}),\n\\begin{equation}\n\\label{eq:F+P}\n\\resizebox{0.95\\hsize}{!}{$\n \\bE_n(F(x_n)+ P_{n+1}) \\leq F(x_{n-1})+P_{n}\n +u_nP_{n+1} -2(a-\\frac{b+\\delta}{4})\\gamma_n P_n\n+C\\gamma_n^2(1+F(x_n)+P_n)\\,.\n$}\n\\end{equation}\nDefine\n$\nV_n\\eqdef (1-C\\gamma_{n-1}^2)F(x_{n-1})+(1-u_{n-1})P_{n}\n$\nwhere the constant $C$ is fixed so that\nEq.~(\\ref{eq:F+P}) holds.\nThen,\n\\begin{equation*}\n  \\bE_n(V_{n+1}) \\leq V_n\n -\\left(2a-\\frac{b+\\delta}{2}-\\frac{u_{n-1}}{\\gamma_n}\\right)\\gamma_n P_n\n+C\\gamma_n^2(1+P_n)+ C\\gamma_{n-1}^2F(x_{n-1})\\,.\n\\end{equation*}\nBy Assumption~\\ref{hyp:stab}, $\\lim\\sup_n u_{n-1}\/\\gamma_n< 2a-b\/2$ and for $\\delta$ small enough, we obtain\n\\begin{equation*}\n  \\bE_n(V_{n+1}) \\leq V_n\n+C\\gamma_n^2(1+P_n)+ C\\gamma_{n-1}^2F(x_{n-1})\\leq (1+ C'\\gamma_n^2)V_n +C\\gamma_n^2\\,.\n\\end{equation*}\nBy the Robbins-Siegmund's theorem  \\cite{robbins1971convergence},\nthe sequence $(V_n)$ converges almost surely to a finite random variable $V_\\infty \\in \\bR^+$.\nIn turn, the coercivity of $F$ implies that $(x_n)$ is almost surely bounded.\nWe now establish the almost sure boundedness of $(m_n)$.\nConsider the martingale difference sequence $\\Delta_{n+1}\\eqdef \\nabla f_{n+1} -\\nabla F(x_n)$.\nWe decompose $m_{n} = \\bar m_n + \\tilde m_n$ where\n$\\bar m_{n+1} = \\alpha_{n+1} \\bar m_n + (1-\\alpha_{n+1})\\nabla F(x_n)$ and\n$\\tilde m_{n+1} = \\alpha_{n+1} \\tilde m_n + (1-\\alpha_{n+1})\\Delta_{n+1}$, setting $\\bar m_0=\\tilde m_0=0$.\nWe prove that both terms $\\bar m_n$ and $\\tilde m_n$ are bounded. Consider the first term:\n$\n\\|\\bar m_{n+1}\\|\\leq \\alpha_{n+1} \\|\\bar m_n\\| + (1-\\alpha_{n+1}) \\sup_k\\|\\nabla F(x_k)\\|\\,.\n$\nBy continuity of $\\nabla F$, the supremum in the above inequality is almost surely finite.\nThus, for every $n$, the ratio $\\|\\bar m_n\\|\/\\sup_k\\|\\nabla F(x_k)\\|$ is upperbounded by the bounded sequence\n$r_n$. Hence, $(\\bar m_n)$ is bounded w.p.1. Consider now the term $\\tilde m_n$:\n\\begin{equation*}\n    \\resizebox{\\hsize}{!}{$\n  \\bE_n(\\|\\tilde m_{n+1}\\|^2)  = \\alpha_{n+1}^2\\|\\tilde m_n\\|^2 +  (1-\\alpha_{n+1})^2\\bE_n(\\|\\Delta_{n+1}\\|^2)\n                               \\leq  (1+(1-\\alpha_{n+1})^2)\\|\\tilde m_n\\|^2 +  (1-\\alpha_{n+1})^2C\\,,\n    $}\n\\end{equation*}\nwhere $C$ is a constant s.t. $\\bE_n(\\|\\nabla f_{n+1}\\|^2) \\leq C$ by Assumption~\\ref{hyp:moment-f}~\\ref{momentegal}.\nHere, we used $\\alpha_{n+1}^2\\leq (1+(1-\\alpha_{n+1})^2)$ and the inequality\n$\\bE_n(\\|\\Delta_{n+1}\\|^2) \\leq \\bE_n(\\|\\nabla f_{n+1}\\|^2)$.\nBy Assumption~\\ref{hyp:stepsizes}, $\\sum_n(1-\\alpha_{n+1})^2<\\infty$. By the Robbins-Siegmund theorem,\nit follows that $\\sup_n\\|\\tilde m_n\\|^2<\\infty$ w.p.1. Finally, it can be shown that $(v_n)$ is\nalmost surely bounded using the same arguments.\n\n\\subsection{Proof of Th.~\\ref{thm:clt}}\n\\label{sec:clt}\n\nWe use~\\cite[Th.~1]{pelletier1998weak}.\nAll the assumptions in the latter can be verified in our case, at the exception of\na positive definiteness condition on the\nlimiting covariance matrix, which corresponds, in our case, to the matrix $Q$\ngiven by Eq.~(\\ref{eq:Q}). As $Q$ is not positive definite, it is strictly speaking not possible\nto just cite and apply \\cite[Th.~1]{pelletier1998weak}. Nevertheless,\na detailed inspection of the proofs of \\cite{pelletier1998weak} shows that only a minor\nadaptation is needed in order to cover the present case.\nTherefore, proving the convergence result of \\cite{pelletier1998weak} from scratch is worthless.\nIt is sufficient to verify the assumptions of \\cite[Th.~1]{pelletier1998weak}\n(except the definiteness of $Q$) and then to point out the specific part of the proof of \\cite{pelletier1998weak}\nwhich requires some adaptation.\n\nLet $z_n=(x_n,m_n,v_n)$ be the output of Algorithm~\\ref{alg:adam-decreasing}.\nDefine $z^*=(x^*,0,S(x^*))$.\nDefine $\\eta_{n+1} \\eqdef (0, a(\\nabla f_{n+1} - \\nabla F(x_n)), b(\\nabla f_{n+1}^{\\odot 2}- S(x_n)))$.\nWe have\n\\begin{equation}\n  \\label{eq:zn_dec}\n  z_{n+1} = z_n + \\gamma_{n+1} h_\\infty(z_n) + \\gamma_{n+1}\\eta_{n+1} + \\gamma_{n+1} \\epsilon_{n+1}\\,,\n\\end{equation}\nwhere $\\epsilon_{n+1} \\eqdef (\\epsilon_{n+1}^1,\\epsilon_{n+1}^2,\\epsilon^3_{n+1})$, whose components are given by\n\\begin{equation*}\n  \\resizebox{\\hsize}{!}{$\n  \\epsilon_{n+1}^1 =  \\frac{m_n}{\\varepsilon+\\sqrt{v_n}} - \\frac{\\hat m_{n+1}}{\\varepsilon+\\sqrt{\\hat v_{n+1}}};\\,\n  \\epsilon_{n+1}^2 = \\left(\\frac{1-\\alpha_{n+1}}{\\gamma_{n+1}}-a\\right)\\left( \\nabla f_{n+1} - m_n\\right);\\,\n  \\epsilon_{n+1}^3 =  \\left(\\frac{1-\\beta_{n+1}}{\\gamma_{n+1}}-b\\right)\\left( \\nabla f_{n+1}^{\\odot 2} - v_n\\right)\\,.\n  $}\n\\end{equation*}\nHere, $\\eta_{n+1}$ is a martingale increment noise and\n$\\epsilon_{n+1} = (\\epsilon_{n+1}^1,\\epsilon_{n+1}^2,\\epsilon^3_{n+1})$ is a remainder term.\nThe aim is to check the assumptions (A1.1) to (A1.3) of\n\\cite{pelletier1998weak}, where the role of the quantities ($h$,\n$\\varepsilon_n$, $r_n$, $\\sigma_n$, $\\alpha$, $\\rho$, $\\beta$) in\n\\cite{pelletier1998weak} is respectively played by the quantities\n($h_\\infty$, $\\eta_n$, $\\epsilon_n$, $\\gamma_n$, $\\kappa$, $1$, $1$) of\nthe present paper.\n\nLet us first consider Assumption (A1.1) for $h_\\infty$.\nBy construction, $h_\\infty(z^*) = 0$. By Assumptions~\\ref{hyp:mean_field_tcl} and \\ref{hyp:S>0},\n$h_\\infty$ is continuously differentiable in the neighborhood of $z^*$ and its\nJacobian at $z^*$ coincides with the matrix $H$ given by Eq.~(\\ref{eq:H}).\nAs already discussed, after some algebra,\nit can be shown that the largest real part of the eigenvalues of $H$ coincides with $-L$ where $L>0$\nis given by Eq.~(\\ref{eq:L}).\nHence, Assumption (A1.1) of \\cite{pelletier1998weak}\nis satisfied for $h_\\infty$. Assumption (A1.3) is trivially satisfied\nusing Assumption~\\ref{hyp:step-tcl}. The crux is therefore to verify Assumption (A1.2).\nClearly, $\\bE(\\eta_{n+1}|\\cF_n)=0$. Using Assumption~\\ref{hyp:moment-f}\\ref{momentreinforce},\nit follows from straightforward manipulations based on Jensen's inequality\nthat for any $M>0$,\nthere exists $\\delta>0$ s.t.\n\n\\sup_{n\\geq 0}\\bE_n\\left(\\|\\eta_{n+1}\\|^{2+\\delta}\\right) \\1_{\\{\\|z_n-z^*\\|\\leq M\\}}<\\infty\\,.\n$\nNext, we verify the condition\n\\begin{equation}\n  \\label{eq:cond-reste}\n  \\lim_{n\\to\\infty} \\bE\\left(\\gamma_{n+1}^{-1}\\|\\epsilon_{n+1}\\|^2\\1_{\\{\\|z_n-z^*\\|\\leq M\\}}\\right) = 0\\,.\n\\end{equation}\nIt is sufficient to verify the latter for $\\epsilon^i_n$ ($i=1,2,3$) in place of $\\epsilon_n$.\nThe map $(m,v)\\mapsto m\/(\\varepsilon+\\sqrt{v})$ is Lipschitz continuous in a neighborhood of\n$(0,S(x^*))$ by Assumption~\\ref{hyp:S>0}. Thus, for $M$ small enough, there exists a constant $C$ s.t.\nif $ \\|z_n-z^*\\|\\leq M$, then\n\n  \\|\\epsilon_{n+1}^1\\| \\leq C\\left\\|r_{n+1}^{-1}m_{n+1}-m_n\\right\\|+C\\left\\|\\bar r_{n+1}^{-1} v_{n+1}-v_n\\right\\|\\,.\n$\nUsing the triangular inequality and the fact that $r_{n+1},\\bar r_{n+1}$ are bounded sequences away from zero, there exists an other constant $C$ s.t.\n\\begin{align*}\n  \\|\\epsilon_{n+1}^1\\| &\\leq C\\left\\|m_{n+1}-m_n\\right\\|+C\\left\\|v_{n+1}-v_{n}\\right\\|\n+C|r_{n+1}-1|+C|\\bar r_{n+1}-1|\\,.\n\\end{align*}\nUsing Lemma~\\ref{lemma:r_n} under Assumption~\\ref{hyp:step-tcl} (note that $\\gamma_0 > 1\/2L \\geq 1\/a$ when $\\kappa=1$),\nwe obtain that the sequence $|r_{n}-1|\/\\gamma_n$ is bounded, thus $|r_{n+1}-1|\\leq C\\gamma_{n+1}$.\n The sequence $(1-\\alpha_{n})\/\\gamma_n$ being also bounded, it holds that\n\\begin{align*}\n  \\|m_{n+1}-m_n\\|^2 \\1_{\\{\\|z_n-z^*\\|\\leq M\\}} \\leq  C\\gamma_{n+1}^2(1+\\|\\nabla f_{n+1}\\|^2 )\\1_{\\{\\|z_n-z^*\\|\\leq M\\}}\\,.\n\\end{align*}\nBy Assumption~\\ref{hyp:moment-f}~\\ref{momentreinforce},\n$\\bE_n(\\|\\nabla f_{n+1}\\|^2|)$ is bounded by a deterministic constant on $\\{\\|z_n-z^*\\|\\leq M\\}$.\nThus, $\\bE_n(\\|m_{n+1}-m_n\\|^2\\1_{\\{\\|z_n-z^*\\|\\leq M\\}})\\leq C\\gamma_{n+1}^2$.\nA similar result holds for $\\|v_{n+1}-v_n\\|^2$. We have thus shown that\n$ \\bE_n\\left(\\|\\epsilon_{n+1}^1\\|^2\\1_{\\{\\|z_n-z^*\\|\\leq M\\}}\\right)\\leq C\\gamma_{n+1}^2$. Hence,\nEq.~(\\ref{eq:cond-reste}) is proved for $\\epsilon_{n+1}^1$ in place of $\\epsilon_{n+1}$.\nUnder Assumption~\\ref{hyp:step-tcl}, the proof uses the same kind of arguments for $\\epsilon_{n+1}^2$, $\\epsilon_{n+1}^3$  and is omitted.\nFinally, Eq.~(\\ref{eq:cond-reste}) is proved.\nContinuing the verification of Assumption (A1.2), we establish that\n\\begin{equation}\n  \\label{eq:lim-cov}\n  \\bE_n(\\eta_{n+1}\\eta_{n+1}^T) \\to Q\\textrm{ a.s. on } \\{z_n\\to z^*\\}\\,.\n\\end{equation}\nDenote by $\\bar Q(x)$ the matrix given by the righthand side of Eq.~(\\ref{eq:Q}) when $x^*$\nis replaced by an arbitrary $x\\in \\cV$. It is easily checked that $\\bE_n(\\eta_{n+1}\\eta_{n+1}^T)=\\bar Q(x_n)$\nand by continuity, $\\bar Q(x_n)\\to Q$ a.s. on $\\{z_n\\to z^*\\}$, which proves (\\ref{eq:lim-cov}).\nTherefore, Assumption (A1.2) is fulfilled, except for the point mentioned at the beginning of this section :\n\\cite{pelletier1998weak} puts the additional condition that the limit\nmatrix in Eq.~(\\ref{eq:lim-cov}) is positive definite. This condition is not satisfied in our case,\nbut the proof can still be adapted. The specific part of the proof where the positive definiteness\ncomes into play is Th.~7 in \\cite{pelletier1998weak}. The proof of \\cite[Th.~1]{pelletier1998weak} can therefore\nbe adapted to the case of a positive semidefinite matrix.\nIn the proof of \\cite[Th.~7]{pelletier1998weak}, we only substitute the inverse of the square root of $Q$ by the Moore-Penrose inverse.\nFinally, the uniqueness of the stationary distribution $\\mu$ and its expression follow from \\cite[Th.~6.7, p. 357]{karatzas-shreve1991}.\n\n\\noindent\\textbf{Proof of Eq.~(\\ref{eq:cov})}.\nWe introduce the $d\\times d$ blocks of the $3d\\times 3d$ matrix\n$\\Sigma = \\left(\\Sigma_{i,j}\\right)_{i,j=1,2,3}$\nwhere $\\Sigma_{i,j}$ is $d\\times d$. We denote by $\\tilde\\Sigma$ the $2d\\times 2d$ submatrix $\\tilde\\Sigma \\eqdef\\left(\\Sigma_{i,j}\\right)_{i,j=1,2}$.\nBy Th.~\\ref{thm:clt}, we have the subsystem:\n\\begin{equation}\n\\tilde H \\tilde\\Sigma + \\tilde\\Sigma\\tilde H^T =\n\\begin{pmatrix}\n  0 & 0 \\\\\n0 & -a^2 \\tilde Q\n\\end{pmatrix}\\qquad\\text{where }\\tilde H \\eqdef\n\\begin{pmatrix}\n  \\zeta I_d & -D \\\\\na\\nabla^2F(x^*) & (\\zeta-a) I_d\n\\end{pmatrix}\\label{eq:lyap-reduced}\n\\end{equation}\nand where $\\tilde Q \\eqdef \\textrm{Cov}\\left(\\nabla\n  f(x^*,\\xi)\\right)$.  The next step is to triangularize the matrix\n$\\tilde H$ in order to decouple the blocks of $\\tilde\\Sigma$.  For\nevery $k=1,\\dots,d$, set\n$\\nu_k^\\pm \\eqdef -\\frac{a}{2}\\pm\\sqrt{a^2\/4-a\\lambda_k}$\nwith the convention that $\\sqrt{-1} =\n\\imath$ (inspecting the characteristic polynomial of $\\tilde H$, these\nare the eigenvalues of $\\tilde H$). Set\n$M^\\pm\\eqdef\\diag{(\\nu_1^\\pm,\n\\cdots,\\nu_d^\\pm)}$ and $R^\\pm\\eqdef\nD^{-1\/2}PM^\\pm P^TD^{-1\/2}$.  Using the identities $M^++M^-=-a I_d$ and\n$M^+M^-=a\\Lambda$ where\n$\\Lambda\\eqdef\\diag{(\\lambda_1,\\cdots,\\lambda_d)}$, it can be checked\nthat\n$$\n\\cR\\tilde H =\\begin{pmatrix}\n  D R^+ + \\zeta I_d & -D \\\\ 0 & R^-D + \\zeta I_d\n\\end{pmatrix}\\cR,\\text{ where }\\cR\\eqdef\n\\begin{pmatrix}\n  I_d & 0 \\\\ R^+ & I_d\n\\end{pmatrix}\\,.\n$$\nSet $\\check \\Sigma \\eqdef \\cR\\tilde\\Sigma\\cR^T$. Denote by\n$(\\check \\Sigma_{i,j})_{i,j=1,2}$ the blocks of $\\check\\Sigma$.\nNote that $\\check\\Sigma_{1,1} = \\Sigma_{1,1}$. By left\/right multiplication of Eq.~(\\ref{eq:lyap-reduced})\nrespectively with $\\cR$ and $\\cR^T$, we obtain\n\\begin{align}\n  &(DR^++\\zeta I_d) \\Sigma_{1,1}+\\Sigma_{1,1}(R^+D+\\zeta I_d) = \\check\\Sigma_{1,2} D+ D\\check\\Sigma_{1,2}^T \\label{eq:lapremiere}\\\\\n& (DR^++\\zeta I_d) \\check\\Sigma_{1,2}+\\check\\Sigma_{1,2} (DR^-+\\zeta I_d) = D\\check\\Sigma_{2,2} \\label{eq:ladeuxieme}\\\\\n& (R^-D+\\zeta I_d) \\check\\Sigma_{2,2} + \\check\\Sigma_{2,2} (DR^-+\\zeta I_d) = -a^2\\tilde Q \\label{eq:laderniere}\n\\end{align}\nSet $\\bar \\Sigma_{2,2} = P^{-1}D^{1\/2}\\check\\Sigma_{2,2} D^{1\/2}P$.\nDefine $C\\eqdef P^{-1}D^{1\/2}\\tilde Q D^{1\/2}P$.\nEq.~(\\ref{eq:laderniere}) yields $(M^-+\\zeta I_d) \\bar\\Sigma_{2,2} + \\bar\\Sigma_{2,2} (M^-+\\zeta I_d) = -a^2C$.\nSet $\\bar \\Sigma_{1,2} = P^{-1}D^{-1\/2}\\check\\Sigma_{1,2} D^{1\/2}P$.\nEq.~(\\ref{eq:ladeuxieme}) rewrites $(M^++\\zeta I_d) \\bar \\Sigma_{1,2}+\\bar \\Sigma_{1,2} (M^-+\\zeta I_d) = \\bar \\Sigma_{2,2}$.\nWe obtain that\n\n\\bar\\Sigma_{1,2}^{k,\\ell} = (\\nu_k^++\\nu_\\ell^-+2\\zeta)^{-1}\\bar\\Sigma_{2,2}^{k,\\ell} = \\frac{-a^2C_{k,\\ell}}{(\\nu_k^++\\nu_\\ell^-+2\\zeta)(\\nu_k^-+\\nu_\\ell^-+2\\zeta)}\\,.\n$\nSet\n$\\bar \\Sigma_{1,1} = P^{-1}D^{-1\/2}\\Sigma_{1,1} D^{-1\/2}P$.\nEq.~(\\ref{eq:lapremiere}) becomes $(M^++\\zeta I_d)\\bar \\Sigma_{1,1} + \\bar \\Sigma_{1,1}(M^++\\zeta I_d) = \\bar \\Sigma_{1,2} + \\bar \\Sigma_{1,2}^T$. Thus,\n\\begin{align*}\n\\bar\\Sigma_{1,1}^{k,\\ell} &\n\\resizebox{.9\\hsize}{!}{$\n= \\frac{\\bar\\Sigma_{1,2}^{k,\\ell}+\\bar\\Sigma_{1,2}^{\\ell,k}}{\\nu_k^++\\nu_\\ell^++2\\zeta}\n=  \\frac{-a^2C_{k,\\ell}}{(\\nu_k^++\\nu_\\ell^++2\\zeta)(\\nu_k^-+\\nu_\\ell^-+2\\zeta)}\\left(\\frac{1}{\\nu_k^++\\nu_\\ell^-+2\\zeta}\n+\\frac{1}{\\nu_k^-+\\nu_\\ell^++2\\zeta}\\right)\n$}\n\\\\ &\n= \\frac{C_{k,\\ell}}{ (1 - \\frac{2\\zeta}{a})(\\lambda_k+\\lambda_\\ell-2\\zeta + \\frac 2a \\zeta^2) +\\frac 1{2(a-2\\zeta)}(\\lambda_k-\\lambda_\\ell)^2}\\,,\n\\end{align*}\nand  the result is proved.\n\n\\bibliographystyle{siamplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{INTRODUCTION}\n\nHigh-mass $\\gamma$-ray\nbinaries (HMGBs) consist of a compact object, either a neutron star (NS) or black  hole (BH), orbiting a\nhot, massive O or B type star.   \nThe number of known HMGBs \nhas been increasing\nin recent years \\citep{2019arXiv190103624P}. However, in all but two HMGBs (LS\\,2883\/B1259--63 and MT91\\,213\/TeV J2031+4130), \nwhich host young pulsars, the nature of the compact object remains unknown. \n Two scenarios\nfor \nhigh-energy emission of HMGBs\n are \n usually discussed: (1) interaction of the\npulsar wind with the wind of the massive donor star leading to the \nformation of an intrabinary shock, or (2) jets powered by accretion\nonto a compact object (likely a BH). All HMGBs that have been observed with VLBI exhibit extended radio  \nemission on milliarcsecond scales, which can be attributed to either a pulsar-wind nebula  or to \njets \nproduced by accretion onto a BH (i.e., the classical microquasar scenario). \n\n\n\nX-ray observations can provide important information for \n understanding the nature of HMGBs. Therefore, \n these objects have been extensively observed in soft and hard X-rays. \n  In particular, {\\sl NuSTAR} observed LS \\,2883 \\citep{2015MNRAS.454.1358C}, MT91\\,213 \\citep{2017ApJ...843...85L}, 1FGL\\,J1018.6--5856  \\citep{2015ApJ...806..166A},  HESS J0632+057 \\citep{2019arXiv190803083P,2020ApJ...888..115A}, LS\\,I\\,+61$^\\circ$303 \\citep{2020MNRAS.tmp.2539M}, and LMC P3 \\citep{2020AAS...23545701C}.  All of these HMGBs were also observed in soft X-rays with {\\sl XMM-Newton}, {\\sl Swift}, {\\sl Suzaku}, and {\\sl Chandra} (see e.g., \\citealt{2009ApJ...697..592T,2013A&ARv..21...64D,2014AN....335..301K}). In all cases, the X-ray spectra appear to be consistent with featureless power-laws (PLs) with photon indices $\\Gamma\\simeq1.3-2$. \n  Joint fits to the soft X-ray and hard X-ray spectra provide no evidence of a spectral cut-off in the {\\sl NuSTAR} band.   No periodicity associated with the compact object spin has been found in any of the X-ray data (including two systems with known radio pulsars -- LS 2883 with PSR B1259--63 and MT91 213 with PSR J2032+4127). Using NuSTAR data, \n  \\cite{2015ApJ...806..166A} \n   and \\cite{2020ApJ...888..115A}\n   found no short-term variability, quasi-periodic oscillations, red noise, or any other temporal or spectral evidence of accretion in 1FGL J1018.6--5856 and HESS J0632+057, respectively.\n\n\n\n\n\n\n\n\n\n\n\n\n LS\\,5039, discovered by \\cite{1998A&A...338L..71M}, is\n a binary at a distance \n $d=2.9\\pm 0.8$ kpc \n composed of a massive ($M_\\ast=23$~M$_\\odot$)\n O6.5V((f)) type star ($V=11.2$ mag) and a compact object \nwith a poorly constrained mass, $\\sim (1$--$4) M_\\odot$. \nThe compact object orbits the star with a period $P_{\\rm orb}\\simeq \n3.9$\ndays, which is the shortest orbital period among all HGMBs.    The binary\norbit inclination angle is $i\\sim 30^\\circ$ \\citep{2005MNRAS.364..899C,Sarty11}. Radio observations have shown a persistent (over many binary periods) AU-scale\nasymmetric extension around LS~5039 whose morphology varies with orbital phase \\citep{2012A&A...548A.103M}.\n  Initially the extension  was interpreted as   jets  from an accreting compact object, \nwhich led to a ``microquasar'' classification \n \\citep{2000Sci...288.2340P}.\nHowever, more recently \n\\cite{2012A&A...548A.103M} attributed the varying extended radio morphology to a pulsar wind nebula whose shape varies due to the interaction with the wind of the massive star. Overall, the debate over\nwhether the compact object is an accreting BH or a pulsar interacting\nwith its surroundings and producing an extended nebula\nresembling ``jets'' is still ongoing (see e.g., \\citealt{2015CRPhy..16..661D}). \n\nObviously, the most direct evidence \n of LS 5039 containing a pulsar would be the \n detection of pulsations.  \n However, among all HMGBs radio\npulsations have only been detected in LS\\,2883 and MT91\\,213 \\citep{2009ApJ...705....1C,2014MNRAS.437.3255S}. \nEven if LS\\,5039 hosts a pulsar, the\nnon-detection of radio pulsations \nis not surprising \nbecause of the tight orbit and correspondingly large\noptical depth to free-free absorption \\citep{2006A&A...456..801D}. Therefore, searching for pulsations in X-rays may be a more promising approach for LS\\,5039 than for the other HMGBs. \n\nPulsations have been searched for in previous X-ray observations of LS 5039. For instance, \\cite{2011MNRAS.416.1514R} found no periodicity in the {\\sl CXO} observation of LS\\,5039. \nHowever, \\cite{Yoneda20}, \n hereafter Y+20, have recently reported the detection of \na period $P\\approx 8.96$ s in the {\\sl Suzaku} HXD data \nfrom 2007, and a potential counterpart at $P\\approx9.05$ s in the \\nustar\\ data from 2016 (the same data we analyze below), at photon energies above 10 keV. \nThe difference in the periods implies a fast spin-down,\nsuggesting that the compact object in LS\\,5039 could be a magnetar (i.e., a NS with a very high magnetic field; \\citealt{2017ARA&A..55..261K}). The statistical significance of this result is, however, questionable, as discussed below.\n\n\nThe shape of the hard X-ray spectrum can also provide critical\ninformation about the nature of the compact object. For instance, some\nNSs in accreting HMXBs show cyclotron resonant scattering features in\nthe range of about 10--100 keV \nand \nmany of them have exponential cutoffs around a few tens of keV\n(e.g., \\citealt{2002ApJ...580..394C}). Such spectral characteristics are very\nefficiently detected with {\\sl NuSTAR} (see e.g., \\citealt{2014ApJ...784L..40F,2016MNRAS.457..258T}). \nLS\\,5039's\nspectrum was  studied in the 0.01--10 MeV range with {\\sl INTEGRAL},\n{\\sl RXTE}, {\\sl Suzaku}, and {\\sl CGRO}. \nMost of these measurements suggest a relatively soft\nhigh-energy spectrum ($\\Gamma\\approx 2$), while the  {\\sl Suzaku} HXD spectrum is more\nconsistent with a simple extrapolation of the \n$\\Gamma=1.4$--1.6 PL spectrum\nmeasured below 10 keV with \n{\\sl CXO}, {\\sl XMM-Newton}, and {\\sl\n  Suzaku} XIS\n  \\citep{2009ApJ...697..592T} \n\n\n\n\nOf all known HMGBs, LS\\,5039 has the shortest orbital period.\nThis prompted us to carry\nout {\\sl NuSTAR} observations of LS\\,5039 over the  entire orbital period to obtain a\ncomplete spectral and temporal portrait of this HMGB. Additionally, we used archival {\\sl Suzaku} XIS observations \nthat also cover the full binary period and extend  spectral coverage to lower energies.\nIn Section \\ref{obs} we\ndescribe the {\\sl NuSTAR} and {\\sl Suzaku} observations and data reduction procedures. In\nSection \\ref{timing_analysis} we describe the binary orbit corrections to the\narrival times caused by the compact object's motion around its massive\ncompanion and present the results of the periodicity\n and  variability search.\n  In Section \\ref{specana} \nwe present the spectrum of LS 5039 and the results of spectral fitting\nas a function of the orbital phase. We discuss our findings and  conclude with a brief summary in Section \\ref{summ}.\n\n\\section{Observations and Data Reduction}\n\\label{obs}\n\n \n\nThe {\\it Nuclear Spectroscopic Telescope Array} (\\nustar, \\citealt{\n  harrison13ApJ:NuSTAR}) consists of two \n similar modules, FPMA and\nFPMB, operating in the energy range 3--79~keV. The \\nustar\\ \n observation  of  LS 5039 (ObsID 30201034002) started on 2016 September 1  \n  (MJD 57632.0972)\nand lasted for $T_{\\rm obs} \\approx 345\\,{\\rm ks}\\,\\,\n\\approx 1.024 P_{\\rm orb}$ ($\\approx60$ consecutive {\\sl NuSTAR} orbits).\n We processed the data using the\n\\nustar\\ Data Analysis Software, \\texttt{nustardas} ver.\\ 1.8.0. The photon arrival times were corrected to the solar system barycenter using the \\texttt{barycorr} tool\\footnote{See \\url{https:\/\/heasarc.gsfc.nasa.gov\/ftools\/caldb\/help\/barycorr.html}.} and the latest clock correction file\\footnote{nuCclock20100101v110.fits.gz \\url{http:\/\/nustarsoc.caltech.edu\/NuSTAR\\_Public\/NuSTAROperationSite\/clockfile.php}}. \nThe timing accuracy of {\\sl NuSTAR} is expected to be 65 $\\mu$s, on average \\citep{2020arXiv200910347B}. \nWe\nreduced the data using the \\texttt{nuproducts} task \nand HEASOFT ver.\\ 6.22.1. \n\nFor spectral analysis and binary lightcurves we used the flags \\texttt{-saacalc=2\n  --saamode=optimized --tentacle=yes} to correct for enhanced\nbackground activity visible at the edges of the good time intervals (GTIs)\nimmediately before entering the SAA. This resulted in a total GTI of about 166~ks.\nWe extracted source events \nfrom the 60\\arcsec radius circle\n around the\nsource position,\n which\nmaximized the S\/N ratio. Background events are extracted from an \nannulus around the source position with the inner and outer radii of\n120\\arcsec\\ and 200\\arcsec.\n\n\n\n\n  \n  \nTo extend \nthe spectral analysis to \n  lower \n   energies, \n we used \n  archival {\\sl Suzaku} data.\n  {\\sl Suzaku} observed LS\\,5039 between 2007 September 9  and 2007 September 15, with a total scientific exposure time of $\\approx 203$ ks (obsID 402015010). The observation, originally reported \n  by \\cite{2009ApJ...697..592T}, provided coverage of about 1.5 orbits of the  LS\\,5039 binary.     In the soft X-ray energy band \n  (0.3--12 keV), {\\sl Suzaku} had four X-ray telescopes \\citep{2007PASJ...59S...9S} each with \n  its own focal plane CCD camera (X-ray Imaging Spectrometer; XIS; \\citealt{2007PASJ...59S..23K}) having an $18'\\times18'$ field-of-view. The XIS2 camera was turned off in November 2006 due to an anomaly and is not used in our analysis. The XIS0 and XIS3 detectors use front-illuminated CCDs, while the XIS1 has a back-illuminated CCD.   \n  \n  We used  HEASOFT ver.\\ 6.25 \n  for {\\sl Suzaku} data reduction. The data were reprocessed using the {\\tt aepipeline} script and were reduced following the standard procedures\\footnote{See \\url{https:\/\/heasarc.gsfc.nasa.gov\/docs\/suzaku\/analysis\/abc\/}.}. The source spectra and light curves were extracted from a $3'$ radius circle centered on the source position, while the background spectra and light curves were extracted from a $3'$ circle placed $7\\farcm5$ south of the source in \n  each of the three XIS images. The response matrix and ancillary response file \n  were made using the {\\tt xisrmfgen} and {\\tt xissimarfgen} tools, respectively. \n  Since there are  known calibration issues near the Si edge in the XIS detectors near 2 keV  (see e.g., \\citealt{2011PASJ...63S.991S,2013ApJ...772...83L}),\n   we exclude the 1.7--2.3 keV energy range from all of our spectral fits.\n    Prior to producing the light curves, the event arrival times were corrected to the solar system barycenter using the {\\tt barycorr} tool. Since the {\\sl Suzaku} XIS time resolution was only 8 s in this observation, we did not use the XIS data for the periodicity search. \n  We \n  also used the (barycentered) {\\sl Suzaku} HXD data \n  to investigate the candidate periodic signal  reported by Y+20, but we did not use them for spectral analysis\n  (because of the strong background contamination in this non-imaging instrument).  All errors quoted throughout the paper are reported \nat the $1\\sigma$ level, unless otherwise noted.\n\n\n\n\n\n\\section{Timing Analysis}\n\\label{timing_analysis}\n\nA fully coherent periodicity search for an observation with a length comparable to the LS~5039 binary period is a ``needle in a haystack''  type problem.   The large uncertainties in the orbital ephemeris (see Table \\ref{tab:orb}) require a \nprohibitively large \ngrid in the multidimensional parameter space to guarantee that the \nperiodic signal \nis not missed (see the discussion in  \\citealt{2012MNRAS.427.2251C}). An alternative approach \nis to segment the observation into multiple time  intervals during  which the radial velocity of the compact object is approximately constant, and \nsearch for periodicity within each segment by analyzing the distribution of Fourier power in the time-frequency domain (a dynamic  power spectrum search; e.g., \\citealt{2012hpa..book.....L}). \n\nSuch an approach \n was employed by Y+20 (who found a period candidate $P=9.05$ s in the \\nustar\\ data). \nThese authors only searched for \n a signal with \n period $P>1$~s. They \n justify this restriction due to the relatively small number of photons\n in the 10--30 keV band chosen for the periodicity search in the \\nustar\\ data. Furthermore, they  \n justify this energy band selection \n by the fact that the 8.96 s period  candidate was seen in {\\sl Suzaku} HXD, which \n is only sensitive above 10 keV.\n However, \n we see no reason for the signal not to be present below 10 keV \n because there is no change in the source spectrum \n (see Section \\ref{specana}). \n Since we \n cannot exclude the possibility\n that the true period is \n different from that claimed by Y+20, we \n perform a period search in a broader range of frequencies and a \n different \n energy range.\n \nTo improve the sensitivity by decreasing the spread of the potential signal in the frequency domain, we (1) use an optimal division into time segments \nthat depends on  the frequency intervals we are searching in, (2) introduce a statistic that provides a higher sensitivity to a signal than the simplistic incoherent summing of Fourier powers from non-overlapping time intervals within the observation,   and (3) apply the R\\\"omer delay correction (e.g., \\citealt{1976ApJ...205..580B}) to the photon arrival times using the best known ephemeris. Then we perform the dynamic pulsation search. \nThis approach allows us to search up to much higher frequencies \n(e.g, 1\\,000 Hz) \nthan the 1 Hz limit used in Y+20. \n\n\n \nSince the nature of the compact object is unknown,  \nwe adopt the approach, \ndescribed in Section \\ref{dynamicfourier}, which allows us to search for both \nperiodic signals and \nquasi-periodic oscillations (QPOs). We also perform a burst-like variability search (on scales from 1 s to 200 s)\nand orbital variability characterization on larger timescales (Section \\ref{bin_lc}). \n  \n\\begin{deluxetable*}{ccccccc}[t!]\n\\tablecaption{Orbital parameters for LS\\,5039 inferred from different observations\n\\label{tab:orb}}\n\\tablewidth{0pt}\n\\tablehead{\n& \\colhead{Aragona09}  &  \\colhead{Casares11} & \\colhead{Sarty11} &\n\\colhead{Yoneda20} & \\colhead{Yoneda20} &\n\\colhead{this work} \\\\\n &   &   &  &\n\\colhead{\\sl NuSTAR} & \\colhead{Suzaku} &\n}\n\\startdata  \n$P_{\\rm orb}$ (d) & $3.90608\\pm0.00010$ & $3.90608\\pm0.00008$   & 3.906           &  3.90608 & 3.90608 & $3.9057\\pm0.28$ \\\\\n$T_0$ (MJD)       &\t$52825.48\\pm0.05$\t& $52477.58\\pm0.06$     & $55016.58\\pm0.06$ \n& $57629.250_{-0.030}^{+0.023}$   & $54352.455_{-0.035}^{+0.05}$ & \n$57633.71\\pm0.06$ \\\\ \n$e$               & $0.337\\pm0.036$     & $0.35\\pm0.03$         & $0.24\\pm0.08$ \t\n& $0.306_{-0.013}^{+0.015}$ & $0.278_{-0.023}^{+0.014}$  & $0.289\\pm0.09$       \\\\\n$W_\\ast$ (deg)       & $236\\pm5.8$         & $212\\pm5$             & $237.3\\pm21.8$    \n& $236.8_{-3.1}^{+2.3}$  & $234.6_{-3.3}^{+5.1}$ \t&\t\t\t\\\\ \n$a_\\ast\\sin i$ (lt-s) & $3.33\\pm0.15$   & $4.06\\pm0.16$         & $4.11\\pm0.35$\t&&&\\\\\t\n$W_p$ (deg)     \t&  \t$56\\pm5.8$        & $32\\pm5$\t        & $57.3\\pm21.8$\t \n& $56.8_{-3.1}^{+2.3}$ & $54.6_{-3.3}^{+5.1}$ & $44.6\\pm4.1$ \\\\\n$a_p\\sin i$ (lt-s)  &  \t$48\\pm14$      & $52_{-19}^{+9}$\t    & $52_{-19}^{+10}$\t\n& $48.1\\pm0.4$ & $53.05_{-0.55}^{+0.7}$ & $48.1\\pm2.7$   \t\t\n\\enddata\n\\tablecomments{Orbital parameters from \\cite{2009ApJ...698..514A}, \\cite{Casares11}, \\cite{Sarty11}, \n Y+20 and this work. Subscripts $\\ast$ and $p$ correspond to the massive star and the compact object (putative pulsar), respectively. For the \\cite{2009ApJ...698..514A}, \\cite{Casares11},  and \\cite{Sarty11} orbital solutions, the projected semi-major axis \n$a_p\\sin i$ \nwas calculated assuming \n $m_\\ast = (23\\pm 3) M_\\odot$\nwhile\nthe compact object's mass\nwas assumed to be $m_p =(1.6\\pm0.4) M_\\odot$ for \\cite{2009ApJ...698..514A} and $m_p=1.8_{-0.6}^{+0.2}M_\\odot$  \nfor \\cite{Casares11} and \\cite{Sarty11}. We use the \\cite{Casares11} orbital parameters \nfor the R\\\"omer delay correction. The Y+20 \nand `this work' parameters were not measured from observations of the massive star but \nobtained from fits maximizing the significance of the putative period near 9.05 s. \nThe large uncertainties of the `this work' parameters take into account the multitude of solutions of about the same significance in the vicinity of the best-fit solution (see Section \\ref{p9s_cand}).\n}\n\n\\end{deluxetable*}\n\n\n\n\n\\subsection{Correction for the R\\\"omer delay caused by the orbital motion}\n\nAs the putative pulsar\nis orbiting a massive star, the distance between the pulsar and the observer changes with  orbital phase, which translates into changing times of photon travel \nto the observer. \nThis effect can be equivalently described in the observer's frame as a \nDoppler shift of the pulsation frequency \nvarying with the binary phase because of\nthe changing radial velocity of the\npulsar.\nDifferent authors have inferred slightly different sets of orbital parameters from optical observations of the massive companion (see examples in Table \\ref{tab:orb}).\nTo correct the event arrival times \nfor this effect (R\\\"omer delay), we adopted the\n`eccentric fit + 1d oscillation' orbital solution from\n \\cite{Casares11},\nwhich included \nmodulation of the radial velocity \nwith a 1 day period, possibly caused by non-radial oscillations of the massive companion in its eccentric orbit.\nThe orbital dependencies of the R\\\"omer delay and Doppler shift \nof the other published binary solutions are within $\\approx \\pm 1\\sigma$ uncertainties of the \\cite{Casares11} curves for these quantities (see Figure \\ref{fig:deltat} \nand Figure \\ref{fig:vel_acc} in the Appendix). \n\nThe projection of the orbit's semi-major axis \nonto the sky plane, $a\\sin i$, and the longitude of periastron, $W$, in \n\\cite{Casares11} (and the other papers quoted in Table \\ref{tab:orb})\npertain to the massive stellar component of the binary.\n They are connected with the corresponding compact object \n parameters as follows, \n\\begin{equation}\n\ta_p\\sin i=\n\t\\frac{m_\\ast}{m_p}\\, a_\\ast\\sin i\\,,\n\t\\quad\\quad\n\tW_p=W_\\ast-180^\\circ,\n\\label{eq1}\n\\end{equation}\nwhere the subscripts $\\ast$\nand $p$ correspond to the massive\nstar and the compact object (putative pulsar), respectively, $m$ is mass, and  $i$ is the orbital inclination (the angle between the orbital plane and the plane of the sky). \nFor the pulsar (NS)  mass we use $m_p=1.8_{-0.6}^{+0.2}M_\\odot$,  from the assumption that $m_p$ should be in the range of 1.2--2.0 solar masses, which gives $a_p\\sin i=52_{-19}^{+9}$\\,lt-s = $(1.55_{-0.6}^{+0.27})\\times 10^{12}$\\,cm.\n\n\n\n\nThe corrections to the photon arrival times  due to orbital motion (R\\\"omer delay)\nare calculated as follows,\n\\begin{eqnarray}\n\tt_\\text{corr}=t-\\frac{a_p\\sin i}{c}\\left[\\sin W_p(\\cos E-e)+\\right.\\nonumber\\\\\\left.\n\t\\sqrt{1-e^2}\\cos W_p\\sin E\\right]\\,.\n\\label{eq:romercorr}\n\\end{eqnarray}\nHere $e$ is the eccentricity of the orbit, and $E$ is the eccentric anomaly,\n\\begin{equation}\n\tE-e\\sin E=\\Omega_{\\rm orb}(t-T_0)\\,,\n\t\\label{exc_anomaly}\n\\end{equation}\nwere $\\Omega_{\\rm orb}=2\\pi\/P_\\text{orb}$, \n$P_\\text{orb}$ is the orbital period, \nand $T_0$ is the epoch of  periastron \\citep{1976ApJ...205..580B}. The right hand side of Equation (\\ref{exc_anomaly}) is commonly called the mean anomaly.\n Depending on the orbital phase, the correction ranges from \n$-$40 s to $+$60 s (see Figure \\ref{fig:deltat}).\n\n\\begin{figure}[t]\n\\centering\n  \\includegraphics[width=0.5\\textwidth]{delta_t.png}\n  \\caption{Arrival time corrections due to R\\\"omer delay during our {\\sl NuSTAR} observation for 6 \n  sets of orbital parameters (see Table \\ref{tab:orb}). Shown is $\\Delta t = t_\\text{corr}-t$ vs.\\ $t$, given by Equation (\\ref{eq:romercorr}). The shaded area shows the  uncertainty of $\\Delta t$ due to the  uncertainties of the orbital parameters for the binary ephemeris from \\cite{Casares11} (the model with 1\\,d oscillations).\n }\n  \\label{fig:deltat}\n\\end{figure}\n \n\n\n\n\\subsection{Periodicity search in the \\nustar\\ data}\n\\label{sec:perser}\n\nWe searched for periodic or quasi-periodic signals up to $f_{\\rm max}=1000$ Hz\nby analyzing   the arrival times of $N=56\\,647$ events, \nregistered during $\\approx190$ ks (when the target was not occulted by the Earth),\nin the photon energy range of 3--20 keV.\nWe excluded higher energies to reduce the background contamination as \\nustar's sensitivity decreases at higher energies. \nWe corrected the event\narrival times \nfor the R\\\"omer delay using the best-fit binary parameters \nfrom Casares et al.\\ 2011\n(Table \\ref{tab:orb}) prior to the search. \n\n\\subsubsection{\nFourier power spectrum for the entire \\nustar\\ observation \nassuming the binary parameters are known with high precision\n}\n\\label{fourier_entire_obs}\nUsing the corrected arrival times from the entire \\nustar\\ observation, we \ncalculated the Fourier power \nspectrum \n(see the top\npanel of \nFigure \\ref{fig:fur} and the Appendix \\ref{appendixA}).\nThe top\npanel of Figure \\ref{fig:fur} shows the values of the Fourier power ${\\cal P}_n$ as a function of frequency $f=n\\,\\Delta f$, where $\\Delta f \\simeq 3\\times 10^{-7}$ Hz is the \nfrequency bin width \n(only 24,448 power values with ${\\cal P}_n>20$ are shown).\nThe power is normalized in such a way that the mean $\\overline{{\\cal P}_n} = 2$ for a Poisson-distributed noise.\nThe \nhigh values of ${\\cal P}_n$ \nat low frequencies, $f\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 0.003$ Hz, are associated with the periodic motion of the \\nustar\\ satellite around the Earth (the frequency of \\nustar\\ revolution, $1.72\\times 10^{-4}$ Hz, and its \n15 harmonics are shown by \nblue vertical lines). \nAt higher frequencies \nnone of \nthe ${\\cal P}_n$ values is  outstanding, and their significances do not exceed $4\\sigma$.\nThus, we conclude that\nno significant periodic signal is detected. \n\nTo look for quasi-periodic signals, we increased the width of frequency bins by factors of 10, 100, and 1000 (compared to the natural width\n$T_{\\rm obs}^{-1}\\approx 2.89$ $\\mu$Hz)\nbut found no outstanding peaks in the binned \nFourier power spectra.  \n\n\\begin{figure*}[]\n\\includegraphics[width=1\\textwidth]{Pn_vs_f.pdf}\n\\includegraphics[width=1\\textwidth]{z2_alpha_t_f.pdf}\n  \\caption{\n  The top panel shows\n   the Fourier power spectrum ${\\cal P}_n$, \n   calculated from the entire \\nustar\\ observation\n    (see \n    Appendix \\ref{appendixA} and Section \\ref{fourier_entire_obs}).\n    Prior to calculating the Fourier power spectrum, the  arrival times of photons (with energies restricted to the 3--20 keV band) were corrected for the R\\\"omer delay \nusing the best-fit orbital parameters from \\cite{Casares11} (see    Table \\ref{tab:orb}).     The vertical blue lines correspond to the \\nustar\\ orbital period and its harmonics. \n    The bottom panels show the \n     dynamic (time-resolved) spectra of Fourier power\n     and significance,\n      calculated individually for each  of the 60  \\nustar\\  orbits, after applying the same R\\\"omer delay correction (see Section \\ref{fourier_spec_sep_orbs} and Appendix \\ref{appendixA}).\n      To improve the visualization quality,\n      we do not show \n      power (and significance) values in each frequency bin\n      but \n      instead divide the entire log-frequency range in 200 segments of equal size \n      and plot (using  color) the maximum Fourier power, ${\\cal P}_k^{\\rm max}$ (left), within the $k$-th segment and its significance, $\\alpha_k$ \n      (right).\n   The vertical black stripes are the gaps due to the occultation of LS 5039 by the Earth. \n    }\n\\label{fig:fur}\n\\end{figure*}\n\n\n\n\n\\subsubsection{\nFourier spectra in separate \\nustar\\ orbits }\n\\label{fourier_spec_sep_orbs}\n\nThe apparent lack of pulsations in the Fourier power spectrum of the entire observation could be due to \na large difference between the actual  binary parameters and \nthe parameters used for the R\\\"omer delay correction.\nBecause of this difference, the binary-phase-dependent frequency shift may not be fully compensated by the R\\\"omer delay correction. As a result, the signal coherence was lost, the power peak corresponding to the (unknown) pulsation frequency was spread over many frequency bins, and the peak's height was strongly reduced. \nTo mitigate the coherence loss, \nwe \nsearch for pulsations in much shorter time segments, corresponding to the intervals of visibility of LS\\,5030 in the 60 consecutive  \\nustar\\ orbits covered by our observation.\nDuring the relatively short intervals, 3.2 ks on average, \nthe difference between the actual and best-fit radial velocities  of the compact companion \ndoes not change as  much as over the entire orbit, and\nthere is a higher chance that the signal coherence is preserved. \n\nSimilar to the search in the entire \\nustar\\ observation, we calculated 60 Fourier power spectra \nand \n  corresponding signal detection significances.\nThe results of this dynamical timing are shown in the bottom\npanels of Figure \\ref{fig:fur} as  time-frequency images in which the  power and detection significance values are shown by the \nbrightness of the image ``pixels''.\nThe vertical light and dark stripes correspond to intervals of visibility and occultation of LS\\,5039, respectively.\n\nIf pulsations were detected, they would be seen as a sequence of brighter (lighter) pixels along the time axis at frequencies around the pulsation frequency. If the actual binary parameters coincided with the \nassumed ones, this sequence would be seen as a horizontal stripe parallel to the time axis. Deflections of the brighter pixels from a horizontal stripe would provide the difference between the \n actual values of the radial velocity and the ones used in the R\\\"omer delay correction, in separate \\nustar\\ orbits. \n\nWe see from the bottom-right panel of Figure \\ref{fig:fur} that the search in separate \\nustar\\ orbits also did not provide periodicity detection. \nAlthough some of the $\\alpha_k$ values may appear marginally significant, one has to keep in mind that $\\alpha_k$ is defined for a single \\nustar\\ orbit (see the Appendix \\ref{appendixA}), and  the true  significance is therefore lower when all orbits are included (due to the larger number of statistical trials). Moreover, we do not see any extended \n(along the horizontal direction) connected clusters of adjacent pixels.   \n\nThe lack of detection could be due to the lower sensitivity of this search. The first reason for the sensitivity loss is the  smaller number of counts in separate orbits \nthan in the entire observation (on average, 994 versus 56,647 counts). Since, for a periodic signal with a given pulsed fraction, the power ${\\cal P}_n$ is proportional to the number of counts, \nweak pulsations would not be detected.\n\nThe second reason for the sensitivity loss is the spread of signal frequency over several frequency bins caused by the Doppler shift.\nIn the $i$-th \\nustar\\ orbit, the \nspread associated with the Doppler shift\nunaccounted  for by the R\\\"omer delay correction can be estimated as $(\\delta f)_i \\sim f\\, |\\Delta \\dot{v}_{\\parallel,i}| T_i\/c$, where $T_i$ is the visibility interval, and $\\Delta \\dot{v}_{\\parallel,i}$ is the difference between the \nassumed and actual radial accelerations of the binary motion in the middle of the $i$-th orbit (see Appendix \\ref{appendixB}). This spread becomes greater than the natural width $T_i^{-1}$ of the frequency bin for $f > \\tilde{f}_i \\sim c\\, (|\\Delta \\dot{v}_{\\parallel,i}| T_i^2)^{-1} = \n3.9\\, [|\\Delta \\dot{v}_{\\parallel,i}|\/(3\\,{\\rm m\\,s}^{-2})]^{-1} [T_i\/(3.15\\,\n{\\rm ks})]^{-2}$ Hz.\nAt a frequency $f$ substantially higher than \n$\\tilde{f}_i$ the  peak in the signal power is spread over $f\/\\tilde{f}_i$ bins, and the peak height will be reduced by about the same amount.\n\n\\subsubsection{Search for pulsations in a dynamic Fourier  spectrum with frequency-dependent time windows}\n\\label{dynamicfourier}\nSince the R\\\"omer delay correction is \n imperfect\ndue to the binary ephemeris uncertainties,\na periodic signal \n with a certain frequency $f_0$ in the reference frame of the pulsar\n can be spread \n by the Doppler effect\n over \n a number of neighboring frequency bins in the observer's reference frame\n(see  the Appendix \n\\ref{appendixB} for details).\nThis spread can be mitigated by \nsplitting the observation duration $T_{\\rm obs}$ into $N_w$ shorter time windows of a length $T_w = T_{\\rm obs}\/N_w$ (hence wider frequency bins $T_w^{-1}$), but then the signal \ncan be shifted to different frequencies $f$ in different time windows.\n  In order to maximize the sensitivity to such a signal, one should  \nselect optimal lengths of the time windows \n  and use an efficient algorithm for detecting the signal \n  in the time-frequency domain.\n\nThe optimal lengths \nof the time \nwindows are determined by the\nrequirement that the maximum possible drift in frequency does not exceed $T_w^{-1}$.  Such  choice of\n$T_w$ is optimal because if one chooses an even smaller $T_w$, then  events  are  lost  (as the  number  of  events  is  proportional to $T_w$) and the Fourier power decreases. \n\nAs we show in Appendix \\ref{appendixB}, the optimal lengths and numbers of time windows depend on the signal frequency\n($T_w\\sim 9.9 f_0^{-1\/2}$ ks, $N_w\\sim 35 f_0^{1\/2}$ in our case). \nFor practical purposes,\nit is convenient to divide the entire 0--1000 Hz frequency range into 7 broad frequency intervals with\ndifferent numbers and lengths of time windows.\nFor each of these time windows we calculate the Fourier power spectrum ${\\cal P}_n$ in the 0--1000 Hz frequency range, with a frequency resolution \nof $T_w^{-1}$.\n\nAs the next step,\nwe split the entire frequency range \ninto \nsegments \n   $ f_m(1-\\beta) < f < f_m(1+\\beta)$, \nwhere the coefficient $\\beta$ is proportional to $\\delta v_\\parallel\/c$,\n$\\delta v_\\parallel$ is the maximum residual uncertainty of the pulsar's radial velocity in the appropriate time window, and the\ncentral frequency $f_m$ of the $m$-th segment\n($m=1, 2, \\ldots$) satisfies Equation (B13). \nThe segment width $2\\beta f_m$ must be large \nenough to ensure that the entire signal,\n  whose Doppler-shifted frequency $f$ is \n  varying with time, \n is contained within this \n segment (see Figure \\ref{fig:mc2} in the Appendix).  \n\nEach of these segments (their total number is about 2880, \nfor the chosen $\\beta=0.0022$\nin the 0.01--1000 Hz range)\nis inspected  for the presence of signal signatures. \nAny chosen segment\nis within one of the seven broad frequency intervals, described in the Appendix \\ref{appendixB}, which determines \nthe number $N_w$ and length $T_w$ of the time windows\n(hence the choice of the precalculated Fourier power spectra) appropriate for the segment analysis.\n\n\n\n \n\n\nIn the time-frequency plane, an $m$-th frequency segment\nconsists of $N_w$ time windows and $N_f = 2\\beta f_m T_w$ frequency bins, i.e., of $N_w\\times N_f= 2\\beta f_m T_{\\rm obs} = 690 (\\beta\/10^{-3}) f_m$ elements\nfor which Fourier powers \n${\\cal P}_{n,j}$ have been calculated ($n$ and $j$ number the frequency bins and time windows, respectively). \nTo locate \npossible signal signatures in the segment, we pinpoint the largest values ${\\cal P}_{j}^{\\rm max}$ \nin the sets of $N_f$ powers ${\\cal P}_{n,j}$ within each of the time windows $j$. \nIf the ${\\cal P}_j^{\\rm max}$ values represent a sufficiently strong signal, then the mean of these values over the entire\ntime of observation\n\\begin{equation}\n\\mu=\\frac{1}{N_w}\\sum_{j=1}^{N_w} {\\cal P}_{j}^{\\rm max},\n\\label{eq:mu}\n\\end{equation}\nshould significantly exceed a similarly defined mean for the noise,\n$\\mu_{\\rm noise}$.\n\nTo characterize the significance of a possible excess of the measured $\\mu$ over $\\mu_{\\rm noise}$,\n   we introduce the following $s$-statistic: \n\\begin{equation}\n\ts=(\\mu-\\mu_\\text{noise})\/\\sigma_\\text{noise}\\,,\n\t\\label{eq:stat}\n\\end{equation}\nwhich  \nprovides the signal significance in units of standard deviation.  Here, $\\sigma_\\text{noise}$ is the standard deviation of the noise.\n\nBecause the observed data contain numerous time gaps, and the count rate \nchanges with time,\nwe used Monte-Carlo simulations to infer $\\mu_\\text{noise}$ and  $\\sigma_\\text{noise}$  for each  $N_w$ \n(when simulating noise,  we included gaps larger than 60 s).\n\n\nIn the right panel of Figure \\ref{fig:stat} \n we plot\nthe $s$-statistic values for\n  for each\n  of 1261 frequency segments in which $s$ is positive\n(the total number of frequency segments is 2496 \nin\nthe 0.017--1000 Hz range). \n  Although there are several peaks  slightly exceeding the $3\\sigma$ level (i.e., $s>3$),\n no signal candidates are passing the $4\\sigma$ threshold.\n\n\n\n\n  \n\n\n\\begin{figure*}[hbt]\n  \\includegraphics[width=0.48\\textwidth]{fig3_1_new.png}\n  \\includegraphics[width=0.48\\textwidth]{fig3_2_new.pdf}\n  \\caption{\n   Left panel shows the $H$ statistic \n   calculated for the frequencies up to 0.017 Hz. \n   Blue vertical lines  correspond to the orbital frequency of \\nustar\\ \n($1.7\\times 10^{-4}$ Hz)\nand its harmonics. \nThe H-statistic peaks are aliases caused by the presence of quasi-periodic gaps (the gap width \nvaries slightly with time) due to the occultation of the target by Earth during the long \\nustar\\ observation\n and SAA passages. \nRight panel shows all positive $s$-statistic values  (see Equation \\ref{eq:stat}) in 1261  frequency segments\nbetween 0.017 Hz and 1000 Hz.\nDots on red vertical lines show the $s$ values for the (artificial) test signals (see the description and Figure (\\ref{fig:mc2}) in the Appendix \\ref{appendixB}); the numbers near the dots are the signal fractions.\n}\n\\label{fig:stat}\n\\end{figure*}\n\n\n\n  \n  \n  \n  \n  This panel also shows the $s$-statistic values computed for six simulated periodic\n  sinusoidal test  signals with varying levels of   \n  signal strength (characterized by the \n signal fraction $p$, with $1-p$ being the   unpulsed\n  fraction)\n  that are imperfectly corrected for the R\\\"omer delay\n  (see the time-frequency images in Figure \\ref{fig:mc2} in Appendix \\ref{appendixB}). \nWe see that the detectability of a signal with a given signal \nfraction strongly depends on  the signal's frequency (the higher the frequency the larger $p$ must be for the signal to be detected). At plausible young pulsar frequencies $\\sim 3$--100 Hz, the signal fractions \n should significantly exceed 0.1--0.2 to be detected with this method in the available \\nustar\\ data. \n   \n   For the low-frequency part of the Fourier power spectrum,\n    (Figure \\ref{fig:stat}, left panel) we \n   use the more conventional $H$-statistic,\n   defined as $H=\\max\\{Z_m^2-4m+4\\}$ for $1 \\le m \\le 20$ \\citep{1989A&A...221..180D}, where $Z_m^2$ is the statistic commonly used in periodicity searches in X-ray and $\\gamma$-ray astronomy \n   (see, e.g., Buccheri et al.\\ 1983). \nThis is possible  because for $f\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 0.01$ Hz the residual drift due to the imperfect R\\\"omer delay correction \n   does not  exceed the natural width ($\\sim T_w^{-1}$) of the peak in $H$-statistic spectrum \n   during the entire observation (i.e., $N_w=1$; see the Appendix \\ref{appendixB}). Although the $H$-statistic values at lower frequencies are high, they mostly coincide with multiple integers of the \\nustar\\ orbital frequency (shown by blue vertical lines) while the others are likely to be  aliases due the visibility gaps that are somewhat varying in their duration. No credible signal is detected.     \n     \n\\subsection{$P=9.05$ s candidate} \n\\label{p9s_cand}\n\nY+20\nsearched the {\\sl Suzaku} HXD data of 2007 September \n for \n periodic \n   signals with \n periods $P>1$~s in the 10--30 keV band. \n They found \n maximum $Z_{4}^2=68.0$ (the other $Z_n^2$ values were not reported)\n at $f_{\\rm HXD}=0.1116510(5)$ Hz, or $P_{\\rm HXD}=8.95648(4)$ s, with the estimated\n significance of 98.8\\%.\n\n \n \nIn a subsequent  search in the \nabove-described {\\sl NuSTAR} data \n(191 ks net observing time, 12,000 events in the 10--30 keV energy band, and assuming $P>1$~s) \nY+20 obtained a maximum $Z_{4}^2$=66.9 at\n$f_{\\rm NuST} = 0.1104507(4)$ Hz or $P_{\\rm NuST}=9.05381(3)$ s,  with an  estimated significance of only 93\\%.\n \n To find these period values, Y+20 varied the orbital parameters in the ranges provided in Table 1 of that work, applied the R\\\"omer delay correction for each parameter set, and chose the set, and the corresponding period, that maximized the $Z_4^2$ value.\nThe uncertainties of the periods given in Y+20  appear to be underestimated because they do not account for the uncertainties in the ephemeris parameters (as we show below).\n Y+20 concluded that two  \n incompatible  sets of  orbital parameters are needed to maximize the  strength of the {\\sl NuSTAR}  and {\\sl Suzaku} candidate periodic signals.\n Moreover, the ephemeris \n that maximizes the periodic candidate signal in the {\\sl NuSTAR} data\n is incompatible with  any previously published ephemeris (within their uncertainties;  see Table \\ref{tab:orb} and Figure \\ref{fig:deltat}).\n \n\n\n \\begin{figure}[t]\n \\centering\n  \\includegraphics[width=0.5\\textwidth]{dynamic_obs.png}\n  \\includegraphics[width=0.5\\textwidth]{dynamic_puls.png}\n  \\caption{ \n  The top panel shows $Z_1^2$  as a function of \n  frequency and \n   time near\n  $f_0=0.110498$ Hz ($P_0=9.04996$ s),  shown by the dashed horizontal line. \n  The $Z_1^2$ was calculated using events extracted \n  from the $r=38''$ aperture in the 10--18 keV band.\n   The red curve shows the \n  Doppler shift due to the orbital motion as a function of the observation time \n  for the \n  orbital parameters that we found (see Table \\ref{tab:orb}).\n    The bottom panel shows the same thing as the top panel but the photon  arrival times  are corrected for the orbital motion with this ephemeris. \n In both cases the color bars show the value of $Z_1^2$ \n per time window.\n }\n  \\label{9ssignal_1}\n\\end{figure}\n  \n   \n\\begin{figure}[t]\n \\centering\n  \\includegraphics[width=0.53\\textwidth]{905sig.pdf}\n  \\includegraphics[width=0.5\\textwidth]{lightcurve_LS5039_all.pdf}\n  \\includegraphics[width=0.5\\textwidth]{lightcurve_LS5039_270-320ks.pdf}\n  \\caption{ The top panel shows $Z_1^2(f)$ \n  in the vicinity of  $f_0=0.110498$ Hz\n  ($P_0=9.04996$ s) calculated \n  for the entire observation \n  with arrival times corrected for the \n  orbital motion with \n  the same ephemeris as in Figure \\ref{9ssignal_1}. The vertical red lines mark the aliases of the main peak  which are offset from  $f_0$\n  by $\\pm 0.017$ mHz, which is the frequency of the \\nustar\\ orbit around the Earth.\n  The middle and bottom panels show the pulse profiles \n  folded on the \n  period $P_0$ for the entire observation and for the 270--320 ks interval \n  (counting from the beginning of the observation), where the signal is particularly strong\n  (see Figure \\ref{9ssignal_1}). \n }\n  \\label{9ssignal_2}\n\\end{figure}   \n   \n\n  \nTo verify the period and the  significance  \nreported by Y+20, we have performed \n an independent period search \nin the {\\sl NuSTAR} data  \naround $P=9.05$ s. \nChoosing a set of 5 orbital parameters within the $2\\sigma$ uncertainties of the Casares et al.\\ (2011) ephemeris,  we applyed the R\\\"omer delay correction to the \ntimes of arrival,\ncalculated $Z_{m}^2(f)$ for $m=1,\\ldots, 20$ (using \nevents from the entire observation) in the frequency interval of (0.109--0.113) Hz, and used the $H$-test \\citep{1989A&A...221..180D} \nto determine the maximum number of significant harmonics.\nVarying the orbital parameters on the 5-dimensional grid, we found the parameter set that maximizes $Z_m^2$ and the corresponding frequency.\nWe repeated these calculations for various energy ranges within 3--40 keV and various source aperture radii up to $r=60''$ to choose an optimal (maximizing $Z_m^2$) energy range and aperture radius. \nAs a result, in  the  10--18 keV energy range and for the aperture radius $r=38''$,\nwe found a  signal (shown in Figure \\ref{9ssignal_2}) with maximum $Z_1^2=60$\nat $f=0.1104977$ Hz ($P=9.049962$ s).\nIn the $Z_m^2(f)$ dependence, the peak at $f=0.1104977$ Hz  \nis surrounded by many other peaks  with slightly lower heights, including a peak at \n$f=0.1104507$ Hz \nreported by Y+20. These \npeaks, appearing at \ndifferent combinations of orbital parameters, look virtually as significant as the highest one, so that we cannot prefer one peak to another. Therefore, the true uncertainty of the putative pulsation frequency (and the fitted orbital parameters) is determined not by the width of a separate peak but by the width of the entire `cluster of peaks', $\\sim 3\\times 10^{-5}$ Hz in our case,  which is about two orders of magnitude larger than the uncertainty claimed by Y+20. \nAccounting for this uncertainty, the frequency and period of the putative pulsations are   \n$f=0.11050(3)$ Hz and $P=9.050(2)$ s), for the orbital parameters listed in the column `this work' of Table \\ref{tab:orb}. \n\n We  note that higher harmonics are not required by the  $H$-test.\nHowever,  for comparison with Y+20, who reports $Z_4^2$, we also calculated it and found  $Z_4^2=71$ for our best fit, at $f= 0.1104977$ Hz.\nWe also note that\n our optimal  \norbital parameters are much closer to those \nfound in the previous papers\nthan the set of parameters \nsuggested by Y+20\n(see Figures \\ref{fig:deltat} and \\ref{fig:vel_acc}).\n\n\n\n\n  \n  To explore the distribution of the Fourier power as a function of \n  time (or binary phase), we calculated\n   a time-resolved Fourier spectrum, \n i.e.\\  the $Z_1^2$\n  distribution in the time-frequency plane around $f=0.110498$ Hz  \n  ($P=9.04996$ s).\n To reduce the effect of time gaps, we used \n a sliding time window with the size of $T_{\\rm obs}\/10$ which is moved by 10\\% at each step \n  (see Figure \\ref{9ssignal_1}). \n  The plots show that the strongest contribution to the signal comes \n  from a time interval close to the end of the \\nustar\\ observation (between 270 and 330 ks, counted from the start of the  observation), near the binary apastron.\n\nWe note that the \n$s$-statistic, introduced in Section \\ref{dynamicfourier}, is not sensitive to such a signal because the Fourier power does not remain constant during the observation time span. \nFigure \\ref{9ssignal_2} shows the $Z_1^2$ distribution calculated from the entire observation \naround $f=0.110498$ Hz ($P=9.04996$) and the corresponding folded pulse profiles from the entire observation and from the part of it with the strongest signal. \n\n\n\n\n In addition,  we re-analyzed the {\\sl Suzaku} HXD data and confirmed   \n the signal candidate reported by Y+20, with maximum $Z_4^2 =67.8$ at $P=8.95648$ s.\n We also analyzed jointly the {\\sl NuSTAR} and {\\sl Suzaku} HXD data, requiring a common binary ephemeris. However, the strongest signal that we were able to find in this case was \n rather insignificant, with  $Z_1^2\\approx50$ \n ($Z_m^2$ with $m>1$ were even less significant). \n\nIt should also be noted that, among other factors (such as the energy range, aperture, and ephemeris choices),  the  significance of the $9.05$ s periodic signal candidate  depends  on the maximum frequency of the frequency range searched. \nThere is no physical a priori reason to limit the search to \n  $f<1$ Hz. \n  Extending the frequency range   to much higher frequencies and accounting for \n  the huge number of trials associated with varying the ephemeris parameters, \n  energy range, and extraction aperture would make the putative $9.05$ s  signal candidate  insignificant.\n  \n\\subsection{\nThe binary light curves and search for nonperiodic variability}\n\\label{bin_lc}\n\nThe background-subtracted light curves of LS\\,5039 \nfor 3 energy bands\nare shown in Figure \\ref{ls5039LC} \nas functions of the binary phase $\\phi={\\rm frac}[(t-T_0)\/P_{\\rm orb}]$, where ${\\rm frac}[X]$ is fractional part of $X$, and \n$T_0$ and $P_{\\rm orb}$ are the best-fit values of the epoch of periastron and binary period taken from   \\cite{2009ApJ...698..514A}.\nThe light curves grow from  minima at $\\phi\\approx 0.1$ (near superior conjunction, $\\phi_{\\rm supc}=0.046$) up to \n$\\phi\\approx 0.4$, then \nremain nearly flat, with short \nfluctuations, around apastron ($\\phi=0.5$), show  narrow peaks at $\\phi=0.6$ (before inferior conjunction; $\\phi_{\\rm infc}=0.67$), and secondary peaks at $\\phi\\approx 0.8$. \nThus, the light curves exhibit  a flat-top maximum, encompassing the  apastron \nand inferior conjunction phases. If, instead of the ephemeris from\n\\citet{2009ApJ...698..514A}, \nwe use the ephemeris from Casares et al.\\ (2011) or Sarty et al.\\ (2011) \nfor folding and\/or  calculating the conjunction phases,\nthe shifts will not exceed 0.1 in phase.\n\nThe overall structure of the light curves does not evolve noticeably with energy, not only within the \\nustar\\ band but also between the  \\suzaku\\  XIS and  \\nustar\\ bands (see \\citealt{2009ApJ...697..592T}). It also \ndoes not show appreciable changes between the  \\suzaku\\ \n XIS and  \\nustar\\ observations,\nseparated by 9 years.\n\n\\begin{figure}[t]\n\\begin{center}\n  \\includegraphics[angle=0,width=0.5\\textwidth]{ls5039_FPMA_lc_3-60keV2500s_phase_backCorr.pdf}\n  \\includegraphics[angle=0,width=0.5\\textwidth]{ls5039_FPMA_lc_3-10keV2500s_phase_backCorr.pdf}\n  \\includegraphics[angle=0,width=0.5\\textwidth]{ls5039_FPMA_lc_10-60keV2500s_phase_backCorr.pdf}\n\\caption{\\nustar\\ background-corrected light curves \nof LS\\,5039\\ \n  as functions of orbital phase. \nThe  upper, middle, and lower\n    panels are for the energy\nbands 3--60~keV, 3--10~keV, and\n  10--60~keV. The superior and inferior conjunctions are \n  shown by\n  the solid and dash-dot\n  lines, while the apastron and periastron\n  are shown by dashed and dotted lines, respectively. }\n\\label{ls5039LC}\n\\end{center}\n\\end{figure}\n\nWe have also searched the data for short aperiodic variability, such as bursts. \nWe \n binned the arrival times\nto produce light curves with  bin sizes varying from 1  to 200 s (so that the largest bins are smaller than the smallest gap in the \\nustar\\  observation). \nWe calculated the Poisson probability, $q$,\nof having a\n number of counts per bin larger than measured.\n Note that the average number of counts per bin (the Poisson distribution parameter) varies with the binary phase.\n This is taken into account  by calculating a  local mean over  $\\approx3000$ s  \n time interval (slightly larger than the largest gap in the \\nustar\\  observation) surrounding the bin for which the probability is calculated (this bin itself is excluded from the mean calculation). \n For the chosen bin size, the Poisson  probability \n should be corrected for the number of trials, $N_{\\rm tr}$,  which is \n  equal to the  number of  bins, \n $N_{\\rm bin}$, in the entire {\\sl NuSTAR} observation:\n  $q_{\\rm corr} =1-(1-q)^{N_{\\rm bin}}$.\n Figure  \\ref{fig:bursts}  shows two most significant bursts that we found. The burst durations are about 1 s and 70 s.  With the above definition of  probability, the significances \n are 3.6$\\sigma$ for the short burst and 3.2$\\sigma$ for the longer burst.  However, we note that this probably  \n was derived for a fixed number of bins. If we account for all possible binning schemes (not just the one that results in the highest significance), then $N_{\\rm tr}>N_{\\rm bin}$, i.e., the \n confidence levels become lower.\n Therefore, these two burst candidates are not\n  truly\n significant. \n\n\\begin{figure*}[hbt]\n  \\includegraphics[width=\\textwidth]{bursts.pdf}\n  \\caption{Light curves of the two most significant bursts (3.6$\\sigma$ and 3.2$\\sigma$, respectively). Bin sizes in the left and right light curves are 1 s and 15 s, respectively.\n  Time \n  (in seconds)\n  is counted from the end of the nearest \n  preceding occultation of the target by Earth.  }\n  \\label{fig:bursts}\n\\end{figure*}\n\n\n\\section{Spectral analysis}\n\\label{specana}\n\n\n\n\n\n\\subsection{Phase-integrated spectroscopy}\n\n\nWe \nanalyzed the {\\sl NuSTAR} and {\\sl Suzaku}\nspectra using XSPEC ver.\\ 12.9.1p\n\\citep{arnaud96conf}.  To account for interstellar absorption, we used the T\\\"{u}bingen-Boulder \nmodel with the photo-electric cross-sections of\n\\citet{1996ADNDT..64....1V} and the abundances of \\citet{2000ApJ...542..914W}.\nWe bin\nthe spectra to have \n$S\/N\\simeq 7$ in each spectral bin,\ncorresponding to 50 counts per bin, and used the $\\chi^2$ statistic \nfor model parameter estimation and error calculation. \nFor all spectral fits, we added \nmultiplicative constants to the FPMA and FPMB normalizations,\nfrozen to\n1 for the former and allowed to vary for the latter, to account for \ncalibration uncertainties between the two \ndetector modules. We found \nthe difference in the normalization factors not exceeding 2\\%.\nWe also \nfound, using the same approach, that the calibration uncertainty \nbetween the \\nustar\\ \n and {\\sl Suzaku}\ninstruments \nis around\n10\\%.\n\n\nLS\\,5039\\ is detected with \\nustar\\ \nup to 70 keV, with a background-corrected count rate of 0.186(1) cts s$^{-1}$ and 0.171(1) cts s$^{-1}$,  in  the FPMA and FPMB detectors, respectively, in the 3--70 keV energy range,\n with a background contribution of $<10\\%$ in each detector. The number of background-corrected counts in the 60--70~keV energy range is about $45\\pm12$ in each\nmodule with a  background contribution of $\\approx70-80\\%$, i.e.,\nthe source becomes \nhardly distinguishable from the background at higher energies.\nAn absorbed PL model gives \na statistically acceptable fit to the phase-integrated \n spectrum in the 3--70 keV band, with $\\chi^2=667$ for\n708 degrees of freedom (dof). We find a hydrogen column density\n$N_{\\rm H}=(0.7\\pm0.4)\\times10^{22}$~cm$^{-2}$, \na photon index\n$\\Gamma=1.61\\pm0.01$,\nand an absorption-corrected  \nenergy flux \n$F_{\\rm 3-70\\,keV} =(2.31\\pm0.02)\\times10^{-11}$~erg~s$^{-1}$~cm$^{-2}$. There are no\nindications of \nspectral features in the fit residuals,\nincluding around\nthe Fe K$\\alpha$ line complex \nat 6--7 keV. Moreover,\nconsidering the \\nustar\\ spectra alone,\nwe find no evidence of a\nhigh-energy cutoff usually present in the hard X-ray spectra of HMXBs\nharboring \naccreting neutron stars \n(see, e.g., \\citealt{2002ApJ...580..394C,2015ApJ...809..140K,2017ApJ...841...35F}). \nThe photon index as inferred from the \\nustar\\\nspectra is \nlarger \nthan $\\Gamma=1.51\\pm0.01$\nfound at \nlower X-ray energies with\n\\suzaku\\ \\citep{2009ApJ...697..592T}.\n\n\\begin{figure}[th]\n\\begin{center}\n\\includegraphics[angle=0,width=0.48\\textwidth]{phAvgSpecLS5039_PL_suzNuS.pdf}\n\\caption{\nFit of the absorbed PL model to the\nphase-integrated \\nustar\\ and \\suzaku\\ XIS data (for clarity we only show XIS0), and the corresponding residuals in \nunits of $\\sigma$ (lower panel).}.\n\\label{suzNuSpec}\n\\end{center}\n\\end{figure}\n\nGiven the \nlong-term stability\nof the \nbinary's periodic X-ray light curve,\n\\citep{2009ApJ...697L...1K}, we fit the \\nustar\\ 3--70~keV and the \\suzaku\\ 0.7--10~keV phase-averaged spectra simultaneously with an absorbed PL model (Figure~\\ref{suzNuSpec}). We linked all parameters among the two spectra, except for a constant normalization to take into account  calibration uncertainties between the instruments. We find a statistically acceptable fit to the data \n($\\chi^2 = 2722$ for 2795 dof), with \n$N_{\\rm H}=(1.18\\pm0.01)\\times10^{22}$~cm$^{-2}$, \n$\\Gamma=1.588\\pm0.007$ (Table~\\ref{specParam}),\nand absorption-corrected  \nfluxes\n$F_{\\rm 0.5-10~keV}=(9.64\\pm0.05)\\times10^{-12}$~erg~s$^{-1}$~cm$^{-2}$, and \n$F_{\\rm\n  10-70~keV}=(18.0\\pm0.2)\\times10^{-12}$~erg~s$^{-1}$~cm$^{-2}$.\nThese fluxes correspond to a luminosity $L_{\\rm 0.5-70\\,keV} \\approx 2.8\\times 10^{34} (d\/2.9\\,{\\rm kpc})^2$~erg~s$^{-1}$.\n\n\n\n\n\\subsection{Phase-resolved spectroscopy}\n\nWe performed broad-band phase-resolved spectroscopy \nusing the \\suzaku\\ XIS\nand \\nustar\\ data\nto look for \nmodulation of spectral parameters  with the orbital \nperiod.\n We used the same definition of binary phase \nas for the \nlight curve shown in Figure \\ref{ls5039LC}. \nWe extract the spectra in orbital phase bins $\\Delta\\phi = 0.1$  \n($\\phi=0$ corresponds to binary periastron).\nBecause of lower count statistics in the \nchosen phase bins, we \nbin the spectra to 5 counts per energy bin and use the Cash statistic. \nWe fit all 10 spectra simultaneously with an absorbed PL. Given that phase-resolved spectroscopy with \\suzaku\\ alone revealed no variability in the hydrogen column density \\citep{2009ApJ...697..592T}, we linked $N_{\\rm H}$ between all spectra, but left the photon index and the normalization of the PL free to vary. We find a good fit to the spectra with a C-stat of 8950 for 9000 dof.\n\nThe fit results are presented in Table \\ref{specParam}.\nFigure~\\ref{PhResSpec} shows the flux and photon index \nvariations \nwith  orbital phase. We find a strong modulation of the photon index $\\Gamma$,\nas similar to those inferred previously \nfrom {\\sl RXTE} and \\suzaku\\ observations \\citep{2005ApJ...628..388B, 2009ApJ...697..592T}. Our \\nustar\\ plus \\suzaku\\ fits have shown harder spectra than {\\sl RXTE} at all phases.\nThe addition of the \\nustar\\ data \nto the \\suzaku\\ data has \nprovided tighter constraints \non the photon index, showing that it varies by $\\Delta\\Gamma\\approx0.1$ \nfrom maximum to minimum.\n\n\n\nA similar tendency was noticed previously in the {\\sl RXTE} phase-resolved spectra (3--30 keV band), but the values of $\\Gamma$ and $\\Delta\\Gamma$ were substantially larger (see Figure 4 in \\citealt{2005ApJ...628..388B}). A similar $\\Gamma$-$F$ anti-correlation in the 1--10 keV range ({\\sl XMM-Newton}, {\\sl Chandra}, {\\sl ASCA} and {\\sl Suzaku} data) \nis shown \nby Figure  1 in \\cite{2009ApJ...697L...1K}. \n\n\n\\begin{figure*}[t]\n\\begin{center}\n  \\includegraphics[angle=0,width=0.494\\textwidth]{ls5039_SuzNuS_specEvol.pdf}\n  \\includegraphics[angle=0,width=0.445\\textwidth]{ls5039_SuzNuS_FvsGam.pdf}\n\\caption{{\\sl Left:} Phase-resolved spectroscopic results with $\\Delta\\phi=0.1$.\n  The blue points show the 3--70~keV unabsorbed flux variation as a function of\n  orbital phase, with periastron at $\\phi=0$. The orange squares show the photon index variation with\n  phase. {\\sl Right}: Anti-correlation of the  photon index \n  and the flux.} \n\\label{PhResSpec}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{deluxetable}{lccc}[t]\n\\tablecaption{Spectral parameters of LS\\,5039 from joint PL fits to the \\nustar\\ and {\\sl Suzaku} data} \\label{specParam}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Phases}\n& \\colhead{$N_{\\rm H}$} & \\colhead{$\\Gamma$} &  \\colhead{Flux (3-70~keV)} \\\\\n\\colhead{} & \\colhead{$10^{22}$\\,cm$^{-2}$}  &   \\colhead{}              & \\colhead{ $10^{-11}$\\,erg\\,s$^{-1}$\\,cm$^{-2}$}\n}\n\\startdata \n$0.0-1.0$ \n& $1.18\\pm0.01$ & $1.588\\pm0.007$ &  $2.31\\pm0.02$\\\\\n\\hline\n$0.0-0.1$  & $0.5\\pm0.4$ & $1.67\\pm0.02$ & $1.24\\pm0.03$\\\\\n$0.1-0.2$      & \\ldots            & $1.65\\pm0.02$ & $0.98\\pm0.03$\\\\\n$0.2-0.3$      & \\ldots            & $1.62\\pm0.02$ &  $1.63\\pm0.04$\\\\\n$0.3-0.4$      & \\ldots            & $1.57\\pm0.02$ &  $2.56\\pm0.06$\\\\\n$0.4-0.5$      & \\ldots            & $1.56\\pm0.01$ &  $3.12\\pm0.06$\\\\\n$0.5-0.6$      & \\ldots            & $1.57\\pm0.02$ & $3.30\\pm0.05$\\\\\n$0.6-0.7$      & \\ldots            & $1.56\\pm0.01$ &  $3.48\\pm0.06$\\\\\n$0.7-0.8$      & \\ldots            & $1.57\\pm0.02$ &  $3.05\\pm0.07$\\\\\n$0.8-0.9$      & \\ldots            & $1.59\\pm0.02$ &  $2.33\\pm0.05$\\\\\n$0.9-1.0$  & \\ldots            & $1.62\\pm0.02$ &  $1.45\\pm0.04$\\\\\n\\enddata\n\\end{deluxetable}\n\n\n\\section{  Discussion and Summary\n}\n\\label{summ}\n\nThe nature of the compact object in LS~5039 has long been elusive. \nThe  two main scenarios \n  that are often used for the interpretation of its  observed properties\n  are the {\\em microquasar scenario} (radiatively-inefficient accretion onto a BH with the nonthermal emission coming from the jets; e.g., \\citealt{2005ApJ...628..388B}), \n and the {\\em colliding winds scenario} \n (the relativistic wind from a young rotation-powered pulsar \n collides with the massive star's wind resulting in an intrabinary shock and particle acceleration (\\citealt{2006A&A...456..801D, 2015CRPhy..16..661D, 2020A&A...641A..84M}).\n In addition, one could also consider \n the `{\\em propeller \n scenario}' in which\n the interaction of \n the massive star's wind with the rotating magnetosphere of a strongly magnetized NS \n can accelerate wind particles to relativistic energies via shocks or magnetic field reconnection.\n Such a scenario was suggested by\n \\citealt{2012ApJ...744..106T} \n for another HMGB, LS\\,I\\,+61$^\\circ$\\,303, from which a magnetar-like burst was likely observed, and it was also mentioned by Y+20 as a possibility for LS\\,5039 if the proposed magnetar \n nature of the compact object  is confirmed.\n The possibility of accretion onto the NS surface (magnetic poles) \n seems to be excluded due to the long-term stability of the orbital light curve in X-rays, featureless PL spectrum from 0.3 to $\\sim$100 keV, low X-ray luminosity,  $L_{X}\\approx2.8\\times 10^{34}(d\/2.9~{\\rm kpc})^2$ erg s$^{-1}$, and a lack of  outbursts.\n\nThe detection of \nfast pulsations, typical for a young rotation-powered pulsar, would \nstrongly support the colliding wind scenario while \ndetection of  slow pulsations \n(e.g., with a few seconds period) could support the \nscenario with a magnetar propeller, as suggested by Y+20. However, our timing analysis did not yield any evidence for a young pulsar. Although we were able to confirm \nthe Fourier power excess \nnear $P=9.05$ s, previously reported by Y+20,\nand find a binary ephemeris \ncompatible with the optical observations\n(contrary to Y+20), the significance of this excess \nis rather low.\nThe excess near $P=8.96$ found by Y+20 in the {\\sl Suzaku} HXD data \nobtained 9 years earlier could strengthen the magnetar hypothesis, if a large enough $\\dot{P}$ is assumed. However, we failed to find a \ncommon binary ephemeris which would \nprovide even marginally acceptable signal detections at both  9.05 s (in \\nustar) and 8.96 s (in {\\sl Suzaku}).  Another  \\nustar\\  observation is needed to decisively confirm or rule out the 9 s period candidate and magnetar scenario  or, if this period is not confirmed, to perform a more sensitive search for pulsations at other frequencies. \n\nOur variability analysis  does not support  the accretion scenario. The binary light curve appears to be rather  stable over timescales of at least 9 years (between the {\\sl Suzaku} and \\nustar\\ observations), including even the fine structure \n(e.g, the narrow spike near the aparstron and inferior conjuction). The stable fine structure cannot be explained by \na clumpy stellar wind or instabilities in the accretion flow.  Understanding the nature of this stable small-scale structure in the light curve may hold the key to  the interaction scenario.\n\nWe also looked for bursts on even shorter time scales ($<200$ s), which could be expected \nin the accretion and magnetar scenarios, but found no significant bursting activity. The binary  light curve \nalso shows little-to-no dependence on energy in the 0.7--70 keV range. Having the light curve maximum around the apastron (which is close to the inferior conjunction in LS 5039) phase is hardly compatible with \nwind accretion (unless the wind has very unusual properties).\n\nThe spectral analysis shows that both the phase-integrated and phase-resolved spectra in the 0.7--70 keV range  can be fitted by a simple PL model modified by the interstellar extinction. We do not find any evidence of a cutoff at higher energies. The photon index values ($\\Gamma\\simeq1.6$ in the phase-integrated spectrum, $1.56\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}}\\Gamma\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 1.67$ in separate phase bins) are typical for such synchrotron sources as  pulsars and PWNe  \\citep{2008AIPC..983..171K}. The value of $\\Gamma$ is also typical for other HMGBs (see e.g., Figure~1 from \\citealt{2014AN....335..301K}) but the spread of $\\Gamma$ with the orbital phase is smaller for LS 5039 than those for the other HMGBs,\nperhaps due to the LS 5039's tighter orbit. \n\nWe \nhave confirmed  a statistically significant anti-correlation between the flux and the photon index observed throughout the binary orbit, \npreviously reported by \\cite{2005ApJ...628..388B},\n\\cite{2009ApJ...697..592T}, and \\cite{2009ApJ...697L...1K}. This phenomenon  may hold another important clue to the nature of the compact object and emission processes in LS\\,5039 and the other HMGBs. A similar anti-correlation has been seen in  at least two other HMGBs: 1FGL J1018.6--5856 \n\\citep{2013ApJ...775..135A} \nand LS\\,I\\,+61$^{\\circ}$303 \n\\citep{2017MNRAS.468.3689M}.\nIn the colliding wind scenario, the anti-correlation could be explained by a model in which the observer sees a harder spectrum of emitted electrons at the phases when the flux is increased by, e.g., the Doppler boosting. \nWe see from Figures \\ref{PhResSpec} (and \\ref{ls5039LC}) that the flux is maximal and the photon index minimal near inferior conjunction, when the compact object is between the observer and the massive star. This suggests that the Doppler boosting occurs in a shocked pulsar wind confined by the dynamically dominant stellar wind in a (hollow) cone (a paraboloid-like shell)  around the star-pulsar direction (see, e.g., left panel in Figures 10  in \\citealt{2013A&ARv..21...64D}).\nThis assumption is supported by\nsimulations by \\cite{2012MNRAS.419.3426B} and \\cite{2019MNRAS.490.3601B} \nwho show that the bulk flow of the shocked pulsar wind \ncan reach a bulk Lorentz factor of a few as the wind is streaming away from the  cone apex.\nIn order to explain the anti-correlation between the flux and photon index,\none has to assume that particle acceleration in the shocked pulsar wind proceeds more efficiently as the bulk flow accelerates, despite the adiabatic and cooling losses. The  mechanism of particle acceleration is unclear but it might be akin to that in extended AGN jets (e.g., the shear acceleration in an expanding flow; \\citealt{2016ApJ...833...34R}). \n\nThe hollow cone configuration of the shocked pulsar wind would naturally give rise to a double peaked structure in the light curve (e.g., the peaks at phases 0.4 and 0.8 in Figure \\ref{ls5039LC}), as \ncone crosses the observers line of sight. This would also naturally explain the relatively flat top in the light curve. However in this scenario it is unclear what could cause the third (and strongest) peak, near phase 0.6.\n\n\n\n\nOverall, we conclude that the colliding wind scenario with the compact object being a young pulsar remains the most plausible option as it can explain the dependence of the flux on the binary phase (by Doppler beaming), the spectrum (by synchrotron emission from particles accelerated by the colliding winds), the longer term stability of the light curve, and the lack of variability on short timescales.   \n\n\n\n{\\em Facilities:} \\facility{{\\sl NuSTAR} },  \\facility{{\\sl Suzaku} (XRT)}\n\n\\begin{acknowledgements}\n\nSupport for this work was provided by the National Aeronautics and\nSpace Administration through  the \\nustar\\ award NNX17AB77G. JH would like to thank John Tomsick for helpful discussions regarding the reduction of Suzaku data.  JH acknowledges support from\nan appointment to the NASA Postdoctoral Program at the\nGoddard Space Flight Center, administered by the Universities Space Research Association under contract with NASA. GY acknowledges support from NASA through Fermi grant 80NSSC20K1571.\n\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Integrating summary data from GWAS and eQTL studies}\n\n\n\nIntegrative genomics aims to  integrate various biological data sets for systematic discovery of genetic basis that underlies and modifies human disease \\citep{giallourakis2005disease}. To realize its full potential in genomic research, methods of both computational efficiency and theoretical guarantee for such integrative analyses are needed in various applications.  This paper proposes a method that combines datasets from genome-wide association studies (\\textsc{gwas}) and expression quantitative trait loci (e\\textsc{qtl}) studies in order to identify genetically regulated disease genes and to provide  an integrative view of the underlying biological mechanism of complex diseases such as heart failure.  Results from \\textsc{gwas}  have revealed that the majority of disease-associated single nucleotide polymorphisms (\\textsc{snp}s) lie in non-coding regions of the genome \\citep{hindorff2009potential}.  These  \\textsc{snp}s likely regulate the expression of a set of downstream genes that may have effects on diseases \\citep{nicolae2010trait}. On the other hand,  e\\textsc{qtl} studies measure the association between both cis- and trans- \\textsc{snp}s and the expression levels of genes, which characterizes how genetic variants  regulate transcriptions.  A key next step in human genetic research is to explore whether these intermediate cellular level e\\textsc{qtl} signals are located in the same loci (``colocalize\") as \\textsc{gwas} signals and potentially mediate  the genetic effects on disease, and to find disease genes whose e\\textsc{qtl}  overlap significantly with the set of loci associated with the disease \\citep{he2013sherlock}.\n\n\n\nThis paper focuses on the integrative analysis of the summary statistics of \\textsc{gwas} and e\\textsc{qtl} studies performed on possibly different set of subjects.  Due to the privacy and confidentiality concerns of \\textsc{gwas}\/e\\textsc{qtl} participants,   the raw genotype data are often not available, instead most of the published papers provide summary statistics that include single \\textsc{snp} analysis results such as the estimated effect size, its p-value and the minor allele frequency. \nBased on these summary statistics, we propose a method that identifies potential disease genes by measuring their genetic overlaps to the disease. In particular, we propose a gene-specific measure, $T$-score, that characterizes the total amount of simultaneous \\textsc{snp} signals that share the same loci in both \\textsc{gwas} and e\\textsc{qtl} study of a relevant normal tissues. Such a measure enables us to prioritize genes whose expression levels may underlie and modify human disease \\citep{zhao2016sparse}. \n\nTreating \\textsc{snp}-specific \\textsc{gwas} and e\\textsc{qtl} summary $z$-score statistics (as obtained for linear or logistic regression coefficients) as two independent sequences  of Gaussian random variables, we define the parameter $T$-score as the sum of the product of the absolute values of two normal means over a given set of $n$  \\textsc{snp}s. Specifically, for any individual gene $g$, we denote $\\mathbf{x}_n^g$ the vector of $z$-scores from e\\textsc{qtl} study, and $\\mathbf{y}_n$ the vector of $z$-scores from \\textsc{gwas}. We assume $\\mathbf{x}_n^g \\sim N(\\theta^g, \\boldsymbol{\\Sigma}_1)$ and $\\mathbf{y}_n \\sim N(\\mu, \\boldsymbol{\\Sigma}_2)$ for some $\\theta^g,\\mu\\in \\mathbb{R}^n$ and covariance matrices $\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_2\\in \\mathbb{R}^{n\\times n}$ with unit diagonals. The $T$-score for gene $g$ is then defined as\n\\begin{equation}\n\\text{$T$-score}(g) = \\sum_{i=1}^n |\\theta_i^g\\mu_i|,\n\\end{equation}\nwhere the summation is over a given set of $n$  \\textsc{snp}s. The $T$-score quantifies the amount of simultaneous signals contained in two Gaussian mean vectors, regardless of the directions of the signals. Intuitively, a large $T$-score would possibly result from a large number of contributing components $i$'s whose means $\\theta^g_i$ and $\\mu_i$ are simultaneously large in absolute values. The supports (nonzero coordinates) of the mean vectors $\\theta$ (hereafter we omit its dependence on $g$ for simplicity) and $\\mu$ are assumed to have sparse overlaps since it has been observed that, for a relatively large set of  \\textsc{snp}s, only a small subset of  \\textsc{snp}s are associated with both disease and gene expression \\citep{he2013sherlock}. By estimating the $T$-scores for all the genes using summary statistics, we would be able to, after proper normalizations that accounts for study sample sizes, the number of \\textsc{snp}s and effect sizes (see Section 2.5), identify and prioritize those genetically regulated candidate disease genes. Besides, the $T$-scores can also be used in the Gene Set Enrichment Analysis to identify the disease associated gene sets and pathways, or to quantify the genetic sharing among different complex traits using the \\textsc{gwas} summary statistics \\citep{Bulik-Sullivan}.\n\n\\subsection{Related works}\n\n\nStatistically, estimation of $T$-score involves estimating a non-smooth functional -- the absolute value function -- of Gaussian random variables. Unlike the problems of estimating smooth functionals such as the  linear or quadratic functionals \\citep{ibragimov1985nonparametric,donoho1990minimax, fan1991estimation,efromovich1994adaptive, cai2006optimal} where some natural unbiased estimators are available, much less is known for estimating the non-smooth functionals. Using approximation theory, \\cite{cai2011testing} established the minimax risk and constructed a minimax optimal procedure for estimating a non-smooth functional. More recently, this idea has been adapted to statistical information theory that also considered estimation of non-smooth functionals such as the R\\'enyi entropy, support size, and $L_1$ distance  \\citep{jiao2015minimax, jiao2016minimax,wu2016minimax,wu2019chebyshev,acharya2016unified}. Nonetheless, how to estimate the absolute inner product of two Gaussian mean vectors ($T$-score) remains unknown.\n\nIn the statistical genetics and genomics literature, several  approaches have been proposed for integrating \\textsc{gwas} and e\\textsc{qtl} data sets. Under the colocalization framework, methods such as \\cite{nica2010candidate} and \\cite{giambartolomei2014bayesian} were developed to detect colocalised  \\textsc{snp}s. However, these methods do not directly identify the potential causal genes. Under the transcriptome-wise association study (TWAS) framework, \\cite{zhu2016integration} proposed a summary data-based Mendelian randomization method for causal gene identification, by posing some structural causality assumptions. \\cite{Pediscan} developed a gene-based association method called PrediXcan that directly tests the molecular mechanisms through which genetic variation affects phenotype. Nevertheless, there is still a need for a quantitative measure of the genetic sharing between genes and the disease that can be estimated from the \\textsc{gwas}\/e\\textsc{qtl} summary statistics. \n\nAs a related but different quantity, the genetic covariance $\\rho$, proposed by \\cite{Bulik-Sullivan} as a measure of the genetic sharing between two traits, can be expressed using our notation as  $\\rho = \\sum_{i=1}^n \\theta_i\\mu_i$. In addition to the difference due to the absolute value function, in the original definition of genetic covariance $\\rho$, the mean vectors $\\theta$ and $\\mu$ represent the conditional effect sizes (i.e., conditional on all other \\textsc{snp}s in the genome), whereas the mean vectors in our $T$-score correspond to the marginal effect sizes, so as to be directly applicable to the standard \\textsc{gwas}\/e\\textsc{qtl} summary statistics. In addition, unlike the \\textsc{ld}-score regression approach considered in \\cite{Bulik-Sullivan}, our proposed method  takes advantage of the fact that the support overlap between $\\theta$ and $\\mu$ are expected to be very sparse. \n\n\n\n\n\n\n\n\\subsection{Main contributions}\n\nIn this paper, we propose an estimator of the $T$-score, based on the idea of thresholding and truncating the best polynomial approximation estimator. To the best of our knowledge, this is the first result concerning estimation of such absolute inner product of two Gaussian mean vectors. Under the framework of statistical decision theory, the minimax lower bounds are obtained and we show that our proposed estimators are minimax rate-optimal  over various parameter spaces. In addition, our results indicate that the proposed estimators are locally adaptive to the unknown sparsity level and the signal strength (Section 2). Our simulation study shows the strong empirical performance and robustness of the proposed estimators in various settings, and provides guidelines for using our proposed estimators in practice (Section 3). Analysis of \\textsc{gwas} and e\\textsc{qtl} data sets of heart failure using the proposed method identifies several  important  genes that are functionally relevant to the etiology of human heart failure (Section 4).\n\n\n\n\\section{Minimax Optimal Estimation of T-score}\n\n\\subsection{Minimax Lower Bounds}\n\nWe start with establishing the minimax lower bounds for estimating $T$-score over various parameter spaces. Throughout, we denote $T(\\theta,\\mu)=\\sum_{i=1}^n|\\theta_i\\mu_i|$. For a vector ${a} = (a_1,...,a_n)^\\top \\in \\mathbb{R}^{n}$, we define the $\\ell_\\infty$ norm $\\| {a}\\|_{\\infty} = \\max_{1\\le j\\le n}  |a_{i}|$. For sequences $\\{a_n\\}$ and $\\{b_n\\}$, we write $a_n\\lesssim b_n$ or $b_n \\gtrsim a_n$ if there exists a constant $C$ such that $a_n \\le Cb_n$ for all $n$. We write $a_n\\asymp b_n$ if $a_n \\lesssim b_n$ and $a_n\\gtrsim b_n$.\n\nAs of both practical and theoretical interest, we focus on the class of mean vector pairs $(\\theta,\\mu)$  with only a small fraction of support overlaps. \nSpecifically,  for any $s< n$, we define the parameter space for $(\\theta,\\mu)$ as $D(s) =\\{(\\theta,\\mu)\\in \\mathbb{R}^n\\times \\mathbb{R}^n : |\\text{supp}(\\theta) \\cap \\text{supp}(\\mu)| \\le s\\}.$\nIntuitively, in addition to the sparsity $s$, the difficulty of estimating $T$-score $T(\\theta,\\mu)$ should also rely on the magnitudes of the mean vectors $\\theta$ and $\\mu$, and the covariance matrices $\\boldsymbol{\\Sigma}_1$ and $\\boldsymbol{\\Sigma}_2$. Towards this end, we define the parameter space for $(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_2)$ as \n\\[\nD^\\infty(s,L_n) = \\big\\{(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_2):  (\\theta,\\mu)\\in D(s),\\max(\\|\\theta\\|_\\infty , \\|\\mu\\|_\\infty )\\le L_n,  \\boldsymbol{\\Sigma}_1=\\boldsymbol{\\Sigma}_2={\\bf I}_n \\big\\},\n\\]\nwhere both $s$ and $L_n$ can growth with $n$.\nIn particular, we calibrate the sparsity $s \\asymp n^{\\beta}$ for some $0<\\beta<1$. Throughout, we consider the normalized loss function as the squared distance scaled by $n^{-2}$ and define the estimation risk for some estimator $\\hat{T}$ as\n\\[\n\\mathcal{R}(\\hat{T})=\\frac{1}{n^2}\\mathbb{E} (\\hat{T}-T(\\theta,\\mu))^2.\n\\]\nTo simplify our statement, we define the rate function\n\\[\n\\psi(s,n) = \\log\\bigg(1+\\frac{n}{s^2} \\bigg)+\\frac{1}{\\log s}.\n\\]\nThe following theorem establishes the minimax lower bound over the parameter space $D^{\\infty}(s,L_n)$.\n\n\\begin{theorem} \\label{sparse.lower3}\nLet $\\mathbf{x_n} \\sim N(\\theta,\\boldsymbol{\\Sigma}_1)$ and $\\mathbf{y_n} \\sim N(\\mu,\\boldsymbol{\\Sigma}_2)$ be multivariate Gaussian random vectors where $(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_2) \\in D^\\infty(s,L_n)$ and $L_n\\gtrsim \\sqrt{\\log n}$. Then \n\\begin{equation} \\label{sparse.lower.equation4}\n\\inf_{\\hat{T}} \\sup_{\\substack{(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_1) \\in D^\\infty(s,L_n)}}\t\\mathcal{R}(\\hat{T}) \\gtrsim \\frac{L_n^2s^2\\psi(s,n)}{n^2}\n\\end{equation}\nwhere $\\hat{T}$ is any estimator based on $(\\mathbf{x_n} ,\\mathbf{y_n} )$.\n\\end{theorem}\nFrom the above theorem and the definition of the rate function $\\psi(s,n)$, apparently, there is a discrepancy between the cases when $\\beta\\in(0,1\/2)$ and when $\\beta\\in [1\/2,1)$. Specifically, when $\\beta\\in (0,1\/2)$, the minimax lower bound (\\ref{sparse.lower.equation4}) becomes\n\\begin{equation} \\label{lb.l}\n\\inf_{\\hat{T}} \\sup_{\\substack{(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_2) \\in D^{\\infty}(s,L_n)}}\t\\mathcal{R}(\\hat{T})\\gtrsim\\frac{L_n^2s^2\\log n}{n^2},\n\\end{equation}\nwhereas when $\\beta\\in [1\/2,1)$, we have\n\\begin{equation} \\label{lb.h}\n\\inf_{\\hat{T}} \\sup_{\\substack{(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_2) \\in D^{\\infty}(s,L_n)}} \t\\mathcal{R}(\\hat{T}) \\gtrsim  \\frac{L_n^2s^2}{n^2\\log n}.\n\\end{equation}\nThese results suggest that the minimax optimal estimators of $T(\\theta,\\mu)$ might not be the same across different regions of $\\beta$.\n\n\\subsection{Optimal estimators of T-score via polynomial approximation}\n\nIn general, the proposed estimators are based on the idea of optimal estimation of the absolute value of normal means studied  by \\cite{cai2011testing}.  Therein, the best polynomial approximation of the absolute value function was applied to obtain the rate-optimal estimator, as well as the minimax lower bound. Specifically, it was shown that, if we define the $2K$-degree polynomial \n\\begin{equation}\n\\label{GK}\nG_K(x) = \\frac{2}{\\pi}T_0(x) +\\frac{4}{\\pi}\\sum_{k=1}^{K} (-1)^{k+1}\n\\frac{T_{2k}(x)}{4k^2-1} \\equiv \\sum_{k=0}^K g_{2k} x^{2k},\n\\end{equation}\nwhere $T_k(x) = \\sum_{j=0}^{\\lfloor k\/2 \\rfloor} (-1)^j \\frac{k}{k-j} {k-j\\choose j} 2^{k-2j-1}x^{k-2j}$ are Chebyshev polynomials, \nthen for any $X \\sim N(\\theta,1)$, if $H_k$ are Hermite polynomials with respect to the standard normal density $\\phi$ such that\n\\begin{equation}\n\\frac{d^k}{dy^k}\\phi(y) = (-1)^k H_k(y) \\phi(y),\n\\end{equation}\nthe estimator\n\\begin{equation} \\label{SK}\n\\tilde{S}_K(X) \\equiv \\sum_{k=0}^K g_{2k}M_n^{-2k+1}H_{2k}(X)\n\\end{equation}\nfor some properly chosen $K$ and $M_n$ has some optimality properties for estimating $|\\theta|$. This important result motivates our construction of the optimal estimators of $T$-score. \n\nWe begin by considering the setting where  $\\mathbf{x}_n=(x_1,...,x_n)^\\top \\sim N(\\theta,{\\bf I}_n)$ and $\\mathbf{y}_n=(y_1,...,y_n)^\\top \\sim N(\\mu,{\\bf I}_n)$. To estimate $T(\\theta,\\mu)$, we first split each sample into two copies, one is used for testing, and the other is used for estimation. Specifically, for $x_i \\sim N(\\theta_i,1)$, one can generate $x_{i1}$ and $x_{i2}$ from $x_{i}$ by letting $z_i \\sim N(0,1)$ and setting $x'_{i1} = x_i+z_i$ and $x'_{i2} = x_i-z_i$. Let $x_{il} =  x'_{il}\/\\sqrt{2}$ for $l=1,2$, then $x_{il} \\sim N(\\theta'_i,1)$ for $l=1,2$ and $i=1,...,n$ with $\\theta_i' = \\theta_i\/\\sqrt{2}$. Similarly, one can construct $y_{il}\\sim N(\\mu'_i,1)$ for $l=1,2$ and $i=1,...,n$ with $\\mu'_i = \\mu_i\/\\sqrt{2}$. Since $T(\\theta,\\mu)=2T(\\theta',\\mu')$, estimating $T(\\theta,\\mu)$ with $\\{ x_i,y_i\\}_{i=1}^n$ is then equivalent to estimating $T(\\theta', \\mu')$ with $\\{x_{il},y_{il} \\}_{i=1}^n, l=1,2$. \n\nIn light of the estimator defined in (\\ref{SK}), we consider a slightly adjusted statistic $S_K(X) = \\sum_{k=1}^K g_{2k}M_n^{-2k+1}H_{2k}(X)$ and define its truncated version\n\\[\n{\\delta}_K(X) = \\min\\{{S}_K(X) ,n^2 \\},\n\\]\nwith $M_n = 8\\sqrt{\\log n}$ and $K\\ge 1$ to be specified later. It will be seen that the above truncation is important in reducing the variance of $\\delta_K(X)$.\nFollowing the sample splitting procedure, we construct an estimator of $|\\theta'_i|$  as\n\\[\n\\hat{V}_{i,K}(x_i)= {\\delta}_K(x_{i1})I(|x_{i2}|\\le 2\\sqrt{2\\log n}) + |x_{i1}| I(|x_{i2}|> 2\\sqrt{2\\log n}),\n\\]\nand a similar estimator of $|\\mu'_i|$ as $\\hat{V}_{i,K}(y_i)$. To further exploit the sparse structure, we also consider their thresholded version \n\\[\n\\hat{V}^S_{i,K}(x_i)= {\\delta}_K(x_{i1})I(\\sqrt{2\\log n}<|x_{i2}|\\le 2\\sqrt{2\\log n}) + |x_{i1}| I(|x_{i2}|> 2\\sqrt{2\\log n})\n\\]\nas an estimator of $|\\theta'_i|$ and similarly $\\hat{V}^S_{i,K}(y_i)$ for $|\\mu'|$. Intuitively, both $\\hat{V}_{i,K}(x_i)$ and $\\hat{V}^S_{i,K}(x_i)$ are hybrid estimators: $\\hat{V}_{i,K}(x_i)$ is a composition of an estimator based on polynomial approximation designed for small to moderate observations (less than $2\\sqrt{2\\log n}$ in absolute value) and the simple absolute value estimator applied to large observations (larger than $2\\sqrt{2\\log n}$ in absolute value), whereas $\\hat{V}^S_{i,K}(x_i)$ has an additional thresholding procedure for small observations (less than $\\sqrt{2\\log n}$ in absolute value). Consequently, we propose two estimators of $T(\\theta,\\mu)$, namely,\n\\begin{equation} \\label{sparse.est}\n\\widehat{T_K(\\theta,\\mu)} = {2}\\sum_{i=1}^n \\hat{V}_{i,K}(x_i)\\hat{V}_{i,K}(y_i),\n\\end{equation}\nas the hybrid non-thresholding estimator and\n\\begin{equation}\n\\widehat{T^S_K(\\theta,\\mu)} = {2}\\sum_{i=1}^n \\hat{V}^S_{i,K}(x_i)\\hat{V}^S_{i,K}(y_i),\n\\end{equation}\nas the hybrid thresholding estimator. Both estimators rely on $K$, which is the tuning parameter to be specified later.\n\n\\subsection{Theoretical properties and minimax optimality}\n\n\nIn this section, we formally study the theoretical properties of $\\widehat{T_K(\\theta,\\mu)}$ and $\\widehat{T^S_K(\\theta,\\mu)}$. \nThe following theorem provides the risk upper bounds and therefore establishes the minimax optimality of $\\widehat{T_K(\\theta,\\mu)}$ and $\\widehat{T^S_K(\\theta,\\mu)}$ over $D^{\\infty}(s,L_n)$ with different sparsity levels.\n\n\\begin{theorem} \\label{sparse.upper3}\nLet $\\mathbf{x_n} \\sim N(\\theta,\\boldsymbol{\\Sigma}_1)$ and $\\mathbf{y_n} \\sim N(\\mu,\\boldsymbol{\\Sigma}_2)$ be multivariate Gaussian random vectors with $(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_2)\\in D^{\\infty}(s,L_n)$, $L_n\\gtrsim \\sqrt{\\log n}$ and $s\\asymp n^\\beta$. Then \n\\begin{enumerate}\n\t\\item for any $\\beta\\in(0,1)$, let $K$ be any finite positive integer, we have,\n\t\\begin{equation} \\label{sparse.upper.equation5}\n\t\\sup_{(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_2)\\in D^{\\infty}(s,L_n)}\t\\mathcal{R}(\\widehat{T^S_K(\\theta,\\mu)} ) \\lesssim \\frac{L^2_ns^2\\log n}{n^2};\n\t\\end{equation}\n\t\\item for any $\\beta\\in (1\/2,1)$, let $K= r\\log n$ for some $0<r<\\frac{2\\beta-1}{12}$, we have,\n\t\\begin{equation} \\label{sparse.upper.equation6}\n\t\\sup_{(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_2)\\in D^{\\infty}(s,L_n)}\t\\mathcal{R}(\\widehat{T_K(\\theta,\\mu)} ) \\lesssim \\frac{L^2_ns^2}{n^2\\log n}.\n\t\\end{equation}\n\\end{enumerate}\n\\end{theorem}\n\nFrom the above theorem, if $0<\\beta<1\/2$, the risk upper bound (\\ref{sparse.upper.equation5})  along with the minimax lower bound (\\ref{lb.l}) implies\n\\begin{equation} \\label{r1}\n\\inf_{\\hat{T}} \\sup_{\\substack{(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_2) \\in D^{\\infty}(s,L_n)}}\t\\mathcal{R}(\\hat{T})\\asymp \\frac{L_n^2s^2\\log n}{n^2},\n\\end{equation}\nwhere the optimal rate can be attained by the thresholding estimator $\\widehat{T^S_K(\\theta,\\mu)}$ with $K$ being any finite positive integer.\nIf $1\/2< \\beta<1$, then the risk upper bound (\\ref{sparse.upper.equation6})  along with the minimax lower bound (\\ref{lb.h}) implies\n\\begin{equation} \\label{r2}\n\\inf_{\\hat{T}} \\sup_{\\substack{(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_2) \\in D^{\\infty}(s,L_n)}} \t\\mathcal{R}(\\hat{T}) \\asymp  \\frac{L_n^2s^2}{n^2\\log n},\n\\end{equation}\nwhere the optimal rate can be attained by the non-thresholding estimator $\\widehat{T_K(\\theta,\\mu)}$ with $K=r\\log n$ for some small $r$. In both cases, the tuning parameter $K$ plays an important role in controlling the bias-variance trade-off. An important consequence of our results concerns the \\emph{local adaptivity} of $\\widehat{T_K(\\theta,\\mu)}$ and $\\widehat{T^S_K(\\theta,\\mu)}$ with respect to $s$ and $L_n$. Specifically, the estimator $\\widehat{T_K(\\theta,\\mu)}$ with $K =r\\log n$ for some $0<r<\\frac{2\\beta-1}{12}$ is simultaneously rate-optimal for any $L_n$ and any $\\beta'\\in(1\/2,1)$ with $\\beta'>\\beta$, whereas the estimator $\\widehat{T_K(\\theta,\\mu)}$ with $K$ being any finite positive integer is simultaneously rate-optimal for any $L_n$ and $\\beta\\in(0,1\/2)$. \n\n\nUnfortunately, even with an appropriate choice of $K$, neither $\\widehat{T^S_K(\\theta,\\mu)}$ nor $\\widehat{T_K(\\theta,\\mu)}$ is simultaneously optimal across all $\\beta\\in(0,1)$. However, since the difference in the optimal rates of convergence between (\\ref{r1}) and (\\ref{r2}) is only of a factor of $\\log n$, we argue that in practice, even when $\\beta\\in(1\/2,1)$, the nearly optimal thresholding estimator $\\widehat{T^S_K(\\theta,\\mu)}$ would perform just as well as the non-thresholding minimax optimal estimator $\\widehat{T_K(\\theta,\\mu)}$.  Moreover, as for the optimal choice of $K$, since the difference between $K\\asymp \\log n$ and $K\\asymp 1$ is minor for most applications with small to moderately large sample sizes (e.g., with $n\\sim 10^5$, we have $\\log n\\approx 11.51$), in practice it suffices to choose $K$ as some fixed moderate constant.\n\n\n\n\\subsection{Sparse estimation via simple thresholding}\n\nAccording to our previous analysis, if the parameter space is very sparse, i.e., $0<\\beta<1\/2$, the proposed estimator $\\widehat{T^S_K(\\theta,\\mu)}$ is minimax optimal if we choose $K$ as any constant positive integer. In other words, any constant degree polynomial approximation suffices for the optimal estimation of $T(\\theta,\\mu)$, including the constant function.  It means that in this case the polynomial approximation is essentially redundant for our purpose. \n\nIn light of the above observation, we consider the following simple thresholding estimator\n\\[\n\\widetilde{T(\\theta,\\mu)}={2}\\sum_{i=1}^n \\hat{U}_i(x_i)\\hat{U}_i(y_i),  \\quad \\text{where} \\quad \\hat{U}_i(x_i)= |x_{i1}| I(|x_{i2}|> 2\\sqrt{2\\log n}).\n\\]\nIn the following, we show that $\\widetilde{T(\\theta,\\mu)}$ has the same rates of convergence as $\\widehat{T_K^S(\\theta,\\mu)}$ over the parameter spaces $D^{\\infty}(n^{\\beta},L_n)$, and, in light of Theorem \\ref{sparse.lower3}, is minimax optimal and adaptive over any sparsity level $\\beta\\in(0,1\/2)$.\n\n\\begin{theorem} \\label{sparse.upper.3}\nLet $\\mathbf{x_n} \\sim N(\\theta,\\boldsymbol{\\Sigma}_1)$ and $\\mathbf{y_n} \\sim N(\\mu,\\boldsymbol{\\Sigma}_2)$ be multivariate Gaussian random vectors with $(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_2) \\in D^{\\infty}(s,L_n)$ and $L_n\\gtrsim \\sqrt{\\log n}$. Then\n\\begin{equation}\n\\sup_{\\substack{(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_1) \\in D^\\infty(s,L_n)}}\\mathcal{R}(\\widetilde{T(\\theta,\\mu)} ) \\lesssim \\frac{L_n^2s^2\\log n}{n^2}.\n\\end{equation}\n\\end{theorem}\n\nSince our simple thresholding estimator $\\widetilde{T(\\theta,\\mu)}$ completely drops the polynomial components in $\\widehat{T^S_K(\\theta,\\mu)}$, its variance is significantly reduced. As a consequence, we find that as long as $\\max(\\|\\theta\\|_\\infty , \\|\\mu\\|_\\infty )\\le \\sqrt{n}$, the isotopic Gaussian assumption $\\boldsymbol{\\Sigma}_1=\\boldsymbol{\\Sigma}_2={\\bf I}_n$ can be removed without changing the rate of convergence. Towards this end, we define the enlarged parameter space\n\\[\nD_0^\\infty(s,L_n) = \\Bigg\\{(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_2): \\begin{aligned} & (\\theta,\\mu)\\in D(s),\\max(\\|\\theta\\|_\\infty , \\|\\mu\\|_\\infty )\\le L_n, \\\\\n&  \\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_2\\succeq 0, \\text{$\\boldsymbol{\\Sigma}_1$ and $\\boldsymbol{\\Sigma}_2$ have unit diagonals}.\n\\end{aligned} \\Bigg\\}.\n\\]\nIn particular, as $\\boldsymbol{\\Sigma}_1$ and $\\boldsymbol{\\Sigma}_2$ have unit diagonals, the sample splitting procedure (Section 2.1) still applies, which only leads to a $1\/2$-scaling of the off-diagonal entries of the covariance matrices.\n\\begin{theorem} \\label{sparse.upper.4}\nLet $\\mathbf{x_n} \\sim N(\\theta,\\boldsymbol{\\Sigma}_1)$ and $\\mathbf{y_n} \\sim N(\\mu,\\boldsymbol{\\Sigma}_2)$ where $(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_2)\\in D_0^{\\infty}(s, \\sqrt{n})$. Then we have\n\\begin{equation}\n\\sup_{\\substack{(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_1) \\in D_0^\\infty(s,\\sqrt{n})}}\\mathcal{R}(\\widetilde{T(\\theta,\\mu)} )\\lesssim \\frac{s^2\\log^2 n}{n^2}.\n\\end{equation}\n\\end{theorem}\n\nBy definition, we have $D^{\\infty}(s,L_n)\\subset D_0^{\\infty}(s,L_n)$. It then follows from Theorem \\ref{sparse.lower3} that for any $s\\lesssim\\sqrt{n}$,\n\\begin{equation}\n\\inf_{\\hat{T}} \\sup_{\\substack{(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_1) \\in D_0^\\infty(s,\\sqrt{ n})}}\\mathcal{R}(\\hat{T}) \\asymp \\frac{s^2\\log^2 n}{n^2},\n\\end{equation}\nwhere the minimax optimal rate can be attained by $\\widetilde{T(\\theta,\\mu)}$. This establishes the minimax optimality and adaptivity of $\\widetilde{T(\\theta,\\mu)}$ over $D_0^{\\infty}(n^\\beta,\\sqrt{n})$ for any $\\beta\\in(0,1\/2)$.\nThe result confirms an important advantage of $\\widetilde{T(\\theta,\\mu)}$ over $\\widehat{T^S_K(\\theta,\\mu)}$, namely, its guaranteed theoretical performance over arbitrary correlation structures, which complies with the fact that in many applications the observations are not necessarily independent.\n\n\n\\subsection{Normalization and linkage disequilibrium}\n\nDealing with linkage disequilibrium  (\\textsc{ld}) among the  \\textsc{snp}s \\citep{reich2001linkage,daly2001high,pritchard2001linkage} is essential in any genetic studies. In this paper, we follow the idea of \\cite{Bulik-Sullivan} and propose  to use the normalized $T$-score\n\\[\n\\text{Normalized $T$-score}(g) = \\frac{\\sum_{i=1}^n|\\theta_i^g\\mu_i|}{\\|\\theta^g\\|_2\\|\\mu\\|_2}\n\\]\nas a measure of genetic overlap between gene $g$ and the outcome disease. In particular, the estimation of the $\\ell_2$ norms $\\|\\theta^g\\|_2$ and $\\|\\mu\\|_2$, or in our context, the \\textsc{snp}-heritability of the traits \\citep{yang2010common}, can be easily accomplished using summary statistics. As a result, every normalized $T$-score lies in between $0$ and $1$, which is scale-invariant (e.g., invariance to study sample sizes and \\textsc{snp} effect sizes) and comparable across many different genes or studies. In addition, as similarly argued in \\cite{Bulik-Sullivan}, the normalized $T$-score is less sensitive  to  the choice of the  $n$-\\textsc{snp} sets used in the analysis. \n\nMoreover, in Theorem 4, we show that the simple thresholding estimator $\\widetilde{T(\\theta,\\mu)}$ does not require the independence of the $z$-scores, which theoretically guarantees its applicability in the presence of arbitrary \\textsc{ld} structure among the  \\textsc{snp}s. However, our theoretical results concerning $\\widehat{T_K(\\theta,\\mu)}$ and $\\widehat{T^S_K(\\theta,\\mu)}$ rely on such an  independence assumption.  In our simulation studies, we found that the empirical performance (including optimality) of $\\widehat{T_K(\\theta,\\mu)}$ and $\\widehat{T^S_K(\\theta,\\mu)}$ is not likely affected by the dependence due to the  \\textsc{ld} structure. As a result, our proposed estimation method, although partially analysed under the independence assumption, can be directly applied to the summary statistics without specifying the underlying \\textsc{ld} or covariance structure.\n\n\n\n\\section{Simulation Studies}\n\nIn this section, we demonstrate and compare the empirical performance of our proposed estimators and some alternative estimators under various settings. Since our goal is to  estimate the T-score for each gene and  to identify important genes that have strong association with  disease, we  generate data that mimic genetic association data, including  block-wise and sparse signals.  In particular, the block-wise correlated structure of the  \\textsc{snp}s and therefore also the corresponding $z$-scores is widely observed in large-scale genetic studies due to  \\textsc{ld} among the  \\textsc{snp}s.  In the following,  we consider the  block-wise signals with both the standard independent errors or  errors with local dependence to show the robustness of our proposed estimator.\n\nWe generate a pair of $n$-dimensional vectors, denoted as $\\mathbf{x}_n$ and $\\mathbf{y}_n$, with $ n = 1.5\\times 10^5,3\\times 10^5$ and $5\\times 10^5$, from a multivariate normal distribution $N(\\theta,\\boldsymbol{\\Sigma})$ and $N(\\mu,\\boldsymbol{\\Sigma})$, respectively. The mean vectors $(\\theta,\\mu)$, and the covariance matrix $\\Sigma$ are generated as follows:\n\\begin{itemize}\n\t\\item Block-wise signals: we first partition the $n$ coordinates of $\\theta$ and $\\mu$ into blocks with  each block consisting  of 10 neighboring coordinates. For any sparsity level $s=50,100,200,300$ or $400$, we randomly pick $s\/10$ blocks as the support. In each of these blocks, we assign coordinates of $\\theta$ and $\\mu$ with symmetric triangle-shaped values, whose maximal value is generated from a uniform distribution over $[3,5]$. For the rest of the $(n-s)\/10$ blocks, we set them as 0.\n\t\\item Global covariance: $\\boldsymbol{\\Sigma}={\\bf I}$. \n\t\\item Block-wise covariances: $\\boldsymbol{\\Sigma}$ is block diagonal where each block is either a $10\\times 10$ Toeplitz matrix (see Supplementary Material for details) or a $1000\\times 1000$ exchangeable covariance matrix whose off-diagonal elements are $0.5$. These two covariance matrices are denoted as $\\boldsymbol{\\Sigma}_1$ and $\\boldsymbol{\\Sigma}_2$, respectively.\n\\end{itemize}\nIn particular, for the  correlated cases, the correlation exists over either a small block of 10 elements whose correlation  decays in distance, or over a  large interval of 1000 elements with a constant correlation. \n\nWe compare the following five estimators of $T(\\theta,\\mu)$: (1) the hybrid thresholding estimator $\\widehat{T^S_K(\\theta,\\mu)}$; (2) the hybrid thresholding estimator without sample splitting procedure, denoted as $\\widehat{T^{S*}_K(\\theta,\\mu)}$; (3) the proposed simple thresholding estimator  $\\widetilde{T(\\theta,\\mu)}$; (4) the hybrid non-thresholding estimator $\\widehat{T_K(\\theta,\\mu)}$; and (5) the estimator that simply calculates the absolute inner product of observed vectors, denoted as Naive.\n\nFor $\\widehat{T^S_K(\\theta,\\mu)}$, $\\widehat{T_K(\\theta,\\mu)}$ and $\\widehat{T^{S*}_K(\\theta,\\mu)}$, we choose $K=8$ (see the last paragraph of Section 2.3). Each setting was repeated 100 times. We compare the estimators using the empirical version of the following rescaled mean square error:\n\\[\n\\textsc{rmse}(\\hat{T}) = \\frac{1}{s}\\sqrt{{\\mathbb{E}(\\hat{T}-{T})^2}}.\n\\]\nIn particular, as the empirical \\textsc{rmse}s are subject to a factor of $s$, small numerical differences in \\textsc{rmse} actually indicate large differences in overall MSE.\nTables 1-3 give the empirical \\textsc{rmse} of the five estimators under the settings with either independent or dependent observations. In general, $\\widehat{T^{S*}_K(\\theta,\\mu)}$ outperforms all the other estimators in all the settings. The performances of $\\widehat{T^S_K(\\theta,\\mu)}$, $\\widetilde{T(\\theta,\\mu)}$ and $\\widehat{T^S_K(\\theta,\\mu)}$ are roughly the same, with $\\widehat{T^S_K(\\theta,\\mu)}$ and $\\widetilde{T(\\theta,\\mu)}$ being slightly better over the more sparse cases. The Naive estimator performs poorly in all the settings. The superiority of $\\widehat{T^{S*}_K(\\theta,\\mu)}$ over $\\widehat{T^S_K(\\theta,\\mu)}$ might be explained by the fact that, by using the sample splitting procedure, we introduced extra variability to our estimator. Since the sample splitting is used only to facilitate our theoretical analysis, in applications we suggest to use $\\widehat{T^{S*}_K(\\theta,\\mu)}$ for its better performance. \n\nAs for the robustness of our estimators to the presence of correlated observations, we see that all of the proposed estimators have roughly the same performance as the independent cases. In real applications, we expect that the observed data are more likely to be ``block-wise signal with block-wise correlation.\"  Our  simulation shows that $\\widehat{T^{S*}_K(\\theta,\\mu)}$ provides the best estimate of the T-scores. \n\n\\begin{table}[t!] \n\t\\caption{\\footnotesize Empirical \\textsc{rmse}  under covariance $\\boldsymbol{\\Sigma}={\\bf I}_n$.   $\\widehat{T^{S*}_K(\\theta,\\mu)}$: the hybrid thresholding estimator without sample splitting procedure;\n\t\t$\\widehat{T^S_K(\\theta,\\mu)}$: the hybrid thresholding estimator;\n\t\t$\\widetilde{T(\\theta,\\mu)}$: the proposed simple thresholding estimator;\n\t\t$\\widehat{T_K(\\theta,\\mu)}$: the hybrid non-thresholding estimator;\n\t\tNaive: the estimator that simply calculates the absolute inner product of observed vectors.}\n\t\\vskip .2cm\n\t\\centerline{\\tabcolsep=3truept\\begin{tabular*}{ 0.78 \\textwidth}{ccccccc}\t\\hline \n\t\t\t$n$&   $s$  &  $\\widehat{T^{S*}_K(\\theta,\\mu)}$  & $\\widehat{T^S_K(\\theta,\\mu)}$   & \t$\\widetilde{T(\\theta,\\mu)}$   &      $\\widehat{T_K(\\theta,\\mu)}$  & Naive \\\\  \n\t\t\t\\hline  \n\t\t\t&100& 4.091 & 5.818 & 5.764 & 5.847 & 954.4 \\\\\n\t\t\t&250&  4.058 & 6.348 & 6.157  & 6.382 & 381.3\\\\\n\t\t\t150,000   & 400&  3.949 & 6.274 & 6.042 & 238.4 & 6.307 \\\\\n\t\t\t& 550  & 4.024 & 6.272 & 6.069  & 6.304 & 173.2\\\\\n\t\t\t& 700& 3.973 & 5.84 & 5.702 & 5.870& 135.9  \\\\\t\t\n\t\t\t\\hline  \n\t\t\t& 100& 4.271 & 6.579 & 6.385  & 6.613& 1909.1 \\\\\n\t\t\t&250& 4.222 & 6.244 & 6.118  & 6.276 & 763.3\\\\\n\t\t\t300,000     & 400&  4.143 & 6.015 & 5.882  & 6.046& 476.8 \\\\\n\t\t\t& 550  & 4.214 & 5.991 & 5.874  & 6.022& 346.6\\\\\n\t\t\t& 700&   4.242 & 6.144 & 6.026& 6.175 & 272.4  \\\\\n\t\t\t\\hline  \n\t\t\t& 100& 4.380 & 6.139 & 6.031  & 6.170 & 3182.7 \\\\\n\t\t\t&250& 4.428 & 6.251 & 6.157 & 6.283& 1272.6  \\\\\n\t\t\t500,000     & 400&  4.374 & 6.109 & 5.997 & 6.140 & 795.3  \\\\\n\t\t\t& 550  &4.395 & 6.149 & 6.056 & 6.180& 578.3 \\\\\n\t\t\t& 700& 4.363 & 6.221 & 6.117  & 6.253& 454.2 \\\\\n\t\t\t\\hline\n\t\\end{tabular*}}\n\t\\label{table:t1}\n\\end{table}\n\n\n\\begin{table}[t!]\n\t\\caption{Empirical \\textsc{rmse} under covariance $\\boldsymbol{\\Sigma}_1$.}\n\t\\vskip .2cm\n\t\\centerline{\\tabcolsep=3truept\\begin{tabular*}{ 0.78 \\textwidth}{ccccccc}\n\t\t\t\\hline \n\t\t\t$n$&   $s$  &  $\\widehat{T^{S*}_K(\\theta,\\mu)}$  & $\\widehat{T^S_K(\\theta,\\mu)}$   & \t$\\widetilde{T(\\theta,\\mu)}$  &       $\\widehat{T_K(\\theta,\\mu)}$  & Naive   \\\\  \n\t\t\t\\hline  \n\t\t\t&100& 4.090 & 6.704 & 6.502  & 7.042& 953.5 \\\\\n\t\t\t&250&  4.134 & 5.816 & 5.702 & 6.104& 381.9  \\\\\n\t\t\t150,000   & 400&  4.068 & 6.087 & 5.896 & 6.396 & 237.9 \\\\\n\t\t\t& 550  & 3.947 & 5.991 & 5.826  & 6.293 & 172.9 \\\\\n\t\t\t& 700& 4.163 & 6.011 & 5.913 & 6.310& 135.8  \\\\\n\t\t\t\\hline  \n\t\t\t& 100&  3.868 & 6.088 & 6.009 & 6.403& 1912.2  \\\\\n\t\t\t&250& 4.196 & 5.963 & 5.797  & 6.262& 764.7 \\\\\n\t\t\t300,000     & 400&  4.215 & 6.171 & 6.09  & 6.478& 476.8\\\\\n\t\t\t& 550  & 4.238 & 6.005 & 5.888  & 6.306& 345.7\\\\\n\t\t\t& 700&   4.409 & 6.056 & 5.965  &  6.136& 272.6 \\\\\n\t\t\t\\hline  \n\t\t\t& 100& 4.288 & 5.385 & 5.345  & 5.651 & 3184.8 \\\\\n\t\t\t&250&  4.353 & 6.227 & 6.027  & 6.531& 1273.0 \\\\\n\t\t\t500,000     & 400&  4.478 & 5.878 & 5.877 & 6.170  & 795.3 \\\\\n\t\t\t& 550  &4.370 & 5.929 & 5.855  & 6.222& 577.7\\\\\n\t\t\t& 700& 4.464 & 6.135 & 6.083  & 6.438 & 455.1\\\\\n\t\t\t\\hline\n\t\\end{tabular*}}\n\t\\label{table:t2}\n\\end{table}\n\n\n\\begin{table}[h!]\n\t\\centering\n\t\\caption{Empirical \\textsc{rmse} under covariance $\\boldsymbol{\\Sigma}_2$.}\n\t\\vskip .2cm\n\t\\centerline{\\tabcolsep=3truept\\begin{tabular*}{ 0.78 \\textwidth}{ccccccc}\n\t\t\t\\hline \n\t\t\t$n$&   $s$  &  $\\widehat{T^{S*}_K(\\theta,\\mu)}$  & $\\widehat{T^S_K(\\theta,\\mu)}$   & \t$\\widetilde{T(\\theta,\\mu)}$  &     $\\widehat{T_K(\\theta,\\mu)}$  & Naive   \\\\  \n\t\t\t\\hline  \n\t\t\t&100& 4.336 & 5.88 & 5.807  & 5.906 & 485.3\\\\\n\t\t\t&250&  4.645 & 5.857 & 5.828  & 5.885& 189.7 \\\\\n\t\t\t150,000   & 400& 4.770 & 6.543 & 6.438 & 6.575 & 120.1 \\\\\n\t\t\t& 550  & 4.566 & 6.035 & 5.985& 6.065 & 87.12  \\\\\n\t\t\t& 700& 4.523 & 5.983 & 5.938  & 6.012 & 68.89\\\\\n\t\t\t\\hline  \n\t\t\t& 100&  4.484 & 6.005 & 5.980  & 6.029 & 961.0\\\\\n\t\t\t&250& 4.718 & 5.86 & 5.865 & 5.887  & 380.6\\\\\n\t\t\t300,000     & 400&  4.753 & 6.26 & 6.241  & 6.291& 241.8\\\\\n\t\t\t& 550  & 4.729 & 5.985 & 5.949  & 6.015& 174.8 \\\\\n\t\t\t& 700&  4.84 & 6.334 & 6.289  & 6.365& 135.7 \\\\\n\t\t\t\\hline  \n\t\t\t& 100& 4.836 & 6.292 & 6.279 & 6.313& 1593.5   \\\\\n\t\t\t&250& 4.749 & 6.629 & 6.569 & 6.658& 633.4  \\\\\n\t\t\t500,000     & 400&  4.889 & 6.409 & 6.401 & 6.438 & 398.1  \\\\\n\t\t\t& 550  &4.867 & 6.343 & 6.309 & 6.373& 288.8 \\\\\n\t\t\t& 700& 4.887 & 6.131 & 6.112  & 6.159& 227.7 \\\\\n\t\t\t\\hline\n\t\\end{tabular*}}\n\t\\label{table:t3}\n\\end{table}\n\n\n\n\\section{Integrative data analysis of human heart failure}\n\nFinally, we apply our proposed estimation procedure to identify genes whose expressions are possibly causally linked to heart failure by integrating \\textsc{gwas} and e\\textsc{qtl} data. \\textsc{gwas} results were obtained from a heart failure genetic association study at the University of Pennsylvania, a  prospective study of patients recruited from the University of Pennsylvania, Case Western Reserve University, and the University of Wisconsin, where genotype data were collected from 4,523 controls and 2,206 cases using the Illumina OmniExpress Plus array. \\textsc{gwas} summary statistics were calculated controlling for age, gender, and the first two principal components of the genotypes. \n\nHeart failure e\\textsc{qtl} data were obtained from the MAGNet e\\textsc{qtl} study (https:\/\/www.med. upenn.edu\/magnet\/index.shtml), where the left ventricular free-wall tissue was collected from 136 donor hearts without heart failure. Genotype data were collected using using Affymetrix genome-wide SNP array 6.0 and only markers in Hardy-Weinberg equilibrium with minor allele frequencies above 5\\% were considered. Gene expression data were collected using Affymetrix GeneChip ST1.1 arrays, normalized using \\textsc{RMA} \\citep{irizarry2003exploration} and batch-corrected using ComBat \\citep{johnson2007adjusting}. To obtain a common set of  \\textsc{snp}s, \n\\textsc{snp}s were imputed using 1000 Genomes Project data.  Summary statistics for the MAGNet e\\textsc{qtl} data were obtained  using the fast marginal regression algorithm of \\citet{sikorska2013gwas} controlling for age and gender.\n\n\\subsection{Ranking of Potential Heart Failure Causal Genes}\nAfter matching the  \\textsc{snp}s of both e\\textsc{qtl} and \\textsc{gwas} data, we had  a total of 347,019  \\textsc{snp}s and 19,081 genes with expression data available.  For each gene, the vector of $z$-scores from e\\textsc{qtl} analysis was obtained and the value of its absolute inner product with the vector of $z$-scores from \\textsc{gwas} data was estimated by our proposed estimator $\\widehat{T_K^{S*}(\\theta,\\mu)}$.  We chose $K=8$ as in our simulation study. After obtaining the estimates of $T$-scores for all the genes, and the corresponding \\textsc{snp}-heritability, we rank these genes in the order of their normalized $T$-scores as defined in Section 2.5. \nAs a result, genes with the highest ranks are considered important in gaining insights into  the biological mechanisms of heart failure.\n\nTo assess that the  top scored genes indeed represent true biological signals, we  calculated the $T$-scores for two ``null datasets\" that  are created using permutations. For the first dataset, we randomly permuted the labels of the  \\textsc{snp}s of the \\textsc{gwas} $z$-scores by sampling without replacement before estimating the normalized $T$-scores  with e\\textsc{qtl} $z$-scores. For the second dataset, we permuted the labels of the  \\textsc{snp}s of the \\textsc{gwas} $z$-scores in a circular manner similar to \\cite{cabrera2012uncovering}.  Specifically,  for each chromosome, we randomly chose one \\textsc{snp} as start of the chromosome and move the \\textsc{snp}s  on the fragment  before this \\textsc{snp}  to the end.  Such a cyclic permutation preserves the local dependence of the $z$-scores. \nBy permuting the data from one phenotype, we break the original matching  of the $z$-scores  between the two phenotypes.  The permutation was performed 50 times and the null distribution of $T$-scores based on the permuted  data was obtained.  Figure \\ref{permute} shows the ranked normalized   $T$-scores based on the original data and the box plots of the ranked $z$-scores based on 50 permutations of the $z$-scores.  We find that all the top ranked genes have larger $T$-scores than the ones based on permutations. In addition, about 30 top ranked genes in the top plot  and about 10 top ranked genes in the bottom plot  have true $T$-scores larger than all the $T$-scores from the permuted datasets.  This further confirms that the top ranked genes based on their estimated normalized $T$-scores are not due to random noise and indeed represent certain sharing of genetic variants between heart failure and gene expression levels. \n\n\\begin{figure}[h!]\n\t\\includegraphics[angle=0,width=14cm]{rank_plot_perm1.pdf}\n\t\\includegraphics[angle=0,width=14cm]{Perm2.pdf}\n\t\\caption{Estimated score for the top 50 genes and the box plots of the top scores based on 50 permutations. Top: random permutation of the \\textsc{gwas} scores; bottom: cyclic permutations of the \\textsc{gwas} scores. }\n\t\\label{permute}\n\\end{figure} \n\n\nTable \\ref{genes}  lists the top eight highest ranked genes along with their biological annotations. All of the genes are either directly or indirectly associated with human heart failure, including \nthose related to fibrotic myocardial degeneration, Wnt signalling activity and heart-valve development. It is interesting that our proposed methods can identify these relevant genes using only the gene expression data measured on normal heart tissues. \n\n\\begin{table}\n\t\\caption{Top eight heart failure associated genes based on the estimated normalized $T$-scores and their functional annotations.}\\label{genes}\n\t\\vskip .2cm\n\t\\centerline{\\tabcolsep=3truept\\begin{tabular*}{ 1 \\textwidth}{lll}\n\t\t\t\\hline \n\t\t\tGene Name &  Annotations  \\\\  \n\t\t\t\\hline  \n\t\t\tTMEM37 &   voltage-gated ion channel activity   \\citep{chen2007calcium} \\\\\n\t\t\tADCY7 &  adenylate cyclase activity; fibrotic myocardial  \\\\\n\t\t\t& degeneration \\citep{nojiri2006oxidative}\\\\\n\t\t\tC1QC &  Wnt signaling activity; associated with heart \\\\\n\t\t\t&  failure \\citep{naito2012complement} \\\\\n\t\t\tFAM98A &  associated with ventricular  septal  defect \\citep{liu2018differential}\\\\\n\t\t\tBMP2 & associated with heart-valve development \\\\  &\\citep{rivera2006bmp2}\\\\\n\t\t\tSLCO2B1   & organic anion transporter; associated with cardiac glycoside \\\\\n\t\t\t&  \\citep{mikkaichi2004organic} \\\\\n\t\t\tC1QA  & Wnt signaling activity; associated with heart \\\\\n\t\t\t&  failure \\citep{naito2012complement} \\\\\n\t\t\tFCGR2B &  intracellular signaling activity; associated with vascular \\\\\n\t\t\t& disease pathogenesis \\citep{tanigaki2015fcgamma}\\\\\n\t\t\t\\hline\n\t\\end{tabular*}}\n\t\\label{table:t4}\n\\end{table}\n\n\n\n\\subsection{Gene Set Enrichment Analysis}\nTo complete our analysis, we finish this section with the gene set enrichment analysis (\\textsc{gsea}) \\citep{subramanian2005gene} using the normalized $T$-scores in order to identify the heart failure associated biological processes.    In the following analysis, we removed genes with low expression and small variability across the samples, which resulted in a total of 6,355 genes. \nFor the normalized $T$-scores $T_j$, $1\\le j\\le J$, given a gene set $S$, the Kolmogorov-Smirnov statistic defined as\n\\[\n\\sup_t \\bigg| \\frac{1}{k}\\sum_{j\\in S}I(T_j \\le t) -\\frac{1}{k'}\\sum_{j\\in S^c}I(T_j \\le t)   \\bigg|,\n\\]\nis calculated, where $k$ and $k'$ are the number of genes in $S$ and $S^c$, respectively. This  tests whether the distribution of  $T_j$ differs between genes in $S$ and genes in $S^c$.  For a given gene set, significance of this test implies that this gene set is enriched by genes that shared similar genetic variants as those for heart failure, suggesting their relevance to the etiology of heart failure. \n\nThis analysis was applied to the gene sets from Gene Ontology (\\textsc{go}) (\\citealt{botstein2000gene}) that contain at least 10 genes.  Specifically, 5,023 biological processes were tested.\nFigure 2 in our supplementary material  presents the directed acyclic graphs of the \\textsc{go} biological processes  that linked to  the most significant \\textsc{go} terms  from the simultaneous signal \\textsc{gsea} analysis.  Table \\ref{table:t5} shows the top six \\textsc{go} biological processes identified from the \\textsc{gsea} analysis. Among these  processes, regulation of skeletal muscle contraction, linoleic acid metabolic process and calcium ion regulation are strongly implicated in  human heart failure. \\cite{murphy2011cardiovascular} showed that skeletal muscle reflexes are essential to the initiation and regulation of the cardiovascular response to exercise, and alteration of this reflex mechanism can happen in disease states such as hypertension and heart failure. In \\cite{farvid2014dietary}, a systematic review of literature and meta-analysis was carried out, which supports a significant inverse association between dietary linoleic acid intake, when replacing either carbohydrates or saturated fat, and risk of coronary heart disease. Moreover,  the importance of calcium-dependent signaling in the heart failure was reported in  \\cite{marks2003calcium}, who suggested that impaired calcium release causes decreased muscle contraction (systolic dysfunction) and defective calcium removal hampers relaxation (diastolic dysfunction).\n\n\n\\begin{table}\n\t\\centering\n\t\\caption{Top six \\textsc{go} biological processes that are associated with heart failure based on the gene set enrichment analysis}\n\t\\vspace{0.3cm}\n\t\\begin{tabular}{lc}\n\t\t\\hline \n\t\t\\textsc{go} term & p-value  \\\\  \n\t\t\\hline  \n\t\t\\emph{Biological Process} &\\\\\n\t\tregulation of skeletal muscle contraction by regulation of release&\\\\\n\t\tof sequestered calcium ion & $7.9\\times 10^{-7}$\\\\\n\t\tlinoleic acid metabolic process & $1.0\\times 10^{-6}$\\\\\n\t\tregulation of skeletal muscle contraction by calcium ion signaling & $3.4\\times 10^{-6}$ \\\\\n\t\tpositive regulation of sequestering of calcium ion & $3.4\\times 10^{-6}$\\\\\n\t\tcellular response to caffeine& $1.0\\times 10^{-5}$\\\\\n\t\tcellular response to purine-containing compound & $1.0\\times 10^{-5}$\\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\label{table:t5}\n\\end{table}\n\n\n\n\\section{Discussion}\n\nThis paper considers the optimal estimation over sparse parameter spaces. However, in Section 2, the minimax rates of convergence were only established for parameter spaces $D^{\\infty}(n^{\\beta},L_n)$ with $\\beta\\in (0,1\/2)\\cup (1\/2,1)$. Apparently, there was a missing gap at $\\beta=1\/2$. Our theoretical analysis suggests a lower bound  (\\ref{sparse.lower.equation4}) with the rate function $\\psi(s,n)\\asymp  \\log^{-1}n$. However, none of our proposed estimators attain such rates, leaving it an open question how to optimality estimate $T(\\theta,\\mu)$ when $s\\asymp \\sqrt{n}$.\n\n\nIn some applications, we may need to consider non-sparse parameter spaces. In this case, our theoretical analysis shows that the estimator $\\widehat{T_K(\\theta,\\mu)}$ with $K=r\\log n$ for some small constant $r>0$ can still be applied. Specifically, from our proof of Theorem 1 and Theorem 2, it follows that, if we define the non-sparse parameter space as $\\mathcal{D}^{\\infty}_U(L_n)=\\big\\{(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_2):  (\\theta,\\mu)\\in \\mathbb{R}^n\\times \\mathbb{R}^n,\\max(\\|\\theta\\|_\\infty , \\|\\mu\\|_\\infty )\\le L_n,  \\boldsymbol{\\Sigma}_1=\\boldsymbol{\\Sigma}_2={\\bf I}_n \\big\\}$ with $L_n\\gtrsim \\sqrt{\\log n}$, then for $\\mathbf{x_n} \\sim N(\\theta,\\boldsymbol{\\Sigma}_1)$ and $\\mathbf{y_n} \\sim N(\\mu,\\boldsymbol{\\Sigma}_2)$, the minimax optimal rate for estimating $T(\\theta,\\mu)$ over $\\mathcal{D}^{\\infty}_U(L_n)$ is\n\\begin{equation} \n\\inf_{\\hat{T}}\\sup_{(\\theta,\\mu,\\boldsymbol{\\Sigma}_1,\\boldsymbol{\\Sigma}_2)\\in \\mathcal{D}^{\\infty}_U(L_n) } \\mathcal{R}(\\hat{T} ) \\asymp \\frac{L_n}{ \\log n},\n\\end{equation}\nwhich can be attained by the above $\\widehat{T_K(\\theta,\\mu)}$. \n\nIn addition to the estimation problem, it is also of interest to conduct hypothesis testing or construct confidence intervals for $T$-score. These problems can be technically challenging due to the non-smooth functional. Moreover, the polynomial approximation technique adopted in the current work may introduce extra complexities to the higher-order asymptotic analysis. We leave these problems for future investigations.\n\n\n\n\n\\vskip 14pt\n\\noindent {\\large\\bf Supplementary Materials}\n\nOur supplementary material includes proofs of the main theorems and the technical lemmas.\n\\par\n\\bibliographystyle{chicago}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Appendix A: Incompatibility robustness SDP and its dual}\n\nThe incompatibility robustness $1+\\mathcal{IR}$, can be cast as the following SDP\n\\begin{eqnarray}\n\\min_{\\tilde{G}_{\\lambda}}\\,&& \\sum_{\\lambda}\\frac{\\tr [\\tilde{G}_{\\lambda} ]}{d} \\\\\n\\label{Eq:constraint1}\\text{s. t.: }&& \\sum_{\\lambda}D(a|x,\\lambda) \\tilde{G}_{\\lambda} \\geq M_{a|x} \\text{ for all }a,x \\\\\n\\label{Eq:constraint2} && \\tilde{G}_{\\lambda}\\geq 0 \\\\\n&& \\sum_{\\lambda} \\tilde{G}_{\\lambda} = \\frac{\\ensuremath{\\mathds{1}}_d}{d} \\sum_{\\lambda} \\tr[\\tilde{G}_{\\lambda}]\\notag,\n\\end{eqnarray}\nwhere $d$ is the dimension of the space of the $G_{\\lambda}$ such that \n$\\sum_{\\lambda} G_{\\lambda} = \\ensuremath{\\mathds{1}}_d$. The number of labels $a$ we denote by $|a|$, and similarly $|x|$ denotes the number of labels $x$. The number of constraints in Eq.~\\eqref{Eq:constraint1} is $|a|\\cdot |x|$ and in Eq.~\\eqref{Eq:constraint2} it is $|a|^{|x|}$.\nThis SDP has both, equalities and inequalities as constraints and it is of the general form\n\\begin{eqnarray}\n\\min_{X}&& \\langle A,X \\rangle \\\\\n\\text{s. t.: }&& \\Phi(X)=B_1 \\notag \\\\\n&& \\Psi(X)\\geq B_2 \\notag \\\\\n&& X\\geq 0. \\notag\n\\end{eqnarray}\nWe choose $A=\\ensuremath{\\mathds{1}}_{d\\cdot |a|^{|x|}} \\qquad\\text{and}\\qquad \nX=\\text{diag}\\qty[\\tilde{G}_{\\lambda}\/d]_{\\lambda}\\in\\mathcal{M}(\\mathbb{C}^{d\\cdot \n|a|^{|x|}})$, which is a block diagnoal matrix with the submatrices \n$\\tilde{G}_{\\lambda}\/d$. The objective function then reads $\\langle A,X \\rangle \n= \\tr[X]=\\sum_\\lambda \\tr [X_\\lambda]$.  For the equality constraint we choose \n$B_1=0$ and define a mapping $\\Phi: \\mathcal{M}(\\mathbb{C}^{d\\cdot |a|^{|x|}})\\mapsto \n\\mathcal{M}(\\mathbb{C}^{d})$ by\n\\begin{equation}\n    \\Phi(X) = \\ensuremath{\\mathds{1}}_d \\tr[X]-d \\sum_\\lambda X_\\lambda.\n\\end{equation}\nFor the inequality constraint we choose $B_2=\\text{diag}\\qty[M_{a|x}]_{a,x}$ and we define $\\Psi(X):  \\mathcal{M}(\\mathbb{C}^{d\\cdot |a|^{|x|}})\\mapsto \\mathcal{M}(\\mathbb{C}^{d\\cdot |a|\\cdot |x|})$ by\n\\begin{equation}\n    \\Psi(X) = \\text{diag}\\qty[d\\sum_{\\lambda}D(a|x,\\lambda) X_{\\lambda}]_{a,x}.\n\\end{equation}\nThe dual problem then reads~\\cite{JWatrous_Notes_2011}\n\\begin{eqnarray}\n\\max_{Z,Y}&& \\langle B_1,Z \\rangle + \\langle B_2,Y \\rangle \\\\\n\\text{s. t.: }&& \\Phi^{\\dagger}(Z) + \\Psi^{\\dagger}(Y) \\leq A \\notag\\\\\n&& Z\\text{ is Hermitian} \\notag \\\\\n&& Y\\geq 0. \\notag\n\\end{eqnarray}\nThe dual $\\Psi^{\\dagger}(Y)$ is straight forward~\\cite{Piani_Watrous_Char_of_EPR_Steering}, that is\n\\begin{eqnarray}\n    \\Psi^{\\dagger}(Y) = \\text{diag}\\qty[d\\sum_{a,x}D(a|x,\\lambda) Y^{a|x}]_{\\lambda}.\n\\end{eqnarray}\nTo find $\\Phi^{\\dagger}(Z)$ we write\n\\begin{eqnarray}\n    \\tr[\\Phi(X)Z]&=&\\tr[\\tr(X) Z]-\\tr[d\\sum_{\\lambda} X_{\\lambda} Z] \\\\\n    &=& \\tr[X\\qty{\\tr(Z)\\ensuremath{\\mathds{1}}_{d\\cdot a^x}-d(Z\\oplus Z\\oplus \\cdots \\oplus Z)}] \\notag \\\\\n    &=& \\tr[X \\Phi^{\\dagger}(Z)]\\notag.\n\\end{eqnarray}\nFrom this we directly obtain the dual of the robustness SDP as\n\\begin{eqnarray}\n\\label{Eq:IRDual}\n\\max_{Y_{a|x}}&& \\sum_{a,x} \\tr[M_{a|x}Y^{a|x}]  \\\\\n\\text{s. t.: }&& \\text{diag}\\qty[d\\sum_{a,x}D(a|x,\\lambda) \nY^{a|x}]_{\\lambda}\\notag\\\\\n&&+ \\tr(Z)\\ensuremath{\\mathds{1}}_{d\\cdot |a|^{|x|}}-d(Z\\oplus Z \\oplus \\cdots \\oplus Z) \\leq \\ensuremath{\\mathds{1}}_{d\\cdot |a|^{|x|}} \\notag\\\\\n&& Z \\text{ is Hermitian} \\notag\\\\\n&& Y\\geq 0. \\notag\n\\end{eqnarray}\n\n\n\\section{Appendix B: Upper bound on the success probability for sets on compatible POVMs}\n\nNext, we show that whenever a set of jointly measurable POVMs is used to discriminate the optimal state assemblage $\\qty{Y^{a|x}}$, we find that\n\\begin{eqnarray}\n    \\sum_{a,x} \\tr[O_{a|x}Y^{a|x}] &=& \\sum_{a,x,\\lambda} D(a|x,\\lambda) \\tr[J_{\\lambda}Y^{a|x}] \\\\\n    &=:& \\sum_{\\lambda} \\tr[J_{\\lambda}\\tilde{Y}^{\\lambda}].\n\\end{eqnarray}\nFrom the first constraint of the dual program in Eq.~\\eqref{Eq:IRDual} we obtain (from each block labeled by $\\lambda$) that $\\tilde{Y}^{\\lambda}\\leq\\frac{\\ensuremath{\\mathds{1}}_d}{d}(1-\\tr Z)+Z$. This leads to\n\\begin{eqnarray}\n    \\sum_{\\lambda} \\tr[J_{\\lambda}\\tilde{Y}^{\\lambda}] &\\leq& \\sum_{\\lambda} \\tr[J_{\\lambda}\\frac{\\ensuremath{\\mathds{1}}_d}{d}(1-\\tr Z)+Z] \\notag\\\\\n    &=& \\tr[\\frac{\\ensuremath{\\mathds{1}}_d}{d}(1-\\tr Z)+Z] = 1.\n\\end{eqnarray}\nHence, for any set of jointly measurable POVMs it holds that\n\\begin{equation}\n\\label{Eq:PsuccJM}\n    \\sum_{a,x} \\tr[O_{a|x}Y^{a|x}] \\leq 1.\n\\end{equation}\n\n\\section{Appendix C: Construction of the state discrimination task with prior information from the optimal dual variable}\n\nFrom the optimal solution of the dual we also construct a state discrimination task with prior information in the following way. First observe that\n\\begin{eqnarray}\nY^{a|x} &=& \\tr[Y]\\frac{\\sum_{a'}\\text{tr}[Y^{a'|x}]}{\\tr[Y]}\\frac{\\tr[Y^{a|x}]}{\\sum_{a'}\\tr[Y^{a'|x}]}\\frac{Y^{a|x}}{\\tr[Y^{a|x}]}\\notag\\\\\n&=& \\tr[Y]p(x)p(a|x)\\varrho_{a|x}.\n\\end{eqnarray}\nInserting this into the objective function of the dual in Eq.~\\eqref{Eq:IRDual} yields\n\\begin{eqnarray}\n    &&\\sum_{a,x} \\tr[M_{a|x}Y^{a|x}] \\notag \\\\\n    &=& \\tr[Y] \\sum_{a,x} p(x)p(a|x) \\tr[M_{a|x}\\varrho_{a|x}] \\notag \\\\\n    &=& \\tr[Y]\\quad p_{\\text{succ}}(M_{a|x}, \\varrho_{a|x}).\n\\end{eqnarray}\nThen, using Eq.~\\eqref{Eq:PsuccJM} the ratio of success probabilities in Eq.~\\eqref{Eq:PsuccRatio} is lower bounded by\n\\begin{eqnarray}\n    &&\\frac{p_{\\text{succ}}(M_{a|x}, \\varrho_{a|x})}{\\max_{O_{a|x}\\in JM} p_{\\text{succ}}(O_{a|x},\\varrho_{a|x})} \\\\\n    &=& \\frac{\\sum_{a,x} \\tr[M_{a|x}Y^{a|x}]}{\\max_{O_{a|x}\\in JM} \\sum_{a,x} \\tr[O_{a|x}Y^{a|x}]}\\\\\n    &\\geq& \\sum_{a,x} \\tr[M_{a|x}Y^{a|x}] =1+\\mathcal{IR}.\n\\end{eqnarray}\nThe inequality follows from Eq.~\\eqref{Eq:PsuccJM}. From this, Observation~\\ref{Obs:JMisusefull} follows.\n\n\n\\section{Appendix D: Robustness of sets of measurements and conic programming}  \n\nDenote any set of free POVMs by $F$. Let $|x|$ be the number of POVMs. The generalized robustness is defined by\n\\begin{equation}\n    \\mathcal{R}_F(M_{a|x}) = \\min\\qty{t\\geq 0 | \\frac{M_{a|x}+tN_{a|x}}{1+t}=O_{a|x}\\in F}.\n\\end{equation}\nThis can be cast as the following conic program\n\\begin{eqnarray}\n1+\\mathcal{R}_F(M_{a|x})  = \\min_t &&\\, 1+t \\\\\n\\text{s. t.: }&& t\\geq 0 \\\\\n&& \\frac{M_{a|x}+tN_{a|x}}{1+t}=O_{a|x}\\in F \\\\\n&& \\qty{N_{a|x}}\\text{ is a POVM}.\n\\end{eqnarray}\nSolving for $N_{a|x}$ one obtains\n\\begin{eqnarray}\n\\min_t &&\\,1+t \\\\\n\\text{s. t.: }&& t\\geq 0 \\\\\n&& (1+t) O_{a|x} - M_{a|x} \\geq 0 \\\\\n&& O_{a|x} \\in F.\n\\end{eqnarray}\nBy defining $\\tilde{O}_{a|x}=(1+t) O_{a|x}$ this can be written as\n\\begin{eqnarray}\n\\min_{\\tilde{O}_{a|x}}&& \\frac{1}{\\abs{x}} \\sum_{a,x} \\frac{\\tr[\\tilde{O}_{a|x}]}{d} \\\\\n\\text{s. t.: }&& \\tilde{O}_{a|x} \\geq M_{a|x} \\\\\n&& \\tilde{O}_{a|x} \\in C_F,\n\\end{eqnarray}\nwhere $C_F$ is a cone with basis $F$. This can be brought into the form of Eq.~(\\ref{Eq:ConicProgram}) by choosing $A=-\\frac{1}{\\abs{x} d}\\ensuremath{\\mathds{1}}$, $X=\\text{diag}(\\tilde{O}_{a|x})_{a,x}$, $B=-\\text{diag}(M_{a|x})_{a,x}$, $\\Lambda=-\\textit{id}$. Then, the dual cone program reads\n\\begin{eqnarray}\n\\max_{Y^{a|x}}&& \\sum_{a,x} \\tr[M_{a|x}Y^{a|x}] \\\\\n\\text{s. t.: }&& -Y+\\frac{1}{|x| d}\\ensuremath{\\mathds{1}} \\in C_F^* \\\\\n&& Y \\geq 0.\n\\end{eqnarray}\nThe first constraint translates to $\\braket{\\frac{1}{|x| d}\\ensuremath{\\mathds{1}}-Y}{T}\\geq 0$. Hence, $\\tr[YT]\\leq \\tr[T\/|x| d]$ for all $T\\in C_F$ or equivalently $\\tr[YT]\\leq 1$ for all $T\\in F$. The final form of the dual then reads\n\\begin{eqnarray}\n\\max_{Y^{a|x}}\\,&& \\sum_{a,x} \\tr[M_{a|x}Y^{a|x}] \\\\\n\\text{s. t.: }&& Y \\geq 0 \\\\\n&& \\tr[YT]\\leq 1 \\text{ for all } T\\in F.\n\\end{eqnarray}\nNote that the last constraint is a typical property of a witness. Similar results hold for state assemblages $\\qty{\\varrho_{a|x}}_{a|x}$ by simply dropping the factor $1\/d$, since the operators $\\sum_a\\varrho_{a|x}$ have a unit trace.\n\n\n\\section{Appendix E: Robustness of state assemblages and conic programming}\n\nFor a given free set of assemblages $F$, an assemblage $\\{\\varrho_{a|x}\\}_{a,x}$, and an subchannel discrimination task $(\\mathbf{\\Lambda},\\mathbf{N})$ we have\n\\begin{align}\n    \\dfrac{p_{\\text{succ}}(\\{\\varrho_{a|x}\\},\\mathbf{\\Lambda},\\mathbf{N})}{\\max_{F} p_{\\text{succ}}(\\{\\sigma_{a|x}\\},\\mathbf{\\Lambda},\\mathbf{N})} \\leq 1+\\mathcal{R}_F(\\varrho_{a|x}).\n\\end{align}\nTo formulate the statement of Theorem~\\ref{Mainresult} for state assemblages we note that the primal problem for the robustness of an assemblage $\\{\\varrho_{a|x}\\}_{a,x}$ with respect to a free set $F$ of assemblages is given as\n\\begin{eqnarray}\n\\min_{\\tilde{\\sigma}_{a|x}}\\, && \\frac{1}{\\abs{x}} \\sum_{a,x}\\tr[\\tilde{\\sigma}_{a|x}] \\\\\n\\text{s. t.: }&& \\tilde{\\sigma}_{a|x} \\geq \\varrho_{a|x}, \\quad \\tilde{\\sigma}_{a|x} \\in C_F, \\notag\n\\end{eqnarray}\nwhere $\\tilde{\\sigma}_{a|x}=(1+t)\\sigma_{a|x}$. The dual program can be written as\n\\begin{eqnarray}\n\\max_{Y^{a|x}}\\,&& \\sum_{a,x} \\tr[\\varrho_{a|x}Y^{a|x}] \\\\\n\\text{s. t.: }&& Y \\geq 0, \\quad \\tr[TY]\\leq 1\\, \\forall T\\in F. \\notag\n\\end{eqnarray}\nWe have again denoted by $Y$ the direct sum of the operators $\\{Y^{a|x}\\}_{a,x}$. Note that Slater's conditions can be verified similarly to the case of measurements for the free sets we are interested in.\n\nUsing the techniques introduced in Ref.~\\cite{Piani_Watrous_Char_of_EPR_Steering} it is clear that any witness $Y$ of the above form can be cast as a subchannel discrimination task with one-way LOCC measurements. Namely, define subchannels and a POVM as $\\Lambda_a^\\dagger(|x\\rangle\\langle x|)=\\alpha Y^{a|x}$ and $N_x=|x\\rangle\\langle x|$, where $\\alpha=\\|\\sum_{a,x} Y^{a|x}\\|_\\infty^{-1}$ and $\\qty{\\ket{x}}_x$ is an \northonormal basis. If these subchannels do not form an instrument, i.e. $\\sum_a\\Lambda_a^\\dagger(\\openone)\\neq\\openone$, the set can be \ncompleted into one by defining an extra subchannel as \n$\\Lambda(\\varrho)=\\tr[(\\openone-\\sum_a\\Lambda_a^\\dagger(\\openone))\\varrho]\\sigma$, \nwhere $\\sigma$ is some quantum state. It is worth noting that we have one more \nsubchannel in the discrimination problem than we have outputs. \n\n\n\\section{Appendix F: Convexity and compactness of the set of assemblages that can be prepared from states with a fixed Schmidt number}\nConvex combinations of such assemblages can be prepared by increasing the size of \nAlice's system. To be more precise, given that Alice's dimension is $d$ and \nthat one assemblage is prepared with measurements $\\{A_{a|x}\\}_{a,x}$ on the \nstate $\\varrho_{AB}$ and another assemblage with measurements $\\{\\tilde \nA_{a|x}\\}_{a,x}$ on the state $\\tilde\\varrho_{AB}$, we can consider the convex \ncombination \n$\\varrho:=\\lambda\\ketbra{0}{0}\\otimes\\varrho_{AB}+(1-\\lambda)\\ketbra{1}{1}\\otimes \n\\tilde\\varrho_{AB}$ and the measurements $\\hat A_{a|x}:=\\ketbra{0}{0}\\otimes \nA_{a|x}+\\ketbra{1}{1}\\otimes \\tilde A_{a|x}$, where $\\{\\ket{0},\\ket{1}\\}$ is \nthe basis of an auxiliary qubit of Alice.\n\nTo prove the compactness of the desired set of assemblages, we first notice that the extremal points are obtained by pure states and that every extremal point can be reached with a finite dimensional Alice. As Bob is assumed to be finite dimensional, any assemblage that is preparable by a Schmidt number $n$ (or smaller) state can be expressed as a finite convex combination of extremal assemblages. Hence, any sequence of assemblages preparable with a Schmidt number $n$ (or smaller) state can be written as\n\\begin{equation}\n(\\varrho_{a|x}^m)_m=(\\sum_{i=1}^k p_{i|m}\\xi_{a|x}^{i|m})_m,\n\\end{equation}\nwhere $k$ is some fixed finite number depending on the number of dimension \n(Carath\u00e9odory's theorem), $p_{i|m}$ is a probability distribution for every \n$m$, and $\\xi_{a|x}^{i|m}$ are assemblages preparable with a pure Schmidt rank \n$n$ or less state. The set of assemblages preparable with pure states is \nclearly compact (as it is the image of a cartesian product of compact sets in \na continuous mapping) and, therefore, for every $i$ we can pick a converging \nsubsequence of $\\xi_{a|x}^{i|m}$.  Picking the subsequences for different \nindices $i$ sequentially (i.e.  subsequences of subsequences) results in \na subsequence in which convergence is guaranteed for every $i$. Repeating the \nprocedure once more to pick a converging sequence of probability distributions \ngives a subsequence of $(\\varrho_{a|x}^m)_m$ that converges. Hence, the set of \nSchmidt number $n$ (or smaller) preparable assemblages is compact.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction}\n\nIn the ongoing quest for unraveling the nonperturbative structure of QCD, considerable effort has been \ndedicated to the study of Green's (correlation) functions by means of both continuous methods~\\cite{Roberts:1994dr,Alkofer:2000wg,Maris:2003vk,Pawlowski:2003hq,Pawlowski:2005xe,Fischer:2006ub,Aguilar:2006gr, Roberts:2007ji,Aguilar:2008xm, Boucaud:2008ky,Fischer:2008uz,Binosi:2009qm,Tissier:2010ts,Campagnari:2010wc,Pennington:2011xs,Aguilar:2011xe,Vandersickel:2012tz,Serreau:2012cg,Fister:2013bh,Cloet:2013jya,Binosi:2014aea,Kondo:2014sta,Aguilar:2015bud,Binosi:2016rxz,Binosi:2016nme,Corell:2018yil,Cyrol:2017ewj,Gao:2017uox,Huber:2018ned,Pelaez:2021tpq}  and large-volume lattice simulations~\\cite{Sternbeck:2005tk,Ilgenfritz:2006he,Cucchieri:2007md,Cucchieri:2007rg,Bogolubsky:2007ud,Bowman:2007du,Cucchieri:2008fc,Cucchieri:2009zt,Bogolubsky:2009dc,Oliveira:2009eh,Oliveira:2010xc,Maas:2011se,Boucaud:2011ug,Oliveira:2012eh,Ayala:2012pb,Bicudo:2015rma}.\nIn this pursuit, the detailed scrutiny of the ghost sector of the theory    \nis particularly important, both because of its direct connection with specific scenarios of color confinement  \\cite{Kugo:1979gm,Nakanishi:1990qm},\nbut also due to its impact on the nonperturbative    \nbehavior of other key Green's functions, such as the gluon propagator and the three-gluon vertex~\\cite{Cucchieri:2006tf,Cucchieri:2008qm,Alkofer:2008jy,\nHuber:2012zj,Aguilar:2013vaa,Pelaez:2013cpa,Blum:2014gna,Eichmann:2014xya,Williams:2015cvx,Blum:2015lsa,Cyrol:2016tym,Duarte:2016ieu,Athenodorou:2016oyh,Boucaud:2017obn,Aguilar:2019jsj,Aguilar:2019uob,Aguilar:2019kxz}.\nIn particular, the nonperturbative masslessness of the ghost is responsible for the vanishing   \nof the gluon spectral density at the origin~\\cite{Cyrol:2018xeq,Haas:2013hpa,Fischer:2020xnb,Horak:2021pfr},  and for the infrared suppression of the three-gluon vertex~\\cite{Huber:2012zj,Aguilar:2013vaa,Athenodorou:2016oyh,Boucaud:2017obn,Aguilar:2019jsj}. \nIn that sense, the ghost dynamics leave their imprint on a variety of fundamental phenomena, \nsuch as chiral symmetry breaking and the generation of quark constituent masses\\mbox{~\\cite{Aguilar:2018epe,Gao:2021wun,Mitter:2014wpa,Aguilar:2010cn,Fischer:2003rp,Roberts:1994dr}},\nthe emergence of a mass gap in the gauge sector of the theory~\\cite{Cornwall:1981zr,Aguilar:2008xm,Aguilar:2019kxz,Aguilar:2020uqw},  and the dynamical formation of hadronic bound states~\\cite{Maris:2003vk,Maris:1997tm,Cloet:2013jya,Eichmann:2009qa,Eichmann:2016yit}  and glueballs~\\cite{Meyers:2012ka,Fukamachi:2016wxf,Souza:2019ylx,Huber:2020ngt}.     \n\n\nIn the framework of the Schwinger-Dyson equations (SDEs),\nthe momentum evolution of the ghost dressing function is governed by a\nrelatively simple integral equation, whose main ingredients are the\ngluon propagator and the fully-dressed ghost-gluon vertex.\nIf one treats the \ngluon propagator as external input obtained from lattice simulations (see {\\it e.g.}, ~\\cite{Aguilar:2013xqa}),\nthen the main technical challenge of this approach is the determination of the ghost-gluon vertex.\nIn the Landau gauge, the ghost-gluon vertex is rather special, because, by virtue of Taylor's theorem, \nits renormalization constant is finite~\\cite{Taylor:1971ff}.  Of the two possible tensorial structures allowed by Lorentz invariance, only that corresponding to the classical (tree-level) tensor survives in the\ncalculations. The form factor associated with it will be denoted by $B_1(r,p,q)$, where\n$r$, $p$, and $q$ are the momenta of the antighost, ghost, and gluon, respectively.\n\n\nThe most complicated aspect of the SDE that determines $B_1(r,p,q)$\nis that, in addition to  $B_1(r,p,q)$ itself, the resulting integral equation,\nderived in the so-called  ``one-loop dressed'' approximation, \ndepends also on the fully-dressed three-gluon vertex.\nThis latter vertex has a rich tensorial structure~\\cite{Ball:1980ax}, \nand a complicated description at the level of the SDEs\\mbox{~\\cite{Schleifenbaum:2004id,Huber:2012kd,Aguilar:2013xqa,Huber:2012zj,Blum:2014gna,Eichmann:2014xya,Williams:2015cvx, Binosi:2016wcx,Hawes:1998cw,Chang:2009zb,Qin:2011dd};} \ntherefore, it is often approximated by resorting to gauge-technique constructions~\\cite{Salam:1963sa,Salam:1964zk,Delbourgo:1977jc,Delbourgo:1977hq}, \nbased on the Slavnov-Taylor identities (STIs) that it \\mbox{satisfies}.\n\nThe comprehensive treatment of the relevant SDEs presented in~\\cite{Aguilar:2018csq}\ngives rise to a $B_1(r,p,q)$  with a mild momentum-dependence and a modest\ndeviation from its tree-level value (see also~\\cite{Aguilar:2013xqa}), and a ghost dressing function, $F(q^2)$, that is in good\n(but not perfect, see, {\\it e.g.},  right panel on \\mbox{Fig.~16} of~\\cite{Aguilar:2018csq}) agreement with the lattice data~\\cite{Bogolubsky:2009dc}.\n\n\nIn the present work,  we take a fresh look at the system of coupled SDEs that determines \nthe ghost dynamics, taking advantage of two recent advances in the area of lattice QCD~\\mbox{\\cite{Aguilar:2021lke,Boucaud:2017ksi, Boucaud:2018xup}}.\nFirst, the simulation of the three-gluon vertex in the ``soft gluon limit'' \\mbox{($q \\to 0$)}~\\cite{Aguilar:2021lke}  \nfurnishes accurate data for a special form factor, denoted by $\\Ls(r^2)$, which\nconstitutes a central ingredient of the SDE for $B_1(r,p,q)$,\nwhen computed in the same kinematic limit, namely $B_1(r,-r,0)$. \nSecond, the lattice two-point functions employed in our study  \nhave been cured from volume and discretization artifacts,\nonce the scale-setting and continuum-limit extrapolation \nput forth in~\\cite{Boucaud:2017ksi,Boucaud:2018xup} have been implemented.\n\nThe way the aforementioned elements are incorporated into the present analysis is as follows.\nThe starting point is the computation of $F(q^2)$ and $B_1(r,p,q)$ from the\ncoupled system of SDEs they satisfy. In the SDE for $B_1(r,p,q)$,  an approximate form of \nthe three-gluon vertex is employed: only the tree-level tensorial structures are retained, and the \nassociated form factors are taken from the STI-based derivation of~\\cite{Aguilar:2019jsj}. \nIn addition, \nthe gluon propagator of~\\cite{Boucaud:2018xup} combined with that of \\cite{Bogolubsky:2009dc}, subjected to the refinements mentioned above, is used in the SDEs as external input.\nThe solution of the system yields a $F(q^2)$ which is in outstanding agreement with the ghost dressing\nfunction of~\\cite{Boucaud:2018xup}. The corresponding\nsolution for $B_1(r,p,q)$, in general kinematics,  displays the salient features known from previous studies~\\cite{Schleifenbaum:2004id,Huber:2012kd,Aguilar:2013xqa,Cyrol:2016tym,Mintz:2017qri,Aguilar:2018csq,Huber:2018ned,Aguilar:2019jsj,Barrios:2020ubx}.\nIn fact, one may extract from it various kinematic limits as special cases, \nand, in particular, the two-dimensional ``slice'' that corresponds to the\nsoft gluon limit, thus obtaining $B_1(r,-r,0)$.  \n\nThe next step is to implement the soft gluon limit \\mbox{($q \\to 0$)} {\\it directly} at the level of the\nSDE for $B_1(r,p,q)$, which is thus converted to a dynamical equation for $B_1(r,-r,0)$.\nBy virtue of this operation, the three-gluon vertex nested in one of the defining Feynman diagrams\nis projected naturally to its soft gluon limit, thus allowing\nus to replace it precisely by the function $\\Ls(r^2)$\nobtained from the lattice analysis of~\\cite{Aguilar:2021lke},\nwithout having to resort to any Ans\\\"atze or simplifying assumptions.\nThe resulting  $B_1(r,-r,0)$ is then compared with the\ncorresponding ``slice'' obtained from the full kinematic analysis of $B_1(r,p,q)$ mentioned above,\nrevealing excellent coincidence. This coincidence, in turn, is indicative of an\nunderlying consonance between elements originating from inherently distinct computational\nframeworks, such as the lattice and the SDEs. \n\n\nThe article is organized as follows. In Sec.~\\ref{sec:back} we present the  \nnotation and theoretical ingredients that are relevant for our analysis.\nIn Sec.~\\ref{sec:coupled} we set up and solve the coupled system of SDEs\nfor the ghost dressing function and the ghost-gluon vertex in general kinematics. \nNext, in Sec.~\\ref{sec:cgsoft} we derive and analyze the SDE\nfor the ghost-gluon vertex in the soft gluon configuration,\ncomparing our results with those obtained in the previous section.\nIn addition, we compare the \nstrong running coupling obtained from the three-gluon vertex with the one constructed from \nthe ghost-gluon vertex, both in the soft gluon configuration.\nIn Sec.~\\ref{sec:conc} we discuss our results and present our conclusions.\nFinally, in Appendix~\\ref{sec:App_renor} we present useful relations between the Taylor and soft gluon\nrenormalization schemes, while in Appendix~\\ref{sec:App_latt} we discuss the treatment\nof finite cut-off effects and lattice scale-setting.\n\n\n\\section{\\label{sec:back}Theoretical background}\n\nIn this section we \nsummarize the main properties of the  two and three-point functions that enter in the nonperturbative determination \nof the ghost-gluon vertex, paying particular attention to the soft gluon limit of\nthe three-gluon vertex. Note that in the present study  we restrict ourselves to a\n\\emph{quenched} version of QCD,  {\\it i.e.},  a pure Yang-Mills theory with no dynamical quarks. \n\n\nThroughout this article we work in the Landau gauge,   where the  gluon propagator \\mbox{$\\Delta^{ab}_{\\mu\\nu}(q) = -i\\delta^{ab} \\Delta_{\\mu\\nu}(q^2)$} assumes the fully transverse form \n\\begin{equation}\n    \\Delta_{\\mu\\nu}(q) = \\Delta(q^2) P_{\\mu\\nu}(q) \\,,\\quad\\quad P_{\\mu\\nu}(q) = g_{\\mu\\nu} - q_\\mu q_\\nu\/q^2 \\,,\\quad\\quad\\, \\Delta(q^2)= \\Dr(q^2)\/q^2.\n\\label{gluon}    \n\\end{equation}\n\nAs has been firmly established by a variety of large-volume simulations and continuous studies, \n$\\Delta(q^2)$ saturates at a finite nonvanishing value, a feature which is widely attributed to the emergence\nof a gluonic mass scale~\\cite{Cornwall:1981zr,Aguilar:2008xm}.  For later convenience,\nthe gluon dressing function,  $\\Dr(q^2)$, has also been defined in \\1eq{gluon}.\n\n \n In addition,  we introduce the ghost propagator,    \\mbox{$D^{ab}(q^2)=i\\delta^{ab}D(q^2)$},  whose \n dressing function, $F(q^2)$, is given by\n\\begin{equation}\nD(q^2) =  F(q^2)\/q^2 \\,,\n\\label{ghost}\n\\end{equation}\nand is known to saturate at a finite value in the deep infrared~\\cite{Boucaud:2008ky,Aguilar:2008xm,Dudal:2008sp,Fischer:2008uz}.\n\n\n\\begin{figure}[h]\n \\centering\n  \\includegraphics[scale=0.65]{fig1}\n  \\caption{Diagrammatic representation of ($a$)~the ghost-gluon vertex, and ($b$)~the three-gluon vertex, with their respective momenta conventions.  All momenta are incoming,  $q+p+r=0$. }\n    \\label{vertices}\n\\end{figure}\n\nTurning to the three-point sector of the theory,\nwe introduce  the ghost-gluon vertex,\n\\mbox{$\\fatg_\\mu^{mna}(r,p,q) = -gf^{mna} \\fatg_\\mu(r,p,q)$}, \nand the three-gluon vertex, \\mbox{$\\fatg^{abc}_{\\alpha\\mu \\nu}(q,r,p) =gf^{abc}\\fatg_{\\alpha\\mu \\nu}(q,r,p)$},\ndepicted diagrammatically in the panels ($a$) and ($b$) of Fig.~\\ref{vertices}, respectively.\n\nIn a series of works~\\cite{Aguilar:2011xe,Binosi:2012sj,Ibanez:2012zk,Binosi:2017rwj,Aguilar:2017dco},\nthe emergence of an infrared finite gluon propagator from the corresponding SDE has been connected \nwith certain outstanding nonperturbative features of the fundamental vertices\n$\\fatg_\\mu(r,p,q)$ and $\\fatg_{\\alpha\\mu \\nu}(q,r,p)$. \nSpecifically, both vertices are composed by two distinct types of terms, according to\n\\begin{equation}\n\\fatg_{\\mu}(r,p,q) = \\Gamma_{\\mu}(r,p,q)+ V_{\\mu}(r,p,q),\\qquad\n\\fatg_{\\alpha\\mu \\nu}(q,r,p) = \\Gamma_{\\alpha\\mu \\nu}(q,r,p)+ V_{\\alpha\\mu \\nu}(q,r,p)\\,.\n\\label{fatG}\n\\end{equation}\nThe terms $V_{\\mu}(r,p,q)$ and $V_{\\alpha\\mu \\nu}(q,r,p)$ are purely nonperturbative\nand contain longitudinally coupled massless poles; \nwhen inserted into the SDE of the gluon propagator, \nthey trigger the\n{\\it Schwinger mechanism}~\\cite{Schwinger:1962tn,Schwinger:1962tp,Jackiw:1973tr,Eichten:1974et}, \ninducing the infrared finiteness of the gluon propagator.\nIt is important to emphasize that these terms drop out from transversely projected Green's functions,  or lattice ``observables'',\ndue to the property\\footnote{\\ Equivalently, the general tensorial structure of the pole vertices is given by  \n\\mbox{$V_{\\mu}(r,p,q) = \\frac{q_\\mu}{q^2} A(r,p,q)$} and \n\\mbox{$V_{\\alpha\\mu\\nu}(q,r,p) = \\frac{q_\\alpha}{q^2}  B_{\\mu\\nu}(q,r,p) +\n\\frac{r_\\mu}{r^2} C_{\\alpha\\nu}(q,r,p) + \\frac{p_\\nu}{p^2} D_{\\alpha\\mu}(q,r,p)$.}} \n\\begin{equation}\n\\label{eq:transvp}\n      {P}_{\\mu'}^{\\mu}(q) V_{\\mu}(r,p,q) =0\\,, \\qquad\\qquad\n      {P}_{\\alpha'}^{\\alpha}(q){P}_{\\mu'}^{\\mu}(r){P}_{\\nu'}^{\\nu}(p) V_{\\alpha\\mu\\nu}(q,r,p) = 0 \\,.\n\\end{equation}\n\nOn the other hand, the terms $\\Gamma_{\\mu}(r,p,q)$ and $\\Gamma_{\\alpha\\mu \\nu}(q,r,p)$ \ndenote the  pole-free components of the two vertices. For large momenta, they\ncapture the standard perturbative contributions, while in the deep infrared they\nmay be finite or diverge logarithmically, depending on whether or not \nthey are regulated by the nonperturbative gluon mass scale~\\cite{Aguilar:2013vaa}.\n\n\nThe most general tensorial decomposition of $\\Gamma_\\mu(r,p,q)$ can be written as \n\\begin{equation}\n \\label{decomp}\n    \\Gamma_\\mu(r,p,q) = B_1(r,p,q)r_\\mu + B_2(r,p,q)q_\\mu\\,,\n\\end{equation}\nwhere  $B_i(r,p,q)$ are the corresponding form factors.\nAt  tree-level, \\mbox{$\\Gamma^{(0)}_\\mu =r_\\mu $},   and so \\mbox{$B_1^{(0)}=1$} and \\mbox{$B_2^{(0)}=0$}. In addition,  by virtue of Taylor's theorem~\\cite{Taylor:1971ff},  the renormalization constant associated with $\\Gamma_\\mu(r,p,q)$ is finite.\n\n\nThe vertex $\\Gamma_{\\alpha\\mu \\nu}(q,r,p)$ is composed by fourteen linearly independent tensors.\nA standard basis, which manifestly reflects the Bose symmetry of $\\Gamma_{\\alpha\\mu \\nu}(q,r,p)$,\nis the one introduced in~\\cite{Ball:1980ax}; see also Eqs.~(3.4) and (3.6) of~\\cite{Aguilar:2019jsj}.\nNote, however, that the explicit form of the basis will not be required in what follows. \n\nAt tree level, $\\Gamma_{\\alpha\\mu \\nu}(q,r,p)$  reduces to the standard expression  \n\\begin{equation}\n\\label{eq:treelevel}\n\\Gamma_{\\!{0}}^{\\alpha\\mu\\nu}(q,r,p) = (q-r)^\\nu g^{\\alpha\\mu} + (r-p)^\\alpha g^{\\mu\\nu} + (p-q)^\\mu g^{\\nu\\alpha}\\,.  \n\\end{equation}\n\nWe next turn to the quantity studied in the lattice simulation of~\\cite{Aguilar:2021lke},  \n\\begin{eqnarray}\n\\Ls(r^2) &=&  \\frac{\\Gamma_0^{\\alpha\\mu \\nu}(q,r,p)\nP_{\\alpha\\alpha'}(q)P_{\\mu\\mu'}(r)P_{\\nu\\nu'}(p) \\Gamma^{\\alpha'\\mu'\\nu'}(q,r,p)}\n{\\rule[0cm]{0cm}{0.45cm}\\; {\\Gamma_0^{\\alpha\\mu\\nu}(q,r,p) P_{\\alpha\\alpha'}(q)P_{\\mu\\mu'}(r)P_{\\nu\\nu'}(p) \\Gamma_0^{\\alpha'\\mu'\\nu'}(q,r,p)}}\n\\rule[0cm]{0cm}{0.5cm} \\Bigg|_{\\substack{\\!\\!q\\to 0 \\\\ p\\to -r}} \\,,  \n\\label{asymlat}\n\\end{eqnarray}\nwhere the external legs have been appropriately amputated\\footnote{In~\\cite{Aguilar:2021lke} and other related lattice works, this quantity has been denominated as the \\emph{asymmetric} kinematic limit. Here\n  we find it more appropriate to employ the term ``soft gluon limit''.}. Note that the starting expression involves the full vertex \n$\\fatg^{\\alpha'\\mu'\\nu'}(q,r,p)$, which, by virtue of \\1eq{eq:transvp}, is reduced to  $\\Gamma^{\\alpha'\\mu'\\nu'}(q,r,p)$,  \n{\\it i.e.},  the term $V^{\\alpha'\\mu'\\nu'}(q,r,p)$ associated with the poles drops out in its entirety. \n\nNow, in the limit of interest, namely $q \\to 0$, the tensorial structure of \nthe three-gluon vertex is considerably simplified, given by\n\\begin{equation}\n\\Gamma^{\\alpha\\mu\\nu}(0,r,-r) = 2 {\\cal A}_1(r^2) \\,r^\\alpha g^{\\mu\\nu} + {\\cal A}_2(r^2)\\, (r^\\mu g^{\\alpha\\nu} + \\,r^\\nu g^{\\alpha\\mu})\n+ {\\cal A}_3(r^2)\\, r^\\alpha r^\\mu r^\\nu  \\,.\n\\label{Gsoft}\n\\end{equation}\nAt tree level,\n\\begin{equation}\n\\Gamma_0^{\\alpha\\mu\\nu}(0,r,-r) = 2 \\,r^\\alpha g^{\\mu\\nu} - (r^\\mu g^{\\alpha\\nu} + \\,r^\\nu g^{\\alpha\\mu})\\,,\n\\label{Gsoft0}\n\\end{equation}\nwhich, in the notation of \\1eq{Gsoft}, means that ${\\cal A}_1^{(0)}(r^2)= 1$,  ${\\cal A}_2^{(0)}(r^2)=-1$,\nand ${\\cal A}_3^{(0)}(r^2)= 0$.\n\nThen, the numerator and denominator of the fraction on the r.h.s. of \\1eq{asymlat}, to be denoted\nby ${\\cal N}$ and ${\\cal D}$, respectively, become \n\\begin{equation}\n{\\cal N} = 4 (d-1) [r^2 - (q\\cdot r)^2\/q^2] {\\cal A}_1(r^2)\\,, \\qquad  {\\cal D} = 4 (d-1) [r^2 - (q\\cdot r)^2\/q^2] \\,.\n\\label{NandD}\n\\end{equation}\nThus, the path-dependent contribution contained in the square bracket drops out when forming the ratio ${\\cal N}\/{\\cal D}$,\nand \\1eq{asymlat} yields simply\n\\begin{equation}\n\\Ls(r^2) =  {\\cal A}_1(r^2) \\,.\n\\label{LisB}\n\\end{equation}\nCombining \\2eqs{Gsoft0}{LisB},  it is immediate to derive one of the key relations of this work, namely \n\\begin{equation}\n\\label{GammaLasym}\nP_{\\mu\\mu'}(r) P_{\\nu\\nu'}(r) \\Gamma^{\\alpha\\mu\\nu}(0,r,-r) = 2 \\Ls(r^2) r^{\\alpha}P_{\\mu'\\nu'}(r) \\,.\n\\end{equation}\n\n\n\\section{\\label{sec:coupled} The system of coupled SDEs}\n\n\nIn this section, we set up and solve  \nthe system of coupled SDEs that governs the ghost dressing function and the ghost-gluon vertex for general space-like momenta. \nThe  external ingredients employed are a fit of the lattice data for the gluon propagator, and certain form factors\nof the three-gluon vertex (in general kinematics), obtained from the nonperturbative Ball-Chiu construction of~\\cite{Aguilar:2019jsj}.\n\n\\begin{figure}[b]\n    \\centering\n        \\vspace{-0.5cm}\n    \\includegraphics[scale=0.7]{fig2}\n    \\vspace{-0.5cm}\n    \\caption{The SDEs for the ghost propagator and the ghost-gluon  vertex (upper and lower panels, respectively). \n      The white circles represent the full gluon and ghost propagators,  while the blue ones\n denote the full ghost-gluon vertex.  The gray ellipse indicates the ``one-particle reducible'' four-point ghost-gluon kernel.}    \n    \\label{fig:coupsys}\n\\end{figure}\n\n\n\\subsection{The ghost gap equation and ghost-gluon SDE}\n\n\nOur starting point is the SDE for the ghost propagator, whose diagrammatic representation is shown in the upper panel of Fig.~\\ref{fig:coupsys}.\nWhen expressed in terms of the ghost dressing function, this SDE acquires the standard form known in the literature, namely \n\\begin{equation}\nF ^{-1}(p^2) =  Z_{\\rm c} + \\Sigma(p^2)\\,,\n\\label{sde_ghost2}\n\\end{equation}\nwith\n\\begin{equation}\n\\Sigma(p^2) =  ig^2C_{\\rm A}Z_1\\!\\! \\int_k f(k,p)B_1(-p,k+p,-k)\\Delta(k)D(k+p)\\,; \\qquad f(k,p):= 1 -\\frac{(k\\cdot p)^2}{k^2p^2}\\,.\n\\label{sde_ghost}\n\\end{equation}\nIn the above equation, $C_\\mathrm{A}$ is the Casimir eigenvalue of the adjoint representation [$N$ for $SU(N)$],\nwhile $Z_{\\rm c}$ and $Z_1$ are the  renormalization constants of $D(p^2)$ and  $\\Gamma^\\mu (r,p,q)$, respectively  [see Eq.~\\eqref{renorm}].\n In addition,   we have introduced the integral measure \n\\begin{equation}\n    \\int_k := \\frac{1}{(2\\pi)^4} \\int\\!\\! \\dd[4]{k} \\,,\n\\end{equation}\nwhere the presence of a symmetry-preserving regularization scheme is implicitly understood.   \n\n\n\nIn this analysis, the renormalization is implemented within the well-known variant of the momentum subtraction\n(MOM) scheme known as ``Taylor scheme''~\\cite{Boucaud:2008gn,Boucaud:2011eh}\\footnote{In the literature this scheme is also known as  minimal momentum subtraction (MiniMOM) scheme~\\cite{vonSmekal:2009ae}, \nand has been employed for a recent determination of $\\alpha_{\\ols{\\srm{MS}}}$ from unquenched lattice simulations~\\cite{Zafeiropoulos:2019flq}, consistent with the experimental \\emph{world average.}}, \nwhich fixes the (finite) vertex renormalization constant at the special value $Z_1=1$.\nAs for $Z_{\\rm c}$, its value is fixed by the standard MOM requirement \n$F ^{-1}(\\mu^2) = 1$, where $\\mu$ is the renormalization scale.\n\nImplementing this condition at the level of \n\\1eq{sde_ghost2} yields\n\\begin{equation}\nZ_{\\rm c} = 1 - \\Sigma (\\mu^2)\\,,\n\\end{equation}\nand \\1eq{sde_ghost2} may be cast in the form\n\\begin{equation}\nF ^{-1}(p^2) =  1 + \\Sigma(p^2) - \\Sigma (\\mu^2)\\,.\n\\label{sde_ghostks}\n\\end{equation}\n\nWe next turn to the SDE for the ghost-gluon vertex, shown diagrammatically in the lower panel of Fig.~\\ref{fig:coupsys}.\nIn the present work,  we will consider the so-called ``one-loop dressed'' approximation of this SDE, which\ncorresponds to keeping only the first two terms in the skeleton expansion of the SDE kernel, shown in Fig.~\\ref{fig:diagvert}.\nNote that the omitted set of contributions is captured by the one-particle irreducible four-point function, represented by the yellow ellipse,\nwhose dynamics has been studied in detail in~\\cite{Huber:2017txg,Huber:2018ned}. As was shown there, this subset of corrections\nis clearly subleading, affecting the ghost-gluon vertex by a mere $2\\%$. \nIt is therefore expected that the above truncation should provide a quantitatively accurate description of\nthe infrared behavior of the ghost-gluon vertex [see also the corresponding discussion in Sec.~\\ref{sec:conc}]. \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=0.7]{fig3}\n    \\caption{The skeleton  expansion of the ``one particle reducible'' four-point ghost-gluon kernel.\n     Only the first two terms will be considered in our analysis.}\n    \\label{fig:diagvert}\n\\end{figure}\n\n\nThus,  the expression for the SDE for the ghost-gluon vertex in the Taylor scheme can  be schematically written as \n\\begin{equation}\n \\label{rvgg}\n \\Gamma_\\mu(r,p,q) = r_\\mu - \\frac{i}{2}g^2 C_\\mathrm{A} [a_{\\mu}(r,p,q) - b_{\\mu}(r,p,q)] ,\n\\end{equation}\nwith\n\\begin{align}\n   \n    a_\\mu(r,p,q) &= r_\\rho \\int_k \\Delta^{\\rho\\sigma}(k) \\fatg_{\\mu\\sigma\\alpha}(q,k,-t) \\Delta^{\\alpha\\beta}(t) \\Gamma_\\beta(-\\ell,p,t) D(\\ell) \\,,  \\nonumber\\\\\n   \n    b_\\mu (r,p,q)&= r_\\alpha \\int_k \\Delta^{\\alpha\\beta}(\\ell) \\Gamma_\\beta(t,p,-\\ell) D(t) \\fatg_\\mu(k,-t,q) D(k)  \\,, \n\\label{d1d2}\n\\end{align}\nwhere $\\ell := k-r$  and $t :=k+q$. Note that we have employed the first of the two relations in \\1eq{eq:transvp} in order to eliminate\nthe terms $V_{\\mu}(r,p,q)$ from the ghost-gluon vertices that are contracted by a transverse gluon propagator\n(Landau gauge). \n\n\nIn order to  isolate  the contribution of the form factor $B_1(r,p,q)$, defined in \\1eq{decomp}, we contract \\1eq{rvgg}\nby the projector~\\cite{Aguilar:2013xqa}\n\\begin{equation} \n \\varepsilon^\\mu(r,q) = \\frac{q^2 r^\\mu - q^\\mu ( q\\cdot r )}{ h(r,q) }\\,, \\qquad h(q,r) = q^2 r^2 - (q\\cdot r)^2 \\,.\n\\label{projector}\n\\end{equation}\nAn immediate consequence of this contraction and the property \\1eq{eq:transvp} is that \n\\begin{eqnarray} \n\\fatg_{\\mu\\sigma\\alpha}(q,k,-t)P^{\\rho\\sigma}(k) P^{\\alpha\\beta}(t)\n&\\to & \\Gamma_{\\mu\\sigma\\alpha}(q,k,-t)P^{\\rho\\sigma}(k) P^{\\alpha\\beta}(t)\\,,\n\\nonumber\\\\\n\\fatg_\\mu(k,-t,q) &\\to & \\Gamma_\\mu(k,-t,q)\\,,\n\\label{subst}\n\\end{eqnarray}\n{\\it i.e.},  the terms associated with the nonperturbative poles are annihilated, and we are only left with the\npole-free components of the two vertices.\n\n\n\n\nThe next step is to carry out in the expressions of \\1eq{d1d2} the substitution\n\\begin{align}\nB_1(-\\ell, p, t) \\to  & \\frac{1}{2}\\left[ B_1(-\\ell, p, t) + B_1(r, \\ell, - k) \\right]\\,,  \\nonumber \\\\\nB_1(t, p, -\\ell) \\to & \\frac{1}{2}\\left[ B_1(t, p, -\\ell) + B_1(r, -k, \\ell) \\right]\\,,\n\\label{sym_sub}\n\\end{align}\nin order to restore the symmetry of $B_1(r,p,q)$ with respect to the interchange of the ghost and antighost momenta,\nwhich has been compromised by the truncation of the SDE~\\cite{Aguilar:2018csq}.\n\nIn addition, the structure of the three-gluon vertex entering in $a_\\mu(r,p,q)$ is approximated\nby retaining only the tensorial structures with a nonvanishing tree-level limit. Specifically,\nin the notation of~\\cite{Aguilar:2019jsj}, we set \n\\begin{equation}\n\\label{eq:vtensortree}\n\\Gamma^{\\alpha\\mu\\nu}(q,r,p) \\approx (q-r)^\\nu g^{\\alpha\\mu}X_1(q,r,p) + (r-p)^\\alpha g^{\\mu\\nu} X_4(q,r,p)\n+ (p-q)^\\mu g^{\\nu\\alpha} X_7(q,r,p)\\,,  \n\\end{equation}\nwhere, due to the Bose symmetry of $\\Gamma^{\\alpha\\mu\\nu}(q,r,p)$, we have \\mbox{$X_1(q,r,p) = X_4(p,q,r) = X_7(r,p,q)$}.\n\n\n\nThus, we  arrive at (Minkowski space)  \n\\begin{equation}\nB_1(r,p,q) = 1 -  \\frac{i}{2}g^2 C_\\mathrm{A} \\left[a(r,p,q)  - b(r,p,q) \\right]\\,,\n \\label{B1general}\n\\end{equation}\nwith\n\\begin{equation}\na(r,p,q) = \\int_k  {\\mathcal K}_1(k,r,q){\\mathcal N}_1 (k,r,q)\\,,  \\qquad \\quad  b(r,p,q) =\\int_k  {\\mathcal K}_2(k,r,q)  {\\mathcal N}_2 (k,r,q) \\,,  \n\\label{d12kernels}\n \\end{equation}\nwhere\n\\begin{eqnarray}\n{\\mathcal K}_1(k,r,q) &=&   \\frac{\\Delta(k^2) \\Delta(t^2) F(\\ell^2)} { k^2 \\ell^2 t^2 h(q,r) } \\left[ B_1(-\\ell, p, t) + B_1(r, \\ell, - k) \\right]\\,,  \\nonumber  \\\\\n{\\mathcal K}_2 (k,r,q)&=&  \\frac{F(k^2) \\Delta( \\ell^2 ) F(t^2) }{ 2\\, k^2 \\ell^2 t^2 h(q,r) }\\left[ B_1(t, p, -\\ell) + B_1(r, -k, \\ell) \\right]B_1(k,-t,q) \\,, \n\\label{kb1}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n{\\mathcal N}_1 &=&  a_1X_1(k, t, q) +a_4X_4(k, t, q)+a_7X_7(k, t, q)\\,,  \\\\ \n{\\mathcal N}_2&=&   [ q^2 ( k \\cdot r ) - ( k\\cdot q )(q\\cdot r ) ] [ (k\\cdot r)(q\\cdot r) + (k\\cdot q )(k\\cdot r) - r^2 (k\\cdot q ) - k^2(q\\cdot r ) - h(k,r) ]\\,.   \\nonumber \n\\end{eqnarray}\nThe coefficients $a_i$ are given by\n\\begin{align}\na_1 =&  \\left[ q^2 ( k\\cdot r ) - ( k\\cdot q ) (q\\cdot r) \\right] \\left\\lbrace k^2 \\left[ ( k\\cdot q ) (k\\cdot r) - ( k\\cdot q )(q\\cdot r)-2 r^2( k\\cdot q ) \\right. \\right. \\nonumber\\\\\n& \\left.\\left. + (k\\cdot r)( q\\cdot r) + (k\\cdot r)^2 - h(q,r) \\right] - k^4 \\left[ (q\\cdot r ) + r^2 \\right] + ( k\\cdot r ) \\left[ q^2 (k\\cdot r) + (k\\cdot q) (k\\cdot r) \\right.\\right. \\nonumber\\\\\n& \\left.\\left. - (k\\cdot q)(q\\cdot r) + (k\\cdot q)^2\\right] \\right\\rbrace \\,, \\nonumber \\\\\na_4 =&  \\left[ k^2 ( q\\cdot r ) - ( k\\cdot q ) ( k\\cdot r ) \\right] \\left\\lbrace \\left(k^2+q^2\\right) h(q,r) - q^2 ( k\\cdot r ) \\left[ q^2 + ( q\\cdot r ) \\right] + (k\\cdot q)^2 (q\\cdot r) \\right.\\nonumber\\\\\n& \\left. + ( k\\cdot q )\\left[ ( k\\cdot r ) (q\\cdot r) - q^2 ( k\\cdot r ) + q^2 (q\\cdot r) + 2 q^2 r^2 - (q\\cdot r)^2 \\right] - q^2 (k\\cdot r)^2 \\right\\rbrace \\,, \\nonumber\\\\\na_7 =& \\left\\lbrace q^2 ( k\\cdot r ) - k^2 \\left[ q^2 + ( q\\cdot r ) \\right] + ( k\\cdot q ) [ ( k\\cdot r ) - ( q\\cdot r ) ] + ( k\\cdot q)^2 \\right\\rbrace \\nonumber\\\\\n& \\times \\left[ k^2 h(q,r) - q^2 (k\\cdot r)^2 + ( k\\cdot q )( k\\cdot r )( q\\cdot r ) \\right]\\,. \n\\label{ai_Mink}\n\\end{align} \n\n\\subsection{\\label{sec:ngene} Numerical analysis}\n\n\nIn order to proceed with the numerical solution, the system of integral equations formed by\n\\2eqs{sde_ghostks}{B1general} must be passed to Euclidean space, \nfollowing standard conventions (see,  {\\it e.g.},  Eq.~(5.1) of~\\cite{Aguilar:2018csq}) and employing spherical coordinates for the\nfinal treatment.\n\nThen, appropriate inputs for the gluon propagator, $\\Delta(q^2)$,  \nand the form factors $X_{1,4,7}(q,r,p)$ of the three-gluon vertex must be furnished.\n\nFor the gluon propagator we employ a fit for the results obtained after a  reanalysis of the lattice data of~\\cite{Bogolubsky:2009dc},\nfollowing the procedure put forth in~\\cite{Boucaud:2017ksi,Boucaud:2018xup}, in order to cure volume and discretization artifacts, \nsee Appendix~\\ref{sec:App_latt} for details. Specifically, the resulting $\\Delta(q^2)$ is shown in the\nleft panel of Fig.~\\ref{fig:X1}, together with the numerical fit given by Eq.~\\eqref{gluonfit}.\n\n\\begin{figure}[t]\n\\begin{minipage}[b]{0.45\\linewidth}\n\\centering\n\\hspace{-1.0cm}\n\\includegraphics[scale=0.24]{fig4a}\n\\end{minipage}\n\\hspace{0.25cm}\n\\begin{minipage}[b]{0.45\\linewidth}\n\\includegraphics[scale=0.9]{fig4b}\n\\end{minipage}\n\\caption{Left panel: Lattice data for the gluon propagator,  $\\Delta(q^2)$,  after performing the continuum extrapolation of~\\cite{Boucaud:2018xup} to the data set of~\\cite{Bogolubsky:2009dc},   together with the corresponding fit  given by Eq.~\\eqref{gluonfit}.   The gluon propagator is renormalized  at \\mbox{$\\mu = 4.3$ GeV}.  Right panel: A representative case of the three-gluon form factor  $X_1(q^2,r^2, \\phi)$  for a fixed value of the angle, $\\phi=0$.} \n\\label{fig:X1}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{minipage}[b]{0.45\\linewidth}\n\\centering\n\\hspace{-1.0cm}\n\\includegraphics[scale=0.26]{fig5a}\n\\end{minipage}\n\\hspace{0.25cm}\n\\begin{minipage}[b]{0.45\\linewidth}\n\\vspace{0.5cm}\n\\includegraphics[scale=0.9]{fig5b}\n\\end{minipage}\n\\caption{ Left panel:  The numerical solution for the  ghost dressing function,  $F(p^2)$,  (red continuous line) compared with the lattice data of~\\cite{Boucaud:2018xup}.   Right panel:  The form factor  $B_1(r^2,p^2,\\theta_1 )$  for a fixed value of the angle $\\theta_1 =\\pi$,  obtained as solution of the coupled system of  \\2eqs{sde_ghost2}{B1general} when  \\mbox{$\\alpha_s(\\mu)= 0.244$}.  }\n\\label{fig:rescoupled}\n\\end{figure}\n\n\n\nFor the determination of the form factors $X_{1,4,7}(q,r,p)$,  we follow the nonperturbative version of the Ball-Chiu construction\ndeveloped in~\\cite{Aguilar:2019jsj}. The general idea of the method is based on reconstructing the longitudinal form factors\nof the three-gluon vertex, such as $X_{1,4,7}(q,r,p)$, from the set of STIs that $\\Gamma_{\\alpha\\mu \\nu}(q,r,p)$ satisfies.\nThis procedures allows us to express $X_{1,4,7}(q,r,p)$ in terms of the ghost dressing function, the ``kinetic'' part of the\ngluon propagator, and three of the form factors of the ghost-gluon kernel.  A representative case of $X_1(q^2,r^2,\\phi=0)$  \nis shown in the right panel of Fig.~\\ref{fig:X1}, where $\\phi$ is the angle formed between the momenta $q$ and $r$. \nNote that the form factor deviates markedly from unity, displaying clearly what is known in the literature\nas ``infrared suppression''~\\cite{Aguilar:2013vaa,Athenodorou:2016oyh,Boucaud:2017obn,Blum:2015lsa,Corell:2018yil,Aguilar:2019jsj}.\n\n\n\n\n\nWith the inputs introduced above, the coupled system is solved numerically by an iterative process.  The external momenta $r^2$ and $p^2$ are distributed on a logarithmic grid, with $96$ points in the interval \\mbox{$[5\\times10^{-5},10^4]$ GeV$^2$},  whereas the angle between them,  $\\theta_1$,   is uniformly distributed in \\mbox{$[0,\\pi]$} with 19 points. The interpolations in three variables, needed for evaluating the $X_i$ and the $B_1$, are performed with B-splines~\\cite{de2001practical}, and the triple integrals are computed with a Gauss-Kronrod method~\\cite{Berntsen:1991:ADA:210232.210234}.\n\n\n\n\nIn Fig.~\\ref{fig:rescoupled},  we show the numerical results for  $F(p^2)$ and $B_1(r^2,p^2, \\theta_1)$,  \nobtained from the solution of the coupled system.\nWe emphasize that the renormalization point has been fixed at \\mbox{$\\mu= 4.3$ GeV},\nwhich coincides with the highest value of the momentum\naccessible by the lattice simulation of~\\cite{Bogolubsky:2009dc}.\nIn particular,   \none can observe that when the gauge coupling assumes the value \\mbox{$\\alpha_s(\\mu):=g^2(\\mu)\/4\\pi = 0.244$},\nthe solution of the system yields a $F(p^2)$ that is in outstanding agreement with the  ghost dressing data of~\\cite{Boucaud:2018xup} (left panel),  which were properly extrapolated to the physical continuum limit, \nas  explained in Appendix~\\ref{sec:App_latt}. \n\n\\begin{figure}[t]\n\\begin{minipage}[b]{0.45\\linewidth}\n\\centering\n\\hspace{-1.0cm}\n\\includegraphics[scale=0.9]{fig6a.pdf}\n\\end{minipage}\n\\hspace{0.25cm}\n\\begin{minipage}[b]{0.45\\linewidth}\n\\includegraphics[scale=0.24]{fig6b}\n\\end{minipage}\n\\caption{Left panel: The form factor $B_1(r^2,q^2,\\theta_2)$ plotted as function of the momenta of antighost,  $r$,   and  gluon,  $q$,   for a fixed value of the angle,  $\\theta_2=\\pi$.\nOn the  3-D surface, three curves are highlighted, representing\nthe soft gluon (red dot-dashed), soft ghost (orange continuous), and symmetric (green dashed) kinematic limits.  Right panel: Direct comparison of the three special configurations (2-D projections)\nidentified in the left panel. } \n\\label{fig:conf}\n\\end{figure}\n\n\nMoreover,  in the right panel of   Fig.~\\ref{fig:conf},    one can see that the solution for  $B_1(r^2,p^2, \\theta_1)$  is symmetric with respect to  the diagonal plane defined by the condition $r=p$.\nThis is a direct consequence of  the ghost-antighost symmetry, and becomes manifest only when $B_1$ is plotted  as a\nfunction of the momenta $r$ and $p$. \n \n\nWe next explore certain special kinematic limits of $B_1$. To that end,\nwe choose  $r$ and $q$ as our reference momenta (antighost and gluon, respectively), denoting by \n$\\theta_2$ the angle between them.  In the left panel of Fig.~\\ref{fig:conf} we plot the corresponding\n3-D plot, for the special value $\\theta_2=2\\pi\/3$; this choice for the angle  \nis particularly convenient,  because  one can identify  \non a unique  3-D surface the following three kinematic limits: \n\n\n{\\it(i)} The  \\emph{soft gluon limit},  obtained by setting \n$q = 0$; then,  the momenta $r$ and $p$ have the same magnitude, \\mbox{$|p|=|r|=|Q|$}, and are anti-parallel, {\\it i.e.},   $\\theta_1=\\pi$.\nThis kinematic configuration is represented by the red dot-dashed curve on the 3-D plot of Fig.~\\ref{fig:conf}.\n\n\n {\\it(ii)} \n The \\emph{soft (anti)ghost limit}, in which $r= 0$  and \nthe momenta \\mbox{$|q|=|p|=|Q|$}; evidently, $|r||q|\\cos\\theta_2= 0$,\nand any dependence on the angle $\\theta_2$ is washed out. This kinematic limit is represented by the orange continuous curve \non the 3-D plot of Fig.~\\ref{fig:conf}.\n\n  \n{\\it(iii)} The \\emph{totally symmetric limit},  defined by \\mbox{$q^2 = p^2= r^2 = Q^2$}; with the\nscalar products given by \n  \\mbox{$(q\\cdot p) = (q\\cdot r) = (p\\cdot r) = -\\frac{1}{2}Q^2$},   and the angles  \\mbox{$ \\widehat{rp} =  \\widehat{rq} =  \\widehat{qp} =2\\pi\/3$},\nrepresented by the green dashed curve  on the 3-D plot of Fig.~\\ref{fig:conf}.\n\n\nThe three 2-D projections described above are plotted together in the right panel of Fig.~\\ref{fig:conf},\nwith all their corresponding momenta denoted by $Q$. As we can see, all cases display a peak around the same region of momenta, {\\it i.e.},  \\mbox{$(0.8-1.2)$ GeV},  with moderate differences in their heights.  In addition,  in the deep infrared, all curves recover\nthe result $B_1(0,0,0) = 1$.\n\n\n\n\n\n\n\\section{\\label{sec:cgsoft}Ghost-gluon vertex in the soft gluon configuration}\n \n\nIn this Section we implement the soft gluon limit,  {\\it i.e.},  ($q \\rightarrow 0$),  directly at the level of the\nSDE for the ghost-gluon vertex, which permits us to \nuse the lattice data for $\\Ls(q^2)$~\\cite{Aguilar:2021lke}\\footnote{In~\\cite{Aguilar:2021lke}, the lattice result for $\\Ls(q^2)$\n  has been reproduced particularly well by means of the STI-based construction of~\\cite{Aguilar:2019jsj}. Nonetheless, in the present analysis we\nemploy directly the best fit to the lattice data, for achieving the highest possible accuracy.}\nin the treatment of the resulting\nintegral equation.\n\nThe basic observation is that, in the soft gluon limit,  \nthe term  \\mbox{$P^{\\rho\\sigma}(k)  \\Gamma_{\\mu\\sigma\\alpha}(q,k,-t) P^{\\alpha\\beta}(t)$} appearing \ninside the $a_\\mu(r,p,q)$ of \\1eq{d1d2} becomes simply  \n\\begin{equation}\nP^{\\rho\\sigma}(k) P^{\\alpha\\beta}(t) \\Gamma_{\\mu\\sigma\\alpha}(q,k,-t) \\xrightarrow[\\text{}]{\\text{$q\\to 0$}}\nP^{\\rho\\sigma}(k)P^{\\alpha\\beta}(k) \\Gamma_{\\sigma\\alpha\\mu}(0,k,-k) = 2 \\Ls(k^2) k_{\\mu}P^{\\rho\\beta}(k)\\,,  \n\\end{equation}\nwhere in the last step \\1eq{GammaLasym} was used. \n\nNote, however, that a final subtlety prevents the immediate use of the lattice results \nfor $\\Ls(k^2)$ into \\1eq{d1d2}. Specifically, the renormalization employed in the lattice  \nanalysis of~\\cite{Aguilar:2021lke} is the ``soft gluon scheme'', which differs from the Taylor scheme\nused in the derivation of the system of coupled SDEs. As a result, the lattice data must\nundergo a finite renormalization, $\\tilde{z}_{3}$, which will convert\nthem from one scheme to the other, according to \\1eq{asytaylor}.  \n\nThen, it is straightforward to implement the soft gluon limit at the level of \\1eq{d1d2}.\nUsing the short-hand notation \\mbox{$B_1(k^2):=B_1(k,-k,0)$}, we arrive at \n %\n\\begin{align}\n  B_1(r^2) & = 1 - \\frac{ig^2C_\\mathrm{A}}{\\tilde{z}_3} \\int_k F(\\ell^2) \\Delta^2(k^2)\n  f(k,r)\\frac{(k\\vdot r)}{\\ell^2} B_1(-\\ell,-r,k)\\Ls(k^2) \\nonumber \\\\\n    & + \\frac{i}{2} g^2C_\\mathrm{A} \\int_k F^2(k^2) \\Delta(\\ell^2)  f(k,r)\\frac{(k\\vdot r)}{k^2\\ell^2} B_1(k,-r,-\\ell) B_1(k^2) \\,, \n\\label{B1exp}    \n\\end{align} \n where the function $f(k,r)$ has been defined in \\1eq{sde_ghost}.\n\nAs a final step, \\1eq{B1exp} will be converted to Euclidean space (spherical coordinates),\nusing standard transformation rules.  Defining \n \\begin{equation}\n    k^2:= y; \\qquad r^2:=x; \\qquad \\ell^2:= z; \\qquad k\\vdot r\\equiv \\sqrt{xy}\\cos{\\theta}; \\qquad \\ell \\vdot r\\equiv \\sqrt{xz}\\cos{\\varphi};\n \\end{equation}\nand setting \n\\begin{eqnarray} \nB_1 (\\ell ,-r, k) \\to  B_1(z,x,\\varphi)\\,,    \\qquad    B_1 (k ,-r, -\\ell) \\to  B_1(y,x,\\pi -\\theta)\\,,   \n\\label{ffeu}\n\\end{eqnarray}\nwe arrive at \n\\begin{eqnarray}\nB_1(x) & =& 1 +\\frac {C_\\mathrm{A}\\alpha_s}{2\\pi^2 \\,\\tilde{z}_{3}} \\int_0^\\infty\\!\\!\\! \\dd{y} y\\sqrt{xy}\\,  \\Ls(y) \\Delta^2(y) \\int_0^\\pi\\!\\!\\! \\dd{\\theta} \\sin^4 \\theta \\cos{\\theta} B_1(z,x,\\varphi) \\, z^{-1} F(z)\\nonumber\\\\   \n      & +& \\frac{C_\\mathrm{A}\\alpha_s}{4\\pi^2}\\int^\\infty_0\\!\\!\\! \\dd{y} \\,\\sqrt{xy} F^2(y) B_1(y) \\int_0^\\pi \\!\\!\\! \\dd{\\theta}\n \\sin^4\\theta \\cos{\\theta} B_1(y,x,\\pi-\\theta)\\, z^{-1} \\Delta(z) \\,, \n\\label{FinalEq}\n\\end{eqnarray}\nwhere we have that $\\cos{\\varphi}= \\sqrt{y\/z} \\cos\\theta - \\sqrt{x\/z}$.\n\n\n\\1eq{FinalEq} will be solved numerically, through an iterative procedure, using the\nfollowing external inputs.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.28]{fig7}\n\\caption{The  lattice data for $\\Ls(q^2)$ (circles)  from~\\cite{Aguilar:2021lke},  together with the fit given by \nEq.~\\eqref{Lasymfit} (blue continuous curve). }\n\\label{fig:Lasym}\n\\end{figure}\n\n{\\it(i)} Throughout the analysis we use \\mbox{$\\mu= 4.3$ GeV} and  $\\alpha_s(\\mu) = 0.244$,  as was determined\nin our numerical study of the  SDE system discussed  in Sec.~\\ref{sec:ngene}.\n\n\n{\\it(ii)} For both $\\Delta(q^2)$ and  $F(q^2)$,  renormalized at the aforementioned $\\mu$,\nwe employ the fits given by \\2eqs{gluonfit}{ghostfit},  respectively.\n\n\n{\\it(iii)} For $\\Ls(q^2)$  we employ an excellent fit to the lattice data of~\\cite{Aguilar:2021lke}.  The curve is shown in Fig.~\\ref{fig:Lasym},  and its functional form is given by\n\\begin{equation}\n\\label{Lasymfit}\n\\Ls(q^2)=F(q^2)T(q^2)+\\nu_1 \\left( \\frac{1}{1+(q^2\/\\nu_2)^2} - \\frac{1}{1+(\\mu^2\/\\nu_2)^2} \\right),     \n\\end{equation}\nwith\n\\begin{equation}\n    T(q^2) = 1 +\\frac{3\\lambda_{\\srm S}}{4\\pi} \\left( 1+ \\frac{\\tau_1}{q^2+\\tau_2} \\right) \\left[ 2\\ln \\left(  \\frac{q^2+ \\eta^2(q^2)}{\\mu^2+ \\eta^2(\\mu^2)}\\right) + \\frac{1}{6}\\ln \\left( \\frac{q^2}{\\mu^2} \\right) \\right],\n\\end{equation}\nand \n\\begin{equation}\n\\label{eta}\n    \\eta^2(q^2) = \\frac{\\eta_1^2}{1 + q^2\/\\eta_2^2}\\,,\n\\end{equation}\nwhere the fitting parameters are given by\n\\mbox{$\\lambda_{\\srm S}=0.27$},\n \\mbox{$\\nu_1=0.179$},  \\mbox{$\\nu_2 =0.830$ GeV$^{2}$},\n \\mbox{$\\tau_1= 2.67$ GeV$^{2}$},\n \\mbox{$\\tau_2  = 1.05$  GeV$^{2}$},\n \\mbox{$\\eta_1^2= 3.10$   GeV$^{2}$},  and \n \\mbox{$\\eta_2^2  = 0.729$  GeV$^{2}$}. \n\n\n\nNote that the above fit incorporates,  by construction, the renormalization condition \\mbox{$\\Ls(\\mu^2) = 1$},  corresponding to the soft gluon MOM scheme employed in the lattice simulation of~\\cite{Aguilar:2021lke}.\nIn addition, the zero crossing of $\\Ls(q^2)$  is located at about 170 MeV.   \n\n{\\it(iv)} \nThe value of $\\tilde{z}_3$  is determined  from the basic relation given by \\1eq{z3from_alphas},\nwhich yields the numerical value \\mbox{$\\tilde{z}_3 \\approx 0.95$}, quoted in \\1eq{z3num}. \n\n\n{\\it(v)}  For the form factors $B_1(\\ell^2,r^2, \\varphi)$ and $B_1(k^2,r^2,\\pi-\\theta)$ we\ninterpolate the results for  $B_1(r^2,p^2,\\theta_1)$ obtained in Sec.~\\ref{sec:ngene} [see  Fig.~\\ref{fig:rescoupled}].\n\n\n\\begin{figure}[t]\n\\begin{minipage}[b]{0.45\\linewidth}\n\\centering\n\\hspace{-1.0cm}\n\\includegraphics[scale=0.26]{fig8a}\n\\end{minipage}\n\\hspace{0.25cm}\n\\begin{minipage}[b]{0.45\\linewidth}\n\\includegraphics[scale=0.26]{fig8b.eps}\n\\end{minipage}\n\\caption{ Left panel:  The $B_1(r^2)$  obtained as solution  of Eq.~\\eqref{FinalEq} (blue continuous)\n together with the lattice data (circles) from~\\cite{Ilgenfritz:2006he,Sternbeck:2006rd}.  Right panel:   The numerical impact of dressing the vertices   $\\Ls(q^2)$  and  $B_1(\\ell^2,r^2, \\varphi)$[$B_1(k^2,r^2,\\pi-\\theta)$]  on $B_1(r^2)$, determined from Eq.~\\eqref{FinalEq}. } \n\\label{fig:B1soft}\n\\end{figure}\n\n\nUsing the inputs  described above, \nthe $B_1(r^2)$ that emerges as a solution of \\1eq{FinalEq} is given by \nthe blue continuous curve in the left panel of Fig.~\\ref{fig:B1soft}, where it is compared \n with the $\\rm SU(3)$ lattice data of~\\cite{Ilgenfritz:2006he,Sternbeck:2006rd}.  \n Although the error bars are rather sizable,  we clearly see that our solution follows the general  trend of the data.  In particular,  notice that both peaks occur in the same intermediate region of momenta.  \n\nThe $B_1(r^2)$ may be accurately fitted with the functional form \n\\begin{equation} \n\\label{fitB1}\n    B_1(r^2) = 1 + \\frac{r^2(a+br^2)}{1+cr ^2 +dr^4\\ln\\left[(r^2+ r_0^2)\/\\rho^2\\right]}\\,,\n\\end{equation}\nwhere the parameters are given by \\mbox{$a=2.07\\, \\mbox{GeV}^{-2}$}, $b=9.85\\, \\mbox{GeV}^{-4}$,  $c=22.3\\, \\mbox{GeV}^{-2}$,  \\mbox{$d=56.4\\,  \\mbox{GeV}^{-4}$},  \\mbox{$r_0^2=1.48\\, \\mbox{GeV}^2$},  and  \\mbox{$\\rho^2=1.0\\, \\mbox{GeV}^2$},\nand the  \\mbox{$\\chi^2\/\\text{d.o.f.} = 1.0 \\times 10^{-6}$}. \n  \n \n \nWe next study the impact that the\namount of ``dressing'' carried by the various vertices  has on $B_1(r^2)$.\nTo  that end,  we solve \\1eq{FinalEq}  considering the three-gluon and ghost-gluon vertices to \nbe either at their tree-level values or fully dressed. The results of the four cases considered are displayed \nin the right panel of Fig.~\\ref{fig:B1soft}. The hierarchy observed in this plot is\ncompletely consistent with the known infrared properties of these two fundamental vertices: \nat low momenta, the ghost-gluon vertex displays a mild enhancement  \nwith respect to its tree-level value, whereas the three-gluon vertex is considerably suppressed.  \n\nBased on this particular combination of facts, one would expect that the solution with the\nmaximal support will be obtained from \\1eq{FinalEq} \nwhen the ghost-gluon vertices are dressed while the three-gluon vertex is kept bare ($ \\Ls=1$);\nthis is indeed what happens, as may be seen from the dot-dashed green curve, which displays\nthe most pronounced peak. By the same logic, the reverse combination, namely bare ghost-gluon vertices  \nand dressed three-gluon vertex, should furnish the most suppressed  $B_1(r^2)$; evidently, this\nis what we find, as shown by the purple dashed curve. The remaining cases, where both vertices\nare bare, or fully dressed, must lie between the two prior cases; this expectation\nis clearly realized within the detailed numerical analysis, as can be seen by the corresponding curves,\nindicated by dotted red and continuous blue, respectively. \n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.26]{fig9}\n\\caption{Comparison of the soft gluon result obtained as  solution of Eq.~\\eqref{FinalEq} (blue continuous curve)  with the one \nextracted from the 3-D plot shown in  Fig.~\\ref{fig:conf} (red dashed curve). }\n\\label{fig:3Dcomp2D}\n\\end{figure}\n\n\n\n\n\nWe conclude our numerical analysis with an instructive  self-consistency check.\nSpecifically, as explained in item {\\it(v)} above, in order to solve \\1eq{FinalEq}\nwe have used as external input the result for the ghost-gluon vertex for general kinematics, \nderived in Sec.~\\ref{sec:ngene}. \nBut, as is clear from Fig.~\\ref{fig:conf}, the input used\nto obtain the soft gluon limit contains already a prediction of what that limit should be,\nnamely the red dot-dashed curve of $B_1(r^2,q^2,2\\pi\/3)$,  shown in Fig.~\\ref{fig:conf}.\nTherefore, a reasonable indication of the self-consistency of the entire procedure\nwould be the degree of coincidence between the latter  2-D projection  and the result\nfor $B_1(r^2)$ obtained from \\1eq{FinalEq},  namely the blue continuous\ncurve in either panel of Fig.~\\ref{fig:B1soft}.\n\nThe direct comparison between these two curves is shown in Fig.~\\ref{fig:3Dcomp2D}, where\nan excellent coincidence may be observed. Specifically, the maximum discrepancy,\nlocated at about 2 GeV, is of the order of 2\\%. The proximity between these results\nsuggests an underlying consistency between the various ingredients entering in the\ncorresponding calculations. Note, in particular, that the insertion of lattice ingredients, such as\nthe gluon propagator and the $\\Ls(q^2)$, into the SDEs appears to be a completely congruous operation.\n\n\nFinally, it is rather instructive to compare  the effective strengths associated with the ghost-gluon and the three-gluon interactions in the soft gluon configuration by means of {\\it renormalization-group invariant}  quantities.\nTo that end, we consider the corresponding effective couplings,\nto be denoted by ${\\alpha}_{\\rm{cg}}(q^2)$ and ${\\alpha}_{\\rm{3g}}(q^2)$,\ndefined as (see, {\\it e.g.},  ~\\cite{Athenodorou:2016oyh,Mitter:2014wpa,Fu:2019hdw})\n\\begin{equation}\n   {\\alpha}_{\\rm{cg}}(q^2) = {\\alpha}_s(\\mu^2) B_1^2(q^2) F^2(q^2)\\Dr(q^2)\\,,\\qquad {\\alpha}_{\\rm{3g}}(q^2)={\\alpha}_s(\\mu^2) [\\TLs(q^2)]^2 \\Dr^3(q^2)\\,,  \n\\label{coup_cg}\n\\end{equation}\nwhere $\\Dr(q^2)$ is the dressing of the gluon propagator, defined in \\1eq{gluon},\nwhile $\\TLs(q^2)$ is the lattice result of~\\cite{Aguilar:2021lke} adjusted to the Taylor scheme, according to \n\\2eqs{asytaylor}{z3num}. Note that, by means of this latter adjustment, all ingredients entering in the definitions of both effective couplings\nare computed in the same renormalization scheme, namely the Taylor scheme. In addition, \naccording to our SDE estimate (see Sec. \\ref{sec:ngene}), we have that ${\\alpha}_s(\\mu) = 0.244$ , at $\\mu = 4.3$ GeV.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.26]{fig10}\n\\caption{ The comparison of the effective couplings, ${\\alpha}_{\\rm{cg}}(q^2)$ (red continuous line) and ${\\alpha}_{\\rm{3g}}(q^2)$   (blue dashed).}\n\\label{fig:coupling}\n\\end{figure}\n\nThe result of the evaluation of the two effective couplings is shown in Fig.~\\ref{fig:coupling}. The main feature, consistent with a variety of previous studies~\\cite{Huber:2012kd,Blum:2014gna,Williams:2014iea,Cyrol:2016tym,Cyrol:2017ewj, Aguilar:2020yni},\nis the considerable suppression displayed by ${\\alpha}_{\\rm{3g}}(q^2)$ in the region below 2 GeV. \n\n\n\n\n\n\n\n\\section{\\label{sec:conc}Conclusions}\n\nIn this work we have carried out a thorough study of the dynamics related with the\nghost sector of pure Yang-Mills theories, incorporating into the\nstandard SDEs pivotal elements stemming from recent lattice studies.\nIn fact, these lattice results serve both as external inputs for some of the quantities that\nare difficult to determine accurately within the SDE framework, such as the gluon propagator and certain\ncomponents of the three-gluon vertex,  as well as refined benchmarks for testing the reliability of our numerical solutions,  such as the ghost dressing function. \nSpecifically, the lattice gluon propagator has been used as a global input in all SDEs considered in the present study, while, in the ``soft gluon'' SDE, the lattice data for the three-gluon vertex \nin the same limit have been employed.\n\n\nThe main results of our analysis are \nsuccinctly captured in  Figs.~\\ref{fig:rescoupled} and~\\ref{fig:3Dcomp2D}. \nIn particular,  in the left panel of Fig.~\\ref{fig:rescoupled},\nthe ghost dressing function obtained as a solution of the coupled SDE system is compared to the \nresults of the lattice simulation of~\\cite{Boucaud:2018xup}. \nIt is important to appreciate that the success of this comparison \nhinges on the optimization for the cure of discretization artifacts, in connection to the scale-setting and continuum extrapolation, imposed on this set of lattice data.\nIndeed, the difference between the latter lattice data and the (non-extrapolated)  \nresults of~\\cite{Bogolubsky:2009dc}, displayed in Fig.~\\ref{fig:ghostdress},\nis rather substantial, affecting a phenomenologically important region of momenta. \nThis difference accounts almost entirely for the discrepancies\nfound in earlier studies~\\cite{Aguilar:2009pp,Aguilar:2013xqa,Aguilar:2018csq},  where the SDE results were compared with the data of~\\cite{Bogolubsky:2009dc}.  \n\nWe next turn to Fig.~\\ref{fig:3Dcomp2D}, where the curves for $B_1(r^2)$,  \nobtained following two distinct procedures, are compared.   \nThe excellent agreement between both results suggests an underlying self-consistency\namong the several elements participating non-trivially in the computation of these results.\nParticularly interesting in this context is the pivotal role played by the three-gluon vertex,\nwhich appears in both computations leading to the results of Fig.~\\ref{fig:3Dcomp2D},\nalbeit in rather dissimilar kinematic arrangements. \nSpecifically, to obtain the result marked by the blue continuous\ncurve, the vertex was approximated by its classical tensor structure,\naccompanied by the corresponding form factors in general kinematics, as explained in Sec.~\\ref{sec:coupled}.\nInstead, the red dashed curve is obtained through the direct use of the\nlattice results in the soft gluon limit, according to the discussion in Sec.~\\ref{sec:cgsoft}. The coincidence between\nthe results indicates that the STI-based construction of~\\cite{Aguilar:2019jsj},\nwhich gave rise to the form factors used for the\ncomputation of the blue continuous curve, is quite reliable.\nIn that sense, it is rather gratifying to see how well the\ndynamical equations respond in this particular set of circumstances; in fact,\nthe use of lattice data as SDE inputs appears to be completely consistent. \n\nNote that the present study is fully compatible with the assertion of~\\cite{Huber:2017txg,Huber:2018ned}\nthat the four-point function represented by the yellow ellipse in Fig.~\\ref{fig:diagvert} \nis numerically rather negligible.\nEvidently, the excellent agreement with the lattice\nfound in the left panel of Fig.~\\ref{fig:rescoupled} indicates that the omission of the corresponding term \nfrom the skeleton expansion of the SDE kernel does not introduce any appreciable error. \nIn fact, it is interesting to observe that an entirely different conclusion about the importance of this\nfour-point function would have been drawn \nif the non-extrapolated lattice results of~\\cite{Bogolubsky:2009dc}  had been used for the\ncomparison in Fig.~\\ref{fig:rescoupled}.\nSpecifically, any attempt to interpret the difference alternatively as a consequence of the kernel truncation would force this four-point function to acquire considerably higher values than those found in the detailed analysis of~\\cite{Huber:2017txg,Huber:2018ned}.\n\nFinally, it would be interesting to extend the considerations of Sec.~\\ref{sec:cgsoft} to the case of the\nquark-gluon vertex, whose SDE and corresponding skeleton expansion\nare given by replacing in Figs.~\\ref{fig:coupsys} and~\\ref{fig:diagvert}, respectively, all ghost lines by quark lines. \nIn particular, the soft gluon limit of the quark-gluon vertex\ninvolves three form factors, whose determination \nhas attracted particular attention over the years.  In fact, up until very recently~\\cite{Kizilersu:2021jen},\nnotable discrepancies existed between the continuous predictions~\\cite{Bhagwat:2004kj,LlanesEstrada:2004jz,Fischer:2003rp,Fischer:2006ub,Aguilar:2014lha,Aguilar:2016lbe,Oliveira:2018fkj, Oliveira:2018ukh,Oliveira:2020yac} and the results of lattice simulations~\\cite{Skullerud:2002ge,Skullerud:2003qu,Skullerud:2004gp,Lin:2005zd,Kizilersu:2006et,Oliveira:2016muq,Sternbeck:2017ntv}.\nIt is likely that the inclusion of $\\Ls$ in the SDE treatment might shed further light on this intricate problem.\n\n\\section*{\\label{sec:acknowledgments}Acknowledgments}\nThe work of  A.~C.~A. is supported by the CNPq grant 307854\/2019-1 and the project 464898\/2014-5 (INCT-FNA).\nA.~C.~A. ,  C. ~O.~A,  and M.~N.~F.    also acknowledge financial support from the FAPESP projects 2017\/05685-2,  2019\/05656-8, and 2020\/12795-1, respectively.\nJ.~P. is supported by the  Spanish MICIU grant FPA2017-84543-P,\nand the  grant  Prometeo\/2019\/087 of the Generalitat Valenciana. \nF.~D.~S. and J.~R.~Q.~ are supported the Spanish MICINN grant PID2019-107844-GB-C2, and regional Andalusian project P18-FR-5057.   This study was financed in part by the Coordena\\c{c}\\~{a}o de Aperfei\\c{c}oamento de Pessoal de N{\\'\\i}vel Superior - Brasil (CAPES) Finance Code 001 (B.~M.~O).\n\n\n\\newpage \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\bf{Introduction}} \\label{sec:intro}\nMagnetars are non-accreting neutron stars with long spin periods ($P\\sim2$--$12\\,{\\rm\ns}$) and the largest spin-down rates ($\\dot{P}\\sim10^{-13}$--$10^{-10}\\,{\\rm\ns}\\,{\\rm\ns}^{-1}$) among the pulsar population. Most of them have spin-down inferred magnetic field\nstrength, $B$,\nup to $\\sim10^{15}\\,{\\rm G}$. It is generally believed that\nmagnetars are young neutron stars and some are found inside Supernova Remnants\n(SNRs).\nMagnetars usually have persistent X-ray luminosity,\n$L_{\\rm X}\\sim10^{34-36}\\,{\\rm erg}\\,{\\rm s}^{-1}$, much larger than\ntheir rotational energy loss rate $\\dot{E}$,\nand they occasionally exhibits violent bursting activities \\citep[see review\nby][]{kb17}.\nIn order to explain the properties of this pulsar class, magnetar models have\nbeen developed. The most popular one is the twisted\nmagnetosphere model \\citep{td95,td01,bel09,bel11}.\nIt suggests that the toroidal magnetic field could exist in the stellar\ncrust.\nIf the internal magnetic field is strong enough,\nit could tear the crust followed by twisting the\ncrust-anchored external field \\citep{td95,tdw00,tyk02}.\nIn addition, a starquake arising from the plastic deformation of the crust would\ncause magnetar\nbursts due to magnetic reconnection \\citep{td95,pbh12,pbh13}.\nPersistent X-ray emission of magnetars could be explained by the magnetic field decay\n\\citep{tyk02,plm07}. Meanwhile, the magneto-thermal evolution theory suggests\nthat the field decay could be\nenhanced due to the changes in the conductivity and the magnetic diffusivity of magnetars\n\\citep{vrp13}. As a consequence, magnetars are observed to have higher surface\ntemperature and X-ray luminosity than canonical pulsars.\nIn general, soft X-ray spectra of magnetars can be\ndescribed by an absorbed blackbody model with temperature\n$kT\\sim0.3$--$0.6\\,{\\rm keV}$ plus an\nadditional power-law with photon index $\\Gamma\\sim2$--$4$ or\nanother blackbody component with $kT\\sim0.7\\,{\\rm keV}$\n\\citep[see][]{ok14,kb17}. It indicates that the\nsoft X-ray emission could be contributed by thermal emission and some\nnon-thermal radiation processes, such as synchrotron or inverse-Compton\nscattering.\\\\\n\\indent \\object{SGR~0501+4516} is a magnetar discovered with the Burst Alert\nTelescope (BAT) on board \\emph{Swift} on 2008\\,August\\,22 due to\na series of short bursts \\citep{bbb08}. X-ray pulsations were detected with a\nperiod of $P\\sim5.7\\,{\\rm s}$ \\citep{rit09}.\nAfter the discovery, the source was subsequently identified in an archival\n\\emph{ROSAT} observation taken in 1992.\nThe soft X-ray flux was $\\sim80$ times higher in the outburst when compared to\nthe 1992 observation \\citep{rit09}. The hard X-ray tail above $10\\,{\\rm keV}$\nwas first discovered with \\emph{INTEGARL} right after the outburst \\citep{rit09}. It had also been\ndetected with \\emph{Suzaku} observation \\citep{ern10}.\nFrom the spin period and spin-down rate, $B$\nwas estimated to be $2\\times10^{14}\\,{\\rm G}$ \\citep{wgk08}. The soft X-ray\nspectrum of SGR~0501+4516~\nbelow $10\\,{\\rm keV}$ could be described by an absorbed blackbody model with a\npower-law component, using\n\\emph{XMM-Newton} observations obtained in the first year after the outburst\n\\citep{rit09,cpr14}.\nThe X-ray spectral properties\nfrom 2008 to 2013 were also measured with four \\emph{Suzaku} observations\n\\citep{esk17}, but it is interesting to note that\nthe results are different from those reported in other literature, including\na smaller hydrogen column density,\nlower blackbody temperature, larger radius, and softer power-law photon index\n\\citep{rit09,gwk10,cpr14}.\\\\\n\\begin{deluxetable*}{ccccc}\n\\tablecaption{Observations of SGR 0501+4516 Used in Our Analysis.\n\\label{tab:obs}}\n\\tablehead{\\colhead{Date} & \\colhead{Observatory\n(Instruments)} & \\colhead{ObsID} &\n\\colhead{Mode} & \\colhead{Net Exposure (ks)}\n}\n\\startdata\n2008 Aug 31 & \\emph{XMM-Newton} (PN) &\n0552971201 &\nSW & $10.2$ \\\\\n2008 Sep 02 & \\emph{XMM-Newton} (PN) &\n0552971301 &\nSW & $20.5$ \\\\\n2008 Sep 25 & \\emph{CXO} (HRC-I) &\n9131 &\n-- & $10.1$ \\\\\n2008 Sep 30 & \\emph{XMM-Newton} (PN\/MOS1\/MOS2) &\n0552971401 &\nLW\/SW\/SW &\n$30.1\/32.3\/32.3$ \\\\\n2009 Aug 30 & \\emph{XMM-Newton} (PN\/MOS1\/MOS2) &\n0604220101 &\nSW\/FF\/SW &\n$53.9\/52.4\/53.1$ \\\\\n2012 Dec 09 & \\emph{CXO} (ACIS-S\\tablenotemark{a})&\n15564 &\nTE & $14.0$ \\\\\n2013 Apr 03 & \\emph{CXO} (ACIS-S\\tablenotemark{a})&\n14811 &\nTE & $13.7$ \\\\\n2013 Aug 31 & \\emph{Suzaku} (XIS0\/XIS1\/XIS3) & 408013010\n& Normal & $36.0\/41.1\/41.2$\n\\enddata\n\\tablecomments{\n\\tablenotetext{a}{Made in the sub-array mode with only one-eighth of CCD\\,7.}}\n\\end{deluxetable*}\n\\indent Until now, there is no accurate distance measurement for SGR~0501+4516.\nAs magnetars are young pulsars, SGR~0501+4516~ is expected to be located\nclose to the\nspiral arm of the Galaxy.\nThe line of sight intercepts the Perseus and Outer arms of the Galaxy, at\ndistances of $\\sim2.5$ and $\\sim5\\,{\\rm kpc}$,\nrespectively.\nIn this paper, we assume the distance $d=5\\,{\\rm kpc}$. In addition, there\nexists a supernova remnant\n(SNR) G160.9+2.6, $\\sim80'$ north of SGR~0501+4516~ \\citep{gc08,gwk10}. The\ndistance and age of the SNR were estimated as\n$800\\pm400\\,{\\rm pc}$ and $4000$--$7000\\,{\\rm years}$ \\citep{lt07}.\n\\citet{gwk10} proposed that SGR~0501+4516~ could be associated\nwith G160.9+2.6. Leaving the distance aside, if\nthis is the case, the magentar should have a large proper motion of\n$0\\farcs7$--$1\\farcs2\\,{\\rm yr}^{-1}$ to the south. \\\\\n\\indent In this paper, we used new X-ray observations to show that\nSGR 0501+4516 had returned to quiescence in 2013,\nfive years after the outburst, and we report on its spectral and timing\nproperties during flux relaxation.\nWe also analyzed\narchival observations to investigate the\nlong-term evolution.\n\\section{\\bf{Observations and data reduction}} \\label{sec:obs}\nThere are eight X-ray observations used in this study (see\nTable\\,\\ref{tab:obs}). We obtained two new\nobservations obtained with the Advanced CCD Imaging Spectrometer (ACIS)\non board \\emph{Chandra X-ray Observatory (CXO)} on 2012\\,December\\,9 and\n2013\\,April\\,3. Both of them were made in the Time Exposure (TE) mode for\n14\\,ks using only one-eighth of the CCD, providing a fast frame time of\n0.4\\,s. This allows us to obtain a crude pulse\nprofile for this $\\sim5.67\\,{\\rm s}$ period pulsar.\nBy inspecting the light curves, no bursts from the source or background flares were detected during\nthe exposures. We checked that pile-up was negligible\nin both observations. In addition to these two ACIS observations, a\n\\emph{Chandra}\nHigh Resolution Camera (HRC) observation taken on 2008\\,September\\,25 was also used\nto measure the source position only.\nAll \\emph{Chandra} data were reprocessed with \\texttt{chandra\\_repro}\nin CIAO 4.8 with CALDB 4.7.4 before performing any analysis.\\\\\n\\indent There were six \\emph{XMM-Newton} observations after the discovery of\nthe source. We only analyzed the latest\nfour from 2008\\,August\\,31 to 2009\\,August\\,30 because\nSGR 0501+4516 showed strong bursting activities during the two earliest\nobservations.\nThe source was still bright 11 days after the outburst; the pile-up effect\nwas an issue in the\nMOS data obtained on 2008\\,August\\,31 and September\\,2 and hence only the PN\ndata were\nused in these two observations.\nWe first reprocessed all the data by the tasks\n\\texttt{epchain}\/\\texttt{emchain}\nin XMMSAS version 1.2. In the analysis,\nonly PATTERN\\,$\\leq4$ events of the PN data and PATTERN\\,$\\leq12$ events in\nthe MOS data were used.\nWe also used the standard screening for\nthe MOS (FLAGS~=~\\#XMMEA\\_EM) and\nPN (FLAGS~=~\\#XMMEA\\_EP) data.\nAfter removal of periods with\nbackground flares, we obtained net exposures ranging from $10.2$ to\n$53.9\\,{\\rm ks}$ (see Table\\,\\ref{tab:obs}).\\\\\n\\indent We also used the latest \\emph{Suzaku} data in the archive taken on\n2013\\,August\\,31,\nto combine with the \\emph{Chandra} data to better constrain the quiescent\nspectral properties.\nIn order to focus on the soft X-ray spectral properties,\nonly the data obtained with the XIS were used (see Table\\,\\ref{tab:obs}).\nThe XIS data were reprocessed using \\texttt{xisrepro} in HEAsoft 6.20\nwith standard screening criteria. We inspected the light curves\nto verify that no bursts were detected throughout the observation with\n$\\sim40\\,{\\rm ks}$.\n\\section{\\bf{ANALYSIS AND RESULTS}}\n\\subsection{Imaging and Astrometry}\nWe measured the position of SGR~0501+4516~ in all\n\\emph{Chandra} data using the\ntask \\texttt{celldetect} and obtained a consistent result of\n$\\alpha$=5:01:06.8,\n$\\delta$=+45:16:34 (J2000) within the uncertainty.\nThe measurement uncertainties in the $90\\%$ confidence level have radii $0\\farcs4$ (HRC) and $0\\farcs5$ (ACIS).\nAs the ACIS images were taken in the sub-array mode with a small field of\nview, we did not find any background sources to align the two images.\nTherefore, we also need to consider the absolute astrometric accuracy of\n\\emph{Chandra}, which is $0\\farcs8$ at the $90\\%$ confidence\nlevel\\footnote{\\url{http:\/\/cxc.harvard.edu\/cal\/ASPECT\/celmon\/}}.\nThis gives an upper limit of the proper motion of\n$0\\farcs32\\,{\\rm yr}^{-1}$ ($90\\%$ confidence level),\nrejecting the suggestion that SGR 0501+4516 was born at the center of SNR\nG160.9+2.6 \\citep{gwk10}.\\\\\n\\indent Finally, we simulated a model point spread function for ACIS data with\nChaRT\\footnote{\\url{http:\/\/cxc.harvard.edu\/ciao\/PSFs\/chart2\/}} using the\nbest-fit spectrum (see Section\\,\\ref{sec:spe_ana} below)\nand confirmed that the radial profile is fully consistent with that of the\nreal data, indicating no extended emission was found near the magnetar.\n\\subsection{Timing Analysis}\\label{sec:tim_ana}\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{pulse_profile}\n\\caption{Pulse profiles of SGR 0501+4516 in the energy range of\n$0.5$--$7\\,{\\rm keV}$ for the latest \\emph{Chandra} observations.\nThe two profiles were aligned manually by matching\nthe brightest bin. The uncertainties are at\nthe $1\\sigma$ level.\n}\\label{fig:pulse_profile}\n\\end{figure}\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{xmm_pn}\n\\caption{\\emph{XMM-Newton} PN spectra of SGR 0501+4516. The solid lines\nindicate the\nbest-fit 2BB+PL model on different epochs. The orange, purple, and black\ndashed\nlines indicate the low temperature BB, high temperature BB, and\nPL components of the 2009 August 30 spectrum, respectively.}\\label{fig:xmm_fits}\n\\includegraphics[width=0.45\\textwidth]{suzaku_chandra}\n\\caption{\\emph{Chandra} and \\emph{Suzaku} spectra of SGR 0501+4516.  All\nsolid lines indicate the same best-fit 2BB+PL model. The different shape is\ndue to different responses of the\ninstruments. The orange, purple, and black dashed lines indicate the low\ntemperature BB, high\ntemperature BB, and PL components, respectively, with the \\emph{Suzaku} XIS\nresponse.}\\label{fig:suzaku_chandra}\n\\end{figure}\nWe extracted the source photons from the two new \\emph{Chandra} observations\nby using a $2\\farcs5$ radius aperture and obtained 4149 and 4043 counts,\nrespectively,\nin the $0.5$--$7\\,{\\rm keV}$ energy range. The estimated\nbackground photon counts in the source region are $\\sim0.6$ for both\nobservations.\nWe then applied a barycentric\ncorrection to the photon arrival times.\nWe employed the $\\chi^2$-test after epoch folding\n\\citep{lea87} and found periods of\n$P=5.76286(8)\\,{\\rm s}$ and $P=5.76299(9)\\,{\\rm s}$ for 2012\\,December\\,9 and\n2013\\,April\\,3 data, respectively.\nThe $1\\sigma$ uncertainties quoted here were estimated using the simulation results from\n\\citet{lea87}.\nWe used the best-fit periods to generate the pulse profiles for\nboth \\emph{Chandra} observations. As the frame time of our\nobservations was $0.4\\,{\\rm s}$, we only divided the pulse period of\n$P=5.76\\,{\\rm s}$ into 13 phase bins.\nFigure\\,\\ref{fig:pulse_profile} shows the pulse profile, which has a\ndouble-peaked shape. The pulse   \nprofile between the two observations did not show any obvious variations,\nsuggesting that\nthe source had already returned to quiescence\nin 2013.\\\\\n\\indent As the dates of the two new \\emph{Chandra}\nobservations were separated by too far apart, we were unable to measure the\nspin-down rate $\\dot{P}$ by phase coherent timing analysis. Meanwhile, the\nuncertainties of individual timing measurements\nwere too large so that $\\dot{P}$ could not be obtained from our \\emph{Chandra}\nobservations.\nWe found that the two periods measured in 2012 and 2013 are formally\nconsistent with each other\nafter accounting for the uncertainties; however, they are different from the value obtained in the 2009 observation\n\\citep{cpr14}.\nComparing our results with the spin period $P=5.7622571(2)$ measured in 2009,\nwe obtained\n$\\dot{P}=6(1)\\times10^{-12}\\,{\\rm s}\\,{\\rm s}^{-1}$ at the \n$1\\sigma$ confidence level from 2009 to 2013, which is\ncompatible with\n$5.94(2)\\times10^{-12}\\,{\\rm s}\\,{\\rm s}^{-1}$ reported by \\citet{cpr14}.\n\\subsection{Spectral Analysis}\\label{sec:spe_ana}\nWe extracted the source spectrum from the \\emph{Chandra} observations using\nthe same $2\\farcs5$ radius apertures as in\nthe timing analysis above.\nFor the \\emph{XMM-Newton} and \\emph{Suzaku} XIS spectra, we used apertures of\n$36''$ and $1\\farcm8$ radius, respectively.\nWe chose a larger region far from the source on the same CCD as the background\nregion. We restricted\nthe analysis in the energy range of $0.5$--$10\\,{\\rm keV}$\nfor \\emph{XMM-Newton} and \\emph{Suzaku} data, and $0.5$--$7\\,{\\rm keV}$ for\n\\emph{Chandra} data to optimize the signal-to-noise ratio. We grouped the\nspectra with a minimum of 30 counts\nper energy bin.\\\\\n\\begin{turnpage}\n\\begin{deluxetable*}{lcccccccccc}[ht!]\n\\centering\n\\tablecaption{Best-Fit Spectral Parameters for SGR~0501+4516 with\nUncertainties\nat the $90\\%$\nConfidence Level\\label{tab:pas_results}}\n\\tablecolumns{10}\n\\tablehead{\\colhead{Date} & \\colhead{$N_{\\rm H}$} &\n\\colhead{$kT_{\\rm low}$} & \\colhead{$R_{\\rm low}$\\tablenotemark{a}}\n& \\colhead{$F_{\\rm low}$\\tablenotemark{b} ($10^{-11}$}\n& \\colhead{$kT_{\\rm high}$} & \\colhead{$R_{\\rm high}$\\tablenotemark{a}} &\n\\colhead{$F_{\\rm high}$\\tablenotemark{b} ($10^{-11}$} & \\colhead{$\\Gamma$}\n& \\colhead{$F_{\\rm PL}$\\tablenotemark{b} ($10^{-11}$} & $\\chi^2\/{\\rm dof}$ \\\\\n & \\colhead{$(10^{22}\\,{\\rm cm}^{-2})$} & \\colhead{(keV)} & \\colhead{(km)} &\n\\colhead{erg\\,cm$^{-2}\\,{\\rm s}^{-1}$)} & \\colhead{(keV)} &\n\\colhead{(km)} & \\colhead{${\\rm erg}\\,{\\rm cm}^{-2}\\,{\\rm s}^{-1}$)} &\n&  \\colhead{erg\\,cm$^{-2}\\,{\\rm s}^{-1}$)} &\n}\n\\startdata\n\\cutinhead{BB+PL model}\n2008 Aug 31\\tablenotemark{c}& $1.34\\pm0.06$ & \\nodata & \\nodata & \\nodata &\n$0.70\\pm0.02$\n& $1.45_{-0.08}^{+0.09}$ & $1.48\\pm0.08$ & $2.9\\pm0.1$ & $1.63\\pm0.08$ & $764.9\/741$\\\\\n2008 Sep 02\\tablenotemark{c} & $1.29_{-0.05}^{+0.04}$ & \\nodata & \\nodata & \\nodata &\n$0.68\\pm0.01$ & $1.46\\pm0.06$ & $1.34\\pm0.05$ &\n$2.85\\pm0.07$ & $1.46\\pm0.06$ & $947.0\/915$ \\\\\n2008 Sep 30\\tablenotemark{d} & $1.36\\pm0.03$ & \\nodata & \\nodata & \\nodata &\n$0.66\\pm0.01$ & $1.03\\pm0.04$ & $0.57\\pm0.02$ & $3.15_{-0.05}^{+0.06}$\n& $0.87\\pm0.02$ & $2367.5\/2169$ \\\\\n2009 Aug 30\\tablenotemark{d} & $1.43\\pm0.04$ & \\nodata & \\nodata & \\nodata &\n$0.56\\pm0.02$ & $0.56\\pm0.05$ & $0.072_{-0.007}^{+0.008}$ &\n$4.0\\pm0.1$ & $0.21\\pm0.01$ & $1308.3\/1227$ \\\\\n2013 Jun 23\\tablenotemark{e} & $1.43_{-0.08}^{+0.09}$ & \\nodata & \\nodata & \\nodata &\n$0.63_{-0.05}^{+0.04}$ & $0.34_{-0.05}^{+0.06}$ & $0.05\\pm0.01$ & $3.9_{-0.2}^{+0.3}$ & $0.15\\pm0.01$\n & $626.9\/603$ \\\\\n\\hline\n\\cutinhead{2BB+PL model}\n2008 Aug 31\\tablenotemark{c} & $0.90\\pm0.02$\\tablenotemark{f} & $0.35_{-0.02}^{+0.03}$ &\n$4.6_{-0.5}^{+0.6}$ & $0.55\\pm0.09$ & $0.75\\pm0.02$ &\n$1.4\\pm0.1$ & $2.14_{-0.09}^{+0.07}$ & $1.33\\tablenotemark{g}$ &\n$0.42\\pm0.07$ & $5911.5\/5653$\\tablenotemark{f} \\\\\n2008 Sep 02\\tablenotemark{c} & $0.90\\pm0.02$\\tablenotemark{f} & $0.31\\pm0.02$ &\n$5.2_{-0.6}^{+0.7}$ & $0.38\\pm0.05$ & $0.71_{-0.02}^{+0.01}$ &\n$1.56_{-0.06}^{+0.10}$ & $1.99_{-0.05}^{+0.04}$ &\n$1.33\\tablenotemark{g}$ & $0.45\\pm0.04$ &\n$5911.5\/5653$\\tablenotemark{f} \\\\\n2008 Sep 30\\tablenotemark{d} & $0.90\\pm0.02$\\tablenotemark{f} & $0.31\\pm0.01$\n& $4.8_{-0.2}^{+0.4}$ & $0.30\\pm0.02$ & $0.69_{-0.01}^{+0.02}$ & $1.13_{-0.06}^{+0.05}$ &\n$0.95\\pm0.02$ &\n$1.33\\tablenotemark{g}$ & $0.20\\pm0.02$ & $5911.5\/5653$\\tablenotemark{f} \\\\\n2009 Aug 30\\tablenotemark{d} & $0.90\\pm0.02$\\tablenotemark{f} & $0.25\\pm0.02$ &\n$4.4_{-0.4}^{+0.8}$ & $0.085\\pm0.008$ & $0.56_{-0.02}^{+0.06}$ & $0.7\\pm0.1$\n& $0.14\\pm0.01$ & $2.6_{-2.5}^{+0.4}$ & $0.06_{-0.02}^{+0.01}$ &\n$5911.5\/5653$\\tablenotemark{f} \\\\\n2013 Jun 23\\tablenotemark{e} & $0.90\\pm0.02$\\tablenotemark{f} & $0.26_{-0.02}^{+0.01}$ &\n$3.7_{-0.7}^{+0.3}$ & $0.07\\pm0.01$ & $0.62_{-0.04}^{+0.03}$ &\n$0.49_{-0.10}^{+0.05}$ &\n$0.10_{-0.02}^{+0.01}$ &\n$2.3_{-2.5}^{+0.7}$ & $0.032_{-0.025}^{+0.012}$ &\n$5911.5\/5653$\\tablenotemark{f}\n\\enddata\n\\tablecomments{\n\\tablenotetext{a}{Assuming a distance of $5\\,{\\rm kpc}$.}\n\\tablenotetext{b}{Absorbed fluxes in the $0.5$--$10\\,{\\rm keV}$ energy range.}\n\\tablenotetext{c}{Only PN data were used.}\n\\tablenotetext{d}{Joint-fit results of both PN and MOS data.}\n\\tablenotetext{e}{Joint-fit results of \\emph{Chandra} and \\emph{Suzaku} data. The date is the weighted-averaged epoch.}\n\\tablenotetext{f}{$N_{\\rm H}$ is linked in the fit for all observations.}\n\\tablenotetext{g}{Fixed at $\\Gamma=1.33$ from the results\nof \\citet{ern10}.}}\n\\end{deluxetable*}\n\\end{turnpage}\nAll spectral analyses were performed in the \\texttt{Sherpa}\nenvironment\\footnote{\\url{http:\/\/cxc.harvard.edu\/sherpa\/}}. We\ntried an absorbed blackbody plus\npower-law (BB+PL) model as in previous studies \\citep{rit09,gwk10,cpr14}. We used\nthe interstellar\nabsorption model \\texttt{tbabs} and the solar abundances\nwere set to\n\\texttt{wilm} \\citep{wam00}.\nThe \\emph{XMM-Newton} spectra from the same epoch were fit with a\nsingle set of parameters.\nWe found that the \\emph{Chandra} and \\emph{Suzaku} spectra share similar\nbest-fit parameters,\nsuggesting the quiescent property. In order to boost the\nsignal-to-noise ratio,\nwe fit them together with the same parameters.\\\\\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{panel}\n\\caption{Evolution trends for the best-fit parameters of SGR 0501+4516 with\nthe 2BB+PL model since the 2008 outburst.}\\label{fig:trends}\n\\end{figure}\n\\indent\nThe best-fit spectral\nparameters are listed in Table\\,\\ref{tab:pas_results}.\nFrom 2008 to 2013, the best-fit blackbody radius shrunk\nsignificantly from $R=1.45\\,{\\rm km}$ to $R=0.34\\,{\\rm km}$ (assuming\n$d=5\\,{\\rm kpc}$) and the power-law photon index\nsoftened from $\\Gamma=2.9$ to $\\Gamma=3.9$.\nOur \\emph{XMM-Newton} results are consistent with those reported by\n\\citet{rit09} and \\citet{cpr14} except with a slightly higher absorption column\ndensity $N_{\\rm H}$ due to the different absorption model we used. While \\citet{cpr14}\nsuggested that the source had already returned\nto its quiescence one year after the 2008 outburst, our new results show that\nthe total absorbed flux was still\ndecreasing from $2.8\\times10^{-12}\\,{\\rm erg}\\,{\\rm cm}^{-2}\\,{\\rm s}^{-1}$\nin 2009 to $2.0\\times10^{-12}\\,{\\rm erg}\\,{\\rm cm}^{-2}\\,{\\rm s}^{-1}$ in\n2013.\nComparing with the previously reported \\emph{Suzaku} results,\nour blackbody component has a higher temperature and smaller size. This could\nbe the\nresult of the much lower column density ($N_{\\rm H}\\sim0.4\\times10^{22}\\,{\\rm\ncm}^{-2}$) reported by \\citet{esk17}. \\\\\n\\indent We noted the best-fit PL component is\nsoft with $\\Gamma\\gtrsim3$. This could indicate the thermal nature of the\nemission. To verify that, we tried to narrow down the energy range to $6\\,{\\rm keV}$ and\ncompared the best-fit results of the BB+PL and the double-blackbody (2BB) models. We\nfound that the latter provided better fits to all spectra, thus, confirming\nour idea. When we fit the entire energy range, the 2BB fit has obvious\nresiduals in the highest energy bins for all \\emph{XMM-Newton} spectra, hinting at an additional PL component.\nIn the final model, we consider the double-blackbody plus power-law (2BB+PL)\nmodel and found that it provides the best fit.\nAssuming that $N_{\\rm H}$ remained unchanged between epochs, we fit\nall spectra simultaneously with a linked absorption model.\nWe found that the PL component dominated only above $\\sim6\\,{\\rm keV}$ for which our\nobservations were not very sensitive. As the first three\n\\emph{XMM-Newton} observations were taken within $\\sim1$ month after the\n2008\\,August\\,26 \\emph{Suzaku} observation, we believe that they should share\na similar spectral property. In order to obtain a better\nfit, we adopted the 2BB+PL result reported by\n\\citet{ern10} and fixed $\\Gamma=1.33$ in the fitting of the 2008\n\\emph{XMM-Newton} spectra. As the photon index could have changed after 2008, we\ndid not fix $\\Gamma$ for all spectra taken after 2009. However, the PL\ncomponent was poorly constrained. We list the best-fit spectral results\nin Table\\,\\ref{tab:pas_results}. The best-fit 2BB+PL model to the \\emph{XMM-Newton} PN spectra at different \nepochs are plotted in Figure\\,\\ref{fig:xmm_fits}, and the fit to the last epoch \\emph{Chandra} and \\emph{Suzaku} \nspectra are plotted in Figure\\,\\ref{fig:suzaku_chandra}. Figure\\,\\ref{fig:trends} shows evolution trends of the\ntwo blackbody components. The\ntemperature of the cooler blackbody component dropped from\n$kT_{\\rm low}=0.35\\,{\\rm keV}$ in 2008 to $0.26\\,{\\rm keV}$ in 2012,\nwhile there was no significant change in the radius, with $R_{\\rm low}$ stayed\n$\\sim4.5\\,{\\rm km}$ among all the observations.\nThe best-fit parameters for the hotter blackbody component, meanwhile,\nare consistent with those from the BB+PL fit. Both the temperature\n$kT_{\\rm high}$ and the radius $R_{\\rm high}$ of this component\ndropped since the outburst.\nWe found that adding the best-fit ${\\rm BB}_{\\rm high}$ component shares\nsimilar parameters as the ${\\rm BB}_{\\rm high}$ in the BB+PL model.\nSimilar to the BB+PL results, Figure\\,\\ref{fig:trends} shows that\n$R_{\\rm high}$ was not lowest in 2009, indicating that the source was\nnot yet in quiescence at that time.\\\\\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{flux}\n\\caption{Decay trends of unabsorbed fluxes of SGR 0501+4516 for all components\nin the 2BB+PL model.}\\label{fig:flux}\n\\end{figure}\n\\indent In Figure\\,\\ref{fig:flux}, we plot the flux evolution of all\ncomponents in the 2BB+PL\nmodel, we see decreasing trends since the 2008 outburst.\nThe plot indicates a significant drop of the ${\\rm BB}_{\\rm high}$ flux after\n2009,\nand we claim that the source had not yet returned to quiescence at that time.\nOn the other hand, we found similar\ncount rates in the 2012 and 2013 \\emph{Chandra} observations, which suggests\nthat\nSGR 0501+4516 had reached a quiescent state five years after the outburst.\nFinally, we note that there is no obvious plateau in the flux evolution,\ncontrary to what the crustal cooling model suggests\n\\citep{let02}.\\\\\n\\indent In addition to the BB+PL and 2BB+PL models,\nwe also tried the resonant cyclotron scattering (RCS) \\citep{rzt08} and the 3D\nsurface thermal emission\nand magnetospheric scattering (STEM3D) models \\citep{wg15} but\nthe fits converged to the boundary of the parameter\nspace. Therefore, we do not believe that the results are physical.\\\\\n\\begin{turnpage}\n\\begin{deluxetable*}{lccccccccc}\n\\centering\n\\tablecaption{Two-Temperature Fits to the Spectra of Magnetars in Quiescence\nwith Uncertainties or Upper Limits at the $90\\%$ Confidence Level\n\\label{tab:low_nh}}\n\\tablewidth{0pt}\n\\tabletypesize{\\scriptsize}\n\\tablehead{ \\colhead{Object} & Instrument (ObsID) & Model & \\colhead{$N_{\\rm H}$} &\n\\colhead{$kT_{\\rm low}$} & \\colhead{$R_{\\rm low}$} & \\colhead{$kT_{\\rm high}$}\n& \\colhead{$R_{\\rm high}$} & \\colhead{$\\Gamma$} & $\\chi^2_\\nu$\\,(dof)\n\\\\& & & \\colhead{$(10^{22}\\,{\\rm cm}^{-2})$} & \\colhead{(keV)} & \\colhead{(km)}\n& \\colhead{(keV)} & \\colhead{(km)} & &\n}\n\\startdata\nCXOU J010043.1--721134 &\\emph{XMM}\\,(see Reference 1) & 2BB & $0.063^{+0.020}_{-0.016}$ & $0.30\\pm0.02$ &\n$12.1^{+2.1}_{-1.4}$ & $0.68^{+0.09}_{-0.07}$ & $1.7^{+0.6}_{-0.5}$ & \\nodata &\n$1.14\\,(100)$ \\\\\nSGR 0526--66 & \\emph{CXO}\\,(10806) & 2BB\\tablenotemark{a} & $0.07_{-0.05}^{+0.06}$ &\n$0.42_{-0.05}^{+0.04}$ & $8.8_{-1.5}^{+1.7}$ &\n$1.1_{-0.3}^{+0.8}$ & $0.8_{-0.5}^{+0.9}$ & \\nodata &1.31\\,(117) \\\\\nXTE J1810--197\\tablenotemark{b}& \\emph{XMM}\\,(see Reference2) & 2BB & $0.60\\pm0.02$\n& $0.167\\pm0.006$ & $9.3\\pm1.1$ &\n$0.33\\pm0.02$ & $0.9\\pm0.2$ & \\nodata\n& $1.21\\,(824)$\\tablenotemark{c}\\\\\n\\emph{Swift} J1822.3--1606 & \\emph{CXO}\\,(15989-15993) & 2BB & $0.62\\pm0.05$ &\n$0.11\\pm0.01$ & $6.3\\pm1.7$ & $0.29\\pm0.03$ & $0.24_{-0.10}^{+0.14}$ & \\nodata &\n1.06\\,(74)\\\\\n4U 0142+61\\tablenotemark{b} &\\emph{XMM}\\,(see Reference 3)& 2BB+PL & $0.70\\pm0.03$ &\n$0.27\\pm0.02$ & $14\\pm3$ &\n$0.50\\pm0.02$ & $2.6\\pm1.1$ & $2.6\\pm0.2$ & $1.11\\,(2350)$\\tablenotemark{c}\\\\\nSGR 0501+4516 & see Table\\,\\ref{tab:pas_results}& 2BB+PL & $0.9\\pm0.2$ &\n$0.26_{-0.02}^{+0.01}$ & $3.7_{-0.7}^{+0.3}$ & $0.62_{-0.04}^{+0.03}$\n& $0.49_{-0.10}^{+0.05}$ & $2.3_{-2.5}^{+0.7}$ &\n$1.05\\,(5653)$\\tablenotemark{c} \\\\\n1E 2259+586 & \\emph{XMM}\\,(0203550701) & 2BB+PL & $1.1\\pm0.2$ & $0.32^{+0.04}_{-0.05}$ &\n$5.6\\pm1.6$ & $0.5^{+0.1}_{-0.2}$ & $1.5^{+1.7}_{-0.7}$ & $3.0^{+0.5}_{-3.2}$\n& 1.03\\,(494)\\\\\n1E 1048.1--5937& \\emph{XMM}\\,(0723330101) &2BB+PL & $1.6^{+0.2}_{-0.6}$ &\n$<0.18$ & $<410$ &\n$0.62\\pm0.01$ & $1.7\\pm0.5$ & $3.2\\pm0.1$ & 0.97\\,(909)\\\\\n1RXS J170849.0--400910 & \\emph{CXO}\\,(4605) & 2BB+PL & $2.72^{+0.04}_{-0.67}$\n& $<0.14$ &\n$<450$ & $0.41\\pm0.05$ & $2.8^{+2.2}_{-1.1}$ & $3.10^{+0.03}_{-0.64}$ &\n1.15\\,(389)\\\\\n1E 1547.0--5408 & \\emph{XMM}\\,(0402910101) &2BB\\tablenotemark{a} & $3.8^{+0.8}_{-0.6}$ &\n$0.39_{-0.08}^{+0.07}$ & $0.9_{-0.4}^{+0.9}$ &\n$0.8_{-0.1}^{+0.3}$ & $0.11_{-0.07}^{+0.12}$ & \\nodata & 1.52\\,(85)\\\\\nSGR 1900+14 & \\emph{XMM}\\,(0506430101) & 2BB+PL & $4.1^{+0.7}_{-0.2}$ & $<0.12$ & $<1080$ &\n$0.39^{+0.01}_{-0.05}$ & $5.5\\pm1.9$ & $2.2^{+0.3}_{-0.1}$ & 1.01\\,(276)\\\\\n1E 1841--045 & \\emph{XMM}\\,(0013340101) & 2BB+PL & $4.2^{+1.8}_{-1.1}$ & $<0.28$ & $<2300$ &\n$0.5^{+0.1}_{-0.2}$ & $6^{+30}_{-4}$ & $1.9^{+0.4}_{-0.7}$ & 1.12\\,(232)\\\\\nCXOU J171405.7--381031 & \\emph{CXO}\\,(11233) & 2BB+PL & $6.6^{+1.1}_{-1.5}$ &\n$<0.28$ & $<90$\n& $0.5\\pm0.1$ & $<4.3$ & $1.3^{+1.8}_{-2.1}$ & 1.08\\,(108)\\\\\nCXOU J164710.2--455216 & \\emph{CXO}\\,(14360) & 2BB+PL & $6.9^{+1.9}_{-1.7}$ &\n$<0.18$ &\n$<3800$ & $0.47^{+0.19}_{-0.12}$ & $<9$ & $3.2^{+0.6}_{-1.1}$ & 0.91\\,(107) \\\\\nSGR 1806--20 & \\emph{CXO}\\,(7612) & 2BB\\tablenotemark{a} & $9.8^{+1.0}_{-0.9}$ &\n$0.67_{-0.09}^{+0.11}$ & $1.8_{-0.5}^{+0.8}$ &\n$2.1^{+0.7}_{-0.3}$ & $0.22\\pm0.08$ & \\nodata &\n1.15\\,(268)\n\\enddata\n\\tablecomments{\n\\tablenotetext{a}{We noted that BB+PL could be a better model (see the text).}\n\\tablenotetext{b}{The uncertainties have been scaled to the\n$90\\%$ confidence level.}\n\\tablenotetext{c}{Joint fits with different observations.}}\n\\tablerefs{\\scriptsize (1) \\citet{tem08}; (2) \\citet{bid09};\n(3) \\citet{gdk10}\n}\n\\end{deluxetable*}\n\\end{turnpage}\n\\section{\\bf{DISCUSSION}}\n\\subsection{Two-Temperature Spectral Model}\\label{sec:2bb_fit}\nOur study showed that the spectrum of SGR 0501+4516 is best described by\na two-temperature model. This is similar to the cases of some magnetars,\nincluding\nCXOU J010043.1--721134, XTE J1810--197, and 4U 0142+61\n\\citep{tem08,bid09,gdk10}.\nThe result motivates us to\ntest this model on a larger sample of the magnetar population. \\\\\n\\indent We identified 15 magnetars with X-ray observations\ntaken a few\nyears after their outbursts.\nThe three sources mentioned above have\npreviously been fit with two-temperature spectral models.\nFor the rest, we reduced \\emph{Chandra} and \\emph{XMM-Newton} observations\nand extracted their spectra with the same\nprocedures as for SGR 0501+4516. We tried both 2BB and\n2BB+PL models on all sources and report the one with lower reduced $\\chi^2$\nvalue ($\\chi^2_\\nu$).\nTable\\,\\ref{tab:low_nh} lists our results and those\nreported in previous studies.\nWe found that, for most magnetars with\nsmall $N_{\\rm H}\\lesssim1\\times10^{22}\\,{\\rm cm}^{-2}$, their spectra are\ngenerally well fit by\nthe two-temperature spectral model.\nThe higher temperature blackbody component always has a smaller radius\n$R\\lesssim3\\,{\\rm km}$ and vice versa. For the sources with large\n$N_{\\rm H}$, the lower temperature blackbody component\nis not well constrained due to heavy absorption by the ISM below $2\\,{\\rm\nkeV}$. There are three exceptional cases: SGR 0526--66, 1E\n1547.0--5408, SGR\n1806--20, for which $kT_{\\rm high}$ seems too high to be physical. We compared the\n$\\chi^2$-statistics between the 2BB and the BB+PL fits and found that\nthey are similar. It is therefore possible that the BB+PL model\nprovides a more physical description of their spectra.\\\\\n\\indent Our results hint that the two-temperature components could be a\ncommon feature among magnetars, although not all could be detected due to\ninterstellar absorption. The physical interpretation\nof the two blackbody components will be discussed below.\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{luminosity_to_hsarea.eps}\n\\caption{Trend of the hotspot X-ray luminosity $L_{\\rm X}$ in\n$0.5$--$10\\,{\\rm keV}$ against the hotspot\narea $4\\pi R_{\\rm high}^2$ for SGR 0501+4516. The blue dashed\nline shows the best-fit correlation\n$L_{\\rm X}=1.8\\times10^{34}A_{11}^{1.42}\\,{\\rm erg}\\,{\\rm s}^{-1}$\nand\nthe black solid lines show the theoretical predicted correlation,\n$L_{\\rm X}=1.3\\times10^{33}KA_{11}^2\\,{\\rm erg}\\,{\\rm s}^{-1}$, with $K=1$ and\n$K=20$.}\\label{fig:l_to_a}\n\\end{figure}\n\\begin{deluxetable*}{lccc}[b!]\n\\tablecaption{Blackbody Temperature and Spin-Inferred Magnetic Field Strength\nof Magnetars\nand Young High Magnetic Field Rotation-Powered Pulsars as Plotted in\nFigure\\,\\ref{fig:kt_to_b}\\label{tab:kt_to_b}}\n\\tablewidth{0pt}\n\\tabletypesize{\\small}\n\\tablehead{\\colhead{Source} & \\colhead{$B$\\tablenotemark{a} ($10^{14}\\,$G)} &\n\\colhead{$kT$\\tablenotemark{b} (keV)} &\n\\colhead{Reference}\n}\n\\startdata\n\\sidehead{Magnetars (entire surface):}\n\\emph{Swift} J1822--1606 & $0.14$ & $0.11\\pm0.01$ & See\nTable\\,\\ref{tab:low_nh}\\\\\n1E 2259+586 & $0.59$ & $0.32_{-0.05}^{+0.04}$ & See Table\\,\\ref{tab:low_nh}\\\\\n4U 0142+61 & $1.3$ & $0.27\\pm0.02$ & 1\\\\\nSGR 0501+4516 & $1.9$ & $0.26_{-0.02}^{+0.01}$ & See Table\\,\\ref{tab:low_nh}\\\\\nXTE J1810--197 & $2.1$ & $0.167\\pm0.006$  & 2\\\\\nCXOU J010043.1--721134 & $3.9$ & $0.30\\pm0.02$  & 3\\\\\n1RXS J170849.0--400910 & $4.7$ & $0.42\\pm0.02$ & 4\\\\\nSGR 0526--66 & $5.6$ & $0.44\\pm0.02$  & 5\\\\\nSGR 1900+14 & $7.0$ & $0.47\\pm0.02$ & 6\\\\\n1E 1841--045 & $7.0$ & $0.45\\pm0.03$ & 7\\\\\nSGR 1806--20 & $11.3$ & $0.55\\pm0.07$ & 8\\\\\n\\sidehead{Magnetars (hotspot):}\nSGR 0418+5729 & $0.061$ & $0.32\\pm0.05$  & 9\\\\\n\\emph{Swift} J1822--1606 & $0.14$ & $0.29\\pm0.03$ & See\nTable\\,\\ref{tab:low_nh}\\\\\n1E 2259+586 & $0.59$ & $0.5_{-0.2}^{+0.1}$ & See Table\\,\\ref{tab:low_nh}\\\\\nCXOU J164710.2--455216 & $1.0$ & $0.53\\pm0.03$ & 10\\\\\n4U 0142+61 & $1.3$ & $0.50\\pm0.02$ & 1\\\\\nSGR 0501+4516 & $1.9$ & $0.62_{-0.04}^{+0.03}$  & See\nTable\\,\\ref{tab:low_nh}\\\\\nXTE J1810--197 & $2.1$ & $0.33\\pm0.02$  & 2\\\\\nSGR 1935+2154 & $2.2$ & $0.47\\pm0.03$ & 11\\\\\n1E 1547.0--5408 & $2.2$ & $0.43\\pm0.05$ & 12\\\\\nPSR J1622--4950 & $2.7$ & $0.5\\pm0.1$  & 13\\\\\nCXOU J010043.1--721134 & $3.9$ & $0.68_{-0.07}^{+0.09}$ & 3\\\\\n1E 1048.1--5937 & $4.5$ & $0.56\\pm0.02$ & 14\\\\\n\\sidehead{High-$B$ rotation-powered pulsars:}\nPSR B1509--58 & $0.15$ & $0.15\\pm0.01$ & 15\\\\\nPSR J1119--6127 & $0.41$ & $0.21\\pm0.04$ & 16\\\\\nPSR J1846--0258 & $0.49$ & $<0.25$ & 17\n\\enddata\n\\tablecomments{\n\\tablenotetext{a}{\\scriptsize Adopted from the Magnetar Catalog\n\\citep{ok14}.}\n\\tablenotetext{b}{\\scriptsize Uncertainties are at the $90\\%$ confidence level.}}\n\\tablerefs{\\scriptsize\n(1) \\citet{gdk10}; (2) \\citet{bid09}; (3) \\citet{tem08};\n(4) \\citet{cri07}; (5) \\citet{phs12}; (6) \\citet{met06b};\n(7) \\citet{ks10};\n(8) \\citet{emt07b}; (9) \\citet{rip13}; (10) \\citet{akac13};\n(11) \\citet{ier16}; (12) \\citet{bis11}; (13) \\citet{ags12};\n(14) \\citet{tgd08}; (15) \\citet{hnt17};\n(16) \\citet{nkh12}; (17) \\citet{ln11}\n}\n\\end{deluxetable*}\n\\subsection{Physical Interpretation of the Hotter Blackbody\nComponent}\\label{sec:high_bb}\nThe best-fit radius of the higher temperature component $R_{\\rm high}$ of SGR~0501+4516\nhad shrunk to $0.49\\,{\\rm km}$ from 2008 to 2013,\nindicating that the thermal emission could come from a hotspot on surface.\nThere were several magnetars with\nblackbody radii that continued to shrink for a few years after their outbursts\n\\citep{bl16}.\n\\citet{bel09} suggested that this\ncould be the observational evidence supporting the $j$-bundle model. When a\ntwisted magnetic field is implanted into the closed\nmagnetosphere, the current ($j$-bundle) would flow along the closed magnetic\nfield lines and return back to the stellar surface,\nheating up the footprints of the $j$-bundle and resulting in hotspots. After\nan outburst, the footprints\nare expected to keep shrinking and the hotspot could be observed as a blackbody\ncomponent with a decreasing radius.\nThis predicts a correlation between the X-ray luminosity and the area of a\nhotspot as\n$L_{\\rm X}=1.3\\times10^{33}KA_{11}^{2}\\,{\\rm erg}\\,{\\rm s}^{-1}$, where $A_{11}$ is\nthe blackbody\narea in units of $10^{11}\\,{\\rm cm}^{2}$ and $K$ is a constant depending on\nthe twisting angle of the $j$-bundle,\nthe surface magnetic field strength, and the discharge voltage\n\\citep{bel09,bel11}.\\\\\n\\indent We plot in Figure\\,\\ref{fig:l_to_a} the hotspot\nluminosity of SGR 0501+4516 against\nits area $A=4\\pi R_{\\rm high}^2$. The distance of the source is assumed to be\n$d=5\\,{\\rm kpc}$\nfor calculating the luminosity. Our result broadly agrees with the theory\nprediction and suggests $K\\sim20$.\nIf we fit the data points with a straight line in the log--log plot in\nFigure\\,\\ref{fig:l_to_a},\nthe best-fit correlation is flatter, with\n$L_{\\rm X}=1.8\\times10^{34}A_{11}^{1.42}\\,{\\rm erg}\\,{\\rm s}^{-1}$.\nSimilar behavior was also found in several other magnetars during flux\nrelaxation after outbursts \\citep{bl16}.\nThe discrepancy could be due to the time variation of the proportionality\nconstant $K$.\\\\\n\\subsection{Physical Interpretation of the Cooler Blackbody\nComponent}\\label{sec:low_bb}\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.43\\textwidth]{kt_to_b}\n\\caption{Blackbody temperature against magnetic field strength of magnetars\nand three young high-$B$ rotation-powered pulsars,\nusing values listed in Table\\,\\ref{tab:kt_to_b}.\nThe red and green dots indicate blackbodies from the entire surface and the\nhotspots, respectively.\nThe blue dots represents the high-$B$ rotation-powered pulsars. The red solid line shows\nthe best-fit correlation\n$kT\\propto B^{0.4}$ using the red data points only. The black dashed line\nrepresents the theoretical prediction of $kT\\propto\nB^{0.5}$.}\\label{fig:kt_to_b}\n\\end{figure}\nIn the two temperature fits, the cooler blackbody always shows a larger radius\n$R_{\\rm low}$ and some values listed in Table\\,\\ref{tab:low_nh} are compatible\nwith the neutron star\nradius. We therefore believe that this blackbody component could originate\nfrom the entire surface. An additional support is that in our case of SGR\n0501+4516, $R_{\\rm low}$ has been relatively stable during flux relaxation.\nTheories suggest that the thermal emission of magnetars\ncould be arise from the decay of the crustal magnetic field\n\\citep{td96,plm07}. If this\nis the only energy source, one expects\na correlation between the surface temperature $kT$ and the magnetic\nfield strength $B$ \\citep{plm07}. The conservation of energy could be\nexpressed as\n\\begin{equation}\n-A\\Delta R\\frac{{\\rm d}E_{\\rm m}}{{\\rm d}t}=A\\sigma T^4, \\label{eq:b_to_t}\n\\end{equation}\nwhere $E_{\\rm m}$ is the magnetic energy density, $A$ is the emission area,\n$\\Delta R$ is the thickness of\nthe neutron star crust, and $\\sigma$ is the Stefan-Boltzmann constant. The\nmagnetic energy density\n$E_{\\rm m}$ could be written as $B^2\/8\\pi$. If the decay of $B$ is in the\nexponential form, it implies a relation $T\\propto \\sqrt{B}$.\nNote that this ignores any age effects that are justified,\nas magnetars are young objects in general \\citep[see][]{vrp13}.\\\\\n\\indent To verify the correlation above,\nwe investigated the trend between $kT$ and $B$ for\nall quiescent magnetars, using the latest results reported in the literature and\nfrom our own analysis (see Table\\,\\ref{tab:low_nh}). These values are listed\nin Table\\,\\ref{tab:kt_to_b}\nand plotted in Figure\\,\\ref{fig:kt_to_b}.\nAs we mentioned, some blackbody components correspond to the hotspot and some\ncorrespond to the entire\nsurface. We show them separately in the plot as two groups, depending on\nwhether the blackbody radius $R$\nis larger or smaller than $3\\,{\\rm km}$. The plot shows an increasing trend for the entire surface $kT$,\nwith a correlation coefficient $r=0.85$. We fit the log--log plot with a\nstraight line and obtained\n$kT\\propto B^{0.4}$, which is a bit flatter than, but generally comparable\nwith the theoretical prediction of $B^{0.5}$.\nOn the other hand, the temperature of the hotspots\nshows no such a correlation, which suggests that they could probably be powered\nby $j$-bundle\ninstead of the decay of the crustal field.\\\\\n\\begin{deluxetable*}{lcccc}\n\\tablecaption{Quiescent X-Ray Luminosity in $2$--$10\\,{\\rm keV}$ and\nSpin-Inferred Magnetic Field Strength\nof Magnetars and High Magnetic Field Rotation-Powered Pulsars as Plotted in\nFigure\\,\\ref{fig:l_to_b}\\label{tab:l_to_b}}\n\\tablewidth{0pt}\n\\tabletypesize{\\small}\n\\tablehead{\\colhead{Source} & \\colhead{$L_{\\rm X}$\\tablenotemark{a} ($10^{35}\\,{\\rm\nerg}\\,{\\rm s}^{-1}$)} &\n\\colhead{$B$\\tablenotemark{b} $(10^{14}\\,{\\rm G})$} &\n\\colhead{Distance\\tablenotemark{b} (kpc)} & \\colhead{Reference}\n}\n\\startdata\n\\sidehead{Magnetars:}\nSGR 0418+5729 & $1.0_{-0.9}^{+1.1}\\times10^{-5}$ & $0.061$ &\n$2.0\\pm0.4$\\tablenotemark{c} & 1\\\\\n\\emph{Swift} J1822.3--1606 & $5_{-4}^{+3}\\times10^{-5}$ & $0.14$ &\n$1.6\\pm0.3$ & See Table\\,\\ref{tab:low_nh} \\\\\n1E 2259+586 & $0.20_{-0.06}^{+0.04}$ & $0.59$ & $3.2\\pm0.2$ & See\nTable\\,\\ref{tab:low_nh} \\\\\nCXOU J164710.2--455216 & ($4.5\\pm3.8)\\times10^{-3}$ & $1.0$ & $3.9\\pm0.7$ &\n2\\\\\n4U 0142+61 & $1.05\\pm0.33$ & $1.3$ & $3.6\\pm0.4$ & 3\\\\\nSGR 0501+4516 & $3.5_{-1.3}^{+1.0}\\times10^{-2}$ & $1.9$ &\n$5.0\\pm1.0$\\tablenotemark{c} & See Table\\,\\ref{tab:low_nh}\\\\\nXTE J1810--197 & $1.3_{-0.9}^{+0.5}\\times10^{-3}$ & $2.1$ & $3.1\\pm0.5$ & 4\\\\\n1E 1547.0--5408 & $1.3_{-0.9}^{+0.5}\\times10^{-2}$ & $2.2$ & $4.5\\pm0.5$ & 5\\\\\nSGR 1627--41 & $2.5_{-1.3}^{+2.3}\\times10^{-2}$ & $2.2$ & $11.0\\pm0.3$ & 6\\\\\nPSR J1622--4950 & $4.4_{-3.6}^{+7.0}\\times10^{-3}$ & $2.7$ &\n$9.0\\pm1.8$\\tablenotemark{c} & 7 \\\\\nCXOU J010043.1--721134 & $0.7_{-0.3}^{+1.7}$ & $3.9$ & $62.4\\pm1.6$ & 8\\\\\n1E 1048.1--5937 & $0.5\\pm0.3$ & $4.5$ & $9.0\\pm1.7$ & 9\\\\\n1RXS J170849.0--400910 & $0.42\\pm0.11$ & $4.7$ & $3.8\\pm0.5$ & 10\\\\\nCXOU J171405.7--381031 & $0.33\\pm0.24$ & $5.0$ & $10.2\\pm3.5$ & 11\\\\\nSGR 0526--66 & $1.9_{-0.4}^{+0.3}$ & $5.6$ & $53.6\\pm1.2$ & 12\\\\\nSGR 1900+14 & $0.7\\pm0.3$ & $7.0$ & $12.5\\pm1.7$ & 13\\\\\n1E 1841--045 & $1.8_{-1.0}^{+0.7}$ & $7.0$ & $8.5_{-1.0}^{+1.3}$ & 14\\\\\nSGR 1806--20 & $1.6_{-0.7}^{+0.8}$ & $11.3$ & $8.7_{-1.5}^{+1.8}$ & 15\\\\\n\\sidehead{High-$B$ rotation-powered pulsars:}\nPSR B1509--58 & $0.96\\pm0.05$ & $0.15$ & $5.2\\pm1.4$ & 16 \\\\\nPSR J1119--6127 & $2.5_{-1.3}^{+3.2}\\times10^{-3}$ & $0.41$ & $8.4\\pm0.4$ & 17\n\\\\\nPSR J1846--0258 & $0.19_{-0.03}^{+0.04}$ & $0.49$ & $6.0_{-0.9}^{+1.5}$ & 18\n\\enddata\n\\tablecomments{\n\\tablenotetext{a}{\\scriptsize $90\\%$ uncertainties in\n$L_{\\rm X}$, derived by\ncombining the errors in flux\nand distance using the standard error propagation formula.}\n\\tablenotetext{b}{\\scriptsize Adopted from the Magnetar Catalog\n\\citep{ok14}. For those with multiple estimated distances, we\nsimply used the most updated or the better measured values.}\n\\tablenotetext{c}{\\scriptsize As the uncertainty in distance is not\nreported, we assumed a relative error of $20\\%$, similar to that of other\nsources.}}\n\\tablerefs{\\scriptsize\n(1) \\citet{rip13}; (2) \\citet{akac13};\n(3) \\citet{rni07}; (4) \\citet{bid09}; (5) \\citet{gg07};\n(6) \\citet{eiz08}; (7) \\citet{ags12}; (8) \\citet{tem08};\n(9) \\citet{tgd08};\n(10) \\citet{rio07}; (11) \\citet{sbni10}; (12) \\citet{phs12};\n(13) \\citet{nmy09}; (14) \\citet{ks10}; (15) \\citet{emt07b};\n(16) \\citet{hnt17}; (17) \\citet{nkh12}; (18) \\citet{ln11}\n}\n\\end{deluxetable*}\n\\indent There is recent evidence showing that both\nyoung high magnetic field rotation-powered pulsars and magnetars share\nsimilar properties, making the division between these two classes blurred\n\\citep[see][]{ggg08,nk11,glk16}. This motivates us to include the three young\nsources with age of $\\sim10^3\\,{\\rm years}\\,$, PSRs B1509--58, J1119--6127, and J1846--0258,\nin Table\\,\\ref{tab:kt_to_b} and Figure\\,\\ref{fig:kt_to_b} for comparison.\nThe thermal emission of PSRs B1509--58 and J1119--6127 has blackbody radii\n$R=10^{+39}_{-5}\\,{\\rm km}$\nand $3^{+4}_{-1}\\,{\\rm km}$, respectively, suggesting that they could be\noriginated from\nthe entire surface (or large area; \n\\citealt{nkh12,hnt17}). However, for PSR B1509--58, the blackbody radius was not\nvery well\nconstrained due to strong non-thermal emission.\nOn the other hand, there is no thermal emission found in PSR J1846--0258 in\nquiescence, with an\nupper limit of $0.25\\,{\\rm keV}$ \\citep{ln11}. From the plot,\nit is interesting to note that all high-$B$ rotation-powered pulsars seem to follow the\nsame $kT$--$B$ trend\nas magnetars. Our results suggest that the energy source, i.e. $B$-field\ndecay, could power the\nentire surface thermal emission of magnetars and high-$B$ rotation-powered pulsars.\\\\\n\\indent\nWhile $kT$ and $B$ appear to show a correlation that is broadly consistent\nwith the theory,\nthere remain some unsolved problems in this picture.\nThe temperature of the cooler blackbody component is typically higher in\noutburst, then decays to a constant value a few years after.\nHence, the outburst could partly contributed to\nthe thermal emission (see Figure\\,\\ref{fig:trends} and also\n\\citealt{bid09} and \\citealt{gdk10}). Also, we note that\nsome radii of the cooler blackbody are\nsmaller than that of a neutron star.\nIt could indicate that the emission regions are smaller than the entire\nsurface or that the temperature\ndistribution is inhomogeneous. It is unclear if the Equation\\,\\ref{eq:b_to_t}\nneeds to be modified in this case.\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.41\\textwidth]{l_to_b}\n\\caption{X-ray luminosity $L_{\\rm X}$ in $2$--$10\\,{\\rm keV}$ against\nmagnetic field strength $B$, using values listed in Table\\,\\ref{tab:l_to_b}. The red dots indicate magnetars in quiescence, and\nthe blue dots indicate high-$B$ rotation-powered pulsars.\nThe red and the green dashed lines show the theoretical predictions of\n$L_{\\rm X}\\propto B^{2}$ and $\\propto B^{4.4}$,\nrespectively.}\\label{fig:l_to_b}\n\\end{figure}\n\\subsection{Correlation Between X-Ray Luminosity and Magnetic\nField}\\label{sec:correlation_L_kT_B}\nWe revisit the correlation between the quiescent X-ray luminosity, $L_{\\rm X}$, and\nthe magnetic field, $B$, of magnetars as reported by \\citet{akt12}, using\nupdated measurements\nlisted in Table\\,\\ref{tab:l_to_b}. The results are plotted in\nFigure\\,\\ref{fig:l_to_b}.\nWe compared the trend with two theoretical predictions of $L_{\\rm X}\\propto\nB^{2}$ deduced from the Equation\\,\\ref{eq:b_to_t} \\citep{plm07} and\n$L_{\\rm X}\\propto B^{4.4}$ based on the ambipolar diffusion model with neutrino\ncooling \\citep{td96}.\nThe plot shows a general trend but with large scatter, particularly for\nmagnetars\nwith $B\\sim10^{14}\\,{\\rm G}$.\nOur updated plot prefers $B^2$, providing some support to the\nsimple magnetic field decay model.\nNote that our result contradicts that reported by \\citet{akt12}.\nThe main discrepancy is due to the updated measurements from two low-field\nmagnetars, SGR 0418+5729 and\n\\emph{Swift} J1822.3--1606. If we fit the log--log plot with a straight line,\nwe obtain a slightly\nflatter correlation of $L_{\\rm X}\\propto B^{1.7}$.\nFrom the plot, 1E~2259+586 and 4U~0142+61 are far more\nluminous than other magnetars with similar $B$. Excluding these two\noutliers gives $L_{\\rm X}\\propto B^{2.8}$, which again prefers $B^2$ to $B^{4.4}$.\\\\\n\\indent Similar to the $kT$--$B$ plot, we also include three young high\nmagnetic field rotation-powered pulsars in Figure\\,\\ref{fig:l_to_b}.\nWe found that only PSR J1119--6127 follows the general trend of magnetars,\nwhile the other two, PSRs B1509--58 and J1846--0258, have luminosities a few\norders of magnitude\nhigher. We believe that their X-ray emission is dominated by\nnon-thermal radiation powered by spin-down, which could be a main difference\nbetween magnetars and high-$B$ rotation-powered pulsars. Although the correlation appears\nto support to the\ntheoretical prediction, there are too few magnetar examples with\n$B<10^{14}\\,{\\rm G}$.\nIncreasing the sample in this magnetic field range in future studies can\nbetter confirm the theory.\n\\section{\\bf{CONCLUSION}}\\label{sec:con}\nWe performed spectral and timing analyses of SGR 0501+4516 using new and\narchival\nX-ray observations taken with \\emph{Chandra}, \\emph{XMM-Newton}, and\n\\emph{Suzaku}. We show that the\nsource has returned to quiescence in 2013, five years after the outburst.\nOur timing results found a spin period of $\\sim5.762\\,{\\rm s}$  with stable\npulse profiles in 2012 and 2013.\nThe \\emph{Chandra} images show no detectable proper motion, with an upper\nlimit of $0\\farcs32\\,{\\rm yr}^{-1}$,\nrejecting the idea that SGR 0501+4516 was born in SNR G160.9+2.6.\nWe found that the soft X-ray spectrum is best described by\na double blackbody plus power-law (2BB+PL) model.\nThe quiescent spectrum has temperatures of $0.26\\,{\\rm keV}$\n(with $R=3.7\\,{\\rm km}$) and $0.62\\,{\\rm keV}$ (with $R=0.49\\,{\\rm km}$).\nWe found a correlation between the X-ray luminosity and the area of the\nevolving hotter blackbody component, which agrees with the prediction of the\n$j$-bundle model.\\\\\n\\indent We further applied the two-temperature spectral model to other\nmagnetars in\nquiescence and found that it provides a good fit to\nmost sources with low column density,\nsuggesting that this could be a common feature.\nWe investigated the correlation between the blackbody temperature $kT$ and the\nspin-inferred\nmagnetic field $B$ of all magnetars in quiescence.\nFor blackbodies with large areas comparable to the entire stellar surface,\nthe correlation generally agrees with the prediction from the simple magnetic\nfield decay model.\nWe found that this simple scenario can also explain the trend between the\nquiescent X-ray luminosity and magnetic field strength of magnetars.\n\n\\acknowledgements\nWe thank the referee for the comments that improved\nthis paper. The scientific results reported in this article are based on observations made\nby the\n\\textit{Chandra X-ray Observatory} and data obtained from the \\textit{Chandra}\nData Archive.\nThis work was based on observations obtained with \\textit{XMM-Newton}, an ESA\nscience mission with instruments and contributions directly funded by ESA\nMember States and NASA.\nThis research has made use of the NASA Astrophysics Data System (ADS) and\nsoftware provided by the Chandra X-ray Center (CXC) in the application package\nCIAO and Sherpa.\nThis work is supported by a GRF grant of Hong Kong Government under HKU\n17300215P.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\blfootnote{\n    \\hspace{-0.65cm} \n    This work is licensed under a Creative Commons \n    Attribution 4.0 International License.\n    License details:\n    \\url{http:\/\/creativecommons.org\/licenses\/by\/4.0\/}.\n}\nThe ability to understand  user's requests is essential to develop effective task-oriented dialogue systems.  For example, in the utterance \"\\textit{I want to listen to Hey Jude by The Beatles}\", a dialogue system should correctly identify that the user's intention is to give a command  to play a song, and that \\textit{Hey Jude} and \\textit{The Beatles} are, respectively, the song's title  and the artist name that the user would like to listen. In a dialogue system this information is typically represented through a \\textit{semantic-frame} structure \\cite{tur2011spoken}, \nand extracting such representation involves two tasks: identifying the correct frame (i.e.  \\textit{intent classification (IC)}) and filling the correct value for the slots of the frame (i.e.  \\textit{slot filling (SF)}). \n\nIn recent years, neural-network based models have achieved the state of the art  for a wide range of natural language processing tasks, including SF and IC. Various neural architectures have been experimented on SF and IC, including RNN-based  \\cite{Mesnil2013InvestigationOR} and attention-based \\cite{Liu2016AttentionBasedRN} approaches, till the more recent transformers models \\cite{Chen2019BERTFJ}.  Input representations have also evolved from  static word embeddings \\cite{Mikolov2010RecurrentNN,Collobert2008AUA,Pennington2014GloveGV} to contextualized word embeddings \\cite{peters2018deep,Devlin2019BERTPO}.  Such progress allows to better address dialogue phenomena involving SF and IC, including  context modeling, handling out-of-vocabulary words, long-distance dependency between words, and to better exploit the  synergy between SF and IC through joint models. \nIn addition to rapid progresses in the research community, the demand for commercial conversational AI is also growing fast, shown by a variety of available solutions, such as Microsoft LUIS, Google Dialogflow, RASA, and Amazon Alexa. These solutions also use various kinds of semantic frame representations as part of their framework.\n\nMotivated by the rapid explosion of scientific progress, and by unprecedented market attention,  we think that a guided map of the approaches on SF and IC  can be useful for a large spectrum of researchers and practitioners interested in dialogue systems.\nThe primary goal of the  survey is to give a broad overview of  recent neural models applied to SF and IC, and to compare their performance in the context of task-oriented dialogue systems.  We also highlight and discuss open issues that still need to be addressed in the future. The paper is structured as follows: Section \\ref{sec:slot_filling_intent_classification} describes the SF and IC tasks,   commonly used datasets and evaluation metrics. Section \\ref{sec:independent_model}, \\ref{sec:joint}, and \\ref{sec:transfer_learning} elaborate on the progress and state of the art of \\textit{independent}, \\textit{joint}, and \\textit{transfer learning models }for both tasks. Section \\ref{sec:discussion} discusses the performance of existing models and   open challenges.\n\n\n\n\\begin{table*}[ht!]\n    \\centering\n    \\small\n    \\begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|c|}\n    \n   \n    \\hline\n    \\textbf{Utterance} & I & want & to & listen  & to & Hey & Jude & by  & The & Beatles       \\\\\n    \\hline\n    \\textbf{Slot}   & \\textsc{O}  & \\textsc{O}       & \\textsc{O}    & \\textsc{O}& \\textsc{O}  & \\textsc{B-SONG} & \\textsc{I-SONG}    & \\textsc{O} & \\textsc{B-ARTIST}  & \\textsc{I-ARTIST}\\\\\n    \\hline\n    \\multicolumn{1}{|l||}{\\textbf{Intent}} & \\multicolumn{10}{|l|}{\\textsc{play\\_song}}  \\\\\n    \\hline\n    \\end{tabular}\n    \\caption{Example of SF and IC output for an utterance. Slot labels are in \\textsc{BIO} format: \\textsc{B} indicates the start of a slot span, \\textsc{I}  the inside of a span while \\textsc{O} denotes that the word does not belong to any slot. }\n    \\label{tab:sf_ic_example}\n\\end{table*}\n\n\\section{Slot Filling and Intent Classification}\n\\label{sec:slot_filling_intent_classification}\nThis section provides some background relevant for SF and IC, sets the scope of the survey with respect to the context in dialogue systems, defines SF and IC as tasks, and introduces the datasets and the metrics that will be used in the rest of the paper.\n\n\n\n\n\\subsection{Background}\n\\label{subsec:background}\nTask-oriented dialogue systems aim to assist users to accomplish a task (e.g. booking a flight, making a restaurant reservation and playing a song) through dialogue in natural language, either in a spoken or written form.\nAs in most of the current approaches, we assume a system involving a pipeline of components \\cite{young2010hidden}, where the user utterance is first processed by an Automatic Speech Recognition (ASR) module and then\nprocessed by a Natural Language Understanding (NLU) component, which interprets the user's needs. Then a Dialogue State Tracker (DST) accumulates the dialogue information as the conversation progresses and may query a domain knowledge base to obtain relevant data. A dialogue policy manager then decides what is the next action to be executed and a Natural Language Generation (NLG) component produces the actual response to the user. \n\nWe focus on the NLU component, and we generalize several recent approaches assuming that the output of the NLU process is a partially filled semantic frame \\cite{wang2005spoken,tur2011spoken}, corresponding to the intent of the user in a certain portion of the dialogue, with a number of slot-value pairs that need to be filled to accomplish the intent.\nThe notion of \\textit{intent} originates from the idea that utterances can be assigned to a small set of \\textit{dialogue acts} \\cite{stolcke-etal-2000-dialogue}, and it is now largely adopted to identify a task or action that the system can execute in a certain domain. \\textit{Slot-value pairs}, on the other end, represent the  domain of the dialogue, and have been actually implemented either as an ontology \\cite{conf\/interspeech\/Bellegarda13}, possibly with reasoning services (e.g. checking the constraints over slot values) or simply trough a list of entity types that the system needs to identify during the dialogue.\n\n\nIntents may correspond either to specific needs of the user (e.g. blocking a credit card, transferring money, etc.), or to general needs (e.g. asking for clarification, thanking, etc.). Slots are defined for each intent: for instance, to block a credit card it is relevant to know the name of the owner and the number of the card. Values for the slots are collected through the dialogue, and can be expressed by the user either in a single turn or in several turns. \nAt each user turn in the dialogue the NLU component has to determine the intent of the user utterance (\\textit{intent classification}) and has to detect the slot-value pairs referred in the particular turn (\\textit{slot filling}).\nTable \\ref{tab:sf_ic_example} shows the expected NLU output for the  utterance \"\\textit{I want to listen to Hey Jude by The Beatles}\".\n\n\n\n\\subsection{Scope of the Survey}\n\nIn Section \\textsection\\ref{subsec:background}, we described a task-oriented system as a pipeline of components, saying that SF and IC are core tasks at the NLU level.  Particularly, IC consists of classifying an utterance with a set of pre-defined intents, while SF is defined as a sequence tagging problem \\cite{Raymond2007GenerativeAD,Mesnil2013InvestigationOR}, where each token of the utterance has to be tagged with a slot label. In this scenario training data for SF typically consist of single utterances in a dialogue where tokens are annotated with a pre-defined set of slot names, and slot values correspond to arbitrary sequences of tokens. In this perspective, it is worth mentioning a research line on dialogue state tracking \\cite{henderson2014word,mrkvsic2015multi,budzianowski-etal-2018-multiwoz}, where the NLU component is usually embedded into DST.  What is relevant for our topic is that in this context SF is defined as a classification problem: given the current utterance and the previous dialogue history, the system has to decide whether a certain slot-value pair defined in the domain ontology is referred or not in the current utterance. Although promising, from the NLU perspective, this research line poses constraints (e.g. all slot-value pairs have to be pre-defined in an ontology,) that limit the SF applicability. For this reason, and because NLU components are the prevalent solution in current task-oriented systems, the focus of our survey will be on SF as a sequence tagging problem, as more precisely defined in the next section.\n\n\n \n\n\n\n\\subsection{Task Definition}\n\n\\label{sec:approach}\nWe formulate SF and IC as follows. \nGiven an input utterance $\\boldsymbol{x} = (x_1,x_2,.., x_T)$, SF consists in a token-level sequence tagging, where the system has to assign a corresponding slot label $\\boldsymbol{y}^{slot} = (y^{slot}_1, y^{slot}_2,.., y^{slot}_T)$ to each token $x_i$ of the utterance. \nOn the other end, IC is defined as a classification task over utterances, where the system has to assign the correct intent label $y^{intent}$ for the whole utterance $\\boldsymbol{x}$. In general, most machine learning approaches learn a probabilistic model to estimate $p(y^{intent},\\boldsymbol{y}^{slot}|\\boldsymbol{x}, \\boldsymbol{\\theta})$ where $\\boldsymbol{\\theta}$ is the parameter of the model. Table \\ref{tab:sf_ic_example} shows an example of the expected output of a model for the SF and IC tasks. In the following sections, we outline the main models that have been proposed for SF and IC, and  categorize the models into three groups, namely \\textit{independent models} (\\textsection \\ref{sec:independent_model}) , \\textit{joint models} (\\textsection \\ref{sec:joint}), and \\textit{transfer learning based models} (\\textsection{\\ref{sec:transfer_learning}}). \n\n\n\n\n\\subsection{Datasets for SF and IC}\n\\label{dataset}\n\nIn this section, according to our task definition, we list available dialogue datasets (most of them are publicly available) where each utterance is assigned to one intent, and tokens are annotated with slot names. Most of such datasets are collections of \\textit{single turn user utterances} (i.e., not multi-turn dialogues). An example of a single-turn utterance annotation is shown in Table \\ref{tab:sf_ic_example}. \n\nThe ATIS (Airline Travel Information System) dataset \\cite{Hemphill1990TheAS} is the most widely used single-turn dataset for NLU benchmarking. The total number of utterances is around 5K utterances that consist of queries related to the airline travel domain, such as searching for a flight, asking for flight fare, etc. While it has a relatively large number slot and intent labels, the distribution is quite skewed; more than 70\\% of the intent is a flight search. The slots are dominated by a slot that expresses location names such as \\textsc{FromLocation} and \\textsc{ToLocation}. The MEDIA dataset \\cite{DBLP:conf\/interspeech\/Bonneau-MaynardRAKM05} is constructed by simulating the conversation between a tourist and a hotel representative in the French language. Compared to ATIS, the MEDIA corpus size is around three times larger; however, MEDIA is only annotated with slot labels. The slots are related to hotel booking scenarios such as the number of people, date, hotel facility, relative distance, etc. The MIT corpus \\cite{DBLP:conf\/icassp\/LiuPCG13} is constructed through a crowdsourcing platform where crowd workers are hired to create natural language queries in English and annotate the slot label in the queries. The MIT corpus covers two domains, namely movie and restaurant, in which the utterances are related to finding information of a particular movie or actor, searching or booking a restaurant with a particular distance and cuisine criteria. The SNIPS dataset \\cite{Coucke2018SnipsVP} was collected by  crowdsourcing through the SNIPS voice platform. Intents include requests to a digital assistant to complete various tasks, such as asking the weather, playing a song, book a restaurant, asking for a movie schedule, etc. SNIPS is now often used as a benchmark for NLU evaluations.\n\nWhile most datasets are available in English, recently there has been growing interest in expanding slot filling and intent classification datasets to non-English languages. The original ATIS dataset has been derived into several languages, namely Hindi, Turkish \\cite{Upadhyay2018AlmostZC}, and Indonesian \\cite{Susanto2017NeuralAF}. The MultiATIS++ dataset from \\newcite{DBLP:journals\/corr\/abs-2004-14353} expands the ATIS dataset to more languages, namely  Spanish, Portuguese, German, French, Chinese, and Japanese. The work from \\cite{DBLP:conf\/clic-it\/BellomariaCFR19} introduces the Italian version of the original SNIPS dataset. The Facebook multi-lingual dataset \\cite{Schuster2018CrosslingualTL},  introduced a dataset on Thai and Spanish languages  across three domains namely weather, alarm, and reminder. \nThe detailed statistics of each dataset are listed in Appendix A.    \n \n\n\\subsection{Evaluation Metrics}\n For the IC task, evaluation is performed on the utterance level. The typical evaluation metric for IC is \\textit{accuracy},  calculated as the number of the correct predictions made by the model divided by the total number of predictions. As for SF, the evaluation is performed on the entity level. The common metrics used is the metric introduced in CoNLL-2003 shared task  \\cite{DBLP:conf\/conll\/SangM03} to evaluate Named Entity Recognition (NER) by computing the F-1 score. The F1-score, is the harmonic mean score between precision and recall. Precision  is the percentage of slot predictions from the model which are correct, while  recall  is the percentage of slots in the corpus that are found by the model. A slot prediction is considered \\textit{correct} when an \\textit{exact} match is found \\cite{DBLP:conf\/conll\/SangM03}. As the slot is annotated in BIO format to mark the boundary of the slot (see Table \\ref{tab:sf_ic_example}), a correct prediction is only counted when the model can predict the correct slot label on the correct token offset. Consequently, the exact match metrics does not reward cases when the model predict correct slot label but get the incorrect slot boundary (\\textit{partial match}).  \n\n\\section{Independent Models for SF and IC}\n\\label{sec:independent_model}\nIndependent models train each task \\textit{separately} and recent neural models typically use RNN as the building block for SF and IC. At each time step \\textit{t}, the encoder transforms the word representation $x_t$ to the hidden state $h_t$. For SF, the output layer predicts the slot label $y^{slot}_t$  condition on $h_t$. For IC, typically the last hidden state $h_T$ is used to predict the intent label $y^{intent}$of the utterance $\\boldsymbol{x}$. Note that, for independent approaches, the models for SF and IC are trained separately.  Most neural models for SF and IC generally consist of several layers, namely an \\textit{input layer}, one or more \\textit{encoder layer}, and an \\textit{output layer}. Consequently, the main differences between models are in the specifics of these layers. The most common dataset used for evaluating independent models is  ATIS. \n\n\nIn the \\textit{input layer} of neural models each word is mapped into embeddings.  \\newcite{Mesnil2013InvestigationOR} compared several embeddings, namely pre-trained SENNA \\cite{Collobert2011NaturalLP}, RNN Language Model (RNNLM) \\cite{Mikolov2011RNNLMR}, and random embeddings. SENNA gives the best result compared to other embeddings, and, typically, further fine-tuning  word embeddings  improves performance. \\cite{Yao2013RecurrentNN} report that  embeddings learned from scratch directly on ATIS data (\\textit{task-specific embeddings}) are better than SENNA. However,  task-specific embeddings are composed not only by words but also by named entities (\\textit{NE}) and syntactic features\\footnote{Gold named entity and syntactic information}. NE improves performance significantly while part-of-speech only adds small benefits. \\newcite{Ravuri2015RecurrentNN} emphasizes the importance of \\textit{character representation} to handle OOV issues. \n\nFor the \\textit{encoder layer}, various RNN architectures have been applied to SF and IC \\cite{Mesnil2013InvestigationOR,Mesnil2015UsingRN,Liu2015RecurrentNN}. \\newcite{Mesnil2013InvestigationOR} compare the Elman \\cite{elman1990finding} and Jordan \\cite{jordan1997serial} RNNs. They observe that the performance of the Jordan RNN is marginally better than Elman. They also experiment  a \\textit{bi-directional} version of Jordan RNN and obtained the best score of 93.89 F1 for SF, performing better than CRF for about +1 absolute F1 improvement. \\newcite{xu2013convolutional} use Convolutional Neural Network (CNN) \\cite{LeCun1998GradientbasedLA} to extract 5-gram features and apply max-pooling to obtain the word representation before passing it to the output layer. Compared with RNN \\cite{Yao2013RecurrentNN,Mesnil2013InvestigationOR},  CNN gives lower performance for SF on ATIS. Other studies \\cite{Yao2014SpokenLU,vu2016bi} adapt Long Short-Term Memory Network (LSTM) \\cite{Hochreiter1997LongSM} to SF. The LSTM model gives better SF performance compared to CRF, CNN, and RNN. \\newcite{Ravuri2015RecurrentNN} compare the performance of vanilla RNN and LSTM for IC. They find that the vanilla RNN works best for shorter utterances, while LSTM is better for longer utterances. \n\n\n\n\n\\begin{table}[t!]\n\\small\n\\begin{center}\n\\begin{tabular}{llllll}\n\\toprule\n\n\\bf   & \\bf Input  & \\textbf{Model (Enc\/Dec)} & \\textbf{Output}  & \\textbf{Slot (F1)} & \\textbf{Intent(Err)}\\\\\n\\toprule\n\\newcite{xu2013convolutional} & lexical & CNN & softmax & 94.35 & 6.65\\\\\n\\newcite{Yao2013RecurrentNN}  & lexical & Elman RNN & softmax & 94.11 &- \\\\\n\\newcite{Yao2013RecurrentNN}  & lexical+NE & Elman RNN & softmax & 96.60 -\\\\\n\n\\newcite{Yao2014SpokenLU}& lexical & LSTM & softmax & 94.85 & -\\\\\n\\newcite{Yao2014RecurrentCR}& lexical+NE & Elman RNN & CRF & 96.65 & -\\\\\n\\newcite{Mesnil2015UsingRN}&  lexical & Hybrid Elman + Jordan RNN & softmax & 95.06 & -\\\\\n\\newcite{Liu2015RecurrentNN}&  lexical & Elman RNN with label sampling & softmax & 94.89 & -\\\\\n\\newcite{vu2016bi} & lexical & bi-directional RNN & softmax &94.92&- \\\\\n\\newcite{Liu2016AttentionBasedRN} & lexical & bi-directional RNN+attention& softmax & 95.75 & 2.35\\\\\n\\newcite{DBLP:conf\/emnlp\/KurataXZY16} & lexical & Encoder-Decoder LSTM& softmax & 95.40 & -\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\caption{ Comparison of independent SF and IC models and their performance on ATIS.}\n\\label{tab:independent_models}\n\\end{table}\n\n \nFor the \\textit{output layer}, typically a \\textit{softmax} function is used for prediction at a particular time step. \\newcite{Yao2014RecurrentCR} propose a \\textsc{R-CRF} model  combining the feature learning power of RNN and the \\textit{sequence level optimization} of CRF for SF. \nThe RNN + CRF scoring mechanism incorporates the features learned from RNN and the transition scores of the slot slot labels. R-CRF outperforms CRF and vanilla RNN on ATIS and on the Bing query understanding dataset. Table \\ref{tab:independent_models} summarizes the performance of independent models on SF and IC. \n\n\n\n\\textbf{Takeaways on independent SF and IC models:}\n\\begin{itemize}[noitemsep,nolistsep, leftmargin=*]\n    \\item Performance of  RNN encoders (\\textit{unidirectional}) are $\\text{Jordan}\\leq\\text{Elman}<\\text{LSTM}$. Bi-directional encoding is additive to the performance of each encoder.  \n    \\item Incorporating more context information is better for SF performance. Using global context information, such as sentence level representation, and attention mechanisms \\cite{DBLP:conf\/emnlp\/KurataXZY16,Liu2016AttentionBasedRN} boosts performance of bi-directional encoder even further. \n    \\item When adding external features is possible, semantic features such as NE are more beneficial than syntactic features for SF. When NE is used, it can boost the model performance for SF significantly.\n    \\item The slot filling task is related to Named Entity Recognition (NER) \\cite{DBLP:conf\/coling\/GrishmanS96} task as slot values can be a named entity such as airline name, city name etc. If the slot filling task is modeled as a sequence tagging problem, basically  recent neural models proposed for NER can be used for slot filling and vice versa. To know more about the recent development of neural NER models, one can consult the survey from \\newcite{yadav-bethard-2018-survey}. \n    \\item The main disadvantage of  independent models is that they do not exploit the interaction between intent and slots and may introduce error propagation when they are used in a pipeline. \n   \n\\end{itemize}\n\n\n\n\\section{Joint Models for SF and IC}\n\\label{sec:joint}\n\n\\begin{figure*}[htb!]\n     \\centering\n     \\includegraphics[scale=0.26]{img\/param_sharing_gating_bert.png}\n    \\caption{\\textit{Left:} Shared Bi-GRU encoder \\protect\\cite{Zhang2016AJM}. \\textit{Middle:} Slot-Gate Mechanism \\protect\\cite{goo2018slot}. \\textit{Right:} BERT Based \\protect \\cite{Chen2019BERTFJ}.}\n    \\label{fig:sharing_gating_mechanism}\n\\end{figure*}\n\nIn Section \\ref{sec:independent_model} we reported  approaches  that treat SF and IC \\textit{independently}. However, as the two tasks always appear together in an utterance and they share information, it is intuitive to think that they can benefit each other. For instance, if the word \"\\textit{The Beatles}\"  is recognized as the slot \\textsc{Artist}, then it is more likely that the intent of the utterance is \\textsc{PlaySong} rather than \\textsc{BookFlight}. On the other hand, recognizing that the intent is \\textsc{PlaySong} would help to recognize \"\\textit{Hey Jude}\" as the slot \\textsc{Artist} rather than \\textsc{MovieName}. \n    \nRecent approaches  model the relationship between SF and IC \\textit{simultaneously} in a \\textit{joint model}.   These approaches promote \\textit{two-way} information sharing between the two tasks instead of a one-way (\\textit{pipeline}). We describe several alternatives to exploit the relation between SF and IC: through \\textit{parameter and state sharing} and \\textit{gate mechanism}.\n\n\n\\subsection{Parameter and State Sharing} \nA pioneering work in joint modeling is \\newcite{xu2013convolutional}, which performs parameter sharing and captures the relation between SF and IC through Tri-CRF \\cite{jeong2008triangular}.  The model uses CNN as a \\textit{shared} encoder for both tasks and the produced hidden states are utilized for SF and IC. In addition to features learned from the NN and from the slot label transition,  Tri-CRF incorporates an additional factor $g$ to learn the correlation between the slot label assigned to each word and the intent  assigned to the utterance, which explicitly captures the dependency between the two tasks.  A similar approach  \\cite{guo2014joint}, shares the node representation produced by Recursive Neural Network (RecNN) which operates on the syntactic  tree of the utterance. The node's representation is \\textit{shared} among SF and IC. \\newcite{Zhang2016AJM} use a \\textit{shared} bi-GRU encoder and a \\textit{joint loss function} between SF and IC (Figure \\ref{fig:sharing_gating_mechanism} \\textit{Left}), in which the loss function has weights associated with each tasks. \n\n\n\\newcite{Liu2016AttentionBasedRN} use  a neural sequence to sequence (encoder-decoder) model with attention mechanism commonly used for neural machine translation. The \\textit{shared} encoder is a bi-directional LSTM, and the last hidden state of the encoder is then used by the decoder to generate a sequence of slot labels, while for IC there is a separate decoder. The attention mechanism is used to learn alignments between  slot labels in the decoder and words in the encoder. \\newcite{HakkaniTr2016MultiDomainJS} also adopt parameter sharing similar to \\newcite{Zhang2016AJM}, but instead of using GRU they use a shared LSTM and perform predictions for slots, intent, and also domain. \n\n\n\n \n In a  recent approach by \\newcite{Wang2018ABB} propose a bi-model based structure to learn the \\textit{cross-impact} between SF and IC. They argue that a single model for two tasks can hurt performance, and, instead of sharing  parameters, they use two-task networks to learn the cross-impact between the two tasks and only share the hidden state of the other task. In the model, every hidden state $h^1_t$ in the first network is combined with the hidden state of the second network $h^2_t$, and vice versa. Training is also done asynchronously, as each model has a separate loss function.  Qin et al. \\newcite{Qin2019ASF}  use a self-attentive shared encoder to produce better context-aware representations, then apply IC at the \\textit{token level} and use this information to guide the SF task. They argue that previous work based on \\textit{single utterance-level} intent prediction is more prone to error propagation. If some token-level intent is incorrectly predicted, the other correct token-level prediction can still be useful for corresponding SF. For the final IC prediction, they use a voting mechanism to take into account the IC prediction on each token.  \n \n\\newcite{Chen2019BERTFJ} use a Transformer \\cite{Vaswani2017AttentionIA} model for joint SF and IC by fine-tuning a pre-trained BERT \\cite{Devlin2019BERTPO} model (Figure \\ref{fig:sharing_gating_mechanism} \\textit{Right}). The input is passed through several layers of transformer encoders and the hidden state outputs are  used to compute slot and intent labels. The hidden state $h^{\\texttt{CLS}}$ is  used for IC\\footnote{\\texttt{[CLS]} is a special token in BERT input format that often used as the sentence representation.} while the rest of the hidden states at each time step $h_i$ serve SF.  \n\n\n\n\\subsection{Slot-Intent Gate Mechanism} \n\nIn addition to parameter and state sharing, a separate network with a \\textit{slot gating mechanism} was  introduced by \\newcite{goo2018slot} to model the interaction between SF and IC more explicitly (Figure \\ref{fig:sharing_gating_mechanism} \\textit{Middle}). In the encoder, a \\textit{slot context vector} for each time step, $\\boldsymbol{c}^{S}_{i}$, and a global intent context vector $\\boldsymbol{c}^{I}$ are computed using an attention mechanism \\cite{DBLP:journals\/corr\/BahdanauCB14}. The slot-gate $\\boldsymbol{g}^{s}$ is computed as a function of $\\boldsymbol{c}^{S}_{i}$ and $\\boldsymbol{c}^{I}$, $\\boldsymbol{g}^{s} = \\sum v \\cdot tanh(\\boldsymbol{c}^{S}_{i} + \\boldsymbol{W}\\cdot \\boldsymbol{c}^{I} )$. Then, $\\boldsymbol{g}^{s}$ is used as a weight between $\\boldsymbol{h}_i$ and $\\boldsymbol{c}^{S}_i$ to compute $y^{slot}_{i}$ as follows: $y^{slot}_i=\\texttt{softmax}(\\boldsymbol{W}(\\boldsymbol{h}_i + \\boldsymbol{g}^{s}\\cdot\\boldsymbol{c}^{S}_i))$. Larger $\\boldsymbol{g}^{s}$ indicates a stronger correlation between $c^{S}_{i}$ and $c^{I}$. \n\n \\newcite{Haihong2019ANB} propose a bi-directional model, SF-ID (SF-Intent Detection) network, sharing ideas with \\newcite{goo2018slot}, with  two key differences. First, in addition to the slot-gated mechanism, they add an intent-gated mechanism as well. Second, they use an iterative mechanism between the SF and ID network, meaning that the gate vector from SF is injected into the ID network and vice versa. This  mechanism is repeated for an arbitrary number of iteration. Compared to \\cite{goo2018slot}, the SF-ID network performs better both in SF and IC on ATIS and SNIPS. The work from \\newcite{Li2018ASM} is also similar to \\newcite{goo2018slot} with two differences. First, they use a self-attention mechanism \\cite{Vaswani2017AttentionIA} to compute $c^{S}_{i}$. Secondly, they use a separate network to compute gate vector $g^{s}$, but the input of this network is the concatenation of $c^{S}_{i}$ and the intent embedding $v$, and $g^s$ is defined as $g^{s} = \\texttt{tanh}(\\boldsymbol{W}^{g}[c^i_{slot}, v^{intent}] + b^s)$. After that, $h_i$ is combined with $g^s$ through element-wise multiplication to compute $y^s_{i}$ as follows: $y^{slot}_i=\\texttt{softmax}(\\boldsymbol{W}^s(h_i \\odot g^s) + b^s$). They report a +0.5\\% improvement on SF over \\newcite{Liu2016AttentionBasedRN}. A recent work by \\newcite{Zhang2019AJL}, further improves the performance of the BERT based model by adding a gate mechanism \\cite{Li2018ASM} to the BERT model. Table \\ref{tab:joint_models} compares the performance of the joint models. \n\n\\begin{table}[htb!]\n\\centering\n\\small\n\\begin{tabular}{llcccc}\n\\toprule\n\\multirow{3}{*}{\\textbf{Method}} & \\multirow{3}{*}{\\textbf{Model}}& \\multicolumn{2}{c}{\\textbf{ATIS}} & \\multicolumn{2}{c}{\\textbf{SNIPS}} \\\\\n & & \\textbf{Slot} & \\textbf{Intent }& \\textbf{Slot} & \\textbf{Intent} \\\\\n & & \\textbf{F1} & \\textbf{Acc\/Err}& \\textbf{F1} & \\textbf{Acc\/Err} \\\\\n \\midrule\n\\textbf{Parameter \\& State Sharing}  & &  &  &  &  \\\\\n \\newcite{xu2013convolutional} &CNN + Tri-CRF & 95.42 & -\/5.91  & - & - \\\\\n\\newcite{guo2014joint} & Recursive NN & 93.96 & 95.40 & - & -\\\\\n\\newcite{Zhang2016AJM} & Joint Multi-Task,Bi-GRU & 95.49 & 98.10 & - & - \\\\\n\\newcite{Liu2016AttentionBasedRN} & Seq2Seq + Attention & 94.20 & 91.10 & 87.80 & 96.70 \\\\\n\\newcite{HakkaniTr2016MultiDomainJS} & Bi-LSTM& 94.30 & 92.60 & 87.30 & 96.90 \\\\\n\\newcite{Qin2019ASF}& Token-Level IC + Self-Attention & 95.90 & 96.90 & 94.20 & 98.00 \\\\\n\n\\newcite{Chen2019BERTFJ} & Transformer (BERT)& 96.10 & 97.50 & 97.00 & 98.60 \\\\\n\\midrule\n\\textbf{State Sharing} & &  &  &  &  \\\\\n\\newcite{Wang2018ABB} &Bi-model, BiLSTM  & 96.89 & 98.99 & - & - \\\\\n\\midrule\n\\textbf{Slot-Intent Gating} & &  &  &  &  \\\\\n\\newcite{goo2018slot}& Slot-Gated Full Attention \\ & 94.80 & 93.60 & 88.80 & 97.70 \\\\\n\\newcite{Li2018ASM}& BiLSTM + Self-Attention & 96.52 & -\/1.23 & -  & - \\\\\n\\newcite{Haihong2019ANB}& SF-ID Network  & 95.75 & 97.76 & 91.43 & 97.43 \\\\\n\\midrule\n\\textbf{Hybrid Param Sharing + Gating}&  &  &  &  &  \\\\\n\\newcite{Zhang2019AJL} & \\textsc{BERT} + Intent-Gate  & 98.75 & 99.76 & 98.78 & 98.96 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Performance comparison of joint models for SF and IC on ATIS and SNIPS-NLU.}\n\\label{tab:joint_models}\n\\end{table}\n\n\n\n\n\\textbf{Takeaways on joint SF and IC models:}\n\\begin{itemize}[noitemsep,nolistsep, leftmargin=*]\n    \\item The overall performance of joint models for SF and IC (Table \\ref{tab:independent_models}) is competitive with independent models (Table \\ref{tab:joint_models}). The advantage of joint models is that they have relatively less parameters than independent models, as both tasks are trained on a single model.   \n    \\item When computational power is not an issue, fine-tuning a pre-trained model such as BERT is the way to go for maximum SF and IC performance. Hybrid methods combining parameter and state sharing + intent gating yield the best performance \\cite{Zhang2019AJL}. \n    \\item For the non BERT-based model, using state sharing \\cite{Wang2018ABB} is the best on ATIS. However, the disadvantage is that it is actually a bi-model and not a single model. \n    \\item Similar to independent models, contextual information is crucial to performance. Adding a self-attention mechanism \\cite{Qin2019ASF,Li2018ASM} to either parameter and state sharing or to slot-intent gating can boost performance even further.\n    \\item When sufficiently large in-domain training data is available, the SF and IC performance in ATIS and SNIPS is already saturated. Therefore, further research on this classic leaderboard chase is not worth it. We discuss more about that in Section 6. \n    \\item Most of the work in joint models and also independent models (Section \\textsection \\ref{sec:independent_model}) reports F1 scores for slot filling performance. However, these scores do not reveal in which specific cases these models behave differently, contributing to overall performance. We leave further analysis on model performance as a potential future work.\n\\end{itemize}\n\n\\section{Scaling to New Domains}\n\\label{sec:transfer_learning}\n\n\n\nSo far, the models that we consider in Section \\textsection \\ref{sec:independent_model} and Section \\textsection \\ref{sec:joint} are  designed to be trained on a \\textit{single domain} (e.g. banking, restaurant reservation) and require relatively \\textit{large labeled data} to perform well. In practice, new intents and slots are regularly added to a system to support new tasks and domains,  requiring data and time intensive processes. Hence, methods to train models for new domains with limited or without labeled data are needed. We refer to this situation as the \\textit{domain scaling} problem. \n\\begin{figure*}[h!]\n     \\centering\n     \\includegraphics[scale=0.3]{img\/transfer_learning.png}\n    \\caption{\\textit{Left:} Data-driven approach \\protect\\cite{Jaech2016DomainAO,HakkaniTr2016MultiDomainJS}. \\textit{Middle:} Model-Driven Approach with expert models \\protect\\cite{Kim2017DomainAW}. \\textit{Right:} Zero-shot model \\protect\\cite{DBLP:conf\/interspeech\/BapnaTHH17}.}\n    \\label{fig:transfer_learning}\n\\end{figure*}\n\\subsection{Transfer Learning Models for SF and IC}\nA common approach to deal with domain scaling is  transfer learning (TF).\\footnote{We do not differentiate between \\textit{domain adaptation} and transfer learning in this paper.}  In the TF setup we have $K$ source domains $\\mathcal{D}^1_{S}, \\mathcal{D}^{2}_{S}, \\ldots, \\mathcal{D}^{K}_{S}$ \nand a target domain $\\mathcal{D}^{K+1}_{T}$, and we assume abundance of data in $\\mathcal{D_S}$ and limited data in $\\mathcal{D_T}$. Instead of \ntraining a target model $\\mathcal{M}_T$ for $\\mathcal{D}_T$ from scratch, TF aims to \\textit{adapt} the learned model $\\mathcal{M}_S$ from $\\mathcal{D}_S$ to produce a model $\\mathcal{M}_T$ trained on $\\mathcal{D}_T$.  TF is typically applied with various parameter sharing and training mechanisms. For SF and IC  two approaches are proposed, namely \\textit{data-driven} and \\textit{model- driven}. As for data-driven techniques, typically we combine data from $\\mathcal{D}_S$ and $\\mathcal{D}_T$ and we partition the parameters in the model into parts that are \\textit{task-specific} and parameters that are shared across tasks.  \nSome studies \\cite{Jaech2016DomainAO,HakkaniTr2016MultiDomainJS,Louvan2019LeveragingNT}  apply this technique using \\textit{multi-task learning} (MTL) \\cite{caruana1997multitask} and the models are trained simultaneously on $\\mathcal{D}_S$ and $\\mathcal{D}_T$ (Figure \\ref{fig:transfer_learning} \\textit{Left}). Results have shown that MTL is particularly effective relative to single-task learning (STL) when the data in $\\mathcal{D_T}$ is scarce and the benefits over STL diminish as more data is available. Another technique that is typically used in  data-driven approaches is based on \\textit{pre-train} and \\textit{fine-tune} mechanisms. \\newcite{DBLP:conf\/naacl\/GoyalMM18} train a joint model of SF and IC, $\\mathcal{M}_S$, on large $\\mathcal{D_S}$, then fine-tune $\\mathcal{M}_S$ by replacing the output layer corresponding with the label space from $\\mathcal{D}_T$ and train the model further on $\\mathcal{D}_T$. \\newcite{DBLP:conf\/aaai\/SiddhantGM19} also uses fine-tuning mechanism, but the main difference with \\newcite{DBLP:conf\/naacl\/GoyalMM18} is they leverage large unlabeled data to learn contextual embedding, ELMo \\cite{peters2018deep}, before fine-tuning on $\\mathcal{D}_T$. \n\nAs we need to train from scratch the whole model when adding a new domain,  data-driven approaches, especially MTL-based, need increasing training time as the number of domains grows. The alternative strategy, the model-driven approach, alleviates the problem by enabling model \\textit{reusability}. Although different domains have different slot schemas,   slots such as \\textit{date}, \\textit{time} and \\textit{location} can be shared. In model driven adaptation \"expert\" models (Figure \\ref{fig:transfer_learning} \\textit{Middle}) are first trained on these reusable slots \\cite{Kim2017DomainAW,Jha2018BagOE} and the outputs of the expert models are used to guide the training of $\\mathcal{M}_T$ for a new target domain. This way the training time of $\\mathcal{M}_T$ is faster, as it is proportional to the $\\mathcal{D}_T $ data size, instead of the larger data size of the whole $\\mathcal{D}_S$ and $\\mathcal{D}_T$. In this model-driven settings, \\newcite{Kim2017DomainAW} do not treat each expert model on each $\\mathcal{D}_S$ equally, instead they use attention mechanism to learn a weighted combinations from the feedback of the expert models. \\newcite{Jha2018BagOE} use a similar model as \\newcite{Kim2017DomainAW}, however they do not use attention mechanism. For training the expert models, instead of using all available  $\\mathcal{D}_S$, they build a repository  consisting of common slots, such as \\textit{date}, \\textit{time}, \\textit{location} slots. The assumption is that these slots are potentially reusable in many target domains. Upon training $\\mathcal{M}_S$ on this reusable repository, the output of $\\mathcal{M}_S$  is directly used to guide the training of $\\mathcal{M}_T$.  \n\n\n\\subsection{Zero-shot Models for SF and IC}\nWhile data-driven and  model-driven approaches can share  knowledge learned on different domains, such models are still trained on a pre-defined set of labels, and can not handle \\textit{unseen} labels, i.e. not mapped to the existing schema. For example, a model trained to recognize a \\textsc{Destination} slot, can not be used directly to recognize the slot \\textsc{Arrival\\_Location} for a new domain, although both slots are semantically similar. For this reason, researchers  have recently been working on \\textit{zero-shot} models,  trained on \\textit{label representations} that leverage  natural language \\textit{descriptions} of the slots \\cite{DBLP:conf\/interspeech\/BapnaTHH17,Lee2019ZeroShotAT}. Assuming that accurate slot descriptions are provided, slots with \\textit{different} names  although semantically similar would have similar description as well. Thus, having trained a model for the \\textsc{Destination} slot with its descriptions,  it is now possible to recognize the slot \\textsc{Arrival\\_Location} without training on it, but  only supplying the corresponding slot description.  \n\nIn addition to slot description, other zero-shot approaches  explore the use of slot value examples \\cite{shah-etal-2019-robust,DBLP:conf\/sigdial\/GueriniMBM18}. \\newcite{shah-etal-2019-robust} showing that a  combination of a small number of slot values examples  with a slot description performs better than \\cite{DBLP:conf\/interspeech\/BapnaTHH17,Lee2019ZeroShotAT} on the SNIPS dataset. Zero-shot models are typically trained on a per-slot basis (Figure \\ref{fig:transfer_learning} \\textit{Right}), meaning that if we have $N$ slots, then the model will output $N$ predictions, therefore, a merging mechanism is needed in case there are prediction overlaps. In order to alleviate the problem of having multiple predictions, \\newcite{DBLP:conf\/acl\/LiuWXF20} propose a \\textit{coarse-to-fine} approach, in which the model learns the slot entity pattern (coarsely) to identify a particular token is an entity or not. After that, the model performs a single prediction of the slot type (fine) based on the similarity between the feature representation and the slot description. \n\n\\paragraph{Takeaways on scaling to new domains:}\n\\begin{itemize}[noitemsep,nolistsep, leftmargin=*]\n    \\item Both data driven methods, MTL and pre-train fine tuning,  improve performance when data in $\\mathcal{D}_T$ is limited. Both are also flexible, as virtually many tasks from different domains can be plugged into these methods. As the number of domains grow, pre-train and fine tuning is more desirable than MTL. However, fine tuning is more prone to the \\textit{forgetting} problem \\cite{He2019AnalyzingTF} compared to MTL.\n    \\item When the number of domain, $K$, is massive, the pre-train fine tuning approach and  model driven approaches, such as expert based adaptation, are preferable with respect of training time.  \n    \\item When there exists $K$ existing domains and no annotation is available in  $\\mathcal{D}_T$, the choice is zero-shot approaches with the expense of providing meta-information such as slot and intent descriptions.\n    \\item As typically zero-shot models perform prediction on a \\textit{per-slot} basis,  potential disadvantages are model accuracy when there is a prediction overlap and the model can also be computationally inefficient when dealing with many slots.\n\\end{itemize}\n\\section{State of the Art and Beyond}\n\\label{sec:discussion}\n\nBased on the results in Table \\ref{tab:independent_models} and \\ref{tab:joint_models}, it is evident that  neural models have achieved outstanding performance on ATIS and SNIPS, showing that it is relatively easy for neural models to capture  patterns that recognize slots and intents. ATIS, in particular, is already overused for SF and IC evaluations and recent analysis \\cite{Bchet2018IsAT,Niu2019RationallyRA} have  shown that the dataset is relatively simple  and the room for performance improvement is tiny. \nA similar trend in performance can be noted for other datasets, such as SNIPS, and it is likely that performance improvement can be quickly saturated. However, it does not mean these models have solved SF and IC, or NLU problems in general,  rather that the model has merely solved the datasets. \nNevertheless, there are still a number of issues in  SF and IC that  need  further investigation:\n\\paragraph{Portable and Data Efficient Models.} Instead of  evaluating  models with the typical \\textit{leaderboard} setup with fixed (train\/dev\/test) splits on a specific target domain, it would be also important to test  models in different scenarios, so that  different aspects of the model can be captured. For example, as neural models are data hungry, more work is still needed on transfer learning scenarios, where evaluation  is carried out with \\textit{less} or \\textit{without} labeled data (\\textit{zero-shot}) for a particular target domain. In addition, most models for SF and IC are evaluated on English, which means that more effort is still needed to make models that work well for other languages. Some recent works have started exploring zero-shot cross lingual methods \\cite{DBLP:conf\/ijcai\/QinN0C20,DBLP:conf\/aaai\/LiuWLXF20,DBLP:conf\/emnlp\/LiuSXWXMF19} and also few-shot scenarios \\cite{DBLP:conf\/acl\/HouCLZLLL20} and the room for improvement for these scenarios is still large. In short, designing a \\textit{data efficient} model that is \\textit{portable} across domains and languages is still a challenging problem for the coming future.\n    \\paragraph{Leveraging unlabeled data from live traffic.} In real situations, personal digital assistants such as Google Home, Apple Siri and Amazon Alexa, receive live traffic data from real users. This large amount of unlabeled data from live traffic is a potential data source for model training, in addition to  in-house annotated data.  Unlabeled live data are likely different from in-house data, as they can contain more diverse utterances and also noisy and irrelevant utterances. In this situation, existing methods to tap on unlabeled data, such as semi-supervised learning, still face unique challenges to  handle live data.\n    It is worth to note that a bottleneck in this direction is that working on live data in academic settings is not trivial. Some recent works explore this line of research by applying semi-supervised learning \\cite{DBLP:conf\/asru\/ChoXLKC19} and also data selection \\cite{DBLP:conf\/emnlp\/DoG19} mechanism.\n    \\paragraph{Generative Models.} Most of the proposed models are \\textit{discriminative}, among the few works  carried out for \\textit{generative models} for SF and IC, \\cite{Raymond2007GenerativeAD,Yogatama2017GenerativeAD} have shown that a generative model is relatively better than a discriminative model in a situation where data is \\textit{scarce}. One possible direction for generative models is to apply data augmentation to automatically create additional training data \\cite{DBLP:conf\/aaai\/YooSL19,DBLP:conf\/emnlp\/ZhaoZY19,DBLP:conf\/coling\/HouLCL18,DBLP:conf\/emnlp\/KurataXZY16,DBLP:journals\/corr\/abs-2004-13952,DBLP:conf\/naacl\/KimRK19}. The main challenge for data augmentation is to generate diverse and fluent synthetic utterance, which \\textit{preserve} the semantics of the original utterance. \n    \n    \\paragraph{Evaluation of SF and IC on more complex dataset.} Existing neural approaches typically evaluated on \\textit{single-intent} utterance, however in a real-world scenario users may indicate \\textit{multiple-intent} in an utterance e.g. \"\\textit{Show me all flights from Atlanta to London and get the cost}\" \\cite{Gangadharaiah2019JointMI} or even expressing multiple sentences in one single turn. While most datasets for slot filling and intent classification are \\textit{single-turn} utterance, there are some recent multi-turn datasets that provide slot annotation on the token-level, namely the \\textsc{Restaurant-8K}, TaskMaster-1 and 2 \\cite{48484}, and Frame \\cite{DBLP:conf\/sigdial\/AsriSSZHFMS17} datasets. The subset of Schema Guided Dialogue (SGD) dataset \\cite{DBLP:conf\/aaai\/RastogiZSGK20} used in DTSC-8 is also annotated with slots in the token-level and covers 16 domains. In addition to that, the TOP dataset \\cite{DBLP:conf\/emnlp\/GuptaSMKL18} introduces datasets annotated with \\textit{hierarchical} representation and MTOP dataset \\cite{DBLP:journals\/corr\/abs-2008-09335} provides both flat and hierarchical representation on 6 languages across 11 domains. \n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nWe have surveyed recent neural-based models applied to SF and IC in the context of task-oriented dialogue systems. We examined three approaches, i.e.  \\textit{independent}, \\textit{joint}, and \\textit{transfer learning based} models. Joint models  exploiting the relation between SF and IC simultaneously shown relatively better performance than independent models. Empirical results have shown that most joint  models nearly \"solve\"  widely used datasets,  ATIS and SNIPS, given \\textit{sufficient in-domain training data}. Nevertheless, there are still several challenges related to SF and IC, especially  improving the scalability of the model to new domains and languages when limited labeled data are available.   \n\n\n\n\\bibliographystyle{coling}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro}\n\nDynamical systems with fast-oscillating conditions are ubiquitous in physics: they  naturally arise in mechanics,\nastrophysics, fluid and air dynamics, and many other domains. They often exhibit surprising  properties, a\nbeautiful example of which is the inverted pendulum, which stabilizes via\nfast vibration of its pivot. The standard way of analyzing such equations \nincludes a procedure of constructing an averaged\nsystem, whose solutions remain close to those of the original system for very long time \n(see e.g. \\cite{kapi, land, ArnMM, arko}).\n\nIn many examples of Hamiltonian  one-frequency oscillating systems\none obtains an additional term, a drift in the averaged  equation. A similar drift (or shift) is observed\nin  hydrodynamical-type systems, including the $\\beta$-plane\nequation in meteorology (see e.g. \\cite{zeit}), infinite-conductivity equation for \nelectron flows \\cite{khch}, and the Craik-Leibovich equation for an ideal\nfluid confined to a domain with oscillating boundary \\cite{crle, vla}.\nIn those hydrodynamical systems\nsuch a shift is often related to the consideration of a central extension of an appropriate Lie algebra \\cite{viz}.\n\nBelow we explain this phenomenon by  building a general connection between the averaging method and symplectic \nreduction in appropriate, possibly nontrivial, $S^1$-bundles. Namely, one starts with\n symplectic reduction of the cotangent bundle over a circle action, which is one of the most studied objects\nin symplectic geometry.\nOne observes two features for the reduction over a nonzero value of the momentum map: the appearance\nof a twisted symplectic structure (similar to how the curvature \narises in the description of gyroscopes on surfaces \\cite{cole2}), \nwhere  a new magnetic term supplements the canonical symplectic form of the reduced space,\nand the appearance of an amended potential function, see Section \\ref{sec:reduction}. It turns out that exactly these two\n phenomena occur in the averaging procedure.  This can be summarized in the following statement (which is a combined version of Theorems \\ref{thm:ave-red} and \\ref{thm:Ave_natHamiltonian}):\n\n\n\n\n\\begin{figure}[H]\n\\tikzstyle{block} = [rectangle, draw, fill=blue!10,\n    text width=6em, text centered, rounded corners, minimum height=6em]\n\\tikzstyle{line} = [draw, -latex']\n\n\\centerline{\n\\begin{tikzpicture}[node distance = 5.1cm,auto]\n   \n    \\node [block] (SDE) {SDE (fast time=new space variable)};\n    \\node [block, right of=SDE] (tilSDE) {$\\widetilde{SDE}$ (fibre-constant objects)};\n    \\node [block, below of=SDE] (aveDE) {$\\overline{DE}_1$ average equation (over fast time)};\n    \\node [block, left of=aveDE] (DE) {DE with fast oscillation (fast time $\\tau=\\omega t$)};\n    \\node [block, below of=tilSDE] (redSDE) {$SDE_{red}$ (symplectic structure with magnetic term; effective potential)};\n    \\node [block, right of=redSDE, node distance=5cm] (Euler) {Euler equation on $\\hat{\\mathfrak{g}}^*$(central extension)};\n   \n    \\path [line] (DE) -- node {suspension $\\dot{\\phi}=\\omega$} (SDE) ;\n    \\path [line] (SDE) -- node [align=center]{space \\\\averaging \\\\for $S^1$-action} (tilSDE) ;\n    \\path [line,transform canvas={yshift=0.1em}]  (DE) -- node [align=center,below]{fast time\\\\averaging\\\\procedure}(aveDE);\n    \\path [ line, decorate, decoration=snake] (SDE) -- node [align=right,left] {Poincar\\'e \\\\approximation\\\\theorem} (aveDE);\n    \\path [line] (tilSDE) -- node [align=right,left]{symplectic reduction  \\\\for a non-zero value \\\\of momentum map }(redSDE);\n    \\path [line,dashed] (tilSDE) -- node [align=right,xshift=-1.5em,yshift=-2em]{configuration space for \\\\$\\widetilde{SDE}$ is the central \\\\extension group $\\widehat G$, \\\\ reduction to $\\hat{\\mathfrak{g}}^*$ }(Euler);\n    \\path [line,dashed] (redSDE) -- node [align=center,below]{for group $G$\\\\ as the base,\\\\ Hamiltonian \\\\~ reduction to $\\hat{\\mathfrak{g}}^*$ }(Euler);\n    \\draw[implies-implies, double distance=0.5em](aveDE) -- (redSDE);\n   \n   \n\\end{tikzpicture}\n}\n\\caption{DIAGRAM}\t\n\\label{fig:diagram}\n\\end{figure}\n\n\\newpage \n\n\\begin{theorem}\nFor a natural slow-fast Hamiltonian system  the resulting slow (averaged) system coincides with the one\nobtained by space averaging over the fibers of an appropriate $S^1$-bundle and performing the symplectic reduction of the corresponding cotangent bundle over $S^1$-action at the momentum value related to the \nfast frequency. The averaged system turns out to be a natural Hamiltonian system with an amended potential function with respect to a twisted (magnetic) symplectic structure.\n\\end{theorem}\n\nFurthermore, central extensions appear whenever the base of the reduction turns out to be a group by itself, as\ndiscussed in Section \\ref{sect:cenExtention}. This can be regarded as a manifestation of the reduction by stages\ndeveloped in \\cite{mars2}.\nThe second main result of the paper is the following abbreviated version of Theorem \\ref{thm:group-ext}:\n\n\\begin{theorem}\nIf the slow manifold is a group $G$ and the perturbed Hamiltonian system is invariant relative to the $G$-action,\nthen the second reduction of such \na fast oscillating system gives an Euler equation, Hamiltonian with respect to  the\nPoisson-Lie bracket on a central extension $\\widehat{\\mathfrak g}$ of the corresponding Lie algebra $\\mathfrak g$.\n\\end{theorem}\n\nThe essence of the paper is described in the diagram in Figure \\ref{fig:diagram}: we show how to view the fast time averaging approximation on the left by going via the averaging on the top and reduction in the right column of\nthe diagram. We describe this averaging-reduction procedure in Section \\ref{sect:averaging}, and compare its result with the one obtained by using the classical fast-time averaging method in  Section \\ref{sec:ODEaveraging}.\n\n\\smallskip\n\nIn Section \\ref{sect:applications} we describe   three examples by using the averaging-reduction procedure developed in this paper: the vibrating pendulum manifests the appearance of the\namended potential, the Craik-Leibovich equation for oscillating boundary is related to the magnetic term in\nthe symplectic structure and a central extension, while  the motion of particles\nin rapidly oscillating potentials has both  magnetic term and additional potential\npresent upon averaging.\n(Note that, instead of the classical approach of applying ingenious canonical transformations  \\cite{cole},\nthe present paper gives an alternative method of averaging natural Hamiltonian systems: one can average the metric, which contains all relevant information, and then obtain the averaged natural system directly\nfrom that  metric.)\n\n\\smallskip\n\nWhile the symplectic reduction  part of this paper is also valid for high-dimensional torus action, i.e. for\nmany-frequency case,   the approximation theorem does not work in this generality, as\nfor several frequencies resonances can appear unavoidably in such systems, \nas e.g. KAM theory manifests.\nNote also that in many examples  the two contributions appearing in averaging, the magnetic term \nand the potential amendment, are of different order in the small parameter of perturbation. \nIt would be interesting to see if it is always the case. \n\n\n\\bigskip\n\n{\\bf Acknowledgments.}\nWe are grateful to Mark Levi and Anatoly Neishtadt for stimulating discussions. A part of this work was done when C.Y. was visiting the Fields Institute in Toronto and B.K. was visiting the Weizmann Institute in Rehovot\nand the IHES in Bures-sur-Yvette. The work was partially supported by an NSERC research grant.\n\n\n\n\n\\medskip\n\n\\section{Symplectic reduction of cotangent bundles}\\label{sec:reduction}\n\nWe start by recalling  (following \\cite{mars2}) general results on the symplectic\n reduction. Consider an action of  an abelian group $\\mathbb T:=S^1$\n(or more generally, a torus $\\mathbb T=T^k$)\ncommon in averaging, while the results   with appropriate amendments hold for a reduction by any Lie group.\nAssume that the group $\\mathbb T$ acts on a configuration space $Q$ (from the right) properly and freely,\nso that the quotient space $Q\/\\mathbb T$ is a manifold.\nOur first goal is to reduce $Q$ by the $\\mathbb T$-action and describe structures on the reduced phase space.\nThe quotient\nprojection $\\pi: Q \\rightarrow B:=Q\/\\mathbb T$ defines a  principal fiber bundle over the base $B$.\nIt turns out that the curvature of this $\\mathbb T$-bundle enters the symplectic structure of the reduced manifold.\nThe gyroscope example below can be regarded as an illustration of the abstract reduction procedure.\n\\smallskip\n\nNamely, the group $\\mathbb T$ acts on $T^* Q$ by cotangent lifts, and we denote the momentum map\n of this action by $J: T^* Q\\rightarrow \\mathfrak{t}^*$.\n The momentum map is a natural  projection \n of  $T_q^* Q$  at any $q\\in Q$ to the  cotangent space to the fiber, $\\mathfrak{t}^*$.\nFor  an arbitrary value  $\\mu\\in \\mathfrak{t}^*$ of the momentum map consider the reduced phase space\\footnote{For an arbitrary Lie group the reduced space is defined as\n$J^{-1}(\\mu)\/\\mathbb T_\\mu$ where $\\mathbb T_\\mu$ is the stationary subgroup of $\\mu$.\nIn this section we use the fact that $\\mathbb T$ is abelian, and hence  the stationary group $\\mathbb T_\\mu$ coincides with the full group: $\\mathbb T_\\mu = \\mathbb T$.}\n$(T^*Q)_\\mu := J^{-1}(\\mu)\/\\mathbb T$.\n\n\n\\begin{theorem} \\label{thm:cotbundle}{\\rm (see e.g. \\cite{mars2})}\n\tLet $\\mathbb T$ be an abelian group acting on a manifold $Q$ so that\n\t$\\pi: Q \\rightarrow Q\/\\mathbb T=:B$ is a principal fiber bundle, and  fix $\\mu \\in \\mathfrak{t}^*$.\n\tLet $\\mathcal{A}: TQ \\rightarrow \\mathfrak{t}$ be a principle connection 1-form on this bundle.\n\tThen\n\t\n\t $i)$ for $\\mu=0$ there is a symplectic diffeomorphism between $(T^* Q)_0$ and\n\t$T^*B=T^* (Q\/\\mathbb T)$ equipped with the canonical symplectic form \t$\\omega_{can}$;\n\t\n\t$ii)$ for $\\mu\\not=0$ there is a symplectic diffeomorphism between $(T^* Q)_\\mu$ and\n\t$T^*B$, where the latter is\n\tequipped with symplectic form $\\omega_\\mu:=\\omega_{can} - \\beta_\\mu$. \tHere  the 2-form\n\t$\\beta_\\mu := \\pi_P^* \\sigma_\\mu$ on $T^*B$ is obtained by the pull-back via the cotangent bundle\n\tprojection $\\pi_P: T^*B \\rightarrow B$ from the 2-form $\\sigma_\\mu$ on $B$. The latter 2-form   is\n\tthe $\\mu$-component of the curvature of the principal fiber bundle $Q$ over $B$, namely\n\t$\\pi^* \\sigma_\\mu = \\mathbf{d}\\left\\langle \\mu, \\mathcal{A}\\right\\rangle.$\n\\end{theorem}\n\n\\begin{proof}[\\textrm{\\textit{Proof outline}}]\nWe just recall an explicit form of the isomorphism between $(T^* Q)_\\mu$ and $T^* (Q\/\\mathbb T)$, see Theorem~2.3.3 in \\cite{mars2} for more detail.\n\nThe isomorphism $\\varphi_0: (T^* Q)_0 \\rightarrow T^* (Q\/\\mathbb T)$ is defined by noting that\n\\[\nJ^{-1}(0) = \\{  p_q \\in T^* Q: \\left< p_q , \\xi_Q(q) \\right> = 0 \\quad \\text{for all $\\xi \\in \\mathfrak{t}$} \\}\\,,\n\\]\nwhere $\\xi_Q$ is the vector field on $Q$ corresponding to the infinitesimal action $\\xi$, i.e. vectors $\\xi_Q(q)$ span the vertical subspace at $q$.\nThus the map $\\Phi : J^{-1}(0) \\rightarrow T^*(Q\/\\mathbb T)$ given by\n\\begin{equation} \\label{barphi}\n\t\\left< \\Phi(p_q) , \\pi_*(v_q) \\right> = \\left<p_q, v_q \\right>\n\\end{equation}\nis well defined.  The map $\\Phi$ is $\\mathbb T$-invariant and surjective, and hence induces a quotient map $\\varphi_0: (T^* Q)_0 \\rightarrow T^* (Q\/\\mathbb T)$.\n\nThe isomorphism $\\varphi_\\mu : (T^* Q)_\\mu \\rightarrow T^* (Q\/\\mathbb T)$ is the composition\n$\\varphi_\\mu = \\varphi_0 \\circ {\\mathrm{shift}}_\\mu$ of $\\varphi_0$ with the \n isomorphism  ${\\mathrm{shift}}_\\mu : (T^* Q)_\\mu  \\rightarrow (T^* Q)_0$  defined as follows.\n Introduce a map ${\\mathrm{Shift}}_\\mu : J^{-1}(\\mu) \\rightarrow J^{-1}(0)$ by\n\\[\n\t{\\mathrm{Shift}}_\\mu(p_q) = p_q - \\left< \\mu, \\mathcal{A}(q) \\right>\n\\]\nfor any $p_q\\in  J^{-1}(\\mu)$. It  is $\\mathbb T$-invariant,  so  it drops to a quotient map ${\\mathrm{shift}}_\\mu: (T^* Q)_\\mu \\rightarrow (T^* Q)_0$.\nThe $\\mathfrak{t}$-valued\n2-form $\\mathbf{d} \\mathcal{A}$ is the curvature of the (abelian) connection $\\mathcal{A}$, while to construct the 2-form  $\\sigma_\\mu$\none uses its $\\mu$-component, cf.  \\cite{mars2}.\n\\end{proof}\n\n\\begin{remark}\nThe isomorphism between $(T^* Q)_\\mu$ and $T^*B=T^* (Q\/\\mathbb T)$ is connection-dependent.\nThe reduced symplectic form on $T^*B$ is modified by the curvature 2-form $\\sigma_\\mu$ on $B$, which is traditionally called a {\\it magnetic term}, since it also appears in the description of  motion of a charged particle in a magnetic field on $B$.\n\\end{remark}\n\n\n\n\\begin{definition}\\label{def:mech-conn}\nLet the space $Q$ be  equipped with a $\\mathbb T$-invariant metric. This metric\ndefines an invariant distribution of horizontal spaces: at each point $q\\in Q$  there is a subspace of $T_qQ$ orthogonal to the fiber (i.e. the $\\mathbb T$-orbit) at $q$. Hence the metric  defines an invariant  connection 1-form\n$\\mathcal{A}: TQ \\rightarrow \\mathfrak{t}$ on this fiber bundle. This 1-form is called a {\\it mechanical connection.}\n\n\\end{definition}\n\nConsider  a natural system on $T^*Q$ with Hamiltonian\n$H(q,p)=(1\/2) (p,\\,p)_q+U(q)$\ninvariant with respect to the $\\mathbb T$-action. (Here and below  $(.\\,,.)_q$ stands for the\nmetric on $Q$, i.e. the inner product \non $TQ$, or the induced one on  $T^*_qQ$, depending on the context. The Euclidean inner product in $\\mathbb R^n$ is denoted by dot.) \nThis system descends to a Hamiltonian system\non the quotient $(T^* Q)_\\mu$ with respect to the symplectic structure  $\\omega_\\mu=\\omega_{can} - \\beta_\\mu$.\nThe new Hamiltonian $H_\\mu$ is obtained from $H$ by applying the map\n${\\mathrm{Shift}}_\\mu$ and the corresponding potential $U(q)$ acquires an additional term, as we discuss below.\n\n\n\n\n\\begin{example}\\label{ex:spinDisk}\nThe following example of a spinning disk (a gyroscope) on a curved surface shed the light on the geometry behind the symplectic reduction above. \n Cox and Levi proved in \\cite{cole2}  that the motion of the disk center coincides with the motion of a charged particle in a magnetic field which is normal to the surface and  equal in magnitude to the Gaussian curvature of the surface. \n \nTo explain their result in the context of  reduction theory,  let $q=(q_1,q_2)$ be orthogonal local coordinates on a surface \n$B\\subset \\mathbb R^3$, so that  metric on the surface is given by $ds^2=a_{11}(q)dq_1^2+a_{22}(q)dq_2^2$. \nWhen the disk is not spinning, its kinetic energy is a function of its position $q$ and linear velocity $\\dot q$, i.e. a function on $TB$. It  is given by\n$$\nE_0=\\frac m2(G\\dot q, \\dot q)+\\frac{\\mathbb I_d}2 h(\\dot q,\\dot q),\n$$\nwhere $G=diag(a_{11},a_{22})$, $h$ is the second fundamental form of $B$, and $\\mathbb I_d$ is the moment of inertia of the disk along its diameter. Denote by $\\mathbb I_a$ the moment of inertia about the disk axis.\n\n\n\\begin{theorem}\\label{thm:spinDisk}{\\rm \\cite{cole2}}\nFor a spinning disk on a surface $B$ the angular momentum $\\mu=\\mathbb I_a\\omega_a$ of the disk about its axis is  constant and  the disk's center satisfies the following equation \n\\begin{equation}\\label{eq:spinDisk} \n\\frac {d}{dt}\\frac{\\partial E_0}{\\partial \\dot q}-\\frac{\\partial E_0}{\\partial q}=\\sqrt{a_{11}a_{22}}\\,\\mu\\,K(q)\\,\\left[\\begin{array}{cc}0&-1\\\\1&0\\end{array}\\right]\\,\\dot q,\n\\end{equation}\nwhere $K(q)$ is the Gaussian curvature of the surface $B$. \n\\end{theorem}\n\n\\begin{remark}\nBefore we provide a different proof of this result via symplectic reduction, note that the configuration space $Q$ of this system is the circle bundle over the surface: the disk position is defined by the position of its center on the surface $B$ \nand the angle of rotation.\nGlobally $Q$ may be a nontrivial $\\mathbb T$-bundle over $B=Q\/\\mathbb T$. However,  for the local consideration below it suffices to consider the case \n$Q=B\\times \\mathbb T$.  The phase space  is the corresponding tangent bundle $TQ$. \nThe metric in $\\mathbb R^3$ allows one to identify $TQ$ and $T^*Q$, while the disk motion is a Hamiltonian system on the cotangent bundle $T^*Q$.\nThe trajectory of the disk center can be obtained as the symplectic reduction  of the system on $T^*Q$ with respect to the\n$\\mathbb T$-action, as we quotient out the disk rotation. Different angular velocities of the disk lead to different values $\\mu$\nof the momentum map, over which  one takes the quotient. \nAccording to Theorem \\ref{thm:cotbundle} the resulting system is a Hamiltonian system on $T^*B$, with two \namendments. The corresponding symplectic structure after the reduction will be twisted by a magnetic term.\nIn this setting it will be proportional to the curvature $K(q)$ of the surface, which one observes in Equation \\eqref{eq:spinDisk}. \nMoreover, the corresponding Hamiltonian undergoes a shift by $\\mu$. However, in the gyroscope case the shift \nreduces to adding a constant to the Hamiltonian and does not appear in the equations.\\footnote{Jumping ahead, in order to see this \none can use an explicit formula of Theorem \\ref{thm:ave-red}, which gives an additional term $\\frac 12 \\langle \\mu,\\,\\mathbb{I}(q)^{-1}\\mu\\rangle$. It is indeed constant, since in the gyroscope case the inertia operator $\\mathbb I$ does not depend on $q$, while $\\mu$ is a constant angular velocity.}\n\\end{remark}\n\n\\begin{proof}\nLet us identify $TB$ and $T^*B$ by means of the metric. \nFirst note that Equation (\\ref{eq:spinDisk}) is the Euler-Lagrange equation for a Lagrangian system, which can be rewritten as a Hamiltonian system with the  Hamiltonian energy function $E_0$ on the cotangent bundle of the surface $B$ (thanks to the metric identification) with a twist symplectic structure given in local coordinates by\n$$\n\\omega_{\\mu}=\\omega_{can}-\\mu\\,\\sqrt{a_{11}a_{22}}\\,K(q)\\,dq_1\\wedge dq_2\\,,\n$$\nwhere $\\omega_{can}$ is the canonical symplectic structure on $T^*B$.\n\nNext, we show how to obtain this system via symplectic reduction.\nDenote by $\\theta$  the angle between a fixed radius on the disk and the positive direction of the line \n$\\{q_2=const\\}$. This gives us a principal\n$\\mathbb T$-bundle $Q$ with the curved surface $B$ as the base. \n\nThe absolute angular velocity of a spinning disk is $\\Omega_a=\\dot \\theta+A(q)\\dot q$, where $A(q)\\dot q$ is the transferred velocity, and $A(q)=(k_1\\sqrt{a_{11}},k_2\\sqrt{a_{22}})$, where $k_1,\\;k_2$ are the geodesic curvatures of coordinate lines\n$\\{q_1=const\\}$ and $\\{q_2=const\\}$.\n\nSo, in local coordinates, the metric on the principal $\\mathbb T$-bundle $Q$ is given by\n$$\n\\left((\\dot q,\\dot \\theta),(\\dot q,\\dot \\theta)\\right)_{(q,\\theta)}=\\mathbb I_a(\\dot\\theta+A(q)\\dot q)^2+m(G\\dot q,\\dot q)+\\mathbb I_dh(\\dot q,\\dot q).\n$$\nNote that this metric is invariant under the $\\mathbb T$-rotations.\n\nTherefore, the momentum map $J:TQ\\rightarrow \\mathfrak t^*=\\mathbb R$ is $J(q,\\theta;\\dot q,\\dot \\theta)=\\mathbb I_a(\\dot\\theta+A(q)\\dot q)$ and the mechanical connection\n$\\mathcal A:TQ\\rightarrow\\mathfrak t=\\mathbb R$ is  $\\mathcal A=d\\theta+A(q)\\,dq$ (here we again identify $TQ$ and $T^*Q$). By Theorem \\ref{thm:cotbundle}, for a fixed value $\\mu=\\mathbb I_a\\Omega_a$ of\nthe momentum map $J$, the system can be reduced to the (co)tangent bundle of the surface $B$ with the magnetic symplectic structure\n$$\n\\omega_{\\mu}=\\omega_{can}-\\mu\\;d(A(q)dq)=\\omega_{can}-\\mu\\;\\sqrt{a_{11}a_{22}}\\,K(q)\\,dq_1\\wedge dq_2,\n$$\nwhere $K(q)$ is the Gaussian curvature of the surface $B$. \n\nThe energy Lagrangian $E=\\frac 12 \\left((\\dot q,\\dot \\theta),(\\dot q,\\dot \\theta)\\right)_{(q,\\theta)}$ \non $Q$ defines the reduced Hamiltonian on $B$, which  turns out to be $E_0=1\/2(m(G\\dot q, \\dot q)+\\mathbb I_d h(\\dot q,\\dot q))$.  Here we omit the constant term $\\mathbb I_a(\\dot\\theta+A(q)\\dot q)^2\n=\\langle \\Omega_a,\\,\\mathbb{I}_a\\Omega_a\\rangle= \\langle \\mu,\\,\\mathbb{I}_a^{-1}\\mu\\rangle$ in the energy expression, \nsince the value $\\mu$ of the momentum map (i.e. the angular momentum of the disk) is conserved. \n\nThis reduced Hamiltonian system with Hamiltonian function $E_0$ \non the cotangent bundle $(T^*B, \\omega_\\mu)$ of the surface describes the motion of the disk center.\n\\end{proof}\n\n\\end{example}\n\n\n\n\\medskip\n\\section{Averaging-Reduction procedure for a natural system}\\label{sect:averaging}\n\\subsection{Averaging}\nLet $\\pi:Q\\to B$ be a principal $\\mathbb T$-bundle. From now on we assume that $\\mathbb T=S^1$ (and occasionally comment on $\\mathbb T=T^n$). The cotangent lift of $\\mathbb T$-action on $Q$ induces $\\mathbb T$-action on $T^*Q$.\nDenote by $\\rho, \\rho^*$, and $\\rho_*$ the $\\mathbb T$-action on $Q, T^*Q$ and $TQ$, respectively.\nLet $d\\eta$ be the standard Euclidean measure on the group $\\mathbb T$.\n\nConsider a natural Hamiltonian system on the cotangent bundle $T^*Q$:\n\\begin{equation}\\label{eq:nature}\nH(q,p)=\\frac 12 (p,\\,p)_q+U(q).\n\\end{equation}\nHere $Q$ is the configuration space of the motion, we assume that this Hamiltonian system has slow motion on the base  $B$ and fast motion on the fibers isomorphic to $\\mathbb T$. \nThe Hamiltonian function $H(q,p)$ is not necessarily invariant\nunder the $\\mathbb T$-action on $T^*Q$. As the first step one passes to the space $\\mathbb T$-average\n$\\overline{H(q,p)}^{\\mathbb T}$, the $\\mathbb T$-invariant function on $T^*Q$ defined by the following formula:\n$$\n \\overline{H(q,p)}^{\\mathbb T}:=\\frac{1}{\\eta(\\mathbb T)}\\int_{g\\in \\mathbb T} H(\\rho_g^*(q,p))\\;d\\eta(g).\n$$\nFor the natural system \\eqref{eq:nature}, one averages both the kinetic and potential parts of the energy:\n$$\n \\overline{H(q,p)}^{\\mathbb T}=\\frac 12 \\overline{(p,p)}_q^{\\mathbb T}+\\overline{U(q)}^{\\mathbb T},\n$$\nwhere  $\\overline{U(q)}^{\\mathbb T} =\\frac{1}{\\eta({\\mathbb T})}\\int_{\\mathbb T} U(\\rho_g(q))\\,d\\eta(g)$\nand $\\overline{(p,p)}_q^\\mathbb T=\\frac{1}{\\eta({\\mathbb T})}\\int_{\\mathbb T}(\\rho_g^*p,\\rho_g^*p)_{\\rho_{g^{-1}}(q)}\\,d\\eta(g)$ \nis defined via the following averaged metric on $Q$:\n\\begin{definition}\n The \\textit{averaged metric} $\\overline{(\\cdot,\\cdot)}^{\\mathbb T}$ on the principal $\\mathbb T$-bundle $Q$ is given by\n$$\n\\overline{(v,v)}_q^{\\mathbb T}:=\\frac{1}{\\eta(\\mathbb T)}\\int_{\\mathbb T}(\\rho_{g*} v,\\rho_{g*} v)_{\\rho_g(q)}\\,d\\eta(g),\n$$\nfor any $v\\in T_qQ$. This defines a  $\\mathbb T$-invariant metric on $Q$.\n\\end{definition}\n\nNow define the connection on $Q$ corresponding to the averaged metric:\n\n\\begin{definition}\n The \\textit{averaged connection} $\\bar{\\mathcal{A}}\\in\\Omega^1(Q,\\mathfrak t)$ on the principal $\\mathbb T$-bundle $Q$ is the  connection  induced by the averaged metric $ \\overline{(\\,,\\,)}_q^{\\mathbb T}$\nby the invariant distribution of horizontal spaces: at each point $q\\in Q$  there is a subspace of $T_qQ$ orthogonal to the fiber (i.e. the $\\mathbb T$-orbit) at $q$.  \n\nThe connection induced by an invariant averaged metric on $Q$ is  the mechanical connection, according to\nDefinition \\ref{def:mech-conn}.\n\\end{definition}\n\n\n\\begin{remark}\nWe would like to give a more explicit description of averaged metrics and connections.\nFirst note that a  $\\mathbb T$-invariant metric $(.\\,,.)_q$ on $Q$ can be defined by means of a metric operator\n$\\mathbb I_Q(q):T_qQ\\to T^*_qQ$ for $q\\in Q$, where $\\mathbb I_Q(q): v\\mapsto v^\\flat$, i.e. \n$(v,v)_q:=\\langle v, \\mathbb I_Q (q)v\\rangle$ for $v\\in T_qQ$.\nThis defines  the ``fiber inertia operator\" $\\mathbb I(q):\\mathfrak t\\to \\mathfrak t^*$ by restricting to\n$\\mathfrak t=T_q\\mathbb T\\subset T_qQ$ the metric operator $\\mathbb I_Q(q)$\nfor $q\\in Q$. (Recall, that for $\\mathbb T=S^1$, we have $\\mathfrak t=\\mathbb R$.) The $\\mathbb T$-invariance of metric implies that the fiber inertia operator $\\mathbb I$ is equivariant, $\\mathbb I(g(q))=Ad_{g^{-1}}^*\\mathbb I(q)$, i.e. it depends on the base point $\\pi(q)\\in B=Q\/\\mathbb T$ only.\n\n\\smallskip\nThe invariant metric on $TQ$ also induces the momentum map $J:T^*Q\\to \\mathfrak t^*$ for the action of the group $\\mathbb T$.\nIn these terms the averaged mechanical connection can be defined explicitly by\n$$\n\\bar{\\mathcal{A}}(v_q)=\\mathbb{I}(q)^{-1}J(p_q),\n$$\nwhere $v_q$ is a tangent vector in $T_qQ$, \\, $p_q:=\\mathbb I_Q(q)v_q=v_q^\\flat\\in T_q^*Q$ is the corresponding metric-dual cotangent vector, and $\\mathbb I$ is the inertia operator on $\\mathfrak t$ in the fiber at $q$.\n\\end{remark}\n\n\\begin{remark}\\label{rem:metric}\nMore specifically, in coordinates for a trivial bundle $Q$  the general form for a $\\mathbb T$-invariant metric on $Q=B\\times \\mathbb T$ is as follows:\n\\begin{equation}\\label{eq:aveMet}\n((u,\\gamma),(u,\\gamma))_{(x,\\tau)}=(u,u)_x+2\\gamma\\, h(x)\\,\\langle A(x),u\\rangle+h(x)\\gamma^2,\n\\end{equation}\nwhere $(u,\\gamma)\\in T_{(x,\\tau)}(B\\times \\mathbb T)=T_x B\\times\\mathfrak t$, $A(x)\\in \\Omega^1(B,\\mathfrak t)=T_x^*B$,   $h(x)\\in \\mathbb R^{+}$, and $\\mathfrak t\\simeq \\mathbb R$.\nFor a non-trivial $Q$ this general form is valid locally on the base.\n\\end{remark}\n\n\\begin{proposition}\\label{prop:momentum}\nFor a trivial bundle $Q=B\\times\\mathbb T$ the averaged connection \n$\n\\bar{\\mathcal{A}}\\in\\Omega^1(B\\times\\mathbb T,\\mathfrak t)= T^*_{(x,\\tau)}(B\\times\\mathbb T)\n$ \ncorresponding to the averaged metric (\\ref{eq:aveMet}) is given by $\\bar{\\mathcal{A}}(x,\\tau)=A(x)+d\\tau$.\nThe summands can be regarded as connections on the base $A(x)\\in T^*_xB$   and in the fiber $d\\tau$.\n\\end{proposition}\n\n\\begin{proof}\nFor a trivial bundle $Q$  the momentum map $J:T_{(x,\\tau)}^*(B\\times \\mathbb T)\\to \\mathfrak t^*$ is given by \n$$\nJ_{(x,\\tau)}(a,\\eta)=h(x)\\langle A(x),u\\rangle+h(x)\\langle d\\tau, \\xi\\rangle,\n$$\nwhere $(a,\\eta)\\in T_{(x,\\tau)}^*(B\\times \\mathbb T)$ and $(u,\\xi)\\in T_{(x,\\tau)}(B\\times \\mathbb T)$ is the image \nof $(a,\\eta)$ under the metric identification. Indeed, by  definition of the momentum map, \nfor any $\\zeta\\in\\mathfrak t$, \n$$\n\\langle J_{(x,\\tau)}(a,\\eta),\\zeta\\rangle=\\langle (a,\\eta),(0,\\zeta)\\rangle=((u,\\xi),(0,\\zeta))\n$$\n$$\n=(u,0)_x+\\zeta h(x)\\langle A(x),u\\rangle+\\xi h(x)\\langle A(x),0\\rangle+\\xi h(x)\\zeta=\\langle h(x)\\langle A(x),u\\rangle+h(x)\\xi,\\zeta\\rangle.\n$$  \nFurthermore, the inertia operator $\\mathbb I(x):\\mathfrak t\\to \\mathfrak t^*$ at $x\\in B$ is given by\n$\\mathbb I(x)\\gamma=h(x)\\gamma $ for any $\\gamma\\in\\mathfrak t,$ hence \nthe average mechanical connection assumes the form\n$\\bar{\\mathcal{A}}(u,\\xi)=\\mathbb{I}(x)^{-1}J(a, \\eta)=\\langle A(x),u\\rangle+\\langle d\\tau, \\xi\\rangle$, as required.\n\\end{proof}\n\n\\smallskip\n\n\n\\subsection{Reduction}\nBy considering the $\\mathbb T$-invariant metric and Hamiltonian (obtained by $\\mathbb T$-averaging)\nwe are now in the framework of Section \\ref{sec:reduction}.\nThe dynamics defined by the averaged Hamiltonian $\\overline{H}^{\\mathbb T}$\non $T^*Q$ can be derived from the corresponding \\textit{averaged} or \\textit{slow motion}, i.e. \nthe dynamics on $T^*B$ of the base space $B=Q\/\\mathbb T$. However, unlike the standard averaging  \nmethod discussed below in Section \\ref{subsec:averHam}, now we obtain this slow motion via symplectic reduction.\n\nRecall that, for a fixed value $\\mu$ of the momentum map,  the reduced space $J^{-1}(\\mu)\/\\mathbb T$ is symplectomorphic to the cotangent bundle $T^*B$ of the base space $B$ with the twisted symplectic form \n $\\omega_\\mu=\\omega_{can}-\\beta_\\mu,$\nwhere $\\omega_{can}$ and $\\beta_\\mu$ are the canonical and magnetic 2-forms\non $T^*B$ (see Theorem \\ref{thm:cotbundle}).\nThe averaged\/slow system turns out to be a Hamiltonian system on the symplectic manifold $(T^*B,\\omega_\\mu)$ with the following reduced Hamiltonian function $\\bar H_\\mu$.\n\n\\begin{theorem}\\label{thm:ave-red}\nFor a natural system on a $\\mathbb T$-bundle $Q$ over the slow manifold $B$\nwith Hamiltonian function $H(q,p)=(1\/2) (p,\\,p)_q+U(q)$\nthe result of the symplectic reduction\nwith respect to the $\\mathbb T$-action of the averaged system is a natural system\nwith the Hamiltonian function $\\bar H_\\mu$,\n\\begin{equation}\\label{aveHam}\n\\bar H_\\mu(q,p)=\\frac 12 (p,\\,p)_B+U_\\mu(q)\\,,\n\\end{equation}\n on the symplectic manifold $(T^*B,\\omega_\\mu)$.\nHere $(q,p)\\in T^*B$, $(\\cdot,\\cdot)_B$ stands for the metric on the base $B=Q\/\\mathbb T$ obtained as a Riemannian\nsubmersion $Q\\to B$ from the metric \n$\\overline{(\\cdot,\\cdot)}^{\\mathbb T}$ on $Q$, while \n$U_\\mu(q):=\\overline{ U(q)}^{\\mathbb T}+\\frac 12 \\langle \\mu,\\,\\mathbb{I}(q)^{-1}\\mu\\rangle$ is the effective potential.\n\\end{theorem}\n\n\\begin{proof}\nWe start by computing the result of averaging and consequent symplectic reduction  on $T^*Q$ with respect to the $\\mathbb T$-action.\nUpon averaging along $\\mathbb T$-orbits one can assume that the Hamiltonian $\\bar H$ on $Q$ is $\\mathbb T$-invariant,\n$\\bar{H}(q,p)=\\overline{ H(q,p)}^{\\mathbb T}$.\nThe reduced Hamiltonian system on the quotient $(T^* Q)_\\mu$ is Hamiltonian with respect to the symplectic structure  \n$\\omega_\\mu=\\omega_{can} - \\beta_\\mu$. The new Hamiltonian is obtained from $\\bar H$ by applying the map\n${\\mathrm{Shift}}_\\mu$.\nNamely, abusing the notation, for $(q,p)\\in T^*B$  and a connection  $\\bar{\\mathcal{A}}$ in the $\\mathbb T$-bundle $Q$ one has\n$$\n\\bar H_\\mu(q,p)=\\bar H (q,\\,p+  \\langle \\mu,\\bar{\\mathcal{A}}(q)  \\rangle)\n=\\frac 12\\overline{( p+  \\langle \\mu, \\bar{\\mathcal{A}}(q)  \\rangle, \\, p+  \\langle \\mu,\\bar{\\mathcal{A}}(q)\\rangle )}_q^{\\mathbb T}+\\overline{U(q)}^{\\mathbb T}\n$$\n$$\n=\\frac 12 (p,\\,p)_B+\\overline{(p,\\,  \\langle \\mu, \\bar{\\mathcal{A}}(q)  \\rangle)}_q^{\\mathbb T}\n+ \\frac 12\\overline{(\\langle \\mu, \\bar{\\mathcal{A}}(q)\\rangle,\\,\\langle \\mu, \\bar{\\mathcal{A}}(q)\\rangle)}_q^{\\mathbb T}+\\overline{U(q)}^{\\mathbb T}\n=\\frac 12 (p,\\,p)_B+\\overline{U_\\mu(q)}^{\\mathbb T}\n$$\nfor $U_\\mu(q):=\\frac 12 \\overline{(\\langle \\mu, \\bar{\\mathcal{A}}(q)\\rangle,\\,\\langle \\mu, \\bar{\\mathcal{A}}(q)\\rangle)}_q^{\\mathbb T}+\\overline{U(q)}^{\\mathbb T}$.\nHere we use that $\\bar{\\mathcal{A}}$ is the mechanical connection corresponding to the averaged metric $\\overline{(\\cdot,\\cdot)_q}^{\\mathbb T}$, and hence we have\n$\\overline{(p,\\,  \\langle \\mu, \\bar{\\mathcal{A}}(q)  \\rangle)}_q^{\\mathbb T}=\\langle \\mu, \\, \\bar{\\mathcal{A}}(q)(v)\\rangle=\\langle \\mu,\\,\\mathbb I(q)^{-1}J(p)\\rangle=0$, since $J(p)=0$, and where $(q,p)\\in T^*B$ is identified with  $(q,v)\\in TB$ by means of\nthe averaged metric. Thus on the reduced symplectic manifold\n$T^*B$ with the twisted symplectic form $\\omega_\\mu=\\omega_{can} - \\beta_\\mu$\nthe new reduced Hamiltonian is\n$$\n\\bar H_\\mu(q,p)=\\frac 12 (p,\\,p)_B+U_\\mu(q)\n$$\nfor $q\\in B$ and $p\\in T^*_qB$. It is  a natural system with a new effective potential\n$$\nU_\\mu(q)=\\frac 12 \\overline{(\\langle \\mu, \\bar{\\mathcal{A}}(q)\\rangle,\\,\\langle \\mu, \\bar{\\mathcal{A}}(q)\\rangle)}_q^{\\mathbb T}+\\overline{U(q)}^{\\mathbb T}\n=\\frac 12 \\langle \\mu, \\mathbb I(q)^{-1}\\mu\\rangle+\\overline{U(q)}^{\\mathbb T}\\,.\n$$\n\\end{proof}\n\nIn Section \\ref{sect:Ave_natHamiltonian} below we will prove the following corollary of Theorem \\ref{thm:ave-red} for averaging\none-frequency fast-oscillating systems: Under certain conditions, solutions of the averaged system and projections to slow manifold of solutions of the actual system with the same initial conditions remain $\\epsilon$-close to each other for $0\\le t\\le 1\/\\epsilon$.\n\n\n\n\\begin{remark}\nThe two features of the averaged-reduced Hamiltonian system are the additional term in the effective potential\n$U_\\mu$ and the magnetic term $-\\beta_\\mu$ in the symplectic structure $\\omega_\\mu$.\nTherefore this averaging-reduction procedure provides a geometrical explanation of these two phenomena,\noften observed in the averaging theory.\n\\end{remark}\n\n\n\\begin{remark}\nIn the classical averaging of fast-oscillating systems  (cf. Section \\ref{subsec:averHam} below)\none starts by fixing the action variable $J$. This  \ncan be regarded as a manifestation of symplectic reduction in flat coordinates, as this means fixing \na certain value of the corresponding momentum map.\nThe bundle averaging-reduction procedure described here is also applicable in that case, but the metric\nin this bundle turns out to be flat. Namely, in the reduction to a submanifold $J=\\mu$ \none chooses a flat connection on the principal bundle which corresponds to the direct product of the base\nand fibres, and hence no twisted symplectic structure appears on the reduced manifold: for\nthe momentum value $J=\\mu$, the averaged Hamiltonian function is $\\epsilon \\,\\bar{H}(Q,P,\\mu)$\non the ``flat\" cotangent bundle  $(T^*\\mathbb R^{\\ell}, dP\\wedge dQ)$.\n\\end{remark}\n\n\n\n\\medskip\n\n\\section{Central extensions in symplectic reduction} \\label{sect:cenExtention}\nAbove we described the reduced phase space $(T^* Q)_\\mu$ for the right  action by the group $\\mathbb T$.\nIn this case, the reduced phase space $(T^* Q)_\\mu$ coincides with $T^* (Q\/\\mathbb T)$, equipped with the magnetic symplectic structure $\\omega_\\mu$ described before.\nNow assume in addition that the base space $Q\/\\mathbb T$ has the structure of another Lie group $G$, which\nacts on itself from the left and leaves the metric on $G=Q\/\\mathbb T$ invariant.\n As a result, $G$ acts on $T^* G=T^* (Q\/\\mathbb T)$ and, as one can check,  this action leaves the symplectic structure $\\omega_\\mu=\\omega_{can} - \\beta_\\mu$ invariant.\n (Recall that the magnetic 2-form $\\beta_\\mu := \\pi_G^* \\sigma_\\mu$\n on $T^* G$ is the pullback of the left-invariant 2-form $\\sigma_\\mu$ on the group $G$.)\n Hence another  reduction  for this $G$-action (``the reduction by stages\") would take this magnetic\n symplectic structure on $T^*G$\n to an appropriate structure on the dual Lie algebra $\\mathfrak{g}^*$, as described below.\n\n\n\n\\begin{theorem}{\\rm (Theorem~7.2.1 in \\cite{mars2})}\\label{thm:poisson}\n  The Poisson reduced space for the left  action  of\n  $G$ on $(T^* G, \\omega_\\mu=\\omega_{can} - \\beta_\\mu)$ is the dual Lie algebra $\\mathfrak{g}^*$ with the\n  Poisson bracket given by\n\\begin{equation} \\label{bracket}\n\\{f, g\\}_\\mu(\\nu) = -\\left< \\nu, \\left[ \\frac{\\delta f}{\\delta \\nu},\n    \\frac{\\delta g}{\\delta \\nu} \\right] \\right> - {\\sigma}_\\mu(e)\\left(\n  \\frac{\\delta f}{\\delta \\nu}, \\frac{\\delta g}{\\delta \\nu} \\right)\n\\end{equation}\nfor $f, g \\in C^\\infty(\\mathfrak{g}^*)$ at any $\\nu\\in \\mathfrak{g}^*$, where ${\\sigma}_\\mu(e)$\nis the value of the  left-invariant  2-form ${\\sigma}_\\mu$ at $e\\in G$ on the pair of tangent vectors  $\\frac{\\delta f}{\\delta \\nu}, \\frac{\\delta g}{\\delta \\nu}\\in T_eG=\\mathfrak{g}$, and $\\beta_\\mu := \\pi_G^* \\sigma_\\mu$ is the pullback\nof ${\\sigma}_\\mu$ to $T^*G$.\n\\end{theorem}\n\n\n\\begin{remark}\nThe above Poisson bracket is the Lie-Poisson bracket of the dual $\\widehat{\\mathfrak{g}}^*$ of the central extension\n$\\widehat{\\mathfrak{g}}$ of the Lie algebra ${\\mathfrak{g}}$ by means of the ${\\mathfrak{t}}$-valued\n2-cocycle $\\sigma$, such that $\\langle \\sigma, \\mu\\rangle :={\\sigma}_\\mu(e)$.\nNamely, the Lie algebra $\\widehat{\\mathfrak{g}}$ is the direct sum ${\\mathfrak{g}}\\oplus {\\mathfrak{t}}$,\nas a vector space, with  the commutator\n$$\n[(u,a), (v,b)]_{\\widehat{\\mathfrak{g}}}:= ([u,v]_{\\mathfrak{g}}, {\\sigma}(u,v))\n$$\nfor  $u,v\\in \\mathfrak{g}$ and $a,b\\in {\\mathfrak{t}}$.\nIt turns out that under certain integrality conditions, the space $Q$ gives a realization of the corresponding\ncentrally extended group $\\widehat G$.\n\\end{remark}\n\nFor the right action of $G$ the bracket changes sign.\n\n\\begin{theorem}\\label{thm:group-ext}\nLet $G$ be a group equipped with a closed integral left-invariant 2-form $\\sigma_\\mu\/2\\pi$. Then the $\\mathbb T$-bundle $Q$\nover the group $G$ with the curvature form $\\sigma_\\mu$ can be canonically identified with the\ncentral extension $\\widehat G$\nof the group $G$ by means of $\\mathbb T$, where the Lie algebra 2-cocycle is ${\\sigma}_\\mu(e)$, i.e.\nits value on a pair of Lie algebra elements $\\xi$ and $\\eta$ is ${\\sigma}_\\mu(e)(\\xi,\\eta)$.\n\\end{theorem}\n\n\\begin{proof} The proof is based on a version of Proposition 4.4.2 of \\cite{segal} adjusted to the setting at hand.\nIn fact, one can explicitly construct $\\widehat G$ and identify it with $Q$, the $\\mathbb T$-bundle over $G$.\nNamely, first for any oriented loop $\\ell$ in $G$  one associates an element\n$C(\\ell)=\\exp (i\\int_{\\partial^{-1}\\ell}\\sigma_\\mu)$, where $\\partial^{-1}\\ell$ is an oriented 2D surface in $G$ bounded\nby $\\ell$. The value $C(\\ell)$ is well-defined, since for two different surfaces with the same boundary\nthe integrals of $\\sigma_\\mu$ for an integral 2-form $\\sigma_\\mu\/2\\pi$ differ by a multiple of $2\\pi$.\n\nThe map $\\ell\\mapsto C(\\ell)$ is independent of parametrization of $\\ell$, additive, and $G$-invariant.\nIt defines a central extension $\\widehat G$ of the group $G$ by $\\mathbb T$ as a set of triples $(g,u, p)$, where\n$g\\in G$, $u\\in \\mathbb T$ and $p$ is a path in $G$ from $e$ to $g$, modulo the following equivalence.\nTwo triples  $(g,u, p)$ and  $(g',u', p')$ are equivalent if $g'=g$ and $u'=C(p'\\cup p^{-1})u$. The composition\nis $(g_1,u_1, p_1)\\circ (g_2,u_2, p_2)= (g_1g_2, u_1u_2, p_1\\cup g_1(p_2)))$.\n\nRecall that the space $Q$ with an invariant metric has a structure of \na $\\mathbb T$-bundle with mechanical connection. Then a triple\n$(g,u, p)$ modulo equivalence can be interpreted as the following point in $Q$: it is the point in the\n$\\mathbb T$-fiber over  $g\\in G$, obtained from the point $(e,u)$ of the $\\mathbb T$-fiber over $e\\in G$\nby a horizontally lifted path $p$ from $e$ to $g$. Then the equivalence of triples stands for their correspondence to the same point in $Q$, since the form $\\sigma_\\mu$ is the curvature of the mechanical connection, while\nthe formula $u'=C(p'\\cup p^{-1})u$ describes the holonomy of the connection over a closed loop.\n\\end{proof}\n\n\\begin{remark}\nTheorems \\ref{thm:poisson} and \\ref{thm:group-ext} can be extended to  \nthe case of a torus $\\mathbb T$-bundle $Q$ over $G$, where $\\mathbb T=T^k$. In Theorem  \\ref{thm:group-ext}\none realizes $Q$ as a group  central extension of $G$ by $\\mathbb T$ by applying the above consideration to \nthe ``coordinate 2-forms\"  $\\sigma_\\mu=\\langle \\sigma, \\mu\\rangle $ of the $\\mathfrak t$-valued 2-form $\\sigma$.\n\\end{remark}\n\n\\begin{remark}\nReturn to the 2-cocycle $\\beta_\\mu$ on the Lie algebra $\\mathfrak g$, which defines\nthe central extension and the magnetic term. In many examples, this 2-cocycle\nis a 2-boundary, i.e. the 2-form $\\sigma_\\mu$ on the Lie algebra can be represented as a linear\nfunctional  of the Lie algebra commutator,  $\\sigma_\\mu(\\xi, \\eta)=L([\\xi,\\eta])$\nfor some element $L\\in \\mathfrak g^*$.\nIn that case, the corresponding Poisson structure on $\\mathfrak g^*$ is the linear Lie-Poisson structure on the dual space $\\mathfrak g^*$ shifted to the point $L$.\nThe associated Euler equation also manifests a certain shift, observed, e.g.\nas a Stokes drift velocity related to surface waves in the  Craik-Leibovich equation, cf. Section \\ref{sect:CL}.\n\\end{remark}\n\n\\begin{remark}\nWhen considering dynamics on the reduced space $T^*G$, in order to use the second reduction\nover the $G$-action one has to confine to the natural systems with effective potential independent of $q$, i.e.\n$U_\\mu(q)=const$. The latter are geodesic flows for the invariant metric on $G$ defined by the inertia operator\n$\\mathbb I_G:\\mathfrak g\\to \\mathfrak g^*$. The second reduction \n defines the Euler equations for the quadratic Hamiltonian\n$H(p):=\\frac 12 (p,p)_e=\\frac 12 \\langle p, \\mathbb I_G^{-1}p\\rangle$\non the dual $\\widehat{\\mathfrak{g}}^*$ of  centrally extended Lie algebra $\\widehat{\\mathfrak{g}}$.\n\\end{remark}\n\n\n\\medskip\n\n\\section{Reminder on averaging and examples}\\label{sec:ODEaveraging}\n\n\\subsection{Averaging in one-frequency Hamiltonian systems}\\label{subsec:averHam}\nConsider a Hamiltonian system with $\\ell+1$ degrees of freedom and Hamiltonian of the form\n$H(q,p, I, \\phi)=H_0(I)+\\epsilon H_1(q,p, I, \\phi)$, where $\\phi ({\\rm mod }\\, 2\\pi)\\in \\mathbb T$, \nwhile $H$ is $2\\pi$-periodic in $\\phi$, and $(q,p, I)\\in D\\subset \\mathbb R^{2\\ell+1}$. (Such perturbations of properly degenerate Hamiltonian  systems are typical in celestial mechanics.)\nThe corresponding Hamiltonian equations for the standard symplectic structure are as  follows:\n\\begin{equation}\\label{eq:Ham_eq}\n\\left\\{\n  \\begin{array}{l}\n        ~ \\dot q = \\quad \\epsilon\\,{\\partial H_1}\/{\\partial p} \\qquad\\quad\n\t\t\\dot I = \\qquad\\qquad  -\\epsilon \\,{\\partial H_1}\/{\\partial \\phi},\\\\\\\n\t\t              \\dot p = -\\epsilon\\,{\\partial H_1}\/{\\partial q}, \n\t\t              \\qquad\\quad\\dot \\phi=\\,{\\partial H_0}\/{\\partial I}+\\epsilon\\,{\\partial H_1}\/{\\partial I}.\n\\end{array} \\right.\n\\end{equation}\n\n\n\\begin{definition}\nThe {\\it averaged system} for the above Hamiltonian $H=H_0+\\epsilon H_1$ is the system of $2\\ell+1$ equations:\n\\begin{equation}\\label{eq:Ave_Ham_eq}\n\\left\\{\n  \\begin{array}{l}\n\t\t\t\t~\\dot Q = \\quad \\epsilon \\,{\\partial \\bar{H}_1}\/{\\partial P}, \\qquad\\dot J = 0,\\\\\\\n        \\dot P =  -\\epsilon \\,{\\partial \\bar{H}_1}\/{\\partial Q},\n\\end{array} \\right.\n\\end{equation}\nwhere $\\bar H_1(Q, P,J):=\\frac{1}{2\\pi}\\int_0^{2\\pi} H_1(Q, P,J, \\phi)\\,d\\phi$.\n\\end{definition}\n\nSince there is no evolution of $J$ in the averaged system, one can  fix it and regard $J$ as a parameter\nfor the Hamiltonian system with $\\ell$ degrees of freedom, where\n$\\bar{H}(Q,P)=\\bar{H}_J(Q,P)=H_0(J)+\\epsilon \\bar{H}_1(Q,P,J)$.\n\\medskip\n\nLet $(q,p, I)$ belong to a domain $D\\subset \\mathbb R^{2\\ell+1}$, and $D_\\delta\\subset D$  stands for\na subdomain whose $\\delta$-neighbourhood belongs to $D$. Assume that the Hamiltonian\n$H$ is $C^3$-bounded for\n$(q, p, I, \\phi) \\in D\\times \\mathbb T$, as well as ${\\partial H_0(I)}\/{\\partial I}>C>0$ in $D$ and\n$(Q(t), P(t), J(t))\\in D_\\delta$ for all $0\\le t\\le 1\/\\epsilon$.\n\n\\begin{theorem}\\label{thm:Ave_Ham}{\\rm (cf. \\cite{ArnMM, arko})}\nFor sufficiently small positive $\\epsilon$ (i.e. $0<\\epsilon<\\epsilon_0$) solutions of the actual system \\eqref{eq:Ham_eq} and averaged system \\eqref{eq:Ave_Ham_eq}\nwith the same initial conditions $(q(0), p(0), I(0))=(Q(0),P(0), J(0))$\nremain $\\epsilon$-close to each other for $0\\le t\\le 1\/\\epsilon$:\n$|I(t)-I(0)|<C_0\\epsilon$ and $|q(t)-Q(t)|+|p(t)-P(t)|<C_0\\epsilon$, where $C_0$ does not depend on $\\epsilon$.\n\\end{theorem}\n\n\nProof of this theorem consists of constructing a canonical transformation $\\epsilon$-close to the identity and mapping the original system to the averaged one modulo $\\epsilon^2$-terms. Then for the time\n$0\\le t\\le 1\/\\epsilon$  solutions of the averaged system remain $\\epsilon$-close to those of\nthe original one, see \\cite{ArnMM}.\n\n\n\n\n\\begin{remark}\nConsider now a fast-oscillating nonautonomous Hamiltonian system with ``$\\ell$ and a half\" degrees of freedom, whose Hamiltonian is $H=H(p,q,\\omega t)$ with  high frequency $\\omega=\\mu\/\\epsilon$. The associated Hamiltonian equations are $\\dot q={\\partial H}\/{\\partial p}$ and $\\dot p= -{\\partial H}\/{\\partial q}$.\nThe fast variable $\\phi=\\omega t$ can be regarded as a new independent space variable by passing to the new autonomous Hamiltonian system for  $\\widetilde H=\\omega I+H(p,q,\\phi)$\nwith $\\ell+1$ degrees of freedom, where variable $I$ is conjugate to $\\phi$.\nCombined with the reparametrization $t\\mapsto \\tau=t\/\\epsilon$ this\nleads to the system of the above type  (where now the upper dot stands for the derivative in the fast time $\\tau$):\n\\begin{equation}\n\\left\\{\n  \\begin{array}{l}\n~ \\dot q=\\quad\\epsilon\\,{\\partial H}\/{\\partial p}\\,, \\quad  \\dot I= -\\epsilon\\,{\\partial H}\/{\\partial \\tau},\\\\\\\n \\dot p= -\\epsilon\\,{\\partial H}\/{\\partial q}\\,, \\quad \\dot \\phi=~ \\mu.\n\\end{array} \\right.\n\\end{equation}\n\\end{remark}\n\n\n\\subsection{Averaging in natural systems}\\label{sect:Ave_natHamiltonian}\nIn this section we study averaging in natural systems on the cotangent bundle of a principle circle bundle $\\pi: Q\\rightarrow B$. \nThe general form of a natural Hamiltonian function on $T^*Q$ is \n$$\nH(x,y)=\\frac 12 (y,\\,y)_x+U(x)\n$$\nfor $(x,y)\\in T^*_xQ$.\n\nWe start by considering natural systems on the cotangent bundle \n$T^*(\\mathbb R^\\ell\\times  \\mathbb T)$ of a direct product with Hamiltonians\n\\begin{equation}\\label{eq:natHamiltonian}\nH(q,\\phi; p,\\gamma)=\\frac 12\\left((p,\\gamma), (p,\\gamma)\\right)_{(q,\\phi)}+U(q,\\phi),\n\\end{equation}\nwhere $(q,\\phi)\\in\\mathbb R^\\ell\\times  \\mathbb T$ and $(p,\\gamma)\\in T_{(q,\\phi)}^*(\\mathbb R^\\ell \\times \\mathbb T)$. \nAssume that the function $H(q,\\phi; p,\\gamma)$ is $2\\pi$-periodic in $\\phi$, \n$U(q,\\phi)=U_0(q)+\\epsilon U_1(q,\\phi),$\n and the metric on $\\mathbb R^\\ell \\times \\mathbb T$ has the form\n\\begin{equation}\\label{eq:metric}\n\\left( (p,\\gamma), (p,\\gamma) \\right)_{(q,\\phi)}=p\\cdot p+2\\gamma a(q,\\phi)\\cdot p+h(q,\\phi)\\,\\gamma^2,\n\\end{equation}\nwhere  dot  stands for the Euclidean inner product and where \nthe corresponding coefficients $a$ and $h$ have the following expansions in $\\epsilon$ as $\\epsilon\\to 0$:\n$$\na(q,\\phi)=a_0(q)+\\epsilon a_1(q,\\phi),\\;\\; h(q,\\phi)=h_0(q)+\\epsilon h_1(q,\\phi),\n$$\nwith  functions $a_1(q,\\phi),\\;h_1(q,\\phi)$ and $U_1(q,\\phi)$ of zero mean with respect to $\\phi$.\n\nThe Hamiltonian equations for this Hamiltonian function $H(q,\\phi; p,\\gamma)$ and \n symplectic structure $\\omega=(1\/\\epsilon)\\,dq\\wedge dp+d\\phi\\wedge d\\gamma$ \non $T^*(\\mathbb R^\\ell\\times \\mathbb T)$ are\n\\begin{equation}\\label{eq:natHamiltonian_eq}\n\\left\\{\n  \\begin{array}{l}\n\t\t\t\t~ \\dot q = \\;\\;\\epsilon \\,(p+\\gamma a(q,\\phi)),\\\\\\\n       \\dot \\phi =  a(q,\\phi)\\cdot p+h(q,\\phi)\\,\\gamma, \\\\\\\n \t\t\t\\dot p =  -\\epsilon\\, \\partial (\\gamma a(q,\\phi)\\cdot p+(1\/2)\\,h(q,\\phi)\\gamma^2 + U(q,\\phi))\/{\\partial q}\\,, \\\\\\\n        \\dot \\gamma = -\\epsilon\\, \\partial (\\gamma a_1(x,\\phi)\\cdot p+(1\/2)\\,h_1(q,\\phi)\\gamma^2 +U_1(q,\\phi))\/{\\partial \\phi}.\n\\end{array} \\right.\n\\end{equation}\nIn these equations $\\phi$ is the fast variable. \nBelow we prove that the averaged Hamiltonian for (\\ref{eq:natHamiltonian}) is\n\\begin{equation}\\label{eq:Ave_natHamiltonian}\n\\bar{H}(Q,P,\\mu)=\\frac 12\\,P\\cdot P+\\mu \\,a_0(Q)\\cdot P+\\frac 12\\,\\mu^2 h_0(Q)+U_0(Q)\\,,\n\\end{equation}\nwhere the part $(1\/2)\\,\\mu^2 h_0(Q)+U_0(Q)$ is related to  an effective potential, while the term $\\mu \\,a_0(Q)\\cdot P$\nlinear in impulses  is related to a magnetic-gyroscopic-like force. Namely, one has the following statement,\nwhich is an adaptation of Theorem \\ref{thm:Ave_Ham} to the system \\eqref{eq:natHamiltonian_eq}.\n\n\\begin{theorem}\\label{thm:Ave_natHamiltonian}\nFor sufficiently small $\\epsilon>0$  solutions for the original Hamiltonian  \\eqref{eq:natHamiltonian} and the averaged Hamiltonian \\eqref{eq:Ave_natHamiltonian}\nwith the same initial conditions $(q(0), p(0), \\gamma(0))=(Q(0),P(0),\\mu(0))$\nremain $\\epsilon$-close to each other for $0\\le t\\le 1\/\\epsilon$:\n$|\\gamma(t)-\\mu(t)|+|q(t)-Q(t)|+|p(t)-P(t)|<C_0\\epsilon$, where $C_0$ does not depend on $\\epsilon$.\n\\end{theorem}\n\\begin{proof}\n The Hamiltonian equations \\eqref{eq:natHamiltonian_eq} can be rewritten in the following form\n\\begin{equation}\\label{eq:natHamiltonian_eq2}\n\\left\\{\n  \\begin{array}{l}\n\t\t\t\t~ \\dot q = \\;\\;\\epsilon \\,(p+\\gamma a_0(q))+\\epsilon^2 \\gamma a_1(q,\\phi)\\,,\\\\\\\n       \\dot \\phi =  a(q,\\phi)\\cdot p+h(q,\\phi)\\,\\gamma, \\\\\\\n \t\t\\dot p =  -\\epsilon\\, \\partial (\\gamma a_0(q)\\cdot p+(1\/2)\\,h_0(q)\\gamma^2 + U_0(q))\/{\\partial q}\\\\\\\n\t~\\qquad\\qquad\\qquad\\qquad \\;\\; -\\epsilon^2 \\partial (\\gamma a_1(q,\\phi)\\cdot p+(1\/2)\\,h_1(q,\\phi)\\gamma^2 + U_1(q,\\phi))\/{\\partial q}\\,, \\\\\\\n        \\dot \\gamma = -\\epsilon\\, \\partial (\\gamma a_1(x,\\phi)\\cdot p+(1\/2)\\,h_1(q,\\phi)\\gamma^2 +U_1(q,\\phi))\/{\\partial \\phi}.\n\\end{array} \\right.\n\\end{equation}\nThese equations differ by $\\epsilon^2$-terms from those for  the averaged Hamiltonian \\eqref{eq:Ave_natHamiltonian} in the symplectic structure $\\Omega=dP\\wedge dQ$:\n\\begin{equation}\\label{eq:Ave_natHamiltonian_eq}\n\\left\\{\n  \\begin{array}{l}\n\t\t\t\t~ \\dot Q = \\epsilon\\, (P+\\gamma a_0(Q))\\,,\\\\\\\n \t\t\\dot P =  -\\epsilon\\, \\partial (\\gamma a_0(Q)\\cdot P+(1\/2)\\,h_0(Q)\\gamma^2 + U_0(q))\/{\\partial Q}\\,,\\\\\\\n       \\dot \\mu = 0,\n\\end{array} \\right.\n\\end{equation}\nso according to Theorem \\ref{thm:Ave_Ham}, we obtain the required proximity of solutions for the original and averaged equations.\n\\end{proof}\n\n\\begin{remark} \\label{rm:Ave_natHamiltonian}\nIn the shifted  coordinates $(Q, P)\\to (Q, P_1:=P+\\mu\\,a_0(Q))$, \nthe averaged Hamiltonian \\eqref{eq:Ave_natHamiltonian} becomes\n\\begin{equation}\\label{eq:Ave_natHamiltonian2}\n\\bar{H}(Q,P_1,\\mu)=\\frac 12\\,P_1\\cdot P_1+\\frac 12\\,\\mu^2 (h_0(Q)-a_0(Q)\\cdot a_0(Q))+U_0(Q),\n\\end{equation}\nwhile the symplectic structure becomes\n$$\n\\Omega=d(PdQ)=d((P_1-\\mu\\,a_0(Q))\\,dQ)=dP_1\\wedge dQ-\\mu\\,d(a_0(Q)\\,dQ).\n$$\nOne notices that  a new (effective) potential  now includes an additional term, \n$$\n\\bar U_{\\mu}(Q):=\\frac 12\\,\\mu^2 (h_0(Q)-a_0(Q)\\cdot a_0(Q))+U_0(Q),\n$$ \nwhile the new symplectic structure acquires the magnetic term $\\mu\\,d(a_0(Q)\\,dQ)$.\n\\end{remark}\n\n\nNow we are ready to prove \n\n\\begin{corollary} \nThe solutions of the reduced Hamiltonian  system and projections to slow manifold of solutions of the actual system with the same initial conditions remain $\\epsilon$-close to each other for $0\\le t\\le 1\/\\epsilon$.\n\\end{corollary}\n\n\n\\begin{proof}\nComparing the result of Theorem \\ref{thm:ave-red} with Remark \\ref{rm:Ave_natHamiltonian} we observe that the Hamiltonian \\eqref{aveHam} in the theorem coincides with the averaged Hamiltonian of the natural system obtained above.\n\nAs a matter of fact, the  consideration of Remark \\ref{rm:Ave_natHamiltonian}\ncan be seen as a local (coordinate) version of the proof of Theorem  \\ref{thm:ave-red}. Indeed, the averaged metric in local coordinates can be expressed as\n$$\n\\overline{((p,\\mu),\\;(p,\\mu))}_{(q,g)}^{\\mathbb T}=p\\cdot p+2\\mu \\,a_0(q)\\cdot p+\\mu^2 h_0(q),\n$$\nwhere $(q,g)\\in B\\times\\mathbb T$ and $(p,\\mu)\\in T_{(q,g)}^*( B\\times\\mathbb T)$,  cf. (\\ref{eq:Ave_natHamiltonian}). Therefore the corresponding mechanical connection $\\bar{\\mathcal{A}} \\in \\Omega^1(B,\\mathbb R)$ in local coordinates is $\\bar{\\mathcal{A}}=a_0(q)\\;dq$. One can see that the symplectic structure and effective potential in Remark \\ref{rm:Ave_natHamiltonian} coincide with those in Theorem~\\ref{thm:ave-red}.\n\nSince one obtains the same averaged system both via the ``local\" proof of Section \\ref{sect:Ave_natHamiltonian} and via the ``global\" proof of Theorem \\ref{thm:ave-red}, then Theorem \\ref{thm:Ave_natHamiltonian} guarantees the required  closeness of averaged  and original solutions.\n\\end{proof}\n\n\n\\begin{remark}\nWhile Theorem \\ref{thm:Ave_natHamiltonian} deals with the topologically trivial \n$\\mathbb T$-bundle $\\mathbb R^\\ell\\times \\mathbb T$,\nthe  result on the existence of an averaged system holds for a topologically nontrivial bundle as well.\nIn the general case of a nontrivial $\\mathbb T$-bundle $\\pi:Q\\to B$ \nwe assume that the Hamiltonian system has fast motions along fibers and slow motions on the base $B$.\nUnder the assumption that  the  oscillatory parts of the Hamiltonian are of order $\\epsilon$, i.e. \n\\begin{equation}\\label{eq:cond}\nU(q)-\\overline{U(q)}^{\\mathbb T}\\sim \\;O(\\epsilon) \\;\\;\\text{and}\\;\\;(p,p)_q-\\overline{(p,p)}_q^{\\mathbb T}\\sim\\; O(\\epsilon),\n\\end{equation}\none can  introduce the averaged kinetic energy\n$(1\/2) \\overline{(p,p)}_q^{\\mathbb T}$ and averaged potential energy $\\overline{U(q)}^{\\mathbb T}$ by\naveraging the system along the fibers, see Section \\ref{sect:averaging}. Then one proves the averaging theorem\nby  considering only the local picture of the principal bundle $\\pi:Q\\to B$,\nsince any bundle is locally trivial. \n\\end{remark}\n\n\n\n\n\n\\subsection{Examples of averaged systems}\\label{subsect:examples}\n\\begin{example}[\\textrm{\\textit{(A pendulum with rapidly oscillating suspension point)}}]\\label{ex:pendulum}\nConsider the motion of a pendulum with vertically vibrating suspension point. Set $\\theta$ to be the angle of\ndeviation of the pendulum from the vertical, $a$ and $\\omega$ are the amplitude and frequency of the\noscillation of the suspension point, $l$ is the length of the pendulum and $g$ is the acceleration of gravity.\nWe assume that the amplitude $\\epsilon a$ is of order $\\epsilon$ and the frequency $\\omega={\\mu}\/{\\epsilon}$ is of order ${1}\/{\\epsilon}$, i.e. the suspension point oscillates with high frequency and small amplitude.\nThe corresponding potential  is $U(\\theta)=-gl \\cos\\theta$ and the \nHamiltonian function is\n$$\nH(\\theta,p,t)=\\frac 12 \\left( \\frac pl-a\\mu \\sin\\omega t \\sin\\theta\\right)^2-gl \\cos\\theta\\,.\n$$\nThis system differs from a natural one because of the shift in $p$, and \nthe standard reasoning goes as follows, see e.g. \\cite{arko}.\n Let $\\phi=\\omega t$ be the fast variable. In order to get rid of the $\\phi$-dependence in the Hamiltonian \n of vibrating pendulum, we seek for a canonical transformation $(\\theta,p)\\mapsto(\\theta_1,p_1)$ \n with a generating function $p_1\\theta+\\epsilon S_1(\\theta,p_1,\\phi)$, \n where the function  $S_1$ is $2\\pi$ periodic in $\\phi$. Then the new Hamiltonian becomes\n$$\n\\mathscr{H}(\\theta_1,p_1,\\phi)=\\mu \\frac{\\partial S_1}{\\partial \\phi}+H(\\theta_1,p_1,\\phi)+O(\\epsilon),\n$$\nas $\\epsilon\\to 0$. By taking $\nS_1(\\theta,p_1,\\phi)=-( p_1\/l)\\, a \\cos\\phi \\sin\\theta+( 1\/8)\\, a^2\\mu \\sin^2\\theta \\sin 2\\phi,\n$ we  obtain the following Hamiltonian averaged to the first order in $\\epsilon$:\n$$\n\\mathscr{H}(\\theta_1,p_1,\\phi)=\\frac {p_1^2}{2l^2}-gl\\cos\\theta_1+\\frac 14 a^2\\mu^2\\sin^2\\theta_1+O(\\epsilon).\n$$\nNotice the appearance of an additional positive definite quadratic term in the effective potential\n$$\nU_{\\mu}(\\theta_1)=-gl\\cos\\theta_1+( 1\/4) a^2\\mu^2\\sin^2\\theta_1\\,.\n$$\nIt causes such an interesting dynamical phenomenon as the stability of the upper position of the pendulum.\n\\end{example}\n\n\n\\begin{example}[\\textrm{\\textit{(A particle in a rapidly oscillating  potential)}}]\\label{ex:rapid}\nConsider the motion of a particle in a rapidly oscillating potential, following \\cite{cole}. The corresponding\nHamiltonian function is $H(q,p, t)=( 1\/2) \\,p\\cdot p+U(q,  t\/{\\epsilon})$, where the potential function $U(q,\\tau)$\nis $2\\pi$-periodic with respect to $\\tau$.\nTo obtain an averaged Hamiltonian modulo the third order  in $\\epsilon$ one needs to iteratively\napply canonical transformations $(q,p)\\to (Q,P)$  four times (see \\cite{cole} for details). The results is\n$$\n\\bar{H}(Q,P)=\\frac 12\\, P\\cdot P+\\bar U(Q)+\\frac{\\epsilon^2}{2}\\overline{V'\\cdot V'}-\\epsilon^3\\,\\overline{S''V'}P,\n$$\nwhere $\\bar U(q)=\\frac 1 {2\\pi}\\int_0^{2\\pi} U(q,\\tau)\\;d\\tau$ is the time \naverage of $U$ over one temporal period,  functions $V$ and $S$ stand for temporal antiderivatives $V(q,\\tau):=\\int^{\\tau} [U(q,\\theta)-\\bar U(q)]\\,d\\theta, \\;S(q,\\tau):=\\int^{\\tau} V(q,\\theta)\\, d\\theta$, with the\nconstants of integration   chosen such that $\\bar V=\\bar S=0$,\nand the  prime denotes the derivative with respect to the new space variable $Q$.\n\\end{example}\n\n\n\\begin{example}[\\textrm{\\textit{(Foucault pendulum)}}]\nBy considering small and rapid oscillations of an ideal pendulum on a  sphere rotating with angular velocity $\\Omega$, one can observe that the plane of oscillation will be rotating with the angular velocity $-\\Omega\\sin\\lambda$, where $\\lambda$ is the pendulum latitude. This is an example of the Foucault pendulum, see \\cite{ArnMM} for a detailed description.\nMore generally, one can consider a curved surface and a pendulum slowly transported along a path on the surface.\nIn this case the plane of oscillation turns out to be parallel transported along the path on the surface, see the discussion \nin \\cite{Klein}, and it can be regarded as a physical interpretation of the Levi-Civita connection.\n\nOne can use the averaging-reduction theory to interpret this phenomenon. First, notice that this system \nhas the same $\\mathbb T$-bundle structure as the one in the gyroscope Example \\ref{ex:spinDisk}. Hence by shifting the momentum similarly to that in Theorem \\ref{thm:cotbundle}, one obtains a Hamiltonian system on (the cotangent bundle of) the tangent bundle of the surface with a twisted symplectic structure, where the twist is given by an additional curvature term related to the Levi-Civita connection. Then the averaging-reduction procedure allows one to descend this system to (the cotangent bundle of) the unit tangent bundle of the surface, on which the parallel transport is observed. We plan to discuss this example in detail elsewhere.\n\\end{example}\n\n\n\n\\medskip\n\n\\section{Applications}\\label{sect:applications}\n\n\\subsection{Pendulum with a vibrating suspension point}\nIn Section \\ref{subsect:examples}, we  obtained the averaged Hamiltonian for a pendulum with a vibrating suspension point using the classical averaging method, and  observed the appearance of an additional quadratic term in the effective potential. In this section,  using the averaging-reduction procedure of Section \\ref{sect:averaging}, we show that this additional term is the result of symplectic reduction.\n\\smallskip\n\nWe start with the following  Hamiltonian function describing a natural mechanical system with a rapidly oscillating potential:\n\\begin{equation}\\label{eq:Ham_Pen1}\n H(x,p, t)=\\frac 12 (p,\\,p)+U(x)+\\epsilon\\,\\omega^2\\;\\widetilde{U}(x,\\omega t),\n\\end{equation}\nwhere the frequency $\\omega=\\mu\/\\epsilon$ is of order $1\/{\\epsilon}$, and the oscillating part of the potential $\\widetilde{U}(x,\\phi)$ is $2\\pi$-periodic and has zero mean with respect to $\\phi$.\n\nIntroduce the fast time $\\tau=t\/\\epsilon$ and fast variable $\\phi=\\omega t=\\mu \\tau$. We split any (vector-) function $f=f(t,\\phi)$ depending\non two times $t$ and $\\tau$ (and $2\\pi$-periodic in $\\phi=\\mu\\tau$)   into the mean and oscillatory parts:\n$$\nf(t,\\phi)=\\overline{f}(t)+\\widetilde{f}(t,\\phi),\n$$\nwhere $\\overline{f}=({1}\/{2\\pi})\\,\\int_{0}^{2\\pi}f(t,\\phi)\\,d\\phi$.\nNow regard the fast variable $\\phi$ as a new coordinate. Note that when the fast time $\\tau$ changes by 1,\nthe slow time $t$  changes only by $\\epsilon$. So for a motion $x(t,\\phi)=\\overline{x}(t)+\\widetilde{x}(t,\\phi)$\ndescribed by the Hamiltonian function (\\ref{eq:Ham_Pen1}), one can fix $\\overline{x}(t)$ and regard\n$\\widetilde{x}(t,\\phi)=\\widetilde{x}(\\overline{x},\\phi)$ as a function of $\\overline{x}$ and $\\phi$ \nmodulo $O(\\epsilon)$ as $\\epsilon\\to 0$. \nIn other words, one can consider  a map from a suspension over $M$, a manifold $M\\times\\mathbb T$,  to $M$ itself,\nwhere  a point $(\\overline{x},\\phi)\\in M\\times \\mathbb T$ is mapped to $x=\\overline{x}(t)+\\widetilde{x}(\\overline{x},\\phi)\\in M$. Here $\\widetilde{x}(\\overline{x},\\phi)$ can be obtained by solving the Hamiltonian system corresponding to the above Hamiltonian  function $H$ in variables $(\\widetilde x, \\widetilde p)$:\n\\begin{equation}\\label{eq:Ham_fib}\n H(\\widetilde{x},\\widetilde{p}, t)=\\frac 12 (\\widetilde{p},\\widetilde{p})+U(\\overline x)+\\epsilon\\,\\omega^2 \\;\\widetilde U(\\overline{x}+\\widetilde{x},\\omega t),\n\\end{equation}\nwith the initial conditions such that the solution $(\\widetilde x(\\bar x,\\phi), \\widetilde p(\\bar x,\\phi))$ has zero mean value with respect to $\\phi$.\n\nIn order to compute the metric on the suspension manifold $M\\times \\mathbb T$ we set  $\\mu=1$ (i.e. $\\omega=1\/\\epsilon$  and $\\phi=\\tau$).  Thinking of   position $\\widetilde{x}$ \nand momentum  $\\widetilde{p}$ as depending on the average position $\\overline{x}$ and fast time $\\tau$, \nwe denote by  $(\\widetilde{x}(\\overline{x},\\tau),\\widetilde{p}(\\overline{x},\\tau))$ \nthe solution of the corresponding Hamiltonian system with initial conditions corresponding to zero mean value  relative  to $\\tau$.\n\\smallskip\n\nAccording to the above consideration, the configuration space is  \na principal $\\mathbb T$-bundle $\\pi:M\\times \\mathbb T\\rightarrow M$, where\nthe $\\mathbb T$-action on $Q=M\\times \\mathbb T$ is the shift in fibres\n$\\phi\\circ(\\overline{x},e^{i\\phi'})=(\\overline{x},e^{i(\\phi+\\phi')}).$\nApplying the averaging-reduction theory of Section \\ref{sect:averaging}, we obtain the following statement.\n\n\\begin{theorem}\\label{thm:Pen_thm1}\nThe Hamiltonian system (\\ref{eq:Ham_Pen1}) averaged using the standard averaging method in Example \\ref{ex:pendulum} coincides with the system obtained\non the reduced symplectic manifold $(T^*M,\\omega_{can})$ with the canonical symplectic structure, the Hamiltonian function\n\\begin{equation}\\label{eq:avePen1}\nH_{slow}(\\overline{x},\\overline{p})=\\frac 12 (\\overline{p},\\overline{p})+U_{\\mu}(\\overline{x}),\n\\end{equation}\nfor $(\\overline{x},\\overline{p})\\in T^*M$, and with the effective potential \n$$\nU_{\\mu}(\\overline{x})=U(\\overline{x})+\\frac{\\epsilon^2\\omega^2}{4\\pi}\\,\\int_0^{2\\pi}(\\widetilde{v}(\\overline{x},\\tau),\\widetilde{v}(\\overline{x},\\tau))\\,d\\tau\\,.\n$$\n\\end{theorem}\n\\begin{proof}\nNow the averaged metric is given by\n$$\n((v,\\gamma),(v,\\gamma))_{(\\overline{x},\\phi)}=(v,v)+2\\pi\\,\\gamma^2\\left(\\int_0^{2\\pi}(\\widetilde{v}(\\overline{x},\\tau),\\widetilde{v}(\\overline{x},\\tau))\\,d\\tau\\right)^{-1},\n$$\nwhere $(v,\\gamma)\\in T_{\\overline{x}}M\\times T_{\\phi}\\mathbb T$. Also, note that the value of $({1}\/{2\\pi})\\int_0^{2\\pi}(\\widetilde{v}(\\overline{x},\\tau),\\widetilde{v}(\\overline{x},\\tau))\\,d\\tau$  depends on $\\overline{x}$ only.\n\nThe corresponding fiber inertia operator $\\mathbb{I}(\\overline{x}):\\mathfrak t=\\mathbb{R}\\rightarrow\\mathfrak t^*=\\mathbb{R}$\nat $\\overline{x}\\in M$  is given by \n$\\mathbb{I}(\\overline{x})\\gamma\n=2\\pi\\gamma\\left(\\int_0^{2\\pi}(\\widetilde{v}(\\overline{x},\\tau),\\widetilde{v}(\\overline{x},\\tau))\\,d\\tau\\right)^{-1}$, while\nthe momentum map $J:T^*(M\\times \\mathbb T)\\rightarrow\\mathfrak t^*=\\mathbb{R}$ is given by\n$J(\\bar x,\\phi,p,\\eta)=\\eta$.\nThe averaged connection $\\widebar{\\mathcal A}\\in\\Omega^1(M\\times \\mathbb T,\\mathbb R)$ \non the principal trivial $\\mathbb T$-bundle $M\\times \\mathbb T$ is given by\n$\\widebar{\\mathcal A}(\\bar x,\\phi,v,\\gamma)=\\gamma.$ This connection is flat, $\\widebar{\\mathcal A}=d\\phi$.\n\\smallskip\n\nRecall that the magnetic term is proportional to the curvature of the bundle $\\pi:Q\\to M$.\nThe flatness of $\\widebar{\\mathcal A}$ implies that the reduced symplectic structure on the  manifold\n$T^*M=J^{-1}(\\mu)\/\\mathbb T$ has no magnetic term, i.e. it coincides with the canonical symplectic structure $\\omega_{can}$.\nBy taking $\\mu=d\\phi\/d\\tau=\\epsilon\\omega$ in\nTheorem \\ref{thm:ave-red}, the effective potential $U_{\\mu}$ in the Hamiltonian function (\\ref{eq:avePen1}) is\n$$\nU_{\\mu}(\\overline{x})=U(\\overline{x})+1\/2\\,\\langle \\mu,\\mathbb I(\\overline{x})^{-1}\\mu\\rangle\n=U(\\overline{x})+({\\mu^2}\/{4\\pi})\\int_0^{2\\pi}(\\widetilde{v}(\\overline{x},\\tau),\\widetilde{v}(\\overline{x},\\tau))\\,d\\tau\n$$\n$$\n=U(\\overline{x})+\\epsilon^2({\\omega^2}\/{4\\pi})\\int_0^{2\\pi}(\\widetilde{v}(\\overline{x},\\tau),\\widetilde{v}(\\overline{x},\\tau))\\,d\\tau.\n$$\n\\end{proof}\n\nNow return to Example \\ref{ex:pendulum} in Section \\ref{subsect:examples}. The Hamiltonian of a pendulum with a vibrating suspension point can be rewritten in the following form:\n\\begin{equation}\\label{eq:Ham_Pen2}\n H(\\theta,p)=\\frac 12 \\left(\\frac pl\\right)^2-(g-\\epsilon\\, a\\, \\omega^2\\sin\\tau)\\,l\\cos\\theta.\n\\end{equation}\n\nRecall our assumption  that the amplitude $\\epsilon a$ is of order $\\epsilon$ \nand the frequency $\\omega=\\mu\/\\epsilon$ is of order\n$1\/{\\epsilon}$, which means that  the above Hamiltonian has the   form  (\\ref{eq:Ham_Pen1}).\nHence we obtain the following result on its slow motion.\n\n\\begin{theorem}\\label{thm:Pen_thm2}\nThe averaged Hamiltonian function reduces to the following Hamiltonian function describing the slow motion:\n\\begin{equation}\\label{eq:avePen2}\nH_{slow}(\\overline{\\theta}, \\overline{p})=\\frac 12 \\left(\\frac{\\overline{p}}{l}\\right)^2+U_{\\mu}(\\overline{\\theta}),\n\\end{equation}\nwhere the effective potential is $U_{\\mu}(\\overline{\\theta})=(1\/4)\\, \\mu^2 a^2\\sin^2\\overline{\\theta}-gl\\cos\\overline{\\theta}$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\tau=t\/\\epsilon$ and $(\\widetilde{\\theta}(\\overline{\\theta},\\tau), \\widetilde{p}(\\overline{\\theta},\\tau))$ be the solution of Hamiltonian system corresponding to the function\n$$\n \\widetilde{H}(\\widetilde{\\theta}, \\widetilde{p})=\\frac 12 \\left(\\frac {\\widetilde{p}}{l}\\right)^2-(g- \\frac a {\\epsilon} \\sin\\tau)\\,l\\cos(\\overline{\\theta}+\\widetilde{\\theta}),\n$$\nwith the initial values determined by the zero mean conditions of $(\\widetilde{\\theta}(\\overline{\\theta},\\tau), \\widetilde{p}(\\overline{\\theta},\\tau))$ with respect to $\\tau$.\n\nOmitting  terms of order $\\epsilon^2$ we obtain the following Newton equation for $\\widetilde{\\theta}$:\n$$\n d^2\\widetilde{\\theta}\/d t^2=-(a\/(\\epsilon l))\\,{\\sin\\tau\\sin\\overline{\\theta}}\\,.\n$$\nBy rewriting it for the fast time $\\tau=t\/\\epsilon$ and integrating we obtain $d\\widetilde{\\theta}\/d\\tau=(\\epsilon a\/l)\\,{\\cos\\tau\\sin\\overline{\\theta}}$ and $\\widetilde{\\theta}=(\\epsilon a\/l)\\,{\\sin\\tau\\sin\\overline{\\theta}}$.\n(In this integration one regards the right-hand side as a function of $\\tau$ modulo higher order terms in $\\epsilon$,\nand uses the zero mean condition on  $d\\widetilde{\\theta}\/d\\tau$ and $\\widetilde{\\theta}$.)\n\nFurthermore, the momentum map\n$J:T^*(\\mathbb T_{\\bar\\theta}\\times \\mathbb T_\\phi)\\rightarrow \\mathbb{R}$, corresponding to the $\\mathbb T$-action on $\\mathbb T_{\\bar\\theta}\\times \\mathbb T_\\phi$ equipped with the averaged metric is $J(\\bar\\theta,\\phi,p,\\mu)=\\mu$, and $({l^2}\/{2\\pi})\\int_0^{2\\pi}(d\\widetilde{\\theta}\/d\\tau,d\\widetilde{\\theta}\/d\\tau)\\,d\\tau=\\epsilon^2(a^2\/2) \\sin^2\\overline{\\theta}$.\n\n Therefore, by Theorem \\ref{thm:Pen_thm1}, the reduced (or slow) symplectic manifold is $(T^*\\mathbb T_{\\bar\\theta},\\omega)$ with symplectic structure  $\\omega=d\\bar\\theta\\wedge d\\bar p$  and\n the Hamiltonian of the slow motion is\n$$\nH_{slow}(\\overline{\\theta}, \\overline{p})=\\frac 12 \\left(\\frac{\\overline{p}}{l}\\right)^2+U_{\\mu}(\\overline{\\theta}),\n$$\nwhere  the effective potential is\n$$\nU_{\\mu}(\\overline{\\theta})=\\frac{\\omega^2 l^2}{4\\pi}\\int_0^{2\\pi}(d\\widetilde{\\theta}\/d\\tau,d\\widetilde{\\theta}\/d\\tau)\\,d\\tau+U(\\bar\\theta)\n$$\n$$\n=\\epsilon^2\\frac{\\omega^2 a^2}{4}\\sin^2\\overline{\\theta}-gl\\cos\\overline{\\theta}=\\frac 14\\, \\mu^2 a^2\\sin^2\\overline{\\theta}-gl\\cos\\overline{\\theta}\\,.\n$$\n\\end{proof}\n\n\n\n\n\\subsection{Craik-Leibovich equation}\\label{sect:CL}\nConsider the motion of an ideal fluid confined to a three-dimensional domain $D$ \nwith  fast oscillating boundary  $\\partial D$. The   averaged fluid motion is described by the following Craik-Leibovich (CL) equation on the fluid velocity field $v$:\n\\begin{equation}\\label{eq:CraLei}\n\\left\\{\n\\begin{array}{ll}\n\\frac{\\partial v}{\\partial t}+(v,\\nabla)v+\\mathrm{curl}\\;v\\times V_s=-\\nabla p,\\\\\n(v+V_s)\\,||\\,\\partial D,\\\\\n\\mathrm{div}\\,v=0,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $V_s$ is a (time-dependent) prescribed Stokes drift velocity related to the average of surface waves.\nWe refer to \\cite{vla} for a derivation of the CL equations via perturbation theory.  \nIn \\cite{yang1, yang2} the Hamiltonian structure of the CL equation was studied, along with \na generalization of the perturbation theory to a principal $\\mathbb T$-bundle over any group $G$ and derivation of the Euler equation associated with a certain central extension of  $G$.\n\n\nIn a more general setting, \nlet $\\mathrm{SDiff}(D)$ be the group of all volume-preserving diffeomorphisms of an $n$-dimensional Riemannian manifold $D$ with boundary $\\partial D$. Its Lie algebra $\\mathfrak g=\\text{SVect}(D)$ consists of\nall the divergence-free vector fields in $D$ tangent to the boundary $\\partial D$, while the regular dual space\n$\\mathfrak g^*=\\Omega^1(D)\/d\\Omega^0(D)$ of this Lie algebra is the space of cosets of 1-forms on $D$ modulo  exact 1-forms.\n\n\\begin{theorem}{\\rm (see \\cite{yang2})}\\label{thm:genCL}\nThe  $n$-dimensional  Craik-Leibovich (CL) equation on the  space\n$\\Omega^1(D)\/d\\Omega^0(D)$ has the form\n\\begin{equation}\\label{eq:genCL}\n \\frac{d}{dt}\\;[u]=-\\mathcal{L}_{v+V_s}\\;[u],\n\\end{equation}\nwhere $v+V_s\\in {\\rm SVect}(D)$, and $[u]=[v^\\flat]\\in\\Omega^1(D)\/d\\Omega^0(D)$ is the coset of the 1-form\n$v^\\flat$ metric-related to the vector field $v$ on $D$.\n\\end{theorem}\n\n\\begin{remark}\nThe requirement $v+V_s\\in {\\rm SVect}(D)$ provides the boundary condition of tangency of   the vector field $v+V_s$  to the boundary $\\partial D$. Although the divergence-free vector fields $v$ and $V_s$ are not necessarily tangent to the boundary, the 1-forms $v^\\flat$ and $V_s^\\flat$, considered up to the differential of a function, are well-defined elements in the dual space $\\Omega^1(D)\/d\\Omega^0(D)$ of Lie algebra ${\\rm SVect}(D)$.\n\\end{remark}\n\nUpon shifting the origin in  $\\mathfrak g^*=\\Omega^1(D)\/d\\Omega^0(D)$ by $[V_s^\\flat]$,\nequation (\\ref{eq:genCL}) becomes\n\\begin{equation}\\label{eq:genCL2}\n\\frac{d}{dt}\\;[w]=-\\mathcal{L}_{\\mathbb{I}^{-1}[w]}\\;\\left([w]-[V_s^\\flat]\\right)\n\\end{equation}\nfor $[w]:=[u+V_s^\\flat]$.\n\n\n\\begin{theorem}{\\rm (see \\cite{yang2})}\\label{thm:cenExt}\nThe equation (\\ref{eq:genCL2}) is the Euler equation on the central extension $\\widehat{\\mathfrak g}$ of the Lie algebra $\\mathfrak{g}={\\rm SVect}(D)$ by means of the 2-cocycle\n$$\n{\\sigma}_{V_s}(X,Y):=-\\left\\langle\\mathcal{L}_{X}\\;V_s^\\flat,Y\\right\\rangle\n$$\nassociated with the vector field $V_s$.\n\\end{theorem}\n\n\\begin{remark}\nMore generally, for a  ``shift 2-cocycle'' \n$$\n{\\sigma}_{V_s}(X,Y):=-\\left\\langle{\\rm ad}_X^*\\mathbb I(V_s),Y\\right\\rangle=-\\left\\langle\\mathbb I(V_s),[X,Y]\\right\\rangle\n$$ \non an arbitrary Lie algebra $\\mathfrak g$,\nconsider the corresponding (possibly trivial) central extension $\\widehat{\\mathfrak g}$ \nof Lie algebra $\\mathfrak g$ by means of this 2-cocycle.  \nThen the Euler equation on the centrally extended Lie algebra $\\widehat{\\mathfrak g}$ is\n\\begin{equation}\\label{eq:cenExtEuler}\n\\frac{d}{dt}\\;\\xi=-{\\rm ad}^*_{\\mathbb{I}^{-1}\\xi}\\left(\\xi-\\mathbb I(V_s)\\right),\n\\end{equation}\nwhere $\\xi\\in \\mathfrak g^*$.\nApplying this to the Lie algebra ${\\rm SVect}(D)$, where ${\\rm ad}_X^*\\xi=\\mathcal{L}_X \\xi$, we obtain the above theorem.\n\\end{remark}\n\nThe averaging-reduction procedure gives us an explanation of the structure of central extension for \nthe CL equation, which  appeared as the result of averaging, as well as the origin of the vector field $V_s$. More specifically, by applying the symplectic reduction procedure to a natural system on the principal \n$\\mathbb T$-bundle $G\\times \\mathbb T\\to G$ for a group $G$ (in the case of  an oscillating flow, $G$ is the group\n$\\text{SDiff}(D)$ of volume-preserving diffeomorphisms of $D$) \none obtains the averaged system on the cotangent bundle $T^*G$ of the base $G$ with  \nthe symplectic structure endowed with a magnetic term.\nFurthermore,  Theorem \\ref{thm:poisson} provides the corresponding central extension.\nFirst we prove the averaging-reduction theorem for a trivial bundle $G\\times \\mathbb T\\to G$, and then explain the necessary changes in the general case. \n\n\\begin{theorem}\\label{thm:aveCL}\nThe averaging of a natural $G$-invariant Hamiltonian system on $T^*(G\\times \\mathbb T)$ reduces to a slow Hamiltonian system on the reduced symplectic manifold $(T^*G,\\omega_\\mu)$ with the Hamiltonian function $H_{slow}(\\bar g,\\bar r)=\\frac 12 (\\bar r,\\,\\bar r),$\nwhere $(\\bar g,\\bar r)\\in T^*G$, and the symplectic structure $\\omega_\\mu$  given by\n$$\n\\omega_\\mu=\\omega_{can}-\\mu\\,\\pi_G^*\\,d\\widetilde{\\alpha},\n$$\nfor the canonical symplectic form   $\\omega_{can}$ on $T^*G$. Here $\\pi_G:T^*G\\rightarrow G$ is \nthe cotangent bundle projection and $\\widetilde{\\alpha}$ is a certain 1-form on the base $G$ depending on the averaged metric.\n\\end{theorem}\n\n\n\n\\begin{proof}\nFor a point $q=(g,x)\\in G\\times \\mathbb T$ let $p=(\\nu, y)$ be  a covector at that point. \nA natural $G$-invariant  Hamiltonian system on $T^*(G\\times \\mathbb T)$ with Hamiltonian \n$H(q,p)=\\frac 12 (p,\\,p)+U(x)$ for a $G$-invariant metric on $G\\times\\mathbb T$  has the form:\n$$\n H(g,x;\\nu,y)=\\frac 12 ((\\nu,y),\\;(\\nu,y))_{(g, x)}+U(x).\n$$\nTo write it more explicitly,\\footnote{This is based on the same consideration as Remark \\ref{rem:metric}.} \nlet $\\mathcal B^*(x): \\mathfrak g^*\\to T_x^*\\mathbb T$ be the linear map associated with the $G$-invariant \nmetric,\\footnote{As a matter of fact, $\\mathcal B^*$ will turn out to be dual of a  flat connection $\\mathcal B$  in the bundle \n$G\\times \\mathbb T\\to \\mathbb T$, see Remark~\\ref{rem:nontriv-bundle}.} and identify covector components $\\nu$ with elements of $\\mathfrak g^*$  \nby right translations. Then the Hamiltonian  can be rewritten as\n\\begin{equation}\\label{eq:Ham_CL2}\n H(g,x;\\nu,y)=\\frac 12 (y,\\;y)_x+(y,\\;\\mathcal B^*\\nu)_x+\\frac 12 (\\nu,\\;\\nu)_x+U(x).\n\\end{equation}\n\n\nNow  fix a position $g$ of the slow motion and  consider the fast motion (i.e. $x$-dependence)\nof the system. In this setting\nwe have $\\nu=0$, since the fast motion has zero mean, omitting  higher order terms in $\\epsilon$. Then the Hamiltonian for the fast motion becomes\n\\begin{equation}\\label{eq:Ham_CL2_fast}\n \\widetilde{H}(\\widetilde x,\\widetilde y)=\\frac 12 (\\widetilde y,\\widetilde y)_{\\widetilde x}+U(\\widetilde x)\\,.\n\\end{equation}\nNote that the  Hamiltonian of fast motion does not depend on the group element $g$.\n\nSuppose that the period of this motion is $2\\pi\/\\omega$ and set the fast variable to be $\\phi=\\omega t$, where the frequency $\\omega=\\mu\/\\epsilon$.\nLet $(\\widetilde{x}(\\phi),\\widetilde{y}(\\phi))$ be any Hamiltonian trajectory for  Hamiltonian function (\\ref{eq:Ham_CL2_fast}) satisfying the following two conditions:  it is $2\\pi$-periodic with respect to the fast variable $\\phi$ and its initial condition is chosen in  a way to provide zero mean for $(\\widetilde{x}(\\phi),\\widetilde{y}(\\phi))$. \nSuch a vanishing condition generically defines a 1-parameter family of solutions parametrized by the energy level\n(e.g. such solutions differ by scaling for a quadratic potential $U$). \nWe use the fast  variable $\\phi$ to define the \n$\\mathbb T$-action on $G\\times \\mathbb T$,  which is given by $\\phi\\circ(g,e^{i\\phi'})=(g,e^{i(\\phi+\\phi')})$. \n\nSince  the  Hamiltonian of fast motion does not depend on the group element $g$,  the averaged kinetic energy $(1\/2\\pi)\\int_0^{2\\pi}(\\widetilde{v}(\\phi),\\widetilde{v}(\\phi))\\,d\\phi$\ndoes not depend on $g$ as well, where $\\widetilde v $ is related to $\\widetilde y$ by means of the metric: $\\widetilde v^\\flat=\\widetilde y$. Take the solution $(\\widetilde{x}(\\phi),\\widetilde{y}(\\phi))$ for which \n the averaged kinetic energy $(1\/2\\pi)\\int_0^{2\\pi}(\\widetilde{v}(\\phi),\\widetilde{v}(\\phi))\\,d\\phi$ is \n $ {1}\/{\\epsilon^2}$.\n\n Then the averaged metric $\\overline{(\\cdot\\,,\\cdot)}^{\\mathbb T}$ on $G\\times\\mathbb T$ \n (cf. Remark \\ref{rem:metric}) is given by\n\\begin{equation}\n\\begin{array}{rcl}\n\\overline{((v,\\gamma),(v,\\gamma))}^{\\mathbb T}&=&(v,v)+2\\pi(2 \\gamma\\,\\langle \\widetilde{\\alpha},v\\rangle+\\gamma^2)\n\\left(\\int_0^{2\\pi}(\\widetilde{v}(\\phi),\\widetilde{v}(\\phi))\\,d\\phi \\right)^{-1}\\\\\n&=&(v,v)+\\epsilon^2 \\,(2\\gamma\\,\\langle \\widetilde{\\alpha},v\\rangle+ \\gamma^2)\\,,\n\\end{array}\n\\end{equation}\nwhere $(v,\\gamma)\\in T_{g}G\\times T_{\\phi}\\mathbb T\\simeq T_{g}G\\times \\mathbb R$, while the\n1-form $\\widetilde{\\alpha}\\in T^*G$ depends on  $\\widetilde y$. This 1-form is right-invariant,  i.e. it satisfies\n$\\widetilde{\\alpha}(hg)=R_{g^{-1}}^*\\widetilde{\\alpha}(h)$. Indeed,\nthe metric on $G\\times \\mathbb T$ is right-invariant under the $G$-action, hence its average with respect to\nthe $\\mathbb T$-action is right-invariant under the $G$-action as well.\n\nThe averaged inertia operator $\\mathbb{I}(g):\\mathfrak t=\\mathbb{R}\\rightarrow\\mathfrak t^*=\\mathbb{R}$ \nis $\\mathbb{I}(g)\\gamma=\\epsilon^2\\gamma$, while the\naveraged momentum map $ J:T(G\\times \\mathbb T)\\rightarrow\\mathfrak t^*=\\mathbb{R}$ is given by\n $ J(g,\\phi, v,\\gamma)=\\langle \\epsilon^2 \\widetilde{\\alpha},v\\rangle+\\epsilon^2\\gamma$, \n cf. Proposition \\ref{prop:momentum}.\nFinally, the averaged connection $\\bar{\\mathcal A}\\in\\Omega^1(G\\times \\mathbb T,\\mathbb R)$ on the principal\n$\\mathbb T$-bundle $G\\times \\mathbb T$ is given by $\\bar{\\mathcal A}(g,\\phi)= \\widetilde{\\alpha}+d\\phi.$\n\n\\smallskip\n\nAccording to Theorem \\ref{thm:ave-red}  the magnetic term in the symplectic structure is $\\beta_\\mu=\\mu\\,\\pi_G^*d \\bar{\\mathcal A}=\\mu\\,\\pi_G^*d \\widetilde{\\alpha}$ for the corresponding \n$ \\mu\\in\\mathbb R$. Therefore, the  symplectic structure  on the reduced manifold $T^*G\\cong  J^{-1}(\\mu)\/\\mathbb T$ \nis the 2-form $\\omega_\\mu=\\omega_{can}-\\mu\\,\\pi_G^*d\\widetilde{\\alpha}$. The correction in the effective potential $U_{\\mu}$ of   Hamiltonian function (\\ref{aveHam}) is  constant,\nand hence   can be omitted.\n\\end{proof}\n\n\n\\begin{remark}\\label{rem:nontriv-bundle}\nFor the general case of a topologically nontrivial $G$-bundle $\\pi:\\,M\\to N$, consider an open subset \n$\\mathcal O\\subset N$. Then  locally in the base  trivialize $M|_{\\pi^{-1}(\\mathcal O)}\\cong \\mathcal O\\times G$ and the cotangent bundle $T^*M|_{\\pi^{-1}(\\mathcal O)}\\cong T^*\\mathcal O\\times G \\times \\mathfrak g^*$. This  decomposition of the tangent\/cotangent spaces gives us a flat connection $\\mathcal B\\in\\Omega^1(\\mathcal O,\\mathfrak g)$ on $ \\mathcal O\\times G$. Therefore, for the local representation $(x,g; y,\\nu)$ of $(q,p)$, we have\n$q=(x,g)$ and $p=(y+\\mathcal B^*\\nu,\\nu)$, and then the Hamiltonian $ H(q,p)=\\frac 12 (p,\\,p)+U(\\pi(q))$\nin $T^*M$  becomes\n$$\n\\begin{array}{rcl}\nH(x,g;y,\\nu)&=&\\frac 12 (y+\\mathcal B^*\\nu,y+\\mathcal B^*\\nu)_x+\\frac 12 (\\nu,\\nu)_x+U(x)\\\\\n                    &=&\\frac 12 (y,y)_x+(\\mathcal B^*\\nu,y)_x+\\frac 12 (\\mathcal B^*\\nu,\\mathcal B^*\\nu)_x+\\frac 12 (\\nu,\\nu)_x+U(x).\n\\end{array}\n$$\nBy combining the third and forth terms in the above expression of the general Hamiltonian and by switching the order of coordinates, we obtain the Hamiltonian (\\ref{eq:Ham_CL2}) in the proof. Finally, by confining ourselves to $x$-periodic solutions in the base lying inside \n$ \\mathcal O$ one can reduce the setting to a Hamiltonian on $T^*(G\\times \\mathbb T)$.\n\\end{remark}\n\n\\begin{remark}\nFor the oscillating flow we\nconsider the   principal $G$-bundle $\\pi:M\\rightarrow N$ with $G=\\text{SDiff}(D)$ to be the group of\nvolume-preserving diffeomorphisms. The infinite-dimensional manifold $M$ is the space of all\nvolume-preserving embeddings of the reference manifold $D$ to $\\mathbb R^n$, while $N$ is the manifold of all  boundaries of such embeddings, i.e. hypersurfaces  diffeomorphic to $\\partial D$ and bounding diffeomorphic manifolds with the same volumes.\n\nFor the averaged metric, the 1-form $\\widetilde{\\alpha}$ on $G$ is metric-related to a vector field $V_s$ on $G$  via $\\widetilde{\\alpha}=V_s^\\flat$. This defines the 2-cocycle\n${\\sigma}_{V_s}(X,Y):=-\\left\\langle\\mathcal{L}_{X}\\;V_s^\\flat,Y\\right\\rangle$ on the Lie algebra ${\\rm SVect}(D)$ in Theorem \\ref{thm:cenExt}.\n\\end{remark}\n\n\\smallskip\n\n\\subsection{Particles in  rapidly oscillating potentials}\n\nIt turns out that the motion of a particle in a rapidly oscillating potential field  \\cite{cole} can also be viewed in the\ncontext of the symplectic averaging-reduction setting. Moreover, in this example one observes\nboth phenomena: additional terms in the effective potential and magnetic correction to the symplectic structure.\n\nNamely, consider the Hamiltonian function\n\\begin{equation}\\label{eq:Ham_Par}\nH(x,p, \\omega t)=\\frac 12 \\,p\\cdot p+\\epsilon^2\\mu^2\\;U(x, \\omega t),\n\\end{equation}\nwhere $x\\in \\mathbb R^n$, the potential function $U(x,\\phi)$ is $2\\pi$-periodic with respect to (the fast variable) $\\phi$ and the\nfrequency $\\omega= \\mu\/\\epsilon$ is of order $1\/\\epsilon$.\nBy ingenious repeated application of  canonical transformations (see \\cite{cole} for details, where $\\mu=1$) one obtains\nan averaged Hamiltonian up to the third order in $\\epsilon$:\n\\begin{equation}\\label{eq:Ham_ave}\n\\overline{H}(\\overline x,\\overline p)=\\frac 12 \\overline p\\cdot\\overline p+\\overline U +\\frac{\\epsilon^2\\mu^2}{2}\\overline{V'\\cdot V'}-\\epsilon^3\\mu\\;\\overline{S''V'}\\,\\overline p,\n\\end{equation}\nwhere $\\overline U(x)=\\frac 1 {2\\pi}\\int_0^{2\\pi} U(x,\\tau)\\;d\\tau,\\;V(x,\\tau)=\\int^{\\tau} U(x,\\theta)-\\overline U(x)\\; d\\theta,$ and \n$ S(x,\\tau)=\\int^{\\tau} V(x,\\theta)\\; d\\theta$ with the constant of integration chosen so\nthat $\\overline V=\\overline S=0$, and where prime denotes the derivative with respect to the space variable $x$.\n\nOn the other hand, the symplectic averaging-reduction procedure applied to the Hamiltonian (\\ref{eq:Ham_Par})\nis as follows. Regard  the fast variable $\\phi\\in \\mathbb T$ as a new (periodic) coordinate, while the quotient along \n$\\phi$-fibers is the slow manifold with coordinates $\\overline x\\in \\mathbb R^n$,  mean values of the solutions.\nThe $\\mathbb T$-action on the principal $\\mathbb T$-bundle $\\pi:\\mathbb R^n\\times \\mathbb T\\rightarrow \\mathbb R^n$\nis given by $\\phi\\circ(\\overline{x},e^{i\\phi'})=(\\overline{x},e^{i(\\phi+\\phi')})$.\n\n\n\\begin{theorem}\\label{thm:Par_thm}\nThe averaged Hamiltonian system \\eqref{eq:Ham_ave} for the natural system    \\eqref{eq:Ham_Par}\n is equivalent to the result of the symplectic averaging-reduction\nprocedure, i.e. the Hamiltonian system\non the reduced symplectic manifold $(T^*\\mathbb R^n,\\omega_{\\mu})$ with the Hamiltonian function:\n\\begin{equation}\\label{eq:avePar}\nH_{slow}(\\overline{x},\\overline{p})=\\frac 12 \\,\\overline{p}\\cdot\\overline{p}+U_{\\mu}(\\overline{x}),\n\\end{equation}\nwhere $(\\overline{x},\\overline{p})\\in T^*\\mathbb R^n$, and the effective potential is $U_{\\mu}(\\overline{x})=\\overline U(\\overline x)+\\frac{\\epsilon^2\\mu^2}{2}\\overline{V'\\cdot V'}$. The reduced symplectic structure is given by\n\\begin{equation}\\label{eq:aveParSym}\n\\omega_{\\mu}=d\\overline x\\wedge d\\overline p-\\epsilon^3\\mu\\; d\\{\\overline{S''V'}^{\\flat}\\}\n\\end{equation}\nwith the above notations for $\\overline U, V, S$ and the prime as above, and $\\flat$ stands for the lifting indices operator ${\\rm Vect}(\\mathbb R^n)\\rightarrow\\Omega^1(\\mathbb R^n)$ corresponding to the Euclidean metric on $\\mathbb R^n$.\n\\end{theorem}\n\n\\begin{proof}\nFor the fast time $\\tau=t\/\\epsilon$ let $(\\widetilde{x}(\\overline{x},\\tau,\\widetilde{p}(\\overline{x},\\tau)))$ be a solution of the Hamiltonian system corresponding to the function\n$$\n \\widetilde{H}(\\widetilde{x},\\widetilde{p})=\\frac 12 \\,\\widetilde{p}\\cdot\\widetilde{p}+\\widetilde U(\\overline x+\\widetilde x,\\tau)\n$$\nand satisfying the periodicity and zero mean requirement in $\\tau$. (Recall that one splits any solution \n$x=\\overline{x}+\\widetilde{x}$  into the mean and periodic parts.)\nUpon discarding higher order terms in $\\epsilon$, the Newton's equation on $\\widetilde{x}$ becomes\n$$\n d^2\\widetilde{x}\/dt^2=-\\widetilde U'(\\overline x+\\widetilde x,\\tau),\n$$\nwhere the prime stands for the derivative with respect to the space variable.\nWe consecutively obtain\n$$\nd\\widetilde{x}\/dt=-\\epsilon\\int^\\tau U'(\\overline x+\\widetilde x,\\tau)\\; d\\tau=-\\epsilon V'\n\\qquad{\\rm and }\\qquad\n\\widetilde{x}=-\\epsilon\\int^\\tau \\epsilon\\, V'(\\overline x+\\widetilde x,\\tau)\\; d\\tau=-\\epsilon^2 S'\\,,\n$$\nwhere one integrates by using the zero mean condition on the constants of integration.\n\nThen the averaged metric is given by\n$$\n((v,\\gamma),(v,\\gamma))_{(\\overline{x},\\phi)}=v\\cdot v\n+2\\,\\epsilon\\gamma\\,\\langle \\overline{S''V'},v\\rangle\\,(\\overline{V'\\cdot V'})^{-1}\n+ {\\gamma^2}(\\epsilon^2\\overline{V'\\cdot V'})^{-1},\n$$\nwhere $(v,\\gamma)\\in T_{\\overline{x}} \\mathbb R^n\\times T_{\\phi}\\mathbb T\\simeq  T_{\\overline{x}}\\mathbb R^n\\times \\mathbb R$. Note that the above metric has the form \\eqref{eq:aveMet}\ndiscussed in Remark \\ref{rem:metric}.\n\n\\smallskip\n\nThe corresponding fiber inertia operator \n$\\mathbb{I}(\\overline{x}):\\mathfrak t=\\mathbb{R}\\rightarrow\\mathfrak t^*=\\mathbb{R},\\;\\overline{x}\\in M$  \nis given by $\\mathbb{I}(\\overline{x})\\gamma={\\gamma}(\\epsilon^2\\overline{V'\\cdot V'})^{-1}$ for  $\\gamma\\in\\mathbb{R}$.\nThen, as follows from Proposition \\ref{prop:momentum},\nthe momentum map $J:T(\\mathbb R^n\\times \\mathbb T)\\rightarrow\\mathfrak t^*=\\mathbb{R}$ \nfor $Q=\\mathbb R^n\\times \\mathbb T$ is given by\n$J(\\bar x,\\phi,v,\\gamma)=\\gamma(\\epsilon^2\\overline{V'\\cdot V'})^{-1}+\\epsilon \\langle \\overline{S''V'},v\\rangle\\,(\\overline{V'\\cdot V'})^{-1},$   where $v$ is the image of $p$ under the metric identification.\n\n\\smallskip\n\nFinally, the averaged connection $\\bar{\\mathcal A}\\in\\Omega^1(\\mathbb R^n\\times \\mathbb T,\\mathbb R)$ on the principal $\\mathbb T$-bundle $\\mathbb R^n\\times \\mathbb T$ is given by\n$\\bar{\\mathcal A}(\\bar x,v,\\phi,\\gamma)=d\\phi+\\epsilon^3 \\overline{S''V'}^{\\flat}.$\n\nWe choose $\\mu$ to be the value of the momentum map. By Theorem \\ref{thm:ave-red}, the reduced symplectic structure on the reduced manifold\n$T^*\\mathbb R^n=J^{-1}(\\mu)\/\\mathbb T$ has a magnetic term\n$$\n\\omega_{\\mu}=d\\overline x\\wedge d\\overline p-\\epsilon^3\\mu\\, d\\{\\overline{S''V'}d\\overline x\\}.\n$$\nThe reduced Hamiltonian function on $(T^*\\mathbb R^n,\\omega_\\mu)$ turns out to be\n$$\nH_{slow}(\\overline{x},\\overline{p})=\\frac 12\\, \\overline{p}\\cdot\\overline{p}+U_{\\mu}(\\overline{x})\n$$\nat any  $(\\overline{x},\\overline{p})\\in T^*M$,  where the effective potential is\n$U_{\\mu}(\\overline{x})=\\overline U(\\overline x)+\\frac{\\epsilon^2\\mu^2}{2}\\overline{V'\\cdot V'}$\\,.\n\\end{proof}\n\nThus the system obtained by applying the symplectic averaging-reduction procedure is equivalent to the one obtained via the classical averaging method in \\cite{cole}.\n\n\\begin{remark}\nIt remains an open question to describe the magnetic term related to the curvature of an appropriate \n$\\mathbb T$-bundle in purely geometric terms, similar to the gyroscope description in \\cite{cole2}.\n\\end{remark}\n\n\\iffalse\n\\medskip\n\n\\section{Appendix: Many-frequency systems and non-abelian groups}\nIn general, the averaging-reduction procedure can be applied to any compact Lie group $G$.\nHowever, to claim that the result will be close to the solutions obtained by the classical averaging requires\nmore assumptions.\n\nLet $G$ be a compact Lie group with the Haar measure $m$ and  $\\pi:Q\\rightarrow P$ is a principal $G$-bundle\nover a manifold $P$. Consider a natural Hamiltonian system along the cotangent bundle $T^*Q$:\n\\begin{equation}\\label{eq:natHam}\n H(q,\\;p)=\\frac 12 (p,\\;p)+U(q).\n\\end{equation}\n Here $Q$ is the configuration space of the motion, we assume that this Hamiltonian system has slow motion on the base space $P$ and fast motion on the fibres.\n\nThe cotangent lift of $G$-action on $Q$ induces a $G$-action on $T^*Q$. We use $\\rho:G\\rightarrow \\text{Map}(Q,Q)$ and $\\Gamma:G\\rightarrow \\text{Map}(T^*Q,T^*Q)$ to denote the $G$-action on $Q$ and $T^*Q$ respectively.\nThe Hamiltonian function $H(q,p)$ {\\it a priori} is not assumed to be invariant under the $G$-action on $T^*Q$. First we start by obtaining a $G$-averaged (and hence $G$-invariant) Hamiltonian  $\\overline{H(q,p)}^G$\ndefined by the following formula:\n\\begin{equation}\n \\overline{H(q,p)}^G=\\frac{1}{m(G)}\\int_G H(\\Gamma(g)((q,p)))\\,dm(g).\n\\end{equation}\nFor the natural system (\\ref{eq:natHam}), we have\n\\begin{equation}\n \\overline{H(q,p)}^G=\\frac 12 \\overline{(p,p)}^G+\\overline{U(q)}^G,\n\\end{equation}\nwhere $\\overline{(p,p)}^G:=\\frac{1}{m(G)}\\int_G (\\rho(g)^*p,\\rho(g)^*p)\\,dm(g)$\nand $\\overline{U(q)}^G :=\\frac{1}{m(G)}\\int_G U(\\rho(g))\\,dm(g)$.\n\nNext we define an averaged metric on $Q$:\n\\begin{definition}\n The {\\it averaged metric} $\\overline{(\\cdot,\\cdot)}^G$ on the principal $G$-bundle $Q$ is given by\n$$\n\\overline{(v,v)}^G=\\frac{1}{m(G)}\\int_G (\\rho(g)_* v,\\rho(g)_* v)\\,dm(g),\n$$\nfor any $v\\in TQ$.\n\\end{definition}\n\nNote that this is a $G$-invariant metric. It induces the corresponding inertia operator and the momentum map.\nThen one can define an averaged connection corresponding to the averaged metric.\n\n\\begin{definition}\n The {\\it averaged connection} $\\mathcal{A}\\in\\Omega^1(Q,\\mathfrak g)$ on the principal $G$-bundle $Q$ is the mechanical connection  induced from the averaged metric by setting an invariant distribution of horizontal spaces: at each point $q\\in Q$ the horizontal space is the subspace of $T_qQ$ orthogonal to the fiber\n (the $G$-orbit) at $q$.\n\\end{definition}\n\nDue to the $G$-invariance of  the averaged Hamiltonian $\\overline{H}^G$, the corresponding dynamics\ncan be described as the motion on the base space $P$ of the principal bundle, called\nthe {\\it averaged or slow motion}. Unlike the classical averaging method, we obtain this averaged\/slow motion as a symplectic reduction.\n\nConsider the reduced space $(T^*Q)_\\mu=J^{-1}(\\mu)\/G_\\mu$, where $G_\\mu$ is the isotropy subgroup of $G$ with respect to $\\mu$. If $G_\\mu=G$, then the reduced space $J^{-1}(\\mu)\/G$ for a fixed value $\\mu$ of the momentum map is symplectically diffeomorphic to the cotangent bundle $T^*P$ of the base space $P$ with the symplectic form $\\sigma_\\mu$ on $T^*P$ is given by $\\sigma_\\mu=\\omega_{can}-B_\\mu,$\nwhere $\\omega_{\\mathrm{can}}$ and $B_\\mu$ are the canonical and magnetic symplectic forms\non $T^*P$ (see Theorem \\ref{thm:cotbundle}).\nThe average\/slow system turns out to be the reduced Hamiltonian system on the symplectic manifold $(T^*P,\\sigma_\\mu)$ with a reduced Hamiltonian function $H_\\mu$:\n\n\\begin{theorem}\\label{thm:ave-red-multi}\nThe averaging procedure reduces the Hamiltonian function $H$ of a natural system \\eqref{eq:natHam}\nto the Hamiltonian function $H_\\mu$ on the symplectic manifold $(T^*P,\\sigma_\\mu)$:\n\\begin{equation}\\label{aveHam-multi}\nH_\\mu(q,p)=\\frac 12 (p,\\,p)_P+U_\\mu(q),\n\\end{equation}\nwhere $(q,p)\\in T^*P$, $(\\cdot,\\cdot)_P$ stands for the restricted metric of $\\overline{(\\cdot,\\cdot)}^{G}$ on $P$, the effective potential $U_\\mu(q)=\\overline{ U(q)}^{G}+\\frac 12 \\langle \\mu,\\,\\mathbb{I}(q)^{-1}\\mu\\rangle$ .\n\\end{theorem}\n\nFinally, we introduce the ergodicity assumption \\bk{What is the meaning of the momentum map in the non-abelian averaging? What is the role of group action?}to guarantee that the averaged system is close to the perturbed system in certain sense. First one can induce a measure $\\nu$ on the fibre from the Haar measure on group $G$ by the following method. Fix a point $x$ in one fibre $F$, then for any other point $y\\in F$, there exists a group element $g$, such that $y=\\rho(g)(x)$. In other words, $x$ can be seen as a map from $G$ to $F$, $x:g\\mapsto\\rho(g)(x)$. The induced measure $\\nu$ on $F$ is the push-forward measure of the Haar measure $m$: $\\nu=x_*m$.\n\nThe ergodicity assumption states that the flow of the fast motion \\bk{How to define this flow? Project to fibre?}on each fibre is ergodic with respect to the measure $\\nu$. Now let $K$ be a compact subset of the base $P$ and domain $\\Gamma=\\pi^{-1}(K)$, the set $D_{\\epsilon}\\in K$ contains the initial points such that for all $t\\in[0,1\/\\epsilon]$, the exact solution remains in $\\Gamma$. For any fixed $\\alpha>0$, subset $D_{\\epsilon}(\\alpha)\\subset D_{\\epsilon}$ contains the initial points such that the difference \\bk{How to measure this difference? Some metric?} between the exact and averaged trajectories does not exceed $\\alpha$ for all $t\\in[0,1\/\\epsilon]$. Then according to the Anosov's theorem \\cite{loch}, under the above ergodicity assumption and assume that the Hamiltonian function is smooth enough, we have\n\\begin{equation}\n\\lim_{\\epsilon\\rightarrow 0} \\lambda(D_{\\epsilon}\\setminus D_{\\epsilon}(\\alpha))=0,\n\\end{equation}\nwhere $\\lambda$ is the volume form on the base.\n\n\n\n\\fi\n\n\n\\medskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently, Ishida et al. have proposed the covariant\n $\\widetilde{U}(12)_{SF}$-classification scheme of hadrons\\cite{IIM2000},\n which gives covariant quark representations for composite hadrons with\n definite Lorentz and chiral transformation properties.\n The $\\widetilde{U}(12)_{SF}$-classification scheme has a unitary symmetry\n in the hadron rest frame, called ``static $U(12)_{SF}$\n symmetry''\\cite{IIYMO2005},\n embedded in the covariant $\\widetilde{U}(12)_{SF}$-representation space,\n of which tensors can be decomposed into representations of\n $\\widetilde{U}(4)_{DS} \\times SU(3)_{F}$,\n $\\widetilde{U}(4)_{DS}$ being the\n pseudounitary homogeneous Lorentz group for Dirac spinors.\n The static $U(12)_{SF}$ contains the Dirac spin group $U(4)_{DS}$\n in its subgroups and\n $U(4)_{DS}$ contains two $SU(2)$ subgroups as\n $U(4)_{DS} \\supset SU(2)_{\\rho} \\times SU(2)_{\\sigma}$,\n where $SU(2)_{\\rho}$ and $SU(2)_{\\sigma}$ are the spin groups\n concerning the boosting and intrinsic-spin rotation, respectively,\n of constituent quarks, being connected with decomposition of Dirac\n $\\gamma$-matrices, $\\gamma \\equiv \\rho \\otimes \\sigma$.\n Thus the static $U(12)_{SF}$ symmetry includes\n the chiral $SU(3)_{L} \\times SU(3)_{R}$ symmetry as\n $U(12)_{SF} \\supset SU(3)_{L} \\times SU(3)_{R} \\times SU(2)_{\\sigma}$.\n This implies\n that the $\\widetilde{U}(12)_{SF}$-classification scheme is able to\n incorporate effectively the effects of chiral symmetry and its\n spontaneous breaking, essential for understanding of properties\n of the low-lying hadrons, into what is called a constituent quark\n model.\n\n\n\\section{Experimental Candidates for the Ground-State\n Quark-Antiquark Mesons}\n\nAn essential feature of the $\\widetilde{U}(12)_{SF}$-classification\n scheme is to have the static $U(4)_{DS}$ symmetry\n for light $u, d, s$ quarks confined inside hadrons.\n The degree of freedom on the $\\rho$-spin, being indispensable\n for covariant description of spin $1\/2$ particles, offers a basis\n to define the rule of chiral transformation for quark-composite hadrons.\n Since we have the $\\rho$-spin degree of freedom, which is discriminated\n by the eigenvalues of $\\rho _{3}$, $r=\\pm$, in addition to the ordinary\n $\\sigma$-spin, the ground states of light-quark $q \\bar{q}$ mesons are\n composed of eight $SU(3)_{F}$ multiplets with respective $J^{PC}$\n quantum numbers, two pseudoscalars\n $(0^{-+}_{\\mathrm{N}}, 0^{-+}_{\\mathrm{E}})$, two scalars\n $(0^{++}_{\\mathrm{N}}, 0^{+-}_{\\mathrm{E}})$, two vectors\n $(1^{--}_{\\mathrm{N}}, 1^{--}_{\\mathrm{E}})$, and two axial-vectors\n $(1^{++}_{\\mathrm{N}}, 1^{+-}_{\\mathrm{E}})$\n (N and E denoting ``normal'' and ``extra''),\n where each N (E) even-parity multiplet is the chiral partner\n of the corresponding N (E) odd-parity multiplet and\n they form linear representations of the chiral symmetry.\n \nSince the eigenstates only with the $\\rho _{3}$-eigenvalue\n $r=+$ are taken for heavy quarks, we have\n for heavy-light meson systems two heavy-spin multiplets,\n $(0^{-}, 1^{-})$ and $(0^{+}, 1^{+})$, which are the chiral partner\n each other, while for heavy-heavy meson systems\n we have the same $(0^{-}, 1^{-})$-spin multiplets as in the\n conventional nonrelativistic quark model.\n\n\n\\subsection{The $\\widetilde{U}(12)_{SF}$-scheme assignments\n for the observed mesons}\n\nWe try to assign some of the observed mesons to the predicted\n $q \\bar{q}$ multiplets, resorting to their $J^{PC}$ quantum\n numbers and masses. The observed meson data are taken from\n the Particle Data Group 2004 edition\\cite{PDG2004},\n except for the following mesons:\n\n\\begin{itemize}\n\n\\item $\\rho(1250)$.\n There are several experimental indications of the existence\n of the $\\rho(1250)$\n reported by the OBELIX\\cite{OBELIX1997} and LASS\\cite{LASS1994}\n Collaborations, and others.\\footnote{See the $\\rho(1450)$\n Particle Listings and the ``Note on the $\\rho(1700)$''\n in \\cite{PDG2004}.}\n \n\\item $\\omega(1200)$.\n The existence of $\\omega(1200)$ is claimed in the analysis\n of the $e^{+}e^{-} \\rightarrow \\pi^{+}\\pi^{-}\\pi^{0}$\n cross section by the SND Collaboration\\cite{SND1999}.\n \n\\end{itemize}\nWe accept the existence of these vector mesons as true\\cite{Komada}.\n The resulting assignments, though some of them are ambiguous,\n are shown in Table 1.\n\\begin{table}\n    \\caption{Experimental candidates for ground-state mesons in the\n             $\\widetilde{U}(12)_{SF}$-classification scheme.}\n\t\\includegraphics[width=.99\\textwidth]{table-1.epsi}\n\\end{table}\nHere we make some comments on these assignments.\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{(\\theenumi)}\n\\item The light scalar mesons\n $\\{a_{0}(980), \\sigma, f_{0}(980), \\kappa\\}$ are assined to\n the $(0^{++}_{\\mathrm{N}})$-nonet as a chiral partner of\n the $\\pi$-meson $(0^{-+}_{\\mathrm{N}})$-nonet.\n\n\\item The low-mass vector mesons\n $\\{\\rho(1250), \\omega(1200), K^{\\ast}(1410)\\}$ are assined\n to the $(1^{--}_{\\mathrm{E}})$-nonet as a chiral partner\n of the $(1^{+-}_{\\mathrm{E}})$-nonet\n $\\{b_{1}(1235), h_{1}(1170),$ $h_{1}(1380),$ $K_{1}(1400)\\}$.\n\n\\item The axial-vector mesons\n $\\{a_{1}(1260), f_{1}(1285), f_{1}(1420), K_{1}(1270)\\}$\n are assined to the $(1^{++}_{\\mathrm{N}})$-nonet\n as a chiral partner of the $\\rho(770)$-meson\n $(1^{--}_{\\mathrm{N}})$-nonet.\n\n\\item The recent observed mesons\n $\\{D_{sJ}^{\\ast}(2317), D_{sJ}(2460)\\}$ are assined to\n the $(0^{+}, 1^{+})$ multiplet as a chiral partner of\n the $(0^{-}, 1^{-})$ multiplet $\\{D_{s}, D_{s}^{*}\\}$\\cite{Ishida2003}.\n These newly observed mesons, together with the $\\sigma$-meson\n nonet, are the best candidates for the hadronic states with $r=-$\n whose existence is expected\n in the $\\widetilde{U}(12)_{SF}$ scheme.\n\n\\item It is noted that the normal (N) and extra (E) states\n with the same $J^{PC}$ generally mix together due to\n the spontaneous as well as explicit breaking of chiral symmetry\n and some other mechanism.\n\n\\end{enumerate}\n\n\n\\section{CHIRAL MASS SPLITTING FOR THE CHARMED AND\n CHARMED-STRANGE MESON SYSTEMS}\n\nIn the $\\widetilde{U}(12)_{SF}$-classification scheme\n heavy-light $(c \\bar{q})$ meson fields, aside from the\n internal space-time wave functions, are given by\n\\begin{equation}\n\t\\Phi(v)=\\frac{1}{2\\sqrt{2}}(1-i v \\cdot \\gamma)\n\t\t(i \\gamma _{5} \\mathbf{D}\n\t\t+i \\tilde{\\gamma} _{\\mu} \\mathbf{D}_{\\mu}^{\\ast}\n\t\t+\\mathbf{D}_{0}\n\t\t+i \\gamma _{5}\\tilde{\\gamma}_{\\mu} \\mathbf{D}_{1 \\mu})\n\\end{equation}\nwith\n\t$v_{\\mu} \\equiv P_{\\mu}\/M,\\ \n\t\\tilde{\\gamma} _{\\mu} \\equiv \\gamma _{\\mu}\n\t\t+ v_{\\mu}(v \\cdot \\gamma)$,\nwhere $(\\mathbf{D},\\mathbf{D}_{\\mu}^{\\ast},\\mathbf{D}_{0},\n\\mathbf{D}_{1 \\mu})$ represent the local fields for the\n $c \\bar{q}$ mesons with $J^{P}=(0^{-},1^{-},0^{+},1^{+})$,\n $P_{\\mu}$ ($M$) is the four-momentum (mass) of meson fields,\n and flavor indices are omitted for simplicity.\nTo describe the light-quark pseudoscalar and scalar mesons\n together with the spontaneous breaking of chiral symmetry,\n we adopt the $SU(3)$ linear sigma model, introducing\n the chiral field $\\Sigma _{5}$ defined by\n\\begin{equation}\n\t\\Sigma _{5} = s-i \\gamma _{5}\\phi\n\\end{equation}\nwith\n\t\\[ s=\\frac{1}{\\sqrt{2}} s^{a} \\lambda^{a}, \\ \\\n\t\\phi=\\frac{1}{\\sqrt{2}} \\phi^{a} \\lambda^{a} \\ \\ \\\n\t(a=0, \\cdots ,8), \\]\nwhere $\\lambda^{0}=\\sqrt{2\/3} \\ \\mathbf{1}$ and $s^{a}$\n ($\\phi^{a}$)\n are the scalar (pseudoscalar) fields. We now write\n a chiral-symmetric effective Lagrangian which gives\n the chiral mass splitting between the heavy-light\n $(0^{-},1^{-})$ and $(0^{+},1^{+})$ multiplets\n through the spontaneous breaking of\n chiral symmetry\\cite{BEH2003,Ishida2003}:\n\\begin{equation}\n\t\\mathcal{L}_{ND}=-g_{ND} \\mathrm{Tr}\n\t[\\Phi \\Sigma _{5} \\bar{\\Phi}] \\label{eq:LND},\n\\end{equation}\nwhere $g_{ND}$ is the dimensionless coupling constant\n of Yukawa interaction in the nonderivative form and\n the trace is taken over the spinor and flavor indices.\n \nWhen the chiral symmetry is spontaneously broken,\n $s$ has the vacuum expectation value,\n $\\langle s \\rangle _{0}=\\mathrm{diag}(a,a,b)$,\n where $a$ and $b$ are related to the pion and kaon decay\n constants by\n\\begin{equation}\n\ta=\\frac{1}{\\sqrt{2}} f_{\\pi}, \\ \\\n\tb=\\frac{1}{\\sqrt{2}} (2f_{k}-f_{\\pi}).\n\\end{equation}\nThen the mass splitting between the two multiplets\n is induced and the mass differences $\\Delta M_{\\chi}(c \\bar q)$\n are given by\n\t$\\Delta M_{\\chi}(c \\bar n)=2g_{ND} a$ and \n\t$\\Delta M_{\\chi}(c \\bar s)=2g_{ND} b$,\nwhich leads to the relation\n\\begin{equation}\n\t\\Delta M_{\\chi}(c \\bar n)=\\Delta M_{\\chi}(c \\bar s) \\frac{a}{b}\n\t=\\Delta M_{\\chi}(c \\bar s) \\left( \\frac{2f_{K}}{f_{\\pi}}-1 \\right)^{-1}.\n\\end{equation}\nFrom this relation with the experimental values\\cite{PDG2004},\n\t$\\Delta M_{\\chi}(c \\bar s) = 348.0 \\pm 0.8 \\ \\mathrm{MeV}$\n\tand\n\t$f_{K^{+}}\/f_{\\pi^{+}} = 1.223 \\pm 0.015$,\nwe obtain\n\t$\\Delta M_{\\chi}(c \\bar n) = 240.8 \\pm 5.4 \\ \\mathrm{MeV}$\nand consequently predict the masses\n\\begin{equation}\n\tM(D_{0}^{\\ast}) = 2.11 \\pm 0.01 \\ \\mathrm{GeV}, \\ \\\n\tM(D_{1}) = 2.25 \\pm 0.01 \\ \\mathrm{GeV}\n\\end{equation}\nfor the $(0^{+},1^{+})$ $c \\bar n$ mesons, using the measured\n mass values\\cite{PDG2004} of the $D(0^{-})$ and $D^{\\ast}(1^{-})$ mesons.\n We hereafter refer to these predicted mesons, respectively, as\n ``$D_{0}^{\\ast}(2110)$'' and ``$D_{1}(2250)$''.\n\n\n\\section{POSSIBLE INDICATIONS OF THE EXISTENCE OF LIGHT SCALAR\n AND AXIAL-VECTOR CHARMED MESONS}\n\nWe could ask experimental data whether there was some evidence\n for the existence of $D_{0}^{\\ast}(2110)$ and $D_{1}(2250)$.\n Here we check on the recent published data on the\n $D \\pi$ and $D^{*} \\pi$ mass distributions in\n $B \\rightarrow (D \\pi)\\pi$, $(D^{*}\\pi)\\pi$ decays from\n the Belle\\cite{Belle2004} and BABAR\\cite{BABAR2003} Collaborations.\n\n\\begin{itemize}\n\\item \\textbf{$D \\pi$ mass spectrum}:\n In the Belle data\\footnote{See the $D \\pi$ mass distribution\n in Figure 3 of \\cite{Belle2004}.}\n we see an excess of events, a single data point of 20 MeV bin,\n at a mass of 2.13 GeV near the predicted mass of the\n $D_{0}^{\\ast}(2110)$, and so might regard it as an indication\n of that resonance, though it is natural to think that\n its data point should be within a statistical error.\n On the other hand, it would seem to us that the BABAR\n data\\footnote{See the $D \\pi$ mass distribution\n in Figure 3 (right) of \\cite{BABAR2003}.} around a mass of\n 2.1 GeV show a typical pattern of interference between\n two or more resonances.\n\n\\item \\textbf{$D^{*}\\pi$ mass spectrum}:\n In the Belle data\\footnote{See the $D^{*} \\pi$ mass distribution\n in Figure 9 of \\cite{Belle2004}.} there is also an excess of events,\n a single data point of 10 MeV bin, at a mass of 2.255 GeV\n near the predicted mass of the $D_{1}(2250)$,\n and so it might be an indication of the resonance.\n Although it is not clear, the BABAR data\\footnote{See the $D^{*} \\pi$\n mass distribution in Figure 3 (left) of \\cite{BABAR2003}.}\n around a mass of 2.26 GeV might show a typical pattern\n of interference.\n\n\\end{itemize}\n\nIf the $D_{0}^{\\ast}(2110)$ and $D_{1}(2250)$ resonances\n really exist, their widths have to be narrow,\n $\\leq 20 \\textrm{-} 30$ MeV, judging from the data mentioned above.\n The dominant decay modes of these resonances are\n $D \\pi$ and $D^{*} \\pi$, respectively, and thus\n we examine their single pion transitions.\n To estimate the widths of $D_{0}^{\\ast}\n \\rightarrow D + \\pi$ and $D_{1}\n \\rightarrow D^{*} + \\pi$ decays, together with\n $D^{\\ast} \\rightarrow D + \\pi$, we set up,\n in addition to the nonderivative interaction $\\mathcal{L}_{ND}$\n in Eq. (\\ref{eq:LND}),\n the chiral-invariant effective interaction\n with the derivative form:\n\\begin{equation}\n\t\\mathcal{L}_{D} = g_{D} \\mathrm{Tr}\n\t[\\Phi (\\partial _{\\mu} \\Sigma _{5}) \\gamma _{\\mu}\n\t(F_{U} \\bar{\\Phi})] \\label{eq:LD},\n\\end{equation}\nwhere\n $F_{U} = \\gamma \\cdot \\partial\n \/ \\sqrt{\\partial \\cdot \\partial}$\n and $g_{D}$ is the coupling constant with a dimension\n of $(mass)^{-1}$, which is related to the axial coupling\n constant $g_{A}$ by\n $g_{D} = g_{A}\/2a = g_{A}\/\\sqrt{2} f_{\\pi}$.\n The pionic decay widths of the $D^{*}$, $D_{0}^{\\ast}$,\n and $D_{1}$ states are derived from $\\mathcal{L}_{ND}$\n and $\\mathcal{L}_{D}$, and the decay widths of\n $D_{0}^{\\ast}$ and $D_{1}$ are identical. Using the measured\n value of $\\Gamma [D^{*+} \\rightarrow D^{0} + \\pi ^{+}]\n =65 \\ \\mathrm{keV}$\\cite{PDG2004}\n and $g_{ND}=1.84$ from $\\Delta M_{\\chi}(c \\bar n)\n = 241 \\ \\mathrm{MeV}$,\n the coupling $g_{D}$ is fixed to 3.96 GeV$^{-1}$\n (corresponding to $g_{A}=0.521$), and then we obtain\n\\begin{equation}\n\t\\Gamma [D_{0}^{*}(2110) \\rightarrow D + \\pi]\n\t=\\Gamma [D_{1}(2250) \\rightarrow D^{*} + \\pi]\n\t\\approx 30 \\ \\mathrm{MeV}.\n\\end{equation}\nThis value is consistent with the speculated widths of\n the $D_{0}^{\\ast}(2110)$ and $D_{1}(2250)$.\n\n\n\\section{CONCLUDING REMARKS}\n\nWe have presented the possible assignments for some of\n the observed mesons in the covariant\n $\\widetilde{U}(12)_{SF}$-classification scheme.\n It is necessary and important to examine the strong- and\n radiative-decay\\cite{Maeda} properties of the assigned states\n in order to establish their assignments.\n On the basis of these assignments we have also predicted\n the existence of the low-mass\n $(0^{+},1^{+})$ $c \\bar n$ mesons with narrow width,\n which might have been seen in the recent published data\n on the $D \\pi$ and $D^{*} \\pi$ mass distributions\n from the Belle and BABAR Collaborations.\n\n\n\\bibliographystyle{aipprocl}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn \n\\cite{Feynman1} \nR. Feynman writes: \"I have pointed out these things because the more you see how strangely Nature behaves, the harder it is to make a model that explains how even the simplest phenomena actually work. So theoretical physics has given up on that.\"\n\n\"A model that explains how phenomena actually work\" is referred to as  a \"functional model\" in this paper. \nA functional model of a given function explicitly describes a sequence of steps for the development of the function's final state. A more detailed explanation of what in this paper is understood as a \"functional model\"  is provided in \\cite{diel2}.\n\nThe need for a functional model of QT was established when the author tried to develop a comprehensive computer model of QT.\nHe discovered that certain computability problems associated with QT\/QFT (see \\cite{diel1}) can be overcome with a functional model of QT\/QFT. A second motivation for a functional model, in particular of QT interactions, was the author's search for a solution to the QT measurement problem, one that did not require modifications of the Schr\\\"odinger equation (or similar equations of motion of QFT) or one that did not rely on assumptions outside the scope of falsifiable physics. In  \\cite{diel4}, a model of a QT measurement is described where measurements are explained in terms of QFT interactions and their functional descriptions.\n\nQuantum field theory (QFT) is the area of QT that addresses interactions among particles. The founders of QFT, including R. Feynman, recognized the need for a process-based theory of QT interactions. Feynman himself used the word \"process\" extensively when describing his quantum electrodynamics (QED) .\n\\footnote{ R. Feynman published a QED related book with the title \"The Theory of Fundamental Processes\" \\cite{Feynman2}.} The most fundamental tool in QFT, Feynman diagrams, give the impression of showing a temporal structure, i.e., a process.\nHowever, upon closer inspection, it turns out that the process orientation of QFT is insufficient and that QFT\/QED must still be considered a declarative (i.e., non-functional)  description. The term \"process\" may be applicable when viewing a QFT interaction as a whole, but the theory does not provide any temporal substructures including intermediate states. \nFeynman diagrams must not be viewed as showing a temporal structure. The fact that multiple diagrams may have to be applied for a (single) scattering process disturbs any temporal interpretation. Additionally, possible loops in Feynman diagrams do not mean temporal loops. \n\n\nThe description of (physical) processes requires a certain specification language. Only very simple dynamic processes can be specified purely in terms of mathematical equations such as differential equations. Unfortunately, there does not exist a generally agreed process (i.e., functional) specification language that is comparably as powerful and compact as the language of mathematical equations used in physics for the description of static relationships (including derivatives). The method of description used in this paper is a mixture of plain English, some semi-formal algorithmic specification language, and the mathematical equations of QFT.\n\nThe model is based on cellular automaton (CA). A CA implies a special structure for the overall dynamic evolution and for space-time. The most important implication is the discreteness with respect to time and space.\n\nThis functional model of interactions in QT is embedded in the overall functional description of QT described in \n\\cite{diel2}.\nThe key features of this overall functional description of QT are listed in Section 2. A functional description of a system describes the dynamical evolution of this system in terms of state changes. Therefore, in Section 3, the structure and components of the system state of the functional model is first described before the steps in interaction processes are described in Sections 5 and 6.\nThere are a number of important QT concepts, such as measurement, entanglement, and decoherence, that are related to interactions. These subjects are also addressed in the present paper in section 7.\n\nIn this paper, interactions are assumed to always occur between two \"in\" particles and to result in two \"out\"  particles (not necessarily identical to the \"in\" particles). Interactions between more than two particles are assumed to be  translatable into a series of interactions between two particles. Special cases, for example,  those in which QFT allows for more than two \"out\" particles, will not be addressed. \n\n\n\n\\section {Key Features of the Overall Functional Description of QT}\n\nThe functional description of interactions in QT, described in this paper, is embedded in the overall QT functional description  \n\\cite{diel2}.\nThe following  features are of primary importance for the overall functional description of QT. \n\n\\subsection{Discreteness of QT attributes}\n\nIn \n\\cite{tHooft} \nG. 't Hooft writes \"Often, authors forget to mention the first, very important, step in this logical procedure: replace the classical procedure one wishes to quantize by a strictly finite theory. Assuming that physical structures smaller than a certain size will not be important for our considerations, we replace the continuum of three-dimensional space by a discrete but dense lattice of points. \". \n\nThe functional description of QT assumes discrete and coarse grain attributes \nnot only for  three-dimensional space, but for most other entities where standard QFT assumes differentiable attributes.\nThis applies to the spatial extension of particles\/waves  and to  their momentum. Also, the wave function is structured into a\ndiscrete set of alternative paths.\n\nOf course, the graining has to be kept fine enough to prevent significant deviations from the predictions obtained\nwith standard QFT.\n\n\\subsection{The Transition from Possibilities to Facts - Handling of Non-Determinism}\n\nThe functional description\/model of QT interactions must demonstrate the evolution of the wave function to generate probability amplitudes in accordance with the predictions of QT\/QFT. However, it does not end with the determination of probability amplitudes but includes\na model for the realization of the predictions represented by the probability amplitude. This process step  is called\n\"the transition from possibilities to facts\". With standard QT, the transition from possibilities to facts is a non-deterministic process step that occurs exclusively with measurements. \n\nOne of the key features of the functional description of QT  is that  the transition from possibilities to facts  is not exclusively tied to measurements. With most interpretations of QT,  the measurement process implies a \"collapse of the wave function\".\nThere is an ongoing debate among QT physicists whether a collapse of the wave function has to be assumed. \nThe functional description assumes that measurements always imply interactions, more specifically, interactions that lead to a collapse of the wave function. The statement that  the transition from possibilities to facts  is not exclusively tied to measurements, first of all, means that the collapse of the wave function is not exclusively observed with measurement interactions. The collapse of the wave function also occurs with other (\"normal\") interactions.\n\n\\subsection{Particle Fluctuations and Interaction Channels Instead of Virtual Particles}\n\nIn the perturbation (Feynman) approach, virtual particles are an essential concept for describing interactions among\nparticles. \nThe functional description reinterprets the role of virtual particles; instead of the original QFT virtual particles, the functional description assumes \"particle fluctuations\" and \"interaction channels\". Particle fluctuations initiate the interaction, if the fluctuation affects multiple (i.e., at least two) particles. These particle fluctuations are assumed to actually occur (with a certain probability), whereas virtual particles are constructs that affect only the probability amplitudes.\nInteraction channels (like those mediated by virtual particles) guide the possible flow of particle transitions during an interaction (see Section 6.2).\n\n\\subsection{Splitting of a Wave Function \\emph{Collection} into Multiple Paths}\n\nThe splitting of a wave function into multiple paths is a constituent part of the perturbation approach to QFT (see  \\cite{Feynman1}). \nThe overall effect of the wave function progression is then determined by the superposition (via path integrals) of the multiple paths.\nWith the QT functional description, the splitting into multiple paths is applied to \\textbf{collections} of particles \nwhich exit an interaction.\nThis allows for the modeling of entanglement (see section 7.6).\n\n\\section{The System State} \n\nBecause the QT functional model does not distinguish between a particle and the (associated) wave, the  term \"particle\/wave\" will be used in the following.\n\\footnote{In the literature on QT some authors used the name \"wavicle\" for what here is called \"particle\/wave\".}\n\nThe description of the evolution of a quantum system, such as a collection of\nparticles\/waves, must be related to the information that makes up this quantum\nsystem. \nFor the functional model it is useful to arrange the information in a certain structure, primarily derived from  the  differing variability and lifetime of the entities. \nThe information that represents the quantum system for the functional\nmodel must encompass all of the entities known from QT\/QFT, such as state\nvectors, wave functions, masses, charges, etc. \nIn addition, the functional model must include objects and state components which are suited for the description of intermediate states. \n\nFor the functional model, the totality of information constituting a quantum\nsystem consists of a set of q-objects plus a set of fields.\n\\small{\n\\begin{verbatim}\nQT-system :=   \n  q-object-set,\n  field-set;\n\\end{verbatim}\n}\n\\normalsize\nFields (i.e., \"field-set\") are not further addressed in the present paper.\n\n\\subsection{The Quantum Object, q-object} \n\nThe most general entity for the description of a quantum system is the q-object.\nA q-object is an aggregate object that can be described by a common wave function and not just the product of the wave functions of the elements of the q-object. A particle\/wave may occur as a separate q-object, or may be part of a q-object.\n\\\\\nFor example, the wave function \n\n(1) $  \\psi =   1\/ \\sqrt{2}\\;  ( | \\; pw1.up, pw2.down > + \\;  | \\;  pw1.down, pw2.up) >   $ \n\\\\\nrefers to a q-object $  \\psi $ with elements (e.g., entangled particle\/waves) pw1 and pw2.\n\nA q-object may be viewed as having a two-dimensional structure. One dimension represents the elements of the q-object (with the above example, pw1 and pw2); the other dimension represents alternatives that may be selected during the evolution of the q-object, for example, by an interaction. In this paper, these alternatives are called \"paths\". Each path has associated a probability amplitude.\n\n\\small{\n\\begin{verbatim}\nq-object :=   \n  path[1],\n  ...\n  path[NPATH];\n\\end{verbatim}\n}\n\\normalsize\n\\begin{table}\n\\caption{\\label{label}Structure of a q-object consisting of two particle\/waves pw1 and pw2.}\n\\begin{tabular} { | c | c | c  | c | }\n\\hline\npaths & pw1-state & pw2-state  & amplitude  \\\\\n\n\\hline\n\npath-1\t  &   pw1-state$_{1}  $    &  pw2-state$_{1} $  & ampl-1  \\\\\n\npath-2\t &   pw1-state$_{2}  $    &  pw2-state$_{2} $   & ampl-2  \\\\\n\n...\t            & ...         & ...    & ...  \\\\\n\npath-N\t &  pw1-state$_{N} $    &  pw2-state$_{N} $  & ampl-N  \\\\\n\\hline\n\\end{tabular}\n\\end{table}  \n\\small{\n\\begin{verbatim}\npath :=   \n  state-element[1], ...,state-element[n], amplitude;\n\\end{verbatim}\n}\n\\normalsize\nWith the above example at least two paths, path[1] := ( pw1.up, pw2.down ) and \npath[2] := ( pw1.down, pw2.up), both with amplitude =  $ 1\/ \\sqrt{2} $ may express the state of a q-object with a specific system evolution. \n\nThe wave function for $ \\psi $, eq. (1), specifies a continuous set of possible measurement results for  $ \\psi $. Conversely, the q-object of the functional model  specifies a discrete set of paths that may be selected with interactions and measurements.\n \nDifferent types of q-objects, containing different types of elements, can be distinguished.\nThe simplest type of q-object is the (single) particle\/wave. In this paper, in addition to the particle\/wave, two other q-objects, the particle\/wave-collection (pw-collection) and the interaction-object, are of particular importance. A pw-collection represents a collection of particles\/waves that are entangled. Table 1 shows the structure of a pw-collection.\nThe interaction object is an interaction internal object created at the beginning of an interaction. At the end of the interaction, the interaction object is transformed to a pw-collection.\n\nIn addition to the q-objects addressed in this paper, there are further entities in QT, such as bound systems, which may fall under the concept of q-objects. This functional model assumes that q-objects are created from interactions and dissipate (i.e., collapse) when they get involved in new interactions.\n\\footnote{As a consequence, the idea of the whole universe constituting a single big q-object (with a common probability amplitude) is not supported by the functional model.}\n\nThe state of an element of a q-object (above denoted state-element []) consists of the state-components known from QFT.\nFor QED, these are, first of all, the parameters used in QFT to specify a matrix element of the scattering matrix\n$ \\Psi_{p1,\\sigma1,n1;p2,\\sigma2,n2, ...} $, i.e., the four-momenta $ p^{\\mu}$, the spin z-component (or for massless particles, helicity) $ \\sigma $, and the particle type n. The Lagrangian, Hamiltonian, and the equations of motion for the typical fields of QFT contain the time derivative $ d\\phi \/ \\partial t $ of the state $ \\phi $. For the functional model, this time derivative therefore  has to be included as part of the state. \nIn addition, the position vector x is part of the state.\n\n\n\\section{The Cellular Automaton}\n\nTo describe the dynamical evolution of the states of a system, some language or description method is required. \nThe method chosen in this paper for the specification of the dynamical evolution of \na QT systems a cellular automaton (CA).\n\n\\subsection{Standard Cellular Automaton}\nThe standard CA consists of a k-dimensional grid of cells. The state of the CA is given by the totality  of the states of the individual cells.\n\n   $    S_{CA} =  \\{ s_{1}, ... , s_{n} \\} $\n\nWith traditional standard CAs, the cell states uniformly consist of the same state components\n\n   $    s_{i} =  \\{ s^{1}_{i}, ... , s^{j}_{i} \\} $\n\\\\\nTypically, the number of state components, j, is 1, and the possible values are restricted to integer numbers.\n\nThe dynamical evolution of the CA is given by the (single general) \"update-function\" which computes the new \nstate of a cell as a function of  its current state and the states of the neighbor cells.\n\\small{\n\\begin{verbatim}\nStandard-CellularAutomaton(initial-state)  :=     \/\/ transition function\nDO FOREVER {\n   state = update-function(state, timestep);\n   IF ( termination-state)  STOP;\n}\n\\end{verbatim}\n}\n\\normalsize\nThe full complexity (if any) of a particular cellular automaton  is concentrated in the update-function. As Wolfram (see \\cite{Wolfram}) and others (see, for example, \\cite{Ilachinski}) showed, a large variety of process types (stable, chaotic, pseudo-random, oscillating) can be achieved with relatively simple update-functions.\n\nFor specific applications of the cellular automaton, the update-function may be derived from application specific specifications.\nIn \\cite{Elze1}, Elze describes a cellular automaton whose update-function is derived from the Hamiltonian (or the equation of motion). \n\n\\subsection{QFTCA, a Cellular Automaton Supporting QFT }\n\nThe CA described in \\cite{Elze1} , \\cite{Elze2}, and  \\cite{Hooft} may be viewed as the starting points for the CA described in the present paper. \nTo support QFT (at least to the extent required to demonstrate a model of interactions), a cellular automaton QFTCA is defined; QFTCA, however, requires certain extensions of the standard cellular automaton and embedding of the cellular automaton in an overall QFT-based structure. The embedding into the QFT-based structure affects two aspects, the structure of the state of the system and the functions for the dynamic evolution of the system.\n\n\\subsubsection{State structure}\n\nThe CA-cells may be viewed as representing the space.\nThe cells have associated with them (i.e., contain) parts of the system state.\nFor the mapping of the overall system state, as described in Section 3, to QFTCA, some \"design decisions\" have to made.\nThe following lists the major characteristics (without supporting arguments):\n\\begin{itemize}\n\\item Time is \\emph{not} a (fourth) dimension for QFTCA. Instead, the time derivatives  $  d\\psi \/ \\partial t $ and $ p^{\\mu}$ are explicit parts of the system state.\n\\item Differing from the standard CA (mentioned in Section 4.1), the states of QFTCA cannot be expressed solely by the collection of cell states. Those parts of the q-object state that are position dependent can be assigned to cell states. Other parts have to be kept besides the cell states. For the functional model described in this paper, left open is the assignment of state components to CA cells as opposed to state components that are kept with the (paths of the) q-objects. We need to ensure that the cells (i.e., space points) belonging to a q-object can be determined and vice versa, that the q-objects occupying a CA cell can be determined.\n\\item A q-object path may cover multiple cells.\n\\item A cell may be covered by multiple q-object paths.\n\\item The above-described state structure and timing considerations result in differing content of CA cells and differing CA update functions. For the proposed model of the QT measurement process, this could be implemented by either  (a) a single relatively complex QFTCA, or (b) by a QFTCA consisting of a collection of  CAs that will partly merge whenever interactions are performed.\nIn the present paper, the decision of (a) or (b) and the details of (a) and (b) are left open.\n\\end{itemize}\n\n\\subsubsection{Evolution of the system state} \nThe structuring of the overall system state into  the CA-cells and  the superimposed q-objects was also motivated by the differing update requirements of the QT-objects.\n\\small{\n\\begin{verbatim}\nQFTCA (initial-state)  :=  \/\/ transition function\nDO FOREVER {\n   state = global-update-function(state, timestep);\n   IF ( termination-state)  STOP;\n}\n\nglobal-update-function(state, timestep)  :=    \nDO PARALLEL {\n  field-state = field-update-function(field-state, timestep);\n  FOR ( all qobjects qobj[i] ) {\n        propertimestep = fx(qobj[i]) * timestep;\n        FOR ( all particle\/waves pw[k] of qobj[i]) {\n              pw[k] = pw-update-function(pw[k], propertimestep);\n             IF ( interaction-occurred( pw[k], pw2 )  \n                   perform-interaction( pw[k], pw2 );\t\n        }\n  }\n}\n\\end{verbatim}\n}\n\\normalsize\nQFT is a relativistic theory. Special relativity distinguishes proper time (or wrist-watch time) of the inertial system and global time of the overall space-time system.\nFor the evolution of the overall system (state), the functional model assumes that the QFTCA proceeds in uniform global time steps. The update-function for the individual q-objects, however, have to proceed in proper time associated with the \nq-objects.\n\nAs a simplified description \"propertimestep = fx(qobj[i]) * timestep;\" indicates the transition from global time to proper time as requested by special relativity. A more detailed and more proper description is outside the scope of this paper.\n\nThe update functions for fields (field-update-function()) and for the normal particle propagation (pw-update-function()) are straightforward and not topics discussed in this paper. The topics discussed are the update functions related to interactions (interaction-occurred() and perform-interaction()). These are addressed in Sections 5 and 6.\n\n\\section{Overall Model of Interactions between Particles\/Waves}\n\nTwo types of interaction between particles\/waves are distinguished: (1) interactions\nthat destroy the superposition among possible multiple paths of a wave function, resulting in a collapse of the wave function, and (2) interactions that only affect the attributes (e.g., momentum) of the involved particles\/waves. For this description of the functional model, interactions resulting in a collapse of the wave function are considered to be the general case. Interactions that do not result in a collapse are exceptions. They are called volatile interactions and will be addressed where appropriate and explicitly in Section 7.2.\n\nWith QFT (see \\cite{Weinberg}) an interaction (e.g., scattering) is described by the scattering  matrix (S-matrix)  which assign a probability amplitude $ S_{\\beta \\alpha}$ to  the transition of a given \"in\"  state $ \\Psi_{\\alpha} $ to an \"out\" state $ \\Psi_{\\beta} $.\n\\\\\n $ S_{\\beta \\alpha} = ( \\Psi_{\\beta}, \\Psi_{\\alpha} )  $\n\\\\\nThe \"in\" and \"out\" states are specified by their state components (see Section 3)\n  $  \\Psi_{\\alpha} = \\Psi_{p1,\\sigma1,n1; p2,\\sigma2,n2, ...};  \\Psi_{\\beta} = \\Psi_{p1',\\sigma1',n1'; p2',\\sigma2',n2', ...}  $\n\\\\\nThe functional model of QFT interactions has to provide a model for the process that transforms an \"in\" state\n$  \\Psi_{\\alpha} = \\Psi_{p1,\\sigma1,n1; p2,\\sigma2,n2, ...}$ into a multitude of possible \"out\" states $ { \\Psi_{\\beta 1}, \n\\Psi_{\\beta 2}, ... }$ .\nThis multitude of possible \"out\" states constitutes a pw-collection as presented in Table 1.\n\nThe overall model of interactions between particles\/waves is based on the following assumptions:\n\\begin{itemize}\n\\item Interactions are process steps that actually occur (with a certain probability) rather than wave function alternatives which are in superposition with \"no interaction occurrence\".\n\\item An interaction always occurs at a definite (discrete) point in space-time.\n\\\\\n(The QFT model of interactions in coordinate space, where the possible results of an interaction are computed by assuming superpositions among all possible interaction positions, is not supported by the functional model.) \n\\item Only those q-object paths that cover  the interaction position affect the outcome of the interaction. If multiple paths of a q-object cover the interaction position, only one of them is \nselected as the interacting path.\n\\item The non-selected paths are discarded.\n\\item The selection of the interaction position and of the significant path represents (a first step in) the transition of probabilities to facts.\n\\item The discarding  of the non-selected paths can be viewed as \"the collapse of the wave function\".\n\\end{itemize}\n\\begin{figure}[ht]\n\\center{\\includegraphics*[scale=0.5] {figure1a.JPG} }\n\\caption{Information flow with the interaction between two particle\/waves}\n\\end{figure}\nFigure 1 shows the overall flow of information with the interaction of two particle\/waves pw1 and pw2. The figure contains four q-objects: pw1, pw2 (the figure does not assume pw1 and pw2 belonging to the same q-object), the interaction-object, and the \"out\" pw-collection containing the \"out\" particle\/waves.\nAt the beginning of the interaction the information contained in pw1 and pw2 is merged into the interaction object. The \ninteraction object performs a sophisticated process, including the formation,  union, and splitting of \"ia-channels\", which finally results in the \"out\" pw-collection.\n\n\\subsection{Occurrence of an interaction}\n\nAlthough QFT provides precise rules for the computation of probabilities (amplitudes) for the occurrence of specific interaction results as a function of the \"in\" states of the involved particle waves, it does not provide further details on the circumstances that must hold for an interaction to occur. In contrast, the functional model of QT interactions has to include a process model for the occurrence of an interaction that supports the above listed basic assumptions. The author claims that measurements imply interactions and therefore interactions occur with probabilities that are equal to the probabilities from the results of measurement.\nThe model chosen by the author for the occurrence of  interactions is the \"particle\/wave fluctuation\" (pw-fluctuation).\nA pw-fluctuation can be thought of as a temporary concentration and amplification of one or several particles\/waves at a certain point in space. The following assumptions are essential in considering pw-fluctuations and their role in the functional model of QT interactions:\n\\begin{itemize}\n\\item With the functional model each interaction is preceded by a pw-fluctuation. \n\\item Only one pw-fluctuation can be active at a given point in time for a  particle\/wave.\n\\item The position where the pw-fluctuation occurs can be anywhere within the space occupied by the involved\nparticles\/waves. The position is determined\nrandomly as a function of the amplitudes of the involved particles\/waves and of the fields involved.\n\\end{itemize}\nThe immediate effect of a pw-fluctuation is the temporary formation of an entity called an interaction-object (see section 6.1). \nWhen the temporary interaction object disappears again, the original particles\/waves may persist (or\nmay be reinstalled), or a different set of particles\/waves may appear. Accordingly, the\nlong-term effect of a pw-fluctuation can be one of the following:\n\\begin{enumerate}\n\\item nothing durable (this may be the case with the majority of pw-fluctuations),\n\\item an interaction \\emph{with} a collapse of the wave function (the major case considered in this paper),\n\\item an interaction \\emph{without} a collapse of the wave function (only roughly addressed in section 7.2),\n\\item a particle decay (not further addressed in this paper).\n\\end{enumerate} \n\n\\subsection{Results of an interaction}\nThis functional model of QT interactions is based on QFT. With QFT, the possible results of an interaction are provided by the S-matrix. It includes probability amplitudes for possible results for a given \"in\" state. The functional model has to deliver the collection of the complete variety of possible \"out\" states and their probability amplitudes in form of the \"out\" pw-collection.\n\n\\section{Functional Steps in an Interaction Process}\n\nThe functional description must show process steps for the dynamic evolution of an interaction. The process steps have to produce results that are compatible with the results predicted by standard QT\/QFT in form of the S-matrix.\n\nFirst thoughts on the mapping of QFT to a functional model may start with looking at the operators and operator equations of QFT. For example, in \\cite{Mandl} the \"\\emph{processes}\" that contribute in QED to the determination of the S-matrix, are expressed as\n\n$ H_{W}(x) = -eN \\{ ( \\bar{\\psi^{+} } +  \\bar{\\psi^{-} }) ( \\not A^{+} + \\not A^{-}) ( \\psi^{+} -  \\psi^{-}) \\}_{x} $\n\\\\\nwhere $ \\bar{ \\psi^{+}}, \\bar{ \\psi^{-}}, \\not A^{+}, \\not A^{-},  \\psi^{+},  \\psi^{-} $ are creation and annihilation operators.\n\nThe further QFT treatment of interactions leads to Feynman diagrams. Although Feynman diagrams already seem to contain certain process-oriented aspects, the QFT operators and Feynman diagrams are not directly usable as  basis for a functional (i.e., process-based) description. \n\nInstead of the QFT creation and annihilation operators, the functional model contains split() and combine()  operators. Similar to the creation and annihilation operators, the split() and combine()  operators have to appear in pairs.\nInstead of the Feynman diagrams, the functional model contains \"interaction channels\". \nOf course, it has to be ensured that the functional model results in QFT compatible predictions.\n\nThe description of the overall interaction process is subdivided into three process steps:\n\\small{\n\\begin{verbatim}\nperform-interaction ::= {\n  Step1: Formation of interaction-object;\n  Step2: Formation and processing of ia-channels;\n  Step3: Generation of \"out\" particle\/wave collection;\n}\n\\end{verbatim}\n}\n\\normalsize\n\n\n\\subsubsection{Example: Bhabha Scattering}\nQFT provides rules and equations for the computation of scattering matrix amplitudes. The equations are derived from the pertinent Feynman diagrams. For Bhabha scattering the equations are\n\n(1)  $ M_{A} =  (-ie)^{2} \\bar{v}(\\vec{p}_{2}, s_{2} ) \\gamma_{\\mu}  u(\\vec{p}_{1}, s_{1} ) (-ig^{\\mu\\nu}\/(p_{1} + p_{2})^{2})\n \\bar{u}(\\vec{p}'_{1}, s'_{1} ) \\gamma_{\\nu}  v(\\vec{p}'_{2}, s'_{2} ) $.\n\\\\and\n\n (2)  $ M_{B} =  (-ie)^{2} \\bar{u}(\\vec{p}'_{1}, s'_{1} ) \\gamma_{\\mu}  u(\\vec{p}_{1}, s_{1} ) ( -ig^{\\mu\\nu}\/(p_{1}-p'_{1})^{2})\n \\bar{v}(\\vec{p}_{2}, s_{2} ) \\gamma_{\\nu}  v(\\vec{p}'_{2}, s'_{2} ) $\n \\\\\nAccording to the usual QFT notation $ u() $  represents the \"in\" electron, $ \\bar{u}()$  the \"out\" electron, $ \\bar{v}$  the \"in\" positron, and $ v() $ the \"out\" positron. $ M_{A} $ and $ M_{B} $  are the probability amplitudes. The total probability amplitude M for  Bhabha scattering (first order perturbation) is $ M = M_{A} - M_{B} $.\n\nThe \"in\" electron  $ u() $ and the positron $ \\bar{v}() $ provide the initial state of pw1 and pw2. Differing from standard QFT computations, no specific value can be assumed for the states of  $ \\bar{u}()$ and $ v() $ when an interaction starts within the functional model. Therefore, the result of an interaction within the functional model will not be a single probability amplitude M, but a set of probability amplitudes embraced in the \"out\" pw-collection. \n\n\\subsection{Formation of interaction object}\n\nAt the beginning of an actual interaction, the information from the interacting particles\/waves is combined into the interaction object. At the end of the interaction, the interaction object is replaced by a new particle\/wave collection\nrepresenting the \"out\" particles\/waves. The interaction object is a special type of q-object.\nWhen the interaction object is initialized, it contains only a single path which contains the information from the selected paths of the two \"in\" particles\/waves.\n\\small{\n\\begin{verbatim}\npw-ia-object :=   path[1];\n\npath :=\n  pw1.selectedPath.attributes,   pw2.selectedPath.attributes,  amplitude;\n\\end{verbatim}\n}\n\\normalsize\nTo support the transition from the \"in\"\nparticles\/waves to the \"out\" particle\/wave collection, the interaction object has\na very dynamic structure.\nDuring the interaction process, the interaction object will be extended by ia-channels which reflect \nintermediate states.\n\\small{\n\\begin{verbatim}\npw-ia-object :=   \n  path[1] := ia-channel[1],\n  ...\n  path[k] := ia-channel[k];\n\\end{verbatim}\n}\n\\normalsize\n\nThe functional model assumes that interaction objects, similarly to virtual\nparticles, have a limited life-time before they decay into the particles\/wave collections that\nare the result of the interaction. \n\n\\subsubsection{Example: Bhabha Scattering}\nFor the formation of the interaction object the mapping of QFT to the functional model is still rather trivial. $u(\\vec{p}_{1}, s_{1} ) $ is mapped to pw1.selectedPath.attributes;  $  \\bar{v}(\\vec{p}_{2}, s_{2} ) $ is mapped to   pw2.selectedPath.attributes. Differing from standard QFT, however, the functional model assumes only single paths to be selected as input for the further processing of the interaction. \n\n\\subsection{Formation and processing of ia-channels}\n\nThe processing of interactions proceeds with the formation of ia-channels. Like the paths of a pw-collection (see section 3), an ia-channel represents one or multiple alternative(s) for the evolution of the \"in\" particles\/waves toward the\ninteraction result. Each ia-channel starts with both \"in\" particles\/waves and\nends with \"out\" particles\/waves. Alternative ia-channels may differ in the set\nof \"out\" particles\/waves and\/or in the sub-channels between the \"in\" and \"out\"\nparticles\/waves.\n\nA specific ia-channel is formed by the combination of the two operators\nsplitl() and combine().\n\n split(a) $ \\rightarrow $  (b,c)  means   that split(a) results in b and c;\n\ncombine(a,b)$  \\rightarrow $  (c)  means that combine(a,b) results in c.\n\\\\\nFor example, starting with two interacting particles\/waves pw1 and pw2, the ia-channel  \n\\\\\n  (pw1,pw2): split(pw1) $  \\rightarrow $ (a,b);  combine( a, pw2 ) $ \\rightarrow $ ( c)   \n\\\\ would result in \"out\" particles\/waves b and c.\n\nThe split() and combine() operators are analogous (although not equal)  to the creation and annihilation operators of QFT. \n\nDerived from QFT the following rules are established for the functional model of interactions:\n\\begin{itemize}\n\\item Rule1: An interaction always starts with two \"in\" particles\/waves and ends with two \"out\" particles\/waves,\n\\item Rule2: An ia-channel always contains one combine() and one split() (in arbitrary sequence).\n\\footnote{In QFT deviations from these rules can be found which, however, will not be addressed here.} \n\\end{itemize}\nGiven the above rules, there are five possible ia-channels which can be constructed from the two interacting particles\/waves pw1 and pw2:\n\\begin{enumerate}\n\\item combine( pw1, pw2 )   $  \\rightarrow $ (a)  split(a)  $  \\rightarrow $ (b,c)\n\\item split(pw1)  $  \\rightarrow $ (a,b) combine( a, pw2 ) $  \\rightarrow $ (c)\n\\item split(pw1)  $  \\rightarrow $  (a,b) combine( b, pw2 ) $  \\rightarrow $ (c)\n\\item split(pw2)  $  \\rightarrow $  (a,b) combine( pw1, a ) $  \\rightarrow $ (c)\n\\item split(pw2)  $  \\rightarrow $  (a,b) combine( pw1, b ) $  \\rightarrow $ (c)\n\\end{enumerate}\nAll these ia-channels end with two particles\/waves (a,c) or (b,c). Depending on\nthe type of particles\/waves involved in the interaction, QFT supports only specific\nkinds of splits and combines. The rules that define what combinations of particle\ntypes may be subject to combine(p1, p2) and what the resulting particle types\nof split(p1) and combine(p1, p2 ) can be are equivalent to the rules regarding the\npossible vertices of Feynman diagrams as described in numerous textbooks on\nQFT\n (see for example, \\cite{Ryder}, \\cite{Griffiths},  \\cite{Mandl}).\n For QED, for example, one of the rules is\n split(  $ \\gamma) \\rightarrow  ( e^{-},  e^{+}) $;   combine($ e^{-}, \\gamma) \\rightarrow ( e^{-}) $ .\n\\\\\nSometimes, QFT rules allow for multiple results for split() and combine(). For example, split(photon) may result in (electron, positron), (muon, antimuon),\nor (tauon, antitauon). Finally, with specific \"in\" particle\/wave combinations, it may turn out that some of the possible\n ia-channels may be considered equivalent, and therefore, only one\nof them has to be included. \n\\footnote{The rules regarding when ia-channels may be considered equivalent are defined with QFT (in terms of equivalent Feynman diagrams) and are not further addressed here.}\nGiven the QFT rules, typically only one or two of the above mentioned five alternative ia-channels are possible for a specific \"in\" particle combination.\n\nThe operator split(a) $\\rightarrow $ (b, c) is a non-bijective function insofar as there\nare many alternatives with respect to the attributes (e.g., momentum and spin)\nof the resulting (b, c). Rather than selecting a particular specific result, QFT\n(and the functional model of QT interactions) requires that the multitude of\npossible results is generated, with differing probability amplitudes assigned.  \n\\footnote{The functional model of QT interactions assumes a certain granularity with respect to the multitude of possible results.}\nThus,   split(a) $ \\rightarrow $ (b, c) results in a two-fold splitting and may be expressed as\n\nsplit(a) $ \\rightarrow  ((b_{1}, c_{1}),  (b_{2}, c_{2}), ... (b_{n}, c_{n})) $. \n\\\\\nFor the interaction object this means that a multitude of paths representing $  ((b_{1}, c_{1}),  (b_{2}, c_{2}), ... (b_{n}, c_{n})) $ has to be generated. As a consequence, at the end,  each ia-channel contains a multitude of paths, because each ia-channel includes a split() operator.\nThe alternative ia-channels that are generated by the varying application of \nthe split() and combine() operators are processed in parallel.  The effects\nof processing the split() and combine() operators is reflected in extensions of and\nchanges in the interaction object. \n\nThe rules governing the computation of the amplitudes of a path of the interaction object must be in accordance with QFT. These rules are well known and described in many textbooks\non QFT\n(see for example, \\cite{Ryder}, \\cite{Griffiths},  \\cite{Mandl}). \nHowever,\nwith QFT the respective rules are defined in terms of (external and internal ) lines and vertices of Feynman diagrams. For the functional model, the QFT rules have been mapped to rules regarding the split() and combine() operators. \n\\footnote{This mapping is not described in this article.} \n\n\\subsubsection{Example: Bhabha Scattering}\nBhabha scattering refers to the electron-positron scattering process  $ ( e^{-}, e^{+} ) \\rightarrow ( e^{-}, e^{+} ) $.\nDerived from the rules of QFT (more specifically, quantum electrodynamics), the following ia-channels are possible:\n\\begin{itemize}\n\\item CA: $ combine1( e^{-}, e^{+} ) \\rightarrow ( \\gamma );$      $   split2( \\gamma ) \\rightarrow (  e^{-}, e^{+} ) $\n\\item CB1:  $ split1( e^{-}) \\rightarrow (  e^{-}, \\gamma );  $   $ combine2( \\gamma,  e^{+} ) \\rightarrow ( e^{+} )$\n\\item CB2: $ split1( e^{+}) \\rightarrow (  e^{+}, \\gamma ); $    $ combine2( \\gamma,  e^{-} ) \\rightarrow ( e^{-} )$\n\\end{itemize}\nCB1 and CB2 can be shown to be equivalent. Therefore in the following only CA and CB (=CB1) will be considered. As can be easily demonstrated CA corresponds to equation (1) for $ M_{A} $ whereas CB  corresponds to equation (2) for $ M_{B} $.\n\\\\\nConsequently, CA $ = combine( e^{-}, e^{+} ) \\rightarrow ( \\gamma );   split( \\gamma ) \\rightarrow (  e^{-}, e^{+} ) $ has to be mapped to \\\\\n $  (-ie)^{2} \\bar{v}(\\vec{p}_{2}, s_{2} ) \\gamma_{\\mu}  u(\\vec{p}_{1}, s_{1} ) (-ig^{\\mu\\nu}\/(p_{1} + p_{2})^{2})\n \\bar{u}(\\vec{p}'_{1}, s'_{1} ) \\gamma_{\\nu}  v(\\vec{p}'_{2}, s'_{2} )  $ and\n\\\\\nCB $ = split( e^{-}) \\rightarrow (  e^{-}, \\gamma ), combine( \\gamma,  e^{+} ) \\rightarrow ( e^{+} )  $ has to be mapped to \\\\\n $ (-ie)^{2} \\bar{u}(\\vec{p}'_{1}, s'_{1} ) \\gamma_{\\mu}  u(\\vec{p}_{1}, s_{1} ) ( -ig^{\\mu\\nu}\/(p_{1}-p'_{1})^{2})\n \\bar{v}(\\vec{p}_{2}, s_{2} ) \\gamma_{\\nu}  v(\\vec{p}'_{2}, s'_{2} ) $.\n\\\\\nThis allows the derivation of the general function logic for the split() and combine() functions.Without going into further details here, the following points are worth mentioning:\n\\begin{itemize}\n\\item Fermion chains known from QFT have to be observed with the computations for  split() and combine(). \n\\item The details of the functions split() and combine() depends on which of these two functions appears first.\n\n\\end{itemize}\n\n\\subsection{Generation of \"Out\" Particle\/Wave Collection}\n\nThe processing of the ia-channel ends with a certain \"out\" particle\/wave combination. With some types of interactions, different \"out\" particle\/wave combinations may occur. The functional model of QT interactions assumes that from the possibly multiple alternative \"out\" particle\/wave combinations only one will actually leave an interaction. This is a further case of \"transition from probability to facts\" which is possibly not in agreement with standard QFT. \n\\footnote{The possible deviation from standard QFT, however, will be difficult to test in experiments.}\nThe determination of the \"out\" particle\/wave combination may be performed somewhere between step 2 (Formation and processing of ia-channels) and (the  present) Step 3.  (Generation of \"Out\" Particle\/Wave Collection ). In this paper the function is addressed in step 3. The detailed mechanism for the selection of the \"out\" particle\/wave combination is beyond the scope of the present paper. Several alternative mechanisms are imaginable.\n\nAfter processing the individual ia-channels (in parallel) and dropping those\nia-channels that do not deliver the selected \"out\" particle\/wave combination,\neach ia-channel contains the same set of paths, however, with different probability amplitudes for the paths. Therefore, the multiple ia-channels can be (re-) united by the summation of the corresponding amplitudes. According\nto QFT rules (usually formulated in terms of Feynman diagrams), the \"summation\" in some cases has to be performed with a negative sign (i.e., $ amplitude1 - amplitude2 $ instead of $ amplitude1 + amplitude2 $).\n\n\\subsubsection{Example: Bhabha Scattering}\n\nElectron-positron scattering may result in lepton pairs (electron, positron), (muon, antimuon), or (tauon, antitauon). For the selection of the \"out\" particle\/wave combination, QFT does not offer any rules besides the equations for the computation of the probability (amplitudes) for the different channels. The selection of the \"out\" particle\/wave combination by the functional model, of course, has to be in accordance with the respective QFT equations.\n\nThe summation (or subtraction) of the probability amplitudes also follows the rules defined by QFT.\n\n\\section{Related topics}\n\n\\subsection{Collapse of the Wave Function} \n\nWhen an \"out\" particle\/wave collection is generated, possibly existing \"in\" particle\/wave collections are obsolete. The \n\"collapse\" of the \"in\" particle\/wave collection consists of two sub-steps:\n\\begin{itemize}\n\\item The particles\/waves that are  involved in the interaction (i.e., pw1 and pw2) will be discarded from their \"in\" particle\/wave collections.\n\\item Those  particles\/waves that are not involved in the interaction will (of course) survive, however, only the paths which caused the interaction.\n\\end{itemize}\nThe fact that this reduction to a single path also affects other particles\/waves that are not (directly) involved in the interaction (but \"entangled\" with the interacting particles\/waves) supports entanglement. \n\n\\subsection{Volatile Interaction - Interactions that Do Not Destroy the Superpositions}\n\n\nNumerous cases are known in the QT concerning\ninteractions between particles\/waves and other matter (particles, atoms, devices, etc.) in which there is no destruction of the superposition before ultimately\na measurement occurs. Typical examples are photons being reflected at mirrors\nor photons passing through transparent materials.\nBased on the process steps described in section 6., which imply the collapse\nof the \"in\" pw-collections, the question becomes under what circumstances will the mechanism described in section 6.1  be suppressed such that the\n\"in\" particle\/wave collections will be preserved. The primary example of this case, which can\nhardly be mapped to the process described in section 6, is the interaction\nbetween a particle\/wave and a bound system, if the bound system has to be\ntreated as a single (large) entity. Bound systems are poorly understood with\nQFT (see below). This functional model of QT, which is mainly a mapping of\nQT\/QFT to a process-oriented model, therefore does not yet contain precise criteria for the determination of when the interaction with a bound system will be\nvolatile. Some widely accepted rules seem to be (1) the higher the energy\/mass\nof the bound system is, the higher the probability of a volatile interaction becomes;\nand (2) the lower the energy\/mass of the scattered particle\/wave (e.g., photon)\nis, the lower the probability of a non-volatile interaction becomes.\n\nThe author expects that the development of a better understanding of bound systems\nmay also provide answers to the question regarding when bound\nsystems must be treated as an entity (resulting in non-volatile interactions).\n\n\n\\subsection{Interaction with a Bound (State) System}\n\nThis functional model of QT interactions is largely derived from the physics\nof interactions as provided by QFT. Predictions of the behaviour of bound (state)\nsystems (henceforth called bound system), such as an atom, a nucleus, or a hadron, can be computed using QFT\nfor special situations only, and only with considerable effort. QFT includes some\ntheory and considerations regarding bound systems \n( see  \\cite{Weinberg},  \\cite{Veltman}, \\cite{Griffiths}), \n however, this does\nnot include a complete and consistent description of the total system in terms\nof QFT constructs such as Feynman diagrams. In \n \\cite{Weinberg} \nS. Weinberg writes, \"It must be said that the theory of relativistic effects and radiative corrections\nin bound states is not yet in entirely satisfactory shape\".\n\nAs a consequence of this QFT weakness, the QT functional model cannot yet offer\na complete model of the interaction between a particle\/wave and a bound system or\nof the interaction between two bound systems. As the major open question, it is not\nunderstood when or to what extent a bound system can\/must be treated as an entity\nor the whole interaction process has to be broken down to interactions among (elementary)\nparticles\/waves. \n\n\\subsection{Model of Measurement Process}\nOne of the motivations for the development of the functional model of QT was the author's suspicion that the apparent nonlinear evolution of the wave function with measurements can get a more plausible explanation, if one relinquishes the requirement that physical processes be describable purely by differential equations. \n\nThe proposed functional model of QT interactions enables a model for the measurement process where there is no explicit \"measurement\" process (step). Measurement is explained in terms of normal interactions as described in the present paper.\nThe functional model of measurements described in \\cite{diel4} is based on the following assumptions:\n\\begin{itemize}\n\\item Measurements require interactions between the measured QT object and part of the measurement apparatus. In general, QT interactions (a) imply transitions from probabilities to facts and (b) have to adhere to the rules and equations of quantum field theory (QFT).\nThus, as described in this paper, the evolution of the wave function during a measurement process is not just a normal linear progression, but a more complicated process which includes the transition from probabilities to facts. \n\\item Interactions, in general, support only a limited exchange of information between the \"in\" QT objects and the \"out\" QT objects. This limited exchange of information is the cause of some of the peculiarities of QT measurements.\n\\end{itemize}\nWith standard QT, the transition of possibilities to facts occurs exclusively\nwith measurements. The functional model of QT interactions assumes non-deterministic actions at various process steps.\n\n\\subsection{Superpositions}\nSuperposition is one of the key concepts of QT. For the functional model it is assumed that superpositions apply to the paths of the same q-object only, and that superpositions become effective only with interactions (including measurement interactions) occurring at space-time points shared by multiple paths.\n\n\\subsection{Entanglement}\n\nAs described in section 6, the most important effect of an interaction is the formation of a particle\/wave collection with \nmultiple paths. \nA path supports correlations and entanglements between the particles\/waves leaving the interaction.\nEach path represents one of the possible outcomes of an interaction (such as a measurement). As described in section 6.1, the interaction discards all paths of the \"in\" particle\/wave collection except the one selected.\n\nWhen particles\/waves are entangled, the entanglement endures for some time\nuntil it becomes \"measured\" as a result of an interaction involving one of the entangled particles\/waves. \"Measured\" indicates that due to an interaction one of the entangled particles\/waves obtains a definite value for the\nmeasured observable. The measurement of a definite value for particle\/wave-1\naffects the possible values that can be measured for particle\/wave-2. With\nthe functional model, the determination of a specific measurement value means\nselection of a path from a pw-collection. The selection of a path means selection of\na specific value for both particle\/wave-1 and particle\/wave-2. The path selected\nis the only one surviving for the further evolution of particle\/wave-2. \n\nThe entanglement concept of the QT functional model maintains the strange\nnon-locality of QT. However, the non-local effect applies to the elimination of\nalternative paths, rather than to direct value changes.\n\n\n\\subsection{Decoherence}\n\nDecoherence theory, like this functional model, discusses process steps such\nas the formation of entanglements, interactions, and the (apparent) collapse of\nthe wave function. Insofar, it may be considered as providing a type of functional\nmodel of QT interactions.\n\nDecoherence is the process of changing the coherent wave function of a local\nquantum system through interactions with the environment to a wave function\nthat is entangled with the environment. H.D. Zeh, one of the founders of \ndecoherence theory, calls decoherence an \"uncontrolled dis-localization of superpositions\" \n(see \\cite{ZehPOR}, page 35).\n\nAccording to Zeh, decoherence occurs constantly as the result of interactions of the observed subsystem with the environment. Even when the observed\nsubsystem may be isolated, decoherence is unavoidable for the measurement apparatus, and thus, the overall system consisting of the observed subsystem, the\nmeasurement apparatus, and the environment is subject to decoherence \n(see \\cite{ZehPOR}, page 34).\n\nDecoherence theory does not assume a collapse of the wave function.  Instead, the many-worlds interpretation \n(see \\cite{Everett})\nis considered a suitable interpretation of measurements and measurement-like\ninteractions. The author considers the denial of the collapse of the wave function as an\nunavoidable consequence of the insistence on differential equations as the only\nphysics-conforming way to describe temporal relationships.\nIn contrast to decoherence theory, the author recognizes a need for an additional\nexplicit process step besides the unitary evolution defined by the Schr\\\"odinger\nequation. Even with the assumption of a many-worlds interpretation, the \"branching\" into new worlds (as a replacement of the collapse of the wave function) is\na non-trivial process step that requires further elaboration and (most importantly)\ncriteria for when it occurs.\n\n\\section{Conclusions}\n\nThe development of the functional model of QT interactions may be considered\nan exercise demonstrating that a functional model of QT that is compatible with\nstandard QT\/QFT is feasible.\n\nAlthough the functional model aims for maximum compatibility with standard QT\/QFT, there are exceptions in the functional model. Examples include some small deviations to standard QFT that have been intentionally included, items where it is not clear what the QT\/QFT conformal behavior exactly is, and items where there is no equivalent QT\/QFT position because the level of detail is below the scope of QT\/QFT. For all these areas, verification by experiments is appropriate.\n\nThere is one area in which the functional model is designed to deviate from standard QT\/QFT, the concept of the \"transition from probabilities to facts\". The functional model assumes the transition from probabilities to facts to occur in multiple steps. As indicated within the paper, it will be very difficult to test possible deviations from standard QT in this area.\n\nIn addition to the areas in which the functional model intentionally deviates from standard QT\/QFT, there may be further accidental deviations. Such accidental deviations should be discovered through computer simulations that compare the predictions of standard QT\/QFT with those of the functional model, as described in \n\\cite{diel3}.\nPreliminary results of these computer simulations show agreement to the extent expected.\n\nThe functional model of QT interactions is not considered complete in all\nareas. The major areas where the specification is incomplete are (1) the definition\nof clear criteria for which interactions do not result in a collapse of the wave\nfunction (see section 7.2) and (2) the treatment of bound systems (see section 7.3).\nIn both areas, which seem to be interrelated, the existing QT\/QFT is not yet in a state that provides much\nhelp for the construction of a functional model.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Causality}\n \nIt is claimed in \\cite{jones} that our analysis of damping of \nneutrino oscillations implies the possibility \nof superluminal signals and thus leads to causality violation. To illustrate \nthis point, the author considers a two-stage thought experiment, in which \ninitially neutrinos \nare produced in \nelectron captures in a low-density gas, so that the interactions of the \ndaughter nuclei with the surrounding particles of the medium can be \nneglected. This leads to relatively long neutrino \nWPs and no \ndecoherence by WP separation observed in the detector. Then, at some \n``decision making time\" $t_0$, one compresses the gas in the source with a \npiston, leading to strong localization of the daughter nucleus $N'$ and much \nshorter neutrino WPs; as a result, an observer at the detector position \nshould see decoherence effects. However, according to the claim in \n\\cite{jones}, this happens outside the future light cone with the origin at \nthe point of $N'$ interaction soon after $t_0$, as the light signal from \nthis point would reach the neutrino detector after the neutrino detection \nprocess has already been over. This would mean causality violation. \n\nThe above argument is based on the incorrect space-time diagrams \nin Figs.~1 and 2 of \\cite{jones} which do not correspond to our \ncalculations. Our analysis in \\cite{us} is based on a consideration of \nmean free times of the particles involved in the neutrino production \nprocess.  We have demonstrated that the production time is determined by \nthe shortest among the mean free times $t_a$ of all the involved particles. \nIn the cases considered in \\cite{jones}, these are the mean free times of \neither electron (Fig.~1\n\\footnote{ \nNote that the light cone in Fig.~1 of \\cite{jones} is plotted \nincorrectly: since the produced electron has in this case the shortest time \nof free propagation, the light cone should originate from the point of  \ninteraction with the surrounding atoms of the electron and not of $N'$.} \nor of the parent nucleus $N$ (both panels of Fig.~2), but not of $N'$. \nThus, in all the diagrams of \\cite{jones} the mean free time $t_{N'}$ of the \ndaughter nucleus corresponds to the times when the neutrino production \nprocess has already been over. In such circumstances the interactions of \n$N'$ are irrelevant; they cannot (and do not) affect the outcome of the \nneutrino detection experiment. \n\nConsider now the situation when $t_{N'}<t_N$ \n(see Fig.~1 of this Reply). \nIn this case the scattering of $N'$ {\\em is} relevant and it determines the \nlength $\\sigma_x$ of the neutrino WP: $\\sigma_x=ct_{N'}$. Simple \ngeometrical considerations then show that, for baselines $L$ exceeding \nthe coherence length $L_{\\rm coh}=(2E^2\/\\Delta m^2)\\sigma_x$ (which means \nthat the WPs of the neutrino mass eigenstates $\\nu_1$ and $\\nu_2$ have \nseparated by more than their length $\\sigma_x$ before reaching the \ndetector), the arrival of the slower neutrino mass eigenstate $\\nu_2$ at \nthe neutrino detector position will be inside the future light cone. \nThis means that no causality violation arises in this case. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.72\\linewidth]{zhe8a.png}\n\\caption{\nSpace-time diagram of neutrino production, propagation and \ndetection for neutrinos produced in $e$-capture in the case of compressed \nsource gas (see text). Wave \npackets of the decaying nucleus $N$, daughter nucleus \n$N'$ and mass-eigenstate components of the emitted  \nneutrino are represented by brown, violet and gray bands, respectively. \n$t_1$, $t_2$ and $t_3$, $t_4$ are the times of arrival at the detector of the \n``front'' and ``rear ends'' of the WPs of $\\nu_1$ and $\\nu_2$, respectively.   \nViolet dashed line corresponds to the case of uncompressed gas in the source. \nRed dashed lines show the borders of the future light cone from the $N'$ \ninteraction point.  The distance of this point from the source \ncorresponds to the mean free path of $N'$. }\n\\label{fig:wp}\n\\end{figure}\n\nFigure~2 of \\cite{jones} actually corresponds to the situations when the \nduration $\\tau$ of the individual processes of scattering of the daughter \nnucleus $N'$ on the surrounding atoms of the source rather than mean free \ntime of $N'$ determines the length of the neutrino WP. (The author does not \nsay this explicitly, but this follows from his figure). \nNote that such an approach is by itself fully legitimate, except that the \ntime intervals $\\tau$ have to be found rather than loosely assumed to be \nsmall or large, depending on the density of the system. This approach was \nmentioned in the end of sec. 3 of our paper \\cite{us} as an alternative to \nthe approach we have elaborated. If implemented, it could in principle give more accurate results for \nthe lengths of the neutrino WPs than simple order-of-magnitude estimates we \nhave obtained. The reason we did not pursue this approach in \\cite{us} is \nthat it would require the knowledge of the lengths of the WPs of all the \nparticles participating in the scattering of $N'$ (including the WPs of the \nscattered states), which in turn would require taking into account their \ninteractions with the surrounding atoms, and so on. Our approach, based on \nthe consideration of mean free times, is in fact a shortcut allowing us to \navoid considering this ladder of interactions. \n\n\\section{Non-orthogonal bases}\nIt has been hypothesized in \\cite{jones} that a possible reason for the \nalleged violation of causality in our approach was our use of WPs \nfor describing propagation of daughter nuclei $N'$, and that WPs  constitute \nnon-orthogonal bases. Using a non-orthogonal basis for summation over \nfinal states may indeed lead to wrong results. The point is, however, that \nwe never performed such a summation, either explicitly or implicitly.  \n\nIt is argued in \\cite{jones} that a summation over all the possible \nfinal states of the daughter nucleus $N'$ has to be performed, as the \nobserver at the neutrino detector has no knowledge about these states. We disagree \nwith this point. The only characteristic of $N'$ we use\nis the total cross-section of its interaction with\nparticles of medium which determines \nthe $N'$ mean free path.   \nWe have shown in \\cite{us} that interactions of $N'$ \nwith medium are only important for the formation of neutrino WPs when they \nhappen on the time scales shorter than those of the interactions of the \nparent nucleus $N$. Therefore, they constitute an important part of the \nneutrino production process, \nand the production is not over until the scattering of $N'$ occurs. \nThis scattering thus cannot be considered as happening ``after the \nneutrino production''. \n\nNotice that if one neglects the effects of $N'$ on neutrino WPs, \nthis would greatly increase the lengths of these WP's, as was shown in \nsec.\\ 3.1.1 of \\cite{us}. This would only suppress decoherence effects \nand make their observation even less feasible. \n\nIn \\cite{jones} several issues related to the entanglement \nof the particles produced at neutrino detection and its connections \nwith the EPR paradox were discussed. These are interesting points \nthat deserve separate consideration. However, they are not directly \nrelated to the issues discussed in our paper \\cite{us}. \n\n\n\\section{Localization due to nucleon interactions within the nucleus}\nIn \\cite{us} we assumed that the localization of the atoms of neutrino \nsource is caused by their scattering on surrounding atoms.\nThe author of \\cite{jones} mentions as an alternative to this \nthe localization of the decaying nucleon with respect to the other  \nnucleons of the same parent nucleus. This would lead to much shorter \nneutrino WPs. \nHowever, such a localization through inter-nucleon interactions does \nnot say anything about the absolute localization of the nucleus as a whole, \nand obviously only the latter is relevant for the formation of the neutrino WP. \nOne might argue that the localization of the atom containing the unstable \nnucleus by its scattering on another atom does not determine the\nabsolute position of such an atomic pair in space either\n\\cite{jones2} (note that in \\cite{us} we considered localization through\natom-atom scattering).\nLet us clarify this issue.\n\n\nConsider a source atom in a given coordinate system. The coordinate \nof the atom can be established (with some uncertainty) through its \nobservable interactions with the other parts of the system. Consider first \nthe situation when the source atom is placed in an empty box of linear \nsize $l$. Then the coordinate uncertainty of the atom and of its nucleus \n(as well as of the constituent nucleons of the nucleus) will be given by \n$l$, not by the size of the atom or by the radius of its nucleus. This is \nbecause there is no way to find out where exactly inside the box the atom \nis, and its localization proceeds only through its collisions with the \nwalls of the box. If the box is filled with a gas, the uncertainty \nof the coordinate of the parent nucleus will be given by the mean \nfree path of its atom, because its scattering on the atoms of the gas \nwill produce measurable recoils and local gas density changes. No \nsuch recoils are produced by the interactions of the decaying nucleon \nwith the other nucleons of the same nucleus because the daughter \nnucleus recoils as a whole. Thus, such inter-nucleon interactions have \nnothing to do with localization of the parent particle in the source.\n \n\\vspace*{1mm}\nTo conclude, the criticism of our results in \\cite{jones} is invalid. \nWe maintain the validity of our results and, in particular, stand by our \nconclusion that WP separation effects cannot be observed in reactor and \nsource experiments. \n\n\\vspace*{1mm}\n{\\em Acknowledgement}. We thank B.J.P.~Jones for sending us his paper \n\\cite{jones} before posting it on the e-print archive and for \ninteresting discussions.  \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nFor the past decade, the authors and their collaborators have\nbeen involved in the development of a theory of large amplitude\ncollective motion.  This enterprise began with a pair of\npapers \\cite{K1,K2} on possible quantum foundations for such a program.\nSoon thereafter \\cite{DK1} it was realized that the leading\napproximation,\nto which most previous studies had been confined,\nwas classical in nature, (with subsequent requantization) and,\nas a consequence, almost all later\nsystematic theoretical development, reviewed in Ref.~\\cite{KWDR},\nwas based on\nthe study of this limit.\n\nQuantum corrections can be important\nin selected circumstances,\nhowever.  Consequently, the elements of a systematic method for\nincluding quantum corrections by expansion about the classical limit\nhas recently been developed \\cite{WKDQ}, distinct from the approaches\nfound in our initial papers.  We consider it worthwhile, nevertheless,\nin the light of subsequent developments,\nespecially within the context of the Berry phase idea \\cite{B1,MSW},\nto update, augment and improve the work described in those initial\nefforts and to tie them, where possible, to more recent work.\nIn a paper currently in preparation, we shall contrast the\nnuclear-physics\nfoundations of the method described in this paper with that found in\nRef.~\\cite{WKDQ}, both approaches requiring elaboration compared\nto what\nhas already been published.\n\nIn Sec.\\ \\ref{sec:oldstuff} we give a condensed but\nat the same time more precise account of the essential theoretical\ncontent of Ref.~\\cite{K1}, namely, a method for separating and\nidentifying\na collective subspace for a specified class of Hamiltonians.\nThis is done in an approximation that suppresses any coupling\nto non-collective degrees of freedom.  As opposed to the corresponding\nproblem in molecular physics, the Hamiltonian is generally\ngiven in a form where the coordinates are not appropriate to make the\nseparation of the Hamiltonian into a collective and a non-collective\npart.\nThe discovery of new\ncoordinates that will effect the separation is an essential part of the\nproblem.  In the regime of large amplitude collective motion, a\nreasonably complete theory that can be implemented \\cite{KWDR,WKDQ}\nhas been developed\nso far only for the case that the transformations to new coordinates\nand momenta are restricted to point transformations.\n\nIn Sec.\\ \\ref{sec:BO}, we describe how the previous theory may be\nextended to\ninclude the interaction between the fast and slow variables.\nThis is done by means of\na standard Born-Oppenheimer (BO) representation of the states of the\nslow variables.   Though some of the ensuing details are similar to\nthose encountered in the molecular problem, leading, for example,\nto the\noccurrence of Berry potentials \\cite{B1,MSW}, others are characteristic\nof the problem of large amplitude collective motion.  In order to\nevaluate the corrections found to the potential energy of the\ncollective variables, it is necessary to obtain wave functions\nfor the fast variables.  This is done by an\nextension of the analysis carried out in Sec.\\ \\ref{sec:oldstuff}.\nAt the same time\nthis development provides a justification for\nsome of the assumptions of the previous analysis.\n\nIn Sec.\\ \\ref{sec:varia}, we present material that has no direct\ncounterpart\nin any previous work\nin this field, outside our own.  One of the results of\nSec.\\ \\ref{sec:BO} was to establish a complete effective\nquantum mechanics in the\ncollective subspace, i.\\ e., we not only computed an effective\nHamiltonian operator in the collective subspace, but also proved\nthat it was expressed in terms of canonical variables.  In\nSec.\\ \\ref{sec:varia}\nwe study variational principles for the associated Heisenberg matrix\nmechanics. If we put aside, momentarily, special difficulties\nassociated\nwith the curved spaces that we allow in our general formulation, we\nfind quantum analogues of variational principles both for Hamilton's\nand for Lagrange's equations.  The connection between the two forms\nis the standard one, even when Berry phase terms are included, in\ncontrast to a result based on the use of path integrals \\cite{MSW}.\nFor the general case of a curved space the situation is more involved,\nand only a variational principle for Hamilton's equations is presented\nand analyzed.\n\nThe corrections to the potential energy of collective motion\nfound in Sec.\\ \\ref{sec:BO} are of three types.  There is first a\n``quantum\"\npotential arising from the curved space (position-dependent mass\ntensor).  This term cannot be evaluated fully with the tools developed\nin our previous work \\cite{KWDR}.  In practical applications to\nsystems with\n$N$ degrees of freedom, this term is of order $N^{-2}$,\nand therefore should be small for a\nmany-body system.  A second term consists of the eigenvalue for the\nstate of the fast variables for a fixed value of the slow variables.\nIn our approximation this state is that of a set of independent\nharmonic oscillators, with frequencies dependent on the instantaneous\nvalue of\nthe collective  coordinates.\nWe are mainly (though not exclusively) interested in the zero-point\nmotion associated with the ground state.  This contribution, which is\nonly of order $N^{-1}$ compared to the leading term has been studied\npreviously in simple examples \\cite{WKDC}.\nA third correction, referred to as the Berry scalar potential,\noccurs together with the Berry vector\npotential, both appearing in the collective Hamiltonian as a\nconsequence of\nthe BO approximation.\n\nExamples in the literature relevant to the approach described in this\npaper are not numerous.  The examples studied by Bulgac \\cite{Bu1,Bu2}\nhave been stimulating, but a bit too simple for our purposes.\nOn the other     hand, the work by\nGirard, LeTourneux, and Vinet \\cite{GTV} on the dipole-quadrupole\nproblem\nin spherical nuclei is too complex to satisfy our initial needs.\nSec.\\ \\ref{sec:models} is devoted to the study of several\n``in-between\" models sufficiently\nsimple that the BO approximation can be carried out,\nincluding the calculation of the Berry potentials, and compared with\nmore\nexact results.\nSome technical details associated with the material of\nSecs.\\ \\ref{sec:oldstuff} and \\ref{sec:BO},\nrespectively, are provided in two appendices.\n\n\\setcounter{equation}{0}\n\\section{Summary of previous results \\label{sec:oldstuff}}\n\nIn this paper we base our study on a Hamiltonian, that with the help\nof the\nsummation convention, takes the form\n\\begin{equation}\nH = \\tilde{H}(\\xi^\\alpha,\\pi_\\alpha)=\\frac{1}{8} \\{\\pi_\\alpha,\n\\{\\pi_\\beta,\\tilde{B}^{\\alpha\\beta}(\\xi)\\}\\} + V(\\xi),\n\\label{eq:2.1}       \\end{equation}\nthat describes $N$ coordinate and momentum pairs,\n$\\xi^\\alpha$ and $\\pi_\\alpha$, $\\alpha=1...N$, that satisfy canonical\ncommutation relations, ($\\hbar=1$),\n\\begin{equation}\n[\\xi^{\\alpha},\\pi_{\\beta}] = i\\delta^\\alpha_\\beta.  \\label{eq:2.100}\n\\end{equation}\nWe thus allow for a curved space described by a (reciprocal)\nmass tensor $\\tilde{B}^{\\alpha\\beta}$.  Not only is this formulation\nof interest for a range of applications outside of nuclear physics\n\\cite{KWDR,WKDC}, but it is also possible \\cite{KM1}\nto transform\nthe large amplitude problem of nuclear collective motion into this form\nas the basis for further development. Associated with this\nHamiltonian is a form for the scalar product in Hilbert space\nthat is discussed in appendix A.\n\nThe problem  of immediate\ninterest is to find a decomposition of the operator\n${\\tilde H}$ into two parts, exactly in rare cases, approximately\nunder most circumstances, that describe slow and fast\n(or collective and\nnon-collective) degrees of freedom.  A basic difficulty of the class\nof problems that interest us is that we cannot assume that this\nseparation occurs for the initial choice of coordinates.\nInstead, we assume that it is possible to effect\nsuch a decomposition, at least approximately, by means of\na point transformation of the form\n\\begin{eqnarray}\n   \\xi^\\alpha &=& g^\\alpha(x), \\nonumber \\\\\n    x&= & \\{Q^i ,q^a\\}, \\nonumber   \\\\\n   i&=&1...K, \\nonumber \\\\\n   a&=&K+1...N,  \\label{eq:2.2}             \\end{eqnarray}\nthat is locally invertible,           \\begin{equation}\nx^\\mu = f^\\mu(\\xi).   \\label{eq:2.3}            \\end{equation}\nThough we have discussed, in the past \\cite{DK1,KWDR},\nthe possible interest\nof going beyond point transformations in the study of large amplitude\ncollective motion, it is only for this simpler class of\ntransformations that a substantial theoretical underpinning can be\nclaimed to exist, a foundation that we wish to widen in the present\nwork.\n(Of course, there is a substantial literature based on transformations\nthat are polynomial in coordinates and momenta that are applicable\nto anharmonic vibrations (see, for instance \\cite{YK1,SMMZ}), but\nthis is not the subject of the current paper.)\n\nA first step is to carry out a formal transformation of the\nHamiltonian,\n$H$, to the new variables.  For the potential energy, we have,\ntrivially enough,                      \\begin{equation}\n{\\tilde V}(\\xi)={\\tilde V}(g^\\alpha(Q,q))\\equiv V(Q,q). \\label{eq:1.1}\n                                      \\end{equation}\nTurning then to the consideration of the kinetic energy, ${\\tilde T}$,\n\\begin{equation}\n{\\tilde T}=\\frac{1}{8}\\{\\pi_\\alpha ,\\{\\pi_\\beta ,\n\\tilde{B}^{\\alpha\\beta}\n(\\xi)\\}\\},        \\label{eq:2.16}             \\end{equation}\nthis expression can be transformed to the new variables with the aid\nof the relations \\cite{K1}\n\\begin{eqnarray}\n\\pi_\\alpha&=& \\mbox{$\\frac{1}{2}$}\\{f^\\mu_{,\\alpha},p_\\mu\\},\\label{eq:2.17} \\\\\nB^{\\mu\\nu}&=& f^\\mu_{,\\alpha}\\tilde{B}^{\\alpha\\beta}f^\\nu_{,\\beta}.\n\\label{eq:2.18}                      \\end{eqnarray}\nThe derivation of (\\ref{eq:2.17}) is given in appendix A, whereas\n(\\ref{eq:2.18}) confirms the tensorial character of the mass tensor.\nIn terms of the new variables, the kinetic energy\nconsequently takes the form               \\begin{equation}\nT=\\frac{1}{8}\\{p_\\mu ,\\{p_\\nu ,B^{\\mu\\nu}(Q,q)\\}\\} + U(Q,q),\n\\label{eq:2.19}                       \\end{equation}\nwhere the second term, which is specifically a quantum potential\narising from\nthe non-commutativity of coordinates and momenta, has the form\n\\begin{equation}\n8U(Q,q)=[f^\\mu_{,\\alpha\\gamma}g^\\gamma_{,\\nu}g^\\alpha_{,\\lambda}\nB^{\\nu\\lambda}]_{,\\mu} - [f^\\mu_{,\\alpha\\gamma}g^\\gamma_{,\\mu}]_{,\\nu}\ng^\\alpha_{,\\lambda}B^{\\nu\\lambda}.  \\label{eq:1.2}  \\end{equation}\nFor further work, the momenta are also divided\ninto collective and non-collective subsets, \\begin{equation}\np_\\mu = \\{P_i,p_a\\}.               \\end{equation}\n\nWe now remind ourselves that the initial, even\nprimary, goal of the present\nconsiderations is to identify a piece of the transformed Hamiltonian\nas collective.  This collective Hamiltonian, $H_C$, should depend only\non the variables $Q^i,P_i$, that we suppose to be canonical pairs.\nThe simplest possible choice would appear to be the restriction of\nthe full Hamiltonian to the values $q^a=0,\\;p_a=0$, and thus (see\nbelow)\n                     \\begin{equation}\nH_C\\cong \\frac{1}{8}\\{P_i,\\{P_j,B^{ij}(Q,0)\\}\\} +V(Q,0).\n\\label{eq:1.3}              \\end{equation}\nThis choice is purely formal until we specify a procedure for\ndetermining the unknown functions $V(Q,0)\\equiv V(Q)$ and $B^{ij}(Q,0)\n\\equiv B^{ij}(Q)$ in terms of the elements of the original Hamiltonian\nand of the point transformation (\\ref{eq:2.2}) and (\\ref{eq:2.3}).  The\ncriteria for such a determination must be chosen so as to somehow\nminimize the coupling between the collective and non-collective\nvariables.\n\nThe simplest approach to this problem is to consider $H_C$,\n(\\ref{eq:1.3}), as the first term of a Taylor expansion in powers\nof the non-collective variables, $q^a$, remembering at the same time\nthat we have a polynomial of degree two in the non-collective momentum\noperators, $p_a$.  Assuming that higher order terms are successively\nof lesser importance, basically the same assumption that governs the\ntreatment of all variables in the small vibrations domain,\nwe adopt the same assumption that defines optimum decoupling\nin the classical limit \\cite{KWDR}, namely\nthat the terms linear in the non-collective\nvariables vanish.  This yields the two sets of conditions\n                             \\begin{eqnarray}\nV_{,a}(Q) +\\frac{1}{8}\\{P_i,\\{P_j,B_{,a}^{ij}(Q)\\}\\}&=&0,\n\\label{eq:1.4}\\\\\nB^{ai}(Q) &=& 0,    \\label{eq:1.5}    \\end{eqnarray}\nrecognizable as the coefficients of $q^a$ and $p_a$, respectively.\nThe operator character of these conditions is apparent only from\nthe second term of (\\ref{eq:1.4}).  However, if we introduce\nthe Wigner transform of these equations, because of the special\nsymmetrization that we have adopted for the kinetic energy, they\nreduce exactly to the classical conditions  studied\nextensively in our previous work\n\\cite{KWDR}.  There, starting from\nthese classical conditions, we have described and illustrated\nsuitable algorithms for the determination of the elements of the\ncollective Hamiltonian.\n\n\nWith the determination of the elements of the Hamiltonian,\n(\\ref{eq:1.3}),\nthe theory is completed by the assumption that the eigenstates of this\noperator are defined on a K-dimensional Hilbert space of\n(round-bracketed) states $|n)$,   \\begin{equation}\n|n)=\\int dQ |Q) (Q|n),     \\label{eq:1.6}    \\end{equation}\nand that these states accurately model a corresponding subspace,\n$|n\\rangle$,\nof states of the full Hilbert space.  The theory described above\nis very closely\nthe theory that has been applied in a number of early applications,\nsuch as \\cite{UK,BKD}. A major aim of the present work is to critique\nand\ngeneralize this procedure.\n\nWe emphasize that in this\nprogram, the elements of the collective Hamiltonian\nare determined from a K-dimensional manifold in configuration\nspace, parametrized by the coordinates $\\{Q\\}$, whereas the quantum\nmechanics is determined in a Hilbert space defined by the\nsame coordinates.\nUpon further reflection, it may strike the reader\nthat this is a strange result, or, at least, a very limited one.\nThe reason is that the true\ncollective states, $|n\\rangle$, though they define\na subspace of the full Hilbert space, are nevertheless states in\nthis space\nand therefore depend also\non the non-collective coordinates, at the very least through the\nzero-point\nmotion of the latter.\nThough we have been able to neglect this dependence in the\napproximation\nconsidered so far, this will no longer be true if we seek\nto include the effect of the coupling to the\nfast variables on the properties of the collective states,\nor in directly describing the motion of the fast variables.\n\n\n\n\\setcounter{equation}{0}\n\\section{Generalization of the formalism to include the effects\nof the fast variables\n \\label{sec:BO}}\n\nIn this section, we set ourselves two tasks.  We shall first find the\nforms of the leading corrections to the collective Hamiltonian\ngiven above, arising\nfrom the coupling of the fast to the slow variables. This will be done\nwith the help of the standard Born-Oppenheimer (BO) approximation.\nIn order to evaluate these corrections explicitly, however, we shall\nthen commit ourselves to further approximations for the\ndependence of the full Hamiltonian\non the fast variables, that yield a normal\nmode description of the latter.\n\nTo generalize the restricted ideas of the previous section,\nwe thus introduce\nthe BO picture into the theory of large\namplitude collective motion.  As stated above\nthis amounts to an extension of the\ncomplexity of structure allowed for the states of the collective\nsubspace so as to take into account the influence of the fast variables.\nWe first assume, more generally,  that the collective states\n$|n\\rangle$ have coordinate space representatives\n\\begin{eqnarray}\n\\langle Q^i q^a|n\\rangle  &\\equiv & \\langle Q,q|n\\rangle \\nonumber \\\\\n        &=&\\sum_\\nu(Q|n\\nu)[q|\\nu\\!:\\!Q],\n\\label{eq:2.4}                      \\end{eqnarray}\nwhere the index $\\nu$ is not to be confused with its previous use as a\n coordinate index.\nHere and for the remainder of this paper, we adopt a notation where\nangular\nbrackets indicate states in the full Hilbert space, square brackets\nstates\nin the space of fast variables (though dependent parametrically on\nthe slow\nvariables. as denoted by $:\\!Q$ in the state vector), and parentheses\nstates    in the collective space.\nWe suppose that for fixed $Q$, the states $[q|\\nu\\!:\\!Q]$ are a\ncomplete set of functions for the fast coordinates,\n\\begin{equation}\n   \\sum_\\nu [q|\\nu\\!:\\!Q][\\nu\\!:\\!Q|q']\n    = \\delta(q-q').     \\label{eq:2.5}         \\end{equation}\nThough for the moment we have not specified the equation\nof which they are the solutions, we shall be able to do so, at least\napproximately, as a consequence of the developments to be carried out\n in this section.\n\nFor the remainder of the current discussion, we shall consider the\nsimplest case in which the fast variables occupy, for any given value\nof the slow variables, their state of lowest energy, which is assumed\nto\nbe non-degenerate.\n For the case to be studied here,\n$\\nu$ takes a single value denoted by zero.\nThe coefficient function, usually\n denoted in this case by\n$(Q|n)$, that appears in (\\ref{eq:2.4}), can then\nbe identified as the wave function\nfor the collective motion, and\nthis identification will agree with the one that has been made in the\nprevious section, where\nthe description of large amplitude collective motion was\nnot tied to the BO approximation.  There we emphasized a connection\nbetween the decoupled Hilbert space of the collective coordinates\nand the crucial existence of a $K$-dimensional decoupled\ncoordinate manifold described by the functions $g^\\alpha(Q,q=0)$.\nThe approximate separability of the associated\nHilbert space is based on the assumption that the functions\n$[q|\\nu\\!:\\!Q]$ describing the fast variables are confined\nto a narrow region in $q$ space in the neighborhood of the collective\nsurface.  If this picture holds, we may expect the mathematical\ndetails to work out reasonably.\n\nAdopting the BO approximation, we set ourselves the task of finding\nan effective Hamiltonian to describe the motion of the collective\nvariables.  This operator is defined, though not yet operationally,\nby means of the equation,                   \\begin{equation}\n(n'|H_{{\\rm eff}}(Q,P)|n)=\\langle n'|\\tilde{H}(\\xi,\\pi)|n\\rangle.\n\\label{eq:2.6}                  \\end{equation}\nThis relation will assume the status of a definition of the collective\nHamiltonian within the space of slow variables and $H_{{\\rm eff}}$\nrecognized as a generalization of $H_C$ only after we\nspecify how to eliminate the fast variables from the right hand side.\nThe procedure that we shall follow is closely akin to the\ntraditional BO approach, with characteristic differences arising\nfrom the facts that at the beginning we cannot specify which are the\nslow and which the fast variables, and that the treatment of the\nfast variables comes as a kind of afterthought, dependent in detail\non the prior treatment of the collective variables.\n\nLet us start with the\npotential energy,                     \\begin{equation}\n\\tilde{V}(\\xi)=\\tilde{V}(g^\\alpha(Q,q))\\equiv V(Q,q),\n\\label{eq:2.7}                       \\end{equation}\nand evaluate the associated piece of (\\ref{eq:2.6})  \\begin{equation}\n\\langle n'|\\tilde{V}(\\xi)|n\\rangle \\equiv (n'|V_{eff}|n)  \\\\\n\\cong\\int dQdq(n'|Q)[ 0\\!:\\!Q|q] \\tilde{V}(\ng^\\alpha(Q,q))[q|0\\!:\\!Q]( Q|n),\n\\label{eq:2.8}                             \\end{equation}\nwithin the BO approximation.\nThe only feasible way, in general,\nof integrating out the fast variables is to expand\n$\\tilde{V}$ in powers of $q^a$,               \\begin{equation}\nV(Q,q)=V(Q) +V_{,a}q^a+\\mbox{$\\frac{1}{2}$} V_{,ab}q^a q^b +... \\;,\n\\label{eq:2.9}                       \\end{equation}\nleading to                            \\begin{equation}\nV_{eff}(Q) = V(Q) + V^{(1)}(Q) + V^{(2)}(Q) +... \\;,\n\\label{eq:2.10}                       \\end{equation}\nwhere assuming that the function $[q|0\\!:\\!Q]$ is normalized,\n\\begin{eqnarray}\nV(Q)&=&V(Q,0),  \\label{eq:2.11}    \\\\\nV^{(1)}(Q)&=&V_{,a}\\int|[q|0\\!:\\!Q] |^2 q^a \\nonumber \\\\\n& \\equiv & V_{,a}(Q)\\langle q^a\\rangle_Q,  \\label{eq:2.12} \\\\\nV^{(2)}(Q) &=& \\mbox{$\\frac{1}{2}$} V_{,ab}(Q)\\langle q^aq^b\\rangle_Q. \\label{eq:2.13}\n\\end{eqnarray}\n\nWe thus see that the leading term is independent of the\nwave function for the fast variables, coinciding with\nthe standard result for the potential energy of large amplitude\ncollective motion \\cite{K1,KWDR} presented in the previous section.\nThe computation only requires the form of the collective\nsubmanifold, $\\xi^\\alpha =g^\\alpha(Q,0)$, which can be determined\nby well-defined procedures \\cite{KWDR}.  To go beyond\nthis lowest order, we need, besides the wave function of the\nfast variables, to rearrange the expansion of V, as explained in\nappendix B.  There it is shown that if we wish to interpret the small\nquantities $q^a$ as components of a vector, we must replace the\nordinary second derivative by a covariant second derivative,\n$V_{;ab}$, that can be computed from known or calculable quantities\naccording to the equations                 \\begin{eqnarray}\nV_{,a}(Q) &=& \\tilde{V}_{,\\alpha}g^\\alpha_{,a}, \\label{eq:2.14} \\\\\nV_{;ab}(Q)&=&\\tilde{V}_{;\\alpha\\beta}g^\\alpha_{,a}g^\\beta_{,b}\n                       \\label{eq:2.15} \\end{eqnarray}\n\nThese equations are simplified by introducing one of the\ndecoupling conditions that follow from Eq.\\ (\\ref{eq:1.4}), namely that\n(\\ref{eq:2.14}) should be zero.  (Here we are\nassuming that the two terms of (\\ref{eq:1.4}) vanish separately.\nConditions for this to be true and modifications necessary when it\nis not have been discussed in Ref.~\\cite{KWDR})\nThe third term $V^{(2)}(Q)$, is the first, $\\Delta^{(1)}V(Q)$,\nof a sequence of contributions that we shall identify as\nthe leading corrections to the potential energy.  Further\ndiscussion of the\nevaluation of this term and of the additional correction terms, to be\nidentified below, will be continued later in this section, following\nthe\nidentification of all the pieces.\n\nThe remaining terms\nwill arise from the study of the kinetic energy, given after\ntransformation of coordinates by Eqs.\\ (\\ref{eq:2.19}) and\n(\\ref{eq:1.2}).\nIn order to integrate out the fast variables in the contribution that\nthese terms make to $H_{eff}$, consider the first term\nof the kinetic energy.  We expand the mass\ntensor in powers of $q$, and keep initially only the leading term\n$B^{\\mu\\nu}(Q,0)\\equiv B^{\\mu\\nu}(Q)$.\nIn this approximation, we first\nrestrict the study to the contribution of\nthose terms where the indices $\\mu,\\nu$ take on values $i,j$\nin the collective subset.\n\nAs a preliminary to this calculation, we study the simpler object\n\\begin{eqnarray}\n(n'|(P_i)_{{\\rm eff}}|n)&=&\\langle n'|P_i|n\\rangle  \\nonumber \\\\\n&     =& (n'|(P_i -A_i)|n)      \\nonumber \\\\\n  & \\equiv& (n'|D_i|n).    \\label{eq:3.50}    \\end{eqnarray}\nHere $P_i$ is the collective momentum operator identified in the\nprevious section, and             \\begin{equation}\nA_i \\equiv i\\int dq [0\\!:\\!Q|q]\\partial_i\n[q|0\\!:\\!Q],\n\\label{eq:3.51}    \\end{equation}\nwhere $\\partial_i$ means partial derivative with respect to $Q^i$.\nNotice, however, that if we calculate straightforwardly,\n\\begin{eqnarray}\n(n'|(P_iP_j)_{{\\rm eff}}|n)&=&\\langle n'|P_iP_j|n\\rangle  \\nonumber \\\\\n&=& (n'|(P_iP_j -A_iP_j -A_jP_i + S_{ij})|n),  \\label{eq:3.52}\n                                    \\end{eqnarray}\nwhere                             \\begin{equation}\nS_{ij} = -\\int dq [0\\!:\\!Q|q](\\partial^2\/\\partial Q^i\n\\partial Q^j) [q|0\\!:\\!Q].  \\label{eq:3.53} \\end{equation}\n\nAs a consequence of this result, we are reassured that the effective\nvalue of zero is zero, i.\\ e.,            \\begin{equation}\n0=\\langle n'|[P_i,P_j]|n\\rangle =(n'|[P_i,P_j]|n).\n\\label{eq:3.54}                        \\end{equation}\nIt is easy to see that the remaining canonical commutators also\nproject without change.\n\nIt is useful to rewrite (\\ref{eq:3.52}) in a form that makes contact\nwith standard results.  We have    \\begin{eqnarray}\n(i\\partial_j)[q|\\nu\\!:\\!Q]i& =&\\sum_{\\nu'}\n{}[q|\\nu'\\!:\\!Q](A_j)(Q))_{\\nu'\\nu},   \\label{eq:3.55}    \\\\\n(A_j)_{\\nu'\\nu}&=&i\\int\\,dq[\\nu'\\!:\\!Q|q]\\partial_j[q|\\nu\\!:\\!Q].\n\\label{eq:3.550}\n                                      \\end{eqnarray}\nBecause of the $Q$ dependence of the matrix element, it now follows\nthat\n                                    \\begin{eqnarray}\nS_{ij}&=& \\sum_\\nu (A_i)_{0\\nu}(A_j)_{\\nu 0} +(i\\partial_i)\nA_j       \\nonumber \\\\\n&=& (i\\partial_i)A_j +A_iA_j +S^{\\prime}_{ij},\n\\label{eq:3.56}                  \\end{eqnarray}        where\n                            \\begin{eqnarray}\n(A_j)_{00} &=&A_j ,      \\label{eq:3.57}   \\\\\nS^{\\prime}_{ij}&=& \\sum_{\\nu\\neq 0}(A_i)_{0\\nu}(A_j)_{\\nu 0}.\n                 \\label{eq:3.58}       \\end{eqnarray}\nConsequently, we may also rewrite Eq.\\ (\\ref{eq:3.52}) as\n                           \\begin{equation}\n(n'|(P_iP_j)_{{\\rm eff}}|n) = (n'|(D_iD_j + S^{\\prime}_{ij})|n).\n              \\label{eq:3.59}        \\end{equation}\nIn the simplest case, where $[0\\!:\\!Q|q]$ is a real\nwave function, and it follows that $A_i$ vanishes, the contribution\n$S^{\\prime}_{ij}$ remains to be taken into account.\n\nWe are now in a position to apply\nto the computation of the collective kinetic energy\nthe same reasoning as just carried out for a product of momentum\noperators.  Making use\nof the analogue of (\\ref{eq:3.59}), the result is\n                           \\begin{equation}\n\\frac{1}{8}\\{D_i ,\\{D_j ,B^{ij}(Q)\\}\\} +\\mbox{$\\frac{1}{2}$} S^{\\prime}_{ij}B^{ij}.\n\\label{eq:2.21}                                 \\end{equation}\nwhere the second term can be incorporated into the\ncollective potential energy as as a second such contribution,\n$\\Delta^{(2)}V(Q)$.\n\nAn additional contribution of this type is\nobtained by setting $q^a=0$ in Eq.\\ (\\ref{eq:1.2}),  \\begin{equation}\n\\Delta^{(3)}V(Q)= U(Q,0),   \\label{eq:1.20}   \\end{equation}\nwhere $U(Q,q)$ is the quantum potential defined in Eqs.\\\n(\\ref{eq:2.19})\nand (\\ref{eq:1.2}).\n\nIt remains for us to discuss the contributions from $T$ that depend\non $B^{ai}(Q)$ that ``mix\" the collective and non-collective indices\nand those that depend on the non-collective mass tensor $B^{ab}(Q)$.\nThe former can be neglected because of one of the decoupling conditions,\nEq.\\ (\\ref{eq:1.5}).\nThe leading contribution of the latter is seen\nto be another contribution to the potential energy,  \\begin{equation}\n\\Delta^{(4)} V =\\mbox{$\\frac{1}{2}$} B^{ab}(Q)\\langle p_a p_b\\rangle_Q.\n\\label{eq:2.26}                               \\end{equation}\n\nTo summarize our findings, we have derived the following effective\nHamiltonian,                               \\begin{equation}\nH_{{\\rm eff}} = \\frac{1}{8}\\{D_i ,\\{D_j ,B^{ij}(Q)\\}\\} + V(Q)\n+\\Delta V(Q),     \\label{eq:2.27}         \\end{equation}\nwhere $\\Delta V(Q)$ is the sum of four terms that summarize the leading\nquantum corrections including the coupling to the fast variables,\n\\begin{equation}\n\\Delta V = \\sum_{i=1}^4 \\Delta^{(i)} V, \\label{eq:2.28} \\end{equation}\ngiven respectively in or in relation to Eqs.\\ (\\ref{eq:2.13}),\n(\\ref{eq:2.21}), (\\ref{eq:1.20}), and (\\ref{eq:2.26}).\nLet us contrast\nthis result with the corresponding form appropriate to the more\nfamiliar molecular case.  The introduction of a curved\nmetric aside, the main difference is that in the molecular case, the\nground state wave function of the fast variables, for a fixed value of\n$Q$, may be assumed known, and its eigenvalue, $\\epsilon_0 (Q)$,\ntogether with\n$\\Delta^{(2)} V$, contained in Eq.\\ (\\ref{eq:2.21}) constitutes the\ncollective  potential\nenergy \\cite{B1,MSW}.\n\nIn the\npresent instance we cannot assume that we know the Hamiltonian of the\nfast variables, except in an approximate sense that we now discuss.\nWe need to extend the considerations of the previous section\nthat led to the definition of the quantum\ncollective Hamiltonian operator $H_{C}$.  Instead\nof specializing the transformed Hamiltonian operator to the\nvalues $q^a=p_a=0$, we now retain terms up to second order in these\nvariables. In this treatment, we may replace $U(Q,q)$ by $U(Q,0)$, since\nthis term is already a small correction, and the remaining terms linear\nin the fast variables may be dropped because of the decoupling\nconditions.  To the specified accuracy, we obtain the following\nquantum Hamiltonian,              \\begin{eqnarray}\nH&=&H_C +U(Q) +H_{NC},   \\label{eq:2.50}   \\\\\nH_{NC}&=&\\mbox{$\\frac{1}{2}$} p_a p_b B^{ab}(Q) +\\mbox{$\\frac{1}{2}$} q^a q^b V_{ab}(Q).\n\\label{eq:2.51}                 \\end{eqnarray}\n\nIn appendix B, we point out that once the elements of $H_C$ have been\ndetermined, one can choose a set of fast variables and\ncalculate the matrices that appear in (\\ref{eq:2.51}).\nFor each value of $Q$, the Hamiltonian $H_{NC}$ represents a standard\nnormal mode problem.  Let $c_\\alpha,c^{\\dag}_\\alpha$ be normal mode\ndestruction and creation operators, $\\Omega_\\alpha$ the corresponding\nfrequencies, and $\\hat{n}_\\alpha=c^{\\dag}_\\alpha c_\\alpha$.  Assuming\nlocal stability, i.\\ e., $\\Omega_\\alpha$ real and positive, we have\n                                        \\begin{equation}\nH_{NC}(Q)=\\sum_\\alpha (\\hat{n}_\\alpha +\\mbox{$\\frac{1}{2}$})\\Omega_\\alpha(Q).\n\\label{eq:2.60}                       \\end{equation}\nThe practical importance of the contribution of (\\ref{eq:2.60}) has\nbeen noted in several applications carried out in the\npast \\cite{WKDC,UK}.\n\nThe quantum Hamiltonian, expressed in terms of the optimum choice of\nvariables, but in a restricted approximation, has thus been found.\nExcept for the term $U(Q)$, it has been expressed in terms of elements\nthat can be calculated.  Evaluation of $U(Q)$ appears to require some\nproperties of the point transformation that have not so far been\nstudied.  Comparison of approximate calculations (with this term\nomitted)\nwith exact calculations indicates that it is probably a small\ncorrection.\nAs previously remarked, it is of order $N^{-2}$  compared to the\nmain term\nin the potential energy, where $N$ is the number of degrees of\nfreedom.         We shall make no further allusion to this term\nin the present paper.\n\nLet us now return to a discussion of the correction terms, $\\Delta V$\nby which $H_{{\\rm eff}}$, Eq.\\ (\\ref{eq:2.27}), differs from $H_C$.\nWe have just discarded\n$\\Delta^{(3)} V$. The sum $\\Delta^{(1)} V + \\Delta^{(4)} V$ has been\nseen in (\\ref{eq:2.60}) to be a sum of oscillator terms in the\napproximation considered, its contribution then depending, naturally,\non the state of motion of the fast variables. As stated above, in\nthis section we shall consider only the lowest energy state for these\nvariables, so that the contribution of this term is just the zero-point\nenergy.\n\nIt remains for us to discuss the contribution of the term\n$\\Delta^{(2)} V$, the Berry scalar potential, associated with the\nprojection of the kinetic\nenergy onto the collective subspace.  This contribution goes together\nwith that from\nthe ``vector potential\", $A_i$.  In Sec.\\ \\ref{sec:models},\nwe shall study\nexamples where these terms contribute.  In fact, the manner in which\n both\nthe vector and scalar potentials contribute is best studied within the\ncontext of these illustrations.\n\nIn this section we have thus gone as far as we shall in defining a\nquantum\ntheory of the slow variables, having derived an effective Hamiltonian\nand shown that it is expressed in terms of canonical variables.  Thus\nwe are also free to study this problem with the help of the Heisenberg\nequations of motion.  In the next section we shall consider variational\nformulations of these equations.\n\n\n\n\\setcounter{equation}{0}\n\\section{Variational principles; Lagrangian formulation\n\\label{sec:varia}}\n\nHaving defined an effective quantum theory within the collective\nsubspace, we can imagine that we study this theory by means of\nHeisenberg's equations of motion,               \\begin{eqnarray}\n-i[Q^i ,H_{{\\rm eff}}] &=& \\frac{\\partial}{\\partial P_i}\nH_{{\\rm eff}}  ,\n\\label{eq:4.1}\\\\\ni[P_i ,H_{{\\rm eff}}] &=& \\frac{\\partial}{\\partial Q^i}\nH_{{\\rm eff}}   .\n\\label{eq:4.2}                        \\end{eqnarray}\nThe purpose of this section is to show that these equations can be\nderived from a variational principle, the so-called trace variational\nprinciple, involving $H_{{\\rm eff}}$, or an operator closely related\nto it;\nit is to show, further,\nthat a reworking of this principle leads to a second version of the\nvariational principle, involving an effective Lagrangian,\n$L_{{\\rm eff}}$, that is defined in the natural way as the Legendre\ntransform of $H_{{\\rm eff}}$.  According to Moody, Schapere, and\nWilzek \\cite{MSW}, when $L_{{\\rm eff}}$ is defined by means of a\npath integral,\nit is not exactly the Legendre transform of $H_{{\\rm eff}}$.  We\nshall not\nencounter any such difficulty as long as the mass tensor is independent\nof coordinates, the only case studied by previous authors.\nWhen the mass tensor\nenters in full generality,\nthe formulation of a variational principle of the type we have in mind\nis not quite as straightforward as in previous instances that we have\nstudied \\cite{KLV,LK2}.\n\nIn order not to have to deal with all subtleties at once, let us first\nconsider the case that the mass tensor does not depend on coordinates.\nIn that case we may utilize the standard version    \\begin{equation}\n\\delta {\\rm Tr}\\{H_{{\\rm eff}} -i\\Lambda[Q^i ,P_i]\\}=0.\n\\label{eq:4.3}                                     \\end{equation}\nHere the trace is taken over a {\\em finite} subspace of the collective\nHilbert space.  This assumption allows us to utilize the invariance\nof the trace under cyclic permutation, a property that plays an\nessential role in the manipulation of the second term of\n(\\ref{eq:4.3}).\nIn this term, $\\Lambda$ is a Lagrange multiplier operator associated\nwith a set of constraints that are the non-vanishing canonical\ncommutation relations.  These, at least, must be imposed because, to\nstart with, the variational quantities are arbitrary matrix elements\nwithin the space included in the trace, and these variations must be\nsubject to the kinematical constraints.  The fact that we can limit\nthe constraints to the commutators for canonical pairs  is, at the\nmoment, an observation.\n\nIn order to come out with the Heisenberg equations of motion upon\nvariation, it is necessary, as well, to choose variations of the\ncoordinates that depend only on the coordinates themselves, so\nas to validate the formula                        \\begin{equation}\n\\delta V(Q) = (\\partial V(Q)\/\\partial Q^k)\\delta Q^k.\n\\label{eq:4.4}                                 \\end{equation}\nThe equations of motion that follow from these assumptions are\n\\begin{eqnarray}\n-i[Q^i ,\\Lambda] &=& \\frac{\\partial H_{{\\rm eff}}}{\\partial P_i},\n\\label{eq:4.5}\\\\\ni[P_i ,\\Lambda] &=& \\frac{\\partial H_{{\\rm eff}}}{\\partial Q^i}.\n\\label{eq:4.6}                      \\end{eqnarray}\nTo make these agree with Heisenberg's equations, the choice\n$\\Lambda=H_{{\\rm eff}}$ is indicated.\n\nWe apply this formalism to the case with non-vanishing Berry\npotentials but constant mass matrix, described by the Hamiltonian\noperator,                        \\begin{equation}\nH_{{\\rm eff}} =\\mbox{$\\frac{1}{2}$} D_iD_jB^{ij} + V,  \\label{eq:4.50}\n\\end{equation}\nwhere it is understood that $V$ may include the correction terms\ndiscussed in the previous section.  The operator equations of motion\nthat follow from (\\ref{eq:4.5}), (\\ref{eq:4.6}), and (\\ref{eq:4.50}),\nafter some standard manipulations, are  \\begin{eqnarray}\n\\dot{Q}^i &=& D_jB^{ji},  \\label{eq:4.51}   \\\\\n\\dot{D}_i &=& -V_{,i}+ \\mbox{$\\frac{1}{2}$}\\{\\dot{Q}^j,{\\cal F}_{ij}\\},\n\\label{eq:4.52}\n       \\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\dot{A}_i &=& \\mbox{$\\frac{1}{2}$} \\{(\\partial A_i\/\\partial Q^j),\\dot{Q}^j\\},\n\\label{eq:4.53} \\\\\n{\\cal F}_{ij} &=& (\\partial A_i\/\\partial Q^j) -(\\partial A_j\/\n\\partial Q^i),      \\label{eq:4.54}       \\end{eqnarray}\nand                                    \\begin{equation}\nD_i = \\dot{Q}^jB_{ji},     \\label{eq:4.55}  \\end{equation}\nwith $B_{ji}$ the matrix inverse to $B^{ji}$, is the solution of\nEq.\\ (\\ref{eq:4.51}) for the gauge-covariant momentum.\n\n From the combination of (\\ref{eq:4.52}) and (\\ref{eq:4.55}), we can\nderive the familiar Lagrangian equations of motion\n\\begin{equation}\n\\ddot{Q}^jB_{ji} +V_{,i} -\\mbox{$\\frac{1}{2}$}\\{{\\cal F}_{ji},\\dot{Q}^j\\} = 0.\n\\label{eq:4.56}                 \\end{equation}\nOf some interest is that the associated variational principle from\nwhich these equations can be derived, namely   \\begin{eqnarray}\n0 &=& \\delta{\\rm Tr}(-L_{{\\rm eff}})   \\nonumber \\\\\n  &=& \\delta{\\rm Tr}(-\\mbox{$\\frac{1}{2}$}\\dot{Q}^i\\dot{Q}^jB_{ji} -\\dot{Q}^iA_i +V),\n\\label{eq:4.57}                \\end{eqnarray}\ncan be obtained\ndirectly from the starting trace variational principle,\n(\\ref{eq:4.3}), simply by eliminating the momenta in favor of the\nvelocities by means of the relations given above.  Instead of\nindependent\nvariations of coordinates and momenta, only the coordinates are to be\nvaried. In this variation, the velocity $\\dot{Q}^i$ is to be replaced\nby\n$-i[Q^i,H_{{\\rm eff}}]$ and $H_{{\\rm eff}}$ is not to be varied.\n               One obtains the\ncorrect equations of motion because algebraic manipulation of the\nresulting trace expressions, containing commutators with\n$H_{{\\rm eff}}$,\nthat is aimed at isolating the coordinate\nvariations, produces the same results as integration by parts of the\ntime derivatives in the corresponding classical variational principle,\nproper attention being paid to the order of non-commuting factors.\nFor the case just concluded, the outcome is thus completely\nsatisfactory.\n\nWhen we reinstate the $Q$ dependence of the mass tensor, an\nadditional problem is encountered.  If we insist on the cyclic\ninvariance of the trace, which has played an essential role in the\nprevious applications of the trace variational principle, then we have\n\\begin{equation}\n{\\rm Tr}\\{P_i ,\\{P_j , B^{ij}\\}\\} = 2{\\rm Tr}\\{P_i P_j, B^{ij}\\}.\n\\label{eq:4.7}                          \\end{equation}\nHowever, direct transformation, using the commutation relations,\nyields\n\\begin{equation}\n\\{P_i ,\\{P_j , B^{ij}\\}\\} = 2\\{P_i P_j, B^{ij}\\} +B^{ij}_{,ij},\n\\label{eq:4.8}                          \\end{equation}\nleading to an apparent contradiction upon formation of the trace.\nOf course, in deriving (\\ref{eq:4.8}), we have used the commutation\nrelations appropriate to an infinite Hilbert space.  The point is the\nsame as that ${\\rm Tr}QP\\neq {\\rm Tr}PQ$ for a truly canonical pair.\nIn order to derive\nthe equations of motion from a trace variational principle, we deal\nwith this problem by adopting\n(\\ref{eq:4.7}), but including the last term in (\\ref{eq:4.8})\nas part of the potential energy. This implies the assumption that the\ntrace is taken, to start with, only over a finite-dimensional vector\nspace and that therefore the invariance of the trace under cyclic\npermutation is a correct operation.  As a consequence, the equations\nof motion are given in matrix form. The operator form of the\nequations of motion is recognized as a limit of these equations.\nIn this limit, the application of (\\ref{eq:4.8})\n{\\em after variation}, can be seen to yield the equations\nof motion in the desired symmetrical (Weyl) structure.\nThus the final variational principle takes the form\n\\begin{equation}\n\\delta {\\rm Tr} \\{H_{{\\rm mod}}-iH_{{\\rm eff}}[Q^i ,P_i ]\\},\n\\label{eq:4.9}\n\\end{equation}                where          \\begin{equation}\nH_{{\\rm mod}} = H_{{\\rm eff}} + \\frac{1}{8}B^{ij}_{,ij}.\n\\label{eq:4.10}\n\\end{equation}\n\nIn the discussion above, we have noted a problem only in connection\nwith the part of the kinetic energy quadratic in the momenta.  It may\nbe verified that a similar difficulty does not occur for the terms\nthat are bilinear in the momenta and the ``vector potential\".\n\nUnfortunately, in the transition from Hamilton's equations to\nLagrange's\nequations for this case additional complications are encountered;\nbecause of the\nnon-commutativity of the mass tensor with the momentum, the Lagrange\nequations do not follow from the expression obtained\nby eliminating the momenta in favor of the velocities in the starting\nvariational principle.  The equations of motion that follow from this\nconjectured Lagrangian variational expression differ from the correct\nequations of motion by additional ``quantum\" potentials.  This line\nof inquiry does not appear to be illuminating and therefore will not\nbe pursued.\n\n\\setcounter{equation}{0}\n\\section{Illustrative models \\label{sec:models}}\n\nIt is simplest to illustrate the main points by choosing models in\nwhich the collective coordinates have already been identified, so that\nwe need not enter into the intricacies of the theory of large amplitude\ncollective motion {\\em per se}.  For instance, in the first model\nstudied\nbelow, there are no terms in the Hamiltonian linear in the fast\nvariables.\nThus the model satisfies the decoupling conditions exactly.\nIn both of the\nmodels to be studied, we break time-reversal invariance,\nand in that sense\nthey are somewhat artificial.\n\n\\subsection{The first model: Berry phase in excited states}\n\nWe study the Hamiltonian\n\\begin{eqnarray}\nH&=& H_{{\\rm core}} + H_{{\\rm sp}} + H_{{\\rm int}}  \\nonumber \\\\    &=&\nH_{{\\rm core}} + H_{{\\rm NC}} \\label{eq:5.1}  \\\\\nH_{{\\rm core}}&=& \\frac{{\\bf P}^2}{2M} + V(Q),  \\label{eq:5.2} \\\\\nH_{{\\rm sp}}&=& \\omega(Q)(a_1^{\\dag}a_1 +a_2^{\\dag}a_2 +1),\n\\label{eq:5.3}\\\\\nH_{{\\rm int}}&=& -G(Q_- a_1^{\\dag}a_2 +Q_+ a_2^{\\dag}a_1).\n\\label{eq:5.4}\n\\end{eqnarray}\nHere the coordinates ${\\bf Q}$ and the canonical momenta ${\\bf P}$\nare both two-dimensional\nvectors.  We use the notation ${\\bf Q}=(Q_1,Q_2)$,\n$Q_{\\pm}=Q_1\\pm iQ_2$, and\n$Q=(Q_1^2 +Q_2^2)^{1\/2}$.  (In this section, we are not\nmaintaining the\ndistinction between upper and lower indices.)\nFurthermore, the $a_i,a_i^{\\dag},i=1,2$\nare boson destruction and creation operators, $G$ is a coupling\nstrength,\nand the frequency, $\\omega(Q)$, of the uncoupled boson modes has\nbeen given\na so far unspecified dependence on $Q$ that will be chosen for\nanalytic and\nnumerical convenience.  Note that the\nHamiltonian, (\\ref{eq:5.1}) conserves the boson number,\n\\begin{equation}\nN = a_1^{\\dag}a_1+ a_2^{\\dag}a_2 = {\\rm constant}.   \\label{eq:5.5}\n\\end{equation}\n\nSince the $N=0$\nproblem is completely trivial,  the first interesting case for the\npresent\nmodel is $N=1$.\nHere, the state vectors may be written exactly as a superposition\n\\begin{equation}\n|n\\rangle = \\int d{\\bf Q}\\{({\\bf Q}|n1)a_1^{\\dag}|0]\n+ ({\\bf Q}|n2)a_2^{\\dag}|0]\\},    \\label{eq:5.6}\n\\end{equation}\nwhere $|0]$ is the vacuum state.  The use of square brackets for the\n vacuum\nstate of the fast variables is consistent with the notation introduced\nin\nSec.\\ \\ref{sec:BO}. The resulting eigenvalue\nequation in the space of the collective variables, ${\\bf Q}$, is\ndetermined\nby the two by two effective Hamiltonian, $H_{eff}$, with matrix\nelements\n\\begin{eqnarray}\n(H_{eff})_{11}&=&H_{core} +2\\omega(Q)=(H_{eff})_{22}, \\nonumber \\\\\n(H_{eff})_{12}&=&-GQ_-=(H_{eff})_{21}^{\\ast}. \\label{eq:5.7}\n\\end{eqnarray}\nBelow we shall describe our solutions of the associated Schr\\\"{o}dinger\nequation.\n\nThese exact solutions are to be compared with the adiabatic\napproximation,\nwith and without the Berry potential terms.  For this approximation, we\nrequire the normal modes of $H_{NC}$, which we calculate in a\nstandard way\nfrom the equations of motion,          \\begin{eqnarray}\n[a_1,H_{NC}]&=& \\omega a_1 - GQ_- a_2,    \\nonumber \\\\\n{}[a_2,H_{NC}]&=& \\omega a_2 - GQ_+ a_1,   \\label{eq:5.8}\n\\end{eqnarray}\nby forming the matrix elements,       \\begin{equation}\n\\psi_i = [0|a_i|\\Psi].   \\label{eq:5.9}   \\end{equation}\nWith $\\Omega$ representing the energy of the state $|\\Psi]$,\nwe obtain the equations         \\begin{eqnarray}\n\\Omega \\psi_1 &=& \\omega \\psi_1 -GQ_-\\psi_2, \\nonumber \\\\\n\\Omega \\psi_2 &=& -GQ_+\\psi_1 +\\omega\\psi_2, \\label{eq:5.10}\n\\end{eqnarray}\nthat yield the eigenvalues        \\begin{eqnarray}\n\\Omega^{(1)}(Q) &=& \\omega(Q) - GQ,  \\nonumber \\\\\n\\Omega^{(2)}(Q) &=& \\omega(Q) + GQ,   \\label{eq:5.11}  \\end{eqnarray}\nthat are degenerate when $Q=0$.\nThe associated normalized solutions of (\\ref{eq:5.10}) are\nrepresented most\nconveniently by introducing the normal-mode creation operators\n\\begin{equation}\nb_i^{\\dag} = \\psi^{(i)}_j a_j^{\\dag}.    \\label{eq:5.12}\n\\end{equation}\nIn detail we have as a possible choice,\n\\begin{eqnarray}\nb_1^{\\dag} &=& \\frac{1}{\\sqrt{2}}a_1^{\\dag}+\n\\frac{1}{\\sqrt{2}}\\exp{i\\phi({\\bf Q})}a_2^{\\dag}, \\label{eq:5.13} \\\\\nb_2^{\\dag} &=& \\frac{1}{\\sqrt{2}}\\exp{i\\phi({\\bf Q})}a_1^{\\dag} -\n\\frac{1}{\\sqrt{2}}a_2^{\\dag}, \\label{eq:5.14}    \\end{eqnarray}\nwhere                  \\begin{equation}\n\\tan\\phi({\\bf Q})=Q_2\/Q_1.   \\label{eq:5.15} \\end{equation}\n\nWe now apply these elementary results to the adiabatic approximation.\nIn this case we represent a suitable subset of the eigenfunctions\n(\\ref{eq:5.6}) in the form                  \\begin{equation}\n|n\\rangle = \\int d{\\bf Q} ({\\bf Q}|n)b_1^{\\dag}|0\\!:\\!Q],\n\\label{eq:5.16}\n\\end{equation}\nwhere the notation $0\\!:\\!Q$ refers to the vacuum for the normal modes.\n(For the current model, it coincides with the uncoupled vacuum.)\nThe considerations of Sec.\\ \\ref{sec:BO} now apply to this\nclass of state vectors and, in particular, we apply\nEq.\\ (\\ref{eq:3.59}).  As a special case of this equation, we have\n\\begin{equation}\n\\sum_i (P_i)^2 \\rightarrow \\sum_i (P_i -A_i)^2 +\\sum_i |(A_i)_{21}|^2,\n\\label{eq:5.160}\n\\end{equation}\nwhere\n\\begin{eqnarray}\nA_i &=& i[0\\!:\\!Q|b_1\\partial_i\nb_1^{\\dag}|0\\!:\\!Q],  \\label{eq:5.17}   \\\\\n(A_i)_{21} &=& i[0\\!:\\!Q|b_2\\partial_i\nb_1^{\\dag}|0\\!:\\!Q],     \\label{eq:5.18}   \\end{eqnarray}\nwhere $\\partial_i$ means the partial derivative with respect to $Q_i$.\nWith the help of Eqs.\\ (\\ref{eq:5.13}) and (\\ref{eq:5.14}), the\nquantities\nof interest are found to take the values       \\begin{eqnarray}\n{\\bf A} &=& \\frac{1}{2Q^2}(Q_2,-Q_1),  \\label{eq:5.19} \\\\\n\\sum_i|(A_i)_{21}|^2 &=& (1\/4Q^2).  \\label{eq:5.20}\n\\end{eqnarray}\nThe singular character of these results was to be expected.\n\nThe collective or adiabatic Hamiltonian, $H_C$, that thus emerges from\nthe assumption that the state vectors of interest can be written in the\nform (\\ref{eq:5.16}), has the structure\n\\begin{equation}\nH_C = (1\/2M)[({\\bf P}-{\\bf A})^2 +({\\bf A}_{21})^2] +V(Q) )\n+(3\/2)\\Omega^{(1)}(Q)+(1\/2)\\Omega^{(2)}(Q). \\label{eq:5.21}\n\\end{equation}\n\n\nWe turn to the problem of solving the associated eigenvalue problem.\nWe wish to compare the results of exact (numerical) calculations\nwith the eigenvalues of the collective Hamiltonian. It is useful\nto choose $\\omega(Q)$ such that the latter is exactly solvable.\nOne such choice is\n\\begin{equation}\n\\omega(Q) = \\frac{1}{2} GQ .\n\\end{equation}\nIn this case the additional contribution to the potential\nis zero,\n\\begin{equation}\n\\frac{3}{2}\\Omega^{(1)}(Q)+\\frac{1}{2} \\Omega^{(2)}(Q)\n= 0.\n\\end{equation}\nWe further take $M=1$ and $V(Q)=\\frac{1}{2}Q^2$.\n\nWe use the definition of the two-dimensional angular momentum,\n\\begin{eqnarray}\nL& =&- i \\frac{\\partial}{\\partial \\phi}\n\\nonumber \\\\ &=&\n-i (Q_1 P_2-Q_2P_1),\n\\end{eqnarray}\nto simplify the ${\\bf P}\\cdot{\\bf A}$ term.\nWe can further simplify the equation\n$H_c \\psi(\\vec Q) = E \\psi(\\vec Q)$\nby substituting\n\\begin{equation}\n\\psi(\\vec Q) = Q^{-1\/2} \\chi(Q) {\\rm e}^{im\\phi}\n\\end{equation}\nand find\n\\begin{equation}\n\\left\\{\n-\\frac{1}{2} \\frac{d^2}{d Q^2}\n+ \\frac{1}{2Q^2}[m^2+m+\\frac{1}{4}]\n+\\frac{1}{2} Q^2\\right\\} \\chi(Q) =\nE\\chi(Q)\n\\end{equation}\nEven though the centrifugal term has changed, this is still\nvery similar to\n the  radial equation for the two-dimensional harmonic\noscillator. It can be solved by the substitution\n\\begin{equation}\n\\chi(q) = q^{\\alpha+1\/2} {\\rm e}^{-q^2\/2} L_n^{(\\alpha)}(q^2).\n\\end{equation}\nAs is well known the right hand side of this equation satisfies the\ncondition\n\\begin{equation}\n- \\frac{d^2}{d q^2} \\chi\n+(q^2+\\frac{1-4\\alpha^2}{4q^2}) \\chi = (4n+2\\alpha+2) \\chi.\n\\end{equation}\nWe thus find that\n\\begin{equation}\n\\alpha(m) = \\sqrt{1\/4+(m+1\/2)^2}\n\\end{equation}\nand\n\\begin{equation}\n E_{nm} = (2n+\\alpha(m)+1).\n\\label{eq:EnmB}\n\\end{equation}\nWithout the Berry's phase terms we would have found\n\\begin{equation}\n E_{nm} = (2n+|m|+1).\n\\label{eq:EnmnoB}\n\\end{equation}\nWhen $G=0$ the solution (\\ref{eq:EnmnoB}) is exact.\nThus Eq.~(\\ref{eq:EnmB}), which is independent\nof $G$, cannot be valid for all $G$. It should be\nvalid in the adiabatic limit, which means that the two\nfrequencies $\\Omega$ must be very different. This\noccurs when $G$ is very large.\n\nThe exact solution of the problem can be calculated using\na spherical harmonic oscillator basis for the $Q$ coordinates.\nThis is coupled to a state containing either one $a^\\dagger_1$ or\none $a^\\dagger_2$ boson,\n\\begin{equation}\n|n,m,n_1,n_2\\rangle = |n,m) (a^\\dagger_1)^{(n_1)}\n(a^\\dagger_2)^{(n_2)}|0].\n\\end{equation}\nThe interaction Hamiltonian only couples states with $n_1=1,m=m_1$\nto states with $n_2=1,m=m_1+1$. If we thus choose the value $m_1$\nwe have only a relatively small matrix to diagonalize. Since we\nwish to obtain accurate results for large $G$, we allow the number of\n$Q$\nharmonic oscillator quanta to be fairly large. We have used\nup to 200 harmonic oscillator states in each block (which leads\nto a 400 by 400 matrix eigenvalue problem).\n\nIn table \\ref{tab:model1.1} we give a selected set of results\nfor $m_1=0$ and a number of values of $G$.\nSimilar results for $m_1$=1 are listed in table \\ref{tab:model1.2}.\nWe clearly see the\nconvergence to the the collective model results including the\nBerry phase for large $G$.\n\n\\begin{table}[htb]\n\\caption{The lowest 10 eigenvalues of the coupled problem for\n$m_1=0$ as a function of $G$. The column labeled Berry lists the\neigen energies of the collective Hamiltonian \\protect{\\ref{eq:5.21}}.\n\\label{tab:model1.1}}\n\\begin{center}\n\\begin{tabular}{rrrrrrrr}\n\\hline\nG &\n0.01     & 0.1      & 1           & 10        & 100       & 1000\n&Berry\\\\\n\\hline\n$E_1$ &\n1.00875  & 1.07849  &   1.36229   &   1.58463 &   1.66532 &   1.69302\n& 1.7071\\\\\n$E_2$ &\n2.01328  & 2.13134  &   2.92007   &   3.48844 &   3.63505 &   3.68298\n& 3.7071\\\\\n$E_3$ &\n3.01540  & 3.14492  &   4.09426   &   5.39473 &   5.60870 &   5.67438\n& 5.7071\\\\\n$E_4$ &\n4.01827  & 4.18125  &   5.30157   &   7.29413 &   7.58443 &   7.66658\n& 7.7071\\\\\n$E_5$ &\n5.01997  & 5.19061  &   6.57000   &   9.17666 &   9.56145 &   9.65932\n& 9.7071\\\\\n$E_6$ &\n6.02221  & 6.22068  &   7.66964   &   11.0262 &   11.5393 &   11.6525\n& 11.7071\\\\\n$E_7$ &\n7.02367  & 7.22768  &   8.90918   &   12.8133 &   13.5177 &   13.6459\n& 13.7071\\\\\n$E_8$ &\n8.02558  & 8.25425  &   10.0230   &   14.4981 &   15.4965 &   15.6396\n& 15.7071\\\\\n$E_9$ &\n9.02687  & 9.25972  &   11.1883   &   16.0926 &   17.4754 &   17.6335\n& 17.7071\\\\\n$E_{10}$ &\n10.0286  & 10.2840  &   12.3410   &   17.7119 &   19.4544 &   19.6276\n& 19.7071\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[htb]\n\\caption{The lowest 10 eigenvalues of the coupled problem for\n$m_1=1$ as a function of $G$. The column labeled Berry lists the\neigen energies of the collective Hamiltonian \\protect{\\ref{eq:5.21}}.\n\\label{tab:model1.2}}\n\\begin{center}\n\\begin{tabular}{rrrrrrrr}\n\\hline\nG &\n0.01     & 0.1      & 1           & 10        & 100       & 1000\n&Berry\\\\\n\\hline\n$E_1$ &\n   2.01308 &   2.11333&   2.41566&   2.54921&   2.57601&   2.58042\n& 2.58114\\\\\n$E_2$ &\n   3.01670 &   3.17369&   4.20360&   4.51926&   4.57057 &   4.57959\n& 4.58114\\\\\n$E_3$ &\n   4.01807 &   4.16340&   5.44213&   6.48766&   6.56479&   6.57866\n& 6.58114\\\\\n$E_4$ &\n   5.02086 &   5.21512&   6.43129&   8.45380&   8.55880&   8.57766\n& 8.58114\\\\\n$E_5$ &\n   6.02202 &   6.20307&   7.86356&   10.4165&   10.5527&   10.5766\n& 10.58114\\\\\n$E_6$ &\n   7.02436 &   7.25000&   8.79958&   12.3739&   12.5464&   12.5755\n& 12.58114\\\\\n$E_7$ &\n   8.02538 &   8.23691&   10.1070&   14.3218&   14.5400&   14.5744\n& 14.58114\\\\\n$E_8$ &\n   9.02745 &   9.28063&   11.1975&   16.2512&   16.5336&   16.5733\n& 16.58114\\\\\n$E_9$ &\n   10.0284 &   10.2669&   12.3141&   18.1346&   18.5270&   18.5721\n& 18.58114\\\\\n$E_{10}$ &\n   11.0302 &   11.3082&   13.5364&   19.8627&   20.5204&   20.5709\n& 20.58114\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsection{The second model: ground state}\n\nWe next study a model that is also exactly decoupled, differing\nfrom the one just investigated only in\nthe form of the interaction.  In this model Eqs.\\\n(\\ref{eq:5.1})-(\\ref{eq:5.3})\nstand as given, but Eq.\\ (\\ref{eq:5.4}) is modified to\n\\begin{eqnarray}\nH_{int} &=& -G_0 Q(a_1^{\\dag}a_2 + a_2^{\\dag}a_1)\n -G_1(Q_+ a_1 a_2 +Q_- a_1^{\\dag}a_2^{\\dag}).  \\label{eq:5.22}\n\\end{eqnarray}\nThe second term of this interaction\nspoils the conservation of boson number used to simplify the\nsolution of the\nprevious problem.  For the present problem, it is already\ninteresting to study\nthe spectrum when the fast variables are in their ground levels,\nsince there\n are now non-trivial ground-state correlations and associated\nnon-trivial\nBerry potentials, that we shall calculate.\n\n\nWe first turn to the study of the BO approximation.  As before we\nneed  the normal modes of the fast variables, as determined from the\nthe equations of motion\n\\begin{eqnarray}\n[a_1,H_{NC}] &=& \\omega a_1 -G_0Q a_2 -G_1Q_- a_2^{\\dag}, \\nonumber \\\\\n{}[a_2,H_{NC}] &=& \\omega a_2 -G_0Q a_1 -G_1Q_- a_1^{\\dag},\n\\label{eq:5.24}\n\\end{eqnarray}\nand their hermitian conjugates.  In terms of the definitions\n\\begin{eqnarray} b_i^{\\dag}&=&\\psi_j^{(i)}a_j^{\\dag}-\\chi_j^{(i)}a_j,\n\\label{eq:5.25} \\\\ \\psi_j^{(i)}&=&[0\\!:\\!Q|a_j|\\Psi^{(i)}],\n\\label{eq:5.26}\\\\ \\chi_j^{(i)}&=&[0\\!:\\!Q|a_j^{\\dag}|\\Psi^{(i)}],\n\\label{eq:5.27}\\\\ |\\Psi^{(i)}] &=&b_i^{\\dag}|0\\!:\\!Q], \\label{eq:5.28}\n\\end{eqnarray}\nwe obtain the eigenvalue equations \\begin{eqnarray}\n\\Omega\\psi_1&=&\\omega\\psi_1 -G_0Q\\psi_2 -G_1Q_-\\chi_2, \\nonumber \\\\\n\\Omega\\psi_2&=&\\omega\\psi_2 -G_0Q\\psi_1 -G_1Q_-\\chi_1, \\nonumber \\\\\n-\\Omega\\chi_1&=&\\omega\\chi_1 -G_0Q\\chi_2 -G_1Q_+\\psi_1,\\nonumber \\\\\n-\\Omega\\chi_2&=&\\omega\\chi_2 -G_0Q\\chi_1 -G_1Q_+\\psi_1.\\label{eq:5.29}\n\\end{eqnarray}\nThe physical eigenvalues are the positive roots of\n\\begin{equation}\n\\Omega^2(Q) = (\\omega \\mp G_0Q)^2 -G_1^2Q^2, \\label{eq:5.30}\n\\end{equation}\nwhich are again degenerate at $Q=0$.\nAs for the first model studied,\nwe label these solutions $\\Omega^{(i)},i=1,2$.\nThe corresponding amplitudes are determined from the equations of\nmotion and\nthe normalization conditions\n\\begin{equation}\n|\\psi_1|^2 +|\\psi_2|^2 -|\\chi_1|^2 -|\\chi_2|^2 =1. \\label{eq:5.31}\n\\end{equation}\nThe simplest forms for the amplitudes are achieved by\nrepeated use of the eigenvalue equation (\\ref{eq:5.30}).  We thus find\n\\begin{eqnarray}\n\\psi_2^{(j)}&=&(-1)^{j+1}\\psi_1^{(j)}={\\rm real}, \\label{eq:5.32} \\\\\n\\chi_i^{(j)}&=&\\exp(i\\phi)\\tilde{\\chi}_i^{(j)}, \\label{eq:5.33} \\\\\n\\tilde{\\chi}_2^{(j)}&=&(-1)^{j+1}\\tilde{\\chi}_1^{(j)}={\\rm real},\n\\label{eq:5.34} \\\\\n\\psi_1^{(j)}&=&\n\\frac{(\\Omega^{(j)}+\\omega)+(-1)^jG_0Q}\n{\\sqrt{2}[(\\omega+\\Omega^{(j)})^2+(-1)^j2G_0Q(\\omega+\\Omega^{(j)})\n+(G_0^2-G_1^2)Q^2]^{1\/2}},\n\\label{eq:5.35} \\\\\n\\chi_1^{(j)}&=&\n (-1)^{j+1}f^{(j)} (Q)\\psi_1^{(j)},\n\\label{eq:5.36}\n\\\\\nf^{(j)}&=&\\frac{G_1Q}{(\\Omega^{(j)}+\\omega)+(-1)^jG_0Q}.\n\\end{eqnarray}\nHere $\\phi$ is the polar angle for $Q$, $\\phi = \\arctan (Q_2\/Q_1)$.\n\n\nThese results are thus available for application to the BO\napproximation studied by means of the assumption\n\\begin{equation}\n|n\\rangle =\\int d{\\bf Q}|{\\bf Q}\\rangle ({\\bf Q}|n)|0\\!:\\!Q].\n\\label{eq:5.39}\n\\end{equation}\nIt follows that the Berry vector potential is given by the formula\n\\begin{equation}\nA_i = i[0\\!:\\!Q|\\partial_i|0\\!:\\!Q].\n\\label{eq:5.40}\n\\end{equation}\nTo carry out this calculation, we need the form of the correlated\nvacuum state,\nas given by the equation \\cite{Sa}\n\\begin{equation}\n|0\\!:\\!Q] = {\\cal N} \\exp[\\mbox{$\\frac{1}{2}$} Z_{ij}a_i^{\\dag}a_j^{\\dag}]|0],\n\\label{eq:5.41}\n\\end{equation}\nwhere ${\\cal N}$ is a real normalization factor, whose value we\nshall not need,\n$a_i |0]=0$, and $Z_{ij}$ is a generally complex array whose\nvalues will\nbe considered below.  Using the constancy of the norm of the\ncorrelated ground\nstate as a function of the values of ${\\bf Q}$, we can rewrite\n(\\ref{eq:5.40})\nas\n\\begin{equation}\nA_i =\\mbox{$\\frac{1}{2}$} i \\{[ 0\\!:\\!Q|\\partial_i|0\\!:\\!Q]\n- [\\partial_i 0\\!:\\!Q|0\\!:\\!Q]\\}, \\label{eq:5.42}\n\\end{equation}\nwhich shows that the (real) normalization factor ${\\cal N}$ does not\ncontribute. This leaves the result\n\\begin{equation}\nA_i = \\frac{1}{4}i[\\langle a_k^{\\dag}a_l^{\\dag}\\rangle\\partial_i\nZ_{kl} - \\langle a_k a_l\\rangle\\partial_i Z_{kl}^{\\ast}],\n\\label{eq:5.43}\n\\end{equation}\nwhere the averages, indicated by angular brackets,\nare with respect to the correlated vacuum.\n\nWe next indicate how (\\ref{eq:5.43}) can be evaluated in terms of the\nsolutions\nwe have found above for the equations of motion.  The quantities\n$Z_{kl}$\nare solutions of the equations\n\\begin{equation}\n\\psi_j^{(i)}Z_{jk}^{\\ast} = \\chi_k^{(i)},    \\label{eq:5.44}\n\\end{equation}\nthat follow from the definition $b_i |0\\!:\\!Q] =0.$  The expectation\nvalues\ncan also be evaluated with the help of the formulas\n\\begin{equation}\na_i = \\psi_i^{(j)}b_j +\\chi_i^{(j)\\ast}b_j^{\\dag},\\label{eq:5.45}\n\\end{equation}\nand their hermitian conjugates, that are the inverse to\nEq.\\ (\\ref{eq:5.25}).\n  An elementary calculation now yields\n\\begin{equation}\n\\langle a_k^{\\dag} a_l^{\\dag}\\rangle = \\langle a_k a_l\\rangle^{\\ast}\n= \\chi_k^{(j)}\\psi_l^{(j)\\ast}.     \\label{eq:5.46}\n\\end{equation}\nThus for the vacuum Berry potential, we obtain the formula\n\\begin{equation}\nA_i = \\frac{1}{4}i[\\chi_k^{(m)}\\psi_l^{(m)\\ast}\\partial_i\nZ_{kl} -\\chi_k^{(m)\\ast}\\psi_l^{(m)}\\partial_i Z_{kl}^{\\ast}].\n\\label{eq:5.47}\n\\end{equation}\n\nAny other matrix element, $(A_i)_{\\nu 0}$ of the Berry potential can be\ncalculated by similar techniques.  In fact, the only non-vanishing\nelements\nof this type are $(A_i)_{11,0}$, $(A_i)_{20,0}$, and $(A_i)_{02,0}$,\nwhere,\nfor example, $\\nu=20$ means a state with two correlated bosons of type 1\nand none of type 2.  We quote formulas for these matrix elements that\nfollow\nfrom the same elementary techniques that furnished Eq.\\ (\\ref{eq:5.47}).\nWe thus have\n\\begin{eqnarray}\n(A_i)_{11,0}&=&i[11\\!:\\!Q|\\partial_i |0\\!:\\!Q] \\nonumber \\\\\n&=& -i[\\partial_i\\; 11\\!:\\!Q|0\\!:\\!Q] \\nonumber \\\\\n&=& -i[0\\!:\\!Q|b_1\\partial_i(b_2 )|0\\!:\\!Q]\\} \\nonumber \\\\\n&=&-i\n[\\sum_{j}\\psi_j^{(1)\\ast}\\partial_i\\chi_j^{(2)\\ast}\n-\\chi_j^{(1)\\ast}\\partial_i\\psi_j^{(2)\\ast}].\n\\label{eq:5.48}   \\end{eqnarray}\nSimilarly, remembering the normalization of the states, we find\n\\begin{equation}\n(A_i)_{20,0}= \\frac{1}{i\\sqrt{2}}[\\sum_{j}\n\\psi_j^{(1)\\ast}\\partial_i\\chi_j^{(1)\\ast}\n-\\chi_j^{(1)\\ast}\\partial_i\\psi_j^{(1)\\ast}].\n\\label{eq:5.490}      \\end{equation}\nFinally, the matrix element $(A_i)_{02,0}$ is obtained from\n\n(\\ref{eq:5.490})\nsimply by replacing all superscripts 1 by 2.\nIn order to implement fully the BO approximation, it is necessary to\nevaluate these matrix elements in terms of the explicit solutions of\nthe\nmodel displayed in Eqs.\\ (\\ref{eq:5.32})-(\\ref{eq:5.35}).  It is also\nappropriate to remark here that in consequence of these relations and\nEq.\\ (\\ref{eq:5.44}) we can write\n\\begin{equation}\nZ_{jk} = \\exp(-i\\phi)\\tilde{Z}_{jk},\\;\\;\\tilde{Z}_{jk}={\\rm real}\n\\label{eq:5.491}\n\\end {equation}\n\nThe calculation of the Berry potentials as defined above is now\nstraightforward.  In terms of purely real quantities, we find,\nfor example,\n\\begin{equation}\nA_i = (\\partial_i \\phi)[(\\tilde{\\chi}_1^{(1)}\\psi_1^{(1)}+\n\\tilde{\\chi}_1^{(2)}\n\\psi_1^{(2)})\\tilde{Z}_{11} + (\\tilde{\\chi}_1^{(1)}\\psi_1^{(1)}\n-\\tilde{\\chi}_1^{(2)}\\psi_1^{(2)})\\tilde{Z}_{12}].  \\label{eq:5.492}\n\\end{equation}     Using the real form of Eq.\\ (\\ref{eq:5.44}), this\nexpression can be simplified to\n\\begin{eqnarray}\nA_i &=& \\partial_i\\phi[(\\tilde{\\chi}_1^{(1)})^2 +\n(\\tilde{\\chi}_1^{(2)})^2]\n\\nonumber\\\\\n&=& \\frac{G_1^2}{2}\\left[\n\\frac{1}{(\\omega+\\Omega^{(1)})(\\omega+\\Omega^{(1)}-2G_0Q)+\n(G_0^2-G_1^2)Q^2}\n\\right. \\nonumber \\\\ && \\left.\n+\\frac{1}{(\\omega+\\Omega^{(2)})(\\omega+\\Omega^{(2)}+2G_0Q)+\n(G_0^2-G_1^2)Q^2}\n\\right](-Q_2,Q_1)_i.\n\\label{eq:5.493}\n\\end{eqnarray}\nIn contrast to the previous model the vector potential is not\nsingular at the origin, but take the finite value\n\\begin{equation}\n{\\bf A}(0) = \\frac{G_1^2}{4 \\omega(0)^2}(-Q_2,Q_1).\n\\end{equation}\n\nTurning to the off-diagonal matrix elements, we find by similar\ncalculations\n\\begin{eqnarray}\n(A_i)_{11,0}&=&0,   \\label{eq:5.494}  \\\\\n(A_i)_{20,0}&=& \\sqrt{2}\\exp(-i\\phi)[(\\partial_i\\phi)\\psi_1^{(1)}\n\\tilde{\\chi}_1^{(1)} +i(\\psi_1^{(1)}\\partial_i\\tilde{\\chi}_1^{(1)}\n- \\tilde{\\chi}_1^{(1)}\\partial_i\\psi_1^{(1)}]\n\\nonumber\\\\ & = &\n-\\sqrt{2}\\exp(-i\\phi)\\left(\\psi_1^{(1)}\\right)^2 f^{(1)}\n\\left[\\partial_i\\phi+i(\\partial_iQ) \\partial_Q \\ln f^{(1)} \\right]\n\\nonumber\\\\ & = &\n\\sqrt{2}\\exp(-i\\phi)\n\\frac{G_1[(\\Omega^{(1)}+\\omega)-G_0Q]}\n{2[(\\omega+\\Omega^{(1)})(\\omega+\\Omega^{(1)}-2G_0Q)+(G_0^2-G_1^2)Q^2]}\n\\nonumber\\\\ && \\times\n\\left[\\frac{1}{Q}(-Q_2,Q_1)+i \\frac{1}{Q}(Q_1,Q_2) Q\\partial_Q\n\\ln f^{(1)} \\right]\n, \\label{eq:5.495}\\\\\n(A_i)_{02,0}&=& -\\sqrt{2}\\exp(-i\\phi)\n\\frac{G_1[G_0Q+(\\Omega^{(2)}+\\omega)]}\n{2[(\\omega+\\Omega^{(2)})(\\omega+\\Omega^{(2)}-2G_0Q)+(G_0^2-G_1^2)Q^2]}\n\\nonumber\\\\ && \\times\n\\left[\\frac{1}{Q}(-Q_2,Q_1)+i \\frac{1}{Q}(Q_1,Q_2) Q\\partial_Q\n\\ln f^{(2)} \\right].\n \\label{eq:5.496}\n\\end{eqnarray}\nThese off-diagonal potentials are also regular as $Q$ goes to zero,\nfor reasonably well behaved $\\omega(Q)$.\n\nWe now study the numerical solution of the adiabatic Hamiltonian,\n\\begin{equation}\nH_{C} = \\frac{1}{2}[({\\bf P}-{\\bf A})^2 + |{\\bf A}_{20,0}|^2 +\n|{\\bf A}_{02,0}|^2]\n+\\frac{1}{2}Q^2 + \\frac{1}{2}[\\Omega^{(1)}(Q)+\\Omega^{(2)}(Q)],\n\\end{equation}\nwhere we have once again made the choice $M=1,V(Q)=\\frac{1}{2}Q^2$.\nFor simplicity we take $G_0=0$, and write $G_1=G$.\nFor this special choice $\\Omega^{(1)}=\\Omega^{(2)}=\\Omega$ and\n$(A_i)_{02,0}=-(A_i)_{20,0}$.\nA simple form for $\\omega(Q)$,\nchosen such that  $\\Omega$ is positive definite,\nis\n\\begin{equation}\n\\omega(Q) = \\omega_0+GQ.\n\\end{equation}\nIn Fig.~\\ref{fig:G1om2} we show some of the relevant quantities\nin the collective Hamiltonian for the choice $G=1$, $\\omega_0=2$.\n\\begin{figure}\n\\epsfysize=12cm\n\\centerline{\\epsffile{plot2.ps}}\n\\caption{Parameters in the collective Hamiltonian for $G=1$,\n$\\omega_0=2$.\nThe lower panel shows the frequencies $\\Omega^{(j)}$, the middle panel\nshows the size of the diagonal Berry potential, and the\nupper panel shows the square of the off-diagonal\nBerry-potentials.\\label{fig:G1om2}}\n\\end{figure}\nWe  diagonalize the collective Hamiltonian by first going to\nspherical coordinates, and use the fact that $m$ is conserved to\nwrite down a radial equation. We then map the $Q$ values\nfrom  the interval $[0,\\infty)$ to the interval $[0,1]$. Finally\nwe make a finite difference approximation to the radial equation, and\nsolve\nthe approximate equation  by matrix diagonalization.\n\nSolving the complete problem, without making the adiabatic\napproximation\nis somewhat involved. Since the Hamiltonian is invariant under the\ninterchange $a^\\dagger_1 \\leftrightarrow a^\\dagger_2$, we introduce\nnew operators that are invariant under this parity transformation,\n\\begin{equation}\na^\\dagger_\\pm = \\frac{1}{\\sqrt 2} (a^\\dagger_1 \\pm a^\\dagger_2).\n\\end{equation}\nThe interaction Hamiltonian takes the simple form\n\\begin{equation}\nH_{int} = -G_0 (a^\\dagger_+a_+ -a^\\dagger_-a_-)\n-\\frac{G_1}{2} (Q_+[a_+^2-a_-^2] + Q_-[a_+^{\\dagger2}-a_-^{\\dagger2}]).\n\\label{eq:Hintnew}\n\\end{equation}\nWe now make the expansion\n\\begin{equation}\n\\Psi = \\sum_{K,\\kappa} \\psi_{K,\\kappa}(Q)\n\\frac{\\left(a^\\dagger_+\\right)^{2K-2\\kappa}\n\\left(a^\\dagger_-\\right)^{2\\kappa}}\n{\\sqrt{(2K-2\\kappa)!(2\\kappa)!}}|0]\n\\end{equation}\n(The even powers in this equation constitute the only combination\nthat includes the vacuum for the fast degrees of freedom.)\n\nAn approximate solution is obtained by limiting the sum over $K$,\nwhile\nsumming over all allowed $\\kappa$. At the same time we expand\n$\\psi_{K,\\kappa}(Q)$ in  a finite number of\nspherical harmonic oscillator eigenfunctions\n$\\langle Q| nm \\rangle$.\nWe denote the $m$ value used in the expansion of\n$\\psi_{K,\\kappa}(Q)$ by $m_{K,\\kappa}$.\nThe interaction does not couple states of different $\\kappa$ for\nfixed $K$.\nWe thus find that\n$m_{K,\\kappa}=m_K=m_{K-1}-1$. Thus the value $m_0$ is\na constant of the motion, and can be used to specify different\nsolutions. This quantity  corresponds directly to the value of $m$\nin the\ncollective Hamiltonian.\nWe have performed matrix diagonalizations for $\\omega_0=2,10$ and\n$G=1$.\nWe have used harmonic oscillator states up to principal\nquantum number 70, and $K$ values up to 20.\n(This corresponds to a 61,401 by 61,401 matrix.)\nThe resulting matrix,\nwhich is very sparse, was diagonalized using a Lanczos algorithm.\nThe eigenvalues were  checked for convergence by comparing\na calculation with smaller cut-offs on $n$ and $K$ against one\nwith larger cutoffs. In tables \\ref{tab:om2} and \\ref{tab:om10}\nwe compare a few selected ground state energies of the complete\ncollective Hamiltonian with the exact solution. The splitting\nbetween states with opposite values of $m$ is completely due\nto the vector potential. For the case $\\omega_0=2$, where the\nadiabatic approximation we see that the size of the splitting\nis very close to the exact value.  The difference is probably due\nto non-adiabatic effects that can not be completely neglected\nfor  $\\omega_0=2$. For the case $\\omega_0=10$ the correspondence is\nmuch closer, but the size of the\nBerry vector potential is much smaller as well.\n\n\\begin{table}[htb]\n\\caption{A comparison between the exact numerical ground state\nenergies and those for the collective Hamiltonian. $\\omega_0=2,G=1$,\nand the values of $m$ are listed in the table.\n\\label{tab:om2}}\n\\begin{center}\n\\begin{tabular}{l|llll}\n           &  $m=-5$ & $m =5$ & $m=-1$ & $m=1$ \\\\\n\\hline\nexact      &  9.54715 & 9.73101 & 4.97309 & 5.03405 \\\\\ncollective &  9.56722 & 9.74656 & 4.98669 & 5.04375\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\begin{table}[htb]\n\\caption{A comparison between the exact numerical ground state\nenergies and those for the collective Hamiltonian. $\\omega_0=10,G=1$,\nand the values of $m$ are listed in the table.\\label{tab:om10}}\n\\begin{center}\n\\begin{tabular}{l|ll}\n           &  $m=-5$ & $m =5$ \\\\\n\\hline\nexact      &  18.0745 & 18.0918 \\\\\ncollective &  18.0750 & 18.0922\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\\section*{Acknowledgement}\nThis work was supported in part by the U.\\ S.\\ Department of Energy\nunder grant number 40264-5-25351.\n\n\\renewcommand{\\theequation}\n{\\Alph{section}.\\arabic{equation}}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nOne of the most common problems encountered in the analysis of differential equations arising\nin applications is to give a qualitative description of the influence \nof parameters $\\mu\\in\\R^p$ on the system dynamics. For ordinary differential equations (ODEs) with\nphase space variables $x\\in \\R^N$, one approach to this problem is to use a standard \nnumerical integration method in combination with Monte-Carlo \nsimulation. For obtaining a basic idea about the system dynamics and exploration of different regimes,\nthis approach is an indispensable tool in modern scientific computing. However, to completely map out\nthe different dynamical regimes in parameter space, this approach requires \na systematic simulation over grids of initial \nvalues as well as over grids in parameter space. Even though there are techniques available to \nalleviate some of the computational effort when $p$ and $N$ are large (formally we write this as \n$p,N\\gg1$), such as multi-level Monte Carlo (MC) \\cite{Heinrich} and quasi-MC \\cite{Caflisch}, \nthe computational effort can still be substantial. In addition, manual inspection of trajectories \nis usually required to detect important invariant sets, metastable states, bifurcations or locally \ninvariant manifolds, which characterize the dynamics. Therefore, direct simulation can be a\ncomputation- and labor-intensive approach to determine dynamical structures in nonlinear systems\\footnote{\nThe MC approach with manual inspection of orbits is actually used very successfully - \nparticularly in many application areas - via a method that could be called gsMC (graduate student \nMonte-Carlo).}. Indeed, this problem is well-known for some time as discussed in \n\\cite[p.1-2]{TuckermanBarkley}:\n\\textit{``However, these [time-stepping] techniques rely on waiting out slow exponential decay,\nor on arduous binary searches, and make highly inefficient use of both machine and human resources.''}\n\\medskip\n\nAs an alternative, numerical continuation methods are frequently used across all the natural \nsciences and engineering to systematically explore parameter space for important dynamical \nstructures \\cite{KrauskopfOsingaGalan-Vioque,KuepperSeydelTroger,RooseDeDierSpence,Seydel}. \nThis approach has been pioneered in the late 1970s, particularly by Keller \n\\cite{Keller,Keller3,Keller2}. The main idea is to use a predictor-corrector scheme in\ncombination with singularity detection to track parametrized curves (or higher-dimensional \nmanifolds) of solutions to nonlinear algebraic systems arising, {e.g.}, from the steady\nstate formulation of the differential equation. The theory of numerical continuation is \nwell-established by now \\cite{AllgowerGeorg,DoedelKellerKernevezI,DoedelKellerKernevezII,Govaerts}. For \nODEs, it has been used to compute, {e.g.}, bifurcation diagrams \\cite{Doedel1,Kuznetsov}, stable\/unstable \nmanifolds \\cite{KrauskopfOsinga1,KrauskopfSurvey}, global homoclinic\/heteroclinic orbits \n\\cite{Beyn2,ChampneysKuznetsovSandstede}, canards and canard values in fast-slow systems \n\\cite{DesrochesKrauskopfOsinga2,KuehnCanLya}, invariant tori \\cite{SchilderOsingaVogt}, \nspectra \\cite{RademacherSandstedeScheel} and higher-dimensional solution manifolds \\cite{Henderson1}. \nThere are also several highly successful software packages available for ODE continuation problems \nsuch as \\texttt{AUTO} \\cite{Doedel_AUTO2007}, \\texttt{XPP} \\cite{XPP}, \\texttt{MatCont} \\cite{DhoogeGovaertsKuznetsov} or \\texttt{PyDSTool} \\cite{ClewleySherwoodLaMarGuckenheimer}. A key reason, why one may provide excellent\nself-contained tools for ODEs, as evidenced by the structure of \\texttt{AUTO} \\cite{Doedel_AUTO2007}, is\nthat many questions may be formulated in terms of classical two-point boundary value problems (BVPs)\n\\cite{AscherMattheijRussell}. Although providing a suitable problem reformulation is not a trivial\ntask, there is substantial practical evidence that the strategy to continue two-point boundary-value \nproblems is very successful\\footnote{One may borrow terminology from physics and describe the \nsituation regarding two-point BVPs as the main ``universality class'' in dynamical systems\ncomputation for ODEs.}.\\medskip\n\nFurthermore, similar approaches as for ODEs have also been implemented for delay differential\nequations (DDEs) \\cite{EngelborghsLuzyaninaSamaey,SzalaiStepanHogan} and have shown promise to also\nwork for stochastic differential equations (SDEs) \\cite{BarkleyKevrekidisStuart,KuehnSDEcont1}.\nHence, it is quite natural to also consider partial differential equations and numerical continuation.\nThis approach has been carried out, {e.g.}, for the Ginzburg-Landau equation \n\\cite{SchloemerAvitabileVanroose}, excitable media \\cite{Barkley1,KneesTuckermanBarkley}, \nthin film equations \\cite{BeltrameThiele}, the Swift-Hohenberg\nequation \\cite{Avitabileetal,LloydSandstedeAvitabileChampneys}, two-dimensional ocean models \n\\cite{KatsmanDijkstraSchmeits}, the Schnakenberg system \\cite{UeckerWetzel}, among certainly many\nother systems. However, there is no uniform approach via a single software package and\/or \ncontinuation toolbox available.\\medskip\n\nIn fact, it is unlikely that we are ever going to reach a similar status in terms\nof stand-alone software, as {e.g.}~\\texttt{AUTO} and \\texttt{MatCont} for ODEs, in the case \nof numerical continuation for PDEs, not\neven for the stationary case. The main reason is that for each class of PDEs, each type of spatial \ndomain and each type of boundary conditions, there is an vast number of different spatial discretization \nmethods available such as finite element methods (FEM) \\cite{BrennerScott}, finite-volume methods (FVM) \n\\cite{LeVeque1}, finite-difference methods (FDM) \\cite{Thomas}, spectral methods \\cite{GottliebOrszag}, \nand many others \\cite{ArnoldBrezziCockburnMarini,BanerjeeButterfield,CottrellHughesBazilevs,Peskin}. \nFurthermore, problems arise on the linear algebra side, when trying to solve large linear systems arising \nfrom various discretizations \\cite{Bindeletal,Dijkstraetal}. Given a PDE from applications, the natural question\nis, what method to use within a numerical continuation framework? The answer clearly is: \nIt depends! Indeed, the various classes of PDEs are so different from a numerical analysis viewpoint,\nthat all we may hope for is to have several software libraries available, one for each class and \neach method. If we want to combine numerical continuation with numerical PDE techniques, there are \ntwo natural approaches. One may just implement the discretization scheme for the PDE ``by-hand'' for\neach problem and then add on top of this scheme ``by-hand'' a numerical continuation approach. This\nhas been used successfully in the PDE continuation examples mentioned above. However, the potential \ndrawback is cost: namely computational cost due to potentially sub-optimal implementation of each \ncomponent and labor cost due to the programming from first principles. The last issue was nicely\nformulated - in a slightly different context - by Bangerth and Heister \\cite{BangerthHeister}; just \none important quote from their opinion \npiece summarizes the situation well: \\textit{``If every graduate student writes the code for a new \ndiscretization from scratch, we will be stuck forever solving `toy problems' (like the proverbial\n`Laplace equation on the square'). Unfortunately, this is no longer sufficient to convince our \ncolleagues in the applied sciences [...] that the new method is also applicable to their vastly\nmore complex problems.''}\\footnote{One may view the described situation as another incarnation of \ngsMC.}\\medskip \n\nIt is one main aspect - among several - of this paper to show that the combination of numerical continuation,\nPDE discretization, scientific software packages and suitable gluing codes, is a\nconvincing strategy to solve relatively complex problems at low computational and low labor costs. \nOf course, one may argue that \\textit{in mathematical theory} there is no problem with this well-known \nidea, it does work. However, \\textit{in numerical practice}, the situation is quite different. Even\nthough software for gluing computations within a certain framework, such as \\texttt{Trilinos} \\cite{Herouxetal}, \nare available, there are still major practical problems to be solved for dynamical systems analysis of \nPDEs. However, to illustrate that we are right now in the situation, where aiming to glue several algorithms and\nsoftware packages in an efficient way, can be successful, we study here various elliptic PDEs:\nthe Lane-Emden-Fowler equation \\cite{Chandrasekhar,Wong} with a linear term, the Lane-Emden-Fowler \nequation with a localized force and the Caginalp \nsystem \\cite{Caginalp,CaginalpFife}. The main computational framework is set up in \\texttt{MatLab} \nas a high-level gluing language with a\nfocus on the recently developed numerical continuation software \\texttt{pde2path} \\cite{UeckerWetzelRademacher,DohnalRademacherUeckerWetzel}.\\medskip \n\nIt is well-known by users of numerical continuation software that, \nduring the practical implementation process, finding starting solutions is a key obstacle as \npointed out nicely by Sandstede and Lloyd\n\\cite{SandstedeLloyd}: \\textit{``Strategies \nfor finding good initial guesses depend strongly on the underlying specific problem: often, \nfinding good starting data is the main obstacle for using [numerical continuation] successfully''}\\footnote{\nThe original quote contains [AUTO] instead of [numerical continuation] but it is clear\nthat the problem will persist from a particular software package to the more general problem of \ndetermining starting solutions.}. Here we address this problem by using a minimax approach to\nvariational elliptic PDE proposed by Zhou and co-workers \\cite{LiZhou,LiZhou1} by employing the \ncode \\cite{Zhou} to generate starting solutions in \\texttt{pde2path}, {i.e.}, instead of programming the\ncontinuation method ``by-hand'' \\cite{ChenZhou} we glue the available combination of FEM, \ncontinuation and variational optimization to minimize the programming effort. The goal is to\ndemonstrate that already today many computational tasks could be simplified substantially if\nbetter interfaces for gluing code would be available, and new code would be designed with this \nproblem in mind.\\medskip \n\nTo cross-validate the gluing approach, we use the Lane-Emden-Fowler equation, which is a standard example exhibiting \nmultiple (unstable) solutions, which are computable by the \\texttt{minimax} method \\cite{LiZhou,Zhou}. We add a linear \nterm term, whose rate is used as the main continuation parameter. Starting solutions are \nfirst determined by the \\texttt{minimax} approach and then continued in \\texttt{pde2path}. As a second approach,\nthe homogeneous zero solution branch is continued numerically, and branch switching is \nperformed at bifurcation points to calculate branches, which each contain a single solution\nto the Lane-Emden-Fowler equation without the added linear term.\\medskip\n\nAs the first main new problem for this paper, we consider the Lane-Emden-Fowler equation\nwith a forcing term. The motivation for modifying the elliptic PDE with a certain forcing term is \ndeveloped by pointing out the analogy to a similar term appearing as a localized microforce \nin the generalized Ginzburg-Landau\/Allen-Cahn equation \\cite{AllenCahn,AransonKramer,Gurtin,Landau},\nwhere the nonlinearity has the opposite sign of the Lane-Emden-Fowler equation. Although the theory \nof the two classes is quite different due to the different signs of operators \\cite{ChenZhouNi}, the \nalgebraic structure is similar and the motivation of external microforces also applies to fluid\ndynamics applications of the Lane-Emden-Fowler equation. As before, \nwe use a \\texttt{minimax} approach to determine a set of starting solutions. In this case, there is \nno homogeneous solution branch available for direct numerical continuation, although a two-parameter homotopy \napproach could be tried to locate starting solutions\\footnote{This approach becomes from a practical implementation\nviewpoint more involved, which is exactly what we aim to avoid here.}. We show that there\nis a quite intricate deformation of localized structures along isolated bifurcation curves. \nThe bifurcation curves are C-shaped curves, which deform onto thinner structures in the $L^\\I$-norm \nas the complexity of the patterns is increased. In particular, the numerical study in this paper illustrates the effect, \nhow symmetry-breaking may occur when asymmetric localized forces are added to elliptic PDEs on symmetric \ndomains. From an analytical viewpoint, special solutions of nonlinear elliptic PDEs on symmetric domains \nare a very active research area, see {e.g.}~\\cite{BartschDAprilePistoia}. The same remark applies to the singularly-perturbed\nelliptic PDE related to Lin-Ni-Takagi type problems \\cite{AoWeiZeng,LinNiTakagi,LinNiWei}, which has the same\nsigns for the operators as our modified Lane-Emden-Fowler equation without the asymmetric microforce. Hence, it\nis critical to understand, for applications and theory, the effect how inhomogeneous bifurcation branches \ncontinue globally under microforce loading.\\medskip\n\nAs a third case study, we consider the \nCaginalp system for phase transitions, such as melting-solidification processes. The domain is chosen as a more\ncomplex, wing-shaped geometry. The motivation for this setup arises directly from (de-)icing problems in\naerospace engineering. As for the previous elliptic PDEs, there are multiple solutions for a fixed parameter \nvalue, which we compute using \\texttt{pde2path} by guessing the right starting solution. \nThe \\texttt{minimax} approach using the code \\cite{Zhou} does not modify in a straightforward way \nto detect the solutions. Therefore, \na more elaborate programming effort would be required to use the variational starting point, which is \nprecisely what we want to avoid here. However, from various recent works, {e.g.}~\\cite{XieYuanZhou}, \none does expect that it\nis possible to modify the approach to the Caginalp system, as well as other variational\nproblems to find starting solutions for continuation; this approach will be considered in \nfuture work. For the Caginalp system we find an interesting bifurcation diagram with \npotential applications to tune parameter values to access different parts of the so-called\nmushy regime between melting and solidification.\\medskip\n\nIn the last part of the paper, we provide a very brief overview of various software\ndeveloping issues for gluing codes in numerical continuation analysis of PDEs. The idea\nis to just record, which aspects have been important to facilitate the simplest possible\ncombination of various algorithms used in this work. The second goal is to provide a \nmore detailed outlook, from the viewpoint of gluing computation and an end-user \nperspective of applied nonlinear dynamics, to help developers improve future software\nreleases. It is hoped that this may help to facilitate the exchange of differential \nequations software between different research communities.\\medskip\n\nThe paper is structured as follows: In Section \\ref{sec:review} we review the basic numerical\ntools very briefly and fix the notation. The cross-validation test problem is considered in\nSection \\ref{sec:test}. The bifurcation analysis for the Lane-Emden-Fowler equation with a\nlinear term and a localized microforce asymmetric load is carried out in Section \n\\ref{sec:microforce}. Numerical accuracy and performance results for the \nalgorithms used in this example, are recorded in Appendix \\ref{ap:performance}. \nThe Caginalp system on an airplane-shaped geometry is numerically \nstudied in Section \\ref{sec:Caginalp}. Section \\ref{sec:software} contains a summary of the\nmain software issues encountered while carrying out the computations. We conclude with a short\nsummary in Section \\ref{sec:summary}.  \n\n\\section{Basic Review of Methods}\n\\label{sec:review}\n\nIn this section, we shall just provide a very quick review of the computational tools we \nare going to use in the gluing-type numerical continuation calculations for elliptic PDEs. Consider\na bounded domain $\\Omega\\subset \\R^2$, $x=(x_1,x_2)^\\top\\in\\Omega$ and $u=u(x)\\in\\R$. We shall \nassume that the boundary $\\partial \\Omega$ is piecewise smooth throughout this paper. In this \npaper, we shall restrict\\footnote{The idea, and most of the software components, would also \nwork for systems of elliptic PDEs. However, it is the goal of this paper to show, how simple \ngluing computation can be, not how difficult it can be made.} to elliptic PDEs \n\\cite{Evans,GilbargTrudinger} of the form\n\\be\n\\label{eq:ellipticPDE}\n-c\\Delta u(x) +au(x)-f(u(x),x,\\mu)=0,\\qquad x\\in \\Omega\\subset \\R^2\n\\ee\nwhere $a\\in\\R$, $c\\in(\\R-\\{0\\})$, $f:\\R^3\\ra \\R$ is sufficiently smooth, $\\mu\\in\\R$ is the main\nbifurcation parameter, and \n$\\Delta=\\frac{\\partial^2}{\\partial x_1^2}+\\frac{\\partial^2}{\\partial x_2^2}$ is the usual\nLaplacian. Furthermore, \\eqref{eq:ellipticPDE} will be augmented with either zero \nDirichlet or Neumann boundary conditions, {i.e.}, we consider\n\\be\n\\label{eq:BC}\nu(x)=0\\quad \\text{for $x\\in\\partial \\Omega$}\\qquad \\text{or}\\qquad \n({\\vec n}\\cdot\\nabla u)(x)=0\\quad \\text{for $x\\in\\partial \\Omega$,} \n\\ee\nwhere ${\\vec n}\\in\\R^2$ denotes the outer unit normal vector to $\\Omega$, \n$\\nabla u=\\left(\\frac{\\partial u}{\\partial x_1},\\frac{\\partial u}{\\partial x_2}\\right)^\\top\\in\\R^2$\nis the gradient of $u$, and `$\\cdot$' denotes the standard Euclidean inner product.\n\n\\subsection{FEM for Elliptic PDE \\& pdetoolbox}\n\\label{ssec:FEM}\n\nThe main underlying PDE solver used for \\eqref{eq:ellipticPDE}, via the numerical continuation\nsoftware \\texttt{pde2path}, is the MatLab \\texttt{pdetoolbox} \\cite{MatLab2013b_base,MatLabPDE}. A standard \nfinite-element method (FEM) is used in the \\texttt{pdetoolbox}, which we very briefly recap here to fix some\nnotation in the case of Dirichlet boundary conditions. Let $H^1_0=H^1_0(\\Omega)$ denote the \nusual Sobolev space\n\\be\nH^1_0=\\left\\{v\\in L^2(\\Omega):\\nabla v\\in L^2(\\Omega),v|_{\\partial \\Omega}= 0\\right\\}.\n\\ee\nLet $\\cT=\\{T_i\\}_{i=1}^{n_t}$ be a triangulation of $\\Omega$ consisting of $n_t$ triangles $T_i$\nand let $\\cN=\\{p_k\\}_{k=1}^{n_p}$ denote the set of $n_p$ associated node\/vertex points $p_k\\in \\R^2$. \nOf course, $n_p$ can potentially be very large if a fine mesh is needed to correctly resolve\nsolutions. Then define the space of piecewise-linear finite elements\n\\be\nV_h:=\\left\\{v\\in C(\\Omega):\\text{$v|_{T_i}$ is affine linear for all $T_i\\in\\cT$}\\right\\},\n\\ee\nwhere the subscript $h$ usually indicates the size of the triangles, {e.g.}, $h$ can be \ntaken as the maximum edge length.\\medskip \n\nThe weak formulation of \\eqref{eq:ellipticPDE}, with a test \nfunction $v_h\\in V_h$, is  \n\\be\n\\label{eq:elliptic_weak}\n\\int_\\Omega \\left[c(\\nabla u\\cdot \\nabla v_h)(x) +au(x) v_h(x)\\right]\\txtd x\n=\\int_\\Omega v_h(x) f(u(x),x,\\mu)~\\txtd x,\\quad \\forall v_h\\in V_h.\n\\ee\nWe want to approximate the true solution $u$ by its representation $u_h$ in $V_h$ given by\n\\be\n\\label{eq:rep_base}\nu_h(x)=\\sum_{k=1}^{n_p}U_k\\phi_k(x)\n\\ee\nusing the nodal basis functions $\\phi_k\\in V_h$, which are defined via the condition \n$\\phi_k(p_j)=\\delta_{k,j}$, where $\\delta_{k,j}$ is the usual Kronecker delta with \n$\\delta_{k,k}=1$ and $\\delta_{k,j}=0$ if $k\\neq j$. Inserting $u_h\\approx u$ into \n\\eqref{eq:elliptic_weak} and using $v_h=\\phi_k$ as test functions for each $k$, \nleads to a nonlinear system \n\\be\n\\label{eq:ellPDE_discrete}\nKU=F(U,\\mu) \\qquad \\text{or}\\qquad KU-F(U,\\mu)=0,  \n\\ee\nwhere $U=\\{U_k\\}_{k=1}^{n_p}\\in \\R^{n_p}$ is the vector of unknowns and \n$K\\in \\R^{n_p\\times n_p}$, $F:\\R^{n_p}\\times \\R\\ra \\R^{n_p}$ are assembled automatically in \nthe \\texttt{pdetoolbox}, {i.e.}, this is the key step, which can be very different for various \nmethods as well as different types of PDEs.\\medskip \n\nIn addition to mesh generation and assembly of \nthe linear system, the \\texttt{pdetoolbox} also provides an adaptive mesh refinement algorithm \n\\cite{JohnsonEriksson} based on the error estimator\n\\be\n\\label{eq:error_estimate}\nE_i=E(T_i)=\\alpha \\|h(f-au_h)\\|_{L^2(T_i)}\n+\\beta c\\left(\\frac{1}{2} \\sum_{\\tau\\in \\partial T_i}h_\\tau^2\\left[{\\vec n_\\tau}\\cdot \\nabla u_h\\right]^2\\right)^{1\/2},\n\\ee\nwhere ${\\vec n_\\tau}$ is the outer unit normal and $h_\\tau$ the length of the edge $\\tau$, \n$h=h(x)$ is the local mesh size for the element $T_i$, the term $\\left[{\\vec n_\\tau}\\cdot \\nabla u_h\\right]$ \ndenotes the flux across the element edge, and $\\alpha,\\beta$ are fixed constants.\nIn practice, one of the most efficient schemes is to equidistribute the error and split \nthose triangles, and relevant adjacent ones, with the highest errors during a mesh refinement step.\nThe error equidistribution principle has been used for a long time in numerical continuation \nproblems for BVPs of ODEs motivated by the COLSYS code \\cite{AscherChristiansenRussell}, which \npre-dates the development of AUTO \\cite{Doedel_AUTO2007}. Of course, we are again faced with a\nsimilar challenge as in the discretization step for PDEs as there is a huge variety of ideas and \nestimators for mesh adaptation, not just the solution \\eqref{eq:error_estimate} implemented in\nthe \\texttt{pdetoolbox}. \n\n\\subsection{Numerical Continuation \\& pde2path}\n\\label{ssec:continuation}\n\nHere we briefly review the numerical continuation algorithm used in \\texttt{pde2path} \n\\cite{UeckerWetzelRademacher}; for more theoretical background see \\cite{Govaerts,Kuznetsov}. \nFor numerical continuation, one views the problem \\eqref{eq:ellipticPDE} as a mapping\n\\be\n\\label{eq:ellipticPDE_cont}\ng(u,\\mu):=-c\\Delta u(x) +au(x)-f(u(x),x,\\mu)=0,\\qquad g:\\cH\\times \\R\\ra \\cH, \n\\ee\nwhere we assume, for simplicity, that $\\cH$ is a Hilbert space, {e.g.}, $\\cH=H^1_0(\\Omega)$. The\ngoal is to trace out a curve (or branch) $z(s):=(u(s),\\mu(s))\\in \\cH\\times \\R$ of solutions to \n$g(u,\\mu)=0$, where $s$ is arclength. However, working in the abstract setting does not lead\nimmediately to an implementable numerical algorithm and one has to consider the problem via\na spatial discretization as in \\eqref{eq:ellPDE_discrete} so one studies\n\\be\n\\label{eq:ellipticPDE_cont1}\nG(U,\\mu):=KU-F(U)=0,\\qquad G:\\R^{n_p}\\times \\R\\ra \\R^{n_p}, \n\\ee\nwhere the Hilbert space is just a, potentially very large, Euclidean space $\\R^{n_p}$. The goal is\nstill to trace out a curve $Z(s):=(U(s),\\mu(s))\\in \\R^{n_p}\\times \\R$. To make $s$ an approximation\nto arclength, one augments \\eqref{eq:ellipticPDE_cont1} by an extra condition and studies \n\\be\nH(U,\\mu):=\\left(G(U,\\mu),p(U,\\mu,s)\\right)^\\top=0,\n\\ee\nwhere $p:\\R^{n_p}\\times \\R\\times \\R\\ra \\R$ will be defined below. Suppose we know a point \n$(U(s_0),\\mu(s_0))=:(U_0,\\mu_0)$ on the curve $Z=Z(s)$ at $s=s_0$ from some initial guess or \notherwise\\footnote{Recall from Section \\ref{sec:intro} that this is already a crucial assumption; \nwe are going to find initial solutions in Section \\ref{ssec:variational} using a specialized \nalgorithm.}. Note that we may compute the tangent vector \n\\be\n\\dot{Z}_0:=(\\dot{U}_0,\\dot{\\mu}_0)^\\top:=\n\\left(\\left.\\frac{\\txtd}{\\txtd s}U(s)\\right|_{s=s_0},\\left.\\frac{\\txtd}{\\txtd s}\\mu(s)\\right|_{s=s_0}\n\\right)^\\top \\in\\R^{n_p+1}\n\\ee\nto $Z(s)$ at $s=s_0$ by differentiating $G(U(s),\\mu(s))=0$ with respect to $s$ at $s_0$ so that\n\\be\n\\label{eq:solve_tangent}\n(\\txtD G)|_{s=s_0}\\dot{Z}_0=0,\n\\ee\nwhere $(\\txtD G)|_{s=s_0}$ is the total derivative of $G$ at $s_0$, {i.e.}, finding $\\dot{Z}_0$\nreduces to solving the linear system \\eqref{eq:solve_tangent}. Then, define\n\\be\np(U,\\mu,s):=\\xi ~\\dot{U}_0\\cdot (U(s)-U_0)+(1-\\xi)~\\dot{\\mu}_0(\\mu(s)-\\mu_0)-(s-s_0),\n\\ee\nwhere $\\xi$ is an algorithm parameter with $0<\\xi<1$. In \\texttt{pde2path}, $\\xi$ is a key parameter \nto ``tune'' or ``control'' the behavior of the numerical continuation algorithm and it is\nassumed that $\\dot{Z}_0$ is normalized to one in the weighted $\\xi$-norm derived from the\ninner product\n\\be\n\\left\\langle (U,\\mu)^\\top,(W,\\nu)^\\top\\right\\rangle_\\xi:=\\xi~ U\\cdot W+(1-\\xi)~\\mu\\nu, \n\\qquad \\|\\dot{Z}_0\\|=\\sqrt{\\langle\\dot{Z}_0,\\dot{Z}_0\\rangle_\\xi}\\stackrel{!}{=}1.\n\\ee\nThen $p(U,\\mu,s)=0$ defines a hyperplane perpendicular, with respect to the inner product \n$\\langle\\cdot,\\cdot\\rangle_\\xi$, to $\\dot{Z}_0$ at a distance $\\delta s:=s-s_0$ from \n$(U_0,\\mu_0)$. Now, one can finally define the main predictor-corrector steps to compute \na new point\n\\be \n(U_1,\\mu_1)^\\top:=(U(s_1),\\mu(s_1))^\\top=(U(s_0+\\delta s),\\mu(s_0+\\delta s))^\\top.\n\\ee\nThe prediction step is to set $(U^1,\\mu^1)^\\top:=(U_0,\\mu_0)^\\top+\\delta s~\\dot{Z}_0$.\nOne possibility to carry out the correction step is to use a straightforward Newton\niteration\n\\be\n\\left(\n\\begin{array}{c}\nU^{j+1}\\\\\n\\mu^{j+1}\n\\end{array}\n\\right)=\n\\left(\n\\begin{array}{c}\nU^{j}\\\\\n\\mu^{j}\n\\end{array}\n\\right)-\n\\left.\\left(\n\\begin{array}{cc}\n\\txtD_uG & \\txtD_\\mu G\\\\\n\\xi \\dot{U}_0 & (1-\\xi)\\dot{\\mu}_0\\\\\n\\end{array}\n\\right)\\right|_{(U^j,\\mu^j)}H(U^j,\\mu^j)\n\\ee\nTerminating Newton's method at some step $j=J_1$ by a suitable error criterion, we set \n$(U_1,\\mu_1):=(U^{J_1},\\mu^{J_1})$. So the key algorithm control parameters for the \\texttt{pde2path} \nimplementation of numerical continuation are the step size $\\delta s$, the criterion for terminating \nNewton's method and the parameter $\\xi$, which basically balances vertical versus horizontal \npreference along bifurcation curves \\cite{UeckerWetzelRademacher}.\\medskip \n\nOf course, there are many \ndifferent variations one may use, {e.g.}, approximating the Jacobian in Newton's method in\ndifferent ways \\cite{Dijkstraetal}, use Moore-Penrose continuation \\cite{Kuznetsov} or replace \nthe Newton method by, a sometimes more robust, secant approach. Another issue are bifurcation\npoints \\cite{Kuznetsov}, where one also wants an algorithm to switch between different \nbifurcation branches; we refer the reader for details of these algorithms to \\cite{Kuznetsov,Govaerts}\nas this is not the main problem considered in this paper.   \n\n\\subsection{Multiple Starting Solutions \\& minimax}\n\\label{ssec:variational}\n\nAs discussed in Section \\ref{sec:intro} and mentioned in Section \\ref{ssec:continuation}, one \nmain challenge in numerical continuation is to find at least one starting solution. Even if\none solution can be guessed, or an algorithm does converge to a solution from a rough initial\nguess, this is often not sufficient. Indeed, frequently not all bifurcation branches connect\nto a single branch on which the single known solution lies, or it is very difficult to trace out\nall solutions from a single branch using standard branch switching approaches \n\\cite{Doedel_AUTO2007}. For example, isolated bifurcation branches, so-called isolas \n\\cite{AvitabileDesrochesRodrigues,Keller4}, appear frequently in applications \\cite{Becketal,\nDesrochesKrauskopfOsinga1,KnoblochLloydSandstedeWagenknecht,McCallaSandstede}. \nIn fact, in two-dimensional spatial problems one may potentially find isola structures \nmore easily \\cite{TaylorDawes}, in comparison to the one-dimensional case\\footnote{Quoting from\n\\cite{TaylorDawes} in the context of a Swift-Hohenberg equation: \u00b4\\textit{``Isolas appear for some \nnon-periodic boundary conditions in one \nspatial dimension but seem to appear generically in two dimensions.''}}.\\medskip \n\nOne approach to systematically look for starting solutions for elliptic PDEs of \nthe form \\eqref{eq:ellipticPDE} is to consider the associated energy variational functional.\nFor zero Dirichlet boundary conditions, consider the space $\\cH=H_0^1$ and the functional \n\\be\n\\label{eq:functional}\nJ(u):=\\int_\\Omega\\left[\\frac{c}{2} \\nabla u(x)\\cdot \\nabla u(x)+\\frac{a}{2}(u(x))^2-F(u(x),x,\\mu)\\right]\\txtd x,\n\\ee \nwhere $F$ is the anti-derivative of $f$ with respect to the $u$-component, {i.e.}, \n\\be\nF(u(x),x,\\mu):=\\int_0^{u(x)} f(w,x,\\mu)~\\txtd w.\n\\ee\nWhen dealing with the energy functional $J(u)$, we shall always assume that the bifurcation \nparameter $\\mu$, as well as the other parameters $a,c$, are fixed so we suppress the dependence\nin the notation; in particular, we are going to search for a set of starting solutions\nfor fixed parameter values.\\medskip\n\nLet $J'(u)\\in \\cH$ denote the Fr\\'echet derivative of $J$ at $u$. Solutions $u$ of the Euler-Lagrange \nequation $J'(u)=0$ are also called critical points. Recall that \n\\be\n\\int_\\Omega J'(u)v~\\txtd x=\\lim_{\\epsilon \\ra 0}\\frac{J(u+\\epsilon v)-J(u)}{\\epsilon}=\n\\int_\\Omega g(u,\\mu)v~\\txtd x\n\\ee \nwhere $g(u,\\mu)=0$ is the original elliptic PDE, {i.e.}, critical points are weak solutions and \nvice versa \\cite{Evans}. By elliptic regularity \\cite{Evans}, weak solutions are smooth \nclassical solutions. In general, critical points can be maxima, minima\nor saddle points. A critical point is non-degenerate if the second Fr\\'echet derivative $J''(u)$\nexists and is invertible. The Morse index (MI) of a non-degenerate critical point is defined as the maximal \ndimension of a subspace on which $J''(u)$ is negative definite\\footnote{Of course, this situation can \nnicely be imagined geometrically for the finite-dimensional case \\cite{JostRG}.}. If $\\textnormal{MI}=0$, \nthen we have a stable minimum for the gradient flow of the energy. For $\\textnormal{MI}>0$ there is at \nleast one unstable direction. \\medskip\n\nNote that minima are relatively easy to find by steepest descent but that saddle points are \nmore challenging, particularly if one wants to find multiple ones for fixed parameters. \nLi and Zhou \\cite{LiZhou,LiZhou1}, motivated by the work in \\cite{ChoiMcKenna,DingCostaChen}, \ndeveloped an algorithm to compute multiple saddle solutions. We shall only outline the basic\nproposed strategy here; for details see \\cite{LiZhou,LiZhou1}. Let $\\tilde{\\cH}\\subset \\cH$\nbe a subspace of the Hilbert space $\\cH$ and consider the unit sphere \n$S_{\\tilde{\\cH}}:=\\{v\\in \\tilde{\\cH}:\\|v\\|_{\\cH}=1\\}$. Let $\\cL$ be a closed subspace in $\\cH$\nwith orthogonal complement $\\cL^\\perp$. For each $v\\in S_{\\cL^\\perp}$, define the closed \nhalf-space\n\\be\n[\\cL,v]:=\\{tv+w:w\\in\\cL,t\\geq 0\\}.\n\\ee\nA set-valued map $P:S_{\\cL^\\perp}\\ra 2^\\cH$ is called the peak mapping of $J$ with respect to\n$\\cL$ if for any $v\\in S_{\\cL^\\perp}$, $P(v)$ is the set of all local maximum points of $J$ in \n$[\\cL,v]$. Essentially, the map $P$ collects local maxima in a half-space. A single-valued \nmap $p:S_{\\cL^\\perp}\\ra \\cH$ is a peak selection of $J$ with respect to $\\cL$ if \n\\be\np(v)\\in P(v),\\qquad \\forall v\\in S_{\\cL^\\perp}.\n\\ee   \nBasically $p$ selects a single maximum parametrized by $v$; the notion of peak selection can be localized \nby intersecting the relevant sets with neighbourhood of $v$ \\cite{LiZhou,LiZhou1}. The main idea\nof finding critical points is to restrict to suitable solution submanifolds\n\\be\n\\cM:=\\{p(v):v\\in S_{\\cL^\\perp}\\}.\n\\ee\nEssentially, one would like to think of the manifold $\\cM$ as (an approximation to) \nthe stable manifold of a saddle critical point.\nThen one may use a descent method on $\\cM$, in combination with a method to stay on $\\cM$ during \nthe iteration, to find the saddle point. Of course, one has also to avoid convergence to a saddle\npoint, which has been found previously. The descent process is a minimization problem and the step\nto return to $\\cM$ is carried out via a maximization problem, leading to a \\texttt{minimax}-type algorithm.\nThe span of the previously found solutions is the space $\\cL$, which one tries to avoid at\nfuture iteration steps, {i.e.}, the search essentially works in the complementary half-space $[\\cL,v]$.\n\nNote that the algorithm just outlined, can be viewed as a descent method with a constraint to stay inside \na certain search space by excluding previously found directions. The entire procedure is easily implemented\nvia FEM as the FEM discretization just reduces the original problem to a discrete finite set of values\nof $u$ at certain points. Then one has to actually solve a large, but finite-dimensional, constrained \noptimization problem. A detailed algorithmic description can be found in \\cite{LiZhou,LiZhou1}, the \nimplementation of the \\texttt{minimax} algorithm we use here is \\cite{Zhou}, while a number of \napplications of the method are discussed, {e.g.}, in \\cite{ChenZhou,ChenZhou1,WangZhou,XieYuanZhou}.\\medskip\n\nIt is desirable to try to glue the variational \\texttt{minimax} approach to a standard continuation\npackage, which itself is glued to a standard FEM package. This is precisely, what is carried out \nin practice here for several problems using \\texttt{pde2path} as the main focal point. For more details on \nthe scientific computing challenges involved, we refer to Section \\ref{sec:software}.\n\n\\section{A Cross-Validation Test Problem}\n\\label{sec:test}\n\nAs a first step, we are going to compare, for a test problem from the class of elliptic PDEs \n\\eqref{eq:ellipticPDE}, two different approaches: \n\n\\begin{itemize}\n \\item[(I)] Use the \\texttt{minimax} approach to calculate at a fixed parameter value $\\mu_0=\\mu(s_0)$ several \n starting solutions $u_0^l=u_0^l(x)$, $l\\in\\{1,2,\\ldots,L\\}$ for some $L\\geq 2$. Then use numerical \n continuation for each starting solution, generating multiple solution branches \n $Z^l(s):=(u^l(s),\\mu^l(s))$.\n \\item[(II)] Start with a single, preferably simple and easy-to-guess, solution at a fixed parameter \n value $(u^*_0,\\mu^*_0)$ and then continue the branch $Z^*(s)=(u^*(s),\\mu^*(s))$. Via branch switching\n at bifurcation points, one can then try to recover the starting set of solutions from (I) as points\n on the branch, {i.e.}, $(u^*(s^l),\\mu^*(s^l))=(u^l_0,\\mu_0)$ for some values $s^l$.   \n\\end{itemize}\n\nIt is helpful to select a test problem, where we may expect that classical continuation ideas using the \nhomotopy approach (II) would suffice, but where the variational problem is nontrivial. As a domain, we choose \na rectangle $\\Omega:=(-l_{x_1},l_{x_1})\\times (-l_{x_2},l_{x_2})\\subset \\R^2$, $l_{x_1},l_{x_2}>0$. Consider \nthe elliptic PDE for $u=u(x)$ given by\n\\be\n\\label{eq:ac_test}\n\\left\\{\n\\begin{array}{ll}\n0=-\\Delta u-\\mu u-u^3=g(u,\\mu),\\quad & x\\in\\Omega,\\\\\n0=u,\\quad & x\\in\\partial \\Omega.\n\\end{array}\\right.\n\\ee\nwhere $\\mu\\in\\R$ is the main bifurcation parameter. For $\\mu=0$, and multiplying the entire equation by $-1$, \nthe PDE \\eqref{eq:ac_test} is also known as the Lane-Emden-Fowler equation \\cite{Chandrasekhar,SerrinZou,Wong}. \nIn fact, $\\mu=0$ is the standard test case from \\cite{Zhou} so we definitely can try to carry out the strategy \n(I) with $\\mu_0=0$. Furthermore, $u\\equiv 0$ is a solution, so setting $(u^*_0,\\mu^*_0)=(0,0)$ is a standard\nstarting point for strategy (II).\\medskip \n\n\\begin{figure}[htbp]\n\\psfrag{lam}{$\\mu$}\n\\psfrag{x}{$x$}\n\\psfrag{y}{$y$}\n\\psfrag{norm}{$\\|u\\|_\\I$}\n\\psfrag{a}{(a)}\n\\psfrag{c1}{(c1)}\n\\psfrag{c2}{(c2)}\n\\psfrag{d}{(d)}\n\\psfrag{e}{(e)}\n\\psfrag{f}{(f)}\n\\psfrag{b1}{(b1)}\n\\psfrag{b2}{(b2)}\n\\psfrag{AC}{$0=-\\Delta u-\\mu u-u^3$}\n\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{.\/fig1}\n\t\t\\caption{\\label{fig:1}Numerical continuation results for \\eqref{eq:ac_test} \n\t\tstarting from variational solutions found via the \\texttt{minimax} algorithm at $\\mu=0$. \n\t\tThe seven starting solutions are shown in (a)-(e). The four double bump solutions\n\t\t(b)-(c) split into two classes with different $L^\\I$-norm, where the norm form\n\t\tfor (b1)-(b2) is smaller than for (c1)-(c2). The norms for (d) and (e) are the two\n\t\tlargest norms. The part (f) of the figure shows the raw output of \\texttt{pde2path} upon \n\t\tcontinuation; see also the description in the text.}\n\\end{figure} \n\nWe start with strategy (I). The variational formulation \\eqref{eq:functional} of \\eqref{eq:ac_test} via \nan energy functional is given, for $u\\in H^1_0(\\Omega)$ and $\\mu_0=0$, by\n\\be\n\\label{eq:var_form}\nJ(u)=\\int_\\Omega\\left[\\frac12 \\|\\nabla u(x)\\|^2-\\frac14 u^4(x)\\right]\\txtd x,\n\\ee \nwhere $\\|\\cdot\\|$ denotes the usual Euclidean norm in $\\R^2$.\nThe results from the \\texttt{minimax} algorithm can be reproduced using \\cite{Zhou} and are shown in \\ref{fig:1}(a)-(e)\nfor $l_{x_1}=0.5=l_{x_2}$. Figure \\ref{fig:1}(a)-(e) shows four distinct classes of solutions. The \nsolution of Morse index $\\textnormal{MI}=1$ is shown in Figure \\ref{fig:1}(a). Two solutions with two peaks\ncentered along the coordinate axis of Morse index $\\textnormal{MI}=2$ are shown in Figures \\ref{fig:1}(c1)-(c2)\nand the corresponding two-peak solutions with the peaks along the diagonal and anti-diagonal are displayed in Figures \n\\ref{fig:1}(b1)-(b2). Then, there are also two different classes of solutions of Morse index \n$\\textnormal{MI}=4$ shown in Figures \\ref{fig:1}(d)-(e). Once the starting solutions have been found, they\nwere saved and re-meshed onto a slightly coarser grid for continuation in \\texttt{pde2path}. Figure \\ref{fig:1}(f)\nshows the resulting bifurcation diagram in $(\\mu,\\|u\\|_\\I)$-space, where $\\|\\cdot\\|_\\I$ denotes the usual \n$L^\\I$-norm. Figure \\ref{fig:1}(f) shows the raw output of the \\texttt{pde2path} continuation, {i.e.}, the actual \npoints computed on the desired seven branches, starting from the seven starting solutions located at $\\mu=0$.\nThe $L^\\I$-norms of the seven starting solutions correspond to lexicographical order of the labels, {e.g.}, \nthe smallest $L^\\I$-norm is in Figure \\ref{fig:1}(a), while the largest is in Figure \\ref{fig:1}(e).\\medskip \n\n\\begin{figure}[htbp]\n\\psfrag{lam}{$\\mu$}\n\\psfrag{norm}{$\\|u\\|_\\I$}\n\\psfrag{a}{(a)}\n\\psfrag{c}{(c)}\n\\psfrag{d}{(d)}\n\\psfrag{e}{(e)}\n\\psfrag{b}{(b)}\n\t\\centering\n\t\t\\includegraphics[width=0.95\\textwidth]{.\/fig2}\n\t\t\\caption{\\label{fig:2}Numerical continuation results for \\eqref{eq:ac_test} \n\t\tstarting from from the zero solution $u\\equiv0$ at $\\mu=0$; the color coding is\n\t\tas in Figure \\ref{fig:1}. Further solutions are\n\t\tfound by branch switching at four detected bifurcation\/branch points (marked as black dots).\n\t\tThe four final ($\\mu=0$) solutions on the nontrivial branches are shown in (a)-(d). \n\t\tThe two double bump solutions (b)-(c) again have different $L^\\I$-norm at $\\mu=0$. \n\t\tThe branch point for the solution in Figure \\ref{fig:1}(e) has not been found in a\n\t\tcontinuation run using the routine \\texttt{cont}. However, the branch point can be \n\t\tfound when using \\texttt{findbif} and is located at $\\mu\\approx 101.218$. The part \n\t\t(e) of this figure shows the raw output of \\texttt{pde2path} upon continuation.}\n\\end{figure} \n\nSome of the important algorithmic parameter values for \\texttt{pde2path} used during the continuation runs were:\n\\be\n\\xi=1\/n_p, \\quad \\texttt{neig}=100, \\quad \\texttt{dsmax}=0.5, \\quad \\texttt{hmax}=0.1,\n\\ee\nwhere \\texttt{neig} is the number of computed eigenvalues for stability\/bifurcation detection, \\texttt{dsmax}\nis the maximum step size for $\\delta s$ and $\\texttt{hmax}$ is the maximum triangle side lengths $h$ for the\n(uniform) global triangulation. Otherwise, we used standard parameter values, where it should be noted that\nusing finite-differencing for the Jacobian seemed to yield a more stable algorithm\\footnote{In the forthcoming \nversion \\texttt{p2p2} of \\texttt{pde2path} \\cite{DohnalRademacherUeckerWetzel}, this effect, which arises\ndue to interpolation error, is resolved. I would like to thank Hannes Uecker for explaining me the\ndetails behind this observation.}. The results in Figure\n\\ref{fig:1}(f) show that all seven initial bifurcation branches connect back to the trivial solution branch\nat $(u,\\mu)=(0,\\mu)$. We also observed that changing some \\texttt{pde2path} continuation parameters it\nwas possible to change the direction the zero branch was continued in once it was reached, either to the\nleft or to the right; Figure \\ref{fig:1} shows all seven branches going to the right, {i.e.}~increasing $\\mu$, once\nthe zero branch was reached. There seems to be a numerical artifact for the branch corresponding to \nthe starting solution in Figure \\ref{fig:1}(e), where a bifurcation point (blue circle) is detected quite a \nbit before the actual zero branch is reached.\\medskip \n\nFor the second approach (II), we continue the trivial zero branch. The results are shown in Figure \n\\ref{fig:2}. Four branch points (black dots) were detected when the continuation was run between $\\mu=0$ \nand $\\mu=110$; again, the last bifurcation point corresponding to the diagonal\/anti-diagonal $\\textnormal{MI}=4$\nsolution from Figure \\ref{fig:1}(e) was not detected using a standard bifurcation run using \\texttt{cont}. However, \nthis branch point is correctly detected using the routine \\texttt{findbif}\\footnote{I would like to thank Hannes \nUecker for making me aware of this, {i.e.}, that \\texttt{findbif} evaluates the eigenvalues, while \\texttt{cont} \nchecks for a vanishing determinant.}. Branch\nswitching was performed using \\texttt{pde2path} at the bifurcation points. The new non-trivial solutions \nwere tracked back up to $\\mu=0$, which yields the same - up to symmetry - solutions as recorded for approach (I).\nIn summary, both approaches essentially lead to the same results and we have definitely cross-validated (I) and (II).   \n\n\\section{Adding a Localized Microforce Load}\n\\label{sec:microforce}\n\nWe have observed for the test problem in Section \\ref{sec:test} that symmetries lead to difficulties\nof detection of certain branches as multiple solution branches come together at a single bifurcation\npoint. Hence, it is natural to break the symmetry by a certain perturbation. In \n\\cite{UeckerWetzelRademacher}, the symmetry is broken for a cubic-quintic Ginzburg-Landau\/Allen-Cahn \ntype equation of the form \n\\be\n\\label{eq:UWR}\n\\left\\{\n\\begin{array}{ll}\n0=-c\\Delta u-\\mu u-u^3+u^5,\\quad & x\\in\\Omega,\\\\\n0=u,\\quad & x\\in\\partial \\Omega,\n\\end{array}\\right.\n\\ee\non a rectangle $\\Omega:=(-l_{x_1},l_{x_1})\\times (-l_{x_2},l_{x_2})\\subset \\R^2$ with $l_{x_1}=1$ and \n$l_{x_2}=0.9$. Instead of breaking the geometry of the domain, one may also ask, how one expects generic \nsymmetry-breaking for Allen-Cahn\/Ginzburg-Landau-type PDEs when the equation itself is perturbed. \nOne possible mechanism can be \nfound in the work of Gurtin \\cite{Gurtin}, who derives the Ginzburg-Landau equation from \nbasic principles of force balance. The basic motivation for Ginzburg-Landau-type models are\ntwo-phase systems, whose evolution is described on a macroscopic level by an order parameter \n$\\rho=\\rho(x,t)$ for $(x,t)\\in\\Omega\\times [0,T]$ for some $T>0$. The force balance equations\nfor a given body lead to the generalized Ginzburg-Landau equation\n\\be\n\\beta\\frac{\\partial \\rho}{\\partial t}=\\nabla \\cdot \\left[\\frac{\\partial \\Psi}{\\partial p}(\\rho,\\nabla \\rho)\\right]\n-\\frac{\\partial \\Psi}{\\partial \\rho}(\\rho,\\nabla \\rho)+\\gamma, \n\\ee\nwhere $p=\\nabla \\rho$, $\\Psi:\\R^2\\ra \\R$ is a given free energy, $\\gamma$ is an external\nmicroforce applied to the body and $\\beta$ is the constitutive modulus as discussed in \\cite{Gurtin}.\nTaking a constant positive $\\beta$ and the free energy as \n$\\Psi(\\rho,\\nabla \\rho)=F(\\rho,\\mu)+\\frac12 \\alpha \\|\\nabla \\rho\\|^2 $ leads to the Ginzburg-Landau\nequation with an applied microforce\n\\be\n\\beta\\frac{\\partial \\rho}{\\partial t}=\\alpha\\Delta \\rho-\\frac{\\partial F}{\\partial \\rho}(\\rho,\\mu)+\\gamma, \n\\ee\nwhere $\\alpha,\\beta$ are positive parameters and $\\gamma=\\gamma(x)$ is an\nexternal microforce on the body. Frequently, the potential $F$ is chosen as a double-well leading to \n$F'(u)=\\mu u-u^3$, which has different signs in comparison to the Lane-Emden-Fowler equation from \nSection \\ref{sec:test}. However, the idea that there is a (microscopic) external force acting on \nthe problem still seems very reasonable. Hence, we propose to study the elliptic problem  \n\\be\n\\label{eq:LEF_force}\n\\left\\{\n\\begin{array}{ll}\n0=-\\Delta u-\\mu u-u^3+\\gamma,\\quad & x\\in\\Omega,\\\\\n0=u,\\quad & x\\in\\partial \\Omega,\n\\end{array}\\right.\n\\ee\nwhere $\\gamma=\\gamma(x)$ depends in a non-trivial way upon the spatial coordinate and $\\Omega:=(-l_{x_1},l_{x_1})\\times (-l_{x_2},l_{x_2})\\subset \\R^2$ with $l_{x_1}=1=l_{x_2}$. \nTo break the symmetry encountered in Section \\ref{sec:test}, we propose to consider a point-type load\nmodeled by the function\n\\be\n\\label{eq:mforce}\n\\gamma=\\gamma(x):=\\left\\{\n\\begin{array}{rl}\n\\gamma_a\\exp\\left(-1\/\\gamma_b[(x_1-\\gamma_1)^2+(x_2-\\gamma_2)^2)] \\right),&\\quad\\text{if \n$x=(x_1,x_2)^\\top \\in\\Omega$,}\\\\\n0,&\\quad \\text{if $x\\in \\partial \\Omega$,}\n\\end{array}\n\\right.\n\\ee\nand we fix $\\gamma_a=10$, $\\gamma_b=\\frac{1}{10}$, $\\gamma_1=0.5=\\gamma_2$. Note that \n$\\gamma:\\Omega\\ra [0,+\\I)$ is basically localized, with respect to numerical accuracy, in a small \nneighbourhood of the point $(x_1,x_2)=(0.5,0.5)$. Note carefully that using \\eqref{eq:mforce} as the forcing \nin \\eqref{eq:LEF_force} means that there are no homogeneous constant steady states $u(x)\\equiv \\text{constant}$ \navailable to start the continuation as in Section \\ref{sec:test}. This makes \\eqref{eq:LEF_force} substantially\nmore difficult.\\medskip\n\n\\begin{figure}[htbp]\n\\psfrag{mu}{$\\mu$}\n\\psfrag{norm}{$\\|u\\|_\\I$}\n\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{.\/fig3}\n\t\t\\caption{\\label{fig:3}Numerical continuation results for \\eqref{eq:LEF_force} with \n\t\texternal force given by \\eqref{eq:mforce}. The starting solution was obtained by \n\t\tthe variational \\texttt{minimax} approach at $\\mu=0$. Then, forward and backward numerical \n\t\tcontinuation from this starting point leads to two branches (black and grey). The \n\t\tstarting solution with Morse index $\\textnormal{MI}=1$ is also shown (on the right side), as well\n\t\tas the two endpoints on the computed branches (on the left side); for a more detailed description\n\t\tsee the main text.}\n\\end{figure} \n\nThe homotopy continuation strategy (II) with branch switching from Section \\ref{sec:test}\nis now extremely complicated to carry out. One option is to set $\\gamma_a=0$, then start with the zero \nsolution, compute the different non-trivial branches using strategy (II), and then continue each branch \nalso in the parameter $\\gamma_a$ up to $\\gamma_a=10$. Albeit certainly possible, it is actually a lot easier \nin this case to follow the gluing strategy (I) as the code can basically be used almost unaltered, just \nby adding $\\gamma$ and noticing that the\nnew energy functional, for $\\mu=0$ ({cf.} equation \\eqref{eq:var_form}), is just given by \n\\be\n\\label{eq:var_form1}\nJ(u)=\\int_\\Omega\\left[\\frac12 \\|\\nabla u(x)\\|^2-\\frac14 u^4(x)+u(x)\\gamma(x)\\right]\\txtd x.\n\\ee \nThe results for the continuation runs are shown in Figures \\ref{fig:3}-\\ref{fig:5} with \n$\\texttt{hmax}=0.07$ and $\\texttt{neig}=200$. The results\nin Figure \\ref{fig:3} show the continuation for the $\\textnormal{MI}=1$ solution. The starting solution\nat $\\mu=0$ has a peak slightly shifted towards the upper right corner of the domain in comparison\nto the solution in Figure \\ref{fig:1}(a). Continuing forward in $\\mu$ leads to the black curve in\nFigure \\ref{fig:3} along which the solution peak shrinks and moves into the top right corner. On this\ncurve, a fold point at $\\mu=\\mu_f\\approx 2.92$ occurs, at which the unstable starting solution stabilizes. \nThe bottom curve below the fold is expected to be the global attractor for non-steady-state starting solutions \nin the range of for all $\\mu\\in(-\\I,\\mu_f]$ for the associated parabolic PDE. \nRunning the continuation backwards, {i.e.} with step size $-\\delta s$, yields the grey curve. Along\nthis curve, the solution peak increases and also moves into the top right corner. We also continued both \nparts of the ``(reflected) C-shaped'' branch up to $\\mu=-100$ and no branching was detected. In fact, the \nbranch shape remains, only the peaks seem to sharpen. In fact, a mesh refinement was helpful for the upper\npart of the curve as discussed in Section \\ref{ssec:FEM}. The sharpening of the peaks is expected from\na formal scaling argument since we can re-write \\eqref{eq:LEF_force} for $\\mu\\neq 0$ as \n\\be\n\\label{eq:LEF_force1}\n\\left\\{\n\\begin{array}{ll}\n0=-\\frac{1}{\\mu}\\Delta u- u-\\frac1\\mu u^3+\\frac1\\mu\\gamma,\\quad & x\\in\\Omega,\\\\\n0=u,\\quad & x\\in\\partial \\Omega,\n\\end{array}\\right.\n\\ee\nwhich is a singularly perturbed elliptic PDE as $|\\mu| \\ra +\\I$ (here $\\mu\\ra -\\I$). For \nsingularly perturbed elliptic PDEs, concentration phenomena, spike- and\/or boundary-layer\nsolutions are a common phenomenon \\cite{LinNiTakagi,Ni,Ni1}. More detailed numerical \nperformance calculations were also carried out for this branch and are discussed in Appendix \n\\ref{ap:performance}.\\medskip \n\n\\begin{figure}[htbp]\n\\psfrag{mu}{$\\mu$}\n\\psfrag{norm}{$\\|u\\|_\\I$}\n\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{.\/fig4}\n\t\t\\caption{\\label{fig:4}Numerical continuation results for \\eqref{eq:LEF_force} with \n\t\texternal force given by \\eqref{eq:mforce}. The starting solution was obtained by \n\t\tthe variational \\texttt{minimax} approach at $\\mu=0$. Then forward and backward numerical \n\t\tcontinuation from this starting point leads to two branches (black and grey). The \n\t\tstarting solution with Morse index $\\textnormal{MI}=2$ is also shown; for a more \n\t\tdetailed description see the main text.}\n\\end{figure} \n\nIn Figure \\ref{fig:4}, a starting solution from the \\texttt{minimax} approach with $\\textnormal{MI}=2$\nand two peaks along the diagonal is considered. Interestingly, along the lower (black) branch of \nsolutions, there is a deformation from a two-peak solution with one minimum and one maximum to a \nsingle minimum solution. Along the upper (grey) branch, the two-peak structure remains to hold.\nAgain, we have a C-shaped branch, which folds even tighter back towards itself in the $L^\\I$-norm.\nContinuing back towards $\\mu=-100$ did not lead to the detection of bifurcation points so we \nmay actually conjecture that the symmetry-breaking leads to various isolas corresponding to different\npeak-structure solutions.\\medskip\n\n\\begin{figure}[htbp]\n\\psfrag{mu}{$\\mu$}\n\\psfrag{norm}{$\\|u\\|_\\I$}\n\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{.\/fig5}\n\t\t\\caption{\\label{fig:5}Numerical continuation results for \\eqref{eq:LEF_force} with \n\t\texternal force given by \\eqref{eq:mforce}. The starting solution was obtained by \n\t\tthe variational \\texttt{minimax} approach at $\\mu=0$. Then forward and backward numerical \n\t\tcontinuation from this starting point leads to two branches (black and grey). The \n\t\tstarting solution with Morse index $\\textnormal{MI}=2$ is also shown. Note that this\n\t\tstarting solutions is different from the one in Figure \\ref{fig:4}; for a more \n\t\tdetailed description see the main text.}\n\\end{figure} \n\nWe also used the \\texttt{minimax} method to find another two-peak saddle-solution shown as the\nstarting solution in Figure \\ref{fig:5}. This starting solutions has $\\textnormal{MI}=2$\nand a slightly broader peak near the top left corner in comparison to the peak in the \nlower right corner. As before, we applied the same forward and backward continuation \nsteps and obtain another C-shaped branch, which does not connect to any other solutions\nup to $\\mu=-100$ and is likely to be isolated from other branches. However, in comparison\nto the case of Figure \\ref{fig:4}, there is no change in the peak structure along the \nbranch and the turning in the $L^\\I$-norm is even tighter.\\medskip\n\nIn summary, the \\texttt{minimax} starting strategy turned out to be computationally simple and \nquite effective to determine starting solutions. Furthermore, we may conclude for the \nLane-Emden-Fowler equation \\eqref{eq:LEF_force} with microforce \\eqref{eq:mforce} that \nsymmetry-breaking by external forces can lead to quite interesting bifurcation branch \nstructures, splitting the various branches, which are connected to $u\\equiv 0$ when \n$\\gamma=0$ as shown in Section \\ref{sec:test}. There seem to be many open questions \nregarding the global bifurcation structure for various forces, even for simple \nelliptic problems in the plane. Although\nthe global structure is easily accessed numerically, one may hope to understand the\nproblem near a bifurcation point of the trivial branch for $0<|\\gamma(x)|\\ll1$ analytically\nby a perturbation argument and local unfolding; again, this questions seems to be open\nfor different microforces.  \n\n\\section{The Caginalp System}\n\\label{sec:Caginalp}\n\nIn addition to the previous two case studies in Sections \\ref{sec:test}-\\ref{sec:microforce},\nwhere the \\texttt{minimax} starting solutions have been used according to approach (I), one may also\nask for an example where the classical continuation approach is preferable but gluing computation\nstill plays an important role.\\medskip\n\nConsider again a compact domain $\\Omega\\subset \\R^2$, which represents a material or body, and\nlet $(x,t)\\in \\Omega\\times [0,T]$. The Caginalp system models the interaction between a phase \nfield (or order parameter) $\\chi=\\chi(x,t)$ and the temperature $\\vartheta=\\vartheta(x,t)$. It\nwas originally proposed in \\cite{Caginalp} and studied using many different techniques \n\\cite{GrasselliMiranvilleSchimperna,MiranvilleQuintanilla}. From the modelling perspective, it\ndescribes, {e.g.}, melting-solidification processes between different pure phases. \nHere we use a version of the Caginalp system in the scaling considered in \n\\cite{GrasselliPetzeltovaSchimperna} given by\n\\be\n\\label{eq:Caginalp}\n\\left\\{\n\\begin{array}{rcll}\n\\frac{\\partial \\vartheta}{\\partial t}+ \\frac{\\partial}{\\partial t}\\lambda(\\chi)  \n& = & \\Delta \\vartheta +f,\\quad & \\text{if $(x,t)\\in\\Omega\\times [0,T]$,}\\\\\n\\frac{\\partial \\chi}{\\partial t} & = & \\Delta \\chi -(\\txtD W)(\\chi)+(\\txtD\\lambda)(\\chi)\\vartheta,\n\\quad & \\text{if $(x,t)\\in\\Omega\\times [0,T]$,}\\\\\n0&=&\\vartheta, \\quad & \\text{if $(x,t)\\in\\partial \\Omega \\times [0,T]$,}\\\\\n0&=&\\vec n \\cdot \\nabla\\chi, \\quad & \\text{if $(x,t)\\in\\partial \\Omega \\times [0,T]$,}\\\\\n\\vartheta &=&\\vartheta_0,\\quad & \\text{if $(x,t)\\in \\Omega \\times \\{0\\}$,}\\\\\n\\chi &=&\\chi_0,\\quad & \\text{if $(x,t)\\in \\Omega \\times \\{0\\}$,}\n\\end{array}\n\\right.\n\\ee \nwhere $W:\\R\\ra \\R$ is a given potential, $f:\\Omega \\ra \\R$ is a heat source\/sink and\n$\\lambda:\\R\\ra \\R$ models the latent heat density. We are just interested in the\nstationary Caginalp system under a given fixed heat source. In this case, the system\nreduces to\n\\be\n\\label{eq:Caginalp1}\n\\left\\{\n\\begin{array}{rcll}\n0  & = & \\Delta \\vartheta +f,\\quad & \\text{if $x\\in\\Omega$,}\\\\\n0 & = & \\Delta \\chi -(\\txtD W)(\\chi)+(\\txtD\\lambda)(\\chi)\\vartheta,\n\\quad & \\text{if $x\\in\\Omega$,}\\\\\n0&=&\\vartheta, \\quad & \\text{if $x\\in\\partial \\Omega $,}\\\\\n0&=&\\vec n \\cdot \\nabla\\chi, \\quad & \\text{if $x\\in\\partial \\Omega$,}\\\\\n\\end{array}\n\\right.\n\\ee \nwhere all unknown functions $\\chi=\\chi(x),\\vartheta=\\vartheta(x)$ only depend upon\nthe spatial variable. Returning to one modelling purpose of the Caginalp system,\nnamely melting-solidification processes, we aim to chose a domain $\\Omega\\subset \\R^2$\non which melting and solidification definitely play a crucial role in applications.\nOne engineering application of interest are airplane wings, where the process of\navoiding ice formation is of critical importance \\cite{ThomasCassoniMacArthur}. So\nwe chose $\\Omega$ as a very basic wing-shaped domain as shown in Figure \\ref{fig:7}.\n\\medskip \n \n\\begin{figure}[htbp]\n\\psfrag{x}{$x$}\n\\psfrag{y}{$y$}\n\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{.\/fig7}\n\t\t\\caption{\\label{fig:7}Solution of the Poisson equation \\eqref{eq:Poisson} computed\n\t\tover the basic wing-shaped domain $\\Omega$ with a constant point heat source \n\t\t$f\\equiv2\\cdot 10^3$. This solution is \n\t\tused as an input to the Caginalp system \\eqref{eq:Caginalp2}.}\n\\end{figure} \n\nA first step for the numerical continuation of \\eqref{eq:Caginalp1} is to note\nthat we may use an offline pre-computing step since we have assumed that $f$ is\ntime-independent, {i.e.}, we may solve a Poisson equation \n\\be\n\\label{eq:Poisson}\n\\left\\{\n\\begin{array}{rcll}\n0  & = & \\Delta \\vartheta +f,\\quad & \\text{if $x\\in\\Omega$,}\\\\\n0&=&\\vartheta, \\quad & \\text{if $x\\in\\partial \\Omega $.}\\\\\n\\end{array}\n\\right.\n\\ee \nIn this context, a gluing framework is very helpful. It is very easy to use the\ngraphical user-interface provided by the \\texttt{pdetoolbox} to define the domain $\\Omega$, \nspecify $f$, the Dirichlet boundary conditions, the PDE itself, and then solve\nthe Poisson equation \\eqref{eq:Poisson}. The solution for $f\\equiv 2000$\nis shown in Figure \\ref{fig:7}. Hence, if we view $\\vartheta(x)=\\vartheta$ as a\nknown temperature input for the Caginalp system \\eqref{eq:Caginalp1} then it\nremains to study\n\\be\n\\label{eq:Caginalp2}\n\\left\\{\n\\begin{array}{rcll}\n0 & = & -\\Delta \\chi +(\\txtD W)(\\chi)-(\\txtD\\lambda)(\\chi)\\vartheta,\n\\quad & \\text{if $x\\in\\Omega$,}\\\\\n0&=&\\vec n \\cdot \\nabla\\chi, \\quad & \\text{if $x\\in\\partial \\Omega$,}\\\\\n\\end{array}\n\\right.\n\\ee    \nas an elliptic PDE for the order parameter $\\chi$.\\medskip\n\n\\begin{figure}[htbp]\n\\psfrag{mu}{$\\mu$}\n\\psfrag{norm}{$\\|\\chi\\|_\\I$}\n\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{.\/fig6}\n\t\t\\caption{\\label{fig:6}Numerical continuation results for \\eqref{eq:Caginalp2} with\n\t\t$\\vartheta=\\vartheta(x)$ from Figure \\ref{fig:7}, latent heat density \\eqref{eq:latentheat}\n\t\tand potential \\eqref{eq:Ceng}. The starting solution was computed for $\\mu=1$ by \n\t\titerating an initial guess. The remaining part of the diagram was computed using \n\t\tcontinuation in \\texttt{pde2path}. For a more detailed description see the main text.}\n\\end{figure} \n\nTo study the reduced\nCaginalp equation \\eqref{eq:Caginalp2} numerically, we still have to specify\nthe form of the latent heat density $\\lambda:\\R\\ra \\R$ and the potential\n$W:\\R\\ra \\R$. The latent heat density is frequently modeled and\/or empirically determined as \nleading-order affine-linear term with higher-order small-prefactor nonlinear polynomial correction \nterms \\cite{RogersYau}. To replicate this modelling, we just consider\n\\be\n\\label{eq:latentheat}\n\\lambda(\\tau):=\\mu \\left(c_0+c_1 \\tau +c_2 \\tau^2\\right),\n\\ee\nwhere $\\mu$ is again the bifurcation parameter and $c_1=1$ and $c_2=0.05$ are\nfixed for the computation; note that $c_0$ is not relevant here as only $\\txtD \\lambda$\nappears in \\eqref{eq:Caginalp2}. For the potential, we select the standard example  \n\\be\n\\label{eq:Ceng}\nW(y):=(y^2-1)^2,\n\\ee\nwhich was already studied by Caginalp \\cite{Caginalp}. Note carefully, that\nthe elliptic PDE \\eqref{eq:Caginalp2} has a variational structure with\nenergy functional\n\\be\n\\label{eq:var_form2}\nJ(\\chi)=\\int_\\Omega\\left[\\frac12 \\|\\nabla \\chi(x)\\|^2+W(\\chi)-\n\\lambda(\\chi(x))\\vartheta(x)\\right]\\txtd x.\n\\ee \nHowever, the quartic term $\\chi^4$ has a positive sign, {i.e.}, the Allen-Cahn\/Ginzburg-Landau\nsign and not the Lane-Emden-Fowler sign. Still one may formally try to apply a\nminimax approach but the \\texttt{minimax} code \\cite{Zhou} we used for Sections \n\\ref{sec:test}-\\ref{sec:microforce} does not yield multiple solutions, even upon\nminor modifications. Hence, a deeper modification, or a completely different approach\nwould be required to scan for multiple solutions at a fixed parameter value.\\medskip\n\nNaturally, we first have to make sure there really are multiple solutions for some \nfixed parameter value. In this case, the continuation approach can be used to find \nthese solutions. In particular, we start at $\\mu=1$ by guessing a constant solution.\nUnder Newton iteration, we actually do reach a true solution of the problem for $\\mu=1$.\nThen we can use numerical continuation to track the solution branch as shown in Figure\n\\ref{fig:6}. We observe that there are at least four different solutions at $\\mu=1$\nso there should definitely be a variational approach to determine them systematically\nand we leave this problem as an open problem for future numerical work\\footnote{It is\nlikely that there is a variational algorithm available to search for multiple solutions\nin this case but not one, which is freely available for the direct gluing computation\napproach proposed here.}.\\medskip\n\nThe bifurcation diagram in Figure \\ref{fig:6} is quite interesting for applications so\nwe briefly comment on the results. We cut off the bifurcation diagram and only consider\nthe region with $\\|\\chi\\|_\\I\\leq 1$ as $\\chi=\\pm1$ represent the pure states and we are interested \nin the transition (or so-called mushy) regime between the pure states. Suppose we start in the \nbifurcation diagram with an (almost) pure \nstate $\\chi(x)\\approx 1$\nfor all $x\\in\\Omega$ and then change $\\mu$. This leads to a transition sequence involving \nfour folds as shown in Figure \\ref{fig:6}. If we interpret $\\chi(x)\\approx 1$ as a melted\n(or liquid) state, then the solution curve at the first fold assumes a mixed\/mushy state\nwhere solidification starts non-uniformly over the domain, with a focus near the airplane body.\nThis focus reverses as the solution branch is traversed further and at the third fold,\nwhen the solidification focuses on the tip. In the last step, the system transitions to the almost fully solid state\nafter the fourth fold has been traversed. Although we shall not pursue these ideas here further,\nit is clear that these observations should have interesting applications in engineering \napplications to build airplane wings and\/or during a de-icing process. We leave a detailed \nparametric numerical continuation study for melting\/solidification processes for future work.     \n\n\\section{Perspectives on Software Development}\n\\label{sec:software}\n\nAlthough we have successfully demonstrated that gluing computing between FEM, continuation\nand minimax can be efficient and lead to very interesting practical results for elliptic PDEs, \nwe have not commented on several\nimportant practical scientific computing issues. In particular, the question is why the computations\nhere worked in a relatively straightforward way, while gluing computation can become very\ncomplicated in many cases when one tries to track patterns via numerical continuation; the thesis \n\\cite{Avitabile} is an excellent example, how challenging such a computational approach can become. \nThe following issues seem to be very helpful to keep in mind for further\nsoftware development\\footnote{Of course, the various issues are written from the viewpoint\nof a user in applied nonlinear dynamics, who is interested in gluing computations.}:\n\\medskip   \n\n\\textbf{High-level language:} A key ingredient to efficiently glue different parts\nrequired for the computation is to use a high-level programming language. Here \\texttt{MatLab} \n\\cite{MatLab2013b_base} is used. However, an excellent non-commercial alternative would be \n\\texttt{Python} \\cite{Python}, which contains aspects designed for gluing computation and is already\nefficiently used in various scientific computing packages for differential equations \n\\cite{Doedel_AUTO2007,Hoffmanetal,GuyerWheelerWarren,Cimrman,Herouxetal}. The problem with working directly \nwith FEM, continuation or variational-PDE packages via a fast-computation, but lower-level, programming \nlanguage, {e.g.}~\\texttt{C}, \\texttt{C++} or \\texttt{Fortran}, is that even apparently simple-looking \ntasks of transferring data, adding on functionality, problem formulation, interlinking of algorithms, \ncross-validation, testing and visualization, become incredibly complicated and \ntime-consuming. To really make a gluing approach work on the basis of a lower-level language, \none has to fully integrate multiple software packages into a single, necessarily constrained,\nenvironment. Although this may be very desirable in standardized industrial applications, it does\nnot adequately represent software development and flexibility requirements in an academic environment.\nOnly a higher-level language provides the required flexibility. Of course, ones has to give up a\nslight bit of computational efficiency but overall, this trade-off seems worthwhile.\\medskip \n\nIn this context, it should be noted that such an integration of input-output via a high-level language\nshould probably be made a design principle from the start. If wrapper-functionality is added\nlater on, it is frequently still necessary to fully comprehend the underlying low-level code\nto actually use the wrapper.\\medskip \n\n\\textbf{Algorithmic blocks:} Based upon the previous point, one should make sure to design \nself-consistent blocks of code, which interface\/communicate directly with a high-level language\nbut run on a lower-level fast language for computational purposes. For example, in the context \nconsidered in this paper one might want to split up the process into the following components:\n\n\\begin{enumerate}\n \\item[(A1)] problem formulation and definitions;\n \\item[(A2)] mesh generation and error-estimator based mesh refinement;\n \\item[(A3)] discretization of the PDE, {i.e.}, conversion to a nonlinear algebraic system;\n \\item[(A4)] algorithms for generating starting solutions sets;\n \\item[(A5)] numerical continuation and bifurcation detection; \n \\item[(A6)] efficient, fast numerical linear algebra;\n \\item[(A7)] data analysis and visualization.\n\\end{enumerate}\n\nOf course, not all the different steps are necessary for a given class of PDEs or chosen discretization\nalgorithm. For example, sometimes one may omit (A4) due to the availability of starting solutions as for approach (II) \nin Section \\ref{sec:test}, or use spectral or other mesh-free methods to generate the nonlinear system\nof algebraic equations to be solved in (A3).\\medskip  \n\n\\textbf{Problem description:} Another natural question is, how we should formulate the PDE meshing, discretization\nand continuation problem in the high-level gluing language? In this regard, \\texttt{pde2path} builds upon the \nsuccessful ideas implemented in \\texttt{AUTO}. The main idea is to have only very few files that specify all the problem \ndetails completely: one (or two) files to specify the elliptic PDE, one file to initialize all the data (domain, initial guess,\nalgorithmic parameters, {etc.}) and one structure to track the current state of the numerical problem. Those ideas turned out, \nagain, to be extremely robust and helpful to carry out the gluing computations in this paper. We also completely\nglued the \\texttt{minimax}-algorithm \\cite{Zhou} to \\texttt{pde2path} by using only data from standard \\texttt{pde2path} problem \ndescription files in \\texttt{minimax}. In principle, this could be implemented permanently into \\texttt{pde2path} but\nthere is a new version released soon \\cite{DohnalRademacherUeckerWetzel}. The variational starting solution approach\nwill be implemented in this new version in future work\\footnote{The new version \\texttt{p2p2} is not backward compatible\nwith \\texttt{pde2path} so we postpone this step to future work. It is expected that compatibility is guaranteed from the version \\texttt{p2p2} onwards \\cite{DohnalRademacherUeckerWetzel}.}.\\medskip\n \n\\textbf{PDE discretization:} By now, there are a large number of different software packages available to automatically\ngenerate meshes, nonlinear equations and adaptive mesh refinements for various classes of PDEs; see Section \\ref{sec:intro}.\nMany of these software packages would provide excellent tools for understanding dynamics and patterns in a wide\nvariety of applications a lot better. In fact, the barrier does not lie in the packages themselves but the difficulty\nto access the problem formulation. For example, it is relatively rare that one can simply \nprovide a problem description and obtain, \\emph{in a simple format}, the discretized nonlinear equations as an output; of course, it is\nalways \\emph{possible} but the goal is to make gluing computation \\emph{easy}. However, there seems significant progress in this \ndirection, {e.g.}~see \\cite{Hoffmanetal}. The demonstration we have given here for three problems also strongly points towards the conjecture\nthat practical gluing computations for dynamical systems analysis of PDEs will become a lot easier in the very near future.\\medskip\n\n\\textbf{Continuation algorithms:} The main barrier to make continuation algorithms more applicable to spatial\nproblems was to move beyond the class of two-point BVPs and provide a more generic package suitable for PDEs.\nThis is one current disadvantage with the \\texttt{pde2path} focus we considered in this paper since \\texttt{pde2path} links \ninternally a continuation algorithm directly to the MatLab \\texttt{pdetoolbox}. Therefore, it is not easy in practice \nto replace the steps (A3) and (A5) as they are tied together in \\texttt{pde2path}. However, this could be \nresolved by using a generic continuation toolbox, such as the recently developed continuation core \\texttt{COCO} \n\\cite{DankowiczSchilder} or certain tools for large-dimensional linear systems such as \\texttt{LOCA} \\cite{Salingeretal} or other recently developed codes \\cite{Bindeletal}. The main point is, that in the future one must \ntake advantage of the advanced numerical analysis discretization schemes to reduce the computational linear algebra\neffort more efficiently. Overall, there is also significant recent progress in this direction supporting the conjecture stated in the last paragraph.\n\n\\section{Summary}\n\\label{sec:summary}\n\nIn this paper we have addressed the issue of gluing computation at the interface between numerical\nPDEs, continuation methods, scientific computing and variational theory in the context of elliptic PDEs for \nthree different equations. We \nhave shown that efficient integration of various algorithms within the framework of a high-level\nprogramming framework can be possible. A combination of \\texttt{pde2path}, the MatLab \\texttt{pdetoolbox} \nand a \\texttt{minimax}-algorithm has been glued and cross-validated using the Lane-Emden-Fowler equation \nwith a linear term as a test problem. As a second step, we argued\nthat a natural symmetry-breaking mechanism, which is not based upon the domain geometry, could be localized asymmetric\nmicroscopic forces which also arise in generalized Ginzburg-Landau\/Allen-Cahn equations. We found\ninteresting inverse-C-shaped bifurcation curves. There is numerical evidence to conjecture that these curves are isolas, which could be analytically captured in the regime of a very small microforce. Along the bifurcation curves, complicated deformation of patterns can already take place under a simple near-localized external microforce. Then we \nproposed to study as a third example the Caginalp system for melting-solidification processes\non a wing-shaped geometry. In this case, the variational \\texttt{minimax}-algorithm could not directly be glued despite\nthe existence of multiple solutions at fixed parameters. This illustrated the limitations of the process. For the Caginalp system, we also computed an interesting phase transition diagram for melting-solidification processes, which pointed towards potentially useful aerospace and engineering applications. Finally, we also \ncommented on scientific computing issues encountered during the process.\\medskip  \n\n\\textbf{Acknowledgments:} I would like to thank the Austrian Academy of Sciences ({\\\"{O}AW})\nfor support via an APART fellowship. I also acknowledge support of the European Commission \n(EC\/REA) via a Marie-Curie International Re-integration Grant. I would like to thank John\nGuckenheimer, Hannes Uecker and Mathieu Desroches for very helpful comments on earlier versions of this \nmanuscript. Furthermore, I would like to thank two anonymous referees, whose comments have helped to improve \nthe paper.\n\n\\section{References}\n\n{\\small \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWith somewhat imprecise boundaries, interstellar clouds are generally\nclassed as diffuse, \\mbox{A$_{\\rm V}$} $\\la$ 1 mag, or dark, \\mbox{A$_{\\rm V}$} $\\ga$ 4-6 mag, with an\nintermediate translucent regime \\citep{SnoMcC06}.  In diffuse clouds the\ndominant form of carbon is C\\mbox{$^+$}\\ and hydrogen is mostly atomic, although\nwith a very significant overall admixture of \\mbox{H$_2$}, 25\\% or more as a global\naverage \\citep{SavDra+77,LisPet+10}.  In dark or molecular clouds the\ncarbon is overwhelmingly in CO with an admixture of C I and the hydrogen\nresides almost entirely in \\mbox{H$_2$}.  The population of diffuse clouds is\nsometimes called H I clouds in radio astronomical terms.\n\nThe shadows of dark clouds are seen outlined against brighter background\nfields and the clouds themselves are often imaged in the mm-wave emission\nof CO and many other species: the {\\emr{^{12}CO}{}(1-0) sky \\citep{DamHar+01} is\n  usually (and in part incorrectly, see below) understood as a map of\n  fully-molecular clouds.  The shadows of H I or diffuse clouds are their\n  absorption-line spectra and for the most part, individual diffuse clouds\n  are known only as kinematic features in optical and\/or radio absorption\n  spectra.  No means exist to image individual diffuse clouds at optical\n  wavelengths and attempts to map individual H I clouds at radio\n  wavelengths are generally frustrated by the blending and overlapping of\n  contributions from multiple clouds and gas phases.  This lack of identity\n  has greatly complicated our ability to define diffuse clouds physically\n  because absorption lines do not generally permit a direct determination\n  of the cloud size or internal density.\n  \n  When diffuse clouds discovered in absorption-line spectra have a\n  sufficiently high complement of molecules they may be imaged at radio\n  wavelengths in species such as OH and CH and, most usefully, CO.  Despite\n  a low fractional abundance of CO relative to \\mbox{H$_2$}, $\\mean{{\\rm X(CO)}} =\n  3\\times 10^{-6}$ \\citep{BurFra+07}, mapping is facilitated by an enhanced\n  brightness of the J=1-0 line in diffuse gas: the temperature is somewhat\n  elevated (\\Tsub K  $\\ga 25$ K), the density is comparatively small at typical\n  ambient thermal pressure \\citep{JenTri11} and the rotation ladder is\n  subthermally excited.  In accord with theory \\citep{GolKwa74}, it is\n  found observationally that there is a simple, linear proportionality\n  between the integrated intensity \\mbox{W$_{\\rm CO}$}\\ of the CO J=1-0 lines and the CO\n  column density, even when the gas is optically thick: N(CO) $\\approx\n  10^{15}~{\\rm cm}^{-2}$ \\mbox{W$_{\\rm CO}$}\/\\emr{\\,K\\,km\\,s^{-1}} for \\mbox{W$_{\\rm CO}$}\\ $\\approx 0.2 - 6$ \\emr{\\,K\\,km\\,s^{-1}}\n  \\citep{LisLuc98,Lis07CO}.  Per molecule, the ratio \\mbox{W$_{\\rm CO}$}\/N(CO) is 30-50\n  times higher in diffuse gas, compared to conditions in dense shielded\n  fully-molecular gas where the rotation ladder is thermalized\n  \\citep{LisPet+10}.  Of course this is of substantial assistance in the\n  present work.  Conversely, the high brightness (5-12 K) of many of lines\n  we detected should \\emph{not} be taken as discrediting their origin in\n  diffuse gas.\n\n\\begin{figure*}\n  \\includegraphics[height=11.75cm]{FindingChart}\n\\caption[]{Outer-galaxy finding chart for the sources studied here except \n  B1730-130 at $l = 12$\\mbox{$^{\\rm o}$}, see Table 1.  The colored background is\n  reddening at 6\\arcmin\\ resolution \\citep{SchFin+98} truncated at a\n  maximum of 2.4 mag as shown on the bar scale at left.  Positions of\n  continuum sources are indicated (see Table 1) along with a few other\n  objects: the high-latitude molecular clouds MBM23 and MBM 53\n  \\citep{MagBli+85}; TMC-1; SAMS 1 and 2 \\citep{Hei04}; and two Perseus\n  stars commonly used for optical absorption line studies.}\n\\end{figure*}\n\n\\begin{table*}\n\\caption[]{Continuum target, line of sight and map field properties$^1$}\n{\n\\small\n\\begin{tabular}{lccccccccccc}\n\\hline\nTarget & ra & dec & l  &  b &  Map & \\mbox{E$_{\\rm B-V}$}$^2$ & N (H I)$^3$ & N(\\mbox{H$_2$})$^4$ & \\mbox {f$_{{\\rm H}_2}$}$^7$ & \\mbox{W$_{\\rm CO}$} & $<$\\mbox{W$_{\\rm CO}$}$>$ \\\\\n       & (J2000)  & (J2000) & & & size & mag   & $10^{20}~{\\rm cm}^{-2}$ & $10^{20}~{\\rm cm}^{-2}$ & & \\emr{\\,K\\,km\\,s^{-1}} & \\emr{\\,K\\,km\\,s^{-1}}\\\\\n\\hline\nB0736+017 & 07:39:18.03 & 01:37:04.6 &  216.99 & 11.38 & 15\\arcmin &0.13 & 7.7  & 3.3 & 0.46 & 0.8 & 0.4\\\\\nB0954+658 & 09:58:47.24 & 65:33:54.7 & 145.75 & 43.13 & 30\\arcmin & 0.12 & 5.3$^5$  & 4.8 & 0.64 & 1.6 & 0.6\\\\\nB1730-130 & 17:33:02.66 & -13:04:49.5 & 12.03  & 10.81 & 30\\arcmin & 0.53 & 28.3 & 4.3 & 0.23 & 0.4 & 0.4\\\\\nB1928+738 & 19:27:48.58 & 73:58:01.6 & 105.63 & 23.54 & 20\\arcmin & 0.13 & 7.2$^5$ & 2.7 & 0.43 & $<$0.2 & 0.0 \\\\\nB1954+513 & 19:55:42.69 & 51.31:48.5 & 85.30 & 11.76 & 30\\arcmin & 0.15 & 12.7$^5$ & 5.0 & 0.44 & 1.6 & 0.3 \\\\\nB2251+158 & 22:53:57.71 & 16:08:53.4 & 86.11 & -38.18 & 30\\arcmin &0.10 & 4.6 & 1.2 & 0.34 & 0.8 & 0.2 \\\\ \n\\hline\nB0528+134 & 05:30:46.41 & 13:31:55.1 & 191.37 & -11.01 & 30\\arcmin & 0.89 & 30.9 & 7.9 & 0.34 & 2.2 & 2.6\\\\\nB2200+420 & 22:02:43.24 & 42:16:39.9 & 92.59 & -10.44 & 30\\arcmin & 0.33 & 9.7 & 8.8 & 0.66 & 5.8 & 3.7\\\\\n\\hline\nB0212+735 & 02:17:30.81 & 73:49:32.6 & 128.93& 11.96 & 30\\arcmin & 0.76 & 32.1 & 18.6 & 0.54 & 5.8 & 1.56\\\\\nB0224+671 & 02:28:50.03 & 67:21:31.3 &132.12 & 6.23 & 30\\arcmin &1.00 & 38.3 & 9.2 & 0.32 &1.9 & 4.0 \\\\\nB0355+508 & 03:59:29.73 & 50:57:50.1 &150.38 &-1.60 &30\\arcmin x 50\\arcmin & (1.50)$^6$ & 111.3 & 24.2 & 0.30 \n& 14.3 & 4.2\\\\\n\\hline\nMean & & & & & & 0.41 & 17.7 & 6.58 & 0.43 &  2.09 & 1.38 \\\\\n\\hline\n\\end{tabular}}\n\\\\\n$^1$Sources are placed in three groups according to their discussion in Sect. 3, 4 and 5 \\\\\n$^2$from \\cite{SchFin+98} \\\\\n$^3$ N(H I) = $2.6 \\times 10^{20}~{\\rm cm}^{-2} \\int{\\tau(H I)}dv$ (see Sect 2) except where noted \\\\\n$^4$ N(\\mbox{H$_2$}) = N(\\hcop)\/$3\\times10^{-9}$ see Sect. 2. \\\\\n$^5$ from \\cite{HarBur97}, N(H I) = $1.823 \\times 10^{18}~{\\rm cm}^{-2} \\int{\\Tsub B (H I)}dv$  \\\\\n$^6$ at such a low galactic latitude \\mbox{E$_{\\rm B-V}$}\\ is not reliably determined \\\\\n$^7$ \\mbox {f$_{{\\rm H}_2}$} = 2N(\\mbox{H$_2$})\/(N(H I) + 2N(\\mbox{H$_2$}))\n\\\\\n\\end{table*}\n\nEarlier we showed that, in the mean, CO-\\mbox{H$_2$}\\ conversion factors are similar\nin diffuse and dense fully molecular gas \\citep{LisPet+10}, because the\nsmall abundance of CO relative to \\mbox{H$_2$}\\ in diffuse gas is compensated by a\nmuch higher brightness per CO molecule.  But the proportionality between\n\\mbox{W$_{\\rm CO}$}\\ and N(CO) in diffuse gas, where CO represents such a small fraction\nof the available gas phase carbon, means that the CO map of a diffuse cloud\nis really an image of the CO chemistry.  Moreover the CO abundance exhibits\nextreme sensitivities to local conditions that are manifested as order of\nmagnitude scatter in N(CO)\/N(\\mbox{H$_2$}) in optical absorption line studies\n\\citep{SonWel+07,BurFra+07,SheRog+07,SheRog+08}, even beyond the\noften-rapid variation of N(\\mbox{H$_2$}) with \\mbox{E$_{\\rm B-V}$}\\ \\citep{SavDra+77} (\\mbox{E$_{\\rm B-V}$}\\ \n$\\approx$ \\mbox{A$_{\\rm V}$}\/3.1).  The net result is that the CO emission map of a\ndiffuse cloud can only indirectly be interpreted as tracing the underlying\nmass distribution, or even that of the \\mbox{H$_2$}.  Nonetheless, it should (we\nhope) provide some impression of the nature of the host gas, especially in\nthe absence of any other means of ascertaining this.\n\nIn this paper we present maps of CO J=1-0 emission at arcminute resolution\nover 11 sky fields, typically 30\\arcmin $\\times$ 30\\arcmin\\ around the\npositions of compact extragalactic mm-wave continuum sources that we have\nlong used as targets for absorption line studies of the chemistry of\ndiffuse clouds.  As is the case for nearly all background sources seen at\ngalactic latitudes $|b| < 15-18$\\mbox{$^{\\rm o}$}, and for some sources at higher\nlatitudes, the current targets were known to show absorption from \\hcop\\ \nand from one or more other commonly-detected species (OH, CO, \\mbox{C$_2$H}, \\mbox{C$_3$H$_2$});\nmost but not all directions also were known to show CO emission in at least\nsome of the kinematic features present in absorption.\n\nThis work is organized as follows. The observational material discussed\nhere is summarized in Sect. 2.  In Sects. 3-5 we discuss the new maps with\nsources grouped in order of kinematic complexity.  Sect. 6 is an\nintermediate summary of the lessons drawn from close scrutiny of the maps.\nSect.  7 briefly discusses the influences of galactic and internal cloud\nkinematics and Sect. 8 presents a comparison of CO intensity and reddening\nwithin a few of the simpler individual fields.  Online Appendix A shows a\nfew position-velocity diagrams that, while of interest, could be considered\nredundant with those shown in the main text in Figs. 13 and 14.  Figures\nB.1 and B.2 in online Appendix B show the target lines of sight in the\ncontext of large-scale galactic kinematics sampled in H I emission.\n\n\\section{Observational material}\n\n\\subsection{CO J=1-0 emission}\n\nOn-the-fly maps of CO J=1-0 emission were made at the ARO 12m telescope in\n2008 December, 2009 January and 2009 December in generally poor weather\nusing filter banks with 100 kHz or 0.260 \\mbox{km\\,s$^{-1}$}{} channel spacing and\nspectral resolution.  System temperatures were typically $450 - 750$ K.\nThe data were subsequently put onto 20\\mbox{$^{\\prime\\prime}$}} \\def\\arcmin{\\mbox{$^{\\prime}$}\\ pixel grids using the AIPS\ntasks OTFUV and SDGRD; the final spatial resolution is 1\\arcmin.  Most maps\nare approximately 30\\arcmin\\ $\\times$ 30\\arcmin\\ on the sky and were\ncompleted in 4-5 hours total observing time.  The new CO emission data are\npresented in terms of the \\mbox{${\\rm T}_{\\rm R}^*$}\\ scale in use at the 12m antenna and all\nvelocites are referred to the kinematic Local Standard of Rest.  The\ntypical rms channel-channel noise in these maps at 1\\arcmin\\ and 0.26 \\mbox{km\\,s$^{-1}$}\\ \nresolution is 0.4-0.5 K; their sensitivity is rather moderate and the\ndetectability limit is of order 1 \\emr{\\,K\\,km\\,s^{-1}}{} for a single line component.\n\nMore sensitive CO J=1-0 line profiles at higher spectral resolution (25\nkHz) had been previously observed toward the continuum sources as part of\nour survey efforts, for instance see \\cite{LisLuc98}.  It is these profiles\nthat are displayed in the Figures shown here representing emission in the\nspecific direction of the background target and used to calculate line\nprofile integrals as quoted in Table 1.\n\nMany interstellar clouds lie at distances of about 150 pc from the Sun,\njust outside the Local Bubble.  At this distance the 1\\arcmin\\ resolution\nof our CO mapping corresponds to 0.041 pc.\n\n\\subsection{H I absorption and emission}\n\nThe $\\lambda$ 21cm H I absorption spectra shown here are largely from the\nwork of \\cite{DicKul+83} augmented by a few spectra taken at the VLA in\n2005 May.  The spectral resolution of this data is 0.4 - 1.0 \\mbox{km\\,s$^{-1}$}.\n\nFigures B.1 and B.2 of the online Appendix B show latitude-velocity\ndiagrams of H I emission drawn from the Leiden-Dwingeloo Survey of\n\\cite{HarBur97}.\n\n\\subsection{Molecular absorption}\n\nAlso shown here are spectra of $\\lambda 18$cm OH absorption from\n\\cite{LisLuc96} and mm-wave absorption spectra of CO \\citep{LisLuc98},\n\\hcop\\ \\citep{LucLis96} \\mbox{C$_2$H}\\ \\citep{LucLis00C2H} and \\mbox{H$_2$CO}\\ \n\\citep{LisLuc+06}.\n\n\\subsection{Reddening}\n\nMaps of reddening were constructed from the results of \\cite{SchFin+98}.\nThis dataset has 6\\arcmin\\ spatial resolution on a 2.5\\arcmin\\ pixel grid.\nThe stated single-pixel error is a percentage, 16\\%, of the pixel value.\nOn average, 1 mag of reddening corresponds to a neutral gas column N(H) $=\n5.8 \\times 10^{21}~{\\rm cm}^{-2}$ \\citep{SavDra+77}.\n\n\\subsection{Target fields}\n\nThe positions and other observational properties are summarized in Table 1\nwhere the sources are grouped according to their order of presentation in\nSect. 3, 4 and 5.  The groups appear in order of increasing reddening and\ngas column density and decreasing distance from the galactic plane.  The\nline profile integrals \\mbox{W$_{\\rm CO}$}\\ quoted in Table 1 result from the more\nsensitive earlier observations noted in Sect. 2.1.  The mean values quoted\nfor \\mbox{W$_{\\rm CO}$}\\ along individual sightlines are averages over the new map data\ntaken for this work.\n\nTable 2 gives some pixel statistics about noise levels and spatial covering\nfactors as discussed in Sect. 8.\n\n\\subsection{Presentation of observations}\n\nA finding chart including all sources except B1730-130 is shown in Fig. 1\nwhere the locations of the background targets are shown on a large-scale\nmap of reddening, along with locations of a few other landmark objects as\nnoted in the figure caption.\n\nMaps and spectra of the target fields and background sources are shown in\nFigs.  2-12.  Within each of three groups, sources appear in order of\nincreasing right ascension. Members of the first group, shown in Figs. 2-7\nand discussed in Sect. 3, are the simplest kinematically.  Figs. 8 and 9\nshow the fields around the background sources B0528+134 and B2200+420 (aka\nBL Lac) that are also kinematically simple but are heavily patterned and\nrather bright in CO emission; these are discussed in Sect. 4.  Figs. 10-12\n(Sect. 5) show the results over three target fields with rather amorphous\nstructure whose kinematics are too complex to fit into the framework in\nwhich the data for the other sources are presented in earlier figures.  Two\nof these sources (B0212+735 and B0224+671) are relatively near each other\non the sky and sample similar galactic structure while the third target\nB0355+508 (aka NRAO150) is the only source within 2\\mbox{$^{\\rm o}$}\\ of the galactic\nequator (see Table 1).\n\nThe format of Figures 2-11 is: at upper left a 90\\arcmin\\ map of \\mbox{E$_{\\rm B-V}$}\\ from\nthe dataset of \\cite{SchFin+98}, with an inset showing the field of view\nmapped in CO, typically 30\\arcmin\\ on a side; at lower left a map of \\mbox{W$_{\\rm CO}$};\nat lower right various atomic (H I) and molecular absorption spectra\nshowing the kinematic structure toward the background source; at upper\nright, CO emission spectra of various sorts as depicted in the figure\ncaptions.  The absorption spectra shown at lower right in these figures are\nsomewhat inhomogeneous because not all sources have the same full\ncomplement of profiles.  In general, H I is at the bottom wherever possible\nand above that are spectra of the most common molecules observed in\nabsorption; \\hcop, observed toward all targets, OH, \\mbox{C$_2$H}\\ and\/or CO.  The\nuppermost spectrum wherever possible is a species like \\mbox{H$_2$CO}\\ or HNC\n\\citep{LisLuc+06,LisLuc01} that is detected less commonly and is indicative\nof greater chemical complexity.\n\n\\begin{figure*}\n  \\includegraphics[height=13cm]{B0736Slide}\n\\caption[]{The sky field around the position of B0736+017. Upper left:\n  reddening at 6\\arcmin\\ resolution \\citep{SchFin+98}.  Lower left:\n  integrated CO emission \\mbox{W$_{\\rm CO}$}\\ in units of K \\mbox{km\\,s$^{-1}$}; contour levels are in\n  1,2,... K \\mbox{km\\,s$^{-1}$}.  Lower right: absorption line profiles, scaled as noted.\n  Upper right: {\\bf CO emission,} toward B0736+017 and at the peak of the\n  nearby CO distribution.}\n\\end{figure*}\n\nAlso shown for all sources are CO emission spectra at various locations in\nthe field mapped, as indicated in the spectra.  More complex aspects of the\npresentation are discussed in the individual figure captions.\n\n\\begin{table}\n\\caption[]{Noise levels and spatial covering factors}\n{\n\\small\n\\begin{tabular}{lccccc}\n\\hline\nTarget & V     & $\\sigma_{\\rm  profile}$ & $\\sigma_{\\rm map}$ & f$_{>1}^a$ & f$_{>2}^b$\\\\ \n       & \\mbox{km\\,s$^{-1}$}  & K  & \\emr{\\,K\\,km\\,s^{-1}} &  &  \\\\\n\\hline\nB0736& 5.1,7.2 & 0.43 & 0.48 & 0.18 & 0.07 \\\\\nB0954& 2.6,5.2 & 0.25 & 0.33 & 0.20 & 0.12 \\\\\nB1730& 4.0,6.1 & 0.36 & 0.52 & 0.20 & 0.03 \\\\\nB1928& -4.2,-0.8 & 0.46 & 0.52 & 0.03 & 0.00 \\\\\nB1954& -1.0,2.4 & 0.33 & 0.35 & 0.20 & 0.13 \\\\\nB2251&-10.8,-7.9 & 0.28 & 0.32 & 0.06 & 0.02 \\\\\n\\hline\nB0528& 0.6.3.7     &  0.23 & 0.36 & 0.07 & 0.05 \\\\\n          & 8.9,11.7    &       & 0.33 & 0.69 & 0.46 \\\\\nB2200& -3.8,2.8 & 0.39 & 0.63 & 0.61 & 0.50 \\\\\n\\hline\nB0212& -14.1,-7.9 & 0.30 & 0.80 & 0.26 & 0.11 \\\\\n          & -1.4,1.0 & & 0.36 & 0.03 & 0.01 \\\\\n          & 1.3,4.9 & & 0.66 & 0.34  & 0.19 \\\\  \nB0224& -17.1,-11.9 & 0.43 & 0.76 & 0.14 & 0.03 \\\\\n          & -11.6,-9.2 &  & 0.60 & 0.33 & 0.20 \\\\\n          & -9.2,-5.1  &  & 0.56 & 0.39 & 0.20 \\\\\n          & -4.9,-2.0 &   & 0.47 & 0.47 & 0.33 \\\\\n          & -2.0,-0.2 &   &0.39 & 0.05 & 0.28 \\\\\n          &  -0.2 .. 2.0 &   &0.42 & 0.09 & 0.03 \\\\\nB0355& -19.2,-15.9 & 0.35 &0.63 &0.18 & 0.07 \\\\\n          & -15.8,-11.7 &  & 0.92 & 0.43 & 0.28 \\\\\n          & -11.1,-9.8  &  &0.59 & 0.34 & 0.18 \\\\\n          & -9.6,-7.3   &   & 0.62 & 0.34 & 0.10 \\\\\n          & -6.0,-1.4   &  & 0.79 & 0.17 & 0.04 \\\\\n\\hline\n\\end{tabular}}\n\\\\\n$^a$ fraction of mapped area with \\mbox{W$_{\\rm CO}$}\\ $\\geq 1$ \\emr{\\,K\\,km\\,s^{-1}} \\\\\n$^b$ fraction of mapped area with \\mbox{W$_{\\rm CO}$}\\ $\\geq 2$ \\emr{\\,K\\,km\\,s^{-1}} \\\\\n\\end{table}\n\n\\subsection{Molecular gas properties in the current sample}\n\nThe sightlines studied here were selected on the basis of their known\n\\hcop\\ absorption spectra, creating the possibility that the sample is\nbiased to large molecular fractions and\/or strong CO emission.  However, it\nwas earlier noticed in a flux-limited survey \\citep{LucLis96} not based on\nprior knowledge of CO emission that very nearly all sightlines at galactic\nlatitudes within about 15\\mbox{$^{\\rm o}$}\\ of the galactic equator show \\hcop\\ \nabsorption.  Our present tally, slightly extending the earlier result, is\nthat \\hcop\\ absorption occurs toward 19 of 19 sources at $|b| \\la 12$\\mbox{$^{\\rm o}$},\ntoward 22 of 25 sources at $|b| \\la 18$\\mbox{$^{\\rm o}$}\\ and toward 4 out of 12\nsources at $|b| \\ga 23$\\mbox{$^{\\rm o}$}\\ including three shown here.  Thus it is a\nnear certainty that \\hcop\\ absorption would be detected over the entirety\nof the sky fields mapped here below about 15\\mbox{$^{\\rm o}$}{}-18\\mbox{$^{\\rm o}$}, no matter what\nis the covering factor of detectable CO emission.  This is discussed in\nSect. 8 immediately following the more descriptive portions of the text.\n\nIf we discuss the mean properties of the ten sightlines in Table 1 having\nreliably determined \\mbox{E$_{\\rm B-V}$}\\ (all except B0355+508 that lies too near the\ngalactic plane) in the same terms that we used earlier to derive the mean\nCO-\\mbox{H$_2$}\\ conversion factor in diffuse gas, \\citep{LisPet+10}, we derive an\nensemble average\n\n\\begin{eqnarray*}\n  \\frac{N(\\mbox{H$_2$})}{\\mbox{W$_{\\rm CO}$}} \n    &=& \\frac{5.8\\times 10^{21}~{\\rm cm}^{-2}<\\mbox{E$_{\\rm B-V}$}>-<N(\\mbox{H I})>}{<2\\mbox{W$_{\\rm CO}$}>} \\\\\n    &=& 1.52 \\times 10^{20}~{\\rm cm}^{-2}(\\emr{\\,K\\,km\\,s^{-1}})^{-1},\n\\end{eqnarray*}\ni.e., 25\\% smaller than the previous result found a larger sample. In the\nsame terms, the mean atomic gas fraction is $<$N(H I)$>$\/ $<$N(H)$> =\n0.74$, as opposed to 0.65 found earlier.\n\nEstimates of N(\\mbox{H$_2$}) based on assuming the ensemble-average mean value\n\\citep{LisPet+10} N(\\hcop)\/N(\\mbox{H$_2$}) $ = 3\\times 10^{-9}$ along each line of\nsight are also given in Table 1.  They indicate higher molecular fractions\nand somewhat higher total column densities N(H) than are found using scaled\n\\mbox{E$_{\\rm B-V}$}\\ for N(H) and using the decomposition discussed just above based on\nsubtracting N(H I) from N(H) determined as the scaled \\mbox{E$_{\\rm B-V}$}.  Specifically\nthe chemistry-based ensemble average is $<$\\mbox {f$_{{\\rm H}_2}$} $>$ = $<$2N(\\mbox{H$_2$})$>$\/\n$<$(N(H I)+2N(\\mbox{H$_2$})$>$ = 0.43.\n\n\\section{Six simple fields at moderate-high latitude}\n\n\\subsection{B0736+0117 $(b \\sim 11\\mbox{$^{\\rm o}$})$}\n\nThe 15\\arcmin\\ sky field around B0736+016 shown in Fig. 2 is the smallest\nand kinematically simplest field studied; the map was made on the spur of\nthe moment in a relatively brief open period between two other larger maps.\nThe reddening is modest over the area of the CO map shown in Fig. 2, \\mbox{E$_{\\rm B-V}$}\n$\\la$ 0.165 mag but the molecular fraction implied by the entries in Table\n1 is of order 30-50\\%.  Toward the source the integrated CO is fairly weak,\n$< 0.8$ \\emr{\\,K\\,km\\,s^{-1}}{} but a very slightly blue-shifted 4.5 K line is found within\njust a few arcminutes.\n\n\\subsection{B0954+658 $(b \\sim 43\\mbox{$^{\\rm o}$})$}\n\n\\begin{figure*}\n  \\includegraphics[height=13cm]{B0954Slide}\n\\caption[] {The sky field around the position of B0954+658, as\n  in Fig. 2.}\n\\end{figure*}\n\n\\begin{figure*}\n  \\includegraphics[height=13cm]{B1730Slide}\n\\caption[]{The sky field around the position of B1730-130, as\n  in Fig. 2.  The emission profile labelled ``nw quadrant'' is an average\n  over that portion of the map.  The emission profile labelled $<$otf {\\bf\n    CO}$>$ is the mean over the entire region mapped in CO. }\n\\end{figure*}\n\n\\begin{figure*}\n  \\includegraphics[height=13cm]{B1928Slide}\n\\caption[]{The sky field around B1928+738. The emission profile at \n  upper right labeled $<$otf {\\bf CO}$>$ is the mean over the 20\\arcmin\n  $\\times$ 20\\arcmin region mapped in CO. }\n\\end{figure*}\n\nThis source (Fig. 3) is seen at the upper tip of the Polaris Flare near the\nlocation of the M81 group and the so-called small area molecular structures\nmapped by \\cite{Hei04} (see the finding chart in Fig. 1).  As toward\nB0736+017, a fairly strong CO line, 5 K, is seen within a few arcminutes of\nthe background target where the CO brightness is more modest, 2 K.\n\nThe reddening is very moderate but a high molecular fraction, above 50\\%,\nis suggested by comparing the value of N(\\mbox{H$_2$}) in Table 1 with N(H I) or\n\\mbox{E$_{\\rm B-V}$}. Consistent with this high molecular fraction and the relative\nsimplicity of a higher-latitude line of sight, this field is the only one\nstudied in which there is a strong proportionality between \\mbox{E$_{\\rm B-V}$}\\ and \\mbox{W$_{\\rm CO}$},\nas discussed in Sect. 8.  The complement of supporting material is\ndisappointingly slender.\n\n\\subsection{B1730-130 $(b \\sim 11\\mbox{$^{\\rm o}$})$}\n\nThe reddening is relatively large over this field (Fig. 4) and the \\hcop\\ \nabsorption is strong, implying a molecular fraction of order one-third, but\nCO emission is very weak toward the background continuum source (\\mbox{W$_{\\rm CO}$}\\ =\n0.4 \\emr{\\,K\\,km\\,s^{-1}}) and absent over most of the field mapped.  Much stronger but\nstill rather weak emission (1.5 K) is seen 15\\arcmin\\ to the northwest as\nindicated in Fig.  4.  All of the absorption profiles shown in Fig. 4 have\na redward wing that is separately visible in the two spatially-averaged CO\nprofiles shown at upper right.\n\n\\subsection{B1928+738 $(b \\sim 23.5\\mbox{$^{\\rm o}$})$}\n\nCO emission is absent over the entire 20\\arcmin\\ field shown in Fig. 5\ndespite the presence of fairly strong \\hcop\\ absorption and a suggested\nmolecular fraction approaching 40\\%.  The reddening is modest,\napproximately 0.15 mag around the target, but the implied CO-\\mbox{H$_2$}\\ \nconversion factor is very large; from the entries in Table 1 we have\nN(\\mbox{H$_2$})\/\\mbox{W$_{\\rm CO}$}\\ $> 2.5\\times\\ 10^{21}~{\\rm cm}^{-2}$ at the $2\\sigma$ level.\n\n\\subsection{B1954+513 $(b \\sim 12\\mbox{$^{\\rm o}$})$}\n\nThe reddening in the field around this source is modest, 0.18 mag (see Fig.\n6 and 17), and CO emission toward the background continuum source is\nunimpressive (2 K) but the \\hcop\\ absorption is strong and the molecular\nfraction is of order 40\\%.  Two kinematic components are found in the field\nwith 4.5 K peak brightness, only one of which is seen toward B1954 in\neither emission or absorption.\n\n\\subsection{B2251+158, aka 3C454.3  $(b \\sim -38\\mbox{$^{\\rm o}$})$}\n\nAs shown in Fig. 1, this very strong continuum source is seen about 3\\mbox{$^{\\rm o}$}\\ \nremoved from an elongated complex of high-latitude clouds that includes the\nobjects MBM 53-55 \\citep{MagBli+85} and new clouds discovered by\n\\cite{YamOni+03}.  B2251+158 lies within the region surveyed by\n\\cite{YamOni+03} in CO but the emission detected here (see Fig. 7) escaped\ntheir notice, presumably because of their 4\\arcmin\\ map sampling of the\n2.7\\arcmin\\ beam.  The reddening is moderate, \\mbox{E$_{\\rm B-V}$}\\ $\\la$ 0.11 mag (see\nFig. 7) and CO emission toward the continuum source is quite weak (0.8 K).\nHowever, much stronger emission (5 K) is seen only 5\\arcmin\\ away, as with\nB0736, B0954 and B1954.  There is no obvious large-scale correlation of the\nCO emission with reddening, as evidenced by the weakness of the CO at the\nposition of highest reddening in the larger field shown at upper left in\nFig. 7 (i.e. the spectrum labelled ``NW'' at upper right there).  The\nrelationship between \\mbox{W$_{\\rm CO}$}\\ and \\mbox{E$_{\\rm B-V}$}\\ is shown in Figs. 16-17.\n\nThe blue wing of the peak emission and the line seen at the northwest\nreddening peak both fall to the blue of the CO emission or absorption seen\ntoward B2251. Nonetheless they overlap a weaker blue wing of the \\hcop\\ \nabsorption that has no counterpart in CO emission, and they fill in a\nportion of the H I absorption spectrum.\n\n\\section{Two unusual fields at moderate latitude}\n\n\\subsection{B0528+134 $(b \\sim -11\\mbox{$^{\\rm o}$})$}\n\nMm-wave absorption toward B0528+134 (Fig. 8) was first discussed by\n\\cite{HogDeG+95}.  This object is viewed against the outer edge of the dark\ncloud B30 in the $\\lambda$ Orionis ring of molecular clouds\n\\citep{MadMor87} that is centered on the H II region S264 and its central\nionizing star Lambda Ori (Fig. 1). There is a very substantial foreground\nreddening \\mbox{E$_{\\rm B-V}$}\\ = 0.86 mag and much more heavily extincted regions in the\nfield to the South.\n\nAlthough CO emission toward B0528+134 is fairly weak, 2.3 K, emission over\nthe surrounding field is characterized by a pronounced quasi-periodic\npattern with some very strong (10-12K) and narrow CO emission lines:\nemission is undetectable over much of the intervening troughs.  A similar\nwavelike pattern may have been observed across the surface of the Orion\nmolecular cloud by \\cite{BerMar+10}.\n\nA weak blue-shifted component of \\hcop\\ absorption that is absent in CO\ntoward B0528 has a very bright CO emission counterpart to the Southeast as\nshown in Fig. 8.  Despite an 8 \\mbox{km\\,s$^{-1}$}\\ velocity difference, the blueshifted\nemision line gives the strong visual impression of being physically\nassociated with the main kinematic component at 10 \\mbox{km\\,s$^{-1}$}, see the map at\nlower left in Fig.  8.  The kinematic span of the CO emission seen at top\nright in Fig. 8 neatly coincides with the extent of the H I absorption\ntoward B0528+134.  Line kinematics in this field are illustrated in more\ndetail in Fig. 14.\n\n\\subsection{B2200+420 aka BL Lac{} $(b \\sim -10.5\\mbox{$^{\\rm o}$})$}\n\nThis target (see Fig. 9) was the first source seen in mm-wave absorption\nfrom diffuse gas \\citep{MarBan+91}, in CO actually, and was also the first\nseen in \\hcop\\ absorption in our work \\citep{LucLis93}.  CO emission toward\nthe source is fairly strong, 4 K or 6 \\emr{\\,K\\,km\\,s^{-1}}{} and the line is quite opaque.\nThe molecular column density indicated by the strong \\hcop\\ absorption is\nabout as large as N(H) inferred from \\mbox{E$_{\\rm B-V}$}\\ = 0.32 mag, given the \\mbox{E$_{\\rm B-V}$}-N(H)\nrelationship N(H) $= 5.8\\times 10^{21}~{\\rm cm}^{-2}\\,\\mbox{E$_{\\rm B-V}$}$ of \\cite{SavDra+77}.\n\nThe CO emission in this field originates from an unusual filamentary\nmorphology (Fig. 9 at lower left) at the edge of an arched pattern in the\nreddening map.  The integrated intensity takes on very large values within\nthe field, up to 20 \\emr{\\,K\\,km\\,s^{-1}}\\ but the profile is compound and relatively\nbroad.  Toward the continuum source only the blue side of the core of H I\nabsorption is seen strongly in molecular absorption or CO emission but a\nred-shifted CO emission component overlaying the red side of the H I line\ncore is present to the Northeast as indicated in Fig 9.\n\n\\section{Three complex fields at low-moderate  latitude}\n\n\\begin{figure*}\n  \\includegraphics[height=15cm]{B1954Slide}\n\\caption[]{The sky field around the position of B1954+513, as in Fig. 2.  \n  In the map at lower left the grayscale represents the total integrated\n  emission at -1$\\le$ v $\\le$ 2 \\mbox{km\\,s$^{-1}$}\\ and the red and blue contours show\n  the individual distributions of red and blue-shifted components,\n  respectively.  Profiles at the peak of the red and blue-shifted emission\n  components are shown at upper right along with the profile toward the\n  continuum source (shaded).}\n\\end{figure*}\n \n\\begin{figure*}\n  \\includegraphics[height=15cm]{B2251Slide}\n\\caption[]{The sky field around the position of B2251+158 (aka 3C454.3), as in \n  Fig. 2.  The map of reddening at upper left is offset to show a separate\n  peak to the Northwest near $22^{\\rm H}50^{\\rm M}$ and a profile at the\n  position of this peak is shown at upper right, shaded green and labeled\n  'NW', along with profiles toward 3C454.3 (shaded) and at the peak of the\n  small clump that is seen immediately adjacent to the continuum source.}\n\\end{figure*}\n\n\\begin{figure*}\n  \\includegraphics[height=15cm]{B0528Slide}\n\\caption[]{The sky field around the position of B0528+134, \n  as in Fig. 2.  The map of CO emission at lower left superposes the\n  integrated intensity at 0-4 \\mbox{km\\,s$^{-1}$}\\ as blue contours against a background\n  grayscale representing emission at v = 8-12 \\mbox{km\\,s$^{-1}$}.  Very strong CO lines\n  are seen in the foreground gas as shown in the upper right panel:\n  positions at which they originate are indicated at lower left.}\n\\end{figure*}\n\n\\subsection{B0212+735 $(b \\sim 12\\mbox{$^{\\rm o}$})$}\n\nB0212+735 (Fig. 10) sits in a mild trough with \\mbox{E$_{\\rm B-V}$}\\ $\\approx$ 0.76 mag in\na region of substantial reddening at moderate galactic latitude b=12\\mbox{$^{\\rm o}$}.\nIt has three molecular absorption components whose balance is entirely\nopposite to that of H I.  Whereas most of the atomic absorption toward\nB0212+725 occurs in a deep and broad feature at v $\\la 10$ \\mbox{km\\,s$^{-1}$}, most of\nthe molecules are concentrated in a narrower-lined feature at v $\\approx$ 4\n\\mbox{km\\,s$^{-1}$}.  An obvious molecular absorption feature at 0-velocity is, very\nunusually, not apparent in H I.  It seems possible that the low velocity\nresolution of the H I profile (1 \\mbox{km\\,s$^{-1}$}) is responsible. The only other\npublished example of this phenomenon is toward B0727-115 \\citep{LeqAll+93}.\n\nThe CO emission line kinematics have been color coded at lower left in Fig.\n10 to display the observed behaviour in one panel.  The gray-scale\nbackground represents the integrated intensity of the gas at 1.5-5 \\mbox{km\\,s$^{-1}$};\nhigher resolution mapping with the IRAM 30m telescope to be discussed in a\nforthcoming paper indicates that the feature is compound but this is not\napparent in the present dataset.  The blue contours represent the CO\nprofile integral at $-16$ \\mbox{km\\,s$^{-1}$}{} $\\leq$ v $\\leq$ $-9.5$ \\mbox{km\\,s$^{-1}$}; consistent\nwith the prominence of this gas in H I, it is almost as widely distributed\nover the field as the stronger emission at 1.5-5 \\mbox{km\\,s$^{-1}$}\\ (Table 2: 26\\% vs.\n34\\%) even if it is barely seen toward the continuum.  The profile labelled\n'A' at upper right is an example.  The black contours represent the profile\nintegral at $-2$ \\mbox{km\\,s$^{-1}$}\\ $\\leq$ v $\\leq$ 1 \\mbox{km\\,s$^{-1}$}{} and an example is shown at\nupper right as profile 'B'.  Emission from this gas occurs only at the\neastern edge of the map area.\n\n\\begin{figure*}\n  \\includegraphics[height=15cm]{B2200Slide}\n\\caption[]{The sky field around the position of B2200+420 (aka BL Lac), \n  as in Fig. 2. The map of CO emission at lower left superposes the\n  integrated intensity in the range v=0-2 \\mbox{km\\,s$^{-1}$}\\ as red contours against a\n  background grayscale representing emission at v $\\leq 0$ \\mbox{km\\,s$^{-1}$}. Molecular\n  absorption and most emission is sequestered in the blue wing of the core\n  of the HI absorption profile but a red-shifted emission component is\n  present to the Northeast as illustrated by the spectrum at position ``C''\n  indicated at lower left.}\n\\end{figure*}\n\n\\begin{figure*}\n  \\includegraphics[height=15cm]{B0212Slide}\n\\caption[]{The sky field around the position of B0212+735,\n  as in Fig. 2.  Contours in the CO emission map at lower left are color\n  coded in blue for emission at $-15 \\leq {\\rm v} \\leq -9.5$ \\mbox{km\\,s$^{-1}$}\\ and in\n  green for emission at $-2 \\leq {\\rm v} \\leq 1$ \\mbox{km\\,s$^{-1}$}.  The gray scale\n  background represents the integrated emission of the strongest emission\n  component seen toward the continuum source, at v = 1.5 - 5 \\mbox{km\\,s$^{-1}$}.  \\mbox{$^{12}$CO}\\ \n  spectra at two locations labelled A and B are shown at upper right along\n  with a strongly-scaled mean profile taken over the full map area.}\n\\end{figure*}\n\n\\subsection{B0224+671 $(b \\sim 6\\mbox{$^{\\rm o}$})$}\n\nThis line of sight toward B0224+671 (Fig. 11) samples the two\nlower-velocity features seen toward B0212+735 but at substantially lower\ngalactic latitude b=6.2\\mbox{$^{\\rm o}$}, see Table 1.  The extinction is large in this\nfield as are the H I and inferred \\mbox{H$_2$}\\ column densities.  CO emission is\nweak on a per-component basis toward the continuum target but fairly total\nlarge values of \\mbox{W$_{\\rm CO}$}\\ are attained overall.\n\nThe integrated CO emission is compact but rather formless because it is the\nsum of many kinematic components.  Paradoxically, the strongest molecular\nfeatures seen toward and near B0224+671 are not widely distributed over the\nmap area as shown in the middle panel at right in Fig. 11 comparing the\nprofile toward B0224+671 with the unweighted average of all profiles\ndenoted '$<$otf$>$': the strongest emission peak, at the red edge of the CO\nemission profile toward B0212+735 is strongly underrepresented in the mean\nprofile.  Examples of profiles seen over the map area are shown at upper\nright in Fig.  11; they were chosen at local peaks in more finely-divided\n(in velocity) moment maps, with velocity increasing from a to g.\nEspecially at v $<$ 0 \\mbox{km\\,s$^{-1}$}\\ the lines shown are much stronger than seen\ntoward the continuum.  Among them, the various CO emission components cover\nthe range of strong H I absorption toward the continuum source, $-15$ \\mbox{km\\,s$^{-1}$}\\ \n$\\leq$ v $\\leq$ 2 \\mbox{km\\,s$^{-1}$}{}. B0224 and B0212 have similar absorption spectra\nin that both have stronger atomic absorption at v $ < -10$ \\mbox{km\\,s$^{-1}$}{} where the\nmolecular absorption is weaker. They thus sample the same large-scale gas\nkinematics although they are separated by about 15 pc, assuming they lie on\na sphere of 150 pc radius.\n\n\\subsection{B0355+508 aka NRAO150 $(b \\sim -1.6\\mbox{$^{\\rm o}$})$}\n\nThis is the only low-latitude source studied here.  The more strongly\nblue-shifted gas seen in this direction is likely to be relatively distant.\nThe actual velocity field is probably affected by galactic streaming\nmotions but a typical velocity gradient due to galactic rotation in this\ndirection is 8\\mbox{km\\,s$^{-1}$}\\kpc$^{-1}$.\n\n\\begin{figure*}\n  \\includegraphics[height=15cm]{B0224Slide}\n\\caption[]{The sky field around the position of B0224+671,\n  much as in Fig. 2. The map at lower left has been integrated over the\n  very wide interval $-15.5 \\leq v \\leq +2$ \\mbox{km\\,s$^{-1}$}.  Shown in the middle\n  panel at right are the CO emission spectrum toward B0224+671 and as\n  averaged over the region of the entire CO emission map.  At top right are\n  example profiles from the positions labelled at lower left, chosen from\n  moment maps over narrow intervals increasing in velocity from a-g.}\n\\end{figure*}\n\n\\begin{figure*}\n  \\includegraphics[height=13.3cm]{B0355Slide}\n\\caption[]{The sky field around the position of B0355+508. At top \n  are maps of integrated CO intensity made over the velocity intervals\n  indicated in each panel, corresponding to the five strong components of\n  the \\hcop\\ absorption profile seen at lower right.  CO emission profiles\n  at various locations indicated in the map panels are shown at lower left.\n  CO emission profiles toward B0355+508 and averaged over the map area are\n  shown above the absorption line profiles.}\n\\end{figure*}\n\nFig. 12 does not display a map of the reddening for this source because the\nreddening maps are not believed to be accurate at such low galactic\nlatitudes.  Taken as a whole the line of sight is very heavily extincted,\nwith large columns of both H I and \\mbox{H$_2$}\\ (Table 1).  However, the high\ninferred \\mbox{H$_2$}\\ column density is the sum of components whose individual\nvalues indicate that they have \\mbox{A$_{\\rm V}$}\\ $\\approx 1$ mag.\n\nThe absorption profiles toward B0355+508 at lower right in Fig. 12 show\nfive obvious \\hcop\\ and CO components having roughly equal N(\\hcop)\n$\\approx 1.2\\times10^{12}~{\\rm cm}^{-2}$ and somewhat more variable N(CO)\n\\citep{LisLuc98}.  Only two have substantial abundances of less-common\nspecies such as HCN.  Less obvious is the fact that the \\hcop\\ absorption\nprofile has a weak broad blue wing extending to $-35$ \\mbox{km\\,s$^{-1}$}{} so that the\nentire core of the H I absorption line is seen in molecular gas\n\\citep[see][]{LisLuc00}.\n\nShown at the top in Fig. 12 are CO moment maps made over velocity intervals\ncorresponding to the obvious \\hcop\\ and CO absorption line features; a map\nintegrated over all velocities is shown in the top right-most panel. The CO\nemission distribution is heavily structured and very complex.  Profiles at\npositions of local peaks in narrower CO moment maps are shown at lower\nleft, along with profiles at the integrated emission peak (see Fig. 12 at\ntop right) and toward the background source.  At the eastern edge of the\nmap at position ``a'' there is a blue-shifted CO emission line that\n(unusually) falls outside the range of the strong molecular absorption\ntoward B0355+508 but is well inside the H I absorption profile).  A very\nbright ($>$ 10 K) line is found at v = $-4.5$ \\mbox{km\\,s$^{-1}$}\\ at the northern edge of\nthe map at position f, corresponding to a prominent molecular absorption\nfeature that has only a very weak emission counterpart toward the\ncontinuum.\n \nThe middle right panel displays the profile toward the continuum background\nsource and the profile averaged over the mapped field of view. The\nkinematics of this region are shown in more detail in Fig. 15 and discussed\nin Sect. 7.2.  The displacement of the two strong CO emission lines about\nthe centroid of the mean profile results from a coherent kinematic\npatttern, perhaps a shell or bubble in the underlying gas distribution.\nRecall, however that absorption at the mean field velocity is not absent\ntoward the continuum source.\n\nThe complexity of the emission distribution makes the division into ranges\nbased on \\hcop\\ absorption quite arbitrary.  Moreover, the emission and\nabsorption profiles show rather different structure even toward the\ncontinuum target.  A very detailed discussion of CO emission within a\n90\\mbox{$^{\\prime\\prime}$}} \\def\\arcmin{\\mbox{$^{\\prime}$}\\ field centered on NRAO150 was given by \\cite{PetLuc+08}.\nRemarkably, the peak emission brightness seen just $6''$ from the\nbackground continuum source is almost 13 K.  As the spatial resolution\nincreases, the CO emission profile toward B0355 more nearly resembles the\nabsorption and the blended emission at v $\\approx$ -10 \\mbox{km\\,s$^{-1}$}\\ resolves into\ntwo distinct components.\n\n\\begin{figure}\n  \\includegraphics[height=7.2cm]{B1954-OneRAV}\n\\caption[]{A right ascension-velocity diagram of CO emission through \n  the position of B1954+513.  The \\hcop\\ (not CO) absorption spectrum\n  toward B1954 is shown with its 0-level at the location of the continuum\n  source; the peak absorption is 90\\%.  Contours are shown at levels 1, 2,\n  3, ... K.}\n\\end{figure}\n\nConversely, in the present dataset, the emission components seen toward\nB0355+508 lose their identity as the resolution degrades below a 4\\arcmin\\ \nhpbw.  Both the peak profile and the mean are broad, largely unstructured\nand centrally peaked about velocities lying between the two strong CO\nemission components seen toward the background.\n\n\\section{Statistical lessons}\n\nFaute de mieux, the first surveys for suitable absorption-line targets were\nconducted in CO emission \\citep{BanMar+91,LisWil93,Lis94} but the discovery\nof yet more common \\hcop\\ absorption \\citep{LucLis96} caused a reversal in\nthe search strategy for diffuse molecular gas.  Thus, all targets studied\nhere were pre-selected to have absorption from \\hcop\\ but only some were\nknown to have CO emission.  However any division between targets with and\nwithout CO emission is misleading because the same sightline may have and\nlack CO emission on a per-component basis.\n\nStronger CO emission is always found somewhere else in the map when CO is\npresent toward the continuum but comparably strong CO emission was found,\nsomewhere on the sky, from absorption components lacking CO emission\ncounterparts toward the continuum background.  This is true with only one\nexception, B1928.  The implication is that CO emission is somewhat more\nubiquitous than is presently believed to be the case because nearly all\n\\hcop\\ absorption components will be found in nearby emission after a small\nsearch.  In terms of numbers, in this work we observed 20 absorption line\ncomponents (and a few distinct line wings) with 13 carbon monoxide emission\ncounterparts toward the continuum targets, and we found 23 CO emission\ncomponents within $15'$ of the background continuum source during the\nmapping.\n\nThe following gives some conclusions that are drawn from the preceding\npresentation; they should generally be understood as applying on a\ncomponent-by-component basis.\n\n{\\bf From the standpoint of CO emission:}\n\n1) In every case where CO emission was detected toward the background\nsource, much stronger emission was also detected within approximately\n15\\arcmin\\ and often much less.\n\n2) Near B0355+508, B0528+134 and B2200+430 the nearby stronger emission was\nvery strong indeed, with peak line temperatures of 10-12 K and\/or line\nprofile integrals as large as 20 K \\mbox{km\\,s$^{-1}$}.\n\n3) 1) and 2) are also generally true for kinematic components that were\npresent in absorption toward the continuum source but {\\it not} detected in\nemission there (7 of 20 components).\n\n4) In one case only (B1928+738), representing 1 of 11 fields and 1 of 20\nkinematic absorption line components, the entire field mapped was devoid of\nemission when emission was not detected toward the continuum source (from a\nkinematic absorption line component).  The field mapped around B1928 was\nonly 20\\arcmin\\ $\\times$ 20\\arcmin\\ as against 30\\arcmin\\ $\\times$\n30\\arcmin\\ or more for all of the other fields.\n\n5) In 2 fields we found an emission feature without a counterpart in\nmolecular absorption toward the background continuum object (B1954 and\nB0355 at -20 \\mbox{km\\,s$^{-1}$}).\n\n{\\bf From the standpoint of absorption:}\n\n1) The same kinematic components are seen in both absorption and emission\nwith angular separations of up to $15'$.\n\n2) The absorption spectra toward a background source are a preview of what\nwill be seen in emission in a larger field about the background source.\n\n3) Molecular absorption components seen toward the continuum source were\nfound in CO emission somewhere in the field except in the smaller region\nmapped around B1928+738\n\n{\\bf From the standpoint of the atomic-molecular transition:}\n\n1) The same kinematic components are seen in both atomic and molecular\ntracers at angular separations between $0'$ and $15'$.\n\n2) The components seen in molecular absorption are present in H I\nabsorption, although somewhat indistinctly in some cases.  For instance,\nthe 0-velocity molecular absorption line in B0212+735 appears only as a\nblue wing of the 4 \\mbox{km\\,s$^{-1}$}\\ H I absorption component.\n\n3) Portions of H I absorption profiles adjacent to molecular features but\nlacking a molecular counterpart are seen in CO emission elsewhere in the\nfield in two cases on (B2200+420 and B0528+134).\n\n4) We saw no molecular features in absorption or emission outside the span\nof the H I absorption (see Appendix B).\n\n\\section{Kinematics}\n\nMolecular gas is generally well-mixed with other components of the ISM\n\\citep{DamTha94,GirBli+94} and does not require exceptional kinematics.\nThis is apparent in our work from the coincidence of molecular and atomic\nabsorption features, even if they do not have precisely the same patterns\nof line depth.  The kinematics are affected by galactic structure and local\nexternal influences such as shocks, but this only becomes apparent on broad\nangular scales.  The targets B0212+735 and B0224+671 (Figs 10-11) are\nrelatively close to each other and both are most strongly absorbed in H I\naround -15 \\mbox{km\\,s$^{-1}$}.  The background target B2251+158 (Fig. 7 and Sect. 3.6) is\nseen in the outskirts of the MBM53-55 cloud complex, which is part of a\nlarge shell that has been extensively mapped in molecular and atomic gas\n\\citep{GirBli+94,YamOni+03}.\n\nIn individual line profiles and over small scales, the kinematics are often\ndominated by the internal structure of individual clouds.  The internal\nmotions of diffuse molecular gas are now understood to reflect turbulent\ngas flows \\citep{PetFal03,HilFal09} that are characterized by unsteady\nprojected velocity fields with strong shears and abrupt reversals of the\nvelocity gradient.  \\cite{SakSun03} show the transition between diffuse and\ndense molecular gas at the edge of TMC1 and \\cite{LisPet+09} discuss gas\nflows in the diffuse cloud occulting $\\zeta$ Oph.\n\nIn this Section we discuss the kinematics of just two of the fields mapped\nhere.  Further examples of CO kinematics in individual sky fields are given\nin Figs. A.1-A.3 of Appendix A (available online) and the galactic context\nfor all fields is given in Figs. B.1 and B.2 of online Appendix B, showing\nlarge-scale latitude-velocity cuts in HI from the Leiden-Dwingeloo H I\nsurvey of \\cite{HarBur97} with the locations of the continuum background\nsources marked in each case.\n\nFig. 13 shows the kinematics in the relatively simple sky field around\nB1954+513 (Sect. 3.5 and Fig. 6) with the spatially-displaced blue and\nred-shifted CO emission components that were illustrated in Fig. 6.  The\nred-shifted component seen toward the continuum has a partially-resolved\nvelocity gradient that carries it just to the midpoint of the associated\n\\hcop\\ absorption profile at the continuum position.  It is certain that\nthe blue-shifted CO emission to the East would have an associated \\hcop\\ \nabsorption at its position but the structure of the redward gas cannot be\ntraced away from the continuum and, regrettably we do not have an H I\nabsorption profile that might show both the red and blue-shifted gas in\natomic absorption as toward BL Lac\\ (Fig. 9 and Sect. 4.2).\n\nFig. 14 shows the more complicated field at low latitude around B0355+508\n(Sect. 5.3, Fig. 12) and illustrates how the partition of a line profile\ninto components, no matter how seemingly obvious, can also be arbitrary and\ncapricious.  None of the well-defined absorption features has an obvious CO\nemission counterpart except perhaps in the immediate vicinity of the\ncontinuum target.  This is not an artifact of taking a cut in declination,\nwhich is actually richer than that in right ascension (see Fig. 12).\n\nNonetheless, mapping the CO emission does help to clarify interpretation of\nthe absorption profiles.  For instance, consider gas near $-9$ \\mbox{km\\,s$^{-1}$}\\ around\nthe location of B0355 in Fig. 14.  In absorption there are two distinct\nkinematic components at $-11$ and $-8$ \\mbox{km\\,s$^{-1}$}\\ that would usually be\ninterpreted as unrelated because, aside from their separation in velocity,\nthey have different patterns of chemical abundances (Fig. 12) However, Fig.\n14 shows that the CO emission line has an appreciable velocity gradient\nacross the position of the continuum source, spanning the two absorption\nlines, making it likely that the two absorption components are part of the\nsame body \\footnote{\\cite{PetLuc+08} show that the overlapping CO emission\n  line is resolved into two kinematic components at 6\\mbox{$^{\\prime\\prime}$}} \\def\\arcmin{\\mbox{$^{\\prime}$}\\ resolution\n  toward the continuum source.}.  Moreover, the CO mapping suggests that\nthe components at $-17$ and $-10$ \\mbox{km\\,s$^{-1}$}\\ may also be part of the same\nstructure (and separated by a velocity gradient), which was actually\nsuggested by several coincidences in our earlier high-resolution CO mapping\n\\citep{PetLuc+08}.  The lines at $-11$ and $-17$ \\mbox{km\\,s$^{-1}$}\\ are very bright (13\nK) at high resolution and have considerable chemical complexity.  There are\nalso some seemingly correlated spatial intensity variations.  The evidence\nfor an association is entirely indirect but has a clear precedent in the\nkinematics around B0528+134 (Sect. 4.1 and Fig.  8) where a similar\nvelocity separation occurs between two emission components that are seen\nsuperposed in an unusual wave-like spatial configuration.\n\n\\section{The brightness of diffuse cloud CO}\n\n\\subsection{\\mbox{W$_{\\rm CO}$}\\ relative to \\mbox{E$_{\\rm B-V}$}\\ and \\mbox {f$_{{\\rm H}_2}$}}\n\nThe large-scale finding chart in Fig. 1 is a map of the total intervening\ngas column density, except where discrete sources of infrared emission\n(often H II regions) ``leak'' into the map (usefully indicating when the\nbackground target may have been observed through disturbed foreground gas).\nLarge-scale surveys of CO emission at 8\\arcmin\\ resolution show a good\ncorrelation with reddening \\citep{DamHar+01}, contributing to the common\ninterpretation of CO sky maps as displaying the global distribution of\ndense, fully-molecular gas.\n\nIn diffuse gas appreciable scatter in the \\mbox{W$_{\\rm CO}$}-\\mbox{E$_{\\rm B-V}$}\\ relationship is\nexpected because the reddening is a sum over atomic and molecular\ncomponents that both make important contributions to N(H), combined with\nthe fact that both N(\\mbox{H$_2$}) and N(CO)\/N(\\mbox{H$_2$}) exhibit order-of-magnitude or\nlarger scatter with respect to \\mbox{E$_{\\rm B-V}$}\\ even when all quantities are measured\nalong the same microscopic sightlines toward nearby bright stars\n\\citep{BurFra+07,RacSno+09}.  The disparity in angular resolution between\nthe reddening data and our 1\\arcmin\\ CO maps presents another sort of\ncomplication that is considered in Sect. 8.2 but does not by itself\ndominate the scatter.  Recall also the discussion in \\cite{LisPet+10} where\na good correlation was shown between \\mbox{E$_{\\rm B-V}$}\\ at 6\\arcmin\\ resolution compared\nwith the integrated H I optical depth measured in absorption at 21cm toward\na larger set of the same kind of point-like radiocontinuum background\ntarget considered here.\n\nSmall-scale maps of reddening are shown in the various Figs. 2-12 detailing\nthe individual fields.  They may visually suggest correlations between\n\\mbox{E$_{\\rm B-V}$}\\ and \\mbox{W$_{\\rm CO}$}, and there is a threshold \\mbox{E$_{\\rm B-V}$}\\ $\\ga $ 0.09 mag for\ndetecting CO emission, consistent with the well-known and quite abrupt\nincrease of N(\\mbox{H$_2$})\/N(H) at comparable reddening \\citep{SavDra+77}.\nHowever, reddening is not a reliable predictor of CO emission in our sky\nfields.  For instance, in the field around B2251+158 in Fig. 7, CO emission\nis much weaker at the peak of the reddening map where \\mbox{E$_{\\rm B-V}$}\\ = 0.14 mag (the\nprofile indicated as ``NW'' at upper right in Fig. 7) than nearer the\ncontinuum source at smaller \\mbox{E$_{\\rm B-V}$}\\ = 0.10 mag.  Around B2200+420 (Fig. 9)\nthe shape of the CO distribution appears to parallel that of the reddening\nbut in detail CO only traces the edge rather than the peak ridge of the\n\\mbox{E$_{\\rm B-V}$}\\ distribution.\n\nIn Fig. 15 we show the relationship between \\mbox{W$_{\\rm CO}$}\\ and \\mbox{E$_{\\rm B-V}$}\\ in the four\nsimple cases discussed in Sect. 3, where the extinction is small and a\nsingle narrow CO spectral component is present at each pixel\n\\footnote{Green diamonds in Figs. 15 and 16 show \\mbox{E$_{\\rm B-V}$}\\ and \\mbox{W$_{\\rm CO}$}\\ toward the\n  continuum target as given in Table 1}. The rms noise levels in these four\ndatasets (Table 2) are 0.48, 0.33, 0.32 and 0.35 \\emr{\\,K\\,km\\,s^{-1}}{} reading clockwise\nfrom upper left so that datapoints with \\mbox{W$_{\\rm CO}$}\\ $\\ga$ 1 \\emr{\\,K\\,km\\,s^{-1}}\\ (the usual\nlast contour on CO sky maps) are detected at or above the 90\\% confidence\nlevel.  To put these brightness and sensitivity levels in context, note\nthat there is a straightforward relationship between \\mbox{W$_{\\rm CO}$}, \\mbox {f$_{{\\rm H}_2}$}, and \\mbox{E$_{\\rm B-V}$}\\ \nonce the CO-\\mbox{H$_2$}\\ and \\mbox{E$_{\\rm B-V}$}\/N(H) conversion factors are fixed; for the\n\\emph{standard} \\mbox{W$_{\\rm CO}$}\/N(\\mbox{H$_2$}) = $2\\times 10^{20} ~{\\rm cm}^{-2}$ \\mbox{H$_2$}\\ (\\mbox{km\\,s$^{-1}$})$^{-1}$ and\nN(H)\/\\mbox{E$_{\\rm B-V}$}\\ = $5.8\\times 10^{21} ~{\\rm cm}^{-2} {\\rm mag}^{-1}$ one has \\mbox{W$_{\\rm CO}$}\\ = 14.5\n\\mbox {f$_{{\\rm H}_2}$}\\ \\mbox{E$_{\\rm B-V}$} \\emr{\\,K\\,km\\,s^{-1}}{}.  At \\mbox{E$_{\\rm B-V}$}\\ = 0.1 mag, emission only slightly exceeding 1\nK \\mbox{km\\,s$^{-1}$}\\ implies a molecular fraction \\mbox {f$_{{\\rm H}_2}$}\\ $>$ 1 and therefore is too\nbright to be accomodated by a CO-\\mbox{H$_2$}\\ conversion factor as large as the\nstandard $2\\times 10^{20} ~{\\rm cm}^{-2}$ \\mbox{H$_2$}\\ (\\mbox{km\\,s$^{-1}$})$^{-1}$.\n\n\\begin{figure}\n  \\includegraphics[height=8.5cm]{B0355-SimpleShell}\n\\caption[]{A declination-velocity diagram of CO emission\n  at the right ascension of B0355+508.  The CO absorption spectrum toward\n  B0355+508 is shown with its baseline level at the declination of the\n  background source.  The strongest CO absorption line is quite opaque, see\n  Fig. 12.}\n\\end{figure}\n\nShown in each panel of Fig. 15 are lines representing the CO emission\nexpected if various fractions \\mbox {f$_{{\\rm H}_2}$}\\ of the total neutral gas column are in\n\\mbox{H$_2$}\\ with a typical galactic \\mbox{W$_{\\rm CO}$}\/N(\\mbox{H$_2$}) conversion factor X$_{\\rm CO} =\n2\\times10^{20}$ \\mbox{H$_2$}\\ $~{\\rm cm}^{-2}$ (\\emr{\\,K\\,km\\,s^{-1}})$^{-1}$.  Much of the CO in Fig. 15\noccurs above the line corresponding to \\mbox {f$_{{\\rm H}_2}$}\\ = 1 and is therefore too\nbright to be accomodated by the usual CO-\\mbox{H$_2$}\\ conversion factor; indeed,\nalmost every CO line with \\mbox{W$_{\\rm CO}$}\\ $\\ga$ 1 \\emr{\\,K\\,km\\,s^{-1}}{} may be described as\noverly-bright in this way if \\mbox {f$_{{\\rm H}_2}$}\\ = 0.5, hence the great majority of all\nthe statistically significant emission represented in Fig.  15 and in the\nmaps shown earlier for these sources.  For the brightest pixels\nN(\\mbox{H$_2$})\/\\mbox{W$_{\\rm CO}$}\\ $< 5\\times10^{19} \\mbox{H$_2$}\\ ~{\\rm cm}^{-2} $ (\\emr{\\,K\\,km\\,s^{-1}})$^{-1}$.\n \nThe same \\mbox{W$_{\\rm CO}$}-\\mbox{E$_{\\rm B-V}$}\\ diagrams are shown for sources with higher \\mbox{E$_{\\rm B-V}$}\\ in\nFig. 16.  Much of the gas around B2200+420 falls above the line for \\mbox {f$_{{\\rm H}_2}$}\\ =\n1, and attains such high brightness that its \\mbox{H$_2$}\/\\mbox{W$_{\\rm CO}$}\\ ratio is 3-4 times\nbelow the standard conversion factor.  However, this case becomes\nincreasingly harder to make toward the other sources having higher \\mbox{E$_{\\rm B-V}$}\\ as\nin the bottom panels of Fig. 16.\n\n\\subsection{Sub-structure in reddening \n  would not eliminate large \\mbox{W$_{\\rm CO}$}\/\\mbox{E$_{\\rm B-V}$}\\ ratios}\n\nCO emission is heavily structured on arcminute scales, well below the\n6\\arcmin\\ angular resolution of the reddening maps, and the high values and\nlarge scatter in \\mbox{W$_{\\rm CO}$}\/\\mbox{E$_{\\rm B-V}$}\\ in Fig. 15 cannot be accomodated with a fixed\nratio of \\mbox{W$_{\\rm CO}$}\/N(\\mbox{H$_2$}) or \\mbox{W$_{\\rm CO}$}\/N(H) except by positing strong unresolved\nvariations, essentially clumping, in \\mbox{E$_{\\rm B-V}$}.  It is important to understand\nthe extent to which this might represent unresolved structure in the total\ncolumn density, for instance with regard to cleaning maps of the cosmic\nmicrowave background \\citep{bernard11}.  Given the extreme sensitivities of\nthe CO abundance and brightness to N(\\mbox{H$_2$}) in diffuse clouds and the fact\nthat even \\mbox {f$_{{\\rm H}_2}$}\\ may vary in diffuse material, it is entirely possible that\nthe large contrasts seen in \\mbox{W$_{\\rm CO}$}\\ do not have strong consequences for the\ndistribution of N(H), \\mbox{E$_{\\rm B-V}$}, or even N(\\mbox{H$_2$}).\n\nShown in Fig. 17 are cumulative distribution functions of the integrated CO\nemission \\mbox{W$_{\\rm CO}$}\\ in the fields around B0954+658 and BL Lac, using the native\nARO data and versions of the data smoothed to lower angular resolution\n3\\arcmin\\ (similar to NANTEN) and 5\\arcmin\\ (similar to Planck).  The\nbrightness distribution of the CO around B0954+658 is compact in Fig. 3 but\nstill sufficiently extended that 4.5 \\emr{\\,K\\,km\\,s^{-1}}{} integrated intensities are\npresent at 5\\arcmin\\ resolution; this is well above the line for \\mbox {f$_{{\\rm H}_2}$}\\ = 1\nin Fig. 15.\n\nThe distribution of strongly emitting CO around BL Lac\\ is sufficiently broad\nin angle that 20-30\\% of the pixels are occupied by CO with \\mbox{W$_{\\rm CO}$}\\ $\\geq$ 5\nK \\mbox{km\\,s$^{-1}$}\\ whether the angular resolution is 1\\arcmin\\ or 5\\arcmin; the very\nstrongest CO lines have \\mbox{W$_{\\rm CO}$}\\ $\\ga$ 15\\emr{\\,K\\,km\\,s^{-1}}\\ at 1-5\\arcmin\\ resolution in\nthe BL Lac\\ field.  This is consistent with our recent observations of CO\nemission in the field around $\\zeta$ Oph\\ \\citep{LisPet+09} where the same peak\nbrightnesses were found in ARO and NANTEN data at 3\\arcmin\\ resolution.\n \n\\begin{figure}\n  \\includegraphics[height=8cm]{FourEBV}\n\\caption[]{ Distribution of \\mbox{E$_{\\rm B-V}$}\\ and \\mbox{W$_{\\rm CO}$}\\ for four fields mapped here\n  in CO.  Each 20\\mbox{$^{\\prime\\prime}$}} \\def\\arcmin{\\mbox{$^{\\prime}$}\\ pixel in the CO maps is plotted as a separate\n  point. The (red) dashed lines in each panel show the CO emission expected\n  if 25\\%, 50\\% and 100\\% of the gas is in molecular form with a typical\n  value of the \\mbox{W$_{\\rm CO}$}-N(\\mbox{H$_2$}) conversion factor, N(\\mbox{H$_2$})\/\\mbox{W$_{\\rm CO}$}\\ =\n  $2\\times10^{20}$\\mbox{H$_2$}$~{\\rm cm}^{-2}$ (\\emr{\\,K\\,km\\,s^{-1}})$^{-1}$.  In each panel a (green) filled\n  diamond is shown at the value given in Table 1 toward the background\n  source.}\n\\end{figure}\n\n\\begin{figure}\n  \\includegraphics[height=8cm]{FourNEBV}\n\\caption[]{As in Fig. 15 for four fields with larger reddening.}\n\\end{figure}\n\n\\begin{figure}\n  \\includegraphics[height=8cm]{Cumulate}\n\\caption[]{Cumulative distribution functions for \\mbox{W$_{\\rm CO}$}\\ at spatial \n  resolutions of 1\\arcmin\\ (gray, thicker), 3\\arcmin\\ (blue) and 5\\arcmin\\ \n  (red) for B0954 (left) and B2200.}\n\\end{figure}\n\nBecause high CO brightness, and, therefore, impossibly high ratios\n\\mbox{W$_{\\rm CO}$}\/\\mbox{E$_{\\rm B-V}$}\\ (requiring \\mbox {f$_{{\\rm H}_2}$}\\ $> 1$ for the mean \\mbox{W$_{\\rm CO}$}\/N(\\mbox{H$_2$}) ratio) persist\nto low angular resolution, \\mbox{W$_{\\rm CO}$}\\ cannot maintain a constant proportionality\nto N(\\mbox{H$_2$}) over the body of these clouds.  The observed variations in \\mbox{W$_{\\rm CO}$}\\ \nare too large to be accomodated by the total amount of material that is\npresent along the line of sight and unresolved structure in reddening\ncannot account for the high values of \\mbox{W$_{\\rm CO}$}\/\\mbox{E$_{\\rm B-V}$}\\ or the range of variation\nin that ratio.\n\n\\subsection{Covering factors and bright and dark CO}\n\nIn Sect. 2.7 we noted the statistical certainty with which \\hcop\\ \nabsorption is found in spectra taken within about 15\\mbox{$^{\\rm o}$}{} - 18\\mbox{$^{\\rm o}$}{} of\nthe galactic plane.  A corollary to this is that molecular gas is certain\nto be omnipresent over the sky fields mapped in CO at such latitudes, no\nmatter how much of sky we actually found to be occupied by CO emission.\n\nTable 2 shows pixel statistics for the CO emission maps made in the course\nof this work; shown are profile channel-channel rms noise values and map\npixel-pixel rms noise in \\mbox{W$_{\\rm CO}$}\\ for each mappable kinematic component.  In\neach case the covering factor was determined by forming a histogram of the\n\\mbox{W$_{\\rm CO}$}\\ values and subtracting a gaussian fit to the component corresponding\nto the noise, and is apparent in that it extends to unphysical negative\nvalues of \\mbox{W$_{\\rm CO}$}.  This is not a large correction; if the noise in \\mbox{W$_{\\rm CO}$}\\ is\nrandom at a level $\\sigma_{\\rm map}$ the covering factor at\/above any given\n\\mbox{W$_{\\rm CO}$}\\ in the absence of signal is dA(\\mbox{W$_{\\rm CO}$})\/A =\n0.5(1-erf(\\mbox{W$_{\\rm CO}$}\/$\\sqrt(2)\\sigma_{\\rm map}$)).  The expected covering factor\ndue to noise at even 2$\\sigma$ significance is already below 3\\%.\n\nTable 2 shows that typical covering factors are 20\\%-40\\%, with a few null\nvalues and only two sky fields (B0528 and B2200) where the majority of the\nmap area is covered.  Very approximately, the covering factors are small to\nabout the same extent that the \\mbox{W$_{\\rm CO}$}\/N(\\mbox{H$_2$}) conversion factors of the\ndetected CO emission are higher than indicated by the standard CO-\\mbox{H$_2$}\\ \nconversion. In the end, some form of spatial averaging over brighter and\ndimmer CO must be responsible for the global mean CO-\\mbox{H$_2$}\\ conversion factor\nwhether in fully molecular or diffuse molecular gas.  The idea that all gas\nparcels would show identically the same \\mbox{W$_{\\rm CO}$}\/N(\\mbox{H$_2$}) is preposterous.\n\n\\subsection{Pressure and density in CO-emitting gas}\n\nPartial pressures of molecular hydrogen p(\\mbox{H$_2$})\/k = n(\\mbox{H$_2$})\\Tsub K \\ were derived\nby \\cite{LisLuc98} for all of the CO-bearing clouds discussed here, in the\ndirections of the background targets.  They generally fall in the range\n$1-5 \\times 10^{3} \\pccc$ K, typical of thermal pressures in the diffuse\nISM sampled by neutral atomic carbon \\citep{JenTri11} , which should share\nthe same volume. The CO derivation depends on recognizing that, when the\ngas is warm and excitation is strongly sub-thermal, the excitation\ntemperature of the J=1-0 transition depends only on p(\\mbox{H$_2$}) and the optical\ndepth of the transition $\\tau(1-0)$.  In the limit of zero optical depth\nthe excitation temperature of the J=1-0 transition \\Tsub exc (1-0) is only\nproportional to p(\\mbox{H$_2$}), not to either the temperature or density\nindependently, making CO a useful barometer. This first became apparent in\nthe work of \\cite{SmiSte+78} and is illustrated in Fig. 11 of\n\\cite{LisLuc98}: it has persisted over several generations of improved\ncollision cross sections.  Moreover, the excitation contribution from\natomic hydrogen should be small in CO-bearing gas where the molecular\nfraction must be appreciable even if \\mbox{H$_2$}\\ does not dominate the overall\natomic-molecular balance \\citep{Lis07CO}.\n\nFor $\\tau(1-0) \\la 3$ and p(\\mbox{H$_2$})\/k $ \\la 2\\times 10^{4} \\pccc$ K the\nbehaviour seen in Fig. 11 of \\cite{LisLuc98} may be straightforwardly\nparameterized to an accuracy of a few percent as\n\n\\begin{equation}\n\\Tsub exc (1-0)-\\Tsub cmb \\ = 0.303 K \\left[ \\frac{p\\left(\\mbox{H$_2$}\\right)}{10^3 \\pccc~K} \\right]^{1.02} \n  e^{\\tau(1-0)^{0.6}\/2.6}\n\\end{equation}\n\nAs examples of the application of this notion, we note:\n\n\\begin{itemize}\n  \n\\item For a typical line with $\\tau(1-0) = 1.5$ and a Rayleigh-Jeans\n  brightness above the CMB of 1.5 K, \\Tsub exc (1-0) = 5.04 K and p(\\mbox{H$_2$})\/k $=\n  5\\times 10^3 \\pccc$ K or n(\\mbox{H$_2$}) $= 200 \\pccc$ at \\Tsub K \\ = 25 K.\n  \n\\item For the strongest absorption line component toward B0355 (-17.8\n  \\mbox{km\\,s$^{-1}$}), $\\tau(1-0) = 3.1$ and \\Tsub exc (1-0) $\\approx 6$ K, so that p(\\mbox{H$_2$})\/k\n  $\\approx\\ 5\\times 10^3 \\pccc$ K once more.\n  \n\\item The $\\approx$ 4.5 K lines observed at the peak positions in several\n  of the simple fields discussed in Section 3 require p(\\mbox{H$_2$})\/k $\\ga\n  8.5\\times 10^3 \\pccc$ K or $\\ga 15\\times 10^3 \\pccc$ K for $\\tau(1-0)$ =\n  3 or 1, respectively.  Such a heavy over-pressure must be transient.\n  \n\\item The very bright 10-12 K lines seen near B0528+134 and B2200+420\n  require excitation temperatures of 15 K or more and lie somewhat beyond\n  the range where \\mbox{W$_{\\rm CO}$}\\ and N(CO) can be shown to be linearly proportional.\n  They will be discussed separately in a forthcoming publication based on\n  observations of \\hcop, HCN, and CS.\n\n\\end{itemize}\n\n\\subsection{Failing to detect \\mbox{H$_2$}\\ when CO emission is weak}\n\nThere are cases where the brightness of the 1-0 line is well below 1 K even\nwhen the CO optical depth is appreciable, as summarized in Table 3;\nunfortunately we do not have a CO absorption profile toward B1928+738 in\nwhose field CO emission was not detected, see Sect. 3.4.  The regions of\nvery low p(\\mbox{H$_2$}) toward B0212 and B0224 somehow manage to produce\nappreciable amounts of CO without exciting it to detectable levels but\nother lines represented in Table 3 do not arise in regions of especially\nlow pressure.\n\nIn Sect. 2.7 we showed that, on the whole, molecular gas is not\nunderrepresented by CO emission in the collection of sightlines comprising\nthis work and earlier we showed that the same is true of the larger sample\nof absorption-cloud sightlines from which the current sample was drawn\n\\citep{LisPet+10}.  Moreover, CO emission from all of the components\nrepresented in Table 3 is detected (usually quite strongly) elsewhere in\nthe mapped fields except around B1928.  However, the fraction of molecular\ngas that is detectable in CO along individual sightlines varies\nsubstantially.  To quantify this we equate the molecular column density\nwith the integrated optical depth measured in \\hcop\\ \\citep{LucLis96}, see\nthe right-most column in Table 3.  In this case the fraction of molecular\ngas that is missed by failing to detect CO emission from particular\nindividual components in four directions is 12\\% toward B0212, 16\\% toward\nB0224, 8\\% toward B0528+134 and 100\\% toward B1928.  Overall the fraction\nof molecular gas represented by the weakly-emitting CO summarized in Table\n3 is 8\\% toward B0528, 16\\% toward B0224, 22\\% toward B0212, 43\\% toward\nB0355 and 100\\% toward B1730 and B1928.\n\n\\begin{table*}\n\\caption[]{Components with weak CO emission toward the continuum target$^a$}\n{\n\\small\n\\begin{tabular}{lcccccccc}\n\\hline\nTarget & V   & $\\tau(1-0)$ & \\mbox{${\\rm T}_{\\rm R}^*$} & dN(CO)\/dV & \\Tsub exc (1-0) & p(\\mbox{H$_2$}) & n(\\mbox{H$_2$})$^b$ & $\\int \\tau(\\hcop)dv$\/total \\\\ \n       & \\mbox{km\\,s$^{-1}$}  &        &   K  & $10^{15} ~{\\rm cm}^{-2}$ (\\mbox{km\\,s$^{-1}$})$^{-1}$  & K     & $10^3 \\pccc$ K & $\\pccc$ & \\\\\n\\hline\nB0212& -10.3  & 0.49 & 0.40         & 0.65 &  3.4-3.6  & 1.9-2.5  & 75-100 &  0.122\\\\\n     &  -0.05 & 0.95 & $<$ 0.14$^c$ & 1.11 & $<$3.1-3.2   & $<$0.9-1.1  & $<$35-45 & 0.102 \\\\\nB0224 & -10.6 & 0.43 & $<$ 0.10     & 0.48 & $<$3.0-3.1 &  $<$0.8-1.0  & $<$30-40 & 0.161 \\\\\nB0355&  -13.9 & 0.45 & 0.31         & 0.52 & 3.8-4.0 & 3.0-3.5  & 120-140 & 0.224\\\\\n     &   -4.0 & 0.86    & 0.37         & 1.10  & 3.6-3.8 & 2.1-2.5  & 85-100 & 0.204\\\\\nB0528&    2.8 & $<$ 0.11 & $<$ 0.16  & &  & &  & 0.080 \\\\\nB1730&    5.1 & 1.15 & 0.24         & 1.42 & 3.5-3.7 & 1.8-2.2& 70-90 & 1.000\\\\\nB1928&    $-3$ & $<$ 0.11 & $<$ 0.11  & &  & &  & 1.000 \\\\\n\\hline\n\\end{tabular}}\n\\\\\n$^a$ Using CO parameters originally derived by \\cite{LisLuc98} and $\\tau(\\hcop)$ from \\cite{LucLis96} \\\\\n$^b$ At \\Tsub K \\ = 25 K \\\\\n$^c$ Upper limits in this column are $2\\sigma$ \\\\\n\n\\end{table*}\n\n\\section{Discussion}\n\nEven at \\mbox{E$_{\\rm B-V}$}\\ = 0.1 - 0.3 mag, the CO emission traced in this work runs the\nfull gamut from undetectable to having brightness comparable to that seen\nin fully-molecular dark clouds. CO emission may be undetectably weak ($<<$\n1K) when molecular gas is present in absorption (including that of CO\nitself) but in other directions it may be so bright that the N(\\mbox{H$_2$})\/\\mbox{W$_{\\rm CO}$}\\ \nratio is 4-5 times smaller than the typical CO-\\mbox{H$_2$}\\ conversion factor\n$2\\times10^{20}~~{\\rm cm}^{-2}$ (\\emr{\\,K\\,km\\,s^{-1}})$^{-1}$.  Under the conditions encountered in\ndiffuse clouds, CO emission is foremost an indicator of the CO chemistry,\nsecondarily an indicator of the rotational excitation (which reflects the\npartial thermal pressure of \\mbox{H$_2$}) and only peripherally a measure of the\nunderlying hydrogen column density distribution as discussed in Sect. 8.\nIndeed, the simulations of CO emission from the interstellar medium by\n\\citet{shetty11} agree with observations for the dense gas. However, a\ndetailed comparison with our results on the diffuse material shows that the\nradiative transfer factor is correct but there are up to 4 orders of\nmagnitude difference in N(\\mbox{H$_2$})\/N(CO). This is linked to the\npoorly-understood polyatomic chemistry in the diffuse gas~\\citep[see][their\nsection 4.3]{shetty11}.\n\nThe over-arching issues most relevant to diffuse cloud CO emission are\nthree-fold: 1) How it may be identified for what it is, originating in\nrelatively tenuous gas that is unassociated with star formation; 2) Whether\nit makes a substantial contribution when CO emission is used as a surrogate\nfor N(\\mbox{H$_2$}) in circumstances where emission contributions from diffuse and\ndense heavily-shielded gas may be blended; 3) How it is related to the\nso-called ``dark'' gas discovered by and FERMI \\citep{grenier10} and PLANCK\n\\citep{bernard11} that is most prominent at moderate extinction where the\ntransition from atomic to molecular gas occurs and is claimed to host\n50\\%-120\\% of the previously-known CO emitting gas in the solar\nneighbourhood.\n\nAs for the identification of diffuse gas, the \\mbox{W$_{\\rm CO}$}\/\\mbox{W$_{^{13}{\\rm CO}}$}\\ ratio is the most\nacessible and direct probe.  When diffuse cloud CO is excited to detectable\nlevels it is generally in the regime where \\mbox{W$_{\\rm CO}$}\\ $\\propto$ N(CO) and \\mbox{W$_{^{13}{\\rm CO}}$}\\ \n$\\propto$ N(\\mbox{$^{13}$CO}) so that the brightness ratio \\mbox{W$_{\\rm CO}$}\/\\mbox{W$_{^{13}{\\rm CO}}$}\\ will be much\nlarger than the values 3-5 that are seen when emission arises from\noptically thick lines from denser gas where the rotation ladder is close to\nbeing thermalized.  Fractionation progressively lowers the\nN(\\mbox{$^{12}$CO})\/N(\\mbox{$^{13}$CO}) column density ratio in diffuse gas at larger N(CO)\n\\citep{LisLuc98,SheRog+07} but not below about 15.  Intensity ratios\n\\mbox{W$_{\\rm CO}$}\/\\mbox{W$_{^{13}{\\rm CO}}$}\\ of 8-10 or higher are a strong indicator that there is a major\ncontribution from diffuse material.\n\nRegarding the contribution of diffuse gas we recently assessed it in the\ncase where an outside observer looked down on the Milky Way disk in the\nvicinity of the Sun \\citep{LisPet+10}.  We compared the mean emission for\nthe ensemble of lines of sight from which the current background targets\nwere drawn with the vertically-integrated emission expected for the\ngalactic disk component at the Solar Circle drawn from galactic plane\nsurveys.  The ensemble mean in our dataset, expressed as an equivalent to\nlooking vertically through the galactic layer, was 2$<$ \\mbox{W$_{\\rm CO}$}\\ sin($|$b$|$)\n$>$ = 0.47 \\emr{\\,K\\,km\\,s^{-1}}{}.  The galactic disk contribution was inferred from\ngalactic plane surveys that find A(CO) = 5 \\emr{\\,K\\,km\\,s^{-1}}{} (kpc)$^{-1}$ and an\nequivalent disk thickness of 150 pc, implying an integrated intensity\nthrough the disk of 5 \\emr{\\,K\\,km\\,s^{-1}}{} (kpc)$^{-1} \\times 0.15$ kpc = 0.75 \\emr{\\,K\\,km\\,s^{-1}}.\n\nEven if viewed as entirely distinct (because it originates at galactic\nlatitudes well above those typically sampled in galactic plane surveys) the\ndiffuse gas contribution to the total seen looking down on the Milky Way\nfrom outside would be 0.47\/(0.47+0.75) = 38\\%, a surprisingly high fraction\ngiven the supposed absence of molecular gas and CO emission at higher\ngalactic latitudes.  The alternative, that the diffuse gas is already\nincorporated in galactic plane surveys, makes the majority of the gas\n(0.47\/0.75) in the galactic plane diffuse.  This is an even more radical\nproposition, but is consistent with finding that the preponderance of the\nmolecular gas seen toward the heavily-extincted line of sight toward\nB0355+508 at $b=-1.6$\\mbox{$^{\\rm o}$}\\ is actually diffuse.\n \nIn fact, the extent of the diffuse and\/or high-latitude contribution to the\nlocal CO emission remains to be determined by wide-field CO surveys whose\ndetection limit is substantially better than 1 \\emr{\\,K\\,km\\,s^{-1}}\\ and perhaps no worse\nthan even 0.25 \\emr{\\,K\\,km\\,s^{-1}}.  Assessing the contribution of diffuse gas at lower\nlatitudes awaits a wider examination of the character of the gas seen in\nthe galactic disk, but the contribution of diffuse molecular gas in the\ninner galactic disk is apparent in recent HERSCHEL\/PRISMAS observations of\nsub-mm absorption spectra toward star-forming regions\n\\citep{GerdeL+10,SonNeu+10}.\n\n\\section{Summary and conclusions}\n\nWe compared maps of CO emission with reddening maps on a typical field of\nview of about $30'\\times30'$ at an angular resolution of $1'$ toward 11\ndiffuse lines of sight for which we already had sub-arcsec molecular and\/or\natomic absorption profiles. This allowed us to draw three kinds of\nconclusions.\n\n\\subsection{ Conclusions about the position-position-velocity\n  structure of the emission}\n\\begin{itemize}\n  \n\\item Although most of the CO emission structure was amorphous or merely\n  blob-like when mapped, the emission around B0528+138 was found to be\n  highly regular and quasi-periodic while that around B2200+420 (aka BL Lac)\n  was seen to be filamentary and tangled.\n  \n\\item Toward B0355+508 and B0528+134, CO mapping suggests that pairs of\n  absorption lines separated by 6-8 \\mbox{km\\,s$^{-1}$}\\ are physically related, not\n  merely accidental superpositions.\n  \n\\item CO mapping shows that partition of an absorption profile into\n  kinematic components, no matter how seemingly obvious, may actually be\n  arbitrary and capricious: the decomposition may have no apparent validity\n  in emission at positions only slightly removed from the continuum\n  background.\n\n\\end{itemize}\n\n\\subsection{Conclusions linking the absorption to the\n  emission kinematics:}\n\\begin{itemize}\n  \n\\item The same clouds were seen in absorption and emission, and in atomic\n  and molecular phases, although not necessarily in the same location.  We\n  failed to find CO emission corresponding to just one out of 20 molecular\n  absorption features, in one relatively small spatial field, i.e\n  20\\arcmin\\ $\\times$ 20\\arcmin\\ vs. 30\\arcmin\\ $\\times$ 30\\arcmin\\ or\n  more.  Conversely, while mapping away from the continuum background we\n  saw only 2 CO emission features lacking molecular absorption\n  counterparts.\n  \n\\item CO emission was sometimes found in the field at velocities\n  corresponding to features seen only in H I absorption toward the\n  continuum.  We saw no molecular features outside the span of the H I\n  absorption.\n  \n\\end{itemize}\n\n\\subsection{Conclusions regarding the CO luminosity of diffuse gas.}\n\\begin{itemize}\n  \n\\item We found relatively bright CO emission at modest reddening in the\n  fields we mapped, with peak brightnesses of 4-5 K at \\mbox{E$_{\\rm B-V}$}\\ $\\la 0.15$ mag\n  and up to 10-12 K at \\mbox{E$_{\\rm B-V}$}\\ $\\simeq$ 0.3 mag (i.e \\mbox{A$_{\\rm V}$} $\\simeq 1$ mag).\n  This was true even for features that were seen only in absorption toward\n  the continuum source in the field center.\n  \n\\item The CO emission lines represent small column densities N(CO) $\\leq\n  10^{16} ~{\\rm cm}^{-2}$, less than 10\\% of the amount of free gas phase carbon\n  expected along a line of sight with \\mbox{E$_{\\rm B-V}$}\\ = 0.15 mag or \\mbox{A$_{\\rm V}$}\\ = 0.5 mag.\n  The dominant form of gas phase carbon is still C\\mbox{$^+$}.\n  \n\\item When CO emission was detected at levels of 1.5 \\emr{\\,K\\,km\\,s^{-1}}{} and higher, it\n  was generally over-luminous in the sense of having a small ratio\n  N(\\mbox{H$_2$})\/\\mbox{W$_{\\rm CO}$}, i.e.  a value of the CO-\\mbox{H$_2$}\\ conversion factor below\n  $2\\times 10^{20}$\\mbox{H$_2$}\\ (\\emr{\\,K\\,km\\,s^{-1}})$^{-1}$.  \\mbox{W$_{\\rm CO}$}\/N(\\mbox{H$_2$}) ratios as small as\n  N(\\mbox{H$_2$})\/\\mbox{W$_{\\rm CO}$}\\ $\\la 5\\times 10^{19} ~{\\rm cm}^{-2}$ \\mbox{H$_2$}\\ (\\mbox{km\\,s$^{-1}$})$^{-1}$ are mandated\n  by the observed reddening in cases where the line of sight was relatively\n  free of extraneous material.\n  \n\\item On the whole, the \\mbox{W$_{\\rm CO}$}\/N(\\mbox{H$_2$}) ratio in diffuse gas is the same as in\n  dense fully molecular clouds despite the presence of strong variations\n  between individual diffuse gas parcels or sightlines.  The global\n  \\mbox{W$_{\\rm CO}$}\/N(\\mbox{H$_2$}) ratio in diffuse gas is the result of averaging over limited\n  regions where CO emission is readily detectable and overly bright (in the\n  sense of having \\mbox{W$_{\\rm CO}$}\/N(\\mbox{H$_2$}) much higher than the mean), and with other\n  regions having a significant molecular component (as seen in absorption)\n  but where CO emission is comparatively weak or simply undetectable.\n  \n\\item Small \\mbox{W$_{\\rm CO}$}\/N(\\mbox{H$_2$}) ratios and sharp variations in the \\mbox{W$_{\\rm CO}$}\/\\mbox{E$_{\\rm B-V}$}\\ ratio\n  are not artifacts of the disparity in resolution between the 1\\arcmin\\ CO\n  emission beam and the 6\\arcmin\\ resolution of the reddening maps, because\n  high CO brightnesses and small \\mbox{W$_{\\rm CO}$}\/\\mbox{E$_{\\rm B-V}$}\\ ratios persist when the\n  resolution of the CO maps is degraded to that of the reddening maps.\n  \n\\item Sharp variations in the CO emission brightness on arcminute scales do\n  not necessarily represent unresolved structure in the reddening maps or\n  in the column density of H or \\mbox{H$_2$}.  Detectable CO emission generally\n  arises in the regime where \\mbox{W$_{\\rm CO}$}\\ $\\propto$ N(CO), and variations in the\n  line brightness represent primarily the CO chemistry with its extreme\n  sensitivity to \\mbox{E$_{\\rm B-V}$}\\ and N(\\mbox{H$_2$}). Secondarily the line brightness is\n  influenced by CO rotational excitation since some features are not seen\n  in emission toward continuum sources where there is CO absorption with\n  appreciable optical depth.\n  \n\\item Only peripherally does the CO brightness represent the underlying\n  mass or \\mbox{H$_2$}\\ column density distribution of diffuse molecular gas.\n\n\\end{itemize}\n\n\\begin{acknowledgements}\n  The National Radio Astronomy Observatory is operated by Associated\n  Universites, Inc.  under a cooperative agreement with the US National\n  Science Foundation.  The Kitt Peak 12-m millimetre wave telescope is\n  operated by the Arizona Radio Observatory (ARO), Steward Observatory,\n  University of Arizona.  IRAM is operated by CNRS (France), the MPG\n  (Germany) and the IGN (Spain).  This work has been partially funded by\n  the grant ANR-09-BLAN-0231-01 from the French {\\it Agence Nationale de la\n    Recherche} as part of the SCHISM project (http:\/\/schism.ens.fr\/).  We\n  thank Edith Falgarone for comments that inspired Sections 8.4 and 8.5 of\n  this work.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDiscontinuous Galerkin methods are a popular approach to discretize PDE\nboundary value problems \\cite{dipietro_ern_2012a}. Similar to conforming finite element methods\n\\cite{ciarlet_2002a,ern_guermond_2019a}, they can handle general meshes, which allows to account for complicated\ngeometries. In addition, the use of discontinuous polynomial shape functions\nfacilitates the presence of hanging nodes and varying polynomial degree.\nAs a result, discontinuous Galerkin methods are especially suited for $hp$-adaptivity\n\\cite{congreve_gedicke_perugia_2019a,houston_schotzau_wihler_2006a}.\nFurthermore, since all their degrees of freedom are attached with mesh cells,\ndiscontinuous Galerkin methods allow linear memory access, which is crucial for\nefficient computer implementations, in particular on GPUs\n\\cite{chan_wang_modave_remacle_warburton_2016a}.\n\nIn the context of wave propagation problems, discontinuous Galerkin methods\nfurther display specific advantages. For time-dependent problems, the resulting\nmass-matrix is block-diagonal \\cite{agut_diaz_2013a,grote_schneebeli_schotzau_2006a},\nwhich enables explicit time-stepping schemes without mass-lumping\n\\cite{cohen_joly_roberts_tordjman_2001a}. For time-harmonic wave propagation too,\ndiscontinuous Galerkin formulations are attractive as they exhibit additional stability\nas compared to conforming alternatives\n\\cite{bernkopf_sauter_torres_veit_2022a,feng_wu_2009a,feng_wu_2011a}.\nThese interesting properties have henceforth motivated a large number of works\nconsidering discontinuous Galerkin discretizations of wave propagation problems,\nand a non-exhaustive list includes \\cite{agut_diaz_2013a,congreve_gedicke_perugia_2019a,%\ndu_zhu_2016a,feng_wu_2009a,feng_wu_2011a,grote_schneebeli_schotzau_2006a,hoppe_sharma_2013a,%\nsauter_zech_2015a}.\n\nIn this work, we consider the acoustic Helmholtz equation, which is probably the\nsimplest model problem relevant to the difficulties of wave propagation.\nSpecifically, given a domain $\\Omega \\subset \\mathbb R^d$, $d=2$ or $3$, and $f: \\Omega \\to \\mathbb C$,\nthe unknown $u: \\Omega \\to \\mathbb C$ should satisfy\n\\begin{equation}\n\\label{eq_helmholtz_strong}\n\\left \\{\n\\begin{array}{rcll}\n-\\omega^2 \\mu u-\\div \\left (\\boldsymbol A\\boldsymbol \\nabla u\\right ) &=& f & \\text{ in } \\Omega,\n\\\\\nu &=& 0 & \\text{ on } \\Gamma_{\\rm D},\n\\\\\n\\boldsymbol A\\boldsymbol \\nabla u \\cdot \\boldsymbol n &=& 0 & \\text{ on } \\Gamma_{\\rm N},\n\\\\\n\\boldsymbol A\\boldsymbol \\nabla u \\cdot \\boldsymbol n - i\\omega\\gamma u &=& 0 & \\text{ on } \\Gamma_{\\rm R},\n\\end{array}\n\\right .\n\\end{equation}\nwhere $\\overline{\\Gamma_{\\rm D}} \\cup \\overline{\\Gamma_{\\rm N}} \\cup \\overline{\\Gamma_{\\rm R}} = \\partial \\Omega$\nis a partition of the boundary and $\\mu,\\boldsymbol A$ and $\\gamma$ are given coefficients.\nPrecise assumptions are listed in Section \\ref{section_setting}.\n\nOur interest lies in interior penalty discontinuous Galerkin (IPDG) discretizations\nof \\eqref{eq_helmholtz_strong}. In particular, we focus on the ``minimal regularity''\ncase, where we do not assume any specific smoothness for the solution $u$. To the best\nof our knowledge, this problem has not been considered in the literature, and available\nworks essentially assumes that the solution belongs to $H^{3\/2+\\varepsilon}$, so that the\ntraces of $\\boldsymbol \\nabla u$ are well-defined on mesh faces, see e.g. \\cite[Eq. (4.5)]{hoppe_sharma_2013a}\nand \\cite[Lemma 2.6]{sauter_zech_2015a}. Unfortunately, such assumptions rule out important\nconfigurations of coefficients and boundary conditions, which may bring the regularity of the\nsolution arbitrarily close to $H^1$ (see the appendix of \\cite{costabel_dauge_nicaise_1999a}\nfor instance).\n\nWhen considering conforming finite element discretizations of \\eqref{eq_helmholtz_strong},\nthe so-called ``Schatz argument'' enables to show that the discrete solution $u_h$ is\nquasi-optimal if the mesh is fine enough \\cite{schatz_1974a}. Assuming for simplicity that\n$\\Gamma_{\\rm R} = \\emptyset$ and introducing the energy norm\n\\begin{equation*}\n\\enorm{v}_{\\omega,\\Omega}^2\n:=\n\\omega^2 \\|v\\|_{\\mu,\\Omega}^2 + \\|\\boldsymbol \\nabla v\\|_{\\boldsymbol A,\\Omega}^2\n\\quad\nv \\in H^1_{\\Gamma_{\\rm D}}(\\Omega),\n\\end{equation*}\nwe have\n\\begin{equation}\n\\label{eq_intro_apriori}\n\\enorm{u-u_h}_{\\omega,\\Omega}\n\\leq\n\\frac{1}{1-\\gamma_{\\rm ba}^2} \\min_{v_h \\in V_h} \\enorm{u-v_h}_{\\omega,\\Omega},\n\\end{equation}\nwhenever the approximation factor\n\\begin{equation}\n\\label{eq_intro_gba}\n\\gamma_{\\rm ba} := 2\\omega \\max_{\\substack{\\psi \\in L^2(\\Omega) \\\\ \\|\\psi\\|_{\\mu,\\Omega} = 1}}\n\\min_{v_h \\in V_h} \\enorm{u_\\psi-v}_{\\omega,\\Omega}\n\\end{equation}\nis strictly less than one (all these notations are detailed in Section \\ref{section_preliminary}\nbelow). Above, $u_\\psi$ solves \\eqref{eq_helmholtz_strong}\nwith right-hand side $\\mu\\psi$ instead of $f$. Similarly, when considering a posteriori\nerror estimation \\cite{chaumontfrelet_ern_vohralik_2021a}, we have\n\\begin{equation}\n\\label{eq_intro_aposteriori}\n\\enorm{u-u_h}_{\\omega,\\Omega} \\leq \\sqrt{1+\\gamma_{\\rm ba}^2} \\eta,\n\\end{equation}\nwhere $\\eta$ is a suitable a posteriori estimator.\n\nEstimates similar to \\eqref{eq_intro_apriori} and \\eqref{eq_intro_aposteriori}\nare available for IPDG discretizations \\cite{melenk_parsania_sauter_2013a,sauter_zech_2015a},\nbut with energy norms invovling the normal trace of the gradient on faces, thus essentially\nrequiring $H^{3\/2+\\varepsilon}$ regularity of $u_\\psi$. Here, in contrast, we extend\n\\eqref{eq_intro_apriori} and \\eqref{eq_intro_aposteriori} to IPDG discretizations without\nadditional regularity assumptions on the solutions $u_\\psi$. As detailed below, our key\nfinding is that it can be achieved by redefining the approximation factor as\n\\begin{equation}\n\\label{eq_intro_gba_IPDG}\n\\gamma_{\\rm ba}^2\n:=\n4\\omega^2\n\\max_{\\substack{\\psi \\in L^2(\\Omega) \\\\ \\|\\psi\\|_{\\mu,\\Omega} = 1}}\n\\left (\n\\min_{v_h \\in V_h} \\enorm{u_\\psi-v_h}_{1,\\dagger,\\mathcal T_h}^2\n+\n\\min_{\\boldsymbol w_h \\in \\boldsymbol W_h} \\enorm{\\boldsymbol A\\boldsymbol \\nabla u_\\psi-\\boldsymbol w_h}_{\\operatorname{div},\\dagger,\\mathcal T_h}^2\n\\right ),\n\\end{equation}\nwhere $\\boldsymbol W_h$ is the BDM finite element space built using the same mesh and polynomial degree\nthan $V_h$, and $\\enorm{{\\cdot}}_{1,\\dagger,\\mathcal T_h}$ and $\\enorm{{\\cdot}}_{\\operatorname{div},\\dagger,\\mathcal T_h}$\nare $H^1_{\\Gamma_{\\rm D}}(\\Omega)$ and $\\boldsymbol H_{\\Gamma_{\\rm N}}(\\operatorname{div},\\Omega)$ norms appropriately\nscaled by the mesh size. The additional term in \\eqref{eq_intro_gba_IPDG}\nas compared to \\eqref{eq_intro_gba} is necessary to account for the non-conformity\nof the scheme.\n\nActually, the interest of the subject exceeds time-harmonic wave propagation.\nIn fact, the a priori and a posteriori error analysis of finite element discretizations\nto \\eqref{eq_helmholtz_strong} rely on duality arguments of Aubin-Nitsche type\n\\cite{schatz_1974a}. Such techniques are crucial in time-harmonic wave propagation\n\\cite{melenk_sauter_2010a,sauter_zech_2015a}, but there also useful in other contexts\nto establish convergence in weak norms, see e.g.\n\\cite[Section 5.1]{arnold_brezzi_cockburn_marini_2002a}.\nTo the best of our knowledge, duality analysis for IPDG discretizations under\nminimal regularity has not been addressed in the literature, and it is our goal\nto do so here.\n\n\nThe remainder of this work is organized as follows. In Section \\ref{section_preliminary},\nwe precise the setting and introduce key notations. Section \\ref{section_duality}\npresents the key argument that enables duality techniques for IPDG under minimal regularity.\nSections \\ref{section_apriori} and \\ref{section_aposteriori} then employ the aforementioned\nreasoning to perform the a priori and a posteriori error analysis of IPDG discretization for\nHelmholtz problems.\n\n\\section{Setting and preliminary results}\n\\label{section_preliminary}\n\n\\subsection{Setting}\n\\label{section_setting}\n\nThroughout this work, $\\Omega \\subset \\mathbb R^d$, with $d=2$ or $3$, is a Lipschitz\npolytopal domain. The boundary of $\\Omega$ is partitioned into three open, Lipschitz\nand disjoint polytopal subsets $\\Gamma_{\\rm D}$, $\\Gamma_{\\rm N}$ and $\\Gamma_{\\rm R}$ such that\n$\\partial \\Omega = \\overline{\\Gamma_{\\rm D}} \\cup \\overline{\\Gamma_{\\rm N}} \\cup \\overline{\\Gamma_{\\rm R}}$.\nWe employ the notation $\\boldsymbol n$ for the unit vector normal to $\\partial \\Omega$\npointing outside $\\Omega$.\n\nWe consider coefficients $\\mu: \\Omega \\to \\mathbb R$, $\\boldsymbol A: \\Omega \\to \\mathbb R^{d \\times d}$\nand $\\gamma: \\Gamma_{\\rm R} \\to \\mathbb R$ satisfying the following properties. We assume that there\nexists a partition $\\mathscr P$ of $\\Omega$ into a finite number disjoint open polytopal\nsubdomains such that $\\mu|_P = \\mu_P \\in \\mathbb R$ and $\\boldsymbol A|_P = \\boldsymbol A_P \\in \\mathbb R^{d \\times d}$ take constant\nvalues for all $P \\in \\mathscr P$. Similarly, there exists a finite partition $\\mathscr Q$ of $\\Gamma_{\\rm R}$\nconsisting of open polytopal subsets such that $\\gamma|_Q = \\gamma_Q \\in \\mathbb R$ is constant\nfor each $Q \\in \\mathscr Q$. For each $P \\in \\mathscr P$, we assume that $\\boldsymbol A_P$ is symmetric, and let\n$\\alpha_P := \\min_{\\boldsymbol \\xi \\in \\mathbb R^d; |\\boldsymbol \\xi| = 1} \\boldsymbol A_P \\boldsymbol \\xi \\cdot \\boldsymbol \\xi$. We then classically\nrequire that $\\min_{P \\in \\mathscr P} \\mu_P > 0$, $\\min_{P \\in \\mathscr P} \\alpha_P > 0$, and\n$\\min_{Q \\in \\mathscr Q} \\gamma_Q > 0$.\n\nWe also fix a real number $\\omega > 0$ representing the (angular) frequency.\n\n\\subsection{Functional spaces}\n\nIf $D \\subset \\mathbb R^d$ is an open set, we denote by $L^2(D)$ the Lebesgue space\nof complex-valued square integrable functions defined over $D$, and we set\n$\\boldsymbol L^2(D) := [L^2(D)]^d$ for vector-valued functions. The notations $(\\cdot,\\cdot)_D$\nand $\\|{\\cdot}\\|_D$ then stand for the usual inner-product and norm of $L^2(D)$ or $\\boldsymbol L^2(D)$.\nIn addition, if $w: D \\to \\mathbb R$ is a measurable function satisfying\n$0 < \\operatorname{ess} \\inf_D w$ and $\\operatorname{ess} \\sup_D w < +\\infty$,\nthen $\\|{\\cdot}\\|_{w,D}^2 := (w \\cdot,\\cdot)_D$ defines a norm on $L^2(D)$ equivalent\nto the standard one. We use the same notation in $\\boldsymbol L^2(D)$ with matrix-valued weights.\nBesides, we employ similar notations for $d-1$ manifolds.\n\n$H^1(D)$ is the Sobolev space of functions $v \\in L^2(D)$ such that $\\boldsymbol \\nabla v \\in \\boldsymbol L^2(D)$,\nwhere $\\boldsymbol \\nabla$ denotes the weak gradient defined in the sense of distributions. If\n$\\Gamma \\subset \\partial D$ is a relatively open subset of the boundary of $D$, then\n$H^1_\\Gamma(D)$ stands for the subset of functions $v \\in H^1(D)$ such that\n$v|_{\\Gamma} = 0$ in the sense of traces. We refer the reader to \\cite{adams_fournier_2003a}\nfor a detailed description of the above spaces. On $H^1_{\\Gamma_{\\rm D}}(\\Omega)$, we will often employ\nthe following ``energy'' norm:\n\\begin{equation}\n\\label{eq_energy_norm}\n\\enorm{v}_{\\omega,\\Omega}^2\n:=\n\\omega^2\\|v\\|_{\\mu,\\Omega}^2 + \\omega\\|v\\|_{\\gamma,\\Gamma_{\\rm R}}^2 + \\|\\boldsymbol \\nabla v\\|_{\\boldsymbol A,\\Omega}^2\n\\quad\n\\forall v \\in H^1_{\\Gamma_{\\rm D}}(\\Omega).\n\\end{equation}\n\nIt turns out that Sobolev spaces of vector-valued functions will be key in the duality\nanalysis we are about to perform. Specifically, we will need the space $\\boldsymbol H(\\operatorname{div},D)$\nof functions $\\boldsymbol v \\in \\boldsymbol L^2(D)$ such that $\\div \\boldsymbol v \\in L^2(D)$ where $\\div$ is the weak\ndivergence operator \\cite{girault_raviart_1986a}. Following, e.g., \\cite{fernandes_gilardi_1997a},\nif $\\Gamma \\subset \\partial D$ is a relatively open set, the normal trace $(\\boldsymbol w \\cdot \\boldsymbol n)|_\\Gamma$\nof $\\boldsymbol w \\in \\boldsymbol H(\\operatorname{div},D)$ can be defined in a weak sense, and $\\boldsymbol H_\\Gamma(\\operatorname{div},D)$\nwill stand for the space of $\\boldsymbol w \\in \\boldsymbol H(\\operatorname{div},D)$ such that $(\\boldsymbol w \\cdot \\boldsymbol n)|_\\Gamma = 0$.\n\nWe are studying ``Robin-type'' boundary conditions which are not naturally handled in\nthe $\\boldsymbol H(\\operatorname{div})$ setting. We follow the standard remedy \\cite{chaumontfrelet_2019a,monk_2003a}\nand introduce the space\n\\begin{equation*}\n\\boldsymbol X(\\operatorname{div},\\Omega)\n:=\n\\left \\{\n\\boldsymbol w \\in \\boldsymbol H(\\operatorname{div},\\Omega) \\; | \\; (\\boldsymbol w \\cdot \\boldsymbol n)|_{\\Gamma_{\\rm R}} \\in L^2(\\Gamma_{\\rm R})\n\\right \\},\n\\end{equation*}\nwhere additional normal trace regularity is enforced on the Robin boundary.\nThe notation $\\boldsymbol X_{\\Gamma_{\\rm N}}(\\operatorname{div},\\Omega) := \\boldsymbol X(\\operatorname{div},\\Omega) \\cap \\boldsymbol H_{\\Gamma_{\\rm N}}(\\operatorname{div},\\Omega)$\nwill also be useful.\n\n\\subsection{Helmholtz problem}\n\nCentral to our considerations will be the sesquilinear form\n\\begin{equation}\n\\label{eq_b}\nb(\\phi,v)\n:=\n-\\omega^2 (\\mu\\phi,v)_\\Omega\n-i\\omega(\\gamma\\phi,v)_{\\Gamma_{\\rm R}}\n+(\\boldsymbol A\\boldsymbol \\nabla \\phi,\\boldsymbol \\nabla v)_\\Omega\n\\qquad\n\\forall \\phi,v \\in H^1_{\\Gamma_{\\rm D}}(\\Omega),\n\\end{equation}\ncorresponding to the weak formulation of \\eqref{eq_helmholtz_strong}.\nWe will assume throughout this work that the considered Helmholtz\nproblem is well-posed, meaning that there exists $\\gamma_{\\rm st} > 0$ such that\n\\begin{equation}\n\\label{eq_well_posed}\n\\min_{\\substack{\\phi \\in H^1_{\\Gamma_{\\rm D}}(\\Omega) \\\\ \\enorm{\\phi}_{\\omega,\\Omega} = 1}}\n\\max_{\\substack{ v   \\in H^1_{\\Gamma_{\\rm D}}(\\Omega) \\\\ \\enorm{ v  }_{\\omega,\\Omega} = 1}}\n\\Re b(\\phi,v)\n=\n\\frac{1}{\\gamma_{\\rm st}}.\n\\end{equation}\nIn our setting, \\eqref{eq_well_posed} always holds as soon as $\\Gamma_{\\rm R}$ has a positive\nmeasure due to the unique continuation principle. On the other hand, if $\\Gamma_{\\rm R} = \\emptyset$,\nthen \\eqref{eq_well_posed} fails to hold if and only if $\\omega^2$ is an eigenvalue of the\nresulting self-adjoint operator. In general, it is hard to quantitatively estimate $\\gamma_{\\rm st}$, but a\nreasonable assumption is that it grows polynomially with the frequency\n\\cite{lafontaine_spence_wunsch_2021a}. It is also worth mentioning that the\nconstant may be explicitly controlled in some specific configurations\n\\cite{%\nbarucq_chaumontfrelet_gout_2017a,%\nchandlerwilde_spence_gibbs_smyshlyaev_2020a,%\nchaumontfrelet_spence_2021a,%\nmoiola_spence_2019a}.\n\nIn the remainder of this work, we fix a right-hand side $f \\in L^2(\\Omega)$,\nand let $u \\in H^1_{\\Gamma_{\\rm D}}(\\Omega)$ be the unique element satisfying\n\\begin{equation}\n\\label{eq_helmholtz_weak}\nb(u,v) = (f,v)_\\Omega \\quad \\forall v \\in H^1_{\\Gamma_{\\rm D}}(\\Omega).\n\\end{equation}\n\n\n\\subsection{Computational mesh}\n\nWe consider a mesh $\\mathcal T_h$ of $\\Omega$ consisting of non-overlapping (closed) simplicial\nelements $K$. We classically assume that $\\mathcal T_h$ is a matching mesh meaning that\nthe intersection of two distinct elements either is empty, or it is a full face,\nedge or vertex of the two elements (see, e.g. \\cite[Section 2.2]{ciarlet_2002a} or\n\\cite[Definition 6.11]{ern_guermond_2019a}).\nThe set of faces of the mesh is denoted by $\\mathcal F_h$. We also employ the standard notations\n$h_K$ and $\\rho_K$ for the diameter of $K \\in \\mathcal T_h$ and the diameter of the largest ball\ncontained in $K$ (see \\cite[Theorem 3.1.3]{ciarlet_2002a} or\n\\cite[Definition 6.4]{ern_guermond_2019a}).\nThen, $\\kappa_K := h_K\/\\rho_K$ is the shape-regularity parameter of $K \\in \\mathcal T_h$, and\n$\\kappa := \\max_{K \\in \\mathcal T_h} \\kappa_K$. We also introduce the global mesh size\n$h := \\max_{K \\in \\mathcal T_h} h_K$. Similarly, $h_F$ stands for the diameter of the face\n$F \\in \\mathcal F_h$.\n\nWe further assume that the mesh $\\mathcal T_h$ conforms with the partition of the boundary\nand coefficients in the sense that, for each $K \\in \\mathcal T_h$, there exists a (unique)\n$P \\in \\mathscr P$ such that $K \\subset \\overline{P}$ and for each $F \\in \\mathcal F_h$ with\n$F \\subset \\partial \\Omega$, we have either $F \\subset \\overline{\\Gamma_{\\rm D}}$,\n$F \\subset \\overline{\\Gamma_{\\rm N}}$ or $F \\subset \\overline{\\Gamma_{\\rm R}}$.\nIf $F \\subset \\overline{\\Gamma_{\\rm R}}$, we additionally require that $F \\subset \\overline{Q}$\nfor some (unique) $Q \\in \\mathscr Q$. We respectively denote by $\\mathcal F_h^{\\rm D}$,\n$\\mathcal F_h^{\\rm N}$ and $\\mathcal F_h^{\\rm R}$ the set of faces $F \\in \\mathcal F_h$ such that\n$F \\subset \\overline{\\Gamma_{\\rm D}}$, $\\overline{\\Gamma_{\\rm N}}$ or $\\overline{\\Gamma_{\\rm R}}$. We also set\n$\\mathcal F_h^{\\rm e} := \\mathcal F_h^{\\rm D} \\cup \\mathcal F_h^{\\rm N} \\cup \\mathcal F_h^{\\rm R}$,\nand $\\mathcal F^{\\rm i} := \\mathcal F_h \\setminus \\mathcal F_h^{\\rm e}$.\nFor $K \\in \\mathcal T_h$, we can then set $\\mu_K := \\mu_P$, $\\alpha_K := \\alpha_P$, and\n$\\vartheta_K := \\sqrt{\\alpha_K\/\\mu_K}$, where $P \\in \\mathscr P$ contains $K$.  Similarly, if\n$F \\in \\mathcal F_h^{\\rm e}$, we set $\\alpha_F := \\alpha_K$ where $K \\in \\mathcal T_h$ is the\nonly element having $F$ as a face. If $F \\in \\mathcal F_h^{\\rm R}$, we also introduce\n$\\gamma_F := \\gamma_Q$ and $\\vartheta_F := \\alpha_F\/\\gamma_F$ where $Q$ is the unique\nsubset in $\\mathscr Q$ containing $F$.\n\nFinally, we associate with each $F \\in \\mathcal F_h$ a unit normal vector $\\boldsymbol n_F$. If\n$F \\in \\mathcal F_h^{\\rm e}$, we require that $\\boldsymbol n_F = \\boldsymbol n$. Otherwise, the orientation\nof $\\boldsymbol n_F$ is arbitrary, but fixed.\n\n\\subsection{Polynomial spaces}\n\nIn the remainder of this work, we fix a polynomial degree $p \\geq 1$.\nFor $K \\in \\mathcal T_h$, $\\mathcal P_p(K)$ is the set of polynomial functions $K \\to \\mathbb C$\nof total degree less than or equal to $q$. For vector-valued functions, we also set\n$\\boldsymbol{\\CP}_p(K) := [\\mathcal P_p(K)]^3$. If $\\mathcal T \\subset \\mathcal T_h$ is a collection of elements\ncovering the (open) domain $\\Theta$, then\n$\\mathcal P_p(K) := \\{ v \\in L^2(\\Theta); \\; v|_K \\in \\mathcal P_p(K) \\; \\forall K \\in \\mathcal T\\}$\nand $\\boldsymbol{\\CP}_p(\\mathcal T) := [\\mathcal P_p(\\mathcal T)]^3$.\n\n\\subsection{Broken Sobolev spaces}\n\nWe define $H^1(\\mathcal T_h)$ as the\nsubset of functions $v \\in L^2(\\Omega)$ such that $v|_K \\in H^1(K)$ for all $K \\in \\mathcal T_h$.\nFor functions in $H^1(\\mathcal T_h)$, we still employ the notations $\\boldsymbol \\nabla$ for the element-wise\nweak gradient, so that $\\boldsymbol \\nabla (H^1(\\mathcal T_h)) \\subset \\boldsymbol L^2(\\Omega)$. To avoid confusions, we\nwill employ the alternative notations $(\\cdot,\\cdot)_{\\mathcal T_h}$ and $\\|{\\cdot}\\|_{\\mathcal T_h}$ for\nthe $L^2(\\Omega)$ and $\\boldsymbol L^2(\\Omega)$ norms and inner-products when working with broken functions\n(we also employ the same notation for weighted norms). Similarly, we write\n\\begin{equation*}\n\\|v\\|_{\\gamma,\\mathcal F_h^{\\rm R}}^2 := \\sum_{F \\in \\mathcal F_h^{\\rm R}} \\|v\\|_{\\gamma,F}^2\n\\quad\n\\forall v \\in L^2(\\Gamma_{\\rm R}).\n\\end{equation*}\nThe notation\n\\begin{equation*}\n(\\phi,v)_{\\mathcal F_h} := \\sum_{F \\in \\mathcal F_h} (\\phi,v)_F\n\\qquad\n\\forall \\phi,v \\in \\bigoplus_{F \\in \\mathcal F_h} L^2(F)\n\\end{equation*}\nwill also be useful.\n\n\\subsection{Jumps, averages, lifting and discrete gradient}\n\nConsidering $\\phi \\in H^1(\\mathcal T_h)$, its jump through a face $F \\in \\mathcal F_h^{\\rm i}$\nis defined by\n\\begin{equation*}\n\\jmp{\\phi}_F := \\phi_+|_F \\boldsymbol n_+ \\cdot \\boldsymbol n_F + \\phi_-|_F \\boldsymbol n_- \\cdot \\boldsymbol n_F\n\\end{equation*}\nwith $K_\\pm \\in \\mathcal T_h$ the two elements such that $F = \\partial K_+ \\cap \\partial K_-$,\n$\\phi|_\\pm := \\phi|_{K_\\pm}$ and $\\boldsymbol n_\\pm := \\boldsymbol n_{K_\\pm}$.\nFor an exterior face $F \\in \\mathcal F_h^{\\rm e}$, we set instead\n\\begin{equation*}\n\\jmp{\\phi}_F := \\phi|_F \\text{ if } F \\in \\mathcal F_h^{\\rm D}\n\\quad\n\\text{ and }\n\\quad\n\\jmp{\\phi}_F := 0 \\text{ otherwise.}\n\\end{equation*}\nSimilarly, if $\\boldsymbol w \\in \\boldsymbol{\\CP}_p(\\mathcal T_h)$, its average on $F \\in \\mathcal F_h$ is given by\n\\begin{equation*}\n\\avg{\\boldsymbol w}_F := \\frac{1}{2} (\\boldsymbol w_+|_F + \\boldsymbol w_-|_F) \\quad \\text{ if } F \\in \\mathcal F_h^{\\rm i}\n\\quad \\text{ and } \\quad\n\\avg{\\boldsymbol w}_F := \\boldsymbol w|_F \\quad \\text{ if } F \\in \\mathcal F_h^{\\rm e},\n\\end{equation*}\nwhere $\\boldsymbol w_\\pm := \\boldsymbol w|_{K_\\pm}$ for the two elements $K_\\pm \\in \\mathcal T_h$ such that\n$F = K_- \\cap K_+$ in the case of an interior face.\n\nA key part of our analysis will be to give a meaning to the terms\n$(\\jmp{\\phi},\\avg{\\boldsymbol A\\boldsymbol \\nabla v} \\cdot \\boldsymbol n_F)_{\\mathcal F_h}$ appearing in the\nIPDG form, for functions $\\phi$ and $v$ only belonging to $H^1(\\mathcal T_h)$.\nThis is subtle, since the normal trace of $\\boldsymbol \\nabla v$ is actually not\ndefined on faces. Following \\cite[Section 4.3]{dipietro_ern_2012a},\nthe solution is to introduce a lifting operator defined as follows.\n\nFor $\\phi \\in H^1(\\mathcal T_h)$ we define the lifting $\\LIFT{\\phi}$ as the\nunique element of $\\boldsymbol{\\CP}_p(\\mathcal T_h)$ such that\n\\begin{equation*}\n(\\LIFT{\\phi},\\boldsymbol w_h)_{\\mathcal T_h}\n=\n(\\jmp{\\phi}_F,\\avg{\\boldsymbol w_h} \\cdot \\boldsymbol n_F)_{\\mathcal F_h}\n\\qquad\n\\forall \\boldsymbol w_h \\in \\boldsymbol{\\CP}_p(\\mathcal T_h).\n\\end{equation*}\nNotice that we can then write $(\\boldsymbol A\\LIFT{\\phi},\\boldsymbol \\nabla v)_{\\mathcal T_h}$\ninstead of $(\\jmp{\\phi},\\avg{\\boldsymbol A\\boldsymbol \\nabla v}\\cdot\\boldsymbol n_F)_{\\mathcal F_h}$\nfor functions $\\phi, v \\in \\mathcal P_p(\\mathcal T_h)$, with the advantage that\nthe second expression is still well-defined for general $H^1(\\mathcal T_h)$ arguments.\n\nThe notion of weak gradient will also be useful. Specifically, we set\n\\begin{equation*}\n\\mathfrak G(\\phi) := \\boldsymbol \\nabla \\phi - \\LIFT{\\phi} \\in \\boldsymbol L^2(\\mathcal T_h)\n\\end{equation*}\nfor all $\\phi \\in H^1(\\mathcal T_h)$. Importantly, we have $\\mathfrak G(\\phi) = \\boldsymbol \\nabla \\phi$\nwhenever $\\phi \\in H^1_{\\Gamma_{\\rm D}}(\\Omega)$, and\n\\begin{equation*}\n(\\mathfrak G(\\phi),\\boldsymbol w_h)_{\\mathcal T_h} + (\\phi,\\div \\boldsymbol w_h)_{\\mathcal T_h} = (\\phi,\\boldsymbol w_h \\cdot \\boldsymbol n)_{\\Gamma_{\\rm R}}\n\\end{equation*}\nfor all $\\phi \\in H^1(\\mathcal T_h)$ and $\\boldsymbol w_h \\in \\boldsymbol{\\CP}_p(\\mathcal T_h) \\cap \\boldsymbol X_{\\Gamma_{\\rm N}}(\\Omega,\\operatorname{div})$.\n\n\\subsection{Broken norms}\n\nFor $v \\in H^1(\\mathcal T_h)$, the broken counterpart of the energy norm introduced in\n\\eqref{eq_energy_norm} is defined by\n\\begin{equation*}\n\\enorm{v}^2_{\\omega,\\mathcal T_h}\n:=\n\\omega^2 \\|v\\|_{\\mu,\\mathcal T_h}^2\n+\n\\omega \\|v\\|_{\\gamma,\\mathcal F_h^{\\rm R}}^2\n+\n\\|\\mathfrak G(v)\\|_{\\boldsymbol A,\\mathcal T_h}^2.\n\\end{equation*}\nNotice that it is equal to the $\\enorm{{\\cdot}}_{\\omega,\\Omega}$ norm if\n$v \\in H^1_{\\Gamma_{\\rm D}}(\\Omega)$. We will also employ the mesh-dependent norms\n\\begin{equation*}\n\\|v\\|_{\\dagger,1,\\mathcal T_h}^2\n:=\n\\sum_{K \\in \\mathcal T_h} \\left \\{\n\\max \\left (\n1,\\frac{\\omega^2 h_K^2}{\\vartheta_K^2}\n\\right )\n\\frac{\\alpha_K}{h_K^2}\\|v\\|_K^2\n+\n\\|\\mathfrak G(v)\\|_{\\boldsymbol A,K}^2\n\\right \\}\n+\n\\sum_{F \\in \\mathcal F_h^{\\rm R}}\n\\max \\left (1,\\frac{\\omega h_F}{\\vartheta_F}\\right )\n\\frac{\\alpha_F}{h_F} \\|v\\|_F^2\n\\end{equation*}\nand\n\\begin{equation*}\n\\|\\boldsymbol w\\|_{\\dagger,\\operatorname{div},\\mathcal T_h}^2\n:=\n\\sum_{K \\in \\mathcal T_h} \\left \\{\n\\|\\boldsymbol w\\|_{\\boldsymbol A^{-1},K}^2\n+\n\\frac{h_K^2}{\\alpha_K} \\|\\div \\boldsymbol w\\|_{K}^2\n\\right \\}\n+\n\\sum_{F \\in \\mathcal F_h^{\\rm R}} \\frac{h_F}{\\alpha_F} \\|\\boldsymbol w \\cdot \\boldsymbol n\\|_F^2\n\\end{equation*}\nfor $\\boldsymbol w \\in \\boldsymbol X_{\\Gamma_{\\rm N}}(\\operatorname{div},\\Omega)$. These norms are ``dual'' to each other in the sense that\n\\begin{equation}\n\\label{eq_norm_dual}\n|(\\mathfrak G(\\phi),\\boldsymbol w)_{\\mathcal T_h} + (\\phi,\\div \\boldsymbol w)_{\\mathcal T_h} - (\\phi,\\boldsymbol w)_{\\mathcal F_h^{\\rm R}}|\n\\leq\n\\|\\phi\\|_{\\dagger,1,\\mathcal T_h}\\|\\boldsymbol w\\|_{\\dagger,\\operatorname{div},\\mathcal T_h}\n\\end{equation}\nfor all $\\phi \\in H^1(\\mathcal T_h)$ and $\\boldsymbol w \\in \\boldsymbol X_{\\Gamma_{\\rm N}}(\\operatorname{div},\\Omega)$.\nFor future references, we notice that the ``$\\max$'' in the definition of the\n$\\|{\\cdot}\\|_{\\dagger,1,\\mathcal T_h}$ are chosen so that\n\\begin{equation}\n\\label{eq_norm_control}\n\\enorm{v}_{\\omega,\\mathcal T_h} \\leq \\|v\\|_{\\dagger,1,\\mathcal T_h} \\qquad \\forall v \\in H^1(\\mathcal T_h).\n\\end{equation}\n\n\\subsection{IPDG form}\n\nFollowing \\cite[Section 4.3.3]{dipietro_ern_2012a}, our DG approximation of the\n$(\\boldsymbol A\\boldsymbol \\nabla\\cdot,\\boldsymbol \\nabla\\cdot)_\\Omega$ form is given by\n\\begin{equation}\n\\label{eq_IPDG_discrete_gradient}\na_h(\\phi,v)\n:=\n(\\boldsymbol A\\mathfrak G(\\phi),\\mathfrak G(v))_{\\mathcal T_h} + s_h(\\phi,v)\n\\qquad\n\\forall \\phi,v \\in H^1(\\mathcal T_h)\n\\end{equation}\nwhere $s_h: H^1(\\mathcal T_h) \\times H^1(\\mathcal T_h) \\to \\mathbb C$ is a sesqulinear form\nsatisfying $s_h(\\phi,v) = 0$ whenever $\\phi \\in H^1_{\\Gamma_{\\rm R}}(\\Omega)$ or\n$v \\in H^1_{\\Gamma_{\\rm R}}(\\Omega)$. Importantly, we have\n\\begin{equation}\n\\label{eq_consistency}\na_h(\\phi,v) = (\\boldsymbol A\\boldsymbol \\nabla\\phi,\\boldsymbol \\nabla v)_\\Omega\n\\end{equation}\nwhen $\\phi,v \\in H^1_{\\Gamma_{\\rm D}}(\\Omega)$.\n\nIt is worthy to note that the IPDG form is usually not presented (and not implemented)\nas it is presented in \\eqref{eq_IPDG_discrete_gradient}. Instead\n\\cite{agut_diaz_2013a,feng_wu_2009a,grote_schneebeli_schotzau_2006a,sauter_zech_2015a},\nthe method is usually written as\n\\begin{multline}\n\\label{eq_IPDG_jump}\na_h(\\phi,v)\n=\n(\\boldsymbol A\\boldsymbol \\nabla\\phi,\\boldsymbol \\nabla v)_{\\mathcal T_h}\n\\\\\n-\n(\\avg{\\boldsymbol A\\boldsymbol \\nabla \\phi}\\cdot \\boldsymbol n_F,\\jmp{v})_{\\mathcal F_h}\n-\n(\\jmp{\\phi},\\avg{\\boldsymbol A\\boldsymbol \\nabla v}\\cdot \\boldsymbol n_F)_{\\mathcal F_h}\n+\n\\sum_{F \\in \\mathcal F_h^{\\rm i} \\cup \\mathcal F_h^{\\rm D}} \\frac{\\beta_F}{h_F}(\\jmp{\\phi},\\jmp{v})_F\n\\end{multline}\nwhere $\\beta_F \\sim p^2$ is a penalty parameter chosen to be large enough\n\\cite{agut_diaz_2013a}.\nHowever, as shown in \\cite[Section 4.3.3]{dipietro_ern_2012a},\nthe formulation of \\eqref{eq_IPDG_jump} can be recast in the framework of\n\\eqref{eq_IPDG_discrete_gradient} by setting\n\\begin{equation}\n\\label{eq_penalization_usual}\ns_h(\\phi,v)\n:=\n\\sum_{F \\in \\mathcal F_h^{\\rm i} \\cup \\mathcal F_h^{\\rm D}} \\frac{\\beta_F}{h_F}(\\jmp{\\phi},\\jmp{v})_F\n-(\\boldsymbol A\\LIFT{\\phi},\\LIFT{v})_\\Omega.\n\\end{equation}\n\n\\subsection{The discrete Helmholtz problem}\n\nConsider the sesquilinear form \n\\begin{equation*}\nb_h(\\phi,v)\n:=\n-\\omega^2 (\\mu\\phi,v)_{\\mathcal T_h}\n-i\\omega (\\gamma\\phi,v)_{\\mathcal F_h^{\\rm R}}\n+a_h(\\phi,v)\n\\qquad\n\\forall \\phi,v \\in H^1(\\mathcal T_h).\n\\end{equation*}\nThen, the discrete problem consists in finding $u_h \\in \\mathcal P_p(\\mathcal T_h)$ such that\n\\begin{equation}\n\\label{eq_helmholtz_ipdg}\nb_h(u_h,v_h) = (f,v_h)_{\\mathcal T_h} \\qquad \\forall v_h \\in \\mathcal P_p(\\mathcal T_h).\n\\end{equation}\nFor any $u_h$ satisfying \\eqref{eq_helmholtz_ipdg}, recalling \\eqref{eq_helmholtz_weak}\nand due to \\eqref{eq_consistency}, the Galerkin orthogonality property\n\\begin{equation}\n\\label{eq_galerkin_orthogonality}\nb_h(u-u_h,w_h) = 0\n\\end{equation}\nholds true for all discrete conforming test functions $w_h \\in \\mathcal P_p(\\mathcal T_h) \\cap H^1_{\\Gamma_{\\rm D}}(\\Omega)$.\nSimilarly, we have\n\\begin{equation}\n\\label{eq_continuity}\n|b_h(\\phi,v)|\n\\leq\n\\enorm{\\phi}_{\\omega,\\mathcal T_h}\\enorm{v}_{\\omega,\\mathcal T_h}\n\\end{equation}\nfor $\\phi,v \\in H^1(\\mathcal T_h)$ whenever $\\phi$ or $v$ belongs to $H^1_{\\Gamma_{\\rm D}}(\\Omega)$.\n\n\n\\subsection{Conforming subspaces and projections}\n\nThe sets $\\mathcal P_p(\\mathcal T_h) \\cap H^1_{\\Gamma_{\\rm D}}(\\Omega)$ and $\\boldsymbol{\\CP}_p(\\mathcal T_h) \\cap \\boldsymbol X_{\\Gamma_{\\rm N}}(\\operatorname{div},\\Omega)$\nare the usual Lagrange and BDM finite element spaces (see, e.g.,\n\\cite[Sections 8.5.1 and 11.5]{ern_guermond_2019a}). Although they are not needed to implement\nthe IPDG discretization, they will be extremely useful for the analysis.\nThe projections\n\\begin{equation*}\n\\pi_h^{\\rm g} v\n:=\n\\arg \\min_{v_h \\in \\mathcal P_p(\\mathcal T_h) \\cap H^1_{\\Gamma_{\\rm D}}(\\Omega)}\n\\|v-v_h\\|_{1,\\dagger,\\mathcal T_h}\n\\end{equation*}\nand\n\\begin{equation*}\n\\pi_h^{\\rm d} \\boldsymbol w\n:=\n\\arg \\min_{\\boldsymbol w_h \\in \\boldsymbol{\\CP}_p(\\mathcal T_h) \\cap \\boldsymbol X_{\\Gamma_{\\rm N}}(\\operatorname{div},\\Omega)}\n\\|\\boldsymbol w-\\boldsymbol w_h\\|_{\\operatorname{div},\\dagger,\\mathcal T_h}\n\\end{equation*}\nare well-defined for all $v \\in H^1_{\\Gamma_{\\rm D}}(\\Omega)$ and $\\boldsymbol w \\in \\boldsymbol X_{\\Gamma_{\\rm N}}(\\operatorname{div},\\Omega)$,\nsince the mesh-dependent norms appearing in the right-hand sides are naturally associated\nwith inner-product. We will also need the conforming projector\n\\begin{equation*}\n\\pi^{\\rm g} v := \\arg \\min_{s \\in H^1_{\\Gamma_{\\rm D}}(\\Omega)} \\|v-s\\|_{1,\\dagger,\\mathcal T_h}\n\\end{equation*}\ndefined for all $v \\in H^1(\\mathcal T_h)$.\n\n\\subsection{Approximation factors}\n\n\\begin{subequations}\n\\label{eq_definition_gba}\nFollowing\n\\cite{chaumontfrelet_ern_vohralik_2021a,\nmelenk_parsania_sauter_2013a,\nmelenk_sauter_2010a,\nsauter_zech_2015a}, our analysis will rely on duality arguments and\napproximation factors will be a central concept.\nFor $\\psi \\in L^2(\\Omega)$ and $\\Psi \\in L^2(\\Gamma_{\\rm R})$,\nlet $u^\\star_\\psi,U^\\star_\\Psi \\in H^1_{\\Gamma_{\\rm R}}(\\Omega)$ solve\n\\begin{equation*}\nb(w,u^\\star_\\psi) = \\omega(\\mu w,\\psi)_\\Omega,\n\\qquad\nb(w,U^\\star_\\Psi) = \\omega^{1\/2}(\\gamma w,\\Psi)_{\\Gamma_{\\rm R}}\n\\qquad\n\\forall w \\in H^1_{\\Gamma_{\\rm R}}(\\Omega).\n\\end{equation*}\nThe approximation factors\n\\begin{equation}\n\\label{eq_definition_gbag}\n\\widecheck \\gamma_{{\\rm ba},{\\rm g}}\n:=\n\\max_{\\substack{\\psi \\in L^2(\\Omega) \\\\ \\|\\psi\\|_{\\mu,\\Omega} = 1}}\n\\|u_\\psi^\\star-\\pi_h^{\\rm g} u_\\psi^\\star\\|_{\\dagger,1,\\mathcal T_h},\n\\quad\n\\widetilde \\gamma_{{\\rm ba},{\\rm g}}\n:=\n\\max_{\\substack{\\Psi \\in L^2(\\Gamma_{\\rm R}) \\\\ \\|\\Psi\\|_{\\gamma,\\Gamma_{\\rm R}} = 1}}\n\\|U_\\Psi^\\star-\\pi_h^{\\rm g} U_\\Psi^\\star\\|_{\\dagger,1,\\mathcal T_h},\n\\end{equation}\nwhere previously employed in \\cite{chaumontfrelet_ern_vohralik_2021a}\nfor the a posteriori error analysis of conforming discretizations.\nHere, we will additionally need divergence-conforming approximation factors\n\\begin{equation}\n\\label{eq_definition_gbad}\n\\begin{aligned}\n\\widecheck \\gamma_{{\\rm ba},{\\rm d}} &:= \\max_{\\substack{\\psi \\in L^2(\\Omega) \\\\ \\|\\psi\\|_{\\mu,\\Omega} = 1}}\n\\|\\boldsymbol A\\boldsymbol \\nabla u_\\psi^\\star-\\pi_h^{\\rm d} (\\boldsymbol A\\boldsymbol \\nabla u_\\psi^\\star)\\|_{\\dagger,\\operatorname{div},\\mathcal T_h},\n\\\\\n\\widetilde \\gamma_{{\\rm ba},{\\rm d}} &:= \\max_{\\substack{\\Psi \\in L^2(\\Gamma_{\\rm R}) \\\\ \\|\\Psi\\|_{\\gamma,\\Gamma_{\\rm R}} = 1}}\n\\|\\boldsymbol A\\boldsymbol \\nabla U_\\Psi^\\star-\\pi_h^{\\rm d} (\\boldsymbol A\\boldsymbol \\nabla U_\\Psi^\\star)\\|_{\\dagger,\\operatorname{div},\\mathcal T_h},\n\\end{aligned}\n\\end{equation}\nto deal with the non-conformity of the IPDG approximation under minimal regularity.\nWe finally introduce the following short-hand notations\n\\begin{align}\n\\label{eq_definition_gba_summed}\n\\gamma_{{\\rm ba},{\\rm g}}^2 &:= 4\\widecheck \\gamma_{{\\rm ba},{\\rm g}}^2 + 2\\widetilde \\gamma_{{\\rm ba},{\\rm g}}^2,\n\\quad\n&\\gamma_{{\\rm ba},{\\rm d}}^2 &:= 4\\widecheck \\gamma_{{\\rm ba},{\\rm d}}^2 + 2\\widetilde \\gamma_{{\\rm ba},{\\rm d}}^2,\n\\\\\n\\widecheck \\gamma_{\\rm ba}^2 &:= \\widecheck \\gamma_{{\\rm ba},{\\rm g}}^2 + \\widecheck \\gamma_{{\\rm ba},{\\rm d}}^2,\n\\quad\n&\\widetilde \\gamma_{\\rm ba}^2 &:= \\widetilde \\gamma_{{\\rm ba},{\\rm g}}^2 + \\widetilde \\gamma_{{\\rm ba},{\\rm d}}^2,\n\\end{align}\nand\n\\begin{equation}\n\\gamma_{\\rm ba}^2 := \\gamma_{{\\rm ba},{\\rm g}}^2 + \\gamma_{{\\rm ba},{\\rm d}}^2 = 4\\widecheck \\gamma_{\\rm ba}^2 + 2\\widetilde \\gamma_{\\rm ba}^2.\n\\end{equation}\n\\end{subequations}\n\nBy combining elliptic regularity and approximation properties of finite element\nspaces, it is easy to see that $\\gamma_{\\rm ba} \\to 0$ as $h\/p \\to 0$. However, the dependency\non the PDE coefficients and on the frequency may be hard to track. In particular,\nthe stability constant $\\gamma_{\\rm st}$ typically plays a central role in estimating $\\gamma_{\\rm ba}$.\nQualitative estimates for $\\gamma_{\\rm ba}$ can be found in\n\\cite{chaumontfrelet_nicaise_2020a,melenk_sauter_2010a}, and explicit estimates\nare available in specific situations \\cite{chaumontfrelet_ern_vohralik_2021a}.\n\n\\section{Duality under minimal regularity}\n\\label{section_duality}\n\nHere, we state in a separate section the key technical result that enables us\nto perform duality techniques under minimal regularity assumptions. We believe\nit will be important in other contexts, so that we make it easy to reference.\nAlthough the arguments are not complicated, we were not able to find a proof\nin the literature.\n\n\\begin{lemma}[Lifting error]\n\\label{lemma_lifting}\nFor all $\\phi \\in H^1(\\mathcal T_h)$ and $\\boldsymbol \\sigma \\in \\boldsymbol X_{\\Gamma_{\\rm N}}(\\operatorname{div},\\Omega)$, we have\n\\begin{multline}\n\\label{eq_nonconf_identity}\n(\\mathfrak G(\\phi),\\boldsymbol \\sigma)_{\\mathcal T_h}\n+\n(\\phi,\\div \\boldsymbol \\sigma)_{\\mathcal T_h}\n-\n(\\phi,\\boldsymbol \\sigma \\cdot \\boldsymbol n)_{\\mathcal F_h^{\\rm R}}\n=\n\\\\\n(\\mathfrak G(\\phi-\\widetilde\\phi),\\boldsymbol \\sigma-\\boldsymbol \\sigma_h)_{\\mathcal T_h}\n+\n(\\phi-\\widetilde \\phi,\\div(\\boldsymbol \\sigma-\\boldsymbol \\sigma_h))_{\\mathcal T_h}\n-\n(\\phi-\\widetilde \\phi,(\\boldsymbol \\sigma-\\boldsymbol \\sigma_h) \\cdot \\boldsymbol n)_{\\mathcal F_h^{\\rm R}}\n\\end{multline}\nand in particular\n\\begin{equation}\n\\label{eq_nonconf_estimate}\n|(\\mathfrak G(\\phi),\\boldsymbol \\sigma)_{\\mathcal T_h}+(\\phi,\\div \\boldsymbol \\sigma)_{\\mathcal T_h}-(\\phi,\\boldsymbol \\sigma \\cdot \\boldsymbol n)_{\\mathcal F_h^{\\rm R}}|\n\\leq\n\\|\\phi-\\widetilde \\phi\\|_{\\dagger,1,\\mathcal T_h}\n\\|\\boldsymbol \\sigma-\\boldsymbol \\sigma_h\\|_{\\dagger,\\operatorname{div},\\mathcal T_h}\n\\end{equation}\nfor all $\\widetilde \\phi \\in H^1_{\\Gamma_{\\rm D}}(\\Omega)$ and\n$\\boldsymbol \\sigma_h \\in \\boldsymbol{\\CP}_p(\\mathcal T_h) \\cap \\boldsymbol X_{\\Gamma_{\\rm N}}(\\operatorname{div},\\Omega)$.\n\\end{lemma}\n\n\\begin{proof}\nWe first observe that if $\\widetilde \\phi \\in H^1_{\\Gamma_{\\rm D}}(\\Omega)$, integration by parts gives\n\\begin{equation*}\n(\\mathfrak G(\\widetilde \\phi),\\boldsymbol \\tau)_{\\mathcal T_h}\n+\n(\\widetilde \\phi,\\div \\boldsymbol \\tau)\n-\n(\\widetilde \\phi,\\boldsymbol \\tau \\cdot \\boldsymbol n)_{\\Gamma_{\\rm R}}\n=\n(\\boldsymbol \\nabla \\widetilde \\phi,\\boldsymbol \\tau)_{\\mathcal T_h}\n+\n(\\widetilde \\phi,\\div \\boldsymbol \\tau)\n-\n(\\widetilde \\phi,\\boldsymbol \\tau \\cdot \\boldsymbol n)_{\\Gamma_{\\rm R}}\n=\n0\n\\end{equation*}\nfor all $\\boldsymbol \\tau \\in \\boldsymbol X_{\\Gamma_{\\rm N}}(\\operatorname{div},\\Omega)$ due to the essential boundary conditions\non $\\Gamma_{\\rm D}$ and $\\Gamma_{\\rm N}$. Thus,\n\\begin{align*}\n(\\mathfrak G(\\phi),\\boldsymbol \\sigma)_{\\mathcal T_h} + (\\phi,\\div \\boldsymbol \\sigma)_{\\mathcal T_h} - (\\phi,\\boldsymbol \\sigma \\cdot \\boldsymbol n)_{\\mathcal F_h^{\\rm R}}\n=\n(\\mathfrak G(\\phi-\\widetilde \\phi),\\boldsymbol \\sigma)_{\\mathcal T_h} + (\\phi-\\widetilde \\phi,\\div \\boldsymbol \\sigma)_{\\mathcal T_h}\n-\n(\\phi-\\widetilde \\phi,\\boldsymbol \\sigma \\cdot \\boldsymbol n)_{\\mathcal F_h^{\\rm R}}.\n\\end{align*}\nIf $\\boldsymbol \\sigma_h \\in \\boldsymbol{\\CP}_p(\\mathcal T_h) \\cap \\boldsymbol H(\\operatorname{div},\\Omega)$, we also have\n\\begin{align*}\n0\n&=\n(\\mathfrak G(\\phi),\\boldsymbol \\sigma_h)_{\\mathcal T_h}\n+\n(\\phi,\\div \\boldsymbol \\sigma_h)_{\\mathcal T_h}\n-\n(\\phi,\\boldsymbol \\sigma_h \\cdot \\boldsymbol n)_{\\mathcal F_h^{\\rm R}}\n\\\\\n&=\n(\\mathfrak G(\\phi)-\\boldsymbol \\nabla \\widetilde \\phi,\\boldsymbol \\sigma_h)_{\\mathcal T_h}\n+\n(\\boldsymbol \\nabla \\widetilde \\phi,\\boldsymbol \\sigma_h)_{\\mathcal T_h}\n+\n(\\phi,\\div \\boldsymbol \\sigma_h)_{\\mathcal T_h}\n-\n(\\phi,\\boldsymbol \\sigma_h \\cdot \\boldsymbol n)_{\\mathcal F_h^{\\rm R}}\n\\\\\n&=\n(\\mathfrak G(\\phi-\\widetilde \\phi),\\boldsymbol \\sigma_h)_{\\mathcal T_h}\n+\n(\\phi-\\widetilde \\phi,\\div \\boldsymbol \\sigma_h)_{\\mathcal T_h}\n-\n(\\phi-\\widetilde \\phi,\\boldsymbol \\sigma_h \\cdot \\boldsymbol n)_{\\mathcal F_h^{\\rm R}},\n\\end{align*}\nand \\eqref{eq_nonconf_identity} follows after summation. Then, \\eqref{eq_nonconf_estimate}\nis a direct consequence of \\eqref{eq_norm_dual}.\n\\end{proof}\n\nA simple consequence of Lemma \\ref{lemma_lifting} is the following result,\nwhich is crucial to estimate the non-conformity error in the context of\nduality arguments.\n\n\\begin{corollary}[Control of the non-confomity]\n\\label{corollary_nonconf}\nFor all $\\phi \\in H^1(\\mathcal T_h)$ and $\\xi \\in H^1(\\Omega)$ with $\\boldsymbol A\\boldsymbol \\nabla \\xi \\in \\boldsymbol X_{\\Gamma_{\\rm N}}(\\operatorname{div},\\Omega)$,\nwe have\n\\begin{equation}\n\\label{eq_nonconf}\n\\left |\na_h(\\phi,\\xi)+(\\phi,\\div(\\boldsymbol A\\boldsymbol \\nabla\\xi))_{\\mathcal T_h}-(\\phi,\\boldsymbol A\\boldsymbol \\nabla \\xi \\cdot \\boldsymbol n)_{\\mathcal F_h^{\\rm R}}\n\\right |\n\\leq\n\\|\\phi-\\widetilde \\phi\\|_{\\dagger,1,\\mathcal T_h}\\|\\boldsymbol A\\boldsymbol \\nabla \\xi-\\boldsymbol \\sigma_h\\|_{\\dagger,\\operatorname{div},\\mathcal T_h}\n\\end{equation}\nfor all $\\widetilde \\phi \\in H^1_{\\Gamma_{\\rm D}}(\\Omega)$ and\n$\\boldsymbol \\sigma_h \\in \\boldsymbol{\\CP}_p(\\mathcal T_h) \\cap \\boldsymbol X_{\\Gamma_{\\rm N}}(\\operatorname{div},\\Omega)$.\n\\end{corollary}\n\n\\section{A priori analysis}\n\\label{section_apriori}\n\nThe purpose of this section is to establish the existence and uniqueness of\na discrete solution $u_h \\in \\mathcal P_p(\\mathcal T_h)$ to \\eqref{eq_helmholtz_ipdg} and\nto derive a priori error estimates controlling $u-u_h$ in suitable norms.\nThe proof follows the line of the Schatz argument \\cite{schatz_1974a}, with\nsuitable modifications to take into account the non-conforming nature of\nthe discrete scheme.\n\nIn this section, we require that there exists $\\rho > 0$ such that\n\\begin{equation}\n\\label{eq_assumption_stabilization_apriori}\n\\Re s_h(v_h,v_h) \\geq \\rho^2 \\|v_h-\\pi^{\\rm g} v_h\\|_{\\dagger,1,\\mathcal T_h}^2\n\\qquad\n\\forall v_h \\in \\mathcal P_p(\\mathcal T_h).\n\\end{equation}\nThis assumption always holds true when\n$s_h(\\cdot,\\cdot) = \\beta\/h(\\jmp{\\cdot},\\jmp{\\cdot})_{\\mathcal F_h}$\nwith $\\beta > 0$. It is also the case when $s_h$ takes the form\n\\eqref{eq_penalization_usual} if the penalization parameter is\nsufficiently large \\cite{agut_diaz_2013a}. We note that this assumption\nrules out some choices of stabilization including complex-value stabilization\nparameters, as proposed e.g. in \\cite{feng_wu_2009a,sauter_zech_2015a}.\nFor the sake of simplicity, we only consider stabilizations satisfying\n\\eqref{eq_assumption_stabilization_apriori}, but slight modifications\nof our proofs could handle more general situations.\n\nWe start with the main duality argument, which is a generalization of the Aubin-Nitsche trick.\n\n\\begin{lemma}[Aubin-Nitsche]\n\\label{lemma_aubin_nitsche}\nAssume that $u_h \\in \\mathcal P_p(\\mathcal T_h)$ satisfies \\eqref{eq_helmholtz_ipdg}.\nThen, we have\n\\begin{equation}\n\\label{eq_aubin_nitsche_omega}\n\\omega\\|u-u_h\\|_{\\mu,\\mathcal T_h}\n\\leq\n\\widecheck \\gamma_{{\\rm ba},{\\rm g}} \\enorm{u-u_h}_{\\omega,\\mathcal T_h}\n+\n\\widecheck \\gamma_{{\\rm ba},{\\rm d}} \\|u_h-\\pi^{\\rm g} u_h\\|_{\\dagger,1,\\mathcal T_h}\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq_aubin_nitsche_GR}\n\\omega^{1\/2}\\|u-u_h\\|_{\\gamma,\\mathcal F_h^{\\rm R}}\n\\leq\n\\widetilde \\gamma_{{\\rm ba},{\\rm g}} \\enorm{u-u_h}_{\\omega,\\mathcal T_h}\n+\n\\widetilde \\gamma_{{\\rm ba},{\\rm d}}\\|u_h-\\pi^{\\rm g} u_h\\|_{\\dagger,1,\\mathcal T_h}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nFor the sake of shortness, we set $e_h := u-u_h$.\nWe first establish \\eqref{eq_aubin_nitsche_omega}. To do so,\nwe define $\\xi$ as the unique element of $H^1_{\\Gamma_{\\rm D}}(\\Omega)$ such that\n$b(\\phi,\\xi) = \\omega (\\mu\\phi,e_h)_\\Omega$ for all $\\phi \\in H^1_{\\Gamma_{\\rm D}}(\\Omega)$.\nIntegrating by parts, we see that the identities\n\\begin{equation*}\n-\\omega^2 \\mu \\xi - \\div\\left (\\boldsymbol A\\boldsymbol \\nabla \\xi\\right ) = e_h\n\\qquad\n\\boldsymbol A\\boldsymbol \\nabla \\xi \\cdot \\boldsymbol n+i\\omega\\xi = 0\n\\end{equation*}\nrespectively hold in $L^2(\\Omega)$ and $L^2(\\Gamma_{\\rm R})$. As a result,\n\\begin{equation*}\n\\omega \\|e_h\\|_{\\mu,\\mathcal T_h}^2\n=\n-\\omega^2 (\\mu e_h,\\xi)\n-i\\omega(e_h,\\phi)_{\\mathcal F_h^{\\rm R}}\n-\n\\left \\{\n(e_h,\\div(\\boldsymbol A\\boldsymbol \\nabla \\xi))_{\\mathcal T_h}\n-(e_h,\\boldsymbol A\\boldsymbol \\nabla \\xi \\cdot \\boldsymbol n)_{\\mathcal F^{\\rm R}}\n\\right \\}\n\\end{equation*}\nand we have from Corollary \\ref{corollary_nonconf} that\n\\begin{equation*}\n\\omega \\|e_h\\|_{\\mu,\\mathcal T_h}^2\n\\leq\n|b_h(e_h,\\xi)|\n+\n\\|e_h-\\widetilde \\phi\\|_{\\dagger,1,\\mathcal T_h}\n\\|\\boldsymbol A\\boldsymbol \\nabla \\xi-\\boldsymbol \\sigma_h\\|_{\\dagger,\\operatorname{div},\\mathcal T_h}\n\\end{equation*}\nfor all $\\widetilde \\phi \\in H^1_{\\Gamma_{\\rm D}}(\\Omega)$ and\n$\\boldsymbol \\sigma_h \\in \\boldsymbol{\\CP}_p(\\mathcal T_h) \\cap \\boldsymbol X_{\\Gamma_{\\rm N}}(\\operatorname{div},\\Omega)$.\nOn the one hand, we select $\\widetilde \\phi = u-\\pi^{\\rm g} u_h$ and\n$\\boldsymbol \\sigma_h = \\pi_h^{\\rm d}(\\boldsymbol A\\boldsymbol \\nabla\\xi)$, so that using \\eqref{eq_definition_gbad},\nwe have\n\\begin{equation*}\n\\omega \\|e_h\\|_{\\mu,\\mathcal T_h}^2\n\\leq\n|b_h(e_h,\\xi)|\n+\n\\widecheck \\gamma_{{\\rm ba},{\\rm d}} \\|u_h-\\pi^{\\rm g} u_h\\|_{\\dagger,1,\\mathcal T_h} \\|e_h\\|_{\\mu,\\mathcal T_h}.\n\\end{equation*}\nOn the other hand, using \\eqref{eq_galerkin_orthogonality} and \\eqref{eq_continuity}, we have\n\\begin{equation*}\n|b_h(e_h,\\xi)|\n=\n|b_h(e_h,\\xi-\\pi_h^{\\rm g}\\xi)|\n\\leq\n\\enorm{e_h}_{\\omega,\\mathcal T_h}\\enorm{\\xi-\\pi_h^{\\rm g}\\xi}_{\\omega,\\Omega}\n\\leq\n\\widecheck \\gamma_{{\\rm ba},{\\rm g}} \\enorm{e_h}_{\\omega,\\mathcal T_h}\\|e_h\\|_{\\mu,\\mathcal T_h}.\n\\end{equation*}\n\nWe then prove \\eqref{eq_aubin_nitsche_GR}. Instead of $\\xi$,\nwe define $\\chi$ as the unique element of $H^1_{\\Gamma_{\\rm D}}(\\Omega)$\nsuch that $b(\\phi,\\chi) = \\omega^{1\/2}(\\gamma\\chi,e_h)_{\\Gamma_{\\rm R}}$ for all\n$\\phi \\in H^1_{\\Gamma_{\\rm D}}(\\Omega)$. We then have\n\\begin{equation*}\n-\\omega^2 \\mu \\chi - \\div\\left (\\boldsymbol A\\boldsymbol \\nabla \\chi\\right ) = 0\n\\qquad\n\\boldsymbol A\\boldsymbol \\nabla \\chi \\cdot \\boldsymbol n+i\\omega\\chi = e_h\n\\end{equation*}\nin $L^2(\\Omega)$ and $L^2(\\Gamma_{\\rm R})$, respectively. We thus have\n\\begin{equation*}\n\\omega^{1\/2} \\|e_h\\|_{\\gamma,\\mathcal F_h^{\\rm R}}^2\n=\n-\\omega^2 (\\mu e_h,\\chi)\n-i\\omega(e_h,\\phi)_{\\mathcal F_h^{\\rm R}}\n-\n\\left \\{\n(e_h,\\div(\\boldsymbol A\\boldsymbol \\nabla \\chi))_{\\mathcal T_h}\n-(e_h,\\boldsymbol A\\boldsymbol \\nabla \\chi \\cdot \\boldsymbol n)_{\\mathcal F^{\\rm R}}\n\\right \\},\n\\end{equation*}\nand applying again Corollary \\ref{corollary_nonconf} we arrive at\n\\begin{equation*}\n\\omega^{1\/2} \\|e_h\\|_{\\gamma,\\mathcal F_h^{\\rm R}}^2\n\\leq\n|b_h(e_h,\\chi-\\pi_h^{\\rm g} \\chi)|\n+\n\\|u_h-\\pi^{\\rm g} u_h\\|_{\\dagger,1,\\mathcal T_h}\n\\|\\boldsymbol A\\boldsymbol \\nabla \\chi-\\pi_h^{\\rm d}(\\boldsymbol A\\boldsymbol \\nabla \\chi)\\|_{\\dagger,\\operatorname{div},\\mathcal T_h},\n\\end{equation*}\nand the result follows from the definitions of $\\widetilde \\gamma_{{\\rm ba},{\\rm g}}$ and $\\widetilde \\gamma_{{\\rm ba},{\\rm d}}$\ngiven in \\eqref{eq_definition_gba_summed}.\n\\end{proof}\n\nWe can now complete the Schatz argument, leading to quasi-optimality\nof the discrete solution for sufficiently refined meshes.\n\n\\begin{theorem}[A priori estimate]\nIf $\\gamma_{{\\rm ba},{\\rm g}} < 1$ and $\\gamma_{{\\rm ba},{\\rm d}} \\leq \\rho$, then there exists a unique\nsolution $u_h \\in \\mathcal P_p(\\mathcal T_h)$ to \\eqref{eq_helmholtz_ipdg} and the estimate\n\\begin{equation}\n\\label{eq_aprio_estimate}\n\\enorm{u-u_h}_{\\omega,\\mathcal T_h}\n\\leq\n\\frac{1}{1-\\gamma_{{\\rm ba},{\\rm g}}^2}\n\\min_{v_h \\in \\mathcal P_p(\\mathcal T_h) \\cap H^1_{\\Gamma_{\\rm D}}(\\Omega)} \\enorm{u-v_h}_{\\omega,\\Omega}\n\\end{equation}\nholds true. In addition, if $\\gamma_{{\\rm ba},{\\rm d}} < \\rho$, we also have\n\\begin{equation}\n\\label{eq_aprio_nonconf}\n\\|u_h-\\pi^{\\rm g} u_h\\|_{\\dagger,1,\\mathcal T_h}\n\\leq\n\\left (\n\\frac{1}{(1-\\gamma_{{\\rm ba},{\\rm g}}^2)(\\rho^2-\\gamma_{{\\rm ba},{\\rm d}}^2)}\n\\right )^{1\/2}\n\\min_{v_h \\in \\mathcal P_p(\\mathcal T_h) \\cap H^1_{\\Gamma_{\\rm D}}(\\Omega)} \\enorm{u-v_h}_{\\omega,\\Omega}.\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nWe first observe that\n\\begin{align*}\n\\Re b_h(e_h,e_h) + \\omega\\|e_h\\|_{\\gamma,\\mathcal F_h^{\\rm R}}^2 + 2\\omega^2\\|e_h\\|_{\\mu,\\mathcal T_h}^2\n&=\n\\enorm{e_h}_{\\omega,\\mathcal T_h}^2\n+\n\\Re s_h(u_h,u_h)\n\\\\\n&\\geq\n\\enorm{e_h}_{\\omega,\\Omega}^2 + \\rho^2 \\|u_h-\\pi^{\\rm g} u_h\\|_{\\dagger,1,\\mathcal T_h}^2.\n\\end{align*}\nAs a result, we have\n\\begin{align*}\n\\enorm{e_h}_{\\omega,\\mathcal T_h}^2 + \\rho^2 \\|u_h-\\pi^{\\rm g} u_h\\|_{\\dagger,1,\\mathcal T_h}^2\n&\\leq\n|b_h(e_h,u-v_h)| + 2\\omega^2\\|e_h\\|_{\\mu,\\mathcal T_h}^2 + \\omega\\|e_h\\|_{\\gamma,\\mathcal F_h^{\\rm R}}^2\n\\\\\n&\\leq\n\\enorm{e_h}_{\\omega,\\mathcal T_h} \\enorm{u-v_h}_{\\omega,\\Omega}\n\\\\\n&+\n2\\left (\\widecheck \\gamma_{{\\rm ba},{\\rm g}} \\enorm{e_h}_{\\omega,\\mathcal T_h} + \\widecheck \\gamma_{{\\rm ba},{\\rm d}} \\|u_h-\\pi^{\\rm g} u_h\\|_{\\dagger,1,\\mathcal T_h}\\right )^2\n\\\\\n&+\n\\left (\\widetilde \\gamma_{{\\rm ba},{\\rm g}} \\enorm{e_h}_{\\omega,\\mathcal T_h} + \\widetilde \\gamma_{{\\rm ba},{\\rm d}} \\|u_h-\\pi^{\\rm g} u_h\\|_{\\dagger,1,\\mathcal T_h}\\right )^2\n\\\\\n&\\leq\n\\enorm{e_h}_{\\omega,\\mathcal T_h} \\enorm{u-v_h}_{\\omega,\\Omega}\n\\\\\n&+\n(4\\widecheck \\gamma_{{\\rm ba},{\\rm g}}+2\\widetilde \\gamma_{{\\rm ba},{\\rm g}}) \\enorm{e_h}_{\\omega,\\mathcal T_h}^2\n+\n(4\\widecheck \\gamma_{{\\rm ba},{\\rm d}}+2\\widetilde \\gamma_{{\\rm ba},{\\rm d}}) \\|u_h-\\pi^{\\rm g} u_h\\|_{\\dagger,1,\\mathcal T_h}^2\n\\end{align*}\nso that\n\\begin{equation*}\n(1-4\\widecheck \\gamma_{{\\rm ba},{\\rm g}}^2-2\\widetilde \\gamma_{{\\rm ba},{\\rm g}}^2)\\enorm{e_h}_{\\omega,\\mathcal T_h}^2\n+\n(\\rho^2-4\\widecheck \\gamma_{{\\rm ba},{\\rm d}}^2-2\\widetilde \\gamma_{{\\rm ba},{\\rm d}}^2)\\|u_h-\\pi_h^{\\rm g} u_h\\|_{\\dagger,1,\\mathcal T_h}^2\n\\leq\n\\enorm{e_h}_{\\omega,\\mathcal T_h}\\enorm{u-v_h}_{\\omega,\\Omega},\n\\end{equation*}\nfrom which \\eqref{eq_aprio_estimate} and \\eqref{eq_aprio_nonconf} follows,\nrecalling the definitions of $\\gamma_{{\\rm ba},{\\rm g}}$ and $\\gamma_{{\\rm ba},{\\rm d}}$ in \\eqref{eq_definition_gba}.\n\\end{proof}\n\n\n\\section{A posteriori analysis}\n\\label{section_aposteriori}\n\nWe now provide a posteriori error estimates under minimal regularity assumptions.\nWe first present an abstract framework, and then show how it can be applied to the\nparticular of residual-based estimators.\n\nThroughout this section, we assume that $u_h$ is a fixed function in\n$\\mathcal P_p(\\mathcal T_h)$ satisfying \\eqref{eq_helmholtz_ipdg}. Notice that unique\nsolvability of \\eqref{eq_helmholtz_ipdg} is not required. In particular,\nthe proposed analysis applies without any restriction on the mesh size\nor polynomial degree. Also, in contrast to the a priori analysis, we do\nnot need any specific positivity assumption on the stabilisation form $s_h$.\n\n\\subsection{Abstract reliability analysis}\n\n\\begin{subequations}\nFollowing \\cite[Theorem 3.3]{ern_vohralik_2015a}, the discretisation error\nmay be bounded using two different terms that respectively control the equation\nresidual, and the non-conformity of the discrete solution. For the equation\nresidual, we introduce the residual functional and its norm\n\\begin{equation}\n\\label{eq_definition_Rr}\n\\langle \\mathscr R_{\\rm r}, v\\rangle := b_h(e_h,v) \\quad \\forall v \\in H^1_{\\Gamma_{\\rm D}}(\\Omega),\n\\qquad\n\\textup R_{\\rm r}\n:=\n\\sup_{\\substack{v \\in H^1_{\\Gamma_{\\rm D}}(\\Omega) \\\\ \\|\\boldsymbol \\nabla v\\|_{\\boldsymbol A,\\Omega} = 1}}\n|\\langle \\mathscr R_{\\rm r},v\\rangle|.\n\\end{equation}\nNotice that this first term only involves conforming test functions.\nTo account for the non-conformity of the discrete solution, we additionally\nintroduce\n\\begin{equation}\n\\label{eq_definition_Rc}\n\\textup R_{\\rm c} := \\|u_h-\\pi^{\\rm g} u_h\\|_{\\dagger,1,\\mathcal T_h}.\n\\end{equation}\nThe total abstract estimator is then the Hilbertian sum of the two above components:\n\\begin{equation}\n\\label{eq_definition_R}\n\\textup R^2 := \\textup R_{\\rm r}^2 + \\textup R_{\\rm c}^2.\n\\end{equation}\n\\end{subequations}\n\nFollowing \\cite{chaumontfrelet_ern_vohralik_2021a,sauter_zech_2015a},\nour a posteriori analysis follows the lines of the Schatz argument \\cite{schatz_1974a}.\nSpecifically, we start by controlling weak norms of the errors with\nthe estimator and the approximation factor using a duality argument.\n\n\\begin{lemma}[Aubin-Nitsche]\nThe following estimates hold true:\n\\begin{equation}\n\\label{eq_aubin_nitsche_apost}\n\\omega \\|u-u_h\\|_{\\mu,\\mathcal T_h} \\leq \\widecheck \\gamma_{\\rm ba} \\textup R,\n\\qquad\n\\omega^{1\/2} \\|u-u_h\\|_{\\gamma,\\mathcal F_h^{\\rm R}} \\leq \\widetilde \\gamma_{\\rm ba} \\textup R.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nLet us set $e_h := u-u_h$.\nWe start with the first estimate in \\eqref{eq_aubin_nitsche_apost}.\nWe define $\\xi$ as the unique element of $H^1_{\\Gamma_{\\rm D}}(\\Omega)$ such that\n$b(w,\\xi) = \\omega (\\mu w,e_h)_{\\mathcal T_h}$ for all $w \\in H^1_{\\Gamma_{\\rm D}}(\\Omega)$.\nFollowing the lines the proof of Lemma \\ref{lemma_aubin_nitsche},\nwe write\n\\begin{align*}\nb_h(e_h,\\xi)\n&=\n-\\omega^2(\\mu e_h,\\xi)_{\\mathcal T_h}\n-i\\omega(\\gamma e_h,\\xi)_{\\mathcal F_h^{\\rm R}}\n+(\\mathfrak G(e_h),\\boldsymbol A\\boldsymbol \\nabla \\xi)_{\\mathcal T_h}\n\\\\\n&=\n(e_h,-\\omega^2 \\mu\\xi-\\div(\\boldsymbol A\\boldsymbol \\nabla \\xi))_{\\mathcal T_h}\n-(\\gamma e_h,\\boldsymbol A\\boldsymbol \\nabla\\xi\\cdot\\boldsymbol n)_{\\mathcal F_h^{\\rm R}}+\n(e_h,\\div(\\boldsymbol A\\boldsymbol \\nabla \\xi))+(\\mathfrak G(e_h),\\boldsymbol A\\boldsymbol \\nabla \\xi)_{\\mathcal T_h}\n\\\\\n&=\n\\omega \\|e_h\\|_{\\mu,\\mathcal T_h}^2\n+\n(\\mathfrak G(\\pi^{\\rm g} u_h -u_h),\\boldsymbol \\sigma-\\boldsymbol \\sigma_h)_{\\mathcal T_h}\n+\n(\\pi^{\\rm g} u_h-u_h,\\div(\\boldsymbol \\sigma-\\boldsymbol \\sigma_h))_{\\mathcal T_h},\n\\end{align*}\nfor all $\\boldsymbol \\sigma_h \\in \\boldsymbol{\\CP}_p(\\mathcal T_h) \\cap \\boldsymbol H(\\operatorname{div},\\Omega)$,\nwhere we employed \\eqref{eq_nonconf_identity} in the last identity. It follows that\n\\begin{equation*}\n\\omega \\|e_h\\|_{\\mu,\\mathcal T_h}^2\n\\leq\n|b_h(e_h,\\xi)| + \\|u_h-\\pi^{\\rm g} u_h\\|_{\\dagger,1,\\mathcal T_h}\\|\\boldsymbol A\\boldsymbol \\nabla \\xi-\\boldsymbol \\sigma_h\\|_{\\dagger,\\operatorname{div},\\mathcal T_h}.\n\\end{equation*}\nOn the one hand, recalling the Galerkin orthgonality property stated\nin \\eqref{eq_galerkin_orthogonality} and the definition of $\\textup R_{\\rm r}$\nat \\eqref{eq_definition_Rr}, we have\n\\begin{equation*}\n|b_h(e_h,\\xi)|\n=\n|b_h(e_h,\\xi-\\pi_h^{\\rm g}\\xi)|\n\\leq\n\\textup R_{\\rm r} \\|\\boldsymbol \\nabla(\\xi-\\pi_h^{\\rm g}\\xi)\\|_{\\boldsymbol A,\\Omega}\n\\leq\n\\widecheck \\gamma_{\\rm ba}^{\\rm g} \\textup R_{\\rm r} \\|e_h\\|_{\\mu,\\mathcal T_h}.\n\\end{equation*}\nOn the other hand, recalling \\eqref{eq_definition_gbad} and \\eqref{eq_definition_Rc}, we have\n\\begin{equation*}\n\\|u_h-\\pi^{\\rm g} u_h\\|_{\\dagger,1,\\mathcal T_h}\\|\\boldsymbol A\\boldsymbol \\nabla \\xi-\\pi_h^{\\rm d}(\\boldsymbol A\\boldsymbol \\nabla \\xi)\\|_{\\dagger,\\operatorname{div},\\mathcal T_h}\n\\leq\n\\widecheck \\gamma_{\\rm ba}^{\\rm d}\\textup R_{\\rm c}\\|e_h\\|_{\\mu,\\mathcal T_h}.\n\\end{equation*}\nThen, the first estimate of \\eqref{eq_aubin_nitsche_apost} follows from \\eqref{eq_definition_R}\nsince\n\\begin{equation*}\n\\widecheck \\gamma_{\\rm ba}^{\\rm g} \\textup R_{\\rm r} + \\widecheck \\gamma_{\\rm ba}^{\\rm d} \\textup R_{\\rm c}\n\\leq\n\\left (\n(\\widecheck \\gamma_{\\rm ba}^{\\rm g})^2 + (\\widecheck \\gamma_{\\rm ba}^{\\rm d})^2\n\\right )^{1\/2}\n\\left (\n\\textup R_{\\rm r}^2\n+\n\\textup R_{\\rm c}^2\n\\right )^{1\/2}\n=\n\\widecheck \\gamma_{\\rm ba} \\textup R.\n\\end{equation*}\n\nFor the sake of shortness, we do not detail the proof of the second estimate\nin \\eqref{eq_aubin_nitsche_apost} as it follows the lines of the first estimate\nbut using the function $\\chi$ from the proof of Lemma \\ref{lemma_aubin_nitsche}\ninstead of $\\xi$.\n\\end{proof}\n\nWe can now conclude the abstract reliability analysis.\nFollowing \\cite{ern_vohralik_2015a}, our proof uses a Pythagorean\nidentity to separate the conforming and non-conforming parts of the error.\n\n\\begin{theorem}[Abstract reliability]\nWe have\n\\begin{equation}\n\\label{eq_abstract_reliability}\n\\enorm{u-u_h}_{\\omega,\\mathcal T_h}\n\\leq\n\\sqrt{1+\\gamma_{\\rm ba}^2} \\textup R.\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nFor the sake of simplicity, we introduce\n\\begin{equation*}\nb^+_h(\\phi,v)\n:=\n\\omega^2 (\\mu\\phi,v)_{\\mathcal T_h}\n+\n\\omega (\\gamma\\phi,v)_{\\mathcal F_h^{\\rm R}}\n+\n(\\boldsymbol A\\mathfrak G(\\phi),\\mathfrak G(v))_{\\mathcal T_h}\n\\qquad\n\\forall \\phi,v \\in H^1(\\mathcal T_h),\n\\end{equation*}\nand we define $\\widetilde u$ as the only element of $H^1_{\\Gamma_{\\rm D}}(\\Omega)$ such that\n\\begin{equation*}\nb^+_h(\\widetilde u-u_h,v) = 0 \\qquad \\forall v \\in H^1_{\\Gamma_{\\rm D}}(\\Omega).\n\\end{equation*}\nNotice that this indeed a well-formed definition, since $b^+_h$ is equivalent\nto the usual inner-product of $H^1_{\\Gamma_{\\rm D}}(\\Omega)$. Classically, we have the Pythagorean identity\n\\begin{align}\n\\label{tmp_rel1}\n\\enorm{u-u_h}_{\\omega,\\mathcal T_h}^2\n&=\n\\enorm{\\widetilde u-u_h}_{\\omega,\\mathcal T_h}^2\n+\n2\\Re b^+(\\widetilde u - u_h,u-\\widetilde u)\n+\n\\enorm{u-\\widetilde u}_{\\omega,\\mathcal T_h}^2\n\\\\\n\\nonumber\n&=\n\\enorm{u_h-\\widetilde u}_{\\omega,\\mathcal T_h}^2\n+\n\\enorm{u-\\widetilde u}_{\\omega,\\mathcal T_h}^2.\n\\end{align}\nThen, on the one hand, it is clear using \\eqref{eq_norm_control} that\n\\begin{equation}\n\\label{tmp_rel2}\n\\enorm{u_h-\\widetilde u}_{\\omega,\\mathcal T_h}\n\\leq\n\\enorm{u_h-\\pi^{\\rm g} u_h}_{\\omega,\\mathcal T_h}\n\\leq\n\\|u_h-\\pi^{\\rm g} u_h\\|_{\\dagger,1,\\mathcal T_h}\n=\n\\textup R_{\\rm c},\n\\end{equation}\nand on the other hand, we have\n\\begin{align*}\n\\enorm{u-\\widetilde u}_{\\omega,\\Omega}^2\n&=\nb^+_h(u-\\widetilde u,u-\\widetilde u)\n=\nb^+_h(e_h,u-\\widetilde u)\n\\\\\n&=\nb_h(e_h,u-\\widetilde u)\n+\n2\\omega^2(\\mu e_h,u-\\widetilde u)_{\\mathcal T_h}\n+\n(1-i)\\omega (\\gamma e_h,u-\\widetilde u)_{\\mathcal F_h^{\\rm R}}\n\\\\\n&\\leq\n\\textup R_{\\rm r}\n\\|\\boldsymbol \\nabla(u-\\widetilde u)\\|_{\\boldsymbol A,\\Omega}\n+\n2\\widecheck \\gamma_{\\rm ba} \\textup R \\omega\\|u-\\widetilde u\\|_{\\mu,\\mathcal T_h}\n+\n\\sqrt{2}\\widetilde \\gamma_{\\rm ba} \\textup R \\omega\\|u-\\widetilde u\\|_{\\gamma,\\mathcal F_h^{\\rm R}}\n\\\\\n&\\leq\n\\left (\n\\textup R_{\\rm r}^2\n+\n(4\\widecheck \\gamma_{\\rm ba}^2 + 2\\widetilde \\gamma_{\\rm ba}^2) \\textup R^2\n\\right )^{1\/2}\n\\enorm{u-\\widetilde u}_{\\omega,\\Omega},\n\\end{align*}\nand recalling \\eqref{eq_definition_gba_summed}, we arrive at\n\\begin{equation}\n\\label{tmp_rel3}\n\\enorm{u-\\widetilde u}_{\\omega,\\Omega}^2\n\\leq\n\\textup R_{\\rm r}^2\n+\n(4\\widecheck \\gamma_{\\rm ba}^2+2\\widetilde \\gamma_{\\rm ba}^2) \\textup R^2\n=\n\\textup R_{\\rm r}^2\n+\n\\gamma_{\\rm ba}^2 \\textup R^2.\n\\end{equation}\nThe estimate in \\eqref{eq_abstract_reliability} then follows from\n\\eqref{tmp_rel1}, \\eqref{tmp_rel2} and \\eqref{tmp_rel3}.\n\\end{proof}\n\n\\subsection{Residual-based a posteriori estimator}\n\nThe residuals $\\textup R_{\\rm r}$ and $\\textup R_{\\rm c}$ can be straightforwardly\ncontrolled using flux and potential reconstructions. We refer the reader\nto \\cite{congreve_gedicke_perugia_2019a,ern_vohralik_2015a} for details\nabout these constructions. Here, we focus instead on residual-based\nestimators, for which the link may be less clear. In this section, the\nletter $C$ refers to a generic constant that may change from one occurrence\nto the other, and that only depends on the contrasts of the coefficients\n$\\mu$, $\\boldsymbol A$ and $\\gamma$ as well as the polynomial degree $p$ and the\nshape-regularity parameter $\\kappa$.\nThe stability result for the lifting operator\n\\begin{equation}\n\\label{eq_stability_lifting}\n\\|\\LIFT{v_h}\\|_{\\boldsymbol A,\\mathcal T_h}^2\n\\leq\nC\n\\sum_{K \\in \\mathcal T_h} \\frac{\\alpha_K}{h_K} \\left \\|\\jmp{v_h}\\right \\|_{\\partial K}^2\n\\quad\n\\forall v_h \\in \\mathcal P_p(\\mathcal T_h)\n\\end{equation}\ncan be found, e.g., in \\cite[Section 4.3]{dipietro_ern_2012a}\nor \\cite[Lemma 4.1]{houston_schotzau_wihler_2006a}.\n\nWe first establish that $\\textup R_{\\rm c}$ can be controlled by the jumps of $u_h$. The notations\n\\begin{equation*}\n\\frac{\\omega h_F^\\star}{\\vartheta_F^\\star}\n:=\n\\max_{F \\in \\mathcal F_h^{\\rm R}} \\frac{\\omega h_F}{\\vartheta_F},\n\\qquad\n\\frac{\\omega h_K^\\star}{\\vartheta_K^\\star}\n:=\n\\max_{K \\in \\mathcal T_h} \\frac{\\omega h_K}{\\vartheta_K},\n\\end{equation*}\nwill be useful.\n\n\\begin{lemma}[Control of the non-conformity]\nWe have\n\\begin{equation}\n\\label{eq_estimate_Rc}\n\\textup R_{\\rm c}^2\n\\leq\nC\n\\max \\left (\n1,\\frac{\\omega h_F^\\star}{\\vartheta_F^\\star},\\left (\\frac{\\omega h_K^\\star}{\\vartheta_K^\\star}\\right )^2\n\\right )\n\\sum_{K \\in \\mathcal T_h} \\frac{\\alpha_K}{h_K} \\|\\jmp{u_h}\\|_{\\partial K \\setminus (\\Gamma_{\\rm N} \\cup \\Gamma_{\\rm R})}^2.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nWe start with two estimates that are easily inferred from\n\\cite[Lemma 4.3]{ern_guermond_2017a}. For all $v_h \\in \\mathcal P_p(\\mathcal T_h)$,\nthere exists a conforming approximation $\\mathcal J v_h \\in \\mathcal P_p(\\mathcal T_h) \\cap H^1_{\\Gamma_{\\rm D}}(\\Omega)$\nsuch that and all $K \\in \\mathcal T_h$, we have\n\\begin{equation*}\n\\|v_h-\\mathcal J v_h\\|_K^2\n\\leq\nC h_K \\sum_{K' \\in \\widetilde \\mathcal T_K} \\|\\jmp{v_h}\\|_{\\partial K'}^2,\n\\qquad\n\\|\\boldsymbol \\nabla(v_h-\\mathcal J v_h)\\|_K^2\n\\leq\nC \\frac{1}{h_K} \\sum_{K' \\in \\widetilde \\mathcal T_K} \\|\\jmp{v_h}\\|_{\\partial K'}^2,\n\\end{equation*}\nwhere $\\widetilde \\mathcal T_K$ collects those elements $K' \\in \\mathcal T_h$ sharing at least\none vertex with $K$. Employing the multiplicative trace inequality\n\\begin{equation*}\n\\|\\theta\\|_{\\partial K}^2\n\\leq\nC \\left \\{\n\\frac{1}{h_K}\\|\\theta\\|_K^2 + \\|\\theta\\|_K\\|\\boldsymbol \\nabla \\theta\\|_K\n\\right \\}\n\\leq\nC \\left \\{\n\\frac{1}{h_K}\\|\\theta\\|_K^2 + h_K\\|\\boldsymbol \\nabla \\theta\\|_K^2\n\\right \\}\n\\qquad\n\\forall \\theta \\in H^1(K),\n\\end{equation*}\nwe also have\n\\begin{equation*}\n\\|v_h-\\mathcal J v_h\\|_{\\partial K}^2\n\\leq\nC \\sum_{K' \\in \\widetilde \\mathcal T_K} \\|\\jmp{v_h}\\|_{\\partial K'}^2.\n\\end{equation*}\nIt is therefore clear that\n\\begin{equation*}\n\\|u_h-\\mathcal J u_h\\|_{\\dagger,1,\\mathcal T_h}^2\n\\leq\nC \\max \\left (\n1,\\frac{\\omega h_F^\\star}{\\vartheta_F^\\star},\\left (\\frac{\\omega h_K^\\star}{\\vartheta_K^\\star}\\right )^2\n\\right )\n\\sum_{K \\in \\mathcal T_h} \\frac{\\alpha_K}{h_K} \\|\\jmp{u_h}\\|_{\\partial K}^2,\n\\end{equation*}\nand \\eqref{eq_estimate_Rc} follows from the definition of $\\textup R_{\\rm c}$ in \\eqref{eq_definition_Rc}.\n\\end{proof}\n\nWe then control the conforming residual $\\textup R_{\\rm r}$.\nThis is classically done by combining element-wise integration\nby parts and a quasi-interpolation operator.\n\n\\begin{lemma}[Control of the residual]\nWe have\n\\begin{multline}\n\\label{eq_estimate_Rr}\n\\textup R_{\\rm r}^2\n\\leq\nC\n\\\\\n\\Bigg \\{\n\\sum_{K \\in \\mathcal T_h}\n\\left (\n\\frac{h_K^2}{\\alpha_K}\n\\|f+\\omega^2 \\mu u_h + \\div (\\boldsymbol A\\boldsymbol \\nabla u_h)\\|_{\\mathcal T_h}^2\n+\n\\frac{h_K}{\\alpha_K}\n\\|\\jmp{\\boldsymbol A\\boldsymbol \\nabla u_h} \\cdot \\boldsymbol n_K\\|_{\\partial K \\setminus \\partial \\Omega}^2\n+\n\\frac{\\alpha_K}{h_K}\n\\|\\jmp{u_h}\\|_{\\partial K \\setminus \\partial \\Omega}^2\n\\right )\n\\\\\n+\\sum_{F \\in \\mathcal F_h^{\\rm D}} \\frac{\\alpha_F}{h_F} \\|u_h\\|_F^2\n+\\sum_{F \\in \\mathcal F_h^{\\rm N}} \\frac{h_F}{\\alpha_F} \\|\\boldsymbol A \\boldsymbol \\nabla u_h \\cdot \\boldsymbol n\\|_F^2\n+\\sum_{F \\in \\mathcal F_h^{\\rm R}} \\frac{h_F}{\\alpha_F} \\|\\boldsymbol A\\boldsymbol \\nabla u_h \\cdot \\boldsymbol n-i\\omega\\gamma u_h\\|_F^2\n\\Bigg \\}.\n\\end{multline}\n\\end{lemma}\n\n\\begin{proof}\nWe consider an arbitrary $v \\in H^1_{\\Gamma_{\\rm D}}(\\Omega)$, and let\n$\\widetilde v := v-\\mathcal Q_h v \\in H^1_{\\Gamma_{\\rm D}}(\\Omega)$, where $\\mathcal Q_h$\nis a quasi-interpolation operator satisfying\n\\begin{equation*}\n\\frac{1}{h_K^2} \\|v-\\mathcal Q_h v\\|_K^2\n+\n\\frac{1}{h_K}\\|v-\\mathcal Q_h\\|_{\\partial K}^2\n+\n\\|\\boldsymbol \\nabla(v-\\mathcal Q_h v)\\|_K^2\n\\leq\nC \\frac{1}{\\alpha_K} \\|\\boldsymbol \\nabla v\\|_{\\boldsymbol A,\\widetilde K},\n\\end{equation*}\nwith $\\widetilde K$ the domain corresponding to all the elements $K' \\in \\mathcal T_h$\nthat share at least one vertex with $K$. We refer the reader to, e.g.,\n\\cite[Theorem 5.2]{ern_guermond_2017a}, for the construction of $\\mathcal Q_h$.\nThanks to Galerkin orthogonality \\eqref{eq_galerkin_orthogonality}, we\ncan write that\n\\begin{align}\n\\label{tmp_rel_Rc0}\n&\\langle \\mathscr R_{\\rm r}, v \\rangle\n\\\\\n\\nonumber\n&=\nb_h(e_h,v)\n=\nb_h(e_h,\\widetilde v)\n\\\\\n\\nonumber\n&=\n(f,\\widetilde v)_\\Omega\n+\n\\omega^2 (\\mu u_h,\\widetilde v)_{\\mathcal T_h}\n+\ni\\omega (\\gamma u_h,\\widetilde v)_{\\mathcal F_h^{\\rm R}}\n+\n(\\boldsymbol A\\mathfrak G(u_h),\\boldsymbol \\nabla \\widetilde v)_{\\mathcal T_h}\n\\\\\n\\nonumber\n&=\n\\underbrace{(f,\\widetilde v)_\\Omega\n+\n\\omega^2 (\\mu u_h,\\widetilde v)_{\\mathcal T_h}\n+\ni\\omega (\\gamma u_h,\\widetilde v)_{\\mathcal F_h^{\\rm R}}\n-\n(\\boldsymbol A\\boldsymbol \\nabla u_h,\\boldsymbol \\nabla \\widetilde v)_{\\mathcal T_h}\n}_{r_1}\n-\n\\underbrace{(\\boldsymbol A\\LIFT{u_h},\\boldsymbol \\nabla \\widetilde v)_{\\mathcal T_h}}_{r_2}.\n\\end{align}\nThen, on the one hand, we see that\n\\begin{align*}\nr_1\n&=\n(f+\\omega^2\\mu u_h+\\div(\\boldsymbol A\\boldsymbol \\nabla u_h),\\widetilde v)_{\\mathcal T_h}\n+\ni\\omega(\\gamma u_h,\\widetilde v)_{\\mathcal F_h^{\\rm R}}\n-\n\\sum_{K \\in \\mathcal T_h} (\\boldsymbol A \\boldsymbol \\nabla u_h \\cdot \\boldsymbol n_K,\\widetilde v)_{\\partial K}\n\\\\\n&=\n(f+\\omega^2\\mu u_h-\\div(\\boldsymbol A\\boldsymbol \\nabla u_h),\\widetilde v)_{\\mathcal T_h}\n+\n\\sum_{F \\in \\mathcal F_h^{\\rm i}} (\\jmp{\\boldsymbol A \\boldsymbol \\nabla u_h} \\cdot \\boldsymbol n_F,\\widetilde v)_{F}\n\\\\\n&\n-(\\boldsymbol A \\boldsymbol \\nabla u_h \\cdot \\boldsymbol n,\\widetilde v)_{\\mathcal F_h^{\\rm N}}\n-(\\boldsymbol A \\boldsymbol \\nabla u_h \\cdot \\boldsymbol n - i\\omega\\gamma u_h,\\widetilde v)_{\\mathcal F_h^{\\rm R}},\n\\end{align*}\nand therefore,\n\\begin{align}\n\\label{tmp_rel_Rc1}\n|r_1|\n&\\leq\n\\sum_{K \\in \\mathcal T_h}\n\\left (\n\\|f+\\omega^2\\mu u_h+\\div\\left(\\boldsymbol A\\boldsymbol \\nabla u_h\\right )\\|_K\n\\|\\widetilde v\\|_K\n+\n\\|\\jmp{\\boldsymbol \\nabla u_h}\\cdot \\boldsymbol n_K\\|_{\\partial K}\\|\\widetilde v\\|_{\\partial K}\n\\right )\n\\\\\n\\nonumber\n&\n+\\sum_{F \\in \\mathcal F_h^{\\rm N}}\\|\\boldsymbol A\\boldsymbol \\nabla u_h\\cdot \\boldsymbol n\\|_F\\|\\widetilde v\\|_F\n+\\sum_{F \\in \\mathcal F_h^{\\rm R}}\\|\\boldsymbol A\\boldsymbol \\nabla u_h\\cdot \\boldsymbol n-i\\omega\\gamma u_h\\|_F\\|\\widetilde v\\|_F\n\\\\\n\\nonumber\n&\\leq\nC \\Bigg \\{\n\\sum_{K \\in \\mathcal T_h}\n\\left (\n\\frac{h_K^2}{\\alpha_K} \\|f+\\omega^2\\mu u_h+\\div\\left(\\boldsymbol A\\boldsymbol \\nabla u_h\\right )\\|_K\n+\n\\frac{h_K}{\\alpha_K}\\|\\jmp{\\boldsymbol \\nabla u_h}\\cdot \\boldsymbol n_K\\|_{\\partial K}\n\\right )\n\\\\\n\\nonumber\n&\n\\qquad\n+\\sum_{F \\in \\mathcal F_h^{\\rm N}}\\frac{h_F}{\\alpha_F}\\|\\boldsymbol A\\boldsymbol \\nabla u_h\\cdot \\boldsymbol n\\|_F\n+\\sum_{F \\in \\mathcal F_h^{\\rm R}}\\frac{h_F}{\\alpha_F}\\|\\boldsymbol A\\boldsymbol \\nabla u_h\\cdot \\boldsymbol n-i\\omega\\gamma u_h\\|_F\n\\Bigg \\} \\|\\boldsymbol \\nabla v\\|_{\\boldsymbol A,\\Omega}.\n\\end{align}\nOn the other hand, we have\n\\begin{align}\n\\label{tmp_rel_Rc2}\n|r_2|\n&\\leq\n\\|\\LIFT{u_h}\\|_{\\boldsymbol A,\\Omega}\\|\\boldsymbol \\nabla \\widetilde v\\|_{\\boldsymbol A,\\Omega}\n\\\\\n\\nonumber\n&\\leq\nC \\left \\{\n\\sum_{K \\in \\mathcal T_h} \\frac{\\alpha_K}{h_K} \\|\\jmp{u_h}\\|_{\\partial K \\setminus \\partial \\Omega}^2\n+\n\\sum_{F \\in \\mathcal F_h^{\\rm D}} \\frac{\\alpha_F}{h_F} \\|\\jmp{u_h}\\|_F^2\n\\right \\}\n\\|\\boldsymbol \\nabla v\\|_{\\boldsymbol A,\\Omega},\n\\end{align}\nand \\eqref{eq_estimate_Rr} follows from \\eqref{tmp_rel_Rc0}, \\eqref{tmp_rel_Rc1}\nand \\eqref{tmp_rel_Rc2} recalling the definition of $\\textup R_{\\rm r}$ in \\eqref{eq_definition_Rr}.\n\\end{proof}\n\nWe therefore define a residual-based estimator by\n\\begin{align*}\n\\eta^2\n&:=\n\\sum_{K \\in \\mathcal T_h}\n\\left (\n\\frac{h_K^2}{\\alpha_K}\n\\|f+\\omega^2 \\mu u_h - \\div (\\boldsymbol A\\boldsymbol \\nabla u_h)\\|_{\\mathcal T_h}^2\n+\n\\frac{h_K}{\\alpha_K}\n\\|\\jmp{\\boldsymbol A\\boldsymbol \\nabla u_h} \\cdot \\boldsymbol n_K\\|_{\\partial K \\setminus \\partial \\Omega}^2\n+\n\\frac{\\alpha_K}{h_K}\n\\|\\jmp{u_h}\\|_{\\partial K \\setminus \\partial \\Omega}^2\n\\right )\n\\\\\n&\n+\\sum_{F \\in \\mathcal F_h^{\\rm D}} \\frac{\\alpha_F}{h_F} \\|u_h\\|_F^2\n+\\sum_{F \\in \\mathcal F_h^{\\rm N}} \\frac{h_F}{\\alpha_F} \\|\\boldsymbol A \\boldsymbol \\nabla u_h \\cdot \\boldsymbol n\\|_F^2\n+\\sum_{F \\in \\mathcal F_h^{\\rm R}} \\frac{h_F}{\\alpha_F} \\|\\boldsymbol A\\boldsymbol \\nabla u_h \\cdot \\boldsymbol n\\|_F^2.\n\\end{align*}\nAs a direct consequence of \\eqref{eq_abstract_reliability},\n\\eqref{eq_estimate_Rc} and \\eqref{eq_estimate_Rr}, we obtain a\nreliability estimate stated in Theorem \\ref{theorem_reliability_residual}.\nIt is to be compared with \\cite[Theorem 2.3]{chaumontfrelet_ern_vohralik_2021a}\nand \\cite[Theorem 3.6]{sauter_zech_2015a}.\n\n\\begin{theorem}[Reliability of the residual estimator]\n\\label{theorem_reliability_residual}\nWe have\n\\begin{equation*}\n\\enorm{e_h}_{\\omega,\\mathcal T_h}\n\\leq\nC\n\\max \\left (\n1,\\frac{\\omega h_F^\\star}{\\vartheta_F^\\star},\\left (\\frac{\\omega h_K^\\star}{\\vartheta_K^\\star}\\right )^2\n\\right )\n(1+\\gamma_{\\rm ba}) \\eta.\n\\end{equation*}\n\\end{theorem}\n\n\\subsection{Efficiency}\n\nFor the sake of shortness, we do not give any efficiency proofs.\nActually, such proofs can be found in \\cite[Section 4]{congreve_gedicke_perugia_2019a}\nfor equilibrated estimators and in \\cite[Section 3.3]{sauter_zech_2015a} for the\nresidual-based estimators considered here.\n\n\\bibliographystyle{amsplain}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nFinancial time series are hard to model, since they are heavily influenced by unpredictable events.\nNatural disasters, uncertainty about public behaviour, statements from governments and central banks, etc., are all events that can drastically affect the market. As a consequence financial data do not behave the same at all times, hence we cannot assume any stationarity property for them. The latter implies that classic techniques used to analyse time series are widely inadequate to model such data, therefore alternative methods have to be developed. The family of \\emph{Markov Switching Models} (MSM) constitutes a possible solution, since these models allow us to effectively address the non-stationarity of financial data.\n\nThe main idea behind the MSM is that, in order to take into account  the changes in the behaviour of the data, we allow the distribution of the observations to change over time. A general MSM model can be written in the following form\n\\begin{equation}\\label{generalMSM}\n\\begin{cases}\ny_t = f(S_t,\\theta,\\psi_{t-1})\\\\\nS_t = g\\left(\\tilde{S}_{t-1},\\psi_{t-1}\\right) \\\\\nS_t \\in \\Lambda\n\\end{cases}\n\\end{equation}\nwhere $S_{t}$ indicates the state of the model at time $t$, $\\theta$ is the vector of the parameters characterizing the model, $\\psi_t:=\\left\\{y_k: k=1,\\dots,t\\right\\}$ is the set of all observations up to time $t$, $\\tilde{S}_{t} := \\{S_{1},...,S_{t}\\}$ is the set of all observed states up to time $t$, $\\Lambda=\\{1,...,M\\}$ is the set of all possible states, and $g$ is the function that governs the transitions between the states. The function $f$ defines how the observation at time $t$ depends on $S_t,\\theta,\\ \\text{and}\\ \\psi_{t-1}$ and finally, $t \\in \\{0,1,...,T\\}$, where $T \\in \\mathbb{N}$, $T < +\\infty$, is the so called \\emph{terminal time}.\nSystem \\eqref{generalMSM} clearly shows the intrinsic richness of the MSM approach. Particular realizations of \\eqref{generalMSM} allow  the treatment of specific problems. Before getting into the details of our study, it is worth mentioning that in most of the dedicated  literature, we can distinguish between two classes of models. The first class consists of models that have complicated distributions for the data or a large number of states, but very simple transition laws, e.g.,  a first order Markov chain, see, e.g., \\cite{DiPMF16,Ham89,Ch98}. The second class is made up of models with simple assumptions and very few states, usually two, but with more complicated transition laws, see, e.g., \\cite{Kim99, DiLeeW94, Pe05}. \n\nThe present paper is structured as follows: in Sections 1 through 4 we cover the mathematical and computational theory needed to establish the techniques that we then develop in subsequent sections; in Section 5 we introduce the jump-diffusion model, while in Section 6 we present a model that uses $\\alpha-$stable distributions;  in Section 7 we explain how the models can be concretely implemented and, in Section 8, we present the related results obtained applying them to a relevant case study which concerns the S\\&P500 index; conclusions and further developments are outlined in Section 9.\n\n\\section{Bayesian Inference}\nBayesian Inference is a branch of statistical inference that assumes the parameters of a probability distribution to be randomly distributed according to a {\\it prior} distribution. In particular the idea is to exploit the observed data, along with the Bayes rule, to generate the {\\it posterior} distribution of the aforementioned parameters. Therefore, the posterior distribution can be interpreted as the distribution of the parameters once we have taken into account both our subjective belief about them, namely the prior, and the data. Such an approach can be rigorously represented as follows\n\\begin{align}\n& \\nonumber \\theta \\sim \\pi(\\theta) \\,;\\\\\n& y|\\theta \\sim f(y|\\theta) \\,;\\\\\n& \\nonumber f(\\theta|y) = \\frac{\\pi(\\theta)f(y|\\theta)}{f(y)} \\, ,\n\\end{align}\nwhere $\\pi(\\theta)$ is the prior distribution, $f(y|\\theta)$ is the distribution of the data depending on the parameter $\\theta$, and $f(\\theta|y)$ is the posterior of $\\theta$. Finally, $f(y)$ is the marginal distribution of $y$, namely\n\\begin{equation*}\nf(y) = \\int f(y,\\theta) d\\theta = \\int \\pi(\\theta)f(y|\\theta) d\\theta\\:.\n\\end{equation*}\nClearly the choice of the prior can have a large impact on the posterior. A particularly convenient form of prior is what is known as a \\emph{conjugate prior}. We say that a prior distribution is conjugate if the posterior distribution derived from it belongs to the same family, as it happens, e.g., for the Beta-Bernoulli pair, namely\n\\begin{equation}\n\\begin{aligned}\n\\pi(\\theta) &= \\theta^{\\alpha-1}  (1-\\theta)^{\\beta-1} \\,;\\\\ \nf(y|\\theta) &= \\binom{n}{y} \\theta^{y} (1-\\theta)^{n-y} \\,;\\\\\nf(\\theta|y) & \\propto \\theta^{(\\alpha + y)-1} (1-\\theta)^{(\\beta+n-y)-1} \\,;\\\\\n& \\propto \\text{Beta}(\\alpha + y,\\beta+n-y)\\,.\n\\end{aligned}\n\\end{equation}\n\nIt follows that if we start with a $\\text{Beta}(\\alpha, \\beta)$ prior and assume that the data are binomially distributed, we end up with a $\\text{Beta}(\\alpha + y,\\beta+n-y)$ posterior. Hence, we do not have to update the distribution for each new observation, just its parameters. We would like to note that the latter is a particularly relevant aspect from the algorithmic point of view since it translates into less computationally expensive code. For the sake of completeness, in the following subsections we list other particularly convenient choices for distribution pairs and, in order to give clear examples, we first start by explaining how the posterior of a set of independent identically distributed (i.i.d) random variables is obtained. Let $\\{y_{1},...,y_{n}\\},\\ n \\in \\mathbb{N}$ be a set of i.i.d. random variables with  \ndensity function $f(y|\\theta)$. Moreover, let $\\theta \\sim \\pi(\\theta)$. Then\n\\begin{equation}\nf(\\theta|y_{1},...,y_{n}) = \\frac{\\pi(\\theta) L_{n}(\\theta)}{\\int\\limits_{\\Theta} \\pi(\\theta) L_{n}(\\theta) d\\theta} \\propto \\pi(\\theta) L_{n}(\\theta)\\;,\n\\end{equation}\nwhere $$L_{n}(\\theta) = \\prod\\limits_{i=1}^{n} f(y_{i}|\\theta)\\;,$$ is the likelihood function of the data, and $\\Theta$ is the set of all possible values of $\\theta$. For the rest of this paper we will denote the vector of observations by $\\boldsymbol y$. \n\n\\subsection{Normal-Normal}\n\\label{nn}\nAssume that we have  $n \\in \\mathbb{N}$ independent observations\n\\begin{align*}\ny_{i}| \\mu \\sim \\mathcal{N}(\\mu,\\sigma^{2}),\\ i \\in \\{1,...,n\\} \\;,\n\\end{align*}\nwhere $\\mu \\in \\mathbb{R}$, and the value of $\\sigma>0$ is known. In order to perform Bayesian inference on the given data, we also need to place a distribution on $\\mu$. Therefore, we set\n\\begin{align*}\n\\mu &\\sim \\mathcal{N} \\left(\\mu_{0},\\frac{1}{k}\\right)\\:.\n\\end{align*}\nThe corresponding likelihood function is\n\\begin{align*}\nL_{n}(\\mu) = \\prod\\limits_{i=1}^{n} \\frac{1}{\\sqrt{2 \\pi \\sigma^{2}}} e^{- \\frac{1}{2} \\left( \\frac{y_{i}-\\mu}{\\sigma} \\right)^{2}} &\\propto e^{- \\frac{1}{2\\sigma^{2}} \\sum\\limits_{i=1}^{n} \\left( y_{i}-\\mu \\right)^{2}} \\\\\n\\propto e^{- \\frac{1}{2\\sigma^{2}} \\left( \\sum\\limits_{i=1}^{n} (y_{i}-\\bar{y})^{2} + n(\\bar{y}-\\mu)^{2} \\right)} &\\propto e^{- \\frac{n}{2\\sigma^{2}}(\\bar{y}-\\mu)^{2}}\n\\:,\n\\end{align*}\nwhile for the posterior we have\n\\begin{align*}\n& f(\\mu|\\boldsymbol y) \\propto e^{- \\frac{n}{2\\sigma^{2}}(\\bar{y}-\\mu)^{2}} e^{- \\frac{k}{2} \\left( \\mu-\\mu_{0} \\right)^{2}} \\\\ \n& = e^{- \\frac{1}{2} \\left( \\tfrac{n}{\\sigma^{2}}(\\bar{y}-\\mu)^{2} + k(\\mu-\\mu_{0})^{2} \\right) } \\\\\n&= e^{-\\frac{1}{2}\\left( \\left(\\frac{n}{\\sigma^{2}}+k\\right) \\left( \\mu - \\frac{\\frac{n}{\\sigma^{2}}\\bar{y} + k \\mu_{0}}{\\frac{n}{\\sigma^{2}}+k} \\right)^2 + \\frac{\\frac{nk}{\\sigma^{2}}}{\\frac{n}{\\sigma^{2}}+k} (\\bar{y}-\\mu_{0})^{2} \\right)  } \\\\\n& \\propto \\mathcal{N}\\left(\\frac{\\frac{n}{\\sigma^{2}}\\bar{y} + k \\mu_{0}}{\\frac{n}{\\sigma^{2}}+k}, \\frac{\\sigma^{2}}{n+k \\sigma^{2}}\\right)\\;,\n\\end{align*}\nhence \n\\begin{equation}\n\\boldsymbol y \\sim \\mathcal{N} \\left(\\frac{n\\bar{y}+\\mu_{0}k\\sigma^{2}}{n+k\\sigma^{2}},\\frac{\\sigma^{2}}{n+k \\sigma^{2}} \\right)\\:.\n\\end{equation} \n\n\\subsection{Inverse Gamma-Normal}\n\\label{ign}\nAssume again that we have  $n \\in \\mathbb{N}$ independent observations\n\\begin{align*}\ny_{i}| \\mu \\sim \\mathcal{N}(\\mu,\\sigma^{2}),\\ i \\in \\{1,...,n\\},\\ n \\in \\mathbb{N}\\;,\n\\end{align*}\nbut, this time, $\\mu \\in \\mathbb{R}$ is known, while $\\sigma>0$ is unknown. Taking $\\sigma^{2}$ to be inverse-gamma distributed with parameters $\\alpha_{0}$ and $\\beta_{0}$, and denoting the distribution by $\\text{inv}\\Gamma(\\alpha_{0},\\beta_{0})$, we can write the density function of $\\sigma$ as follows\n\\begin{equation}\nf(\\sigma) = \\frac{\\beta_{0}^{\\alpha_{0}}}{\\Gamma(\\alpha_{0})} \\left( \\sigma^{2} \\right)^{-\\alpha_{0}-1} e^{-\\frac{\\beta_{0}}{\\sigma^{2}}}\\;,\n\\end{equation}\nwhere $\\Gamma(t)$ is the extension of the factorial to the set of positive real numbers, known as the {\\it Gamma function}, and defined by\n\\begin{equation*}\n\\Gamma(t) := \\int\\limits_{0}^{\\infty} x^{t-1} e^{-x} dx\\:.\n\\end{equation*}\nTherefore, the associated likelihood function is\n\\begin{equation*}\nL_{n}(\\sigma) \\propto \\left( \\sigma^{2} \\right)^{-\\frac{n}{2}} e^{- \\frac{1}{2\\sigma^{2}} \\sum\\limits_{i=1}^{n} \\left( y_{i}-\\mu \\right)^{2}}\\;,\n\\end{equation*}\nhence, the posterior is\n\\begin{equation}\n\\begin{aligned}\nf(\\sigma| \\boldsymbol y)  \\propto \\left( \\sigma^{2} \\right)^{-\\frac{n}{2}-\\alpha_{0}-1} e^{- \\frac{1}{2\\sigma^{2}} \\sum\\limits_{i=1}^{n} \\left( y_{i}-\\mu \\right)^{2}} e^{-\\frac{\\beta_{0}}{\\sigma^{2}}} \\\\\n\\propto \\text{inv}\\Gamma \\left(\\frac{n}{2}+\\alpha_{0}, \\frac{1}{2} \\sum\\limits_{i=1}^{n} \\left( y_{i}-\\mu \\right)^{2} + \\beta_{0} \\right)\n\\:.\n\\end{aligned}\n\\end{equation}\n\\begin{remark}\n\tUnfortunately, not all distribution pairs are as convenient as the previously mentioned ones, especially from the point of view of the parameter simulation needed by concrete computational studies. When the posterior is a well known distribution, as in the {\\it normal-normal} and {\\it inverse gamma-normal} cases, we can simulate the parameters using, e.g., existing R libraries. Otherwise, {\\it ad hoc} sampling algorithms have to be developed. The next section addresses these problems.\n\n\\end{remark}\n\n\\section{Markov Chain Monte Carlo}\nIn this section, we describe two methods that will be used to sample the parameters, namely, the \\emph{Gibbs Sampling Method} and the \\emph{Metropolis-Hastings Algorithm}. The latter will be used in situations where the posterior distribution is non-standard, while the former will be used when the distribution can be simulated using an existing software.\n\n\\subsection{Gibbs Sampling}\nAssume that we have a model with a finite number $k$ of parameters, $\\boldsymbol \\theta = (\\theta_{1},...,\\theta_{k})$, and that we want to find the full posterior distribution $f(\\theta_{1},...,\\theta_{k}|\\boldsymbol y)$. This goal can be quite difficult to reach, since the multivariate simulation of distributions is much more tangled and computationally heavy than its univariate counterpart. The Gibbs sampling approach allows the sampling of $f(\\theta_{1},...,\\theta_{k}|\\boldsymbol y)$, knowing only the conditional distributions $f(\\theta_{i}|\\theta_{1},...,\\theta_{i-1},\\theta_{i+1},...,\\theta_{k}, \\boldsymbol y), i \\in \\{1,...,k\\}$.\n\nLet $N$ be the number of simulations we want to perform. We assign arbitrary starting values $(\\theta_{1}^{0},...,\\theta_{k}^{0})$ to each of the parameters. Then, for every $j \\in \\{1,...,N\\}$, we perform  the following steps\n\\begin{align*}\n\\textbf{Step 1:}\\ & \\text{Draw}\\ \\theta_{1}^{j}\\ \\text{from}\\ f(\\theta_{1}^{j}|\\theta_{2}^{j-1},...,\\theta_{k}^{j-1}, \\boldsymbol y)\\,; \\\\\n\\textbf{Step 2:}\\ & \\text{Draw}\\ \\theta_{2}^{j}\\ \\text{from}\\ f(\\theta_{2}^{j}|\\theta_{1}^{j}, \\theta_{3}^{j-1},...,\\theta_{k}^{j-1}, \\boldsymbol y)\\,; \\\\\n\\vdots \\\\\n\\textbf{Step k:}\\ & \\text{Draw}\\ \\theta_{k}^{j}\\ \\text{from}\\ f(\\theta_{k}^{j}|\\theta_{1}^{j},...,\\theta_{k-1}^{j}, \\boldsymbol y)\\,;\n\\end{align*}\nhence we can simulate each of the model parameters. The first $J$ simulations are discarded, being part of what is called the \\emph{burn in period}, in order to get rid of the dependence on the arbitrary choice of the starting point $(\\theta_{1}^{0},...,\\theta_{k}^{0})$, while the remaining $N-J$ values are assumed to be a suitable approximation of the real distribution. It is worth mentioning that the number of iterations, as well as the length of the {\\it burn in period}, should be chosen carefully, since for larger values of $N$ the simulations become too time consuming, while small values might not provide enough iterations for the sampler to converge.\n\n\\subsection{Metropolis-Hastings Algorithm}\n\\label{sbsec:metropolis_hastings}\nThe Gibbs sampler is rather easy to implement, but its major drawback is that it requires each $f(\\theta_{i}| \\boldsymbol \\theta_{-i}, \\boldsymbol y)$ to be readily samplable, where $\\boldsymbol \\theta_{-i}$ is the vector $\\boldsymbol \\theta \\backslash \\{\\theta_{i}\\}$. The Metropolis-Hastings algorithm allows for a solution to such an inconvenience. In particular, it only requires a function $f^{\\star}(\\theta_{i}|\\boldsymbol \\theta_{-i}, \\boldsymbol y)$  proportional to the density function $f(\\theta_{i}|\\boldsymbol \\theta_{-i}, \\boldsymbol y)$, and a \\emph{proposal distribution} $q(\\cdot| \\boldsymbol \\theta)$ which denotes a proper probability density function defined on the space $\\Theta$ of all possible values of $\\boldsymbol \\theta$.\nIn what follows we provide the description of the general Metropolis-Hastings algorithm, which uses the full parameter vector $\\boldsymbol \\theta$, as it is reported in \\cite{Car}. We underline that the algorithm remains unchanged when  $\\boldsymbol \\theta$ is a scalar. \n\nLet $N$ be the number of simulations we want to perform. We assign an arbitrary starting value $\\boldsymbol \\theta_{0}$ to the parameter vector. Then, for every $j \\in \\{1,...,N\\}$, we perform  the following steps\n\\begin{align*}\n& \\textbf{Step 1:}\\ \\text{Draw}\\ \\boldsymbol \\theta_{new}\\ \\text{from}\\ q(\\cdot| \\boldsymbol \\theta^{j-1})\\,; \\\\\n& \\textbf{Step 2:}\\ \\text{Compute the ratio}\\ r =\\frac{f^{\\star}(\\boldsymbol \\theta_{new}) q(\\boldsymbol \\theta^{j-1}|\\boldsymbol \\theta^{new})}{f^{\\star}(\\boldsymbol \\theta^{j-1})q(\\boldsymbol \\theta_{new}|\\boldsymbol \\theta^{j-1})} \\,;\\\\\n& \\textbf{Step 3:}\\ \\text{Define}\\ p = \\min \\left\\lbrace 1,r \\right\\rbrace \\,;\\\\\n& \\textbf{Step 4:}\\ \\text{Set}\\ \\boldsymbol \\theta^{j} := \n\\begin{cases}\n\\boldsymbol \\theta^{new}, &\\text{with probability}\\ p \\\\\n\\boldsymbol \\theta^{j-1}, &\\text{with probability}\\ 1-p\n\\end{cases}\\,.\n\\end{align*}\nHaving defined a method to sample the parameters, we now have the task of simulating the states of the models we will be using. This problem is the subject of the next section.\n\n\\section{State Simulation}\n\\label{chap:states}\nIn this section our goal is to simulate the state vector $\\tilde{S}_{T}$. In order to accomplish this, we first need to obtain the values $\\mathbb{P}(S_{1}|\\tilde{y}_{1}),...,\\mathbb{P}(S_{T}|\\tilde{y}_{T})$. We start by setting arbitrary values for the parameters, and then we use the following expression\n\\begin{equation}\n\\begin{aligned}\ng(\\tilde{S}_{T}|\\tilde{y}_{T}) &= \ng(S_{T}|\\tilde{y}_{T}) \\prod_{t=1}^{T-1} g(S_{t}|S_{t+1}, \\tilde{y}_{t}) \\\\\n& = g(S_{T}|\\tilde{y}_{T}) \\prod_{t=1}^{T-1} g(S_{t+1}|S_{t}) g(S_{t}| \\tilde{y}_{t})\\:.\n\\end{aligned}\n\\end{equation}\n\nNotice  that, $\\forall t \\in \\{1,...,T-1\\}$, we can sample from $\\tilde{S}_{T}$ if we have $g(S_{t+1}|S_{t})$, which is nothing more than the transition probability from one state to another,  and $g(S_{t}| \\tilde{y}_{t})$. The latter can be obtained, $\\forall t \\in \\{1,...,T\\}$, exploiting the Hamilton filter, see below.\n\\subsection*{Hamilton filter}\nThe basic Hamilton filter, see \\cite{Ham89}, can be described as input-output-byproduct.\n\n\\begin{flushleft}\n\t\\textbf{input:} $g(S_{t-1}=s_{t-1}|\\tilde{y}_{t-1})\\;;$\n\t\n\t\\textbf{output:} $g(S_{t}=s_{t}|\\tilde{y}_{t})\\;;$\n\t\n\t\\textbf{byproduct:} $f(y_{t}|\\tilde{y}_{t-1})\\:.$\n\\end{flushleft}\nRunning the Hamilton filter for $t \\in \\{1,...,T\\}$, we get the desired values $g(S_{1}| \\tilde{y}_{1}),..., g(S_{T}|\\tilde{y}_{T})$, which can be used  to generate $\\tilde{S}_{T}$, as described in what follows\n\\begin{align*}\n\\mathbb{P} (S_{T}=i|\\tilde{y}_{T}) &= \\frac{g(S_{T}=i|\\tilde{y}_{T})}{\\sum\\limits_{j=1}^{4} g(S_{T}=j|\\tilde{y}_{T})}\\;,\n\\end{align*}\nsuch a  probability is used to draw a sample of $S_{T}$, i.e.\n\\begin{align*}\n& \\mathbb{P} (S_{T-1}=i|\\tilde{y}_{T-1}) = \\frac{g(S_{T-1}=i|\\tilde{y}_{T-1})}{\\sum\\limits_{j=1}^{4} g(S_{T-1}=j|\\tilde{y}_{T-1})} \\\\ &= \\frac{g(S_{T}|S_{T-1}=i) g(S_{T-1}=i| \\tilde{y}_{T-1})}{\\sum\\limits_{j=1}^{4} g(S_{T}|S_{T-1}=j) g(S_{T-1}=j| \\tilde{y}_{T-1})}\\;,\n\\end{align*}\nthen, the above probability together with the previously simulated $S_{T}$, are both used to simulate $S_{T-1}$, and, proceeding iteratively, we have\n\\begin{align*}\n&\\ \\ \\vdots \\\\\n\\mathbb{P} (S_{1}=i|\\tilde{y}_{1}) & = \\frac{g(S_{1}=i|\\tilde{y}_{1})}{\\sum\\limits_{j=1}^{4} g(S_{1}=j|\\tilde{y}_{1})} \\\\ & = \\frac{g(S_{2}|S_{1}=i) g(S_{1}=i| \\tilde{y}_{1})}{\\sum\\limits_{j=1}^{4} g(S_{2}|S_{1}=j) g(S_{1}=j| \\tilde{y}_{1})}\\;,\n\\end{align*}\ntherefore,\nwe can simulate $S_{1}$ obtaining  the last component of $\\tilde{S}_{T}$. The latter implies  that, for every $t \\in \\{1,...,T\\}$, we know what the distribution of $y_{t}$ is, because we know what the state we are in is. In the next two sections we present the models that will be used later.\n\n\\section{Jump Diffusion Model}\\label{JumpDiffusionModel}\nIn the paper {\\it The Variation of Certain Speculative Prices}, see \\cite{Mb63}, Benoit Mandelbrot draws attention to the fact that the normal distribution is inadequate when it comes to describing economic and financial data. \n\\begin{figure}[!h]\n\t\\includegraphics[width = \\columnwidth, height = .2\\paperheight]{\"normal_inad\"}\n\t\\caption{ \\small{The histogram of the weekly log returns of the S\\&P500 and the density of the normal distribution obtained by the maximum likelihood method. We see that the normal density has to {\\it sacrifice} values around the mean  to cover the values at the tails.} }\n\\end{figure}\nHe argues that although the histograms of price changes seem to behave according to a Gaussian distribution,  a more careful analysis reveals that the large number of outliers makes the normal distribution fitted to the data much flatter than the actual data are, and with not enough density at the tails to include all the extreme values. If one  tries to manipulate the variance of the Gaussian distribution  to accommodate the values around the mean, then the result is a distribution that is even worse than the previous one where the extreme values are concerned. \n\\begin{figure}[!h]\n\t\\includegraphics[width = \\columnwidth, height = .2\\paperheight]{\"normal_peak\"}\n\t\\caption{ \\small{The histogram of the weekly log returns of the S\\&P500 and the density of the normal distribution with modified variance. In this case the larger values, in absolute terms,  are severely underrepresented by the density.} }\n\\end{figure}\nIn what follows we will show how to solve the aforementioned issue by using a Gaussian distribution, to model the values around the mean, plus  jumps of stochastic intensity, to include outlying values. Specifically, our model is the following\n\\begin{equation}\\label{model}\n\\begin{cases}\ny_t = \\epsilon_{t} + \\delta \\sum\\limits_{i=1}^{N_{t}} z_{i} \\\\\n\\epsilon_{t} \\sim \\mathcal{N}(\\mu_{S_{t}}, \\sigma_{S_{t}}^{2}) \\\\\nz_{i} \\stackrel{i.i.d.}{\\sim} \\mathcal{E}(b) \\\\\nN_{t} \\sim \\mathcal{P}(\\theta_{S_{t}}) \\\\\n\\delta \\in \\{-1,1\\},\\ \\mathbb{P} (\\delta=-1) = \\mathbb{P} (\\delta=1) = \\frac{1}{2} \\\\\n\\mu_{S_{t}} = \\mu_{j}\\ \\text{if}\\ S_{t}=j,\\ \\forall j \\in \\{1,...,M\\} \\\\\n\\sigma_{S_{t}} = \\sigma_{j}\\ \\text{if}\\ S_{t}=j,\\ \\forall j \\in \\{1,...,M\\} \\\\\nb >0 \\\\\n\\theta_{S_{t}} = \\theta_{j}\\ \\text{if}\\ S_{t}=j,\\ \\forall j \\in \\{1,...,M\\} \\\\\nS_{t} \\in \\{1,...,M\\} \\\\\np_{ij} = \\mathbb{P} (S_{t}=j|S_{t-1}=i) \\\\\n\\pi_{0} = [\\mathbb{P} (S_{0}=1),...,\\mathbb{P} (S_{0}=M)]\n\\end{cases}\n\\end{equation}\nWe divide the analysis of the model defined in \\eqref{model} into two components, the {\\it Gaussian component} and the {\\it jump component}.\n\n\\subsection{Gaussian Element}\\label{GaussianElement}\nWe will use the Gaussian distribution to model most of the data by means of the random variable $\\epsilon_{t} \\sim \\mathcal{N}(\\mu_{S_{t}}, \\sigma_{S_{t}}^{2})$, where both the mean and the variance of $\\epsilon_{t}$ are state dependent. In particular, we define the state dependence of the mean as follows\n\\begin{equation}\n\\mu_{S_{t}} = \\mu_{j}\\ \\text{, if}\\ S_{t}=j, \\, \\forall j \\in \\{1,...,M\\}\\;,\n\\end{equation}\nhence each state has its own, constant mean, without further restrictions. Concerning the variance, we assume that it increases depending on the state, namely\n\\begin{equation}\n\\sigma_{S_{t}}^{2} =\n\\begin{cases}\n\\sigma^{2}_{1} , &S_{t} = 1 \\\\\n\\sigma^{2}_{1} \\prod\\limits_{i=2}^{S_{t}} (1+h_{i}), &S_{t} \\in \\{2,...,M\\}\n\\end{cases}\n\\;,\n\\end{equation}\nwhere, $\\forall i \\in \\{2,3,...,M\\}$, $h_{i}>0$, which gives us\n\\begin{equation}\n\\label{sigmainequality}\n\\sigma_{1}^{2} < \\sigma_{2}^{2} < ... < \\sigma_{M}^{2}\\;,\n\\end{equation}\nhence, by \\eqref{sigmainequality}, as we go up in states we also go up in volatility.\n\n\\subsection{Jump Element}\\label{JumpElement}\nJump diffusion models, first introduced into finance by Robert C. Merton in \\cite{Mert75}, are currently widely accepted as an effective way to model the behaviour of financial data, see, e.g., \\cite{Bates96, It14}. \nIn order to incorporate the jump feature in our model, we have to deal with two major difficulties. First, we have to find a distribution under which the sum of independent random variables behaves well, at least from the point of view of real statistical applications. This task is not as straightforward as it may seem, since even the sum of i.i.d. uniform random variables has a distribution that rapidly grows in complexity with the number of addends. To overcome this particular problem, we have chosen to exploit the exponential distribution to model  the i.i.d. jump amplitudes, as the sum of i.i.d. exponential random variables follows a {\\it Gamma distribution}, namely\n\\begin{equation}\\label{Gamma}\nz_{i} \\stackrel{i.i.d.}{\\sim} \\mathcal{E}(b),\\ 1 \\leq i \\leq N \\Rightarrow \\sum\\limits_{i=1}^{N}z_{i} \\sim \\Gamma(N,b)\\;,\n\\end{equation}\nwhere $\\Gamma(\\alpha, \\beta)$ is the Gamma distribution in the $(\\alpha, \\beta)$ parameterization, while $b>0$ and $N$ are given natural numbers.\n\nThe aforementioned choice leads us to the second problem, which concerns the sign of the jumps. Obviously, financial shocks  can have both positive  and negative values, while the Gamma distribution allows only for the positive ones. The issue can be solved by multiplying the  sum in \\eqref{Gamma} by a random variable $\\delta$ taking values in $\\{-1,1\\}$, with equal probability. We refer to the resulting distribution as the \\emph{symmetric Gamma distribution} and we denote it by $\\text{sym}\\Gamma(\\alpha, \\beta)$. Assuming now that $X \\sim \\text{sym}\\Gamma(\\alpha, \\beta)$, the probability density function of $X$ is given by\n\\begin{equation}\nf_{X}(x) = \\frac{\\beta^{\\alpha}}{2 \\Gamma(\\alpha)} |x|^{\\alpha-1} e^{-\\beta |x|},\\ x \\in \\mathbb{R}\\;,\n\\end{equation}\nhence $X$ has mean equal to zero, and variance\n\\begin{equation}\\label{GammaVar}\n\\begin{aligned}\n\\mathbb{V}(X) & = \\mathbb{E}(X^{2}) - \\mathbb{E}(X)^{2} = \\mathbb{E}(X^{2}) \\\\ \n& = \\int\\limits_{-\\infty}^{\\infty} x^{2} \\frac{\\beta^{\\alpha}}{2 \\Gamma(\\alpha)} |x|^{\\alpha-1} e^{-\\beta |x|} dx \\\\\n& = \\int\\limits_{0}^{\\infty} \\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)} \\underbrace{x^{(\\alpha+2)-1} e^{-\\beta x} }_{\\propto \\Gamma(\\alpha+2, \\beta)} dx \\\\\n& = \\dfrac{\\beta^{\\alpha}}{\\Gamma(\\alpha)} \\frac{\\Gamma(\\alpha+2)}{\\beta^{\\alpha+2}} = \\frac{\\alpha(\\alpha+1)}{\\beta^{2}}\n\\end{aligned} \n\\end{equation}\n\n\n\\begin{figure}[!h]\n\t\\includegraphics[width = \\columnwidth, height = .25\\paperheight]{\"krc\"}\n\t\\caption{ \\small{Comparison of four of the variants of the symmetric Gamma probability density function.} }\n\\end{figure}\n\nLooking at eq. \\eqref{GammaVar} one can see that $\\beta$ can be used to control how much $\\alpha$ influences the variance of the distribution. For example, if we take two random variables $X_{i} \\sim \\text{sym}\\Gamma(i,1),\\ i \\in \\{1,2\\}$, we have $\\mathbb{V}(X_{1})=2$ and $\\mathbb{V}(X_{1})=6$ which is a drastic increase in variance. Taking $X_{i} \\sim \\text{sym}\\Gamma(i,30),\\ i \\in \\{1,2\\}$ on the other hand gives us the variances $\\mathbb{V}(X_{1}) \\approx 0.0022$ and $\\mathbb{V}(X_{2}) \\approx 0.0066$, which is a much smaller increase, for the same change in $\\alpha$. The previous, rather straightforward observation, will be useful later since in our model $\\alpha$ will represent the number of jumps at a certain point in time. Taking a large $\\beta$ means that every extra jump only slightly increases the variance of the model, hence  allowing for a finer analysis of the data.\nThe next step consists in determining the length $N$ of the sum in eq. \\eqref{Gamma}. In particular we assume that such a sum has a state dependent length represented by a state-dependent Poisson random variable $N_{t} \\sim \\mathcal{P}(\\theta_{S_{t}})$, hence we have to determine the values of $\\theta_{S_{t}}$. In keeping with the interpretation of the states, see eq. \\eqref{sigmainequality}, we want the number of jumps to increase as the state the data are in increases. This can be done by ordering the parameters $\\theta_{S_{t}}$. Moreover, in order to also allow the parameters to be sufficiently flexible for our purposes, we assume them to be distributed as follows\n\\begin{equation}\n\\label{poissonprior}\n\\theta_{S_{t}} \\sim \n\\begin{cases}\n\\mathcal{U}(0,u_{1}), &{S_{t}}=1 \\\\\n\\mathcal{U}(u_{1},u_{2}), &{S_{t}}=2 \\\\\n& \\vdots \\\\\n\\mathcal{U}(u_{M-2},u_{M-1}), &{S_{t}}=M-1 \\\\\n\\mathcal{U}(u_{M-1},u_{M}), &{S_{t}}=M\n\\end{cases}\n\\end{equation}\nwhich clearly guarantees that $\\theta_{1} \\leq \\theta_{2} \\leq ... \\leq \\theta_{M}$.\n\n\\subsection{Full Model}\nSumming up the definitions stated in subsections \\ref{GaussianElement} and \\ref{JumpElement}, we can write the full model as follows\n\\begin{equation}\n\\begin{cases}\ny_{t} &= \\epsilon_{t} + \\zeta_{t} \\\\\n\\epsilon_{t} &\\sim \\mathcal{N}(\\mu_{S_{t}}, \\sigma_{S_{t}}^{2}) \\\\\n\\zeta_{t} &\\sim \\text{sym}\\Gamma(N_{t},b)\n\\end{cases}\n\\:.\n\\end{equation}\nThere is no analytic expression for the distribution of $y_{t}$, but we can obtain an integral form of it using the following well known fact.\nLet $X$ and $Y$ be two independent random variables with density functions $f_{X}(x)$ and $f_{Y}(x)$, defined for $x \\in \\mathbb{R}$. Then the sum $Z = X + Y$ is a random variable with density function $f_{Z}(z)$ given by\n\\begin{align}\\label{convolution}\nf_{Z}(z) = \\int\\limits_{-\\infty}^{\\infty} f(z - y)g(y) dy = \\int\\limits_{-\\infty}^{\\infty}f(y) g(z - y) dy\\:.\n\\end{align}\nTherefore, by the convolution formula in \\eqref{convolution}, we have\n\\begin{equation}\\label{FMdensity}\nf_{y_{t}}(z) = \\int\\limits_{-\\infty}^{\\infty} \\frac{b^{N_{t}} |y|^{N_{t}-1}}{\\sqrt{8 \\pi \\sigma_{S_{t}}^{2}}\\Gamma(N_{t})} e^{-\\frac{1}{2} \\left( \\frac{z-y-\\mu_{S_{t}}}{\\sigma_{S_{t}}} \\right)^{2} -b |y|} dy\n\\end{equation}\nAlthough not very useful in general, the expression in eq. \\eqref{FMdensity} can be computationally handled with  little difficulty, a crucial fact for the concrete case study we will consider in Section \\ref{CaseStudy}. In the next section we consider the $\\alpha$-stable distribution model.\n\n\\section{$\\boldsymbol \\alpha$-Stable Distribution Model}\\label{AlfaStableModel}\nIn Section \\ref{JumpDiffusionModel}, we pointed out that the Gaussian distribution is not adequate  to model  financial data, mainly because of its slim tails, which we offset by adding jumps. In what follows, we will consider a different approach, namely we will model the data using a distribution that has fatter tails than the Gaussian one, but still preserves its most important characteristics.\n\n\\subsection{$\\alpha$-Stable Distribution}\nThere are multiple equivalent ways to define a stable distribution. We will consider the two most common ones,  the interested reader can refer to, e.g.,  \\cite{SamTaq}, for the others.\n\\begin{definition}\n\tA random variable $X$ is said to have a stable distribution if, for every $A$ and $B$ positive, there exists a positive number $C$ and a real number $D$ such that\n\\end{definition}\n\\begin{equation}\nAX_{1}+BX_{2} \\stackrel{\\text{ d}}{=} CX+D\\;,\n\\end{equation}\nwhere $X_{1}$ and $X_{2}$ are independent copies of $X$ and $\\stackrel{\\text{ d}}{=}$ stands for {\\it equal in distribution}. This implies that the sum of two stable independent identically distributed random variables is still a stable random variable, with the same distribution, up to a {\\it scale factor} $C$, and a shift component $D$. As an example, we can consider  two Gaussian random variables $X_{1}$ and $X_{2}$, assumed to be independent copies of $ X \\sim \\mathcal{N}(\\mu, \\sigma^{2})$. Then, $X_{1}+X_{2} \\sim \\mathcal{N}(2\\mu, 2\\sigma^{2})$, which means that $X_{1}+X_{2} \\stackrel{\\text{ d}}{=} \\sqrt{2}X + (2-\\sqrt{2})\\mu$.\nAlternatively, we can define the stable distribution using characteristic functions, namely\n\n\\begin{definition}\n\tA random variable $X$ is said to have a stable distribution if there exist parameters $0 < \\alpha \\leq 2,\\ \\sigma \\geq 0,\\ |\\beta| \\leq 1\\ \\text{and}\\ \\mu \\in \\mathbb{R}$, such that its characteristic function has the form\n\t\\begin{equation}\n\t\\mathbb{E}(e^{i\\theta X}) =\n\t\\begin{cases}\n\te^{\\left[ - \\sigma^{ \\alpha } | \\theta |^{\\alpha} (1 - i \\beta \\text{sgn}(\\theta) \\tan(\\frac{\\pi \\alpha}{2})) + i \\mu \\theta \\right]}, &\\alpha \\neq 1 \\\\\n\te^{\\left[ - \\sigma | \\theta | (1 + i \\beta \\frac{2}{\\pi} \\text{sgn}(\\theta) \\ln |\\theta| ) + i \\mu \\theta \\right]}, &\\alpha = 1\n\t\\end{cases}\n\t\\:.\n\t\\end{equation}\n\\end{definition}\nWe call $\\alpha$ the \\emph{stability} parameter, $\\beta$ the \\emph{skewness} parameter, $\\gamma$ the \\emph{scale} parameter and $\\mu$ the \\emph{location} parameter. For $\\alpha=2$ we obtain the normal distribution, which is the only member of the stable distribution family that has finite variance. For $\\alpha \\in (1,2)$ we have infinite variance and mean $\\mu$, while, for $\\alpha \\in (0,1]$, both the mean and the variance are undefined. We note that in general there is not a solution in closed form  for the probability density function of a stable distribution. The stable distribution will be denoted by $\\mathcal{S}_{\\alpha,\\beta}(\\gamma,\\mu)$ for the remainder of the paper. \n\n\\begin{figure}[!h]\n\t\\includegraphics[width = \\columnwidth, height = .25\\paperheight]{\"alphaComp\"}\n\t\\caption{ \\small{Comparison of the alpha-stable distribution for different values of $\\alpha$. In the special case $\\alpha=2$ we have a normal distribution.} }\n\\end{figure}\n\n\n\\subsection{The Model}\nIn the model we propose, the data are assumed to follow a symmetric $\\alpha$-stable distribution, more precisely $y_{t} \\sim\\mathcal{S}_{\\alpha,0}(\\gamma_{S_{t}},\\mu_{S_{t}})$. The full model is presented in the following\n\\begin{equation}\\label{alfamodel}\n\\begin{cases}\ny_{t} \\sim\\mathcal{S}_{\\alpha,0}(\\gamma_{S_{t}},\\mu_{S_{t}}) \\\\\n\\gamma_{S_{t}} = \\gamma_{j}\\ \\text{if}\\ S_{t}=j,\\ \\forall j \\in \\{1,...,M\\} \\\\\n\\mu_{S_{t}} = \\mu_{j}\\ \\text{if}\\ S_{t}=j,\\ \\forall j \\in \\{1,...,M\\} \\\\\n\\alpha \\in (1,2) \\\\\nS_{t} \\in \\{1,...,M\\} \\\\\np_{ij} = \\mathbb{P} (S_{t}=j|S_{t-1}=i) \\\\\n\\pi_{0} = [\\mathbb{P} (S_{0}=1),...,\\mathbb{P} (S_{0}=M)]\n\\end{cases}\n\\:.\n\\end{equation}\nThe motivation behind the choice of the model represented by \\eqref{alfamodel}, mainly relies on empirical observations of financial data which exhibit fat tails that  can not be well described using the Gaussian approach. In particular we believe that such phenomenon can be suitably addressed by exploiting $\\alpha$-stable distributions with $\\alpha \\in (1,2)$. Moreover, financial data often exhibit structural breaks because of abrupt changes in the market, e.g. as during the sub-prime mortgage credit crisis of 2008, which is the reason why  we consider both the scale and the location parameters, to be state-dependent.\nAs we  previously mentioned, in general there is no closed form for the density of an $\\alpha$-stable distribution. Nevertheless, this problem can be circumvented using the fact that $y_{t}$ can be conditionally represented as a Gaussian random variable, see, e.g.,  \\cite{GoKuRu10,SamTaq}, by  introducing a random variable $\\lambda$ and using the property\n\\begin{align}\n\\label{conditionalnormal}\n\\begin{split}\n\\text{If:}\\ \\qquad &\\lambda \\sim \\mathcal{S}_{\\tfrac{\\alpha}{2},1}(2 \\left( \\cos(\\frac{\\pi \\alpha}{4}) \\right)^{\\frac{2}{\\alpha}} ,0) \\\\\n\\text{then:}\\ \\qquad &y_{t}|\\lambda \\sim \\mathcal{N}(\\mu_{S_{t}}, \\lambda\\gamma^{2}_{S_{t}})\\;,\n\\end{split}\n\\end{align}\nwhich allows us to have an analytic likelihood function which significantly speeds up the sampling process. \n\n\\begin{figure}[!h]\n\t\\includegraphics[width = \\columnwidth, height = .3\\paperheight]{\"microbenchmark\"}\n\t\\caption{ \\small{Comparison between the time needed for the jump diffusion model (top) and the stable distribution model (bottom) to draw 1000 samples. The best performance of the first model was $156$ seconds, its worst performance $1103$ seconds, with a mean of $207$ and a median of $160$. The  best performance of the second model was $24$ seconds, its worst performance $868$ seconds, with a mean of $68$ and a median of $25$ seconds.} }\n\t\\label{fig:benchmark}\n\\end{figure}\n\nAnalogously to what we  considered in Section \\ref{JumpDiffusionModel}, we have one mean for each state, without further restrictions, namely\n\\begin{equation}\n\\mu_{S_{t}} = \\mu_{j}\\ \\text{if}\\ S_{t}=j,\\ \\forall j \\in \\{1,...,M\\}\\;.\n\\end{equation}\nWe also want the scale parameter to be increasing with respect to the state, namely\n\\begin{equation}\n\\gamma_{S_{t}} =\n\\begin{cases}\n\\gamma_{1} , &S_{t} = 1 \\\\\n\\gamma_{1} \\prod\\limits_{i=2}^{S_{t}} (1+h_{i}), &S_{t}=j,\\ 2 \\leq j \\leq M\n\\end{cases}\n\\;,\n\\end{equation}\nwhere $\\forall i \\in \\{2,3,...,M\\},\\ h_{i}>0$, which leads to the property\n\\begin{equation}\n\\gamma_{1} < \\gamma_{2} < ... < \\gamma_{M}\\;,\n\\end{equation}\nso that  an increase in the state number indicates an increase in volatility.\n\n\\section{Implementation}\nIn this section we get into the specifics of our two models. In particular we provide the details regarding the likelihood functions, the priors and the posteriors, for both the {\\it Jump Diffusion Model}, described in Section \\ref{JumpDiffusionModel}, and for the {\\it $\\alpha-$Stable Model}, defined in Section \\ref{AlfaStableModel}.\nThe concept of duration analysis is also explained, along with its importance. Both of the models will be characterized by four states, with the states being interpreted as \\emph{low, medium, high} and \\emph{very high} volatility {\\it regime}. Let us start by defining the following quantities\n\\begin{align}\ny^{j} & = \\{y_{t} \\in \\tilde{y}_{T} :S_{t}=j\\},\\ j \\in \\{1,2,3,4\\} \\;,\\\\\nn_{j} & = \\# y^{j}\\:.\n\\end{align}\nFor the rest of this section we will suppress unneeded parameters. Therefore, e.g., the conditional posterior $f(\\mu_{j}|\\boldsymbol y, \\sigma_{j}, N_{j},...)$, will be denoted by $f(\\mu_{j}|\\boldsymbol y)$, the general rule being that the parameters that are not being inferred on are considered known.\n\\subsection{Jump Diffusion Model}\n\\label{subsec:jump_diffusion_model}\nThe description  of the implementation is divided into three parts, namely: the first part deals with the form of the likelihood function, the second with the priors while the third part provides a detailed analysis of the different types of obtained posteriors. \n\\subsubsection{Likelihood}\n\\label{likelihood1}\nWe have to take into account whether there are jumps in the model or not, as well as the state of each observation. Hence, if $N_{j} = 0$, we define\n\\begin{equation}\n\\begin{aligned}\nL^{j}\\left( \\cdot \\right) &= \\prod\\limits_{y_{t} \\in y^{j}} \\frac{e^{- \\frac{1}{2} \\left( \\frac{y_{t}-\\mu_{j}}{\\sigma_{j}} \\right)^{2}} }{\\sqrt{2 \\pi \\sigma_{j}^{2}}} \\\\\n& = \\left( 2 \\pi \\sigma_{j}^{2} \\right)^{-\\frac{n_{j}}{2}} e^{-\\frac{1}{2 \\sigma_{j}^{2}} \\sum\\limits_{y_{t} \\in y^{j}}\\left( y_{t}-\\mu_{j} \\right)^{2} }\\;,\n\\end{aligned}\n\\end{equation}\nwhile, if $N_{j} \\geq 1$, we define $L^{j}\\left( \\cdot \\right)$ as\n\\begin{equation}\n\\prod\\limits_{y_{t} \\in y^{j}} \\frac{b^{N_{j}}}{\\sqrt{8 \\pi \\sigma_{j}^{2}}\\Gamma(N_{j})} \\int\\limits_{-\\infty}^{\\infty} |y|^{N_{j}-1}e^{-\\frac{1}{2}\\left( \\frac{y_{t}-y-\\mu_{j}}{\\sigma_{j}} \\right)^{2}}dy\\:.\n\\end{equation}\nThen, the full likelihood function is \n\\begin{equation}\nL_{n} = \\prod\\limits_{j=1}^{4} L^{j}\\:,\n\\end{equation}\nwhich has a standard form only if $N_{j}=0$, for every $j \\in \\{1,2,3,4\\}$. As this very rarely happens, we will use the Metropolis-Hastings algorithm in this model.\n\n\\subsubsection{Priors}\n\\begin{raggedleft}\n\t\\textbf{Mean:}\n\\end{raggedleft} \nWe take the mean to be normally distributed. Moreover, we give the same prior to the means of all the states, namely\n\\begin{equation}\n\\label{meanprior}\n\\mu_{j} \\sim \\mathcal{N}\\left(0, \\frac{1}{k}\\right),\\ 1 \\leq j \\leq 4 \\ \\text{and}\\ k>0\\:.\n\\end{equation}\n\n\\begin{raggedleft}\n\t\\textbf{Variance:}\n\\end{raggedleft} \nThe variance $\\sigma_{1}^{2}$ will have an inverse-gamma prior.\n\\begin{equation*}\n\\sigma_{1}^{2} \\sim\\ \\text{inv}\\Gamma(\\alpha_{0}, \\beta_{0}),\\ \\alpha_{0}, \\beta_{0}>0\\:.\n\\end{equation*}\n\n\\begin{raggedleft}\n\t\\textbf{H parameters:}\n\\end{raggedleft}\nWe previously saw that in order for \\eqref{sigmainequality} to hold, we need $h_{j} > 0,\\ \\forall j \\in \\{2,3,4\\}$, hence we define $h_{j}^{\\star} := 1+h_{j}$, for all $j$, and make these parameters Fr\\'{e}chet distributed, namely \n\\begin{equation}\nh_{j}^{\\star} \\sim \\mathcal{F}(h_{j}^{\\star}|1, \\alpha_{F}, s_{F})\\;,\n\\end{equation}\nthen the density function of $h_{j}^{\\star}$ reads as follows\n\\begin{equation}\nf(h_{j}^{\\star}) = \\frac{\\alpha_{F}}{s_{F}} \\left( \\frac{h_{j}^{\\star}-1}{s_{F}} \\right)^{1-\\alpha_{F}} e^{- \\left( \\frac{h_{j}^{\\star}-1}{s_{F}} \\right)^{- \\alpha_{F}}}\\;,\n\\end{equation}\nwhere $\\alpha_{F},s_{F}>0$, and $f$ is defined for $h_{j}^{\\star}>1$.\n\n\\begin{raggedleft}\n\t\\textbf{Poisson parameters:}\n\\end{raggedleft} \nFor the  priors of the Poisson parameters we refer to \\eqref{poissonprior}.\n\n\\begin{raggedleft}\n\t\\textbf{Transition probabilities:}\n\\end{raggedleft}\nFor the transition probabilities we will use a Dirichlet prior, namely \n\\begin{equation}\n(p_{1j},p_{2j},p_{3j},p_{4j}) \\sim \\mathcal{D}(m_{1j},m_{2j},m_{3j},m_{4j})\\;,\n\\end{equation}\nfor every $j \\in \\{1,2,3,4\\}$. The density function of this particular Dirichlet distribution is given by\n\\begin{align}\nf(p_{1j},p_{2j},p_{3j},p_{4j}) = \\frac{\\Gamma\\left(\\sum\\limits_{i=1}^{4} m_{ij} \\right)}{\\prod\\limits_{i=1}^{4}\\Gamma(m_{ij})} \\prod\\limits_{i=1}^{4} p_{ij}^{m_{ij}-1}\n\\;,\n\\end{align}\nand it is defined on the simplex\n\\begin{align}\n\\nonumber & p_{1j},p_{2j},p_{3j} > 0 \\;, \\\\\n& p_{1j}+p_{2j}+p_{3j} < 1 \\;,\\\\\n\\nonumber & p_{4j} = 1 - p_{1j}+p_{2j}+p_{3j}\n\\;,\n\\end{align}\nwhile, everywhere else, its value is zero. Finally, the parameter $b$ is a constant.\n\n\\subsubsection{Posteriors}\n\\begin{raggedleft}\n\t\\textbf{Mean:}\n\\end{raggedleft}\nBecause the likelihood function depends on whether or not jumps have occurred, we have two different posteriors for the mean. In particular, if $N_{j}=0$, by \\eqref{nn}, we have\n\\begin{equation}\n\\label{meanposterior}\n\\mu_{j}|y^{j} \\sim \\mathcal{N} \\left(\\frac{n_{j}\\bar{y}^{j}}{n_{j}+k\\sigma_{j}^{2}},\\frac{\\sigma_{j}^{2}}{n_{j}+k \\sigma_{j}^{2}} \\right)\\;,\n\\end{equation}\nfor all $j \\in \\{1,2,3,4\\}$, while, if $N_{j} \\geq 1$, we obtain\n\\begin{equation}\nf(\\mu_{j}|y^{j}) \\propto f(\\mu_{j}) L^{j}(\\mu_{j}) \\:.\n\\end{equation}\n\n\\begin{raggedleft}\n\t\\textbf{Variance:}\n\\end{raggedleft}\nSimilarly to the previous point, we have to differentiate between the jump and no-jump cases. Therefore, if $N_{j}=0$, by \\eqref{ign}, we have\n\\begin{equation}\n\\sigma_{1}^{2} \\sim\\ \\text{inv}\\Gamma \\left(\\frac{n_{1}}{2}+\\alpha_{0}, \\frac{1}{2} \\sum\\limits_{y_{t} \\in y^{1}} \\left( y_{t}-\\mu_{1} \\right)^{2} + \\beta_{0} \\right)\\;,\n\\end{equation}\notherwise, we obtain\n\\begin{equation}\nf(\\sigma_{1}|y^{1}) \\propto f(\\sigma_{1}) L^{1}(\\sigma_{1})\\:.\n\\end{equation}\n\n\\begin{raggedleft}\n\t\\textbf{H parameters:}\n\\end{raggedleft}\nIn order to obtain $h_{i},\\ i \\in \\{2,3,4\\}$, we need to transform the data, also taking into account the  different states. \n\nIn particular we have the following cases\n\n\\begin{raggedleft}\n\t$\\boldsymbol{S_{t}=2:}$\n\\end{raggedleft}\n\\begin{align*}\ny_{t} \\sim \\mathcal{N}(\\mu_{2}, \\sigma_{1}^{2} h_{2}^{\\star}) \\Rightarrow  \\psi_{t} := \\frac{y_{t}-\\mu_{2}}{\\sigma_{1}} \\sim \\mathcal{N}(0, h_{2}^{\\star})\\:,\n\\end{align*} \nwhere we have used the transformed data set $\\psi_{t}$  to obtain a posterior for $h_{2}^{\\star}$ when $N_{2}=0$, namely\n\\begin{equation}\nf(h_{2}^{\\star}|y^{2}) \\propto \\left( h_{2}^{\\star} \\right)^{- \\frac{n_{2}}{2}} e^{- \\frac{1}{2 h_{2}^{\\star}} \\sum\\limits_{i=1}^{n_{2}} \\psi^{2}_{i} } \\mathcal{F}(h_{2}^{\\star}|1,\\alpha_{F},s_{F})\\:.\n\\end{equation}\n\n\\begin{raggedleft}\n\t$\\boldsymbol{S_{t}=3:}$\n\\end{raggedleft}\n\\begin{equation*}\ny_{t}  \\sim \\mathcal{N}(\\mu_{3}, \\sigma_{1}^{2} h_{2}^{\\star} h_{3}^{\\star}) \\Rightarrow  \\zeta_{t} := \\frac{y_{t}-\\mu_{3}}{\\sqrt{\\sigma_{1}^{2} h_{2}^{\\star}}} \\sim \\mathcal{N}(0, h_{3}^{\\star})\\;,\n\\end{equation*} \nwhich gives us the posterior\n\\begin{equation}\nf(h_{3}^{\\star}|y^{3}) \\propto \\left( h_{3}^{\\star} \\right)^{- \\frac{n_{3}}{2}} e^{- \\frac{1}{2 h_{3}^{\\star}} \\sum\\limits_{i=1}^{n_{3}} \\zeta^{2}_{i} } \\mathcal{F}(h_{3}^{\\star}|1,\\alpha_{F},s_{F})\\;,\n\\end{equation}\nhence, when $N_{3}=0$, the posterior is analogous to the one for $h_{2}^{\\star}$, with the only difference being that we use $\\zeta_{t}$ instead of $\\psi_{t}$. \n\\begin{flushleft}\n\t$\\boldsymbol{S_{t}=4:}$\n\\end{flushleft}\n\\begin{align*}\ny_{t} &\\sim \\mathcal{N}(\\mu_{4}, \\sigma_{1}^{2} h_{2}^{\\star} h_{3}^{\\star} h_{4}^{\\star})\\\\ \\Rightarrow  \\theta_{t} &:= \\frac{y_{t}-\\mu_{4}}{\\sqrt{\\sigma_{1}^{2} h_{2}^{\\star} h_{3}^{\\star}}} \\sim \\mathcal{N}(0, h_{4}^{\\star})\\;,\n\\end{align*} \nwhich yields the posterior\n\\begin{equation}\nf(h_{4}^{\\star}|y^{4}) \\propto \\left( h_{4}^{\\star} \\right)^{- \\frac{n_{4}}{2}} e^{- \\frac{1}{2 h_{4}^{\\star}} \\sum\\limits_{i=1}^{n_{4}} \\theta^{2}_{i} } \\mathcal{F}(h_{4}^{\\star}|1,\\alpha_{F},s_{F})\\:.\n\\end{equation}\n\nIn the case where we have jumps, i.e. $N_{j} \\geq 1,\\ j \\in \\{2,3,4\\}$, there is no analytic expression, therefore \n\\begin{equation*}\nf(h_{j}^{\\star}|y^{j}) \\propto \\mathcal{F}(h_{j}^{\\star}|1,\\alpha_{F},s_{F}) L^{j}(h_{j}^{\\star})\\:.\n\\end{equation*}\n\n\\begin{raggedleft}\n\t\\textbf{Poisson parameters:}\n\\end{raggedleft}\nConcerning the posterior of the theta parameters, for $j \\in \\{1,2,3,4\\}$, we have\n\\begin{equation}\n\\begin{aligned}\nf(\\theta_{j}|y^{j}) \\propto f(y^{j}|\\theta_{j}) = \\sum_{N_{j}=0}^{\\infty} f(y^{j},N_{j}|\\theta_{j}) \\\\ \\propto \\sum_{N_{j}=0}^{\\infty} f(N_{j}|\\theta_{j})f(y^{j}|N_{j},\\theta_{j}) = \\sum_{N_{j}=0}^{\\infty} f(N_{j}|y^{j},\\theta_{j})\n\\end{aligned}\n\\end{equation}\n\n\n\\begin{raggedleft}\n\t\\textbf{Transition probabilities:}\n\\end{raggedleft}\nThe transition probabilities differ from the other parameters in that they do not depend directly on the observations $y_{1},...,y_{T}$. Instead, they depend on the vector of states $\\tilde{S}_{T}$. Assuming that the vector $\\tilde{S}_{T}$ is known, the posterior distribution of the transition probability vector $(p_{1j},p_{2j},p_{3j},p_{4j})$, $j \\in \\{1,2,3,4\\}$, has the Dirichlet distribution\n\\begin{equation}\n\\mathcal{D}(m_{1j}+n_{1j},m_{2j}+n_{2j},m_{3j}+n_{3j},m_{4j}n_{4j})\\;,\n\\end{equation}\nwhere $n_{ij}$ is the number of transitions from state $j$ to state $i$.\n\n\\subsection{$\\alpha$-Stable Distribution Model}\nIn what follows, we proceed analogously to subsection \\ref{subsec:jump_diffusion_model}.\n\\subsubsection{Likelihood}\nUsing the fact that, in the present setting, our data are conditionally normal, see \\eqref{conditionalnormal},  the likelihood function reads as follows\n\\begin{align*}\nL \\left( \\cdot|\\boldsymbol y \\right) &= \\prod\\limits_{j=1}^{4} \\prod\\limits_{y_{t} \\in y^{j}} \\frac{e^{- \\frac{1}{2} \\left( \\frac{y_{t}-\\mu_{j}}{\\lambda\\gamma_{j}}\\right)^{2}} }{\\sqrt{2 \\pi \\lambda \\gamma_{j}^{2}}} \\\\\n& = \\prod\\limits_{j=1}^{4} \\left( 2 \\pi \\lambda \\gamma_{j}^{2} \\right)^{-\\frac{n_{j}}{2}} e^{-\\frac{1}{2 \\lambda \\gamma_{j}^{2}} \\sum\\limits_{y_{t} \\in y^{j}}\\left( y_{t}-\\mu_{j} \\right)^{2} }\\;,\n\\end{align*}\nand, unlike in the previous model, we do not have to worry about multiple cases.\n\n\\subsubsection{Priors}\n\\begin{raggedleft}\n\t\\textbf{Mean:}\n\\end{raggedleft}\nThe prior of the means is the same as in eq.~\\eqref{meanprior}.\n\n\\begin{raggedleft}\n\t\\textbf{Scale:}\n\\end{raggedleft}\nThe distribution of the scale is analogous to that of the variance in the previous model, namely\n\\begin{align*}\n\\gamma^{2} \\sim\\ \\text{inv}\\Gamma(\\alpha_{1}, \\beta_{1}), \\alpha_{1}, \\beta_{1}>0\\:.\n\\end{align*}\n\n\\begin{raggedleft}\n\t\\textbf{H parameters:}\n\\end{raggedleft}\nThese parameters are exactly the same as they were in the previous model, in fact their role remains unchanged, since they allow for the volatility to increase as the states increase. \n\n\\begin{raggedleft}\n\t\\textbf{Lambda:}\n\\end{raggedleft}\nThe lambda parameter follows a stable distribution, hence\n\\begin{align*}\n\\lambda \\sim S_{\\tfrac{\\alpha}{2},1}(2 \\left( \\cos\\left(\\frac{\\pi \\alpha}{4}\\right) \\right)^{\\frac{2}{\\alpha}} ,0)\\:.\n\\end{align*}\n\n\\subsubsection{Posteriors}\n\\begin{raggedleft}\n\t\\textbf{Mean:}\n\\end{raggedleft}\nThe posterior of the mean is analogous to that of the one in eq. \\eqref{meanposterior}, with the only difference being the form of the variance. In particular, we have\n\\begin{equation}\n\\mu_{j}|y^{j} \\sim \\mathcal{N} \\left(\\frac{n_{j}\\bar{y}^{j}}{n_{j}+k \\lambda \\gamma_{j}^{2}},\\frac{\\lambda \\gamma_{j}^{2}}{n_{j}+k \\lambda \\gamma_{j}^{2}} \\right)\n\\:.\n\\end{equation}\n\n\\begin{raggedleft}\n\t\\textbf{Scale:}\n\\end{raggedleft}\nThe posterior of the scale is \n\\begin{equation}\n\\gamma^{2} \\sim\\ \\text{inv}\\Gamma \\left(\\frac{n_{1}}{2}+\\alpha_{1}, \\frac{1}{2} \\sum\\limits_{y_{t} \\in y^{1}} \\left( y_{t}-\\mu_{1} \\right)^{2} + \\beta_{1} \\right)\n\\:.\n\\end{equation}\n\n\\begin{raggedleft}\n\t\\textbf{H parameters:}\n\\end{raggedleft}\nIn what follows we limit ourselves to listing the needed transformations, therefore we have\n\n\\begin{raggedleft}\n\t$\\boldsymbol{S_{t}=2:}$\n\\end{raggedleft}\n\\begin{align*}\ny_{t} \\sim \\mathcal{N}(\\mu_{2},\\lambda \\gamma^{2} h_{2}^{\\star}) \\Rightarrow  \\psi_{t} := \\frac{y_{t}-\\mu_{2}}{\\sqrt{\\lambda \\gamma^{2}}} \\sim \\mathcal{N}(0, h_{2}^{\\star})\\:.\n\\end{align*} \n\n\\begin{raggedleft}\n\t$\\boldsymbol{S_{t}=3:}$\n\\end{raggedleft}\n\\begin{equation*}\ny_{t}  \\sim \\mathcal{N}(\\mu_{3}, \\lambda \\gamma^{2} h_{2}^{\\star} h_{3}^{\\star}) \\Rightarrow  \\zeta_{t} := \\frac{y_{t}-\\mu_{3}}{\\sqrt{\\lambda \\gamma^{2} h_{2}^{\\star}}} \\sim \\mathcal{N}(0, h_{3}^{\\star})\\:.\n\\end{equation*} \n\n\\begin{raggedleft}\n\t$\\boldsymbol{S_{t}=4:}$\n\\end{raggedleft}\n\\begin{align*}\ny_{t} &\\sim \\mathcal{N}(\\mu_{4}, \\lambda \\gamma^{2} h_{2}^{\\star} h_{3}^{\\star} h_{4}^{\\star}) \\\\\n\\Rightarrow  \\theta_{t} &:= \\frac{y_{t}-\\mu_{4}}{\\sqrt{\\lambda \\gamma^{2} h_{2}^{\\star} h_{3}^{\\star}}} \\sim \\mathcal{N}(0, h_{4}^{\\star})\\:.\n\\end{align*} \nThe posteriors are obtained as in the previous case.\n\n\\begin{raggedleft}\n\t\\textbf{Lambda:}\n\\end{raggedleft}\nSince there is no closed form for the posterior distribution of the lambda parameter,  we only write\n\\begin{equation}\n\\label{eq:lambda_posterior}\nf(\\lambda|\\boldsymbol y) \\propto S_{\\tfrac{\\alpha}{2},1}(2 \\left( \\cos\\left(\\frac{\\pi \\alpha}{4}\\right) \\right)^{\\frac{2}{\\alpha}} ,0) L(\\lambda|\\boldsymbol y)\n\\:.\n\\end{equation}\n\nThe prior and posterior of the transition probabilities are the same in both proposed models. In fact, the transition probabilities do not depend by any of the parameters, but only by the state vector $S_{T}$.\n\n\\subsection{Duration Analysis}\n\\label{subsubsec:duration}\nThe expected duration of each state for a MSM is a quantity of significant interest. Having an estimate of how long a certain data set remains in a particular state can give us useful insights into how the model will behave for a certain period of time. In this section we are going to explain how the expected duration can be  calculated exploiting the transition probabilities.\n\nThe expected duration, denoted by $\\hat{d}_j$, is  defined as follows\n\\begin{equation}\n\\label{eq:expdur_def}\n\\hat{d}_j := \\mathbb{E}\\left[d_j\\right]\\;,\n\\end{equation}\nwhere $d_j$ is the random variable that models the length of the time interval for which the time series is in state $j$. The first thing we have to consider is $\\mathbb{P}\\left(d_j = d\\right)$, the probability of the data being in state $j$, meaning\n\\begin{equation}\n\\nonumber\n\\mathbb{P}\\left(d_j = d\\right) = p_{jj}^{d-1} \\left(1-p_{jj}\\right)\\;,\n\\end{equation}\nwhere $p_{ij} = \\mathbb{P}\\left(S_t = j| S_{t-1} = i\\right)$. It just so happens that $\\hat{d}_j$ has a very simple closed form, in particular\n\\begin{align}\n\\nonumber\\mathbb{E}\\left(d_j\\right) &= \\sum_{m=1}^\\infty m p_{jj}^{m-1} \\left(1-p_{jj}\\right) \\\\\n\\nonumber&= \\left(1-p_{jj}\\right) \\lim_{n\\to\\infty} \\sum_{m=1}^{n} m\\, p_{jj}^{m-1} \\\\\n\\nonumber&= \\left(1-p_{jj}\\right) \\lim_{n\\to\\infty} \\frac{1}{p_{jj}} \\sum_{m=1}^{\\infty} m\\, p_{jj}^m \\\\\n\\nonumber&= \\left(1-p_{jj}\\right) \\lim_{n\\to\\infty} \\frac{1-(n+1)p_{jj}^n+n p_{jj}^{n+1}}{(1-p_{jj})^2}\\\\\n\\nonumber&= \\left(1-p_{jj}\\right) \\frac{1}{(1-p_{jj})^2}\\\\\n\\label{eq:expdur}&= \\frac{1}{1-p_{jj}}.\n\\end{align}\n\nWe will use the previous expression later on, when we  compare the state durations obtained in the present paper, with  those provided in \\cite{DiPMF16}. We will see that there is a significant difference in the state durations, showing that  the models developed in this paper perform better than the one proposed in \\cite{DiPMF16}  to model the time series of the Chicago Board Options Exchange Volatility Index, better known as VIX.\n\n\\section{Case Study}\\label{CaseStudy}\nOur case study is concerned with the application of the above theory to developing an indicator that has a role similar to the one played by the VIX. In particular we use the set of S\\&P500 weekly prices, considering a time interval that runs from the 3rd of January, 2007 to 29th of December, 2014. We picked this interval to include the sub-prime mortgage crash of 2008 as well as the subsequent period of relative calm. This choice allows us to analyse how our approach  performs in both situations. We will show that our techniques  improve the results stated in \\cite{DiPMF16}, where the model was very effective in periods of high volatility, but also too smooth in case of low volatility. Our results are summarised below with respect to both the {\\it Jump Diffusion Model}, defined in Section \\ref{JumpDiffusionModel}, and the \n{\\it $\\alpha-$Stable Model}, provided in Section \\ref{AlfaStableModel}.\n\n\\subsection{Jump Diffusion Model}\n\\label{jump_diffusion_case}\nFor the jump diffusion model we model the data as a zero mean process in order to make the framework more parsimonious. We take the exponential distribution parameter $b$ to be equal to 40, in order to make the contribution of each extra jump to the variance relatively small. This choice of b allows for a finer analysis. We first present the histograms of the sampled variances, see Fig. \\ref{fig:variance_comparison}. \n\nAs we can see, the algorithm is rather accurate in sampling the variances. In particular, we recall that the theoretical posterior of the variances is an inverse-Gamma distribution, which is exactly what we can observe in the histograms. \n\\begin{figure}[!h]\n\t\\includegraphics[width = \\columnwidth, height = .35\\paperheight]{\"variance_comparison\"}\n\t\\caption{ \\small{Comparison between the histograms of each of the variances belonging to the four states.} }\n\t\\label{fig:variance_comparison}\n\\end{figure}\n\nMoreover, in Table \\ref{tbl:variance1}, we report the point estimates of each variance value.\n\\begin{table}[h!]\n\t\\caption{Gaussian Variance Point Estimates}\n\t\\centering\n\t\\begin{tabular}{c|c}\n\t\t\\label{tbl:variance1}\n\t\tEstimator & Value \\\\\n\t\t\\hline\n\t\t$\\hat{\\sigma}_{1}^{2}$ & $0.155709$ \\\\\n\t\t$\\hat{\\sigma}_{2}^{2}$ & $0.2336135$ \\\\\n\t\t$\\hat{\\sigma}_{3}^{2}$ & $0.254559$ \\\\\n\t\t$\\hat{\\sigma}_{4}^{2}$ & $0.2816716$\n\t\\end{tabular}\n\t\\label{tab:variance1}\n\\end{table}\n\nConcerning the distribution of the jumps, we refer to Fig. \\ref{fig:jump_comparison} which highlights that the jumps, like the Gaussian variances,  they have an amplitude that increases according with the number of state. The first state almost never has jumps. The latter means that, when we are in the first state, the description of the variance of the observations  is left mainly to the parameter $\\sigma_{1}^{2}$. For the other states we see an increase in non-zero jumps and also an increase in the average number of jumps, as we go up in states.\n\n\\begin{figure}[h!]\n\t\\includegraphics[width = \\columnwidth, height = .35\\paperheight]{\"jump_histograms\"}\n\t\\caption{ \\small{Comparison between the histograms of each of the variances belonging to the four states.} }\n\t\\label{fig:jump_comparison}\n\\end{figure} \n\nNoticing how close the sample variances $\\hat{\\sigma}_{2}^{2}, \\hat{\\sigma}_{3}^{2},\\ \\text{and}\\ \\hat{\\sigma}_{4}^{2}$ are in value (Table \\ref{tab:variance1}), we see that the distinction between the states is in the jumps. This is exactly the result that we have aimed  to obtain by defining this model. In \\eqref{eq:transition_matrix_jump}, we report the matrix with the transition probabilities of the model, while eq. \\eqref{eq:transition_matrix_jump} indicates the average number of transitions from one state to another. In particular, the element  $\\hat P_{ij}$ is the transition probability $\\hat p_{ij}$ and $\\hat M_{ij}$ is the number of transitions from state $i$ to state $j$.\n\n\\begin{equation}\n\\label{eq:probability_matrix_jump}\n\\hat P = \\begin{bmatrix}\n0.6147 &  0.2522 &  0.0894 &  0.0435\\\\\n0.5326 &  0.3073 &  0.1082 &  0.0518\\\\\n0.1491 &  0.0805 &  0.7250 &  0.0453\\\\\n0.1671 &  0.0904 &  0.0487 &  0.6936\n\\end{bmatrix}\n\\end{equation} \n\n\\begin{equation}\n\\label{eq:transition_matrix_jump}\n\\hat M = \\begin{bmatrix}\n113.62 & 46.61 & 16.53 & 8.04 \\\\\n46.35 & 26.74 & 9.41 & 4.51 \\\\\n15.50 & 8.37 & 75.37 & 4.71 \\\\\n9.38 & 5.08 & 2.74 & 38.97\n\\end{bmatrix}\n\\end{equation} \n\nThe last thing we want to list before comparing our indicator with the VIX index, is the expected values of the state durations. Using eq. \\eqref{eq:expdur}, we obtain the following result\n\\begin{align}\n\\label{duration1}\n\\hat{d_1} = 2.6501, \\quad \\hat{d_2}=1.4227\\;,\\\\\n\\nonumber \\hat{d_3}=3.6518, \\quad \\hat{d_4}=3.3432\\;,\n\\end{align}\nwhich are coherent with the choice we made of exploiting  the Dirichlet prior  to make the model more resistant on states 3 and 4, while allowing more transitions between states 1 and 2.\n\n\\begin{remark}\n\tIn~\\cite{DiPMF16} the authors underlined that one of the main  issues that the proposed model cannot fix was the over-smoothing effect. In particular, they obtained the following durations: $\\hat{d_1}=45.2489 $, $\\hat{d_2}=23.9808$ $\\hat{d_3}=26.4550$ $\\hat{d_4}=3.3356$. One can notice that the duration of the highest state is conserved while the others changed, which was exactly our goal. Moreover such results clearly show an important development in the solution of the aforementioned issue, since duration can be used as a quantitative indicator of smoothness.\n\\end{remark}\n\nFinally, we compare our results with the VIX index data. In particular, our volatility indicator, denoted by $I_{t}^{J}$, will  indicates the expected standard deviation of the data at time $t$, namely\n\\begin{equation}\nI_{t}^{J} := \\sqrt{\\sum\\limits_{j=1}^{4} \\mathbb{P} (S_{t}=j|\\psi_{t}) \\left(\\hat{\\sigma}_{j}^{2} + \\frac{\\hat{N}_{j} \\left( \\hat{N}_{j}+1 \\right)}{40^{2}} \\right)} \\;, \n\\end{equation} \nwhere $J$ stands for {\\it jump},  we also present a visual comparison in the figure below.\n\n\\begin{figure}[!h]\n\t\\includegraphics[width = \\columnwidth, height = .25\\paperheight]{\"jump_b40\"}\n\t\\caption{ \\small{Visual comparison between the VIX and the volatility indicator $I_{t}^{J}$. We have applied a linear scaling function to $I_{t}^{J}$.} }\n\\end{figure}\n\nSince we want to compare this model to the $\\alpha-$stable distribution model that follows, we need some way to quantify what is a {\\it good} estimate of the volatility. Following a standard approach, we take into consideration the sum of the squares of the difference between our volatility indicator and the VIX, namely we define \n\\begin{equation}\n\\label{eq:squares1}\n\\mathbf{S}^{J} = \\sum\\limits_{t=1}^{T} \\left( I_{t}^{J} - y_{t, \\text{\\tiny VIX}} \\right)^{2}\\;,\n\\end{equation}\nwhere $y_{t, \\text{\\tiny VIX}}$ is the value of the VIX at time $t$. Using eq. \\eqref{eq:squares1}, we obtain $\\mathbf{S}^{J}=26.23$. In the next section we analyse the performances provided by  the second model.\n\n\\subsection{$\\alpha$-Stable Distribution Model}\n\\label{alpha_stable_case}\nIn this model we do not have to sample the jumps $N_{t}$, or their parameters $\\theta_{S_{t}}$. We use this decrease in the number of parameters  to infer on the mean $\\mu_{S_{t}}$ without making the simulations too cumbersome, or inaccurate. In Fig. \\ref{fig:stable_variance_histograms}, we present the histograms of the values $\\lambda \\gamma_{j}^{2},\\ j \\in \\{1,2,3,4\\}$. We choose these values, instead of just $\\gamma_{j}^{2}$, since the indicate the variance of $y_{t}|\\lambda$. Related estimates can be found in Table \\ref{tbl:variance2}. \n\n\\begin{figure}[!h]\n\t\\includegraphics[width = \\columnwidth, height = .3\\paperwidth]{\"stable_variance_histograms\"}\n\t\\caption{ \\small{The histograms of the variances $\\lambda \\gamma_{S_{t}}^{2}$.} }\n\t\\label{fig:stable_variance_histograms}\n\\end{figure}\n\nLooking at  Table \\ref{tbl:variance2}, we notice that there is a much bigger difference between the last three scale estimators, than there one between the last three Gaussian variance estimators. The latter result is due to the fact that the present model lacks of a jump component, therefore all the volatility has to be {\\it explained} by mean of the scale parameters. \n\n\\begin{table}[h!]\n\t\\caption{Parameter Point Estimates}\n\t\\centering\n\t\n\t\\begin{tabular}{c|c}\n\t\t\\label{tbl:variance2}\n\t\tEstimator & Value \\\\\n\t\t\\hline\n\t\t$\\hat{\\lambda}$ & $0.00390925$ \\\\\n\t\t$\\hat{\\gamma}_{1}^{2}$ & $0.1017084$ \\\\\n\t\t$\\hat{\\gamma}_{2}^{2}$ & $0.1841539$ \\\\\n\t\t$\\hat{\\gamma}_{3}^{2}$ & $0.3254673$ \\\\\n\t\t$\\hat{\\gamma}_{4}^{2}$ & $0.9113278$\n\t\\end{tabular}\n\\end{table}\n\nWe would like to underline that the simulations of $\\lambda$ are not robust. In particular, there is a very low acceptance rate in the exploited Metropolis-Hastings. We explain why this happens by an example, all the notation used in what follows, being the same as in subsection \\ref{sbsec:metropolis_hastings}.\n\nWe are in the special case of the Metropolis-Hastings algorithm where $\\lambda = \\boldsymbol \\theta$ is the only parameter. If we take $\\lambda^{j-1}=0.01$, where $\\lambda^{j-1}$ stands for the $(j-1)$-st sample of $\\lambda$, hence it is not its $(j-1)$-st power, and plot the graph of the \\emph{nonzero acceptance probabilities} around that value, see Fig. \\ref{fig:acceptance_probabilities}, we clearly see where the  simulations fail to be robust. \n\\begin{figure}[!h]\n\t\\includegraphics[width = \\columnwidth, height=.25 \\paperheight ]{\"acceptance_probabilities\"}\n\t\\caption{ \\small{Acceptance probabilities of candidates around the value 0.01.} }\n\t\\label{fig:acceptance_probabilities}\n\\end{figure}\nIn particular, if $\\lambda_{new} \\in \\{0.013,0.0014,...,0.0099\\}$, then its acceptance probability is 1, namely we automatically take $\\lambda^{j}=\\lambda_{new}$. There are only three values that $\\lambda_{new}$ can take and that are larger than $0.01$, i.e.  $0.0101,\\ 0.0102,\\ \\text{and}\\ 0.0103$, with acceptance probabilities $0.23,\\ 0.053,\\ \\text{and}\\ 0.012$, respectively. The latter implies that the samples of $\\lambda$ will converge towards zero, which poses a numerical problem since then the values of $\\gamma_{S_{t}}$ will blow up, since  $V(y_{t}|\\lambda) = \\lambda \\gamma_{S_{t}}^{2}$, in order to compensate. At some point the values of $\\gamma_{S_{t}}$ become too large to be numerically handled, subsequently the simulation crashes. We can get around small values for $\\lambda$, e.g. bounding it from below by some small constant, but then, once $\\lambda$ reaches such a  value, the algorithm rarely  accept a larger value as a sample, hence leading to the problem of too few samples of $\\lambda$ being accepted. Despite the aforementioned shortcoming, the proposed model still works quite effective, as we will see further down. \n\nIn Fig. \\ref{fig:mean_histograms}, we present the histograms of the different means, while Table \\ref{tbl:means} reports their point estimates.\n\\begin{table}[!h]\n\t\\caption{Mean Point Estimates}\n\t\\centering\n\t\\begin{tabular}{c|c}\n\t\t\\label{tbl:means}\n\t\tEstimator & Value \\\\\n\t\t\\hline\n\t\t$\\hat\\mu_{1}$ & $0.007381561$ \\\\\n\t\t$\\hat\\mu_{2}$ & $0.008574574$ \\\\\n\t\t$\\hat\\mu_{3}$ & $0.002191889$ \\\\\n\t\t$\\hat\\mu_{4}$ & $-0.02942247$\n\t\\end{tabular}\n\\end{table}\n\nOne thing that stands out in the mean point estimates, is the sign of the mean of the fourth state, which is negative. The latter should not come as a surprise since it refers to highest volatility value in the time series, namely the one related to the mortgage crisis of 2008. We recall that, during a severe financial crisis, most price movements are downward, resulting in a negative drift.\n\n\\begin{figure}[!h]\n\t\\includegraphics[width = \\columnwidth, height=.25 \\paperheight]{\"mean_comparisons\"}\n\t\\caption{ \\small{The histograms of the means $\\mu_{S_{t}}$.} }\n\t\\label{fig:mean_histograms}\n\\end{figure}\n\nIn what follows, we present the transition probability matrix, see eq. \\eqref{eq:probability_matrix_stable}, and the transition matrix, see eq. \\eqref{eq:transition_matrix_jump}, namely\n\n\\begin{equation}\n\\label{eq:probability_matrix_stable}\n\\hat P = \\begin{bmatrix}\n0.3492 & 0.2719 & 0.2102 & 0.1686\\\\\n0.3655 & 0.2728 & 0.1993 & 0.1622\\\\\n0.0157 & 0.0144 & 0.9518 & 0.0179\\\\\n0.0264 & 0.0222 & 0.0220 & 0.9292\n\\end{bmatrix}\n\\end{equation} \n\n\\begin{equation}\n\\label{eq:transition_matrix_stable}\n\\hat M = \\begin{bmatrix}\n29.10 & 15.65 & 7.75 & 3.62\\\\\n16.16 & 13.76 & 5.82 & 2.95\\\\\n6.61 & 5.54 & 270.39 & 4.17\\\\\n4.02 & 3.49 & 3.22 & 39.68\n\\end{bmatrix}\n\\end{equation} \n\nAs in the previous case, we note the expected values of the state durations obtained using eq. \\eqref{eq:expdur}, namely\n\\begin{align}\n\\label{duration2}\n\\hat{d_1} &= 1.5365, \\quad \\hat{d_2}=1.3751\\;,\\\\\n\\nonumber \\hat{d_3}&=20.746, \\quad \\hat{d_4}=14.124\\;.\n\\end{align}\nWe can note how the difference between the results in \\eqref{duration2} and those in \\eqref{duration1}, is significant. In order to better  explain the latter datum, let us define the volatility indicator within the present framework, and make a comparison with the VIX index. In particular we define a second volatility indicator, denoted by $I_{t}^{\\alpha}$, which, analogously to the previous case, will stand for the expected standard deviation of the data at time $t$, i.e.\n\\begin{equation}\nI_{t}^{\\alpha} = \\sqrt{ \\hat{\\lambda} \\sum\\limits_{j=1}^{4} \\mathbb{P} (S_{t}=j|\\psi_{t}) \\hat{\\gamma}_{j}^{2} }  \\;.\n\\end{equation} \nIn Fig. \\ref{fig:stable_comparison}, we can see a visual comparison between the two values.\n\\begin{figure}[!h]\n\t\\includegraphics[width = \\columnwidth, height = .25\\paperheight]{\"stable_vix\"}\n\t\\caption{ \\small{Visual comparison between the VIX and the volatility indicator $I_{t}^{\\alpha}$. We have applied a linear scaling function to $I_{t}^{\\alpha}$.} }\n\t\\label{fig:stable_comparison}\n\\end{figure}\nUsing eq. \\eqref{eq:squares1}, we obtain $\\mathbf{S}^{\\alpha}=39.69$, which is a significant increase over $\\mathbf{S}^{J}$. This leads us to conclude that the estimate obtained from the jump diffusion model is closer to the VIX than the one obtained from the $\\alpha$-stable distribution model.\n\nWe now briefly explain the difference between the results in \\eqref{duration1} and \\eqref{duration2}. The expected duration of state 1 falls while at the same time there is a drastic increase in the expected durations of states 3 and 4. Looking at the way the estimators behave in Fig. \\ref{fig:jump_comparison} and in Fig. \\ref{fig:stable_comparison},  it is to note that the estimator obtained from the jump diffusion model is much more {\\it jagged}, because of the regular transition from one state to another; while the one obtained from the stable distribution model is much smoother, seeing as the time series tends to stay in the high volatility states much longer. Furthermore, when the stable distribution model has to place observations, that should be in the low volatility states, in the high volatility ones, so that to solve the  variance underestimation problem mentioned in \\cite{DiPMF16}, the jump diffusion model can simply add a few jumps to make up for the missing variance. This is why, despite its attempts to increase the variance by staying in the higher states, we see the indicator of the stable model {\\it drooping} and underestimating the low volatility, while, in this situation, the jump model stays much closer to the VIX.\n\n\\section{Conclusion and Future Developments}\nIn the present paper we have presented  two novel techniques to implement a   Markov Switching Model  (MSM) type approach to  non-stationary data,  namely  a jump diffusion-MSM and an $\\alpha-$stable-MSM. \nIn Sec. \\ref{jump_diffusion_case}, we have shown that the first one is  very effective in mimic the VIX index, moreover its implementation can be smoothly done without sacrificing its theoretical peculiarities, see Sec. \\ref{JumpDiffusionModel}.\nA slightly different situation concerns the implementation of the second approach, see Sec. \\ref{alpha_stable_case}, since eeven if the  $\\alpha$-stable-MSM approach turns to be quite effective, we have to consider sampling problems of one of its parameters, implying that computational results do not  behave exactly the way the are meant to. \n\nWe would like to underline that the achieved tractability of the jump diffusion model is a crucial point, and it  witnesses how such technique can be fruitfully used   to model any kind of time series presenting pronounced tails, not just financial ones.\n\nAs far as the  issues of over-smoothing and excessive state duration, which have been stated in  \\cite{DiPMF16} as the main deficiencies of the MSM approach to financial data,  we have shown that, using the models here presented, the state durations can been significantly reduced, see subsection \\ref{jump_diffusion_case}, and the problem of over-smoothing can be solved, see subesctions \\ref{jump_diffusion_case} and \\ref{alpha_stable_case}.\nConcerning future developments, we aim at improving our jump diffusion-MSM model by considering , instead of a simple first-order Markov transition law,  a $k$-th order Markov transition law. Other possibilities consist in dealing with a transition law that is state duration dependent,  or allowing the law to depend on other observable quantities used as indicators  of the economy behaviour, e.g., real personal income, industrial production index, rate of private credit growth, etc. \n\\linebreak \\\\\n\\textbf{Acknowledgements:} We would like to sincerely thank Matteo Frigo for his insightful comments and his fundamental suggestions, which have helped us a lot in preparing the present work, especially with respect to the duration analysis.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:intro}\nIn this paper we consider efficient computational approaches to compute approximate\nsolutions of a linear inverse problem,\n\\begin{equation}\nb = Ax_\\text{true} + \\eta, \\quad \\quad A\\in\\mathbb{R}^{m\\times n},\\label{eq:inverse_problem}\n\\end{equation}\nwhere $A$ is a known matrix, \nvector $b$ represents known acquired data, $\\eta$ represents noise, and vector $x_\\text{true}$ \nrepresents the unknown quantity that needs to be approximated.\nWe are particularly interested in imaging applications where $x_\\text{true} \\geq 0$ and $Ax_\\text{true} \\geq 0$.\nAlthough this basic problem has been studied extensively \n(see, for example, \\cite{Engl2000Regularization,Hansen2010Discrete,Mueller2012Linear,Vogel2002Computational}\nand the references therein), the noise is typically assumed to come from a single source (or to be\nrepresented by a single statistical distribution) and the data to contain no outliers.\nIn this paper we focus on a practical situation that arises in many imaging applications, and for which relatively\nlittle work has been done, namely when the noise is comprised of a mixture of Poisson and\nGaussian components {\\em and}\nwhen there are outliers in the measured data.  While some research has been done\non the two topics separately (i.e., mixed Poisson--Gaussian noise models {\\em or} \noutliers in measured data), to our knowledge no work has been done when the measured\ndata contains both issues. In the following, we review some of the approaches used to handle each of the issues.\n\\subsubsection*{Poisson--Gaussian noise}\nA Poisson--Gaussian statistical model for the measured data takes the form\n\\begin{equation}\nb_i = n_\\text{obj}(i) + g(i), \\ i = 1,\\ldots,m, \\quad \\ n_\\text{obj}(i) \\sim \\text{Pois}([Ax_\\text{true}]_i), \\ g(i) \\sim \\mathcal{N}(0,\\sigma^2), \\label{eq:noise}\n\\end{equation}\nwhere $b_i$ is the $i$th component of the vector $b$ and $[Ax_\\text{true}]_i$ the $i$th component of the true noise-free \ndata $Ax_\\text{true}$. We assume that the two random variables $n_\\text{obj}(i)$ and $g(i)$ are independent. \nThis mixed noise model arises in many important applications, such as when using charged coupled device (CCD) arrays,\nx-ray detectors, and infrared \nphotometers \\cite{Bardsley2003nonnegatively,Gravel2004method,Luisier2011Image,Makitalo2013Optimal,Snyder1993Image}.\nThe Poisson part (sometimes referred to as shot noise) can arise\nfrom the accumulation of photons over a detector, and the Gaussian part usually is due to {\\em read-out} noise from\na detector, which can be generated by thermal fluctuations in interconnected electronics.\n\nSince the log-likelihood function for the mixed Poisson--Gaussian model (\\ref{eq:noise}) has an infinite series representation \\cite{Snyder1993Image}, we assume a simplified model, where both random variables have the same type of distribution. \nThere are two main approaches one can take to generate a simplified model.\nThe first approach is to add $\\sigma^2$ to each component of the vector $b$, and from (\\ref{eq:noise}) it then follows that\n\\begin{equation}\n\\mathbb{E}(b_i + \\sigma^2) = [Ax_\\text{true}]_i + \\sigma^2 \\quad \\text{and} \\quad \\text{var}(b_i + \\sigma^2) = [Ax_\\text{true}]_i + \\sigma^2. \\label{eq:poiss_approx_noise}\n\\end{equation}\nFor large $\\sigma$, the Gaussian random variable $g(i) + \\sigma^2$ is well-approximated by a Poisson random variable with the Poisson parameter $\\sigma^2$, and therefore $b_i + \\sigma^2$ is also well approximated by a Poisson random variable with the Poisson parameter $[Ax_\\text{true}]_i + \\sigma^2$. The data fidelity function corresponding to the negative Poisson log-likelihood then has the form\n\\begin{equation}\n\\sum_{i=1}^m ([Ax_\\text{true}]_i + \\sigma^2) - (b_i +\\sigma^2)\\,\\log([Ax_\\text{true}]_i + \\sigma^2);\\label{eq:poisson}\n\\end{equation}\nsee also \\cite{Snyder1993Image}. An alternative approach is to approximate the true negative log-likelihood by a weighted least-squares function, where the weights correspond to the measured data, i.e.,\n\\begin{equation}\n \\sum_{i=1}^m \\frac{1}{2}\\left(\\frac{[Ax]_i - b_i}{\\sqrt{b_i + \\sigma^2}}\\right)^2;\\label{eq:wls_basic}\n\\end{equation}\nsee \\cite[Sec. 1.3]{Hansen2013Least}. \nA more accurate approximation can be achieved by replacing the measured data by the computed data \n(which depends on $x$), i.e., replace the fidelity function (\\ref{eq:wls_basic}) by\n\\begin{equation}\n \\sum_{i=1}^m \\frac{1}{2}\\left(\\frac{[Ax]_i - b_i}{\\sqrt{[Ax]_i + \\sigma^2}}\\right)^2;\\label{eq:wls}\n\\end{equation}\nsee \\cite{Stagliano2011Analysis} for more details. \nAdditional additive Poisson noise (e.g., background emission) can be incorporated into the model in a straightforward way. \n\\subsubsection*{Outliers}\nFor  data corrupted solely with Gaussian noise, i.e., \n\\begin{equation}\nb_i = [Ax]_i + g(i), \\ i = 1,\\ldots,m, \\quad g(i) \\sim \\mathcal{N}(0,\\sigma^2), \\label{eq:gausssian_noise}\n\\end{equation} employing the negative log-likelihood leads to the standard least-squares functional\n\\begin{equation}\n \\sum_{i=1}^m \\frac{1}{2}\\left([Ax]_i - b_i\\right)^2.\\label{eq:ls}\n\\end{equation} \nIt is well known however that a computed solution based on least squares is not robust if outliers occur, meaning that even a small number of components with gross errors can cause a severe deterioration of our estimate. Robustness of the least squares fidelity function can be achieved by replacing the loss function $\\frac{1}{2}t^2$ used in \\eqref{eq:ls} by a function $\\rho(t)$  as\n\\begin{equation}\n \\sum_{i=1}^m \\rho\\left([Ax]_i - b_i\\right),\\label{eq:robust}\n\\end{equation}where the function $\\rho$ is less stringent towards the gross errors and satisfies the following conditions:\n\\begin{enumerate}\n\\item $\\rho(t) \\geq 0$;\n\\item $\\rho(t) = 0 \\Leftrightarrow t =0$;\n\\item $\\rho(-t) = \\rho(t)$;\n\\item $\\rho(t^\\prime) \\geq \\rho(t)$, for $t^\\prime \\geq t \\geq 0$;\n\\end{enumerate}\nsee also \\cite[Sec. 1.5]{Hansen2013Least}. A list of eight most commonly used loss functions $\\rho$ can be found in  \\cite{Coleman1980system} or in MATLAB under \\texttt{robustfit}; some of them are discussed in Section~\\ref{sec:conv_anal}. Each of these functions also depends on a parameter $\\beta$ (see Section~\\ref{sec:beta}) defining the trade-off between the robustness and efficiency. Note that if we use this robust regression approach, in order to reduce the influence of possible outliers, we always sacrifice some efficiency at the model.\n\n\\bigskip\n\nIn this paper, we focus on combining these two approaches to suppress the influence of outliers for data with mixed noise \\eqref{eq:noise}. Our work has been motivated by O'Leary \\cite{OLeary1990Robust}, and more recent work by Calef \\cite{Calef2013Iteratively}.  The initial ideas of our work were first outlined in the \nconference paper \\cite{Kubinova2015Iteratively}.\n\nThe paper is organized as follows. In Section~\\ref{sec:robust} we introduce a data-fidelity function suitable for data corrupted both with mixed Poisson--Gaussian noise and outliers. In Section~\\ref{sec:reg_param} we propose a regularization parameter choice method for the regularization of the resulting inverse problem, and in Section~\\ref{sec:optim} we focus on the optimization algorithm and the solution of the linear subproblems. Section~\\ref{sec:num_exp} demonstrates the performance of the resulting method on image deblurring problems with various types of outliers.\n\nThroughout the paper, $D$ (or $D$ with an accent) denotes a general real diagonal matrix, $e_i$ denotes the $i$th column of the identity matrix of a suitable size.\n\n\n\\section{Data-fidelity function}\\label{sec:robust}\nIn Section \\ref{sec:intro}, we reviewed fidelity functions \\eqref{eq:poisson}, \\eqref{eq:wls_basic}, and \\eqref{eq:wls}, commonly used for problems with mixed Poisson--Gaussian noise and also robust loss  functions used to handle problems with Gaussian noise and outliers \\eqref{eq:robust}. Since we need to deal with both issues simultaneously here, we propose combining both approaches. More specifically, combining a robust loss function with the weighted least squares problem \\eqref{eq:wls}, so that the data fidelity function becomes\n\\begin{equation}\n J(x) = \\sum_{i=1}^m \\rho\\left(\\frac{[Ax]_i - b_i}{\\sqrt{[Ax]_i + \\sigma^2}}\\right).\n\\label{eq:robust_wls}\n\\end{equation}\nIn the remainder of this section, we investigate the properties of the proposed data-fidelity function \\eqref{eq:robust_wls} and the choice of the robustness parameter $\\beta$, which is defined in the next subsection.\n\\subsection{Choice of the loss function -- convexity analysis} \\label{sec:conv_anal}\nFor ordinary least squares, functions known under names Huber, logistic, Fair, and Talwar, shown in Figure~\\ref{fig:loss_functions}, lead to an interval-wise convex data fidelity function, see \\cite{OLeary1990Robust}, i.e., positive-semidefinite Hessian, which is favorable for Newton-type minimization algorithms. This however does not \nalways hold in our\ncase where the weighted least squares formulation \\eqref{eq:robust_wls} has solution-dependent weights. \n\nTo see this, let us begin by denoting the components of the residual as $r_i \\equiv [Ax]_i - b_i$ and the \nsolution-dependent weights as $w_i \\equiv \\frac{1}{\\sqrt{[Ax]_i + \\sigma^2}}$. Then the gradient and the Hessian of \\eqref{eq:robust_wls} can be rewritten as\n\n\\begin{align}\n\\text{grad}_J(x) &= A^Tz, & z_i &= \\left(w^\\prime_ir_i + w_i\\right)\\rho^{\\prime}(w_ir_i);\\\\\n\\text{Hess}_J(x) &= A^TDA, & \\quad D_{ii} &= (w^{\\prime\\prime}_ir_i + 2w_i^{\\prime})\\rho^{\\prime}(w_ir_i) + (w^{\\prime}_i r_i + w_i)^2\\rho^{\\prime\\prime}(w_ir_i). \\label{eq:hess_and_grad}\n\\end{align}\nWe investigate the entries $D_{ii}$ in order to examine the positive semi-definiteness of the Hessian $\\text{Hess}_J(x)$. Recall that $A^TDA$ is positive semi-definite, if $D_{ii}\\geq 0$.\n\nAssuming $\\rho^{\\prime\\prime}\\geq 0$, the signs of the diagonal entries $D_{ii}$ in \\eqref{eq:hess_and_grad} are\n\\begin{equation}\n\\text{sign}(D_{ii}) = \\left(\\circlearound{+}\\cdot\\text{sign}(r_i) + 2\\circlearound{-}\\right)\\cdot\\text{sign}(\\rho^{\\prime}(w_ir_i)) + \\circlearound{+}\\circlearound{+}, \\label{eq:dii_sign}\n\\end{equation}\nwhere we have replaced some of the quantities in the\nexpression for $D_{ii}$ shown in equation \\eqref{eq:hess_and_grad} with the symbol $\\circlearound{-}$ when the\nvalue it replaces is always a negative number and with $\\circlearound{+}$ when the value it replaces is always nonnegative. We will now investigate all possible cases with respect to $\\text{sign}(\\rho^{\\prime}(w_ir_i))$:\n\\begin{itemize}\n\\item \\textit{Case 1: $\\rho^\\prime(w_ir_i)< 0$} \\\\\n$\\rho^\\prime(w_ir_i)< 0$  yields $r_i< 0$, and therefore $D_{ii} > 0$.\n\\item \\textit{Case 2: $\\rho^\\prime(w_ir_i)= 0$} \\\\\n$\\rho^{\\prime}(w_ir_i) = 0 $ yields $D_{ii} = 0$.\n\\item \\textit{Case 3: $\\rho^\\prime(w_ir_i)> 0$} \\\\\n Substituting for $w_i$ and $r_i$ in \\eqref{eq:hess_and_grad}, we obtain\n\\begin{align*}\\label{eq:dii}\n D_{ii} &= \\left(\\frac{3}{4}([Ax]_i-b_i)([Ax]_i+\\sigma^2)^{-5\/2} - ([Ax]_i+\\sigma^2)^{-3\/2})\\right)\\rho^{\\prime}(w_ir_i)\\\\ \n\\nonumber & \\quad + \\left(-\\frac{1}{2}([Ax]_i-b_i)([Ax]_i+\\sigma^2)^{-3\/2} + ([Ax]_i+\\sigma^2)^{-1\/2}\\right)^2\\rho^{\\prime\\prime}(w_ir_i).\n\\end{align*} \nFor $[Ax]_i\\gg b_i+\\sigma^2$, to achieve $D_{ii}\\geq 0$, \n\\[\n\\sqrt{[Ax]_i}\\cdot\\rho^{\\prime\\prime}(\\sqrt{[Ax]_i})\\gtrsim \\rho^{\\prime}(\\sqrt{[Ax]_i}),\\\\\n\\]\nmust hold. This corresponds to \n\\[\n\\rho^{\\prime}(t) \\gtrsim t \\quad \\text{yielding} \\quad  \\rho(t) \\gtrsim t^2\/2,\\\\\n\\]\ni.e., for large $[Ax]_i$, the loss function $\\rho$ has to grow at least quadratically.\n\\end{itemize}\n\nConcluding, for large $t$, the loss function $\\rho(t)$ has to be either constant or grow at least quadratically, which is in contradiction with the idea of robust regression. Therefore, considering the functions from \\cite{Coleman1980system}, the only loss function $\\rho$ for which the data fidelity function (\\ref{eq:robust_wls}) has positive semidefinite Hessian, is the function Talwar:\n\\begin{equation}\n\\rho(t) = \\left\\{\\begin{array}{ll}\nt^2\/2, & |t|\\leq\\beta,\\\\\n\\beta^2\/2, & |t| > \\beta.\n\\end{array} \\right.\\label{eq:talwar}\n\\end{equation} \n\\subsection{Selection of the robustness parameter}\\label{sec:beta}\nParameters $\\beta$ for $95\\%$ asymptotic efficiency with respect to the standard loss function $\\frac{1}{2}t^2$ when the disturbances come from the unit normal distribution can again be found in \\cite{Coleman1980system}. For Talwar, the 95\\% efficiency tuning parameter is \n\\begin{equation}\n\\beta_{95} = 2.795.\\label{eq:beta_opt}\n\\end{equation} \nNote that in our specific case, the random variable inside the function $\\rho$ in (\\ref{eq:robust_wls}) is already rescaled to have unit variance and therefore approximately unit normal distribution. We may therefore apply the parameter $\\beta_{95}$ without any further rescaling based on estimated variance, which is usually required in case of ordinary least squares with unknown variance of noise. Function Talwar with  $\\beta = \\beta_{95}$ is shown in Figure~\\ref{fig:rho_talwar}. \n\\begin{figure}\n\\centering\n\\begin{subfigure}[b]{.4\\textwidth}\n\\centering\n\\includegraphics[width = .8\\textwidth]{loss_function_fair} \n\\caption{Fair}\n\\end{subfigure}\n\\begin{subfigure}[b]{.4\\textwidth}\n\\centering\n\\includegraphics[width = .8\\textwidth]{loss_function_huber} \n\\caption{Huber}\n\\end{subfigure}\n\n\\begin{subfigure}[b]{.4\\textwidth}\n\\centering\n\\includegraphics[width = .8\\textwidth]{loss_function_logistic} \n\\caption{logistic}\n\\end{subfigure}\n\\begin{subfigure}[b]{.4\\textwidth}\n\\centering\n\\includegraphics[width = .8\\textwidth]{loss_function_talwar} \n\\caption{Talwar}\\label{fig:rho_talwar}\n\\end{subfigure}\n\n\\caption{Loss functions Fair, Huber, logistic and Talwar for the tuning parameter $\\beta$ corresponding to 95\\% efficiency (solid line)  together with the standard loss function $t^2\/2$ (dashed line).\n}\\label{fig:loss_functions}\n\\end{figure}\n\n\\subsection{Non-negativity constraints}\nIn many applications, such as imaging, the reconstruction will benefit from taking into account the prior information about the component-wise non-negativity of the true solution $x_\\text{true}$. Here, however, imposing non-negativity is not just a question of visual appeal, it also guarantees the two estimates (\\ref{eq:poisson}) and (\\ref{eq:wls}) of the negative log-likelihood will provide similar results; see \\cite{Stagliano2011Analysis}. Therefore, the component-wise non-negativity constraint is an integral part of the resulting optimization problem. However, employment of the non-negativity constraint results in the need of more sophisticated optimization tools. The use of one of the possible algorithms is discussed in Section~\\ref{sec:optim}.\n\n\\section{Regularization and selection of the regularization parameter}\\label{sec:reg_param}\nAs a consequence of noise and ill-posedness of the inverse problem (\\ref{eq:inverse_problem}), some form of regularization needs to be employed in order to achieve a reasonable approximation of the true solution $x_\\text{true}$. For computational convenience, we use Tikhonov regularization with a quadratic penalization term, i.e., we minimize the functional of the form\n\\begin{equation}\nJ_\\lambda(x) \\equiv \\sum_{i=1}^m \\rho\\left(\\frac{[Ax]_i - b_i}{\\sqrt{[Ax]_i + \\sigma^2}}\\right) + \\frac{\\lambda}{2}\\|Lx\\|^2, \\qquad x\\geq 0.\\label{eq:functional}\n\\end{equation}\nWe assume that a good regularization parameter $\\lambda$ with respect to $L$ is used so that the penalty term is reasonably close to the prior and the residual therefore is close to noise. In case of robust regression, it is particularly important not to over-regularize, since this would lead to large residuals and too many components of the data $b$ would be considered outliers.  Methods for\nchoosing $\\lambda$ are discussed in this section.\n\n\\subsection{Morozov's discrepancy principle}\nSince the residual components are scaled, and for data without outliers we have the expected value\n \\begin{equation}\nE\\left\\{\\frac{1}{n}\\sum_{i=1}^{n}\\frac{([Ax]_i - b_i)^2}{[Ax]_i + \\sigma^2}\\right\\} = 1,\\label{eq:discrepancy_estimate}\n\\end{equation}\nan obvious choice would be to use the Morozov's discrepancy principle \\cite{Morozov1966solution,Vogel2002Computational}. However, as reported in \\cite{Stagliano2011Analysis}, even without outliers, the discrepancy principle based on \\eqref{eq:discrepancy_estimate} tends to provide unsatisfactory reconstructions for problems with large signal-to-noise ratio. Therefore we will not consider this approach further.\n\n\\subsection{Generalized cross validation}\\label{sec:gcv}\nThe generalized cross validation method \\cite{Golub1979Generalized}\\cite[chap. 7]{Vogel2002Computational} is a method derived from the standard leave-one-out cross validation. To apply this method for linear Tikhonov regularization, one selects the regularization parameter $\\lambda$ such that it minimizes the GCV functional\n\\begin{equation}\n\\text{GCV}(\\lambda) = \\frac{n\\|r_\\lambda\\|^2}{(\\text{trace}(I-A_\\lambda))^2},\\label{eq:gcv_fun}\n\\end{equation}\nwhere $r_\\lambda = Ax_\\lambda - b = (A_\\lambda - I)b$ is the residual, $n$ is its length, and the influence matrix $A_\\lambda$ takes the form $A_\\lambda = A(A^TA + \\lambda L^TL)^{-1}A^T$. Here, due to the non-negativity constraints and the weights, the residual and the influence matrix have a more complicated form. An approximation of the influence matrix for problems with mixed noise, but without outliers, has been proposed in \\cite{Bardsley2009Regularization}. There the numerator of the GCV functional\ntakes the form  $n\\|Wr_\\lambda\\|^2$ and the approximate influence matrix\n\\begin{equation}\nA_\\lambda = WA(D_\\lambda(A^TW^2A + \\lambda L^TL)D_\\lambda)^{\\dagger}D_\\lambda A^TW,\\label{eq:gcv_fun_new} \n\\end{equation}\nwhere $W$ and $D_\\lambda$ are diagonal matrices\n\\begin{align}\nW_{ii} &=\n\\frac{1}{\\sqrt{[Ax_\\lambda]_i + \\sigma^2}};\n\\label{eq:W_ii_old}\\\\\n[D_\\lambda]_{ii} &= \\left\\{\n\\begin{array}{cc}\n1 & [x_\\lambda]_i > 0, \\\\ \n0 &\\text{otherwise,}\n\\end{array} \\right. \\nonumber\n\\end{align}\nand \\,${}^\\dagger$ denotes the Moore-Penrose pseudoinverse. Matrix $D_\\lambda$ only handles the non-negativity constraints, \nand therefore can be adopted directly. The matrix $W$ needs a special adjustment, due to the change of the loss function to Talwar. The aim is to construct a matrix $W$ satisfying\n\\[\\|Wr_\\lambda\\|^2 = \\sum_{i=1}^m \\rho\\left(\\frac{[Ax_\\lambda]_i - b_i}{\\sqrt{[Ax_\\lambda]_i + \\sigma^2}}\\right).\\]\nSubstituting for $\\rho$ from the definition of the function Talwar \\eqref{eq:talwar}, we redefine the scaling matrix as\n\\begin{align}\nW_{ii} &\\equiv \\left\\{\n\\begin{array}{cc}\n\\frac{1}{\\sqrt{[Ax_\\lambda]_i + \\sigma^2}} & \\quad \\left\\vert\\frac{[Ax_\\lambda]_i - b_i}{\\sqrt{[Ax_\\lambda]_i + \\sigma^2}}\\right\\vert\\leq \\beta, \\\\ \n\\frac{\\beta}{[Ax_\\lambda]_i - b_i} &\\text{otherwise};\n\\end{array} \\right.\\label{eq:W_ii}\n\\end{align}\n\nIn order to make the evaluation of \\eqref{eq:gcv_fun_new} feasible for large-scale problems, we approximate the trace of a matrix using the random trace estimation \\cite{Hutchinson1990stochastic,Vogel2002Computational} as $\\text{trace}(M)\\approx v^TMv$, where the entries of $v$ take values $\\pm 1$ with equal probability. Applying the random trace estimation to \\eqref{eq:gcv_fun_new}, we obtain\n\\[\n(\\text{trace}(I-A_\\lambda))^2 \\approx (v^Tv - v^TA_\\lambda v)^2.\n\\]\nFinally, $A_\\lambda v$ is approximated by $WAy$, with $y$ obtained applying truncated conjugate gradient iteration to\n\\begin{equation}\n(D_\\lambda(A^TW^2A + \\lambda L^TL)D_\\lambda)y = D_\\lambda A^TWv.\\label{eq:gcv_lin_syst}\n\\end{equation}\n\n\\section{Minimization problem}\\label{sec:optim}\n\nIn this section we discuss numerical methods to compute a minimum of \\eqref{eq:functional}.\nWe consider incorporating a non-negative constraint and solution of linear subproblems, including\nproposing a preconitioner.  \n\n\\subsection{Projected Newton's method}\nVarious methods for constrained optimization have been developed over the years, some related to image deblurring can be found in \\cite{Bardsley2003nonnegatively,Bonettini2009scaled,Hanke2000Quasi,More1991solution,Nagy2000Enforcing}. \nFor our computations, we chose a projected Newton's method\\footnote{In \\cite{Haber2015Computational}, the method was derived as the Projected Gauss--Newton method. Here, since the evaluation of the Hessian does not represent a computational difficulty, we use it as a variant of Newton's method. Therefore, in the remainder of the text, the method is referred to as the Projected Newton's Method.}, combined with projected PCG to compute the search direction in each step, see \\cite[sec. 6.4]{Haber2015Computational}. The convenience of this method lies in the fact that the projected PCG does not require any special form of the preconditioner and a generic conjugate gradient preconditioner can be employed. Besides lower bounds, upper bounds on the reconstruction can also be enforced. For completeness, we include the projected Newton method in Algorithm~\\ref{alg:projGNCG}, and projected PCG in\nAlgorithm~\\ref{alg:projPCG}.\n\n\n\\begin{algorithm}\n\\caption{Projected Newton's method \\cite{Haber2015Computational}}\n\\label{alg:projGNCG}\n\\begin{algorithmic}\n\\STATE{$k = 0$}\n\\WHILE{not converged} \n\\STATE{$\\text{Active} = (x^{(k)} \\leq 0$)} \n\\STATE{$g = \\text{grad}_{J_\\lambda}(x^{(k)})$}\n\\STATE{$H = \\text{Hess}_{J_\\lambda}(x^{(k)})$}\n\\STATE{$M = \\texttt{prec}(H)$} \\COMMENT{setup preconditioner for the Hessian}\n\\STATE{$s = \\texttt{projPCG}(H,-g,\\text{Active},M)$} \\COMMENT{compute the search direction for inactive cells}\\STATE{$g_a = g(\\text{Active})$} \n\\IF{$\\max(\\text{abs}(g_a)) >  \\max(\\text{abs}(s))$} \n\\STATE {$g_a = g_a\\cdot\\max(\\text{abs}(s))\/\\max(\\text{abs}(g_a))$} \\COMMENT{rescaling needed} \\ENDIF\n\\STATE{$s(\\text{Active}) = g_a$} \\COMMENT{take gradient direction in active cells}\n\\STATE{$x^{(k+1)} = \\texttt{linesearch}(s,x^{(k)},J_\\lambda,\\text{grad}_{J_\\lambda})$}\n\\STATE{$k = k+1$}\n\\ENDWHILE\n\\RETURN $x^{(k)}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}[H]\n\\caption{Projected PCG \\cite{Haber2015Computational}}\n\\label{alg:projPCG}\n\\begin{algorithmic}\n\\STATE{input: $A$, $b$, Active, $M$}\n\\STATE{$x_0 = 0$}\n\\STATE{$D_\\mathcal{I} = \\text{diag}(1-\\text{Active})$} \\COMMENT{projection onto inactive set}\n\\STATE{$r_0 = D_\\mathcal{I}b$}\n\\STATE{$z_0 = D_\\mathcal{I}(M^{-1}r_0)$}\n\\STATE{$p_0 = z_0$}\n\\STATE{$k = 0$}\n\\WHILE{not converged} \n\\STATE{$\\alpha_k = \\frac{r_k^Tz_k}{p^TD_\\mathcal{I}Ap_k}$} \n\\STATE{$x_{k+1} = x_k + \\alpha_kp_k $}\n\\STATE{$r_{k+1} = x_k - \\alpha_kD_\\mathcal{I}Ap_k $}\n\\STATE{$z_{k+1} = D_\\mathcal{I}(M^{-1}r_k)$}\n\\STATE{$\\beta_{k+1} = \\frac{z_{k+1}^Tr_{k+1}}{z_{k}^Tr_{k}}$}\n\\STATE{$p_{k+1} = z_{k+1} + \\beta_kp_k$}\n\\STATE{$k = k+1$}\n\\ENDWHILE\n\\RETURN $x_{k}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\subsection{Solution of the linear subproblems}\\label{sec:lin_systems}\nEach step of the projected Newton method (Algorithm \\ref{alg:projGNCG}) \nrequires solving a linear system with the Hessian:\n\\begin{align}\n\\text{Hess}_{J_\\lambda}(x^{(k)})s &= -\\text{grad}_{J_\\lambda}(x^{(k)})\\nonumber \\\\\n(A^TD^{(k)}A + \\lambda L^TL)s &= -\\left(A^Tz^{(k)} + \\lambda L^TLx^{(k)}\\right).\\label{eq:lin_system}\n\\end{align}\nFor the objective functional \\eqref{eq:functional}, the diagonal matrix $D^{(k)}$ and the vector $z^{(k)}$ have the form:\n\\begin{align}\nz_{i} &= \\left\\{\\begin{array}{ll}\n\\frac{1}{2}\\left(1 - \\frac{\\left(b_i + \\sigma^2\\right)^2}{\\left([Ax]_i + \\sigma^2\\right)^2}\\right), & \\left\\vert\\frac{[Ax]_i - b_i}{\\sqrt{[Ax]_i + \\sigma^2}}\\right\\vert\\leq \\beta,\\\\\n0, & \\text{otherwise}.\n\\end{array}\n\\right.\\\\\nD_{ii} &= \\left\\{\\begin{array}{ll}\n\\frac{\\left(b_i + \\sigma^2\\right)^2}{\\left([Ax]_i + \\sigma^2\\right)^3}, & \\left\\vert\\frac{[Ax]_i - b_i}{\\sqrt{[Ax]_i + \\sigma^2}}\\right\\vert\\leq \\beta,\\\\\n0, & \\text{otherwise}.\\label{eq:row_scaling}\n\\end{array}\n\\right.\n\\end{align}\nNote that in case of constant weights, robust regression represents extra computational cost in comparison with standard least squares because it leads to a sequence of weighted least squares problems, while standard least squares problems are solved in one step. In our setting, the weights in \\eqref{eq:wls} themselves have to be updated and therefore employing a different loss function does not change the type of the problem we need to solve. \n\nWithout preconditioning, the convergence of projected PCG can be rather slow, and\nit is therefore important to consider preconditioning.  The idea of many preconditioners, such as constraint \\cite{Keller2000Constraint,Dollar2007Using}, constraint-type \\cite{Dollar2007Constraint} or Hermitian and skew-Hermitian \\cite{Benzi2006Preconditioned} preconditioners is based on the fact that in many cases it is possible to efficiently solve the \nlinear system in \\eqref{eq:lin_system} if the diagonal matrix $D^{(k)}$ is the identity matrix; that is, if the linear system\ninvolves the matrix\n\\begin{equation}\nA^TA + \\lambda L^TL \\label{eq:without_D}.\n\\end{equation}\nFor example, in the case of image deblurring, it is well known that linear systems involving the matrix\n\\eqref{eq:without_D} can be solved efficiently using fast trigonometric or \nfast Fourier transforms (FFT).\n\nAlthough the constraint-type, and Hermitian and skew-Hermitian preconditioners seem to perform well for problems with a random \nmatrix $D^{(k)}$ (i.e., a random row scaling), see \\cite{Benzi2006Preconditioned}, they performed unsatisfactorily for problems of the form \\eqref{eq:lin_system}, \\eqref{eq:row_scaling}.\n\nA preconditioner based on a similar idea of fast computations with matrices of type \\eqref{eq:without_D} for imaging problems was proposed in \\cite{Fessler1999Conjugate}. In this case, the row scaling is approximated by a column scaling; that is, \nwe find $\\hat{D}^{(k)}$ such that \n\\begin{equation}\nA^TD^{(k)}A \\approx \\hat{D}^{(k)}(A^TA)\\hat{D}^{(k)},\\label{eq:pullout_prec}\n\\end{equation}\nwhere\n\\begin{equation}\n\\hat{D}_{ii}^{(k)} \\equiv \\sqrt{\\frac{e_i^T(A^TD^{(k)}A)e_i}{e_i^T(A^TA)e_i}}.\\label{eq:out_diag_def}\n\\end{equation}\nNote that for $\\hat{D}^{(k)}$ defined in \\eqref{eq:out_diag_def}, the diagonals of the matrices on the two sides of approximation \\eqref{eq:pullout_prec} are exactly equal. \n\nSince for large-scale problems, matrix $A$ is typically not formed explicitly, exact evaluation of the entries of $\\hat{D}^{(k)}$ might become too expensive. To get around this restriction, note that\n\\begin{equation}\ne_i^T(A^TD^{(k)}A)e_i = ((A^T)\\,.^{2}\\,\\text{diag}(D^{(k)}))_i \\quad \\text{and} \\quad  e_i^T(A^TA)e_i =((A^T)\\,.^2\\,\\mathbf{1})_i,\n\\label{eq:AT_squared}\n\\end{equation}\nwhere $\\mathbf{1}$ a vector of all ones, and \nwe use MATLAB notation $.^2$ to mean component-wise squaring.\nIn some cases it may be relatively easy to compute both the entries of $(A^T).^2$ and the vector\n$(A^T).^2\\mathbf{1}$; this is the case for image deblurring, and is\ndiscussed in more detail in Section~\\ref{sec:num_exp}.\n\n\nUsing \\eqref{eq:pullout_prec}, we define the preconditioner for the linear system \\eqref{eq:lin_system} as\n\\begin{equation}\nM \\equiv\\hat{D}^{(k)}  \\left( A^TA + \\hat{\\lambda} L^TL \\right) \\hat{D}^{(k)},\\label{eq:prec}\n\\end{equation}\nwith \n\\[\\hat{\\lambda} \\equiv \\lambda\/\\text{mean}\\left(\\text{diag}(\\hat{D}^{(k)})\\right)^2 .\\]\nMore details on the computational costs involved in constructing and applying the preconditioner\nin the case of image deblurring are provided in Section~\\ref{sec:num_exp}.\n\n\n\\section{Numerical tests}\\label{sec:num_exp}\n\nThe Poisson--Gaussian model arises naturally in image applications, so in this section\nwe present numerical examples from image deblurring.  Specifically, we consider the \ninverse problem \\eqref{eq:inverse_problem} with data model \\eqref{eq:noise}, where\nvector $b$ is an observed image that is corrupted by blur and noise, matrix $A$ models\nthe blurring operation, vector $x_\\text{true}$ is the true image, and $\\eta$ is noise.\nAlthough an image is naturally represented as an array of pixel values, when we \nrefer to `vector' representations, we assume the pixel values have been reordered\nas vectors.  For example, if we have a $p \\times p$ image of pixel values, these can\nbe stored in a vector of length $n = p^2$ by, for example, lexicographical ordering\nof the pixel values.\n\nIn many practical image deblurring applications, the blurring is spatially invariant,\nand $A$ is structured matrix defined by a {\\em point spread function} (PSF).\nIn this situation, image deblurring can also be referred to as image decovolution,\nbecause the operation $Ax_\\text{true}$ is the convolution of $x_\\text{true}$ and the PSF.\nAlthough the PSF may be given as an actual function, the more common situation is\nto compute estimates of it by imaging point source objects.  Thus, the PSF can be \nrepresented as an image; we typically display the PSF as a mesh plot, which makes\nit easier to visualize how a point in an image is spread to its neighbors because of\nthe blurring operation.\nThe precise structure of the matrix $A$ depends on the imposed boundary condition;\nsee \\cite{Hansen2006Deblurring} for details.  In this section we impose periodic boundary\nconditions, so that $A$ and $L$ are both diagonalizable by FFTs.\n\nSo far we have only described what we refer to as the {\\em single-frame} situation, where\n$b$ is a single observed image.  It is often the case, especially in astronomical imaging,\nto have multiple observed images of the same object, but with each having\na different blurring matrix associated with it.  We refer to this as the\nmulti-frame image deblurring problem.  Here, $b$ represents all observed images, stacked\none on top of each other, and similarly $A$ is formed by stacking the various blurring matrices.\n\nBefore describing the test problems used in this section, we first summarize the computational\ncosts.\nFrom the discussion around equation \\eqref{eq:AT_squared}, to construct the preconditioner we need to \nbe able to efficiently square all entries of the matrix $A^T$, or equivalently those of $A$; this can\neasily be approximated by squaring the point-spread function component-wise before forming the operator, i.e.,\n\\begin{equation}\n(A_\\text{PSF}).^2 \\approx A_{\\text{PSF}.^2}\\, .\\label{eq:PSF_squaring}\n\\end{equation}\nUsing this approximation, in each Newton step we only need to perform one multiplication by a matrix,\none component-wise multiplication, and one component-wise square-root to obtain the entries of the diagonal matrix \\eqref{eq:out_diag_def}. \nWith the assumption that $A$ and $L$ are both diagonalizable by FFTs, \nefficient multiplication by the Hessian \\eqref{eq:lin_system} involves two \ntwo-dimensional forward and inverse FFTs, which we refer to as \\texttt{fft2} and \\texttt{ifft2}, respectively. \nSolving systems with matrix \\eqref{eq:prec} involves only  one \\texttt{fft2} and one \\texttt{ifft2}. \nIn addition to the \\texttt{fft2} requirements, multiplication by the Hessian \\eqref{eq:lin_system} involves 4 pixel-wise multiplications and 1 addition. Solving systems with the preconditioner \\eqref{eq:prec} involves 3 pixel-wise multiplications (component-wise reciprocals are assumed to be computed only once at the beginning). The total counts for each operation are shown in Table \\ref{tab:op_counts}.\n\\begin{table}\n\\caption{Operation counts for single-frame case.}\\label{tab:op_counts}\n\\centering\n{\\small\n\\begin{tabular}{llcccc}\n\\toprule\n& operation& \\texttt{fft2} & \\texttt{ifft2} & mults & adds \\\\ \n\\midrule\nHessian \\eqref{eq:lin_system} &multiply & 2 & 2 & 4 & 1 \\\\ \npreconditioner \\eqref{eq:prec} &solve & 1 & 1 & 3 & 0 \\\\ \n\\bottomrule\n\\end{tabular}} \n\\end{table}\n\n\n\n\n\nThe robustness and the efficiency of the proposed method is demonstrated on two test problems adopted from \\cite{Nagy2004Iterative}:\n\\paragraph{Satellite}\nAn atmospheric seeing problem with spatially invariant atmospheric blur (moderate seeing conditions with the Fried parameter 30). We also consider a multi-frame case, where the same object is blurred by three different PSFs. \nThese PSFs are generated by transposing and flipping the first PSF. \nThe setting is shown in Figures \\ref{fig:satellite} and \\ref{fig:PSF_mesh_1}.\n\\paragraph{Carbon ash}\nAn image deblurring problem with spatially invariant non-separable Gaussian blur, where the PSF has the\nfunctional definition\n\\begin{equation}\n\\label{eq:GaussianPSF}\n  \\mbox{PSF}(s,t) = \\frac{1}{2\\pi \\sqrt{\\gamma}} \\exp\\left\\{\n  -\\frac{1}{2} \\left[ \\begin{array}{cc} s & t \\end{array} \\right] C^{-1}\n   \\left[ \\begin{array}{c} s \\\\ t \\end{array} \\right] \\right\\}\\, ,\n\\end{equation}\nwhere \n$$\n  C = \\left[ \\begin{array}{cc} \\gamma_1^2 & \\tau^2 \\\\[3pt] \\tau^2 & \\gamma_2^2 \\end{array} \\right]\\,, \\quad\n  \\mbox{and} \\quad \\gamma_1^2 \\gamma_2^2 - \\tau^4 > 0\\,.\n$$\nThe shape of the Gaussian PSF depends on the parameters $\\gamma_1, \\gamma_2$ and $\\tau$;\nwe use\n$\\gamma_1 = 4$, $\\gamma_2 = 2$; $\\tau = 2$. We also consider a multi-frame case, where the same object is blurred by three different PSFs. The other two PSFs are Gaussian blurs with parameters $\\gamma_1 = 4$, $\\gamma_2 = 2$, $\\tau = 0$, and $\\gamma_1 = 4$, $\\gamma_2 = 2$, $\\tau = 0$. The setting is shown in Figures \\ref{fig:carbon_ash} and \\ref{fig:PSF_mesh_2}.\n\nAs previously mentioned, in the multi-frame case, the vector $b$ in (\\ref{eq:inverse_problem}) is concatenation of the vectorized blurred noisy images, the matrix $A$ is concatenation of the blurring operators, i.e., $A\\in\\mathbb{R}^{3n\\times n}$. For the test problems all true images are $256 \\times 256$ arrays of pixels (with intensities scaled to $[0,255]$), and thus\n$n = 65536$.\n\nComputation was performed in MATLAB R2015b. Noise is generated artificially using MATLAB functions \\texttt{poissrnd} and \\texttt{randn}. Unless specified otherwise, the standard deviation $\\sigma$ is set to $5$. We use the discretized Laplacian, see \\cite[p. 95]{Hansen2006Deblurring}, as the regularization matrix $L$. The Projected Newton method (Algorithm \\ref{alg:projGNCG}) is terminated when the relative size of the projected gradient\n\\begin{equation}\n\\mathcal{P}(\\text{grad}_{J_{\\lambda}}(x^{(k)})), \\quad \\text{where} \\quad \\mathcal{P}(v) \\equiv v.*(1 - \\text{Active}) + \\text{Active}.*(v < 0),\n\\label{eq:proj_grad_def}\n\\end{equation}\nreaches the tolerance $10^{-4}$ or after 40 iterations. We use MATLAB notation $.^*$ to mean component-wise multiplication. Projected PCG (Algorithm \\ref{alg:projPCG}) is terminated when the relative size of the projected residual (denoted in Algorithm \\ref{alg:projPCG} by $r_i$) reaches $10^{-1}$, or the number of iterations reaches $100$. \nWe use the preconditioner given in \\eqref{eq:prec} as the default preconditioner. Given a search direction $s_k$, we apply a projected backtracking linesearch, with the initial step length equal to 1, which we terminate when \n\\[J_\\lambda(x^{(k+1)}) < J_\\lambda(x^{(k)}).\\]  \n\\begin{figure}[!th]\n\\centering\n\\includegraphics[width=.2\\textwidth]{SatelliteSingle_true}\n\\hspace*{.5cm}\n\\includegraphics[width=.2\\textwidth]{SatelliteMulti_noisy_1}\n\\includegraphics[width=.2\\textwidth]{SatelliteMulti_noisy_2}\n\\includegraphics[width=.2\\textwidth]{SatelliteMulti_noisy_3}\n\\caption{Test problem Satellite: true image (left) together with three blurred noisy images (right). \n}\\label{fig:satellite}\n\\end{figure}\n\\begin{figure}[!th]\n\\centering\n\\includegraphics[width=.2\\textwidth]{CarbonAshSingle_true}\n\\hspace*{.5cm}\n\\includegraphics[width=.2\\textwidth]{CarbonAshMulti_noisy_1}\n\\includegraphics[width=.2\\textwidth]{CarbonAshMulti_noisy_2}\n\\includegraphics[width=.2\\textwidth]{CarbonAshMulti_noisy_3}\n\\caption{Test problem Carbon ash: true image (left) together with three blurred noisy images (right). \n}\\label{fig:carbon_ash}\n\\end{figure}\n\n\\begin{figure}[!th]\n\\centering\n\\begin{subfigure}[b]{.45\\textwidth}\n\\includegraphics[width=.95\\textwidth]{SatelliteSingle_PSFmesh_1}\n\\caption{Satellite}\\label{fig:PSF_mesh_1}\n\\end{subfigure}\n\\begin{subfigure}[b]{.45\\textwidth}\n\\includegraphics[width=.95\\textwidth]{CarbonAshSingle_PSFmesh_1}\n\\caption{Carbon ash}\\label{fig:PSF_mesh_2}\n\\end{subfigure}\n\\caption{Point-spread functions for the first frame of each test problem.\n}\\label{fig:setting}\n\\end{figure}\n\n\n\n\\subsection{Robustness with respect to various types of outliers}\n\nIn this section, we consider several types of outliers, whose choice was motivated by \\cite{Calef2013Iteratively}, and demonstrate the robustness of the proposed method. Note that the difference between \\cite{Calef2013Iteratively} and the proposed approach lies, among others, in the fact that while in \\cite{Calef2013Iteratively}, the approximation of the solution is computed in order to update the outer (robust) weights associated with the components of residual. Here, the weights are represented by the loss function $\\rho$ and are updated implicitly in each Newton step and therefore our approach does not involve any outer iteration.\n\n\\subsubsection*{Random corruptions}\nFirst we consider the most simple case of the outliers -- a given percentage of pixels is corrupted at random. These corruptions are generated artificially by adding a value randomly chosen between $0$ and $\\max(Ax_\\text{true})$ to the given percentage of pixels. The location of these pixels is also chosen randomly. Figures \\ref{fig:sliding_curves_1}, \\ref{fig:sliding_curves_2}, \\ref{fig:sliding_curves_3}, and \\ref{fig:sliding_curves_4}  show semiconvergence curves\\footnote{For ill-posed problems,\nthe relative error of an iterative method generally does not decrease monotonically. Instead, unless the problem is highly over-regularized, the relative errors decrease in the early iterations, but at later iterations the noise and\nother errors tend to corrupt the approximations.  This behavior, where the relative errors decrease to a certain\nlevel and then increase at later iterations, is referred to as {\\em semiconvergence}; for\nmore information, we refer readers to \\cite{Engl2000Regularization,Hansen2010Discrete,Mueller2012Linear,Vogel2002Computational}.}, representing the dependence of the error on the regularization parameter $\\lambda$, when we increase the percentage of corrupted pixels.  \n\\begin{figure}[!ht]\n\\centering\n\\begin{subfigure}[b]{\\textwidth}\n\\includegraphics[width=.31\\textwidth]{semiconv_curve_SatelliteSingle_0}\n\\hspace*{.2cm}\n\\includegraphics[width=.31\\textwidth]{semiconv_curve_SatelliteSingle_2}\n\\hspace*{.2cm}\n\\includegraphics[width=.31\\textwidth]{semiconv_curve_SatelliteSingle_10}\n\\caption{Satellite single-frame}\\label{fig:sliding_curves_1}\n\\end{subfigure}\n\n\\vspace*{.3cm}\n\n\\begin{subfigure}[b]{\\textwidth}\n\\includegraphics[width=.31\\textwidth]{semiconv_curve_SatelliteMulti_0}\n\\hspace*{.2cm}\n\\includegraphics[width=.31\\textwidth]{semiconv_curve_SatelliteMulti_2}\n\\hspace*{.2cm}\n\\includegraphics[width=.31\\textwidth]{semiconv_curve_SatelliteMulti_10}\n\\caption{Satellite multi-frame}\\label{fig:sliding_curves_2}\n\\end{subfigure}\n\n\\vspace*{.3cm}\n\n\\begin{subfigure}[b]{\\textwidth}\n\\includegraphics[width=.31\\textwidth]{semiconv_curve_CarbonAshSingle_0}\n\\hspace*{.2cm}\n\\includegraphics[width=.31\\textwidth]{semiconv_curve_CarbonAshSingle_2}\n\\hspace*{.2cm}\n\\includegraphics[width=.31\\textwidth]{semiconv_curve_CarbonAshSingle_10}\n\\caption{Carbon ash single-frame}\\label{fig:sliding_curves_3}\n\\end{subfigure}\n\n\\vspace*{.3cm}\n\n\\begin{subfigure}[b]{\\textwidth}\n\\includegraphics[width=.31\\textwidth]{semiconv_curve_CarbonAshMulti_0}\n\\hspace*{.2cm}\n\\includegraphics[width=.31\\textwidth]{semiconv_curve_CarbonAshMulti_2}\n\\hspace*{.2cm}\n\\includegraphics[width=.31\\textwidth]{semiconv_curve_CarbonAshMulti_10}\n\\caption{Carbon ash multi-frame}\\label{fig:sliding_curves_4}\n\\end{subfigure}\n\\caption{Semiconvergence curves -- dependence of the relative error of the reconstruction on the size of the regularization parameter $\\lambda$ for various percentages of outliers: Talwar \\eqref{eq:robust_wls} - \\eqref{eq:beta_opt} (solid line) and the standard data fidelity function \\eqref{eq:wls} (dashed line). \n}\n\\end{figure}\nIt is no surprise that when outliers occur, more regularization is needed in order to obtain a reasonable approximation of the true image $x_\\text{true}$. This is however not the case if we use loss function Talwar, for which the semiconvergence curve remains the same even with increasing percentage of outliers, and therefore no adjustment of the regularization parameter is needed. In Figures \\ref{fig:reconstructions_satellite} and \\ref{fig:reconstructions_carbonash}, we show the reconstructions corresponding to $10 \\%$ outliers. The regularization paramter is chosen as a close-to-the-optimal regularization parameter for the same problem with no outliers. Note that Figures \\ref{fig:reconstructions_satellite} and \\ref{fig:reconstructions_carbonash} show only one frame for illustration. In the multi-frame case, the corruptions look similar for all frames, except that the random locations of the outliers is different. For random outliers like this, robust regression is clearly superior to standard weighted least squares. The influence of the outliers in the multi-frame case is less severe, due to intrinsic regularization of the overdetermined system (\\ref{eq:inverse_problem}). A more comprehensive comparison of the standard and robust approach is shown in Table \\ref{tab:percent}, giving the percentage of cases in which the robust approach provides better reconstruction. The robust approach provides better reconstruction in all cases except for the test problem Satellite with no outliers, where the standard approach gave sometimes slightly better reconstructions. However, even in these cases we observed that the difference between the errors of the reconstructions is rather negligible, about $3\\%$.\n\\begin{figure}[!th]\n\\centering\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{data_SatelliteSingleRand10} \n\\caption{data\\newline}\n\\end{subfigure}\n\\hspace*{.1cm}\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{reconstruction_standard_SatelliteSingleRand10} \n\\caption{single-frame\\\\ standard}\n\\end{subfigure}\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{reconstruction_robust_SatelliteSingleRand10} \n\\caption{single-frame\\\\ robust}\n\\end{subfigure}\n\\hspace*{.1cm}\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{reconstruction_standard_SatelliteMultiRand10} \n\\caption{multi-frame,\\\\ standard}\n\\end{subfigure}\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{reconstruction_robust_SatelliteMultiRand10} \n\\caption{multi-frame \\\\robust}\n\\end{subfigure}\n\\caption{Random corruptions: (a) blurred noisy image with $10 \\%$ corrupted pixels (only first frame is shown); (b) - (d) reconstructions corresponding to $\\lambda = 10^{-4}$. \n}\\label{fig:reconstructions_satellite}\n\\end{figure}\n\\begin{figure}[!th]\n\\centering\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{data_CarbonAshSingleRand10} \n\\caption{data\\newline}\n\\end{subfigure}\n\\hspace*{.1cm}\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{reconstruction_standard_CarbonAshSingleRand10} \n\\caption{single-frame\\\\ standard}\n\\end{subfigure}\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{reconstruction_robust_CarbonAshSingleRand10} \n\\caption{single-frame\\\\ robust}\n\\end{subfigure}\n\\hspace*{.1cm}\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{reconstruction_standard_CarbonAshMultiRand10} \n\\caption{multi-frame,\\\\ standard}\n\\end{subfigure}\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{reconstruction_robust_CarbonAshMultiRand10} \n\\caption{multi-frame \\\\robust}\n\\end{subfigure}\n\\caption{Random corruptions: (a) blurred noisy image with $10 \\%$ corrupted pixels (only first frame is shown); (b) - (d) reconstructions corresponding to $\\lambda = 10^{-3}$. \n}\\label{fig:reconstructions_carbonash}\n\\end{figure}\n\\begin{table}[!th]\n\\centering\n\\caption{Comparison of the quality of reconstruction for the standard vs. the robust approach. For each test problem and each percentage of outliers, the results are averaged over 100 independent positions and sizes of random corruptions. Regularization parameters are chosen identically as in Figures \\ref{fig:reconstructions_satellite} and \\ref{fig:reconstructions_carbonash}. Reconstructions are considered to be of the same quality if the difference between the corresponding relative errors is smaller than 1\\%. \n}\\label{tab:percent}\n{\\small \\begin{tabular}{lcccc} \\toprule \n\\multicolumn{5}{c}{better reconstruction: robust\/same\/standard} \\\\ \nproblem\/\\% outliers & \\multicolumn{1}{c}{0\\%} & \\multicolumn{1}{c}{1\\%} & \\multicolumn{1}{c}{2\\%} & \\multicolumn{1}{c}{5\\%} \\\\ \\midrule\nSatellite single-frame& 0\/93\/7& 100\/0\/0& 100\/0\/0& 100\/0\/0\\\\ \nSatellite multi-frame& 0\/94\/6& 100\/0\/0& 100\/0\/0& 100\/0\/0\\\\ \nCarbon ash single-frame& 0\/100\/0& 100\/0\/0& 100\/0\/0& 100\/0\/0\\\\ \nCarbon ash  multi-frame& 0\/100\/0& 100\/0\/0& 100\/0\/0& 100\/0\/0\\\\ \n\\bottomrule\\end{tabular}}\n\\end{table}\n\n\n\\subsubsection*{Added object with different blurring}\nWe also consider a situation when a small object appears in the scene, but is blurred by a different PSF than the main object (satellite). The aim is to recover the main object, while suppressing the influence of the added one. In our case, the added object is a small satellite in the left upper corner that is blurred by a small motion blur. In the multi-frame case, the small satellite is added to the first frame only. The difference between the reconstructions using standard and robust approach is shown in Figure \\ref{fig:alien}. For the single frame problem, reconstructions obtained using the standard loss function is fully dominated by the small added object. For the multi-frame situation, the influence of the outlier is somewhat compensated by the two frames without outliers. In both cases, however, robust regression provides better reconstruction, comparable to the reconstruction from the data without outliers.\n\\begin{figure}[!th]\n\\centering\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{data_SatelliteSingleAlien} \n\\caption{data\\newline}\n\\end{subfigure}\n\\hspace*{.1cm}\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{reconstruction_standard_SatelliteSingleAlien} \n\\caption{single-frame\\\\ standard}\n\\end{subfigure}\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{reconstruction_robust_SatelliteSingleAlien} \n\\caption{single-frame\\\\ robust}\n\\end{subfigure}\n\\hspace*{.1cm}\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{reconstruction_standard_SatelliteMultiAlien} \n\\caption{multi-frame,\\\\ standard}\n\\end{subfigure}\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{reconstruction_robust_SatelliteMultiAlien} \n\\caption{multi-frame \\\\robust}\n\\end{subfigure}\n\\caption{Added object: (a) blurred noisy image with a small object added to the first frame (only first frame is shown); (b) - (d) reconstructions corresponding to $\\lambda = 10^{-4}$. \n}\\label{fig:alien}\n\\end{figure}\n\n\n\\subsubsection*{Outliers introduced by boundary conditions}\nDefining the boundary conditions plays an important role in solving image deblurring problems. As is well known, see e.g. \\cite{Hansen2006Deblurring}, unless some strong a priori information about the scene outside the borders is available, any choice of the boundary conditions may lead to artefacts around edges in the reconstruction.  Similarly as in \\cite{Calef2013Iteratively}, we may expect that the robust objective functional (\\ref{eq:functional}) can to some extent compensate for these edge artifacts, i.e., the outliers are represented by the `incorrectly' imposed boundary conditions. In our model we assume periodic boundary conditions, which is computationally very appealing, since it allows evaluating the multiplication by $A$ very efficiently using the fast Fourier transform. However, if any of the objects in the scene is close to the boundary, these boundary conditions will most probably cause artifacts. In order to demonstrate the ability of the proposed scheme to eliminate influence of this type of outlier, we shifted the satellite to the right edge of the image. Other settings remain unchanged. Reconstructions using standard and robust approach are shown in Figure \\ref{fig:bc}. We see that, although not spectacular, robust regression can reduce the artifacts caused by incorrectly imposed boundary conditions and therefore provide better reconstruction of the true image. Quantitative results for this and all the previous types of outliers are shown in Tables~\\ref{tab:2a} and \\ref{tab:2b}.\n\\begin{figure}[!th]\n\\centering\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{data_SatelliteSingleBC} \n\\caption{data\\newline}\n\\end{subfigure}\n\\hspace*{.1cm}\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{reconstruction_standard_SatelliteSingleBC} \n\\caption{single-frame\\\\ standard}\n\\end{subfigure}\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{reconstruction_robust_SatelliteMultiBC} \n\\caption{single-frame\\\\ robust}\n\\end{subfigure}\n\\hspace*{.1cm}\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{reconstruction_standard_SatelliteMultiBC} \n\\caption{multi-frame,\\\\ standard}\n\\end{subfigure}\n\\begin{subfigure}[b]{.18\\textwidth}\\captionsetup{justification=centering}\n\\includegraphics[width=.98\\textwidth]{reconstruction_robust_SatelliteSingleBC} \n\\caption{multi-frame \\\\robust}\n\\end{subfigure}\n\\caption{Incorrectly imposed periodic boundary conditions: (a) blurred noisy image close to the edge (only first frame is shown); (b) - (d) reconstructions corresponding to $\\lambda = 10^{-4}$. \n}\\label{fig:bc}\n\\end{figure}\n\n\\begin{table}[!th]\n\\caption{Comparison of the standard and robust approach in terms of relative error of the reconstruction. Each row contains results for the standard and robust approach. Abbreviation `\\# it' stands for the number of Newton steps performed before the relative size of the projected gradient reached the tolerance $10^{-4}$. Corresponding reconstructions are shown in Figures~\\ref{fig:reconstructions_satellite}--\\ref{fig:bc}. \n}\n\\centering\n{\\small\n\\begin{subtable}[h]{\\textwidth}\n\\subcaption{single-frame}\n\\centering\n\\begin{tabular}{lcccc}\\toprule \n & \\multicolumn{2}{c}{standard} & \\multicolumn{2}{c}{robust} \\\\ \n\\multicolumn{1}{c}{problem} & \\multicolumn{1}{c}{\\# it} & \\multicolumn{1}{c}{rel. error} &  \\multicolumn{1}{c}{\\# it} & \\multicolumn{1}{c}{rel. error} \\\\ \\midrule\nSatellite& 15& 3.40$\\times 10^{-1}$& 16& 3.42$\\times 10^{-1}$\\\\ \nSatellite random corr. 10\\%& 14& 6.78$\\times 10^{-1}$& 14& 3.57$\\times 10^{-1}$\\\\ \nCarbon ash& 10& 3.10$\\times 10^{-1}$& 11& 3.08$\\times 10^{-1}$\\\\ \nCarbon ash random corr. 10\\% & 11& 3.80$\\times 10^{-1}$& 14& 3.10$\\times 10^{-1}$\\\\ \nSatellite added object& 15& 4.72$\\times 10^{-1}$& 15& 3.43$\\times 10^{-1}$\\\\ \nSatellite boundary conditions& 15& 5.48$\\times 10^{-1}$& 25& 4.51$\\times 10^{-1}$\\\\ \n\\bottomrule\\end{tabular}\\label{tab:2a}\n\\end{subtable}\n\n\\medskip\n\n\\begin{subtable}[h]{1\\textwidth}\n\\subcaption{multi-frame}\n\\centering\n\\begin{tabular}{lcccc} \\toprule\n & \\multicolumn{2}{c}{standard} & \\multicolumn{2}{c}{robust} \\\\ \n\\multicolumn{1}{c}{problem} & \\multicolumn{1}{c}{\\# it} & \\multicolumn{1}{c}{rel. error} & \\multicolumn{1}{c}{\\# it} & \\multicolumn{1}{c}{rel. error} \\\\ \\midrule\nSatellite& 12& 2.89$\\times 10^{-1}$& 11& 2.89$\\times 10^{-1}$\\\\ \nSatellite random corr. 10\\%& 11& 6.45$\\times 10^{-1}$& 13& 3.00$\\times 10^{-1}$\\\\ \nCarbon ash& 12& 3.07$\\times 10^{-1}$& 11& 3.05$\\times 10^{-1}$\\\\ \nCarbon ash random corr. 10\\%& 9& 3.70$\\times 10^{-1}$& 19& 3.06$\\times 10^{-1}$\\\\ \nSatellite added object& 13& 3.33$\\times 10^{-1}$& 11& 2.90$\\times 10^{-1}$\\\\ \nSatellite boundary conditions& 14& 5.26$\\times 10^{-1}$& 14& 4.27$\\times 10^{-1}$\\\\ \n\\bottomrule\\end{tabular}\\label{tab:2b}\n\\end{subtable}}\n\\end{table} \n\n\\subsection{Generalized cross-validation}\\label{sec:gcv_tests}\nFor the remainder of this section we will only assume the robust approach, i.e., functional \\eqref{eq:functional} with the loss function Talwar. In Section~\\ref{sec:gcv} we described a regularization parameter selection rule based on leave-one-out cross validation. Since GCV belongs to standard methods, we focus here mainly on the influence of the outliers on its reliability. To obtain various noise levels, we scale the original true scene (with maximum intensity = 255) by 10 and by 100, which results in a decrease of the relative size of Poisson noise. The standard deviation $\\sigma$ for the additive Gaussian noise is scaled accordingly by $\\sqrt{10}$ and $10$. We compute the resulting signal-to-noise ratio as the reciprocal of the coefficient of variation, i.e.,\n\\[\n\\text{SNR} = \\frac{\\|Ax\\|}{\\sqrt{\\sum_{i=1}^n([Ax]_i + \\sigma^2)}}.\n\\]\n\nFor our computations, we use CG to solve \\eqref{eq:gcv_lin_syst}, which we terminate if the relative size of the residual reaches $10^{-4}$ or if the number of iterations reaches 150. \nTo minimize the GCV functional, we use the MATLAB built-in function \\texttt{fminbnd}, for which we set the lower bound to $0$ and the upper bound to $10^{-1}$, $10^{-2}$, $10^{-4}$, depending on the maximum intensity of the image. The tolerance \\texttt{TolX} was set to $10^{-8}$. \n\nFor test problem Satellite, we show the semiconvergence curves including the minimum error and the error obtained using GCV in Figure~\\ref{fig:GCV_satellite}. Quantitative results (averaged over 10 realizations of outliers) for both test problems are shown in Table~\\ref{tab:gcv}. We observe that the proposed rule is rather stable with respect to the increasing number of outliers and generally better for the Carbon ash than for the Satellite. As expected, the method provides better approximation of the optimal regularization parameter for smaller noise levels  (larger $Ax_\\text{true}$), where the functional \\eqref{eq:wls} approximates better the maximum likelihood functional for the mixed Poisson--Gaussian model. Occasionally, GCV provides slightly worse reconstruction for the highest percentage (10\\%) of outliers.\n\\begin{figure}[!ht]\n    \\centering\n    \\begin{subfigure}[b]{\\textwidth}\n    \\includegraphics[width=.28\\textwidth]{GCV_SatelliteSingle_Laplacian_1_wls_talwar} \n    \\quad\n\t\\includegraphics[width=.28\\textwidth]{GCV_SatelliteSingle_Laplacian_5_wls_talwar} \n    \\quad\n\t\\includegraphics[width=.28\\textwidth]{GCV_SatelliteSingle_Laplacian_10_wls_talwar} \n       \t\\caption{Satellite single-frame, max. intensity 255 (SNR = 5).}\\end{subfigure}\n    \n\\vspace*{.3cm}\n\n\n    \\begin{subfigure}[b]{\\textwidth}\n    \\includegraphics[width=.28\\textwidth]{GCV_SatelliteSingle2550_Laplacian_1_wls_talwar} \n    \\quad\n\t\\includegraphics[width=.28\\textwidth]{GCV_SatelliteSingle2550_Laplacian_5_wls_talwar}\n    \\quad\n\t\\includegraphics[width=.28\\textwidth]{GCV_SatelliteSingle2550_Laplacian_10_wls_talwar} \n       \t\\caption{Satellite single-frame, max. intensity 2550 (SNR = 17).}\\end{subfigure}\n    \n\\vspace*{.3cm}\n\n\n    \\begin{subfigure}[b]{\\textwidth}\n    \\includegraphics[width=.28\\textwidth]{GCV_SatelliteSingle25500_Laplacian_1_wls_talwar} \n    \\quad\n\t\\includegraphics[width=.28\\textwidth]{GCV_SatelliteSingle25500_Laplacian_5_wls_talwar}\n    \\quad\n\t\\includegraphics[width=.28\\textwidth]{GCV_SatelliteSingle25500_Laplacian_10_wls_talwar} \n       \t\\caption{Satellite single-frame, max. intensity 25500 (SNR = 52).}\\end{subfigure}\n    \n\\vspace*{.3cm}\n\n   \t\n    \\begin{subfigure}[b]{\\textwidth}\n    \\includegraphics[width=.28\\textwidth]{GCV_SatelliteMulti_Laplacian_1_wls_talwar} \n    \\quad\n\t\\includegraphics[width=.28\\textwidth]{GCV_SatelliteMulti_Laplacian_5_wls_talwar}\n    \\quad\n\t\\includegraphics[width=.28\\textwidth]{GCV_SatelliteMulti_Laplacian_10_wls_talwar} \n       \t\\caption{Satellite multi-frame, max. intensity 255 (SNR = 5).}\\end{subfigure}\n    \n    \t\\caption{GCV for data with outliers. \n} \\label{fig:GCV_satellite}\n\\end{figure}\n\n\n\\begin{landscape}\n\\begin{table}\n\\centering\n\\caption{Relative errors of the reconstruction: optimal $\\lambda$ vs. $\\lambda$ obtained by minimizing the GCV functional \\eqref{eq:gcv_fun}. \n}\\label{tab:gcv}\n\\begin{subtable}[h]{1.5\\textwidth}\n\\subcaption{Satellite}\n\\centering\n{\\scriptsize\\begin{tabular}{llcllclcclc} \\toprule \n\\multicolumn{2}{r}{}& \\multicolumn{4}{r}{error min} & \\multicolumn{4}{l}{($\\lambda$ optimal)} \\\\ \n\\multicolumn{2}{r}{} & \\multicolumn{4}{r}{error GCV} & \\multicolumn{4}{l}{($\\lambda$ GCV)} \\\\ \n\\multicolumn{1}{r}{} & \\multicolumn{9}{c}{\\% outliers}\\\\ \nproblem & max. int. & \\multicolumn{2}{c}{1\\%} & & \\multicolumn{2}{c}{5\\%} & & \\multicolumn{2}{c}{10\\%} \\\\ \\midrule \n\\multirow{2}{*}{single-frame} & \\multirow{2}{*}{255}&  $3.39\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 2.1\\times 10^{-4}$) & & $3.43\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 2.5\\times 10^{-4}$) & & $3.49\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 3.2\\times 10^{-4}$)\\\\ \n & &  $3.60\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 2.1\\times 10^{-3}$) & & $3.68\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 3.1\\times 10^{-3}$) & & $3.80\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 5.6\\times 10^{-3}$)\\\\ \\addlinespace[.2cm] \n\\multirow{2}{*}{single-frame} & \\multirow{2}{*}{2550}&  $2.46\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 1.5\\times 10^{-6}$) & & $2.48\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 1.5\\times 10^{-6}$) & & $2.50\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 1.7\\times 10^{-6}$)\\\\ \n & &  $2.48\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 1.5\\times 10^{-6}$) & & $2.57\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 4.5\\times 10^{-6}$) & & $2.69\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 8.4\\times 10^{-6}$)\\\\ \\addlinespace[.2cm] \n\\multirow{2}{*}{single-frame} & \\multirow{2}{*}{25500}&  $1.78\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 1.0\\times 10^{-8}$) & & $1.79\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 1.0\\times 10^{-8}$) & & $1.81\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 1.0\\times 10^{-8}$)\\\\ \n & &  $1.79\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 8.8\\times 10^{-9}$) & & $1.81\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 1.5\\times 10^{-8}$) & & $2.42\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 7.5\\times 10^{-6}$)\\\\ \\addlinespace[.2cm] \n\\multirow{2}{*}{mutli-frame} & \\multirow{2}{*}{255}& $2.89\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 1.8\\times 10^{-4}$) & & $2.92\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 1.8\\times 10^{-4}$) & & $2.97\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 1.8\\times 10^{-4}$)\\\\ \n & & $3.16\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 1.6\\times 10^{-3}$) & & $3.13\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 1.2\\times 10^{-3}$) & & $3.49\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 5.8\\times 10^{-3}$) \\\\ \\addlinespace[.2cm] \n\\bottomrule\\end{tabular}\n}\n\\end{subtable}\n\n\\begin{subtable}[h]{1.5\\textwidth}\n\\subcaption{Carbon ash}\n\\centering\n{\\scriptsize\\begin{tabular}{llcllclcclc} \\toprule \n\\multicolumn{2}{r}{}& \\multicolumn{4}{r}{error min} & \\multicolumn{4}{l}{($\\lambda$ optimal)} \\\\ \n\\multicolumn{2}{r}{} & \\multicolumn{4}{r}{error GCV} & \\multicolumn{4}{l}{($\\lambda$ GCV)} \\\\ \n\\multicolumn{1}{r}{} & \\multicolumn{9}{c}{\\% outliers}\\\\ \nproblem & max. int. & \\multicolumn{2}{c}{1\\%} & & \\multicolumn{2}{c}{5\\%} & & \\multicolumn{2}{c}{10\\%} \\\\ \\midrule \n\\multirow{2}{*}{single-frame} & \\multirow{2}{*}{255}& $3.08\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 1.8\\times 10^{-3}$) & & $3.09\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 1.8\\times 10^{-3}$) & & $3.10\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 1.7\\times 10^{-3}$)\\\\ \n& & $3.12\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 8.9\\times 10^{-4}$) & & $3.11\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 1.5\\times 10^{-3}$) & & $3.11\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 2.2\\times 10^{-3}$) \\\\ \\addlinespace[.2cm] \n\\multirow{2}{*}{single-frame} & \\multirow{2}{*}{2550}& $2.91\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 1.9\\times 10^{-5}$) & & $2.92\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 2.1\\times 10^{-5}$) & & $2.92\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 1.9\\times 10^{-5}$)\\\\ \n& & $2.97\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 9.7\\times 10^{-6}$) & & $2.95\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 1.2\\times 10^{-5}$) & & $2.93\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 2.2\\times 10^{-5}$)\\\\ \\addlinespace[.2cm] \n\\multirow{2}{*}{single-frame} & \\multirow{2}{*}{25500}& $2.77\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 3.0\\times 10^{-7}$) & & $2.78\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 2.7\\times 10^{-7}$) & & $2.79\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 2.6\\times 10^{-7}$)\\\\ \n& & $3.00\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 5.2\\times 10^{-8}$) & & $2.84\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 9.4\\times 10^{-8}$) & & $2.82\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 1.3\\times 10^{-7}$)\\\\ \\addlinespace[.2cm] \n\\multirow{2}{*}{multi-frame} & \\multirow{2}{*}{255}& $3.03\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 1.8\\times 10^{-3}$) & & $3.04\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 1.8\\times 10^{-3}$) & & $3.04\\times 10^{-1}$ & ($\\lambda_\\text{opt} = 1.8\\times 10^{-3}$)\\\\ \n & & $3.14\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 6.8\\times 10^{-4}$) & & $3.07\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 9.8\\times 10^{-4}$) & & $3.05\\times 10^{-1}$ & ($\\lambda_\\text{GCV} = 1.9\\times 10^{-3}$)\\\\ \\addlinespace[.2cm] \n\\bottomrule\\end{tabular}\n}\n\\end{subtable}\n\\end{table} \n\\end{landscape}\n\n\n\n\\subsection{Linear subproblems}\n\nAs mentioned earlier, various types of preconditioners have been developed to speed up convergence of iterative methods applied to systems of type \\eqref{eq:lin_system} or its saddle-point counterpart \n\\[\n\\begin{pmatrix}\n  D^{-1} & A  \\\\\n  A^T & \\lambda L^TL\\\\\n \\end{pmatrix} = \n \\begin{pmatrix}\n  -z \\\\\n  \\lambda L^TLx\\\\\n \\end{pmatrix}.\\label{eq:saddle_point}\n\\]\nThe Hermitian and skew-Hermitian (HSS) preconditioner, as well as constraint precondtioner belong to the best known preconditioners for this type of linear system. Both were incorporated in GMRES and tested on deblurring problems with random diagonal scaling $D$ in \\cite{Benzi2006Preconditioned}. Using random $D$, they indeed accelerate convergence also in our case, as shown in Figure \\ref{fig:prec1}. However, our preconditioner \\eqref{eq:prec} provides a much better speedup. Moreover, for real computations, e.g., when the matrix $D$ is actually generated during the Projected Newton computation, the HSS and constraint preconditioners did not perform well, and even slowed down the convergence, see Figure~\\ref{fig:prec2}. This is fortunately not the case for \nour proposed preconditioner. In this experiment, we did not assume projection on the non-negative half-plane and since in \\eqref{eq:saddle_point}, we need to evaluate $D^{-1}$, if some component $D_{ii}=0$, we replaced it by $2\\sqrt{\\epsilon_\\text{mach}}$, see also \\cite{Mastronardi2008Fast}.\nWe also did not incorporate any outliers for these initial experiments with the preconditioners; \nthese results are intended to show that \nour proposed preconditioning for these problems\noften performs much better than the well-known standard preconditioners.\nIn fact, we see that the behavior of the constraint and HSS preconditioner depends heavily on the actual setting of the problem. In the remainder of this section we will therefore focus on the preconditioner given in \\eqref{eq:prec}.\n\\begin{figure}[!ht]\n    \\centering\n        \\begin{subfigure}[b]{0.85\\textwidth}\n        \t\\includegraphics[width=.47\\textwidth]{preconditioners_unconstrained_CG_random} \n        \t\\hspace*{.1cm}\n        \t\\includegraphics[width=.47\\textwidth]{preconditioners_unconstrained_GMRES_random} \n        \t\\caption{Satellite single-frame, random diagonal}\n    \t\\end{subfigure}\n    \t\\caption{Preconditoner defined in \\eqref{eq:prec}, constraint preconditioner (CP), and Hermitian and skew-Hermitian splitting preconditioner (HSSP) performance for $(A^TDA + \\lambda L^TL)s = -A^Tb$, where $A$ and $b$ are adopted from the test problem Satellite, and $D$ is a diagonal with random entries uniformly distributed in $(0,1)$. \n}\\label{fig:prec1}\n\\end{figure}\n\\begin{figure}[!ht]\n    \\centering\n        \\begin{subfigure}[b]{0.85\\textwidth}\n        \t\\includegraphics[width=.47\\textwidth]{preconditioners_unconstrained_CG_real} \n        \t\\hspace*{.1cm}\n        \t\\includegraphics[width=.47\\textwidth]{preconditioners_unconstrained_GMRES_real} \n        \t\\caption{Satellite single-frame, Newton it = 3}\n    \t\\end{subfigure}\n    \t\\caption{Preconditoner defined in \\eqref{eq:prec}, constraint preconditioner (CP), and Hermitian\/skew-Hermitian preconditioner (HSSP) performance for $(A^TD^{(k)}A + \\lambda L^TL)s = -\\left(A^Tz^{(k)} + \\lambda L^TLx^{(k)}\\right)$. \n}\\label{fig:prec2} \n\\end{figure}\n\nIn Figure \\ref{fig:prec_levels}, we investigate the overall speedup of the convergence by plotting the number of projected PCG steps needed in each Newton iteration to reach the desired tolerance on the relative size of the projected gradient. Even for the most generous tolerance $10^{-1}$, preconditioner \\eqref{eq:prec} significantly reduces the number of projPCG iterations. Note that in this experiment, the linear subproblems solved in each Newton iteration are generally not identical, since the subproblems are not solved exactly and therefore the approximations $x^{(k)}$ are not the same. We set the outer tolerance to $0$ in order to perform always at least $15$ Newton iterations. \n\\begin{figure}[!ht]\n    \\centering\n    \\begin{subfigure}[b]{\\textwidth}\n    \\includegraphics[width=.28\\textwidth]{precond_levels_SatelliteSingle_5_1} \n    \\quad\n\t\\includegraphics[width=.28\\textwidth]{precond_levels_SatelliteSingle_5_2} \n    \\quad\n\t\\includegraphics[width=.28\\textwidth]{precond_levels_SatelliteSingle_5_3} \n       \t\\caption{Satellite single-frame}\\end{subfigure}\n    \n     \n    \\vspace*{.3cm}   \t\n    \n   \\begin{subfigure}[b]{\\textwidth}\n    \\includegraphics[width=.28\\textwidth]{precond_levels_SatelliteMulti_5_1} \n    \\quad\n\t\\includegraphics[width=.28\\textwidth]{precond_levels_SatelliteMulti_5_2}\n    \\quad\n\t\\includegraphics[width=.28\\textwidth]{precond_levels_SatelliteMulti_5_3} \n       \t\\caption{Satellite multi-frame}\\end{subfigure}\n    \n     \n    \\vspace*{.3cm}   \t\n    \n   \\begin{subfigure}[b]{\\textwidth}\n    \\includegraphics[width=.28\\textwidth]{precond_levels_CarbonAshSingle_5_1} \n    \\quad\n\t\\includegraphics[width=.28\\textwidth]{precond_levels_CarbonAshSingle_5_2} \n    \\quad\n\t\\includegraphics[width=.28\\textwidth]{precond_levels_CarbonAshSingle_5_3} \n       \t\\caption{Carbon Ash single-frame}\\end{subfigure}\n    \n    \\vspace*{.3cm}   \t\n    \n    \\begin{subfigure}[b]{\\textwidth}\n    \\includegraphics[width=.28\\textwidth]{precond_levels_CarbonAshMulti_5_1} \n    \\quad\n\t\\includegraphics[width=.28\\textwidth]{precond_levels_CarbonAshMulti_5_2} \n    \\quad\n\t\\includegraphics[width=.28\\textwidth]{precond_levels_CarbonAshMulti_5_3} \n       \t\\caption{Carbon Ash multi-frame}\\end{subfigure}\n    \n    \n    \t\\caption{The effect of preconditioning by preconditioner defined in \\eqref{eq:prec}: number of projPCG iterations performed in each Newton iteration to achieve the desired tolerance. 5 \\% outliers.\n} \\label{fig:prec_levels}\n\\end{figure}\nThe choice of projPCG tolerance is a difficult question, but from the average number of Newton iterations\/projPCG iterations\/fast Fourier transforms shown in Table \\ref{tab:numit}, we observe that raising the tolerance does not considerably increase the number of Newton steps we need to perform here. Therefore larger tolerance, here $10^{-1}$, leads to a smaller total number of projPCG iterations. This is independent of the percentage of outliers. For each setting, the number of projPCG iterations is significantly smaller for the preconditioned version. This is not always the case for the total count of the fast Fourier transforms, since we need to perform 6 \\texttt{fft2}\/\\texttt{ifft2} in each iteration vs. 4 for the unpreconditioned iterations; see Table \\ref{tab:op_counts}. For large scale problems, however, the computational complexity of fast Fourier transform, which is $\\mathcal{O}(n\\log n)$ is comparable to other operations performed in projPCG, such as the inner products, whose complexity is $\\mathcal{O}(n)$, and therefore the number of projPCG iterations seems to be the more important indicator of efficiency of the preconditioner.  Recall here that\n$n$ is the number of pixels in the image, so if we have a $256 \\times 256$ array of pixels, then $n = 65535$.\n\\begin{table}[!ht]\n\\centering\n\\caption{Average number of Newton iterations, projPCG iterations, and (inverse) 2D Fourier transforms for projPCG with and without preconditioning, and two tolerances on the relative size of the projPCG residual. Results are averaged over 10 independent realization of noise and outliers.  \n}\\label{tab:numit}\n\\begin{subtable}[h]{\\textwidth}\n\\subcaption{projPCG tol = $10^{-1}$}\n\\centering\n{\\small\\begin{tabular}{lcccc} \\toprule \n \\multicolumn{5}{c}{average count: Newton\/CG\/\\texttt{fft2}s } \\\\ \n& & \\multicolumn{3}{c}{\\% outliers} \\\\ \n\\cmidrule(r){3-5}\n\\multicolumn{1}{c}{problem} & precond & \\multicolumn{1}{c}{0\\%} & \\multicolumn{1}{c}{2\\%} & \\multicolumn{1}{c}{10\\%} \\\\ \\midrule\n\\multirow{2}{*}{Satellite single-frame}& no& 14\/290\/1383& 14\/274\/1329& 14\/283\/1374\\\\ \n & yes& 15\/161\/1280& 16\/172\/1362& 14\/158\/1252\\\\ \n\\addlinespace[.2cm]\n\\multirow{2}{*}{Satellite multi-frame}& no& 12\/250\/2398& 12\/216\/2104& 13\/241\/2364\\\\ \n& yes& 12\/107\/1545& 11\/107\/1535& 12\/103\/1507\\\\ \n\\addlinespace[.2cm]\n\\multirow{2}{*}{Carbon ash single-frame}& no& 11\/190\/939& 10\/179\/891& 11\/184\/915\\\\ \n& yes& 10\/71\/641& 11\/72\/654& 13\/82\/753\\\\ \n\\addlinespace[.2cm]\n\\multirow{2}{*}{Carbon ash multi-frame}& no& 14\/221\/2200& 14\/219\/2179& 16\/254\/2542\\\\ \n& yes& 16\/88\/1510& 15\/85\/1419& 17\/99\/1654\\\\ \n\\bottomrule\n\\end{tabular}}\n\\end{subtable}\n\n\\vspace*{.5cm}\n\n\\begin{subtable}[h]{1\\textwidth}\n\\subcaption{projPCG tol = $10^{-2}$}\n\\centering\n{\\small\\begin{tabular}{lcccc} \\toprule \n \\multicolumn{5}{c}{average count: Newton\/CG\/\\texttt{fft2}s } \\\\ \t\n& & \\multicolumn{3}{c}{\\% outliers} \\\\ \n\\cmidrule(r){3-5}\n\\multicolumn{1}{c}{problem} & precond & \\multicolumn{1}{c}{0\\%} & \\multicolumn{1}{c}{2\\%} & \\multicolumn{1}{c}{10\\%} \\\\ \\midrule\n\\multirow{2}{*}{Satellite single-frame}& no& 13\/536\/2359& 13\/539\/2373& 14\/641\/2819\\\\ \n& yes& 14\/284\/2001& 15\/302\/2117& 14\/296\/2091\\\\ \n\\addlinespace[.2cm] \n\\multirow{2}{*}{Satellite multi-frame}& no& 11\/457\/4082& 11\/460\/4130& 12\/499\/4511\\\\ \n& yes& 11\/201\/2536& 11\/197\/2499& 12\/213\/2747\\\\ \n\\addlinespace[.2cm]\n\\multirow{2}{*}{Carbon ash single-frame}& no& 11\/393\/1754& 11\/432\/1912& 13\/526\/2315\\\\ \n& yes& 10\/121\/934& 11\/129\/1004& 13\/155\/1201\\\\ \n\\addlinespace[.2cm]\n\\multirow{2}{*}{Carbon ash multi-frame}& no& 13\/426\/3813& 14\/461\/4127& 16\/498\/4467\\\\ \n& yes& 13\/144\/1954& 14\/150\/2038& 16\/173\/2359\\\\ \n\\bottomrule\n\\end{tabular}}\n\\end{subtable}\n\n\\end{table} \n\n\\section{Conclusion}\nWe have presented an efficient approach to compute approximate\nsolutions of a linear inverse problem that is contaminated with mixed Poisson--Gaussian noise, and \nwhen there are outliers in the measured data. \nWe investigated the convexity properties of various robust regression functions and\nfound that the Talwar function was the best option.  We proposed a preconditioner,\nand illustrated that it was more effective than other\nstandard preconditioning approaches on the types of problems studied in this paper.  Moreover, we showed that a variant of the GCV method\ncan perform well in estimating regularization parameters in robust regression.\nA detailed discussion of computational\ncosts, and extensive numerical experiments illustrate the approach proposed in this paper is effective and efficient on\nimage deblurring problems.\n\n\\section{Acknowledgment}\nThe authors would like to thank Lars Ruthotto for pointing them to the Projected PCG algorithm and providing them with a Matlab code. The first author would like to thank Emory University for the hospitality offered in academic year 2014-2015, when part of this work was completed.\n\n\n\\FloatBarrier\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe phenomena of influence propagation through social networks have attracted a great body of research works. A key function of an online social network (OSN), besides sharing, is that it enables users to express their personal opinions about a product or trend of news by means of posts, likes\/dislikes, etc. Such opinions are propagated to other users and might make a significant influence on them, either positive or negative. \n \nReal world is full of imprecision and uncertainty and this fact necessarily impacts on OSN data. In fact, social interactions can not be always precise and certain, also, OSN allows only a limited access for their data which generates more imprecision and uncertainty. Then, if we ignore this imperfection, we may  be confronted to erroneous analysis results. In such a situation, the theory of belief functions \\cite{Dempster67a,Shafer76} have been widely applied. Furthermore, this theory was used for analyzing social networks \\cite{Jendoubi14a,Jendoubi2015,Zhou2015}. \n\n\nInfluence maximization (IM) is the problem of finding a set of $k$ seed nodes that are able to influence the maximum number of nodes in the social network. In the literature, we find many solutions for the IM problem. \\textit{Kempe et al.} \\cite{Kempe03} propose two propagation simulation models which are the \\textit{Linear Threshold Model (LTM)} and the \\textit{Independent Cascade Model (ICM)}. Besides, the credit distribution (CD) model \\cite{Goyal12} is a data based approach that investigates past propagation to detect influencers. However, these solutions does not consider the user's opinion. Zhang et al. \\cite{Zhang2013} propose an opinion based cascading model that considers the user's opinion about the product. However, their work is not based on real word data to estimate user's opinion and influence.  \n\nIn this paper, we propose a new data based model for influence maximization\nin online social networks that searches to detect influencer users\nthat adopt a positive opinion about the product. \nThe proposed model is data based because we use past propagation to estimate the influence, and users messages to estimate the opinion. Besides, it uses the theory of belief functions to estimate the influence to deal with\ndata imprecision and uncertainty. To the best of our knowledge, the proposed model\nis the first evidential data based model that maximizes the influence\non OSN, detects influencer users having a positive opinion about\nthe product and uses the theory of belief functions to process the\ndata imperfection.\n\nThe remainder of this paper is organized as follows: section 2 introduces the proposed model\nfor maximizing the positive opinion influence, section 3\nshows the performance of our model through some relevant\nexperiments. Finally, the paper is concluded in section 4.\n\n\\vspace{-0.05cm}\n\n\\section{Maximizing positive opinion influence}\n\nIn this section, we present our positive opinion influence measure and the proposed influence maximization\nalgorithm.\n\n\n\\subsection{Influence measure}\n\nGiven a social network $G=\\left(V,E\\right)$, a frame of discernment expressing opinion $\\Theta=\\left\\{ Pos,\\, Neg,\\, Obj\\right\\} $, $Pos$ for positive, $Neg$ for negative and $Obj$ for objective, a frame of discernment expressing influence and passivity $\\Omega=\\left\\{ I,P\\right\\} $, $I$ for influencer and $P$ for passive user, a probability distribution $\\Pr^{\\Theta}\\left(u\\right)$ defined on $\\Theta$ that expresses the opinion of the user $u\\in V$ about the product and a basic belief assignment (BBA) function~\\cite{Shafer76}, $m^{\\Omega}\\left(u,v\\right)$, defined on $\\Omega$ that expresses the influence that exerts the user $u$ on the user $v$. The first step of the influence maximization process is to measure the influence of each user in the network. Then we propose an influence measure to estimate the positive influence of each user in the network. \n\nThe mass value $m^{\\Omega}\\left(u,v\\right)\\left(I\\right)$ measures the influence of $u$ on $v$ but without considering the opinion of $u$ about the product. We define the positive opinion influence of $u$ on $v$ as the positive proportion of $m^{\\Omega}\\left(u,v\\right)\\left(I\\right)$ and we measure this proportion as: \n\\begin{equation}\nInf_{Pos}\\left(u,v\\right)=\\textrm{Pr}^{\\Theta}\\left(u\\right)\\left(Pos\\right).m^{\\Omega}\\left(u,v\\right)\\left(I\\right)\n\\end{equation}\n\nNext, we define the amount of influence given to a set of nodes $S\\subseteq V$\nfor influencing a user $v\\in V$. We estimate the influence of $S$\non a user $v$ as follows:\n\n\\begin{equation}\nInf_{Pos}\\left(S,v\\right)=\\begin{cases}\n1 & \\textrm{if} \\, v\\in S \\\\\n{\\displaystyle \\sum_{u\\in S}{\\displaystyle }\\sum_{x\\in D_{IN}\\left(v\\right)\\cup\\left\\{ v\\right\\} }Inf_{Pos}\\left(u,x\\right).Inf_{Pos}\\left(x,v\\right)} & \\textrm{Otherwise}\n\\end{cases}\n\\end{equation}\nsuch that $Inf_{Pos}\\left(v,v\\right)=1$ and $D_{IN}\\left(v\\right)$ is\nthe set of nodes in the indegree of $v$. Finally, we define the influence\nspread $\\sigma\\left(S\\right)$ under the evidential model as the total\ninfluence given to $S\\subseteq V$ from all nodes in the social network\nas $\\sigma\\left(S\\right)=\\sum_{v\\in V}Inf_{Pos}\\left(S,v\\right)$. In the\nspirit of the IM problem, as defined by \\textit{Kempe et al.} \\cite{Kempe03}, $\\sigma\\left(S\\right)$\nis the objective function to be maximized.\n\n\n\\subsection{Influence maximization}\n\nIn this section, we present the evidential positive opinion influence\nmaximization model. Its purpose is to find a set of nodes $S$ that\nmaximizes the objective function $\\sigma\\left(S\\right)$. Given a\ndirected social network $G=\\left(V,\\, E\\right)$, an integer $k\\leq |V|$, the goal is to find a set of users $S\\subseteq V,$ $|S|=k$,\nthat maximizes $\\sigma\\left(S\\right)$. We proved that $\\sigma\\left(S\\right)$\nis monotone and sub-modular, also the influence maximization under\nthe proposed model is NP-Hard. However, the number of pages limitation\nprevents us to present proofs in detail. \n\nThe influence maximization under the evidential positive opinion influence\nmaximization model is NP-Hard, consequently, the greedy algorithm performs\ngood approximation for the optimal solution especially when we use\nit with this formula: \n\\begin{equation}\n\\label{eq:mg}\n\\sigma_{Bel} \\left(S\\cup\\left\\{ x\\right\\} \\right)-\\sigma_{Bel}\\left(S\\right)=1+\\sum_{v\\in V\\setminus S\\,}\\sum_{a\\in D_{IN}\\left(v\\right)\\cup\\left\\{ v\\right\\} }Inf\\left(x,a\\right).Inf\\left(a,v\\right)\n\\end{equation}\nthat computes the marginal gain of a candidate node $x$. We choose\nthe cost effective lazy-forward algorithm (CELF) \\cite{Leskovec07b}\nwhich is a two pass modified greedy algorithm. It exploits the sub-modularity\nproperty of the objective function, also, it is about 700 times faster\nthen the basic greedy algorithm.\nThe CELF based evidential influence maximization algorithm starts by estimating the marginal gain of all users in the network and sorts them according to their marginal gain, then, it selects the user that have the maximum marginal gain and add it to $S$ (seed set). After that, the algorithm iterates on the following steps until getting $|S| = k$: 1) Choose the next user in the list, 2) Update its marginal gain (formula (\\ref{eq:mg})), and 3) If the chosen node keeps its position in the list (it still the maximum)\nthen add it to $S$\n\n\n\\section{Experiments}\n\nIn this section, we conduct some experiments on real world data. We used the library Twitter4j\\footnote{http:\/\/twitter4j.org\/en\/index.html} which is a java implementation of the Twitter API to collect Twitter data. We crawled the Twitter network for the period between 08\/09\/2014 and 03\/11\/2014, and we filtered our data by keeping only tweets that talk about smartphones and users that have at least one tweet in the data base. To estimate the opinion polarity of each tweet in our data set, first, we used the java library ``Stanford POS Tagger'' \\footnote{http:\/\/nlp.stanford.edu\/software\/tagger.shtml} with the model ``GATE Twitter part-of-speech tagger''\\footnote{https:\/\/gate.ac.uk\/wiki\/twitter-postagger.html} that were designed for tweets. This step gives a tag (verb, noun, etc) to each word in the tweet. After, we estimated the opinion polarity of each tweet using the SentiWordNet 3.0\\footnote{http:\/\/sentiwordnet.isti.cnr.it\/} dictionary and tags from the first step. We estimated $m^{\\Omega}\\left(u,v\\right)$ using the network structure and past propagation between $u$ and $v$. First, we calculated the number of common neighbors between $u$ and $v$, the number of tweets where $u$ mentions $v$ and the number of tweets where $v$ retweets from $u$. After we used the process defined by \\textit{Wei et al.} \\cite{Wei13} to estimate a BBA for each defined variable. Finally we combine the resulting BBAs to obtain $m^{\\Omega}\\left(u,v\\right)$. In this section, we call belief model: our model in which we use $Inf\\left(u,v\\right)=m^{\\Omega}\\left(u,v\\right)\\left(I\\right)$ as influence measure, CD model: the credit distribution model and opinion model: the proposed positive opinion based model.\n\nThe goal of the first experiment is to show that the proposed model\ndetects well influencer spreaders. To examine the quality of the selected\nseeds, we fixed four comparison criteria which are: the number of\nfollowers, \\#Follow, the number of tweets, \\#Tweet, the number of\ntimes the user was mentioned and retweeted, \\#Mention and \\#Retweet.\nIn fact, we assume that if a user is an influencer on Twitter he would be necessarily:\nvery active so he has a lot of tweets, he is followed by many users\nin the network, he is frequently mentioned and his tweets are retweeted several times. In Figure\n(\\ref{fig:Comparison}), we compare the maximization results of the proposed opinion model with CD model and belief model according to the fixed criteria. Figure (\\ref{fig:Comparison}) shows the performance\nof the proposed model against CD model and belief model. In fact, we see that\nthe proposed opinion model detects influencer that \nhave many followers (more than 8000 for 50 influencer), many tweets (over 250 for 50 users), many mentions (about 1200) and many retweets (about 800).\nHowever, users detected using the belief model have only two good criteria, \\textit{i.e.} \\#Follow (over 8000 follower for 50 users) and \\#Tweet (over 150 tweets for 50 users), and the CD model does not satisfy any criteria. This shows that, the opinion model is the best in detecting influencers.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[scale=0.45]{IphoneDataBaseFollowCompare.png} \\includegraphics[scale=0.45]{IphoneDataBaseMentionCompare.png}\n\\par\\end{centering}\n\n\\begin{centering}\n\\includegraphics[scale=0.45]{IphoneDataBaseRetweetCompare.png} \\includegraphics[scale=0.45]{IphoneDataBaseTweetCompare.png}\n\\par\\end{centering}\n\n\\caption{Comparison between opinion model, belief model and CD model according to \\#Follow,\n\\#Mention, \\#Retweet and \\#Tweet\\label{fig:Comparison}}\n\\end{figure}\n\n\nIn a second experiment, we calculated the mean positive opinion of the first 100 influencer user. The proposed model performed well by selecting influencers that have a positive opinion about the product. In fact, it gives a mean positive opinion equals to 0.89 ($\\pm0.04$, $95\\%$ confidence interval). However, the belief model gives 0.34 ($\\pm0.05$) and the CD model gives only 0.09 ($\\pm0.04$). These results show the performance of the proposed model in selecting influencer users that have a positive opinion against the belief and the CD models that have not.\n\n\n\\section{Conclusion}\n\nIn this paper, we proposed a new influence measure that estimates the positive opinion\ninfluence of OSN users. We used the theory of belief functions to deal with the problem of data imperfection. In future works, we will search to improve the proposed influence maximization model by considering other parameters like the user's profile and the propagation time.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNearly five decades since the publication in 1974 of Allen Shweenk's article \\cite{Schwenk1974}, the determination\nof the spectrum and characteristic polynomial of the generalized composition $H[G_1, \\dots, G_p]$ (recently\ndesignated $H$-join of $\\mathcal{G}=\\{G_1, \\dots, G_p\\}$ \\cite{Cardoso_et_al2013}), in terms of the spectra\n(and characteristic polynomials) of the graphs in $\\mathcal{G}$ and an associated matrix, where all graphs\nare undirected, simple and finite, was limited to families of regular graphs. Very recently in\n\\cite{SaravananMuruganArunkumar2020}, as an application of a new generalization of Fiedler's Lemma, the\ncharacteristic polynomial of the universal adjacency matrix of the $H$-joim of a family of arbitrary graphs\nis determined in terms of the characteristic polynomial and a related racional function of each component, and\nthe determinant of an associated matrix. In this work, using a distincy approach, the determination of the spectrum\n(and characteristic polynomial) of the $H$-join, in terms of its components and associated matrix, is extended to\nfamilies of arbitrary graphs (which should be undirected, simple and finite).\\\\\n\nThe generalized composition $H[G_1, \\dots, G_p]$, introduced in \\cite[p. 167]{Schwenk1974} was rediscovered in\n\\cite{Cardoso_et_al2013} under the designation of $H$-join of a family of graphs $\\mathcal{G}=\\{G_1, \\dots, G_p\\}$,\nwhere $H$ is a graph of order $p$. In \\cite[Th. 7]{Schwenk1974}, assuming that $G_1, \\dots, G_p$ are all regular graphs\nand taking into account that $V(G_1) \\cup \\dots \\cup V(G_p)$ is an equitable partition $\\pi$, the characteristic\npolynomial of $H[G_1, \\dots, G_p]$ is determined in terms of the characteristic polynomials of the graphs\n$G_1, \\dots, G_p$ and the matrix associated to $\\pi$. Using a generalization of a Fiedler's result\n\\cite[Lem. 2.2]{Fiedler1974} obtained in \\cite[Th. 3]{Cardoso_et_al2013}, the spectrum of the $H$-join of a family of\nregular graphs (not necessarily connected) is determined in \\cite[Th. 5]{Cardoso_et_al2013}.\n\nWhen the graphs of the family $\\mathcal{G}$ are all isomorphic to a fixed graph $G$, the  $H$-join of $\\mathcal{G}$\nis the same as the lexicographic product (also called the composition) of the graphs $H$ and $G$ which is denoted as\n$H[G]$ (or $H \\circ G$). The lexicographic product of two graphs was introduced by Harary in \\cite{harary1} and\nSabidussi in \\cite{sabidussi} (see also \\cite{harary2, hammack_et_al}). From the definition, it is immediate that\nthis graph operation is associative but not commutative.\n\nIn \\cite{abreu_et_al}, as an application of the $H$-join spectral properties, the lexicographic powers of a graph\n$H$ were considered and their spectra determined, when $H$ is regular. The $k$-th lexicographic power of $H$, $H^k$,\nis the lexicographic product of $H$ by itself $k$ times (then $H^2=H[H], H^3=H[H^2]=H^2[H], \\dots$). As an example,\nin \\cite{abreu_et_al}, the spectrum of the $100$-th lexicographic power of the Petersen graph, which has a gogool\nnumber (that is, $10^{100}$) of vertices, was determined. With these powers, $H^k$, in \\cite{Cardoso_et_al2017} the\nlexicographic polynomials were introduced and their spectra determined, for connected regular graphs $H$, in terms\nof the spectrum of $H$ and the coefficients of the polynomial.\n\nOther particular $H$-join graph operations appear in the literature under different designations, as it is the case\nof the mixed extension of a graph $H$ studied in \\cite{haemers_et_al}, where special attention is given to the mixed\nextensions of $P_3$. The mixed extension of a graph $H$, with vertex set $V(H)=\\{1, \\dots, p\\}$, is the $H$-join of\na family of graphs $\\mathcal{G}=\\{G_1, \\dots, G_p\\}$, where each graph $G_i \\in \\mathcal{G}$ is a complete graph or\nits complement. From the $H$-join spectral properties, we may conclude that the mixed extensions of a graph $H$ of\norder $p$ has at most $p$ eigenvalues unequal to $0$ and $-1$.\\\\\n\nThe remaining part of the paper is organized as follows. The focus of Section~\\ref{sec_2} is the preliminaries.\nNamely, the notation and basic definitions, the main spectral results of the $H$-join graph operation and the more\nrelevant properties, in the context of this work, of the main characteristic polynomial and walk-matrix of a graph.\nIn Section~\\ref{sec_3}, the main result of this artice, the determination of the spectrum of the $H$-join\nof a family of arbitrary graphs is deduced. Section~\\ref{sec_4} includes some final remarks. Namely, about\nparticular cases of the $H$-join such as the lexicographic product and determination of the eigenvectors of the\nadjacency matrix of the $H$-join in terms of the eigenvectors of the adjacency matrices of the components and an\nassociated matrix.\n\n\\section{Preliminaries}\\label{sec_2}\n\n\\subsection{Notation and basic definitions}\n\nThroughout the text we consider undirected, simple and finite graphs, which are just called graphs. The vertex set\nand the edge set of a graph $G$ is denoted by $V(G)$ and $E(G)$, respectively. The order of $G$ is the cardinality\nof its vertex set and when it is $n$ we consider that $V(G) = \\{1, \\dots, n\\}$. The eigenvalues of the adjacency\nmatrix of a graph $G$, $A(G)$, are also called the eigenvalues of $G$.\nFor each distinct eigenvalue $\\mu$ of $G$, ${\\mathcal E}_G(\\mu)$ denotes the eigenspace of $\\mu$ whose dimension is\nequal to the algebraic multiplicity of $\\mu$, $m(\\mu)$. The spectrum of a graph $G$ of order $n$ is denoted by\n$\\sigma(G)=\\{\\mu^{[m_1]}_1,\\dots,\\mu^{[m_s]}_s,\\mu^{[m_{s+1}]}_{s+1}, \\dots, \\mu_t^{[m_t]}\\}$, where\n$\\mu_1,\\dots,\\mu_s,\\mu_{s+1}, \\dots, \\mu_t$ are the distinct eigenvalues of $G$, $\\mu_i^{[m_i]}$ means that\n$m(\\mu_i)=m_i$ and then $\\sum_{j=1}^{t}{m_j}=n$. When we say that $\\mu$ is an eigenvalue of $G$ with zero multiplicity\n(that is, $m(\\mu)=0$) it means that $\\mu \\not \\in \\sigma(G)$. The distinct eigenvalues of $G$ are indexed in such way\nthat the eigenspaces ${\\mathcal E}_G(\\mu_i)$, for $1 \\le i \\le s$, are not orthogonal to ${\\bf j}_n$, the all-$1$\nvector with $n$ entries (sometimes we simple write ${\\bf j})$. The eigenvalues $\\mu_i$, with $1 \\le i \\le s$, are\ncalled main eigenvalues of $G$ and the remaining distinct eigenvalues non-main. The concept of main (non-main)\neigenvalue was introduced in \\cite{cvetkov70} and further investigated in several publications. As it is well known,\nthe largest eigenvalue of a connected graph $G$ is main and, when $G$ is regular, all its remaining distinct\neigenvalues are non-main \\cite{cds79}. A survey on main eigenvalues was published in \\cite{rowmain}.\n\n\\subsection{The $H$-join operation}\n\nNow we recall the definition of the $H$-join of a family of graphs \\cite{Cardoso_et_al2013}.\n\n\\begin{definition}\\label{def_h-join}\nConsider a graph $H$ with vertex subset $V(H)=\\{1, \\dots, p\\}$ and a family of graphs\n$\\mathcal{G} = \\{G_1, \\dots, G_p\\}$ such that $|V(G_1)|=n_1, \\dots , |V(G_p)|=n_p$.\nThe $H$-join of $\\mathcal{G}$ is the graph\n$$\nG = \\bigvee_{H}{\\mathcal{G}}\n$$\nin which $V(G) = \\bigcup_{j=1}^{p}{V(G_j)}$ and\n$E(G) = \\left(\\bigcup_{j=1}^{p}{E(G_j)}\\right) \\cup \\left(\\bigcup_{rs \\in E(H)}{E(G_r \\vee G_s)}\\right)$,\nwhere $G_r \\vee G_s$ denotes the join.\n\\end{definition}\n\n\\begin{theorem}\\label{H-Join_Spectra} \\cite{Cardoso_et_al2013}\nLet $G$ be the $H$-join as in Definition~\\ref{def_h-join}, where $\\mathcal{G}$ is a family of regular graphs\nsuch that $G_1$ is $d_1$-regular, $G_2$ is $d_2$-regular, $\\dots$ and  $G_p$ is $d_p$-regular. Then\n\\begin{equation}\\label{h-join-spectra}\n\\sigma(G) = \\left(\\bigcup_{j=1}^{p}{\\left(\\sigma(G_j) \\setminus \\{d_j\\}\\right)}\\right) \\cup \\sigma(\\widetilde{C}),\n\\end{equation}\nwhere the matrix $\\widetilde{C}$ has order $p$ and is such that\n\\begin{equation}\\label{matrix_c}\n\\left(\\widetilde{C}\\right)_{rs} = \\left\\{\\begin{array}{ll}\n                                          d_r          & \\hbox{if } r=s,\\\\\n                                         \\sqrt{n_rn_s} & \\hbox{if } rs \\in E(H),\\\\\n                                          0            & \\hbox{otherwise,} \\\\\n                                         \\end{array}\\right.\n\\end{equation}\nand the set operations in \\eqref{h-join-spectra} are done considering possible repetitions of elements of the multisets.\n\\end{theorem}\n\nFrom the above theorem, if there is $G_i \\in \\mathcal{G}$ which is disconnected, with $q$ components, then  its\nregularity $d_i$ appears $q$ times in the multiset $\\sigma(G_i)$. Therefore, according to \\eqref{h-join-spectra},\n$d_i$ remains as an eigenvalue of $G$ with multiplicity $q-1$.\n\nFrom now on, given a graph $H$, we consider the following notation:\n$$\n\\delta_{i,j}(H) = \\left\\{\\begin{array}{ll}\n                          1 & \\hbox{if } ij \\in E(H), \\\\\n                          0 & \\hbox{otherwise.}\n                          \\end{array}\n                          \\right.\n$$\n\nBefore the next result, it is worth observe the following. Considering a graph $G$, it is always possible to extend\na basis of the eigensubspace associated to a main eigenvalue $\\mu_j$, ${\\mathcal E}_G(\\mu_j) \\cap {\\bf j}^{\\perp}$,\nto one of ${\\mathcal E}_G(\\mu_j)$ by adding an eigenvector, $\\hat{\\bf u}_{\\mu_j}$, which is orthogonal to\n${\\mathcal E}_G(\\mu_j) \\cap {\\bf j}^{\\perp}$ and uniquely determined, without considering its multiplication by a\nnonzero scalar. The eigenvector $\\hat{\\bf u}_{\\mu_j}$ is called the main eigenvector of $\\mu_j$. The subspace with\nbasis $\\{\\hat{\\bf u}_{\\mu_1}, \\dots, \\hat{\\bf u}_{\\mu_s}\\}$ is the main subspace of $G$ and is denoted as $Main(G)$.\nNote that for each main eigenvector $\\hat{\\bf u}_{\\mu_j}$ of the basis of $Main(G)$,\n$\\hat{\\bf u}_{\\mu_j}^T{\\bf j}  \\ne 0$.\n\n\\begin{lemma}\\label{main_and_nom-main_eigenvalues-join}\nLet $G$ be the $H$-join as in Definition~\\ref{def_h-join} and $\\mu_{i,j} \\in \\sigma(G_i)$. Then\n$\\mu_{i,j} \\in \\sigma(G)$ with multiplicity\n$$\n\\left\\{\\begin{array}{ll}\n              m(\\mu_{i,j})    & \\hbox{whether $\\mu_{i,j}$ is a non-main eigenvalue of } G_i, \\\\\n              m(\\mu_{i,j})-1 & \\hbox{whether $\\mu_{i,j}$ is a main eigenvalue of } G_i.\n \\end{array}\\right.\n$$\n\\end{lemma}\n\n\\begin{proof}\nDenoting $\\delta_{i,j} = \\delta_{i,j}(H)$, then $\\delta_{i,j} {\\bf j}_{n_i}{\\bf j}^T_{n_j}$ is an $n_i \\times n_j$\nmatrix whose entries are 1 if $ij \\in E(H)$ and $0$ otherwise. Then the adjacency matrix of $G$ has the form\n$$\nA(G) = \\left(\\begin{array}{ccccc}\n        A(G_1)                            & \\delta_{1,2}{\\bf j}_{n_1} {\\bf j}^T_{n_2} & \\cdots & \\delta_{1,p-1}{\\bf j}_{n_1} {\\bf j}^T_{n_{p-1}}&\\delta_{1,p}{\\bf j}_{n_1} {\\bf j}^T_{n_p}\\\\\n       \\delta_{2,1}{\\bf j}_{n_2} {\\bf j}^T_{n_1} & A(G_2)                            & \\cdots & \\delta_{2,p-1}{\\bf j}_{n_2} {\\bf j}^T_{n_{p-1}}&\\delta_{2,p}{\\bf j}_{n_2} {\\bf j}^T_{n_p}\\\\\n       \\vdots                              &   \\vdots                          & \\ddots & \\vdots                               & \\vdots \\\\\n       \\delta_{p-1,1}{\\bf j}_{n_{p-1}}{\\bf j}^T_{n_1} &\\delta_{p-1,2}{\\bf j}_{n_{p-1}} {\\bf j}^T_{n_2} & \\cdots & A(G_{p-1})&\\delta_{p-1,p}{\\bf j}_{n_{p-1}} {\\bf j}^T_{n_p}\\\\\n       \\delta_{p,1}{\\bf j}_{n_p}{\\bf j}^T_{n_1}       &\\delta_{p,2}{\\bf j}_{n_p} {\\bf j}^T_{n_2}& \\cdots &\\delta_{p,p-1}{\\bf j}_{n_p} {\\bf j}^T_{n_{p-1}} & A(G_p)\\\\\n  \\end{array}\\right).\n$$\nLet $\\hat{\\bf u}_{i,j}$ be an eigenvector of $A(G_i)$ associated to an eigenvalue $\\mu_{i,j}$ whose sum of its\ncomponents is zero (then, $\\mu_{i,j}$  is non-main or it is main with multiplicity greater than one). Then,\n\\begin{equation}\\label{eigenvalue-equation}\nA(G) \\left(\\begin{array}{c}\n                 0 \\\\\n                 \\vdots \\\\\n                 0 \\\\\n                 \\hat{\\bf u}_{i,j} \\\\\n                 0 \\\\\n                 \\vdots \\\\\n                 0\n        \\end{array}\\right) = \\left(\\begin{array}{c}\n                                    \\delta_{1,i}\\left({\\bf j}^T_{n_i}\\hat{\\bf u}_{i,j}\\right){\\bf j}_{n_1}  \\\\\n                                    \\vdots \\\\\n                                    \\delta_{i-1,i}\\left({\\bf j}^T_{n_i}\\hat{\\bf u}_{i,j}\\right){\\bf j}_{n_{i-1}} \\\\\n                                    A(G_i)\\hat{\\bf u}_{i,j} \\\\\n                                    \\delta_{i+1,i}\\left({\\bf j}^T_{n_i}\\hat{\\bf u}_{i,j}\\right){\\bf j}_{n_{i+1}} \\\\\n                                    \\vdots \\\\\n                                    \\delta_{p,i}\\left({\\bf j}^T_{n_i}\\hat{\\bf u}_{i,j}\\right){\\bf j}_{n_p}\n                             \\end{array}\\right).\n\\end{equation}\nIt should be noted that when $\\mu_{i,j}$ is main, there are $m(\\mu_{i,j})-1$ linear independent eigenvectors\nbelonging to ${\\mathcal E}_G(\\mu_{i,j}) \\cap {\\bf j}^{\\perp}$.\n\\end{proof}\n\n\\subsection{The main characteristic polynomial and the walk-matrix}\nIf $G$ has $s$ distinct main eigenvalues $\\mu_1, \\dots, \\mu_s$, then the main characteristic polynomial of $G$\nis the polynomial of degree $s$ \\cite{rowmain}\n\\begin{eqnarray}\nm_G(x) &=& \\Pi_{i=1}^{s}{(x-\\mu_i)} \\nonumber\\\\\n       &=& x^s - c_{0} - c_{1}x - \\cdots - c_{s-2} x^{s-2} - c_{s-1} x^{s-1}. \\label{mcpG}\n\\end{eqnarray}\nAs referred in \\cite{rowmain} (see also \\cite{CvetkovicPetric84}), if $\\mu$ is a main eigenvalue of $G$, so is its\nalgebraic conjugate $\\mu^*$ and then the coefficients of $m_G(x)$ are integers. Furthermore, it is worth to recall\nthe next result which follows from \\cite[Th. 2.5]{Teranishi2001} (see also \\cite{rowmain}).\n\n\\begin{theorem}\\cite[Prop. 2.1]{rowmain}\\label{minimal_p}\nFor every polynomial $f(x) \\in \\mathbb{Q}[x]$, $f(A(G)){\\bf j}={\\bf 0}$ if and only if $m_G(x)$ divides $f(x)$.\n\\end{theorem}\n\nIn particular, it is immediate that $m_G(A(G)){\\bf j}={\\bf 0}$. Therefore,\n\\begin{equation}\\label{main_polynomial}\nA^s(G){\\bf j} = c_{0}{\\bf j} + c_{1}A(G){\\bf j} + \\cdots + c_{s-2} A^{s-2}(G){\\bf j} + c_{s-1} A^{s-1}(G){\\bf j}.\n\\end{equation}\n\nGiven a graph $G$ of order $n$, let us consider the $n \\times k$ matrix \\cite{harscwk2, posu}\n$$\n{\\bf W}_{G;k} = \\left({\\bf j}, A(G) {\\bf j}, A^{2}(G){\\bf j}, \\ldots, A^{k-1}(G){\\bf j} \\right ).\n$$\nThe vector space spanned by the columns of ${\\bf W}_{G;k}$ is denoted as $ColSp{\\bf W}_{G;k}$. The matrix\n${\\bf W}_{G;k}$ with largest integer $k$ such that the dimension of $ColSp{\\bf W}_{G;k}$ is equal to $k$, that is,\nsuch that its columns are linear independent, is referred to be the walk-matrix of $G$ and is just denoted as\n${\\bf W}_G$. Then, as a consequence of Theorem~\\ref{minimal_p} and equality \\eqref{main_polynomial}, it follows a\ntheorem which appears in \\cite{hagos1}.\n\n\\begin{theorem}\\cite[Th. 2.1]{hagos1} \\label{hagos}\nThe rank of $W_G$ is equal to the number of main eigenvalues of the graph $G$.\n\\end{theorem}\n\nFrom Theorem~\\ref{hagos}, we may conclude that the number of distinct main eigenvalues is\n$s = \\max \\{k: \\{{\\bf j}, A(G){\\bf j}, A^2(G){\\bf j}, \\ldots, A^{k-1}(G){\\bf j}\\} \\text{ is linearly\nindependent}\\}.$ \\\\\n\nThe equality \\eqref{main_polynomial} also implies the next corollary.\n\n\\begin{corollary} \\label{ap}\nThe $s$-th column of $A(G){\\bf W}_G$ is $ A^{s}(G){\\bf j} = {\\bf W}_G\\left(\\begin{array}{c}\n                                                                                c_{0} \\\\\n                                                                                \\vdots \\\\\n                                                                                c_{s-2} \\\\\n                                                                                c_{s-1} \\\\\n                                                                                \\end{array} \\right ),$\nwhere $c_j$, for $0 \\le j \\le s-1$, are the coefficients of the main characteristic polynomial $m_G$,\ngiven in \\eqref{mcpG}.\n\\end{corollary}\n\nThis corollary allows the determination of the coefficients of the main characteristic polynomial, $m_G$, by solving\nthe linear system ${\\bf W_G \\hat{x}} = A^{s}(G){\\bf j}$.\\\\\n\nFrom \\cite[Th. 2.4]{rowmain} we may conclude the following theorem.\n\n\\begin{theorem}\\label{colspw}\nLet $G$ be a graph with adjacency matrix $A(G)$. Then $ColSp{\\bf W_G}$ coincides with $Main(G)$. Moreover $Main(G)$\nand the vector space spanned by the vectors orthogonal to $Main(G)$, $\\left(Main(G)\\right)^{\\perp}$, are both\n$A(G)$--invariant.\n\\end{theorem}\n\nFrom the above definitions, if $G$ is a $r$-regular graph of order $n$, since its largest eigenvalue, $r$,\nis the unique main eigenvalue, then $m_G(x) = x - r$ and $W_G = \\left( {\\bf j}_n \\right)$.\n\n\\section{The spectrum of the $H$-join of a family of arbitrary graphs}\\label{sec_3}\n\nBefore the main result of this article, we need to define a special matrix ${\\bf \\widetilde{W}}$ which will be called\nthe $H$-join associated matrix.\n\n\\begin{definition}\\label{main_def}\nLet $G$ be the $H$-join as in Definition~\\ref{def_h-join} and denote $\\delta_{i,j}=\\delta_{i,j}(H)$. For each\n$G_i \\in \\mathcal{G}$, consider the main characteristic polynomial \\eqref{mcpG},\n$m_{G_i}(x)=x^{s_i} - c_{i,0} - c_{i,1}x - \\cdots - c_{i,s_i-1} x^{s_i-1}$ and its walk-matrix ${\\bf W}_{G_i}$.\nThe $H$-join associated matrix is the $s \\times s$ matrix, with $s= \\sum_{i=1}^{p}{s_i}$,\n$$\n\\widetilde{\\bf W} = \\left(\\begin{array}{ccccc}\n                       {\\bf C}(m_{G_1})    & \\delta_{1,2}{\\bf M}_{1,2}& \\dots & \\delta_{1,p-1}{\\bf M}_{1,p-1}& \\delta_{1,p}{\\bf M}_{1,p} \\\\\n                        \\delta_{2,1}{\\bf M}_{2,1}& {\\bf C}(m_{G_2})   & \\dots & \\delta_{2,p-1}{\\bf M}_{2,p-1}& \\delta_{2,p}{\\bf M}_{2,p} \\\\\n                        \\vdots             &    \\vdots          &\\ddots & \\vdots                 & \\vdots \\\\\n                        \\delta_{p,1}{\\bf M}_{p,1}& \\delta_{p,2}{\\bf M}_{p,2}& \\dots & \\delta_{p,p-1}M_{p,p-1}& {\\bf C}(m_{G_p})\n                  \\end{array}\\right), \\text{ where}\n$$\n${\\bf C}(m_{G_i}) = \\left(\\begin{array}{ccccc}\n                           0 &   0   &\\dots &   0   & c_{i,0} \\\\\n                           1 &   0   &\\dots &   0   & c_{i,1} \\\\\n                           0 &   1   &\\dots &   0   & c_{i,2} \\\\\n                           \\vdots &\\vdots &\\ddots&\\vdots &\\vdots\\\\\n                           0 &   0   &\\dots &   1   &c_{i,s_i-1} \\\\\n                          \\end{array}\\right)$ and ${\\bf M}_{i,j}=\\left(\\begin{array}{c}\n                                                                   {\\bf j}^T_{n_j}{\\bf W}_{G_j} \\\\\n                                                                         0    \\; \\dots \\; 0 \\\\\n                                                                   \\vdots \\; \\ddots\\; \\vdots\\\\\n                                                                         0    \\; \\dots \\; 0 \\\\\n                                                                   \\end{array}\\right)$, for $1 \\le i,j \\le p$.\n\\end{definition}\n\nNote that the ${\\bf C}(m_{G_i})$ is the Frobenius companion matrix of the main characteristic polynomial $m_{G_i}$\nand ${\\bf M}_{i,j}$ is a $s_i \\times s_j$ matrix such that its first row is\n${\\bf j}_{n_i}^T{\\bf W}_{G_i} = (N_0^i, N_1^i, \\dots, N_{s_i-1}^i)$, where $N_k^i$ is the number of walks of length $k$ in\n$G_i$, for $0 \\le k \\le s_i-1$ (considering $N^i_0=n_i$) and the entries of the remaining rows are equal to zero.\n\n\\begin{theorem}\\label{main_theorem}\nLet $G$ be the $H$-join as in Definition~\\ref{def_h-join}, where $\\mathcal{G}$ is a family of arbitrary graphs.\nIf for each graph $G_i$, with $1 \\le i \\le p$,\n\\begin{equation}\\label{spectrum_Gi}\n\\sigma(G_i)=\\{\\mu_{i,1}^{[m_{i,1}]}, \\dots, \\mu_{i,s_i}^{[m_{i,s_i}]}, \\mu_{i,s_i+1}^{[m_{i,s_i+1}]}, \\dots, \\mu_{i,t_i}^{[m_{i,t_i}]}\\},\n\\end{equation}\nwhere $m_{i,j}=m(\\mu_{i,j})$ and $\\mu_{i,1}, \\dots, \\mu_{i,s_i}$ are the main distinct eigenvalues of $G_i$, then\n\\begin{eqnarray}\n\\sigma(G) &=& \\bigcup_{i=1}^{p}{\\{\\mu_{i,1}^{[m_{i,1}-1]}, \\dots, \\mu_{i,s_i}^{[m_{i,s_i}-1]}\\}} \\cup\n              \\bigcup_{i=1}^{p}{\\{\\mu_{i,s_i+1}^{[m_{i,s_i+1}]}, \\dots, \\mu_{i,t_i}^{[m_{i,t_i}]}\\}} \\cup\n              \\sigma({\\bf \\widetilde{W}}), \\label{G_spectrum}\n\\end{eqnarray}\nwhere the union of multisets is considered with possible repetitions.\n\\end{theorem}\n\n\\begin{proof}\nFrom Lemma~\\ref{main_and_nom-main_eigenvalues-join} it is immediate that\n$$\n\\bigcup_{i=1}^{p}{\\{\\mu_{i,1}^{[m_{i,1}-1]}, \\dots, \\mu_{i,s_i}^{[m_{i,s_i}-1]}\\}} \\cup\n              \\bigcup_{i=1}^{p}{\\{\\mu_{i,s_i+1}^{[m_{i,s_i+1}]}, \\dots, \\mu_{i,t_i}^{[m_{i,t_i}]}\\}} \\subseteq \\sigma(G).\n$$\nSo it just remains to prove that $\\sigma(\\widetilde{{\\bf W}}) \\subseteq \\sigma(G)$.\\\\\n\nLet us define the vector\n\\begin{eqnarray}\n\\hat{\\bf v}      &=& \\left(\\begin{array}{c}\n                            \\hat{\\bf v}_{1} \\\\\n                            \\vdots \\\\\n                            \\hat{\\bf v}_{p} \\\\\n                     \\end{array}\\right), \\text{ such that } \\label{vector_v}\\\\\n\\hat{\\bf v}_{i}  &=& \\sum_{k=0}^{s_i-1}{\\alpha_{i,k}A^k(G_i){\\bf j}_{n_i}}\n                      = {\\bf W}_{G_i}\\hat{\\mathbf{\\alpha}}_{i},\\label{main_vector_Gi}\n\\end{eqnarray}\nwhere $\\hat{\\mathbf{\\alpha}}_{i} = \\left(\\begin{array}{c}\n                                    \\alpha_{i,0} \\\\\n                                    \\alpha_{i,1} \\\\\n                                    \\vdots      \\\\\n                                    \\alpha_{i,s_i-1} \\\\\n                                    \\end{array} \\right ), \\text{ for } 1 \\le i \\le p.$\nFrom \\eqref{main_vector_Gi}, each $\\hat{\\bf v}_{i} \\in Main(G_i)$ and then all vectors $\\hat{\\bf v}$ defined in\n\\eqref{vector_v} are orthogonal to the eigenvectors of $A(G)$ in \\eqref{eigenvalue-equation}. Moreover,\n\n\\begin{equation}\\label{main_subspace}\nA(G_i)\\hat{\\bf v}_{i} = A(G_i){\\bf W}_{G_i}\\hat{\\mathbf{\\alpha}}_{i} = \\sum_{k=0}^{s_i-1}{\\alpha_{i,k}A^{k+1}(G_i){\\bf j}_{n_i}}, \\text{ for } 1 \\le i \\le p.\n\\end{equation}\n\nTherefore,\n\n\\begin{eqnarray}\nA(G)\\hat{\\bf v} &=& \\left(\\begin{array}{cccc}\nA(G_1)                                         & \\delta_{1,2}{\\bf j}_{n_1} {\\bf j}^T_{n_2}& \\cdots & \\delta_{1,p}{\\bf j}_{n_1} {\\bf j}^T_{n_p}\\\\\n\\delta_{2,1}{\\bf j}_{n_2} {\\bf j}^T_{n_1}      & A(G_2)                                   & \\cdots & \\delta_{2,p}{\\bf j}_{n_2} {\\bf j}^T_{n_p}\\\\\n       \\vdots                                  &   \\vdots                                 & \\ddots & \\vdots \\\\\n\\delta_{p,1}{\\bf j}_{n_p}{\\bf j}^T_{n_1}       &\\delta_{p,2}{\\bf j}_{n_p} {\\bf j}^T_{n_2} & \\cdots & A(G_p)\\\\\n  \\end{array}\\right) \\left(\\begin{array}{c}\n                     \\hat{\\bf v}_{1}\\\\\n                     \\hat{\\bf v}_{2}\\\\\n                     \\vdots \\\\\n                     \\hat{\\bf v}_{p}\n  \\end{array}\\right) \\nonumber \\\\\n&=& \\left(\\begin{array}{c}\n                   A(G_1)\\hat{\\bf v}_{1} + \\left(\\sum_{k \\in [p] \\setminus \\{1\\}}{\\delta_{1,k}{\\bf j}^T_{n_k}\\hat{\\bf v}_{k}}\\right){\\bf j}_{n_1}\\\\\n                   A(G_2)\\hat{\\bf v}_{2} + \\left(\\sum_{k \\in [p] \\setminus \\{2\\}}{\\delta_{2,k}{\\bf j}^T_{n_k}\\hat{\\bf v}_{k}}\\right){\\bf j}_{n_2}\\\\\n                   \\vdots \\\\\n                   A(G_p)\\hat{\\bf v}_{p} + \\left(\\sum_{k \\in [p] \\setminus \\{p\\}}{\\delta_{p,k}{\\bf j}^T_{n_k}\\hat{\\bf v}_{k}}\\right){\\bf j}_{n_p}\n         \\end{array}\\right) \\label{walk_1}\\\\\n&=& \\left(\\begin{array}{c}\n                   A(G_1)\\hat{\\bf v}_{1} + \\left(\\sum_{k \\in [p] \\setminus \\{1\\}}{\\delta_{1,k}{\\bf j}^T_{n_k}{\\bf W}_{G_k}\\hat{\\mathbf{\\alpha}}_{k}}\\right){\\bf j}_{n_1}\\\\\n                   A(G_2)\\hat{\\bf v}_{2} + \\left(\\sum_{k \\in [p] \\setminus \\{2\\}}{\\delta_{2,k}{\\bf j}^T_{n_k}{\\bf W}_{G_k}\\hat{\\mathbf{\\alpha}}_{k}}\\right){\\bf j}_{n_2}\\\\\n                   \\vdots \\\\\n                   A(G_p)\\hat{\\bf v}_{p} + \\left(\\sum_{k \\in [p] \\setminus \\{p\\}}{\\delta_{p,k}{\\bf j}^T_{n_k}{\\bf W}_{G_k}\\hat{\\mathbf{\\alpha}}_{k}}\\right){\\bf j}_{n_p}\n          \\end{array}\\right), \\label{walk_2}\n\\end{eqnarray}\nwhere \\eqref{walk_2} is obtained applying \\eqref{main_vector_Gi} in \\eqref{walk_1}. Defining\n\\begin{equation*}\n\\beta_{i,0}=\\sum_{k \\in [p] \\setminus \\{i\\}}{\\delta_{i,k}{\\bf j}^T_{n_k}{\\bf W}_{G_k}\\hat{\\mathbf{\\alpha}}_{k}},\n           \\text{ for } 1 \\le i \\le p\n\\end{equation*}\nand taking into account \\eqref{main_subspace}, the $i$-th row of \\eqref{walk_2} can be written as\n{\\small \\begin{eqnarray}\n\\hspace{-0.5cm}\\beta_{i,0}{\\bf j}_{n_i} + A(G_i)\\hat{\\bf v}_{i}  &=& \\left(\\underbrace{\\sum_{k \\in [p]\\setminus\\{i\\}}{\\delta_{i,k}{\\bf j}^T_{n_k}{\\bf W}_{G_k}\\hat{\\bf \\alpha}_{k}}}_{\\beta_{i,0}}\\right){\\bf j}_{n_i} + \\sum_{k=0}^{s_i-1}{\\alpha_{i,k}A^{k+1}(G_i){\\bf j}_{n_i}} \\nonumber\\\\\n\\hspace{-0.5cm}  &=& \\beta_{i,0}{\\bf j}_{n_i} + \\sum_{k=1}^{s_i-1}{\\alpha_{i,k-1}A^k(G_i){\\bf j}_{n_i}}\n                                      + \\alpha_{i,s_i-1}A^{s_i}(G_i){\\bf j}_{n_i} \\label{ith-row_1}\\\\\n\\hspace{-0.5cm}  &=& \\beta_{i,0}{\\bf j}_{n_i} + \\sum_{k=1}^{s_i-1}{\\alpha_{i,k-1}A^k(G_i){\\bf j}_{n_i}}\n                                      + \\alpha_{i,s_i-1}{\\bf W}_{G_i}\\left(\\begin{array}{c}\n                                                                                c_{i,0} \\\\\n                                                                                c_{i,1} \\\\\n                                                                                \\vdots \\\\\n                                                                                c_{i,s_i-1} \\\\\n                                                                            \\end{array} \\right) \\label{ith-row_2}\n\\end{eqnarray}}\n\\begin{eqnarray}\n\\qquad \\qquad &=& {\\bf W}_{G_i}\\left(\\begin{array}{c}\n                        \\beta_{i,0} + \\alpha_{i,s_i-1}c_{i,0} \\\\\n                        \\alpha_{i,0} + \\alpha_{i,s_i-1}c_{i,1}\\\\\n                        \\vdots \\\\\n                        \\alpha_{i,s_i-2} + \\alpha_{i,s_i-1}c_{i,s_i-1} \\\\\n                        \\end{array}\\right). \\label{ith-row_3}\n\\end{eqnarray}\nObserve that \\eqref{ith-row_2} is obtained applying Corollary~\\ref{ap} to \\eqref{ith-row_1}. Taking into account the\ndefinition of $\\beta_{i,0}$, \\eqref{ith-row_3} can be replaced by the expression\n\n{\\tiny\n$$\n\\hspace{-1.5cm}{\\bf W}_{G_i}\\underbrace{\\left(\\begin{array}{ccccccccccc}\n\\overbrace{\\delta_{i,1}{\\bf j}^T_{n_1}{\\bf W}_{G_1}}^{s_1\\text{ columns}}&\\cdots&\\overbrace{\\delta_{i,{i-1}}{\\bf j}^T_{n_{i-1}}{\\bf W}_{G_{i-1}}}^{s_{i-1}\\text{ columns}}&      0    &   0  &\\cdots&  0   &  c_{i,0} & \\overbrace{\\delta_{i,{i+1}}{\\bf j}^T_{n_{i+1}}{\\bf W}_{G_{i+1}}}^{s_{i+1}\\text{ columns}} &\\cdots&\\overbrace{\\delta_{i,p}{\\bf j}^T_{n_p}{\\bf W}_{G_p}}^{s_p\\text{ columns}}\\\\\n                  {\\bf 0}                                          &\\cdots& {\\bf 0}&      1    &   0  &\\cdots&  0   &  c_{i,1}  &{\\bf 0}&\\cdots&      {\\bf 0}    \\\\\n                  {\\bf 0}                                          &\\cdots&{\\bf 0}&      0    &   1  &\\cdots&  0   &  c_{i,2}  &{\\bf 0}&\\cdots&      {\\bf 0}    \\\\\n                  \\vdots                                           &\\ddots&\\vdots&   \\vdots  &\\vdots&\\ddots&\\vdots&  \\vdots   &\\vdots&\\ddots&      \\vdots     \\\\\n                  {\\bf 0}                                          &\\cdots&{\\bf 0}&      0    &   0  &\\cdots&  1   &c_{i,s_i-1} &{\\bf 0}&\\cdots&      {\\bf 0}\n\\end{array}\\right)}_{\\widetilde{\\bf W}_i}\\left(\\begin{array}{c}\n                        \\hat{\\bf \\alpha}_{1}\\\\\n                        \\vdots \\\\\n                        \\hat{\\bf \\alpha}_{i-1}\\\\\n                        \\hat{\\bf \\alpha}_{i}\\\\\n                        \\hat{\\bf \\alpha}_{i+1}\\\\\n                        \\vdots \\\\\n                        \\hat{\\bf \\alpha}_{p}\n                        \\end{array}\\right)\n$$}\nwhich is equivalent to the expression\n$$\n{\\bf W}_{G_i}\\underbrace{\\left(\\begin{array}{ccccccc}\n                   \\delta_{i,1}{\\bf M}_{i,1} & \\dots & \\delta_{i,i-1}{\\bf M}_{i,i-1} & {\\bf C}(m_{G_i}) & \\delta_{i,i+1}{\\bf M}_{i,i+1} & \\dots & \\delta_{i,p}{\\bf M}_{i,p}\\\\\n                        \\end{array}\\right)}_{\\widetilde{\\bf W}_i}\\left(\\begin{array}{c}\n                                                                     \\hat{\\bf \\alpha}_{1}\\\\\n                                                                     \\vdots \\\\\n                                                                     \\hat{\\bf \\alpha}_{i-1}\\\\\n                                                                     \\hat{\\bf \\alpha}_{i}\\\\\n                                                                     \\hat{\\bf \\alpha}_{i+1}\\\\\n                                                                     \\vdots \\\\\n                                                                     \\hat{\\bf \\alpha}_{p}\n                                                                     \\end{array}\\right).\n$$\nFrom the above analysis\n\\begin{eqnarray*}\nA(G)\\hat{\\bf v} & = & \\left(\\begin{array}{cccc}\n                            {\\bf W}_{G_1} &    {\\bf 0}    & \\cdots &  {\\bf 0} \\\\\n                            {\\bf 0}   & {\\bf W}_{G_2} & \\cdots &  {\\bf 0} \\\\\n                            \\vdots    &    \\vdots     & \\ddots &  \\vdots  \\\\\n                            {\\bf 0}   &    {\\bf 0}    & \\cdots & {\\bf W}_{G_p}\n                            \\end{array}\\right) \\left(\\begin{array}{c}\n                                                           {\\bf \\widetilde{W}}_1\\\\\n                                                           {\\bf \\widetilde{W}}_2\\\\\n                                                           \\vdots\\\\\n                                                           {\\bf \\widetilde{W}}_p\n                                                      \\end{array}\\right)\\left(\\begin{array}{c}\n                                                                              \\hat{\\bf \\alpha}_{1}\\\\\n                                                                              \\hat{\\bf \\alpha}_{2}\\\\\n                                                                              \\vdots \\\\\n                                                                              \\hat{\\bf \\alpha}_{p}\n                                                                              \\end{array}\\right)\n\\end{eqnarray*}\nand, according to \\eqref{main_vector_Gi},\n\\begin{eqnarray*}\n\\hat{\\bf v} &=& \\left(\\begin{array}{cccc}\n                           {\\bf W}_{G_1} &    {\\bf 0}    & \\cdots &  {\\bf 0} \\\\\n                           {\\bf 0}   & {\\bf W}_{G_2} & \\cdots &  {\\bf 0} \\\\\n                           \\vdots    &    \\vdots     & \\ddots &  \\vdots  \\\\\n                           {\\bf 0}   &    {\\bf 0}    & \\cdots & {\\bf W}_{G_p}\n                           \\end{array}\\right)\\left(\\begin{array}{c}\n                                                       \\hat{\\bf \\alpha}_{1}\\\\\n                                                       \\hat{\\bf \\alpha}_{2}\\\\\n                                                       \\vdots \\\\\n                                                       \\hat{\\bf \\alpha}_{p}\n                                                       \\end{array}\\right). \\label{vecto_v}\n\\end{eqnarray*}\nTherefore, $A(G)\\hat{\\bf v} = \\rho \\hat{\\bf v}$ if and only if\n\\begin{eqnarray}\n\\underbrace{\\left(\\begin{array}{cccc}\n       {\\bf W}_{G_1} &    {\\bf 0}    & \\cdots &  {\\bf 0} \\\\\n       {\\bf 0}       & {\\bf W}_{G_2} & \\cdots &  {\\bf 0} \\\\\n           \\vdots    &    \\vdots     & \\ddots &  \\vdots  \\\\\n       {\\bf 0}       &    {\\bf 0}    & \\cdots & {\\bf W}_{G_p}\n      \\end{array}\\right)}_{{\\bf (*)}}\\left(\\left(\\begin{array}{c}\n                               {\\bf \\widetilde{W}}_1\\\\\n                               {\\bf \\widetilde{W}}_2\\\\\n                                    \\vdots          \\\\\n                               {\\bf \\widetilde{W}}_p\n                              \\end{array}\\right) - \\rho I_s\\right)\\left(\\begin{array}{c}\n                                                             \\hat{\\bf \\alpha}_{1}\\\\\n                                                             \\hat{\\bf \\alpha}_{2}\\\\\n                                                             \\vdots \\\\\n                                                             \\hat{\\bf \\alpha}_{p}\n                                                             \\end{array}\\right) &=& {\\bf 0}. \\label{main_equality}\n\\end{eqnarray}\nIt is immediate that ${\\bf \\widetilde{W}} = \\left(\\begin{array}{c}\n                                                  {\\bf \\widetilde{W}}_1\\\\\n                                                  {\\bf \\widetilde{W}}_2\\\\\n                                                       \\vdots          \\\\\n                                                  {\\bf \\widetilde{W}}_p\n                                                  \\end{array}\\right)$ and since the columns of each matrix\n${\\bf W}_{G_i}$ are linear independent, the columns of the matrix $(*)$ are also linear independent. Consequently,\n\\eqref{main_equality} is equivalent to\n\\begin{equation}\n\\left({\\bf \\widetilde{W}} - \\rho I_s\\right)\\hat{\\bf \\alpha} = {\\bf 0}, \\label{w_eigenvetor}\n\\end{equation}\nwhere $\\hat{\\bf \\alpha} = \\left(\\begin{array}{c}\n                                \\hat{\\bf \\alpha}_{1}\\\\\n                                \\vdots \\\\\n                                \\hat{\\bf \\alpha}_{p}\n                                \\end{array}\\right).$\nFinally, we may conclude that $(\\rho,\\hat{\\bf v})$ is an eigenpair of $A(G)$ if and only if $(\\rho, \\hat{\\bf \\alpha})$,\nis an eigenpair of the $H$-join associated matrix ${\\bf \\widetilde{W}}$.\n\\end{proof}\n\nBefore the next corollary of Theorem~\\ref{main_theorem}, it is convenient to introduce the notation\n$\\phi(G)$ and $\\phi({\\bf A})$ which, from now on, will be used for the characteristic polynomial of a graph $G$\nand a matrix ${\\bf A}$, respectively.\n\n\\begin{corollary}\\label{cor_charact_poly}\nLet $G$ be the $H$-join as in Definition~\\ref{def_h-join} with associated matrix ${\\bf \\widetilde{W}}$. Assuming\nthat $\\mathcal{G}=\\{G_1, \\dots, G_p\\}$ is  a family of arbitrary graphs for which $\\sigma(G_1), \\dots, \\sigma(G_p)$\nare defined as in \\eqref{spectrum_Gi} and $m_{G_1}, \\dots, m_{G_p}$ are their main characteristic polynomials, then\n$$\n\\phi(G) = \\left(\\prod_{i=1}^{p}{\\frac{\\phi(G_i)}{m_{G_i}}}\\right)\\phi({\\bf \\widetilde{W}}).\n$$\n\\end{corollary}\n\n\\begin{proof}\nSince, for $1 \\le i \\le p$,\n$\\sigma(G_i)=\\{\\mu_{i,1}^{[m_{i,1}]}, \\dots, \\mu_{i,s_i}^{[m_{i,s_i}]}, \\mu_{i,s_i+1}^{[m_{i,s_i+1}]}, \\dots, \\mu_{i,t_i}^{[m_i,t_i]}\\}$,\nwhere the first $s_i$ eigenvalues are main, and the roots of $m_{G_i}$ are all simple main eigenvalues of $G_i$,\nit is immediate that $\\phi(G_i)=m_{G_i}\\phi'(G_i)$, where the roots of the polynomial $\\phi'(G_i)$ are the eigenvalues\nof $G_i$,\n$$\n\\{\\mu_{i,1}^{[m_{i,1}-1]}, \\dots, \\mu_{i,s_i}^{[m_{i,s_i}-1]}, \\mu_{i,s_i+1}^{[m_{i,s_i+1}]}, \\dots, \\mu_{i,t_i}^{[m_i,t_i]}\\}.\n$$\nTherefore, according to \\eqref{G_spectrum}, the roots of $\\prod_{i=1}^{p}{\\frac{\\phi(G_i)}{m_{G_i}}}$ are the\neigenvalues in $\\sigma(G) \\setminus \\sigma({\\bf \\widetilde{W}})$.\n\\end{proof}\n\n\n\\begin{example}\nConsider the graph $H \\cong P_3$, the path with three vertices, and the graphs\n$K_{1,3}$, $K_2$ and $P_3$ depicted in the Figure~\\ref{figura_1}. Then\n$\\sigma(K_{1,3})=\\{\\sqrt{3},-\\sqrt{3},0^{[2]}\\},$ $\\sigma(K_2)=\\{1,-1\\}$,\n$\\sigma(P_3)=\\{\\sqrt{2},-\\sqrt{2},0\\}$\nand their main characteristic polynomials are $m_{K_{1,3}}(x) = x^2 - 3$, $m_{K_2}(x) = x - 1$\nand $m_{P_3}(x) = x^2 - 2$, respectively.\n\n\\begin{figure}[h]\n\\begin{center}\n\\unitlength=0.25 mm\n\\begin{picture}(400,120)(60,60)\n\\put(50,110){\\line(2,1){50}} \n\\put(50,110){\\line(2,-1){50}}\n\\put(50,110){\\line(1,0){50}} \n\\put(150,85){\\line(0,1){50}} \n\\put(225,110){\\line(0,1){25}}\n\\put(225,110){\\line(0,-1){25}\n\\put(50,110){\\circle*{5.7}} \n\\put(100,135){\\circle*{5.7}}\n\\put(100,110){\\circle*{5.7}}\n\\put(100,85){\\circle*{5.7}} \n\\put(150,135){\\circle*{5.7}}\n\\put(150,85){\\circle*{5.7}} \n\\put(225,135){\\circle*{5.7}}\n\\put(225,110){\\circle*{5.7}}\n\\put(225,85){\\circle*{5.7}}\n\\put(40,110){\\makebox(0,0){\\footnotesize 1}}\n\\put(100,145){\\makebox(0,0){\\footnotesize 2}}\n\\put(110,110){\\makebox(0,0){\\footnotesize 3}}\n\\put(100,75){\\makebox(0,0){\\footnotesize 4}}\n\\put(75,50){\\makebox(0,0){\\footnotesize $K_{1,3}$}}\n\\put(150,145){\\makebox(0,0){\\footnotesize 5}}\n\\put(150,75){\\makebox(0,0){\\footnotesize 6}}\n\\put(150,50){\\makebox(0,0){\\footnotesize $K_2$}}\n\\put(225,145){\\makebox(0,0){\\footnotesize 7}}\n\\put(235,110){\\makebox(0,0){\\footnotesize 8}}\n\\put(225,75){\\makebox(0,0){\\footnotesize 9}}\n\\put(225,50){\\makebox(0,0){\\footnotesize $P_3$}}\n\\put(300,110){\\line(2,1){50}} \n\\put(300,110){\\line(2,-1){50}}\n\\put(300,110){\\line(1,0){50}} \n\\put(400,85){\\line(0,1){50}}\n\\put(300,110){\\line(4,1){100}}\n\\put(300,110){\\line(4,-1){100}\n\\put(350,135){\\line(1,0){125}}\n\\put(350,135){\\line(1,-1){50}}\n\\put(350,110){\\line(2,1){50}}\n\\put(350,110){\\line(2,-1){50}}\n\\put(350,85){\\line(1,0){125}} \n\\put(350,85){\\line(1,1){50}}  \n\\put(400,135){\\line(3,-1){75}}\n\\put(400,135){\\line(3,-2){75}}\n\\put(400,85){\\line(3,1){75}}  \n\\put(400,85){\\line(3,2){75}}  \n\\put(475,135){\\line(0,-1){50}}\n\\put(300,110){\\circle*{5.7}}\n\\put(350,135){\\circle*{5.7}}\n\\put(350,110){\\circle*{5.7}} \n\\put(350,85){\\circle*{5.7}} \n\\put(400,135){\\circle*{5.7}\n\\put(400,85){\\circle*{5.7}}\n\\put(475,135){\\circle*{5.7}}\n\\put(475,110){\\circle*{5.7}}\n\\put(475,85){\\circle*{5.7}} \n\\put(295,110){\\makebox(0,0){\\footnotesize 1}}\n\\put(350,145){\\makebox(0,0){\\footnotesize 2}}\n\\put(360,110){\\makebox(0,0){\\footnotesize 3}}\n\\put(350,75){\\makebox(0,0){\\footnotesize 4}}\n\\put(400,145){\\makebox(0,0){\\footnotesize 5}}\n\\put(400,75){\\makebox(0,0){\\footnotesize 6}}\n\\put(475,145){\\makebox(0,0){\\footnotesize 7}}\n\\put(485,110){\\makebox(0,0){\\footnotesize 8}}\n\\put(475,75){\\makebox(0,0){\\footnotesize 9}}\n\\put(400,50){\\makebox(0,0){{\\footnotesize $G=\\bigvee_{P_3}{\\{K_{1,3}, K_2, P_3\\}}$}}}\n\\end{picture}\n\\end{center}\n\\caption{The $P_3$-join of the family of graphs $K_{1,3}$, $K_2$ and $P_3$.}\\label{figura_1}\n\\end{figure}\nSince\n$$\n\\begin{array}{lclcl}\n{\\bf \\widetilde{W}}_1 &=& \\left(\\begin{array}{ccccc}\n                            0  & c_{1,0} &  \\delta_{1,2}2 & \\delta_{1,3}3 & \\delta_{1,3}4 \\\\\n                            1 & c_{1,1} &         0       &         0     &          0    \\\\\n                           \\end{array}\\right) & = & \\left(\\begin{array}{ccccc}\n                                                           0  & 3 & 2 & 0 & 0 \\\\\n                                                           1  & 0 & 0 & 0 & 0 \\\\\n                                                          \\end{array}\\right),\\\\\n{\\bf \\widetilde{W}}_2 &=& \\left(\\begin{array}{ccccc}\n                           \\delta_{2,1}4 & \\delta_{2,1}6 & c_{2,0} & \\delta_{2,3}3 & \\delta_{2,3}4\\\\\n                         \\end{array}\\right) & = & \\left(\\begin{array}{ccccc}\n                                                         4 & 6 & 1 & 3 & 4\\\\\n                                                        \\end{array}\\right),\\\\\n{\\bf \\widetilde{W}}_3 &=& \\left(\\begin{array}{ccccc}\n                           \\delta_{3,1}4 & \\delta_{3,1}6 & \\delta_{3,2}2 &  0  & c_{3,0} \\\\\n                                   0     &          0    &          0    &  1  & c_{3,1}\n                         \\end{array}\\right) & = & \\left(\\begin{array}{ccccc}\n                                                         0 & 0 &  2 &  0 & 2 \\\\\n                                                         0 & 0 &  0 &  1 & 0\n                                                        \\end{array}\\right),\n\\end{array}\n$$\nit follows that\n\\begin{equation*}\n{\\bf \\widetilde{W}} = \\left(\\begin{array}{c}\n                              {\\bf \\widetilde{W}}_1\\\\\n                              {\\bf \\widetilde{W}}_2\\\\\n                              {\\bf \\widetilde{W}}_3\n                              \\end{array}\\right) = \\left(\\begin{array}{ccccc}\n                                                                0 & 3 & 2 & 0 & 0 \\\\\n                                                                1 & 0 & 0 & 0 & 0 \\\\\n                                                                4 & 6 & 1 & 3 & 4 \\\\\n                                                                0 & 0 & 2 & 0 & 2 \\\\\n                                                                0 & 0 & 0 & 1 & 0\n                                                          \\end{array}\\right).\n\\end{equation*}\nTherefore, the characteristic polynomial of ${\\bf \\widetilde{W}}$ is the polynomial\n$$\n\\phi({\\bf \\widetilde{W}}) = -42 - 40 x + 15 x^2 + 19 x^3 + x^4 - x^5\n$$\nand, applying Corollary~\\ref{cor_charact_poly}, we obtain the characteristic polynomial of $G$,\n$$\n\\phi(G) = x^3(x+1)\\phi({\\bf \\widetilde{W}}) = x^3(x+1)(-42 - 40 x + 15 x^2 + 19 x^3 + x^4 - x^5).\n$$\n\\end{example}\n\n\\section{Final remarks}\\label{sec_4}\n\nWhen all graphs of the family $\\mathcal{G}$ are regular, that is, $G_1$ is $d_1$-regular, $G_2$ is $d_2$-regular,\n$\\dots$, $G_p$ is $d_p$-regular, the walk matrices are ${\\bf W}_{G_1}=\\left({\\bf j}_{n_1}\\right)$,\n${\\bf W}_{G_2}=\\left({\\bf j}_{n_2}\\right)$, $\\dots$, ${\\bf W}_{G_p}=\\left({\\bf j}_{n_p}\\right)$, respectively.\nConsequently, the main polynomials are $m_{G_1}(x) = x - d_1$, $m_{G_2}(x) = x - d_2$, $\\dots$, $m_{G_p}(x) = x - d_p$.\nAs direct\nconsequence, for this particular case, the $H$-join associated matrix is\n{\\small $$\n{\\bf\\widetilde{W}}=\\left(\\begin{array}{cccc}\n                          d_1 &  \\delta_{1,2}{\\bf j}_{n_2}^T{\\bf W}_{G_2}  & \\cdots & \\delta_{1,p}{\\bf j}_{n_p}^T{\\bf W}_{G_p}\\\\\n                          \\delta_{2,1}{\\bf j}_{n_1}^T{\\bf W}_{G_1} &  d_2  & \\cdots & \\delta_{2,p}{\\bf j}_{n_p}^T{\\bf W}_{G_p}\\\\\n                          \\vdots                             &\\vdots & \\ddots & \\vdots \\\\\n                          \\delta_{p,1}{\\bf j}_{n_1}^T{\\bf W}_{G_1} & \\delta_{p,2}{\\bf j}_{n_2}^T{\\bf W}_{G_2}  & \\cdots &d_p \\\\\n                        \\end{array}\\right) = \\left(\\begin{array}{cccc}\n                                                    d_1 &  \\delta_{1,2}n_2  & \\cdots & \\delta_{1,p}n_p \\\\\n                                                    \\delta_{2,1}n_1 &  d_2  & \\cdots & \\delta_{2,p}n_p \\\\\n                                                    \\vdots          &\\vdots & \\ddots & \\vdots \\\\\n                                                    \\delta_{p,1}n_1 &\\delta_{p,2}n_2 & \\cdots & d_p \\\\\n                                               \\end{array}\\right).\n$$}\nTherefore, it is immediate that when all the graphs of the family $\\mathcal{G}$ are regular, the matrix\n${\\bf\\widetilde{W}}$ and the matrix $\\widetilde{C}$ in \\eqref{matrix_c} are similar matrices. Note that\n$\\widetilde{C} = D {\\bf\\widetilde{W}}D^{-1}$, where\n$D = \\text{diag}\\left(\\sqrt{n_1}, \\sqrt{n_2}, \\dots, \\sqrt{n_p}\\right)$ and thus\n${\\bf\\widetilde{W}}$ and $\\widetilde{C}$ are cospectral matrices as it should be.\\\\\n\nIn the particular case of the lexicographic product $H[G]$, which is the $H$-join of a family of graphs\n$\\mathcal{G}$, where all the graphs in $\\mathcal{G}$ are isomorphic to a fixed graph $G$, consider that the graph $H$\nhas order $p$ and the graph $G$ has order $n$. Let\n$\\sigma(G)=\\{\\mu_1^{[m_1]}, \\dots, \\mu_{s}^{[m_s]}, \\mu_{s+1}^{[m_{s+1}]}, \\dots, \\mu_{t}^{[m_t]}\\}$, where\n$\\mu_1, \\dots, \\mu_s$ are the distinct main eigenvalues of $G$ and $\\sum_{i=1}^{t}{m_{i}}=n$. Then, according to\nthe Definition~\\ref{main_def}, the $H$-join associated matrix is\n$$\n{\\bf \\widetilde{W}} = \\left(\\begin{array}{ccccc}\n                       {\\bf C}(m_{G}) & \\delta_{1,2}{\\bf M} & \\dots & \\delta_{1,p-1}{\\bf M} & \\delta_{1,p}{\\bf M} \\\\\n                        \\delta_{2,1}{\\bf M} & {\\bf C}(m_{G})& \\dots & \\delta_{2,p-1}{\\bf M} & \\delta_{2,p}{\\bf M} \\\\\n                        \\vdots        &    \\vdots     &\\ddots & \\vdots          & \\vdots \\\\\n                        \\delta_{p,1}{\\bf M} & \\delta_{p,2}{\\bf M} & \\dots & \\delta_{p,p-1}{\\bf M} & {\\bf C}(m_{G})\n                      \\end{array}\\right),\n$$\nwhere ${\\bf C}(m_{G})$ is the Frobenius companion matrix of $m_G$ and\n${\\bf M} = \\left(\\begin{array}{c}\n            {\\bf j}_n^{\\top}{\\bf W}_G\\\\\n            {\\bf 0}\\\\\n           \\end{array}\\right)$ (both are $s \\times s$ matrices). Applying Theorem~\\ref{main_theorem}, it follows\n\\begin{eqnarray*}\n\\sigma(G) &=& p{\\{\\mu_{1}^{[m_{1}-1]}, \\dots, \\mu_{s}^{[m_{s}-1]}\\}} \\cup\n              p{\\{\\mu_{s+1}^{[m_{s+1}]}, \\dots, \\mu_{t}^{[m_{t}]}\\}} \\cup\n              \\sigma({\\bf \\widetilde{W}}),\n\\end{eqnarray*}\nwhere the multiplication of $p$ by a set $X$ means the union of $X$ with himself $p$ times. Therefore, from\nCorollary~\\ref{cor_charact_poly}, the characteristic polynomial of $H[G]$ is\n\\begin{equation}\n\\phi(H[G]) = \\left(\\frac{\\phi(G)}{m_G}\\right)^p \\phi({\\bf \\widetilde{W}}). \\label{charact_poly_lex}\n\\end{equation}\n\nIn \\cite[Th. 2.4]{WangWong2018} a distinct expression for $\\phi(H[G])$ is determined. Such expression is not\nonly related with $H$ and $G$ but also with eigenspaces of the adjacency matrix of  $G$.\\\\\n\nFrom the obtained results, we are able to determine all the eigenvectors of the adjacency matrix of the $H$-join of\na family of arbitrary graphs $G_1, \\dots, G_p$ in terms of the eigenvectors of the adjacency matrices $A(G_i)$, for\n$1 \\le i \\le p$, and the eigenvectors of the $H$-join associated matrix ${\\bf \\widetilde{W}}$, as follows.\n\\begin{enumerate}\n\\item Let $G$ be the $H$-join as in Definition~\\ref{def_h-join}, where $\\mathcal{G}=\\{G_1, G_2, \\dots, G_p\\}$ is a\n      family of arbitrary graphs.\n\\item For $1 \\le i \\le p$, consider $\\sigma(G_i)$ as defined in \\eqref{spectrum_Gi}. For each eigenvalue\n      $\\mu_{i,j} \\in \\sigma(G_i)$, every eigenvetor\n      $\\hat{\\bf u}_{i,j} \\in \\mathcal{E}_{G_i}(\\mu_{i,j}) \\cap {\\bf j}_{n_i}^{\\perp}$ defines an eigenvector for\n      $A(G)$ as in \\eqref{eigenvalue-equation}.\n\\item The remaining eigenvectors of $A(G)$ are the vectors $\\hat{\\bf v}$ defined in \\eqref{vector_v}-\\eqref{main_vector_Gi}\n      from the eigenvectors of ${\\bf \\widetilde{W}}$, $\\hat{\\bf \\alpha}$, obtained as linear independent solutions\n      of \\eqref{w_eigenvetor}, for each $\\rho \\in \\sigma({\\bf \\widetilde{W}})$.\n\\end{enumerate}\n\n\\medskip\\textbf{Acknowledgments.}\nThis work is supported by the Center for Research and Development in Mathematics and Applications (CIDMA) through\nthe Portuguese Foundation for Science and Technology (FCT - Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia), reference\nUIDB\/04106\/2020.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this erratum to \\cite{F}, we point out that there is a gap in the proof of\nTheorem 1 of that paper. Specifically, the estimate for $\\sup_{x\\in E_{n}%\n}\\left\\Vert \\overline{f}_{n}^{\\prime}\\left(  x\\right)  -F_{n}^{\\prime}\\left(\nx\\right)  \\right\\Vert $ in \\cite{F} does not hold (as the inductive proof\nfails here), and as a consequence the conclusion of Theorem 1 does not follow.\nNevertheless, using a construction from \\cite{J}, techniques from\n\\cite{AFGJL}, and employing a similar proof as originally, we are able to\nestablish all the results of \\cite{F} under the additional assumption that the\nsubset $Y\\subset X$ is convex (see Theorem 1 below.)\n\nWe note that the main motivation for this work was to find an analogous result\nto that of \\cite{AFM} for not necessarily bounded functions. Let us recall in\n\\cite{AFM} it was shown, in particular, that for a separable Banach space $X$\nadmitting a Lipschitz, $C^{p}$ smooth bump function, that given $\\varepsilon\n>0$ and a bounded, uniformly continuous function $f:X\\rightarrow\\mathbb{R}$,\nthere exists a Lipschitz, $C^{p}$ smooth function $K$ with $\\left\\vert\nf-K\\right\\vert <\\varepsilon$ on $X.$ We remark that to establish our result\nhere, we need to further assume that our Banach space $X$ has an unconditional\nbasis. However, in addition to relaxing the boundedness condition on $f$, when\n$f$ is also Lipschitz, unlike the result of \\cite{AFM}, we are able to find\nLipschitz, $C^{p}$ smooth approximates $K$ where the Lipschitz constants do\nnot depend on the $\\varepsilon$-degree of precision in the approximation. We\nalso note that the results of \\cite{AFM} are restricted to real-valued maps.\n\n$\\smallskip$\n\nThe notation we employ is standard, with $X,Y,$ etc. denoting Banach spaces.\nWe write the closed unit ball of $X$ as $B_{X}.$ The G\\^{a}teaux derivative of\na function $f$ at $x$ in the direction $h$ will be denoted $D_{h}f\\left(\nx\\right)  ,$ while the Fr{\\'{e}}chet derivative of $f$ at $x$ on $h$ is\nwritten $f^{\\prime}\\left(  x\\right)  \\left(  h\\right)  .$ We note that a\n$C^{p}$-smooth function is necessarily Fr{\\'{e}}chet differentiable (see e.g.,\n\\cite{BL}.)\n\n$\\smallskip$\n\nA $C^{p}$\\textbf{-smooth bump function} $b$ on $X$ is a $C^{p}$-smooth,\nreal-valued function on $X$ with bounded, non-empty support, where\n\\[\n\\text{support}\\left(  b\\right)  =\\overline{\\left\\{  x\\in X:b\\left(  x\\right)\n\\neq0\\right\\}  }.\n\\]\nIf $f:X\\rightarrow Y$ is Lipschitz with constant $\\eta,$ we will say that $f $\nis $\\eta$-Lipschitz. Most additional notation is explained as it is introduced\nin the sequel. For any unexplained terms we refer the reader to \\cite{DGZ} and\n\\cite{FHHMPZ}. For further historical context see the introduction in \\cite{F}.\n\n\\section{Main Results}\n\nWe first introduce some notation which will be used throughout the paper. Let\n$\\left\\{  e_{j},e_{j}^{\\ast}\\right\\}  _{j=1}^{\\infty}$ be an unconditional\nSchauder basis on $X,$ and $P_{n}:X\\rightarrow X$ the canonical projections\ngiven by $P_{n}\\left(  x\\right)  =P_{n}\\left(  \\sum_{j=1}^{\\infty}x_{j}%\ne_{j}\\right)  =\\sum_{j=1}^{n}x_{j}e_{j},$ and where we set $P_{0}=0.$ By\nrenorming, we may assume that the unconditional basis constant is $1.$ In\nparticular, $\\left\\Vert P_{n}\\right\\Vert \\leq1$ for all $n.$ We put\n$E_{n}=P_{n}\\left(  X\\right)  ,$ and $E^{\\infty}=\\cup_{n}E_{n},$ noting that\n$\\dim E_{n}=n,$ $E_{n}\\subset E_{n+1},$ and $E^{\\infty}\\ $is a dense subspace\nof $X.$ It will be convenient to denote the closed unit ball of $E_{n}$ by\n$B_{E_{n}}.$\n\n$\\smallskip$\n\nThe proof of our main theorem is a modification of some techniques found in\n\\cite{M} and \\cite{AFGJL}, where $C^{p}$-fine approximation on Banach spaces\nis considered. We also rely on the main construction from \\cite{J}. We follow\nthe original proof of \\cite{F} closely, and have decided to reproduce the\ndetails so that this note is self contained.\n\n\\begin{theorem}\nLet $X$ be a Banach space with unconditional basis which admits a Lipschitz,\n$C^{p}$-smooth bump function. Let $Y\\subset X$ be a convex subset and\n$f:Y\\rightarrow\\mathbb{R}$ a uniformly continuous map. Then for each\n$\\varepsilon>0$ there exists a Lipschitz, $C^{p}$-smooth function\n$K:X\\rightarrow\\mathbb{R}$ such that for all $y\\in Y,$%\n\n\\[\n\\left\\vert f(y)-K(y)\\right\\vert <\\varepsilon.\n\\]\n\n\n$\\smallskip$\n\nIf $Z\\ $is any Banach space, $Y\\subset X$ is any subset, and $f:X\\rightarrow\nZ$ (respectively $f:Y\\rightarrow\\mathbb{R}$) is Lipschitz with constant\n$\\eta,$ then we can choose $K:X\\rightarrow Z$ (respectively $K:X\\rightarrow\n\\mathbb{R}$) to have Lipschitz constant no larger than $C_{0}\\eta,$ where\n$C_{0}>1$ is a constant depending only on $X$ (in particular, $C_{0}$ is\nindependent of $\\varepsilon.$)\n\\end{theorem}\n\n\\medskip\n\n\\textbf{Proof\\ \\ }As noted before, the main idea of the proof is a\nmodification of the proof of \\cite[Lemma 5]{AFGJL} using ideas from \\cite{J}.\n\n\\medskip\n\n\\noindent We will need to use the following result, and refer the reader to\n\\cite[Proposition II.5.1]{DGZ} and \\cite{L} for a proof.\n\n\\medskip\n\n\\begin{proposition}\nLet $Z$ be a Banach space. The following assertions are equivalent.\n\n(a).$\\ Z$ admits a $C^{p}$-smooth, Lipschitz bump function.\n\n\\smallskip\n\n(b). There exist numbers $a,b>0$ and a Lipschitz function $\\psi:Z\\rightarrow\n\\lbrack0,\\infty)$ which is $C^{p}$-smooth on $Z\\setminus\\{0\\}$, homogeneous\n(that is $\\psi(tx)=|t|\\psi(x)$ for all $t\\in\\mathbb{R},x\\in Z$), and such that\n$a\\Vert\\cdot\\Vert\\leq\\psi\\leq b\\Vert\\cdot\\Vert$.\n\\end{proposition}\n\n\\medskip\n\nFor such a function $\\psi$, the set $A=\\{z\\in Z:\\psi(z)\\leq1\\}$ is what we\ncall a $C^{p}$-smooth, Lipschitz \\textbf{starlike body}, and the Minkowski\nfunctional of this body, $\\mu_{A}(z)=\\inf\\{t>0:(1\/t)z\\in A\\}$, is precisely\nthe function $\\psi$ (see \\cite{AD} and the references therein for further\ninformation on starlike bodies and their Minkowski functionals).\n\nWe will denote the open ball of center $x$ and radius $r$, with respect to the\nnorm $\\Vert\\cdot\\Vert$ of $X$, by $B(x,r).$ If $A$ is a bounded starlike body\nof $X$, we define the \\textbf{open}\\textit{\\ }$A$\\textit{-}\\textbf{pseudoball}\nof center $x$ and radius $r$ as%\n\n\\[\nB_{A}(x,r):=\\{y\\in X:\\mu_{A}(y-x)<r\\}.\n\\]\n\n\nAccording to Proposition 1 and the preceding remarks, because $X$ has a\n$C^{p}$-smooth, Lipschitz bump function, there is a bounded starlike body\n$A\\subset X$ (which we fix for the remainder of the proof) whose Minkowski\nfunctional $\\mu_{A}=\\psi$ is Lipschitz and $C^{p}$-smooth on $X\\setminus\n\\{0\\}$, and there is a number $M\\geq1$ such that $\\frac{1}{M}\\Vert x\\Vert\n\\leq\\mu_{A}(x)\\leq M\\Vert x\\Vert$ for all $x\\in X$, and $\\Vert\\mu_{A}^{\\prime\n}(x)\\Vert\\leq M$ for all $x\\in X\\setminus\\{0\\}$. Notice that in this case we\nhave,\n\\begin{equation}\nB(x,\\frac{r}{M})\\subseteq B_{A}(x,r)\\subseteq B(x,Mr)\n\\end{equation}\nfor every $x\\in X$, $r>0$. This fact will sometimes be used implicitly in what follows.\n\nFor the proof, we shall first define a function $\\overline{f}:E^{\\infty\n}\\rightarrow\\mathbb{R},$ then a map $\\Psi:X\\rightarrow E^{\\infty},$ and\nfinally our desired function $K$ will be given by $K=\\overline{f}\\circ\\Psi. $\n\nTo begin the proof, first note that as $f$ is real-valued and $Y$ is convex,\nby \\cite[Proposition 2.2.1 (i)]{BL} $f$ can be uniformly approximated by a\nLipschitz map, and so we may and do suppose that $f$ is Lipschitz with\nconstant $\\eta$. Using an infimal convolution, we extend $f$ to a Lipschitz\nmap $F$ on $X$ with the same constant $\\eta$ by defining, $F\\left(  x\\right)\n=\\inf\\left\\{  f\\left(  y\\right)  +\\eta\\left\\Vert x-y\\right\\Vert :y\\in\nY\\right\\}  .$\n\nLet $\\varepsilon>0$ and $r\\in\\left(  0,\\varepsilon\/3M\\eta\\right)  .$ We shall\nrequire the main construction from \\cite{J} (see also \\cite{FWZ}), and for the\nsake of completeness we outline this here. Let $\\left\\{  h_{i}\\right\\}\n_{i=1}^{\\infty}$ be a dense sequence in $B_{X},$ and $\\varphi_{i}\\in\nC^{\\infty}\\left(  \\mathbb{R},\\mathbb{R}^{+}\\right)  $ with $\\int_{\\mathbb{R}%\n}\\varphi_{i}=1$ and support$\\left(  \\varphi_{i}\\right)  \\in\\left[\n-\\frac{\\varepsilon}{6\\eta2^{i}},\\frac{\\varepsilon}{6\\eta2^{i}}\\right]  .$\n\n\\medskip\n\n\\noindent Now we define functions $g_{n}:X\\rightarrow\\mathbb{R}$ by,%\n\n\\[\ng_{n}\\left(  x\\right)  =\\int_{\\mathbb{R}^{n}}F\\left(  x-\\sum_{i=1}^{n}%\nt_{i}h_{i}\\right)  \\prod_{i=1}^{n}\\varphi_{i}\\left(  t_{i}\\right)  dt,\n\\]\n\n\n\\noindent where the integral is $n$-dimensional Lebesgue measure.\n\n\\medskip\n\n\\noindent It is proven in \\cite{J} that the following hold:\n\n\\medskip\n\n\\begin{enumerate}\n\\item There exists $g$ with $g_{n}\\rightarrow g$ uniformly on $X,$\n\n\\item $\\left\\vert g-F\\right\\vert <\\varepsilon\/3$ on $X,$\n\n\\item The map $g$ is $\\eta$-Lipschitz,\n\n\\item The map $g$ is uniformly G\\^{a}teaux differentiable\n\\end{enumerate}\n\n\\medskip\n\n\\noindent Next, following \\cite[Lemma 5]{AFGJL}, let $\\varphi:\\mathbb{R}%\n\\rightarrow\\left[  0,1\\right]  $ be a $C^{\\infty}$-smooth function such that\n$\\varphi\\left(  t\\right)  =1$ if $\\left\\vert t\\right\\vert <1\/2$,\n$\\varphi\\left(  t\\right)  =0$ if $\\left\\vert t\\right\\vert >1,$ $\\varphi\n^{\\prime}([0,\\infty))\\subseteq\\left[  -3,0\\right]  ,$ $\\varphi(-t)=\\varphi(t)$.\n\n\\medskip\n\n\\noindent Let us define a function G\\^{a}teaux differentiable on $X,$ and\n$C^{p}$-smooth on $E_{n},$ by%\n\n\\[\nF_{n}\\left(  x\\right)  =\\frac{(a_{n})^{n}}{c_{n}}\\int_{E_{n}}g(x-y)\\varphi\n(a_{n}\\mu_{A}\\left(  y\\right)  )dy\n\\]\nwhere\n\\[\nc_{n}=\\int_{E_{n}}\\varphi\\left(  \\mu_{A}\\left(  y\\right)  \\right)  dy,\n\\]\nand (keeping in mind (2.1) and $\\left(  3\\right)  $) we have chosen the\nconstants $a_{n}$ large enough that\n\\begin{equation}\n\\sup_{x\\in E_{n}}\\left\\vert F_{n}\\left(  x\\right)  -g\\left(  x\\right)\n\\right\\vert <\\frac{\\varepsilon}{6}2^{-n}.\n\\end{equation}\n\n\n\\noindent As pointed out to us by P. H\\'{a}jek, since $g$ is Lipschitz and\nuniformly G\\^{a}teaux differentiable, by \\cite[Lemma 4]{HJ} for each $h$ the\nmap $x\\rightarrow D_{h}g\\left(  x\\right)  $ is uniformly continuous. From\nthis, the Lipschitzness of $g,$ and compactness of $B_{E_{n}},$ we can choose\nthe $a_{n}$ larger if need be so that for all $h\\in B_{E_{n}}$ we have,%\n\n\\begin{equation}\n\\sup_{x\\in E_{n}}\\left\\vert D_{h}F_{n}\\left(  x\\right)  -D_{h}g\\left(\nx\\right)  \\right\\vert <\\frac{\\eta}{2}2^{-n}.\n\\end{equation}\n\n\n\\smallskip\n\n\\noindent Note that for any $x,x^{\\prime}\\in X$,%\n\n\\begin{align*}\n\\left\\vert F_{n}\\left(  x\\right)  -F_{n}\\left(  x^{\\prime}\\right)\n\\right\\vert  &  \\leq\\frac{(a_{n})^{n}}{c_{n}}\\int_{E_{n}}\\left\\vert\ng(x-y)-g\\left(  x^{\\prime}-y\\right)  \\right\\vert \\varphi(a_{n}\\mu_{A}\\left(\ny\\right)  )dy\\\\\n& \\\\\n&  \\leq\\eta\\left\\Vert x-x^{\\prime}\\right\\Vert \\frac{(a_{n})^{n}}{c_{n}}%\n\\int_{E_{n}}\\varphi(a_{n}\\mu_{A}\\left(  y\\right)  )dy=\\eta\\left\\Vert\nx-x^{\\prime}\\right\\Vert ,\n\\end{align*}\n\n\n\\noindent that is, $F_{n}$ is $\\eta$-Lipschitz.\n\n\\medskip\n\n\\noindent We next define a sequence of G\\^{a}teaux differentiable functions\n$\\overline{f}_{n}:X\\rightarrow\\mathbb{R},$ $C^{p}$-smooth on $E_{n},$ as\nfollows. Put $\\bar{f}_{0}=f\\left(  0\\right)  ,$ and supposing that\n$\\overline{f}_{0},...,\\overline{f}_{n-1}$ have been defined, we set\n\n\\medskip%\n\n\\[\n\\bar{f}_{n}\\left(  x\\right)  =F_{n}\\left(  x\\right)  +\\bar{f}_{n-1}\\left(\nP_{n-1}\\left(  x\\right)  \\right)  -F_{n}\\left(  P_{n-1}\\left(  x\\right)\n\\right)  .\n\\]\n\n\n\\medskip\n\n\\noindent One can verify by induction, using $\\left\\Vert P_{n}\\right\\Vert\n\\leq1,\\ \\left(  2.2\\right)  $ and $\\left(  2.3\\right)  ,$ that,\n\n\\medskip\n\n(i). The $\\bar{f}_{n}$ are G\\^{a}teaux differentiable, the restriction of\n$\\bar{f}_{n}$ to $E_{n}$ is $C^{p}$-smooth, and $\\bar{f}_{n}$ extends $\\bar\n{f}_{n-1}$,\n\n\\medskip\n\n(ii).\\ $\\sup_{x\\in E_{n}}\\left\\vert \\bar{f}_{n}\\left(  x\\right)  -g\\left(\nx\\right)  \\right\\vert <\\frac{\\varepsilon}{3}\\left(  1-\\frac{1}{2^{n}}\\right)\n$,\n\n\\medskip\n\n(iii). $\\sup_{x\\in E_{n}}\\left\\vert D_{h}\\bar{f}_{n}\\left(  x\\right)\n-D_{h}g\\left(  x\\right)  \\right\\vert \\leq\\eta\\left(  1-\\frac{1}{2^{n}}\\right)\n$, for all $h\\in B_{E_{n}}.$\n\n\\medskip\n\n\\noindent We now define the map $\\bar{f}:E^{\\infty}\\rightarrow\\mathbb{R}$ by\n\n\\medskip%\n\n\\[\n\\bar{f}\\left(  x\\right)  =\\lim_{n\\rightarrow\\infty}\\bar{f}_{n}\\left(\nx\\right)  .\n\\]\n\n\n\\noindent For $x\\in E^{\\infty}=\\cup_{n}E_{n},$ define $n_{x}\\equiv\\min\\left\\{\nn:x\\in E_{n}\\right\\}  ,$ and note that we have for any $m\\geq n_{x}$,%\n\n\\begin{equation}\n\\bar{f}\\left(  x\\right)  =\\lim_{n\\rightarrow\\infty}\\bar{f}_{n}\\left(\nx\\right)  =\\bar{f}_{m}\\left(  x\\right)  .\n\\end{equation}\n\n\n\\medskip\n\n\\noindent In particular, for any $n,$ $\\overline{f}\\mid_{E_{n}}=\\overline\n{f}_{n}.$ One can verify using $\\left(  2.4\\right)  ,$ (i), (ii), and (iii)\nabove that $\\bar{f}$ has the following properties:\n\n\\medskip\n\n(i)$^{\\prime}$. The restriction of $\\bar{f}$ to every subspace $E_{n}$ is\n$C^{p}$-smooth,\n\n\\medskip\n\n(ii)$^{\\prime}$. $\\sup_{x\\in E^{\\infty}}\\left\\vert \\bar{f}\\left(  x\\right)\n-g\\left(  x\\right)  \\right\\vert \\leq\\frac{\\varepsilon}{3}.$\n\n\\medskip\n\n(iii)$^{\\prime}$. $\\sup_{x\\in E_{n}}\\left\\vert D_{h}\\bar{f}\\left(  x\\right)\n-D_{h}g\\left(  x\\right)  \\right\\vert \\leq\\eta$, for all $h\\in B_{E_{n}}.$\n\n\\medskip\n\n\\noindent The proof now closely follows \\cite[Lemma 5]{AFGJL}, and we provide\nsome of the details for the sake of completeness.\n\n\\medskip\n\n\\noindent Next let $x=\\sum_{n}x_{n}e_{n}\\in X$ and define the maps\n\\[\n\\chi_{n}\\left(  x\\right)  =1-\\varphi\\left[  \\frac{\\mu_{A}\\left(\nx-P_{n-1}\\left(  x\\right)  \\right)  }{r}\\right]  ,\n\\]\nand\n\\[\n\\Psi\\left(  x\\right)  =\\sum_{n}\\chi_{n}\\left(  x\\right)  x_{n}e_{n}.\n\\]\n\n\nFor any $x_{0},$ because $P_{n}\\left(  x_{0}\\right)  \\rightarrow x_{0}$ and\nthe $\\Vert P_{n}\\Vert$ are uniformly bounded, there exist a neighbourhood\n$N_{0}$ of $x_{0}$ and an $n_{0}=n_{x_{0}}$ so that $\\chi_{n}\\left(  x\\right)\n=0$ for all $x\\in N_{0}$ and $n\\geq n_{0}$ and so $\\Psi\\left(  N_{0}\\right)\n\\subset E_{n_{0}}.$ Thus, $\\Psi:X\\rightarrow E^{\\infty}$ is a $C^{p}$-smooth\nmap whose range is locally contained in the finite dimensional subspaces\n$E_{n}$. Using the fact that $\\left\\{  e_{n}\\right\\}  $ is unconditional with\nconstant $C=1,$ one can show that (see \\cite[Fact 7]{AFGJL})\n\\begin{equation}\n\\left\\Vert x-\\Psi\\left(  x\\right)  \\right\\Vert <Mr.\n\\end{equation}\n\n\nWe next consider the derivative of $\\Psi.$ A straightforward calculation,\nusing the facts:\\ $|\\varphi^{\\prime}(t)|\\leq3,\\;\\Vert\\mu_{A}^{\\prime}\\left(\nx\\right)  \\Vert\\leq M$ and $\\Vert(I-P_{n-1})^{\\prime}(x)\\Vert\\leq2$ for all\n$x,t,$ verifies that $\\left\\Vert \\chi_{n}^{\\prime}\\left(  x\\right)\n\\right\\Vert \\leq6Mr^{-1}.$ Also, since $\\left(  \\chi_{n}\\left(  x\\right)\nx_{n}\\right)  ^{\\prime}=\\chi_{n}^{\\prime}\\left(  x\\right)  x_{n}+\\chi\n_{n}\\left(  x\\right)  e_{n}^{\\ast},$ it follows that%\n\n\\begin{equation}\n\\Psi^{\\prime}\\left(  x\\right)  \\left(  \\cdot\\right)  =\\sum_{n}\\chi_{n}%\n^{\\prime}\\left(  x\\right)  \\left(  \\cdot\\right)  x_{n}e_{n}+\\sum_{n}\\chi\n_{n}\\left(  x\\right)  e_{n}e_{n}^{\\ast}\\left(  \\cdot\\right)  .\n\\end{equation}\n\n\n\\smallskip\n\nNow, using (2.7), the estimate for $\\left\\Vert \\chi_{n}^{\\prime}\\right\\Vert $\nabove, and again using the fact that $\\left\\{  e_{n}\\right\\}  $ is\nunconditional with constant $C=1$, it is shown in \\cite[Fact 7]{AFGJL} that\nfor all $x\\in X,$\n\\[\n\\left\\Vert \\Psi^{\\prime}\\left(  x\\right)  \\right\\Vert \\leq8M^{2}.\n\\]\n\n\n\\medskip\n\nWe define $K\\left(  x\\right)  =\\bar{f}\\left(  \\Psi\\left(  x\\right)  \\right)\n.$ Note that $K$ is $C^{p}$-smooth on $X,$ being the composition of $C^{p}%\n$-smooth maps, and noting that $\\Psi$ maps locally into some $E_{n}.$ Now, for\n$x\\in X,$ by (ii)$^{\\prime}$, choice of $r,$ using $\\left(  2\\right)  $,\n$\\left(  2.6\\right)  $ and that $g$ is Lipschitz with constant $\\eta$, we have,%\n\n\\begin{align*}\n\\left\\vert F\\left(  x\\right)  -K\\left(  x\\right)  \\right\\vert  &\n\\leq\\left\\vert g\\left(  x\\right)  -F\\left(  x\\right)  \\right\\vert +\\left\\vert\ng\\left(  x\\right)  -g\\left(  \\Psi\\left(  x\\right)  \\right)  \\right\\vert \\\\\n& \\\\\n&  +\\left\\vert \\bar{f}\\left(  \\Psi\\left(  x\\right)  \\right)  -g\\left(\n\\Psi\\left(  x\\right)  \\right)  \\right\\vert \\\\\n& \\\\\n&  \\leq\\varepsilon\/3+\\eta Mr+\\varepsilon\/3<\\varepsilon.\n\\end{align*}\n\n\n\\medskip\n\n\\noindent In particular, since $F\\mid_{Y}=f,$ we have for $y\\in Y$ that\n$\\left\\vert f\\left(  y\\right)  -K\\left(  y\\right)  \\right\\vert <\\varepsilon.$\n\n\\smallskip\n\n\\noindent Finally we consider $K^{\\prime}\\left(  x\\right)  =\\overline\n{f}^{\\prime}\\left(  \\Psi\\left(  x\\right)  \\right)  \\Psi^{\\prime}\\left(\nx\\right)  .$ Fix $x\\in X,$ and $h\\in X$ with $\\left\\Vert h\\right\\Vert\n\\leq\\frac{1}{8M^{2}}.$ Note that $\\left\\Vert \\Psi^{\\prime}\\left(  x\\right)\n\\left(  h\\right)  \\right\\Vert \\leq1.$\n\n\\smallskip\n\n\\noindent Now $\\Psi$ maps a neighbourhood of $x$ into $E_{n_{x}},$ and hence\nalso $\\Psi^{\\prime}\\left(  x\\right)  \\left(  h\\right)  \\in E_{n_{x}};$ in\nparticular, $\\Psi^{\\prime}\\left(  x\\right)  \\left(  h\\right)  \\in B_{E_{n_{x}%\n}}.$ Now using this fact, $\\left(  3\\right)  ,\\ $(iii)$^{\\prime},$ and our\nestimates above, we have,%\n\n\\begin{align*}\n\\left\\vert K^{\\prime}\\left(  x\\right)  \\left(  h\\right)  \\right\\vert  &\n=\\left\\vert \\bar{f}^{\\prime}\\left(  \\Psi\\left(  x\\right)  \\right)  \\left(\n\\Psi^{\\prime}\\left(  x\\right)  \\left(  h\\right)  \\right)  \\right\\vert \\\\\n& \\\\\n&  =\\left\\vert \\bar{f}_{n_{x}}^{\\prime}\\left(  \\Psi\\left(  x\\right)  \\right)\n\\left(  \\Psi^{\\prime}\\left(  x\\right)  \\left(  h\\right)  \\right)  \\right\\vert\n\\\\\n& \\\\\n&  <\\left\\vert D_{\\left(  \\Psi^{\\prime}\\left(  x\\right)  \\left(  h\\right)\n\\right)  }g\\left(  \\Psi\\left(  x\\right)  \\right)  +\\eta\\right\\vert \\\\\n& \\\\\n&  \\leq\\eta+\\eta=2\\eta.\n\\end{align*}\n\n\n\\medskip\n\n\\noindent As $K^{\\prime}\\left(  x\\right)  :X\\rightarrow\\mathbb{R}$ is\ncontinuous and linear, from the above estimate we have, $\\left\\vert K^{\\prime\n}\\left(  x\\right)  \\left(  h\\right)  \\right\\vert \\leq16M^{2}\\eta$ for all\n$h\\in B_{X},$ and hence $\\left\\Vert K^{\\prime}\\left(  x\\right)  \\right\\Vert\n\\leq16M^{2}\\eta$ for all $x\\in X.$\n\n\\smallskip\n\n\\noindent This proves the first statement of the theorem. For the second\nstatement, we observe that for Lipschitz functions $f:X\\rightarrow Z$ into an\narbitrary Banach space $Z,$ we do not require the real-valued assumption on\n$f$ that was used to apply the result from \\cite{BL}, and the methods of\n\\cite{J} and \\cite{AFGJL} apply equally well to arbitrary Banach space valued\nmaps. For Lipschitz functions $f:Y\\rightarrow\\mathbb{R},$ we can directly\nextend $f$ to a Lipschitz map on $X$ via an infimal convolution as before.\nHence the theorem follows with $C_{0}=16M^{2}.\\ \\ \\blacksquare$\n\n\\medskip\n\n\\noindent We have the following characterization slightly extending the one\nfrom \\cite{F}.\n\n\\begin{corollary}\nLet $X$ have unconditional basis. Then the following are equivalent.\n\n\\smallskip\n\n\\begin{enumerate}\n\\item $X$ admits a Lipschitz, $C^{p}$-smooth bump function.\n\n\\medskip\n\n\\item For every convex subset $Y\\subset X,$ uniformly continuous map\n$f:Y\\rightarrow\\mathbb{R},$ and $\\varepsilon>0,$ there exists a Lipschitz,\n$C^{p}$-smooth map $K:X\\rightarrow\\mathbb{R}$ with $\\left\\vert f-K\\right\\vert\n<\\varepsilon$ on $Y.$\n\n\\item For every subset $Y\\subset X,$ Lipschitz function $f:Y\\rightarrow\n\\mathbb{R},$ and $\\varepsilon>0,$ there exists a Lipschitz, $C^{p}$-smooth map\n$K:X\\rightarrow\\mathbb{R}$ with $\\left\\vert f-K\\right\\vert <\\varepsilon$ on\n$Y.$\n\n\\item For every Banach space $Z,$ Lipschitz map $f:X\\rightarrow Z,$ and\n$\\varepsilon>0,$ there exists a Lipschitz, $C^{p}$-smooth map $K:X\\rightarrow\nZ$ with $\\left\\Vert f-K\\right\\Vert <\\varepsilon$ on $X.$\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\nThat $\\left(  1\\right)  \\Rightarrow\\left(  2\\right)  ,\\left(  3\\right)  ,$ and\n$\\left(  4\\right)  $ is Theorem 1. For $\\left(  2\\right)  \\Rightarrow\\left(\n1\\right)  ,$ choose $Y=X,$ and $f=\\left\\Vert \\cdot\\right\\Vert .$ Let\n$K:X\\rightarrow\\mathbb{R}$ be a $C^{p}$-smooth, Lipschitz map with $\\left\\vert\nf-K\\right\\vert <1$ on $X.$ Let $\\xi:\\mathbb{R\\rightarrow R}$ be $C^{\\infty}%\n$-smooth and Lipschitz with, $\\xi\\left(  t\\right)  =1$ if $t\\leq1$ and\n$\\xi\\left(  t\\right)  =0$ if $t\\geq2.$ Then $b=\\xi\\circ K$ is a $C^{p}%\n$-smooth, Lipschitz map with $b\\left(  0\\right)  =1$ and $b\\left(  x\\right)\n=0$ when $\\left\\Vert x\\right\\Vert \\geq3.$ The remaining implications are similar.\n\\end{proof}\n\n\\medskip\n\n\\textbf{Remark }The Lipschitz constant of $K$ obtained for the second\nstatement of Theorem 1 is not the best possible. By using better derivative\nestimates$,$ one can show that for any $\\delta>0,$ we may arrange $\\left\\Vert\nK^{\\prime}\\right\\Vert \\leq\\left(  \\eta+\\delta\\right)  \\left(  2\\left(\n2+\\delta\\right)  M^{2}+1\\right)  .$ This should be compared with the recent\nresult in \\cite{AFLR}, where it is shown in particular that for separable\nHilbert spaces $X$, any Lipschitz, real-valued function on $X$ can be\nuniformly approximated by $C^{\\infty}$ smooth functions with Lipschitz\nconstants arbitrarily close to the Lipschitz constant of $f.$ It is open\nwhether such a result holds outside the Hilbert space setting.\n\n\\medskip\n\n{\\small Acknowledgement\\ \\ The author wishes to thank D. Azagra and J.\nJaramillo for bringing to his attention the problems addressed in this note\nduring the rsme-ams 2003 conference in Sevilla. Also we want to thank P.\nH\\'{a}jek for pointing out \\cite[Lemma 4]{HJ} which enabled us to correct and\nsimplify an earlier version of this corrigendum. }\n\n\\medskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduciton}\n\nThere are increasing interests in seeking  novel electro-optic\nmaterials analogous to the famous Sr$_{x}$Ba$_{1-x}$Nb$_{2}$O$_6$\n(SBN, $0.32\\leq x\\leq 0.75$) that  belongs to the\n(A$_1)_2$(A$_2)_4$(B$_1)_2$(B$_2)_8$O$_{30}$ tetragonal tungsten\nbronze (TTB) structure  with the non-centrosymmetry of P4{\\it bm} at\nroom temperature (see  Fig.\\protect\\ref{Fig0})\n\\protect\\cite{Podlozhenov,Chernaya1,Chernaya2} and shows excellent\nelectro-optic and pyroelectric properties \\protect\\cite{Glass} and typical\nrelaxor behaviors.\\protect\\cite{Kleemann1,Kleemann2} Recently,\nCa$_{x}$Ba$_{1-x}$Nb$_{2}$O$_6$ (CBN, $0.2\\leq x\\leq 0.4$) single\ncrystals have received considerable critical attention due to their\nexcellent electro-optic properties and  higher working temperature\nin comparison with SBN.\\protect\\cite{Ebser,Burianek,Muehlberg,Qi,Song} The\nferroelectricity of CBN is first discovered  by Smolenskii et\nal..\\protect\\cite{Smolenskii} CBN alloys also have  the ferroelectric TTB\nstructure at room temperature, however,  contrary to Sr in SBN where\nSr ions randomly occupy both the two large A$_1$ and A$_2$ sites in\nthe tungsten bronze framework of NbO$_6$ octahedra, smaller Ca ions\nin CBN only occupy the A$_1$ sites in the structure. It has been\ndemonstrated experimentally that smaller A$_1$ site is almost\nexclusively occupied by Ca while  larger A$_2$ site is predominately\nby Ba in CBN single crystal with congruent melting composition\n($x$=0.28).\\protect\\cite{Graetsch1,Graetsch2} In ferroelctric CBN alloys,\nniobium atoms are displaced from the centers of their coordination\npolyhedra and shifted along the tetragonal $c$-axis, which is\nconsidered to be the origin of  spontaneous electric\npolarization.\\protect\\cite{Chernaya2, Abrahams}\n\nCurrently, most of investigations on CBN alloys are focused on the\ncrystal growth\\protect\\cite{Ebser,Song}, dielectric,\\protect\\cite{Qi,Niemer}\nferroelectric,\\protect\\cite{Qi} optic,\\protect\\cite{Ebser2,Heine,Sheng} and elastic\n\\protect\\cite{Pandey,Pandey2,Pandey3,Suzuki} properties  of CBN single\ncrystal with congruent melting composition $x$=0.28. Despite these\ninvestigations on CBN, there is still a lack of full understanding\non the basic issues in CBN alloys. One important problem is that CBN\nalloys are   relaxors (such as their isostructural compounds SBN\nalloys\\protect\\cite{Kleemann2,Blinc,Banys,Miga}), which is characterized by\na broad peak of dielectric susceptibility with strong frequency\ndispersion in the radio frequency over a large temperature range\n\\protect\\cite{Miga,Levstik,Fu1,Fu2} and  smear ferroelectric phase\ntransition\\protect\\cite{Fu1,Taniguchi} without an obvious heat capacity\npeak,\\protect\\cite{Moriya} or they are  ferroelectrics (such as the normal\nferroelectric BaTiO$_3$), which have a well-defined\npara-ferroelectric  thermal phase transition at Curie point $T_{\\rm\nc}$ \\protect\\cite{Fu3} but show polarization precursor dynamics before\ntransition into the ferroelectric\nphase.\\protect\\cite{Burns,Tai,Ziebinska,Ko,Dulkin,Pugachev} Since CBN is an\nisostructural compound of SBN that typical relaxor phenomena have\nbeen clearly demonstrated,\\protect\\cite{Banys,Miga} it is naturally expected\nthat CBN should show the relaxor behaviors such as a broad peak of\ndielectric susceptibility with strong frequency dispersion  due to\nthe  dynamics of polar-nano-regions (PNRs) occurring in the\nparaelectric mother phase. Following this scenario, several attempts\nhave been made to confirm the relaxor behaviors in\nCBN.\\protect\\cite{Pandey,Pandey2,Pandey3,Suzuki} From the investigations on\nlattice strain and thermal expansion of CBN single crystal with\ncongruent melting composition $x$=0.28, Pandey et al. found the\ndeviation from the linear temperature dependence of lattice strain\nand the anomalous thermal expansion in this single crystal.\\protect\\cite{Pandey} They\nattribute these anomalous elastic behaviors to the relaxor phenomena\noccurring in CBN crystal. They further suggest  Burns\ntemperature \\protect\\cite{Burns} $T_{\\rm B}$ to be 1100 K, which\ncharacterizes the initiation of dynamic PNRs, and the intermediate\ntemperature $T^*$ to  be 800 K,  which indicates the beginning of\nPNRs freezing,\\protect\\cite{Roth,Dkhil}  for CBN with $x=0.28$.\\protect\\cite{Pandey}\nOn the other hand, in  a recent Brillouin scattering study, $T_{\\rm\nB}$ and $T^*$ are proposed to be approximately 790 K and 640 K,\nrespectively, for CBN with the same composition.\\protect\\cite{Suzuki} The\ndifference in     $T_{\\rm B}$ and $T^*$ values estimated by\ndifferent techniques for a same compound is surprisingly large.\nApparently, the existence and  exact values of $T_{\\rm B}$ and $T^*$\nin CBN need to be clarified in further investigations.\n\nIt should be noticed that the existence of $T_{\\rm B}$ and $T^*$ is\nnot the characteristic properties of relaxor. In a normal\nferroelectric such as BaTiO$_3$, the existence of these two\ncharacteristic temperatures have been clearly\ndemonstrated.\\protect\\cite{Burns,Dulkin} In\n early investigations, Burns et al.\\protect\\cite{Burns} have demonstrated that\nthe temperature dependence of the optic index of refraction, $n(T)$,\ndeviates from the high temperature extrapolated value between\n$T_{\\rm c}$ and $T_{\\rm c} + 180$ K in the paraelectric phase of\nBaTiO$_3$ crystal. This deviation is commonly accepted to be due to\nthe local polarization precursor dynamics  before the\npara-ferroelectric phase transition in BaTiO$_3$, thus the\ntemperature where this deviation occurs is generally called as Burns\ntemperature $T_{\\rm B}$, which is widely used to characterize the\nappearance of PNRs in the paraelectric phase of ferroelectric or\nrelaxor. In additional to the optical\nmeasurements,\\protect\\cite{Burns,Ziebinska,Pugachev} various other\nexperiments such as pulsed X-Ray laser\nmeasurements,\\protect\\cite{Tai,Namikawa} acoustic emission\\protect\\cite{Dulkin} and\nBrillouin light scattering\\protect\\cite{Ko} all show the existence of local\npolarization in BaTiO$_3$ before transition into the ferroelectric\nphase. Theoretically, it has been proposed that local dynamical\nprecursor domains are common to perovskite ferroelectrics, and these\npolar domains grow upon approaching $T_{\\rm c}$ to coalesce into a\nhomogeneously polarized state at $T_{\\rm\nc}$.\\protect\\cite{Bussmann-Holder1,Bussmann-Holder2} Similarly, PNRs develop\nfrom $T_{\\rm B}$ in relaxor, and cause the very strong frequency\ndispersion of dielectric susceptibility. In contrast to relaxor,\nprecursor domains in the perovskite ferroelectric such as BaTiO$_3$\ndo not need to give rise to a frequency-dependent dielectric\nresponse in the radio frequency.\\protect\\cite{Bussmann-Holder1}\n\nFurthermore, a sharp peak of heat capacity has been observed at the\npara-ferroelectric phase transition in CBN  crystal with\n$x=0.311$.\\protect\\cite{Muehlberg} Also, in the dielectric measurements on\nCBN single cyrstal with congruent melting composition $x$=0.28, it\nseems that the para-ferroelectric phase transition temperature is\nnot dependent on  frequency.\\protect\\cite{Qi} All these findings in CBN  are\nin sharp contrast to the characteristic properties of relaxor such\nas  smear phase transition \\protect\\cite{Fu1,Taniguchi} without an evident heat\ncapacity peak,\\protect\\cite{Moriya} and very strong frequency dependence\nof dielectric responses in the radio frequency around a temperature,\n$T_{\\rm m}$, where the maximum of dielectric response occurs. These\nresults suggest that CBN alloys may be classified as a\nferroelectric with thermal phase transition associating with\npolarization precursor dynamics rather than a relaxor.\n\n\nIn this study, we firstly determined the solid solution limit of CBN\nferroelectric alloys. We then investigated the para-ferroelectric\nphase transition in these  alloys by the dielectric measurements and\nthe differential scanning calorimetry (DSC). Contrary to the relaxor\npicture what have been expected in previous\ninvestigations,\\protect\\cite{Pandey,Pandey2,Pandey3} we clearly demonstrated that CBN\nalloys can be classified as ferrolectrics with a first-order thermal\nphase transition associating with precursor dynamics over a large\ntemperature range above $T_{\\rm c}$ such as\nBaTiO$_3$.\\protect\\cite{Burns,Ziebinska,Pugachev} We also found that the\nlocal polarizations grown exponentially for a temperature range of\n$T_{\\rm c}<T<T_{\\rm c}+ 88\\sim140$ K on cooling in CBN alloys. These findings are\nof benefit in understanding the nature of ferroelectric phase\ntransition in  CBN crystals that are currently developed for the\nelectro-optic applications.\n\n\n\\section{Experimental}\n\nWe used a conventional solid-state reaction method to prepare  CBN\npolycrystals, which allows us to easily control composition of CBN\nsamples. To obtain dense ceramics for electrical measurements, CBN ceramics samples were fired\n3 hours at a temperature of 1693 K (approximately 50 K lower than\nits melting point\\protect\\cite{Muehlberg}). This firing allows us to obtain\nceramics samples with a relative density larger than 93 \\% of the\ndensity calculated by the structural data. Grain size in the\nceramics samples was observed to be  approximately 10$\\mu$m $\\sim$ $50\\mu$m by Scanning Electron Microscopy. To determine the solid\nsolution limit, ceramics samples with composition of $0.15\\leq\nx\\leq0.40$ were prepared and investigated. Phase formation and\nlattice parameters of CBN were determined by powder X-ray\ndiffraction technique using the Cu-K$\\alpha$ radiation\n($\\lambda=1.5418$ {\\AA}). Ceramics samples coated with gold electrode\nwere used for the dielectric measurements by using Agilent 4980A LCR\nmeter. The dielectric measurements were carried out in a\ntemperature range of 300 K - 870 K with a heating\/cooling rate of 2\nK\/min for the exact determination of   ferroelectric transition\ntemperature in CBN alloys. To confirm the thermal phase transition\nin CBN, we also performed DSC measurements by using SII DSC6220\ndifferential scanning calorimetry.\n\n\n\\section{Results and Discussions}\n\n\\subsection{Ferroelectric solid solution limit and  lattice parameters}\n\nTo determine the  solid solution limit of CBN ferroelectric alloys,\nwe have prepared the samples with  composition ranging from $x=0.15$ to\n$x=0.40$ at a firing temperature of 1693 K, and examined the phase\nformation by powder x-ray diffractions. As shown in\nFig.\\protect\\ref{fig1}, single phase of CBN with ferroelectric TTB structure\nis  only available in a composition range of $0.19\\leq x\\leq0.32$, beyond\nwhich CaNb$_2$O$_6$- or BaNb$_2$O$_6$-type phase with\nnon-ferroelectric orthorhombic structure was  found in the\ncompounds. The change of lattice parameters or Curie temperature\nwith composition also show a saturation for $x<0.19$ or $x>0.32$.\nThese facts indicate that single phase of CBN ferroelectric alloys\nis available only in a composition range of $0.19\\leq x\\leq0.32$.\n\nThe lattice parameters of CBN  alloys at room temperature were then\ndetermined by the method of least squares using  twelve reflections\nwith 2$\\theta>50$ degree. The results are shown in Fig.\\protect\\ref{fig2}. In\nCBN  alloys with  ferroelectric TTB structure, $a$-axis lattice\nis nearly unchanged with composition within the error range. In contrast, the polar\n$c$-axis lattice is shorten with an increase in Ca concentration,\nwhich results in a similar decrease of unit cell volume. This result\nis expectable because the ionic radii of Ca and Ba are 1.34 {\\AA}\nand 1.61 {\\AA},\\protect\\cite{Shannon} respectively, and Ca has a smaller\nionic radius. In spite of the large change in composition $x$ ($\\Delta\nx=0.13$), the corresponding change in the unit cell volume is\nsurprisingly small (approximately 0.65\\%). This situation is\ndifferent significantly from the case of substitution of Ca for Ba\nin Ba$_{1-x}$Ca$_x$TiO$_3$ perovskite oxides, which shows an\napproximate 3.5\\%-reduction in the unit cell volume at a same level\nof Ca-substitution.\\protect\\cite{Fu3} This fact seems to suggest that the\nstacking structure in TTB-type CBN crystal is mainly determined by\nthe framework of NbO$_6$ octahedra as shown in Fig.(\\protect\\ref{Fig0}).\n\n\n\\subsection{Thermal phase transition}\n\nTo investigate the para-ferroelectric phase transition in CBN\nalloys, we observed the variation of dielectric susceptibility as a\nfunction of temperature in a frequency range from 100 Hz to 100 kHz.\nThe results were shown in Fig.\\protect\\ref{fig3} and Fig.\\protect\\ref{fig4},\nrespectively. The dielectric loss for lower frequencies ($\\leq$ 1 kHz) was\nlarge for temperatures higher than approximately 500 K and was not shown here,  which is\nvery likely due to the thermal activation of vacancies in the\nsamples. The large dielectric loss made difficult to determine a\nreliable value of dielectric susceptibility for these low frequencies for\nthe high temperatures. However, as shown in Fig.\\protect\\ref{fig4}, we\nindeed observed that (1) Curies point is not dependent on frequency  ranging from 100 Hz to 100 kHz,\nand (2) the dielectric susceptibility measured at frequency between\n10 KHz and 100 kHz is nearly independent of frequency in the\ntemperature range of 300 K - 870 K. We thus consider that the\ndielectric susceptibility measured at frequency between 10 KHz and 100 kHz is free from\nthe effects of vacancies and reflects the intrinsic dielectric\nresponse of  CBN alloys, which will be used in the later analysis.\nApparently,  the dielectric response with frequency and\ntemperature in CBN alloys  is in sharp contrast to  that observed in\nrelaxor.\\protect\\cite{Fu1} Instead, the dielectric behaviors in CBN are\nsimilar to those observed in  normal ferroelectric such as\nBaTiO$_3$.\\protect\\cite{Fu3}\n\nThe temperature dependence of dielectric susceptibility shown in\nFig.\\protect\\ref{fig4} clearly indicates that the para-ferroelectric phase\ntransition has a thermal hysteresis.  Curies points on heating and\ncooling  have different values. The difference of $T_{\\rm c}$\nbetween heating and cooling is 12.4 K for $x=0.19$ and it increases\nto 25.2 K for $x=0.32$. This indicates that the substitution of Ba\nwith Ca in CBN enhances the thermal hysteresis of para-ferroelectric\nphase transition. On the other hand, this substitution lowers\nCuries point $T_{\\rm c}$ of CBN alloys as demonstrated clearly in\nFig.\\protect\\ref{fig3}.\n\nTo further confirm the nature of thermal phase transition in CBN\nferroelectric alloys, we also observed the change in enthalpy during\nthe phase transition by the DSC measurements. The results are shown\nin Fig.\\protect\\ref{fig5} Although our equipment doesn't allow us to\ndetermine the heat capacity of phase transition, we really\nobserved a change in enthalpy during the phase transition. DSC\nmeasurements also clearly  indicate that the phase transition has a thermal\nhysteresis. This result is in good agreements with that observed in\nthe dielectric measurements shown in Fig.\\protect\\ref{fig3}. Our result is\nalso well accorded with that reported for CBN crystal with $x=0.31$\n showing a sharp peak of heat capacity during the phase\ntransition.\\protect\\cite{Muehlberg}\n\nAll facts showed above indicate that CBN alloys undergo a thermal\nphase transition. It can be concluded that the para-ferroelectric\nphase transition in CBN alloys is of first-order and has a thermal\nhysteresis of 12.4 K to 25.2 K depending on  Ca-concentration.\n\n\n\\subsection{Precursor behaviors}\n\nIn a recent investigation on CBN single crystal with congruent\nmelting composition $x$=0.28, the birefringence has been\ndemonstrated to occur in a temperature region of  $T_{\\rm\nc}<T<T_{\\rm c}+ 40 $K.\\protect\\cite{Ebser2} The birefringence of CBN with\nTTB structure might have arisen due to the anisotropy of\npolarization fluctuation $\\langle P_c^2\\rangle- \\langle\nP_a^2\\rangle$. In  this case,  the birefringence\nreads\\protect\\cite{Lehnen,Takagi}\n \\begin{equation}\\label{eq1}\n\\Delta n=-(n^3_0\/2)( g_{11}-g_{12})\\langle P^2 \\rangle,\n \\end{equation}\nwhere $g_{11}$ and  $g_{12}$  are the electro-optic coefficients.\nThe occurrence of birefringence provides a direct evidence to the\nexistence of local polarization in the paraelectric phase of CBN\ncrystal.\n\nTo get further insights on the polarization precursor dynamics\noccurring for $T>T_{\\rm c}$, we performed an analysis on the\ntemperature dependence of dielectric susceptibilities of CBN alloys\nwithin its solid solution limit. As mentioned in the  above section,\nhere, we only used the data obtained at 100 kHz since these data are\nconsidered to be free from the effects of vacancies on the\ndielectric response. As shown in Fig.\\protect\\ref{fig6}(b), the dielectric\nsusceptibility $\\chi'$ obeys Curie law,\n\\begin{equation}\\label{eq2}\n    \\chi'=\\pm C\/(T-T_0)\\;\\; {\\rm for}\\;\\; T<T_{\\rm c}\\;\\;  {\\rm or}\\;\\; T>T_{\\rm\n    B},\n\\end{equation}\nwhere $C$ is Curie constant and $T_0$ is Curie-Weiss temperature.\nUpon cooling, the  dielectric susceptibility deviates from  Curie\nlaw at a characteristic temperature. Since the existence of Burns\ntemperature $T_{\\rm B}$ and  the intermediate temperature $T^*$ are\nnot well established for CBN,\\protect\\cite{Pandey,Suzuki} Here, we\ntentatively consider this characteristic temperature as Burns\ntemperature $T_{\\rm B}$. The values of $T_{\\rm B}$ were estimated to\nbe approximately $T_{\\rm c}$+88 K for $x=0.19$ and $T_{\\rm c}$+143 K\nfor $x=0.32$, respectively, and $T_{\\rm B}$ showed a nearly linear change\nwith composition as shown\nin the phase diagram (Fig.\\protect\\ref{fig9}(a)). The fitting parameters\nobtained from Curie law are summarized in Fig.\\protect\\ref{fig7}. It should\nbe noticed that Curie constant in the paralectric phase ($T>T_{\\rm\nB}$) of CBN alloys has a similar magnitude of that of BaTiO$_3$\nsingle crystal ($1.5 \\times10^5$ K).\\protect\\cite{Shiozaki}\n\n\n\nFor temperatures higher than $T_{\\rm B}$, the dielectric\nsusceptibility of  CBN alloys exactly follows Curie law, which\nindicates that the dielectric response in these high temperatures\ncan be considered to be due to the lattice dynamics. However, on\ncooling, deviation from Curie law  was observed from $T\\approx\nT_{\\rm B}$. As mentioned above, this deviation from Curie law of the\ndielectric susceptibility can be reasonably attributed to the\npolarization precursors dynamics occurring before para-ferroelectric\nphase transition in CBN alloys. Therefore, it can be considered that\nthe total dielectric susceptibility is contributed by  both lattice\nand precursor dynamics for $T_{\\rm c}<T<T_{\\rm B}$. The contribution\nof lattice dynamics to dielectric responses can be estimated from\nCurie law, here, we denote this part of dielectric susceptibility as\n$\\chi'_{\\rm Curie}$. The contribution of  precursor dynamics to the\ndielectric response was then estimated by subtracting $\\chi'_{\\rm\nCurie}$ from the total dielectric susceptibility $\\chi'$, and has\nthe form of $\\Delta\\chi'=\\chi'-\\chi'_{\\rm Curie}$. The temperature\nvariation of $\\Delta\\chi'$ is shown in Fig.\\protect\\ref{fig6}(a) and (c) for\nthe two end compositions of CBN alloys. Upon cooling, $\\Delta\\chi'$\nincreases rapidly, for example, for $x=0.32$, it increases from\nnearly zero at $T_{\\rm B}$ to a large value that is about 34 \\% of\nthe total dielectric response (2240) at $T_{\\rm c}$. For another end\ncomposition $x=0.19$ , precursor dynamics  contributes approximately\n25\\% of the total dielectric response at $T_{\\rm c}$. As shown in\nFig.\\protect\\ref{fig8}(a), Ca-substitution  enhances the contribution of\nprecursor dynamics to the total dielectric response in CBN alloys.\n\n\nSurprisingly, it was found that $\\Delta\\chi'$ increases\nexponentially upon cooling from $T_{\\rm B}$ to $T^+_{\\rm c}$. This\ncan be clearly seen in Fig.\\protect\\ref{fig6}(c) where the {\\it y}-axis was\nplotted in a logarithmic scale. The data of CBN alloys within the\nsolid solution limit can be well analyzed by the following\nexponential law,\n \\begin{equation}\\label{eq3}\n    \\Delta\\chi'=\\Delta\\chi'_{T_{\\rm c}}e^{-k_B(T-T_{\\rm c})\/E_a}\n     \\end{equation}\nwhere $k_B$ is Boltzmann constant, $\\Delta\\chi'_{T_{\\rm c}}$ and\n$E_a$ are constants. It should  be noticed that the exponential law of  dielectric response\nwith temperature also has been observed in Pb-based ferroelectric\noxides in our recent investigation.\\protect\\cite{Zhao} The obtained\nparameters for equation (\\protect\\ref{eq3}) were shown in Fig.\\protect\\ref{fig8}(a)\nand (b).\n\nFrom textbook, we know the relation of polarization $P$ and\ndielectric susceptibility $\\chi'$,\n \\begin{equation}\\label{eq4}\n    P=\\chi'\\epsilon_0E,\n     \\end{equation}\nwhere $E$ is an electric field and $\\epsilon_0$ is the dielectric\npermittivity  of vacuum. With the combination of this relationship and\nthe exponential law  shown in equation (\\protect\\ref{eq3}), it can be\npredicted that  local polarizations grow exponentially upon cooling\nfrom $T_{\\rm B}$ to $T^+_{\\rm c}$ in CBN alloys. Here, $E_a$ is\nconsidered to be thermal activation energy required for the  local\npolarization growth before  transition into the ferroelectric phase.\n$E_a$ was evaluated to be 2.2 mev $\\sim$ 3 mev for CBN alloys, which\ncorresponds to a temperature of 25 K $\\sim$ 30 K. This barrier\nrequired for the precursor growth seems to be reasonable although\nthere are no available  data in the literatures to our\nknowledge.\n\nWe further used equation (\\protect\\ref{eq4}) to make an estimation of local polarization growing at $T=T_{\\rm c}^+$ using the very\nweak electric field ($E=10^{-3}$ kV\/cm) used in the dielectric\nmeasurements. The estimated values were given in Fig.\\protect\\ref{fig8}(c)\nas a function of composition $x$. At $T=T_{\\rm c}^+$, the local\npolarization is predicted to grow to a magnitude of 0.03 nC\/cm$^2$ $\\sim$\n0.07 nC\/cm$^2$ in CBN ceramics.  Ca-substitution seems to enhance\nthe local polarization growth in CBN alloys. This is reasonable\nsince Ca-substitution reduces  Cuire point to a lower temperature,\nresulting in a larger temperature range ($ T_{\\rm c}<T<T_{\\rm B}$)\nfor the local polarization growth as shown in Fig.\\protect\\ref{fig6}(c) and\nFig.\\protect\\ref{fig9}(a).\n\n\\subsection{Phase diagram}\n Having discussed the thermal phase transition and the precursor\n behaviors in the above sections, we can establish a phase diagram\n for CBN alloys now. The proposed phase diagram was shown in\n Fig.\\protect\\ref{fig9}(a), in which the transition temperature $T_{\\rm c}$ on cooling\n or heating was determined from the temperature dependence of\n dielectric susceptibility and $T_{\\rm B}$ was estimated from\n deviation of Curie law in the dielectric susceptibility obtained\n upon cooling.\n\n In our phase diagram,  CBN alloys can be considered as\n paralectrics with a centrocymmetric tetragonal structure as\n that of isostructural compounds SBN for $T>T_{\\rm B}$. On cooling to $T\\approx T_{\\rm\n B}$, polarization precursors emerge in the paraelectric mother\n phase of CBN alloys. On further cooling, these local\n polarizations growth exponentially with the temperature before\n transition into a ferrolectric phase with a non-centrocymmetric tetragonal  structure  at\n $T=T_{\\rm c}$. This ferroelectric thermal phase transition is of\n first-order, and the transition temperature upon heating is\n $12.4\\sim 25.2$ K higher than that on cooling. As shown in the\n phase diagram, increase in Ca-concentration leads to a larger thermal\n hysteresis together with the lowering of $T_{\\rm c}$ and $T_{B}$. On the\n other hand, the temperature range of precursor existence becomes\n larger as increasing the amount of substitution  of  smaller Ca ions for Ba\n ions. It is still unclear why Ca-substitution enlarges the precursor growth region, the\n local polarization developed at $T_{\\rm c}^+$, and  the thermal\n hysteresis of phase transition in CBN alloys.\n A likely reason is relative to the increase of vacancies in\n larger A$_2$ site in the TTB structure due to the increase of  substitution amount of\n Ca for Ba. Because Ca almost exclusively occupied smaller A$_1$\nsite in CBN TTB structure, increase of Ca-concentration naturally\nresults in the increase of vacancies in\n A$_2$ site occupied initially by Ba.\\protect\\cite{Graetsch1,Graetsch2} Increase of vacancies in larger A$_2$ site\nwith $x$ may give rise to larger fluctuation of local lattice\ndistortion in CBN alloys, which is though to  be the sources of\nlocal polarizations. This suggestion is supported by  recent\ninvestigations on the variation of lattice strain with $x$ in CBN\nalloys, in which the thermal expansion along the polar axis becomes\nlarger as increasing  Ca-concentration.\\protect\\cite{Pandey3}\n\nOn the other hand, substitution of Ca for Ba leads to a nearly\nlinear reduction of $T_{\\rm c}$ and $T_{B}$ with $x$. As well known\nin many normal ferroelectrics, pressure can significantly reduce the\nferroelectric transition temperature. In solid solution, pressure\ncan be due to chemical substitution of smaller ions for  original\nlarger one, and this kind of pressure is normally called as chemical\npressure. Chemical-pressure effects on the reduction of $T_{\\rm\nc}$ have been well studied in many ferroelectric alloys with\nperovskite structure such as\n(Ba$_{1-x}$Sr$_x$)TiO$_3$,\\protect\\cite{Shiozaki} in which  $T_{\\rm\nc}$-reduction is generally considered to be due to the reduction of\nunit cell volume by the chemical pressure effects.  As mentioned in\nthe above section, in CBN alloys, substitution of smaller Ca ions\nfor Ba ions only lead to an extremely small change  in its unit cell\nvolume ($\\Delta=(V(x)-V(0.19))\/V(0.19)$), and  $\\Delta$ is less than\n0.65\\% within the solid solution limit. Thus, it can be considered\nthat the chemical pressure effects on the reduction of $T_{\\rm c}$\nwas vanishingly small in CBN alloys.\n\nThe spontaneous polarization in CBN has been reported to be\noriginated from  Nb-displacement along the $c$-axis in the TTB\nstructure.\\protect\\cite{Graetsch1} It can is expected that the lattice\ndistortion along the $c$-axis has influence on  Nb-displacement.\nFigure\\protect\\ref{fig9}(b) showed the variation of $T_{\\rm c}$ with\n$c$-axis lattice constant. Indeed, one can see that $T_{\\rm c}$ is\nreduced with the shrink of  $c$-axis lattice. This fact indicates\nthat  the shrink of  $c$-axis lattice should reduce the space\navailable for  Nb-displacement, thus leading to the reduction of\nferroelectricity and its $T_{\\rm c}$. However, since the reduction\nin  $c$-axis lattice constant is approximately 0.03 ${\\AA}$\nwithin the solid solution, this lattice shrink is not enough to\ncompletely explain the large reduction in $T_{\\rm c}$ in CBN alloys.\nIt seems that other reasons may be existed for larger reduction of\n$T_{\\rm c}$ with Ca-concentration in CBN alloys. The exact reason for this large reduction in $T_{\\rm c}$ with Ca-substitution in CBN alloys remains to be\nclarified in  further structural investigations.\n\n\n\n\\section{Summary}\n\nIn summary, we have studied the phase transition in CBN alloys\nwithin its solid solution limit ranging from $x=0.19$ to $x=0.32$.\nIn contrast to their isostructural compounds SBN that  normally show\ntypical relaxor behaviors, CBN alloys exhibit  behaviors such as\nnormal ferroelectric BaTiO$_3$, but show precursors dynamics before\ntransition into the ferroelectric phase. For $T>T_{\\rm B}$, CBN\nalloys obey classical Curie law and can be considered to be\nparaelectric. On further cooling toward $T_{\\rm c}$, local\npolarizations occurs in the paraelectric mother phase, and these\npolarization precursors grow exponentially as temperature lowering.\nFinally, a ferroelectric phase transition was realized at $T_{\\rm c}$. This thermal\nferroelectric phase transition is essentially of first-order. A\nphase diagram was then establish for CBN ferroelectric alloys. These\nfindings  provide new insights on understanding the underlying\nphysics in CBN ferroelectric alloys.\n\n\n\n\n\n\n\\newpage\n\n\n\\section{Introduction}\n\nThis is the author's guide to \\revtex~4, the preferred submission\nformat for all APS journals. This guide is intended to be a concise\nintroduction to \\revtex~4. The documentation has been separated out\ninto smaller units to make it easier to locate essential\ninformation.\n\nThe following documentation is also part of the APS \\revtex~4\ndistribution. Updated versions of these will be maintained at\nthe \\revtex~4 homepage located at \\url{http:\/\/publish.aps.org\/revtex4\/}.\n\\begin{itemize}\n\\item \\textit{APS Compuscript Guide for \\revtex~4}\n\\item \\textit{\\revtex~4 Command and Options Summary}\n\\item \\textit{\\revtex~4 Bib\\TeX\\ Guide}\n\\item \\textit{Differences between \\revtex~4 and \\revtex~3}\n\\end{itemize}\nThis guide assumes a working \\revtex~4\ninstallation. Please see the installation guide included with the\ndistribution.\n\nThe \\revtex\\ system for \\LaTeX\\ began its development in 1986 and has\ngone through three major revisions since then.  All versions prior to\n\\revtex~4 were based on \\LaTeX2.09 and, until now, \\revtex\\ did not\nkeep pace with the advances of the \\LaTeX\\ community and thus became\ninconvenient to work with. \\revtex~4 is designed to remedy this by\nincorporating the following design goals:\n\n\\begin{itemize}\n\\item\nMake \\revtex\\ fully compatible with \\LaTeXe; it is now a \\LaTeXe\\\ndocument class, similar in function to the standard\n\\classname{article} class.\n\n\\item\nRely on standard \\LaTeXe\\ packages for common tasks, e.g,\n\\classname{graphicx},\n\\classname{color}, and\n\\classname{hyperref}.\n\n\\item\nAdd or improve macros to support translation to tagged formats such as\nXML and SGML. This added markup will be key to enhancing the\npeer-review process and lowering production costs.\n\n\\item\nProvide a closer approximation to the typesetting style used in\n\\emph{Physical Review}.\n\n\\item\nIncorporate new features, such as hypertext, to make \\revtex\\ a\nconvenient and desirable e-print format.\n\n\\item\nRelax the restrictions in \\revtex\\ that had only been necessary for\ntypesetting journal camera-ready copy.\n\\end{itemize}\n\nTo meet these goals, \\revtex~4 is a complete rewrite with an emphasis\non maintainability so that it will be easier to provide enhancements.\n\nThe \\revtex~4 distribution includes both a template\n(\\file{template.aps}) and a sample document (\\file{apssamp.tex}).\nThe template is a good starting point for a manuscript. In the\nfollowing sections are instructions that should be sufficient for\ncreating a paper using \\revtex~4.\n\n\\subsection{Submitting to APS Journals}\n\nAuthors using \\revtex~4 to prepare a manuscript for submission to\n\\textit{Physical Review} or \\textit{Reviews of Modern Physics} \nmust also read the companion document \\textit{APS Compuscript Guide\nfor \\revtex~4}\ndistributed with \\revtex\\ and follow the guidelines detailed there.\n\nFurther information about the compuscript program of the American\nPhysical Society may be found at \\url{http:\/\/publish.aps.org\/ESUB\/}.\n\n\\subsection{Contact Information}\\label{sec:resources}%\nAny bugs, problems, or inconsistencies should reported to\n\\revtex\\ support at \\verb+revtex@aps.org+.\nReports should include information on the error and a \\textit{small}\nsample document that manifests the problem if possible (please don't\nsend large files!).\n\n\\section{Some \\LaTeXe\\ Basics}\nA primary design goal of \\revtex~4 was to make it as compatible with\nstandard \\LaTeXe\\ as possible so that authors may take advantage of all\nthat \\LaTeXe\\ offers. In keeping with this goal, much of the special\nformatting that was built in to earlier versions of \\revtex\\ is now\naccomplished through standard \\LaTeXe\\ macros or packages. The books\nin the bibliography provide extensive coverage of all topics\npertaining to preparing documents under \\LaTeXe. They are highly recommended.\n\nTo accomplish its goals, \\revtex~4 must sometimes patch the underlying\n\\LaTeX\\ kernel. This means that \\revtex~4 requires a fairly recent version of\n\\LaTeXe. Versions prior to 1996\/12\/01 may not work\ncorrectly. \\revtex~4 will be maintained to be compatible with future\nversions of \\LaTeXe.\n\n\\subsection{Useful \\LaTeXe\\ Markup}\n\\LaTeXe\\ markup is the preferred way to accomplish many basic tasks.\n\n\\subsubsection{Fonts}\n\nBecause \\revtex~4 is based upon \\LaTeXe, it inherits all of the\nmacros used for controlling fonts. Of particular importance are the\n\\LaTeXe\\ macros \\cmd{\\textit}, \\cmd{\\textbf}, \\cmd{\\texttt} for changing to\nan italic, bold, or typewriter font respectively. One should always\nuse these macros rather than the lower-level \\TeX\\ macros \\cmd{\\it},\n\\cmd{\\bf}, and \\cmd{\\tt}. The \\LaTeXe\\ macros offer\nimprovements such as better italic correction and scaling in super-\nand subscripts for example. Table~\\ref{tab:fonts}\nsummarizes the font selection commands in \\LaTeXe.\n\n\\begin{table}\n\\caption{\\label{tab:fonts}\\LaTeXe\\ font commands}\n\\begin{ruledtabular}\n\\begin{tabular}{ll}\n\\multicolumn{2}{c}{\\textbf{Text Fonts}}\\\\\n\\textbf{Font command} & \\textbf{Explanation} \\\\\n\\cmd\\textit\\marg{text}  & Italics\\\\\n\\cmd\\textbf\\marg{text}  & Boldface\\\\\n\\cmd\\texttt\\marg{text}  & Typewriter\\\\\n\\cmd\\textrm\\marg{text}  & Roman\\\\\n\\cmd\\textsl\\marg{text}  & Slanted\\\\\n\\cmd\\textsf\\marg{text}  & Sans Serif\\\\\n\\cmd\\textsc\\marg{text}  & Small Caps\\\\\n\\cmd\\textmd\\marg{text}  & Medium Series\\\\\n\\cmd\\textnormal\\marg{text} & Normal Series\\\\\n\\cmd\\textup\\marg{text}  & Upright Series\\\\\n  &\\\\\n\\multicolumn{2}{c}{\\textbf{Math Fonts}}\\\\\n\\cmd\\mathit\\marg{text}  & Math Italics\\\\\n\\cmd\\mathbf\\marg{text}  & Math Boldface\\\\\n\\cmd\\mathtt\\marg{text}  & Math Typewriter\\\\\n\\cmd\\mathsf\\marg{text}  & Math Sans Serif\\\\\n\\cmd\\mathcal\\marg{text}  & Calligraphic\\\\\n\\cmd\\mathnormal\\marg{text} & Math Normal\\\\\n\\cmd\\bm\\marg{text}& Bold math for Greek letters\\\\\n                  & and other symbols\\\\\n\\cmd\\mathfrak\\marg{text}\\footnotemark[1]  & Fraktur\\\\\n\\cmd\\mathbb\\marg{text}\\footnotemark[1] & Blackboard Bold\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\footnotetext[1]{Requires \\classname{amsfonts} or \\classname{amssymb} class option}\n\\end{table}\n\n\\subsubsection{User-defined macros}\n\\LaTeXe\\ provides several macros that enable users to easily create new\nmacros for use in their manuscripts:\n\\begin{itemize}\n\\footnotesize\n\\item \\cmd\\newcommand\\marg{\\\\command}\\oarg{narg}\\oarg{opt}\\marg{def} \n\\item \\cmd\\newcommand\\verb+*+\\marg{\\\\command}\\oarg{narg}\\oarg{opt}\\marg{def}\n\\item \\cmd\\renewcommand\\marg{\\\\command}\\oarg{narg}\\oarg{opt}\\marg{def}\n\\item \\cmd\\renewcommand\\verb+*+\\marg{\\\\command}\\oarg{narg}\\oarg{opt}\\marg{def}\n\\item \\cmd\\providecommand\\marg{\\\\command}\\oarg{narg}\\oarg{opt}\\marg{def}\n\\item \\cmd\\providecommand\\verb+*+\\marg{\\\\command}\\oarg{narg}\\oarg{opt}\\marg{def}\n\\end{itemize}\nHere \\meta{\\\\command} is the name of the macro being defined,\n\\meta{narg} is the number of arguments the macro takes,\n\\meta{opt} are optional default values for the arguments, and\n\\meta{def} is the actually macro definiton. \\cmd\\newcommand\\ creates a\nnew macro, \\cmd\\renewcommand\\ redefines a previously defined macro,\nand \\cmd\\providecommand\\ will define a macro only if it hasn't\nbeen defined previously. The *-ed versions are an optimization that\nindicates that the macro arguments will always be ``short'' arguments. This is\nalmost always the case, so the *-ed versions should be used whenver\npossible.\n\nThe use of these macros is preferred over using plain \\TeX's low-level\nmacros such as\n\\cmd\\def{},\\cmd\\edef{}, and \\cmd\\gdef{}. APS authors must follow the\n\\textit{APS Compuscript Guide for \\revtex~4} when defining macros.\n\n\\subsubsection{Symbols}\n\n\\LaTeXe\\ has added some convenient commands for some special symbols\nand effects. These are summarized in Table~\\ref{tab:special}. See\n\\cite{Guide} for details.\n\n\\begin{table}\n\\caption{\\label{tab:special}\\LaTeXe\\ commands for special symbols and effects}\n\\begin{ruledtabular}\n\\begin{tabular}{lc}\nCommand & Symbol\/Effect\\\\\n\\cmd\\textemdash & \\textemdash\\\\\n\\cmd\\textendash & \\textendash\\\\\n\\cmd\\textexclamdown & \\textexclamdown\\\\\n\\cmd\\textquestiondown & \\textquestiondown\\\\\n\\cmd\\textquotedblleft & \\textquotedblleft\\\\\n\\cmd\\textquotedblright & \\textquotedblright\\\\\n\\cmd\\textquoteleft & \\textquoteleft\\\\\n\\cmd\\textquoteright & \\textquoteright\\\\\n\\cmd\\textbullet   & \\textbullet\\\\\n\\cmd\\textperiodcentered & \\textperiodcentered\\\\\n\\cmd\\textvisiblespace & \\textvisiblespace\\\\\n\\cmd\\textcompworkmark & Break a ligature\\\\\n\\cmd\\textcircled\\marg{char} & Circle a character\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\LaTeXe\\ also removed some symbols that were previously automatically\navailable in \\LaTeX 2.09. These symbols are now contained in a\nseparate package \\classname{latexsym}. To use these symbols, include\nthe package using:\n\\begin{verbatim}\n\\usepackage{latexsym}\n\\end{verbatim}\n\n\\subsection{Using \\LaTeXe\\ packages with \\revtex}\\label{sec:usepackage}%\n\nMany \\LaTeXe\\ packages are available, for instance, on CTAN at\n\\url{ftp:\/\/ctan.tug.org\/tex-archive\/macros\/latex\/required\/}\nand at\n\\url{ftp:\/\/ctan.tug.org\/tex-archive\/macros\/latex\/contrib\/}\nor may be available on other distribution media, such as the \\TeX\\\nLive CD-ROM \\url{http:\/\/www.tug.org\/texlive\/}.  Some of these packages\nare automatically loaded by \\revtex~4 when certain class options are\ninvoked and are, thus, ``required''.  They will either be distributed\nwith \\revtex\\ or are already included with a standard \\LaTeXe\\\ndistribution.\n\nRequired packages are automatically loaded by \\revtex\\ on an as-needed\nbasis.  Other packages should be loaded using the\n\\cmd\\usepackage\\ command. To load the\n\\classname{hyperref} package, the document preamble might look like:\n\\begin{verbatim}\n\\documentclass{revtex}\n\\usepackage{hyperref}\n\\end{verbatim}\n\nSome common (and very useful) \\LaTeXe\\ packages are \\textit{a priori}\nimportant enough that \\revtex~4 has been designed to be specifically\ncompatible with them. \nA bug stemming from the use of one of these packages in\nconjunction with any of the APS journals may be reported by contacting\n\\revtex\\ support.\n\\begin{description}\n\\item[\\textbf{AMS packages}] \\revtex~4 is compatible with and depends\n upon the AMS packages\n\\classname{amsfonts},\n\\classname{amssymb}, and\n\\classname{amsmath}. In fact, \\revtex~4 requires use of these packages\nto accomplish some common tasks. See Section~\\ref{sec:math} for more.\n\\revtex~4 requires version 2.0 or higher of the AMS-\\LaTeX\\ package.\n\n\\item[\\textbf{array and dcolumn}]\nThe \\classname{array} and \\classname{dcolumn} packages are part of\n\\LaTeX's required suite of packages. \\classname{dcolumn} is required\nto align table columns on decimal points (and it in turn depends upon\nthe \\classname{array} package).\n\n\\item[\\textbf{longtable}]\n\\file{longtable.sty} may be used for large tables that will span more than one\npage. \\revtex~4 dynamically applies patches to longtable.sty so that\nit will work in two-column mode.\n\n\\item[\\textbf{hyperref}] \\file{hyperref.sty} is a package by Sebastian Rahtz that is\nused for putting hypertext links into \\LaTeXe\\ documents.\n\\revtex~4 has hooks to allow e-mail addresses and URL's to become\nhyperlinks if \\classname{hyperref} is loaded.\n\\end{description}\n\nOther packages will conflict with \\revtex~4 and should be\navoided. Usually such a conflict arises because the package adds\nenhancements that \\revtex~4 already includes. Here are some common\npackages that clash with \\revtex~4:\n\\begin{description}\n\\item[\\textbf{multicol}] \\file{multicol.sty} is a package by Frank Mittelbach\nthat adds support for multiple columns. In fact, early versions of\n\\revtex~4 used \\file{multicol.sty} for precisely this. However, to\nimprove the handling of floats, \\revtex~4 now has its own macros for\ntwo-column layout. Thus, it is not necessary to use \\file{multicol.sty}.\n\n\\item[\\textbf{cite}] Donald Arseneau's \\file{cite.sty} is often used to provide\nsupport for sorting a \\cmd\\cite\\ command's arguments into numerical\norder and to collapse consecutive runs of reference numbers. \\revtex~4\nhas this functionality built-in already via the \\classname{natbib} package.\n\n\\item[\\textbf{endfloat}] The same functionality can be accomplished\nusing the \\classoption{endfloats} class option.\n\n\\item[\\textbf{float}] \\revtex~4 already contains a lot of this\nfunctionality.\n\\end{description}\n\n\\section{The Document Preamble}\n\nThe preamble of a \\LaTeX\\ document is the set of commands that precede\nthe \\envb{document} line. It contains a\n\\cmd\\documentclass\\ line to load the \\revtex~4 class (\\textit{i.~e.},\nall of the \\revtex~4 macro definitions), \\cmd\\usepackage\\ macros to\nload other macro packages, and other macro definitions.\n\n\\subsection{The \\emph{documentclass} line}\nThe basic formatting of the manuscript is controlled by setting\n\\emph{class options} using\n\\cmd\\documentclass\\oarg{options}\\aarg{\\classname{revtex4}}.\nThe macro \\cmd\\documentclass\\ \nreplaces the \\cmd\\documentstyle\\ macro of \\LaTeX2.09. The optional\narguments that appear in the square brackets control the layout of the\ndocument. At this point, one only needs to choose a journal style\n(\\classoption{pra}, \\classoption{prb},\n\\classoption{prc}, \\classoption{prd},\n\\classoption{pre}, \\classoption{prl}, \\classoption{prstab},\nand \\classoption{rmp}) and either \\classoption{preprint} or\n\\classoption{twocolumn}. Usually, one would want to use\n\\classoption{preprint} for draft papers. \\classoption{twocolumn} gives\nthe \\emph{Physical Review} look and feel. Paper size options are also\navailable as well. In particular, \\classoption{a4paper} is available\nas well as the rest of the standard \\LaTeX\\ paper sizes. A\nfull list of class options is given in the \\textit{\\revtex~4 Command\nand Options Summary}.\n\n\\subsection{Loading other packages}\nOther packages may be loaded into a \\revtex~4 document by using the\nstandard \\LaTeXe\\ \\cmd\\usepackage\\ command. For instance, to load\nthe \\classoption{graphics} package, one would use\n\\verb+\\usepackage{graphics}+.\n\n\\section{The Front Matter}\\label{sec:front}\n\nAfter choosing the basic look and feel of the document by selecting\nthe appropriate class options and loading in whatever other macros are\nneeded, one is ready to move on to creating a new manuscript. After\nthe preamble, be sure to put in a \\envb{document} line (and put\nin an \\enve{document} as well). This section describes the macros\n\\revtex~4 provides for formatting the front matter of the\narticle. The behavior and usage of these macros can be quite\ndifferent from those provided in either \\revtex~3 or \\LaTeXe. See the\nincluded document \\textit{Differences between \\revtex~4 and \\revtex~3} for an\noverview of these differences.\n\n\\subsection{Setting the title}\n\nThe title of the manuscript is simply specified by using the\n\\cmd\\title\\aarg{title} macro. A \\verb+\\\\+ may be used to put a line\nbreak in a long title.\n\n\\subsection{Specifying a date}%\n\nThe \\cmd\\date\\marg{date} command outputs the date on the\nmanuscript.  Using \\cmd\\today\\ will cause \\LaTeX{} to insert the\ncurrent date whenever the file is run:\n\\begin{verbatim}\n\\date{\\today}\n\\end{verbatim}\n\n\\subsection{Specifying authors and affiliations}\n\nThe macros  for specifying authors and their affiliations have\nchanged significantly for \\revtex~4. They have been improved to save\nlabor for authors and in production. Authors and affiliations are\narranged into groupings called, appropriately enough, \\emph{author\ngroups}. Each author group is a set of authors who share the same set\nof affiliations. Author names are specified with the \\cmd\\author\\\nmacro while affiliations (or addresses) are specified with the\n\\cmd\\affiliation\\ macro. Author groups are specified by sequences of\n\\cmd\\author\\ macros followed by \\cmd\\affiliation\\ macros. An\n\\cmd\\affiliation\\ macro applies to all previously specified\n\\cmd\\author\\ macros which don't already have an affiliation supplied.\n\nFor example, if Bugs Bunny and Roger Rabbit are both at Looney Tune\nStudios, while Mickey Mouse is at Disney World, the markup would be:\n\\begin{verbatim}\n\\author{Bugs Bunny}\n\\author{Roger Rabbit}\n\\affiliation{Looney Tune Studios}\n\\author{Mickey Mouse}\n\\affiliation{Disney World}\n\\end{verbatim}\nThe default is to display this as \n\\begin{center}\nBugs Bunny and Roger Rabbit\\\\\n\\emph{Looney Tune Studios}\\\\\nMickey Mouse\\\\\n\\emph{Disney World}\\\\\n\\end{center}\nThis layout style for displaying authors and their affiliations is\nchosen by selecting the class option\n\\classoption{groupedaddress}. This option is the default for all APS\njournal styles, so it does not need to be specified explicitly.\nThe other major way of displaying this\ninformation is to use superscripts on the authors and\naffiliations. This can be accomplished by selecting the class option\n\\classoption{superscriptaddress}. To achieve the display\n\\begin{center}\nBugs Bunny,$^{1}$ Roger Rabbit,$^{1,2}$ and Mickey Mouse$^{2}$\\\\\n\\emph{$^{1}$Looney Tune Studios}\\\\\n\\emph{$^{2}$Disney World}\\\\\n\\end{center}\none would use the markup\n\\begin{verbatim}\n\\author{Bugs Bunny}\n\\affiliation{Looney Tune Studios}\n\\author{Roger Rabbit}\n\\affiliation{Looney Tune Studios}\n\\affiliation{Disney World}\n\\author{Mickey Mouse}\n\\affiliation{Disney World}\n\\end{verbatim}\n\nNote that \\revtex~4 takes care of any commas and \\emph{and}'s that join\nthe author names together and font selection, as well as any\nsuperscript numbering. Only the author names and affiliations should\nbe given within their respective macros.\n\nThere is a third class option, \\classoption{unsortedaddress}, for\ncontrolling author\/affiliation display. The default\n\\classoption{groupedaddress} will actually sort authors into the\napproriate author groups if one chooses to specify an affiliation for\neach author. The markup:\n\\begin{verbatim}\n\\author{Bugs Bunny}\n\\affiliation{Looney Tune Studios}\n\\author{Mickey Mouse}\n\\affiliation{Disney World}\n\\author{Roger Rabbit}\n\\affiliation{Looney Tune Studios}\n\\end{verbatim}\nwill result in the same display as for the first case given\nabove even though Roger Rabbit is specified after Mickey Mouse. To\navoid Roger Rabbit being moved into the same author group as Bugs\nBunny, use the\n\\classoption{unsortedaddress} option instead. In general, it is safest\nto list authors in the order they should appear and specify\naffiliations for multiple authors rather than one at a time. This will\nafford the most independence for choosing the display option. Finally,\nit should be mentioned that the affiliations for the\n\\classoption{superscriptaddress} are presented and numbered \nin the order that they are encountered. These means that the order\nwill usually follow the order of the authors. An alternative ordering\ncan be forced by including a list of \\cmd\\affiliation\\ commands before\nthe first \\cmd{\\author} in the desired order. Then use the exact same\ntext for each affilation when specifying them for each author.\n\nIf an author doesn't have an affiliation, the \\cmd\\noaffiliation\\\nmacro may be used in the place of an \\cmd\\affiliation\\ macro.\n\n\n\\subsubsection{Collaborations}\n\nA collaboration name can be specified with the \\cmd\\collaboration\\\nmacro. This is very similar to the \\cmd\\author\\ macro, but it can only\nbe used with the class option \\classoption{superscriptaddress}. The\n\\cmd\\collaboration\\ macro should appear at the end of the list of\nauthors. The collaboration name will be appear centered in parentheses\nbetween the list of authors and the list of\naffiliations. Because collaborations\ndon't normally have affiliations, one needs to follow the\n\\cmd\\collaboration\\ with \\cmd\\noaffiliation.\n\n\\subsubsection{Footnotes for authors, collaborations, affiliations or title}\\label{sec:footau}\n\nOften one wants to specify additional information associated with an\nauthor, collaboration, or affiliation such an e-mail address, an\nalternate affiliation, or some other anicillary information. \n\\revtex~4 introduces several new macros just for this purpose. They\nare:\n\\begin{itemize}\n\\item\\cmd\\email\\oarg{optional text}\\aarg{e-mail address}\n\\item\\cmd\\homepage\\oarg{optional text}\\aarg{URL}\n\\item\\cmd\\altaffiliation\\oarg{optional text}\\aarg{affiliation}\n\\item\\cmd\\thanks\\aarg{miscellaneous text}\n\\end{itemize}\nIn the first three, the \\emph{optional text} will be prepended before the\nactual information specified in the required argument. \\cmd\\email\\ and\n\\cmd\\homepage\\ each have a default text for their optional arguments\n(`Electronic address:' and `URL:' respectively). The \\cmd\\thanks\\\nmacro should only be used if one of the other three do not apply. Any\nauthor name can have multiple occurences of these four macros. Note\nthat unlike the\n\\cmd\\affiliation\\ macro, these macros only apply to the \\cmd\\author\\\nthat directly precedes it. Any \\cmd\\affiliation\\ \\emph{must} follow\nthe other author-specific macros. A typical usage might be as follows:\n\\begin{verbatim}\n\\author{Bugs Bunny}\n\\email[E-mail me at: ]{bugs@looney.com}\n\\homepage[Visit: ]{http:\/\/looney.com\/}\n\\altaffiliation[Permanent address: ]\n                     {Warner Brothers}\n\\affiliation{Looney Tunes}\n\\end{verbatim}\nThis would result in the footnote ``E-mail me at: \\texttt{bugs@looney.com},\nVisit: \\texttt{http:\/\/looney.com\/}, Permanent address: Warner\nBrothers'' being attached to Bugs Bunny. Note that:\n\\begin{itemize}\n\\item Only an e-mail address, URL, or affiliation should go in the\nrequired argument in the curly braces.\n\\item The font is automatically taken care of.\n\\item An explicit space is needed at the end of the optional text if one is\ndesired in the output.\n\\item Use the optional arguments to provide customized\ntext only if there is a good reason to.\n\\end{itemize}\n\nThe \\cmd\\collaboration\\ , \\cmd\\affiliation\\ , or even \\cmd\\title\\ can\nalso have footnotes attached via these commands. If any ancillary data\n(\\cmd\\thanks, \\cmd\\email, \\cmd\\homepage, or\n\\cmd\\altaffiliation) are given in the wrong context (e.g., before any\n\\cmd\\title, \\cmd\\author, \\cmd\\collaboration, or \\cmd\\affiliation\\\ncommand has been given), then a warning is given in the \\TeX\\ log, and\nthe command is ignored.\n\nDuplicate sets of ancillary data are merged, giving rise to a single\nshared footnote. However, this only applies if the ancillary data are\nidentical: even the order of the commands specifying the data must be\nidentical. Thus, for example, two authors can share a single footnote\nindicating a group e-mail address.\n\nDuplicate \\cmd\\affiliation\\ commands may be given in the course of the\nfront matter, without the danger of producing extraneous affiliations\non the title page. However, ancillary data should be specified for\nonly the first instance of any particular institution's\n\\cmd\\affiliation\\ command; a later instance with different ancillary\ndata will result in a warning in the \\TeX\\ log.\n\nIt is preferable to arrange authors into\nsets. Within each set all the authors share the same group of\naffiliations. For each author, give the \\cmd\\author\\ (and appropriate\nancillary data), then follow this author group with the needed group\nof \\cmd\\affiliation\\ commands.\n\nIf affiliations have been listed before the first\n\\cmd\\author\\ macro to ensure a particular ordering, be sure\nthat any later \\cmd\\affiliation\\ command for the given institution is\nan exact copy of the first, and also ensure that no ancillary data is\ngiven in these later instances.\n\n\nEach APS journal has a default behavior for the placement of these\nancillary information footnotes. The \\classoption{prb} option puts all\nsuch footnotes at the start of the bibliography while the other\njournal styles display them on the first page. One can override a\njournal style's default behavior by specifying explicitly the class\noption\n\\classoption{bibnotes} (puts the footnotes at the start of the\nbibliography) or \\classoption{nobibnotes} (puts them on the first page).\n\n\\subsubsection{Specifying first names and surnames}\n\nMany APS authors have names in which either the surname appears first\nor in which the surname is made up of more than one name. To ensure\nthat such names are accurately captured for indexing and other\npurposes, the \\cmd\\surname\\ macro should be used to indicate which portion\nof a name is the surname. Similarly, there is a \\cmd\\firstname\\ macro\nas well, although usage of \\cmd\\surname\\ should be sufficient. If an\nauthor's surname is a single name and written last, it is not\nnecessary to use these macros. These macros do nothing but indicate\nhow a name should be indexed. Here are some examples;\n\\begin{verbatim}\n\\author{Andrew \\surname{Lloyd Weber}}\n\\author{\\surname{Mao} Tse-Tung}\n\\end{verbatim}\n\n\\subsection{The abstract}\nAn abstract for a paper is specified by using the \\env{abstract}\nenvironment:\n\\begin{verbatim}\n\\begin{abstract}\nText of abstract\n\\end{abstract}\n\\end{verbatim}\nNote that in \\revtex~4 the abstract must be specified before the\n\\cmd\\maketitle\\ command and there is no need to embed it in an explicit\nminipage environment.\n\n\\subsection{PACS codes}\nAPS authors are asked to supply suggested PACS codes with their\nsubmissions. The \\cmd\\pacs\\ macro is provided as a way to do this:\n\\begin{verbatim}\n\\pacs{23.23.+x, 56.65.Dy}\n\\end{verbatim}\nThe actual display of the PACS numbers below the abstract is\ncontrolled by two class options: \\classoption{showpacs} and\n\\classoption{noshowpacs}. In particular, this is now independent of\nthe \\classoption{preprint} option. \\classoption{showpacs} must be\nexplicitly included in the class options to display the PACS codes.\n\n\\subsection{Keywords}\nA \\cmd\\keywords\\ macro may also be used to indicate keywords for the\narticle. \n\\begin{verbatim}\n\\keywords{nuclear form; yrast level}\n\\end{verbatim}\nThis will be displayed below the abstract and PACS (if supplied). Like\nPACS codes, the actual display of the the keywords is controlled by\ntwo classoptions: \\classoption{showkeys} and\n\\classoption{noshowkeys}. An explicit \\classoption{showkeys} must be\nincluded in the \\cmd\\documentclass\\ line to display the keywords.\n\n\\subsection{Institutional report numbers}\nInstitutional report numbers can be specified using the \\cmd\\preprint\\\nmacro. These will be displayed in the upper lefthand corner of the\nfirst page. Multiple \\cmd\\preprint\\ macros maybe supplied (space is\nlimited though, so only three or less may actually fit). \n\n\\subsection{maketitle}\nAfter specifying the title, authors, affiliations, abstract, PACS\ncodes, and report numbers, the final step for formatting the front\nmatter of the manuscript is to execute the \\cmd\\maketitle\\ macro by\nsimply including it:\n\\begin{verbatim}\n\\maketitle\n\\end{verbatim}\nThe \\cmd\\maketitle\\ macro must follow all of the macros listed\nabove. The macro will format the front matter in accordance with the various\nclass options that were specified in the\n\\cmd\\documentclass\\ line (either implicitly through defaults or\nexplicitly).\n\n\\section{The body of the paper}\n\nFor typesetting the body of a paper, \\revtex~4 relies heavily on\nstandard \\LaTeXe\\ and other packages (particulary those that are part\nof AMS-\\LaTeX). Users unfamiliar with these packages should read the\nfollowing sections carefully. \n\n\\subsection{Section headings}\n\nSection headings are input as in \\LaTeX.\nThe output is similar, with a few extra features.\n\nFour levels of headings are available in \\revtex{}:\n\\begin{quote}\n\\cmd\\section\\marg{title text}\\\\\n\\cmd\\subsection\\marg{title text}\\\\\n\\cmd\\subsubsection\\marg{title text}\\\\\n\\cmd\\paragraph\\marg{title text}\n\\end{quote}\n\nUse the starred form of the command to suppress the automatic numbering; e.g.,\n\\begin{verbatim}\n\\section*{Introduction}\n\\end{verbatim}\n\nTo label a section heading for cross referencing, best practice is to\nplace the \\cmd\\label\\marg{key} within the argument specifying the heading:\n\\begin{verbatim}\n\\section{\\label{sec:intro}Introduction}\n\\end{verbatim}\n\nIn the some journal substyles, such as those of the APS,\nall text in the \\cmd\\section\\ command is automatically set uppercase.\nIf a lowercase letter is needed, use \\cmd\\lowercase\\aarg{x}.\nFor example, to use ``He'' for helium in a \\cmd\\section\\marg{title text} command, type\n\\verb+H+\\cmd\\lowercase\\aarg{e} in \\marg{title text}.\n\nUse \\cmd\\protect\\verb+\\\\+ to force a line break in a section heading.\n(Fragile commands must be protected in section headings, captions, and\nfootnotes and \\verb+\\\\+ is a fragile command.)\n\n\\subsection{Paragraphs and General Text}\n\nParagraphs always end with a blank input line.  Because \\TeX\\\nautomatically calculates linebreaks and word hyphenation in a\nparagraph, it is not necessary to force linebreaks or hyphenation.  Of\ncourse, compound words should still be explicitly hyphenated, e.g.,\n``author-prepared copy.''\n\nUse directional quotes for quotation marks around quoted text\n(\\texttt{``xxx''}), not straight double quotes (\\texttt{\"xxx\"}).\nFor opening quotes, use one or two backquotes; for closing quotes,\nuse one or two forward quotes (apostrophes).\n\n\\subsection{One-column vs. two-column}\\label{sec:widetext}\n\nOne of the hallmarks of \\textit{Physical Review} is its two-column\nformatting and so one of the \\revtex~4 design goals is to make it easier to\nacheive the \\textit{Physical Review} look and feel. In particular, the\n\\classoption{twocolumn} option will take care of formatting the front matter\n(including the abstract) as a single column. \\revtex~4 has its own\nbuilt-in two-column formatting macros to provide well-balanced columns\nas well as reasonable control over the placement of floats in either\none- or two-column modes.\n\nOccasionally it is necessary to change the formatting from two-column to\none-column to better accomodate very long equations that are more\neasily read when typeset to the full width of the page. This is\naccomplished using the \\env{widetext} environment:\n\\begin{verbatim}\n\\begin{widetext}\nlong equation goes here\n\\end{widetext}\n\\end{verbatim}\nIn two-column mode, this will temporarily return to one-column mode,\nbalancing the text before the environment into two short columns, and\nreturning to two-column mode after the environment has\nfinished. \\revtex~4 will also add horizontal rules to guide the\nreader's eye through what may otherwise be a confusing break in the\nflow of text. The\n\\env{widetext} environment has no effect on the output under the \n\\classoption{preprint} class option because this already uses\none-column formatting.\n\nUse of the \\env{widetext} environment should be restricted to the bare\nminimum of text that needs to be typeset this way. However short pieces\nof paragraph text and\/or math between nearly contiguous wide equations\nshould be incorporated into the surrounding wide sections.\n\nLow-level control over the column grid can be accomplished with the\n\\cmd\\onecolumngrid\\ and \\cmd\\twocolumngrid\\ commands. Using these, one\ncan avoid the horizontal rules added by \\env{widetext}. These commands\nshould only be used if absolutely necessary. Wide figures and tables\nshould be accomodated using the proper \\verb+*+ environments.\n\n\\subsection{Cross-referencing}\\label{sec:xrefs}\n\n\\revtex{} inherits the \\LaTeXe\\ features for labeling and cross-referencing\nsection headings, equations, tables, and figures. This section\ncontains a simplified explanation of these cross-referencing features.\nThe proper usage in the context of section headings, equations,\ntables, and figures is discussed in the appropriate sections.\n\nCross-referencing depends upon the use of ``tags,'' which are defined by\nthe user.  The \\cmd\\label\\marg{key} command is used to identify tags for\n\\revtex. Tags are strings of characters that serve to label section\nheadings, equations, tables, and  figures that replace explicit,\nby-hand numbering.\n\nFiles that use cross-referencing (and almost all manuscripts do)\nneed to be processed through \\revtex\\ at least twice to\nensure that the tags have been properly linked to appropriate numbers.\nIf any tags are added in subsequent editing sessions, \n\\LaTeX{} will display a warning message in the log file that ends with\n\\texttt{... Rerun to get cross-references right}.\nRunning the file through \\revtex\\ again (possibly more than once) will\nresolve the cross-references.  If the error message persists, check\nthe labels; the same \\marg{key} may have been used to label more than one\nobject.\n\nAnother \\LaTeX\\ warning is \\texttt{There were undefined references},\nwhich indicates the use of a key in a \\cmd\\ref\\ without ever\nusing it in a \\cmd\\label\\ statement.\n\n\\revtex{} performs autonumbering exactly as in standard \\LaTeX.\nWhen the file is processed for the first time,\n\\LaTeX\\ creates an auxiliary file (with the \\file{.aux} extension) that \nrecords the value of each \\meta{key}.  Each subsequent run retrieves\nthe proper number from the auxiliary file and updates the auxiliary\nfile.  At the end of each run, any change in the value of a \\meta{key}\nproduces a \\LaTeX\\ warning message.\n\nNote that with footnotes appearing in the bibliography, extra passes\nof \\LaTeX\\ may be needed to resolve all cross-references. For\ninstance, putting a \\cmd\\cite\\ inside a \\cmd\\footnote\\ will require at\nleast three passes.\n\nUsing the \\classname{hyperref} package to create hyperlinked PDF files\nwill cause reference ranges to be expanded to list every\nreference in the range. This behavior can be avoided by using the\n\\classname{hypernat} package available from \\url{www.ctan.org}.\n\n\\subsection{Acknowledgments}\nUse the \\env{acknowledgments} environment for an acknowledgments\nsection.  Depending on the journal substyle, this element may be\nformatted as an unnumbered section title \\textit{Acknowledgments} or\nsimply as a paragraph. Please note the spelling of\n``acknowledgments''.\n\\begin{verbatim}\n\\begin{acknowlegments}\nThe authors would like to thank...\n\\end{acknowlegments}\n\\end{verbatim}\n\n\\subsection{Appendices}\nThe \\cmd\n\\section{Introduction}\nThis document gives a brief summary of how \\revtex~4 is different from\nwhat authors may already be familiar with.  The two primary design\ngoals for \\revtex~4 are to 1) move to \\LaTeXe\\ and 2) improve the\nmarkup so that infomation can be more reliably extracted for the\neditorial and production processes. Both of these goals require that\nauthors comfortable with earlier versions of \\revtex\\ change their\nhabits. In addition, authors may already be familiar with the standard\n\\classname{article.cls} in \\LaTeXe. \\revtex~4 differs in some\nimportant ways from this class as well. For more complete\ndocumentation on \\revtex~4, see the main \\textit{\\revtex~4 Author's\nGuide}.  The most important changes are in the markup of the front\nmatter (title, authors, affiliations, abstract, etc.). Please see\nSec.~\\ref{sec:front}.\n\n\\section{Version of \\LaTeX}\nThe most obvious difference between \\revtex~4 and \\revtex~3 is that\n\\revtex~4 works solely with \\LaTeXe; it is not useable as a \\LaTeX2.09 package.\nFurthermore, \\revtex~4 requires an up-to-date \\LaTeX\\ installation\n(1996\/06\/01 or later); its use under older versions is not supported.\n\n\\section{Class Options and Defaults}\nMany of the class options in \\revtex~3 have been retained in\n\\revtex~4. However, the default behavior for these options can be\ndifferent than in \\revtex~3. Currently, there is only one society\noption, \\classoption{aps}, and this is the default. Furthermore, the\nselection of a journal (such as \\classoption{prl}) will automatically\nset the society as well (this will be true even after other societies\nare added).\n\nIn \\revtex~3, it was necessary to invoke the \\classoption{floats}, but\nthis is the default for \\classoption{aps} journal in\n\\revtex~4. \\revtex~4 introduces two new class options,\n\\classoption{endfloats} and \\classoption{endfloats*} for moving floats\nto the end of the paper.\n\nThe preamble commands \\cmd{\\draft} and \\cmd{\\tighten} have been replaced\nwith new class options \\classoption{draft} and\n\\classoption{tightenlines}, respectively. The \\cmd{\\preprint} command\nis now used only for specifying institutional report numbers (typeset\nin the upper-righthand corner of the first page); it no longer\ninfluences whether PACS numbers are displayed below the abstract. PACS\ndisplay is controlled by the \\classoption{showpacs} and\n\\classoption{noshowpacs} (default) class options.\n\nPaper size options (\\classoption{letter}, \\classoption{a4paper}, etc.)\nwork in \\revtex~4. The text ``Typeset by \\revtex'' no longer appears\nby default - the option \\classoption{byrevtex} will place this text in\nthe lower-lefthand corner of the first page.\n\n\\section{One- and Two-column formatting}\n\n\\revtex~4 has excellent support for achieving the two-column\nformatting in the \\textit{Physical~Review} and \\textit{Reviews of\nModern Physics} styles. It will balance the columns\nautomatically. Whereas \\revtex~3 had the \\cmd{\\widetext} and\n\\cmd{\\narrowtext} commands for switching between one- and two-cloumn\nmodes, \\revtex~4 simply has a \\env{widetext} environment,\n\\envb{widetext} \\dots \\enve{widetext}. One-column formatting can be\nspecified by choosing either the \\classoption{onecolumn} or\n\\classoption{preprint} class option (the \\revtex~3 option\n\\classoption{manuscript} no longer exists). Two-column formatting is\nthe default for most journal styles, but can be specified with the\n\\classoption{twocolumn} option. Note that the spacing for\n\\classoption{preprint} is now set to 1.5, rather than full\ndouble-spacing. The \\classoption{tightenlines} option can be used to\nreduce this to single spacing.\n\n\n\\section{Front Matter Markup}\n\\label{sec:front}\n\n\\revtex~4 has substantially changed how the front matter for an article\nis marked up. These are the most significant differences between\n\\revtex~4 and other systems for typesetting manuscripts. It is\nessential that authors new to \\revtex~4 be familiar with these changes.\n\n\\subsection{Authors, Affiliations, and Author Notes}\n\\revtex~4 has substantially changed the markup of author names,\naffiliations, and author notes (footnotes giving additional\ninformation about the author such as a permanent address or an email\naddress).\n\\begin{itemize}\n\\item Each author name should appear separately in\nindividual \\cmd\\author\\ macros. \n\n\\item Email addresses should be marked up using the \\cmd\\email\\ macro.\n\n\\item Alternative affiliation information should be marked up using\nthe \\cmd\\altaffiliation\\ macro.\n\n\\item URLs for author home pages can be specified with a\n\\cmd\\homepage\\ macro.\n\n\\item The \\cmd\\thanks\\ macro should only be used if one of the above\ndon't apply.\n\n\\item \\cmd{\\email}, \\cmd{\\homepage}, \\cmd{\\altaffiliation}, and\n\\cmd{\\thanks} commands are grouped together under a single footnote for\neach author. These footnotes can either appear at the bottom of the\nfirst page of the article or as the first entries in the\nbibliography. The journal style controls this placement, but it may be\noverridden by using the class options \\classoption{bibnotes} and\n\\classoption{nobibnotes}. Note that these footnotes are treated\ndifferently than the other footnotes in the article.\n\n\\item The grouping of authors by affiliations is accomplished\nautomatically. Each affiliation should be in its own\n\\cmd{\\affiliation} command. Multiple \\cmd{\\affiliation},\n\\cmd{\\email}, \\cmd{\\homepage}, \\cmd{\\altaffiliation}, and \\cmd{\\thanks}\ncommands can be applied to each author. The macro \\cmd\\and\\ has been\neliminated.\n\n\\item \\cmd\\affiliation\\ commmands apply to all previous authors that\ndon't have an affiliation already declared. Furthermore, for any\nparticular author, the \\cmd\\affilation\\ must follow any \\cmd{\\email},\n\\cmd{\\homepage}, \\cmd{\\altaffiliation}, or \\cmd{\\thanks} commands for\nthat author.\n\n\\item Footnote-style associations of authors with affilitations should\nnot be done via explicit superscripts; rather, the class option\n\\classoption{superscriptaddress} should be used to accomplish this\nautomatically.\n\n\\item A collaboration for a group of authors can be given using the\n\\cmd\\collaboration\\ command.\n\n\\end{itemize}\n\nTable~\\ref{tab:front} summarizes some common pitfalls in moving from\n\\revtex~3 to \\revtex~4.\n\\begin{table*}\n\\begin{ruledtabular}\n\\begin{tabular}{lll}\n\\textbf{\\revtex~3 Markup} & \\textbf{\\revtex~4 Markup} & \\textbf{Explanation}\\\\\n& & \\\\\n\\verb+\\author{Author One and Author Two}+ & \\verb+\\author{Author One}+ & One name per\\\\\n& \\verb+\\author{Author Two}+  & \\verb+\\author+ \\\\\n& & \\\\\n\\verb+\\author{Author One$^{1}$}+ & \\verb+\\author{Author One}+& Use \\classoption{superscriptaddress}\\\\\n\\dots &\\dots &  class option \\\\\n\\verb+\\address{$^{1}$APS}+ &\\verb+\\affiliation{APS}+ & \\\\\n& & \\\\\n\\verb+\\thanks{Permanent address...}+ & \\verb+\\altaffiliation{}+& Use most\nspecific macro \\\\\n\\verb+\\thanks{Electronic address: user@domain.edu}+ &\n\\verb+\\email{user@domain.edu}+& available\\\\\n\\verb+\\thanks{http:\/\/publish.aps.org\/}+ &\n\\verb+\\homepage{http:\/\/publish.aps.org\/}+& \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\caption{Common mistakes in marking up the front matter}\n\\label{tab:front}\n\\end{table*}\n\n\n\\subsection{Abstracts}\n\\revtex~4, like \\revtex~3, uses the \\env{abstract} environment\n\\envb{abstract} \\dots \\enve{abstract} for the abstract. The\n\\env{abstract} environment must appear before the \\cmd{\\maketitle}\ncommand in \\revtex~4. The abstract will be formatted\nappropriately for either one-column (preprint) or two-column\nformatting. In particular, in the two-column case, the abstract will\nautomatically be placed in a single column that spans the width of the\npage. It is unnecessary to use a \\cmd{\\minipage} or any other macro to\nachieve this result.\n\n\n\\section{Citations and References}\n\n\\revtex~4 uses the same \\cmd{\\cite},\\cmd{\\ref}, and \\cmd{\\bibitem}\ncommmands as standard \\LaTeX\\ and \\revtex~3. Citation handling is\nbased upon Patick Daly's \\classname{natbib} package. The\n\\env{references} environment is no longer used. Instead, use the\nstandard \\LaTeXe\\ environment \\env{thebibliography}.\n\nTwo new \\BibTeX\\ files have been included with \\revtex~4,\n\\file{apsrev.bst} and \\file{apsrmp.bst}. These will format references\nin the style of \\textit{Physical Review} and \\textit{Reviews of Modern\nPhysics} respectively. In addition, these \\BibTeX\\ styles\nautomatically apply a special macro \\cmd{\\bibinfo} to each element of the\nbibliography to make it easier to extract information for use in the\neditorial and production processes. Authors are strongly urged to use\n\\BibTeX\\ to manage their bibliographies so that the \\cmd{\\bibinfo}\ndirectives will be automatically included.  Other bibliography styles\ncan be specified by using the \\cmd\\bibliographystyle\\ command, but\nunlike standard \\LaTeXe, you must give this command \\emph{before} the\n\\envb{document} statement.\n\nPlease note that the package \\classname{cite.sty} is not needed with\n\\revtex~4 and is incompatible.\n\n\\section{Footnotes and Tablenotes}\n\\label{sec:foot}\n\n\\revtex~4 uses the standard \\cmd{\\footnote} macro for\nfootnotes. Footnotes can either appear on the bottom of the page on\nwhich they occur or they can appear as entries at the end of the\nbibliography. As with author notes, the journal style option controls\nthe placement; however, this can be overridden with the class options\n\\classoption{footinbib} and \\classoption{nofootinbib}.\n\nWithin a table, the \\cmd{\\footnote} command behaves differently. Footnotes\nappear at the bottom of the table. \\cmd{\\footnotemark} and\n\\cmd{\\footnotetext} are also available within the table environment so\nthat multiple table entries can share the same footnote text.  There\nis no longer a need to use a \\cmd{\\tablenote}, \\cmd{\\tablenotemark},\nand \\cmd{\\tablenotetext} macros.\n\n\\section{Section Commands}\n\nThe title in a \\cmd\\section\\marg{title} command will be automatically\nuppercased in \\revtex~4. To prevent a particular letter from being\nuppercased, enclose it in curly braces.\n\n\\section{Figures}\n\nFigures should be enclosed within either a \\env{figure} or \\env{figure*}\nenvironment (the latter will cause the figure to span the full width\nof the page in two-column mode). \\LaTeXe\\ has two convenient packages\nfor including the figure file itself: \\classname{graphics} and\n\\classname{graphicx}. These two packages both define a macro\n\\cmd{\\includegraphics} which calls in the figure. They differ in how\narguments for rotation, translation, and scaling are specified. The\npackage \\classname{epsfig} has been re-implemented to use these\n\\classname{graphicx} package. The package \\classname{epsfig} provides\nan interface similar to that under the \\revtex~3 \\classoption{epsf}\nclass option. Authors should use these standard\n\\LaTeXe\\ packages rather than some other alternative.\n\n\\section{Tables}\n\nShort tables should be enclosed within either a \\env{table} or \\env{table*}\nenvironmnent (the latter will cause the table to span the full width\nof the page in two-column mode). The heart of the table is the\n\\env{tabular} environment. This will behave for the most part as in\nstandard \\LaTeXe. Note that \\revtex~4 no longer automatically adds\ndouble (Scotch) rules around tables. Nor does the \\env{tabular}\nenvironment set various table parameters as before. Instead, a new\nenvironment \\env{ruledtabular} provides this functionality. This\nenvironment should surround the \\env{tabular} environment:\n\\begin{verbatim}\n\\begin{table}\n\\caption{...}\n\\label{tab:...}\n\\begin{ruledtabular}\n\\begin{tabular}\n...\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\\end{verbatim}\n\nUnder \\revtex~3, tables automatically break across pages. \\revtex~4\nprovides some of this functionality. However, this requires adding the\ntable a float placement option of [H] (meaning put the table\n``here'') to the \\envb{table} command.\n\nLong tables are more robustly handled by using the\n\\classname{longtable.sty} package included with the standard \\LaTeXe\\\ndistribution (put \\verb+\\usepackage{longtable}+ in the preamble). This\npackage gives precise control over the layout of the table. \\revtex~4\ngoes out of its way to provide patches so that the \\env{longtable}\nenvironment will work within a two-column format. A new\n\\env{longtable*} environment is also provided for long tables that are\ntoo wide for a narrow column. (Note that the \\env{table*} and\n\\env{longtable*} environments should always be used rather than\nattempting to use the \\env{widetext} environment.)\n\nTo create tables with columns of numbers aligned on decimal points,\nload the standard \\LaTeXe\\ \\classname{dcolumn} package and use the\n\\verb+d+ column specifier. The content of each cell in the column is\nimplicitly in math mode: Use of math delimiters (\\verb+$+) is unnecessary\nin a \\verb+d+ column.\n\nFootnotes within a table can be specified with the\n\\cmd{\\footnote} command (see Sec.~\\ref{sec:foot}). \n\n\\section{Font selection}\n\nThe largest difference between \\revtex~3 and \\revtex~4 with respect to\nfonts is that \\revtex~4 allows one use the \\LaTeXe\\ font commands such\nas \\cmd{\\textit}, \\cmd{\\texttt}, \\cmd{\\textbf} etc. These commands\nshould be used in place of the basic \\TeX\/\\LaTeX\\ 2.09 font commands\nsuch as \\cmd{\\it}, \\cmd{\\tt}, \\cmd{\\bf}, etc. The new font commands\nbetter handle subtleties such as italic correction and scaling in\nsuper- and subscripts.\n\n\\section{Math and Symbols}\n\n\\revtex~4 depends more heavily on packages from the standard \\LaTeXe\\\ndistribution and AMS-\\LaTeX\\ than \\revtex~3 did. Thus, \\revtex~4 users\nshould make sure their \\LaTeXe\\ distributions are up to date and they\nshould install AMS-\\LaTeX\\ 2.0 as well. In general, if any fine control of\nequation layout, special math symbols, or other specialized math\nconstructs are needed, users should look to the \\classname{amsmath}\npackage (see the AMS-\\LaTeX\\ documentation).\n\n\\revtex~4  provides a small number of additional diacritics, symbols,\nand bold parentheses. Table~\\ref{tab:revsymb} summarizes this.\n\n\\begin{table}\n\\caption{Special \\revtex~4 symbols, accents, and boldfaced parentheses \ndefined in \\file{revsymb.sty}}\n\\label{tab:revsymb}\n\\begin{ruledtabular}\n\\begin{tabular}{ll|ll}\n\\cmd\\lambdabar & $\\lambdabar$ &\\cmd\\openone & $\\openone$\\\\\n\\cmd\\altsuccsim & $\\altsuccsim$ & \\cmd\\altprecsim & $\\altprecsim$ \\\\\n\\cmd\\alt & $\\alt$ & \\cmd\\agt & $\\agt$ \\\\\n\\cmd\\tensor\\ x & $\\tensor x$ & \\cmd\\overstar\\ x & $\\overstar x$ \\\\\n\\cmd\\loarrow\\ x & $\\loarrow x$ & \\cmd\\roarrow\\ x & $\\roarrow x$  \\\\\n\\cmd\\biglb\\ ( \\cmd\\bigrb ) & $\\biglb( \\bigrb)$ &\n\\cmd\\Biglb\\ ( \\cmd\\Bigrb )& $\\Biglb( \\Bigrb)$ \\\\\n& & \\\\\n\\cmd\\bigglb\\ ( \\cmd\\biggrb ) & $\\bigglb( \\biggrb)$ &\n\\cmd\\Bigglb\\ ( \\cmd\\Biggrb\\ ) & $\\Bigglb( \\Biggrb)$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nHere is a partial list of the more notable changes between \\revtex~3\nand \\revtex~4 math:\n\\begin{itemize}\n\\item Bold math characters should now be handle via the standard\n\\LaTeXe\\ \\classname{bm} package (use \\cmd{\\bm} instead of \\cmd{\\bbox}).\n\\cmd{\\bm} will handle Greek letters and other symbols.\n\n\\item Use the class options \\classoption{amsmath},\n\\classoption{amsfonts} and \\classoption{amssymb} to get even more math\nfonts and symbols. \\cmd{\\mathfrak} and \\cmd{\\mathbb} will, for instance, give\nFraktur and Blackboard Bold symbols.\n\n\\item Use the \\classoption{fleqn} class option for making equation\nflush left or right. \\cmd{\\FL} and \\cmd{\\FR} are no longer provided.\n\n\\item In place of \\cmd{\\eqnum}, load the \\classname{amsmath} package\n[\\verb+\\usepackage{amsmath}+] and use \\cmd{\\tag}.\n\n\\item In place of \\cmd{\\case}, use \\cmd{\\textstyle}\\cmd{\\frac}.\n\n\\item In place of the \\env{mathletters} environment, load the\n\\classname{amsmath} package and use \\env{subequations} environment.\n\n\\item In place of \\cmd{\\slantfrac}, use \\cmd{\\frac}.\n\n\\item The macros \\cmd{\\corresponds}, \\cmd{\\overdots}, and\n\\cmd{\\overcirc} have been removed. See Table~\\ref{tab:obsolete}.\n\n\\end{itemize}\n\n\\section{Obsolete \\revtex~3.1 commands}\n\nTable~\\ref{tab:obsolete} summarizes more differences between \\revtex~4\nand \\revtex~3, particularly which \\revtex~3 commands are now obsolete.\n\n\\begin{table*}\n\\caption{Differences between \\revtex~3.1 and \\revtex~4\nmarkup}\\label{tab:diff31}\n\\label{tab:obsolete}\n\\begin{ruledtabular}\n\\begin{tabular}{lp{330pt}}\n\\textbf{\\revtex~3.1 command}&\\textbf{\\revtex~4 replacement}\n\\lrstrut\\\\\n\\cmd\\documentstyle\\oarg{options}\\aarg{\\classname{revtex}}&\\cmd\\documentclass\\oarg{options}\\aarg{\\classname{revtex4}}\n\\\\\noption  \\classoption{manuscript}& \\classoption{preprint}\n\\\\\n\\cmd\\tighten\\ preamble command & \\classoption{tightenlines} class option\n\\\\\n\\cmd\\draft\\ preamble command   & \\classoption{draft} class option\n\\\\\n\\cmd\\author                    & \\cmd\\author\\marg{name} may appear\nmultiple times; each signifies a new author name.\\\\\n                               & \\cmd\\collaboration\\marg{name}:\nCollaboration name (should appear after last \\cmd\\author)\\\\\n                               & \\cmd\\homepage\\marg{URL}: URL for preceding author\\\\\n                               & \\cmd\\email\\marg{email}: email\naddress for preceding author\\\\\n                               & \\cmd{\\altaffiliation}: alternate\naffiliation for preceding \\cmd\\author\\\\\n\\cmd\\thanks                    & \\cmd\\thanks, but use only for\ninformation not covered by \\cmd{\\email}, \\cmd{\\homepage}, or \\cmd{\\altaffilitiation}\\\\\n\\cmd\\and                       & obsolete, remove this command\\\\\n\\cmd\\address                   & \\cmd\\affiliation\\marg{institution}\\ gives the affiliation for the group of authors above\\\\\n                               & \\cmd\\affiliation\\oarg{note} lets you specify a footnote to this institution\\\\\n                               & \\cmd\\noaffiliation\\ signifies that the above authors have no affiliation\\\\\n\n\\cmd\\preprint                  & \\cmd\\preprint\\marg{number} can appear multiple times, and must precede \\cmd\\maketitle\\\\\n\\cmd\\pacs                      & \\cmd\\pacs\\ must precede \\cmd\\maketitle\\\\\n\\env{abstract} environment     & \\env{abstract} environment must precede \\cmd\\maketitle\\\\\n\\cmd\\wideabs                       & obsolete, remove this command\\\\\n\\cmd\\maketitle                 & \\cmd\\maketitle\\ must follow\n\\emph{all} front matter data commands\\\\\n\\cmd\\narrowtext                & obsolete, remove this command\\\\\n\\cmd\\mediumtext                & obsolete, remove this command\\\\\n\\cmd\\widetext                  & obsolete, replace with \\env{widetext} environment\\\\\n\\cmd\\FL                        & obsolete, remove this command\\\\\n\\cmd\\FR                        & obsolete, remove this command\\\\\n\\cmd\\eqnum                     & replace with \\cmd\\tag, load \\classname{amsmath}\\\\\n\\env{mathletters}              & replace with \\env{subequations}, load\n\\classname{amsmath}\\\\\n\\env{tabular} environment      & No longer puts in doubled-rules. Enclose \\env{tabular} in \\env{ruledtabular} to get old behavior.\\\\\n\\env{quasitable} environment   & obsolete, \\env{tabular} environment no longer\nputs in rules\\\\\n\\env{references} environment   & replace with \\env{thebibliography}\\verb+{}+\\\\\n\\cmd\\case                      & replace with \\cmd\\textstyle\\cmd\\frac\\\\\n\\cmd\\slantfrac                 & replace with \\cmd\\frac\\\\\n\\cmd\\tablenote                 & replace with \\cmd\\footnote\\\\\n\\cmd\\tablenotemark             & replace with \\cmd\\footnotemark\\\\\n\\cmd\\tablenotetext             & replace with \\cmd\\footnotetext\\lrstrut\\\\\n\\cmd\\overcirc                  & Use standard \\LaTeXe\\ \\cmd\\mathring\\ \\\\\n\\cmd\\overdots                  & Use \\cmd\\dddot\\ with \\classoption{amsmath}\\\\\n\\cmd\\corresponds               & Use \\cmd\\triangleq\\ with \\classoption{amssymb}\\\\\n\\classoption{epsf} class option & \\verb+\\usepackage{epsfig}+\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\n\n\\section{Converting a \\revtex~3.1 Document to \\revtex~4}\\label{sec:conv31}%\n\n\\revtex~3 documents can be converted to \\revtex~4 rather\nstraightforwardly. The following checklist covers most of the major\nsteps involved.\n\n\\begin{itemize}\n\\item Change \\cmd\\documentstyle\\verb+{revtex}+ to\n\\cmd\\documentclass\\verb+{revtex4}+, and run the document under\n\\LaTeXe\\ instead of \\LaTeX2.09.\n\n\\item\nReplace the \\cmd\\draft\\ command with the \\classoption{draft} class option.\n\n\\item\nReplace the \\cmd\\tighten\\ command with the \\classoption{tightenlines}\nclass option.\n\n\\item\nFor each \\cmd\\author\\ command, split the multiple authors into\nindividual \\cmd\\author\\ commands. Remove any instances of \\cmd\\and.\n\n\\item For superscript-style associations between authors and\naffiliations, remove explicit superscripts and use the\n\\classoption{superscriptaddress} class option.\n\n\\item\nUse \\cmd\\affiliation\\ instead of \\cmd\\address.\n\n\\item\nPut \\cmd\\maketitle\\ after the \\env{abstract} environment and any\n\\cmd\\pacs\\ commands.\n\n\\item If double-ruled table borders are desired, enclose \\env{tabular}\nenviroments in \\env{ruledtabular} environments.\n\n\\item\nConvert long tables to \\env{longtable}, and load the\n\\classname{longtable} package. Alternatively, give the \\env{table}\nan [H] float placement parameter so that the table will break automatically.\n\n\\item\nReplace any instances of the \\cmd\\widetext\\ and \\cmd\\narrowtext\\\ncommands with the \\env{widetext} environment.\nUsually, the \\envb{widetext} statement will replace the \\cmd\\widetext\\\ncommand, and the \\enve{widetext} statement replaces the matching\n\\cmd\\narrowtext\\ command.\n\nNote in this connection that due to a curious feature of \\LaTeX\\\nitself, \\revtex~4 having a \\env{widetext} environment means that it\nalso has a definition for the \\cmd\\widetext\\ command, even though the\nlatter cammand is not intended to be used in your document.\nTherefore, it is particularly important to remove\nall \\cmd\\widetext\\ commands when converting to \\revtex~4.\n\n\\item\nRemove all obsolete commands: \\cmd\\FL, \\cmd\\FR, \\cmd\\narrowtext, and\n\\cmd\\mediumtext\\ (see Table~\\ref{tab:diff31}).\n\n\\item\nReplace \\cmd\\case\\ with \\cmd\\frac. If a fraction needs to be set\nin text style despite being in a display equation, use the\nconstruction \\cmd\\textstyle\\cmd\\frac.  Note that \\cmd\\frac\\ does not\nsupport the syntax \\cmd\\case\\verb+1\/2+.\n\n\\item\nReplace \\cmd\\slantfrac\\ with \\cmd\\frac.\n\n\\item\nChange \\cmd\\frak\\ to \\cmd\\mathfrak\\marg{char}\\index{Fraktur} and\n\\cmd\\Bbb\\ to \\cmd\\mathbb\\marg{char}\\index{Blackboard Bold}, and invoke\none of the class options \\classoption{amsfonts} or\n\\classoption{amssymb}.\n\n\\item\nReplace environment \\env{mathletters} with environment\n\\env{subequations} and load the \\classname{amsmath} package.\n\n\\item\nReplace \\cmd\\eqnum\\ with \\cmd\\tag\\ and load the \\classname{amsmath} package.\n\n\\item\nReplace \\cmd\\bbox\\ with \\cmd\\bm\\ and load the \\classname{bm} package.\n\n\\item\nIf using the \\cmd\\text\\ command, load the \\classname{amsmath} package.\n\n\\item\nIf using the \\verb+d+ column specifier in \\env{tabular} environments,\nload the \\classname{dcolumn} package. Under \\classname{dcolumn}, the\ncontent of each \\verb+d+ column cell is implicitly in math mode:\nremove any \\verb+$+ math delimiters appearing in cells in a \\verb+d+\ncolumn.\n\n\\item\nReplace \\cmd\\tablenote\\ with \\cmd\\footnote, \\cmd\\tablenotemark\\ with\n\\cmd\\footnotemark, and \\cmd\\tablenotetext\\ with \\cmd\\footnotetext.\n\n\\item\nReplace \\envb{references} with \\envb{thebibliography}\\verb+{}+;\n\\enve{references} with \\enve{thebibliography}.\n\\end{itemize}\n\\end{document}\n\n\\section{<heading>}+, \\verb+\\subsection{<heading>}+,\n\\verb+\\subsubsection{<heading>}+ & Start a new section or\nsubsection.\\\\\n\\verb+\\section*{<heading>}+ & Start a new section without a number.\\\\\n\\verb+\n\\section{\\label{sec:level1}First-level heading:\\protect\\\\ The line\nbreak was forced \\lowercase{via} \\textbackslash\\textbackslash}\n\nThis sample document demonstrates proper use of REV\\TeX~4 (and\n\\LaTeXe) in mansucripts prepared for submission to APS\njournals. Further information can be found in the REV\\TeX~4\ndocumentation included in the distribution or available at\n\\url{http:\/\/publish.aps.org\/revtex4\/}.\n\nWhen commands are referred to in this example file, they are always\nshown with their required arguments, using normal \\TeX{} format. In\nthis format, \\verb+#1+, \\verb+#2+, etc. stand for required\nauthor-supplied arguments to commands. For example, in\n\\verb+\\section{#1}+ the \\verb+#1+ stands for the title text of the\nauthor's section heading, and in \\verb+\\title{#1}+ the \\verb+#1+\nstands for the title text of the paper.\n\nLine breaks in section headings at all levels can be introduced using\n\\textbackslash\\textbackslash. A blank input line tells \\TeX\\ that the\nparagraph has ended. Note that top-level section headings are\nautomatically uppercased. If a specific letter or word should appear in\nlowercase instead, you must escape it using \\verb+\\lowercase{#1}+ as\nin the word ``via'' above.\n\n\\subsection{\\label{sec:level2}Second-level heading: Formatting}\n\nThis file may be formatted in both the \\texttt{preprint} and\n\\texttt{twocolumn} styles. \\texttt{twocolumn} format may be used to\nmimic final journal output. Either format may be used for submission\npurposes; however, for peer review and production, APS will format the\narticle using the \\texttt{preprint} class option. Hence, it is\nessential that authors check that their manuscripts format acceptably\nunder \\texttt{preprint}. Manuscripts submitted to APS that do not\nformat correctly under the \\texttt{preprint} option may be delayed in\nboth the editorial and production processes.\n\nThe \\texttt{widetext} environment will make the text the width of the\nfull page, as on page~\\pageref{eq:wideeq}. (Note the use the\n\\verb+\\pageref{#1}+ to get the page number right automatically.) The\nwidth-changing commands only take effect in \\texttt{twocolumn}\nformatting. It has no effect if \\texttt{preprint} formatting is chosen\ninstead.\n\n\\subsubsection{\\label{sec:level3}Third-level heading: References and Footnotes}\nReference citations in text use the commands \\verb+\\cite{#1}+ or\n\\verb+\\onlinecite{#1}+. \\verb+#1+ may contain letters and numbers.\nThe reference itself is specified by a \\verb+\\bibitem{#1}+ command\nwith the same argument as the \\verb+\\cite{#1}+ command.\n\\verb+\\bibitem{#1}+ commands may be crafted by hand or, preferably,\ngenerated by using Bib\\TeX. REV\\TeX~4 includes Bib\\TeX\\ style files\n\\verb+apsrev.bst+ and \\verb+apsrmp.bst+ appropriate for\n\\textit{Physical Review} and \\textit{Reviews of Modern Physics},\nrespectively. REV\\TeX~4 will automatically choose the style\nappropriate for the journal specified in the document class\noptions. This sample file demonstrates the basic use of Bib\\TeX\\\nthrough the use of \\verb+\\bibliography+ command which references the\n\\verb+assamp.bib+ file. Running Bib\\TeX\\ (typically \\texttt{bibtex\napssamp}) after the first pass of \\LaTeX\\ produces the file\n\\verb+apssamp.bbl+ which contains the automatically formatted\n\\verb+\\bibitem+ commands (including extra markup information via\n\\verb+\\bibinfo+ commands). If not using Bib\\TeX, the\n\\verb+thebibiliography+ environment should be used instead.\n\nTo cite bibliography entries, use the \\verb+\\cite{#1}+ command. Most\njournal styles will display the corresponding number(s) in square\nbrackets: \\cite{feyn54,witten2001}. To avoid the square brackets, use\n\\verb+\\onlinecite{#1}+: Refs.~\\onlinecite{feyn54} and\n\\onlinecite{witten2001}. REV\\TeX\\ ``collapses'' lists of\nconsecutive reference numbers where possible. We now cite everyone\ntogether \\cite{feyn54,witten2001,epr}, and once again\n(Refs.~\\onlinecite{epr,feyn54,witten2001}). Note that the references\nwere also sorted into the correct numerical order as well.\n\nWhen the \\verb+prb+ class option is used, the \\verb+\\cite{#1}+ command\ndisplays the reference's number as a superscript rather than using\nsquare brackets. Note that the location of the \\verb+\\cite{#1}+\ncommand should be adjusted for the reference style: the superscript\nreferences in \\verb+prb+ style must appear after punctuation;\notherwise the reference must appear before any punctuation. This\nsample was written for the regular (non-\\texttt{prb}) citation style.\nThe command \\verb+\\onlinecite{#1}+ in the \\texttt{prb} style also\ndisplays the reference on the baseline.\n\nFootnotes are produced using the \\verb+\\footnote{#1}+ command. Most\nAPS journal styles put footnotes into the bibliography. REV\\TeX~4 does\nthis as well, but instead of interleaving the footnotes with the\nreferences, they are listed at the end of the references\\footnote{This\nmay be improved in future versions of REV\\TeX.}. Because the correct\nnumbering of the footnotes must occur after the numbering of the\nreferences, an extra pass of \\LaTeX\\ is required in order to get the\nnumbering correct.\n\n\\section{Math and Equations}\nInline math may be typeset using the \\verb+$+ delimiters. Bold math\nsymbols may be achieved using the \\verb+bm+ package and the\n\\verb+\\bm{#1}+ command it supplies. For instance, a bold $\\alpha$ can\nbe typeset as \\verb+$\\bm{\\alpha}$+ giving $\\bm{\\alpha}$. Fraktur and\nBlackboard (or open face or double struck) characters should be\ntypeset using the \\verb+\\mathfrak{#1}+ and \\verb+\\mathbb{#1}+ commands\nrespectively. Both are supplied by the \\texttt{amssymb} package. For\nexample, \\verb+$\\mathbb{R}$+ gives $\\mathbb{R}$ and\n\\verb+$\\mathfrak{G}$+ gives $\\mathfrak{G}$\n\nIn \\LaTeX\\ there are many different ways to display equations, and a\nfew preferred ways are noted below. Displayed math will center by\ndefault. Use the class option \\verb+fleqn+ to flush equations left.\n\nBelow we have numbered single-line equations; this is the most common\ntype of equation in \\textit{Physical Review}:\n\\begin{eqnarray}\n\\chi_+(p)\\alt{\\bf [}2|{\\bf p}|(|{\\bf p}|+p_z){\\bf ]}^{-1\/2}\n\\left(\n\\begin{array}{c}\n|{\\bf p}|+p_z\\\\\npx+ip_y\n\\end{array}\\right)\\;,\n\\\\\n\\left\\{%\n \\openone234567890abc123\\alpha\\beta\\gamma\\delta1234556\\alpha\\beta\n \\frac{1\\sum^{a}_{b}}{A^2}%\n\\right\\}%\n\\label{eq:one}.\n\\end{eqnarray}\nNote the open one in Eq.~(\\ref{eq:one}).\n\nNot all numbered equations will fit within a narrow column this\nway. The equation number will move down automatically if it cannot fit\non the same line with a one-line equation:\n\\begin{equation}\n\\left\\{\n ab12345678abc123456abcdef\\alpha\\beta\\gamma\\delta1234556\\alpha\\beta\n \\frac{1\\sum^{a}_{b}}{A^2}%\n\\right\\}.\n\\end{equation}\n\nWhen the \\verb+\\label{#1}+ command is used [cf. input for\nEq.~(\\ref{eq:one})], the equation can be referred to in text without\nknowing the equation number that \\TeX\\ will assign to it. Just\nuse \\verb+\\ref{#1}+, where \\verb+#1+ is the same name that used in\nthe \\verb+\\label{#1}+ command.\n\nUnnumbered single-line equations can be typeset\nusing the \\verb+\\[+, \\verb+\\]+ format:\n\\[g^+g^+ \\rightarrow g^+g^+g^+g^+ \\dots ~,~~q^+q^+\\rightarrow\nq^+g^+g^+ \\dots ~. \\]\n\n\\subsection{Multiline equations}\n\nMultiline equations are obtained by using the \\verb+eqnarray+\nenvironment.  Use the \\verb+\\nonumber+ command at the end of each line\nto avoid assigning a number:\n\\begin{eqnarray}\n{\\cal M}=&&ig_Z^2(4E_1E_2)^{1\/2}(l_i^2)^{-1}\n\\delta_{\\sigma_1,-\\sigma_2}\n(g_{\\sigma_2}^e)^2\\chi_{-\\sigma_2}(p_2)\\nonumber\\\\\n&&\\times\n[\\epsilon_jl_i\\epsilon_i]_{\\sigma_1}\\chi_{\\sigma_1}(p_1),\n\\end{eqnarray}\n\\begin{eqnarray}\n\\sum \\vert M^{\\text{viol}}_g \\vert ^2&=&g^{2n-4}_S(Q^2)~N^{n-2}\n        (N^2-1)\\nonumber \\\\\n & &\\times \\left( \\sum_{i<j}\\right)\n  \\sum_{\\text{perm}}\n \\frac{1}{S_{12}}\n \\frac{1}{S_{12}}\n \\sum_\\tau c^f_\\tau~.\n\\end{eqnarray}\n\\textbf{Note:} Do not use \\verb+\\label{#1}+ on a line of a multiline\nequation if \\verb+\\nonumber+ is also used on that line. Incorrect\ncross-referencing will result. Notice the use \\verb+\\text{#1}+ for\nusing a Roman font within a math environment.\n\nTo set a multiline equation without \\emph{any} equation\nnumbers, use the \\verb+\\begin{eqnarray*}+,\n\\verb+\\end{eqnarray*}+ format:\n\\begin{eqnarray*}\n\\sum \\vert M^{\\text{viol}}_g \\vert ^2&=&g^{2n-4}_S(Q^2)~N^{n-2}\n        (N^2-1)\\\\\n & &\\times \\left( \\sum_{i<j}\\right)\n \\left(\n  \\sum_{\\text{perm}}\\frac{1}{S_{12}S_{23}S_{n1}}\n \\right)\n \\frac{1}{S_{12}}~.\n\\end{eqnarray*}\nTo obtain numbers not normally produced by the automatic numbering,\nuse the \\verb+\\tag{#1}+ command, where \\verb+#1+ is the desired\nequation number. For example, to get an equation number of\n(\\ref{eq:mynum}),\n\\begin{equation}\ng^+g^+ \\rightarrow g^+g^+g^+g^+ \\dots ~,~~q^+q^+\\rightarrow\nq^+g^+g^+ \\dots ~. \\tag{2.6$'$}\\label{eq:mynum}\n\\end{equation}\n\nA few notes on \\verb=\\tag{#1}=. \\verb+\\tag{#1}+ requires\n\\texttt{amsmath}. The \\verb+\\tag{#1}+ must come before the\n\\verb+\\label{#1}+, if any. The numbering set with \\verb+\\tag{#1}+ is\n\\textit{transparent} to the automatic numbering in REV\\TeX{};\ntherefore, the number must be known ahead of time, and it must be\nmanually adjusted if other equations are added. \\verb+\\tag{#1}+ works\nwith both single-line and multiline equations. \\verb+\\tag{#1}+ should\nonly be used in exceptional case - do not use it to number all\nequations in a paper.\n\nEnclosing single-line and multiline equations in\n\\verb+\\begin{subequations}+ and \\verb+\\end{subequations}+ will produce\na set of equations that are ``numbered'' with letters, as shown in\nEqs.~(\\ref{subeq:1}) and (\\ref{subeq:2}) below:\n\\begin{subequations}\n\\label{eq:whole}\n\\begin{equation}\n\\left\\{\n abc123456abcdef\\alpha\\beta\\gamma\\delta1234556\\alpha\\beta\n \\frac{1\\sum^{a}_{b}}{A^2}\n\\right\\},\\label{subeq:1}\n\\end{equation}\n\\begin{eqnarray}\n{\\cal M}=&&ig_Z^2(4E_1E_2)^{1\/2}(l_i^2)^{-1}\n(g_{\\sigma_2}^e)^2\\chi_{-\\sigma_2}(p_2)\\nonumber\\\\\n&&\\times\n[\\epsilon_i]_{\\sigma_1}\\chi_{\\sigma_1}(p_1).\\label{subeq:2}\n\\end{eqnarray}\n\\end{subequations}\nPutting a \\verb+\\label{#1}+ command right after the\n\\verb+\\begin{subequations}+, allows one to\nreference all the equations in a subequations environment. For\nexample, the equations in the preceding subequations environment were\nEqs.~(\\ref{eq:whole}).\n\n\\subsubsection{Wide equations}\nThe equation that follows is set in a wide format, i.e., it spans\nacross the full page. The wide format is reserved for long equations\nthat cannot be easily broken into four lines or less:\n\\begin{widetext}\n\\begin{equation}\n{\\cal R}^{(\\text{d})}=\n g_{\\sigma_2}^e\n \\left(\n   \\frac{[\\Gamma^Z(3,21)]_{\\sigma_1}}{Q_{12}^2-M_W^2}\n  +\\frac{[\\Gamma^Z(13,2)]_{\\sigma_1}}{Q_{13}^2-M_W^2}\n \\right)\n + x_WQ_e\n \\left(\n   \\frac{[\\Gamma^\\gamma(3,21)]_{\\sigma_1}}{Q_{12}^2-M_W^2}\n  +\\frac{[\\Gamma^\\gamma(13,2)]_{\\sigma_1}}{Q_{13}^2-M_W^2}\n \\right)\\;. \\label{eq:wideeq}\n\\end{equation}\n\\end{widetext}\nThis is typed to show the output is in wide format.\n(Since there is no input line between \\verb+\\equation+ and\nthis paragraph, there is no paragraph indent for this paragraph.)\n\\section{Cross-referencing}\nREV\\TeX{} will automatically number sections, equations, figure\ncaptions, and tables. In order to reference them in text, use the\n\\verb+\\label{#1}+ and \\verb+\\ref{#1}+ commands. To reference a\nparticular page, use the \\verb+\\pageref{#1}+ command.\n\nThe \\verb+\\label{#1}+ should appear in a section heading, within an\nequation, or in a table or figure caption. The \\verb+\\ref{#1}+ command\nis used in the text where the citation is to be displayed.  Some\nexamples: Section~\\ref{sec:level1} on page~\\pageref{sec:level1},\nTable~\\ref{tab:table1}, and Fig.~\\ref{fig:epsart}.\n\n\\section{Figures and Tables}\nFigures and tables are typically ``floats'' which means that their\nfinal position is determined by \\LaTeX\\ while the document is being\ntypeset. \\LaTeX\\ isn't always successful in placing floats\noptimally.\n\nFigures may be inserted by using either the \\texttt{graphics} or\n\\texttt{graphix} packages. These packages both define the\n\\verb+\\includegraphics{#1}+ command, but they differ in how optional\narguments for specifying the orientation, scaling, and translation of the\nfigure. Fig.~\\ref{fig:epsart} shows a figure that is small enough to\nfit in a single column. It is embedded using the \\texttt{figure}\nenvironment which provides both the caption and the imports the figure\nfile.\n\\begin{figure}\n\\includegraphics{fig_1\n\\caption{\\label{fig:epsart} A figure caption. The figure captions are\nautomatically numbered.}\n\\end{figure}\n\nFig.~\\ref{fig:wide} is a figure that is too wide for a single column,\nso instead the \\texttt{figure*} environment has been used.\n\\begin{figure*}\n\\includegraphics{fig_2\n\\caption{\\label{fig:wide}Use the figure* environment to get a wide\nfigure that spans the page in \\texttt{twocolumn} formatting.}\n\\end{figure*}\n\nThe heart of any table is the \\texttt{tabular} environment which gives\nthe rows of the tables. Each row consists of column entries separated\nby \\verb+&+'s and terminates with \\textbackslash\\textbackslash. The\nrequired argument for the \\texttt{tabular} environment\nspecifies how data are displayed in the columns. For instance, entries\nmay be centered, left-justified, right-justified, aligned on a decimal\npoint. Extra column-spacing may be be specified as well, although\nREV\\TeX~4 sets this spacing so that the columns fill the width of the\ntable. Horizontal rules are typeset using the \\verb+\\hline+\ncommand. The doubled (or Scotch) rules that appear at the top and\nbottom of a table can be achieved enclosing the \\texttt{tabular}\nenvironment within a \\texttt{ruledtabular} environment. Rows whose\ncolumns span multiple columns can be typeset using the\n\\verb+\\multicolumn{#1}{#2}{#3}+ command (for example, see the first\nrow of Table~\\ref{tab:table3}).\n\nTables~\\ref{tab:table1}-\\ref{tab:table4} show various effects. Tables\nthat fit in a narrow column are contained in a \\texttt{table}\nenvironment. Table~\\ref{tab:table3} is a wide table set with the\n\\texttt{table*} environment. Long tables may need to break across\npages. The most straightforward way to accomplish this is to specify\nthe \\verb+[H]+ float placement on the \\texttt{table} or\n\\texttt{table*} environment. However, the standard \\LaTeXe\\ package\n\\texttt{longtable} will give more control over how tables break and\nwill allow headers and footers to be specified for each page of the\ntable. A simple example of the use of \\texttt{longtable} can be found\nin the file \\texttt{summary.tex} that is included with the REV\\TeX~4\ndistribution.\n\nThere are two methods for setting footnotes within a table (these\nfootnotes will be displayed directly below the table rather than at\nthe bottom of the page or in the bibliography). The easiest\nand preferred method is just to use the \\verb+\\footnote{#1}+\ncommand. This will automatically enumerate the footnotes with\nlowercase roman letters. However, it is sometimes necessary to have\nmultiple entries in the table share the same footnote. In this case,\nthere is no choice but to manually create the footnotes using\n\\verb+\\footnotemark[#1]+ and \\verb+\\footnotetext[#1]{#2}+.\n\\texttt{\\#1} is a numeric value. Each time the same value for\n\\texttt{\\#1} is used, the same mark is produced in the table. The\n\\verb+\\footnotetext[#1]{#2}+ commands are placed after the \\texttt{tabular}\nenvironment. Examine the \\LaTeX\\ source and output for\nTables~\\ref{tab:table1} and \\ref{tab:table2} for examples.\n\n\\begin{table}\n\\caption{\\label{tab:table1}This is a narrow table which fits into a\nnarrow column when using \\texttt{twocolumn} formatting. Note that\nREV\\TeX~4 adjusts the intercolumn spacing so that the table fills the\nentire width of the column. Table captions are numbered\nautomatically. This table illustrates left-aligned, centered, and\nright-aligned columns.  }\n\\begin{ruledtabular}\n\\begin{tabular}{lcr}\nLeft\\footnote{Note a.}&Centered\\footnote{Note b.}&Right\\\\\n\\hline\n1 & 2 & 3\\\\\n10 & 20 & 30\\\\\n100 & 200 & 300\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\begin{table}\n\\caption{\\label{tab:table2}A table with more columns still fits\nproperly in a column. Note that several entries share the same\nfootnote. Inspect the \\LaTeX\\ input for this table to see\nexactly how it is done.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccccccc}\n &$r_c$ (\\AA)&$r_0$ (\\AA)&$\\kappa r_0$&\n &$r_c$ (\\AA) &$r_0$ (\\AA)&$\\kappa r_0$\\\\\n\\hline\nCu& 0.800 & 14.10 & 2.550 &Sn\\footnotemark[1]\n& 0.680 & 1.870 & 3.700 \\\\\nAg& 0.990 & 15.90 & 2.710 &Pb\\footnotemark[2]\n& 0.450 & 1.930 & 3.760 \\\\\nAu& 1.150 & 15.90 & 2.710 &Ca\\footnotemark[3]\n& 0.750 & 2.170 & 3.560 \\\\\nMg& 0.490 & 17.60 & 3.200 &Sr\\footnotemark[4]\n& 0.900 & 2.370 & 3.720 \\\\\nZn& 0.300 & 15.20 & 2.970 &Li\\footnotemark[2]\n& 0.380 & 1.730 & 2.830 \\\\\nCd& 0.530 & 17.10 & 3.160 &Na\\footnotemark[5]\n& 0.760 & 2.110 & 3.120 \\\\\nHg& 0.550 & 17.80 & 3.220 &K\\footnotemark[5]\n&  1.120 & 2.620 & 3.480 \\\\\nAl& 0.230 & 15.80 & 3.240 &Rb\\footnotemark[3]\n& 1.330 & 2.800 & 3.590 \\\\\nGa& 0.310 & 16.70 & 3.330 &Cs\\footnotemark[4]\n& 1.420 & 3.030 & 3.740 \\\\\nIn& 0.460 & 18.40 & 3.500 &Ba\\footnotemark[5]\n& 0.960 & 2.460 & 3.780 \\\\\nTl& 0.480 & 18.90 & 3.550 & & & & \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\footnotetext[1]{Here's the first, from Ref.~\\onlinecite{feyn54}.}\n\\footnotetext[2]{Here's the second.}\n\\footnotetext[3]{Here's the third.}\n\\footnotetext[4]{Here's the fourth.}\n\\footnotetext[5]{And etc.}\n\\end{table}\n\n\\begin{table*}\n\\caption{\\label{tab:table3}This is a wide table that spans the page\nwidth in \\texttt{twocolumn} mode. It is formatted using the\n\\texttt{table*} environment. It also demonstates the use of\n\\textbackslash\\texttt{multicolumn} in rows with entries that span\nmore than one column.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccc}\n &\\multicolumn{2}{c}{$D_{4h}^1$}&\\multicolumn{2}{c}{$D_{4h}^5$}\\\\\n Ion&1st alternative&2nd alternative&lst alternative\n&2nd alternative\\\\ \\hline\n K&$(2e)+(2f)$&$(4i)$ &$(2c)+(2d)$&$(4f)$ \\\\\n Mn&$(2g)$\\footnote{The $z$ parameter of these positions is $z\\sim\\frac{1}{4}$.}\n &$(a)+(b)+(c)+(d)$&$(4e)$&$(2a)+(2b)$\\\\\n Cl&$(a)+(b)+(c)+(d)$&$(2g)$\\footnotemark[1]\n &$(4e)^{\\text{a}}$\\\\\n He&$(8r)^{\\text{a}}$&$(4j)^{\\text{a}}$&$(4g)^{\\text{a}}$\\\\\n Ag& &$(4k)^{\\text{a}}$& &$(4h)^{\\text{a}}$\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\n\\begin{table}\n\\caption{\\label{tab:table4}Numbers in columns Three--Five have been\naligned by using the ``d'' column specifier (requires the\n\\texttt{dcolumn} package). Non-numeric entries (those entries without\na ``.'') in a ``d'' column are aligned on the decimal point. Use the\n``D'' specifier for more complex layouts. }\n\\begin{ruledtabular}\n\\begin{tabular}{ccddd}\nOne&Two&\\mbox{Three}&\\mbox{Four}&\\mbox{Five}\\\\\n\\hline\none&two&\\mbox{three}&\\mbox{four}&\\mbox{five}\\\\\nHe&2& 2.77234 & 45672. & 0.69 \\\\\nC\\footnote{Some tables require footnotes.}\n  &C\\footnote{Some tables need more than one footnote.}\n  & 12537.64 & 37.66345 & 86.37 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\textit{Physical Review} style requires that the initial citation of\nfigures or tables be in numerical order in text, so don't cite\nFig.~\\ref{fig:wide} until Fig.~\\ref{fig:epsart} has been cited.\n\n\\begin{acknowledgments}\nWe wish to acknowledge the support of the author community in using\nREV\\TeX{}, offering suggestions and encouragement, testing new versions,\n\\dots.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nA \\emph{repetition} of \\emph{size} $t\\geq 1$ in a sequence \nS=s_{1}s_{2}\\ldots s_{n}$ is a subsequence of the form $s_{i+1}s_{i+2}\\ldots\ns_{i+2t}$ satisfying $s_{i+j}=s_{i+t+j}$ for all $j=1,2,\\ldots ,t$. A\nsequence $S$ is \\emph{nonrepetitive} if there is no repetition (of any size)\nin $S$. A celebrated theorem of Thue \\cite{Thue}\\ (cf. \\cite{BerstelThue})\nfrom 1906 asserts that there are arbitrarily long nonrepetitive sequences\nover the set of just $3$ symbols. This result has lots of applications and\ngeneralizations (cf. \\cite{AlloucheShallit}, \\cite{BerstelPerrin}, \\cit\n{GrytczukDM}, \\cite{Lothaire}, \\cite{PezarskiZmarz}). Recently, some game\ntheoretic variants has been introduced leading to new challenges in the area \n\\cite{Pegden}. A basic idea is that a nonrepetitive sequence is created by\ntwo players, Alice and Bob, say, but only Alice cares of avoiding\nrepetitions. For instance, they may be picking alternately symbols from a\nfixed set $A$ and appending them at the end of the existing sequence.\nWhenever a repetition occurs, its second part is being erased immediately.\nIt is proved in \\cite{GKM}\\ that Alice can still create arbitrarily long\nnonrepetitive sequences (no matter what Bob is doing) provided the size of $A\n$ is at least $8$.\n\nIn this paper we introduce another Thue type game, which we call the \\emph\nonline Thue game}. In one round of the game Bob chooses a position in the\nexisting sequence $S$, which is specified by a number $i\\in \\{0,1,\\ldots ,n\\}\n$, and Alice is picking a symbol $x\\in A$ which is inserted right after \ns_{i}$, thereby giving a new sequence $S^{\\prime }=s_{1}\\ldots\ns_{i}xs_{i+1}\\ldots s_{n}$, with $i=0$ meaning that $x$ is placed at the\nbeginning of $S$. The goal of Bob is to force Alice to create a repetition,\nwhile Alice will try to avoid that for as long as possible. For instance, if \n$A=\\{a,b,c\\}$ and $S=acbc$, then Bob catches Alice in one move by choosing \ni=1$. Indeed, picking any $x\\in A$ results in a repetition in $S^{\\prime }$: \n$\\boldsymbol{aa}cbc$, $a\\boldsymbol{bcbc}$, $a\\boldsymbol{cc}bc$. Actually,\nit is easy to check that Alice has no chance for a longer play over $3$\nsymbols. A bit more effort is needed to check that the same is true when $A$\nis of size $4$, and one may start thinking that there is no finite bound at\nall. However we shall prove the following theorem.\n\n\\begin{theorem}\nThere is a strategy guaranteeing Alice arbitrarily long play in the online\nThue game on the set of $12$ symbols.\n\\end{theorem}\n\nThe proof is based on a former result of K\\\"{u}ndegen and Pelsmajer \\cit\n{KundgenPelsmajer}, and Bar\\'{a}t and Varj\\'{u} \\cite{BaratVarju}, on\nnonrepetitive coloring of outerplanar graphs. A vertex coloring of a graph $G\n$ is \\emph{nonrepetitive} if no repetition can be found along any simple\npath of $G$.\n\n\\begin{theorem}\n\\emph{(\\cite{BaratVarju}, \\cite{KundgenPelsmajer}) }Every outerplanar graph\nhas a nonrepetitive coloring using $12$ colors.\n\\end{theorem}\n\nThe proof of Theorem 1, in a slightly different setting, is given in the\nnext section. The last section contains discussion and some open problems.\n\n\\section{Proof of the main result}\n\nWe start with introducing an equivalent setting for the online Thue game\nthat will be more convenient for our purposes. First notice that this game\ncan be played on the real line in such a way that Bob is choosing points of\nthe line and Alice is coloring them using $A$ as the set of colors. Then\nafter $n$ rounds we have a sequence of points $B=b_{1}b_{2}\\ldots b_{n}$, \nb_{i}\\in \\mathbb{R}$, written in increasing order $b_{1}<b_{2}<\\ldots <b_{n}\n, and the corresponding sequence of colors $S=c(b_{1})c(b_{2})\\ldots c(b_{n})\n$, $c(b_{i})\\in A$, assigned to the points $b_{i}$ by Alice. The goal of\nAlice is to avoid repetitions in $S$.\n\n\\begin{proof}\nWithout loss of generality we may assume that Bob starts with $0$, and\nwhenever he wants to extend the sequence $B$ to the left or to the right, he\nalways chooses the closest integer point. So, in the second move he picks\neither $-1$ or $1$. In consequence the extreme points $b_{1}$ and $b_{n}$ of \n$B$ are always integers. Furthermore we assume that when Bob decides to\ninsert a point between $b_{i}$ and $b_{i+1}$, then the new point lies\nexactly in the middle of these two. So, if he has chosen $1$ in the second\nround, then he may now choose $-1,1\/2,$ or $2$. The set of points accessible\nin this way is just the set of \\emph{dyadic rationals} $D$ which consists of\nall numbers of the form $a\/2^{k}$, where $a$ is any integer, and $k$ is any\nnonnegative integer. We assume that $(a,2^{k})=1$, and we call the exponent \nk$ the \\emph{depth} of $a\/2^{k}$. Denote by $D_{k}$ the set of all dyadic\nrationals of depth $k$. Notice that $D_{0}$ is just the set\\ of all integers.\n\nNow, for each integer $k\\geq 0$ consider a graph $P_{k}$ whose vertex set is\nthe union $D_{0}\\cup D_{1}\\cup \\ldots \\cup D_{k}$, with two points joined by\nan edge if and only if there is no other point of the union between them on\nthe line. Clearly $P_{k}$ is a bi-infinite path. Consider finally a graph $G$\non the vertex set $D$ whose set of edges is the union of sets of edges of\nall paths $P_{k}$. It is not hard to see that $G$ is an outerplanar graph.\nIndeed, we may draw the edges of each path $P_{k}$ as half-circles of\ndiameter $1\/2^{k}$ lying above the line. Then no two edges of $G$ can cross,\nand clearly all vertices of $G$ touch the outer face of this embedding.\n\nA simple (but crucial) observation is that any sequence $B=b_{1}b_{2}\\ldots\nb_{n}$ that can be formed by Bob during the game is a path in $G$. Actually, \n$B$ is a path in a subgraph $G_{n}$ of $G$ which is the union of paths \nP_{0},P_{1},\\ldots ,P_{n}$. Indeed, assume inductively that this holds for $n\n$ and consider a sequence $B^{\\prime }$ from the next round. Suppose first\nthat a new point $x$ is placed at the beginning of $B$, that is $B^{\\prime\n}=xB$. Then, accordingly to the rules, $x$ must be the closest integer to \nb_{1}$, which itself is an integer. Hence $xb_{1}$ is an edge in $P_{0}$,\nand consequently $B^{\\prime }$ is a path in $G_{n+1}$. The same holds for \nB^{\\prime }=Bx$. Now, suppose $x$ has been inserted between $b_{i}$ and \nb_{i+1}$. By inductive assumption these two points lie consecutively on some\npath $P_{j}$, $0\\leq j\\leq n$. Hence, $b_{i},x,b_{i+1}$ lie consecutively on\npath $P_{j+1}$. In other words, the edge $b_{i}b_{i+1}$ has been subdivided,\nand therefore $B^{\\prime }$ is a path in $G_{n+1}$.\n\nTo complete the proof we need only to demonstrate that graph $G$ has a\nnonrepetitive coloring using $12$ colors. But this follows easily from\nTheorem 2 by invoking the compactness principle.\n\\end{proof}\n\n\\section{Remarks and open problems}\n\nFirst notice that we have actually proved a stronger result asserting that\nAlice has a strategy to play \\emph{infinitely} long, not just to play \\emph\narbitrarily} long. The strategy seems nonconstructive, as it is based on\nnonrepetitive coloring of an infinite graph obtained via compactness\nprinciple. However, looking into details of the proof of Theorem 2, one has\nan impression that it can be adapted to infinite graphs so that an explicit\nformula for the nonrepetitive coloring of $G$ is perhaps possible. \n\nNotice also that in fact we need a weaker coloring in which only\nforward-directed paths are nonrepetitive. We suspect that this can be\nachieved with smaller number of colors, perhaps $9$ are sufficient.\n\nAnother observation leads to a major open problem of the area of\nnonrepetitive colorings of graphs. Notice that graph $G$ from the proof of\nTheorem 1 has \\emph{page number} $1$. What about nonrepetitive coloring of\ngraphs with higher page number?\n\n\\begin{conjecture}\nThere is a finite constant $N$ such that every graph of page number $2$ has\na nonrepetitive coloring using $N$ colors.\n\\end{conjecture}\n\nThis innocently looking question was propounded some years ago by Idziak.\nHowever, now we know that it is strong enough to imply the following\nstatement.\n\n\\begin{conjecture}\nThere is a finite constant $M$ such that every planar graph has a\nnonrepetitive coloring using $M$ colors.\n\\end{conjecture}\n\nThe best result to date \\cite{Dujmovic2} gives a logarithmic upper bound (in\nthe number of vertices of a graph). Perhaps it would be easier to verify\ndirected versions of the above problems.\n\nFinally, one may consider online versions of other theorems in combinatorics\non words. Particularly interesting seems the case of \\emph{abelian\nrepetitions}, that is subsequences of the form $r_{1}\\ldots r_{t}r_{\\sigma\n(1)}\\ldots r_{\\sigma (t)}$, where $\\sigma $ is any permutation of the set \n\\{1,2,\\ldots ,t\\}$. Answering a question of Erd\\H{o}s \\cite{Erdos}, Ker\\\"{a\nnen \\cite{Keranen} proved that there exist arbitrarily long sequences over $4\n$ symbols avoiding abelian repetitions of any size. Is the online version of\nthis result possible?\n\n\\begin{acknowledgement}\nJ. Grytczuk acknowledges a support of the Polish Ministry of Science and\nHigher Education, grant N206257035.\n\\end{acknowledgement}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the Hamiltonian formulation of classical mechanics, the canonical transformations, given locally by expressions of the form $Q_{i} = Q_{i}(q_{j}, p_{j}, t)$, $P_{i} = P_{i}(q_{j}, p_{j}, t)$, act on the extended phase space (the Cartesian product of the phase space and the time axis) and, therefore, there is a naturally defined action of a canonical transformation on any function defined on the extended phase space (that is, on any function of $q_{i}$, $p_{i}$, and $t$). On the other hand, Hamilton's principal function, usually denoted by an $S$, is a function defined on the extended configuration space (that is, a function of $q_{i}$ and $t$), just like the wave function in the elementary Schr\\\"odinger equation, but there is no obvious way to define the action of a canonical transformation on a function of $q_{i}$ and $t$.\n\nHowever, in Ref.\\ \\cite{DI} it was shown that, under certain conditions, one can define an action of a time-independent canonical transformation [that is, a canonical transformation that does not involve the time, $Q_{i} = Q_{i}(q_{j}, p_{j})$, $P_{i} = P_{i}(q_{j}, p_{j})$] on functions defined on the configuration space, in such a way that if a (time-independent) Hamiltonian is invariant under the canonical transformation, a solution of the corresponding time-independent Hamilton--Jacobi (HJ) equation is mapped into another solution. Furthermore, it was shown that the action of the one-parameter group of canonical transformations generated by an arbitrary function $G(q_{i}, p_{i})$ is determined by an equation similar to the HJ equation.\n\nIn this paper, we consider systems with a Hamiltonian that can depend on the time, we show that one can define the action of the one-parameter group of canonical transformations generated by an arbitrary function $G(q_{i}, p_{i}, t)$ on functions defined on the extended configuration space, in such a way that if $G$ is a constant of motion, then a solution of the corresponding HJ equation is mapped into a one-parameter family of solutions of this equation. In this manner, each constant of motion adds a continuous parameter to a given solution of the HJ equation. We also show that any complete solution of the HJ equation can be obtained from a solution without parameters by means of the action of the groups of canonical transformations generated by $n$ constants of motion in involution.\n\nThroughout this paper, several examples are given in order to illustrate the definitions and results presented here.\n\n\\section{The action of a one-parameter group of canonical transformations on functions defined on the extended configuration space}\nWe start by reviewing the HJ equation, which will give us the pattern to follow in the definition of the action of any one-parameter group of canonical transformations on functions defined on the extended configuration space.\n\n\\subsection{The Hamilton--Jacobi as an evolution equation}\nGiven a Hamiltonian $H(q_{i}, p_{i}, t)$ of a system with $n$ degrees of freedom, the corresponding HJ equation is the first-order partial differential equation\n\\begin{equation}\nH(q_{i}, \\partial S\/\\partial q_{i}, t) + \\frac{\\partial S}{\\partial t} = 0. \\label{hj}\n\\end{equation}\nUsually one is interested in {\\em complete solutions}\\\/ of the HJ equation (\\ref{hj}), that is, solutions $S(q_{i}, t, \\alpha_{i})$ of Eq.\\ (\\ref{hj}), containing $n$ arbitrary parameters $\\alpha_{i}$, such that\n\\begin{equation}\n\\det \\left( \\frac{\\partial^{2} S}{\\partial q_{i} \\partial \\alpha_{j}} \\right) \\not= 0, \\label{compl}\n\\end{equation}\nbecause they yield the solution of the Hamilton equations (see, {\\em e.g.}, Refs.\\ \\cite{Wi} and \\cite{Ga}). However, one can consider the HJ equation as an {\\em evolution equation}, which determines the function $S(q_{i}, t)$ that reduces to a given function $f(q_{i})$, for $t = 0$ (or any other initial value, $t_{0}$, of $t$). (This fact does not sound very strange if we consider the relationship between the HJ equation and the Schr\\\"odinger equation.) In fact, if we have the solution of the Hamilton equations, we can use it to find the solution of the HJ equation that satisfies any initial condition $S(q_{i}, t_{0}) = f(q_{i})$. (Similar ideas are often mentioned in the standard textbooks on analytical mechanics, without presenting, however, a clear statement or an explicit procedure.)\n\nFor example, in the case of the Hamiltonian\n\\begin{equation}\nH = \\frac{p_{1}{}^{2} + p_{2}{}^{2}}{2m} + mg q_{2}, \\label{hamex}\n\\end{equation}\nwhere $m$ and $g$ are constants, the solution of the corresponding Hamilton equations can be readily obtained and is given by\n\\begin{equation}\nq_{1} = (q_{1})_{0} + \\frac{(p_{1})_{0} t}{m}, \\qquad q_{2} = (q_{2})_{0} + \\frac{(p_{2})_{0} t}{m} - \\frac{gt^{2}}{2}, \\qquad p_{1} = (p_{1})_{0}, \\qquad p_{2} = (p_{2})_{0} - mgt, \\label{solex}\n\\end{equation}\nwhere $(q_{1})_{0}$ denotes the value of $q_{1}$ at $t = 0$, and so on. As the initial condition we choose\n\\begin{equation}\nS(q_{1}, q_{2}, 0) = \\alpha_{1} q_{1} + \\alpha_{2} q_{2}, \\label{icex}\n\\end{equation}\nwhere $\\alpha_{1}$, $\\alpha_{2}$ are arbitrary constants. We want to find the solution of the HJ equation [see Eqs.\\ (\\ref{hj}) and (\\ref{hamex})]\n\\begin{equation}\n\\frac{\\partial S}{\\partial t} = - \\frac{1}{2m} \\left[ \\left( \\frac{\\partial S}{\\partial q_{1}} \\right)^{2} + \\left( \\frac{\\partial S}{\\partial q_{2}} \\right)^{2} \\right] - mg q_{2}, \\label{hjex}\n\\end{equation}\nthat satisfies the initial condition (\\ref{icex}). Recalling that $\\partial S\/\\partial q_{i} = p_{i}$, making use of Eqs.\\ (\\ref{solex}) and (\\ref{icex}) we have\n\\[\n\\frac{\\partial S}{\\partial q_{1}} = \\left. \\frac{\\partial S}{\\partial q_{1}} \\right|_{t = 0} = \\alpha_{1}, \\qquad \\frac{\\partial S}{\\partial q_{2}} = \\left. \\frac{\\partial S}{\\partial q_{2}} \\right|_{t = 0} - mgt = \\alpha_{2} - mgt.\n\\]\nSubstituting these expressions into the right-hand side of Eq.\\ (\\ref{hjex}) we have\n\\[\n\\frac{\\partial S}{\\partial t} = - \\frac{1}{2m} \\big[ \\alpha_{1}{}^{2} + (\\alpha_{2} - mgt)^{2} \\big] - mg q_{2}.\n\\]\nCombining the last three equations one readily finds that, choosing the integration constant so that Eq.\\ (\\ref{icex}) is satisfied,\n\\begin{equation}\nS(q_{1}, q_{2}, t) = \\alpha_{1} q_{1} + \\alpha_{2} q_{2} - mgtq_{2} - \\frac{\\alpha_{1}{}^{2} t}{2m} + \\frac{(\\alpha_{2} - mgt)^{3} - \\alpha_{2}{}^{3}}{6m^{2}g}. \\label{prinex}\n\\end{equation}\nThe expression (\\ref{prinex}) is a (complete, R-separable) solution of the HJ equation that reduces to the specified function (\\ref{icex}) for $t = 0$. (A solution is R-separable if it is the sum of a function of two or more variables and functions of one variable.)\n\nWe close this subsection with a second example. The solution of the Hamilton equations for the time-dependent Hamiltonian\n\\begin{equation}\nH = \\frac{p^{2}}{2m} - ktq, \\label{n1}\n\\end{equation}\nwhere $m$ and $k$ are constants, is given by\n\\begin{equation}\nq = q_{0} + \\frac{tp_{0}}{m} + \\frac{kt^{3}}{6m}, \\qquad p = p_ {0} + \\frac{1}{2} kt^{2}, \\label{n2}\n\\end{equation}\nwhere $q_{0}$ and $p_{0}$ denote the values of $q$ and $p$ at $t = 0$, respectively. Letting\n\\begin{equation}\nS(q,0,\\alpha) = \\alpha q, \\label{n3}\n\\end{equation}\nwhere $\\alpha$ is an arbitrary constant, with the aid of Eqs.\\ (\\ref{n2}), we have\n\\begin{equation}\n\\frac{\\partial S}{\\partial q} = \\left. \\frac{\\partial S}{\\partial q} \\right|_{t = 0} + \\frac{1}{2} kt^{2} = \\alpha + \\frac{1}{2} kt^{2}. \\label{n4}\n\\end{equation}\nThus, from Eqs.\\ (\\ref{hj}), (\\ref{n1}), and (\\ref{n4}), we have\n\\begin{equation}\n\\frac{\\partial S}{\\partial t} = - \\frac{1}{2m} \\left( \\frac{\\partial S}{\\partial q} \\right)^{2} + ktq = - \\frac{1}{2m} \\left( \\alpha + \\frac{1}{2} kt^{2} \\right)^{2} + ktq. \\label{n5}\n\\end{equation}\nThen, from Eqs.\\ (\\ref{n3})--(\\ref{n5}), we readily obtain\n\\[\nS(q, t, \\alpha) = \\alpha q + \\frac{1}{2} kt^{2} q - \\frac{1}{2m} \\left( \\alpha^{2} t + \\frac{1}{3} \\alpha k t^{3} + \\frac{k^{2} t^{5}}{20} \\right),\n\\]\nwhich is an R-separable complete solution of the HJ equation. (We might start with expressions more complicated than (\\ref{icex}) and (\\ref{n3}), but these simple expressions are enough to obtain complete solutions of the HJ equation.)\n\nIt should be clear that a similar construction can be devised using an arbitrary function of $(q_{i}, p_{i}, t)$ instead of a Hamiltonian.\n\n\\subsection{Families of solutions of the HJ equation and constants of motion}\nAny differentiable function, $G(q_{i}, p_{i}, t)$, defines a (possibly local) one-parameter group of canonical transformations, determined by the (autonomous) system of first-order ordinary differential equations\n\\begin{equation}\n\\frac{{\\rm d} q_{i}}{{\\rm d} \\alpha} = \\frac{\\partial G(q_{j}, p_{j}, t)}{\\partial p_{i}}, \\qquad\n\\frac{{\\rm d} p_{i}}{{\\rm d} \\alpha} = - \\frac{\\partial G(q_{j}, p_{j}, t)}{\\partial q_{i}}, \\label{odes}\n\\end{equation}\nwhich are of the form of the Hamilton equations.\n\nFor example, the function\n\\begin{equation}\nG(q_{1}, q_{2}, p_{1}, p_{2}, t) = \\frac{p_{1} p_{2}}{m} + mgq_{1}, \\label{a}\n\\end{equation}\nwhere $m$ and $g$ are constants, leads to the system of equations\n\\begin{equation}\n\\frac{{\\rm d} q_{1}}{{\\rm d} \\alpha} = \\frac{p_{2}}{m}, \\qquad \\frac{{\\rm d} q_{2}}{{\\rm d} \\alpha} = \\frac{p_{1}}{m}, \\qquad \\frac{{\\rm d} p_{1}}{{\\rm d} \\alpha} = - mg, \\qquad \\frac{{\\rm d} p_{2}}{{\\rm d} \\alpha} = 0. \\label{sim}\n\\end{equation}\nFrom the last two equations we find that\n\\begin{equation}\np_{1} = (p_{1})_{0} - mg \\alpha, \\qquad p_{2} = (p_{2})_{0}, \\label{apes}\n\\end{equation}\nwhere $(p_{i})_{0}$ denotes the value of $p_{i}$ for $\\alpha = 0$. Substituting these expressions into the first pair of equations (\\ref{sim}) we get\n\\[\n\\frac{{\\rm d} q_{1}}{{\\rm d} \\alpha} = \\frac{(p_{2})_{0}}{m}, \\qquad \\frac{{\\rm d} q_{2}}{{\\rm d} \\alpha} = \\frac{(p_{1})_{0} - mg \\alpha}{m}.\n\\]\nHence,\n\\begin{equation}\nq_{1} = (q_{1})_{0} + \\frac{(p_{2})_{0} \\alpha}{m}, \\qquad q_{2} = (q_{2})_{0} + \\frac{(p_{1})_{0} \\alpha}{m} - \\frac{g \\alpha^{2}}{2}. \\label{aqs}\n\\end{equation}\n\nThe action of the canonical transformations defined by a function $G(q_{i}, p_{i}, t)$, on functions of\n$(q_{i}, t)$, will be defined imitating the HJ equation.\n\n\\noindent{\\bf Definition.} The image of a given arbitrary function $f(q_{i}, t)$ under the one-parameter group of canonical transformations generated by $G(q_{i}, p_{i}, t)$ is the function $S(q_{i}, t, \\alpha)$ such that\n\\begin{equation}\nG(q_{i}, \\partial S\/\\partial q_{i}, t) + \\frac{\\partial S}{\\partial \\alpha} = 0, \\label{action}\n\\end{equation}\nwith the initial condition $S(q_{i}, t, 0) = f(q_{i}, t)$. ({\\em Cf.}\\ Eq.\\ (\\ref{hj}).)\n\nThe following example shows that this definition produces the expected effect in the case of transformations that only affect the coordinates of the configuration space. Indeed, the one-parameter group of canonical transformations generated by, {\\em e.g.}, $G = p_{1}$ is given by [see Eqs.\\ (\\ref{odes})]\n\\[\nq_{1} = (q_{1})_{0} + \\alpha, \\qquad q_{i} = (q_{i})_{0}, \\quad {\\rm for \\ } i \\geqslant 2, \\qquad p_{i} = (p_{i})_{0}, \\]\nthat is, ``translations'' along the $q_{1}$-axis. Then, Eq.\\ (\\ref{action}) takes the form\n\\[\n\\frac{\\partial S}{\\partial q_{1}} + \\frac{\\partial S}{\\partial \\alpha} = 0,\n\\]\nwhose general solution is\n\\[\nS(q_{i}, t, \\alpha) = F(q_{1} - \\alpha, q_{2}, \\ldots, q_{n}, t),\n\\]\nwhere $F$ is an arbitrary function. By imposing the initial condition $S(q_{i}, t, 0) = f(q_{i}, t)$, we find that\n\\[\nS(q_{i}, t, \\alpha) = f(q_{1} - \\alpha, q_{2}, \\ldots, q_{n}, t),\n\\]\nwhich is the expected effect of a translation along the $q_{1}$-axis.\n\nAs a second example, the images of the function\n\\begin{equation}\nf(q_{i}, t) = (\\alpha_{2} - mgt)q_{2} + \\frac{(\\alpha_{2} - mgt)^{3} - \\alpha_{2}{}^{3}}{6m^{2}g} \\label{icex2}\n\\end{equation}\n(which is the function (\\ref{prinex}) with $\\alpha_{1} = 0$) under the transformations generated by (\\ref{a}) can be obtained proceeding as in the examples of Sec.\\ 2.1. Making use of Eqs.\\ (\\ref{apes}) and (\\ref{icex2}) we have [with $S(q_{i}, t, 0) = f(q_{i}, t)$]\n\\begin{equation}\n\\frac{\\partial S}{\\partial q_{1}} = \\left. \\frac{\\partial S}{\\partial q_{1}} \\right|_{\\alpha = 0} - mg \\alpha = - mg \\alpha, \\qquad \\frac{\\partial S}{\\partial q_{2}} = \\left. \\frac{\\partial S}{\\partial q_{2}} \\right|_{\\alpha = 0} = \\alpha_{2} - mgt. \\label{partial}\n\\end{equation}\nSubstituting these expressions into Eq.\\ (\\ref{action}), taking into account Eq.\\ (\\ref{a}), we obtain\n\\[\n\\frac{\\partial S}{\\partial \\alpha} = - \\frac{1}{m} (- mg \\alpha) (\\alpha_{2} - mgt) - mg q_{1}.\n\\]\nCombining the last equations and the initial condition (\\ref{icex2}) one finds that\n\\begin{equation}\nS(q_{1}, q_{2}, t, \\alpha_{2}, \\alpha) = (\\alpha_{2} - mgt)q_{2} - mg \\alpha q_{1} + \\frac{1}{2} (\\alpha_{2} - mgt)g \\alpha^{2} + \\frac{(\\alpha_{2} - mgt)^{3} - \\alpha_{2}{}^{3}}{6m^{2}g}. \\label{prinex2}\n\\end{equation}\nNote that $t$ is treated here as a constant because the canonical transformations do not affect $t$. Since the function (\\ref{icex2}) is a particular case of (\\ref{prinex}), is a solution of the HJ equation (\\ref{hjex}), and one can readily verify that (\\ref{prinex2}) is {\\em also}\\\/ a solution of Eq.\\ (\\ref{hjex}), for all values of the parameter $\\alpha$ [which is essentially the two-parameter family of solutions (\\ref{prinex})].\n\nThus, we have obtained a two-parameter family of solutions of the HJ equation (\\ref{hjex}), starting from a one-parameter solution of this equation. As we shall show in the following Proposition, this is a consequence of the fact that the function (\\ref{a}) is a constant of motion for the Hamiltonian (\\ref{hamex}).\n\n\\noindent {\\bf Proposition 1.} The image of a solution of the HJ equation, $S_{0}(q_{i}, t)$, under the one-parameter group of canonical transformations generated by a constant of motion $G(q_{i}, p_{i}, t)$ is a one-parameter family of solutions of the same HJ equation.\n\n\\noindent {\\em Proof.} Assuming that $S(q_{i}, t, \\alpha)$ is a solution of Eq.\\ (\\ref{action}), making use of the chain rule repeatedly, we have (here, and in what follows, there is summation over repeated indices)\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial \\alpha} \\left[ H(q_{i}, \\partial S\/\\partial q_{i}, t) + \\frac{\\partial S}{\\partial t} \\right] & = & \\frac{\\partial H}{\\partial p_{j}} \\frac{\\partial^{2} S}{\\partial \\alpha \\partial q_{j}} + \\frac{\\partial^{2} S}{\\partial \\alpha \\partial t} \\nonumber \\\\\n& = & - \\frac{\\partial H}{\\partial p_{j}} \\frac{\\partial G(q_{i}, \\partial S\/\\partial q_{i}, t)}{\\partial q_{j}} - \\frac{\\partial G(q_{i}, \\partial S\/\\partial q_{i}, t)}{\\partial t} \\nonumber \\\\\n& = & - \\frac{\\partial H}{\\partial p_{j}} \\left( \\frac{\\partial G}{\\partial q_{j}} + \\frac{\\partial G}{\\partial p_{i}} \\frac{\\partial^{2} S}{\\partial q_{j} \\partial q_{i}} \\right) - \\frac{\\partial G}{\\partial p_{i}} \\frac{\\partial^{2} S}{\\partial t \\partial q_{i}} - \\frac{\\partial G}{\\partial t}. \\label{des1}\n\\end{eqnarray}\nAccording to the hypothesis, $G$ is a constant of motion, that is\n\\begin{equation}\n\\frac{\\partial G}{\\partial t} + \\frac{\\partial G}{\\partial q_{j}} \\frac{\\partial H}{\\partial p_{j}} - \\frac{\\partial G}{\\partial p_{j}} \\frac{\\partial H}{\\partial q_{j}} = 0, \\label{const}\n\\end{equation}\nthus, the last line of Eq.\\ (\\ref{des1}) amounts to\n\\begin{eqnarray*}\n- \\frac{\\partial G}{\\partial p_{i}} \\left( \\frac{\\partial H}{\\partial q_{i}} + \\frac{\\partial H}{\\partial p_{j}} \\frac{\\partial^{2} S}{\\partial q_{j} \\partial q_{i}} + \\frac{\\partial^{2} S}{\\partial t \\partial q_{i}} \\right) & = & - \\frac{\\partial G}{\\partial p_{i}} \\frac{\\partial}{\\partial q_{i}} \\left[ H(q_{i}, \\partial S\/\\partial q_{i}, t) + \\frac{\\partial S}{\\partial t} \\right] \\\\\n& = & - \\frac{{\\rm d} q_{i}}{{\\rm d} \\alpha} \\frac{\\partial}{\\partial q_{i}} \\left[ H(q_{i}, \\partial S\/\\partial q_{i}, t) + \\frac{\\partial S}{\\partial t} \\right]\n\\end{eqnarray*}\n[see Eqs.\\ (\\ref{odes})]. Hence, we obtain the equation\n\\[\n\\frac{{\\rm d}}{{\\rm d} \\alpha} \\left[ H(q_{i}(\\alpha), \\partial S(q_{i}(\\alpha), t, \\alpha)\/\\partial q_{i}, t) + \\frac{\\partial S(q_{i}(\\alpha), t, \\alpha)}{\\partial t} \\right] = 0,\n\\]\nwhich involves the images of $S$ and the coordinates $q_{i}$ under the transformations generated by $G$. Since the expression inside the brackets in the last equation vanishes for $\\alpha = 0$, it is equal to zero for all values of $\\alpha$, thus proving the assertion.\n\nA partial converse of this Proposition is true. The steps in the proof show that if the images of a solution of the HJ equation under the transformations generated by $G$ are solutions of the same HJ equation, and $S$ is a complete solution of the HJ equation, then $G$ is a constant of motion. The assumption that $S$ is a complete solution of the HJ equation assures that Eq.\\ (\\ref{const}) holds everywhere. In fact, the following stronger result holds.\n\n\\noindent {\\bf Proposition 2.} A complete solution of the HJ equation, $S(q_{i}, t, \\alpha_{i})$ (not necessarily separable or R-separable), is the image of $S(q_{i}, t, 0)$ under the transformations generated by the $n$ constants of motion\n\\begin{equation}\nG_{j}(q_{i}, p_{i}, t) = - \\frac{\\partial S}{\\partial \\alpha_{j}}(q_{i}, t, \\alpha_{i}(q_{k}, p_{k}, t)) \\label{prop3}\n\\end{equation}\n($j = 1, 2, \\ldots, n$), obtained by eliminating the $\\alpha_{i}$ by means of\n\\begin{equation}\np_{j} = \\frac{\\partial S}{\\partial q_{j}} (q_{i}, \\alpha_{i}, t). \\label{pes}\n\\end{equation}\nFurthermore, the functions $G_{i}$ are in involution ({\\em i.e.}, the Poisson bracket of $G_{i}$ and $G_{j}$ is equal to zero). (Note that, according to Eq.\\ (\\ref{compl}), in principle, one can find the $\\alpha_{i}$ from Eqs.\\ (\\ref{pes}).)\n\n\\noindent {\\em Proof.} We note that, by virtue of Eqs.\\ (\\ref{pes}), Eq.\\ (\\ref{prop3}) is equivalent to Eq.\\ (\\ref{action}). Since, by hypothesis, $S(q_{i}, t, \\alpha_{i})$ is a complete solution of the HJ equation, each function $G_{i}$, defined by (\\ref{prop3}), is a constant of motion. In order to prove that the functions $G_{i}$ are in involution, making use of Eq.\\ (\\ref{prop3}), we calculate the mixed partial derivative\n\\[\n\\frac{\\partial^{2} S}{\\partial \\alpha_{i} \\partial \\alpha_{j}} = - \\frac{\\partial}{\\partial \\alpha_{i}} G_{j}(q_{i}, \\partial S\/\\partial q_{i}, t, \\alpha_{k}) = - \\frac{\\partial G_{j}}{\\partial p_{k}} \\frac{\\partial}{\\partial \\alpha_{i}} \\frac{\\partial S}{\\partial q_{k}} = \\frac{\\partial G_{j}}{\\partial p_{k}} \\frac{\\partial G_{i}}{\\partial q_{k}}.\n\\]\nThen we see that the commutativity of the partial derivatives of $S$ implies that the Poisson brackets of the $G_{i}$ are equal to zero. (The assumption that $S(q_{i}, t, \\alpha_{i})$ is a complete solution of the HJ equation implies that the Poisson brackets $\\{ G_{i}, G_{j} \\}$ vanish everywhere.) Since the Poisson brackets $\\{ G_{i}, G_{j} \\}$ are equal to zero, the one-parameter groups of transformations generated by the functions $G_{i}$ commute. (In fact, the groups of canonical transformations generated by two functions $F$ and $G$ commute if and only if $\\{ F, G \\}$ is a trivial constant, not necessarily zero \\cite{DM}.)\n\nLet us consider, for example, the function\n\\begin{equation}\nS(q_{i}, t, \\alpha_{i}) = \\alpha_{1} q_{1} + \\alpha_{2} q_{2} - mgtq_{2} - \\frac{\\alpha_{1}{}^{2} t}{2m} + \\frac{(\\alpha_{2} - mgt)^{3} - \\alpha_{2}{}^{3}}{6m^{2}g}, \\label{prinexo}\n\\end{equation}\nwhich is the complete solution (\\ref{prinex}) of the HJ equation constructed in Sec.\\ 2.1. One finds that\n\\[\n\\frac{\\partial S}{\\partial q_{1}} = \\alpha_{1}, \\qquad \\frac{\\partial S}{\\partial q_{2}} = \\alpha_{2} - mgt,\n\\]\nwhich lead to [see Eqs.\\ (\\ref{pes})]\n\\begin{equation}\n\\alpha_{1} = p_{1}, \\qquad \\alpha_{2} = p_{2} + mgt. \\label{inversi}\n\\end{equation}\nOn the other hand,\n\\[\n\\frac{\\partial S}{\\partial \\alpha_{1}} = q_{1} - \\frac{\\alpha_{1} t}{m}, \\qquad \\frac{\\partial S}{\\partial \\alpha_{2}} = q_{2} + \\frac{(\\alpha_{2} - mgt)^{2} - \\alpha_{2}{}^{2}}{2 m^{2} g},\n\\]\nwhich, taking into account Eqs.\\ (\\ref{inversi}), are of the form (\\ref{prop3}) with\n\\begin{equation}\nG_{1} = - q_{1} + \\frac{p_{1} t}{m}, \\qquad G_{2} = - q_{2} + \\frac{p_{2} t}{m} + \\frac{1}{2} gt^{2}. \\label{ges}\n\\end{equation}\nOne can readily verify that $G_{1}$ and $G_{2}$ are constants of motion in involution, and that the function (\\ref{prinexo}) can be obtained by means of the transformations generated by $G_{1}$ and $G_{2}$ from\n\\begin{equation}\nS(q_{i},t) = - mgt q_{2} - \\frac{mg^{2}t^{3}}{6}, \\label{init}\n\\end{equation}\nwhich is obtained from (\\ref{prinexo}) setting $\\alpha_{1} = \\alpha_{2} = 0$.\n\nIndeed, the function\n\\begin{equation}\nG \\equiv \\alpha_{1} G_{1} + \\alpha_{2} G_{2} = \\alpha_{1} \\left( - q_{1} + \\frac{p_{1} t}{m} \\right) + \\alpha_{2} \\left( - q_{2} + \\frac{p_{2} t}{m} + \\frac{1}{2} gt^{2} \\right) \\label{gee}\n\\end{equation}\nis a constant of motion for all values of the constants $\\alpha_{1}$ and $\\alpha_{2}$. The one-parameter group of canonical transformations generated by $G$ is given by (denoting by $s$ the corresponding parameter)\n\\[\nq_{1} = (q_{1})_{0} + \\frac{\\alpha_{1} t}{m} s, \\quad q_{2} = (q_{2})_{0} + \\frac{\\alpha_{2} t}{m} s, \\quad p_{1} = (p_{1})_{0} + \\alpha_{1} s, \\quad p_{2} = (p_{2})_{0} + \\alpha_{2} s,\n\\]\nhence, making use of the initial condition (\\ref{init}),\n\\[\n\\frac{\\partial S}{\\partial q_{1}} = 0 + \\alpha_{1} s, \\qquad \\frac{\\partial S}{\\partial q_{2}} = - mgt + \\alpha_{2} s,\n\\]\nand substituting these expressions into Eq.\\ (\\ref{action}), with $G$ given by (\\ref{gee}),\n\\[\n\\frac{\\partial S}{\\partial s} = \\alpha_{1} q_{1} - \\frac{\\alpha_{1}t}{m}(\\alpha_{1} s) + \\alpha_{2} q_{2} - \\frac{\\alpha_{2} t}{m} (-mgt + \\alpha_{2} s) - \\frac{1}{2} \\alpha_{2} gt^{2}.\n\\]\nThus,\n\\[\nS(q_{1}, q_{2}, t) = \\alpha_{1} s q_{1} + \\alpha_{2} s q_{2} - mgtq_{2} - \\frac{(\\alpha_{1} s)^{2} t}{2m} + \\frac{(\\alpha_{2} s - mgt)^{3} - (\\alpha_{2} s)^{3}}{6m^{2}g},\n\\]\nwhich reduces to the expression (\\ref{prinexo}) when $s = 1$.\n\n\\section{Concluding remarks}\nAccording to Proposition 2, any complete solution of the HJ equation is, locally, the image of a particular solution (without free parameters) under the action of an Abelian $n$-dimensional group (that is, a group locally isomorphic to the additive Lie group $\\mathbb{R}^{n}$). As pointed out above, these solutions need not be separable or R-separable. It should be noted, however, that some functions are invariant under the transformations generated by a function $G$, even if these transformations are different from the identity. For instance, a function that does not depend on $q_{1}$ is invariant under the translations generated by $p_{1}$.\n\nOne can readily see that, at least in the case of time-independent operators, an analog of Proposition 1 holds in the case of the solutions of the Schr\\\"odinger equation.\n\nIn addition to the R-separable complete solution (\\ref{prinex}), the HJ equation for the Hamiltonian (\\ref{hamex}) also admits complete separable solutions in the same coordinate system (see also Ref.\\ \\cite{Li})\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nA maximally oxidised theory  is a Lagrangian theory of  gravity coupled to forms\nand dilatons defined in the highest possible space-time dimension $D$ which\nupon dimensional reduction  to three dimensions exhibits the symmetry of\na simple Lie group ${\\cal G}$ realised on a coset space\n${\\cal G}\/{\\cal H}$. Here\n$\\cal H$ is the maximal compact subgroup of $\\cal G$. The maximally\noxidised actions  have been constructed for all   $\\cal\nG$ \\cite{cjlp}.  They  comprise in particular pure gravity in $D$\ndimensions and the low energy effective actions of the bosonic string and\nof  M-theory.\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=12cm]{dynk.eps}\n \\caption { \\small The\nnodes labelled 1,2,3 define the Kac-Moody extensions of the  Lie\nalgebras. The horizontal line\nstarting at 1 defines the `gravity line', which is the\nDynkin diagram  of a\n$A_{D-1}$ subalgebra. Note that for some Kac-Moody algebras the choice of a\ngravity line is\n not unique, in which case we take here the one shown in the figure.}\n   \\label{first}\n\\end{figure}\n\n\n\n\nIt\nhas been conjectured that these theories, or some extensions of them,\npossess the much larger very-extended Kac-Moody symmetry\n${\\cal G}^{+++}$. ${\\cal G}^{+++}$ algebras are defined by the Dynkin diagrams depicted\nin Fig.1,  obtained from those of $\\cal G$ by adding three\nnodes~\\cite{ogw}. One first adds the affine node, labelled 3 in the\nfigure, then a second node, 2,  connected to it by a single line and\ndefining the overextended ${\\cal G}^{++}$ algebra,   then  a third\none, 1, connected  by a single line to the overextended node. Such\n${\\cal G}^{+++}$ symmetries were first conjectured in the aforementioned\nparticular cases\n\\cite{west01,lw} and the extension to all\n${\\cal G}^{+++}$ was proposed in\n\\cite{ehtw}. In a different development, the study of the properties\nof cosmological solutions in the vicinity of a space-like singularity,\nknown as cosmological billiards\n\\cite{damourhn00},  revealed an  overextended symmetry ${\\cal G}^{++}$\nfor all maximally oxidised theories\n\\cite{damourh00, damourbhs02}.\n\n\nTo explore the possible fundamental significance of these huge\nsymmetries two Lagrangian formulations \\cite{damourhn02, eh}  {\\it\nexplicitly} invariant under such infinite-dimensional Kac-Moody\nalgebras  have been proposed.\n\nFirst  \\cite{damourhn02} an\n$E_{10}\\equiv E_8^{++}$-invariant action was constructed in the\ncontext of M-theory as a reparametrisation invariant $\\sigma$-model of\nfields depending on one parameter $t$ and living on the coset space\n$E_8^{++}\/K_8^{++}$. Here $K_8^{++}$ is the subalgebra of $E_8^{++}$\ninvariant  under the Chevalley involution. The action is built in a\nrecursive way  by a level expansion of $E_8^{++}$ with respect to its \nsubalgebra $A_9$ whose Dynkin diagram is the `gravity line' defined in Fig. 1,\nwith the node 1 deleted.  The level of an irreducible representation of\n$A_9$ occurring in the decomposition of the adjoint  representation of\n$E_8^{++}$ counts the number of  times the simple root  not\npertaining to the gravity line appears in\nthe decomposition.  The $\\sigma$-model, limited to the\nreal roots up to level 3, reveals  a perfect  match with the bosonic\nequations of motion of 11-dimensional supergravity in the vicinity of\nthe  space-like singularity of the $E_8^{++}$ cosmological billiards,\nwhere the fields depend only on time\\footnote{See also \\cite{damourn04} for an\nanalysis in a different formulation, and\n\\cite{nike} for a discussion of $E_8^{++}$ adapted to $IIB$\nsupergravity.}. In the dictionary relating the\n$\\sigma$-model to supergravity, the  parameter $t$\n is identified with time.  It was conjectured that   space derivatives\nare hidden  in some higher level fields of the $\\sigma$-model.\n This approach is straightforwardly generalised, as seen\nbelow, to   actions $S_{{\\cal G}_C^{++}}$ related to cosmological\nbilliards and invariant under the overextended ${\\cal G}^{++}$. Such\n${\\cal G}^{++}$-invariant formulation  singles out naturally a time coordinate.\n\nThe second  approach \\cite{eh}  yields  a ${\\cal G}^{+++}$-invariant action $S_{{\\cal G}^{+++}}$\nwhich puts  all the space-time coordinates on the same footing. This\nis achieved by   a reparametrisation invariant $\\sigma$-model with\nfields depending on a parameter $\\xi$ spanning a world-line a priori\nunrelated to space-time. One uses a level decomposition of ${\\cal G}^{+++}$ with\nrespect to the subalgebra $A_{D-1}$ of its gravity line\\footnote{ Level expansions of very-extended\nalgebras in terms of the subalgebra $A_{D-1}$ have been\nconsidered in\n\\cite{west02, nifi, weke}.}  and\n$D$ is identified to  the space-time dimension. The $\\xi$-dependent\nfields live in a coset space ${\\cal G}^{+++} \/K^{+++}$ where the subalgebra\n$K^{+++}$ is invariant under a temporal involution\n\\cite{eh} which preserve the Lorentz algebra $SO(1,D-1)$ and ensures\nthat the action $S_{{\\cal G}^{+++}}$ is Lorentz invariant\nat each level.\n\nIn this paper, we analyse the ${\\cal G}^{++}$ content of the\n${\\cal G}^{+++}$-invariant action $S_{{\\cal G}^{+++}}$. We find that the ${\\cal G}^{++}$-invariant action\n$S_{{\\cal G}_C^{++}}$  is obtained by putting to zero in $S_{{\\cal G}^{+++}}$,\nconsistently with all its equations of motion, all the fields\nwhich do not appear in $S_{{\\cal G}_C^{++}}$. This consistent\ntruncation of the ${\\cal G}^{+++}$ theory implies that all the solutions of the equations of\nmotion of $S_{{\\cal G}_C^{++}}$ are also solutions of the\nequations of motion of $S_{{\\cal G}^{+++}}$. We then find that $S_{{\\cal G}^{+++}}$ contains another\n action $S_{{\\cal G}_B^{++}}$ obtained by performing the\nconsistent truncation {\\em after}  conjugation by a Weyl reflection in\n${\\cal G}^{+++}$. The action $S_{{\\cal G}_B^{++}}$ is also invariant under a ${\\cal G}^{++}$\nalgebra but is nonequivalent to $S_{{\\cal\nG}_C^{++}}$ and singles out naturally a space direction. Through Weyl\ntransformations in ${\\cal G}^{++}$ expressed in terms of fields,\n$S_{{\\cal G}_B^{++}}$ can be formulated\n in various equivalent forms endowed with\nvarious Lorentzian signatures $(-\\dots -, +\\dots +)$,  a consequence of\nthe non-commutativity of the temporal involution with Weyl reflections\n\\cite{keu1,keu2}. Exact solutions of $S_{{\\cal G}_B^{++}}$ are obtained in\naccordance with references \\cite{eh,eh2}.  These solutions   are   {\\em\nidentical} to those of the covariant Einstein and field equations describing,\ndepending on the signature, conventional or exotic intersecting extremal branes\nsmeared in all directions but one\n\\cite{aeh, ah}. They  transform into each other by\n`duality' Weyl transformations. As all these brane solutions of\n$S_{{\\cal G}_B^{++}}$  are also solutions of the\n${\\cal G}^{+++}$-invariant action $S_{{\\cal G}^{+++}}$, the latter necessarily contains solutions\nof exotic counterparts of the original maximally oxidised theories. In\nparticular the $E_{11}\\equiv E_8^{+++}$-invariant action contains\nin addition to the M-theory solutions, the exotic branes of the\nrelated M' and M* theories \\cite{hull1,hull2,ah, exop}.\nOf course $S_{{\\cal G}^{+++}}$ can, as $S_{{\\cal\nG}_B^{++}}$, be formulated with different signatures related\nby field transformations.\n\nThe fact that $S_{{\\cal G}_B^{++}}$, and hence $S_{{\\cal G}^{+++}}$, contains\n{\\em exact} intersecting extremal  brane solutions of space-time\ncovariant theories compactified to all dimensions but one provides a\nlaboratory to analyse the significance of at least some subset of the\ninfinite many fields appearing in the Kac-Moody invariant theories.\nExtremal branes in   more non-compact dimensions differ from the\none dimensional ones only by the dependance\nof a harmonic function on the number of non-compact dimensions. For\nsuch  decompactified solutions to exist in\nthe Kac-Moody theory,   higher level fields must provide the\nderivatives needed to obtain  higher dimensional harmonic\nfunctions. This test is crucial to settle the issue of whether or not\nthe Kac-Moody theories discussed here can really describe uncompactified\nspace-time covariant theories.\n\nThe paper is organised as follows. In section 2, we recall  the\nconstruction of the ${\\cal G}^{+++}$-invariant actions $S_{{\\cal G}^{+++}}$.  In section 3, we\nshow how the `cosmological' ${\\cal G}^{++}$-invariant action $S_{{\\cal G}_C^{++}}$\nis obtained from $S_{{\\cal G}^{+++}}$ by a consistent truncation.  In section 4, we\nobtain the\n`brane' action $S_{{\\cal G}_B^{++}}$ which arises from a Weyl\ntransformation in ${\\cal G}^{+++}$ and a consistent truncation.\nWe show how to formulate $S_{{\\cal G}_B^{++}}$ in various equivalent\nways to exhibit various Lorentzian signatures. Differential\nequations for the Weyl  transformations of fields ensuring the\nequivalence of the different formulations are obtained. The equivalence \nproves that, in addition to \nconventional ones, exotic dualities are present in  ${\\cal G}^{+++}$. This is made\nexplicit in the particular case of $E_{11}$ and is  in agreement\nwith the results of reference \\cite{keu1, keu2}, linking in\n${\\cal G}^{+++}$ M-theory to M' and M*-theories. In Section 5, we derive the exact solutions\nof $S_{{\\cal G}_B^{++}}$, and hence of $S_{{\\cal G}^{+++}}$, which are identical to the\nspace-time solutions describing intersecting extremal branes, conventional and\nexotic, smeared in all directions but one. We illustrate by a specific\nnon-trivial example how dualities transforming these solutions into one\nanother operate. In Section 6 we stress  perspectives suggested by our\nresults.\n\n\n\n\\section{${\\cal G}^{+++}$-invariant action}\n\n\n\nActions $S_{{\\cal G}^{+++}}$ invariant under non-linear\ntransformations of\n${\\cal G}^{+++}$ are constructed recursively from a level\ndecomposition with\nrespect to a subalgebra\n$A_{D-1}$ where $D$ is interpreted as the space-time dimension. The\naction is defined in a reparametrisation invariant way  on a\nworld-line, a priori unrelated to space-time, in terms of fields\n$\\varphi(\\xi)$ where $\\xi$ spans the world-line. The fields $\\varphi(\\xi)$\nlive in a coset space\n${\\cal G}^{+++}\/K^{+++}$ where the subalgebra $K^{+++}$ is invariant under a\n`temporal involution'\n preserving at each level a Lorentz algebra\\footnote{In Section 4, we shall see\nthat $G^{+++}$  contains other $A_{D-1}$ subalgebras intersecting\ndifferently with\n$K^{+++}$.}\n$SO(1,D-1)= A_{D-1} \\cap K^{+++}$.\n\nWe now recall in more detail the construction of these\n${\\cal G}^{+++}$-invariant theories\n\\cite{eh}.\n\n${\\cal G}^{+++}$ contains a  subalgebra $GL(D)$ such that $SL(D) (=A_{D-1})\n\\subset GL(D) \\subset\n{\\cal G}^{+++}$. The generators of the $GL(D)$ subalgebra are taken to be\n$K^a_{~b}\\ (a,b=1,2,\\ldots ,D)$   with commutation relations\n\\begin{equation}\n\\label{Kcom} [K^a_{~b},K^c_{~d}]   =\\delta^c_b\nK^a_{~d}-\\delta^a_dK^c_{~b}\\,  .\n\\end{equation}  The $K^a_{~b}$ along with the  abelian generator $R$,\nwhich is present  when the corresponding maximally oxidised action\n$S_{\\cal G}$ has one\ndilaton\\footnote{All the maximally oxidised  theories have at most one dilaton except\nthe\n$C_{q+1}$-series.  The maximally oxidised theory\n$C_{q+1}$ is a four dimensional theory   containing $q$ dilatons and\n $C_{q+1}^{+++}$  is constructed with\n$q$ abelian generators $R_i \\quad i=1 \\dots q$ (see for instance ref\n\\cite{eh} appendix A3). To avoid crowding of indices, we consider\nhere  only the theories with at most one dilaton.}, are the level zero\ngenerators. The step operators of level greater than zero are tensors\n$R^{\\quad c_1\\dots c_r}_{ d_1\\dots d_s}$ of the\n$A_{D-1}$ subalgebra. The lowest levels contain antisymmetric tensor\nstep  operators\n$R^{a_1a_2 \\dots a_r}$ associated with  electric and magnetic\nroots arising from the dimensional reduction of field strength forms in $S_{\\cal\nG}$. They satisfy the tensor and scaling relations\n\\begin{eqnarray}\n\\label{root} &&[K^a_{~b},R^{a_1\\dots a_r}]   =\\delta^{a_1}_b R^{aa_2\n\\dots a_r} +\\dots +\n\\delta^{a_r}_b R^{a_1 \\dots a_{r-1}a}\\, ,\\\\\n\\label{root2} && [R, R^{a_1\\dots a_r}\n] =   -\\frac{\\varepsilon_A\na_A}{2}\\,  R^{a_1\\dots a_r}\\, ,\n\\end{eqnarray} where $a_A$ is the dilaton coupling constant to the\n field strength form   and\n$\\varepsilon_A$ is $+1\\, (-1)$ for an\nelectric (magnetic) root \\cite{ehtw}. The generators obey the invariant\nscalar product relations\n\\begin{eqnarray}\n\\label{killing}\n&&\\langle K_{~a}^a K_{~b}^b\\rangle =G_{ab}\\, ,\\quad \\langle\nK^b_{~a}K_{~c}^d\\rangle=\n\\delta_c^b\\delta_a^d \\ a\\neq b\\, , \\quad\\langle R R\\rangle\n=\\frac{1}{2}\n\\,,\\\\\n\\label{step}\n&&\\langle\nR^{\\quad a_1\\dots a_r}_{ b_1\\dots b_s} , \\bar  R_{ d_1\\dots   d_r}^{\\quad\nc_1\\dots c_s}\\rangle\n=\\delta^{c_1}_{b_1}\\dots\\delta^{c_s}_{b_s}\\delta^{a_1}_{d_1}\\dots\\delta^\n{a_r}_{d_r}\\,.\n\\end{eqnarray}\nHere $G= I_D -\n\\frac{1}{2}\\Xi_D$ where $\\Xi_D$ is a D-dimensional matrix with all\nentries   equal to unity and   $\\bar  R_{ d_1\\dots   d_r}^{\\quad\nc_1\\dots c_s}$ designates the negative step operator conjugate to $ R^{\\quad\nd_1\\dots   d_r}_{ c_1\\dots c_s}$.\n\nThe temporal involution $\\Omega_1$ generalises the Chevalley\ninvolution to allow identification of the index 1 to a time coordinate in\n$SO(1,D-1)$. It is  defined by\n\\begin{equation}\n\\label{map} K^a_{~b}\\stackrel{\\Omega_1}{\\mapsto}\n-\\epsilon_a\\epsilon_b K^b_{~a}\\quad R\\stackrel{\\Omega_1}{\\mapsto} -R\\quad\n, \\quad R^{\\quad c_1\\dots c_r}_{ d_1\\dots d_s}\n\\stackrel{\\Omega_1}{\\mapsto}\n-\\epsilon_{c_1}\\dots\\epsilon_{c_r}\\epsilon_{d_1}\\dots\\epsilon_{d_s}\n  \\bar R_{ c_1\\dots c_r}^{\\quad d_1\\dots d_s}\\, ,\n\\end{equation}\n with $\\epsilon_a =-1$ if $a=1$ and\n$\\epsilon_a=+1$ otherwise. It leaves invariant a subalgebra $K^{+++}$ of\n${\\cal G}^{+++}$.\n\n The fields $\\varphi(\\xi)$ living in the coset space ${{\\cal G}^{+++}}\/K^{+++}\n$ parametrise the Borel group  built out of Cartan and positive\nstep operators in\n${\\cal G}^{+++}$. Its elements $\\cal V$  are written as\n\\begin{equation}\n\\label{positive} {\\cal V(\\xi)}= \\exp (\\sum_{a\\ge b}\nh_b^{~a}(\\xi)K^b_{~a} -\n\\phi(\\xi) R) \\exp (\\sum\n\\frac{1}{r!s!} A^{\\quad a_1\\dots a_r}_{ b_1\\dots b_s}(\\xi) R_{\na_1\\dots   a_r}^{\\quad b_1\\dots b_s} +\\cdots)\\, ,\n\\end{equation}\n where the first exponential contains only  level zero  operators and\nthe second one the positive step operators of levels strictly greater\nthan zero. Defining\n\\begin{equation}\n\\label{sym}\ndv(\\xi)= d{\\cal V} {\\cal V}^{-1}\\quad d\\tilde\nv(\\xi)=   -\\Omega_1\n\\, dv(\\xi)\n\\qquad\\quad dv_{sym}=\\frac{1}{2} (dv+d\\tilde v)\\, ,\n\\end{equation}\none obtains, in terms of the\n$\\xi$-dependent fields, an  action $S_{\\cal {\\cal G}^{+++}}$ invariant under\nglobal ${\\cal G}^{+++}$ transformations, defined on  the\ncoset ${{\\cal G}^{+++}}\/K^{+++}$\n\\begin{equation}\n\\label{actionG} S_{\\cal {\\cal G}^{+++}}=\\int d\\xi  \\frac{1}{n(\\xi)}\\langle\n(\\frac{dv_{sym}(\\xi)}{d\\xi})^2\\rangle\\, ,\n\\end{equation} where\n$n(\\xi)$ is an arbitrary lapse function ensuring reparametrisation\ninvariance on the world-line. One has\n\\begin{equation}\n\\label{sumsym}\ndv_{sym}= dv_{sym}^0+\\sum_A dv_{sym}^{(A)}\\, ,\n\\end{equation}\nwhere $dv_{sym}^0$ contains all the level zero contributions. One gets\n\\begin{equation}\n\\label{v0} dv_{sym}^0=-\\frac{1}{2}\\sum_{a\\ge b} [e^  h (de^{-\nh})]_b^{~a} (K^b_{~a}- \\Omega_1 K^{b}_{~a}) -\nd\\phi  R\\, ,\n\\end{equation}\n\\begin{equation}\n\\label{vform} dv^{(A)}_{sym}=\\frac{1}{2r!s!}DA_{\\mu_1\\dots\n\\mu_r}^{\\quad\n\\nu_1\\dots\n\\nu_s} \\, \\exp (-\n\\lambda \\phi)\\, e^{~\\mu_1}_{{a}_1}...\ne^{~\\mu_r}_{{a}_r}e_{\\nu_1}^{~{b}_1}\n... e_{\\nu_s}^{~{b}_s}\\, (R^{\\quad a_1\\dots a_r}_{ b_1\\dots\nb_s}-\\Omega_1 R^{\\quad a_1\\dots a_r}_{ b_1\\dots\nb_s})\\,  ,\n\\end{equation}\nwhere $e_\\mu^{~a}=(e^{-h(\\xi)})_\\mu^{~a}$, $\\lambda$ is the\ngeneralisation of the scale parameter\n$-\\varepsilon_A a_A\/2$ to all roots\nand $D\/D\\xi$ is a  covariant derivative generalising\n$d\/d\\xi$ through  non-linear terms arising from  non-vanishing\ncommutators  between  positive  step operators.\n\nWriting\n\\begin{equation}\n\\label{full} S_{{\\cal G}^{+++}} =S_{{\\cal G}^{+++}}^{(0)}+\\sum_A\nS_{{\\cal G}^{+++}}^{(A)}\\, ,\n\\end{equation} where $S_{{\\cal G}^{+++}}^{(0)}$ contains all level\nzero contributions, one obtains\n\\begin{equation}\n\\label{fullzero} S_{{\\cal G}^{+++}}^{(0)} =\\frac{1}{2}\\int d\\xi\n\\frac{1}{n(\\xi)}\\left[\\frac{1}{2}(g^{\\mu\\nu}g^{\\sigma\\tau}-\n\\frac{1}{2}g^{\\mu\\sigma}g^{\\nu\\tau})\\frac{dg_{\\mu\\sigma}}{d\\xi}\n\\frac{dg_{\\nu\\tau}}{d\\xi}+\n\\frac{d\\phi}{d\\xi}\\frac{d\\phi}{d\\xi}\\right],\n\\end{equation}\n\\begin{equation}\n\\label{fulla} S_{{\\cal G}^{+++}}^{(A)}=\\frac{1}{2 r! s!}\\int d\\xi\n\\frac{ e^{- 2\\lambda\n\\phi}}{n(\\xi)}\\left[\n\\frac{DA_{\\mu_1\\dots \\mu_r}^{\\quad \\nu_1\\dots\n\\nu_s}}{d\\xi} g^{\\mu_1{\\mu}^\\prime_1}...\\,\ng^{\\mu_r{\\mu}^\\prime_r}g_{\\nu_1{\\nu}^\\prime_1}...\\,\ng_{\\nu_s{\\nu}^\\prime_s}\n\\frac{DA_{{\\mu}^\\prime_1\\dots {\\mu}^\\prime_r}^{\\quad\n{\\nu}^\\prime_1\\dots {\\nu}^\\prime_s}}{d\\xi}\\right].\n\\end{equation}  The $\\xi$-dependent fields $g_{\\mu\\nu}$ are defined as\n$g_{\\mu\\nu} =e_\\mu^{~a}e_\\nu^{~b}\\eta_{ab}$. The appearance of the\nLorentz metric $\\eta_{ab}$ with $\\eta_{11}=-1$ is a consequence of the\ntemporal involution $\\Omega_1$. The metric $g_{\\mu\\nu}$ allows a\nswitch from  the Lorentz  indices  $(a,b)$ of the fields appearing  in\nEq.(\\ref{positive}) to\n$GL(D)$ indices $(\\mu,\\nu)$.\n\nExact solutions have been found for all ${\\cal G}^{+++}$ theories. In \\cite{eh}\nsolutions describing the algebraic properties of the BPS extremal\nbranes have been discovered.  Each single extremal p-brane is\ncharacterised by  {\\it one\\\/} non-zero field multiplying a  component\nof an antisymmetric tensor step operator of low level.  Each extremal\nbrane is related in this way to a  {\\em real positive root} in ${\\cal G}^{+++}$. In\n\\cite{eh2} these results have been extended and exact solutions of\n$S_{{\\cal G}^{+++}}$ for all  the extremal intersecting brane solutions\n\\cite{aeh,ah} of the maximally oxidised theory  were found. The\nintersection rules determining such  configurations are neatly encoded\nin the ${\\cal G}^{+++}$ algebra. They are expressed as\northogonality  conditions on the real roots characterising  the intersecting\nbranes.\n\n\n\\setcounter{equation}{0}\n\\section{From ${\\cal G}^{+++}$ to  the cosmological ${\\cal G}^{++}_C$-invariant action }\n\nConsider the overextended algebra ${\\cal G}^{++}_C$ obtained from the very-extended\nalgebra ${\\cal G}^{+++}$  by deleting the  node labelled 1 from the Dynkin\ndiagrams of ${\\cal G}^{+++}$ depicted in Fig.1. This algebra is realised in the space of\nKasner solutions of the action $S_{{\\cal G}^{+++}}$ and more generally in the\ncosmological billiards solutions\\footnote{In what follows, we use the notation\n${\\cal G}^{++}$ when referring to the multiplication table of the overextended algebra and\nadd an index to ${\\cal G}^{++}$  when we want to keep track of the\nembedding in ${\\cal G}^{+++}$.}. In this\nsection we show how  an action invariant under\n${\\cal G}^{++}_C$  emerges from the\n${\\cal G}^{+++}$-invariant action $S_{{\\cal G}^{+++}}$   Eq.(\\ref{full}) through a {\\it\nconsistent} truncation. This `cosmological' action $S_{{\\cal G}^{++}_C}$ generalises to\nall ${\\cal G}^{++}$  the\n$E_{10}\\, (\\equiv E_8^{++})$ action of reference \\cite{damourhn02}. It is \nobtained below from\n$S_{\\cal\n{\\cal G}^{+++}}$ by putting  to zero  in the coset representative Eq.(\\ref{positive}) the\nfield multiplying the Chevalley generator\n$H_1=  K_{~1}^1- K_{~2}^2$ and all the\nfields multiplying the positive step operators associated to  roots whose\ndecomposition in terms of simple roots contains  the deleted root\n$\\alpha_1$.  Such a truncation of the action\n$S_{\\cal {\\cal G}^{+++}}$ will be shown to be consistent with all its equations of motion.\n\n\nConsider first the fields $A^{\\quad a_1\\dots a_r}_{ b_1\\dots\nb_s}(\\xi)$ multiplying the positive step\noperators $R_{\na_1\\dots   a_r}^{\\quad b_1\\dots b_s}$.  The $A^{\\quad a_1\\dots a_r}_{\nb_1\\dots b_s}(\\xi)$ equated to zero fall into two classes.\nFirst the set of all  fields appearing in  irreducible\nrepresentations of $A_{D-1}$ whose lowest weight contains $\\alpha_1$.\nSecond, in the other representations of\n$A_{D-1}$, the set of  tensor components with at least one time index. The\nconsistency of the truncation of the fields from the first class is obvious as\nall  terms in the action Eq.(\\ref{full}) either contain no such field or\ncontain at least two of them. Fields of the second class may occur linearly but\nare then multiplied by non diagonal metrics\n$g_{1\\hat{\\mu}}$ where the hatted indices  $\\hat{\\mu}$ run from\n$2$ to $D$. In the triangular gauge for $h_\\mu^{~a}$ stemming from the Borel\ngroup defined in Eq.(\\ref{positive}), the linear terms in the second class\n$A$-fields are necessarily multiplied by at least\none $h_1^{~a} $-field $(a>1)$ associated through $K^1_{~a}$ to\na root $\\alpha_1$. Taking\n\\begin{equation}\n\\label{ofdiag} g_{1\\hat{\\mu}}=0\\, ,\n\\end{equation}\nwe ensure the consistency of putting to zero the fields of the second\nclass and in fact of all the fields multiplying step operators associated\nto a root\n$\\alpha_1$, including those of level zero. Indeed, the equation of\nmotions of the second class $A$-fields are now satisfied. Consider now\nthe equations of motion of the $g_{1\\hat{\\mu}}$. All the terms\ncontaining $g_{1\\hat{\\mu}}$  contain necessarily at least\none other\n$g_{1\\hat{\\nu}}$ in Eq.(\\ref{fullzero}) or at least one \n$A$-field component put to zero in\nEq.(\\ref{fulla}). The equations of motion of the $g_{1\\hat{\\mu}}$\nare thus also consistent with the truncation.\n\nWe now turn to the  fields multiplying Cartan generators. Labelling  $q_1$\n the field multiplying $H_1$ in the\nChevalley basis we express the equation $q_1=0$ in terms of the metric\n$g_{\\mu\\nu}$. We define\n\\begin{equation}\n\\label{pdef}\np_a =  - h^{~a}_a\\, ,\n\\end{equation}\nwhere $h^{~a}_a$ is the\ncoefficient of $K^a_{~a}$ in Eq.(\\ref{positive}) . One gets for all ${\\cal G}^{+++}$\n\\cite{ehtw} the equivalence\n\\begin{equation}\n\\label{embed}\n q_1=0\\qquad\\Longleftrightarrow\\qquad\n p_1 = \\sum_{a=2}^{D} p_a\\, .\n\\end{equation}  Using the fact that the coset model Eq.(\\ref{actionG})\nwas computed in  the triangular Borel  gauge and using Eq.(\\ref{ofdiag}),\nwe  rewrite Eq.(\\ref{embed}) as\n\\begin{eqnarray}\n\\label{det1} g_{11}&=&{\\bf g}\\, ,\\\\\n\\label{det2} {\\bf g}&\\equiv&\\det{g_{\\hat{\\mu}\\hat{\\nu}}}\\, .\n\\end{eqnarray}\n\n\nConsistency of the truncation from ${\\cal G}^{+++}$ to ${\\cal G}^{++}$ requires\nthat Eq.(\\ref{det1}) satisfy the equation of motion of\n$g_{11}$. Only variations with respect to $g_{11}$ in the action\nEq.(\\ref{fullzero}) have to be considered, as the\nvariations in Eq.(\\ref{fulla}) are automatically satisfied because\n$g_{11}$ multiplies  $A$-fields already equated to zero.\nUsing Eq.(\\ref{ofdiag})  we get\n\\begin{equation}\n\\label{eomzero}\n\\frac{d^2}{d\\xi^2}\\ln |g_{11}|-\\frac{d^2}{d\\xi^2}\\ln{\\bf g} = 0\\, ,\n\\end{equation} which admits  as solution Eq.(\\ref{det1}).\n\nTo obtain the  actions $S_{{\\cal G}^{++}_C}$  whose\nequations of motion are all contained in the equations of motion\nobtained from\n$S_{{\\cal G}^{+++}}$ given by Eq.(\\ref{actionG}), it thus suffices to substitute in\nEq.(\\ref{fullzero}) $g_{11}$ by its value given in Eq.(\\ref{det1}) and\nequate to zero in Eq.(\\ref{fulla}) all first and second\nclass $A$-fields. Denoting the remaining $A$-fields by $B$ and using the\nhatted indices $\\hat\\mu , (\\mu=2,\\dots D)$, we obtain\n\n\\begin{equation}\n\\label{fullp} S_{{\\cal G}^{++}_C} = S_{{\\cal G}^{++}_C}^{(0)}+\\sum_B\nS_{{\\cal G}^{++}_C}^{(B)}\\, ,\n\\end{equation} where\n\\begin{eqnarray}\n\\label{fullop} S_{{\\cal G}^{++}_C}^{(0)}& =&\\frac{1}{2}\\int dt\n\\frac{1}{n(t)}\\left[\\frac{1}{2}(g^{\\hat{\\mu}\\hat{\\nu}}g^{\\hat{\\sigma}\\hat{\\tau}}-\ng^{\\hat{\\mu}\\hat{\\sigma}}g^{\\hat{\\nu}\\hat{\\tau}})\\frac{dg_{\\hat{\\mu}\n\\hat{\\sigma}}}{d\\tau}\n\\frac{dg_{\\hat{\\nu}\\hat{t}}}{dt}+\n\\frac{d\\phi}{dt}\\frac{d\\phi}{dt}\\right] ,\\\\\n\\label{fullc} S_{{\\cal G}^{++}_C}^{(B)}&=&\\frac{1}{2 r! s!}\\int dt\n\\frac{ e^{- 2\\lambda\n\\phi}}{n(t)}\\left[\n\\frac{DB_{\\hat{\\mu}_1\\dots \\hat{\\mu}_r}^{\\quad \\hat{\\nu}_1 \\dots\n\\hat{\\nu}_s}}{dt} g^{\\hat{\\mu}_1{\\hat{\\mu}}^\\prime_1}...\\,\ng^{\\hat{\\mu}_r{\\hat{\\mu}}^\\prime_r}g_{\\hat{\\nu}_1{\\hat{\\nu}}^\\prime_1}\n...\\ ,    g_{\\hat{\\nu}_s{\\hat{\\nu}}^\\prime_s}\n\\frac{DB_{{\\hat{\\mu}}^\\prime_1\\dots {\\hat{\\mu}}^\\prime_r}^{\\quad\n{\\hat{\\nu}}^\\prime_1\\dots {\\hat{\\nu}}^\\prime_s}}{dt}\\right] .\n\\end{eqnarray} Here we  renamed $\\xi$ as $t$.\nEqs.(\\ref{fullp}),(\\ref{fullop}) and(\\ref{fullc}) follow from \nEq.(\\ref{actionG}) by replacing\n$dv_{sym}$ by\n$dv{_{sym}}_{(+)}$,   which describes a motion on the coset\n${\\cal G}^{++}\/K^{++}_{(+)}$ where  $K^{++}_{(+)}$ is the subalgebra of ${\\cal G}^{++}$\ninvariant under the Chevalley involution. In particular, we\nsee that for ${\\cal G}^{++} = E_8^{++}$, we recover as a consistent truncation of\n$S_{E_8^{+++}}$  given by Eq.(\\ref{actionG}) the action for the\n$E_{10}\\equiv E_8^{++}$-invariant theory proposed in reference\n\\cite{damourhn02}.\n\nThe fact that the solutions of the ${\\cal G}^{++}$-equations of motion are\nalso solutions of the ${\\cal G}^{+++}$-equations of motion corresponding to a\nparticular choice of the initial conditions can be viewed in a\ndifferent way. Consider the general solution of the ${\\cal G}^{+++}\/K^{+++}$\nnon linear sigma-model. In the notations of \\cite{damourhn00}, it\nreads\n\\begin{equation} {\\cal M} (\\xi) = {\\cal M} (0) \\cdot \\exp (\\xi J)\\, , \\label{solu}\n\\end{equation}\nwhere ${\\cal M}$ is defined as ${\\cal M} = {\\cal V}^{T'} {\\cal V}$ in terms\nof the ${\\cal G}^{+++}$-group element ${\\cal V}$.  Here, the generalised\ntransposition is defined in terms of the involution $\\Omega_1$ as\n$E^{T'} = - \\Omega_1(E)$ for any Lie algebra element $E$ (and\nextended by the rule $(AB)^{T'} = B^{T'} A^{T'}$). In\n(\\ref{solu}), the $J$'s are the conserved currents and belong\nto ${\\cal G}^{+++}$.  If we choose the initial conditions in such a way\nthat ${\\cal M} (0)$ belongs to the ${\\cal G}^{++}$-subgroup and the current $J$\nis in the subalgebra ${\\cal G}^{++}$, then we see from (\\ref{solu}) that\n${\\cal M} (\\xi)$ is in the ${\\cal G}^{++}$-subgroup for all ``times\" $\\xi$.\nThis proves consistency of the truncation at the level of the\nequations of motion.\n\n\n\\setcounter{equation}{0}\n\\section{From ${\\cal G}^{+++}$ to  the brane ${\\cal G}^{++}_B$-invariant action}\n\n\\subsection{Weyl reflections of the gravity line}\nThe Weyl reflection $W_{\\alpha_1}$ in the hyperplane perpendicular to\n$\\alpha_1$ generates a subalgebra ${\\cal G}^{++}_2$ conjugate to ${\\cal G}^{++}_C$ in\n${\\cal G}^{+++}$. More generally, performing  Weyl reflections from roots of the\ngravity line, we generate in this way $(D-1)$  subalgebras ${\\cal G}^{++}_a,\n(a = 2,\\dots, D)$, which are all conjugate in ${\\cal G}^{++}$.  While the generators of\n${\\cal G}^{++}_C$ were labelled by  spatial indices only, the\nindex\n$a$ in\n${\\cal G}^{++}_a$ must   be interpreted as  a time coordinate. The fact\nthat we have $D-1$ different identifications of the  time\ncoordinate follows from the non-commutativity of  the time\ninvolution with the Weyl reflections \\cite{keu1} as seen\nbelow. This non-commutativity will be exploited in the \nmore general context of Section 4.2 but is well illustrated by\nits effect on the gravity line.\n\nExpressing a Weyl transformation $W$  as a conjugation by a group\nelement\n$U_W$ of ${\\cal G}^{+++}$, we define the involution $\\Omega^\\prime$ operating on\nthe conjugate elements by\n\\begin{equation}\n\\label{newinvolve}\nU_W\\, \\Omega T\\, U^{-1}_W=\\Omega^\\prime \\, U_W  T U^{-1}_W\\, ,\n\\end{equation}\nwhere $T$ is any generator of ${\\cal G}^{+++}$.  Applying Eq.(\\ref{newinvolve}) to\nthe Weyl reflection $W_{\\alpha_1}$  generates the subalgebra\n${\\cal G}^{++}_2$ conjugate to\n${\\cal G}^{++}_C$. One gets\n\\begin{eqnarray}\n\\label{permute}\n&&U_1\\, \\Omega K^2_{\\ 1} \\, U^{-1}_1= \\rho K^2_{\\ 1}= \\rho\\Omega^\\prime\n K^1_{\\ 2}\\nonumber\\, ,\\\\\n&&U_1\\, \\Omega K^1_{\\ 3} \\, U^{-1}_1= \\sigma K^3_{\\ 2}=\n\\sigma\\Omega^\\prime\n K^2_{\\ 3} \\, ,\\\\\n&&U_1\\, \\Omega K^i_{\\ i +1} \\, U^{-1}_1= -\\tau K^{i+1}_{\\ \\, i}=\n\\tau\\Omega^\\prime  K^i_{\\ i +1}\\quad i >2\\, .\\nonumber\n\\end{eqnarray}\nHere $\\rho,\\sigma,\\tau$ are plus or minus signs which may arise\nas step operators are representations of the Weyl group \nup to signs. Eq.(\\ref{permute}) illustrate the general result\nthat such signs always cancel in the determination of\n$\\Omega^\\prime$ because they are identical in the Weyl transform of\ncorresponding positive and negative roots, as\ntheir commutator is in the Cartan subalgebra which  forms a true\nrepresentation of the Weyl group. The content of Eq.(\\ref{permute}) is\nrepresented in Table 1. The signs below the generators of the gravity\nline indicate the sign in front of the\n negative step operator obtained by the involution: a\nminus sign is in agreement with the conventional Chevalley involution\nand indicates that the indices in\n$K^a_{\\ a +1}$ are both either space or time indices while a plus sign\nindicates that one index must be time and the other  space.\n\\begin{table}[h]\n\\caption{\\small Involution switch from $\\Omega$ to\n$\\Omega^\\prime$ due to the Weyl reflection $W_{\\alpha_1}$}\n\\begin{center}\n\\begin{tabular}{|c|ccccc|c|}\n\\hline\ngravity line&$K^1_{\\ 2}$&$K^2_{\\ 3}$&$K^3_{\\ 4}$&$\\cdots$&$K^{D-1}_{\\\nD}$&time coordinate\\\\\n\\hline\\hline\n$\\Omega$&$+$&$-$&$-$&$-$&$-$&1\\\\\n\\hline$\\,\\Omega^\\prime$&$+$&$+$&$-$&$-$&$-$&2\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{gravity}\n\\end{table}\n\n\\noindent\nTable 1 show that\nthe  time coordinates in\n${\\cal G}^{++}_2$ must now be identified either with 2, or with all indices\n$\\neq 2$. We choose the first description, which leaves\nunaffected coordinates attached to planes invariant under the Weyl\ntransformation. Similarly the time coordinate in\n${\\cal G}^{++}_a$ is taken to be\n$a$.\n\nOne may now apply the analysis of Section 3 to obtain  actions\n$S_{{\\cal G}^{++}_a}$ invariant under ${\\cal G}^{++}_a$, namely we equate as before to zero\nall the fields in the ${\\cal G}^{+++}$-invariant action Eq.(\\ref{full}) which\nmultiply generators not involving the root $\\alpha_1$, but this\ntruncation is performed {\\em after} the Weyl transformation which\ntransmutes the time index 1 to a space index. The actions $S_{{\\cal G}^{++}_a}$ are then\nformally identical to the one given by Eqs.(\\ref{fullp}),\n(\\ref{fullop}) and (\\ref{fullc}) but with a Lorentz signature for the\nmetric, which in the flat coordinates amounts to a negative sign for the\nLorentz metric component $\\eta_{aa}$, and with $\\xi$ identified to the missing\nspace coordinate instead of $t$.\n\nThe $D-1$ actions $S_{{\\cal G}^{++}_a}$ for $a = 2,\\dots, D$  differ only by the\nindex identifying the time coordinate and the concomitant identification of\n$\\xi$ with a space coordinate. They are thus equal up to a trivial redefinition\nof all tensor fields by an interchange of indices. This redefinition may be\nviewed as the Weyl transformations generated by the roots of the subalgebra\n$A_{D-2}$ on the space of fields. Equivalence under Weyl transformations from\nroots which do not belong to the gravity line are far less trivial and will now\nbe examined.\n\n\n\n\n\\subsection{General Weyl reflections}\n\nThe $D-1$  actions $S_{{\\cal G}^{++}_a}$ differ by the labelling of the\ntime coordinate and have all the same global signature $(1,9)$.   They\nare related through Weyl transformations of ${\\cal G}^{++}$ from roots of\nthe gravity line. Consider now the Weyl transformations of ${\\cal G}^{++}$\ngenerated in addition by the simple roots not belonging to $A_{D-2}$.\nThese will yield  actions with different global signatures.\nAll such actions will be shown to be equal and related by field\nredefinitions. This equivalence realises in the action\nformalism  the general analysis of Weyl transformations by Keurentjes\n\\cite{keu1,keu2}. It will be related to the existence, in\naddition to conventional duality symmetries, to exotic dualities, which\nin the particular case of M-theory, are relating it to M* and\nM'-theories \\cite{hull1,hull2}. To illustrate how Weyl transformation change\nglobal signatures, we first consider explicitly  the signature changes for\n$E_8^{++}\\equiv E_{10}$.\n\nThe Weyl reflection from the root\nassociated to any generator $R^{abc}$ multiplying the three-form\npotential $A_{abc}$ can always be mapped to the Weyl reflection\n$W_{\\alpha_{11}}$ from the simple root $\\alpha_{11}$ by products of\nWeyl reflections of the gravity line, thereby changing the  time\ncoordinate, originally positioned at 1, to any position.\n\nLet us choose the time coordinate to be 9 and consider the Weyl\nreflection $W_{\\alpha_{11}}$. The only generator of the gravity line\naffected by the Weyl transformation is $K^8_{\\ 9}$. One has from\nEq.(\\ref{newinvolve})\n\\begin{equation}\n\\label{weyla}\nU_{11}\\, \\Omega R^{8\\,10\\,11} \\, U^{-1}_{11}= -\\rho K^9_{\\ 8}=\n\\rho\\Omega^\\prime\n K^8_{\\ 9}\\, .\n\\end{equation}\nImposing that the coordinate 2 remains unaffected by the transformation,\nwe see that both 10 and 11 become time coordinates, as\nillustrated in Table 2. The transformation of the  generator\n$R^{9\\,10\\,11}$ yields\n\\begin{equation}\n\\label{weylb}\nU_{11}\\, \\Omega R_{9\\,10\\,11} \\, U^{-1}_{11}= \\sigma R_{9\\,10\\,11}=\n\\sigma\\Omega^\\prime\n R^{9\\,10\\,11}.\n\\end{equation}\nThe action of the involution\n$\\Omega^\\prime$  on the simple root  not\npertaining to the gravity line, $\\Omega^\\prime R^{9\\,10\\,11}=\nR_{9\\,10\\,11}$, differs by a sign from the action of a temporal\ninvolution  defined according to Eq.(\\ref{map}) when the times are\n10 and\n11.   This shift of sign is shown\nin the last column of Table~\\ref{e10}. It will lead to negative kinetic\nenergy terms in the corresponding actions\n$S_{{\\cal G}{}_{(10\\,11)}^{++}}^{(2,8,-)}$ below. Table~\\ref{e10}\nshows how by repeated use of Eqs.(\\ref{weyla}) and (\\ref{weylb}) one\nreaches the signatures $(5,5,+)$,\n$(6,4,-)$ and\n$(9,1,+)$.\n\n\\begin{table}[h]\n\n\\caption{\\small Involution switches from $\\Omega$ to\n$\\Omega^\\prime$ due to the Weyl reflection $W_{\\alpha_{11}}$}\n\n\\begin{center}\n\\begin{tabular}{|ccccccccc|c|c|}\n\\hline\n$K^2_{\\ 3}$&$K^3_{\\ 4}$&$K^4_{\\ 5}$&$K^5_{\\ 6}$&$K^6_{\\\n7}$&$K^7_{\\ 8}$&$K^8_{\\ 9}$&$K^9_{\\ 10}$&$K^{10}_{\\ 11}$&\ntimes & $(t,s,\\pm)$ \\\\\n\\hline\\hline\n$-$&$-$&$-$&$-$&$-$&$-$&$+$&$+$&$-$&9&$(1,9,+) $\\\\\n\\hline $-$&$-$&$-$&$-$&$-$&$-$&$-$&$+$&$-$&10,11&$(2,8,-)\n$\\\\\n\\hline\\hline\n$-$&$-$&$-$&$-$&$+$&$-$&$+$&$-$&$-$&7,8&$(2,8,-) $\\\\\n\\hline $-$&$-$&$-$&$-$&$+$&$-$&$-$&$-$&$-$&7,8,9,10,11&$(5,5,+)\n$\\\\\n\\hline\\hline$-$&$-$&$-$&$+$&$-$&$-$&$+$&$+$&$-$&2,3,4,5,9&$(5,5,+)\n$\\\\\n\\hline$-$&$-$&$-$&$+$&$-$&$-$&$-$&$+$&$-$&2,3,4,5,10,11&$(6,4,-)\n$\\\\\n\\hline\\hline$-$&$-$&$-$&$-$&$-$&$+$&$-$&$-$&$-$&2,3,4,5,6,7&$(6,4,-)\n$\\\\\n\\hline$-$&$-$&$-$&$-$&$-$&$+$&$+$&$-$&$-$&2,3,4,5,6,7,9,10,11&$(9,1,+)\n$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{e10}\n\\end{table}\n\nIt is interesting to illustrate the string interpretation of the\nWeyl transformations of Table 2  in terms of signature changing\ndualities discussed in \\cite{hull1,hull2}. Consider the type IIA interpretation\nof $E_8^{+++}$. In the Dynkin diagram of\n$E_8^{+++}$ in Fig. 1 the node labelled 10 is no longer on\nthe gravity line and should be redrawn as perpendicular to it at the node\nlabelled 9. Recall that in the embedding of  $E_8^{++}$ in\n$E_8^{+++}$ the coordinate 1 is  a space coordinate and that the string\ninterpretation of the Weyl reflection $W_{\\alpha_{11}}$ is a double\n$T$-duality in the directions 9 and 10 plus an exchange\nof these  directions \\cite{weylt,banks,ehtw}.\n\nThe first Weyl\nreflection of Table 2  maps  the\n$E_8^{++}$ signature\n$(1,9,+)$ to $(2,8,-)$. To get the corresponding string theories we  ignore the\ndirection $11$ and take into account the direction\n$1$ which is spacelike. The $(1,9,+)$ signature corresponds to type\n$IIA$ theory. \nThe action of $W_{\\alpha_{11}}$ renders 11  timelike. Consequently the\nstring interpretation of the $(2,8,-)$ signature is a string theory with\nsignature $(1,9)$ and wrong signs for the kinetic terms in the $RR$ sector.\nThis defines the type $IIA^*$ theory consistently  with \nstring dualities. Indeed a double T-duality with respect to one time\ndirection, 9, and one space direction, 10, maps type $IIA$ onto\ntype $IIA^*$ \\cite{hull1,hull2}. The second Weyl\nreflection of Table 2  maps  the\n$E_8^{++}$ signature\n$(2,8,-)$ with  time directions $7,8$ to $(5,5,+)$ with time directions\n$7,8,9,10,11$. The string interpretation is obtained by the same argument\nas above and one gets in this way a mapping of  $IIA_{8+2}$ to\n$IIA_{6+4}$ theories, in the notation of\n\\cite{hull2}, where the index $s+t$ designates a theory with $s$ space\n and $t$ time directions. This result is in agreement with the string\ndualities. Indeed a double spacelike\n$T$-duality in the direction $9,10$ maps $IIA_{8+2}$ to\n$IIA_{6+4}$ (see Figure 5 of \\cite{hull2}). The third Weyl\nreflection of Table 2  maps  the\n$E_8^{++}$ signature\n$(5,5,+)$ with  time directions $2,3,4,5,9$ on $(6,4,-)$ with time\ndirections\n$2,3,4,5,10,11$. The string interpretation maps accordingly type $IIA_{5+5}$\nto  type $IIA^*_{5+5}$. These two theories are indeed related by a double\n$T$-duality in the time direction 9 and in the space direction\n10. The last Weyl reflection of Table 2 maps the $E_8^{++}$ signature\n$(6,4,-)$ with  time directions $2,3,4,5,6,7$ to $(9,1,+)$ with time\ndirections\n$2,3,4,5,6,7,9,10,11$, in accordance with     the string interpretation\nwhere type $IIA_{4+6}$ is mapped to type $IIA_{2+8}$ by\ndouble spacelike T-duality.\n\n\\subsection{Weyl equivalence of different space-time\nsignatures}\n\nThe action $S_{{\\cal G}^{++}_2}$ is, as $S_{{\\cal G}^{++}_C}$, given by\nEqs.(\\ref{fullp}),(\\ref{fullop}) and (\\ref{fullc}) but with\n$\\eta_{22}=-1$. It can be expressed in the general form \nEq.(\\ref{actionG}) by replacing\n$dv_{sym}$ by $dv{_{sym}}_{(-)}$   which describes a motion on the coset\n${\\cal G}^{++}\/K^{++}_{(-)}$.  $K^{++}_{(-)}$ is the subalgebra of ${\\cal G}^{++}$\ninvariant under the  time involution $\\Omega_2$ defined as in\nEq.(\\ref{map}) with 2 as the time coordinate and restricted to ${\\cal G}^{++}$. To\ncompute\n$dv{_{sym}}_{(-)}$ one specifies a\ncoset  representative of ${\\cal G}^{++}\/K^{++}_{(-)}$ and a level expansion\nabout a gravity line endowed with the involution\n$\\Omega_2$. We shall refer to this embedding of ${\\cal G}^{++}$ in ${\\cal G}^{+++}$ by ${\\cal G}^{++}_B$\nto distinguish it from ${\\cal G}^{++}_C$.\n\n\n\nThe action $S_{{\\cal G}^{++}_2}$  is characterised by a global signature $(1,9, +)$ and\ntime coordinate~2. We may reparametrise the coset space ${\\cal G}^{++}\/K^{++}_{(-)}$ by\nsubjecting its elements to  a Weyl conjugation by an element $U$ of ${\\cal G}^{++}$.\nThis selects a new gravity line in ${\\cal G}^{++}$ which may be endowed with a\ndifferent involution than $\\Omega_2$, as the temporal involution does\nnot in general commute with Weyl transformations. The Borel group and the\nlevel expansion accommodating the new involution will then not coincide with\nthe old one, although the new $\\xi$-dependant fields could\nalways in principle be expressed in terms of the old ones. One can\nconstruct in this way apparently many distinct actions in terms of these\ndifferent set of fields.  Weyl\ntransformations from the gravity line roots permute the tensor indices and\nhence may change  time indices while Weyl reflections from roots which do\nnot belong to the gravity line may change  the global signature, as\nexemplified for\n$E_8^{++}$ in Table 2.  We denote these actions by\n$S_{{\\cal G}{}_{(i{}_1i{}_2\\dots i{}_t)}^{++}}^{(t,s,\\varepsilon)}$, where\nthe global signature is $(t,s,\\varepsilon)$ with\n$\\varepsilon$ denoting  a set of $+1$ or $-1$ signs associated to each\nsimple root which does not pertain to the gravity line, and\n$i{}_1i{}_2\\dots i{}_t$ are the time indices. We shall verify that all these\nactions are equivalent and that their corresponding Lagrangians are equal\nfor all $\\xi$. We show the equivalence by deriving the differential equations\n relating the fields parametrising the different coset representatives.\n\n\n\nWe write  $S_{{\\cal G}{}_{(i{}_1i{}_2\\dots\ni{}_t)}^{++}}^{(t,s,\\varepsilon)}$  in terms of\n$dv_{sym \\, (i{}_1i{}_2\\dots i{}_t)}^{(t,s,\\varepsilon)}$ as\n\\begin{equation}\n\\label{newaction}\n S_{{\\cal G}{}_{(i{}_1i{}_2\\dots\ni{}_t)}^{++}}^{(t,s,\\varepsilon)} =\\int d\\xi  \\frac{1}{n(\\xi)}\\langle\n(\\frac{dv_{sym \\, (i{}_1i{}_2\\dots\ni{}_t)}^{(t,s,\\varepsilon)}(\\xi)}{d\\xi})^2\\rangle\\, ,\n\\end{equation}\nwhere $dv_{sym \\, (i{}_1i{}_2\\dots i{}_t)}^{(t,s,\\varepsilon)}$ is\ncomputed  as in Eqs.(\\ref{positive}) and (\\ref{sym}), but using a\nBorel  representative  of\n${\\cal G}^{++}\/K^{++}_{(-)}$,  built  from a gravity line endowed\nwith the involution\n$\\widetilde\n\\Omega$ defining the signature $(t,s,\\varepsilon)$ with\n$\\{i{}_1i{}_2\\dots i{}_t\\}$ as time coordinates.\nThis yields the same expressions as in Eqs.(\\ref{fullp}),\n(\\ref{fullop}) and (\\ref{fullc}) but with a different space-time\nsignature and a change of sign for all terms in levels generated by an\nodd number of simple roots not pertaining to the gravity line and\ncarrying a negative contribution to $\\varepsilon$.\n\n\nWe denote the Cartan\ngenerators  $K^a_{~a}$ and $R$  as $H^{\n i}$ and the tensor positive step operators as $R^{\nj}$. In terms of the differential forms $\\{X_i, Y_j\\}$ built from\nthe Borel group fields , one has\n\\begin{equation}\n\\label {new}\n dv_{sym \\, (i{}_1i{}_2\\dots i{}_t)}^{(t,s,\\varepsilon)}=\n\\sum_i X_i H^i + \\sum_j  Y_j\n (R^j -\n\\widetilde\\Omega R^j)\\, .\n\\end{equation}\nOn the other hand $S_{{\\cal G}{}_{(i{}_1i{}_2\\dots\ni{}_t)}^{++}}^{(t,s,\\varepsilon)}$ can be obtained by\nconjugation from $S_{{\\cal G}^{++}_2}$ in the following way. Write\n$dv{_{sym}}_{(-)}$ as\n\\begin{equation}\n\\label{sym-}\ndv{_{sym}}_{(-)} = \\sum_i X_i^\\prime H^{\\prime\\, i} + \\sum_j Y_j^\\prime\n(R^{\\prime\\, j} -\n\\Omega_2 R^{\\prime\\, j})\\, ,\n\\end{equation}\nwhere $\\{X_i^\\prime, Y_j^\\prime\\}$ are the differential forms built   \n   from the Borel fields defined by a gravity line endowed with\nthe involution\n$\\Omega_2$. Select\n generators $R^{\\prime\\prime\\, j}$  whose  Weyl transform\n$R^i=\\widetilde U R^{\\prime\\prime\\, j} \\widetilde U^{-1},\\,\n\\widetilde U\n\\subset{\\cal G}^{++}$,\n spans the\npositive step generators in the level expansion defining $dv_{sym \\,\n(i{}_1i{}_2\\dots i{}_t)}^{(t,s,\\varepsilon)}$ and  maps the\ninvolution\n$\\Omega_2$ to\n$\\widetilde \\Omega$. We have\n\\begin{equation}\n\\label{correspond}\n\\widetilde U (R^{\\prime\\prime\\, j} -\n\\Omega_2 R^{\\prime\\prime\\, j}) \\widetilde U^{-1}=\\rho^j (R^j -\n\\widetilde\\Omega R^j)\\, ,\n\\end{equation}\nand $\\rho^j$ is a\nsign. Note that\n$R^{\\prime\\prime\\, j} $ need not be a positive step operator\n$R^{\\prime\\, j}$ in Eq.(\\ref{sym-})  but one  always has\n\\begin{equation}\n\\label{negative}\n (R^{\\prime\\prime\\, j}  -\n\\Omega_2 R^{\\prime\\prime\\, j} ) =\\lambda^j ( R^{\\prime\\, j} -\n\\Omega_2  R^{\\prime\\, j})\\, ,\n\\end{equation}\n  where $\\lambda^j$ is a\nsign. Performing the conjugation  on $dv{_{sym}}_{(-)}$ one gets\n\\begin{equation}\n\\label {conjugate}\n\\widetilde U dv{_{sym}}_{(-)}\\widetilde\nU^{-1}= \\sum_i X_i^\\prime H^i + \\sum_j  Y_j^\\prime\n\\lambda^j\\rho^j (R^j -\n\\widetilde\\Omega R^j)\\, .\n\\end{equation}\nComparing Eq.(\\ref{new}) with Eq.(\\ref{conjugate}) we see that the\nintegrable differential equations relating the different Borel fields\n\\begin{equation}\n\\label{field}\n\\hbox{\\framebox{$ X_i^\\prime= X_i$\\, ,\\, $  Y_j^\\prime \\lambda^j\\rho^j=\nY_j$}}\\,\n\\end{equation}\n ensures that $dv_{sym \\, (i{}_1i{}_2\\dots i{}_t)}^{(t,s,\\varepsilon)}$\nand $dv{_{sym}}_{(-)}$ define the same Lagrangian for all $\\xi$ and\nprove the equivalence of\n $S_{{\\cal G}{}_{(i{}_1i{}_2\\dots\ni{}_t)}^{++}}^{(t,s,\\varepsilon)}$ and  $S_{{\\cal G}^{++}_2}$.\nThe resulting field transformations realise\nthe Weyl transformation $\\widetilde U$ in field space. We denote the\nset of equivalent actions  by $S_{{\\cal G}^{++}_B}$.\nThe two distinct  actions $S_{{\\cal G}^{++}_B}$ and $S_{{\\cal G}^{++}_C}$ are both contained\nas consistent distinct truncations of the unique action $S_{{\\cal G}^{+++}}$, which\nencompasses different signatures and can be formulated, as was\ndone for $S_{{\\cal G}^{++}_B}$, as separate actions identified through field\nredefinitions.\n\n\nWe illustrate the field transformations obtained from Eq.(\\ref{field}) by\nconsidering for ${\\cal G}^{++} =E_8^{++}$ the equivalence of $S_{{\\cal\nG}{}_{(9)}^{++}}^{(1,9,+)}$ and $ S_{{\\cal G}{}_{(10\\,\n11)}^{++}}^{(2,8,-)} ,$ under the Weyl transformation $W_{\\alpha_{11}}$\ngiven in the first block of Table 2. First consider the non-trivial\nrelation $ Y_j^\\prime \\lambda^j\\rho^j= Y_j$ at the lowest level. Taking into\naccount Eq.(\\ref{weyla}) and Eqs.(\\ref{v0})-(\\ref{vform}) we get the following\nnon-linear relations (up to the sign $\\lambda^j\\rho^j$ not taken into account here)\n\\begin{eqnarray}\n\\label{relaa}\n\\frac{1}{3!}dA^\\prime_{\\mu \\nu\n\\rho}\\,\n (e^{h^\\prime})^{~\\mu}_{8} (e^{h^\\prime})^{~\\nu}_{10}\n(e^{h^\\prime})^{~\\rho}_{11}   &=& -[e^  h\n(de^{-h})]_8^{~9}\\, ,\\\\\n\\label{relab}-[e^{h^\\prime}\n(de^{-{h^\\prime}})]_8^{~9}\n &=&\\frac{1}{3!}dA_{\\mu \\nu\n\\rho}\\, (e^h)^{~\\mu}_{8} (e^h)^{~\\nu}_{10}\n(e^h)^{~\\rho}_{11}\\, .\n\\end{eqnarray}\nWe now turn to the relation $X_j^\\prime= X_j$ between the Cartan\ngenerators. From $\\sum_{i=2}^{11} X_i^\\prime H^{\\prime\\, i}\n\\equiv -\\sum_{a=2}^{11} dp^{\\prime a} K^a_{~a}~(p^{\\prime a}=- h^{~\\prime a}_a)$ \none expresses the Weyl reflection\n$W_{\\alpha_{11}}$ as\n\\cite{ehtw}\n\\begin{eqnarray}\\nonumber K^{\\prime a}{}_a&=&K^a_{~a}\\quad a=2,\\ldots ,8\n\\\\ \\nonumber K^{\\prime a}{}_{a}&=&K^a_{~a}+{1\\over\n3}(K^2_{~2}+\\dots+K^8_{~8})-{2\\over\n3}(K^{9}_{~9}+K^{10}_{~10}+K^{11}_{~11})~~ a=9,10,11 \\,\n\\end{eqnarray} to recover the known result\n\\begin{eqnarray}\n\\label{WeylMe2} p^{\\prime a}+\\frac{1}{3}(p^{\\prime\n9}+p^{\\prime 10}+p^{\\prime 11})&=&p^a\n\\qquad\na=2,\\dots, 8\\\\\n\\label{WeylMe1} p^{\\prime a} -\\frac{2}{3}(p^{\\prime\n9}+p^{\\prime 10}+p^{\\prime 11}) &=&p^a\n\\qquad\na=9,10,11\\, .\n\\end{eqnarray}\nWe shall apply in the next section Eqs. (\\ref{relaa}),(\\ref{relab}),\n(\\ref{WeylMe2}), and (\\ref{WeylMe1}) to exact \nextremal brane solutions of ${\\cal G}^{++}_B$.\n\n\\setcounter{equation}{0}\n\\section{Intersecting extremal branes in ${\\cal G}^{++}$ theories}\nExact solutions of  $S_{{\\cal G}^{+++}}$  giving the algebraic properties\nof  intersecting  brane configurations have been constructed in\n\\cite{eh2}. We  show in this section that in $S_{{\\cal G}^{++}_B}$,\nthese coincide with the solutions of the Einstein and field equations of\nthe maximally oxidised theories describing  the intersecting branes\n smeared in all directions but one. $\\xi$ is  identified to the\nnon-compact coordinate.\n\n\n\nWe consider the level decomposition in ${\\cal G}^{++}_B$ for a  signature\n$(t,s,\\varepsilon)$ with $D-1=t+s$ connected by  Weyl\nreflections  to the `phase' $(1,D-2,+)$ with time coordinate 2 of the\n${\\cal G}^{++}_B$ theory. This is the phase described by the fields in the action\n$S_{{\\cal G}^{++}_2}\\equiv S_{{\\cal G}{}_{(2)}^{++}}^{(1,9,+)}$.\n\nThe equations of motion yielding the intersecting brane\nconfiguration solutions from $S_{{\\cal G}^{+++}}$ in \\cite{eh2} can immediately be\nread in $S_{{\\cal G}^{++}_2}$ provided the embedding relation\nEq.(\\ref{embed}) is satisfied with  $p_1 = \\sum_{a=2}^{D} p_a$ \nlabelling  a space direction orthogonal to the branes. The solutions are\nunaltered for\n$a=2,3\\dots D$. As pointed out in\n$\\cite{eh2}$, these solutions can be extended to exotic branes with\n$t_A$ longitudinal timelike and $s_A$ spacelike directions connected in\n${\\cal G}^{+++}$ by Weyl reflections. The results of Section 4 yield therefore the\nintersecting brane configuration solutions, exotic or not, from the fields\ndefining \n$S_{{\\cal G}{}_{(i{}_1i{}_2\\dots\ni{}_t)}^{++}}^{(t,s,\\varepsilon)}$. For each of the $\\cal N$ branes\npresent in the configuration and characterised by\n$\\tau_1 \\dots \\tau_{t_A}$ longitudinal timelike directions and\n$\\lambda_1\\dots\n\\lambda_{s_A}$ longitudinal spacelike directions, one has\n\\begin{equation} A_{\\tau_1 \\dots \\tau_{t_A}\\lambda_1\\dots \\lambda_{s_A}}=\n\\epsilon_{\\tau_1 \\dots \\tau_{t_A}\\lambda_1\\dots \\lambda_{s_A}}\n[\\frac{2(D-2)}{\\Delta_A}]^{1\/2}H_A^{-1}(\\xi)\n\\qquad A=1 \\dots {\\cal N}\\, ,\n\\label{aequ}\n\\end{equation} and\n\\begin{eqnarray}\n &&p^a= \\sum_{A=1}^{\\cal N} p_A^a=\\sum_{A=1}^{\\cal N}\n\\frac{\\eta_A^a}{\\Delta_A}\n\\ln H_A(\\xi)\\qquad a=2,3,\\dots,D  \\label{pequ}\\\\\n\\label{phiequ} &&\\phi =\\sum_{A=1}^{\\cal N} \\, \\phi_A = \\sum_{A=1}^{\\cal\nN}\\frac{D-2}{\\Delta_A}\\varepsilon_A a_A\n\\ln H_A(\\xi) \\, .\n\\end{eqnarray} Here $\\eta_A^a=s_A+t_A$ or $-(D-2-s_A-t_A)$ depending on\nwhether  the direction $a$ is perpendicular or parallel to the\n$q_A$-brane and\n$\\Delta_A= (s_A+t_A)(D-2-s_A-t_A)+\\frac{1}{2}a_A^2(D-2)$.  The factor\n$\\varepsilon_A$ is $+1$ for an electric brane and $-1$ for  a magnetic one. Each\nof the  branes in the configuration is thus described as electrically charged and\nis characterised by one positive harmonic function  in\n$\\xi$-space, namely one has\n\\begin{equation}\n\\frac{d^2H_A(\\xi)}{d\\xi^2}=0 \\qquad A=1 \\dots {\\cal N}\\, .\n\\label{harmo}\n\\end{equation}\nFrom Eq.(\\ref{pequ}), the embedding relation Eq.(\\ref{embed}) yields for\nthe spatial  direction 1 the result\n\\begin{equation}\n\\label{dir1}\np^1= \\sum_{A=1}^{\\cal N}\n\\frac{s_A+t_A}{\\Delta_A}\\,\n\\ln H_A(\\xi)\\, ,\n\\end{equation}\nidentifying it to a direction transverse to all branes.\nThe Eqs. (\\ref{pequ}), (\\ref{phiequ}) and (\\ref{dir1})\nare solutions provided the generalised intersection rules\n\\cite{ah}\n\\begin{equation}\n\\bar{s}+\\bar{t}=\\frac{(s_A+t_A)(s_B+t_B)}{D-2}-\\frac{1}{2}\n\\varepsilon_A a_A \\varepsilon_B a_B \\label{exorule}\n\\end{equation}\nare satisfied. The intersection rules Eq.(\\ref{exorule}) can be expressed\n as an orthogonality condition between the real positive roots of\n${\\cal G}^{++}_B$ (and\n${\\cal G}^{+++}$) for all  branes present in the configuration\n\\cite{eh2}. This orthogonality condition is in fact the input that\npermits the derivation of the exact solutions  Eqs.\n(\\ref{aequ}), (\\ref{pequ}),  (\\ref{phiequ}) and (\\ref{harmo}) by\nallowing a reduction of $S_{{\\cal G}^{++}_B}$ to  quadratic\nterms.\n\nWe may   verify from the above equations  that the  lapse constraint in\n$S_{{\\cal G}^{++}_B}$ is satisfied and takes the form\n\\begin{equation}\n\\label{xiextremal}\n\\sum_{a=2}^D ( d p^{a})^2 -(\\sum_{a=2}^D\ndp^{a})^2 +\n\\frac{1}{2} (d\\phi)^2-  \\sum_{A=1}^{\\cal N}\\frac{D-2}{\\Delta_A} (d\\ln\nH_A)^2=0\\, ,\n\\end{equation}\nwhere the differentials are taken in $\\xi$-space.\nEq.(\\ref{xiextremal}) expresses the vanishing of the action for\nsolutions involving fields associated to orthogonal roots. This\ncondition is preserved under a Weyl reflection, whether or not the\nlatter induces a signature change. It thus follows from\nthe invariance of $S_{{\\cal G}^{++}_B}$, as expressed by Eq.(\\ref{field}), that the lapse\nequation is invariant under  Weyl reflections. In addition the term $\\Lambda= -\n\\sum_{A=1}^{\\cal N}(D-2\/\\Delta_A) (d\\ln H_A)^2$ is separately invariant because\nthe other terms in Eq.(\\ref{xiextremal}) are the  quadratic form of ${\\cal G}^{++}$\nrestricted to its Cartan subalgebra, which is Weyl invariant. This\ninvariance relates different intersecting branes, Kaluza-Klein momenta\nand monopoles \\cite{eh,eh2} of different phases, conventional or exotic.\n\n\nThese exact solutions of $S_{{\\cal G}^{++}_B}$ describe  not only the\nalgebraic properties of all corresponding solutions, exotic or not,\nof all the maximally oxidised theories and of their exotic counterparts\nrelated to them by `dualities' encoded in the Weyl group of ${\\cal G}^{++}_B$.\nThey are now  identical to the solutions of the Einstein and field\nequations describing  these\nintersecting branes smeared in all directions but the spatial\ndimension 1. This follows on the one hand from the fact that the\nEq.(\\ref{harmo}) describes a harmonic function in one dimension as\nwould be required by the latter solutions when $\\xi$ is identified with\nthe single non-compact space coordinate $x^1$. On the other hand such\nidentification is consistent with Eq.(\\ref{dir1}) which shows that this\nnon-compact direction is indeed transverse to all branes.\n\n\nIt is instructive to  illustrate by a specific example how the\naforementioned  solutions of the Einstein and field\nequations are transformed into themselves by the Weyl transformations\nensuring the uniqueness of $S_{{\\cal G}^{++}_B}$ for conventional and exotic\nsolutions. We consider the mapping of the Kaluza-Klein momentum in\n11-dimensional supergravity, smeared in all transverse directions but\n$x^1$ to the exotic smeared membrane with 2 longitudinal times. We\ntake, as in Table 2, the time to be 9 and the momentum in the direction\n8.  The metric is\n\\begin{equation} ds^2=- \\widetilde H^{-1}\\, (dx^9)^2+ \\widetilde H\n\\left[ dx^8 +(\\widetilde H^{-1}-1)dx^9 \\right]^2 +(dx^1)^2\n+\\sum_{\\mu=2\\dots7, 10, 11}   (dx^\\mu)^2,\n  \\label{kkmetric}\n\\end{equation} where $\\widetilde H(x^1)$  is\na positive   harmonic function in one dimension\n$\\widetilde H(x^1)= A+Q |x^1|\n$.  In the triangular gauge the relevant vielbein are given by\n\\begin{eqnarray}\n\\label{vielkk}e_8{}^{8} &=&  (2-\\widetilde H)^{-{1\\over 2}}= H^{-{1\\over\n2}}\\, ,\\nonumber\\\\ e_9{}^9 &=& (2-\\widetilde H)^{1\\over 2}=H^{1\\over\n2}\\, ,\\nonumber\\\\ e_8{}^9 &=&  (2-\\widetilde H)^{-{1\\over\n2}}-(2-\\widetilde H)^{1\\over 2} =H^{-{1\\over 2}}-H^{1\\over 2}\\, .\n\\end{eqnarray}\nHere we have defined $H=2-\\widetilde H$ and the range of $x^1$ is restricted\nto $H > 0$. These results, when expressed in terms of $H$, differs from\nthe vielbein computed in Appendix B1 of\n\\cite{eh}, where the time direction was 1, by a  sign in the\nexponents of the diagonal time and space vielbein. This difference, which is a\nconsequence of the triangular gauge when the time index is bigger than\nthe space one, is crucial in Eq.(\\ref{weylbrane}) below.\nThe $p^{\\prime a}$ parametrising the Cartan subalgebra  are from\nEq (\\ref{vielkk})\n\\begin{equation}\n\\label{cartanmkk} p^{\\prime 8}=-{1 \\over 2} \\ln H\\, , \\qquad   p^{\\prime\n9}={1\n\\over 2} \\ln H\\, ,\n\\end{equation} while, using Eq.(\\ref{vielkk}), one may compute the non-zero\nnon-Cartan field (see Appendix B1  of\n\\cite{eh})\n\\begin{equation}\n\\label{stepmkk} h_8{}^{\\prime 9}= \\ln H\\, .\n\\end{equation}\nThe  vielbein equation Eq.(\\ref{vielkk}) also yields\n\\begin{equation}\n\\label{vielk}\n-[e^{h^\\prime}\n(de^{-{h^\\prime}})]_8^{~9} =d\\ln H\\, .\n\\end{equation}\nWe perform the Weyl reflection $W_{\\alpha_{11}}$ using Eq.(\\ref{relab}),\n(\\ref{WeylMe2}) and (\\ref{WeylMe1}) . For the Cartan we find  \n\\begin{eqnarray}\n\\nonumber\np^a&=&\\frac{1}{6}\\ln H \\qquad a=2,3,4,5,6,7,9\\\\\n\\label{weylbrane}\np^a&=&-\\frac{1}{3}\\ln H \\qquad a=8,10,11\\, .\n\\end{eqnarray}\nFor the non-Cartan fields we find from Eqs.(\\ref{relab}), (\\ref{vielk}) and\n(\\ref{weylbrane}), up to a sign,\n\\begin{equation}\n\\label{Afield}\ndA_{8 \\, 10\n\\, 11} = dH^{-1}.\n\\end{equation}\nTaking into account the relation Eq.(\\ref{dir1}) this solution describes\nan exotic `membrane' in the directions 8, 10 and 11,  in a phase with\nsignature $(2,8,-)$ with time directions 10 and 11,  characterised by\nthe harmonic function\n$H=2-\\widetilde H$.\nUnder this Weyl transformation the smeared $KK$-momentum of M-theory is\nthus mapped onto an exotic membrane with two longitudinal times of\n$M^*$-theory. The exotic membrane obtained is in perfect agreement\nwith the T-duality interpretation as it can be checked by applying\nthe Buscher transformations \\cite{buscher}\non the KK-momentum solution in 10 dimensions\nobtained by reducing  Eq.(\\ref{kkmetric}) along the direction 11 and\nthen uplifting back to  $M^*$.\n\n\nWhile Eq.(\\ref{weylbrane}) illustrate the invariance of the  quadratic\nform of ${\\cal G}^{++}$ restricted to its Cartan subalgebra, \nEq.(\\ref{Afield})  confirms the invariance of the lapse constraint,\nyielding\n\\begin{equation}\n\\label{branex}\n\\Lambda= -\\frac{1}{2} (d\\ln H)^2\\, ,\n\\end{equation}\nwhere, despite the fact that the membrane has two longitudinal times 10\nand 11, a minus sign does arise in the quadratic lapse constraint from\nthe negative kinetic term in the\n$(2,8,-)$ signature.\n\n\nWe point out that the Einstein solutions required  the additional\nconditions\n\\begin{equation}\n\\Theta_E (-1)^{t_A+1}=1\\qquad \\Theta_E=(-1)^{T+1} \\Theta_M\\, ,\n\\label{theta}\n\\end{equation}\nwhere $\\Theta_E$ and $\\Theta_M$ are the signs of the kinetic\nterms for the electric and magnetic potentials  in the exotic actions.\nThese conditions are trivially satisfied in the conventional phase\n$S_{{\\cal G}^{++}_2}\\equiv S_{{\\cal G}{}_{(2)}^{++}}^{(1,9,+)}$ where $\\Theta_E =1$. This\nphase  is characterised by one time which is always longitudinal to the\nbranes. Therefore they are automatically satisfied in all phases as the\nsolutions in these phases can be obtained from Weyl transformations and\nhence do exist with $\\Theta_E$ identified with the corresponding sign\nencoded in $\\varepsilon$.\n\n\\setcounter{equation}{0}\n\\section{Perspectives}\n\nWe have shown that all solutions of the cosmological and the brane\noverextended invariant actions $S_{{\\cal G}^{++}_C}$ and $S_{{\\cal G}^{++}_B}$ are solutions\nof the very-extended invariant actions $S_{{\\cal G}^{+++}}$. The variable $\\xi$\nused to parametrise the motion of the fields on the coset space\n${\\cal G}^{+++}\/K^{+++}$ in the non-linear realisation $S_{{\\cal G}^{+++}}$ of ${\\cal G}^{+++}$ is then\nidentified respectively to a time  or to a space coordinate and a\nset of fields are consistently made to\nvanish. This suggests that the generators of ${\\cal G}^{+++}$ contain a huge gauge\nredundancy. These generators are indeed\nassociated to fields which must be interpreted as potentials rather\nthan field strengths and equivalent descriptions may be possible with\nboth space and time components of gauge fields. The\nconventional intersecting extremal branes,  easily described by the\n$S_{{\\cal G}^{++}_B}$, could perhaps also be obtained from $S_{{\\cal G}^{++}_C}$ in a\ncomplicated way, while the latter action describes trivially   the\nKasner-like cosmological solutions.\n\nThus the fact that ${\\cal G}^{+++}$ contains on equal footing time and space allows\na selection of the best gauge potentials to describe a\nparticular solution but this `covariant' approach need not  in\nprinciple comprehend a larger gauge invariant content than a\nnon-covariant ${\\cal G}^{++}$,  as least as long as only conventional `phases' of\ntheories are considered. The existence of exotic solutions is a specific\nfeature of ${\\cal G}^{+++}$ which has no counterpart in the overextended action\n$S_{{\\cal G}^{++}_C}$ and is a necessary concomitant of ${\\cal G}^{+++}$-invariance and\nthe temporal involution, as suggested in reference \\cite{keu1}. The only way\nexotic phases could decouple in  very-extended algebra would be by restricting\nthe signature to be Euclidean, a rather problematic alternative.\n\nAn essential result of this work is the existence of solutions of\n$S_{{\\cal G}^{+++}}$ and $S_{{\\cal G}^{++}_B}$ which are identical to the space-time covariant\nsolutions of intersecting extremal branes smeared in all directions but\none. This was made possible by the interpretation of  $\\xi$ as a\nspatial coordinate in the context of $S_{{\\cal G}^{++}_B}$. In this way  we do not\nhave to restrict the interpretation of  $S_{{\\cal G}^{+++}}$ to  algebraic\nconsiderations as in reference \\cite{eh}. The reason of the\nimportance of this result is that the exact solutions of the space-time\ncovariant theories exist with the same algebraic structure but\ndifferent, although simple, functional dependance in more uncompactified\ndimensions. Thus we have a laboratory to check whether or not the\nKac-Moody theories can reach, at least in these simple cases,\nuncompactified gravity from the information contained in higher levels.\nIf this turns out not to be the case, then the content of these\napproaches would probably only be a consistent dimensional reduction of\nthe covariant space time theories down to one dimension through an\ninfinity of equivalent fields. If uncompactified theories are reached,\neven at this elementary level, one has attained the first step of a\nformulation of gravity coupled to some matter fields, which is\nconceptually completely different from the Einstein approach, and which\npossibly includes new degrees of freedom which are hinted at by the\nstring perturbative approaches. We hope to be able to provide the\nanswer to this question in future work.\n\n\\section*{Acknowledgments}\n\n\nWe thank Riccardo Argurio, Hermann Nicolai and Philippe Spindel  for\nfruitful discussions. \n\nThis work was supported in part  by the NATO grant PST.CLG.979008, by\nthe ``Interuniversity Attraction Poles Programme -- Belgian Science\nPolicy '', by IISN-Belgium (convention 4.4505.86),   by Proyectos\nFONDECYT 1020629, 1020832 and 7020832 (Chile) and by the   European\nCommission FP6 programme MRTN-CT-2004-005104, in which F.~E., M.~H. and\nL.~H. are associated to the V.U.Brussel (Belgium).\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Notes}\n\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\n\n\n\n\\subsection{The Multiple Point Principle}\n\\label{sec:MPP}\n\n\nThe origin of the values for the 19 parameters of the Standard Model is unknown. These parameters are the three lepton masses, six quark masses,\nthe three coupling strengths, the QCD vacuum angle, the CKM mixing angles along with a phase, \nthe electroweak vacuum-expectation value and the Higgs-boson mass.\nAs an underlying mechanism to generate parameter values, the so-called Mulitple Point Principle (MPP) ~\\cite{Bennett:1993pj,Bennett:1996hx,Bennett:1996vy} has been proposed.\nThis principle may explain how parameters of the Standard Model, like the masses of the Higgs boson and the top quark, come about.\nThe basic idea of this principle may be illustrated by the triple point of water (as pointed out for \ninstance in~\\cite{Bennett:2003yr}): At the triple point of water we observe\n three coexisting phases, that is, its solid, liquid, and gaseous form.  \nThe triple point occurs at specific, that is, fine-tuned values for\nthe intensive parameters temperature and pressure of about 273.16 K and 0.612 kPa. \nSince the transitions between the three phases are all of first order,\nenergy and volume can be varied in a certain range without changing neither\ntemperature nor pressure of the triple point. \nIn an analogous way coexisting phases of the Higgs potential may determine\nthe fine-tuned values of the masses of the bosons and fermions.\nBesides one minimum at the electroweak scale of  ${\\cal O}(100) \\text{ GeV}$ there\nshould be a degenerate minimum or degenerate minima at a scale $\\Lambda$ far above the electroweak scale up to the Planck scale. \nBased on this principle\nthe Higgs boson mass has been predicted more than 20 years ago to be \n$135 \\pm 9 \\text{ GeV}$~\\cite{Froggatt:1995rt}!\nA more refined analysis yielded a mass $129.4 \\pm 2$ GeV \\cite{Degrassi:2012ry}\nto be compared to the observed mass $125.10 \\pm 0.14$ GeV \\cite{Tanabashi:2018oca} of the Higgs boson\nwhich was discovered by the CMS and ATLAS collaborations in 2012 \\cite{Chatrchyan:2012xdj,Aad:2012tfa}.\n\nLet us briefly sketch the derivation of this remarkable result following closely~\\cite{Froggatt:1995rt}:\nThe MPP states that\n\\begin{enumerate}\n\\item The Higgs-boson doublet $\\varphi$ has at least two coexisting vacua 1 and 2: with the same potential value, that is,\n\\begin{equation} \\label{Cond1}\nV_{\\text{eff}}(\\langle \\varphi \\rangle_1 )= V_{\\text{eff}}(\\langle \\varphi \\rangle_2 ).\n\\end{equation}\n\\item The additional minimum or minima should appear at the high scale $\\Lambda$ with $100 \\text{ GeV}\\ll \\Lambda <  M_{\\text{Planck}}$, \n\\begin{equation} \\label{Cond2}\n\\langle \\varphi \\rangle_2 = {\\cal O}(\\Lambda).\n\\end{equation}\n\\end{enumerate}\nIn the Standard Model with only one Higgs-boson doublet, the\neffective gauge-invariant potential is written, where we define  $\\phi = \\sqrt{ \\varphi^\\dagger \\varphi}$,\n\\begin{equation}\nV_{\\text{eff}} = \\mu^2(\\phi) \\phi^2 + \\frac{\\lambda(\\phi)}{8} \\phi^4 \\;,\n\\end{equation}\nand where the dependence of the parameters on the scale is written explicitly.\nClose to the second vacuum the quartic term is dominant,\n\\begin{equation}\nV_{\\text{eff}} \\approx \\frac{\\lambda(\\phi)}{8} \\phi^4 \\;.\n\\end{equation}\nThe two conditions (1) and (2) above then give, using the fact that \nthe degeneracy of the potential values requires that $\\lambda(\\Lambda) \\approx 0$,\n\\begin{equation}\n\\frac{d V_{\\text{eff}} }{d \\phi} \\bigg |_{\\text{vac 2}} \n= 0 \n = \\frac{1}{2} \\lambda (\\Lambda) \\Lambda^3\n + \\frac{1}{8} \\frac{d \\lambda(\\phi)}{d \\phi}  \\bigg |_{\\text{vac 2}}  \\Lambda^4 \n = \\frac{1}{8} \\frac{d \\lambda(\\phi)}{d \\phi}  \\bigg |_{\\text{vac 2}}  \\Lambda^4 \n = \\frac{1}{8} \\beta_\\lambda(\\Lambda)  \\Lambda^3\\;.\n\\end{equation}\nThis means that in addition to the quartic parameter $\\lambda$ also\nits $\\beta_\\lambda$ function has to vanish at the scale $\\Lambda$. \nThe $\\beta_\\lambda$ function depends in the following way on the Higgs-boson field $\\phi$:\n\\begin{equation}\n\\beta_\\lambda (\\phi) = \\frac{d \\lambda(\\phi)}{d \\ln (\\phi)}\n=\\phi \\frac{d \\lambda(\\phi)}{d (\\phi)}\n =\\beta_\\lambda ( \\lambda (\\phi), g_t(\\phi), g_1(\\phi), g_2(\\phi), g_3(\\phi) )\n\\end{equation}\nwith $g_t(\\phi)$ the top-Yukawa coupling and $g_{1\/2\/3}(\\phi)$ the scale-dependent gauge couplings. From\nthe explicit form of the $\\beta_\\lambda$ function in the Standard Model, Nielsen and Froggatt evaluate the renormalization group equation numerically, using two loop beta functions and plot $\\lambda(\\phi)$. \nThe evolution depends on the masses of the top-quark and the Higgs boson.\nRequiring a vanishing quartic parameter $\\lambda(\\phi)$  as well as a vanishing $\\beta_\\lambda$ function  at the \nhigh scale $\\Lambda$, the masses of the top quark and the Higgs boson are predicted.\\\\\n\nThis prediction for two of the Standard Model parameters is clearly remarkable and raises the question whether the particular form of the Higgs effective potential \nat the high scale is just an accident or whether there is a deeper law of nature represented by the MPP. \nFor this reason, it is worthwhile to have a closer look how the MPP has been motivated in the literature. \nOriginally, the MPP was justified using thermodynamical arguments \\cite{Bennett:1993pj,Bennett:1996hx,Bennett:1996vy} considering a micro-canonical\nensemble in which an extensive quantity (energy, volume, number of particles) is fixed to a given value. It is then argued that this leads to the MPP.\nThe system is described by a set of intensive quantitiies (temperature, pressure, chemical potential) and it is assumed that more than one\nphase exists with a strong (first order) phase transition between different phases.\n\nIt is then argued that in such a system with a fixed extensive variable\nthe probability is high that intensive quantities (temperature, pressure, chemical potential) take on critical values corresponding to a state with two or more coexisting phases.\n\nMotivated by this result, the question arises  what are the consequences of the MPP in the Two-Higgs-Doublet Model (THDM)?\nThe original motivation by T. D.~Lee~\\cite{Lee:1973iz} to study the two-Higgs-doublet extension\nhas been to have another source for CP violation -- one of the shortcomings of the Standard Model, where violation of CP only arises from the CKM matrix (and the PMNS matrix) and is too small to explain the observed baryon asymmetry dynamically. Another motivation has been given by supersymmetric models which require to have more than one Higgs-boson doublet in order to give masses to up- and down type fermions. A more pragmatic reason is that there is nothing which prevents the introduction of more copies of Higgs-boson doublets. \nIn particular, the $\\rho$ parameter, relating the masses of the electroweak gauge bosons with the weak mixing angle (see~\\cite{Bernreuther:1998rx} for details) is measured close to one in agreement with the Standard Model.\nThe $\\rho$ parameter is known to keep unchanged at tree level with respect to additional copies of Higgs-boson doublets.\nEventually, let us mention that compared to the two real parameters of the Higgs potential of the Standard Model, the potential of the THDM has a much richer structure allowing for different phases.\nFor a review of the THDM we refer to~\\cite{Branco:2011iw}.\n\nThe most general gauge-invariant potential with two Higgs-boson doublets\n\\begin{equation} \\label{doublets}\n\\varphi_1 = \\begin{pmatrix} \\varphi_1^{(+)}\\\\ \\varphi_1^{(0)} \\end{pmatrix},\n\\qquad\n\\varphi_2 = \\begin{pmatrix} \\varphi_2^{(+)}\\\\ \\varphi_2^{(0)} \\end{pmatrix},\n\\end{equation}\nin the convention with both Higgs doublets carrying hypercharge $y=+1\/2$, \nreads~\\cite{Gunion:1989we}\n\\begin{equation}\n\\label{Vconv}\n\\begin{split}\nV^{\\text{THDM}} (\\varphi_1, \\varphi_2) =~& \nm_{11}^2 (\\varphi_1^\\dagger \\varphi_1) +\nm_{22}^2 (\\varphi_2^\\dagger \\varphi_2) -\nm_{12}^2 (\\varphi_1^\\dagger \\varphi_2) -\n(m_{12}^2)^* (\\varphi_2^\\dagger \\varphi_1)\\\\\n& +\\frac{1}{2} \\lambda_1 (\\varphi_1^\\dagger \\varphi_1)^2 \n+ \\frac{1}{2} \\lambda_2 (\\varphi_2^\\dagger \\varphi_2)^2 \n+ \\lambda_3 (\\varphi_1^\\dagger \\varphi_1)(\\varphi_2^\\dagger \\varphi_2) \\\\ \n&+ \\lambda_4 (\\varphi_1^\\dagger \\varphi_2)(\\varphi_2^\\dagger \\varphi_1)\n+ \\frac{1}{2} [\\lambda_5 (\\varphi_1^\\dagger \\varphi_2)^2 + \\lambda_5^* \n(\\varphi_2^\\dagger \\varphi_1)^2] \\\\ \n&+ [\\lambda_6 (\\varphi_1^\\dagger \\varphi_2) + \\lambda_6^* \n(\\varphi_2^\\dagger \\varphi_1)] (\\varphi_1^\\dagger \\varphi_1) + [\\lambda_7 (\\varphi_1^\\dagger \n\\varphi_2) + \\lambda_7^* (\\varphi_2^\\dagger \\varphi_1)] (\\varphi_2^\\dagger \\varphi_2).\n\\end{split}\n\\end{equation}\nThe parameters $m_{12}^2$, $\\lambda_{5\/6\/7}$ are complex, whereas all other parameters have to be real \nin order to yield a real potential. Therefore we count in total 14 real parameters in contrast to two real parameters of the Standard Model.\\\\\n\nIn the work~\\cite{Froggatt:2004st} the MPP has been studied with respect to the general THDM. \nThe argumentation in this work has been developed in the following way:\nSupposing that the potential\nhas a second minimum at a high scale $\\Lambda$, after an appropriate \n\\mbox{$SU(2)_L \\times U(1)_Y$}  transformation the vacuum expectation values of the two Higgs-boson doublets are parametrized as \\cite{Froggatt:2004st}\n\\begin{equation} \\label{Holgpara}\n\\langle \\varphi_1 \\rangle = \\phi_1 \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix}, \n\\qquad\n\\langle \\varphi_2 \\rangle = \\phi_2 \\begin{pmatrix} \\sin(\\theta) \\\\ \\cos (\\theta) e^{i \\omega} \\end{pmatrix},\n\\end{equation}\nwhere $\\Lambda^2= \\phi_1^2+\\phi_2^2$.\nThen the conditions are studied for the potential \\eqref{Vconv} and its derivatives with respect to $\\phi_{1\/2}$  to be independent of the phase $\\omega$ at the scale $\\Lambda$. This leads to conditions for the quartic couplings as well as their derivatives, that is, the $\\beta$ functions, evaluated at the scale $\\Lambda$.\nIn particular it is shown that these conditions originating from the MPP yield a CP conserving potential, obeying\nin addition a softly broken $Z_2$ symmetry giving an argument for the absence of flavor-changing neutral currents.\nTherefore, they continue their analysis in the framework of models with natural flavor conservation, e.g.\\ the THDM type II.\n\n{In \\cite{McDowall:2018ulq} a detailed phenomenological study of the MPP is carried out, starting from the results of \\cite{Froggatt:2004st}, and applying them to the THDM type II as well as the Inert Doublet Model. It is found that in both cases, the MPP is incompatible with the requirement of providing simultaneously the experimental value of the top quark mass, electroweak symmetry breaking and stability.\n}\n\nHere we show that the study of the MPP in the THDM can be implemented \nconcisely in the bilinear formalism~\\cite{Nagel:2004sw,Maniatis:2006fs,Nishi:2006tg}.\nThe advantage of the bilinear formalism is that all unphysical gauge degrees of freedom are eliminated systematically and the potential and all parameters are real. Basis transformations, that is, \na unitary mixing of the two doublets, for instance, are given by simple rotations.\nSimilar to the case of the Standard Model, where we have to have a vanishing quartic coupling together with its $\\beta$ function at the high scale $\\Lambda$, we find conditions among the potential parameters and its derivatives in order to satisfy the MPP.\nWe present a classification of all possible realizations of the MPP in the THDM.  \nThe conditions for these classes of realizations are given in a basis-invariant way and can be checked easily for any THDM. \n\nIn order to arrive at the conditions for the MPP we present the $\\beta$ functions\nof the potential parameters in the bilinear formalism (see also~\\cite{Ma:2009ax}).\nWe demonstrate the conditions of different realizations of the MPP in examples\nand we show in particular that the results of \\cite{Froggatt:2004st} can be recovered in the presently developed formalism as one possible realization of the MPP.\nIt should be noted that other MPP solutions have been discussed in \\cite{Froggatt:2008am} in the conventional formalism.\nHere, we will present a complete classification of the MPP solutions in a transparent way using the bilinear formalism.\n\n\\subsection{Brief review of bilinears in the THDM}\n\\label{sec:bilinears}\n\nHere we briefly review the bilinears in the THDM~\\cite{Nagel:2004sw, Nishi:2006tg, Maniatis:2006fs} in order to make this article \nself contained. We will also discuss briefly basis transformations.\nBilinears systematically avoid unphysical gauge degrees of freedom and are defined in the following way: \nAll possible gauge-invariant scalar products of the two doublets $\\varphi_1$ and $\\varphi_2$ which may appear\n in the potential can be arranged in one matrix \n\\begin{equation}\n\\twomat{K} = \n\\begin{pmatrix}\n \\varphi_1^\\dagger \\varphi_1 &  \\varphi_2^\\dagger \\varphi_1\\\\\n \\varphi_1^\\dagger \\varphi_2 &  \\varphi_2^\\dagger \\varphi_2\n \\end{pmatrix}.\n \\end{equation}\n This  hermitian matrix can be decomposed into a basis of \n the unit matrix and the Pauli matrices,\n \\begin{equation}\n \\twomat{K} = \\frac{1}{2} \\left(\nK_0 \\mathbbm{1}_2 + K_a \\sigma_a \\right), \\qquad a=1,2,3,\n\\end{equation}\nwith four real coefficients $K_0$, $K_a$, called bilinears.\nBuilding traces on both sides of this equation (also with products of Pauli matrices) \nwe get the four real bilinears explicitly,\n\\begin{align}\n\\label{eqKphi}\n&K_0 = \\varphi_1^\\dagger \\varphi_1 + \\varphi_2^\\dagger \\varphi_2,\n&&K_1 = \\varphi_1^\\dagger \\varphi_2 + \\varphi_2^\\dagger \\varphi_1, \\nonumber \\\\\n&K_2 = i\\varphi_2^\\dagger \\varphi_1 - i\\varphi_1^\\dagger \\varphi_2, \n&&K_3 = \\varphi_1^\\dagger \\varphi_1 - \\varphi_2^\\dagger \\varphi_2. \n\\end{align}\nThe matrix $\\twomat{K}$ is positive semi-definite. \nFrom $K_0 = \\tr (\\twomat{K})$  and $\\det(\\twomat{K}) = \\tfrac{1}{4} (K_0^2 -K_a K_a)$\nwe get \n\\begin{equation} \\label{Kdom}\nK_0 \\ge 0, \\qquad K_0^2 - K_a K_a \\ge 0.\n\\end{equation}\nAs has been shown in~\\cite{Maniatis:2006fs} there is a one-to-one correspondence between the original\ndoublet fields and the bilinears apart from unphysical gauge-degrees of freedom. \nIn terms of bilinears\nwe can write any THDM (a constant term can always be dropped),\n\\begin{equation}\n\\label{pot}\nV^{\\text{THDM}} (K_0, K_a)=\n \\xi_0 K_0 + \\xi_a K_a + \\eta_{00} K_0^2 + 2 K_0 \\eta_a K_a + K_a E_{ab} K_b,\n\\end{equation}\nwith real parameters $\\xi_0$, $\\xi_a$, $\\eta_{00}$, $\\eta_a$, $E_{ab} = E_{ba}$, $a,b \\in \\{1,2,3\\}$.\nExpressed in terms of the conventional parameters~\\eqref{Vconv}, these new parameters read\n\\begin{align}\n\\xi_0 &= \\frac{1}{2}\\left(m_{11}^2+m_{22}^2\\right) ,\n\\quad\n\\tvec{\\xi} = (\\xi_\\alpha)=\\frac{1}{2}\n\\begin{pmatrix}\n- 2 \\re(m_{12}^2), &\n 2 \\im(m_{12}^2), &\n m_{11}^2-m_{22}^2\n\\end{pmatrix}^\\trans,\n\\label{eq:para1}\n\\\\\n\\eta_{00} &= \\frac{1}{8}\n(\\lambda_1 + \\lambda_2) + \\frac{1}{4} \\lambda_3  ,\n\\quad\n\\tvec{\\eta} = (\\eta_{a})=\\frac{1}{4}\n\\begin{pmatrix}\n\\re(\\lambda_6+\\lambda_7), & \n-\\im(\\lambda_6+\\lambda_7), & \n\\frac{1}{2}(\\lambda_1 - \\lambda_2)\n\\end{pmatrix}^\\trans, \n\\label{eq:para2}\n\\\\\nE &= (E_{ab})= \\frac{1}{4}\n\\begin{pmatrix}\n\\lambda_4 + \\re(\\lambda_5) & \n-\\im(\\lambda_5) & \\re(\\lambda_6-\\lambda_7) \n\\\\ \n-\\im(\\lambda_5) & \\lambda_4 - \\re(\\lambda_5) & \n-\\im(\\lambda_6-\\lambda_7) \\\\ \n\\re(\\lambda_6-\\lambda_7) & \n-\\im(\\lambda_6 -\\lambda_7) & \n\\frac{1}{2}(\\lambda_1 + \\lambda_2) - \\lambda_3\n\\end{pmatrix}.\n\\label{eq:para3}\n\\end{align}\n\nA unitary basis transformations of the doublets,\n\\begin{equation}\n\\varphi'_i = U_{ij} \\varphi_j, \\quad \\text{with } U = (U_{ij}),  \\quad U^\\dagger U = \\mathbbm{1}_2\\, ,\n\\end{equation}\ncorresponds to a transformation of the bilinears,\n\\begin{equation} \\label{b1}\nK_0' = K_0, \\quad K_a' = R_{ab}(U) K_b \\;,\n\\end{equation}\nwith $R_{ab}(U)$ defined by\n\\begin{equation}\nU^\\dagger \\sigma^a U = R_{ab}(U) \\sigma^b \\;.\n\\end{equation}\nIt follows that $R(U) \\in SO(3)$, that is, $R(U)$ is a proper rotation in three dimensions. \nWe see that the potential~\\eqref{pot} stays invariant under a \nchange of basis of the bilinears \\eqref{b1}\nif we simultaneously transform the parameters~\\cite{Maniatis:2006fs}\n\\begin{equation} \\label{b2}\n\\xi'_0 = \\xi_0, \\quad\n\\xi_a' = R_{ab} \\xi_b, \\quad\n\\eta_{00}' = \\eta_{00}, \\quad\n\\eta_{a}' = R_{ab} \\eta_{b}, \\quad\nE_{cd}' = R_{ca}  E_{ab} R_{bd}^\\trans \\, .\n\\end{equation}\nNote that by a change of basis we \ncan always diagonalize the real symmetric matrix~$E$. \n\n{As an illustration, the following parametrization for a unitary transformation~\\cite{Maniatis:2006fs}}\n\\begin{align}\\label{EWbasisChange}\n    \\begin{pmatrix} \\varphi_1' \\\\ \\varphi_2' \\end{pmatrix} &= \n    \\begin{pmatrix} \\cos({\\beta}) && \\sin({\\beta})\\,e^{- i \\zeta} \\\\ -\\sin({\\beta})\\, e^{i \\zeta} && \\cos({\\beta}) \\end{pmatrix}  \n    \\begin{pmatrix} \\varphi_1 \\\\ \\varphi_2 \\end{pmatrix} \\equiv U \\begin{pmatrix} \\varphi_1 \\\\ \\varphi_2 \\end{pmatrix}\n\\end{align}\n{corresponds, in terms of bilinears, to the rotation matrix}\n\\begin{equation} \\label{unitR}\nR(U) = \\begin{pmatrix}\n \\cos ^2(\\beta )-\\sin ^2(\\beta ) \\cos (2 \\zeta ) & -\\sin ^2(\\beta ) \\sin (2 \\zeta ) & -\\sin (2 \\beta ) \\cos (\\zeta ) \\\\\n -\\sin ^2(\\beta ) \\sin (2 \\zeta ) & \\sin ^2(\\beta ) \\cos (2 \\zeta )+\\cos ^2(\\beta ) & -\\sin (2 \\beta ) \\sin (\\zeta ) \\\\\n \\sin (2 \\beta ) \\cos (\\zeta ) & \\sin (2 \\beta ) \\sin (\\zeta ) & \\cos (2 \\beta )\n\\end{pmatrix}\n\\end{equation}\n{\nand is useful to relate a given general basis to the so-called Higgs basis (see appendix~\\ref{EWbreak}) if the doublets acquire non-zero \nvacuum-expectation values. In this case, the angle $\\beta$ fulfils $|v_1^0| \\sin{\\beta} = |v_2^0| \\cos{\\beta}$ (or $\\tan{\\beta} = |v_2^0|\/|v_1^0|$). \\\\\n}\n\nLet us also briefly recall that (standard) CP transformations, that is, $\\varphi_i \\to \\varphi_i^*$, $i=1,2$, have a simple geometric picture in terms of bilinears~\\cite{Maniatis:2007vn}. With view on~\\eqref{eqKphi} we see that a (standard) CP transformation corresponds to $K_2 \\to - K_2$ keeping all other bilinears invariant in addition to the\nparity transformation which flips the sign of the arguments not written explicitly.\nNow let us assume for simplicity, that by a change of basis, the parameter matrix $E$ is diagonal. \nFor the general case of arbitrary matrices $E$ we refer to~\\cite{Maniatis:2007vn}. \nWith $E$ diagonal we see that\nthe potential~\\eqref{pot}  is invariant under the (standard) \nCP transformation if the parameters $\\xi_2$ and  $\\eta_2$ vanish. \nEventually we note that by a basis change \\eqref{b1}, \\eqref{b2} this is equivalent to any commonly vanishing\nentries of the parameter vectors  $\\tvec{\\xi}=(\\xi_1, \\xi_2, \\xi_3)^\\trans$ and \n$\\tvec{\\eta}=(\\eta_1, \\eta_2, \\eta_3)^\\trans$ at the same position.\n\nLet us also prepare the analysis of the THDM for the case of large $K_0$.\nFirst we note that $K_0 \\ge 0$ and for $K_0=0$ the potential is trivially vanishing.\nWe define for $K_0 > 0$~\\cite{Maniatis:2006fs}\n\\begin{equation}\n\\label{eq-ksde}\nk_a = \\frac{K_a}{K_0}, \\quad a \\in \\{1,2,3\\} \\quad \\text{and} \\quad \n\\tvec{k} = \n\\begin{pmatrix} k_1, & k_2, & k_3 \\end{pmatrix}^\\trans \\;.\n\\end{equation}\nWith \\eqref{eq-ksde}\nwe can write the potential~\\eqref{pot}\nin the form\n\\begin{equation}\n\\label{eq-vk}\nV^{\\text{THDM}} = K_0\\, J_2(\\tvec{k}) + K_0^2\\, J_4(\\tvec{k})\n\\end{equation}\nwith the functions\n\\begin{equation}\n\\label{J2J4}\nJ_2(\\tvec{k}) = \\xi_0 + \\tvec{\\xi}^\\trans \\tvec{k},\\qquad\nJ_4(\\tvec{k}) = \\eta_{00} \n  + 2 \\tvec{\\eta}^\\trans \\tvec{k} + \\tvec{k}^\\trans E \\tvec{k},\n\\end{equation}\ndefined \non the compact domain, as follows from~\\eqref{Kdom},\n\\begin{equation}\n|\\tvec{k}| \\leq 1 \\;.\n\\end{equation}\n\nIn appendix \\ref{EWbreak} we recap some parts of\n electroweak symmetry breaking in the THDM in terms of bilinears.\n \n\n\n\n\\section{Classification of the vacua}\n\\label{class}\nWe now apply the MPP to the THDM potential, that is, we study \n the two conditions (1) and (2) as mentioned in section~\\ref{sec:MPP} for the case of the THDM.\nAnalogously to the Standard Model case we consider the parameters as scale dependent. \nIn the bilinear formalism, advantageous in the \ndescription of the THDM potential, large field configurations correspond to a\n large bilinear\n$K_0$ which itself is bilinear in the Higgs-doublet fields, see \\eqref{eqKphi}.\nTherefore the \nbilinears depend quadratically on the mass scale $\\Lambda$.\nThe THDM potential is \nconsidered as an effective parametrization $V_{\\text{eff}}^{\\text{THDM}}$ of the form \\eqref{eq-vk}, \\eqref{J2J4}.\nHigher-dimensional order operators are neglected since \nwe consider a  \nscale $\\Lambda$ much larger than the electroweak scale but also sufficiently below\nthe Planck mass.\nFor large fields close to the high scale the quartic terms are dominant,\nso we have\n\\begin{equation}\n\\label{Veff}\nV_{\\text{eff}}^{\\text{THDM}} \\approx K_0^2\\, J_4, \\quad \\text{ for large $K_0$}\n\\end{equation}\nwith \n\\begin{equation}\nJ_4 = \\eta_{00} (K_0)\n  + 2 \\eta_a (K_0) k_a + k_a E_{ab}(K_0) k_b,\n\\end{equation}\nwhere we write explicitly the dependence of the parameters on the\nscale $K_0$. In appendix~\\ref{appendixJ4} we briefly discuss the condition of \na vanishing function $J_4$ at the high scale.\n Note that the bilinear dimensionless field $\\tvec{k}$ is \ndefined on the domain $|\\tvec{k}| \\le 1$. \nWith respect to the MPP we are looking for a potential\n which has a second minimum at the high scale, that is, \n $\\langle K_0 \\rangle_2 = \\Lambda^2$ at a \ncorresponding ``direction'' of the second minimum, see \\eqref{eq-ksde}-\\eqref{J2J4},\n\\begin{equation}\n\\langle \\tvec{k} \\rangle_2 \\;.\n\\end{equation}\n\nIn order to have a degenerate vacuum at the high scale with the same\npotential value we find with respect to \\eqref{Veff} for large $K_0$ the condition\n\\begin{equation}   \\label{cond1THDM}\nJ_4 ( K_0 = \\Lambda^2, \\langle \\tvec{k} \\rangle_2 ) = \n\\eta_{00} (\\Lambda^2)\n  + 2 \\eta_a (\\Lambda^2) \\langle k_a \\rangle \n  + \\langle k_a \\rangle E_{ab}(\\Lambda^2) \\langle k_b \\rangle\n= 0 \\;.\n\\end{equation}\nAs a function of $K_0$, $J_4( K_0, \\langle \\tvec{k}\\rangle_2 )$\nshould provide a minimum with $J_4 = 0$. \nThis requires also\n\\begin{equation}\n\\left .\\frac{ \\partial V_{\\text{eff}}^{\\text{THDM}} } { \\partial K_0} \\right |_{ \\text{vac 2}}\n= 0 \n= \n 2 \\Lambda^2 J_4 \\bigg |_{ \\text{vac 2}}\n+\n\\Lambda^4\n \\frac{ \\partial J_4 } { \\partial K_0} \\bigg |_{ \\text{vac 2}}\n=\n\\Lambda^4 \n\\left . \\frac{ \\partial J_4 } { \\partial K_0} \\right |_{ \\text{vac 2}} .\n\\end{equation}\nWith \n\\begin{equation}\n2 K_0\n\\frac{ \\partial J_4 } { \\partial K_0}\n= \n2 K_0\n\\left(\n\\frac{ \\partial \\eta_{00}(K_0) } { \\partial K_0}\n+\n2 \\frac{ \\partial \\eta_a (K_0) } { \\partial K_0} \nk_a\n+\nk_a\n\\frac{ \\partial E_{ab}(K_0) } { \\partial K_0} \nk_b\n\\right)\n=\n\\beta_{\\eta_{00}} + 2 \\beta_{\\eta_{a}} \nk_a\n+ \nk_a\n \\beta_{E_{ab}} \nk_b\n \\, ,\n\\end{equation}\nwe find the condition for the beta functions at the Planck scale:\n\\begin{equation} \\label{cond2THDM}\n\\beta_{\\eta_{00}} (\\Lambda^2) + 2 \\beta_{\\eta_{a}}(\\Lambda^2)   \\langle {k}_a  \\rangle_2 + \n\\langle {k}_a \\rangle_2 \\beta_{E_{ab}}(\\Lambda^2)  \\langle {k}_b \\rangle_2 = 0.\n\\end{equation}\nFor the THDM the conditions \\eqref{cond1THDM} and \\eqref{cond2THDM}  \nreplace the ones for the SM. \n\n\\subsection{Stationary points at the high scale $\\Lambda$}\n\\label{statPoints}\n\nSo far we have found that a minimum $\\langle \\tvec{k} \\rangle_2$ should satisfy\nthe conditions \\eqref{cond1THDM} and \\eqref{cond2THDM}.\nWe now study the stationarity structure of the dominant quartic terms.\nThis study is quite analogous to the stability study in~\\cite{Maniatis:2006fs}. \nWe have $|\\tvec{k} | \\le 1$ and we  consider the cases\n $|\\tvec{k} | < 1$ and  $|\\tvec{k} | = 1$ separately.\n\nFor $|\\tvec{k} | < 1$, stationarity of the potential requires,\nexpressed in terms of $J_4$ (since $K_0 >0$),\n\\begin{equation}\n\\nabla_{\\tvec{k}} J_4 (\\tvec{k})\n=\n 2 \\tvec{\\eta}^\\trans + 2 \\tvec{k}^\\trans E  = 0, \n\\end{equation}\nthat is, since $E$ is symmetric\n\\begin{equation} \\label{con}\n\\tvec{\\eta} + E  \\tvec{k} =0\\;.\n\\end{equation}\nNote that we do not write explicitly the scale dependence of the parameters which \nimplicitly is given by $\\Lambda^2$.\nWith the condition \\eqref{con} for a vanishing gradient we can \nin the case $|\\tvec{k} | < 1$ write the condition for vanishing $J_4$ \\eqref{cond1THDM} \nnow in the form\n\\begin{equation} \\label{cond12}\n\\tvec{\\eta}^\\trans \\langle \\tvec{k} \\rangle_2 = - \\eta_{00} \\;.\n\\end{equation}\n\nFor $\\det (E) \\neq 0$ the regular solution of \\eqref{con} is\n\\begin{equation} \\label{solreg}\n\\langle \\tvec{k} \\rangle_2 = - E^{-1} \\tvec{\\eta}\n\\end{equation}\nor we have for $\\det (E) = 0$ exceptional solutions. \nWe check that for the regular solutions we have with $1-\\tvec{k}^\\trans \\tvec{k} > 1$ indeed\n\\begin{equation} \\label{J4reg}\nJ_4 (\\langle \\tvec{k} \\rangle_2)\n=\n \\eta_{00} - \\tvec{\\eta}^\\trans E^{-1} \\tvec{\\eta}  = 0, \n \\quad\n \\text{if }\n 1 - \\tvec{\\eta}^\\trans E^{-2} \\tvec{\\eta} >0 .\n\\end{equation}\n\nFor $|\\tvec{k} | = 1$ we impose a Lagrange multiplier,\nand the stationary solutions follow from \n\\begin{equation}\n\\nabla_{\\tvec{k}, u} (J_4 (\\tvec{k}) + u (1-\\tvec{k}^2) )\n= 0 \\;,\n\\end{equation}\nthat is,\n\\begin{equation} \\label{eqbord}\n(E-  \\mathbbm{1}_3 u) \\tvec{k} + \\tvec{\\eta} = 0, \\quad  \n(1-\\tvec{k}^\\trans \\tvec{k})=0 \\;.\n\\end{equation}\nWith these conditions for a vanishing gradient we can \nin the case $|\\tvec{k} | = 1$ write the condition for a vanishing $J_4$ \\eqref{cond1THDM} \nnow in the form\n\\begin{equation} \\label{cond12u}\n\\tvec{\\eta}^\\trans \\langle \\tvec{k} \\rangle_2 = - \\eta_{00} -u \\;.\n\\end{equation}\n\nThe regular solution of \\eqref{eqbord}, that is, a solution \nwith $\\det(E- \\mathbbm{1}_3 u) \\neq 0$ is\n\\begin{equation} \\label{solbord}\n\\langle \\tvec{k} \\rangle_2 = - (E- \\mathbbm{1}_3 u)^{-1} \\tvec{\\eta}\\;,\n\\end{equation}\nwhere $u$ follows from $(1-\\tvec{k}^\\trans \\tvec{k})=0$ with \\eqref{solbord} from\n\\begin{equation} \\label{solu}\n 1 - \\tvec{\\eta}^\\trans (E- \\mathbbm{1}_3 u)^{-2} \\tvec{\\eta} = 0 \\;.\n\\end{equation}\n We check that for the regular solution we have indeed\n \\begin{equation}\n J_4 (\\langle \\tvec{k} \\rangle_2)\n=\n u + \\eta_{00} - \\tvec{\\eta}^\\trans E^{-1} \\tvec{\\eta}  = 0, \n\\end{equation}\nwith $u$ the solution of \\eqref{solu}.\nAlternatively, we may have for $\\det(E- \\mathbbm{1}_3 u) = 0$\nexceptional solutions of \\eqref{eqbord}.\n\n\nEventually we note that we have to ensure that the \nstationary solutions of \\eqref{con}\nfor $|\\tvec{k}| < 1$, respectively, \\eqref{eqbord} for $|\\tvec{k}| = 1$ are local minima. \nAs usual this can be \ndone by considering the (bordered) Hessian matrix. Alternatively, in \ncase of a stable potential, that is, a potential which is bounded from below, we \nmay look for the deepest stationary solution or in the degenerate case, solutions,\nwhich are then of course minima. \n\n\n\n\n\\subsection{Classification of the MPP in the THDM}\n\\label{classification}\n\nLet us now study the vacuum structure with respect to the MPP in detail.\nEspecially, we derive the conditions to have isolated points, respectively, continuous\nstationarity regions, corresponding to the MPP in a weaker or \na stronger sense. \nFirst we recall that we can, by a basis change, \\eqref{b1}, \\eqref{b2}, diagonalize the real symmetric matrix $E$ and therefore we suppose to have\n\\begin{equation}\nE' = \\diag (E_{11}', E_{22}', E_{33}' ) .\n\\end{equation}\nWe emphasize that $E'$ diagonal is assumed to hold at the scale $\\Lambda^2$. This in particular means that in principle the matrix $E'$ may be non-diagonal at a different scale. We discuss the running\nof the parameters of the THDM in the next section.\n\nIn order to distinguish the parameters in the new basis, where the matrix $E'$ is diagonal, from the\noriginal ones, we denote them with a prime symbol.\nIn conventional notation the potential with a diagonal matrix $E$ corresponds\nto arbitrary parameters with  \n$\\lambda_6=\\lambda_7$ general complex and $\\lambda_5$ real.\n\nFor $K_0 \\neq 0$ the bilinear space is defined on the domain $|\\tvec{k}| \\le 1$. \nLet us first consider the case $|\\tvec{k}| < 1$. \n\nAs pointed out above, for $\\det (E) = \\det (E') \\neq 0$ the regular solution of\nthe gradient equation \\eqref{con}\nis a single point \\eqref{solreg} and requires that $\\eta_{00}$ also satisfies \\eqref{cond12},\nproviding a degenerate value of the potential. \nWe emphasize that the condition $\\det (E) \\neq 0$ as well as the parameter $\\eta_{00}$ \nare invariant under a change of basis.\n\nIf one of the eigenvalues of $E$ is zero, say in the \ndiagonalized matrix $E'$ its upper component, \nwe get from \\eqref{con} \n\\begin{multline}\nE'_{11} = 0,\\; E'_{22} \\neq 0,\\; E'_{33} \\neq 0  :\\\\\n\\quad \\text{solution } \\langle  \\tvec{k} \\rangle_2 = \n\\begin{pmatrix} x, &  -\\frac{\\eta_2'}{E_{22}'}, & - \\frac{\\eta_3'}{E_{33}'} \\end{pmatrix}^\\trans\n\\text{ with } \nx^2 < 1 -  \\frac{\\eta_2'^2}{E_{22}'^2}  -  \\frac{\\eta_3'^2}{E_{33}'^2}  \\text{ for } \\eta_1' =0.\n\\end{multline}\n This is a line segment. \nIn the case that $\\eta_1'$ together with $E_{11}'$ are vanishing, that is,\n the two zero components of $\\tvec{\\eta'}$ and $E'$  \n are aligned, \n we may have a degenerate line of solutions satisfying \\eqref{con}. \nFor $\\eta_1' \\neq 0$  there is no solution of \\eqref{con}.\n Besides, the\n $\\eta_{00}$ parameter has in any case to satisfy \\eqref{cond12}.\n We want to derive these conditions in a basis-invariant way. \nFirstly, we remark that one vanishing eigenvalue of the matrix $E$ (note that $E$ is \nthe original parameter matrix and not necessarily diagonal) corresponds to \n$\\rank(E) = 2$ and this in turn gives the basis-invariant conditions\n\\begin{equation} \\label{Erank2}\n \\rank(E) = 2: \\qquad  \\det(E) = 0 \\text{ and } \\tr^2(E) - \\tr(E^2)  \\neq 0 \\;.\n\\end{equation}\nNow we can construct the conditions to have one zero in the parameter vector $\\tvec{\\eta}'$\naligned with $E'$ in a basis-invariant way:\n\\begin{equation} \\label{1align}\n(\\tvec{\\eta} \\times (E \\tvec{\\eta}))^\\trans E^2 \\tvec{\\eta} = 0,\n\\quad \\text{and} \\quad\n(E^2 \\tvec{\\eta} \\times (E \\tvec{\\eta}))^\\trans (E^2 \\tvec{\\eta} \\times (E \\tvec{\\eta})) \\neq 0 \\;.\n\\end{equation}\nWe get the statement, that for a matrix $E$ of rank 2, that is, a matrix $E$ fulfilling the conditions \\eqref{Erank2},\nthere is a line of second degenerate vacua possible if in addition the two conditions\n\\eqref{1align} hold and the basis-invariant parameter \n$\\eta_{00}$ satisfies \\eqref{cond12}. This corresponds to the MPP in the stronger sense.\n In case that only the rank conditions \\eqref{Erank2} hold but not the conditions \\eqref{1align} there is no realization \n of the MPP possible. \n\nSimilarly we can treat the case that two eigenvalues of $E$ vanish. Going to a basis where $E'$ is diagonal, we suppose\nthat the two upper components of the diagonal matrix $E'$ vanish,  \nthen we find from \\eqref{con} the solutions\n\\begin{multline} \\label{2zero}\nE'_{11} =  E'_{22} = 0,\\; E'_{33} \\neq 0:\\\\\n\\text{solution } \\langle  \\tvec{k} \\rangle_2 = \n\\begin{pmatrix} x, & y , &  -\\frac{\\eta_3'}{E_{33}'} \\end{pmatrix}^\\trans\n\\text{ with } x^2 + y^2  < 1 -  \\frac{\\eta_3'^2}{E_{33}'^2},\\text{ for }\n\\eta_1'=\\eta_2'=0.\n\\end{multline}\nThis solution is a disk. In case \nthat not both components of $\\tvec{\\eta'}$ aligned with\nthe vanishing diagonal entries of  $E'$ vanish,  there is no solution.\nWe note that for a solution also the parameter $\\eta_{00}$ has to satisfy \\eqref{cond12}.\n\nThe formulation in a basis-invariant way is as follows:\nTwo vanishing eigenvalues correspond to a matrix $E$ of $\\rank(E) =1$, that is,\n\\begin{equation} \\label{Erank1}\n \\rank(E) = 1:\\qquad  \\det(E) = 0 ,\n \\text{ and } \\tr^2(E) - \\tr(E^2)  = 0 ,\n \\text{ and }\n\\tr(E) \\neq 0\\;.\n\\end{equation}\nSince two eigenvalues vanish, we can by a basis change always achieve \nthat also one of the components of $\\tvec{\\eta}$ vanish, aligned with one of the vanishing\nentries of $E$. This can be written in a basis-invariant way:\n\\begin{equation} \\label{cond2}\n\\tvec{\\eta}^\\trans E \\tvec{\\eta} \\neq 0, \n\\quad\n\\text{and}\n\\quad\n(E \\tvec{\\eta} \\times \\tvec{\\eta})^\\trans (E \\tvec{\\eta} \\times \\tvec{\\eta}) = 0 \\;.\n\\end{equation}\nOnly in case that in addition to \\eqref{Erank1} \nthe conditions \\eqref{cond2} are satisfied,\nwe can have the MPP in form of a disk in bilinear space, that is, the MPP in the stronger sense.\nWe note, that in this case it is required that the parameter \n$\\eta_{00}$ satisfies \\eqref{cond12}.\nOtherwise, if the rank 1 conditions are fulfilled, but \\eqref{cond2} do not both hold, there is no solution as a multiple point.\n\nEventually, we consider the case $E=0$. Note that a vanishing\nmatrix $E$ does not depend on the chosen basis. Now, we find from \\eqref{cond12},\nthat is, the condition of a vanishing potential at the second minimum,\n that there is for $\\tvec{\\eta} \\neq 0$ no solution with respect to the MPP.\n However, if we have in addition to $E=0$ also $\\tvec{\\eta} = 0$ and $\\eta_{00}=0$ \nwe have a sphere of solutions, that is, the MPP realized in the stronger sense.\n\n\n\n\nThe case $|\\tvec{k}| =1$ can be treated analogously to the previous one. \nWe look for solutions of the gradient equation \\eqref{eqbord} instead of \n\\eqref{con}.\nThis system of four equations has in general solutions for the indeterminants\n$\\langle \\tvec{k} \\rangle_2$ as well as the Lagrange multiplier $u$.\nThe solutions of \\eqref{eqbord} and in particular the degeneracy of the solutions depend on the determinant of the matrix\n\\begin{equation}\nM=E - \\mathbbm{1}_3 u \\;.\n\\end{equation}\nAgain we get for  $\\det{M} \\neq 0$\n from \\eqref{eqbord} solution points \\eqref{solbord}. \nThe determinant is of course invariant under basis changes.\n \nLet us now turn to the exceptional cases, with at least one  \neigenvalue of the matrix $M$ vanishing.\nThe argumentation is quite analogously to the previous study where\nwe have to replace the matrix $E$ by $M$ and have to\ntake into account the condition $\\tvec{k}^\\trans \\tvec{k} =1$. \n\nIf one of the eigenvalues of $M$ is zero, say, without loss of generality in the \ndiagonalized matrix $M'$ its upper component, \nwe get from \\eqref{con} \n\\begin{multline}\nM'_{11} = 0,\\; M'_{22} \\neq 0,\\; M'_{33} \\neq 0  :\\\\\n\\text{solution } \\langle  \\tvec{k} \\rangle_2 = \n\\begin{pmatrix} x, &  -\\frac{\\eta_2'}{M_{22}'}, & - \\frac{\\eta_3'}{M_{33}'} \\end{pmatrix}^\\trans\n\\quad \\text{with } \nx^2 = 1 -  \\frac{\\eta_2'^2}{M_{22}'^2}  -  \\frac{\\eta_3'^2}{M_{33}'^2},\\text{ for }\n\\eta_1' =0 .\n \\end{multline}\n This gives at most two points, supposed that $\\eta_1' =0$, and\n otherwise there is no solution. \n In addition, the $\\eta_{00}$ parameter has to satisfy \\eqref{cond12u} in order to \n give a degenerate second vacuum.\n We want to find the conditions independent of the chosen basis. \nOne vanishing eigenvalue of the matrix $M$ corresponds to \n$\\rank(M) = 2$, hence, basis-invariantly written,\n\\begin{equation} \\label{Erank2u}\n \\rank(M) = 2: \\qquad  \\det(M) = 0 \\text{ and } \\tr^2(M) - \\tr(M^2)  \\neq 0 \\;.\n\\end{equation}\nThe conditions to have one zero in the parameter vector $\\tvec{\\eta'}$\naligned with the vanishing eigenvalue in $M$ are\n\\begin{equation} \\label{1alignM}\n(\\tvec{\\eta} \\times (M \\tvec{\\eta}))^\\trans M^2 \\tvec{\\eta} = 0,\n\\quad \\text{and} \\quad\n(M^2 \\tvec{\\eta} \\times (M \\tvec{\\eta}))^\\trans (M^2 \\tvec{\\eta} \\times (M \\tvec{\\eta})) \\neq 0 \\;.\n\\end{equation}\nIn case there is a solution of  a vacuum with $u$ from \\eqref{eqbord} satisfying the conditions \\eqref{Erank2u}\nand also the conditions \\eqref{1alignM},\nthere are points as a second degenerate vacuum possible\nsupposed that $\\eta_{00}$ satisfies \\eqref{cond12u}.\n\nSuppose now that the two components, say, the upper components of the diagonalized matrix $M'$ vanish, then we find from \\eqref{eqbord} the solutions\n\\begin{multline} \\label{2zerou}\nM'_{11} =  M'_{22} = 0,\\; M'_{33} \\neq 0:\\\\\n\\text{solution } \\langle  \\tvec{k} \\rangle_2 = \n\\begin{pmatrix} x, & y , &  -\\frac{\\eta_3'}{M_{33}'} \\end{pmatrix}^\\trans\n\\quad \\text{with } x^2 + y^2  = 1 -  \\frac{\\eta_3'^2}{M_{33}'^2},\\text{ for }\n\\eta_1' = \\eta_2' =0.\n\\end{multline}\nOnly in case that we have also the two vanishing components of $\\tvec{\\eta'}$ \naligned with the vanishing eigenvalues of $M'$, we \nget a circle of degenerate solutions; otherwise there\nis no solution.\n\nThe formulation in a basis-invariant way is as follows:\nTwo zero eigenvalues correspond to a matrix $M$ of rank one, that is,\n\\begin{equation} \\label{Erank1u}\n \\rank(M) = 1:\\qquad  \\det(M) = 0 ,\n \\text{ and } \\tr^2(M) - \\tr(M^2)  \\neq 0 ,\n \\text{ and }\n\\tr(M) \\neq 0\\;.\n\\end{equation}\nSince two eigenvalues vanish, we can by a further change of basis always achieve \nthat one of the components of $\\tvec{\\eta}'$ vanishes, aligned with one of the vanishing\nentries of $M'$. \nTo this purpose we construct two additional conditions, manifestly basis invariant,\n\\begin{equation} \\label{cond2u}\n\\tvec{\\eta}^\\trans M \\tvec{\\eta} \\neq 0, \n\\quad\n\\text{and}\n\\quad\n(M \\tvec{\\eta} \\times \\tvec{\\eta})^\\trans (M \\tvec{\\eta} \\times \\tvec{\\eta}) =0.\n\\end{equation}\nOnly in the case that both conditions \\eqref{Erank1u} and \\eqref{cond2u} hold,\nwith $\\eta_{00}$ satisfying \\eqref{cond12u} we can have the MPP realized in form of a \ncircle in bilinear space.\nIn all other cases there is no second vacuum possible.\n\nEventually, we consider the case $M=0$. Then, we find from \\eqref{cond12u},\nthat is, a vanishing potential at the second minimum,\n that there is for $\\tvec{\\eta} \\neq 0$ no solution with respect to the MPP and\nin case of $\\tvec{\\eta} = 0$ we have a surface of a sphere of solutions, that is, \nthe MPP in the stronger sense.\nNote that these conditions are already basis invariant. \n\n\nLet us mention that we have seen, that\nin addition to a vanishing eigenvalue of the matrix $E$, respectively, \n$M=E - u \\mathbbm{1}_3$, also the corresponding component \nof $\\tvec{\\eta'}$ has to vanish (in a basis where $E'$, respectively $M'$, is diagonal). \nThis in turn means that we do have CP conservation in\nthis case~\\cite{Maniatis:2007vn}.\nWe thus confirm the result~\\cite{Froggatt:2004st}\nthat the MPP in the THDM in the stronger sense corresponds\nto a CP conserving potential.\nIn the strongest case where the second vacuum is a degenerate sphere we have \nto have\n$E = 0$ together with $\\tvec{\\eta}$ and $\\eta_{00}$ vanishing. \nThis means that the potential has $J_4 =0$ for all $\\tvec{k}$. \nIn conventional notation this\ngives $\\lambda_i = 0$, $i=1,\\ldots,7$.\n\nMoreover, let us note the interesting aspect of solutions corresponding \nto $|\\tvec{k}| < 1$ (see \\cite{Maniatis:2006fs} for details),\nwhich give charge-breaking minima and solutions corresponding to\n$|\\tvec{k}| = 1$ \nwhich give  electroweak symmetry breaking \\mbox{$SU(2)_L \\times U(1)_Y$}  $\\to$ \\mbox{$U(1)_{em}$} , \nhowever, for a second vacuum at\na high vacuum expectation scale $\\Lambda$, there is no reason to discard the possibility of charge-breaking minima.\n\nWe summarize our findings in table \\ref{tabMPP}.\n\n\\begin{table}[h!]\n\\begin{tabular}{lllll}\n\\hline\n\\hline\n$\\det(E) \\neq 0$ & & & & point\\\\\n\\hline\n\\multirow{12}{*}{$\\det(E) = 0$}\n\t & \\multirow{6}{*}{$\\tr^2(E)-\\tr(E^2) \\neq 0$}\n\t\t\t& & \\multicolumn{2}{l}{\\hspace{-24pt}$(\\tvec{\\eta} \\times (E \\tvec{\\eta}))^\\trans E^2 \\tvec{\\eta} = 0$}\n\\\\\n& & & and & line\\\\\n& & & \n\\multicolumn{2}{l}{\\hspace{-24pt}$(E^2 \\tvec{\\eta} \\times (E \\tvec{\\eta}))^\\trans (E^2 \\tvec{\\eta} \\times (E \\tvec{\\eta})) \\neq 0$}\n\\\\\n\t \t\t\\cline{4-5}\n& & & \\multicolumn{2}{l}{\\hspace{-24pt}$(\\tvec{\\eta} \\times (E \\tvec{\\eta}))^\\trans E^2 \\tvec{\\eta} \\neq 0$}\\\\\n& & &  or \t &  no \\\\\n& & & \\multicolumn{2}{l}{\\hspace{-24pt}$(E^2 \\tvec{\\eta} \\times (E \\tvec{\\eta}))^\\trans (E^2 \\tvec{\\eta} \\times (E \\tvec{\\eta})) = 0$}\t\n\\\\\n\t \\cline{2-5}\n\t &  \\multirow{8}{*}{$\\tr^2(E)-\\tr(E^2) = 0$}\n\t \t& \\multirow{2}{*}{$\\tr(E) \\neq 0$}\n\t \t\t&\n\t \t\t$\\tvec{\\eta}^\\trans E \\tvec{\\eta} \\neq 0$\\\\\n& & &\t\\text{ and } & disk\\\\\n& & & $(E \\tvec{\\eta} \\times \\tvec{\\eta})^\\trans (E \\tvec{\\eta} \\times \\tvec{\\eta}) = 0$ \\\\\n\t \t\t\\cline{4-5}\n\t \t\t& & \t& \n \t\t$\\tvec{\\eta}^\\trans E \\tvec{\\eta} = 0$ \\\\\n& & &\t\t\t\\text{ or } & no\\\\\n& & &\t\t\t$(E \\tvec{\\eta} \\times \\tvec{\\eta})^\\trans (E \\tvec{\\eta} \\times \\tvec{\\eta}) \\neq 0$\\\\ \n\t \t\\cline{3-5}\n\t\t& &  \\multirow{2}{*}{$E=0$} & $\\tvec{\\eta} \\neq 0$  & no\\\\\n\t\t\\cline{4-5}\n \t\t& &  & $\\tvec{\\eta} = 0$ & sphere\\\\\n\\hline\n\\hline\n$\\det(M) \\neq 0$ & & & & point\\\\\n\\hline\n\\multirow{5}{*}{$\\det(M) = 0$}\n\t & \\multirow{6}{*}{$\\tr^2(M)-\\tr(M^2) \\neq 0$}\n\t\t\t& & \\multicolumn{2}{l}{\\hspace{-24pt}$(\\tvec{\\eta} \\times (M \\tvec{\\eta}))^\\trans M^2 \\tvec{\\eta} = 0$}\n\\\\\n& & & and & point\\\\\n& & & \n\\multicolumn{2}{l}{\\hspace{-24pt}$(M^2 \\tvec{\\eta} \\times (M\\tvec{\\eta}))^\\trans (M^2 \\tvec{\\eta} \\times (M \\tvec{\\eta})) \\neq 0$}\n\\\\\n\t \t\t\\cline{4-5}\n& & & \\multicolumn{2}{l}{\\hspace{-24pt}$(\\tvec{\\eta} \\times (M \\tvec{\\eta}))^\\trans M^2 \\tvec{\\eta} \\neq 0$}\\\\\n& & &  or \t &  no \\\\\n& & & \\multicolumn{2}{l}{\\hspace{-24pt}$(M^2 \\tvec{\\eta} \\times (M \\tvec{\\eta}))^\\trans (M^2 \\tvec{\\eta} \\times (M \\tvec{\\eta})) = 0$}\t\\\\\n\t \\cline{2-5}\n\t &  \\multirow{8}{*}{$\\tr^2(M)-\\tr(M^2) = 0$}\n\t \t& \\multirow{2}{*}{$\\tr(M) \\neq 0$}\n\t \t\t&\n\t \t\t$\\tvec{\\eta}^\\trans M \\tvec{\\eta} \\neq 0$\\\\\n& & &\t\t\t\\text{ and } & circle\\\\\n& & &\t\t\t$(M \\tvec{\\eta} \\times \\tvec{\\eta})^\\trans (M \\tvec{\\eta} \\times \\tvec{\\eta}) = 0$\\\\\n\t \t\t\\cline{4-5}\n\t \t\t& & \t& \n\t \t\t$\\tvec{\\eta}^\\trans M \\tvec{\\eta} = 0$\\\\\n& & &\t\t or & no\\\\\n& & &\t\t$(M \\tvec{\\eta} \\times \\tvec{\\eta})^\\trans (M \\tvec{\\eta} \\times \\tvec{\\eta}) \\neq 0$\n\t\t  \\\\\n\t\t \t\\cline{3-5}\n\t\t& &  \\multirow{2}{*}{$M=0$} & $\\tvec{\\eta} \\neq 0$  & no\\\\\n\t\t\\cline{4-5}\n \t\t& &  & $\\tvec{\\eta} = 0$ & surface\\\\\n \\hline\n \\hline\n\\end{tabular}\n\\caption{\n\\label{tabMPP} \nClassification of possible realizations of the MPP in the THDM. \nThe last column gives the kind of realization of the MPP or ``no'' in \ncase the MPP is not realized. The upper part gives solutions for \nthe case $|\\tvec{k}|<1$ and the lower part for the case $|\\tvec{k}|=1$.\nIn all cases where the MPP is realizable, the parameter $\\eta_{00}$ has\nto fulfill the condition \\eqref{cond12}, respectively, \\eqref{cond12u}. The solutions\nof the vacuum vector $\\langle \\tvec{k} \\rangle$ follow from \\eqref{con}, respectively,\nthe solutions of $\\langle \\tvec{k} \\rangle$ and $u$ from \\eqref{eqbord}.\nThe conditions are given in a basis-invariant way and\nare therefore directly applicable to any THDM.\n}\n\\end{table}\n\n\n\n\\subsection{Constraints from the quantum potential}\n\\label{quantum}\n\nThanks to the bilinear formalism, the 1-loop $\\beta$-functions of the general THDM can be put in a concise tensor form (see appendix~\\ref{rges}), allowing one to perform an analytical study of the renormalization group. In the case where $|\\tvec{k}| < 1$, using~\\eqref{con} and~\\eqref{cond12}, the constraint~\\eqref{cond2THDM} can be put into \nthe following form:\n\\begin{align}\\label{nullDK0}\n    \\begin{split}\n    0 ={}& 8\\left[\\left( \\tvec{k}^\\trans E \\tvec{k}\\right)^2 - 2\\, \\tvec{k}^\\trans E^2 \\tvec{k} + \\tr \\left( E^2 \\right) \\right] + g \\tvec{k}^\\trans \\tvec{k} + G\\\\\n    &-\\frac{1}{2}\\Big\\{ \\tensor{\\left(\\sigma_0 + \\tvec{k}^\\trans \\tvec{\\sigma}\\right)}{^a_b}\\tensor{\\left(\\sigma_0 + \\tvec{k}^\\trans\\tvec{\\sigma}\\right)}{^c_d} \\Big\\}\\,\\tensor{\\mathcal{T}}{_a^b_c^d} + \\left(1 - \\tvec{k}^\\trans \\tvec{k} \\right) \\mathcal{T}_{\\mathcal{U}\\mathcal{D}}\\,,\n    \\end{split}\n\\end{align}\nwhere we defined for convenience the strictly positive quantities \n\\begin{equation}\n    g \\equiv \\frac{9}{20} g_1^2 g_2^2 \\quad\\text{and}\\quad G \\equiv \\frac{9}{8}\\left(\\frac{3 g_1^4}{25}+g_2^4\\right)\n\\end{equation}\nand where the definition of the $\\mathcal{T}$-tensors is given in~\\eqref{tTrace}.\nIf $|\\tvec{k}| = 1$, we must use \\eqref{eqbord} and~\\eqref{cond12u} and the constraint reads \n\\begin{align}\\label{nullDK0u}\n    \\begin{split}\n    0 ={}& 8\\left[ 2 u^2 + \\left( \\tvec{k}^\\trans M \\tvec{k}\\right)^2 - 2\\, \\tvec{k}^\\trans M^2 \\tvec{k} + \\mathrm{Tr}\\left( M^2 \\right) \\right] + g + G \\\\\n    & -\\frac{1}{2}\\Big\\{ \\tensor{\\left(\\sigma_0 + \\tvec{k}^\\trans \\tvec{\\sigma}\\right)}{^a_b}\\tensor{\\left(\\sigma_0 + \\tvec{k}^\\trans \\tvec{\\sigma}\\right)}{^c_d} \\Big\\}\\,\\tensor{\\mathcal{T}}{_a^b_c^d} \\,.\n    \\end{split}\n\\end{align}\n\nLet us first consider a simplified situation where the theory does not contain Yukawa couplings. In that case, eq.~\\eqref{nullDK0} becomes\n\\begin{align}\n    0 ={}& 8\\left[\\left( \\tvec{k}^\\trans E \\tvec{k}\\right)^2 - 2\\, \\tvec{k}^\\trans E^2 \\tvec{k} + \\mathrm{Tr}\\left( E^2 \\right) \\right] + g \\tvec{k}^\\trans \\tvec{k} + G \\,,\n\\end{align}\nand has in fact no solutions. This can be seen by working in a basis where $\\tvec{k} = (k_1,0,0)^\\trans$, and where the term in the brackets now reads\n\\begin{align}\n\\begin{split}\n    \\left( \\tvec{k}^\\trans E \\tvec{k}\\right)^2 - 2\\, \\tvec{k}^\\trans E^2 \\tvec{k} + \\mathrm{Tr}\\left( E^2 \\right) ={}& k_1^4 E_{11}^2 -2 k_1^2 \\left(E_{11}^2+ E_{12}^2+E_{13}^2\\right) \\\\\n   &+E_{11}^2+E_{22}^2+E_{33}^2 + 2 (E_{12}^2+ E_{13}^2 + E_{23}^2) \\\\\n   ={}&(1-k_1^2)^2 E_{11}^2 + E_{22}^2 + E_{33}^2 + 2(1-k_1^2)\\big[ E_{12}^2 + E_{13}^2 + E_{23}^2 \\big]\\,.\n\\end{split}\n\\end{align}\nClearly, since $\\tvec{k}^\\trans \\tvec{k} <1$ the right-hand side is a positive quantity. This holds in any basis, and implies in turn that \n\\begin{align}\n    8\\left[\\left( \\tvec{k}^\\trans E \\tvec{k}\\right)^2 - 2\\, \\tvec{k}^\\trans E^2 \\tvec{k} + \\tr \\left( E^2 \\right) \\right] + g \\tvec{k}^\\trans \\tvec{k} + G > 0\\,.\n\\end{align}\n\nWe therefore have proven that in absence of Yukawa couplings, and if $\\tvec{k}^\\trans \\tvec{k} < 1$, eq.~\\eqref{cond2THDM} cannot be satisfied, which means that the MPP cannot be realized. Applying the above reasoning to the case $\\tvec{k}^\\trans \\tvec{k}=1$ we get the same conclusion.\\\\\n\nThe situation changes if we consider Yukawa couplings. For the known fermions we will see that only the dominant \ncontribution from the top quark allows for a realization of the MPP. Alternatively, lower bounds on the Yukawa couplings\ncould be given in order to have the MPP realized.\n\n\\subsection{Comparison with previous work on the MPP in the THDM}\n\\label{comparisonPreviousWorks}\n\nAs a concrete application of the results derived in sections~\\ref{classification} and~\\ref{quantum}, let us take the example of a \nTHDM type II\\footnote{In fact, the following discussion remains valid for any THDM where $\\lambda_6 = \\lambda_7 = 0$ and where \nnone of the fermions couple simultaneously to both doublets.}. \nThus we will be able to compare our results with the ones from \\cite{Froggatt:2004st} and \\cite{McDowall:2018ulq}.\\\\\n\nThe scalar potential \nconsidered in \\cite{Froggatt:2004st,McDowall:2018ulq} is the general potential but with the restriction $\\lambda_6 = \\lambda_7 = 0$. \nWith these assumptions the quartic couplings \\eqref{eq:para2}, \\eqref{eq:para3} simplify as:\n\\begin{align}\n\\eta_{00} &= \\frac{1}{8}\n(\\lambda_1 + \\lambda_2) + \\frac{1}{4} \\lambda_3  ,\n\\quad\n\\tvec{\\eta} = (\\eta_{a})=\\frac{1}{4}\n\\begin{pmatrix}\n0, & \n0, & \n\\frac{1}{2}(\\lambda_1 - \\lambda_2)\n\\end{pmatrix}^\\trans, \n\\\\\nE &= (E_{ab})= \\frac{1}{4}\n\\begin{pmatrix}\n\\lambda_4 + \\re(\\lambda_5) & \n-\\im(\\lambda_5) & 0\n\\\\ \n-\\im(\\lambda_5) & \\lambda_4 - \\re(\\lambda_5) & \n0 \\\\ \n0 & \n0 & \n\\frac{1}{2}(\\lambda_1 + \\lambda_2) - \\lambda_3\n\\end{pmatrix}.\n\\end{align}\n\n\nBy a basis change we can diagonalize the matrix $E$ without changing $\\eta_{00}$, $\\tvec{\\eta}$. In the new basis the parameter $\\lambda_5$ is real,\n\\begin{align}\nE &= (E_{ab})= \\frac{1}{4}\n\\begin{pmatrix}\n\\lambda_4 + \\lambda_5 & \n0 & 0\n\\\\ \n0 & \\lambda_4 - \\lambda_5 & \n0 \\\\ \n0 & \n0 & \n\\frac{1}{2}(\\lambda_1 + \\lambda_2) - \\lambda_3\n\\end{pmatrix}.\n\\end{align}\n\nThe most general form for the vacuum expectation values of the doublets may be parametrized as \\cite{Froggatt:2004st}\n\\begin{equation}\n\\langle \\varphi_1 \\rangle = \\phi_1 \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix}, \n\\qquad\n\\langle \\varphi_2 \\rangle = \\phi_2 \\begin{pmatrix} \\sin(\\theta) \\\\ \\cos (\\theta)\\, e^{i \\omega} \\end{pmatrix},\n\\end{equation}\nwhere $\\Lambda^2= \\phi_1^2+\\phi_2^2$. In terms of the bilinear fields, this corresponds to \n\\begin{align}\n\t\\begin{split}\\label{Ks}\n\t\tK_0 &= \\phi_1^2 + \\phi_2^2 = \\Lambda^2,\\\\\n\t\tK_1 &= 2 \\phi_1 \\phi_2  \\cos(\\theta) \\cos(\\omega),\\\\\n\t\tK_2 &= 2 \\phi_1 \\phi_2  \\cos(\\theta) \\sin(\\omega),\\\\\n\t\tK_3 &= \\phi_1^2 - \\phi_2^2\\,,\n\t\\end{split}\n\\end{align}\nso there is a direct mapping between $(\\phi_1, \\phi_2, \\theta, \\omega)$ and $(K_0, K_1, K_2, K_3)$. Since this will be useful in the following, we note that\n\\begin{equation}\n\t\\tvec{k}^\\trans \\tvec{k} \n\t= \\frac{\\tvec{K}^\\trans \\tvec{K}}{K_0^2} \n\t= \\frac{1}{\\left(\\phi_1^2 + \\phi_2^2\\right)^2}\\left( \\phi_1^4 + 2 \\cos(2\\theta) \\phi_1^2 \\phi_2^2 + \\phi_2^4 \\right )\\,.\n\\end{equation}\nTherefore we find that for $\\cos(2\\theta) = 1$ or, equivalently, $\\cos(\\theta) = \\pm 1$ we have \n$\\tvec{k}^\\trans \\tvec{k} =1$.\\\\\n\nWe can now begin the comparison between the present work and the previous analysis of the MPP by Froggatt and Nielsen. The study in \\cite{Froggatt:2004st} results in two possible vacuum configurations, namely\n\\begin{subequations}\n\t\\begin{equation}\\label{chargeCons}\n\t\t\\langle \\varphi_1 \\rangle = \\begin{pmatrix} 0 \\\\  \\phi_1 \\end{pmatrix}, \n\\qquad\n\\langle \\varphi_2 \\rangle = \\begin{pmatrix} 0 \\\\  e^{i \\omega} \\phi_2 \\end{pmatrix},\n\t\\end{equation}\n\t\\begin{equation}\\label{chargeBr}\n\t\t\\langle \\varphi_1 \\rangle =\\begin{pmatrix} 0 \\\\  \\phi_1  \\end{pmatrix}, \n\\qquad\n\\langle \\varphi_2 \\rangle = \\begin{pmatrix}  \\phi_2 \\\\ 0 \\end{pmatrix},\n\t\\end{equation}\n\\end{subequations}\n\nwhich respectively correspond to a charge-conserving, CP-violating and a charge-breaking CP-conserving minimum.\\\\\n\nIn the first case, \\eqref{chargeCons}, that is, $\\cos(\\theta) = \\pm 1$, we have $\\tvec{k}^\\trans \\tvec{k} =1$ at the vacuum. \nAlso Froggatt and Nielsen require that the value of the potential at the minimum is independent of $\\omega$, meaning that, with view on \\eqref{Ks}, in the approach developed in the present work, this corresponds to a circle-shaped vacuum.\nThis in turn requires, as can be seen from table \\ref{tabMPP}, that there is a basis where the parameters must satisfy\n\\begin{equation}\n\tM = E - u \\mathbbm{1}_3 = \\diag(0, 0, E_{33}-u) \\quad\n\\text{yielding} \\quad \\lambda_5 = 0 \\quad\\text{and}\\quad u = \\frac{1}{4}\\lambda_4\\, .\n\\end{equation}\n\nThe value of $u$ being fixed, we may solve equations~\\eqref{eqbord} and \\eqref{cond12u}, giving\n\\begin{gather}\n\t\\eta_3^2 = (\\eta_{00} + u)(E_{33}-u)\\,,\n\\end{gather}\nwhich in terms of the conventional parameters gives\n\\begin{equation}\n\t\\pm \\sqrt{\\lambda_1 \\lambda_2} + \\lambda_3 + \\lambda_4 = 0\\,,\n\\end{equation}\nRequiring that $|k_3| < 1$, it can be shown that the choice of the solution depends on the sign of $\\lambda_1$ and $\\lambda_2$, thus giving\n\\begin{gather}\n\t\\lambda_1 > 0\\quad,\\quad\\lambda_2 > 0 \\quad,\\quad +\\sqrt{\\lambda_1 \\lambda_2} + \\lambda_3 + \\lambda_4 = 0\\quad,\\quad \\lambda_5 = 0 \\label{minimum1}\\\\[.1cm]\n\t\\textbf{or} \\nonumber\\\\[.1cm]\n\t\\lambda_1 < 0\\quad,\\quad\\lambda_2 < 0 \\quad,\\quad -\\sqrt{\\lambda_1 \\lambda_2} + \\lambda_3 + \\lambda_4 = 0\\quad,\\quad \\lambda_5 = 0\\,. \n\t\\label{minimum2}\n\\end{gather}\nIn addition, ensuring that the extremum is a minimum rules out the second solution \\eqref{minimum2}. \nThe only remaining set of constraints \\eqref{minimum1} corresponds to the one derived in \\cite{Froggatt:2004st}.\\\\\n\nWe now turn to the second case~\\eqref{chargeBr} where $\\cos(\\theta) = 0$. Making use of~\\eqref{Ks}, this means that $k_1 = k_2 = 0$ and, in the case \nwhere neither $\\phi_1$ nor $\\phi_2$ vanish, $|k_3| < 1$.\\footnote{Note that for a vacuum at a large scale we cannot have $\\phi_1 = \\phi_2 = 0$.}\nIn any case, from a geometric point of view, the solution is a point, meaning that none of the eigenvalues of the matrix $E$ should vanish. \nApplying the constraints~\\eqref{con} and~\\eqref{cond12} gives:\n\\begin{equation}\n\t\\eta_3^2 = E_{33} \\eta_{00}\n\\end{equation}\nor, in conventional parameters,\n\\begin{equation}\n\t\\lambda_3 = \\pm \\sqrt{\\lambda_1 \\lambda_2}\\,,\n\\end{equation}\nwhere obviously the quantity $\\lambda_1 \\lambda_2$ must be positive. In addition to the above constraint, we can work out the condition $|\\tvec{k}| < 1$ to give in the conventional formalism:\n\\begin{equation}\n\t\\lambda_1 < 0\\;,\\; \\lambda_2 < 0\\;,\\; \\lambda_3 = \\sqrt{\\lambda_1 \\lambda_2} \\quad\\textbf{or}\\quad \\lambda_1 > 0\\;,\\; \\lambda_2 > 0\\;,\\; \n\t\\lambda_3 = -\\sqrt{\\lambda_1 \\lambda_2} \\, .\n\\end{equation}\n\nOnce again we want to ensure that the solution is a minimum, meaning here that $E_{33}$ must be positive. This rules out the first solution while the second set of conditions corresponds to the one in \\cite{Froggatt:2004st}. Thus the present formalism agrees with the results of Froggatt and Nielsen in both cases.\\\\\n\nFinally, relation~\\eqref{cond2THDM} must hold if the MPP is to be realized. This results in an additional constraint among the beta-functions, \nthat is\\footnote{The fact that $\\beta_{\\eta_1}$ and $\\beta_{\\eta_2}$ vanish is due to the property of the Yukawa sector that each fermion couples to only one doublet.}:\n\\begin{equation}\n\t\\beta_{\\eta_{00}} + 2 k_3 \\beta_{\\eta_3} + k_1^2 \\beta_{E_{11}} + k_2^2 \\beta_{E_{22}} + k_3^2 \\beta_{E_{33}} = 0\\,.\n\\end{equation}\nInjecting the expression of the bilinear couplings in terms of the conventional parameters gives:\n\\begin{equation}\\label{betaFN}\n\t(1 + k_3)^2 \\beta_{\\lambda_1} + (1 - k_3)^2 \\beta_{\\lambda_2} + 2(1-k_3^2) \\beta_{\\lambda_3} + 2(k_1^2 + k_2^2)\\beta_{\\lambda_4} + 2(k_1^2 - k_2^2)\\beta_{\\lambda_5} = 0\\,.\n\\end{equation}\n\nIn the case of the charge-conserving vacuum~\\eqref{chargeCons} we have $\\lambda_5 = \\beta_{\\lambda_5} = 0$ and $k_1^2 + k_2^2 + k_3^2 = 1$. \nUsing these relations as well as the expression of $k_3$ in \\eqref{cond12u} in terms of the conventional parameters we find\n\\begin{equation}\n\t\\frac{1}{2}\\sqrt{\\frac{\\lambda_2}{\\lambda_1}} \\beta_{\\lambda_1} + \\frac{1}{2} \\sqrt{\\frac{\\lambda_1}{\\lambda_2}} \\beta_{\\lambda_2} + \\beta_{\\lambda_3}  + \\beta_{\\lambda_4} = 0 \\, .\n\\end{equation}\nIn the other case \\eqref{chargeBr}, we have $k_1 = k_2 = 0$ and with \\eqref{cond12} we find\n\\begin{equation}\n\t\\frac{1}{2}\\sqrt{\\frac{\\lambda_2}{\\lambda_1}} \\beta_{\\lambda_1} + \\frac{1}{2} \\sqrt{\\frac{\\lambda_1}{\\lambda_2}} \\beta_{\\lambda_2} + \\beta_{\\lambda_3} = 0 \\, .\n\\end{equation}\n\nThese two expressions exactly match the condition (26) from ref.\\ \\cite{Froggatt:2004st}, with the only difference that here we did not need to consider the sign of $\\lambda_4$.\n Instead, what distinguishes the two cases is the shape of the MPP vacuum.\\\\\n\nTo summarize this section, we have shown that the constraints derived by Froggatt and Nielsen \\cite{Froggatt:2004st, Froggatt:2006zc} and \nlater reused by McDowall and Miller \\cite{McDowall:2018ulq} in the framework of a THDM type II constitute in fact a very particular case of the \napplication of the formalism developed in this paper and can be easily recovered. \nWe stress that numerous different realizations of the MPP may be derived in the same manner using the classification from table~\\ref{tabMPP}, which \nmight lead to various phenomenological implications of this principle at the EW scale.\nLet us emphasize that the conclusion in the works \\cite{Froggatt:2004st, Froggatt:2006zc, McDowall:2018ulq}, that the MPP in the THDM cannot be realized \nfor a SM-like Higgs-boson mass and the observed top-quark mass is based on a special case of the MPP.\nHere we have seen in the geometric approach in terms of bilinears that the THDM may develop many more\ndifferent kinds of realizations of the MPP.\n\n\n\\subsection{Example potential}\n\\label{examplePotential}\n\nAs an additional example we study a CP conserving THDM potential \nin which the parameters in conventional notation \\eqref{Vconv}\nsatisfy\n\\begin{equation}\n\\lambda_1 = \\lambda_2 = \\lambda_3, \\quad \n\\lambda_4 = \\lambda_5 = \\lambda_5^*, \\quad\n\\lambda_6 = \\lambda_6^* = \\lambda_7, \\quad \n\\text{ with } \\lambda_4 \\neq - \\lambda_5.\n\\end{equation}\nLet us recall that these relations are assumed to hold at the scale $\\Lambda$.\nIn bilinear space \\eqref{pot} this corresponds\nto the quartic parameters in the form\n\\begin{equation}\nE = \\diag (E_{11}, 0, 0 ), \\text{ with } E_{11} \\neq 0, \n\\qquad\n\\tvec{\\eta} = \\begin{pmatrix} \\eta_1, & 0, & 0 \\end{pmatrix}^\\trans, \\text{ with } \\eta_1 \\neq 0.\n\\end{equation}\nWith view on table \\ref{tabMPP} we have the case $\\det(E)=0$, \n$\\tr^2(E)-\\tr(E^2)=0$, but with $\\tr(E) \\neq 0$.\ntogether with $\\tvec{\\eta}^\\trans E \\tvec{\\eta} = E_{11} \\eta_1^2 \\neq 0$\nand also \n$(E \\tvec{\\eta} \\times \\tvec{\\eta})^\\trans (E \\tvec{\\eta} \\times \\tvec{\\eta}) = 0$,\nthat is the MPP is realizable as a disk supposed $\\eta_{00}$ satisfies \nthe condition \\eqref{cond12}. \n\nLet us look into the solutions in detail. \nFirst we note that we can, by a change of basis \\eqref{b2}, with the rotation matrix \n\\begin{equation}\nR = \n\\begin{pmatrix}\n0 & \\phantom{+}0 & - 1\\\\\n0 & \\phantom{+}1 & \\phantom{+}0\\\\\n1 & \\phantom{+}0 & \\phantom{+}0\n\\end{pmatrix}\n\\end{equation}\nshift both,\nthe diagonal entry as well as the corresponding entry of $\\tvec{\\eta}$.\nThis case corresponds therefore to the case \\eqref{2zero} with \ntwo vanishing eigenvalues of $E$.\n\nIn order to study the MPP we consider first the case $|\\tvec{k}|<1$. \nThe second vacuum follows from \\eqref{con} \n\\begin{equation} \\label{vac21}\n\\langle  \\tvec{k} \\rangle_2 = \n\\begin{pmatrix} -\\frac{\\eta_1}{E_{11}}, & y, & z \\end{pmatrix}^\\trans\n\\quad \\text{with } y^2 + z^2  < 1 -  \\frac{\\eta_1^2}{E_{11}^2} \n\\text{ for } \\eta_1^2 < E_{11}^2.\n\\end{equation}\nThis is indeed an open disk in the $y-z$ direction with \nborder radius $\\sqrt{  1 - \\eta_1^2\/E_{11}^2}$.\nThe parameter $\\eta_{00}$ has thereby to fulfill the \ncondition \\eqref{cond12},\n\\begin{equation} \\label{sol1eta00}\n\\eta_{00} =  \\frac{\\eta_1^2}{E_{11}}\\;.\n\\end{equation}\nMoreover, the $\\beta$ functions have to satisfy \\eqref{cond2THDM}.\nIn the case that we only consider the potential without any Yukawa couplings \nthese conditions read\n\\begin{equation}  \\label{exabeta}\n\\beta_{\\eta_{00}} =  2 \\frac{\\eta_1}{E_{11}} \\beta_{\\eta_1}  - \\left( \\frac{\\eta_1}{E_{11}}\\right)^2 \\beta_{E_{11}}\n\\end{equation}\nagain at the scale $\\Lambda^2$. \n\nFor $|\\tvec{k}|=1$ the stationarity condition is \ngiven by \\eqref{eqbord} with a Lagrange multiplier $u$. \nFor $u=0$ we get an exceptional solution at the border of\nthe solution \\eqref{vac21},\n\\begin{equation} \\label{vac21u}\n\\langle  \\tvec{k} \\rangle_2 = \n\\begin{pmatrix} -\\frac{\\eta_1}{E_{11}}, & y, & z \\end{pmatrix}^\\trans\n\\quad \\text{with } y^2 + z^2  = 1 -  \\frac{\\eta_1^2}{E_{11}^2}\n\\text{ for } \\eta_1^2 \\le E_{11}^2.\n\\end{equation}\nwhich is, as to be expected from table \\ref{tabMPP} a circle\nin case that $\\eta_{00}$ fulfills \\eqref{cond12u} which equals \\eqref{sol1eta00}\nin this case. Also the $\\beta$ functions have to fulfill \\eqref{cond2THDM}\nwhich give, neglecting Yukawa interactions \\eqref{exabeta}.\n\nFor $u \\neq 0$ we immediately get the solution from \\eqref{eqbord}\n\\begin{equation} \\label{vac22}\n \\langle  \\tvec{k} \\rangle_2 = \n\\begin{pmatrix}  \\pm 1  & 0, & 0 \\end{pmatrix}^\\trans ,\n\\end{equation}\nthat is, two possible points \nwith corresponding values for the Lagrange multiplier \n\\begin{equation}\nu = E_{11} \\pm \\eta_1 \\;.\n\\end{equation}\nThe condition \\eqref{cond12u} restricts the parameter $\\eta_{00}$, \n\\begin{equation} \\label{soleta002}\n\\eta_{00} = - E_{11} \\mp 2 \\eta_1 \\;.\n\\end{equation}\ncorresponding to one of the two discrete vacua in \\eqref{vac22}.\nBesides, we have to satisfy the condition for the beta functions \\eqref{cond2THDM},\nthat is,\n\\begin{equation} \\label{ex1beta}\n\\beta_{\\eta_{00}}  \\pm  2 \\beta_{\\eta_{1}}   + \n\\beta_{E_{11}}  = 0.\n\\end{equation}\n\nWe note that the potential is CP conserving \\cite{Maniatis:2007vn} (see section \\ref{sec:bilinears})\nand we conclude that the MPP is realizable \nin the stronger sense with a continuous disk of degenerate stationary points with \nthe parameters and its $\\beta$ functions satisfying the discussed constraints. \nIf these constraints are not fulfilled we may get at most an isolated point, that is,\nthe MPP in the weaker sense, where the parameters and $\\beta$ functions\n have to have in particular\nto satisfy \\eqref{soleta002} and \\eqref{cond2THDM}.\n\n\n\\section{Application of the MPP: From the high scale to the EW scale}\n\\label{analysisMPP}\n\nHaving classified the possible types of vacua at the high scale $\\Lambda$, we now want to study the MPP and its low-energy phenomenological implications. \nThe method we use in this analysis can be summarized by the following steps:\n\\begin{itemize}\n\t\\item At the high scale $\\Lambda$ we encounter 7 real parameters from the quartic part of the potential besides\n\tthe parameters of the Yukawa couplings. The three gauge couplings can be run up from their known values\n\tat the electroweak scale. \n\t\n\t\\item We consider the constraints provided by the MPP at the high scale $\\Lambda$, as given by Tab.~\\ref{tabMPP}.\n\t\n\t\\item We run all couplings down to the electroweak scale by the evolution equations at one-loop accuracy. \n\tAt the electroweak scale we have to consider in addition the \n\tquadratic parameters of the potential. These quadratic parameters are constrained since the model should provide\n\t the observed spontaneous electroweak symmetry breaking.\n\t\n\t\\item\n\tWe scan over all remaining free parameters.\n\t\n\t\\item For every parameter set, we compute the masses of the physical Higgs bosons of the THDM and the masses of the fermions.\n\\end{itemize}\n\n\nThe purpose of this analysis is to study the implications of the MPP on the masses of the physical states.\nThe main goal is to determine whether the application of the MPP to the THDM may yield correct (\\textit{i.e.} observed) masses of a Standard-Model-like Higgs boson and the top-quark. \nIn appendix \\ref{EWbreak} we recall the mechanism of spontaneous symmetry breaking in the THDM in\nthe bilinear formalism. In particular we present in this appendix the mass matrices of the Higgs bosons and\nthe mass of the pair of charged Higgs bosons.\n\n\n\\subsection{First example: MPP as a spherical vacuum}\n\nAs an application of our methods, \nwe first study a THDM potential with the MPP realized as \na spherical vacuum characterized by the potential parameter matrix $E=0$ (respectively $M=0$ for $|\\tvec{k}|=1$) (see table~\\ref{tabMPP}). We consider for simplicity only the top Yukawa coupling and we stay in the framework of a THDM type III to keep the discussion general. Note however that other types of THDM may always be obtained as special cases when either $y_t$ or $\\epsilon_t$ is set to 0 in some well-chosen basis. In appendix~\\ref{top} we show the Lagrangian of the top quark Yukawa coupling and its behavior under Higgs-basis changes.\n\nFollowing the discussion in section~\\ref{class} we have to consider the cases $|\\tvec{k}| < 1$\nand $|\\tvec{k}| = 1$. In the first case, equation~\\eqref{nullDK0} reads\n\\begin{align}\n\\begin{split}\n\tg \\tvec{k}^\\trans \\tvec{k} + G &= \\frac{3}{2}\\left[ Y^\\dagger\\left(\\sigma_0 + \\tvec{k}^\\trans \\tvec{\\sigma}\\right) Y \\right]^2\\\\\n\t&= \\frac{3}{2}\\left[(1+k_3) \\abs{y_t}^2 + (1-k_3)\\abs{\\epsilon_t}^2 + k_1(\\epsilon_t^* y_t + y_t^*\\epsilon_t) + i k_2(\\epsilon_t^* y_t - y_t^*\\epsilon_t)\\right]^2\\,. \\label{sphereCase1}\n\\end{split}\n\\end{align}\n\nIn order to simplify the evaluation of this relation, we may first perform a change of basis making the top quark Yukawa couplings real. Defining $y_t = \\abs{y_t} e^{i \\theta_y}$ and $\\epsilon_t = \\abs{\\epsilon_t} e^{i \\theta_\\epsilon}$, the associated unitary transformation is (see appendix \\ref{top})\n\\begin{equation}\n\\label{Ureal}\n\tU = \\begin{pmatrix}\n\t\te^{-i \\theta_y} & 0 \\\\\n\t\t0 & e^{-i \\theta_\\epsilon}\n\t\\end{pmatrix}\\,,\n\\end{equation}\nand corresponds, in terms of bilinears, to a rotation matrix $R(U)$ \\eqref{unitR}. The latter is in fact a rotation around the $z$-axis which in our case can be performed without loss of generality. Equation \\eqref{sphereCase1} can be further simplified after rotation to a basis where \\mbox{$Y = (y'_t, \\epsilon'_t)^\\trans = (y'_t,  0)^\\trans$}. The associated transformation $U$ is a 2D rotation matrix and corresponds, in the bilinear space, to a rotation around the $y$-axis. In this new basis, \\eqref{sphereCase1} now reads \n\\begin{equation}\n\tg \\tvec{k}^\\trans \\tvec{k} + G = \\frac{3}{2} (1+k_3)^2 y_t^4\\,,\n\\end{equation}\nwhere we dropped the primes for clarity. We see that choosing a point $\\tvec{k}$ within the sphere fixes the value of the top Yukawa coupling. Furthermore, it can be shown that a necessary condition for the above equation to be satisfied is\\footnote{To obtain this inequality, we used the fact that $g < G$ at all energy scales.} \n\\begin{equation}\n\t\\abs{y_t} \\geq \\left(\\frac{g + G}{6} \\right)^{1\/4}\\, ,\n\\end{equation}\nwhich can be reformulated in a basis-invariant way:\n\\begin{equation}\n\t\\label{yukInequality}\n\t\\sqrt{\\abs{y_t}^2 + \\abs{\\epsilon_t}^2} \\geq \\left(\\frac{g + G}{6}  \\right)^{1\/4} \\approx 0.38 \\quad \\text{for} \\quad \\Lambda=\\SI[parse-numbers=false]{10^{18}}{\\GeV}\\, .\n\\end{equation}\n\nGiven the above inequality, and under the assumption that the left hand-side is roughly of order $\\abs{y_t^{SM}}$, we expect the top quark to be the only fermion that couples strongly enough to the Higgs doublets to satisfy it. However the situation may not be as clear in some limiting cases, e.g. the THDM type II with a high value of $\\tan \\beta$.\\\\\n\n\nWe now turn to the case $|\\tvec{k}| = 1$, where the evaluation of~\\eqref{nullDK0u} in the same basis as above gives a constraint involving the Lagrange multiplier $u$, namely \n\\begin{equation}\n\t16 u^2 + g + G = \\frac{3}{2} (1+k_3)^2 y_t^4\\, .\n\\end{equation}\nWe note that in this case, condition \\eqref{yukInequality} must be satisfied as well.\n\n\nChoosing a specific value for $u$ and $k_3$ we can fix all the relevant parameters at the high scale: In case $|\\tvec{k}| < 1$ all the quartic couplings vanish whereas in case $|\\tvec{k}| = 1$ the non-zero parameters are related to the Lagrange multiplier $u$ via\n\\begin{subequations}\n\t\\begin{align}\n\t\t\\eta_{00} &= -u\\,,\\\\\n\t\tE_{ii} &= u\\:\\:,\\:\\: i = 1,2,3\\,.\n \t\\end{align}\n\\end{subequations}\n\nIt is remarkable that in both cases this set of couplings implies CP conservation at the level of the quartic part of the scalar potential \\cite{Maniatis:2006fs}. The only remaining possible source of CP violation is a non-zero value for the scalar mass coupling $\\xi_2$ in the basis chosen above.\\\\\n\nThe next step is to perform the running of the couplings down to the electroweak scale, where the study of spontaneous symmetry breaking will eventually allow for the determination of the masses of the Higgs bosons and the fermions. \nAt the electroweak scale we have to consider the quadratic parameters $\\xi_0$, $\\xi_{i,\\ i=1,2,3}$  of the Higgs potential.\nThese couplings are however subject to constraints in order to give a proper $SU(2)_L\\times U(1)_Y\\rightarrow U(1)_{em}$ symmetry breaking pattern.\n Using~\\eqref{EWbasisChange}, we can trade these four parameters for $u_\\text{EW}$, $\\beta$, $\\zeta$ and $v_0$. The latter is known since it corresponds to the Standard Model vacuum-expectation value $v_0 \\approx \\SI{246}{\\GeV}$.\\\\\n\nThe angles $\\beta$ and $\\zeta$ \nparametrize the basis transformation \\eqref{EWbasisChange} which allows to achieve the form of the Higgs basis \\eqref{higgsBasis}.\nWe note that a non-zero value for $\\zeta$ will in our case generate CP violation at the level of the scalar potential, and accordingly imply a mixing between the scalar and pseudo-scalar physical states. For simplicity reasons\nwe consider the case $\\zeta = 0$, in which we will identify the lightest neutral scalar\nwith the CP-even Standard-Model like Higgs boson.\\\\\n\n\\subsection{Second example: MPP as a disk-shaped vacuum}\n\nWe now briefly discuss the next-to-maximal symmetric vacuum, namely the disk-shaped one. This includes in particular the CP-conserving potential discussed in section~\\ref{examplePotential}.\n\nIn the following, we will work in a basis where $E = \\diag (0, 0, E_{33})$, respectively $M = \\diag (0, 0, M_{33})$ if $\\abs{\\tvec{k}} = 1$. In the latter case, the MPP vacuum reduces to a circle. Using the results from section~\\ref{statPoints}, the quartic parameters at the high scale satisfy the following constraints in order to provide stationary points:\n\\begin{align}\n\t\\abs{\\tvec{k}} < 1 &: \\begin{cases}\n\t\t\\eta_3 + E_{33} k_3 = 0 \\, ,\\\\\n\t\t\\eta_{00} + \\eta_3 k_3 = 0\\, ,\\\\\n\t\tk_1^2 + k_2^2 < 1 - k_3^2\\, ,\n\t\\end{cases}\\\\[.15cm]\n\t\\abs{\\tvec{k}} = 1 &: \\begin{cases}\n\t\t\\eta_3 + M_{33} k_3 = 0\\, , \\\\\n\t\tu + \\eta_{00} + \\eta_3 k_3 = 0\\, ,\\\\\n\t\tk_1^2 + k_2^2 = 1 - k_3^2\\, .\n\t\\end{cases}\n\\end{align}\nNote that, as discussed earlier, all other quartic parameters have to vanish at the high scale. The constraints from the quantum potential \\eqref{nullDK0}, \\eqref{nullDK0u} respectively simplify as\n\\begin{gather}\n\\begin{split}\n\\label{diskQpot1}\n\t8 E_{33}^2 \\left(1 - k_3^2 \\right)^2 + g\\tvec{k}^\\trans \\tvec{k} + G &= \\frac{3}{2}\\left[ Y^\\dagger\\left(\\sigma_0 + \\tvec{k}^\\trans \\tvec{\\sigma}\\right) Y \\right]^2 \\, ,\n\\end{split}\\\\\n\\begin{split}\n\\label{diskQpot2}\n\t16 u^2 + 8 M_{33}^2 \\left(1 - k_3^2 \\right)^2 + g + G &= \\frac{3}{2}\\left[ Y^\\dagger\\left(\\sigma_0 + \\tvec{k}^\\trans \\tvec{\\sigma}\\right) Y \\right]^2 \\, .\n\\end{split}\n\\end{gather}\n\nAlthough it is still possible to make the Yukawa couplings real using transformation~\\eqref{Ureal}, we cannot rotate to a basis where either $y_t$ or $\\epsilon_t$ vanishes without introducing a mixing between $k_1$ and $k_3$. In this context, equations~\\eqref{diskQpot1} and~\\eqref{diskQpot2} can be respectively rewritten as\n\\begin{gather}\n\\begin{split}\n\t8 E_{33}^2 \\left(1 - k_3^2 \\right)^2 + g\\tvec{k}^\\trans \\tvec{k} + G &= \\frac{3}{2}\\left[(1+k_3) y_t^2 + (1-k_3)\\epsilon_t^2 + 2 k_1 \\epsilon_t y_t\\right]^2\\, ,\n\\end{split}\\\\\n\\begin{split}\n\t16 u^2 + 8 M_{33}^2 \\left(1 - k_3^2 \\right)^2 + g + G &= \\frac{3}{2}\\left[(1+k_3) y_t^2 + (1-k_3)\\epsilon_t^2 + 2 k_1 \\epsilon_t y_t\\right]^2\\, .\n\\end{split}\n\\end{gather}\n\nNote that the number of free parameters at the high scale is increased here compared to the spherical case. We also emphasize that the previous comments about CP conservation are still valid in this case.\n\n\n\\subsection{Results and discussion}\n\nWe now present the results of the numerical analyses for the MPP for \na Higgs potential providing a spherical and disk-shaped vacuum. \nIn order to detect the minima at the electroweak scale we have to solve the \ncorresponding gradient equation (see appendix~\\ref{EWbreak} and \\cite{Maniatis:2006fs}). In principle, we may encounter regular and\nirregular solutions of these equations. In the spherical case, the irregular solutions provide very low values of $u_{EW}$. Consequently, either the scalar mass matrix \\eqref{scalarMasses} develops negative eigenvalues or leads to situations where the lightest scalar is much lighter than the SM-like Higgs boson. On the other hand, in the disk and circle cases, these solutions may provide some acceptable values for the masses of the physical states. In any case, for simplicity, we consider here only the regular solutions of eq.~\\eqref{eqEW}.\n\nFor these solutions, we show in  Fig.~\\ref{BManalysis} the results of the numerical analysis in the $(m_h, m_t)$ plane for $\\Lambda = \\SI[parse-numbers=false]{10^{18}}{\\GeV}$. The list of free parameters at the high scale along with their allowed range is presented in Table~\\ref{freeParams}. We systematically excluded from the parameter scan the points violating perturbativity and unitarity bounds \\cite{Branco:2011iw}. We also studied the conditions of a bounded from below potential \\cite{Branco:2011iw, McDowall:2018ulq} at all energy scales as illustrated in Fig.~\\ref{BManalysis}. We finally note that the ranges of $u$, $y_t$ and $\\epsilon_t$ at the high scale were chosen based on the observations that $\\abs{u} > 0.5$ sytematically violate perturbativity and unitarity bounds and that for Yukawa couplings greater than $1$ the RG flow tends to develop Landau poles.\n\nAt the EW scale, we have to scan over two remaining free parameters, namely $\\beta$ and $u_{EW}$. The former is taken in the range $[\\shortminus\\frac{\\pi}{2}, \\frac{\\pi}{2}]$ and the latter\\footnote{In order to give a physical meaning to $u_{EW}$, we note that $m_{H^\\pm} =v_0\\sqrt{ 2 u_{EW} }$ takes values between $0$ and $\\SI{1.1}{\\TeV}$ when $0 < u_{EW} < 10$.} in the range $[0, 10]$. We emphasize that the classical potential is ensured to develop a global minimum at the EW scale since we require Theorem 3 of \\cite{Maniatis:2006fs} to be satisfied.\n\n\\begin{table}[h]\n{\n\\hfill\n\\begin{tabular}{|C{1cm}|C{1.5cm}|C{1.5cm}|}\n\\hline\n\\multicolumn{3}{|c|}{\\large Spherical vacuum}                                       \\\\ \\hline\n $\\abs{\\tvec{k}}$ & $\\left[0, 1\\right[$ & $1$                                 \\\\ \\hline\n $u$              & \/                   & $\\left[\\shortminus 0.5, 0.5\\right]$ \\\\ \\hline\n $k_3$            & $\\left]\\shortminus1, 1\\right[$ & $\\left[\\shortminus1, 1\\right]$                 \\\\ \\hline\n\\end{tabular}\n\\hfill\n\\begin{tabular}{|C{1cm}|C{1.5cm}|C{1.5cm}|}\n\\hline\n\\multicolumn{3}{|c|}{Disk-shaped vacuum}                                                \\\\ \\hline\n$\\abs{\\tvec{k}}$ & $\\left[0, 1\\right[$            & $1$                                 \\\\ \\hline\n$u$              & \/                              & $\\left[\\shortminus 0.5, 0.5\\right]$ \\\\ \\hline\n$k_3$            & $\\left]\\shortminus1, 1\\right[$ & $\\left[\\shortminus1, 1\\right]$      \\\\ \\hline\n$k_1$            & $\\left]\\shortminus1, 1\\right[$ & $\\left[\\shortminus1, 1\\right]$      \\\\ \\hline\n$y_t$            & \\multicolumn{2}{c|}{$\\left[0, 1\\right]$}                             \\\\ \\hline\n$\\epsilon_t$     & \\multicolumn{2}{c|}{$\\left[0, 1\\right]$}                             \\\\ \\hline\n\\end{tabular}\n\\hfill\n}\n\\caption{Ranges of the free parameters at the high scale, in the spherical and disk case. The value of the $k_i$'s are always chosen such that $\\abs{\\tvec{k}} \\leq 1$.}\n\\label{freeParams}\n\\end{table}\n\n\n\\begin{figure}[h]\n\t\\centering\n\t\\begin{subfigure}[h]{0.49\\linewidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{Figures\/case1.png}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[h]{0.49\\linewidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{Figures\/case2.png}\n\t\\end{subfigure}\\\\\n\t\\begin{subfigure}[h]{0.49\\linewidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{Figures\/case3.png}\n         \\caption{}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[h]{0.49\\linewidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{Figures\/case4.png}\n         \\caption{}\n\t\\end{subfigure}\n\t\\caption{Results of the parameter scans in the $(m_h, m_t)$ plane in cases (a) $|\\tvec{k}| < 1$ and (b) $|\\tvec{k}| = 1$ for $\\Lambda = \\SI[parse-numbers=false]{10^{18}}{\\GeV}$. The two upper plots are obtained in the case of a spherical vacuum, and the lower ones in the case of a disk\/circle-shaped vacuum. For the points shown in red, the scalar potential is bounded from below at all energy scales between $M_Z$ and $\\Lambda$. The red stars indicate the position of $(m_h, m_t^{\\overline{\\mathrm{MS}}}) \\approx (\\SI{125}{\\GeV}, \\SI{163}{\\GeV})$. \\label{BManalysis}}\n\\end{figure}\n\n\nIn all four considered cases, there are regions of the parameter space for which the MPP constraints are compatible with the observed values of $m_h$ and $m_t$. However, if we also take the constraint into account that the potential has to be bounded from below, the spherical vacua are ruled out. In contrast, the disk- and circle-shaped vacua provide acceptable values for the masses along with the requirement of a stable potential. Although the question of identifying the precise regions of the parameter space yielding correct $(m_h, m_t)$ values goes beyond the scope of this work, the conclusion of this simple analysis is that the realization of the MPP is \\textit{a priori} not incompatible with a SM-like Higgs boson and top quark in the context of the general THDM\n\n\n\nEventually, we would like to mention that we varied in our numerical analysis the value of the high scale $\\Lambda$ in the range $10^{15} - \\SI[parse-numbers=false]{10^{20}}{\\GeV}$ and found that our predictions on the masses show a negligible dependence on this parameter.\n\n\\section{Conclusion and outlook}\n\nThe MPP \\cite{Froggatt:1995rt} forces the Higgs potential to provide degenerate vacua with the same potential value.\nA long time before the discovery of the Higgs boson, its mass has been predicted to be $135 \\pm 9 \\text{ GeV}$\nbased on this principle applied to the Standard Model -- a quite remarkable result.\\\\\n \nSome effort \\cite{Froggatt:2004st, Froggatt:2006zc, McDowall:2018ulq}\nhas been spent to apply the MPP to the TDHM. \nThis has been done in the conventional \nformalism, where the gauge degrees of freedom appear explicitly. Here we have studied the MPP\nin the THDM applying the bilinear formalism \\cite{Nagel:2004sw, Maniatis:2006fs, Nishi:2006tg} . \nThis formalism turns out to be quite powerful to \nstudy models with additional Higgs-boson doublets.\\\\\n\nWe have classified all different types of degenerate vacua \n in the THDM. In particular, we find degenerate vacua which realize the MPP in a weaker sense, providing\nadditional isolated points, but also realizations in  a stronger sense of line segments, circles, \nsurfaces of spheres, as well as spheres.\nWe have presented the classification in a basis-invariant way.\nFor any THDM the corresponding conditions for the different types of realizable MPP's can be easily checked. \\\\\n\nWe have studied the $\\beta$-functions of the THDM in detail in terms of bilinears and have shown that the MPP, considering only the THDM potential, is not realizable.\nThis changes if we consider the Yukawa couplings.\\\\\n\nWe recover the MPP cases studied in the literature but in addition can identify\ndifferent realizations of the MPP in the THDM. \nWe explored in section~\\ref{analysisMPP} four different realizations of the MPP in the context of a general THDM with a simplified matter content. Two of them (the disk- and circle-shaped vacua) are compatible with the measured values of the Higgs and top quark masses, satisfying simultaneously the constraints of a correct electroweak symmetry breaking, perturbativity and unitarity as well as the requirement of a stable scalar potential at all energy scales.\\\\\n\n\n\n\nOur analysis is done at the one-loop order of the $\\beta$-functions and using the tree-level RG-improved potential. In the future we would like to extend this analysis to higher orders of the effective potential, the RGEs, as well as the matching conditions between the $\\overline{\\mathrm{MS}}$ and pole masses. Also we would like to consider the experimental constraints from negative searches of additional Higgs bosons. Eventually, we plan to extend this study by considering all three families of fermions also in the context of THDMs with natural flavour conservation.\n\n\n\\acknowledgments\nThe project was supported in part by the UBB projects ''Materia Obscura y los bosones de Higgs''\nwith No. DIUBB 193209 1\/R and Fondecyt with No. 1200641.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{The isentropic compressible Navier-Stokes system and the inviscid limit problem}\n\nWe consider the motion of a compressible fluid in the half-space $\\Omega=\\mathbb{R}^{2}\\times\\mathbb{R}^{+}$. The density of the fluid is a scalar function $\\rho(t,x)$ with $t$ being the time variable and $x \\in\\Omega$, its velocity is $u(t):\\Omega \\rightarrow \\mathbb{R}^{3}$: these will be the unknowns of the system. \nThe temperature of the fluid, $\\theta$, is constant throughout the paper, and the pressure $P$ will follow a barotropic law: $P=P(\\rho)=\\frac{k}{\\gamma}\\rho^{\\gamma}$, with $k>0$ and the adiabatic constant $\\gamma> 1$. \nThe motion of the fluid is governed by the isentropic compressible Navier-Stokes system, which consists of two conservation laws:\n\\begin{itemize}\n\\item conservation of mass\n\\begin{equation} \\derp{t}\\rho + \\div(\\rho u) = 0, \\label{NSdiv} \\end{equation}\n\\item and conservation of momentum\n\\begin{equation} \\derp{t}(\\rho u) + \\div(\\rho u\\otimes u) = \\div \\Sigma + \\rho F, \\label{NS} \\end{equation}\n\\end{itemize}\nwith $F(t,x)$ a force term. To avoid technical complications with compatibility conditions, we consider that the force is smooth and that fluid is at rest for negative times: $F(t,x)=0$ and $(\\rho,u)(t,x)=(1,0)$ for $t<0$.\n\nIn these equations, $\\Sigma$ is the internal stress tensor,\n$$ \\Sigma = 2\\varepsilon\\mu Su + (\\varepsilon\\lambda_0 \\div u - P(\\rho)) I_{3}, $$\nin which $\\varepsilon>0$ will be arbitrarily small, $Su=\\frac{1}{2}(\\nabla u+\\nabla u^{T})$ is the symmetric gradient, and $I_{3}$ is the $3\\times 3$ identity matrix. The parameters $\\mu>0$ (dynamic viscosity) and $\\lambda:=\\lambda_0 + \\mu >0$ (bulk viscosity) will be given regular functions of $\\rho$. In equation (\\ref{NS}), we will write\n$$ \\div \\Sigma = \\varepsilon(\\mu \\Delta u +\\lambda \\nabla\\div u) - \\nabla (P(\\rho)) + \\varepsilon\\sigma(\\nabla \\rho,\\nabla u). $$\nThe final term $\\sigma$ is non-zero only when $\\lambda'(\\rho)$ and $\\mu'(\\rho)$ are non-zero ($\\lambda$, $\\mu$ not constant).\n\\newline\n\nWe shall not go into detailed historics about the isentropic system; a survey of existence and regularity results up to 1998 was put together by B.~Desjardins and C-K.~Lin, \\cite{DL}. \nWe shall quickly cite the emblematic result on the subject: the global existence theorem for weak solutions, as proved by P-L.~Lions in the 1990s (\\cite{Lpl} or \\cite{Lplbook2}), a result that has since been improved, for instance, by E.~Feireisl, A.~Novotn\\'y and H.~Petzeltov\\'a, \\cite{FNP}, \nand extended to some cases with density-dependent (variable, therefore) viscosity coefficients by D.~Bresch, B.~Desjardins and D.~G\\'erard-Varet \\cite{BDG}. \nIn the latter paper, a nonlinear drag force, written as $F=F(u)=r_0 |u| u$, is considered. \nWe also refer to the local strong existence theory, initiated in the 1960s and 70s by J.~Nash and then V.A.~Solonnikov \\cite{Nj, Sva} and improved on by R.~Danchin (for instance, see survey \\cite{Drsurv}). In this framework, blowup can occur \\cite{Xz}.\n\\newline\n\nIn this article, we will aim to obtain local-in-time existence of strong solutions to the isentropic Navier-Stokes system, with a lower bound for the time of existence that does not depend on the viscosity parameters when these are small, and work on the inviscid limit problem. \nAs we are on the half-space, we will have some boundary conditions.\n\n\\begin{nota} Throughout the paper, we will use the notation $x=(y,z)$, with $y\\in\\mathbb{R}^{2}$ and $z\\in\\mathbb{R}^{+}$; the boundary is therefore the set $\\{x=(y,z)\\in\\mathbb{R}^{3}~|~z=0\\}$.\n\nMoreover, for a vector field $v(x)$, the tangential part to the boundary is, for $x$ on the boundary, $v_\\tau(x)=v(x)+(v(x)\\cdot\\vec{n}(x))\\vec{n}(x)$, where $\\vec{n}(x)$ is the outer normal vector to $\\partial\\Omega$ at point $x$. As $\\vec{n}(x)\\equiv(0,-1)$, we extend the notation to all $\\Omega$: $v_\\tau(x)=(v_{1}(x),v_{2}(x))$. \\end{nota}\n\nThe boundary conditions on $u$ are the standard non-penetration condition on the boundary,\n\\begin{equation} u\\cdot\\vec{n}|_{z=0}=u_{3}|_{z=0}=0 , \\label{NPB} \\end{equation}\nand the Navier (slip) boundary condition $\\left[\\left(\\frac{1}{\\varepsilon}\\Sigma\\vec{n}+ au\\right)_\\tau\\right]|_{z=0} = 0$ in which $a>0$. In our case, with a flat boundary, the Navier condition can be rewritten as\n\\begin{equation} [\\mu(\\rho) \\derp{z}u_\\tau]|_{z=0} = 2a u_\\tau |_{z=0} \\label{NBC} \\end{equation}\nfor $t>0$ and $y\\in\\mathbb{R}^{2}$. As opposed to the Dirichlet or no-slip condition, which, in our setting, would be $u|_{z=0}=0$, the Navier condition, proposed by H.~Navier himself in the XIX\\textsuperscript{th} century \\cite{Nh}, allows the fluid to slip along the boundary, and this occurs wherever interaction at the boundary is non negligeable. \nFor instance, the slip phenomenon can be observed on the contact line of two immiscible flows \\cite{QWS}, and in capillary blood vessels, which are the microscopic, tissue-irrigating vessels where molecular exchanges with the neighbouring cells take place \\cite{PRD}. \nIt also appears when homogenising rough and porous boundaries (\\cite{JM} and \\cite{GVM}), and can be derived mathematically from a Boltzmann microscopic model with a Maxwell reflection boundary condition \\cite{MSR}. \nTo be physically pertinent, the slip coefficient $a$ should be chosen positive, but our results do not technically require $a$ to have a specific sign, so we take $a\\in \\mathbb{R}$.\n\nWe also impose the limit condition\n\\begin{equation} U(t,x):=(\\rho(t,x)-1,u(t,x))\\stackrel{|x|\\rightarrow+\\infty}{\\rightarrow} 0 \\label{limcond} \\end{equation}\nso that $U(t) \\in L^{2}(\\Omega)$.\n\\newline\n\nFormally, taking $\\varepsilon=0$ leads to the compressible Euler equations\n\\begin{equation} \\left\\{ \\begin{array}{rcl} \\derp{t}\\rho + \\div(\\rho u) & = & 0 \\\\\n\\rho \\derp{t}u + \\rho u\\cdot\\nabla u + \\nabla P(\\rho) & = & \\rho F . \\end{array} \\right. \\label{Euler} \\end{equation}\nThe Euler equation is of order one, so it only requires one boundary condition equation, which is (\\ref{NPB}). \nThis leads to the appearance of boundary layers: if the solutions of low-viscosity Navier-Stokes equations are expected to behave like a solution of the Euler equation far away enough from the boundary, solutions of Navier-Stokes are still required to satisfy a second boundary condition, whereas the reference solution of the Euler equation is not. \nA typical boundary layer expansion for solutions to Navier-Stokes will read\n\\begin{equation} u^\\varepsilon(t,y,z) = u^E(t,y,z) + V \\left(t,y,\\frac{z}{\\sqrt{\\varepsilon}}\\right) , \\label{blayer} \\end{equation}\nwith $u^E$ solving the Euler equation, and $V$, acting on a shorter scale, picking up the boundary condition.\n\nThe inviscid limit problem is a major challenge for mathematicians, whether one considers compressible or incompressible fluids. We remind the reader of the (non-forced) incompressible system:\n$$ \\left\\{ \\begin{array}{rcl} \\div u & = & 0 \\\\\n\\derp{t}u + u\\cdot\\nabla u - \\varepsilon\\nu\\Delta u + \\nabla q & = & 0 \\\\ u|_{t=0} & = & u_0 , \\end{array} \\right. $$\nin which $\\nu=\\frac{\\mu}{\\rho}$, with $\\rho$ constant, is the kinematic viscosity and $q$ is the kinematic pressure. \nRegarding the inviscid limit results on the incompressible system with Navier boundary conditions, the problem is solved in $L^2$ framework in 2D (see \\cite{Bc}, \\cite{CMR}, \\cite{Kjp}), and convergence of weak solutions of the Navier-Stokes equation towards a strong solution of the Euler equation, when the limit initial condition is regular enough, has been obtained for a range of Navier slip coefficients of the form $a=a'\\varepsilon^{-\\beta}$: \nstarting with D.~Iftimie and G.~Planas \\cite{IP} ($\\beta=0$), \\cite{WWX} ($\\beta<1\/2$) and \\cite{Pm11} ($\\beta<1$ for positive slip coefficients and $\\beta\\leq 1\/2$ regardless of sign) have extended the range of numbers $\\beta$ for which convergence occurs. \nC.~Bardos, F.~Golse and L.~Paillard recently obtained weak convergence results for Leray solutions of Navier-Stokes towards dissipative solutions of the Euler equation \\cite{BGP}.\n\nThe solutions to the 3D incompressible Navier-Stokes equations with a Navier boundary condition that does not depend on $\\varepsilon$ have a better asymptotic expansion than (\\ref{blayer}); D.~Iftimie and F.~Sueur showed in \\cite{IS} that the boundary layer $V$ has a smaller amplitude:\n\\begin{equation} u^\\varepsilon(t,y,z)=u^E(t,y,z) + \\sqrt\\varepsilon V(t,y,\\varepsilon^{-1\/2}z) . \\label{blayern} \\end{equation}\nThe problems of local existence of strong solutions on a time interval that does not depend on $\\varepsilon$, and of the corresponding inviscid limit, showing behaviour in agreement with this ansatz, have been solved by N.~Masmoudi and F.~Rousset \\cite{MR}. \nTheir approach is based on energy estimates in conormal Sobolev spaces, and the same technique has allowed them to prove similar results for the corresponding free-boundary system \\cite{MRfs}, and we will see that a similar approach for the isentropic system is valid. We refer the reader to \\cite{XX} and \\cite{BC2} for other studies in 3D.\n\nIn the incompressible case also, a variable viscosity coefficient, $\\nu=\\nu(q,|Su|^2)$, can be considered. This appears in elastohydrodynamics or the mechanics of granular materials for example. \nExistence of weak solutions for such equations on a bounded domain with the Navier boundary condition has been obtained by M.~Bul\\'i\\v{c}ek, J.~M\\'alek and K.~Rajagopal \\cite{BMR}.\n\\newline\n\nOn the 3D isentropic Navier-Stokes system with Navier boundary conditions that we are interested in, F.~Sueur recently showed the convergence of weak solutions to a strong solution of the Euler equation when the limit initial condition is smooth, and for slip coefficients that can depend on $\\varepsilon$, such as $a\/\\varepsilon^\\beta$ with $\\beta<1$ \\cite{Sf13}. \nIn the case where the slip coefficient does not depend on $\\varepsilon$, D.~Hoff obtained global solutions with intermediate regularity (more regularity than weak solutions \\textit{\\`a-la-}Lions, but not classical solutions) in 2005 \\cite{Hoff}, while Y-G.~Wang and M.~Williams justified a WKB expansion for strong solutions \\cite{WW} in 2012. \nIn particular, Wang and Williams show that solutions to the isentropic Navier-Stokes system behave similarly to their incompressible counterparts in the inviscid limit, in the sense that we have the asymptotic expansion (\\ref{blayern}). \nAs we aim to get existence of strong solutions of the Navier-Stokes equation through uniform \\textit{a priori} estimates, we will make use of conormal derivatives that we introduce below.\n\\newline\n\nTo complete the references on the inviscid limit problem for compressible fluids, we cite F.~Huang, Y.~Wang and T.~Yang (ideal gas, \\cite{HWY}), and Feireisl and Novotn\\'y (weak-strong convergence in the Navier-Stokes-Fourier setting, \\cite{FN13}), for advances on the full compressible Navier-Stokes equations, with an extra equation on the internal energy, temperature or entropy. \nWe also refer to \\cite{XY} for results on the linearised 2D system, \\cite{Rf05} and \\cite{GMWZ} for boundary layer analysis with characteristic and non-characteristic boundary conditions respectively, and \\cite{MZ}, \\cite{AB} for results on more general parabolic-hyperbolic systems.\n\n\\subsection{The conormal functional setting}\n\nIf we consider a boundary-layer expansion of the form (\\ref{blayern}), which the solutions of the equation we will study satisfy, we see that we can expect uniform control of $u^\\varepsilon$, its derivatives, but not its second derivatives: a factor $\\varepsilon^{-1\/2}$ results from the differentiation of the boundary layer $V$. \nThus, the functional setting used in this paper will be that of conormal Sobolev spaces. Introduced in the mid-60s \\cite{Hl}, these spaces have been used to work on hyperbolic systems with characteristic boundaries (see, for example, \\cite{Rj}, \\cite{Go}, \\cite{Sp95}). \nSuch spaces on a domain $\\Omega$, which has a boundary, are constructed by differentiating functions following a finite set of generators of vector fields that are tangent to the boundary of $\\Omega$. Namely, in the case of the half-space, we can choose\n$$ Z_{1,2} = \\derp{y_{1},y_{2}},~Z_{3} = \\phi(z)\\derp{z} , $$\nwith $\\phi$ a smooth, positive, bounded function of $\\mathbb{R}^{+}$ such that $\\phi(0)=0$ and $\\phi'(0)\\neq 0$ - typically, consider $\\phi(z)=\\frac{z}{1+z}$. \n\nConormal derivatives will allow us to get high-order uniform-in-$\\varepsilon$ estimates. Considering an expansion like (\\ref{blayern}), if we look at the conormal derivatives of $\\derp{z}u^\\varepsilon$, the boundary-layer term is written as\n$$ Z^3 (\\partial V(\\varepsilon^{-1\/2}z)) = \\varepsilon^{-1\/2}\\phi(z)\\partial^2 V(\\varepsilon^{-1\/2}z) , $$\nand this is of amplitude ${\\cal O}(1)$ in a neighbourhood of the boundary of size $\\sqrt\\varepsilon$ thanks to the factor $\\phi(z)$. Thus the conormal setting is the only one in which we can expect uniform bounds on a large number of derivatives.\n\nThe conormal Sobolev space on $\\Omega$, $W^{m,p}_{co}(\\Omega)$, is then naturally defined as the set of functions $f(x) \\in L^p(\\Omega)$ such that the conormal derivatives of order at most $m$ of $f$ are also in $L^p(\\Omega)$. \n\\newline\n\nAs part of their estimation process in \\cite{MR}, Masmoudi and Rousset used the incompressibility equation to express $\\derp{z}u_3$ as a combination of conormal derivatives:\n$$ \\derp{z}u_3 = -\\derp{y_1}u_1 -\\derp{y_2}u_2 = -Z_1 u_1 - Z_2 u_2. $$\nWe will be able to use a similar trick, but with the equation of conservation of mass (\\ref{NSdiv}), in which $\\derp{t}\\rho$ intervenes. Also, we will regularly use equation (\\ref{NS}) to replace terms with two normal derivatives ($\\derp{zz}u$), and there, $\\derp{t}u$ is involved. \nFor these reasons, we add $Z_0 = \\derp{t}$ for functions that depend on $(t,x)$, and introduce the conormal Sobolev spaces on $[0,T]\\times \\Omega$ in the sense of O.Gu\\`es \\cite{Go}, for a set time $T$. \nFor $\\alpha\\in\\mathbb{N}^{4}$, we write $Z^{\\alpha}=Z_{0}^{\\alpha_{0}}Z_{1}^{\\alpha_{1}}Z_{2}^{\\alpha_{2}}Z_{3}^{\\alpha_{3}}$, and $|\\alpha|=\\sum_{i=0}^{3} \\alpha_i$: the conormal Sobolev space $W^{m,p}_{co}([0,T]\\times \\Omega)$ is the set of functions $f:\n[0,T]\\times \\Omega\\rightarrow\\mathbb{R}^{d}$ such that $Z^{\\alpha}f\\in L^{p}([0,T]\\times \\Omega)$, for every $\\alpha$ with $|\\alpha|\\leq m$. We will only use $p=+\\infty$ and $p=2$, with the notation $H^m_{co}=W^{m,2}_{co}$. We therefore have\n$$ H^m_{co}([0,T]\\times\\Omega) = \\{ f(t,x)~|~ \\forall ~0\\leq k\\leq m, ~ \\derp{t}^k f \\in L^2([0,T],H^{m-k}_{co}(\\Omega))\\} . $$\nCompared to $W^{m,p}_{co}(\\Omega)$, the notation is slightly abusive, in that $W^{m,p}_{co}([0,T]\\times\\Omega)$ is not a space whose conormal derivatives are tangent to the boundary ($Z_0=\\derp{t}$ is not tangent to the boundary of $[0,T]$).\n\nIn our \\textit{a priori} estimation process, we will be interested in the following space:\n$$ X^m_T(\\Omega) = \\{ f(t,x) ~|~ \\forall ~0\\leq k\\leq m,~ \\derp{t}^k f\\in L^\\infty([0,T],H^{m-k}_{co}(\\Omega)\\}. $$\nThis is more restrictive than asking for $f\\in H^{m}_{co}([0,T]\\times\\Omega)$. For a set $t\\geq 0$, we introduce the semi-norms\n$$ \\norme{m}{f(t)}^2 = \\sum_{|\\alpha|\\leq m} \\norme{L^2(\\Omega)}{Z^\\alpha f(t)}^2 ~~\\mathrm{and}~~ \\norme{m,\\infty}{f(t)} = \\sum_{|\\alpha|\\leq m} \\norme{\\infty}{Z^\\alpha f(t)}. $$\nNote that these semi-norms coincide with the $H^m_{co}(\\Omega)$ and $W^{m,\\infty}_{co}(\\Omega)$ norms if $f$ is stationary. Based on these semi-norms, we construct two norms on $X^m_T$ and $W^{m,\\infty}_{co}([0,T]\\times\\Omega)$ respectively, which are essentially $L^\\infty$-in-time norms,\n$$ \\trinorme{m,T}{f} := \\sup_{t\\in[0,T]} \\norme{m}{f(t)} ~~\\mathrm{and}~~ \\trinorme{m,\\infty,T}{f} := \\sup_{t\\in[0,T]} \\norme{m,\\infty}{f(t)} , $$\nthe latter of which coincides with the $W^{m,\\infty}_{co}([0,T]\\times\\Omega)$ norm, and $L^2$-in-time norms\n$$ \\int_0^T \\norme{m}{f(t)}^2~dt ~~\\mathrm{and}~~ \\int_0^T \\norme{m,\\infty}{f(t)}^2~dt , $$\nthe former of which is the natural norm for $H^m_{co}([0,T]\\times\\Omega)$. We will prefer not to abbreviate these last norms, as the forms we have given will make the \\textit{a priori} estimation process clearer.\n\nWe add the following abbreviations: $\\norme{\\infty}{f(t)}:=\\norme{0,\\infty}{f(t)}$, $\\trinorme{\\infty,T}{f}:=\\trinorme{0,\\infty,T}{f}$, and we denote by $\\trinorme{\\mathrm{Lip},T}{f}$ the standard Lipschitz norm on $[0,T]\\times\\Omega$.\n\n\\subsection{Results and proof strategy}\n\nWe introduce the notation $U(t,x)=(\\rho-1,u)(t,x)$, and will consider the class of solutions satisfying the following property:\n$$ {\\cal E}_m(T,U) := \\trinorme{m,T}{U}^2 + \\trinorme{m-1,T}{\\derp{z}u_{\\tau}}^2+\\int_0^T \\norme{m-1}{\\derp{z}u_{3}(s)}^2+\\norme{m-1}{\\derp{z}\\rho(s)}^2~ds \\hspace{50pt} $$\n\\begin{equation} \\hspace{130pt} +\\trinorme{1,\\infty,T}{\\derp{z}u_{\\tau}}^2+\\int_0^T \\norme{1,\\infty}{\\derp{z}\\rho(s)}^2+\\norme{1,\\infty}{\\derp{t}\\derp{z}\\rho(s)}^2~ds <+\\infty . \\label{emtu} \\end{equation}\nIn terms of our functional setting in the previous section, if we have $U\\in X^m_T$ and $\\derp{z} U\\in X^{m-1}_T \\cap W^{2,\\infty}_{co}([0,T]\\times\\Omega)$ (with $L^\\infty$ in time norms only), then ${\\cal E}_m(T,U)$ is also finite. \n\nNote that in ${\\cal E}_m(T,U)$ we only have control of $L^2$-in-time norms on the derivatives of $u_3$ and $\\rho$. Simply using $|f(t,x)|^2 = |f(0,x)|^2 + \\int_0^t \\derp{t}(|f(s,x)|^2)~ds$, we get that\n\\begin{equation} \\trinorme{1,\\infty,T}{\\derp{z}\\rho}^2 \\leq \\norme{1,\\infty}{\\derp{z}\\rho(0)}^2 + C\\int_0^t \\norme{1,\\infty}{\\derp{z}\\rho(s)}^2+\\norme{1,\\infty}{\\derp{t}\\derp{z}\\rho(s)}^2~ds , \\label{l2timeinf} \\end{equation}\nthus the final integral in ${\\cal E}_m(T,U)$ acts to control the $W^{1,\\infty}_{co}$ norm of $\\derp{z}\\rho$. \nRequiring control of the $W^{1,\\infty}_{co}$ norm of $\\nabla U$ is typical of characteristic hyperbolic problems, see \\cite{Go}.\n\nThe force must be smooth on $\\mathbb{R}\\times \\Omega$, hence we introduce the notation:\n$$ {\\cal N}_m(T,F) = \\sup_{t \\in [-T,T]} \\norme{m}{F(t)}^2 + \\norme{m-1}{\\nabla F(t)}^2 + \\norme{2,\\infty}{\\nabla F(t)}^2 . $$\n\n\\begin{theo} \\underline{Uniform existence of solutions to the Navier-Stokes system.}\n\n\\noindent Let $m\\geq 7$, $F=(0,F)$ be such that ${\\cal N}_m(t,F)<+\\infty$ for any $t>0$, and $\\mu$ and $\\lambda$ be positive and bounded ${\\cal C}^m$ functions of $\\rho$. \nThen, for $\\varepsilon_0>0$, there exists $T^*>0$ such that, for every $0<\\varepsilon<\\varepsilon_0$, there is a unique $U^\\varepsilon$ satisfying ${\\cal E}_m(T^*,\\cdot)<+\\infty$, solution to (\\ref{NSdiv})-(\\ref{NS})-(\\ref{NPB})-(\\ref{NBC})-(\\ref{limcond}), the isentropic compressible Navier-Stokes system on $(0,T^*)\\times\\Omega$ with Navier boundary conditions. \nMoreover, there is no vacuum on this time interval: there exists $c>0$ such that $\\rho(t,x)\\geq c$ for $t\\in [0,T^*]$ and $x\\in\\Omega$. \\label{unifexist} \\end{theo}\n\n\\begin{theo} \\underline{Inviscid limit.}\n\nUnder the same conditions as above, the family $(U^\\varepsilon=(\\rho^\\varepsilon-1,u^\\varepsilon))_{0<\\varepsilon<\\varepsilon_0}$ of solutions to the Navier-Stokes system converges in $L^2([0,T^*]\\times\\Omega)$ and $L^\\infty([0,T^*]\\times\\Omega)$, towards $V=(\\rho-1,u)$, the unique solution to the isentropic compressible Euler system, (\\ref{Euler})-(\\ref{NPB})-(\\ref{limcond}), that satisfies ${\\cal E}_m(T^*,V)<+\\infty$. \\label{inviscid} \\end{theo}\n\nNote that there are no restrictions on the viscosity parameters other than positiveness and sufficient regularity (${\\cal C}^m$). It seems physically justified to ask $\\mu(\\rho)$ and $\\lambda(\\rho)$ to be increasing with the density, but, like the sign of the slip coefficient, the signs of $\\mu'$ and $\\lambda'$ do no intervene technically. \nAlso, the results are shown for a barotropic pressure law, but we can extend them to any positive, increasing ${\\cal C}^1$ pressure law.\n\nFinally, we expect that our results are also valid in any domain of $\\mathbb{R}^3$ with a ${\\cal C}^{m'}$ boundary, for $m'$ large enough, locally characterised by equations of type $z=\\psi(y)$. As shown in \\cite{MR}, the differences are technical, and extra difficulties arise only because the normal vector to the boundary is no longer constant.\n\\newline\n\nOur results are obtained by classical arguments once a uniform estimate is shown. The key bound is the following.\n\\begin{theo} \\underline{Uniform energy bound.}\n\nLet $m\\geq 7$ and $M_0>0$. We assume that ${\\cal N}_m(t,F)<+\\infty$ for every $t>0$, and that the initial value of $U$ satisfies ${\\cal N}_m(0,U) \\leq M_0$. Then, there exist $\\varepsilon_0>0$, $T^*>0$ and a positive increasing function $Q: \\mathbb{R}^{+}\\rightarrow \\mathbb{R}^{+}$, with $Q(z)\\geq z$, such that, for $\\varepsilon\\leq \\varepsilon_0$ and $0\\leq t\\leq T^*$,\n\\begin{equation} {\\cal E}_m(t,U)+\\trinorme{1,\\infty,t}{\\derp{z}u_3}^2 + \\varepsilon\\int_0^t \\norme{m}{\\nabla u(s)}^2 + \\norme{m-1}{\\nabla^2 u_\\tau(s)}^2~ds \\leq Q(2M_0) \\label{th13} \\end{equation} \\label{thEE} \\end{theo}\n\nLet us outline the proof of \\thref{thEE}. It is proved by showing that the left-hand side of (\\ref{th13}) is bounded by\n$$ Q(M_0) + (t+\\varepsilon) Q({\\cal E}_m(t,U)+{\\cal N}_m(t,F)). $$\nThe energy function we consider, ${\\cal E}_m$, contains $L^2$ in space norms with many derivatives, on the first line of (\\ref{emtu}), and conormal-Lipschitz norms on the second line. The first type of term is dealt with by performing energy estimates, in which we will have to control the commutators between the conormal derivatives and the operators that appear in the equation. \nIn particular, we will make use of the symmetrisable hyperbolic-parabolic structure of the compressible model to get the energy estimates on $U$. For the $L^\\infty$ norms, we will widely use an anisotropic Sobolev embedding theorem (\\thref{sobo}), which is the main contributor to the restriction on $m$. \nFor the terms in the second line of (\\ref{emtu}), a maximum principle will provide us with bounds on $\\trinorme{1,\\infty,t}{\\derp{z}u_\\tau}$, while the Duhamel formula for the equation satisfied by $\\derp{z}\\rho$, which is obtained by combining $\\varepsilon\\times(\\ref{NSdiv})$ with the third component of (\\ref{NS}), will give us the bounds for the $W^{1,\\infty}_{co}$ norms on $\\derp{z}\\rho$. \nA bootstrap argument closes the proof.\n\nNote that, in the context of \\tworef{unifexist}{inviscid}{Theorems}, $M_0$ can be arbitrarily small, as ${\\cal N}_m(0,U)=0$. We will be able to prove the energy estimate taking initial conditions into account; this allows one to extend our results to less regular force terms or different initial values, providing the compatibility conditions yield uniform bounds on the norms of $U$ at $t=0$.\n\\newline\n\n\\textbf{\\underline{Organisation of the paper.}} In the next section, we prove \\tworef{unifexist}{inviscid}{Theorems}, assuming \\thref{thEE}. The remaining sections will all be dedicated to proving this uniform estimate. \nStarting with some important commutator estimates in section 3, we then proceed to prove the bound \\textit{a priori}, looking at each component of $(U,\\derp{z}U)$ separately, and getting the required conormal and $L^\\infty$ bounds on each of them: $U$ in section 4, the normal derivative of $u_\\tau$ in section 5, $\\derp{z}u_3$ in section 6, and the normal derivative of the density in section 7. \nWe conclude the proof of \\thref{thEE} in section 8.\n\n\\section{Proof of \\tworef{unifexist}{inviscid}{Theorems}}\n\nIn this section, we assume the uniform estimate in \\thref{thEE}. To obtain \\thref{unifexist}, we can use a classical fixed-point iteration method to get existence of solutions for any fixed $\\varepsilon>0$ on a time interval $[0,T^\\varepsilon]$ depending on the viscosity (see \\cite{WW}, section 4.1). A bootstrap argument with the uniform bounds then yields a uniform existence time.\n\\newline\n\nWe will further detail the proof of \\thref{inviscid}. We first get local convergence by using a standard compactness argument.\n\n\\begin{propo} \\underline{Conormal compact embedding theorem.}\n\nLet $T>0$ and $(U_n)_{n\\in\\mathbb{N}}$ be a bounded sequence of $H^m_{co}([0,T]\\times\\Omega)$, such that the sequence $(\\nabla U_n)_n$ is bounded in $H^{m-1}_{co}([0,T]\\times\\Omega)$. \nThen we can extract a sub-sequence $(U_{n_j})_{j\\in\\mathbb{N}}$ such that, for every $\\alpha\\in\\mathbb{N}^4$ with $|\\alpha|\\leq m-1$, $Z^\\alpha U_{n_j}$ converges in $L^2_{loc}([0,T]\\times\\Omega)$ - we will say that $(U_n)$ is \\underline{locally compact} in $H^{m-1}_{co}([0,T]\\times\\Omega)$.\n\\label{rellich} \\end{propo}\n\nThe bound on the gradient of $U_n$ is crucial here. In other contexts in which conormal Sobolev spaces have been used, such as in \\cite{Go} and \\cite{Sp}, one normal derivation costs two conormal derivations, thus their $H^m_{co}$ spaces locally and compactly embed in $H^{m-2}_{co}$. \nBut in our energy estimates on the isentropic Navier-Stokes system, we find that one normal derivative can be controlled by the same number of conormal derivatives. The proof of \\propref{rellich} is similar to that in \\cite{Sp}.\n\\newline\n\nFor a given $\\varepsilon>0$, let $U^\\varepsilon=(\\rho^\\varepsilon-1,u^\\varepsilon)$ be the solution to (\\ref{NSdiv})-(\\ref{NS})-(\\ref{NPB})-(\\ref{NBC})-(\\ref{limcond}) with viscosity coefficient $\\varepsilon$, given by \\thref{unifexist}. \nThe energy estimate in \\thref{thEE} tells us that the family $({\\cal E}_m(T^*,U^{\\varepsilon}))_{0<\\varepsilon<\\varepsilon_{0}}$ is bounded, so we can immediately state that the sequence $(U^\\varepsilon)_{0<\\varepsilon<\\varepsilon_0}$ is locally compact in $H^{m-1}_{co}([0,T^*]\\times \\Omega)$ by \\propref{rellich}. \nAs $H^{m-1}_{co}([0,T^*]\\times \\Omega) \\hookrightarrow {\\cal C}([0,T^*],L^2(\\Omega))$, we can consider a sequence $\\varepsilon_{n}\\stackrel{n\\rightarrow+\\infty}{\\rightarrow}0$ such that $U^{\\varepsilon_{n}}$ converges locally in $H^{m-1}_{co}([0,T^*]\\times \\Omega)$ and in ${\\cal C}([0,T^*],L^2(\\Omega))$ towards a function $V=(\\rho-1,u)$, which is easily seen to be a weak solution to the compressible Euler system. \nThanks to the uniform bounds, we see that $V$ has the same regularity as $U^{\\varepsilon}$, and in particular $V$ is Lipschitz-class, which yields uniqueness of solutions for the Euler equation in the space of functions satisfying ${\\cal E}_m(T^*,V)<+\\infty$. \nThe whole sequence then converges towards $V$, strongly and locally in $H^{m-1}_{co}([0,T^*]\\times\\Omega)$, and $U^\\varepsilon(t)$ converges weakly in $L^2([0,T^*]\\times\\Omega)$ towards $V(t)$.\n\\newline\n\nWe now prove strong convergence in $L^2$. For $t\\leq T^*$, we start with the classical energy inequality for the Navier-Stokes system, as in \\cite{LM},\n\\begin{equation} \\left| E(t,U^\\varepsilon) - E(0,U^\\varepsilon) - \\int_0^t \\int_\\Omega \\rho^\\varepsilon F(s)\\cdot u^\\varepsilon (s) ~dx~ds \\right| \\leq C\\varepsilon\\int_0^t \\norme{H^1(\\Omega)}{u^\\varepsilon(s)}^2~ds , \\label{clasen} \\end{equation}\n$$ \\mathrm{where}\\hspace{10pt} E(t,U^\\varepsilon) = \\int_\\Omega \\frac{1}{2}\\rho^\\varepsilon(t)|u^\\varepsilon(t)|^2 + \\frac{k}{\\gamma(\\gamma-1)}\\int_\\Omega (\\rho^\\varepsilon)^\\gamma(t) - 1 - \\gamma(\\rho^\\varepsilon(t)-1)~dx. $$\nNote that $E(0,U^\\varepsilon) = 0$ for every $\\varepsilon$. As $\\norme{H^1([0,T^*]\\times\\Omega)}{u^\\varepsilon}$ is bounded by \\thref{thEE} and $\\rho^\\varepsilon u^\\varepsilon$ converges weakly towards $\\rho u$ in $L^2$ (as that is the case in the sense of distributions by local strong convergence), we get that, for $t\\in [0,T^*]$,\n$$ E(t,U^\\varepsilon) \\stackrel{\\varepsilon\\rightarrow 0}{\\longrightarrow} \\int_0^t \\int_\\Omega \\rho(s,x) F(s,x)\\cdot u(s,x)~dx~ds . $$\nBut $\\int_{[0,t]\\times\\Omega} \\rho F\\cdot u$ is equal to $E(t,V)$, as (\\ref{clasen}) with $\\varepsilon=0$ yields the energy equality for the compressible Euler equation. So, we have $E(t,U^\\varepsilon) \\stackrel{\\varepsilon\\rightarrow 0}{\\longrightarrow} E(t,V)$ for every $t\\leq T^*$.\n\nWe get $L^2$ convergence of the density by using the strict convexity of $z\\mapsto z^\\gamma$ for $z>0$ and $\\gamma>1$. On one hand, we have\n$$ \\rho^\\varepsilon|u^\\varepsilon|^2-\\rho|u|^2 - 2 \\sqrt{\\rho}u \\cdot (\\sqrt{\\rho^\\varepsilon}u^\\varepsilon - \\sqrt{\\rho}u) = |\\sqrt{\\rho^\\varepsilon}u^\\varepsilon - \\sqrt{\\rho}u|^2 , $$\nand on the other hand, there exists $c>0$ such that $\\rho(t,x)>c>0$, thus the Taylor expansion of $z\\mapsto z^\\gamma$ yields\n$$ (\\rho^\\varepsilon)^\\gamma - \\rho^\\gamma - \\gamma \\rho^{\\gamma-1}(\\rho^\\varepsilon-\\rho) \\geq c\\gamma(\\gamma-1) (\\rho^\\varepsilon-\\rho)^2 . $$\nAdding the two together with the coefficients that appear in $E$, and taking the integral on $\\Omega$, we get\n\\begin{equation} E(t,U^\\varepsilon)-E(t,V) - R(t,U^\\varepsilon,V) \\geq c_0 \\left(\\norme{L^2(\\Omega)}{(\\sqrt{\\rho^\\varepsilon}u^\\varepsilon - \\sqrt{\\rho}u)(t)}^2 + \\norme{L^2(\\Omega)}{(\\rho^\\varepsilon-\\rho)(t)}^2\\right) \\label{strconv} \\end{equation}\nwhere $R(t,U^\\varepsilon,V) = \\int_\\Omega \\sqrt{\\rho}u \\cdot (\\sqrt{\\rho^\\varepsilon}u^\\varepsilon - \\sqrt{\\rho}u) + \\frac{k}{\\gamma-1}(\\rho^\\varepsilon-\\rho)(\\rho^{\\gamma-1}-1)~dx$, which converges to zero. \nIndeed, the uniform boundedness of the energy means that $\\norme{L^2}{\\sqrt{\\rho^\\varepsilon(t)}u^\\varepsilon(t)}$ is bounded, so we can extract a weakly converging sub-sequence in $L^2(\\Omega)$, $\\sqrt{\\rho^{\\varepsilon_n}(t)}u^{\\varepsilon_n}(t)$, whose limit is necessarily $\\sqrt{\\rho(t)} u(t)$, thus the whole sequence converges weakly in $L^2(\\Omega)$. \nSo the first term of $R(t,U^\\varepsilon,V)$ goes to zero. Likewise, we have $\\rho^\\varepsilon(t) \\rightharpoonup \\rho(t)$ in $L^2(\\Omega)$, and $\\rho^{\\gamma-1}-1$ is seen to be in $L^2(\\Omega)$ by use of the order-one Taylor expansion of $z\\mapsto z^{\\gamma-1}$ at $z=1$, so the second term also converges to zero.\n\nMoreover, $E(t,U^\\varepsilon)-E(t,V) \\stackrel{\\varepsilon\\rightarrow 0}{\\longrightarrow} 0$, so (\\ref{strconv}) gives us the global, strong $L^2$ convergence of $\\rho^\\varepsilon$ towards $\\rho$. Now, by remarking that\n\\begin{equation} \\left|\\norme{L^2(\\Omega)}{(\\sqrt{\\rho^\\varepsilon}u^\\varepsilon)(t)}^2 - \\norme{L^2(\\Omega)}{(\\sqrt{\\rho}u^\\varepsilon)(t)}^2\\right| \\leq \\norme{L^\\infty}{u^\\varepsilon(t)}\\norme{L^2}{u^\\varepsilon(t)}\\norme{L^2}{\\rho^\\varepsilon(t)-\\rho(t)} , \\label{normu} \\end{equation}\nwe get that $\\displaystyle \\lim_{\\varepsilon\\rightarrow 0} \\norme{L^2(\\Omega)}{(\\sqrt{\\rho} u^\\varepsilon)(t)} = \\lim_{\\varepsilon\\rightarrow 0} \\norme{L^2(\\Omega)}{(\\sqrt{\\rho^\\varepsilon}u^\\varepsilon)(t)} = \\norme{L^2(\\Omega)}{\\sqrt{\\rho}(t) u(t)}$, as $u^\\varepsilon$ is uniformly bounded in $L^2$ and $L^\\infty$. \nSo $u^\\varepsilon$ converges towards $u$ in $L^2([0,T^*]\\times \\Omega,dt~\\rho dx)$, and $\\rho dx$ is an equivalent measure to the Lebesgue measure, hence we conclude that $U^\\varepsilon$ converges towards $V$ in $L^2([0,T^*]\\times\\Omega)$. \n$L^\\infty$ convergence is obtained by using the Sobolev embedding inequality and the uniform bounds. $\\square$\n\n\\section{Preliminary properties of conormal derivatives}\n\nWe begin this section by reminding the reader of some important properties of our functional setting, and that will be used throughout the \\textit{a priori} estimation process. Set $T>0$.\n\n\\begin{propo} \\underline{Generalised Sobolev-Gagliardo-Nirenberg inequality (or tame estimate).} \\cite{Go}\n\nThere exists a constant $C(T)$, which does not blow up as $T\\rightarrow 0$, such that, for $f,g\\in L^{\\infty}([0,T]\\times\\Omega)\\cap H^{m}_{co}([0,T]\\times\\Omega)$, and $\\alpha_{1},~\\alpha_{2}\\in\\mathbb{N}^{4}$ such that $|\\alpha_{1}|+|\\alpha_{2}|=m$,\n$$ \\int_0^T \\norme{0}{(Z^{\\alpha_{1}}f Z^{\\alpha_{2}}g)(s)}^{2}~ds \\leq C(T)\\left[\\trinorme{\\infty,T}{f}^{2}\\int_0^T \\norme{m}{g(s)}^{2}~ds +\\trinorme{\\infty,T}{g}^{2}\\int_0^T \\norme{m}{f(s)}^{2}~ds \\right].$$ \\label{tame} \\end{propo}\n\n\\begin{propo} \\underline{Trace inequality.}\n\nIf $f\\in L^2([0,T]\\times\\Omega)$ and $\\nabla f\\in L^2([0,T]\\times\\Omega)$, then\n$$ \\int_0^T \\int_{\\partial\\Omega} |f(t,y,0)|^{2}~dt~dy \\leq C \\int_0^T \\norme{0}{f(t)}\\norme{0}{\\derp{z}f(t)}~dt $$ \\label{trace} \\end{propo}\n\n\\begin{propo} \\underline{Anisotropic Sobolev embedding theorem.}\n\nIf $f\\in H^3_{co}([0,T]\\times\\Omega)$, $\\nabla f\\in H^2_{co}([0,T]\\times\\Omega)$, then $f\\in L^\\infty([0,T]\\times\\Omega)$ and\n$$ \\trinorme{\\infty,T}{f}^{2} \\leq C\\left(\\norme{2}{f(0)}^2+\\norme{1}{\\derp{z}f(0)}^2 + \\int_0^T \\norme{3}{f(t)}^{2}+\\norme{2}{\\derp{z}f(t)}^2 ~dt\\right) $$ \\label{sobo} \\end{propo}\n\nThis last theorem is a direct application of the $H^m_{co}(\\Omega)$ Sobolev embedding, used in \\cite{MR} and \\cite{MRfs}: for a given $t$, we have\n$$ \\norme{\\infty}{f(t)}^2 \\leq C(\\norme{H^2_{co}(\\Omega)}{f(t)}^2 + \\norme{H^1_{co}(\\Omega)}{\\nabla f(t)}^2) \\leq C(\\norme{2}{f(t)}^2+\\norme{1}{\\derp{z}f(t)}^2). $$\nWe combine this with the following property: for $f\\in H^{m+1}_{co}([0,T]\\times\\Omega)$,\n\\begin{equation} \\trinorme{m,t}{f}^2 \\leq \\norme{m}{f(0)}^2 + C \\int_0^t \\norme{m+1}{f(s)}^2~ds . \\label{l2time} \\end{equation}\nThis is shown by writing $n(t)=n(0)+\\int_0^t n'(s)~ds$ for $n(s)=\\norme{m}{f(s)}^2$; given that\n$$ n'(s) = 2\\sum_{|\\alpha|\\leq m} \\int_\\Omega (\\derp{t}Z^\\alpha f(s,x))(Z^\\alpha f(s,x))~dx, $$\nwe easily see that $n'(s)\\leq 2\\norme{m+1}{f(s)}^2$. \nMorally, the bound (\\ref{l2time}) means that we can exchange an $L^\\infty$-in-time norm for an $L^2$-in-time norm for the cost of one conormal derivative, similarly to (\\ref{l2timeinf}).\n\nWe will now show the important commutator properties we will need.\n\n\\subsection{Commuting with $\\derp{z}$}\n\nThe first technical key to the proof in the subsequent parts of the paper is how to estimate the commutators that will appear when applying $Z^{\\alpha}$ to equations (\\ref{NSdiv}), (\\ref{NS}). \nA lot of the commutators are trivial, since all $Z_{i}$'s commute with $\\derp{t}$, $\\derp{y_{1}}$ and $\\derp{y_{2}}$, but $Z_{3}$ does not commute with $\\derp{z}$. \nSpecifically, we have\n\\begin{equation} [Z_{3},\\derp{z}] = \\phi(z)\\derp{zz}-\\derp{z}(\\phi(z)\\derp{z}) = -\\phi'(z)\\derp{z} \\label{commu1} \\end{equation}\nLikewise, we can observe the commutator of $Z_3$ with $\\derp{zz}$, which will come from the $\\div\\Sigma$ term of (\\ref{NS}). This time, we have\n\\begin{equation} [Z_{3},\\derp{zz}] = -2\\phi'\\derp{zz} - \\phi''\\derp{z} . \\label{commul} \\end{equation}\n\nWhen commutating with a higher order operator, $Z_3^m$ for $m>1$, we show the following:\n\\begin{propo} (a) For $m\\geq 1$, there exist two families of bounded functions $(\\varphi_{\\beta,m})_{0\\leq \\beta <m}$ and $(\\varphi^{\\beta,m})_{0\\leq\\beta<m}$, such that\n\\begin{equation} [Z_{3}^{m},\\derp{z}] = \\sum_{\\beta=0}^{m-1} \\varphi_{\\beta,m}(z)Z_{3}^{\\beta}\\derp{z} = \\sum_{\\beta=0}^{m-1} \\varphi^{\\beta,m}(z)\\derp{z}Z_{3}^{\\beta} \\label{commum} \\end{equation}\n\n(b) For $m\\geq 1$, there exist four families of bounded functions $(\\psi_{1,\\beta,m})$, $(\\psi_{2,\\beta,m})$, $(\\psi^{1,\\beta,m})$ and $(\\psi^{2,\\beta,m})$, for $0\\leq \\beta <m$, such that\n\\begin{eqnarray*} [Z_{3}^{m},\\derp{zz}] & = & \\sum_{\\beta=0}^{m-1} \\psi_{1,\\beta,m}(z) Z_{3}^{\\beta}\\derp{z} + \\psi_{2,\\beta,m}(z)Z_{3}^{\\beta}\\derp{zz} \\\\\n & = & \\sum_{\\beta=0}^{m-1} \\psi^{1,\\beta,m}(z)\\derp{z}Z_{3}^{\\beta} + \\psi^{2,\\beta,m}(z)\\derp{zz}Z_{3}^{\\beta}. \\end{eqnarray*}\n\\label{comprop} \\end{propo}\nIn practice, we can therefore choose to place the normal derivatives as the first or last derivative to be applied in all the terms of the commutator. We can deduce from the proposition the basic estimate for the commutators with $\\derp{z}$:\n\\begin{coro} For any $f\\in H^{m}_{co}([0,T]\\times \\Omega)$ such that $\\derp{z}f\\in H^{m-1}_{co}([0,T]\\times \\Omega)$, and for any $|\\alpha|\\leq m$,\n$$ \\int_0^T \\norme{0}{[Z^\\alpha,\\derp{z}]f(t)}^2~dt \\leq C\\int_0^T \\norme{m-1}{\\derp{z}f(t)}^2~dt $$ \\label{comcoro} \\end{coro}\nWe will deal with the commutators with $\\derp{zz}$ directly in context: they often appear with a factor $\\varepsilon$, and equation (\\ref{NS}) will allow us to substitute the difficult terms.\n\\newline\n\n\\preu{Proof of \\propref{comprop}}: equation (\\ref{commu1}) shows the case $m=1$, and we continue by induction. Let us just explain the case $m=2$ to show the mechanism; the rest is left to the reader. We have\n\\begin{eqnarray*} [Z_{3}^{2},\\derp{z}] & = & Z_{3}[Z_{3},\\derp{z}]+[Z_{3},\\derp{z}]Z_{3} \\\\\n & = & -\\phi\\phi''\\derp{z}-\\phi'(Z_{3}\\derp{z}+\\derp{z}Z_{3}). \\end{eqnarray*}\nUsing (\\ref{commu1}), we can write the second part of the last line as either $\\phi'(z)(2Z_{3}\\derp{z}+\\phi'\\derp{z})$ or $\\phi'(z)(2\\derp{z}Z_{3}-\\phi'\\derp{z})$, which proves the proposition for $m=2$.\n\nThe proof of (b) is also an elementary induction. $\\square$\n\n\\subsection{Commuting with a function}\n\n\\begin{nota} For $\\alpha,~\\beta\\in\\mathbb{N}^{4}$, we write $\\beta \\leq \\alpha$ if, for every $i$, $\\beta_{i}\\leq\\alpha_{i}$. \\end{nota}\n\n\\begin{propo} Let $\\alpha\\in\\mathbb{N}^{4}$, $|\\alpha|=m>0$, be fixed, and $f\\in H^m_{co}([0,T]\\times\\Omega)\\cap L^\\infty([0,T]\\times\\Omega)$ and $g\\in L^\\infty$ such that $\\derp{t}g,~\\nabla g\\in H^{m-1}_{co}([0,T]\\times\\Omega)\\cap L^\\infty([0,T]\\times\\Omega)$. Then we have the following inequality:\n\\begin{eqnarray} \\int_{0}^{T}\\norme{0}{[Z^{\\alpha},g]f}^{2}~dt & \\leq & C\\sum_{j=0}^3\\int_{0}^{T}\\trinorme{\\infty,T}{Z_j g}^{2}\\norme{m-1}{f}^{2}+\\trinorme{\\infty,T}{f}^{2}\\norme{m-1}{Z_{j}g}^{2}~dt, \\label{funcom} \\\\ & \\leq & C\\int_{0}^{T}\\trinorme{1,\\infty,T}{g}^{2}\\norme{m-1}{f(s)}^{2}+\\trinorme{\\infty,T}{f}^{2}\\norme{m}{g(s)}^{2}~dt, \\nonumber \\end{eqnarray}\nif, moreover, $g\\in L^2([0,T]\\times\\Omega)$. \\label{funcprop} \\end{propo}\n\nThis proposition is easily proved, using the Leibniz formula and \\propref{tame}. We will require formulation (\\ref{funcom}) whenever $g\\notin L^2$ (for example when $g=\\rho$).\n\\newline\n\nWe prove one more commutator estimate, which we will need when estimating the normal derivatives of $u_\\tau$ in the conormal spaces (section 5), as directly using the above would lead to $\\norme{m-2}{\\derp{zz} u}$ appearing, which we cannot bound uniformly in $\\varepsilon$ by using the equation.\n\\begin{propo} Let $f,~g,~\\alpha$ be as in \\propref{funcprop}, with $g$ scalar such that $g|_{z=0}=0$. Then there exists $C$ which does not depend on $T$ such that\n$$ \\int_{0}^{T}\\norme{0}{[Z^{\\alpha},g\\derp{z}]f}^{2}~dt \\leq C\\int_{0}^{T}(\\trinorme{\\mathrm{Lip},T}{g}^{2}+\\trinorme{1,\\infty,T}{\\nabla g}^{2})\\norme{m}{f(s)}^{2}+\\trinorme{1,\\infty,T}{f}^{2}\\norme{m-1}{\\nabla g(s)}^{2}~dt $$ \\label{comlem} \\end{propo}\n\n\\preu{Proof:} we decompose the commutator as $[Z^{\\alpha},g\\cdot\\nabla]f = [Z^{\\alpha},g]\\derp{z} f + g [Z^{\\alpha},\\derp{z}]f$, and start by taking a closer look at the second term. \nBy \\propref{comprop}, it is equal to a sum of terms of the form $\\varphi^{\\beta}(z)g~\\derp{z}Z^{\\beta}f$, with $\\beta\\leq\\alpha$, $\\beta\\neq\\alpha$. As $g|_{z=0}=0$, we have\n\\begin{equation} |g(t,x)|\\leq \\phi(z)\\trinorme{\\infty,T}{\\derp{z}g}, \\label{u3phi} \\end{equation}\nso, as $Z_{3}=\\phi(z)\\derp{z}$,\n\\begin{equation} \\norme{0}{g[Z^{\\alpha},\\derp{z}]f} \\leq C\\trinorme{\\infty,T}{\\nabla g}\\norme{m-1}{\\phi(z)\\derp{z}f} \\leq C\\trinorme{\\mathrm{Lip},T}{g}\\norme{m}{f}. \\label{hmcom3} \\end{equation}\n\nNow we look at the first term of the decomposition. We prove that\n\\begin{equation} \\int_{0}^{T}\\norme{0}{[Z^{\\alpha},g]\\derp{z}f}^{2} \\leq C\\int_{0}^{T}(\\trinorme{\\mathrm{Lip},T}{g}^{2}+\\trinorme{1,\\infty,T}{\\nabla g}^{2})\\norme{m}{f}^{2} + \\trinorme{1,\\infty,T}{f}^{2}\\norme{m-1}{\\nabla g}^{2}~dt, \\label{comfinal} \\end{equation}\nwhich would end the proof of the proposition.\n\nWe can write $[Z^{\\alpha},g]\\derp{z}f$ as a sum, on $\\beta$ and $\\delta$, of terms of the form $\\varphi^{\\delta}(z) Z^{\\beta}g~\\derp{z}Z^{\\delta} f$, where $\\beta\\leq\\alpha$ and $|\\beta|>0$, and $\\delta\\leq(\\alpha-\\beta)$. The idea is to insert $\\frac{1}{\\phi(z)}\\times\\phi(z)$, thus we have to estimate\n$$ \\norme{0}{\\frac{1}{\\phi(z)}(Z^{\\beta}g)(Z_{3}Z^{\\delta}f)}. $$\nThis will obviously be done by using the tame estimate, but we also have to deal with the first factor. We have $\\phi^{-1}Z^{\\beta}g = Z^{\\beta}(\\phi^{-1}g) - [Z^{\\beta},\\phi^{-1}]g$, and we need to write this last term more explicitly.\n\\begin{lemma} For every $b\\in\\mathbb{N}$, $\\phi Z_{3}^{b}(\\phi^{-1})$ is smooth and bounded, then there exists a family of smooth bounded functions $(\\sigma_{p})_{0\\leq p\\leq b}$ such that for every $f\\in H^{m}_{co}([0,T]\\times\\Omega)$,\n$$ [Z^{m}_{3},\\phi^{-1}]f = \\sum_{p=0}^{m} \\sigma_{p}(z) Z^{p}_{3}(\\phi^{-1}f) $$ \\label{phicom} \\end{lemma}\n\n\\preu{Proof:} as $\\phi$ only depends on $z$, we are only interested in $[Z_3^m,\\phi^{-1}]$ for $m\\geq 1$. The Leibniz formula yields that the commutator is a linear combination of terms written as $(Z_3^b(\\phi^{-1}))(Z^{m-b}f)$, for $0< b\\leq m$. \nIn the case of $\\phi(z)=\\frac{z}{1+z}$, we notice that $Z_3^b (\\phi^{-1})$ has the same following properties as $\\phi^{-1}$: it is smooth on $]0,+\\infty[$, bounded at infinity and has the same blow-up rate at $z=0$ as $\\phi^{-1}$ (blows up like $z^{-1}$). So, for each $b$, $\\phi Z^b_3(\\phi^{-1})$ is a bounded function on $[0,+\\infty[$, and if we write\n$$ (Z^b_{3}(\\phi^{-1}))(Z^{m-b}f) = \\phi Z^b_{3}(\\phi^{-1})\\left[Z^{m-b}\\left(\\frac{1}{\\phi}f\\right) - [Z^{m-b},\\phi^{-1}]f\\right], $$\nwe have $m-b<m$, and we can reiterate the process. As $m$ is fixed, we conclude the proof with a finite number of iterations. $\\square$\n\\newline\n\nApplying the lemma, we have that, in any norm,\n\\begin{equation} \\norme{}{\\phi^{-1}Z^{\\beta}g} \\leq C\\sum_{\\beta'\\leq\\beta} \\norme{}{Z^{\\beta'}(\\phi^{-1}g)} \\label{phicom2} \\end{equation}\nWith that, the tame estimate gives us\n$$ \\int_{0}^{T}\\norme{0}{\\phi^{-1}(Z^{\\beta}g)(Z_{3}Z^{\\delta}f)}^{2}~dt \\leq C \\int_{0}^{T} \\trinorme{\\infty,T}{Z_{3}f}^{2} \\sum_{j=0}^{3} \\norme{m-2}{Z_j\\left(\\frac{g(s)}{\\phi}\\right)}^{2} $$\n$$ \\hspace{70pt} + \\left(\\trinorme{\\infty,T}{\\phi^{-1}g}^2 +\\sum_{j=0}^{3} \\trinorme{\\infty,T}{Z_{j}(\\phi^{-1}g)}^2\\right)\\norme{m-1}{Z_3 f(s)}^{2}~dt $$\nand the $\\trinorme{\\infty,T}{\\phi^{-1}g_3}$ comes from the terms in (\\ref{phicom2}) with $\\beta'=0$.\n\nIt remains to deal with the terms involving $\\phi^{-1}g$, and the key fact here is that $Z^{\\beta}g|_{z=0}=0$. We easily have by (\\ref{u3phi}),\n\\begin{equation} \\norme{\\infty}{\\phi^{-1}g} \\leq \\norme{\\mathrm{Lip}}{g} ~~\\mathrm{and}~~ \\norme{\\infty}{Z_{j}(\\phi^{-1}g)}\\leq C\\norme{1,\\infty}{\\nabla g}, \\label{hardyinf} \\end{equation}\nas $Z_{3}(\\phi^{-1}g) = \\phi^{-1}Z_{3}g+\\phi'\\phi^{-1}g$, with $\\phi'$ bounded, and likewise we can use the Hardy inequality to get\n$$ \\norme{m-2}{Z_{j}(\\phi^{-1}g)} \\leq C\\norme{m-1}{\\derp{z}g}, $$\nwhich ends the proof of (\\ref{comfinal}). $\\square$\n\n\\section[Proof of \\thref{thEE}, part I: \\textit{a priori} estimates on $U$]{Proof of \\thref{thEE}, part I \\\\ \\textit{A priori} estimates on $U$\n \\sectionmark{Proof of \\thref{thEE} part I: a priori estimates on $U$}}\n\\sectionmark{Proof of \\thref{thEE} part I: a priori estimates on $U$}\n\n\\begin{assu} Throughout the \\textit{a priori} estimation process, we will assume that there is no vacuum on the time of study: there exists $0<c_{0}<1$ such that $c_{0}\\leq\\rho(t,x)$ for $x\\in\\Omega$. \nAlso, as $u_3|_{z=0}=0$, for any $\\delta>0$, we assume that there exists $z_\\delta>0$, independent of $\\varepsilon$, such that $|u_3(t,x)|\\leq \\delta$ for $x\\in \\sdel{F} := \\mathbb{R}^2\\times [0,\\sdel{z}]$.\n\nAfter getting the estimates, we will show in section 8 that the final bounds actually imply these two properties on $[0,T^*]$, with $T^*$ such that ${\\cal E}_m(T^*,U)\\leq M$, and prove by a bootstrap argument that $T^*$ does not depend on $\\varepsilon$.\n\\label{assu1} \\end{assu}\n\n\\begin{nota} As of now, $0<c<1$ will designate a small constant, $C>1$ a large constant, and $Q(z)$ a positive increasing function of $\\mathbb{R}^{+}$, with polynomial growth and $Q(z)\\geq z$. \nAll three can change from one line to the next, and can depend on any of the system's parameters (constants $a$, $k$ and $\\gamma$, or bounds of the viscosity functions $\\mu(\\rho)$ and $\\lambda(\\rho)$ - we will often omit the dependence on $\\rho$), on the order of derivation $m$ or on $\\varepsilon_0$. \\end{nota}\n\n\\subsection{Conormal energy estimates}\n\nWe start with energy estimates on $U=(\\rho-1,u)$ and its conormal derivatives: this will estimate the $\\trinorme{m}{U}$ part of ${\\cal E}_{m}(t,U)$, and it will uncover other terms of ${\\cal E}_{m}(t,U)$ that we will need to estimate later.\n\nWe will be able to estimate $\\rho-1$ and $u$ simultaneously by taking full advantage of the symmetrisable hyperbolic structure of the order-one part of the isentropic Navier-Stokes system $(\\ref{NSdiv}),~ (\\ref{NS})$. We rewrite it as\n\\begin{equation} A_{0}(\\rho)\\derp{t}U + \\sum_{j=1}^{3} A_{j}(U)\\derp{x_{j}}U - \\varepsilon(0,\\mu(\\rho) \\Delta u + \\lambda(\\rho) \\nabla\\div u)^t = (0,\\rho F-\\varepsilon\\sigma(\\nabla U))^t , \\label{matrixform} \\end{equation}\nwhere $\\lambda=\\mu+\\lambda_0>0$, $\\sigma$ has the following expression\n$$ \\sigma(\\nabla \\rho,\\nabla u) = 2Su\\cdot\\nabla (\\mu(\\rho)) + \\div u \\nabla (\\lambda_0(\\rho)) = 2\\mu'(\\rho) Su\\cdot \\nabla \\rho + \\lambda'(\\rho)\\div u \\nabla \\rho. $$\nThe matrices $A_{j}(U)$ are: $A_{0}(\\rho)=\\mathrm{diag}(1,\\rho,\\rho,\\rho)$,\n$$ A_{1}(U)=\\left(\\begin{array}{cccc} u_{1} & \\rho & 0 & 0 \\\\ k\\rho^{\\gamma-1} & \\rho u_{1} & 0 & 0 \\\\ 0 & 0 & \\rho u_{1} & 0 \\\\ 0 & 0 & 0 & \\rho u_{1} \\end{array} \\right) ~,~ A_{2}(U)=\\left(\\begin{array}{cccc} u_{2} & 0 & \\rho & 0 \\\\ 0 & \\rho u_{2} & 0 & 0 \\\\ k\\rho^{\\gamma-1} & 0 & \\rho u_{2} & 0 \\\\ 0 & 0 & 0 & \\rho u_{2} \\end{array} \\right) $$\n$$ \\mathrm{and} ~ A_{3}(U)=\\left(\\begin{array}{cccc} u_{3} & 0 & 0 & \\rho \\\\ 0 & \\rho u_{3} & 0 & 0 \\\\ 0 & 0 & \\rho u_{3} & 0 \\\\ k\\rho^{\\gamma-1} & 0 & 0 & \\rho u_{3} \\end{array} \\right). $$\nThis system is symmetrisable: multiplying these matrices on the left by the positive diagonal matrix $D(\\rho)=\\mathrm{diag}(k\\rho^{\\gamma-2},1,1,1)$, we get:\n\\begin{equation} DA_{0}\\derp{t}U + \\sum_{j=1}^{3} DA_{j}\\derp{x_{j}}U -\\varepsilon(0,\\mu\\Delta u +\\lambda\\nabla\\div u)^t = (0,\\rho F-\\varepsilon\\sigma(\\nabla U))^t \\label{sym} \\end{equation}\n\n$DA_{0}(\\rho)$ is symmetric, so\n$$ \\frac{1}{2} \\frac{d}{dt}\\left(\\int_{\\Omega} DA_{0}U\\cdot U\\right) = \\int_{\\Omega} DA_{0}\\derp{t}U\\cdot U + \\frac{1}{2}\\int_{\\Omega} \\derp{t}(DA_{0})U\\cdot U. $$\nIntegrating this in time between $0$ and $t$, and since $\\rho$ is uniformly bounded from below by $c_{0}$ and from above by $c_{1}$, there exist $0<c<C$ such that\n\\begin{equation} c \\norme{0}{U(t)}^{2} \\leq C\\norme{0}{U(0)}^{2}+C\\trinorme{\\infty,t}{\\derp{t}DA_0}\\int_{0}^{t}\\norme{0}{U(s)}^{2} + \\int_{0}^{t}\\int_{\\Omega} DA_{0}\\derp{t}U(s)\\cdot U(s)~ds \\label{l2ee1} \\end{equation}\n\nWe replace $DA_{0}\\derp{t}U$ by its expression in (\\ref{sym}), and we use integration by parts on the integrals with order-one derivatives of $U$:\n\\begin{equation} \\int_{\\Omega} DA_{j}\\derp{x_{j}}U \\cdot U = -\\frac{1}{2} \\int_{\\Omega} \\derp{x_{j}}(DA_{j}) U\\cdot U . \\label{l2ee2} \\end{equation}\nIndeed, $DA_{j}(U)$ is symmetric, so $DA_{j}U\\cdot \\derp{x_{j}}U = U\\cdot DA_{j}\\derp{x_{j}}U$, and we notice that, for each $j$, $DA_{j}U\\cdot U = (3\\rho P'(\\rho)+\\rho|u|^2)u_j$, which means that there is no boundary term (at $z=0$ for $j=3$) when we integrate by parts, which gives us\n\\begin{eqnarray} c\\norme{0}{U(t)}^2 & \\leq & C\\norme{0}{U(0)}^2 + C\\left(\\sum_{j=0}^3 \\trinorme{\\mathrm{Lip},t}{DA_j}\\right)\\int_0^t \\norme{0}{U(s)}^2 ~ds \\nonumber \\\\\n & & \\hspace{10pt} + \\varepsilon\\int_0^t \\int_\\Omega \\mu\\Delta u\\cdot u + \\lambda \\nabla \\div u \\cdot u + \\sigma\\cdot u~ds + \\int_0^t \\int_\\Omega \\rho F\\cdot u~ds \\nonumber \\\\\n & \\leq & C\\norme{0}{U(0)}^2 + C(1+\\trinorme{\\mathrm{Lip},t}{U}^2)\\int_0^t \\norme{0}{U(s)}^2 ~ ds \\nonumber \\\\\n & & \\hspace{10pt} + \\varepsilon \\int_0^t \\int_\\Omega \\mu\\Delta u \\cdot u + \\lambda \\nabla\\div u\\cdot u + \\sigma\\cdot u ~ds + \\int_0^t \\int_\\Omega \\rho F\\cdot u~ds \\label{l2ee2b} \\end{eqnarray}\n\nWe use integration by parts on the order-two derivatives and the Navier boundary condition (\\ref{NBC}) to deal with the boundary term:\n\\begin{eqnarray} \\int_\\Omega \\mu\\Delta u\\cdot u + \\lambda \\nabla\\div u\\cdot u & = & -\\int_\\Omega (\\mu|\\nabla u|^2 + \\lambda|\\div u|^2) -\\int_{z=0} 2a |u_\\tau|^2 \\hspace{50pt} \\nonumber \\\\\n & & \\hspace{20pt} - \\int_\\Omega \\left(\\nabla (\\mu(\\rho)) \\cdot \\nabla u\\cdot u + \\div u~ \\nabla (\\lambda(\\rho))\\cdot u\\right) , \\label{l2ee3} \\end{eqnarray}\nwith the notation $\\int_{z=0}f = \\int_{\\mathbb{R}^{2}} f(y,0)~dy$. The first term of (\\ref{l2ee3}) is moved the left-hand side of (\\ref{l2ee2b}), as is the second if $a>0$. \nWhen $a<0$, we can use the trace theorem, \\propref{trace}, and absorb the norm of $\\nabla u$ in the left-hand side by using Young's inequality, $\\nu\\zeta \\leq \\frac{\\eta}{2} \\nu^{2} + \\frac{1}{2\\eta} \\zeta^{2}$ for any $(\\nu,\\zeta)\\in\\mathbb{R}^2$ and $\\eta>0$, with an adequate parameter $\\eta$ (same as in \\cite{Pm11}). \nYoung's inequality is also used on the term containing the derivatives of $\\mu$ and $\\lambda$, which turns (\\ref{l2ee3}) into\n\\begin{equation} \\int_{\\Omega} (\\mu\\Delta u\\cdot u+\\lambda\\nabla \\div u\\cdot u) \\leq \\frac{c(1+\\trinorme{\\mathrm{Lip},t}{\\rho})}{\\eta}\\norme{0}{u}^2 - (c-(|a|+1)\\eta) \\norme{0}{\\nabla u}^2 , \\label{l2ee3b} \\end{equation}\nso we choose $\\eta$ so that $c-(|a|+1)\\eta = c\/2$, which allows us to move the $\\norme{0}{\\nabla u}^2$ term to the left-hand side and absorb it with the first term of (\\ref{l2ee3}) that we moved there earlier. The remainder $\\int_\\Omega \\sigma\\cdot u$ is bounded the same way.\n\nCombining (\\ref{l2ee2b}) and (\\ref{l2ee3b}), and the assumption that $\\rho$ is uniformly bounded, there exist $0<c<C$ such that\n$$ c\\left[\\norme{0}{U(t)}^{2} + \\varepsilon\\int_{0}^{t}\\norme{0}{\\nabla u}^{2} \\right] \\leq C\\left(\\norme{0}{U(0)}^{2}+(1+\\trinorme{\\mathrm{Lip},t}{U}^2) \\int_{0}^{t}\\norme{0}{U(s)}^{2} + \\norme{0}{F(s)}^2~ds\\right), $$\nas $\\norme{\\infty}{\\nabla(DA_{j})}\\leq C\\norme{\\infty}{U}\\norme{\\infty}{\\nabla U}$.\n\\newline\n\nWe have shall now show the following, higher order estimate.\n\n\\begin{esti} For every $m\\geq 0$,\n$$ c\\left[\\norme{m}{U(t)}^{2} + \\varepsilon\\int_{0}^{t}\\norme{m}{\\nabla u(s)}^{2} ~ds\\right] \\hspace{250pt} $$\n$$ \\hspace{10pt} \\leq C\\left[\\norme{m}{U(0)}^{2}+(1+\\trinorme{\\mathrm{Lip},t}{U}^2+\\trinorme{\\infty}{F}^2)\\int_{0}^{t} \\norme{m}{U}^{2} + \\norme{m-1}{\\nabla U}^{2} + \\norme{m}{F}^2 ~ds\\right] $$\n(the negative index that appears when $m=0$ is ignored). \\label{hmee} \\end{esti}\n\n\\preu{Proof:} higher-order estimates work exactly the same way as above, only we will have to estimate the commutators between $Z^{\\alpha}$, with $|\\alpha|\\leq m$, and the operators in the equation. \nWe apply $Z^{\\alpha}$ to equation (\\ref{matrixform}), and isolate the highest-order terms as follows:\n\\begin{equation} A_0 \\derp{t}Z^{\\alpha}U + \\sum_{j=1}^{3} A_{j}\\derp{x_{j}}(Z^{\\alpha}U) - \\varepsilon(0,\\div Z^\\alpha(\\mu\\nabla u)+\\lambda\\nabla\\div (Z^{\\alpha}u))^t = (0,Z^\\alpha (\\rho F+\\varepsilon\\tilde{\\sigma}))^t-{\\cal C}^{\\alpha}. \\label{matrixformalpha} \\end{equation}\n${\\cal C}^{\\alpha}$ contains the commutators:\n$$ {\\cal C}^{\\alpha} = [Z^\\alpha,A_0\\derp{t}]U + \\sum_{j=1}^{3} [Z^{\\alpha},A_{j}\\derp{x_{j}}]U -\\varepsilon(0, [Z^{\\alpha},\\div] (\\mu \\nabla u)+[Z^{\\alpha},\\lambda(\\rho)\\nabla \\div]u)^t. $$\nNotice the peculiar formulation for the laplacian term. \nThis allows us to avoid difficult commutator terms on the boundary when $\\mu$ is not constant; indeed, the Navier boundary condition for $Z^\\alpha u$ is $Z^\\alpha(\\mu\\derp{z}u_\\tau) = 2a Z^\\alpha u_\\tau$, thus, when we multiply equation (\\ref{matrixformalpha}) by $Z^\\alpha u$ and integrate by parts, we can use this boundary condition immediately. \nThe remainder of the order-two part of the equation $\\tilde{\\sigma}$ is the equivalent of $\\sigma$ for this formulation.\n\nWe then multiply by the matrix $D$, which is uniformly bounded in $L^\\infty$, and can repeat the above to obtain:\n$$ c\\left[\\norme{0}{Z^{\\alpha}U(t)}^{2} + \\varepsilon\\int_{0}^{t}\\left(\\norme{0}{Z^\\alpha \\nabla u}^{2} + \\int_{z=0}|Z^{\\alpha}u|^{2}\\right)\\right] \\leq C\\norme{0}{Z^{\\alpha}U(0)}^{2} \\hspace{100pt} $$\n$$ \\hspace{80pt} +C\\int_0^t (1+\\trinorme{\\mathrm{Lip},t}{U}^2+\\trinorme{\\infty,t}{F})\\norme{m}{U}^{2} + \\varepsilon \\norme{m-1}{\\nabla U}^2 + \\norme{m}{F}^2 ~ds $$\n\\begin{equation} \\hspace{120pt} + C\\int_0^t\\int_{\\Omega} |(\\varepsilon Z^\\alpha \\sigma+{\\cal C}^{\\alpha})\\cdot Z^{\\alpha}U| + |{\\cal I}^\\alpha_\\nabla| ~ds. \\label{hmee1} \\end{equation}\nThe term ${\\cal I}^\\alpha_\\nabla$ contains extra commutators arising from the following integration by parts:\n\\begin{eqnarray*} -\\int_\\Omega \\div Z^\\alpha(\\mu\\nabla u)\\cdot Z^\\alpha u & = & \\int_{\\partial\\Omega} Z^\\alpha(\\mu\\derp{z}u)\\cdot Z^\\alpha u + \\int_\\Omega Z^\\alpha(\\mu \\nabla u):\\nabla Z^\\alpha u \\\\\n & = & \\int_{\\partial\\Omega} 2a |Z^\\alpha u_\\tau|^2 + \\int_\\Omega \\mu|Z^\\alpha \\nabla u|^2 \\\\\n & & \\hspace{40pt} +\\int_\\Omega ([Z^\\alpha,\\mu]\\nabla u) : Z^\\alpha \\nabla u - Z^\\alpha(\\mu\\nabla u):([Z^\\alpha,\\nabla] u), \\end{eqnarray*}\nso ${\\cal I}^\\alpha_\\nabla$ is the final integral multiplied by $\\varepsilon$. Here, we have used the contracted matrix product: $A:B = \\sum_{i,j} a_{i,j} b_{i,j}$.\n\nTo deal with the commutator terms, we will use the tools shown in section 3. Let us first look at ${\\cal I}^\\alpha_\\nabla$. In the first term, we have\n$$ \\varepsilon \\left| \\int_\\Omega ([Z^\\alpha,\\mu]\\nabla u) : Z^\\alpha \\nabla u \\right| \\leq \\norme{0}{[Z^\\alpha,\\mu]\\nabla u} \\times \\varepsilon\\norme{0}{Z^\\alpha \\nabla u}. $$\nBy estimate (\\ref{funcom}) in \\propref{funcprop}, the integral in time of the square of the first norm is easily bounded, whereas the second can be moved to the left-hand side (absorbed) by using Young's inequality with an adequate parameter $\\eta$ to have\n\\begin{equation} \\left|\\int_0^t \\varepsilon \\norme{m}{\\nabla u} \\norme{0}{[Z^\\alpha,\\mu]\\nabla u}\\right | \\leq C\\varepsilon (1+\\trinorme{\\mathrm{Lip},t}{U}^2)\\int_0^t (\\norme{m}{\\rho-1}^2+\\norme{m-1}{\\nabla u}^{2}) + \\frac{c}{4}\\varepsilon\\int_0^t \\norme{0}{Z^\\alpha \\nabla u}^{2} . \\label{hmeeb} \\end{equation}\nThe second term of ${\\cal I}^\\alpha_\\nabla$ is dealt with in exactly the same fashion, with \\cororef{comcoro} controlling the side of the product containing the commutator, and Young's inequality allowing to absorb the other.\n\\newline\n\nNow to the commutators in ${\\cal C}^\\alpha$ relative to the order-one part of the equation. We notice that, for $j\\in\\{0,1,2,3\\}$,\n$$ [Z^{\\alpha},A_{j}\\derp{x_j}]U = [Z^{\\alpha},A_{j}]\\derp{x_j} U+A_{j}[Z^{\\alpha},\\derp{x_j}]U, $$\nwhere $x_0$ is understood to be $t$. The first term is estimated using inequality (\\ref{funcom}) of \\propref{funcprop} (we cannot write $\\norme{m}{A_{j}}$ because $\\rho \\notin L^{2}$), and the second, which is either 0, if $j\\neq 3$, or $A_{3}[Z^{\\alpha},\\derp{z}]U$, is estimated using \\cororef{comcoro}, so estimating this commutator yields\n\\begin{equation} \\int_{\\Omega} |A_{3}[Z^{\\alpha},\\derp{z}]U\\cdot Z^{\\alpha}U | \\leq C(1+\\trinorme{\\mathrm{Lip},t}{U}^{2})\\norme{m-1}{\\nabla U}^{2}+\\norme{m}{U}^{2}. \\label{hmee2} \\end{equation}\n\nThe commutators on the order-two part of the equation can be split into terms of two types as follows: commutators on the viscosity parameters (functions of $\\rho$) and commutators on the differential operators, for instance\n\\begin{equation} [Z^\\alpha,\\lambda\\nabla\\div] = [Z^\\alpha,\\lambda]\\nabla \\div + \\lambda[Z^\\alpha,\\nabla\\div] . \\label{lapcomsplit} \\end{equation}\n\nWe begin with the first term. Notice that we can write this commutator as a sum of the following type of integral,\n$$ I_{i,j} = \\int_0^t \\int_\\Omega \\varepsilon [Z^\\alpha,\\lambda]\\derp{x_i,x_j}u \\cdot Z^\\alpha u, $$\nwith $i,~j \\in \\{1,2,3\\}$. Using \\tworef{funcprop}{comprop}{Propositions}, we write\n\\begin{equation} I_{i,j} = \\varepsilon \\sum_{|\\beta|+|\\delta|\\leq m, |\\delta|< m} \\int_0^t \\int_\\Omega \\psi_{\\beta,\\delta}(z) Z^\\beta(\\lambda(\\rho))  ~ \\derp{x_i}Z^\\delta\\derp{x_j}u\\cdot Z^\\alpha u ~ ds , \\label{801} \\end{equation}\nwhere the $\\psi_{\\beta,\\delta}$ are ${\\cal C}^\\infty$ bounded functions of $z$. If $|\\beta|< m$, we can integrate by parts on the $x_i$ variable:\n\\begin{equation} I_{i,j} \\leq \\sum_{|\\beta|+|\\delta|\\leq m} c_{\\beta,\\delta} \\varepsilon \\int_0^t \\int_\\Omega |\\derp{x_i}Z^\\beta(\\lambda(\\rho))~Z^\\delta\\derp{x_j}u\\cdot Z^\\alpha u + Z^\\beta(\\lambda(\\rho))~Z^\\delta\\derp{x_j}u\\cdot \\derp{x_i}Z^\\alpha u|~ds . \\label{802} \\end{equation}\nWe use Young's inequality, then we use the tame estimate on the left of the scalar products to get\n$$ I_{i,j} \\leq \\frac{c\\varepsilon}{2}\\int_0^t \\norme{m}{\\nabla u}^2~ds + \\varepsilon Q(\\trinorme{\\mathrm{Lip},t}{\\rho}^2) \\int_0^t \\norme{m-1}{\\nabla u}^2 + \\norme{m}{u}^2~ds \\hspace{50pt} $$\n$$ \\hspace{100pt} + \\varepsilon\\trinorme{\\infty,t}{\\nabla u}^2 Q\\left(\\int_0^t \\norme{m-1}{\\nabla \\rho}^2~ds\\right) . $$\nThis allows us to absorb the first term. Note that we have used the tame estimate on the pair $(Z \\rho,\\derp{x_j}u)$ rather than $(\\rho,\\derp{x_j}u)$ on the second term of (\\ref{802}) to get $H^{m-1}_{co}$ norms of $\\nabla u$ with a factor $\\varepsilon$, which can be controlled by induction on $m$.\n\nThere is a slight difference when $\\beta=\\alpha$ in (\\ref{801}). In this case, $\\delta=0$, but after integrating by parts, we would need to control $\\norme{m}{\\nabla \\rho}$. \nSo we cannot integrate by parts and must estimate $J_{i,j}:=\\varepsilon\\int_0^t \\int_\\Omega Z^\\alpha(\\lambda(\\rho)) ~\\derp{x_i,x_j}u\\cdot u$ directly. \nThis is not a problem if $(i,j)\\neq (3,3)$: we bound $J_{i,j}$ by\n$$ J_{i,j}\\leq \\varepsilon \\int_0^t \\trinorme{\\infty}{u}^2 \\norme{1}{\\nabla u}^2 Q\\left(\\norme{m}{\\rho-1}^2\\right)~ds . $$\nAt this stage, if $m=1$, we use Young's inequality to absorb $\\varepsilon\\int_0^t \\norme{1}{\\nabla u}^2$, and if $m\\geq 2$, we can use \\propref{hmee} with $m=1$. When $(i,j)=(3,3)$, we instead replace $\\varepsilon\\derp{zz}u$ by using the equation. We have\n$$ \\varepsilon(\\mu \\derp{zz}u_\\tau, (\\mu+\\lambda)\\derp{zz}u_3) = \\rho\\derp{t}u + (\\rho u\\cdot\\nabla u) + \\nabla P - \\rho F -\\varepsilon v_2, $$\nwhere order-two derivatives appear in $v_2$. The terms that appear are therefore either controlled in $L^\\infty$ by $\\norme{\\mathrm{Lip}}{U}$, or, in the case of $v_2$, dealt with using Young's inequality as above. \nThe term $\\int_\\Omega \\varepsilon Z^\\alpha\\sigma\\cdot Z^\\alpha u$ in (\\ref{hmee1}) is dealt with the same techniques as these commutators (the terms it contains ressemble those in (\\ref{802})).\n\\newline\n\nFor the remaining commutator in (\\ref{lapcomsplit}) (the estimation of the last term $[Z^\\alpha,\\div] (\\mu\\nabla u)$ follows the same lines), we assume that $\\alpha_3>0$, as these commutators are zero otherwise. \nThe terms of the commutator $[Z^{\\alpha},\\nabla\\div]u$ of the form $[Z^{\\alpha},\\nabla_\\tau\\derp{z}]u_{3}$ (the others are either trivial or contain $\\derp{zz}$), can then be estimated with \\cororef{comcoro}:\n$$ \\int_{\\Omega} \\varepsilon\\lambda|[Z^{\\alpha},\\derp{z}]\\nabla_\\tau u_{3}\\cdot Z^{\\alpha}u_\\tau| \\leq C\\varepsilon\\norme{m}{\\derp{z}u}\\norme{m}{u} \\leq C\\varepsilon\\norme{m}{u}^{2}+\\frac{c}{4}\\varepsilon\\norme{m}{\\derp{z}u}^{2}, $$\nagain using Young's inequality to allow us to absorb $\\varepsilon\\int_0^t \\norme{m}{\\derp{z}u}^{2}$. On to the remaining term $\\varepsilon[Z^{\\alpha},\\derp{zz}]u$. By \\propref{comprop}, there exist two families of functions, $(\\varphi_{\\beta})_{\\beta\\leq\\alpha}$ and $(\\psi^\\beta)_{\\beta\\leq\\alpha}$, such that\n$$ [Z^{\\alpha},\\derp{zz}]u = \\sum_{\\beta\\leq\\alpha,~\\beta\\neq\\alpha} \\varphi_\\beta(z) Z^{\\beta}\\derp{z}u+\\psi^\\beta(z) \\derp{zz}Z^{\\beta}u $$\nThe first part of the right-hand side is simply bounded by $\\norme{m-1}{\\derp{z}u}$, so it only remains to control $J:=\\varepsilon\\int_{\\Omega} |\\derp{zz}Z^{\\beta}u\\cdot Z^{\\alpha}u|$. As $\\alpha_{3}>0$, we have $Z^{\\alpha}u=0$ on the boundary, so integrating by parts and again using Young's inequality provides us with\n$$ J = \\varepsilon\\int_{\\Omega} |\\derp{z}Z^{\\beta}u\\cdot\\derp{z}Z^{\\alpha}u| \\leq C\\varepsilon\\norme{m-1}{\\derp{z}u}\\norme{m}{\\derp{z}u} \\leq C\\varepsilon\\norme{m-1}{\\derp{z}u}^{2}+\\frac{c}{4}\\varepsilon\\norme{m}{\\derp{z}u}^{2}. $$\n\nCombining this last inequality with (\\ref{hmee1}), (\\ref{hmeeb}) and (\\ref{hmee2}), then summing for all $|\\alpha|\\leq m$, we get the result. $\\square$\n\n\\subsection{$L^\\infty$ estimates}\n\nWith conormal energy estimates, we obtain inequalities for any order of derivation $m$, and these inequalities contain $L^\\infty$ or Lipschitz norms of $U$, and, as we shall see soon, $W^{1,\\infty}_{co}$ norms of $\\nabla U$. \nThese need to be controlled, and the goal of this control is to get the conormal energy estimates to be closed for $m$ large enough, by using the Sobolev embedding inequality in \\propref{sobo}. \nPutting the $L^\\infty$ norm of the normal derivative $\\derp{z}U$ to one side for now (this will be dealt with in the next sections), we can apply \\propref{sobo} to $\\trinorme{1,\\infty,t}{U}$, and (\\ref{l2time}) again on the terms involving the derivatives of $\\rho$ and $u_3$.\n\n\\begin{esti} We have the following bound for $\\trinorme{1,\\infty,t}{U}$:\n\\begin{eqnarray*} \\trinorme{1,\\infty,t}{U}^2 & \\leq & C(\\norme{4}{U(0)}^2+\\norme{4}{\\derp{z}U(0)}^2) \\\\\n & & \\hspace{15pt} + Ct\\left(\\trinorme{5,t}{U}^2+\\trinorme{4,t}{\\derp{z}u_\\tau}^2+\\int_0^t \\norme{5}{\\derp{z}u_3}^2+\\norme{5}{\\derp{z}\\rho}^2~ds\\right) \\end{eqnarray*} \\label{linfu} \\end{esti}\n\nWe can inject this into \\propref{hmee} and partially close the energy estimate for $m-1\\geq 5$. We widely use the Young inequality to separate product terms, and we get the following.\n\n\\begin{esti} For every $m\\geq 6$, there exists a constant $C>1$ and a positive increasing function $Q:\\mathbb{R}^{+}\\rightarrow \\mathbb{R}^{+}$ such that\n\\begin{eqnarray*} \\norme{m}{U(t)}^2  & \\leq & C\\norme{m}{U(0)}^2 + t Q(\\trinorme{m,t}{U}^2+\\trinorme{m-1,t}{\\derp{z}u_\\tau}^2+\\trinorme{m,t}{F}^2+\\trinorme{m-1,t}{\\nabla F}^2) \\\\\n & & \\hspace{15pt} +C \\trinorme{\\infty,t}{\\derp{z}U}^2\\int_0^t \\norme{m-1}{\\derp{z}\\rho(s)}^2 + \\norme{m-1}{\\derp{z}u_3(s)}^2 ~ds. \\end{eqnarray*}\nThus, under the conditions of \\assuref{assu1}, we have, for $t\\leq T^*$,\n$$ \\trinorme{m,t}{U}^2 + \\varepsilon \\int_0^t \\norme{m}{\\nabla u}^2 \\leq M_0 + Q(M+M_F)\\left(t + \\trinorme{\\infty,t}{\\derp{z}U}^2\\int_0^t \\norme{m-1}{\\derp{z}\\rho(s)}^2+\\norme{m-1}{\\derp{z}u_3(s)}^2~ds\\right). $$ \\label{hmeeinf} \\end{esti}\n\n\\section[Proof of \\thref{thEE}, part II: \\textit{a priori} estimates on $\\derp{z}u_\\tau$]{Proof of \\thref{thEE}, part II \\\\ \\textit{A priori} estimates on $\\derp{z}u_\\tau$\n \\sectionmark{Proof of \\thref{thEE} part II: a priori estimates on $\\derp{z}u_\\tau$}}\n\\sectionmark{Proof of \\thref{thEE} part II: a priori estimates on $\\derp{z}u_\\tau$}\n\n\\subsection{Conormal energy estimates}\n\nIn this section, we get estimates on the tangential components of $\\derp{z}u$. We will perform conormal energy estimates on the first two components of the equation of the vorticity\n$$ \\omega=\\overrightarrow{\\mathrm{rot}~} u = \\left( \\begin{array}{c} \\derp{y_{2}}u_{3}-\\derp{z}u_{2} \\\\ \\derp{z}u_{1}-\\derp{y_{1}}u_{3} \\\\ \\derp{y_{1}}u_{2}-\\derp{y_{2}}u_{1} \\end{array} \\right). $$\nBy applying derivations to equation (\\ref{NS}), we get that $w=\\omega_\\tau$ solves the following equation:\n\\begin{equation} \\rho\\derp{t}w + \\rho u\\cdot\\nabla w - \\varepsilon\\mu\\Delta w = {\\cal M} , \\label{roteqn} \\end{equation}\n$$ \\mathrm{where} \\hspace{20pt} {\\cal M}=-\\rho\\omega\\cdot\\nabla u + \\rho(\\div u)\\omega + \\overrightarrow{\\mathrm{rot}~} (\\rho F) + {\\cal M}^I + \\varepsilon {\\cal M}^{II}. $$\nThe remainder ${\\cal M}^I$ is a sum of terms written as $\\derp{x_i}\\rho(\\derp{t}u_j+u\\cdot\\nabla u_j)e_j$, and ${\\cal M}^{II}$ is a sum of terms written as $\\kappa'(\\rho) \\derp{x_j}\\rho (\\derp{x_k,x_l}^2 u_j)e_j$, with $i,j,k,l\\in\\{1,2,3\\}$, $i\\neq j$, $(e_1,e_2,e_3)$ is the canonical basis of $\\mathbb{R}^3$, and $\\kappa$ is either $\\lambda$ or $\\mu$. \nThe terms in ${\\cal M}^{II}$ come from the derivation of the laplacian terms, but also from $\\overrightarrow{\\mathrm{rot}~} \\sigma$, in which there are no terms with two derivatives on $\\rho$ thanks to $\\overrightarrow{\\mathrm{rot}~}\\nabla=0$. \nBut the boundary condition on $\\omega_\\tau$, according to (\\ref{NBC}) and the non-penetration condition $u_{3}|_{z=0}=0$, is\n$$ \\mu(\\rho)\\omega_\\tau|_{z=0} = 2a u_\\tau^{\\bot}|_{z=0}, $$\nwhere for $v=(v_{1},v_{2})\\in\\mathbb{R}^{2}$, $v^{\\bot} = (-v_{2},v_{1})$, which makes integrations by parts difficult. So we introduce a modified vorticity:\n$$ W = \\omega_\\tau - \\frac{2a}{\\mu(\\rho)} u_\\tau^{\\bot} $$\nBy the tame estimate, we can harmlessly identify conormal Sobolev norms of $W$ with those of $\\omega_\\tau$ and $\\derp{z}u_\\tau$. The modified vorticity satisfies $W=0$ on the boundary, and solves the equation\n\\begin{equation} \\rho\\derp{t}W + \\rho u\\cdot\\nabla W - \\varepsilon\\mu\\Delta W = H, \\label{vorteqn} \\end{equation}\n$$ \\mathrm{with} \\hspace{10pt} H=2a[(\\rho\\derp{t}+\\rho u\\cdot\\nabla - \\varepsilon\\mu\\Delta),\\mu^{-1}] u_\\tau^\\bot-2a\\varepsilon\\mu^{-1}\\lambda\\nabla^{\\bot}_\\tau\\div u \\hspace{70pt}  $$\n$$ \\hspace{70pt} + 2ak\\mu^{-1}\\rho^{\\gamma-1}\\nabla^{\\bot}_\\tau\\rho -2a\\mu^{-1}\\rho F_\\tau^\\bot + {\\cal M}_\\tau . $$\nWe now prove the following.\n\n\\begin{esti} The modified vorticity $W$ satisfies the following conormal energy estimate for every $m\\geq 1$:\n$$ c\\left[\\norme{m-1}{W(t)}^{2} + \\varepsilon\\int_{0}^{t} \\norme{m-1}{\\nabla W(s)}^{2}~ds\\right] \\leq C\\left[\\norme{m-1}{W(0)}^{2} + \\norme{m}{U(0)}^{2}\\right] \\hspace{110pt} $$\n\\begin{equation} \\hspace{20pt} + Q(\\trinorme{\\mathrm{Lip},t}{U}^2+\\trinorme{1,\\infty,t}{\\nabla U}^2+\\trinorme{\\infty,t}{\\nabla F}^2) \\int_0^t (\\norme{m}{U}^{2} + \\norme{m-1}{\\derp{z}U}^{2} + \\norme{m}{F}^2 + \\norme{m-1}{\\nabla F_\\tau}^2)~ds, \\label{hmvfinal} \\end{equation}\nfor some positive increasing function $Q(z)$. \\label{hmvort} \\end{esti}\n\n\\preu{Proof:} we repeat the reasoning of section 4.1 on the above equation. We start by taking the $L^{2}$ scalar product of (\\ref{vorteqn}) with $W$ and integrate by parts to get\n$$ c\\norme{0}{W(t)}^{2} + \\varepsilon\\mu\\int_{0}^{t}\\norme{0}{\\nabla W(s)}^{2}~ds \\leq C\\norme{0}{W(0)}^{2} \\hspace{125pt} $$\n\\begin{equation} \\hspace{80pt} +C\\int_{0}^{t} \\left[\\trinorme{\\infty,t}{\\div(\\rho u)}\\norme{0}{W(s)}^{2} + \\int_\\Omega H(s)\\cdot W(s) \\right]~ ds. \\label{l2vort} \\end{equation}\nWe notice that, thanks to the compressibility equation (\\ref{NSdiv}), $\\trinorme{\\infty,t}{\\div(\\rho u)}$ can be replaced by $\\trinorme{\\infty,t}{\\derp{t}\\rho}\\leq \\trinorme{1,\\infty,t}{\\rho}$. It remains to estimate $\\int_\\Omega H\\cdot W$. \nThe only terms we need this scalar product form for are the order-two terms in ${\\cal M}^{II}$, and $\\varepsilon \\int_\\Omega \\mu^{-1} (\\Delta \\mu) u_\\tau\\cdot W$, which comes from the commutator $\\varepsilon \\mu [\\Delta,\\mu^{-1}]u_\\tau$. After integrating $\\int_\\Omega \\mu^{-1} (\\Delta \\mu) u_\\tau\\cdot W$ by parts, we can simply bound $\\mu$ and $\\nabla \\mu$ in $L^\\infty$ and use Young's inequality on the terms involving $u$ and $W$. \nThe term $\\int_\\Omega {\\cal M}^{II} \\cdot W$ is bounded by integration by parts on the variable $x_k$ with $k\\neq 3$ and the use of Young's inequality with an adequate to absorb $\\norme{0}{\\nabla W}$ whenever it appears, while the terms containing $\\derp{x_k,x_i}^2\\rho$ are controlled by bounding the order-two part on $\\rho$ in $L^\\infty$ (thus getting $\\\nnorme{1,\\infty}{\\nabla \\rho}$). When $(k,l)=(3,3)$, we use (\\ref{NS}) to replace $\\varepsilon\\derp{zz}u$. For example,\n$$ \\varepsilon\\mu\\derp{zz}u_1 = \\rho\\derp{t}u_1 + (\\rho u \\cdot\\nabla) u_1 + \\derp{y_1}P(\\rho) - \\rho F_1 - \\varepsilon [\\mu (\\derp{y_1 y_1}+\\derp{y_2 y_2})u_1 + \\lambda \\derp{y_1}\\div u] , $$\nand the only difficulty is to control $\\varepsilon\\int_0^t \\norme{1}{\\derp{z}u_1}^2$, but this is given by \\propref{hmee}.\n\nWe can use the Cauchy-Schwarz inequality and handle the norm of the rest of $H$ as follows:\n\\begin{itemize}\n\\item the other commutators with $\\mu^{-1}$ are controlled by $Q(1+\\norme{\\mathrm{Lip}}{U})(\\norme{1}{U}+\\norme{0}{W})$,\n\\item $\\norme{0}{\\rho^{\\gamma-1}\\nabla_\\tau^{\\bot}\\rho} \\leq C\\norme{1}{U}$ as $\\rho$ is assumed to be uniformly bounded,\n\\item $\\norme{0}{\\rho\\omega\\cdot\\nabla u_\\tau + \\rho(\\div u)\\omega} \\leq C\\norme{\\infty}{\\nabla u}(\\norme{0}{W}+\\norme{0}{u})$,\n\\item ${\\cal M}^I$ is bounded using $\\norme{0}{\\derp{x_{i}}\\rho(\\derp{t}u_j+u\\cdot\\nabla u_j)} \\leq \\trinorme{\\infty,t}{\\nabla \\rho}(\\norme{1}{u}+\\trinorme{\\infty,t}{\\nabla u}\\norme{0}{u})$,\n\\item and finally, we have $\\varepsilon\\norme{0}{\\nabla_\\tau \\div u}\\norme{0}{W} \\leq \\varepsilon(\\norme{1}{\\nabla u}^{2}+\\norme{0}{W}^{2})$ and use \\estiref{hmee} to control $\\varepsilon\\norme{1}{\\nabla u}$,\n$$ \\varepsilon\\int_{0}^{t}\\norme{1}{\\nabla u(s)}^{2}~ds \\leq C\\left(\\norme{1}{U(0)}^{2}+(1+\\trinorme{\\mathrm{Lip},t}{U}^{2})\\int_{0}^{t}(\\norme{1}{U(s)}^{2} + \\norme{0}{\\nabla U(s)}^{2})~ds\\right). $$\n\\end{itemize}\n\nWe thus get the final $L^{2}$ estimate on $W$:\n$$ c\\left[\\norme{0}{W(t)}^{2}+\\varepsilon\\int_{0}^{t}\\norme{0}{\\nabla W(s)}^{2}~ds\\right] \\leq C(\\norme{0}{W(0)}^{2} + \\norme{1}{U(0)}^{2}) \\hspace{120pt} $$\n$$ \\hspace{60pt} +Q(1+\\trinorme{\\mathrm{Lip},t}{U}^{2}+\\trinorme{1,\\infty,t}{\\nabla U}^2)\\int_{0}^{t} \\norme{1}{U}^{2} + \\norme{0}{\\derp{z}U}^{2}+\\norme{1}{F}^2 + \\norme{0}{\\nabla F}^2~ds . $$\n\nNow we move on to the $H^{m-1}_{co}$ estimate with $m>1$, which will also follow the same pattern as the previous section. We apply $Z^{\\alpha}$ to (\\ref{vorteqn}), for $|\\alpha|\\leq m-1$ and isolate the maximum order terms:\n$$ \\rho\\derp{t}Z^{\\alpha}W + \\rho u\\cdot \\nabla Z^{\\alpha} W - \\varepsilon\\mu\\Delta Z^{\\alpha}W = Z^{\\alpha}H - {\\cal C}^{\\alpha}_{W}, $$\nwhere ${\\cal C}^{\\alpha}_{W}= [Z^{\\alpha},\\rho]\\derp{t}W+[Z^{\\alpha},\\rho u\\cdot\\nabla]W - \\varepsilon[Z^\\alpha,\\mu\\Delta]W$ contains the commutators. Multiplying the above equation by $Z^\\alpha W$ in $L^{2}$, we have\n$$ c\\left[\\norme{0}{Z^{\\alpha}W(t)}^{2} + \\varepsilon\\int_{0}^{t}\\norme{0}{Z^{\\alpha}\\nabla W}^{2}~ds\\right] \\leq C\\left[\\norme{0}{Z^\\alpha W(0)}^{2} + \\trinorme{1,\\infty,t}{\\rho}\\int_0^t \\norme{m-1}{W}^2~ds\\right] \\hspace{50pt} $$\n\\begin{equation} \\hspace{60pt} + C\\varepsilon\\int_{0}^{t} \\norme{m-2}{\\nabla W}^2~ds + C\\int_0^t \\left[\\norme{m-1}{H}^2+\\int_\\Omega |{\\cal C}^{\\alpha}_{W}\\cdot Z^\\alpha W|\\right] ~ds . \\label{hmvort1} \\end{equation}\n\nThe terms on the second line of this inequality are the ones we need to control. \nFirst, $\\varepsilon\\int_0^t \\norme{m-2}{\\nabla W}^2$, which comes from changing $\\nabla Z^\\alpha W$ into $Z^\\alpha \\nabla W$ (using \\propref{comprop}) in the integration by parts on the laplacian, is dealt with by induction, by using (\\ref{hmvfinal}) at a lower order. \nThen, $\\norme{m-1}{H}$ is easily estimated as above, also using the tame estimate in \\propref{tame}:\n\\begin{eqnarray*} \\int_0^t \\norme{m-1}{H(s)}^2~ds & \\leq & \\varepsilon\\int_0^t \\norme{m}{\\nabla u}^{2}~ds + Q(\\trinorme{\\infty,t}{U}^2+ \\trinorme{1,\\infty,t}{\\nabla U}^2+\\trinorme{\\infty,t}{\\nabla F}^2) \\\\\n & & \\hspace{10pt} \\times \\left(\\int_0^t \\norme{m}{U}^{2}+\\norme{m-1}{\\derp{z}U}^{2} + \\norme{m}{F}^2+ \\norme{m-1}{\\nabla F_\\tau}^2~ds\\right) . \\end{eqnarray*}\nUsing \\estiref{hmee} to cover the worst term $\\varepsilon\\int_{0}^{t}\\norme{m}{\\nabla u(s)}^{2}~ds$, we have\n\\begin{eqnarray} \\int_0^t \\norme{m-1}{H(s)}^2~ds & \\leq & C\\norme{m}{U(0)}^2 + Q(\\trinorme{\\infty,t}{U}^2+\\trinorme{1,\\infty,t}{\\nabla U}^2+\\trinorme{\\infty,t}{\\nabla F}^2) \\nonumber \\\\\n & & \\hspace{10pt} \\times \\int_0^t \\norme{m}{U}^2+\\norme{m-1}{\\derp{z}U}^2 + \\norme{m}{F}^2 + \\norme{m-1}{\\nabla F}^2~ds . \\label{hmvrhs} \\end{eqnarray}\n\nWe finally need to estimate the commutators in $\\norme{0}{{\\cal C}^{\\alpha}_{W}}$. The first and last terms of ${\\cal C}^{\\alpha}_{W}$ are easily estimated as in the previous section,\n\\begin{equation} \\int_0^t \\norme{0}{[Z^{\\alpha},\\rho]\\derp{t}W}^2~ds \\leq C\\int_0^t (\\trinorme{\\infty,t}{\\rho}^2+\\trinorme{\\infty,t}{\\derp{t}W}^2)(\\norme{m-1}{W}^2+\\norme{m-1}{U}^2)~ds \\label{hmcom1} \\end{equation}\nby \\propref{funcprop}, and again using the decomposition in \\propref{comprop} (b) and integrating by parts, we have\n\\begin{equation} \\varepsilon \\int_{\\Omega} |[Z^\\alpha,\\mu\\Delta]W\\cdot Z^{\\alpha} W| \\leq \\frac{1+\\trinorme{\\mathrm{Lip},t}{\\rho}}{c}\\varepsilon\\norme{m-2}{\\derp{z}W}^{2}+\\frac{c}{4}\\varepsilon\\norme{m-1}{\\derp{z}W}^{2}, \\label{hmcom2} \\end{equation}\nin which the first term is estimated by induction (using (\\ref{hmvfinal}) at order $m-2$) and the second is absorbed by the left-hand side of (\\ref{hmvort1}). \nFinally, instead of applying the commutator results we have used so far to $\\int_0^t \\norme{0}{[Z^\\alpha, \\rho u\\cdot\\nabla] W}^2$, which would yield a $\\norme{m-2}{\\derp{zz}^2 U}$ term that we do not expect to control uniformly in $\\varepsilon$, we use \\propref{comlem}:\n$$ \\int_{0}^{t}\\norme{0}{[Z^{\\alpha},(\\rho u)(s)\\cdot\\nabla]W(s)}^{2}~ds \\leq \\int_{0}^{t}Q(\\trinorme{\\mathrm{Lip},t}{U}^{2}+\\trinorme{1,\\infty,t}{\\nabla U}^{2})\\norme{m-1}{W(s)}^{2}~ds $$\n$$ \\hspace{135pt} +\\int_{0}^{t}Q(\\trinorme{1,\\infty,t}{\\nabla U}^{2})\\norme{m-1}{\\nabla U(s)}^{2}~ds. $$\nThis finishes off the proof of \\estiref{hmvort}. $\\square$\n\n\\subsection{$L^\\infty$ estimates}\n\nTo deal with $\\norme{1,\\infty}{\\derp{z}u_\\tau}$, we examine $\\omega_\\tau$. The result is:\n\\begin{esti} There exists a positive, increasing function on $\\mathbb{R}^{+}$, $Q$, such that\n$$ \\norme{1,\\infty}{\\omega_\\tau(t)}^2 \\leq Q(\\norme{6}{U(0)}^2+\\norme{5}{\\omega_\\tau(0)}^2+\\norme{1,\\infty}{\\omega_\\tau(0)}^2) \\hspace{170pt} $$\n$$ \\hspace{30pt} + Q(\\trinorme{\\mathrm{Lip},t}{U}^2+\\trinorme{1,\\infty,t}{\\nabla U}^2+{\\cal N}_6(t,F))\\int_0^t 1+\\norme{6}{U(s)}^2+\\norme{6}{\\omega_\\tau(s)}^2+\\norme{5}{\\derp{z}\\rho(s)}^2~ds. $$  \\label{linfvort} \\end{esti}\n\nWe can now deduce the following update of \\propref{hmeeinf}.\n\\begin{coro} Under the conditions of \\assuref{assu1}, for $m\\geq 7$ and $t\\leq T^*$,\n$$ \\trinorme{m,t}{U}^2+\\trinorme{m-1,t}{\\nabla u_\\tau}^2+\\trinorme{1,\\infty,t}{\\nabla u_\\tau}^2 + \\varepsilon\\int_0^t \\norme{m}{\\nabla u}^2 + \\norme{m-1}{\\nabla W}^2~ds \\hspace{70pt} $$\n$$ \\hspace{80pt} \\leq Q(M_0) + Q(M+M_F)\\left(t+ \\trinorme{1,\\infty,t}{\\derp{z}(\\rho,u_3)}^2 \\int_0^t \\norme{m-1}{\\derp{z}\\rho}^2+\\norme{m-1}{\\derp{z}u_3}^2~ds\\right). $$\n\\label{uvortbd} \\end{coro}\n\nThe first tool to prove \\propref{linfvort} is the following version of the maximum principle.\n\\begin{propo} Consider $X$ a Lipschitz-class solution to the following hyperbolic-parabolic system on the half-space $\\Omega\\subset\\mathbb{R}^3$:\n\\begin{equation} \\left\\{ \\begin{array}{rcll} a\\derp{t}X + b\\cdot\\nabla X - \\varepsilon\\mu(a)\\Delta X & = & G & \\mathrm{in}~\\Omega \\\\\nX & = & h & \\mathrm{on}~\\partial\\Omega \\\\ X|_{t=0} & = & X_0\\in L^\\infty(\\Omega), & \\end{array} \\right. \\label{mpsys} \\end{equation}\nin which $X:\\mathbb{R}^{+}\\times\\Omega\\rightarrow \\mathbb{R}^d$, $a$ is a scalar function, bounded and positive uniformly in time ($\\inf_{(t,x)} a(t,x)>c>0$), $b:\\mathbb{R}^{+}\\times\\Omega\\rightarrow\\mathbb{R}^3$ is tangent to the boundary, $G:\\mathbb{R}^{+}\\times\\Omega\\rightarrow\\mathbb{R}^d$ and $h:\\mathbb{R}^{+}\\times \\mathbb{R}^{2} \\rightarrow \\mathbb{R}^d$ are in $L^\\infty$ and $\\mu$ is a positive regular function of $a$. \nWe assume that $a$ and $b$ satisfy $\\derp{t}a+\\div b = 0$. Then we have, for $t\\in [0,T]$,\n$$ \\norme{\\infty}{X(t)} \\leq \\norme{\\infty}{X(0)} + \\trinorme{\\infty,T}{h} + \\frac{1}{\\inf a}\\int_0^t \\norme{\\infty}{G(s)} + \\varepsilon\\norme{\\infty}{\\mu'(a(s))\\nabla a(s) \\cdot \\nabla X(s)} ~ds. $$ \\label{mp1} \\end{propo}\n\nNote that when $\\mu$ is constant, the term involving $\\nabla(\\mu(a))$ vanishes on the right-hand side. \nIn fact, in that case we can directly apply the result to the modified vorticity $W$ instead of $\\omega$, which has the further advantage of satisfying a homogeneous boundary condition ($h=0$). \nIn the non-constant case however, if we consider $W$, we cannot deal with the term $\\derp{zz}\\rho$ that emerges from the commutator $[\\Delta,\\mu^{-1}]$, so we will use the proposition on $\\omega_\\tau$. \nHaving $\\nabla X$ in the right-hand side looks disastrous, but in the context of the estimation process, the factor $\\varepsilon$ intervenes crucially.\n\\newline\n\n\\preu{Proof:} we look for a function $g(t)$ that will control $\\norme{\\infty}{X(t)}$. To do so, we perform energy estimates on the functions $(X-g)_+$ and $(X+g)_-$ ($g$ will indifferently designate the scalar function $g$ and the vector $g\\mathbf{1}\\in\\mathbb{R}^d$), with the convention that, for a given scalar function $f$, $f=f_+ + f_-$ (so $f_-\\leq 0$).\n\nWe concentrate on $(X-g)_+$, the procedure on $(X+g)_-$ being identical. We want to find a function $g(t)$ that satisfies\n$$ \\norme{0}{(X(0)-g(0))_+} = 0 ~~\\mathrm{and}~~ \\frac{d}{dt}\\norme{0}{(X(t)-g(t))_+}^2 \\leq 0,~t> 0; $$\nsuch a function $g$ is an upper bound for $\\norme{\\infty}{X_+}$. Choosing $g(0)=\\norme{\\infty}{X(0)}$, we examine\n\\begin{eqnarray} \\frac{1}{2}\\frac{d}{dt}\\left(\\int_\\Omega a|(X(t)-g(t))_+|^2\\right) & = & \\int_\\Omega a\\derp{t}X\\cdot(X-g)_+ - \\int_\\Omega a\\derp{t}g\\cdot(X-g)_+ \\nonumber \\\\\n & & \\hspace{10pt} + \\frac{1}{2} \\int_\\Omega \\derp{t}a |(X-g)_+|^2 \\nonumber \\\\\n & = & \\int_\\Omega (-b\\cdot\\nabla X + \\varepsilon\\mu\\Delta X + G)\\cdot(X-g)_+ \\nonumber \\\\\n & & \\hspace{10pt} + \\int_\\Omega -a\\derp{t}g\\cdot(X-g)_+ + \\frac{1}{2}\\derp{t}a|(X-g)_+|^2 \\label{mpee} \\end{eqnarray}\nBecause of the scalar product with $(X-g)_+$, which is zero wherever $X-g\\leq 0$, we can replace $\\nabla X$ by $\\nabla ((X-g)_+)$ and $\\Delta X$ by $\\Delta((X-g)_+)$. Then, integration by parts gives us\n$$ -\\int_\\Omega b\\cdot\\nabla X\\cdot(X-g)_+ = \\frac{1}{2}\\int_\\Omega \\div b |(X-g)_+|^2 = -\\frac{1}{2} \\int_\\Omega \\derp{t}a|(X-g)_+|^2, $$\nby the first line of (\\ref{mpsys}), so this term cancels out with the final term of (\\ref{mpee}), and it remains to look at the term involving the laplacian. When integrating by parts, we need to guarantee that $(X-g)_+$ vanishes on the boundary. We therefore impose the condition\n\\begin{equation} g(t) \\geq \\trinorme{L^\\infty(\\partial\\Omega),T}{h} \\label{gbc} \\end{equation}\nand write\n\\begin{eqnarray*} \\varepsilon\\mu\\int_\\Omega \\Delta X\\cdot(X-g)_+ & = & -\\varepsilon\\mu\\norme{0}{\\nabla ((X-g)_+)}^2 - \\varepsilon \\int_\\Omega \\nabla (\\mu(a))\\cdot\\nabla X\\cdot (X-g)_+ \\\\\n & \\leq & -\\varepsilon \\int_\\Omega \\mu'(a) \\nabla a\\cdot \\nabla X\\cdot (X-g)_+ \\end{eqnarray*}\nTherefore, setting $\\til{G} = G-\\varepsilon \\mu'(a) \\nabla X\\cdot\\nabla a$, of (\\ref{mpee}) there only remains\n$$ \\frac{d}{dt}\\left(\\int_\\Omega a|(X(t)-g(t))_+|^2\\right) \\leq 2\\int_\\Omega (\\til{G}-a\\derp{t}g)\\cdot(X-g)_+, $$\nwhich is negative if, for each $j\\in\\{1,\\cdots,p\\}$, $\\til{G}_j-a\\derp{t}g\\leq 0$ on $\\Omega$. Integrating this leads to $g(t)\\geq g(0)+\\int_0^t \\frac{\\til{G}_j(s,x)}{a(s,x)}~ds$ for every $x\\in\\Omega$.\n\nIdentical estimates on $(X+g)_-$ lead to $g(t)\\geq g(0)-\\int_0^t \\frac{\\til{G}_j(s,x)}{a(s,x)}~ds$, so, also taking into account (\\ref{gbc}), we choose\n$$ g(t) = \\norme{\\infty}{X(0)} + \\trinorme{L^{\\infty}(\\partial\\Omega),T}{h} + \\int_0^t \\norme{\\infty}{\\frac{\\til{G}(s)}{a(s)}}~ds, $$\nand this controls $\\norme{\\infty}{X(t)}$ as desired. $\\square$\n\\newline\n\n\\preu{Proof of \\propref{linfvort}:} we immediately apply \\propref{mp1} to $w:=\\omega_\\tau$ and $Z_j w$, to get\n$$ \\norme{1,\\infty}{w(t)} \\leq \\norme{1,\\infty}{w(0)} + 2a\\trinorme{1,\\infty,t}{\\mu^{-1} u_\\tau} + \\int_0^t \\norme{1,\\infty}{\\frac{{\\cal M}_\\tau(s)}{\\rho(s)}} + \\sum_{j=0}^3\\norme{\\infty}{{\\cal C}^j_w(s)}~ds, $$\nwhere ${\\cal C}_w^j=[Z_j,\\rho]\\derp{t}w+[Z_j,\\rho u\\cdot\\nabla] w - \\varepsilon[Z_j,\\mu\\Delta]w$ are the commutator terms that appear in the equations on $Z_j w$. Thus, by using the Cauchy-Schwarz and Young inequalities,\n\\begin{equation} \\norme{1,\\infty}{w(t)}^2 \\leq \\norme{1,\\infty}{w(0)}^2 + 4a^2\\trinorme{1,\\infty,t}{\\mu u_\\tau}^2 + t\\int_0^t \\norme{1,\\infty}{\\frac{{\\cal M}_\\tau(s)}{\\rho(s)}}^2+\\sum_{j=0}^3 \\norme{\\infty}{{\\cal C}^j_w(s)}^2~ds . \\label{inftyv1} \\end{equation}\nNote that a factor $t$ has been extracted in front of the source terms and commutators, so we can be satisfied with fairly crude bounds on these. The $W^{1,\\infty}_{co}$ term stemming from the boundary is easily dealt with using \\propref{sobo}.\n\\newline\n\nLet us start with the control of the commutators. Given that, by (\\ref{commul}),\n$$ \\varepsilon\\mu[Z_3,\\derp{zz}]w=-2\\varepsilon\\mu\\phi'\\derp{zz}w-\\varepsilon\\mu\\phi''\\derp{z}w , $$\nwe replace $\\varepsilon\\mu\\derp{zz}w$ by its expression in the equation. So we write, using the notation $\\Delta_\\tau = \\derp{y_1 y_1}+\\derp{y_2 y_2}$,\n\\begin{eqnarray*} \\varepsilon^2 \\mu^2\\norme{\\infty}{[Z_3,\\derp{zz}]w}^2 & \\leq & C\\varepsilon^2\\norme{\\infty}{\\derp{z}w}^2 + C\\norme{\\infty}{\\rho\\derp{t}w+\\rho u\\cdot\\nabla w - \\varepsilon\\mu\\Delta_\\tau w - {\\cal M}_\\tau}^2 \\\\\n & \\leq & C\\varepsilon^2\\norme{\\infty}{\\derp{z}w}^2 + C(\\norme{1,\\infty}{\\rho}^2+\\norme{\\mathrm{Lip}}{u}^2\\norme{1,\\infty}{w}^2) \\\\\n & & \\hspace{30pt} + C\\varepsilon^2\\norme{2,\\infty}{w}+C\\norme{\\infty}{{\\cal M}_\\tau}^2, \\end{eqnarray*}\nby using the fact that $\\norme{\\infty}{u_3\\derp{z}w} \\leq C\\norme{\\infty}{\\derp{z}u_3}\\norme{\\infty}{Z_3 w}$. \nWe see that $\\norme{\\infty}{{\\cal C}_w^3}$ is bounded, among other terms, by $\\norme{\\infty}{{\\cal M}_\\tau}$; we'll examine the source term last. \nIn this commutator, it remains to control\n$$ J:=\\int_0^t \\varepsilon^2 (\\norme{\\infty}{\\derp{z}w(s)}^2+\\norme{2,\\infty}{w(s)}^2)~ds . $$\nOn both parts of this term, we start by applying \\propref{sobo}, the Sobolev embedding theorem, to get\n$$ J \\leq C\\int_0^t \\varepsilon^{2}(\\norme{3}{\\derp{zz}w(s)}^2+\\norme{5}{\\derp{z}w(s)}^2+\\norme{5}{w(s)}^2)~ds. $$\nThe second term, $\\int_0^t \\varepsilon^2\\norme{5}{\\derp{z}w}^2$, is controlled by using \\estiref{hmvort}, and, like above, the first term, $\\int_0^t \\norme{3}{\\varepsilon\\derp{zz}w}^2$, is controlled by replacing $\\varepsilon\\derp{zz}w$ by its expression in (\\ref{roteqn}), and by using the tame estimate:\n\\begin{eqnarray*} \\int_0^t \\norme{3}{\\varepsilon\\derp{zz}w(s)}^2~ds & \\leq & C \\int_0^t (\\trinorme{\\infty,t}{\\rho}^2+\\trinorme{\\infty,t}{\\rho u}^2)\\norme{4}{w}^2+\\trinorme{1,\\infty,t}{w}^2\\norme{3}{U}^2 \\\\\n& & \\hspace{50pt} + \\norme{3}{\\rho u_3 \\derp{z} w} + \\norme{5}{w}^2 + \\norme{3}{{\\cal M}_\\tau}^2 ~ ds \\end{eqnarray*}\nThe source term norm $\\int_0^t \\norme{3}{{\\cal M}_\\tau(s)}^2~ds$ is given by (\\ref{hmvrhs}), and $\\int_0^t \\norme{3}{\\rho u_3 \\derp{z} w}^2~ds$ is controlled in the same way as is done in \\propref{comlem}:\n\\begin{eqnarray*} \\int_0^t \\norme{3}{(\\rho u_3 \\derp{z} w)(s)}^2~ds & = & \\int_0^t \\norme{3}{\\frac{(\\rho u_3)(s)}{\\phi} Z_3 w(s)}^2~ds \\\\\n & \\leq & C \\int_0^t \\trinorme{1,\\infty,t}{w}^2 \\norme{4}{\\derp{z}(\\rho u)}^2+(\\trinorme{\\mathrm{Lip},t}{\\rho u}^2+\\trinorme{1,\\infty,t}{\\nabla (\\rho u)}^2)\\norme{4}{w}^2~ds \\end{eqnarray*}\nThe commutator $\\varepsilon[Z_3,\\mu]\\Delta w=Z_3(\\mu(\\rho)) \\Delta w$ is controlled by replacing $\\varepsilon\\derp{zz}w$ by its expression. Therefore, in total, we have\n$$ \\varepsilon^2\\int_0^t \\norme{\\infty}{[Z_3,\\mu\\Delta]w}^2~ds \\leq C[\\norme{6}{U(0)}^2+\\norme{5}{w(0)}^2] + C\\int_0^t \\norme{\\infty}{{\\cal M}_\\tau(s)}^2~ds \\hspace{50pt} $$\n\\begin{equation} \\hspace{60pt} + C\\int_0^t Q(\\trinorme{\\mathrm{Lip},t}{U}^2+\\trinorme{1,\\infty,t}{\\nabla U}^2)(1+\\norme{6}{U}^2+\\norme{5}{w}^2+\\norme{5}{\\derp{z}\\rho}^2)~ds+{\\cal N}_6(t,F) \\label{lapcomwinf} \\end{equation}\nIt is now straightforward to notice that the whole commutator satisfies (\\ref{lapcomwinf}), the main tool to bound $\\norme{\\infty}{[Z_j,\\rho]\\derp{t}w+[Z_j,\\rho u \\cdot\\nabla]w}$ being property (\\ref{hardyinf}), as used in \\propref{comlem}.\n\\newline\n\nMoving on to the source term, we now focus on the terms in $\\varepsilon {\\cal M}^{II}$ which arise when the viscosity coefficients are not constant. First, we examine terms involving one normal derivative on $u_1$ or $u_2$. These are linked to $\\nabla w$, which we split into two parts:\n\\begin{eqnarray*} \\varepsilon \\nabla w & = & \\varepsilon\\nabla ((\\overrightarrow{\\mathrm{rot}~} u)_\\tau) = \\varepsilon \\nabla (\\derp{z} u_\\tau^\\bot - \\nabla_\\tau^\\bot u_3) \\\\\n & = & \\varepsilon \\left(\\begin{array}{c} \\nabla_\\tau \\\\ \\derp{z} \\end{array} \\right) \\derp{z}u_\\tau^\\bot - \\varepsilon \\nabla (\\nabla_\\tau^\\bot u_3) \\end{eqnarray*}\nWe are interested in the $L^2$-in-time $W^{1,\\infty}_{co}$ norm of this. On the terms with only one normal derivative, which are the second and the $\\nabla_\\tau$ components of the first, we apply the Sobolev embedding, \\propref{sobo}, leading to having to bound $\\norme{5}{\\varepsilon \\derp{zz}u_j}^2$ for a certain $j$; once again we can do so by replacing $\\derp{zz}u_j$ by its expression in (\\ref{NS}). \nThe remains of the first term, $\\derp{zz}u_\\tau$, are no more difficult: having replaced $\\varepsilon\\derp{zz}u_\\tau$ using (\\ref{NS}), all the subsequent terms are dealt with easily using the Sobolev embedding in most places, including on $\\varepsilon\\lambda \\nabla_\\tau \\derp{z}u_3$, which comes from the $\\nabla\\div$term, and another replacement of $\\varepsilon\\derp{zz}u_3$ completes the estimate.\n\nIn the remaining terms of $\\varepsilon {\\cal M}^{II}$, terms with two normal derivatives on $u$ can appear: we then replace them using the equation. Terms with two conormal derivatives are dealt with by using the Sobolev embedding inequality.\n\\newline\n\nEstimating the $L^2$-in-time conormal-Lipschitz norm of the rest of the source term, ${\\cal M}_\\tau$, which, we remind the reader, is the tangential component of $-\\rho\\omega\\cdot\\nabla u + \\rho(\\div u)\\omega + \\overrightarrow{\\mathrm{rot}~} (\\rho F) + {\\cal M}^I$, is straight-forward. $\\square$\n\n\\section[Proof of \\thref{thEE}, part III: \\textit{a priori} estimates on $\\derp{z}u_3$]{Proof of \\thref{thEE}, part III \\\\ \\textit{A priori} estimates on $\\derp{z}u_3$\n \\sectionmark{Proof of \\thref{thEE} part III: a priori estimates on $\\derp{z}u_3$}}\n\\sectionmark{Proof of \\thref{thEE} part III: a priori estimates on $\\derp{z}u_3$}\n\nTo begin with, we bound $\\int_0^t \\norme{m-1}{\\derp{z}u_3(s)}^2~ds$, which will essentially only need the tame estimate in \\propref{tame}. The compressibility equation (\\ref{NSdiv}) gives us\n$$ \\rho Z^\\alpha \\derp{z} u_3 = - Z^\\alpha \\derp{t}\\rho - Z^\\alpha(u\\cdot\\nabla \\rho) - Z^\\alpha(\\rho \\mathrm{div}_\\tau u_\\tau) - [Z^\\alpha,\\rho]\\derp{z}u_3, $$\nwith $|\\alpha|\\leq m-1$ and introducing the notation $\\mathrm{div}_\\tau u_\\tau = \\derp{y_1}u_1 + \\derp{y_2}u_2$. Integrating the square of this equality in time and space, we quickly get\n$$ c\\int_0^t \\norme{0}{Z^\\alpha \\derp{z}u_3(s)}^2~ds \\leq C (1+\\trinorme{\\mathrm{Lip},t}{U}^2) \\int_0^t \\norme{m}{U(s)}^2~ds \\hspace{75pt} $$\n$$ \\hspace{50pt} + C\\trinorme{\\infty,t}{u_3}^2 \\int_0^t \\norme{m-1}{\\derp{z}\\rho(s)}^2~ds + C\\trinorme{1,\\infty,t}{\\rho}^2 \\int_0^t \\norme{m-2}{\\derp{z}u_3(s)}^2~ds, $$\nusing \\propref{funcprop} on the commutator term. With this, we can perform an induction on $m>0$ and conclude that there exists a positive increasing polynomial function $Q$ such that\n$$ \\int_0^t \\norme{m-1}{\\derp{z}u_{3}(s)}^2~ds \\leq Q(1+\\trinorme{\\mathrm{Lip},t}{U}^2) \\int_0^t \\norme{m}{U(s)}^2 + \\norme{m-1}{\\derp{z}\\rho(s)}^2~ds . $$\nEssentially, this means that $H^{m-1}_{co}([0,T]\\times\\Omega)$ norms of $\\derp{z}u_3$ can be replaced by the same norms of $\\derp{z}\\rho$ and $W^{1,\\infty}_{co}$ norms of $\\derp{z}u_3$, which, in turn, will be bounded by $H^{m_0}_{co}$ norms for a certain $m_0$, so we need to extract a small parameter, in this case $t$, to close the estimate. We already have\n\\begin{equation} \\int_0^t \\norme{m-1}{\\derp{z}u_3(s)}^2~ds \\leq Q(1+\\trinorme{\\mathrm{Lip},t}{U}^2)\\left(t\\trinorme{m,t}{U} + \\int_0^t \\norme{m-1}{\\derp{z}\\rho(s)}^2~ds \\right) , \\label{hmdzu3} \\end{equation}\nso we need the estimates on $\\norme{1,\\infty}{\\derp{z}u_3}$ and $\\norme{1,\\infty}{\\derp{z}\\rho}$ to yield a small parameter, $t$ or $\\varepsilon$, as we cannot use (\\ref{l2time}) on the $L^2$-in-time norm present here (by \\estiref{linfu}, $\\norme{1,\\infty}{U}$ already yields the desired parameter).\n\\newline\n\nFor now, we focus on $\\norme{1,\\infty}{\\derp{z}u_3}$. Simply reading (\\ref{NSdiv}), we have\n$$ \\trinorme{\\infty,t}{\\derp{z}u_3}^2 \\leq \\trinorme{\\infty,t}{\\derp{t}\\rho + \\mathrm{div}_\\tau (\\rho u_\\tau)}^2 + \\trinorme{\\infty,t}{u_3}^2\\trinorme{\\infty,t}{\\derp{z}\\rho}^2, $$\nto which we can apply (\\ref{l2timeinf}), and we get\n\\begin{eqnarray} \\trinorme{\\infty,t}{\\derp{z}u_3}^2 & \\leq & Q(1+\\norme{\\mathrm{Lip}}{U(0)}^2) + \\int_0^t Q(1+\\norme{2,\\infty}{U(s)}^2+\\norme{1,\\infty}{\\derp{z}\\rho(s)}^2)~ds \\nonumber \\\\\n & \\leq & Q(1+\\norme{\\mathrm{Lip}}{U(0)}^2) + tQ(1+\\trinorme{5,t}{U}^2+\\trinorme{4,t}{\\derp{z}U}^2 + \\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2) \\label{linfnd1} \\end{eqnarray}\nWe can now do the same for $\\norme{1,\\infty}{\\derp{z}u_3}$, bearing in mind that there is a commutator, $[Z_j,\\rho]\\derp{z}u_3 = (Z_j\\rho)\\derp{z}u_3$, and that we do not want to lose derivatives on $\\norme{1,\\infty}{\\derp{z}\\rho}$ this time:\n\\begin{eqnarray*} \\trinorme{1,\\infty,t}{\\derp{z}u_3}^2 & \\leq & \\trinorme{1,\\infty,t}{\\derp{t}\\rho + \\mathrm{div}_\\tau(\\rho u_\\tau )}^2 + \\trinorme{1,\\infty,t}{u_3}^2\\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2 + \\trinorme{1,\\infty}{\\rho}^2\\trinorme{\\infty}{\\derp{z}u_3}^2 \\\\\n & \\leq & Q(1+\\norme{2,\\infty}{U(0)}^2) + \\int_0^t \\norme{2,\\infty}{\\derp{t}\\rho+ \\mathrm{div}_\\tau (\\rho u_\\tau )}^2 \\\\\n & & \\hspace{30pt} +\\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2\\left(\\norme{1,\\infty}{u_3(0)}^2+\\int_0^t \\norme{2,\\infty}{u_3(s)}~ds\\right) \\\\\n & & \\hspace{30pt} + \\trinorme{\\infty,t}{\\derp{z}u_3}^2\\left(\\norme{1,\\infty}{\\rho(0)}^2+\\int_0^t \\norme{2,\\infty}{\\rho(s)}^2~ds\\right) \\\\\n & \\leq & Q(1+\\norme{\\mathrm{Lip}}{U(0)}^2+\\norme{1,\\infty}{\\nabla U(0)}^2) + C\\trinorme{1,\\infty,t}{\\derp{z}\\rho}^4 \\\\\n & & \\hspace{30pt}  + tQ(\\trinorme{6,t}{U}^2+\\trinorme{5,t}{\\derp{z}U}^2+\\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2) , \\end{eqnarray*}\nby (\\ref{linfnd1}), which, by applying (\\ref{l2time}) provides us with the following.\n\n\\begin{esti} (a) There exists an increasing, positive polynomial function $Q$ on $\\mathbb{R}^{+}$ such that\n$$ \\trinorme{1,\\infty,t}{\\derp{z}u_3}^2 \\leq Q(1+\\norme{\\mathrm{Lip}}{U(0)}^2+\\norme{1,\\infty}{\\nabla U(0)}^2) + Q(\\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2) \\hspace{110pt} $$\n\\begin{equation} \\hspace{40pt} + t Q\\left( 1+\\trinorme{6,t}{U}^2+\\trinorme{5,t}{\\nabla u_\\tau}^2+\\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2+\\int_0^t \\norme{6}{\\derp{z}u_3(s)}^2+\\norme{6}{\\derp{z}\\rho(s)}^2~ds \\right) \\label{linfnd} \\end{equation}\n\n(b) Combining (\\ref{hmdzu3}) and (\\ref{linfnd}), we get that, for $m\\geq 7$,\n$$ \\int_0^t \\norme{m-1}{\\derp{z}u_3(s)}^2~ds \\leq Q(1+\\norme{\\mathrm{Lip}}{U(0)}^2+\\norme{1,\\infty}{\\nabla U(0)}^2) \\hspace{150pt} $$\n$$ \\hspace{35pt} + t Q\\left(1+\\trinorme{m,t}{U}^2+\\trinorme{m-1,t}{\\nabla u_\\tau}^2 + \\trinorme{1,\\infty,t}{\\nabla u_\\tau}^2 + \\int_0^t \\norme{m-1}{\\derp{z}u_3(s)}^2~ds \\right) $$\n\\begin{equation} + Q\\left(\\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2 + \\int_0^t \\norme{m-1}{\\derp{z}\\rho(s)}^2~ds \\right) \\hspace{130pt} \\label{hmnddef} \\end{equation}\n\\label{hmnd} \\end{esti}\n\nAs a result, we can update \\cororef{uvortbd}. Set\n\\begin{eqnarray*} \\til{E}^\\varepsilon _{m}(t,U)^2 & = & \\trinorme{m}{U}^2+\\trinorme{m-1,t}{\\nabla u_\\tau}^2 + \\int_0^t \\norme{m-1}{\\derp{z}u_3(s)}^2~ds+\\trinorme{1,\\infty,t}{\\nabla u}^2 \\\\\n & & \\hspace{40pt} + \\varepsilon\\int_0^t \\norme{m}{\\nabla u}^2 + \\norme{m-1}{\\nabla W}^2~ds . \\end{eqnarray*}\nThis contains ${\\cal E}_m(t,U)$ except the terms involving $\\derp{z}\\rho$, plus the $W^{1,\\infty}_{co}$ norm of $\\derp{z}u_3$ and the gradient terms from the energy estimates. The combination of \\cororef{uvortbd} and \\propref{hmnd} (b) gives us the following.\n\\begin{coro} Under the conditions of \\assuref{assu1}, for $m\\geq 7$ and $t\\leq T^*$,\n$$ \\til{E}^\\varepsilon_m(t,U) \\leq Q(M_0) + Q(M+M_F)\\left(t + Q\\left(\\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2 + \\int_0^t \\norme{m-1}{\\derp{z}\\rho(s)}^2~ds\\right)\\right) $$ \\label{endp3} \\end{coro}\nWe see on this estimate that it remains to look at the normal derivative of the density.\n\n\\section[Proof of \\thref{thEE}, part IV: \\textit{a priori} estimates on $\\derp{z}\\rho$]{Proof of \\thref{thEE}, part IV \\\\ \\textit{A priori} estimates on $\\derp{z}\\rho$\n \\sectionmark{Proof of \\thref{thEE} part IV: a priori estimates on $\\derp{z}\\rho$}}\n\\sectionmark{Proof of \\thref{thEE} part IV: a priori estimates on $\\derp{z}\\rho$}\n\n\\subsection{Conormal energy estimates}\n\nIn this section, we examine $R:=\\derp{z}\\rho$. The equation satisfied by $R$ is given by the differentiation of (\\ref{NSdiv}):\n$$ \\derp{t}R + u\\cdot\\nabla R + R(\\div u+\\derp{z}u_3) + \\rho\\derp{zz}u_3 = - \\rho\\derp{z}\\mathrm{div}_\\tau u_\\tau - \\derp{z}u_\\tau \\cdot\\nabla_\\tau \\rho. $$\nA very problematic term appears in this equation: $\\derp{zz}u_3$. The idea is to multiply the equation by $l\\varepsilon:=(\\mu+\\lambda)\\varepsilon$, in order to replace $l\\varepsilon\\derp{zz}u_3$ by its expression in the equation,\n\\begin{equation} l\\varepsilon\\derp{zz} u_3 = \\rho \\derp{t}u_3 + \\rho u\\cdot\\nabla u_3 - \\mu\\varepsilon\\Delta_\\tau u_3 - \\lambda\\varepsilon\\derp{z}\\mathrm{div}_\\tau u_\\tau + R. \\label{NSrwR} \\end{equation}\nThis brings us to\n\\begin{equation} l\\varepsilon(\\derp{t}R + u\\cdot\\nabla R) + \\rho P'(\\rho) R = h, \\label{eqnR} \\end{equation}\nwhere we remind the reader that $\\rho P'(\\rho) = k\\rho^\\gamma$ (and $P'$ is positive; we can thus expect our results to extend to other, strictly increasing and positive pressure laws), and $h$ will be treated as a source term:\n\\begin{eqnarray*} h & = & \\varepsilon[\\rho(\\mu\\Delta_\\tau u_3+\\lambda\\derp{z}\\mathrm{div}_\\tau u_\\tau+\\sigma(\\nabla U)_3) \\\\\n & & \\hspace{20pt} -l\\derp{z}u_\\tau\\cdot\\nabla_\\tau\\rho-l(\\div u+\\derp{z}u_3)R]-\\rho^2(\\derp{t}u_3+u\\cdot\\nabla u_3)+\\rho^2 F_3. \\end{eqnarray*}\n\nIn these estimates, we will strongly use \\assuref{assu1}: for $t\\leq T^*$, we have $\\rho(t,x)\\geq c_0>0$, $|u_3(t,x)|$ is uniformly bounded near the boundary on the same time interval, and ${\\cal E}_m(t,U)\\leq M$ for some $m\\geq 7$. We use the latter assumption to simplify the presentation of what follows.\n\n\\begin{esti} Under the conditions of \\assuref{assu1}, for $m\\geq 1$, there exists a positive, increasing function on $\\mathbb{R}^{+}$, $Q$, such that, for $t\\leq T^*$ and $\\varepsilon\\leq \\varepsilon_0$,\n$$ \\int_0^t \\norme{m-1}{R(s)}^2~ds \\leq Q(\\norme{m-1}{R(0)}^2 +\\norme{m}{U(0)}^2) + (t+\\varepsilon)Q(M+M_F) + Q(\\trinorme{1,\\infty,t}{R}^2). $$\n\\label{hmeeR} \\end{esti}\n\n\\preu{Proof:} we start with the $L^2$ estimate. We multiply (\\ref{eqnR}) by $R$ and integrate in space, and, as usual, we integrate the term containing $u\\cdot\\nabla R$ by parts and get\n\\begin{equation} \\frac{l\\varepsilon}{2}\\frac{d}{dt}(\\norme{0}{R}^2) + \\gamma \\int_\\Omega \\rho P'(\\rho) R^2 = \\int_\\Omega \\left(\\frac{l\\varepsilon}{2} R^2 \\div u + \\frac{l'(\\rho)\\varepsilon}{2} R^2 u\\cdot\\nabla \\rho + hR\\right). \\label{l2eeR} \\end{equation}\nThe coefficients $l$ and $\\rho P'$ are bounded from below, so there exist $0<c<1<C$ such that the left-hand side, integrated in time, is greater than\n$$ c\\left(\\varepsilon \\norme{0}{R(t)}^2 + \\int_0^t \\norme{0}{R(s)}^2~ds\\right) -C\\norme{0}{R(0)}^2 . $$\nThen, using the Young inequality on the right-hand side of (\\ref{l2eeR}) with the parameter $\\eta=\\frac{c}{2}$, we get\n$$ c\\varepsilon \\norme{0}{R(t)}^2 + \\frac{c}{2}\\int_0^t \\norme{0}{R(s)}^2~ds \\leq C\\norme{0}{R(0)}^2 + C\\int_0^t \\varepsilon \\trinorme{\\mathrm{Lip},t}{U}\\norme{0}{R(s)}^2 + \\norme{0}{h(s)}^2~ds. $$\n\nIt only remains to bound $\\int_0^t \\norme{0}{h}^2$, and this is simple:\n$$ \\int_0^t \\norme{0}{h(s)}^2~ds \\leq C\\varepsilon^2\\int_0^t \\trinorme{\\infty,t}{\\nabla u}^2\\norme{0}{R(s)}^2 + \\norme{1}{\\nabla u(s)}^2+\\norme{1}{\\rho(s)-1}^2~ds \\hspace{50pt} $$\n$$ \\hspace{150pt} + C\\int_0^t (1+\\norme{\\infty}{\\derp{z}u_3}^2)\\norme{1}{u_3}^2 + \\norme{0}{F}^2~ds, $$\nin which $\\varepsilon\\int_0^t \\norme{1}{\\nabla u(s)}$ is bounded using \\propref{hmee}, so we have proved the energy estimate on $R$ for $m=1$.\n\\newline\n\nWe now move to the $H^{m-1}_{co}$ estimates with $m\\geq 2$, and once again, the main focus will be the commutators. Given $\\alpha$ of length $m-1$, the equation on $Z^\\alpha R$ is\n$$ l\\varepsilon(\\derp{t}Z^\\alpha R + u\\cdot\\nabla Z^\\alpha R) + \\rho P'(\\rho) Z^\\alpha R = Z^\\alpha h - {\\cal C}^\\alpha_R, $$\nwith ${\\cal C}^\\alpha_R = l\\varepsilon[Z^\\alpha, u\\cdot\\nabla] R + [Z^\\alpha, \\rho P'(\\rho)]R$. Repeating the above procedure, we reach the estimate\n$$ c\\varepsilon\\norme{0}{Z^\\alpha R(t)}^2 + c\\int_0^t \\norme{0}{Z^\\alpha R(s)}^2~ds \\leq C\\norme{m-1}{R(0)}^2 \\hspace{100pt} $$\n\\begin{equation} \\hspace{100pt} + C\\int_0^t \\varepsilon\\trinorme{\\mathrm{Lip},t}{U}\\norme{m-1}{R(s)}^2+\\norme{m-1}{h(s)}^2 + \\norme{0}{{\\cal C}^\\alpha_R(s)}^2~ds. \\label{hmndr1} \\end{equation}\n\nControlling $\\int_0^t \\norme{m-1}{h}^2$ is immediate, except for the term $\\rho^2 u_3 \\derp{z}u_3$, which figures in $\\rho^2 u\\cdot\\nabla u_3$. We start by replacing $\\rho\\derp{z}u_3$ by using (\\ref{NSdiv}),\n$$ \\rho^2 u_3 \\derp{z}u_3 = -\\rho u_3(\\derp{t}\\rho + \\rho \\mathrm{div}_\\tau u_\\tau + u_\\tau\\cdot\\nabla_\\tau \\rho + u_3\\derp{z}\\rho), $$\nand the problematic term here is clearly $\\rho u_3^2 \\derp{z}\\rho$. Directly using the tame estimate basically yields $\\int_0^t \\norme{m-1}{R}^2$, but there is no factor $\\varepsilon$ in this term to allow us to hope to absorb it. \nHowever, we have a factor $u_3$, and we use the assumption that, for a given $\\delta>0$, there exist $\\sdel{z}>0$ which does not depend on $\\varepsilon$, such that $|u_3(x)|<\\delta$ for $x$ in the strip $\\sdel{\\omega} = \\mathbb{R}^2\\times [0,\\sdel{z}]$. \nNow, we split the $H^{m-1}_{co}$ norm of $\\rho u_3^2 R$ into two parts,\n$$ \\int_0^t \\norme{m-1}{\\rho u_3^2 R(s)}^2 ~ds = \\int_0^t \\norme{m-1,\\sdel{\\omega}}{\\rho u_3^2 R(s)}^2 + \\norme{m-1,\\Omega\\backslash \\sdel{\\omega}}{\\rho u_3^2 R(s)}^2~ds. $$\nHere, for $\\omega\\subset \\Omega$, we set $\\norme{m-1,\\omega}{f(s)}$ as a sort of semi-norm of $f(s)$ restricted to $\\omega$:\n$$ \\norme{m-1,\\omega}{f(s)}^2 = \\sum_{|\\beta|\\leq m-1} \\norme{0}{(Z^\\beta f)(s) |_{\\omega}}^2 $$\nWe apply the tame estimate to both norms:\n\\begin{eqnarray*} \\int_0^t \\norme{m-1}{\\rho u_3^2 R}^2 & \\leq & C\\int_0^t \\trinorme{L^\\infty(\\sdel{\\omega}),t}{\\rho u_3^2}^2\\norme{m-1}{R}^2 + \\trinorme{\\infty,t}{R}^2\\norme{m-1}{\\rho u_3^2}^2 \\\\\n& & \\hspace{50pt} + \\trinorme{\\infty,t}{\\rho u_3^2}^2\\norme{m-1,\\Omega\\backslash\\sdel{\\omega}}{\\derp{z}\\rho}^2~ds. \\end{eqnarray*}\nThe two key terms are the first and the last. To deal with the last term, we use \\propref{comprop} to write\n$$ \\norme{m-1,\\Omega\\backslash\\sdel{\\omega}}{\\derp{z}\\rho}^2 \\leq C\\sum_{|\\beta|< m} \\norme{0}{\\derp{z} Z^\\beta \\rho}^2, $$\nand we note that, for $z\\geq \\sdel{z}$, $\\phi(z)\\geq \\phi(\\sdel{z})$, therefore\n$$ |\\derp{z} Z^\\beta \\rho(x)|\\leq \\frac{1}{\\phi(\\sdel{z})}|Z_3 Z^\\beta \\rho(x)| $$\nfor $x\\neq \\sdel{\\omega}$. This means that conormal derivatives are equivalent to standard derivatives away from the boundary, thus, $\\norme{m-1,\\Omega\\backslash\\sdel{\\omega}}{\\derp{z}\\rho}^2 \\leq \\norme{H^m_{co}(\\Omega)}{\\rho-1}^2$. \nThe first term is led by $\\norme{L^\\infty(\\sdel{\\omega})}{\\rho u_3^2}^2$, which, given the boundedness of $\\rho$ and the properties of $u_3$ on $\\sdel{\\omega}$, is bounded by $c_1^2\\delta^4$, which is smaller that $\\frac{c}{2}$ if $\\delta$ is small enough, where $c$ is the coefficient on the left-hand side of (\\ref{hmndr1}): we therefore choose $\\delta$ so that this term can be absorbed.\n\nThe other terms in $h$ are straight-forward, as we can use \\estiref{hmee} on the $\\varepsilon^2\\int_0^t \\norme{m}{\\nabla u}^2$ term that comes from the order-two terms of $h$ - and the factor $\\varepsilon^2$ is essential, as it leaves a factor $\\varepsilon$ which will allow us to close the complete estimate for $\\varepsilon$ and $t$ small. In total, we get\n$$ \\int_0^t \\norme{m-1}{h(s)}^2~ds \\leq C\\norme{m}{U(0)}^2 + \\left(\\varepsilon Q(1+\\trinorme{\\mathrm{Lip},t}{U}^2) + \\frac{c}{2}\\right)\\int_0^t \\norme{m-1}{R(s)}^2~ds $$\n\\begin{equation} \\hspace{50pt} + C\\int_0^t Q(1+\\trinorme{\\mathrm{Lip},t}{U}^2 + \\trinorme{1,\\infty,t}{\\nabla u}^2) (\\norme{m}{U(s)}^2+\\norme{m-1}{\\nabla u_\\tau(s)}^2+\\norme{m-1}{F(s)}^2)~ds \\label{hmndr2} \\end{equation}\n\nThe estimation of the commutators is also mostly straight-forward. $[Z^\\alpha,u\\cdot\\nabla]R$ can be controlled by using \\propref{comlem}, bearing in mind that there is a factor $\\varepsilon$ in front of it:\n$$ \\varepsilon^2\\int_0^t \\norme{0}{[Z^\\alpha,u\\cdot\\nabla]R}^2~ds \\leq C\\varepsilon \\int_{0}^{T}(\\trinorme{\\mathrm{Lip},t}{u}^{2}+\\trinorme{1,\\infty,t}{\\nabla u}^{2})\\norme{m-1}{R(s)}^{2}~ds \\hspace{30pt} $$\n$$ \\hspace{110pt} +C\\varepsilon \\int_{0}^{t}\\trinorme{1,\\infty,t}{R}^{2}\\norme{m-1}{\\nabla u(s)}^{2}+\\trinorme{\\infty,t}{R}^{2}\\norme{m}{u(s)}^{2}~ds. $$\nThe other commutator, $[Z^\\alpha,\\rho P'(\\rho)]$ does not have a factor $\\varepsilon$, so we need to gain a derivative on $R$ by using \\propref{funcprop},\n$$ \\int_{0}^{t}\\norme{0}{[Z^{\\alpha},\\rho^\\gamma]R}^{2}~ds \\leq C\\trinorme{1,\\infty,t}{\\rho}^{2\\gamma}\\int_{0}^{t}\\norme{m-2}{R(s)}^{2}~ds+\\trinorme{\\infty,t}{R}^{2}\\int_0^t \\norme{m-1}{(\\rho-1)(s)}^{2}~ds , $$\nand use (\\ref{l2time}) to extract a factor $t$ in the first term. This leaves us with an isolated $t\\trinorme{1,\\infty,t}{\\rho}$, to which we apply the anistropic Sobolev embedding, \\propref{sobo}. Thus,\n$$ t\\trinorme{1,\\infty,t}{\\rho}^2 \\leq Ct\\left(1+\\norme{4}{R(0)}^2 +\\trinorme{4,t}{U}^2+ \\int_0^t \\norme{5}{R(s)}^2~ds\\right) $$\nNote that it is this last inequality that restricts us to $m-1\\geq 5$. We conclude the proof of \\estiref{hmeeR} by combining (\\ref{hmndr1}), (\\ref{hmndr2}) and these bounds on the commutators. $\\square$\n\n\\subsection{$L^\\infty$ estimates}\n\nAs stated in the introduction, we will control $L^\\infty$ norms of $R$ with $L^2$-in-time bounds by virtue of (\\ref{l2timeinf}),\n\\begin{equation} \\trinorme{1,\\infty,t}{R}^2 \\leq \\norme{1,\\infty}{R(0)}^2 + C\\int_0^t \\norme{1,\\infty}{\\derp{t}R(s)}^2+\\norme{1,\\infty}{R(s)}^2~ds . \\label{l2timeinfr} \\end{equation}\nLet us define $Y_t$ the space of functions $f$ satisfying $\\norme{Y_t}{f}^2 := \\int_0^t \\norme{1,\\infty}{f(s)}^2+\\norme{1,\\infty}{\\derp{t}f(s)}^2~ds$ finite.\n\n\\begin{esti} Under the conditions of \\assuref{assu1}, there exists an increasing, positive function $Q$ on $\\mathbb{R}^{+}$ such that, for $t\\leq T^*$ and $\\varepsilon\\leq \\varepsilon_0$,\n$$ \\norme{Y_t}{R}^2 \\leq Q(M_0) + (t+\\varepsilon)Q(M+M_F) $$ \\label{infR} \\end{esti}\nThis ends the proof of \\thref{thEE} providing we can pick up \\assuref{assu1}.\n\\newline\n\n\\preu{Proof:} the main tool in this proof will be the Duhamel formula for the ordinary differential equation $\\varepsilon f' + \\til{p}f = \\til{h}$. We reach this ODE by considering $R$ along the characteristics of the transport equation $\\derp{t}R+u\\cdot\\nabla R = 0$, in other words\n$$ f(t,x) = R^X(t,x) := R(t,X(t,x)), $$\nwhere $X(t,x)$ satisfies $\\derp{t}X(t,x) = u(t,X(t,x))$ and $X(0,x)=x$. We extend the notation $g^X$ to any function $g$ as above. \nThus, we have the identity $\\derp{t}(R^X) = (\\derp{t}R+u\\cdot\\nabla R)^X$, so (\\ref{eqnR}) becomes,\n$$ \\varepsilon l(\\rho^X) \\varepsilon\\derp{t}(R^X) + \\rho^X P'(\\rho^X) R^X = h^X. $$\n\nFor the higher-order estimates, it is important to apply the conormal derivatives first, then follow the flow of $u$. So, the equation we are interested in is\n$$ \\varepsilon l(\\rho^X)\\varepsilon\\derp{t}((Z^\\alpha R)^X) + \\rho^X P'(\\rho^X) (Z^\\alpha R)^X = (Z^\\alpha h)^X - ({\\cal C}_{R}^\\alpha)^X, $$\nwith ${\\cal C}_R^\\alpha=l\\varepsilon[Z^\\alpha,u\\cdot\\nabla]R+ [Z^\\alpha,\\rho P'(\\rho)]R$ as in the previous paragraph, and $\\alpha$ is either of length $\\leq 1$, or of length 2 with $\\alpha_0\\geq 1$ (these are the $\\alpha$ that intervene in $Y_t$).\n\nTo lighten the load, we introduce the following notations: $g_\\alpha = Z^\\alpha h - {\\cal C}^\\alpha_R$,\n$$ j(s',s,x) = \\int_s^{s'} \\frac{[\\rho^X P'(\\rho^X)](\\sigma,x)}{\\varepsilon l(\\rho^X)(\\sigma,x)} ~d\\sigma ~,~\\mathrm{and}~ J(s,x) = \\int_0^s \\frac{g_\\alpha^X (\\sigma,x)}{\\varepsilon l(\\rho^X)(\\sigma,x)} e^{-j(s,\\sigma,x)}~d\\sigma. $$\nThe Duhamel formula for this equation then reads:\n$$ (Z^\\alpha R)^X(s,x)=Z^\\alpha R(0,x)e^{-j(s,0,x)} + J(s,x) . $$\nWe integrate the square of this equality in time between $0$ and $t$, which yields\n\\begin{equation} \\int_0^t [(Z^\\alpha R)^X(s,x)]^2~ds \\leq C\\left[\\norme{\\infty}{Z^\\alpha R(0)}^2\\int_0^t e^{-2j(s,0,x)}~ds + \\int_0^t J^{2}(s,x)~ds\\right] . \\label{duhamel2} \\end{equation}\nAgain, we use the uniform bounds on $\\rho$ to get, for any $s,~s' \\in \\mathbb{R}^{+}$, $x\\in\\Omega$,\n\\begin{equation} j(s',s,x) \\geq \\int_s^{s'} \\frac{c}{\\varepsilon} = \\frac{c(s'-s)}{\\varepsilon}. \\label{jbelow} \\end{equation}\nThus,\n\\begin{equation} \\int_0^t e^{-cj(s,0,x)}~ds \\leq C\\varepsilon \\label{jest} , \\end{equation}\nwhich deals with the first term of the right-hand side of (\\ref{duhamel2}). For the second term, we use (\\ref{jbelow}) again and write the integral to reveal a convolution in the time variable:\n\\begin{eqnarray*} \\int_0^t J^2(s,x)~ds & \\leq & \\int_0^t \\left[\\int_\\mathbb{R} |g_\\alpha^X|(\\sigma,x)\\mathbf{1}_{(0,t)}(\\sigma) \\frac{e^{-c\\varepsilon^{-1}(s-\\sigma)}}{\\varepsilon}\\mathbf{1}_{(0,t)}(s-\\sigma)~d\\sigma\\right]^2 ~ds \\\\\n & \\leq & \\norme{L^2(0,t)}{\\left(g_\\alpha^X(\\cdot,x)\\mathbf{1}_{(0,t)}\\right) \\ast \\left(\\varepsilon^{-1}e^{-c\\varepsilon^{-1}(\\cdot)}\\mathbf{1}_{(0,t)}\\right)}^2 \\\\\n & \\leq & \\norme{L^2(0,t)}{g_\\alpha^X(\\cdot,x)}^2 \\norme{L^1(0,t)}{\\varepsilon^{-1}e^{-c\\varepsilon^{-1}(\\cdot)}}^2 \\\\\n & \\leq & C\\int_0^t \\norme{\\infty}{g_\\alpha (s)}^2~ds, \\end{eqnarray*}\nby (\\ref{jest}) and standard convolution inequalities.\n\\newline\n\nSo now we only need to control the $Y_t$ norm of $h$ and the $L^\\infty$ norm of the commutators. We remind the reader that\n\\begin{eqnarray*} h & = & \\varepsilon[\\rho(\\mu\\Delta_\\tau u_3+\\lambda\\derp{z}\\mathrm{div}_\\tau u_\\tau+\\sigma(\\nabla U)_3) \\\\\n & & \\hspace{20pt} -l\\derp{z}u_\\tau\\cdot\\nabla_\\tau\\rho-l(\\div u+\\derp{z}u_3)R]-\\rho^2(\\derp{t}u_3+u\\cdot\\nabla u_3)+\\rho F_3. \\end{eqnarray*}\nThe starting point here is to notice that, for two functions $f$ and $g$,\n\\begin{equation} \\norme{2,\\infty}{fg} \\leq C(\\norme{2,\\infty}{f}\\norme{\\infty}{g} + \\norme{1,\\infty}{f}\\norme{2,\\infty}{g}). \\label{fg2inf} \\end{equation}\nIn the case of $h_1:=\\rho(\\mu\\Delta_\\tau u_3 + \\lambda\\derp{z}\\mathrm{div}_\\tau u_\\tau)$, $f=\\rho$, so\n$$ \\norme{2,\\infty}{h_1}^2 \\leq C(\\norme{2,\\infty}{\\rho}^2(\\norme{2,\\infty}{u}^2+\\norme{1,\\infty}{\\nabla u_\\tau}^2) + \\norme{1,\\infty}{\\rho}(\\norme{4,\\infty}{u}^2 + \\norme{3,\\infty}{\\nabla u_\\tau}^2)), $$\nin which we apply the Sobolev inequality to all terms except $\\norme{1,\\infty}{\\nabla u_\\tau}^2$ (it is part of the total quantity we wish to bound), and use the Young inequality to split the products up. \nIn the process, we obtain $(\\varepsilon^2 \\int_0^t \\norme{4}{\\derp{z}\\nabla u_\\tau(s)}^2~ds)^2$, which is controlled by using \\estiref{hmvort}. Therefore, in total, using all the estimates available to us,\n$$ \\int_0^t \\varepsilon^2\\norme{2,\\infty}{h_1}^2 ~ds \\leq \\varepsilon Q(M_0) + \\varepsilon \\int_0^t Q(1+\\norme{7}{U}^2+\\norme{6}{\\nabla u_\\tau}^2+\\norme{6}{R}^2 +\\norme{1,\\infty}{\\nabla u_\\tau}^2 + \\norme{1,\\infty}{R}^2)~ds, $$\n\\begin{equation} \\mathrm{thus}~~\\int_0^t \\varepsilon^2\\norme{2,\\infty}{h_1}^2 ~ds \\leq Q(M_0) + C\\varepsilon(t+1)Q(M+M_F) + C\\varepsilon t \\trinorme{1,\\infty,t}{R}^2. \\label{densinfh1} \\end{equation}\nLikewise, setting $h_2:= \\derp{z}u_\\tau\\cdot\\nabla_\\tau\\rho+(\\div u + \\derp{z}u_3)R + \\sigma_3 = \\derp{z}u\\cdot\\nabla \\rho + R\\div u + \\sigma_3$, we use (\\ref{fg2inf}) again, applied to the space $Y_t$ instead of $W^{2,\\infty}_{co}$, and we get, after applying the Sobolev embedding to all the terms except $\\norme{1,\\infty}{\\nabla u_\\tau}^2$, and $\\norme{Y_t}{R}^2$, to get\n$$ \\varepsilon^2\\norme{Y_t}{h_2}^2 \\leq Q(M_0) + \\varepsilon(t+1)Q(M+M_F) + C\\varepsilon^2 \\norme{Y_t}{R}^2 + C\\varepsilon^2 \\int_0^t \\norme{6}{\\derp{zz}u_3}^2~ds . $$\nWe have once again used \\estiref{hmvort} to control the $H^4_{co}([0,T]\\times\\Omega)$ norm of $\\varepsilon \\derp{z}\\nabla u_\\tau$, and we have used (\\ref{l2time}) to get an $L^2$-in-time norm of $\\derp{z}u_3$ (which can then be controlled using \\propref{hmee}) and $\\derp{zz}u_3$. \nThus, we replace $\\varepsilon\\derp{zz}u_3$ by its expression in (\\ref{NSrwR}), use the tame estimate, take the supremum inside the integral and extract a factor $t$, and this yields\n\\begin{equation} \\varepsilon^2\\norme{Y_t}{h_2}^2 \\leq Q(M_0) + \\varepsilon(t+1)Q(M+M_F) + C\\varepsilon \\norme{Y_t}{R}^2. \\label{densinfh2} \\end{equation}\n\nIn the source term $h$, it remains to examine $h_3:= \\rho^2(\\derp{t}u_3+u\\cdot\\nabla u_3+F_3)$. The difference here is that there is no factor $\\varepsilon$ ready to provide us with a small parameter. Instead, we once again extract $t$ from the integral to use as the small parameter. \nBy simply using the Sobolev embedding inequality and breaking the result down almost completely with the tame estimate, we get\n\\begin{eqnarray} \\int_0^t \\norme{2,\\infty}{h_3(s)}^2~ds & \\leq & Q(M_0)+ C\\int_0^t \\norme{5}{h_3(s)}^2+\\norme{4}{\\derp{z}h_3(s)}^2~ds \\nonumber \\\\\n & \\leq & Q(M_0) + \\int_0^t Q(1+\\trinorme{1,\\infty,t}{U}^2+\\trinorme{1,\\infty,t}{\\derp{z}U}^2+M_F) \\nonumber \\\\\n & & \\hspace{35pt} \\times \\left(1+\\norme{6}{U}^2+\\norme{5}{\\derp{z}U}^2+\\norme{4}{u_3\\derp{zz}u_3}^2~ds\\right) . \\label{h31} \\end{eqnarray}\nThe only product we do not split using the tame estimates and the Young inequality is $u_3\\derp{zz}u_3$, which appears in $\\derp{z}h_3$. Indeed, we need the factor $u_3$ to be able to compensate for the two $z$-derivatives. We have already used in (\\ref{h31}) the fact that\n$$ \\trinorme{\\infty,t}{u_3\\derp{zz}u_3}^2 \\leq \\trinorme{1,\\infty,t}{\\derp{z}u_3}^4 , $$\ndue to (\\ref{u3phi}), and $\\norme{3}{u_3\\derp{zz}u_3}$ is dealt with in same way as in the proof of (\\ref{comfinal}) in \\propref{comlem}: multiply and divide by $\\phi$, which means we are actually looking at $\\norme{3}{\\phi^{-1}u_3 Z_3\\derp{z}u_3}$, and use the Hardy inequality to get that this quantity satisfies (\\ref{comfinal}), with $g=u_3$ and $f=\\derp{z}u_3$, so\n$$ \\int_0^t \\norme{2,\\infty}{h_3(s)}^2 ~ds \\leq Q(M_0) + Q(1+\\trinorme{1,\\infty,t}{U}^2+\\trinorme{1,\\infty,t}{\\derp{z}U}^2)\\int_0^t \\norme{6}{U}^2+\\norme{5}{\\derp{z}U}^2~ds . $$\nNow we extract $t$ and use (\\ref{l2time}) to get $L^2$-in-time norms on $R$ and $\\derp{z}u_3$, so\n\\begin{equation} \\int_0^t \\norme{2,\\infty}{h_3(s)}^2~ds \\leq Q(M_0) + tQ(M+M_F) . \\label{densinfh3} \\end{equation}\n\nFinally, we examine the $L^{\\infty}$ norms of the commutators, ${\\cal C}^{\\alpha}_R$, with $|\\alpha|= 1$, or $|\\alpha|=2$ and $\\alpha_0\\geq 1$. We begin with the case $|\\alpha|=1$, so $Z^\\alpha=Z_j$ for a certain $j$, and\n$$ {\\cal C}^{\\alpha}_{R} = l\\varepsilon[Z_j,u\\cdot\\nabla] R + [Z_j,\\rho P'(\\rho)]R = l\\varepsilon(Z_j u)\\cdot\\nabla R + l\\varepsilon u_3[Z_j,\\derp{z}]R + \\gamma(Z_j P)R , $$\nin which the second term is either 0 ($j\\neq 3$) or $-l\\varepsilon\\phi' u_3\\derp{z}R$. Bounding this is straight-forward, using (\\ref{hardyinf}) along the way:\n\\begin{equation} \\int_0^t \\norme{\\infty}{{\\cal C}^\\alpha_R(s)}^2~ds \\leq Ct(1+\\varepsilon^2)\\trinorme{1,\\infty,t}{\\nabla u}^2\\trinorme{1,\\infty,t}{R}^2 . \\label{densinfc1} \\end{equation}\nThe second case, $|\\alpha|=2$ and $\\alpha_0>0$ is also simple. We can write $Z^\\alpha = Z_j \\derp{t}$ for a certain $j$, so\n\\begin{eqnarray*} {\\cal C}^{\\alpha}_R & = & l\\varepsilon\\left((Z_j\\derp{t} u)\\cdot\\nabla R + (Z_ju)\\cdot\\nabla(\\derp{t}R) + (\\derp{t}u)\\cdot\\nabla (Z_j R) + (\\derp{t}u_3)[Z_j,\\derp{z}]R \\right. \\\\\n & & \\hspace{25pt} \\left. + u_3[Z_j,\\derp{z}]\\derp{t}R\\right) + \\gamma((Z_j\\derp{t}P)R+(Z_j P)(\\derp{t}R)+(\\derp{t} P)(Z_jR)) \\end{eqnarray*}\nThe $L^\\infty$ norm of this is, for most terms, bounded using only the Sobolev embedding and\/or (\\ref{hardyinf}), wherever $u_3\\derp{z}$ or conormal derivatives of $u_3 \\derp{z}$ appear. One can then take the supremum in time inside the integral and integrate to get a factor $t$. For instance, the last term satisfies\n\\begin{eqnarray*} \\norme{\\infty}{(\\derp{t}r)(Z_jR)}^2 & \\leq & C(\\norme{2,\\infty}{u_\\tau}^2+\\norme{1,\\infty}{\\derp{z}u_3}^2+\\norme{1,\\infty}{\\rho}^2)\\norme{1,\\infty}{R}^2 \\\\\n & \\leq & C(1+\\norme{5}{U}^2+\\norme{4}{\\nabla u_\\tau}^2+\\norme{3}{R}^2+\\norme{1,\\infty}{\\derp{z}u_3}^2)\\norme{1,\\infty}{R}^2 , \\end{eqnarray*}\nthen we apply \\estiref{hmnd} (a) on $\\norme{1,\\infty}{\\derp{z}u_3}$. Two specificities do appear though in the terms containing $u_3 \\derp{z}R$ and the like.\n\\begin{itemize}\n\\item The term $l \\varepsilon u_3[Z_3,\\derp{z}]\\derp{t}R = -l\\varepsilon\\phi' u_3\\derp{zt}R$ leads to two conormal derivatives on $R$. We cannot extract a factor $t$ from the integral here, since we want $L^2$-in-time norms of $\\norme{1,\\infty}{\\derp{t}R}$. However, we do have a factor $l\\varepsilon$, which will act as the small parameter. So,\n$$ \\int_0^t \\norme{\\infty}{l\\varepsilon\\phi' u_3\\derp{zt}R}^2~ds \\leq C\\varepsilon^2\\trinorme{\\infty,t}{\\derp{z}u_3}^2\\int_0^t \\norme{1,\\infty}{\\derp{t}R(s)}^2~ds . $$\n\\item The term $l \\varepsilon(Z_j\\derp{t}u)\\cdot\\nabla R$ contains $\\varepsilon(Z_j\\derp{t}u_3)\\derp{z}R$, which, after using (\\ref{hardyinf}), leads to a term with two conormal derivatives on $\\varepsilon\\derp{z}u_3$ (and no more than one on $R$, which is therefore not problematic), so we re-use the trick we used when estimating $\\varepsilon h_2$: we use the Sobolev inequality on the term with two conormal derivatives, to get\n$$ \\int_0^t \\varepsilon^2\\norme{2,\\infty}{\\derp{z}u_3(s)}^2~ds \\leq M_0 + C\\int_0^t \\varepsilon^2\\norme{5}{\\derp{z}u_3(s)}^2 + \\norme{5}{\\varepsilon\\derp{zz}u_3(s)}^2~ds , $$\nand we replace $\\varepsilon\\derp{zz}u_3$ by its expression in (\\ref{NS}), use the tame estimate, take the supremum inside the integral and extract a factor $t$.\n\\end{itemize}\n\nIn total, combining the above consideration with inequalities (\\ref{densinfh1}) to (\\ref{densinfc1}), we conclude that\n$$ \\int_0^t \\norme{\\infty}{g_\\alpha(s)}^2~ds \\leq Q(M_0) + (t+\\varepsilon) Q(M+M_F) , $$\nwhich ends the proof of the estimate. $\\square$\n\n\\section[Proof of \\thref{thEE}, part V: conclusion]{Proof of \\thref{thEE}, part V \\\\ Conclusion\n \\sectionmark{Proof of \\thref{thEE} part V: conclusion}}\n\\sectionmark{Proof of \\thref{thEE} part V: conclusion}\n\nLet us consider times $t\\leq T'$, where\n$$ T' = \\sup\\{ t\\leq T_0 ~|~ {\\cal E}_m(t,U) + \\trinorme{1,\\infty,t}{\\derp{z}u_3}^2 \\leq Q(2M_0)\\} . $$\nThe time $T'$ depends \\textit{a priori} on $\\varepsilon$. With this, we pick up the uniform boundedness from below of $\\rho$ and the uniform smallness of $u_3$ near the boundary that we have so far assumed.\n\\begin{itemize}\n\\item Let $\\rho(0,x)\\geq c_{0}$. Equation (\\ref{NSdiv}) provides us with the differential equation\n$$ \\derp{t}\\rho^{X}(t,x) = - (\\rho\\div u)^{X}(t,x), $$\nwhere $f^{X}$ is once again $f$ following the flow of $u$. Thus,\n$$ \\rho^{X}(t,x) = \\rho(0,x)\\exp\\left(-\\int_{0}^{t}\\div u(s,x)~ds\\right), $$\nand $\\div u$ is uniformly bounded on $[0,T']\\times \\Omega$, so\n$$ |\\rho(t,x)|\\geq c_{0}e^{-Q(2M_0)T'} := c_0', $$\nfor $t\\leq T'$, and $c_0'>0$ can be chosen independent of $\\varepsilon$ (if $T'=+\\infty$ for some $\\varepsilon$, we replace with some finite $T'_0$ independent of $\\varepsilon$). The density $\\rho$ is therefore uniformly bounded from below on $[0,T']$.\n\\item For $t\\leq T'$, we can repeat the proof of (\\ref{linfnd1}),\n$$ \\norme{\\infty}{\\derp{z}u_3}^2 \\leq Q(1+\\norme{\\mathrm{Lip}}{U(0)}^2) + tQ(1+\\trinorme{4,t}{U}^2+\\trinorme{3,t}{\\derp{z}U}^2 + \\trinorme{1,\\infty,t}{\\derp{z}\\rho}^2), $$\nwhich is only a reading of (\\ref{NSdiv}) combined with the Sobolev embedding and property (\\ref{l2timeinf}), hence $\\derp{z}u_3(t)$ has a uniform bound for $t\\leq T'$, thus $|u_3(t,z)|\\leq M'z$ for some $M'$ when $t\\leq T'$. \nAs a result, we get uniform smallness of $u_3$ near the boundary: $|u_3(t,z)|\\leq M'\\delta$ when $z<\\delta$. \\end{itemize}\n\nFor $t\\leq T'$, we have the bounds required to make the \\textit{a priori} estimation process valid, so we have\n$$ {\\cal E}_m(t,U) \\leq Q(M_0)+(t+\\varepsilon)Q(Q(2M_0)+M_F). $$\nFor $\\varepsilon\\leq \\varepsilon_0$ small enough, we see that the right-hand side is smaller than $Q(2M_0)$ for $t\\leq T^*\\leq T'$, with $T^*$ depending on $\\varepsilon_0$ but not on $\\varepsilon$. Thus $T^*$ can be chosen independent of $\\varepsilon$, and the theorem is proved. $\\square$\n\\vspace{12pt}\n\n\\textbf{\\underline{Acknowledgements.}} This is one of the results from my Ph.D thesis, prepared at the University of Rennes 1. Therefore, I am deeply grateful to my supervisor Fr\\'ed\\'eric Rousset for the opportunity to work on this topic, for his guidance and patience throughout the preparation of the present article. \nI would also like to thank Mark Williams for the discussions that have helped improve the clarity of the paper.\n\n\\begin{small}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{\\label{Intro}Introduction}\n   In $1930$, Tolman \\cite{Tolman:1930zza,Tolman:1930ona} showed that the proper temperature of a system, composed of matter in thermodynamic equilibrium with a gravitational field, as measured by a local observer, satisfies the relation: $T\\sqrt{-g_{tt}}$ = constant, where $T$ is the proper temperature and the other one is the square root of the $dt^2$ coefficient of the background metric. The last one is known as redshift factor. This implies that the temperature of a black hole horizon, measured by an observer which is very near to the horizon, diverges as $g_{tt}=0$ in this regime. It turns out that such an observation is consistent with the Unruh effect \\cite{Unruh:1976db}. The temperature calculated from the Rindler frame, which is passing through a hyperbola very near to the light cone, is infinite as the acceleration of the fame for such path is very large. Moreover, for Hawking radiation \\cite{Hawking:1974rv}, the constant on the right hand side of the relation is identified as the Hawking temperature as in the asymptotic infinity limit the red shift factor goes to unity. The importance of this result is that it helps us to relate the temperatures at each point of spacetime. Till then this remains as an integral part of gravitational theories.\n   \n   In the original derivations \\cite{Tolman:1930zza,Tolman:1930ona}, the information about the external matter (radiation) plays an important role.  In one approach, the conservation of the energy-momentum tensor for the ideal fluid has been used while in others they came through {\\it explicit} use of Einstein' s equations of motion. Also the Stefan-Boltzmann law for radiation was needed {\\footnote{Very recently, in \\cite{Gim:2015era} a correction to Toman temperature has been introduced for the ($1+1$) dimensional case in presence of trace anomaly in the enery-momentum tensor for external fields. In this analysis a modification to the usual Stefan-Boltzmann law has been found.}}. Moreover, a derivation, based on the extremum principle, was done by extremizing the entropy of the radiation.  In summary, such an analysis used both the information of gravity as well as matter and is an on-shell calculation in the sense that the Einstein's equations of motion were needed. \n   \n   I already mentioned that the Unruh temperature obeys the Tolman relation and also note that the Rindler metric is not a solution of Einstein's equation. So the expression is not only restricted to a particular theory of gravity, rather it has much more applicability. Therefore one must expect an derivation of such relation which does not use the equations of motion. Moreover, for the Hawking effect the constant is just the Hawking temperature ($1\/8\\pi M$ for Schwarzschild black hole), which is purely gravitational contribution. It is the temperature of the event horizon as measured from infinity. In addition, the redshift factor is also a contribution from gravitational field. All these indicate that the Tolman relation might has a completely gravitational interpretation and if that is so, it is possible to find a independent derivation which deals only gravity; i.e. the external matter part should not play any role. \nThe aim of the present paper is precisely in these directions.  Here I shall obtain the required relation purely from the surface term of the gravitational action. The action will be taken as the Gibbons-Hawking-York (GHY) surface term. \n\n   Motivation for considering only the surface term is as follows. It must be noted that the Einstein-Hilbert (EH) action, in a local frame, reduces to a total derivative term. Moreover, the equations of motion can be obtained purely from GHY term under certain prescription \\cite{Padmanabhan:2004fq}. Most importantly, it has been observed that the calculation of this action on the horizon leads to horizon entropy of a black hole \\cite{Paddy} (see also \\cite{Padmanabhan:2012bs,Majhi:2013jpk}). This is not a new observation. Earlier, when Gibbons-Hawking  \\cite{Gibbons:1976ue} obtained the entropy in the action principle, it must be noted that the contribution came from only GHY term as the EH part vanishes after imposition of the on-shell condition. Very recently, the same was obtained from the Noether charge corresponding to surface action \\cite{Majhi:2012tf,Majhi:2012nq} and also a entropy function has been defined to find the entropy of an near horizon extremal black hole \\cite{Majhi:2015pra}. All these indicate that the surface may contain all the information about the bulk term. Keeping in mind the earlier aim and the present motivation, in this paper I shall give an analysis to find Tolman temperature distribution by just considering the GHY action.  In this analysis no information about the external matter and the Einstein's equations of motion will be used. We shall see that such a derivation is much more simple and give more insight of the relation. \n   \n   Let me now summarise the procedure to be adopted in this paper. In the original derivation the entropy of the radiation was extremized. Since, the horizon entropy is related to the GHY term, here I shall extremize it to achieve the goal. The background will be chosen as a static, spherically symmetric metric. The action will be first expanded in terms this metric coefficients and then the usual extemization procedure will be applied. This will lead to equations for metric coefficients. Finally, using them the Tolman expression will be derived. Here the Einstein's equations of motion will not be used explicitly. \n   \n   The organization of the paper is as follows.  In section \\ref{GHY}, a short introduction about the GHY action will be given. Next section is the main part of the paper which will contain a detailed analysis to reach to the Tolman form by extremizing the surface action. Finally in section \\ref{Conclusions}, I shall conclude. An appendix, with details of the calculation, will also be added at the end.  \n\n\n\\section{\\label{GHY}GHY surface term: a brief introduction}\n     Let us first introduce the GHY surface action. It must be noted that the EH action consists of both the first order and the second order derivatives of the metric $g_{ab}$ as the Ricci scalar is made of square of connection and its derivative. Therefore an arbitrary variation of the action leads to a boundary term containing both the variations of the metric as well as its derivative. So in order to obtain the equations of motion, using the least action principle, we need to fix both the metric and its derivative at the boundary. But it must be mentioned that use of both the boundary conditions is not a correct prescription. This has been avoided by adding a surface term to action in such a way that the total variation (i.e. EH plus the surface action) is free of total derivative of variation of the derivative of $g_{ab}$. Then one has to fix only the metric at the boundary. \nAlthough there is no unique choice of the boundary term \\cite{Charap:1982kn}, people, in most of the time, choose the GHY surface term, introduced by Gibbons, Hawking and York. Such a term is foliation dependent; defined either on the timelike surface or on the spacelike surface {\\footnote{An attempt has been made recently in \\cite{Parattu:2015gga} to define a boundary term on an arbitrary null surface.}}. It has been observed that in this case one has to fix only the induced metric of the foliated surface at the boundary to obtain the Einstein's equations of motion \\cite{Paddy}. \n     \n     The GHY surface action is defined as\n\\begin{equation}\n\\mathcal{A}_{GHY} = -\\frac{1}{8\\pi G}\\int_{\\partial \\mathcal{V}}\\sqrt{|h_{(i)}|}d^3x \\epsilon K_{(i)}~,\n\\label{2.01}\n\\end{equation}\nwhere $\\partial \\mathcal{V}$ is the boundary surface of the full manifold $\\mathcal{M}$. As stated earlier $\\partial\\mathcal{V}$ can be either timelike or spacelike, depending on the value of $\\epsilon$. Here $h_{(i)}$ is the determinant of the induced metric of the surface $\\partial\\mathcal{V}$ and $K_{(i)}$ is the second fundamental. The expression of $K_{(i)}$, in terms of the unit normal $N^a_{(i)}$ to the boundary, is  \n\\begin{equation}\nK_{(i)}=-\\nabla_aN^a_{(i)}~.\n\\label{2.02}\n\\end{equation}\nRemember that ``$i$'' in the subscript denotes the kind of surface (spacelike or timelike) we are choosing.  Also keep in mind that $\\epsilon =+1$ refers to the timelike surface while $\\epsilon=-1$ denotes a spacelike surface.\nFor our future purpose, we shall express the GHY action in the following form. Gauss's theorem helps us to write Eq. (\\ref{2.01}) as\n\\begin{equation}\n\\mathcal{A}_{GHY} = -\\int_{\\mathcal{M}}d^4x\\sqrt{-g}\\mathcal{L}_{GHY} =-\\frac{1}{8\\pi G}\\int_{\\mathcal{M}}d^4x\\sqrt{-g} \\nabla_a\\Big(K_{(i)}N^a_{(i)}\\Big)~,\n\\label{2.03}\n\\end{equation}\nwhere one has to consider $N^a_{(i)}N_{a(i)}=\\epsilon$. I shall use the above form in the next section to achieve the main goal.\n\n\n\\section{\\label{Tolman}Tolman temperature}\n   The present section will show that it is possible to obtain the Toman temperature distribution from the surface part of the whole gravitational action. We first calculate the action under a static, spherically symmetric background and the we shall see that the extremization of it will lead to the required relation. For that let us consider a general spherically symmetric, static metric:\n\\begin{equation}\nds^2 = -e^{\\nu}dt^2+e^{\\mu}(dr^2+r^2d\\theta^2+r^2\\sin^2\\theta d\\phi^2)~,\n\\label{3.01}\n\\end{equation}\nwhere $\\mu$ and $\\nu$ are functions of radial coordinate $r$ only. Below we shall now calculate the GHY action, given by the form (\\ref{2.03}), for the above metric. \n\n    Note that for the present metric the relevant surfaces, which are denoted by subscript ``$i$'' in (\\ref{2.03}), are $t=constant$ surface which is spacelike and $r=constant$, $\\theta=constant$, $\\phi=constant$ surfaces which are timelike.\nTherefore, we first write the Lagrangian density in the following form:\n\\begin{eqnarray}\n\\sqrt{-g}\\mathcal{L}_{GHY} &=& \\frac{1}{8\\pi G}\\partial_a\\Big(\\sqrt{-g}K_{(i)}N^a_{(i)}\\Big)\n\\nonumber\n\\\\\n&=& \\frac{1}{8\\pi G}\\Big[\\partial_a\\Big(\\sqrt{-g}K_{(t)}N^a_{(t)}\\Big)+\\partial_a\\Big(\\sqrt{-g}K_{(r)}N^a_{(r)}\\Big)\n\\nonumber\n\\\\\n&+& \\partial_a\\Big(\\sqrt{-g}K_{(\\theta)}N^a_{(\\theta)}\\Big)+\\partial_a\\Big(\\sqrt{-g}K_{(\\phi)}N^a_{(\\phi)}\\Big)\\Big]~.\n\\label{3.02}\n\\end{eqnarray}\nNow for the metric (\\ref{3.01}), the unit normal vectors to different surfaces turn out to be\n\\begin{eqnarray}\n&&N^a_{(t)}=\\Big(e^{-\\nu\/2},0,0,0\\Big); \\,\\,\\,\\ N^a_{(r)} = \\Big(0,e^{-\\mu\/2},0,0\\Big); \n\\nonumber\n\\\\\n&&N^a_{(\\theta)} = \\Big(0,0,\\frac{e^{-\\mu\/2}}{r},0\\Big); \\,\\,\\,\\ N^a_{(\\phi)} = \\Big(0,0,0,\\frac{e^{-\\mu\/2}}{r\\sin\\theta}\\Big)~.\n\\label{3.03}\n\\end{eqnarray}\nMoreover, the metric coefficients are independent of $t$ and $\\phi$ coordinates. Therefore, the first and last terms on the right hand side of (\\ref{3.02}) will not contribute and it reduces to\n\\begin{equation}\n\\sqrt{-g}\\mathcal{L}_{GHY} = \\frac{1}{8\\pi G}\\Big[\\partial_r\\Big(\\sqrt{-g}K_{(r)}N^r_{(r)}\\Big) + \\partial_\\theta\\Big(\\sqrt{-g}K_{(\\theta)}N^\\theta_{(\\theta)}\\Big)\\Big]~.\n\\label{3.04}\n\\end{equation}\nNext using $K_{(i)}=-\\nabla_aN^a_{(i)}=-(1\/\\sqrt{-g})\\partial_a(\\sqrt{-g}N^a_{(i)})$ we rewrite the above as\n\\begin{equation}\n\\sqrt{-g}\\mathcal{L}_{GHY} = -\\frac{1}{8\\pi G}\\Big[\\partial_r\\Big\\{N^r_{(r)}\\partial_r\\Big(\\sqrt{-g}N^r_{(r)}\\Big)\\Big\\} + \\partial_\\theta\\Big\\{N^{\\theta}_{(\\theta)}\\partial_{\\theta}\\Big(\\sqrt{-g}N^\\theta_{(\\theta)}\\Big)\\Big\\}\\Big]~.\n\\label{3.05}\n\\end{equation}\nSubstituting (\\ref{3.03}) and $\\sqrt{-g}=(r^2\\sin\\theta)\\exp[(3\\mu+\\nu)\/2]$ in the above and after doing some simplifications we obtain\n\\begin{equation}\n\\sqrt{-g}\\mathcal{L}_{GHY} = -\\frac{\\sin\\theta}{8\\pi G} e^{\\frac{\\mu+\\nu}{2}}\\Big[1+r\\Big(3\\mu'+2\\nu'\\Big)+r^2\\Big(\\mu''+\\frac{\\nu''}{2}\\Big)+r^2\\Big(\\frac{\\mu'^2}{2}+\\frac{\\nu'^2}{4}+\\frac{3\\mu'\\nu'}{4}\\Big)\\Big]~,\n\\label{3.06}\n\\end{equation}\nwhere the prime denotes the derivative with respect to radial coordinate.\nUsing this in (\\ref{2.03}) and performing the integration in $\\theta$ and $\\phi$, one obtains the GHY action as\n\\begin{equation}\n\\mathcal{A}_{GHY}=\\frac{1}{2G}\\int dt dr  e^{\\frac{\\mu+\\nu}{2}}\\Big[1+r\\Big(3\\mu'+2\\nu'\\Big)+r^2\\Big(\\mu''+\\frac{\\nu''}{2}\\Big)+r^2\\Big(\\frac{\\mu'^2}{2}+\\frac{\\nu'^2}{4}+\\frac{3\\mu'\\nu'}{4}\\Big)\\Big]~.\n\\label{3.07}\n\\end{equation}\nSince the metric is static, the integration on time coordinate can also be done. In this case we shall choose the limits of integration by the following argument. It is well known that if the usual time coordinate is Euclideanised first and then it is assumed to be periodic, the zero temperature theory leads to the finite temperature theory where the inverse temperature of the system is the periodicity of the Euclidean time. Following the same path, let us assume that the time has periodicity $\\beta_0=1\/T_0$. Then the above will lead to\n\\begin{equation}\n\\mathcal{A}_{GHY}=\\frac{1}{2G}\\int dr  \\frac{r^2e^{\\frac{\\mu+\\nu}{2}}}{T_0}\\Big[\\frac{1}{r^2}+\\frac{3\\mu'+2\\nu'}{r}+\\mu''+\\frac{\\nu''}{2}+\\frac{\\mu'^2}{2}+\\frac{\\nu'^2}{4}+\\frac{3\\mu'\\nu'}{4}\\Big]~.\n\\label{3.08}\n\\end{equation}\nIn the below, I shall vary the above action for arbitrary variations of $\\mu$ and $\\nu$ to obtain equations of motion for these two variables.\n\n      The variation of the GHY action with respect to $\\mu$ and $\\nu$ yields\n\\begin{eqnarray}\n&&\\delta\\mathcal{A}_{GHY} = \\frac{1}{2G}\\int dr  \\frac{r^2e^{\\frac{\\mu+\\nu}{2}}}{T_0}\\Big(\\frac{\\delta\\mu+\\delta\\nu}{2}\\Big)\\Big[\\frac{1}{r^2}+\\frac{3\\mu'+2\\nu'}{r}+\\mu''+\\frac{\\nu''}{2}+\\frac{\\mu'^2}{2}+\\frac{\\nu'^2}{4}+\\frac{3\\mu'\\nu'}{4}\\Big]\n\\nonumber\n\\\\\n&&+\\frac{1}{2G}\\int dr \\frac{r^2e^{\\frac{\\mu+\\nu}{2}}}{T_0}\\Big[\\frac{3\\delta\\mu'+2\\delta\\nu'}{r}+\\delta\\mu''+\\frac{\\delta\\nu''}{2}+\\mu'\\delta\\mu'+\\frac{\\nu'\\delta\\nu'}{2}+\\frac{3\\nu'\\delta\\mu'}{4}+\\frac{3\\mu'\\delta\\nu'}{4}\\Big]\n\\label{3.09}\n\\end{eqnarray}\nIgnoring the total derivative terms; i.e. imposing the conditions $\\delta\\mu=\\delta\\nu=0$ and $\\delta\\mu'=\\delta\\nu'=0$ at the boundary $r=constant$, in the above we find the following expression for variation of the surface action:\n\\begin{eqnarray}\n\\delta\\mathcal{A}_{GHY} &=& \\frac{1}{2G}\\int dr \\Big[\\frac{d^2}{dr^2}\\Big(\\frac{r^2e^{\\frac{\\mu+\\nu}{2}}}{T_0}\\Big)-\\frac{d}{dr}\\Big\\{\\frac{r^2e^{\\frac{\\mu+\\nu}{2}}}{T_0}\\Big(\\frac{3}{r}+\\mu'+\\frac{3\\nu'}{4}\\Big)\\Big\\}+\\mathcal{B}\\Big]\\delta\\mu\n\\nonumber\n\\\\\n&+& \\frac{1}{2G}\\int dr \\Big[\\frac{d^2}{dr^2}\\Big(\\frac{r^2e^{\\frac{\\mu+\\nu}{2}}}{2T_0}\\Big)-\\frac{d}{dr}\\Big\\{\\frac{r^2e^{\\frac{\\mu+\\nu}{2}}}{T_0}\\Big(\\frac{2}{r}+\\frac{\\nu'}{2}+\\frac{3\\mu'}{4}\\Big)\\Big\\}+\\mathcal{B}\\Big]\\delta\\nu~,\n\\label{3.10}\n\\end{eqnarray}\nwhere $\\mathcal{B}$ is given by\n\\begin{equation}\n\\mathcal{B} = \\frac{r^2e^{\\frac{\\mu+\\nu}{2}}}{2T_0}\\Big(\\frac{1}{r^2}+\\frac{3\\mu'+2\\nu'}{r}+\\mu''+\\frac{\\nu''}{2}+\\frac{\\mu'^2}{2}+\\frac{\\nu'^2}{4}+\\frac{3\\mu'\\nu'}{4}\\Big)~.\n\\label{3.11}\n\\end{equation}\nFor shake of completeness and clarity, a detailed derivation of Eq. (\\ref{3.10}) has been presented in Appendix \\ref{AppA}.\nTherefore, setting the variation of the action equal to zero one obtains the equations of motion for $\\mu$ and $\\nu$ as\n\\begin{eqnarray}\n&&\\frac{d^2}{dr^2}\\Big(\\frac{r^2e^{\\frac{\\mu+\\nu}{2}}}{T_0}\\Big)-\\frac{d}{dr}\\Big\\{\\frac{r^2e^{\\frac{\\mu+\\nu}{2}}}{T_0}\\Big(\\frac{3}{r}+\\mu'+\\frac{3\\nu'}{4}\\Big)\\Big\\}+\\mathcal{B} = 0~;\n\\label{3.12}\n\\\\\n&&\\frac{d^2}{dr^2}\\Big(\\frac{r^2e^{\\frac{\\mu+\\nu}{2}}}{2T_0}\\Big)-\\frac{d}{dr}\\Big\\{\\frac{r^2e^{\\frac{\\mu+\\nu}{2}}}{T_0}\\Big(\\frac{2}{r}+\\frac{\\nu'}{2}+\\frac{3\\mu'}{4}\\Big)\\Big\\}+\\mathcal{B} = 0~.\n\\label{3.13}\n\\end{eqnarray}\nSubtracting (\\ref{3.13}) from (\\ref{3.12}) we obtain\n\\begin{equation}\n\\frac{d^2}{dr^2}\\Big(\\frac{r^2e^{\\frac{\\mu+\\nu}{2}}}{2T_0}\\Big) - 2\\frac{d}{dr}\\Big(\\frac{re^{\\frac{\\mu+\\nu}{2}}}{T_0}\\Big) - \\frac{d}{dr}\\Big(\\frac{r^2e^{\\frac{\\mu+\\nu}{2}}}{T_0}\\frac{\\mu'}{2}\\Big) - \\frac{d}{dr}\\Big(\\frac{r^2e^{\\frac{\\mu+\\nu}{2}}}{T_0}\\frac{\\nu'}{2}\\Big) = 0~.\n\\label{3.14}\n\\end{equation}\nNow the inverse of the proper temperature of the system can be defined as\n\\begin{equation}\n\\frac{1}{T} = \\int_0^{\\beta_0} dt e^{\\frac{\\nu}{2}} = \\frac{e^{\\frac{\\nu}{2}}}{T_0}~.\n\\label{3.15}\n\\end{equation}\nUse of this and after some rearrangements, Eq. (\\ref{3.14}) can be expressed in the following form:\n\\begin{equation}\n\\frac{d}{dr}\\Big[r^2e^{\\frac{\\mu}{2}}\\frac{d}{dr}\\Big(\\frac{1}{T}\\Big)\\Big] = \\frac{d}{dr}\\Big[\\frac{r^2e^{\\frac{\\mu}{2}}}{T}\\frac{\\nu'}{2}\\Big]~.\n\\label{3.16}\n\\end{equation}\nNext I shall solve the differential equation under a boundary condition to reach my ultimate goal.\n\n    Before doing so let me make some comments about it. In the original work of Tolman \\cite{Tolman:1930zza}, the same equation was obtained (see, Eq. (41) and Eq. (51) of \\cite{Tolman:1930zza}) by two ways. In one case he found this by extremizing the entropy of the radiation, whereas the same was also derived by extremizing the entropy of a perfect fluid.  In the second approach, the first law of thermodynamics was used. It also must be noted that for both the approaches, Tolman used Einstein's equations of motion as well as Stefan-Boltzmann law for radiation to achieve the above result. On the contrary, here I derived the relation by just extremizing the GHY surface action where no used of the above stated informations was needed. So, in this sense, my analysis is {\\it off}-shell. \n    \n   Now the solution of the differential equation (\\ref{3.16}) turns out to be\n\\begin{equation}\n\\frac{d\\ln T}{dr} = -\\frac{1}{2}\\frac{d\\nu}{dr}+C_0\\frac{Te^{-\\frac{\\mu}{2}}}{r^2}~,\n\\label{3.17}\n\\end{equation}\nwhere $C_0$ is the constant of integration. To fix this constant impose the condition that the variation of the proper temperature; i.e. $dT\/dr$ vanishes at $r=0$, although $T$ is not equal to zero. It yields $C_0=0$. Finally integrating once again we find\n\\begin{equation}\nT=Ce^{-\\frac{\\nu}{2}}~,\n\\label{3.18}\n\\end{equation}\nwith $C$ is a constant. This is the well known Tolman relation for temperature distribution.\n    \n   As promised earlier, I showed that the Tolman temperature can be derived purely from the surface term of the action. Let me now discuss the significance and importance of the present analysis. As we know an accelerated observer also measures temperature of the Rindler horizon, known as Unruh effect.  Note that such a metric is not a solution of Einstein's equations. Furthermore, one can verify that the Unruh temperature obeys the Tolman relation. Therefore, it is quite apparent that Tolman relation is much more fundamental than the Einstein's equation and hence one should expect an {\\it off}-shell derivation of such relation. This has precisely been shown here. In the whole analysis no {\\it explicit} information of Einstein's equations of motion has been borrowed. Moreover, as I mentioned earlier, the surface action is related to the horizon entropy. So it is obvious that we are actually extremizing the gravitational entropy, instead of the radiation entropy. Hence the present analysis signifies that the Tolman relation is more than a temperature distribution of the external fields under gravitational background; rather it is actually a relation for gravity itself. It is consistent with the Hawking effect in which case the constant $C$ is identified as the Hawking temperature. This is the temperature of the event horizon as measured from infinity. But if an observer radially moves towards the horizon, the measured value is redshifted by the relation (\\ref{3.18}). Note that in this case the observer dependent value depends only on the gravitational fields. That is all the entities, appearing in the relation, are the properties of gravity itself; no properties of external matter are playing any role.\n   \n   The last point I want to mention is that in the calculation the contribution from the GHY action for all the surfaces has been taken into account which led to the correct expression for Tolman temperature. The same has also been observed earlier in \\cite{Padmanabhan:2002sha} and \\cite{Majhi:2015pra} in the context of thermodynamics of horizon from the surface term.\n\n\n\n\\section{\\label{Conclusions}Conclusions}   \n       In this short paper, I showed that it is possible to find the well know Tolman temperature from the surface term of the gravitational action. Here, GHY has been considered. The difference between the original one and the present one is that we do not need any explicit information about the evaluation of spacetime; i.e. the Einstein's equation of motion as well as the existence of external matter. In this sense this analysis is an {\\it off}-shell derivation. As I stated earlier, this has an important consequence in the context of Unruh effect. Moreover, since the GHY action is related to entropy of horizon and as the extremization of it led to the Tolman relation, one can argue that the expression is a consequence of the extremization of gravitational entropy. This completely a new observation as in earlier case \\cite{Tolman:1930zza} the same was obtained by extremizing the radiation entropy.\n       \n       Finally, so far only the surface term has been taken into account. It would be interesting to look into the same problem by considering the EH action only and also in presence of both EH action and GHY term. In addition, let me mention that the present analysis is confined within the general theory of relativity (GR). But we know that the Tolman relation is beyond any particular theory; it is applicable to any theory of gravity. Therefore, one needs to clarify this point by considering the surface term beyond GR theory. Let me also point out that even the original like derivation for other theories of gravity is absent in literature. The investigations in these directions are going on.\n   \n    \n \n\n\\vskip 9mm\n\\noindent\n{\\bf{Acknowledgements}}\\\\\n\\noindent\nThe research of the author is supported by a START-UP RESEARCH GRANT (No. SG\/PHY\/P\/BRM\/01) from Indian Institute of Technology Guwahati, India.\n\n\\vskip 9mm\n\\section*{Appendix}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn many relevant scientific and engineering fields one is presented with time series data switching back and forth between different regimes. A classical estimation problem in this setting is the well-studied change point detection problem where one tries to estimate when some properties of the sequence of the random variables changes. This local property is of prime importance in many learning tasks such as signal segmentation \\cite{Abou-Elailah16,Kim09}, change point detection in comparative genomics for early cancer diagnosis \\cite{Lai05}, and modeling and forecasting of changes in financial data \\cite{Lavielle06,Spokoiny09}.\n\nFor other applications, one needs more than this local answer and is interested in a more general overview of the time series where for instance earlier data samples behave like new ones. Examples of this are found in: electricity market data, where prices might have different behavior corresponding to different price regimes that might reappear depending on some triggering events; signal partitioning with some parts of the signal sharing similar properties; and speech segmentation with different alternating sources. Generally speaking, it is of interest in these situations to determine the change points and the clusters for a more precise description of the inhomogeneous time series. \n\nParametric models for solving the change point detection problem have been proposed in \\citet{Cleynen14} and \\citet{Rigaill12}. However, in dealing with the change point and clustering problem we would naturally require that our solution does not assume any knowledge of the number of changes nor the actual number of clusters, as these numbers would evolve over time, so we expect new changes in the process to happen and new clusters to form as $N$, the number of samples, grows. Thus, any practical procedure should be able to estimate these numbers and also have adaptive guarantees with respect to how fast these numbers grow. Similar setups for change point detection have been the subject of study by \\citet{Harchaoui07a}, \\citet{Arlot16} and \\citet{Garreau17} who use characteristic kernels for detecting changes in the distribution, while from a computational stand point a more effective implementation has been proposed by \\citet{Celisse17}. In this study, we will restrict ourselves to an \\emph{iid} (independent and identically distributed) Gaussian sequence model of the data with known variance, noting that the same study can be done using kernels and that the algorithm we develop can be effectively implemented using the same procedure as in \\cite{Celisse17}, as explained later in the paper.\n\nTwo other related lines of research, but which we do not explore here, are on-line algorithms for segmentation and $L_1$-regularized segmentation. We refer the reader to \\cite{Tartakovsky14} for an extensive review of on-line algorithm for change point detection. Data segmentation using the $L_1$-penalty, also known as total variation denoising, was introduced by \\citet{Rudin92}. The one dimensional case, corresponding to the Fussed LASSO, has been studied in \\citep{Tibshirani05} and \\citep{Rennie69} and an efficient algorithm  has been proposed by \\citet{Arnold16}. More recent results can be found in \\citep{Dalalyan17} for the one dimensional case and \\citep{H\u00fctter16} for two dimensional case.\n\n\\textbf{Main contribution}: The change point detection and clustering for sequences of data points does not seem to have been previously studied. In this work we propose a two-pass dynamic programming algorithm for selecting an adequate model from a collection of candidate models. We motivate the choice of the algorithm computationally by showing that it runs in $\\mathcal{O}(N^2D+D^4)$ time (where $D$ is an upper bound on the number of change points), statistically by showing that it can be seen as an approximation of a computationally hard MAP optimization problem for which we can derive an oracle inequality that guarantees low sample complexity, consistency and adaptivity, and practically by testing the model on simulation data.\n\n\\textbf{Structure of the paper}: In Section~2 we formulate the problem as one of nonparametric model selection from a family of models over all partitions of the data set. After some preliminaries and notations are given in Section~3, we propose in Section~4 a two-pass dynamic programing algorithm as a computationally effective relaxation of the optimization criterion and analyze its computational cost. We then put the model selection problem in a  Bayesian framework in Section~5, and use a Laplace-type approximation to derive as optimization criterion the maximum a-posteriori probability. In Section~6 we derive an oracle inequality for the criterion that our algorithm is approximating, and study its properties. Experimental results showing that the clusters and segments can be obtained effectively estimated are presented in Section~7 using simulation data.\n\n\\section{Problem formulation}\n\nLet $\\mathcal{Y}$ be a measurable space and $Y_1,Y_2,\\dots,Y_N \\in \\mathcal{Y}$ denote random variables with distributions $P_{Y_i}$. Our goal is on one hand to detect changes in the sequence of distribution measures $(P_{Y_i})_{i=1}^N$ and on the other hand to cluster the data points coming from the same process. Hence we put random variables between two consecutive changes in the same segment, and we think of random variables of the same segment or different segments as belonging to the same cluster, if they are the realization of the same process. \n\nOne important case both in theory and in practice is the uniform constant design model were the $Y_i$s depend on deterministic variables uniformly spaced on a grid $X_i=i$ for $i \\in \\llbracket 1,N \\rrbracket := \\{1, \\dots, N\\}$ through a regression function $f^*$ with an additive \\emph{iid} random noise $(\\epsilon_i)_{i=1}^N$. Taking the distribution of the $\\epsilon_i$'s as $\\mathcal{N}(0,\\sigma^2)$ with known variance, we end up with the following Gaussian sequence model:\n\\begin{equation} \\label{regression}\nY_i=f_i^*+\\epsilon_i, \\quad \\text{ for } \\ i \\in \\llbracket 1,N \\rrbracket.\n\\end{equation}\nHere we are placed in a regression setting of the form $Y=f^*+\\epsilon$, where $Y = [Y_1\\,\\,\\cdots\\,\\,Y_N]^T$, $f^* = [f_1^*\\,\\,\\cdots\\,\\,f_N^*]^T$ and $\\epsilon = [\\epsilon_1\\,\\,\\cdots\\,\\,\\epsilon_N]^T \\sim \\mathcal{N}(0,\\sigma^2I_N)$, and we are interested in estimating $f^*$ as a piecewise constant function that takes a limited number of values. \n\nWe emphasize that it is unlikely that the data correspond exactly to a piecewise constant function plus independent random Gaussian noise and that we are in this low dimensional hidden structure exactly, yet there might exist a good sparse linear approximation. Hence our search is not for an exact model, rather we are trying to select the best model in a collection of candidates, as we explain in the next section.\n\n\\section{Preliminaries and notation}\n\nWe would like to perform dimensionality reduction by exploiting the hidden structure on the data sequence $Y_1,Y_2,\\dots,Y_N$. To do this we split it into different segments while also putting the segments sharing same mean into the same cluster. Hence if we knew the clusters our problem reduces to fitting a constant to a set of observations over each cluster. Observe that if $f^*$ is constant over parts of $\\llbracket 1, N \\rrbracket$, the it determines a clustering of $Y_1,Y_2,\\dots,Y_N$ over the values where it is constant. Hence, we can think about the problem as, first determining the clustering of the $Y_1, Y_2, \\dots, Y_N$ which would result in a partition $\\pi$ of $\\llbracket 1,N \\rrbracket$, and then choosing the best value of $\\hat{f}$ over each part as our estimate. So $f^*$ belong to the subspace $\\mathcal{F}_{\\pi}$: subspace of functions that are constant over the parts of the partition $\\pi$. \n\nTo formalize this, let $\\mathcal{M}$ be an index set over the collection of partitions $\\Pi_N$ of $\\llbracket 1,N \\rrbracket$; given $m \\in \\mathcal{M}$, denote by $\\mathcal{F}_m$ the subspace of functions that are constant over the parts of $\\pi_m$. Our goal is two-fold: find $\\hat{m}$ as the index estimate of $\\mathcal{F}_{\\hat{m}}$, the subspace where the estimate of $f^*$ lives, and from $\\mathcal{F}_{\\hat{m}}$ compute $\\hat{f}_{\\hat{m}}$ as our estimate. We represent a partition $\\pi$ as an unordered collection of its subsets $\\pi=\\{[1],[2],\\dots,[|\\pi|]\\}$ with $[k]$ being the $k^{th}$-equivalent class, -part or -cluster, and $|\\pi|$ the cardinality of the partition. Every part $[k]$ can be seen as the union of segments $[k]=\\{[k_1],[k_2],\\dots,[k_{|[k]|}]\\}$ where $(k_i)_{i=1}^{|[k]|}$ is the collection of maximal intervals in $[k]$ that we call segments of the $k^{th}$-cluster. The last element in each segment $[k_i]$ is called a change point. We define $d'_m := |\\pi_m|-1 = \\dim(\\mathcal{F}_m)-1$ as the clustering dimension. Even though this choice might create some confusion it will be consistent the notations used in the proofs of sections 5 and 6. Also we define $d''_m= | \\pi_m|_0 := |\\cup_{k=1}^{d_m'+1}[k]|$ as the change point dimension. \n\nTo link partitions to subspaces let $e_l := (0,\\dots,1,\\dots,0)$ be the $l^{th}$-component of the standard orthonormal basis of $\\mathbb{R}^N$, and define for a subset $A$ of $\\llbracket 1,N \\rrbracket$ the vector $\\mathbbm{1}_A:=\\sum_{l \\in A} e_l$. For $[k]$, the $k^{th}$ cluster of $\\pi_m$, with a slight abuse of notation we define $\\mathbbm{1}_{[k]} := \\sum_{i=1}^{|[k]|} e_{k_i}$, and observe that $\\mathcal{F}_m=\\spn\\{\\mathbbm{1}_{[k]}\\colon k \\in \\pi_m\\}$, which is consistent with the definition of the clustering dimension $d'_m := |\\pi_m|-1 = \\dim(\\mathcal{F}_m)-1$.\n\nWe define $\\spanf{f^*} := \\spn\\{f^*\\}$, $\\mathcal{S}_1 \\oplus \\mathcal{S}_2$ as the direct sum of the two vector space $\\mathcal{S}_1 $ and $ \\mathcal{S}_2$, and $\\mathcal{S}_1 \\ominus \\mathcal{S}_2$ as their direct difference. $\\Proj_{\\mathcal{S}}$ denotes the (orthogonal) projection operator onto the subspace $\\mathcal{S}$. We also define the partitions inclusion as $m_1 \\subset m_2$ if $\\mathcal{F}_{m_1} \\subset \\mathcal{F}_{m_2}$, or equivalently if $\\pi_{m_2}$ is finer than $\\pi_{m_1}$. \n\n\\begin{figure}[t]\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{ICML_example}}\n\\caption{Example of a piecewise constant signal $f^*$ (black line) and observed signal $Y$ (pink line) with clustering values  (blue circles) and change points (green circles).}\n\\label{fig1}\n\\end{center}\n\\end{figure}\n\n\\textbf{Example 1.} \\begin{small} \\it\nConsider the signal $f^*$ of Figure~\\ref{fig1}, whose partition is \n\\begin{align} \\label{example}\n\\pi = \\{ [ 1 ];[ 2 ];[ 3 ];[ 4 ];[ 5 ] \\},\n\\end{align}\nwhere\n\\begin{align*}\n[1] &=\\llbracket 615,678 \\rrbracket \\cup \\llbracket 821,926 \\rrbracket \\cup \\llbracket 1019,1211 \\rrbracket\\ \\cup \\llbracket 1753,2000 \\rrbracket \\\\\n[2] &=\\llbracket 1,100 \\rrbracket \\cup \\llbracket 679,820 \\rrbracket \\cup \\llbracket 1212,1280 \\rrbracket\\ \\\\\n[3] &=\\llbracket 101,214 \\rrbracket\\ \\cup \\llbracket 505,614 \\rrbracket \\cup \\llbracket 926,1018 \\rrbracket \\cup \\llbracket 1281,1600 \\rrbracket \\\\\n[4] &=\\llbracket 215,504 \\rrbracket\\\\\n[5] &=\\llbracket 1601,1752 \\rrbracket.\n\\end{align*}\nHence, $d_{\\pi}'= 4$ and $d_{\\pi}''= 12$ for this signal.\n\\end{small}\n \nWe also denote by $C_k^N$ the binomial coefficient that gives the number of ways, disregarding order, that $k$ objects can be chosen from among $N$ objects. This is given by\n\\begin{align}\nC_k^N := \\frac{N!}{k!(N-k)!}. \\label{binomial coefficient}\n\\end{align}\nThe Stirling numbers of the second kind, $S(N,k)$, correspond to the number of ways to partition a set of $N$ objects into $k$ non-empty subsets, or, similarly, to the number of different equivalence relations with precisely $k$ equivalence classes that can be defined on an set of $N$ elements.\n\nWe are precisely interested in the case where the element set is $\\llbracket 1,N \\rrbracket$ and the distance between every two elements in each equivalence class is at least 2; we denote the number of such equivalent classes by $S^2(N,k)$. $S(N,k)$ and $S^2(N,k)$ satisfy the following recurrence relations:\n\\begin{align}\nS(N,k) &= S(N-1,k-1) + kS(N-1,k), \\quad N \\geq k, \\nonumber \\\\\nS^2(N,k) &= S(N-1,k-1), \\quad N, k \\geq 2. \\label{recursive Stirling}\n\\end{align}\nFor the proofs of these results, we refer the reader to \\cite{ConcreteMathematics} and \\cite{AMohr}.\n\n\\section{Two-pass dynamic programming for change point detection and clustering}\nTo solve the change point and clustering problem, a natural approach is to consider the minimization of a criterion of the form\n\\begin{align} \\label{Criterion}\n\\Crit(m) = \\|y-\\hat{f}_m\\|_2^2 + \\sigma^2K\\pen(m),\n\\end{align}\nwith a penalty term $\\pen(m)$ depending only on $d'$ and $d''$ and a multiplicative tuning parameter $K$. Indeed, as we shall see the penalty can be chosen such that the minimizer $\\hat{f}_{\\hat{m}}$ of \\eqref{Criterion} behaves like an approximation to a maximum a-posteriori estimator (MAP), and also, the average expected risk $\\frac{1}{N}\\mathbb{E}[\\|\\hat{f}_{\\hat{m}}-f^*\\|_2^2] \\to 0$ for a large class of signals $f^*$, namely, those corresponding to models with $d'\\leq d''= o(N\/\\ln N)$, \\emph{i.e.}, $f^*$ is a consistent estimator for those signals. The specific form of $\\pen(m)$ will be derived in the next section, based on an oracle inequality that will guarantee consistency and adaptivity of our estimator.\n\nAlthough the estimator $\\hat{f}_{\\hat{m}}$ enjoys good statistical properties, from a computational stand it would involve the exploration of $\\mathcal{M}$. The set $\\mathcal{M}$ is identified with the collection of all the partitions of $\\llbracket 1,N \\rrbracket$, whose number asymptotically behaves like $\\mathcal{O}(Ne^N\/\\ln N )$, rendering the minimization of the criterion~\\eqref{Criterion} computationally challenging. A way to bypass this issue for the change point only detection problem is via dynamic programming~\\cite{Harchaoui07a}; this approach works in this simplified setup since there is a natural ordering for exploring the subproblems, which does not hold here. To overcome this, we will relax the criterion in such a way to create a subproblem ordering and thus derive a computationally feasible approximation. The proposed new method is outlined in Algorithm~\\ref{two passes Dynamic programming algorithm}.\n\n\\begin{algorithm}\n\\begin{small}\n   \\caption{Two-Pass Dynamic Programming Algorithm}\n   \\label{two passes Dynamic programming algorithm}\n\\begin{algorithmic}[1]\n   \\INPUT data points $(y_i)_{i=1}^N$, maximum number of changes $D$ and penalty strength $K$.\n   \\STATE {\\begin{align*}\n   \\\\[-1.2cm] \\bar{y}_{[k,l]} &:= \\frac{\\sum_{i=k}^{l} Y_i}{k-l+1} \\\\\n   R_{[k,l]} &:= \\sum_{i=k}^{l} (y_i-\\bar{y}_{[k,l]})^2, \\quad 1 \\leq k \\leq l \\leq N.\n   \\end{align*}}\n   \\FOR {$d=1$ {\\bfseries to} $D$}\n   \\STATE \\label{step:1} use the dynamic programing recurrence in  \\eqref{dynamic programming 1} and a backtracking step to compute \n   \\begin{align}\n   C_d(N) &:= \\min \\limits_{|\\bar{m}|=d} \\|Y - \\Proj_{\\mathcal{F}_{\\bar{m}}} Y\\|^2, \\label{change point detection} \\\\\n   \\tilde{m}_d &\\in \\arg \\min \\limits_{|\\bar{m}|=d} \\|Y - \\Proj_{\\mathcal{F}_{\\bar{m}}} Y\\|^2. \\nonumber\n   \\end{align}\n    \\ENDFOR\n   \\FOR{$d=1$ {\\bfseries to} $D$}\n   \\STATE \\begin{align*}\n   \\\\[-1.2cm] \\tilde{m}_d &=: \\{0 \\leq i_1 < i_2 < \\dots < i_d < N\\} \\\\\n   (\\alpha_k)_{k=0}^{d} &:= (i_{k+1}-i_k)_0^{d}, \\quad (i_0=0, i_{d+1}=N).\n   \\end{align*}\n   \\STATE sort $(\\bar{y}_{[1,i_1]}, \\bar{y}_{[i_1+1,i_2]}, \\dots, \\bar{y}_{[i_d+1,N]})$.\\label{step:2}\n   \\STATE $\\left(\\bar{y}_{(k)}\\right)_{k=0}^d := $ ordered sequence of $(\\bar{y}_{[i_k+1,i_{k+1}]})_{k=0}^{d}$ $(\\alpha_{(k)})_{k=0}^{d} := $ corresponding permuted $(\\alpha_k)_{k=0}^{d}$ according to permutation $\\phi_d$.\n   \\STATE $\\bar{\\bar{y}}_{(k,l)} := \\frac{\\sum_{i=k}^{l} \\alpha_{(i)}\\bar{y}_{(i)}}{\\sum_{i=k}^{l-1} \\alpha_{(i)}}$ and $\\bar{R}_{[k,l]} := \\sum_{i=k}^{l} \\alpha_{(i)}(\\bar{y}_{(i)}-\\bar{\\bar{y}}_{(k,l)})^2$, $1 \\leq k \\leq l \\leq d$.\n\\FOR{$\\delta=1$ {\\bfseries to} $d$}\n   \\STATE \\label{step:3} use the dynamic programing recurrence in  \\eqref{dynamic programming 2} and a backtracking step to compute  \n   \\begin{align} \\label{clustering}\n   G_{(d,\\delta)} &:= \\min \\limits_{m \\in \\mathcal{M}_{\\bar{y}_{\\tilde{m}},\\delta}} \\| \\Proj_{\\mathcal{F}_{\\bar{m}}} Y - \\Proj_{\\mathcal{F}_{m}}\\Proj_{\\mathcal{F}_{\\bar{m}}}  Y\\|^2, \\\\\n   \\tilde{\\tilde{m}}_{(d,\\delta)} &\\in \\arg \\min \\limits_{m \\in \\mathcal{M}_{\\bar{y}_{\\tilde{m}}\\delta}} \\| \\Proj_{\\mathcal{F}_{\\bar{m}}} Y - \\Proj_{\\mathcal{F}_{m}}\\Proj_{\\mathcal{F}_{\\bar{m}}} Y\\|^2. \\nonumber\n   \\end{align} \n     \\ENDFOR\n   \\ENDFOR\n   \\STATE $B_{(d,\\delta)} := C_d + G_{(d,\\delta)} + \\sigma^2K \\pen((d,\\delta))$, $1 \\leq \\delta \\leq d \\leq D$.\n   \\STATE \\label{step:4} $(\\hat{d},\\hat{\\delta}) := \\arg \\min \\limits_{1 \\leq \\delta \\leq d \\leq D }B_{(d,\\delta)}$.\n   \\STATE \\label{step 5} reconstruct $m_{(\\hat{d},\\hat{\\delta})}$ from $\\tilde{m}_{\\hat{d}}$ and $\\tilde{\\tilde{m}}_{(\\hat{d},\\hat{\\delta})}$.\n   \\OUTPUT value of criterion $\\Crit(m_{(\\hat{d},\\hat{\\delta})}) =B_{(\\hat{d},\\hat{\\delta})}$ and selected model for change points and clusters $m_{(\\hat{d},\\hat{\\delta})}$.\n\\end{algorithmic}\n\\end{small}\n\\end{algorithm}\n\n\n\nLet $\\bar{y}_{[k]} := (\\sum_{i \\in [k]} Y_i)\/|[k]|$, the average of the elements of $Y$ in the $[k]$-th part. Notice that, given $\\pi_m = \\{ [ 1 ];[ 2 ];\\dots;[ d'_m -1] \\}$, we have\n\\begin{align*}\n\\Proj_{\\mathcal{F}_m}Y = \\sum \\limits_{k=1}^{d'_m-1}  \\frac{\\inner{Y}{\\mathbbm{1}_{[k]}}}{\\|\\mathbbm{1}_{[k]}\\|^2}\\mathbbm{1}_{[k]}\n=\\sum \\limits_{k=1}^{d'_m-1} \\bar{y}_{[k]} \\mathbbm{1}_{[k]}. \n\\end{align*}\n\nThe minimization of criterion~\\eqref{Criterion} can then be equivalently written as \n\\begin{equation*}\n\\begin{aligned}\n\\min \\limits_{m \\in \\mathcal{M}} \\Crit(m) \n&= \\min \\limits_{m \\in \\mathcal{M}} \\{ \\|y-\\Proj_{\\mathcal{F}_m}Y\\|_2^2 + \\sigma^2K\\pen(d'_m,d''_m) \\} \\\\\n&= \\min \\limits_{0 \\leq d' \\leq d'' \\leq D} \\left\\{ \\min   \\limits_{\\substack{|m|=d'\\\\ |m|_0=d''}} \\|y-\\Proj_{\\mathcal{F}_m}Y \\|_2^2 + \\sigma^2K\\pen(d',d'') \\right\\},\n\\end{aligned}\n\\end{equation*}\nwhere $D$ is a reasonable upper bound on the number of change points. As we shall see later, from a statistical point of view there is no need to explore all possible values of $d'$ and $d''$, since the statistical guarantees only hold in a regime where $d' \\leq d'' = o(N \/ \\ln N)$.\n\nWe define $\\pi_{\\bar{m}}$ to be the partition having as elements all the segments of $\\pi_{m}$ and instead of computing the minimum exactly we will take a greedy step by defining $$\\tilde{m} := \\arg \\min_{|\\bar{m}|=d''} \\|Y - \\Proj_{\\mathcal{F}_{\\bar{m}}} Y\\|^2$$ and defining $\\mathcal{M}_{\\tilde{m},d'}: = \\{ m \\in \\mathcal{M}\\colon m \\subset \\tilde{m},|m|=d'\\} $, which can be identified with the collection of all partitions of $\\llbracket 1,d'' \\rrbracket$ into $d'$ sets. We restrict further this collection to partitions $\\pi$ satisfying what we call the \\textbf{clustering property}, which states that if $\\mathbb{I}_1$, $\\mathbb{I}_2$ and $\\mathbb{I}$ are segments in some (possibly different) parts of $\\pi$, then \n\n\\begin{align}\\label{clutering property}\n    \\begin{cases*}\n     \\mathbb{I}_1, \\mathbb{I}_2 \\in [k] \\\\\n     \\bar{y}_{\\mathbb{I}_1} \\leq \\bar{y}_{\\mathbb{I}} \\leq \\bar{y}_{\\mathbb{I}_2}\n    \\end{cases*}\n    \\quad \\Rightarrow \\quad \\mathbb{I} \\in [k].\n\\end{align}\nThis sub-collection will be denoted as $\\mathcal{M}_{\\bar{y}_{\\tilde{m}},d'}$. Simply put, this property says that the partitions considered are those that respect the ordering of $(\\bar{y}_{[i_k+1,i_{k+1}]})_{k=0}^{d''}$, since if two segments \n$\\mathbb{I}_1,\\mathbb{I}_2$  belong to $[k]$, and the segment $\\mathbb{I}$ satisfies $\\bar{y}_{\\mathbb{I}_1} \\leq \\bar{y}_{\\mathbb{I}} \\leq \\bar{y}_{\\mathbb{I}_2}$, then it should also be in cluster $[k]$.\n\nThis leads to the following upper bound, whose detailed derivation is given in appendix B:\n\\begin{align*}\n\\min \\limits_{m \\in \\mathcal{M}} \\Crit(m)  &\\leq \\min \\limits_{0 \\leq d'' \\leq D} \\bigg\\{ \\min \\limits_{|m|=d''} \\|Y - \\Proj_{\\mathcal{F}_{m}} Y\\|^2 \\\\ \n&\\qquad + \\min \\limits_{\\substack{0 \\leq d' \\leq d'' \\\\ m \\in \\mathcal{M}_{\\bar{y}_{\\tilde{m}},d''}}} \\| \\Proj_{\\mathcal{F}_{\\tilde{m}}} Y - \\Proj_{\\mathcal{F}_{m}}\\Proj_{\\mathcal{F}_{\\tilde{m}}} Y\\|^2   + \\sigma^2K\\pen(d',d'') \\bigg\\}.\n\\end{align*}\n\nTherefore, we can define the following relaxation for the minimization of the criterion in ~\\eqref{Criterion}:\n\\begin{align} \\label{equation for the algorithm}\n\\Crit_r(d'') &:=   \\min \\limits_{|m|=d''} \\|Y - \\Proj_{\\mathcal{F}_{m}} Y\\|^2 \\nonumber \\\\ \n&+ \\min \\limits_{\\substack{0 \\leq d' \\leq d''\\\\ m \\in \\mathcal{M}_{\\bar{y}_{\\tilde{m}},d''}}} \\Bigg\\{ \\| \\Proj_{\\mathcal{F}_{\\tilde{m}}} Y - \\Proj_{\\mathcal{F}_{m}}\\Proj_{\\mathcal{F}_{\\tilde{m}}} Y\\|^2  + \\sigma^2K\\pen(d'_m,d''_m) \\bigg\\}.\n\\end{align}\nand our algorithm computes $\\min \\limits_{0 \\leq d'' \\leq D} \\Crit_r(d'')$ and returns $m_{(\\hat{d},\\hat{\\delta})}$. From this last definition we observe that\n\\begin{align*}\n\\min \\limits_{m \\in \\mathcal{M}} \\Crit(m) \\leq \\Crit(m_{(\\hat{d},\\hat{\\delta})})=\\min \\limits_{0 \\leq d'' \\leq D} \\Crit_r(d'').\n\\end{align*}\nThus, obtaining $m_{(\\hat{d},\\hat{\\delta})}$ ensures making progress toward the minimization of $\\Crit(m)$. The Two-Pass Dynamic Programming Algorithm~\\ref{two passes Dynamic programming algorithm} is aimed at doing this by computing the value of the minimum in ~\\eqref{equation for the algorithm} and returning a solution $\\hat{m}=m_{(\\hat{d},\\hat{\\delta})}$ in the following way:\n\n\\medskip\n\\textbf{Details of Main Steps in Algorithm~\\ref{two passes Dynamic programming algorithm}}\n\\begin{small}\n\\begin{itemize}\n\\item \\textbf{Step~3:} It computes $C_d(n)$ defined in ~\\eqref{dynamic programming 1} for all $d$ and $n$ to obtain $C_d(N)$ for all $d \\in \\llbracket 1,N \\rrbracket$. It does so by using a dynamic programming algorithm that computes recursively for all $ 2 \\leq d \\leq D$ and $d \\leq n \\leq N$ the following recurrence, similar to the one in \\citet{Jackson05}:\n\\begin{align} \\label{dynamic programming 1}\nC_1(n) &:= R_{[1,n]} \\\\\nC_d(n) &:= \\min \\limits_{i \\in \\llbracket d,n \\rrbracket} \\{ C_{d-1}(i-1) + R_{[i,n]}\\}, \\; d \\geq 2. \\nonumber\n\\end{align}\n\\item \\textbf{Step~7:} For all values of $d$, it sorts the obtained segments according to their levels to yield $\\left(\\bar{y}_{(k)}\\right)_0^d$, and it keeps track of the segments' sizes as $(\\alpha_k)_{k=0}^{d}=(i_{k+1}-i_k)_0^{d}$. \n\\item \\textbf{Step~11:} It runs a modified dynamic programming recurrence on $\\left(\\bar{y}_{(k)}\\right)_0^d$ that uses weights according to the sizes $(\\alpha_{(k)})_{0}^{d}$. It does so using the following recurrence for all $1 \\leq \\delta \\leq t \\leq d$:\n\\begin{align}\\label{dynamic programming 2}\nG_{(t,1)} &:=  \\bar{R}_{[1,t]}, \\\\\nG_{(t,\\delta)} &:= \\min \\limits_{i \\in \\llbracket \\delta,t \\rrbracket} \\{ G_{(i-1,\\delta-1)} + \\bar{R}_{[i,t]}\\}, \\; \\delta \\geq 2. \\nonumber\n\\end{align}\n\\item \\textbf{Step~15:} It computes the minimum in ~\\eqref{equation for the algorithm} and finds for which model it is attained by solving the minimization problem:\n\\begin{align*}\n(\\hat{d},\\hat{\\delta}) := \\arg \\min \\limits_{1 \\leq \\delta \\leq d \\leq D }B_{(d,\\delta)}.\n\\end{align*}\n\\item \\textbf{Step~16:} It finally reconstructs $m_{(\\hat{d},\\hat{\\delta})}$ from $\\tilde{m}_{\\hat{d}}$ and $\\tilde{\\tilde{m}}_{(\\hat{d},\\hat{\\delta})}$ using the permutation $\\phi(\\hat{d})$.\n\\end{itemize}  \n\\end{small}\n\nThis algorithm can be thought of as an efficient way to compute the relaxation in ~\\eqref{equation for the algorithm}, based on solving the change point detection problem in ~\\eqref{change point detection} using the dynamic programing recurrence of ~\\eqref{dynamic programming 1}, followed by a solving a clustering problem in ~\\eqref{clustering} using the dynamic programing recurrence of ~\\eqref{dynamic programming 2}.\n\nThe next theorem shows that Algorithm~\\ref{two passes Dynamic programming algorithm} correctly solves the minimization problem in ~\\eqref{equation for the algorithm} and explicits its time and space complexity.\n\\begin{theorem}\\label{computaional complexity}\nLet $(y_i)_{i=1}^N \\subset \\mathbb{R}$, $D \\in \\mathbb{N}$ and $K>0$. Then,\n\\begin{itemize}\n\\item for all $1 \\leq d \\leq D$,\n\\begin{align*}\n\\tilde{m}_d \\in \\arg \\min \\limits_{|\\bar{m}|=d} \\|Y - \\Proj_{\\mathcal{F}_{\\bar{m}}} Y\\|^2,\n\\end{align*}\n\\item for all $1 \\leq \\delta \\leq d \\leq D$,\n\\begin{align*} \n\\tilde{\\tilde{m}}_{(d,\\delta)} \\in \\arg \\min \\limits_{m \\in \\mathcal{M}_{\\bar{y}_{\\tilde{m}}\\delta}} \\| \\Proj_{\\mathcal{F}_{\\bar{m}}} Y - \\Proj_{\\mathcal{F}_{m}}\\Proj_{\\mathcal{F}_{\\bar{m}}}  Y\\|^2.\n\\end{align*}  \n\\end{itemize}\nFurthermore, Algorithm~\\ref{two passes Dynamic programming algorithm} correctly solves the minimization problem in ~\\eqref{equation for the algorithm}, with time and space complexity $\\mathcal{O}(N^3+D^4)$ and $\\mathcal{O}(N^2+D^3)$, respectively. \n\\end{theorem}\n\\begin{proof}\nSee Appendix~B.\n\\end{proof}\n\nThe time and space complexity can be improved to $\\mathcal{O}(N^2D+D^4)$ and $\\mathcal{O}(DN+D^3)$, respectively. We refer the reader to the discussion after the proof in Appendix~B for the derivation of this result. In this way we obtain a computationally feasible algorithm that finds the minimum in \\eqref{equation for the algorithm} and returns an approximation to the criterion in \\eqref{Criterion}. In the next section we will motivate the use of Algorithm~\\ref{two passes Dynamic programming algorithm} from a statistical point of view by showing that the minimization of criterion~\\eqref{Criterion} can be viewed as an approximate maximum a-posteriori estimator. \n\n\\section{Model selection criterion for change point detection and clustering}\n\nIn this part we provide a derivation of the optimization criterion in ~\\eqref{Criterion}. We start by proposing a Bayesian model selection scheme, which is later inverted to arrive at an integral form of the maximum a-posteriori probability (MAP) estimator. Then we use a Laplace approximation to derive turn the MAP into an optimization problem of the desired form.\n\nHere we show that the proposed selection criterion in ~\\eqref{Criterion} follows naturally from a Bayesian reasoning. For this, we model the data as being the outcome of the following sampling model. The observation $Y$ is generated from a multivariate Gaussian of mean $F$ and variance $\\sigma^2I_N$ as described by ~\\eqref{regression}. For the random variable $F$, given that it belongs to a subspace $\\mathcal{F}_m$, we choose an absolutely continuous measure $\\mathcal{L}^{d'_m}$ with respect to $\\mathcal{\\lambda}^{d'_m}$, the Lebesgue measure on $\\mathbb{R}^{d'_m+1}$, such that $d\\mathcal{L}^{d'_m}=  l_{f\/m}d\\mathcal{\\lambda}^{d'_m+1} = \\prod \\limits_{k=1}^{d'_m+1}(l_{f_k\/m}d\\mathcal{\\lambda})$ with $l_{f_1\/m} = \\dots =l_{f_{d'_m+1}\/m}$. Later we will see that the choice $l_{f\/m}$ will not matter in comparison to the order of approximation, nevertheless we would like it to be a bounded continuous prior satisfying some additional conditions given in Lemma~\\ref{approx lemma}, even though we might be chosen as an improper prior. On the family of models $\\mathcal{M}$ we impose a categorical distribution measure $\\mathbb{P}_\\mathcal{M}$ as prior, with a weight $p_m$ for model $m$. Thus, we obtain the following sampling model for the data\\footnote{Here and in the sequel, the dependence of $p_m$ and $\\mathbb{P}_\\mathcal{M}$ on the number of samples $N$ is omitted, for simplicity of notation.}:\n\\begin{align}\nY\/F &\\sim \\mathcal{N}(F,\\sigma^2I_N) \\nonumber \\\\ \nF\/m &\\sim \\mathcal{L}^{d'_m} \\label{sampling scheme} \\\\  \nm &\\sim \\mathbb{P}_\\mathcal{M}=\\Categorical((p_m)_{m\\in \\mathcal{M}}). \\nonumber\n\\end{align}\nSince $Y$, $F$ and $m$ are now random variables, it makes sense to compute $\\mu_{m\/Y}$, the posterior distribution of $m$ given $Y$, and maximize it, to arrive at a MAP estimate of $m$ given bellow. \n{\\small\n\\begin{align} \\label{apposteriori}\np_{m\/Y}=\\frac{p_m \\displaystyle \\int_{f \\in \\mathcal{F}_m} \\phi_N \\left(\\frac{Y-f}{\\sigma}\\right) l_{f\/m}(f)df}{\\sum_{m' \\in \\mathcal{M}} p_{m'} \\displaystyle \\int_{f' \\in \\mathcal{F}_m} \\phi_N \\left(\\frac{Y-f'}{\\sigma}\\right) l_{f\/m'}(f')df'}. \n\\end{align}}\nFor the complete derivation of the formula in  \\ref{apposteriori} we refer you to appendix B.\n\nStarting from the a-posteriori distribution~\\eqref{apposteriori} we can derive an approximation for the MAP as follows:\n\\begin{align}\np_{m\/Y} &\\propto p_m \\int_{f \\in \\mathcal{F}_m} \\phi_N \\left(\\frac{Y-f}{\\sigma}\\right) l_{f\/m}(f)df \\nonumber \\\\\n&= p_m \\prod_{k=1}^{d_m'+1} \\frac{1}{(2\\pi\\sigma^2)^{\\frac{|[k]|}{2}}}   \\cdot \\int_{\\mathbb{R}} \\exp\\left(-\\frac{\\|y_{[k]}-f_k\\mathbbm{1}_{[k]}\\|_2^2}{2\\sigma^2}\\right) l_{f_k\/m}(f_k)df_k. \\label{for approx}\n\\end{align}\n\nIn the last step of \\eqref{for approx} we define $y_{[k]}$ as the vector obtained from the entries of $y$ corresponding to cluster $[k]$. To obtain an approximation of the MAP estimate as a solution of a criterion of the form~\\eqref{Criterion} we need  the result of lemma \\ref{approx lemma} stated and proved in Appendix C using a Laplace approximation type of argument. We then obtain the following upper bound for the MAP for all $K \\geq 1$ :\n\\begin{multline} \\label{upper bound for the MAP}\n\\Crit_\\text{MAP}(m) \\leq \\frac{\\|y-\\Proj_{\\mathcal{F}_m}y\\|_2^2}{2\\sigma^2} + K\\left(\\ln \\frac{1}{p_{m}} +\\frac{1}{2}(d_m'+1)\\ln\\frac{N}{d_m'}\\right)  +\\mathcal{O}(d_m').\n\\end{multline}\nThe complete derivation of \\eqref{upper bound for the MAP} can be found in Appendix~C. Now we define our approximate MAP criterion as:\n\\begin{align}\\label{Crit equation}\n\\Crit(m) &= \\|y-\\Proj_{\\mathcal{F}_m}y\\|_2^2 +\\sigma^2K \\pen(m) , \\nonumber \\\\\n\\pen(m) &= \\left(2 \\ln \\frac{1}{p_{m}} +(d_m'+1)\\ln\\frac{N}{d_m'}\\right).\n\\end{align}\nIn the next section we finish the specification of the penalty term by providing the probabilities $p_m$ over the space of models. To do so we will exhibit an oracle inequality satisfied by the estimator that minimizes \\eqref{Criterion}, and choose a probability mass function $(p_m)$ that gives a reasonable upper bound on the expected quadratic risk defined below.\n\n\\section{Oracle inequality and upper bound for the risk}\nThe standard way of assessing the performance of a statistical algorithm is by comparing its performance to a reasonable oracle. For this we use as a measure of performance of an estimator $\\hat{f}$ the expected quadratic risk:\n\\begin{align*}\n\\mathcal{R}_n(\\hat{f}) = \\mathbb{E}[\\|\\hat{f}-f^*\\|_2^2].\n\\end{align*}\nIn the case of the change point detection and clustering problem the comparison should be non-asymptotic, reflecting our lack of knowledge about both the clustering dimension and the change point dimension. For this we state below a non-asymptotic oracle inequality for $\\Crit(m)$ using an oracle with remainder of the form:\n\\begin{align*}\n\\inf_{m \\in \\mathcal{M}} \\{ \\mathcal{R}_n(\\Proj_{\\mathcal{F}_m}y) + o_m(1) \\}.\n\\end{align*}\nThis type of oracle has access to $f^*$ and chooses the $m$ that minimizes the risk criterion up to a remainder term.\n\nTo derive this we finish the specification of $\\Crit(m)$ by providing an appropriate prior $p_m$. The intuition behind our choice is the following. Defining $\\hat{r}_m =\\|y-\\hat{f}_m\\|_2^2$ and $\\pen(m)=2\\sigma^2 \\ln \\frac{1}{p_{m}} +\\sigma^2 (d_m'+1)\\ln\\frac{N}{d_m'}$ we see that the criterion~\\eqref{Crit equation} is of the form:\n\\begin{align*}\n\\Crit(m)=\\hat{r}_m + \\pen(m).\n\\end{align*}\nThe number of models in the family $\\mathcal{M}$ having the same values of $d_m'$ and $d_m''$ grows exponentially with those dimensions. Thus for fix $d_m'$ and $d_m''$ we might find a model with low $\\hat{r}_m$ just because of randomness since some of them will deviate largely from their means, which would correspond to an over-fitting case. Therefore, we need to penalize models of high dimensions more by taking into account the number of models with same dimensions. On the other hand we want this penalty to be as small as possible this way we give more importance to the fitting term $\\hat{r}_m$. In particular we would prefer the term $2\\sigma^2 \\ln \\frac{1}{p_{m}}$ to stay close to $\\sigma^2 (d'+1)\\ln\\frac{N}{d'}$ at least for values of $d_m'$ close to $d_m''$. \n\nOur choice for $p_m$, useful inequalities and a complete discussion of the role of $p_m$ as a prior and tuning parameter for the risk can be found in Appendix D. From Lemmas \\ref{lem:B_N} and \\ref{important bounds}, the following oracle inequality can be derived for $\\hat{f}_{\\hat{m}}$:\n\n\\begin{theorem}[Oracle inequality for $\\hat{f}_{\\hat{m}}$]\\label{thm:1}\nWith $\\mathcal{M}$ restricted to models such that $ed'_m \\leq N$ and for the choice of $m  \\in \\mathcal{M}$   corresponding to\n\\begin{align}\n\\hat{m} &\\in \\arg \\min \\limits_{m \\in \\mathcal{M}} \\|y-\\hat{f}_m\\|_2^2 + \\sigma^2K\\pen(m), \\\\\n\\pen(m) &:= 2\\ln \\frac{1}{p_{m}} +(d'+1)\\ln\\frac{N}{d'}, \\label{eq:pen}\n\\end{align}\nwith $K=3a$, we obtain for all $a>1$,\n\\begin{multline}\\label{oracle inequality} \nE_{f^*}[\\|  \\Proj \\limits_{\\mathcal{F}_{\\hat{m}} }Y  - f^* \\|^2]  \\leq \\arg\\min \\limits_{m \\in \\mathcal{M}} \\bigg\\{ \\frac{a}{a-1} \\mathbb{E}_{f^*}[\\|  \\Proj \\limits_{\\mathcal{F}_{m} }Y - f^*\\|^2] \\\\\n+ \\frac{a^2 \\sigma^2}{a-1} \\left(7 + 3(d_m'+1)\\ln\\frac{N}{d_m'}+6\\ln\\frac{1}{p_m} \\right) \\bigg\\}.\n\\end{multline}\n\\end{theorem}\n\\begin{proof}\nSee Appendix~D.\n\\end{proof}\nBy investigating the oracle inequality, one notices that for an optimal choice of $a$ one has to make a trade-off between the performance of the oracle part and the bias part of the inequality. In general this trade-off is not possible to optimize since the value of the oracle part is not available to us and depends on the variance of the noise. In practice, one can use the SLOPE heuristic introduced in \\citet{Lebarbier02} and described in \\citet{Baudry12} and  in \\cite{Arlot09}. In our case, the value of the tuning parameter can be chosen independently of the variance of the noise and we can use the value of $a$ for which we know that our estimator $\\hat{f}_{\\hat{m}}$ will perform well.\n\\begin{corollary}\\label{useful regime}\nFor the set of models described in ~\\eqref{regression} with $f^* \\in \\mathcal{F}_{m^*}$ the following properties hold:\n\\begin{itemize}\n\\item Adaptation and Risk Upper bound: The following adaptive upper bound in terms of $d_{m^*}'$ and $d_{m^*}''$ holds for $a=2$:\n{\\small \\begin{align*}\n&E_{f^*}[\\|  \\Proj \\limits_{\\mathcal{F}_{\\hat{m}} }Y  - f^* \\|^2] \\leq 4 \\sigma^2 \\bigg( 7 + 3(d_{m^*}'+1)\\ln\\frac{N}{d_{m^*}'} \\\\ \n& \\qquad \\qquad +6\\bigg(d_{m^*}' \\ln[d_{m^*}''e^{\\frac{13}{6}}] + d_{m^*}'' \\ln[d_{m^*}'e^2] + d_{m^*}'' \\ln \\frac{N}{d_{m^*}''} \\bigg) \\bigg).\n\\end{align*}}\n\\item Consistency: If $d_{m^*}'' = o(N \/ \\ln N)$, then: $$\\lim_{N \\to \\infty } \\frac{1}{N} \\mathbb{E}_{f^*}[\\| \\hat{f}_{\\hat{m}}- f^*\\|^2] = 0. $$\n\\end{itemize}\n\\end{corollary}\n\\begin{proof}\nSee Appendix~D.\n\\end{proof}\nWe notice that the consistency condition $d_{m^*}'' = o(N \/ \\ln N)$ is within the restriction on the models in theorem \\ref{thm:1}, hence there is no loss of generality of having only models with $ed'_m \\leq N$ in $\\mathcal{M}$ since for other models we cannot guarantee convergent mean square risk anyway. In the next section we validate these theoretical guarantees by a series of tests on simulated data to get a sense of how tight the oracle inequality is, which signals  are difficult to estimate and how the algorithm behaves in practice. \n\n\\section{Experimental results}\nConsider first an experiment based data generated randomly according to the setup of \\eqref{regression} with the same change points of Example~1. This is considered to be an easy case since $d'_{m^*}=4 < d''_{m^*}= 12 \\ll N = 2000$, which is within the range of signals for which the consistency result of Corollary~\\ref{useful regime} holds.\n\n\\begin{figure}[h] \n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{signal_to_noise}}\n\\caption{Estimates $\\hat f$ (blue line) of $f^*$ (doted black line) obtained by Algorithm~\\ref{two passes Dynamic programming algorithm} using the observed signal $Y$ (pink line), with $3$ different levels of signal-to-noise ratio.}\n\\label{fig2}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\nThe experiments in Figure~\\ref{fig2} show that the algorithm is quite robust to the level of noise as measured by the signal-to-noise ratio $S\/N = \\frac{\\textmd{magnitude of smallest jump in }f^*}{\\sigma^2}$. We observe that the difference between the ground truth $f^*$ and $\\hat{f}_{\\hat{m}}$ is quite small even for small $S\/N$ levels such as $S\/N=0.5$ and the change point locations do not vary appreciably; in fact, for this experiment, $S\/N =0.3$ seems to be the limiting case for which the algorithm performs well, and for lower values the risk upper-bound in Corollary~\\ref{useful regime} becomes loose when $\\sigma$ increases. Also, we note that an $S\/N$ of $0.5$ is quite low for these kind of problems. In particular, algorithms relaying on the $L_1$-penalty such as Fussed LASSO do not achieve this kind of performance on the simpler task of change point only detection, while on the other hand they are more computational efficient \\citep{Xin14}.\n\n\\begin{figure}[h]\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{difficult_signal}}\n\\caption{Estimate $\\hat f$ (blue line) of $f^*$ (black line) obtained by Algorithm~\\ref{two passes Dynamic programming algorithm} from a difficult observation sample $Y$ (pink line) with high signal-to-noise ratio ($1.5$) and few observations per segment ($N=500$ and $d''_{m^*}= 13$).}\n\\label{fig3}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\nFigure~\\ref{fig3} illustrates a difficult case, where we reduced the number of observation by segment by scaling down the signal $f^*$ to a support of size $N=500$. Now we are outside of the useful regime of Corollary~\\ref{useful regime} and we notice that the second segment $\\llbracket 15,53 \\rrbracket$ is wider than what it should since the first change point at $25$ was detected at $14$; also the segment $\\llbracket 206,237\\rrbracket$ belongs to cluster $[4]$ while it is actually in cluster $[3]$ in the original signal $f^*$. Nevertheless we can observe an interesting property for segment $\\llbracket 324,346 \\rrbracket$, namely, that the end point $346$ does not correspond to any real change point, yet this segment belongs to the optimal solution of the $1^{st}$ dynamic programming pass. On the other hand the $2^{nd}$ dynamic programming pass puts it in the same cluster $[3]$ as $\\llbracket 347,399 \\rrbracket$, turning them into one single segment of cluster $[3]$. This behavior actually is the norm for the algorithm, where false changes are often detected in difficult signals in the $1^{st}$ dynamic programming pass but are removed after the $2^{nd}$ pass. These kinds of false discoveries are actually one of the weaknesses of many change point only detection algorithms like Fussed LASSO, and they have been studied in \\citep{Harchaoui07b}, \\citep{Rinaldo09} and \\citep{Rojas14}. \n\\begin{figure}[h]\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{hist_icml_fig}}\n\\caption{Top histogram: location of estimated (black) and true (black) change points in 300 simulations. Middle histogram: location of estimated (red) and true (blue) clusters in 300 simulations. Bottom histogram: 300 estimates of the average mean square error (red) and its theoretical upper bound (blue).}\n\\label{fig4}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\nIn the last experiment, we run Algorithm~\\ref{two passes Dynamic programming algorithm} 300 times with the parameter values $d'_{m^*}=4 < d''_{m^*}= 12 \\ll N=2000$ and signal-to-noise ratio $S\/N = 1$; Figure~\\ref{fig4} summarizes the results. In the top histogram we notice that the algorithm successfully detects the change points most of the time; in fact, the achieved accuracy was $\\frac{\\textrm{number of change points correctly detected}}{\\textrm{number of change points detected}} \\approx 0.8528$. The middle histogram shows the placement of estimated clusters and the true values of the clusters; we observe that the true values lie in a small neighborhood of the estimated values for every cluster. In the bottom histogram we observe that the theoretical upper bound on the average mean square error --in this case $12.1575$-- found in Corollary~\\ref{useful regime} is very conservative and most of the 300 estimates --given by $\\frac{\\|\\hat{f}_{\\hat{m}}-f^*\\|^2}{N}$-- are significantly smaller.\n\n\\section{Conclusions}\nIn this work we considered a novel problem related to change point detection where we have to address the simultaneous task of segmenting and clustering the observed signal. Our approach has been to view this problem as a non parametric model selection problem on the set of all possible partitions. We derived for this the computationally tractable Algorithm~\\ref{two passes Dynamic programming algorithm}, that computes a relaxation of the penalized minimization of criterion~\\eqref{Criterion}, and we justified it from a statistical standpoint by showing that this minimization can be viewed as an approximate MAP. This approximate MAP estimate enjoys the properties of being adaptive and consistent in the sense of Corollary~\\ref{useful regime}. We finally justified the use of Algorithm~\\ref{two passes Dynamic programming algorithm} by simulation data that shows some useful properties of the resulting estimate and validates the theoretical guarantees.\n\nOne extension of this work concerns developing a more complete analysis of Algorithm~\\ref{two passes Dynamic programming algorithm}, to obtain consistency results on the number and locations of the change points and clusters. Another possible extension relates to the use of Algorithm~\\ref{two passes Dynamic programming algorithm} in the non scalar case; this was already explored for change point only detection in \\cite{Arlot16} through the use of characteristic kernels~\\citep{Sriperumbudur11}. We believe that the same approach can be adopted here except that we cannot perform the sorting step; this can be overcome using a Kernel clustering algorithm~\\cite{Filipponea08} or a spectral version of it \\cite{Sch\u00f6lkopf98} for the second stage. Finally, the remark after Figure~\\ref{fig3} hints to the possibility of using a combined algorithm starting with the sparse solution of Fussed LASSO and running the $2^{nd}$ dynamic programming pass of our algorithm as a way to boost the performance of Fussed LASSO to get rid of false discoveries. This would be still computationally attractive according to the comment after Theorem~\\ref{computaional complexity}, since the solution of Fussed LASSO has a small number of changes.  \n\\newpage\n\n\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgement}\nThis work was supported by the National Science Foundation of China (NSFC No. 62206194), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20220488), and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\nNeural language models (LMs) are one of the frontier research fields of deep learning.\nWith the explosion of model parameters and data scale, these language models demonstrate superior generalization capability, which can enhance many downstream tasks even under the few-shot and zero-shot settings~\\cite{radford2018improving,radford2019language,brown2020language,zhang2021commentary}.\nAlthough these models have achieved remarkable success, they are trapped by the time-agnostic setting in which the model is trained and tested on data with significant time overlap.\nHowever, real-world applications usually adopt language models pre-trained on past data (e.g., BERT \\cite{devlin2019bert} and RoBERTa~\\cite{liu2019roberta}) to enhance the downstream task-specific models for future data, resulting in a temporal misalignment~\\cite{luu2021time}. \nRecent works have empirically demonstrated that such a misalignment hurts the performance of both the upstream language models and downstream task-specific methods~\\cite{lazaridou2021mind,rottger2021temporal}. \n\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=1\\linewidth]{picture\/0church_consistent.pdf}\n  \\caption{Examples of lexical semantic change across time, where the semantic of word \\textit{church} is stable as it often refers to a building or a local congregation for Christian religious activities. However, the semantic of word \\textit{consistent} varies dramatically at different times.}\n  \\label{fig:type distribution}\n\\end{figure}\n\nTo better understand and solve the temporal misalignment problem, a series of studies have been launched on pre-trained language models (PLMs) and downstream tasks.\nThe analysis on PLMs~\\cite{lazaridou2021mind} revealed that PLMs (even with larger model sizes) encounter a serious temporal generalization problem, and the misalignment degree increases with time.\nThey also found that continually pre-training PLMs with up-to-the-minute data does mitigate the temporal misalignment problem but suffers from catastrophic forgetting and massive computational cost since further pre-training the converged PLMs is as difficult as pre-training from scratch.\nThe study on downstream tasks further indicates that temporal adaptation (i.e., continually pre-training with unlabelled data that is mostly overlapped in time), while effective, has no apparent advantages over domain adaptation~\\cite{rottger2021temporal} (i.e., continually pre-training with domain-specific unlabelled data) and fine-tuning on task-specific data from the target time~\\cite{luu2021time}.\n\n\nTo analyze the reason behind the limited performance of temporal adaptation, we launch a study from the lexical level, which also matches the token-level masking operation in advanced PLMs.\nUnlike existing research that launches analysis on part-of-speech (POS), topic words, and newly emerging words,\nwe mainly explore the correlations between language model performance and tokens\/words with salient lexical semantic change, which is also an extensively studied concept in computational linguistics~\\cite{dubossarsky2015bottom,hamilton2016diachronic,giulianelli2020analysing} to investigate how the semantics of words change over time.\nExperimental results demonstrate that tokens\/words with salient lexical semantic change do contribute much more than the rest of tokens\/words to the temporal misalignment problem, manifested as their significantly higher perplexity~(\\textit{ppl.}) over randomly sampled tokens from the target time.\nHowever, the widely-adopted masked language model (MLM) objective in state-of-the-art PLMs uniformly deals with each token\/word, letting the salient lexical-level semantic change information over time being overwhelmed by other tokens\/words, which can also explain why temporal adaptation has no obvious advantage compared with domain adaptation.\n\nBased on the above findings, we propose a lexical-based masked Language Model (LMLM) objective to capture the lexical semantic change between different temporal splits.\nExperimental results demonstrate that our proposed method yields significant performance improvement over domain adaptation methods on two different PLMs and four benchmark datasets. \nExtensive studies also show that LMLM is effective when utilizing different lexical semantic change metrics.\n\nIn a nutshell, our contributions are shown below:\n\\begin{itemize}[leftmargin=*]\n\\setlength{\\itemsep}{0pt}\n\\item We empirically study the temporal misalignment of PLMs at the lexical level and reveal that the tokens\/words with salient lexical semantic change contribute much more to the misalignment problem than other tokens\/words. We also disclose that such lexical temporal misalignment information can be overwhelmed by the masked language model training objective of PLMs, resulting in limited performance improvement over temporal and domain adaptation methods.\n\\item We propose a simple yet effective Lexical-based Masked Language Model (LMLM) objective to improve the temporal generalization of PLMs.\n\\item Experiments on two PLMs and four different benchmark datasets confirm that our proposed method is extensively effective in addressing the temporal misalignment problem for downstream tasks, which can significantly outperform existing temporal and domain adaptation methods.\n\\end{itemize}\n\n\\section{Conclusion \\& Future Work}\nIn this paper, we investigate the temporal misalignment of the PLMs from the lexical level and observe that the words with salient lexical semantic change contribute significantly to the temporal problems.\nWe propose a lexical-based masked Language Model (LMLM) objective based on the above observation.\nExperiments on two PLMs with the sequence classification task on three datasets under the specific domain setting and one name entity recognition task under the open-domain setting confirm that our proposed method performs better than the previous temporal adaptation methods and the state-of-the-art domain adaptation methods.\nIn the future, we will keep discovering such temporal misalignment problems in the text generation tasks, e.g., machine translation, and improve our method by reducing the extra offline computational cost on procedures like Semantic Change Detection.\n\n\n\\section{Implementation of the K-Means Algorithm}\n\\label{appdix:k_means}\nGiven a representation matrix $\\mathcal{R}^{t}_{i} = \\{r^{t}_{i}\\}_{i\\in (1, \\cdots, N)}$ for the word $w^{t}_{i}$ we utilize the silhouette score~\\cite{rousseeuw1987silhouettes} to obtain the optimal $K$ for K-Means algorithm.\nWe experiment with $K \\in [2,10]$ in a heuristic way.\nFor each $K$, the clustering result is the one that yields the minimal distortion value, i.e., the minimal sum of squared distances of each data point from its closest centroid, and we execute ten iterations to alleviate the influence of different initialization values~\\cite{vassilvitskii2006k}.\nSince there are several monosemous words, i.e., the number of $K$ is 1, we filter those words with a threshold $d$.\nSpecifically, if the intra-cluster dispersion value of a word is below $d$, we would allocate $K = 1$, otherwise, $K \\geq 2$.\nThe optimal $K$ is the one that can simultaneously minimize the dispersion score and maximize the silhouette score.\n\n\n\\section{Implementation of the Masking Strategies}\n\\label{appdix:semantic}\nIn this section, we illustrate the frequency masking strategy and importance masking strategy in detail.\nGiven a dataset that contains $n$ texts, we firstly utilize the NLTK tool\\footnote{\\url{https:\/\/github.com\/nltk\/nltk}} to tokenize each text and follow the below processes:\n\\paragraph{Frequency Masking Strategy}\nWe add each tokenized token into the dictionary $\\mathcal{D}$ and record the number of its occurrence.\nWe sort the tokens in $\\mathcal{D}$ according to the occurrence times and select the tokens to mask in descending order until the masking ratio is satisfied.\n\n\\paragraph{Importance Masking Strategy}\nWe utilize the YAKE! method as mentioned above to sort the tokenized tokens according to the scores calculated with the task label, e.g., the label of CS in the ARXIV dataset.\nFinally, we select the tokens to mask in descending order until the masking ratio is satisfied.\n\n\\section{Model Training \\& Parameters}\n\\label{appdix:para}\n\\paragraph{Architecture}\nWe utilize the BERT-base uncased model pretrained on a large corpus of English data with the MLM objective.\nThe model contains 12 transformer layers, 12 attention heads, and the hidden layer size is 768.\nThe total number of parameters is 110 million.\nWe add a linear layer after the last BERT layer for the downstream classification task and generate the output with softmax.\nThe maximum input sequence length is 512.\n\n\\paragraph{Training Details}\nWe utilize cross-entropy loss in the pre-training and fine-tuning stages and apply AdamW~\\cite{loshchilov2017decoupled} as the optimizer.\nSpecifically, the learning rate is 5e-5, and the weight decay is 0.01.\nMoreover, we set a 10\\% dropout probability for regularisation,\nWe pre-train the model for one epoch and fine-tune the model until convergency.\nWe set the batch size as 128 and conduct the experiments on eight NVIDIA GTX3090 GPUs.\n\n\\paragraph{Evaluation Metric}\nWe utilize the F1 score\\footnote{\\url{https:\/\/scikit-learn.org\/stable\/modules\/generated\/sklearn.metrics.f1_score.html}} as the evaluation metric in all the experiments.\n\n\n\\section{Implementation of the Baselines}\n\\label{appdix:methods}\nThis section will elaborate on how to apply the domain adaptation methods under the temporal adaptation settings.\n\\paragraph{PERL} ~\\cite{ben2020perl} This method  model\nparameters using a pivot-based ~\\cite{blitzer2006domain, blitzer2007biographies} variant of the MLM object with unlabeled datasets from both the source and target temporal split. \nInstead of masking each token with the same probability, we divided token into pivots and non-pivots to learn the pivot\/non-pivot distinction on unlabeled data from the source and target time span. The encoder weights are frozen during training for the downstream task.\nSpecifically, we rank those frequent features (occurs at least 20 times in the unlabeled data from the source and target time split) based on the mutual information with the task label according to source domain labeled data.\nThen, we select top 100 which have relative high scores as pivot features.\nThe non-pivot feature subset consists of features that do not match the two requirements.\n\n\n\\paragraph{DILBERT} \n~\\cite{lekhtman2021dilbert} is the SOTA in Aspect Extraction while using a fraction of the unlabeled data.\nDifferent from PERL, they challenge the \"high MI with the task label\" criterion in the pivot definition. In our settings, we harness the information about the golden label(physics, cs., etc) in the source and target temporal split to mask words that are more likely to bridge the gap between the different periods. Specifically, we compute the cosine similarity between each input word and the label from both the source and the target. We keep the highest similarity score for each word and mask the top 0.15\\% of the input words. For the downstream task, they add a logistic regression head on top of all outputs and fine-tune the model on the source period labeled data.\n\n\\section{Feature Extracting}\n\\label{appdix:upper_layer}\nOne point that worth discussing is the hidden states from the last layer of the BERT model ($\\mathrm{LMLM_{\\texttt{LAST}}}$) contains massive amounts of contextual information, which may overwhelm the lexical information.\nThus we turn to the representation from the shallow layer of the BERT model, e.g., representation from the second BERT layer ( $\\mathrm{LMLM_{\\texttt{SECOND}}}$).\nSpecifically, we conduct the experiment on the ARXIV testing set in 2013, and the results are shown in Table~\\ref{tab:rep_layer}.\n\n\\begin{table}[h]\n    \\centering\n    \\begin{tabular}{l|l}\n    \\toprule\n        Model & F1 \\\\\n    \\midrule\n        $\\mathrm{LMLM_{\\texttt{LAST}}}$ & \\bf 87.45 \\\\\n        $\\mathrm{LMLM_{\\texttt{SECOND}}}$ & 86.67 \\\\\n        TAda & 85.04 \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\caption{Results on the ARXIV testing set.}\n    \\label{tab:rep_layer}\n\\end{table} \n\nAs we can observe from the table that the $\\mathrm{LMLM_{\\texttt{SECOND}}}$ can achieve a better result than TAda, which indicates that the representations from the earlier layer are strong enough to help achieve a decent performance improvement.\nBesides, the $\\mathrm{LMLM_{\\texttt{LAST}}}$ achieves a better result than $\\mathrm{LMLM_{\\texttt{SECOND}}}$, which means that the hidden states which contain sentence-level information can help promote the accuracy in the detection process. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Linking Temporal Misalignment with Lexical Semantic Change}\nRecent work on temporal adaptation~\\cite{rottger2021temporal} has found that post-tuning the converged PLMs with unlabeled time-specific data by reusing the MLM objective can make the PLMs perceive related event-driven changes in language usage.\nSuch adaptation can achieve decent performance because the widely-adopted MLM objective can capture the overall changes in the data distribution by randomly masking a specific ratio of the whole sequence. \nHowever, such a training objective makes the lexical-level temporal information ignored or overwhelmed by the time-agnostic tokens\/words, resulting in little to no performance superiority over domain adaptation methods.\nBased on the above background, it is natural to explore the role of lexical-level temporal information in temporal adaptation, i.e., whether these tokens\/words with salient lexical-semantic changes\\footnote{The concept of semantic change is also essential in computational linguistics~\\cite{gulordava2011distributional,bamler2017dynamic,rosenfeld2018deep,del2018short,giulianelli2020analysing}.} over time impair the temporal adaptation performance. \nAs a result, we launch a thorough study from the perspective of lexical semantic change to figure out the reason behind the limited performance of temporal adaptation in the specific domain.\nTo the best of our knowledge, this is the first study that explores the correlation between the lexical-semantic change and the temporal adaptation of PLMs.\nWe will firstly illustrate our methods to find those semantic changed words in Section~\\ref{sec:dectection} and introduce the discovery experiment as well as analyze the results in Section~\\ref{sec:lsc_ana}.\n\n\n\n\n\\subsection{Lexical Semantic Change Detection}\n\\label{sec:dectection}\nTo obtain the semantic changed words, we design a lexical semantic change detection process.\nFor better illustration, we decompose the process into three steps: candidate words selection, feature extraction \\& Clustering, and semantic change quantification, which are correspond with the step~\\ding{172}$\\sim$~\\ding{174} in Figure~\\ref{fig:pipeline}.\n\n\\paragraph{Candidate Words Selection}\nBefore obtaining the representation of each word, we sample a certain number of candidate words $\\mathcal{W}^{t} = \\{w_{1}^{t}, w_{2}^{t}, \\cdots, w_{k}^{t}\\}$ from the texts $\\mathcal{D}^{t}$ of time $t$.\nConsidering that different texts have different domains (politics, culture, history), most keyword extraction methods either heavily rely on dictionaries and a fussy training  process~\\cite{witten2005kea,rose2010automatic} or are too simple to handle such intricate domain changes, i.e., TF-IDF~\\cite{ramos2003using}.\nInstead, we turn to YAKE!~\\cite{campos2018yake}, a feature-based and unsupervised system to extract keywords in one document.\nSince the goal is to measure the lexical semantic change among different time splits, we further filter the $\\mathcal{W}^{t}$ by calculating the number of the candidate words in different periods and removing the words that are repetitive, too few, or have no real meanings, e.g., pronouns, particles, mood words, etc.\n\n\\paragraph{Feature Extraction and Clustering}\nGiven a word $w_{i}$ and one text $d_{i}^{t}=(t_{1},\\cdots,t_{i},\\cdots,t_{n})$, where $d_{i}^{t}\\in \\mathcal{D}^{t}$ and $t_{i}=w_{i}$, we utilize a pre-trained language model BERT~\\cite{devlin2019bert} to contextualise each text as the representation $r_{i}^{t}$.\nSpecifically, we look up the sentences in $\\mathcal{D}^{t}$ which contain the same candidate words in $\\mathcal{W}^{t}$ and feed them into BERT to extract the corresponding word representations followed by aggregating them together~\\cite{giulianelli2020analysing}.\nIt is worth noting that we extract the representations from the last layer of the BERT model in all experiments, but we also consider extracting the features from the shallow layers of the BERT model. More details can be referred to in Appendix~\\ref{appdix:upper_layer}. \n\nTo prevent too much information brought by the long sequences overwhelming the meaning of the candidate words, we specify 128 as the size of occurrence window around the word $w_{i}$, i.e., truncating the redundant part of each sentence.\nAfter obtaining $N$ usage representations for each word, we combine them together as representation matrix $\\mathcal{R}_{i}^{t}=(r_{1}^{t},r_{2}^{t},\\cdots ,r_{N}^{t})$ and normalise it.\n\nTo distinguish the different semantic representations of each word, we utilize the $K$-Means algorithm, which can automatically cluster the similar word usage type into $K$ groups after $p$ turns according to the representation matrix of each word.\nDetails about the $K$-Means algorithm is elaborated in Appendix~\\ref{appdix:k_means}.\nAfter clustering, we count the number of sentences in each cluster and calculate the frequency distribution for the candidate word $w_{i}$.\nWhen normalized, the frequency distribution can be viewed as the probability distribution $p^{t}_{i}$ over usage types for the candidate word $w_{i}$ at the time $t$.\nTo meet our temporal settings, we should get the probability distributions for the same candidate word in different periods for comparison.\n\n\\paragraph{Semantic Change Quantification}\nTo measure the difference between the probability distributions $p^{t}_{i}$ and $p^{t^{\\prime}}_{i}$ of the same candidate words in different periods over word usages, we utilize the Jensen-Shannon divergence~\\cite{lin1991divergence} metric:\n\\begin{equation}\n\\begin{aligned}\n    {\\rm JSD}(p^{t}_{i}, p^{t^{\\prime}}_{i})& =  \\mathbb{H}\\left[\\frac{1}{2}(p^{t}_{i}+p^{t^{\\prime}}_{i})\\right] \\\\\n    &-\\frac{1}{2}\\left[\\mathbb{H}(p^{t}_{i})-\\mathbb{H}(p^{t^{\\prime}}_{i})\\right],\n\\end{aligned}\n\\end{equation}\nwhere $\\mathbb{H}$ is the Boltzmann-Gibbs-Shannon entropy~\\cite{ochs1976basic}.\nHigh JSD represents the different frequency distributions, i.e., significant lexical semantic change of the word $t_{i}$, and visa versa.\nWe utilize a hyper-parameter $k$ to control the degree of the lexical semantic change. \nSpecifically, we rank the candidate words according to their JSD values and sample the top-$k$ words as the salient semantic changed words.\nSeveral other metrics can also quantify the lexical semantic change, e.g., Entropy Difference (ED)~\\cite{nardone2014entropy} and Average pairwise distance (APD)~\\cite{bohonak2002ibd}, and we will compare the performance among them below.\n\n\\subsection{Discovery Experiment \\& Analysis}\n\\label{sec:lsc_ana}\nTo highlight the influence of the salient semantic changed words, we design a special masked language modeling objective LMLM, which first masks the candidate words $\\mathcal{W}^{t}$ in the texts.\nDetails of the LMLM objective are elaborated in section~\\ref{sec:LMLM}.\nAll the experiments in this section are conducted with the ARXIV dataset\\footnote{\\url{https:\/\/arxiv.org\/help\/oa\/index}}, which contains the abstracts of five subjects in different periods, e.g., CS, Math, etc.\nWe apply the pre-trained BERT-base model\\footnote{\\url{https:\/\/github.com\/google-research\/bert}} which has been pre-trained on a large corpus and evaluate it with the latter-released testing sets\\footnote{The BERT model is pre-trained with the data in 2015, while the three testing sets are after 2017.} by reporting the Perplexity (\\textit{ppl.}) value.\nAll the above data are tokenized with Moses\\footnote{\\url{ https:\/\/github.com\/alvations\/sacremoses}}, and non-English documents are removed.\n\n\\paragraph{Influence of the Semantic Changed Tokens}\nFor comparison, we introduce four masking strategies: random masking, frequency masking, importance masking, and LMLM.\nThe masking ratio of the strategies above is 15\\%.\nThe random masking strategy, as mentioned above, masks the tokens in the texts randomly, while the frequency masking strategy masks the tokens according to the lexical occurrence frequency, and the importance masking strategy masks the tokes according to the YAKE! scores.\nDetails of the masking strategies are illustrated in Appendix~\\ref{appdix:semantic}.\nThe results are shown in the figure~\\ref{fig:sub1}.\nWe can observe that the \\textit{ppl.} of the LMLM (blue dotted curve) is much higher than the others, which indicates that it is hard for the PLM to predict the lexical semantic changed tokens. \nThe rising trend of four curves shows that the PLM performs increasingly worse when predicting future utterances further away from their training period.\n\n\\paragraph{Influence of the Quantification Metric}\nFurthermore, We utilize three current popular metrics: Jensen-Shannon divergence (JSD),  Entropy Difference (ED), and Average Pairwise Distance (APD) to measure the semantic change.\nThe results are shown in the figure~\\ref{fig:sub2}.\nSince the slope of the red curve (JSD) is much higher than the others, which means the candidate words selected with the JSD metric are hard to predict, i.e., the semantic change phenomenon of those words is more significant, we apply the JSD metric in the later experiments to find the candidate words $\\mathcal{W}^{t}$.\n\n\\begin{figure}[t]\n\\centering\n\\label{fig:prob}\n\\subfigure[]{\n\\begin{minipage}[t]{0.225\\textwidth}\n  \\includegraphics[width=1\\linewidth]{picture\/0ablation_pretrain_strategy.pdf}\n  \\label{fig:sub1}\n\\end{minipage}\n}\n\\subfigure[]{\n\\begin{minipage}[t]{0.225\\textwidth}\n  \\includegraphics[width=1\\linewidth]{picture\/0PPL_pretrain_analysis.pdf}\n  \\label{fig:sub2}\n\\end{minipage}\n}\n\\caption{Results of the \\textit{ppl.}~value. Figure~(a) and (b) shows the effect of the semantic changed words and the results of different quantification metrics respectively.}\n\\end{figure}\n\\section{LMLM Objective}\n\\label{sec:LMLM}\nMasked Language Model (MLM) objective is a widely-adopted unsupervised task proposed by BERT~\\cite{devlin2019bert}, which randomly masks proportional tokens and predicts them. \nSince the degradation of the PLM in a specific domain over time is mainly attributed to the words with salient semantic change, we should make the PLM more aware of them.\nThus, we propose our Lexical-based Masked Language Model (LMLM) objective.\nContrary to the traditional random masking strategy (MLM), LMLM preferentially masks the words with salient semantic change over time.\nFormally, given the text set $\\mathcal{D}^{t}=\\{d_{1}^{t}, d_{2}^{t}, \\cdots, d_{n}^{t}\\}$ at time $t$, we select the candidate words $\\mathcal{W}^{t}$ with the aforementioned detection method and rank them according to their JSD value.\nThen, we select $k~(k \\in \\{100, 200, \\cdots, 1000\\})$ words which have relative high scores as the masking candidates $\\mathcal{W}_{mask}^{t}$.\nGiven masking ratio $\\alpha$, LMLM firstly selects the words in the $\\mathcal{W}_{mask}^{t}$ to mask.\nIf there are not enough candidates to meet the total number of masking tokens, LMLM masks the other words in the text randomly.\nThe whole process is corresponding to the step \\ding{175} in Figure~\\ref{fig:pipeline}.\nAssuming it masks $m$ tokens in total and the sentence after masking is $d_{i}^{t^{\\prime}}$.\nThe optimization objective of LMLM can be formulated as:\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{LMLM}} = - \\sum\\limits_{j=1}^{m} \\log P(x=w_{j}|d_{i}^{t^{\\prime}};\\theta).\n\\label{bert_mlm}\n\\end{equation}\n\\section{Limitation}\nThere are still some limitations in our work which are listed below:\n\\begin{itemize}[leftmargin=*]\n    \\setlength{\\itemsep}{0pt}\n    \\setlength{\\parskip}{0pt}\n    \\item The other tokens in the text influence the meaning of the target word to some extent since we utilize sentence contextualization to represent the meaning of the target word. \n    To this end, it is hard to interpret why some candidate words are selected by the detection step, e.g., name entities or numbers, whose meaning remains unchanged.\n    We will design a better unsupervised word selection strategy in the future. \n    \\item We utilize the lexical-level masking strategy, while the semantic change can also be reflected with the whole sequence, e.g., the topic of ``Malaysia Airlines crashed into the sea'' may be one hypothesis before 2014, but it became a severe accident in 2014.\n    Current famous MLM objectives like span masking objective~\\cite{raffel2020exploring} or sentence masking objective~\\cite{tay2022unifying} have observed that the performance of denoising the whole sequence is better than denoising the single token in some NLU tasks.\n    In the future, we will explore whether the sequence masking objective mentioned above can mitigate the temporal misalignment problem inherent in the PLMs.\n\\end{itemize}\n\\section{Experiments}\nWe conduct experiments on the classification task by employing the pre-trained BERT model implemented with the Hugging-Face transformers package\\footnote{\\url{https:\/\/huggingface.co}} in all experiments.\nFurther details about model training and parameters can be found in Appendix~\\ref{appdix:para}.\nWe will introduce the datasets and the time-stratified settings in Section~\\ref{sec:settings}, the baselines in Section~\\ref{sec:baseline}, and show the results in Section~\\ref{sec:res}.\n\n\\input{tabel\/data}\n\n\\subsection{Basic Settings}\n\\label{sec:settings}\n\\paragraph{Datasets}\nTo ensure the PLM is trained with the data in the specific domain, we select the data with the same or similar distributions between the upstream and downstream stages.\nWe choose the ARXIV dataset for the scientific domain and Reddit Time Corpus (RTC) dataset\\footnote{\\url{https:\/\/github.com\/paul-rottger\/temporal-adaptation}} for the political domain.\nWe also turn to two different datasets with a similar distribution for pre-training and fine-tuning, respectively.\nSpecifically, we select WMT News Crawl (WMT)\\footnote{\\url{https:\/\/data.statmt.org\/news-crawl\/}} dataset, which contains news covering various topics, e.g., finance, politics, etc, as unlabeled data and PoliAff\\footnote{\\url{https:\/\/github.com\/Kel-Lu\/time-waits-for-no-one}} dataset  in politic domain as labeled data.\n\n\\paragraph{Time-Stratified Settings}\nGenerally, the PLM is adapted to temporality using unlabelled data, fine-tuned with the downstream labeled data, and then evaluated with the testing data which has the same time as the pre-training data.\nWe set the $k$ as 500 in all experiments.\nAs for the ARXIV dataset, we utilize the unlabeled data in 2021 for pre-training and extract five years of data from 2011 to 2019 on a four-year cycle for fine-tuning as well as the data in 2021 for testing.\nSimilarly, we collect the data in 2015 and 2016 from the PoliAff dataset as the fine-tuning data and test the model with the data in 2017.\nFor the RTC dataset, we follow the previous work~\\cite{rottger2021temporal} to select the unlabeled News Comments dataset for post-training and the political subreddit subset for fine-tuning.\nHowever, the number of masking candidates $k$ is less than 500 in most RTC fine-tuning sets of different time splits, which could make the LMLM strategy be regarded as the random masking strategy. \nThus, we select the data in April 2017 for fine-tuning (where $k \\geq 500$ in this subset) and the data in April 2018 and August 2019 for testing.\nDetailed data statistics are shown in Table~\\ref{tab:data}.\n\n\n\\subsection{Baselines}\n\\label{sec:baseline}\nTo meet the time-stratified settings, we select the temporal adaptation \\textbf{TAda} method~\\cite{rottger2021temporal} as baseline, which first incorporates the temporal information into the PLM by utilizing the time-specific unlabeled data for pre-training and then adapt the PLM to the downstream task with the supervised data.\nBesides the temporal adaptation method, we also turn to some up-to-date domain adaptation methods since previous work~\\cite{rottger2021temporal} points out that such method can mitigate the temporal misalignment problems to some extent.\nSpecifically, we select PERL~\\cite{ben2020perl} and DILBERT~\\cite{lekhtman2021dilbert} methods, and implement them under the time-stratified settings.\nDetails of the domain adaptation methods are shown in the Appendix~\\ref{appdix:methods}.\nWe calculate the F1 score as the testing results for all the experiments.\n\n\\input{tabel\/arxiv_res}\n\\input{tabel\/poliaff_res}\n\n\\subsection{Main Results}\n\\label{sec:res}\n\\paragraph{ARXIV Dataset}\nThe results of the ARXIV dataset are shown in Table~\\ref{tabel:arxiv-main-result}, and we can observe that applying domain adaptation methods under the time-stratified settings aggravate the temporal misalignment problem as the scores of the PERL and DILBERT methods are not as high as those for utilizing the TAda directly.\nHowever, the performance of LMLM is much better than the other three methods.\n\n\n\\paragraph{PoliAff Dataset}\nWe report the results of the PoliAff dataset in Table~\\ref{poli}. \nAlthough there is a slight domain difference between the pre-training and fine-tuning data, i.e., news and politic, the LMLM can still achieve the best results, and the domain adaptation methods still perform worse than the temporal adaptation methods.\n\n\n\\paragraph{RTC Dataset}\nThe results of the RTC dataset is shown in Table~\\ref{tabel:rtc-main-result}, and we can find the similar tendency as the previous results, i.e., LMLM still achieve the best performance.\nHowever, the differences among the four methods in RTC dataset are much smaller compared with the previous results, which is largely due to the slight dynamic temporality of the RTC dataset.\n\n\n\\input{tabel\/rtc_res}\n\\section{Related Work}\n\\subsection{Temporal Misalignment}\nPrevious studies have shown that models trained on texts from one time period perform poorly when tested on texts in later periods for NLP tasks like machine translation~\\cite{levenberg2010stream}, review and news article classification~\\cite{huang2019neural,huang2018examining}, named entity recognition~\\cite{rijhwani2020temporally} and so on.\nWithin the current paradigm of using PLMs~\\cite{devlin2019bert}, studies have focused more on the expansion of dataset~\\cite{liu2019roberta, lewis2020bart, yang2019xlnet} and model capacity~\\cite{raffel2019exploring, lan2019albert, brown2020language} to achieve better performance but ignore the temporal effects.\nFew studies focus on such problem, \\citet{lazaridou2021mind} have empirically studied the degraded performance of PLMs over time, and \\citet{rottger2021temporal} focus on post-tuning BERT with the data in specific periods to mitigate the temporal misalignment problems.\nFurthermore, \\citet{amba2021dynamic} propose sampling methods to help PLMs achieve better performance on the evolving content.\nIn this paper, we conduct a detailed investigation from the perspective of lexical semantic change to figure out the reason behind the limited performance of the PLMs under the time-stratified settings. \n\n\n\\subsection{Lexical Semantic Change}\nLexical semantic change is an extensively studied concept in the computational linguistics, which mainly focuses on deciding whether the concept of a word has changed over time (semantic change detection)~\\cite{gulordava2011distributional,kulkarni2015statistically,dubossarsky2015bottom,hamilton2016diachronic} or discovering the instances with high semantic change (semantic change discovery)~\\cite{hengchen2021data,kurtyigit2021lexical,jatowta2021computational}.\nAmong them, most studies utilize contextualized word representations~\\cite{turney2010frequency,giulianelli2020analysing} and measure the distance among them in different periods~\\cite{cook2010automatically, gulordava2011distributional, hamilton2016diachronic} to detect or discover the instances with salient semantic change.\nPrevious studies mainly concentrate on applying PLMs to discover the semantic change phenomena, while our work focuses on solving such problems intrinsic in the PLMs.\nThus, besides observing such semantic changed phenomenon, we aim to find the corresponding words and apply the LMLM objective to make the PLMs more aware of them to mitigate the temporal misalignment problem.\nMost studies focused on obtaining those words are under the supervised settings~\\cite{kim2014temporal,basile2016diachronic,basile2018exploiting,tsakalidis2019mining} by scoring and selecting the top-ranked words through author intuitions or known historical data~\\cite{kurtyigit2021lexical}.\nWhile \\citeauthor{giulianelli2020analysing} propose one unsupervised method, adding one clustering process to the traditional selecting methods.\nTo our best knowledge, this is the first work that links semantic change with temporal adaptation.\n\n\n\\section{Study}\nIn this section, we conduct extensive studies to help better understand our method.\nIt is worth noting that all the experiments in this section are conducted on the BERT model with the ARXIV dataset unless there is a clear explanation.\n\n\\subsection{Effect of Pre-training Data Selection}\n\\label{sec:eff_pretrain_data}\nTo explore the temporal impact brought by pre-training data, we launch experiments with MLM objective under two different pre-training settings:\n\\begin{itemize}[leftmargin=*]\n    \\setlength{\\itemsep}{0pt}\n    \\setlength{\\parskip}{0pt}\n    \\item \\textbf{Source Year Consistent Pre-training (SYCP)} We keep the time of pre-training data consistent with that of fine-tuning data to ensure the consistency between the two stages.\n    \\item \\textbf{Target Year Consistent Pre-training (TYCP)} Following the previous work \\cite{rottger2021temporal, lazaridou2021mind}, we utilize the pre-training data in consistent with the evaluation data in temporal dimension, i.e., the time of pre-training data and evaluation data is same.\n\\end{itemize}\n\nWe also implement our LMLM objective in the pre-training stage for comparison, where the masking ratio is 15\\%, and $k$ is 1000.\nThe results are shown in the table \\ref{tabel:different-pretrain-data}.\nWe can find that the performance of SYCP gradually overwhelms the TYCP as the time passes towards the target year. \nWhen the PLM is pre-trained with MLM objective under the SYCP setting, it can even outcome the performance of TYCP, and the PLM pre-trained with LMLM objective under the TYCP setting can achieve the best performance.\nOn the one hand, we can infer that the temporal adaptation method is effective since TYCP beats the SYCP.\nOn the other hand, the LMLM objective can make the PLM pay more attention to the salient semantic changed words as pre-training with the LMLM objective under the SYCP settings (SYCP+LMLM) can even surpass the original temporal adaptation method (TYCP).\n\n\n\\begin{table}[t]\n\\centering\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{c|cccc|c}\n\\toprule\n\\multirow{2}{*}{Method} & \\multicolumn{5}{c}{Fine-tuning Data} \\\\\n\\cmidrule{2-6}\n& 2007 & 2011 & 2015 & 2019 & Avg. \\\\ \n\\midrule\nSYCP & 82.81  & 84.62  & 85.54 & 85.17 & 84.54 \\\\\nTYCP (TAda) & 82.97 & 84.72 & 84.82 & 84.99 & 84.38 \\\\\n\\midrule\nSYCP + LMLM & 83.52 & 85.41 & 86.36 & 87.03 & 85.58 \\\\\nTYCP + LMLM & \\textbf{84.93} &  \\textbf{86.52} & \\textbf{86.49} & \\textbf{87.22} & \\bf 86.29 \\\\\n\\bottomrule\n\\end{tabular}}\n\\caption{Results of different pre-training strategies.}\n\\label{tabel:different-pretrain-data}\n\\end{table}\n\n\n\\subsection{Hyper-Parameter Analysis}\n\\label{sec:hyp_LMLM}\nSince there is a strong relationship between the masking ratio and the model's performance, we conduct experiments to look for the best masking strategy for the LMLM objective.\nFurthermore, we also want to know whether the temporal misalignment problem can be better mitigated by masking more salient semantic changed words. \nThus, we select the data in 2021 for pre-training with our LMLM objective and the data in 2009 for fine-tuning, followed by testing the model with the data in 2021. \nFor better comparison, we utilize a heat map (Figure~\\ref{fig:heat}) to display the results, where the vertical axis of this graph represents the masking ratio $\\alpha$, and the horizontal axis represents the number of masked salient semantic changed words $k$.\n\n\\paragraph{The Influence of $\\alpha$}\nWe calculate the average value of the results of each masking ratio under different settings of $k$ and observe that when the masking ratio is around 30\\%, the PLM can achieve the best performance.\n\n\\input{tabel\/distribution}\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=1.1\\linewidth]{picture\/0heatmap.pdf}\n    \\caption{Results of the different masking strategies of LMLM. \nThe horizontal axis indicates the number of masked semantic changed words $k$ and the vertical axis represents for the masking ratio $\\alpha$.}\n    \\label{fig:heat}\n\\end{figure}\n\n\\paragraph{The Influence of $k$}\nNo doubt forcing the model to predict more high semantic change words can better mitigate the temporal problem generally.\nHowever, it is surprising to observe that the improvement is slight across different settings of $k$.\nThus, we quantify the semantic change of 1,000 random sampled words from the candidates $\\mathcal{W}^{t}_{mask}$ according to the JSD value and the distribution of those words is shown in Table~\\ref{tab:dis_seman_words}.\nWe can find that only around $10\\%$ words have relative significant semantic change (JSD value $\\geq 0.05$) while around $72\\%$ words have little or no semantic change (JSD value $\\approx 0.00$).\nWe can conclude that the improvement mainly comes from predicting a few keywords, i.e., topic words and newly emerging words, which have relatively salient semantic change.\n\n\\subsection{Pre-trained Language Model Analysis}\nTo verify the generalization of our methods on different PLMs, we implement our method on two PLMs, i.e., BERT and RoBERTa, and utilize the temporal adaptation method~\\cite{rottger2021temporal} as the baseline for comparison.\nAs shown in table~\\ref{tabel:pre_train_model}, we find there is a dramatic improvement of each PLM, i.e., 1.42 points improvement of the BERT model and 0.48 points improvement of the RoBERTa model on average.\n\n\n\\input{tabel\/PLM-analysis}\n\\input{tabel\/quantification_analysis}\n\\input{tabel\/conll_res}\n\\input{tabel\/error_analysis_1}\n\n\\subsection{Quantification Metrics}\nAs mentioned above, there are several metrics to quantify the semantic change, and we primarily conduct the experiment to select the JSD metric \nand we compare three commonly used metrics, i.e., ED, APD, and JSD, in this section.\nAs shown in the table~\\ref{tabel:Comparison_of_metric}, we can find that although different metrics have their advantages, the differences among them are slight.\nFor example, the maximum difference is 0.24 points among three metrics on average.\n\n\\subsection{Open-Domain Temporal Adaptation}\nAs mentioned above, we conduct all the experiments under the domain-specific setting.\nIn this section, we explore the effect of the LMLM objective with the name entity recognition task under the open-domain setting, i.e., the downstream dataset has no specific domain.\nSpecifically, we select the WMT dataset in 2017 as the unlabeled data and the subset in 2015 and 2016 from the CoNLL dataset\\footnote{\\url{https:\/\/github.com\/shrutirij\/temporal-twitter-corpus}} as the fine-tuning data.\nIn the end, we evaluate the model with the data in 2017.\nThe results are shown in Table~\\ref{tab:NER}.\nWe can find that the LMLM method outperforms the original temporal adaptation method with around 1 point improvement.\n\n\\subsection{Error Analysis}\nWe also conduct fine-grained experiments to study why our method fails with some examples.\nSpecifically, we utilize the ARXIV dataset from 2007 to 2017 to fine-tune the model and the data in 2021 for testing.\nWe first select the top 100 lexical semantic changed tokens for each testing set.\nThen, we divide the testing data into two parts: a subset with temporal information and a subset without temporal information by judging whether the texts contain the selected tokens.\nAs shown in the first group of Table~\\ref{tab:error_analysis}, the LMLM method can achieve better results than TAda on both testing subsets, and the improvement on the subset with temporal information is more significant than that on the subset without the  temporal information.\nA possible explanation for why LMLM performs better on the subset without temporal information is that there is still some temporal information left in this data since we distinguish the subset with only 100 lexical semantic changed tokens.\n\nBesides, we collect the failed testing sets, i.e., the model predicting wrong labels on those data, and mask those mentioned above top 100 lexical semantic changed tokens in the texts with two strategies: (1) replace those tokens with special placeholder \\texttt{<MASK>} or \\texttt{<PAD>}, and (2) randomly utilize other tokens in the vocabulary (except the aforementioned lexical semantic changed tokens) for substitution.\nThe results are shown in the second group of Table~\\ref{tab:error_analysis}, where we can observe that TAda surpasses our method in general\\footnote{It is worth noting that LMLM surpasses the TAda on the REP testing set in 2007, which can be attributed to the possibility of replacing the original tokens with lexical semantic changed tokens.}.\nWe think those masked\/replaced lexical semantic changed tokens, which LMLM pays more attention to, may be the critical messages for the model to help the decision.\nThe missing of that important information can cause a negative impact on the model, which leads to the performance decreasing.\n\n\\section{Introduction}\n\nNeural language models (LMs) are one of the frontier research fields of deep learning.\nWith the explosion of model parameters and data scale, these language models demonstrate superior generalization capability, which can enhance many downstream tasks even under the few-shot and zero-shot settings~\\cite{radford2018improving,radford2019language,brown2020language,zhang2021commentary}.\nAlthough these models have achieved remarkable success, they are trapped by the time-agnostic setting in which the model is trained and tested on data with significant time overlap.\nHowever, real-world applications usually adopt language models pre-trained on past data (e.g., BERT \\cite{devlin2019bert} and RoBERTa~\\cite{liu2019roberta}) to enhance the downstream task-specific models for future data, resulting in a temporal misalignment~\\cite{luu2021time}. \nRecent works have empirically demonstrated that such a misalignment hurts the performance of both the upstream language models and downstream task-specific methods~\\cite{lazaridou2021mind,rottger2021temporal}. \n\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=1\\linewidth]{picture\/0church_consistent.pdf}\n  \\caption{Examples of lexical semantic change across time, where the semantic of word \\textit{church} is stable as it often refers to a building or a local congregation for Christian religious activities. However, the semantic of word \\textit{consistent} varies dramatically at different times.}\n  \\label{fig:type distribution}\n\\end{figure}\n\nTo better understand and solve the temporal misalignment problem, a series of studies have been launched on pre-trained language models (PLMs) and downstream tasks.\nThe analysis on PLMs~\\cite{lazaridou2021mind} revealed that PLMs (even with larger model sizes) encounter a serious temporal generalization problem, and the misalignment degree increases with time.\nThey also found that continually pre-training PLMs with up-to-the-minute data does mitigate the temporal misalignment problem but suffers from catastrophic forgetting and massive computational cost since further pre-training the converged PLMs is as difficult as pre-training from scratch.\nThe study on downstream tasks further indicates that temporal adaptation (i.e., continually pre-training with unlabelled data that is mostly overlapped in time), while effective, has no apparent advantages over domain adaptation~\\cite{rottger2021temporal} (i.e., continually pre-training with domain-specific unlabelled data) and fine-tuning on task-specific data from the target time~\\cite{luu2021time}.\n\n\nTo analyze the reason behind the limited performance of temporal adaptation, we launch a study from the lexical level, which also matches the token-level masking operation in advanced PLMs.\nUnlike existing research that launches analysis on part-of-speech (POS), topic words, and newly emerging words,\nwe mainly explore the correlations between language model performance and tokens\/words with salient lexical semantic change, which is also an extensively studied concept in computational linguistics~\\cite{dubossarsky2015bottom,hamilton2016diachronic,giulianelli2020analysing} to investigate how the semantics of words change over time.\nExperimental results demonstrate that tokens\/words with salient lexical semantic change do contribute much more than the rest of tokens\/words to the temporal misalignment problem, manifested as their significantly higher perplexity~(\\textit{ppl.}) over randomly sampled tokens from the target time.\nHowever, the widely-adopted masked language model (MLM) objective in state-of-the-art PLMs uniformly deals with each token\/word, letting the salient lexical-level semantic change information over time being overwhelmed by other tokens\/words, which can also explain why temporal adaptation has no obvious advantage compared with domain adaptation.\n\nBased on the above findings, we propose a lexical-based masked Language Model (LMLM) objective to capture the lexical semantic change between different temporal splits.\nExperimental results demonstrate that our proposed method yields significant performance improvement over domain adaptation methods on two different PLMs and four benchmark datasets. \nExtensive studies also show that LMLM is effective when utilizing different lexical semantic change metrics.\n\nIn a nutshell, our contributions are shown below:\n\\begin{itemize}[leftmargin=*]\n\\setlength{\\itemsep}{0pt}\n\\item We empirically study the temporal misalignment of PLMs at the lexical level and reveal that the tokens\/words with salient lexical semantic change contribute much more to the misalignment problem than other tokens\/words. We also disclose that such lexical temporal misalignment information can be overwhelmed by the masked language model training objective of PLMs, resulting in limited performance improvement over temporal and domain adaptation methods.\n\\item We propose a simple yet effective Lexical-based Masked Language Model (LMLM) objective to improve the temporal generalization of PLMs.\n\\item Experiments on two PLMs and four different benchmark datasets confirm that our proposed method is extensively effective in addressing the temporal misalignment problem for downstream tasks, which can significantly outperform existing temporal and domain adaptation methods.\n\\end{itemize}\n\n\\section{Conclusion \\& Future Work}\nIn this paper, we investigate the temporal misalignment of the PLMs from the lexical level and observe that the words with salient lexical semantic change contribute significantly to the temporal problems.\nWe propose a lexical-based masked Language Model (LMLM) objective based on the above observation.\nExperiments on two PLMs with the sequence classification task on three datasets under the specific domain setting and one name entity recognition task under the open-domain setting confirm that our proposed method performs better than the previous temporal adaptation methods and the state-of-the-art domain adaptation methods.\nIn the future, we will keep discovering such temporal misalignment problems in the text generation tasks, e.g., machine translation, and improve our method by reducing the extra offline computational cost on procedures like Semantic Change Detection.\n\n\n\\section{Implementation of the K-Means Algorithm}\n\\label{appdix:k_means}\nGiven a representation matrix $\\mathcal{R}^{t}_{i} = \\{r^{t}_{i}\\}_{i\\in (1, \\cdots, N)}$ for the word $w^{t}_{i}$ we utilize the silhouette score~\\cite{rousseeuw1987silhouettes} to obtain the optimal $K$ for K-Means algorithm.\nWe experiment with $K \\in [2,10]$ in a heuristic way.\nFor each $K$, the clustering result is the one that yields the minimal distortion value, i.e., the minimal sum of squared distances of each data point from its closest centroid, and we execute ten iterations to alleviate the influence of different initialization values~\\cite{vassilvitskii2006k}.\nSince there are several monosemous words, i.e., the number of $K$ is 1, we filter those words with a threshold $d$.\nSpecifically, if the intra-cluster dispersion value of a word is below $d$, we would allocate $K = 1$, otherwise, $K \\geq 2$.\nThe optimal $K$ is the one that can simultaneously minimize the dispersion score and maximize the silhouette score.\n\n\n\\section{Implementation of the Masking Strategies}\n\\label{appdix:semantic}\nIn this section, we illustrate the frequency masking strategy and importance masking strategy in detail.\nGiven a dataset that contains $n$ texts, we firstly utilize the NLTK tool\\footnote{\\url{https:\/\/github.com\/nltk\/nltk}} to tokenize each text and follow the below processes:\n\\paragraph{Frequency Masking Strategy}\nWe add each tokenized token into the dictionary $\\mathcal{D}$ and record the number of its occurrence.\nWe sort the tokens in $\\mathcal{D}$ according to the occurrence times and select the tokens to mask in descending order until the masking ratio is satisfied.\n\n\\paragraph{Importance Masking Strategy}\nWe utilize the YAKE! method as mentioned above to sort the tokenized tokens according to the scores calculated with the task label, e.g., the label of CS in the ARXIV dataset.\nFinally, we select the tokens to mask in descending order until the masking ratio is satisfied.\n\n\\section{Model Training \\& Parameters}\n\\label{appdix:para}\n\\paragraph{Architecture}\nWe utilize the BERT-base uncased model pretrained on a large corpus of English data with the MLM objective.\nThe model contains 12 transformer layers, 12 attention heads, and the hidden layer size is 768.\nThe total number of parameters is 110 million.\nWe add a linear layer after the last BERT layer for the downstream classification task and generate the output with softmax.\nThe maximum input sequence length is 512.\n\n\\paragraph{Training Details}\nWe utilize cross-entropy loss in the pre-training and fine-tuning stages and apply AdamW~\\cite{loshchilov2017decoupled} as the optimizer.\nSpecifically, the learning rate is 5e-5, and the weight decay is 0.01.\nMoreover, we set a 10\\% dropout probability for regularisation,\nWe pre-train the model for one epoch and fine-tune the model until convergency.\nWe set the batch size as 128 and conduct the experiments on eight NVIDIA GTX3090 GPUs.\n\n\\paragraph{Evaluation Metric}\nWe utilize the F1 score\\footnote{\\url{https:\/\/scikit-learn.org\/stable\/modules\/generated\/sklearn.metrics.f1_score.html}} as the evaluation metric in all the experiments.\n\n\n\\section{Implementation of the Baselines}\n\\label{appdix:methods}\nThis section will elaborate on how to apply the domain adaptation methods under the temporal adaptation settings.\n\\paragraph{PERL} ~\\cite{ben2020perl} This method  model\nparameters using a pivot-based ~\\cite{blitzer2006domain, blitzer2007biographies} variant of the MLM object with unlabeled datasets from both the source and target temporal split. \nInstead of masking each token with the same probability, we divided token into pivots and non-pivots to learn the pivot\/non-pivot distinction on unlabeled data from the source and target time span. The encoder weights are frozen during training for the downstream task.\nSpecifically, we rank those frequent features (occurs at least 20 times in the unlabeled data from the source and target time split) based on the mutual information with the task label according to source domain labeled data.\nThen, we select top 100 which have relative high scores as pivot features.\nThe non-pivot feature subset consists of features that do not match the two requirements.\n\n\n\\paragraph{DILBERT} \n~\\cite{lekhtman2021dilbert} is the SOTA in Aspect Extraction while using a fraction of the unlabeled data.\nDifferent from PERL, they challenge the \"high MI with the task label\" criterion in the pivot definition. In our settings, we harness the information about the golden label(physics, cs., etc) in the source and target temporal split to mask words that are more likely to bridge the gap between the different periods. Specifically, we compute the cosine similarity between each input word and the label from both the source and the target. We keep the highest similarity score for each word and mask the top 0.15\\% of the input words. For the downstream task, they add a logistic regression head on top of all outputs and fine-tune the model on the source period labeled data.\n\n\\section{Feature Extracting}\n\\label{appdix:upper_layer}\nOne point that worth discussing is the hidden states from the last layer of the BERT model ($\\mathrm{LMLM_{\\texttt{LAST}}}$) contains massive amounts of contextual information, which may overwhelm the lexical information.\nThus we turn to the representation from the shallow layer of the BERT model, e.g., representation from the second BERT layer ( $\\mathrm{LMLM_{\\texttt{SECOND}}}$).\nSpecifically, we conduct the experiment on the ARXIV testing set in 2013, and the results are shown in Table~\\ref{tab:rep_layer}.\n\n\\begin{table}[h]\n    \\centering\n    \\begin{tabular}{l|l}\n    \\toprule\n        Model & F1 \\\\\n    \\midrule\n        $\\mathrm{LMLM_{\\texttt{LAST}}}$ & \\bf 87.45 \\\\\n        $\\mathrm{LMLM_{\\texttt{SECOND}}}$ & 86.67 \\\\\n        TAda & 85.04 \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\caption{Results on the ARXIV testing set.}\n    \\label{tab:rep_layer}\n\\end{table} \n\nAs we can observe from the table that the $\\mathrm{LMLM_{\\texttt{SECOND}}}$ can achieve a better result than TAda, which indicates that the representations from the earlier layer are strong enough to help achieve a decent performance improvement.\nBesides, the $\\mathrm{LMLM_{\\texttt{LAST}}}$ achieves a better result than $\\mathrm{LMLM_{\\texttt{SECOND}}}$, which means that the hidden states which contain sentence-level information can help promote the accuracy in the detection process. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Linking Temporal Misalignment with Lexical Semantic Change}\nRecent work on temporal adaptation~\\cite{rottger2021temporal} has found that post-tuning the converged PLMs with unlabeled time-specific data by reusing the MLM objective can make the PLMs perceive related event-driven changes in language usage.\nSuch adaptation can achieve decent performance because the widely-adopted MLM objective can capture the overall changes in the data distribution by randomly masking a specific ratio of the whole sequence. \nHowever, such a training objective makes the lexical-level temporal information ignored or overwhelmed by the time-agnostic tokens\/words, resulting in little to no performance superiority over domain adaptation methods.\nBased on the above background, it is natural to explore the role of lexical-level temporal information in temporal adaptation, i.e., whether these tokens\/words with salient lexical-semantic changes\\footnote{The concept of semantic change is also essential in computational linguistics~\\cite{gulordava2011distributional,bamler2017dynamic,rosenfeld2018deep,del2018short,giulianelli2020analysing}.} over time impair the temporal adaptation performance. \nAs a result, we launch a thorough study from the perspective of lexical semantic change to figure out the reason behind the limited performance of temporal adaptation in the specific domain.\nTo the best of our knowledge, this is the first study that explores the correlation between the lexical-semantic change and the temporal adaptation of PLMs.\nWe will firstly illustrate our methods to find those semantic changed words in Section~\\ref{sec:dectection} and introduce the discovery experiment as well as analyze the results in Section~\\ref{sec:lsc_ana}.\n\n\n\n\n\\subsection{Lexical Semantic Change Detection}\n\\label{sec:dectection}\nTo obtain the semantic changed words, we design a lexical semantic change detection process.\nFor better illustration, we decompose the process into three steps: candidate words selection, feature extraction \\& Clustering, and semantic change quantification, which are correspond with the step~\\ding{172}$\\sim$~\\ding{174} in Figure~\\ref{fig:pipeline}.\n\n\\paragraph{Candidate Words Selection}\nBefore obtaining the representation of each word, we sample a certain number of candidate words $\\mathcal{W}^{t} = \\{w_{1}^{t}, w_{2}^{t}, \\cdots, w_{k}^{t}\\}$ from the texts $\\mathcal{D}^{t}$ of time $t$.\nConsidering that different texts have different domains (politics, culture, history), most keyword extraction methods either heavily rely on dictionaries and a fussy training  process~\\cite{witten2005kea,rose2010automatic} or are too simple to handle such intricate domain changes, i.e., TF-IDF~\\cite{ramos2003using}.\nInstead, we turn to YAKE!~\\cite{campos2018yake}, a feature-based and unsupervised system to extract keywords in one document.\nSince the goal is to measure the lexical semantic change among different time splits, we further filter the $\\mathcal{W}^{t}$ by calculating the number of the candidate words in different periods and removing the words that are repetitive, too few, or have no real meanings, e.g., pronouns, particles, mood words, etc.\n\n\\paragraph{Feature Extraction and Clustering}\nGiven a word $w_{i}$ and one text $d_{i}^{t}=(t_{1},\\cdots,t_{i},\\cdots,t_{n})$, where $d_{i}^{t}\\in \\mathcal{D}^{t}$ and $t_{i}=w_{i}$, we utilize a pre-trained language model BERT~\\cite{devlin2019bert} to contextualise each text as the representation $r_{i}^{t}$.\nSpecifically, we look up the sentences in $\\mathcal{D}^{t}$ which contain the same candidate words in $\\mathcal{W}^{t}$ and feed them into BERT to extract the corresponding word representations followed by aggregating them together~\\cite{giulianelli2020analysing}.\nIt is worth noting that we extract the representations from the last layer of the BERT model in all experiments, but we also consider extracting the features from the shallow layers of the BERT model. More details can be referred to in Appendix~\\ref{appdix:upper_layer}. \n\nTo prevent too much information brought by the long sequences overwhelming the meaning of the candidate words, we specify 128 as the size of occurrence window around the word $w_{i}$, i.e., truncating the redundant part of each sentence.\nAfter obtaining $N$ usage representations for each word, we combine them together as representation matrix $\\mathcal{R}_{i}^{t}=(r_{1}^{t},r_{2}^{t},\\cdots ,r_{N}^{t})$ and normalise it.\n\nTo distinguish the different semantic representations of each word, we utilize the $K$-Means algorithm, which can automatically cluster the similar word usage type into $K$ groups after $p$ turns according to the representation matrix of each word.\nDetails about the $K$-Means algorithm is elaborated in Appendix~\\ref{appdix:k_means}.\nAfter clustering, we count the number of sentences in each cluster and calculate the frequency distribution for the candidate word $w_{i}$.\nWhen normalized, the frequency distribution can be viewed as the probability distribution $p^{t}_{i}$ over usage types for the candidate word $w_{i}$ at the time $t$.\nTo meet our temporal settings, we should get the probability distributions for the same candidate word in different periods for comparison.\n\n\\paragraph{Semantic Change Quantification}\nTo measure the difference between the probability distributions $p^{t}_{i}$ and $p^{t^{\\prime}}_{i}$ of the same candidate words in different periods over word usages, we utilize the Jensen-Shannon divergence~\\cite{lin1991divergence} metric:\n\\begin{equation}\n\\begin{aligned}\n    {\\rm JSD}(p^{t}_{i}, p^{t^{\\prime}}_{i})& =  \\mathbb{H}\\left[\\frac{1}{2}(p^{t}_{i}+p^{t^{\\prime}}_{i})\\right] \\\\\n    &-\\frac{1}{2}\\left[\\mathbb{H}(p^{t}_{i})-\\mathbb{H}(p^{t^{\\prime}}_{i})\\right],\n\\end{aligned}\n\\end{equation}\nwhere $\\mathbb{H}$ is the Boltzmann-Gibbs-Shannon entropy~\\cite{ochs1976basic}.\nHigh JSD represents the different frequency distributions, i.e., significant lexical semantic change of the word $t_{i}$, and visa versa.\nWe utilize a hyper-parameter $k$ to control the degree of the lexical semantic change. \nSpecifically, we rank the candidate words according to their JSD values and sample the top-$k$ words as the salient semantic changed words.\nSeveral other metrics can also quantify the lexical semantic change, e.g., Entropy Difference (ED)~\\cite{nardone2014entropy} and Average pairwise distance (APD)~\\cite{bohonak2002ibd}, and we will compare the performance among them below.\n\n\\subsection{Discovery Experiment \\& Analysis}\n\\label{sec:lsc_ana}\nTo highlight the influence of the salient semantic changed words, we design a special masked language modeling objective LMLM, which first masks the candidate words $\\mathcal{W}^{t}$ in the texts.\nDetails of the LMLM objective are elaborated in section~\\ref{sec:LMLM}.\nAll the experiments in this section are conducted with the ARXIV dataset\\footnote{\\url{https:\/\/arxiv.org\/help\/oa\/index}}, which contains the abstracts of five subjects in different periods, e.g., CS, Math, etc.\nWe apply the pre-trained BERT-base model\\footnote{\\url{https:\/\/github.com\/google-research\/bert}} which has been pre-trained on a large corpus and evaluate it with the latter-released testing sets\\footnote{The BERT model is pre-trained with the data in 2015, while the three testing sets are after 2017.} by reporting the Perplexity (\\textit{ppl.}) value.\nAll the above data are tokenized with Moses\\footnote{\\url{ https:\/\/github.com\/alvations\/sacremoses}}, and non-English documents are removed.\n\n\\paragraph{Influence of the Semantic Changed Tokens}\nFor comparison, we introduce four masking strategies: random masking, frequency masking, importance masking, and LMLM.\nThe masking ratio of the strategies above is 15\\%.\nThe random masking strategy, as mentioned above, masks the tokens in the texts randomly, while the frequency masking strategy masks the tokens according to the lexical occurrence frequency, and the importance masking strategy masks the tokes according to the YAKE! scores.\nDetails of the masking strategies are illustrated in Appendix~\\ref{appdix:semantic}.\nThe results are shown in the figure~\\ref{fig:sub1}.\nWe can observe that the \\textit{ppl.} of the LMLM (blue dotted curve) is much higher than the others, which indicates that it is hard for the PLM to predict the lexical semantic changed tokens. \nThe rising trend of four curves shows that the PLM performs increasingly worse when predicting future utterances further away from their training period.\n\n\\paragraph{Influence of the Quantification Metric}\nFurthermore, We utilize three current popular metrics: Jensen-Shannon divergence (JSD),  Entropy Difference (ED), and Average Pairwise Distance (APD) to measure the semantic change.\nThe results are shown in the figure~\\ref{fig:sub2}.\nSince the slope of the red curve (JSD) is much higher than the others, which means the candidate words selected with the JSD metric are hard to predict, i.e., the semantic change phenomenon of those words is more significant, we apply the JSD metric in the later experiments to find the candidate words $\\mathcal{W}^{t}$.\n\n\\begin{figure}[t]\n\\centering\n\\label{fig:prob}\n\\subfigure[]{\n\\begin{minipage}[t]{0.225\\textwidth}\n  \\includegraphics[width=1\\linewidth]{picture\/0ablation_pretrain_strategy.pdf}\n  \\label{fig:sub1}\n\\end{minipage}\n}\n\\subfigure[]{\n\\begin{minipage}[t]{0.225\\textwidth}\n  \\includegraphics[width=1\\linewidth]{picture\/0PPL_pretrain_analysis.pdf}\n  \\label{fig:sub2}\n\\end{minipage}\n}\n\\caption{Results of the \\textit{ppl.}~value. Figure~(a) and (b) shows the effect of the semantic changed words and the results of different quantification metrics respectively.}\n\\end{figure}\n\\section{LMLM Objective}\n\\label{sec:LMLM}\nMasked Language Model (MLM) objective is a widely-adopted unsupervised task proposed by BERT~\\cite{devlin2019bert}, which randomly masks proportional tokens and predicts them. \nSince the degradation of the PLM in a specific domain over time is mainly attributed to the words with salient semantic change, we should make the PLM more aware of them.\nThus, we propose our Lexical-based Masked Language Model (LMLM) objective.\nContrary to the traditional random masking strategy (MLM), LMLM preferentially masks the words with salient semantic change over time.\nFormally, given the text set $\\mathcal{D}^{t}=\\{d_{1}^{t}, d_{2}^{t}, \\cdots, d_{n}^{t}\\}$ at time $t$, we select the candidate words $\\mathcal{W}^{t}$ with the aforementioned detection method and rank them according to their JSD value.\nThen, we select $k~(k \\in \\{100, 200, \\cdots, 1000\\})$ words which have relative high scores as the masking candidates $\\mathcal{W}_{mask}^{t}$.\nGiven masking ratio $\\alpha$, LMLM firstly selects the words in the $\\mathcal{W}_{mask}^{t}$ to mask.\nIf there are not enough candidates to meet the total number of masking tokens, LMLM masks the other words in the text randomly.\nThe whole process is corresponding to the step \\ding{175} in Figure~\\ref{fig:pipeline}.\nAssuming it masks $m$ tokens in total and the sentence after masking is $d_{i}^{t^{\\prime}}$.\nThe optimization objective of LMLM can be formulated as:\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{LMLM}} = - \\sum\\limits_{j=1}^{m} \\log P(x=w_{j}|d_{i}^{t^{\\prime}};\\theta).\n\\label{bert_mlm}\n\\end{equation}\n\\section{Limitation}\nThere are still some limitations in our work which are listed below:\n\\begin{itemize}[leftmargin=*]\n    \\setlength{\\itemsep}{0pt}\n    \\setlength{\\parskip}{0pt}\n    \\item The other tokens in the text influence the meaning of the target word to some extent since we utilize sentence contextualization to represent the meaning of the target word. \n    To this end, it is hard to interpret why some candidate words are selected by the detection step, e.g., name entities or numbers, whose meaning remains unchanged.\n    We will design a better unsupervised word selection strategy in the future. \n    \\item We utilize the lexical-level masking strategy, while the semantic change can also be reflected with the whole sequence, e.g., the topic of ``Malaysia Airlines crashed into the sea'' may be one hypothesis before 2014, but it became a severe accident in 2014.\n    Current famous MLM objectives like span masking objective~\\cite{raffel2020exploring} or sentence masking objective~\\cite{tay2022unifying} have observed that the performance of denoising the whole sequence is better than denoising the single token in some NLU tasks.\n    In the future, we will explore whether the sequence masking objective mentioned above can mitigate the temporal misalignment problem inherent in the PLMs.\n\\end{itemize}\n\\section{Experiments}\nWe conduct experiments on the classification task by employing the pre-trained BERT model implemented with the Hugging-Face transformers package\\footnote{\\url{https:\/\/huggingface.co}} in all experiments.\nFurther details about model training and parameters can be found in Appendix~\\ref{appdix:para}.\nWe will introduce the datasets and the time-stratified settings in Section~\\ref{sec:settings}, the baselines in Section~\\ref{sec:baseline}, and show the results in Section~\\ref{sec:res}.\n\n\\input{tabel\/data}\n\n\\subsection{Basic Settings}\n\\label{sec:settings}\n\\paragraph{Datasets}\nTo ensure the PLM is trained with the data in the specific domain, we select the data with the same or similar distributions between the upstream and downstream stages.\nWe choose the ARXIV dataset for the scientific domain and Reddit Time Corpus (RTC) dataset\\footnote{\\url{https:\/\/github.com\/paul-rottger\/temporal-adaptation}} for the political domain.\nWe also turn to two different datasets with a similar distribution for pre-training and fine-tuning, respectively.\nSpecifically, we select WMT News Crawl (WMT)\\footnote{\\url{https:\/\/data.statmt.org\/news-crawl\/}} dataset, which contains news covering various topics, e.g., finance, politics, etc, as unlabeled data and PoliAff\\footnote{\\url{https:\/\/github.com\/Kel-Lu\/time-waits-for-no-one}} dataset  in politic domain as labeled data.\n\n\\paragraph{Time-Stratified Settings}\nGenerally, the PLM is adapted to temporality using unlabelled data, fine-tuned with the downstream labeled data, and then evaluated with the testing data which has the same time as the pre-training data.\nWe set the $k$ as 500 in all experiments.\nAs for the ARXIV dataset, we utilize the unlabeled data in 2021 for pre-training and extract five years of data from 2011 to 2019 on a four-year cycle for fine-tuning as well as the data in 2021 for testing.\nSimilarly, we collect the data in 2015 and 2016 from the PoliAff dataset as the fine-tuning data and test the model with the data in 2017.\nFor the RTC dataset, we follow the previous work~\\cite{rottger2021temporal} to select the unlabeled News Comments dataset for post-training and the political subreddit subset for fine-tuning.\nHowever, the number of masking candidates $k$ is less than 500 in most RTC fine-tuning sets of different time splits, which could make the LMLM strategy be regarded as the random masking strategy. \nThus, we select the data in April 2017 for fine-tuning (where $k \\geq 500$ in this subset) and the data in April 2018 and August 2019 for testing.\nDetailed data statistics are shown in Table~\\ref{tab:data}.\n\n\n\\subsection{Baselines}\n\\label{sec:baseline}\nTo meet the time-stratified settings, we select the temporal adaptation \\textbf{TAda} method~\\cite{rottger2021temporal} as baseline, which first incorporates the temporal information into the PLM by utilizing the time-specific unlabeled data for pre-training and then adapt the PLM to the downstream task with the supervised data.\nBesides the temporal adaptation method, we also turn to some up-to-date domain adaptation methods since previous work~\\cite{rottger2021temporal} points out that such method can mitigate the temporal misalignment problems to some extent.\nSpecifically, we select PERL~\\cite{ben2020perl} and DILBERT~\\cite{lekhtman2021dilbert} methods, and implement them under the time-stratified settings.\nDetails of the domain adaptation methods are shown in the Appendix~\\ref{appdix:methods}.\nWe calculate the F1 score as the testing results for all the experiments.\n\n\\input{tabel\/arxiv_res}\n\\input{tabel\/poliaff_res}\n\n\\subsection{Main Results}\n\\label{sec:res}\n\\paragraph{ARXIV Dataset}\nThe results of the ARXIV dataset are shown in Table~\\ref{tabel:arxiv-main-result}, and we can observe that applying domain adaptation methods under the time-stratified settings aggravate the temporal misalignment problem as the scores of the PERL and DILBERT methods are not as high as those for utilizing the TAda directly.\nHowever, the performance of LMLM is much better than the other three methods.\n\n\n\\paragraph{PoliAff Dataset}\nWe report the results of the PoliAff dataset in Table~\\ref{poli}. \nAlthough there is a slight domain difference between the pre-training and fine-tuning data, i.e., news and politic, the LMLM can still achieve the best results, and the domain adaptation methods still perform worse than the temporal adaptation methods.\n\n\n\\paragraph{RTC Dataset}\nThe results of the RTC dataset is shown in Table~\\ref{tabel:rtc-main-result}, and we can find the similar tendency as the previous results, i.e., LMLM still achieve the best performance.\nHowever, the differences among the four methods in RTC dataset are much smaller compared with the previous results, which is largely due to the slight dynamic temporality of the RTC dataset.\n\n\n\\input{tabel\/rtc_res}\n\\section{Related Work}\n\\subsection{Temporal Misalignment}\nPrevious studies have shown that models trained on texts from one time period perform poorly when tested on texts in later periods for NLP tasks like machine translation~\\cite{levenberg2010stream}, review and news article classification~\\cite{huang2019neural,huang2018examining}, named entity recognition~\\cite{rijhwani2020temporally} and so on.\nWithin the current paradigm of using PLMs~\\cite{devlin2019bert}, studies have focused more on the expansion of dataset~\\cite{liu2019roberta, lewis2020bart, yang2019xlnet} and model capacity~\\cite{raffel2019exploring, lan2019albert, brown2020language} to achieve better performance but ignore the temporal effects.\nFew studies focus on such problem, \\citet{lazaridou2021mind} have empirically studied the degraded performance of PLMs over time, and \\citet{rottger2021temporal} focus on post-tuning BERT with the data in specific periods to mitigate the temporal misalignment problems.\nFurthermore, \\citet{amba2021dynamic} propose sampling methods to help PLMs achieve better performance on the evolving content.\nIn this paper, we conduct a detailed investigation from the perspective of lexical semantic change to figure out the reason behind the limited performance of the PLMs under the time-stratified settings. \n\n\n\\subsection{Lexical Semantic Change}\nLexical semantic change is an extensively studied concept in the computational linguistics, which mainly focuses on deciding whether the concept of a word has changed over time (semantic change detection)~\\cite{gulordava2011distributional,kulkarni2015statistically,dubossarsky2015bottom,hamilton2016diachronic} or discovering the instances with high semantic change (semantic change discovery)~\\cite{hengchen2021data,kurtyigit2021lexical,jatowta2021computational}.\nAmong them, most studies utilize contextualized word representations~\\cite{turney2010frequency,giulianelli2020analysing} and measure the distance among them in different periods~\\cite{cook2010automatically, gulordava2011distributional, hamilton2016diachronic} to detect or discover the instances with salient semantic change.\nPrevious studies mainly concentrate on applying PLMs to discover the semantic change phenomena, while our work focuses on solving such problems intrinsic in the PLMs.\nThus, besides observing such semantic changed phenomenon, we aim to find the corresponding words and apply the LMLM objective to make the PLMs more aware of them to mitigate the temporal misalignment problem.\nMost studies focused on obtaining those words are under the supervised settings~\\cite{kim2014temporal,basile2016diachronic,basile2018exploiting,tsakalidis2019mining} by scoring and selecting the top-ranked words through author intuitions or known historical data~\\cite{kurtyigit2021lexical}.\nWhile \\citeauthor{giulianelli2020analysing} propose one unsupervised method, adding one clustering process to the traditional selecting methods.\nTo our best knowledge, this is the first work that links semantic change with temporal adaptation.\n\n\n\\section{Study}\nIn this section, we conduct extensive studies to help better understand our method.\nIt is worth noting that all the experiments in this section are conducted on the BERT model with the ARXIV dataset unless there is a clear explanation.\n\n\\subsection{Effect of Pre-training Data Selection}\n\\label{sec:eff_pretrain_data}\nTo explore the temporal impact brought by pre-training data, we launch experiments with MLM objective under two different pre-training settings:\n\\begin{itemize}[leftmargin=*]\n    \\setlength{\\itemsep}{0pt}\n    \\setlength{\\parskip}{0pt}\n    \\item \\textbf{Source Year Consistent Pre-training (SYCP)} We keep the time of pre-training data consistent with that of fine-tuning data to ensure the consistency between the two stages.\n    \\item \\textbf{Target Year Consistent Pre-training (TYCP)} Following the previous work \\cite{rottger2021temporal, lazaridou2021mind}, we utilize the pre-training data in consistent with the evaluation data in temporal dimension, i.e., the time of pre-training data and evaluation data is same.\n\\end{itemize}\n\nWe also implement our LMLM objective in the pre-training stage for comparison, where the masking ratio is 15\\%, and $k$ is 1000.\nThe results are shown in the table \\ref{tabel:different-pretrain-data}.\nWe can find that the performance of SYCP gradually overwhelms the TYCP as the time passes towards the target year. \nWhen the PLM is pre-trained with MLM objective under the SYCP setting, it can even outcome the performance of TYCP, and the PLM pre-trained with LMLM objective under the TYCP setting can achieve the best performance.\nOn the one hand, we can infer that the temporal adaptation method is effective since TYCP beats the SYCP.\nOn the other hand, the LMLM objective can make the PLM pay more attention to the salient semantic changed words as pre-training with the LMLM objective under the SYCP settings (SYCP+LMLM) can even surpass the original temporal adaptation method (TYCP).\n\n\n\\begin{table}[t]\n\\centering\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{c|cccc|c}\n\\toprule\n\\multirow{2}{*}{Method} & \\multicolumn{5}{c}{Fine-tuning Data} \\\\\n\\cmidrule{2-6}\n& 2007 & 2011 & 2015 & 2019 & Avg. \\\\ \n\\midrule\nSYCP & 82.81  & 84.62  & 85.54 & 85.17 & 84.54 \\\\\nTYCP (TAda) & 82.97 & 84.72 & 84.82 & 84.99 & 84.38 \\\\\n\\midrule\nSYCP + LMLM & 83.52 & 85.41 & 86.36 & 87.03 & 85.58 \\\\\nTYCP + LMLM & \\textbf{84.93} &  \\textbf{86.52} & \\textbf{86.49} & \\textbf{87.22} & \\bf 86.29 \\\\\n\\bottomrule\n\\end{tabular}}\n\\caption{Results of different pre-training strategies.}\n\\label{tabel:different-pretrain-data}\n\\end{table}\n\n\n\\subsection{Hyper-Parameter Analysis}\n\\label{sec:hyp_LMLM}\nSince there is a strong relationship between the masking ratio and the model's performance, we conduct experiments to look for the best masking strategy for the LMLM objective.\nFurthermore, we also want to know whether the temporal misalignment problem can be better mitigated by masking more salient semantic changed words. \nThus, we select the data in 2021 for pre-training with our LMLM objective and the data in 2009 for fine-tuning, followed by testing the model with the data in 2021. \nFor better comparison, we utilize a heat map (Figure~\\ref{fig:heat}) to display the results, where the vertical axis of this graph represents the masking ratio $\\alpha$, and the horizontal axis represents the number of masked salient semantic changed words $k$.\n\n\\paragraph{The Influence of $\\alpha$}\nWe calculate the average value of the results of each masking ratio under different settings of $k$ and observe that when the masking ratio is around 30\\%, the PLM can achieve the best performance.\n\n\\input{tabel\/distribution}\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=1.1\\linewidth]{picture\/0heatmap.pdf}\n    \\caption{Results of the different masking strategies of LMLM. \nThe horizontal axis indicates the number of masked semantic changed words $k$ and the vertical axis represents for the masking ratio $\\alpha$.}\n    \\label{fig:heat}\n\\end{figure}\n\n\\paragraph{The Influence of $k$}\nNo doubt forcing the model to predict more high semantic change words can better mitigate the temporal problem generally.\nHowever, it is surprising to observe that the improvement is slight across different settings of $k$.\nThus, we quantify the semantic change of 1,000 random sampled words from the candidates $\\mathcal{W}^{t}_{mask}$ according to the JSD value and the distribution of those words is shown in Table~\\ref{tab:dis_seman_words}.\nWe can find that only around $10\\%$ words have relative significant semantic change (JSD value $\\geq 0.05$) while around $72\\%$ words have little or no semantic change (JSD value $\\approx 0.00$).\nWe can conclude that the improvement mainly comes from predicting a few keywords, i.e., topic words and newly emerging words, which have relatively salient semantic change.\n\n\\subsection{Pre-trained Language Model Analysis}\nTo verify the generalization of our methods on different PLMs, we implement our method on two PLMs, i.e., BERT and RoBERTa, and utilize the temporal adaptation method~\\cite{rottger2021temporal} as the baseline for comparison.\nAs shown in table~\\ref{tabel:pre_train_model}, we find there is a dramatic improvement of each PLM, i.e., 1.42 points improvement of the BERT model and 0.48 points improvement of the RoBERTa model on average.\n\n\n\\input{tabel\/PLM-analysis}\n\\input{tabel\/quantification_analysis}\n\\input{tabel\/conll_res}\n\\input{tabel\/error_analysis_1}\n\n\\subsection{Quantification Metrics}\nAs mentioned above, there are several metrics to quantify the semantic change, and we primarily conduct the experiment to select the JSD metric \nand we compare three commonly used metrics, i.e., ED, APD, and JSD, in this section.\nAs shown in the table~\\ref{tabel:Comparison_of_metric}, we can find that although different metrics have their advantages, the differences among them are slight.\nFor example, the maximum difference is 0.24 points among three metrics on average.\n\n\\subsection{Open-Domain Temporal Adaptation}\nAs mentioned above, we conduct all the experiments under the domain-specific setting.\nIn this section, we explore the effect of the LMLM objective with the name entity recognition task under the open-domain setting, i.e., the downstream dataset has no specific domain.\nSpecifically, we select the WMT dataset in 2017 as the unlabeled data and the subset in 2015 and 2016 from the CoNLL dataset\\footnote{\\url{https:\/\/github.com\/shrutirij\/temporal-twitter-corpus}} as the fine-tuning data.\nIn the end, we evaluate the model with the data in 2017.\nThe results are shown in Table~\\ref{tab:NER}.\nWe can find that the LMLM method outperforms the original temporal adaptation method with around 1 point improvement.\n\n\\subsection{Error Analysis}\nWe also conduct fine-grained experiments to study why our method fails with some examples.\nSpecifically, we utilize the ARXIV dataset from 2007 to 2017 to fine-tune the model and the data in 2021 for testing.\nWe first select the top 100 lexical semantic changed tokens for each testing set.\nThen, we divide the testing data into two parts: a subset with temporal information and a subset without temporal information by judging whether the texts contain the selected tokens.\nAs shown in the first group of Table~\\ref{tab:error_analysis}, the LMLM method can achieve better results than TAda on both testing subsets, and the improvement on the subset with temporal information is more significant than that on the subset without the  temporal information.\nA possible explanation for why LMLM performs better on the subset without temporal information is that there is still some temporal information left in this data since we distinguish the subset with only 100 lexical semantic changed tokens.\n\nBesides, we collect the failed testing sets, i.e., the model predicting wrong labels on those data, and mask those mentioned above top 100 lexical semantic changed tokens in the texts with two strategies: (1) replace those tokens with special placeholder \\texttt{<MASK>} or \\texttt{<PAD>}, and (2) randomly utilize other tokens in the vocabulary (except the aforementioned lexical semantic changed tokens) for substitution.\nThe results are shown in the second group of Table~\\ref{tab:error_analysis}, where we can observe that TAda surpasses our method in general\\footnote{It is worth noting that LMLM surpasses the TAda on the REP testing set in 2007, which can be attributed to the possibility of replacing the original tokens with lexical semantic changed tokens.}.\nWe think those masked\/replaced lexical semantic changed tokens, which LMLM pays more attention to, may be the critical messages for the model to help the decision.\nThe missing of that important information can cause a negative impact on the model, which leads to the performance decreasing.\n\\section*{Acknowledgement}\nThis work was supported by the National Science Foundation of China (NSFC No. 62206194), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20220488), and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nIn recent years there has been an enormous breakthrough in research of\ncomplex networks due to the application of statistical\nphysics methodology \\cite{albert:review,dorogovtsev:review,dorogovtsev03:book}. Many different complex\nsystems instead of being completely random prove to have signatures\nof organization such as clustering and power-law distribution of\nlinks. Together with the small-world property \\cite{milgram:small} these are the inherent features\nof an extremely wide variety of systems such as the World-Wide\nWeb \\cite{albert:diameter,kleinberg:web99,kumar:web00},\nInternet \\cite{satorras:internet}, collaboration networks of movie\nactors \\cite{watts:1998,newman:random} and scientists \\cite{newman:random}, the web of human\nsexual contacts \\cite{liljeros:2001} and many others. In spite of the\nfact that some concepts of complex networks theory were originally\nintroduced in sociology the statistical study of social networks is\ncomplicated by the difficulty in reliable data collection due to certain privacy and ethical reasons.\n One of the solutions for this problem is the analysis of collaboration\nnetworks \\cite{watts:1998,newman:random}, e-mail\ninteractions \\cite{arenas:email,arenas:community,newman:email}, instant\nmessaging \\cite{smith:2002} and online blogging \\cite{kumar:bursty,kumar:structure,nowell:phd,nowell:pnas}.  \n\\comment{\nNowell {\\em et al.} recently studied geographic aspects of LiveJournal\n(www.livejournal.com) blog space  and they reported parabolic shape of\nfriendship degrees distributions \\cite{nowell:pnas}. }\nHere we studied basic structural properties of LiveJournal blog service\nsocial network and demonstrated the diffusion-motivated method to\ndiscover communities on the case of this network.\n\n\\section{LIVEJOURNAL NETWORK}\n\nLiveJournal (LJ) is an online web-based journal service\nwith an emphasis on users interactions \\cite{lj:faq}. In January 2006 it had $9.3 \\cdot 10^6$\nusers in total, $2.0\n\\cdot 10^6$ of them were {\\em active in some way} according to\nofficial  LiveJournal statistics \\cite{lj:stat}.\nThe essential feature of LJ service is the ``friends'' concept which helps\nusers to organize their reading preferences and \nprovides security regulations for their journal entries and personal data. Friends list\nis an open information and can be accessed through a conventional WWW\ninterface or through a dedicated bot interface provided by LJ system.\n\nData collection was performed by crawler programs running simultaneously\non two computers and exploring the LJ space by following directional\nfriendship links starting from two users with a large number of\nincoming friendship links. For each user the crawler was obtaining his friends list (outgoing links) and the\nnumber of users who have the given user in their friends list (incoming\nlinks). Each user from the friends list which was not yet explored by\nthe crawler was added to the end of the processing queue if he was not already\nthere. If the user was in the queue his queue score was\nincremented every time he was found in someones' friends list. Users\nwith higher queue scores were processed first. This ensured fast\ncollection of the essential part of the network. Basically this algorithm \nis a modification of Tarjan's depth-first search algorithm for\nfinding the connected component of a graph \\cite{tarjan:alg72,hopcroft:alg73}.\nTotal time of collection was 14 days with the total number of \ndiscovered users $3\\:746\\:264$ found in a connected component. We are aware\nthat during the time of collection the network was \nundergoing continuous changes. We estimated the number of users deleted from the LJ\ndatabase but still present in the friends lists was less than 0.1\\%\nwhich makes us believe that the evolution of LJ network did not\ninfluence our statistics much.\n\nThe estimated probability distribution functions of in- and out-degree are presented in\nlog-log scale in the Fig.~\\ref{fig:degrees}.  The estimated mean\nof the numbers of  outgoing  and incoming friendship links is $\\langle k_{out}\\rangle =\n15.91$ and $\\langle k_{in}\\rangle = 16.07$, correspondingly. The\naverage in-to-out ratio $\\langle k_{in} \/ k_{out}\\rangle =\n1.157$. The number of incoming links is slightly larger than the\nnumber of outgoing due to the fact that only the outgoing links were\nused for crawler navigation so some of the LJ users were unreachable by\ndirectional links but they were listed in the users pages. \n\nThere are also several technical restrictions for the degrees: maximum\nnumber of friends per user is 750 and only 150 of them can be listed on the users' info\npage and can be effortlessly accessed by the LJ users. From our experience LJ bots interface\ndoes have some problems listing the users who consider a certain\nuser as a friend if there are more than 2500 of them hence we cut the\ndata at $k_{out\\: max} = 2500$. \n\n As one can see from the Fig.~\\ref{fig:degrees} in- and out-degree distributions\nreveal a power-law decay $P(k) \\sim k^{-\\gamma} $ for $k > 100$ with \nthe value of the exponent $\\gamma_{in} \\approx \\gamma_{out} =  3.45 \\pm 0.05$ which is\nsurprisingly close to the values $\\gamma_{in} \\approx \\gamma_{out} \\approx\n3.4$ obtained by Liljeros {\\em et al.} for sexual\ncontacts \\cite{liljeros:2001}. Scaling of the distributions contradicts the\nresults of Nowell {\\em et al.} \\cite{nowell:pnas}\nwho reported parabolic shape of LJ degrees distributions.\nThe skewness of \nthe distributions in our case can be explained by the social origin of\nLJ network. As it is pointed out by Jin {\\em et\n  al.} \\cite{jin:structure} degree distribution for social\nnetworks does not appear to follow power-law distribution due to the\ncost in terms of time and efforts to support friendship. In the case of\nLJ network the cost of friendship is the size of friends feed which\naccumulates all the recent entries of the user's friends. We can also\nseparate two classes of LJ users: ``readers'' and ``writers''. The first are\nmainly using their accounts to read the journals of others. They update\njournals only episodically and are not deeply involved in LJ\ncommunity life. They do not have many incoming and outgoing links and\nthey are responsible for skewness of the distributions for $k <\n100$. Meanwhile active ``writers'', representing minority of the registered users \nexploit full capacity of LJ system. They spend much time participating\nin LJ community life, and they have a larger number of incoming and\noutgoing links which are distributed by power-law.\n\nThe origin of power-law region in the distributions can be explained by\ncontinuous evaluation and self-organization of the LJ network and preferential attachment\nmechanism similar to the general WWW growth mechanism\n\\cite{barabasi:1999}. One an interesting journal gets popular it will\nbe cited and promoted in \nthe journals of its readers which will help to further increase its\npopularity which leads to a ``rich-get-richer'' effect occurring\nin many network systems\n\\cite{barabasi:1999,dorogovtsev:review}. However linear growth\nwith linear preferential attachment protocol leads to a power-law\ndegree distribution with $\\gamma = 3$ which is smaller than the exponent\nobtained for our study. Larger values of exponent can be\nexplained by alternative growth mechanisms: preferential attachment\nwith rewiring \\cite{albert:topology00} and copying mechanism\n\\cite{kleinberg:web99,kumar:web00}. Rewiring in LJ system implies that\nusers are not only establishing new friendship links but also breaking\nthe old ones while copying occurs when the user inherits part of the\nfriendship connections of his friends. The latter effect is called\n''transitivity'' in sociology \\cite{wasserman:94} and is responsible for\nusers cliques formation or clustering.\n\nWe characterize clustering of LJ users by calculating the clustering\ncoefficient as introduced by Watts and Strogatz \n\\cite{watts:1998,albert:review}. It is defined as the number of links between user's friends divided\nby the maximum possible number of links between them averaged over all\nusers in the network. If the user $i$ has $k_i$\nfriends with $E_i$ links between them the maximum possible number of\ndirected links is $k_i (k_i - 1)$ and the clustering coefficient for the\nuser $i$ in the case of directed network can be defined as:\n\\begin{equation}\nC_i = \\frac{E_i}{k_i (k_i - 1)}.\n\\label{eq:clustering}\n\\end{equation}\nThe average clustering coefficient for the whole network as calculated\nfrom our data is: $C = \\langle C_i \\rangle_{i=1..N} \\approx 0.3302$. It is worth to\ncompare this value to the clustering coefficient of a random\ndirectional Erd\\H{o}s-R\\'{e}nyi graph which can be found as $C_{rand}\n= \\langle k \\rangle \/ (N - 1)$ which for LJ network is ca. $4.24 \\cdot 10^{-6}$. The fact that actual \nclustering coefficient for LJ network is nearly five orders of magnitude larger than\nit would be expected from randomly linked network with the same degree and\nsize is a clear indication of high user clustering.\n\n\nThe peculiar feature of the LJ network is the high \nreciprocity \\cite{wasserman:94} of friendship links. We found that 79.26\\% of links \nare bi-directional which means that this percentage of outgoing links \nis returned as incoming and {\\em vice versa}\nthe same percentage of incoming links originates from users friends.\nThis value is higher than reciprocity 57\\% found for the WWW\n\\cite{newman:email02} which is the technical environment of LJ. Increasing\nof reciprocity may be explained by social origin of LJ network. Due to\nthe rules of social interactions user $A$ usually feels obliged to establish a\nfriendship connection to the user $B$ if such a connection was already\nestablished by $B$ to $A$. Another explanation for high reciprocity is\nthat often relations in  \nLJ space is based on real-life people relations which means that\nLJ users are linking to the other users which are their friends in the real\nworld. In this case the LJ network directly inherits the undirectional\nstructure of the underlying social network.\n\n\n\\begin{figure}\n\\centering\n  \\includegraphics[width=\\linewidth]{fig2.eps}\n  \\caption{Probability distribution function of the minimum path length\n    between LiveJournal users through the directional friends links.}\n  \\label{fig:path_distance}\n\\end{figure}\n\n\nIn order to characterize small-world properties of LJ network we estimated the\nprobability distribution function $P_\\ell(\\ell)$ of the minimum path distance\nor hopcount between the nodes through directional links. The results are presented in\nthe Fig.~\\ref{fig:path_distance}. The average distance estimated for\nour set of data is $\\langle \\ell \\rangle = 5.86$. \n\\comment{\nAccording to the\ngeneral approach developed by Newman \\textit{et al.}\n\\cite{newman:random} an average path length can be estimated using the\nfollowing expression:\n\\begin{equation}\n\\mathcal{\\ell} = \\frac{ln(N \/ z_1)}{ln(z_2\/z_1)} + 1,\n\\end{equation}\nwhere $N$ is the size of the network and $z_1 = \\langle k_{out} \\rangle $ and $z_2$ is the number\nof the first and the second neighbours. From this we obtained $\\ell \\approx\n4.3$ which is significantly smaller than the value obtained from\nthe distribution. We are considering this as a first sign of structure\nwithin LJ network. \n}\nBased on the recently obtained expression for the mean distance between\nthe nodes in scale-free networks by Hooghiemstra \\textit{et al.} \\cite{hoog:mean05}\nwho improved the widely used result of Newman \\textit{et\n  al.} \\cite{newman:random} the value of $\\langle \\ell \\rangle$ can be estimated\nas the following:\n\\begin{equation}\n\\langle \\mathcal{\\ell} \\rangle_{th} \\approx \\frac{ln N}{ln \\nu} + \\frac{1}{2} -\n\\left ( \\frac{\\gamma_e + ln \\mu - ln (\\nu - 1)}{ln \\nu} \\right ) - 2\n\\frac{\\epsilon}{log \\nu},\n\\label{eq:hoog}\n\\end{equation}\nwhere $N$ is the size of the network, $\\mu = \\langle k \\rangle$,\n$\\nu = \\langle k (k - 1)\\rangle \/ \\langle k \\rangle$, $\\gamma_e\n\\approx 0.577$ is the Euler-Mascheroni constant, and $\\epsilon$ is the\nexpectation of the logarithm of the limit of a super-critical\nbranching process which depends on the scaling exponent $\\gamma$ \nand belongs to the half-open interval $(-1,0]$, where the lower\n  boundary is the numerical extrapolation of the results from\n  \\cite{hoog:mean05} and the upper boundary the theoretical prediction\n  for $\\gamma > 3$.\n\nFor LJ data the equation \\eqref{eq:hoog} gives the following range of the mean distance:\n$ 4.53 \\le \\langle \\mathcal{\\ell} \\rangle_{th} < 5.05$ which is in any case\nsmaller than statistically obtained value. This theoretical prediction\nassumes the homogeneity of the graph, and  we believe the possible reason\nfor such an underestimation of the mean path length is the \nmacroscopic structuring of the network which is discussed further.\n\n\\begin{figure}\n\\centering\n  \\includegraphics[width=\\linewidth]{fig3.eps}\n  \\caption{Illustration of the community detection algorithm. After\n    diffusion process starts from the initiator node virtual ink\n  propagates through network links. Communities can be recognized as the\n    groups of nodes with similar amount of ink.}\n  \\label{fig:scheme}\n\\end{figure}\n\n\n\\section{COMMUNITY DISCOVERING METHOD}\n\nIt seems to be quite natural for the nodes of the complex networks\nto aggregate into macroscopic structures with high internal links\ndensity and weak connection to the rest of the network. Such groups are\noften referred to as communities. Particular reasons for communities \nformation may depend on the type of the network but this feature\nproved to be quite universal and can be found in social, biological and\ncomputer networks \\cite{girvan:pnas02,newman:fast}. Finding these\nstructures within the network is the major step towards understanding its topology.\n\nThis problem is known as a graph-partitioning problem in graph theory\nand has a nondeterministic polynomial (NP) complexity  which makes it\nalmost inapplicable for large networks. \n\nRecent advances in the study of complex networks stimulated the search\nof alternative techniques for community discovering and many original solutions\nwere proposed\n\\cite{girvan:pnas02,newman:mixing,newman:community,newman:fast,clauset:2004,pons:rw05,wu:physics04,simonsen:diff04}. \nThese algorithms can be divided into two main classes: \\textit{divisible}, which\nhierarchically split the network by removing edges with the highest\nbetweenness \\cite{girvan:pnas02,newman:community} and\n\\textit{agglomerative} which start from the maximal community\ndivision when each node belongs to its own separate community and\ncontinuously merge these communities basing on some parameter of\nnodes similarity \\cite{wu:physics04,pons:rw05} or optimizing\nthe partitioning. In their recent work Clauset \\textit{et al.}\n\\cite{clauset:2004} used the greedy optimization in order to maximize\nthe \\textit{modularity} measure of partitioning quality\n\\cite{newman:community,newman:fast}. Currently this method is one of the\nfastest and runs in time $O (M H ln N)$, where $M = \\langle k \\rangle N$ is the number\nof edges in the network and $H$ is the number of decomposition levels\nwhich is usually small ($H = O (ln N)$)\n\\cite{clauset:2004,pons:rw05}. In a sparse network the degree is limited\nand $M = O (N)$ and so the complexity is $O (N ln^2 N)$ which makes it\nfastest nowadays.\n\nHere we propose a method to find communities based oh the principles\nof thermodynamics. When the system gets large enough so that the behavior\nof its microscopic constituents can be successfully averaged to give\nbasis for a scientific descriptions of phenomena with avoidance of\nmicroscopic details. Since in thermodynamics behavior of the system can\nbe described without solving the equation of motion of every\nconstituent molecule we believe that structure of the large complex\nnetwork can be explored without explicit solution of graph\npartitioning problem.\n\nOur current study is based on the simulation of a mass diffusion process in the complex network\nas in a multi-dimensional porous system with directional links following\nphysical laws. The diffusion process initiated at one of the nodes by\naddition of the virtual ink produces a non-uniform mass distribution at the intermediate state\nwhich can be used to reveal well-interconnected communities within the\ncomplex network by selecting the nodes with similar concentration\nvalues. In this sense our method falls in the class of agglomerative\ntechniques with the concentration as the similarity measure. However, it\ncan be shown that the quantity $r_{AB} = | ln \\phi_A - ln \\phi_B |$,\nwhere $\\phi_A$ and $\\phi_B$ are two values of concentration  in the nodes $A$\nand $B$, as the measure of distance between these nodes. Thus\nedge betweenness, characterized as the drop of the logarithm of concentration\nalong the edge, can be used for hierarchical decomposition of the\nnetwork.\n\nThe similar measure of distance between nodes based on the random walk\nhas been recently introduced by Pons \\textit{et al.} \\cite{pons:rw05}\nfor the class of undirected networks. It is defined as the difference in probabilities\nfor a random walker to reach nodes the $A$ \nand $B$ in certain number of steps $t$ starting from some node\n$Z$. As these probabilities for a large $t$ are mainly determined by\nthe in-degrees of the nodes the values of distance should be normalized\nA short number of steps $t$ may depend on a particular\nnetwork and should be known in advance. Pons \\textit{et al.} also pointed out\nconceptual difficulties of the random walk scheme application for the directed\nnetworks \\cite{pons:rw05}. Several other diffusion motivated\napproaches proposed recently (\\textit{e.g.}\n\\cite{wu:physics04,simonsen:diff04,fouss:novel05}) are more or\nless consistent with random-walk analogy.\n\nIn our model we break the similarity with classical random\nwalks and the theory of flows in the graph \\cite{diestel:graph} in\nfavour of a realistic physical picture. First, we allow nodes to\naccumulate substance by assigning to them infinite maximum capacity. \nThe direct flow from the node $A$ to the node $B$ is possible if there is\na directed link from $A$ to $B$ and $\\phi_A > \\phi_B$. The flow rate\nin this case depends on the concentration difference $\\phi_B - \\phi_A > 0$ and the out degree\n$k_{out}$ of the node $A$. In the case of $ A < B$ no mass is\ndelivered directly from $A$ to $B$. Such rules in the limit of\ninfinite time lead to equilibrium state with equal mass distribution\nwhich meets the physical expectations. \n\nNetwork links in our realization represent pipes (Fig.~\\ref{fig:scheme}), directed\nlinks  act as pipes allowing mass to pass in one direction. Mass\npropagation within the network system is driven by Flick's law of diffusion:\n\\begin{equation}\ndM = - D \\frac{\\delta \\phi}{\\delta x} dS dt, \n\\end{equation}\nwhere $dM$ is mass change, $\\delta \\phi \/ \\delta x$ is\nconcentration gradient and $dS$ is an area element.\n\nFor our discrete system this implies that the rate of mass\nexchange between the neighbouring nodes is proportional to the difference of masses in these\nnodes. Every node uses its outgoing links to deliver mass to its neighbors with\na smaller amount of ink. The amount of ink $\\Delta_{out} M_i$\ndelivered by the node to its $i$th neighbour is:\n\\begin{equation}\n\\Delta_{out} M_i = - \\frac{\\alpha}{k_{out}} (M_0 - M_i),\n\\label{eq:main}\n\\end{equation}\nwhere $M_0 > M_i$ and $\\alpha$ is the coefficient determining the\ntransfer rate and is constant for all\nnodes. We analyze the mass $M$ contained in the node instead of\nthe concentration $\\phi$ assuming that all nodes have the same\ngeometrical volume.\nThe total delivered mass for a node is the following:\n\\begin{multline}\n\\Delta_{out} M = \\sum_{i=1}^{k_{out}} \\Delta_{out} M_i =  - \\alpha \\left ( M_0 - \\frac{1}{k_{out}}\n\\sum_{i=1}^{k_{out}} M_i \\right) =  \\\\\n- \\alpha ( M_0 - \\overline{M} ),\n\\end{multline}\nwhere $\\overline{M}$ is the mean ink mass in the neighbouring\nnodes with smaller masses. Mass transfer in the pipe happens\ninstantaneously.\n Thus we can apply mass conservation\nlaw and increase mass in the neighbouring nodes by the amount\ntaken from the node:\n\\begin{eqnarray}\n\\Delta_{out} M & = & - \\sum_{i=1}^{k_{out}} \\Delta_{in} M_i \\\\\n\\Delta_{in} M  & = & - \\sum_{i=1}^{k_{in}} \\Delta_{out} M_i \n\\end{eqnarray}\n\nThe total change of mass at a certain node is composed of the loss of mass due\nto diffusion to the neighbours through outgoing links and gain of mass\nby the\namount delivered from neighbors through\nincoming links: $\\Delta M = \\Delta_{in} M + \\Delta_{out} M$. This\nconservation law is the extension of Kirchhoff's\nlaw \\cite{diestel:graph} for the node with non-zero capacity.\n\nIn order to prevent inequality due to sequential nodes processing, mass changes\nfor all nodes were calculated without actual changing the masses and then\nvalues of the masses in all nodes were updated. For the special case of absence of outgoing\nlinks $\\Delta_{out} M = 0$ the specific node acts as a virtual ink absorber which can\nonly gain ink from the neighbours but does not have ways to deliver it\nback. Nodes without incoming links are not \nconsidered due to their invisibility for the data collecting crawler and\nthus are absent in our database.\n\nWe start by putting an initial amount of ink of $M_0 = N$ mass\n units in one of the nodes which we call the \\textit{initiator}. Subsequently system is\nallowed to proceed to the equilibrium state by continuous mass\nredistribution within the network according to our rules. The\n expectation for an \nequilibrium state for a connected network system is equal\ndistribution of mass $M_0$ among the nodes so that each of \nthem ends up having $M_0 \/ N = 1$ mass units. While evolving to this state the system\npasses through non-equilibrium states with non-uniform mass\ndistributions.\n\nImagine a cluster of well connected nodes inside the network\nconnected to the outside world only by few outgoing and\nincoming links. The ink diffusion inside the cluster is relatively fast due to the\npresence of a large number of exchange channels between the \nmembers and a high conductivity of the channels\nensemble. Limited number of channels going outside the cluster forms the\nbottleneck for mass delivery. Under these conditions the flow rate between\nthe members is much higher than between the members and non-members and dispersed ink will\nlikely form an equi-concentrational \\comment{Phys. Rev. E 49, 5431--5437\n  (1994)} volume within the cluster. \nEach cluster in this system with\nspecific connection properties such as flow rate and distance from\nthe initiator would have in each of its\nnodes the same concentration of ink with the value specific to the \nparticular cluster. Thus by estimating the probability distribution\nfunction of concentration one can analyze non-uniformity of ink\ndistribution and reveal separated clusters by determining the\nsignatures of equi-concentration volumes.\n\n\\begin{figure}\n\\centering\n  \\includegraphics[width=1.05\\linewidth]{fig4.eps}\n  \\caption{Dynamics of relative  concentration change in the\n  initiator node {\\em doctor\\_livsy} for different flow rates\n  $\\alpha$. Inset shows rescaled data. Oscillatory parts were cut away.}\n  \\label{fig:concentration_decay}\n\\end{figure}\n\nThe flow rate $\\alpha$ from the equation \\ref{eq:main} can be selected\nfrom the half-interval (0;1] and defines the speed of\n  simulation. Values larger than 0.5 are not desirable because they\n  can cause concentration waves or back-reflections in some cases.\n\nThe proposed method does not aim to decompose the whole network on\nminimal clusters but to reveal significant clusters within the\nnetwork. As we regard the network as an open system which does\nnot have to be fully described by existing database we do not assign\nmeasure of clustering of the whole network like modularity proposed by Newman\n\\cite{newman:mixing,newman:community}. However we can quantify the\nisolation of the individual community $i$ by parameter of \nconfinement $K_i$ which is the characterization of assortative mixing of\nindividual community. We can define $K_i$ using notation of\nNewman \\cite{newman:mixing} as following:\n\\begin{equation}\nK_i = \\frac{e_{ii}}{\\sum_j e_{ij}} = \\frac{e_{ii}}{b_i},\n\\end{equation}\nwhere $e_{ij}$ is the fraction of network edges connecting nodes of\nthe community $i$ to the community $j$ and $\\sum_j e_{ij} = b_i$ is\nthe fraction of edges starting from the members of $i$. Thus \nparameter $K_i$ defines the number of links connecting the nodes\ninside the community $i$ as a fraction  of the total number of links\noriginating from the members of $i$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{fig5.eps}\n\\caption{Probability distribution functions of virtual ink\n  concentration $M$ at two stages of the diffusion process with $\\alpha =\n  0.1$ and {\\em doctor\\_livsy} as the initiator node. Inset represents\n  the same data in linear scale. Two well pronounced peaks of two\n  separated communities are clearly seen.}\n \\label{fig:profiles}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{fig6.eps}\n    \\caption{Dynamics of virtual ink distribution within LJ\n      network as a logarithmically color coded probability\n      distribution function of the ink concentration (vertical \n      axis) and simulation step (horizontal axis). Separation of\n      Russian-speaking community (thin upper line, high concentration\n      values) from general English-speaking (thicker lower line, lower\n      concentration values) can be clearly seen.}\n    \\label{fig:dynamics}\n\\end{figure}\n\n\\section{RESULTS AND DISCUSSION}\n\nTo test our method we performed ink diffusion simulations using our\nLJ database starting from different initiator nodes.\nFig.~\\ref{fig:concentration_decay} shows the relative mass decay as a \nfunction of simulation step number $T$ for the flow rates $\\alpha =\n0.1$, 0.25 and 0.5. User {\\em doctor\\_livsy} with a high number\nof incoming links was chosen as the initiator node. As we will show later\nthis user belongs to extremely confined Russian-speaking community.\nThe inset of Fig.~\\ref{fig:concentration_decay} shows \nthe same data rescaled with respect \nto $\\alpha$. As one can see from the match of rescaled curves the\ndynamics of the process does not depend on the flow rate $\\alpha$ in this\nrange. The striking feature of the presented data is the obvious\nstep-like form of the curves which is the effect of non-homogeneous\nstructure of the LJ network. Flat parts of the $\\Delta M \/ M$ curves\ncorrespond to the exponential decays of $M$ which is the\nsign of non-restricted diffusion of ink. The first significant drop of the\ndecay rate happens when $T \\alpha \\approx 5$ which is equal to the\ndouble radius of the community to which our initiator belongs. This\ncorresponds to the moment when virtual ink \nfills the whole community and further expansion of filled area is\nimpeded by the limited number of links going outside the community.\nSo if it takes $T_0$ simulation steps for the virtual ink to reach the\nborders of the community it also takes $T_0$ simulation steps for the\ndecay of concentration gradient to reach the initiator node and together this\ngives double size of the community.\nThe second drop at $T \\alpha \\approx 22$ is not well pronounced and\n corresponds to the filling of the whole network. \n\nAs our community discovering algorithm is based on the detection of\nequi-concentration volumes we performed the calculation of the\nprobability distribution function of $M$ at two stages of\nvirtual ink diffusion for $\\alpha = 0.1$ (Fig.\\ref{fig:profiles}). One\ncan see two well \npronounced peaks on all plots which occurred to be the Russian speaking\ncommunity (larger values of mass $M$) and the rest of LJ network (broader peak at\nsmaller values of $M$). \n\nThe dynamics of virtual ink distribution is presented in\nthe Fig.\\ref{fig:dynamics}. As it can be seen a distinct separation of the\nRussian community peak from the main peak is formed before step\n$T \\alpha = 50$. At the latter stage it is quite stable and easily distinguishable up to\niteration $T \\alpha = 10^3$ which gives quite a long quasi-stationary stage\nthat can be used for communities detection. It also demonstrates that the\nprocess of equi-concentrational volumes formation is much faster than the\nrelaxation of the whole system. \n\nIf the initiator node is selected somewhere outside the community the\nsplitting of the distribution peak is also observed but for this case\naverage concentration within the Russian community is smaller compared\nto the\nrest of the LJ nodes. This supports the expectations that if the\ncommunity has a limited number of outgoing links it also lacks \nincoming links.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{fig7.eps}\n    \\caption{Two-dimensional map of LJ users network obtained by\n    concentration configurations of independent diffusion processes\n    from two initiator nodes on the stage $T \\alpha = 100$.}\n    \\label{fig:mapping}\n\\end{figure}\n\n\\begin{table*}[t]\n\\caption{Examples of discovered communities within LiveJournal\n  userspace.}\n\\label{tab:comms}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccl}\nRepresenting node & Number of users & Specificity & Confinement $K$ &\nComments \\\\\n\\hline\n{\\em doctor\\_livsy} & 227314 & 99.89\\% & 98.34\\% & Russian speaking\ncommunity\\footnote{92\\% of users have Cyrillic letters in their\n  information pages or journals} \\\\\n{\\em future\\_visions} & 421 & 98.36\\% & 96.22\\% & Fandom High Role-Playing\nGame community \\\\\n{\\em alected } & 262 &  99.21\\% & 99.10\\%  & Leviosa Role-Playing Game community \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\n\nThe accuracy of community discovering scheme can be improved by\nsimultaneous simulation of the diffusion from two or more initiator \nnodes. Here we assigned two  independent concentration values to a\nsingle node. All diffusion processes proceed without\ninfluencing each other. The LJ network can now be mapped as a \nprobability distribution function of two concentrations and thus the\ncommunity can be localized on a two dimensional plot \nas shown in the Fig.~\\ref{fig:mapping} for {\\em doctor\\_livsy} and\n{\\em future\\_visions} as the initiator nodes. One can see two main separated\npeaks corresponding to the major part of LJ network and the Russian-speaking\ncommunity. The abundance of noise-like spots on the map corresponds to\nthe small well-separated and well linked communities existing in the\nnetwork which are well localized.  \n\nThe selection of nodes from a certain community can be performed by simple\nthresholding the values of both concentrations. The group of nodes with the\nconcentration values within the selected range which form the  connected\ncomponent in the network can be identified as the community.\nThe ratio of the number of connected nodes to the total number of\nusers with concentrations within the range defines the\n\\textit{specificity} of the method. \n\nAs the complete analysis of LJ community structure as well as the\nreasons of their formation is out of the scope of the current paper we\nwill not list all user cliques found. However in the Tab.~\\ref{tab:comms} we\nlist the largest LJ community and two smaller\nones together with their parameters. The size of discovered\nRussian-speaking community is of the order of the total number of LJ \nusers from the Russian Federation according to LJ database statistics \\cite{lj:stat}\n($232\\;241$ users in January 2006). The obvious reason for the separation of\nthis community with a very high value of confinement $K = 98.34$\\% is\nthe prevailing usage\nof Russian language. We found by separate analysis of info pages and\njournal entries that 92\\% of the users within this community are using\nCyrillic alphabet. The fact that the Russian LJ community differs from\nthe rest of LJ network has been already\npointed out by Internet observers (e.g. Ref.~\\cite{gorny:RLJ}).\nThe two other listed communities are the examples of surprisingly popular class of Role-Playing Game\n communities formed by the virtual users playing characters and \nwriting their journals on behalf of these characters.  \n\n\\section{CONCLUSIONS}\n\nThe LiveJournal friendship network was studied with the general approach\ndeveloped for the complex networks and a power-law tail with exponent\n$\\gamma = 3.45$ was found in the degree distributions. This network\nalso demonstrated small-world property and high clustering.\n\nTo study the community structure we utilized the original thermodynamic approach.\nWe found that diffusion in an essentially non-euclidean geometry\nof a complex network with community structure leads to a peculiar\nphenomenon of formation of quasi-stationary equi-concentration volumes\nas shown by our simulation. This proves to be very useful\nfor the detection of well-interconnected groups of nodes. With a limited number of\nparallel diffusion processes sufficient for a rough decomposition our method has an $O(N ln\nN)$ complexity  (each simulation step analyzes\n$M = \\langle k \\rangle N$ edges which for a sparse matrix $M = O(N)$\nand the required number of steps is proportional to the \ndiameter of the network which is $O (ln N) $). It is currently one of\nthe fastest algorithms and was applied for a huge directed network of LJ users \ncontaining several millions of nodes. To obtain results presented in this \npaper it takes only one or two hours of \ndesktop computer time. Moreover this method can be applied locally to\na specific part of the network even with the lack of complete information about distant\nparts of the network. The sensitivity of decomposition can be tuned by\nincreasing the number of initiator nodes with the limit of complete decomposition\nwhen every node acts like initiator of its own diffusion process.\n\n\\acknowledgments\n\nFinancial support by the Swiss National Science Foundation is\ngratefully acknowledged. We thank Frank Scheffold for helpful\ndiscussion. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe measurement of the Li abundance of turnoff and dwarf stars in OCs of different ages and metallicities is a very powerful \ntool that can be used to investigate a number of topics: the possibility of sampling stars of different evolutionary status and original mass allows to\nunderstand how this element is destroyed and to test different evolutionary models and mixing mechanisms \n(Boesgaard \\& Tripicco 1986; Pasquini 2000; Randich et al. 2000; Randich et al. 2003; Smiljanic et al. 2010). \nThis is possible because for OC stars we can obtain reliable ages, metallicities, masses, and evolutionary stages from precise \nphotometry and high-resolution spectroscopy. Cluster stars are the most powerful targets that can be studied to disentangle the effect of each of these parameters \non mixing phenomena. Since Li is burnt through $(p,\\alpha)$ reactions at a temperature of $\\simeq 2.5 \\times 10^{6}$ K, differences in \nLi abundance in stars of different masses and temperatures can be used to constrain the physical processes that bring the surface \nmaterial to the interior of the star. \n\nThe cluster we have studied, NGC~2243, is particularly interesting because it is one of the most metal-poor OCs accessible for \nhigh-resolution spectroscopy. It is almost as metal-poor as the globular cluster 47 Tucanae, considered  ``metal rich'' among  globular clusters with a metallicity of [Fe\/H]=-0.76 $\\pm$ 0.04 dex (\\citet{Koch08}). \nThe comparison of the Li abundances in the two clusters is very interesting, since, in spite of the similar \nmetallicity, NGC~2243 is much younger than 47 Tuc. The so-called ``Li dip'', i.e., the disappearance of Li in the spectra of stars over \na narrow temperature range, is a feature that is present in OCs, but in none of the globular clusters. It is interesting  \nto investigate whether it is present also in an OC with a low metallicity comparable to that of ``metal-rich'' globular clusters. \nAnother interesting characteristic of OCs  that are young enough to have hot main sequence stars on the blue side of the dip (see, e.g., \\citealt{Sestito05}, and references therein) is that hot stars out of the ``Li dip'' show a high level of \nLi abundance, close to the abundance present in the interstellar medium when they formed. This  \ninitial Li abundance in metal-poor clusters will provide precious data to be compared with \nmodels of Li evolution in the Galaxy (e.g., Matteucci 2010)  and primordial nucleosynthesis \\citep{CFO08}.\n\nThe first determination of the Li abundance in this cluster was performed by Hill \\& Pasquini (2000). They found a mean \nLi abundance of $\\log n{\\rm (Li)}=2.35$ in the four stars for which they could detect Li. These authors hinted at the presence of a Li \ndip, but could not confirm it, given their limited sample. \n\nHere, we present a new study of elemental abundances in NGC~2243, based on high-resolution spectra obtained with FLAMES on the VLT. \nIn Sect.~\\ref{sec:ngc2243} we briefly describe the cluster. The observations and data reduction are given in Sect.~\\ref{sec:Obs}, \nwhile the data analysis is discussed in Sect.~\\ref{sec:analysis}. The results, discussion, and conclusions are presented in \nSects.~\\ref{sec:results}, \\ref{sec:discussion}, and \\ref{sec:concl}.\n\n\\label{sec:Intro}\n\n\n\\section {NGC 2243: an old metal-poor open cluster}\n\\label{sec:ngc2243}\n\nNGC~2243 is an old and metal-poor OC located towards the anti-center at $R_{\\rm G}=10.76$ kpc and $z=1.1$ kpc away from the plane \nof the Milky Way ($l_{II}=239\\degr.478$, $b_{II}=-18\\degr.014$). The first CCD photometry of NGC~2243 was obtained by Bonifazi et al. (1990) \nwho found an age of $4\\pm1$ Gyr, a metallicity of [Fe\/H]$=-0.80\\pm0.10$, and a distance modulus $(m-M)_{0}=12.8\\pm0.2$. From $BV$ CCD \nphotometry, Bergbusch et al. (1991) derived a distance modulus of 13.05 (assuming $E(B-V)=0.06$), a metallicity of $-0.47$ dex, an \noverabundance of oxygen [O\/Fe]=+0.23, and an age of 5 Gyr. \nGratton \\& Contarini (1994)  confirmed this low metallicity using high-resolution spectra. \nThey determined  the iron abundance in two giants of the cluster and found [Fe\/H]$=-0.48\\pm0.15 $ dex \nand  [Ca\/Fe]$=0.18 \\pm0.17 $ dex.\nFitting the data of Bergbusch et al. (1991) to new isochrones, assuming core overshooting, \nVandenberg et al. (2006) obtained [Fe\/H]$=-0.61$, [$\\alpha$\/Fe]=0.3, an age of 3.1 Gyr and $(m-M)_{V}=13.15$, assuming $E(B-V)=0.062$. \nFrom $VI$ CCD photometry, Kaluzny et al. (1996) found $E(V-I)=0.10\\pm0.04$. Kaluzny et al. (2006) extended their study of NGC~2243 to \nthe analysis of detached eclipsing binaries. They derived a distance modulus $(m-M)_{V}=13.24\\pm0.08$. Using model age-luminosity and \nage-radius relations for the binary system NV CMa, a member of the cluster, they obtained an age of $4.35\\pm0.25$ Gyr assuming a metallicity \n[Fe\/H]$=-0.525$. However, these values are sensitive to the adopted metallicity, a change of $-0.08$ dex in the metallicity leading to \nan age decrease of $\\simeq 0.5$ Gyr. \n\n\\section{Observations and data reduction}\n\\label{sec:Obs}\nThe spectra were obtained in four hours in four observing nights during the FLAMES+GIRAFFE Science Verification program. \nFLAMES (Pasquini et al. 2002) is mounted at the Nasmyth A platform of \nthe 8.2m UT2\/Kueyen of the VLT. Observations were carried out in MEDUSA mode with two GIRAFFE high-resolution (HR) settings  and with a single \n2K$\\times$4K EEV CCD (15 $\\mu$m pixels). In particular, the employed settings were Filter N. 14 ({\\it \\'echelle} grating order \nN. 9) with central wavelength 651.5 nm and wavelength range between 638.3 and 662.6 nm, and Filter N. 15 ({\\it \\'echelle} order \nN. 8) with the central wavelength 679.7 nm and wavelength range between 659.9 and 695.5 nm. \nThe resolution was $R\\sim$28\\,800 and $R\\sim$19\\,300, respectively. The log of the observations is given in Table~\\ref{tab:observations}. \n\nWe selected from the catalog of \\cite{Kalu1996} the main-sequence (MS), turnoff (TO), and (sub-)giant stars with $14\\fm0\\le V \\le17\\fm5$ and, \nthanks to the GIRAFFE configuration, we were able to observe 100 targets (Tables~\\ref{tab:LiAb}, \\ref{tab:LiAbUp}, \\ref{tab:binaries}). \nSeparate exposures were obtained to be able to identify short and intermediate period binaries by comparing the radial velocities \nat different epochs. The spectra have a typical signal-to-noise ($S\/N$) ratio of 30--40. \n\nThe observations were reduced using the version 1.12 of the GIRAFFE Base-Line Data Reduction Software \n(girBLDRS\\footnote{http:\/\/girbldrs.sourceforge.net\/}; \\citealt{Blecha2000}), which is a set of python \nscripts\/modules and a C library. With it we removed the instrumental signature from the observed data, \nsubtracting the bias and dividing by the normalized flat-field. Flat-field acquisitions were also \nused to trace the position of all the fibers, and to derive the parameters for the optimal extraction \nof the science exposures. Finally, the wavelength calibration was obtained using the day-time Th-Ar lamp exposure. \n\nRadial velocities were measured using the IRAF\\footnote{IRAF is distributed by the National Optical Astronomy \nObservatory, which is operated by the Association of the Universities for Research in Astronomy, Inc. (AURA) under \ncooperative agreement with the National Science Foundation.} package FXCOR, which cross-correlates the observed \nspectrum with a template. As a template, we used a solar spectrum acquired with FLAMES\/GIRAFFE. Finally, the \nheliocentric correction was applied. We considered only the spectra acquired with the HR14 setting, as it leads to \nthe radial velocity measurements with the lowest rms uncertainties (around 1 km\\,s$^{-1}$),  compared to the 2.5 km\\,s$^{-1}$ obtained with the HR15 setting. The two spectra\/star\/setting were finally co-added to perform the spectroscopic determination of elemental abundance (and effective temperature). \n\nThe GIRAFFE solar spectra\\footnote{http:\/\/www.eso.org\/observing\/dfo\/quality\/GIRAFFE\/pipeline\/solar.html}, \ntaken with the same setups of our observations, are used throughout this work for spectroscopic comparison with the stars \nand the synthetic spectra. Each solar spectrum was obtained by averaging most of the GIRAFFE \nspectra (some spectra show clear flat field problems and were not used) and it has a nominal $S\/N$ ratio above 400.\n \n\\begin{table}  \n\\caption{Log of the observations.}\n\\label{tab:observations}\n\\begin{center}  \n\\begin{tabular}{cccccc}\n\\hline\n\\hline\n$\\alpha$    & $\\delta$  &  Date      &  UT      & $t_{\\rm exp}$ & Filter \\\\\n(\\degr)     &  (\\degr)  & (d\/m\/y)    & (h:m:s)  &  (s)          &        \\\\ \n\\hline\n97.40327     & $-$31.28795   & 27\/01\/2003 & 04:12:26 & 3600\t& HR15\t \\\\\n97.40331     & $-$31.28800   & 28\/01\/2003 & 05:49:49 & 3600\t& HR14\t \\\\\n97.40326     & $-$31.28808   & 29\/01\/2003 & 03:29:14 & 3600\t& HR15     \\\\\n97.40333     & $-$31.28806   & 31\/01\/2003 & 02:14:15 & 3600\t& HR14     \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}  \n\n\n\n\\section{Data analysis}\n\\label{sec:analysis}\n\n\\subsection{Radial velocities}\nThe evolutionary status of the analyzed stars is indicated by their position in the color-magnitude diagram (CMD) shown in \nFig.\\ref{fig:cmd_ngc2243}, in which our sample is shown with circles. The stars are mostly MS, TO, post-TO, subgiants, \nand giants, where the region around the TO is very well sampled. \n\nHaving several exposures for each object allows the comparison of the single radial velocity (RV) for each star. \nWe have retained as most-probable single members all those showing RV variations smaller than 2 km\\,s$^{-1}$~  \nin the two exposures acquired and having a mean velocity within about 3 sigma ($\\approx 3$ km\\,s$^{-1}$) from the median \ncluster RV. The final values of the difference in the computed RV is given in Fig.~\\ref{fig:delta_vrad_V}, \nas well as the distribution, which is well represented by a Gaussian with a $\\sigma$ of 0.8 km\\,s$^{-1}$. The final mean\n radial velocities are shown in Fig.~\\ref{fig:vrad_VI} as a function of the $V-I$ color. Most of the RV \nare concentrated at around 62 km\\,s$^{-1}$, close to the expected radial velocity of the cluster. A total of 82 stars are retained as {\\it bona fide} RV single members because their RV distribution can be well approximated by a Gaussian with central RV of $61.9\\pm0.8$ km\\,s$^{-1}$  as shown in Fig.~\\ref{fig:vrad_distribution_fin_selNEW}. In Tables~\\ref{tab:LiAb}, \\ref{tab:LiAbUp}, and \\ref{tab:binaries} the RV values of \nthe stars with well-measured Li abundances, upper-limited Li abundances, and most-probable binaries\n or non-members are listed.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{cmd_NGC2243_zoom_bis.eps}\n\\caption{Portion of the CMD of NGC~2243. Small dots represent $V$ and $V-I$ data taken from \\cite{Kalu1996}, \nwhile squares, triangles, and diamonds mark the positions of red giants, variables, and blue stragglers \ntaken from the literature (WEBDA database). Circles refer to our targets, where \nthe position of the most-probable single stars and binaries or non-members \n(from our radial velocity analysis) are marked in green and blue, \nrespectively, while red giants from the literature (WEBDA database) are in red.} \n\\label{fig:cmd_ngc2243}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{delta_vrad_V.eps}\n\\caption{Final radial velocity variations versus $V$ magnitude. The right part of the figure represents the histogram of the computed mean RV differences. \nOpen symbols refer to the final selection of binaries or non-members.} \n\\label{fig:delta_vrad_V}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{vrad_VI.eps}\n\\caption{Mean radial velocities as a function of the $V-I$ color. The dashed line represents the final mean RV $\\sim 62$ km\\,s$^{-1}$, \nwhile the filled area is the \n$\\pm 3 \\sigma$ level ($\\sim 3$ km\\,s$^{-1}$). Open symbols refer to the final selection of binaries or non-members.} \n\\label{fig:vrad_VI}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{vrad_distribution_fin_selNEW.eps}\n\\caption{Histogram of the radial velocity distribution of the 82 most-probable single members selected in NGC~2243 (continuous line). \nA Gaussian fit to the distribution of member stars  is also displayed, with a mean \nvalue of $<V_{\\rm rad}>=61.9\\pm0.8$ km\\,s$^{-1}$ (dashed line).} \\label{fig:vrad_distribution_fin_selNEW}\n\\end{figure}\n\n\n\\subsection{Atmospheric parameters}\nWe used the photometric results for $V-I$ colors from \\cite{Kalu1996} to derive the effective temperatures\n from the relations of \\cite{Casa2010}. We also attempted to measure effective temperatures from \n our GIRAFFE spectra, using the H$\\alpha$ wings, but the results were not convincing when comparing \nthese effective temperatures with photometry. Most likely the flat field procedure adopted did not\n succeed in fully removing the rather strong blaze present in the spectra, possibly because of the relatively\n  close position of the H$\\alpha$ line to the spectrum edge. \n\nThe surface gravities for the program stars were obtained from the photometric data quoted above and computed using the standard relation given in \nEq.~\\ref{eq:logg}. We adopted the following solar values: $\\log g_{\\odot} = 4.44 dex, T_{\\rm eff, \\odot} = 5790$ K, $M_{\\rm bol, \\odot} = 4.72$ mag. \nThe bolometric corrections were computed using the relation of \\cite{Alon1999}. We assumed a TO mass of $1.15 M_{\\odot}$ and a distance modulus of 13.15. We note that even if the adopted turnoff mass  was wrong by 25$\\%$, the errors in computing $\\log g$ would be less than 0.1 dex.\n\n\\begin{equation}\n\\label{eq:logg}\n \\log g_{*} = \\log g_{\\odot} + \\log \\frac{M_{*} }{M_{\\odot}} + \\log \\frac {T_{\\mathrm{eff},*}}{T_{\\mathrm{eff},\\odot}} + 0.4 (M_{\\mathrm{bol},*} - M_{\\mathrm{bol},\\odot}). \n\\end{equation}\nThe microturbulence velocities were computed using the relation of \\cite{Edva1993}. \n\n\n\n\\section{Results}\n\\label{sec:results}\n\n\\subsection{Lithium abundance}\nThe equivalent widths of the Li resonance doublet at $\\lambda=$670.78 nm was measured using the code \n{\\it FITLINE} written by P. Fran\\c cois. Details on the algorithm can be found in \\cite{Lema2007}. The line was \nmeasurable in 27 stars. When the lines were not detectable, an upper limit for the equivalent width of the line was \nestablished via the Cayrel formula \\citep{Cayrel1988} and translated into an upper limit for the Li abundance. \n\nWe then derived Li abundances for the measured line equivalent widths  using {\\it Turbospectrum} \n\\citep{Alva1998} taking the most updated version of the OSMARCS atmosphere models  \n(e.g., \\citealt{Gusta2008}, \\citealt{Plez2008}). Plane parallel 1D models have been used for the computations. We adopted the \nGrevesse \\& Sauval (1998) solar abundances. \nIn Table~\\ref{tab:LiAb} the list of stars for which the Li abundances could be measured is given, \nwhile Table~\\ref{tab:LiAbUp} lists the estimation of the \nLi abundance upper limits for the remaining stars of the  sample.\n\n\n\\begin{table*}\n\\scriptsize\n\\caption{Stars with well-measured lithium abundance.}\n\\label{tab:LiAb}\n\\begin{center}  \n\\begin{tabular}{lcccccccccc}\n\\hline\nObject & $V$ & $V-I$ & $T_{\\rm eff}$ & $\\log g$ & [\\ion{Fe}{i}\/H] & [\\ion{Fe}{ii}\/H] &$\\log n{\\rm (Li)}$ & $V_{\\rm rad}^{1}$ & $V_{\\rm rad}^{2}$ & $<V_{\\rm rad}>$\\\\\n       & (mag) &  & (K) &  &  &  & & (km\\,s$^{-1}$)& (km\\,s$^{-1}$)& (km\\,s$^{-1}$)\\\\\n\\hline\n\n 148 &\t17.037\t& 0.504 & 6646 & 4.39 &\t$-$0.56 &\t$-$0.62 &\t2.46  & 60.8$\\pm$1.4& 60.9$\\pm$1.2& 60.8$\\pm$0.9\\\\\n 231 &\t17.400\t& 0.559 & 6355 & 4.45 &\t$-$0.64 &\t$-$0.62 &\t2.44  & 64.3$\\pm$1.3& 64.4$\\pm$1.5& 64.3$\\pm$1.0\\\\\n 244 &\t17.467\t& 0.549 & 6405 & 4.49 &\t$-$0.61 &\t        &\t2.64  & 62.5$\\pm$1.7& 63.1$\\pm$1.5& 62.8$\\pm$1.1\\\\\n 403 &\t14.394\t& 0.935 & 5051 & 2.77 &\t$-$0.33 &\t$-$0.42 &\t1.51  & 62.7$\\pm$0.8& 62.6$\\pm$0.5& 62.7$\\pm$0.4\\\\\n 414 &\t17.412\t& 0.540 & 6452 & 4.48 &\t$-$0.59 &\t$-$0.54 &\t2.71  & 63.4$\\pm$1.6& 62.7$\\pm$1.2& 63.0$\\pm$1.0\\\\\n 655 &\t16.969\t& 0.482 & 6773 & 4.39 &\t$-$0.56 &\t$-$0.59 &\t2.66  & 61.9$\\pm$1.9& 62.3$\\pm$1.1& 62.1$\\pm$1.1\\\\\n 728 &\t16.987\t& 0.500 & 6668 & 4.37 &\t$-$0.55 &\t$-$0.40 &\t2.58  & 61.8$\\pm$1.0& 62.3$\\pm$1.3& 62.0$\\pm$0.8\\\\\n 766 &\t17.315\t& 0.540 & 6452 & 4.44 &\t$-$0.55 &\t        &\t2.47  & 61.1$\\pm$1.7& 60.8$\\pm$1.4& 60.9$\\pm$1.1\\\\\n 873 &\t17.377\t& 0.524 & 6536 & 4.49 &\t$-$0.58 &\t$-$0.63 &\t2.55  & 60.2$\\pm$2.5& 60.5$\\pm$1.6& 60.3$\\pm$1.5\\\\\n 1106&\t17.477\t& 0.518 & 6568 & 4.54 &\t$-$0.50 &\t        &\t2.73  & 63.5$\\pm$1.9& 63.1$\\pm$1.5& 63.3$\\pm$1.2\\\\\n 1161&\t17.194\t& 0.524 & 6536 & 4.42 &\t$-$0.76 &\t$-$0.87 &\t2.33  & 61.6$\\pm$1.6& 62.0$\\pm$1.5& 61.8$\\pm$1.1\\\\\n 1183&\t17.213\t& 0.510 & 6612 & 4.45 &\t$-$0.43 &\t        &\t2.83  & 62.5$\\pm$3.0& 61.0$\\pm$2.3& 61.8$\\pm$1.9\\\\\n 1189&\t17.366\t& 0.535 & 6478 & 4.47 &\t$-$0.56 &\t        &\t2.57  & 61.5$\\pm$1.5& 61.6$\\pm$1.5& 61.6$\\pm$1.1\\\\\n 1241&\t15.824\t& 0.507 & 6629 & 3.90 &\t$-$0.58 &\t$-$0.52 &\t2.39  & 60.9$\\pm$0.8& 60.8$\\pm$0.5& 60.8$\\pm$0.5\\\\\n 1273&\t15.796\t& 0.459 & 6913 & 3.96 &\t$-$0.52 &\t$-$0.58 &\t2.69  & 61.2$\\pm$1.0& 61.5$\\pm$0.6& 61.4$\\pm$0.6\\\\\n 1284&\t17.284\t& 0.526 & 6525 & 4.45 &\t$-$0.66 &\t        &\t2.36  & 61.1$\\pm$2.6& 61.9$\\pm$2.1& 61.5$\\pm$1.7\\\\\n 1317&\t17.153\t& 0.508 & 6624 & 4.43 &\t$-$0.57 &\t        &\t2.84  & 62.1$\\pm$2.0& 63.4$\\pm$1.5& 62.7$\\pm$1.3\\\\\n 1323&\t17.134\t& 0.503 & 6652 & 4.43 &\t$-$0.61 &\t$-$0.63 &\t2.63  & 62.9$\\pm$1.0& 62.7$\\pm$1.2& 62.8$\\pm$0.8\\\\\n 1405&\t17.263\t& 0.528 & 6515 & 4.44 &\t$-$0.73 &\t        &\t2.80  & 62.7$\\pm$1.2& 61.4$\\pm$3.5& 62.1$\\pm$1.8\\\\\n 1469&\t17.434\t& 0.545 & 6426 & 4.48 &\t$-$0.42 &\t$-$0.36 &\t2.39  & 63.4$\\pm$1.9& 62.4$\\pm$1.4& 62.9$\\pm$1.2\\\\\n 1506&\t16.944\t& 0.499 & 6674 & 4.36 &\t$-$0.50 &\t        &\t2.63  & 63.5$\\pm$1.5& 64.3$\\pm$1.8& 63.9$\\pm$1.2\\\\\n 1634&\t15.734\t& 0.581 & 6249 & 3.75 &\t$-$0.49 &\t$-$0.47 &\t2.88  & 62.7$\\pm$0.4& 62.0$\\pm$0.5& 62.4$\\pm$0.3\\\\\n 1663&\t15.729\t& 0.561 & 6345 & 3.78 &\t$-$0.48 &\t$-$0.51 &\t2.92  & 61.1$\\pm$2.6& 62.1$\\pm$1.8& 61.6$\\pm$1.6\\\\ \n 1776&\t17.031\t& 0.512 & 6601 & 4.37 &\t$-$0.74 &\t$-$0.62 &\t2.66  & 62.0$\\pm$0.7& 62.0$\\pm$0.4& 62.0$\\pm$0.4\\\\\n 1792&\t17.491\t& 0.537 & 6467 & 4.52 &\t$-$0.66 &\t        &\t2.85  & 62.9$\\pm$3.2& 62.9$\\pm$2.0& 62.9$\\pm$1.9\\\\\n 1801&\t15.632\t& 0.518 & 6568 & 3.80 &\t$-$0.49 &\t$-$0.57 &\t2.61  & 61.4$\\pm$1.3& 61.1$\\pm$1.6& 61.2$\\pm$1.0\\\\\n 1992&\t15.141\t& 0.864 & 5231 & 3.15 &\t$-$0.29 &\t$-$0.30 &\t1.24  & 63.1$\\pm$0.8& 63.1$\\pm$0.5& 63.1$\\pm$0.5\\\\\n\n\n\n\n\\hline\t\t\t\t       \t\t\t          \n\\end{tabular}\n\\end{center}\n\\end{table*}\n\\normalsize\t\t\t       \n\n\\begin{table*}\n\\scriptsize\n\\caption{Stars with upper limits in lithium abundances.}\n\\label{tab:LiAbUp}\n\\begin{center}  \n\\begin{tabular}{lcccccccccc}\n\\hline\nObject & $V$ & $V-I$ & $T_{\\rm eff}$ & $\\log g$ & [\\ion{Fe}{i}\/H] & [\\ion{Fe}{ii}\/H] &$\\log n{\\rm (Li)}$ & $V_{\\rm rad}^{1}$ & $V_{\\rm rad}^{2}$ & $<V_{\\rm rad}>$\\\\\n       & (mag) &  & (K) &  &  &  & & (km\\,s$^{-1}$)& (km\\,s$^{-1}$)& (km\\,s$^{-1}$)\\\\\n\\hline\n\n33   & 17.223 & 0.524 & 6536 & 4.43 & $-$0.58 &         & 2.05   & 61.5$\\pm$1.5& 61.1$\\pm$1.8& 61.3$\\pm$1.2\\\\\n80   & 16.587 & 0.475 & 6815 & 4.25 & $-$0.54 & $-$0.48 & 1.71   & 62.6$\\pm$0.9& 62.7$\\pm$1.2& 62.6$\\pm$0.8\\\\\n120  & 15.791 & 0.510 & 6612 & 3.88 & $-$0.50 & $-$0.59 & 2.12   & 64.9$\\pm$0.7& 64.6$\\pm$0.5& 64.8$\\pm$0.4\\\\\n213  & 16.371 & 0.482 & 6773 & 4.16 & $-$0.75 & $-$0.58 & 2.22   & 62.0$\\pm$1.2& 62.5$\\pm$1.3& 62.3$\\pm$0.9\\\\\n274  & 17.243 & 0.534 & 6483 & 4.42 & $-$0.58 & $-$0.61 & 1.79   & 63.9$\\pm$1.9& 62.7$\\pm$1.4& 63.3$\\pm$1.2\\\\\n354  & 16.636 & 0.478 & 6797 & 4.27 & $-$0.57 & $-$0.70 & 2.08   & 62.8$\\pm$1.0& 62.0$\\pm$1.2& 62.4$\\pm$0.8\\\\\n397  & 17.136 & 0.517 & 6574 & 4.41 & $-$0.66 &         & 2.13   & 63.5$\\pm$2.7& 63.5$\\pm$2.1& 63.5$\\pm$1.7\\\\\n439  & 16.937 & 0.487 & 6743 & 4.37 & $-$0.47 &         & 2.28   & 62.3$\\pm$1.5& 63.6$\\pm$1.2& 62.9$\\pm$1.0\\\\\n456  & 16.640 & 0.470 & 6845 & 4.28 & $-$0.62 &         & 1.96   & 61.9$\\pm$1.0& 62.9$\\pm$1.4& 62.4$\\pm$0.9\\\\\n485  & 16.870 & 0.489 & 6732 & 4.34 & $-$0.63 & $-$0.60 & 2.03   & 61.9$\\pm$1.0& 61.7$\\pm$1.3& 61.8$\\pm$0.8\\\\\n520  & 16.688 & 0.465 & 6875 & 4.31 & $-$0.66 &         & 2.29   & 62.7$\\pm$1.9& 61.8$\\pm$2.3& 62.2$\\pm$1.5\\\\\n523  & 16.274 & 0.491 & 6720 & 4.10 & $-$0.56 & $-$0.53 & 1.96   & 64.0$\\pm$1.0& 64.5$\\pm$0.8& 64.2$\\pm$0.7\\\\\n529  & 16.131 & 0.484 & 6761 & 4.06 & $-$0.47 & $-$0.51 & 2.09   & 62.8$\\pm$1.0& 63.2$\\pm$0.8& 63.0$\\pm$0.6\\\\\n547  & 16.240 & 0.486 & 6749 & 4.10 & $-$0.36 & $-$0.36 & 1.90   & 62.5$\\pm$0.8& 62.5$\\pm$0.6& 62.5$\\pm$0.5\\\\\n577  & 15.641 & 0.596 & 6179 & 3.69 & $-$0.39 &\t$-$0.61 & 1.74   & 61.7$\\pm$0.4& 61.9$\\pm$0.5& 61.8$\\pm$0.3\\\\\n612  & 16.332 & 0.473 & 6827 & 4.15 & $-$0.44 & $-$0.60 & 2.26   & 62.2$\\pm$1.2& 62.2$\\pm$1.3& 62.2$\\pm$0.9\\\\\n631  & 15.272 & 0.770 & 5509 & 3.32 & $-$0.51 & $-$0.56 & 0.86   & 64.1$\\pm$0.6& 64.2$\\pm$0.4& 64.1$\\pm$0.3\\\\\n657  & 16.374 & 0.511 & 6607 & 4.11 & $-$0.68 & $-$0.67 & 2.15   & 61.0$\\pm$1.0& 61.3$\\pm$1.0& 61.1$\\pm$0.7\\\\\n684  & 17.033 & 0.507 & 6629 & 4.38 & $-$0.69 &         & 1.81   & 61.1$\\pm$2.1& 61.6$\\pm$1.7& 61.3$\\pm$1.4\\\\\n700  & 17.212 & 0.503 & 6652 & 4.46 & $-$0.52 &         & 2.03   & 61.6$\\pm$2.2& 62.0$\\pm$1.7& 61.8$\\pm$1.4\\\\\n770  & 16.734 & 0.480 & 6785 & 4.30 & $-$0.61 & $-$0.45 & 1.69   & 62.6$\\pm$1.1& 62.4$\\pm$1.2& 62.5$\\pm$0.8\\\\\n784  & 16.726 & 0.479 & 6791 & 4.30 & $-$0.56 & $-$0.50 & 1.92   & 61.1$\\pm$1.3& 60.8$\\pm$1.3& 61.0$\\pm$0.9\\\\\n838  & 15.869 & 0.464 & 6882 & 3.98 & $-$0.54 & $-$0.59 & 2.30   & 61.2$\\pm$0.8& 61.7$\\pm$0.7& 61.5$\\pm$0.5\\\\\n863  & 15.794 & 0.457 & 6925 & 3.96 & $-$0.49 &         & 2.12   & 61.9$\\pm$1.1& 62.2$\\pm$1.0& 62.0$\\pm$0.8\\\\\n875  & 17.190 & 0.504 & 6646 & 4.45 & $-$0.65 & $-$0.58 & 2.13   & 62.7$\\pm$2.0& 62.4$\\pm$1.4& 62.6$\\pm$1.2\\\\\n1035 & 16.464 & 0.458 & 6919 & 4.23 & $-$0.51 & $-$0.54 & 2.01   & 61.8$\\pm$0.8& 62.1$\\pm$0.8& 61.9$\\pm$0.6\\\\\n1100 & 16.390 & 0.465 & 6875 & 4.19 & $-$0.60 & $-$0.73 & 2.29   & 62.9$\\pm$1.3& 62.3$\\pm$1.3& 62.6$\\pm$0.9\\\\\n1253 & 16.370 & 0.486 & 6749 & 4.15 & $-$0.68 & $-$0.64 & 1.98   & 61.4$\\pm$1.4& 61.8$\\pm$0.9& 61.6$\\pm$0.8\\\\\n1299 & 16.868 & 0.488 & 6738 & 4.35 & $-$0.52 & $-$0.48 & 2.28   & 62.1$\\pm$1.2& 62.1$\\pm$1.1& 62.1$\\pm$0.8\\\\\n1321 & 16.605 & 0.470 & 6845 & 4.27 & $-$0.45 & $-$0.40 & 2.27   & 62.4$\\pm$1.2& 61.8$\\pm$1.0& 62.1$\\pm$0.8\\\\\n1339 & 16.454 & 0.481 & 6779 & 4.19 & $-$0.59 & $-$0.62 & 1.91   & 65.3$\\pm$1.5& 65.4$\\pm$1.2& 65.4$\\pm$1.0\\\\\n1375 & 16.471 & 0.476 & 6809 & 4.21 & $-$0.59 & $-$0.51 & 2.25   & 62.0$\\pm$2.0& 62.4$\\pm$1.4& 62.2$\\pm$1.2\\\\\n1566 & 16.375 & 0.482 & 6773 & 4.16 & $-$0.44 & $-$0.38 & 2.22   & 61.7$\\pm$2.3& 61.9$\\pm$1.1& 61.8$\\pm$1.3\\\\\n1590 & 16.317 & 0.495 & 6697 & 4.11 & $-$0.54 & $-$0.50 & 1.86   & 62.8$\\pm$0.8& 63.0$\\pm$0.8& 62.9$\\pm$0.6\\\\\n1597 & 16.567 & 0.465 & 6875 & 4.26 & $-$0.51 & $-$0.57 & 2.29   & 61.8$\\pm$1.6& 62.5$\\pm$1.0& 62.1$\\pm$0.9\\\\\n1635 & 17.046 & 0.499 & 6674 & 4.40 & $-$0.51 &         & 2.15   & 61.6$\\pm$1.9& 60.7$\\pm$1.5& 61.1$\\pm$1.2\\\\\n1648 & 15.058 & 0.855 & 5255 & 3.13 & $-$0.16 & $-$0.33 & 0.71   & 62.0$\\pm$0.3& 62.1$\\pm$0.2& 62.1$\\pm$0.2\\\\\n1687 & 16.520 & 0.465 & 6875 & 4.24 & $-$0.47 & $-$0.48 & 1.98   & 62.9$\\pm$1.3& 62.0$\\pm$0.9& 62.4$\\pm$0.8\\\\\n1709 & 16.481 & 0.463 & 6888 & 4.23 & $-$0.47 & $-$0.53 & 1.99   & 61.9$\\pm$1.4& 61.8$\\pm$1.1& 61.9$\\pm$0.9\\\\\n1751 & 16.461 & 0.469 & 6851 & 4.21 & $-$0.42 & $-$0.48 & 2.27   & 63.1$\\pm$1.2& 62.7$\\pm$0.9& 62.9$\\pm$0.7\\\\\n1752 & 16.359 & 0.493 & 6709 & 4.13 & $-$0.62 &         & 1.87   & 62.0$\\pm$1.7& 61.6$\\pm$1.1& 61.8$\\pm$1.0\\\\\n1839 & 16.816 & 0.481 & 6779 & 4.34 & $-$0.62 & $-$0.54 & 2.22   & 62.1$\\pm$1.3& 61.0$\\pm$1.5& 61.5$\\pm$1.0\\\\\n1874 & 16.496 & 0.454 & 6944 & 4.25 & $-$0.52 & $-$0.44 & 2.33   & 63.6$\\pm$1.6& 62.1$\\pm$1.2& 62.8$\\pm$1.0\\\\\n\n\n\n\\hline\t\t\t\t       \t\t\t          \n\\end{tabular}\n\\end{center}\n\\end{table*}\n\\normalsize\t\t\t       \n\n\\begin{table}[b]\n\\scriptsize\n\\caption{Radial velocities of the most-probable binary stars and non-members.}\n\\label{tab:binaries}\n\\begin{center}  \n\\begin{tabular}{lccccc}\n\\hline\nObject & $V$ &  $V_{\\rm rad}^{1}$ & $V_{\\rm rad}^{2}$ & $<V_{\\rm rad}>$\\\\\n       & (mag)         &(km\\,s$^{-1}$)     & (km\\,s$^{-1}$)& (km\\,s$^{-1}$)\\\\\n\\hline\n 343 & 14.790 &    52.4$\\pm$0.7&   51.8$\\pm$0.7&   52.1$\\pm$0.5\\\\\n  92 & 15.751 &     15.5$\\pm$0.7&   15.7$\\pm$0.5&   15.6$\\pm$0.4\\\\\n 671 & 16.644 &     56.4$\\pm$1.2&  109.4$\\pm$2.2&   82.9$\\pm$1.2\\\\\n 182 & 15.136 &     37.2$\\pm$0.4&   37.2$\\pm$0.4&   37.2$\\pm$0.3\\\\\n 153 & 14.583 &     17.9$\\pm$0.2&   17.7$\\pm$0.2&   17.8$\\pm$0.2\\\\\n 495 & 17.268 &     96.4$\\pm$2.3&   95.5$\\pm$2.2&   95.9$\\pm$1.6\\\\\n1020 & 17.332 &     57.3$\\pm$4.2&   55.5$\\pm$5.8&   56.4$\\pm$3.6\\\\\n1906 & 14.420 &     11.4$\\pm$0.7&   11.3$\\pm$0.5&   11.3$\\pm$0.4\\\\\n 370 & 17.409 &     67.3$\\pm$1.5&   67.7$\\pm$1.8&   67.5$\\pm$1.2\\\\\n2003 & 16.872 &    64.8$\\pm$5.0&   53.4$\\pm$8.0&   59.1$\\pm$4.7\\\\\n2018 & 15.573 &     79.8$\\pm$0.9&   80.0$\\pm$0.6&   79.9$\\pm$0.5\\\\\n1438 & 16.662 &    67.6$\\pm$1.5&   67.2$\\pm$1.4&   67.4$\\pm$1.0\\\\\n1410 & 16.760 &      ...$^{*}$  &     ...$^{*}$ &    ...$^{*}$ \\\\\n1631 & 16.687 &     55.8$\\pm$2.8&   55.9$\\pm$1.5&   55.9$\\pm$1.6\\\\\n 794 & 15.132 &     78.1$\\pm$0.2&   78.0$\\pm$0.6&   78.1$\\pm$0.3\\\\\n1831 & 15.566 &     44.1$\\pm$0.9&   44.0$\\pm$0.6&   44.1$\\pm$0.5\\\\\n1930 & 15.626 &  $-$18.0$\\pm$1.1&$-$17.9$\\pm$0.8&$-$17.9$\\pm$0.7\\\\\n1454 & 17.030 &    67.5$\\pm$1.4&   67.5$\\pm$2.0&   67.5$\\pm$1.2\\\\\n\\hline\t\t\n\\end{tabular}\n\\end{center}\n$^{*}$ Almost continuum-type spectra. \t\t       \t\t\t          \n\\end{table}\n\\normalsize\t\t\t       \n                                                                    \n\n\\subsubsection{Elemental abundance of other elements}\n\nThe line list used to compute the iron abundance ([Fe\/H]) was taken from \\cite{Pace2010}. We used this list to measure well-defined unblended \\ion{Fe}{i} and \\ion{Fe}{ii} lines.  As for the determination of the Li abundances,  we measured the equivalent widths of the   Fe lines (from 3 to 25 lines for \\ion{Fe}{i} lines and  up to 4 lines  for \\ion{Fe}{ii} lines) using the code {\\it FITLINE}. We then derived elemental  abundances for the measured line equivalent widths  using {\\it Turbospectrum}  \\citep{Alva1998} taking the latest version of the OSMARCS atmosphere models  \n(e.g., \\citealt{Gusta2008}, \\citealt{Plez2008}). Plane parallel 1D models were used for the computations. We adopted the \nGrevesse \\& Sauval (1998) solar abundances. \n\n These lines have been used to compute the Fe abundance but also to check that the ionization equilibrium is consistent \nwith the adopted gravity as shown in Fig.~\\ref{fig:dfe_logg}. The average [Fe\/H] from \\ion{Fe}{i} line is of $-0.54\\pm0.10$ dex, while the average from \n\\ion{Fe}{ii} line is of $-0.53\\pm0.14$ dex.\nFor the three brightest stars, we found a higher Fe abundance than  the rest of the sample. \nThis result is in line with the ongoing discussion about the systematic slight differences which \ncan found between Fe abundances determined in dwarfs and giants. \n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.5 \\textwidth ]{plot_dfe_logg.eps}\n\\caption{[\\ion{Fe}{i}\/H] - [\\ion{Fe}{ii}\/H]  abundance as a function of the stellar surface gravity.} \n\\label{fig:dfe_logg}\n\\end{figure}\n\nWe also determined the abundance of  Ca, Ti and Si  in stars where lines were detectable using the linelist of \n\\cite{Pace2010}. We finally obtained  the  abundance of Ca  in 76 stars, the abundance of Ti  in 33 stars,  and the abundance of Si in 13 stars.  \n\nFor calcium, we also measured the equivalent width of four to six  lines found in this list to determine the mean [$\\alpha$\/Fe] for the cluster. \nThe mean [Ca\/Fe] abundance for NGC~2243 based on the determination of Ca in 76 stars gives [Ca\/Fe]$=0.00\\pm0.14$ dex, in agreement within the error bars with the slightly higher value found by \\cite{grat1994}. \n\n The origin of this low value is probably related to the difference in the oscillator strength determination used in\n\\cite{grat1994} and \\cite{Pace2010}.\nThe aim of this work was to determine the Li abundance in a sample of stars  around the TO and it was not focused  on the determination of the [$\\alpha$\/Fe] ratios, so the instrumental setup was not optimized for it. \nBecause of the limited wavelength range used in this study (however for a large number of stars), we could not find a single Ca line in common with other abundance studies on NGC~2243. Therefore, this solar [Ca\/Fe] should be considered with caution. \nWe derived a mean Ti abundance of [Ti\/Fe] $=0.20\\pm0.22$ dex and a Si mean abundance [Si\/Fe] $=0.12\\pm0.20$ dex. In conclusion, our results (i.e., $[\\alpha\/Fe]$ abundances ranging from 0.00 to 0.2 dex) are \n in fair agreement with the  low $[\\alpha\/Fe]$ enhancement  found in other open clusters or thin disk stars of this metallicity ([Fe\/H] $\\simeq -0.5$ dex).\n\n\n\n\n\\subsection{Error estimates}\n\n The error in the determination of the Li abundances are a combination of the error in the measurement of the equivalent widths and the errors induced by uncertainties of the atmospheric parameters.  Errors in the determination of the equivalent widths have been estimated by changing the continuum position. The resulting errors range from 0.05 dex to 0.10 dex. \nThe errors  due to the uncertainties in the atmospheric parameters have been computed by evaluating the impact on the Li abundance of a $T_{\\rm eff}$ variation by 100 K, $\\log g$ by 0.2 dex, and $\\xi$ by 0.5 km\/s on the adopted atmospheric parameters. \nThese values are based on the  uncertainties found in the literature for the color excess and the distance modulus of the cluster.\nAn error of 0.02 on the color index corresponds to an error of $\\simeq$ 100 K in the temperature scale.\nAn error on the distance modulus of 0.3 corresponds to a change in the surface gravity of 0.2 dex.\nWe added a typical uncertainty of 0.5 km\/s for the microturbulent velocity.  As all the measured  lines are on the linear part of the curve of growth, the\nerror associated to this parameter is negligible. \n\nThe resulting errors have been summed quadratically and give a typical error on the Li abundance of 0.14 dex. The final errors on lithium are of the order of 0.18 dex.  Errors on the Fe abundances, estimated by the line-to-line scatter, give values of 0.1 to 0.2~dex. Summing quadratically the errors\n coming from uncertainties in atmospheric parameters leads to errors ranging from 0.15 to 0.22~dex. \n\nThe abundance uncertainties are dominated by the error in effective temperature. The chosen reddening from Kaluzny et al. (1996) is on the high side of the different \nestimations and our effective temperatures are in fact higher by about 200 K compared to the low temperature scale adopted, for instance by Hill \\& Pasquini (2000). A difference of this magnitude would shift our abundances by accounting for a lower Li of about 0.15 dex and a lower [Fe\/H] of 0.15 dex and a change in the ionization balance \\ion{Fe}{i}  - \\ion{Fe}{ii}  by 0.12 dex. Instead the effect on the [Ca\/Fe] ratio is negligible. Our temperature scale is consistent with that of Kaluzny et al. (1996) ; this is not surprising \ngiven the adoption of the same photometry and reddening.\n\n\\section{Discussion}\n\\label{sec:discussion} \nFigure~\\ref{fig:mV_Teff} shows the position of the stars on a $T_{\\rm eff}-V$ space, similar to a color magnitude diagram. \nIt should be remembered that the \\cite{Hill2001} points are on a lower \ntemperature scale than that adopted. As explained above, their points would be $\\sim$0.15 dex higher and $\\sim$200 K moved to the \nleft if they were on the same temperature scale adopted in this work. \n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.5\\textwidth]{plot_V_Teff.eps}\n\\caption{$V$ magnitude as a function of  stellar effective temperature. Black squares are measurements for our targets. Red arrows represent upper \nlimits of our Li abundance measurements. Blue symbols are results from \\cite{Hill2001}, where arrows represent upper limits. } \n\\label{fig:mV_Teff}\n\\end{figure}\nFig.~\\ref{fig:Abun_Li_V} shows the Li abundances as a function of the $V$ magnitude of the star. \n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.5 \\textwidth ]{plot_Li_V.eps}\n\\caption{Lithium abundance  as a function of $V$ magnitude. Symbols are the same as in Fig.~\\ref{fig:mV_Teff}.}\n\\label{fig:Abun_Li_V}\n\\end{figure}\n\n\\subsection{The lithium dip}\n\nThe Li dip is a feature first observed  in the Hyades OC. In a narrow temperature range, the Li abundance \nis strongly depleted by factors up to 100, as first shown by  Boesgaard \\& Tripicco (1986), and subsequently\n studied by several authors. \\cite{Bala1995} compared several clusters in a homogeneous \nanalysis concluding that the mass to which the dip occurs depends on the stellar metallicity, while \nthe ZAMS $T_{\\rm eff}$ does not. With NGC~2243 we have the opportunity to greatly enlarge the metallicity range explored. \n\nFor many years the cause of the ``extra mixing'' producing the Li dip have been debated (\\citealt{Talon2010} and references therein). \nRecently, Smiljanic et al. (2010) succeeded in reproducing the Li and Be \nbehavior of the hot side of the Li dip in the IC~4651 cluster by using intermediate mass stars as\n low as $\\sim 1.2 M_{\\odot}$ and rotating models (Charbonnel \\& Lagarde 2010). \n\nFigure~\\ref{fig:Abun_Li_Teff} shows the results of the abundance determination in NGC~2243 as a \nfunction of the temperature of the star. This figure reveals the clear \npresence of upper limits in a narrow range of temperature starting around $T_{\\rm eff}$ = 6700 K. \nThe range of temperature $6700-6900$ K represents the Li dip \nposition for this cluster. When informations from Fig.~\\ref{fig:Abun_Li_Teff} and Fig.~\\ref{fig:mV_Teff}\n are put together, one can see that the stars without a Li \ndetection are located in the upper (hottest) part of the ZAMS. Some non detections are also found for \ncooler (so fainter) stars. We attribute the lack of detection in the faint stars \nto the resulting low $S\/N$ ratio in their spectra. Star 1273, with a temperature $T_{\\rm eff}$= 6913 K and a magnitude of \n$V=15.8$ mag, shows a Li abundance of 2.70 dex and seems to be the first star located just on the blue side of the dip. \n\nOnce proven beyond any doubt the presence of the dip in this metal-poor cluster, we may discuss some\n additional points. The effective temperature of the dip is higher than that observed in other clusters. \n The mass of the dip can be estimated with the help of evolutionary isochrones. Using the Girardi \n(web page\\footnote{http:\/\/stev.oapd.inaf.it\/~lgirardi\/}) tracks, a cluster with a turnoff of 6900 K and a \nmetal content of [Fe\/H]=$-0.7$ is expected to have an age of 3.5 Gyr and a turnoff mass of $1.12 M_\\odot$. \nUsing the same isochrone, we could derive approximative masses for the eclipsing binary studied by \nKaluzny et al. (2006), finding 0.97 $M_\\odot$ for their components, in fair agreement with the \nestimated mass by Kaluzny et al. (2006) of  1.089 and 1.069 $M_\\odot$. Positioning the red side\n of the Li dip at 6700 K, the expected mass of the red edge of the dip would be of 1.03 $M_\\odot$. \n\nThe Li dip in this metal-poor cluster is therefore characterized by higher temperatures and substantially\n lower masses compared to  more metal-rich clusters. \nIn the Hyades and Praesepe clusters, the dip center is at about 1.4 $M_\\odot$ (Balachandran 1995). In IC~4651, with [Fe\/H]=0.11, Smiljanic et al. (2010) did not \ncompute the mass of the center, but estimated the red edge of the dip to be at 1.2 $M_\\odot$. In general, the decrease of the Li dip mass with metallicity is \nvery clear, and the effect is clearly visible in NGC~2243, given its low metallicity. \nAdopting a temperature scale cooler by $\\simeq 200 K$ degrees would move the dip towards cooler stars, which means also lower masses. Thus the use of a cooler  \n$T_{\\rm eff}$ scale would not change this result.\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.5\\textwidth]{plot_Li_Teff.eps}\n\\caption{Abundance of Li as a function of the effective temperature. Symbols are the same as in  Fig.~\\ref{fig:mV_Teff}} \n\n\\label{fig:Abun_Li_Teff}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.5\\textwidth]{plot_Mdip_FeH.eps}\n\\caption{ Correlation of the dip cool-side position in mass and the cluster's [Fe\/H] ratio.  The black symbol\nrepresents our result for NGC2243; red symbols are data from the literature. The black line represents a linear\nregression fit to the data. The slope value is 0.33 and the correlation coefficient is 0.98.} \n\\label{fig:mdip_feh}\n\\end{figure}\n\nFigure~\\ref{fig:mdip_feh} shows the position of the dip cool-side  expressed in mass  as a function of the metallicity [Fe\/H] for a sample of OCs (\\citealt{Cum2012} and references therein). \nWith the addition of NGC~2243, the [Fe\/H] range covered has a metallicity of almost 1 dex.\nThis figure reveals a rather tight correlation over the whole range of metallicity.\nThis dependence of the Li dip mass from the metallicity seems to be a well-established feature that models should reproduce. \n\n\n\\subsection{Li chemical evolution} \nObservations of many clusters have shown that the regions in the color diagram at the edge of the Li dip show the highest level of Li abundances, \nand probably indicate the Li abundance of the pristine gas. \nNGC~2243 is not an exception and the five stars bluer to the dip show an average Li abundance of 2.70 $\\pm0.20$ dex. \nWe may assume this value to be a conservative lower limit of the pristine Li abundance in the cluster. This value is consistent, within errors, with \nthe upper envelope of Li abundances observed in thin disk stars of metallicity similar to that of NGC~2243. \n\\cite{lambert2004} in their Table 2 provide\nthe mean Li abundance for the six stars richest  in Li in each metallicity bin and for the bin --0.4 to --0.6 this is 2.64$\\pm 0.07$ dex. The recent large sample of \n\\cite{rami2012} is consistent with this value. The models for Li evolution of \\cite{Prantzos2012}  predict a value of Li at this metallicity that is \nabout a factor of two higher. Thus, in order to make these models compatible with the observations, it is necessary to assume that even the stars with the \nhighest observed Li abundances at this metallicity have depleted Li by a factor of two. This is a direct consequence of the fact that these models assume \nthat the primordial level of Li is provided by the baryonic density measured from the fluctuations of the Cosmic Microwave Background by the WMAP satellite \n\\citep{LDH11,KSD11}, coupled with standard Big Bang nucleosynthesis. The same models, assuming the level of the Spite plateau \n\\citep{spite1982,SboBC10} as primordial Li abundance, would be in good agreement with the upper envelope of Li abundances in \nthin disk stars at all metallicities, without the need of extra Li depletion. For a review on the evolution of Li see also \\cite{mat10}.\n\\cite{Howk2012} estimate from the insterstellar medium (ISM)  the Li abundance of the Small Magellanic Cloud equal to   $2.68 \\pm 0.16 $ dex for a metallicity of [Fe\/H]=$-0.59$ dex.\nOur  estimate is  $\\log n{\\rm (Li)}= 2.70 \\pm 0.20$ dex for [Fe\/H] = $-0.54$ dex, therefore in close agreement with the ISM measurements  at the same metallicity. \nOur results would tend to confirm the finding by Howk et al. (2012) that the Li evolution is extremely constrained when the Li primordial abundance from WMAP\nis assumed. \n\n\n\\subsection{Comparison with 47 Tuc}\n\nAs already mentionned in the introduction, one of the interesting characteristics of NGC~2243 is that its metallicity almost overlaps with that of the well-studied globular cluster 47 Tuc. \n\nThe highest Li abundance in NGC 2243 is, as in all young OCs, found at the blue edge of the Li dip and is \nof  2.70 $\\pm$ 0.2 dex, with values potentially as high as 2.9 dex.  47 Tuc has been studied in the past by several groups.\nIt was first observed  by \\cite{pas97} who found three stars at a level of $\\log n{\\rm (Li)}$=2.35 dex, but these observations, \nmade with the 3.5m NTT telescope, had a low $S\/N$ ratio. With the advent of the VLT, higher quality observations and more objects became available. \n\\cite{bon07} observed four stars, finding evidence of strong  Li variability and a possible correlation with other light elemental abundances, \ntypical of globular clusters. This result was later confirmed by \\cite{doraz10} who have used FLAMES\/GIRAFFE data for  one hundred stars, showing that a large Li variation is present. \nInterestingly, this Li variation is not related to the Na-O correlation found in this cluster.\n\nThe extensive work of D'Orazi et al. (2010) confirms the presence of a very large Li variation in this cluster, with  Li between 2.4 and 2.5 dex and one object as high as 2.78 dex. Moreover, the maximum observed Li abundance \nin  47~Tuc stars is about 0.2 dex lower than in NGC 2243.  It is not easy, however, to firmly establish whether these abundances are representative of the pristine Li abundance of the cluster gas. While this is the case for NGC2243 (stars bluer than the gap do not show evidenc, either observational or theoretical, for Li depletion),  this is not the case for 47 Tuc. The turnoff stars of this cluster are rather cool (T$_{\\rm{eff}}$ less than $\\simeq$ 5800 K), more than 500 degrees K cooler than the faintest stars observed in NGC2243.  At this effective temperature, old OC stars already show evidence of main sequence Li depletion\n\\citep{pasq00,Sestito05}. It is important to note that none of the OCs studied so far  is as metal-poor as 47 Tuc or NGC~2243; it would be very important to observe fainter stars in NGC 2243 to confirm the Li trend  with effective temperature. Still, the possibility that the Li observed in 47 Tuc stars does not correspond to the pristine Li, but to slightly depleted Li, cannot be excluded. \nKeeping this possibility in mind, it is worth analyzing the case that the Li abundance in the two clusters was not initially the same.\n\nGiven its old age, in spite of its  high \nmetallicity, it would be natural to assume that 47 Tuc started from the primordial Li abundance, $\\log n{\\rm (^{7}Li)} = 2.72$ dex, from WMAP+SBBN \n\\citep{CFO08}. Instead, NGC~2243, with a much younger age, should have incorporated the Li produced by the Galactic sources, \nthus $\\log n{\\rm (^{7}Li)} \\sim 3.0$ dex, according to the models of \\cite{Prantzos2012} at this metallicity. If this is the case, one has to conclude \nthat the Li depletion with time is a highly non-linear process. In fact, the two clusters started with a difference in Li of a factor of 2 \nand they retain this difference, in spite of the fact that in 47 Tuc the Li has had 8 Gyrs more to accomplish its depletion. Another intriguing aspect \nof this scenario is that the amount of depletion (initial\/present) is again roughly a factor of two in both cases. The conclusion could be that Li \nis depleted very rapidly, by a factor of two, from whatever value it starts from, and then remains constant for all of the star's lifetime\nwithout taking into account the ``extra mixing'' that enters into play to explain the ``Li dip'' and the cooler stars. This scenario could be \nconsistent with the suggestion that the Li depletion takes place during the pre-main sequence  (\\cite{molaro2012}.\nHowever, the observational evidences show that pre-main Sequence (PMS) depletion in solar stars is not really effective (Randich et al. 2001; \nJeffries 2006). Considering the relatively low metal abundance of these clusters compared to the Sun, even less PMS depletion would be theoretically expected.\n\nThis promising picture is, however, troubled if we note that the star with the highest level of Li abundance in the \\cite{doraz10} sample has \n$\\log n{\\rm (^{7}Li)}=2.78$, i.e.,  close to the highest value also observed in NGC~2243.  This ``upper limit\" of the Li content in 47 Tuc stars  can be considered  the closest value  to the original Li content of the cluster. \n\nThis value would then suggest that the two clusters started with similar Li abundance. The implications of this are puzzling. We see three possible situations:\n\\begin{enumerate}\n\\item Li is depleted both in 47 Tuc and NGC~2243, 47 Tuc already benefitted from an initial Li that was a factor of two higher than the \nprimordial value and thereafter the Li abundance was constant up to the time of the NGC~2243 cluster formation; \n\\item neither 47 Tuc nor NGC~2243 have depleted any Li and were formed with the primordial abundance. Again no Li is produced \nbetween 12 Gyr ago and 4 Gyr ago;\n\\item 47 Tuc started with the primordial Li abundance, but its highest Li stars suffered no Li depletion. NGC 2243 started with a \nhigher Li-abundance and its highest Li stars suffered depletion by a factor of two.  \n\\end{enumerate}\n\nAll of these contradict our current understanding of Li evolution and stellar structure.  It would certainly be useful to \nextend the work of \\cite{doraz10}  and have new determinations of the  Li abundance  in 47 Tuc, to check if  this very high  Li abundance \nfound in one star is  real and is not simply a spurious result due to a low S\/N spectrum or incorrect stellar parameters. \n\nIf the Li initial abundance were the same between the two clusters, then the Li enrichment would  basically be determined by the Fe content rather than by the star formation history of the two clusters, which was quite different, as shown by their different [$\\alpha$\/Fe] enrichment. \n\n\\section{Conclusion}\n\n\\label{sec:concl}\n\n{VLT-FLAMES observations of a sample of stars belonging to NGC2243 has permitted us to determine the Li \nabundances in 27 stars   and the Fe and Ca abundances in 76 stars.  We found  a metallicity of [Fe\/H] =\n$-0.54 \\pm 0.10$ dex and  [Ca\/Fe] = $0.00 \\pm 0.14$ dex.  The Li dip is well defined with a center \ncorresponding to a mass of $M = 1.06~M_{\\odot}$, a value much smaller than the value found for solar \nmetallicity OCs. \nWe confirm a correlation between the Li dip cool-side position in mass as a function of the cluster's [Fe\/H] ratio, \nextending the relation towards metallicity down to [Fe\/H] $\\simeq -0.55 $ dex.\nA Li abundance of $2.70 \\pm 0.2$ dex found at the blue edge of the Li dip of  NGC 2243 is in good agreement with results obtained  for the Li of  the interstellar medium in the SMC  having similar low metallicity. This value of Li in metal-poor objects is comparable to the Li abundance deduced from WMAP measurements putting strong constraints on the models of Li enrichment during the early history of our Galaxy.}\n\n\n\n\\begin{acknowledgements}\n\nThe authors are grateful to the anonymous referee for a very careful reading of the paper and valuable comments. \nP.F. acknowledges support from the ESO Visitor program. \nK.B. was supported by the ESO DGDF 2008, and by the Italian {\\em Ministero dell'Istruzione, Universit\\`a e Ricerca} (MIUR) fundings. \nWe thank Simone Zaggia for the discussions about the GIRAFFE code girBLDRS. PF and PB acknowledge support from the Programme National \nde Physique Stellaire (PNPS) and the Programme National de Cosmologie et Galaxies (PNCG) of the Institut National de Sciences de \nl'Universe of CNRS. This research made use of WEBDA, operated at the Institute for Astronomy of the University of Vienna, and of \nSIMBAD\/VIZIER databases operated at the CDS (Strasbourg, France). \n\n\n\\end{acknowledgements}\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\chapter[The topological types of length bounded multicurves]{The topological types of length bounded multicurves}\n\n\n\\author{Hugo Parlier}\n\\authormark{H. Parlier}\n\n\n\\keywords{hyperbolic surfaces, curves, Teichm\\\"uller spaces, moduli spaces}\n\\subjclass{Primary 32G15, 52K20; secondary 53C22, 30F60}\n\n\n\n\n\\begin{abstract}\nThis article discusses inequalities on lengths of curves on hyperbolic surfaces. In particular, a characterization is given of which topological types of curves and multicurves always have a representative that satisfies a length inequality that holds over all of moduli space.\n\\end{abstract}\n\n\n\\section{Introduction}\n\nLength inequalities for curves play an important role in the understanding of hyperbolic surfaces and their moduli spaces. A prime example of this is a theorem of Bers, which states that any closed hyperbolic surface admits a pants decomposition of length bounded by a constant which only depends on the topology of the surface, and not its geometry. Short pants decompositions have been very useful in the understanding of the underlying Teichm\\\"uller space, and its large to medium scale geometry. This result generalizes bounds on the length of the shortest non-trivial closed curve (the systole). These are both examples of families of curves or multicurves that admit upper bounds over Teichm\\\"uller or moduli space of a given topological surface. More specifically, these results tell you that any hyperbolic surface of given topology admits a curve, or a multicurve, taken among a family of topological types, and which satisfies a certain length bound.\n\nThe goal of this short note is to characterize which families of curves or multicurves admit such upper bounds on their lengths. By multicurve, we mean a finite union of curves, all of them considered up to free homotopy, and by length we mean the length of a minimum length, thus geodesic, representative. The bounds are only allowed to depend on the topology of the underlying surface. Because of the nature of the game, we are only interested in the topological type of the curves, or said differently, in its mapping class group orbit. In order to try and satisfy the length inequality, we are allowed to choose curves of minimal length in the orbit. \n\nThe result is stated in terms of length functions. To a multicurve, we can associate the function, which takes a hyperbolic surface in the Teichm\\\"uller $\\mathcal{T}(\\Sigma)$ of a finite-type surface orientable $\\Sigma$, and which associates to it the length of the (unique) geodesic in the free homotopy class of the multicurve. The existence of the upper bounds described depends on whether certain length functions are bounded over Teichm\\\"uller or moduli space.\n\n\\begin{theorem}\\label{thm:main}\nLet $\\Gamma$ be a multicurve on $\\Sigma$. Then the quantity\n$$\\max_{X\\in \\mathcal{T}(\\Sigma)} \\min_{\\phi \\in \\mathrm{Mod}(\\Sigma)} \\ell_{\\phi(\\Gamma)}(X)$$\nis finite if and only if, for every pants decomposition $P$, there exists $\\phi$ in the mapping class group  $\\mathrm{Mod}(\\Sigma)$ such that $i(P, \\phi(\\Gamma))) = 0$.\n\\end{theorem}\nFor simplicity, it is stated only for multicurves but more generally it holds for families of multicurves (see Theorem \\ref{thm:maingeneral}). In more colloquial terms, what the result says is that if you are given a type of multicurve, then the function that associates to any hyperbolic surface the multicurve of minimal length of that type, is a bounded function over moduli space if and only if there is a multicurve of that type disjoint from any pants decomposition. \n\nNote that, given a lower bound on the systole, by compactness of pinched moduli space, there is similar conditional length inequality statement (Proposition \\ref{prop:conditionallength}) that holds for any multicurve, but the implied constants depend on topology, curve type and the lower bound of the systole.\n\nNote that here we are only interested in the very first value in a subset of the length spectrum. This is obviously very different from recent results about the asymptotic growth of curves of a given type (see \\cite{Erlandsson-Souto} and references therein). This is also very different from the related problems of finding precise constants, and in particular, exploring surfaces that are extremal for different geometric quantities. The focus here is on understanding what type of inequalities are possible. This is of course inspired by the many uses that have been made of these, or related, inequalities in the study of the large or medium scale of Teichm\\\"uller spaces with different metrics \\cite{Brock,Cavendish-Parlier} for the Weil-Petersson metric, \\cite{Rafi,Rafi-Tao,Papadopoulos-Theret} for the Teichm\\\"uller and Thurston metrics. \n\nFinally, note that this note is about upper bounds of length functions. If one replaces the $\\max$ with a $\\min$ in the above theorem, the quantity is strictly positive if and only if the multicurve has intersection (coming from a non-simple closed curve or pairwise intersecting curves). This is a consequence of the collar lemma \\cite{Keen} and generalisations \\cite{BasmajianCollar}. Lower bounds that depend on curve type have been studied in some detail by Basmajian \\cite{BasmajianUniversal}.\\\\\n\n\\noindent{\\bf Organization.} The article is organized as follows. The second section is mainly notation and definitions, and includes Proposition \\ref{prop:conditionallength}. The third and final section is dedicated to Theorem \\ref{thm:maingeneral} and ends with a discussion of which previously known length inequalities fall within its framework.\n\n\\section{Preliminaries and setup}\nThroughout $\\Sigma$ will be an orientable finite-type surface with $\\chi(\\Sigma) <0$. $\\Sigma$ is entirely determined by its genus $g$ and number of ends $n$, and $\\chi(\\Sigma) = 2-2g -n$.The space of marked complete hyperbolic structures on $\\Sigma$, that is Teichm\\\"uller space, will be denoted $\\mathcal{T}(\\Sigma)$, and should be thought of as a continuous deformation space of hyperbolic metrics. For the purpose of simplicity, we ask the metrics to be geodesically complete, and so the ends of $X\\in \\mathcal{T}(\\Sigma)$ are realized as cusps. The underlying moduli space, that is the space of hyperbolic structures on $\\Sigma$ up to isometry, will be denoted $\\mathcal{M}(\\Sigma)$. $\\mathrm{Mod}(\\Sigma)$ will be the (full) mapping class group of $\\Sigma$, that is the group of self-homeomorphisms of $\\Sigma$ up to isotopy. This group acts on $\\mathcal{T}(\\Sigma)$ and its quotient is $\\mathcal{M}(\\Sigma)$. \n\nFormally a curve is the continuous image of a circle on $\\Sigma$, but we will only be interested in a curve up to free homotopy. In particular, we only consider essential curves, that is those non-homotopic to a point or a boundary. A multicurve is a (finite) collection of curves. Associated to a multicurve $\\Gamma$ is a function which associates to $X\\in \\mathcal{T}(\\Sigma)$ the quantity $\\ell_\\Gamma(X)$, the length of the unique geodesic representative of $\\Gamma$ on $X$. These length functions are continuous, analytic and in fact convex \\cite{Kerckhoff, Wolpert}. \n\nIntersection between curves is defined as minimal intersection among representatives, and is denoted $i(\\cdot,\\cdot)$. A curve is simple if it has no self-intersections. A pants decomposition is a maximal collection of disjoint and distinct simple closed curves, and decomposes the surface into three-holed spheres (pairs of pants). The boundary curves of a pair of pants are sometimes called cuffs.\n\nThe following result \\cite{Keen} is stated here in non-quantitative terms for simple closed geodesics. Note that a version also holds for non-simple closed geodesics as well \\cite{BasmajianCollar}, with the notable difference being that you cannot pinch a non-simple closed curve. \n\n\\begin{lemma}[Collar lemma]\\label{lem:collar}\nA simple closed geodesic of length $\\ell$ on a hyperbolic surface $X$ admits a cylindrical neighborhood (its collar) of positive width $w(\\ell)$ which only depends on $\\ell$ and such that $w(\\ell) \\to \\infty$ when $\\ell \\to 0$. \n\\end{lemma}\n\nOur main use of the above result will be to consider hyperbolic structures where curves of a pants decomposition have length tending towards $0$, and hence all curves that are not among the cuffs of the given pants decomposition have length that tend to infinity. \n\nGiven $X\\in \\mathcal{T}(\\Sigma)$, the length of its shortest essential curve is its systole and is denoted $\\mathrm{sys}(X)$. Unless $X$ is a pair of pants, the systole is realized by a simple closed geodesic. For $\\epsilon>0$, the $\\epsilon$-thick part of Teichm\\\"uller space is the subset $\\mathcal{T}^{\\epsilon}(\\Sigma)\\subset \\mathcal{T}(\\Sigma)$ consisting of surfaces with $\\mathrm{sys}(X)\\geq \\epsilon$. By Mahler's compactness criterion, the corresponding thick part of moduli space $\\mathcal{M}^{\\epsilon}(\\Sigma)$ is compact.\n\nIn this paper, we study length inequalities, which here will be upper bounds for lengths of curves or multicurves with given properties. Generally these inequalities will be about a topological type of curve or multicurve: two multicurves $\\Gamma$ and $\\Gamma'$ are of the same type if there exists $\\phi\\in \\mathrm{Mod}(\\Sigma)$ such that $\\phi(\\Gamma)=\\Gamma'$. Given a multicurve $\\Gamma$, we can consider its mapping class group orbit. These orbits divide the space of all multicurves into equivalence classes\n$$\n[\\Gamma]:= \\{ \\phi(\\Gamma) \\mid \\phi \\in \\mathrm{Mod}(\\Sigma)\\}\n$$\nsorted by type. Now given $X\\in \\mathcal{T}(\\Sigma)$, we can study the length of a minimal representative of an equivalence class:\n$$\n\\ell_{[\\Gamma]}(X) = \\min_{\\phi \\in \\mathrm{Mod}(\\Sigma)} \\big\\{ \\ell_X(\\phi(\\Gamma)\\big\\}\n$$\nThe existence of a minimum follows from the discreteness of the length spectrum. This function, due to its mapping class group invariance, descends nicely to $\\mathcal{M}(\\Sigma)$. Note that although the function $\\ell_{[\\Gamma]}(\\cdot)$ remains continuous over $\\mathcal{M}(\\Sigma)$, it is no longer smooth. This is due to possible changes of homotopy classes realizing the minimum length of in an equivalence class.\n\nThe following result is a non-explicit general bound that holds for any topological type of multicurve. The proof is a compactness argument. The constant $K$ in the statement depends on the topology of $\\Sigma$, the topological type of $\\Gamma$ and a lower bound on the systole.\n\n\\begin{proposition}\\label{prop:conditionallength}\nLet $[\\Gamma]$ be a type of multicurve on $\\Sigma$. For any $\\epsilon>0$, there exists a constant $K=K(\\epsilon, \\Sigma, [\\Gamma])$ such that for any $X\\in \\mathcal{T}^{\\epsilon}(\\Sigma)$, we have\n$$\n\\ell_{[\\Gamma]}(X) \\leq K.\n$$\n\\end{proposition}\n\n\\begin{proof}\nThe space $\\mathcal{M}^{\\epsilon}(\\Sigma)$ is compact, hence the continuous function $\\ell_{[\\Gamma]}(\\cdot)$ admits a maximum on $\\mathcal{M}^{\\epsilon}(\\Sigma)$. This maximum value is exactly $K$. \n\\end{proof}\n\nAs an explicit example of the above result, consider the following result due to Buser and S\\\"eppala \\cite{Buser-Seppala}. For a closed surface $\\Sigma$ of genus $g\\geq 2$, they consider {\\it canonical homology bases}, that is collections of $2g$ simple closed curves $\\{\\alpha_i,\\beta_i\\}_{1\\leq i \\leq g}$ that satisfy:\n\\begin{enumerate}[a)]\n\\item\n$i(\\alpha_i,\\beta_j)=\\delta_{ij}$ for all $i,j\\in \\{1,\\hdots,g\\}$ (where $\\delta_{ij}$ is the Kronecker delta),\n\\item\n$i(\\alpha_i,\\alpha_j)=i(\\beta_i,\\beta_j)=0$ for $i\\neq j$. \n\\end{enumerate}\nNote that such a system of curves automatically generate integer homology. See Figure \\ref{fig:homology} for an illustration in genus $2$.\n\n\\begin{figure}[htbp]\n\\leavevmode \\SetLabels\n\\L(.23*.5) $\\alpha_1$\\\\%\n\\L(.32*.27) $\\beta_1$\\\\%\n\\L(.74*.5) $\\alpha_2$\\\\%\n\\L(.66*.27) $\\beta_2$\\\\%\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=8cm]{Figures\/N-homology.pdf}}}\n\\caption{A canonical homology basis in genus $2$}\n\\label{fig:homology}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\n\nThey prove that any $X\\in \\mathcal{T}^{\\epsilon}(\\Sigma)$ admits a canonical homology basis with all curves of length at most\n$$\n(g-1)\\left(45+6\\,\\mathrm{arcsinh}\\frac{1}{\\epsilon}\\right).\n$$\nThis is an improvement over an earlier quantification in \\cite{Buser-Seppala2}. The above bound shows that the $K$ in this instance is bounded above by \n$$\n2g(g-1)\\left(45+6\\,\\mathrm{arcsinh}\\frac{1}{\\epsilon}\\right)\n$$\nbecause there are $2g$ curves in the family. Of course, for this to be an exact quantification of the constant $K$, it would have to be sharp (which it is not). Exact quantifications are rarely known but, as proved in \\cite{Buser-Seppala}, the real constant must depend on $\\epsilon$. Alternatively, this dependency on $\\epsilon$ can also be deduced from Theorem \\ref{thm:main}.\n\nNaturally, this leads to the existence of upper bounds which only depend on topology and curve type, but not on systole length. A multicurve type $[\\Gamma]$ is said to satisfy a strong length inequality if $\\ell_{[\\Gamma]}(\\cdot)$ is upper bounded over $\\mathcal{M}(\\Sigma)$. By continuity of the length function $\\ell_{[\\Gamma]}(\\cdot)$, this is equivalent to the existence of a surface $X_{\\max}\\in \\mathcal{M}(\\Sigma)$ such that \n$$\n\\max_{X\\in \\mathcal{M}(\\Sigma)} \\ell_{[\\Gamma]}(X) = \\ell_{[\\Gamma]}(X_{\\max}).\n$$\nTo see this, one must show that the supremum of the length function cannot be reached on the boundary of moduli space, that is on a noded surface. As it turns out, as a consequence of Lemma \\ref{lem:stretchpants} from the next section, a (finite) supremum is always reached in the ''thick\" part of moduli space, and so in particular the $\\sup$ is indeed a $\\max$.\n\n\\section{Stretching pants, Bers' theorem and consequences}\n\nThe following lemma is by now well-known, but a sketch proof is provided for completeness. The proof uses strip maps, introduced by Thurston \\cite{Thurston}, and used to great effect by many authors \\cite{Danciger-Gueritaud-Kassel, Gueritaud, Papadopoulos-Theret, Parlier} to study of deformations of hyperbolic structures. Recall that a hyperbolic pair of pants is uniquely determined by its cuff lengths. In order to allow pants with cusp boundary, we use the convention that a cusp is a cuff of $0$ length. \n\n\\begin{lemma}[Pants stretching lemma]\\label{lem:stretchpants}\nLet $Y_{x,y,z}$ be the unique hyperbolic pair of pants with cuff lengths $x,y,z \\geq 0$. Then, for any (non-boundary) homotopy class of closed curve $\\gamma$ on a pair of pants, and any $\\epsilon>0$, we have\n$$\n\\ell_\\gamma\\left(Y_{x,y,z}\\right) < \\ell_\\gamma\\left(Y_{x+\\epsilon,y,z}\\right)\n$$\n\\end{lemma}\n\\begin{proof}[Sketch proof]\nConsider on $Y=Y_{x+\\epsilon,y,z}$, the simple orthogeodesic $a$ with both endpoints on the boundary curve of length $x+\\epsilon$ (see Figure \\ref{fig:arc}). \n\n\\begin{figure}[htbp]\n\\leavevmode \\SetLabels\n\\L(.445*.5) $a$\\\\%\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=5cm]{Figures\/N-arc.pdf}}}\n\\caption{The orthogeodesic $a$}\n\\label{fig:arc}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\n\nNote the closed geodesic $\\gamma$ intersects $a$ at least once. Exactly like in the collar lemma, $a$ admits an embedded neighborhood, topologically a strip (see Figure \\ref{fig:strip}). The idea is to now remove at least part of this strip to reduce the length of the boundary curve. \n\n\\begin{figure}[htbp]\n\\leavevmode \\SetLabels\n\\L(.445*.5) $a$\\\\%\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=5cm]{Figures\/N-strip.pdf}}}\n\\caption{A strip surrounding $a$}\n\\label{fig:strip}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\n\nTo do this properly, it is more convenient to consider the complete pair of pants by adding funnels. The arc $a$ can now be extended into a complete simple and infinite length geodesics. In addition, because the boundary curve of length $x+\\epsilon$ is not $0$, there is a family of simple complete geodesics parallel to each other and to $a$. We can take any two of these, cut away the strip between them, and paste them together to obtain a (complete) hyperbolic metric, say $Y'$ (see Figure \\ref{fig:operation}). (The slightly sketchy part is here: in fact it is possible by a variational argument to show that this can be done such that the length of the new boundary component of $Y'$ is exactly $x$, but we won't dwell on this, the main point being that the boundary length has been reduced.)\n\n\\begin{figure}[H]\n\\leavevmode \\SetLabels\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=10cm]{Figures\/N-construction.pdf}}}\n\\caption{Removing the strip neighborhood $\\mathcal{S}(a)$}\n\\label{fig:operation}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\n\nWe denote the strip enclosed by these geodesics by $\\mathcal{S}(a)$.\n\nNow can analyse the result of this operation on the length of $\\gamma$. Note that, due to the topology of the strip neighborhood $\\mathcal{S}(a)$ of $a$, $\\gamma \\cap \\mathcal{S}(a)$ is a union of simple geodesic arcs. We look at each one. In order to find a representative of $\\gamma$ on $Y'$, we replace each simple geodesic arc with its projection to $a$ (see Figure \\ref{fig:project}). The point is that the projection strictly reduces lengths.\n\\begin{figure}[H]\n\\leavevmode \\SetLabels\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=5cm]{Figures\/N-project.pdf}}}\n\\caption{A local picture of $\\gamma$ under the operation}\n\\label{fig:project}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\nThus, this results in a curve on $Y'$, in the same homotopy class, and of length strictly smaller. The corresponding geodesic is thus of length strictly smaller than it was previously. Hence we have $\\ell_\\gamma\\left(Y\\right) > \\ell_\\gamma\\left(Y'\\right)$ as required. \n\\end{proof}\nTo make the proof fully rigorous \nWe will mainly need an immediate corollary of the above result.\n\\begin{corollary}\\label{cor:longpants}\nFor $L>0$, let $Y$ be a pair of pants with cuff lengths between $0$ and $L$. Let $Y_L$ be the pair of pants with all cuff lengths exactly $L$. Then, for any interior homotopy class of curve $\\gamma$ on the pair of pants, we have $\\ell_\\gamma(Y_L) \\geq \\ell_\\gamma(Y)$ with equality only occurring if $Y=Y_L$. \n\\end{corollary}\n\nPants decompositions play an essential role in this story. The following result, originally due to Bers \\cite{Bers1,Bers2}, has been since quantified by different authors \\cite{BuserBook,BPS,ParlierShort}.\n\n\\begin{theorem}[Bers' constants]\\label{thm:bers}\nThere exists a constant $B(\\Sigma)$ such that any $X\\in \\mathcal{T}(\\Sigma)$ admits a pants decomposition with all curves of length at most $B(\\Sigma)$.\n\\end{theorem}\n\nNote that this result does not fall in the framework of either Proposition \\ref{prop:conditionallength} or Theorem \\ref{thm:main}. Indeed, in order to find a short pants decomposition, we are allowed to choose any topological type of pants decomposition. The number of these grows with topology (for instance there are roughly $\\sim g^g$ different types when $\\Sigma$ is of genus $g$ and $g$ is large). So here we are not only minimizing among mapping class group orbits of a fixed multicurve, but we are minimizing among multicurves that belong to a family. The following result holds for all such families of multicurves. A family will be denoted $\\{\\Gamma_\\alpha\\}_{\\alpha\\in I}$ where $I$ is an index set (possibly infinite, but countable as there are only countably many topological types of finite multicurve on a finite type surface). \n\n\\begin{theorem}\\label{thm:maingeneral}\nLet $\\{\\Gamma_\\alpha\\}_{\\alpha\\in I}$ be a family of multicurves. Then the quantity \n$$\\max_{X\\in \\mathcal{M}(\\Sigma)} \\min_{\\alpha\\in I} \\ell_{[\\Gamma_\\alpha]}(X)$$\nis finite if and only if, for every pants decomposition $P$, there exists $\\alpha\\in I$ and $\\phi \\in \\mathrm{Mod}(\\Sigma)$ such that $i(P, \\phi(\\Gamma_\\alpha)) = 0$. \n\\end{theorem}\n\nThe statement might seem a little confusing at first, due to the fact that we are taking a maximum among hyperbolic structures of a double minimum (over a family and then over the mapping class group orbit). If the family is reduced to a single curve, the statement becomes Theorem \\ref{thm:main} from the introduction. The proofs of both statements are identical, so we prove the more general statement above.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:maingeneral}]\n\nWe begin with the more straightforward direction, showing that if there is a pants decomposition $P$ which is intersected by any mapping class group orbit of any multicurve in the family $\\{\\Gamma_\\alpha\\}_{\\alpha\\in I}$, then there is no upper bound on the length. This follows directly from Lemma \\ref{lem:collar} (the collar lemma). Indeed, by considering a sequence of hyperbolic structures with all curves in the pants decomposition of $P$ that converge to $0$, the length of any curve that intersects one of the pants curve necessarily goes to $+\\infty$. As, by hypothesis, there is at least one curve in every multicurve $\\phi(\\Gamma_\\alpha)$, for all $\\phi \\in \\mathrm{Mod}(\\Sigma)$ and all $\\alpha \\in I$, that intersects a curve in $P$, the result follows. \n\nThe other direction will follow from Bers' theorem (Theorem \\ref{thm:bers}) and Corollary \\ref{cor:longpants} above.\n\nTake any $X\\in \\mathcal{T}(\\Sigma)$. By Bers' theorem, there exists a pants decomposition of $X$, say $P$, with all curves of length at most $B(\\Sigma)$. Now, by hypothesis, there exists $\\Gamma_\\alpha$ such that $i(\\Gamma_\\alpha,P)= 0$. Note that $\\Gamma_\\alpha$ can contain curves from $P$, but none that intersect curves of $P$ transversally. \n\nBy hypothesis all curves that belong to both $\\Gamma_\\alpha$ and $P$ are upper bounded by $B(\\Sigma)$. We now the corollary to all remaining curves of $\\Gamma_\\alpha$. Indeed, any such a curve $\\gamma \\in \\Gamma_\\alpha$ is contained in a pair of pants. By hypothesis, the cuff lengths of this pair of pants are at most $B(\\Sigma)$. By Corollary \\ref{cor:longpants}, $\\ell_X(\\gamma)$ is at most the length when the pair of pants has all cuff lengths equal to $B(\\Sigma)$, which is some finite number that depends on $B(\\Sigma)$ and the topological type of $\\gamma$. As there are a finite number of such curves, the result follows.\n\\end{proof}\n\nThere are many instances of the above theorem, each obtained by changing the multicurve or family of multicurves. Note that only the topological type of a multicurve matters in the statement. Some of these results are in the positive direction: the theorem implies that there is an upper bound on a minimal length representative that only depends on the topology of $\\Sigma$. Others are in the negative direction, that is that certain multicurves, or families of multicurves, do not admit such upper bounds. We give a (non-exhaustive) list of results of this type.\n\n\\begin{enumerate}[I.]\n\\item Even though it was used in the proof, Bers' theorem is an example obtained by taking the family of multicurves to be the full set of pants decompositions (or simply one pants decomposition for each topological type). To put in the framework of the theorem, the length of a pants decomposition should be defined as the sum, and not the maximum value. \n\nThe quantification of the implied constants - for both the sum and the max - has attracted some attention over the years, but it does not seem to be an easy problem. In fact, even the rough growth in terms of genus is not known \\cite{BuserBook,ParlierShort,GPY}. On the positive side, the exact constant in genus $2$ is known by a result of Gendulphe \\cite{Gendulphe}, as is the rough growth in terms of the number of cusps \\cite{Balacheff-Parlier,BPS}. \n\n\\item One can also take the full set of all topological types of all multicurves: this boils down to the systolic inequality. Quantifying the exact constants that appear is an arduous task. For orientable closed surfaces, the only constant known is again in genus $2$ \\cite{Jenni}. Buser and Sarnak \\cite{Buser-Sarnak} showed that the constants must grow logarithmically in genus, and since then there have been multiple variations and refinements of this (see for instance \\cite{BFR,KSV,Fanoni-Parlier, SchmutzSchaller}), including generalizations in the world of variable curvature and higher dimensional manifolds \\cite{Gromov}.\n\n\\item A slight variation of the above is to look at the homological systole of a closed surface $\\Sigma$. That is, the shortest curve that is not only non-trivial in homotopy, but also in homology. By cut and paste arguments, the homological systole is always realized by a non-separating simple closed curve. Hence, it is always in the same mapping class orbit. As above, few optimal constants are known. However, for closed surfaces, it is known that the optimal constants are equal to those from the systolic inequality. \\cite{ParlierPapa} Said differently, the systole of a maximal surface is homologically non-trivial.\n\n\\item In addition to the results of Buser and Sepp\\\"al\\\"a mentioned previously, one can try and bound families of homologically independent curves, but that do not necessarily form (part of) a canonical basis. If one requests a full homology basis, by the theorem above, there is no upper bound on its length over moduli space. However, by Bers' theorem, and the observation that any pants decomposition contains $g$ homologically independent curves, one can find an upper bound on the length of up until $g$ curves by a function of topology. So what about $g+1$ curves?\n\nGromov observed \\cite[Section 5]{GromovFilling} that any minimal length homology basis consists of simple curves that pairwise intersect at most once. Now given $g+1$ homologically independent and simple curves, there must be at least a pair that intersect. And a pair of intersecting simple curves necessarily intersects {\\it all} pants decompositions. Hence by the theorem above, there is no upper bound for such a family and so strong length inequality stops at exactly $g$ homologically independent curves. \n\nMore precise quantifications of the constants have also attracted attention. In particular the Buser-Sarnak logarithmic bound can be extended to roughly $ag$ curves for any $a<1$ \\cite{BPS}. \n\nFinally note that Gromov's observation above (on the intersection properties of minimal bases) still forces one to consider multiple, although finite, topological types of multicurves.\n\n\\item In a somewhat opposite direction, consider $\\{\\Gamma\\}_\\alpha$ to be the set of separating simple closed curves on a closed surface $\\Sigma$ of genus $g$. Then, as there exists a pants decomposition consisting only of non-separating curves, it will essentially intersect any element of the mapping class group orbit of any element of $\\Gamma$. Hence, for all $\\alpha\\in I$, the function $\\ell_{[\\Gamma_\\alpha]}(\\cdot)$ admits no upper bound over $\\mathcal{M}(\\Sigma)$. If however one takes the larger set $\\{\\Gamma\\}_\\alpha$ of all homologically trivial curves (but not homotopically trivial), then it is a consequence of a theorem of Sabourau \\cite{Sabourau} that this admits an upper bound that only depends on genus. This subtle difference lead to a small gap in Mirzakhani's study of the expected value of the shortest {\\it simple} homologically trivial curve in \\cite{Mirzakhani}, but this was not the difficult part of her work and has since been cleared up \\cite{PWX}.\n\n\\item For multicurves containing more than one curve, many of these problems admit variations. Here we suppose the length of a multicurve is the sum of lengths of its components, but of course one could also consider other variations, such as the $\\max$ or the product. For instance, the product of lengths of homologically distinct curves was studied in \\cite{Balacheff-Karam-Parlier}. Replacing the sum of the lengths with the maximum will also satisfy the same boundedness condition of Theorem \\ref{thm:maingeneral}, but products - or other functions of the lengths of the components - need to be checked on a case-by-case basis. \n\\end{enumerate}\n\n\\ack{This work was supported by the Luxembourg National Research Fund OPEN grant O19\/13865598.}\n\n\n\n\\chapter[The topological types of length bounded multicurves]{The topological types of length bounded multicurves}\n\n\n\\author{Hugo Parlier}\n\\authormark{H. Parlier}\n\n\n\\keywords{hyperbolic surfaces, curves, Teichm\\\"uller spaces, moduli spaces}\n\\subjclass{Primary 32G15, 52K20; secondary 53C22, 30F60}\n\n\n\n\n\\begin{abstract}\nThis article discusses inequalities on lengths of curves on hyperbolic surfaces. In particular, a characterization is given of which topological types of curves and multicurves always have a representative that satisfies a length inequality that holds over all of moduli space.\n\\end{abstract}\n\n\n\\section{Introduction}\n\nLength inequalities for curves play an important role in the understanding of hyperbolic surfaces and their moduli spaces. A prime example of this is a theorem of Bers, which states that any closed hyperbolic surface admits a pants decomposition of length bounded by a constant which only depends on the topology of the surface, and not its geometry. Short pants decompositions have been very useful in the understanding of the underlying Teichm\\\"uller space, and its large to medium scale geometry. This result generalizes bounds on the length of the shortest non-trivial closed curve (the systole). These are both examples of families of curves or multicurves that admit upper bounds over Teichm\\\"uller or moduli space of a given topological surface. More specifically, these results tell you that any hyperbolic surface of given topology admits a curve, or a multicurve, taken among a family of topological types, and which satisfies a certain length bound.\n\nThe goal of this short note is to characterize which families of curves or multicurves admit such upper bounds on their lengths. By multicurve, we mean a finite union of curves, all of them considered up to free homotopy, and by length we mean the length of a minimum length, thus geodesic, representative. The bounds are only allowed to depend on the topology of the underlying surface. Because of the nature of the game, we are only interested in the topological type of the curves, or said differently, in its mapping class group orbit. In order to try and satisfy the length inequality, we are allowed to choose curves of minimal length in the orbit. \n\nThe result is stated in terms of length functions. To a multicurve, we can associate the function, which takes a hyperbolic surface in the Teichm\\\"uller $\\mathcal{T}(\\Sigma)$ of a finite-type surface orientable $\\Sigma$, and which associates to it the length of the (unique) geodesic in the free homotopy class of the multicurve. The existence of the upper bounds described depends on whether certain length functions are bounded over Teichm\\\"uller or moduli space.\n\n\\begin{theorem}\\label{thm:main}\nLet $\\Gamma$ be a multicurve on $\\Sigma$. Then the quantity\n$$\\max_{X\\in \\mathcal{T}(\\Sigma)} \\min_{\\phi \\in \\mathrm{Mod}(\\Sigma)} \\ell_{\\phi(\\Gamma)}(X)$$\nis finite if and only if, for every pants decomposition $P$, there exists $\\phi$ in the mapping class group  $\\mathrm{Mod}(\\Sigma)$ such that $i(P, \\phi(\\Gamma))) = 0$.\n\\end{theorem}\nFor simplicity, it is stated only for multicurves but more generally it holds for families of multicurves (see Theorem \\ref{thm:maingeneral}). In more colloquial terms, what the result says is that if you are given a type of multicurve, then the function that associates to any hyperbolic surface the multicurve of minimal length of that type, is a bounded function over moduli space if and only if there is a multicurve of that type disjoint from any pants decomposition. \n\nNote that, given a lower bound on the systole, by compactness of pinched moduli space, there is similar conditional length inequality statement (Proposition \\ref{prop:conditionallength}) that holds for any multicurve, but the implied constants depend on topology, curve type and the lower bound of the systole.\n\nNote that here we are only interested in the very first value in a subset of the length spectrum. This is obviously very different from recent results about the asymptotic growth of curves of a given type (see \\cite{Erlandsson-Souto} and references therein). This is also very different from the related problems of finding precise constants, and in particular, exploring surfaces that are extremal for different geometric quantities. The focus here is on understanding what type of inequalities are possible. This is of course inspired by the many uses that have been made of these, or related, inequalities in the study of the large or medium scale of Teichm\\\"uller spaces with different metrics \\cite{Brock,Cavendish-Parlier} for the Weil-Petersson metric, \\cite{Rafi,Rafi-Tao,Papadopoulos-Theret} for the Teichm\\\"uller and Thurston metrics. \n\nFinally, note that this note is about upper bounds of length functions. If one replaces the $\\max$ with a $\\min$ in the above theorem, the quantity is strictly positive if and only if the multicurve has intersection (coming from a non-simple closed curve or pairwise intersecting curves). This is a consequence of the collar lemma \\cite{Keen} and generalisations \\cite{BasmajianCollar}. Lower bounds that depend on curve type have been studied in some detail by Basmajian \\cite{BasmajianUniversal}.\\\\\n\n\\noindent{\\bf Organization.} The article is organized as follows. The second section is mainly notation and definitions, and includes Proposition \\ref{prop:conditionallength}. The third and final section is dedicated to Theorem \\ref{thm:maingeneral} and ends with a discussion of which previously known length inequalities fall within its framework.\n\n\\section{Preliminaries and setup}\nThroughout $\\Sigma$ will be an orientable finite-type surface with $\\chi(\\Sigma) <0$. $\\Sigma$ is entirely determined by its genus $g$ and number of ends $n$, and $\\chi(\\Sigma) = 2-2g -n$.The space of marked complete hyperbolic structures on $\\Sigma$, that is Teichm\\\"uller space, will be denoted $\\mathcal{T}(\\Sigma)$, and should be thought of as a continuous deformation space of hyperbolic metrics. For the purpose of simplicity, we ask the metrics to be geodesically complete, and so the ends of $X\\in \\mathcal{T}(\\Sigma)$ are realized as cusps. The underlying moduli space, that is the space of hyperbolic structures on $\\Sigma$ up to isometry, will be denoted $\\mathcal{M}(\\Sigma)$. $\\mathrm{Mod}(\\Sigma)$ will be the (full) mapping class group of $\\Sigma$, that is the group of self-homeomorphisms of $\\Sigma$ up to isotopy. This group acts on $\\mathcal{T}(\\Sigma)$ and its quotient is $\\mathcal{M}(\\Sigma)$. \n\nFormally a curve is the continuous image of a circle on $\\Sigma$, but we will only be interested in a curve up to free homotopy. In particular, we only consider essential curves, that is those non-homotopic to a point or a boundary. A multicurve is a (finite) collection of curves. Associated to a multicurve $\\Gamma$ is a function which associates to $X\\in \\mathcal{T}(\\Sigma)$ the quantity $\\ell_\\Gamma(X)$, the length of the unique geodesic representative of $\\Gamma$ on $X$. These length functions are continuous, analytic and in fact convex \\cite{Kerckhoff, Wolpert}. \n\nIntersection between curves is defined as minimal intersection among representatives, and is denoted $i(\\cdot,\\cdot)$. A curve is simple if it has no self-intersections. A pants decomposition is a maximal collection of disjoint and distinct simple closed curves, and decomposes the surface into three-holed spheres (pairs of pants). The boundary curves of a pair of pants are sometimes called cuffs.\n\nThe following result \\cite{Keen} is stated here in non-quantitative terms for simple closed geodesics. Note that a version also holds for non-simple closed geodesics as well \\cite{BasmajianCollar}, with the notable difference being that you cannot pinch a non-simple closed curve. \n\n\\begin{lemma}[Collar lemma]\\label{lem:collar}\nA simple closed geodesic of length $\\ell$ on a hyperbolic surface $X$ admits a cylindrical neighborhood (its collar) of positive width $w(\\ell)$ which only depends on $\\ell$ and such that $w(\\ell) \\to \\infty$ when $\\ell \\to 0$. \n\\end{lemma}\n\nOur main use of the above result will be to consider hyperbolic structures where curves of a pants decomposition have length tending towards $0$, and hence all curves that are not among the cuffs of the given pants decomposition have length that tend to infinity. \n\nGiven $X\\in \\mathcal{T}(\\Sigma)$, the length of its shortest essential curve is its systole and is denoted $\\mathrm{sys}(X)$. Unless $X$ is a pair of pants, the systole is realized by a simple closed geodesic. For $\\epsilon>0$, the $\\epsilon$-thick part of Teichm\\\"uller space is the subset $\\mathcal{T}^{\\epsilon}(\\Sigma)\\subset \\mathcal{T}(\\Sigma)$ consisting of surfaces with $\\mathrm{sys}(X)\\geq \\epsilon$. By Mahler's compactness criterion, the corresponding thick part of moduli space $\\mathcal{M}^{\\epsilon}(\\Sigma)$ is compact.\n\nIn this paper, we study length inequalities, which here will be upper bounds for lengths of curves or multicurves with given properties. Generally these inequalities will be about a topological type of curve or multicurve: two multicurves $\\Gamma$ and $\\Gamma'$ are of the same type if there exists $\\phi\\in \\mathrm{Mod}(\\Sigma)$ such that $\\phi(\\Gamma)=\\Gamma'$. Given a multicurve $\\Gamma$, we can consider its mapping class group orbit. These orbits divide the space of all multicurves into equivalence classes\n$$\n[\\Gamma]:= \\{ \\phi(\\Gamma) \\mid \\phi \\in \\mathrm{Mod}(\\Sigma)\\}\n$$\nsorted by type. Now given $X\\in \\mathcal{T}(\\Sigma)$, we can study the length of a minimal representative of an equivalence class:\n$$\n\\ell_{[\\Gamma]}(X) = \\min_{\\phi \\in \\mathrm{Mod}(\\Sigma)} \\big\\{ \\ell_X(\\phi(\\Gamma)\\big\\}\n$$\nThe existence of a minimum follows from the discreteness of the length spectrum. This function, due to its mapping class group invariance, descends nicely to $\\mathcal{M}(\\Sigma)$. Note that although the function $\\ell_{[\\Gamma]}(\\cdot)$ remains continuous over $\\mathcal{M}(\\Sigma)$, it is no longer smooth. This is due to possible changes of homotopy classes realizing the minimum length of in an equivalence class.\n\nThe following result is a non-explicit general bound that holds for any topological type of multicurve. The proof is a compactness argument. The constant $K$ in the statement depends on the topology of $\\Sigma$, the topological type of $\\Gamma$ and a lower bound on the systole.\n\n\\begin{proposition}\\label{prop:conditionallength}\nLet $[\\Gamma]$ be a type of multicurve on $\\Sigma$. For any $\\epsilon>0$, there exists a constant $K=K(\\epsilon, \\Sigma, [\\Gamma])$ such that for any $X\\in \\mathcal{T}^{\\epsilon}(\\Sigma)$, we have\n$$\n\\ell_{[\\Gamma]}(X) \\leq K.\n$$\n\\end{proposition}\n\n\\begin{proof}\nThe space $\\mathcal{M}^{\\epsilon}(\\Sigma)$ is compact, hence the continuous function $\\ell_{[\\Gamma]}(\\cdot)$ admits a maximum on $\\mathcal{M}^{\\epsilon}(\\Sigma)$. This maximum value is exactly $K$. \n\\end{proof}\n\nAs an explicit example of the above result, consider the following result due to Buser and S\\\"eppala \\cite{Buser-Seppala}. For a closed surface $\\Sigma$ of genus $g\\geq 2$, they consider {\\it canonical homology bases}, that is collections of $2g$ simple closed curves $\\{\\alpha_i,\\beta_i\\}_{1\\leq i \\leq g}$ that satisfy:\n\\begin{enumerate}[a)]\n\\item\n$i(\\alpha_i,\\beta_j)=\\delta_{ij}$ for all $i,j\\in \\{1,\\hdots,g\\}$ (where $\\delta_{ij}$ is the Kronecker delta),\n\\item\n$i(\\alpha_i,\\alpha_j)=i(\\beta_i,\\beta_j)=0$ for $i\\neq j$. \n\\end{enumerate}\nNote that such a system of curves automatically generate integer homology. See Figure \\ref{fig:homology} for an illustration in genus $2$.\n\n\\begin{figure}[htbp]\n\\leavevmode \\SetLabels\n\\L(.23*.5) $\\alpha_1$\\\\%\n\\L(.32*.27) $\\beta_1$\\\\%\n\\L(.74*.5) $\\alpha_2$\\\\%\n\\L(.66*.27) $\\beta_2$\\\\%\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=8cm]{Figures\/N-homology.pdf}}}\n\\caption{A canonical homology basis in genus $2$}\n\\label{fig:homology}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\n\nThey prove that any $X\\in \\mathcal{T}^{\\epsilon}(\\Sigma)$ admits a canonical homology basis with all curves of length at most\n$$\n(g-1)\\left(45+6\\,\\mathrm{arcsinh}\\frac{1}{\\epsilon}\\right).\n$$\nThis is an improvement over an earlier quantification in \\cite{Buser-Seppala2}. The above bound shows that the $K$ in this instance is bounded above by \n$$\n2g(g-1)\\left(45+6\\,\\mathrm{arcsinh}\\frac{1}{\\epsilon}\\right)\n$$\nbecause there are $2g$ curves in the family. Of course, for this to be an exact quantification of the constant $K$, it would have to be sharp (which it is not). Exact quantifications are rarely known but, as proved in \\cite{Buser-Seppala}, the real constant must depend on $\\epsilon$. Alternatively, this dependency on $\\epsilon$ can also be deduced from Theorem \\ref{thm:main}.\n\nNaturally, this leads to the existence of upper bounds which only depend on topology and curve type, but not on systole length. A multicurve type $[\\Gamma]$ is said to satisfy a strong length inequality if $\\ell_{[\\Gamma]}(\\cdot)$ is upper bounded over $\\mathcal{M}(\\Sigma)$. By continuity of the length function $\\ell_{[\\Gamma]}(\\cdot)$, this is equivalent to the existence of a surface $X_{\\max}\\in \\mathcal{M}(\\Sigma)$ such that \n$$\n\\max_{X\\in \\mathcal{M}(\\Sigma)} \\ell_{[\\Gamma]}(X) = \\ell_{[\\Gamma]}(X_{\\max}).\n$$\nTo see this, one must show that the supremum of the length function cannot be reached on the boundary of moduli space, that is on a noded surface. As it turns out, as a consequence of Lemma \\ref{lem:stretchpants} from the next section, a (finite) supremum is always reached in the ''thick\" part of moduli space, and so in particular the $\\sup$ is indeed a $\\max$.\n\n\\section{Stretching pants, Bers' theorem and consequences}\n\nThe following lemma is by now well-known, but a sketch proof is provided for completeness. The proof uses strip maps, introduced by Thurston \\cite{Thurston}, and used to great effect by many authors \\cite{Danciger-Gueritaud-Kassel, Gueritaud, Papadopoulos-Theret, Parlier} to study of deformations of hyperbolic structures. Recall that a hyperbolic pair of pants is uniquely determined by its cuff lengths. In order to allow pants with cusp boundary, we use the convention that a cusp is a cuff of $0$ length. \n\n\\begin{lemma}[Pants stretching lemma]\\label{lem:stretchpants}\nLet $Y_{x,y,z}$ be the unique hyperbolic pair of pants with cuff lengths $x,y,z \\geq 0$. Then, for any (non-boundary) homotopy class of closed curve $\\gamma$ on a pair of pants, and any $\\epsilon>0$, we have\n$$\n\\ell_\\gamma\\left(Y_{x,y,z}\\right) < \\ell_\\gamma\\left(Y_{x+\\epsilon,y,z}\\right)\n$$\n\\end{lemma}\n\\begin{proof}[Sketch proof]\nConsider on $Y=Y_{x+\\epsilon,y,z}$, the simple orthogeodesic $a$ with both endpoints on the boundary curve of length $x+\\epsilon$ (see Figure \\ref{fig:arc}). \n\n\\begin{figure}[htbp]\n\\leavevmode \\SetLabels\n\\L(.445*.5) $a$\\\\%\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=5cm]{Figures\/N-arc.pdf}}}\n\\caption{The orthogeodesic $a$}\n\\label{fig:arc}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\n\nNote the closed geodesic $\\gamma$ intersects $a$ at least once. Exactly like in the collar lemma, $a$ admits an embedded neighborhood, topologically a strip (see Figure \\ref{fig:strip}). The idea is to now remove at least part of this strip to reduce the length of the boundary curve. \n\n\\begin{figure}[htbp]\n\\leavevmode \\SetLabels\n\\L(.445*.5) $a$\\\\%\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=5cm]{Figures\/N-strip.pdf}}}\n\\caption{A strip surrounding $a$}\n\\label{fig:strip}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\n\nTo do this properly, it is more convenient to consider the complete pair of pants by adding funnels. The arc $a$ can now be extended into a complete simple and infinite length geodesics. In addition, because the boundary curve of length $x+\\epsilon$ is not $0$, there is a family of simple complete geodesics parallel to each other and to $a$. We can take any two of these, cut away the strip between them, and paste them together to obtain a (complete) hyperbolic metric, say $Y'$ (see Figure \\ref{fig:operation}). (The slightly sketchy part is here: in fact it is possible by a variational argument to show that this can be done such that the length of the new boundary component of $Y'$ is exactly $x$, but we won't dwell on this, the main point being that the boundary length has been reduced.)\n\n\\begin{figure}[H]\n\\leavevmode \\SetLabels\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=10cm]{Figures\/N-construction.pdf}}}\n\\caption{Removing the strip neighborhood $\\mathcal{S}(a)$}\n\\label{fig:operation}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\n\nWe denote the strip enclosed by these geodesics by $\\mathcal{S}(a)$.\n\nNow can analyse the result of this operation on the length of $\\gamma$. Note that, due to the topology of the strip neighborhood $\\mathcal{S}(a)$ of $a$, $\\gamma \\cap \\mathcal{S}(a)$ is a union of simple geodesic arcs. We look at each one. In order to find a representative of $\\gamma$ on $Y'$, we replace each simple geodesic arc with its projection to $a$ (see Figure \\ref{fig:project}). The point is that the projection strictly reduces lengths.\n\\begin{figure}[H]\n\\leavevmode \\SetLabels\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=5cm]{Figures\/N-project.pdf}}}\n\\caption{A local picture of $\\gamma$ under the operation}\n\\label{fig:project}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\nThus, this results in a curve on $Y'$, in the same homotopy class, and of length strictly smaller. The corresponding geodesic is thus of length strictly smaller than it was previously. Hence we have $\\ell_\\gamma\\left(Y\\right) > \\ell_\\gamma\\left(Y'\\right)$ as required. \n\\end{proof}\nTo make the proof fully rigorous \nWe will mainly need an immediate corollary of the above result.\n\\begin{corollary}\\label{cor:longpants}\nFor $L>0$, let $Y$ be a pair of pants with cuff lengths between $0$ and $L$. Let $Y_L$ be the pair of pants with all cuff lengths exactly $L$. Then, for any interior homotopy class of curve $\\gamma$ on the pair of pants, we have $\\ell_\\gamma(Y_L) \\geq \\ell_\\gamma(Y)$ with equality only occurring if $Y=Y_L$. \n\\end{corollary}\n\nPants decompositions play an essential role in this story. The following result, originally due to Bers \\cite{Bers1,Bers2}, has been since quantified by different authors \\cite{BuserBook,BPS,ParlierShort}.\n\n\\begin{theorem}[Bers' constants]\\label{thm:bers}\nThere exists a constant $B(\\Sigma)$ such that any $X\\in \\mathcal{T}(\\Sigma)$ admits a pants decomposition with all curves of length at most $B(\\Sigma)$.\n\\end{theorem}\n\nNote that this result does not fall in the framework of either Proposition \\ref{prop:conditionallength} or Theorem \\ref{thm:main}. Indeed, in order to find a short pants decomposition, we are allowed to choose any topological type of pants decomposition. The number of these grows with topology (for instance there are roughly $\\sim g^g$ different types when $\\Sigma$ is of genus $g$ and $g$ is large). So here we are not only minimizing among mapping class group orbits of a fixed multicurve, but we are minimizing among multicurves that belong to a family. The following result holds for all such families of multicurves. A family will be denoted $\\{\\Gamma_\\alpha\\}_{\\alpha\\in I}$ where $I$ is an index set (possibly infinite, but countable as there are only countably many topological types of finite multicurve on a finite type surface). \n\n\\begin{theorem}\\label{thm:maingeneral}\nLet $\\{\\Gamma_\\alpha\\}_{\\alpha\\in I}$ be a family of multicurves. Then the quantity \n$$\\max_{X\\in \\mathcal{M}(\\Sigma)} \\min_{\\alpha\\in I} \\ell_{[\\Gamma_\\alpha]}(X)$$\nis finite if and only if, for every pants decomposition $P$, there exists $\\alpha\\in I$ and $\\phi \\in \\mathrm{Mod}(\\Sigma)$ such that $i(P, \\phi(\\Gamma_\\alpha)) = 0$. \n\\end{theorem}\n\nThe statement might seem a little confusing at first, due to the fact that we are taking a maximum among hyperbolic structures of a double minimum (over a family and then over the mapping class group orbit). If the family is reduced to a single curve, the statement becomes Theorem \\ref{thm:main} from the introduction. The proofs of both statements are identical, so we prove the more general statement above.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:maingeneral}]\n\nWe begin with the more straightforward direction, showing that if there is a pants decomposition $P$ which is intersected by any mapping class group orbit of any multicurve in the family $\\{\\Gamma_\\alpha\\}_{\\alpha\\in I}$, then there is no upper bound on the length. This follows directly from Lemma \\ref{lem:collar} (the collar lemma). Indeed, by considering a sequence of hyperbolic structures with all curves in the pants decomposition of $P$ that converge to $0$, the length of any curve that intersects one of the pants curve necessarily goes to $+\\infty$. As, by hypothesis, there is at least one curve in every multicurve $\\phi(\\Gamma_\\alpha)$, for all $\\phi \\in \\mathrm{Mod}(\\Sigma)$ and all $\\alpha \\in I$, that intersects a curve in $P$, the result follows. \n\nThe other direction will follow from Bers' theorem (Theorem \\ref{thm:bers}) and Corollary \\ref{cor:longpants} above.\n\nTake any $X\\in \\mathcal{T}(\\Sigma)$. By Bers' theorem, there exists a pants decomposition of $X$, say $P$, with all curves of length at most $B(\\Sigma)$. Now, by hypothesis, there exists $\\Gamma_\\alpha$ such that $i(\\Gamma_\\alpha,P)= 0$. Note that $\\Gamma_\\alpha$ can contain curves from $P$, but none that intersect curves of $P$ transversally. \n\nBy hypothesis all curves that belong to both $\\Gamma_\\alpha$ and $P$ are upper bounded by $B(\\Sigma)$. We now the corollary to all remaining curves of $\\Gamma_\\alpha$. Indeed, any such a curve $\\gamma \\in \\Gamma_\\alpha$ is contained in a pair of pants. By hypothesis, the cuff lengths of this pair of pants are at most $B(\\Sigma)$. By Corollary \\ref{cor:longpants}, $\\ell_X(\\gamma)$ is at most the length when the pair of pants has all cuff lengths equal to $B(\\Sigma)$, which is some finite number that depends on $B(\\Sigma)$ and the topological type of $\\gamma$. As there are a finite number of such curves, the result follows.\n\\end{proof}\n\nThere are many instances of the above theorem, each obtained by changing the multicurve or family of multicurves. Note that only the topological type of a multicurve matters in the statement. Some of these results are in the positive direction: the theorem implies that there is an upper bound on a minimal length representative that only depends on the topology of $\\Sigma$. Others are in the negative direction, that is that certain multicurves, or families of multicurves, do not admit such upper bounds. We give a (non-exhaustive) list of results of this type.\n\n\\begin{enumerate}[I.]\n\\item Even though it was used in the proof, Bers' theorem is an example obtained by taking the family of multicurves to be the full set of pants decompositions (or simply one pants decomposition for each topological type). To put in the framework of the theorem, the length of a pants decomposition should be defined as the sum, and not the maximum value. \n\nThe quantification of the implied constants - for both the sum and the max - has attracted some attention over the years, but it does not seem to be an easy problem. In fact, even the rough growth in terms of genus is not known \\cite{BuserBook,ParlierShort,GPY}. On the positive side, the exact constant in genus $2$ is known by a result of Gendulphe \\cite{Gendulphe}, as is the rough growth in terms of the number of cusps \\cite{Balacheff-Parlier,BPS}. \n\n\\item One can also take the full set of all topological types of all multicurves: this boils down to the systolic inequality. Quantifying the exact constants that appear is an arduous task. For orientable closed surfaces, the only constant known is again in genus $2$ \\cite{Jenni}. Buser and Sarnak \\cite{Buser-Sarnak} showed that the constants must grow logarithmically in genus, and since then there have been multiple variations and refinements of this (see for instance \\cite{BFR,KSV,Fanoni-Parlier, SchmutzSchaller}), including generalizations in the world of variable curvature and higher dimensional manifolds \\cite{Gromov}.\n\n\\item A slight variation of the above is to look at the homological systole of a closed surface $\\Sigma$. That is, the shortest curve that is not only non-trivial in homotopy, but also in homology. By cut and paste arguments, the homological systole is always realized by a non-separating simple closed curve. Hence, it is always in the same mapping class orbit. As above, few optimal constants are known. However, for closed surfaces, it is known that the optimal constants are equal to those from the systolic inequality. \\cite{ParlierPapa} Said differently, the systole of a maximal surface is homologically non-trivial.\n\n\\item In addition to the results of Buser and Sepp\\\"al\\\"a mentioned previously, one can try and bound families of homologically independent curves, but that do not necessarily form (part of) a canonical basis. If one requests a full homology basis, by the theorem above, there is no upper bound on its length over moduli space. However, by Bers' theorem, and the observation that any pants decomposition contains $g$ homologically independent curves, one can find an upper bound on the length of up until $g$ curves by a function of topology. So what about $g+1$ curves?\n\nGromov observed \\cite[Section 5]{GromovFilling} that any minimal length homology basis consists of simple curves that pairwise intersect at most once. Now given $g+1$ homologically independent and simple curves, there must be at least a pair that intersect. And a pair of intersecting simple curves necessarily intersects {\\it all} pants decompositions. Hence by the theorem above, there is no upper bound for such a family and so strong length inequality stops at exactly $g$ homologically independent curves. \n\nMore precise quantifications of the constants have also attracted attention. In particular the Buser-Sarnak logarithmic bound can be extended to roughly $ag$ curves for any $a<1$ \\cite{BPS}. \n\nFinally note that Gromov's observation above (on the intersection properties of minimal bases) still forces one to consider multiple, although finite, topological types of multicurves.\n\n\\item In a somewhat opposite direction, consider $\\{\\Gamma\\}_\\alpha$ to be the set of separating simple closed curves on a closed surface $\\Sigma$ of genus $g$. Then, as there exists a pants decomposition consisting only of non-separating curves, it will essentially intersect any element of the mapping class group orbit of any element of $\\Gamma$. Hence, for all $\\alpha\\in I$, the function $\\ell_{[\\Gamma_\\alpha]}(\\cdot)$ admits no upper bound over $\\mathcal{M}(\\Sigma)$. If however one takes the larger set $\\{\\Gamma\\}_\\alpha$ of all homologically trivial curves (but not homotopically trivial), then it is a consequence of a theorem of Sabourau \\cite{Sabourau} that this admits an upper bound that only depends on genus. This subtle difference lead to a small gap in Mirzakhani's study of the expected value of the shortest {\\it simple} homologically trivial curve in \\cite{Mirzakhani}, but this was not the difficult part of her work and has since been cleared up \\cite{PWX}.\n\n\\item For multicurves containing more than one curve, many of these problems admit variations. Here we suppose the length of a multicurve is the sum of lengths of its components, but of course one could also consider other variations, such as the $\\max$ or the product. For instance, the product of lengths of homologically distinct curves was studied in \\cite{Balacheff-Karam-Parlier}. Replacing the sum of the lengths with the maximum will also satisfy the same boundedness condition of Theorem \\ref{thm:maingeneral}, but products - or other functions of the lengths of the components - need to be checked on a case-by-case basis. \n\\end{enumerate}\n\n\\ack{This work was supported by the Luxembourg National Research Fund OPEN grant O19\/13865598.}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n Repulsive gravitational effects are\nan essential ingredient of the Standard Cosmological Model (SCM), where they are evoked to overcome the difficulties faced by the Friedmann model, such as the horizon, flatness, and origin of the primordial density fluctuations problems \\cite{GUTH_1981, LINDE_1982, ALBRECHT_STEINHARDT_1982, LINDE_1983, LINDE_2005}, as well as the observed late-time accelerated expansion of the universe \\cite{RIESS_1998, PERLMUTTER_1999}. According to General Relativity (GR), when the minimal coupling principle is enforced, repulsive gravitational effects can only be generated by fluids with negative pressure. Such fluids are invariably behind inflationary models (see \\cite{LINDE_2005} and references therein) and dynamical dark energy models \\cite{DE_PDG_2013, COPELAND_2006, FRIEMAN_TURNER_HUTERER_2008}.\n\n A series of attempts to overcome the cosmological singularity problem, arguably the most severe difficulty faced by the Friedmann model (see \\cite{NOVELLO_BERGLIAFFA_2008} and references therein), were proposed in the past in which the alternative concept of repulsive gravitational effects generated by fields non-minimally coupled to gravity was introduced \\cite{LINDE_1979,NOVELLO_SALIM_1979, NOVELLO_1980}. In \\cite{LINDE_1979}, in particular, a conformally coupled Higgs-like field with a potential $V(\\phi) = m^2\\phi^2 + \\sigma\\phi^4$ ($\\sigma >0$) was responsible for a reversal of the effective gravitational constant in the early universe, when the density reached a critical value $\\rho_c$. Such model was later ruled-out on stability grounds, since as $\\rho \\to \\rho_c$ the effective gravitational constant diverges, $\\kappa_{eff} \\to \\infty$ \\cite{STAROBINSKY_1981}. Despite that, this idea paved the way for inflationary models based on non-minimally coupled Higgs-like scalar fields \\cite{SALOPEK_BOND_BARDEEN_1989, FAKIR_UNRUH_1990, KAISER_1995, KOMATSU_FUTAMASE_1999, BEZRUKOV_SHAPOSHNIKOV_2008, BARVINSKY_KAMENSHCHIK_STAROBINSKY_2008} (see \\cite{LINDE_2005} and references therein), and was the source of other gravitational repulsion models (see \\cite{HOHMANN_WOHLFARTH_2010} and references therein). As we shall see, that criticism does not apply here.\n\n\n In the present work we revisit a model \\cite{NOVELLO_1980}\nin which ordinary matter (radiation) can generate repulsive gravitational effects by intervention of a conformally coupled scalar field. The main idea can be synthesized in the following steps:\n\n\\begin{itemize}\n \\item[(i)] We assume the existence of a scalar field $\\phi$ which has a quartic self interaction potential with the form $V(\\phi) = m^2\\phi^2 -\\sigma \\phi^4$ (we call attention to the relative sign);\n \\item[(ii)] This scalar field couples non-minimally to gravity;\n \\item[(iii)] We select among all possible candidates of such a coupling a conformal one;\n \\item[(iv)] Matter interacts with the scalar field only through gravity;\n \\item[(v)] The symmetry of the theory is broken by a state $\\phi = \\phi_0 = $ constant;\n \\item[(vi)] When the field is in the broken symmetry state, the net result\non the equation for the metric is to produce an effective gravitational constant which, under certain conditions, has the opposite sign of the bare Newton's\ngravitational constant, provided that the energy-momentum tensor for the matter fields is traceless ($T=0$).\n \\item[(vii)] It will be shown that a necessary consequence of this process is that a ordinary radiation ($T=0$) dominated universe exhibits a bounce. \n\\end{itemize}\n\n\n\n\n\\section{The model}\n\n\n \\subsection{Non-minimally coupled scalar field}\n\n\n\nIn what follows, we adopt the space-time metric with signature $(+---)$.\nThe theory considered here is defined by a Lagrangian with a scalar field non-minimally (conformally) coupled to gravity \n\\begin{equation}\nL = \\sqrt{-g}\\left[ \\frac{1}{\\kappa}R + g^{\\alpha\\beta}\\partial_{\\alpha}\\phi^{\\ast}\\partial_{\\beta}\\phi - V(\\phi^{\\ast}\\phi)\n- \\frac{1}{6}R\\phi^{\\ast}\\phi + \\mathcal{L}_m \\right],\n\\label{lagrangian}\n\\end{equation}\nwhere $g_{\\alpha\\beta}$ are the components of the metric field, $g = \\mbox{det}(g_{\\alpha\\beta})$, $R$ is the Ricci scalar,\n$\\kappa = 8\\pi G$ is the reduced gravitational constant ($c=\\hslash = 1$), $\\mathcal{L}_m$ is the matter Lagrangian density,\nand the self-interaction potential of the scalar field is given by \\cite{NOVELLO_1980}\n\\begin{equation}\nV(\\phi^{\\ast}\\phi) = m^2\\phi^{\\ast}\\phi - \\sigma(\\phi^{\\ast}\\phi)^2 - 2V_0.\n\\end{equation}\nWe call attention for the fact that this potential differs from the usual ones employed in \\cite{LINDE_1979} or in inflation models (see \\cite{LINDE_2005}) by the relative sign between the mass term, $m^2\\phi^{\\ast}\\phi$, and the quartic self-interaction term, $\\sigma(\\phi^{\\ast}\\phi)^2$. \n\nThe set of field equations obtained from the Lagrangian (\\ref{lagrangian}) are\n\\begin{widetext}\n\\begin{subequations}\n \\begin{align}\n   R_{\\alpha\\beta} - \\frac{1}{2}Rg_{\\alpha\\beta} = & -\\frac{1}{{\\frac{1}{\\kappa} - \\frac{1}{6}\\phi^2}}\\left[ \\tau_{\\alpha\\beta}(\\phi) + T_{\\alpha\\beta} \\right],\n\\label{grav_field_eq}\n \\end{align}\n\\begin{equation}\n \\Box\\phi + m^2\\phi - 2\\sigma\\phi^3 + \\frac{1}{6}R\\phi = 0, \\label{scalar_field_eq}\n\\end{equation}\n\\end{subequations}\n\\end{widetext}\nwhere $\\Box \\equiv g^{\\alpha\\beta}\\nabla_{\\alpha}\\nabla_{\\beta}$, $\\phi^2 \\equiv \\phi^{\\ast}\\phi$, $\\phi^3 = (\\phi^{\\ast}\\phi)\\phi$, $T_{\\alpha\\beta}$ is the energy-momentum tensor of the matter fields, and\n\\begin{widetext}\n\\begin{align}\n \\tau_{\\alpha\\beta}(\\phi) \\equiv \\partial_{\\alpha}\\phi^{\\ast}\\partial_{\\beta}\\phi\n- \\frac{1}{2}\\Big( \\partial_{\\mu}\\phi^{\\ast}\\partial^{\\mu}\\phi - m^2\\phi^2 + \\sigma\\phi^4 \\Big)g_{\\alpha\\beta}\n- \\frac{1}{6}\\Big( \\nabla_{\\alpha}\\nabla_{\\beta}\\phi^2 - \\Box\\phi^2g_{\\alpha\\beta} \\Big) - V_0 g_{\\alpha\\beta} \\label{em_scalar}\n\\end{align}\n\\end{widetext}\nis the ``improved\" energy-momentum tensor of the scalar field.\nTaking the trace of the field equation (\\ref{grav_field_eq}),\nand using equation (\\ref{scalar_field_eq}), it follows that the Ricci scalar can be expressed as follows\n\\begin{equation}\n R = m^2\\phi^2 - 4V_0 + T,\n\\end{equation}\nwhere $T\\equiv g^{\\alpha\\beta}T_{\\alpha\\beta}$. This enables us to rewrite equation (\\ref{scalar_field_eq}) in the form\n\\begin{equation}\n  \\Box\\phi + \\left( m^2 - \\frac{2}{3}V_0 + \\frac{1}{6} T\\right)\\phi + \\left(\\frac{1}{6} m^2 - 2\\sigma\\right)\\phi^3 = 0. \\label{scalar_field_eq_b}\n \\end{equation}\n\n\n\n\n \\subsection{Broken symmetry and gravitational repulsion from ordinary matter}\n\n\n\nIt is clear from equation (\\ref{grav_field_eq}) that in the case where the ground state of the scalar field potential vanishes, $V_0=0$,\nthe improved energy-momentum tensor of the scalar field $\\tau_{\\alpha\\beta}(\\phi)$\nis irremediably not conserved. Following Callan et al. \\cite{CALLAN_COLEMAN_JACKIW_1970}, however, we can define the conserved energy-momentum tensor\n\\begin{equation}\n E_{\\alpha\\beta}(\\phi) \\equiv \\frac{1}{{\\frac{1}{\\kappa} - \\frac{1}{6}\\phi^2}}\\tau_{\\alpha\\beta}(\\phi). \\label{em_complex}\n\\end{equation}\nWe now look for a constant solutions of equation (\\ref{scalar_field_eq_b}) \nwhich correspond to stable vacua of the scalar field. Clearly, equation (\\ref{scalar_field_eq_b}) only admits a constant solution in the special case where the trace of the energy momentum-tensor of matter fields is a constant. \nWe stress the fact that the symmetry breaking process is only possible in this case, \notherwise no nontrivial ground state solution of equation (\\ref{scalar_field_eq_b}) exists.\nLet us concentrate, from now on, on\nmatter fields described by a traceless energy-momentum tensor, $T=0$.\n\nAccording to expressions (\\ref{em_complex})-(\\ref{em_scalar}), for a constant solution $\\phi=\\phi_0$ the energy density $E_{00}(\\phi) \\equiv E(\\phi)$\ncorresponding to the energy-momentum tensor (\\ref{em_complex}) has the form\n\\begin{equation}\nE(\\phi_0) = \\frac{1}{{\\frac{1}{\\kappa} - \\frac{1}{6}\\phi_0^2}}(m^2\\phi_0^2 - \\sigma\\phi_0^4 - 2V_0). \\label{energy_func}\n\\end{equation}\nOn the other hand, for $T= 0$ equation (\\ref{scalar_field_eq_b}) admits, besides the trivial solution $\\phi_0 = 0$, two constant solutions\n\\begin{equation}\n \\phi_0^2 = \\frac{6m^2 - 4V_0}{12\\sigma - m^2}. \\label{relation}\n\\end{equation}\nThe constant nontrivial solutions which minimizes the energy density functional (\\ref{energy_func}), and satisfies relation (\\ref{relation}),\nare given by\n\\begin{equation}\n \\phi_0 = \\pm \\frac{2\\sqrt{V_0}}{m}, \\ \\ \\ \\  \\sigma = \\frac{m^4}{8V_0}. \\label{const_sol}\n\\end{equation}\nThe resulting behavior of the energy density $E(\\phi_0)$ exhibits three uncommunicating regions, two of them containing the stable local minima\nat $\\phi_0 = \\pm 2\\sqrt{V_0}\/m$, to which correspond $E(\\phi_0) = 0$, the other\ncontaining two unstable maxima and a metastable minimum at $\\phi_0 = 0$ (see Fig \\ref{energy_fig}).\n\n\n\n\nConsequently, when the energy-momentum tensor for the matter fields is traceless ($T=0$), and the field is in a non-trivial stable ground state (\\ref{const_sol}), the gravitational field equation (\\ref{grav_field_eq}) assumes the form\n\\begin{align}\n  R_{\\alpha\\beta} = -\\kappa_{eff}T_{\\alpha\\beta} \\label{grav_field_eq_2},\n \\end{align}\nwhere\n\\begin{equation}\n\\kappa_{eff} \\equiv \\frac{1}{\\frac{1}{\\kappa} - \\frac{1}{6}\\phi_0^2} =  \\left(\\frac{3m^2\\kappa}{3m^2 - 2\\kappa V_0}\\right)\n\\end{equation}\nis the renormalized gravitational constant.\nThe term multiplying the energy-momentum tensor of matter (ordinary radiation) in equation (\\ref{grav_field_eq_2}) can, thus,\nbe viewed as a ``renormalized'' gravitational constant. Clearly, in the regime where the relation\n\\begin{equation}\n\\kappa V_0 > \\frac{3}{2}m^2 > 0 \\label{condition_mass}\n\\end{equation}\nis satisfied ({\\it i.e.} either the mass $m$ of the scalar field has to be very small, or $V_0$ has to be very large), gravity is reversed, {\\it i.e.} $\\kappa_{eff} < 0$. \n\n \\begin{figure}[H]\n   \\centering\n \\vspace{15mm}\n  \\includegraphics[scale=\n0.8 \n]{ENplots.eps} \n   \\caption{Plot of the energy density $E(\\phi_0)$, for $\\kappa V_0>3m^2\/2>0$.\nThe figure shows the solutions $\\phi_0 = \\pm 2\\sqrt{V_0}\/m$ corresponding to the nontrivial stable vacua,\nand also two asymptotes at $\\phi_0 = \\pm \\sqrt{6}$ separating three uncommunicating regionsa are indicated (dashed lines).} \\label{energy_fig}\n   \\end{figure} \n \n \n\\section{Bouncing cosmological model} \n\n\n  \\subsection{Scalar field induced bounce}\n  \n\nWe now investigate what cosmological solutions are compatible with the model discussed above. We assume spatial homogeneity and isotropy, and adopt a Friedmann metric for the space-time\n\\begin{equation}\nds^2 = dt^2 - a^2(t)\\left[ \\frac{dr^2}{1- \\epsilon r^2} + r^2 \\big(d\\theta^2 + \\sin^2\\theta d\\varphi^2\\big) \\right], \\label{friedmann_metric}\n\\end{equation}\nwhere $a(t)$ is the scale factor and $\\epsilon$ determine the geometry of the spatial section. In order to be the source of the Friedmann metric (\\ref{friedmann_metric}) the scalar field must depend on the cosmic time only, $\\phi = \\phi(t)$. We consider a radiation-dominated universe, $\\rho = 3p$ ($T= 0$).\n In the regime $\\phi = \\phi_0$ the field equations reduce to\n\\begin{subequations}\\label{cosm_eq}\n\\begin{equation}\n\\left( \\frac{\\dot{a}}{a} \\right)^2 + \\frac{\\epsilon}{a^2} = \\frac{1}{3}\\kappa_{eff} \\rho, \n\\end{equation}\n\\begin{equation}\n\\frac{\\ddot{a}}{a} = -\\frac{1}{6}\\kappa_{eff}(\\rho + 3p), \n\\end{equation}\n\\end{subequations}\nwhere $\\rho(t)$ and $p(t)$ are the density and pressure of the radiation fluid. In this case, if condition (\\ref{condition_mass}) holds, gravity is reversed, $\\kappa_{eff}<0$, and the field equations (\\ref{cosm_eq}) only admit a solution for an open spatial section $\\epsilon = -1$. Form energy conservation we have $\\rho(t) \\propto \\rho_0 a^{-4}(t)$, where $\\rho_0$ is a constant, so that the system of equations (\\ref{cosm_eq}) yield\n\\begin{equation}\n\\dot{a} = \\sqrt{ 1 - \\frac{1}{3}|\\kappa_{eff}| \\rho_0 a^{-2}}.\n\\end{equation}\nThis equation can be readily integrated, and we obtain the following form for the scale factor\n\\begin{equation}\na(t) = \\sqrt{ t^2  + a_0^2}. \\label{scale_fac_bounce}\n\\end{equation}\nTherefore, the universe in this model exhibits a bounce around $t = 0$, the constant $a_0 = \\sqrt{|\\kappa_{eff}| \\rho_0\/3}$ being the minimum value attainable by the scale factor. \n\nInterestingly, the bouncing solution obtained above coincides with a model based on a non-minimally coupled vector field proposed by one of the authors \\cite{NOVELLO_SALIM_1979}, even though the two models differ considerably (see appendix A).\n\n\n\n\n\\section{Final Remarks}\n\n\nIn the present work we revisited a model \\cite{NOVELLO_1980} in which a scalar field conformally coupled to gravity can generate\nrepulsive gravitational effects when only ordinary matter with traceless energy-momentum tensor (radiation) is coupled to gravity. It was shown that, in a radiation-dominated universe, when the scalar field is in a non-trivial ground (broken symmetry) state the only solution admissible by the field equations is a bouncing universe. \n\n\n\n\n\n\n\n \\begin{appendix}\n  \n \\section{Bouncing model generated by a vector field non-minimally coupled to gravity \\label{app_ns}}\n \n \n We include here a short review of the bouncing cosmological model proposed in \\cite{NOVELLO_SALIM_1979} and the one presented above for an easy comparison between them. The Lagrangian describing a vector field non-minimally coupled to gravity employed in \\cite{NOVELLO_SALIM_1979} is\n \\begin{equation}\n L = \\sqrt{-g}\\left[ \\frac{1}{\\kappa}R + \\beta RA_{\\mu}A^{\\mu} - \\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu} \\right],\n \\label{lagrangian_d}\n \\end{equation}\nwhere $\\beta$ is a constant, and $F_{\\mu\\nu} = \\nabla_{\\mu}A_{\\nu} - \\nabla_{\\nu}A_{\\mu}$. The field equation are\n\\begin{widetext}\n\\begin{subequations} \\label{field_eq_app}\n \\begin{align}\n   R_{\\alpha\\beta} - \\frac{1}{2}Rg_{\\alpha\\beta} = & -\\frac{1}{{\\frac{1}{\\kappa} + \\beta A_{\\mu}A^{\\mu}}}\\tau_{\\alpha\\beta}(A^{\\mu}) ,\n \\end{align}\n\\begin{equation}\n \\nabla_{\\nu}F^{\\mu\\nu} + \\beta A^{\\mu} = 0, \n\\end{equation}\n\\end{subequations}\n\\end{widetext}\nwhere $\\tau_{\\alpha\\beta}(A^{\\mu})$ is the improved energy-momentum tensor of the vector field. \n\nIn a Friedmann geometry determined by the metric (\\ref{friedmann_metric}), making the choice $A_{\\mu} = A(t)\\delta^0_{\\mu}$, and defining $\\Omega(t) = \\frac{1}{\\kappa} + \\beta A^2(t)$, the set of field equations (\\ref{field_eq_app}) assume the form\n\\begin{subequations}\n\\begin{equation}\n3\\frac{\\ddot{a}}{a} = -\\frac{\\ddot{\\Omega}}{\\Omega},\n\\end{equation}\n\\begin{equation}\n\\frac{\\ddot{a}}{a} + 2\\left( \\frac{\\dot{a}}{a} \\right)^2 + 2\\frac{\\epsilon}{a^2} = -\\frac{\\dot{a}}{a}\\frac{\\dot{\\Omega}}{\\Omega},\n\\end{equation}\n\\begin{equation}\n\\Box\\Omega = 0.\n\\end{equation}\n\\end{subequations}\nA particular solution for an open spatial section, $\\epsilon = -1$, furnishes a scale factor with the form\n\\begin{equation}\na(t) = \\sqrt{ t^2  + a_0^2},\n\\end{equation}\nwhere $a_0$ is the minimum value of the scale factor. Although this is the same solution (\\ref{scale_fac_bounce} ) obtained for the scalar field, the two models differs in crucial points. First, the quantity $\\Omega(t)$, which is analogous to the effective gravitational constant of the model discussed in the main text, is not a constant but a function of the cosmic time (although a model similar to the one considered in the main text would arise in the special case $A_{\\mu}A^{\\mu} =$ constant). Second, here the non-minimally coupled vector field alone can be the source of the geometry, while in the bouncing model induced by the scalar field, in the absence of matter fields and for the scalar field in the non-trivial (broken symmetry) ground state, the gravitational field equations reduce to the vacuum Einstein equations.\n\n\n\\end{appendix}\n\n\n\n\\vspace{5mm}\n\n\\section*{Acknowlegments}\n\nThe authors would like to thank the Brazilian National Council of Technological and Scientific Development (CNPq) and the Research Support Foundation of the State of Rio de Janeiro (FAPERJ) for a grant. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Bar formation: nature vs. nurture}  \nThe observational proofs about the influence of the environment on bar\nformation and\/or  evolution are still rare.   In \\citet{aguerri09}, we\ninvestigate  a volume-limited  sample  of 2106  disc  galaxies in  the\nnearby universe to derive the bar  fraction as a function of the local\ngalaxy density.  The local  density was\ncalculated for  every sample galaxy using the  fifth nearest neighbor\nmethod,  obtaining that  80\\% of  our  galaxies were  located in  very\nlow-density  environments  ($\\Sigma_5   <$  1  Mpc$^{-2}$),  and  20\\%\n(corresponding to  more than 400  galaxies) covers mostly  the typical\nvalues  measured for  loose ($\\Sigma_5  >$ 1  Mpc$^{-2}$)  and compact\ngalaxy groups  ($\\Sigma_5 \\sim$  10 Mpc$^{-2}$). We  did not  find any\ndifference  between the local  galaxy density  of barred  and unbarred\ngalaxies in our  range of densities.  \n\nTo  extend this  conclusion to  higher density  environments,  we have\ninvestigated the fraction of barred galaxies present in the two nearby\n\\emph{benchmark}  clusters: Virgo  (Zarattini et  al., in  prep.)  and\nComa (M\\'endez-Abreu et al., in  prep.).  In both clusters, we found a\nbar fraction in their bright galaxy population consistent with that in\nthe  field. Similar  results have  been recently  found in  cluster at\nhigher   redshift  by  Marinova   et  al.    (2009)  and   Barazza  et\nal.  (2009). However,  the errors  in the  clusters bar  fractions are\nlarge  and  the  results  might   be  affected  by  the  small  number\nstatistics. To deal with this  problem, we are undergoing an ambitious\nproject  to  study the  bar  fraction in  a  large  sample of  cluster\ngalaxies drawn from the WINGS project.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nModels of complex scalar matter fields coupled to gauge fields have\nbeen much studied in condensed matter physics, since they are believed\nto describe several interesting systems, such as superconductors and\nsuperfluids, quantum Hall states, quantum SU($N$) antiferromagnets,\nunconventional quantum phase transitions, etc., see, {\\em e.g.},\nRefs.~\\cite{SSNHS-03,SSS-04,BSA-04,MV-04,SBSVF-04,KHI-11} and\nreferences therein.  Scalar electrodynamics, or Abelian-Higgs (AH)\nmodel, is a paradigmatic model, in which a $N$-component complex\nscalar field ${\\bm \\Phi}$ is minimally coupled to the electromagnetic\nfield $A_\\mu$. The corresponding continuum Lagrangian reads\n\\begin{equation}\n{\\cal L} = \n|D_\\mu{\\bm\\Phi}|^2  + r\\, |{\\bm \\Phi}|^2 + \n{1\\over 6} u \\,(|{\\bm \\Phi}|^2)^2\n+ {1\\over 4 g^2} \\,F_{\\mu\\nu}^2 \n\\,,\n\\label{abhim}\n\\end{equation}\nwhere $F_{\\mu\\nu}\\equiv \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu$, and\n$D_\\mu \\equiv \\partial_\\mu + i A_\\mu$.  The renormalization-group (RG)\nanalysis of the continuum AH model~\\cite{HLM-74,MZ-03} should provide\ninformation on the nature of the finite-temperature phase transitions\noccurring in $d$-dimensional systems characterized by a global U($N$)\nsymmetry and a local U(1) gauge symmetry.\n\nIn this paper we consider the multicomponent AH model, in which the\nscalar field ${\\bm\\Phi}$ has $N\\ge 2$ components. Such a model has a\nlocal U(1) gauge invariance and a global U($N$) invariance. We assume\nthat the field belongs to the fundamental representation of the U(1)\ngroup, i.e., it has charge 1. The one-component AH model has been\nextensively discussed in the literature\n\\cite{KKLP-98,KNS-02,MHS-02,WBJSS-05,CIS-06}. In three dimensions,\nthese systems may undergo continuous transitions in the XY\nuniversality class.\n\nLattice formulations of the three-dimensional AH model are obtained by\nassociating complex $N$-component unit vectors ${\\bm z}_{\\bm x}$ with\nthe sites ${\\bm x}$ of a cubic lattice, and U(1) variables\n$\\lambda_{{\\bm x},\\mu}$ with each link connecting the site ${\\bm x}$\nwith ${\\bm x}+\\hat\\mu$ (where $\\hat\\mu=\\hat{1},\\hat{2},\\ldots$ are\nunit vectors along the lattice directions).  The partition function of\nthe system reads\n\\begin{equation}\nZ = \\sum_{\\{{\\bm z}\\},\\{\\lambda\\}} e^{-H}\\,,\n\\label{partfun}\n\\end{equation}\nwhere the Hamiltonian is~\\cite{footnoteH}\n\\begin{eqnarray}\nH &=& - \\beta N\n\\sum_{{\\bm x}, \\mu}\n\\left( \\bar{\\bm{z}}_{\\bm x} \n\\cdot \\lambda_{{\\bm x},\\mu}\\, {\\bm z}_{{\\bm x}+\\hat\\mu} \n+ {\\rm c.c.}\\right)\n\\label{gllf}\\\\\n&-& \\beta_g \\sum_{{\\bm x},\\mu\\neq\\nu} \n\\left(\n\\lambda_{{\\bm x},{\\mu}} \\,\\lambda_{{\\bm x}+\\hat{\\mu},{\\nu}} \n\\,\\bar{\\lambda}_{{\\bm x}+\\hat{\\nu},{\\mu}}  \n  \\,\\bar{\\lambda}_{{\\bm x},{\\nu}} \n+ {\\rm c.c.}\\right)\\,,\n\\nonumber\n\\end{eqnarray}\nwhere the first sum is over all links of the lattice, while the second\none is over all plaquettes.  \n\nFor $\\beta_g=0$ we recover a particular lattice formulation of the\nCP$^{N-1}$ model, which is quadratic with respect to the spin\nvariables and contains explicit gauge link variables. The CP$^{N-1}$\nmodel has been extensively studied.  In spite of several\nfield-theoretical and numerical studies for $N=2,3,4$ and\n$N\\to\\infty$, there are still some controversies on the nature of its\ntransition \\cite{MZ-03,KHI-11,NCSOS-11,NCSOS-13,DPV-15,PTV-17,PV-19}.\nFor $\\beta_g\\to\\infty$ the gauge link variables are all equal to 1\nmodulo gauge transformations and the AH model becomes equivalent to\nthe standard O($n$) vector model with $n=2N$, whose critical behavior\nis well understood \\cite{PV-02}.  We also mention that some numerical\nresults for the AH lattice model (\\ref{gllf}) have been reported in\nRefs.~\\cite{KHI-11,ODHIM-07,TIM-05}, but a definite picture has not\nbeen achieved yet~\\cite{footnoteold}.\n\nIt is important to stress that we consider the lattice compact version\nof electrodynamics (the so-called Wilson lattice formulation of gauge\ntheories).  In the absence of matter fields, its behavior \\cite{QED}\nis controlled by topological excitations, the monopoles, which are\ninstead suppressed in noncompact formulations. Therefore, the critical\nproperties of the AH lattice model that we consider might differ from\nthose of the model in which gauge fields are noncompact.\n\nIn this paper we investigate the phase diagram and the nature of the\nphase transitions of the three-dimensional AH model (\\ref{gllf}).  We\nconsider systems with $N=2$ and $N=4$, and investigate the nature of\nthe transition line by varying $\\beta$ at fixed gauge coupling\n$\\beta_g$, for some values of $\\beta_g$.  In both cases the phase\ndiagram of the AH lattice model (\\ref{gllf}) turns out to present two\nphases: for small $\\beta$ there is a disordered confined phase, while\nfor large values of $\\beta$ there is an ordered Higgs phase in which\ncorrelations of the gauge-invariant hermitian operator $Q_{\\bm x}^{ab}\n= \\bar{z}_{\\bm x}^a {z}_{\\bm x}^b - \\delta^{ab}\/N$ show long-range\norder.  In both phases, and also along the transition line, the\ncorrelations of gauge variables do not show critical behaviors. The\ngauge coupling $\\beta_g$ does not play any significant role: the\nfeatures of two phases are the same for any finite $\\beta_g$.  The two\nphases are separated by a single transition line, which connects the\nCP$^{N-1}$ transition point ($\\beta_g = 0$) to the O($2N$) transition\npoint ($\\beta_g = \\infty$) in the space of the two parameters $\\beta$\nand $\\beta_g$.  For $\\beta_g=0$ the transition is continuous for $N=2$\n(belonging to the Heisenberg universality class) and of first order\nfor $N=4$.  We conjecture that the nature of the transitions along the\nline separating the Higgs and confined phases does not change with\n$\\beta_g$.  Therefore, the transition is always continuous\n(discontinuous) for $N=2$ ($N=4$).  We also observe significant\ndeviations for $\\beta_g$ large ($\\beta_g\\gtrsim 1$), i.e., when gauge\nfluctuations are suppressed. They are interpreted as a crossover\nphenomenon due to the presence of a O($2N$) vector transition in the\nlimit $\\beta_g\\to\\infty$.\n\nThe paper is organized as follows. In Sec.~\\ref{FTanalysis} we review\nthe field-theoretical results for the AH model. In Sec.~\\ref{rgeps} we\nreview the $\\varepsilon$-expansion predictions obtained in the\ncontinuum AH model, and in Sec.~\\ref{GLW} we present instead the\nresults of the Landau-Ginzburg-Wilson (LGW) approach based on a\ngauge-invariant order parameter. The two approaches are critically\ncompared in Sec.~\\ref{compFT}.  The numerical results are presented in\nSec.~\\ref{numres}.  The definitions of the quantities we consider are\ngiven in Sec.~\\ref{nusim}, while Sec.~\\ref{N2res} and \\ref{N4res}\npresent our results for $N=2$ and 4, respectively, focusing on the\nbehavior of the gauge-invariant order parameter.  Results for vector\nand gauge observables are presented in Sec.~\\ref{Vector}.  In\nSec.~\\ref{Conclusions} we summarize and present our conclusions.  In\nApp.~\\ref{FSSON} we discuss the limit $\\beta_g\\to\\infty$. More details\non the behavior of the different observables in this limit are given\nin the supplementary material \\cite{suppl-mat}.\n\n\\section{Field theoretical approaches}\n\\label{FTanalysis}\n\nIn this section we outline some apparently alternative\nfield-theoretical approaches which can be employed to infer the nature\nof the phase transitions in systems characterized by a U($N$) global\nsymmetry and a local U(1) gauge symmetry, such as the AH lattice\nmodel.\n\n\\subsection{Renormalization-group flow \nin the AH model close to four dimensions}\n\\label{rgeps}\n\nWe now summarize the main features of the RG flow in the continuum AH\nmodel (\\ref{abhim}), which has been analyzed close to four dimensions\nin the $\\varepsilon\\equiv 4-d$ expansion\nframework~\\cite{HLM-74,FH-96,IZMHS-19}, and in the large-$N$\nlimit~\\cite{MZ-03}.\n\nClose to four dimensions, the RG flow in the space of the renormalized\ncouplings $u$ and $f\\equiv g^2$ (we rescale them as $u\\to u\/(24\\pi^2)$\nand $f\\to f\/(24\\pi^2)$ to simplify the equations) can be computed in\nperturbation theory. At one loop, the $\\beta$ functions\nread~\\cite{HLM-74}\n\\begin{eqnarray}\n&&\\beta_u \\equiv \\mu {\\partial u\\over \\partial \\mu}\n= - \\varepsilon u + \n(N+4) u^2 - 18 u f + 54 f^2\\,,\n\\nonumber \\\\\n&&\\beta_f \\equiv \\mu {\\partial f \\over \\partial \\mu}\n= - \\varepsilon f + N f^2\\,.\n\\label{betafunc}\n\\end{eqnarray} \nOne can easily verify that a stable fixed point exists only for $N >\nN_c(\\varepsilon)$,  with \n\\begin{equation}\nN_c(\\varepsilon) =  N_{4}\n+ O(\\varepsilon)\\,,\\quad N_{4} = 90 +\n24\\sqrt{15} \\approx 183\\,.\n\\label{nceps}\n\\end{equation}\nThe corresponding zero of the $\\beta$ functions is \n\\begin{eqnarray}\n&&f^* = {\\varepsilon \\over N}\\,,\\label{fstar}\\\\\n&&u^*  = \n{N+18 + \\sqrt{N^2-180N-540}\\over 2 N(N+4)}\\,\\varepsilon\n\\approx {\\varepsilon\\over N}\\,.\n\\label{ufstar}\n\\end{eqnarray}\nThe presence of a stable fixed point indicates that these systems may\nundergo a continuous transition if $N$ is large enough [$N>N_c(1)$ in\n  three dimensions], in agreement with the direct large-$N$\nanalysis~\\cite{MZ-03}.  The qualitative picture obtained in the\none-loop calculation is not changed by higher-order calculations.  The\nperturbative expansion has been recently extended to four loops\n\\cite{IZMHS-19}, obtaining $N_c(\\varepsilon)$ to $O(\\varepsilon^3)$,\n\\begin{equation}\nN_c(\\varepsilon) = N_{4}\\left[1 - 1.752 \\,\\varepsilon + 0.789\\,\n\\varepsilon^2 + 0.362 \\,\\varepsilon^3+ O(\\varepsilon^4)\\right] .\n\\end{equation}\nThe large coefficients make a reliable three-dimensional\n($\\varepsilon=1$) estimate quite problematic.  Nevertheless, by means\nof a resummation of the expansion, Ref.~\\cite{IZMHS-19} obtained the\nestimate $N_c = 12.2(3.9)$ in three dimensions, which confirms the\nabsence of a stable fixed point for small values of $N$.\n\nIn the limit $\\beta_g \\to\\infty$, the lattice AH model (\\ref{gllf}) is\nequivalent to the symmetric O$(2N)$ vector theory. Therefore, for\nlarge $\\beta_g$ one expects significant crossover effects, which\nincrease as $\\beta_g$ increases, due to the nearby O$(2N)$ critical\nbehavior.  In the continuum AH model, the crossover is controlled by\nthe RG flow in the vicinity of the O$(2N)$ fixed point\n\\begin{equation}\nu^*_{{\\rm O}(2N)}= {1\\over N+4}  \\varepsilon\\,,\\qquad\nf=0\\,.\n\\label{O2nfp}\n\\end{equation}\nThis fixed point exists for any $N$ and is always unstable.  The\nanalysis of the stability matrix $\\Omega_{ij} = \\partial\n\\beta_i\/\\partial g_j$ shows that it has a positive eigenvalue\n$\\lambda_u=\\omega$, where $\\omega>0$ is the exponent controlling the\nleading scaling corrections in O$(2N)$ vector models~\\cite{PV-02}, and\na negative eigenvalue, which gives the dimension of the operator that\ncontrols the crossover behavior,\n\\begin{equation}\n\\lambda_f = \\left. {\\partial \\beta_f \\over \\partial f} \\right|_{f=0,u=u^*}\n \\; .\n\\end{equation}\nSince the $\\beta$-function $\\beta_f(u,f)$ \nassociated with $f$ has the general form\n\\begin{equation}\n\\beta_f = -\\varepsilon f + f^2 F(u,f)\n\\label{betafform}\n\\end{equation}\n where $F(u,f)$ has a regular perturbative expansion (see, e.g., the\n four-loop expansion reported in Ref.~\\cite{IZMHS-19}), we obtain\n\\begin{equation}\n\\lambda_f = -\\varepsilon\n\\label{lambdafest}\n\\end{equation}\nto all orders in perturbation theory. Therefore, the crossover exponent \n$y_f=-\\lambda_f$ is 1 in\nthree dimensions.  Note that these crossover features related to the\nunstable O($2N$) fixed point are independent of the existence of the\nstable fixed point, which is only relevant to predict the eventual\nasymptotic behavior.\n\n\\subsection{Gauge-invariant Landau-Ginzburg-Wilson framework}\n\\label{GLW}\n\nAn alternative field-theoretical approach is the LGW\nframework~\\cite{Landau-book,WK-74,Fisher-75,Ma-book,PV-02,PV-19}, in\nwhich one assumes that the relevant critical modes are associated with\nthe gauge-invariant local composite site variable\n\\begin{equation}\nQ_{{\\bm x}}^{ab} = \\bar{z}_{\\bm x}^a z_{\\bm x}^b - {1\\over N}\n\\delta^{ab},\n\\label{qdef}\n\\end{equation}\nwhich is a hermitian and traceless $N\\times N$ matrix.  As discussed\nin Refs.~\\cite{PTV-17,PTV-18,PV-19}, this is a highly nontrivial\nassumption, as it postulates that gauge\nfields do not play an important role in the effective theory.  \nThe order-parameter field in\nthe corresponding LGW theory is therefore a traceless hermitian matrix\nfield $\\Psi^{ab}({\\bm x})$, which can be formally defined as the\naverage of $Q_{\\bm x}^{ab}$ over a large but finite lattice domain.\nThe LGW field theory is obtained by considering the most general\nfourth-order polynomial in $\\Psi$ consistent with the U($N$) global\nsymmetry:\n\\begin{eqnarray}\n{\\cal H}_{\\rm LGW} &=& {\\rm Tr} (\\partial_\\mu \\Psi)^2 \n+ r \\,{\\rm Tr} \\,\\Psi^2 \\label{hlg}\\\\\n&+&   w \\,{\\rm tr} \\,\\Psi^3 \n+  \\,u\\, ({\\rm Tr} \\,\\Psi^2)^2  + v\\, {\\rm Tr}\\, \\Psi^4 .\n\\nonumber\n\\end{eqnarray}\nAlso in this framework continuous transitions may only arise if the RG\nflow in the LGW theory has a stable fixed point.\n\nFor $N=2$, the cubic term in Eq.~(\\ref{hlg}) vanishes and the two\nquartic terms are equivalent.  Therefore, one recovers the\nO(3)-symmetric vector LGW theory, leading to the prediction that the\nphase transition may be continuous and, in this case, that it belongs to\nthe Heisenberg universality class. For $N\\ge 3$, the cubic term is\ngenerally expected to be present.  Its presence is usually taken as an\nindication that phase transitions occurring in this class of systems\nare generally of first order.  Indeed, a straightforward mean-field\nanalysis shows that the transition is of first order in four\ndimensions where mean field applies. If statistical fluctuations are\nsmall---this is the basic assumption---the transition should be of\nfirst order also in three dimensions.  In this scenario, continuous\ntransitions may still occur, but they require a fine tuning of the\nmicroscopic parameters leading to the effective cancellation of the\ncubic term.  These arguments were originally \\cite{DPV-15,PV-19}\napplied to predict the behavior of CP$^{N-1}$ models. However, as they\nare only based on symmetry considerations, they can be extended to AH\nlattice models, as well.\n\n\\subsection{Comparison of the alternative field-theoretical approaches}\n\\label{compFT}\n\nThe two field-theoretical approaches outlined above give inconsistent\npredictions both for small and large values of $N$.  The contradiction\nis quite striking for the two-component $N=2$ case.  For this value of\n$N$, the continuum AH model predicts the absence of continuous\ntransitions, due to the absence of a stable fixed point. On the other\nhand, a stable fixed point---it is the usual Heisenberg O(3) fixed\npoint---exists in the effective LGW theory based on a gauge-invariant\norder parameter, leaving open the possibility of observing continuous\ntransitions (first-order transitions are never excluded as the\nstatistical model may be outside the attraction domain of the fixed\npoint). The numerical results for the CP$^1$ lattice\nmodels~\\cite{NCSOS-11,PV-19}, as well as the AH lattice results we\nshall present below, confirm the existence of continuous transitions\nin models with $N=2$: the LGW theory provides the correct description\nof the large-scale behavior of these systems. There are at least two\npossible explanations for the failure of the continuum AH model. A\nfirst possibility is that it does not encode the relevant degrees of\nfreedom at the transition. A second possibility is that the problem is\nnot in the continuum AH model, but rather in the perturbative\ntreatment around four dimensions. The three-dimensional fixed point\nmay not be analytically related to a four-dimensional fixed point, and\ntherefore it escapes any perturbative analysis in powers of\n$\\varepsilon$.\n\nWe also recall that the perturbative AH approach of Sec.~\\ref{rgeps}\nalso fails for $N=1$. Although no stable fixed point is identified in\nthe $\\varepsilon$ expansion, see Sec.~\\ref{rgeps}, these models may\nundergo continuous transitions in the XY universality class\n\\cite{KKLP-98,KNS-02,MHS-02}.  It is worth mentioning that there are\nalso other systems in which the $\\varepsilon$ expansion fails to\nprovide the correct physical picture in three dimensions. We mention\nthe $\\phi^4$ theories describing frustrated spin models with\nnoncollinear order~\\cite{CPPV-04,NO-14} and the $^3$He superfluid\ntransition from the normal to the planar phase~\\cite{DPV-04}.\n\nFor large values of $N$, the continuum AH theory and the effective LGW\napproach give again contradictory results. Indeed, the former approach\nindicates that continuous transitions are possible, a prediction which\nis supported by the large-$N$ analysis of lattice models, see, e.g.,\nRef.~\\cite{PV-19}.  If one trusts the argument based on the relevance\nof the cubic term, the LGW approach predicts instead a first-order\ntransition, unless a fine tuning of the microscopic parameters is\nperformed.  Again, there are two possible explanations for the\ndifferent conclusions obtained in the LGW approach.  A first\npossibility is that the critical modes at the transition are not\nexclusively associated with the gauge-invariant order parameter $Q$\ndefined in Eq.~(\\ref{qdef}). Other features, for instance the gauge\ndegrees of freedom, may become relevant, requiring an effective\ndescription different from that of the LGW theory (\\ref{hlg}). If this\ninterpretation is correct, the continuum AH model would be the correct\ntheory as it includes the gauge fields explicitly.  A second\npossibility is that the presence of a cubic term in the LGW\nHamiltonian does not necessarily imply the absence of continuous\ntransitions in three dimensions, as it is usually assumed. It might be\nthat statistical fluctuations soften the transition as one moves from\nfour to three dimensions; see, e.g., Refs.~\\cite{NCSOS-11,NCSOS-13} for\na discussion of this issue.\n\nWhile the two field-theoretical approaches give different predictions\nfor $N=1,2$ and $N$ large (more precisely, for $N>N_c$, see\nSec.~\\ref{rgeps}), for $3 \\le N < N_c$ they both predict that all\nmodels undergo a first-order transition. For $N=3$ simulation results\ndo not presently confirm it.  Indeed, while numerical results for the\nlattice model with Hamiltonian (\\ref{gllf}) and $\\beta_g = 0$ show a\nrobust indication that the transition is of first order~\\cite{PV-19},\nthe results for the loop model considered in\nRefs.~\\cite{NCSOS-11,NCSOS-13} apparently favor a continuous\ntransition.  The available numerical results for lattice CP$^3$\nmodels, i.e., for $N=4$, are generally consistent with first-order\ntransitions~\\cite{KHI-11,NCSOS-13,PV-19}.  We also mention that\nRef.~\\cite{KHI-11} claims that the AH lattice model (\\ref{gllf})\nundergoes a continuous transition for $\\beta_g=1$ and $N=4$, a result\nwhich is at odds with the above arguments. However, as we shall show,\nthe numerical results that we present later do not confirm their\nconclusions, but are instead consistent with a relatively weak\nfirst-order transition.\n\n\n\\section{Numerical results}\n\\label{numres}\n\n\\subsection{Numerical simulations and observables}\n\\label{nusim}\n\n\\begin{figure}[tbp]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{betac.eps}\n\\caption{ The phase diagram of the AH lattice model in the space of\n  parameters $\\beta$ and $\\beta_g$, for $N=2$ and $N=4$. The points\n  are the MC estimates of the critical points, the dotted lines that\n  connect them are only meant to guide the eye.  The horizontal lines\n  indicate the limiting values of $\\beta_c$ for $\\beta_g\\to \\infty$\n  for $N=2$ ($\\beta_c\\approx 0.23396$, dashed) and $N=4$\n  ($\\beta_c\\approx 0.24084$, dot-dashed): they correspond to the\n  critical $\\beta_c$ for the standard three-dimensional O$(4)$ and\n  O$(8)$ vector models, respectively. For $\\beta_g = 0$, we have \\cite{PV-19}\n  $\\beta_c = 0.7102(1)$ ($N=2$) and $\\beta_c = 0.5636(1)$ ($N=4$).\n}\n\\label{betac}\n\\end{figure}\n\n\nIn this section we present a finite-size scaling (FSS) analysis of\nnumerical results of Monte Carlo (MC) simulations for $N=2$ and $N=4$.\nFor this purpose we consider cubic lattices of linear size $L$ with\nperiodic boundary conditions. We study the behavior of the system as a\nfunction of $\\beta$ at fixed $\\beta_g$.\n\nThe linearity of Hamiltonian (\\ref{gllf}) with respect to each lattice\nvariable allows us to employ an overrelaxed algorithm for the updating\nof the lattice configurations.  It consists in a stochastic mixing of\nmicrocanonical and standard Metropolis updates of the lattice\nvariables~\\cite{CRV-92,DMV-04,Hasenbusch-17}.  To update each lattice\nvariable, we randomly choose either a standard Metropolis update,\nwhich ensures ergodicity, or a microcanonical move, which is more\nefficient than the Metropolis one but does not change the energy. On\naverage, we perform three\/four microcanonical updates for every\nMetropolis proposal.  In the Metropolis update, changes are tuned so\nthat the acceptance is 1\/3.\n\nWe compute the energy density and the specific heat, defined as\n\\begin{eqnarray}\nE = {1\\over N V} \\langle H \\rangle,\\qquad\nC ={1\\over N^2 V}\n\\left( \\langle H^2 \\rangle \n- \\langle H  \\rangle^2\\right),\n\\label{ecvdef}\n\\end{eqnarray}\nwhere $V=L^3$.  We consider correlations of the hermitean gauge\ninvariant operator (\\ref{qdef}). Its two-point correlation function is\ndefined as\n\\begin{equation}\nG({\\bm x}-{\\bm y}) = \\langle {\\rm Tr}\\, Q_{\\bm x} Q_{\\bm y} \\rangle,  \n\\label{gxyp}\n\\end{equation}\nwhere the translation invariance of the system has been taken into\naccount.  The susceptibility and the correlation length are defined as\n$\\chi=\\sum_{\\bm x} G({\\bm x})$ and\n\\begin{eqnarray}\n\\xi^2 \\equiv  {1\\over 4 \\sin^2 (\\pi\/L)}\n{\\widetilde{G}({\\bm 0}) - \\widetilde{G}({\\bm p}_m)\\over \n\\widetilde{G}({\\bm p}_m)},\n\\label{xidefpb}\n\\end{eqnarray}\nwhere $\\widetilde{G}({\\bm p})=\\sum_{{\\bm x}} e^{i{\\bm p}\\cdot {\\bm x}}\nG({\\bm x})$ is the Fourier transform of $G({\\bm x})$, and ${\\bm p}_m =\n(2\\pi\/L,0,0)$ is the minimum nonzero lattice momentum.  We also\nconsider the Binder parameter\n\\begin{equation}\nU = {\\langle \\mu_2^2\\rangle \\over \\langle \\mu_2 \\rangle^2} \\,, \\qquad\n\\mu_2 = \n\\sum_{{\\bm x},{\\bm y}} {\\rm Tr}\\,Q_{\\bm x} Q_{\\bm y}\\,.\n\\label{binderdef}\n\\end{equation}\nWe consider correlations of the fundamental variable ${\\bm z}_x$.\nTo obtain a gauge-invariant quantity, we consider correlations with\n$\\lambda$ strings, i.e., averages like\n\\begin{equation}\n \\hbox{Re } \\left\\langle  \n  \\bar{\\bm z}_{\\bm x}\\cdot {\\bm z}_{\\bm y} \\prod_{\\ell \\in \\cal C} \\lambda_\\ell\n   \\right\\rangle,\n\\end{equation}\nwhere the product extends over the link variables that belong to a\nlattice path $\\cal C$ connecting points $\\bm x$ and $\\bm y$. To\ndefine quantities that have the correct FSS, the path ${\\cal C}$ must\nbe chosen appropriately, as discussed in Ref.~\\cite{APV-08}. Here, to\nsimplify the calculations, we only consider correlations between\npoints that belong to lattice straight lines.  We define\n\\begin{eqnarray}\nG_V(d,L) ={1\\over V} \n   \\sum_{{\\bm x}} \n  \\hbox{Re}\\left\\langle\n  \\bar{\\bm z}_{\\bm x}\\cdot {\\bm z}_{{\\bm x}+d \\hat{\\mu}}  \n   \\prod_{n=0}^{d-1} \\lambda_{{\\bm x}+n \\hat{\\mu},\\mu}\n   \\right\\rangle,\n   \\label{Gd}\n\\end{eqnarray}\nwhere all coordinates should be taken modulo $L$ because of the\nperiodic boundary conditions.  Note that in the definition of $G_V$ we\naverage over all lattice sites ${\\bm x}$ exploiting the translation\ninvariance of systems with periodic boundary conditions, and select a generic\nlattice direction $\\hat{\\mu}$ (in our MC simulations we also average\nover the three equivalent directions).  Note also that $G_V(0,L) = 1$\nand that $G_V(L,L)$ is the average value $P(L)$ of the Polyakov loop,\n\\begin{equation}\n   P(L) = {1\\over V} \\sum_{\\bm x} \n   \\hbox{Re } \\left\\langle \\prod_{n=0}^{L-1} \\lambda_{{\\bm x}+n \\hat{\\mu},\\mu}\n   \\right\\rangle\\,.\n   \\label{Polyakov}\n\\end{equation}\nFinally, we consider the so-called Wilson loop\ndefined as\n\\begin{equation}\n    W(m,L) = \\hbox{Re } \\left\\langle \\prod_{\\ell\\in \\cal C}\n    \\lambda_\\ell \\right\\rangle, \n    \\label{Wilson}\n\\end{equation}\nwhere the path $\\cal C$ is a square of linear size $m$.\n\nIn the following we present a FSS analysis of the above observables,\nfor $N=2$ and $N=4$ and some values of $\\beta_g>0$. In\nFig.~\\ref{betac} we anticipate the resulting phase diagrams.  For both\n$N=2$ and $4$, $\\beta_c$ decreases as $\\beta_g$ increases and,\neventually, it converges to the value appropriate for the $n$-vector\nmodel with $n=4$ and 8, $\\beta_c=0.233965(2)$~\\cite{BFMM-96,CPRV-96}\nand $\\beta_c = 0.24084(1)$~\\cite{DPV-15}.\n\nAs we shall discuss, our numerical data are consistent with a simple\nscenario in which the nature of the transitions along the line\nseparating the confined and Higgs phases is unchanged for any finite\n$\\beta_g\\ge 0$.  Therefore, for $N=2$ the phase transitions are continuous and\nbelong to the Heisenberg universality class as it occurs in the CP$^1$\nmodel.  The O(4) critical behavior occurs only for $\\beta_g$ strictly\nequal to $\\infty$. For $N=4$ instead, transitions are of first order,\nexcept for $\\beta_g=\\infty$, where the system develops an O(8)\nvector critical behavior.\n\n\n\n\\subsection{Continuous transitions for $N=2$}\n\\label{N2res}\n\n\nAs already mentioned, lattice versions of the three-dimensional CP$^1$\nmodel undergo continuous transitions belonging to the Heisenberg\nuniversality class, i.e., that of the standard $N=3$ vector model.\nThis has been also shown~\\cite{PV-19} for model\n(\\ref{gllf}) with $\\beta_g=0$ [$\\beta_c = 0.7102(1)$ in this case].\nOn the other hand, for $\\beta_g=\\infty$ the model is equivalent \nto the standard O(4) vector model that has a continuous transition for \n\\cite{BFMM-96,CPRV-96} $\\beta_c=0.233965(2)$. \nAs already inferred\nfrom the RG flow of the AH continuum theory, the $\\beta_g=\\infty$ O(4)\ncritical behavior is expected to be unstable against perturbations\nassociated with nonzero values of $\\beta_g^{-1}$. Therefore, the most\nnatural hypothesis is that the all transitions for finite $\\beta_g\\ge\n0$ belong to the Heisenberg universality class.  However, a\nsubstantial crossover from the O(4) to the O(3) behavior is expected\nto characterize the transition for relatively large values of\n$\\beta_g$, $\\beta_g\\gtrsim 1$ say.\n\n\\begin{figure}[tbp]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{rxi-n2-c1.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{rxi-n2-c1o2.eps}\n\\caption{ $R_\\xi$ versus $\\beta$ for the $N=2$ AH lattice model, for\n  $\\beta_g=0.5$ (bottom) and $\\beta_g=1$ (top).  In both cases the\n  data for different values of $L$ show a crossing point, whose\n  position provides an estimate of the critical point:\n  $\\beta_c=0.4145(5)$ and $\\beta_c=0.276(1)$ for $\\beta_g=0.5$ and\n  $\\beta_g=1$, respectively.  }\n\\label{rxi-n2}\n\\end{figure}\n\nTo provide evidence of this scenario, we perform MC simulations for\n$\\beta_g=0.5$ and 1. As in our previous work \\cite{PV-19}, we study\nthe FSS behavior of the Binder parameter $U$ and of $R_\\xi = \\xi\/L$.\nAt continuous transitions the FSS limit is obtained by taking\n$\\beta\\to \\beta_c$ and $L\\to\\infty$ keeping \n\\begin{equation}\nX \\equiv (\\beta-\\beta_c)L^{1\/\\nu}\n\\label{Xdef}\n\\end{equation}\nfixed. Any RG invariant quantity $R$, such\nas $R_\\xi\\equiv \\xi\/L$ and $U$, is expected to asymptotically behave\nas\n\\begin{eqnarray}\nR(\\beta,L) = f_R(X) + O(L^{-\\omega})\\,,\n\\label{rsca}\n\\end{eqnarray}\nwhere $\\omega>0$ is the leading scaling correction\nexponent~\\cite{PV-02}, and $f_R(X)$ is universal apart from a\nnormalization of its argument.  The function $f_R(X)$ only depends on\nthe shape of the lattice and on the boundary conditions.  In the case\nof the Heisenberg universality we\nhave~\\cite{GZ-98,PV-02,CHPRV-02,HV-11} $\\nu=0.7117(5)$ and\n$\\omega=0.78(1)$.  As $R_\\xi$ is monotonically increasing as a\nfunction of $X$, Eq.~(\\ref{rsca}) implies that\n\\begin{equation}\nU = F(R_\\xi) + O(L^{-\\omega})\\,,\n\\label{r12sca}\n\\end{equation}\nwhere $F(x)$ is a universal scaling function.  As in our previous work\n\\cite{PV-19}, we will use Eq.~(\\ref{r12sca}) to perform a direct check\nof universality, because no model-dependent normalizations enter: If\ntwo models belong to the same universality class, the data for both of\nthem should collapse onto the same curve as $L$ increases.  The only\ndifficulty in the approach is that one should be careful in\nidentifying corresponding operators in the two models.\n\nTo identify the correct operators, one may reason as follows. In the AH\nlattice model the basic quantity that we consider is the local\noperator (\\ref{qdef}).  To identify the corresponding operator in the\nHeisenberg model, we use the explicit relation between the CP$^1$ and\nthe O(3) vector model. Under the mapping, the parameter $U$ and $\\xi$\ncorrespond to the usual O(3) vector Binder parameter and correlation\nlength (i.e., computed from correlations of the fundamental spin\nvariable ${\\bm s}_{\\bm x}$). The mapping of the large-$\\beta_g$ limit\nof the AH lattice model into the O(4) vector model is instead more\ncomplex and is discussed in detail in App.~\\ref{FSSON} and in the\nsupplementary material \\cite{suppl-mat}. The correspondence is not\ntrivial and $U$ is identified with a combination of suitably defined\nO(4) tensor Binder parameters.\n\n\\begin{figure}[H]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{bi-rxi-n2-c0.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{bi-rxi-n2-c1o2.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{bi-rxi-n2-c1.eps}\n\\caption{ The Binder parameter $U$ versus $R_\\xi$ for the $N=2$ AH\n  lattice model, and for $\\beta_g=0$ (top, data from\n  Ref.~\\cite{PV-19}), $\\beta_g=0.5$ (middle) and $\\beta_g=1$ (bottom).\n  In all panels the dashed line is the Heisenberg curve, as obtained\n  from MC simulations of the O(3) vector model.  The dot-dashed line\n  in the lower panel is the limiting curve for\n  $\\beta_g\\to\\infty$ (see App.~\\ref{FSSON}).  The inset enlarges the\n  region $R_\\xi<0.25$, showing that the nonmotonic behavior that\n  characterizes the small-size data (similar to the one of the O(4)\n  curve) for $R_\\xi\\approx 0.1$ disappears with increasing $L$.  The\n  horizontal dashed line shows the asymptotic value $U(R_\\xi\\to 0) =\n  5\/3$.  }\n  \\label{bi-rxi-n2}\n\\end{figure}\n\n\nIn Fig.~\\ref{rxi-n2} we plot $R_\\xi$ versus\n$\\beta$, for several values of $L$.  The data for different values of\nthe size $L$ show crossing points, which provide estimates of the\ncritical point: $\\beta_c = 0.4145(5)$ and $\\beta_c=0.276(1)$ for\n$\\beta_g=0.5$ and $\\beta_g=1$, respectively.  Data are consistent with\na continuous transition. \n\nWe now argue that the transitions are consistent with the expected\nasymptotic Heisenberg behavior.  The best evidence is provided by the\nplots of $U$ versus $R_\\xi$, see Fig.~\\ref{bi-rxi-n2}. In all panels\nwe report the data and the corresponding O(3) curve. If our simple\nscenario is correct, the data for all values of $\\beta_g$ must\napproach the O(3) curve with increasing $L$.  For $\\beta_g = 0$ we\nobserve very good agreement, as already discussed in\nRef.~\\cite{PV-19}. For $\\beta_g = 0.5$ convergence is slower,\nindicating that scaling corrections increase with increasing\n$\\beta_g$.  For $R_\\xi \\lesssim 0.25$ we observe a good collapse of\nthe data, while in the opposite case, we observe a clear upward trend,\nconsistent with an asymptotic O(3) behavior.  For $\\beta_g = 1$, for\nsmall values of $L$ we observe significant differences between data\nand O(3) curve.  These discrepancies can be explained by scaling\ncorrections. For $R_\\xi\\lesssim 0.25$, the results for $L=64$ fall on\ntop of the O(3) scaling curve, as predicted. For larger values of\n$R_\\xi$, crossover effects are stronger, but the trend of the data is\nthe expected one.\n\nTo be more quantitative, let us note that, for large values of $L$,\nthe Binder parameter $U$ should behave as~\\cite{PV-02}\n\\begin{equation}\nU(\\beta,\\beta_g,L) = F(R_\\xi) + a(\\beta_g)\\ G(R_\\xi)\\ L^{-\\omega} + \\ldots\n\\label{corrections}\n\\end{equation}\nwhere $F(R_\\xi)$ is the O(3) scaling function, $G(R_\\xi)$ is a\nuniversal function, and $a(\\beta_g)$ is a constant that encodes the\n$\\beta_g$-dependent size of the leading scaling corrections decaying\nas $L^{-\\omega}$.  We have verified that our data for $\\beta_g = 0.5$\nand 1 are consistent with Eq.~(\\ref{corrections}), if we take $\\omega\n= 0.78$ (the leading correction-to-scaling exponent in Heisenberg\nsystems \\cite{GZ-98,PV-02,CHPRV-02,HV-11}) and $a(1)\/a(0.5) \\approx\n5$.  This can be checked from Fig.~\\ref{differenze}, where we report\n\\begin{equation}\n\\Delta(\\beta,\\beta_g,L) = \n    {1\\over a(\\beta_g)} L^\\omega [U(\\beta,\\beta_g,L) - F(R_\\xi)]\\,,\n\\label{deltadef}    \n\\end{equation}\nwhere $F(R_\\xi)$ has been determined in the O(3) vector model, $\\omega\n= 0.78$, $a(1) = 5$, and $a(0.5) = 1$. All data reported in the figure\nare consistent with a single scaling curve that would be identified\nwith the function $G(R_\\xi)$ in Eq.~(\\ref{corrections}).  The\nexistence of similar crossover effects for $\\beta_g = 0.5$ and 1 is\nanother demonstration of universality.\n\n\\begin{figure}[!ht]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{scalcorr-n2.eps}\n\\caption{The quantity $\\Delta(\\beta,\\beta_g,L)$ defined in \nEq.~(\\ref{deltadef}) versus $R_\\xi$. We report data for $\\beta_g=1$ \nand 0.5 and several values of $L$.}\n\\label{differenze}\n\\end{figure}\n\nIt is interesting to note that the behavior of the data for small $L$\nat $\\beta_g=1$ can be interpreted as due to the presence of O(4) fixed\npoint that controls the critical behavior for $\\beta_g\\to\\infty$.  In\nthe lower panel of Fig.~\\ref{bi-rxi-n2}, we also plot the O(4) scaling\ncurve.  The data for $\\beta_g=1$ apparently follow the O(4) curve for\nsmall lattice sizes, and then move toward the O(3) curve with\nincreasing $L$.  In particular, note the nonmotonic behavior of the\ndata for small lattice sizes and $R_\\xi\\approx 0.1$, similar to the\none that characterizes the O(4) curve.  Such a behavior disappears\nwith increasing $L$ (see the inset in the lower panel of\nFig.~\\ref{bi-rxi-n2}).\n\nOn the basis of the above numerical results we argue that the\nfinite-temperature transition is continuous for any finite $\\beta_g\\ge\n0$ and belongs to the Heisenberg universality class.  However, for\nrelatively large values of $\\beta_g$, say $\\beta_g\\gtrsim 1$, notable\ncrossover effects emerge.  They are apparently related to the presence\nof the O(4) fixed point, which is the relevant one $\\beta_g\\to\\infty$.\nFor large values of $\\beta_g$, such effects may hide the asymptotic\nHeisenberg behavior. For intermediate sizes, data are expected to show\nan effective O(4) critical behavior, converging to the Heisenberg\nbehavior only for very large lattices.\n\n\n\n\\subsection{First-order transitions for $N=4$}\n\\label{N4res}\n\n\\begin{figure}[tbp]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{u-n4-c0.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{u-n4-c1o2.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{u-n4-c1.eps}\n\\caption{Plot of the Binder parameter $U$ versus $\\beta$, for\n  $\\beta_g=0$ (top, from Ref.~\\cite{PV-19}), $\\beta_g=1\/2$ (middle),\n  and $\\beta_g=1$ (bottom), for $N=4$.  The vertical lines correspond\n  to the estimates of the transition points.  The horizontal dashed\n  lines show the values $U(\\beta \\to 0) = 17\/15$ and\n  $U(\\beta\\to \\infty) = 1$.  } \n\\label{U-N4}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{bi-rxi-n4-c0.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{bi-rxi-n4-c1o2.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{bi-rxi-n4-c1.eps}\n\\caption{ The Binder parameter $U$ versus $R_\\xi$ versus $\\beta$, for\n  $\\beta_g=0$ (top, from Ref.~\\cite{PV-19}), $\\beta_g=1\/2$ (middle),\n  and $\\beta_g=1$ (bottom) for $N=4$.  The dashed line in the lower\n  panel is the O(8) limiting scaling curve, see App.~\\ref{FSSON}.  The\n  horizontal dashed lines show the asymptotic values $U(R_\\xi\\to 0) =\n  17\/15$ and $U(R_\\xi\\to \\infty) = 1$.  }\n\\label{UvsR-N4}\n\\end{figure}\n\n\nWe now discuss the behavior of the $N=4$ AH lattice model, providing\nevidence that the transitions along the line separating the Higgs and\nconfined phases are of first order for any finite $\\beta_g$. Only when\n$\\beta_g$ is strictly infinity is the transition continuous: it\nbelongs to the O(8) vector universality class.\n\nAs shown in Ref.~\\cite{PV-19}, the transition is of first order for\n$\\beta_g=0$.  To show that the nature of the transition is unchanged\nfor $\\beta_g>0$, we first consider the specific heat and the Binder\nparameter $U$. Both of them are expected to increase as the volume at\na first-order transition.  Indeed, according to the standard\nphenomenological theory~\\cite{CLB-86}, for a lattice of size $L$ there\nexists a value $\\beta_{{\\rm max},C}(L)$ of $\\beta$ where $C$ takes its\nmaximum value $C_{\\rm max}(L)$, which asymptotically increases as\n\\begin{eqnarray}\n&&C_{\\rm max}(L) = V\\left[ {1\\over 4} \\Delta_h^2 + O(1\/V)\\right]\\,,\n\\label{cmaxsc}\\\\\n&&\\beta_{{\\rm max},C}(L)-\\beta_c\\approx c\\,V^{-1}\\,, \\label{betamax}\n\\end{eqnarray}\nwhere $V=L^d$ and $\\Delta_h$ is the latent heat\n[defined as $\\Delta_h = E(\\beta\\to\\beta_c^+) - E(\\beta\\to\\beta_c^-)$].\nAnalogously, the\nbehavior of the Binder parameter $U(\\beta,L)$ is expected to show a\nmaximum $U_{\\rm max}(L)$ at fixed $L$ (for sufficiently large $L$) at\n$\\beta = \\beta_{{\\rm max},U}(L) < \\beta_c$\nwith~\\cite{VRSB-93,CPPV-04,PV-19}\n\\begin{eqnarray}\n&&U_{\\rm max} \\sim a\\,V + O(1)\\,,\\label{umaxsca}\\\\ &&\\beta_{{\\rm\n    max},U}(L) - \\beta_c \\approx b \\,V^{-1}\\,.\n\\label{bpeakU}\n\\end{eqnarray}\nThe previous relations are valid in the asymptotic limit and, for weak\ntransitions, require data on large lattices.  As we discussed in\nRef.~\\cite{PV-19}, one can identify first-order transitions on\nsignificantly smaller lattices from the analysis of the behavior of\nthe Binder parameter $U$. In the presence of a first-order transition,\none observes large violations of the scaling relation (\\ref{r12sca})\nfor values of $L$ that are significantly smaller than those at which\nrelations (\\ref{cmaxsc}) and (\\ref{bpeakU}) hold. We will follow this\napproach here, considering again two values of $\\beta_g$, 0.5 and 1.\n\nIn Fig.~\\ref{U-N4} we report numerical estimates of $U$ at $\\beta_g=0$\n(taken from Ref.~\\cite{PV-19}), $\\beta_g=0.5$, and\n$\\beta_g=1$. Clearly, the maximum $U_{\\rm max}$ increases with\nincreasing $L$, as expected for a first-order transition. However,\nwith increasing $\\beta_g$, the rate of increase becomes smaller,\nindicating that the transition becomes weaker. The specific heat\nbehaves analogously.\n\nTo obtain a better evidence that the finite-size behavior is not\ncompatible with a continuous transition, we plot $U$ versus $R_\\xi$,\nsee Fig.~\\ref{UvsR-N4}. Data do not show any scaling behavior, as\nexpected at a first-order transition.\n\n\\begin{figure}[tbp]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4},angle=0]{bi-rxi-n4-c4.eps}\n\\caption{ The Binder parameter $U$ versus $R_\\xi$ for $\\beta_g = 4$\n  and $N=4$.  The full line connects the data with $L=16$.  The dashed\n  line is the O(8) limiting scaling curve, see App.~\\ref{FSSON}.  }\n\\label{U-N4-c4}\n\\end{figure}\n\n\nNote that for $\\beta_g = 1$ the small size data show an apparent\nscaling behavior for small values of $R_\\xi$ and small $L$, which may\nlead to erroneous conclusions when limiting the FSS analysis to small\nlattices (as in Ref.~\\cite{KHI-11}).  To clarify the origin of the\ntransient effects, and understand whether they can be interpreted as\ndue to the O(8) fixed point that controls the behavior for $\\beta_g =\n\\infty$, we have performed MC simulations for $\\beta_g =\n4$. This value is so large that, for our range of values of $L$, we do\nnot expect to observe effects related to the first-order nature of the\ntransition and therefore all data should be in the crossover\nregion. The analysis of $U$ as a function of $\\beta$ allows us to\nestimate $\\beta_c = 0.2484(2)$, which is close to the O(8) value,\n$\\beta_c \\approx 0.2408$. At the transition, gauge fields are\nsignificantly ordered and indeed, the average value of the product of\nthe gauge fields along an elementary plaquette (a Wilson loop of size\n1) is 0.95780(5) (for comparison, such a product is equal to 0.8235(5)\nat the transition for $\\beta_g = 1$). In Fig.~\\ref{U-N4-c4} we report\n$U$ versus $R_\\xi$ and compare it with the O(8) curve. The numerical\ndata with $16\\le L \\le 48$ apparently fall onto a single scaling\ncurve, while the data corresponding to $L=64$ and 96 begin to show the\ndrift that characterizes the results at $\\beta_g \\le 1$ and which is\nrelated to the asymptotic first-order nature of the transition.  The\napparent scaling curve for small values of $L$ is different from the\nO(8) one, indicating that for $\\beta_g = 4$ we are observing a sizable\ncontribution due the relevant operator that destabilizes the O(8)\nfixed point for finite $\\beta_g$. We can also infer from the\nsubstantial stability of the results for $L\\le 48$ that it has a very\nsmall (positive) scaling dimensions $y$.  Indeed, close to the O(8)\nfixed point, we expect\n\\begin{equation}\n  U(\\beta,\\beta_g) = F(R_\\xi,b(\\beta_g,\\beta) L^y) \\,,\n\\end{equation}\nwhere $b(\\beta_g,\\beta)$ is a nonuniversal amplitude, which vanishes\nfor $\\beta_g \\to \\infty$. For each $\\beta_g$ the crossover region is\nthe one in which $b(\\beta_g,\\beta_c) L^y \\ll 1$. If this condition\nholds, $U$ can be written as\n\\begin{equation}\n  U(\\beta,\\beta_g) = F(R_\\xi,0) + b(\\beta_g,\\beta) L^y G(R_\\xi)\\,,\n\\end{equation}\nwhere the first term is the O(8) scaling function.  This equation\nwould imply that the deviations from the O(8) behavior scale, at least\nfor $\\beta_g$ very large, as $L^y$. Our results therefore imply that\n$y$ should be small enough, so that $L^y$ does not change\nsignificantly as $L$ varies from 16 to 48.\n\nIn conclusion, the numerical results favor a phase diagram based on a\nfirst-order transition line for $\\beta_g>0$, starting from the\nfirst-order transition of the CP$^3$ models, corresponding to\n$\\beta_g=0$.  With increasing $\\beta_g$ the first-order transition\nbecomes weaker.  We observe substantial crossover phenomena for\n$\\beta_g\\gtrsim 1$.  They may be explained in terms of the O(8) fixed\npoint controlling the behavior for $\\beta_g\\to\\infty$, perturbed by a\nrelevant operator with a relatively small scaling dimension.\n\n\n\n\\subsection{Vector and gauge observables} \\label{Vector}\n\nIn the previous sections we discussed the behavior of quantities\ndefined in terms of the gauge-invariant order parameter $Q_{\\bm\n  x}^{ab}$. Here we discuss instead the vector correlation function\n(\\ref{Gd}) and the gauge observables (\\ref{Polyakov}) and\n(\\ref{Wilson}). We focus on $N=4$.\n\n\\begin{figure}[tbp]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4},angle=0]{gdcrit-beta0p8.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4},angle=0]{xiz-beta0p8.eps}\n\\caption{Top: Vector correlation function $G_V(x,L)$ versus $x$ for\n  $\\beta=0.8$ and several values fo $\\beta_g$. Bottom: Vector\n  correlation length $\\xi_z$ as a function of $\\beta_g$ for $\\beta =\n  0.8$. The line shows that $\\xi_z$ scales as $\\beta_g$ for large\n  $\\beta_g$ (the parameters have been determined by performing a\n  linear fit of the data with $\\beta_g \\ge 1.6$).  For $x=L$, $G_V(L,L)$ \n  corresponds to the average of the Polyakov loop.\n}\n\\label{Gdbeta0p8}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4},angle=0]{gdcrit-c0p5.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4},angle=0]{gdcrit-c1.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4},angle=0]{gdcrit-c4.eps}\n\\caption{The vector correlation function $G_V(x,L)$ in the critical\n  region for $\\beta_g = 0.5$ (top), 1 (middle) and 4 (bottom). For\n  $\\beta_g = 0.5$ we report the estimate for $\\beta = 0.365$\n  (high-temperature phase) and for $\\beta = 0.370$ (low-temperature\n  phase). For $\\beta_g = 1$ and 4, we report an effective estimate of\n  the correlation function in the coexistence region (see text for a\n  discussion), computed at $\\beta = 0.279$ for $\\beta_g = 1$, and at\n  $\\beta = 0.248$ for $\\beta_g = 4$.  }\n\\label{Gdcrit}\n\\end{figure}\n\nLet us first discuss their behavior in the two different phases. In\nthe high-temperature phase $\\beta < \\beta_c$, we find that the\ncorrelation function can be approximated as ($x > 0$)\n\\begin{equation}\n  G_V(x,L) = A e^{-x\/\\xi_z}\\,,\n\\label{Gd-exp}\n\\end{equation}\nas soon as $x$ is 2 or 3. Moreover, for small $\\beta$, $\\xi_z$ is very\nlittle dependent on $\\beta_g$.  For instance, for $\\beta = 0.1$, the\nstrong-coupling behavior $G_V(x,L) \\sim (N\\beta)^x$ holds quite precisely\nfor all values of $\\beta_g$. Wilson loops behave in a very similar\nfashion. We find $W(m,L) \\approx B \\exp(-4 m\/\\xi_w)$ with $\\xi_w\n\\approx \\xi_z$ as long as $m\\gtrsim 2$.  Clearly, in the\nhigh-temperature phase a single gauge mode controls the behavior of\nall observables that involve gauge degrees of freedom.\n\nThe behavior in the low-temperature phase is analogous.  The\ncorrelation function $G_V(x,L)$ behaves as in Eq.~(\\ref{Gd-exp}); see\nthe upper panel of Fig.~\\ref{Gdbeta0p8} for results at $\\beta = 0.8$.\nMoreover, Polyakov and Wilson loops satisfy\n\\begin{equation}\nP(L) = A e^{-L\/\\xi_z}\\,, \\qquad W(m) = B e^{-4 m \/\\xi_z}\\,,\n\\label{P-exp}\n\\end{equation}\nwith the same correlation length and $A,B\\approx 1$.\nFig.~\\ref{Gdbeta0p8} also shows that $G_V(x,L)$ has a very precise\nexponential decay even when $\\xi_z \\gtrsim L$. Clearly, it couples to\na single isolated mode and hence there are no corrections to the\nleading exponential behavior.  In this phase the correlation length\nincreases with $\\beta_g$ (see the lower panel of\nFig.~\\ref{Gdbeta0p8}): in agreement with perturbation theory, it\nscales linearly with $\\beta_g$ in the limit $\\beta_g \\to \\infty$. Note\nthe $\\xi_z$ is also expected to diverge in the limit $\\beta\\to \\infty$\nat fixed $\\beta_g$. Indeed, for $\\beta\\to \\infty$, the relevant\nconfigurations are those that minimize the Hamiltonian term that\ndepends on the fields ${\\bm z}$. If we perform a local minimization on\neach link, we find the constraint\n\\begin{equation}\n    z_{{\\bm x} + \\hat{\\mu}} = \\bar\\lambda_{{\\bm x},\\mu} z_{\\bm x}\\,.\n\\end{equation}\nThis constraint can be satisfied simultaneously on the four links\nbelonging to a plaquette only if the product of the gauge fields along\nthe plaquette is 1. Analogously, the constraint is satisfied on the\nlinks that belong to a loop that wraps around the lattice only if the\nPolyakov operator is 1. It follows that gauge configurations are\ntrivial---$\\lambda_{{\\bm x},\\mu}$ is 1 on all links modulo gauge\ntransformations---and $\\xi_z$ is infinite in this limit.\n\nThese results for the gauge observables indicate that gauge and vector\nobservables are noncritical in both phases and that their behavior is\nanalogous for small and large values of $\\beta$.  Only the limit\n$\\beta_g\\to \\infty$ distinguishes the two sectors. If\n$\\beta_{c,\\infty}$ is the transition point for $\\beta_g\\to \\infty$\n[therefore in the O($2N$) theory], for $\\beta_g\\to \\infty$, the\ncorrelation length $\\xi_z$ is finite for $\\beta < \\beta_{c,\\infty}$\nand infinite in the opposite case. This guarantees that vector\ncorrelations are critical in the O($2N$) theory with a finite\nlow-temperature magnetization.  But this only occurs for $\\beta_g$\nstrictly equal to infinity. For finite $\\beta_g$, only $Q$\ncorrelations display criticality.\n\nFinally, let us discuss the behavior of vector and gauge quantities\nalong the transition line.  For $N=4$, as we are dealing with\nfirst-order transitions, we expect $G_V(x,L)$ to depend on the phase\none considers. In the CP$^3$ model ($\\beta_g = 0$) the transition is\nstrong and therefore the high-temperature (HT) and low-temperature\n(LT) correlation functions can be easily computed by fixing $\\beta$ in\nthe coexistence region and starting the simulation from a random or an\nordered configuration.  We find that in both cases the correlation\nfunction decays very rapidly and estimate $\\xi_z \\approx 1.9$ and\n$\\xi_z \\approx 1.7$ in the LT and HT phase, respectively.  Clearly,\nvector modes are not critical. Similar results hold for the CP$^1$ and\nCP$^2$ models. In the first case, we obtain $\\xi_z \\approx 2.2 $.  For\n$N=3$ the transition is so weak that we cannot identify the two phases\nand we are only able to compute an effective correlation function,\nwhich is a linear combination of those appropriate for the two phases.\nThis quantity still allows us to compute the largest of the two\ncorrelation lengths, i.e., $\\xi_z$ in the LT phase, obtaining $\\xi_z\n\\approx 2.2$.  Results for finite $\\beta_g$ are reported in\nFig.~\\ref{Gdcrit}. The first distinctive feature is that, for the\ncases we consider, the correlation function does not behave as a\nsingle exponential, although an exponential behavior sets when $x \\gg\n\\xi_z$.  Clearly, at the critical point several modes are playing an\nimportant role and an exponential behavior is only observed when\n$\\xi_z$ is significantly less than $L$.  Second, the correlation\nlength $\\xi_z$ increases with increasing $\\beta_g$ along the\ntransition line. For $\\beta_g = 0.5$ we obtain $\\xi_z \\approx 3.8$,\n3.6, 3.1 for $L=32$, 48, and 64 in the LT phase (runs at $\\beta =\n0.370$ with ordered start).  In the HT phase (runs at $\\beta = 0.365$)\nwe obtain $\\xi_z \\approx 2.2$ with a small $L$ dependence.  Although\n$\\xi_z$ is small, it is larger than the value it takes in the CP$^3$\nmodel, i.e. the AH model with $\\beta_g=0$.  For $\\beta_g = 1$, we are\nnot able to distinguish the two phases and, therefore, we only compute\nthe LT estimate of $\\xi_z$. Results for $L=32,48,64$ essentially agree\nand give $\\xi_z \\approx 6.9$. This estimate is confirmed by the\nanalysis of the Polyakov loop.  A fit to Eq.~(\\ref{P-exp}) gives\n$\\xi_z = 6.9(1)$, in very good agreement with the results obtained\nfrom $G_V(x)$.  For $\\beta_g = 4$, even for $L = 96$ we are not yet in\nthe regime in which one can reliably identify a range of distances in\nwhich the correlation function decays exponentially. If we fit the\ncorrelation function to Eq.~(\\ref{Gd-exp}) in the range $L\/3 \\le 2\nL\/3$, we obtain $\\xi_z = 17.1(1)$ and 17.6(1) for $L=64$ and 96,\nrespectively.  The analysis of the Polyakov loop gives a somewhat\nlarger value $\\xi_z = 20.9(3)$. Whatever the exact asymptotic result is,\ndata confirm that, for $\\beta_g = 4$, we are deep in the crossover\nregion, where vector and gauge excitations compete with\ngauge-invariant excitations associated with $Q_{\\bm x}$ (for\ncomparison note that $\\xi = 20.3(3)$ for $L=96$).  These results\nprovide us a physical explanation of the crossover effects we\nobserve. The asymptotic first-order behavior is only observed when the\ncorrelation length $\\xi(L)$ at the transition point is significantly\nlarger than $\\xi_z$. When $\\xi(L) \\sim \\xi_z$ we observe an apparent\nscaling behavior in which both the (gauge-field independent) degrees\nof freedom associated with $Q$ and the (gauge-field dependent) ones,\nthat are encoded in the gauge observables and in the vector\ncorrelations, are both relevant.\n\n\\section{Conclusions} \\label{Conclusions}\n\nWe have studied the phase diagram and critical behavior of\nmuticomponent AH lattice models, in which an $N$-component complex\nfield ${\\bm z}_{\\bm x}$ is coupled to quantum electrodynamics.  We\nconsider the compact Wilson formulation of Abelian lattice gauge\ntheories in which the fundamental gauge fields are complex numbers of\nunit modulus, see Eq.~(\\ref{gllf}).  For the scalar fields, we\nconsider the unit-length limit and fix $|{\\bm z}_{\\bm x}|^2 =\n1$. Finally, we fix $q=1$ for the charge of the matter fields. We\nfocus on systems with a small number of components, considering $N=2$\nand $N=4$.\n\n\nWe investigate the phase diagram of the model as a function of the\ncouplings $\\beta$ and $\\beta_g$. The phase diagram is characterized by\ntwo phases: a low-temperature phase (large $\\beta$) in which the order\nparameter $Q^{ab}$ condenses, and a high-temperature disordered phase\n(small $\\beta$).  The gauge coupling does not play any particular role\nin the two phases: gauge observables and vector observables do not\nshow long-range correlations for any finite $\\beta$ and $\\beta_g$. The\ntwo phases are separated by a transition line that connects the\nCP$^{N-1}$ transition point ($\\beta_g = 0$) with the O($2N$)\ntransition point ($\\beta_g = \\infty$).  Concerning the nature of the\ntransition line, our numerical data are consistent with a simple\nscenario, in which the nature of the transition line is independent of\n$\\beta_g$.  Therefore, we predict Heisenberg critical behaviors along\nthe whole line for $N=2$, and first-order transitions for $N=4$. Note\nthat, for $\\beta_g\\to \\infty$ the model becomes equivalent to the\nO($2N$) vector model, and therefore one expects strong crossover\neffects controlled by the O($2N$) fixed point.  These crossover\neffects are related to the presence of a second length scale $\\xi_z$\nassociated with the vector correlations, which is finite for any\n$\\beta_g$ and diverges in the limit $\\beta_g\\to\\infty$ in the whole\nlow-temperature phase. \n\nThe scenario supported by our numerical data is fully consistent with\nthe LGW approach that assumes a gauge-invariant order parameter. On\nthe other hand, at least for $N=2$, it disagrees with the\n$\\varepsilon$-expansion predictions obtained using the standard\ncontinuum AH model: for $N=2$ this approach does not predict a\ncontinuous transition.  Numerical results allow us to understand why\nthe LGW approach is more appropriate than the continuum AH model for\nthese values of $N$.  At the transition (for both $N=2$ and $N=4$)\nonly correlations of the gauge-invariant operator $Q^{ab}$ display\nlong-range order. Gauge modes represent a background that gives only\nrise to crossover effects and indeed, the asymptotic behavior sets in\nonly when the correlation length of the gauge fluctuations is\nnegligible compared to that of the $Q$ correlations.  It is important\nto note that a LGW approach based on a gauge-invariant order parameter\nhas also been applied to the study of phase transitions in the\npresence of nonabelian gauge symmetries, and, in particular, to the\nstudy of the finite-temperature transition of hadronic matter as\ndescribed by the theory of strong interactions, quantum chromodynamics\n\\cite{PW-84,BPV-03,PV-13}. Our results for the AH lattice model lend\nsupport to the correctness of the approach and of the predictions\nobtained.\n\nWe expect that AH lattice models with higher (but not too large)\nvalues of $N$ have a phase diagram similar to the one obtained for\n$N=4$, with a first-order transition line separating the ordered and\ndisordered phases. The phase diagram may change for large values of\n$N$. In this regime, the system may undergo continuous transitions\ncontrolled by the stable fixed point of the continuum AH model.  This\nissue requires additional investigations.\n\nIt is important to stress that we have considered here a compact\nversion of electrodynamics. Other models of interest in\ncondensed-matter physics consider complex fields (spinons) coupled to\nnoncompact electrodynamics \\cite{SVBSF-04,SBSVF-05,SBSVF-04}. Such a\nmodel may have a different critical behavior due the suppression of\nmonopoles \\cite{MHSF-08,BMK-13,SP-15,DMPS-15}.  Numerical studies have\nidentified the transition, but at present there is no consensus on its\norder.  The same is true for loop models which supposedly belong to\nthe same universality class (if it exists), see, e.g.,\nRefs.~\\cite{MV-04,KMPST-08,NCSOS-11,NCSOS-13,NCSOS-15}.  Clearly,\nadditional work is needed to settle the question.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe classic Poisson formula naively says that a harmonic function on the unit disk in the complex plane, that is a function whose Laplacian vanishes, can be represented as an integral transform of its values on the  boundary of the disk. The integral transform is with respect to a kernel known as the Poisson kernel.  Probabilistically, the Poisson kernel is the distribution of the position where Brownian motion exits the open disk. The idea of representing a harmonic function as an integral transform of its boundary values extends beyond Laplacian to other contexts. In particular, in the theory of Markov chains, the study of notion of Poisson formula  goes back to the works of  Blackwell \\cite{Blackwell1955} and Feller \\cite{Feller1956}. \n\nLet $\\EuScript{X}$ be an infinite, countable set. This will serve as our state space. Let $p:\\EuScript{X} \\times \\EuScript{X} \\to [0,1]$  be a transition function, that is $\\sum_y p(x,y)=1$ for all $x\\in \\EuScript{X}$.   A function $f:\\EuScript{X} \\to {\\mathbb R}$ is $p$-harmonic if $\\sum_y p(x,y) |f(y)|<\\infty$ for all $x\\in \\EuScript{X}$ and it also satisfies the  mean value property\n$f(x)=\\sum_y p(x,y)f(y)$ for all $x\\in\\EuScript{X}$.  Note that the constant functions are bounded and  $p$-harmonic, and that the set of $p$-harmonic functions is a linear space. It is easy to see that the space of bounded $p$-harmonic functions is a Banach space with respect to the  sup-norm.  By Rohlin's theory of measurable partitions  \\cite{Rohlin52} and Doob's theory of martingales \\cite{Doob53}, there exists a probability space called the Poisson boundary such that its $L^\\infty$ as a Banach space is isometrically isomorphic to the space of bounded $p$-harmonic functions, see \\cite{Kaimanovich1996}. \n\n\nThere are  extensive developments of the theory of Poisson boundaries whenever the state space $\\EuScript{X}$ has some special structure, e.g. a group, a Riemannian manifold,  see \\cite{Furstenberg1963}, \\cite{Kaimanovich2002}, \\cite{Kaimanovich-Woess2002} and the references therein. \n\nFurstenberg \\cite{Furstenberg1973} showed that the Poisson boundary of a random walk on a group is isomorphic to the induced random walk to a recurrent subgroup. Later, Kaimanovich \\cite{Kaimanovich83}, Muchnik \\cite{Muchnik2006}, Willis \\cite{Willis1990} provided more constructions on a random walk on group that preserve the Poisson boundary. The most general method up to date to construct random walks on groups with a common  Poisson boundary was recently introduced  by  Kaimanovich and the second author in \\cite{Forghani-Kaimanovich2016}, \\cite{Behrang2016}.  Their method is based on applying a randomized stopping time to the space of sample paths to obtain a new random walk with identical  Poisson boundary.  A crucial step in the proof is that each countable group can be viewed as a quotient space of some free semigroup. Hence the proofs are  applicable only  to random walks on countable groups. This approach was employed to study the space of positive harmonic functions on a countable group with respect to a stopping time \\cite{Forghani-Mallahi2016}.\n\nIn this paper, we consider the case when the state space has \\emph{no additional structure}.  Given a Markov chain $\\boldsymbol{x} = (x_0,x_1,\\dots)$ on a countable state space  $\\EuScript{X}$ with transition function $p$ and a stopping time $\\tau$ which is almost surely finite (see equation \\eqref{eq:finiteness}), \nwe consider the process obtained by looking at the Markov chain corresponding to $p$ along iterations of the stopping time, a sequence we denote by $\\langle \\tau (\\boldsymbol{x}) \\rangle$.   Through the strong Markov property, see \\cite{Revuz1984}, this  process is a Markov chain on $\\EuScript{X}$ which we call the transformed chain, and whose transition function we denote by $p^\\tau$, given by $p^\\tau (x,y) = \\mbox{\\bf P}_x (x_\\tau = y)$.  We say  $\\tau$ can be asymptotically recovered (Assumption~\\ref{as:asymp}) when there exists a positive integer-valued map $\\rho$ on the space of sample paths such that  \n$$\n\\displaystyle\\limsup_{n\\to\\infty}  \\inf_{x} \\mbox{\\bf P}_x \\bigg(n+\\rho (x_n,x_{n+1},\\cdots) \\in \\langle \\tau(\\boldsymbol{x}) \\rangle\\bigg)=1.\n$$\n\nA simple example for a stopping time that can be asymptotically recovered is that of a hitting time. More examples are given in  Section \\ref{sec:examples}.\n\n\nWe investigate how the Poisson boundary (bounded harmonic functions) of the original and transformed Markov chains are related. We do this by showing the intuitively clear fact that  if the stopping time can be asymptotically recovered  (Assumption~\\ref{as:asymp}), then the space of bounded harmonic functions for the transformed process is  embedded in the space of space-time harmonic functions for the original chain.  This is the statement  of our main result,  Theorem \\ref{th:newmain}:\n\\begin{theo}\n Let  $p$ be  a transient  transition function on $\\EuScript{X}$, and suppose that $o\\in \\EuScript{X}$ is such that all $x\\in \\EuScript{X}-\\{o\\}$ are accessible from $o$.  Let $\\tau$ be a stopping time for which is finite a.s. under any initial distribution and can  be   asymptotically recovered (Assumption~\\ref{as:asymp}).\nThen for any positive bounded $p^\\tau$--harmonic function $u$ there exists an extension $\\bar{u}$ to   $\\EuScript{X}\\times {\\mathbb Z}_+$,  ${\\bar u}(x,0)=u(x)$, such that\n  \\begin{enumerate} \n \\item  $\\bar{u}$ is a positive bounded $p$-harmonic sequence. \n \\item $\\|{\\bar u} \\|_\\infty =\\|u\\|_\\infty$. \n \\end{enumerate} \n \\end{theo} \n We note that in general, and unlike the case of random walks on groups, the Poisson boundaries of the original and transformed  chains may be fundamentally  different, see the example in Section \\ref{splitting}.   In Theorem \\ref{th:martin} we extend the scope to positive harmonic functions under some additional conditions. \n\nOur proof of the theorem is based on the construction of the \\emph{Martin boundary}, which is one the main qualitative space in boundary theory and potential theory associated to Markov chains. The Martin boundary of a Markov chain is the  topological counterpart of the Poisson boundary which is responsible for representation of positive harmonic functions. If the Martin boundary as a Borel space equipped with an appropriate probability measure, then it is isomorphic (as a measure space) with the Poisson boundary, see \\cite{Dynkin69}. \n\nThe organization of the paper is as follows.  In Section \\ref{sec:preliminaries} we recall the theory of  Poisson, tail, and Martin boundaries. Section 3 devoted to constructing Markov chains via transformation.  In  Section 4 we show how the tools from the theory of Martin boundary can be applied to the transformed Markov chains via stopping times. \nOur main result is  proved in Section \\ref{sec:main}, and in Section \\ref{sec:examples} we present a number of examples. Two standard approximation results used in the proof of Theorem \\ref{th:martin} are  proved in the Appendix.\n\n\\subsection*{Acknowledgements}\nWe would like to thank the anonymous referee for reading  the manual very carefully and suggesting improvements.\n\n\\section{Preliminaries}\\label{sec:preliminaries} \n\\subsection{Markov chains}\n\\label{sec:MC_defn} \nLet $\\EuScript{X}$ be a countable set. The set $\\EuScript{X}$ and its power set form a measurable space.  Let $\\EuScript{X}^{{\\mathbb Z}_+}=\\{\\boldsymbol{x}=(x_0,x_1,\\dots):x_n \\in \\EuScript{X},n\\in{\\mathbb Z}_+\\}$, the set of $\\EuScript{X}$-valued sequences indexed by ${\\mathbb Z}_+$. For every $n\\in{\\mathbb Z}_+$ define the coordinate function $\\omega_n(\\boldsymbol{x})=x_n$. Denote by $\\cal{F}_k^\\infty(\\EuScript{X})=\\sigma(\\omega_k,\\omega_{k+1},\\cdots)$ the sigma-algebra generated by the coordinate functions $\\omega_i,~i\\ge k$. Let ${\\cal F^\\infty}=\\cal F^\\infty_0(\\EuScript{X})$.\nThe measurable  space  $(\\s^{\\mathbb{Z}_+},{\\cal F^\\infty})$ is called the  space of sample paths. For our work, we will also need to define an auxiliary process  $\\boldsymbol{z}=(z_n:n\\in {\\mathbb Z}_+)$, as follows. Given $\\boldsymbol{x}$ and $t \\in{\\mathbb Z}_+$, we let $ z_n = (x_n,t+n)$. That is, $\\boldsymbol{z}$ also keeps track of time. \n\nLet $p$ be a transition function on $\\EuScript{X}$. That is $p:\\EuScript{X} \\times \\EuScript{X} \\to [0,1]$ and $\\sum_y p(x,y) =1$ for all $x\\in \\EuScript{X}$. Let $m$ be a probability measure on $\\EuScript{X}$.  For $n\\in {\\mathbb Z}_+$, define the $n$-th iteration of $p$, denoted by $p^n$, through \n\\begin{equation}\n\\label{eq:p_iterated} \np^0(x,y) = \\delta_{x}(y), \\mbox{ and }p^{n+1} (x,y) = \\sum_z p(x,z) p^n (z,y)\n\\end{equation}\n(note that $p^1=p$, and that $p^n$ is also a transition function). By Kolmogorv's extension theorem, there exists a unique probability measure  $\\mbox{\\bf P}_m$ on the space of sample paths satisfying\n$$\n\\mbox{\\bf P}_m(a_0,a_1,\\cdots,a_n)=m(a_0)p(a_0,a_1)\\cdots p(a_{n-1},a_n)\n$$ \nwhere $(a_0,a_1,\\cdots,a_n)=\\{\\boldsymbol{x}\\in\\s^{\\mathbb{Z}_+}\\ :\\ \\omega_i(\\boldsymbol{x})=a_i,\\ i=0,\\cdots,n\\}$. The probability measure \n$m$ is usually referred to as the initial distribution under $\\mbox{\\bf P}_m$. As usual, we write $\\mbox{\\bf E}_m$ for the expectation operator associated with $\\mbox{\\bf P}_m$, also for $x\\in \\EuScript{X}$ we abbreviate and write $\\mbox{\\bf P}_x,\\mbox{\\bf E}_x$ instead of $\\mbox{\\bf P}_{\\delta_x},\\mbox{\\bf E}_{\\delta_x}$. Note then that \n$$\n\\mbox{\\bf P}_m=\\sum_x m(x)\\mbox{\\bf P}_x,\\ \\ \\ \\  \\mbox{\\bf E}_m=\\sum_xm(x)\\mbox{\\bf E}_x.\n$$\nThe triple $(\\EuScript{X},p,m)$ is called a Markov chain on the state space $\\EuScript{X}$ with the transition function $p$ and the initial distribution $m$.\n\\subsection{Harmonic functions}\nSuppose that $f:\\EuScript{X} \\to {\\mathbb R}$ satisfies $\\sum_y p(x,y) |f(y)|< \\infty$ for all $x\\in \\EuScript{X}$. Then we can define a function $pf:\\EuScript{X}\\to {\\mathbb R}$ through \n$$pf(x)=\\sum_yp(x,y)f(y).$$ \nIf $pf =f$, then $f$ is called $p$--harmonic.  We denote the set of all bounded $p$--harmonic functions and the set of all positive $p$--harmonic functions by $H^{\\infty}(\\EuScript{X},p)$ and $H_{+}(\\EuScript{X},p)$, respectively.  We also write $H^\\infty_+(\\EuScript{X},p)$ for the convex cone of bounded positive harmonic functions.  \n\\subsection{Harmonic sequences}\\label{Z}\nA sequence of functions $(f_n:n\\in {\\mathbb Z}_+)$, where  $f_n: \\EuScript{X}\\to{\\mathbb R}$, is called a $p$--harmonic sequence whenever \n$$pf_{n+1}=f_n.$$\nThe space $p$--harmonic sequence is denoted by $S(\\EuScript{X},p)$, while the subspace of  bounded $p$--harmonic sequences, and nonnegative $p$--harmonic  sequences are denoted by  $S^{\\infty}(\\EuScript{X},p)$ and  $S_+(\\EuScript{X},p)$, respectively.  If $f$ is a nonnegative $p$--harmonic, then  $(f,f,\\dots)$ is a nonnegative $p$--harmonic sequence.  Harmonic sequences sometimes are called space--time harmonic functions. Indeed, given a $p$--harmonic sequence $(f_n:n\\in{\\mathbb Z}_+)$, define a function $f:\\EuScript{X}\\times {\\mathbb Z}_+\\to {\\mathbb R}$ by letting $f(x,n)=f_n(x)$.  Now if one defines a transition function  $p^+$ on $\\EuScript{X}\\times {\\mathbb Z}_+$ by letting\n$$p^+((x,n),(y,n+1))=p(x,y),$$\nthen $p^+ f = f$. Conversely, if $f$ is $p^+$--harmonic, then $(f(x,0),f(x,1),\\dots)$ is a $p$-harmonic sequence. In terms of notation, ${\\cal S}(\\EuScript{X},p)=H(\\EuScript{X}\\times {\\mathbb Z}_+,p^+)$. For $(x,t)\\in \\EuScript{X}\\times{\\mathbb Z}_+$, write  $\\mbox{\\bf P}_{x,t}$ for the probability measure on the space of sample paths on $\\EuScript{X}\\times {\\mathbb Z}_+$ induced by  $p^+$ with initial distribution $\\delta_{x,t}$.  A sample path in that space will be written as  $\\boldsymbol z=(z_0,z_1,\\cdots)$.\n\\subsection{Poisson boundary}\\label{sec:Poisson}\nLet $S:\\s^{\\mathbb{Z}_+}\\to\\s^{\\mathbb{Z}_+}$ be the time-shift, that is \n$$\nS(x_0,x_1,\\cdots)=(x_1,x_2,\\cdots),\n$$\nand for $k\\ge 1$ write $S^k$ for the $k$-th iteration of $S$, i.e, $S^k (x_0,x_2,\\cdots) =(x_k,x_{k+1},\\cdots)$.\nThe sigma-algebra  $\\mathcal{I}=\\{A\\in\\mathcal F^\\infty\\ :\\ S^{-1}A=A\\}$ is called the  invariant sigma-algebra. Let $\\theta$ be a probability measure on $\\EuScript{X}$ with full support, that is $\\theta(x)>0$ for all $x\\in \\EuScript{X}$. Let $\\overline{\\mathcal{I}}(\\EuScript{X},p)$ denote the completion of ${\\cal I}$ with respect to the probability measure $\\mbox{\\bf P}_{\\theta}$. As can be easily seen, this completion is  equivalent to requiring that whenever $A\\subseteq \\s^{\\mathbb{Z}_+}$ is such that $A\\subseteq B$ for some $ B \\in{\\cal F^\\infty}$, satisfying $\\mbox{\\bf P}_x (B) = 0$ for all $x\\in \\EuScript{X}$, then $A$ is measurable with respect to the completion. Therefore $\\overline{\\cal I}(\\EuScript{X},p)$ is independent of the particular choice of $\\theta$.   The Poisson boundary is defined as the  restriction of $(\\s^{\\mathbb{Z}_+},\\cal F^\\infty,\\mbox{\\bf P}_{\\theta})$ to  $\\overline{\\mathcal{I}}(\\EuScript{X},p)$.  Denote the Poisson boundary with respect to the transition function $p$ as $\\mathcal{PB}(\\EuScript{X},p):=(\\EuScript{X}^{{\\mathbb Z}_+}, \\overline{\\mathcal{I}}(\\EuScript{X},p), \\mbox{\\bf P}_\\theta)$.  As an ergodic theoretic object, the Poisson boundary is identified with  the space of ergodic components of the time-shift on the space of sample paths \\cite{Kaimanovich91}.  \n\nThe Poisson boundary identifies the space of bounded harmonic functions. More precisely, let $f$ be a bounded  $p$--harmonic function, define \n\n\\begin{equation}\\label{eq:martin-approach}\n\\begin{array}{rcl}\nH^\\infty(\\EuScript{X}, p) &\\longrightarrow &L^\\infty(\\mathcal{PB}(\\EuScript{X},p))\\\\\nf  &\\longmapsto&  \\phi_f(\\boldsymbol{x}):= \\displaystyle\\lim_{n\\to\\infty} f(x_n)  \\ \\mbox{ a.e.} \n\\end{array}\n\\end{equation}\n\nBecause $f$ is a bounded $p$--harmonic function, the sequence $(f(x_n))_{n}$ is $\\mbox{\\bf P}_\\theta$--martingale, therefore $\\phi_f(x)$ almost surely exists. Because $\\overline{\\mathcal{I}}(\\EuScript{X},p)$ is a complete sigma--algebra, $\\phi_f$ is measurable with respect to the probability space $\\mathcal{PB}(\\EuScript{X},p)$.  We can define the inverse as follows.\n\\begin{equation}\\label{eq: bounded-infinity}\n\\begin{array}{rcl}\nL^\\infty(\\mathcal{PB}(\\EuScript{X},p)) &\\longrightarrow &H^\\infty(\\EuScript{X}, p) \\\\\n\\phi  &\\longmapsto&  f_\\phi(x):= \\int \\phi(\\boldsymbol{x}) \\ d\\mbox{\\bf P}_x(\\boldsymbol{x})\n\\end{array}\n\\end{equation}\nMoreover the definitions of $\\phi_f$ and $f_\\phi$ imply that $\\|f\\|_\\infty=\\|\\phi_f\\|_\\infty$, hence $H^\\infty(\\EuScript{X},p)$\nis isometrically isomorphic to $L^\\infty(\\mathcal{PB}(\\EuScript{X},p))$, see also Corollary~\\ref{cor:isometry}.\n\n\n\\subsection{Tail boundary}\nConsider the  tail sigma--algebra $\\mathcal{T}=\\displaystyle\\bigcap_{k=0}^{\\infty}S^{-k}(\\cal F^\\infty)$  on the space of sample paths. The completion of $\\mathcal{T}$ with respect to $\\mbox{\\bf P}_\\theta$, where $\\theta$ is as in the last paragraph, is denoted by $\\overline\\mathcal{T}(\\EuScript{X},p)$. The tail boundary is the  restriction of $(\\s^{\\mathbb{Z}_+},\\cal F,\\mbox{\\bf P}_{\\theta})$  to the sigma-algebra  $\\overline\\mathcal{T}(\\EuScript{X},p)$. The tail boundary associated with the transition function $p$ is denoted by $\\mathcal{TB}(\\EuScript{X},p):=(\\EuScript{X}^{{\\mathbb Z}_+}, \\overline{\\mathcal{T}}(\\EuScript{X},p), \\mbox{\\bf P}_\\theta))$. Similarly to the Poisson boundary, the space of bounded  $p$--harmonic sequences $S^\\infty(\\EuScript{X},p)$ is isometrically isomorphic to $L^\\infty(\\mathcal{TB}(\\EuScript{X},p))$:\n$$\n\\begin{array}{rcl}\nS^\\infty(\\EuScript{X}, p) &\\longrightarrow &L^\\infty(\\mathcal{TB}(\\EuScript{X},p))\\\\\nF=(f_n)_n  &\\longmapsto&  \\psi_F(\\boldsymbol{x}):= \\displaystyle\\lim_{n\\to\\infty} f_n(x_n)\\ a.e\n\\end{array}\n$$\nUsing the fact that the space of bounded $p^+$--harmonic functions can be viewed as the space of bounded $p$--harmonic functions implies $H^\\infty(\\EuScript{X}\\times {\\mathbb Z}_+,p^+)$ is isometrically isomorphic to $L^\\infty(\\mathcal{TB}(\\EuScript{X},p))$. On the other hand, $H^\\infty(\\EuScript{X}\\times {\\mathbb Z}_+,p^+)$ is isometrically isomorphic to \n$L^\\infty(\\mathcal{PB}(\\EuScript{X}\\times{\\mathbb Z}_+,p^+))$. Summarizing: we have $L^\\infty(\\mathcal{TB}(\\EuScript{X},p))$  is isometrically isomorphic to $L^\\infty(\\mathcal{PB}(\\EuScript{X}\\times{\\mathbb Z}_+,p^+))$. Therefore,  the tail boundary associated with $p$ is isomorphic (as a probability space) to the Poisson boundary associated with $p^+$, for more details see \\cite{Ka92,Kaimanovich1996}\n\n$$ \n \\begin{tikzcd}\nH^\\infty(\\EuScript{X}\\times{\\mathbb Z}_+,p^+)\\arrow[r, \"\\cong\"]  & S^\\infty(\\EuScript{X},p)  \\dar{\\cong} \\\\  \nL^\\infty(\\mathcal{PB}(\\EuScript{X}\\times{\\mathbb Z}_+,p^+)) \\arrow[u, \"\\cong\"]   & L^\\infty(\\mathcal{TB}(\\EuScript{X},p))\n\\end{tikzcd}\n$$\n\n\n Viewing the Poisson boundary as representing the  bounded harmonic functions,  the tail boundary as representing the  bounded harmonic sequences, and using the fact that  every harmonic function uniquely extends to a harmonic sequence, we can think of the Poisson boundary as a subset on the tail boundary. We will not get into any details here.   However, we have the following:\n\n$$ \n \\begin{tikzcd}\nH^\\infty(\\EuScript{X},p)\\arrow[r, hook]  & S^\\infty(\\EuScript{X},p)  \\dar{\\cong} \\\\  \nL^\\infty(\\mathcal{PB}(\\EuScript{X},p)) \\arrow[u, \"\\cong\"]   & L^\\infty(\\mathcal{TB}(\\EuScript{X},p))\n\\end{tikzcd}\n$$\n \n A transition function of a Markov chain is called steady whenever the tail boundary  coincides mod $\\mbox{\\bf P}_x$ with  the Poisson boundary  for any $x$ in $\\EuScript{X}$, and, in particular, all bounded harmonic sequences are bounded harmonic functions.  The ``0-2'' law  determines whether a transition function is steady, see  \\cite{Ka92} for more details. Here is one sufficient condition: \n \n\\begin{example}[\\cite{Ka92}]\nFor $x\\in \\EuScript{X}$, let  $g_x = \\inf \\{n\\ge 1: p^n(x,x)>0\\}$. Then $p$ is steady if the greatest common divisor of $\\{g_x>0:x\\in \\EuScript{X}\\}$ is $1$.  \n\\end{example}\n\n\\subsection{Martin boundary}\n\\label{sec:martin_boundary}\nThe relation between the Poisson and the tail boundaries to the invariant and tail sigma-algebras allow  to characterize the space of bounded harmonic functions and bounded harmonic sequences, respectively.  We now introduce the Martin boundary, a topological boundary also used to characterize positive harmonic functions (Theorem \\ref{th:doob59} below), and which is more suitable for our purposes.  We comment that the Poisson boundary can be identified as a subset of the Martin topology equipped with an appropriate probability measure \\cite{Kaimanovich1996}, \\cite{Sawyer1997}, and \\cite{Woess09}, also see Corollary \\ref{cor:isometry} below.  One can refer to \\cite{Derriennic1976}, \\cite{Kaimanovich1996}, \\cite{Sawyer1997} and  \\cite{Woess09} for the construction of Martin boundary.  In this section, we remind the reader about the definition of the Martin boundary and related results which will be used  later.\n\nLet $p$ be a transition function on $\\EuScript{X}$.  The  transition function is called transient whenever the Green's function $G^p(x,y)=\\sum_{n\\geq0}p^n(x,y)$ is finite for all $x,y\\in \\EuScript{X}$, $p^n$ is the $n$-th iteration of $p$, defined in \\eqref{eq:p_iterated}. \n\nWe always make the following assumption on $p$: \n\\begin{assum}~\n\\label{as:required} \n\\begin{enumerate}\n\\item $p$ is transient. \n\\item There exists a state $o$ such that all $y \\in \\EuScript{X}-\\{o\\}$ are accessible from $o$:  \n$$G^p(o,y)>0\\mbox{ for all }y\\in \\EuScript{X}.$$  \n\\end{enumerate}\n\\end{assum} \n\nWe also define  the {\\it Martin kernel} on $\\EuScript{X}\\times \\EuScript{X}$\n$$K^p(x,y)=\\begin{cases} \\frac{G^p(x,y)}{G^p(o,y)} & G^p(o,y)>0 \\\\ 0 & \\mbox{otherwise}\\end{cases}$$ \n The  {\\it Martin compactification} of $\\EuScript{X}$ is the topological space $M(\\EuScript{X},p)$\n\n satisfying the following requirements: \n \\begin{enumerate}\n \\item Every singleton $\\{x\\},~x\\in \\EuScript{X}$, is open. \n \\item $\\EuScript{X}$ is dense. \n\\item For  $x\\in \\EuScript{X}$, the function $K^p(x,\\cdot)$ extends to a continuous function on $M(\\EuScript{X},p) $, and the  set of extensions separate points in $M(\\EuScript{X},p) \\backslash \\EuScript{X}$. \n\\end{enumerate}\nThese  requirements uniquely determine a compact topological space (up to homeomorphism). Furthermore, the resulting space is metrizable \\cite{Woess09}. The compact topological space $\\partial(\\EuScript{X},p)=M(\\EuScript{X},p) \\backslash \\EuScript{X}$ is called the \\emph{Martin boundary} of the Markov chain with respect to the transition function $p$. \n\nThe \\emph{minimal Martin boundary} is the Borel subset $\\partial_m(\\EuScript{X},p)$ of $\\partial(\\EuScript{X},p)$  consisting of all $\\xi$ satisfying \n\\begin{enumerate} \n\\item $K^p(\\cdot, \\xi) \\in H_+(\\EuScript{X},p)$. \n\\item $K^p(\\cdot,\\xi)$ is {\\it minimal harmonic}: if $u\\in H_+(\\EuScript{X},p)$ and  $u \\le K^p(\\cdot,\\xi)$, then  $u = c K^p(\\cdot, \\xi)$ for some $c\\le 1$. \n\\end{enumerate} \n\\begin{theorem}\\cite{Doob59}\\label{th:doob59}\nLet $u\\in H_+(\\EuScript{X},p)$. Then  there exists a unique finite measure $\\mu_u$ on the Borel sigma-algebra on $\\partial_m(\\EuScript{X},p)$ such that \n$$\nu(x)=\\int_{\\partial_m(\\EuScript{X},p)}K^p(x,\\xi)d\\mu_u(\\xi).\n$$\n\\end{theorem}\nThe measure $\\mu_u$ is called the representation of $u$. Since    $K^p(o,\\xi)=1$ for all $\\xi\\in \\partial_m (\\EuScript{X},p)$, we have that  $u(o) = \\mu_u (\\partial_m(\\EuScript{X},p))$. Note that we can consider $\\mu_u$ as a finite measure on the compact metric space $\\partial (\\EuScript{X},p)$, by letting $\\mu_u(\\partial (\\EuScript{X},p)-\\partial_m(\\EuScript{X},p))=0$. \n A special role is reserved for $\\mu_{\\bf 1}$ the representation of the constant function ${\\bf 1}$. This is due to the following two results: \n\\begin{cor}\\label{cor:isometry}\nThe mapping  $T$ given by \n\\begin{equation}\\label{eq:mapping}  (T f)(x) = \\int_{\\partial_m(\\EuScript{X},p)} K^p(x,\\xi) f (\\xi) d\\mu_{\\bf 1}.\n\\end{equation}  \ndefines a linear isometry from $L^\\infty(\\mu_1)$ onto $H^\\infty (\\EuScript{X},p)$ and also $L^\\infty(\\mathcal{PB}(\\EuScript{X},p))$.\n\\end{cor} \n\\begin{proof}\nThe right-hand side of \\eqref{eq:mapping} defines a  linear mapping from $L^\\infty(\\mu_{\\bf 1})$ to the linear space of bounded real-valued functions on $\\EuScript{X}$ equipped with the $\\sup$-norm. Also,  \n $$|Tf(x) | \\le \\|f\\|_\\infty \\int_{\\partial_m(\\EuScript{X},p)} K^p(x,\\xi) d \\mu_{\\bf 1}(\\xi) = \\|f\\|_\\infty.$$ \n Therefore $\\|Tf \\|_\\infty \\le \\|f\\|_\\infty$. \n  By dominated convergence, \n$$ p (T f)(x) =  \\int_{\\partial_m(\\EuScript{X},p)}\\sum_y p(x,y)  K^p(y,\\xi) f (\\xi) d \\mu_{\\bf 1}(\\xi)=  \\int_{\\partial_m(\\EuScript{X},p)} K^p(x,\\xi) f(\\xi) d \\mu_{\\bf 1} (\\xi)=(Tf)(x),$$ \ntherefore $Tf \\in H^\\infty (\\EuScript{X},p)$. Next we show that $T$ is an isometry. Suppose first that $f\\in L^\\infty(\\mu_{\\bf 1})$ is nonnegative, and let $u = Tf$. Then since $\\|u\\|_{\\infty}{\\bf 1}-u$ and $u$ are both in $H_+(\\EuScript{X},p)$, it follows from the uniqueness assertion in Theorem \\ref{th:doob59}, that $\\|u\\|_\\infty \\mu_{\\bf 1} =\\mu_{ \\|u\\|_{\\infty}{\\bf 1}-u}+ \\mu_u$, the sum of two positive measures. Therefore, not only $\\mu_u \\ll \\mu_{\\bf 1}$, but also, $0\\le  \\frac{ d \\mu_u}{d \\mu_{\\bf 1}} \\le \\|u\\|_\\infty$. Since $\\frac{d \\mu_u}{d\\mu_{\\bf 1} } =f$, we have $\\|f\\|_\\infty \\le \\|u\\|_\\infty$. In the case of signed $f$, this means \n$$\n\\| \\left .  \\|f\\|_\\infty{\\bf 1} \\pm f\\right.  \\|_\\infty \\le \\| \\left. \\|f\\|_\\infty {\\bf 1} \\pm  Tf \\right.\\|_\\infty.\n$$ \nThe righthand side is bounded above by $\\|f\\|_\\infty + \\|Tf \\|_\\infty$. As for the left hand side, we can choose the sign so that the norm is equal to $2\\|f\\|_\\infty$. Therefore, $\\|Tf \\|_\\infty \\ge \\|f\\|_\\infty$, and since the reverse inequality is already established,  $T$ is an isometry. Finally, we show that $T$ is onto. If  $v\\in  H^\\infty_+(\\EuScript{X},p)$, then as already seen,  $\\mu_v \\ll \\mu_{\\bf 1}$, and $\\frac{d\\mu_v}{d\\mu_{\\bf 1}} \\in L^\\infty(\\mu_{\\bf 1})$, so that $v$ is in the range of $T$.  If $u \\in H^\\infty(\\EuScript{X},p)$, then we can write it as a difference of two elements in $H^\\infty_+(\\EuScript{X},p)$, i.e, $u=(\\|u\\|_\\infty{\\bf 1}  + u) - \\|u\\|_{\\infty}$.  Therefore, $u$ is also in the range of $T$. \n\\end{proof} \n\\begin{theorem}\\cite{Dynkin69,Woess09}\n\\label{th:limits}\n\\begin{enumerate} \n\\item There exists a $\\partial(\\EuScript{X},p)$-valued random variable $x_\\infty$ such that for $\\mbox{\\bf P}_x$-a.s. sample path $\\lim_{n\\to\\infty} x_n=x_\\infty $ is in the Martin topology for all $x$ in $\\EuScript{X}$. \n\\item  The random variable $x_\\infty$ is supported on $\\partial_m(\\EuScript{X},p)$ and for every measurable set $A$ in $\\partial_m(\\EuScript{X},p)$,   \n$$\n\\mbox{\\bf P}_x (x_\\infty \\in A) = \\int_A K^p(x,\\xi) d \\mu_{\\bf 1} (\\xi).\n$$\n\\end{enumerate} \n\\end{theorem} \n\n\n\\section{Transformed Markov chains}\\label{sec:transformed}\n\\subsection{Stopping time}\n\\label{sec:stopped_process}\nA measurable function $\\tau:\\EuScript{X}^{{\\mathbb Z}_+}\\to {\\mathbb Z}_+\\cup\\{\\infty\\}$ is called stopping time, if for every $k\\in {\\mathbb Z}_+$, the set $\\{ \\boldsymbol{x}: \\tau(\\boldsymbol{x})=k\\}$ is a measurable set in the sigma-algebra generated by the first $k+1$ coordinate function $\\sigma(\\omega_0,\\omega_1,\\cdots\\omega_k)$. In what follows, we will assume that \n\\begin{equation} \n\\label{eq:finiteness} \n\\mbox{\\bf P}_x(\\tau < \\infty)=1 \\mbox{  for all }x\\in \\EuScript{X}.\n\\end{equation}\nGiven a stopping time $\\tau$ satisfying \\eqref{eq:finiteness}, a nondecreasing sequence is induced by iteration:  \n$$\n\\tau_0 =0,\\ \\ ~\\tau_1=\\tau,\\ \\  \\tau_{n+1} = \\begin{cases} \\tau_n + \\tau \\circ  S^{\\tau_n} & \\tau_n <\\infty; \\\\ \\infty & \\mbox{otherwise.} \\end{cases}$$ \nWith this sequence, we obtain a transformed process $\\boldsymbol{x}^{\\tau}=(y_n(\\boldsymbol{x}):n\\in{\\mathbb Z}_+)$ given by $y_n = x_{\\tau_n}$. \nBy the strong Markov property, see \\cite{Revuz1984}, \n$$ \\mbox{\\bf P}( x_{\\tau_{n+1}}=z | \\sigma(x_{\\tau_1}\\cdots,x_{\\tau_n} )) = \\mbox{\\bf P}_{y_n} (x_{\\tau} = z).$$\nTherefore $\\boldsymbol{x}^{\\tau}$ is a Markov chain with  the transition function \n$$\np^{\\tau}(x,y)=\\mbox{\\bf P}_x (x_{\\tau} = y).\n$$\n\n\n\nNote that Doob's optional stopping theorem implies that for any stopping time $\\tau$, we can write\n$$\nH^{\\infty}(\\EuScript{X},p)\\subseteq H^{\\infty}(\\EuScript{X},p^\\tau).$$\nSimilarly,   \n$$H^{\\infty}(\\EuScript{X}\\times{\\mathbb Z}_+,p^+)\\subseteq H^{\\infty}(\\EuScript{X}\\times{\\mathbb Z}_+,(p^{+})^\\tau).\n$$\n\nLet $\\mbox{\\bf P}^\\tau$ denote the probability measure on the space of sample paths with respect to the transition function $p^\\tau$. We could map almost every sample path with respect to $p$ to  a sample path with respect to $p^\\tau$:\n$$\n\\begin{array}{rcl}\n(\\EuScript{X}^{{\\mathbb Z}_+},\\mbox{\\bf P}_\\theta)&\\longrightarrow &(\\EuScript{X}^{{\\mathbb Z}_+},\\mbox{\\bf P}^\\tau_\\theta)\\\\\n\\boldsymbol{x}=(x_n)_n &\\longmapsto & \\boldsymbol{x}_\\tau:=(x_{\\tau_n})_n,\n\\end{array}\n$$\nwhich implies $L^\\infty(\\mathcal{PB}(\\EuScript{X},p))$ is isomorphic to a subspace of $L^\\infty(\\mathcal{PB}(\\EuScript{X},p^\\tau))$.\n\nConsider the following:   \n\\begin{assum} \\label{as:asymp} \nThere exists a mapping $\\rho:{\\EuScript{X}}^{{\\mathbb Z}_+}\\to {\\mathbb Z}_+$ such that\n$$\n\\displaystyle\\limsup_{n\\to\\infty}  \\inf_{x} \\mbox{\\bf P}_x (\\rho_n (\\boldsymbol{x}) \\in \\langle \\tau(\\boldsymbol{x}) \\rangle)=1,\n$$\nwhere $\\rho_n({\\boldsymbol{x}}) = n + \\rho \\circ S^n({\\boldsymbol{x}})$ and  $\\langle \\tau(\\boldsymbol{x}) \\rangle=(\\tau_0(\\boldsymbol{x}),\\tau_1(\\boldsymbol{x}),\\cdots)$. \n\\end{assum} \nIn order to be able to employ the tools from the last section, we need to insure that $p^\\tau$ satisfies the conditions of Assumption~\\ref{as:required}. \nWe observe that if $\\boldsymbol{x}$ (equivalently, $p$) is transient then so is $\\boldsymbol{x}^\\tau$ (equivalently $p^\\tau$). \n\n\n\t\n\\begin{lemma}\nIf $p$ is transient, then so is  $p^\\tau$.  Furthermore, for all $x$ and $y\\in \\EuScript{X}$, we have $G^{p^\\tau}(x,y) \\le G^p(x,y)$. \n\\end{lemma}\n\\begin{proof}\n\tTransience of $p^\\tau$ is equivalent to $\\boldsymbol{x}^\\tau$ visiting each state finitely often under $\\mbox{\\bf P}_x$ for all $x\\in \\EuScript{X}$. Since this holds for $\\boldsymbol{x}$, and the paths of $\\boldsymbol{x}^\\tau$ are subsequences of $\\boldsymbol{x}$ both statements hold.  \n\\end{proof}\t\n\nNote that $p^\\tau$ may, in general, not satisfy the second condition in Assumption \\ref{as:required}. For example let $\\EuScript{X}={\\mathbb Z}_+$, the set of nonnegative integers and let  $p(n,n+1)=1$, then $G^p(0,n)=1>0$ for any natural number $n$. If we consider the stopping time $\\tau=2$, then for any $n\\in{\\mathbb Z}_+$, $G^{p^{\\tau}}(m,n)>0$ if and only if $n-m \\in 2{\\mathbb Z}_+$. Therefore there does not exist $m\\in{\\mathbb Z}_+$ such that $G^{p^{\\tau}}(m,n)>0$ for all $n\\in{\\mathbb Z}_+$.  In Section~\\ref{sec : extension}, we will remedy this by expanding the state space.  \n\n\\section{The extension} \\label{sec :  extension}\nAssumption \\ref{as:asymp} does not warrant that  $p^\\tau$ satisfies the second condition of Assumption \\ref{as:required} which is required for defining the Martin boundary. If it does, we need not do anything. Otherwise, we need to introduce the following completion. \n\nOur starting point is a transition function $p$ on $\\EuScript{X}$ satisfying Assumption~\\ref{as:required},  and a stopping time $\\tau$ for $p$ satisfying Assumption~\\ref{as:asymp}.\n\nThe first step is to append a state to $\\EuScript{X}$, and extend $p$ to the new resulting extended state space. Let $\\EuScript{X}^* = \\EuScript{X} \\cup\\{*\\}$, where $*$ is a state not in $\\EuScript{X}$. Let $\\theta$ be any probability measure on $\\EuScript{X}^*$ with $\\theta(x)>0$ for all $x\\in \\EuScript{X}$ and $\\theta(*)=0$.  Extend $p$ to $\\EuScript{X}^*$ by letting\n$$  p^*(x,y) = \\begin{cases} p(x,y)& ~x,y\\in \\EuScript{X}\\\\ \n\\theta(y)& x=* \\\\ \n0 & \\mbox{otherwise}. \n\\end{cases}$$\nWe write  $\\boldsymbol{x}^*$ for the corresponding Makrov chain. \nClearly $p^*$ satisfies both conditions in Assumption~\\ref{as:required} with $\\EuScript{X}$ replaced by $\\EuScript{X}^*$ and $o=*$. We also write $\\mbox{\\bf P}^*_x $ and $\\mbox{\\bf E}^*_x$ for the distribution and corresponding expectation associated with $\\boldsymbol{x}^*$, starting from $x$ in $\\EuScript{X}^*$. Note that for any $x\\in \\EuScript{X}$, we have $\\mbox{\\bf P}^*_x$ is supported on $\\EuScript{X}$-valued sequences and coincides with $\\mbox{\\bf P}_x$. It could be therefore viewed as an extension of $\\mbox{\\bf P}_x$. \n\nThis extension preserves the space of bounded harmonic functions: \n\\begin{lemma}\nLet $f$ be a bounded  function on $\\EuScript{X}$. Then, $f$ is $p$--harmonic if and only there exists a unique bounded function $f^*$ on $\\EuScript{X}^*$ such that \n\\begin{enumerate} \n\\item $f^*$ is an extension of $f$, that is $f^*(x)=f(x)$ for all $x$ in $\\EuScript{X}$,\n\\item $f^*$ is $p^*$--harmonic. \n\\end{enumerate} \n\\end{lemma}\n\\begin{proof}\nIt is enough to define $f^*(*) = \\sum_{y\\in \\EuScript{X}} \\theta(y) f(y)$ and $f^*(x)=f(x)$ for any $x$ in $\\EuScript{X}$. Them $f^*$ is an extension of $f$ and a bounded $p^*$--harmonic. The reverse direction is clear.\n\\end{proof}\nNext we extend the stopping time $\\tau$ to   $\\EuScript{X}^*$-valued sequences by setting \n \\[ \\tau^*(\\boldsymbol{x}^*) = \\begin{cases} \\tau(\\boldsymbol{x}^*)&\\mbox{if } x_0^*,x_1^*,\\dots \\in \\EuScript{X} \\\\\n  1+ \\tau(x_1^*,x_2^*,\\dots)&\\mbox{if }  x_0^*=*,x_1^*,x_2^*,\\dots \\in \\EuScript{X}\\\\\n  \\min\\{j:x_j^*= *\\}&  \\mbox{ otherwise} \n \\end{cases} \n \\]\n\nIt immediately follows that  $\\tau^*$ is a stopping time for $\\boldsymbol{x}^*$. Note that under $\\mbox{\\bf P}_x^*$ for  $x\\in \\EuScript{X}$, we have $\\tau^*=\\tau$ a.s. Furthermore,   if $x\\in \\EuScript{X}^*$,  then $\\tau^*$ is finite a.s. $\\mbox{\\bf P}^*_x$. As a result, and similarly to the definition of $p^\\tau$, we have an induced transition function $(p^*)^{\\tau^*}$, defined as follows: \n$$ \n(p^*)^{\\tau^*} (x,y) = \\mbox{\\bf P}_x^* (\\boldsymbol{x}^*_{\\tau^*}=y) = \\begin{cases} p^{\\tau}(x,y) & x,y \\in \\EuScript{X} \\\\ \\sum_{x'}\\theta(x') p^\\tau(x',y) & x =*,y \\in \\EuScript{X}\\\\\n0 & \\mbox{otherwise} \\end{cases}\n$$\n\nWe will write $\\boldsymbol{y}^*$ for the induced chain, that is $y^*_n = \\boldsymbol{x}^*_{\\tau^*_n},~n\\in{\\mathbb Z}_+,$ \nwhere, \n$$\n\\tau^*_0=0\\ \\mbox{ and }\\ \\tau^*_{n+1} = \\tau^*_n+\\tau^*\\circ S^{\\tau^*_n}$$ \nFinally, we restrict $(p^*)^{\\tau^*}$ to $\\widehat \\EuScript{X}$, the subset of  all states which can be reached from any state under $(p^*)^{\\tau^*}$ and the state $*$. More precisely, $\\widehat \\EuScript{X}$, defined as follows: \n$$\\widehat\\EuScript{X}  = \\{y \\in \\EuScript{X}^*: (p^*)^{\\tau^*}(x,y) >0\\mbox{ for some }x\\in \\EuScript{X}^* \\}\\cup\\{*\\}.$$\nEquivalently, \n\\begin{align*} \\widehat \\EuScript{X} \n & = \\{y\\in \\EuScript{X} : p^{\\tau}(x,y)>0\\mbox{ for some }x \\in \\EuScript{X}\\}\\cup \\{*\\}.\n \\end{align*} \n \n Denote the restriction of $(p^*)^{\\tau^*}$ to $\\widehat \\EuScript{X}$ by $\\widehat{p^\\tau}$.  The sample paths with respect to the transition function $\\widehat{p^\\tau}$ will be denoted by $\\widehat\\boldsymbol{y}=(\\widehat y_0,\\widehat y_1,\\widehat y_3,\\cdots)$. By construction, the Markov chain associated with the transition function $\\widehat{p^\\tau}$ on $\\widehat \\EuScript{X}$ satisfies Assumption~\\ref{as:required}.\n \n\n \\section{Main Result}\\label{sec:main}\n We are ready to state our main results. \n \\begin{theorem}\n \\label{th:newmain}\n  Suppose that $p$ is a transition function on $\\EuScript{X}$ satisfying Assumption \\ref{as:required} and that $\\tau$ is a stopping time on $\\EuScript{X}$-valued sequences satisfying Assumption \\ref{as:asymp}. \n  Then for any $0\\le u \\in H^\\infty (\\EuScript{X},p^\\tau)$, there exists a function  $\\bar{u}$ on   $\\EuScript{X}\\times {\\mathbb Z}_+$ such that\n  \\begin{enumerate} \n  \t\\item ${\\bar u}(x,0)=u(x),~x\\in \\EuScript{X}$. \n  \\item  $0\\le {\\bar u}\\in S^\\infty (\\EuScript{X},p)$. \n  \\item $\\|{\\bar u} \\|_\\infty =\\|u\\|_\\infty$. \n  \\end{enumerate} \n \\end{theorem} \n Note that if $p$ is steady, then $u(x)= {\\bar u}(x,t)$ for all $t\\in{\\mathbb Z}_+$  hence the embedding in the theorem gives $u\\in H^\\infty(\\EuScript{X},p)$. \n\nTo introduce the next result, recall that that the support of a Borel measure $\\mu$ on a metric space $(M,d)$, $\\mbox{Supp}(\\mu)$,  is defined as \n$$ \\mbox{Supp}(\\mu) = \\{y\\in M: \\mu (U) >0 \\mbox{ if } U\\mbox{ is open and }y\\in U\\}.$$ \nBy definition, the support is closed and its complement is a $\\mu$-null set.\n\n   We say that a transition function $p$ on $\\EuScript{X}$ has a locally finite range if for every $x\\in\\EuScript{X}$, the set $\\{y\\in\\EuScript{X}:p(x,y)>0\\}$ is finite.  \n \\begin{theorem}\n \t\\label{th:martin}\n Suppose that $p$ is a transition function on $\\EuScript{X}$ with locally finite range satisfying  Assumption \\ref{as:required}, and  that \n  $\\tau$ is  a stopping time on $\\EuScript{X}$-valued sequences satisfying Assumption \\ref{as:asymp}. \n  \n  Let  $u\\in H_+(\\widehat{\\EuScript{X}},{\\widehat {p^\\tau}})$ be such that  $\\mbox{Supp}(\\mu_u^{\\widehat{p^\\tau}})\\subseteq \\mbox{Supp}(\\mu_{\\bf 1}^{\\widehat{p^\\tau}})$. \n\nThen there exists a function ${\\bar u}$ on $\\EuScript{X}\\times {\\mathbb Z}_+$ such that \n\\begin{enumerate}\n\t\\item ${\\bar u}(x,0)=u(x),~x \\in \\EuScript{X}$. \n\t\\item  ${\\bar u}\\in S_+(\\EuScript{X},p)$. \n\t\\end{enumerate}\n \\end{theorem}\n The assumption on the support of $u$ is needed to ensure that one can approximate $u$ through bounded harmonic functions. We  need the local finite range assumption  to show that pointwise  limits of the approximating sequence are indeed harmonic sequences, avoiding a strict inequality in Fatou's lemma.  As for the assumption on the support of the measures, it is known that for random walks on regular trees the support of any harmonic function coincides with the minimal Martin boundary (\\cite[Section 8]{Sawyer1997}). Nevertheless, the  assumption on the support of a positive harmonic function does not hold in general, even for transition functions with locally finite range  (\\cite[Sections 6,7]{Sawyer1997}).\n\t\n\nWe now prove  two lemmas we will use to prove Theorem \\ref{th:newmain}. The lemmas will be followed by the proof of the Theorem~ \\ref{th:newmain} and the proof of Theorem~\\ref{th:martin}.\n\n  \\begin{lemma}\n\\label{lem:inv_to_tail}\nUnder the conditions of Theorem \\ref{th:newmain}, for  $A \\in \\partial({\\widehat \\EuScript{X}}, \\widehat {p^\\tau})$ there exists $I_A\\in {\\cal T} (\\EuScript{X}^*,p^*)$ such that \n\\begin{equation} \n\\label{eq:inv_in_tail} \\{\\lim_{n\\to\\infty} y^*_n \\in  A\\} = I_A,~\\mbox{\\bf P}_x-a.s. \\mbox{ for all }x\\in {\\widehat \\EuScript{X}}-\\{*\\}.\n\\end{equation}  \nFurthermore, if $A$ and $A'$ are disjoint,  $I_A$ and $I_{A'}$ are disjoint. \n\\end{lemma} \n\\begin{proof} \nWe split the proof of the lemma into two parts. In the first part, we show that \\eqref{eq:inv_in_tail} holds for all $A$ which are intersection of $\\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau})$ with an open ball (in the Martin topology on $\\widehat \\EuScript{X}$ relative to the transition function $\\widehat{p^\\tau}$),  centered at a point in $\\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau})$. Once this is proved, we show how the lemma extends to all Borel (in the subspace topology)  subsets of $\\partial(\\widehat \\EuScript{X}, \\widehat{p^\\tau})$. \n\nWe begin with the first part. Let $B_\\epsilon(\\zeta)$ be a neighborhood of $\\zeta$ in  $\\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau})$\n with radius $\\epsilon$. Since the topology on $\\partial(\\widehat \\EuScript{X}, \\widehat{p^\\tau})$ is the induced topology from the compact metric space $\\widehat\\EuScript{X}\\cup \\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau})$, a basis for the topology on $\\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau}) $ is the collection of sets of the form\n\n\\begin{equation}\n\\label{eq:special_As} \nA = B_\\epsilon(\\zeta)\\cap \\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau})\\mbox{ for some } \\epsilon>0\\mbox{  and }\\zeta \\in \\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau}).\n\\end{equation} \n \n Fix  $\\zeta$ in $\\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau})\n$ and $\\epsilon>0$. Let  $A = B_\\epsilon(\\zeta)  \\cap \\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau})$ and let  $B = B_\\epsilon(\\zeta) \\cap \\widehat \\EuScript{X}$.  Clearly, \n$$\\{ \\widehat y_\\infty  \\in A\\}=\\{x^*_{\\tau_n}\\in B \\mbox{ eventually}\\}$$ because by definition, $x^*_{\\tau_n} = \\widehat y_n$ for $n\\in{\\mathbb Z}_+$. Denote the event on the right hand side by $B_\\infty$. Therefore instead of dealing with the event on the left hand side, we will work with $B_\\infty$. Let \n$$\nK_n = \\left\\{\\boldsymbol{x} : \\rho_n(\\boldsymbol{x}) \\not\\in \\langle \\tau(\\boldsymbol{x})\\rangle\\right\\}.\n$$\n By assumption,  there exists a subsequence $(n_i:i\\in{\\mathbb N})$ such that \n$$ \\sum_{i} \\mbox{\\bf P}_x(K_{n_i})<\\infty,~x\\in \\EuScript{X}.$$ \nTherefore the event $\\Gamma=\\{K_{n_i} \\mbox{ finitely often}\\}$ has $\\mbox{\\bf P}_x (\\Gamma)=1$ for all $x\\in \\EuScript{X}$. Write $\\Gamma_x = \\Gamma \\cap \\{x_0=x\\}$. For each $i$, let $\\rho_{i,0} = \\rho_{n_i}$, and continue inductively, \n$$\n\\rho_{i,j+1} = \\rho_{i,j} + \\tau^* \\circ S^{\\rho_{i,j}}.\n$$\n Observe then that $\\rho_{i,j}$ are all ${\\cal F}_{n_i}^\\infty(\\EuScript{X}^*)$-measurable. Let \n$$\nC_{i} = \\bigcap_{j\\in{\\mathbb Z}_+} \\left\\{x^*_{\\rho_{i,j}} \\in B\\right\\}\n$$\nand \n$C=\\limsup C_i =  \\cap_{k=1}^\\infty \\cup_{i\\ge k} C_i.$\n Therefore $\\cup_{i\\ge k} C_i \\in {\\cal F}_{n_i}^\\infty(\\EuScript{X}^*)$, and since this union is decreasing in $k$, it follows that $C$ belongs to ${\\cal F}_{n_k}^\\infty(\\EuScript{X}^*)=S^{-n_k} ({\\cal F}^\\infty(\\EuScript{X}^*))$ for all $k\\in{\\mathbb N}$, that is $C\\in {\\cal T} (\\EuScript{X}^*)$. Clearly, on $\\Gamma_x$, $C$ implies $B_\\infty$, and conversely, on $\\Gamma_x$, $B_\\infty$ implies $C$. Therefore, \n$$ \nB_\\infty = C,~\\mbox{\\bf P}_x-\\mbox{ a.s.,}~x \\in\\widehat\\EuScript{X}-\\{*\\}.\n$$ \nThis proves \\eqref{eq:inv_in_tail} for the particular choice of $A$, with $I_A = C$. Note that by construction, if $A$ and $A'$ are disjoint, so are the corresponding $C$ and $C'$. \n \nWe continue to the second part. Under the subspace topology,  $\\partial(\\widehat \\EuScript{X},\\widehat{p^\\tau})$ is a compact metric space. Therefore  it is separable and  every open set is a countable union of such sets from this basis, and every compact subset is a complement of such a countable union.  \nIn particular, it follows from the first stage that for any compact set $K\\in \\partial(\\widehat \\EuScript{X},\\widehat{p^\\tau})$, there exists $I_K\\in \\overline\\mathcal{T}(\\EuScript{X}^*,{p^*})$, such that \n$$  \\{\\widehat y_\\infty \\in K\\}= I_K,~\\mbox{\\bf P}_x-\\mbox{a.s., for all }x\\in \\widehat\\EuScript{X}-\\{*\\}.$$  \n\nNote that  $\\mu_{\\bf 1}^{\\widehat{p^\\tau}}$ is a probability measure on a compact metric space (due to extension explained below Theorem \\ref{th:doob59}),  it is regular. So for a fixed  Borel set $A \\subseteq\\partial(\\widehat \\EuScript{X},\\widehat{p^\\tau})$, there exists an increasing sequence $(Q_j:j\\ge 1)$ such that for any $j$ the set $Q_j$ is a compact subsets of  $\\partial( \\widehat \\EuScript{X}, \\widehat{p^\\tau})$ and $\\mu_{\\bf 1}^{\\widehat{p^\\tau}} (A-Q_j) \\to 0$. Now \n$ \\{\\widehat y_\\infty \\in \\cup Q_j \\}=\\cup \\{\\widehat y_\\infty \\in Q_j\\}$, and therefore, there exists $I_{H} \\in {\\cal T} (\\EuScript{X}^*,p^*)$ such that \n$$ \\{\\widehat y_\\infty \\in \\cup Q_j\\} = I_H,\\quad \\mbox{\\bf P}_x-\\mbox{a.s.}\\quad x \\in \\widehat \\EuScript{X} - \\{*\\}.$$ \nWe also observe that \n $$\\mbox{\\bf P}_x ( \\widehat y_\\infty \\in A -\\cup Q_j) = \\int_{A-\\cup Q_j} K^{\\widehat{p^\\tau}}(x,\\zeta)d \\mu_{\\bf 1}^{\\widehat{p^\\tau}} (\\zeta)=0,$$ \n for   $x \\in \\widehat\\EuScript{X}$. \nNote that for $x=*$ or $x\\in \\EuScript{X} - \\widehat \\EuScript{X}$, then \n$$\\mbox{\\bf P}_x (y^*_\\infty \\in A - \\cup Q_j) = \\sum_{y\\in \\widehat\\EuScript{X}-\\{*\\}} {\\widehat{p^\\tau}}(x,y)\\mbox{\\bf P}_y(\\widehat y_\\infty \\in A -\\cup Q_j) =0.$$ \nBy completeness, it follows that the event $\\{y^*_\\infty \\in A - \\cup Q_j\\}$ is in $\\overline{{\\cal T}}(\\EuScript{X}^*,p^*)$. From this we conclude that \n$$ \\{\\widehat y_\\infty \\in A\\}= I_H\\quad \\mbox{\\bf P}_x \\mbox{-a.s.}\\quad x \\in \\widehat\\EuScript{X}-\\{*\\}.$$ \nThis completes the proof of the second part, and of the lemma. \n\\end{proof} \n\\begin{lemma}\n\\label{lem:supremum}\nFor $x$ in $\\EuScript{X}$ and $t$ in ${\\mathbb Z}_+$, let $v(x,t) = \\sum_{i=1}^N c_i \\mbox{\\bf P}_{x,t} (I_{A_i})$,  \nwhere $c_i\\ge 0$ and $A_i\\in \\partial( \\widehat \\EuScript{X},\\widehat{p^\\tau})$ are disjoint and  $I_{A_i}$ is as in Lemma \\ref{lem:inv_to_tail}. Then \n$$\\|v\\|_\\infty = \\sup_{x\\in \\widehat \\EuScript{X}-\\{*\\}}\\|v(x,0)\\|_{\\infty}.$$ \n\\end{lemma}\n\\begin{proof}\n  Clearly, $\\|v \\|_\\infty \\ge \\|v(\\cdot,0)\\|_\\infty$. \n  On the other hand,\n  $$ v(x,0) = \\sum c_i \\mbox{\\bf P}_x (y^*_\\infty \\in A_i),$$ therefore by Theorem \\ref{th:limits}, $\\sup_{x \\in \\widehat \\EuScript{X}-\\{*\\}} v(x,0)=\\max c_i$,\n  while\n  $$ v(x,t) = \\sum c_i \\mbox{\\bf P}_{x,t} (I_{A_i}) \\le \\max c_i,$$ \n  because $I_{A_i}$ are disjoint, $\\mbox{\\bf P}_{x,t}$-a.s.  for all $(x,t)\\in \\EuScript{X}^*\\times {\\mathbb Z}_+$.   \n\\end{proof} \n\\begin{proof}[Proof of Theorem \\ref{th:newmain}] \nWithout loss of generality, we can assume that $\\theta$ is such that  $\\sum_{x\\in \\EuScript{X}} \\theta (x) u(x)<\\infty$. Thus, extend $u$ to $\\EuScript{X}^*$ by letting $u(*) = \\sum_{x\\in \\EuScript{X}} \\theta (x) u(x)$, and let $\\widehat u$ be the restriction of $u$ to $\\widehat \\EuScript{X}$. By assumption, there exists $0\\le f\\in L^1(\\mu_1^{\\widehat {p^\\tau}})$ such that \n$$\n\\widehat u (x) = \\int_{\\partial_m( \\widehat{\\EuScript{X}} , \\widehat{p^\\tau})}  K^{\\widehat{p^\\tau}}(x,\\zeta) f (\\zeta) d \\mu_1^{\\widehat {p^\\tau}}(\\zeta).\n$$ \nThere exists  a nondecreasing sequence of nonnegative simple functions   $(f_n:n\\in{\\mathbb N})$ on $\\partial_m( \\widehat{\\EuScript{X}} , \\widehat{p^\\tau})$ such that $f_n \\nearrow f$. Letting \n$$\\widehat u_n (x) =  \\int_{\\partial_m( \\widehat{\\EuScript{X}} , \\widehat{p^\\tau})} K^{\\widehat{p^\\tau}}(x,\\zeta) f_n (\\zeta) d \\mu_1^{\\widehat{p^\\tau}}(\\zeta),$$\nit follows from Lemma \\ref{lem:inv_to_tail} that there exists a combination $\\sum c_i {\\bf 1}_{I_{A_i}}$, $I_{A_i}\\in {\\cal T} (\\EuScript{X}^*,p^*)$, such that\n$$ \\widehat u_n (x) = \\sum c_i \\mbox{\\bf P}_{x,0} (I_{A_i}),\\quad x \\in \\widehat \\EuScript{X}-\\{*\\}.$$\n\nLet $v_n(x,t) = \\sum c_i \\mbox{\\bf P}_{x,t} (I_{A_i}),~(x,t)\\in \\EuScript{X}^*\\times {\\mathbb Z}_+$. Then $v_n \\in  S^\\infty(\\EuScript{X}^*,p^*)$.  \nSince by Lemma \\ref{lem:supremum},  \n$\\|v_n\\|_\\infty\\le \\sup_{x\\in \\widehat \\EuScript{X}-\\{*\\}} \\|v_n(x,0)\\|_\\infty\\le \\|\\widehat u\\|_\\infty$, we can extract a subsequence $(v_{n_k})$ which converges pointwise. Clearly, $v_\\infty (x,0)=u(x)$ on $\\widehat \\EuScript{X}-\\{*\\}$. However, it also follows from dominated convergence that  $v_\\infty \\in S^\\infty(\\EuScript{X}^*,p^*)$. Furthermore, \n$\\|v_\\infty\\|_\\infty \\le \\|\\widehat u\\|_\\infty$.\n\\end{proof} \n\n\\begin{remark} Here is an outline of an alternative proof to Theorem  \\ref{th:newmain}, based entirely on the construction of the Poisson boundary presented in Section \\ref{sec:Poisson} and  avoiding the notion of Martin boundary. The proof was suggested  by the referee. \\\\ \n\nLet $u\\in H^\\infty(\\EuScript{X},p^{\\tau})$, and let $\\phi_u$ be the element in $L^\\infty({\\cal PB}(\\EuScript{X},p^\\tau))$ obtained through \\eqref{eq:martin-approach}:\n\\begin{equation}\n\\label{eq:y_embed}\n\\phi_u ({\\boldsymbol{y}}) = \\lim_{n\\to\\infty} u(y_n), \\quad \\mbox{\\bf P}_{\\theta}\\mbox{-a.s.},\n\\end{equation}\nNow $\\boldsymbol{y}$ is a deterministic function of $\\boldsymbol{x}$: $\\boldsymbol{y} = \\boldsymbol{y}(\\boldsymbol{x})$ through $y_n = x_{\\tau_n}$, and therefore one can rewrite the lefthand side as a function of $\\boldsymbol{x}$, ${\\widetilde{\\phi_u}}(\\boldsymbol{x}) = \\phi_u(\\boldsymbol{y}(\\boldsymbol{x}))$. \nThe  Borell-Cantelli argument in the heart of Lemma \\ref{lem:inv_to_tail} gives a sequence $(\\rho_{n_i}:i=1,2,\\dots)$ of ${\\cal F}_{n_i}^\\infty(\\EuScript{X})$-measurable random variables with $n_i \\nearrow \\infty$, and $\\rho_{n_i}  \\in \\langle \\tau \\rangle$, eventually $\\mbox{\\bf P}_{\\theta}$-a.s.  Therefore the righthand side of \\eqref{eq:y_embed} is  $\\lim_{i\\to\\infty} u(x_{\\rho_{n_i}})$,  $\\mbox{\\bf P}_{\\theta}$-a.s., and is therefore  $\\overline{{\\cal T}}(\\EuScript{X},p)$-measurable. Thus, we have obtained an embedding \n$$H^{\\infty}(\\EuScript{X},p^\\tau) \\ni u \\hookrightarrow \\widetilde{\\phi_u} \\in L^\\infty ({\\cal TB}(\\EuScript{X},p)).$$\n\\end{remark} \n\nWe now prove Theorem \\ref{th:martin}. \n\\begin{proof}[Proof of Theorem \\ref{th:martin}]\n\tBy the assumption on  the support of $\\mu_u^{\\widehat p^\\tau}$, there exists a sequence $f_n\\in L^\\infty (\\mu_{\\bf 1}^{\\widehat {p^\\tau}})$ such that $f_n d\\mu_1^{\\widehat{p^\\tau}}$ converges weakly to $\\mu^{\\widehat {p^{\\tau}}}_{u}$ (see Proposition \\ref{pr:approx}). Since for each $x\\in \\widehat \\EuScript{X}$, the  mapping $\\zeta \\to  K^{\\widehat{p^\\tau}}(x,\\zeta)$ is bounded and continuous on the compact metric space $\\partial ({\\widehat \\EuScript{X}},{\\widehat {p^\\tau}} )$, it follows that each of the functions \n\t$$u_n (x) = \\int K^{\\widehat{p^\\tau}}(x,\\zeta) f_n (\\zeta) d\\mu_{\\bf 1}^{\\widehat {p^\\tau}},~x\\in{\\widehat \\EuScript{X}}$$ \n\tis in $H^\\infty(\\widehat{\\EuScript{X}},\\widehat{p^\\tau})$ and  the sequence $(u_n :n\\in{\\mathbb N})$  converges pointwise to $u$. By Theorem \\ref{th:newmain}, there exists a function $0\\le \\overline{u}_n \\in S^\\infty(\\EuScript{X},p)$ such that  $\\overline{u}_n(x,0)=u_n (x),~x \\in \\EuScript{X}$.   \n\tSince $p \\overline{u}_n(x,t+1)=\\overline{u}_n (x,t),~x\\in \\EuScript{X}$,  it follows that $p(x,y) \\overline{u}_n(y,t+1)\\le \\overline{u}_n(x,t)$ whenever $p(x,y)>0$, or \n\t$$ \\overline{u}_n (y,t+1)\\le \\inf \\{ \\frac{\\overline{u}_n(x,t)}{p(x,y)}: x~, p(x,y)>0\\}.$$\n\tIterating, and using the fact that $p$ is irreducible, we have that for every $(y,t)\\in \\EuScript{X}\\times {\\mathbb Z}_+$ there exist $x=x(y,t)\\in \\EuScript{X}$ and a constant $c(y,t)>0$, both not depending on $n$, such that \n\t$\\overline{u}_n(y,t) \\le c(y,t)\\overline{u}_n(x(y,t),0)$. Since $\\overline{u}_n (x,0)$ converges to $u(x)$ as $n\\to\\infty$ for every $x\\in \\EuScript{X}$,  it follows that the sequence of nonnegative numbers $(\\overline{u}_n (y,t):n\\in{\\mathbb N})$ is bounded. As a result, there exists a subsequence $(n_j:j\\in{\\mathbb N})$ such that $\\overline{u}_{n_j}(x,t)$ converges to a finite limit at all $(x,t)\\in \\EuScript{X}\\times{\\mathbb Z}_+$. Denote this limit function by $\\overline{u}$. Clearly, $\\overline{u}(x,0)=u(x)$. Since $p$ is locally finite, \n\t$$ p \\overline{u} (x,t+1)= \\sum_{y} p(x,y)\\lim_{j\\to\\infty} \\overline{u}_{n_j} (y,t+1) = \\lim_{j\\to\\infty} p \\overline{u}_{n_j}(x,t+1)=\\lim_{j\\to\\infty}\\overline{u}_{n_j}(x,t)= \\overline{ u}(x,t),$$ \n\tcompleting the proof. \t \t  \n\\end{proof} \n\n\n\n\n\\section{Examples}\\label{sec:examples}\nIn this section, we provide some examples.\n\\subsection{Deterministic Stopping times} \nSuppose that $\\tau=c$. Let $\\rho = 1$, we have that $\\mbox{\\bf P}_x (\\rho_n \\in \\langle \\tau \\rangle) =1$ whenever $n+1$ is multiple of $c$.\n\n\\subsection{First Passage times}\\label{sec:hitting}\nLet $A$ be a recurrent set for $p$ and $\\tau = \\inf\\{n \\ge1: x_n \\in A\\}$. Setting $\\rho = \\tau$ satisfies the condition of Assumtption \\ref{as:asymp}. This is the generalization of Furstenberg's result for random walks on groups,  when $A$ is a recurrent subgroup  \\cite{Furstenberg1973}.\nWe will now show how this can be used to study equivalence of bounded harmonic functions on product spaces. Suppose that $p$ is an irreducible and  transient  transition function on $X \\times Y$, where $X$ and $Y$ are nonempty countable sets. $X$ may be finite. We will  assume that there exists $o\\in X$ such that the set  $A=\\{o\\}\\times Y$ is $p$--recurrent. Write $\\boldsymbol{x}=(\\boldsymbol{x}(1),\\boldsymbol{x}(2))$ for the corresponding Markov chain,  where $\\boldsymbol{x}(1)$ is an $X$-valued process, and $\\boldsymbol{x}(2)$ is a $Y$-valued process. Note that in general neither $\\boldsymbol{x}(1)$ nor $\\boldsymbol{x}(2)$ are Markov chains. Let $\\tau$ denote the first passage time to $A$: \n$$ \\tau = \\inf\\{t\\ge 1: {x}_t(1) = o\\}.$$ \nThen $\\rho=\\tau$ satisfies Assumption \\ref{as:asymp}.\nObserve next that $p^\\tau$ induces a transition function $\\gamma$  on $Y$ through the relation $$\\gamma(y,y') = p^{\\tau}((o,y),(o,y')),~x,y\\in Y.$$\n Given $x\\in X$ and $y \\in Y$, let $v((x,y)) = \\mbox{\\bf E}_{(x,y)} [ u({y}_{\\tau})]$. Clearly,  $v((o,y)) = \\gamma u (y) = u(y)$. Therefore, $p^{\\tau}v((o,y))=\\gamma u(y) = u(y) = v ((o,y))$. Next, if $x\\ne o$, we have \n$$p^\\tau v ((x,y)) = \\sum_{y'}p^\\tau((x,y),(o,y')) \\mbox{\\bf E}_{(o,y')} [ u({ y}_{\\tau})]=\\sum_{y'}p^{\\tau}((x,y),(o,y')) u(y')=v((x,y)).$$  \n      \nOne easy example is $X={\\mathbb Z}$,  $Y={\\mathbb Z}^{d-1}$, $d\\ge 3$,  and   $p$ being the simple symmetric random walk on $X \\times Y = {\\mathbb Z}^d$ and $o=0$.  It is known that the Markov chain is steady, and that all bounded harmonic functions are constants, so in particular, $S^\\infty(\\EuScript{X},p)$ consists only of constant functions. Furthermore, the first (or any component) is recurrent. Thus  $\\gamma$ is a transition function on  ${\\mathbb Z}^{d-1}$ which is symmetric, but not nearest neighbor (in fact, it is easy to see that $\\sum_{y\\in {\\mathbb Z}^{d-1}} \\gamma(0,y)|y|=\\infty$). From Theorem \\ref{th:newmain} we therefore obtain that all bounded harmonic functions for $\\gamma$ are constants. \n\n\\subsection{Additive functionals} \nRecall that an additive functional for a Markov chain is a real-valued process $I= (I_n:n \\ge 0)$, such that $I_{n+k} = I_n + I_k \\circ S^n$ and $I_k$ is measurable with respect to the sigma-algebra generated by the first $k+1$ coordinate functions for all $k$ (enough for $k=0,1$). An example for an additive functional is $I_n = \\sum_{k\\le n} f(x_k)$ where $f:{\\EuScript{X}}\\to {\\mathbb R}$ is any function. \\\\\nLet $(I_n:n \\ge 0)$ be an additive functional for $\\boldsymbol{x}$ that satisfies $\\displaystyle\\lim_{n\\to\\infty} I_n = \\infty$ $\\mbox{\\bf P}_x$-a.s. Let $\\tau = \\inf \\{n\\ge 1: I_n > I_{n-1}\\}$. Setting $\\rho = \\tau$ clearly satisfies the condition. Note that the above example is a special case, with the additive functional counting the number of visits to $A$. On the other hand, letting $\\tau = \\inf \\{n: I_n \\ge a\\}$ for some fixed $a$, in general does not satisfy the condition.\n\n\\subsection{A generic choice for $\\rho$} \nLet $\\EuScript{X}$ be a  free semigroup generated by the finite nonempty set ${\\cal G}$. That is, the elements of ${\\EuScript{X}}$ are finite sequences of elements in ${\\cal G}$, the empty sequence included, denoted by $\\emptyset$. Given $x \\in {\\EuScript{X}}$, we write $ x g$ for the sequence in ${\\EuScript{X}}$ obtained by concatenating  $g$ to $x$ from the right.  If $y=xg$, we write $g=x^{-1}y$  and refer to $g$ as the increment. Assume that for any $x \\in \\EuScript{X}$ and $g\\in {\\cal G}$ the transition function $p$ is invariant under the action of semigroups that is  $p(x,xg) = p_g$, where $g\\to p_{g}$ is any probability measure on ${\\cal G}$. \nWe will also assume that $\\tau$ is a stopping time invariant under the action of semigroup in the following sense\n\\begin{equation} \n\\label{eq:invariant_tau}\\tau (\\omega_0,\\omega_1, \\dots) =\\tau (x\\omega_0,x\\omega_1, \\dots) = \\tau(\\emptyset, \\omega_0^{-1}\\omega_1,\\omega_0^{-1} \\omega_2,\\dots)\n\\end{equation}\nfor all $x$ in the semigroup $\\EuScript{X}$.\nThat is, $\\tau$ is  a function of the consecutive increments rather than the actual path.\nLet $\\tau$ satisfy the following two additional conditions: \n\n\\begin{itemize}\n\\item[i)] $\\tau$ is bounded by $M \\in {\\mathbb Z}_+$; \n\\item[ii)] $\\mbox{\\bf P}_\\emptyset (\\tau =1)> 0$. \n\\end{itemize}\nLet $A=\\{g\\in {\\cal G}:\\tau(\\emptyset,g)=1\\}$. \nObserve that from assumption (ii), $\\sum_{g\\in A} p_g>0$, and therefore $\\mbox{\\bf P}_x$-a.s., given  a path $(\\omega_0,\\omega_1,\\dots)$, its associated sequence of increments $(\\omega_0^{-1}\\omega_1,\\omega_1^{-1}\\omega_2,\\dots)$ contains infinitely many  runs (of consecutive increments) in $A$ longer than any fixed $k$. By (i), any ``time'' interval of length longer than $M$ contains at least one element in $\\langle \\tau\\rangle$ before its last element. If, in addition, all increments corresponding to this time interval are in $A$, it necessarily follows that the last element is in $\\langle \\tau \\rangle$. This simple idea translates to the following definition of $\\rho$:\n$$\\rho(\\boldsymbol{x}) = \\inf\\left\\{n>3M: x_{i-1}^{-1}x_i \\in A,~\\mbox{ for all } i =n-2M,\\dots,n  \\right\\}.$$\nFrom the definition of the random walk, $\\rho$ is finite $\\mbox{\\bf P}_x$-a.s. for any $x\\in \\EuScript{X}$. Also, because $\\tau \\le M$, between time   $\\rho_n - 2M$ and $\\rho_n$ there exists at least one element of $\\langle \\tau \\rangle$, call the first such element $\\tau_m$. Since all consecutive increments until time $\\rho_n$ are in $A$,  all elements of the sequence $(\\tau_m, \\tau_m+1, \\dots, \\rho_n)$ are in $\\langle \\tau \\rangle$.  Thus  $\\mbox{\\bf P}_x (\\rho_n \\in \\langle \\tau \\rangle )=1$ and Assumption \\ref{as:asymp} is satisfied. \n\nThe second author and Kaimanovich show that the Poisson boundary of random walks on countable groups preserved under any stopping times. Moreover, they show that the results hold for randomized stopping times with finite logarithmic moment. They first lift random walks on a countable group to a random walk on a free finitely or infinitely countable generated) semigroup whose first step of the random walk is distributed on the generators of the free semigroup, and show their results in this setup and then apply them to prove the results for any countable groups, see \\cite{Behrang} and \\cite{Forghani-Kaimanovich2016} for more details. However, our result in the above example provides a different proof for special stopping times on finitely generated free semigroups.\n\n\\subsection{ $\\boldsymbol\\rho$ is not $\\boldsymbol\\tau$}\nWe want to show that also in general choosing $\\rho=\\tau$ will not satisfy Assumption~\\ref{as:asymp}.  Let $\\EuScript{X}$ be the free  semigroup generated by ${\\cal G}=\\{a,b\\}$. Then for every $x$ we define  transition function   on ${\\EuScript{X}}$ by $$ p(x,xa)=p(x,xb)=\\frac12.$$ \nFor any sample path $\\boldsymbol{x}=(x_0,x_1,x_2,\\cdots)$ and $x$ in $\\EuScript{X}$, define the stopping time  \n$$\\tau(\\boldsymbol{x})=\\begin{cases}\n1 & x_0=x,\\ x_1=xa\\\\\n2 & x_0=x,\\ x_1=xb.\n\\end{cases}$$\n\n\nWe claim that $\\tau$ does not satisfy Assumption~\\ref{as:asymp}.  By contradiction,  suppose that  $\\rho=\\tau$. \n Then \n\\begin{align*} \n\t \\mbox{\\bf P}_x (\\rho_n \\in \\langle \\tau \\rangle )&=\\mbox{\\bf P}_x(\\rho_n \\in  \\langle \\tau \\rangle | n \\in \\langle \\tau \\rangle  )\\mbox{\\bf P}_x( n \\in \\langle \\tau \\rangle)+\\mbox{\\bf P}_x(\\rho_n \\in  \\langle \\tau \\rangle | n \\not \\in \\langle \\tau \\rangle  )\\mbox{\\bf P}_x( n \\not\\in \\langle \\tau \\rangle)\\\\\n\t &= 1 \\times \\mbox{\\bf P}_x( n \\in \\langle \\tau \\rangle) + \\frac 34 (1- \\mbox{\\bf P}_x( n \\in \\langle \\tau \\rangle) )\\\\\n\t & = \\frac 14(3+\\mbox{\\bf P}_x( n \\in \\langle \\tau \\rangle)),\n\t\\end{align*}\n\twhere the $\\frac 34$ factor on the second line is because if $n \\not\\in \\langle \\tau \\rangle$, then $n+1 \\in \\langle \\tau \\rangle$ and so  $\\rho_n\\in  \\langle \\tau \\rangle$ if and only if either: i) $\\rho_n=n+1$, that is $x_n^{-1}x_{n+1}=a$; or ii)  $\\rho_n=n+2$ and $n+2\\in \\langle \\tau \\rangle$, that is  $x_{n}^{-1}x_{n+1} = b$ and $x_{n+1}^{-1}x_{n+2}=a$. Finally, letting $\\alpha_n = \\mbox{\\bf P}_x (n \\in \\langle \\tau \\rangle)\n$, we have $\\alpha_1=\\frac 12$, $\\alpha_2= \\frac 34$ and by conditioning on the first increment, for $n\\ge 3$, $\\alpha_n=\\frac 12 \\alpha_{n-1}+ \\frac 12 \\alpha_{n-2}$. As a result, $\\lim_{n\\to\\infty}\\alpha_n = \\frac 23$. \tIt follows that \n$$ \\displaystyle\\lim_{n\\to\\infty}\\mbox{\\bf P}_x (\\rho_n \\in \\langle \\tau \\rangle )= \\frac{11}{12}<1.$$ \n\n\\subsection{Delayed stopping} \nWe construct a Markov chain and a corresponding stopping time with the following property. There exist states such that for every choice of  $\\rho$, $\\mbox{\\bf P}_x (\\rho_n \\in \\langle \\tau \\rangle)<1$, for all $n$ large. Nevertheless,  we can define   $\\rho$ so that   $\\displaystyle\\lim_{n\\to\\infty} \\mbox{\\bf P}_x (\\rho_n \\in \\langle \\tau\\rangle) =1$ for all $x\\in \\EuScript{X}$. \\\\\n Let ${\\EuScript{X}}={\\mathbb N}\\times \\{0,1\\}$. Let $r$ be any irreducible and transient transition function on ${\\mathbb N}$. We will also assume that $r$ is lazy, that is $r(x,x) =\\frac 12$ for all $x$.  Let $s$ be a transition function on $\\{0,1\\}$ such that  $0$ is an absorbing state, $s(0,0)=1$, and $s(1,0)=s(0,1)=\\frac12$.\n For $n_1,n_2 \\in {\\mathbb N}$ and $d_1,d_2 \\in \\{0,1\\}$, let \n$$\np((n_1,d_1),(n_2,d_2)) = r(n_1,n_2)s(d_1,d_2).\n$$\nThe space of sample paths of the Markov chain  with transition function  $p$ can be expressed in terms of the component processes. That is, if $\\boldsymbol{x}$ denotes that chain, then  $\\boldsymbol{x}=(\\boldsymbol{x}(1),\\boldsymbol{x}(2))$, with $\\boldsymbol{x}(1)$ and  $\\boldsymbol{x}(2)$ are two independent sample paths with respect to $r$ and $s$, receptively.  Assume that $A$ and $B$ are disjoint recurrent sets for $r$. Then clearly, both $A$ and $B$ are infinite.  Let $\\sigma = \\inf\\{n\\ge 1: x_n(1) \\in A\\}$, and continue inductively $\\sigma_1=\\sigma,~ \\sigma_{n+1} = \\sigma\\circ {S^{\\sigma_n}}$.  Define  \n$$\n\\tau = \\begin{cases} \n\\sigma & x_0(2) =0\\\\ \\sigma_{\\sigma} & x_0(2)=1 \n\\end{cases}\n$$ \nIn other words, if $x_0(2)=0$, we stop when $\\boldsymbol{x}(1)$ hits $A$. Otherwise, we wait until $\\sigma$, and then stop after $\\boldsymbol{x}(1)$ hits  $A$ an additional $\\sigma-1$ more times. Let $m$ be in $B$. Suppose that $\\rho:{\\EuScript{X}}^{{\\mathbb Z}_+}\\to {\\mathbb Z}_+$. \nWe begin by observing that \n$$\\mbox{\\bf P}_{(m,1)}(\\rho_n \\in \\langle \\tau\\rangle) \\le \\mbox{\\bf P}_{(m,1)}(\\sigma \\le n) + \\mbox{\\bf P}_{(m,1)} (\\rho_n \\in \\langle \\tau\\rangle,\\sigma>n).$$\nNow since $x_0(2)=1$, it follows from the definition of $\\tau$, that under $\\mbox{\\bf P}_{(m,1)}$, we have $\\tau=\\sigma_{{\\sigma}}$. On the event $\\sigma>n$, this automatically implies $\\tau_1>n$, and in particular, if $\\rho_n \\in \\langle \\tau\\rangle $, we must have $\\rho_n > \\sigma+ (\\sigma-1) > 2n$. As a result, the event $\\{\\rho_n \\in \\langle \\tau \\rangle\\}\\cap  \\{\\sigma>n \\} $ is contained in the event $\\{\\rho \\circ S^n > n\\}\\cap \\{\\sigma>n \\}$. This, with the Markov property imply:  \n\\begin{align} \\nonumber \\mbox{\\bf P}_{(m,1)}(\\rho_n \\in \\langle \\tau\\rangle) &\\le  \\mbox{\\bf P}_{(m,1)}(\\sigma \\le n)  + \\mbox{\\bf P}_{(m,1)}(\\rho\\circ S^n>n,\\sigma>n) \\\\\n\\label{eq:as} \n& =  \\mbox{\\bf P}_{(m,1)}(\\sigma \\le n) +  \\mbox{\\bf E}_{(m,1)} [\\mbox{\\bf P}_{\\boldsymbol{x}_n}(\\rho>n),\\sigma>n].\n\\end{align}\nSince we assume that $\\rho$ is finite $\\mbox{\\bf P}_{(m,0)}$ a.s., there exists $n_0\\in{\\mathbb N}$ such that  for all $n\\ge n_0$,  $\\mbox{\\bf P}_{(m,0)}(\\rho>n) <\\frac 12$. Under $\\mbox{\\bf P}_{m,1}$, the event \n$$\nC_n=\\{x_0(1)=x_1(1)=\\dots=x_n(1)\\}\\cap \\{x_n(2) = 0\\}\n$$ has probability $2^{-n} \\times (1-2^{-n})$, and is  contained in  $\\{\\sigma>n\\}$. Thus for $n\\ge n_0$, \n\\begin{align*} \n \\mbox{\\bf E}_{(m,1)} [\\mbox{\\bf P}_{\\boldsymbol{x}_n}(\\rho>n), \\sigma>n] &\\le  \\mbox{\\bf P}_{(m,1)} ( C_n)\\times  \\mbox{\\bf P}_{(m,0)}(\\rho>n)+ \\mbox{\\bf P}_{(m,1)} (C_n^c,\\sigma>n) \\\\\n & <\\frac 12 \\mbox{\\bf P}_{(m,1)}(C_n) +  \\mbox{\\bf P}_{(m,1)} (C_n^c ,\\sigma>n)\\\\\n & =  \\frac 12 \\mbox{\\bf P}_{(m,1)}(C_n) + \\mbox{\\bf P}_{(m,1)} (\\sigma>n) - \\mbox{\\bf P}_{(m,1)}(C_n,\\sigma>n)\\\\\n & =\\mbox{\\bf P}_{(m,1)} (\\sigma>n)  - \\frac 12 \\mbox{\\bf P}_{(m,1)}(C_n),\n \\end{align*}\n where the last equality is due to the fact that under $\\mbox{\\bf P}_{(m,1)}$, $C_n \\subseteq \\{\\sigma>n\\}$. Plug this inequality into  \\eqref{eq:as} to obtain \n $$  \\mbox{\\bf P}_{(m,1)}(\\rho_n \\in \\langle \\tau\\rangle) \\le 1 - \\frac 12 2^{ -n} (1-2^{-n}),~n\\ge n_0.$$ \n \nNext we show that choosing $\\rho=\\sigma$ satisfies $\\displaystyle\\lim_{n\\to\\infty} \\mbox{\\bf P}_{x} (\\rho_n \\in \\langle \\tau \\rangle) =1$, for all $x$ (note:  we can also choose $\\rho=\\tau$). Let $N_{i,j}$ count the number of times $\\boldsymbol{x}(1)$ visits $A$ between times $i$ and $j$. That is, \n$$\nN_{i,j} = \\sum_{i \\le n \\le j} {\\bf 1}_A (x_n(1)).\n$$\nObserve that from the definition of $\\tau$, if $x^2_k =0$, then the times of the $k$-th, $k+1$-th, etc. hits of $A$ by $\\boldsymbol{x}(1)$ will all be in $\\langle \\tau \\rangle$. Letting $T = \\inf \\{n: \\boldsymbol{x}_n(2) =0\\}$, we have \n$$ \\mbox{\\bf P}_x (\\rho_n \\in \\langle \\tau \\rangle) \\ge \\sum_{k} \\mbox{\\bf P}_x ( T = k,N_{k,n}\\ge k).$$ \nSince $A$ is recurrent, $ N_{k,n}\\underset{n\\to\\infty}{ \\nearrow}  \\infty$, and so the result follows from monotone convergence. \n\n\\subsection{Splitting of Poisson Boundary}\\label{splitting}\nThis final example is of a transition function $p$ and a stopping time  where $S^\\infty (\\EuScript{X},p)$ consists only of constant functions, yet $H^\\infty(\\EuScript{X},p^{\\tau})$ contains at least two linearly independent elements. In particular, there does not exist $\\rho$ satisfying Assumption \\ref{as:asymp}.\\\\\n Let $p$ be the transition function of the nearest neighbor symmetric random walk on $\\EuScript{X}={\\mathbb Z}^d$, $d\\ge 3$ (the assumption on the dimension is to ensure transience of $p$). By Ney and Spitzer \\cite{Ney-Spitzer}, both the  Poisson and Martin boundary with respect to $p$ are  trivial: all harmonic functions are constants. Write ${ x}=({ x}(1),\\dots, { x}(d))$. Define stopping times $T_{+},T_-,T_0$ as follows: \n$$T_+ = \\inf\\{n \\ge 1:{ x}_n(1)={x}_{0}(1)+ 1\\} \\hspace{1cm} T_- = \\inf\\{n \\ge 1:{ x}_n(1)={x}_{0}(1)- 1\\}$$\nand \n$$ T_0=\\inf\\{n\\ge 1:{x}_n(1) ={ x}_0(1)\\}.$$ \nFinally, let $T_{+,2} = T_{0}\\circ S^{T_{+}}$ and $ T_{-,2} = T_{0}\\circ S^{T_{-}}$.  In words, $T_\\pm$ is the first time ${x}(1)$ is one unit to the right (for $+$) or to the left (for $-$) of its starting point, and $T_{\\pm,2}$ is the second time ${x}(1)$ is one unit to the right ($+$) or one unit to the left ($-$) from its starting location. \n\nSet \n$$\n\\tau=\\begin{cases} T_+ \\wedge T_{-,2}\\mbox{ if } x_0(1)\\ge 0 \\\\ T_-\\wedge T_{+,2}\\mbox{ if } x_0(1) < 0. \\end{cases}\n$$\n\nObserve that by symmetry, $\\mbox{\\bf P}(T_+<T_-)=\\frac 12$, and $\\mbox{\\bf P}(T_+< T_{-,2})= \\frac 12+ \\frac 12 \\times \\frac 12 \\times \\frac 12=\\frac 58$. It immediately follows that $x_{\\tau}(1)$ is a nearest neighbor Markov chain on ${\\mathbb Z}$ with the following transition probabilities: \n$$ \nr(x,y)= \\begin{cases} \\frac 58 & x \\ge 0,~ y= x+1 \\mbox{ or } x<0, ~y=x-1 \\\\ \\frac 38 & x\\ge 0,~ y=x-1\\mbox{ or } x<0,~ y=x+1.\\end{cases}\n$$ \n\nIn particular, ${ x}^{\\tau}(1)$ is transient with positive drift on the positive half line and negative drift on the negative half line. Clearly,  $\\{\\lim  { x}_{\\tau_n}(1)=+\\infty\\}$ is invariant with respect to $\\boldsymbol{x}^{\\tau}$, therefore the function $\\mbox{\\bf P}_x (\\lim {x}_{\\tau_n}(1) = +\\infty)$ is a $p^{\\tau}$-harmonic function. It is easy to see that this function is not constant. Therefore the conclusion of Theorem \\ref{th:newmain} does not hold, which in turn implies that there does not exist a $\\rho$ satisfying Assumption \\ref{as:asymp}. Note that with this construction, $p^\\tau$ is irreducible.   As simpler, yet not irreducible example can be obtained by  defining $\\tau$ as the first time $x(1)$ is one unit away from $0$ than where it started at. In this case, $x_{\\tau}(1)$ is the Markov chain on ${\\mathbb Z}$ which jumps from $0$ to $\\pm 1$ with probability $1\/2$ each, and jumps from $x$ to $x+\\mbox{sgn}(x)$ with probability $1$ for all $x\\ne 0$.\n\\section{Appendix}\n\\begin{lemma}\n\t\\label{lem:partition}\n\tLet $\\mu$ be a Borel measure on a compact metric space $(M,d)$. Then for  every $\\delta$, there exists a number $n\\in{\\mathbb N}$ and Borel subsets  $N_0,N_1,\\dots,N_n$ of $M$ such that \n\t\\begin{enumerate} \n\t\t\\item $\\cup_{j=0}^n N_j = M$. \n\t\t\\item $N_j \\cap N_i =\\emptyset$ for $i\\ne j$. \n\t\t\\item $N_0$ is contained in the complement of $\\mbox{Supp}(\\mu)$. \n\t\t\\item For $j=1,\\dots,n$, the diameter of $N_j$ is $< \\delta$.  \n\t\t\\item $N_j,~j=1,\\dots,n$ contains a neighborhood of an element in the support of $\\mu$. \n\t\\end{enumerate} \n\\end{lemma}\n\\begin{proof}\n\tLet $B(x)$ denote the open ball of center $x$ with radius $\\delta\/2$. \n\tThe collection of balls $(B(x):x\\in \\mbox{Supp}(\\mu))$ is an open cover of the compact set $\\mbox{Supp}(\\mu)$. Therefore it possesses a finite subcover indexed by centers $x_1,\\dots,x_n$.  \n\tLet $N_1 =\\{y\\in B(x_1):d(y,x_1)\\le \\min_{j\\ne 1} d(y,x_j)\\}$. Then $x_1\\in N_1$, $N_1$ has nonempty interior, and $x_j\\not \\in N_1$ for all $j\\ne 1$. Continue inductively, letting \n\t$$N_{i+1}=\\{y\\in B(x_{i+1}):d(y,x_{i+1})\\le \\min_{j\\ne i+1} d(y,x_j)\\}-\\cup_{j\\le i}N_i.$$\n\tThus, $x_{i+1}\\in N_{i+1}$, $N_{i+1}$ has nonempty interior, and $x_j\\not \\in N_{i+1}$ for all $j\\le i+1$. Finally, let $N_0=M- \\cup_{j=1}^n N_j$. \n\\end{proof}  \n\\begin{prop}\n\t\\label{pr:approx}\n\tLet $\\mu$ and $\\nu$ be two probability measures on the Borel sets of  a compact metric space $(M,d)$, satisfying $\\mbox{Supp}(\\nu)\\subseteq \\mbox{Supp}(\\mu)$. Then there exists a sequence $(f_k:k\\in{\\mathbb N})$ of nonnegative simple functions such that $f_n d\\mu$ converges weakly to $\\nu$. That is, \n\t$$ \\lim_{n\\to\\infty} \\int g f_n d\\mu = \\int g d\\nu$$ \n\tfor every continuous $g$ on $M$. \n\\end{prop}\n\\begin{proof} \n\tBy lemma \\ref{lem:partition} for every $k\\in{\\mathbb N}$ we can find $n=n(k)$ and a partition  $N_0,N_1,\\dots,N_n$ of $M$ such that the diameter of $N_1,\\dots,N_n$ is $<\\frac{1}{k}$, $\\nu(N_0)=\\mu(N_0)=0$ and $\\mu(N_j)>0$ for $j=1,\\dots,n$. \n\tLet $f_k$ be the simple function equal to $0$ on $N_0$ and to $\\nu(N_j)\/\\mu(N_j)$ on $N_j$ for  $j=1,\\dots,n$. Then $\\int f_k d \\mu =1$. Next fix a continuous function $g$ on $M$. Then by compactness, $g$ is uniformly continuous. Fix $\\epsilon>0$, and choose $k$ to such that \n\t$|g(x) - g(y)|<\\epsilon$ whenever $d(x,y)<\\frac{1}{k}$.  Let $N_0,\\dots,N_n$ as above, and let $x_1,\\dots,x_n$ be arbitrary elements in $N_1,\\dots,N_n$, respectively.  Then \n\t$$| \\int g d\\nu - \\sum_{i=1}^n g(x_i) \\nu(N_i)| <\\epsilon$$ \n\tand  \n\t$$ |\\int g f_k d\\mu - \\sum_{i=1}^n g(x_i) \\nu(N_i) | <\\epsilon,$$ \n\tand therefore \n\t$$ |\\int gf_k d \\mu - \\int g d\\nu|<2\\epsilon,$$ \n\tcompleting the proof. \n\\end{proof} \n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:Introduction}\n\n\nThe quark model of Gell-Mann~\\cite{GellMann:1964nj} and Zweig~\\cite{Zweig:1964} classifies mesons \n({\\ensuremath{\\quark\\quarkbar}}\\xspace) and baryons ({\\ensuremath{\\Pq}}\\xspace\\quark{\\ensuremath{\\Pq}}\\xspace) into multiplets, and also allows for hadrons \nwith more than the minimal quark contents.\nIn 2015, LHCb observed two pentaquark states in the decay\n${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{{\\PJ\\mskip -3mu\/\\mskip -2mu\\Ppsi\\mskip 2mu}}}\\xspace {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\kaon^-}}\\xspace}$\\cite{LHCb-PAPER-2015-029}. In the decay  \nchannel  \n${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$,\\footnote{Unless explicitly noted, charge conjugate decays are implied.}  \ncharmed dibaryon resonant states \ncould be present. As discussed \nin Ref.~\\cite{MAIANI201537}, such states could manifest via the decay\n${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace + [cd][ud][ud] = {\\ensuremath{\\overline \\proton}}\\xspace + \\mathscr{D}_c^+}$, where\n${\\mathscr{D}_c^+}$ is the dibaryon state with a mass below 4682\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace.\nThe subsequent decay of the ${\\mathscr{D}_c^+}$ dibaryon could proceed either via\nquark rearrangement to the final state ${{\\ensuremath{\\Pp}}\\xspace\\PSigma_c^0}$, with\n${\\PSigma_c^0\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$, or via string breaking to the final state $\\mathscr{P}_c^0(\\bar{u}[cd][ud])$, \nwhich could involve\na lighter, yet undiscovered $\\mathscr{P}_c^0$  pentaquark state,  \n$\\mathscr{D}_c^+\\ensuremath{\\rightarrow}\\xspace \\mathscr{P}_c^0(\\bar{u}[cd][ud]) {\\ensuremath{\\Pp}}\\xspace$,  \nwith ${\\mathscr{P}_c^0 \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$~\\cite{MAIANI201537}.\nThe discovery of any of these decay modes would test the predictions \nof quantum chromodynamics and the fundamental workings of the Standard Model.\n\nIn this Letter, the first observation of the decay ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$, \nreferred to as the signal channel, is reported. \nA measurement is made of its  \nbranching fraction relative to the normalisation channel ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$.  \nResonance structures within the ${{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ system are also investigated. \nWhile no evidence for dibaryon resonances is found, \nsignificant contributions from the $\\PSigma_c(2455)^0$ and $\\PSigma_c^{*}(2520)^0$ resonances \nare found in the ${{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ invariant mass spectrum. \nThe ratios of branching fractions between decays via these resonances, \nhereinafter denoted as $\\PSigma_c^0$ and $\\PSigma_c^{*0}$, and the \n${{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ inclusive decay are also reported.\nThe measurements in this Letter  are based on a data sample of $pp$ collisions collected  \nwith the LHCb detector at centre-of-mass energies of $\\sqrt{s}=$\n7\\ifthenelse{\\boolean{inbibliography}}{\\ensuremath{~T\\kern -0.05em eV}}{\\ensuremath{\\mathrm{\\,Te\\kern -0.1em V}}}\\xspace in 2011 and $\\sqrt{s}=$ 8\\ifthenelse{\\boolean{inbibliography}}{\\ensuremath{~T\\kern -0.05em eV}}{\\ensuremath{\\mathrm{\\,Te\\kern -0.1em V}}}\\xspace in 2012, corresponding to an integrated luminosity of\n3\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace.\n\n\n\n\\section{Detector and simulation}\n\\label{sec:Detector}\n\nThe \\mbox{LHCb}\\xspace detector~\\cite{Alves:2008zz,LHCb-DP-2014-002} is a single-arm forward\nspectrometer covering the \\mbox{pseudorapidity} range ${2<\\eta <5}$,\ndesigned for the study of particles containing {\\ensuremath{\\Pb}}\\xspace or {\\ensuremath{\\Pc}}\\xspace\nquarks. The detector includes a high-precision tracking system\nconsisting of a silicon-strip vertex detector surrounding the $pp$\ninteraction region, a large-area silicon-strip detector located\nupstream of a dipole magnet with a bending power of about\n$4{\\mathrm{\\,Tm}}$, and three stations of silicon-strip detectors and straw\ndrift tubes placed downstream of the magnet.\nDifferent types of charged hadrons are distinguished using information\nfrom two ring-imaging Cherenkov (RICH) detectors.\nPhotons, electrons and hadrons are identified by a calorimeter system consisting of\nscintillating-pad and preshower detectors, an electromagnetic\ncalorimeter and a hadronic calorimeter. Muons are identified by a\nsystem composed of alternating layers of iron and multiwire\nproportional chambers.\nThe online event selection is performed by a trigger~\\cite{LHCb-DP-2012-004},\nwhich consists of a hardware stage, based on information from the calorimeter and muon\nsystems, followed by a software stage, in which all charged particles\nwith ${\\mbox{$p_{\\mathrm{ T}}$}\\xspace>500\\,(300)\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c}}\\xspace}$ are reconstructed for 2011\\,(2012) data, where \\mbox{$p_{\\mathrm{ T}}$}\\xspace is the transverse momentum~\\cite{LHCb-DP-2012-004}.  \nAt the hardware trigger stage, events are required to contain a \nmuon or dimuon pair with high \\mbox{$p_{\\mathrm{ T}}$}\\xspace, or a hadron, photon or electron \nwith high transverse energy deposited in the calorimeters.\nThe software trigger requires a two-, three- or four-track\nsecondary vertex with a significant displacement from \nany primary proton-proton interaction vertices (PVs).  \nAt least one charged particle\nmust have a ${\\mbox{$p_{\\mathrm{ T}}$}\\xspace > 1.7~(1.6)\\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V\\!\/}c}}\\xspace}$ for 2011 (2012) data, and be\ninconsistent with originating from a PV.\nA multivariate algorithm~\\cite{BBDT} is used for\nthe identification of secondary vertices consistent with the decay\nof a {\\ensuremath{\\Pb}}\\xspace hadron.\n\n\nSimulated samples of the signal, the normalisation channels and \nbackgrounds produced in $pp$ collisions  are generated using\n\\mbox{\\textsc{Pythia}}\\xspace~\\cite{Sjostrand:2007gs,*Sjostrand:2006za}\nwith a specific \\mbox{LHCb}\\xspace\nconfiguration~\\cite{LHCb-PROC-2010-056}. Decays of hadronic particles\nare described by \\mbox{\\textsc{EvtGen}}\\xspace~\\cite{Lange:2001uf}, in which final-state\nradiation is generated using \\mbox{\\textsc{Photos}}\\xspace~\\cite{Golonka:2005pn}. The\ninteraction of the generated particles with the detector, and its response,\nare implemented using the \\mbox{\\textsc{Geant4}}\\xspace\ntoolkit~\\cite{Allison:2006ve, *Agostinelli:2002hh} as described in\nRef.~\\cite{LHCb-PROC-2011-006}.\n\n\n\\section{Candidate selection}\nThe ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ and ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ candidates \nare reconstructed using the decay ${{\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace}$. \nAn offline selection is applied,  \nbased on a loose preselection, followed by a multivariate analysis. \nTo minimize the systematic uncertainty on the ratio of efficiencies  \nbetween the signal and the normalisation channels, the selection criteria on the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidates   \nare similar  between the two channels. \n\n\nReconstructed final-state particles in ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ and \n${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ candidate decays are required to have \na momentum ${p>1\\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V\\!\/}c}}\\xspace}$ and ${\\mbox{$p_{\\mathrm{ T}}$}\\xspace>100\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c}}\\xspace}$.  \nProtons and antiprotons are required to \nhave ${p>10\\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V\\!\/}c}}\\xspace}$ to improve particle identification.  \nAll final-state particles are also required to be inconsistent with originating from any PV, \nby rejecting the tracks with a small \\ensuremath{\\chi^2_{\\text{IP}}}\\xspace, where \n\\ensuremath{\\chi^2_{\\text{IP}}}\\xspace is the difference in the vertex-fit \\ensuremath{\\chi^2}\\xspace of a given PV \nwith or without the track considered, requiring ${\\ensuremath{\\chi^2_{\\text{IP}}}\\xspace > 4}$. \nCandidate {\\ensuremath{\\Lz^+_\\cquark}}\\xspace decays are required to have at least one decay product with ${\\mbox{$p_{\\mathrm{ T}}$}\\xspace>500\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c}}\\xspace}$ and \n${p>5\\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V\\!\/}c}}\\xspace}$, a good vertex-fit quality, and an invariant mass \nwithin $\\pm$15\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace of the known {\\ensuremath{\\Lz^+_\\cquark}}\\xspace mass~\\cite{PDG2016}. The scalar sum of the transverse momenta of \nthe {\\ensuremath{\\Lz^+_\\cquark}}\\xspace decay products is required to be greater than 1.8\\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V\\!\/}c}}\\xspace. \n\nThe {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace candidate is reconstructed by combining a {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidate \nwith a pion, and the signal candidate is reconstructed by combining a {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidate \nwith a pion, a proton and an antiproton.  \nThese combinations must form a {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate with a good-quality vertex and  \nbe consistent with \noriginating from the associated PV,  \ndefined as that for which the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate has the least \\ensuremath{\\chi^2_{\\text{IP}}}\\xspace. \nFurthermore, the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidate is required to decay downstream of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace \ndecay vertex. The {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decay time, calculated as ${t = m_{\\ensuremath{\\Lz^0_\\bquark}}\\xspace L\/p}$, is required to be \ngreater than 0.2\\ensuremath{{\\mathrm{ \\,ps}}}\\xspace, where $m_{\\ensuremath{\\Lz^0_\\bquark}}\\xspace$ is the mass, $L$ is the decay length and $p$ is \nthe momentum of the ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace$ candidate.\nThe {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate is also required to have at least one final-state particle in the decay chain with \n${\\mbox{$p_{\\mathrm{ T}}$}\\xspace>1.7\\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V\\!\/}c}}\\xspace}$, ${p>10\\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V\\!\/}c}}\\xspace}$, and have at least one track significantly \ninconsistent with originating from the associated PV by requiring the track to have  ${\\ensuremath{\\chi^2_{\\text{IP}}}\\xspace > 16}$. \nFinal-state tracks of signal and normalisation channel candidates  \nmust pass strict particle-identification requirements based on the RICH\ndetectors, calorimeters and muon stations. \nA constrained fit~\\cite{Hulsbergen:2005pu} is applied to the candidate decay chain for \nboth the signal and the normalisation channels,  \nrequiring the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate to come from the associated PV and constraining the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace particle to its known \nmass~\\cite{PDG2016}. In the case of the search of the resonant  contributions, the mass of the \n{\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate is also constrained to the known mass~\\cite{PDG2016}. \n\nTrigger signals are associated with reconstructed\nparticles from the decays of the signal channel or of the normalisation channel. \nSelection requirements can therefore be made on the trigger\nselection itself and on whether the decision was due to the reconstructed candidate decay, \nother particles produced in the $pp$ collision, or a combination of the two.\nThis association makes it possible to use a data-driven method for the correction\nand systematic uncertainty estimation on the trigger efficiencies~\\cite{LHCb-DP-2012-004}.\nTo take advantage of the similarity between the signal and the normalisation channels,  \nwhich helps to  minimize the systematic uncertainty on the ratio of their efficiencies,\ncandidates are classified in one of the following two hardware trigger categories.\nIn the first category, called Triggered On Signal ({TOS}), the candidate must include a hadron consistent with originating from\nthe decay of a ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace$ candidate and which deposited enough transverse energy  in the calorimeter\nto satisfy the hardware trigger requirements. The typical value of the transverse energy\nthreshold is around 3.5\\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V\\!\/}c^2}}\\xspace.\nAs the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace baryon is a {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decay product for both the signal  and the \nnormalisation channels, this choice minimizes the difference between the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decay modes. \nThe second category,\ncalled Triggered Independent of Signal ({TIS}), comprises events\nwhich satisfied the hardware trigger through signatures unassociated with the complete {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decay chains, \neither due to a muon with high \\mbox{$p_{\\mathrm{ T}}$}\\xspace,\nor a hadron, photon, or electron with high transverse\nenergy deposited in the calorimeters.\nThe efficiencies of the TIS and TOS requirements are different, so\nthe data are divided into two\nstatistically independent samples, one TIS, and the other TOS and not TIS,\nwhich will be referred to as TOS for the rest of this Letter.\n\nThe so-called cross-feed backgrounds, contributing under the peak of the \ninvariant mass of the normalisation channel or of the signal channel  \nfrom the  ${{\\ensuremath{\\Bbar{}^0}}\\xspace({\\ensuremath{\\Bbar{}^0_\\squark}}\\xspace)\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\D^+}}\\xspace({\\ensuremath{\\D^+_\\squark}}\\xspace) {\\ensuremath{\\pion^-}}\\xspace}$ and  \n${{\\ensuremath{\\Bbar{}^0}}\\xspace({\\ensuremath{\\Bbar{}^0_\\squark}}\\xspace) \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\D^+}}\\xspace({\\ensuremath{\\D^+_\\squark}}\\xspace) {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace {\\ensuremath{\\pion^-}}\\xspace}$ decays, respectively, with ${{\\ensuremath{\\D^+}}\\xspace({\\ensuremath{\\D^+_\\squark}}\\xspace) \\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace}$ \nor ${{\\ensuremath{\\D^+}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip}$,  \nwhere either the kaon or pion is misidentified as a proton,  \nare explicitly vetoed when both of the following two conditions are satisfied. First, the mass hypothesis \nof the proton from the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidate is replaced with either the kaon or pion hypothesis,  \nand the resulting invariant mass of the combination is consistent  \nwith the known ${\\ensuremath{\\D^+}}\\xspace({\\ensuremath{\\D^+_\\squark}}\\xspace)$ mass~\\cite{PDG2016} within $\\pm$15\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace.\nSecond, the invariant mass of the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidate \nis set to the known ${\\ensuremath{\\D^+}}\\xspace({\\ensuremath{\\D^+_\\squark}}\\xspace)$ mass~\\cite{PDG2016}, and the resulting invariant mass of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate is consistent with  \nthe known ${\\ensuremath{\\Bbar{}^0}}\\xspace({\\ensuremath{\\Bbar{}^0_\\squark}}\\xspace)$ mass~\\cite{PDG2016} within $\\pm$25\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace for ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ decays, and within  \n$\\pm$45\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace for ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ decays.\n\nFurther background reduction is achieved using a multivariate\nanalysis based on a gradient boosted decision tree (BDTG)\\cite{Breiman}. The BDTG\nis trained using twelve variables: the vertex-fit quality of the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace \nand {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidates, the decay-vertex displacement along the beamline between the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace and {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidates, \nthe displacement between the decay vertex of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate and the associated PV, the \n\\ensuremath{\\chi^2_{\\text{IP}}}\\xspace of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate, the angle between the reconstructed {\\ensuremath{\\Lz^0_\\bquark}}\\xspace momentum and \nthe direction of flight from the associated PV to the decay vertex, \nthe smallest \\mbox{$p_{\\mathrm{ T}}$}\\xspace  \nand smallest \\ensuremath{\\chi^2_{\\text{IP}}}\\xspace among the three {\\ensuremath{\\Lz^+_\\cquark}}\\xspace decay products, the \\mbox{$p_{\\mathrm{ T}}$}\\xspace and \\ensuremath{\\chi^2_{\\text{IP}}}\\xspace of the pion originating directly  \nfrom the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decay, and the smallest \\mbox{$p_{\\mathrm{ T}}$}\\xspace and smallest \\ensuremath{\\chi^2_{\\text{IP}}}\\xspace between  \nthe {\\ensuremath{\\Pp}}\\xspace and {\\ensuremath{\\overline \\proton}}\\xspace originating directly from the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decay.   \nThe BDTG training is performed using simulated samples for the signal,  \nand data distributions for the background, \nwith reconstructed invariant mass \nwell above the known {\\ensuremath{\\Lz^0_\\bquark}}\\xspace mass~\\cite{PDG2016}. Cross-feed backgrounds from the\ndecays ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$, ${{\\ensuremath{\\Bbar{}^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ and ${{\\ensuremath{\\Bbar{}^0_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$\nare explicitly vetoed during the BDTG-training process by requiring the difference between the    \nreconstructed {\\ensuremath{\\Pb}}\\xspace-hadron mass and its known mass to be  \ngreater than $\\pm$30\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace. The BDTG selection is optimized for the figure of merit  \n$S\/\\sqrt{S+B}$, where $S$ and $B$ are the expected signal and background yields  \nwithin $\\pm$30\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace of the known {\\ensuremath{\\Lz^0_\\bquark}}\\xspace mass~\\cite{PDG2016}. The initial value of $S$ and $B$  \nwithout BDTG selection is obtained from the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace mass spectrum in data.   \nNo improvement in the normalisation channel\nis found using a similar procedure, therefore no BDTG selection is applied. \nA systematic uncertainty is assessed for this choice in Section~\\ref{sec:syst}.   \n\nDue to the large number of final-state particles in the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decays, particles   \nwith the same charge may share track segments, representing a possible background. \nThese  tracks are referred to as clones, \nand are suppressed by requiring    \nthat the opening angle between any same-charged tracks in the candidate is larger than 0.5 mrad. \nThis selection\nremoves 2\\% of candidates in the signal sample and 0.1\\% in the normalisation sample. \nIf multiple {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidates are reconstructed in one single event, \none candidate is chosen at random in the following two cases. \nFirst, if the proton from the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace decays is exchanged with that directly from \nthe {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decays, forming two candidates with nearly the same {\\ensuremath{\\Lz^0_\\bquark}}\\xspace mass. \nSecond, if a track from one candidate shares a segment with a track from another  \ncandidate. \nWith these criteria, 2.5\\% of candidates \nin the signal sample and 0.1\\% in the normalisation sample are vetoed. \nAfter these selections, 0.8\\% of events in the signal sample  \nand 0.2\\% in the normalisation sample contain multiple {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidates. \nThese remaining multiple candidates mainly originate from the random \ncombinations of the final-state tracks, and have a negligible influence on the \nestimation of the signal yields.  \nNo further vetoes on these candidates are applied. \n\n\n \n \n\n\n\\section{Efficiencies}\n\\label{sec:eff}\nThe total efficiencies of the signal and the normalisation decays are given by\n\\begin{equation}\n\\epsilon_{\\rm total} = \\epsilon_{\\rm a}\\cdot\\epsilon_{\\rm rec\\&sel|a}\\cdot\\epsilon_{\\rm trig|sel}\\cdot\\epsilon_{\\rm PID},\n\\end{equation}\nwhere $\\epsilon_{\\rm a}$ represents the geometrical acceptance of the \\mbox{LHCb}\\xspace detector,\n$\\epsilon_{\\rm rec\\&sel|a}$ is the efficiency of reconstruction and selection  \ncalculated on candidates in the acceptance, $\\epsilon_{\\rm trig|sel}$ is the trigger efficiency  \nof the selected candidates, and $\\epsilon_{\\rm PID}$ is the particle-identification efficiency.\nAll efficiencies except $\\epsilon_{\\rm PID}$ and $\\epsilon_{\\rm trig|sel}$\nare determined from simulation. The particle-identification efficiency is determined  \nfrom calibration data specific to each data-taking year, binned in momentum and pseudorapidity\nof the track in question, as well as in the multiplicity of the event~\\cite{LHCb-PROC-2011-008}.\nThe trigger efficiency is determined from a combination of\nsimulation and data-driven techniques where the agreement between data and simulation\nis explicitly verified using the normalisation sample satisfying the TIS requirement. \nAll efficiencies are calculated separately for the TIS and TOS\ntrigger samples, and for data-taking year, due to the difference in\ncentre-of-mass energies. Agreement between data and simulation is improved by applying\na per-candidate weight to the \\mbox{$p_{\\mathrm{ T}}$}\\xspace and rapidity, $y$, of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace baryon in simulated events to match\nthe normalisation sample in the TIS category, which is largely independent of trigger\nconditions. \nThe \\mbox{$p_{\\mathrm{ T}}$}\\xspace and $y$ distributions of {\\ensuremath{\\Lz^0_\\bquark}}\\xspace produced in $pp$ collision \nare identical for the signal and the normalisation channels, so the same per-candidate weights \nare applied to the signal sample. \nThe simulated \\ensuremath{\\chi^2_{\\text{IP}}}\\xspace of the final-state particles and the vertex-fit \\ensuremath{\\chi^2}\\xspace of {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidates  \nare weighted to reproduce the data distributions. \nThe ratio between the efficiencies of the signal and the normalisation channels,  $\\epsilon_r$, \nis $(10.00\\pm0.12)\\%$ for the TIS sample and $(11.39\\pm0.22)\\%$ for the\nTOS sample, including uncertainties due to the limited size of the simulated sample.\n\n\n\\section{Fit model and the ratio of branching fractions}\n\\label{sec:eff}\nThe yields in both the signal and  \nthe normalisation channels are determined  \nfrom an unbinned extended maximum-likelihood fit to the corresponding invariant-mass spectra \nwith both the TIS and TOS samples combined.  \nThe signal  \nis modelled by a sum of two Crystal Ball functions\\cite{Skwarnicki:1986xj} with\na common mean of the Gaussian core, and with the tail parameters fixed from simulation. For both the\nsignal and the normalisation channels, the background from random combinations of final-state  \nparticles is described by an exponential function, whose parameters are left free in the fits \nand are independent between the signal and the normalisation channels.  \nFor the\nnormalisation channel, background from the ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\rhomeson^-}}\\xspace$ decays, with\n${\\ensuremath{\\rhomeson^-}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$ is modelled by the convolution of an empirical threshold function  \nwith a Gaussian resolution. The contribution due to misidentification of the kaon to pion from ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$\nis modelled by a sum of two Crystal Ball functions. The parameters of these two background sources are\ntaken from simulation. The fits to the invariant-mass distributions for the signal \nand the normalisation channels are shown in Figure~\\ref{fig:fits}. In this figure, the TIS and TOS \nsamples are combined. From these fits, $926\\pm43$\n${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ and $(167.00\\pm0.50)\\times10^3$   \n${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ decays are observed.\n\n\\begin{figure}[!tbp]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{Fig1a.pdf}\n\\includegraphics[width=0.48\\textwidth]{Fig1b.pdf}\n\\caption{Invariant mass distributions of the (a) ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ \nand (b) ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ candidates. Fit results are overlaid as a solid blue line.\nFor (a), the red dotted line represents the signal component and the green dotted\nline the background due to random combinations. For (b), the red dotted line is the\nsignal component, the green dotted line is the random combination background,\nthe purple dashed line is the contribution from ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\rhomeson^-}}\\xspace$ and the brown\ndashed-dotted line represents the contribution from ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$.}\n\\label{fig:fits}\n\\end{center}\n\\end{figure}\n\n\nTo determine the ratio of  branching fractions  \n${\\frac{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace  p \\overline{p}\\pi^-)}{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace \\pi^-)} }$, \nindicated \u00a0in the  following by $\\mathcal{B}_r$,  \na simultaneous fit is performed to the signal and the normalisation channels, each divided into the two  \nindependent trigger categories. The yield of the normalisation sample, $N({\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)$, is \na free parameter in the fits, whereas the yield of the signal sample is calculated as\n${N({\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace) = \\mathcal{B}_r\\times\\epsilon_r\\times N({\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)}$, where\n$\\epsilon_r$ is the ratio between the total efficiency of the ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ and ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ decays.  \nThe ratio of branching fractions $\\mathcal{B}_r$  \nis the same for the TIS and TOS subsamples and is measured to be  \n${\\mathcal{B}_r = 0.0542 \\pm 0.0023}$. The corresponding\nsignal yields are 677 $\\pm$ 29 for the TIS subsample and 259 $\\pm$ 11 for the TOS subsample; the yields in the normalisation\nsample are ${(124.9 \\pm 0.4)\\times 10^3}$ for the TIS subsample and ${(41.9 \\pm 0.2) \\times10^3}$ for the TOS subsample.\n\n\n\\section{Systematic uncertainties}\n\\label{sec:syst}\nThe systematic uncertainties on the measurement of the ratio of branching \nfractions are listed in Table~\\ref{tab:systemerr}. The total systematic uncertainty  \nis determined from the sum in quadrature of all terms. \n\nFirst, the uncertainty\nrelated to the background modelling is considered.\nIn the signal sample, \nthe exponential function is replaced with a second-order polynomial for the background component. \nFor the normalisation\nchannel, the model is varied by using the sum of\ntwo exponential functions. The resulting uncertainty on the ratio of branching  \nfractions is 0.6\\%. The uncertainties due to the ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace}$ shape parameters are assessed\nby increasing the width of the Crystal Ball functions  \nby 10\\%, corresponding to two standard deviations, resulting in a change of 0.1\\%.\nThe uncertainty due to the ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\rhomeson^-}}\\xspace}$ contribution is estimated by varying\nthe shape parameters by one standard deviation, resulting in an uncertainty of 0.4\\%.\nThe total uncertainty on the ratio of the branching fractions due to the background modelling \nis 0.7\\%.  \n\nThe signal-model parameterization is changed to a single Hypatia function\\cite{Santos:2013gra},\nwhere the mean and width are allowed to float and all other parameters are taken  \nfrom simulation, resulting in an uncertainty of 0.1\\%. \n\n\n\\begin{table}[!tbp]\n\\centering\n\\caption{Summary of systematic uncertainties and correction factors\nto the ratio of branching fractions measurement. \nAll uncertainties \nare given as a percentage of the ratio  \nof branching fractions.}\n\\label{tab:systemerr}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{lSS}\\hline\\hline\n\\multicolumn{1}{l}{Source} & \\multicolumn{1}{c}{Uncertainty (\\%)}  & \\multicolumn{1}{c}{Correction factor} \\\\\\hline\nBackground fit model & 0.7 & $\\rm{-}$ \\\\\nSignal fit model & 0.1 & $\\rm{-}$ \\\\\nPID efficiency &0.3 & $\\rm{-}$ \\\\\nTracking efficiency calibration &0.8 & 0.985\\\\\nKinematic range of final-state tracks&0.7 & $\\rm{-}$ \\\\\nHadron interaction &4.4 & $\\rm{-}$ \\\\\n$\\mbox{$p_{\\mathrm{ T}}$}\\xspace, ~y$ weighting  & 1.0 & $\\rm{-}$ \\\\\nTrigger efficiency  &2.9 & $\\rm{-}$\\\\\nSimulated sample size  &1.3 & $\\rm{-}$\\\\\nCandidates with clone tracks and multiple candidates & 0.2&$\\rm{-}$ \\\\\nVeto of the reflection background & 0.4 & $\\rm{-}$ \\\\\n{\\ensuremath{\\Lz^+_\\cquark}}\\xspace Dalitz weighting &0.2  &0.984\\\\\n{\\ensuremath{\\Lz^+_\\cquark}}\\xspace polarization & 0.3  & 0.987\\\\\nResonant structures & 1.8& 1.041\\\\\n\\hline\nTotal&6.0 &0.996 \\\\\\hline\n\\end{tabular}\n}\n\\end{table}\n\nThe uncertainty on the relative efficiency of the particle identification is assessed by\ngenerating pseudoexperiments. For each pseudoexperiment, efficiencies in different \nmomentum, pseudorapidity and multiplicity bins are  \ndetermined from independent Gaussian distributions with mean values equal to the nominal efficiencies and widths \ncorresponding to their uncertainties. \nThis procedure is repeated 1000 times, and\nthe width of the resulting efficiency is taken as the\nsystematic uncertainty. This procedure, performed separately for the TIS and TOS\nsamples, results in a 0.13\\% uncertainty for both samples. Binning effects on \nthe efficiency are estimated by halving the bin size of the momentum distributions, resulting\nin a relative change of 0.2\\% for the TIS sample and 0.1\\% for the TOS sample.\nThe total uncertainty on the relative efficiency for the TIS and TOS samples is   \n0.24\\% and 0.16\\%, respectively, \ncorresponding to an uncertainty of 0.3\\% on the ratio of the branching fractions. \n\nTracking efficiencies are determined with simulated\nevents weighted to match the kinematic properties of dedicated calibration samples.\nThe weights are determined  \nas a function of the kinematic variables,  \nseparately for each data-taking year~\\cite{LHCb-DP-2013-002}. \nThe kinematic properties of the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace\ndecay products are similar for\nthe signal and the normalisation samples and therefore provide minor    \ncontributions to the total tracking efficiency ratio. The dominant contribution to the systematic uncertainty  \ncomes from the knowledge of the {\\ensuremath{\\Pp}}\\xspace and {\\ensuremath{\\overline \\proton}}\\xspace tracking efficiencies, \nwhose systematic uncertainties are fully correlated. \nThe efficiency correction procedure gives a change in efficiency\nof 2.0\\% for the TIS sample and 1.4\\% for the TOS sample, yielding a total\ncorrection factor of 0.985 for the ratio of branching fractions, and \na systematic uncertainty of 0.4\\% for each of the {\\ensuremath{\\Pp}}\\xspace and {\\ensuremath{\\overline \\proton}}\\xspace \n mainly stemming from the finite size of the calibration sample~\\cite{LHCb-DP-2013-002}. \n\nDue to distinct trigger requirements, the kinematic acceptance of the\ncalibration samples differs slightly from the signal and the normalisation channels.\nA nonnegligible  fraction of candidates have final-state  \nparticles in a kinematic range  outside of the regions covered \nby the calibration samples.  \nAbout 20\\% of the candidates from both channels fall in this category \ndue to the low-momentum pion from the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace decay. In addition, 10\\% \nof the candidates from the signal channel are also affected, mainly due to the pion originating from the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decay. \nFor all of these outside-range candidates, the efficiency correction in the nearest available \nbin is used. As the effects for {\\ensuremath{\\Lz^+_\\cquark}}\\xspace decays cancel in the relative efficiency, only the additional \n10\\% candidates in the signal channel contribute a 0.7\\% uncertainty on the relative efficiency.   \n\nHadronic interactions with the \\mbox{LHCb}\\xspace\ndetector contribute an additional uncertainty of 2.2\\% on the ratio \nof the branching fractions for each {\\ensuremath{\\Pp}}\\xspace or\n{\\ensuremath{\\overline \\proton}}\\xspace (4.4\\% in total), which is obtained from simulation, accounting for the imperfect knowledge of  \nmaterial budget of the \\mbox{LHCb}\\xspace detector\\cite{LHCbVELOGroup:2014uea}.\n\nPer-candidate weights depending on \\mbox{$p_{\\mathrm{ T}}$}\\xspace and $y$ of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace baryon are applied in \nsimulated events to improve the agreements between data and simulation.  \nSystematic uncertainties for the weighting due to the finite\nsize of the normalisation sample are assessed with pseudoexperiments.\nIn each pseudoexperiment, the weights are varied within their uncertainties, \nand the results are propagated  to the ratio of branching fractions. The\nstandard deviation of the obtained distributions is taken as a systematic uncertainty,\nresulting in 0.65\\% for the TIS sample and 0.65\\% for the TOS sample. \nThe systematic uncertainties due to the binning scheme of the weighting in \\mbox{$p_{\\mathrm{ T}}$}\\xspace and $y$ are   \nestimated by halving the bin size, or using the gradient boosting~\\cite{friedman2001greedy}\\cite{Belov:2010xm}, which is an unbinned method of weighting, \nto check the changes on the relative efficiencies.  \nThe resulting systematic uncertainties\nare 0.43\\% for the TIS sample and 1.5\\% for the TOS sample.  \nAfter propagation through the entire fit procedure, this results\nin an uncertainty of 1.0\\% on the ratio of the branching fractions.\n \nTrigger efficiencies for the TOS samples are also assessed using pseudoexperiments which are propagated\nto the final measurement, resulting in a final uncertainty of 0.1\\%.\nThe trigger efficiency of the TIS sample is taken from simulation. \nIts systematic uncertainty is computed from the difference between \nthe \nTIS efficiency taken from data and simulation for events which are \ntriggered both on the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidate  and also on other tracks unassociated to the signal decay.  \nAs a result, a systematic uncertainty of 3.9\\% is assigned for the relative trigger efficiency of the TIS sample, \ncorresponding to an uncertainty of 2.9\\% on the ratio of the branching fractions.\n\n\n\n\nThe effect of the finite size of the simulated samples is assessed by considering\nthe possible variation of the efficiency with weighted samples in a bin of \\mbox{$p_{\\mathrm{ T}}$}\\xspace and\nrapidity of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate, and the corresponding systematic uncertainty  \non the efficiency of the signal or normalisation channel, TIS or TOS sample, is given by\n\\begin{equation}\n\\sigma_\\epsilon = \\sqrt{\\sum_{i}\\epsilon_{i}(1-\\epsilon_{i})N_{i}w_{i}}\/\\sum_{i}N_{i}w_{i},\n\\end{equation}\nwhere\nfor each bin $i$, $N_{i}$ is the number of candidates, $w_{i}$ is the single event weight,\nand $\\epsilon_{i}$ is the single event efficiency. \nThe total uncertainty on the relative efficiency for the TIS and TOS samples is  \n1.2\\% and 1.9\\%, respectively,\ncorresponding to an uncertainty of 1.3\\% on the ratio of the branching fractions.\n\n\n\nThe uncertainty due to the removal of candidates reconstructed \nwith clone tracks and multiple candidates is assessed by applying\nthe same procedure to simulation, resulting in a difference of 0.2\\%.\n\nVetoes on the invariant mass of possible cross-feed backgrounds may bias the signal mass distributions.\nAn uncertainty of 0.4\\% is determined \nby changing the fit range of the normalisation\nsample to begin at 5450\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace, instead of 5350\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace.  \n\nThe agreement between data and simulation in the ${{\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace}$ decay\nis also tested by comparing the Dalitz plot distributions. The normalisation sample  is\nweighted in the $m^2({\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace)$ versus $m^2({\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace)$ plane. Due to the smaller\nsample size of the signal channel, weights obtained from the normalisation channel are\napplied to the signal. The resulting procedure renders all distributions consistent\nwithin one statistical standard deviation. The difference in the ratio of  branching\nfractions is 1.3\\% smaller than the nominal result, providing a correction\nfactor of 0.984. \nAn uncertainty of 0.2\\% is determined by using an alternative binning scheme \n and varying the Dalitz-plot weights by their statistical uncertainties. \n\nThe polarization of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace particles has been measured to be consistent with zero\\cite{LHCb-PAPER-2012-057}, but   \nthe weak decay of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace baryon may induce a polarization in the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace system.  \nIn the simulation, it is assumed that the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace particle is unpolarized, leading  \nto a difference in angular distributions between simulation and data. A possible effect \ndue to the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace polarization is assessed by applying\na weighting procedure to the distribution of the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace helicity angle,\nwhich is defined as the angle between the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace flight direction in the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace rest frame and the\ndirection of the ${\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ pair in the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace rest frame. \nThis weight is obtained through a comparison between the angular distributions in simulation \nand data for the signal and the normalisation channels individually. \nApplying this weight\nto both the signal and the normalisation channels does not change the efficiency with\nrespect to any of the other possible angles, and leads to a change of 1.1\\% in the\nrelative efficiency for the TOS sample and 1.4\\% for the TIS sample. Propagation of these\nuncertainties leads to \na correction factor of 0.987 on the ratio of the branching fractions. \nAn uncertainty of 0.3\\% is determined by using an alternative binning scheme\n and varying the single-candidate weights by their statistical uncertainties.\n\nSimulated data are generated using a phase-space model for the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decay,\nwhich does not take into\naccount possible resonances in the ${{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ system. Upon\ninspection, clear signals from the $\\PSigma_c^0$ and $\\PSigma_c^{*0}$\nresonances are found, as described in Section~\\ref{sec:sigc}.  \nTo assess the\neffect of these resonances, the simulation is weighted to reproduce the data.\nWeights are applied in two invariant mass dimensions, namely the ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ invariant mass\nand another invariant mass of any two or three body combination. \nAmong these weighting strategies, applying weights in $m({\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)$ and $m({\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)$ (option 1) leads \nto the smallest $\\mathcal{B}_r$, while weights in $m({\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)$ and $m({\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)$ (option 2) leads to the \nlargest $\\mathcal{B}_r$. \nA correction factor is computed as the average of the central values of the ratio of  branching fractions \nfor the two options divided by the nominal branching fraction, with an \nuncertainty determined by half the difference between the two ratios of branching fractions.\nThis leads to a correction factor of 1.041 and a resulting systematic uncertainty of\n1.8\\%.\n\n\nUncertainties due to the use of the BDTG are tested by repeating the BDTG  training  \nand selection procedure to the normalisation channel \nwithout variables related to the {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace pair; \nthe ratio of branching fractions is found to be consistent.\n\n\n\\section{\\boldmath Resonance structures in the ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ mass spectrum}\n\\label{sec:sigc}\n\nAs the resonant structure of ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ decays is  \nunexplored, the \nresonances in the ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ system are studied.\nAn unbinned maximum-likelihood fit of the ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ mass is performed \nfor those candidates which pass all the selection criteria for the signal\n${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ decays, to determine if there are \nresonant contributions. In this case the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate is constrained to its known\nmass~\\cite{PDG2016} when obtaining the ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ invariant mass spectrum.  \n\n\nThe signal shapes of the $\\PSigma_c^0$ \nand $\\PSigma_c^{*0}$ resonances are given as the modulus squared of the relativistic \nBreit-Wigner function\\cite{PDG2016}, \n\\begin{equation}\n\\left|{\\rm BW}(m|M_{0},\\Gamma_{0})\\right|^2 = \\left|1\/(M_{0}^2-m^2 -iM_{0}\\Gamma(m))\\right|^2, \n\\end{equation}\nmultiplied by $m \\Gamma(m)$, and convolved with a Gaussian resolution determined from simulation.\nHere, $M_0$ is the known value of the $\\PSigma_c^0$ or $\\PSigma_c^{*0}$ mass~\\cite{PDG2016}, \n$m$ is the ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ invariant mass, \nand $\\Gamma_0$ is the mass-independent width of the resonance, namely 1.83\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace for the \n$\\PSigma_c^0$ and 15.3\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace for the $\\PSigma_c^{*0}$ resonance. The mass-dependent width\nis given by \n\\begin{equation} \n\\Gamma(m) = \\Gamma_0 \\times \\left(\\frac{q}{q_0}\\right)^{2L+1}\\frac{M_0}{m}B_L(q,q_0,d)^2, \n\\end{equation}\nwhere $L$ is the angular momentum in the resonance decay, $q$ is the momentum of the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace baryon\nin the $\\PSigma_c^{(*)0}$ rest frame, ${q_0 \\equiv q(m=M_0)}$\nand $d$ stands for the size of the $\\PSigma_c^{(*)0}$ particles. From parity\nand angular momentum conservation, it follows that $L=1$. The width also\ndepends on the Blatt-Weisskopf factor $B_L(q,q_0,d)$\\cite{VonHippel:1972fg}, where the value of $d$\nis set to be 1\\ensuremath{\\mathrm{ \\,fm}}\\xspace (5 GeV$^{-1}$ in natural units).  \nThe ratio of widths of the Gaussian resolution functions for the $\\PSigma_c^0$ and\n$\\PSigma_c^{*0}$ resonances is fixed from simulation to be 1.96.\nThe background is described with an empirical\nthreshold function. The fit shown in Figure~\\ref{fig:sigcres}\nyields  $59\\pm10$ ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\PSigma_c^0{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace}$ decays and $104\\pm17$\n${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\PSigma_c^{*0}{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace}$ decays.\n\n\\begin{figure}[!tbp]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{Fig2.pdf}\n\\caption{Invariant mass of the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace system from the decay ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$.\nThe $\\PSigma_c^0$ and $\\PSigma_c^{*0}$ resonances are indicated. The fit to the\ndata is shown as a blue continuous line, with the background component shown as a green dotted line, \nthe $\\PSigma_c^{0}$ shape shown as a dashed red line, and \n the $\\PSigma_c^{*0}$ shape shown as a dash-dotted magenta line.}\n\\label{fig:sigcres}\n\\end{center}\n\\end{figure}\n\nThe relative efficiencies for the decays  ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\PSigma_c^0{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace}$, with ${\\PSigma_c^0\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ \nand ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\PSigma_c^{*0}{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace}$, with ${\\PSigma_c^{*0}\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$  \nwith respect to ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ decays are determined\nwith an analogous  procedure as that for the ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ decays relative\nto the ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ decays, but with the trigger samples combined due to\nlimited sample size. The efficiencies are $0.685\\pm0.021$\nfor the $\\PSigma_c^0$ mode and $0.904\\pm0.021$ for the $\\PSigma_c^{*0}$ mode, relative to\n${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$.\n\nMany of the systematic uncertainties cancel out in the measurement\nof the ratio of  branching fractions, with the remaining systematic\nuncertainties stemming from the yield determination. The value of $d$  \n in the Blatt-Weisskopf factor is varied between 1.5 and 0.5\\ensuremath{\\mathrm{ \\,fm}}\\xspace, with\nthe largest variation for each resonance taken as the systematic uncertainty,\nresulting in 3.4\\% for the $\\PSigma_c^0$ resonance and 1.9\\% for the $\\PSigma_c^{*0}$ resonance. The\nbackground shape is changed to a third-order polynomial, with a relative difference\nof 1.7\\% for the $\\PSigma_c^0$ resonance and 10.6\\% for the $\\PSigma_c^{*0}$ resonance taken as the systematic\nuncertainty. The masses and widths of the $\\PSigma_c^{(*)0}$ resonances are allowed\nto float within one standard deviation of their known values~\\cite{PDG2016}, resulting in a 3.8\\%\ndifference of the raw yield for the $\\PSigma_c^0$ resonance and 2.2\\% difference for the\n$\\PSigma_c^{*0}$ resonance. All uncertainties in the relative efficiency cancel, except for\nthose related to the weighting due to resonant structures in the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace system. The scaling factor of\n1.041, with an uncertainty of 1.8\\% on the relative efficiency, which is shown in Table~\\ref{tab:systemerr}, \nis therefore used here as well. The resulting ratios of branching fractions are\n\n\\begin{align*}\n  \\frac{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\PSigma_c^0 p\\overline{p})\\times\\mathcal{B}(\\PSigma_c^0\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace \\pi^-)}{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace  p \\overline{p}\\pi^-)} = 0.089\\pm0.015\\pm0.006,\\\\\n  \\frac{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\PSigma_c^{*0} p\\overline{p})\\times\\mathcal{B}(\\PSigma_c^{*0}\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace \\pi^-)}{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace  p \\overline{p}\\pi^-)} = 0.119\\pm0.020\\pm0.014,\n\\end{align*}\nwhere the first uncertainty is statistical and the second is systematic.\n\n\n\\section{Search for dibaryon resonances}\nThe existence of dibaryon resonances, \n${\\mathscr{D}_c^+\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Pp}}\\xspace\\PSigma_c^0}$, is investigated in the \n${{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\Pp}}\\xspace}$ mass spectrum of background-subtracted data.  \nThe full $m({\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)$ spectrum is considered, while the signal regions of \n$\\PSigma_c^0$ and $\\PSigma_c^{*0}$ resonances are defined  by the ranges \n${ 2450 < m({\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace) <  2458\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace }$ and  ${2488 < m({\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace) <  2549\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace}$, respectively.   \nThe background is subtracted with the \\mbox{\\em sPlot}\\xspace technique\\cite{Pivk:2004ty}.\nNo peaking structures are observed in the \ndistributions shown in Figure~\\ref{fig:dibaryon}. \nThe two-dimensional distribution of $m({\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)$ versus $m({\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)$ \nhas been checked and does not exhibit  \nany clear structure. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.32\\textwidth]{Fig3a.pdf}\n\\includegraphics[width=0.32\\textwidth]{Fig3b.pdf}\n\\includegraphics[width=0.32\\textwidth]{Fig3c.pdf}\n\\caption{Background-subtracted mass spectrum of the ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\Pp}}\\xspace$ system from the decay\n${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ in (a) the full ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ mass spectrum,\n(b) the  signal region of the $\\PSigma_c^0$ resonance, and (c) \n  the signal region of the $\\PSigma_c^{*0}$ resonance. In all figures, the\n  black points are data and the red points  \n  are simulated events where the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace baryon decays to the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace final state \n  (a) based on a uniform-phase-space model, (b) through the $\\PSigma_c^0$ resonance and (c) \n  through the $\\PSigma_c^{*0}$ resonance.  \n  No evident peaking shapes are visible.}\n\\label{fig:dibaryon}\n\\end{center}\n\\end{figure}\n\n\n\\section{Conclusion}\nThe first observation of the decay ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ is presented. The\nratio of the branching fractions using the decay ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace {\\ensuremath{\\pion^-}}\\xspace}$ \nas the  normalisation  channel  \nis measured to be\n\\begin{equation*}\n    \\frac{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace {\\ensuremath{\\pion^-}}\\xspace)}{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace {\\ensuremath{\\pion^-}}\\xspace)} = 0.0540 \\pm 0.0023 \\pm 0.0032,\n\\end{equation*}\nusing data corresponding to an integrated luminosity of 3\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace collected during 2011 and 2012 with the \\mbox{LHCb}\\xspace detector.\nContributions from the $\\PSigma_c(2455)^0$ and $\\PSigma_c^{*}(2520)^0$ resonances are observed,\nand the ratios of their branching fractions with respect to the ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace  {\\ensuremath{\\pion^-}}\\xspace}$ decays  are measured to be\n\\begin{align*}\n  \\frac{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\PSigma_c^0 {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace )\\times\\mathcal{B}(\\PSigma_c^0\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace {\\ensuremath{\\pion^-}}\\xspace)}{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace  {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace {\\ensuremath{\\pion^-}}\\xspace)} = 0.089\\pm0.015\\pm0.006,\\\\\n  \\frac{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\PSigma_c^{*0} {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace )\\times\\mathcal{B}(\\PSigma_c^{*0}\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace {\\ensuremath{\\pion^-}}\\xspace)}{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)} = 0.119\\pm0.020\\pm0.014.\n\\end{align*}\nIn all of the above results, the first uncertainty is statistical and the second is systematic.\n\nThe mass spectra of the ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ final state are also inspected for possible dibaryon\nresonances, but no evidence of peaking structures is observed.\n\n\n\\section*{Acknowledgements}\n\\noindent We express our gratitude to our colleagues in the CERN\naccelerator departments for the excellent performance of the LHC. We\nthank the technical and administrative staff at the LHCb\ninstitutes. We acknowledge support from CERN and from the national\nagencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST and NSFC\n(China); CNRS\/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN\n(Italy); NWO (The Netherlands); MNiSW and NCN (Poland); MEN\/IFA\n(Romania); MinES and FASO (Russia); MinECo (Spain); SNSF and SER\n(Switzerland); NASU (Ukraine); STFC (United Kingdom); NSF (USA).  We\nacknowledge the computing resources that are provided by CERN, IN2P3\n(France), KIT and DESY (Germany), INFN (Italy), SURF (The\nNetherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex\nLLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil),\nPL-GRID (Poland) and OSC (USA). We are indebted to the communities\nbehind the multiple open-source software packages on which we depend.\nIndividual groups or members have received support from AvH Foundation\n(Germany), EPLANET, Marie Sk\\l{}odowska-Curie Actions and ERC\n(European Union), ANR, Labex P2IO and OCEVU, and R\\'{e}gion\nAuvergne-Rh\\^{o}ne-Alpes (France), Key Research Program of Frontier\nSciences of CAS, CAS PIFI, and the Thousand Talents Program (China),\nRFBR, RSF and Yandex LLC (Russia), GVA, XuntaGal and GENCAT (Spain),\nHerchel Smith Fund, the Royal Society, the English-Speaking Union and\nthe Leverhulme Trust (United Kingdom).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe growing importance of simulation tools in analysis and design has directed\nthe interest of the research community towards building accurate,  yet \ninexpensive,  mathematical models.\nThe need of  accuracy  leads  invariably to the incorporation of a large number of state variables. This, in turn, leads to large scale models which in general may be difficult to simulate owing to time or storage constraints. Model reduction methods alleviate this issue by constructing simplified models which retain prescribed features of the original system~\\citep{antoulas2005approximation}.\n\nModel reduction methods for linear systems belong broadly to two main classes~\\citep{antoulas2005approximation}. The first class makes use of the  singular value decomposition~\\citep{moore1981principal,glover1984all},  while the second class is based on the concept of moment matching or on Krylov projectors~\\citep{georgiou1983partial,antoulas1990solution,georgiou1999interpolation}. While moment matching methods are generally more efficient and reliable from a numerical point of view, methods based on the singular value decomposition not only preserve important features of the original system, but also offer error bounds~\\citep{antoulas2005approximation}.   \n\nOver the past decades, several nonlinear counterparts of linear model reduction methods have been developed~\\citep{berkooz1993proper,scherpen1993balancing,hahn2002improved,astolfi2010model}. \nThis line of research has mostly focused on the construction of reduced order models for \\textit{local} behaviors around equilibria. However, the problem of approximating the \\textit{global} behavior of a nonlinear system is still open. Besselink and co-authors have addressed this problem in~\\citep{besselink2009error,besselink2013model} and shown that (incremental) stability properties can be preserved for systems that can be decomposed as the feedback interconnection of a linear system and a nonlinear system. The  main idea is to reduce the linear dynamics using standard model reduction methods and to impose (incremental) small gain conditions to guarantee existence, uniqueness, and incremental stability   of a  steady-state equilibrium solution.\n\n\nThe present work extends the approach of~\\citep{besselink2009error} to multistable and oscillatory Lure systems \nusing dominance theory~\\citep{forni2018differential,felix2018analysis,padoan2019feedback,padoan2019norm}.\nThe goal is to develop a model reduction theory for non-equilibrium behaviors.\n The proposed framework   is based on two key ingredients: \n(i) the dynamics is split into dominant and non-dominant components, and standard model reduction methods are used to reduce the non-dominant one; \n(ii) the asymptotic behavior of the closed-loop system is characterized using small gain conditions from dominance theory. These conditions guarantee that the behavior of the original system is well captured by that of the reduced order model when the approximation error is small. \nThe framework is illustrated by developing a balanced truncation method for dominant Lure systems inspired by the method in~\\citep{besselink2009error}.  \n\n\nThe remainder of the paper is organized as follows. \nSection~\\ref{sec:problem-formulation} provides the problem formulation.\nSection~\\ref{sec:preliminaries} recalls  some   preliminary results  from   dominance theory. \nSection~\\ref{sec:main-results} contains the main results of the paper, where  a general model reduction framework for dominant Lure systems is presented. \nSection~\\ref{sec:mr}  discusses a balanced truncation method for dominant Lure systems inspired by the method presented in~\\citep{besselink2009error}.   \nSection~\\ref{sec:example} provides an illustrative example,   in which\nthe oscillatory behavior of a discretized heat flow control system\nneeds to be approximated.   \nSection~\\ref {sec:conclusion} summarizes our results  and outlines future research directions.   \n\n\n\n\\section{Problem formulation} \\label{sec:problem-formulation}\n\nConsider a continuous-time, single-input, single-output, time-invariant  \\textit{Lure system} described by the equations \n\\begin{equation} \\label{eq:system-Lure-mr}\n\\dot{x} = Ax +B_u u +B_w w, \\, y = C_y x,  \\, z = C_z x,  \\, w = - \\varphi(z),  \\! \\!\n\\end{equation}\nin which ${x\\in\\R^n}$, ${u\\in\\R}$, ${y\\in\\R}$,  ${w\\in\\R}$, ${z\\in\\R}$, ${A\\in\\R^{n \\times n}}$, ${B_u\\in\\R^{n \\times 1}}$, ${B_w\\in\\R^{n \\times 1}}$, ${C_y\\in\\R^{1\\times n}}$, and ${C_z\\in\\R^{1\\times n}}$ are constant matrices, and ${\\varphi:\\R \\to \\R}$ is a continuously differentiable function\\footnote{Local Lipschitz continuity would be sufficient. Continuous differentiability is assumed to simplify the exposition.} such that ${\\varphi(0) =0}$. Let \n\\begin{equation} \\nn\nG(s) \n=\nC (sI-A)^{-1} B\n= \n\\bma\n\\begin{array}{cc}\nG_{yu}(s) & G_{yw}(s)\\\\\nG_{zu}(s) & G_{zw}(s)\n\\end{array}\n\\ema\n,\n\\end{equation}\nin which ${B = [\\, B_{u} \\ B_{w} \\,]}$ and  ${C = [\\, C_{y}^{\\transpose} \\ C_{z}^{\\transpose} \\,]^{\\transpose}}$, respectively.\n\nSuppose we wish to construct a reduced order model   of   order ${\\nu < n}$ of system~\\eqref{eq:system-Lure-mr} described by the equations\n\\begin{equation} \\label{eq:system-ROM}\n\\dot{\\hat{x}} = \\hat{A}\\hat{x} +\\hat{B}_u  \\hat{u}   +\\hat{B}_w \\hat{w}, \\, \\hat{y} = \\hat{C}_y \\hat{x},  \\, \\hat{z} = \\hat{C}_z \\hat{x},  \\, \\hat{w} = - \\varphi(\\hat{z}),  \\! \\!\n\\end{equation}\nwith ${\\hat{x}\\in\\R^\\nu}$, ${ \\hat{u} \\in\\R}$, ${\\hat{y}\\in\\R}$,  ${\\hat{w}\\in\\R}$, ${\\hat{z}\\in\\R}$, ${\\hat{A}\\in\\R^{\\nu \\times \\nu}}$, ${\\hat{B}_u\\in\\R^{\\nu \\times 1}}$, ${\\hat{B}_w\\in\\R^{\\nu \\times 1}}$, ${\\hat{C}_y\\in\\R^{1\\times \\nu}}$, and ${\\hat{C}_z\\in\\R^{1\\times \\nu}}$. Let \n\\begin{equation} \\nn\n\\hat{G}(s) \n=\n\\hat{C} (sI-\\hat{A})^{-1} \\hat{B}\n= \n\\bma\n\\begin{array}{cc}\n\\hat{G}_{yu}(s) & \\hat{G}_{yw}(s)\\\\\n\\hat{G}_{zu}(s) & \\hat{G}_{zw}(s)\n\\end{array}\n\\ema\n,\n\\end{equation}\nin which ${\\hat{B} = [\\, \\hat{B}_{u} \\ \\hat{B}_{w} \\,]}$ and  ${\\hat{C} = [\\, \\hat{C}_{y}^{\\transpose} \\ \\hat{C}_{z}^{\\transpose} \\,]^{\\transpose}}$, respectively.  \nThe transfer function of the error system is defined as\n\\begin{equation} \\nn\n\\tilde{G}(s) = G(s) - \\hat{G}(s). \n\\end{equation}\nThe reduced order model~\\eqref{eq:system-ROM} is constructed by replacing the linear dynamics of the original system\n\\begin{equation} \\label{eq:system-Lure-mr-linear}\n\\dot{x} = Ax +B_u u +B_w w, \\ y = C_y x,  \\ z = C_z x, \n\\end{equation}\nwith a linear system described by the equations\n\\begin{equation} \\label{eq:system-ROM-linear}\n\\dot{\\hat{x}} = \\hat{A}\\hat{x} +\\hat{B}_u  \\hat{u}   +\\hat{B}_w \\hat{w}, \\ \\hat{y} = \\hat{C}_y \\hat{x},  \\ \\hat{z} = \\hat{C}_z \\hat{x}, \n\\end{equation}\nand by leaving the static nonlinearity ${\\varphi}$ unchanged, as illustrated in Fig.~\\ref{fig:model_reduction}. \n\n\n\\begin{figure}[h!]\n\\centering\n\\begin{tikzpicture}[scale=.75, black,every node\/.style={transform shape}]\n \n\\draw[line width = .5 pt]  (10.3125,3.5) rectangle (11.3125,2.5);\n\\node at (10.8125,3) {$G$};\n\\draw[rounded corners,line width = .5 pt]  (10.3125,2.3125) rectangle (11.3125,1.3125);\n\\node at (10.8125,1.8125) {$-\\varphi(\\cdot)$};\n\\draw[-latex, line width = .5 pt] (11.3125,2.75) -- (12.3125,2.75) -- (12.3125,1.8125) -- (11.3125,1.8125);\n\\draw[-latex, line width = .5 pt] (10.3125,1.8125) -- (9.3125,1.8125) -- (9.3125,2.75) -- (10.3125,2.75) ;\n\\draw [-latex, line width = .5 pt](9.3125,3.25) -- (10.3125,3.25);\n\\node at (9.8125,3.5) {$u$};\n\\node at (11.8125,3.5) {$y$};\n\\node at (9.8125,2.5) {$w$};\n\\node at (11.8125,2.5) {$z$}; \n\\draw [-latex,line width = .5 pt](11.3125,3.25) -- (12.3125,3.25);\n\n  \n\\draw[line width = .5 pt]  (16.6875,3.5) rectangle (17.6875,2.5);\n\\node at (17.1875,3) {$\\hat{G}$};\n\\draw[rounded corners,line width = .5 pt]  (16.6875,2.3125) rectangle (17.6875,1.3125);\n\\node at (17.1875,1.8125) {$-\\varphi(\\cdot)$};\n\\draw[-latex, line width = .5 pt] (17.6875,2.75) -- (18.6875,2.75) -- (18.6875,1.8125) -- (17.6875,1.8125);\n\\draw[-latex, line width = .5 pt] (16.6875,1.8125) -- (15.6875,1.8125) -- (15.6875,2.75) -- (16.6875,2.75) ;\n\\draw [-latex, line width = .5 pt](15.6875,3.25) -- (16.6875,3.25);\n\\node at (16.1875,3.5) {$\\hat{u}$};\n\\node at (18.1875,3.5) {$\\hat{y}$};\n\\node at (16.1875,2.5) {$\\hat{w}$};\n\\node at (18.1875,2.5) {$\\hat{z}$}; \n\\draw [-latex,line width = .5 pt](17.6875,3.25) -- (18.6875,3.25);\n\n\\draw [thick, -latex](12.8125,2.5) -- (15.1875,2.5);\n\\node at (14,3) { Model reduction};\n \n\\end{tikzpicture} \n\\centering\n\\caption{Diagrammatic illustration of the original system~\\eqref{eq:system-Lure-mr} (left) and of the reduced order model~\\eqref{eq:system-ROM} (right).\n}\n\\label{fig:model_reduction}\n\\end{figure\n\n\nThis work addresses basic, yet open, questions about the \\textit{global} behavior of the reduced order model~\\eqref{eq:system-ROM}. For example, \nsuppose that the original system~\\eqref{eq:system-Lure-mr} is multistable or oscillatory. Further, suppose an error bound is available for the approximation of the linear dynamics. One can expect that~\\eqref{eq:system-ROM} is indeed a good reduced order model if the error is small, but how does one \\textit{quantify} this? Can one construct  better  reduced order models in a systematic and computationally appealing fashion? Is it sensible to use the reduced order model for analysis and design?\n\nThe paper~\\citep{besselink2009error} has shown that these questions can be addressed for (incrementally) \\textit{stable} Lure systems using sector conditions and small gain conditions.\nThe next sections show that similar conclusions can be drawn for \\textit{dominant} Lure systems by combining existing linear model reductions methods with dominance theory.\n\n\n\n\n\\section{Dominance theory} \\label{sec:preliminaries}\n\n\nConsider a continuous-time, nonlinear, time-invariant system and its linearization described by the equations\n\\begin{align}\n\\,\\Sigma : \\quad \\ \\dot{x} &= f(x) +Bu, \\quad \\quad \\quad \\ \\ \\ y = \\ Cx , \\label{eq:system-nonlinear} \\\\\n\\delta\\Sigma :\\quad \\dot{\\delta x} &= \\partial f(x) \\delta x + B \\delta u, \\quad ~\\, \\delta y =~ C \\delta x,   \\label{eq:system-nonlinear-linearization}\n\\end{align}\nin which ${x\\in\\R^n}$, ${u\\in\\R^m}$,  ${y\\in\\R^l}$,  \n${f:\\R^{n} \\to \\R^{n}}$ is a continuously differentiable vector field,   ${B\\in\\R^{n\\times m}}$,   and ${C\\in\\R^{m\\times n}}$ are constant matrices, ${\\delta x\\in\\R^n}$, ${\\delta u\\in\\R^m}$,   ${\\delta y\\in\\R^l}$   (identified with the respective tangent spaces), and ${\\partial f}$ is the Jacobian of the vector field $f$.\n\n\n\n\n\\begin{definition} \\label{def:dominance}\nThe system~\\eqref{eq:system-nonlinear} is $p$-dominant with rate ${\\lambda \\in \\Rge}$ (for ${u=0}$) if there exist ${\\varepsilon \\in \\Rge}$ and a symmetric matrix ${P\\in\\R^{n\\times n}}$, with inertia\\footnote{The inertia of the matrix ${A \\in \\R^{n \\times n}}$ is defined as $\\operatorname{In}(A)=(n_{-},n_{0},n_{+})$, where $n_{-}$ is the number of eigenvalues of $A$ in the open left half-plane, $n_{0}$ is the number of eigenvalues of $A$ on the imaginary axis, and $n_{+}$ is the number of eigenvalues of $A$ in the open right half-plane, respectively.  \n} $\\operatorname{In}(P) = ( p, 0,n-p)$, such that the prolonged system~\\eqref{eq:system-nonlinear}-\\eqref{eq:system-nonlinear-linearization}\nsatisfies\n\\begin{equation}  \\label{eq:dominance-LMI}\n\\bma\n\\begin{array}{c}\n \\dot{\\delta x}  \\\\\n\\delta x\n\\end{array}\n\\ema^{\\transpose}\n\\bma\n\\begin{array}{cc}\n0 & P \\\\\nP &  2\\lambda P + \\varepsilon I\n\\end{array}\n\\ema\n\\bma\n\\begin{array}{c}\n\\dot{\\delta x} \\\\\n\\delta x\n\\end{array}\n\\ema \\le 0 \n\\end{equation} \nfor every $(x,\\delta x) \\in \\R^n \\times \\R^n$. The property is strict if ${\\varepsilon >}0$.\n\\end{definition}\n\n\nThe property of $p$-dominance ensures the existence of a splitting between $p$ dominant modes and $n-p$ transient modes.  In particular, the matrix ${\\partial f}(x) +\\lambda I$ has $p$ unstable eigenvalues and $n-p$ stable eigenvalues for every ${x\\in\\R^n}$. For linear, time-invariant systems, such eigenvalues are referred to as \\textit{dominant and non-dominant}, respectively.\n\n\n\\begin{definition} \\label{def:pgain}\nThe system~\\eqref{eq:system-nonlinear} is said to have \\textit{finite (differential) ${\\L_{2,p}}$-gain (from $u$ to $y$) less than ${\\gamma \\in \\Rge}$ with rate ${\\lambda \\in \\Rge}$} if there exist ${\\varepsilon \\in \\Rge}$ and a symmetric matrix ${P\\in\\R^{n\\times n}}$, with inertia   $(p, 0, n-p)$,   such that the conic constraint\n\\begin{equation} \\label{eq:system-linear-MIMO-conic-IO}\n\\begingroup \n\\setlength\\arraycolsep{1.3pt}\n\\bma\n\\begin{array}{c}\n\\dot{\\delta x} \\\\\n\\delta x\n\\end{array}\n\\ema^{\\transpose} \\!\n\\bma\n\\begin{array}{cc}\n0 & P \\\\\nP &  2\\lambda P + \\varepsilon I\n\\end{array}\n\\ema \\!\n\\bma\n\\begin{array}{c}\n  \\dot{\\delta x} \\\\\n\\delta x\n\\end{array}\n\\ema \n\\le\n\\bma\n\\begin{array}{c}\n\\delta y \\\\\n\\delta u\n\\end{array}\n\\ema^{\\transpose} \\!\n\\bma\n\t\\begin{array}{cc}\n\t-I & 0 \\\\\n\t 0 & \\gamma^2 I\n\t\\end{array}\n\t\\ema \\!\n\\bma\n\\begin{array}{c}\n\\delta y \\\\\n\\delta u\n\\end{array}\n\\ema  \\! \\! \n\\endgroup\n\\end{equation}\nholds along the solutions of the prolonged system~\\eqref{eq:system-nonlinear}-\\eqref{eq:system-nonlinear-linearization}.\nThe \\textit{(differential) $\\L_{2,p}$-gain of system~\\eqref{eq:system-nonlinear}  (from $u$ to $y$) with rate $\\lambda$} is defined as ${\\gamma_{\\lambda} = \\inf \\left\\{ \\gamma \\in \\Rge  :  \\text{\\eqref{eq:system-linear-MIMO-conic-IO} holds} \\right\\}}.$ The properties are strict if $\\varepsilon >0$. \n\\end{definition}\n\nThe asymptotic behavior of a $p$-dominant system is $p$-dimensional and, hence, its attractors\nare severely constrained for small values of $p$~\\citep{forni2018differential}.\n\n\\begin{theorem}  \\label{thm:asymptotic}\nAssume system~\\eqref{eq:system-nonlinear} is strictly $p$-dominant with rate ${\\lambda \\in \\Rge}$. Then every bounded solution of~\\eqref{eq:system-nonlinear} converges asymptotically to\n\\begin{description}\n\\item[{\\small $\\bullet$}] the unique equilibrium  point if ${p=0}$; \n\\item[{\\small $\\bullet$}] a (possibly non-unique) equilibrium  point if ${p=1}$;\n\\item[{\\small $\\bullet$}] a simple attractor if ${p=2}$, \\textit{i.e.} an equilibrium  point, a set of equilibrium  points and their connected arcs or a limit cycle.\n\\end{description} \n\\end{theorem} \n\nThe dominance properties of an interconnected system can be studied\nusing small ${\\L_{2,p}}$-gain conditions~\\citep{forni2018differential,padoan2019norm}.\n\n\n\\begin{theorem}[Small-gain theorem for $p$-dominance] \\label{thm:small-gain}\nLet $\\Sigma_i$ be a system with input ${u_i\\in\\R^{m_i}}$, output ${y_i\\in\\R^{l_i}}$, and (strict) $\\L_{2,p_i}$-gain less than ${\\gamma_i\\in\\Rge}$ with rate ${\\lambda \\in \\Rge}$, with ${i \\in \\{1,2\\}}$. Then the closed-loop system defined by the feedback interconnection equations  ${u_1 = -y_2}$ and ${u_2 = y_1}$  is strictly ${(p_1+p_2)}$-dominant with rate $\\lambda$ if ${\\gamma_1 \\gamma_2 <1}$. \n\\end{theorem}\n\n\\subsection{Dominant systems in the frequency domain}\n\nConsider a continuous-time, linear, time-invariant system described by the equations \n\\begin{equation} \\label{eq:system-linear-MIMO}\n\\quad \\dot{x} = Ax+Bu, \\quad y=Cx,\n\\end{equation}\nin which ${x \\in\\R^n}$, ${u\\in\\R^m}$, ${y\\in\\R^l}$, and ${A\\in\\R^{n \\times n}}$, ${B\\in\\R^{n \\times m}}$, and ${C\\in\\R^{l\\times n}}$, are constant matrices, respectively. Let $G(s) = C(sI-A)^{-1} B$ be its transfer function and, given ${\\lambda \\in \\Rge}$, let ${G_\\lambda(s) = G (s-\\lambda)}$ be its shifted transfer function, provided $G$ is well-defined on the axis ${\\Re(s) = - \\lambda}$.\n\nThe ${\\L_{2,p}}$ gain is intimately connected to the \\textit{${\\H_{\\infty,p}}$ norm} of the transfer function of system~\\eqref{eq:system-linear-MIMO}, defined as\n\\begin{equation} \\label{eq:Hq-norm-lambda-strip} \n\\norm{G}_{\\infty,p} = \\displaystyle \\sup_{\\omega \\in  \\R } {\\sigma_{\\max}} (G_{\\lambda}(i\\omega)) , \n\\end{equation} \nwhere ${\\sigma_{\\max}}(M)$ is the largest singular value of the matrix ${M\\in\\Co^{l\\times m}}$ and ${G}$ has $p$ poles in the open half-plane ${\\Re(s)>-\\lambda}$, respectively. The ${\\H_{\\infty,p}}$ norm is not uniquely defined by the integer $p$, as it depends on the parameter $\\lambda$.\n\n\n\\begin{theorem} \\citep{padoan2019norm} \\label{thm:pgain}\nAssume system~\\eqref{eq:system-linear-MIMO} has (strict) $\\L_{2,p}$-gain  ${\\gamma_{\\lambda} }$ with rate ${\\lambda \\in \\Rge}$. Then ${\\gamma_{\\lambda}  =  \\norm{G}_{\\infty,p}.}$\n\\end{theorem}\n\n\n \n\n\n\nThe property of $p$-dominance can be also characterized \\textit{graphically} in the frequency domain~\\citep{felix2018analysis}.  Consider a single-input, single-output, Lure system described by the equations \n\\begin{equation} \\label{eq:system-Lure}\n\\dot{x} = Ax +Bu, \\quad y = Cx,  \\quad u =  -   \\varphi(y),\n\\end{equation}\nin which ${x\\in\\R^n}$, ${u \\in\\R}$, ${y \\in\\R}$, ${A\\in\\R^{n \\times n}}$, ${B\\in\\R^{n \\times 1}}$, and ${C\\in\\R^{1\\times n}}$  are constant matrices, and ${\\varphi:\\R \\to \\R}$ is a continuously differentiable function which satisfies \nthe \\textit{differential sector condition} ${\\partial \\varphi \\in  [\\alpha, \\beta]}$, defined as\n\\begin{equation} \\label{eq:sector-condition} \n(\\partial \\varphi (y) \\delta y - \\alpha \\delta y) (\\partial \\varphi (y) \\delta y - \\beta \\delta y) \\leq 0, \\  \\forall \\, y \\in \\R,\n\\end{equation}\nwith ${-\\infty \\le \\alpha < \\beta \\le \\infty}$. Let \n${G(s) = C(sI-A)^{-1}B}$.\n\n\\begin{theorem}[Circle criterion for $p$-dominance] \\label{thm:circle_criterion}\nConsider \\linebreak system~\\eqref{eq:system-Lure} and let $\\lambda \\in \\Rge$. Assume \n$\\partial \\varphi \\in [\\alpha, \\beta]$, $G_{\\lambda}$ has $q$ unstable poles and no poles along the imaginary axis,\nthe Nyquist diagram of $G_{\\lambda}$ encircles ${(p-q)}$ times the point $-\\tfrac{1}{\\alpha}$  in the clockwise direction, and one of the following  mutually exclusive   conditions hold\\footnote{$D(\\alpha, \\beta)$  denotes the closed disk in the complex plane associated with the sector $[\\alpha, \\beta]$, with ${\\alpha < \\beta}$. The notation is standard and is defined, \\textit{e.g.}, in~\\citep[p.82]{felix2018analysis}.}\n\\begin{itemize}\n\\item[(a)] ${\\alpha \\beta \\ge 0}$ and the Nyquist diagram of $G_{\\lambda}$ lies \\textit{outside} the disk $D(\\alpha, \\beta)$.\n\\item[(b)] ${\\alpha \\beta <0}$ and the Nyquist diagram of $G_{\\lambda}$ lies \\textit{inside} the disk $D(\\alpha, \\beta)$.\n\\end{itemize}\nThen system~\\eqref{eq:system-Lure} is strictly $p$-dominant with rate $\\lambda$.\n\\end{theorem}\n\n\n\\section{Main results} \\label{sec:main-results}\n\nThe first step of our model reduction approach for dominant Lure systems is to split the linear dynamics into dominant and non-dominant components. Standard model reduction methods are then applied to the non-dominant component, as illustrated in Fig.~\\ref{fig:model_reduction_detail_new}. The main steps are summarized in Algorithm~\\ref{alg:mr-Lure}.  This approach preserves by construction the dominance properties of the linear dynamics of the original system. Furthermore, taking advantage of small-gain conditions for $p$-dominance, it guarantees that the behavior of the original original system is well captured by that of the reduced order model, when the approximation error is small. This is formalized in Theorem~\\ref{thm:1}.\n \n\\begin{algorithm}[h!] \n\\algnewcommand\\algorithmicto{\\textbf{to}}\n\\algnewcommand\\algorithmicbreak{\\textbf{break}}\n\\algnewcommand\\algorithmicstop{\\textbf{stop}}\n\\algnewcommand\\algorithmicass{\\textbf{Assumption}}\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output}}\n\\caption{}\n \\label{alg:mr-Lure}  \n\\algorithmicrequire{: The system~\\eqref{eq:system-Lure-mr}}\\\\\n\\algorithmicensure{: The reduced order model~\\eqref{eq:system-ROM}}\\\\\n\\algorithmicass{: {Strict\\,$p$-dominance\\,with\\,rate\\,$\\lambda$\\,of\\,system\\,\\eqref{eq:system-Lure-mr}}} \n\\begin{algorithmic}[1] \n\\State Consider the linear dynamics~\\eqref{eq:system-Lure-mr-linear} and, upon a possible change of coordinates, let\n\\begin{equation}  \\nn\nA =\n\\bma\n\\begin{array}{cc}\nA^+ & 0 \\\\\n0 & A^-\n\\end{array}\n\\ema, \\\nB =\n\\bma\n\\begin{array}{c}\nB^+ \\\\\nB^-\n\\end{array}\n\\ema, \\\nC =\n\\bma\n\\begin{array}{cc}\nC^+  &\nC^- \n\\end{array}\n\\ema, \n\\end{equation}\nwhere ${\\spectrum{A^+}}$ and ${\\spectrum{A^-}}$ contain $p$ dominant eigenvalues and $n-p$ non-dominant eigenvalues,  respectively.  \n\\State Construct a reduced order model \n\\begin{equation} \\nn\n\\dot{\\hat{x}}^- = \\hat{A}^-\\hat{x}^- +\\hat{B}^-  \\hat{v}^-, \\quad \\hat{y}^- = \\hat{C}^- \\hat{x}^-, \n\\end{equation}\nof the non-dominant linear dynamics \n\\begin{equation} \\nn\n\\dot{x}^- = {A}^- {x}^- + {B}^-{v}^-, \\quad {y}^- = {C}^- {x}^-, \n\\end{equation}\nusing a stability-preserving model reduction method. \n\\State Define the reduced order model~\\eqref{eq:system-ROM} as\n\\begin{equation}  \\nn\n\\hat{A} =\n\\bma\n\\begin{array}{cc}\nA^+ & 0 \\\\\n0 & \\hat{A}^-\n\\end{array}\n\\ema, \\\nB =\n\\bma\n\\begin{array}{c}\nB^+ \\\\\n\\hat{B}^-\n\\end{array}\n\\ema, \\\nC =\n\\bma\n\\begin{array}{cc}\nC^+  &\n\\hat{C}^- \n\\end{array}\n\\ema .\n\\end{equation} \n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\\begin{figure}[h!]\n\\centering\n\\begin{tikzpicture}[scale=.75, black,every node\/.style={transform shape}]\n      \n\\draw[line width = .5 pt]  (10.6875,-0.375) rectangle (11.6875,-1.375);\n\\node at (11.1875,-0.875) {$\\hat{G}_{zw}^{-}$};\n\\draw[rounded corners,line width = .5 pt]  (10.6875,-2.6964) rectangle (11.6875,-3.6964);\n\\node at (11.1875,-3.1964) {$-\\varphi(\\cdot)$};\n\\draw[-latex, line width = .5 pt] (12.1875,-2.0625)   -- (12.1875,-3.1964) -- (11.6875,-3.1964);\n\\draw[-latex, line width = .5 pt] (10.6875,-3.1964) -- (10.1875,-3.1964) -- (10.1875,-2) -- (10.6875,-2) ;\n\\draw [-latex, line width = .5 pt] (10.1875,-0.875) --(10.1875,0.4466) -- (10.6875,0.4466);\n\\draw [-latex,line width = .5 pt](11.6875,0.4466) -- (12.1875,0.4466)  -- (12.1875,-0.8194);\n\\draw[fill = white] (12.1875,-0.875) circle [radius=0.05];\n\\draw[semithick, dashed] (10,1.0716) rectangle (12.6875,-0.1159);\n\\draw[semithick, dashed] (9.9992,-0.3) rectangle (12.6883,-2.6076);\n\\draw  (10.6875,0.9466) rectangle (11.6875,-0.0534);\n\\node at (11.1875,0.4466) {$\\tilde{G}_{zw}^{-}$};\n\\draw[-latex, line width = .5 pt] (11.6875,-0.875) -- (12.1319,-0.875); \n\\draw[-latex, line width = .5 pt]  (10.6875,-1.5) rectangle (11.6875,-2.5);\n\\node at (11.1875,-2) {${G_{zw}^+}$};\n\\draw [-latex, line width = .5 pt](10.1875,-2) -- (10.1875,-0.875) -- (10.6875,-0.875);\n\\draw[fill = white] (12.1875,-2) circle [radius=0.05];\n  \\draw [-latex, line width = .5 pt](12.1875,-0.9375) -- (12.1875,-1.9375);\n\\draw [-latex, line width = .5 pt](11.6875,-2) -- (12.125,-2);\n\\node at (12.4375,0.8216) {$\\tilde{G}$};\n\\node at (12.4375,-2.2951) {$\\hat{G}$};\n\n \n\\draw [thick, -latex](12.875,-1.25) -- (15.25,-1.25);\n\\node at (14.0625,-0.75) { Model reduction};\n  \n\\draw[line width = .5 pt]  (16.1875,-0.375) rectangle (17.1875,-1.375);\n\\node at (16.6875,-0.875) {$\\hat{G}_{zw}^{-}$};\n\\draw[rounded corners,line width = .5 pt]  (16.1875,-2.6964) rectangle (17.1875,-3.6964);\n\\node at (16.6875,-3.1964) {$-\\varphi(\\cdot)$};\n\\draw[-latex, line width = .5 pt] (17.6875,-2.0625)   -- (17.6875,-3.1964) -- (17.1875,-3.1964);\n\\draw[-latex, line width = .5 pt] (16.1875,-3.1964) -- (15.6875,-3.1964) -- (15.6875,-2) -- (16.1875,-2) ;\n\n\n\\draw[semithick, dashed] (15.4992,-0.3) rectangle (18.1883,-2.6076);\n  \n\\draw[-latex, line width = .5 pt]  (16.1875,-1.5) rectangle (17.1875,-2.5);\n\\node at (16.6875,-2) {${G_{zw}^+}$};\n\\draw [-latex, line width = .5 pt](15.6875,-2) -- (15.6875,-0.875) -- (16.1875,-0.875);\n\\draw[fill = white] (17.6875,-2) circle [radius=0.05];\n  \\draw [-latex, line width = .5 pt](17.1875,-0.875) -- (17.6875,-0.8661) -- (17.6875,-1.9375);\n\\draw [-latex, line width = .5 pt](17.1875,-2) -- (17.625,-2);\n \n\\node at (17.9375,-2.2951) {$\\hat{G}$};\n\n\\end{tikzpicture} \n\\centering\n\\caption{Diagrammatic illustration of the model reduction method described in Algorithm~\\ref{alg:mr-Lure}.\n}\n\\label{fig:model_reduction_detail_new}\n\\end{figure\n\n\n\\begin{theorem} \\label{thm:1}\nConsider system~\\eqref{eq:system-Lure-mr} and a reduced order model~\\eqref{eq:system-ROM} constructed as  in Algorithm~\\ref{alg:mr-Lure}.  Assume\n\\begin{description}\n\\item[(A1)]  \\eqref{eq:system-Lure-mr-linear} is strictly   $p$-dominant with rate ${\\lambda\\in\\Rge}$,\n\\item[(A2)] ${\\partial \\varphi \\in  [-\\mu, \\mu]}$, with  ${\\mu\\in \\Rge}$,\n\\item[(A3)] ${\\norm{\\tilde{G}}_{\\infty,0} \\le \\epsilon}$, with ${0<\\epsilon<\\mu^{-1}}$,\n\\item[(A4)] ${\\norm{\\hat{G}_{zw}}_{\\infty,p} < \\mu^{-1} - \\epsilon}$.\n\\end{description}\nThen system~\\eqref{eq:system-Lure-mr} is strictly $p$-dominant with rate ${\\lambda}$.\n\\end{theorem}\n\n\n\n\n\n\n\\begin{pf}\nBy the circle criterion for $p$-dominance, system~\\eqref{eq:system-Lure-mr} is $p$-dominant with rate ${\\lambda}$ if \n\\begin{equation} \\label{eq:thm1-proof-1}\n\\norm{G_{zw}}_{\\infty,p} < \\mu^{-1},\n\\end{equation}\nsince, by assumptions (A1) and (A2), \\eqref{eq:system-Lure-mr} is $p$-dominant with rate ${\\lambda}$ (and, thus, the transfer function ${G(s-\\lambda)}$ has $p$ unstable poles and no poles on the imaginary axis) and ${\\partial \\varphi \\in  [-\\mu, \\mu]}$. However, \\eqref{eq:thm1-proof-1} follows directly from assumptions (A3) and (A4), as\n\\begin{align*}\n\\norm{G_{zw}}_{\\infty,p}\n\\stackrel{\\text{def}}{=} \\norm{\\hat{G}_{zw}  + \\tilde{G}_{zw}}_{\\infty,p} \n\\stackrel{(\\text{A3})}{\\le} \\norm{\\hat{G}_{zw}}_{\\infty,p} + \\epsilon  \n\\stackrel{(\\text{A4})}{<} \\mu^{-1}.\n\\end{align*}\n\\end{pf} \n\n \n\n\nTheorem~\\ref{thm:1} establishes that the dominance properties of the \noriginal closed-loop system can be inferred from those of the reduced order model if the approximation error between the original linear component $G$ and the reduced linear component $\\hat{G}$ is sufficiently small.  \nFrom Algorithm~\\ref{alg:mr-Lure}, by construction, Assumption (A1) guarantees that the reduced linear dynamics $\\hat{G}$ is $p$-dominant with rate $\\lambda$. \nAssumption (A2) and the circle criterion for $p$-dominance \nensure that the closed-loop reduced order model is also $p$-dominant \nwith the same rate  if ${\\norm{\\hat{G}_{zw}}_{\\infty,p} < \\mu^{-1} }$ (which, in turn, is implied by assumption (A4)). The approximation error bound of assumption (A3) expresses the fact that the  linear   dynamics of the reduced order model is ``close'' to that of the original system in the ${\\H_{\\infty,0}}$ norm.\nFinally, the small ${\\LTp}$ gain condition of assumption (A4) ensures that the dominance properties of the original system are captured by those of the the reduced order model when the approximation error is regarded as a perturbation.\n\n\n\\begin{remark}\nTheorem~\\ref{thm:1} may be conservative as it requires only minimal information about the static nonlinearity $\\varphi$; it applies to \\textit{any} static nonlinearity $\\varphi$ such that ${\\varphi(0) =0}$ and ${\\partial \\varphi \\in  [-\\mu, \\mu]}$. The model reduction framework is therefore robust to nonlinear uncertainty.\n\\hspace*{\\fill} $\\blacktriangle$\n\\end{remark}\n\n\n\n\nTheorem~\\ref{thm:1} can be seen an application the small ${\\LTp}$ gain theorem.  Assumptions (A3) and (A4) imply the small ${\\LTp}$ condition ${\\gamma_1 \\gamma_2 < 1}$, in which ${\\tilde{\\gamma} = \\norm{\\tilde{G} }_{\\infty,0}}$ and ${\\hat{\\gamma} = (\\mu^{-1} - \\norm{\\hat{G}}_{\\infty,p} )^{-1}}.$ This guarantees that the closed-loop system in Fig.~\\ref{fig:theorem_new} (left) is $p$-dominant, as the product of the differential  gain  ${\\tilde{\\gamma}}$ (from $w$ to $\\tilde{z}$) and the differential gain ${\\hat{\\gamma}}$ (from $\\tilde{z}$ and $w$) is strictly less than one. \\hspace*{\\fill} \n\n  \nThe small ${\\LTp}$ condition (A4)    admits a nice graphical interpretation. From a geometrical viewpoint, it implies that the distance between the critical circle and the Nyquist diagram of the shifted transfer function ${\\hat{G}_{\\lambda}}$ must be at least ${\\epsilon}$, as illustrated in Fig.~\\ref{fig:theorem_new} (right). This implies that $D(-\\hat{\\mu}, \\hat{\\mu})$, with ${\\hat{\\mu} = (\\mu^{-1} - \\epsilon)^{-1}}$, is a $p$-disk margin for the reduced order model~\\citep{padoan2019feedback}. This suggests that claim (i) can be established by a homotopy argument  (given, \\textit{e.g.}, in   \\cite[Theorem 3]{glover1983robust}).\n\n \n\n\n\n\\begin{figure}[h!]\n\\centering\n\\usetikzlibrary{arrows.meta}\n\\tikzset{>={Latex[width=3pt,length=3pt]}}\n\\usetikzlibrary{decorations.pathreplacing}\n\\begin{tikzpicture}[black, scale=.7, every node\/.style={transform shape}]\n \n \n\\draw[line width = .5 pt]  (11.8125,3.3125) rectangle (12.8125,2.3125);\n\\node at (12.3125,2.8125) {$\\hat{G}_{zw}^{-}$};\n\\draw[rounded corners,line width = .5 pt]  (11.8125,0.9375) rectangle (12.8125,-0.0625);\n\\node at (12.3125,0.4375) {$-\\varphi(\\cdot)$};\n\\draw[-latex, line width = .5 pt] (13.3125,1.625)   -- (13.3125,0.4375) -- (12.8125,0.4375);\n\\draw[-latex, line width = .5 pt] (11.8125,0.4375) -- (11.3125,0.4375) -- (11.3125,1.6875) -- (11.8125,1.6875) ;\n\\draw [-latex, line width = .5 pt] (11.3125,2.8125) --(11.3125,4.5625) -- (11.8125,4.5625);\n\\node at (11.125,3.6875) {$w$};\n\\node at (13.5625,3.6875) {$\\tilde{z}$};\n\\draw [-latex,line width = .5 pt](12.8125,4.5625) -- (13.3125,4.5625)  -- (13.3125,2.8681);\n\\draw[fill = white] (13.3125,2.8125) circle [radius=0.05];\n\\draw[semithick, dashed] (11.125,5.1875) rectangle (13.8125,4);\n\\node at (15,4.5625) {{ Gain $\\tilde{\\gamma}$}};\n\\draw[semithick, dashed] (11.1242,3.3875) rectangle (13.8133,1.0625);\n\\node at (15,1.75) {{ Gain ${\\hat{\\gamma}}$}};\n\\draw  (11.8125,5.0625) rectangle (12.8125,4.0625);\n\\node at (12.3125,4.5625) {$\\tilde{G}_{zw}^{-}$};\n\\draw[-latex, line width = .5 pt] (12.8125,2.8125) -- (13.2569,2.8125); \n\\draw[-latex, line width = .5 pt]  (11.8125,2.1875) rectangle (12.8125,1.1875);\n\\node at (12.3125,1.6875) {${G_{zw}^+}$};\n\\draw [-latex, line width = .5 pt](11.3125,1.6875) -- (11.3125,2.8125) -- (11.8125,2.8125);\n\\draw[fill = white] (13.3125,1.6875) circle [radius=0.05];\n  \\draw [-latex, line width = .5 pt](13.3125,2.75) -- (13.3125,1.75);\n\\draw [-latex, line width = .5 pt](12.8125,1.6875) -- (13.25,1.6875);\n\\node at (13.5625,4.9375) {$\\tilde{G}$};\n\\node at (13.5625,1.375) {$\\hat{G}$};\n\n\\draw [-latex](18.0625,0.125) -- (18.0625,5);\n\\draw [-latex](16.0625,2.5) -- (21,2.5);\n\\draw[thick] plot[smooth cycle, tension=.7] coordinates {(18.6733,2.5) (18.5069,2.8888) (18.0625,2.9444) (17.8405,2.5832) (18.0625,2.4168) (18.2289,2.4168)(18.2289,2.5832) (18.0625,2.5832) (17.8405,2.4168) (18.0625,2.0556) (18.5069,2.1112)} ;\n\\node at (20.5625,2.125) {{ $\\mathbf{Re} \\,\\hat{G}_{\\lambda}(i\\omega)$}};\n\\node at (17.0625,4.75) {{ $\\mathbf{Im} \\,\\hat{G}_{\\lambda}(i\\omega)$}};\n\\node at (16.1875,2.125) {{\\large $-\\tfrac{1}{\\mu}$}}; \n\\node at (19.25,3.75) {{\\large${\\mathbf{{\\epsilon}}} $}}; \n\n\\begin{scope}[even odd rule, shift={(2.0625,0)}] \n\\clip (16,2.5) ellipse (1.5 and 1.5) ellipse (1.0 and 1.0);\n\\draw[thick, pattern=north west lines,  pattern color=gray!50]  (16,2.5) ellipse (1.9 and 1.9);\n\\end{scope} \n\\draw[semithick,color=gray!80 ]  (18.0625,2.5) ellipse (1.0 and 1.0);\n\\draw[semithick,color=gray!80 ]  (18.0625,2.5) ellipse (1.5 and 1.5);\n\\draw[<->,thick] (18.7301,3.2232) -- (19.0901,3.5832);\n\n\\draw [thick, decorate, decoration={brace, amplitude=5pt}](14.0625,5.1875) -- (14.0625,4);\n\\draw [thick, decorate, decoration={brace, amplitude=5pt}](14.0625,3.5) -- (14.0625,0);\n\\end{tikzpicture} \n\\centering\n\\caption{Theorem~\\ref{thm:1}: a small ${\\LTp}$ gain intepretation (left) and a graphical interpretation (right), respectively.\n}\n\\label{fig:theorem_new}\n\\end{figure\n\n\nA reduced order model can be very useful for analysis. As such, it should be able to reproduce faithfully the behavior of the original system. This point  is  addressed by the next result, which provides conditions under which the dominance properties of the original system are preserved by the reduced order model. The proof is omitted as it closely parallels that of Theorem~\\ref{thm:1}, with  ${G}$ and ${-\\tilde{G}}$ playing the roles of ${\\hat{G}}$ and ${\\tilde{G}}$, respectively. \n\n\\begin{corollary} \\label{corollary:1}\nConsider system~\\eqref{eq:system-Lure-mr} and  a  reduced order model~\\eqref{eq:system-ROM}\nconstructed as in Algorithm~\\ref{alg:mr-Lure}. Assume\n\\begin{description}\n\\item[(A1)$^{\\star}$]\\eqref{eq:system-ROM-linear} is  strictly   $p$-dominant with rate ${\\lambda\\in\\Rge}$,\n\\item[(A2)$^{\\star}$] ${\\partial \\varphi \\in  [-\\mu, \\mu]}$, with  ${\\mu\\in \\Rge}$,\n\\item[(A3)$^{\\star}$] ${\\norm{\\tilde{G}}_{\\infty,0} \\le \\epsilon}$, with ${0<\\epsilon<\\mu^{-1}}$;\n\\item[(A4)$^{\\star}$] ${\\norm{G_{zw}}_{\\infty,p} < \\mu^{-1} - \\epsilon}$.\n\\end{description}\nThen the reduced order model~\\eqref{eq:system-ROM} is strictly $p$-dominant with rate ${\\lambda}$.\n\\end{corollary}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Balanced truncation of Lure systems} \\label{sec:mr}\n \nThe results developed in the previous section require a stable  reduced order model of the non-dominant linear dynamics   together with an \\textit{a priori} error bound. In general, any model reduction method that produces a reduced order model with these properties can be used to reduce the original Lure system. \n\nModel reduction by balanced truncation~\\citep[Section 7]{antoulas2005approximation} is an obvious candidate, as it preserves stability and provides an error bound that is easy to compute. We briefly recall this method for completeness.\nGiven a linear, time-invariant, minimal, stable system~\\eqref{eq:system-linear-MIMO},  balancing consists in finding a coordinates transformation ${\\bar{x} = T^{-1} x }$  such that the reachability gramian $P\\in\\R^{n \\times n}$ and the observability gramian $Q\\in\\R^{n \\times n}$ of the system, defined implicitly by the Lyapunov equations\n\\begin{align}\nA P +PA^{\\transpose} +BB^{\\transpose} = 0, \\\\\nA^{\\transpose} Q +QA  +C^{\\transpose}C = 0, \n\\end{align} \nare both diagonal and equal. A \\textit{balancing transformation} $T$ acts on the reachability and observability gramians as \n\\begin{equation}\nT^{-1} P T^{-\\transpose}  = T^{\\transpose} Q T =   \\operatorname{diag}(\\sigma_1, \\ldots, \\sigma_n), \n\\end{equation}\nin which case the corresponding realization is said to be \\textit{(principal-axis) balanced}. The elements ${\\sigma_1, \\ldots, \\sigma_n}$ are the \\textit{Hankel singular values} of the system. These are system invariants  which  measure the  influence  of each state  on  the overall input-output behavior of the system.\nModel reduction by balanced truncation consists in ordering the Hankel singular values and in eliminating ${n-\\nu}$ state variables by truncation. The resulting reduced order model  is of order $\\nu$, stable,   and satisfies the error bound\n\\begin{equation}\n\\norm{G - \\hat{G}}_{\\infty} \\le 2 \n\\sum_{j=\\nu+1}^{n} \\sigma_{j} =\\epsilon ,\n\\end{equation}\nin which $G$ and $\\hat{G}$ are the transfer functions of the original system and  of  the reduced order model, respectively.\n\n\nAs anticipated above, balanced truncation can be applied to  a  dominant Lure system as follows. First, one decomposes the linear dynamics of the original system~\\eqref{eq:system-Lure} into the parallel interconnection of two subsystems described by the transfer functions $G^+$ and  $G^-$, which are strictly $p$-dominant and $0$-dominant with rate $\\lambda$, respectively. Then a reduced order model of order $\\nu^{-}$  of system ${G}^-$ is obtained by balanced truncation using the Lyapunov equations\n\\begin{align}\n(A^{-}+\\lambda I) P +P(A^{-}+\\lambda I)^{\\transpose} +B^{-}(B^{-})^{\\transpose} = 0, \\label{eq:Lyapunov-reachability}\\\\\n(A^{-}+\\lambda I)^{\\transpose} Q +Q(A^{-}+\\lambda I) +(C^{-})^{\\transpose}C^{-} = 0, \\label{eq:Lyapunov-observability}\n\\end{align} \nwith $(A^{-}, B^{-}, C^{-})$ a minimal realization of order $n^{-}$ of ${G}^-$. The resulting reduced order model $\\hat{G}^-$  is strictly $0$-dominant with rate $\\lambda$ and satisfies the error bound\n\\begin{equation} \\label{eq:error-dominant}\n\\norm{G^{-} - \\hat{G}^{-}}_{\\infty,0} \\le 2 \n\\sum_{j=\\nu^{-}+1}^{n^{-}} \\sigma_{j} =\\epsilon .\n\\end{equation}\nFinally, a reduced order model is constructed by considering first the parallel interconnection of two subsystems $G^+$ and  $\\hat{G}^-$ and then the feedback interconnection of the resulting system with the static nonlinearity $\\varphi$, as illustrated in Fig.~\\ref{fig:model_reduction_detail_new}. Balanced truncation allows for direct control over the error bound $\\epsilon$ according to~\\eqref{eq:error-dominant} by appropriately selecting the order $\\nu^{-}$\nof the reduced model $\\hat{G}^-$.  In the context of dominant systems, the error bound $\\epsilon$   is related to the rate $\\lambda$ selected to compute the  $H_{\\infty,0}$ norm. In principle, a careful selection of this additional parameter could lead to tighter reduction errors. This requires additional investigation.  \n\n\n\n\n\n\n \n \n \n\\section{An illustrative example}  \\label{sec:example}\n\nThe approximation of large-scale systems arising from the spatial discretization of the heat equation is a paradigmatic model reduction problem~\\citep{chahlaoui2002collection,antoulas2005approximation}. The goal is often to capture the behavior of these systems around equilibrium. This example, instead, focuses on oscillatory regimes, motivated by the fact that the heat equation describes key physical phenomena involving transport, which play a fundamental role in fluid dynamics and neuroscience~\\citep{keener1998mathematical}.\n\nConsider the problem of regulating heat flow oscillations in a homogeneous rod of unitary length via saturated\nproportional control, as illustrated in Fig.~\\ref{fig:setup}.\n\\begin{figure}[h!]\n\\centering\n\\usetikzlibrary{decorations.markings}\n\\usetikzlibrary{patterns}\n\\usetikzlibrary{arrows, arrows.meta}\n\\tikzset{>={Latex[width=3pt,length=3pt]}}\n\\begin{tikzpicture}[black, scale=.8, every node\/.style={transform shape}]\n\\draw  (1.5,3) rectangle (2.5,2);\n\\node at (2,2.5) {$-k_P$};\n\\draw[rounded corners]  (3.5,3) rectangle (4.5,2);\n\n\\draw[ line width = 1 pt] (3.7,2.35) -- (3.85,2.35) -- (4.15,2.65) -- (4.35,2.65);\n\\draw[-latex, line width = 0.5 pt] (5.5556,2.5) -- (6.35,2.5);\n\\draw [-latex, line width = 0.5 pt](2.5,2.5) -- (3.5,2.5);\n\\draw [-latex, line width = 0.5 pt](9.5,2.5) -- (10.5,2.5) -- (10.5,1.5) -- (0.5,1.5) -- (0.5,2.5) -- (1.5,2.5);\n\\draw[fill = white] (5.5,2.5) circle [radius=0.05];\n\\draw [-latex, line width = 0.5 pt](5.5,3.5) -- (5.5,2.5556);\n\\node at (10.25,2.75) {$y =z$};\n\\node at (5.7776,3.1112) {$u$};\n   \n\\draw[top color=white,bottom color=black!70] (6.5,28mm) arc (90:270:0.15 and 0.3)--++(3cm,0) arc (-90:-270:0.15 and 0.3)-- cycle;\n\\draw (9.5cm,28mm) arc (90:-90:0.15 and 0.3);\n\\draw (7cm,28mm) arc (90:270:0.15 and 0.3);\n\\draw (7.5cm,28mm) arc (90:270:0.15 and 0.3);\n\\draw (9cm,28mm) arc (90:270:0.15 and 0.3);\n\\node at (8.2,2.5) {$\\cdots$};\n \\node at (7.05,3) { $x_1$};\n\\node at (7.55,3) { $x_2$};\n\\node at (9.2,3) { $x_n$};\n\\node at (6.75,1.85) {\\footnotesize $h$};\n\\draw [<->, thick](6.5,2) -- (7,2);\n\\draw [-latex, line width = 0.5 pt](4.5,2.5) -- (5.4444,2.5);\n\\node at (5,2.7776) {$w$};\n\\end{tikzpicture}\n\\centering\n\\caption{Heat flow control system for a homogeneous rod of unitary length via saturated proportional control.\n}\n\\label{fig:setup}\n\\end{figure\n\nThe rod is insulated along its length and at the end. All temperature changes result from heat transfer at one end of the rod and by heat conduction along the rod.  By Fourier's law of heat conduction, the propagation of heat along the rod is described by the equation\n\\begin{equation}\n\\frac{\\partial T}{\\partial t} = \\kappa \\frac{\\partial^2 T}{\\partial \\xi^2},\n\\end{equation}\nin which ${T \\in \\Rge}$ is the temperature field on the rod, ${t \\in \\Rge}$ is the time variable, ${\\xi \\in (0,1)}$ is the spatial variable, and ${\\kappa > 0}$ is the thermal diffusivity, respectively. The temperature is measured at the end of the rod, \\textit{i.e.} ${y(t) = z(t) = T(1,t)}$. The temperature is initially zero and the heat flow source is placed at the beginning of the rod. The initial condition is\n\\begin{equation}\n T(\\xi,0) = 0, \\quad \\xi \\in (0,1)\n\\end{equation}\nand the (Neumann) boundary conditions are\n\\begin{equation}\n\\frac{\\partial T}{\\partial \\xi}(0,t) = u(t)+w(t) , \\quad  \\frac{\\partial T}{\\partial \\xi}(1,t) = 0,\n\\end{equation} \nin which ${u \\in \\R}$ is the exogenous input and ${w \\in \\R}$ is defined as $w = -\\sat(k_P z )$, where ${k_P>0}$ is the proportional gain and ${{\\sat(z) = \\tanh(z)}}$ is the saturation function.  The spatial domain is discretized into segments of length ${h = \\tfrac{1}{n+1}}$ using the second order central difference scheme\n\\begin{equation}\n\\frac{\\partial^2 T}{\\partial z^2}(kh,t) \\approx \\frac{1}{h^2}(T_{k+1}(t) - 2T_{k}(t) + T_{k+1}(t)),\n\\end{equation} \nwhere ${T_{k}(t) = T(kh,t)}$. This yields a system described by the equations~\\eqref{eq:system-Lure}, with \n${x = [\\, T_{1} \\ \\cdots \\ T_{n} \\,]^{\\transpose}}$, ${x(0) = 0}$, and \n\\begin{equation}  \\!\n\\scalebox{0.925}{$\n\\setlength{\\arraycolsep}{1.25 pt}\n\\renewcommand{\\arraystretch}{0.5}\nA =\n\\tfrac{\\kappa}{h^{2}}\n\\bma\n\\begin{array}{rrcrr} \n-1 &1 & & & \\\\\n1 &-2 &\\ddots & &\\  \\\\\n &\\ddots &\\ddots & \\ddots & \\\\ \n & &\\ddots &-2  &1 \\\\\n &  \t\t & &1  &-1 \n\\end{array}\n\\ema  \\! , \\, \nB=\n\\tfrac{\\kappa}{h}\n\\bma\n\\begin{array}{cc} \n1 & 1\\\\\n0 & 0 \\\\\n\\vdots & \\vdots\\\\\n0 & 0\n\\end{array}\n\\ema \\! , \\,\nC=\n\\bma\n\\begin{array}{cc} \n0 & 0\\\\\n0 & 0 \\\\\n\\vdots & \\vdots\\\\\n1 & 1\n\\end{array}\n\\ema^{\\transpose} \\!\\!  . $} \\!\\!\\! \n\\end{equation}\nThe transfer function of the system is\n\\begin{equation} \\label{eq:example-transfer-function}\nG(s) \n= \n\\bma\n\\begin{array}{cc}\nG_{yu}(s) & G_{yw}(s)\\\\\nG_{zu}(s) & G_{zw}(s)\n\\end{array}\n\\ema\n=\n\\bgroup\n\\setlength{\\arraycolsep}{3 pt}\n\\renewcommand{\\arraystretch}{1}\n\\tfrac{\\gamma_0}{\\prod_{j=1}^n \\left(s+\\lambda_k\\right)} \n\\bma\n\\begin{array}{cc}\n1 & 1 \\\\\n1 & 1 \n\\end{array}\n\\ema\n\\egroup\n,\n\\end{equation}\nwith ${\\gamma_0 = \\frac{ \\kappa^n}{h^{2n-1}}}$ and ${\\lambda_k = 2\\tfrac{\\kappa}{h^2} -2\\tfrac{\\kappa}{h^2}\\cos\\tfrac{(k-1)\\pi}{n}}$ for every integer ${ k \\in [1,n]}$. The static nonlinearity is ${\\varphi(z) = \\text{sat}(k_Pz)}$, which satisfies the differential sector condition ${\\partial {\\varphi} \\in [0,k_P]}$.\n\nThe system captures the main features of the Goodwin model~\\citep{murray2002mathematical},  a classical model in cellular physiology described by a cascade of reactions modelled by first-order lags and a negative feedback modelled by a Hill function. Thus, while approximating the heat equation away from equilibrium might not have significant engineering relevance, this could be crucial to understand, simulate and control biological rhythms.\n\n\nTo illustrate our model reduction framework and to ensure that assumptions (A1) and (A2) of Theorem~\\ref{thm:1} hold, we consider the loop transformation defined as \n\\begin{equation} \\label{eq:loop-transformation}\n\\bar{\\varphi}(z) = \\varphi(z) - \\tfrac{k_P}{2} z .\n\\end{equation}\nThis converts the differential sector condition ${\\partial {\\varphi} \\in [0,k_P]}$ into ${\\partial \\bar{\\varphi} \\in [- \\mu, \\mu]}$, with  ${\\mu = \\tfrac{k_P}{2}}$, and the transfer function $G_{zw}$ into\n\\begin{equation} \\label{eq:example-transfer-function-loop}\nH_{zw}(s) = \\frac{G_{zw}(s)}{1+ \\tfrac{k_P}{2} G_{zw}(s)}.\n\\end{equation} \nThe root locus of the transfer function $H_{zw}$ reveals that for any choice of the step size $h$ it is possible to establish strict $2$-dominance for ${k_P\\in[\\underline{k}_P,\\overbar{k}_P]}$, with \n${0<\\underline{k}_P <\\overbar{k}_P}$. For example, let ${\\kappa = 1}$, ${n=29}$, ${k_P=20}$  and ${\\lambda = 12}$. Then $H_{zw}$ has exactly two unstable poles for ${k_P = 20}$.  Direct inspection of the Nyquist diagram and the circle criterion for $p$-dominance imply that the system is strictly $2$-dominant with the same rate, as illustrated in Fig.~\\ref{fig:nyquist-P}. The system has therefore a unique limit cycle, since all solutions are bounded and the equilibrium at the origin is unstable.\n\n\\begin{figure}[h!]\n\\centering \n\\begin{tikzpicture}[black, every tick label\/.append style={font=\\scriptsize}]\n\\matrix{\n\\begin{axis}[\nheight = 0.23\\textwidth,\nwidth = 0.27\\textwidth,\nxmin=-.1,xmax=.1,\nymin=-.1,ymax=.1,\nx tick label style={\n        \/pgf\/number format\/.cd,\n            fixed,\n            fixed zerofill,\n            precision=2,\n        \/tikz\/.cd\n    },\ny tick label style={\n        \/pgf\/number format\/.cd,\n            fixed,\n            fixed zerofill,\n            precision=2,\n        \/tikz\/.cd\n    }    ,\ngrid\n]\n\\draw[name path =A] (axis cs:-0.1,-0.1) -- (axis cs:-0.1,0.1) -- (axis cs:0.1,0.1) -- (axis cs:0.1,-0.1) -- cycle;\n\\addplot[name path =B,no markers, black] table [x index = {0}, y index = {1},  col sep=comma]{circle.csv};\n\\tikzfillbetween[of=A and B]{pattern=north west lines};  \n\\addplot [black, very thick,  %\npostaction={decorate}, %\ndecoration={\n  markings,\n    mark=at position 0.5*\\pgfdecoratedpathlength-7.5pt\n  with {\\arrow[xshift=2.5\\pgflinewidth,>=stealth]{>}},\n  mark=at position 0.5*\\pgfdecoratedpathlength+7.5pt\n  with {\\arrow[xshift=2.5\\pgflinewidth,>=stealth]{>}}\n} %\n]  table [x index = {0}, y index = {1},  col sep=comma]{nyquist_system.csv};\n\\end{axis}\n~&~\n\\begin{axis}[\nheight = 0.23\\textwidth,\nwidth = 0.27\\textwidth,\nxmin=-10,xmax=10,\nymin=-20,ymax=20,\ngrid\n]\n\\fill [color=gray!50,fill=gray!50,fill opacity=0.5] (axis cs:-10,-10) rectangle (axis cs:10,10);\n\\addplot[black, very thick]  table [x index = {0}, y index = {1},  col sep=comma]{sector_condition.csv};\n\\end{axis}\n\\\\[1 pt]\n};\n\\end{tikzpicture}\n\\vspace{-.5cm}\n\\centering\n\\caption{{Left}: Nyquist diagram of the shifted transfer function ${H_{zw}}$ defined by~\\eqref{eq:example-transfer-function-loop} for ${\\kappa = 1}$, ${n=29}$, ${k_P=20}$ and ${\\lambda = 12}$. The disk $D(-\\mu,\\mu)$, with  ${\\mu =  \\tfrac{k_P}{2} =  10}$, is represented by diagonal lines. {Right}: Derivative of the static nonlinearity ${\\bar{\\varphi}}$. The differential sector condition ${ \\partial \\bar{\\varphi} \\in [-\\mu,\\mu]}$  is represented by the shaded area.\n}\n\\label{fig:nyquist-P}\n\\end{figure\n \n\nSimulations have been run to assess the properties of different reduced order models. Algorithm~\\ref{alg:mr-Lure} and classical balanced truncation have been used to obtain reduced order models of the original system of order ${\\nu = 3, 4, 5}$.  Assumption (A3) of Theorem~\\ref{thm:1} holds, as the linear dynamics of the resulting reduced models are such that ${\\norm{\\tilde{H}_{zw}}_{\\infty,0} \\le \\epsilon_{\\nu}}$, with ${\\epsilon_3 = 3.27\\cdot 10^{-3}}$,    ${\\epsilon_4 = 2.55\\cdot10^{-4}}$, ${\\epsilon_5 = 2.28\\cdot10^{-5}}$, indicating that the approximation error ${\\epsilon_{\\nu}}$ is a strictly decreasing function of $\\nu$.  Figure~\\ref{fig:bode_bt} shows the magnitude of the shifted frequency response of the transfer function ${H_{zw}}$ defined by~\\eqref{eq:example-transfer-function-loop}, with ${\\kappa = 1}$, ${n=29}$, and  ${\\lambda = 12}$, (solid) and the shifted frequency response of the transfer function ${\\hat{H}_{zw}}$ of the reduced order model obtained via balanced truncation for ${\\nu = 3}$ (dashed), ${\\nu = 4}$ (dashdotted), ${\\nu = 5}$ (dotted), respectively. Note that assumption (A4) is satisfied by all reduced order models, as the magnitude of the shifted frequency response of the transfer function ${\\hat{H}_{zw}}$ is below the lines defined  by ${\\mu^{-1} - \\epsilon_{\\nu}}$. Figure~\\ref{fig:bode_bt} only shows the (solid) line defined by ${\\mu^{-1} = 0.1}$ as this is almost overlapped with the lines defined  by ${\\mu^{-1} - \\epsilon_{\\nu}}$. Theorem~\\ref{thm:1} guarantees that the original system is also strictly $2$-dominant, since balanced truncation preserves $2$-dominance of the linear dynamics and since ${\\norm{H_{zw}}_{\\infty,p}<\\mu^{-1} - \\epsilon_\\nu }$ for ${\\nu = 3, 4, 5}$. This is confirmed by the fact that the magnitude of the shifted frequency response of the transfer function ${{H}_{zw}}$ is below the (solid) line defined by ${\\mu^{-1} = 0.1}$. \n \n\n\n\\begin{figure}[t!]\n\\centering\n\\usetikzlibrary{spy}\n\\tikzset{new spy style\/.style={spy scope={%\n magnification=5,\n size=1.25cm, \n connect spies,\n every spy on node\/.style={\n   circle,\n   draw,\n   },\n every spy in node\/.style={\n   draw,\n   circle,\n   fill=gray!10,\n   }\n  }\n }\n} \n\\begin{tikzpicture}[new spy style,every tick label\/.append style={font=\\scriptsize}]\n\\begin{loglogaxis}[\nheight = 0.225\\textwidth,\nwidth = \\columnwidth,\nxmin=1e-0, xmax=1e03,\nymin=1e-6, ymax=1e0,\nxlabel={{$\\omega$} \\textrm{[rad\/s]}},          \nxlabel near ticks,\ngrid\n]\n\\node[left]  \t\t\tat (axis cs: 9*1e2,3*1e-2) {{\\footnotesize ${\\tfrac{1}{\\mu}=0.1}$}};\n\\node[left, rotate=-10]  at (axis cs: 9*1e2,6*1e-4) {{\\footnotesize ${\\nu=3}$}};\n\\node[left, rotate=-11]  at (axis cs: 9*1e2,7*1e-5) {{\\footnotesize ${\\nu=4}$}};\n\\node[left, rotate=-12]  at (axis cs: 9*1e2,8*1e-6) {{\\footnotesize ${\\nu=5}$}};\n\n\\addplot [black, thick]  table [x index = {0}, y index = {1},  col sep=comma]{bode_system.csv};\n\\addplot [black, thick,  dashed]  table [x index = {0}, y index = {1},  col sep=comma]{bode_rom1.csv};\n\\addplot [black, thick, dashdotted]  table [x index = {0}, y index = {1},  col sep=comma]{bode_rom2.csv};\n\\addplot [black, thick, densely dotted]  table [x index = {0}, y index = {1},  col sep=comma]{bode_rom3.csv};\n\\addplot [blue, thick ]  table [x index = {0}, y index = {1},  col sep=comma]{bode_mu_bound.csv};\n\\end{loglogaxis}\n\\spy[size=1.5cm,magnification=7] on (0.125,2.1) in node at (1.5,0.9);\n\\end{tikzpicture}\n\\vspace{-.25cm}\n\\centering\n\\caption{Magnitude  of the frequency response of the transfer function ${H_{zw}(s -\\lambda)}$  defined by~\\eqref{eq:example-transfer-function-loop},  with ${\\kappa = 1}$, ${n=29}$, ${k_P=20}$, ${\\lambda = 12}$, and   the frequency response of the transfer function ${\\hat{H}_{zw}(s -\\lambda)}$  of the reduced order model obtained via balanced truncation. \n}\n\\label{fig:bode_bt}\n\\end{figure}%\n\n\n\n\n\n\\section{Conclusion} \\label{sec:conclusion}\n\n\n\nA model reduction framework geared towards the analysis and design of systems that switch and oscillate has been described. Classical balanced truncation for linear, time-invariant systems has been revisited to develop a model reduction method for Lure systems which preserves dominance properties. The theory has been illustrated by the approximation of an oscillatory Lure system arising from the discretization of a heat flow control problem. \nA future research direction is to extend the proposed framework to systems that can be decomposed as the feedback interconnection of a linear system and a dominant nonlinear system using the small $\\LTp$ gain theorem.  \n\n\n\n \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{introduction}\n\nLow mass X-ray binaries (LMXBs) are interacting binaries where a low mass \ndonor transfers matter onto a neutron star or black hole. \n4U 1636-536 (=V801 Ara) and 4U 1735-444 (=V926 Sco) are among the optically \nbrighter members in the class of {\\it persistent} LMXBs, characterized by  \n$L_{\\rm x} \\simeq 10^{37-38}$ erg s$^{-1}$ and blue spectra with weak\nhigh-excitation emission lines (mainly HeII $\\lambda$4686 and the \nCIII\/NIII Bowen blend at $\\lambda$4640). They also share similar \nproperties: they are both atoll sources (as based on the pattern traced \nin X-ray color-color diagrams, see e.g.\\citealt{hasin89}) with frequent \nburst activity and short orbital periods (3.80 and 4.65 hrs respectively) \nrevealed through optical photometry (\\citealt{corbet86}, \\citealt{pedersen81}). \nTheir lightcurves display shallow sinusoidal modulations which have been \ninterpreted as due to the geometrically varying visibility of the irradiated \ndonor star (e.g. \\citealt{van88}). Therefore, the photometric maxima \nsupposedly define orbital phase 0.5 i.e. inferior conjunction of the \ncompact object. Note, however, that this assumption requires confirmation \nbecause photometric maxima can sometimes be associated with asymmetries in the \ndisc structure such as the visibility of the irradiated inner disc bulge \nat phase $\\sim 0.3$ (e.g. 4U 1822-371 \\citealt{hell89}) or superhump\nactivity (see \\citealt{haswell01}). \n\nOnly a few \nspectroscopic studies have been presented on these two binaries up to now. \nFor example, \\cite{smale91} report H$_{\\alpha}$ spectroscopy of V926 Sco \nshowing that  the line core is dominated by emission from the disc bulge or\nsplash region where the gas stream interacts with the outer disc rim. \nOn the other hand, \\cite{aug98} (A98 hereafter) present radial velocity curves \nof HeII $\\lambda$4686 and the Bowen blend at $\\lambda$4640 for both V926 Sco \nand V801 Ara. They conclude that these high excitation lines are also tracing \nthe motion of the disc bulge.  \n\nV801 Ara is particularly remarkable since it is one of only 14 bursters where \n``burst oscillations\" (i.e. nearly coherent high-frequency pulsations) \nhave been detected during several thermonuclear X-ray bursts \n(\\citealt{giles02}, G02 hereafter). \nFurthermore, a train of burst oscillations was also discovered in a \n13 min  interval during a ``superburst\", showing a frequency drift which has \nbeen interpreted as the orbital motion of the neutron star (\\citealt{stro02},\nSM02 hereafter). \nBy fitting the frequency evolution with a circular orbit \nSM02 constrain the radial velocity amplitude of the neutron star to \nthe range $90<K_1<175$ km s$^{-1}$. These constraints on $K_1$ \nwill be readdressed and improved in this paper. \nFurther constraints on the system parameters and the component masses, \nhowever, require dynamical information on the donor\nstar which, unfortunately, is normally overwhelmed by the optical emission \nfrom the X-ray irradiated disc.    \n                                                                                \nA new indirect method to extract dynamical information from donor stars in \npersistent LMXBs was proposed by \\cite{stee02} (SC02 hereafter). They detected \nnarrow high excitation emission lines from the irradiated surface of the donor \nstar in Sco X-1 which led to the first determination of its radial velocity \ncurve and mass function. These lines are strongest in the Bowen blend, a \ncombination of C\\,III and N\\,III lines which are powered by photoionization and \nfluorescence (respectively) due to UV photons arising from the hot inner \ndisk (\\citealt{mc75}). This technique has been amply confirmed by subsequent \nstudies of the eclipsing ADC (Accretion Disc Corona) and X-ray pulsar \n4U 1822-371 (\\citealt{casa03}), the black hole soft X-ray transient \nGX 339-4 during its 2002 outburst (\\citealt{hynes03}) and   \nAql X-1 during its 2004 outburst (\\citealt{cornel06}).\n \nIn this paper we apply this method to the X-ray bursters \n4U 1636-536 (=V801 Ara) and 4U 1735-444 (=V926 Sco) and present the first \ndetection of the donor stars in these two LMXBs. The paper is organized as\nfollows: Section 2 summarizes the observation details and data reduction. \nThe average spectra and main emission line parameters are presented in Sect. \n3, with multigaussian deconvolution of the Bowen blend. In Section 4 we analyse \nthe radial velocities and orbital variability of the strong Bowen blend and \nHeII $\\lambda$4686 emission lines. Estimates of the systemic velocities are \nobtained through the Double Gaussian technique applied to the wings of the \nHeII line. Using these systemic \nvelocities we compute Doppler tomograms of HeII $\\lambda$4686 and the Bowen \nfluorescence NIII $\\lambda$4640 which are presented in Section 5. The NIII maps \ndisplay evidence of irradiated emission from the donor star, which is used to\nrefine the absolute phasing and systemic velocities. Finally, in Section \n6 we provide an improved determination of $K_1$  for V801 Ara and present our \nconstraints on the masses in the two binaries.\n\n\\section{Observations and Data Reduction} \\label{observations}\n\nV801 Ara and V926 Sco were observed on the nights of 23 and 25 June 2003 using \nthe FORS2 Spectrograph attached to the 8.2m Yepun Telescope (UT4) at the \nObservatorio Monte Paranal (ESO). A total of 42 spectra of 600s of V801 \nAra and 102 exposures of 200s of V926 Sco were \nobtained with the R1400V holographic grating, covering a complete \norbital cycle per night for each target. We used a 0.7 arcsec slit width which \nrendered a wavelength coverage of $\\lambda\\lambda$4514-5815 at \n70 km s$^{-1}$ (FWHM) resolution, as measured from Gaussian fits to the arc \nlines. The seeing was variable between 0.6\" -- 1.2\" during \nour run. The flux standard Feige 110 was also observed with the \nsame instrumental configuration to correct for the instrumental \nresponse of the detector.\n\nThe images were de-biased and flat-fielded, and the spectra subsequently \nextracted using conventional optimal extraction techniques in order to  \noptimize the signal-to-noise ratio of the output spectra (Horne 1986). A \nHe+Ne+Hg+Cd \ncomparison lamp image was obtained in daytime to provide the wavelength \ncalibration scale. This was obtained by a 4th-order polynomial fit to \n19 lines, resulting in a dispersion of 0.64 \\AA~pix$^{-1}$ and rms scatter \n$<$ 0.05 \\AA. Instrumental flexure was monitored through cross-correlation \nbetween the sky spectra and was found to be very small, always within 14 km\ns$^{-1}$ (0.4 pix.) on each night. These velocity drifts were removed from each \nindividual spectrum, and the zero point of the final wavelength scale \nwas established from the position of the strong OI $\\lambda$5577.338 sky line.  \nAll the spectra were calibrated in flux using observations of the flux \nstandard Feige 110. However, due to light loss caused by our narrow slit and \nvariable seeing conditions, our flux calibration is only accurate to $\\sim$ 50\\%. \n\n\\section{Average Spectra and Emission Line Parameters}\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig1.eps}\n\\caption{Summed spectra of V801 Ara and V926 Sco with the principal emission \nlines indicated.}\n\\label{fig1}\n\\end{figure}\n\n\nFigure~\\ref{fig1} presents the average spectra of V801 Ara and V926 Sco in \n$f_{\\lambda}$ flux units. They show a blue continuum with broad high excitation \nemission lines of HeII $\\lambda$4686, $\\lambda$5411, the Bowen blend at \n$\\lambda\\lambda$4630-50 and H$\\beta$, which are typical of X-ray active \nLMXBs. Possible HeI lines at $\\lambda$4922 and $\\lambda$5015 \nare also identified but these are significantly weaker. Table 1 summarizes the \nFWHM, EW and centroid $\\lambda_{\\rm c}$ of the main emission lines obtained \nthrough simple Gaussian fits. \nWe note that the average spectra (including the line strengths) look very \nsimilar to those presented by A98, which seems to imply that there has been no \nlarge, long-term variations between the two data epochs. \nIncidentally, the EWs and FWHMs of all lines do not show any significant \nnigh-to-night variability nor modulation with the orbital period.\nAside from the intrinsically broad Bowen blend, which is \na blend of at least three CIII\/NIII transitions, we note that emission lines \nare a factor $\\sim$2 narrower in V926 Sco than in V801 Ara. Given  the \nsimilarities in their orbital periods, this suggests a projection effect with a \nlower inclination angle for V926 Sco. In addition, the emission line centroids \nin V926 Sco are significantly blueshifted, probably due to a larger \n(approaching) systemic velocity. \nThis also applies to the Bowen complex since the difference in\n  wavelength between V801 Ara and V926 Sco (-110 km\/s) is consistent with \n  the difference in velocities found for the other lines. \nThe blue continuum is also steeper for \nV926 Sco, as expected because of its lower reddening (see A98).  \n\n\\begin{table}\n\\centering\n\\caption[]{Emission line Parameters} \n\\begin{tabular}{lcccc}\n\\hline \n{\\em Line} & {\\em FWHM} & {\\em EW} & {\\em Centroid} &  {\\em $\\Delta V$} \\\\\n &  {\\em (km s$^{-1}$)} & {\\em (\\AA)} & {\\em (\\AA)} & {\\em (km s$^{-1}$)} \\\\\n\\hline\n{\\bf V801 Ara} & & & \\\\\nBowen               & 1848 $\\pm$ 65 & 4.25 $\\pm$ 0.03 & 4643.2 $\\pm$ 0.4 & -- \\\\ \nHeII $\\lambda$4686  & 1216 $\\pm$ 45 & 3.31 $\\pm$ 0.03 & 4685.6 $\\pm$ 0.3 & -9  $\\pm$ 11 \\\\\n$H\\beta$            &  963 $\\pm$ 56 & 1.67 $\\pm$ 0.02 & 4860.9 $\\pm$ 0.3 & -25 $\\pm$ 21  \\\\\n\\hline\n\\hline\n{\\bf V926 Sco} & & & \\\\\nBowen               & 1662 $\\pm$ 65 & 3.99 $\\pm$ 0.02 & 4641.5 $\\pm$ 0.4 & -- \\\\ \nHeII $\\lambda$4686  &  657 $\\pm$ 26 & 2.18 $\\pm$ 0.02 & 4683.9 $\\pm$ 0.2 & -118 $\\pm$ 12 \\\\\n$H\\beta$            &  507 $\\pm$ 39 & 0.43 $\\pm$ 0.02 & 4859.2 $\\pm$ 0.4 & -134 $\\pm$ 25  \\\\\n\\hline\n\\label{lines}\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig2.eps}\n\\caption{Combined fit to HeII $\\lambda$4686 and the Bowen blend using three \nGaussians at $\\simeq$4634\\AA~ (NIII), $\\simeq$4641\\AA~(NIII) and \n$\\simeq$4651\\AA~ (CIII). See text for details.}\n\\label{fig2}\n\\end{figure}\n \nThe Bowen blend mainly consists of emissions corresponding to the NIII \ntransitions at $\\lambda\\lambda$4634,4641,4642 and CIII at  \n$\\lambda\\lambda$4647,4651,4652 (\\citealt{scha89}). In an attempt to estimate \nthe relative contribution of the different transitions we have performed a \ncombined multi-Gaussian fit to \nthe average emission profiles of HeII $\\lambda$4686 and the Bowen blend for \nthe two LMXBs. The fit consists of four Gaussians, one for the HeII line, two \nfor the NIII emissions at $\\lambda$4634 and $\\lambda$4641-2 and another for \nthe CIII emissions at $\\lambda$4647-52. Free parameters are the \nline widths, which are set to be equal for all the lines, individual line \ncentroids and intensities. The line widths are mainly driven by the fit to the\nunblended HeII line and hence the latter is effectively used as a template to \nconstrain the line profiles within the Bowen blend. Figure~\\ref{fig2} presents \nthe results of the fit. Because widths are set to be the same for all lines, \nflux ratios are given by simple peak ratios. Our best fit yields NIII ratio \n$I(\\lambda4634)\/I(\\lambda\\lambda4641-2)= 0.31 \\pm 0.10$ and $0.59 \\pm 0.02$ \nfor V801 Ara and V926 Sco, respectively. This is a factor $\\leq$ 2 lower than \nthe theoretical NIII ratio of 0.71, computed by \\cite{nuss71} for Bowen \nfluorescence, but consistent with the results of \\cite{scha89} for \nSco X-1. Similarly, Hynes (private communication) finds a NIII ratio in the \nrange 0.36-0.56 (depending on time) for GX 339-4 during its 2002 outburst. \nWe can also calculate the CIII\/NIII ratio \n$I(\\lambda\\lambda4647-52)\/I(\\lambda4634 + \\lambda\\lambda4641-2)$ \nand find $0.38 \\pm 0.06$ and $0.35 \\pm 0.01$ for V801 Ara and V926 Sco \nrespectively. For comparison, we have \nperformed the same analysis on average spectra of Sco X-1 and 4U 1822-371, \nusing data presented in SC02 and \\cite{casa03}, and find \n$0.44 \\pm 0.03$ and $0.49 \\pm 0.01$ respectively. The lower CIII\/NIII ratio in \nV801 Ara and V926 Sco may indicate possible evidence of CNO processed material \nbut we have to be cautious here because of the oversimplification in our \nfitting model. In particular, we cannot rule \nout other possible transitions (most likely OII lines at \n$\\lambda$4641.8 and $\\lambda$4649.1; see \\citealt{mc75}) \ncontributing to our components at $\\sim$4641\\AA~and $\\sim$4651\\AA, \nwhich so far we have assumed to be dominated by NIII and CIII transitions. \nFurthermore, since the CIII and \nOII lines are not powered by Bowen fluorescence but photoionization, \ndifferent line ratios may stem from differences in efficiency of these two \nexcitation mechanisms rather than true CNO abundance variations. \n\n\\section{Radial Velocity Study and Orbital Variability}\n\nFigure~\\ref{fig3} displays the radial velocities of the Bowen blend and HeII \n$\\lambda$4686 line for V801 Ara and V926 Sco, obtained by cross-correlating \nthe individual spectra with Gaussians of fixed FWHM as given in Table 1. \nWe have assumed $\\lambda_c = 4643.0$ for the central \nwavelength of the Bowen blend. \nThe velocities are folded in orbital phase, using the\nephemerides of G02 and A98 for V801 Ara \nand V926 Sco but shifted in phase by +0.5 to make phase zero \ncoincide with the inferior conjunction of the donor star. These will be  \ncalled the {\\it photometric ephemerides} and will be used throughout this \nSection. The accumulated phase uncertainty at the time of our observations is \n$\\pm 0.06$ for both systems. Note that this phase convention assumes that \nlightcurve maxima are driven by the visibility of the irradiated donor star.  \n\nThe radial velocity curves for V801 Ara show maxima at phase $\\sim$ 1. \nThis phasing suggests that the velocity variations contain a significant \ncomponent arising in the disc bulge (which has its maximum visibility at \nphase $\\sim$ 0.75), caused by the \ninteraction of the gas stream with the outer edge of the disc, as \nis typically seen in persistent LMXBs (e.g. 4U 1822-371; \\citealt{cowley82}). \nOn the other hand, the radial velocity curves for V926 Sco \nshow maxima at phase $\\sim$ 0.7. \nBy comparing the HeII curves in the two binaries we clearly see  \nevidence for a large systemic velocity of $\\sim$ -150 km s$^{-1}$ in \nV926 Sco, in good agreement with \\cite{smale91}. We also note that the \nCIII\/NIII blend in V801 Ara is modulated with \na much larger amplitude than in V926 Sco i.e. 280 km s$^{-1}$ versus \n70 km s$^{-1}$, respectively. \n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig3.eps}\n\\caption{Radial velocity curves of the Bowen blend (top) and HeII \n$\\lambda$4686 in V801 Ara and V926 Sco.}\n\\label{fig3}\n\\end{figure}\n\nFigure~\\ref{fig4} presents the trailed spectra of the Bowen blend and \nHeII$\\lambda$4686 for V801 Ara and V926 Sco, after co-adding our individual \nspectra in 15 phase bins to improve statistics. The core of the emission lines \nin V801 Ara show clear S-wave components. The S-wave in the Bowen blend is \nquite narrow and it may arise from the heated face of the donor star, as \nobserved in Sco X-1 (SC02). \nThe S-wave in HeII is rather extended and is likely produced in a \nregion with a large velocity dispersion such as the disc bulge. \nOn the other hand, the Bowen blend in V926 Sco is rather noisy and does not \nexhibit any clear components that are visible by eye. The HeII trailed \nspectra do show a clear orbital modulation with complex structure. \n \n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig4.eps}\n\\caption{Trailed spectra showing the orbital evolution of the Bowen blend \nand HeII $\\lambda$4686 in 15 phase bins using the {\\it photometric ephemerides}}\n\\label{fig4}\n\\end{figure}\n    \nFollowing our work on Sco X-1 (SC02) and 4U 1822-371\n(\\citealt{casa03}), we have applied the \ndouble-Gaussian technique (\\citealt{sch80}) to the wings of the HeII \n$\\lambda$4686 line in an attempt to estimate the radial velocity curve of \nthe compact object. The trailed spectra presented in Fig.~\\ref{fig4} \ndemonstrate that the line cores are dominated by \nstrong, complex, low-velocity components associated with asymmetric emission \nfrom the outer disc and\/or the donor star, which we want to avoid. \nTherefore, by convolving the emission line with a double-Gaussian filter of \nsufficiently large Gaussian \n separation we can extract radial velocity curves from the wings of \nthe profile, which are expected to follow the motion of the compact star. We \nhave used a double-Gaussian bandpass with $FWHM=100$ km s$^{-1}$ and \nrelative Gaussian separations in the range  \n$a= 400-1400$ km s$^{-1}$ in steps of 100 km s$^{-1}$. In order to \nimprove statistics, we have co-added our spectra in 15 phase bins using the \n{\\it photometric ephemerides}. The radial velocity curves\nobtained for different Gaussian separations are subsequently fitted with \nsine-waves of the form $V(\\phi)=\\gamma + K \\sin 2\\pi (\\phi-\\phi_{0})$, fixing \nthe period to the orbital value. The best fitting parameters \nare displayed as a function of the Gaussian separation $a$ in a Diagnostic \nDiagram (see Fig.~\\ref{fig5}). \n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0,width=84mm]{fig5.eps}\n\\caption{Diagnostic Diagram for HeII $\\lambda$4686 in V801 Ara (solid circles) \nand V926 Sco (open circles). The dotted horizontal line marks $\\phi_{0}=0.5$ \nor the expected inferior conjunction of the compact object, according to the\n{\\it photometric ephemerides}.}\n\\label{fig5}\n\\end{figure}\n\nIn both cases we see how the line cores are dominated by high-amplitude \n($K\\sim 200$ km s$^{-1}$) S-waves which fade as we move to the line wings,\nwhere lower $K-$amplitudes of a few tens of km s$^{-1}$ are found. On the other\nhand, the $\\gamma$ velocities are very steady throughout the profiles, with \naverage values of~$\\sim -30$ (V801 Ara) and~$\\sim -130$ km s$^{-1}$ (V926 Sco). \nThe blue-to-red crossing phase for V926 Sco \ndisplays a smooth decreasing trend from the line core to the wings, \nwhereas it is rather constant for V801 Ara, with an average value of \n$\\phi_{0} \\sim 0.7$. \nWe estimate that the velocity points start to be\ncorrupted by continuum noise for Gaussian separations larger than \n$a\\sim1000$ as indicated by the diagnostic parameter $\\sigma(K)\/K$ \n(see \\citealt{shafter86}). Therefore, we decided to adopt the average values \nfor the parameters obtained from the last two separations before $\\sigma(K)\/K$ \nstarts to rise i.e. $a=900-1000$ km s$^{-1}$ for V801 Ara and 1000-1100 km \ns$^{-1}$ for V926 Sco. \nThis yields $K_1 = 93 \\pm 25$ km s$^{-1}$, $\\gamma = -42 \\pm 4$ km \ns$^{-1}$,  $\\phi_{0}=0.68 \\pm 0.03$ for V801 Ara and $K_1 = 96 \\pm 9$ \nkm s$^{-1}$, $\\gamma = -121 \\pm 7$ km s$^{-1}$, $\\phi_{0}=0.20 \n\\pm 0.01$ for V926 Sco. The errorbars have been adjusted to incorporate the\nscatter between different values within our preferred range. Note that \n$\\phi_{0}$ of V801 Ara is delayed by \n$\\sim0.18$ with respect to the inferior conjunction of the compact \nobject (dotted line in Fig.~\\ref{fig5}), as predicted by the {\\it photometric \nephemerides}. \nThis is classically observed in interacting binaries and interpreted as \ncontamination of the line wings by residual emission from the \ndisc-bulge\/hot-spot (\\citealt{marsh98}).   \nOn the other hand, $\\phi_{0}$ of V926 Sco leads the inferior \nconjunction of the compact star by $\\sim0.30$, according to the \n{\\it photometric ephemerides}. Such a large \nshift is unexpected and a full discussion is diverted to Sect. 6.\nFurthermore, we also note the very asymmetric distributions obtained \nin the HeII Doppler maps (see next Sect.) which may invalidate the \ndouble-Gaussian technique. Therefore, the constraints on zero-phase, \n$\\gamma$-velocity and $K_1$ (for V801 Ara) will be superseded in the \nfollowing sections. \n\n\\section{Bowen Fluorescence from the Irradiated Companion}   \n\nDoppler tomography enables us to map the brightness \ndistribution of a binary system in velocity space through combining \norbital phase spectra (see details in \\citealt{marsh01}). This is \nparticularly effective when dealing with weak emission\nfeatures which are barely detected or embedded by noise in individual \nspectra, such as our narrow Bowen emission lines. In order to compute Doppler \nimages of the NIII $\\lambda$4640 fluorescence line in V801 Ara and V926 Sco \nwe have rectified the individual spectra by subtracting a low order \nspline fit to the continuum regions and rebinned them into a uniform \nvelocity scale of 37 km s$^{-1}$ per pixel. Doppler images were subsequently \ncomputed using the {\\it photometric ephemerides} and $\\gamma=-42$ and $-121$ \nkm s$^{-1}$ for V801 Ara and V926 Sco, respectively. The maps show compact \nspots shifted in phase by $+0.03$ and $-0.18$ with respect to the expected \nlocation of the donor star in  the velocity maps (i.e. along the vertical\n$V_{\\rm y}$ axis)\\footnote{Note that the greyscale in the NIII maps have \nbeen set to enhance the contrast of the sharp components. This is the reason \nwhy these maps do not show any trace of emission from the underlying broad \ncomponent.}. Assuming that these spots \nare produced on the irradiated hemisphere of the donor star, as has been\nshown to be the case in Sco X-1 (SC02) and 4U 1822-371 \n(\\citealt{casa03}), we correct the previous {\\it photometric ephemerides} and \nderive the following {\\it spectroscopic ephemerides}, which will be used in the \nremainder of the paper:   \n\n\\begin{equation} \nT_{0}(HJD) = 245 2813.531(2) + 0.15804693(16) E    \n\\end{equation}\n\n\\begin{equation} \nT_{0}(HJD) = 245 2813.495(3) + 0.19383351(32) E      \n\\end{equation}\n\n\\noindent\nwhere the zero-phase error comes from the uncertainty in the centroid\nposition of the spots. Equation (1) corresponds to the ephemerides of  \nV801 Ara and equation (2) to V926 Sco.\nWith these ephemerides, the HeII $\\lambda$4686 maps of the two binaries show \na crescent shape brightness distribution pointing towards emission from an \nextended disc bulge.  \n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig6.eps}\n\\caption{Doppler maps of HeII $\\lambda$4686 (left panels) and NIII \n$\\lambda$4640 (right panels) for V801 Ara and V926 Sco, computed using our \n{\\it spectroscopic ephemerides}. Systemic velocities \n$\\gamma = -34$ km s$^{-1}$ and $\\gamma = -140$ km s$^{-1}$ were adopted for \nV801 Ara and V926 Sco, respectively. For comparison we also plot the gas\nstream trajectory (in units of 0.1 R$_L1$) and Roche lobe of the donor star \nfor the case of $K_2=360$ km s$^{-1}$ (V801 Ara) and $K_2=298$ km \ns$^{-1}$  (V926 Sco). A mass ratio $q=0.27$ has been assumed for the two\nbinaries. \n }\n\\label{fig6}\n\\end{figure}\n\nFollowing SC02 we have used the HeII Doppler maps to refine the \nsystemic velocities derived in the previous section. The \n$\\chi^2$ value of the map was calculated for a range of $\\gamma$'s, and the \nbest fit in terms of mimimal $\\chi^2$ was achieved for $\\gamma\\simeq -39 $  \nkm s$^{-1}$ (V801 Ara) and $\\gamma \\simeq -132 $  km s$^{-1}$ (V926 Sco). \nThe use of an incorrect systemic velocity has the effect  \nof blurring bright spots into , as opposed to V801 Ara,elongated ``defocused\" \nfeatures in Doppler \nimages. Therefore, as a further test, we have also computed NIII  \n$\\lambda$4640 maps for a set of $\\gamma$-velocities in the range \n-200 -- +200 km s$^{-1}$ and we looked for the best focused NIII spots by \ncomputing the\nskewness using box sizes of 5, 10 and 20 pixels. We find that the most \nsymmetric and compact spots are found for $\\gamma$ values in the\nrange -29 -- -39 km s$^{-1}$ (V801 Ara) and 137-143  km s$^{-1}$ (V926 Sco). \nTherefore, we decided to adopt $\\gamma=-34 \\pm 5$ km s$^{-1}$ (V801Ara) \nand $\\gamma=-140 \\pm 3$ km s$^{-1}$ (V926 Sco) and these are the \nvalues that we have used in the final Doppler maps, which are presented in\nFig.~\\ref{fig6}. The NIII maps reveal compact bright spots \nwhich we interpret as irradiated emission from the inner hemisphere of\nthe donor star. These spots are obviously located along the vertical axis\nbecause of our ephemerides definition. We have calculated the position of \nthese spots by computing their centroids using the centering algorithm \nCENTROID in IRAF\\footnote{IRAF is distributed by the National Optical \nAstronomy Observatories, which are operated by the Association of Universities \nof Research in Astronomy, Inc., under cooperative agreement with the\nNacional Science Foundation.} \nand find $K_{\\rm em}= 277 \\pm 22$ km \ns$^{-1}$ (V801 Ara) and $226 \\pm 22$ km s$^{-1}$ (V926 Sco). We note that \n$K_{\\rm em}$ is very weakly dependent on $\\gamma$, with only a 10 km s$^{-1}$ \ndrift when $\\gamma$ varies by $\\pm 20$ km s$^{-1}$ around our central value. \nTherefore, our $K_{\\rm em}$ determinations seem very robust and they represent \nstrict lower limits to the true radial velocity semiamplitude $K_2$ of the \ndonor stars because they must arise from the irradiated hemisphere facing the \nneutron star. Further confirmation of our system parameters is provided by \nthe appearance of the sharp CIII\/NIII transitions after coadding all the \nspectra in the rest frame of the NIII $\\lambda$4640 emission line region \n(see Fig.~\\ref{fig7}). \nNote that we have not attempted to compute Doppler images of the H$\\beta$ \nlines because these are contaminated by phase-variable absorption components, \nas was also the case in 4U 1822-371 (e.g. \\citealt{casa03}). Absorption \nviolates the principles of Doppler Tomography and makes it very difficult to\ninterpret the corresponding Doppler images.     \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig7.eps}\n\\caption{Average spectra of V801 Ara (left panel) and V926 Sco (right panel).\nTop spectra have been Doppler corrected in the rest frame of the donor using \nthe spectroscopic ephemerides and the $K$-velocity amplitudes derived from the \nbright spots in the NIII $\\lambda4640$ Doppler images. The narrow NIII and \nCIII lines are clearly seen in the Bowen blend. These are smeared out and cannot\nbe detected in the straight averages shown at bottom. Noise has been filtered \nusing a 2 pixel boxcar smoothing.}\n\\label{fig7}\n\\end{figure}\n\n\\section{DISCUSSION}\n\nThe detection of NIII $\\lambda$4640 fluorescence emission from the irradiated \ndonor in V801 Ara and V926 Sco provides new absolute (spectroscopic) \nephemerides and opens the possibility to derive the first constraints on the \ndynamical masses of these two LMXBs. In the light of the new ephemerides, the\nphotometric lightcurve maxima of V801 Ara take place at phase $0.47 \n\\pm 0.06$ (with the uncertainty dominated by error propagation of the G02 \nephemerides) and, therefore, they are consistent with X-ray irradiation of the \ndonor star. On the other hand, lightcurve maxima in V926 Sco are located at\norbital phase $0.82 \\pm 0.06$. This is in between the maximum visibility of \nthe irradiated donor and the disc-bulge and hence it would imply that \nlightcurve maxima arise by a combination of these two emitting regions. \nHowever, we note that the uncertainty in the photometric ephemerides of V926 \nSco is likely to be higher than stated in A98 due to the limited number of\narrival times fitted and their large scatter (Augusteijn, private\ncommunication). Therefore, it is possible that lightcurve maxima in V926 Sco are\nalso consistent with superior conjunction of the irradiated donor star. This is\nthe most likely scenario because it would be hard to understand an emission\nsource peaking at phase $\\sim 0.8$ in an irradiation dominated environment\nwithout invoking some strange (and stable) disc configuration.  \n\nWe have also determined the systemic velocities for the two binaries i.e. \n$-34\\pm5$ km s$^{-1}$ (V801 Ara) and $-140\\pm3$ km s$^{-1}$ (V926 Sco). \nThese can be compared with $V_{\\rm r}$, the radial velocity,  \nrelative to the Local Standard of Rest, due to Galactic \nrotation at the location of the binaries. For the case of \nV801 Ara, we take $l=332.9^{\\circ}$, $b=-4.8^{\\circ}$ \nand $d=6-7$ kpc, based on peak fluxes of radius-expansion \nX-ray bursts for $M_1=1.4-2$ M$_{\\odot}$ (\\citealt{galloway05}). Assuming \nthe Galactic rotation curve of \\cite{naka03} we find \n$V_{\\rm r}$ in the range -103 to -111 km s$^{-1}$. This is much larger \nthan our systemic velocity which could be explained through the recoil \nvelocity gained by the binary during the supernova explosion which formed \nthe neutron star. Thus, we can set a lower limit to the kick velocity in \nV801 Ara of $\\simeq70$ km s$^{-1}$. Regarding V926 Sco, we take \n$l=346.1^{\\circ}$, $b=-7.0^{\\circ}$ and a distance $d=9.1$ kpc \n(for a canonical 1.4 M$_{\\odot}$ neutron \nstar, see A98). The Galactic rotation curve at the position of \nV926 Sco yields $V_{\\rm r}= -111$ km s$^{-1}$, close but \nsignificantly lower than our systemic velocity. The difference might also be \nascribed to a kick velocity received by the neutron star at birth or,\nalternatively, a neutron star mass of $\\simeq$1.7 M$_{\\odot}$. Note that the \nlatter implies a larger distance of $d=10$ kpc and hence $V_{\\rm r}=-140$ km \ns$^{-1}$.  \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0,width=84mm]{fig8.eps} \n\\caption{Top panel: Dynamic Fourier power contours showing the excursion \nin oscillation frequency during the 2001 February 22 superburst from 4U \n1636-53 (V801 Ara) with the best fitting orbit model overlayed. Time zero for \nthe x-axi is MJD 51962.7102530038 (TDB) \nBottom panel: Pulse phase residuals for the best fitting model.} \n\\label{fig8}\n\\end{figure}\n\n\\subsection{System Parameters for V801 Ara}   \n\nThe NIII $\\lambda$4640 spot in the Doppler map yields $K_{\\rm em}= 277 \\pm \n22$ km s$^{-1}$. It must arise on the inner hemisphere of the donor\nand hence it sets a lower limit to the velocity semi-amplitude of the \ncompanion's center of mass $K_2$. In order to find the real $K_2$ we need to\ncalculate the $K-correction$ or $K_{\\rm em}\/K_2$ for the case of emission \nlines in illuminated atmospheres, which depends mainly on the binary mass \nratio $q=M_2\/M_1$ (with $M_2$, $M_1$ the masses of the donor and compact \nstar respectively) and the disc flaring angle $\\alpha$. The $K-correction$ has \nbeen calculated by \\cite*{munoz05} using an irradiation binary code which \nincludes shadowing by an axisymmetric flared disc. The results are \ntabulated as a function of $q$, $\\alpha$ and the inclination angle $i$ in \nTable 1 of their paper, although the dependence on $i$ is very weak. The \n$K-correction$ is constrained between $\\alpha=0$ (i.e. we neglect disc \nshadowing) and the geometric limit set by emission from the irradiated limb of \nthe donor, i.e. $K_{\\rm em}\/K_2 < 1-0.213 q^{2\/3} (1+q)^{1\/3}$ (see \n\\citealt{munoz05}). \n\nAn upper limit on $q$ is established by taking $M_1\\ge1.4 $M$_{\\odot}$ and \n$M_2\\le 0.48$ M$_{\\odot}$, the largest possible zero-age main sequence (ZAMS) \nstar fitting in a 3.79 hr period Roche lobe (\\citealt{tout96}), which leads to \n$q\\le 0.34$. As a first approach, one can assume the empirical Mass-Radius \nrelation for low-mass stars in cataclysmic variables and LMXBs (see e.g. \n\\citealt{smith98}, \\citealt{warner95}) which yields $M_2 \\simeq 0.32$. This, \ncombined with a canonical neutron star of $M_1\\simeq1.4$ M$_{\\odot}$, \nwould lead to \na plausible $q\\sim 0.23$. The $K-correction$ for $K_{\\rm em}=277\\pm22$ km \ns$^{-1}$ and $q\\sim0.23$ would yield $303<K_2<404$ km s$^{-1}$, where we have \nadopted the coefficients for the case $i=40^{\\circ}$ in Table 1 of \n\\cite{munoz05} because V801 Ara is not eclipsing.\n  \nA more refined $K-correction$ requires a knowledge of $q$. This can be\nconstrained by the rotational broadening $V_{\\rm rot} \\sin i$ of the companion\nstar which, for the case of synchronous rotation, is related to $q$ through \n\n\\begin{equation}\nV_{\\rm rot} \\sin i \\simeq 0.462 K_2 q^{1\/3} (1+q)^{2\/3}\n\\end{equation}\n\n\\noindent\n(\\citealt{wadehorne88}). \nA lower limit to $V_{\\rm rot} \\sin i$ can be estimated from the width \nof the sharp NIII fluorescence emission in the Doppler corrected average \nspectrum. This is because emission lines arise not from the entire\nRoche lobe but from the irradiated part only. \nA multigaussian fit to the Bowen profile presented in Fig.~\\ref{fig7} gives \n$FWHM (NIII \\lambda4640) = 140 \\pm 21$ km s$^{-1}$,  which includes the effect \nof our intrinsic instrumental resolution (70 km s$^{-1}$). This has been \naccounted for by broadening a Gaussian template of $FWHM=70$ km s$^{-1}$ \nbetween 10 and 200 km s$^{-1}$, in steps of 10 km s$^{-1}$, using a Gray \nrotational profile (\\citealt{gray92}) without limb-darkening, because \nfluorescence lines arise in optically thin conditions. Our simulation \nindicates that $V_{\\rm rot} \\sin i=92\\pm16$ km s$^{-1}$ \nis equivalent to $FWHM = 140 \\pm 21$ km s$^{-1}$. On the other hand, by \nsubstituting $V_{\\rm rot} \\sin i \\ge 76$ km s$^{-1}$ and the $K-correction$ for \n$\\alpha>0^{\\circ}$ into equation (3), one finds a secure lower limit $q\\ge \n0.08$. \n\nAn upper limit to the rotational broadening of the donor star is set by \n$V_{\\rm rot} \\sin i=2\\pi R_2 \\sin i \/P$. By assuming that the donor \nmust be more evolved than a ZAMS star within a 3.79 hr Roche lobe, then \n$R_2\\le 0.44 R_{\\odot}$ (\\citealt{tout96}). This, together with  \n$i\\le 78^{\\circ}$ (lack of X-ray eclipses for $q\\ge0.08$), yields \n$V_{\\rm rot} \\sin i \\le 138$ km s$^{-1}$. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig9.eps} \n\\caption{Constraints on $K_2$ and $q$ for V801 Ara. The disc flaring angle \nmust be between $\\alpha=0^{\\circ}$ and $\\alpha=max$, whereas the rotational \nbroadening is constrained between the observed $V_{\\rm rot} \\sin i \\simeq 76$ \nkm s$^{-1}$ and 138 km s$^{-1}$, the maximum allowed for a ZAMS star in a 3.8hr \nperiod and for $i\\le 78^{\\circ}$. \nThe shaded area represents the $K_1$ solution obtained from a reanalysis of \nthe superburst pulsations. These constrain the binary mass ratio to be \n$q=0.21-0.34$ and $K_2=286-433$ km s$^{-1}$. For comparison we also mark  \n(dashed line) the solution for a hypothetic disc flaring angle of \n$\\alpha=12^{\\circ}$.}\n\\label{fig9}\n\\end{figure}\n\nFurther constraints are provided by the study of burst oscillations which \ncan set limits to the neutron star's projected velocity $K_1 (= q K_2)$.  \nSM02 derived $90\\le K_1\\le 175$ km s$^{-1}$ by fitting the frequency\ndrift of highly coherent X-ray pulsations observed in an 800s interval\nduring a superburst. They used a circular orbit model and fixed the zero phase \nto the ephemeris of G02. \nWe have used our new spectroscopic ephemeris to reanalyse the superburst\npulsation data in order to better constrain $K_1$.  Based on our spectroscopic\nephemeris, the pulsation interval during the superburst from V801 Ara \ncomes slightly earlier in orbital phase by 0.032 cycles as compared to the\nG02 ephemeris. We fit the pulsation data (see SM02 for details on the phase \ntiming analysis) to a circular orbit model with the\nreference epoch fixed to our new $T_0$, and we also fixed the orbital period\n(using the G02 value). This\nleaves two free parameters, the projected neutron\nstar velocity, $K_1$, and the rest-frame spin frequency, $\\nu_0$.  We find\nacceptable fits  with a reference epoch within the $\\pm 1 \\sigma$ range for\nour spectroscopic $T_0$.  The inferred $K_1$ velocity ranges from 90 - 113 km \ns$^{-1}$ as $T_0$ ranges over $\\pm 1\\sigma$.  \nThe best fit $\\chi^2$ values range from 14.6 to 10.6 (with 7 dof) over this \nsame range, that is, higher velocities are modestly favored, but given the \nshortness of the pulse\ntrain compared to the orbital period, we do not consider these differences \nas significant. With the reference epoch fixed, the statistical error on the\nvelocity is much smaller than the range given above, so 90 - 113 km s$^{-1}$ is \na robust range for $K_1$ at the $\\pm 1\\sigma$ limits of $T_0$.  The best fit \n$\\nu_0$ ranges from 582.04143 to 582.09548 Hz, and which \nmay help constrain future searches for a persistent millisecond pulsar in V801  \nAra. Figure ~\\ref{fig8} summarizes the results of the new timing analysis. \n\nFig.~\\ref{fig9} summarizes all our restrictions in the $K_2-q$ parameter \nspace i.e. $0^{\\circ}\\le \\alpha\\le max$, $V_{\\rm rot} \\sin i =76-138$ km \ns$^{-1}$ and $K_1=90-113$ km s$^{-1}$. The shaded area \nindicates the region allowed by our constraints, which yields $q=0.21-0.34$ \nand $K_2=286-433$ km s$^{-1}$. These imply a mass function $f(M)=M_1 \n\\sin^3 i \/(1+q)^2=0.76\\pm0.47$ M$_{\\odot}$ and hence $M_1 \n\\sin^3 i=1.24\\pm0.77$ M$_{\\odot}$. \nClearly the large uncertainties are dominated by the error in \n$K_{\\rm em}$ and the uncertainty in the disc flaring angle.\n\n\n\\subsection{System Parameters for V926 Sco}   \n\nThe same reasoning can be applied to V926 Sco. The spot in the NIII \n$\\lambda$4640 map yields $K_{\\rm em}=226 \\pm 22$ km s$^{-1}$ whereas \nthe ZAMS Mass-Radius relation for a 4.65 hr Roche lobe leads to $M_2 \n\\le 0.58 M_{\\odot}$ (\\citealt{tout96}) and hence $q\\le 0.41$.\nOn the other hand, we measure $FWHM (NIII \\lambda4640) = 115 \\pm 23$ km \ns$^{-1}$ which, after deconvolution of the instrumental resolution using Gray\nrotation profiles as above, yields $V_{\\rm rot} \\sin i \\ge 71 \\pm 21$ km \ns$^{-1}$. This lower limit, combined with the $K-correction$ for \n$\\alpha>0^{\\circ}$ and equation (3), yields $q\\ge 0.05$ and, hence \n$i\\le 80^{\\circ}$. And, under the assumption that the donor star is more \nevolved than a ZAMS star, the Roche lobe geometry implies $R_2 \\le 0.54 \nR_{\\odot}$ which, for $i\\le 80^{\\circ}$, leads to $V_{\\rm rot} \\sin i \n\\le 137$ km s$^{-1}$. \n\nAll these restrictions translate into constraints on the $K_2-q$ plane \nwhich are presented in Fig.~\\ref{fig10}. The allowed region results in \n$q=0.05-0.41$ and $K_2=215-381$ km s$^{-1}$, depending on the value of \n$\\alpha$. These numbers imply  $f(M)=0.53\\pm0.44$ \nM$_{\\odot}$ and, thus, $M_1 \\sin^3 i= 0.80 \\pm 0.71$ M$_{\\odot}$.  \nUnfortunately we do not have any contraints on $K_1$ from  \nburst oscillations and hence our mass restrictions are not as \nwell constrained as for V801 Ara.  \nOur best estimates of the system parameters for both objects \nare presented in Table 2. \n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig10.eps} \n\\caption{Same as Fig.~\\ref{fig9} but for V926 Sco, but with no pulsation \nconstraints yet.}\n\\label{fig10}\n\\end{figure}\n\n \n\\begin{table}\n\\centering\n\\caption[]{System Parameters.~$T_0$ indicates zero phase or inferior conjunction\nof the donor star. Orbital periods $P_{\\rm orb}$ are from G02 and A98.} \n\\begin{tabular}{lcc}\n\\hline \nParameter  & {\\em 4U 1636-536} & {\\em 4U 1735-444} \\\\\n           & {\\em V801 Ara}  & {\\em V926 Sco}  \\\\\n\\hline\n$P_{\\rm orb}$ (days)       & 0.15804693(16) & 0.19383351(32)         \\\\\n$T_0$ (HJD$-$2\\,452\\,000)  & 813.531 $\\pm$ 0.002    & 813.495 $\\pm$ 0.003 \\\\\n$\\gamma$ (km~s$^{-1}$)     & -34     $\\pm$ 5        & -140    $\\pm$ 3     \\\\\n$K_{\\rm em}$ (km~s$^{-1}$) & 277     $\\pm$ 22       & 226     $\\pm$ 22    \\\\\n$q$ ($M_2\/M_1$)            &      0.21-0.34         &       0.05-0.41     \\\\\n$K_1$ (km~s$^{-1}$)        & 90-113         &     ---    \\\\\n$K_2$ (km~s$^{-1}$)        & 360     $\\pm$ 74       & 298     $\\pm$ 83    \\\\\n$f(M)$ (M$_\\odot$)         & 0.76    $\\pm$ 0.47     & 0.53    $\\pm$ 0.44  \\\\\n$i$ (deg)                  & 36-60                  & 27-60               \\\\\n\\hline\n\\label{param}\n\\end{tabular}\n\\end{table}\n \n\\subsection{Constraints on the inclination}\n\nFurther constraints on the stellar masses requires a knowledge of the \nbinary inclination. While strict upper limits are set by the absence \nof X-ray eclipses, lower limits can be established by combining \n$M_1 \\le 3.1$ M$_{\\odot}$ (the maximum mass allowed for a stable \nneutron star, \\citealt{rho74}) with our mass function and $q$ restrictions. \nThis leads to $i=36^{\\circ}-74^{\\circ}$ and $i=27^{\\circ}-80^{\\circ}$ for \nV801 Ara and V926 Sco, respectively. In addition, the physical model of \n\\cite{frank87} indicates $i\\le60^{\\circ}$ due to the lack of X-ray dips. \n\nOn the other hand, we mentioned in Section 3 that the factor $\\sim$2 narrowness \nof the HeII $\\lambda$4686 profile in V926 Sco with respect to V801 Ara \nindicates a lower inclination. To test this hypothesis\nwe have produced Doppler corrected averages for V801 \nAra and V926 Sco by coadding all the spectra in two bins centered at orbital \nphases 0.0 and 0.5, when the visibility of the irradiated face of the donor\nis minimum and maximum, respectively. These spectra are presented in \nFig.~\\ref{fig11} and show a marked difference for the two binaries: the narrow \nCIII\/NIII lines become significantly enhanced around phase 0.5 (top spectra) \nfor V801 Ara, but not much difference is seen in V926 Sco between phase 0.5 and\nphase 0. This clearly supports a higher inclination angle in V801 Ara. \nFurthermore, the relative contibution of the NIII $\\lambda$4640 line with \nrespect to the broad base, as estimated through a multigaussian fit, is a \nfactor $\\sim 2$ larger for V801 Ara than for V926 Sco. This can be taken \nas the relative contributions of the heated donor and disc, and hence, \nstrongly suggests that V801 Ara is seen at a higher inclination angle than \nV926 Sco. This seems at odds with the fact that the optical lightcurves \nin V801 Ara and V926 Sco display similar amplitudes, $A\\simeq$ 0.2 mags  \n(\\citealt{vanamer87}, \\citealt{van90}). However, \\cite{dejong96} have shown \nthat lightcurve amplitudes are more sensitive to $\\alpha$ than $i$, which  \nmight simply imply a thicker disc in V926 Sco  and \nhence lightcurve amplitudes cannot be taken as a simple indication of the \ninclination angle. \n\n\\subsection{Mass Estimate and Evolutionary Scenarios} \n\n\\cite{meyer82} analyzed the\neffects of X-ray heating in the vertical structure of discs and demonstrated \nthat these are strongly correlated. Even in the absence of irradiation, they\nfind a disc opening angle of $\\sim6^{\\circ}$. De Jong et al. (1996) \nmodelled the effects of irradiation in optical lightcurves and, by\ncomparing with observations of LMXBS, they infer a mean disc opening angle of \n12$^{\\circ}$. Most of their systems are short period LMXBs, just like  \nV801 Ara and V926 Sco, and hence it seems justified to speculate what our  \nsystem parameters would be in this particular case. \n\nRegarding V801 Ara, $\\alpha=12^{\\circ}$ leads to  \n$q\\simeq0.25-0.30$, $K_2\\simeq356-375$ km s$^{-1}$ (see dashed line \nin Fig.~\\ref{fig9}) and hence $f(M)\\simeq 0.81 \\pm 0.07$ M$_{\\odot}$ and \n$M_1 \\sin^3 i \\simeq 1.32 \\pm 0.13$ M$_{\\odot}$. Asssuming that the donor star \nis a 0.48 M$_{\\odot}$ \nZAMS, then $M_1\\simeq1.6-1.9$ M$_{\\odot}$ and $i\\simeq 60-71^{\\circ}$. On the\nother hand, a canonical 1.4 M$_{\\odot}$ would imply an evolved \ndonor with $M_2\\simeq0.35-0.42$ M$_{\\odot}$ and $i\\ge71^{\\circ}$, which \nis difficult to reconcile with the absence of X-ray  eclipses, dips and the\nlow amplitude of the optical modulation (see Sect. 6.3). Therefore, a flaring \nangle of $\\alpha\\simeq12^{\\circ}$ seems to support scenarios with ZAMS donors \nand massive neutron stars. These masses can be reproduced by recent \nevolutionary scenarios where some LMXBs are suposed to descend from binaries\nwith high mass donors (\\citealt{schenker02}). These are significantly evolved \nat the begining of the mass transfer phase and mass is transferred in a thermal \ntimescale until mass ratio reverses, which can result in the accretion of \nseveral tenths solar masses by the neutron star. This would make V801 Ara \nsimilar to other LMXBs such as 4U 1822-371 (\\citealt{munoz05}). \n\nHowever, it is also possible to accommodate a \ncanonical neutron star with a MS donor and a plausible inclination angle by \npushing the disc flaring angle to higher values.   \nFor instance, $\\alpha=16^{\\circ}$ leads to $q\\simeq0.30-0.34$,   \n$f(M)\\simeq 0.51$ M$_{\\odot}$ and $M_1 \\sin^3 i \\simeq \n0.88$ M$_{\\odot}$. Then, for $M_1=1.4$ M$_{\\odot}$, \n$M_2\\simeq0.45$ M$_{\\odot}$ and $i\\simeq57^{\\circ}$. These masses would be \nconsistent with standard evolutionary pictures where donors in LMXBs descend\nfrom slightly evolved low-mass stars through dynamically stable mass transfer\nand result in slightly undermassive stars.  \n        \nNote also that $\\alpha<12^{\\circ}$ are virtually ruled out \nbecause this would imply higher $K_2$ and $f(M)$ values which result in massive\nneutron stars and very high inclinations. For instance, $\\alpha=8^{\\circ}$ \nleads to $f(M)\\simeq 0.96$ M$_{\\odot}$ and $M_1 \\sin^3 i \n\\simeq 1.5$ M$_{\\odot}$. Then, for $i\\leq76^{\\circ}$ (lack of X-ray \neclipses), $M_1\\ge1.7$ M$_{\\odot}$ \nwhereas a plaussible $i\\simeq57^{\\circ}$ yields $M_1\\simeq2.6$ M$_{\\odot}$.   \n\nThe case of V926 Sco is unfortunately unconstrained because of the lack \nof an X-ray mass function. For example, assuming $\\alpha=12^{\\circ}$ leads to \n$q\\simeq0.09--0.38$, $f(M)\\simeq 0.22--0.68$ M$_{\\odot}$ and $M_1 \\sin^3 i \n\\simeq 0.29--1.07$ M$_{\\odot}$. The allowed range in $M_1$ and $M_2$ is too\nwide as to impose any useful restriction on possible evoultionary scenarios.    \nMore, higher resolution data are required to measure the \nNIII $\\lambda$4640 flux and $V_{\\rm rot} \\sin i$ as a function of orbital \nphase, from which tighter constraints on the inclination and disc flaring\nangle can be set. This, \ntogether with the determination of the X-ray mass function for V926 Sco \nthrough pulse delays of burst oscillations and smaller errors in the \n$K_{\\rm em}$ determinations, is expected to provide stronger  \nlimits on the stellar masses and the evolutionary models for these two LMXBs \n(see e.g. \\citealt{munoz05}).\n   \n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig11.eps}\n\\caption{Doppler corrected averages of V801 Ara (left panel) and V926 Sco \n(right panel) in the rest frame of the donor star. Top spectra present\naverages between orbital phases 0.25-0.75 and bottom spectra between \n0.75-0.25. The sharp CIII\/NIII components in V801 Ara are clearly more  \nintense around phase 0.5 than 0, suggesting a higher inclination\nbinary. Spectra have been smoothed by a 2 pixel boxcar.}\n\\label{fig11}\n\\end{figure}\n\n\\section{Conclusions} \n\nThe main results of the paper are summarized as follows:\n\n\\begin{itemize}\n\n\\item We have presented the first detection of the donor stars in the bursters \nLMXBs V801 Ara (=4U 1636-536) and V926 Sco (=4U 1735-444) through NIII\n$\\lambda$4640 fluorescent emission caused by irradiation. \n\n\\item The narrow NIII $\\lambda$4640 spots in the Doppler maps define \n$K_{\\rm em}=277\\pm22$ km s$^{-1}$ (V801 Ara) and $K_{\\rm em}=226\\pm22$ km\ns$^{-1}$ (V926 Sco), and a new set of {\\it spectroscopic} ephemerides which \nlend support to the assumption that photometric modulation is driven by the\nvisibility of the irradiated donor stars. On the other hand, the phasing of the\nradial velocity curves of HeII $\\lambda$4684 and the Bowen blend suggest\nthat these lines are mainly associated with the disc bulge. Our spectroscopic \nephemerides, combined with burst oscillations data for V801 Ara, enables us \nto refine the neutron star's projected velocity to $K_1=90-113$ km s$^{-1}$ \nfor this particular system.\n\n\\item Following \\cite{munoz05}, we have computed the $K-corrections$ for \nthe two LMXBs and obtain $K_2=360\\pm74$ km s$^{-1}$, $q=0.21-0.34$, \n$f(M)=0.76\\pm0.47$ M$_{\\odot}$ (V801 Ara) and $K_2=298\\pm83$ km s$^{-1}$, \n$q=0.05-0.41$, $f(M)=0.53\\pm0.44$ M$_{\\odot}$ (V926 Sco). \nBoth systems are seen at intermediate inclination angles in the range \n$i\\simeq30-60^{\\circ}$, with V801 Ara having the higher inclination of the two.\n\n\\item Regarding V801 Ara, disc flaring angles $\\alpha \\le 8^{\\circ}$ seem to be\nruled out because of the high inclinations implied. Opening angles \n$\\alpha\\simeq 12^{\\circ}$ support massive neutron stars and main-sequence \ndonors which may descend from intermediate-mass X-ray binaries as predicted by \nsome evolutionary models (\\citealt{schenker02}). Alternatively, higher opening \nangles $\\alpha\\simeq16^{\\circ}$ are consistent with canonical neutron stars and   \nmain-sequence (or slightly evolved) donors which have evolved from standard\nLMXBs through a dynamically stable mass tranfer phase. \n\n\\item The lack of an X-ray mass function in V926 Sco prevents to set tighter \nconstraints to the $K-correction$, mass estimates and evolutionary history in\nthis LMXB. \n\n\\end{itemize}\n\n \n  \n\n\n\\section*{Acknowledgments}\n\nWe thank the referee Thomas Augusteijn for helful comments to the manuscript. \nMOLLY and DOPPLER software developed by T.R. Marsh is gratefully \nacknowledged. JC acknowledges support from the Spanish Ministry of Science and\nTechnology through the project AYA2002-03570. DS acknowledges a Smithsonian \nAstrophysical Observatory Clay Fellowship as well as support through NASA GO \ngrants NNG04GG96G and NNG04G014G. Based on data collected \nat the European Southern Observatory, Monte Paranal, Chile.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTaking the flat-space limit (zero cosmological constant limit)\nof asymptotically AdS spacetimes results in asymptotically flat geometries.\nThis procedure can be done by taking\nthe $\\ell\\to\\infty$ limit where $\\ell$ is the radius of AdS spacetime.\nFrom the field theory perspective,\none could expect that the $\\ell\\to\\infty$ limit in the bulk theory\nhas a holographic description at the boundary.\nRecently, it has been argued that the flat-space limit of AdS gravity\nis dual to the $\\dot {\\rm I}$n$\\ddot {\\rm o}$n$\\ddot {\\rm u}$-Wigner\ncontraction of the boundary conformal field theory (CFT) \\cite{Bagchi:2010zz,Bagchi:2012cy}.\\footnote{It is worth noticing\nthat another interesting approach to understanding\nflat space quantum gravity is given in \\cite{Krishnan:2013wta}.\nTherein, the authors showed that interpreting the inverse AdS$_{3}$ radius $1\/\\ell$\nas a Grassmann variable leads to a map from gravity in AdS$_{3}$ to gravity in flat space.}\n\nThe so-called flat\/contracted conformal field theory (CCFT) correspondence has received a great deal of attention recently.\nFor example, in \\cite{Bagchi:2012xr} a Cardy-like formula has been obtained for the\ntwo-dimensional CCFT which yields the correct entropy of the three-dimensional cosmological solution.\nThese asymptotically flat spacetimes can be obtained by taking the flat-space limit,\nas in \\cite{Cornalba:2002fi}, of nonextremal Ba\\~{n}ados-Teitelboim-Zanellii (BTZ) black holes.\nAfter taking the flat-space limit, the outer horizon of BTZ is mapped to infinity;\nhowever, the value of the inner horizon remains finite and defines the cosmological horizon. \nThe entropy of the cosmological solution has been identified with the area of the cosmological horizon.\nIn the literature (see for example \\cite{Detournay:2012ug}), a modified Cardy formula has been introduced\nwhich reproduces the entropy of the inner horizon of BTZ black holes.\nThe CFT origin of this formula has not been well understood yet, but the observation of \\cite{Fareghbal:2014qga,Riegler:2014bia}\nis that if we accept the modified Cardy formula related to the inner horizon of the BTZ\nand contract it by using appropriate parameters of CCFT, the final result\nis exactly the CCFT Cardy-like formula which yields the entropy of the cosmological horizon.\n\nFurthermore, in \\cite{Fareghbal:2013ifa}, the authors found the correlation functions\nof CCFT energy-momentum tensor by using the contraction of CFT correlation functions\nfor finding the quasilocal stress tensor of the asymptotically\nflat spacetimes which gives the correct conserved charges of these geometries.\n\nThe Flat\/CCFT correspondence can also propose a dual field theory\nwhich lives at the horizon of nonextreme black holes.\nThe idea begins from the appearance of Rindler spacetime in the near horizon limit of\nnonextreme black holes.\nIf one starts with the Rindler-AdS\/CFT correspondence \\cite{Czech:2012be,Parikh:2012kg} and takes\nthe flat-space limit in the bulk, which results in the Rindler spacetime,\nthe boundary field theory is given by the contraction of the parent CFT.\nThis proposal has been used in \\cite{Fareghbal:2014oba}.\nTherein, the authors found the Bekenstein-Hawking entropy of the nonextreme BTZ black hole\nby counting the CCFT microsates.\n\nMoreover,   Bagchi {\\it et al.}\\,\ncalculated the entanglement entropy of a two-dimensional field theory with\nGalilean conformal symmetry\\footnote{The group of symmetries of CCFT$_{2}$ is isomorphic to Galilean Conformal Algebra\nbut in higher dimensions these are not the same.} recently \\cite{Bagchi:2014iea}.\nThe authors used the Wilson lines approach and found the holographic entanglement entropy\nthey computed is in precise agreement with the ones obtained in the field theory side. \nFor an almost complete list of papers related to the Flat\/CCFT correspondence, see the references of \\cite{Bagchi:2014iea}.\n\nIn (2+1)-dimensional Einstein gravity, black holes can exist only in the\npresence of a negative cosmological constant \\cite{Ida:2000jh}.\nIn order to find asymptotically flat black holes in three dimensions one has  to consider higher-derivative gravity theories.\nThe entropy of these black holes can be obtained by Wald\\rq{}s formula \\cite{Wald:1993nt}.\nA successful theory of quantum gravity should be able to give a microscopic description of this semiclassical entropy.\nAn alternative approach for this study is using holography.\nThe Falt\/CCFT correspondence as a duality between quantum gravity in the asymptotically flat backgrounds\nand a field theory with contracted conformal symmetry can be an appropriate context to study this problem.\nThe current paper is focused in this direction.\n\nWe consider a theory of gravity which is given by taking the flat-space limit of  new massive gravity (NMG) \\cite{Bergshoeff:2009hq}.\nThis theory possesses remarkable properties.\nIn \\cite{Deser:2009hb}, it was shown that it is a ghost-free and power-counting UV finite, three-dimensional gravity.\nWe use the dictionary of the Flat\/CCFT correspondence for finding quasi local stress tensor\nof the new type of asymptotically flat black hole \\cite{Oliva:2009ip}.\nUsing the holographic stress tensor along with the Brown and York\\rq{}s method \\cite{Brown:1992br},\nwe compute the conserved charges of this black hole.\nWe also take the limit from the Cardy formula and find a Cardy-like formula for the\nCCFT and show that this gives agreement with the Wald\\rq{}s semiclassical approach.\n\nThe next sections are devoted to two main parts.\nIn Sec.(\\ref{Bulk Solutions}) we introduce the bulk solutions.\nWe start from NMG and review its  asymptotically AdS rotating hairy black hole.\nThen we take the flat-space limit from the\naction and its black hole solution and introduce the asymptotically\nflat rotating hairy black hole with some novel properties.\nWe calculate its entropy using Wald\\rq{}s formula and verify for it the first law\nof black hole thermodynamics.\nIn Sec.(\\ref{Dual Boundary Theory}) we argue about the dual boundary theory of the bulk solution.\nWe shortly review the known results about the dual CFT of NMG\nand then try to contract these results and find a CCFT which is dual to the\nhigher-derivative gravity theory of \\cite{Deser:2009hb}.\nThis work is another check for the correctness of the Flat\/CCFT correspondence.\n\n\\section{Bulk Solutions}\n\\label{Bulk Solutions}\n\n\\subsection{Rotating hairy black hole of NMG}\nWe consider the three-dimensional higher-derivative gravity theory known as NMG.\nThis theory is described by the parity-invariant action \\cite{Bergshoeff:2009hq}\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{BHTaction}\nS=\\frac{1}{16\\pi G}\\int d^{3}x \\sqrt{-g}\\left[R-2\\Lambda+\\frac{1}{m^{2}}K\\right],\n\\ee\nwhere\n\\begin{equation}}   \\def\\ee{\\end{equation}\nK=R_{\\mu\\nu}R^{\\mu\\nu}-\\frac{3}{8}R^{2}.\n\\ee\n\nThe above theory (\\ref{BHTaction}) for the special case $m^{2}\\ell^{2}=1\/2$\nhas the following rotating black hole solution \\cite{Oliva:2009ip}\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{BHTbh}\nds^{2}=-NFdt^{2}+\\frac{dr^{2}}{F}+r^{2}\\left(d\\phi+N^{\\phi}dt\\right)^{2},\n\\ee\nwhere $N$, $N^{\\phi}$, and $F$ are  given by\n\\begin{align}\nN  &  =\\left[  1+\\frac{b\\ell^{2}}{4H}\\left(  1-\\Xi^{\\frac{1}{2}}\\right)  \\right]\n^{2}\\ ,\\nonumber\\\\\nN^{\\phi}  &  =-\\frac{a}{2r^{2}}\\left(  4GM-bH\\right)  \\ ,\\label{Ns&F}\\\\\nF  &  =\\frac{H^{2}}{r^{2}}\\left[  \\frac{H^{2}}{\\ell^{2}}+\\frac{b}{2}\\left(\n1+\\Xi^{\\frac{1}{2}}\\right)  H+\\frac{b^{2}\\ell^{2}}{16}\\left(  1-\\Xi^{\\frac{1}{2}\n}\\right)  ^{2}-4GM\\ \\Xi^{\\frac{1}{2}}\\right]  \\ ,\\nonumber\n\\end{align}\nand\n\\begin{align}\n\\nonumber H&=\\left[  r^{2}-2GM\\ell^{2}\\left(  1-\\Xi^{\\frac{1}{2}}\\right)  -\\frac{b^{2}\\ell^{4}\n}{16}\\left(  1-\\Xi^{\\frac{1}{2}}\\right)  ^{2}\\right]  ^{\\frac{1}{2}},\n\\label{H}\\\\\n\\Xi&=1-a^{2}\/\\ell^{2}.\n\\end{align}\nIt is labeled by three parameters:\nthe mass $M$, the angular momentum $J=M a$, and an additional \\lq\\lq{}gravitational hair\\rq\\rq{} parameter $b$.\nThe rotation parameter $a$ is bounded according to $-\\ell\\leq a\\leq \\ell$.\n\nThe angular velocity of the horizon is\n\\begin{equation}}   \\def\\ee{\\end{equation}\n\\Omega_{+}=\\frac{1}{a}\\left(  \\Xi^{\\frac{1}{2}}-1\\right)  \\ . \\label{Omega+}\n\\ee\nWe can associate a Hawking temperature and entropy to it\n\\begin{align}\nT&=\\frac{\\Xi^{\\frac{1}{2}} }{\\pi \\ell}\\sqrt{2G\\Delta M \\left( 1+\\Xi^{\\frac{1}{2}} \\right) ^{-1}},  \\label{Temperature}\\\\\nS&=\\pi \\ell\\sqrt{\\frac{2}{G}\\Delta M\n\\left( 1+\\Xi^{\\frac{1}{2}} \\right) },  \\label{Entropy}\n\\end{align}\nwhere\n\\begin{equation}}   \\def\\ee{\\end{equation}\n\\Delta M=M+\\frac{b^{2}\\ell^{2}}{16G}\\ .\n\\ee\nThese quantities fulfill the relation\n\\begin{equation}}   \\def\\ee{\\end{equation}\nTdS=\\Xi^{\\frac{1}{2}} \\text{ }dM+\\frac{b\\ell^{2}}{8G}\\Xi^{\\frac{1}{2}} \n\\text{ }db-\\frac{1}{a}\\left( 1-\\Xi^{\\frac{1}{2}} \\right) \\Delta M \\text{ }da\\ .  \\label{calentate}\n\\ee\nNot much more work would bring this equation to the familiar form of the first law of black hole thermodynamics\n\\begin{equation}}   \\def\\ee{\\end{equation}\nd(\\Delta M)=TdS-\\Omega_{+}d(\\Delta J)\\, ,\n\\ee\nwhere we defined\n\\begin{equation}}   \\def\\ee{\\end{equation}\n\\Delta J=a \\Delta M\\, .\n\\ee\n\n\\subsection{The flat-space limit of NMG and its black hole solution}\nIn order to have a well-defined flat-space limit ($\\Lambda\\to 0$ or $\\ell\\to\\infty$)\nfor \\eqref{BHTaction} in the special point $m^2\\ell^2=1\/2$, we also need to scale Newton\\rq{}s constant to infinity while keeping fixed $\\kappa=\\ell^{2}\/G$ .\nThus, the flat-space limit of NMG action \\eqref{BHTaction} becomes\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{Flataction}\nS=\\frac{\\kappa}{8\\pi}\\int d^{3}x \\sqrt{-g} K.\n\\ee\nMoreover, a well-defined flat-space limit for the black hole solution \\eqref{BHTbh}\nneeds a scaling of the mass parameter $M$ such that $\\mu=M G$ remains fixed.\nThe final line element for the asymptotically flat rotating hairy black hole is given by \n \\begin{equation}}   \\def\\ee{\\end{equation}\\label{FlatBH}\nds^{2}=-{\\cal F}dt^{2}+\\frac{r^{2}}{{\\cal F}\\Delta}dr^{2}+a{\\cal F}dtd\\phi+r^{2}d\\phi^{2},\n\\ee\nwhere $\\Delta(r)$ and ${\\cal F}(r)$  are functions of the radial coordinate $r$, given by\n\\begin{align}\n\\Delta &  = r^{2}- \\mu a^{2}-\\left(\\frac{a^{2}b}{8}\\right)^{2} \\ ,\\nonumber\\\\\n{\\cal F} &  = b\\sqrt{\\Delta}-4 \\mu\\,.\n\\end{align}\nThe Ricci scalar of this asymptotically flat black hole can be written as\n\\begin{equation}}   \\def\\ee{\\end{equation}\nR=-\\frac{16\\, b}{a^2 b+8\\sqrt{\\Delta}}\\, .\n\\ee\nOne can verify that \\eqref{FlatBH} satisfies the\nequations of motion resulting from the action \\eqref{Flataction}.\nIt is worth noting that in \\cite{Deser:2009hb}, it was argued that the three-dimensional gravity\ntheory described by \\eqref{Flataction} is  ghost free and finite.\n\nHorizons of \\eqref{FlatBH} are at\n\\begin{equation}}   \\def\\ee{\\end{equation}\nr_{+}= \\frac{a^2 b}{8}+\\frac{4\\mu}{b}\\, ,\n\\qquad r_{-}=\\sqrt{\\left(\\frac{a^{2} b}{8}\\right)^{2}+\\mu a^{2}}\\, ,\n\\ee\nand one can calculate the entropy of the outer horizon using the Wald\\rq{}s formula.\nThis formula gives the black hole entropy in an arbitrary diffeomorphism invariant theory and is given by\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{Wald}\nS=-\\frac{2\\pi}{16\\pi G}\\int_{\\Sigma_{h}}\\,\\frac{\\delta L}{\\delta R_{\\alpha\\beta\\gamma\\delta}}\\,\n\\epsilon_{\\alpha\\beta}\\,\\epsilon_{\\gamma\\delta}\\,\\bar{\\epsilon}\\ ,\n\\ee\nwhere $L$ is the Lagrangian, and $\\bar{\\epsilon}$, $\\epsilon_{\\mu\\nu}$,\ndenote the volume form and the binormal vector to the spacelike\nbifurcation surface $\\Sigma_{h}$, respectively.\n$\\epsilon_{\\mu\\nu}$ is normalized as $\\epsilon^{\\mu\\nu}\\epsilon_{\\mu\\nu}=-2$.\nFor the action \\eqref{Flataction} and the asymptotically flat rotating hairy black hole solution \\eqref{FlatBH} we obtain \n\\begin{subequations}\n\\begin{align}\n\\frac{\\partial L}{\\partial R_{\\alpha\\beta\\gamma\\delta}}\n&=\\frac{3}{8}R\n\\left(g^{\\alpha\\delta}g^{\\beta\\gamma}-g^{\\alpha\\gamma}g^{\\beta\\delta}\\right)\n+\\frac{1}{2}\\left(g^{\\alpha\\gamma}R^{\\beta\\delta}\n-g^{\\alpha\\delta}R^{\\beta\\gamma}\n-g^{\\beta\\gamma}R^{\\alpha\\delta}\n+g^{\\beta\\delta}R^{\\alpha\\gamma}\\right),\\\\\n\\epsilon_{\\alpha\\beta}&=-\n\\left(\\frac{a^{2}{\\cal F}+4r^{2}}{\\Delta}\\right)^{\\frac{1}{2}}\n\\delta^{t}_{[\\alpha}\\delta^{r}_{\\beta]}\\, .\n\\end{align}\n\\end{subequations}\nTherefore, the Wald\\rq{}s entropy for the new type of asymptotically flat black hole becomes\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{flat entropy}\nS_{{\\rm flat}}=\\frac{\\pi\\kappa b}{2}\\, .\n\\ee\n\nIt is instructive to derive the above entropy by taking the flat-space limit of the entropy \\eqref{Entropy}.\nThe entropy can, therefore, be computed as follows:\n\\begin{align}\\label{FEntropy}\n\\lim_{\\ell\\to\\infty} S&=\\lim_{\\ell\\to\\infty}\\pi \\ell\\sqrt{\\frac{2}{G}\\Delta M\\left(  1+\\Xi^{\\frac{1}{2}}\\right)  }\\nonumber\\\\\n&=\\lim_{\\ell\\to\\infty}\\frac{\\pi\\kappa b}{2}\\sqrt{1+\\frac{16\\mu}{b^{2}\\ell^{2}}}\\nonumber\\\\\n&=\\frac{\\pi\\kappa b}{2}=S_{{\\rm flat}}\\, .\n\\end{align}\nNow, consider the Hawking temperature.\nFrom (\\ref{Temperature}) it follows that\n\\begin{align}\\label{FTemperature}\n\\lim_{\\ell\\to\\infty}T  &  =\\lim_{\\ell\\to\\infty}\\frac{1}{\\pi\\ell}\\Xi^{\\frac{1}{2}}\\sqrt{2G\\Delta M\\left(  1+\\Xi^{\\frac\n{1}{2}}\\right)  ^{-1}}\\ ,\\nonumber\\\\\n&=\\lim_{\\ell\\to\\infty}\\frac{b}{4\\pi}\\sqrt{1+\\frac{16\\mu}{b^{2}\\ell^{2}}}\\nonumber\\\\\n&=\\frac{b}{4\\pi}=T_{{\\rm flat}}\\, .\n\\end{align}\nThese quantities fulfill the relation\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{Fcalentate}\nT_{\\text{flat}}\\,dS_{\\text{flat}}=\\frac{b \\kappa}{8}\\,db\\, .\n\\ee\nThis agrees precisely with the $\\ell\\to\\infty$ limit of (\\ref{calentate}).\nA direct calculation by using \\eqref{FlatBH} or taking the flat-space limit of \\eqref{Omega+}\nshows that the angular velocity of the black hole \\eqref{FlatBH} at the outer horizon\nis zero ($\\Omega_{{+}_{\\rm flat}}=0$) though it has a nonvanishing angular momentum.\n\nFrom \\eqref{flat entropy} and \\eqref{FTemperature}, it is clear that the hair parameter $b$\ndetermines the entropy and the temperature of the outer horizon.\nIn the $b\\to 0$ limit the hairy black hole \\eqref{FlatBH} is reduced to the cosmological solution of \\cite{Cornalba:2002fi}.\nIn this limit, $r_{+}$ is mapped to infinity; however, $r_{-}$ remains finite and defines the cosmological horizon. \n\n\\section{Dual Boundary Theory}\n\\label{Dual Boundary Theory}\n\n\\subsection{CFT dual to NMG}\nIn \\cite{Bergshoeff:2009aq, Giribet:2009qz}, it was proposed that NMG has a dual description in terms of a CFT.\\footnote{In 2013, de Buyl {\\it et al.}\nconsidered the asymptotically dS case, $\\Lambda=+1\/\\ell^{2}>0$,\nwithin the context of dS\/CFT correspondence \\cite{deBuyl:2013ega}.\nWe would like to thank the referee for bringing this paper to our attention.}\nThe charges associated to the asymptotic symmetries enhance the isometry of asymptotically $AdS_3$ spacetimes\nto two copies of the Virasoro algebra.\nThe central charges are given by\n\\begin{equation}}   \\def\\ee{\\end{equation}\nc_{\\pm}=c=\\frac{3\\ell}{2G}\\left(1+\\frac{1}{2m^{2}\\ell^{2}}\\right)\\ .\\label{central charge}\n\\ee\nAt the spacial point  $m^{2}\\ell^{2}=1\/2$, the central charges are twice\nthe values proposed by Brown and  Henneaux for the Einstein gravity\nwith negative cosmological constant \\cite{Brown:1986nw}, i.e.,\n\\begin{equation}}   \\def\\ee{\\end{equation}\nc=\\frac{3\\ell}{G}\\ .\\label{central charge}\n\\ee\nThe entropy of the black hole \\eqref{BHTbh} can be given by the Cardy formula\n\\begin{equation}}   \\def\\ee{\\end{equation}\nS=2\\pi\\sqrt{\\frac{c_{+}\\Delta_{+}}{6}}+2\\pi\\sqrt{\\frac{c_{-}\\Delta_{-}}{6}\n}\\ ,\\label{Cardy formula}\n\\ee\nwhere $\\Delta_\\pm$ are the eigenvalues of the left and right Virasoro generators $L_{0}^{\\pm}$ and are given by\n\\begin{equation}}   \\def\\ee{\\end{equation}\n\\Delta_{\\pm}=\\frac{1}\n{2}\\Delta M\\left(  \\ell\\pm a\\right)  \\ .\\label{Delta+-}\n\\ee\nUsing (\\ref{central charge}) and (\\ref{Delta+-}), this is\n\\begin{equation}}   \\def\\ee{\\end{equation}\nS =\\pi \\ell\\sqrt{\\frac{2}{G}\\left(  1+\\Xi^{\\frac{1}{2}}\\right)  \\Delta M}\\ ,\n\\ee\nin precise agreement with  (\\ref{Entropy}).\n\n\\subsection{CCFT dual to the flat-space limit of NMG}\nIn this section we want to propose a dual description for the theory of gravity given by \\eqref{Flataction}.\nTo do so, we will use the idea which was first proposed in papers \\cite{Bagchi:2010zz,Bagchi:2012cy}.\nThat is, if we start from the AdS\/CFT correspondence,\nthe large AdS radius limit in the bulk is equivalent to a contraction\nof spacetime coordinates in the boundary CFT.\\footnote{We refer to \\cite{Bagchi:2012cy} for\na full-scale investigation into the CCFT representation.}\n\nWe shall first show how one obtains the appropriate coordinate which must be contracted in the parent CFT.\nLet us look at the conformal boundary of the black hole \\eqref{BHTbh} for an arbitrary large $\\ell$.\nIt could be written as follows:\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{CB}\nds_{{\\rm C.B.}}^{2}=\\frac{r^{2}}{\\kappa^{2}}\\left(-\\frac{\\kappa^{2}}{\\ell^{2}}dt^{2}+\\kappa^{2}d\\phi^{2}\\right)\\, .\n\\ee\nWe have  used $\\kappa$ in the conformal factor to make it dimensionless.\nMoreover, the fact that $\\kappa$ is fixed in our flat-space limit makes\nthe conformal factor well defined for all large values of $\\ell$.\nNow, $\\ell$ can be absorbed by defining new time coordinate as $\\tau=\\kappa t\/\\ell$.\nThe dual CFT lives on a cylinder with coordinates $(\\tau, \\phi)$ and radius $\\kappa$.\nTaking the $\\ell\\to\\infty$ limit (or $\\kappa\/\\ell\\to 0$ limit),\nit is obvious that the flat-space limit in the bulk induces a contraction in\n$t$ of the boundary CFT reducing it to the two-dimensional CCFT.\n\n\\subsubsection{Symmetries of CCFT}\nAccording to the proposal of \\cite{Bagchi:2010zz,Bagchi:2012cy}, the symmetries of CCFT\nrealize the group of  asymptotic symmetries of the asymptotically flat  spacetimes at null infinity,\nnamely the Bondi-Metzner-Sachs (BMS) group \\cite{BMS,Barnich aspects}.\nThere is a very precise procedure, called the $\\dot {\\rm I}$n$\\ddot {\\rm o}$n$\\ddot {\\rm u}$-Wigner contraction, by which one\ncan obtain the CCFT algebra from the relativistic conformal algebra of the parent CFT.\nLet us consider two copies of the Virasoro algebra\\footnote{The relativistic conformal algebra\nconsists of two copies of the Virasoro algebra.}\n\\begin{align}\\label{Virasoro}\n  \\nonumber [L^+_m,L^+_n]&=(m-n)L^+_{n+m}+{c^+\\over 12}m(m^2-1)\\delta_{m+n,0}\\, ,\\\\\n  \\nonumber [L^-_m,L^-_n]&=(m-n)L^-_{n+m}+{c^-\\over 12}m(m^2-1)\\delta_{m+n,0}\\, ,\\\\ \n            [L^+_m,L^-_n]&=0\\, .\n  \\end{align}  \nFor a small parameter $\\epsilon$, at the level of the algebra, if we define\\footnote{It is clear from (3.8)\nthat other positive powers of $\\epsilon$ are meaningless\nsince one can redefine $\\epsilon$ and rewrite them finally as in (3.8).\nThis definition is consistent with the observation of \\cite{bar-comp}.\nThat is, one can obtain the BMS$_{3}$ algebra by taking the flat-space limit\nfrom the asymptotic symmetry algebra of three-dimensional asymptotically AdS spacetimes.}\n\\begin{align}\\label{def of BMS generators}\n\\nonumber L_n &=L^+_n-L^-_{(-n)}\\, ,\\\\\n          M_n &=\\epsilon\\left(L^+_n+L^-_{(-n)}\\right), \n\\end{align}\nwe can see that the CCFT algebra is generated from the Virasoro algebras, on taking the $\\epsilon\\to0$ limit, i.e.,\n\\begin{align}\\label{BMS}\n  \\nonumber [L_m,L_n]&=(m-n)L_{n+m}+{c_L\\over 12}m(m^2-1)\\delta_{m+n,0}\\,,\\\\\n  \\nonumber [L_m,M_n]&=(m-n)M_{n+m}+{c_M\\over 12}m(m^2-1)\\delta_{m+n,0}\\, ,\\\\ \n            [M_m,M_n]&=0\\, ,    \n  \\end{align}\nwhere the central charges $c_L$ and $c_M$ are given by the linear combination of the parent relativistic central charges\n\\begin{equation}\\label{def of cC of CCFT}\nc_L=\\lim_{\\epsilon\\to 0 }(c^+-c^-)\\, ,\\qquad c_M=\\lim_{\\epsilon\\to 0 }\\epsilon(c^++c^-)\\, .\n\\end{equation}\nThe algebra \\eqref{BMS} which is given by the contraction of the Virasoro algebra in the boundary theory is exactly\nthe (centrally extended) BMS$_{3}$ algebra \\cite{bar-comp}.\n\nWe would expect the same symmetry group for the CCFT dual to\nthe theory described by the action \\eqref{Flataction} at the special point $m^2\\ell^2=1\/2$.\nAs we stated earlier, the $\\epsilon\\to 0$ limit in the boundary corresponds to the flat-space limit\nor, more precisely, the $\\kappa\/\\ell\\to 0$ limit in the bulk side.\nUsing \\eqref{central charge} and \\eqref{def of cC of CCFT}, for the problem in hand we find\n\\begin{equation}}   \\def\\ee{\\end{equation}\n c_L=c_M=0\\, .\n \\ee\n\nWe will show that although the CCFT algebra has vanishing central charges,\nit is possible to find a Cardy-like formula for the asymptotic growth of the number of states\nwhich reproduces the entropy of the black hole \\eqref{FlatBH}.\nTo add strength to this claim, let us find more evidence about\nthe correctness of our proposal using the CCFT energy-momentum tensor.\n\n\\subsubsection{Quasi local stress tensor}\nThe one-point function of the CCFT energy-momentum operator\ncorresponds to the quasilocal stress tensor of the bulk theory.\nIt was argued in \\cite{Fareghbal:2013ifa} that the definition \\eqref{def of BMS generators} provides a recipe\nto calculate the components of the stress tensor in the asymptotically flat spacetimes.\nTherefore, we can write\n\\begin{eqnarray}\\label{def of energy-momentum }\n \\nonumber \\tilde T_{++}+\\tilde T_{--}&=&\\lim_{\\epsilon\\to 0}\\epsilon\\left(T_{++}+T_{--}\\right)\\, ,\\\\\n  \\nonumber\\tilde T_{++}-\\tilde T_{--}&=&\\lim_{\\epsilon\\to 0}\\left(T_{++}-T_{--}\\right)\\, ,\\\\\n  \\tilde T_{+-}&=&\\lim_{\\epsilon\\to 0} T_{+-}\\, ,\n  \\end{eqnarray}\nwhere $T_{ij}$ and $\\tilde T_{ij}$ are, respectively, the stress tensor of the asymptotically AdS and flat spacetimes\nand $x^{\\pm}$ are the light-cone coordinates constructed from the nonradial coordinates of the metrics.\nIn the above definition it was assumed that both the asymptotically AdS\nand flat spacetimes are given in the BMS gauge \\cite{Fareghbal:2013ifa}. \n\nThe nonzero components of the stress tensor at the boundary\nof the asymptotically AdS black hole \\eqref{BHTbh} are given by \\cite{Kwon:2011jz} \n\\begin{align}\n\\nonumber T_{tt}&=\\frac{1}{8\\pi G \\ell }\\left(\\frac{b^{2}\\ell^{2}}{4}+4 M G\\right), \\\\\n\\nonumber T_{t\\phi}&=-\\frac{a}{8\\pi G \\ell}\\left(\\frac{b^{2}\\ell^{2}}{4}+4 M G\\right), \\\\\n          T_{\\phi\\phi}&=\\frac{\\ell}{8\\pi G }\\left(\\frac{b^{2}\\ell^{2}}{4}+4 M G\\right).\n\\end{align}\nThe formula \\eqref{def of energy-momentum } results in a stress tensor $\\tilde T_{ij}$\nfor the asymptotically flat black hole \\eqref{FlatBH} as follows:\n\\begin{equation}\\label{flat stress tensor}\n\\tilde T_{tt}={b^2\\over 32\\pi}\\, ,\\qquad \\tilde T_{\\phi\\phi}={\\kappa^2 b^2\\over 32 \\pi}\\, , \\qquad \\tilde T_{t\\phi}=-{a b^2\\over 32\\pi}\\, .\n\\end{equation}\nUsing $\\tilde T_{ij}$ we can calculate the conserved charges of the black hole \\eqref{FlatBH}.\n\n Let us denote the hypersurface of the spacetime where CCFT lives with $\\partial {\\cal M}$.\nIts line element is given by taking the $\\ell\\to \\infty$ limit of the conformal boundary \\eqref{CB},\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{bundary of flat}\nds_{\\partial {\\cal M}}^{2}=\\frac{r^{2}}{\\kappa^{2}}\\left(-dt^{2}+\\kappa^{2}d\\phi^{2}\\right).\n\\ee\nFollowing Brown and York\\rq{}s method \\cite{Brown:1992br},\nthe charges associated to a boundary Killing vector $\\xi^\\mu$ are given by\n\\begin{equation}\\label{BY formula}\nQ_{\\xi}=\\int_{\\Sigma} d\\phi \\sqrt{\\sigma}\\xi^\\mu n^\\nu \\tilde T_{\\mu\\nu}\\, ,\n\\end{equation}\nwhere $\\Sigma$ is the spacelike surface embedded in $\\partial {\\cal M}$ with induced metric $\\sigma_{\\mu\\nu}$.\nMoreover, $n^{\\mu}$ is the timelike unit normal to $\\Sigma$.\nUsing \\eqref{bundary of flat} and \\eqref{BY formula},\nthe mass and the angular momentum of the asymptotically flat black hole \\eqref{FlatBH} are \n\\begin{equation}\\label{charges}\n{\\cal M}=Q_{\\partial_t}={\\kappa b^2\\over 16}\\, ,\\qquad {\\cal J}=Q_{\\partial_\\phi}=-{\\kappa a b^2\\over 16}\\, .\n\\end{equation}\nIt is clear that $|{\\cal J}|\/{\\cal M}=a$ as expected.\n\nGiven the expressions above, together with \\eqref{Fcalentate},\nit is straightforward to check that the first law of black hole thermodynamics is satisfied, i.e.\n\\begin{equation}}   \\def\\ee{\\end{equation}\nd{\\cal M}=T_{\\rm flat}\\,dS_{\\rm flat}-\\Omega_{{+}_{\\rm flat}}\\,d{\\cal J}\\, .\n\\ee\n\n\n\\subsubsection{Cardy-like formula}\\label{entropy}\nIf the gravity theory \\eqref{Flataction} has a dual description in terms of a CCFT,\nthen the entropy of the black hole \\eqref{FlatBH} must be given by\nthe asymptotic growth of the number of states in the boundary theory.\nIn \\cite{Bagchi:2012xr}, the authors found a Cardy-like formula\nby computing the CCFT partition function using the saddle-point approximation.\nHowever, in the recent papers \\cite{Fareghbal:2014qga,Riegler:2014bia}\nit was shown that the Cardy-like formula of \\cite{Bagchi:2012xr} can be obtained\nif one writes the Cardy formula in terms of CCFT parameters and then takes the $\\epsilon\\to 0$ limit.\nIn the current work we will use the same approach and take the $\\epsilon\\to 0$ limit from the Cardy formula in the parent CFT.\n  \nThe CCFT algebra is given by \\eqref{BMS}.\nWe denote the eigenvalues of $L_{0}$ and $M_{0}$ by $\\Delta_L$ and $\\Delta_M$, respectively.\nFor the current problem $c_{L}=c_{M}=0$, however, the eigenvalues of $L_{0}$ and $M_{0}$ are nonzero.\nFrom the viewpoint of the limit \\eqref{def of BMS generators} we see that the two labels $\\Delta_L$ and $\\Delta_M$\nare related to the conformal weights in the two-dimensional CFT as\n\\footnote{\nThe Hilbert space construction of a CCFT is analogous to that of the relativistic 2D CFT.\nNow, the states are labeled by the eigenvalues under $L_{0}$ and $M_{0}$.\nWe shall use the cylinder representation of the CCFT algebra \\cite{Bagchi:2012cy}.\nThe Hilbert space of the 2D CCFT are constructed by considering the states having definite scaling dimensions.\nWe define primary states by demanding that the states in the theory be annihilated by all generators with $n>0$.\nOne can build up a tower of operators by acting on a primary state with the creation operators $L_{-n}$ and $M_{-n}$ ($n>0$).\n}\n\\begin{align}\\label{def of D_LM}\n\\Delta_{L}&=\\lim_{\\epsilon\\to 0}\\left(\\Delta_{+}-\\Delta_{-}\\right)\\, ,\n\\qquad \\Delta_{M}=\\lim_{\\epsilon\\to 0}\\epsilon\\left(\\Delta_{+}+\\Delta_{-}\\right)\\, .\n\\end{align}\nWe would like to remind the reader that the $\\epsilon\\to 0$ limit in the boundary field theory\ncorresponds to the $\\kappa\/\\ell\\to 0$ limit in the bulk.\nTherefore, using \\eqref{Delta+-} and \\eqref{def of D_LM} one can easily find\n\\begin{equation}\\label{CCFT quantities}\n\\Delta_{L}=\\frac{a\\kappa b^{2}}{16}\\, ,\\qquad \\Delta_{M}=\\frac{\\kappa^2 b^{2}}{16}\\, .\n\\end{equation}\nLet us consider the Cardy formula \\eqref{Cardy formula} and try to take its $\\epsilon\\to 0$  limit.\nFor our current problem the relativistic central charges are $c_+=c_-=3\\epsilon$.\nUsing \\eqref{CCFT quantities}, we obtain\n\\begin{align}\\label{CCFT cardy}\n\\nonumber \\lim_{\\epsilon\\to 0} {S_{\\text{CFT}}}&= \\lim_{\\epsilon\\to 0} 2\\pi\\left(\\sqrt{c_+\\Delta_+\\over 6}+\\sqrt{c_-\\Delta_-\\over 6}\\right)\\\\\n\\nonumber &=\\lim_{\\epsilon\\to 0} \\pi\\left[\\sqrt{{\\epsilon}\\left({\\Delta_M\\over\\epsilon}+\\Delta_L\\right)}+\\sqrt{{\\epsilon}\\left({\\Delta_M\\over\\epsilon}-\\Delta_L\\right)}\\right]\\\\\n&=2\\pi\\sqrt{\\Delta_M}=S_{\\text{CCFT}}\\, .\n\\end{align}\nThis is  the Cardy-like formula for the CCFT dual to the flat-space limit of NMG.\nInserting (\\ref{CCFT quantities}) into (\\ref{CCFT cardy}), we finally recover the entropy (\\ref{flat entropy}),\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{Flat\/CCFT}\nS_{\\text{CCFT}}=S_{\\text{flat}}\\, ,\n\\ee\nas we wanted to show.\nIt is a quite nontrivial result since the theory has vanishing central charges.\n\n\\section{Conclusions}\n\nIn this paper, we have proposed a flat space generalization of the AdS$_3$\/CFT$_{2}$ holographic correspondence.\\footnote{We note that in \\cite{Hasanpour:2011ji},\nthe authors considered an asymptotically flat geometry which is a solution to\nthree-dimensional Einstein gravity conformally coupled to a scalar field\nand discussed gravity\/CFT correspondence for this background.}\nWe have provided the first example of a holographic dual of an asymptotically flat black hole solution.\nDue to the absence of black hole solutions in three-dimensional Einstein gravity with vanishing cosmological constant,\nwe have considered higher-derivative gravity theories which admit asymptotically flat black hole solutions.\nThe theory we have investigated is given by taking the flat-space limit ($\\Lambda\\to 0)$ of NMG.\nWe argued that the dual field theory of the black hole solution of this theory is a CCFT.\nFor this purpose, we have constructed a stress tensor for the\nasymptotically flat black hole solution and computed the conserved charges.\nWe then verified it using the first principle of black hole thermodynamics.\nFurthermore, we have used the Flat\/CCFT correspondence to find the black hole entropy\nin terms of the asymptotic growth of the number of CCFT states.\n\nIt is interesting to note that the symmetry algebra of the corresponding CCFT had vanishing central charges though\nthe asymptotic growth of  states were nonzero.\nThis remarkable point can be used for finding holographic duals of four-dimensional asymptotically flat spacetimes.\nAccording to the proposal of Flat\/CCFT correspondence, the dual of four-dimensional asymptotically flat black holes\nare field theories with BMS$_{4}$ symmetry  \\cite{Barnich:2009se, Barnich aspects}.\nIn \\cite{Barnich:2011mi}, the authors constructed the field-dependent central extension of BMS$_{4}$ algebra\nand found that for the Kerr black hole some of the charges involved divergent integrals on the 2-sphere\nif they used extended BMS algebra with both supertranslations and superrotations.\nThus, at first sight, it seems that counting CCFT$_{3}$ states would be a problematic issue, but our current work\nshows that counting the asymptotic growth of CCFT states can be done whatever the central charges are.\n\nAlthough our current study gives a holographic description of asymptotically flat black holes in three-dimensional higher-derivative gravity,\nwe believe that the Flat\/CCFT correspondence can be extended to find a holographic description of\nblack holes in higher dimensions and, specifically, the four-dimensional Kerr black hole.\nWe hope to explore other intriguing aspects of the relation between asymptotically flat spacetimes and CCFTs in our future works.\n\n\\section*{Acknowledgments}\nWe would like to thank Ali Naseh for useful discussions.\nWe are grateful to \\'{A}lvaro V\\'{e}liz-Osorio and Arjun Bagchi for their useful\ncomments on the revised version of the manuscript.\nS.M.H. is supported in part by INFN.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThis paper is the first part of our project to integrate\nrepresentations up to homotopy of Lie algebras (algebroids). Our\noriginal motivation is to integrate the standard Courant algebroid\n$TM\\oplus T^*M$ since it is this Courant algebroid that is much used\nin Hitchin and Gualtieri's program of generalized complex geometry.\nCourant algebroids are Lie 2-algebroids in the sense of Roytenberg\nand \\v{S}evera \\cite{royt, s:funny}. The general procedure to\nintegrate Lie $n$-algebras (algebroids) is already described in\n\\cite{getzler, henriques, s:funny}. We want to pursue some explicit formulas\nfor the special case of the  standard Courant algebroid. It turns\nout that the sections of the Courant algebroid $TM \\oplus T^*M$ form\na semidirect product of a Lie algebra with a representation up to\nhomotopy.   Abad and Crainic \\cite{abad-crainic:rep-homotopy}\nrecently studied the representations up to homotopy of Lie algebras,\nLie groups, and even Lie algebroids, Lie groupoids, in general. Just\nas one can form the semidirect product of a Lie algebra with a\nrepresentation, one can form the semidirect product with\nrepresentations up to homotopy too. In our case, the semidirect\nproduct coming from the standard Courant algebra is a Lie 2-algebra.\nBut using the fact that it is also a semidirect product, the\nintegration becomes easier. The integration result is related to the\nsemidirect product of Lie groups with its representation up to\nhomotopy, which will be discussed in Section \\ref{sec:gp}. However\nit turns out that the concept of representation up to homotopy of\nLie groups of Abad and Crainic will not be general enough to cover\nall the integration results. This we will continue in a forthcoming\npaper \\cite{sheng-zhu:II}.\n\nIn this paper we focus on exhibiting more examples of representation\nup to homotopy and  their semidirect products to demonstrate the\nimportance of our integration procedure. The examples are all sorts\nof variations of Courant algebroids. One is Chen and Liu's omni-Lie\nalgebroids, which generalizes Weinstein's omni-Lie algebras. Hence\nwe expect to give an integration to Weinstein's omni-Lie algebras\nvia Lie 2-algebras in the next paper \\cite{sheng-zhu:II}.\n\n\nAnother example comes from the so called string Lie 2-algebra. It is\nessentially  a Courant algebroid over a point (see Section\n\\ref{sec:string}),  namely  a Lie algebra with an adjoint-invariant\ninner product. This sort of Lie algebra is usually called a {\\em\nquadratic Lie algebra}. This concept also appears in the context of\nManin triples and double Lie algebras. The example  $\\mathbb R}\\newcommand{\\Z}{\\mathbb Z \\to \\g\n\\oplus \\g^*$ that we consider in this paper is an analogue of the\nstandard Courant algebroid, and is basically a special case taken\nfrom \\cite{lu-weinstein} \\footnote{private conversation\n  with Jiang-Hua Lu}. We give an integration of\nthe string Lie 2-algebra $\\mathbb R}\\newcommand{\\Z}{\\mathbb Z \\to \\g \\oplus \\g^*$  at the end.\n\nUsually people require the base Lie algebra of a string Lie algebra\nto be semisimple and of compact type (see Remark\n\\ref{rk:semisimple}). For such usual sort of string Lie 2-algebras,\nBaez et al \\cite{baez:2gp} have proved a no-go theorem, namely such\nstring Lie 2-algebras can not be integrated to finite dimensional\nsemi-strict Lie 2-groups. Here a {\\em semi-strict Lie 2-group} is a\ngroup object in $\\rm DiffCat$, where  $\\rm DiffCat$ is the\n2-category consisting of categories, functors, and natural\ntransformations in the category of differential manifolds, or\nequivalently $\\rm DiffCat$ is a 2-category with Lie groupoids as\nobjects, strict morphisms of Lie groupoids as morphisms, and\n2-morphisms of Lie groupoids as 2-morphisms.  Our semi-strict Lie\n2-group is actually called a Lie 2-group by the authors in\n\\cite{baez:2gp}. However, we call it a semi-strict Lie 2-group\nbecause  compared to the Lie 2-group in the sense of Henriques\n\\cite{henriques}, or equivalently the stacky group in the sense of\nBlohmann \\cite{blohmann},  it is stricter. Basically,  their Lie\n2-group is a group object in the 2-category with objects as Lie\ngroupoids, morphisms as Hilsum-Skandalas bimodules (or generalized\nmorphisms), 2-morphisms as 2-morphisms of Lie groupoids.\nSchommer-Pries realizes the string 2-group as such a Lie 2-group\nwith a finite dimensional model \\cite{schommer:string-finite-dim}\nand the integration of a string Lie 2-algebra to such a model is a\nwork in progress \\cite{ccc:integration-string}.\n\nIt is not needed in the\ndefinition of the string Lie 2-algebra for the base Lie algebra to be\nsemisimple of compact type. One only needs a quadratic Lie algebra.\nAs soon as we relax this condition on compactness, we find out that\none can integrate  $\\mathbb R}\\newcommand{\\Z}{\\mathbb Z \\to \\g \\oplus \\g^*$ to a finite dimensional\nsemi-strict Lie\n2-group in the sense of Baez et al. The integrating object is actually\na special Lie 2-group (very close to a strict Lie 2-group) in the\nsense of Baez et al.\n\nThen of course, as we relax the\ncondition, we are in danger that the class corresponding to this\nLie 2-algebra in $H^3(\\g\\oplus \\g^*, \\mathbb R}\\newcommand{\\Z}{\\mathbb Z)$ might be trivial, and\nconsequently our Lie 2-algebra might be strict. Then what\nwe have done\nwould not have been a big surprise  because  a strict\nLie 2-algebra corresponds to a crossed module of Lie algebras, and it\neasily integrates to a strict Lie 2-group by integrating the crossed module.  However,\nwe verified that when $\\g$ itself (not $\\g \\oplus \\g^*$) is semisimple, this Lie 2-algebra is not strict.\n\n\n\n{\\bf Acknowledgement:} We give warmest thanks to  Zhang-Ju Liu,\nJiang-Hua Lu, Giorgio Trentinagia and Marco Zambon for\nuseful comments and discussion. Y. Sheng gives his warmest thanks to Courant Research Centre ``Higher Order Structures'', G\\\"{o}ttingen University, where this work was\ndone when he visited there.\n\n\\section{Representations up to homotopy of Lie algebras}\nIn this section, we first consider the 2-term representation up to\nhomotopy of Lie algebras. We give explicit formulas of the\ncorresponding 2-term $L_\\infty$-algebra, which is their semidirect\nproduct. Then we give several interesting examples including Courant\nalgebroids and omni-Lie algebroids.\n\n\\subsection{Representation up to homotopy of Lie algebras and their semidirect products}\n\n$L_\\infty$-algebras,  sometimes  called strongly homotopy Lie\nalgebras,  were introduced by Drinfeld and Stasheff\n\\cite{stasheff:shla} as a model for ``Lie algebras that satisfy\nJacobi up to all higher homotopies''. The following convention of\n$L_\\infty$-algebras has the same grading as in \\cite{henriques} and\n\\cite{rw}.\n\n\\begin{defi}\nAn $L_\\infty$-algebra is a graded  vector space $L=L_0\\oplus\nL_1\\oplus\\cdots$ equipped with a system $\\{l_k|~1\\leq k<\\infty\\}$ of\nlinear maps $l_k:\\wedge^kL\\longrightarrow L$ with degree\n$\\deg(l_k)=k-2$, where the exterior powers are interpreted in the\ngraded sense and the following relation with Koszul sign ``Ksgn'' is\nsatisfied for all $n\\geq0$:\n\\begin{equation}\n\\sum_{i+j=n+1}(-1)^{i(j-1)}\\sum_{\\sigma}\\mathrm{sgn}(\\sigma)\\mathrm{Ksgn}(\\sigma)l_j(l_i(x_{\\sigma(1)},\\cdots,x_{\\sigma(i)}),x_{\\sigma(i+1)},\\cdots,x_{\\sigma(n)})=0,\n\\end{equation}\nwhere the summation is taken over all $(i,n-i)$-unshuffles with\n$i\\geq1$.\n\\end{defi}\n\n\nLet $n=1$, we have\n$$\nl_1^2=0,\\quad l_1:L_{i+1}\\longrightarrow L_i,\n$$\nwhich means that $L$ is a complex and we usually write $\\mathrm{d}=l_1$.\nLet $n=2$, we have\n$$\n\\mathrm{d} l_2(x,y)=l_2(\\mathrm{d} x,y)+(-1)^pl_2(x,\\mathrm{d} y),\\quad \\forall~x\\in L_p,\ny\\in L_q,\n$$\nwhich means that $\\mathrm{d} $ is a derivation with respect to $l_2$. We\nusually view $l_2$ as a bracket $[\\cdot,\\cdot]$. However, it is not\na Lie bracket, the obstruction of Jacobi identity is controlled by\n$l_3$: \\begin{eqnarray} \\nonumber\n&&l_2(l_2(x,y),z)+(-1)^{(p+q)r}l_2(l_2(y,z),x)+(-1)^{qr+1}l_2(l_2(x,z),y)\\\\\n&&=-\\mathrm{d} l_3(x,y,z)-l_3(\\mathrm{d} x,y,z)+(-1)^{pq}l_3(\\mathrm{d}\ny,x,z)-(-1)^{(p+q)r}l_3(\\mathrm{d} z, x,y),\n\\end{eqnarray}\nwhere $x\\in L_p,~ y\\in L_q,~z\\in L_q$ and $l_3$ also satisfies\nhigher coherence laws.\n\nIn particular, if the $k$-ary brackets are zero for all $k>2$, we\nrecover the usual notion of {\\bf differential graded Lie algebras}\n(DGLA). If $L$ is concentrated in degrees $<n$, $L$ is called an\n {\\bf $n$-term $L_\\infty$-algebra}.\n\nIn this paper, we mainly consider\n 2-term $L_\\infty$-algebras, which are equivalent  to Lie 2-algebras\n given in \\cite{baez:2algebras} by John Baez and Alissa Crans.   In\nthis special case, $l_4$ is always zero. Thus restricting the coherence\nlaw satisfied by $l_3$ on degree-0, we obtain\n\\begin{equation}\nl_3(l_2(x,y),z,w)+c.p.-\\big(l_2(l_3(x,y,z),w)+c.p.\\big)=0,\\quad\\forall~x,y,z,w\\in\nL_0.\n\\end{equation} Lie 2-algebras are categorified\n version of Lie algebras. In a Lie 2-algebra, the Jacobi identity\n is replaced by an isomorphism which is called the {\\bf\n Jacobiator}. The Jacobiator  satisfies a certain law of its own.  Given a 2-term\n $L_\\infty$-algebra $L_1\\stackrel{\\mathrm{d}}{\\longrightarrow}L_0$, the\n underlying 2-vector space of the corresponding Lie 2-algebra is\nmade up by $L_0$  as the vector space of objects and $L_0\\oplus\n L_1$ as the vector space of morphism. Please see \\cite[Theorem\n 4.3.6]{baez:2algebras} for more details.\n\n\nRecall from \\cite{baez:2algebras} that a Lie 2-algebra is skeletal\nif isomorphic objects are equal. Viewing a Lie 2-algebra as a 2-term\n$L_\\infty$-algebra, explicitly we have\n \\begin{defi}\\label{defi:skeletal L 2}\nA 2-term $L_\\infty$-algebra $L_1\\stackrel{\\mathrm{d}}{\\longrightarrow}L_0$\nis called {\\bf skeletal} if $\\mathrm{d}=0$.\n \\end{defi}\n\nThey also prove the following theorem\n\\begin{thm}{\\rm\\cite{baez:2algebras}}\\label{thm:skeletal Lie 2 a}  There is a\n one-to-one correspondence between 2-term skeletal $L_\\infty$-algebra\n $L_1\\stackrel{\\mathrm{d}}{\\longrightarrow}L_0$ and quadruples\n $(\\mathfrak k_1,\\mathfrak k_2,\\phi,\\theta)$ where $\\mathfrak k_1$ is a Lie algebra, $\\mathfrak k_2$ is a\n vector space, $\\phi$ is  a representation of $\\mathfrak k_1$ on $\\mathfrak k_2$ and\n $\\theta$ is a 3-cocycle on $\\mathfrak k_1$ with values in $\\mathfrak k_2$.\\end{thm}\n\n\n\nWe briefly recall that given a 2-term skeletal $L_\\infty$-algebra\n $L_1\\stackrel{\\mathrm{d}}{\\longrightarrow}L_0$,\n $\\mathfrak k_1$ is $L_0$, $\\mathfrak k_2$ is $L_1$, the representation $\\phi$ comes from $l_2$\n and the 3-cocycle $\\theta$ is obtained from $l_3$.\n\n Please see \\cite{abad-crainic:rep-homotopy, abad} for\nthe general theory of  representation up to homotopy of Lie\nalgebroids. In this paper we only consider the 2-term representation\nup to homotopy of Lie algebras.\n\\begin{defi}{\\rm\\cite{abad-crainic:rep-homotopy}}\nA 2-term representation up to homotopy of a Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$\nconsists of\n\\begin{itemize}\n\\item[\\rm 1.] A 2-term complex of vector spaces\n$V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0$.\n\n\\item[\\rm 2.] Two linear maps $\\mu_i: \\g \\to End(V_i)$, which are compatible with $\\mathrm{d}$, i.e. for\nany $X\\in\\mathfrak g}\\newcommand{\\g}{\\mathfrak g,~\\xi\\in V_1$, we have\n\\begin{equation}\\label{eqn:d mu}\n\\mathrm{d}\\circ\\mu_1(X)(\\xi)=\\mu_0(X)\\circ\\mathrm{d}(\\xi).\n\\end{equation}\n\n\\item[\\rm 3.] A linear map $\\nu:\\wedge^2\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\to\\mathrm{Hom}(V_0,V_1)$ such that\n\\begin{eqnarray}\n\\label{eqn:mu0}\\mu_0[X_1,X_2]-[\\mu_0(X_1),\\mu_0(X_2)]&=&\\mathrm{d}\\circ\n\\nu(X_1,X_2),\\\\\n\\label{eqn:mu1}\\mu_1[X_1,X_2]-[\\mu_1(X_1),\\mu_1(X_2)]&=&\n\\nu(X_1,X_2)\\circ\\mathrm{d},\n\\end{eqnarray}\nas well as\n\\begin{equation}\\label{eqn:d k}\n[\\mu_0(X_1) + \\mu_1(X_1),\\nu(X_2,X_3)]+c.p.= \\nu([X_1,X_2],X_3)+c.p.,\n\\end{equation}\nwhere $c.p.$ stands for  cyclic permutation.\n\\end{itemize}\nWe usually write $\\mu=\\mu_0+\\mu_1$ and denote a 2-term\nrepresentation up to homotopy of a Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ by\n$(V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0,\\mu,\\nu)$.\n\\end{defi}\n\n\nIn \\cite[Example 3.25]{abad-crainic:rep-homotopy}, the authors\nproved that associated to any representation up to homotopy\n$V_\\bullet$ of a Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$, one can form a new $L_\\infty\n$-algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\ltimes V_\\bullet$, which is their semidirect\nproduct. Here we make this construction explicit in the 2-term case.\n\nLet $(V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0,\\mu,\\nu)$ be a 2-term\nrepresentation up to homotopy of $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$, then we can form a new\n2-term complex\n$$\n(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\ltimes\nV_\\bullet,\\mathrm{d}):V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus V_0).\n$$\nDefine  $l_2:\\wedge^2(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\ltimes\nV_\\bullet)\\longrightarrow\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\ltimes V_\\bullet $ by setting\n\\begin{equation}\\label{2bracket}\n\\left\\{\\begin{array}{l}l_2(X+\\xi,Y+\\eta)=[X,Y]+\\mu_0(X)(\\eta)-\\mu_0(Y)(\\xi),\\\\\nl_2(X+\\xi,f)=\\mu_1(X)(f),\\\\\nl_2(f,g)=0,\\end{array}\\right.\n\\end{equation}\nfor any $X+\\xi,Y+\\eta\\in\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus V_0$ and $f,~g\\in V_1$.  One\nshould note that $l_2$ is not a Lie bracket, but we have\n$$\nl_2(l_2(X+\\xi,Y+\\eta),Z+\\gamma)+c.p.=\\mathrm{d}(\\nu(X,Y)(\\gamma))+c.p..\n$$\nDefine $l_3:\\wedge^3(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\ltimes\nV_\\bullet)\\longrightarrow\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\ltimes V_\\bullet $ by setting:\n\\begin{equation}\\label{3bracket}\nl_3(X+\\xi,Y+\\eta,Z+\\gamma)=-\\nu(X,Y)(\\gamma)+c.p.,\n\\end{equation} then we have\n\\begin{pro}\\label{pro:Lie 2}\nWith the above notation, if\n$(V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0,\\mu,\\nu)$ is a 2-term\nrepresentation up to homotopy of a Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$, then\n$(V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus V_0),l_2,l_3)$ is a\n2-term $L_\\infty$-algebra.\n\\end{pro}\n\n\n\n\\subsection{Example I: Courant algebroids $TM\\oplus T^*M$}\n\nCourant algebroids were first introduced in \\cite{lwx} to study the\ndouble of Lie bialgebroids. It is a vector bundle $E\\longrightarrow\nM$ equipped with a nondegenerate symmetric bilinear form\n$\\pairing{\\cdot,\\cdot}$ on the bundle, an antisymmetric bracket\n$\\Courant{\\cdot,\\cdot}$ on the section space $\\Gamma(E)$ and a\nbundle map $\\rho:E\\longrightarrow TM$ such that a set of axioms are\nsatisfied. It can be viewed \\cite{royt} as a Lie 2-algebroid with a\n``degree 2 symplectic form''. The first example  is the standard\nCourant algebroid $(\\mathcal{T}=TM\\oplus\nT^*M,\\pairing{\\cdot,\\cdot},\\Courant{\\cdot,\\cdot},\\rho)$ associated\nto a manifold $M$, where $\\rho:\\mathcal{T}\\longrightarrow TM$ is the\nprojection, the canonical pairing $\\pairing{\\cdot,\\cdot}$ is given\nby\n\\begin{equation}\\label{eqn:pair}\n\\pairing{X+\\xi,Y+\\eta}=\\frac{1}{2}\\big(\\xi(Y)+\\eta(X)\\big),\\quad\\forall\n~X,Y\\in\\mathfrak X(M),~\\xi,\\eta\\in\\Omega^1(M),\n\\end{equation}\nthe antisymmetric bracket $\\Courant{\\cdot,\\cdot}$ is given by\n\\begin{equation}\\label{eqn:bracket}\n\\Courant{X+\\xi,Y+\\eta}\\triangleq[X,Y]+L_X\\eta-L_Y\\xi+\\frac{1}{2}d(\\xi(Y)-\\eta(X)),\\quad\\forall~X+\\xi,~Y+\\eta\\in\\Gamma(\\mathcal{T}).\n\\end{equation}\nIt is not a Lie bracket, but we have\n\\begin{equation}\\label{eqn:jacobi}\n\\Courant{\\Courant{e_1,e_2},e_3}+c.p.=d T(e_1,e_2,e_3),\\quad\n\\forall~e_1,e_2,e_3\\in\\Gamma(\\mathcal{T}),\n\\end{equation}\nwhere $T(e_1,e_2,e_3) $ is given by\n\\begin{equation}\\label{T}\nT(e_1,e_2,e_3)=\\frac{1}{3}(\\pairing{\\Courant{e_1,e_2},e_3}+c.p.).\n\\end{equation}\n\n\nNow we realize the section space of $\\mathcal{T}$ as the semidirect\nproduct of the Lie algebra $\\mathfrak X(M)$ of vector fields with the\nnatural 2-term deRham complex\n\\begin{equation}\\label{eq:natural-cx}\nC^\\infty(M)\\stackrel{d}{\\longrightarrow}\\Omega^1(M).\n\\end{equation}\nFor this we need to define a representation up to homotopy of $\\mathfrak X(M)$\non this complex. For any $X\\in\\mathfrak X(M)$, define linear actions $\\mu_0$ and $\\mu_1$ by\n\\begin{eqnarray}\n\\label{C 1}\\mu_0(X)(\\xi)&\\triangleq &\\Courant{X,\\xi}=L_X\\xi-\\frac{1}{2} d(\\xi(X)),\\quad\\forall ~\\xi\\in\\Omega^1(M)\\\\\n\\label{C 2}\\mu_1(X)(f)&\\triangleq &\\pairing{X,d\nf}=\\frac{1}{2}X(f),\\quad\\forall~f\\in C^\\infty(M).\n\\end{eqnarray}\nDefine $\\nu:\\wedge^2\\mathfrak X(M)\\to\\mathrm{Hom}(\\Omega^1(M),C^\\infty(M))$ by\n\\begin{equation}\\label{C 3}\n\\nu(X,Y)(\\xi)=T(X,Y,\\xi),\\quad\\forall\n~X,Y\\in\\mathfrak X(M),~\\xi\\in\\Omega^1(M). \\end{equation}\n\\begin{pro}\\label{pro:rep up to homotopy}\nWith the above notations,\n$(C^\\infty(M)\\stackrel{d}{\\longrightarrow}\\Omega^1(M),\\mu=\\mu_0+\\mu_1,\\nu)$\n is a representation up to homotopy of the Lie algebra $\\mathfrak X(M)$.\n\\end{pro}\n\\noindent{\\bf Proof.}\\  For any $f\\in C^\\infty(M)$, we have\n$$\n\\mu_0(X)(d f)=L_X df-\\frac{1}{2}d X( f)=\\frac{1}{2}d X( f),\n$$\nwhich implies $\\mu_0\\circ d=d\\circ\\mu_1$, i.e. $\\mu_i$'s are compatible\nwith the differential $d$. By straightforward computations, we have\n\\begin{eqnarray*}\n\\mu_0[X,Y](\\xi)-[\\mu_0(X),\\mu_0(Y)](\\xi)&=&\\Courant{\\Courant{X,Y},\\xi}+c.p.\\\\\n&=&d T(X,Y,\\xi) =d \\big(\\nu(X,Y)(\\xi)\\big),\\\\\n\\mu_1[X,Y](f)-[\\mu_1(X),\\mu_1(Y)](f)&=&\\frac{1}{2}[X,Y](f)-\\frac{1}{4}\\big(X(Y(f))-Y(X(f))\\big)\\\\\n&=&\\frac{1}{4}[X,Y](f),\\\\\n\\nu(X,Y)(d f)&=&T(X,Y,d\nf)=\\frac{1}{3}\\big(\\frac{1}{2}[X,Y](f)+\\frac{1}{4}\\big(X(Y(f))-Y(X(f))\\big)\\\\\n&=&\\frac{1}{4}[X,Y](f),\n\\end{eqnarray*}\nwhich implies (\\ref{eqn:mu0}) and (\\ref{eqn:mu1}). At last we need\nto prove (\\ref{eqn:d k}), which is obviously equivalent to\n\\begin{equation}\\label{c temp}\n\\mu_1(X)T(Y,Z,\\xi)-T(Y,Z,\\mu_0(X)(\\xi))+c.p.(X,Y,Z)=T([X,Y],Z,\\xi)+c.p.(X,Y,Z).\n\\end{equation}\nObserve that since $\\mu_0(X)(\\xi)=\\Courant{X,\\xi}$, we have\n$$T(Y,Z,\\mu_0(X)(\\xi))+c.p.(X,Y,Z)+T([X,Y],Z,\\xi)+c.p.(X,Y,Z)=T([X,Y],Z,\\xi)+c.p.(X,Y,Z,\\xi).$$\nFurthermore, since for any $f\\in C^{\\infty}(M)$, $\\mu_1(X)(f)=\\pairing{X,df}$\nand the cotangent bundle $T^*M$ is isotropic under the pairing\n$(\\ref{eqn:pair})$, we have\n$$\n\\mu_1(X)T(Y,Z,\\xi)+c.p.(X,Y,Z)=\\pairing{X,dT(Y,Z,\\xi)}+c.p.(X,Y,Z,\\xi).\n$$\n\nThus, (\\ref{c temp}) is equivalent to\n$$\n\\pairing{X,d T(Y,Z,\\xi)}+c.p.(X, Y, Z, \\xi)=T([X,Y],Z,\\xi)+c.p.(X, Y,\nZ, \\xi),\n$$\nwhich holds by Lemma 4.5 in \\cite{rw}. \\hfill ~\\vrule height6pt width6pt depth0pt\\vspace{3mm}\n\nBy Proposition \\ref{pro:Lie 2}, we have\n\n\\begin{cor}\\label{cor:courant}\n$\\big(C^\\infty(M) \\xrightarrow{\\mathrm{d}=0\\oplus d }(\\mathfrak X(M)\\oplus\\Omega^1(M)),l_2,l_3\\big)$\nis a 2-term $L_\\infty$-algebra, where $l_2$ and $l_3$ are given by\n(\\ref{2bracket}) and (\\ref{3bracket}), in which $\\mu_0$, $\\mu_1$ and\n$\\nu$ are given by (\\ref{C 1}),~(\\ref{C 2}) and (\\ref{C 3})\nrespectively.\n\\end{cor}\n\\begin{rmk}\\label{rmk:L infty}\n\n\nIn \\cite{rw}  the authors proved  that  the sections of  a Courant\nalgebroid $({\\mathcal{C}}}%{\\mathcal{C},\\pairing{\\cdot,\\cdot},\\Courant{\\cdot,\\cdot},\\rho)$\nform an $L_\\infty$-algebra. In the case when ${\\mathcal{C}}}%{\\mathcal{C}=\\mathcal{T}$ the\nstandard Courant algebroid, the 2-term $L_\\infty$-algebra is given\nby\n$$\nC^{\\infty}(M) \\xrightarrow{\\mathrm{d}=0\\oplus d } \\mathfrak X(M) \\oplus\n\\Omega^1(M)=\\Gamma(\\mathcal{T}),\n$$\nwith brackets  given by\n$$\nl_2(e_1,e_2)=\\Courant{e_1,e_2},\\quad l_2(e_1,f)=\\pairing{e_1,\\mathrm{d}\nf},\\quad l_3(e_1,e_2,e_3)=-T(e_1,e_2,e_3), \\quad l_{i\\ge 4} =0,\n$$\nfor any $e_1,e_2,e_3\\in\\Gamma(\\mathcal{T}),~f\\in C^{\\infty}(M)$. Here $T$ is\ndefined by (\\ref{T}). It is easy to verify that this is the same as\nour 2-term $L_\\infty$-algebra in Corollary \\ref{cor:courant}.\n\\end{rmk}\n\n\n\n\nWe can also modify our complex \\eqref{eq:natural-cx} to\n$$\n\\Omega^1(M)\\stackrel{\\rm{Id}}{\\longrightarrow}\\Omega^1(M).\n$$\nFollowing the same procedure, we obtain another representation up to\nhomotopy of the Lie algebra $\\mathfrak X(M)$ and therefore obtain another\n2-term $L_\\infty$-algebra which is also totally determined by the\nCourant algebroid\n$(\\mathcal{T},\\pairing{\\cdot,\\cdot},\\Courant{\\cdot,\\cdot},\\rho)$. More\nprecisely, for any $X\\in \\mathfrak X(M),~\\xi\\in\\Omega^1(M)$, $\\mu_0=\\mu_1$\nis given by\n$$\n\\mu_0(X)(\\xi)\\triangleq\\Courant{X,\\xi},\n$$\nand\n$\\nu:\\mathfrak X(M):\\longrightarrow\\Omega^2(\\mathfrak X^2(M),\\mathrm{End}(\\Omega^1(M),\\Omega^1(M)))$\nis given by\n$$\n\\nu(X,Y)(\\xi)\\triangleq d T(X,Y,\\xi).\n$$\n\\begin{pro}\nWith the above notations,\n$(\\Omega^1(M)\\stackrel{\\rm{Id}}{\\longrightarrow}\\Omega^1(M),\\mu=\\mu_0=\\mu_1,\\nu)$\nis a representation up to homotopy of the Lie algebra $\\mathfrak X(M)$.\n\\end{pro}\n\n\\subsection{Example \\textup{II}: Courant algebroids over a point and\n  string Lie 2-algebras} \\label{sec:string}\n\nA Courant algebroid over a point is literally  a quadratic Lie\nalgebra, namely a Lie algebra $\\mathfrak k$ together with nondegenerate\ninner product $\\pairing{\\cdot,\\cdot}$ which is invariant under the\nadjoint action.  People often think that a Courant algebroid over a\npoint  is a string Lie 2-algebra \\footnote{private conversation with\n  John Baez and Urs Schreiber}. Here we justify this thinking.\n\\begin{defi}\\label{defi:string-alg}\nThe  string Lie 2-algebra associated to a quadratic Lie algebra\n$(\\mathfrak k, \\pairing{\\cdot,\\cdot})$, is a  2-term $L_\\infty$-algebra\n$\\mathbb R\\stackrel{0}{\\longrightarrow}\\mathfrak k$, whose degree-$0$\npart is $\\mathfrak k$, degree-$1$ part is $\\mathbb R$,  and $l_2,~l_3$ are\ngiven by\n\\begin{eqnarray*}\n~l_2(e,c)&=&0,\\\\\n~l_2((e_1,c_1),(e_2,c_2))&=&([e_1,e_2],0),\\\\\n~l_3((e_1,c_1),(e_2,c_2),(e_3,c_3))&=&(0,\\pairing{[e_1,e_2],e_3}),\n\\end{eqnarray*}\nwhere $e,~e_1,~e_2,~e_3\\in\\mathfrak k,~c,~c_1,~c_2,~c_3\\in\\mathbb R$.\n\\end{defi}\n\nThe representation $\\phi $ of $\\mathfrak k$ on $\\mathbb R$ and the\n3-cocycle $\\theta:\\wedge^3\\mathfrak k\\longrightarrow \\mathbb R$ in the\ncorresponding quadruple in Theorem \\ref{thm:skeletal Lie 2 a} is\ngiven by\n\\begin{eqnarray}\n\\nonumber\\rho(e)(c)&=&l_2(e,c)\\\\\n\\label{3cocycle}\\theta(e_1,e_2,e_3)&=&\\pairing{[e_1,e_2],e_3}.\n\\end{eqnarray}\n\n\\begin{rmk}\\label{rk:semisimple}\nIn the definition of string Lie 2-algebras \\cite{baez:string,\n  henriques} the base Lie algebra $\\mathfrak k$ is usually required\nto be semisimple and of compact type, such that the Jacobiator gives\nrise to the generator of $H^3(\\mathfrak k, \\mathbb Z)=\\mathbb Z$. This is\nbecause  Witten's original motivation is to obtain a $3$-connected\ncover of $Spin(n)$, and $\\mathfrak{so}(n)$ is simple and of compact\ntype. However, to write down the structure of the string Lie\n2-algebra, we only need a quadratic Lie algebra. Then $H^3(\\mathfrak k,\n\\mathbb Z)$ is not necessarily $\\mathbb Z$ for a general quadratic\nLie algebra $\\mathfrak k$. For example, for the abelian Lie algebra $\\mathbb R}\\newcommand{\\Z}{\\mathbb Z$,\nany inner product is adjoint-invariant, and $H^3(\\mathbb R}\\newcommand{\\Z}{\\mathbb Z, \\Z)=0$.  We\nthus face the danger that sometimes, the string Lie 2-algebra is\ntrivial, that is the Jacobiator corresponds to the trivial element\nin $H^3(\\mathfrak k, \\mathbb Z)$. Then what we have is a strict Lie\n2-algebra, which is a crossed module of Lie algebras. Then the\nintegration of a crossed module of Lie algebras is simply a crossed\nmodule of Lie groups.  However, we will verify that the example we\nconsider is not such a case.\n\\end{rmk}\n\n\n\nThe standard Courant algebroid\nmotivates us to consider the case of  the direct sum\n$\\mathfrak k=\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$ of  a Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$\nand its dual with  the\nsemidirect product  Lie algebra structure:\n$$\n[X+\\xi,Y+\\eta]=[X,Y]_\\g+\\mathrm{ad}_X^*\\eta-\\mathrm{ad}_Y^*\\xi,\n$$here $[\\cdot,\\cdot]_\\g$ is the Lie bracket of $\\g$. The nondegenerate invariant pairing\n$\\pairing{\\cdot,\\cdot}$ on $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$  is given by\n$$\n\\pairing{X+\\xi,Y+\\eta}=\\frac{1}{2}(\\eta(X)+\\xi(Y)),\\quad\\forall~X+\\xi,Y+\\eta\\in\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*.\n$$ With these definitions,\n$(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,[\\cdot,\\cdot],\\pairing{\\cdot,\\cdot})$ is a\nquadratic Lie algebra. In fact, we have\n\\begin{eqnarray*}\n\\pairing{[X_1+\\xi_1,X_2+\\xi_2],X_3+\\xi_3}&=&\\pairing{[X_1,X_2]_\\g+\\mathrm{ad}_X^*\\xi_2-\\mathrm{ad}_Y^*\\xi_1,X_3+\\xi_3}\\\\\n&=&\\pairing{[X_1,X_2]_\\g,\\xi_3}+c.p..\n\\end{eqnarray*}\nSimilarly, we have\n$$\n\\pairing{X_2+\\xi_2,[X_1+\\xi_1,X_3+\\xi_3]}=\\pairing{[X_1,X_3]_\\g,\\xi_2}+c.p.,\n$$\nwhich implies that\n$\\pairing{X_2+\\xi_2,[X_1+\\xi_1,X_3+\\xi_3]}+\\pairing{[X_1+\\xi_1,X_2+\\xi_2],X_3+\\xi_3}=0$,\ni.e. the nondegenerate inner product $\\pairing{\\cdot,\\cdot}$ is\ninvariant under the adjoint action. This example is a special case\nof \\cite[Theorem 1.12]{lu-weinstein} with $\\g^*$ equipped with $0$\nLie bracket. Thus $(\\g, \\g^*)$ forms a Lie bialgebra or equivalently\n$(\\g\\oplus \\g^*, \\g, \\g^*)$ is a Manin triple. However, honestly we\nhave not found other Lie bialgebras (Manin triples) giving rise to\nLie 2-algebras of the form of semidirect products.\n\nWe  denote the corresponding string Lie 2-algebra of $(\\g \\oplus \\g^*,\n[\\cdot, \\cdot], \\pairing{\\cdot,\\cdot})$ by\n\\begin{equation}\\label{Lie 2 g g dual}\n\\mathbb R\\stackrel{0}{\\longrightarrow}\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*.\n\\end{equation}\nThe corresponding 3-cocycle (see \\eqref{3cocycle}), which we denote\nby $\\widetilde{\\nu}:\\wedge^3(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*)\\longrightarrow\n\\mathbb R$ is\n\\begin{eqnarray}\\label{eqn:nu3}\n\\widetilde{\\nu}(X_1+\\xi_1,X_2+\\xi_2,X_3+\\xi_3)&=&\\pairing{[X_1+\\xi_1,X_2+\\xi_2],X_3+\\xi_3}\\\\\n\\nonumber&=&\\pairing{[X_1,X_2]_\\g,\\xi_3}+c.p\n\\end{eqnarray}\n\n\\begin{pro}  $(\\mathbb\nR\\stackrel{0}{\\longrightarrow} \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,\\mu_1=0,\n\\mu_0=\\mathrm{ad}^*,\\nu=[\\cdot,\\cdot]_\\g)$ is a 2-term representation up to\nhomotopy of the Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$. Moreover the string Lie\n2-algebra $ \\mathbb R\\stackrel{0}{\\longrightarrow}\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\n\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^* $ is the semidirect product of $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g $ and the complex\n$\\mathbb R\\stackrel{0}{\\longrightarrow} \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$.\n\\end{pro}\n\\begin{proof}\nSince $\\mathrm{d}=0$, we only need to verify that $\\mu_0$ and $\\mu_1$ are\nLie algebra morphisms and \\eqref{eqn:d k}. $\\mathrm{ad}^*: \\g \\to\n\\mathrm{End}(\\g^*)$ and $0$ are both Lie algebra morphisms. \\eqref{eqn:d k}\nfollows from Jacobi identity of $[\\cdot,\\cdot]_\\g$.\n\nThen  it is not hard to see that the string Lie 2-algebra $\\mathbb\nR\\stackrel{0}{\\longrightarrow}\\g\\oplus \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$ with formulas in\nDefinition \\ref{defi:string-alg} is exactly the semidirect product\nof $\\g$ with the complex  $(\\mathbb R\\stackrel{0}{\\longrightarrow}\n\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,\\mu_1=0, \\mu_0=\\mathrm{ad}^*,\\nu=[\\cdot,\\cdot]_\\g)$ with the\nformulas \\eqref{2bracket}, \\eqref{3bracket}.\n\\end{proof}\n\n\n\\begin{pro}\\label{pro:nondegenerate}\nIf the Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ is semisimple, the Lie algebra 3-cocycle\n$\\widetilde{\\nu}$ which is given by (\\ref{eqn:nu3}) is not exact.\n\\end{pro}\n\\noindent{\\bf Proof.}\\  Let $\\pairing{\\cdot,\\cdot}_k$ be the Killing form on $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$.\nThe proof follows from the fact that the Cartan 3-form\n$\\pairing{[\\cdot,\\cdot],\\cdot}_k$   on a semisimple Lie algebra is\nnot exact.\nSince $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ is semisimple, the Killing form\n$\\pairing{\\cdot,\\cdot}_k$ is nondegenerate. Identify $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$ and\n$\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ by using the Killing form $\\pairing{\\cdot,\\cdot}_k$ and let\n$\\mathcal{K}$ be the corresponding isomorphism,\n$$\n\\pairing{\\mathcal{K}(\\xi),X}_k=\\pairing{\\xi,X}.\n$$\nAssume that $\\widetilde{\\nu}=d\\phi$ for some\n$\\phi:\\wedge^2(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*)\\longrightarrow \\mathbb R$, define\n$\\varphi:\\wedge^2(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\\mathfrak g}\\newcommand{\\g}{\\mathfrak g)\\longrightarrow \\mathbb R$ by\n$$\n\\phi(X+\\xi,Y+\\eta)=\\varphi(X+\\mathcal{K}(\\xi),Y+\\mathcal{K}(\\eta)),\n$$then we have\n\\begin{eqnarray*}\n\\widetilde{\\nu}(X,Y,\\xi)&=&d\\phi(X,Y,\\xi)\\\\\n&=&-\\phi([X,Y],\\xi)+\\phi(\\mathrm{ad}^*_X\\xi, Y)-\\phi(\\mathrm{ad}^*_Y\\xi, X)\\\\\n&=&-\\varphi([X,Y],\\mathcal{K}(\\xi))+\\varphi([X,\\mathcal{K}(\\xi)],\nY)-\\varphi([Y,\\mathcal{K}(\\xi)], X)\\\\\n&=&d\\varphi(X,Y,\\mathcal{K}(\\xi)).\n\\end{eqnarray*}\nOn the other hand, we have\n$$\n\\widetilde{\\nu}(X,Y,\\xi)=\\pairing{[X,Y],\\xi}=\\pairing{[X,Y],\\mathcal{K}(\\xi)}_k,\n$$\nwhich implies that the Cartan 3-from\n$$\n\\pairing{[X,Y],\\mathcal{K}(\\xi)}_k=d\\varphi(X,Y,\\mathcal{K}(\\xi)),\n$$ is exact.\nThis is a contradiction. \\hfill ~\\vrule height6pt width6pt depth0pt\\vspace{3mm}\n\n\nIn the last section, we will give  the integration of the string Lie\n2-algebra $ \\mathbb R\\stackrel{0}{\\longrightarrow}\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\n\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^* $ by using the semidirect product of a Lie group with its\n2-term representation up to homotopy. It turns out that this string\nLie 2-algebra can be integrated to a special Lie 2-group with a\nfinite dimensional model.\n\n\n\n\\subsection{Example III: Omni-Lie algebroids $\\mathfrak{D} E\\oplus \\mathfrak{J} E$}\n\nThe notion of  omni-Lie algebroids, which was introduced by Chen and\nLiu in \\cite{clomni}, is a generalization of Weinstein's omni-Lie\nalgebras. Just as Dirac structures of an omni-Lie algebra\ncharacterize Lie algebra structures on a vector space, Dirac\nstructures of an omni-Lie algebroid  characterize Lie algebroid\nstructures on a vector bundle. See \\cite{clsdirac} for more details.\nIn fact, the role of omni-Lie algebroids $\\mathfrak{D} E\\oplus \\mathfrak{J} E$ in\n$E$-Courant algebroids, which was introduced in \\cite{clsecourant},\nis the same as the role of standard Courant algebroids $TM\\oplus\nT^*M$ in Courant algebroids.\n\n\nNow we briefly recall the notion of omni-Lie algebroids and we will\nsee that it gives rise to a 2-term $L_\\infty$-algebra which is a\nsemidirect product. In this subsection $E$ is a vector bundle over a\nsmooth manifold $M$, $\\Gamma(E)$ is the section space of $E$.\n\nFor a vector bundle $E$, let $\\mathfrak{D} E$ be its covariant differential\noperator bundle. The associated Atiyah sequence is given by\n\\begin{equation}\\label{Seq:DE}\n\\xymatrix@C=0.5cm{0 \\ar[r] & \\mathfrak {gl}(E)  \\ar[rr]^{\\mathbbm{i}} &&\n                \\mathfrak{D}{E}  \\ar[rr]^{a} && TM \\ar[r]  & 0.\n                }\n\\end{equation}\nThe associated $1$-jet vector bundle ${\\mathfrak{J}} E$ is defined as\nfollows. For any $m\\in M$, $({\\mathfrak{J}}{E})_m$ is defined as a quotient\nof local sections of $E$. Two local sections $u_1$ and $u_2$ are\nequivalent and we denote this by $u_1\\sim u_2$ if\n$$ u_1(m)=u_2(m) ~~\\mbox{ and }~~ d\\pairing{u_{1\n},\\xi}_m=d\\pairing{u_{2 },\\xi}_m, \\quad\\forall~ \\xi\\in\\Gamma(E^*).$$\nSo any $\\mu\\in ({\\mathfrak{J}}{E})_m$ has a representative $u\\in\\Gamma(E)$\nsuch that\n  $\\mu=[u]_m$.\nLet $\\mathbbm{p}$ be the projection which sends $[u]_m$ to $u(m)$.  Then\n$\\mathrm{Ker}\\mathbbm{p} \\cong  \\mathrm{Hom}(TM,E)$ and there is a short exact sequence,\ncalled the jet sequence of $E$,\n\\begin{equation}\\label{Seq:JetE}\n\\xymatrix@C=0.5cm{0 \\ar[r] & \\mathrm{Hom}(TM,E)  \\ar[rr]^{~\\qquad \\mathbbm{e}~} &&\n                {\\mathfrak{J}}{E} \\ar[rr]^{\\mathbbm{p}} && E \\ar[r]  & 0,\n                }\n\\end{equation}\nfrom which it is straightforward to see that $\\mathfrak{J} E$ is a finite\ndimensional vector bundle. Moreover, $\\Gamma(\\mathfrak{J}{E})$ is isomorphic\nto $\\Gamma(E) \\oplus \\Gamma (T^*M \\otimes E)$ as an $\\mathbb R$-vector\nspace,\n   and any\n$u\\in \\Gamma (E)$ has a lift $\\mathbbm{d} u\\in\\Gamma ({\\mathfrak{J}} E)$ by taking\nits equivalence class, such that\n\\begin{equation}\\label{Eqt:CWMmoduleGammaE}\n\\mathbbm{d}(fu)=f \\mathbbm{d} u+ d f\\otimes u, \\quad \\forall~ f\\in C^{\\infty}(M).\n\\end{equation}\nIn \\cite{clomni} the authors proved that\n\\begin{eqnarray*}  {\\mathfrak{J} E} \\cong\n\\set{\\nu\\in \\mathrm{Hom}(\\mathfrak{D}{E},E)\\,|\\,\n\\nu(\\Phi)=\\Phi\\circ\\nu(\\rm{Id}_{E}),\\quad\\forall~ ~ ~\\Phi\\in \\mathfrak {gl}(E)}.\n\\end{eqnarray*}\n\nTherefore, there is   an $E$-pairing between $\\mathfrak{J}{E}$ and $\\mathfrak{D}{E}$\nby setting:\n\\begin{equation}\n\\label{Eqt:conpairingE} \\conpairing{\\mu,\\mathfrak d}_E~ \\triangleq\n\\mathfrak d(u),\\quad\\forall~ ~~ \\mu\\in (\\mathfrak{J}{E})_m,~\\mathfrak d\\in(\\mathfrak{D}{E})_m,\n\\end{equation}\nwhere $u\\in \\Gamma(E)$ satisfies $\\mu=[u]_m$. Particularly, one has\n\\begin{eqnarray}\\label{conpairing1}\n\\conpairing{\\mu,\\Phi}_E &=& \\Phi\\circ \\mathbbm{p}(\\mu),\\quad\\forall~ ~~ \\Phi\\in \\mathfrak {gl}(E),~\\mu\\in\\mathfrak{J}{E};\\\\\n\\label{conpairing2} \\conpairing{{\\mathfrak y},\\mathfrak d}_{E} &=& {\\mathfrak y}\\circ\na(\\mathfrak d),\\quad\\forall~ ~~ \\mathfrak y\\in \\mathrm{Hom}(TM,E),~\\mathfrak d\\in\\mathfrak{D}{E}.\n\\end{eqnarray}\n Furthermore, we claim that $\\Gamma (\\mathfrak{J} E)$ is an invariant\nsubspace of the Lie derivative $\\frkL_{\\mathfrak d}$ for any\n $\\mathfrak d \\in\\Gamma(\\mathfrak{D}{E})$, which is defined by the\n Leibniz rule as follows:\n\\begin{eqnarray}\\nonumbe\n\\conpairing{\\frkL_{\\mathfrak d}\\mu,\\mathfrak d^\\prime}_{E}&\\triangleq&\n\\mathfrak d\\conpairing{\\mu,\\mathfrak d^\\prime}_{E}-\\conpairing{\\mu,[\\mathfrak d,\\mathfrak d^\\prime]_{\\mathfrak{D}}}_{E},\n\\quad\\forall~ \\mu \\in \\Gamma(\\mathfrak{J}{E}), ~\n~\\mathfrak d^\\prime\\in\\Gamma(\\mathfrak{D}{E}).\n\\end{eqnarray}\n\nIntroduce a nondegenerate symmetric $E$-valued $2$-form\n$\\ppairingE{\\cdot,\\cdot}$ on $ \\mathcal{E}\\triangleq\\mathfrak{D}{E}\\oplus \\mathfrak{J}{E}$\nby:\n$$\n\\ppairingE{\\mathfrak d+\\mu,\\mathfrak r+\\nu}\\triangleq \\frac{1}{2}(\\conpairing{\\mathfrak d,\\nu}_E\n+\\conpairing{\\mathfrak r,\\mu}_E),\\quad\\forall~ ~~\n\\mathfrak d,\\mathfrak r\\in\\mathfrak{D}{E},~\\mu,\\nu\\in\\mathfrak{J}{E}.\n$$\nFurthermore, we define an antisymmetric bracket\n$\\Courant{\\cdot,\\cdot}$ on $\\Gamma(\\mathcal{E})$ by:\n\\begin{eqnarray*}\n\\Courant{\\mathfrak d+\\mu,\\mathfrak r+\\nu}&\\triangleq&\n[\\mathfrak d,\\mathfrak r]_{\\mathfrak{D}}+\\frkL_{\\mathfrak d}\\nu-\\frkL_{\\mathfrak r}\\mu +\n\\frac{1}{2}\\big(\\mathbbm{d}\\conpairing{\\mu,\\mathfrak r}_E-\\mathbbm{d}\\conpairing{\\nu,\\mathfrak d}_E\\big).\n\\end{eqnarray*}\n\nIn \\cite{clomni} the authors  call  the quadruple\n$(\\mathcal{E},\\Courant{\\cdot,\\cdot},\\ppairingE{\\cdot,\\cdot},\\rho_{\\varepsilon})$ the\n{\\bf omni-Lie algebroid} \\footnote{It is slightly different from the\nnotion given in \\cite{clomni}, where the bracket is not\nskewsymmetric.} associated to the vector bundle $E$,  where $\\rho$\nis the projection of $\\mathcal{E}$ onto $\\mathfrak{D}{E}$. Even though\n$\\Courant{\\cdot,\\cdot}$ is antisymmetric,  it is not a Lie bracket.\nMore precisely, for any $X,Y,Z\\in \\Gamma(\\mathcal{E})$, we have\n$$\\Courant{\\Courant{X,Y},Z}+c.p.=\\mathbbm{d} T(X,Y,Z),$$\nwhere $T:\\Gamma(\\wedge^3\\mathcal{E})\\longrightarrow \\Gamma(E)$ is defined\nby \\be\\label{T}\nT(X,Y,Z)=\\frac{1}{3}\\big(\\ppairingE{\\Courant{X,Y},Z}+c.p.\\big). \\ee\n\n\nNow let us construct a 2-term $L_\\infty$-algebra from the omni-Lie\nalgebroid $\\mathcal{E}$. Obviously, $\\Gamma(\\mathfrak{D} E)$ is a Lie algebra and\nthere is a natural 2-term complex\n$$\n\\Gamma(E)\\stackrel{\\mathrm{d}=0\\oplus\\mathbbm{d}}{\\longrightarrow}\\Gamma(\\mathfrak{J} E).\n$$\n\n\nFor any $\\mathfrak d\\in\\Gamma(\\mathfrak{D} E)$, define linear actions $\\mu_0$ and\n$\\mu_1$ by\n\\begin{equation}\\label{O 1}\\left\\{\\begin{array}{l}\n\\mu_0(\\mathfrak d)(\\mu)\\triangleq \\Courant{\\mathfrak d,\\mu}=\\frkL_\\mathfrak d\\mu-\\mathbbm{d}\\ppairingE{\\mu,\\mathfrak d},\\quad\\forall~\\mu\\in \\Gamma(\\mathfrak{J} E),\\\\\n\\mu_1(\\mathfrak d)(u)\\triangleq \\ppairingE{\\mathfrak d,\\mathbbm{d}\nu}=\\frac{1}{2}\\mathfrak d(u),\\quad\\forall~u\\in\\Gamma(E).\\end{array}\\right.\n\\end{equation}\nDefine $\\nu:\\wedge^2\\Gamma(\\mathfrak{D} E)\\longrightarrow\\mathrm{Hom}(\\Gamma(\\mathfrak{J}\nE),\\Gamma( E))$ by\n\\begin{equation}\\label{O 2}\n\\nu(\\mathfrak d,\\mathfrak t)(\\xi)=T(\\mathfrak d,\\mathfrak t,\\mu),\\quad\\forall\n~\\mathfrak d,\\mathfrak t\\in\\Gamma(\\mathfrak{D} E),~\\mu\\in\\Gamma(\\mathfrak{J} E).\n\\end{equation}\nSimilar to Proposition \\ref{pro:rep up to homotopy}, we prove\n\\begin{pro}\nWith the above notations,\n$(\\Gamma(E)\\stackrel{\\mathrm{d}=0\\oplus\\mathbbm{d}}{\\longrightarrow}\\Gamma(\\mathfrak{J}\nE),\\mu=\\mu_0+\\mu_1,\\nu)$\n is a representation up to homotopy of the Lie algebra $\\Gamma(\\mathfrak{D}\nE)$.\n\\end{pro}\n\\begin{cor}\n$\\big(\\Gamma(E)\\stackrel{\\mathrm{d}=0\\oplus\\mathbbm{d}}{\\longrightarrow}\\Gamma(\\mathfrak{D}\nE)\\oplus\\Gamma(\\mathfrak{J} E),l_2,l_3\\big)$ is a 2-term $L_\\infty$-algebra,\nwhere $l_2$ and $l_3$ are given by (\\ref{2bracket}) and\n(\\ref{3bracket}), in which $\\mu$ and $\\nu$ are given by (\\ref{O 1})\nand (\\ref{O 2}) respectively.\n\\end{cor}\n\n\n\\begin{rmk}\nIf the base manifold $M$ is a point, i.e. $E$ is a vector space, for\nwhich  we use a new notation $V$, then $\\mathfrak{D} V=\\mathfrak {gl}(V),~\\mathfrak{J} V=V$,\nand we recover the notion of omni-Lie algebras. The complex\n$\\Gamma(V)\\stackrel{0\\oplus\\mathbbm{d}}{\\longrightarrow}\\Gamma(\\mathfrak{J} V)$\nreduces to $V\\stackrel{\\rm{Id}}{\\longrightarrow}V$, which is a\nrepresentation up to homotopy of $\\mathfrak {gl}(V)$ with $(\\mu_0=\\mu_1, \\nu)$\ngiven by\n\\begin{equation}\\label{omni mu nu}\n\\mu_0(A)(u)=\\frac{1}{2} Au,\\quad\n\\nu(A,B)=\\frac{1}{4}[A,B],\\quad\\forall~A,B\\in\\mathfrak {gl}(V),~u\\in V.\n\\end{equation}\nHence even though an omni-Lie algebra   $\\mathfrak {gl}(V)\\oplus V$ is not a\nLie algebra,  we can extend it to a  2-term $L_\\infty$-algebra, of\nwhich $L_0=\\mathfrak {gl}(V)\\oplus V,~L_1=V$, $l_2$ and $l_3$ are given by\n(\\ref{2bracket}) and (\\ref{3bracket}), in which $\\mu$ and $\\nu$ are\ngiven by (\\ref{omni mu nu}). This 2-term $L_\\infty$-algebra is a\nsemidirect product of $\\mathfrak {gl}(V)$ with $V\\xrightarrow{\\rm{Id}} V$.\n\nWe will study the global object of the 2-term $L_\\infty$-algebra\nassociated to an omni-Lie algebra in the forthcoming paper \\cite{sheng-zhu:II}.\n\\end{rmk}\n\n\\section{Representation up to homotopy of Lie groups and semidirect\n  product} \\label{sec:gp}\n\nThe representation up to homotopy of a Lie group was introduced in\n\\cite{abad}. In this section we define the semidirect product of a\nLie group with a 2-term representation up to homotopy and prove that\nthe semidirect product is a Lie 2-group. Thus we first recall some\nbackground on Lie 2-groups.\n\nA group is a monoid where every element has an inverse. A 2-group is\na monoidal category where every object has a weak inverse and every\nmorphism has an inverse. Denote the category of smooth manifolds and\nsmooth maps by $\\rm Diff$, a semistrict\\footnote{Please see the\n  introduction for the reason why we call it a semistrict Lie 2-group.} Lie 2-group is  a 2-group\nin $\\rm DiffCat$, where  $\\rm DiffCat$ is the 2-category consisting\nof categories, functors, and natural transformations in $\\rm Diff$.\nFor more details, see \\cite{baez:2gp} and here  we only recall the\nexpanded definition:\n\\begin{defi}{\\rm\\cite{baez:2gp}}\nA semistrict Lie 2-group consists of an object $C$ in $\\rm\nDiffCat$, i.e. \\[ \\xymatrix{ C_1 \\ar@<1ex>[r]^{s} \\ar[r]_{t} & C_0},\n\\]\nwhere $C_1,~C_0$ are objects in $\\rm Diff$, $s,~t$ are the source\nand target maps, and there is a vertical multiplication\n$\\cdot_\\mathrm{v}:C\\times C\\longrightarrow C$, together with\n\\begin{itemize}\n\\item[$\\bullet$] a  functor (horizontal\nmultiplication) $\\cdot_\\mathrm{h}:C\\times C\\longrightarrow C$,\n\\item[$\\bullet$] an identity object $1$,\n\\item[$\\bullet$]a contravariant  functor  $\\mathrm{inv}:C\\longrightarrow C$\n\\end{itemize}\nand the following natural isomorphisms:\n\\begin{itemize}\n\\item[$\\bullet$] the {\\bf associator}\n$$\na_{x,y,z}:(x\\cdot_\\mathrm{h} y)\\cdot_\\mathrm{h} z\\longrightarrow x\\cdot_\\mathrm{h}\n(y\\cdot_\\mathrm{h} z),\n$$\n\\item[$\\bullet$]the {\\bf left} and {\\bf right unit}\n$$\nl_x:1\\cdot_\\mathrm{h} x\\longrightarrow x,\\quad r_x:x\\cdot_\\mathrm{h}\n1\\longrightarrow x,\n$$\n\\item[$\\bullet$]the  {\\bf  unit} and {\\bf  counit}\n$$\ni_x:1\\longrightarrow x\\cdot_\\mathrm{h} \\mathrm{inv}(x),\\quad e_x:\\mathrm{inv}(x)\\cdot_\\mathrm{h}\nx\\longrightarrow 1,\n$$\n\\end{itemize}\nsuch that the following diagrams commute:\n\\begin{itemize}\n\\item[$\\bullet$] the {\\bf pentagon identity } for the associator\n\\[\n \\xymatrix{ & (w\\cdot_\\mathrm{h} x)\\cdot_\\mathrm{h}(y\\cdot_\\mathrm{h} z)\\ar[dr]^{a_{w,x, y\\cdot_\\mathrm{h} z}}&  \\\\\n((w\\cdot_\\mathrm{h} x)\\cdot_\\mathrm{h} y)\\cdot_\\mathrm{h} z\\ar[ur]^{a_{(w\\cdot_\\mathrm{h}\nx),y,z}}\\ar[dr]_{a_{w,x,y}\\cdot_\\mathrm{h} 1_z}&&w\\cdot_\\mathrm{h} (x\\cdot_\\mathrm{h}\n(y\\cdot_\\mathrm{h} z))\\\\\n&(w\\cdot_\\mathrm{h} (x\\cdot_\\mathrm{h} y))\\cdot_\\mathrm{h} z\\stackrel{a_{w,x\\cdot_\\mathrm{h}\ny,z}}{\\longrightarrow} w\\cdot_\\mathrm{h} ((x\\cdot_\\mathrm{h} y)\\cdot_\\mathrm{h}\nz)\\ar[ur]^{1_w\\cdot_\\mathrm{h} a_{x,y,z}}&}\n\\]\n\\item[$\\bullet$] the {\\bf triangle identity} for the left and right\nunit lows:\n\\[\n \\xymatrix{\n( x\\cdot_\\mathrm{h} 1)\\cdot_\\mathrm{h} y\\ar[rr]^{a_{x,1,y}}\\ar[dr]^{r_x\\cdot_\\mathrm{h} 1_y}&&x\\cdot_\\mathrm{h}(1\\cdot_\\mathrm{h} y)\\ar[dl]^{1_x\\cdot_\\mathrm{h} l_y}\\\\\n&x\\cdot_\\mathrm{h} y& }\n\\]\n\\item[$\\bullet$]the {\\bf first zig-zag identity}:\n\\[\n \\xymatrix{\n&(x\\cdot_\\mathrm{h} \\mathrm{inv}(x))\\cdot_\\mathrm{h} x\\stackrel{a_{x,\\mathrm{inv}(x),x}}{\\longrightarrow}x\\cdot_\\mathrm{h}(\\mathrm{inv}(x)\\cdot_\\mathrm{h} x)\\ar[dr]^{1_x\\cdot_\\mathrm{h} e_x}&\\\\\n1\\cdot_\\mathrm{h} x\\ar[dr]^{l_x}\\ar[ur]^{i_x\\cdot_\\mathrm{h} 1_x}&&x\\cdot_\\mathrm{h} 1\\\\\n&x\\ar[ur]^{r_x^{-1}}&}\n\\]\n\\item[$\\bullet$]the {\\bf second zig-zag identity}:\n\\[\n \\xymatrix{\n&\\mathrm{inv}(x)\\cdot_\\mathrm{h} (x\\cdot_\\mathrm{h} \\mathrm{inv}(x))\\stackrel{a_{\\mathrm{inv}(x),x,\\mathrm{inv}(x)}}{\\longrightarrow}(\\mathrm{inv}(x)\\cdot_\\mathrm{h} x)\\cdot_\\mathrm{h}\\mathrm{inv}(x)\\ar[dr]^{e_x\\cdot_\\mathrm{h} 1_{\\mathrm{inv}(x)}}&\\\\\n\\mathrm{inv}(x)\\cdot_\\mathrm{h} 1\\ar[dr]^{r_{\\mathrm{inv}(x)}}\\ar[ur]^{1_{\\mathrm{inv}(x)}\\cdot_\\mathrm{h} i_x}&&1\\cdot_\\mathrm{h} \\mathrm{inv}(x).\\\\\n&\\mathrm{inv}(x)\\ar[ur]^{l_{\\mathrm{inv}(x)}^{-1}}&}\n\\]\n\\end{itemize}\n\\end{defi}\n\nIn the special case where $a_{x,y,z},~l_x,~r_x,~i_x,~e_x$ are all\nidentity isomorphisms, we call such a Lie 2-group a {\\em strict Lie\n  2-group}\\footnote{The notion of strict Lie 2-groups is the same as \\cite{baez:2gp}.}.\n\nIn the same reference, they also defined the notion of special\n2-groups, which we recall here,\n\\begin{defi}\\label{defi:special 2 G}\nA {\\bf special Lie 2-group}  is a Lie 2-group of which the source\nand target coincide and  the left unit law $l$, the right unit law\n$r$, the unit $i$ and the counit $e$ are identity isomorphisms.\n\\end{defi}\n\nFor classification of special Lie 2-groups,  we need the group cohomology with smooth cocycles, that is we\nconsider the cochain complex with smooth morphisms $G^{\\times\nn}\\to M$ with $G$ a Lie group, $M$ its module. And the differential is defined as usual for\ngroup cohomology. We denote this cohomology by $H_{sm}^\\bullet(G, M)$.\n\n\\begin{thm}{\\rm \\cite[Theorem 8.3.7]{baez:2gp}}\\label{thm:special 2 G}  There is a\none-to-one correspondence between special Lie 2-groups and\nquadruples $(K_1,K_2,\\Phi,\\Theta)$ consisting a Lie group $K_1$, an\nabelian group $K_2$, an action $\\Phi$ of $K_1$ as automorphisms of\n$K_2$ and a normalized smooth 3-cocycle $\\Theta:K_1^3\\longrightarrow\nK_2$. Two special Lie 2-groups are isomorphic if and only if they\ncorresponds to the same\\footnote{up to isomorphisms of groups of\n  course} $(K_1, K_2, \\Phi)$ and the\ncorresponding 3-cocycles represent the same element in $H^3_{sm}(K_1,\nK_2)$.\n\\end{thm}\n\\begin{rmk}\nWe briefly recall that given a quadruple $(K_1,K_2,\\Phi,\\Theta)$ the\ncorresponding semistrict Lie 2-group has the Lie group $K_1$ as the space\nof objects and the semidirect product Lie group $K_1 \\ltimes_{\\Phi}\nK_2$ as the space of morphisms. The associator is given by $\\Theta$.\n\\end{rmk}\n\\begin{defi}\nA unital 2-term representation up to homotopy of a Lie group $G$\nconsists of\n\\begin{itemize}\n\\item[\\rm 1.] A 2-term complex of vector spaces\n$V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0$.\n\n\\item[\\rm 2.] A nonassociative action $F_1$ on $V_0$ and $V_1$\n  satisfying $$ \\mathrm{d} F_1 = F_1 \\mathrm{d}, \\quad F_1(1_G)=\\rm{Id}.$$\n\n\\item[\\rm 3.] A smooth map $F_2:G\\times G\\longrightarrow\\mathrm{End}(V_0,V_1)$ such that\n\\begin{equation}\\label{eqn:F fail}\nF_1(g_1)\\cdot F_1(g_2)-F_1(g_1\\cdot g_2)=[\\mathrm{d},F_2(g_1,g_2)],\n\\end{equation}\nas well as\n\\begin{equation}\\label{eqn:F closed}\nF_1(g_1)\\circ F_2(g_2,g_3)-F_2(g_1\\cdot g_2,g_3)+F_2(g_1, g_2\\cdot\ng_3)-F_2(g_1, g_2)\\circ F_1(g_3)=0.\n\\end{equation}\n\\end{itemize}\n\\end{defi}\nWe denote this 2-term representation up to homotopy of the Lie group\n$G$ by $(V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0,F_1,F_2)$. One should\nbe careful that even if $F_1$ is a usual associative action,\n\\eqref{eqn:F closed} is not equivalent to $F_2$ being a 2-cocycle. This is strangely different from the Lie algebra case (see Section\n\\ref{sec:int}). Define $\\widetilde{F_2}:(G\\ltimes\nV_0)^3\\longrightarrow V_1$ by\n\\begin{equation}\\label{F2tuta}\n\\widetilde{F_2}((g_1,\\xi_1),(g_2,\\xi_2),(g_3,\\xi_3))=F_2(g_1,\ng_2)(\\xi_3).\n\\end{equation}\nIf  $F_1$ is a usual associative action, we form  $G\\ltimes V_0$\nthe semidirect product. Then\n$V_1$ is a  $G\\ltimes V_0$-module with an associated action\n$\\widetilde{F_1}$ of $G\\ltimes V_0$ on $V_1$\n$$\n\\widetilde{F_1}(g,\\xi)(m)=F_1(g)(m),\\quad\\forall~m\\in V_1.\n$$\n\n\n\\begin{pro}\\label{pro:3cocycle} If  $F_1$ is the usual associative action of the Lie group $G$  on the complex\n$V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0$, then $\\widetilde{F_2}$\ndefined by (\\ref{F2tuta}) is a group 3-cocycle representing an\nelement in $H^3_{sm}(G\\ltimes V_0,V_1)$.\n\\end{pro}\n\\noindent{\\bf Proof.}\\  By direct computations, we have\n\\begin{eqnarray*}\n&&d\\widetilde{F_2}((g_1,\\xi_1),(g_2,\\xi_2),(g_3,\\xi_3),(g_4,\\xi_4))\\\\&=&\\widetilde{F_1}(g_1,\\xi_1)\\widetilde{F_2}((g_2,\\xi_2),(g_3,\\xi_3),(g_4,\\xi_4))\\\\\n&&-\\widetilde{F_2}((g_1,\\xi_1)\\cdot(g_2,\\xi_2),(g_3,\\xi_3),(g_4,\\xi_4))+\\widetilde{F_2}((g_1,\\xi_1),(g_2,\\xi_2)\\cdot(g_3,\\xi_3),(g_4,\\xi_4))\\\\\n&&-\\widetilde{F_2}((g_1,\\xi_1),(g_2,\\xi_2),(g_3,\\xi_3)\\cdot(g_4,\\xi_4))+\\widetilde{F_2}((g_1,\\xi_1),(g_2,\\xi_2),(g_3,\\xi_3))\\\\\n&=&F_1(g_1)F_2(g_2,g_3)(\\xi_4)-F_2(g_1\\cdot g_2,g_3)(\\xi_4)+F_2(g_1,\ng_2\\cdot g_3)(\\xi_4)\\\\\n&&-F_2(g_1,g_2)(\\xi_3+F_1(g_3)(\\xi_4))+F_2(g_1,g_2)(\\xi_3)\\\\\n&=&\\big(F_1(g_1)\\circ F_2(g_2,g_3)-F_2(g_1\\cdot g_2,g_3)+F_2(g_1,\ng_2\\cdot g_3)-F_2(g_1, g_2)\\circ F_1(g_3)\\big)(\\xi_4).\n\\end{eqnarray*}\nBy (\\ref{eqn:F closed}), $\\widetilde{F_2}$ is a Lie group 3-cocycle.\n\\hfill ~\\vrule height6pt width6pt depth0pt\\vspace{3mm}\n\n\n\nSimilar to the fact that associated to any representation of a Lie\ngroup, we can form a new Lie group which is their semidirect\nproduct, for the 2-term representation up to homotopy of a Lie\ngroup, we can form a Lie 2-group.\n\n\\begin{thm}\\label{thm:main 1}\nGiven a 2-term representation up to homotopy\n$(V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0,F_1,F_2)$ of a Lie group\n$G$, its semidirect product with $G$ is defined to be\n\\begin{equation}\\begin{array}{c}\nG\\times V_0\\times V_1\\\\\n\\vcenter{\\rlap{s }}~\\Big\\downarrow\\Big\\downarrow\\vcenter{\\rlap{t }}\\\\\nG\\times V_0.\n \\end{array}\\end{equation}\nThen it is a Lie 2-group with the following structure maps:\n\n The source and target are given by\n\\be\\label{s t} s(g,\\xi,m)=(g,\\xi),\\quad t(g,\\xi,m)=(g,\\xi+\\mathrm{d} m).\n\\ee The vertical multiplication $\\cdot_\\mathrm{v}$ is given by\n$$\n(h,\\eta,n)\\cdot_\\mathrm{v}(g,\\xi,m) =(g,\\xi,m+n),\\quad \\mbox{where}~\nh=g,\\eta=\\xi+\\mathrm{d} m.\n$$\nThe horizontal multiplication $\\cdot_\\mathrm{h}$ of objects is given by\n\\be\\label{m o} (g_1,\\xi)\\cdot_\\mathrm{h} (g_2,\\eta)=(g_1\\cdot\ng_2,\\xi+F_1(g_1)(\\eta)), \\ee the horizontal multiplication\n$\\cdot_\\mathrm{h}$ of morphisms is given by \\be\\label{m m}\n(g_1,\\xi,m)\\cdot_\\mathrm{h} (g_2,\\eta,n)=(g_1\\cdot\ng_2,\\xi+F_1(g_1)(\\eta),m+F_1(g_1)(n)). \\ee The inverse map $\\mathrm{inv}$ is\ngiven by \\be \\mathrm{inv}(g,\\xi)=(g^{-1},-F_1(g^{-1})(\\xi)). \\ee The\nidentity object is $(1_G,0)$.\\\\\nThe  associator\n$$\na_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)}:\\big((g_1,\\xi)\\cdot_\\mathrm{h}(g_2,\\eta)\\big)\\cdot_\\mathrm{h}(g_3,\\gamma)\\longrightarrow\n(g_1,\\xi)\\cdot_\\mathrm{h}\\big((g_2,\\eta)\\cdot_\\mathrm{h}(g_3,\\gamma)\\big)\n$$\nis given by \\be\\label{associator}\na_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)}=(g_1\\cdot g_2\\cdot\ng_3,\\xi+F_1(g_1)(\\eta)+F_1(g_1\\cdot\ng_2)(\\gamma),F_2(g_1,g_2)(\\gamma)). \\ee The  unit\n$i_{(g,\\xi)}:(1_G,0)\\longrightarrow (g,\\xi)\\cdot_\\mathrm{h} \\mathrm{inv}(g,\\xi)$ is\ngiven by \\be\\label{unit} i_{(g,\\xi)}=(1_G,0,-F_2(g,g^{-1})(\\xi)).\n\\ee All the other natural isomorphisms are identity isomorphisms.\n\\end{thm}\n\\noindent{\\bf Proof.}\\  By (\\ref{s t}), (\\ref{m o}) and (\\ref{m m}), it is\nstraightforward to see that\n\\begin{eqnarray*}\ns\\big((g_1,\\xi,m)\\cdot_\\mathrm{h} (g_2,\\eta,n)\\big)&=&s(g_1,\\xi,m)\\cdot_\\mathrm{h} s(g_2,\\eta,n),\\\\\nt\\big((g_1,\\xi,m)\\cdot_\\mathrm{h} (g_2,\\eta,n)\\big)&=&t(g_1,\\xi,m)\\cdot_\\mathrm{h}\nt(g_2,\\eta,n).\n\\end{eqnarray*}\nThus the multiplication $\\cdot_\\mathrm{h}$ respects  the source and target\nmap. Furthermore, it is not hard  to check that the horizontal and\nvertical multiplications commute, i.e.\n$$\n\\big((g,\\xi+\\mathrm{d} m,n)\\cdot_\\mathrm{h}(g^\\prime,\\eta+\\mathrm{d}\np,q)\\big)\\cdot_\\mathrm{v}\\big((g,\\xi,m)\\cdot_\\mathrm{h}(g^\\prime,\\eta,p)\\big)=\\big((g,\\xi+\\mathrm{d}\nm,n)\\cdot_\\mathrm{v}(g,\\xi,m)\\big)\\cdot_\\mathrm{h}\\big((g^\\prime,\\eta+\\mathrm{d}\np,q)\\cdot_\\mathrm{v}(g^\\prime,\\eta,p)\\big)\n$$\n\n\n\n\n\\be\\label{m commute with v} \\xymatrix@C+2em{\n  \\bullet &\n  \\ar@\/_2pc\/[l]_{(g,\\xi)}_{}=\"0\"\n  \\ar[l]|{(g,\\xi+\\mathrm{d} m)}^{}=\"1\"_{}=\"1b\"\n  \\ar@\/^2pc\/[l]^{(g,\\xi+\\mathrm{d} (m+n))}^{}=\"2\"\n  \\ar@{=>} \"0\";\"1\"^{m}\n  \\ar@{=>} \"1b\";\"2\"^{n}\n  \\bullet &\n  \\ar@\/_2pc\/[l]_{(g^\\prime,\\eta)}_{}=\"3\"\n  \\ar[l]|{(g^\\prime,\\eta+\\mathrm{d} p)}^{}=\"4\"_{}=\"4b\"\n  \\ar@\/^2pc\/[l]^{(g^\\prime,\\eta+\\mathrm{d} (p+q))}^{}=\"5\"\n  \\ar@{=>} \"3\";\"4\"^{p}\n  \\ar@{=>} \"4b\";\"5\"^{q}\n  \\bullet.\n} \\ee\n\nIt follows from (\\ref{eqn:F fail}) that the associator\n$a_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)}$ defined by\n(\\ref{associator}) is indeed a morphism from\n$\\big((g_1,\\xi)\\cdot_\\mathrm{h}(g_2,\\eta)\\big)\\cdot_\\mathrm{h}(g_3,\\gamma)$ to\n$(g_1,\\xi)\\cdot_\\mathrm{h}\\big((g_2,\\eta)\\cdot_\\mathrm{h}(g_3,\\gamma)\\big)$. To see that\nit is natural, we need to show that \\be\\label{left} a_{(g_1,\\xi+\\mathrm{d}\nm),(g_2,\\eta+\\mathrm{d} n),(g_3,\\gamma+\\mathrm{d} k)}\\cdot_\\mathrm{h}\n\\Big(\\big((g_1,\\xi,m)\\cdot_\\mathrm{h} (g_2,\\eta,n)\\big)\\cdot_\\mathrm{h}\n(g_3,\\gamma,k)\\Big) \\ee is equal to \\be\\label{right}\n\\Big((g_1,\\xi,m)\\cdot_\\mathrm{h} \\big((g_2,\\eta,n)\\cdot_\\mathrm{h}\n(g_3,\\gamma,k)\\big)\\Big)\\cdot_\\mathrm{h}\na_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)}, \\ee i.e. the following\n diagram commutates:\n\\[\n\\xymatrix{\n\\big((g_1,\\xi)\\cdot_\\mathrm{h}(g_2,\\eta)\\big)\\cdot_\\mathrm{h}(g_3,\\gamma)\\ar[d]\\ar[r]^{a}&(g_1,\\xi)\\cdot_\\mathrm{h}\\big((g_2,\\eta)\\cdot_\\mathrm{h}(g_3,\\gamma)\\big)\\ar[d]\\\\\n\\big((g_1,\\xi+\\mathrm{d} m)\\cdot_\\mathrm{h}(g_2,\\eta+\\mathrm{d}\nn)\\big)\\cdot_\\mathrm{h}(g_3,\\gamma+\\mathrm{d} k)\\ar[r]^{a}&(g_1,\\xi+\\mathrm{d}\nm)\\cdot_\\mathrm{h}\\big((g_2,\\eta+\\mathrm{d} n)\\cdot_\\mathrm{h}(g_3,\\gamma+\\mathrm{d} k)\\big). }\n\\]\nBy straightforward computations, we  obtain that (\\ref{left}) is\nequal to\n$$\n\\big(g_1\\cdot g_2\\cdot g_3,\\xi+F_1(g_1)(\\eta)+F_1(g_1\\cdot\ng_2)(\\gamma),m+F_1(g_1)(n)+F_1(g_1\\cdot\ng_2)(k)+F_2(g_1,g_2)(\\gamma+\\mathrm{d} k)\\big),\n$$\nand (\\ref{right}) is equal to\n$$\n\\big(g_1\\cdot g_2\\cdot g_3,\\xi+F_1(g_1)(\\eta)+F_1(g_1\\cdot\ng_2)(\\gamma),m+F_1(g_1)(n)+F_1(g_1)\\cdot\nF_1(g_2)(k)+F_2(g_1,g_2)(\\gamma)\\big).\n$$\n Hence (\\ref{left}) is equal to (\\ref{right}) by (\\ref{eqn:F fail}).\nThis implies that $a_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)}$ defined by\n(\\ref{associator}) is a natural isomorphism.\n\nBy (\\ref{eqn:F fail}) and the fact that $F_1(1_G)=\\rm{Id}$, the unit\ngiven by (\\ref{unit}) is indeed  a morphism from $(1_G,0)$ to\n$(g,\\xi)\\cdot_\\mathrm{h} \\mathrm{inv}(g,\\xi)$. To see that it is natural, we need to\nprove\n$$\n\\big((g,\\xi,m)\\cdot_\\mathrm{h}\\mathrm{inv}(g,\\xi,m)\\big)\\cdot_\\mathrm{h}\ni_{(g,\\xi)}=i_{(g,\\xi+\\mathrm{d} m)},\n$$\ni.e. the following commutative diagram:\n\n\n \n \n$$\n\\xymatrix{\n  &  (1_G,0)\\ar[dr]^{i_{(g,\\xi)}} \\ar_{i_{(g,\\xi+\\mathrm{d} m)}}[dl]\\\\\n(g,\\xi+\\mathrm{d} m)\\cdot_\\mathrm{h} \\mathrm{inv}(g,\\xi+\\mathrm{d} m)\n&&\\ar[ll]_{\\quad\\qquad\\qquad~(g,\\xi,m)\\cdot_\\mathrm{h}\\mathrm{inv}(g,\\xi,m)}(g,\\xi)\\cdot_\\mathrm{h}\\mathrm{inv}(g,\\xi)\n}\n$$\n This follows from\n $$\nF_2(g,g^{-1})(\\mathrm{d} m)=F_1(g)\\cdot F_1(g^{-1})(m)-F_1(g\\cdot\ng^{-1})(m)=F_1(g)\\cdot F_1(g^{-1})(m)-m,\n $$ which is a special case of (\\ref{eqn:F fail}).\n\nSince $F(1_G)=\\rm{Id}$, we have $$(1_G,0)\\cdot_\\mathrm{h} (g,\\xi)=(g,\\xi),\\quad\n(g,\\xi)\\cdot_\\mathrm{h}(1_G,0)= (g,\\xi).$$ Hence the left unit and the right\nunit can also be taken as the identity isomorphism.\n\n The counit\n$e_{(g,\\xi)}:\\mathrm{inv}(g,\\xi)\\cdot_\\mathrm{h} (g,\\xi)\\longrightarrow (1_G,0)$ can\nbe taken as the identity morphism since we have\n$$\n\\mathrm{inv}(g,\\xi)\\cdot_\\mathrm{h} (g,\\xi)=(g^{-1},-F_1(g^{-1})(\\xi))\\cdot_\\mathrm{h}\n(g,\\xi)=(1_G,0).\n$$\nAt last, we need to show\n\\begin{itemize}\n\\item[$\\bullet$] the  pentagon identity  for the associator,\n\\item[$\\bullet$] the  triangle identity for the left and right\nunit laws,\n\\item[$\\bullet$]the  first zig-zag identity,\n\\item[$\\bullet$]the  second zig-zag identity.\n\\end{itemize}\nWe only give the proof of the  pentagon identity, the others can be\nproved similarly and we leave them to the readers. In fact, the\npentagon identity is equivalent to\n\\begin{eqnarray*}\n&&a_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)\\cdot_\\mathrm{h}(g_4,\\theta)}\\cdot_\\mathrm{h}\na_{(g_1,\\xi)\\cdot_\\mathrm{h}(g_2,\\eta),(g_3,\\gamma),(g_4,\\theta)}=\\\\\n&&\\big((g_1,\\xi)\\cdot_\\mathrm{h}\na_{(g_2,\\eta),(g_3,\\gamma),(g_4,\\theta)}\\big)\\cdot_\\mathrm{h}\na_{(g_1,\\xi),(g_2,\\eta)\\cdot_\\mathrm{h}(g_3,\\gamma),(g_4,\\theta)}\\cdot_\\mathrm{h}\n\\big(a_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)}\\cdot_\\mathrm{h}(g_4,\\theta)\\big)\n\\end{eqnarray*}\nBy straightforward computations, the left hand side is equal to\n$$\n\\big(g_1\\cdot g_2\\cdot g_3\\cdot g_4,\\xi+F_1(g_1)(\\eta)+F_1(g_1\\cdot\ng_2)(\\gamma)+F_1(g_1\\cdot g_2\\cdot g_3)(\\theta),F_2(g_1\\cdot g_2,\ng_3)(\\theta)+F_2(g_1, g_2)(\\gamma+F_1(g_3)(\\theta))\\big),\n$$\nand the right hand side is equal to\n\\begin{eqnarray*}\n&\\big(g_1\\cdot g_2\\cdot g_3\\cdot g_4,\\xi+F_1(g_1)(\\eta)+F_1(g_1\\cdot\ng_2)(\\gamma)+F_1(g_1\\cdot g_2\\cdot g_3)(\\theta),\\\\&F_2(g_1,\ng_2)(\\gamma)+F_2(g_1, g_2\\cdot g_3)(\\theta)+F_1(g_1)\\circ F_2(g_2,\ng_3)(\\theta)\\big).\n\\end{eqnarray*}\nBy (\\ref{eqn:F closed}), they are equal. \\hfill ~\\vrule height6pt width6pt depth0pt\n\n\\section{Integrating string Lie 2-algebra $\\mathbb R\\longrightarrow \\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$}\\label{sec:int}\n\nAs an application of Theorem \\ref{thm:main 1}, we consider the\nintegration of the string Lie 2-algebra $\\mathbb R\\longrightarrow\n\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$ given by (\\ref{Lie 2 g g dual}). Now we\nrestrict to the case that $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ is finite dimensional. Obviously,\ngiven a quadruple $(K_1,K_2,\\Phi,\\Theta)$ which represents a special\nLie 2-group (see Theorem \\ref{thm:special 2 G}), by differentiation,\nwe obtain a quadruple $(\\mathfrak k_1,\\mathfrak k_2,\\phi,\\theta)$, which\nrepresents a 2-term skeletal $L_\\infty$-algebra.\n\\begin{defi}\nA special Lie 2-group which is represented by\n$(K_1,K_2,\\Phi,\\Theta)$ is an integration of a 2-term skeletal\n$L_\\infty$-algebra which is represented by\n$(\\mathfrak k_1,\\mathfrak k_2,\\phi,\\theta)$ if the differentiation of\n$(K_1,K_2,\\Phi,\\Theta)$ is $(\\mathfrak k_1,\\mathfrak k_2,\\phi,\\theta)$.\n\\end{defi}\n\nIf the differential $\\mathrm{d}$ in a 2-term complex\n$V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0$ is 0, a representation up to\nhomotopy of Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ on\n$V_1\\stackrel{0}{\\longrightarrow}V_0$ consists of two strict\nrepresentations $\\mu_1$ and $\\mu_0$, and  a liner map\n$\\nu:\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\wedge\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\longrightarrow\\mathrm{Hom}(V_0,V_1)$ satisfying\nequation (\\ref{eqn:d k}).  This equation implies that $\\nu$ is a Lie\nalgebra 2-cocycle representing an element in\n$H^2(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g,\\mathrm{Hom}(V_0,V_1))$, with the representation\n$[\\mu(\\cdot),\\cdot]$ of  $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ on $\\mathrm{Hom}(V_0,V_1)$  defined by\n$$\n[\\mu(\\cdot),\\cdot](X)(A)\\triangleq[\\mu(X), A]=\\mu_1(X)\\circ A-A\\circ\n\\mu_0(X),\\quad \\forall ~X\\in \\mathfrak g}\\newcommand{\\g}{\\mathfrak g,~A\\in\\mathrm{Hom}(V_0,V_1).\n$$\n\n\\begin{lem}\nDefine $\\widetilde{\\nu}:\\wedge^3(\\g\\oplus V_0)\\longrightarrow V_1$\nby\n$$\n\\widetilde{\\nu}(X_1+\\xi_1,X_2+\\xi_2,X_3+\\xi_3)=\\nu(X_1,X_2)(\\xi_3)+c.p.\n$$\nthen $\\nu$ is a 2-cocycle if and only if $\\widetilde{\\nu}$ is a\n3-cocycle where the representation $\\widetilde{\\mu}$ of $\\g\\oplus\nV_0$ on $V_1$ is given by\n$$\n\\widetilde{\\mu}(X+\\xi)(m)=\\mu(X)(m).\n$$\n\\end{lem}\n\\noindent{\\bf Proof.}\\  By direct computations, for any $X_i+\\xi_i\\in\\g\\oplus\nV_0,~i=1,2,3,4,$ we have\n$$d\\widetilde{\\nu}(X_1+\\xi_1,X_2+\\xi_2,X_3+\\xi_3,X_4+\\xi_4)=d\\nu(X_1,X_2,X_3)(\\xi_4)+c.p.,$$\nwhich implies the conclusion.\\hfill ~\\vrule height6pt width6pt depth0pt\n\n\n The  Lie algebra homomorphism $\\mu$ from $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ to\n$\\mathrm{End}(V_0)\\oplus\\mathrm{End}(V_1)$ integrates to a Lie group homomorphism\n$F_1$ from the simply connected Lie group $G$ of $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ to\n$GL(V_0)\\oplus GL(V_1)$ with\n$$\n\\mu(X)=\\frac{d}{dt}\\Big|_{t=0}F_1(\\exp tX),\\quad\\forall~X\\in\\mathfrak g}\\newcommand{\\g}{\\mathfrak g.\n$$\n Consequently, $\\mathrm{Hom}(V_0,V_1)$ is a $G$-module with $G$ action\n$$\ng\\cdot A=F_1(g)\\circ A \\circ F_1(g)^{-1},\\quad\\forall ~g\\in G,~A\\in\n\\mathrm{Hom}(V_0,V_1).\n$$ The Lie algebra 2-cocycle\n$\\nu:\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\wedge\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\longrightarrow\\mathrm{Hom}(V_0,V_1)$ can integrate to\na smooth Lie group 2-cocycle $\\overline{F_2}:G\\times G\\longrightarrow\n\\mathrm{Hom}(V_0,V_1)$, satisfying\n\\begin{equation}\\label{eqn:F2 closed}\nF_1(g_1)\\circ(\\overline{F_2})(g_2,g_3)\\circ\nF_1(g_1)^{-1}-(\\overline{F_2})(g_1\\cdot\ng_2,g_3)+(\\overline{F_2})(g_1, g_2\\cdot\ng_3)-(\\overline{F_2})(g_1,g_2)=0,\n\\end{equation}\nand $\\overline{F_2}(1_G,1_G)=0.$ Let us explain how.\n\nThe classical theory of cohomology of discrete groups tells us that\nthe equivalence classes of extensions of $G$ by a $G$ module $M$,\none to one correspond to the elements in $H^2(G, M)$. In our case,\nthe same theory tells us that  $H^2_{sm}(G, \\mathrm{Hom}(V_0, V_0))$\nclassifies  the equivalence classes of splitting extensions of $G$\nby the $G$-module $\\mathrm{Hom}(V_0, V_1)$, which is a splitting short exact\nsequence of Lie groups, with $\\mathrm{Hom}(V_0, V_1)$ endowed with an\nabelian group structure,\n\\begin{equation}\\label{eq:extension} \\mathrm{Hom}(V_0, V_1) \\to \\hat{G} \\to G. \\end{equation}\nIn a general extension $\\hat{G}$ is a principal bundle over $G$,\nthus it usually does not permit a smooth lift $G\n\\xrightarrow{\\sigma} \\hat{G}$. It permits such a lift if and only if\nthe sequence splits. However in our case, since the abelian group\n$\\mathrm{Hom}(V_0, V_1) $ is a vector space, we have  $H^1(X,\\mathrm{Hom}(V_0, V_1)\n)=0$ for any manifold $X$. The proof makes use of a partition of\nunity and similar to the proof showing that $H^1(X,\n\\underline{\\mathbb R}\\newcommand{\\Z}{\\mathbb Z})=0$ for the sheaf cohomology. Hence all $ \\mathrm{Hom}(V_0,\nV_1)$ principal bundles are trivial. Therefore \\eqref{eq:extension}\nalways splits. On the other hand\n it is well-known that when $G$ is simply connected, there is a\n one-to-one correspondence between extensions of $G$\n \\cite[Theorem 4.15]{brahic} and extensions of its Lie algebra\n $\\g$, which in turn are classified by the Lie algebra cohomology\n $H^2(\\g, \\mathrm{Hom}(V_0, V_1))$. Hence in our case the differentiation\n map\n\\[ H^2_{sm} (G, \\mathrm{Hom}(V_0, V_1)) \\to H^2(\\g, \\mathrm{Hom}(V_0, V_1)) \\]\nis an isomorphism. Hence $\\nu$  always integrates to a smooth Lie\ngroup 2-cocycle unique up to exact 2-cocycles. Then\n$\\overline{F_2}(1_G,1_G)=0$ can be arranged too, because we can\nalways modify the section $\\sigma: G \\to \\hat{G}$ to satisfy\n$\\sigma(1_G)=1_{\\hat{G}}$ and the modification of sections results\nin an exact term. Then combined with (\\ref{eqn:F2 closed}), it is\nnot hard to see that\n\\begin{equation}\\label{temp1}\n\\overline{F_2}(1_G,g)=\\overline{F_2}(g,1_G)=0,\\quad\\forall~g\\in\nG.\\end{equation} Thus $\\overline{F_2}$ is a normalized 2-cocycle.\n\n\n\n\\begin{pro}\\label{pro:int mu nu }\nFor any 2-term representation up to homotopy $(\\mu,\\nu)$ of a Lie\nalgebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ on the complex\n$V_1\\stackrel{0}{\\longrightarrow}V_0$, there is an associated\nrepresentation up to homotopy $(F_1,F_2)$ of the Lie group $G$ on\nthe complex $V_1\\stackrel{0}{\\longrightarrow}V_0$, where $F_1$ is\nthe integration of $\\mu$ and $F_2:G\\times G\\longrightarrow\n\\mathrm{End}(V_0,V_1)$ is defined by\n\\begin{equation}\\label{eqn:F2}\nF_2(g_1,g_2)=\\overline{F_2}(g_1,g_2)\\circ F_1(g_1\\cdot g_2).\n\\end{equation}\n\\end{pro}\n\\noindent{\\bf Proof.}\\  Obviously, (\\ref{eqn:F fail}) is satisfied. To see (\\ref{eqn:F\nclosed}) is also satisfied, combine  (\\ref{eqn:F2}) with\n(\\ref{eqn:F2 closed}). By the fact that $F_1$ is a homomorphism, we\nobtain\n\\begin{eqnarray*}\n&F_1(g_1)\\circ F_2(g_2,g_3)\\circ F_1(g_2\\cdot g_3)^{-1}\\circ\nF_1(g_1)^{-1}-F_2(g_1\\cdot g_2,g_3)\\circ F_1(g_1\\cdot g_2\\cdot\ng_3)^{-1}\\\\&+F_2(g_1, g_2\\cdot g_3)\\circ F_1(g_1\\cdot g_2\\cdot\ng_3)^{-1}-F_2(g_1,g_2)\\circ F_1(g_1\\cdot g_2)^{-1}=0.\n\\end{eqnarray*}\nComposed with $F_1(g_1\\cdot g_2\\cdot g_3)$ on the right hand side,\nwe obtain (\\ref{eqn:F closed}). \\hfill ~\\vrule height6pt width6pt depth0pt\n\nBy Proposition \\ref{pro:int mu nu } and Theorem \\ref{thm:main 1}, we\nhave\n\\begin{thm}\\label{thm:main 2}\nLet $G$ be the simply connected Lie group integrating $\\g$, then the\nstring Lie 2-algebra $\\mathbb\nR\\stackrel{0}{\\longrightarrow}\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$ given by (\\ref{Lie\n2 g g dual}) integrates to the following Lie 2-group,\n\\begin{equation}\\label{2 group}\\begin{array}{c}\nG\\times \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*\\times \\mathbb R\\\\\n\\vcenter{\\rlap{s }}~\\Big\\downarrow\\Big\\downarrow\\vcenter{\\rlap{t }}\\\\\nG\\times \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,\n \\end{array}\\end{equation}\nin which the source and target are given by \\be\ns(g,\\xi,m)=t(g,\\xi,m)=(g,\\xi)\\ee the vertical multiplication\n$\\cdot_\\mathrm{v}$ is given by\n$$\n(h,\\eta,n)\\cdot_\\mathrm{v}(g,\\xi,m) =(g,\\xi,m+n),\\quad \\mbox{where}~\nh=g,\\eta=\\xi,\n$$\nthe horizontal multiplication $\\cdot_\\mathrm{h}$ of objects is given by\n$$\n (g_1,\\xi)\\cdot_\\mathrm{h} (g_2,\\eta)=(g_1\\cdot\ng_2,\\xi+\\mathrm{Ad}^*_{g_1}\\eta),\n$$\nthe horizontal multiplication $\\cdot_\\mathrm{h}$ of morphisms is given by\n$$ (g_1,\\xi,m)\\cdot_\\mathrm{h} (g_2,\\eta,n)=(g_1\\cdot\ng_2,\\xi+\\mathrm{Ad}^*_{g_1}\\eta,m+n),\n$$\nthe inverse map $\\mathrm{inv}$ is given by $$\n\\mathrm{inv}(g,\\xi)=(g^{-1},-\\mathrm{Ad}^*_{g^{-1}}\\xi), $$ The\nidentity object is $(1_G,0)$,\\\\\nthe  associator\n$$\na_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)}:\\big((g_1,\\xi)\\cdot_\\mathrm{h}(g_2,\\eta)\\big)\\cdot_\\mathrm{h}(g_3,\\gamma)\\longrightarrow\n(g_1,\\xi)\\cdot_\\mathrm{h}\\big((g_2,\\eta)\\cdot_\\mathrm{h}(g_3,\\gamma)\\big)\n$$\nis given by \\be\\label{associator}\na_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)}=(g_1\\cdot g_2\\cdot\ng_3,\\xi+\\mathrm{Ad}^*_{g_1}\\eta+\\mathrm{Ad}^*_{g_1\\cdot\ng_2}\\gamma,F_2(g_1,g_2)(\\gamma)), \\ee\nAll the other structures are identity isomorphisms.\n\\end{thm}\n\\noindent{\\bf Proof.}\\  Since $F_1$ is a usual associative action, we may modify the\nunit (\\ref{unit}) given in Theorem \\ref{thm:main 1} to be the\nidentity natural transformation. It turns out that (\\ref{2 group})\nis a special Lie 2-group and is represented by $(G\\ltimes\n\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,\\mathbb R,\\rm{Id},\\widetilde{F_2})$, where $G\\ltimes \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$ is\nthe semidirect product with the coadjoint action of $G$ on $\\g^*$,\n$\\rm{Id}$ is the constant map $G\\ltimes \\g^* \\to \\mathrm{Aut}(\\mathbb R}\\newcommand{\\Z}{\\mathbb Z)$ by mapping\neverything to $\\rm{Id} \\in \\mathrm{Aut}(\\mathbb R}\\newcommand{\\Z}{\\mathbb Z)$, and  $\\widetilde{F_2}$ is given by\n\\begin{equation}\\label{eq:tf2}\n\\widetilde{F_2}((g_1,\\xi_1),(g_2,\\xi_2),(g_3,\\xi_3))=F_2(g_1,\ng_2)(\\xi_3)=\\overline{F_2}(g_1, g_2)\\circ F_1(g_1\\cdot g_2)(\\xi_3).\n\\end{equation} Since $\\overline{F_2}$ is normalized, $\\widetilde{F_2}$ is also\nnormalized.  The string Lie 2-algebra $\\mathbb\nR\\stackrel{0}{\\longrightarrow}\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$ is  skeletal and\nis represented by $(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,\\mathbb\nR,0,\\widetilde{\\nu})$, where $\\widetilde{\\nu}$ is given by\n(\\ref{eqn:nu3}). Thus to show that our Lie 2-group is an integration\nof the string Lie algebra, we only need to show that the\ndifferential of the Lie group 3-cocycle $\\widetilde{F_2}$ is the Lie\nalgebra 3-cocycle $\\widetilde{\\nu}$.  By direct computations\n\\cite[Lemma 7.3.9]{brylinski}, we have\n\\begin{eqnarray*}\n&&\\frac{\\partial^3}{\\partial_{t_1}\\partial_{t_2}\\partial_{t_3}}\\Big|_{t_i=0}\\sum_{\\sigma\\in\nS_3}\\epsilon(\\sigma)\\widetilde{F_2}\\big((e^\n{t_{\\sigma(1)}X_{\\sigma(1)}},t_{\\sigma(1)}\\xi_{\\sigma(1)}),(e^\n{t_{\\sigma(2)}X_{\\sigma(2)}},t_{\\sigma(2)}\\xi_{\\sigma(2)}),(e^\n{t_{\\sigma(3)}X_{\\sigma(3)}},t_{\\sigma(3)}\\xi_{\\sigma(3)})\\big)\\\\\n&=&\\frac{\\partial^3}{\\partial_{t_1}\\partial_{t_2}\\partial_{t_3}}\\Big|_{t_i=0}\\Big(\\overline{F_2}(e^{\nt_1X_1},e^ {t_2X_2})\\circ F_1(e^{ t_1X_1}\\cdot e^\n{t_2X_2})(t_3\\xi_3)\\Big)+c.p.\\\\\n&=&\\frac{\\partial^2}{\\partial_{t_1}\\partial_{t_2}}\\Big|_{t_i=0}\\Big(\\overline{F_2}(e^{\nt_1X_1},e^ {t_2X_2})\\circ F_1(e^{ t_1X_1}\\cdot e^ {t_2X_2})(\\xi_3)\\Big)+c.p.\\\\\n&=&\\frac{\\partial}{\\partial_{t_1}}\\Big|_{t_1=0}\\Big(\\frac{\\partial}{\\partial_{t_2}}\\big|_{t_2=0}\\overline{F_2}(e^{\nt_1X_1},e^ {t_2X_2})\\circ F_1(e^{\nt_1X_1})(\\xi_3)+\\overline{F_2}(e^{t_1X_1},1_G)\\circ\\frac{\\partial}{\\partial_{t_2}}\\big|_{t_2=0}F_1(e^{\nt_1X_1}\\cdot e^{ t_2X_2})\\Big)\\\\&&+c.p.\\\\\n&=&\\frac{\\partial}{\\partial_{t_1}}\\frac{\\partial}{\\partial_{t_2}}\\Big|_{t_i=0}\\overline{F_2}(e^{\nt_1X_1},e^\n{t_2X_2})(\\xi_3)+\\frac{\\partial}{\\partial_{t_2}}\\big|_{t_2=0}\\overline{F_2}(1_G,e^\n{t_2X_2})\\circ \\frac{\\partial}{\\partial_{t_1}}\\Big|_{t_1=0}F_1(e^{\nt_1X_1})(\\xi_3)\\\\\n&&+c.p.\\quad\\mbox{by (\\ref{temp1})}\\\\\n&=&\\nu(X_1,X_2)(\\xi_3)+c.p.\\\\\n&=&\\widetilde{\\nu}(X_1+\\xi_1,X_2+\\xi_2,X_3+\\xi_3)  \\quad\\mbox{by\n(\\ref{eqn:nu3})},\n\\end{eqnarray*}\nwhich completes the proof. \\hfill ~\\vrule height6pt width6pt depth0pt\n\\begin{cor}\nIf Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ is semisimple, the Lie group 3-cocycle\n$\\widetilde{F_2}$ is not exact, i.e. $[\\widetilde{F_2}]\\neq0$\nin $H^3_{sm}(G\\ltimes \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,\\mathbb R)$.\n\\end{cor}\n\\noindent{\\bf Proof.}\\  By Theorem \\ref{thm:main 2}, the differentiation of the Lie\ngroup 3-cocycle $\\widetilde{F_2}$ is the Lie algebra 3-cocycle\n$\\widetilde{\\nu}$. We only need to show that when $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ is\nsemisimple, the Lie algebra 3-cocycle $\\widetilde{\\nu}$ is not\nexact. This fact is proved in Proposition \\ref{pro:nondegenerate}.\n\\hfill ~\\vrule height6pt width6pt depth0pt\n\n\\begin{remark}Since $G\\ltimes \\g^*$ is a fibration over $G$, the spectral sequence with $E_2^{p,q}=H^p_{sm}(G, H^q_{sm}( \\g^*,\n  \\mathbb R}\\newcommand{\\Z}{\\mathbb Z))$ calculates the group cohomology $H^3_{sm}(G\\ltimes\n  \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,\\mathbb R)$. Since $\\g^*$ is an abelian group, $H^q_{sm}( \\g^*,\n  \\mathbb R}\\newcommand{\\Z}{\\mathbb Z)=\\wedge^q\\g^*$. Thus when $G$ is compact, $ H^p_{sm}(G, H^q_{sm}( \\g^*,\n  \\mathbb R}\\newcommand{\\Z}{\\mathbb Z))  = (\\wedge^q\\g^*)^G$ if $p=0$ and 0 otherwise, where\n  $(\\wedge^q\\g^*)^G$ denotes the set of invariant elements of\n  $\\wedge^q \\g^*$  under the coadjoint action of $G$. Thus when $G$ is compact, $H^3_{sm}(G\\ltimes \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,\\mathbb R)=(\\wedge^3\ng^*)^G \\neq 0$ because the Cartan 3-form is an element of\n$(\\wedge^3 g^*)^G$.\n\\end{remark}\n\nNotice that our 2-cocycle $\\overline{F_2}$ is unique only up to\nexact terms. Hence by Theorem \\ref{thm:special 2 G},  to verify that our\nconstruction is unique up to isomorphism, we need the following\nlemma,\n\n\\begin{lemma}\nIf $\\overline{F_2}= \\mathrm{d} \\alpha$ is exact, then $\\widetilde{F_2}=\\mathrm{d} \\beta$ is also exact\nwith\n\\[\\beta((g_1, \\xi_1), (g_2, \\xi_2))=\\alpha(g_1)F_1(g_1)(\\xi_2). \\]\n\\end{lemma}\n\\begin{proof}\nIt is a direct calculation. Since $\\overline{F_2}= \\mathrm{d} \\alpha$, we have\n\\[\\overline{F_2}(g_1, g_2) = F_1 (g_1) \\alpha(g_2) F_1(g_1)^{-1} - \\alpha(g_1\ng_2) + \\alpha(g_1). \\]\nFrom the definition of $F_2$, we know that\n\\[ F_2(g_1, g_2)= F_1(g_1) \\alpha(g_2) F_1(g_1)^{-1} F_1(g_1 g_2) -\n\\alpha(g_1 g_2) F_1(g_1 g_2) + \\alpha(g_1) F_1(g_1g_2). \\]\nBy \\eqref{eq:tf2}, we have\n\\[\n\\begin{split}\n\\widetilde{F_2}((g_1, \\xi_1), (g_2, \\xi_2),(g_3, \\xi_3))= &F_1(g_1) \\alpha(g_2)\nF_1(g_1)^{-1} F_1(g_1g_2) (\\xi_3) - \\alpha(g_1g_2) F_1(g_1g_2) (\\xi_3)\n+ \\alpha(g_1) F_1(g_1g_2) (\\xi_3) \\\\\n= &\\mathrm{d} \\beta((g_1, \\xi_1), (g_2, \\xi_2),(g_3, \\xi_3)),\n\\end{split}\n \\] since $F_1$ is a group homomorphism.\n\\end{proof}\n\n\\begin{remark}\nOur Lie 2-group as a stacky group has the underlying differential\nstack $G \\times \\g^* \\times B\\mathbb R}\\newcommand{\\Z}{\\mathbb Z$. Thus it is  0,1,2-connected (i.e.\nit has $\\pi_0=\\pi_1=\\pi_2=0$) since $\\pi_2(B\\mathbb R}\\newcommand{\\Z}{\\mathbb Z)=\\pi_1(\\mathbb R}\\newcommand{\\Z}{\\mathbb Z)=0$ and\n$\\pi_1(B\\mathbb R}\\newcommand{\\Z}{\\mathbb Z)=\\pi_0(\\mathbb R}\\newcommand{\\Z}{\\mathbb Z)=0$. Thus it is the unique 0,1,2-connected\nstacky Lie group integrating the string Lie 2-algebra $\\mathbb R}\\newcommand{\\Z}{\\mathbb Z\n\\xrightarrow{0} \\g\\oplus \\g^*$ in the sense of \\cite{z:lie2}.\n\\end{remark}\n\\bibliographystyle{habbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n\n\nProduced by Tyler Garrett, Mormon Texts Project Intern\n(http:\/\/mormontextsproject.org)\n\n\n\n\n\n\nBlood Atonement and the Origin of Plural Marriage\n\nA DISCUSSION\n\nCorrespondence between ELDER JOSEPH F. SMITH, JR. of the Church of\nJesus Christ of Latter-day Saints\n\nAND\n\nMR. RICHARD G. EVANS, Second Counselor in the Presidency of the\n\"Reorganized\" Church\n\n* * * * *\n\n\n\"To correct misrepresentation, we adopt self representation.\"\n\n--John Taylor.\n\n\nCorrespondence between ELDER JOSEPH F. SMITH, (JR.,) of the Church of\nJesus Christ of Latter-day Saints, and MR. RICHARD C. EVANS, second\ncounselor (1905) in the Presidency of the \"Reorganized\" Church. A\nconclusive refutation of the false charges persistently made by\nministers of the \"Reorganized\" Church against the Latter-day Saints and\ntheir belief. Also a supplement containing a number of affidavits and\nother matters bearing on the subjects.\n\nSALT LAKE CITY, UTAH\n\nPRINTED IN U.S.A.\n\n\n\nINTRODUCTION\n\nThe correspondence in this pamphlet was brought about through the\nwilful misrepresentation of the doctrines of the Latter-day Saints and\nthe unwarranted abuse of the authorities of the Church by Mr. Richard\nC. Evans, in an interview which appeared in the Toronto (Canada) _Daily\nStar_ of January 28, 1905. A copy of the interview was placed in the\nhands of the writer, who, on February 19th following, replied to Mr.\nEvans in an open letter which was published in the Toronto Star on or\nabout the 25th of the month.[1] This open letter was answered by Mr.\nEvans in a personal letter, and on the 23rd of May, a rejoinder to his\nreply was sent to Mr. Evans at his home in London, Ontario, Canada.\nIn all, four communications--including the interview--have passed\nbetween us, and all of these four communications are here reproduced\n_in full_. A copy of the open letter which appeared in the _Star_,\nwas also sent to Mr. Evans who acknowledged its receipt. Nothing more\nwas done in regard to this correspondence until August 17th and 24th,\nwhen an article containing a portion of it appeared in the _Zion's\nEnsign_, published by the \"Reorganized\" church at Independence, Jackson\nCounty, Missouri, under the title: \"Statements Authenticated,\" in which\nit was made to appear that the full and complete communications were\nreproduced. But this, however, was not the case.\n\nIn a letter from Mr. Evans to the editor of the _Ensign_ which\naccompanied the above mentioned article, he said:\n\nBelieving that good will be accomplished by the publication of the\nentire matter, I herewith mail you the referred to matter.\n\nFrom this it would naturally be supposed that the _complete_\ncorrespondence would be given. However I was not surprised to see that\nMr. Evans' side of the controversy was _in full_, while a large portion\nof my first communication had been purposely suppressed; and that\nmy second letter _did not appear at all_! And thus was the \"_entire\nmatter_\" given to the readers of the _Ensign_ that \"good\" might be\n\"accomplished.\" (?)\n\nThe parts that were purposely left out of my communication by Mr.\nEvans, were most vital to the subject and have been indicated as they\nappear in the body of this work by being placed in italics, excepting\na few minor matters which he omitted that I have not mentioned,\nnevertheless matters that throw light upon the subject.\n\nOne of these quotations was in relation to two articles in the first\nvolume of the _Saints' Herald_ which were important, coming, as they\ndid from the \"enemy's\" camp. Here is the omitted part:\n\n    If you believe your statement to be true, will you kindly explain\n    the following passage in the _Saints' Herald_, your official organ,\n    volume I, page 9,--it would be well for you to read the entire\n    chapter, which is entitled \"Polygamy.\" The quotation is as follows:\n\n    \"The death of the Prophet is one fact that has been realized,\n    although he abhorred and repented of this iniquity (meaning\n    \"polygamy\") before his death. This branch of the subject we shall\n    leave to some of our brethren, who are qualified to explain it\n    satisfactorily.\"\n\n    In the same volume, page 27, what is meant by the following: \"He,\n    (Joseph Smith) caused the revelation on the subject (polygamy) to\n    be burned, and when he voluntarily came to Nauvoo and resigned\n    himself into the arms of his enemies he said that he was going to\n    Carthage to die. At that time he also said that if it had not been\n    for that accursed spiritual wife doctrine he would not have come to\n    that.\" Kindly read the context.\n\n    There is more evidence that can be produced, but if you will\n    explain this it may suffice.\n\nThe first half of the succeeding paragraph was quoted but the second\nhalf was omitted. I quote in full with the part suppressed in italics:\n\n    In the light of the knowledge I have received and the evidence\n    at my command, I know that the Prophet Joseph Smith made no such\n    statement as the above, and that he did not have the revelation\n    burned. _There is, however, value in the above statements from your\n    \"Herald,\" for they bear witness to the origin and introduction of\n    the principle of plural marriage and revelation concerning the\n    same_.\n\nIt is easy to perceive that Mr. Evans felt \"that good will be\naccomplished by the publication of the 'entire matter'\"; and for\nthat reason he omitted this evidence which the leaders of the\n\"Reorganization\" have been trying so successfully to destroy for lo\nthese many years. The two articles in the _Saints' Herald_ have caused\nthe leaders of that sect no end of trouble, and today they are in\nthe same fix in regard to plural marriage that the first editor of\nthat paper was when he wrote, for they cannot explain the Prophet's\nconnection with the principle \"satisfactorily,\" and never will be able\nto until they acknowledge the truth.\n\nAnother of Mr. Evans' ommissions that \"good\" might be \"accomplished\"\n(?) is the following paragraph in reference to President Brigham Young:\n\n    It is true that President Young was elected president at\n    Kanesville; but on what grounds do you charge him with holding\n    the office in trust for the \"dead president's son?\" Do you not\n    know that such a statement --contrary to the written word--was\n    antagonistic to the teachings of President Young, as recorded in\n    the _Times and Seasons_, as well as since that time?\n\n    Will you please explain on what grounds you charge President Young\n    with being \"under suspicion at the time of Joseph Smith's death?\"\n    Am I to infer by this that you mean to convey the idea that Brigham\n    Young was in any way responsible for the death of Joseph Smith? The\n    Prophet never had a truer friend. You know that at the time of the\n    martyrdom Brigham Young was on a mission away from home. If this is\n    the inference you wish to convey, it is not only contemptible but\n    viciously false.\n\nIt appears from the actions of many of those who fight the Latter-day\nSaints, that they fully realize their inability to successfully oppose\nthe doctrines of the Church with truth as a weapon of attack, and,\ntherefore, resort to falsehood, vilification and abuse, attempting to\nblind those who are not acquainted with the facts. The doctrine of\nthe Church has survived all such onslaughts and continues to spread\nthroughout the earth, as a witness against those who have adopted such\nbase methods for its overthrow. It will continue to spread, bless\nmankind and prepare all who accept it, and follow its teachings in\nrighteousness, for an inheritance in the kingdom of God.\n\nThe Reorganite ministers are generally in the front rank among those\nwho oppose the Church and resort to tactics of a doubtful character.\nThey travel from place to place, never losing an opportunity in\nprivate, on the rostrum or through the press, to \"explain the radical\ndifference\" between their organization and that of the Church of Jesus\nChrist of Latter-day Saints, and in denouncing \"the Utah Mormon and his\niniquities.\" On such occasions they will quote garbled and isolated\nextracts from sermons and from writings by Elders of the Church, taking\nparticular pains to cover up the context in order to prejudice the\nuninformed mind. In this way many a harmless, inoffensive passage has\nbeen made to do great execution in some quarters and among a certain\nclass. Nor is this all. Nearly every crime that was committed within a\nthousand miles of Utah in early days and many that were invented out of\nwhole cloth, are brought to bear against the \"dreadful Mormons,\" the\nChurch and the Gospel, that they may be stigmatized and made to appear\nvile and hateful before the world. So much of their time is spent in\nthis way that they can surely have but little left in which to tell the\nworld what they themselves believe.\n\nNo reason except that of misrepresentation and jealousy can be assigned\nfor actions of this kind. These men oppose the truth in a spirit\nof jealousy and to cover up their own false position, and by such\nan attitude prove that they are ashamed of their own faith, being\nconscious of its weakness.\n\nThe supplement following the correspondence is composed of a number\nof affidavits and other testimony bearing on the subjects under\ndiscussion, which, it is hoped, will be of interest and perhaps of\nvalue to the reader.\n\n                                              JOSEPH F. SMITH, JR.\n\nSalt Lake City, Utah, September 5, 1905.\n\n\nFootnotes\n\n1. As I did not receive a copy of the _Toronto Star_ I cannot\npositively say that my article appeared in full, but if it did not Mr.\nEvans is still without excuse for not considering the _entire matter_\nfor he received personally a duplicate copy of the article sent the\n_Star_ which contained those portions he has failed to include in his\n\"entire matter\" in the _Zion's Ensign_.\n\n\n\nMR. R. C. EVANS' INTERVIEW IN THE TORONTO, CANADA, \"DAILY STAR,\"\nJAN. 28, 1905\n\nLATTER-DAY SAINT VISITING TORONTO--MR. R. C. EVANS, WHO IS\nPROMOTING THE GROWTH OF HIS CHURCH IN CANADA, NOT A BELIEVER IN\nPOLYGAMY--DENOUNCES THE UTAH MORMONS.\n\nThe name Mormon does not please Toronto's six hundred baptized\nLatter-day Saints, not to mention the fifty thousand others scattered\nover the globe.\n\nThis fact was emphasized today, when R. C. Evans, one of the three\nmembers of the Presidency, explained the radical difference between the\ntwo denominations. Mr. Evans, who reached Toronto a few days ago to\nspend a month here, denounces the \"Utah Mormon and his iniquities.\"\n\n\"We do not believe in polygamy, blood atonement, and kindred evils,\"\nhe said to the _Star_ last night at 142 Peter street, where he is\nvisiting, \"They are an abomination to the Lord. The term Mormon is\noffensive to us, because it is associated in the public mind with the\npractices that I have specified. The other night, while I was holding a\nservice here, four Utah Elders came to me. I referred to polygamy, and\nthey defended it. 'We endorse it,' they told me, 'but we don't practice\nit.' Three women were with them, and I said to one, 'Do you believe in\npolygamy?' 'I do,' she replied, 'and I know that God will punish the\nUnited States for prohibiting it.' I understand that there are five\nUtah elders in Toronto at the present time, and in addresses here I\nwill expose polygamy and blood atonement.\"\n\n\nBORN NEAR MONTREAL\n\nMr. Evans is forty-three years old, but doesn't look his age. He is\nrather below medium height, strongly built, wears his black hair short,\nand his round, slightly olive face is clean shaven. He is animated in\nmanner, and though his English is occasionally at fault, he speaks\nfluently and well. He was born at St. Andrew's near Montreal, but his\nancestry is not confined to any one country, Irish, Welsh and German\nblood flows in his veins and his somewhat nasal voice is typically\nAmerican.\n\n\"I was baptized in 1876,\" he said, \"ordained a priest in 1882, became\nan elder in 1884, entered the quorum of seventy in 1886, was chosen\none of the twelve apostles in 1897; and in 1902, was selected one of\nPresident Joseph Smith's two counselors, the other being his eldest\nson, Frederick M. Smith. I was the pastor of the London, Ontario,\nchurch from 1882 to 1886, and have given particular attention to\nCanada. We occupy a rented church on the corner of Sumac and St. David\nstreets, a new church on Camden street, and another at Humber Bay,\npractically three congregations in Toronto.\"\n\nThe Latter-day Saints and the Utah Mormons, according to Mr. Evans, are\nfrequently confused, greatly to his regret.\n\n\nTROUBLES OF THE SECT\n\n\"My President Joseph Smith,\" he explained, \"is the oldest son of Joseph\nSmith, who, when a boy of fifteen, was directed to the mound wherein he\nfound the golden plates from which he compiled the Book of Mormon.\n\n\"He organized his church in 1830, when 25 years old, and between\n1830 and 1844 his following numbered 200,000. In 1844 he was shot\nand killed for his anti-slavery sympathies,[1] and with him died his\nbrother Hyrum. John Taylor, a Toronto convert of 1838, was wounded, but\nrecovered. Joseph Smith's city of Nauvoo, Illinois, was wrecked, and in\n1847, at Kanesville, Iowa, Brigham Young was elected president, though\nhe still professed to hold the office in trust for the dead president's\neldest son, also, Joseph, whom the father had consecrated as his\nsuccessor.[2] Brigham Young reorganized[3] the church, rebaptized every\nmember, including himself, and in 1848 (1847) he reached Salt Lake\nCity. With him went the widow and children of Hyrum Smith, whose son\nJoseph F., is now president of the Utah church. The widow of the first\npresident had refused to follow Young, and her boy Joseph was brought\nup in his father's footsteps, hating polygamy and other impurities.\n'Young Joseph,' as he was called, connected himself with the Saints,\nwho had rejected Brigham Young, and was elected their president. He\nwas then 28 years old. In 1872 he was called to Washington, a report\nhaving reached the Government that Mormonism had again sprung up in\nIllinois. He disproved the charge of polygamy and blood atonement,\nand demonstrated that Latter-day Saintism was in keeping with the law\nand supported by the Bible. Incorporation was granted, and we have\nprospered.\n\n\nUPHELD DEATH\n\n\"Brigham Young, who had been under suspicion at Joseph Smith's death,\nintroduced polygamy and blood atonement at Salt Lake City. Blood\natonement meant death to anyone who left his church. Brigham Young's\nargument was that the apostate whose throat was cut from ear to ear,\nthe favorite way, saved his soul, but his object was to keep his people\nunder his iron heel. Young was a shrewd, bad man.\n\n\"I spent a day and a half with Joseph F. Smith at Salt Lake City three\nyears ago, and he gave me a group photo of himself, his surviving five\nwives, and thirty-six children. His first wife was dead. She died\nbroken-hearted and insane. Personally, Joseph F. Smith is a genial,\nkindly man, but he and I differed on Polygamy. I told him it was vile\nand wicked, always had been, and always would be. In appearance he\nresembles his cousin, my own president.\"\n\nMr. Evans is married, and has two children. The three faces look at\nyou from his watch case. He has recently returned from the northwest.\nHis faith has several thriving churches there, he says, while the Utah\nMormons are settled in one part of Alberta.\n\n\nFootnotes\n\n1. Mr. Evans' declaration that the Prophet was killed for his anti-\nslavery sympathies is rather surprising, when we consider that he was\nin one of the anti-slave states, and the mob at Carthage was largely\ncomposed of men with very strong \"anti-slavery sympathies.\" The fact\nis he and his brother Hyrum were martyred for their religion of which\nCelestial Marriage, (including Plural Marriage) formed a part. One of\nthe charges made against them was that of teaching \"polygamy.\"\n\n2. In proof that the Prophet did not ordain or consecrate his son as\nhis successor, the reader is referred to the affidavits of John W.\nRigdon and Bathsheba W. Smith.\n\n3. As the Church was never disorganized, it could not be reorganized.\nMr. Evans has made a mistake. It was the Quorum of the First\nPresidency that was disorganized at the Prophet's death and which was\n_reorganized_ when Brigham Young was elected President, and not the\nChurch.\n\n\n\nREPLY TO R. C. EVANS\n\nThe following letter was published in the Toronto _Daily Star_ in\nanswer to the false charges which appeared in Mr. Evans' interview.\n\nSalt Lake City, Feb. 19, 1905.\n\n_Mr. R. C. Evans_,\n\n_Counselor in Presidency of Reorganized Church_.\n\nSir:--I have before me a copy of the Toronto _Daily Star_, bearing date\nof January 28, last, in which there is a column on the front page,\npurporting to be an interview, by a representative of that paper with\nyou, in which I desire to call your attention.\n\nIn doing so I desire to be fair and dispassionate, and also candid, and\nI would like it if you would receive and reply to this communication in\nthe same spirit and manner to me personally.\n\nYou are reported as not being \"pleased,\" nor Toronto's six hundred\nbaptized members, with the name \"Mormon.\" \"This fact,\" says the _Star_,\n\"was emphasized today when R. C. Evans, one of the three members\nof the Presidency explained the radical difference between the two\ndenominations. Mr. Evans * * * denounced the Utah Mormon and his\niniquities.\" Then you are made to say: \"The term Mormon is offensive\nto us, because it is associated in the public mind with the practices\nthat I have specified.\" That is, the alleged practices of the Utah\n\"Mormons,\" namely, \"polygamy and blood atonement.\"\n\nDid you know that \"the term Mormon\" has always been applied to the\nChurch of Jesus Christ of Latter-day Saints? That the name attached\nto the Church with the publication and promulgation of the Book of\nMormon? That it was first applied by the enemies of the Church as an\nopprobrium; but that during the lifetime of Joseph Smith the Martyr,\nand ever since it has been a term accepted by the Church because of\npopular custom, as an appellation?\n\nIf, then, the name is so distasteful to you and your fellows in\nCanada and throughout the world, although it be on the grounds you\nhave named, why do you not discard the Book of Mormon, from whence\nthe name is derived, as well as the name. Is not the term _Book of\nMormon_ as closely associated in the public mind with \"polygamy and\nblood atonement,\" as is the _name_ of the Book? How are you going to\ndisassociate the book itself from the name as commonly applied to\nthe Church, since this name has been attached to the Church from the\nbeginning, and before the alleged \"practices\" of the \"Utah Mormon\"\ngained such publicity? _Really, I think it would be quite proper for\nthose holding the view which you are said to have expressed, not only\nto renounce the name \"Mormon\" as applied to the Church but also the\nBook itself_.[1]\n\nYou do not believe in blood atonement. Is not this the more reason why\nyou should discard the Book of Mormon? Are you not at issue with the\nteachings not only of that book, but also with those of the Bible on\nthis matter? If so, why not discard the Bible, and while you are about\nit, the Book of Doctrine and Covenants also? Both of these, as well as\nthe Book of Mormon, teach the doctrine of \"blood atonement,\" and they\nare all \"associated in the public mind\" with the alleged \"practices\" of\nthe Church of Jesus Christ of Latter-day Saints.\n\nLet us consider this subject of \"blood atonement.\"\n\nBook of Mormon:\n\n    Mosiah 3:11.--His blood atoneth for the sins of those who have\n    fallen by the transgression of Adam. Verse 15.--And understood not\n    that the law of Moses availeth nothing except it were through the\n    atonement of his blood. Verse 16.--Even so the blood of Christ\n    atoneth for their sins.\n\n    Alma 21:9.--Now Aaron began to open the Scriptures unto them\n    concerning the coming of Christ, and also concerning the\n    resurrection of the dead, and that there could be no redemption for\n    mankind, save it was through the death and suffering of Christ, and\n    the atonement of his blood.\n\n    I Nephi 12:10.--Their garments are made white in his blood.\n\n    II Nephi 9:7.--And if so, (not an infinite atonement) this flesh\n    must have laid down to rot and to crumble to its mother earth, to\n    rise no more.\n\nFrom the Bible:\n\n    Mark 14:22-25.--And as they did eat, Jesus took bread and blessed\n    and brake it, and gave to them, and said: Take, eat; this is my\n    body.\n\n    And he took the cup, and when he had given thanks, he gave it to\n    them: and they all drank of it.\n\n    And he said unto them, This is my blood of the new testament which\n    is shed for many.\n\n    Verily I say unto you, I will drink no more of the fruit of the\n    vine, until that day that I drink it new in the Kingdom of God.\n\nFrom the Doctrine and Covenants:\n\n    Section 45:4.--(Utah edition) Saying, Father, behold the sufferings\n    and death of him who did no sin, in whom thou wast well pleased;\n    behold the blood of thy Son which was shed--the blood of him whom\n    thou gavest that thyself might be glorified.\n\n    Section 74:7.--But little children are holy, being sanctified\n    through the atonement of Jesus Christ, and this is what the\n    scriptures mean.\n\n    Section 76:39-41.--For all the rest shall be brought forth by the\n    resurrection of the dead, through the triumph and the glory of the\n    Lamb, who was slain, who was in the bosom of the Father before the\n    worlds were made. And this is the gospel, the glad tidings which\n    the voice out of the heavens bore record unto us. That he came into\n    the world, even Jesus, to be crucified for the world, and to bear\n    the sins of the world, and to sanctify the world, and to cleanse it\n    from all unrighteousness.\n\n    Section 29:1.--Listen to the voice of Jesus Christ, your Redeemer,\n    the Great I AM, whose arm of mercy hath atoned for your sins. Verse\n    17.--And it shall come to pass, because of the wickedness of the\n    world, that I will take vengeance upon the wicked, for they will\n    not repent; for the cup of mine indignation is full; for behold, my\n    blood shall not cleanse them if they hear me not.\n\n\nSTATEMENT OF AN ENEMY\n\nBut the report says: \"This doctrine was introduced by Brigham Young\"\nand that it meant \"death to anyone who left the Church * * * that\nthe apostate whose throat was cut from ear to ear * * * saved his\nsoul.\" Why you made this statement you best know; but were you not\naware that it was but the repetition of the ravings of enemies of the\nChurch, without one grain of truth? Did you not know that not a single\nindividual was ever \"blood atoned,\" as you are pleased to call it, for\napostasy or any other cause? Were you not aware, in repeating this\nfalse charge, that it was made by the most bitter enemies of the Church\nbefore the death of the Prophet Joseph Smith? Do you know of anyone\nwhose blood was ever shed by the command of the Church, or members\nthereof, to \"save his soul?\" Did you not know that you were embittering\nthe people against the \"Mormon\" Elders, and that just such malicious\ncharges and false insinuations have made martyrs for the Church, whose\nblood does not \"cease to come up into the ears of the Lord of Sabaoth?\"\n\nNever in the history of this people can the time be pointed to when\nthe Church ever attempted to pass judgment on, or execute an apostate\nas per your statement. There are men living in Utah today who left the\nChurch in the earliest history of our State who feel as secure, and are\njust as secure and free from molestation from their former associates\nas you or any other man could be.\n\n\nEFFICACY OF THE BLOOD OF CHRIST\n\nThe Latter-day Saints believe in the efficacy of the blood of Christ.\nThey believe that through obedience to the laws and ordinances of the\nGospel they obtain a remission of sins; but this could not be if Christ\nhad not died for _them_. If you did believe in blood atonement, I might\nask you why the blood of Christ was shed? and _in whose stead was it\nshed_? I might ask you to explain the words of Paul: \"Without shedding\nof blood is no remission.\"\n\n\nUNPARDONABLE SINS\n\nAre you aware that there are certain sins that man may commit for which\nthe atoning blood of Christ does not avail? Do you not know, too, that\nthis doctrine is taught in the Book of Mormon? And is not this further\nreason why you should discard the Book as well as the name? Is it not\nsafe for us to rely upon the scriptures for the solution of problems of\nthis kind? Let me quote:\n\nFrom the Book of Mormon:\n\n    II Nephi 9:35.--Wo unto the murderer who deliberately killeth, for\n    he shall die.\n\n    Alma 1:13, 14.--And thou hast shed the blood of a righteous man,\n    yea, a man who has done much good among this people; and were we to\n    spare thee, his blood would come upon us for vengeance.\n\n    Alma 42:19.--Now, if there were no law given--if a man murdered he\n    should die, would he be afraid he would die if he should murder?\n\nFrom the Bible:\n\n    Genesis 9:12, 13.--And whoso sheddeth man's blood, by man shall his\n    blood be shed; for man shall not shed the blood of man.\n\n    For a commandment I give, that every man's brother shall preserve\n    the life of man, for in mine own image have I made man. (Inspired\n    translation.)\n\n    Luke 11:50.--That the blood of all the prophets, which was\n    shed from the foundation of the world, may be required of this\n    generation.\n\n    Hebrews 9:22.--And almost all things are by the law purged with\n    blood; and without shedding of blood is no remission.\n\n    Hebrews 10:26-29.--For if we sin wilfully, after that we have\n    received the knowledge of the truth, there remaineth no more\n    sacrifice for sins.\n\n    * * * *\n\n    He that despised Moses' law died without mercy under two or three\n    witnesses;\n\n    Of how much sorer punishment, suppose ye, shall he be thought\n    worthy, who hath trodden under foot the Son of God, and hath\n    counted the blood of the covenant, wherewith he was sanctified, an\n    unholy thing.\n\n    (I commend to you the careful reading of these two chapters:)\n\n    I John 3:15.--No murderer hath eternal life abiding in him.\n\n    I John 5:16.--If any man see his brother sin a sin which is not\n    unto death, he shall ask, and he shall give him life for them that\n    sin not unto death. There is a sin unto death: I do not say that he\n    shall pray for it.\n\nFrom the Doctrine and Covenants:\n\n    Section 87:7.--That the cry of the saints, and of the blood of\n    the saints, shall cease to come up into the ears of the Lord of\n    Sabbath, from the earth, to be avenged of their enemies.\n\n    Section 101:80.--And for this purpose have I established the\n    constitution of this land, by the hands of wise men, whom I raised\n    up unto this very purpose, and redeemed the land by the shedding of\n    blood.\n\n    Section 42:18, 19.--And now, behold, I speak unto the church. Thou\n    shalt not kill; and he that kills shall not have forgiveness in\n    this world, nor in the world to come.\n\n    And again, I say, thou shalt not kill; but he that killeth shall\n    die.\n\n    Verse 79.--And it shall come to pass, that if any persons among you\n    shall kill, they shall be delivered up and dealt with according to\n    the laws of the land; for remember that he hath no forgiveness, and\n    it shall be proved according to the laws of the land.\n\n\nTHE LAW OF CAPITAL PUNISHMENT\n\nIn pursuance of, and in harmony with this scriptural doctrine, which\nhas been the righteous law from the days of Adam to the present\ntime, the founders of Utah incorporated in the laws of the Territory\nprovisions for the capital punishment of those who wilfully shed the\nblood of their fellow man. This law, which is now the law of the State,\ngranted unto the condemned murderer the privilege of choosing for\nhimself whether he die by hanging, or whether he be shot, and thus have\nhis blood shed in harmony with the law of God; and thus atone, so far\nas it is in his power to atone, for the death of his victim. Almost\nwithout exception the condemned party chooses the latter death. This\nis by the authority of the law of the land, not that of the Church.\nThis law was placed on the statutes through the efforts of the \"Mormon\"\nlegislators, and grants to the accused the right of jury trial. It is\nfrom this that the vile charge, which you are pleased to repeat, has\nbeen maliciously misconstrued by the enemies of the Church, who prefer\nto believe a lie. When men accuse the Church of practicing \"blood\natonement\" on those who deny the faith, or, for that matter, on any\nliving creature, they know that they bear false witness, and they shall\nstand condemned before the judgment seat of God.\n\n\nPLURAL MARRIAGE\n\nSince the action taken by the United States government, and also by the\nChurch, in regard to plural marriage, I shall not discuss its virtues\nnor answer arguments in opposition to that principle as a principle of\nour faith. As you, however, are reported to have said that \"Brigham\nYoung introduced\" that doctrine \"in Salt Lake City,\" I would be pleased\nif you would explain, as a matter of history, why Sidney Rigdon, before\n\"President Young introduced\" the doctrine, declared that the principle\nof plural marriage was introduced, to his knowledge, by Joseph Smith\nthe Prophet, and that he, Sidney Rigdon, rejected that doctrine and\n\"warned Joseph Smith and his family\" that it would bring ruin upon\nthem. You will find this in the _Messenger and Advocate_, published\nin June, 1846, volume 2, page 475, number 6. Will you kindly explain\nwhy this same Sidney Rigdon practiced polygamy, which he so fervently\ncondemns? Will you kindly explain why Lyman Wight, James J. Strang,\nGladden Bishop, William Smith, and others, none of whom had much love\nfor President Young and did not follow him, also taught and practiced\npolygamy _before plural marriage was \"introduced by President Young_.\"\nIf you doubt this, I will gladly furnish you with the proof. Indeed,\nyou may find a great deal of it in the third volume of your church\nhistory.\n\n\nTHE \"SAINTS' HERALD\" AS A WITNESS\n\nIf you believe your statement to be true, will you kindly explain the\nfollowing paragraph in the _Saints Herald_, your official organ, volume\n1, page 9. It would be well for you to read the entire chapter, which\nis entitled \"polygamy.\" The quotation is:\n\n\"_The death of the prophet is one fact that has been realized, although\nhe abhorred and repented of this iniquity (meaning 'polygamy,') before\nhis death. This branch of the subject we shall leave to some of our\nbrethren, who are qualified to explain it satisfactorily_.\"\n\nIn the same volume, page 27, what is meant by the following?\n\n\"_He (Joseph Smith) caused the revelation on the subject ('polygamy')\nto be burned, and when he voluntarily came to Nauvoo and resigned\nhimself into the arms of his enemies he said that he was going to\nCarthage to die. At that time he also said that if it had not been for\nthat accursed spiritual wife doctrine, he would not have come to that.\"\nKindly read the context_.\n\n_There is more evidence that can be produced, but if you will explain\nthis it may suffice_.\n\nIn the light of the knowledge I have received and the evidence at my\ncommand, I know that the Prophet Joseph Smith made no such statement\nas the above, and that he did not have the revelation burned. _There\nis, however, value in the above statements from your \"Herald,\" for they\nbear witness to the origin and introduction of the principle of plural\nmarriage, and the revelation concerning the same_.[2]\n\n\nTHE UTAH VISIT\n\nIn connection with this, let me call your attention to your visit to\nSalt Lake City some three years ago. At that time you met President\nLorenzo Snow, a man whose veracity cannot justly be questioned; you\nheard him bear his testimony to the effect that he was taught that\nprinciple by the Prophet Joseph Smith, and that the Prophet declared\nto Lorenzo Snow that he had married his sister, Eliza R. Snow. You\nmet and conversed with Lucy Walker Smith, and she told you that she\nwas married to the Prophet Joseph Smith on the first day of May,\n1843, in Nauvoo, Elder William Clayton performing the ceremony. You\nmet Catherine Phillips Smith, who told you she was married in August,\n1843, in Nauvoo, to the Patriarch Hyrum Smith, his brother Joseph the\nProphet officiating in that ceremony. You will remember that the first\nwives of both these men were living at the time. I hardly think these\ntestimonies have passed from your memory in so brief a time. I am\npersonally acquainted with these women, and know that they are truthful\nand honest--honorable women, whose testimonies should be believed.\n\nIn the face of all this evidence, do you think it fair and consistent\nfor you and your fellow believers to constantly lay at the door of\nPresident Young the responsibility for the \"introduction of plural\nmarriage\" and the \"authorship\" of the above mentioned revelation?\n\nMy letter is already long, but I desire to briefly mention another item\nor two.\n\n\nPRESIDENT SMITH'S DENIAL\n\nIn the interview you are made to say that while on your visit to Salt\nLake City, you spent a day and a half with Joseph F. Smith; that you\nand he \"differed on polygamy,\" and that you \"told him it was vile and\nwicked, always had been, and always would be.\" I took occasion to ask\nmy father if you and he had discussed polygamy at that time and if\nyou had uttered that above expression or any other of like nature. He\nreplied that he had no discussion with you on that subject; that you\ndid not say one word to him in relation to polygamy, either favorable\nor otherwise; that your visit was a social one, and friendly, and\nwas not occupied by the discussion of any differences which may have\nexisted.\n\n_It is true that President Young was elected president at Kanesville,\nbut on what grounds do you charge him with holding the office in\ntrust for the \"dead president's son?\" Do you not know that such a\nstatement--contrary to the written word--was antagonistic to the\nteachings of President Young, as recorded in the \"Times and Seasons,\"\nas well as since that Time_?\n\n\nPRESIDENT YOUNG THE PROPHET'S FRIEND\n\n_Will you please explain on what grounds you charge President Young as\nbeing \"under suspicion at the time of Joseph Smith's death?\" Am I to\ninfer by this that you mean to convey the idea that Brigham Young was\nin any way responsible for the death of Joseph Smith? The Prophet never\nhad a truer friend. You know that at the time of the martyrdom Brigham\nYoung was on a mission away from home. If this is the inference you\nwish to convey, it is not only contemptible but viciously false_.[3]\n\nWith reference to my father's first wife, you say she died \"broken\nhearted and insane.\" If you mean to insinuate that this condition, if\ntrue, was the result of any act whatever on the part of my father, it\nis also scandalously false. I have good reason to believe that she died\nneither broken hearted nor insane. If it were true, I would still think\nthat you, as a professed minister of the Gospel, might employ your time\nto better advantage than as an aspersor or a scandal-monger.\n\n                                          Respectfully,\n                                             Joseph F. Smith, Jr.\n\n\nFootnotes\n\n1. This sentence in italics was omitted in Mr. Evans' publication of\nthe _entire matter_ in the _Zion's Ensign_, August 17th, 1905.\n\n2. The quotations from the _Saints' Herald_ which are in Italics were\npurposely omitted from Mr. Evans' \"publication of the entire matter,\"\nas it appeared in the _Zion's Ensign_ of August 7, 1905. The reason for\nthe suppression of this evidence is easy to discern. The authorities of\nthe \"Reorganization\" have tried to destroy the evidence, that it could\nnot be circulated among their church members, therefore very few copies\nof this particular _Herald_ can today be found.\n\n3. These paragraphs in italics were also omitted from Mr. Evans'\n\"publication of the entire matter,\" as it appeared in the _Zion's\nEnsign_ August 17, 1905.\n\n\n\nMR. EVANS' LETTER\n\n_Mr. Joseph F. Smith, Jr.:_\n\nSir:--Your open letter published in the Toronto _Star_ for February 25,\nis before me. You say: \"I desire to be fair, dispassionate and also\ncandid.\" Those who read your letter will see plainly that you have\nmispresented the interview, my faith and the facts concerning my visit\nto Salt Lake, and that you are guilty of a labored effort to cover up\nthe _true facts_ regarding \"blood atonement,\" \"polygamy,\" etc., and my\nfaith in the Book of Mormon. So much for those desires.\n\nMy position with regard to the Book of Mormon, and the name \"Mormon,\"\nis too well known for you to blind the people concerning it. The\ninterview shows plainly in what sense \"the term 'Mormon' is offensive\nto us.\" Read it again, sir: \"Because it is associated in the public\nmind with the practices that I have specified.\" The abominations of\n_Brighamism_; namely, polygamy, blood atonement, Adam-God,[1] and other\nevils that have disgraced the name throughout civilization.\n\nThe true Church never has adopted the name \"Mormon\" as being the\nproper name of the church. The Latter-day Saints were sometimes called\n\"Mormons\" in derision, as you admit, because they believed in the\ndivine authenticity of the Book of Mormon, and some church members may\nhave been willing to be called \"Mormon\"; yet you \"candidly (?) fairly,\ndispassionately\" ask me, \"Why do you not discard the Book of Mormon\nfrom whence the name is derived?\" Now, sir, I profess to believe in the\ndivine authenticity of the Holy Bible; as well call me a Bible, because\nI believe in the Bible,[2] as call me a Mormon because I believe in the\nBook of Mormon.\n\nThe church that I have the honor to represent is incorporated under the\nlaws of the United States as \"The Reorganized Church of Jesus Christ of\nLatter Day Saints.\"\n\n\nBLOOD ATONEMENT\n\nThere is not an honest thinking person on earth who is acquainted with\nthe faith of the church regarding the atonement of Jesus Christ but\nthat will say your attempt to misrepresent my faith in this regard is\ndiametrically opposite to your stated desire to be \"fair, dispassionate\nand candid.\" You know that a prominent article in the Epitome of\nthe Faith and Doctrine of the _true church_ reads as follows: \"We\nbelieve that through the atonement of Christ, all men may be saved by\nobedience to the laws and ordinances of the gospel.\" You know that the\ntrue church believes in the atoning blood of Christ as stated in the\nscriptures you cite in your letter, and yet you try to make out that\nbecause we do not believe in the doctrine of blood atonement as taught\nby Brigham Young and his successors in \"Utah Mormonism,\" that we do not\nbelieve in the atonement of our Lord and Savior Jesus Christ.\n\nThe doctrine of the atonement of Christ is far above the doctrine of\nblood atonement as taught by Brighamism. To prove this, I submit the\nstatements as made by Brigham Young and other leading members of the\nUtah Church, as found in their sermons, printed by your church:\n\nBrigham Young said, October 9, 1852: \"What shall be done with the sheep\nthat stink the flock so? We will take them, I was going to say, and\ncut off their tails two inches behind their ears; however I will use a\nmilder term, and say cut off their ears.\"--Journal of Discourses, vol.\n1:213.\n\nBrigham said again, March 27, 1853: \"I say, rather than that apostates\nshould flourish here, I will unsheath my bowie knife, and conquer or\ndie. (Great commotion in the congregation and a simultaneous burst\nof feeling, assenting to the declaration.) Now, you nasty apostates,\nclear out, or judgment will be put to the line and righteousness to\nthe plummet. (Voices generally, 'Go it, go it.') If you say it is all\nright, raise your hands (all hands up). Let us call upon the Lord to\nassist us in this and every good work.\"--Journal of Discourses, vol.\n1:83.\n\nEchoing what Brigham said, P. P. Pratt said, on March 27, 1853, \"My\nfeelings are with those who have spoken, decidedly and firmly so. * *\n* I need not repeat their doom, it has been told here today, they have\nbeen faithfully warned. * * * It is too late in the day for _us_ to\nstop and inquire whether such an outcast has the truth.\"--Journal of\nDiscourses, vol. 1, pp. 84, 86.\n\nElder Orson Hyde said April 9, 1853: \"Suppose the shepherd should\ndiscover a wolf approaching the flock, what would he be likely to do?\nWhy, we would suppose, if the wolf was within proper distance, that he\nwould kill him at once * * * kill him on the spot. * * * It would have\na tendency to place a terror on those who leave these parts, that may\nprove their salvation when they see the heads of thieves taken off, or\nshot down before the public.\"--Journal of Discourses, vol. 1:72, 73.\n\nPresident Brigham Young preached, February 8, 1857, as follows \"All\nmankind love themselves; and let these principles be known by an\nindividual and he would be glad to have his blood shed. That would be\nloving themselves even to an eternal exaltation. Will you love your\nbrothers and sisters likewise when they have committed a sin that\ncannot be atoned for without the shedding of blood? That is what Jesus\nChrist meant. He never told a man or woman to love their enemies in\ntheir wickedness. He never intended any such thing.\n\n\"I could refer you to plenty of instances where men have been\nrighteously slain in order to atone for their sins. I have seen scores\nand hundreds of people for whom there would have been a chance in the\nlast resurrection if their lives had been taken and their blood spilled\nupon the ground, as a smoking incense to the Almighty, but who are now\nangels to the devil, until our elder brother, Jesus Christ, raises them\nup, conquers death, hell and the grave.[3] I have known a great many\nmen who have left this church, for whom there is no chance whatever for\nexaltation; but if their blood had been spilt it would have been better\nfor them. The wickedness and ignorance of the nations forbid this\nprinciple being in full force, but the time will come when the law of\nGod will be in full force.\n\n\"This is loving our neighbor as ourselves; if he needs help, help\nhim; and if he wants salvation and it is necessary to spill his blood\nupon the ground in order that he may be saved, spill it.\"--Journal of\nDiscourses, vol. 4, p. 220, or Deseret News, vol. 6, p. 397.\n\nPresident J. M. Grant said, September 21, 1856: \"I say there are men\nand women here that I would advise to go to the president immediately,\nand ask him to appoint a committee to attend to their case, and\nthen let a place be selected, and let that committee shed their\nblood.\"--_Deseret News_, vol. 6, p. 235.\n\nPresident Heber C. Kimball said; July 19, 1854: \"It is believed in the\nworld that our females are all common women. Well, in one sense they\nare common--that is, they are like all other women, I suppose, but they\nare not unclean, for we wipe all unclean ones out of our midst; we not\nonly wipe them from our streets, but we wipe them out of existence. And\nif the world wants to practice uncleanness, and bring their prostitutes\nhere, if they do not repent and forsake their sins, we will wipe the\nevil out. We will not have them in this valley unless they repent, for\nso help me God, while I live I will lend my hand to wipe such persons\nout, and I know this people will.\"--_Deseret News_, August 16, 1854,\nand _Millennial Star_, vol. 16, pages 738-9.\n\nThe above statements speak for themselves, and these were what I read\nto the reporter. You ask, \"Do you _know_ of anyone whose blood was\never shed by the command of the church or members thereof to save his\nsoul?\" To _know_ by hearing such a command given, or seeing a murder\ncommitted, is one thing, to believe the evidence of many who have\ntestified is another. No sir, I was never present when such a command\nwas given, nor when murder was committed; but I have read that which\nleads me to believe that under Brighamism, Utah was for years a land\nof assassination and a field of blood. What of the Mountain Meadow\nmassacre--the destruction of the Aiken party; the dying confession of\nBishop J. D. Lee; the Hickman butcheries; the Danties? Alfred Henry\nLewis, writing in _Collier's Weekly_ for March 26, 1904, states:\n\"Brigham Young invented his destroying angels, placed himself at their\nhead, and when a man rebelled, _he had him murdered_, if one fled the\nfold he was pursued and slain.\"\n\nThe world has recently read the testimony of persons under oath, in\nWashington, who testified concerning the endowment oaths, so I will\nforbear any further remarks on this subject.\n\n\nPOLYGAMY\n\nSpeaking of \"plural marriage,\" you say, \"I shall not discuss its\n_virtues_.\" Surely that is kind. Let civilization give ear, Mr. Smith\ncalls that a virtue which wrecks the happiness of every woman who is\nenslaved by it, that doctrine which permits Brighamites to live in what\nthey call marriage with three sisters at one time, with mother and\ndaughter at the same time. Your father, Joseph F. Smith, married and is\nnow living with _two sisters as wives_. I refer to Julina Lambson and\nEdna Lambson, both bearing children to him; yet you call that system a\n_virtue_.\n\nI have no evidence that those men you refer to, as having practiced\npolygamy _before Young was guilty, as stated by you_. But the following\nevidence shows clearly that Brigham Young was under suspicion before\nJoseph's death, and that he has since admitted that he had a revelation\non polygamy before the church knew anything of the doctrine:\n\nIn a speech of Brigham Young on June 21, 1874, (see _Deseret News_ of\nJuly 1, 1874), we read the following statement relative to the origin\nof this doctrine of polygamy:\n\n    While we were in England (in 1839 and 1840, I think) the Lord\n    manifested to me by vision and His Spirit, things that I did not\n    then understand. I never opened my mouth to anyone concerning them,\n    until I returned to Nauvoo; _Joseph had never mentioned this; there\n    had never been a thought of it in the church_ that I ever knew\n    anything about at that time;--but I had this for myself and kept it\n    for myself.--The Messenger, volume 1, page 29.\n\nWell, no one need blame Joseph any more, Brigham is the self-confessed\nchannel through which polygamy was given to his people.\n\nI here submit the testimony of Brigham Young's legal wife, who left him\nafter he was untrue to her. Testimony of Major Thomas Wanless, given to\nR. C. Evans, his nephew, in the presence of Mrs. Wanless, Mrs. Evans\nand her daughter, in St. Louis, Missouri, September 7, 1904:\n\n    I met Brigham Young's first and legal wife and her daughter in the\n    winter of 1860 and 1861, at Central City, Colorado; she told me\n    that Joseph Smith had nothing to do with polygamy; that he did not\n    teach, practice, or in any way endorse the doctrine of polygamy,\n    that he had nothing to do with the so-called revelation on\n    celestial marriage; that he had but one wife. My husband, Brigham\n    Young, Orson Pratt (she gave the name of another man whose name I\n    have forgotten) made up the revelation on celestial marriage.\n\n    Before they left Illinois some of them practiced polygamy. Brigham\n    Young went to Utah to reorganize the church and publicly introduced\n    polygamy, or to reorganize the Church on a polygamous basis.\n\n    She left Brigham Young, finally obtained a divorce from him, and\n    was then living with her daughter. Brigham sent the daughter money\n    according to an agreement. She told me they ought to have shot\n    Brigham Young in place of Joseph Smith.\n\nThis statement of Major Wanless that she was Brigham's first wife is\na mistake. Brigham married Miriam Works, October 8, 1824; she died\nSeptember 8, 1832. In February, 1834, he married May Ann Angel; she\nwas his _legal wife_, and perhaps is the one referred to by the Major.\nIt is quite pardonable in Major Wanless in getting Brigham's wives\nmixed up. We opine poor Brigham was at his wit's end to keep the family\nrecord correct himself.\n\nChambers' encyclopedia, volume 8, students' edition, confirms Mrs.\nYoung's statement, in part. It says, speaking of the practice of\npolygamy: \"Young, Pratt and Hyde are its true originators. Emma, wife\nand widow of the prophet, stoutly denied that her husband had any wife\nbut herself. Young's revelation she declared to be a fraud.\"\n\nFrom a host of other witnesses who testify that Brigham Young was the\nman that introduced polygamy in the Church, I submit the statement of\nanother broken-hearted woman from the ranks of Brigham's Church. Fanny\nStenhouse says: \"Polygamy was unheard of among the (English) Saints in\n1849.\" (pages 45, 47, 48) \"Tell It All,\" by Fanny Stenhouse. \"In June\n1850, I heard the first whisper of polygamy. In January, 1853, I first\nsaw the revelation on Polygamy; it was published in the _Millennial\nStar_,\" (page 132).\n\n\"Out of thirty thousand Saints in England in 1853, 1776 had been\nexcommunicated for apostasy through polygamy, the president of the\nconference was cut off,\" (page 160). When speaking regarding polygamy\nshe says: \"They know that the only source of all their revelations is\nthe man BRIGHAM YOUNG,\" (page 190).\n\n\"Brigham has outraged decency and driven asunder the most sacred ties,\nby his shameless introduction of polygamy,\" (page 273).\n\n\"There have been many apostates from the teachings of Joseph Smith in\nearly days, but of all apostates, Bro. Brigham is the chief,\" (page\n614).\n\nIt is reported by Fanny Stenhouse, and many others, that Joseph Smith\nsaid, \"If ever the Church had the misfortune to be led by Bro. Brigham,\nhe would lead it to hell,\" (page 268).\n\nWhy did Joseph Smith a short time prior to his death make the above\nand similar statements regarding the man Brigham Young? The reason is\nplain. He too had doubtless heard some rumors as to his conduct and\nsecret teachings, and the evidence would seem to indicate that just\nbefore his death he made a move to bring the guilty to judgment. We\nwill let William Marks, who was president of the Nauvoo Stake at the\ntime of Joseph Smith's death testify:\n\n\"A few days after this occurrence, I met with Bro. Joseph, he said\nthat he wanted to converse with me on the affairs of the Church, and\nwe retired by ourselves; I will give his words _verbatim_ for they are\nindelibly stamped upon my mind. He said he had desired for a long time\nto have a talk with me on the subject of polygamy. He said it would\neventually prove the overthrow of the Church, and we should soon be\nobliged to leave the United States, unless it could be speedily put\ndown. He was satisfied that it was a cursed doctrine, and that there\nmust be every exertion to put it down. He said that he would go before\nthe congregation and proclaim against it, and I must go into the High\nCouncil, and he would prefer charges against those in transgression,\nand I must sever them from the Church unless they made ample\nsatisfaction. There was much more said, but this was the substance. The\nmob commenced to gather about Carthage in a few days after, therefore\nthere was nothing done concerning it.\" (_Saints' Herald_, vol. 1, pp.\n22, 23.)\n\nPresident Marks, after Joseph Smith's death, made mention of the above\nconversation; it was soon rumored that he was about to apostatize, and\nthat his statement was a tissue of lies.\" (See _Saints' Herald_, vol.\n1, pp. 22, 23.)\n\nSpeaking of the revelation on polygamy, Marks said, \"I never heard of\nit during Joseph's life. It was evidently gotten up by Brigham Young\nand some of the Twelve, after Joseph's death.\" (Briggs' Autobiography;\n_Herald_ 1901.)\n\nNow I propose to produce evidence showing that Joseph Smith and the\nChurch during his lifetime condemned polygamy in the strongest terms.\nFirst, I submit the testimony of thirty-one witnesses as published by\nthe Church on October the 1st, 1842. We deem this sufficient to show\nyou where Joseph and Hyrum Smith stood on this question of polygamy.\n\n\"We, the undersigned members of the Church of Jesus Christ of\nLatter-day Saints, and residents of the city of Nauvoo, persons of\nfamilies, do hereby certify and declare, that we know of no other\nrule or system of marriage than the one published from the Book of\nCovenants, and we give this certificate to show that Dr. John C.\nBennett's secret wife system is a creature of his own make, as we know\nof no such society in this place, nor never did.\"\n\nThis is signed by a number of the leading men of the Church, some of\nthe Twelve Apostles, some of the First Presidency of the Utah Church,\nand a number of the leading men of the Church. A similar document is\nsigned by Emma Smith the wife of Joseph Smith, and a number of the\nleading women of the Church, thirty-one witnesses in all.\n\nNow I submit for your consideration a statement made by Joseph Smith\nand his Brother Hyrum just a few months prior to their assassination.\nThey learned that a man up here in the state of Michigan was teaching\npolygamy, and this is what they said about it: \"As we have lately\nbeen credibly informed that a member of the Church of Jesus Christ of\nLatter-day Saints, a man by the name of Hyrum Brown, has been teaching\npolygamy and other false and corrupt doctrines, in the county of\nLapeer, state of Michigan, this is to notify him and the Church in\ngeneral that he has been cut off from the Church for his iniquity.\"\nSigned, Joseph Smith, Hyrum Smith, Presidents of the Church.\n\nThis was given in February, 1844. Joseph was killed four months after\nthat. Here he declares that polygamy is a crime, and the man was\nexcommunicated from the Church for preaching it. Now I want to give\nyou the testimony of George Q. Cannon, whom I met in Salt Lake City,\nas one of the presidency of the Salt Lake Mormon Church: \"A prevalent\nidea has been that this prejudice against us owes its origin and\ncontinuation to our belief in a plurality of wives. * * * Joseph and\nHyrum Smith were slain in the Carthage Jail, and hundreds of persons\nwere persecuted to death previous to the Church having any knowledge of\nthis doctrine.\"--_Journal of Discourses_, vol. 14, pages 165, 166.[4]\n\nThis being true, Joseph Smith was not guilty of the practice of\npolygamy; he was killed before the people knew anything about polygamy.\nThis is the statement of George Q. Cannon. Let me strengthen this now\nby the son-in-law of Brigham Young, H. B. Clawson:\n\n\"Polygamy at that time (that is at the time of Joseph Smith's death)\nwas not known among those of the Mormon faith. * * * The doctrine of\npolygamy was not promulgated until they got to Salt Lake; not, in\nfact, until some little time after they had arrived there.\" Salt Lake\n_Herald_, February 9, 1882.[5]\n\nJoseph Smith was killed in 1844. They arrived in Salt Lake the 24th of\nJuly, 1847, and he says not until some little time after that was it\nintroduced. The little time was the 29th of August, 1852, eight years\nand two months after the assassination of Joseph Smith.\n\nWe have Brigham Young himself on this. He being interviewed by\nSenator Trumbull in 1869, said: \"It (polygamy) was adopted by us as a\n_necessity_ after we came here.\" Ah, there never was a greater truth\ntold in all the world than that. Polygamy was not an original tenet\nof the Church, and Brigham Young says it was adopted as a _necessity_\nafter \"we came here.\" The real facts are, Brigham Young, as I will show\nfrom their own evidence, and a few other Elders were living vile lives\nsecretly, and to cover up the consequences of their bad conduct, as he\ntruthfully says in this \"as a necessity\"; yea, as a necessity polygamy\nwas introduced. But who will dare to blame Joseph Smith for their\nintroducing polygamy eight years after his death?\n\nI have been careful to take these clippings right from their own\npapers, so that they cannot say that we have changed the words or\nanything of that kind. Here is another statement; this is found from\nElder Ephraim Jenson:\n\n\"Polygamy was not practiced by the Mormons prior to and at the time of\nthe execution of Joseph Smith, who was executed at Nauvoo, Illinois.\n* * * Fourth, that only three per cent of the Mormon men practiced\npolygamy, a proof itself that it was not essential to the creed.\"--_The\nYeoman's Shield._\n\nHere is another one:\n\n    \"Go back to the foundation of our Church, April 6, 1830, there was no\n    polygamy practiced or taught in Mormon literature until five years\n    after that band of persecuted Saints reached Utah.\" _New York Herald_,\n    January 8, 1900.[6]\n\nThis is by Elder Whitaker, who knew who _did_ introduce this polygamy.\nNow I might introduce dozens and dozens of witnesses to prove that\nJoseph Smith had nothing to do with it. Well, who did it? Here is\nwhat the Apostle's wife says of it: \"How then, asked the reader, did\npolygamy originate? It was born in the vile and lustful brain of\nBrigham Young, and was grafted on the faith to gratify his sensual\nbestiality.\"[7] (Mysteries of Mormonism, pp. 16, 17.)\n\nOne of the Mormon wives said that, and she ought to know whereof she\naffirms.\n\nWe have learned from the above statements that polygamy was not taught\nor practiced by Joseph Smith, but was introduced into an apostate\nbranch of the church, after his death, as is admitted by Brigham Young\nand others of his followers.\n\nHaving read the works of the church for over a quarter of a century. I\nconfidently affirm that there is not a single word, in a single sermon,\nlecture, statement, newspaper or church publication printed during the\nlifetime of the Prophet Joseph Smith wherever he, by word, has endorsed\nthe doctrine of plurality of wives; not a single statement; and there\nis no Salt Lake Mormon breathing who can produce one and prove its\nauthenticity.\n\nBut suppose you could prove that Joseph Smith secretly taught and\npracticed polygamy, that would not make it a Christian doctrine. If\nJoseph Smith secretly taught, practiced, or endorsed the doctrine of\npolygamy, he did it contrary to all the revelations given for the\ngovernment of the church in the Bible, Book of Mormon, and Doctrine and\nCovenants; contrary to all his sermons, speeches, and public teachings;\nand he was a criminal before the law of his country, a base hypocrite\nbefore the God whom he openly worshiped, a despicable traitor to the\nwoman whom he claimed to love and cherish as his wife, and was untrue\nto all the sacred principles of fidelity and integrity which he evinced\nin all his public utterances and conduct.\n\nIn the face of all this, the wife and children of Joseph Smith,\ntogether with thousands of people who knew him in life, refuse to\nbelieve the contradictory statements of Brigham Young and others who\nare wallowing in the mire of polygamy.\n\n\nMY VISIT TO UTAH\n\nIf your father denies that he and I discussed the doctrine of polygamy,\nall I have to say about it is, that what he states is untrue. Here are\na few points that may help him to remember what was said and done:\nWhen talking with Joseph F. Smith in Salt Lake City two years ago,\nhe brought up a number of witnesses and I examined them--that is, he\nrepeated the testimony of some who had testified. He finally said, \"I\ncan produce a living woman who will testify that Joseph Smith was a\npolygamist, and she knew it.\" I said, \"Bring her along here and let\nus examine her.\" Well, I met \"Aunt Lucy\" Walker Kimball, to whom you\nrefer, and we talked the matter over, and here is the one point to\nwhich I want to draw your attention, to show how these poor dupes of\nBrigham Young may be led. Coming to the testimony of Emma Smith, I\nsaid, \"You were personally acquainted with Emma Smith?\" \"Yes.\" \"What\nhave you to say as to her integrity, as to her fidelity and honor?\" The\nold woman looked me fair in the face and said, \"Emma Smith was one of\nGod's noble women--she was truth personified; and anything that Emma\nSmith may say you can bank on it until the day of your death.\" \"Well,\"\nI said, \"she testifies that her husband never had any wife but her; she\ntestifies that she never heard of that revelation on polygamy until you\nfolks had gone to Salt Lake; she testifies she never saw it, and she\ntestifies that it is an unmitigated falsehood manufactured by Brigham\nYoung; that he stated that she had the revelation and burned it. Now\nwhat have you to say to that?\" I said. She looked me fair in the face\nand said, \"You can afford to build on anything that Emma Smith has to\nsay.\" \"Thank you,\" said I.\n\nIt is true that she told me she was married to Joseph Smith May 1,\n1843; but when I showed her that the so-called revelation permitting a\nplurality of wives was dated July 12, 1843, and referred to her former\ntestimony as given in the _Historical Record_, and that given under\noath in the Temple Lot suit, she was confounded. I felt sorry for the\nold lady as she sat silent and confounded.\n\nIt is true that I saw a very old lady in your father's parlor, as\nshe came slowly in for prayers. Your father said, \"This is Catherine\nPhillips Smith. She was married to my father, Hyrum Smith, and she has\nnever married since. I am not sure that the old lady heard a word. It\nis certain that _she did not testify to me_, but it was your father who\nmade the statement, and at once called us to prayer, thus preventing me\nfrom speaking to the old lady.\n\nLorenzo Snow did testify to me, as stated; but then and there, in\nthe presence of Joseph F. Smith and George Q. Cannon, I showed _his\ntestimony to be false, by his own evidence_, when given _under oath_,\nand _by his sister's statement signed in 1842_. At this, Snow, Cannon\nand Smith were all much annoyed. So much for your father's statement,\nwhich says \"you did not say one word to him in relation to polygamy.\"\n\n\nYOUR FATHER'S FIRST WIFE\n\nYou seem to feel sore over the statement that your father's \"first wife\ndied broken hearted and insane\"; and you add, \"If you mean to insinuate\nthat this condition, if true, was the result of any act whatever on\nthe part of my father, it is also slanderously false.\" I insinuate\nnothing; let the public judge the facts. Your father's first wife was\nhis cousin; she refused to consent to additional wives, and when he\npersisted in marrying the Lambson sisters, she obtained a divorce in\nCalifornia. Julina and Edna Lambson were sisters and were married to\nJoseph F. Smith on the same day.[8]\n\nNumber of wives married to Joseph F. Smith since 1865: 6\n\nNumber of children born to him in 38 years: 42\n\nNumber of children born since plural marriage was prohibited in 1890: 13\n\nChildren of Julina Lambson Smith: 2\n\nChildren of Sarah Richards Smith: 2\n\nChildren of Edna Lambson Smith: 2\n\nChildren of Alice Kimball Smith: 3\n\nChildren of Mary Schwartz Smith: 4\n\nEstimated income available for supporting five establishments: $75,000\n\nCorporations, banks and factories of which Joseph F. Smith is a\ndirector: 20\n\nThe only Mormon Apostle who surpasses the record of President\nSmith is M. W. Merrill, with 8 wives, 45 children, and 156\ngrandchildren.--_Collier's_ for March 26, 1894 [1904].\n\n* * * * *\n\nWhile in Utah I was informed that your father's first wife died broken\nhearted and insane. God and civilization know that a woman who loved\nher husband from youth up has enough to break her heart and send her\ninsane when her husband will marry two other women, both sisters, in\none day.\n\nPerhaps you will be assisted to view the matter as I do, should you\nread the following in the Book of Mormon, Jacob 2:6, 7. Here it is\nstated, in consequence of polygamy, \"ye have broken the hearts of\nyour tender wives.\" Does this make the prophet an asperser or a\nscandalmonger?\n\nI have answered your letter as it appeared in the Toronto _Star_ as\nfully as space would permit.\n\n                                               Respectfully,\n                                                       R. C. Evans.\n\nToronto, Ontario, March 1, 1905.[9]\n\n\nFootnotes\n\n1. The teachings of the Latter-day Saints in relation to the doctrine\nof the Godhead are clearly set forth in Elder B. H. Roberts' valuable\nwork, \"Mormon Doctrine of Deity.\" For the belief of the \"Mormon\" people\nregarding Adam and his place in the universe, attention is called\nespecially to chapters one, five and six of that work; also to Doctrine\nand Covenants, sec. 78:15-18, sec. 107:53-57 and Daniel 7:9-14. In\nrelation to this matter I quote the following from the remarks of\nPresident Anthon H. Lund delivered at the General Conference, October\n6, 1902.\n\n\"Some there are who follow our Elders, and after they have preached the\nprinciples of salvation, these men get up and charge that the Elders\ndo not believe in God, but that they believe in Adam as their God,\nand they will bring up a few passages from sermons delivered by this\nor that man in the Church to substantiate this charge. Now, we are\nnot ashamed of the glorious doctrine of eternal progression, that man\nmay attain the position of those to whom came the word of God, that\nis gods. When Jesus was preaching unto the Jews on one occasion they\nstoned Him, and He wanted to know if they stoned Him for the good works\nHe had been doing. Oh, no, they say, 'for the good work we stone thee\nnot; but for blasphemy; and because that thou, being a man, makest\nthyself God.'\"\n\nHe quoted the 33rd to 37th verses of the 10th chapter of the Gospel of\nSt. John, and said:\n\n\"We believe that there are gods as the Savior quoted. He repeated what\nwas written in the law, and he did not say that it was wrong, but used\nit as an argument against them (The Jews.) While, however, we believe\nas the scripture states, that there are more gods, to us there is but\none God. We worship the God that created the heavens and the earth.\nWe worship the same God that came to our first parents in the Garden\nof Eden. In the revelation contained in section 116 of the Book of\nDoctrine and Covenants the Lord speaks concerning Adam-ondi-Ahman, 'the\nplace where Adam shall come to visit his people, or the ancient of days\nshall sit, as spoken of by Daniel the Prophet.' In the 107th section\nthe Lord speaks of Adam as Michael, the Prince, the Archangel, and\nsays that he shall be a prince over the nations forever. We may with\nperfect propriety call him Prince, the Ancient of Days, or even God in\nthe meaning of the words of Christ, which I have just quoted. When our\nmissionaries are met with these sophistries and with isolated extracts\nfrom sermons we say to them anything that is a tenet of our religion\nmust come through revelation and be sustained by the Church, and they\nneed not do battle for anything outside of the works, that have been\naccepted by the Church as a body.\"\n\n2. If popular custom had designated the true believers of the Bible as\n\"Bibles\" as a term of distinction from other worshippers, there is no\nreason why a true believer should be offended even at that appellation\nbut rather honored. Mr. Evans, without doubt, is not ashamed of the\nname \"Christian,\" yet this term, like that of \"Mormon\" was first\napplied to the followers of Christ in derision, \"because it was\nassociated in the public mind with the practices\" of the early Saints,\nwhich practices in that day were looked on as \"abominations.\"\n\n3. This is a misquotation, it should be: \"I could refer you to plenty\nof instances where men have been righteously slain, in order to atone\nfor their sins. I have seen scores and hundreds of people for whom\nthere would have been a chance (in the last resurrection there will be)\nif their lives had been taken and their blood spilled on the ground as\na smoking incense to the Almighty, but who are now angels to the devil,\nuntil our elder brother Jesus Christ raises them up--conquers death,\nhell and the grave.\"\n\nIn that same discourse President Young declares that those who were\n\"righteously slain\" were the wicked that the \"Lord had to slay\" in\nancient Israel. There is not one word in that discourse to indicate\nthat those who were slain to \"atone for their sins\" were killed in\nUtah; but to the contrary they were ancient inhabitants of the earth,\nviz., the antediluvians who perished in the flood, the inhabitants\nof Sodom and Gomorrah, of Jericho and the cities destroyed by the\nIsraelites; the prophets of Baal whom Elijah slew (I Kings 18:40) and\na host of others of that class and the class to whom the one belonged\nof whom the Savior said: \"It were better for him that a millstone\nwere hanged about his neck, and that he were drowned in the depth of\nthe sea.\" President Young's remarks agree with those of Peter when he\ndeclared that the Jews who were guilty of assenting to the crucifixion\nof Christ could not be baptized nor have their \"sins blotted out\" until\nthe \"times of refreshing shall come,\" which was at the time of the\n\"restitution of all things.\"--Acts 3:19-21.\n\n4. In extreme haste here to make a point, Mr. Evans left in the\nmiddle of a sentence and hurried on to the next page to complete the\nexpression he desired to convey. This is what President Cannon said:\n\"A prevalent idea has been that this prejudice against us owes its\norigin and continuation to our belief in a plurality of wives; but when\nit is recollected that the mobbings, drivings, and expulsions from\ncities, counties and states which we have endured, and our exodus to\nthese mountains all took place before the revelation of that doctrine\nwas PUBLICLY known, it will be seen at once that our belief in it has\nnot been the cause of persecution.\" Now, I ask, is it not plain to\nsee why his quotation stopped in the middle of a sentence? The Saints\nall know that President George Q. Cannon was always faithful to his\ntestimony that plural marriage was introduced by the Prophet Joseph\nSmith. Latter-day Saints generally declare that this doctrine was not\n_publicly_ known in the days of Joseph the Seer, but that it was taught\nby him to his trusted friends. When this fact is known the alleged\nquotations which follow, purported to be from H. B. Clawson, Ephraim\nJensen and \"Elder Whitaker\" lose their force.\n\n5. This is not in the Salt Lake Herald of February 9, 1852.\n\n6. The following is the Brooklyn _Citizen's_ report of that same\ndiscourse from which Mr. Evans quotes his passage as given in the New\nYork _Herald_: Elder Whitaker said: \"The people of the East have been\nled to believe that polygamy was alone responsible for all the troubles\nof the Mormons, but the fact remains, that as the fight was waged\nagainst Jesus Christ, against his followers, and against all great men\nfor declaring the truth, so the same spirit is manifest now; but the\nMormons will humbly seek those willing to accept the truths inspired\nof God, leaving the justice of their cause to be vindicated by honest\ninvestigation and time. The fight is directed against the doctrine of\nthe Mormon Church, though polygamy has done such yeoman service in\narousing public sentiment, to attain certain ends unworthy of honest\nmen. The crusaders have kept the public mind from the real cause of\nthe attack. From the time the Church was organized in 1830-47, when\nthe people, after many previous drivings, persecutions, mobbings and\ncruel mockings, were driven to Utah, the cry of polygamy was never made\na cause of their persecutions; indeed, that subject was not committed\nin writing until 1843, never published to the world until 1852, and\nwas abandoned by the issuance of the 'Manifesto' of President Wilford\nWoodruff, in 1890, since which time not one polygamous marriage has\nbeen solemnized; but those having wives at that time were never asked,\nand it was never expected they would abandon them, and when death\nbrings such relations to a close, there will be no polygamy among the\nMormons.\" The Brooklyn _Citizen_, Monday, January 8, 1900.\n\nWhy Mr. Evans accepted the brief extract from the New York _Herald_ in\npreference to the full account in the Brooklyn _Citizen_ will require\nno comment, but it certainly does appear that Elder Whitaker _did_ know\nwho introduced \"polygamy.\"\n\nAs I do not have the Yeoman's _Shield_ and am not in communication\nwith Elder Ephraim Jenson, I cannot vouch for his remarks, but feel\nsafe in saying that if the whole report were published, his testimony\nwould agree with that of Elder Whitaker as published in the Brooklyn\n_Citizen_.\n\n7. In quoting from \"The Mysteries of Mormonism, by an Apostle's\nWife,\" Mr. Evans reveals the character of his \"dozens and dozens of\nwitnesses.\" The reader will perceive that he depends largely on the\nmost bitter anti-\"Mormons\" and apostates for his \"evidence,\" but in\nquoting from \"The Mysteries of Mormonism, by an Apostle's Wife,\" he\ncertainly reaches the climax of this base testimony. This work was\npublished in 1882, by Richard K. Fox, proprietor of the notorious\n_Police Gazette_. The author of these \"Mysteries,\" undoubtedly a\nman, assumes the title of \"An Apostle's Wife,\" in order to hide his\nperfidy. The work is one of the vilest and most contemptible of all\nanti-\"Mormon\" publications, and is most bitter in its denunciation of\nthe Prophet Joseph Smith. In it he is called a \"lusty toper,\" \"the\nworst of a bad breed,\" \"an ignorant, brutal loafer,\" \"immoral, false\nand fraudulent,\" and the author says, \"_this_ is the man who founded\nwhat he dared to call a faith, and grafted on the United States the\nreligion of licentiousness and bodily lust known as Mormonism.\" An\napology is perhaps due for even referring to this matter, but since Mr.\nEvans makes this work one of the chief of his \"dozens and dozens of\nwitnesses,\" I feel that he should be exposed. He professes to believe\nin the divine mission of Joseph Smith, and yet calls upon us to accept\nthe wicked falsehoods of this disreputable witness, whom he declares\n\"_ought to know whereof she affirms_.\" Shame upon the man who draws his\ninspiration from such a source!\n\n8. This whole statement is absolutely false, and there was not the\nleast shadow of reason for uttering it. President Smith's first wife\ndid not refuse to consent to additional wives. He did not marry two\nsisters on the same day. In depending on the unreliable Alfred Henry\nLewis for his argument, Mr. Evans shows the desperate weakness of his\nposition. It would be a hard matter to squeeze more falsehoods in the\nspace occupied by the article of A. H. Lewis, from which Mr. Evans\nquotes so faithfully.\n\n9. This letter is dated March 1, 1905, but was not written until\nsometime after April 19, 1905, for on the latter date Mr. Evans wrote:\n\"You may look for reply to your letter as it appeared in the Toronto\n_Star_, as soon as I have time to reply thereto.\" This reply was\nreceived May 5, 1905.\n\n\n\nA REJOINDER TO MR. R. C. EVANS' LETTER\n\n                                           Salt Lake City, May 23, 1905.\n\n_Mr. R. C. Evans_,\n\n_Counselor in Presidency of Reorganized Church_.\n\nSir:--Your reply to my open letter of February 17 was received May\n5. Whether I was \"fair, dispassionate and also candid\" in my letter,\nor, as you seem to think, \"guilty of a labored effort to cover up the\ntrue facts regarding 'blood atonement, polygamy, etc.'\" and \"your\nfaith\"--which was not discussed--I am perfectly willing to leave to the\njudgment of \"those who read\" the same in the Toronto _Star_. So on this\npoint we may both rest satisfied.\n\n\nBLOOD ATONEMENT\n\nI will now consider your \"labored effort to cover up the _true facts_\nregarding blood atonement.\"\n\nIn my letter I candidly placed the true belief and teachings of the\nLatter-day Saints in relation to this doctrine before you. This fact\nappears to be displeasing to you, as it overturns your conclusions\nand accusations against our people. If you desire to know the correct\nposition of the Church on this doctrine, I would recommend a careful\nstudy of John Taylor's _Meditation and Atonement_ and Charles W.\nPenrose's _Blood Atonement_, which was published in answer to such\nwicked misrepresentations as I claim you have made in relation to this\nprinciple and our belief in relation thereto. There is no reason for\nany person to misunderstand our position, unless he desires to do so.\nI claim, too, that we are in a better position to teach that which\nwe believe than is the stranger who attempts to present our case,\nespecially if he is antagonistic or unfriendly.\n\nIf you do not believe the doctrine of blood atonement as that doctrine\nis taught by the Church of Jesus Christ of Latter-day Saints, which\nchurch you are pleased to call \"Utah Mormonism,\" then I say that you\n_do not_ believe in the atonement of our Lord and Savior Jesus Christ.\nTo this I will refer later.\n\nYou delight--as all anti-\"Mormons\" do--in referring to statements\nmade by President Brigham Young, Jedediah M. Grant and others during\nthe troublous times preceding the advent of Johnston's army into\nUtah. I see, too, that like many others, you place your own _desired_\ninterpretation on their remarks, place them before the public in a\ngarbled state, taking care to give the darkest interpretation possible\nfrom which the public may gather false conclusions. You take great\npains to cover up the conditions prevailing which called forth such\nextreme and in some instances unwise remarks. Conditions in some\nrespects akin to those surrounding the Saints in Missouri in 1838-39\nwhen other unwise remarks were made by members of the leading quorums\nof the Church, but in a sense justifiable and which should be condoned\nunder the trying circumstances that called them forth.[1]\n\n\nTHE CHURCH JUDGED FROM ITS ACCEPTED STANDARDS\n\nWriting on this subject Elder B. H. Roberts, in his criticism on Harry\nLeon Wilson's plagarisms in his _Lions of the Lord_, declares the\nposition taken by members of the Church and all fair-minded men in\nthese words:\n\n    \"The justice of Burke's assertion has never been questioned, and\n    without any wresting whatever it may be applied to \"Mormon\" leaders\n    who sometimes spoke and acted under the recollection of rank\n    injustice perpetrated against themselves and their people; or to\n    rebuke rising evils against which their souls revolted.\"\n\nEven the president of the Reorganized Church recognized this fact in\nhis answer to _The American Baptist_, wherein he said:\n\n    \"Whoever counseled or did evil in those times (in Missouri) are\n    responsible, personally, therefor; but the church, as such is no\n    more responsible for it than were the early Christians for Peter's\n    attempt to kill the high priest's servant when he cut off his\n    ear with his sword. The church, as such, should be judged by its\n    authorized doctrines and deeds, and not by the unauthorized sayings\n    or doings of some or many of its members or ministers.\n\n    It is not to be wondered at that in those times when the embryo\n    authors and abettors of the \"Border Ruffianism\" that reigned in\n    Missouri and Kansas from 1854 to 1865 had matters all their own\n    way, that some of the Saints, vexed, confused and excited, should\n    have done many things unwisely and wrongfully, and contrary to the\n    law of God.\"--_Saints' Herald_, 37:51.\n\nWith this I heartily agree.\n\nNow, when the statements were made, which you in a garbled manner both\nquote and misquote, there was in Utah a class of individuals who spent\nthe greater part of their time in circulating wicked and malicious\nreports about the Saints, threatening their lives, committing crimes\nand attempting to make the Saints their scape-goats. The officers of\nthe law were General Government officials appointed by the President\nof the United States, and I am sorry to say, some of these were among\nthe chief villifiers of the people. The most damnable and bloodthirsty\nfalsehoods were concocted and served up to the people of the United\nStates to stir them up to anger against the \"despised Mormons.\" Almost\nevery crime that was committed within a thousand miles of Salt Lake\nCity was charged to the leaders of the \"Mormon\" people and became the\nfoundation of a multitude of anti-\"Mormon\" publications that still\nflood the world. Because of these false and highly colored tales, in\n1857--one year later than the time that most of the utterances were\ngiven on which you so delight to dwell--the Government of the United\nStates sent an army to suppress in Utah a rebellion that never existed,\nand forced the Saints to defend themselves. When the Government found\nout how it had blundered it was humiliated.\n\nNow, in brief, these were the conditions at the time, and is it any\nwonder that unwise and even harsh things were said? The wonder is\nthat the people bore it as patiently as they did. The officers were\nnon-\"Mormons,\" the Territory was under Federal control and contained\nmany Gentiles, many of whom were most bitter in their feelings and\never ready to accuse the Saints of crime. The government was strong\nenough to enforce the law if broken. Now, I ask you if you believe the\nhorrors, as they have been pictured, could have existed under such\nconditions?\n\nSuch a state of affairs would have been a reproach and a shame to the\nAmerican government. And no such state of affairs existed.\n\nThe conditions at the time led Jacob Forney, superintendent of Indian\naffairs in Utah, to declare in 1869:\n\n    I fear, and I regret to say it, that with certain parties here\n    there is a greater anxiety to connect Brigham Young and other\n    Church dignitaries with every criminal offense than dilgent\n    endeavor to punish the actual perpetrators of crime.\n\n    Bancroft's History of Utah, p. 561.\n\n    Whitney's History of Utah, p. 108, vol. 1.\n\nMr. Forney was a Gentile official and the truth of this statement can\nbe relied upon.\n\nThis being the case, Brigham Young and the \"Mormon\" people could not\nhave engaged in the crimes charged against them.\n\nIn connection with this let me quote from Bancroft:\n\n    It is not true that Mormons are not good citizens, lawabiding\n    and patriotic. Even when hunted down, and robbed and butchered\n    by the enemies to their faith, they have not retaliated. On this\n    score they are naturally very sore. When deprived of those sacred\n    rights given to them in common with all American citizens, when\n    disfranchised, their homes broken up, their families scattered,\n    their husband and father seized, fined and imprisoned, they have\n    not defended themselves by violence but have left their cause to\n    God and their country.--History of Utah, pp. 390-392.\n\nAgain, I repeat, that the presence in Utah of apostates and\nanti-\"Mormons\" from the beginning and \"that there are men living in\nUtah today who left the Church in the earliest history of our State,\nwho feel as secure and are just as secure and free from molestation\nfrom their former associates as you or any other man could be,\" proves\nthe falseness of the malicious accusation that \"Utah was for years a\nland of assassination and a field of blood.\"\n\n\nMR. EVANS' FALSE QUOTATIONS\n\n    \"What shall be done with the sheep that stink the flock so? We will\n    take them, I was going to say, and cut off their tails two inches\n    behind their ears; however I will use a milder term, and say cut\n    off their ears.\"\n\nYour conclusion is most certainly far fetched. Had you continued the\nquotation your attempt would have appeared even more ridiculous. The\nnext sentence is:\n\n    \"But instead of doing this, we will try to cleanse them; and will\n    wash them with soap; that will come nigh taking off the skin; we\n    will then apply a little Scotch snuff, and a little tobacco, and\n    wash them again until we make them clean.\"\n\nAnd you try to make this appear as threatening life! It is apparent\nthat your sense of humor has been sadly neglected. This whole passage\nis humorous and you make yourself ridiculous by not having discovered\nit.\n\nAgain from Parley P. Pratt, you quote:\n\n    \"My feelings are with those who have spoken, decidedly and firmly\n    so.\"\n\nThis from page 84. Then you skip to page 86 and add:\n\n    \"I need not repeat their doom, it has been told here today, they\n    have been faithfully warned.\"\n\nThen three paragraphs off, the following:\n\n    \"It is too late in the day for us to stop and inquire whether such\n    an outcast has the truth.\"\n\nThis method of proving things reminds me of the reason why you should\nbe hanged:\n\n    And Judas \"went out and hanged himself.\"\n\n    \"Go thou and do likewise.\"\n\nNow let me quote some extracts from this discourse which you purposely\nleft out.\n\n    \"Sooner than be subjected to a repetition of these wrongs, _I\n    for one_, would rather march out today and be shot down. These\n    are my feelings, and have been for some time. Talk about liberty\n    of conscience! Have not men liberty of conscience here? Yes. The\n    Presbyterian, Methodists, Quakers, etc., have _here_ the liberty to\n    worship God in their own way, and so has every man in the world.\n    People have the privilege of apostatizing from this Church and\n    worshiping devils, snakes, toads, or geese, if they please, and\n    only let their neighbors alone. But they have not the privilege\n    to disturb the peace, nor to endanger life or liberty; that is\n    the idea. If they will take that privilege, _I need not repeat\n    their doom, it has been told here today, they have been faithfully\n    warned_.\"\n\nAgain:\n\n    \"He (Gladden Bishop) was disfellowshiped, and received on his\n    professions of repentance, so often, that the Church at length\n    refused to admit him any more as a member. These apostates talk of\n    proof. Have we not proved Joseph Smith to be a prophet, a restorer,\n    standing at the head of this dispensation? Have we not proved the\n    priesthood which he placed upon others by the command of God?\n\n    \"I see no ground, then, to prove or to investigate the calling of\n    an apostate, who has always been trying to impose upon this people.\n    _It is too late in the day for us to stop and inquire whether such\n    an outcast has the truth_.\n\n    \"We have truths already developed, unfulfilled by us--unacted upon.\n    There are more truths poured out from the eternal fountain, already\n    than our minds can contain, or that we have places or preparations\n    to carry out. And yet we are called upon to prove--what? _Whether\n    an egg that was known to be rotten fifteen years ago, has really\n    improved by reason of age_!\n\n    \"'_You are going to be destroyed_,' say they. '_Destruction awaits\n    this city_.' Well! what if we are? We are as able to be destroyed\n    as any people living. What care we whether we are destroyed or not?\n    These old tabernacles will die of themselves, if left alone.\n\n    \"We have nothing to fear on that head, for we are as well prepared\n    to die as to live. One thing we have heard today, and I am glad\n    to hear it. We shall not be destroyed in the old way--as we have\n    been heretofore. We shall have a change in the manner, at least.\n    We shall probably be destroyed _standing, this time_, and not in\n    a _sitting_, or _lying position_. We can die as well as others\n    who are not as well prepared! I am glad that while we do live we\n    shall not submit to be yoked or saddled like a dumb ass. We shall\n    not stand still to see men, women, and children murdered, robbed,\n    plundered, and driven any more, as in the States heretofore. Nor\n    does God require it at our hands. That is the best news we have\n    heard today. * * *\n\n    \"It is the policy not to wait till you are killed, but act on\n    the defensive while you still live. I have said enough on this\n    subject.\"--pp. 86-87.\n\nThe vicious malignancy of a depraved mind is made so apparent in this\ncontrast between your garbled quotations and the whole truth, that it\nscarcely deserves further comment.\n\nI have quoted quite extensively in order to show the reason for these\nremarks of which you quote such brief and disjointed extracts. You\nshould remember that the Saints had but a short time before being\ndriven from their homes at the cannon's mouth, and were forced to\ntraverse a desert under the most trying circumstances to find a new\nabode where they could rest in peace and call their souls their own.\nWhen followed, as they were, by a miserable class that were determined\nto again have them driven, where heaven only knows, in their might\nand righteous indignation they firmly took their stand for home and\nliberty. I for one, say that they were justified in this course, the\nprotection of their liberty, honor and lives. Had the threats of\ntheir enemies here in Utah been carried out as they boasted that they\nwould be, and as they were carried out in Missouri and Illinois, then\nBrigham Young and his people would have been as thoroughly justified in\nunsheathing the bowie knife, to conquer or die, as were the patriots at\nLexington and Bunker Hill!\n\nHome and liberty and life, with the right to worship God, are just as\ndear to a \"Mormon\" as to members of any other denomination or even an\napostate \"Mormon,\" and when the \"Mormons\" are persecuted, driven and\nslain and forced to seek a home in the savage wilds, would any honest\nman blame them if they declined to move again?\n\nWhy is it worse for \"Utah Mormons\" to defend themselves than for\n\"Mormons\" at Crooked river and Nauvoo? Even the noble Prophet Joseph\nSmith, when dragged from home and persecuted by wicked men, solemnly\ndemurred. Said he to the Saints at Nauvoo on the 30th day of June,\n1843, after his escape from Missourian assassins:\n\n    \"Before I will be dragged away again among my enemies for trial,\n    _I will spill the last drop of blood in my veins and will see all\n    my enemies in hell_! To bear it any longer would be a sin, and I\n    will not bear it any longer. Shall we bear it any longer? (one\n    universal, No! ran through all the vast assembly like a loud peal\n    of thunder.) * * * If mobs come upon you any more here, dung your\n    gardens with them. We don't want any excitement; but after we have\n    done all, we will rise up Washington-like and break off the hellish\n    yoke that oppresses us, and will not be mobbed!\"\n\nI have copied this from the manuscript history of the Prophet Joseph\nSmith, as it was recorded at the time. I have learned also that it is\ncorroborated by the journal of Wilford Woodruff of the same date--June\n30th, 1843.\n\n\nUTAH NOT A FIELD OF BLOOD\n\nYou say, \"I have read that which leads me to believe that under\nBrighamism\"--as you slurringly remark--\"Utah was for years a land of\nassassination and a field of blood,\" and then you ask me, \"what of\nthe Mountain Meadows massacre,--the destruction of the Aiken party;\nthe dying confession of Bishop J. D. Lee; the Hickman butcheries; the\nDanties?\"\n\nWell, that which you have read counts for but little when the source is\nconsidered. Your case is most certainly desperate when you are forced\nto accept the statements of murderers.\n\nIt's a strange thing that you and many of your elders accept all the\nblood-curdling tales from Beadle, Stenhouse and other apostate sources\n_when_ they happen to refer to Brigham Young and \"Utah Mormons,\"\nand denounce the same sources when they refer to the Prophet Joseph\nSmith. Yet, I repeat, the same class of charges--in many respects\nidentical--that you charge against Brigham Young, of murder, bloodshed,\nadultery, and even Danties, were first made by bitter enemies of the\nChurch before the death of the Prophet Joseph Smith, and that just such\nfalsehoods brought about the bitterness that resulted in his death.\n\nYou resort to sources that even the editor of your official paper\ndenounces as \"Idle and vicious stories gathered from the awful files\nof terrible tales told about the Mormons, by those at enmity with\nthem.\"--_Saints Herald_ 52:2.\n\nIf you desire to know the character of Christ do you accept the\nstatements of the Roman guard at the sepulchre? the Jew with\nblood-stained hands who rejoices in his death? and the anti-Christian?\nWherein then, is your consistency in asking me to accept the testimony\nof those whose hands are imbrued in blood, apostates and bitter enemies\nof my people?\n\nVery well then, I return your question. What about them? Pray tell,\nwhat about the Mountain Meadows massacre? the Aiken party? the\nconfessions of Lee? (by the way, the fact that you call him a \"Bishop\"\nproves the source of your information); what about Hickman and above\nall, the Danties?\n\nWhen Alfred Henry Lewis, in _Collier's Weekly_ of March 26, 1964,\nstated, \"Brigham Young invented his destroying angels, placed himself\nat their head, and when a man rebelled had him murdered, if one fled\nthe fold, he was pursued and slain,\" he repeated one of the most\ncolossal falsehoods ever uttered. Nor is that the only falsehood in his\narticle you are pleased to quote.\n\nBrigham Young was _not_ a man of blood. The \"Mormon\" people were _not_\nguilty of the Mountain Meadows massacre.[2] There was no destruction\nof an Aiken party. Hickman and Lee are not worth the mention; and the\nDanties! Had you not better read Church history of 1838? In Utah there\nnever were destroying angels or Danties, except in the imagination of\nbitter anti-\"Mormons\" and I am satisfied that Mr. R. C. Evans knows\nthat fact.\n\n\nCHARACTER OF THE \"MORMONS\"\n\nIn answer to your many charges about Utah and the \"Mormons,\" I desire\nto refer to credible references from witnesses who understood the truth\nand were bold enough to express it.\n\n    Last winter there was a census taken of the Utah Penitentiary\n    and the Salt Lake City and county prisons with the following\n    result:--In Salt Lake City there are about 75 Mormons to 25\n    non-Mormons; in Salt Lake County there are about 80 Mormons to\n    20 non-Mormons; yet in the city prison there were 29 convicts,\n    all non-Mormons. In the county prison there were 6 convicts all\n    non-Mormons. The jailer stated that the county convicts for the\n    five years past were all anti-Mormons except _three_! * * *\n\n    Out of the 200 saloon, billiard, bowling alley and pool table\n    keepers not over a dozen even profess to be Mormons. All of\n    the bagnios and other disreputable concerns in the territory\n    are run and sustained by non-Mormons. Ninety-eight per cent of\n    the gamblers in Utah are of the same element. * * * Of the 250\n    towns and villages in Utah, over 200 have no \"gaudy sepulchre of\n    departed virtue,\" and these two hundred and odd towns are almost\n    exclusively Mormon in population. Of the suicides committed in Utah\n    ninety odd per cent are non-Mormons, and of the Utah homicides and\n    infanticides over 80 per cent are perpetrated by the 17 per cent of\n    \"outsiders.\"--Phil Robinson, in _Sinners and Saints_, p. 72.\n\n    The Logan police force is a good-tempered looking young man.\n    There is another to help him, but if they had not something\n    else to do they would either have to keep arresting each other,\n    in order to pass the time, or else combine to hunt gophers and\n    chipmunks.--_Sinners and Saints_, p. 142.\n\n    Whence have the public derived their opinions about Mormonism?\n    From _anti-Mormons_ only. I have ransacked the literature of the\n    subject, and yet I really could not tell any one where to go for\n    an impartial book about Mormonism, later in date than Burton's\n    \"City of the Saints,\" published in 1862. * * * But put Burton on\n    one side and I think I can defy any one to name another book about\n    the Mormons worthy of honest respect. From that truly _awful_\n    book, \"The History of the Saints,\" published by one Bennet (even\n    an anti-Mormon has styled him \"the greatest rascal that ever came\n    to the west\") in 1842, down to Stenhouse's in 1873, there is not,\n    to my knowledge a single Gentile work before the public that is\n    not utterly unreliable from distortion of facts. Yet it is from\n    these books--for there are no others--that the American public has\n    acquired nearly all its ideas about the people of Utah.--_Sinners\n    and Saints_, p. 245.\n\n    And in relation to opposing evidence, almost every book that has\n    been put forth respecting the people of Utah by one not a Mormon,\n    is full of calumny, each author apparently endeavoring to surpass\n    his predecessor in the libertinism of abuse. Most of these are\n    written in a sensational style, and for the purpose of deriving\n    profit by pandering to a vitiated public taste, and are wholly\n    unreliable as to facts.--_Bancroft's History of Utah_, preface page\n    7.\n\n    It is only fair to state that no Gentile, even the unprejudiced,\n    who are rare aves, however long he may live or intimately he may\n    be connected with Mormons, can expect to see anything but the\n    superficies. * * *\n\n    The Mormons have been represented, and are generally believed to\n    be, an intolerant race. I found the reverse far nearer the fact.\n    The best proof of this is that there is hardly one anti-Mormon\n    publication, however untruthful, violent, or scandalous, which\n    I did not find in Great Salt Lake City.--Burton's _City of the\n    Saints_, p. 203.\n\n    I have not yet heard the single charge against them as a community,\n    against their habitual purity of life, their integrity of dealing,\n    their toleration of religious differences in opinion, their regard\n    for the laws, or their devotion to the Constitutional government\n    under which we live, that I do not from my own observation, or the\n    testimony of others know to be unfounded.--General Thomas L. Kane,\n    U. S. A., _The Mormons_, p. 83.\n\n    The Mormons are sober, industrious and thrifty.--Bishop Spaulding,\n    of the Episcopalian Church, in the _Forum_, March, 1887.\n\n    Had the Mormons been a low, corrupt or shiftless people they never\n    would or could have done what they did in Utah. * * * When they\n    controlled their own city of Salt Lake it contained no saloons,\n    gambling houses or places of ill repute, and when the town had\n    grown to be a goodly city order was kept by two constables. If by\n    their fruits we may know them, the Mormons deserve our confidence\n    and praise.--_The Brooklyn Eagle_, editorial of Aug. 12, 1897.\n\n    I shall not arraign the Mormon people as wanting in comparison with\n    other people in religious devotion, virtue, honesty, sobriety,\n    industry, and the graces and qualities that adorn, beautify\n    and bless life.--Caleb W. West, Governor of Utah (and a strong\n    anti-Mormon) in report to Secretary of the Interior for 1888.\n\n    I know the people of the east have judged the Mormons unjustly.\n    They have many traits worthy of admiration. I know them to be\n    honest, faithful, prayerful workers.--D. S. Tuttle, Bishop\n    Episcopalian Church.\n\n    I never met a people so free from sensualism and immorality of\n    every kind as the Mormons are. Their habits of life are a thousand\n    per cent superior to those who denounce them so bitterly.--Mrs.\n    Olive N. Robinson. (I recommend this to you.)\n\nI assure you there are many others of equal force but this should be\nsufficient to prove the scandalous effusions false that you profess to\nbelieve true.\n\n\nGAGGING AT A KNAT\n\nI am glad you profess to believe the Bible. There is one other\nthing which appears strange to me, that is, why you are continually\ndenouncing Brigham Young and \"Utah Mormonism,\" and calling Utah a\n\"land of assassination and a field of blood,\" because vile men without\nconscientious scruples have accused the people of many false and lurid\ntales of blood, and at the same time with sanctimonious countenance and\nupturned eyes you swallow the following without a gulp:\n\n    \"Thus saith the Lord of hosts. * * * Now go up and smite Amalek,\n    and utterly destroy all that they have, and spare them not; but\n    slay both man and woman, infant and suckling, ox and sheep, camel\n    and ass.\" I Samuel 15:3 (I. T.)\n\nHaven't you swallowed the camel and gagged at his tail?\n\n\nTHE DOCTRINE OF BLOOD ATONEMENT\n\nJust a word or two now, on the subject of blood atonement. _What\nis that doctrine_? Unadulterated if you please, laying aside the\npernicious insinuations and lying charges that have so often been\nmade. It is simply this: Through the atonement of Christ all mankind\nmay be saved, by obedience to the laws and ordinances of the Gospel.\nThis salvation is two-fold; General,--that which comes to all men\nirrespective of a belief in Christ--and Individual,--that which man\nmerits through his own acts through life and by obedience to the laws\nand ordinances of the Gospel. But man may commit certain grievous\nsins--according to his light and knowledge--that will place him beyond\nthe reach of the atoning blood of Christ. If then he would be saved\nhe must make sacrifice of his own life to atone--so far as in his\npower lies--for that sin, for the blood of Christ alone under certain\ncircumstances will not avail.\n\nDo you believe this doctrine? If not, then I do say you do not believe\nin the true doctrine of the atonement of Christ! This is the doctrine\nyou are pleased to call the \"blood atonement of _Brighamism_.\" This is\nthe doctrine of Christ our _Redeemer_, who died for us. This is the\ndoctrine of Joseph Smith, and I accept it.\n\nIn whose stead did Christ die? I wish your church members could be fair\nenough to discuss this subject on _its merits_.\n\nI again recommend you to a careful reading of the quotations in my\nopen letter. You will find them as follows: Book of Mormon,--II Nephi\n9:35. Alma 1:13, 14, and 42:19. Bible,--Genesis 9:12, 13, (I. T.) Luke\n11:50. Hebrews 9:22 and 10:26-29. I John 3:15 and 5:16. Doctrine and\nCovenants,--87:7. 101:80. 42:18, 19, 79. (Utah edition.)\n\nTo these I will add:\n\n    \"Whoso killeth any person, the murderer shall be put to death by\n    the mouth of witnesses; but one witness shall not testify against\n    any person to cause him to die.\n\n    Moreover ye shall take no satisfaction for the life of a murderer,\n    which is guilty of death; but he shall be surely put to death.\n\n    So ye shall not pollute the land wherein ye are; for blood\n    it defileth the land; and the land cannot be cleansed of the\n    blood that is shed therein, but by the blood of him that shed\n    it.\"--Numbers 35:30, 31, 33. (I. T.)[3]\n\nDo you want a few references of where men were righteously slain to\natone for their sins? What about the death of Nehor? (Alma 1:15)\nZemnariah and his followers (III Nephi 4:27-28). What about Er and\nOnan, whom the Lord slew? (Gen. 38:7, 10), of Nadab and Abihu? (Lev.\n10:2) and the death of Achan? (Joshua 7:25.)\n\nWere not these righteously slain to atone for their sins? And it was of\nthis class of cases that President Young referred in his discourse you\nmisquote (_Journal of Discourses_ 4:220). He tells us so, in the same\ndiscourse in the portion which you _did not quote_. It is:\n\n\"Now take the wicked, and I can refer you to where the Lord had to slay\nevery soul of the Israelites that went out of Egypt except Caleb and\nJoshua. He slew them by the hand of their enemies, by the plague and\nby the sword. Why? Because he loved them and promised Abraham he would\nsave them.\"\n\n\nPOLYGAMY\n\nIn using the term \"polygamy\" in reference to the principle that was\ntaught and practiced by the Saints, I desire it distinctly understood\nthat I use it in the sense of a man having more than one wife.\nPolygamy, in the sense of plurality of husbands and of wives never was\npracticed in the Church of Jesus Christ of Latter-day Saints in Utah or\nelsewhere; but Celestial marriage--including a plurality of wives--was\nintroduced by the Prophet Joseph Smith and was practiced more generally\nby the saints under the administration of President Brigham Young.\n\nYou say that you have no evidence that those men, _viz_. Lyman Wight,\nJames J. Strang, Gladden Bishop, William Smith and others that I\nmentioned to you \"practiced _polygamy_\" before plural marriage was\n\"introduced\" (as claimed by you) by Brigham Young. You said polygamy\nwas \"introduced\" eight years after the Prophet's death by Brigham\nYoung. If so, then why did these men practice it before that time? I\nwas satisfied that you would not exert yourself in seeking for this\nknowledge and tried to help you find the information.\n\n\nPOLYGAMY IN THE \"FACTIONS\"\n\nIn a letter written by the President of the Reorganized church by Mr.\nJoseph Davis of Wales, dated Lamonia, Oct. 13, 1899, I read:\n\n    \"Nearly all the factions into which the church broke had plural\n    marriage in some form. None in the form instituted by President\n    Young. Sidney Rigdon had one form practiced by but a few, and that\n    spasmodically, as an outburst of religious fervor rather than as a\n    settled practice. William Smith had a sort of Priestess Lodge, in\n    which it was alleged there was a manifestation of licentiousness.\n    This he denied, and I never had actual proof of it. Gladden Bishop\n    taught something like it, but I believe he was himself the only\n    practioner. James J. Strang had a system something like Mohamet,\n    four I think, being allowed the king. Lyman Wight had a system but\n    it had no very extended range. President Young's system you may\n    know of.\"\n\nIt is true that William Smith denied that he taught \"polygamy\" but that\nhe practiced plural marriage he cannot deny. Jason W. Briggs said he\n(William) did, and that is why Mr. Briggs left his church. Plaintiff's\nAbstract, Temple Lot suit, p. 395. Hist. of Reorg. Ch. vol. 3:200 and\n_The Messenger_, vol. 2. William entered into plural marriage in the\nProphet's day and his wives lived here in Utah. They are Precilla M.\nSmith, Sarah Libby and Hannah Libby. One of these is still living.\n\nThe third volume of your church history says of Lyman Wight:\n\n    \"Lyman Wight lived and died an honorable man, respected well by\n    those who knew him best. The only thing that can be urged against\n    his character is that about 1845 or 1846 he entered into the\n    practice of polygamy, but we have seen no record of any teaching of\n    his upon the subject.\"\n\nThe fact is that Lyman Wight entered into that relation before the time\nhere mentioned. Affidavits in this regard can be produced but it will\nbe unnecessary.\n\nThat John E. Page practiced \"polygamy\" I have the testimony of his\nwife, Mrs. Mary Eaton of Independence, who told me and others, in\nAugust 1904, that she _gave her husband_, John E. Page, other wives.\n\nThese men did not follow Brigham Young, but denounced him, yet they\npracticed plural marriage and did not get that doctrine from him.\n\n\nTHE TESTIMONY OF A BOGUS WIFE\n\nThe \"testimony\" you submit from President Young's \"legal wife\" is\nspurious. It matters not if you did receive the \"information\" from\nyour uncle. The poor man was tricked and deceived. Bogus \"wives\" and\n\"daughters\" of President Young have \"worked\" the public before. Mary\nAnn Angel Young, President Young's legal wife, was not in Colorado in\n1860 and 1861. She never was divorced and died in this city true to\nher husband, his family and the faith, on the 27th day of June, 1882.\n(_News_, July 5, 1882.) So much for this \"bogus\" testimony.\n\n\nTESTIMONY IMPEACHED\n\nThe testimony of T. B. H. and Fanny Stenhouse is sufficiently impeached\nin the _Saints' Herald_, vol. 52, p. 2; 20, p. 602, and _Sinners and\nSaints_, p. 245. The woman's bitterness would condemn her writings.\nHowever I will mention one statement--you make Mrs. Stenhouse say:\n\"It is reported by Fanny Stenhouse and many others, that Joseph\nSmith said, 'If ever the Church had the misfortune to be led by Bro.\nBrigham, he would lead it to hell.'\" She gives this as a rumor that is\n\"reported,\" so do the \"many others\" who are mostly from your church.\nOh, yes, I have heard of this before. But do you know where the report\noriginated? It originated with the apostate and would-be assassin,\nRobert D. Foster, who threatened the Prophet Joseph's life in 1844,\nand who was one of the incorporators and advocates of the notorious\n_Nauvoo Expositor_, and one of the chief actors in bringing about the\nmartyrdom, June 27, 1844. In a toadying letter to your president, dated\nFebruary 14, 1874, he said the prophet \"remarked, in the presence of\nMr. Law, Bishop Knight, John P. Greene, Reynolds Cahoon, and some\nothers, that if ever Brigham Young became the leader of the Church, he\nwould lead them down to hell.\"\n\n\nMARVELOUS GROWTH OF THE CHURCH\n\nI decline to accept the statements of such a character; besides,\nPresident Young did not lead the Church to hell, but preserved\nit, and under his direction it grew, expanded, and accomplished a\nwonderful, even a miraculous work. In the reclamation of the arid\nwest, the permanent establishment of prosperous communities in the\ndesert wilds, and for their unity, strength, and industrial and\ntemporal independence, the \"Mormon\" people are today the marvel, if\nnot the admiration of the thinking world. They came here with nothing\nbut the good will of God. They began in poverty, and \"having almost\nnothing to invest,\" says Mr. William E. Symthe in _The Conquest of\nArid America_, \"except the labor of their hands and brains, and that\nall they have expended in a period of fifty years for all classes of\nimprovements--from the first shanty to the last turret of the last\ntemple--came primarily from the soil.\"\n\nAgain he says in the same work:\n\n\nTESTIMONY OF MR. SMYTHE\n\n    Nowhere else has the common prosperity been reared upon firmer\n    foundations. Nowhere else are institutions more firmly buttressed\n    or better capable of resisting violent economic revolutions. The\n    thunder cloud that passed over the land in 1893, leaving a path of\n    commercial ruin from the Atlantic to the Pacific, was powerless to\n    close the door of a single Mormon store, factory or bank. Strong\n    in prosperity, the co-operative industrial and commercial system\n    stood immovable in the hour of widespread disaster. The solvency\n    of these industries is scarcely more striking than the solvency\n    of the farmers from whom they draw their strength. No other\n    governor, either in the West or in the East, is able to say what\n    the Honorable Heber M. Wells said in assuming the chief magistracy\n    of the new state in January, 1896, \"We have in Utah,\" said the\n    young governor. \"19,816 farms, and 17,584 of them are absolutely\n    free from incumbrance.\" A higher percentage in school attendance\n    and lower percentage of illiterates than even in the State of\n    Massachusetts, is another of Utah's proud records. P. 71.\n\n\nTHE GUIDANCE OF JEHOVAH\n\nWithout the divine guidance and the constant watchcare of Jehovah\nover the destinies of the \"Mormon\" pioneers, with Brigham Young at\ntheir head, the West today would be but a barren wilderness. Under the\nleadership of Brigham Young the \"Mormon\" people prospered, and he left\nthem in a better condition temporally and physically, and spiritually\nmore united and more firmly established in the faith than they ever\nwere before. Where among the so-called \"factions\" can you point to\none that has accomplished the hundredth part of what the followers of\nBrigham Young have accomplished? They have all practically disappeared\nbut one--gone to their destruction. And the one that remains will\ndissolve and disappear as surely as the sun shines. You cannot fight\nthe work of God and prosper.\n\n\nWILLIAM MARKS\n\nThe testimony of William Marks--a man who was out of harmony with the\nProphet before the latter's death! This testimony of William Marks\nsounds too suspicious, given as it was, when it was, and describing an\nalleged conversation which never could have taken place. \"The reader\nwill please notice,\" said David Whitmer in his _Address_ (p. 41), \"this\nfact in regard to William Marks' statement; and that is, the time when\nBrother Joseph told him that polygamy must be put down in the Church.\"\nThat time was a \"few days\" before the Prophet's death.\n\nTrue, the Prophet was no \"fool\" (_Herald_ 51:74), and such a\n\"conversation\" as this related by William Marks would have stamped him\n\"foolish, irrational and a moral suicide,\" _because_ he could not bring\na charge against others for that for which he was himself responsible.\nThe Prophet had plural wives, and had officiated in the ceremony of the\nsealing of plural wives to others. I have conversed with the principals\nin these cases, and know that they told the truth. Furthermore, Mr.\nMarks' testimony condemns itself. He proves--if he proves anything at\nall--that the Prophet was responsible for this doctrine. This thought\nis in harmony with the early teachings of the original elders of the\nReorganization, for the time was when even your elders acknowledged\nthat the Prophet received the revelation on celestial (including\nplural) marriage. On this point David Whitmer says:\n\n    As time rolled on, many of the Reorganization saw that to\n    _continue_ to acknowledge that Brother Joseph received the\n    revelation would bring bitter persecution upon themselves, as the\n    public feeling at that time was very bitter. * * * The leaders\n    of the Reorganized church, after a time, began to suppress their\n    opinions concerning this matter. They would answer the question\n    when asked about it \"_I do not know whether Joseph Smith received\n    the revelation or not_.\"\n\n\nTHE \"SAINTS' HERALD\" A WITNESS OF \"POLYGAMY\"\n\nNow, if it is true--and I claim it is--that the leaders of the\nReorganized church acknowledged that the Prophet received the\nrevelation and practiced that principle, there must be some proof. Turn\nto the first volume of the _True L.D.S. Herald_ and read the editorial\non pages 6 to 11. It is on polygamy. After trying to explain the reason\nwhy the Prophet taught and practiced this doctrine, the editor said:\n\n    And if the prophet be deceived when he hath spoken a thing, I, the\n    Lord, have deceived the prophet, and I will stretch out my hand\n    upon him and will destroy him from the midst of my people Israel. *\n    * * We have here the facts as they have transpired and as they will\n    continue to transpire in relation to this subject. The death of the\n    prophet is one fact that has been realized, although he abhorred\n    and repented of this iniquity before his death. Page 9.\n\nAnd on page 27:\n\n    He (Joseph Smith) caused the Revelation to be burned, and when\n    he voluntarily came to Nauvoo and resigned himself into the arms\n    of his enemies, he said that he was going to Carthage to die. At\n    that time he also said that if it had not been for that accursed\n    spiritual wife doctrine, he would not have come to that. By his\n    conduct at that time he proved the sincerity of his repentance,\n    and of his profession as a prophet. If Abraham[4] and Jacob, by\n    repentance, can obtain salvation and exaltation, so can Joseph\n    Smith.\n\nMark you, we have the evidence of the revelation from your own side and\nyou well remember that but _one_ could and did receive revelations. I\ndo not accept the apology of your editor; I do not believe that the\nProphet had the revelation burned, or called the doctrine accursed. My\nfaith in Joseph Smith is such that if he had the revelation--which your\nwitnesses declare he did--that it was from God as much as any other\nrevelation he received!\n\n\nTESTIMONY OF JASON BRIGGS\n\nJason W. Briggs, one of the founders of your church, in the Temple Lot\nsuit, said:\n\n    I heard something about a revelation on polygamy, or plural\n    marriage, when I was in Nauvoo, in 1842. I heard there was one:\n    there was talk going on about it at that time, and continued to be;\n    but it was not called plural marriage; it was called sealing.\n\n    You ask me what I understood this sealing to be, at the time the\n    talk was going on. What I understood it to be was sealing a woman\n    to a man to be his wife, to be his wife hereafter, his wife in the\n    spirit world.\n\n    I was asked in my direct examination if I did not hear of the\n    doctrine of polygamy, etc., and I answered that I talked with\n    members with reference to sealing, and I understood that the\n    doctrine of sealing, was for eternity; it was sealing a man's wife\n    to him for eternity, or wives, either. Record pp. 349, 431, 505.\n\n\nTESTIMONY OF JAMES WHITEHEAD\n\nJames Whitehead said:\n\n    There was an ordinance in the Church for sealing, as early as 1842\n    or 1843.\n\n    They would be married according to the law of God, not only for\n    time but for eternity as well.\n\nThese men were among the founders of your church.\n\n\nSIDNEY RIGDON'S TESTIMONY\n\nSidney Rigdon, in a lengthy letter to his official paper, _The\nMessenger and Advocate_, in 1845 declared that the Prophet was\nresponsible for the plural marriage doctrine, and said:\n\n    This system was introduced by the Smiths some time before their\n    death, and was the thing which put them in the power of their\n    enemies, and was the immediate cause of their death. P. 475, vol. 2.\n\nHe says he \"warned Joseph Smith and his family,\" and told them that\ndestruction would come upon them if they continued in their course.\n\n\nORIGINAL RECORDS OF PLURALITY OF WIVES\n\nYou \"confidently affirm that there is not a single word in a single\nsermon, lecture, statement, newspaper or Church publication _printed_\nduring the life of Joseph Smith, wherein he by word has endorsed the\ndoctrine of plurality of wives, not a single statement.\" Whether any\nsuch statement was ever _printed_ in his lifetime or not I am not\nprepared to say. But I do know of such evidence being recorded during\nhis lifetime, for I have seen it.\n\nI have copied the following from the Prophet's manuscript record of\nOct. 5, 1843, and know it is genuine:\n\n    \"Gave instructions to try those persons who were preaching,\n    teaching or practicing the doctrine of plurality of wives; for\n    according to the law, I hold the keys of this power in the last\n    days; for there is never but one on earth at a time on whom this\n    power and its keys are conferred; and I have constantly said no\n    man shall have but one wife at a time unless the Lord directs\n    otherwise.\"\n\nThere is also at the Historian's office in this city, a Bible, which\nI have before me, containing the record of the marriage of Melissa\nLott to the Prophet Joseph Smith, which was recorded at the time,\nSeptember 20, 1843. This Bible also contains the record of the sealing\nof Cornelius P. and Parmelia Lott, parents of Melissa, which was\ndone by Patriarch Hyrum Smith in the Prophet's presence and with his\n\"seal\" or sanction. The president of your church has seen this record,\nand it matters not what he may say _now_ he _then_ acknowledged the\ngenuineness of the record.\n\nThe following is also copied from the journal of William Clayton which\nis in the Historian's office:\n\n    May 1st, (1843) A.M. At the Temple. At 10 married Joseph to Lucy\n    Walker. P.M. at Prest. Joseph's; he has gone out with Woodsworth.\n\nThis is the same William Clayton who wrote the revelation at the\ndirection and from the dictation of the Prophet July 12, 1843. However,\nthis principle was first revealed to the Prophet several years before\nthat time, as you learned in your conversation with President Lorenzo\nSnow, when you were in his office.\n\n\nMORE EVIDENCE CONSIDERED\n\nRight here we will consider the \"evidence\" you produce to show that\n\"Joseph Smith and the Church during his lifetime condemned polygamy\nin the strongest terms.\" The testimony of the thirty-one witnesses\nyou \"produce\" was against the \"secret wife system\" of the vile John\nC. Bennett who was excommunicated for betraying female virtue. This\nBennett system had nothing to do with the system of celestial marriage\nintroduced by the Prophet Joseph Smith, and was no more like the\nProphet's doctrine than darkness is like daylight. The certificate\nof these parties that you mention was given in October 1842 (T. & S.\n3:939), nearly one year before the revelation on celestial marriage\nwas recorded. At that time the law of marriage in the Church was\nthat adopted in 1835, and was binding on all who had accepted the\nhigher law, and they were few in number.[5] The best proof that these\n\"witnesses\" did not condemn the celestial marriage doctrine of the\nprophet in this communication, is that out of the thirty-one, at least\nsixteen have testified that the Prophet introduced that system. One of\nthis number of witnesses became the Prophet's wife, one performed a\nmarriage ceremony in which the Prophet was married to a plural wife,\nand one other was a witness to such a marriage ceremony. At least six\ntestify that the Prophet taught them the principle of plural marriage\nand the others, so far as I know, are not on record. That these\nwitnesses were the dupes of Brigham Young cannot truthfully be said,\nfor three of them left the Church and never followed Brigham Young, yet\nthey testify of these things.\n\nThe action of Joseph and Hyrum Smith, as recorded in the _Times\nand Seasons_ (5:3), wherein Hyrum Brown was cut off the Church for\npreaching polygamy and other false doctrines, was just and timely. The\nsame action would have been taken at any other period of the existence\nof the Church. Polygamy never was a doctrine of the Church, and the\nsystem introduced by the Prophet Joseph Smith was not called by that\nname in his day. Nor was the system of the Prophet the same as that of\nHyrum Brown; and if it had been, the ruling of the Prophet of October\n5, 1843, would have cost Brown his standing in the Church, the polygamy\nof Brown and John C. Bennett was of their own make. In relation to this\nsubject, I will quote from the _Life of John Taylor_, pages 223-224:\n\n    The polygamy and gross sensuality charged by Bennett and repeated\n    by those ministers in France, had no resemblance to celestial or\n    patriarchal marriage which Elder Taylor knew existed at Nauvoo, and\n    which he had obeyed. Hence in denying the false charges of Bennett,\n    he did not deny the existence of that system of marriage that God\n    had revealed; no more than a man would be guilty of denying the\n    legal, genuine currency of the country by denying the genuineness\n    and denouncing what he knew to be a mere counterfeit of it.\n\n    Another illustration: Jesus took Peter, James and John into the\n    mountain, and there met with Moses and Elias, and the glory of God\n    shone about them, and these two angels talked with Jesus, and the\n    voice of God was heard proclaiming Him to be the Son of God. After\n    the glorious vision, as Jesus and His companions were descending\n    the mountain, the former said: \"Tell the vision to no man, until\n    the Son of Man be risen from the dead.\" Suppose one of these\n    apostles had turned from the truth before the Son of Man was risen\n    from the dead and under the influence of wicked, lying spirit,\n    should charge that Jesus and some of his favorite apostles went up\n    into a mountain, and there met Moses and Elias,--or some persons\n    pretending to represent them--together with a group of voluptuos\n    courtesans, with whom they spent the day in licentious pleasure.\n    If the other apostles denounced that as an infamous falsehood,\n    would they be untruthful? No; they would not. Or would they be\n    under any obligations when denying the falsehoods of the apostate\n    to break the commandments the Lord had given them by relating\n    just what had happened in the mountain? No; it would have been a\n    breach of the Master's strict commandment for them to do that. So\n    with Elder Taylor. While he was perfectly right and truthful in\n    denying the infamous charges repeated by his oponents, he was under\n    no obligation and had no right to announce to the world, at that\n    time the doctrine of celestial marriage. It was not the law of the\n    Church, or even the law of the Priesthood of the Church; the body\n    thereof at the time knew little or nothing of it, though it had\n    been revealed to the Prophet and made known to some of his most\n    trusted followers. But today, now that the revelation on celestial\n    marriage is published to the world, if the slanderous charges\n    contained in the writings of John C. Bennett should be repeated,\n    every Elder in the Church could truthfully and consistently do just\n    what Elder Taylor did in France--he could deny their existence.\"\n\n\nTHAT UTAH VISIT\n\nAfter receiving your letter, I requested of my father that he give me\na written statement in answer to your charge that he \"discussed\" the\ndoctrine of \"polygamy\" with you, and received the following:\n\n    _Joseph F. Smith, Jr_.\n\n    Dear Son:--You have submitted to me some statements made by Mr.\n    R. C. Evans of the Reorganized church, and desire to know what I\n    have to say about them. He says: \"If your father denies that he\n    and I discussed the doctrine of polygamy, all I have to say about\n    it is, that what he states is untrue.\" Perhaps I could dismiss\n    this statement precisely in the same way he has. I could certainly\n    do so far more truthfully. He and I did not discuss the doctrine\n    of \"polygamy\" at all. It is true I did introduce him to President\n    Lorenzo Snow, to Aunt Lucy W. Smith, to Aunt Catherine P. Smith, to\n    Heber J. Grant and a few others. Whatever \"discussion\" he had on\n    the \"doctrine of polygamy\" may have been with these parties, but\n    not with me. While in my company he was my guest by introduction\n    from my cousin Joseph Smith, president of the Reorganized church,\n    and I carefully avoided any discussion with him upon any and all\n    differences of opinion which existed between us, the discussion of\n    which could only have resulted in ill feeling and perhaps extreme\n    bitterness. I treated him as any gentleman should treat another,\n    not as an antagonist but as a stranger within my gates, indeed,\n    as my guest; and when we parted it was with mutual good feelings\n    and interchange of kindly wishes, without the slightest breath\n    or suspicion of unpleasantness, which must have existed had we\n    indulged in a \"discussion of the doctrine of polygamy,\" or any\n    other points of difference.\n\n    Aunt Catherine P. Smith was making us a short visit at the time,\n    and I introduced her to Mr. Evans as the wife of my father, Hyrum\n    Smith. They had some conversation, in which I took no part, and\n    to the best of my recollection he drew out from her the fact that\n    she was married to Hyrum Smith, by Joseph Smith the Prophet, in\n    August 1843, in the brick office of Hyrum Smith, at Nauvoo, in the\n    presence of her mother, Sarah Godshall Phillips, Mrs. Julia Stone\n    and her daughter Hettie.\n\n    Mr. Evans attempted to cross-question her on her statement, but she\n    stoutly and unequivocally affirmed the truth of what she had said.\n    Mrs. Lizzie Wilcox, your mother and two or three other members of\n    the family were present and heard what was said.\n\n    With reference to Mr. Evans' alleged interview with Aunt Lucy W.\n    Smith at the Theatre, I need only say I occupied a seat adjoining\n    them, and heard the conversation between them, and I have not the\n    slightest recollection of the statement he has made about that\n    interview. The strong point which he attempts to make is the fact\n    that Lucy was married to the Prophet Joseph Smith, on May 1, 1843,\n    while the revelation on plural marriage was dated \"July 12, 1843,\"\n    and her consequent embarrassment, was far-fetched; for no one knew\n    better than she did that the revelation was given as far back as\n    1834, and was first reduced to writing in 1843. And on one could\n    have been better prepared to state that fact than Aunt Lucy W.\n    Smith. There could not be, therefore, any cause for embarrassment\n    on her part on that score, and I apprehend she would have been one\n    of the last persons to \"sit silent and confused\" under such an\n    implied impeachment.\n\n    That she bore testimony to the good character of Aunt Emma Smith\n    with reference to other matters than plural marriage is true;\n    but not to her conduct toward that principle. Aunt Lucy is still\n    living, and sound mentally and physically. She can, and no doubt\n    will, fully clear away any sophistry and falsehood of Mr. Evans'\n    statement of the alleged interview.\n\n    Referring to the interview with President Snow, Mr. Evans says:\n    \"Lorenzo Snow did testify to me as stated. But then and there, in\n    the presence of Joseph F. Smith and George Q. Cannon, I showed\n    his testimony to be false by his own evidence when given under\n    oath, and his sister's statement signed in 1842. At this, Snow,\n    Cannon and Smith were much annoyed. So much for your father's\n    statement, which says 'you did not say one word to him in relation\n    to polygamy.'\" The fact is, President Snow gave Mr. Evans, in my\n    presence and hearing, a plain, simple narration of the instructions\n    he received from Joseph Smith in regard to the doctrine of plural\n    marriage, including almost word for word the statement he had\n    previously made under oath, and testified that Joseph informed\n    him that his sister Eliza R. Snow had been sealed to him as his\n    wife. This much and more in this line I distinctly heard and as\n    distinctly remember, but I did not hear the alleged arraignment\n    of President Snow's testimony by Mr. Evans, nor did I witness or\n    experience any \"annoyance\" on the part of myself or anyone present\n    because of the said arraignment. Indeed, I am prepared to affirm\n    that Mr. Evans did not \"then and there\" in my presence and that of\n    Geo. Q. Cannon, nor in the presence of any one there, \"show his\n    (Snow's) testimony to be false,\" either \"by his own evidence when\n    given under oath,\" or \"by his sister's statement signed in 1842,\"\n    or at any other time.\n\n    I am here constrained to say that Mr. Evans was treated by\n    President Snow, as also by President George Q. Cannon and\n    myself, in the most courteous and respectful manner, and so\n    far as I observed his demeanor towards us was reciprocal and\n    gentlemanly--and not one word was said to him by anyone nor by\n    him to anyone in my presence that was in any degree discourteous,\n    contentious or embarrassing.\n\n    I conclude, therefore, that the foregoing statements made by\n    Mr. Evans, were after thoughts uttered by him with a view to\n    misrepresent the truth and the facts, on the lines of the bitter\n    and relentless opposition of himself and associates to the Church\n    of Jesus Christ of Latter-day Saints in general, and the doctrine\n    of plural marriage in particular, as revealed, taught and practiced\n    by Joseph Smith himself, from whom Brigham Young and many others\n    received it. On these matters they are so surcharged with animus\n    that they will not receive, admit, or tell the truth.\n\n    With reference to Mr. Evans' allusion to my first wife I will\n    simply say: She was most intimately acquainted from her childhood\n    with the young lady who became my second wife, and it was with\n    their full knowledge and consent that I entered into plural\n    marriage, my first wife being present as a witness when I took my\n    second wife, and freely gave her consent thereto. Our associations\n    as a family were pleasant and harmonious.\n\n    It was not until long after the second marriage that my first wife\n    was drawn away from us, not on account of domestic troubles, but\n    for other causes which I do not care to mention. In eight years of\n    wedded life we had no children. She constantly complained of ill\n    health and was as constantly under a doctor's care. She concluded\n    to go to California for her health and before going procured a\n    separation. This all occurred previous to 1867. On March 1, 1868\n    I married Sarah E. Richards, and January 1, 1870, I married Edna\n    Lambson, from one to three years after my first wife separated\n    from me, and had become a resident of California. She subsequently\n    returned to Utah and later went to St. Louis where she died.\n\nYour self-exaltation in classing yourself with Jacob is most\nstupendous, to say the least. He was above accepting idle rumors, from\nsuch sources as those given by the writer of the article of _Collier's_\nwhich you quote, and which are false. Jacob was no aspersor.\n\nAunt Catherine Phillips Smith also declares that she did testify to you\nin regard to her marriage and that you questioned her quite closely.\nMy mother declares the same for she was present at the conversation.\nPresidents Snow and Cannon are not here to speak in their defense, but\nI am satisfied that they would bear witness to the foregoing letter.\nAunt Lucy may testify for herself.\n\n\nTESTIMONY OF LUCY W. SMITH\n\nThe day I received a copy of the _Ensign_ containing your discourse\nfrom which you give extracts in your \"reply,\" in relation to your\n\"conversation\" with Aunt Lucy W. Smith, I sent her a copy of your\nremarks with the request that she tell me if you had correctly reported\nher testimony. In the course of a few days I received this:\n\n    My Dear Boy: I very much regret not feeling able to answer your\n    request at an earlier date. I am, however, much improved in health\n    since coming to Logan, and take pleasure in declaring to you that\n    the infamous discourse delivered 16th Feb. 1905 (the date of the\n    _Ensign_) at St. Louis, Missouri, by Mr. Evans, is a fabrication of\n    falsehoods and misrepresentations. I confess that I was not only\n    surprised, but shocked beyond measure. Now one of the presidency\n    of the Reorganized church, just think of it! And at the time he\n    came to Salt Lake City three years ago, he claimed to be one of\n    \"young Joseph's apostles; came with a letter of introduction from\n    cousin Joseph to his cousin Joseph F., saying that any courtesy\n    shown him would be appreciated. Accordingly, Mr. Evans was shown\n    every consideration. He accepted the generous hospitality of our\n    President and his model family. Having expressed a desire to meet\n    Mrs. Lucy W. Kimball, who was engaged that afternoon, arrangements\n    were made to meet at the theatre, as he had to leave next day.\n    He asked me many questions which I answered frankly--some very\n    offensive hearsay questions that aroused my indignation, but I\n    bore the ordeal as a martyr should. And from this opportunity\n    sprang the wonderful discourse of wicked falsehood and malicious\n    misrepresentation. O, shame! Where canst thou hide thy brazen face!\n    How dare he resort to such infamy unless to satiate a morbid desire\n    for notoriety among sensation-mongers, who seek not for light or\n    truth! If so he only gratified the cravings of the basest and\n    lowest caste.\n\n    I cannot believe that the once highly and beloved Emma who was so\n    loyal and true to her husband in all the early trials and hardships\n    to which he was subject, when in chains and bondage, when he was\n    dragged from his bed, tarred and feathered, imprisoned and mocked\n    and scoffed at, ridiculed and abused, and his life threatened by\n    infuriated mobs and she stood by him and comforted him in all of\n    his afflictions--I cannot believe after enduring all this for\n    his sake, that Emma Smith ever denied seeing the revelation on\n    celestial marriage after receiving it in good faith and accepting\n    it as a command from God, _knowing_ as I do, that she taught it\n    to Eliza and Emily Partridge, Maria and Sarah Lawrence, and urged\n    them to accept it by being sealed to her husband. She treated them\n    kindly and considerately and knew they were associated with him as\n    his wives. She was then a happy woman, until the tempter came in\n    human form, and she partook of the apostate spirit so rife in those\n    days. She could not deny these facts without sinning against her\n    husband, sinning against his wives, against the truth, and against\n    her God!\n\n    If her son insists that this denial was her last testimony he\n    fastens a stigma on her once noble character in the estimation of\n    her former friends and associates, who were familiar with the facts\n    of the period referred to. This misguided son, young and without\n    experience, was surrounded by his father's most wicked enemies who\n    had betrayed his father, and had been instrumental in taking his\n    life; and who, after they had accomplished this foul act, through\n    sinister policies, determined to destroy the work his father was\n    commanded to do, and had laid a permanent foundation on which to\n    build up his church--the Church of Christ. They sought to influence\n    his son against the teachings of his father, call him forth as a\n    \"leader\" with promises of success, and good backing. Poor boy was\n    flattered and led on and on, by crafty men, until he became an\n    unbeliever of the principles his father had taught; and I cannot\n    but believe that through such influences his mother has been\n    misrepresented. I am unwilling to believe otherwise.\n\n    I expressed regrets to Mr. Evans in relation to the course taken\n    by \"young Joseph\" through the influence of the bitter opponents of\n    his father. I said he had closed his eyes to anything that would\n    cast a ray of light on the vexed question: \"Did my father have\n    more [other] wives than my mother?\" I answered truthfully without\n    hesitation. Afterwards he went to Lehi, called on Melissa Lott,\n    with whom he had been associated from early childhood and asked:\n    \"Will you answer me one question, I come to you knowing you will\n    tell me the truth, were you my father's wife?\" \"Yes, Joseph, I\n    was.\" \"Where is your proof?\" She stepped to the stand and took the\n    family Bible opened to the family record, placed it on his knee and\n    asked: \"Do you recognize the handwriting?\" \"Certainly that is your\n    father's (Cornelius P. Lott's) handwriting, know it as well as my\n    own.\" Then read the marriage certificate of the Prophet Joseph and\n    Melissa Lott.\n\n    Oliver Huntington who is still living testifies that they were very\n    intimate as boys, and when together had often talked the matter\n    over.\n\n    Referring to Mr. Evans again. I said: \"Does this prove him (Joseph)\n    an honest man?\" Now does this cover the ground of your inquiry? I\n    have so often been interrupted by callers, that I may not have been\n    explicit enough. My personal testimony you already have, if not you\n    can get it by referring to \"Reminiscences of Latter-day Saints,\"\n    by L. O. Littlefield, which you will find at the President's\n    (Historian's) office.\n\nDoes this read much like she had been correctly represented?\n\n\nBRIGHAM YOUNG UPHELD BY THE LORD\n\nIn reference to the wicked charge you make in your discourse mentioned\nin Aunt Lucy's letter, against President Young of practicing gross\nimmorality while on his mission in England in 1840 and winter of 1841,\na sufficient answer will be found in the revelation of January 19,\n1841, wherein the Lord, by revelation through the Prophet Joseph Smith\ndeclares:\n\n    I give to you _my servant_ Brigham Young, to be a President over\n    the Twelve traveling Council,\n\n    Which Twelve hold the keys to open up the authority of my kingdom\n    upon the four corners of the earth, and after that to send my word\n    to every creature.\n\nAnd the revelation of July 9, 1841, given after his return from England:\n\n    * * * Verily thus saith the Lord unto you, my servant Brigham, it\n    is no more required at your hand to leave your family as in times\n    past, for your offering is acceptable to me.\n\nIn this abusive charge against President Young you are striking at\nJehovah, and accusing Him, either of condoning such a grievous sin, or\nfailing to discover it. Such a charge as that is ridiculously absurd,\nI feel safe in accepting the word of the Lord in preference to the\nribald, indecent statements of those who speak forth the vulgar desires\nof their own minds.\n\n                                         Respectfully,\n                                                Joseph F. Smith, Jr.\n\n\nFootnotes\n\n1. I am not so blind in my admiration of the \"Mormon\" people or so\nbigoted in my devotion to the \"Mormon\" faith as to think there are\nno individuals in the Church chargeable with fanaticism, folly,\nintemperate speech, and wickedness; nor am I blind to the fact that\nsome in their over-zeal have lacked judgment; and that in times of\nexcitement, under stress of special provocation, even \"Mormon\" leaders\nhave given utterances to ideas that are indefensible. But I have yet to\nlearn that it is just in a writer of history, or of \"purpose fiction,\"\nthat \"speak truly,\" to make a collection of these things and represent\nthem as the essence of that faith against which said writer draws an\nindictment.\n\n\"No one would measure the belief of 'Christians,'\" says a truly great\nwriter, \"by certain statements in the Fathers, nor judge the moral\nprinciples of Roman Catholics by prurient quotations from the Casuist;\nnor yet estimate Lutherans by the utterances and deeds of the early\nsuccessors of Luther, nor Calvinists by the burning of Servetus. In all\nsuch cases the general standpoint of the times has to be first taken\ninto account.\"--Edeshiem's Life and Times of Jesus the Messiah, preface\np. 8.\n\nA long time ago the great Edmund Burke in his defense of the rashness\nexpressed in both speech and action of some of our patriots of the\nAmerican revolution period said: \"It is not fair to judge of the temper\nof the disposition of any man or any set of men when they are composed\nand at rest from their conduct or their expressions in a state of\ndisturbance and irritation.\"\n\n2. Writing of the Mormon Meadows massacre Hubert H. Bancroft, in his\nHistory of Utah, page 544 says: \"Indeed it may well be understood at\nthe outset that this horrible crime, so often and so persistently\ncharged upon the Mormon church and its leaders, was the crime of an\nindividual, the crime of a fanatic of the worst stamp, one who was a\nmember of the Mormon church, but of whose intentions the church knew\nnothing, and whose bloody acts the members of the church, high and\nlow, regard with as much abhorrence as any out of the church. Indeed,\nthe blow fell upon the brotherhood with threefold force and damage.\nThere was the cruelty of it, which wrung their hearts; there was the\nodium attending its performance in their midst; and there was the\nstrength it lent their enemies further to malign and molest them. The\nMormons denounce the Mountain Meadows massacre, and every act connected\ntherewith, as earnestly and as honestly as any in the outside world.\nThis is abundantly proved, and may be accepted as a historical fact.\"\n\n3. See also Doctrine and Covenants section 101:80, on this point.\n\n4. A polygamist the friend of God, whose praise you sing, and the man\nyou are _glad_ to call the father of the faithful.--_Saints' Herald_\n52:437.\n\n5. Those thirty-one witnesses were: S. Bennett, George Miller, Alpheus\nCutler, Reynolds Cahoon, Wilson Law, Wilford Woodruff, Newel K.\nWhitney, Albert Petty, Elias Higbee, John Taylor, Ebenezer Robinson,\nAaron Johnson, Emma Smith, Elizabeth A. Whitney, Sarah M. Cleveland,\nEliza R. Snow, Mary C. Miller, Lois Cutler, Thirza Cahoon, Ann Hunter,\nJane Law, Sophia Marks, Polly Z. Johnson, Abagail Works, Catharine\nPetty, Sarah Higbee, Phebe Woodruff, Leonora Taylor, Sarah Hillman,\nRosanna Marks, and Angeline Robinson.\n\n\n\nTHE SAINTS' HERALD ON THE ORIGIN OF PLURAL MARRIAGE\n\nIn both replies to Mr. Evans, mention is made of two articles in the\n_Saints' Herald_, volume one, that were written by Isaac Sheen, the\nfirst editor of that paper. These references were ignored by Mr. Evans\nin his publication of a portion of the foregoing correspondence. It\nwould occupy too much space to copy these articles in full as they\nare quite lengthy, but I feel that the gist of the matter should be\npresented in more detail than it is given in the replies.\n\nMr. Sheen's argument is that the Saints at Nauvoo \"set up their idols\nin their heart,\" and went to the Prophet Joseph Smith and asked him to\ninquire of the Lord and ascertain from Him if it would not be proper\nfor them to practice plural marriage. This the Prophet Joseph did and\nin answer the Lord gave him the revelation on celestial marriage,\ngranting the practice of plural marriage, and then, after giving this\nrevelation the Lord smote the Prophet for his 'iniquity' in asking for\nthe revelation, and poured out wrath and indignation upon the Saints\nfor their participation in what he calls \"abominations.\"\n\nReference is also made to the prophecies of Ezekiel, Balaam and Micaiah\nto substantiate his theory which Mr. Sheen admits he is unable to\n\"satisfactorily explain.\" An extensive quotation from the first article\nfollows, which will give an idea of the position in which the members\nof the Reorganized church regard the Prophet Joseph Smith and the\nculmination of his most glorious mission.\n\n\nSTATEMENT OF ISAAC SHEEN\n\n    We might call your attention to many prophecies in the Bible which\n    these backsliders[1] have fulfilled by their abominations. Ezekiel\n    appears to have had a very clear manifestation of the wickedness of\n    these men and the plan pursued by them, by which they embark into\n    polygamy. In Ezekiel 14 c. 1, 5, v, the prophet says, \"Then came\n    certain elders of Israel unto me, saying, Son of man, these men\n    have set up their idols in their heart and put the stumblingblock\n    of their iniquity before their face: should I be inquired of at all\n    by them? Therefore speak unto them, and say unto them, Thus saith\n    the Lord God; Every man of the house of Israel that setteth up his\n    idols in his heart, and putteth the stumblingblock of his iniquity\n    before his face, and cometh to the prophet; I the Lord, will\n    answer him that cometh according to the multitude of his idols;\n    that I may take the house of Israel in their own heart, because\n    they are all estranged from me through their idols.\" We have shown\n    you that God gave a revelation unto us in which he commanded that\n    every man should \"cleave unto his wife and none else,\" and that\n    he commanded us saying, \"Repent and remember the Book of Mormon\n    and the former commandments which I have given them, not only to\n    say, but to do according to that which I have written,\" and that\n    in that book there is much testimony against polygamy. All these\n    instructions were sufficient for our guidance, but \"men have set\n    up their idols in their hearts, and put the stumblingblock of\n    their iniquity before their faces.\" This adulterous spirit had\n    captivated their hearts and they desired a license from God to\n    lead away captive the fair daughters of His people, and in this\n    state of mind they came to the Prophet Joseph. Could the Lord do\n    anything more or less than what Ezekiel hath prophesied? The Lord\n    hath declared by Ezekiel what kind of an answer he would give them,\n    therefore he answered them according to the multitude of their\n    idols. Paul had also prophesied that \"for this cause God shall send\n    them strong delusion, that they should believe a lie; that they\n    all might be damned who believed not the truth, but had pleasure\n    in unrighteousness.\" Both these prophecies agree. In Ezekiel's\n    prophecy the Lord also says, \"I will set my face against that\n    man, and will make him a sign and a proverb, and I will cut him\n    off from the midst of my people; and ye shall know that I am the\n    Lord. And if the prophet be deceived when he hath spoken a thing, I\n    the Lord have deceived that prophet,[2] and I will stretch out my\n    hand upon him and I will destroy him from the midst of my people\n    Israel. And they shall bear the punishment of their iniquity;\n    the punishment of the prophet shall be even as the punishment of\n    him that seeketh unto him; that the house of Israel may go no\n    more astray from me, neither be polluted any more with all their\n    transgression; but that they may be my people, and I may be their\n    God, saith the Lord God,\" 8c., 11 v. We have here the facts as they\n    have transpired and as they will continue to transpire in relation\n    to this subject. The death of the prophet is one fact that has\n    been realized although he abhorred and repented of this iniquity\n    before his death. This branch of the subject we shall leave to some\n    of our brethren, who are qualified to explain it satisfactorily.\n    Those who have practiced these abominations have become \"a sign\n    and a proverb\" among men in accordance with this prophecy. These\n    are the \"false teachers\" prophesied of by Peter, of whom he said\n    \"many shall follow their pernicious ways; by reason of whom the way\n    of truth shall be evil spoken of. And through covetousness shall\n    they with feigned words make merchandise of you; whose judgment\n    now of a long time lingereth not, and their abomination slumbereth\n    not.\" The reason why the Lord destroyed the prophet and made those\n    who \"set up their idols in their heart,\" a sign and a proverb,\n    made them bear the punishment of their iniquity is worthy of our\n    earnest attention. We are informed that the reason why the Lord\n    would perform all these things was this, \"that the house of Israel\n    may go no more astray from me, neither be polluted any more with\n    all their transgressions; but that they may be my people, and I\n    may be their God.\" Here is positive evidence that this prophecy\n    was to be fulfilled in the last days, for there has only been a\n    small part of the house of Israel (at any time since this prophecy\n    was given) that were obedient to the Lord. The time is not fully\n    come when Israel shall \"go no more astray,\" and not \"be polluted\n    any more with all their transgressions,\" therefore the punishment\n    of these men who have committed these sins must continue until\n    that happy day shall come. But as the Lord says in this prophecy,\n    \"repent and turn yourselves from your idols; and turn away your\n    faces from your abominations, so say we, and return unto the fold\n    from whence you have strayed.\" As some may yet doubt whether God\n    would act in this way toward men who set up their idols in their\n    heart, we will see how God dealt with Balaam. In Numbers 22 c. we\n    are informed that Balak, king of the Moabites, sent the elders of\n    Moab and Midian unto Balaam with the rewards of divination in their\n    hands to entreat him that he would curse Israel, but God said unto\n    Balaam, \"Thou shalt not go with them; thou shalt not curse the\n    people, for they are blessed.\" And Balaam rose up in the morning,\n    and said unto the Princes of Balak, \"Get you unto your land; for\n    the Lord refuseth to give me leave to go with you.\" And Balak\n    sent yet again princes, more, and more honorable than they. And\n    they came to Balaam and said to him, \"Thus sayeth Balak, the son\n    of Zippor, let nothing, I pray thee, hinder thee from coming unto\n    me: For I will promote thee unto very great honor, and I will do\n    whatsoever thou sayest unto me; come, therefore, I pray thee, curse\n    me this people.\" Now although the Lord had said unto Balaam, \"Thou\n    shalt not go with them; thou shalt not curse the people, for they\n    are blessed,\" yet the great honor that was offered him, allured\n    him, and he inquired of the Lord again, and said unto the princes,\n    \"Tarry ye also here this night, that I may know what the Lord will\n    say unto me more.\" And God came unto Balaam at night, and said unto\n    him, \"If the men come to call thee, rise up and go with them: but\n    yet the word which I shall say unto thee, that shalt thou do.\" And\n    Balaam rose up in the morning and saddled his ass, and went with\n    the princes of Moab. And God's anger was kindled because he went;\n    and the angel of the Lord stood in the way for an adversary against\n    him. So we find that the Lord told him not to go, but afterwards,\n    having \"set up his idol in his heart\" he inquired of the Lord\n    again whether he might not go and curse Israel and God's anger was\n    kindled against him because he did so, although God had commanded\n    him to go. This is, therefore, a parallel case with Ezekiel's\n    prophecy.[3]\n\n    In I Kings, 22 c. we are informed that the King of Israel wanted\n    Jehoshaphat, king of Judah, to go up with him to Ramoth-Gilead\n    to battle, and there were four hundred prophets who said \"Go up,\n    for the Lord shall deliver it into the hands of the king.\" And\n    Jehoshaphat said, \"Is there not here a prophet of the Lord besides,\n    that we might inquire of him?\" And the king of Israel said unto\n    Jehoshapat, \"There is yet one, Micaiah, the son of Imlah, by\n    whom we may inquire of the Lord; but I hate him, for he doth not\n    prophesy good concerning me, but evil.\" And Jehoshaphat said, \"Let\n    not the king say so.\" So he was sent for. The messenger that was\n    gone to call Micaiah spake unto him, saying, \"Behold now the words\n    of the prophets declare good unto the king with one mouth: let thy\n    word, I pray thee, be like the word of one of them, and speak that\n    which is good.\" And Micaiah said, \"As the Lord liveth, what the\n    Lord saith unto me, that will I speak.\" We are then informed that\n    Micaiah prophesied like the false prophets,[4] and then against\n    them. And he said, \"I saw the Lord sitting on his throne, and all\n    the hosts of heaven standing by him on his right hand and on his\n    left. And the Lord said, Who shall persuade Ahab that he may go up\n    and fall at Ramoth-Gilead? And one said on this matter, and another\n    said on that manner. And there came forth a spirit and stood before\n    the Lord and said, I will persuade him. And the Lord said unto him\n    wherewith? And he said, I will go forth, and I will be a lying\n    spirit in the mouth of all his prophets. And he said, thou shalt\n    persuade him, and prevail also; go forth and do so. Now therefore\n    behold the Lord hath put a lying spirit in the mouth of all these\n    thy prophets, and the Lord hath spoken evil concerning thee.\" This\n    doctrine was extensively preached in the Church before iniquity\n    overthrew the Church, and by this doctrine the Church might have\n    been saved, if men had not \"set up their idols in their heart.\"\n\n\nFootnotes\n\n1. The Prophet Joseph Smith, Brigham Young and the Saints.\n\n2. The inspired translation reads: \"I the Lord have not deceived that\nprophet.\"\n\n3. Mr. Sheen forgets that the Lord said, \"Thou shalt not curse the\npeople, for they are blessed,\" which command Balaam hearkened to.\n\n4. The prophecy was; \"Go and prosper; for the Lord shall deliver it\ninto the hands of the king,\" v. 15. This was uttered in mockery, if not\nwhy did the king reply: \"How many times shall I adjure thee that thou\ntell me nothing but that which is true in the name of the Lord,\" v. 16.\n_Then_ Micaiah told the king that he should fall at Ramoth-Gilead, so\nthe king acted with full knowledge of the word of the Lord concerning\nhis death when he went forth to battle. Therefore the Lord did not\ndeceive Ahab in this matter.\n\n\n\nINTRODUCTION OF CELESTIAL AND PLURAL MARRIAGE\n\nAdditional testimony of a few out of the multitude[1] of witnesses who\nwere taught these principles by the Prophet Joseph Smith, and who knew\nthat he received the revelation known as section 132 in the Book of\nDoctrine and Covenants.\n\n\nAFFIDAVIT OF PRESIDENT LORENZO SNOW\n\nIn the month of April, 1843, I returned from my European mission. A\nfew days after my arrival at Nauvoo, when at President Joseph Smith's\nhouse, he said he wished to have some private talk with me, and\nrequested me to walk out with him. It was toward evening. We walked a\nlittle distance and sat down on a large log that lay near the bank of\nthe river. He there and then explained to me the doctrine of plurality\nof wives; he said that the Lord had revealed it unto him, and commanded\nhim to have women sealed to him as wives; that he foresaw the trouble\nthat would follow, and sought to turn away from the commandment; that\nan angel from heaven then appeared before him with a drawn sword,\nthreatening him with destruction unless he went forward and obeyed the\ncommandment.\n\nHe further said that my sister Eliza R. Snow had been sealed to him as\nhis wife for time and eternity. He told me that the Lord would open the\nway, and I should have women sealed to me as wives. This conversation\nwas prolonged, I think one hour or more, in which he told me many\nimportant things.\n\nI solemnly declare before God and holy angels, and as I hope to come\nforth in the morning of the resurrection, that the above statement is\ntrue.\n\nLorenzo Snow.\n\nTerritory of Utah, Box Elder County. ss.\n\nPersonally came before me J. C. Wright, Clerk of the County and Probate\nCourts in and for the County and Territory aforesaid, Lorenzo Snow, and\nwho being duly sworn deposeth and says that the foregoing statement by\nhim subscribed is true of his own certain knowledge.\n\nWitness my hand and seal of Court, at my office in Brigham City, Box\nElder County, Utah Territory, this 28th day of August, A.D. 1869.\n\n[Seal.]\n\nJ. C. Wright, Clerk.\n\n\nAFFIDAVIT OF LUCY WALKER\n\nUnited States of America,\n\nState of Utah.\n\nCounty of Salt Lake.\n\nLucy Walker Smith Kimball, being first duly sworn, says:\n\nI was a plural wife of the Prophet Joseph Smith, and was married for\ntime and eternity in Nauvoo, State of Illinois, on the first day of\nMay, 1843, by Elder William Clayton. The Prophet was then living with\nhis first wife, Emma Smith, and I know that she gave her consent to the\nmarriage of at least four women to her husband as plural wives, and she\nwas well aware that he associated and cohabited with them as wives. The\nnames of these women are Eliza and Emily Partridge, and Maria and Sarah\nLawrence, all of whom knew that I too was his wife.\n\nWhen the Prophet Joseph Smith mentioned the principle of plural\nmarriage to me I felt indignant, and so expressed myself to him,\nbecause my feelings and education were averse to anything of that\nnature. But he assured me that this doctrine had been revealed to him\nof the Lord, and that I was entitled to receive a testimony of its\ndivine origin for myself. He counseled me to pray to the Lord, which\nI did, and thereupon received from Him a powerful and irresistible\ntestimony of the truthfulness and divinity of plural marriage, which\ntestimony has abided with me ever since.\n\nOn the 8th day of February, 1845, I was married for _time_ to President\nHeber C. Kimball, and bore to him nine children. And in this connection\nallow me to say to his everlasting credit that during the whole of my\nmarried life with him he never failed to regard me as the wife for\neternity of his devoted friend, the Prophet Joseph Smith.\n\n                                       Lucy Walker Smith Kimball.\n\nSubscribed and sworn to before me, this 17th day of December, 1902.\n\n[Seal.]\n\nJames Jack, Notary Public.\n\n\nAFFIDAVIT OF CATHERINE PHILLIPS SMITH\n\nUnited States of America,\n\nState of Utah.\n\nCounty of Salt Lake.\n\nCatherine Phillips Smith,[2] being first sworn, says:\n\nI am the daughter of Thomas Denner and Sarah Godshall Phillips, and\nwas born in Philadelphia, State of Pennsylvania, on the first day of\nAugust, 1819. My present residence is East Jordan, Salt Lake County,\nUtah.\n\nI was married to Hyrum Smith, brother of the Prophet Joseph Smith,\nas his plural wife, and lived with him as his wife. The sealing was\nperformed by the Prophet Joseph Smith himself, in Nauvoo, State of\nIllinois, in August, 1843, in the brick office belonging to my husband,\nand occupied at the time as a dwelling by Brother and Sister Robert\nand Julia Stone, and was witnessed by my mother, Sister Stone and her\ndaughter Hettie.\n\nIn consequence of the strong feeling manifested at the time against\nplural marriage and those suspected of having entered into it, I, with\nmy mother, moved to St. Louis near the close of the year, where I was\nliving when the Prophet Joseph and my husband were martyred.\n\nThe purpose of this affidavit is that my testimony to the truthfulness\nand divinity of plural marriage may live after I shall have passed\naway; and in this spirit I commend it to all to whom it may come.\n\n                                         Catherine Phillips Smith.\n\nSubscribed and sworn to before me, this 28th day of January, 1903.\n\n[Seal]\n\n                                      L. John Nuttall, Notary Public.\n\n\nAFFIDAVIT OF ALMIRA W. JOHNSON SMITH BARTON\n\nTerritory of Utah, County of Iron. ss.\n\nBe it remembered on this first day of August A.D. 1883, personally\nappeared before me John W. Brown a notary public in and for said\ncounty, Almira W. Johnson Smith Barton, who was by me sworn in due\nform of law, and upon her oath says: I am a citizen in the Territory\nof Utah, over the age of twenty-one years, and I am the daughter of\nEzekiel Johnson and Julia Hills Johnson his wife; that I was born at\nWestford, in the State of Vermont on the 22nd day of October A.D. 1813;\nthat I had nine brothers who were named respectfully Joel H., Seth,\nDavid, Benjamin F., Joseph E., Elmer, George W., William D., and Amos;\nand six sisters named respectfully Nancy, Dulcena, Julia, Susan, Mary\nand Esther, all of whom, with myself, were baptized into the Church of\nJesus Christ of Latter-day Saints with the exception of Elmer, who died\nin infancy.\n\nDeponent further says, that in the years 1842 and 1843, I resided most\nof the time at Macedonia, in the County of Hancock, State of Illinois,\nsometimes with my sister who was the wife of Almon W. Babbitt, and\nsometimes with my brother Benjamin F. Johnson. During that time the\nProphet Joseph Smith taught me the principle of celestial marriage\nincluding plurality of wives and asked me to become his wife. He first\nspoke to me on this subject at the house of my brother Benjamin F. I\nalso lived a portion of the time at Brother Joseph Smith's in Nauvoo,\nwhen many conversations passed between him and myself on this subject.\nOn a certain occasion in the spring of the year 1843, the exact date of\nwhich I do not now recollect, I went from Macedonia to Nauvoo to visit\nanother of my sisters, the one who was the widow of Lyman R. Sherman,\ndeceased, at which time I was sealed to the Prophet Joseph Smith. At\nthe time this took place Hyrum Smith, Joseph's brother, came to me and\nsaid I need not be afraid. I had been fearing and doubting about the\nprinciple and so had he, but he now knew it was true. After this time\nI lived with the Prophet Joseph as his wife, and he visited me at the\nhome of my brother Benjamin F. at Macedonia.\n\nDeponent further says that I had many conversations with Eliza Beaman\nwho was also a wife of Joseph Smith, and who was present when I was\nsealed to him, on the subject of plurality of wives, both before and\nafter the performance of that ceremony. And also that since the death\nof the Prophet Joseph Smith I was married for time to Reuben Barton of\nNauvoo, Hancock Co., Ill., by whom I have had five daughters, one only\nof whom is now living.\n\n                                        Almira W. Johnson Smith Barton.\n\nSubscribed and sworn to by the said Almira W. Johnson Smith Barton the\nday and year first above written.\n\n[Seal.]\n\n                                          John W. Brown, Notary Public.\n\n\nAFFIDAVIT OF MARTHA McBRIDE KIMBALL\n\nTerritory of Utah, County of Millard. ss.\n\nBe it remembered that on this eighth day of July, A.D. 1869, personally\nappeared before me Edward Partridge, Probate Judge in and for said\ncounty, Martha McBride Kimball, who was by me sworn in due form of law,\nand upon her oath saith that sometime in the summer of the year 1842,\nat the city of Nauvoo, county of Hancock, state of Illinois, she was\nmarried or sealed to Joseph Smith, President of the Church of Jesus\nChrist of Latter-day Saints, by Heber C. Kimball, one of the Twelve\nApostles in said Church, according to the laws of the same regulating\nmarriage.\n\n                                                Martha McBride Kimball.\n\nSubscribed and sworn to by said Martha McBride Kimball the day and year\nfirst above written.\n\n[Seal.]\n\n                                       Edward Partridge, Probate Judge.\n\n\nAFFIDAVIT OF MELISSA LOTT WILLES\n\nTerritory of Utah, County of Salt Lake. ss.\n\nBe it remembered that on this twentieth day of May, A.D. 1869,\npersonally appeared before me, James Jack a notary public in and for\nsaid county, Melissa Lott Willes, who was by me sworn in due form of\nlaw, and upon her oath saith that on the twentieth day of September,\nA.D. 1843, at the city of Nauvoo, county of Hancock, state of Illinois,\nshe was married or sealed to Joseph Smith, President of the Church of\nJesus Christ of Latter-day Saints, by Hyrum Smith, Presiding Patriarch\nof said Church, according to laws of the same, regulating marriage, in\nthe presence of Cornelius P. Lott and Parmelia Lott.\n\n                                                  Melissa Lott Willes.\n\nSubscribed and sworn to by the said Melissa Lott Willes, the day and\nyear first above written.\n\n[Seal.]\n\n                                            James Jack, Notary Public.\n\n\nLOVINA SMITH WALKER'S TESTIMONY\n\nI, Lovina Walker, hereby certify that while I was living with Aunt Emma\nSmith, in Fulton City, Fulton Co., Illinois, in the year 1846, that she\ntold me that she, Emma Smith, was present and witnessed the marrying or\nsealing of Eliza Partridge, Emily Partridge, Maria Lawrence and Sarah\nLawrence to her husband, Joseph Smith, and that she gave her consent\nthereto.\n\n                                                         Lovina Walker.\n\nWe hereby witness that Lovina Walker made and signed the above\nstatement on this 16th day of June, A.D. 1869, at Salt Lake City, S. L.\nCounty, Utah Territory, of her own free will and record.\n\n                                                       Hyrum S. Walker,\n                                                       Sarah E. Smith,\n                                                       Joseph F. Smith.\n\n\nAFFIDAVIT OF SARAH A. KIMBALL\n\nTerritory of Utah, County of Salt Lake. ss.\n\nBe it remembered that on this nineteenth day of June, A.D. 1869,\npersonally appeared before me Elias Smith, Probate Judge for said\ncounty, Sarah Ann Kimball, who was by me sworn in due form of law, and\nupon her oath saith that on the twenty-seventh day of July, A.D. 1842,\nat the city of Nauvoo, county of Hancock, state of Illinois, she was\nmarried or sealed to Joseph Smith, President of the Church of Jesus\nChrist of Latter-day Saints, by Newell K. Whitney, Presiding Bishop of\nsaid Church, according to the laws of the same regulating marriage, in\nthe presence of Elizabeth Ann Whitney her mother.\n\n                                                      Sarah A. Kimball.\n\nSubscribed and sworn to by the said Sarah Ann (Whitney) Kimball, the\nday and year first above written.\n\n                                               E. Smith, Probate Judge.\n\n\nAFFIDAVIT OF ELIZABETH A. WHITNEY\n\nTerritory of Utah, County of Salt Lake. ss.\n\nBe it remembered that on this thirtieth day of August, A.D. 1869,\npersonally appeared before me, James Jack, a notary public in and for\nsaid county, Elizabeth Ann Whitney, who was by me sworn in due form\nof law, and upon her oath saith that on the twenty-seventh day of\nJuly, A.D. 1842, at the city of Nauvoo, county of Hancock, state of\nIllinois, she was present and witnessed the marrying or sealing of her\ndaughter Sarah Ann Whitney to the Prophet Joseph Smith, for time and\nall eternity, by her husband Newel K. Whitney then Presiding Bishop of\nthe Church.\n\n                                                        E. A. Whitney.\n\nSubscribed and sworn to by the said Elizabeth Ann Whitney the day and\nyear first above written.\n\n                                            James Jack, Notary Public.\n\n\nAFFIDAVIT OF ORSON HYDE\n\n                                         Springtown, Sept. 15, 1869.\n\nI, Orson Hyde, do hereby certify and declare according to my best\nrecollection that on the fourth day of September I was married to Miss\nMarinda N. Johnson, in Kirtland, Ohio, in the year of our Lord 1834,\nand in the month of February or March, 1843, I was married to Miss\nMartha R. Browitt, by Joseph Smith, the martyred prophet, and by him\nshe was sealed to me for time and for all eternity in Nauvoo, Ill.,\nand in the month of April of the same year, 1843, I was married by the\nsame person to Mrs. Mary Ann Price, and by him she was sealed to me for\ntime and for all eternity, in Nauvoo, Ill., while the woman to whom I\nwas first married was yet living, and gave her cordial consent to both\ntransactions, and was personally present to witness the ceremonies.\n\n                                                         Orson Hyde.\n\nSworn to and subscribed to before me this the 15th day of September,\n1869, at Springtown, Sanpete County, UT.\n\n                                George Brough, Justice of the Peace.\n\nI hereby certify that the above named George Brough is a justice of the\npeace for the precinct of Springtown in the county of Sanpete, UT., and\nthat he is duly qualified in accordance with law; in testimony whereof,\nI hereunto set my hand and official seal of the County Court of Sanpete\nCounty, at my office, Manti City, this Sept. 16, 1869.\n\n[Seal.]\n\n                                         William T. Reed, County Clerk.\n\n\nAFFIDAVIT OF JOSEPH BATES NOBLE\n\nTerritory of Utah, County of Salt Lake. ss.\n\nBe it remembered that on the 26th day of June, A.D. 1869, personally\nappeared before me, James Jack, a notary public in and for said county,\nJoseph Bates Noble, who was by me sworn in due form of law, and upon\nhis oath saith, that on the fifth day of April, A.D. 1841, at the city\nof Nauvoo, County of Hancock, State of Illinois, he married or sealed\nLouisa Beaman to Joseph Smith, President of the Church of Jesus Christ\nof Latter-day Saints, according to the order of celestial marriage\nrevealed to the said Joseph Smith.\n\n                                                   Joseph B. Noble.\n\nSubscribed and sworn to by the said Joseph Bates Noble, the day and\nyear first above written.\n\n[Seal.]\n\n                                         James Jack, Notary Public.\n\n\nAFFIDAVIT OF RHODA RICHARDS SMITH\n\nTerritory of Utah, County of Salt Lake. ss.\n\nBe it remembered that on this first day of May, A.D. 1869, personally\nappeared before me, Elias Smith, Probate Judge for said county, Rhoda\nRichards, who was by me sworn in due form of law and upon her oath\nsaith that on the twelfth day of June A.D. 1843, at the city of Nauvoo,\nCounty of Hancock, State of Illinois, she was married or sealed to\nJoseph Smith, President of the Church of Jesus Christ of Latter-day\nSaints, by Willard Richards, one of the Twelve Apostles of said Church,\naccording to the laws of the same regulating marriage.\n\n                                                     Rhoda Richards.\n\nSubscribed and sworn to by the said Rhoda Richards, the day and year\nabove written.\n\n[Seal.]\n\n                                          Elias Smith, Probate Judge.\n\n\nTESTIMONY OF BENJAMIN F. JOHNSON\n\n                                 Mesa City, Arizona, 9th March, 1904.\n\n_President Joseph F. Smith_,\n\n_Washington, D. C_.\n\nMy Dear Brother:--\n\nIn reading reports from the Senate Committee on the Reed Smoot case,\nI see that witnesses are subpoenaed to prove that the Prophet Joseph\nSmith did not authorize or practice polygamy; and I do know that he did\nteach plural marriage, and that he did give to me a plural wife who is\nstill living with me, and that I saw one of my sisters married to him.\n* * *\n\nAnd I do know that at his Mansion House was living Mariah and Sarah\nLawrence and one of Cornelius P. Lott's daughters as his plural wives\nwith the full knowledge of his wife, Emma, of the married relations to\nhim.\n\nAt that time I was his legal business agent at Macedonia or Ramtis,\nand was familiar with his family or domestic affairs; and occupying,\nas I did, the family mansion often in a business way with Emma, the\nProphet's first wife, who at no time did ever in my hearing deny the\nplural character of her husband's family.\n\nAnd now with this and much more knowledge relating to this subject,\ncould my evidence before the Senate Committee be of any real value to\nthe cause of truth? If so, although too infirm to travel alone I would\nwillingly try to be there, if according to your counsel and wish.\n\nLoyal to the truth, I am,\n\n                             Always brother,\n\n                                                 B. F. Johnson.\n\n\nTHE CELESTIAL AND PLURAL MARRIAGE REVELATION\n\nThe following letter was written by Elder William Clayton who wrote the\nrevelation known as section 132 in the Book of Doctrine and Covenants,\nat the direction of the Prophet Joseph Smith, July 12, 1843.[3]\n\n    Salt Lake City, Nov, 11, 1871.\n\n    _Madison M. Scott, Esq_.\n\n    Dear Sir:\n\n    Your letter of 23rd of June last, was received by due course of\n    mail, but owing to my being so very closely confined with public\n    duties, which has almost destroyed my health, I have not answered\n    your letter so promptly as is my practice. My health is yet very\n    poor, but I have resigned the office which was bearing so heavy\n    upon me, and am in hopes to regain my usual sound health.\n\n    Now, in regard to the subject matter of your letter, it appears to\n    me that the principal topic is what is commonly called polygamy,\n    but which I prefer to call celestial marriage. As to young Joseph\n    saying that the Church here have apostatized; that _we_ have\n    introduced polygamy, denying bitterly that his father ever had a\n    revelation on the subject, that is all mere bosh! I _believe_ he\n    knows better, and I have often felt sorry to learn that the sons\n    of the Prophet should spend their time in contending against a\n    pure and holy principle which their father's blood was shed to\n    establish. They will have a heavy atonement to make when they meet\n    their father in the next world. They are in the hands of God, and\n    my respect for their father will not permit me to say much about\n    the wicked course of his sons.\n\n    _Now, I say to you, as I am ready to testify to all the world, and\n    on which testimony I am most willing to meet all the Latter-day\n    Saints and all apostates, in time and through all eternity, I did\n    write the revelations on celestial marriage given through the\n    Prophet Joseph Smith, on the 12th of July, 1843_.\n\n    When the revelation was written there was no one present except the\n    Prophet Joseph, his brother Hyrum and myself. It was written in the\n    small office upstairs in the rear of the brick store which stood\n    on the banks of the Mississippi river. It took some three hours to\n    write it. Joseph dictated sentence by sentence, and I wrote it as\n    he dictated. After the whole was written Joseph requested me to\n    read it slowly and carefully, which I did, and he then pronounced\n    it correct. The same night a copy was taken by Bishop Whitney,\n    which copy is now here (in the Historian's office) and which I know\n    and testify is correct. The original was destroyed by Emma Smith.\n\n    I again testify that the revelation on polygamy was given through\n    the prophet Joseph on the 12th July, 1843; and that the Prophet\n    Joseph both taught and practiced polygamy I do positively know,\n    and bear testimony to the fact. In April, 1843, he sealed to me my\n    second wife, my first wife being then living. By my said second\n    wife I had two sons born in Nauvoo. The first died; the second is\n    here now, and is married.\n\n    I had the honor to seal one woman[4] to Joseph under his direction.\n    I could name ten or a dozen of his wives who are now living in\n    this territory, so that for any man to tell me that Joseph did not\n    teach polygamy, he is losing his time, for I know better. It is\n    not hearsay, nor opinion with me, for I positively know of what I\n    speak, and I testify to the truth, and shall be willing to meet all\n    opponents on the subject through all eternity.\n\n    As to the Church here having apostatized that is all a mere\n    matter of assertion, destitute of truth. President Young and his\n    associates are, and have been doing everything they can to carry\n    out the plans and instructions of the Prophet Joseph, and so\n    eternity will prove to the condemnation and confusion of all their\n    enemies. Any one who says to the contrary does not know Joseph nor\n    the mission the Lord gave him to fulfill. * * *\n\n                                            Truly yours,\n                                                   William Clayton.\n\n\nAFFIDAVIT OF HOWARD CORAY\n\nTerritory of Utah, County of Salt Lake. ss.\n\nAs many false statements have been made in relation to the authorship\nof the revelation on celestial marriage, I deem it but justice to all\nlovers of truth for me to express what I know concerning this very\nimportant matter.\n\nOn the 22nd day of July, A.D. 1843, Hyrum Smith, the martyred\nPatriarch, came in a carriage to my house in Nauvoo; he invited me\nand my wife to take a ride with him; accordingly, as soon as we could\nmake ourselves ready, we got into his carriage and he set off in the\ndirection of Carthage. Having gone a short distance, he observed to us\nthat his brother Joseph Smith, the Prophet, had received a revelation\non marriage, that was not for the public yet, which he would rehearse\nto us, as he had taken pains to commit it to memory. He then commenced\nrehearsing the revelation on celestial marriage not stopping till he\nhad gone quite through with the matter. After which he reviewed that\npart pertaining to plurality of wives, dwelling at some length upon the\nsame, in order that we might clearly understand the principle. And on\nthe same day (July 22, 1843,) he sealed my wife, formerly Martha Jane\nKnowlton, to me; and when I heard the revelation on celestial marriage\nread on the stand in Salt Lake City, in 1852, I recognized it as the\nsame as that repeated to me by Brother Hyrum Smith. Not long after\nthis I was present when Brother David Fullmer and wife were sealed by\nBrother Hyrum Smith, the martyred Patriarch, according to the law of\ncelestial marriage. And, besides the foregoing, there was quite enough\ncame within the compass of my observation to have fully satisfied my\nmind that plural marriage was practiced in the city of Nauvoo.\n\n                                                        Howard Coray.\n\nSubscribed and sworn to before me, this 18th day of June, A.D. 1882.\n\n[Seal.]\n\n                                          James Jack, Notary Public.\n\n\nAFFIDAVIT OF DAVID FULLMER[5]\n\nTerritory of Utah, County of Salt Lake. ss.\n\nBe it remembered that on this fifteenth day of June, A.D. 1869,\npersonally appeared before me, James Jack, a notary public in and for\nsaid county, David Fullmer, who was by me sworn in due form of law,\nand upon his oath saith, that on or about the 12th day of August,\nA.D. 1843, while in meeting with the High Council [he being a member\nthereof] in Hyrum Smith's brick office, in the City of Nauvoo, County\nof Hancock, State of Illinois, Dunbar Wilson made inquiry in relation\nto the subject of plurality of wives, as there were rumors about\nrespecting it, and he was satisfied there was something in those\nrumors, and he wanted to know what it was. Upon which Hyrum Smith\nstepped across the road to his residence, and soon returned bringing\nwith him a copy of the revelation on celestial marriage given to Joseph\nSmith July 12, 1843, and read the same to the High Council, and bore\ntestimony to its truth. The said David Fullmer further saith that,\nto the best of his memory and belief, the following named persons\nwere present: William Marks, Austin A. Cowles, Samuel Bent, George\nW. Harris, Dunbar Wilson, William Huntington, Levi Jackman, Aaron\nJohnson, Thomas Grover, David Fullmer, Phineas Richards, James Allred\nand Leonard Soby. And the said David Fullmer further saith that William\nMarks, Austin A. Cowles and Leonard Soby were the only persons present\nwho did not receive the testimony of Hyrum Smith, and that all the\nothers did receive it from the teachings and testimony of the said\nHyrum Smith; and further, that the copy of said revelation on celestial\nmarriage published in the _Deseret News_ extra of September 14, A.D.,\n1852, is a true copy of the same.\n\n                                                        David Fullmer.\n\nSubscribed and sworn to by the said David Fullmer the day and year\nfirst above written.\n\n[Seal.]\n\n                                            James Jack, Notary Public.\n\n\nAFFIDAVIT OF LEONARD SOBY[6]\n\nBe it remembered that on the 23rd day of March, in the year 1886,\nbefore, Joshua W. Roberts, notary public for the City of Beverly,\nCounty of Burlington, State of New Jersey, Leonard Soby, of said city,\ncounty and state, was by me duly sworn, and upon his oath saith:\n\nThat on or about the 12th day of August, 1843, I was a resident of\nNauvoo, Hancock County, State of Illinois, and being a member of the\nHigh Council of the Church of Jesus Christ of Latter-day Saints, was\npresent at a meeting of said council at the time herein above stated;\nThomas Grover, Alpheus Cutler, David Fullmer, William Huntington and\nothers; when Elder Hyrum Smith, after certain explanations, read the\nrevelation on celestial marriage.\n\nI have read and examined carefully said revelation, since published\nin the Book of Doctrine and Covenants of said Church, and say to the\nbest of my knowledge and belief it is the same, word for word, as the\nrevelation then read by Hyrum Smith.\n\nThe deponent says further, that the revelation did not originate with\nBrigham Young, as some persons have falsely stated, but was received by\nthe Prophet Joseph Smith, and read in the High Council by his authority\nas a revelation to the Church of Jesus Christ of Latter-day Saints.\n\nWhen read to this deponent and said High Council, I believed it was a\nrevelation from Jesus Christ, and I believe so now.\n\n                                                          Leonard Soby.\n\nSubscribed and sworn to by the said Leonard Soby the day and year first\nabove written.\n\n                                      Joshua W. Roberts, Notary Public.\n\nWitnessed by:\n\nJames H. Hart,\n\nSamuel Harrison.\n\n\nAFFIDAVIT OF JOHN W. RIGDON\n\nState of Utah, County of Salt Lake. ss.\n\nJohn W. Rigdon, being duly sworn, says: I am the son of Sidney Rigdon,\ndeceased. Was born at Mentor, in the State of Ohio, in the year 1830,\nand am now over seventy-five years of age. My father, Sidney Rigdon,\njoined the Church of Jesus Christ of Latter-day Saints that year,\nand was in 1833 ordained to be Joseph Smith's first counselor which\nposition he held up to the time Joseph the Prophet was killed, at\nCarthage jail, in 1844. That Joseph Smith and Sidney Rigdon moved from\nKirtland, with their families, to the State of Missouri, during the\nwinter of 1837, but Rigdon did not reach Far West, in the State of\nMissouri, until the last of April, 1838. That during the troubles in\nMissouri, in the year 1838, Joseph Smith, Hyrum Smith, his brother,\nSidney Rigdon, Lyman Wight and others, whose names I do not now\nremember were arrested and imprisoned in Liberty jail, about forty\nmiles from the village of Far West, in Caldwell County, Missouri, where\nthey all remained incarcerated for several months. That while said\nJoseph Smith, Hyrum Smith, Sidney Rigdon, Lyman Wight and others were\nprisoners in said Liberty jail, as aforesaid I, with my mother, wife\nof Sidney Rigdon, Emma Smith, wife of said Joseph Smith, and Joseph\nSmith, son of Joseph and Emma Smith, went to see the said prisoners\nduring the latter part of the winter of 1838. We all went together in\nthe same carriage and came home together. We stayed at Liberty jail\nwith the prisoners three days and then left for home. The story that\nis being told by some of the members of the Reorganized Church, at\nLamoni, that young Joseph Smith, now president of the said Reorganized\nChurch, was ordained by his father, Joseph Smith, to be the leader of\nthe Church of Jesus Christ of Latter-day Saints after his father's\ndeath, is not true, for I know that no such ordination took place while\nwe were at Liberty jail; that if any such ordination had taken place I\nmost certainly should have known it and remembered it, as I was with\nyoung Joseph, the Prophet's son, all the time we were there. If Joseph\nSmith had ordained his son Joseph to be the leader of the Church at his\ndeath, he would have done so in a manner that there could have been\nno doubt about it. Both of his counselors were then in prison with\nhim, namely, Sidney Rigdon and Hyrum Smith, and it would have been in\norder for the prophet to have called upon them to assist him in such\nan ordination had it taken place, and a record of the same made in the\nChurch books, so that all members of the Church might have known that\nsuch an ordination had taken place. But nothing of the kind appears in\nthe Church books. My father and mother lived a good many years after\nthe incarceration at Liberty jail, and I, who lived near my father,\nnever heard my father or my mother mention that such an ordination\never took place in Liberty jail; and as I know myself that no such\nordination took place in Liberty jail, and inasmuch as it is not\nclaimed that an ordination of this character was bestowed at any other\nplace, therefore I deny it as an untruth and a story gotten up by the\nReorganized Church for effect.\n\nBesides all this, if Joseph Smith, the President of the Reorganized\nChurch was ordained while in Liberty jail, why did he, sixteen years\nafter his father's death, receive an ordination under the hands of\nWilliam Marks, William W. Blair, and Zenas H. Gurley? Would it not seem\nthat one ordination (and that too, said to have been by his own father,\nthe President of the Church) should have been sufficient? But further\nWm. Marks, Wm. W. Blair and Zenas H. Gurley had all been excommunicated\nfrom the Church of Jesus Christ of Latter-day Saints (excepting William\nW. Blair, who never belonged to it) before they \"ordained\" young\nJoseph to be President of the Reorganized Church, and therefore they\ndid not have the authority to ordain him. The whole story of his being\nordained by anyone having authority to do so is too preposterous to be\nentertained for a single moment, and should be rejected by all who hear\nsuch a story mentioned.\n\nAs to the truth of the doctrine of polygamy being introduced by the\nProphet Joseph Smith, deponent further says: Joseph Smith was absolute\nso far as spiritual figures were concerned, and no man would have dared\nto introduce the doctrine of polygamy or any other new doctrine into\nthe \"Mormon\" Church at the city of Nauvoo during the years 1843 and\n1844, or at any other place or time, without first obtaining Joseph\nSmith's consent. If anyone had dared to have done such a thing he would\nhave been brought before the High Council and tried, and if proven\nagainst him, he would have been excommunicated from the Church, and\nthat would have ended polygamy forever, and would also have ended the\nman who had dared to introduce such a doctrine without the consent of\nthe Prophet Joseph.\n\nAnd deponent further says: Joseph the Prophet, at the City of Nauvoo,\nIllinois, some time in the latter part of the year 1843, or the\nfirst part of the year 1844, made a proposition to my sister, Nancy\nRigdon, to become his wife. It happened in this way: Nancy had gone to\nChurch, meeting being held in a grove near the temple lot on which the\n\"Mormons\" were then erecting a temple, an old lady friend who lived\nalone invited her to go home with her, which Nancy did. When they got\nto the house and had taken their bonnets off, the old lady began to\ntalk to her about the new doctrine of polygamy which was then being\ntaught, telling Nancy, during the conversation, that it was a surprise\nto her when she first heard it, but that she had since come to believe\nit to be true. While they were talking Joseph Smith the Prophet came\ninto the house, and joined them, and the old lady immediately left\nthe room. It was then that Joseph made the proposal of marriage to my\nsister. Nancy flatly refused him, saying if she ever got married she\nwould marry a single man or none at all, and thereupon took her bonnet\nand went home, leaving Joseph at the old lady's house. Nancy told\nfather and mother of it. The story got out and it became the talk of\nthe town that Joseph had made a proposition to Nancy Rigdon to become\nhis wife, and that she refused him. A few days after the occurrence\nJoseph Smith came to my father's house and talked the matter over with\nthe family, my sister, Mrs. Athalia Robinson also being present, who\nis now alive. The feelings manifested by our family on this occasion\nwere anything but brotherly or sisterly, more especially on the\npart of Nancy, as she felt that she had been insulted. A day or two\nlater Joseph Smith returned to my father's house, when matters were\nsatisfactorily adjusted between them, and there the matter ended. After\nthat Joseph Smith sent my father to Pittsburgh, Pa., to take charge of\na little church that was there, and Ebenezer Robinson, who was then the\nChurch printer, or at least had been such, as he was the printer of the\npaper in Kirtland, Ohio, and a printer by trade, was to go with him\nto print a paper there, and nine days before Joseph Smith was shot at\nCarthage we started, reaching Pittsburgh the day before he was killed.\n\nDeponent further says: I have in my possession a paper called the\n_Nauvoo Expositor_, bearing date, Nauvoo, Illinois, Friday, June 7th,\n1844, which said paper's printing plant was destroyed by the City\nCouncil at Nauvoo a night or two after that issue. There never was but\none issue of this paper. Joseph Smith the Prophet was then Mayor of the\nCity of Nauvoo. In the afternoon of the day on which the printing plant\nwas destroyed, Henry Phelps, a son of W. W. Phelps, came down Main\nStreet selling this paper, the _Nauvoo Expositor_, and everyone who\ncould raise five cents bought a copy. In that paper the three following\naffidavits appeared, which I reproduce herewith.\n\n\nAFFIDAVITS\n\n    I hereby certify that Hyrum Smith did (in his office) read to me a\n    certain written document which he said was a revelation from God.\n    He said that he was with Joseph when it was received. He afterwards\n    gave me the document to read and I took it to my house and read\n    it and showed it to my wife and returned it the next day. The\n    revelation (so called) authorized certain men to have more wives\n    than one at a time in this world and in the world to come. It said\n    this was the law, and commanded Joseph to enter into the law. And\n    also that he should administer to others. Several other items were\n    in the revelation, supporting the above doctrines.\n\n                                                              Wm. Law.\n\n    State of Illinois,\n\n    Hancock County.\n\n    I, Robert D. Foster, certify that the above certificate was sworn\n    to before me as true in substance, this fourth day of May, A.D.\n    1844.\n\n                                               Robert D. Foster, J. P.\n\n\n    I certify that I read the revelation referred to in the above\n    affidavit of my husband. It sustained in strong terms the doctrine\n    of more wives than one at a time in this world and in the next. It\n    authorized some to have to the number of ten, and set forth that\n    those women who would not allow their husbands to have more wives\n    than one should be under condemnation before God.\n\n                                                             Jane Law.\n\n    Sworn and subscribed before me this 4th day of May, A.D. 1844.\n\n                                               Robert D. Foster, J. P.\n\n\n    To all whom it may concern:\n\n    Forasmuch as the public mind hath been much agitated by a course of\n    procedure in the Church of Jesus Christ of Latter-day Saints by a\n    number of persons declaring against certain doctrines and practices\n    therein (among whom I am one) it is but meet that I should give my\n    reasons at least in part as a cause that hath led me to declare\n    myself. In the latter part of the summer of 1843, the Patriarch\n    Hyrum Smith did in the High Council, of which I was a member,\n    introduce what he said was a revelation given through the Prophet,\n    that the said Hyrum Smith did essay to read the said revelation in\n    the said council; that according to his reading there was contained\n    the following doctrines: 1st. The sealing up of persons to eternal\n    life, against all sins save that of shedding innocent blood or of\n    consenting thereto; 2nd. The doctrine of plurality of wives or\n    marrying virgins; that David and Solomon had many wives, yet in\n    this they sinned not, save in the matter of Uriah. This revelation\n    with others, evidence that the aforesaid heresies were taught and\n    practiced in the Church, determined me to leave the office of first\n    counselor to the President of the Church at Nauvoo, inasmuch as I\n    dared not teach or administer such laws. And further deponent saith\n    not.\n\n                                                       Austin Cowles.\n\n    State of Illinois,\n\n    Hancock County.\n\n    To all whom it may concern: I hereby certify that the above\n    certificate was sworn and subscribed before me, this fourth day of\n    May, 1844.\n\n                                              Robert D. Foster, J. P.\n\n                                                      John W. Rigdon.\n\n    Sworn to before me this 28th day of July, 1905.\n\n    [Seal.]\n\n                                           James Jack, Notary Public.\n\n\nSTATEMENT OF ORANGE L. WIGHT\n\nThe following confirmation of John W. Rigdon's affidavit is copied from\nthe _Deseret News_ of Saturday, August 12, 1905:\n\n    Bunkerville, Lincoln County, Nev., August 4, 1905:--Seeing the\n    testimony of J. W. Rigdon in the semi-weekly _News_ of July 31,\n    and being much interested in the subject, and knowing that there\n    lived in this place a man that was quite familiar with the early\n    scenes of church history, especially those in and about Far West,\n    Missouri, and having heard him say that he had many times visited\n    his father and the Prophet Joseph, while they were incarcerated in\n    Liberty jail, I went and interviewed Orange L. Wight (eldest son of\n    former Apostle Lyman Wight), who is now 82 years old and resides\n    with his daughter, Sister Harriet M. Earl. Brother Wight is quite\n    feeble in body, but his mind seems to be as bright as ever.\n\n    I found Brother Wight in his usual good humor, and seemed quite\n    willing to talk, in fact, was pleased to do so. \"Elder Wight,\"\n    said I, \"are you willing to make a statement for publication in\n    regard to what you know about Joseph Smith, son of the Prophet\n    Joseph, being ordained while in Liberty jail to lead the Church?\"\n    \"Certainly I am.\" \"Then,\" said I, \"just write me out a brief\n    statement covering those points, and I will give it in your own\n    words.\" Following is Brother Wight's statement:\n\n    \"In regard to the statement of John W. Rigdon, I endorse it in\n    every point. Brother John W. Rigdon speaks of being in Liberty\n    prison when the Prophet Joseph Smith, Sidney Rigdon, Hyrum Smith,\n    Lyman Wight, and others were there (the others were Caleb Baldwin\n    and Alexander McRae). I also visited the prisoners at or about the\n    same time, and slept with them many times at different periods, and\n    I cannot recollect of ever hearing the subject of an ordination\n    mentioned.\n\n    \"My father, Lyman Wight, nor my mother, never alluded to it during\n    their lifetime in my presence; so I take it for granted that\n    Joseph, the son of the Prophet Joseph Smith, was not ordained to\n    fill the place of his father, in the Liberty jail. I was born in\n    the State of New York, November 29, 1823, hence am about seven\n    years older than Brother John W. Rigdon. And if an ordination of\n    Young Joseph had occurred in the prison, I would likely have heard\n    it, and would certainly recollect it.\n\n    \"Previous to this, while I was several years younger, the Twelve\n    Apostles were organized and commissioned to assist in leading and\n    governing the Church. I can recollect every detail distinctly.\n    My acquaintance with the Prophet was from the year 1830 to his\n    martyrdom, and I can truly say he was a Prophet of God, and was\n    appointed to the divine mission to organize the Church of Jesus\n    Christ of Latter-day Saints in this last dispensation.\n\n    \"As to the Prophet's believing and practicing polygamy, I have as\n    near a certain knowledge of the fact, I may say, as any man living.\n    I was well acquainted with most or all of his wives, and talked\n    with them on the subject, at the same time my wife also talked with\n    them.\n\n    \"If there is anything further that is necessary for me to\n    communicate in regard to my recollection, I will willingly do so.\n\n                                           \"Respectfully,\n                                                \"Orange L. Wight.\"\n\nFurther talk with Brother Wight brought out the following facts: He was\nbaptized into the Church in the spring of 1832; was with the Church\nthrough all their troubles in the State of Missouri. Brother Wight\nfilled a thirteen months' mission in the State of Virginia in company\nwith Jedediah M. Grant and others; was in Nauvoo at the time the\nProphet was captured at Dixon, Ill., and was one of those who went up\nthe Illinois river on the steamer \"Maid of Iowa,\" to assist in rescuing\nthe Prophet.\n\n                                                      Joseph I. Earl.\n\n\nAFFIDAVIT OF BATHSHEBA W. SMITH\n\nState of Utah, County of Salt Lake. ss.\n\nBathsheba W. Smith, being first duly sworn on oath, deposes and says:\n\nI was a resident of Nauvoo, State of Illinois, from 1840 to 1846. I\nwas married to George A. Smith July 25, 1841, Elder Don Carlos Smith\nperforming the ceremony. Near the close of the year 1843, or in the\nbeginning of the year 1844, I received the ordinance of anointing in\na room in Sister Emma Smith's house in Nauvoo, and the same day, in\ncompany with my husband, I received my endowment in the upper room\nover the Prophet Joseph Smith's store. The endowments were given under\nthe direction of the Prophet Joseph Smith, who afterwards gave us\nlectures or instructions in regard to the endowment ceremonies. There\nhas been no change, to my certain knowledge, in these ceremonies. They\nare the same today as they were then. A short time after I received my\nanointing, I was sealed to my husband, George A. Smith, for time and\neternity, by President Brigham Young, in the latter's house, according\nto the plan taught, to my knowledge, by the Prophet Joseph Smith. When\nI was married in 1841, I was married for time, and not for eternity.\n\nAt the time I was anointed in Sister Emma Smith's house, she (Emma\nSmith) said in my presence, to me and to others who were present upon\nthat occasion, \"Your husbands are going to take more wives, and unless\nyou consent to it, you must put your foot down and keep it there.\" Much\nmore was said in regard to plural marriage at that time by Sister Emma\nSmith, who seemed opposed to the principle.\n\nIn the year 1840, at a meeting held in Nauvoo, at which I was present,\nI heard the Prophet Joseph Smith say that the ancient order would be\nrestored as it was in the days of Abraham. In the year 1844, a short\ntime before the death of the Prophet Joseph Smith, it was my privilege\nto attend a regular prayer circle in the upper room over the Prophet's\nstore. There were present at this meeting most of the Twelve Apostles,\ntheir wives, and a number of other prominent brethren and their wives.\nOn that occasion the Prophet arose and spoke at great length, and\nduring his remarks I heard him say that he had conferred on the heads\nof the Twelve Apostles all the keys and powers pertaining to the\nPriesthood, and that upon the heads of the Twelve Apostles the burden\nof the Kingdom rested, and that they would have to carry it.\n\nIt has been, and is, necessary for me to make this statement, as\ncontrary reports have been circulated as coming from me. Any statements\npurporting to come from me that have been made, or that may be made\nby any party or parties, in opposition or conflicting with this sworn\nstatement, are false, as I have never, to my knowledge, deviated one\niota from this statement.\n\n                                                    Bathsheba W. Smith.\n\nSigned in the presence of\n\nJoseph F. Smith, Jr.,\n\nB. Morris Young.\n\nSubscribed and sworn to before me this 19th day of November, 1903.\n\n[Seal.]\n\n                                      Martin S. Lindsay, Notary Public.\n\n\nFootnotes\n\n1. One hundred or more affidavits in relation to the introduction of\ncelestial and plural marriage are on file in the historian's Office,\nSalt Lake City, and are the expressions of eye and ear witnesses, who\nknow that the Prophet Joseph Smith introduced and taught celestial and\nplural marriage. Most of these witnesses are members of the Church,\nbut some of them are not, and have not been connected with the Church\nfrom before the martyrdom of the Prophet and Patriarch. It would\nbe impracticable and even unnecessary to produce all this evidence\nhere. A portion should suffice, in order that the truth regarding the\nintroduction of these principles should be established; for, in this\ncase as in all others, the testimony of two or three reliable witnesses\nshould establish the truth of these things. Celestial marriage, which\nis marriage for eternity, should not be confused with plurality of\nwives, as is often done by those not acquainted with these teachings.\n\n2. Some time during the month of September four members of the\nReorganized Church called on Catherine Phillips Smith at her home in\nEast Jordan, with the object in view of having her deny her testimony\nregarding her marriage to the Patriarch Hyrum Smith, which she\nresolutely refused to do.\n\nIn a statement given on September 24th, two days before her death, she\nsaid: \"They tried to get me to tell a lie and deny that I was married\nto the Patriarch Hyrum Smith; but I would not do it. I never have lied\nand will not now; my affidavit is true. They asked me if my mother knew\nof my marriage, and I told them that the Patriarch asked my mother if\nshe was willing for him to marry her daughter, and she said he could\nask the daughter, and she could do as she pleased. I told them that\nthe Prophet Joseph sealed me to the Patriarch Hyrum Smith as his wife\nfor time and all eternity, and they tried to get me to deny it, and I\nwould not do it, for it is true. I told them the truth. They annoyed me\nvery much, and I finally told them to leave my house and never enter it\nagain.\"\n\n3. This, however, was not the time this principle was first made known\nto the Prophet Joseph Smith, for as early as 1831 the Lord revealed the\nprinciple of celestial and plural marriage to him and he taught it to\nothers.\n\n4. See affidavit of Lucy Walker Smith.\n\n5. Similar affidavits by most of the members of this High Council at\nNauvoo are also on file.\n\n6. Leonard Soby was at first opposed to this revelation, and shortly\nafter the martyrdom he left the Church. When this statement was given\nhe was not a member of the Church.\n\n\n\nTHE REORGANIZED CHURCH--SOME FACTS REGARDING ITS ORIGIN\n\nThe ministers of the \"Reorganized\" Church, or the \"New Organization,\"\nas it was first called,[1] declare that the Church at the death of\nthe Prophet Joseph and Patriarch Hyrum Smith, was badly divided, its\nmembers scattered to the four winds, and that the Church was rejected\nwith its dead. They also claim that the \"Reorganization\" is composed of\nthe faithful who did \"not bow the knee to Baal,\" but remained true to\nthe \"original faith\" as revealed and practiced by the Prophet Joseph\nSmith. In the words of their president: \"The individuals who kept this\ncovenant (the new and everlasting covenant) were accepted of Him and\nwere not rejected, nor their standing before God put in jeopardy by\nthe departure of others from the faith. Whatever the office in the\npriesthood each held, under the ordinations ordered by the call of\nGod and vote of the Church, would remain valid. They could as elders,\npriests, etc., pursue the duties of warning, expounding, and inviting\nall to come to Christ, and by command of God could build up the Church\nfrom any single branch, which, like themselves, had not bowed the knee\nto Baal, or departed from the faith of the Church as found in the\nstandard works of the body at the death of Joseph and Hyrum Smith.\"[2]\n\nIt is strongly implied in this quotation from the writings of the\npresident of the \"Reorganization\" that all those who followed\nPresident Brigham Young and the Twelve Apostles, lost their Priesthood\nand standing before the Lord, and that the founders of the \"New\nOrganization\" and their followers were the only ones who remained true\nand steadfast to the Truth. The evidence in this regard is against\nthem. The truth is that the founders of the \"Reorganized\" church were\nthe ones who followed every will-o-the-wisp, bowed the knee to Baal and\ndeparted from the faith, while the Twelve and the Saints on the other\nhand, pursued an even course and were steadfast under all trials and\ndifficulties even to the end.\n\nIt is not true that the Church was broken, scattered and rejected\nfollowing the martyrdom and that the \"Reorganization\" is a portion of\nthe original church. Their organization did not come into existence\nuntil some sixteen years after the death of the Prophet and Patriarch\nand was an outgrowth of the movement under James J. Strang.\n\nThere was a movement on foot to divide the Church, following the\nassassination of the Prophet and Patriarch, but its range was not as\nextensive as has generally been supposed. The chief actors in this\nmovement were Sidney Rigdon, James J. Strang and William Smith, each\nof whom aspired to lead the Church. Mr. Rigdon based his claim to the\npresidency on the fact that he had been the first counselor to the\nProphet Joseph Smith, and therefore by right should be the \"guardian\"\nof the Church. His claim was in conflict with the position of the\nChurch and the teachings of the Prophet. He laid his case before the\nconference of the Church August 8, 1844, and his claim was rejected\nby the Saints almost unanimously. At the same conference the Twelve\nApostles were sustained as the presiding quorum of the Church. Mr.\nStrang's claim to the presidency was based on his statement that the\nProphet had appointed him as his successor by letter, a few days before\nthe martyrdom. William Smith claimed the right of presidency by virtue\nof being the brother of the Prophet.\n\nEach of these men gathered around him a few followers, principally of\nthat class of restless, erratic individuals, who never remain contented\nvery long in any one place or under any circumstances; but none of them\ngathered many followers. Their organizations barely existed for a few\nyears and then disappeared; the fragments becoming the nucleus of the\n\"Reorganization.\"\n\nThe movement which resulted in the bringing forth of the \"Reorganized\"\nchurch, was of more recent date and was due principally to the efforts\nof two men, viz., Jason W. Briggs and Zenas H. Gurley. Mr. Briggs was\nborn June 25, 1821, at Pompey, Oneida County, New York. He joined the\nChurch June 6, 1841, and members of the \"Reorganization\" declare that\nhe was ordained an Elder in 1842. His home was in Beloit, Wisconsin,\nfrom 1842 to 1854. After the death of the Prophet, Mr. Briggs sustained\nthe Twelve Apostles and the Church and was apparently true to them\nuntil the exodus in 1846. At that time he lost heart, turned from the\nChurch in its darkest hour and sought the favor of the world. Some\ntime subsequent to this he joined the movement under James J. Strang.\nIn Strang's organization he did missionary work, received honors\nand organized a branch. In 1850 he renounced Mr. Strang and joined\nwith William Smith, in the latter organization he was \"ordained\" an\n\"apostle.\" He soon tired of William Smith, and in 1851 joined with\nZenas H. Gurley who was at that time a follower of James J. Strang.\nThese two men then organized a church of their own which afterwards was\nknown as the \"Reorganized\" church. In 1886 Jason W. Briggs withdrew\nfrom this organization of his own begetting, declaring that it was not\nthe Church of Christ.\n\nZenas H. Gurley was just as unstable as Mr. Briggs. He was born at\nBridgewater, New York, May 29, 1801, joined the Church in April, 1838,\nand moved to Far West, from whence he was driven with the Saints in the\nexpulsion of 1838-39. After this expulsion he settled in Nauvoo, where,\nin 1844, he was ordained a Seventy,[3] under the direction of President\nJoseph Young, and on the 6th day of April, 1845, he was ordained senior\npresident of the twenty-first quorum of Seventy. He sustained the\nTwelve and followed their teachings and remained with the Church until\nFebruary, 1846, (the month of the exodus) when he also left the Church\nand shortly afterwards joined with James J. Strang. Mr. Gurley was\nendowed in the Nauvoo Temple with his wife January 6, 1846, and of that\nevent the record of Seventies states under date of January 10, 1846:\n\n    President Zenas H. Gurley arose and said that the Presidents of the\n    quorum (21st) had received their endowment. He observed that it was\n    remarkable for the unusual outpouring of the Holy Spirit.--Page 29.\n\nAgain speaking of the authorities of the Church he said:\n\n    He remembered forcibly the sayings of the First Presidents of\n    Seventy, that we should so live that no charge can be brought\n    against us. A few years ago the men in high standing in this Church\n    were as little as we are. They obtained their exaltation by patient\n    submission to right, and minding their own business.--Page 29.\n\nOn January 25th, 1846, he said:\n\n    The Saints who have passed through the trials of the Church were\n    generally rooted and grounded in love and have a witness in their\n    own hearts or they would not have remained.--Page 33.\n\nWithin a very few days of this time Zenas H. Gurley deserted the Church\nbecause he was unable to face the trials and hardships the Saints were\nforced to undergo. The \"Mormon\" people were journeying in a strange\nland, the prospects before them were dark and some of the members\nbecame faint-hearted and were unable to endure to the end. Of this\nnumber Jason W. Briggs and Zenas H. Gurley were two who turned back and\nsought refuge in the apostate organization of James J. Strang. Indeed\nit required a strong heart and a firm-rooted faith for men and women\nto give up all earthly comforts and undertake a journey of that kind.\nDeath stared the Saints in the face, they were poorly clothed, without\nshelter, save their ragged tents that would not shed the rain, and\nalmost destitute of food; yet with the exception of the few who sought\nthe \"flesh-pots of Egypt,\" they patiently and determinedly pursued\ntheir way until crowned with the victory. The opinion of the world\nat that time was that the exodus meant the end of \"Mormonism,\" and\nthat the Latter-day Saints had gone to their destruction; for without\nthe necessary means to support life, and isolated as they were from\nthe rest of civilization, they must surely perish in the barren and\ndistant West. Such, too, would doubtless have been the case had not the\nprotecting hand of Jehovah guided them. Is it any wonder under such\ntrying conditions that the hearts of those weak in the faith should\nfail them?\n\nIn 1849 Mr. Gurley filled a mission for Mr. Strang and made a number of\nconverts to that faith. In 1850 he organized the \"Yellowstone branch,\"\nfor the Strangite church. In 1852 he rejected the claim of Mr. Strang\nand joined with Mr. Jason W. Briggs, and these two men united their\nrespective Strangite branches, those of Yellowstone and Beloit, and\norganized themselves into a new religious movement known today as the\n\"Reorganized\" church. In 1853, the leaders of this movement called a\nnumber of men to the ministry, \"ordained\" seven \"apostles\" and began\na proselyting movement. For several years they tried to get \"young\nJoseph,\" the son of the Prophet Joseph Smith, who had never affiliated\nwith the Saints since the exodus from Nauvoo, to join them and become\ntheir president. In this they failed, but were diligent and finally,\nthrough their continued efforts and the persuasion of his mother, he\naccepted that position in 1860, was \"ordained\" president of their\nchurch by William Marks, Zenas H. Gurley, and William W. Blair, and has\ncontinued in that position ever since.\n\nMr. Gurley remained with this movement till his death, but his family,\ntogether with Jason W. Briggs, voluntarily withdrew in 1886.\n\nIn 1852, when Jason W. Briggs and Zenas H. Gurley combined their\nStrangite forces the membership was about one hundred souls, most of\nwhom were converts made for Mr. Strang. In 1860, when \"young Joseph\"\nassumed the leadership, the membership was three hundred souls, most\nof whom were converts that had never belonged to the Church of Jesus\nChrist of Latter-day Saints.\n\nOf the members of the Church who were in fellowship in 1844-46, the\n\"Reorganization\" has received no more, and likely less than one\nthousand converts, which fact shows that the apostasy was not so\ngreat in 1844-46, as has been pictured. These statements are based on\nthe testimony of original members of the \"Reorganization,\" as they\ntestified before the U. S. Court of Appeals for the Western District of\nMissouri, in 1894, in the Temple Lot suit, which was for the possession\nof property in the hands of the \"Church of Christ\" or \"Hedrickites.\"\n\nBefore that court Mr. William W. Blair, who for many years was a member\nof the presidency of the \"Reorganization\" and who was one of its oldest\nmembers, testified that \"one thousand was probably too high an estimate\nfor the members of the original church, that had joined the Reorganized\nchurch.\" He could \"approximately say\" that one thousand had joined the\n\"Reorganized church, and possibly that estimate was too large.\" Record\npp. 180, 181.\n\nWilliam Marks, whose testimony is referred to by Mr. Evans, was also\none of those who joined the \"Reorganization\" in an early day. At the\ntime of the martyrdom he was president of the Nauvoo Stake, but was\ndisfellowshipped for transgression at the October conference, 1844,\nand finally excommunicated. Afterwards he joined the organization\nunder James J. Strang. In that organization he became a \"bishop,\"\nwas a member of the \"high council,\" and later a member of the \"first\npresidency.\" After the death of James J. Strang, he joined the\norganization of Charles B. Thompson, another apostate. This is the same\nWilliam Marks who \"ordained\" Joseph Smith, of Lamoni, president of\nthe \"Reorganization.\" In that ordination he was assisted by Zenas H.\nGurley and William W. Blair. Mr. Blair never belonged to the Church. It\nis almost needless to add that these men held no divine authority and\ncould not bestow the Priesthood and officiate in the ordinances of the\nGospel, and, therefore, the pretentions of the \"Reorganized\" church are\nfraudulent. Judged by its history, doctrines and the unstable character\nof its founders it is proved to be a counterfeit and nothing more.\n\nConsidering the conditions under which the \"Reorganization\" came\ninto existence, and the fact that in the beginning the original one\nhundred members came from the Strangite church, and that during the\nexistence of that organization from its foundation to 1894, not more\nthan one thousand members of the \"original church\" (i.e. the Church\nof Jesus Christ of Latter-day Saints as it stood in 1844) had joined\nit, we are not to be blamed if we declare that that church is not the\nsuccessor, a faction or a portion of the \"original church\" founded by\nJoseph Smith the Prophet through the command of God, April 6, 1830.\nAnd after following the history of its founders and pointing out their\ninstability and the manner in which they followed after false leaders,\nreceiving \"ordinations\" and honors under their hands, we can most\nemphatically declare that they were not the faithful who did \"not bow\nthe knee to Baal,\" and who kept the \"everlasting covenant.\"\n\n\nFootnotes\n\n1. _Saints' Herald_, Vol. one.\n\n2. See article in _Era_, Vol. 7, No. 11, entitled, \"The Church\nRejected--When?\"\n\n3. The \"Reorganized\" Church History states that Z. H. Gurley was\nordained a Seventy in Far West in 1838. This is an error, they have\nno original record of such an ordination. The original records of\nthe Seventies in the Historian's Office, Salt Lake City, give his\nordination as stated here.\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Blood Atonement and the Origin of\nPlural Marriage, by Joseph Fielding Smith, Jr.\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nTable of Contents\n\nTitle Page\n\nCopyright Page\n\nAcknowledgements\n\nINTRODUCTION\n\nASCENT\n\nAPPROACHING SIXTY\n\nBUT WORLD ENOUGH\n\nCONSIGNED\n\nMY GIRLFRIEND FATE\n\nA RAT'S TALE\n\nTHE BONES OF MAMMOTH MALONE\n\nDEATH AND TAXES\n\nFATE DOGS\n\nTHE MAN WITH ONE BRIGHT EYE\n\nA TAPESTRY OF SOULS\n\nTHE FINAL CHOICE\n\nTHE PROPHECY OF SYMON THE INEPT\n\nCHOICE OF THE ORACLES\n\nCAMELOT'S GREATEST HITS\n\nJACK\n\nAbout the Author\n_**''Thou hast freed Excalibur!''**_\n\n_''Whoa!''_ said the lad, examining the gleaming sword. ''Awesome!''\n\n''By pulling Excalibur from the stone, Arthur,'' I said, ''you have set your foot upon the road you were destined to travel.''\n\n''I figured I'd take it to the market, see if I could sell it,'' Arthur said.\n\n'' _Sell_ Excalibur?''\n\n''Uh-huh.''\n\n''For _money_?'' I cried.\n\n''Well, unless someone's got a really good lute to trade for it.''\n\n''A lute? To _trade_?''\n\n''But I'd rather go for the money,'' Arthur said.\n\n''This sword looks pretty valuable. I'll bet I could get enough dough to outfit the whole band, don't you think?''\n\n'' _What_ band?''\n\n''Well, okay, I don't have a band yet. That's in the future. I figured I'd go on the road alone first, build my rep, get some steady gigs, and then acquire the band. I'm going to call it Arthur Pen and the Dragons. What do you think?''\n\n_''What?''_\n\n''Or maybe just Arthur and the Dragons. Or Arthur Pen's Dragons? I dunno yet.''\n**Also Available from DAW Books:**\n\n_**Places to Be, People to Kill**_ **, edited by Martin H. Greenberg and Brittiany A. Koren**\n\nAssassins\u2014are they born or made? And what does an assassin do when he or she isn't out killing people? There are just some of the questions you'll find answered in this all-original collection of tales. From Vree, the well-known assassin from Tanya Huff's _Quarters_ novels . . . to a woman whose father's vengeful spirit forced her down dark magic's bloody path . . . to an assassin seeking to escape his Master's death spell . . . to the origins of the legendary nin-sha and the ritual of the hundredth kill . . . here are spellbinding stories of murder and mayhem of shadowy figures who strike from night's concealment or find their way past all safeguards to reach their unsuspecting victims. With stories by Jim C. Hines, S. Andrew Swann, Sarah A. Hoyt, Ed Gorman, and John Marco.\n\n_**Pandora's Closet**_ **, edited by Martin H. Greenberg and Jean Rabe** When Pandora's Box was opened, so the ancient tale goes, all the evils that would beset humanity were released into the world, and when the box was all but empty, the only thing that remained was hope. Now some of fantasy's finest, such as Timothy Zahn, Kevin J. Anderson & Rebecca Moesta, Louise Marley, and Sarah Zettel have taken on the task of opening Pandora's closet, which, naturally, is filled with a whole assortment of items that can be claimed by people, but only at their own peril. From a ring that could bring its wearer infinite wealth but at a terrible cost . . . to a special helmet found in the most unlikely of places . . . to a tale which reveals what happened to the ruby slippers...to a mysterious box that held an ancient, legendary piece of cloth . . . to a red hoodie that could transform one young woman's entire world, here are unforgettable stories that will have you looking at the things you find in the back of your own closet in a whole new light. . . .\n\n_**Army of the Fantastic**_ **, edited by John Marco and John Helfers** How might the course of WWII have changed if sentient dragons ran bombing missions for the Germans? This is just one of the stories gathered in this all-original volume that will take you to magical place in our own world and to fantasy realms where the armies of the fantastic are on the march, waging wars both vast and personal. With stories by Rick Hautala, Alan Dean Foster, Tanya Huff, Tim Waggoner, Bill Fawcett, and Fiona Patton.\n\nCopyright \u00a9 2007 by Tekno Books and Daniel M. Hoyt.\n\nAll Rights Reserved.\n\nDAW Book Collectors No. 1419.\n\nDAW Books is distributed by Penguin Group (USA).\n\nAll characters and events in this book are fictitious.   \nAny resemblance to persons living or dead is coincidental.\n\nIf you purchase this book without a cover you should be aware that this book may have been stolen property and reported as ''unsold and destroyed'' to the publisher. In such case neither the author nor the publisher has received any payment for this ''stripped book.''\n\nThe scanning, uploading and distribution of this book via the Internet or any other means without the permission of the publisher is illegal, and punishable by law. Please purchase only authorized electronic editions, and do not participate in or encourage the electronic piracy of copyrighted materials. Your support of the author's rights is appreciated.\n\nFirst Printing, October 2007   \n1 2 3 4 5 6 7 8 9\n\nDAW TRADEMARK REGISTERED   \nU.S. PAT. OFF. AND FOREIGN COUNTRIES   \n\u2014MARCA REGISTRADA   \nHECHO EN U.S.A.\n\nPRINTED I NTHE U .S. A.\n\n<http:\/\/us.penguingroup.com>\nACKNOWLEDGMENTS\n\n_Introduction_ \u00a9 2007 by Daniel M. Hoyt\n\n_Ascent_ \u00a9 2007 by Julie E. Czerneda\n\n_Approaching Sixty_ \u00a9 2007 by Mike Resnick & Barry N. Malzberg\n\n_But World Enough_ \u00a9 2007 by Sarah A. Hoyt\n\n_Consigned_ \u00a9 2007 by Thranx, Inc.\n\n_My Girlfriend Fate_ \u00a9 2007 by Darwin A. Garrison\n\n_A Rat's Tale_ \u00a9 2007 by Barbara Nickless\n\n_The Bones of Mammoth Malone_ \u00a9 2007 by Esther M. Friesner\n\n_Death and Taxes_ \u00a9 2007 by Kristine Kathryn Rusch\n\n_Fate Dogs_ \u00a9 2007 by Robert A. Hoyt\n\n_The Man With One Bright Eye_ \u00a9 2007 by Joseph E. Lake Jr.\n\n_A Tapestry of Souls_ \u00a9 2007 by Paul Crilley\n\n_The Final Choice_ \u00a9 2007 by Phyllis Irene Radford\n\n_The Prophecy of Symon the Inept_ \u00a9 2007 by Rebecca Lickiss\n\n_Choice of the Oracles_ \u00a9 2007 by Kate Paulk\n\n_Camelot's Greatest Hits_ \u00a9 2007 by Laura Resnick\n\n_Jack_ \u00a9 2007 by Dave Freer\nINTRODUCTION\n\n_Daniel M. Hoyt_\n\nI was fated to write this introduction. I think the first time I became aware of it, I was sixteen. I remember it quite clearly. I was at a carnival one night, under a full moon, there was a palpable tension to the air, and then this wizened old woman with straggly white hair and a jagged scar slashed across one eye walked right up to me\u2014\n\nWell, you know the story.\n\n''It was inevitable.''\n\n''An ironic twist of fate.''\n\n''Self-fulfilling prophecy.''\n\nFor denizens of high-tech societies that generally eschew the notion of mysticism, modern humans seem to have an odd fascination with fate. The concept itself is rooted in ancient man's desire to explain things and events he couldn't reconcile intellectually. After thousands, maybe millions, of years, we still haven't shuffled off the pervasive instinct to fall back on fate as a last-resort explanation.\n\nEven prophecies, cousins of fate that are essentially stories of preemptive rationalization, seem to survive well beyond their inception. Maybe they're just fated to survive. Or maybe once we acknowledge them as prophecies, we _make_ them survive, _because_ they're prophecies. Whatever the reason, they survive, sometimes for several millennia. Many ancient religions\u2014 and even new ones, for that matter\u2014have prophecies central to their belief systems. Nostradamus, less than five hundred years ago, charmed the French court with prophecies that are still believed by many today.\n\nIronically, even when prophecies fail to come true, we refuse to reject fate. Instead, we twist the results and rationalize them in some other way, yet cling to the underlying concept of fate.\n\nEqually ingrained in our subconscious seems to be the notion that fate may not always be fixed. Every now and then, we're allowed to reject our fate, to twist it around to suit our own needs. Once. Just once.\n\nAnd it's there that we find the truly interesting stories (for we are storytellers, and always have been) about fate. It's right there, when we allow ourselves to believe that we're not merely a society of automata absentmindedly marching toward our deaths according to some cosmic blueprint. There we find the stories that make us believe in life, in free will, and\u2014maybe because they're exceptions to it\u2014in fate.\n\nThere we find the stories that make us human.\n\nOkay, maybe I never met a woman at a carnival when I was sixteen imploring me to write an introduction about fate. What _did_ happen, though, is that sixteen very talented fantasy and science fiction short-story writers have come together to explore fate, as only they can.\n\nFrom Julie Czerneda's sublimely multilayered reflection of our relationship with fate, to an academic consideration of fate and consequences as only Kris Rusch can present it, to Dave Freer's thoughtful collision of mythology and morality, you'll find fate examined from unique and unexpected angles.\n\nOn the lighter side, consider Darwin Garrison's amusing take on dating one of the three Fates, or Robert Hoyt's tongue-in-cheek look at hot dog street vendors through a stockbroker's eyes, or Laura Resnick's retelling of the Arthurian legend, set to music.\n\nYou can't go wrong with Mike Resnick or Barry Malzberg; together, they take on gambling and the Kabbalah. Or try Alan Dean Foster's thought-provoking ascent into Hell. Esther Friesner's trademark wit shines with her tale of a soothsayer, lemmings, and a frozen tundra. Irene Radford reminds us that even Fate has rules.\n\nIn these pages, you'll find wonderful stories that will delight you and challenge your own notion of fate. I know you will.\n\nIt's fate.\nASCENT\n\n_Julie E. Czerneda_\n\nI was born to kill.\n\nThere was no other purpose to my existence. No other was required. I knew my life's role as few ever did and it brought me, if not joy, then a steadfast measure. That which took me to that destiny I sought and embraced. That which did not, or worse, could distract, I avoided or ignored.\n\nUntil I met Zephyr on the Dedmas Cliff.\n\n''Oh, do come on!'' The shout skipped along the cliff face. Those seabirds still unaroused by our ascent rioted in a blur of wing, feather, and raucous cry.\n\nI tucked my head toward my chest to shield my eyes from beak and justified temper, my fingertips digging into stone, holding still until the last bird launched from its hole. Spitting guano from my dry mouth, I waited another breath, then lifted my head.\n\nFour round downy heads looked back at me from their crack, eyes unblinking.\n\nI smiled at the chicks.\n\n''C'mooonnn!''\n\nThree chicks shrieked as I climbed after my companion.\n\n_Companion?_\n\nThorn. Irritant. Bully. Prod.\n\n_Companion,_ I told myself, easing up the crumbling cliff.\n\nNimble where I was ponderous. Agile where I was methodical and slow. Above all, that gentle touch where I was force incarnate.\n\nAs if reacting to the thought, my fingers closed too firmly and my left hand fell away from the cliff, filled with pulverized stone.\n\nHanging for an endless moment from my other, more reasoned right hand, I contemplated the problem.\n\n''Use your feet.''\n\nMy companion was swift of thought, too, while I needed to view a thing from all its sides, consider consequences, and only then act. He dropped to my side as though we laid on a floor and only pretended waves and jagged rock teeth waited so far below.\n\nHe tapped my legs sharply. ''You've two of them. Remember?''\n\n_He was life._\n\n_I was death._\n\nDestinies neither of us chose.\n\nMy feet, obeying him before me, found a deep crack\u2014and the chicks I'd spared, I realized too late, licking any evidence from my teeth. Zephyr turned strange when I did such things. Strange and cold and wary.\n\nIt wasn't fair, I thought, my free hand finding and holding\u2014more carefully now\u2014the stone. There was nothing of Zephyr that changed how I regarded him.\n\n''Keep moving, lummox,'' he urged with a chuckle. ''There's a ledge broad enough for even your haunches almost in reach. I'll go ahead and start some jaff.''\n\nI was bred to end a threat.\n\nThey'd tipped me overboard in the cold shadow of the cliff, their small ship more fragile than I, more at risk within the surge and froth of ocean. I'd been buffeted too, slammed by dark boiling waves as my out-reached hands fought to take hold. But they'd known I could endure what they could not, so they'd turned their ship toward safety with cries of relief that echoed the birds wheeling far above.\n\nThey'd never reached it. The hold I found had been their ship. Wood was nothing to the hands I'd been given. When a wave heaved me up and away, my hands had stayed clenched around a strip of newly splintered plank, the small ship already listing well to one side and beyond salvation.\n\nTheir cries had turned to futile curses.\n\nI'd reached the base of the cliff and begun my climb without witness.\n\nHow had Zephyr known to find me there?\n\n''Nice view.''\n\nHe made conversation. I made the jaff myself, putting a thick pinch of leaves from the pouch at my waist into the flattened sphere that was both pot and cup, holding it open against the nearest of many rivulets draining down the cliff. The water had rained on an unseen land, from clouds as far above us now as the ocean lay below. I wondered if the result would taste foreign.\n\nI crouched over the pot, rocking it from side to side to draw the most from the leaves. There wasn't room to move otherwise. With my back against the continent, my feet dangled out over what I'd climbed these long hours.\n\n''You're planning to share, I hope?''\n\nWithout a word I held out the battered pot. Zephyr laughed and shook his head, eyes agleam with mischief. ''I couldn't. From the look of you, you need it more than I. Go on. If I get as slow and weary as you, I'll take my turn.''\n\nHe was anything but weary, restlessly pacing the part of the ledge not covered by me. His hands, delicate and sure, moved like the birds whirling through the air below. As if he thought he could fly.\n\n''So, what's it like up there?''\n\nI put my lips to the spout and let the cool, bitter liquid drip into my mouth. Slow was better. The virtue within the leaves was absorbed best by the tongue. Mine was swollen, its cracks taking fire from the spice. I persisted, drop by drop. I'd been taught to embrace pain.\n\nTaught by it.\n\n_Taught to embrace it,_ I told myself, staring out at the ocean. Pain was a tool. I understood its use.\n\nZephyr straddled my legs with his. I couldn't see past his slender form and frowned at this trivial victory. ''What's it like?'' He was pointing up. ''Surely you know.''\n\nI didn't need to know, so I kept the spout to my lips.\n\n''Aren't you the least bit curious?''\n\nI was made invincible.\n\nI'd pulled myself from the surf at the base of the infamous cliff and begun my ascent. The battering waves had tried to brush me aside; they found easier work eroding the layers of softer stone. The wind, rising up the wall with its load of icy spray, had tried to numb my flesh. Easier to chill the birds riding the updraft.\n\nI had looked no farther than my next grip. My destination would matter when I reached it.\n\nAs we'd approached through morning mist, hearing the roar before seeing our goal, the Dedmas Cliff had suddenly punched through the clouds, an immensity of black tilted seaward as if about to crash down on the ship. Those who'd brought me had exclaimed in awe. Up close, its surface was rotten with crumbling, newer stone. Through those gaps the exposed bones of the world jutted at sharp angles, defying the ocean's power.\n\nFor now. Other such bones lay wave-washed and defeated at the cliff's foot.\n\nThe cliff was nothing more than my road. I'd ignored the exclamations of the soon-to-be dead and studied the surf. And as I'd begun my climb, I'd never looked to the top or thought of what was to come.\n\nUntil Zephyr's soft cheery voice had whispered in my ear, ''Well met, lummox!'' Then he'd asked, for the first time, ''Do you know what's up there?''\n\n''Wake up! C'mon!''\n\nI blinked open my eyes, scrubbed flakes of dried jaff from my lips with the back of one hand. I must have dozed. No matter. Even the body they'd given me appreciated rest after a day of such climbing. Another day to go. Or rather, night.\n\nI blinked, then squinted. The sun had dropped below the sullen clouds, riding the horizon. Its light skipped and danced over the waves. When I rose to my feet with a grunt and turned to face the cliff, the sun peered over my shoulder and found glints of crystal in the rock. They looked like eyes.\n\nZephyr was on the move, his quick toes nipping into the smallest crevice, his delicate fingertips caressing the cliff. He'd shaken his head at the scars I left in my passing.\n\nI hadn't asked for such mammoth hands and feet, with their ever-sharp talons that made it impossible to keep from tearing my own clothes. I hadn't asked for arms to match, or shoulders too wide for the doors others used.\n\nI hadn't asked for a nightmare's face.\n\n''Hope you can see in the dark, friend lummox,'' Zephyr called from above.\n\nI was destruction.\n\nWhen Zephyr had appeared beside me on the cliff, nimble and happy and sly, I'd thought him a dream. Not that I'd been encouraged to fancy, but the play-ground of my thoughts had outlasted the rest. He'd acted like a dream, too. Unsurprised.\n\n_Unafraid._\n\nI'd forgotten what it was like, to have company who didn't fear me.\n\n_Or hurt me, when they still could._\n\nThe novelty had stayed my horrible hands, not that I could have touched him. From that meeting, he ascended beside me with the grace of a swimmer, every gesture with purpose, every reach without flaw.\n\nWhile I climbed with raw power, ruining the stone.\n\n''Over here!''\n\nZephyr balanced on a thrust of sheer rock, waving one hand vigorously at me. I didn't like it when he risked himself, though he had the skill to do so. I'd learned, however, that he wouldn't be ignored. With a sigh, I abandoned the straight ascent and edged sideways.\n\n''Hurry! I tell you it's remarkable!'' He disappeared from view.\n\n_Distraction._\n\nYet I found a crack to take my right foot, forced my right hand into another, eased myself over the dark spire. My belly, scaled and rough, left its mark on the softer stone. This, a rib of the world, ignored my passing.\n\nZephyr's so-remarkable find was another fall of water, nothing more. The cliff was laced with such. Elsewhere, water clung to the rock in seeping stains or evaporated into a cloying spray. This ribbon was wider than others; judging by the growth of small, feathered green through the stone, more permanent. Its crystal strands made a sound as they writhed past and down, like the cold chatter of teeth.\n\nMy companion leaned precariously close to its near side. ''Stick your head und\u2014'' The exhortation became a gurgling laugh as Zephyr took his own suggestion. I waited in dread for the waterfall to wash him from the cliff. The spray-washed surface was treacherous and slick. But he might have been a fish bounding upstream for all the water's plunge affected him. After a moment, he surfaced and shook his head, drops flying like pearls from his hair. His teeth flashed in a smile. ''C'mon! You stink. You don't want to get to the top like that, do you?''\n\nI looked into the cloud-hidden heights where this waterfall had been spawned.\n\nI didn't want to get to the top at all.\n\nHaving said it, if only to my innermost self, I trembled.\n\nI was taught to obey.\n\nWhen old enough to understand such things, I was told my life's purpose: to kill our enemy, those who dwelled above us, who wove spells within the clouds for this land's ultimate destruction. Until that moment, I would live in the arena where training would hone the physical tools given me. I did my best, always. In the early years, the broken bodies of my instructors would be dragged away after each session. More precisely, what remained of those bodies, since I ate what I could before they were taken from me.\n\nThere was no other food given me; I asked for none.\n\nThe time came when there were bodies, but no longer instruction. I recall a vague curiosity. Had I learned all they could teach or was I being sent those who would not themselves learn?\n\nNot that I asked or was answered.\n\nI was tested; it was always the same. Each time the moon hid her face, regardless of the weather, I was to climb the wall of the arena, using only my hands, until I reached the mark scratched in the stone. Halfway up. There, I was to stay. Moon after moon, year after year, eventually, inevitably, my hands would grow numb, my arms spasm, and I would fall to the sand. I wasn't punished. No need.\n\nFailure meant I'd stay here.\n\nInevitably, the time came when I held to the wall, conscious only that my body would die before my hands would let go. I remember being content if that were so.\n\nThey'd looked down on me from above; faces that came and went with the days. Then one came who was different from the others. He'd stared down at me with hollow eyes and told me my goal. I would ascend the unclimbable Dedmas Cliff and kill all who'd believed themselves safe living atop it.\n\nThey'd brought me models of the cliff face. I'd studied them until I knew its stoneworks more intimately than those of my arena. They'd provided rocks like those I would encounter. I'd slept with them under my body and held them in my hands until that grip felt more natural than an empty palm.\n\nThe time came when they'd opened the gate of my arena and my journey began.\n\nI recall a vague joy it would soon be over.\n\n''Sure you don't need help?''\n\nZephyr's breath entered my nostrils and mouth, warm and fresh. I saw my eyes reflected in the dark gleam of his. ''You look,'' he added consideringly, ''stuck.''\n\nWhat had appeared in the models as a vertical gash in the cliff, a promising path through a section that leaned out over the surf, had narrowed into a trap. I leaned my head against the wall behind me, twisting my neck to assess the possibilities. If the gash had been a chimney, the sharp bend overhead would pin and collect smoke, let alone a climber. Maybe I could reach outside, try to pull myself out and past the obstruction . . .\n\nZephyr had followed my gaze. ''That won't work,'' he decided. ''You'll have to go down and try another path.''\n\nMy feet were against the other wall; the force of my legs pressed my back and shoulders against this one. I'd inched my way in similar fashion with success until now. I tested the cracks above with my fingers; rock crumbled at my touch. The fragments clicked and tapped and spun away into the dark. We were halfway up the cliff now. Had the moon been shining, I might have seen the distant blur of white that marked the waves now far below.\n\nUndeterred by our height or my situation, Zephyr used me as his personal stair. Standing on my bent knees, he reached to slap his slender palm against the inward lean of stone. ''Face it. They made you too big.'' A rebuke, as if he'd expected better.\n\nI sighed. I was as I was. I didn't belong here; he did. Where I struggled, he moved without effort. On passing the mark I'd set myself, the height above which I knew my body could not survive a fall, he'd become even more careless. No, not careless. Casual, as if there was no place safer.\n\nNow he bounced on my knees. ''You taking a nap? I tell you, friend lummox, you're stuck. The only way out is down.''\n\nThe only way I couldn't go. Down was away from my goal. To move in that direction was to negate my life's purpose.\n\nDid Zephyr guess what that was?\n\nI became patience.\n\nThe beslimed lower reach of the cliff had almost defeated me before I'd properly begun my ascent. What use was strength denied a grip? I'd clung to the first hold I'd found, able to resist the suck of wave and the flood of spray rushing down the wall. Knowing I'd be vulnerable to their force when I reached for another, I'd taken my time, and stared through the froth to study the cliff, considering this path and that. I'd ignored the rain when it began, drunk my fill, welcomed the flashes of lightning. Finally, satisfied, I'd chosen cracks for my fingers, felt barnacles crush under my palms and the soles of my feet, and climbed.\n\nOnce past the domain of water dwellers, I'd moved steadily, with growing confidence. The rain had ended, leaving its tears along the stone. Soon I'd reached the lowest of the narrow sloped ledges crammed with nests. The debris and guano made for treacherous holds, the adult birds had screamed their rage, but I'd smiled. I'd grabbed what warm life couldn't fly away, made short work of it.\n\n_Been distracted._\n\nBreath, soft on my ear. ''Well met, friend lummox.''\n\nBlood had dripped from my chin and coated my chest, made my hands wet and slippery as I'd turned to find the source of the voice. My first thought had been to destroy. My second, to feast.\n\nUntil I'd looked into his face.\n\n''You're being stubborn.''\n\nThe gash was deep, so deep my outstretched fingers couldn't feel its end. I flexed my knees to dislodge Zephyr, then shifted sideways, into the shadow.\n\nThe air was stilled here, sound muffled, so I heard the deep steady thud of my own heart. My training had not included medicine or other knowledge, but I knew hearts, how to stop them, how to pull apart their lean dense meat, their taste. I knew lungs and intestines and the startling softness of brain within its bone case.\n\nI possessed such parts. _That didn't make me alive._\n\n''And what have we here?'' I felt Zephyr squeeze by me, his hands feather-soft and warm on my forearms and shoulders. For an instant I could see him, despite the shadows, as light from some unseen source found his face. ''Clever lummox. You found a passage!'' A surge of movement and he was gone.\n\nI braced myself within the rock, alone in the dark. The cliff engulfed me, as if I rested within its cold, hard womb. I could stay here, listening to my heart, until I starved and died.\n\n_Would that be murder enough for my makers?_\n\n''Are you coming?''\n\n_Why did he insist I survive?_\n\nI stirred myself to follow Zephyr.\n\nI was intended to prevail.\n\nThey'd never told me whom I would face atop the cliff. Either they hadn't known, or they'd believed it was a foe who could be defeated by talon and fang, overwhelmed by brute force. I'd been given no other weapons, no defense except the body they'd made me and taught me to use.\n\nI remember wondering if I was their first attempt.\n\n''Now that's a view!''\n\nI rocked my little pot of jaff, pretending, to myself at least, my hands were no longer shaking, trusting the rest and spice would restore the strength I'd left in that dreadful hole.\n\nWhat had been for Zephyr a quick and easy scurry up a shaft, its sides roughened into a natural ladder, had been for my larger, heavier, stiffer self a traumatic birth. I'd had to dig my way through the last of it, force a passage with all the power this body possessed, losing ground with each exhalation as the tightening walls refused my lungs room to expand, leaving blood and scale behind as the rock resisted my every move.\n\nI didn't quite believe I'd escaped.\n\nI put my lips to the spout, unable to hold it steady as the first drips entered my mouth. I licked the burning liquid from my lips and understood something for the first time.\n\nI could fail.\n\n''There!'' Zephyr's slender arm shot out like a spear. ''The harbor lights at Darphos! Come, lummox. Take a look.''\n\nThe harbor I'd left the night before held no lure. I took my time finishing the jaff, then put the sphere back in the waist pouch. I'd lost flesh, but not this, during my struggle.\n\nA struggle that had brought me to this wide ledge beneath an overhang of stone. Close to a cave, but the floor was polished smooth, warning it would offer no protection from wind-driven rain or runoff from above.\n\n''It must be a wonderful place, this Darphos.'' Zephyr continued, gazing outward into the night. ''What do they make there? You know a people by what they make.''\n\n_They'd made me._\n\nHad the people above made him?\n\nI was terror.\n\nMy first instructor had dropped to his knees before me, young and new as I'd been, pleading in a language I didn't know. I recall being uncertain. It was the last time I'd felt thus. They'd descended from the walls with prods and hooks to teach me the cost of indecision.\n\nMy next instructor had cried out at the sight of me, but she'd been proud and strong. She'd taught me the cost of failing to anticipate an attack.\n\nThe other faces had blurred over the years; perhaps someone else had kept count.\n\nWhen they were to take me to Darphos, to board the ship that would carry me to my fate, I'd felt a stir of curiosity. The world outside my arena was a mystery, populated my dreams with wonders.\n\nMy first steps into that world had taken me through the deep, pitted gates to a dusty road. The road had sliced a barren landscape to the horizon, straight as the flight of a bird. At the beginning of that road had waited a pair of great horned beasts cloaked in gleaming metal. They'd been harnessed to an ebony cart set with flags whose searing colors twisted and flailed in the wind.\n\nThe beasts had screamed at the sight of me and tried to bolt.\n\nSo I'd traveled to Darphos with my head and taloned hands wrapped in black cloth, my huge feet hidden in bags, my grotesque body crouched within a box. The world's wonders remained no more than dreams.\n\nSafe from me.\n\n''Rest, friend lummox, while I name the constellations for you.''\n\nI leaned my head against the cliff, my feet hanging in the air, and listened to the warm lilt of Zephyr's voice. I knew the names well enough. One of those who'd walked the upper walls of my arena had taught me, having nothing else to do. No one escaped my arena; no one tried to enter. I'd taken the names in silence and, each night when I'd laid my aching body on the bloodied sand to sleep, cursed them in the privacy of my thoughts. I'd cursed their freedom, their clarity, and most of all, their unblinking witness.\n\nThey'd glowed brightest on those moonless nights, while I held to the wall by my talons; they'd watched me fail.\n\nYet, as Zephyr pointed and named, reciting some fool's tale for each, those shards embedded in the body of the sky became unfamiliar. His voice had a strange power over me, erasing the past, offering . . .\n\n_Distraction._\n\nI stood and reached for the overhang, driving my fingertips into its silent stone. By that grip alone, I pulled myself upward and away.\n\nMy life had a purpose. He made me forget that.\n\n''Ready to go, are we?'' Zephyr laughed. He used my body as his ladder, my head as a step. It was like being climbed by a mouse. Once above me, he leaned down, eyes like stars. ''View's better higher up anyway.''\n\nI was given a purpose.\n\nIt had been so clear at the bottom. Climb to the top of the Dedmas Cliff and kill those I found. I'd assumed I was to feed as well, though I'd wondered if the cliff dwellers would taste foreign or if all blood and flesh was the same once dead.\n\nWhen Zephyr had arrived, my self-appointed companion, nothing was clear except his presence.\n\n_Had I been lonely?_\n\nHis presence and the towering cliff. They were one in my mind; two halves of a whole. I would have climbed forever, to have him with me.\n\n_Was I happy?_\n\nQuestions I'd never considered. Dangerous, distracting questions.\n\nI began to fear the people of Dedmas had a defense after all, and it climbed at my side.\n\n''There you go. Morning. Time for some jaff, I'd say.''\n\nI didn't stop, despite Zephyr's cheery suggestion. The sun was rising, its rays touching the ocean beyond, the island of Darphos, other lands beyond that. It did nothing to brighten or warm the cliff's face.\n\nI climbed faster, reckless, losing my grip and regaining it, driven.\n\nZephyr easily kept pace beside me. He looked less happy. ''You should stop, friend lummox,'' he urged. ''A drink, at least.''\n\nI paused to tilt my head into one of the many rivulets streaming from above, catching the water in my mouth. It tasted of soil and mineral; I spat out grit.\n\n''Good. Good. Now take a look. Don't waste this light.''\n\nI pressed my face against the rough stone, eyes closed, and reached for a new hold.\n\nHis voice, soft in my ear. ''For me. Please. Look. So few have seen this.''\n\nI could endure any pain, but not his plea. I lifted my head and looked at him, my cheek scraping the rock.\n\nHe smiled and flung out his arm as if to encompass all of existence. To forestall an even wilder gesture over nothing, I wedged my leg and shoulder into a crevice and turned so I could look.\n\nWe were higher than I'd imagined. Pinwheels of white flashed over the ocean\u2014birds, too distant to see as individuals, dancing in the air. Puffs and streaks of gray hung above them, below us\u2014clouds, I realized.\n\nI looked beyond, and saw that Darphos and what I'd thought to be other islands were connected, one land. Connected to this cliff as well. And this cliff . . .\n\nIt was the base of a mountain, a mountain with sheer, white-clad shoulders that extended as far as I could see.\n\nNearer was a cup of verdant green, laced to the black rock by vine and artful stonework and a glittering river of white that tumbled into the air. Other cups, like inlaid emeralds, studded the upper rim of the cliff where it joined the mountain. A vertical world, complete unto itself. Beautiful and inviolate.\n\nWhere I was to kill.\n\nZephyr leaned close, blocking my view.\n\n''You do remember, don't you?'' he asked.\n\nI had no past.\n\nThey'd taken that before they'd taken my body. My earliest memories were of heat and fumes and darkness. I remembered fighting to breathe, then pain coursing through me with each beat of my heart, as if my blood had become fire. My first conscious act had been to scream.\n\nThough I hadn't heard my voice over theirs.\n\nSome unknowable time later there'd been light. I remembered it as dim and moving, as if seen from the depths of the ocean. But no ocean burned like this. Even then, young as I'd been, I'd understood I was being remade and wondered why.\n\nWhen they gave me a reason, I remembered being grateful.\n\n''You can't stay here.''\n\nI ignored Zephyr, who'd tried everything short of falling to gain my attention. I needed to think.\n\nI ignored him, but all my thoughts were about him. My odd companion was the key. Somehow, I knew it.\n\nI puzzled at the shape of his face, its smooth curved lips and wide, gleaming eyes. I considered the free and joyous way he climbed, the strength that must be innate within those deceptively slender limbs.\n\nI no longer wondered why he was here, with me. He belonged to that world, to those emeralds tucked within rock. I was a threat; he was their response.\n\nBut he was not here, I decided, to delay me. Or even harm me. He was prince of the cliff. At any time, had it been his intent, he could have led me to a trap or tricked my hands loose from the wall. He had not. In fact, I realized, frowning, he'd saved me twice.\n\n''You're thinking very hard, friend lummox,'' Zephyr commented. He let go with one hand, swinging idly from the other, his toes tapping against the rock. A perilous game, this high, unless somehow sure of safety.\n\nOr without need of it.\n\n''Clever lummox,'' he smiled, patting my knee.\n\nI didn't feel clever. I felt as though every reach and hold and pull of the ascent to this moment had ground itself into my flesh, torn my hands and feet, worn my body to the breaking point. My purpose, the goal of my life? The scheming of distant specks. They hadn't seen this place. They hadn't seen themselves from it. They'd feared what they couldn't know.\n\nI felt unmade.\n\n''Almost there,'' soft on my ear.\n\nI clutched the cliff, flattened my body against it, afraid to take too deep a breath and fall.\n\n''C'mon, friend lummox,'' Zephyr coaxed. ''You're too near to stop now.'' His fingers collected drops from the wall and flicked them at me. ''Thirsty?''\n\nI licked the moisture from my lips and froze. The water fell from above, from those cups of green. How simple to poison it . . . add a spell of confusion . . . cause attackers to have visions . . .\n\nMine laughed. ''So now I'm a vision?'' Zephyr poked me with his toes. ''Take that.''\n\nNot a vision. A manifestation. _But why this clever, dear sprite of a being?_\n\nZephyr cupped my face with both hands, brought his so close I could see myself in his eyes.\n\n_It was impossible; he stood on nothing._\n\nYet I didn't fear he'd fall.\n\n''You know,'' he said gently. ''You've known from the first moment you saw me.''\n\nHe came closer and closer, until the moment came when there was no difference between us and I felt his life fill this body they'd made.\n\n''My life,'' I said, knowing it was true.\n\nMy life as it would have been, should have been, had they not stolen me from this cliff and made me a weapon against my own.\n\nI gazed at the nearest emerald cup, far overhead, imagining its wonders.\n\nOne last time, I tightened the fingers they'd given me, crushing the rock held by their hands, hands that dropped free.\n\nThen, being Zephyr, I spread my arms and laughed as I fell.\nAPPROACHING SIXTY\n\n_Mike Resnick and Barry N. Malzberg_\n\n\"Kabbalah'' is a word I can barely pronounce, let alone spell, something like the name ''Artismo,'' which was the appellation of a fancy allowance horse more years ago than I would like to think, or ''Secretariat,'' which was a great horse but had the letter _a_ where the _e_ should be or vice-versa.\n\nThis book containing the Kabbalah is one composed of signs and wonders, mysteries and omens, as my brother-in-law Jake tells me when he hands it to me at a family dinner. This is just as the losing streak approaches sixty, which probably qualifies for the _Guinness Book of World Records._\n\nAt thirty I gave up the _Form_ and the _Telegraph_ and went to the _Green Sheet_. At forty I started phoning Creepy Conrad and giving him my credit card number in exchange for what he promised was surefire information. At fifty, I even went to _shul_ and nagged Jehovah a bit. But as I approached sixty, I was out of alternatives and ready to grasp at anything.\n\n''It could be called an apostasy, Demetrius,'' says Jake, who spent seven months studying to be a rabbi, before they threw him out when they found him in bed with not one but two blond shiksa bimbos, ''to offer you a book of instruction in Kabbalah for a mere losing streak. But I can see that your distress has gone beyond the obvious to the malignant. Following the sport of kings is a dangerous profession and I intuit that you are weak with need.'' Probably he also intuits it from the two thousand bucks I owe him and can't pay.\n\nThere is nothing for me to say in response to this. I could berate Jake for his condescension (but not until I pay him the two large). I could object to his patronizing, to the fact that he has always treated my presence in the family as something of an embarrassment and that he considers my profession with the horses as a disgrace (at least during losing seasons).\n\nBut I can also understand that in many ways I have caused Jake and Nate at least as much concern and trouble as they have caused me. So I take the book, a rather limp number, from his extended hand and say, ''I am always grateful for advice, Jake, even though I can make no commitment.''\n\n''You listen to me,'' he says, a long-lost rabbinical fervor seeming to crease his features. ''You're not supposed to even study this unless you are a man and you are over forty. Women and immature males are excluded. Children are excluded. This is considered too dangerous for a younger man, one whose flames have not been properly coaxed and controlled.''\n\n''I am over forty,'' I reply. ''I am in fact forty-nine years of age, as well you know, since I've been part of this family for more than twenty years. My flames are coaxed and controlled''\u2014his sister the yenta can testify to this, and often does, whenever she can find an audience. ''In fact, at this moment my flames are in need of spontaneous fuel. Have you ever lost fifty-nine of _anything_ in a row? Can you even imagine how this feels?''\n\nIt is odd that I make the remarks at that time in that way, because the book itself, this mysterious Kabbalah text which Jake has handed me, seemed to be emitting a strange warmth of its own, an uncomfortable heat that passed from the binding into my hand the way that a rein can slip into the hand of a jockey.\n\n''Do I understand consequence?'' replies Jake, and I can tell he's gearing up for a long-distance oratory. ''My entire life is consequence. So is yours. So is everyone's, as the Kabbalah makes clear. We are the creations and extensions of a celestial order we can barely understand.''\n\nThere is much more of this, Jake the accountant often returning to his rabbinical roots when properly encouraged (or even discouraged), but I will pass on the remainder of that conversation, and also on the remonstrations of my wife, Sylvia, and my other brother-in-law, Nate, the nearsighted and color-blind custom tailor, who are both highly displeased when they obtain knowledge of what Jake has placed in my hands.\n\nSylvia of course has lived with low and high disgust for many decades and has learned resignation, but Nate is blunter and more direct. ''Give that back to me,'' he said, spying the book in my hand and instantly deducing the reasons for its presence.\n\n''It's _my_ Kabbalah!'' I say. ''Go get your own!''\n\n''You're such a _meshuggener pisher_ you think the Kabbalah's a book, like the Bible or something by Abba Eban. It's the _knowledge,_ the mystic system _within_ those pages.''\n\n''The book, the knowledge, it's all the same,'' I say. ''It's a system, like betting claimers who are moving down in class, or doubling up on front runners when the track comes up muddy.''\n\n''Jake again!'' mutters Nate. ''What does he think he's doing? Give it back at once!''\n\n''Stay calm, Nate,'' says Jake, overhearing the conversation. ''Demetrius is in need of aid at the present time and this may be a way of bringing him back to the fold.''\n\n''Didn't you hear him?'' screams Nate. ''He thinks it's God's betting system! You are as crazy as you were when you tried to crawl into the ark during Yom Kippur fifteen years ago!''\n\nWell, it goes on like this for hours, but I do not return the Kabbalah, and eventually Nate and Jake decide they have fulfilled today's quota of acrimony and they go home to their wives, and I will now engage in what the Kabbalah text\u2014which I look at briefly in the late evening before putting it to the other side of the bed\u2014calls a necessary pause, an important transition, and point to subsequent events at Aqueduct Race Track in South Ozone Park in the borough of Queens, New York, which followed from my acquaintance with the Kabbalah as the night does the day.\n\nAs one of only two or three thousand people in this era of off-track betting who still attends Aqueduct on a regular basis, actually entering the premises, I carry my burden of being exceptional with fair grace and no little trepidation. Fifty-nine consecutive losses will not induce the humility which already comes from being one of only two or three thousand.\n\nBook in hand, eighteen hours after my conversation with Jake, I am standing by the rail watching the horses, nonwinners of two, stagger onto the track. I consult the book denied flaming men and all women and use it to induce a psychic moment.\n\nI pour over the words. I try to make sense of the commands. I attempt to order the mathematics.\n\nAnd then, suddenly, a very clear voice within my head says: _Five._\n\nI look around. ''Who said that?''\n\nBut there is no one near me.\n\nI look at the tote board in the infield and rub the book for good luck.\n\n_Five._\n\nThe number 5 horse is 17-to-1. I borrow a _Form_. His name is Quanto La Gusta, and he has lost fourteen in a row since winning a claimer at Finger Lakes, which hosts such poor horses that you could throw a saddle on Jake and _he_ could come in third if the track wasn't too muddy.\n\nI reach into my pocket. I've got $1,650 left, all the money I have in the world. When I lose this, my profession is done, and I will either have to get an honest job or go to Big-Hearted Ernie, who charges 20 percent interest per day. I look at the board again. The number 5 horse is up to 22-to-1.\n\n''You're sure?'' I whisper to no one in particular.\n\n_Five,_ says the voice with the confidence of one who has never lost fifty-nine in a row,\n\nThe horses are coming onto the track, and I decide I can wait no longer. I leave the rail, make my way through the crowd, and stand in line at the $50 window. And as I do so, I look at the parallel line to the next $50 window, and I see a Hassidic Jew standing right opposite me, wearing his signature black _shtrieml,_ a round fur-trimmed hat decorated with a feather. He is humbly reading a Kabbalah text, and he is wearing the humblest $900 black alpaca coat I have ever seen, and there is an incredible shine on his humble $300 shoes, and I realize that I have been wandering in a fool's paradise, that the true answer was here all along. I want to tell him to hide the text, that we don't want anyone else figuring out how to get a direct line to God, because if even half a dozen big plungers learn the secret, Quanto La Gusta could go down to 4-to-1 by the time they reach the gate.\n\nI avert my eyes, because I don't want anyone to see me staring at the book and wondering what I am staring at, and I whistle to myself and gaze alternately at the ceiling and my feet until I am finally first in line.\n\n''What do you want?'' says the clerk at the window.\n\n''I want number 5, thirty-three times,'' I say.\n\nHe looks at me like I'm crazy.\n\n''You're sure?''\n\n''Thirty-three tickets on Number 5, right,'' I say.\n\nHe shrugs and punches a button and out pop the tickets. I share a secret winner's smile with the Hassidic Jew, and then I go back to my spot at the rail, right by the sixteenth pole, and wait.\n\n''The horses are at the post!'' announces Marshall Cassidy over the public address system, trying to put a little excitement into his voice even though this is just a nothing race for nonwinners of two, and two thousand pairs of binoculars are lifted into position.\n\n''And they're off!'' yells Cassidy.\n\nQuanto La Gusta breaks in the middle of the pack, but Jose Santos quickly hustles him up to the front, and as they hit the far turn with half a mile to go he's seven lengths ahead of the field, and I am cursing myself for not finding this remarkable book ten years sooner.\n\nBy the head of the stretch he's ten lengths in front, and I can see that he's not even breathing hard\u2014and then, as quickly as he opened up on the field, he begins to shorten stride, and his lead goes from ten lengths to six to three to one to nothing in less than a furlong. As they pass me, with a sixteenth of a mile to run, he's already eighth, and by the time they hit the finish line he is eleventh, and the only horse he has beaten is one that broke down on the backstretch.\n\nI stare dumbfounded at Quanto La Gusta as he trots back on his way to the barn, and I cannot believe it. God _told_ me he couldn't lose.\n\nI decide to ask the old Hassidic Jew what went wrong, but he is nowhere to be seen. Finally, on a hunch, I go to the cashier's window. Sure enough, the old man is walking away with a wad of hundreds that would choke Quanto La Gusta.\n\nI walk up to him.\n\n''Excuse me,'' I say, ''but I notice you use the Kabbalah text . . .''\n\n''Yes, I use a Kabbalah text,'' he answers with a twinkle in his eye.\n\n''And you just won?''\n\n_''Baruch Hashem!''_ he says happily. Praise the Lord.\n\n''Did He whisper a number to you?''\n\n''That's how it works,'' says the old Jew. ''He whispered _seven,_ plain as day.''\n\nI frown. ''You're _sure?_ ''\n\nHe smiles and holds up his bankroll.\n\n''I don't understand it,'' I say. ''I used the book, and He told me to bet the number 5 horse.''\n\nThe old man extends his hand toward my book. ''May I?''\n\nI hand it over to him.\n\n''Well,'' he says, still smiling. ''That explains it.''\n\n''Explains what?'' I demand.\n\n''I use the _Sefer Yetzirah,_ the Book of Creation,'' he explains. _This,_ '' he adds, handing the book back to me and trying unsuccessfully to hide his amusement, ''is the _Sefer ha Mafli,_ the Book That Astounds.''\n\n''There's a difference?'' I ask.\n\n''The _Sefer ha Zohar_ is the Kabbalah for _trotters_.''\n\nHe is still chuckling to himself while I try to remember Big-Hearted Ernie's phone number.\nBUT WORLD ENOUGH\n\n_Sarah A. Hoyt_\n\n_''The Gods sell all that they give.\"_\n\n_\u2014Fernando Pessoa_\n\nThe one-eyed man sat at the table and rolled the knucklebones of the sacrificed ox. Outside, the wind blew from the mountains and the dry and cold winter of Bythinia settled upon the land. Hannibal Barca's house was deserted. Hannibal had made his way down from his bedroom, with its wide balcony and easy point of access to the vast, empty kitchen, where he sat at the much scrubbed pine table, and rolled his knucklebones by the light of the dying fire. And still he couldn't get an answer.\n\nHe'd sent the servants out, long ago. No reason to endanger them as well as himself. At any rate, they were foreigners who could not speak the Punic language and who had no personal loyalty to him, but served only for coin.\n\nAnd yet, it seemed like only yesterday that Hannibal had commanded the loyalty of crowds, multitudes, of armies and followers. If he closed his eye now, he could almost see behind it, Carthaginians and Africans\u2014 and Celts and Greeks too\u2014marching side by side, in massed ranks, a hundred thousand of them marching, with horses and elephants over the Alps, attempting the impossibly daring feat of conquering Rome itself.\n\nWas it the echo of their footsteps he heard when he rolled the knucklebones and heard them hit the scrubbed pine table with a hollow sound? Was it the sound of their shouts in the wind?\n\nNo. He pulled his cloak tighter around himself. He knew his hair was streaked with more white than black, his once sharp features made blurry by age and wrinkles. And he knew his arms could no longer hold the sword throughout endless combat. And his legs would no longer allow him to ride over half a continent . . .\n\nAnd yet, he thought, frowning into the dying fire in his fireplace, and yet, the gods had promised him a great empire. How could he achieve it now? What did it all mean?\n\nHe rolled the knucklebones and stared in confusion at the result. The combination of bones, the way they'd fallen, meant nothing. Or nothing he could understand.\n\nHe gathered them again into their ivory cup, and flung them once again onto the table. Tanit had promised. And now she must answer him.\n\n''Hannibal,'' Hamilcar's voice echoed, anguished, somewhere above his nine-year-old son.\n\nHannibal felt his father's calloused hand on his arm. The emotion in his father's voice shook him. He fought back from the intense light and confusion in his mind, struggled for control of the legs that had given out under him. He held on to his father's hand, drew himself up. He opened his eyes.\n\nAnd faced the sacrifice\u2014a young bullock that the priest of Tanit had butchered upon the altar. The altar was in a courtyard, where the whitewashed walls of the other rooms of the temple\u2014the reception room, the holy inner room, the priest's area, formed three sides of it. The fourth side was open, on the side of the mountain that faced the city of Carthage.\n\nBrilliant sunlight of summer shone from whitewashed wall upon whitewashed wall of the great African city, making it glare like a shameless jewel. Amid the houses, people moved, dark haired and light of limb. Farmers and merchants and scholars. Phoenician and Libyan and Carthaginian and half a dozen other nationalities.\n\nBeyond the city, in the sparkling blue bay, boats of many lands sat at anchor, come to sell and buy and trade with the greatest traders on Earth.\n\nIn that moment, a great engulfing love for Carthage came to Hannibal. He understood, just enough, dimly, to know it wasn't his, but the love of something . . . something that linked to him, that was in his mind, that made him long for Carthage as men longed for women, as a hungry child might long for a sweet. _All this,_ a voice in his mind said. _All this might have been lost. And more._\n\nIt wasn't a voice, really, more of a buzz, a tension, a feeling like . . . like what happened when thunderbolt threatened and your hair stood out from your head. Hannibal's head seemed to vibrate with it, to become filled by it. And his legs failed him again.\n\n''Hannibal,'' his father, Hamilcar, said. This time he was grasping his son by both arms, trying to make him stand.\n\nAs from a great distance, Hannibal heard the priest speak. ''It will be demons. You didn't have a child pass through the fire and now demons will take your oldest son. You mark my words.''\n\n_Nonsense,_ the voice-feeling in Hannibal's mind buzzed. _Nonsense. Not demons._\n\nThings that Hannibal's too-young mind\u2014more schooled, as well-born Carthaginians were, in the traditions of Greece than those of his own people\u2014had never known flooded him. He looked around, the scene diffuse in his sight as though a dazzling light shone behind his eyes. His head pounded with an ache that wasn't an ache.\n\nThis morning he'd come to the sacrifice glad only that his father thought him old enough to go with him. Now he knew everything about it in a way he could never have dreamed. They were here to offer sacrifice and beg the lady to grant his father, Hamilcar, and his brother-in-law Hasdrubal a good start to their new colonies\u2014what they hoped would be their new empire\u2014in Spain. Hannibal had just been admitted to the rites, though perhaps a little too young for them.\n\nThe linen robe they'd made him wear, so different from the Greek chiton he normally wore at home, seamless and long-sleeved, had made him feel stupid when they'd walked to the temple. It no longer did. He knew it was proper to wear Punic attire to address a Punic goddess, come to Carthage from ancestral Phoenicia with Queen Dido centuries ago. And he knew the voice in his head was the goddess addressing him\u2014him, personally.\n\nHe would have been overwhelmed by the honor, only the goddess didn't seem to be trying to honor him. She, whose image was so sacred that she was usually represented by a circle with two lines for up-raised arms, had a message for him. A message she'd been trying to convey when, as the bull lay dying, she'd entered his mind and made his legs weak and his head spin.\n\nBut the message wasn't real words, and Hannibal had to struggle to make them words. It was, Hannibal suddenly understood, the problem of an eternal being trying to speak to a mortal one.\n\n''Now he'll twist in a fit,'' the priest said. He was a small, bald man, his eyes narrow and spiteful. ''And he'll forever be possessed by demons.''\n\n''Not . . . demons,'' Hannibal managed. The words seemed to take up his whole breath and he had to struggle for more.\n\n''. . . too young for this,'' Hasdrubal said. ''He's just too young.''\n\n_And no doubt,_ Hannibal thought, running his gaze over his brother-in-law's black-bearded countenance, middle-aged and already running to fat, _it would suit Hasdrubal very well if I were too young. It would suit him even better if I were truly in the grip of a demon_. Because Hannibal was the oldest son of the great general Hamilcar. The real heir to the great fortune of the Barca family. His brother Hanno was but an infant, and yet subject to the many forms of demon likely to take a baby suddenly in the night. Until their late births, Hasdrubal, handpicked by Hamilcar to marry his oldest daughter, had been Hamilcar's right hand and stood in place of an heir to him.\n\nThis Hannibal knew, suddenly, without surprise, though there should have been surprise. Up till now, it had never occurred to the boy that Hasdrubal would benefit by having him out of the way. He'd always thought Hasdrubal merely a kindly older man, with sons close to Hannibal's age.\n\nNow, as he struggled for breath under the weight of revelations, Hannibal looked at his father\u2014an older man, tanned and beaten by a thousand storms, a million battles. Hamilcar had fought the endless, punishing war against the Romans. And he'd come home to quell a military revolt which had cost him the life of his second in command and best friend, Gisgo. Now, his hair white, his eyes burning with bitterness amid a nest of wrinkles, he looked at his oldest son, the hope of his old age.\n\nHamilcar was setting out, away from Carthage, away from the Senate of Carthage which had thwarted his campaign against the Romans through insufficient funds. The Senate, which had caused the mercenaries to revolt by trying to cheat them of their pay at the end of the war. He was setting out, at an age when most of his peers would be settling into contented senescence, cradling grandchildren on their arms, to try yet again to create a place where he could stand free and defend himself without the constraints of the weak-kneed Senate.\n\nAnd Hannibal knew what the goddess had told him, in the words that were no words. _You must go to Spain with your father. You must not leave him. You must learn from him the art of war. You must attack Rome, relentlessly._\n\n''Why?'' he asked the clear air, the voice inside his head, the terrace-built houses of Carthage. ''Why?''\n\n_Because you will create a great empire, an empire that will mold all of the future,_ the voice in his head said. _Because you will be remembered forever. Immortal. Like Dido._\n\n''Why?'' the boy asked again even as his elders looked at him, with varying degrees of amusement and shock. The priest was babbling something about speaking to no one.\n\n_Because it is your fate,_ the voice said.\n\nAnd with that, Hannibal's mind and eyes cleared, his legs stood beneath him, his once more. And he looked at his brother-in-law, at his father, at the still babbling priest.\n\n''Quiet,'' he said to the priest. His voice echoed with the steel and fire of battle, with the command of a great general. It was his father's voice in the piping tones of a young boy. The priest stopped, mouth agape. Hannibal turned to his father, whose hands were still on Hannibal's shoulders, holding him up. ''I am well, Father, it is only that in the moment the sacrifice died I knew my fate.''\n\n''Your fate?'' Hamilcar asked, letting go of Hannibal's shoulders.\n\nHannibal nodded. ''My fate is to go with you to Spain, to learn to command your army from you. And when I'm a man, I shall make war on Rome. Rome shall never know peace while I live. And I shall make Carthage a great empire that will echo to the end of history and make men remember\u2014''\n\nHannibal realized suddenly what moved his father, and he smiled.\n\n''\u2014your name forever.''\n\nThere was a moment of silence, and Hamilcar looked at his son, eyes wide. For just a moment, Hannibal wondered if his father was going to order him put to the fire\u2014his throat slit, his body burned, his ashes buried in the shrine of the goddess\u2014as was often done to children who proved defective. They were given as sacrifice, to keep the other children healthy.\n\nBut then Hamilcar threw his head back and laughed, an uproarious laughter rare in the man who had spent his entire life battling ruthless enemies, half of them within his own city. ''The goddess has given you a grand fate, has she not, Hannibal?'' he said.\n\nHasdrubal huffed impatiently. ''He is but a child,'' he said. ''And we are not taking him to Spain with us. He will stay here, with his mother, learn the traditions of Carthage, and become a proper man before he joins us.''\n\nAnd by then, Hannibal thought, Hamilcar would be dead, Hasdrubal would have seized control of the colonies in Spain, and his brother-in-law would be Hamilcar's heir for good.\n\n''No,'' Hannibal said, the force of his new knowledge behind him. ''No. You will take me.''\n\nAnd his father, a smile playing on his lips, looked at Hasdrubal. ''Do not forget,'' he said, ''who is the head of the family and the general of all-mighty Carthage.'' Then he smiled at his son. ''I think I will take you with me, Hannibal. I think I will.''\n\nHannibal bowed solemnly. ''Then I can promise you I'll always be an enemy of Rome,'' he said. As he spoke, he could feel the goddess withdrawing from his mind, leaving behind the feeling of a job completed.\n\n''You are out of your head,'' twenty-five-year-old Hannibal told Hasdrubal. ''To be accepting Roman treaties.''\n\nThey were in Hasdrubal's house, a broad house, built of stone and whitewashed, like the houses of Carthage, with a vast terrace open to the breezes of the Mediterranean.\n\nHasdrubal sat on this terrace, upon a low-slung chair with broad arms. He wore Greek clothing\u2014as did they all except when addressing the gods\u2014which in his case consisted of a very fine chiton of what seemed like a most delicate peach color and\u2014from the bit showing on his left shoulder\u2014either silk from the orient or the finely spun byssos from the isle of Amorgos, for which the ladies of Greece pined and swooned. The himation was of just as fine stuff and embroidered all over with tiny red flowers.\n\nHe'd been reading a scroll when Hannibal stormed in, and now he let the scroll fall from his fingers and looked up at Hannibal, a half smile with which an adult might indulge the tantrum of a spoiled child on his lips. ''Hannibal,'' he said in a voice of soft censure.\n\nHannibal threw his head back and squared his shoulders. ''No, answer me this, _brother_.'' The title, which under the law he was supposed to use for his brother-in-law, dripped from his tongue laced with sarcasm, reminding Hasdrubal that however close to the family he might be, he was no Barca. ''I heard you received Roman envoys. And you agreed to conquer nothing north of the river Ebro. Why are you accepting limitations from the Romans on the building of your own empire, _brother_?''\n\nHasdrubal sighed. He rose from his chair with a jangle of the silver bracelets on his arm. He pulled the himation looser over his broad paunch. ''Hannibal,'' he said. ''You are young.''\n\nHannibal glared. He could feel the glare burning out of his eyes. ''I might be young, but I'm my father's son,'' he said. ''My father would not want us to deal with the Romans. Ever.''\n\n''Your father,'' Hasdrubal said, speaking softly, ''dealt with the Romans, Hannibal. He signed the treaty of Catulus, ending the war.''\n\n''But you would sign a treaty without even warring.'' Hannibal paced in front of his brother-in-law, who stood stock-still and watched him, frowning. ''You signed the treaty in vile submission before anyone demanded that you submit.''\n\n''Hannibal,'' Hasdrubal said, and put his hand on Hannibal's shoulder where his short soldier's garment\u2014his chalmys\u2014fastened, leaving his skin uncovered and his shoulder and arm free to wield his sword. ''We have enough of an army to worry the Romans. We have enough of an army that they're not demanding we leave Spain. That, I know, is in great part your doing. You've trained them. You've led them. And the Romans are not trying to attack us. They merely want assurances we will not attack their precious Saguntum, the city they consider a friend of Rome; all else, we're free to do. Unspoken in the treaty is the understanding that we're allowed to control the rest of the territory south of the Ebro. They will not hinder us in this. And then, perhaps, we can strike at Rome. Or perhaps simply squeeze her out for commerce.''\n\nHannibal glared at Hasdrubal and shook Hasdrubal's hand from his shoulder. He could understand the truth of his brother-in-law's statement. He could understand the facts behind his talk. In some ways what Hasdrubal proposed was more of a Punic way. To slowly strangle the enemy with commerce. To reduce the enemy to dependence with smart bargaining.\n\nBut it was not what Hannibal wanted. And it did not accord with what he remembered of the goddess inside his head. _You will create a great empire,_ she'd said. _Your name will be remembered._\n\nIf they went with Hasdrubal's plan they would eventuallywin out. Or at least, perhaps they would. The Romans were in many ways a plodding people, less flexible and certainly less adept at winning the loyalty of those in whose lands they had colonies.\n\nCarthage stood at least an even chance of taking over Spain, of colonizing the vast land and converting the Greeks and Celts already living here to the Carthaginian way of life and Carthaginian culture.\n\nBut by the time that soft victory came, Hannibal would be long dead, his name forever forgotten. ''It is not what my father would want,'' Hannibal said. ''It is not what my brothers would want. My brothers and I\u2014''\n\n''Your brother Hanno is sixteen, Hannibal, and your brother Mago is twelve and still with his teachers. What can they have to do with this?''\n\nAnd now Hannibal lost the temper he had long held back, and stomped hard with his sandal-clad foot upon the polished rock of Hasdrubal's terrace. ''They are sons of Hamilcar, as I am! And they know he would want us to make incessant war upon the Romans!''\n\n''No,'' Hasdrubal said softly. ''No, Hannibal. You want that. Your father never did. And Hannibal,'' Hasdrubal drew himself up, pulling himself so that despite his growing corpulence he affected a military and proud bearing. ''I am the commander of the armies, by election. And you are just my younger brother-in-law.''\n\nHannibal took a good five breaths, in offended silence, before he could collect himself enough to storm out of Hasdrubal's house in a temper.\n\nThe temper carried him all the way into his own house, where he ignored the soft voice of his wife, Imilce, calling his name, and ran up the stone stairs to his own quarters, his own room. There he took deep breaths to calm himself, but in vain.\n\nInstead, he started pacing, hands clenched. He'd made the army. He'd made it what it was. An amalgamation of mercenaries, of Libyans, of recruits from nearby friendly Celtic tribes, it stood at near forty thousand men and it was far from the ragtag mercenary force his father had once commanded against the Romans. These men knew Hannibal, had seen him grow up and trusted him, were loyal to him with their own lives.\n\nHe'd always thought that was the gift of the goddess to him, that he had the ability to lift men's hearts and souls, to win them to his side with his words, and they would follow him with enthusiasm. He certainly was better than his father at it.\n\nAnd yet the one man that Hannibal couldn't persuade was Hasdrubal. And Hasdrubal had been elected to command the army when Hamilcar had died, because Hannibal had been only seventeen and was too young to be in charge. Now Hasdrubal was in charge until he died. And though he looked older, and going to fat, he was only twenty years older than Hannibal. By the time he died, Hannibal himself could be well advanced in middle age, and not in the state to make any of the war on Rome he'd promised the goddess he would.\n\nLeaning on the cool wall of his chamber, he pressed his closed fists hard against his temples, where pain burned as though something were trying to burrow in. Or his anger were trying to crawl out. He pushed his hands hard, knuckles biting into his skin. _Please,_ he thought. _Please. Lady. I want but to serve you._ _But how can I when this . . . soft merchant bars my way and would barter away our chances at an empire?_\n\nFrom outside the house came the sound of screaming, a sound that grew nearer. Someone pounded on the front door. It was opened, then closed again, hurried steps upon the stairs. Imilce screamed, a short scream, and then, ''I'll tell him.''\n\nHis room door opened. ''Hannibal, Hannibal,'' Imilce said. ''Your brother Hasdrubal is dead.''\n\nHannibal opened his eyes and brought his hands down. ''Hasdrubal?'' Hasdrubal had looked no more likely to die soon than he'd looked likely to take flight. ''Dead?''\n\n''A Celt killed him,'' Imilce said, her dark eyes brimming with tears, her dark hair disheveled as though she'd torn at it, though she'd had little contact with Hasdrubal. ''With a silver knife.''\n\n''Hasdrubal?'' Hannibal asked again. ''Dead?'' In his mind he thought of the goddess reaching out her powerful arm. Now Hannibal would be the commander of the armies. And he could do her will.\n\n''There are Romans,'' Imilce told Hannibal as he returned, muddied and bone-tired from his campaign north, as far as the river Douro, to consolidate his holdings and solidify his domain over the loose confederation of tribes that occupied that wild land.\n\nIt was fall now and the steady-falling rain of that region, with the bone-chilling cold that accompanied it, had got into Hannibal's body so much that even the week back to his more southerly domains, as he approached Cartagena, hadn't seemed to dispel the chill. Indeed he'd ridden back faster than he surely should, more desirous of warmth than even of his wife's company.\n\nNow he stood in the entrance room to his house, his brother Hanno just behind him, still chilled and covered in mud. He wanted his bath and his bed. He wanted Imilce and calm. Instead there was this.\n\n''Romans?'' he asked.\n\n''A delegation of them. All very proper,'' Imilce said. ''Important people.''\n\n''What do want they with me?'' he asked. That the Romans might be alarmed at his relentless campaigns since Hasdrubal had died, he didn't doubt. That they might find this sudden Carthaginian obsession with war and fighting\u2014when before they'd been ready to deal\u2014disquieting, Hannibal could well imagine. But why would they send him a delegation? Did they not know, was it not the word everywhere, that he was a sworn enemy of Rome?\n\n''They will wait,'' he said, ''until I've had my bath.'' He walked past Imilce before she could object, and all the way up the stairs to his chambers, where the servants had set warm water and oils and his clothing. He unwound his cloak from his chilled body and bathed, with the aid of two body servants.\n\nWhen he was done, he rejected the clothes laid out for him\u2014a chiton and soft himation, which he might very well have worn in the house. In their place he demanded just the rough wool chiton that he wore in battle, and which he fastened to his shoulder and side with a serviceable fibula. He'd go to see these Romans dressed as a soldier.\n\nOutside his door, already bathed and similarly dressed, Hanno waited.\n\nTogether they went to face the Romans, who looked like fat merchants and who introduced themselves as Publius Quintus Sanguntinus and Marcus Illius Sextus.\n\nImilce, ever the housewife and hostess, had served them olives and pomegranates served in silver dishes mined from the region. One of the Romans had dripped pomegranate juice down the front of his tunic, staining it purple.\n\n''You wish to see me,'' he said, sitting down.\n\n''Yes, my lord,'' Sanguntinus said. ''We come at the behest of the Roman people.''\n\n''The Roman people will always have my ear,'' Hannibal said. And said it without irony. After all, hatred was, much like love, a devotion, a constant attention to the other.\n\nSextus fidgeted. ''We would like to request, my lord, that you not . . . that you leave Sanguntum in peace. For it is a friend of Rome.''\n\nSanguntum. Several discordant thoughts ran through Hannibal's mind. Sanguntum was, after all, on the south side of the Ebro, and Hasdrubal had said that the contract left that to the Carthaginians. And the Romans were asking for peace for Sanguntum. That meant that they believed he would be within his rights to attack it.\n\nHe stood up and smiled. ''I will grant you, gentlemen, that I've not in the least changed my intentions toward Sanguntum this last year.''\n\nOne of the Romans opened his mouth as though to ask what those intentions were and what it all meant, but Hannibal was smiling, the smile so broad on his face that it almost made his features crack. ''But you've waited long, and you're probably tired. Let me call my wife, and she will make sure slaves bring you wine and food and whatever else you desire.''\n\nThey couldn't ask while he was being hospitable, and in two hours they were too drunk to ask. By the time they sobered all they would remember was that Hannibal had been hospitable, and therefore that he could not possibly mean to attack Roman interests. They would only remember he was a friend.\n\n''And what mean you to do about Sanguntum?'' Hanno asked while he and Hannibal walked amid Hannibal's olive trees.\n\n''I mean to take it,'' Hannibal said. ''It is to the south of the Ebro, and even by Hasdrubal's dealing, ours to take.''\n\nHanno gave Hannibal a worried look. ''The Romans aren't likely to take it calmly.''\n\n''The Romans,'' Hannibal said, ''can take it as they will.''\n\n''Carthage has been given an ultimatum by Rome,'' Hanno said, breathless, coming into Hannibal's study, where Hannibal was playing with his year-old son, Hamilcar. ''They said they can let fall from their breast either war or peace.''\n\nHannibal scarcely looked up. His decision, if such it was, had been made long ago, when the goddess had told him that his fate was to always be an enemy of Rome. ''Well, then we'll snatch at the war and let the peace go.'' He positioned Hamilcar on his knee and bounced him up and down gently. The baby smiled, letting a thin tendril of drool fall, which Hannibal wiped with the back of his hand.\n\nEven while doing it, he was aware that his brother had stopped, unnaturally still, like a statue.\n\n''But,'' Hanno said, ''you can't mean it. We've lost two wars against Rome. We are not prepared . . .''\n\nHannibal smiled. ''But we are. I've had it all planned a long time. We will do what no one has ever done. We will march into Italy and attack Rome.''\n\n''Into Italy?'' Hanno asked. ''What ships will take us there?''\n\n''None. We will march by land, over the mountains.''\n\n''You are mad, brother.''\n\nHannibal looked up. ''If madness is greatness, then I am. The Celts cross the mountains all the time to raid the northern part of Italy. Don't fret. It can be done.'' He grinned wide. ''We shall do it with elephants.''\n\nHanno backed up blindly, to sit on a chair near Hannibal. ''With elephants?'' he said.\n\nHannibal laughed at his tone of surprise, and on Hannibal's knee Hamilcar giggled the rounded giggle of a contented baby.\n\n_My son will be master of an empire,_ Hannibal thought. The goddess had promised, and he trusted.\n\nHannibal crossed the mountains with horses and elephants and forty thousand followers. Some elephants and some men had died in the crossing, but Hannibal could hardly regard that as important. His decision had been made long ago, and if the goddess was with him, then these losses were inconsequential.\n\nAs though to prove his blessing from Tanit, almost immediately men from nearby tribes attached themselves to him. They more than compensated for the ones he'd lost. He pushed on.\n\nIn the marches of the Arno, he lost an eye to a chill that became a raging fever. This, though marring forever the countenance that his wife had loved, also meant nothing. He must continue. It was his fate.\n\nHe plundered Arretium. The city of Capua, second only to Rome in glory, deserted Rome to stand by Hannibal's side. Latins, Umbrians, Sabines, and Picen-tines stood against him in vain.\n\nIn Rome, human sacrifice was practiced for the first time in centuries\u2014two Gauls and two slaves were killed to appease the gods. It meant nothing. They could not stand against the power of Tanit.\n\nOn the shores of Lake Trasimene, he won a great victory and near Cannae in early August\u2014with Magus's help\u2014he defeated the Roman legions and cut them to pieces.\n\nBut the Romans were in Spain now, and laying waste to the careful empire that Hannibal had started building there. And Hannibal didn't have enough men. Never enough men.\n\nIt was then that he started using the knucklebones, from an animal sacrificed to Tanit. He would roll the bones and read the result. Since the goddess was no longer in his head, he had to roll the dice to find what she meant him to do.\n\nHe rolled the dice when, early one summer, news came to him that his wife and son had been killed by a fever\u2014a result of the chaos of war in Spain. And the knucklebones told him to press on. He expounded to the goddess on his lack of men, on the fact that not enough Celts were crossing over to his side and that the Italianate tribes remained deaf to his appeals.\n\nBut the knucklebones said, _Press on._\n\nHe rolled them again when Syracuse rebelled and aligned itself with Carthage. And again when news of his brothers' deaths reached him. And again when the pestilence swept the Carthaginian camp and destroyed their fighting force, leaving Hannibal with even fewer men.\n\nAnd he rolled them, yet again, when, trying to protect Capua, he attempted to divert troops from laying siege to his allied city by making sorties on Rome itself, with his elephants. This failed. The Roman Senate apparently knew that the walls were far too strong to be breached.\n\nAt long last, bitter at heart, Hannibal left Italy. He left it to defend Carthage itself. The knucklebones could not be lying. As they could not be lying when they told him that he must sue for peace after his defeat.\n\nHannibal hadn't understood\u2014he still didn't understand\u2014but he knew the goddess would not deceive him.\n\nAn enemy of Rome, not forgotten by the wrath of her people and her rulers, he moved from principality to principality, trying to hide, trying to live a little longer, trying to spy an opportunity from which to charge again. Because it was his fate to make war on Rome and to win and to create a name for himself.\n\nIn Bythinia, at the end of the world, the winters were as cold as the summers were parched, and before the snows fell, blanketing the mountains, the wind came, carrying just a few flakes, biting to the bone of the old man Hannibal had become.\n\nFor a couple of years\u2014he disdained to count the years of his exile\u2014he'd lived here, with his comfortable house that contained nothing of himself, served by men who worked for the money alone and did not know, nor did they care, whom they served.\n\nAnd then there had been rumors of Romans. Romans had tracked him down and with Bythinian complicity\u2014or indifference\u2014paving their way, they had gathered in the nearby city. One by one they'd trickled up, over the last few days. There was a veritable detachment of them surrounding his home.\n\nThey thought\u2014and he couldn't say they were wrong\u2014that he had been encouraging the king of Bythinia to rebellion.\n\nAnd now there were dozens of them outside. _Dozens to kill an old man._\n\nHannibal had sent his servants away. A great general, as he remembered his father teaching him, risked men when risk was necessary. The rest of the time, he tried to save the lives of those who served him, even if their service involved lighting fires and cooking meat, and their loyalty was to gold and not to him.\n\nAll alone here, in his empty kitchen, he rolled the knucklebones. And he got no answer. Or no answer that he could understand. The symbols the bones formed on the scrubbed pine were always the same. The symbol for completion. The image of something done.\n\nHannibal shook his head. He wasn't done. If he were done, Rome would be defeated and Carthage would be the great power in the world. Or at least that offshoot of Carthage which was his beloved Cartagena in Spain.\n\nBut he wasn't done. He got up and poured himself a cup of wine from a flagon hanging on the wall of his kitchen. It tasted sour, not like the wine he'd drunk in Italy while he commanded his troops. But he drank it anyway, against the bone-biting cold.\n\nWithout the goddess and her promises, he would have stayed in Cartagena, by the Mediterranean Sea. He could imagine it now\u2014even in this season\u2014bathed by mild breezes and the warm waters of the ocean. He could imagine the terraced houses\u2014which might now be all destroyed by the Romans\u2014reflecting the sunlight. And children running and playing in the dusty streets.\n\nIf he hadn't fulfilled his promise to the goddess, his son might have been one of those children. But no. By now Hamilcar the younger would have had children, and perhaps given his father a cloud of grandchildren to play with and bounce on his aged knee.\n\nHannibal surveyed his knee where it showed, bony and scarred, beneath his chiton. He drank another cup of wine and tossed the knucklebones again.\n\nAnd again the symbol was for completed. And for accomplishment. He'd accomplished nothing. From outside, he could hear the steps of his enemies and voices calling as they walked around the shuttered house, seeking entrance.\n\nWas that the front door giving, as they applied shoulders to it? And how long till they decided to set fire the house and roast their old enemy alive? And what could the goddess mean by all of it?\n\n_I mean you're done,_ a voice said. _And well done. You have fulfilled your fate, Hannibal Barca._\n\nHis head hurt, as it had those many years ago in Carthage. His eyes seemed blinded by a light that shone behind them. ''I have done nothing,'' he said, his voice reedy and wavering. ''I have done nothing, save cause the loss of Carthage and of all my family.''\n\nThe light resolved itself into a figure. A female figure, drapery enveloping the voluptuous curves, the arms raised as the arms of the goddess were in the only sketchy representations of her that Carthage allowed.\n\nHannibal's mouth felt dry and tight. _The wine,_ he thought. _One of the hired servants poisoned my wine._ It was such a Roman trick to buy the demise of enemies by bribing their servants that he could almost have laughed. Only he couldn't laugh past the tightness at his throat, and it was too late, and his chest hurt when he inhaled.\n\n_Oh, but you have done everything I told you,_ Tanit said, her voice sweet and soft in Hannibal's ears like the voice of his Imilce in the short years of their life together.\n\nHe tried to shake his head, though it hurt him. He fought with all his might against the workings of the poison in his body. He wouldn't die. He didn't have an empire. He was not ready to die.\n\n_By your actions Carthage was lost,_ Tanit said. _By your actions Rome learned the power of final conquest, of total destruction of an enemy. It has shown them how to be ruthless. And how to win. From now on, nation after nation will fall under the heel of Rome. And Rome will last long enough to forever be part of the civilization of men._\n\nIn his mind's eye, Hannibal saw Roman statues, Roman clothing, Roman poetry in lands beyond those he could even imagine. He saw many languages derived from Latin.\n\n_But you promised I . . ._ he thought. He could no longer talk.\n\n_I promised you would never be forgotten. And you won't. Wherever Rome is mentioned, so will be the name of her greatest enemy._\n\nHannibal wanted to laugh or cry or scream at the depth of Tanit's betrayal. But his mind, working fast, and with all it had left, reminded him that he was Carthaginian. Carthaginians were first of all merchants.In any deal, you had to make sure that you were getting what you thought you were.\n\nHannibal and Carthage itself, Hannibal now realized, had been nothing but a sacrifice in the gods' games of power.\n\nBut if he'd lost a bargain, he'd lose it like a Carthaginian. He shrugged off his flesh like an ill-fitting cloak and crossed over to the arms of the radiant goddess, laughing at his own foolishness.\n\nHe'd got exactly what he'd bargained for.\nCONSIGNED\n\n_Alan Dean Foster_\n\n\"Why are you so determined to go to hell?'' Crouched down behind the fold of reef, Menno kept his bubbles small and deliberately unobtrusive as he whispered a reply to Codan. \"All my life I was told that I was destined to go to hell. It was a determination that I accepted willingly. What has always astonished me is that no one else shares my desire to do so.''\n\nHolding his spear steady as he floated parallel to the sandy bottom, his friend spoke while intently eyeing the parade of passing polosto. ''How awkward it must be for you to find that the rest of the world is so terribly afflicted with sanity.''\n\n''Always the sarcastic one.'' Menno joined his friend in patiently watching and waiting as the school of unaware polosto continued to drift nearer. ''Sarcastic and apathetic.''\n\n''I am not at all apathetic,'' Codan assured the other hunter. ''I am adamantly opposed to the very notion. Didn't you ever realize that the suggestion was not to be taken literally? People are often told to go to hell. None of them ever takes it as a personal challenge. At least, none until you, my friend.'' Upon concluding this observation he exploded forward over the top of the reef and thrust madly with his spear. With their cover thus broken, Menno had little choice but to follow.\n\nEven so, he had to admit that Codan had chosen the moment well. The polosto had come in very close to the reef, enabling both hunters to spear good-sized specimens before the school could flee. As the impaled prey writhed on the polished lengths of zek shell, unable to flee or reach around far enough to bite the hunters, the two Tyry enjoyed eating their prey alive. There was, after all, nothing a true Tyry enjoyed more than eating. Beginning with the tentacles, a true delicacy, the pair soon worked their way down to the polostos' internal support, a single expanding spiral of bone.\n\nThe rest of the carcasses and all of the polostos' internal organs were saved and stowed in drag-sacks of woven absab weed. These would be towed home to be distributed among members of both males' extended families. Once the two hunters had eaten their fill, they turned away from the outer reef and began the long, steady swim homeward. Timing their departure from the reef to the changing of the current allowed them to make steady and relatively easy progress.\n\n''Why don't you come with me?'' Using a right-side tentacle to flick aside a persistent cluster of parasitic keleth worms, Menno swam a little closer to his friend. His strong, broad, four-lobed tail made short work of those keleth who persisted.\n\n''What, to hell?'' Using one webbed tentacle, Codan gestured disparagingly. ''I wish you would drop this enduring mania, Menno. It was already becoming tiresome years ago.'' Using his spear, he prodded a likely looking hole as he swam past the opening. If it had been home to a tasty foudan, the angry invertebrate would have displayed itself and its anger immediately, an appearance that would have led to its becoming an instant appetizer. ''Much as certain members of the community might wish it, you are not going to hell, however much you persist in pretending to continue to humor their requests.''\n\n''Well, don't say I didn't give you the chance.'' With a half flip of his tail, Menno moved out ahead of his friend.\n\nTurning his head slightly to the right so that he could focus one eye completely on the other Tyry, Codan flicked his own tail and caught up. ''What are you talking about? What 'chance'?''\n\nMenno slowed deliberately, dropping down beneath his friend in a gesture of polite deference. They were very close now to the place he had chosen for the attempt. ''I'm going. This evening.'' Raising two of his four webbed tentacles, he extended them rightward. ''There.''\n\nBackfinning, Codan halted. ''Now you're scaring me, Menno. A joke's a joke, but you're not bubbling laughter.'' Having turned, he now found himself staring in the direction his friend had designated.\n\nInstead of being composed of sand or reef or storm debris, the indicated bottom was smooth and solid; all rock, with nothing growing or hiding on it. Other than a distinct greenish tinge, its most notable characteristic was a remarkably unvarying upward slope.\n\n''I have been all the way to the end,'' Menno declared proudly. ''To the terminus. To the place where the real world ends. The slope continues on and on, the greenstone ascending steadily until it breaks the sky\u2014and passes beyond it to enter hell. I am certain of this.''\n\nTurning away from the unvarying slope, Codan stared at his lifelong friend and hunting partner, turning to regard him first out of his left eye, then the right. ''You're not joking, are you?''\n\n''I am quite serious.'' Menno gazed back at him.\n\nComing close, Codan studied his companion's upper body. ''Your eyes are clear and unfogged, so you have not been eating the purple kalis that grows in the lower caves. Your words are precise. Only your thoughts, it seems, betray any madness.'' Having rendered this opinion, Codan resumed swimming.\n\nMenno caught up to him easily. ''Since others told me that I was destined to go there, I have always dreamed of being the first Tyry to visit hell and return.''\n\n''Easy to visit.'' Codan gestured with his left tentacles. ''Impossible to return.''\n\n''Not any longer. I have devised an apparatus.''\n\nThat brought Codan up short. ''An apparatus?'' He held up his spear. '' _This_ is an apparatus, and a very effective one. But it cannot get me to hell and back. No artificial construct can do that.''\n\n''I have prepared for this all my life.'' Menno's tone was confident, assured. ''You'll see. I will show you tomorrow morning. If you won't join me, will you at least come to observe? Without a credible witness there will be none to believe me.''\n\n''Bereave you, you mean.'' Reaching out, Codan draped a tentacle around his friend's upper body. ''Be a sensible fellow. We've made a good catch, we're bringing back ample food to share, and tomorrow is another bright and beautiful day. Enough of this talk of going to hell.'' He made a gesture that was reflective of his innate sardonicism. ''We all get there sooner or later anyway.''\n\n''If you won't come . . .'' Menno's voice trailed off tersely as he kicked hard with his tail lobes.\n\n''I didn't say that.'' Codan rushed to catch up. ''I suppose I must. As a friend and hunter-brother. Someone has to be there to revive you.''\n\n''Thanks.'' Menno entwined one of his own tentacles around that of his friend. ''As a moment of historical importance it will be in need of recording.''\n\n''A moment of hysterical importance,'' Codan corrected him\u2014but gently, for he was truly and honestly concerned.\n\nThe morning of the following day dawned clear and sunny. The atmosphere was bright, cool, and devoid of current. A perfect day for visiting hell, Menno assured his friend. Or rather, friends, for Codan had insisted on bringing Kedef along. In addition to being senior to both of the hunters, among the Tyry Kedef had an impeccable reputation for honesty. Whatever he reported would be believed. This was the reason Codan offered for inviting him, anyway. In reality it was a subterfuge, a way for him to enlist the aid of a respected elder in trying to talk Menno out of what was seeming to be more and more an inescapable insanity.\n\n''Even if you could somehow enter hell and survive,'' Kedef was muttering as the three males swam parallel to one another in the direction of the greenstone slope, ''you are sure to be eaten by a demon.'' He turned reflective. ''I commend to you the tale of Ses-Haban.''\n\nSes-Haban was a mythical Tyry of ancient times who had dared to swim high enough to gesture contemptuously at the inhabitants of hell. According to the legend, he had thrust two tentacles completely out of the real world and into that terrible nether realm. What he sensed there was unknown, because as soon as he had done the unbelievable deed, a demon had promptly bitten off both offending limbs.\n\nSingled-minded of purpose and determined to fulfill the urge that had driven him since childhood, Menno would not be dissuaded. ''Ses-Haban, if he existed, was unlucky.''\n\n''Ses-Haban, if he existed,'' Codan suggested brusquely, ''was stupid.''\n\n''Where is this apparatus of which you have spoken?'' the elder inquired. Codan did not try to conceal his surprise. Was Kedef showing interest?\n\n''You will see it soon,'' Menno assured him. ''I've had to work on it in secret, when away from the community.'' Having reached the flow of greenstone, he was leading them downslope. ''Jeers and jibes do not bother me, but there are those whose fears exceed their common sense. Had they known what I was about they might have tried to interfere with my work.'' He twisted his upper body to look back the way they had come. ''They might have tried to stop me.''\n\n''You can hardly blame people for fearing hell,'' Codan reminded him.\n\n''People fear that which they do not understand,'' the other hunter riposted.\n\n''Often with good reason,'' Codan added. But this time, Menno did not respond.\n\nThe entrance to the small cave had been camouflagedwith loose reef growth and rock. Working with all four of his tentacles, Menno pulled the rubble aside. From the dark interior he extracted a thick, perfectly transparent tuzaca shell. But it was not the classic spiral tuzaca shape. A deformity had produced a shell that took the shape of an elongated bubble.\n\nKedef's tone as he studied the product of the younger Tyry's labors was admiring. Codan realized to his horror that the elder's presence was starting to have the opposite effect from what he had intended.\n\n''You have modified this,'' Kedef declared with conviction as he eyed the unusually formed shell. ''In several places, it would appear.''\n\nGratification suffused Menno's reply. ''Yes. See here, where I have smoothed the base after removing the animal?''\n\nKedef's tentacle ends tiptoed appraisingly over the exterior of the crystal clear shell, stopping at a round opening. ''And what is this hole for? It is not natural.''\n\n''No. I made it. That is where the voyoupa intestine goes.''\n\nShells and intestines. Codan was beginning to wonder if he was going to have to use force to put a stop to this lunacy.\n\nAs he and Kedef looked on, Menno extracted another shell from the cavity in the rocks. This one had a lid. When opened, both sight and smell identified the shell's contents. Codan's tentacles recoiled. The shell contained chisith innards. Sticky and stinking, they were violently expelled by a chisith whenever it was attacked or disturbed. Entwined in the viscous strands, a preoccupied predator would pay no further attention to the chisith, allowing that simple creature to scoot safely on its way. What possible use could Menno have for such a pot of scavenged goo?\n\nHe proceeded to show them.\n\nWorking with a silicate spatula, he daubed chisith innards all over the upper part of his body, careful to stay above his tentacles but below his head. Taking up the large tuzaca, he then carefully pulled it down over his head so that its base rested against the painted line of chisith guts that now encircled his upper torso. The gummy chisith innards gripped the base of the shell. Menno then brought out a long coil of voyoupa intestine. Additional chisith innards were pasted both inside and outside the round opening he had made in the tuzaca shell. One end of the intestine was passed through the hole. More chisith innards were added to the outside. Within seconds it had been tightly sealed in place.\n\n''You perceive how this apparatus works?'' he asked boldly. Though muted by the intervening shell that now enclosed his head and upper body, his words remained intelligible.\n\nCodan had composed a choice response, but Kedef spoke first. ''I think so, somewhat, but still . . .'' The elder's voice trailed off into uncertainty. ''I am not sure I understand the proper function of the length of intestine.''\n\n''The principle is very simple.'' Clutching the coil of intestine in two tentacles, Menno used another to hold up the end. ''There is no atmosphere in hell. Only searing heat. Studies suggest that a Tyry can endure the increased temperature for some time before it begins to adversely affect the body. But this point is moot in the absence of breathable atmosphere.'' With his remaining free tentacle he reached up to tap the outside of the transparent tuzaca shell.\n\n''The adhesive chisith insides form an atmospherically tight seal between my upper body and the tuzaca shell. The latter is presently full of perfectly breathable atmosphere. When I venture into airless hell, I will carry my breathable atmosphere with me.''\n\n''It won't last you more than couple of minutes,'' Codan pointed out relentlessly. ''Studies suggest _that,_ too.''\n\n''Which difficulty brings us to the purpose of the voyouda intestine.'' Once again, Menno waved the open end. ''There is a very small hole near the base of the tuzaca shell. Once in hell, it will leak atmosphere. It is designed to do so. As exhausted atmosphere bleeds out, I will use the tube formed by this intestine to suck fresh atmosphere into the shell. Bad atmosphere out, good atmosphere in.'' He eyed the two other Tyry; one old and thoughtful, the other young and alarmed.\n\n''What if this crazy setup fails on you?'' Codan demanded to know. ''What if the chisith seal fails, letting the atmosphere inside the shell all out at once? What if you can't suck in fresh atmosphere fast enough to replace the old?''\n\n''Then I will still have time enough to return to the real world.'' Menno's confidence had not deserted him. Indeed, Codan thought, the closer his friend came to suicide the more assured he appeared to become. ''I will only need a minute or two. If it comes to it, I can hold my breath longer than that.''\n\n''It is an astounding and bold notion.'' Kedef hovered in the water above the smooth greenstone, his tail moving lazily back and forth to hold him in position against the current.\n\nCodan turned on the elder. ''You are not actually going to give your blessing to this madness?''\n\nTurning to one side, Kedef eyed the younger Tyry with his left eye. ''It is not a matter of giving the blessing or the withholding of it. Menno is of age. I cannot and would not try to stop him. Besides, everyone in the community knows he has spoken of a desire to fulfill the requests of many ever since his tail lobes were first formed.'' He looked back at the obviously proud Menno. ''Triumph rewards the audacious.''\n\n''There is audacity,'' Codan pointed out, ''and then there is folly.'' But by this time Kedef was paying him no more attention than was his friend.\n\nSo thoroughly had things turned around that Kedef helped Menno to secure the length of intestine to the hole in the tuzaca shell. Codan refused to lend a tentacle to the final preparations. Even at the speed he could swim, it was too far to race back to the community to raise the alarm. He could only hope that the foredoomed experiment would not destroy his friend.\n\nWhen the last of the preparations were concluded, Menno turned a slow circle in the water. ''How does it look?''\n\n''Everything appears tight and secure, young boldness.'' Kedef's tone was full of encouragement. ''How is your breathing?''\n\nHolding up the open end of the voyouda intestine, Menno waggled it back and forth. ''No problem.''\n\n''Of course there's no problem,'' Codan growled. ''He's still in the real world.'' But it would not be long until that boundary was tested.\n\nHe and Kedef followed as far as they dared. They trailed behind the self-assured Menno as the latter swam upslope with even, powerful strokes of his tail. Soon they were in the shallows, and still Menno continuedswimming. Onward and upward. Codan realized that further pleading was of no use. His friend was determined to go through with this. It was, after all, the fulfillment of a lifelong impulse.\n\nDrifting together side by side, he and Kedef watched in fascination as Menno kept swimming, swimming, just kept swimming, until all they could see was the hunter's lower body, then just his tail, and finally not even that. He was well and truly gone. Gone into hell itself. But the end of the long coil of voyouda intestine trailed behind, whipping slowly back and forth in the slight current that marked the boundary between hell and the real world.\n\n''Do you think he's still alive?'' Codan finally asked after an interminable several moments had passed.\n\n''I think he must be.'' There was wonderment in Kedef's voice. ''If he had been eaten by a demon, then I would think the piece of voyouda would have fallen back into the water by now. A demon would either pull it out or let it go, but not maintain a constant length. Think of it! A Tyry, moving about in hell itself, observing and recording for the first time! What tales your friend will have to tell us when he returns. What unprecedented sights he will have seen!''\n\nCodan said nothing.\n\nMore time passed. It seemed impossible that Menno could have survived so long beyond the sky. The waiting was taking its toll on both him and Kedef. They had been half rocked to sleep by the current when his eyes suddenly dilated sharply.\n\n''There! By the Line of Otos, he's coming back!''\n\nUpslope was indeed the familiar shape of Menno that was coming in their direction. But something was wrong. The atmosphere-retaining tuzaca shell was still in place, glued securely to the hunter's upper body by the gooey chisith innards. The upper end of the voyouda intestine was still sealed to the hole Menno had artfully drilled in the shell. Despite this, the intrepid explorer was not swimming normally. His tail was not moving. Tentacles as limp as strands of ncasa weed, he drifted partway toward them before being caught by the current.\n\nCodan and Kedef did not hesitate. Taking deep breaths in case hell should come after them, they swam rapidly upward, grabbed hold of the motionless hunter, and dragged him down into deeper, safer water.\n\n''Menno!'' Codan was tugging and flailing at the tuzaca shell. When it would not come loose, he pulled his knife from his waistwrap and used it to cut through the spongy, solidified chisith. With Kedef pulling, they soon had the shell off. Still Menno did not respond.\n\nWhen he saw why, Codan whirled and lost the undigested remnants of his morning meal. Even the venerable Kedef was forced to momentarily turn away. Only when both had managed to overcome the initial shock were they able to turn back to regard Codan's friend.\n\nMenno was dead. That had been immediately apparent. It was the particularly hellish means of his passing that had so shocked them. The driven explorer's entire upper body was burned dark, as if he had been shoved into one of the underwater volcanic vents that dotted the far boundaries of the community's territory. His eyes had been boiled away. Codan tried to imagine his friend under that kind of assault, reeling from the unexpected fiery pain as he struggled to make his way back into the water. With his eyes gone he had probably lost his way, and had only stumbled back into the real world when it was too late. Stumbled, or fallen and rolled downslope.\n\nBut the tuzaca shell, full of cooling, soothing atmosphere, had survived intact and still sealed in place. What had gone wrong?\n\n''I cannot imagine,'' Kedef murmured in response to Codan's query. ''I have not the kind of specialized knowledge necessary to posit an answer to that question.'' Drifting in the water, the somber elder considered the dead, burned body. ''Yet his lower body, beneath the shell, shows signs of only slight wrinkling. It is almost as if in some way the tuzaca shell that was necessary to sustain life somehow contributed to its destruction. If this attempt is to be repeated, the matter will require considerable further study.''\n\n''Further . . . ?'' Codan gaped at the elder. ''You don't mean that you're going to suggest that another Tyry try this?''\n\nKedef responded with a gesture that was difficult to interpret. ''You are yet young, Codan. A thing once tried is destined to be tried again. Wiser Tyry than you or I will discuss what has happened here today and seek ways of overcoming it. There will always be those who feel they have no choice but to push at the limits of the possible. There will always be those who from an early age will be urged to go to hell, to the point where they come to regard it as their destiny.''\n\nCodan was quiet for a long moment. When he spoke again, it was with renewed admiration for his deceased companion. ''Well, all is not lost today. In his passing Menno has provided me with the opportunity to fulfill a long-held desire of my own. To eat a close friend.''\n\nKedef gestured understandingly. ''A laudable, if not especially revolutionary desire. I will assist with the hauling.''\n\nTogether they towed the body of the dead explorer back to the community. The story of his valiant effort, temporary success, and subsequent demise was the occasion of much discussion and debate. In the end, the effort was accounted a modest success. After all, there were leftovers.\nMY GIRLFRIEND FATE\n\n_Darwin A. Garrison_\n\nNelson Wong glanced around the edge of the heavy steel fire door into the sunlit Seattle alley that skirted the uphill side of the parking garage. Less than twenty feet away, partially concealed by a Dumpster in a shaded corner, a pile of cardboard and restaurant garbage moved fitfully in time with a wet snuffling sound.\n\n''Ghoul, you think?'' Given the soft, feminine nature of the voice behind him, most men would have felt a warm thrill of excitement. All Nelson could feel at the moment was a light pain behind his eyes and a stomach-churning irritation . . . that, and the overwhelming _wrongness_ of a dark fae in the wrong place at the Wong time.\n\n''Well, duh,'' he whispered back after forcing a derisive snort. ''I can feel the disruption from here. Can't you?''\n\nA piqued sniff came from behind his head. ''I don't do _vibes,_ Nel.''\n\nNelson allowed himself a tiny victory smile. Moira hated being reminded that he was better at this aspect of his part-time job than she could ever be. Consideringwhat a pleasant mess she'd made of his life since turning up, he had to take his satisfaction wherever he could find it.\n\n''You could have stayed in the car.''\n\n''All you have are those saccharine anime soundtrack CDs. I'd tear my hair out in less than five minutes.''\n\n''There's the radio.''\n\n''Thirty seconds. Ten if I accidentally tune Stern.'' A pair of slender arms encircled him, followed by the feel of a well-endowed feminine chest pressing against the back of his oilcloth duster. ''Besides,'' a warm breath whispered into his right ear, ''I have to make sure nothing happens to you. Don't I, lover?''\n\nThe tongue that darted into his ear worked like a detonator in the plastic explosive of his temper.\n\n''Sweet Jesus!'' He threw off Moira's arms and spun around to face her. Without resistance, the springloaded heavy door slammed home with a hollow _clang_. ''I am _working_ here! Can't you be serious for even five minutes?''\n\n''Never,'' she told him with her trademark half smile, half leer on her full lips. ''You're just too damned sexy.'' It was that same expression, along with her striking violet eyes, glittering golden hair, and sensuous curves that had started the fiasco that currently passed as his life. Now he knew better, although if she had been wearing the same bikini as she had back on that beach in Greece . . .\n\nHe shook his head violently to clear that particular memory out of his mind before tilting his head down and pressing the fingertips of both hands to his forehead.\n\n''You are so frustrating sometimes!'' When he swept his hands away from his face, he was able to see Moira's feet poking out from beneath her designer jeans. As in, really see them, painted toenails and all, through the spaghetti-thin crimson straps of her shoes. He looked back up into her eyes, disbelieving.\n\n''You wore _heels_ on a hunt?''\n\nHer smile was dazzling.\n\n''Aren't they the most darling thing? I mean, sure, they don't show my calves off as well as the Giorgios but they're so much more comfortable. And the red. I mean, they just draw attention.''\n\n''Attention?'' Nelson said, still stuck between disbelief and detonation. ''You generally don't _want_ to draw the attention of the things that I have to deal with. It's bad karma. What the hell were you thinking?''\n\nShe pulled her left foot back behind her right calf, dropped her chin, and twisted up in her very best shy-little-girl pose. ''I was hoping that . . . after the hunt and all . . . well . . . dancing.''\n\n''Dancing?'' Nelson wanted to yell, but when Moira did the whole pose thing with the lower lip and the fluttering eyelashes and . . . ''O, sweet Mother Mary. Dancing.''\n\n''We haven't gone in a long, long time, honey,'' she cooed as she reached out to take his elbow. ''There's this new club in Kirkland. I hear about it all the time. C'mon. Take me after this. Please.''\n\n''Look,'' he began reasonably, ''I'm probably going to end up smelling like cordite and sweat before we're done here. That's why I always like to shower after a job. Besides, I don't dance unless\u2014''\n\nHe stopped and cocked his head quizzically at her. ''Tell me you weren't going to slip me some more of that crap you got from your aunt.''\n\nThe way Moira's eyes darted to one side along with the flush on her cheeks spoke volumes. Then her eyes flashed back to his as another disarming smile curved her lips.\n\n''Aphrodite's such a dear.''\n\n''For crying out loud!'' he bellowed, his temper finally fracturing. ''How can you justify doing that to me again?''\n\n''It was fun!'' she shouted back. ''Besides, you can be such a stick in the mud without a little _help_.''\n\n''It was a week! When I finally came to my senses, I couldn't walk! You're damned lucky we aren't parents!''\n\n''Like I'd mind!'' she snarled back with a vehemence that rocked Nelson back on his heels. ''You should try listening to your biological clock tick for three thousand years! There's nothing wrong with\u2014''\n\nA tentative scratching came from the door.\n\nThey turned in perfect synchronization and stared.\n\n''Oopsie,'' Moira whispered. ''Was there a handle on the other side?''\n\n''Um, not sure.''\n\n''Car, you think?''\n\n''Oh, yeah.''\n\nNelson reached out to guide Moira away from the door and back through the garage toward their car. As soon as he heard the click of her heels against the concrete bustling away from him, he backpedaled himself, swinging his SPAS assault shotgun up from beneath his duster as he moved.\n\n''This is all about to go pear-shaped,'' he muttered as he jacked a shell into the chamber and aimed the weapon at the door, ''I just know it.''\n\nTen steps of attempted running were all that it took to convince Moira that Nelson had a point about the heels. She slipped out of her shoes, snatched them up by the straps in a smooth twirl, and jogged toward the car at a much faster pace. Seconds later, Nelson's SPAS roared to life in the darkness behind her and she sprinted.\n\n''This is going to be murder on my pedicure!'' she snarled aloud and dodged a roof pillar to follow the down ramp back to the car. They had parked halfway between the downhill street entrance and the uphill side alleyway where they had just spotted the ghoul, which was the same place where the paper said the murders had occurred. She had to admit his ability to pinpoint disruptions was getting better and better.\n\nNow if only he could get paid for doing this Boy Scout crap.\n\nBut as she turned onto the ramp and the car came into sight, her train of thought shifted radically. The sight of two female figures standing casually by the silver Magnum slowed her to a walk. Stopping well outside of arm's reach, she nodded her head.\n\n''Addie. Cleo. This, um, isn't a real good time.''\n\nAddie, the elder of Moira's two sisters, quirked a wry eyebrow in reply. With her silver hair pulled back into a severe braid and dressed in a dark blue Armani business suit-skirt combo, her trim figure practically screamed authority and competency. Cleo, on the other hand, was far more comforting in her stylish torn jeans and jacket, with a bright red Che Guevera T-shirt peeking out. She had chosen to dye one half of her normally dark brown bangs neon purple as well. Moira's little sister gave her a cheery wave along with an impish teenaged grin.\n\nLooking back to Addie, Moira shook her head. ''Really. I mean it. This isn't a good time.'' Accenting her statement, the SPAS barked again on the level behind them, followed by a string of Chinese expletives and the sound of running bootsteps on concrete. Moira winced as both of Addie's eyebrows lifted and Cleo covered her mouth in wide-eyed shock.\n\n''Does he talk like that all the time?'' Addie asked mildly.\n\nMoira shrugged. ''Depends. Like I said, bad timing.''\n\n''Well, all you have to do is give me the scissors and we'll leave you to your business.''\n\nMoira shook her head and crossed her arms. ''Sorry. Not happening.''\n\nCleo tugged on Addie's sleeve. ''You said we were going to dinner as a family. You promised.''\n\nThe elder sister brushed Cleo's hand away. ''Not now.''\n\n''And you promised not to fight.''\n\n''I promised to _try,_ '' Addie said in exasperation. ''Moira's not making that easy.''\n\n''No, I'm not. Now excuse me.'' With that she strode directly at her sisters and bulled between them.\n\n''Rude!'' Cleo snapped. ''You could have asked.''\n\n''I'm a bit pinched for time,'' Moira shot back as she tossed her shoes onto the roof of the car and beat on the side of the muscular-looking station wagon with her fist. ''Yo! Gunner! Wake the hell up!''\n\n''Gunner?'' Addie asked as she stepped away from the car. She gave the darkened windows a sidelong glance. ''There's someone in there?''\n\nMoira, busy patting down her pockets, nodded. ''You could call him that. Damn it! I think I left my keys in my purse.'' She reached out to smack the tailgate with her open hand. ''Gunner! Up and at 'em! He's gonna need shells!''\n\nThe tailgate popped open without warning, almost clipping Moira and the curious Cleo both in the chin.\n\n''Put a sock in it, blondie,'' said a sullen, gravelly voice from within. A web belt bandolier filled with crimson shotgun shells shot out into Moira's waiting hands. ''Your boyfriend's gonna be fine.'' With that the tailgate slammed shut again.\n\n''Rude!'' Cleo shouted back before turning back to Moira. ''Who the hell was that?''\n\n''Like Addie said, it's _somebody,_ '' Moira replied with a sigh. ''And _rude_ is just his way. You get used to it . . . sort of.'' Cleo turned a doubtful expression on the tailgate while Moira spun back toward the ramp. Before she got two steps away, though, a vise-like hand clamped on her left bicep.\n\n''The scissors,'' Addie said in a soft, cold voice, ''Now.''\n\nMoira shook off her sister's grasp and spun to face her. ''What part of _no_ don't you understand, sis? You never use the damn things anymore, for pity's sake. Why are you so hot to have them back?''\n\n''They're my responsibility.''\n\n''Your symbol of power, more like. Just get used to sharing for a while!''\n\nAddie's hand drew back and a spinning ball of purple lightning snapped to life beneath her fingertips. Moira shouldered the bandolier and crossed her arms, bringing a flickering wall of translucent cerulean to life between the two.\n\n''No fighting!'' Cleo shouted at the pair, followed immediately by a panicked shout from up the ramp.\n\n''Moira! Shells!''\n\nA faint smile crossed Addie's lips as she slowly lowered her hand. ''Your master calls.''\n\nMoira straightened. ''Get your vocabulary right,'' she said sweetly. ''The correct term is _lover._ ''\n\nNelson let the SPAS dangle on the shoulder rig so he could dig into his coat pocket for the car keys as he ran down the ramp. The thumping distraction was ill-timed. By the time he became aware of the tense little scene near his car, he had already lost any reasonable opportunity to run back toward the ghouls.\n\n''Oh,'' he stammered as three pairs of stressed eyes turned toward him. ''Cleo. Addie. Hi. Um, this is a really bad time.'' No doubt Addie's stare would have frozen him solid if she had retained any significant power in his world.\n\n''So I've been told,'' she practically growled. Cleo's expression, on the other hand, turned from misery to heartfelt relief. She darted around Moira and sprinted forward to grab him in a fierce hug.\n\n''Nel! Make them stop! They're fighting over those damned scissors!''\n\n''Again?'' he snapped as tried to pry Cleo off. ''This really isn't the time or place for that!'' Glancing down at her face, he spied a sparkle of silver in her right nostril. ''Is that a nose stud?''\n\n''You're not going to give me crap about that right now, are you? This is serious! I told Addie not to fight, but she won't listen to me. She never listens to me!''\n\n''They're not sanitary, you know? Besides, aren't you allergic to silver?''\n\n''It's nickel!''\n\nAn angry hissing echoed down the ramp from the upper level.\n\n''Oh, crap,'' Nelson muttered as he finally got Cleo loose. He pushed her toward Addie and Moira and reached under his duster to grab a canister off his web harness. ''You guys will want to get behind the car, cover your ears, and open your mouths.'' Moira grabbed a sister in each hand and dragged them behind the Dodge. Seeing that she had things well in hand, he turned back to the ramp and pulled the pin on the flash-bang grenade.\n\n''God,'' he prayed as he let the paddle spring free, ''please let these bastards be really, really stupid.'' With that he hurled the smoking green cylinder up the ramp toward the gathering cloud of ghouls. Before the bomb had even cleared the ramp, Nelson was on the move. He ran back to the car and rolled across the hood, landing with a bone-jarring _thud_ behind the front wheel just as the grenade detonated.\n\nThe blast shook free dust and flaked concrete from above them, coating everyone with a fine grit. Fortunately, they were far enough away that none of the vaporized silver nitrate, iron filings, or saltwater reached any of his fae charges.\n\nGetting up on his hands and knees, he took quick stock of the little group. Addie and Cleo both looked a bit like surprised owls, wide-eyed and dusted with white. Glancing up, he felt a little thrill of pride when he saw that Moira was already back on her stride, systematically dusting off her hair, peasant blouse, and jeans while wearing an expression of vague irritation.\n\n''You're really something, you know that?'' he told her, the warmth he felt washing away his earlier frustration.Her irritated expression disappeared into a private smile.\n\n''Dancing. I'm serious now.''\n\n''Bath first?''\n\n''Kinky. I like it.''\n\n''If you two are done verbally feeling each other up,'' Addie broke in, ''I think your new friends are regrouping.''\n\nNelson let out another string of his grandfather's choicest words. Catching sight of Cleo's shocked expression, instinct kicked in and he just managed to jerk backward out of range of Addie's open-handed swing.\n\n''What the hell?'' he snapped at her.\n\n''Watch your foul mouth, you heathen! There's a child present!''\n\n''Heathen? Look who's talking, you refugee from a Greek\u2014'' Moira's wildly waving arms and vigorous head shaking caught his attention. Gritting his teeth, Nelson choked his temper back under control. Best not to tempt Fate . . . any of them.\n\nHe stood up with a jerk and pulled his keys out of his pocket. ''Fine,'' he said in a flat voice before pressing the unlock button on his key fob. Reaching between Moira and Addie, he opened the back door. ''Get in.''\n\n''What?'' Addie huffed in affronted shock.\n\n''Get in. The damn. Car. We're leaving.''\n\nThe hissing was starting to gather up the ramp again as Addie and Cleo scrambled into the backseat. An odd echo caught Nelson's ear and he turned his head down the lower ramp to sort it out. There, a second ball of darkness was gathering just outside of a puddle of sunlight near the exit tollbooth.\n\n''Moira,'' he said softly when he took the bandolier off her shoulder.\n\n''Hmm?'' she answered as she gave him a warm kiss on the cheek. He held up the keys in front of her nose.\n\n''You're going to have to drive.''\n\n''Ooh!'' She snatched the keys from his hand with a gleeful expression on her face. ''I finally get to drive the hemi!''\n\nNelson focused on trying to keep an encouraging smile on his face while his stomach was busy crawling into his boots. Somehow, despite Moira's gleeful chuckling and rush to adjust the driver's seat, he suspected that it came off forced.\n\n_Oh, my aching insurance premium,_ he mourned silently.\n\nSix liters of Detroit's best iron grumbled to life next to him. Before the engine could even settle into a smooth idle, Moira opened the throttle up twice in quick succession, revving the motor to an ear-splitting howl each time.\n\n''At least give it a couple of seconds to get the oil pressure up!'' he snapped while dashing around the front to get to the passenger door.\n\n''Yeah, yeah,'' she shouted, adding a flippant hand wave. ''Listen, studly, get in, sit down, shut up, and hang on. I am _so_ up for this!''\n\nThe moment his butt hit the passenger seat, Moira shifted into reverse and floored the accelerator. The tail of the Magnum spun back and to the left, almost flinging Nelson out onto the concrete. She stabbed the brakes, bringing the abbreviated power slide to a door-slamming halt that left the nose of the car pointed up the ramp.\n\n''She calls this driving?'' Addie wailed from the backseatas she struggled off the floorboard in front of Cleo and back onto her seat.\n\n''That was cool!'' chimed in the securely seat-belted Cleo. ''Do a doughnut next!''\n\n''Don't encourage her!'' Nelson snapped as he pulled shells from the bandolier loops and stuffed them into the SPAS. ''Drop the passenger window, Moira!''\n\nThe movement of the Magnum had drawn the attention of the ghouls. Their massed darkness was already flowing down the ramp toward the momentarily stationary car. Nelson could only assume that the second ball of darkness below was flowing upward toward them as well. He disconnected the SPAS from his shoulder rig, jacked a fresh round into the chamber, and shoved the gun through the passenger window opening.\n\n''Go!'' he yelled to Moira, then pulled the trigger.\n\nThe SPAS jumped in his hand and a ballistic path opened in the dark wall ahead. He jerked back into his seat, fully expecting a V-8 roar and a sudden surge of acceleration.\n\nInstead, he heard Moira saying, ''Hi, Timothy! What's up?''\n\nHis head ratcheted left, inexorably drawn to stare at Moira, slack-jawed, as she finished fitting her wireless earset, her BlackBerry carefully slotted in the center console.\n\n''Well, I told you that the prices would do that. Why can't you ever just do what I say the first time?'' She looked at Nelson with a raised eyebrow, _What?_ expression.\n\nHe bunched his shoulders and gestured out the front window, his brows pulled together in his best _Remember this?_ expression. Moira rolled her eyes and dropped her hand to the gearshift, pulling the lever into drive with a click.\n\n_Business,_ she mouthed, her eyes flashing dangerously.\n\nNelson drew the index and middle finger of his right hand across his throat then pointed back out the windshield where the ghouls were closing the gap in their ranks.\n\n_So?_ she mouthed. With that, she floored the accelerator. The engine roared through his open window. Smoke clouded the air to the sides and behind as the Magnum broke completely free and spun up to speed. Tires squealing, the car surged forward up the ramp, Moira smoothly cranking the steering wheel left to offset the drift of the tail.\n\n''Look, Timothy,'' she shouted over the mechanical cacophony, driving the Magnum practically sideways up the ramp toward the swirling mass of ghouls, ''it's a simple option-out deal. If you can't handle a little variation in the details then I'll find someone who can.''\n\n''Moira!'' shrieked Addie as the side of the car began to blow through the diaphanous substance of the ghouls. Vague faces with sharp teeth and disembodied clawed hands slammed into the windows then splashed into smoking goo. The hissing that had previously characterized their gathering foes changed to a howling wail of confusion and horror.\n\n''Gross!'' Despite her disgusted pronouncement, Cleo was watching the ghoul-smashing show through the window with obvious glee.\n\n''We left the Beyond to get away from creatures like this!'' Addie snapped back at her while clawing her way across the seat toward her little sister.\n\nMoira feathered the throttle just a tad, allowing the tires to hook back up with the concrete. The Magnum straightened out and surged forward, bouncing Addie off the seat back then onto the floorboard with an outraged squeak.\n\nFor a moment, the ghoul-splatting continued on the windshield. Then a white face twisted with anger passed before them and Moira was spinning the steering wheel right to avoid the onrushing wall of the parking garage.\n\n''Where to?'' she shouted. ''No, not you, Timothy!''\n\nNelson jerked his thumb up. ''Head for the roof!''\n\n''Was that her?''\n\n''I think so, yeah. We need some room, so get us topside.''\n\n''Her who?'' Addie asked from between the front seats.\n\n''Best guess is she used to be Carmen Voistra,'' Nelson told her as he slipped another shell out of the bandolier. ''Bit of a local legend.''\n\nAnother corner loomed and Moira put the Magnum into an abbreviated power slide. As she cranked the steering wheel, a clunking noise came from the roof above Addie and Cleo.\n\n''Gunner!'' Moira howled. ''My shoes!''\n\nThe tailgate lifted even before Moira had finished. A dark shape swung out from under the cargo cover to land on the roof with a thump as Moira straightened the Magnum out for its charge up the final ramp to the rooftop lot. Then the process reversed itself as the shape returned and slammed the tailgate shut again.\n\n''Did you get them?''\n\n''Yeah, yeah,'' came Gunner's rumbling reply. ''Try to keep your head screwed on next time, toots.''\n\nThe shoes appeared between the two front seats next to Nelson's head, dangling by the heel straps from a jet-black clawed hand.\n\n''Yo, N-man,'' Gunner chuckled, ''your custom silver paint job's hosed . . . again.''\n\n''Gee. Great news. Thanks,'' Nelson said as he turned to take the shoes. At that moment, he caught sight of Addie's horrified, blood-drained expression. She took in a deep breath and he knew without a doubt what was coming next. ''Oh, crap.''\n\n''Imp!'' Addie screamed at the top of her lungs, thus kicking off a World Fae Federation cage match in Nelson's backseat.\n\nIf Gunner had just tossed the shoes up front, he probably could have gotten away clean with only minor hearing loss. However, in his enthusiasm to tweak Nelson about the damage to the Magnum's finish, he had inadvertently exposed his stocking-capped, knobbly head out from under the cargo tonneau. A born scrapper, Addie took advantage of not being restrained by a seat belt to curl up on her back and start kicking the living bejeezus out of the imp's head and shoulder with her black patent leather pumps. Not to be outdone in either panic or reaction time by her elder sister, Cleo let out a battle scream of terror that ululated in time with the overhead blows of her clenched hands smacking the imp in the face.\n\n''Ahh!'' the two demi-goddesses shrieked in unison, over and over again, at ear-splitting volume.\n\n''Help!'' howled Gunner. ''Get 'em off of me! Get off! Oof!''\n\n''No magic in the car! Ixnay on the agicmay!'' Nelson yelled when he caught a flicker of purple coming from Addie's fingertips.\n\n''No, Timothy, we're not having an orgy!'' Moira snarled, ''And you most certainly may _not_ bring your boyfriend to join in!''\n\nAll things considered, Nelson could have just let the three fae in the backseat duke it out until the car had at least stopped. Even as he twisted around to restrain Cleo's fists of doom, he had a vague suspicion that he should stay out of reach and let things come to a natural balance. The moment he got hold of Cleo's wrist on the upswing, though, the entire situation spiraled out of his control.\n\nWith the bludgeoning to his head at least temporarily restrained, Gunner took advantage of the change in action to use his free hand to grab one of Addie's ankles. The elder sister screeched in outrage and twisted to throw her arms around the headrest of the driver's seat . . . and Moira's throat.\n\n''Get it off! Don't let it eat me!''\n\n_Gurgle!_ Moira would have said more, but air is generally required for conversation and Addie had just cornered the market on her younger sister's personal supply. Nelson heard the engine roar and knew his girlfriend had stiffened up against the accelerator while trying to escape Addie's choke hold.\n\n''Lady, you're too damned stringy to eat! Just quit kicking me!''\n\n''Addie! Let go!'' Nelson shouted as he tried to pull her arms away from Moira's throat. ''She's driving!''\n\nNelson believed that in every knock-down, drag-out, fur-ball fight there came a point where a single moment of clarity could change things for the better or the worse . . . or maybe just point out how incredibly stupid all the participants really were for forgetting certain key facts. That instant of epiphany for this particular scrap came when the Magnum launched full throttle off the top of the roof ramp and, for a fleeting second, everyone involved felt the unmistakable stomach lurch of ballistic weightlessness, thus simultaneously remembering that they were on the top floor of a five-story parking building.\n\nFour separate voices and one choked gasp combined into a single howling chorus of panic.\n\nThe screaming stopped when the Magnum slammed back down on the rooftop parking lot. The front left tire hit first, slewing the vehicle sideways into a spin as the other bits of its suspension made contact. During their brief moment of flight, Moira had somehow managed to get her foot off the gas and onto the brake, so the sound of squealing rubber and the smells of hot iron and friction pads suffused everything inside the car. Nelson, Gunner, and Addie bounced around the interior like Ping-Pong balls in a washing machine as the dynamics of the spins had their way with them. Features of the roof lot flashed by, repeating themselves once, twice, three times before the big Dodge rocked to a final groaning halt mere inches from the far retaining wall. The rumble of the V-8 coughed to a stop with the click of a key.\n\nNelson counted five heartbeats of silence before the mutters and moaning began. At that point, he was willing to concede that they had, in fact, cheated death. Since he had somehow ended up facedown on the front passenger seat floorboard, he used his right hand to open the door before rolling out onto the pavement in a relieved heap.\n\n''Timothy,'' he heard Moira wheeze from inside the car, ''now is not a good time. Just deal with it.'' There was a soft beep and then the earset sailed out Nelson's door and clattered across the concrete. He pulled himself up on the rocker panel and gave a disheveled Moira an ironic grin.\n\n''Rough day at the office?'' Her answering glare only made her that much cuter in his eyes.\n\n''Oh, shut up.''\n\nChuckling, Nelson pushed himself off the ground as the door behind him swung open. Cleo staggered out, took two steps, and then collapsed onto her hands and knees.\n\n''You okay, kiddo?'' When she looked up at him, Nelson got the distinct impression that the green of her complexion and the purple of her dye job clashed.\n\n''If I wasn't me,'' she told him miserably, ''I'd puke.''\n\n''Hang in there, Cleo. At least the ride is over.'' He turned to pound on the roof of the car. ''Yo, Gunner! You still with us or did you take an express back to hell?''\n\n''I'm good, Nel,'' came an entirely too cheerful gravelly chuckle. ''Real good, in fact. Soft landing you might say.''\n\n_That doesn't bode well,_ said a little voice in the back of Nelson's mind right before a low feminine moan transformed into a squeal of outrage.\n\n''Get your face off my chest, you pervert!''\n\n''Oh, honey,'' Gunner said in what, for him at least, passed for a seductive voice, ''for an old broad, you've got one hell of a rack. I think I'm in love.''\n\n''Are you moving your hips?'' Addie shrieked. ''Get off right now!''\n\n''Gunner,'' sighed Nelson, ''quit scre . . . messing around. Get your polished black ass in the back and get me a claymore.''\n\n''Killjoy,'' muttered the imp. Still, Nelson could hear him moving to get in the back, followed by a relieved female sigh. Seconds later, Addie's door popped open as the elder sister escaped into the late afternoon sun. She stood up, straightened her skirt, brushed a stray strand of hair out of her eyes, and then shuddered violently. With that, a bit of her composure returned and she was able to turn and glare at Nelson.\n\n''I. Need. A. Bath.''\n\nHe pulled a stick of gum from his inner coat pocket, peeled the paper off, and popped it in his mouth.\n\n''Join the club. You can have it after Moira and I are done.'' He banged on the roof again. ''Hurry it up, Gunner. You're burning daylight!''\n\nThe tailgate popped open and a familiar black arm deposited an obviously homemade square box by the side of the car. ''Yeah, yeah. There it is. Like letting me have a bit of fun would have mattered.''\n\nNelson closed Cleo's door and walked back to pick up the box. ''You never know,'' he said calmly toward the still open tailgate. ''Not too much longer until the shadows of these buildings get long enough. I think we're about to be hip deep in soul eaters. Hey, don't those guys report back to your old boss?''\n\nThe tailgate slammed shut and Nelson couldn't help but chuckle. For a political refugee from hell, Gunner had all the self-preservation instinct of a lemming unless you pointed things out to him. Gigging the imp was yet another of the little joys that made his day complete.\n\n''Nel?'' came Cleo's tremulous voice. He looked up and saw her standing several feet away with a none-too-sure expression on her face.\n\n''Yeah?''\n\n''Something doesn't feel right about this.''\n\n''She's right,'' Addie chimed in from the front of the car. ''This isn't some random tear in the fabric between worlds.''\n\nNelson sighed as he hooked up the ''clacker'' switch to the wires leading from his homemade anti-fae claymore mine. ''I know.''\n\n''And I know, too,'' Moira chimed in from the driver's side of the car. ''That's why you asked me to come with you today, isn't it?''\n\nHe grabbed hold of the handle on top of the box and stood up, turning to look at her. The concern in her eyes touched him to his very core. Despite the fact they bickered and shot sarcastic little jabs at each other every day, he knew that she loved him every bit as much as he loved her.\n\n''Why do we fight so much?'' he asked her.\n\n''Balance,'' she answered without hesitation. ''Otherwise we'd form a singularity and black holes are bad for tourism.''\n\n''Touching little love scenes aside,'' Addie broke in, ''what's causing this?''\n\nNelson shook his head to clear his thoughts and began walking back toward the ramp. ''I think we've got a binding going on.''\n\n''Eh?'' Cleo asked in surprise. ''You mean some human cut a deal?''\n\n''Looks like it,'' Nelson said as he started counting paces. When he reached twenty-five, he set the box down on the concrete, turning the side with the words THIS SIDE TOWARD SPOOKS in the direction of the ramp. He then began walking back toward the car, spooling wire off the back of the box as he went.\n\n''This Carmen Voistra wasn't a very nice person,'' he told Cleo and Addie as he walked back. ''In fact, she was pretty damned near the devil incarnate.''\n\n''Hey!'' came a muffled protest from the interior of the Magnum.\n\n''Relatively speaking. No offense, Gunner.''\n\nA haughty sniff echoed out of the car. ''None taken.''\n\nNelson chuckled as he opened his door. He looked up at Addie. ''You'll want to be either inside the car or behind a door. Your choice.'' He turned his head to look at Cleo. ''You too, squirt.\n\n''Anyway, during the course of a rather colorful criminal career, she managed to acquire quite a collection of enemies, most of whom shared her more . . . peculiar tastes in violence and revenge. They caught up with her in an alley right next to this parking garage.''\n\nHe sat down on the edge of the passenger seat and looked up at the buildings surrounding them as the shadows lengthened in the waning daylight. ''The cops found twelve dead men and women clustered around a systematically dismembered female corpse that had been raped and abused multiple times before it was carved up. The twelve surrounding bodies showed no discernable cause of death.'' He paused to blow a bubble with his gum and pop it.\n\n''I think Carmen reached through the veil while she was being tortured and made a deal with a dark fae boss of some kind. She became a gate in exchange for revenge against her enemies.''\n\n''That's not good, Nel,'' Cleo said. ''She'll be tied to his power.''\n\n''Yup. Plus she's using that ball of ghouls like a shield. We're going to have to strip them away before we can get at her.''\n\nThe shadows that had been creeping across the roof moved closer and closer to the ramp entrance line like the shadowed arms of night reaching to open a door. A familiar hissing sound started to grow from the darkness of the ramp.\n\n''Showtime,'' Nelson said as he stood up. He looked back across at Moira. ''You ready honey?'' She winked and blew him a kiss.\n\n''Piece of cake.''\n\nThe shadows closed on the entrance at last and the black cloud that defined Voistra's shield of spirits flowed up and out of the ramp. Nelson watched the white dashed lines disappear beneath the boiling mass faster and faster as they approached the claymore. Finally, he dropped behind the door and lifted the clacker.\n\n''Fire in the hole!'' he shouted and smacked the switch three times out of habit. The mine detonated with a muffled thump and thousands of fragments of rock salt mixed with silver and iron shot into the black cloud. The concussion rocked the car and set his ears ringing but, by the time he got up, Moira was already on the move.\n\nA wind crossed the roof and drove the lingering smoke sideways along the parking lot as Moira dashed forward. She reached behind her back with her right hand and shifted through the veil between the worlds to where she hid the sheathed scissors of Fate.\n\nShe could see Voistra clearly now. Nelson's bomb had cleared away the swirling cloud of ghouls the bitch had called to defend herself. Beyond that, Moira could see the twined silver threads that bound both Voistra and her unknown fae sponsor in their deadly pact.\n\n''Gotcha!'' she cried as she pulled the scissors from the Beyond into the Now of Nelson's world. With her left hand she twisted and bound the threads in her fingers and pulled them tight. The ethereal fibers hummed and strained against her grasp as she swung the bronze shears into play. The look of surprise and horror that replaced the rage on Voistra's face brought a grim laugh to her lips.\n\n''My sisters and I may not hold sway in the world of men anymore but you're not from around here now, are you?''\n\nWith that she snipped both threads with a single cut.\n\nA pair of agonized shrieks rattled across the rooftop, one from a disembodied human soul, the other from a far more distant fae throat. Neither, however, would play any part in either world ever again.\n\n''Give Uncle Hades my regards, you pissants!'' she called after them.\n\nAfter favoring the now-clear ramp with one final desultory glance, she turned around to saunter back to the car, the bulky archaic scissors bouncing jauntily on her right shoulder. Nelson grinned at her as she neared the car and Moira could not help but return his smile.\n\n''You're really something, you know?'' he called to her as she approached.\n\nShe walked up to his door and kissed him hard on the lips, holding his head with her free hand for good measure. After much tongue wrestling, they broke apart with a gasp.\n\n''I have to be to keep up with you.'' She nipped his nose lightly then chewed her stolen gum twice before blowing a bubble.\n\n''Gross!'' Cleo exclaimed from behind her own car door. ''You two can be so _childish_!'' Moira waggled her eyebrows at her younger sister.\n\n''If you don't like it then don't watch.'' Leaving Cleo spluttering in indignation, she turned back to Nelson with a grin. ''Bath, then dancing?''\n\nHe jerked his head over his shoulder to indicate her flustered little sister and the just-emerging Addie. ''What about these two?''\n\n''Let them get their own dates.''\n\n''I'm available,'' came a gravelly voice from inside the car, ''and I love a good m\u00e9nage \u00e0 trois.''\n\nIgnoring the twin shrieks of outrage that rose to meet Gunner's suggestion, Nelson smiled and reached out to pull Moira into another kiss. When she leaned forward to accept it, she felt a pinch between her breasts when the car door pressed against the jeweled ampoule that dangled from a necklace beneath her shirt.\n\n_Ah yes,_ she thought as she opened her lips to Nelson's, dear Aphrodite's tears. _I think I'll only use two drops tonight. Three_ was _a little much last time._\nA RAT'S TALE\n\n_Barbara Nickless_\n\nWalking the path to the shifting field on the sur-face of Denar is like stuffing yourself inside a nail-lined barrel and then rolling around in it. The wind will peel off any unprotected skin and turn your lungs inside out.\n\nEven in pressure suits it's not pleasant.\n\nI followed Darro and Fahler down the tube to the changing room. Johannes, the new guy, trailed after me. He'd shown up at the rec room an hour ago, the size of an elephant with the voice of a mouse.\n\n''I'm your new teammate,'' he'd announced, ''here to help protect the future for everyone.''\n\nHe said it like he meant it.\n\nWe all laughed.\n\n''Guarding the Past to Save the Future,'' the recruitment brochure said. We called it Sooner hunting. As in, the tourists would sooner die than let us drag them back to Earth.\n\nThe pressure doors dialed shut behind us as we stepped onto the surface. Around us rose the gray-black rock of Denar, tinged crimson as the planet rolled toward its sun. Our shadows streaked across the landscape.\n\nJohannes hesitated, letting the wind buffet him, looking for all the worlds like he was a tourist admiring the scenery. This guy was going to be trouble. Why do they send me the optimists and the losers? I tapped my radio.\n\n''Jo\u2014let's go.''\n\nHis gaze rested unsteadily on my face. ''Sure, Nathalie.''\n\n''It's Gratz,'' I snapped.\n\nWe huddled together on the shifting field and took out our black boxes. We needed only one box to shift. But in case something freaky happened, like one of us being sucked into the wrong wormhole, we all carried our own. Denar sits at a space-time nexus, where shifting is efficient and precise. But the nexus is a maelstrom more warped than I am. We knew Denar's coordinates better than we knew our mother's faces.\n\nThose of us who had mothers.\n\nMy shifter box fed our destination's four-dimensional coordinates to the other boxes.\n\n''Jo, you done this before?'' I asked.\n\nThe giant of a man shook his head.\n\n''You'll probably puke,'' Darro said. ''And when you land, your eyeballs might be growing out of your toes. Don't let it bother you.''\n\n''Shut up, Darro. Jo, hold on to Fahler and don't let go of her. Fahler, make sure he's got the right coordinates on screen.'' I lifted my shifter. ''We're going to the lovely vacation world of Rocaste, for those of you who care.''\n\nI opened my shifter and was sucked into a nightmare that lasted forever.\n\nAnd for no time at all.\n\nThat's the nature of plummeting into a time-well. The shifter grabs the time-well, then shoves you along the time axis and simultaneously compresses space. It does weird things to your body while it's at it. Sometimes you really do come out with all your toes and fingers in the wrong places.\n\nSometimes you come out dead.\n\nWe thumped down.\n\nJohannes didn't throw up, from what I could see inside his helmet.\n\nOur Sooners, a couple named George and Carrie Nace, sat twenty meters in front of us, dressed in coats and hiking boots, warming their hands over a camp-fire. Tall pines soared around us. Yellow-white morning sunlight filtered through gray-green needles. I couldn't smell or feel the wind, but I knew just what it would be like. I'd sampled the sims of long-ago Earth, so like the Rocaste of this time-well. A Rocaste that didn't exist in our space-time. For only a minute, I imagined living here for the length of my natural life and understood the temptation the Sooners faced.\n\nGeorge and Carrie stood.\n\n''Damn,'' Carrie whispered. Still young, still beautiful. Rocaste was world enough for someone like her, if only she'd had the time. There was never enough time.\n\nGeorge kicked at the fire. ''We had two weeks.'' In his wire-framed glasses and brown sweater he looked like a college professor.\n\nTransmitters, courtesy of World Time-Well Association and hidden from Carrie and George in their implanted shifter-boxes, carried their voices into our skulls.\n\nNone of the tourists ever caught on that we monitored their timelines in case they decided to become Sooners and shift illegally. Every time a tourist tried to run, he risked emerging from the well before he went in. Maybe before he was born. No one had ever proved that the laws of physics provided chronology protection. WTA had people calculating time lines and tracking the shifters. Whenever there was a ripple, the trackers sent us in to keep it from happening.\n\nWe knew whenever the tourists went.\n\n''It wasn't enough time,'' Carrie said.\n\n''It was all we could afford.'' George must have been wondering why we had shown up. He hadn't done anything wrong. Not yet.\n\n''I'm not going back to Earth.'' Carrie's voice broke on a sob. ''Nothing is worth _that_!''\n\n''But they'd find us,'' George said. ''When could we go where they couldn't find us?''\n\nWe were advancing without getting too close, proffering bottles of water and smiling. Poor George didn't have a chance. Carrie had already convinced him to flee a million times over, across every permutation of the time line. The trackers had spotted it and sent us back three minutes before the Naces jumped.\n\n''Darro, do your thing,'' I said. We were still ten meters away.\n\nDarro drew his gun. ''Stop right there, folks.'' His voice was flat, bored. ''Come with us.''\n\nFahler and I pulled our weapons. Fahler smiled inside her helmet.\n\nCarrie shrieked and reached behind her ear for the shifter controls.\n\nDarro blew her arm off just above the elbow.\n\nGeorge lunged for Carrie. Maybe to protect her. Maybe to keep her with him. With her other hand, Carrie clawed for the shifter.\n\nFahler shot them both.\n\nGood-bye, George. Good-bye, Carrie. Their time line lay flat.\n\nThere was a noise behind me. I looked around.\n\nJohannes was throwing up in his helmet.\n\nHours later, in the heart of the guards' barracks on Denar, there came a tap on my door. I grunted.\n\n''It's Johannes. May I come in?''\n\nI rolled over and snapped off the snuff-porn. ''Sure.'' Johannes ducked as he entered the room. I offered him a chair and a glass of vodka. He took the seat but shook his shaggy head at the vodka. I poured myself a drink and studied his face.\n\n''Fahler says you wouldn't go with her. You should give her a chance. I've heard she knows some moves\u2014''\n\n''Nobody told me I might have to kill people on this job, Gratz.''\n\nI resettled on my bunk. I could feel the hum of the generators through the mattress.\n\n''It's not exactly a well-advertised point. 'Join the Time Guards and Kill Sooners.' Wouldn't go over well back on Earth.''\n\n''I think I can bring them back.''\n\n''Huh?''\n\n''I'm good with people. I could talk them out of running.''\n\n''Yeah? Well.'' I drank my vodka. After five years it no longer burned going down. ''The good folks of WTA don't care about that.''\n\n''WTA has a tough guy image. But that's probably not what they really feel.''\n\n''What planet are you from, Johannes? Nobody gives a damn about Sooners.''\n\n''Just let me do the talking before you shoot.''\n\nI stood to get some ice for my drink.\n\nI believed in the fundamental stupidity of people. I'd heard every breathless cry about the nobility of man. But mostly I'd noticed that if you feed them, give them something to screw, and stick a VR box in front of them, they cluck like chickens in a henhouse. The Sooners were smart enough to try to get out of the coop, and for that I could almost admire them. But they weren't smart enough to escape a rat like me.\n\nStill, Darro had been my point man for three years, and he hadn't convinced a single Sooner not to run. What could it hurt to give Johannes a chance?\n\n''Okay. You can talk. But we know precisely when they're going to shift. You either convince them before then or we blow them away. That's the job. Any longer than that, and we mess with the future.''\n\n''It's not enough time. We could shift in a minute or two earlier.''\n\nI frowned. ''Nobody likes blowing people up. But it's my job to make sure those people don't alter the time line. They'll drop me in vacuum if I fail. You know we're restricted to the time slot we're given.''\n\nI'd talked to the time-physicists, and I knew we had a little more time to play with. But I was curious to see how much Johannes knew.\n\n''I've read the journals,'' he said. ''We have an additional three minutes before the time disturbance is large enough that we can't resynch the Sooners to flat time. And that's just with the shallower wells. We usually have more time than that.''\n\nJohannes leaned forward. The cheap plastic table creaked under his weight. His eyes were a startlingly light shade of gray, somber and hopeful. Given the way things happened around here, in a few months only Johannes's sorrow would still be showing.\n\nBut that hope had me hooked.\n\n''Go on,'' I said.\n\n''Let's go in a little early, and if I don't talk them out of it, you can do what you have to do, right when their time is up. I'm not asking you to risk anything. Just give me a chance.''\n\n''Aw, hell. Okay. Three tries.'' I raised my glass. ''Congratulations. You're my new point man.''\n\nMy reward was a billion-watt smile.\n\n''Get out of here.''\n\nThe next call was a family of five.\n\nI hate it when kids are involved.\n\n''Hello!'' Johannes called in his high voice, yelling across twenty meters of emerald sand. A sapphire ocean hissed on our right. ''We're here to talk you out of running.''\n\nSo now the Sooners knew we spied on them. Hell.\n\n''I don't know what you're talking about,'' the father said in the cautious tone of the would-be innocent. ''We're just getting ready to return to Alpha Base.''\n\nJohannes shook his head and, against all regulations, removed his helmet. His blond hair fell past his shoulders. In the silvery sunlight his suit reflected diamond points on the sand. He walked forward.\n\n''It's not so bad on Earth, is it? You've got your job. A nice apartment. Your kids should have a chance to grow up. Is this where you really want it to end, Mr. Wakeman?''\n\n''Jerry?'' The wife hugged the youngest of the children. A baby, really.\n\n''It's okay, Sue,'' Jerry said. ''Mister, we're shifting back to base. Just like we're supposed to.''\n\n''You've got two minutes, Jo,'' I said.\n\nHe took a few more steps until he was only seven meters from the family. I had to fight the urge to drag him back. If one of the family shifted, they'd take pieces of Johannes with them.\n\nFahler snarled a curse.\n\n''Think of your kids, Mr. Wakeman,'' Johannes said. ''You can't get away with this. We'll find you. We always do. We have eternity to look for you. I know Earth isn't the best, man. I've been there. But you want your kids to grow up, don't you?''\n\n''Jerry.'' Mrs. Wakeman started to cry.\n\n''I\u2014'' The father was at a loss. He studied Johannes, and I imagined him seeing that same look of sorrow and hope in those gray eyes. ''I\u2014I don't know what you mean.''\n\n''Trust me,'' Johannes said. ''You can trust me.''\n\n''Sixty seconds,'' I hissed.\n\n''Listen to me. I'm one of you. You can't make it. These worlds, these vacation planets, they aren't for the likes of us. We're of Earth. Earth is our home, our fate. We don't belong anywhere else.'' His voice was both lullaby and command. His words vibrated at some sweet level, and I think he could have talked the angels down from heaven. ''Take care of your family, Mr. Wakeman. Return home.''\n\nThe father made a decision. I could see it in the way his shoulders fell and his face lost all hope.\n\nI heard the voice of one of the trackers on Denar in my ear. ''They're shifting back to Alpha,'' she said before it happened.\n\nJohannes backed out of range just in time.\n\nThat night we celebrated. Word had gotten out that we'd saved a family, and we were heroes.\n\n''You did great,'' I said to Johannes when he joined me at the bar. I slapped him on the back. It was like hitting an oak tree. ''Why don't you have a drink? You deserve one.''\n\n''I already feel good.'' He smiled at me and the bartender and the mirror of people swirling by.\n\n''Jesus, you got a nice smile.''\n\nHe smiled again.\n\nI pulled back a little. I didn't want to be a member of the Johannes fan club. Friendships aren't healthy here. You can't afford to trip up and worry about someone else. Besides, I knew he couldn't be perfect. He was a guard. We all had that special _something_ , that edge of darkness, to earn us this kind of crap job.\n\n''What the hell got you here, Jo? Darro's here because he's got the empathy of a cockroach. Fahler killed her own parents. You don't strike me as guard material.''\n\nThe lights twinkled off the bottles behind the bar. A sail of laughter drifted past.\n\n''We all have our secrets, Gratz.''\n\nI tossed down my vodka and waved my hand for another.\n\n''You're a spy, right?'' I said after the bartender had deposited my drink on the counter and moved away.\n\n''You could call me a revolutionary.''\n\nI glanced around. The room echoed with voices. The bartender was at the other end, watching a drone wash the glasses. No one even looked our way. ''Very funny. But don't say that around here. Some people might not know you're joking.''\n\nHe lifted his big frame from the stool. ''Righto, Gratz. I was just fooling.''\n\nI watched him leave. The guards cheered and clapped him on the back as he passed.\n\nI drank an extra vodka for him.\n\nAfter three more no-kill runs, other guards started coming to Johannes for advice. A month after his arrival, he organized classes in the rec room on Wednesday nights. The number of deaths fell. We grew friendlier with each other. A few relationships became permanent. The owner of the pharmacy complained that his drug sales were down.\n\nDon't ask me to explain it. I didn't think these guards could love anyone. But they loved Johannes. Fahler might have been one of the few who didn't fall for him. Fahler's self-esteem came from the notches on her bedpost, and\u2014near as I could tell\u2014Johannes was the only straight male guard who didn't sleep with her. But if she resented him, she never let on.\n\nI saw a lot of Johannes because, crazy son of a bitch, he liked me.\n\n''What makes you so angry?'' he asked me one day.\n\nWe were in my room, drinking. Vodka for me. Water for Johannes.\n\nI spun my glass on the table. ''I grew up in an Earth slum before WTA took me away at the tender age of thirteen. All us slum kids are angry and paranoid. We can't help it. When the recruiters came through, Jesus, we used to watch the sims of vacation worlds and dream about getting out.''\n\n''Is that why you're here? You wanted to see the worlds?''\n\nI barked a laugh. ''Get real, Jo. Everyone is here cause the other choice was a life doped up behind bars. It's not like we're _enjoying_ those worlds.''\n\nHe closed his eyes and opened them again. In the dim light, his gray eyes looked almost black. ''What did you do?''\n\nI thought about it awhile. You didn't have to commit a crime to become guard material. You just had to have a Psycho-An that showed you were capable of violence. I could have lied to Johannes. I was surprised when I didn't.\n\n''I was a whore. One of my johns tried to slit my throat. I turned the tables.'' I'd never confessed to anyone. I jammed my hands on my hips and batted my eyes. ''It just played havoc with my Psycho-An.''\n\n''Did you get help?''\n\n''For the murder or the whoring?''\n\n''Either.''\n\n''Sure. Trained psychiatrists came down once a week to make sure all the child prostitutes were happily adjusted.''\n\n''I grew up on a farm. A converted asteroid.'' There was wistfulness in his voice.\n\n''Gee, I never would have guessed. You probably harvested the fields by yourself and carried injured cows back to the barn where you healed them by laying your hands on them.''\n\n''You're joking,'' he said with that smile. ''But you're right.''\n\nI don't know how much longer our no-kill record might have stood if Fahler hadn't talked. Maybe until we were all able to retire to the far side of Denar, rich and a bit less guilty than we would have been if Johannes hadn't come into our lives. My team hadn't killed anyone since I made Johannes the point man.\n\nBut one night at dinner, Fahler said, ''They're killing them, you know.''\n\nDarro looked up. ''What are you talking about?''\n\nWe were off duty, but Darro wasn't using many stupors these days. His words weren't even slurred.\n\n''The Sooners. WTA kills them after they go back to base.''\n\nFahler watched Johannes from the corner of her eye. The giant had frozen with a bite of food halfway to his mouth. His face turned ashen.\n\n''Shut the hell up, Fahler,'' I said. ''You don't know what you're talking about.''\n\n''Why would they kill them?'' Darro asked.\n\nJohannes dropped his fork. It hit the edge of the table and ricocheted to the floor.\n\nI slammed the table with both palms. ''Shut up, both of you. They're not killing anyone.''\n\n''Anyone who tries to run is obviously trouble, part of the criminal element,'' Fahler went on. ''WTA can't take the chance they might try to run again, wind up in the past where they could wipe out the Association. So they gas 'em. Really, gassing is worse than if they let us shoot them. At least that's fast.'' She folded her hands on the table and arched a brow at Johannes. ''They do the kids first.''\n\nI slapped her, hard, across the face.\n\nShe fell back in her seat, a trickle of blood running from her nose.\n\nShe licked at the blood with a pink tongue. ''The truth hurts, Gratz.''\n\nJohannes dragged me back to my room and made me swallow some stupor pills, washed down with vodka. Then he took off my boots and put me in bed. I think he kissed me on the forehead, like my mother used to do. I felt better. Like one of his cattle, I supposed.\n\nI woke in darkness.\n\nSleep had burned the fuzzy edges in my brain. I thought about Sooners and Johannes, and finally I sat up and went over to the computer.\n\nA farm boy who healed cows didn't wind up on Denar. It was time to learn the truth. I was taking a risk if they caught me, but I could always say that as Johannes' team led, I felt justified in checking his Psycho-An. Never mind how I got the passwords.\n\nWTA, operating as the Democracy of Aligned Worlds, kept psychoanalysis sheets on everyone, starting at birth. Psycho-Ans we called them. The tests were subtle, and most people weren't even aware of them. WTA monitored you until you were thirteen, then they encouraged you to move where they wanted you. If you were smart, you took their advice.\n\nWell, maybe I was drunker than I thought, and I punched the wrong combinations. Or maybe\u2014and I think this is right\u2014Johannes wasn't all he claimed to be. There was no Psycho-An on Johannes. There was no DOB certificate. There wasn't even a goddamn driver's license.\n\nI hit more keys. I found Fahler and Darro. I found my mother and my dead brother. And I found my own Psycho-An and spent a couple of minutes staring at my picture, wondering if I was really that ugly.\n\nBut, according to the computer, Johannes didn't exist.\n\nThis didn't even fit into the realm of the theoretically possible. WTA did a psychoanalysis on _everyone_. Who the hell was he?\n\nI spent half an hour chewing on all the possibilities before coming up blank and stumbling back to bed.\n\nI dreamed that Johannes carried me to a place far from Denar and time-wells and guards. I struggled in his arms, because I'd always walked on my own. But I was glad\u2014glad that I was leaving. As a bright yellow sun set behind Johannes' massive shoulder, I wept into his shirt.\n\nHe shook me awake. The vodka and drugs had worn off, and I felt like crap. I was too hungover to get up.\n\n''Screw this headache,'' I said when my eyes focused.\n\nHe sat on the chair. ''It's true, Nath.''\n\n''What's true?'' I struggled to sit up. Lights ricocheted off the insides of my eyelids.\n\n''I checked with a couple of sources. They kill the people as soon as they return to base. Fahler was right.''\n\n''You don't know crap.''\n\nI got up and poured a drink.\n\nJohannes put his face in his hands, and after a while I realized he was crying.\n\n''Hey,'' I said, as gently as I could through the hangover. ''Maybe it's for the best. If they ran, they could really mess things up.''\n\nHe looked at me. ''I'm going, Gratz.''\n\nI got a chill. ''What?''\n\n''I'm going to run. I can't do this anymore. I'd rather die.''\n\n''You _will_ die, if you run. You think you're the first guard to get this idea? WTA's tracked them all down, and we've killed them.''\n\n''I thought I was here to save the Sooners,'' he went on as if he hadn't heard me. ''But now I know it's something else, something bigger. It's time for me to run. I'm here to prove it can be done.''\n\n''It _can't_ be done.''\n\n''They won't know where I've gone, or when. They don't have a transmitter on me.''\n\n''Why the hell not?''\n\n''Friends in the right places, Gratz.''\n\n''That's why you're not in the computer.''\n\n''The revolution is bigger than anyone realizes. WTA is so intent on protecting the past, they forgot to look ahead. We've responded to their plans before they know what their plans are.''\n\n''They'll find you. They'll see the ripple when you shift, and they'll hunt you down.'' I paused. ''And what if you do succeed? You'll cause a paradox.''\n\nThere. The worst kind of crime.\n\nJohannes smiled. My heart thumped.\n\n''I _am_ a paradox, Nathalie,'' he said.\n\nI gaped at him. ''You're saying you're from the future?''\n\nHe took my small, cold hand in his large one. ''Do you think it's possible?''\n\nI looked away. ''What do you want me to do?''\n\nWe took the elevator shaft in the hurricane-eye quiet of night. I helped Johannes into his suit. He smelled of sweat and endorphins. He hugged me before he put on his helmet.\n\n''Come with me,'' he said.\n\n''Sorry. I don't want to die.''\n\n''Good-bye, Gratz.'' He started toward the door.\n\n''Johannes?''\n\nHe turned back. He looked supernaturally large in the dim blue light of the suiting room.\n\nI walked forward until his shadow swallowed mine. ''Please don't let me find you.''\n\n''I don't think it can be helped.'' He snapped his helmet. ''When you find me, Gratz, kill me.''\n\n''What?''\n\n''For the future.''\n\nI stayed in the suiting room long after he left, wishing I'd had the guts to go with him. But Johannes was born larger than life. I was born small and hungry, a rat in a maze.\n\nJohannes managed to avoid us. He _was_ smarter than he looked, after all. He took to stealing some of the Sooners before we got to them. We heard he'd organized a base deep in a time-well. He and the revolutionaries had blown up some space stations and Alpha One. David against Goliath.\n\nFahler, Darro and I had a new guy, Winger, and he'd worked out pretty well. We were back to killing all the Sooners who ran. Darro took stupor pills, Winger drank more than I did, and Fahler still shacked up every chance she got.\n\nMe, well, I was still a rat. Devious, sneaky, skulking for scraps. There are some things you can't escape.\n\nEveryone was surprised to see Johannes.\n\nHe'd shifted in to help the Sooner, and he must have miscalculated. I caught a flash of the Sooner's long, dark hair and wide eyes before he stepped in front of her.\n\n''Well, well,'' Fahler said. ''If it ain't our old friend.''\n\nDarro lifted his gun. ''Johannes. What the hell?''\n\n''This the great Johannes?'' Winger grinned.\n\nCuriosity must have gotten to the Sooner. She peeked around Johannes's shoulder.\n\nFahler got her with a single shot through the eye and then aimed at Johannes.\n\nI pulled my gun and pointed it at Fahler. ''Hold your fire.''\n\n''We got orders,'' Darro said.\n\n''To bring him in alive.''\n\n''Screw that,'' Winger said. He chambered a round.\n\nI whirled and fired, a single round through his heart. Fahler aimed at me, but I was faster. I shot her and\u2014 when his shot ricocheted off my helmet\u2014Darro as well.\n\nI forgot about Johannes as I stared at my dead companions.\n\nHe moved up beside me. ''Nath,'' he said, reaching for me.\n\nI ducked away. I was covered with blood.\n\nHe hadn't aged nearly as much as I. Of course not. He'd been hiding in time-wells, living on borrowed time.\n\nBut I loved him for being what he was. For escaping. ''Get out of here,'' I said. ''I'll blame you for their deaths and say I barely got out with my life.''\n\n''They won't believe you.''\n\n''They will.''\n\n''The time lines are written, Nat. But I landed in the only one where you didn't kill me. You were supposed to kill me.''\n\n''And I suppose I was supposed to kill them, too?''\n\nHe looked at the bodies. ''It was self-defense.''\n\n''So what? Get out of here.''\n\nHe grabbed my gun and jumped away from me. For a second, I thought he was going to shoot me. But he touched the barrel to his chin.\n\n''I'm sorry, Nath. We need a martyr. I'm more useful to my people dead.''\n\n''You can't die. They need you. _I_ need you. How can you help anyone if you're dead?''\n\n''Read about Joan of Arc sometime.'' He smiled, but it was sad. ''Get out of the maze, Nath Rat.''\n\nHe blew his head off.\n\nAfter Johannes died, I lied at the inquest and then retired to the far side of Denar. From time to time, guards came to me, asking questions about time-wells, needing advice. The ones who were serious, I helped escape.\n\nTen years after everyone thought I killed Johannes, the revolutionaries toppled WTA. A week after that, they came for me.\n\nThey're going to execute me for the murder of their leader. The mob has demanded it, and their martyr can't be a suicide. My own words framed me for his death and now all of history will curse me. Jo knew this, and though he loved me, he loved them more.\n\nBut even a rat can look over the edge of the maze and see the world beyond, maybe become part of that world. I knew what my fate would be when the rebels won, but I'm okay with dying. I'm even okay with being reviled for the rest of time.\n\nLike Jo, I came to love those he tried to save.\n\nFor a while, I was more than a rat.\nTHE BONES OF MAMMOTH MALONE\n\n_Esther M. Friesner_\n\nThe name's Malone; Mammoth Malone. You got questions? I got answers. You got problems? I got solutions. I keep them in my bones.\n\nIt was a cold day out on the tundra. It's _always_ a cold day out on the tundra. If I had any sense I'd move the hell _off_ the $%@*& tundra, but where else is there, these days? It's _all_ tundra, baby, except where it's ice, and Mammoth Malone doesn't do glaciers. I'd just poured myself a double shot of breakfast when _she_ came into my yurt like she owned the place.\n\nShe looked like trouble. Most broads do, but not until after they take their coats off. Until then, they look like most guys. This one was different: It takes a lot of woman to fill out a blargh-skin coat like that, and believe me, she filled it out like the blargh never did. (I don't know what they used to call blarghs back in the times before the Big Chill hit and hit hard, but these days we call 'em like we see 'em. A big, shaggy, horned, nasty-tempered beast looks you in the eye, snorts so hard your _grandkids_ are covered in nose-goo, and says ''Blargh,'' that's what you call him. Saves time chipping through the ice to reach an encyclopedia.)\n\n''Mr. Malone, I need help,'' she said. No intro, no hi, how are you, how're they hanging (Like icicles, sweetheart!), just slapped her cards down on the table, straight out. I liked that in a broad.\n\nI thumbed back the hood of my parka and said, ''That's my business, sister: helping people. I'm just one big freakin', frostbitten heap of altruism. Keep talking.''\n\n''Would you mind if I sat down?'' she asked.\n\nI glanced around the yurt. It was a mess: piles of storage boxes, blargh hides, a giant rack of reindeer antlers, and over by the firepit a heap of bones from about a dozen different kinds of animals. I could even I.D. some of them\u2014like the reindeer\u2014by their names P.B.C. (Pre-Big Chill). I was proud of that.\n\n''Sister,'' I said, ''if you can find somewhere to sit in this dump that won't freeze your _farfegn\u00fcgen_ to the ground, help yourself.''\n\nShe gave me a smile that went through me like a polar bear's claw. ''Not exactly a gentleman, are you, Mr. Malone?''\n\n''This 'gentleman' thing of which you speak, what's that? Can you eat it? Can you sell it? Can you\u2014''\n\n''Never mind.'' I expected her to drag one of my storage boxes up to the desk, but instead she sashayed over to the big rack of reindeer antlers, tossed a blargh hide over them, and made herself a pretty impressive chair. Pleased with herself, she patted the bony armrests and said, ''Anyone ever tell you that you've got a nice rack, Mr. Malone?''\n\n''You didn't come here just to tell me what I already know, did you, angel?'' I said. ''Because if you did, this is going to be the easiest job I've had in a long, long time.''\n\nShe stopped smiling. ''I'll get to the point then, shall I?'' She peeled off her mittens and reached into the pouch at her waist, pulling out something small and furry and squeaky. She tossed it onto my desk and the two of us watched it scamper around in bewildered circles for a while. Then it seemed to get its bearings. It stopped, looked around, saw the edge of my desk, and tore off at blinding speed. There was no mistaking what it was or what it had in mind, if you like to think that a lemming _has_ a mind to start with.\n\nA lemming. Of all the desks in all the yurts across all the tundra, why did she have to bring that rodent into mine? Little buggers creep me out, for some reason. Never trust anything small enough to run up your pants' leg and rearrange your outlook, that's what I always say. I reached for the bottle. Kumiss isn't the answer to everything, but sometimes it lets you ignore the question.\n\nBefore I could yank the stopper, the lemming had zipped across the desk all the way to the very edge and there . . . he stopped. No legendary leap into the void, no minor splat when he hit the floor, no comical bounce-on-impact, nothing. Not even a glance at the precipice and the rodent equivalent of a _What the_ **** _was I thinking?_ look on his face; he just _stopped_.\n\n''Well,'' I said, taking a long swig straight from the bottle. ''That's different.''\n\n''I'll say it is.'' She took a deep breath. ''Mr. Malone, my name is Randi Vixen, hereditary freelance zoologist. Ever since the Big Chill, my tribe has made it our business to keep an eye on the world's fauna.''\n\n''And don't think we're not grateful,'' I told her. That was true: I still remembered my grandpa telling us kids the stories _his_ grandpa used to tell about how it was when the Big Chill first hit and took hold. The ice sheets cometh, er, came, and came fast. Scientists couldn't explain it. Some blamed fluoride in the water, others the latest crooked elections, but as far as coming up with anything that was truly _helpful,_ no dice. While civilization took a tumble down the cosmic crevasse, people everywhere were waking up to the realization that the salad bar was closed and that their continued survival depended on how many of their fellow species were catchable and edible. (I never did understand what a salad bar was, but every time I asked the old man for an explanation, he forgot where he was in his story and had to start over from the beginning. It was either stop asking questions or have _both_ of us die of old age before the story was over. I made my choice.)\n\nThe freelance zoologist tribe took on the responsibility of policing the rest of us, making sure no one got greedy and hunted a particularly slow\/stupid\/tasty species into extinction (and out of the global pantry). They also looked out for the animals on a more direct level, keeping tabs on their birthrate, migratory patterns, and behavior, any one of which could do a number on continued human well-being.\n\nWhen something weird cropped up in the animal kingdom, the freelance zoologists always took it seriously. They were a proud tribe who guarded their status jealously. They moved secretly among us ordinary Joe Floes, watching from the shadows, troubleshooting, averting crises. Sometimes we didn't even know there'd _been_ a crisis averted; we only heard rumors. That suited them just fine: They didn't like outsiders.\n\nWhich kind of made Miss Vixen's visit all the more intriguing.\n\n''Mr. Malone, I won't lie to you,'' she said. (I knew what that meant: She was going to lie like a wolfskin rug.) ''My tribe doesn't know I'm here. We're never supposed to get help from Outlanders.''\n\n''Then why'd you do it, dollface?'' I asked. ''Why come to see me? Rodents aren't my specialty, unless you count my ex-wife's lawyer. Sure, I can ask the bones, but they're mostly good for coughing up the will of the gods, savvy? You got a question that needs a yes\/no answer or starts with who, what, where, or when, and I can make a shinbone sing pretty. But questions that start with _why_ . . . no can do, sweetheart.''\n\nShe glared at me. ''You're not exactly a sympathetic character, Mr. Malone,'' she said.\n\n''Sympathy doesn't put any blubber in the bank account, princess. Anyway, what's the big deal if this lemming's acting weird? It's not like you should go jump off a cliff just because _he_ won't.''\n\nHer pretty little eyebrows drew together until she was wearing a scowl fit to stop a charging blargh in its tracks. ''I'm sorry if I didn't make myself clear before, Mr. Malone,'' she said, her voice hoarse and strained. ''It's not just _this_ lemming. It's all of them. For some reason, the entire species is becoming more . . . cautious. They still take part in the mass migrations for which they are famed, but when they come to the edge of a precipice, or anything even vaguely resembling a precipice, they stop. All of them.''\n\n''Well, excuse me, sister, but I find that a little tough to swallow,'' I said. ''If you're talking a bunch of lemmings on the move and the little squeakers up front reach the edge of a cliff and stop, the furballs behind them\u2014way, _way_ behind them\u2014can't see the drop-off. They've got no reason to stop, and if they don't, they're gonna shove the ones ahead of them until you get the domino effect. The front lemmings aren't jumping, but they're sure as hell gonna get _pushed_.''\n\nMiss Vixen's scowl softened to a look that as good as said _You poor, dumb, ignorant bastard,_ only in a nice way. ''You'd think that, wouldn't you?'' she said sadly. ''You'd _like_ to think that. _I'd_ like to think that. But it's not true. When I say that the lemmings all stop when they reach the edge of a cliff, that's exactly what I mean. They _all_ stop. It's as if the herd was being guided by a single mind, a single will.''\n\nI rubbed my chin in thought. ''Okay, now _that's_ bizarre, I'll admit it. But I still don't get why you're coming to _me_ about this.''\n\nShe pursed her luscious lips and let out a sigh that calved my glacier. ''Are you at all familiar with the hierarchy of the freelance zoologist tribe?''\n\n''If it's like other tribes, you're probably split up into clans, and the clans are divvied into kinship units, right?''\n\nShe nodded. ''Well, my clan's totem beast is the lemming.'' Then she burst into tears.\n\nI'm a tough guy. In this racket, you've got to be tough. Sometimes you get a client who doesn't like what the bones tell him, so he decides maybe he'll take out his disappointment on _your_ bones. Or try. I can take a spear to the gut, a club to the noggin, and a bowl of bad dolphin au gratin to the belly, but I just can't take a weepy dame.\n\n''Here, sister,'' I said, handing her the lemming. She was so broken up, she dabbed it at her eyes, blew her nose in its fur, and tossed it into the wastebasket without noticing anything out of the ordinary. As for the beast, it took what she dished out without the smallest squeak or squirm until after it hit the wastebasket. Then it climbed to the rim, glowered at us, and made low, chittering sounds that sounded somehow threatening. I'd never looked closely at a lemming before\u2014 like I said, they creep me out\u2014but in the short time I'd shared yurt space with this one I'd noticed that it had worn a blank, mindless expression right up until the instant it hit the wastebasket. Now, in used-handkerchief mode, there was actually a gleam of _awareness_ in its eye; angry awareness.\n\n_Definitely weird,_ I thought. But before I could get what it all meant, Miss Vixen was talking again, and I was distracted from the hostile rodent.\n\n''I'm sorry to be such a weakling, Mr. Malone,'' she began.\n\n''Call me Mammoth,'' I said. ''And you're no weakling; you're just upset. Can't say I blame you. Bad enough for a freelance zoologist when _any_ animal goes nuts on your watch, but your clan's totem beast? _Big_ trouble. Major consequences, and I don't mean just to the ecosphere. This makes it _personal_. Am I wrong?''\n\nShe smiled gratefully. She had a pair of killer dimples to match her pair of killer . . . eyes. ''You're a sympathetic person after all, Mr. . . . Mammoth. And you're quite right, this phenomenon does have personal repercussions for me. Very deep, very real ones. You see, among the hereditary freelance zoology tribe, there is a great prophecy that says\u2014''\n\nOh, murder. Prophecies. A _great_ prophecy yet. If I had a walrus's _oosik_ (hey, look it up!) for every great prophecy that's loused up someone's life, I'd be up to my eyeballs in pinniped dingdongs.\n\n''Hold it right there, sister.'' I raised one hand, silencingthe delectable Miss Vixen. ''Let me take a stab at this 'great prophecy' of yours: If the lemmings start acting wacky, your clan has to set things right by making a huge sacrifice to fix things. Well?''\n\nShe didn't respond. I presumed her silence meant she agreed with me, so I forged on: ''And I bet I can tell you who made that 'great prophecy,' while I'm at it. Someone from the tribe of hereditary _seers_ ; the tribe that the rest of you have to buy off\u2014 I mean, _thank_ for making nice-nice between gods and mortals.''\n\nThis time Miss Vixen did react: She looked horrified. ''How can you say such blasphemous things?''\n\n''Easy, dollface: I _am_ a seer, remember?''\n\n''Of course. That's why I came to you. But the way you're talking, it's as if\u2014''\n\n''\u2014I've divorced myself from the rest of the tribe? Bingo. No hater like a self-hater, as my dear old dead grandpa used to say, right before he walloped my skinny behind for doubting the chief of the bone-readers' clan. Yeah, our clan-chief, good ol' Marmot Eddie, now _there_ was a bone-reader's bone-reader. Man, he could tell you your fate from the cracks in an anchovy's rib! Funny thing, though\u2014it seemed like every time he tossed a femur in the fire, the cracks all came up telling us that it was time to give him the rrrrowr's share of dinner, or the prettiest maiden within sniffing distance to be his bedmate. Like my sister, Lutefiskina. Trouble was, when she said she'd rather die than embrace the fate Marmot Eddie swore was in the bones for her, she wasn't kidding. Hey, someday you really ought to get a load of the view from Lutefiskina's Leap, Miss Vixen. Spectacular.''\n\n''Just because _you_ had some bad experiences with your fellow bone-readers does not mean that all who interact with the gods are corrupt,'' she said stiffly. Her face was pale. Something in my story had touched a nerve. Either that, or she'd shifted her weight unwisely in the makeshift chair and one of the prongs on the reindeer rack under her rump was getting personal.\n\n''Maybe not, but I bet _you'd_ like that to be true,'' I said. ''I've got you figured out, angel. You come to my yurt, you show me your rodent, you start telling me about this 'great prophecy,' but you don't need to tell me that whatever it prophesies is curdling your chai. _That's_ as obvious as a pile of rrrrowr droppings. Yeah, you're upset, unhappy, but you're _here_. Why? Because we both know that one thing in this crazy world that trumps prophecy is a last-minute reprieve from the gods themselves.''\n\nHer eyes weren't just dry by now, they were hard and cold as polar pebbles. ''You're a very astute man,'' she said, her mouth tight and small.\n\n''What I am, angel, is a _seer_. It's our business to know things. Things like how the _real_ money isn't in passing along the will of the gods as it comes to us, hot off the bones, but in making a few little editorial changes along the way.''\n\nShe frowned, puzzled. ''What's 'editorial'?''\n\nI shrugged. ''Obsolete word from P.B.C.; got it from Grandpa. He said it referred to a legendary monster that changed whatever it touched past all hope of recognition. But isn't that what _you're_ after? A bought-off seer to editorify the 'great prophecy' for you? Easy enough: My people work solo. When little Bobby Bone-reader hands down the verdict from on high, folks _have_ to take his word for it that he's telling it like it is. Who's gonna tell 'em that he's crooked and there's no way the bones say what he claims they do?\n\n''But why am I telling _you_ all about shady shinbone shysters, angel? That's why you're here. It's not about the leapless lemmings, it's to buy a bone-reading that'll save your own sweet skin from that 'great prophecy' fallout.''\n\n'' _Mister_ Malone, I don't think I care for your tone.'' Her voice made the temperature inside my yurt drop a good ten degrees. ''You're partly correct: The great prophecy states that if our clan's totem beast should ever behave in a way that goes against all natural law, it is a sign that the Great Winter is upon us and we must\u2014''\n\n''Whoa, whoa, whoa!'' My hands went up in surrender. ''You people have a _great_ prophecy about the _Great_ Winter coming? Baby, has anyone in your clan bothered to stick his nose out the doorflap in the past umpty-ump years? If you're waiting for the Great Winter, what do you folks call the meat-locker we're living in right now? The _Pretty Good_ Winter?''\n\nMy would-be client stood up and looked down her nose at me. ''Is that all you can do, Mr. Malone? Doubt? Question? Blaspheme? Sneer at things you don't understand?''\n\n''No, sweetcakes,'' I replied, pulling open the top drawer of my desk. ''I can do this, too.''\n\nBefore she could say another word, I reached into the open drawer and pulled out a little brass saucer, a cube of charcoal, a flint, and a piece of steel. Two quick strikes and the charcoal was blazing. Smoke filled the yurt, making the adorable-if-snippy Miss Vixen cough. (There were even a few backup barks from the lemming still holding on to the lip of the wastebasket.) By the time she'd wiped her stinging eyes, I'd opened yet another desk drawer and plucked out a tiny bone that I dropped onto the miniature brazier. Sparks flew while I muttered the proper incantations. At the very instant that I fell silent, there was a sound like the crackle of chain lightning; the flames and smoke rising from the bronze saucer froze, then were sucked back down into the ashes of the charcoal square and vanished in a heartbeat.\n\nTaking my second-best pair of chopsticks from the dirty yak-butter tea mug on the desk, I picked out the bone and turned it this way and that, studying it. Miss Vixen watched, fascinated. After a time, I spoke:\n\n''Who's Harvey?'' I asked.\n\nMiss Vixen's eyes rolled back in her head and she collapsed in a heap. I got her onto a pile of skins before she froze to the floor. I rubbed her wrists, patted her face, and held the remains of an ancient sardine sandwich under her nose, trying to revive her. Her eyelids fluttered, and the first words out of her mouth were: ''How\u2014how did you know about Harvey?''\n\n''The bones talk, I listen. It's what I do. And I never talk _for_ them, got it? Not at any price. So if you're still hoping the get me to play along with whatever you've got in mind\u2014''\n\n''No.'' She bowed her head. ''I'm sorry, Mammoth. You were right, I did come here hoping to convince you to return home with me, fake a bone-reading, and tell my people something that would make them drop all thought of the great prophecy and\u2014and what I'll have to do if we want to stave off the Great Winter. I've been wrong, trying to fight my fate. What will be, must be. There's no escape. I should just be a good girl and yield to my destiny.''\n\n''A good girl?'' I repeated. ''Don't you mean a good princess . . . princess?''\n\nHer eyes flew wide open. ''How did you . . .'' Then she glanced at the burned bone, still pinched between my chopsticks. _''Ohhhh!''_ came the dawn.\n\n''Want to know what else this little splinter has to tell me?'' I asked. She nodded. ''That you and my poor sister Lutefiskina have a lot more in common than I ever wanted to know.''\n\nThe holt of the lemming clan was hidden away in a narrow gorge where natural hot springs gave the people a little taste of temperate paradise lost. Even though Randi and I had muckponies to ride, the place was so far removed from any other human settlement that by the time we reached it we were on a first-name, fifty-third mukluk-knocking basis, if you get my drift, and I'm not talking snow. If the rest of the women of her tribe were even _half_ as hot as Randi, the men of _my_ tribe wouldn't need fires to put a few cracks into the ol' bones.\n\nYeah, I said that to her, too, after about our fifth go-round. I even backed it up by giving her my crowd-pleasing boyish grin. She gave it back in the palm of her hand, smacking it smartly across my cheek. But that didn't stop her from coming back for helpings six-through-fifty-three of that famous Malone ohbabyohbaby magic. Let's just say that there's more than one reason they call me _Mammoth,_ sweetheart.\n\nAs we rode into the lemming clan holt, every doorflap was thrown back as Randi's clan emerged to check us out. I didn't know how well she got along with her kin, but even if she was voted Most Likely to Give Everyone Gingivitis, I expected _some_ words of welcome, at least.\n\nWhat she got was silence and surly stares. From the way everyone was glaring at her, you'd think she'd come home with a big basket of Mad Blargh Disease for the kiddies. She ignored the battery of belligerent looks and rode on, sitting proud and tall in the saddle, until she reined in her muckpony in front of one particular yurt. A plump, silver-haired woman stood in front of the doorflap, arms folded, face stern. I didn't need any bones to catch wise to the family resemblance: Mama Vixen wasn't real pleased with little Randi, or I was a penguin's uncle.\n\n''Young woman, do you have any idea of the anxiety you've been putting your clan through?'' she snapped. ''The wedding is supposed to take place tomorrow, or the great prophecy won't be fulfilled and the Great Winter will descend on all the land! Where have you _been_?''\n\n''Out,'' said Randi. I could've hugged her for having that kind of backbone in facing down one formidable-looking old lady. I could've done more to her than that, but not in front of Mama.\n\nMama Vixen's eyes slewed from her sassy daughter to me. ''And _what_ did you bring back with you?'' she demanded. ''For the last time, no, you may _not_ have a pet yeti, I don't care if you do promise to feed it and walk it and pet it and squeeze it and call it George.''\n\nI moved fast, before Randi could reply. Slinging one leg over my muckpony's back, I dropped easily to the ground and flashed the same mini-bone-reading kit I'd used back at the yurt. (I had it packed in one of the leather pouches at my belt for the trip. You never know when you're going to need to ask a mouse-skull for directions to the nearest tavern.) By the time Mama Vixen said _George,_ I had a charcoal square on fire and a bone dropping into the flames.\n\nThe thunderous sound a good bone makes when it gives up its secrets is a real attention-grabber, and the plume of black smoke that spouts, then drops back and disappears isn't too shabby either. When the air was clear, I had Mama Vixen and every other soul in the lemming clan holt standing in a spellbound circle around me\u2014at a safe distance.\n\n''People of the lemmings!'' I declared, holding the bronze saucer high above my head with two hands. ( _Damn,_ that sucker was hot!) ''I am Mammoth Malone of the bone-reader clan, a son of the tribe of seers! I come to you with the word of the gods! Hear me now, before you make the worst mistake of your lives, in the name of your great prophecy!''\n\n_''Who dares invoke the great prophecy of the lemming clan?''_ The voice that rang out over the heads of Randi's clanfolk was so loud and ugly it would've made a blargh wet itself. It blared from a hulking, wrinkled, balding, saggy old guy who came plowing through the crowd like he owned the place.\n\n''Chief Harvey!'' Mama Vixen gasped. Well, whaddaya know? He _did_ own the place! Or at least he ruled it. The whole clan whirled away from me and bowed to him so low, so fast, that suddenly I was staring at nothing but . . . butt.\n\nI sized up the lemming clan chief. He towered over me by at least three hand-spans. The waves of stink coming off him made me want to find a beached whale carcass and do deep-breathing exercises just to clear my head, fighting reek with reek. He may've been old, but he still had the remains of some pretty impressive muscles roping his body. Too bad said muscles had gotten into a tug-o'-war with gravity. (By the way, gravity was winning.) At least he hadn't gone for Old-Guy-Who's-Losing-His-Hair's Stupid Trick #1, a.k.a. The Comb-Over. No, he went for O.G.W.L.H.H. Stupid Trick #2 instead: The Pity Ponytail. He might've been a looker in his heyday, but hey! Not _this_ day.\n\nBy the time Harvey reached me, everyone had stopped bowing. A low buzz ran through the massed clanfolk, a buzz he swatted into oblivion with one frosty glance. Then he turned to me.\n\n''Outlander, why are you here?'' he demanded. ''I could kill you where you stand.''\n\n''Tsk. And I thought you'd be happy to see me, big guy,'' I drawled. ''I bring you _good_ news: The Great Winter's been called off on account of lemmings on the playing field. Or should I say _no_ lemmings?''\n\nChief Harvey's getting-higher-by-the-falling-follicle brow creased into the mother of all scowls. ''You _dare_ to take the name of the sacred rodent upon your blasphemous lips, Outlander scum?''\n\n''Hey, just doing my job,'' I replied, holding out the bronze saucer. ''I've been told all about what's going on here, how the lemmings aren't acting normal. How your 'great prophecy' says that when that happens, it is the fate of the princess of the lemming clan to wed the chief of the lemming clan, and if she denies that fate, the Great Winter will come.'' I nodded to where Randi still sat astride her muckpony. ''Now I'm not even going to bother figuring out how come you people have a clan princess _without_ having a king and queen, but I will tell you this: Those lemmings are _not_ acting weird on account of any prophecy, great or small. I know: I asked the bones and the bones don't lie. The lemmings are all higgledy-piggledy because someone's been tampering with th\u2014AK!''\n\nChief Harvey's huge paw shot out, clenched itself around my windpipe, and cut me off in mid-wiseguy. The bronze saucer and the burned bone in it went tumbling across the tundra. Slowly he lifted me until I just brushed the ground with the tips of my mukluks. A cruel smile curled his lipless mouth.\n\n''Now when I put you down again, Outlander\u2014 _if_ I choose to put you back down instead of snapping your neck like a dry grass stalk\u2014you will speak to me with _respect_. And then I'm going to snap your neck anyway. Got it?''\n\nI tried to nod, but it was damned hard to do with that meathook collaring my throat. His smile got a little wider and he let me drop. I hunched on the ground, coughing and trying to get my eyes to focus again.\n\nThe lemming clan chief chuckled, then looked at Randi. ''Does this pitiful specimen belong to you, my dear?'' he asked. ''I hope you didn't pay him _too_ much. You should've asked me for a recommendation: I know where you can get corrupt bone-readers cheap. Which they are.''\n\n''I didn't pay him _anything_.'' Randi leaped lightly off her muckpony and knelt beside me, holding me tenderly to her bosom. I didn't know if she really had protective feelings for me or if she was only trying to piss off Chief Harvey, and I didn't much care: I was just enjoying the view. ''And he's _not_ corrupt. He can prove it.''\n\n''Yeah,'' I croaked, getting back on my feet with Randi's help. ''Unlike some people I could mention.'' I thrust my hand into my belt pouch and yanked out a small, brown, wriggling beast that chittered angrily at the massed clan. ''Does _this_ look familiar?'' I cried, holding the lemming high above my head.\n\nThe clan gave a shared gasp of horror and outrage. ''The Outlander has touched the holy rodent!'' someone called out. ''He must die!''\n\n''Or at the very least, apologize,'' someone else added.\n\nBefore the mob could act, Randi snatched the lemming from my hands and confronted her outraged kin-folk. ''You fools, don't you know our own lore? No Outlander may lay hands upon the holy rodents except those gifted by the gods! This man is a bone-reader, a seer. No one is closer to the gods than he!''\n\n''That remains to be seen,'' Chief Harvey said dryly. ''My dear Princess Randi, it seems that when you told me you would rather die sooner than marry me, you weren't just whistling Dixie\u2014whatever _that_ means. The gods are benevolent, but they are also notoriously cranky. Hence, the penalty for blasphemy is death, lest the all-merciful ones take umbrage. To aid and abet one who utters blasphemy is to share his guilt and his punishment. Are you willing to take that risk?''\n\n''Mammoth Malone is not a false seer!'' Randi replied hotly. ''His powers are true; I'll stake my life on it.''\n\n''You just did.''\n\nSome time later, I found myself standing in the center of the lemming clan holt with a huge brazier to my right and a table to my left. On top of the table was a basket and in that basket were a baker's dozen of prime lemmings.\n\nAt my feet were the bones I'd brought along with me. They came from a bull blargh and their spiritual power was in direct proportion to the physical power the beast had enjoyed in life. While the lemming clan watched, I recited the proper incantations, held a blargh thigh-bone high, and called out, ''O gods, as we strip the changing flesh from the solid bone, strip away the shadows of deception from the substance of fact! Behold, against their nature, the lemmings no longer plunge into the depths. What's up with that? Speak to us! Give us a sign to show that what I read in the fire-born cracks in this bone is the truth! And make it a good one, okay? I'll owe you one.''\n\nI threw the bone into the fire and everyone witnessed the usual routine of _snap-crackle-BOOM-WHOOSH_ . As soon as the smoke vanished and the flames went out, I removed the bone and studied the cracks that fire had made. It was just as I'd suspected.\n\n''People of the lemming clan!'' I cried. ''The gods reveal that someone has been tampering with your rodents. I'm not one to name names, but if you check out Chief Harvey's yurt, you'll find a _very_ interesting stash of herbal mind-control potions and kind of a girly atomizer. They're hidden under a pile of _Hot Clan Princess Action_ scrolls.''\n\n''Lies!'' the clan chief thundered. ''I'll slay any man or woman who dares cross my yurt's threshold running this fool's errand!'' He gestured dramatically in my direction, a foot-long knife suddenly in his hand. ''If what you say is truly from the gods, I demand that you prove it here and now!''\n\n''Hey, no sweat,'' I said calmly. ''Not in this weather. Behold!'' I picked up the basket full of lemmings and dumped it onto the tabletop. The little creatures scurried around in a panic, then formed up ranks and headed right for the edge. The lemming clan held its collective breath, watching, waiting, and\u2014\n\nThe lemmings reached the table's edge and stopped dead.\n\n''You see?'' Chief Harvey bared his teeth in a triumphantgrin. ''It is even as the great prophecy says! The fate of the world is\u2014''\n\nI picked up one of the frozen lemmings and blew my nose on the critter's back, hard. Then I set it down again. The lemming blinked, looked around in bewilderment, then focused on the table's edge and with a joyous cry of a rodent embracing its destiny, raced forward and did a swan dive into the void. Before he hit the ground and took off for the horizon I managed to grab his fellow lemmings by twos and give them the same treatment. One rather slimy snort to the fur and they all went through the same wake-up call routine. As the last of them made the leap, the clans-folk cheered.\n\nChief Harvey did not. ''That proves nothing,'' he growled. ''Except that you are one of the most unbelievably disgusting seers I've ever met.''\n\n''Flatterer.'' I winked at him. I reached into my biggest belt pouch and pulled out one more lemming. It was the same one that Randi had brought to my yurt, the one she'd used as an improvised hankie, the one I'd toted all the way back here. I was getting kind of attached to the little guy, all the more so since I'd read the bones one more time before we'd left my yurt and they'd confirmed my theory: That to make the great prophecy serve his own ends, Chief Harvey had somehow contrived to dope as many of the local lemmings as possible with his mind-control spray, but that the effects of that potion were shattered by a big enough shock to the rodents' system. Having someone blow his nose on you? Yeah, I'd be shocked, too.\n\nI talked fast, bringing the lemming clan up to speed. Before I could finish, a couple of enterprising young clansmen came running up, their arms full of waterskins.While their chief had been distracted, they'd been busy boys.\n\n''We found these in Chief Harvey's yurt, just where the bone-reader said we would!'' one of them exclaimed. Harvey let out an enraged roar and lunged for him, knife held high, but a bunch of his kin used sheer numbers to overpower, disarm, and immobilize him.\n\nArms pinioned by half a dozen angry clansmen, the lemming chief could only growl, ''So you found a couple of waterskins in my yurt. So what? I get thirsty.''\n\n''For this?'' I snatched one of the waterskins and squirted a little of the contents into Mrs. Vixen's face, then commanded, ''You're a chicken!'' When Randi's mother began to cluck, Chief Harvey scoffed, claiming I'd bought the old doll off. What'd he want? _Eggs?_ I had to give seven more lemming clan tribesfolk the same treatment and get a whole flock going before I proved my point.\n\n''What're they going to do with him?'' I asked Randi as we watched the men of the lemming clan drag their former chief away.\n\nHer eyes could've given chill lessons to a glacier. ''He sleeps with the narwhals,'' she said somberly. Then she giggled, because it's impossible to say the word _narwhal,_ even as part of a death sentence, and keep a straight face.\n\nI watched until Chief Harvey was hauled out of sight. I felt kind of sorry for him, but the feeling faded fast when I reminded myself how far he'd been willing to go to satisfy his own desires. It was a big, wide, frozen-solid world out there and it'd be a better one for having one fewer ruthless, self-serving _oosik_ -heads in it. I turned my back and left him to the narwhals.\n\nHeh. _Narwhals._\n\nI stopped snickering in time to see Randi smile at me. Then I heard her give the command to her kin to go forth, find all the lemmings affected by Chief Harvey's evil plot, and free their minds in the only way possible.\n\n_''Ewwwwww!''_ the lemming clan replied as one. Some of Randi's clanmates touched their noses and sniffled self-consciously, yet in spite of their reluctance, they knew that if they ever hoped to get things back to normal, they had no choice but to find, catch, and sneeze on every lemming they could find.\n\nYeah, _ewwwwww,_ but what did they expect? Like my grandpa always said: Fate's nothing to sneeze at.\n\nIt's not.\nDEATH AND TAXES\n\n_Kristine Kathryn Rusch_\n\nSixteen years and half a continent away from the great American Midwest, Patrick saw Keri. She was running out of the market across the street from his favorite coastal caf\u00e9, a bottle of wine in her hand.\n\nAt first he thought it couldn't be her. Her long brown hair caught the sun, reflecting it in golden highlights. She was slender, and the blue sundress she wore hung off her as if she hadn't grown into it yet.\n\nPerpetually twenty. That was what he thought as he sipped his mocha and returned to _The Wall Street Journal_. Keri would always be twenty and coltish, not quite grown into her body.\n\nHe smiled at himself, at his romantic nature. Proof, perhaps, that he had loved her because he saw her in every gangly twenty-year-old with the promise of great beauty.\n\nThen a car horn made him look up. The woman was standing in the middle of the street, staring at him, cars stopped all around her. The bottle of wine had shattered at her feet.\n\nHis gaze met hers.\n\nShe hadn't changed.\n\nAnd there was a look of abject horror on her face.\n\nSixteen years and half a continent away, he'd been twenty-five, callous, and certain of his own future. The son of a prominent lawyer, he'd become a lawyer too\u2014not with the thought of practicing law, but with the thought of creating it. He studied politics like it was a religion, and decided that he had to be in the seat of government. So, with his newly minted certificate from the bar, he headed downstate thinking the capital would welcome him.\n\nInstead, he learned that any state capital had its share of locally grown lawyers. With his pedigree, the partners at the large local firms said, he could get a job anywhere. The following question\u2014why here?\u2014 had an underlying meaning: what's wrong with you? How come you haven't gone to your father's firm?\n\nHe couldn't very well say he had come because he thought getting into politics would be easier here. It wasn't. He didn't know anyone, and the art of politics was the managing of connections.\n\nEventually, he got a job as a junior staff lawyer at the Fair Housing Coalition, a job he saw as beneath him both financially and politically. Yes, yes, he believed everyone should have a home and everyone should be treated fairly, but most of the people he saw were too dumb to realize that a lease agreement was a legal document and that their behavior had put them in trouble with their landlord and the local laws.\n\nHe could have, he later supposed, joined the interoffice coalition that was working to change some of the more egregious landlord-tenant laws, but his heart wasn't in it. Instead, he gravitated to the local university,spending his time in the student union, drinking with people who reminded him of his friends back home, talking philosophy and planning to change the world, one little decision at a time.\n\nThat was how he learned about the professors Simmons and their interdisciplinary study\u2014financed by any number of government agencies and private corporations\u2014and extended, theoretically, over decades.\n\nThe study only made it through the first five months of its existence.\n\nIt caused two deaths, and derailed any hopes he had of politics\u2014at least out front. Only fast-talking and the excellent attorneys of his father's firm had saved Patrick from being disbarred.\n\nBy then, he didn't care. He'd already met and lost Keri.\n\nAnd had his belief in everything shattered.\n\nHe grabbed his mocha as he headed out of the caf\u00e9. Interesting, he would later think, that he'd left his PDA and his newspaper, but took his beverage.\n\nIt was a clear sign that he wasn't thinking, just reacting, running through the closely set tables to the double doors, pushing them open and hurrying into the street.\n\nA VW bug swerved past him, and the driver shouted an obscenity. A sedan, following, leaned on its horn.\n\nBut he didn't move. He stared at the broken bottle, the red wine running like blood down the empty sidewalk.\n\nKeri was gone\u2014as if she had never been.\n\nThe professors Simmons were not related. There were four of them, all in different disciplines. They met at a large university faculty gathering where everyone had been asked to clump alphabetically. Their common last names, their common ages, and their uncommon interests held them together a lot longer than the meeting had.\n\nProfessor Abigail Simmons taught philosophy. She had two seminars in which she tortured undergraduates, forcing them to challenge the realities in the world around them. She also taught three graduate seminars to the same twenty grad students, the courageous few who thought majoring in philosophy was a good idea, no matter how badly it ruined them for the job market. She had grown frightened for her own job, discovering that publishing occasional articles in philosophical and religious journals wasn't enough to impress her dean. Apparently, she had to do some sort of breakthrough research to justify her salary. But, she would argue, breakthrough research and philosophy were by definition incompatible, something her dean believed she\u2014of all people\u2014could overcome.\n\nProfessor Roderick Simmons taught political science. He was the right-wing guru of the poli-sci department, the man that local media always called to give a reliable\u2014and seemingly balanced\u2014view of local elections. Roderick Simmons specialized in political systems and, in addition to his well-received books, he spent a lot of time away from campus, consulting with various groups, many of them tied to the Republican Party. He was tenured and secure, which made him perfect for this joint project.\n\nProfessor Marilyn Simmons was a biologist. Her teaching work involved occasional lectures to overcrowded 101 classes (with the day-to-day work done by teaching assistants) and supervising the research of sleep-deprived graduate students. Her seat at the university had funding from outside grants; she was a star professor who felt her own area of expertise had grown a bit stale. She was looking for a new challenge, one that would improve her prestige even more, and this, she felt, was it.\n\nProfessor Nash Simmons was the youngest and the most professionally insecure of the group. Even his specialty reflected his insecurities: His professorial bio said that he focused on cognitive analysis and behavioral theories\u2014a lot of words, he liked to joke, that meant he had no idea what he was doing. He did whatever it was that he did from the behavioral science department, where he taught upper level psychology classes and graduate seminars about the brain. He supervised almost no graduate students and his thesis, a trailblazing work on cognitive theory that had been published to great acclaim, was now several years old. He had to produce something new, and in the way of all who were acclaimed when they were too young, he felt that the something new had to be trailblazing as well.\n\nPatrick had no idea how the multidisciplinary study went from cocktail party talk to grant-writing to grant-winning, but by the time he had encountered Simmons-N, as Nash Simmons had been designated by those involved in the work, the study was looking for willing bodies. That Patrick wasn't a student and had an understanding of the body politic made him an unusual choice.\n\nThat he was willing to step into the real world in the name of science made him even more unusual.\n\nBut it was his willingness to apply experimental techniques to that real world that made him the most desirable candidate the professors Simmons had found.\n\nPatrick walked into the market. It smelled of garlic and fish overlaid with the faint scent of roses from a display near the door. The place was dark compared to the street and cramped, which instantly made him uncomfortable. He preferred the large chain grocery store at the end of town, where the lights were bright and the products were displayed according to dictates of some corporate official in another state.\n\nAs his eyes adjusted, he saw six different aisles heading toward the seafood department along the back wall. The seventh aisle, which started behind the cashier, carried wines, beers, and hard liquor. Cigarettes were stacked high, where no one could get them without help from the staff.\n\nHe waited in line, noting that everyone ahead of him had fresh produce and canned products with the words _healthy_ or _organic_ or _natural_ on the label. He shuddered, hating the pretense, remembering when he used to do the same thing just to fit in with his university friends.\n\nWhen he made it to the front of the line, he reached into the back pocket of his jeans and pulled out his badge. Most people in this small town knew their sheriff, but he was cautious for the handful that didn't.\n\n''The woman who just left,'' he said. ''The one with the wine. Can I see the copy you made of her license?''\n\nThe clerk flushed and for a moment, he thought the gambit wouldn't work. Keri still looked twenty; she should have been carded. Oregon law stated that anyone who looked thirty-five and younger had to show identification to buy liquor.\n\nBut the clerk nodded and called for a manager, who took Patrick to the back office where he could look at the fuzzy identification that had been scanned into the computer system.\n\nKerissa Simon, the I.D. said, the last name dangerously close to Simmons\u2014so close that it made his head hurt.\n\n''Any idea where she's staying?'' he asked, knowing the store didn't need a record of that, but often took it to avoid problems later on.\n\nHe got the name of a roadside motel, cheap but comfortable, and somehow it didn't surprise him, just like her appearance in his refuge hadn't surprised him.\n\nAlthough it should have.\n\nThe meeting room was an old lecture hall in one of the campus's earliest buildings. The building was now used primarily for offices, but this room had clearly been too big to give to just any professor.\n\nRadiators ran along the walls beneath the single-paned windows, and despite the constant heat blowing into the room, there was still a draft. Patrick sat near the door in a wooden desk chair that was at least eighty years older than he was. Some of the names carved into the desk's surface had been there so long that their edges had worn smooth.\n\nHe traced them, feeling out of place among the students, knowing he looked out of place in the suit his father had purchased for him before his first moot court appearance. Patrick had taken off the tie and stuffed it into his briefcase, but the fact that he had a briefcase instead of a backpack and a suit coat instead of a sweater already showed that he wasn't One of Them.\n\nA few stared, and a couple kept glancing at him like they expected him to get up front and talk about the various studies.\n\nHe had had some preliminary meetings with the professors Simmons and the assisting graduate students; he assumed these other participants had as well. Now, though, they were getting together for their first official meeting. They would have four such meetings before splitting into various subsets, four meetings in which the professors Simmons would lay out the purpose of the studies as best they could, without tainting the results.\n\nThe professors stood in the hallway, heads bent, conferring, while a graduate student with a clipboard checked off the names of each attendee. Finally, a young woman, snowflakes melting on her hair and collar, stopped near the graduate student, gesturing an apology as she gave her name. Then she slipped inside the room, and took the only remaining chair, right next to Patrick.\n\n''The snowstorm they predicted came, huh?'' he asked.\n\nShe leaned away from him and finger-combed the moisture from her brown hair. Then she peeled off her coat, meticulously hanging it on the back of her seat.\n\n''The roads are a mess,'' she said. ''I had to park six blocks away.''\n\nHe was in one of the private lots, courtesy of the Fair Housing Council. He hadn't really noticed the snow until he started climbing the hill. Then he worried about the swiftness of the storm, knowing that the sidewalks could get buried during the few short hours of the meeting.\n\n''I'm Patrick,'' he said as she sat down across from him.\n\n''Keri.'' She stuck her mohair scarf inside her coat sleeve, then smiled at him. ''You need the money, too?''\n\n_No,_ he wanted to say but didn't, _I just need the company._ He knew this study paid the highest of any conducted on campus, and he thought he knew why. The interdisciplinary approach allowed for even more grant money than usual, and the professors decided to use that money to pay the subjects extra, so that they'd stick for the duration rather than leave when the semester ended.\n\n''Money's always nice,'' he said, which was as much of a dodge as he wanted to give her. He wasn't sure why he felt this odd need for honesty. She was a bit thin for his taste\u2014all elbows and knees and sharp angles. She was also at least five years younger than he was, an undergraduate when he'd been out of school for a year now.\n\nShe smiled at him, then pulled an older laptop from her backpack. The laptop barely fit on the desk. Several other participants had laptops or AlphaSmarts or PDAs with keyboards.\n\nHe hadn't even thought of taking notes, which suddenly showed him how far he had come from the student mentality. He leaned to the right, opened his briefcase, and pulled out both a legal pad and his BlackBerry, not sure which would work best in this situation.\n\nThen the door opened one more time, and the professors Simmons came in. Their appearance was as varied as their disciplines. Simmons-A was short and dumpy, her curly hair a mixture of gray and grayer. Simmons-R wore a suit as expensive as Patrick's. His black hair had a precision cut, and his hands looked manicured. Simmons-M was slender and wore her long red hair in some sort of upswept 'do that looked like it took time and three other people to create. Simmons-N had the prerequisite professorial ponytail and wispy goatee. His glasses fell to the edge of his nose, making him seem even more absentminded than he probably was.\n\nPatrick's stomach turned. Studies, waivers, payment by the hour, altering his behavior because he had agreed to do so, not because he wanted to do so.\n\nWas he that lonely? Was he that lost?\n\nHe glanced around the room, at the stressed, pimply faces around him, and realized he probably was.\n\nThe motel had been built in the late 1950s, when this coastal community had been known as the Disneyland of the Pacific Northwest. Once there'd been a theme park (although in those days, they'd called it something else) on the outskirts of town. Only a few remnants remained\u2014a red and white store downtown that made its own candy; a go-cart park across from a restaurant once known as (and still referred to by locals as) the Pixie Kitchen; and a five-story resort hotel built in the Cape Cod style where presidents had stayed but which had become, in the intervening years, an old-folks' home.\n\nThis motel, unoriginally called the Beach-Goer, still advertised that it had television and clean, comfortable rooms. It stood on a bluff overlooking the ocean, prime real estate that the elderly owners refused to sell to all sorts of development firms.\n\nThe main entrance was off a narrow drive that barely fit today's SUVs; he had no idea how the large automobiles of forty years ago had negotiated the same road.\n\nHe drove a truck van combination with an engine modified for high speeds. The county owned the vehicle, and if he ever lost a local election, he would have to give the thing back. Sometimes he thought he might miss it\u2014in the back was all sorts of life-saving equipment mixed with weaponry\u2014but mostly he saw it as a burden of his job, one of many he hadn't understood when he learned that his checkered past mattered less to the people here than it probably should have.\n\nHe parked just outside the entrance, making sure that the official decals were facing away from the street, so as not to interfere with any walk-in business. Then he went inside.\n\nThe desk clerk was a local gal who played bingo at the casino every Wednesday night. He didn't know her name, but they'd seen each other around. It was hard to miss the other locals in a town of seven thousand.\n\nShe smiled at him with recognition. He didn't have to flash his badge. He just asked for Keri Simons's room, and the clerk gave him a room key.\n\nHe weighed it in his hand as he walked along the concrete sidewalk. The key was a kind of power: if she wasn't there, he could wait inside her room, surprise her, let her know who was really in charge.\n\nThat he even had the thought surprised and appalled him at the same time. He had never thought of control in connection to Keri before.\n\nBut the study itself, the reason they met, was all about control.\n\nAnd hubris.\n\nAnd the belief that somehow, humankind had the power to alter its own destiny.\n\n''For thousands of years, mankind has felt it has a destiny.'' Simmons-A stood in front of the long wooden desk beneath the chalkboard. She had taken a piece of chalk before beginning her welcoming remarks, almost as if the chalk provided a kind of comfort. All during her talk, she kept the piece in her palm, alternately rolling it and clenching her fingers around it.\n\nPatrick found himself watching the chalk instead of her face, partly because she reminded him of every professor he'd ever disliked, and he wasn't exactly sure why.\n\n''Not just a species destiny,'' Simmons-A was saying, ''but individual destinies as well. We can turn to almost any early document based on the oral tradition and find evidence. Genesis tells us that God created Man in His own image, and just that sentence alone implies that God had a purpose for Man, a purpose that Woman screwed up, of course.''\n\nThe group laughed, but it sounded dutiful. Patrick made himself smile, even though he hadn't felt like it, but Keri crossed her arms.\n\n''Mythology gives us story after story of people confronting their destinies, from the Christ story to the Greek story of Oedipus.''\n\nPatrick shifted in his chair. He didn't need the history lesson if that was what it could be called. He just wanted to get on with the actual business of the study, whatever it would be.\n\n''Fighting destiny is one of the greatest themes mankind has.'' Simmons-A tossed the chalk into the air and caught it. ''Look at Joseph Campbell's _The Hero's Journey._ Look at our own fiction from popular tales of Harry Potter to _Star Wars_. Even romance fiction flirts with destiny. Romance hints that every person on earth has a soul mate\u2014someone they're destined to be with. If we take the time to find that person\u2014 or if we recognize that person (apparently some of us do not)\u2014then, the theory goes, we shall live happily ever after.''\n\nKeri bit her lower lip. Patrick didn't know when he started looking at her instead of Simmons-A.\n\n''From time immemorial,'' Simmons-A said, ''mankind has tried to fight its destiny, whatever that destiny might be. Few are born with that proverbial silver spoon in their mouths. Even fewer accept that spoon with grace. If you do not believe me, look at the remaining royal families of the world. Such tales we hear of debauchery and rebellion.''\n\nA few people smiled, but most stirred, just like Patrick had.\n\n''What has this to do with us?'' Simmons-A asked. ''Simple.''\n\nShe then gave a capsule summary of the faculty meeting, the conversation the professors Simmons had in lieu of listening to faculty debate.\n\n''The only thing that the four of us could agree on that evening,'' she said with a smile that transformed her face from sour and discouraged to slightly pretty, ''was that Benjamin Franklin was right: in this world, nothing is certain but death and taxes.''\n\n''And being born,'' someone said from the front row.\n\nShe looked at him in surprise. He had broken her rhythm. Simmons-M, the biologist, came forward at that moment, rescuing her colleague.\n\n''We can't change that,'' she said. ''We've all been born and we've survived. So once we're here, all we can be certain of are death and taxes.''\n\nShe had a powerful voice with a touch of music to it. She also had a great deal of charisma, and Patrick found himself wishing that she had been in charge of the opening speech instead of Simmons-A.\n\nBut Simmons-M knew her place, at least in this beginning. She made a little bow to Simmons-A and returned to the cluster of Simmonses near the blackboard.\n\n''Precisely,'' Simmons-A said, attempting to recover. ''Death and taxes. We argued about that flip remark for weeks. And somehow, we went from a philosophical discussion of certainties and uncertainties in this world to what we're calling a multidisciplinary study. According to our grants, we're attempting to see if humankind can change its known destiny. But between us\u2014''\n\nAnd she grinned again, looking over her shoulder at her colleagues like a schoolgirl. Only Simmons-N smiled back.\n\n''\u2014we decided to have a race. We have four things to prove. That we do indeed have destinies. That we can change them. That human-made systems\u2014in this case, taxes\u2014can be changed. That biological systems\u2014 in this case, death\u2014can be changed.''\n\nThen she stepped back with a little nod, and Simmons-R came forward. Patrick slouched. He'd never realized until this moment how much Simmons-R reminded him of his father.\n\n''You'd think,'' Simmons-R boomed and half the room sat up as if they'd dozed off and been rudely awakened, ''that human systems would be the easiest to change. But I have my doubts. In a cursory search of governmental systems throughout human history, I cannot readily find an example of a society without taxation. Once again, let me turn to the Bible. The Egyptians . . .''\n\nAnd as he discussed the levies that the Egyptians placed on their subjects, the taxes that built the Roman roads, the demands the medieval Japanese put on families, he made an eloquent, if familiar, point.\n\nPatrick assumed the point probably wasn't as familiar to the undergraduates\u2014few people in the room besides the professors Simmons were as overeducated as he was. He glanced at Keri. This time her gaze caught his, and she smiled.\n\nHe felt twelve again, and actually had to resist the urge to write a note on his legal pad and pass it to her. So far, she hadn't typed anything into her laptop, and he'd only written down ''Death and Taxes'' as if it were the topic sentence of an essay exam.\n\nIn fact, he never wrote anything else down that night. The remainder of the evening was a jumble of lecture\u2014Simmons-M discussing the necessity of death, not just on an individual scale, but on a worldwide one (species death; death of ecosystems; the eventual death of the planet itself), and the arrogance of humankind to think it can alter death, even on a small scale; and Simmons-N referring to behavioral studies that suggest humankind's perceptions of the world have led humans to misunderstand it\u2014and an increasingly shared but silent intimacy with Keri, who seemed to find the whole thing as pretentious and amusing as Patrick did.\n\n''One of the things that we're going to examine,'' Simmons-N said in his nasal voice, ''is whether time actually exists or is just a matter of perception. Because if it is a matter of perception, then nothing around us is real\u2014or everything is real, from the primordial soup that the earth once was to this moment to the heat death of the universe, all happening at once.''\n\nSimmons-A smiled through that entire speech, as if she agreed. Indeed, it seemed to Patrick that she would have been better off saying it than the cognitive and behavioral scientist.\n\nPatrick said that later to Keri, at an all-night coffee shop just off the main drag. They'd ducked inside on their way back to their cars\u2014or, more accurately, on the way back to hers; he'd passed his blocks before, but hadn't told her, enjoying her company enough to hazard the ice pellets and heavy wind that the storm had become.\n\nThey found a booth in a warm corner away from the door, where they spent the next few hours laughing about the pretension, about the silly race between the disciplines _(which implied,_ Patrick said, _that they would all succeed in areas of study where no one had succeeded before),_ about the ironic coincidence that led the professors Simmons to each other in the first place.\n\nSometime during the evening, Keri postulated that the winner of the entire thing might end up being the philosopher, who had somehow gotten a group of diverse people together to reexamine their beliefs in a way that seemed as irrational as the most screwball religious cult.\n\nPatrick had laughed at that remark. And it was his own laughter that he thought of most often when he thought of Keri. Not of those nights at his apartment, not of the horrible last day. Just the laughter.\n\nAnd the professors' fight against a complacency that he didn't then understand.\n\nHe understood it now, even felt it on days when the sunlight hit the ocean, and his small town was bathed in a clear, almost unworldly light. He would tell himself, as he looked at that beauty, that he had done the best with what he had.\n\nBut there was always an itchy restlessness underneath\u2014 a what-if chorus that continued to sing: What if he had gone to his father's law firm first? What if he hadn't been interested in politics? What if he had never met Keri?\n\nWhat if, what if, what if.\n\nHe played the scenarios in his mind as if he were a screenwriter finishing a script for a time-travel movie.\n\nWhat if . . .\n\nHe didn't know. He would never know.\n\nHe only knew that if the ancient Greeks had written his life story, the what-ifs didn't matter. Destiny was destiny. The Greeks always showed that no matter what changes mere mortals tried to make, destiny would win out.\n\nSomehow, the Greek version seemed to tell him, he would meet Keri anyway, she would die, and everyone would be sued. Careers would end. Lives would be ruined. Simmons-N would commit suicide all over again.\n\nAnd Patrick would end up here, carrying a little sheriff's badge in an unimportant town on the Oregon coast, living alone, and wishing none of it had ever happened.\n\nHe paused before knocking on the door to her room. Only now, with a key in hand, and the memories fresh, did he realize how silly he was being.\n\nKeri Andreeson was dead. He'd seen her corpse. They all had. They had clustered around it in the biology lab, her mouth slack, her tongue protruding ever so slightly, her eyes bulging, and her skin an unnatural clay color, and they had stared.\n\nNo one had said a word. He wasn't sure, even then, if anyone completely understood how much her death would change everything.\n\nHe wasn't sure he understood even now.\n\nHe swallowed against a dry throat. Was standing here a sign of a growing insanity? The fact that he was willing to believe that some girl\u2014coincidentally named Keri (spelling the same)\u2014whose driver's license claimed she was twenty-two and from Illinois (Keri had been from North Dakota, complete with a melodic Fargo accent)\u2014the fact that he was willing to believe she was the same person as the girl whose body he'd seen, the fact that he was willing to believe she was alive, and looked the same, and was terrified of him\u2014showed just how far he had fallen intellectually, how little he believed in reality anymore, how much he hoped for miracles.\n\nWhich made him no better than the people who had placed their faith in those studies.\n\nOr put their faith in anything, for that matter. For what was faith, but a belief in the impossible? An irrational belief in something unbelievable.\n\nHe clutched the key in his fist, tempted to open the door and scare the girl, whoever she was. Who would she report her fear to? The sheriff?\n\nHe felt a bitter smile cross his lips. Then he turned away.\n\nBetter to leave the past in the past. Better to leave destiny or fate\u2014or the lack thereof\u2014to the philosophers and the professors and the dreamers.\n\nBetter to return to the realities of traffic accidents and one murder a year and a lonely house on a cliff face overlooking the ocean, a house with a television as large as his bookshelves, a place where he went when he couldn't stand reality anymore.\n\nHe had just stepped into the parking lot when he heard a lock turn and a door open behind him.\n\nAnd before he had time to think\u2014or maybe he lied about that: maybe he did have time to think and he chose this\u2014he turned, and stared Keri Simons\u2014Keri Andreeson\u2014in the face.\n\nThey'd become lovers even though the professors Simmons had cautioned against fraternizing. That alone might have skewed the study\u2014or one of the studies\u2014had any of them been completed. But the thing had barely gotten off the ground when it all ended. Patrick had just received his work orders from Simmons-R the week before, work orders that included an overall personal plan that extended for five years.\n\nThe breadth of the study surprised him, even then.\n\nPatrick was to ally himself with a local political group\u2014any political group would do, so long as it worked on the grass-roots level\u2014and slowly ease them to a new vision: that taxation was a scourge, that government needed fiscal responsibility, and that required budget-tightening, reduced spending, and no new taxes. Over time, the no-new-taxes pledge would become a no-tax pledge, depending on how high up the political ladder he could climb, how much power he could attain, and how many followers he could convert to his\u2014actually, Simmons-R's\u2014 way of thinking.\n\nPatrick, in his n\u00e4\u00efvet\u00e9, had thought it possible. Much as he believed politics was the art of compromise, he also knew it lived in the realm of argument. A charismatic man with the right argument could change the playing field\u2014make compromise happen on the one-yard line instead of the fifty-yard line, and yet convinceeveryone that they had attained a middle ground.\n\nHe'd actually see it happen, years later. The political center moved farther and farther right as he moved farther and farther west. When he finally stopped long enough to look at what America had become while he'd tried to outrun his past, he found himself wondering if some of Simmons-R's other subjects hadn't continued with the experiment, working their way up the political ranks until they reached the national level, influencing everyone from senators to the president himself.\n\nThen Patrick would shake that feeling off\u2014surely he would recognize someone from the bad old days, right?\u2014and he would remind himself that taxes still existed, that the United States went through cycles of heavy taxation followed by cycles of light taxation, but never, in its two-hundred-plus-year history had the United States ever gone without taxing someone for something.\n\nHe found that vaguely reassuring, just like he found the obituary columns reassuring. People continued dying. Humankind kept fulfilling their destinies, one grave at a time.\n\nShe was twenty. That was the first thought which reached his brain as he stared at her, framed in that cheap wooden doorway, sunlight peaking over the building's eaves and the shush-shush of the ocean beyond.\n\nIn no way could this woman be in her mid-thirties, stretched by time and loss and years on the run.\n\nShe put both hands on the doorframe as if bracing herself or blocking his entrance or simply holding herselfup. She was as thin as ever, coltish, all angles and lines, a girl who had not yet fulfilled her physical potential, whatever that might be.\n\n''Can I help you?'' she asked in a voice he wasn't sure he remembered.\n\nHe flushed. She had seen him pause in front of her door, maybe even seen his hand raise slightly, his fist clench the key. She'd certainly seen his indecision, and, ultimately, his retreat.\n\n''You bought a bottle of wine today,'' he said, finally choosing an official approach.\n\n''Is that illegal?'' She tilted her head slightly as if she were interested in the answer. The movement was familiar. Keri used to do it when she was flirting.\n\nHis heart literally contracted. He'd only felt that squeezed sensation once before, when he saw her on the cot in the lab, her arm dangling to one side, the IV still taped into it but listing, as if it had died with her.\n\nIt took a physical effort to bring himself to the present.\n\n''No,'' he said. ''Buying wine isn't illegal. Neither is dropping it. But you could have picked up the glass.''\n\n''Is this town so poor that it has sends someone out to get its littering fees?'' Then he heard it: the Scandinavian music behind the Fargo accent. The accent existed in the up-and-down cadence of the words as much as the long-vowel pronunciation.\n\nShe had cured the long vowels, but not the melodious intent behind them.\n\n''I'm not here to collect any fees,'' he said, ''even though I am the county sheriff.''\n\n''I would have thought that a man who read _The Wall Street Journal_ had higher ambitions.''\n\nShe had seen him then, drinking his mocha and reading his paper, taking his afternoon break and pretending he was someone else.\n\nIf she had seen him, then that look of horror had been real.\n\nAnd if that look of horror had been real, did that mean she had recognized him?\n\nAnd if she had recognized him, did that mean she was Keri Andreeson masquerading as Keri Simons?\n\n''I did have higher ambitions once.'' He felt odd discussing them with a woman he thought dead in the parking lot of a cheap motel. ''I left the caf\u00e9 to talk with you, but you'd already vanished.''\n\n''Vanished.'' She smiled. That smile belonged to a woman, not a girl. It was learned. It held a wisp of sadness as well as a touch of irony. And through it all, her eyes hadn't changed. ''Leaving broken glass behind.''\n\nHe should have brought a bottle of wine. He saw that now. It would have eased the moment, given it some symmetry. But he wasn't that kind of thinker.\n\nOr maybe he was\u2014a man who knew better than to tempt fate.\n\n''We got it cleaned up,'' he said as if he had something to do with it.\n\nShe nodded. She didn't ask who he was. She just studied him in the odd light filtering over the building.\n\nFinally, he had to become the supplicant, even though he didn't want to. ''Is there somewhere we can talk?''\n\nShe shrugged a single shoulder, her hands remaining in place. ''This is fine.''\n\nIt wasn't fine. Even though the motel was sheltered by the trees, there were other doors, other windows, other rooms where people might be. They might listen. The desk clerk might be listening, and later she'd mention the odd conversation to her friend in the bingo hall, telling them how strange the sheriff seemed on that sun-dappled afternoon.\n\n''It's not very private,'' he said.\n\n''I don't see other cars,'' she said, as if she'd expected his objection.\n\nHe sighed, and walked back toward her. She locked her arms, and he had the sense she had done that instead of flinching. Why would she be afraid of him? If they hadn't met, then it was something about her. If they had, then she was afraid he'd recognize her. He'd know that she hadn't died, that people had gone to jail for no reason.\n\nBut she had died. He had touched her waxy skin. He had cried for her.\n\nHe'd loved her.\n\nHe hadn't thought of any way to approach this conversation, and now he felt tongue-tied. Did he ask her if she'd known a Keri Andreeson? Wouldn't someone who had changed her name deny it? Or should he ask if she had gone to the university? Or simply ask what brought her here, to the literal end of the earth?\n\nFinally, he settled on, ''Have we met before?'' Her mouth opened as if she planned to answer him, then closed as if she thought better of it. ''You mean besides now.''\n\nHe nodded, not willing to play any more word games.\n\n''Outside the market, you looked at me like I frightened you,'' he said, then wished he hadn't.\n\n''That's why you ran outside?'' she asked, her voice rising. ''That's why you tracked me down?''\n\n''I was already thinking you looked familiar,'' he said, letting the implication hang that yes, he had sought her out because he wanted to find out what terrified her.\n\n''A lot of people say that.'' She gave another one-shoulder shrug. ''I have one of those faces.''\n\nBut not one of those bodies. Not in combination. But he didn't dare say anything like that lest she think it improper. Not that she would have any recourse here, in this small town, where he normally was the recourse.\n\n''Still,'' he said. ''Something you saw frightened you.''\n\nShe studied him for a moment. ''I don't think we have met,'' she said, answering the earlier question. ''You seem like a man a woman would remember.''\n\nHe felt his breath catch. The other Keri had described him that way. When he had asked her why she had gone with him that first night, she had said she would have regretted not going. He had asked why. She had smiled. _Because,_ she said, _you're the kind of man a woman would remember._\n\nThe echo bothered him. Everything about this meeting bothered him.\n\n''You came to see me because you thought I was frightened,'' she said.\n\n''I came because I wanted to find out if you're the woman I remembered,'' he said, noting the echo in his own language.\n\n''Am I?'' she asked.\n\nHe swallowed, his throat still dry. The movement was painful.\n\n''No,'' he said after a moment. ''I don't think you are.''\n\nShe died testing the equipment. That was the official story. She was lying on the cot, taking a bit of fluid in the IV, seeing if the heart monitors worked, when somehow, she went into cardiac arrest.\n\nExperiments on human beings, whether in government-funded labs or university trials, were forbidden in the United States. Tests could be performed\u2014trial runs of pharmaceuticals, for example, or psychological batteries\u2014 all with waivers, properly signed, and the risks carefully laid out.\n\nFor the death study, administered by Simmons-M with help from Simmons-N, the risks hadn't been properly laid out. The implication\u2014never proven\u2014 was that the participants would be brought to the brink of death and brought back. At the brink of death they would attempt to prolong life, through perception changes or medications or some other procedure.\n\nBut unlike the tax part of the study, none of this was written down. They didn't dare, although the grant for this part of the study had been explicit enough to bring the two professors Simmons to criminal court and drag the university into a system-wide scandal. Simmons-R got brought in when it became clear he had lobbied the institution that issued the grant money, but Simmons-A remained untouched.\n\nSimmons-A had only her grant proposal to delineate her involvement, and her participants were going to examine the philosophical underpinnings of both death and taxes, with a touch of psychological attribution.\n\nShe claimed betrayal by the other Simmonses, and that was how she parlayed her involvement into best-selling nonfiction books, while the other professors spent years in court.\n\nArguing over Keri's death. Accident? Possibly. The administering nurse was really a nursing grad student, not through her pharmacological classes. Perhaps she had put a sedative into the IV in error, or grabbed the wrong IV in error. But there was too much verbal testimony otherwise.\n\nToo many indicators that the Simmons Three, as the press had started to call them, had become arrogant enough to believe they could conquer death. Simmons-N's suicide, shortly after he had been let out of jail on bond, led to jokes in the local media\u2014that the Simmonses were again trying to prove they could conquer the state, if not death itself.\n\nThere was no sympathy.\n\nNot even from Patrick.\n\nHe had stayed for the trials, even though his father told him not to. He had stayed, even though he (and the other tax participants) were classified as non-involved.\n\nNo one discovered his relationship with Keri, and he didn't confess it.\n\nHe watched as Simmons-M's brilliant career dissolved, as Simmons-R went from being an authority to being a blowhard, as the two of them sat across from a jury and waited for judgment.\n\n_What's your destiny now?_ The reporters would ask as the two of them and their lawyers hurried out of the courtroom every night.\n\nTheir destiny, it turned out, was a plea bargain. Negligent homicide for Simmons-M. Conspiracy for Simmons-R. A few years' time in a minimum security prison, followed by community service.\n\nNone of this brought Keri back.\n\nSimmons-A didn't even attend the trials. When Patrick went to see her, after the trial, she grew rude and frightened when he said he wanted to discuss the study. But he didn't leave.\n\n_Did you really want to change destinies?_ he asked.\n\n_I told them it couldn't be done,_ she said. _It's the one thing philosophers agree on. That in life, some things cannot be changed._\n\nHe almost fell for it. Then he realized that she was wrong. The Hindu system was based on knowledge\u2014 reincarnation as learning, improving, changing, growing\u2014and, by implication, changing destiny. Not accepting it as the Christians taught. Not bowing to its inevitability, like the Greeks.\n\nBut he didn't challenge her. He no longer had the energy.\n\nHe couldn't change his destiny. But he could change his life.\n\nSo he headed west.\n\nWhere, he thought as he got into his truck, he had become a man who drowned in taxes. They created his job, provided his ride, paid his salary. In an odd, and completely unplanned way, taxes were his destiny.\n\nJust as death would be someday.\n\nHe started to pull away, and then he stopped. None of that explained Keri, her look of fright, her resemblance to the other Keri, the one he thought he had loved.\n\nHe couldn't leave. Not yet.\n\nHe rested his head on the steering wheel and sighed. Then he got out of the truck one final time.\n\nHe rehearsed what he was going to say as he crossed the parking lot.\n\n_Do you believe in destiny?_ he'd ask. _Do you believe in soul mates? In love that doesn't die?_\n\nHe didn't know what he'd do if she said yes.\n\nBut he was willing to find out.\nFATE DOGS\n\n_Robert A. Hoyt_\n\nHe was a somber gentleman, of the sort who might be a CEO, and as it happened that afternoon, he was manning one of the many roach coaches in New York on my lunch break. He had in his hands a copy of the _Times_ but for some reason the picture on the front page wasn't quite visible.\n\nAs I walked by, he lifted his head, and in the practiced voice of a street salesman said, ''Fate in a bun, sir?''\n\nI paused in mid-step, and turned to look at him.\n\n''What?''\n\n''Fate in a bun, sir. I can add mustard if you like. No extra charge.''\n\nI looked up at his sign, which sure enough said, clear as day, FATE, in big bold letters, printed on gaudy yellow and red stripes.\n\nThe man behind the cart, clean-cut and wearing an expensive designer suit, was as out of place as a cabbie speaking Oxford English. Part of the act?\n\nI had to admit, it was original. After a moment I reached discretely for my wallet.\n\n''What the heck. I admire the gimmick. How much for a 'fate'?'' I asked, a crisp five-dollar bill between my fingers.\n\n''Five dollars even will get you a Business Fate, sir. Would you like anything with that?'' His expression didn't seem to contain even a hint of irony. I guessed it was for the believability.\n\n''Do you have any relish?''\n\n''Always, sir.'' And he reached behind the mysterious semi-counter that all of the hot dog vendors have on their cart.\n\nA couple of moments later, he handed me a steaming hot dog. At least I imagined it was a hot dog, although I couldn't really tell through the steam.\n\n''And, as ever, all of our fates our guaranteed, or your money back,'' he said, while carefully filing my bill away in a cash register that I didn't remember seeing before.\n\nI decided that it would be prudent not to attempt to interfere with his business approach, and took a bite of my hot dog.\n\nFor a second, I didn't taste anything. There was a sort of faint, watered-down impression of a sensation, as if I were remembering eating a hot dog before.\n\nAnd then, through the mists of my mind, came things that I had only half heard.\n\nThe grain company that my boss was about to drop all of the stock on due to a twenty-point decrease in the last few days. The market analysis that I glanced over briefly on the subject.\n\nAnd the twelve years of market investment experience coming into sharp focus between the two.\n\nA sweet taste flooded my mouth, like dollar bills dipped in honey.\n\nSuccess.\n\nLaboring with the weight of my sudden realization on my shoulders, I wolfed down the remainder of the hot dog, taking off in a running sprint for my office.\n\nThe lobby of the building was empty, as it always was at lunch hour. The air felt heavy with the scent of fresh coffee in twelve stories of offices, and the faint muffled tone of groundbreaking meetings being held in the remote reaches of the building.\n\nBriefly, I paused by the elevator. But the instructions in my mind burned like a red-hot poker inserted into my forehead, and I couldn't wait.\n\nHalf crazed, I charged through the door to the stairs, and pushed my legs to their limits as I ran the seven flights separating me and my office.\n\nI burst out into the silent reception area, and frantically greeted the secretary.\n\nShe was a stunning blonde named Sophie, who had surprisingly not developed a junkyard dog personality in so many years of playing mother hen for a little over a hundred disgruntled employees.\n\nShe was beyond beautiful to me, but then if I had to tell the truth, I was in love with her. Unfortunately, I knew she was out of my league.\n\n''Sophes, will you buzz me in please? I forgot something on the computer.'' My voice wavered a little. She smiled.\n\nI would have smiled, too, but it was all I could do not to rip down the door. I had to get to my computer.\n\nShe took my keycard and buzzed me in.\n\nThe screen saver lazily taunted me, a glaringly colored bouncy ball striking the edges of the screen and bouncing through the middle again. I moved the mouse, and with a speed that I had never had before, struck keys in rapid succession, first bringing up the stock reports, and then the information for buying the stocks.\n\nThe stock was running at a lowly twelve dollars a share.\n\nI paused as I decided where to find crazy investors. Finally, I bought a large block of stock from my own account, but not before I called up a few friendly clients who liked taking risks.\n\nAs my own little gift to her, I sent Sophie an e-mail with my hot tip.\n\nWith a final sort of fortitude, I pressed the enter key to accept the transaction.\n\nThree days later, the grain company announced a major breakthrough in genetic engineering. At the press conference, the representative revealed that their long awaited zero-carb flour called Puff was finally developed, tested, and slated for rapid approval. With luck, it would be hitting the shelves within the year.\n\nThe representatives from the FDA, for once, agreed.\n\nThe ticker price skyrocketed. In less than a week, the company was running strong high seventies, and was expected to continue to increase.\n\nIn the meantime, the adrenaline rush of investing my life savings on a whim had caught up with me. I even considered confessing to the boss.\n\nAnd I did, when the timing was correct.\n\nI hadn't had time to eat lunch for a while, and I hadn't gotten another fate dog since the first. But when the ticker price on the grain company went through the ceiling, I did tell my boss about my little personal venture.\n\nThere was a silence that lasted about two minutes, as he seemed to ponder the situation in a serious manner. Finally, he looked up, put his fingers around the cigar that wreathed his face in smoke, and said ''You took a risk . . . you followed your gut. And it seems one of your tip buddies told a friend at one of our biggest clients, so they're mighty happy with us, too. Far be it for me to criticize one of the few workers around here with balls.''\n\nI thought I had just barely survived, until a week later a promotion came through. I was suddenly a whole new level of business broker, and my paycheck rescued my bleeding savings account from the incident.\n\nEven Sophie pulled out a tidy little sum, her somewhat shy smile as I walked past in the mornings an ever-present reward.\n\nAnd the more that I considered this unimaginable and unforeseeable run of good luck, the more trouble I had with telling myself it was all a coincidence.\n\nBut the extra wad of money in my wallet wasn't enough to justify it as true for my subconscious. And the more I thought, the more I realized that only one thing would satisfy my curiosity on the subject.\n\nI had to get another fate dog.\n\nSo when my lunch period came around that Thursday, I used the elevator, and walked out on the street to find the vendor.\n\nI laughed at myself when I saw him in the same place he had been then. I was almost afraid that he would have left.\n\nAs I walked up, he carefully put down his newspaper, folding it down the middle, and without yet having made eye contact said, ''Fate in a bun, sir?''\n\n''Yes,'' my mouth said, before my brain could attend to the matter. Wrestling for control, I added, ''But could you tell me the menu first?''\n\n''Business for five, Gambling and Dispute for ten, twenty for Love, and, of course, our famous fifty-dollar Chance, sir. Chance covers what the others don't.'' he said dryly. His businessman's tone could have fit in without a single turned head at a board meeting. ''Difficult to get prepared, though, sir. That's why it's more expensive.''\n\nI thought about it for a second. In the end, Chance seemed the most likely prospect. But something about his tone put me off the idea. I wasn't really sure what he meant, but it didn't seem entirely pleasant. As for Dispute, that was something that the newly rich do well to avoid.\n\nBut if Chance and Dispute were out, then the most likely one would be . . .\n\n''Gambling. I'll have a Gambling dog, please.''\n\nOnce again, the same complicated technique with reaching behind the counter, and a steaming bun whose contents were just barely obscured emerged.\n\nI stared down at it. For a second I felt somewhat apprehensive.\n\n''No, it's just a silly urge. A lucky strike. Nothing to be afraid of,'' I told myself.\n\nI rammed the hot dog in my mouth quickly, before I could think of a rebuttal and let my good sense get in the way.\n\nOnce more, the watered-down hot dog taste, like eating something extremely far away. And then, without warning, I knew more than I had ever realized about the world around me.\n\nI could taste the things around me happening, and I knew at once where the money was being made and lost.\n\n''No!'' This was in my best Telling-a-Dog-It's-Done-Something-Wrong voice, and aloud, which fortunately didn't make me stand out from all the street crazies around.\n\nBut I had to interject some willpower.\n\nToo little, too late. My legs were off on their own. Like a deranged maverick, I cut through the crowd of lunchtime commuters.\n\nI would have felt some sort of fatigue, I suppose, after the first mile, if I had paid attention, but I was more concerned with this urge to keep going.\n\nFinally, my legs drew me to a halt in front of a local gambler's haven. It was a grand and exceptionally notorious establishment in my circles, hidden behind a grubby iron door in a back alley, ignored by police who didn't want to get certain people angry.\n\nMy legs took me through the front door. The floor of the gambling den bristled in my senses. And then, I saw the roulette table.\n\nI couldn't hold myself back. I ran past the bouncer, who vouched for me to the boss with a nod, and entered the game with ten thousand dollars.\n\nThe wheel spun hypnotically and the whirl drew me in. A man standing next to me took me for a tourist, snickering under his breath.\n\nHe didn't expect my money to last too long.\n\nBut my brain screamed at me. Red 32.\n\nWith no choice but to obey, I placed all of my cash on Red 32. The wheel whirled, the little white ball clicking along rhythmically.\n\nTime slowed down for a moment as the ball came to a stop balanced between two numbers, finally landingon Red 32 with a little plunk that took up the majority of my consciousness.\n\nThree spins later, a couple of men came up to me and ominously requested that I retire for the evening. I knew better than to say no.\n\nI cashed out, getting a standard tin suitcase to hold my phenomenal winnings\u2014less my original stake, which I pocketed for ready cash\u2014hailed a cab, and took it home.\n\nBut as I stepped out of the cab I heard a click behind my head.\n\nMy thoughts cleared as only a cocked hammer on a powerful handgun can make them.\n\n''And now, sir, if you would be so kind as to hand over the suitcase, then I believe that we can both call it a night,'' the barrel pressed into my neck.\n\nI tried to calculate my chances in my head, and realized that my ability to gauge odds had deserted me, like my heightened ability to sense the air around me.\n\n''Very interesting,'' I said, to no one in particular, as I handed the suitcase to the man who had come up behind me. ''Disappears under high duress.''\n\nBy this time, my cab driver had taken the opportunity to get far away from us both.\n\n''Much obliged. And now, as I can see I have made you uneasy, I shall be glad to repay you by helping you sleep.''\n\nA gun handle clocked me like a cavalry charge, and I fell to the ground.\n\nThe early morning sun shone on my face. I was in the suit that I had been wearing the day before, slumped over on my front doorstep.\n\nThe quick run over my senses acknowledged that I had the sort of deep-seated headache that seeps into your brain and permeates every single nerve with pain.\n\nI looked at my watch, which luckily I still had.\n\nBut then again, in my state, I didn't look the sort of person who would own a watch anyway.\n\nMy spine went as straight as a rod. I was an hour late for work.\n\nNot even bothering to dress in fresh clothes, I stood up, and felt in my inside pocket. A sheaf containing ten thousand dollars\u2014less last night's cab fare\u2014came to hand.\n\nI hailed a taxi. Ten minutes later, I was standing in the reception lobby of the ninth floor, where my promotion had booted me.\n\nOf course, it looked like the same lobby as the seventh, and more importantly, Sophie was at her desk.\n\nShe looked at me in surprise, ''Are you all right? You look horrible.'' She seemed genuinely concerned.\n\n''Fine, fine. Could you just check me in, Sophes, and hold my calls for a while? My head hurts a little. Nothing more.''\n\n_A little meaning horribly,_ I added to myself.\n\n''Normally, I would. But the boss asked to see you as soon as you came in. You didn't check back in yesterday.''\n\nDamn. I had forgotten to come back to work in my starstruck haze.\n\nI would lay even money now that the boss was not about to offer me a CEO position. If I was lucky, he might offer me references after he fired me.\n\nI should be so hopeful.\n\n''Good luck.'' She said, as she buzzed the private elevator with the button under the desk.\n\nMy humiliation was complete. I would be fired after showing up to the office and attempting to get the girl that I loved\u2014I supposed it was high time I admitted it\u2014to check me in, immediately after I disappeared randomly . . . all the while looking like a bum.\n\nGambling fate. Lovely.\n\nI stepped into the elevator. The doors shut with a ding and it began to whir along quietly, passing floor after floor.\n\nFinally, the moment to face the music came. The elevator doors opened, revealing the boss, looking stern behind the thick, smelly cigar smoke.\n\n''Take a seat,'' he said, hand gesturing toward the leather guest chair.\n\nI did so, the red carpet thick like molasses under my feet.\n\n''I hear that you disappeared yesterday. Care to elaborate?'' he asked, with a sort of trap taunting me in his voice. You could almost hear a sort of half-hearted chuckle in his question, as if he were enjoying a fine irony at my expense.\n\nI considered telling him about the fate dogs, and the inexplicable urge to go and gamble impossibly large amounts of money, and the whole shebang.\n\nAnd then I asked myself why I felt it necessary to get into therapy, and the loony bin, when I was already getting fired.\n\n''I ran into a client,'' I said, my mind spinning an elaborate yarn. His eyebrows rose.\n\n''Who?''\n\n''I don't remember. We went out for drinks, and then I came back around. But a man mugged me, and clocked me on the head, so I don't feel so well, as you may imagine.''\n\nA complicated alibi needed more memory than I had.\n\nHe stared at me for a moment.\n\n''I see,'' he said, and then after a minute asked, ''Can I get you a drink?''\n\nWhy was he toying with me?\n\n''No, thank you.''\n\n''I didn't think so.'' He shook his head and leaned forward. The desk creaked under his weight.\n\n''How about you just tell the truth, kid? You met some blonde piece who you helped invest, and you have a hangover from drinking enough alcohol to stock an infirmary. I was young once, too.''\n\nHe got up, poured himself a gin, and downed it in a single shot.\n\n''Look, just don't let it happen again. We can't have you disappearing in the middle of an important deal. Now go on. Use the executive gym and get yourself cleaned up.''\n\nI couldn't believe it. I walked over to the door of the elevator. It made a little ding, and opened. The boss stopped me by the door.\n\n''Remember, kid, I have hopes for you. Do me and yourself a favor and don't piss it away.''\n\nI just nodded.\n\nAfter the ride down, I was greeted with Sophie's face. She was waiting expectantly.\n\n''Which way to the hotshot's gym, Sophes? I need to freshen up a little.''\n\nI handed her my access card, and she swiped it for mods with a smile I hoped was because I had not been fired.\n\n''Down the hall and to the left. Congratulations on surviving!'' she called to me as an afterthought.\n\nI just smiled halfheartedly, and headed for the shower.\n\nFor a day or two, I was too scared to think about the hot dog stand.\n\nBut hope springs eternal.\n\nThe more I thought about it, the more I fell for Sophie. I even occasionally toyed with the idea that she might feel the same way about me.\n\nAnd if there was anything that might give me the fortitude to charm her, I had to try it. And then I thought of the Love option the man had mentioned.\n\nIt was insanity, I knew. I was throwing good money after bad, I knew.\n\nBut . . . I knew I would regret it if I never tried.\n\nI waited carefully, biding my time until Friday came.\n\nThat way, if something went wrong, then at least it wouldn't interfere with work. I wasn't taking any chances. I waited until the day ended to find the hot dog vendor, sitting in the usual place.\n\n''May I ask you something?'' I asked the man in the booth, and waiting only long enough for him to glance up at me, plowed ahead ''I know that this is a crazy question, but will a 'Love' make me able to charm a girl whom I already love?''\n\nHe stared at me blankly. ''Yes,'' he replied flatly, without thought.\n\nI forked over a twenty-dollar bill.\n\n''One Love, please. Mind you, remember the ketchup.'' I was half deranged with precautions.\n\nSeconds later, I held in my hand a Love dog. I didn't even think about it. There was no argument, I just ate it.\n\nAnd even as I bit into it, I realized that I had only asked if I would be _able_ to charm her. And it struck me very hard that I likely wouldn't _want_ to.\n\nHowever, I could already feel the beginning of the craziest night of my life pressing itself into my mind.\n\nI knew where to go, and I knew what to do for the night.\n\nMy first stop was a local nightclub.\n\nAlmost involuntarily, I greased the doorman with a hundred-dollar bill, and slipped through the door.\n\nI was there for half an hour. In that time I got two drinks and two girls who had had far too many drinks.\n\nI don't recall their names.\n\nI don't even recall offhand whose place we went to.\n\nTwo hours later, I woke up in the bed of whomever's place we had gone to, in between the two girls.\n\nI resisted the urge to peek beneath the covers, even though I knew that if they were in the same state I was in, I would regret that for the rest of my life.\n\nI got dressed hurriedly, took a quiet shower, and left the apartment with the other two still sleeping.\n\nIt was early in the morning. The majority of the city was undoubtedly still asleep.\n\nI walked beneath the city lamps, the early morning chill clearing my head. It was a quiet part of the city, and there were no people on the streets.\n\n_Well, so much for saving myself for marriage._ Or, somewhat more urgently, just facing Sophie. The worst part was that I remembered everything because I kept thinking about Sophie. I had had a night that other men only dreamed of, and I had actually hated every minute of it.\n\n_Congratulations,_ I said to myself. _You have officially gone crazy._\n\nBut who cared? I had pretty much blown it anyway. My last chance to get Sophie was gone. I had used the fate dogs, and they had reacted perfectly.\n\nBut not the right ''Perfect'' for my purposes.\n\nI was ready to cry. I saw worst-case scenarios rain down, each one with her finding out in more and more uncomfortable ways.\n\nIt would take a miracle.\n\nThe man said Chance dogs handled ''everything else.''\n\n''Forget it,'' I said aloud. ''It isn't going to happen.''\n\nI wasn't worried about looking insane. There was no one around, and besides, I might well be insane anyway.\n\nBut that prying little voice that had gotten me into this situation in the first place persisted.\n\nAnd little by little, I broke down, until I found myself standing on the street where the vendor normally was, in spite of my better judgment.\n\nFor a moment, I peered into the night. I told myself that I wouldn't see him.\n\nFor a second I believed it. But I knew what I wanted.\n\nAnd then, even in the fog of the predawn glow, I could see the gaudy yellow and red stripes.\n\nThey wrapped around me like a blanket, and I took a step foreword.\n\nI would just buy a Chance dog, and then I could have Sophie.\n\nThat was how the world worked, I knew it.\n\nI walked forward, and came to a stop in front of the vendor.\n\nHe looked up in anticipation. The air grew heavy and thick, as if he were simply waiting for me to give a request that he already knew.\n\nEven in the city that never sleeps, there is a lag in pedestrian traffic at five in the morning on this street.\n\nBut looking around, I suddenly realized that I was standing alone on an empty street. Not even a car in the road.\n\n''Can I help you?'' he said. His slate gray eyes fixed cold and unfeeling on mine.\n\nMy skin crawled.\n\nIt didn't feel right.\n\nIt wasn't right. But I needed to do it. It was my only chance with Sophie.\n\nBut that wasn't right, either. Even I knew that. Even before, we had always talked. We knew each other. Maybe I could have a chance anyway.\n\nA chance at a little more than a one-night stand. That was really what I had been after, even if I hadn't admitted it.\n\nI felt my hand go to the money in my pocket.\n\nI was about to spend fifty dollars. I was about to lose control of my decisions again.\n\nCould I really afford that?\n\nI paused. The tiny little itch in my mind suddenly had a megaphone, and was screaming in my ear, in none too diplomatic terms.\n\nI never really could control the fates these dogs gave me, and telling myself I could was madness. I knew in my heart that another fate dog would just make me lose control again.\n\nAnd at the same instant I realized that none of them could possibly act as I would. It was like becoming someone else, someone who loved other people, knew other things.\n\nWanted other things.\n\nAnd none of them were Sophie.\n\nFinally, defiantly, I seized the reins of my own willpower.\n\nI stared the man at the stall in the eyes.\n\n''Good morning,'' I said, with a slightly ironic smile on my lips, and walked away.\n\nThe man in the stall went back to reading his paper. The sun peeked over the horizon, showering the streets with gold.\n\nAfter all, if I still had a chance, then I wanted to make the most of it.\n\nI thought about where she lived.\n\nIt wasn't a long walk, really.\n\n''Besides,'' I chuckled to myself, ''I feel lucky.'\nTHE MAN WITH ONE BRIGHT EYE\n\n_Jay Lake_\n\nThis is a story about the Lord Douglas-Ouyang, who was born to a life ordained by the group-mothers of the worlds. This is a story about how the Lord Douglas-Ouyang met the Queen of Roads and defeated her knife golem. This is a story about how the Lord Douglas-Ouyang vanished beyond the rainbow wall. Most of all, this is a story about how the Lord Douglas-Ouyang cheated fate.\n\nNow, like life itself, the story begins.\n\n''He is born,'' proclaimed group-mother Anna-Chao to her colleagues arrayed in council at the Bright Oval. Her hair streamed on a time-wind, blown back so far no one save a Kroniate adept could have seen the other end. Some said she had lost\u2014or perhaps hidden away\u2014her favorite lovers in the years-long amber tresses. Whatever the fate of her hair, and those who might have climbed it down the well of entropy, her eyes were firmly in the present. Blazing with chips of congohelium, in point of fact, so that her gaze cast shadows.\n\nThe other group-mothers bowed their heads in varyingcombinations of ill-suppressed resentment and ophthalmic self-preservation. None objected.\n\nAnna-Chao reached a few hours into the past and unswaddled a sleeping baby from an eddy in her hair. The child was not much bigger than the palms of her two hands but he was fat in that way that babies are. His pate was bald, lacking even the ordinary fuzz clouding an infant's scalp. Some of the group-mothers noticed he had no knot upon his belly, just smooth skin from groin to nipple, but they had forgone their right of objection.\n\n''I give him vision,'' Anna-Chao said. She used her left pinkie to hook her left eye out of her head, then popped the same eye from the sleeping child. He woke with a pained screech, but Anna-Chao slid her blazing orb of congohelium into the baby's socket, and he settled down again. His natural eye, pathetic, bloody thing that it was, she tucked back into her hair. ''He shall be the Lord Douglas-Ouyang.''\n\n''Homage to the group, homage to the worlds, homage to the Lord Douglas-Ouyang,'' the group-mothers intoned. One by one they came to kiss the child, then filed out of the Bright Oval to scuttle off for coffee or sex or some other diversion, and to gossip over how Anna-Chao was passing ever farther out of phase with the realities of life.\n\nThe little Lord Douglas-Ouyang was seven before he understood that not everything he saw was real. Or even imaginary. His one bright eye frightened other children, though its tendency to occasionally set fires brought them giggling back to him again. (He learned to stay away from the bracken of his forest home when the weather was dry.) With that eye he saw seas of honey-colored hair, the fate of men, and sometimes what was going to be served for dessert later in the week. With his other eye, colored the gleaming blue of a spring sky, he saw the usual things of childhood: hammock bowers, teachers clanking and chuffing up their morning head of steam, Improbables springing out of traps in the soil to go about their odd business.\n\nBoth eyes saw rainbows, of course.\n\nHe was still bald, though wrapped in a dignity that no amount of hair might have given him.\n\nMaristella was the child the little Lord Douglas-Ouyang was closest to. She had eyes the color of their cr\u00e8che-forest, hair the hue of rhododendron bark, and could run faster than any of them. The little lord and Maristella often slept in the same hammock. ''You will be a hero,'' she said, cradling him close and stroking his scalp. ''Just like Vuang.'' She whispered stories of champions from ages past: Vuang of course, as well as Rose-Kolodny, Churchill, Piotr the Great, Alaric\u2014 the list ran on and on.\n\nDouglas-Ouyang knew that she was right. His one bright eye saw futures of gleaming swords and chanting crowds and honors given on tall reviewing stands. ''Please,'' he asked her. ''What must I do to be a proper hero?''\n\n''Be like the great ones out of time.''\n\nSo, fighting with sticks among the deep green, the Lord Douglas-Ouyang made it his business to be like the great ones out of time.\n\nWhen the Lord Douglas-Ouyang was fourteen, liveried men came and took him away from his weapon and his women. He was dancing naked and armed on the edge of a cliff, his hairless body gleaming with olive oil. Far below his feet a frothing fall flickered with winged salmon making their way up the gravity gradient. His three lovers of that week, Althea, Naomi, and Merope, sat at the edge of the grove, legs tucked beneath them, their pert breasts in no way distracting the lord from his sword dance.\n\nThree men arrived without warning through a door of air, which instantiated to reveal gleaming lights in darkness beyond. They were tall, broad-chested, with the heads of dogs above their scarlet collars with the silver piping. One snarled at the girls, who fled screaming into the flicker-leaved aspens beyond the lord's dancing ground. The other two moved to seize the Lord Douglas-Ouyang.\n\nThe lord stepped into their attack and pierced the shoulder of one, a proper hit. The man did not yelp, but simply ran Douglas-Ouyang down, snatched away his foil, and broke his arm in the three places.\n\nHis fellow lashed restraints to the lord's feet.\n\n''That was _not_ sporting,'' Douglas-Ouyang yelled as they dragged him to the door of air.\n\n''You will trouble me no more today,'' growled the one he'd wounded. ''If you do not fight to win, why fight at all?''\n\nIndeed, he thought, but he kept his bright eye shut, for he wanted to know who had sent him this attack more than he wanted to burn away the faces of these miserable miscreants.\n\nIn darkness reeking of machines and stale air, the Lord Douglas-Ouyang's captors dropped him before six women seated in an arc. He could smell their sex upon them, and hear their breaths quickening as they looked him over.\n\n''You are much changed since we saw you last,'' one said.\n\nDouglas-Ouyang knew by the catch in her voice that she felt desire for him. He rolled onto his left hip, favoring his broken arm and allowing them to gaze upon the beauty of his youth. Let them be distracted\u2014 he knew who these women were now. His mother's friends.\n\n''Greetings, group-mothers.'' His bright eye saw them in a different place, filled with light and water, being harangued by Anna-Chao. ''You are most brave to lay hands on me so.''\n\n''We have already destroyed the dogs.'' A different voice this, but also eager. Distracted as well, judging from the cadences. ''No one else will mark this moment.''\n\n''Save me,'' said the Lord Douglas-Ouyang. ''I do not expect my memory to lapse in the foreseeable future.''\n\n''A threat?'' This, from a third.\n\n''I do not threaten.'' Douglas-Ouyang kept his voice mild. ''I merely remember.''\n\n''Enough,'' said the first. ''You were made for a purpose. This is why you are the Lord Douglas-Ouyang, and not merely a man in your own right. The time has come to prepare you for the next stage of that fate.''\n\nHe opened his bright eye and gazed upon them with the fire of congohelium. Though none of the group-mothers had forgotten what gift Anna-Chao had bestowed upon her son, they had not realized his control of the forces behind the door of his eyelid.\n\nThe Lord Douglas-Ouyang moved his gaze from group-mother to group-mother. Each shrieked or moaned, covered her face, cowered in her robes. Their chairs were not thrones, merely folding metal seats. The great machines behind them showed in the brightnessas nothing more than racks lined with broken automata, some still struggling weakly or mouthing forgotten speech.\n\nAnd they were dumpy, pale, flowing with flesh. Nothing like his sweet girls back in the forest, who were lithe as fawns, able to wrap themselves round him and each other like grapevines.\n\nHe shrugged off his bonds with the power of his bright eye, healing his arm in the same moment. Then he stood, and walked close to each seat in turn. His nakedness swung before the group mothers as the Lord Douglas-Ouyang stared each down, then moved on.\n\n''No one tells me what I will or will not be. No one tells me what I am capable of.'' He let them see, and scent him, before bounding off naked into the darkness, away from his fate and toward whatever chance might bring him next.\n\nWhen the Lord Douglas-Ouyang was twenty-one, he went traveling among the worlds of man. The Second Great Road ran through all of the major cities of all the worlds, and many of the minor ones. To use that storied highway one was generally required to travel in the company of a Kroniate adept, or else purchase a ticket on one of the \u03b4-coaches, great silver and copper Faraday cages powered by force-grown whale muscles immersed in nutrient tubes. Kroniates were scarce and rarely for hire, while the \u03b4-coaches were the bane of ethicists everywhere; they also possessed a distressing tendency to leave their passengers with unscratchable itches deep inside their backs and buttocks.\n\nBecause of the congohelium embedded in his bright eye, the Lord Douglas-Ouyang had found he could simply walk the Second Great Road. So he adventured, dressed in the simple azure robes of a _na_ paladin. Some cities taxed his weapons, others set men to bar his way or fight him. All bowed before him when the lord grew bored with struggle and presented his parentage.\n\nThere came a day when the sky above the lord's head had just segued from a pale violet to a creamy mauve. The Second Great Road had been surrounded by the spare windworn rocks near the desert city of Port Desire. In that moment the highway made transit into the redrock highlands above the ancient pleasure domes of Eura-m\u00e9l-Khos, where even the very air is perfumed with the scent of youth in ecstasy. The Lord Douglas-Ouyang smiled to see Eura-m\u00e9l-Khos\u2014many travelers never reached those fabled precincts, and he had a mind to sample such delights as ''three virgins in wine.''\n\nThe creature that rose from the fractured pavers before him was a nightmare out of darkest legend. It was all metal, with a thousand edges, bundled blades and whipcord saws and glittering scalpels, a whirling serrated disk for its cyclopean eye, and a mouth made of needles flexing open and shut.\n\n''Who are you?'' the Lord Douglas-Ouyang asked, drawing his chromed yatagan with the circuits printed on the blade. (He had taken up the yatagan which his mother had prepared for him when the foil proved too predictable, not to mention easily broken.)\n\n''He is mine,'' said a woman behind him.\n\nThe Lord Douglas-Ouyang turned to see the most beautiful woman in all the worlds. She was of moderate height, with a good set of hips beneath the fall of an orange robe, hair stubbled short and brown. But she carried within herself a soul-stealing beauty that nearly drew him down.\n\nThe lord cut a charm through the air with his yatagan, the circuits glowing as the blade whipped. Her glamour faded some, though his body still strained with desire.\n\n''All men are yours, lady,'' Douglas-Ouyang said with a clumsy attempt at gallantry.\n\n''Most do not draw blades in homage.'' Her voice was mild, but still she nodded past her shoulder. ''Kill him for me.''\n\nThe Lord Douglas-Ouyang did not know whether she spoke to him or to the creature of blade and edge, but he supposed it did not matter. In either case, the command was the same. This was the first great battle on his hero's journey. He moved even before she was done speaking, his _na_ -paladin's robes hardening into ballistic fabric as the yatagan began singing a death song in three voices\u2014the whistle of a microscopically sharp edge, the hiss of cloven air, and the hum of ancient circuits brought fully to life.\n\nHe was made and born to be the greatest fighter of his age. He was blessed by a parentage both magical and mighty. He carried a weapon which gods would have squabbled over. Still the Lord Douglas-Ouyang could do little better than stand his ground against a creature that had more blades than a sunflower has seeds, and that spread them wide as a fisherman's net. Every twisting block the lord made was surrounded by a cage of steel. Each canny, curving stroke met a counter that seemed to have been woven from his own thoughts. Every low thrust was caught with a lower parry and a dozen replies in the same moment.\n\nThey fought till sweat stood in the lord's eyes. They fought till the edges of his azure robes were lacy with cuts, while sparks striking off his opponent had set the roadside grass to smoldering. They fought till the shadows walked from one side of the highway to the other. The most beautiful woman in all the worlds watched with lips pursed and eyes glittering.\n\nFinally, as the lord's arms tired and his blade began to waver, she called, ''Halt.''\n\nHe and his opponent both stopped in the same breath.\n\n''My lady,'' the Lord Douglas-Ouyang said. He lowered the tip of his blade and surreptitiously took a moment's rest.\n\n''No one has ever stood before my knife golem so long.'' She walked around him, inspecting him. ''You are that child, are you not? Of the renegade Anna-Chao.''\n\n''Yes, she is my mother,'' he said, thinking of years-long hair the color of honey. ''And no renegade at all.''\n\n''The perfect man. Well, I am the perfect woman.'' The smile was far more disturbing than perfected. ''You may address me as 'your majesty.' I am the Queen of Roads. Bend a knee, sir.''\n\nThe Lord Douglas-Ouyang had bent a knee to no one in his life, not even the group-mothers. He would not start now. Instead he turned back to the knife golem and hurled the yatagan high into the air, trusting its circuits to know what to do next. He stepped close into the blades, trusting his _na_ -paladin's robes to accept their edges and turn their hurts. He opened his one bright eye and stared close into the whirling disk of the knife golem's eye.\n\nThe congohelium's translight flared off the spinning metal and fractured into more colors than the world could know, blinding both the knife golem and the Queen of Roads. The yatagan dropped from the sky right between the bent blades of the golem's head, shattering the engraved metal chem within.\n\nThe Lord Douglas-Ouyang snatched free his trusted blade and ran for the next transit of the Second Great Road, forgoing the pleasure domes of Eura-m\u00e9l-Khos and its wine-soaked virgins. Even as he plunged into the wet night beyond, he wondered what the Queen of Roads had intended for him.\n\nAt the age of twenty-eight, the Lord Douglas-Ouyang was finally ready to become a man. He had found his way back to the high castle of the group-mothers, above the Sea of Indolence. There he studied the law and the sciences and the arts. He killed a free man, and raised a slave to life. Word would come from time to time of the rampages of the Queen of Roads, but even the great Kroniate proctors could not seem to catch her at her crimes.\n\nHe sat with Anna-Chao on a terrace overlooking the worldforest that stretched east and south from the quiet shores of the sea. Her hair reached some hours into the past here before being folded by a mouse-bodied servitor who was doing maintenance.\n\n''I have come to understand more of you, Mother,'' the Lord Douglas-Ouyang said. ''You command secrets within secrets, but you are not too subtle.''\n\n''Not at all,'' said Anna-Chao. ''Others make what they will.''\n\n''You made _me_.''\n\n''That is what parents do.''\n\nThey both watched the doghawks bark and quarrel above the trees awhile, and enjoyed the breeze scented with lemons. Finally, the Lord Douglas-Ouyang spoke again. ''I do not understand one thing.''\n\n''Only one? You are to be congratulated.''\n\nHe refused to be diverted. ''I do not understand why you made me.''\n\nShe sucked air through her teeth a moment, in his experience a sure sign she was about to lie. ''I reach backward through time to the uttermost years, when men cowered from the tiger-haunted night in their primitive zeppelins and mud huts, praying to some lost god through vacuum tubes. But I only reach forward as far as I can see, and I am tethered by the present.'' One hand slipped up to pat her hair. ''You as my child, my son, are also my spear cast forward through the years to come. It is your fate to carry my fate with you. Ever it has been so with parents and children.''\n\nThe Lord Douglas-Ouyang stood and went to the railing of the terrace, leaning on the low limestone with his back to his mother. ''In earlier years, I would have rejected your weird out of hand. 'Say what you will,' I might have told you, 'but my fate is my own.' I know better since I met the Queen of Roads, who commanded me with words. I am not my own.'' He turned back to his mother. ''But when I step through the Chalcedony Gate and become a man, then I will be my own.''\n\nShe opened her one bright eye, twin to his, and seared him with the light of her regard, until the Lord Douglas-Ouyang saw only rainbows inside the translight of congohelium. He could not counter her attack, not his own mother's, so instead he let himself tumble backward over the railing. Unfortunately, he fell into the rainbow wall within the translight.\n\nThe Queen of Roads waited there for him.\n\n''You killed my knife golem,'' she said from atop a surge of bright maroon. ''You owe me a servant, or servitude.''\n\nOnce more her words possessed him, but the Lord Douglas-Ouyang had not wasted his studies. He closed his ears with snatches torn from yellow and smiled at her. ''I owe you nothing,'' he said, ''but I will give to you freely nonetheless.''\n\nBlind to her voice, he could not hear her response, but the lord took her by the hand. ''Walk with me awhile,'' he shouted, knowing his words to be over-loud from his temporary deafness.\n\nHer own beauty did not suffice to armor the Queen of Roads from Douglas-Ouyang's charms, and so she followed him.\n\nA hairless man with a strange gleam in one eye and a woman with plain but extraordinarily beautiful features stepped out of a door of air onto a corner of an ordinary street. He looked about thirty-five, she a little older. Both had a strange pallor to their skin.\n\nNo one noticed. Not in this city of pale stone and black roads and gray skies.\n\n''Here,'' he said. ''We will never need to go back. Or forward either.''\n\nShe tapped the side of his head. He pulled something bright yellow from his ear, that vanished amid a belch of diesel exhaust from a passing bus. ''Sorry.''\n\n''What about her?''\n\nAnna-Chao, the Lord Douglas-Ouyang knew she meant. He reached back into nowhere and retrieved his gleaming yatagan, circuits bright with the heat of metal thought. ''I can cut her hair with this, should it reach across the hours and years.''\n\nThey both looked around. Honey-colored tendrils would stand out in this place better than fresh blood on wedding sheets. She shrugged and picked a direction. He followed, the two of them shedding unnoticed rainbows as they went, until both were gray as the people around them.\n\nNo one even noticed the sword.\n\nNow, like life itself, the story ends.\n\nNot with death, or the grand agony of families and generations. Someone else was found to work the will of the group-mothers in the name of fate. Someone else patrolled the Second Great Road with a mystic weapon. Someone else fit what was needed, for someone else always does.\n\nBut the man with one bright eye and the oddly beautiful woman had their time together. When he was done with his life, he sent the sword back to his mother's latest hero, to tempt fate once more.\nA TAPESTRY OF SOULS\n\n_Paul Crilley_\n\nWe fought a war against God today.\n\nWhen I remember the events of this terrible afternoon, two images will stand out from all the others, stark against my mind like the black branches of a tree against a gray winter sky. The first was a brief glimpse I had of the Goddess, caught through a gap in the screaming hordes. She was felling seraphs with glances from her terrible, beautiful eyes, sending them plummeting to the ground with their wings ablaze, leaving trails of greasy smoke as they fell.\n\nThe other image . . . well, the other is the reason for all of this.\n\nYou see, it is my fault, this war. I bear the burden, but I feel no guilt.\n\nFor it was love that led me here, and love is pure, is it not?\n\nShe came to our village one summer's afternoon when I was twelve. I sat in the shade of the stables watching the horses flick their tails lazily against the heat, their sides quivering with irritation at the flies. My father was working in the forge across the street. I could hear the pounding of hammer on metal, its usually loud echo muted in the stifling air. The heat was a massive hand pressing everything down, dulling everything with its weight. I felt a twinge of guilt. How hot must it be in the forge? I was supposed to be helping him, but again I had managed to escape the work. It was not that I shied from any kind of hard labor; it was just that I did not want to learn the craft of smithing. We had argued, like we always did, and I had stormed out feeling frustrated and angry, because he asked the question he always asked me when we fought: What did I want to do with my life then? I didn't know, but he couldn't understand that. He was a practical man, my father.\n\nI had just resolved to head home and fetch him a tankard of ale by way of apology when I saw the air above the crest of the hill outside the village begin to shimmer.\n\nI wiped the sweat from my eyes and peered hard, thinking it nothing more than the heat waves rising from the ground.\n\nBut then I watched as the shimmer drew closer, floating slowly above the ground.\n\nA voice shouted suddenly behind me. Stef, whose turn it was to make sure the talismans around the village were still intact, stood in the middle of the dusty street staring over my shoulder. He shouted again and ran off. I could not tell whether it was fear or excitement. Possibly both. I stood up and sprinted after him.\n\nThe town square was filling rapidly by the time I arrived, panting and sweating and nursing a stitch in my side that was like a needle point under my ribs. A discordant clashing of noise crashed around the square, people jostling and shoving each other as they vied for positions they thought would be the most advantageous. I hung back, uncomfortable, not understanding the barely suppressed violence I seemed to sense in people who were normally the best of friends. Others my age were huddling in the doorways of the buildings surrounding the square. I saw my friend Kian being pushed through the mob by his red-faced mother.\n\nOther things, images and sounds, assailed my senses until I felt myself suffocating beneath their weight: Kian's mother slapping Mrs. Kenndir, raking her cheek with ragged nails when she failed to move out of the way. Sela's father raising his arms in the air as he tried to calm people down. A feeling of building pressure, like the air before a thunderstorm.\n\nMy ears popped. I felt myself falling, my vision blurred behind a sudden veil of tears I did not understand.\n\nThen someone propped me up. I wiped my eyes and saw my father smiling down at me, his eyes excited. He ruffled my hair, his way of saying sorry. Then he cradled his hands together and boosted me up onto one of the empty ale barrels that the oldsters used as gaming tables outside the tavern.\n\n''They notice everyone, young one. No need to get crushed with the fools.''\n\nThen silence fell, utter and complete. I looked up and saw everyone staring behind me. She was coming toward us, no longer a vague presence in the air, but a woman, the most wondrous woman I had ever seen, and floating four feet above the ground. I squinted at her and saw that the shimmering I had seen earlier formed the shape of half-seen wings, extending ten feet to either side of her body. She drew abreast of me. Her long black hair floated in the air behind her, drifting in silent, undulating eddies. A strand touched my face, a spark of energy. I must have cried out, for she turned to me and smiled. Her hand came up. She stroked my face, wiping my forgotten tears away.\n\nMy life changed in that instant.\n\nI stared at her in awe. ''I love you,'' I said simply, so low that even my father directly below me could not hear. But she did. Her smile widened. Her deep brown eyes looked into mine.\n\n''Thank you,'' she said. Her lips did not move as she spoke, though I heard her words clear as the clang of metal on a winter's morning. ''I love you, too.''\n\nHer hand lingered for a moment, and then left my face. She turned her eyes forward and drifted higher into the air, over the heads of those watching me jealously.\n\nMy heart felt empty. I wanted to cry out for her to return, to put her hand back on my face and leave it there forever. I wanted to look into her eyes again, for I was sure there was something hidden there, some unknown truth that I would discover could I but stare long enough.\n\nI watched her all afternoon. My father told me she was a messenger from the Goddess who lived in a thousand-foot tower somewhere to the north of us. She carried petitions and grievances back with her so that all could be judged fairly. He said the tower was made from the frozen tears the Goddess wept when her lover deserted her and brought night to the world, the time when evil walks and good must hide.\n\nHe also told me of a man who thought himself powerful enough to rule over a section of countryside in his own name instead of that of the Goddess. Her anger was terrible, and she rained fire down upon the man and his subjects as a lesson to all.\n\nI shivered in the cooling afternoon, my earlier sweat now clammy on my body. My father left to close up the forge and stoke the fire so it would not die. I watched her face as each person stepped up to speak to her. I don't think it was my vanity, but I did not see her smile at any of them the way she smiled at me.\n\nThere were few people left now. She would be leaving soon. My heart ached, my stomach clenching unpleasantly at the thought. I was scared I would never see her again.\n\nTwilight faded into night as I made my way up the road outside the village. I walked for half an hour until I came to a large tree growing by the side of the road. I was relying on my hope that she would carry on through the village and follow the road northward.\n\nI sat down with my back to the tree's rough bark, staring up at the stars and listening to the insects chirruping in the bushes. I stroked the feathered talisman I had taken from our home. It was a strange feeling to be out at night. I'd heard stories of the horrors that scoured the land every time the Goddess went to sleep and her Betrayer awoke, but the fear I expected did not come. I think it was the fact that I felt safe with her close by rather than any benefits the talisman might bring. It was as if she held the evil back, that while I thought of her I could not be harmed.\n\nA cool breeze brushed my cheek and I thought again of her touch caressing my face. I closed my eyes and settled in to wait, but it was not long before something made me sit up again.\n\nI peered out from behind the tree and saw her drifting slowly up the road. I stepped out from my place of concealment and waited as she drew closer.\n\nShe stopped in front of me, her wings more clearly visible now as they picked up silver highlights from the stars. They vibrated gently in the air.\n\nI performed a clumsy bow, then saw the leaves and grass that stuck to my leggings as I did so. I flushed and tried to brush them away. I could see a small (beautiful!) frown of puzzlement etching her forehead.\n\n''What are you doing so far from the village?'' she asked, and again her mouth did not move.\n\n''I . . . I came to see you. One last time. Before you go. You see . . . I really do love you. I didn't just say that.''\n\nShe stared at me for some time. She drifted to the ground, then bent her knees so that she was kneeling before me. Horrified, I tried to get her to stand up, then realized that I shouldn't even be touching her. I snatched my hands back.\n\n''Please!'' I said. ''You mustn't kneel before me. Please stand up. Please!''\n\n''Hush,'' she whispered, and put a finger to my lips. Startled, I looked at her again and saw a single tear slipping from the corner of her eye, white and glowing like mother-of-pearl in the moonlight. Instinctively, I reached out and wiped it away with my finger. It rested there awhile, still glowing, then it slowly faded to nothingness.\n\nShe reached out and hugged me. I hesitated, then hugged her back, feeling my own tears flowing onto her chest.\n\nWe stayed like that for a long time, the stars winking at us, the moon moving gently in the Heavens.\n\nFinally, she pushed me gently away.\n\n''I must go,'' she said. ''Keep you compassion, Crispin. It serves you and others well.''\n\n''I . . . I will,'' I said, watching as she lifted gently off the ground. ''Good-bye.''\n\nShe smiled. ''Good-bye, my little love.''\n\nI followed her, of course. I'm not sure why. I think I had some foolish notion of watching over her, protecting her on her journey. I felt somehow that _this_ was what I was meant to be doing. I didn't know what _this_ was, just that it meant something to me in a way that hitting bits of metal into tools and weapons never could. I waited until the moon on her wings was no more than a distant glimmer, then set off in her wake. I think she knew I was following, for she set a pace that I could easily follow. She even stopped after a few hours to rest. I crept up the road and hid behind a bush. She floated in the air above a field off to the side of the road, to all intents and purposes asleep. I tried to stay awake but my body refused. Soon I was asleep and soaring through the stars with this mysterious beauty I had so gravely and naively fallen in love with.\n\nI awoke with a start.\n\nThe sky was gray in the predawn, my clothes soaking wet with dew. My heart pounded erratically in my chest, my breath coming in short gasps. Something had startled me. A noise heard in sleep but forgotten upon waking.\n\nA scream of pain erupted suddenly behind me. I shot upright and scrabbled around the bush. There I froze, crying out in horror.\n\nShe was twisting in midair, her limbs flicking spasmodically, bending in ways they were not meant to bend. Her face was clenched in a hideous mask of pain. I ran to her.\n\n''No!'' Her eyes found mine. ''Flee, Crispin. He has found me. Go!''\n\n''I will not!'' I screamed, not understanding her words. I tried to reach up to her but she was too high. Her limbs twisted unnaturally, like someone was trying to push and pull them out of joint. I heard a horrible cracking sound like dogs crunching on meat bones.\n\n''What's happening?'' I shouted, tears streaming from my eyes.\n\nHer eyes had closed. She opened them again. We stared _into_ each other.\n\nThen I saw the life fade from her eyes, as if a piece of gauze had been pulled across her vision.\n\nI stroked her hair. I knelt by her body for hours, waiting, willing her to wake up. My mind would not accept that she was gone. I stared at her chest, watching for the rise and fall that breath would bring.\n\nI would convince myself that there was movement, the smallest inhalation of air, and I would wait expectantly for her to open her eyes again.\n\nShe didn't, though, and I finally realized that she never would.\n\nI picked her up. She was as light as a bird. I looked, once, back toward the village. Then I turned my back on it and headed up the road in the opposite direction. It seemed the right thing to do.\n\nTo take her home.\n\nI stayed far from the roads, walking through the forests when I could and avoiding villages. I knew how to fend for myself. All the children in our village took turns going out on hunts with the elders. We were shown how to make traps and forage for food.\n\nShe did not decompose. After a couple of days in the summer heat I was dreading what it would be like carrying her body back to the Goddess. But her skin remained marble smooth.\n\nI would lay her down each night we stopped, fold her arms over her chest, and watch her by the light of the stars and moon. I still found it hard to believe she was dead. I think deep down I was hoping the Goddess would bring her back to life, that she would open her eyes and I would see her smile once again.\n\nMy father told me the Goddess's tower was a week's ride to the north. It took me almost four weeks. Days of bathing in streams and hiding at the sound of hoofbeats or voices. I grew to know every angle of her face. The lines around her mouth where she smiled. The small mole just below her lip. She was the last thing on my mind when I slept and the first thing when I awoke.\n\nA few days before I reached the tower I woke up to find a small girl watching me. She looked a few years younger than I, and stared at me with an expression of puzzlement.\n\n''I was trying to decide if you were like the lady. But you're not.''\n\n''No,'' I said, rising into a sitting position.\n\n''What's wrong with her?''\n\n''She's dead.''\n\nThe girl frowned. ''How?'' she asked.\n\n''I don't know. I'm taking her back to the Goddess.''\n\n''Maybe she'll think you did it.''\n\n''Why would she think that?''\n\nThe girl shrugged. I stood up.\n\n''Are you going now?'' the girl asked.\n\n''I am. And you should be getting back to your family.''\n\n''They're gone.''\n\nI paused. I noticed for the first time how disheveled the girl looked. Her hair appeared knotty and uncombed. ''What do you mean gone?''\n\n''Just gone. Some people came to our farm and took them away. I've been following them,'' she added proudly.\n\nI frowned, not understanding. ''What, you mean they took your family against their will?''\n\nShe looked down at the ground. ''I'm not sure.''\n\n''Well, were they struggling?'' I asked, exasperated.\n\n''I don't know!'' the girl screamed, then abruptly burst into tears.\n\n''Oh, here, look, I'm sorry.'' I hesitated, not knowing what to do. I couldn't leave her alone. ''You can come with me,'' I said, after some thought. ''The first village we come to we'll ask about your family, yes? It's probably all just a mistake.''\n\n''Do you think so?'' she asked, sniffing.\n\n''Sure. Come, let's go. What's your name?''\n\n''Sara,'' she said.\n\nWe reached the tower before we reached a village. I had been staring at it for a long time before I realized what I was looking at. An afternoon storm that had been building all day, sucking every drop of moisture from my body, had broken. Purple bruised clouds spat lightning and rain. Thunder bellowed, like the cracking of dry wood. We stood on the edge of a copse of trees looking out onto miles of grassland, the shin-high grass swaying and flicking violently in the warm wind. I noticed that the falling rain formed a glistening shape far off in the distance. It took a while for my brain to realize I was looking at the home of the Goddess. I had finally arrived.\n\nI shifted her weight in my arms, then looked down at Sara.\n\n''This is it,'' I said. I felt curiously deflated. I wasn't sure why. I had reached my goal. I had brought her home. Maybe that was the reason. I was handing her over. I would never see her again.\n\n''I want to walk with you,'' Sara said.\n\nI glanced down at her.\n\n''We can look for a village after. You said.''\n\nI smiled. ''Yes, I did. Let's go then.''\n\nWe walked into the refreshing rain, the warm wind soothing our cheeks, the wet grass dragging over our legs.\n\nWe reached the tower as nightfall descended. As we slowly drew closer I had looked for some means of gaining entrance, but I could find none. It seemed we would be forced to walk its circumference until we came to some way inside.\n\n''I'll take her now,'' said a woman's voice.\n\nI looked around and saw only Sara standing beside me.\n\n''I thank you for bringing her home to me,'' said Sara. At least, the voice came from her mouth, but the voice was not hers. ''Even though her soul is lost we can still say good-bye. Good-byes are important.'' She reached out her arms. I looked down at the body I had carried for the past month. When I looked up again, Sara had gone. In her place stood the Goddess. She was tall and pale and had bright red hair. Black feathered wings jutted from her back and rose gracefully above her. I fell to my knees and gently laid the body before her feet.\n\n''Come, Crispin, rise. Have we not become friends over the past days?''\n\nI looked up at what I thought was the humor I heard in her voice. ''All that time . . .''\n\n''It was I.'' She picked up the body, staring at the smooth features. ''So . . . it has come to this,'' she said softly. ''Things will change, young Crispin. He darkens the weave with his actions. Bad times are coming.''\n\n''What do you mean?''\n\n''He has done this. He attacks my servants, seeking to possess them. This one fought him. In a way she won.''\n\n''I don't understand.''\n\n''He has her soul, Crispin. He wanted her body, to add to his army. She fought him off, but he has her soul.''\n\n''What will he do with it?''\n\n''Imprison it. Toy with it. Torture it. Whatever is his whim. He has gone mad, I fear. He seeks to destroy the Tapestry.''\n\n''Will there be war?''\n\nShe looked at me with a strange expression on her face. ''Maybe. When he gathers enough strength.''\n\n''Why do you not strike first, before he can gather his strength?''\n\n''You are young, Crispin,'' she said. ''In time you will see that if I did that, I would be no better than him. Worse, because I know better.''\n\nShe turned to the tower, carrying the body.\n\n''Goddess,'' I called. She turned back to me.\n\n''What was her name?'' I asked.\n\n''Ser'ashen.''\n\nI returned to my village, ignoring the looks from those I passed. Obviously, my parents were furious. They demanded to know where I had been, but I couldn't find it in myself to tell them. I felt terrible about deceiving them. I think they'd thought me dead, taken on the road by some creature or another.\n\nThe days and months passed, and they eventually stopped asking. I wandered listlessly through the village, avoiding my father in the mornings when he set off to the forge, barely speaking to him when he returned. Something had been taken from me with Ser'ashen's death. A part of me had gone with her, so that I could no longer muster up enthusiasm for anything. My life stretched ahead of me, lonely and frightening. It was as if I had glimpsed a possibility of what could be, that my life might have some meaning, but then it was yanked away again, leaving me worse off than before, because now I knew there was something better out there.\n\nI dreamed of her every night.\n\nThe rumors increased over the next few years. Of demonlike creatures attacking villages and towns in the middle of the night, then even during the day. They were crossing over into the Goddess's time, slaughtering man, woman, and child.\n\nI traveled to the tower once a year, on the anniversary of her death. I never saw the Goddess again, but I didn't mind. I did it for her, for Ser'ashen.\n\nI lost my childhood because of that fateful day. People my own age seemed ludicrously childish and irritating.I think they found me pompous and boring, too serious to partake in their experiences.\n\nOne cold afternoon in winter none of that would come to matter. I was sitting with my back against my father's forge, reveling in the warmth and staring into the cold gray sky. Dusk was approaching. I seemed to recall my mother saying that today was the shortest day of the year. Wood smoke, for me the very smell of winter, drifted from the roofs of all the houses and huts, the peaceful scent mingling into the sharpness in the air.\n\nThe attack came suddenly. One second I was living in a mundane world wondering whether I should return home for some hot soup to warm my bones, the next I was staring into a street suddenly filled with bestial screaming and hordes of nightmarish creatures.\n\nI felt a hand tug at my shoulder and jerked frantically away. It was only my father. He took a quick look up and down the street and then yanked me into the forge. He ran to the back wall and pulled two old blades from his reject bin. He wrapped cloth around the hilt end and handed me one. I looked up at him, expecting to see the fear that was washing through my body in great shivering spasms, but there was no fear there. Only determination, and anger. Lots of anger.\n\n''There's no way around this, son. Kill or be killed. Those things out there, they're not taking prisoners. Understand?''\n\nI nodded.\n\n''The talismans must have failed. We have to get through to your mother. She needs us now.'' He hefted the blade in his hand, testing its balance. He breathed in deeply, let the air out through his nostrils. ''You ready?''\n\nNo.\n\n''Yes,'' I whispered.\n\nHe bent down and hugged me. There was no time for extra words. Later, you always wished you'd said things, things that you know will be remembered long after the moment they are uttered, but in real life there's not always the chance.\n\nMy father opened the door. The cold air flurried in and touched my face. I shivered, imagining it was Death himself caressing my cheek. Then we were out in the street, running toward home, and everything became confused.\n\nI saw my father slice and hack his way through a group of the hollering creatures. My mother had told me he'd once been in the army but he didn't like talking about it. After that day was over, I could understand why. He swung his blade and a gray piglike head went tumbling in a splash of red. Purple-white intestines spilled between slippery club fingers trying to hold them inside. I swung the sword and caught one of the creatures in the back of the leg. It collapsed and tried to grab hold of me with black claws. I waved my sword blade, an instinctive action, and the hand landed to my left, the claws slowly closing like a dying spider.\n\nOther villagers were joining the fight, taking up rusty arms against the attackers. My father glanced back to check on me, a split second, then turned back to fight a winged monstrosity with four arms and as many swords.\n\nWe moved on, taking cuts and slices as we went, our home ever closer and now in sight. My left arm hung useless by my side, paralyzed by a lucky back-hand swing. I looked down at it once, saw a red-lipped wound grinning at me with teeth made of bone, and didn't look again.\n\nAnd then my father was at the open door. He ran inside, fearful that some of the creatures had beaten us, shouting my mother's name. I think I heard her reply. I'm not sure, because at that moment I was looking down at a sword point that had just punched through my stomach.\n\nI thought to myself, _That's my blood,_ then the sword was yanked from my back with a curious sucking sound and blood bubbled from the wound.\n\nI couldn't speak. I looked up at the empty door, thinking that if I could just get inside everything would be all right, that my mother would fix everything. I tried to walk, but my knees gave out and my face hit the rocky ground.\n\nThe last sound I heard was a guttural grunting noise from somewhere behind me.\n\nI woke to the sound of creaking harnesses and the quiet _chuffing_ of horses breathing into the air. I opened my eyes and stared up at the gray sky moving slowly above me. I opened my mouth to call for my mother, but all that came out was a dry croak. I felt the horses slow down, then stop completely. My father's face appeared above me. I was shocked by how worn he looked. His face was deeply lined, as if charcoal had been ingrained into all his wrinkles. Dark shadows ringed his eyes. I'd never seen him looking so frail. He tried to smile at me, but it faltered and he had to catch his breath.\n\n''Welcome back, son,'' he whispered, and that was when I felt the pain. It was the color of fire and felt like ragged glass that crunched and splintered with every breath. I cried out and my father held a water skin to my lips. I felt the moisture sucked away like a droplet on drought-stricken earth. He poured some more, then took it away. ''Easy,'' he said. ''Not too much.''\n\n''What\u2014?'' I croaked\n\n''Don't talk. They're gone, son. We beat them.''\n\n''Oh.'' I licked dry lips. ''Did I do good?'' I whispered.\n\n''You made me proud, my boy. You made us both proud.''\n\nI stared into his eyes. It was as if I saw his fears written plain upon his face. I knew then that things were looking bad for me.\n\n''They got me.''\n\n''Aye, son,'' he said softly. ''They got you.''\n\n''Where are we going?''\n\n''To the city. A healer there will take care of you.''\n\nI nodded wearily and closed my eyes. I slept again and woke to my father changing the dressing on my wound. I saw him wince at the acidic smell it gave off.\n\n''It's getting worse, isn't it?'' I whispered.\n\n''No!'' he said, too quickly. ''Another day and we'll be there. That healer will fix you right up.''\n\nI didn't say anything. I was scared that if I opened my mouth I would say what I was really thinking. That it was too late for healing, that I could feel that the poison had already gone too deep.\n\nI must have dropped off again, because I opened my eyes to darkness and the most intense pain I had ever felt. It was like someone had taken a hot coal from my father's forge and laid it in my wound. I cried out. My father was there instantly, holding my hand. He made meaningless noises and stroked my brow. When the worst of the pain had gone he went to the fire and brewed me some kind of foul-smelling concoction that seemed to dull the pain. I watched him walk the perimeter of the camp, checking on the talismans he'd laid out. I don't know why he bothered, though. The ones at the village hadn't worked.\n\nI could actually smell the poison through the bandages the next morning. I watched my father stoke the fire in the early morning mist and surreptitiously probed my stomach. The pain had spread well beyond the initial area. I sighed. I'd heard about stomach wounds before. They took a while to kill you, but kill you they usually did. Anyone who thought otherwise was fooling himself.\n\nI watched my father collect the talismans, then sit down and pull out some hard bread and cheese for breakfast.\n\n''Dad,'' I said. He put down his breakfast and looked at me. ''I know this is hard, but you know and I know that I won't recover from this.''\n\nHe just stared at me, then lowered his eyes to the ground.\n\n''I think I'm all right, though,'' I said. ''I think this was meant to happen. I would never have been content with life. Not . . . not after what happened.''\n\nHe looked up at that, and I went on to tell him of the events that had so drastically changed my life.\n\n''I want to go back, Dad. To the Goddess. I want to say good-bye.''\n\nHe said nothing for a while. Then he slowly shook his head. I could see the pain he felt. ''Boy, you don't know what you ask.''\n\n''I do. What good will traveling to the city do? I don't want to die scared in a place I don't know. I want her to be there.''\n\n''How do you know she will be?''\n\nI hesitated, then. How did I know? I wasn't sure, but it just felt right. ''She'll be there,'' I said. ''I feel . . . I feel like she's calling me.''\n\nBy the time we arrived at the copse of woods where I first sighted the tower my wound was so hot that it made my whole body sweat.\n\n''We have to walk from here.''\n\n''Crispin, you can hardly sit upright. How can you walk?''\n\nI started to climb from the cushions and blankets. I slipped and fell. The pain shot up my spine so hard that I blacked out for a moment. When I came round again my father had his arm around me and we were walking through the grass.\n\nHe was making strange noises, and when I looked over I saw that he was crying, tears streaming down his face. He sensed my movement and looked down at me.\n\n''I'm losing you today, boy. By the Goddess, I'll never see you again.''\n\nThat nearly broke me, right there. Seeing him cry like that. My father, the one person who was always supposed to be strong for everyone else. I tightened my grip around his waist, and in some strange way, we almost seemed to switch places. It felt like I was holding _him_ up as walked through the still, winter-brown grass. Like I was the one who had to be strong. It was there that I knew for truth what before I had only suspected. Whatever happened today, it was what I had been born for.\n\nShe was waiting for us when we arrived. Somehow, I knew she would be. I left my father standing some way back. He refused to come any closer, staring at the ground instead of laying eyes on his Goddess.\n\n''I've been waiting,'' she said as I approached.\n\n''You knew.''\n\n''I saw the possibility.'' She reached out and touched my stomach. The pain fled instantly. I stood upright again, took my first clean breath in days. ''I can heal you, Crispin. You can go back to your family.''\n\nI looked down at my wound. It was still there. She had only taken away the pain. I looked up at her again, stared into her gray eyes. ''No,'' I said. ''I feel this is . . . right. Something has been missing ever since . . . what happened. I think that there is a reason for what happened to me, then _and_ now. But I thank you for giving me the choice.''\n\n''I cannot see your future, Crispin. I know you are an important thread in the Weave but I cannot see how. The choices you make will affect not just yourself, but others also.''\n\nI bowed my head. I did not say anything. It was her decision now. She knew what I desired, had known even before I had.\n\nI felt her hand on my head, and then I was _inside_ her mind. I saw the potential held in every decision ever made. I saw the world as a tapestry, and every thought, every action, every word, made up its thread. Here was blackness, but it was surrounded by gold. Here was silver and love, but it was cut off by angry red. It was confusing and meaningless up close, but then I saw the whole thing as if I was suddenly miles and miles away. I saw the Tapestry as a whole, saw how it all balanced out.\n\nI felt the wings grow from my back. They thrummed gently in the air, like the wings of a newly born butterfly. My earthly life, my past, seemed to fade. It did not disappear, because it made me what I was, but it was pushed to the background. I felt my clothes slide from my limbs like old skin, felt the wind tingle my body and tell me stories of distant lands. I looked up again. The Goddess leaned down and kissed me.\n\n''It is done,'' she said.\n\nI turned to the being who had once been my father.\n\nHis head was still bowed. I stood before him. ''Look upon me,'' I said gently. ''Tell your wife I have not died. I have gone on to a better place.'' He cried, great sobs that he was unable to control. I wiped his tears with my hands. ''I will always watch out for you both,'' I said softly. ''I'm sorry you were never able to be proud of me.''\n\n''Oh, son!'' he whispered. ''You're just a boy. Of course we're proud of you.'' He reached out and gripped my arm, looking hard into my eyes. ''Know that, if you know anything. I could not be prouder.''\n\nThen he turned and walked back the way we had come, an old man bent with his loss.\n\nI watched him until he mounted one of the horses and disappeared into the trees, then I flexed my wings and floated slowly into the sky, higher and higher until the earth below me looked like the tapestry I had so recently glimpsed. I closed my eyes and saw the Weave overlaid on my vision. I could see the glow that represented my mother.\n\nI saw the black hole that was the creature who had run me through.\n\nI concentrated, and I saw the glow that was Ser'ashen, gold and silver surrounded by a halo of darkness. I reached out and touched it. I had a sense of fire, walls of pulsating flesh, of wind tearing through the hollow eye sockets of thousands upon thousands of skulls, then I _pulled_. I heard a scream of absolute hate, and then something more, something that froze my heart: a laugh of triumph.\n\nI opened my eyes and looked at my hands. I held a softly glowing bead.\n\n''I think I should take that.''\n\nI handed it over to the Goddess, unsurprised that she was there. She looked at it carefully. ''With one act the future of the world is changed,'' she said. She searched my eyes. ''I think I knew what you planned, even though you yourself did not. He was waiting for this, Crispin. I have trespassed onto his lands now. All the rules have changed.''\n\n''I only wanted to help her.''\n\n''I'm not saying you were wrong. In fact, I think you could have acted no other way. Your ancestors, your parents, your upbringing, all of it brought you to this precise moment where you acted the way you did. The one thing that makes me think this will work out is that you acted out of love. That is something he does not understand anymore.''\n\nShe drifted slowly to the ground. ''Come. We must prepare.''\n\n''For what?''\n\nShe gestured with her free hand. ''For that.''\n\nI turned my head to see what she was pointing at. Darkness filled the sky to the south, a foul wind blowing before it. I could make out individuals in the mass, horrible, twisted faces and broken wings. I felt fear then, such as I had never felt before, but there was something else as well. I felt hope. When this was over, somehow I knew the Tapestry would be pure again, that the Weave would once again be as it should be.\n\nWe fought a war against God today . . .\n\nThe other image, the second one I will always recall when I remember the events of this terrible afternoon . . .\n\nIt was a brief glimpse I had through the thousands of fighting seraphs. A brief glimpse of Ser'ashen as she fought to cleanse the Weave of the world.\nTHE FINAL CHOICE\n\n_Irene Radfford_\n\nDeath sat at the bar wondering what he had forgot-ten to do. Eleven-oh-two p.m. December 31. There was something he had to do before midnight or the New Year would not arrive. Time would stop. Life would be frozen in an endless cold sleep. Souls would have no home.\n\nChange would not continue to shape the universe.\n\nFates would not be fulfilled.\n\nDeath took a sip of his drink and concentrated on his duty.\n\nThe potential suicide in the corner vacillated in her decision. Her well-cut red suit looked too bright and cheerful for her mood. She twisted a diamond wedding set around and around her heart finger. Death shouldn't leave until she made up her mind.\n\nSuicides always disrupted the schedule of appointments. He didn't like _last minute_ changes.\n\nBut there was something else . . .\n\nHe checked his appointment book. The potential suicide wasn't listed anywhere in the last few pages. In two minutes, a man with a heart condition would run out of time. Death grabbed his staff of office and left the bar. If the woman in the corner made her decision in the next two minutes, she'd still need two more to find a means and a place.\n\nThe little black appointment book with magnificent gold calligraphy on the cover burned in the pocket of his flannel shirt beneath a down parka. His staff of office, half again as tall as he, shrank to the length of a walking stick. The ebony end that curved back on itself to form a window for a huge black crystal dissolved into a knob with the winking crystal set into the end. No flowing black cape and skeletal hands for the heart attack victim. This candidate for death needed the reassurance of a familiar personage to make the transition quietly.\n\nDeath sidled through crowded Times Square. He appeared to be just another reveler on New Year's Eve.\n\nHis candidate jumped up and down, waving to friends and strangers alike. He paused in his excited dance only long enough to chugalug the whiskey in his hip flask.\n\nDeath tapped his shoulder.\n\n''Hi! I'm George. Who are you?'' The candidate greeted Death.\n\n''Hello, George. You have an appointment.''\n\nAn overweight, middle-aged body collapsed on the sidewalk. George turned to look at his former shell. ''I guess I have to leave now. Before the New Year.''\n\n''Yes, you do.''\n\n''Pity. I've never actually been here on New Year's Eve when the ball dropped.'' He looked wistfully at the great ball of light atop a nearby building. ''I guess now I never will. Can't I stay a little longer, just until the ball drops?''\n\n''Sorry, George. Eleven-oh-seven. You are precisely on time. You can't linger, even to see the New Year.''\n\nGeorge looked back at his former self, one last time. A Good Samaritan had already begun CPR on the limp body.\n\n''He might revive me.''\n\nAn ambulance siren wailed in the distance.\n\n''He can't revive you, I have touched you. Your fate is determined. If you choose to wait, or refuse my escort now, you will wander aimlessly as a lost soul for all time. Your choice.''\n\n''Some choice.'' George looked back on his body with longing in his eyes and posture. Then he nodded in quiet acquiescence. Death took George's elbow and led him out of the crowd.\n\nTwo minutes later, Death blinked his eyes and transported back to the bar. Little Miss Indecision was still dithering, still twisting her rings, occasionally tugging at them. They wouldn't come off easily. She'd worn them long enough that her finger and knuckle had grown thicker. Death pulled out the appointment book once more. A little book now with only a few names left. The page with George's name dissolved under his gaze.\n\n''Who's next?'' he asked the book.\n\nAt year's beginning the book had been a huge tome that had gradually dissipated to this thin reminder. Not many names left. Not much time before one year faded into the next. Choices and change had to continue. Unless . . .\n\nDeath ordered a drink. He took a sip, remembering when alcohol tasted good, made him feel good.\n\nStrange, he wasn't supposed to remember life, only his duties as Death.\n\nTime. Eleven-twenty-six. There was something he had to do. The appointment book heated up again. A child dying of cancer. A child ready for the release of pain. Too bad his parents weren't ready to let him go. They had made all of the child's decisions for him. This last choice had to be his alone.\n\nDeath walked into pediatric intensive care at Mercy Hospital dressed as a teenage candy striper, the staff of office now only a small syringe with a black crystal plunger on a tray. The family of the candidate hovered around the bed. Tears and aching hearts filled the room with an aura of misery.\n\nThe candidate smiled at Death. ''Hi, I'm Mike. About time you showed up.''\n\n''Hi, yourself, Mike,'' Death replied in his feminine voice.\n\nMike's body convulsed and gasped for breath.\n\n''No. You can't die. I won't let you.'' Mike's mother threw herself onto the little boy's body, oblivious of tubes and machines. The woman looked up directly into the eyes of Death. ''Take me instead. He's so little. He hasn't had a chance to live yet. Take me!''\n\nTime stopped until a choice was made. Fate required a death.\n\n''Can you do that? Change the appointment?'' Mike asked, eyes wide and wondering. Momentarily he was free of the constraints of his body. Only his soul knew what transpired.\n\n''Fate dictates that my appointment is in this room, at this time,'' Death announced to all those present. ''The name of the candidate is not known until the actual moment of death. Anyone here may accept the fated death.''\n\n''Take me,'' the mother said resolutely. ''Spare my little boy.''\n\n''No, Mom.'' Mike looked around at his loving family, frozen in time until the choice was made. ''Think of Dad and the family. Julie and Tom need a mom. Dad needs his wife. They'll learn to get by without me. Life will be a lot harder on them without you than me. Let me go. I'm tired of hurting. I'm tired of watching you hurt because of me.''\n\nEmotions flooded Death. He remembered pain and how love lessened it. Getting involved with his candidates was not a part of the job. But Mike was so strong, so adult, he reminded Death of . . .\n\nDeath refused to remember life. Change and a choice of fates belonged to others now, not him.\n\nMike climbed off the bed, leaving his body behind. ''Time to go. Good-bye, Mom. Bye, Dad, Julie, Tom.''\n\nTime resumed. The family clung to each other in sorrow.\n\nDeath and Mike walked slowly toward a swirling circle of light, leaving life behind.\n\nA car accident diverted Death's attention on his transport back to the bar. Smashed metal, flying glass, the smell of hot gasoline ready to ignite. Agony! Desperate pleas. Death put aside the memory. Those terrible things must have happened to someone else.\n\nHe checked the most recent victims. Serious injuries, but no one needed his guidance to the other side yet. He made a note in his appointment book to check back with the five passengers and three drivers once they reached the hospital.\n\nThe little black book didn't accept the note. The accident victims would not face a final choice within the remaining minutes of this year. What would happento the book at one second past midnight? Death did not know.\n\nOr could not remember.\n\nWhich?\n\nDeath looked at the last page of his book curiously. He saw an appointment listed for eleven-fifty-nine and fifty-nine seconds. The assignation hadn't been there earlier in the evening. No place, cause, or name followed the time. Strange. What did he have to do in the last second of the Old Year to make certain the New Year came? The thought of all the souls of humanity drifting homeless for eternity made him shudder. Time must continue. People must experience change and make choices. Fates must be fulfilled.\n\nHe returned to the bar, drawn there as if by a magnet. His staff glowed brighter as he approached. The appointment book suddenly felt heavy. He checked, but no new appointments had been added past the cryptic one in the last second of the Old Year.\n\nThe bartender took Death's order for another drink. Automatically Death assessed the man's condition: arthritis, right shoulder and knee; weak arteries; and swollen feet. Six months tops.\n\nA newcomer opened the door of the bar. The noise of the New Year celebration in Times Square filled the shabby drinking establishment with a moment of lively joy. The potential suicide wavered a moment in her decision. The door closed and the noise died. As the Old Year was dying.\n\nDeath hefted his little black book once more. ''I need time to look in on the pope. I have to keep tabs on all the assassination attempts and a few great musicians and artists. There isn't enough time. Something I have to do . . .''\n\nDeath sipped at the drink that was supposed to taste good or make him feel good but did neither. He watched himself in the mirror behind the bar. He looked like any other generic, middle-aged male, not too prosperous, nor too downtrodden, his staff of office hiding as a black umbrella propped against his stool. The persona fit this neighborhood. He was used to the instant changes in his appearance. He didn't like terrorizing people\u2014except some of the truly evil personalities. When Mother Teresa finally passed on, Death had chosen to be another elderly nun so as not to frighten the woman. But that determined lady hadn't been frightened by life. Why should Death in any guise scare her?\n\nEleven-forty-three. A sense of desperate need tickled his senses. The potential suicide in the corner passed her crisis and decided to give Life one more year.\n\nDeath followed her onto the street. He had too much to do in the last seventeen minutes of the year.\n\nHis long staff appeared in his hand, keeping its proper shape and size\u2014nine feet of shining ebony, slender top curved into a full circle. Thousands of facets from the crystal reflected tiny pinpricks of light. His black-hooded cloak folded around him. He became one with the shadows, seeking the source of that last appointment. Only when the candidate, location, and circumstances were chosen would his guise take shape.\n\nHis hands tingled with the power encased in his trappings. His staff glowed in the reflection of street lamps. Aware, not fully active. Yet.\n\nOut on the street, Death turned the staff right and left, seeking. A faint glow emanated from the crystal when it faced right, up ahead. A very dark alley. Streetlights shot out, garbage piled high. A haven for vagrants, criminals, and violence.\n\nThe appointment book burned with impatience.\n\n''Just another mugging,'' Death sighed. ''I'd hoped for something spectacular to close out the year.''\n\nThe crystal glowed brighter, taking on red tones. ''Odd. Red indicates a death of great importance, someone who will stop time if his, or her, destiny goes unfulfilled.'' That had happened with Princess Diana as she clung to life for agonizing moments, but others had stepped in to continue her work. One instance when the victim became more powerful dead than alive.\n\nWhile the world mourned her passing, people continued to make choices and grow through change.\n\nDeath followed the crystal with increasing urgency. For the sake of all lost souls, time had to continue.\n\nThe woman who had chosen life over death walked ahead of him, head high, shoulders back. She had chosen life and her posture reflected reawakened joy and confidence. Her high heels tapped a rhythm onto the sidewalk akin to the song of life.\n\nGrunts. Cries for help. Scuffling feet and thumping bodies. The woman gone.\n\nDeath hurried.\n\nHe rounded the corner into the alley. Three Lives standing. One desperate Life sprawled amid piles of junk and empty boxes, right leg twisted unnaturally beneath her. The skirt of her red power suit hiked up immodestly, torn at the side seam. Blood spilled on the pavement.\n\nEleven-fifty-eight and forty-seven seconds. In the distance the shouts from Times Square increased. Close up, one of the standing Lives lifted a gun and took aim at the Life who waited. A feral smile grew around broken and rotted teeth. All four Lives were fully conscious. All four knew that Death awaited one of them.\n\n''Should have given us the diamond ring along with the purse right off, yuppie bitch. We'd have let you off with a sore head,'' the youth with the gun sneered at his victim.\n\nThe diamond on the woman's left hand winked in the weak light, almost as brightly as Death's black crystal. A cherished wedding ring. A promise of love. The muggers had broken her leg while she struggled to protect the ring.\n\nThere was still time for Death to offer choices.\n\nAll four Lives froze in a tableau that screeched of man's violation of his covenant with Life. The black crystal in the staff passed from red to blinding white. The appointment book grew heavier and hotter yet.\n\nEleven-fifty-nine and forty-five seconds. Time stopped.\n\nDeath looked anxiously from the crystal to the Life who awaited his touch. Time awaited the next candidate. Who? The book didn't tell him.\n\n''Don't kill me!'' the woman who had left the bar filled with renewed purpose yelled at the three muggers. ''You've got my purse and my jewelry. I can't run away. My leg is broken. Leave me alone.''\n\nNo one moved. Nothing moved, not even the freezing wind.\n\nDeath waited. A curious sensation of warmth engulfed him. He'd been cold so long he'd forgotten what warmth was. Not exactly warmth, an absence of heart-chilling cold. But with the warmth came pain too. Sharp pains filled his leg in empathetic sharing with the woman. Curiosity and dread warred with fear for mastery within him. His heart raced and then seemed to stop. This was the last appointment in his book and he would be an active participant instead of an escort after all the choices were made.\n\nFate had caught up with him at last.\n\n''Help me, please. I don't want to die,'' the woman called to Death. Her hand reached out in entreaty.\n\nDeath heard himself issuing the same plea a year ago. He remembered fear and its copper taste on his tongue.\n\nHe shook off the memory and the residual tremors. He had a duty to perform.\n\n''I have an appointment with someone in this alley. One of you must go with me.'' Death's voice echoed around the alley, like a bronze bell. The three muggers remained frozen in time. Not so much as an eyelash twitched among them.\n\n''Take one of them.'' The woman pointed to the tableau of criminals frozen in the act of theft and murder. Her hand wavered and almost pointed directly at Death.\n\nDeath tried to retreat within the folds of his hood. ''They are outside this decision, ma'am. Only you and I are here.'' Her name eluded him. Why? This had never happened to him before . . . before he became Death.\n\nA year ago he had wanted so desperately to live that he had chosen to become Death rather than accept his fate. And now he was faced with another Life in the same dilemma. One of them must die.\n\nHe planted his staff in front of him. The glowing black wood gave him authority and confidence. Someonein the alley had to die. Time would not resume unless Death escorted a candidate to the other side. He still had a choice.\n\n''I'm not volunteering to die,'' the woman screamed. ''I'm not ready to die! I just decided to live. Please let me live!''\n\n''I can't give you that choice,'' Death lied.\n\n''Do something.'' The woman grabbed the staff and shook it in desperation.\n\nDeath jerked back on the length of wood in panic. ''The staff is my badge of office. Only I may touch it.'' His hood fell back. This time he knew his appearance was the classic personification of Death, a skeletally thin face, pasty white. Deep-set eyes that looked into eternity.\n\nThe woman held tight to the staff, shaking it again.\n\n''You. Must. Let. Go.'' Death grabbed the black staff with both hands, trying to wrest his tool away from her. ''You. May. Not. Touch. It.''\n\n''If you won't help me, then let me have it to save myself.'' She clung to the staff as if it were Life itself; a Life she desperately wanted. Now. A few moments ago she'd almost thrown it away. ''I can use this as a weapon to save myself!''\n\nPower raced up and down the wood binding her to the staff and to Death. He almost let go. Desperation kept him glued to the wood.\n\nIf he let her live, what would happen to him? Someone had to die or time would not resume.\n\nWho would it be?\n\nWhat choices were left?\n\n_The only way to cheat fate and Death is to become Death,_ another voice had told him a year ago.\n\nDeath stumbled. The woman twisted the staff and tripped him with it. Death dropped to his knees. His cloak fell away, revealing a red woolen suit, the same cut as woman's. Same blouse. Same scarf around the neck. His skeleton took on flesh but remained pasty white.\n\n''Who are you?'' The woman rolled to her left, away from the collapsing body of Death. She stood up in one fluid motion with the staff in hand, as if her leg hadn't been broken a moment ago. Her shoulders hunched and she aged a thousand years in a moment.\n\n''There is something I have to do before midnight.'' Death's voice remained deep and solemn, echoing and reverberating around the alley. Which one would die? His right leg twisted unnaturally beneath him.\n\nThe muggers came back to life.\n\nA mighty roar rose from Times Square. _''Ten.''_\n\nThree shots rang out in rapid succession.\n\n_''Nine.''_\n\nThe muggers turned and bolted from the alley. ''What do you have to do? There isn't much time.'' Shock made the standing woman's words weak and squeaky. She bent low to catch Death's words, feeling for a pulse, trying to stop the flow of blood from his chest. She didn't have enough hands, or medical knowledge to save him.\n\n_''Eight.''_\n\nDeath grabbed her lapel and pulled her closer yet. His clawlike hands seemed incredibly strong for someone who'd just been shot in the belly, the heart, and the lung.\n\n_''Seven.''_\n\n''Tell me what you have to do. I can help,'' she cried.\n\n''There is a way for you to survive this encounter.''\n\n_''Six.''_\n\n''I have survived; you're the one who is dying.''\n\n''One of us must die at the stroke of midnight. You have taken the choice away from me. The only way to cheat death is to become Death.'' He repeated the words spoken to him a year ago. A lifetime ago.\n\nEveryone was fated to die. The choice of when fell to only a few.\n\n_''Five.''_\n\n''Become Death? You mean I'll die, too. Who are you?''\n\n_''Four.''_\n\n''I am your destiny, your fate. Life or Death. You must choose. As I chose a year ago. I loved life too much to give it up. I still do. But I no longer have the right to make that choice.'' Choice and change belonged to the living. Everyone had to die. Fate determined when and where. No choice.\n\nExcept for the last death of the year.\n\n_''Three.''_\n\n''If it means living, I'll become Death, I'll become Santa Claus or whoever it takes. Just so I can live. I decided not to kill myself over my husband's infidelity and a mangled career because I realized that life is too beautiful to waste.''\n\n''I thought the same thing last year at this time.'' A fiery car crash, pain beyond enduring, and still he had clung to life rather than let Death take him. ''And now I know that all Life is beautiful. If one of us does not die then time will cease, taking all Life with it. Life must be preserved.''\n\n_''Two.''_\n\n''Then why must I become Death? I'd rather be alive.''\n\n''Death, like change, is a part of life. If Death does not walk the streets then all Life will cease. The choice is yours.''\n\n''What is that supposed to mean?''\n\n''You'll find out.''\n\n_''One! Yeah! Whoopee. Yahoo!''_\n\nThe body of a young man, who had refused to die in a car crash the year before, took on the last vestiges of the woman wearing a red suit. He\/she collapsed in the alley. The last page of the old appointment book dissolved.\n\nA skeletally thin old hag, dressed in tattered red and black draperies, with eyes that burned clear through to eternity stood up and retrieved Death's cloak, without dropping the staff. All memory of her life, her decision to live, her wrestling match with Death, faded. She was Death now, with duties to perform.\n\nFrom the folds of black cloth fell an enormous book bound in black leather. The gold calligraphy on the front was fresh and new, spelling out one word.\n\n''Appointments.''\n\n''Let's see. Victims of an automobile accident. Mercy Hospital,'' the old hag cackled. ''Five passengers. Three drivers. Four of the eight need an escort in two minutes. A musician is shooting a bad batch of drugs in Central Park in six minutes. The pope can wait a little longer.''\n\nShe morphed into a young nurse wearing bright turquoise scrub pants and a white tunic with tumbling turquoise teddy bears. The staff of office coiled around her neck like a stethoscope, the black crystal blocked the metal bell at one end.\n\nDeath popped into the emergency room of Mercy Hospital, ready to escort those who needed her.\n\nA diamond on her left hand winked in the bright hospital lights as she escorted the first death of the New Year into the swirl of light.\nTHE PROPHECY OF SYMON THE INEPT\n\n_Rebecca Lickiss_\n\n''No!'' she bellowed. ''Bastard son of a whoring slug, I will not.'' She emphasized her point by planting her foot solidly in the midsection of the man on her right, her brother Mark, bending him neatly in half. In the gathering gloom her brother straightened himself and stood rubbing his midsection for a moment.\n\nLord Leopold Smallpenny watched from a safe distance. ''Sorry, dear, but this must be.''\n\nHe was a smallish man, rail thin, with weasely features. Lady Patience Smallpenny, n\u00e9e Tillman, was a woman of large and generous proportions. Her current delicate condition would have prevented any other woman from physically domineering all and sundry, but not her. She wasn't the sort to allow anyone or anything to get the best of her. Lord Smallpenny had learned to fear her more than anything else in the world.\n\nHaving managed to get this far seemed rather remarkable. Lord Smallpenny suspected that she wasn't entirely set against the necessities of this night. It was just that certain aspects of the coming ordeal, while romantic to contemplate over a cup of wine by a roaring fire, were decidedly unpleasant when confronted in substantiality.\n\n''You fool-swilling, toad-warted, beetle-witted, mangled, bootless miscreant!'' Lady Smallpenny lay half in the wagon, clinging to the side with fierce determination. ''I will NOT give birth in a stable.''\n\n''Yes, dear,'' he responded automatically without thinking or meaning. He motioned to her brothers to continue their efforts to get her into the stable.\n\n''Cowardly curs. I'll see you flogged for this!'' She struggled against her four brothers as they slowly dragged her enormous, resisting bulk from the wagon.\n\nTo Lord Smallpenny's way of thinking, his wife's brothers were all cut from the same generous mold, big as moose, strong as oxen, fierce as bears, and about as intelligent as your average goat. With odds of four against one you'd have thought they were a sure bet to win. Problem was, they were unaccustomed to contradicting the females of their family. For the last ten months of Lord Smallpenny's acquaintance with them they'd shown him no sign of any thoughts or ambitions in their minds, other than what was put there by their mother or sister. They were willing enough to go along with his schemes, as long as Mother said, or their sister supported the idea. Now with all her shouting and cursing, that looked in doubt. The youngest, John, looked to Lord Smallpenny for support.\n\n''Get her inside!'' Lord Smallpenny had spared nothing for this moment, and he wasn't about to have it ruined by a gaggle of foolish louts. ''Remember what we are doing this for. You'll be princes. Hurry. It won't be long.''\n\n''You sniveling, beslubbering, pribbling, addle-pated, beetle-brained, sparrow-headed boar-pigs,'' she shouted at her brothers as she slid from the wagon to the ground. Her kicks had knocked Mark to the dirt, where he nicely cushioned her fall.\n\nMatthew, the eldest brother, in a sudden fit of intelligence, grasped his sister's knees, one in each hand. He pulled her behind him as if she were a wheelbarrow he was dragging. Mark and Luke each grabbed up an arm. John covered her eyes in an attempt to calm her. However, what works for a horse generally doesn't on an extremely enraged woman. She nearly knocked them over at the door of the stable.\n\nThe midwife, Prudence Marblepot, looked on disapprovingly. ''She'll be wanting her mother.''\n\nLord Smallpenny smiled. ''Mother was . . . unavoidably detained. It's unfortunate that she won't be able to join us.''\n\nHer mother was currently locked in a windowless tower of the castle. Lord Smallpenny could not have controlled the situation if both women were loose. More important, the Tillman boys would've long since lost their nerve. He could hear the vile things she would currently be shrilly howling without even being there. Pity she usually freed herself in less than a day.\n\n''It won't be long, will it?'' Lord Smallpenny asked Prudence pointedly.\n\n''Not long at all. She's been trying to deliver for a week now.'' Prudence's mouth puckered, as if tasting something sour. ''I doubt you'll be able to get a pint to drink before the child is born, let alone finish it. But know this, the child may be a half-wit. What with all the potions used to keep the lady from confinement till now. You've only got yourself to blame.''\n\nLord Smallpenny fixed her with an icy stare. ''Go. In. Deliver my son. Make sure he is born before midnight. Do you understand?''\n\nShe fixed him with an icy stare of her own. ''Oh, I understand. I understand all too well.'' She snorted, adding under her breath, ''Midnight,'' then motioned for her assistants.\n\nPrudence Marblepot could adequately deliver a child with nothing more than a paring knife to cut the cord. However, she believed in the mystique of midwifery, and anything that would make her position more important. For an ordinary peasant birth she would have three assistants: one to boil four gallons of water and heat three blankets, another to sharpen to a razor's edge two knives and stand around looking important while handing over the contents of the Midwife's Bag, the last was to comfort the mother. For this birth she had fifteen assistants. Following directly behind her was her chief apprentice, carrying the Midwife's Bag. Behind her chief apprentice were two other, younger apprentices, each carrying a large stack of blankets. Behind them, two ornately dressed bourgeois gentlewomen each carrying a large red velvet, gold fringed pillow cushioning a sharpened saber. Following were two of the village young women carrying swaddling clothes. Next, another pair of young women with a woven baby basket between them. After them were two stout goodwives each hefting a large steaming bucket. The next two matrons carried bowls and basins. Bringing up the rear were two girls carrying woven baskets containing herbs. Prudence led the procession to the stable, muttering about foolish lords and silly prophecies.\n\nShortly after the midwife entered the newly built stable, the lady's brothers hurried out. Lady Smallpenny's shout followed them, ''Fetid drool of a pox-faced donkey! Get away from me!''\n\nLord Smallpenny hoped that the sounds that followed could be associated with birth, though the last time he'd heard most of them had been during battle. One noise particularly reminded him of the sound of a fist pounding flesh. The only good thing about his wife's currently enlarged swelling bulk was that she wouldn't be able to leave the stable under her own power. She'd finally become too cumbersome even for herself.\n\n''Let's get a drink,'' Lord Smallpenny said to his brothers-in-law. ''And see if the midwife's prediction is correct.''\n\nAs they walked into the Vanishing Moon tavern, the dusk gave way to dark. On the far horizon clouds banded together in preparation for an assault on the ground. This pleased Lord Smallpenny greatly. He gave a happy sigh, and with great satisfaction pulled the cord to make the moon on the inn's sign vanish. All would be well.\n\nMatthew went behind the bar, nodding to his pleasant wife, Anne. She smiled back at him from the hearth where she tended the fire. Mark, Luke, and John sat uneasily at a table not far away, nursing their drinks. Lord Smallpenny appropriated a table away from the heat of the fire, and motioned for food and drink to be brought him.\n\nThe door opened, and Lady Smallpenny could be heard saying, ''\u2014puke-eyed, empty-chested, yellow-gutted, slug-limbed, swamp-reeking\u2014''\n\nTwo girls from the midwife's entourage hurried in carrying the buckets and some of the blankets. As they spoke with Anne, Lord Smallpenny knew the delivery of his meal would be delayed.\n\nThe women set about getting the water on to boil, setting the blankets by the fire to warm, chatting and laughing and doing whatever other dawdling they could think of. Lord Smallpenny considered pulling rank, but decided in the end that since the interruption aided the birth of his son he would allow it, and bank his carefully cultivated patience. He'd been working toward this goal for a long, long time.\n\nAs he sat stoically he compared his preparations to the prophecy, the old soothsayer's prediction of the birth of the one who would unite the warring factions of Killgully. Even though he was a child when the prophecy was announced, his ambitions had caused him to immediately seize upon it. He had memorized it, treasured it, and manipulated it.\n\nHe could still hear his father reading the prophecy when it was published.\n\nWhen Lording takes a peasant shrew   \nA common thread of life to hew   \nThey a royal line will make anew\n\nThe first fruit of the Goodwife's labor   \nBorn in the stable of her kin and neighbor   \nA warrior child's life-cord cut by saber.\n\nRoyal birth under the vanishing moon   \nOn the night of the midsummer June   \nHope for the kingdom, to be united soon.\n\nFor the child of this birth so royal   \nAll princes, lords, and vassals loyal   \nShall all enemies bury in soil.\n\nSymon the Inept, Soothsayer\n\n''Old fool,'' his father had laughed. ''The meter is off, the rhymes are forced . . . it's complete and utter nonsense. I've said for a while old Symon was not what he used to be. In his younger days he would have been ashamed to predict such drivel. He's getting old and senile, and while we must tolerate his lunatic ravings, a man would have to be insane to believe this hogwash.'' He had looked down his patrician nose in disgust as he tossed the notice in the fire.\n\nLeopold Smallpenny knew his father better than to indicate he took the prophecy seriously. By waiting until his father's death he managed to choose his own bride. He had scoured the countryside for a shrewish peasant girl, with the intention of marrying the worst he found. His quest had been endless and unfruitful until the day his manservant Kermit returned from a little hamlet at the western edge of the country.\n\n''Oh, she's the one. Meaner than a wolf, with a mouth a harpy would envy. Her four brothers live in fear of her tongue. She scolds and chides and nags the whole day through. Her mother is the same and taught her well. Life with her would be a miserable, eternal hell.''\n\nHer name had been Patience Tillman. His courtship had lasted long enough for the flowers he'd brought to end up in the muddy street. Her mother, however, realized his intentions and what coming up in the world would mean for her. She had a short talk with her daughter, and the engagement began. He insisted that the marriage take place in September. His son had to be born at midsummer.\n\nOf course, there were a few other prophetic demands that had to be arranged.\n\nMatthew was set up at a local inn. Marrying him to the old innkeeper's daughter, and bribing them to rename the inn the Vanishing Moon was actually the easiest part of the whole plan. Especially since his mother-in-law had insisted on new positions for the whole family as befitted their new rank. He'd even had two new signs made, one for the inn and one for the stable attached to it. That way the stable belonged to her kin and neighbor, and the boy would be born under the sign of the Vanishing Moon.\n\nThen his wife, recalcitrant termagant, started delivery much too early. He'd had to force the midwife to use her skills and her herbs to delay the birth. It was now the night of midsummer. The midwife had been cautioned to cut the cord with the saber he'd provided, and nothing else, under pain of death. There was nothing more he could do but wait for the prophecy to be fulfilled. His son would be king.\n\nThunder heralded the arrival of the storm. Peasants streamed into the inn from their midsummer revelry, dripping rivulets of water on the floor. Lord Smallpenny always provided generously for the midsummer festival. It insured the continued cooperation of his peasantry. He hushed their grumbling at the interruption of the festivities by buying a round of drinks for all.\n\nHis pint finally arrived without the midwife. Lord Smallpenny considered quaffing its contents just to prove her wrong, but drink had a tendency to go right to his head, and he wanted to be particularly alert tonight.\n\nRain beat down on the inn's roof and thunder shook the walls. While the storm may have ruined the festival for the common folk, it made Lord Smallpenny's heart glad. For now the moon really had vanished behind a curtain of clouds. There was nothing now that could tarnish this night. The omens for the birth of his son, who would one day be king, were perfect.\n\nAt the height of the storm, about an hour before midnight, one of the midwife's apprentices hurried in to say the birth was imminent, and to gather up the hot water and warmed blankets waiting at the fire. Lord Smallpenny was forced to endure the sly grins, goodwill toasts, and overfamiliar back slaps of the excited horde. Even knowing everything was perfect couldn't reassure Lord Smallpenny. Why was it taking so long?\n\nIn the midst of the excited merrymaking an inhuman scream could be heard from the stables. It was followed by something rather like, ''Day-old dung of a putrid skunk! He'll never touch me again!''\n\nAnne turned to Matthew and winked. ''It won't be long now.''\n\nLord Smallpenny sat impatiently at his table, nursing his pint. Waiting. His son would be king. His son would be king! His SON WOULD BE KING.\n\nAs the storm subsided, so did the crowd's excitement. Eventually, some drifted out into the disappearing drizzle for other, more private, camaraderie. At midnight, Lord Smallpenny headed toward the door closest to the stable, intending violence.\n\nWhen he opened the door he could see the midwife's assistants gathered in the open stable door. They looked like a group looking for a messenger to deliver bad news. As he approached them some of the braver ones ran into the covering darkness of the village.\n\n''Well, what news?'' he hollered as he approached them.\n\n''The lady is well,'' one volunteered, then disappeared.\n\n''And . . . ?'' he bellowed as he had heard his wife do.\n\n''And she's recovering quite nicely.'' She quickly followed the first.\n\n''And . . . ?'' He had reached the group by now.\n\n''And . . .'' this one twisted her hands round and round, ''she's very happy.'' Several looked like they wished they were somewhere else, like running off in the dark, but none of them dared.\n\n''And my son?'' he roared.\n\n''And the child is fine,'' the midwife pushed through her few remaining loyal, but nervous assistants. ''A strong and healthy brat. Would you like to see?''\n\nThere was something about her smile that unnerved him. He pushed several assistants aside and sauntered into the stable. Behind him he could hear running feet, and the midwife's laugh.\n\nHe walked down the aisle of stalls until he came to the last one.\n\nIt had been specially prepared for this occasion with a low couch. Strewn about were the midwifery tools and extraneous implements brought by the midwife's entourage. The reek of the bloodstained straw overpowered the other ordinary stable smells. Reclining on the couch was the Lady Patience Smallpenny. At her breast was a baby, squirming and sucking vigorously. The baby had managed to wriggle out of its swaddling clothes. Lord Smallpenny could see its pink wrinkled feet and small hands pumping in the air. Fat, healthy, vigorous, everything everyone claimed to want in a baby.\n\n''I've done it. The prophecy has come true.'' Lord Smallpenny was invigorated by his victory. He couldn't help grinning and pumping his own fists into the air. ''Yes, my son will be king over all!''\n\nLady Smallpenny smiled at him in the same way the midwife had. ''Would you like to meet _my_ daughter?'' The baby shrieked its rage as Lady Smallpenny detached her, and rolled her over for him to see.\n\nDaughter! Ruin crumbled Lord Smallpenny, literally, to the ground. As he knelt in the filthy straw, shocked, his mind would not accept defeat. He had come too far, worked too hard to surrender now. This could not be.\n\nThe words of the prophecy flowed through his mind. Each one standing out clear and crisp. He had satisfied every particular; he would not, could not, be thwarted at this point.\n\nRage overfilled him. He stood up, raised his fist to his wife, and . . .\n\n''Don't you dare!'' she shouted, and flung a wad of bloodstained rags at him. ''You bird-brained, snake-faced hog's rump.''\n\nHe collapsed again, ruined, defeated. This couldn't be happening. He'd done everything right, arranged everything. He looked up at the obstinate, captious woman who ruined his world.\n\nLady Smallpenny bundled up the screaming baby, and gave it what it wished.\n\n''Think on it, you old goat-faced, spit-spattered, mouse-mucked, bewoozled, sorry excuse for a husband,'' Patience Smallpenny said. ''She will rule all. As a woman she will have advantages in uniting the kingdom that a man would not.''\n\nHe rose again. ''Yes, yes, dear. Whatever you say.'' Then he turned, and walked out the door. As he stood just outside the stable he whispered to himself, ''What have I damned this kingdom to?''\n\nHe could hear her inside, crooning to the misbegotten wench.\n\n''Eat, my puny-footed baby. Your father is a silly one, but then again, he is a man. No matter, _I_ see the greatness in you, and after all you do fulfill the prophecy.'' Lady Smallpenny cooed quietly for a moment. ''When we get back to the castle, you will have silks and furs. Your grandmother and I will teach you how to rule over men. You will rule with an iron fist.''\n\nFar off in the forest, an old soothsayer laid down his looking glass and his life with a long and hearty laugh.\nCHOICE OF THE ORACLES\n\n_Kate Paulk_\n\nThe great looms stood silent, strands of Scylla-silk glimmering in semidarkness above the completed weaves. Air heavy with the acrid tension of electrical discharge sparked and crackled with each of my cautious steps.\n\nThe rounded shape of a cable met my probing toes, sending a thrill of fear through my body and clenching my stomach. Delicately, I twisted my foot, my toes questing for the next safe step.\n\nWithout Scylla-silk, Delphi died, and the Oracles with her. The Founders would not thank me for dooming their descendants, not when they had risked worse than death to escape the Old Earth Empire.\n\nAs loom engineer, I had the responsibility of repairing the great looms after each firestorm. The fate of my people rested on my shoulders.\n\nMy foot found clear floor. I eased forward, transferring my weight to my right leg. My body flowed into its new posture, the crackling air transforming the normally simple act of walking into a prickling dance. Too fast, and I would generate a current that could destroy the great looms.\n\nI carried no metal, and Scylla-silk covered me head to toe. Still, an accidental discharge could fry any number of critical components in a heartbeat. The prickling on my skin, the crackling air . . . I needed no instruments to tell me I walked in the heart of a firestorm barely quieted.\n\nBarely five meters separated me from the main consoles. They stood in the island between the looms, lit only by shafts of light from viewplates in the wall behind me. Even in that faint light, I could see enough.\n\nThe boxy shapes of the Weavers were gone, replaced by an amorphous heap of melted, scorched metal and plastics. The Weavers were dead, and my life's purpose with them.\n\nCool, sterile air met me as I passed through the Scylla-silk iris into decontamination. The taste of ash lingered in my mouth as every trace of the spores was sterilized and destroyed.\n\nI shifted from foot to foot, restless, anxious. What future would the Oracles ordain, now that the Weavers were gone?\n\nWithout the Weavers, Scylla-silk was nothing, mere strands of filament that shone and glittered in the sunlight. Only when woven and sterilized did it become valuable. Nothing else worked as well for electrical and EMP insulation. Not even Delphi's firestorms could touch something wrapped in Scylla-silk.\n\nWith a brush of air, the decontamination unit opened to Delphi's habitat dome, spilling me into a crowd of people anxiously waiting to know what the storm had done to their livelihood. My livelihood.\n\nThe answer must have shown on my face as I stumbledthrough the iris into the corridor beyond, for the crowd drew back, their buzzing speculation silenced in my presence. I looked for Diarran among the familiar faces, found him. His square face held nothing I could read. As always, he was a mystery to me, but he was the one the Oracles had determined I should wed.\n\nThe cool voice of the Oracles scattered my thoughts. ''Engineer. Report.'' As always, it was impossible to guess which of them had initiated communication.\n\nI swallowed, pressing my hands against my thighs to keep them from curling into fists. My head rose to face the nearest monitor as though I faced one of the Oracles themselves. ''The Weavers are beyond repair.'' My voice cracked. I dared not say anymore.\n\nSilence. I could hear every breath, every tiny shift of fabric. My hands ached with tension. Fear crackled in the air, as potent as any firestorm.\n\nFinally, the Oracles spoke again. ''This is unacceptable. You are fated to maintain the looms.''\n\nMy lips drew back into a snarl. ''So tell me how to repair melted sludge!'' For all their power, all their monitoring links, the Oracles needed human eyes and hands to work with the great looms. They could not even communicate with the Weavers.\n\nDelphi's ever-present spores and the static they carried blocked communications lines, and risked discharge. Scylla-silk lined the habitat dome to protect all of us within it, but the thread could not be woven inside the dome. Attempts on small looms produced cloth. Nothing more.\n\nThe Oracles' analyses determined that sterilized dead spores were integral to Scylla-silk's properties. Unless the thread was woven in the presence of spores, it was of no more use than any other cloth.\n\n''Insubordination is not merited, Engineer.'' The Oracles never expressed any emotion. ''You have twenty-four hours to reconsider.''\n\nMy mother's voice was the first to rise after the Oracles fell silent. ''Chandra! How could you?'' She was somewhere to my left, beyond my range of sight.\n\nI turned to my right, and stalked away, my heart a tight pounding knot. Each footfall measured my fate. If I apologized to the Oracles when they next asked for my report, I might receive nothing worse than flavorless rations for a time, or a pain sequence. If I did not . . . the Oracles ruled. They could sentence me to anything they considered suitable for my offense.\n\nIt was only fair that they should govern our actions, for they protected us from the dangers of the spores and the firestorms. Without them, the human settlement on Delphi would have died in the first firestorm to thunder past the dome.\n\nI saw no one in the too long walk to my quarters. They were adequate to my station, three small, stark rooms with bare fiberpress walls. The main room held nothing more than the minimum furnishing allotment\u2014 a chair, a low table, and a stool, all fiberpress. The bench along the right wall had a catering unit set into the wall beside the personal terminal.\n\nThe terminal blinked green, indicating that I had unread messages. I ignored it.\n\nThe door to my bedroom slid into its recess in the wall as I approached, then closed behind me with little more than a whisper. I saw the bareness, the bleakness as if with new eyes. The narrow bed with its undyed coverings, the tiny closet holding my uniforms and the two threadbare dresses that were my only leisure outfits. Both had faded to the same gray as the walls. The chest of drawers holding my underwear. My two pairs of gray work boots and one pair of faded strap sandals on the floor of the closet.\n\nI pulled off my insulation suit's gloves and tossed them on the bed. After unwrapping the tail of my hood, I pulled it off and threw it on top of the gloves. The rest of the suit joined it as quickly as my fingers could pull the ties loose. The Scylla-silk piled on the bed, shimmering with iridescent colors that seemed to light the room. Strange that something so lovely could cause such grief.\n\nWith a sigh, I reached for the coarse polysynth fabric of my uniform shirt, and pulled it over my head without bothering to unbutton it. It hit the wall by the bathroom door and dropped into the laundry hamper with a satisfying thump. The rest of my clothes followed as quickly as I could strip them off and throw them.\n\nNaked, I padded the handful of steps to the bathroom. My fingers traced the well-known outline of scars as I walked. The long ridged line from left breast to knee I gained when a laser-guidance cable came loose and scored its path in my skin. The hashing of hair-thin stripes earned when a snapped frame sent its threads scything into anything unfortunate enough to be in the way. Each scar was a mark of honor, a peril of my fate I had survived and conquered.\n\nAnd now, lost.\n\nDiarran sat in my main room when I emerged from the bedroom. I wore a clean uniform, but my feet were bare, and I had not tied back my hair. It curled around my face and into my eyes though I tucked it behind my ears to keep it from getting too much in the way.\n\nHe stood when he saw me. ''Chandra! I . . . are you well?'' His dark hair stood upright, as though he had been running his fingers through it.\n\n''Well enough.'' I shrugged. ''Thank you.'' I moved toward the stool so that he could have the chair.\n\nDiarran did not look away from me. His dark eyes seemed too bright against his pale skin. His hands clenched into his uniform pants. ''Are the Weavers really . . .''\n\n''Yes.'' There was little else I could say. Diarran's destiny lay in the hydroponics section: he knew little of the great looms.\n\nHe swallowed. The muscles of his jaw tightened. He raised fisted hands, let them drop. After a long moment, he turned away from me. ''You must go to the Oracles, and beg their advice.''\n\n''I know.'' The Oracles guided every aspect of life. If they could not offer an answer, there was no answer. Questioning their authority as I had done might be insubordination, but asking their advice was not. Perhaps if I asked, even begged, they might show mercy toward me.\n\nThey would need my skills to repair the looms, even if the Weavers could not be rebuilt. I had no apprentice, and my own teacher had died three years before. The Oracles said there was ample time to choose the right parents and select sperm and egg. I was young and healthy, and there were more important places to fill.\n\nPlaces like Diarran's, maintaining the hydroponics systems that kept us all fed. Or like my mother's work in the atmospherics unit, ensuring that the air we breathed remained clean. Delphi was only barely self-sufficient, so the Oracles rarely chose to increase our population.\n\nDiarran rubbed his chin. He turned to look at me with troubled eyes. ''Chandra . . . please think before you speak. The Oracles . . . they prefer considered, reasoned words.''\n\nI nodded. Emotion was alien to the Oracles; they knew only the benefit of Delphi. My outspoken ways had earned me punishment cycles before. Blurting out the unwanted truth damaged morale. ''I promise I will.''\n\nHe stepped forward, caught my hands briefly in his. ''And say nothing of the Founders.'' His intense gaze held me. ''Father tells me they dislike the old legends, especially those of the Watchers.''\n\nI blinked, startlement showing on my face before I regained proper control. ''But why? What harm can legends do?'' Old, old stories said that the Founders had left treasures in safe storage near their colony ship, with sleeping crew members to guard them and watch over us.\n\nEven if such a treasure had been left, it had surely been lost to the firestorms long ago. There could be no reason for the Oracles to dislike the stories.\n\nDiarran's stare seemed to hold me in a steel vise. ''Are legends logical? That alone is reason enough.'' He shook his head. ''Be safe, Chandra.'' With that, he released my hands, and left me to my thoughts.\n\nWithin an hour I stood as close to the Oracles as any human ever came. Here in the bare metal Request Room I could place my petitions and beg for mercy. I took a deep breath and knelt facing the blank screen that was the room's only decoration.\n\n''I seek advice and offer apology for my words earlier, Oracles. I should not have risked morale.''\n\nSilence. The air in the room seemed to press heavily on my shoulders as I waited for the Oracles to determine the priority of my request.\n\nFinally, the cool, sexless voice spoke. ''Your apology will be taken into consideration when punishment is determined, Engineer.'' The Oracles never inflected their speech. ''State the items for which you desire advice.''\n\nI swallowed again. My hands sweated against my uniform pants as I stared at the viewscreen. ''Can the Weavers be rebuilt? Can the great looms?'' My voice broke and a shudder ran through my body. ''Delphi's economy depends on Scylla-silk\u2014if we are unable to weave it, how shall we survive?''\n\nWith my interwoven fears spoken, a weight seemed to lift from me. Whatever happened now, it was the decision of the Oracles. My fate was in their hands: I had done my part.\n\nThen the Oracles gave their verdict. ''Delphi lacks the materials to replace the Weavers or the looms. Survival will require reduction in personnel to bring consumption to self-sustaining levels. Nonessential personnel will be scheduled for termination.''\n\nMy eyes burned, my heart tightening around a knot in my chest. I had asked: now I knew. I could see the future of Delphi closing around me, its fate determined by the logic of the Oracles. Population would fall, driven by the ever-decreasing ability to produce the necessities of life, until at last only the Oracles would survive. All for the good of Delphi.\n\nI could not believe that the Oracles could see only that choice. ''Surely that will eventually destroy Delphi?'' The words fell from my mouth before I could stop them. I hastened to minimize my error. ''Each new problem will force further reductions with fewer personnel to mend them. Is there no alternative?''\n\nThe pause before the Oracles replied told me they had at least considered my question. ''There is no alternative. Current stocks of Scylla-silk are insufficient to purchase replacement units. Continued existence of the settlement dictates the selected course of action.''\n\nI could not sleep. The knowledge that my fate was decided, that I along with all other surplus colonists would soon be dead . . . there could be no rest for me.\n\nDiarran's words about the colony ship, the legendary treasures guarded by sleepers, worked upon me, driving me to wonder if there might be some miracle there. I could not wait quietly for death\u2014I had to act, even if that meant a foolhardy chase for a legend none truly believed.\n\nThe evening firestorms had passed and the stillness of full night lay heavy in the air as I stole through empty corridors to the Weaving annex. The iris opened for me, admitting me to the decontamination unit though the Oracles were undoubtedly aware that I was breaking curfew. Perhaps they thought it considerate of me to remove myself from their aegis and spare them the resources required for termination.\n\nMy Scylla-silk insulation suit whispered with each movement as I walked through the inner iris to the Weaving annex.\n\nMy breath caught in my throat. Spores danced in the air, light sparking between them in a flickering dance of miniature lightning. The half-finished Scylla-cloth glittered with spore light, making the huge room almost as light as it was when the lights were powered up.\n\nI felt my way through the room, not caring if I sparked a firestorm. There was nothing left to destroy. The greasy feel to the air and actinic taste told me how much static had built up, static the disconnected grounding rods no longer drew away from the precious looms.\n\nAs I passed beneath the massive plas-steel structures, I did not even look up. They had been my life: now they were nothing but twisted shapes in the night.\n\nAt the far end of the annex, I felt along the wall until my gloved hands found the door to the cocoon bay. The plastic latch snicked as I turned it, the door scratching along the floor when I pulled it open. Electricity jumped between the floor and the nearest spores in a shower of blue-white light.\n\nA step, and I was outside, or nearly. The cocoon bay was no more than an enclosure of stone walls set against the Weaving annex. Perhaps thirty man-sized cocoons lay scattered about the enclosure, each the home of a Scylla-grub partway through its transformation into one of the immense butterfly-like Scyllae.\n\nHere, grubs could sleep safe from whatever predators roamed the rest of the world. The noises that came from the annex combined with the two-meter walls to ensure that few creatures other than the grubs ever entered the cocoon bay. Until they slept, there was an ample supply of decaying fiberpress to feed them and give them the energy to spin their cocoons.\n\nThe cocoons glittered in the night; their coating of spores sending out tiny sparks as new spores drifted closer.\n\nThe wall was almost as old as the dome, and lacked any protection from firestorms. Charred black streaks marred the pale stone blocks, while deep shadows showed where stone had crumbled away.\n\nI set my teeth, and began to climb.\n\nIt was easier than I had anticipated\u2014I was not even out of breath when I reached the top of the wall and looked out over Delphi.\n\nA seething moil of fungi met my eyes. From the edge of a wide swath of seared, glassy rock, fungi grew so fast I could see weird shapes swelling into familiar bulbous spore pods, spore-catching vanes spreading and thinning, even fungal spires as tall as the dome. Chittering and hooting sounds that surely came from animals offset the crackling rustle of fungal growth. I could see no animals, though in that great sea of life even a giant might be hidden.\n\nI swallowed and dropped from the wall to glassy stone. This must be the perimeter the first settlers had made, back when they still had working plasma weapons.\n\nA darker line winding through the forest of fungi caught my eye. Curious, I walked toward the line, my nose wrinkling and my eyes watering when the smell of vegetable decay caught me.\n\nI paused to rearrange my hood so that a fold of the silk covered my nose and mouth. It could not protect me from that awful stench, but it did soften the effect. I swallowed down nausea as I tied the hood in place.\n\nThe dark line was a path of seared rock. It led into the seething forest, apparently without end.\n\nTo the Ship? Or perhaps it led to the legendary treasures of the Watchers. Either destination was worthwhile and taking the path was a far better choice than circling the dome in search of a clearer way. I swallowed, and stepped forward.\n\nFungal growths made fans and umbrellas, partly shadowing the glassy stone on which I walked. Glittering light from insects kept me from utter darkness. Despite the stench, I stared around, entranced.\n\nSurely there was hope to be found here. Surely the Oracles were wrong. That such beauty could be of no use was unthinkable.\n\nThe explosive opening of a spore pod warned me of approaching dawn. Around me, spore pods filled, straining.\n\nI dropped to the ground and covered my head with my arms. Each pod blasted open as though explosives had been set within it, fungi straining to send their spores as far as they could. Meaty ripping sounds marked where spores were driven through fungal flesh, then another explosion as a fast-flung spore tore through an unopened pod sent the whole spore release cascading on.\n\nAll I could do was hope I was below the spore range. Spores that could not escape the shading fans drifted to the path, layering the smooth surface of the path with shimmering light.\n\nFinally, I could hear no more explosions. I climbed to my feet, trembling. The gray light robbed the fungi of their magical lights and colors, drove them into themselves, to shrink and shrivel as sunlight approached.\n\nAt least it was safe to walk. The spores could only grow on dead vegetable matter, and then only at night.\n\nAs I trudged forward, the fungi receded, wasting away, until I could see clear to a distant silvery speck. The Ship? Curiosity stirred within me, and I walked a little faster.\n\nThe first rays of sunlight gave no warmth, only clear, golden light. That light triggered another transformation in the Delphian wildlife: plants began to push aside the withering fungi, straining toward the light.\n\nMy mouth sagged open. The sound of plants growing crackled through the air, bringing with it a rich, earthy taste. I watched as a single round leaf pushed through the dying fungi, then shot sprouts as it grew to top my height in less than a minute.\n\nNot until I saw the growth of flower buds did I realize my danger. Exploding spore pods would not clog my lungs and swell my throat shut. The pollen likely to erupt from the plants would.\n\nI ran.\n\nI staggered up a rise, my feet slipping on rock slick with pollen and spores. My eyes were too swollen for me to see, though they still streamed tears. Every wheezing breath burned my chest. My nose streamed, and I could not pull enough air through my mouth.\n\nStumbling, I fought my way to the crest of the rise. The ground tilted beneath me and I began to descend. My feet skidded, the Scylla-silk insulation suit offering no purchase on pollen-coated stone.\n\nMy arms flailed as I fought to keep my balance, to keep from tipping back. If I fell now, I would die.\n\nThe ground tilted further, the path growing steeper. My feet shot out from under me, and I landed hard, pain shooting from my tailbone. Even that was not enough to slow my slide. I scrambled to try to stop myself, to grasp fast-growing plants I could not see. Nothing met my straining fingers.\n\nThe ground beneath me ceased to exist.\n\nI reached for something, anything that could stop the inevitable fall. My hand brushed something hard, but it was gone before I could grasp it.\n\nAir I could not breathe rushed past me as I fell. My heart throbbed once, twice.\n\nSolid ground slammed into my body, and I knew nothing more.\n\nI awakened to warmth and comfort. When I opened my eyes, I saw that I lay in a cave of treasures. Golden light filled the cave, radiating from globe-shaped lamps attached to the walls. Computer parts lay scattered around as though worthless, sharing the floor with hydroponics equipment and things I could not name.\n\nI lay on a thick mattress of some synthetic material, a white computer beside me. It had a multitude of appendages: grippers, scanners, fine knives, needles, and other things that seemed purposeless to me. The arms were motionless and folded.\n\nIt was not until I sat that I realized I was in no pain. My eyes were no longer swollen and I could breathe without difficulty. Was the system beside me a legendary autodoc?\n\nI shook my head, but my confusion did not clear. Nothing in this cave of parts gave me any indication how I had come to be here. I could not have landed on the mattress. This cave had no holes in the roof, no gaps I could have slid through. The only entrance I could see was a door at the far end of a cleared path from the mattress. It stood open, silent.\n\nThere was only one thing I could say with certainty: this was not the colony ship. There were signs that at least part of this cave had been smoothed out with plasma cutters, but it remained indisputably stone. The air had a faint tang of oxidation, far weaker than I was accustomed to.\n\nFootsteps echoed, growing closer.\n\nMy chest and stomach tightened, and I pressed my hands against my thighs, striving to appear properly calm. I should not be here.\n\nAll pretense of calm fled when a giant of a man filled the doorway. He had to duck his head to enter. I stared as he approached, realizing first that he was no one I knew, then that his clothing could not have come from Delphi. We could not produce such deep blues, such brilliant gold ornamentation.\n\nHe smiled as openly as an infant. ''Ah, good. You're awake.'' His voice was open, too, lacking any of the restraint I had thought was normal for adults. Despite his accent, he was easy to understand. ''The autodoc had to wake me to put together an evac unit for you.''\n\nI blinked, trying to make sense of his words. He had found me and brought me here?\n\n''It was a close thing,'' he added as though he had not noticed my reaction. ''You did yourself plenty of damage with a massive allergic reaction and then the fall into the caves.'' He shook his head. ''Why in the world didn't they give you a breather?''\n\nI swallowed. My eyes burned, and I blinked several times. The damning emotions would not be held back. ''Why would the Oracles do that? I am surplus to requirements.'' Nothing could keep bitterness from my voice. I needed all my control just to keep from crying.\n\nThe man's eyes opened wide and his mouth fell open. ''Surplus to . . .'' He swallowed. A muscle along his jawline twitched. His hands clenched into his shirt, unclenched slowly. ''I think perhaps you need to tell me what's been happening at the dome.''\n\n''You do not know?'' I gulped, my hands clenching into my pants. I should never have allowed myself such an undisciplined outburst.\n\nHe raised one eyebrow. ''I've been in suspended animation since a few months after we made landfall. It's a little hard to keep up with current events when you're on ice.'' Amusement laced his voice, telling me he either had not noticed my transgression, or was choosing to ignore it. ''So, what's this with the Oracles making the rules? They were supposed to help the colony, not control it.''\n\nThe cave and its treasures seemed to spin around my head. That the Oracles ruled for the greater good was a fact of life as solid as the rock beneath my mattress. And yet . . .\n\nI shuddered, feeling tears scalding my face, the man's arms holding me as a father might comfort a child. What had my people done? What had we become?\n\nEventually the man\u2014Medical Officer Jared Aldreth\u2014 brought me from the treasure cave to a much smaller cave furnished with chairs and tables, and set a meal in front of me.\n\nWhile we ate, he asked question after question of me, his mood growing darker with each answer I gave. The pent-up anger within him held until I told him why I had fled the dome.\n\n''Damn it, are you all sheep?'' His hands clenched the tabletop. ''Your ancestors risked _everything_ to escape being controlled like this.''\n\nI flinched from his words, shame burning within me as I bowed my head. ''I apologize, officer.''\n\nHe shook his head. ''Just Jared, Chandra. I don't need fancy titles.''\n\nI looked up in time to see him close his eyes briefly. ''And it's me who should apologize. This mess isn't your fault.''\n\nI picked up a grape and popped it in my mouth, savoring the cool smooth shape of it for a moment before biting down to release the sweet juice and flesh. Food like this was a rare luxury: most of the hydroponics section was given over to the enhanced soy plants we processed into normal rations. Plants like grapes, which gave so little nutritional return, could only be considered when the needs of the colony were met.\n\nThoughts spun through my mind. The colony was doomed if the Oracles were not stopped. They had saved Delphi, but they had also destroyed us, draining our souls until we were mere shadows beside our ancestors.\n\nI swallowed another grape before I asked, ''So what happens now?''\n\nJared sighed. When he laid his hands flat on the table they seemed so much larger and stronger than any of the people I knew that I could not help feeling like a child beside him.\n\nI swallowed, tasting the last of the fruit in my saliva. ''I can't solve this one for you.'' His eyes were very blue and open, so different from Diarran. I could not look away. ''You and the other colonists are going to have to work that out for yourselves.'' His shoulders dropped and he leaned forward, his eyelids falling to free me from his gaze. ''If I fix it for you, I'll just be another thing for you to worship, you see?''\n\nDisappointment ebbed as his words sank in. The Oracles had indeed conditioned us to worship them, to leave all decisions for them to make. But we did not have to stay slaves to our own weaknesses.\n\nI sat a little straighter. ''Can you help me do something?'' I had already defied the Oracles once. Once more could not be so difficult.\n\nThey would not welcome my interference. Reasoning with them would surely fail\u2014I was certain now that the Oracles knew of this place and chose not to use the resources here\u2014but they held so much knowledge, so much power ''Could you help me shut them down, and replace them with ordinary computers?''\n\nHe smiled, the corners of his eyes crinkling. ''Now that, I can do.''\n\nMy stomach clenched tight as I tied my insulation suit closed. The once-sleek lines of the skimmer beside me now bulked out with the assortment of computers, hydroponics equipment, and seed stock Jared and I had loaded onto it.\n\nAn ancient insulation sheet covered and partially concealed the load, less effective than Scylla-silk but better than nothing. It could not conceal the plasma cannon we had mounted on the steering mechanism.\n\nThe plasma pistol I wore on my belt bulged against my insulation suit. An uneasy thrill slid through my body each time my arm brushed past it. I tried not to think about having to use it.\n\n''The storms have passed.'' Jared's soft voice did nothing to calm me. I pressed my hands flat against my thighs, feeling the silky smoothness of my suit.\n\nA deep breath, in, out. ''I should start.'' My voice trembled. ''Thank you.''\n\nHe nodded, weariness showing in his slumped posture, the strain around his eyes. ''Good luck.'' Jared blinked and wiped his arm across his face. ''I need to go back to sleep. They warned me before I volunteered that any forced wakes would play hell with my system, but I didn't realize they'd be this bad.''\n\n''Being brought out of suspended animation for me?'' I hazarded. While Jared's accent and speech patterns were not difficult to follow, his terminology confused me.\n\n''Yeah.'' He leaned against the wall. ''Waking up's supposed to be done over two to three days. The drugs they pump in to bring me up in under an hour . . . once they wear off, I'm useless.''\n\nI winced. ''The systems here can close up after me, yes?'' If Jared fell unconscious, I doubted I could carry him anywhere, much less to wherever he was supposed to sleep. He was starting to tremble.\n\nHe nodded. ''Yeah, the systems handle that without a hassle.''\n\n''Then please, go.'' There would be enough on my conscience by the time this night ended. ''I can manage here.''\n\nI watched as Jared stumbled away. If I dared take the time, I would have helped him. My eyes burned as I reminded myself that if I did not act, and soon, the Oracles would begin killing my people. They might have already begun.\n\n''Be well, Jared,'' I whispered.\n\nWith a deep breath, I climbed onto the skimmer. Its controls were simple even for one who had never used such equipment. A power button, a thumb tab on the right to control speed, another on the left for height, and the steering arm for direction. The synthetic fabric on the handlebars held even my Scylla-silk gloves in place.\n\nI pushed the power button. Green light shone on the steering arm, a gauge that seemed to indicate that the skimmer was fully powered. It rose gently, so gently I felt nothing and only the movement of the walls relative to me told me I was rising. A soft chime sounded, and another section of the steering arm glowed with light. This one seemed to be an indicator of speed, for when I touched the right thumb tab, it rose as the skimmer moved forward, the color changing from dark blue to bright green as my speed increased.\n\nCool night air caressed my face as the skimmer passed from the confines of the holding area and into open air. I pushed the height tab up, and the skimmer angled upward, climbing until I released the tab.\n\nBelow me, fungi writhed and flourished, filling the air with their magical light. There was no greasy overload of static in the air, no sparkle of discharge around me. Unlikely as it might have seemed, I was above the static, and the air smelled fresh and clean.\n\nI breathed deeply, throwing back my hood that I might fully experience it. As I thumbed the skimmer's speed up, the breeze grew until it lifted my hair and plastered my clothes to my body. I grinned, heedless as a child. My heart seemed to lift and swell with pleasure until I thought I might cry for joy.\n\nThe silvery blue glitter of the habitat dome shattered my joy as surely as the storms had destroyed the Weavers. Now it was time for me to decide my fate, and the fate of my people.\n\nI left the bulk of my cargo on the skimmer. The insulation cloth concealed most of my treasures, though nothing could conceal the hole I had made in the wall. The Weaver annex had not been designed to allow vehicles to enter.\n\nFor tonight, I needed only the pistol in my belt and a handheld electromagnetic pulse generator and circuit locator. The most important knowledge the Oracles held had come from the original colonists. The datacells from the systems on the skimmer held that same knowledge.\n\nI slipped through the glittering darkness to the entry iris. A single pulse fried the circuits, allowing me to pull the iris apart and step through into the quiescent decontamination unit. The same process took me from the decontamination unit into the habitat dome.\n\nThe still, sterile air seemed lifeless as I padded through featureless corridors. In my insulation suit, I ghosted through pathways familiar only because my journeys through them had become habit.\n\nThe Oracles no doubt monitored my every move, though it was unlikely they divined my intention. Perhaps they thought I had returned with a solution to restart Scylla-silk production.\n\nThe corridor opened to the market square, a large, open room where we bartered for the few luxuries Delphi possessed. Here, the dim night-lights shone on atrocity.\n\nBodies lay sprawled on the floor, over tables and chairs, wherever they had fallen. The light was too dim for me to see their faces. My chest tightened and my eyes burned. Too many of the fallen were children, too young to be productive.\n\nI swallowed in a throat tight with suppressed grief. The Oracles could not be allowed to take any more lives. They had stolen too much from us already.\n\nMy feet shuffed against the floor as I ran from the market square, half blinded by unshed tears. My lips curled back from my teeth in a grimace. Never again.\n\nOnly crashing into a wall stopped me. I blinked; catching my breath while my head stopped spinning. This was nowhere familiar to me. Dust lay thick on the ground, inactive spores glimmering with faint static.\n\nMy left hand tightened around the EMP generator.\n\nThe spores sparked and faded. Beside me, a door I had not seen slid open, its lock deactivated by the generator.\n\nI rubbed my nose to keep from sneezing, and stepped inside.\n\nAn alarm flashed red. ''Unauthorized intrusion.'' The words must be intended to warn any intruder to flee. I took another step, peering into the flashing darkness. If alarms were going off, I was close to the Oracles.\n\nDistant alarms wailed, intruder and attack alerts. The possibility that Delphi's tiny security force had been terminated as surplus to requirements made my lips curl in the tiniest of smiles.\n\nMetallic shapes emerged from the shadows, great looming bots with cutting tools on their arms. I hit the EMP generator again and ran forward, not waiting to see if I had fried their circuits.\n\nThe toneless voice of the Oracles echoed from more speakers than I could locate. ''Rebellion is not tolerated. Surrender now, or face the consequences.''\n\nI sped into darkness, trusting my instincts to keep me from falling. The circuit location function of the EMP generator showed a massive cluster of circuitry not far from me. It had to be the Oracles.\n\nVoices, human voices, behind me. ''Great Oracles! How can we serve you?''\n\nSo the Oracles had not terminated security. I was less surprised than I might have been just days before.\n\nLight blazed, making me squint into a room suddenly brighter than daylight. The Oracles stood perhaps twenty meters from me: too far for the EMP generator to damage them. A great chasm gouged from the stone beneath the dome surrounded the Oracles, a protective moat too wide to jump. If I did not jump, I would not be able to kill the Oracles.\n\nMy right hand fumbled for the pistol in my belt. My gloved fingers closed around smooth metal, and I drew the pistol out.\n\n''The intruder must be terminated.''\n\nMy death sentence hung in the air. With no hope in any other direction, I sprinted for the gap, wincing as blasts of superheated air shot past me. I would destroy the Oracles as I fell.\n\n''Aim for the _woman,_ fool!''\n\n''This is no firing range!''\n\nMy feet touched the edge of the chasm and I leaped. Mid-leap, something crashed into my back, pushing me forward. I smelled charred fabric, and the air around me burned. I had a moment to realize that I had been shot, that only my insulation suit had saved me, then I slammed into the floor, skidding along the dust-slicked surface. Sparks flew around me, spores giving off their static.\n\nI wondered how many years had passed since spore-bornestatic first corrupted the Oracles. How long had we followed the orders of insane AIs?\n\nBreathing hurt. My lungs burned, but there was something I must do.\n\nMy left hand tightened around the EMP generator, and held.\n\nThe death of the Oracles was not accomplished in a shower of sparks, as storytellers like to relate. They simply fell silent, the purr of their drives silenced. Around them, the bots froze, and all through the habitat dome lights flickered on and doors opened as ancient emergency backups powered on.\n\nI hauled myself to my feet to find the security crew kneeling, their heads bowed. When I tried to speak, I managed only a croak.\n\nOne of the men lifted his head. I recognized Diarran's father, but his face was twisted with emotions I could not begin to name. We stared at each other for what seemed like eternity. I wanted to speak, to apologize for destroying everything he had known, but the words would not come.\n\n''The Oracles,'' he said at last. ''You . . . you destroyed them.''\n\nI nodded. ''They would have killed us all, like the children in the market square.'' Though I ached from head to toe, I found strength to glare at him. ''Is that what you wanted?''\n\nHe winced with his whole body. ''No! But . . . you _destroyed_ them! Our only hope.''\n\n''No!'' My voice cracked, echoing in the still air. ''We still have hope.'' I blinked to clear my eyes. ''I brought old datacells from the cache the Founders left for us. Everything we need is there. We have generators,Scylla-silk, hydroponics. We do not need the Oracles.''\n\nHis jaw worked. He looked so like Diarran that my heart tightened. Surely Diarran had not been among those terminated. Surely not. For a moment, it seemed that he would argue, then his shoulders sagged and he bowed his head in submission.\n\nI longed to scream, to protest what I could see about to occur, what Jared had refused to become. Weariness and emotion stilled my tongue.\n\nHis voice was dull, empty, when he asked, ''What is your bidding, Great Oracle?'\nCAMELOT'S GREATEST HITS\n\n_Laura Resnick_\n\nThe boy who pulled the sword out of the stone that summer's day was, if I may be candid, something of a disappointment to me.\n\n(Don't look at me like that. He had ample opportunity to tell the story _his_ way, after all.)\n\nHe was young at the time, just as the most popular songs and stories about him (most of which _he_ wrote, by the way) indicate; but he was hardly the stalwart, fresh-faced lad of noble spirit and unstained soul that most people suppose.\n\nHowever, in the interests of accuracy, as well as whatever honesty I may owe to posterity and to the heavenly rest that awaits me (God willing), I will admit that the spin put on his image over the years was not entirely devoid of my influence. In the beginning, as you will no doubt realize, it made sense. Indeed, it seemed downright necessary. It was only later on that the fiction took on a life of its own, hurtling through time and space in a thousand-legged body of exaggerations and fabrications that were not, to put it mildly, always in the best possible taste.\n\nBut I digress.\n\nThe day he pulled the sword out of the stone, I foresaw no hint of what was to come.\n\nSince I am a famous prophet, you can perhaps imagine what a huge admission this is for me to make. However, I merely foretell the future as it is _meant_ to be. What actually happens, what flawed mortals choose to _do_ with this information\u2014that's another department entirely. In my defense, I say only that I did my best with the material that I found in my path. Can any wizard do more?\n\nAnd such _un_ promising material it was, too, on that fateful day! He was skinny and pale, with poor posture and long, unkempt black hair. His clothes were slovenly and ill-fitting, and his overall manner was that of an unprepossessing adolescent of the lower orders.\n\nNonetheless, fate cannot be denied. So when I recognized that my destiny had come for me, my disappointed hesitation lasted only, I assure you, a brief moment.\n\n''Hail, young squire!'' I cried after that (truly) brief moment, my long white hair and dark robes flowing in the summer's breeze as I made my way across the village square to the lad who had just performed the long-awaited and supposedly impossible feat of withdrawing that famous sword of incalculable value, known as Excalibur, from the massive stone in which it had been firmly lodged up to the hilt for so long.\n\n_''Whoa!''_ said the lad, examining the gleaming sword. ''Awesome!''\n\n''Thou hast freed Excalibur!'' I cried. ''As foretold in prophecy\u2014er, mostly _my_ prophecies, if I may forego modesty in favor of truth\u2014a young man of noble but obscure birth has come forth to seize Excalibur, which he will now hoist overhead to drive our enemies from British soil forever!''\n\n''Huh?'' said the boy.\n\n''You are the great war leader we have awaited, lo these many dark years!''\n\n''You talkin' to _me_?'' He looked around, as if thinking there might be _two_ boys in the square who had just withdrawn swords from stones.\n\n''You have come at last to lead us and to save the Britons from annihilation!'' I proclaimed.\n\n''Whoa, dude,'' said the boy, blinking. ''Get a hold of yourself. You're, like, totally tripping out.''\n\n''Your coming was foretold. I realize this may be hard to absorb all at once.'' I took him by the shoulders and leaned close to him, peering hard into his vacuous blue eyes.\n\nHe gripped the sword more firmly and backed away rather quickly. ''Hey, bro, look, if you're one of those Greek types, that's cool, no problem, but that's just not my scene. Dig?''\n\nI glared at him, offended.\n\nHe added, ''Anyhow, you're a little old for me, dontcha think?''\n\n''I do not have abhorrent proclivities,'' I said frostily.\n\n''Hey, I wasn't trying to imply anything about your health, dude. Just saying I don't play that side of the street. So are we cool now?''\n\n''Let's start over,'' I suggested after a long moment of consternation.\n\n''Sure, man, we got off on the wrong foot,'' Arthur said amiably. ''No hard feelings?''\n\n''I would not be sane if I had not by now learned to overlook the mistakes of mere mortals.''\n\n''Great. Put 'er there.'' He offered me a none-too-clean hand. ''Name's Arthur. Arthur Pen.''\n\n''And I am Merlin, the prophet and sorcerer,'' I announced.\n\n''Prophet? No kidding! Is there any dough in that?''\n\n''Dough?''\n\n''You know. Bread. Moolah. Shekels.''\n\n''It's a living,'' I said. ''And, as it happens, I'm custodian of one of the most important prophecies in history.''\n\n''So I guess you've got job security?''\n\n''You and I were destined to meet.''\n\n''Is this, like, a fortune-telling?'' he asked warily. ''Because, if it is, dude, I can't pay you. I'm skint.''\n\n''By pulling Excalibur from the stone, Arthur,'' I said patiently, ''you have set your foot upon the road you were destined to travel.''\n\n''Yeah?''\n\n''Indeed.''\n\n''I had a feeling,'' the boy said, looking excited.\n\n''You sensed your future embracing you?'' I asked eagerly, hoping to get things on track at last.\n\n''Yeah! I saw this sword, that's been left sitting here, like, _forever_. And I thought, hey, you know, that is one _good_ -looking sword, I'll bet it's worth something.''\n\n''It is worth a nation's fate and history's adulation,'' I assured him.\n\n''So I can get something for it?''\n\n''Pardon?''\n\n''I figured I'd take it to the market, see if I could sell it,'' Arthur said.\n\n'' _Sell_ Excalibur?''\n\n''Uh-huh.''\n\n''For _money?_ '' I cried.\n\n''Well, unless someone's got a really good lute to trade for it.''\n\n''A lute? To _trade?_ ''\n\n''But I'd rather go for the money,'' Arthur said. ''This sword looks pretty valuable. I'll bet I could get enough dough to outfit the whole band, don't you think?''\n\n'' _What_ band?''\n\n''Well, okay, I don't have a band yet. That's in the future. I figured I'd go on the road alone first, build my rep, get some steady gigs, and then acquire the band. I'm going to call it Arthur Pen and the Dragons. What do you think?''\n\n_''What?''_\n\n''Or maybe just Arthur and the Dragons. Or Arthur Pen's Dragons? I dunno yet.''\n\n''We have to talk,'' I said urgently.\n\n''Another time, bro. I need to go down to the market and sell this sword.''\n\n''Wait!'' I cried in my most thundering voice. ''You cannot sell that sword! You're making a grave mistake!''\n\n''No time to talk now, dude. Catch you on the flip side.''\n\n_''Stop!''_ I used my mighty power to freeze time and space for a few moments, so that I could talk some sense into this boy.\n\nArthur looked around at animals frozen in mid-stride and villagers frozen in mid-sentence as the universe came to a sudden halt upon my bidding.\n\n''Whoa, dude! _Awesome!_ ''\n\n''Well, first of all,'' the boy said, after I had explained his destiny to him, ''I don't think I like what you're implying about my mom.''\n\n''But Igraine was beloved of a great king!''\n\n''If he was so great, how come the country's in such a mess?'' Before I could respond, the lad continued, ''Besides, that's not the point. The point is, you're saying she was, like, you know, _untrue_ to my dad.''\n\n''Uther _was_ your dad.''\n\n''Unfaithful to her husband, then.''\n\n''Whom you've never even met, so what does it matter?''\n\n''Dude, wise up. She's my _mom_. So you really want to quit that whole 'she lay in adulterous union with a king' rap, and you want to quit it right _now_. Dig?''\n\nI sighed. ''All right, fine, I di . . . er, I comprehend.'' I had neglected to consider how touchy boys his age could be about their parents. God forbid that his mother had ever been a young woman with passionate needs. _No._ In a teenage boy's worldview, his mother was necessarily an asexual being who existed only to clean up after him and feed him.\n\nBut I asked curiously, ''What did Igraine tell you about your father?''\n\nArthur shuffled his feet. ''He's doing ten to twenty upstate for stealing horses.''\n\nIt seemed a rather unsatisfactory alternative to telling the boy he was the illegitimate spawn of a liaison with the late King Uther, but I decided to let it pass.\n\n''In any event,'' I said, ''you are the stalwart young man who hath pulled Excalibur from the stone, so you're definitely\u2014''\n\n''No _way_ am I this war leader you're looking for, man,'' Arthur interrupted. ''I, like, _hate_ war. Totally! I'm a lover, not a fighter.''\n\n''But\u2014''\n\n''I'm a _musician,_ bro. An artiste!''\n\n''You _were_ a musician, but now you're destined to unite Britain under a strong monarchy and usher in a Golden Age,'' I explained.\n\n''No, I want to be the court minstrel _singing_ about the Golden Age. Which, frankly, can't come soon enough, as far as I'm concerned. With things being so bad these days, it is _incredibly_ hard to start a career as a bard.''\n\n''Arthur, there is no escaping your destiny.''\n\n''Mom wants me to settle down to something steady, like agriculture or brewing. But what can I say? I've got ballads in my blood.''\n\n''And I'm sure music will make a nice hobby,'' I said firmly, ''while you're changing the world and carving your name into history for all time.''\n\n''How about this?'' Arthur suggested. ''What if I give you the sword and you go find someone who _wants_ to wage war against the Saxons? I'll write a song about every battle! How about that? You could give me a nice new lute in exchange for the sword.''\n\n''Arthur, no, _you_ have to wield the\u2014''\n\n''Or you could take the sword for free,'' he said, pushing it into my arms and trying to get rid of me. ''I'll pick up some extra work at harvest time to pay for a new lute. Or maybe I'll just go on the road with the old lute, it's not really that bad. One string is missing, and the\u2014''\n\n''I will _create_ the finest lute in the land right now and give it to you, if you'll just shut up and cooperate,'' I snapped.\n\nArthur paused. ''Really? You can create a good lute?''\n\nI sighed and muttered, ''The things I do for destiny.''\n\nI am forced to admit that Arthur was a most creative and versatile musician, and it cannot be denied that he had considerable technical skill. His music was not to _my_ taste, of course. I prefer the classics. But there is no denying that, once I put a worthy lute in his hands, his musical influence quickly grew\u2014and, indeed, _grew_.\n\nHis war skills, however, were pathetic beyond the power of mere words to describe. When it came to slaughtering Saxons, his heart simply wasn't in it. Our first battle was an embarrassment. Our second was a total disgrace.\n\nThen Arthur came up with his master plan for the third battle against the Saxons. We had very few warriors left alive and in one piece, and I was thoroughly demoralized and all out of ideas. So, although I thought his plan was half-mad and doomed to certain failure, I didn't oppose him. If we were going to go down, we might as well go down singing, I supposed.\n\nThus it was that, on the eve of Arthur's third battle against the Saxons, he and a select group of warriors, whom he now called the Dragons, sang the epic ballad that Arthur had spent the previous month composing and rehearsing with the men. It was entitled ''The Bloody, Gory Tragedy of the Doomed Saxons'' and was performed in four-part harmony, with a nice little lute solo for Arthur just before the big finish. It told the tale of Saxons perishing by the thousands in their first and second battles against Arthur, only to return foolishly for a third battle, in which the rest of them died most horribly.\n\nI thought the fact that _we_ had died by the thousands during our first two battles against the Saxons, while they had come out of those confrontations with most of their army intact, would make that ballad the laughingstock of the Saxon infantry that night. However, Arthur assured me that there is nothing more powerful than a good story well told\u2014particularly when it's accompanied by a catchy tune.\n\nAnd he proved to be right. Arthur Pen's Dragons sang about the bloody, gory, could-have-been-avoided - if- only - they'd- used - their- heads - and- turned-back-while-there-was-still-time deaths of thousands of Saxons during the third battle for Britain . . . and by morning, we discovered we couldn't actually _fight_ the third battle, because the Saxons were busy fleeing back to Saxony as fast as their bony feet would carry them. The only Saxons who had remained to fight us were a few tone-deaf Bavarians, whom we quickly took as prisoners and held for ransom.\n\nThe ransom we got helped finance Arthur's purchase of some superior musical instruments from the Continent, and he used these in an elaborate composition that ensured victory in our fourth battle against the Saxons. It was a mourning song wherein (supposedly) Saxon women wail for the men who would never return home from this battle, so mighty and undefeatable was the massive British army. The (real) Saxons, listening to this ballad, grew so overcome with grief at their own imminent deaths and guilt over the women they had abandoned back home in search of vainglory, they surrendered at dawn, weeping copiously and crying phrases which our translator interpretedas, ''Tell Mom I love her,'' and ''I left my heart in Freiberg.''\n\nThis was actually the end of the Saxon wars. But you know how it is once you've got a hit on your hands. The public just keeps demanding more. Thus it was that Arthur wrote several more ballads, right up through his ''ninth'' battle with the Saxons. By then, he felt burned out on ballads of military victory and Saxon doom.\n\n''I'm an artiste, Merlin,'' he said to me. ''If I keep covering the same ground over and over, I'll grow stale and my muse will desert me.''\n\nI didn't object, despite the tremendous acclaim the immortal ''battle ballads'' had won for the young King Arthur (who was crowned, by popular demand, after his first battle trilogy went double platinum). I had too many responsibilities on my hands, since Arthur was not interested in _ruling_ the country of which he was king, to quibble about his music. It's only in retrospect that I realize I should have taken a more active interest in the band.\n\nHow was I to know that two new additions to the group would eventually change everything? Their names were Lancelot and Guinevere.\n\nGuinevere had auditioned to be in the wailing chorus for ''O Ye Women of Saxony, Lament Your Lost Loves,'' the smash hit that had ensured our victory in the fourth Saxon battle. Arthur had been so taken with her lilting voice (not to mention her long blond hair and generous curves) that he wrote a solo especially for her. After the battle was over, they became inseparable. Before long, they were living together.\n\nBeing very _modern,_ they never actually married, though she was crowned queen. She also brought some mismatched furniture to the relationship, including a large round table that came from her ancestral home. Yes, _that_ Round Table. Arthur composed most of his ballads at that table, though its real source of fame can be found in one ballad in particular, ''The Knights of the Round Table,'' a wholly fictional composition in which well-meaning aristocrats (hah!) pass laws and mete out justice while assembled around the Round Table. (It wasn't round to create the illusion of equality, a nonsensical bit of legend that lingers from the song, but rather because it was cut from a massive fallen tree in Wales and, in its natural state, fitted perfectly in some lesser king's watchtower along the border.)\n\nArthur and Guinevere bought an estate in the West Country which they called Camelot, and they began an ambitious series of renovations that they never did manage to finish. The rehearsal studio was the only portion of the whole place that was ever fully completed, and they spent most of their time there. Arthur was entering a new phase of his development as a troubadour, and he spent several years working on a complex and obtuse set of compositions collectively known as the ''Green Knight'' ballads. (You may recall the work? His nephew Gawain plays a major role in one of the pieces. He wrote the boy into the song as a sort of christening present. However, Arthur's half-sister Morgan, whose musical taste was always conservative, promptly feuded with him over this, and things were never again the same between them.)\n\nI, meanwhile, kept trying to interest Arthur in things like, oh, _ruling the nation_. But, well\u2014have you ever tried reasoning with a musician? Sometimes I simply tore my beard out in sheer frustration.\n\nIt was during the ''Green Knight'' years that Lancelot appeared in our midst. Or as he laughingly called it, our ''mist.'' He never did get used to British weather.\n\nHe'd been experimenting with percussion instruments that had made their way into the Aquitaine via the Iberian peninsula, and when he heard about the interesting work going on at Camelot, he packed up his drums and came to our little corner of the world to join Arthur's band. Arthur was immensely taken with the young man, hugely enthused about his musical input, and filled with admiration for his improvisational abilities.\n\n''I'm telling you, Merlin,'' Arthur said to me one day, ''this French fellow is the finest drummer in all the land!'' This undisguised favoritism caused some resentment among the king's musicians, sour feelings which would eventually contribute to the disintegration of the band.\n\nArthur and Lancelot became close friends and started writing new ballads together. Since many of these songs required only male voices, Guinevere felt a little left out; consequently, she and Arthur started growing apart. Meanwhile, in his solo work, Arthur created a new character in his ballads, a noble knight named Lancelot who won every joust and could slay every beast. This new cycle of ballads turned out to be Arthur's most popular work yet, and his dear friend became a huge national star, second in popularity only to the king himself. Several of the chorus members started drifting away from the band, feeling their own talents were being neglected, their own contributions overlooked.\n\nHowever, the unparalleled success of Arthur's ''Lancelot'' ballads led to a fresh burst of raw energy in the King's creative drive, and he soon began work on his most ambitious song cycle ever, the ''Quest for the Holy Grail.'' It was such a huge hit that I agreed with Arthur's decision to go on tour for the first time since his coronation.\n\n''My fans are clamoring to see me, Merlin,'' he said. ''I can't disappoint them.''\n\n''I'll run the country until you get back,'' I offered. Why not? I'd already been running it for _years_. Though no one gives me credit for that, thanks to all those spin-off ballads about what a dedicated monarch Arthur was.\n\nWell, the ''Holy Grail'' tour was such a success that he extended it, then extended it again, and then _again_. Lancelot, however, had lost some popularity after a much-publicized breakup with the Lady of Astolat, whom he had treated rather shabbily. There was even some booing in the crowd when the band played in villages near Astolat. So when Lance came down with food poisoning in Scotland, Arthur saw that the long tour was taking a heavy toll on his favorite musician. He decided to send him home to Camelot to recuperate.\n\nCall me prophetic, but I thought that any supposed ''genius of the human heart'' (as one critic described our king, after seeing his standing-room-only performance in Londinium) should have been able to foretell what would happen when he sent a drummer to stay with his disenchanted wife while he remained on tour.\n\nArthur always suspected that Morgan started the gossip, never having forgiven him for whatever unspeakablethings she thought the ''Green Knight'' cycle implied about Gawain's soul. Other people said the source was Mordred, her youngest son. I've always suspected a third party, possibly the spiteful Lady of the Lake, since I cannot imagine that either Morgan or Mordred, difficult personalities though they were, would have started that (wholly untrue) rumor about incest, which somehow became inextricably linked with the whole tale. And that's all I'm going to say on that distasteful subject.\n\nLance himself said the rumors were started by jealous members of the band, musicians who thought his own popularity had prevented them from gaining the limelight. And maybe that was so.\n\nIn any case, it wasn't long before Arthur, still on the road and playing to sold-out houses, heard that Guinevere was involved in an illicit liaison with Lancelot. Fame and fortune and coronation cannot change the human heart, and Arthur was crushed by the news. He even canceled a portion of his tour and retired to Sir Galahad's estate, where he alternated between drinking himself senseless and writing angry, accusatory letters to Guinevere.\n\nAnd although I don't believe that Mordred started the rumors, there is no denying that he quickly took advantage of the tale. He'd wanted to join Arthur's ''Holy Grail'' tour, but the king had said that he wasn't ready, that he needed more rehearsal. An impatient and rebellious lad, Mordred now formed his own band, Lot's Sons, and went on the road. They had a new kind of sound, harsh, aggressive, antiestablishment, and atonal. And their first big hit was a direct challenge to the monarchy: ''The Queen Is Having It Off with the King's Favorite Drummer.'' This discordant and treasonous music appealed to a new generation, to young people who suddenly started saying that Arthur's best work was already behind him, that the ''Holy Grail'' cycle, until now the king's biggest hit ever, was just rehashed old stuff from his early years, dressed up with a little foreign percussion.\n\nRealizing that his career\u2014nay, his entire musical legacy!\u2014hung in the balance, Arthur pulled himself together enough to take his dispirited band back on the road to finish their tour. But the tide was turning in favor of Lot's Sons. Arthur Pen's Dragons were now playing to half-empty houses, and the new reviews were tepid.\n\nMeanwhile, the Saxons, having heard of the crisis in the British monarchy, were again marching against us. (Those damn Saxons! Don't they _ever_ quit?) I tried to rally the king to write another battle ballad, but he was in creative freefall. With no feasible alternative, I went to Mordred, the uncrowned king of music in our country now, and begged him to write a song to defeat the Saxons.\n\n''Don't be silly,'' Mordred said. '' _Singing_ the Saxons into submission is the old way. We need a fresh approach. I say we raise an army and kill them all.''\n\nRecalling our previous battles against the Saxons, I said, ''That may not be such a good idea. Arthur is no war leader or military general, but he did invent an effective strat\u2014''\n\n''He's also no king or ruler,'' Mordred said contemptuously. ''I mean, after twenty years of the king devoting himself entirely to his band, look where we are today! The king is on tour with some rehashed old material\u2014''\n\n''Actually, the 'Grail' cycle is really rather\u2014''\n\n''\u2014while the Saxons march on our country! It's a disgrace! And has the king even, oh, passed a _law_ in twenty years of rule? Has he ever bothered to show up at court since his coronation? Does anyone even remember where the Treasury _is_ anymore?''\n\nThere was no denying that Mordred had a point. Arthur was not precisely the king I had foretold. He was more like the court minstrel who should have sung _about_ the king.\n\n''Oh, dear,'' I said, realizing that perhaps I had not managed destiny quite as well as I might have done. ''I wonder if it could have been some _other_ sword that some _other_ lad was supposed to pull out of a stone?'' I mused.\n\n''What are you babbling about?'' Mordred said.\n\n''Never mind. It's the Saxons we need to focus on right now.''\n\n''I'll go raise an army and kill them all.''\n\n''All right,'' I said, giving in. ''Let's try it your way.''\n\n''And when I come back, I insist on being crowned king.''\n\n''Er, um . . .''\n\n''Face it, Merlin. My 'King Arthur Is a Cuckold' cycle is already more popular than anything Arthur's ever written. My time has come.''\n\n''We'll talk about it after the Saxons are all dead,'' I said.\n\n''You bet we will.''\n\nUnfortunately, Mordred was not much more of a war leader than his uncle was. He _was_ every bit as influential a musician, which turned out to be a huge problem. Mordred's music was mean-spirited and divisive. It didn't bother the Saxons at all, but it _was_ successfullydividing the Britons and speeding up the collapse of our society. After Arthur inexplicably disappeared from the touring circuit, Mordred's ''This Monarchy Sucks'' became the most popular ballad in Britain, followed immediately by his darkest composition yet, ''Your Parents Never Really Loved You'' (in which the repeated refrain was, ''Get over it, get over it, get oh-oh-oh-oh-OH-ver it!''). Clearly we were entering a very Dark Age in Britain.\n\nThen one day, as we were packing up the court and preparing to flee from the Saxon invaders, the king showed up.\n\n''Where have you been?'' I demanded.\n\n''Rehearsing the new battle ballad!'' Arthur said.\n\n''Oh, thank God! You know Mordred's losing the war?''\n\nArthur sighed and shook his head. ''I keep telling him he's not ready for public performance yet. But does he listen? _Teenagers._ Was _I_ ever that irritating, Merlin?''\n\n''Never mind that. Can you win this war without Lancelot and Guinevere?''\n\n''You don't need to worry about that. We're all back together again, and the band is better than ever!''\n\n''Really?'' This was a surprise.\n\n''I went home to Camelot and worked on my relationships.''\n\n''While the country was being _invaded_?'' I demanded. ''You're the king, damn it!''\n\n''Hey, I'm no good to the country if I've got writer's block,'' Arthur said. ''I had to get my personal life in order before I could focus on the Saxon invasion.''\n\n_Musicians._\n\n''Well, now that you're focused,'' I said, ''please, go out onto the battlefield and start singing! Otherwise, Britain is doomed!''\n\nWell, you now how it ended. Or, that is to say, you probably know as much as I do.\n\nSubsequent reports from the front were very confused, since we were busy being conquered at the time. But it seems that I made a fatal error when I put two rival bands on the battlefield at the same time. When Arthur started singing, Mordred couldn't stand the challenge to his artistry. He stopped fighting and launched his new ballad right then and there. Arthur lost his focus and, instead of singing the Saxons into submission, he concentrated on outsinging Mordred. This battle of the bands fully occupied all the Britons, who scarcely seemed to notice that the Saxons took the opportunity to slay or enslave everyone, burn everything in sight, overrun the country, and make us all start speaking Saxon.\n\nOf course, by the time I was learning to conjugate Saxon verbs so that I could negotiate our unconditional surrender, Mordred was dead, Lancelot had been shredded by the critics for his subpar performance, Guinevere had fled to a nunnery, and Arthur, mortally wounded, had been carried to Avalon. The Golden Age was over, our court lay in ruins, Camelot was abandoned, and the fickle public had already embraced a new musical form called the polka that the Saxons had brought with them.\n\nOnce the surrender was completed, I went to Avalon to visit Arthur on his deathbed. Despite his pain and delirium, he'd been working on a new composition.\n\n''I don't think this polka trend can last for long,'' he said to me as he gasped for his final breaths.\n\n''In case you were wondering,'' I said, ''we're ruled by Saxons now.''\n\n''I think there'll eventually be a revival of a truly Celtic sound,'' Arthur rasped. ''And I swear to you, I'm going to make a comeback. I'll reunite the band, and . . .'' The thought ended in his death rattle.\n\n''Prophecy or no,'' I said, ''I should _never_ have tried to work with a musician.''\nJACK\n\n_Dave Freer_\n\n_Now you see me ..._\n\n_Now you don't._\n\n_But I'm out there, oh, yes. You all just keep looking for me, my little darlings. Peer out into the dark. There are always things that skitter in the night. Twigs that crack in the silence. A hint of movement where all should be grave-still. Ha. As if a grave is ever really still. I've spent enough time in one to know that. You haven't. Yet._\n\n_A muffled shriek at midnight. It could be an owl._\n\n_Or it could be me._\n\n_Tch. Keep looking out, silly ones. Stare hard into the shadows under the wind-gnarled trees. It's very black over there._\n\n_Now you see me . . ._\n\n_Now you don't._\n\n_Which is all really very odd because I'm in here, in the circle of firelight, with you. I DO so like it when you look for me out there. Stopping myself from giggling is the hardest part. And it is nice and warm here by your fire. The things people will believe. As if that_ _would keep me away. Your cold iron could kill me, but that is a chance I must take._\n\n_Gray dawn is a fair way off, still, and before then my mischief and damage must be done. I'll have to slip off and to go to earth then, before the sunlight comes. It is such a lovely foxy way of putting it, ''to go to earth.'' Appropriate, too, for me, even though I can't say that it is too accurate. But it has a better ring to it than ''to go to the mound.'' A pity, that. My kind are rather obsessed with accuracy._\n\n_Magic requires it._\n\nShe huddled into her hooded cloak and leaned against him for security. The child was shivering slightly. Well, they all were, probably, and not from the cold. Terrible tales were told of Gnita Heath. Of the dragon, of the ring of old misshapen rocks which were strangely bare of the lichen that grew on the other stones out here. Of the doom that overtook those who wandered too far onto it.\n\nOr, in other words, about Jack.\n\n''It'll be all right. We're ready for anything. We'll see him coming,'' said Hrolf, trying to keep even the hint of a quaver out of his voice. ''And there are lots of us.''\n\nShe smiled at him. There was a quick flash of white teeth in the shadows of the hood. ''You're so brave.''\n\nIn his heart of hearts, Hrolf Ragnarsson did not feel it. He had come here to die, seeking the death that had been foretold when he was barely three months old. The wise women, the chanter of _galdr,_ and mistress of _seid_ had spoken with his _fylgja\u2014_ his fetch. He would die on Gnita Heath. A part of him wanted that death but not yet, not now. Not at the hands of a night-monster. Not overtaking the others with him. ''We warriors are not afraid of anything,'' he said with all the stoutness he could muster. The poor mite was so frail. What did the king think he was doing, sending a waif of a girl out here?\n\n''Not even a ghost that drinks blood and sends men mad?'' she asked.\n\nHer teeth were very even and very white.\n\nDid she have to mention that story? That was almost worse than Jack.\n\n_I suppose it is rather nasty, enjoying watching him shake. But some things need to be kept from humans. Some things are holy, and fragile. Anyway, I like doing my job._\n\n_Jack . . ._\n\n_Jack o' lantern . . ._\n\n_Jack o' shadows . . ._\n\n_Jack o' bedlam._\n\n_The redcap of the heath. It's quite a job. Quite a responsibility._\n\n_I do wonder why the humans think I'm male. My kind never are. There are some tasks we females do better than male-kind. The princes of Faerie find that an affront too. But thrones and powers mean little to_ our charge. _And she is our sacred trust._\n\nThe girl-child snuggled into him. Looked up at him with big green hazel-flecked eyes. ''We could slip off somewhere together,'' she said quietly. ''Just you and I. There's a little dell just beyond the firelight. They'll never know we're gone.''\n\nHrolf blinked. ''Are you mad? We're keeping watch for Jack.''\n\n''You said that there were lots of us keeping watch. And you're surely not afraid?'' she said archly, rubbing her cheek against his shoulder. ''Not a great warrior like you. You could protect me.'' She smiled ever so . . . admiringly. Or was that a hint of mockery? A hint that Hrolf Ragnarsson was maybe not man enough, and should prove it?\n\n''You're a child,'' said Hrolf roughly. ''You should still be thinking of children's games, playing with dolls, not slipping off into the darkness with men.''\n\nIt was her turn to blink those big eyes of hers. They really were too big for that elfin face. ''I'm not that young.''\n\nHe could see the soft swelling of her breasts now. He hadn't noticed them before. Perhaps she'd pushed back her cloak or something. It was a very fine cloak, he noticed. What was this girl-child doing here? ''You're younger than my daughter would be if she were still alive. Go and lie down by the fire and behave yourself,'' he said, turning away. Looking out into the darkness again. He wouldn't show this little temptress the tear starting down his cheek, brought by memories that had been best left undisturbed. Little Bryn was dead, and Helga, his precious Helga, with her. She'd died trying to protect their only chick.\n\nSo, now he was out here on Gnita Heath, seeking out his fate, with this bunch of hirelings and gutter-sweepings that King Heimir had sent after the rest of the treasure. There wasn't a decent shield-brother among them. It had seemed fitting to go and meet that fate now that there was nothing left to live for. Now he found that, despite the sorrow, he was not ready to die.\n\n_I am really not accustomed to this! Do my powers fade? Dolls? Dolls! I was old when he was playing with a wooden sword. Still . . . there is something about him that made me choose him first. If he had been taken, the rest of them would have broken and fled. I'll try another. The vain-looking blond one. This Hrolf is only one man. But then Sigurd was only one man. And between him and Reginn, they killed our guardian. Fafnir the dragon failed. Reginn, who would have done well to guard the hoard, fell to Sigurd. I cannot fail. Only Sigurd's greed and stupidity saved us and our duty last time. This Hrolf may not be so obliging._\n\n_Perhaps blond Thorvard will do._\n\nHrolf looked out into the darkness. Right now it was a shield hiding his face from the others. His eyes were so blurred he couldn't have seen J\u00f6rmanrekk's horde, let alone the redcap. There'd been nothing but night-noises, really. And you couldn't see much anyway. The night-mist had crept in and was shrouding everything.\n\n''Thorvard is missing,'' said Vikar, from behind him. ''Have you seen him?''\n\nHrolf rubbed his eyes hastily. ''He was here just a little while ago. Maybe he went to make water.''\n\n''Stupid fool,'' said Vikar irritably. ''If I find he's just gone off to sleep, I'll have his guts for bra . . .''\n\nA terrible quavering scream came out of the darkness. Far off in the darkness. Those few who had managed to get a little sleep were all awake now. Swords in hand, eyes peering into the dark, uneasy words on lips. More kindling on the fire made it flare a little. Not that it made much difference in the misty dark. Hrolf saw that the girl was among them. She looked suitably scared now. Too scared to try her charms. Just as well. That could have been her, and him, if she'd not been so young and reminded him so much of Bryn.\n\nShe looked at Vikar and said, in a quiet voice that somehow cut through the fearful hubbub. ''Why is there blood on your collar?''\n\nTo his horror Hrolf realized there was also blood on Erp's cheek, and Bikki's neck.\n\n''The ghost who drinks the blood of men and sends them mad,'' she said. ''Which of you is as big as an ox, and has a baleful eye? I saw such a shape stoop over those three.''\n\n_The hardest part is always not to snigger. All that panic for a nick with a bronze knife, no worse than a man might give himself shaving. Well, they'll not sleep anymore tonight. And in the morning they'll find Thorvard's blood-stained clothes. And no trace of him._\n\n_Well, no trace, unless, like that fool a few years ago, he gets lost. Instead of running away that one had somehow blundered in a full circle in the mist. He'd come running into camp, screaming, as naked as the day his mother birthed him, with no more than the few little cuts it had taken to provide me with a bit of blood for my work. It was a good thing that his companions had thought him a_ fylga _\u2014a fetch\u2014and killed him. Then, of course, when they'd found out that they'd killed their own, they'd blamed Jack._\n\n_That suited me perfectly._\n\n_And now the others would be eyeing the three of them out of the corners of their eyes, waiting for the_ _moment they went berserk-mad from the bite of the blood-drinking_ draugr _. Sooner or later, the victims would snap and flee. If they managed to get away from their own companions they'd be lucky. Or, as I'd seen often enough before, one of them would make some innocent but untoward movement. A hand trailed on a sword hilt, a hand too close to a dagger, or something like that. And then it would be bloody mayhem. Sometimes they'd bind them and leave them here, usually promising to return. Ha. It was always left to me to deal with them in the end. Not that they appreciated it much. Some really were mad by then._\n\nHrolf saw how the frightened eyes of his companions darted around, staring at the _draugr-_ victims and then to the darkness of the heath. And how they paused, just for a moment, as they looked at him.\n\nHe was not as big as an ox.\n\nBut he was the largest man there.\n\nDawn was still a fair way off, and it didn't look as if it would bring much cheer.\n\nHe still longed for it.\n\nThey huddled in their cloaks, backs to the fire, swords out, waiting. Swords out except for Vikar, Erp, and Bikki. ''We can't take a chance on you three,'' Erik Big-Belly had declared. ''Give us your swords. You tend the fire.''\n\nFor a tense moment it had looked as if the three would choose not to. But, reluctantly, Vikar had agreed. He was the nearest thing to a leader that this group had had. So now they tended the hungry flames while the rest of them stared into the darkness. With occasional nervous glances at the fire tenders, who kept that fire burning bright. Hrolf knew that they had precious few more bundles of firewood on the pack-ponies. Fuel was scarce out here, on the high and bleak heath. What trees there were, were little more than bushes, twisted by the almost relentless wind. Mind you, when they'd had a lull in the wind, the midges nearly drove them all mad, never mind the blood-drinking _draugr_.\n\nIt was a long wait, made longer by the fear, until Hrolf realized that the mists were paling. Dawn had crept up on them like a thief\u2014or a ghost\u2014in the night.\n\n''What are we going to do with these three?'' asked Red Gunnar, named thus for the color of his hair rather than his sword, pointing at the victims of the blood drinker. In the pale light of the new morning the three men looked wan and fearful. The wounds on their necks were still there, although they'd all tried to rub the dried gore away. Vikar's once white shirt, that he'd boasted of taking from a chieftain's hall on a Viking raid to Ireland, was the worst testimony to the attack. The rusty brown stain was an abject reminder of the terrors of the darkness. Hrolf wondered how the girl had weathered it. He couldn't see her right now.\n\n''Kill them before they kill us,'' said one of the men. The Southerner. Hrolf hadn't got his name and didn't care if he never learned it.\n\n''We can't do that,'' protested Big-Belly. ''Bikki's a friend of mine.''\n\n''A better reason to kill him, if you ask me,'' muttered one of the others, but not too loudly. Big-Belly was a kin-slayer who'd sought refuge and employment at King Heimir's halls. There were not a few of them here that Heimir might just be grateful to rid his kingdom of.\n\n''Bind them and leave them here,'' said one of the others. ''We can take them with us on our way back.''\n\nHrolf knew that would be as much of a death sentence as they'd been given by the Southerner. Worse, it was a cowardly way of imposing that death sentence. ''No.'' he said grimly, stilling the nodding heads, turning them to look at him. ''Sigurd was six days traveling across Gnita Heath from Fafnir's lair before he came to Heimir's halls. We cannot leave them here, on the heath, bound and unarmed, at the mercy of Jack. What kind of men are you?''\n\nThe doubtful looks told Hrolf that whatever else they were, they were men who did not want three men infected with _draugr_ madness to be in their midst. Looking at the three, all a little wild-eyed and tense, ready to fight, Hrolf could understand their point.\n\n''Give us our swords and our horses and let us go,'' said Vikar. ''If we ride west we should be a long way from this accursed spot by nightfall, and into King Gjuki's lands before morning. That's where I should have run to when I first heard about being sent here.''\n\nHrolf was shocked. But he had been born and bred under King Heimir's rule; Vikar had not. But surely this hire-sword had some loyalty? It would appear that it was not so.\n\n''And have you turn back when the madness takes you, and attack us? I think not,'' said Red, standing well back from what would be knife-work at the least.\n\n''Bind their hands and put them on their horses,'' said Hrolf. ''We'll take them along.''\n\nNo one had any better ideas, so they agreed to that. But they laughed when he suggested getting the woman to watch them.\n\n''No bites on his neck are there?'' asked Big-Belly. ''He's mad enough. Let's have some ring-bread and some ale and then ride on. Not that I don't know if Vikar's not right about the direction.''\n\nIt was then that they discovered that they had ale . . . but no food. One pack was missing. Stolen in the night.\n\n_In the cool darkness, here, where in the daylight hours I take my rest, I pondered that big man. Hrolf. I'd set doubt among them. It could be that by nightfall he'd have been attacked and pulled down by the man-pack he ran with. He would have to go. He saw too much, even through the glamour. Perhaps the old blood ran in his veins. Such men were dangerous._\n\n_Anyway, I would see about it at nightfall. I was tired after my labors. We can go far and fast compared to humans, walking the low roads that they cannot see. But I had traveled far to get to the resting place I shared. My host didn't mind my using his tomb. He had been the sort of man I'd not minded sharing with. We'd raised a good mound for him, but there were more of us around here then. He'd placed honor before ring-gold, which is why we laid him here in that honor, with his sword, his dagger, and a coat of mail. I'm not fond of cold iron, but I don't let it keep me from my sleep. It's not as if he's going to use it anymore. A pity, actually. We could use such a defender again._\n\n''We need a defensible camp,'' said Hrolf, pointing at the stones. There weren't many other features out here. Just grass, heather, and a little gorse. Little boggy gullies. Occasionally a few scrubby trees in hollows, where they'd found some shelter from this wind. Unfortunately the midges also found shelter there.\n\nBig-Belly shook his head. ''That did us no good. We should just ride on. Maybe we can outrun Jack and the blood-drinker.''\n\nHrolf shook his head. ''That's the ale talking, not common sense. The horses need to rest.'' Ale, tiredness, and lack of food was making for a fractious party.\n\n''True,'' said Big-Belly. He turned in the saddle to shout to the others. ''Ho. Where is Red? And . . . and Hogni and little Erik?''\n\nTurning, Hrolf realized that he was right. The three tail enders in the column were missing. ''I'd guess at half way to Huldaland by now,'' said Vikar sourly. ''Where we should be. Or among the Jordanes.''\n\n''Shut up. You can stay sitting in the saddle for that,'' said Big-Belly, dismounting at the stones. Climbing off his horse, Hrolf saw that they were worked stone. It was a ruin of some kind. Perhaps a shepherd's croft. Not that anyone had kept sheep here for many centuries. Fafnir had seen to that. Dragons liked sheep. And shepherds, too, for that matter. There was a twisted lime tree here in the shelter of the ruin. It made a good place to tie the horses. And to off-load some of the ale. They said that beer was liquid bread, but Hrolf would rather have had the solid kind. Anyway, this was well-bounced ale, and none too good to start with. He had a feeling that more than one of them would go down with the flux. They'd seen\u2014and brought down\u2014only one hare all day, and it wasn't going to go far between the remaining fifteen men. That Hogni was a better archer than Hrolf would have guessed.\n\n''Hoy! Stop them!'' yelled Big-Belly, looking up, and wetting his boots. Vikar, Bikki, and Erp, left sitting in their saddles, had taken the chance to flee, tied hands or no. You couldn't blame them. Things looked black for them. Actually, things looked black for all of them.\n\nHogni had taken his bow with him when he dismounted. He drew and loosed while the other men were still untying horses.\n\nHe was as good at shooting men as he was hares. The last of the three peeled out of the saddle. No living man bounced like that.\n\n''That was Bikki! You've killed Bikki,'' said Big-Belly, grabbing Hogni's bow.\n\n''You told me to stop him,'' said the jowl-faced Hogni sullenly.\n\n''Not by shooting him, you _nithing_. Why didn't you shoot the horse? But you've no brains above your balls.'' Big-Belly smashed the bow against the rocks.\n\nWith a choked snarl Hogni pulled his knife from his belt and stabbed upward into the largest part of the man who had just broken his bow. Big-Belly seemed to have trouble believing it. He grabbed Hogni by the throat, as the man stabbed again and again. He had big, meaty powerful hands, and by the time the two were pulled apart, it was too late for Hogni. Too late also for Big-Belly. His face was ashen, and he slumped against the gray stones, coughing blood. He looked up at them, but it was plain by what he said that all that he saw was his own murdered kin. The others backed away. Only Hrolf stood close, holding his sword-hilt, wondering if he should give the man battle-mercy. But death spared him that decision.\n\nThe bodies were laid around the far side of the scrap of wall, out of sight, if not out of mind. They were down to ten now, and three more bundles of firewood, with a long night ahead. She poured foaming ale for all of them. There was more ale than fire-roasted hare.\n\nHrolf blinked at the slight woman handing out ale horns. Where had she been earlier? It was on the tip of his tongue to ask when he had to knock two heads together to stop another fight. Fighting like rats, when they faced death. He shook his head. It was almost as if they were being torn apart from within, let alone by the redcap of Gnita Heath. He shook his head again. She really was very like his Bryn, or Helga, when she was younger. And, like the taste of this ale, there was something wrong with that. He poured the ale out, quietly.\n\n_Well, my little ones, you have got farther into Jack's lands than most people do. Most blunder around in circles. There are less of you than there used to be, though. Still peering faithfully into the dark, I see. And choosing such a good place for me to work my glamours. So kind. Even though I am sure you don't mean to be, hee hee. Beside cursed ruins and a holy tree. The lime has always been good for my kind._\n\n_I must bind the dead that you have provided for me onto their horses. The horses will go home, and not along the winding routes you humans send them on. It does do such a lot for my reputation when the bodies come back from the heath. I am strong, far stronger_ _than most mortal men, but the task of lifting that one onto a horse will be hard. Poor beast. At least he won't be using spurs on you._\n\n_I think it must be tonight. You are too close. Yes, I will act tonight, in the small hours, when you are full of ale and fullest of fears. It is always odd that they never notice the ale I add to the supply. A triumph of human optimism, I suppose: the hope that the drink will not run out, against the certain knowledge that it will. With my help, hope wins, for once. And the buck-bean I put into that ale will send some off to squat in the dark. I do find it funny to take them with their breeks down around their ankles. Childish, I know. But in some ways we never really grow up._\n\nJack came just before moonset. Just after they had found out that Nidud and the Southerner had taken the choice between soiling themselves and just stepping out of the circle of firelight, and had made the wrong choice. The fire was low now. They had very little fuel left, and the green lime branches would not burn. Now there were eight of them left. The drink the girl kept plying them with might have given them some courage, but the tiredness seemed to cancel it. No one, no matter how mazed, dared sleep.\n\nJack was as bad as all the tales of the evening made him out to be. The moon shone on his red cap, dyed\u2014 and still dripping\u2014with the blood of men. He stood as tall as a mounted man. Sharp tusky teeth snaggled in his cruel smile. He was clad in a mail shirt cobbled from the flat-beaten breastplates of dead warriors, hung with the pelts of wolf and lynx. His arms were long and covered in a mat of black hair. In one hand he held an iron-studded club and in the other a cruel hook-bladed knife. He laughed as he strode forward out of the dark, and a rank foxy odor blew off him.\n\nHis laughter was as cold and sharp as the cracking of ice.\n\nHrolf raised his shield, and held his sword before him . . . It had come at last. And facing it was better than the waiting.\n\nAnd his companions screamed, and, as one, turned, and ran for the horses.\n\nFor a moment, Hrolf nearly joined them. Then he raised the sword point as Jack strode closer out of the dark. ''Come back you cowards!'' he yelled. ''Come and die like men, at least.''\n\nHis answer was the sound of iron-shod hooves. He wondered briefly if they'd even left his horse or the pack-ponies. Well, no matter, now. He took a step toward Jack.\n\nThe redcap halted. Lowered the club slightly. In a gravelly voice full of menace it asked, ''Is the gold worth dying for?''\n\nHrolf looked at the monster, its bulk that of four men. ''No,'' he said, truthfully. There was no one to lie to here anymore. ''I don't think I really came for the gold. But if I fight you, then the others and the girl-child may get away.''\n\nRedcap snorted. ''They're oath-breakers and hire-swords. Not worth your dying for either.''\n\n''True.'' Hrolf nodded. ''But there is the girl. She may be a slut, or even a _draugr_ of Gnita Heath. But she reminded me of my daughter. And _that_ is worth dying for. I was not there to defend my own child when the raiders came. I have looked for meaning and purpose since, which I have not found in war or in the service of the king. This may not be much of a purpose. But it is the first one I have found in all the years since Bryn and my Helga died. And I am fated to die here.''\n\nRedcap put the club on the ground. ''Run,'' it said in a quiet voice. ''I will not hurt them. And the woman is not what you think she is.''\n\nHrolf did not move. He knew that this was his chance. He should attack now, lop off the thing's ugly head before it could raise its treelike club again. He would be the hero of Gnita Heath. Renowned, like Sigurd, in poem and song. Wealthy and famous. Instead he too put his sword point to the ground. ''What is she, then?''\n\nThe redcap looked at him in silence. In the distance he could hear the rest of the party riding hard. Away. ''She is Jack,'' answered the Redcap, finally.\n\n''But you are Jack. The Redcap.''\n\n''We are many. The monster who dyes his cap in the blood of murdered victims. The _draugr_ that drinks blood and brings madness.'' The huge figure blurred and faded in the last of the moonlight and the girl-child stood there, eyes large and sad in her elfin face. ''But we are also only one. I have no need of your defense of me, Hrolf Ragnarsson. Go now. I will not harm you or even the others. I never do.''\n\nHrolf felt his legs go weak. He sat down. It was better than falling, even if it made her smile. At least the ground felt solid. ''You have bespelled me. You are the monster, Jack.''\n\nNow her smile turned to a snort of laughter. ''I wish I could. My skills are small. A minor glamour is all that is really in my power. And it works badly on one such as you, Hrolf. You see me much as I am, unless I exert every grain of my will on you. I can only do that for a short time. Not one of your party questioned what a young woman was doing in your midst. Or should I say none of them except you. None of them remembered me by daylight. Except you. Go now. I will not harm you or even the others. As I said, I never do.''\n\nHrolf shook his head. He began to see his duty here, as little as he liked it, with Jack in this form. ''You have killed many. All that dare to venture here.''\n\n''I have never killed any man. Have I killed any on this expedition?''\n\n''Thorvard.''\n\n''Did you see his body?''\n\nHonesty compelled Hrolf to admit that he had not. ''But then what happened to him?''\n\n''He fled from me onto the heath. After that I don't know. The cold or his own fear may have killed him. But not me. That would be against our compact and my nature. Besides, you are all shielded and armed with cold iron that would kill me. I only have my illusions and your fear to protect my sacred charge.''\n\n''But many expeditions have set out to seek the rest of the dragon's hoard ever since Sigurd came back with part of it. They have all disappeared. Bodies . . . and ponies have returned, with the riders savaged and killed.''\n\n''Some of them were killed by each other. I have always returned the bodies. As for the others, those who disappear, well there are a dozen little kingdoms all around here. The chances are against them all ending up in the same one. And humans being what they are, they're unlikely to tell a true tale when they get back. I am a precise Jack. There have been twenty-three expeditions to find the rest of Fafnir's gold. There must be ten 'sole survivors' of each of them out there, and many more who do not admit that they ever ventured onto Gnita Heath.''\n\n''Then why do you do it?'' asked Hrolf. Something about it all rang true. After all, what she had said about their own expedition and her glamour did seem correct.\n\n''It is my work,'' she answered. ''A part of my appointed task. I do enjoy it a lot, I admit. They're many. I am one and weak against the daylight and cold iron. It's a challenge.'' She smiled, her face foxy with mischief. ''It keeps me from being too bored. The watch is a long and lonely one.''\n\nWell. As a bored young guard he'd once tied his fellow warriors' boots together with thongs, and then woken them. ''So you keep those who search for Fafnir's hoard away by trickery and illusions?''\n\nShe pulled a face. ''Not all, no. But greed killed the one group that did find their way there. They fell to fighting among themselves. Their bones lie with those Fafnir devoured.''\n\nShe was a creature of night and darkness. But she looked like his daughter might have done, if she'd lived. ''So why do you guard the dragon's hoard, or what is left of it?''\n\nShe shook her head. ''Say rather that the dragon guarded things for us. A dragon sits on his hoard. He does not use it. And Sigurd took that which he could. He had to choose, and he chose gold and rings, and the gold-hilted sword Hr\u00f3tti, the gold-embossed helmet of terror, and the golden mail shirt. But he was too blinded by gold to see quite what we had given the Dragon to guard. There was a more dangerous thing than the helmet of terror, and a more precious thing than the cursed ring-gold. Now there is no more dragon. Only me.''\n\n''Why are you telling me this?''\n\nShe stood and looked intently at him with those hazel-flecked green eyes. ''Because, just as I remind you of one you once loved, you too remind me of someone. Someone I trusted. I had forgotten how much he meant to me. It has been many years.'' She took a deep breath. Gestured with a hand, and his horse, with all its tack in place came out of the darkness. ''Gnita Heath is bigger and less big than it seems, especially by night. Come. Ride and follow me and I will take you to the cave of Fafnir and the mound of Hedin Vidarsson.''\n\nWhatever magic she could, or could not do, she could get a restive horse tacked up. And Grani trusted her. That was enough testimony for Hrolf. He mounted and followed as she ran through landscapes that blurred and swirled around him as if dimly seen through the mist. Sometimes he heard the sounds of the chase in the distance, and the far-off blowing of horns. But he paid them no mind and followed her as she ran, light of foot, fleet as any fox.\n\nThe sky was paling when they came at last to a valley through which a stream laughed its way across the rocks and a black cave-mouth looked out from under the twin hills, just as Sigurd had described. In the valley stood a long barrow-mound, mightier even than that which had been raised to rest King Budli.\n\nShe led him past the bones of Fafnir, into the cave, past the unburied and still helm-clad skulls. She kindled a brand and Hrolf gazed on the fabled hoard. Sigurd had left what he could not carry. He would not have needed many more pack-ponies. There was very little left, and not much of it was gold. The dream of Fafnir's hoard was little more than that. Words spoken with the bragarful cup, and taken as truth, that had sent men to scour Gnita Heath.\n\nShe pointed. ''There are two things here. One must lie here until the Time. The end itself. We cannot unmake it, we cannot return it, and we cannot let it loose again without its rightful wielder. The other waits until after the Time. She waits for Baldr. Her father. She will not wake until he comes.''\n\nHrolf looked at the great plain steel sword which was called Head, and once the weapon of Frey. A sword which would make any king more mighty than the Huns themselves, more mighty than any king. The other\u2014a sleeping child in a simple cradle.\n\nHe nodded. ''There are things more precious than gold. I suppose there is a need for a day guard too.''\n\nShe nodded. ''That was Hedin's task, before Fafnir. Men who can see that truth are rarer than dragons, and more valued here, in the place where all times and all places meet.''\n\n_He's soft, really. Few of them die anymore. I question the wisdom that says that only one of my sex is a fitting guardian in case the child wakes. I can wet-nurse her, yes. But the dog-wolf watches over the sleeping pups as diligently, even if he has no milk to give them._\n\n_Fewer come searching these days, as the memories grow dim, and the legend of Jack o' the Heath grows. And I must admit that I like the company, even if he's not as given to nasty practical jokes as I am. Besides, with us all being female, we do have to find mates from outside. Perhaps, this time, there will be children. A new redcap\u2014we grow old too, eventually\u2014or, well,_ _there are seldom boy children. It does happen. I think Hrolf is of the line of one such. Not exactly the line of kings, as is sometimes claimed among men. They make poor kings. They lack the greed and the ambition, and have too clear a vision. But they make good men as a result._\n\n_Yes, he will die on Gnita Heath, as foretold. I will raise a mound for him as I and my sisters did for Hedin. The_ fylgjur _did not lie._\nABOUT THE AUTHORS\n\n**Daniel M. Hoyt** 's influence in the science fiction\/fantasy field is not yet seminal, but he hopes it will be one day. After Dan's first short story appeared in _Analog,_ he sold several more to other magazines and anthologies. Most recently, his work appeared in _Cosmic Cocktails_. Along the way, Dan also found time to be a small press publisher and editor and an anthology editor. Dan takes a break from his writing and editing with rocket science, math and piano, all of which he inflicts on his family\u2014and any unsuspecting guests in his Colorado Springs home\u2014with surprising regularity.\n\nCanadian author and editor **Julie E. Czerneda** has been a finalist for the John W. Campbell Award for Best New Writer and the Philip K. Dick Award for Most Distinguished Science Fiction Novel, as well as the winner of three Prix Aurora Awards, the Canadian version of SF's Hugo, for her novel _In the Company of Others,_ her short story ''Left Foot on a Blind Man,'' published in _Silicon Dreams,_ edited by Martin H. Greenberg and Larry Segriff, and as editor of the anthology _Space Inc._ Julie has published eleven science fiction novels with DAW Books, the most recent being _Reap the Wild Wind_ , first of her Stratification trilogy. She's edited _Polaris: A Celebration of Polar Science_ and, with Jana Paniccia, _Under Cover of Darkness_. Her next anthologies will be _Misspelled_ and, with Rob St. Martin, _Ages of Wonder_.\n\n**Mike Resnick** currently stands first on the all-time short fiction award list, according to _Locus_. He has won five Hugos and a Nebula, plus other major awards in the United States, France, Japan, Spain, Poland, and Croatia. He is the author of more than fifty novels, two hundred short stories, fourteen collections, and two screenplays, and is the editor of more than forty anthologies. His work has been translated into twenty-three languages.\n\n**Barry N. Malzberg** began publishing short stories in 1967, novels in 1970, and became known as a prolific writer of fiction that took a sardonic view of the meaning\u2014or lack thereof\u2014in individuals' lives and undertakings, to the point of occasionally being labeled anti-science fiction in his outlook. Notable novels include _Beyond Apollo_ (1972), winner of the first John W. Campbell Award. He lives with his wife, Joyce, in Teaneck, New Jersey.\n\n**Sarah A. Hoyt** is the author of the acclaimed fantasy trilogy consisting of _Ill Met By Moonlight, All Night Awake,_ and _Any Man so Daring_ that undertakes a magical reconstruction of Shakespeare's life. She has also published over three dozen short stories in magazines that include _Analog, Asimov's,_ and _Amazing_. Her shifter novel, _Draw One in the Dark,_ has recently been published as has _Death of a Musketeer,_ the first of her Musketeer Mysteries (written as Sarah D'Almeida). An upcoming trilogy takes place in the far-flung reaches of a magical British Empire. Sarah lives in Colorado with her husband, two teen males, and a pride of cats and is furiously at work on her next dozen books.\n\n**Alan Dean Foster** 's sometimes humorous, occasionally poignant, but always entertaining short fiction has appeared in all the major science fiction magazines as well as in original anthologies and several ''Best of the Year'' compendiums. His published oeuvre includes more than one hundred books. The Fosters reside in Prescott, Arizona, in a house built of brick salvaged from a turn-of-the-century miners' brothel, along with assorted dogs, cats, fish, several hundred houseplants, visiting javelina, porcupines, eagles, red-tailed hawks, skunks, coyotes, bobcats, and the ensorceled chair of the nefarious Dr. John Dee. He is presently at work on several new novels and media projects.\n\n**Darwin A. Garrison** lives in northeast Indiana with his wife and three children, a dog, and a cat (the squirrels don't count as they feed themselves and don't require lawn scooping). He began writing in his teens but gave it up as a bad bet after getting one story published in his college journal. Two decades later, after almost dying of boredom while treading water in the great cesspool that is corporate America, he again took up the word processor. Thanks to the existence of the Internet, this led to meeting many wonderful people and not a few outright scary ones. On the whole, however, it allowed him to network with some wonderful mentors and learn to write fiction that people could actually read without vomiting (occasionally and with adequate preventative measures such as prophylactic Dramamine).\n\n**Barbara Nickless** sold her first short story to _Pulphouse_ and has since appeared regularly in other magazines and anthologies. Barb holds a BA in English with additional degree work in physics. She is a classically trained pianist who hopes someday to master Grieg's Concerto in A Minor. She lives in Colorado Springs with her husband and two children.\n\nNebula Award-winner **Esther M. Friesner** is the author of thirty-one novels and over one hundred fifty short stories, in addition to being the editor of seven popular anthologies. Her works have been published in the United States, the United Kingdom, Japan, Germany, Russia, France, Poland, and Italy. She is also a published poet, a produced playwright, and once wrote an advice column, ''Ask Auntie Esther.'' Her articles on fiction writing have appeared in _Writer's Market_ and _Writer's Digest_ books. Besides winning two Nebula Awards in succession for Best Short Story (1995 and 1996), she was a Nebula finalist three times and a Hugo finalist once. She received the Skylark Award from NESFA and the award for Most Promising New Fantasy Writer of 1986 from _Romantic Times_. Her latest publications include _Tempting Fate_ ; a short story collection, _Death and the Librarian and other Stories,_ and _Turn the Other Chick_ , fifth in the popular ''Chicks in Chainmail'' series that she created and edits. She is currently working on two YA novels about young Helen of Troy, _Nobody's Princess_ and _Nobody's Prize,_ as well as continuing to write and publish short fiction. Educated at Vassar College, receiving a BA in both Spanish and drama, she went on to receive her MA and PhD in Spanish from Yale University, where she taught for a number of years. She is married, the mother of two, harbors cats, and lives in Connecticut.\n\n**Kristine Kathryn Rusch** is an award-winning fiction writer. Her novella _The Gallery of His Dreams_ won the Locus Award for Best Short Fiction. Her body of fiction work won her the John W. Campbell Award, in 1991. She has been nominated for several dozen fiction awards, including the Mystery Writers' Association Edgar Award for both short fiction and novel, and her short work has been reprinted in six _Year's Best_ collections. Before that, she and her husband, Dean Wesley Smith, started and ran Pulphouse Publishing, a science fiction and mystery press in Eugene, Oregon. She lives and works on the Oregon coast.\n\nWith a name like Robert Anson, it was fated that **Robert A. Hoyt** would eventually write fantastic fiction. He lives in Colorado, where he enjoys reading classic science fiction and designing strange machinery. He is of the firm opinion that everything and anything can be found in New York City.\n\n**Jay Lake** lives in Portland, Oregon, with his books and two inept cats, where he works on numerous writing and editing projects, including the World Fantasy Award-nominated _Polyphony_ anthology series. Current projects include _Rocket Science_ and _TEL: Stories_. His next novel, _Mainspring,_ will be released in 2007. Jay is the winner of the 2004 John W. Campbell Award for Best New Writer, and a multiple nominee for the Hugo and World Fantasy awards.\n\n**Paul Crilley** was born in Scotland in 1975 and moved to South Africa when he was eight years old. He was rather disappointed to discover that Africa was not at all like the Tarzan movies he watched on Sunday afternoons and that he would not, in fact, have elephants and lions strolling through his backyard. When he was eighteen he met Caroline, and they have been together ever since. They have a two-year-old daughter, Isabella-Rose, who is a stubborn little bundle of joy who they both love to bits. He and his family live in what was once a small village (but is now rapidly being invaded by rich people from Jo'burg) called Hillcrest. They have two dogs and seven cats. He is currently trying to finish the first book in his fantasy series, The Sundered Land Cycle.\n\nA member of an endangered species, a native Oregonian who lives in Oregon, **Irene Radford** and her husband make their home in Welches, Oregon, where deer, bear, coyote, hawks, owls, and woodpeckers feed regularly on their back deck. As a service brat, she lived in a number of cities throughout the country before returning to Oregon in time to graduate from Tigard High School. She earned a BA in history from Lewis and Clark College, where she met her husband. In her spare time, Irene enjoys lacemaking and is a long-time member of an international guild.\n\nA passionate reader, **Rebecca Lickiss** began telling stories at an early age. She finally decided to write them down for publication, since it was better than cleaning house again. Her husband and her children humor her; otherwise they're making their own dinner. Her husband also writes, because he doesn't want to clean house either. In addition to her short story publications, Rebecca's most recent novels are _Never After_ and _Remember Me_. Just to keep herself busy, Rebecca has gone back to school to get her master's degree.\n\n**Kate Paulk** takes interesting medication. This explains her compulsion to write science fiction and fantasy and also means you'll be seeing a lot more of her in the future. Her friends would fear for her sanity, but she claims not to have any. She's been published in _Crossroads_ and is hard at work on a novel. She lives in semi-urban Pennsylvania with her husband and two bossy lady cats. Whether this has any effect on her sanity is not known.\n\nA music lover and longtime fan of Arthuriana, **Laura Resnick** 's current fantasy novels include _Disappearing Nightly, Doppelgangster, The White Dragon,_ and _The Destroyer Goddess_. You can find her on the Web at www.LauraResnick.com.\n\n**Dave Freer** is a South African of half Scots, half Yorkshire and half Afrikaans descent, which explains a number of things: his logic, his mathematical skill, and his attitude. Dave lives in KwaZulu Natal, Mount West (which is a white painted rock three miles farther in from the middle of nowhere), a remote, high altitude extensive farming and timber area. It's all dirt roads and very few people. Around there ''log on'' is adding fuel to the fireplace. And ''Internet'' is what happens to fish that do not swim away from the net. The locals\u2014those that matter anyway\u2014mostly speak Zulu and regard computers with distrust or as very boring TV. Dave has fourteen novels written, collaborated on (with Eric Flint or Mercedes Lackey), or upcoming. He can be found hunched, red-eyed and gnomelike, staring at his computer screen while working on his next novel.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nAlso by Brandon Mull\n\nSeries:\n\nFablehaven\n\nThe Candy Shop War\n\nBeyonders\n\nFive Kingdoms\n\nDragonwatch\n\nBooks:\n\nThe Caretaker's Guide to Fablehaven\n\nFablehaven Book of Imagination\n\nSpirit Animals\n\nSmarter Than a Monster\n\u00a9 2019 Brandon Mull\n\nAll rights reserved. No part of this book may be reproduced in any form or by any means without permission in writing from the publisher, Shadow Mountain\u00ae, at permissions@shadowmountain.com. The views expressed herein are the responsibility of the author and do not necessarily represent the position of Shadow Mountain.\n\nAll characters in this book are fictitious, and any resemblance to actual persons, living or dead, is purely coincidental.\n\nVisit us at shadowmountain.com\n\nLibrary of Congress Cataloging-in-Publication Data\n\nNames: Mull, Brandon, 1974\u2013 author. | Dorman, Brandon, illustrator. | Mull, Brandon, 1974\u2013 Dragonwatch (Series) ; bk. 3.\n\nTitle: Master of the phantom isle \/ Brandon Mull.\n\nDescription: Salt Lake City : Shadow Mountain, [2019] | Series: Dragonwatch; book 3 | Summary: As the evil Celebrant conquers more and more dragon sanctuaries, Kendra must raise an army of friends and allies on her own to fight him because her brother Seth and Bracken are missing.\n\nIdentifiers: LCCN 2019024701 | ISBN 9781629726045 (hardback) | eISBN 978-1-62973-791-1 (eBook)\n\nSubjects: LCSH: Young adult fiction. | CYAC: Dragons\u2014Fiction. | Brothers and sisters\u2014Fiction. | Missing children\u2014Fiction. | Magic\u2014Fiction. | LCGFT: Fantasy fiction.\n\nClassification: LCC PZ7.M9112 Mas 2019 | DDC [Fic]\u2014dc23\n\nLC record available at https:\/\/lccn.loc.gov\/2019024701\n\nPrinted in the United States of America 6\/2019\n\nPublishers Printing, Salt Lake City, UT\n\n10 9 8 7 6 5 4 3 2 1\n\nCover illustration by Brandon Dorman\n\nBook design \u00a9 Shadow Mountain\n\nArt direction: Richard Erickson\n\nDesign: Sheryl Dickert Smith\n\nAuthor photo by Busath Photography\n\nFor Chase, forever my little pal\nContents\n\nUnder\n\nRestless\n\nHero\n\nInnocent\n\nJibarro\n\nDefenses\n\nMortals and Magic\n\nThreat\n\nLiberation\n\nFalling\n\nExecution\n\nQuestions\n\nGiant\n\nPrisoner\n\nOld Friends\n\nMission\n\nTree House\n\nLandfall\n\nCrypt\n\nTriclops\n\nCyclopses\n\nHeartstone\n\nShipwreck Key\n\nDesperation Beach\n\nNegotiations\n\nDry Depths\n\nMerfolk\n\nRemulon\n\nSage\n\nAlone\n\nAssignment\n\nWishing\n\nPhantom Isle\n\nEverbloom\n\nUnderking\n\nUnicorns\n\nVoyage\n\nVictories\n\nAcknowledgments\n\nReading Guide\n\nAbout the Author\n\nRonodin opened the door without knocking. \"The Underking wishes to see you.\"\n\nStartled, Seth looked up from the desk, where he had been making a coin spin by flicking it. Ronodin had assigned him to read a thick book about phantoms, which had sounded like an interesting topic until he encountered long-winded passages written in archaic language. Even skipping ahead a few times, he had failed to find an interesting part.\n\n\"Who is the Underking?\" Seth asked.\n\n\"Our host,\" Ronodin said. \"We're currently residing in a small section of his lair. You weren't reading.\"\n\n\"I tried,\" Seth said.\n\n\"For how long?\"\n\n\"Every minute felt like an hour,\" Seth said. \"You should be amazed that I'm still awake.\"\n\nRonodin scowled at the book. \"It may take some time to reach the interesting parts.\"\n\n\"Maybe you could summarize?\" Seth asked hopefully.\n\n\"Later,\" Ronodin said, folding his arms. A handsome young man dressed in black, Ronodin always seemed sure of himself. Seth hadn't known the guy very long. They had no history. In fact, Seth had no history with anyone. His memories went back less than a day, to a room in a castle with a girl named Kendra who claimed to be his sister. Ronodin had appeared shortly thereafter, as had some other people, including a magical dwarf. After a confusing confrontation, Seth had been teleported to another castle, where a big wooden puppet had grabbed him and brought him through a barrel to this underground location. He had been here for only a few hours.\n\n\"This matter of the Underking is a real problem,\" Ronodin continued. \"I had hoped you could avoid the meeting entirely.\"\n\n\"He won't like me?\" Seth asked.\n\nRonodin gave him a smirk. \"He'll more than like you. He'll crave you. He hungers for all living creatures, let alone someone with dark powers.\"\n\n\"He's king of the underground?\" Seth asked.\n\n\"King of the Under Realm,\" Ronodin said in a solemn tone. \"He rules over all of the undead.\"\n\n\"Like zombies?\" Seth asked. \"And phantoms?\" He glanced at the black book on the desk. \"Maybe I should have read more.\"\n\n\"If it should be dead but isn't, the Underking probably governs it,\" Ronodin said. \"Entering his presence is incredibly dangerous. I have done so only five times, and I've known him for centuries.\"\n\n\"Then why should I risk it?\" Seth asked.\n\n\"He asked for you,\" Ronodin said.\n\n\"What if I just slip out of here?\" Seth asked.\n\nRonodin sighed. \"For one, we couldn't sneak away. The Underking is enormously powerful here in his domain. His attention is already on you. In addition, we need access to his tunnels. The Under Realm connects to your world in many surprising places, simplifying travel.\"\n\nSeth stared at Ronodin. How was he supposed to respond? Where could he even start? Apparently he was the hostage of an underground king. And what about Ronodin? Was he his ally or his captor? Seth sighed. Maybe he deserved this predicament. There was no way to be sure. Whoever he had been in his forgotten past, he must have led an interesting life.\n\n\"Don't look at me that way,\" Ronodin said. \"Survival is uncomfortable sometimes. You've been through worse.\"\n\n\"I don't remember,\" Seth said.\n\n\"Sounds like a relief to me,\" Ronodin said. \"Start over with a clean slate, unburdened by past mistakes.\"\n\n\"I could be anyone,\" Seth said.\n\n\"Not anyone,\" Ronodin corrected. \"You are a shadow charmer. You wield arcane power. You can speak to wraiths and quench fire.\"\n\nSeth knew he had the powers Ronodin was describing because, in their short time together, Ronodin had given him opportunities to use them. Supposedly Ronodin could help his powers increase.\n\n\"And you're a unicorn?\" Seth asked.\n\n\"In human form,\" Ronodin said. \"You mustn't keep the Underking waiting. It could make things worse for you.\"\n\nSeth tried to calm himself. Without memories, it was hard to make sense of his circumstances. It felt unfair to be facing consequences he did not understand. How had he gotten into this mess? There didn't seem to be any way out. For now, his best chance was to learn what he could and try to stay alive.\n\n\"I have no choice,\" Seth said.\n\n\"In this matter, no,\" Ronodin said.\n\n\"I'm not a bad guy,\" Seth asserted.\n\n\"How do you know?\" Ronodin asked.\n\n\"I just know,\" Seth said.\n\n\"I'm not a bad guy either,\" Ronodin said. \"But we're both outsiders.\"\n\n\"Who live in the subterranean lair of the king of zombies?\" Seth asked.\n\n\"We're visitors here,\" Ronodin said. \"Follow me.\"\n\nRonodin walked out without looking back. Seth hesitated, then followed him, jogging to catch up. If this meeting needed to happen, he had better follow the unicorn's advice.\n\n\"You must not look at the Underking,\" Ronodin instructed. \"To gaze upon him is to join the undead.\"\n\n\"I'd become a zombie?\" Seth asked.\n\n\"Or worse,\" Ronodin said. \"You will walk into the room with your back to him. You will keep your eyes closed. You will never face him or open your eyes. Not for any reason. Understand?\"\n\n\"What if he sneaks up on me?\" Seth asked.\n\n\"We're in his domain,\" Ronodin said. \"If the Underking wants you dead, you will die. If he wants you undead, little could stop him. Your job is to cause no unnecessary harm to yourself. Remember, don't look at him.\"\n\n\"What about in a mirror?\" Seth asked.\n\n\"That might work with a gorgon,\" Ronodin said. \"Not the Underking\u2014eyes closed, back to him.\"\n\n\"You've never seen him?\" Seth asked.\n\n\"I would be undead if I had,\" Ronodin said.\n\n\"What if he isn't there?\" Seth asked. \"What if it's just a sound system?\"\n\n\"He could be a hamster for all I know,\" Ronodin said. \"Or he could be a monster the size of a mountain. I suspect he looks somewhat like his subjects\u2014he rules them because he is one of them.\"\n\n\"Why does he want to see me?\"\n\n\"You are a guest in his domain,\" Ronodin said. \"You are a shadow charmer in a time when that gift has become exceedingly rare. Beyond that, I cannot pretend to fathom his mind.\"\n\n\"Won't it seem rude if I walk in backwards?\" Seth asked.\n\n\"He knows what is required to communicate with him and remain mortal,\" Ronodin said.\n\nThey turned a corner and moved down another subterranean corridor. Seth tried to imagine what it would feel like to become a zombie or a wraith. He had recently met a few wraiths with Ronodin, and they seemed like miserable creatures, cold and empty.\n\n\"Have I talked to the Underking before?\" Seth asked.\n\n\"Never,\" Ronodin said. \"Few have.\"\n\n\"I wish I could remember who I am,\" Seth said.\n\n\"It might prove better this way,\" Ronodin said, \"at least for our present purposes. You'll be more of an enigma to him. Trust your instincts. You're not like other mortals. You know the basics about the magical world.\"\n\n\"I know magical creatures are real,\" Seth said. \"Seeing the wraiths was no surprise. I generally remember them. In theory, I mean. I just don't recall having any experiences with them. I don't feel anything about them. I have a bunch of information but no opinions about it. There are no associations to rely on. How can I have reached this age with no connections to anything?\"\n\n\"Seth, you lost your memories,\" Ronodin said. \"If that erased your opinions too, find the benefit. Our assumptions often blind us. Now you have a chance to experience life through new eyes.\"\n\n\"I guess,\" Seth said. \"What else should I know about the Underking?\"\n\n\"Honestly, the less you know, the better,\" Ronodin said. \"Be respectful. Be humble. Be meticulously honest. Don't make any foolish bargains. Agree to what you must. Don't look at him.\"\n\n\"That sounds awkward,\" Seth said. \"Talking without looking.\"\n\n\"The Underking would be the last thing you beheld with mortal eyes,\" Ronodin warned. \"Three of the great monarchs wear much of their power on the outside: the Dragon King, the Giant Queen, and the Demon King. The Fairy Queen and the Underking are more subtle, and all the more dangerous for it. If you want strategy, my advice is to be content. Desire nothing. Ask for nothing. The undead are full of insatiable cravings. More than with most, contentment looks like power to the Underking.\"\n\n\"Could he help me get my memory back?\" Seth asked.\n\nRonodin stopped walking. \"What did I just say? Desire nothing.\"\n\n\"Which makes me think about what I want,\" Seth said.\n\n\"Put that longing away,\" Ronodin said. \"Pack all of your desires away, lock them up, and throw out the key. Portray yourself as a young shadow charmer whom I have offered to teach. You are here to learn from me. You do not desire accelerated learning. You don't desire anything. You are the simplest, most content boy in the world.\"\n\n\"I'll try,\" Seth said.\n\n\"Once he knows what you want, he'll figure out how to tempt you,\" Ronodin said. \"Beguile you. Ensnare you. Destroy you.\"\n\n\"I do want to remember my identity,\" Seth said, \"whether I admit it to him or not.\"\n\n\"Try to change your thoughts about that,\" Ronodin said, continuing along the corridor. \"Your survival could depend on it.\"\n\n\"Can the Underking read my mind?\" Seth asked.\n\n\"Possibly,\" Ronodin said. \"But voicing a thought gives it added power. Makes it more deliberate. Deny all desires in his presence, and you may survive.\"\n\n\"Do you think I'll pull through?\" Seth asked.\n\n\"Much of that depends on you,\" Ronodin said. He stopped at a large black door ornamented with bleached skulls of varying size, few of them human. \"Ready?\"\n\n\"No,\" Seth said. \"Does that matter?\"\n\n\"Not really,\" Ronodin said, extending a hand to rest a finger on the forehead of a small, narrow skull with pointy teeth. Ronodin closed his eyes for a moment, and, as he removed his finger, the door vanished. Cold darkness flowed from the doorway, washing over Seth and quickly blackening the hallway.\n\nPale blue light flared at the end of a short, gnarled wand, held by Ronodin, the tip bright enough to drive back the inky darkness, but not very far. Though it lacked any tangibility that Seth could discern, this particular darkness seemed especially resistant to light. Ronodin handed the holly wand to Seth.\n\n\"Why take a light if I don't want to see him?\" Seth asked.\n\n\"Even without light,\" Ronodin said, \"you would still see the Underking if you gazed in his direction. But you will need some light to stay on the tiled path. This holly wand can drive back even empowered darkness. Step gently, avoid the pits, and do not stray off into the bones.\"\n\n\"And I'm supposed to walk backward,\" Seth said. \"Without looking up.\"\n\nRonodin shook his hand with formality. \"You're catching on. Off you go.\" Ronodin released his hand. \"Remember, if something goes wrong, if you need anything at all, you're absolutely on your own.\"\n\nSeth had no reply. It was possible Ronodin was trying to use humor to relax him, but Seth suspected he was simply apprenticed to a jerk. Ronodin gestured for him to proceed. Seth turned his back to the gaping doorway and passed through it in reverse with careful steps. Holding the wand low to better illuminate the ground, Seth kept his eyes downward to see where he was going without peering too far ahead. He had not taken many steps before darkness swallowed all view of Ronodin and the doorway.\n\nThe plain black tiles reflected little light. Seth backed along quietly. Jagged heaps of bones appeared off to one side of the floor, starkly white at the edge of the light from his wand. He reached a pit of impenetrable blackness, no bigger around than a hula hoop. No matter how close Seth brought the wand, no light brightened the pit.\n\nSeth felt a creeping suspicion that something behind him was watching, perhaps even sneaking up on him, but he couldn't look in case he might accidentally see the Underking. There seemed to be movement in the darkness beyond his light, unseen, unheard, but convincingly present. Seth felt unsettled, then scared, and though part of him knew the impulse was irrational, it took an effort not to run off screaming in a random direction.\n\nAs he inched along the tile floor, messy piles of bones appeared off to one side. Seth wound around several smallish pits, wondering how long he would be traveling. Was he crossing a normal-sized room, or might this go on for miles?\n\nWalking backward in a crouch was tedious. He felt entombed by the incredible quiet and stillness surrounding him. The shuffling of his feet seemed to desecrate an otherwise perfect silence. Seth resisted the mounting, suspenseful terror.\n\nOccasionally he reached an individual bone, which he avoided in case it was cursed or diseased. The air was so cold that Seth could see his breath pluming in the wandlight.\n\nDry words reached Seth with no warning, slithering into his ears while chilling his nerves. \"Who brings light into my darkness?\" inquired a voice that seemed ancient, all rasp and whisper.\n\nGoosebumps erupted all over Seth and his muscles tensed. The fear of anticipation was over\u2014doom had arrived. Seth had to resist the urge to look over his shoulder toward that desiccated voice. Whoever had spoken was awaiting a response.\n\nSeth stood up straighter, keeping his back to the speaker and his eyes closed. \"I hardly know anymore,\" he said. \"I'm told my name is Seth.\"\n\n\"You are warm and alive, young one,\" the voice observed. Each wheezy word carried the impression of skeletal hands reaching for him.\n\n\"I've been warmer,\" Seth said, trying to sound calm. \"Alive for now, I guess. Who are you?\"\n\n\"I am master here,\" the voice affirmed. \"Ruler of all who dwell down below.\"\n\n\"The Underking,\" Seth said.\n\n\"What brings one so warm to such a cold place? Why does one who bears a light seek the shadows?\"\n\n\"It wasn't my choice,\" Seth said. \"Ronodin brought me here.\"\n\n\"The living who venture into my realm must join the undead,\" the Underking said.\n\n\"I don't want that,\" Seth said.\n\n\"What is it you want?\" the Underking asked.\n\n\"Really?\" Seth asked. \"To leave.\"\n\nThe Underking laughed softly. \"The moment you entered my domain, you belonged to me. Have you no purpose here?\"\n\nSeth decided he had better fall back on Ronodin's instructions. \"I'm a shadow charmer. Ronodin is teaching me.\"\n\nSeth waited in silence. There came a faint scraping sound from the direction of the voice, as if the speaker were stirring.\n\n\"I could teach you so much more,\" the Underking rasped softly.\n\n\"I bet,\" Seth said.\n\n\"The unicorn knows a type of darkness,\" the Underking said. \"I dwell where the darkness deepens, Seth\u2014where it continues for time immeasurable. All light is brief, Seth, so frail and fleeting, a feeble flash against the steady night.\"\n\n\"I just want to train with Ronodin,\" Seth said, sticking to the plan. \"Can I stay here and do that?\"\n\n\"Ah,\" the Underking said. \"You speak the sacred words.\"\n\n\"I mean, stay temporarily, while I train,\" Seth said.\n\n\"You are alive,\" the Underking said. \"Living is about change. Temporary states. You wish to remain under my protection for a season, like the unicorn. Are you brave?\"\n\n\"I'm here,\" Seth said.\n\n\"Are you true?\" the Underking asked.\n\n\"I think so,\" Seth replied.\n\n\"There are limits to what the unicorn can make of you,\" the Underking said. \"What more do you want?\"\n\n\"Just to learn from Ronodin,\" Seth said, remembering the warning to keep his needs simple. \"I don't want anything else.\"\n\n\"Speak truly.\"\n\nSeth sighed. \"I'm trying to remember who I am. But I trust I'll figure it out.\"\n\n\"You are cursed,\" the Underking said. \"My realm welcomes wayfarers such as you. And mortal shadow charmers have grown so very scarce. Turn and look at me.\"\n\n\"I better not,\" Seth said.\n\nThe Underking gave a hoarse chuckle. \"Would some illumination help?\"\n\nSeth could feel new light through his closed lids. With his back to the Underking, was it safe to peek? By slitting his eyes, Seth glimpsed cold white radiance filling the room, revealing all without brightening anything too much. He squeezed his eyes tightly closed again just in case.\n\n\"If I look at you, I can never leave,\" Seth said.\n\n\"Gaze upon me and you will endure,\" the Underking whispered, the words penetrating deeper than if he had shouted them. \"You will outlast the stars.\"\n\n\"I just need a place to train with Ronodin,\" Seth said.\n\n\"What about my crown?\" the Underking asked, the words reaching Seth as if whispered from close to his ears, private communication meant for him alone. \"You could replace me, young one, and the splendors and powers of the night would be yours. You could be master over sightless reaches of virgin darkness undefiled by light, endless voids, lifeless and deathless, behind, beneath, and above the petty stages of the sunlit worlds.\"\n\n\"I'm just trying to keep out of the rain,\" Seth said. \"I don't need more than a place to stay for a time.\"\n\n\"I will grant you sanctuary here,\" the Underking rasped. \"All boons come at a price. I do you a favor by sparing the scant flicker of your life, and I do another by granting refuge. You will agree to two conditions in return, or your spark will be quenched here and now. You will not leave my domain until permission is granted by me, and you will assist Ronodin in freeing some of my captive subjects. Is that agreeable?\"\n\nSeth wasn't sure he had a choice, if he wanted to survive. \"Will you ever give me permission to leave?\"\n\n\"I will let you leave temporarily to perform errands for me,\" the Underking said. \"Eventually, if you so desire, I will devise a task for you to perform that will grant you freedom.\"\n\n\"How long will I have to wait before you give me the task?\" Seth asked.\n\n\"I shall devise it when I devise it,\" the Underking answered.\n\nSeth felt sure he would get no better offer than this. \"All right.\"\n\n\"Sealed,\" the Underking said.\n\nSeth jumped as he felt something cold close around his wrist. Peeking, he saw a ghostly manacle clamped just beneath his hand, the translucent chain fading into invisibility after three links. The manacle was weightless, and the initial cold sensation quickly faded. Seth tried to touch the manacle with his free hand, but his fingers passed right through it. When he shook the arm to which the manacle was affixed, it refused to come loose.\n\n\"What's on my arm?\" Seth asked.\n\n\"A reminder of our arrangement,\" the Underking said. \"Ronodin will now represent my will to you. Follow his instructions as you would mine.\"\n\nSeth knew some sort of trap was closing around him. But at least he was alive. Hopefully he could stay that way long enough to figure out what battle he was actually fighting. \"I understand.\"\n\n\"Away with you then,\" the Underking rasped, the slithery words diminishing. \"Take the light with you.\"\n\nDespite the sunlight outlining the edges of the heavy curtains in her bedroom at Blackwell Keep, Kendra was supposed to be asleep. Everyone agreed she needed rest. After spending all night fighting for her life at Stormguard Castle, she had succeeded in breaking the enchantment imposed by the dwarf Humbuggle and at keeping the Wizenstone from the dragons. For much of the time the battle had not seemed winnable. Victory had felt like a miracle. A major threat to the safety of the world had been averted.\n\nAnd her brother was gone.\n\nSeth had sacrificed his identity to open a magical door. In the room beyond the door, Kendra had used a magical staff to send the Wizenstone away. Had Celebrant and his fellow dragons claimed the stone, they would have become too powerful to stop. Against staggering odds, she and Seth had done their duty as caretakers and protected Wyrmroost.\n\nBut her brother was gone.\n\nAfter they had returned to Blackwell Keep, when the battle had seemed over, the animated limberjack Mendigo had snatched Seth and disappeared through a barrel. Originally, the barrel had been linked to another one at Fablehaven. The connected barrels had served as their portal to return home. Disturbingly, the barrel at Fablehaven had gone missing. The loss of that barrel meant her brother could be almost anywhere.\n\nKendra kicked off the covers. Her body felt weary, but her mind refused to rest. She had already shared all the clues she could think of with Grandma and Grandpa Sorenson. They had ordered her to go to bed, assuring her that they would keep pursuing all possible leads about Seth.\n\nKendra exited her room. Bleak stone walls kept the corridor cool and gloomy. Unsure of a destination, she walked to the door at the end of the hall. When she stepped out into the daylight, the heavy buildings and towers around her felt like ruins. Why not abandon this place? Should people sacrifice their lives to protect these dusty courtyards? Was it worth losing her brother?\n\nShuffling down a flight of stairs to the courtyard, Kendra wondered where she would go. Maybe to the farthest corner of the darkest dungeon. Or perhaps to the Blackwell\u2014a deep pit teeming with the undead might match her mood.\n\nBattle-ax resting on one shaggy shoulder, Brunwin the minotaur paced along the top of a nearby wall. Kendra averted her gaze\u2014it was hard to imagine getting any real understanding from him. Up ahead, Newel was crossing the courtyard. The sight of the satyr lightened her mood instantly. He loved Seth. Though Newel wasn't the best at taking things seriously, he might be able to appreciate what she had lost.\n\nShe quickened her pace and called, \"Newel.\"\n\nThe satyr glanced her way and stopped walking. \"Hello.\"\n\n\"My grandparents want me to sleep, but I need to talk,\" she said.\n\n\"I've never believed in artificial bedtimes,\" the satyr said. \"If you can't sleep, make the best of it\u2014eat, revel with friends, watch a show.\"\n\n\"Can we go to your room?\" Kendra asked.\n\n\"Proper concealment is an important aspect of taking control of your sleep,\" Newel said, waving her to follow him. \"Cozying up to the quiet side of a haystack can help you snooze through an otherwise exhausting afternoon.\"\n\n\"But I don't want to sleep,\" Kendra said.\n\n\"Nobody wants to get stuck in bed wide awake by someone else's decree,\" Newel said as they reached his door. He used a key to open it. \"I'll shelter you.\"\n\nKendra entered and Newel followed, closing the door behind them. She crossed to a couch and sat down.\n\n\"Hungry?\" Newel asked.\n\n\"I can't stop worrying about Seth,\" Kendra said, tears squeezing from her eyes.\n\n\"Me too, Kendra,\" Newel said with concern. \"He lost his memory?\"\n\n\"He looked at me like a stranger,\" Kendra said. \"Like he was meeting me for the first time.\"\n\n\"That's terrible.\"\n\n\"He was just . . . gone. He had been there a moment before. He was being so brave. And then he didn't know me. He's been taken someplace. We don't know where. What if Ronodin has him? Or some other evil person? What if Seth gets filled with lies? My brother is blank right now. Empty. How can he know what to believe?\"\n\nNewel crouched down in front of her. \"Seth became who he is once. Can't we trust him to do it again?\"\n\n\"I hope so,\" Kendra said. \"And I hope we can find a way to reverse his memory loss. But what if we can't? And what if starting over with lots of bad examples and wrong information makes him regrow all crooked?\" Kendra shied away from the next words\u2014even unuttered, they produced horror. But she couldn't resist. \"What if we've really lost him?\"\n\nNewel laid a hand on hers.\n\n\"What if we can't get him back?\" Kendra went on, unable to stop herself. \"What if by the time we find him, he doesn't want to come back?\" The thought was too terrible for tears. Kendra felt numb and desolate.\n\n\"We can't let that happen,\" Newel said. \"Do we have any leads on where he might be?\"\n\n\"Grandpa says the wizard Vernaz who helped re-create Mendigo has a relationship with the Sphinx. Mendigo was revived from splinters of his former self, and he has needed additional work a few times since then as problems surfaced. Grandpa contacted Agad, who is trying to see what clues he can uncover. But really, Seth could be anywhere. The Sphinx might be involved. Ronodin or Humbuggle could be part of it. Or maybe somebody else.\"\n\n\"Isn't the Sphinx on our side now?\" Newel asked.\n\n\"They made him an Eternal,\" Kendra said. \"Supposedly he has been playing nice since then. He would have to die for the demon prison to open, so he's probably not up to the same old mischief. But that doesn't mean he can't get involved with the dragons. Or help other private causes of his own. We know he has operated in secret before.\"\n\n\"The Sphinx has always been shifty,\" Newel said. \"Did I hear correctly that Celebrant is no longer a caretaker?\"\n\n\"The Somber Knight stripped him of that title after Celebrant declared war,\" Kendra said. \"With Seth gone, I'm the only caretaker left for Wyrmroost. We went from three to one in a couple of days.\"\n\n\"It will be hard for you to look for Seth without leaving Wyrmroost,\" Newel said.\n\nTears blurred her vision. Could she tell Newel everything she felt? That her brother was more important to her than Wyrmroost? That part of her would rather all the preserves around the world fell into disarray if it meant she could have Seth back? And yes, that would probably mean the end of the world, with magical creatures running wild across the globe, in which case she would lose all of her family, Seth included, so maybe it wasn't what she really wanted, but it almost felt true right now.\n\nShe fingered her caretaker's medallion. \"I wish this wasn't my job. I want to go find Bracken and Seth, not babysit a bunch of dragons who want to kill me.\"\n\n\"You're doing an important service,\" Newel said. \"You're protecting the people you love along with everyone else.\"\n\nKendra studied the goat-man and saw real concern in his eyes. \"I know you really care about Seth.\"\n\nHe smiled. \"Of course I do. Satyrs don't pretend. We do what we want when we want. No matter where you have to be, I'll not rest until Seth is safe and sound. But first I have to get back to Doren.\"\n\n\"Grandma mentioned that Doren and Tess are at Terrabelle,\" Kendra said.\n\n\"True,\" Newel said. \"Before the drama at Stormguard Castle, we went looking for Seth with Knox and Tess. Maybe not our best idea . . . but we made it safely to Terrabelle. Knox left us with Tess when he took off with Seth to find you. After some deliberating, Doren and I realized your grandparents would be worried, so I ran back to Blackwell Keep with an update. I arrived right before Midsummer Eve began. Had I reached the gates ten minutes later, you would have one less half-goat friend.\"\n\nKendra looked him up and down. \"I sometimes forget that you're half goat.\"\n\nNewel shrugged. \"We satyrs say it's the top half that counts.\" He leaned in and whispered conspiratorially, \"Don't tell the minotaurs.\"\n\n\"I'm glad you're going back to help Tess,\" Kendra said. \"We'll send extra help. Maybe Henrick. Seth may be out of reach for now, but we can take care of those who are still here.\"\n\n\"What about Knox?\" Newel said.\n\n\"We left him inside the Quiet Box at Stormguard Castle,\" Kendra said. \"Tanu and Lomo were with him, and the stingbulb of Patton was nearby, along with Calvin. But we need to make sure Knox gets out. Grandma and Grandpa Sorenson are on it.\"\n\n\"They talked to me about retrieving Tess,\" Newel said, cracking his knuckles. \"Shouldn't be hard. The roads are protected again. Celebrant has less authority after losing his caretaker status.\"\n\n\"Be careful not to underestimate the danger,\" Kendra said.\n\n\"Don't overestimate it either,\" Newel said, removing an apple from a bowl and taking a big bite. \"Takes all the fun out of the adventure.\"\n\n\"Fun?\" Kendra asked, incredulous.\n\n\"Sure,\" Newel said, wiping his lips on his forearm. \"We have to make the best of things.\"\n\n\"Since when has this been fun?\" Kendra asked. \"Scary, and challenging, and maybe necessary, but not fun.\"\n\nNewel scowled in confusion. \"Adventuring is often fun. And profitable and thrilling and satisfying. It's what I like about it. Seth too.\"\n\nKendra felt anger rising inside of her. \"Do you think Seth is having fun right now?\"\n\n\"I hope so,\" Newel said.\n\n\"Fun as a prisoner?\" Kendra asked. \"Fun without his identity? Fun becoming corrupted in ways that may never be undone?\"\n\nNewel raised his eyebrows. \"Maybe you do need some sleep.\"\n\nKendra leaped to her feet. \"Why did I think for one second that you might understand how it feels to lose somebody you love?\"\n\n\"I miss Seth too,\" Newel said, almost taking another bite of his apple, then lowering the fruit. \"What good does it do to get all worked up over it?\"\n\nKendra clenched both hands into fists. \"I'm not trying to do any good! I know I can't solve the problem right now! But I do have feelings! And none of this is fun!\"\n\nNewel reached back into the bowl and held out some fruit to Kendra. \"Orange?\"\n\n\"No, thank you!\" Kendra replied icily.\n\nNewel replaced the orange. \"Banana?\"\n\nKendra laughed a little, largely in disbelief. \"I'm not hungry.\" She grabbed the sides of her head and started pacing. Newel probably wasn't trying to be rude or antagonistic. Seth found adventuring fun, and so did she\u2014sometimes. But not right now. Not one bit. \"I better go,\" she said.\n\n\"You can stay if you want,\" Newel said. \"You don't have to eat.\"\n\n\"I shouldn't be around anyone right now,\" Kendra said, trying to speak calmly. \"I am tired and hurting inside.\"\n\nNewel gave a nod. \"We'll get your brother back. Just you wait.\"\n\nKendra looked at the satyr. She hoped he was right. But it didn't seem likely. This felt different from any problem they had ever faced. \"We'll try.\"\n\nShe walked to the door, and Newel opened it.\n\n\"Can I give you some advice you may not want?\" he asked.\n\n\"I guess,\" Kendra said, bracing herself.\n\n\"Find a haystack. You're exhausted.\"\n\nShe did feel stretched. But she doubted she could sleep. \"We'll see.\"\n\n\"Your choice,\" Newel said. \"Your grandparents weren't trying to punish you by sending you to bed. They were trying to help.\"\n\n\"I know,\" Kendra said.\n\n\"Where are you going to go?\"\n\n\"Someplace I can be alone.\"\n\nKnox had barely settled into the weightless nothingness within the Quiet Box, dimly wondering how long he would drift in stasis, vaguely relieved that he could no longer feel his shoulder turning to gold, when the door swung open. Instead of floating, he was suddenly standing, upholstered walls close at either side and behind him. His first reaction was to wonder what had prompted Kendra and Seth to remove him so quickly. His second reaction was to question how Patton was standing before him.\n\nThe real Patton Burgess had died many years ago. This was a replica created by a plant called a stingbulb. Supposedly, only mortals could enter Stormguard Castle, or creatures like dragons that could temporarily become mortal.\n\nBehind Patton stood Lomo, a rebel member of the Fair Folk who had left the city of Terrabelle to assist them, and the potion master Tanu, who had reportedly been turned to silver. Knox looked to his formerly golden hand to find it was now entirely made of flesh. He gratefully flexed fingers that had lost most of their feeling moments before.\n\nBut it had not actually been a matter of moments. If his hand was healed, and Tanu was restored, and Patton was here, then the trials of the castle must be over. They had won!\n\n\"Where are Seth and Kendra?\" Knox asked.\n\n\"No longer here,\" Patton said.\n\n\"They left me behind?\" Knox asked.\n\n\"They were gone when I regained my senses,\" said a lean man with brown stubble on his chin. \"I'm Lockland. I came out of that box when you went in.\"\n\n\"Who did you put inside to get me out?\" Knox wondered.\n\n\"The same criminal who I removed when I entered,\" Lockland said. \"A thief named Jasmine.\"\n\n\"She won't steal much in there,\" Knox said with a glance back at the Quiet Box. Several people in the dingy room laughed. Besides Tanu, Lomo, and Patton, there were a dozen attractive men and women assembled, dressed in courtly outfits\u2014their finery incongruent with a dungeon cell. One breathtaking girl looked only a year or two older than Knox. \"Kendra and Seth got away?\" he checked.\n\n\"We're not sure,\" Tanu said. \"We hope so.\"\n\n\"When we awoke from the spell, there was no sign of them,\" Lomo said.\n\n\"Nor Celebrant,\" Lockland said. \"Nor Tregain.\"\n\n\"Good,\" Knox said.\n\n\"We hope Tregain is unharmed and simply missing for now,\" Lockland said. \"He is my brother.\"\n\n\"And he was trying to turn us to silver,\" Knox said. \"Who ended up with the Wizenstone?\"\n\n\"We're not sure,\" Lomo said. \"Perhaps the hunt has moved elsewhere.\"\n\n\"The dragon Jaleesa fled in human form,\" said a solid man with a glossy brown beard. \"She was missing an arm.\"\n\n\"May I introduce my brother Heath,\" Lockland said.\n\n\"You were the one turning people to gold,\" Knox said.\n\n\"Yes, I possessed the golden glove for many years,\" Heath said. \"After all that time, the challenge ended in a mess. We don't know what became of the prize.\"\n\n\"You were changed to gold before the end,\" Lockland said.\n\n\"I only hope Tregain managed to keep the prize in the family,\" Heath said. \"Doom will soon follow should the dragons claim it.\"\n\nKnox noticed a distinguished older man wearing a conspicuous crown. He stood beside an older woman with a jeweled tiara. \"Wait, is that the king and queen?\" Knox asked.\n\n\"My parents,\" Lockland said. \"King Hollorix and Queen Satilla.\"\n\n\"I must congratulate you for your efforts undoing the curse,\" the king said to Knox. \"We languished for years trying to break the stalemate.\"\n\n\"Just trying to help,\" Knox said with a glance at Tanu and Lomo.\n\n\"We wish to show our gratitude by throwing a feast in your honor,\" the king said, his gaze taking in Tanu and Lomo.\n\n\"Lockland gave essential help as well,\" Tanu said. \"Along with Seth and Kendra Sorenson, the caretakers of Wyrmroost.\"\n\n\"It was Lockland's duty to help,\" the king said. \"You three came to our rescue of your own accord.\"\n\n\"I don't want to rain on the parade,\" Knox said, \"but most of your food might be spoiled.\"\n\nThe king gave a laugh. \"We checked our stores. The spell held most of the castle in stasis. Our larders remain well provisioned with recommendable fare.\"\n\nKnox smiled. \"I am pretty hungry.\"\n\nMany chuckled at the comment.\n\n\"Unfortunately, we must depart in a hurry,\" Tanu said. \"We need to check in with Blackwell Keep and learn what happened to Kendra and Seth.\"\n\n\"Nonsense,\" King Hollorix said. \"If you dine with us, this will have been a diplomatic mission, meaning our coach can take you back to the keep. There is no surer or faster way.\"\n\n\"He's right,\" Lomo said. \"Their flying coach has full immunity. A reliable ride home is just what you need.\"\n\n\"I won't argue against that,\" Tanu said.\n\n\"The castle is still safe from dragons?\" Knox asked.\n\n\"No location in Wyrmroost is more secure,\" Lockland said. \"It's why Jaleesa fled. She could not have transformed back into her dragon state until off this property. And once out, she would not be able to get back in without an invitation.\"\n\n\"Preparations for the feast are already under way,\" the king said. \"You will have seats of honor.\"\n\n\"How about some help against the dragons?\" Knox asked.\n\nThe atmosphere in the room grew more somber.\n\n\"Our neutrality policy stands as always, young man,\" King Hollorix said.\n\n\"But the dragons declared war,\" Knox said.\n\n\"Not on us,\" the king replied.\n\n\"The boy is right that the Fair Folk could make a difference in this fight,\" Lomo said.\n\n\"The last time the Fair Folk went to war, all who participated were lost,\" the king said. \"And the world was no better for it.\"\n\n\"What if we had stayed neutral about your castle?\" Knox asked. \"You would still be cursed.\"\n\n\"Do not overstate your generosity,\" the king said. \"We are grateful but we are not fools. Like the dragons, you came to this castle for the Wizenstone. My sons and I wanted it as well. The stone is evidently gone, and, as a happy accident, the curse is broken. Our gratitude is not an obligation. I recommend you accept it gracefully.\"\n\n\"Good food and a ride home,\" Tanu said. \"I've had worse offers.\"\n\n\"There must be more comfortable rooms than this one,\" Knox said.\n\nFriendly laughter greeted the statement. \"Agreed,\" King Hollorix said. \"We came here only to fetch you. Let us retire upstairs.\"\n\nAs they filed from the room into the dungeon hallway, Knox ended up beside the gorgeous girl. She was a little taller than him, with fine features, frosty blue eyes, and pale blonde hair.\n\n\"You were brave to come here,\" the girl said.\n\nKnox shrugged. \"I like saving people.\"\n\n\"I'm Regina,\" the girl said.\n\n\"Knox. Were you turned to metal?\"\n\n\"Silver,\" she said.\n\n\"How was it?\" Knox asked.\n\n\"Like a dream,\" Regina said. \"Not a nice one. I faded in and out of awareness, but I always felt like a passenger in my body, everything muted and numbed. I think I was more aware on the festival nights, especially when in motion. My mother, Bethany, was turned to gold. She was the one who touched you.\"\n\n\"I'm sorry,\" said a woman from behind Knox. He recognized her as the human version of the golden figure who had started his transformation to gold.\n\n\"No problem,\" Knox said. \"It helped get Lockland out of the Quiet Box.\"\n\n\"My brother,\" Bethany said.\n\n\"I'm just sad I missed the action,\" Knox said, stretching his formerly golden arm. \"Would have been nice to give those dragons some payback.\"\n\n\"I'm happy I got to dream away most of it,\" Regina said.\n\n\"I wish I knew what happened to my cousins,\" Knox said.\n\n\"The caretakers are your cousins?\" Regina asked.\n\n\"Yeah,\" Knox said. \"I think my grandparents want me to get more experience before I take over. I already saved Wyrmroost once. Went up against a bunch of dragons alone. In their dragon form, of course. Human form is a lot easier.\"\n\n\"You sound like quite a hero,\" Bethany said.\n\n\"I don't use that word,\" Knox said. \"People around me do sometimes. I guess they think it fits.\"\n\n\"You are difficult to describe,\" Tanu said over his shoulder. \"So many words fit.\"\n\nKnox avoided meeting eyes with Tanu. He hadn't been aware the potion master was listening. Knox cleared his throat softly. \"I couldn't have done any of it without my teammates.\"\n\n\"You performed bravely, Knox,\" Tanu said.\n\nKnox glanced at Regina. \"It's what I do.\"\n\nAre you sure nobody will see us?\" Tess asked as she marched through the tall grass behind Eve.\n\n\"We made a clean getaway,\" Eve said, not looking back. \"And we look like sheep to any onlookers.\"\n\nTess looked around. Woolly sheep grazed here and there, bright bells on their collars. No people were in view. Nearby, the sails of a large windmill slowly turned in the breeze.\n\n\"How far away are the fairies?\" Tess asked.\n\n\"We're almost there,\" Eve said. \"I can't promise we'll see any, but I usually find one or two.\"\n\nTess picked up her pace to stay near her new friend. Eve was one of the Fair Folk, the daughter of Lord Dalgorel. She was so pretty and nice, and Tess was delighted to be treated as an equal by a girl who looked a couple of years older than her. It had been Eve's idea to come see the fairies after Tess had shown her the cloak of innocence. Eve had been unfairly grounded by her father, so they had to sneak out of the castle. The escape hadn't been too difficult\u2014Eve could disguise them with magic, and she knew lots of secret ways. Some climbing had been required, and a crawl through one low tunnel, but since they had left the town, the journey had been easy.\n\nEve led them into a stand of trees, where they found a trickling creek that in places became almost narrow enough to jump across. A small, ornate bridge spanned it at one point, and near the bridge fluttered a fairy with dragonfly wings.\n\n\"Hello!\" Tess called, dashing ahead to the fairy. \"I'm Tess! Am I glad to see you!\"\n\nThe blue-skinned fairy zipped toward Tess and paused in front of her face, looking her up and down. \"How did a little human like you get fairystruck?\" the fairy asked.\n\n\"I love fairies,\" Tess said. \"I never saw one with blue skin before.\"\n\n\"You're not one of the Folk,\" the fairy said.\n\n\"I'm a visitor,\" Tess said.\n\n\"You understand her?\" Eve asked Tess.\n\n\"Don't you?\" Tess asked. \"Do you need to drink some milk?\"\n\n\"I'm part of the magical world, silly,\" Eve said. \"The milk makes no difference for me. But she's speaking a dialect of Silvian that I don't know.\"\n\n\"Why are you with her?\" the fairy asked Tess, waving a hand at Eve. \"She comes here a lot and pesters us.\"\n\n\"She means no harm,\" Tess assured the fairy.\n\n\"Ask her about the peaceful dragon,\" Eve said.\n\n\"What dragon?\" Tess asked.\n\n\"Does she mean Raxtus?\" the fairy asked.\n\n\"I understood that name,\" Eve said. \"We're not looking for Raxtus. Seth met a peaceful dragon named Dromadus. He used to be the Dragon King.\"\n\n\"Ah,\" the fairy said. \"Do you know Kendra?\"\n\n\"She's my cousin,\" Tess said.\n\n\"She's my friend,\" Eve said.\n\n\"She is bossy!\" the fairy said.\n\n\"You don't have to tell me,\" Tess said. \"But I like her.\"\n\n\"Kendra asked some of us to keep an eye on Dromadus,\" the fairy said. \"On all of the dragons, really.\"\n\nTess looked at Eve. \"Did you understand?\"\n\n\"No.\"\n\n\"Kendra asked her to watch that dragon, and the other dragons too,\" Tess said.\n\n\"The fairy knows where to find Dromadus?\" Eve asked.\n\n\"I do,\" the fairy said.\n\n\"Now she speaks English,\" Eve said.\n\nTess did not hear the difference.\n\n\"Can you lead us to Dromadus?\" Eve asked the fairy.\n\n\"I could,\" the fairy said.\n\n\"Won't we get in trouble?\" Tess asked. \"Tanu told me to stay put until they came for me. Is Dromadus far?\"\n\n\"It would take some time,\" the fairy said.\n\n\"As long as you stay with me, you're following your instructions,\" Eve said. \"They just didn't want you wandering off alone. Dromadus is a peaceful dragon. Now that the dragons declared war, maybe he'll help us. We should go ask him.\"\n\n\"I heard the dragons are all really dangerous,\" Tess said.\n\n\"Most are,\" Eve agreed. \"This one is the exception. With your cloak of innocence, we can travel there without a problem. In a time of war, we need to do all we can to help.\"\n\n\"I heard something about a war,\" the fairy said.\n\n\"Don't you want to see a dragon?\" Eve asked Tess. \"A nice one, I mean?\"\n\n\"Of course,\" Tess said. \"It's what I want most of all.\" She turned her attention to the fairy. \"Will you take us there?\"\n\nThe fairy yawned, patting her dainty lips. \"Wingless creatures are so slow.\"\n\n\"We can't help that,\" Tess said. \"We'll walk fast and won't complain.\"\n\n\"I hope not,\" the fairy said. \"I can't stand complainers.\"\n\n\"You'll take us?\" Tess asked.\n\nThe fairy jerked a thumb at Eve. \"For that one? No way! For you, though? I suppose.\"\n\nTess clapped her hands and grinned. \"Thank you. How could fairies get any better?\"\n\n\"It would be a challenge,\" the fairy said.\n\n\"Do you have a name?\" Tess asked.\n\n\"Poza,\" the fairy said. \"When would you like to go?\"\n\nTess looked to Eve.\n\n\"Now, if possible,\" Eve said.\n\n\"Aha!\" called a male voice. Tess turned to find Doren storming out of some bushes. \"Oh no you don't, not under my watchful eye.\"\n\nEve covered her face with one hand and shook her head.\n\n\"Who is this?\" Poza asked.\n\n\"Our goat-man friend,\" Tess said.\n\n\"More like your temporary guardian,\" Doren said.\n\n\"How did you find us?\" Eve asked wearily.\n\n\"Followed you from the castle,\" Doren said. \"Your illusions made my mind a little fuzzy, but I stuck with you.\"\n\n\"Why wait until now to interrupt?\" Eve asked.\n\nDoren blinked. \"I thought you might be up to some innocent girlish mischief. I'm not out to spoil all your fun. But seeing this dragon would take you away from Terrabelle, and I can't have that. I was charged with watching over Tess.\"\n\n\"I didn't even notice you,\" Tess said.\n\n\"I'm stealthy,\" Doren said. \"It's a talent. Not too hard to stay low in tall grass.\"\n\n\"This is a disaster,\" Eve said. \"We need to talk to Dromadus. He might be able to help. With Wyrmroost at war, only the cloak of innocence gives us a realistic chance of getting to him.\"\n\nDoren looked back and forth between the two girls. \"Newel and I had planned to use the cloak to poke around a bit.\"\n\n\"This isn't just curiosity,\" Eve said. \"This could be a chance to secure key help.\"\n\nDoren looked in the direction of the castle. \"It is kind of dull back that way. The Fair Folk women are so stuck up.\"\n\n\"Then you should come with us,\" Eve said. \"Unlike those neutral cowards, we're actually trying to help. With a big, strong satyr like you helping us, what could go wrong?\"\n\n\"You make a valid point,\" Doren said.\n\n\"A satyr was never part of the agreement,\" Poza said.\n\n\"Satyrs add to any group we join,\" Doren said, offended. \"We're good luck! We're fun! And we appreciate beauty. You, my dear fairy, are absolutely exquisite.\"\n\n\"All right, I like him,\" the fairy said.\n\n\"Let's go,\" Eve said.\n\n\"I didn't agree to come,\" Doren said. \"Or to let you go.\"\n\n\"But we're trying to save Wyrmroost,\" Eve said.\n\n\"I think you're more interested in going on an adventure,\" Doren said.\n\n\"What's wrong with an adventure?\" Eve asked.\n\n\"The timing is wrong,\" Doren said. \"We're at war. With dragons.\"\n\n\"We're going to a nice dragon,\" Tess said.\n\n\"Seth and Kendra visited him,\" Eve said. \"Dromadus could help us in the war. He used to be the king of the dragons.\"\n\n\"You're telling me a former Dragon King is nice?\" Doren clarified. \"How is he even alive? How reliable are your sources?\"\n\n\"Seth and Kendra got the scepter from Dromadus,\" Eve said. \"Seth told me about it while we were walking away from Skyhold together. Dromadus is famously the only Dragon King to renounce his position rather than lose it in combat. He got sick of all the fighting after he had killed his own nephew in a challenge.\"\n\n\"What makes you think he would help you?\" Doren asked.\n\n\"It's common knowledge that Dromadus has become a hermit,\" Eve said. \"Seth told me he is also a pacifist. I'm hoping he'll be displeased that the dragons needlessly declared war.\"\n\n\"Sounds like a long shot,\" Doren said.\n\n\"It will be fun to meet an actual dragon,\" Tess said. \"He might even have treasure.\"\n\nDoren folded his arms. \"Treasure is always of interest. But making a dragon angry by taking his treasure might not be wise.\"\n\n\"We wouldn't steal anything,\" Eve said. \"That would forfeit the power of the cloak. Don't forget the cloak. This will be safe.\"\n\n\"You three are exasperating,\" Poza said. \"Are we going or not?\"\n\nDoren scratched his head. \"I may get busted for this, but I do want to see more of the sanctuary, and I like your initiative. Let's do it.\"\n\n\"Hooray,\" Tess cheered. \"Lead the way, Poza!\"\n\nDoren sidled nearer to Eve. \"Think your dad will be angry?\"\n\n\"He's already furious,\" Eve said. \"I'd rather be punished for having met Dromadus than not.\"\n\n\"How long before he notices you're gone?\" Doren asked.\n\n\"I'm sure he already has.\"\n\nDoren glanced around nervously. \"Will he come down hard on us?\"\n\n\"Me, yes. Tess is too young. And satyrs, well, your kind have a reputation for frivolity.\"\n\n\"Play dumb. Got it.\"\n\n\"Or maybe skip the castle on the way back,\" Eve suggested.\n\nDoren stroked his chin. \"I prefer avoiding awkward conversations to having them. This is shaping into a good plan. Are you sure it's worth the trouble from your father?\"\n\n\"To talk to a dragon?\" Eve said. \"My father can't take away that experience. I want to do my part to help Wyrmroost. I'm fascinated by dragons. And I'm a student of history.\"\n\n\"A punishment isn't so bad if accompanied by the right memories,\" Doren said.\n\nEve gave a nod. \"Exactly.\"\n\n\"Lead on, fairy!\" Doren called.\n\n\"There is nothing lucky about a satyr,\" the fairy muttered.\n\nDark water sloshed against the rocky shore of the grotto, the churning surface rising and falling as it surged and splashed. Seth could smell salt in the moist air.\n\n\"Is this ocean water?\" Seth asked. \"Are we in a sea cave?\"\n\nRonodin used his torch to light a trio of cressets, each one nearly as tall as Seth, the slender stands topped by metal bowls to hold the flames. \"The domain of the Underking is large. Right now we're topside, not far from the sea, though the water here runs very deep.\"\n\n\"Does the Underking let you roam wherever you want?\" Seth asked.\n\n\"I come and go as I please,\" Ronodin said. \"There are areas of his domain that he keeps private, and regions where only a fool would venture, but otherwise I am without limits. It seems like your conversation with his majesty went well.\"\n\n\"How can you tell?\"\n\n\"You're alive,\" Ronodin said.\n\n\"I had to promise to do things,\" Seth said, glancing at the spectral manacle on his wrist. He held it up. \"Do you see that?\"\n\n\"Your hand?\"\n\n\"I'm cuffed to a ghost chain,\" Seth said.\n\n\"Some sort of bargain was bound to happen,\" Ronodin said. \"Hard to get something for nothing.\"\n\n\"Why did you bring me here?\"\n\n\"You need a place where you can regain your lost talents,\" Ronodin said. \"Many would try to keep you from them.\"\n\n\"Why do you get to come and go as you please?\" Seth asked.\n\n\"Because I earned the right.\"\n\n\"How?\"\n\nRonodin grinned in the firelight. \"We all have our trade secrets.\"\n\n\"The Underking assigned me to free some of the undead,\" Seth said.\n\n\"No big task,\" Ronodin said. \"But you need to learn to manage your fear or they will devour you, body and soul.\"\n\n\"I wasn't afraid of the wraiths,\" Seth pointed out.\n\n\"The wraiths here are under orders not to harm you,\" Ronodin said. \"And there are more dreadful beings than wraiths. Not all creatures who generate fear are undead. Few cause more paralyzing horror than dragons.\"\n\n\"I've heard that,\" Seth said.\n\n\"We need to learn how you fare against one,\" Ronodin said.\n\nSeth looked around. \"Here? Is it a baby?\"\n\nRonodin gave a chuckle. \"Hardly.\"\n\n\"This room isn't very big,\" Seth said.\n\n\"Not above the water,\" Ronodin said. He pointed with his torch toward an enormous drum. \"Go strike that three times.\"\n\nSeth reconsidered the turbulent water. \"To call something?\"\n\n\"Not all dragons dwell in the sky,\" Ronodin said.\n\n\"Can we handle a dragon?\" Seth asked.\n\n\"In these close quarters?\" Ronodin asked. \"We would need an arrangement with the brute to survive.\"\n\n\"Do we have an arrangement?\" Seth asked.\n\n\"Well, I do,\" Ronodin said. \"Smite the drum.\"\n\nHeld in place on a wooden scaffold, the drum was shaped like a huge barrel tipped on its side. The circular drumhead was taller than Seth. Crossing to stand before the drum, he noticed a few bones scattered in front of it.\n\n\"That's right,\" Ronodin said. \"Use a bone.\"\n\nSeth picked up what looked like a large thigh bone. Holding it in both hands, he took a shortened practice swing. The prospect of hitting the drum was exciting. The thought of summoning a dragon was thrilling. The possibility of being devoured was less comfortable.\n\n\"Three times,\" Ronodin encouraged.\n\nSeth could have asked more questions, but he had already learned that Ronodin did not stray from his intentions. Hitting the drum would bring answers soonest and was probably the inevitable outcome.\n\nSeth did not believe that Ronodin wanted him dead. Still, his palms were sweaty as he swung the bone hard. The impact produced a satisfying boom that seemed to sink into the roots of the cavern. Seth hit the drum again, feeling the resultant vibrations in his teeth. After the final stroke, Seth stepped away from the thrumming drumhead and set down the bone, then looked up to find Ronodin backing out of the room.\n\n\"Go stand in the middle of the cressets,\" Ronodin called, pointing toward them.\n\n\"Where are you going?\" Seth asked, panicked.\n\n\"See how they form a triangle?\" Ronodin asked. \"Stand at the center.\"\n\n\"Nice try,\" Seth said, hurrying away from the water toward Ronodin. \"I'm coming with you.\"\n\n\"Do as you're told if you want to live,\" Ronodin said. \"You have only seconds. You're safe within the triangle.\"\n\nSeth hesitated. \"Why aren't you in there?\"\n\n\"The cressets can protect only one person,\" Ronodin said. \"Face the water.\"\n\nSeth ran over to the triangle of cressets. \"What now?\" he cried.\n\n\"Talk to him,\" Ronodin called.\n\nThe brine foamed and roiled as never before, and the ground trembled. Gritty dust sprinkled down from the stalactites overhead. Mouth dry, Seth glanced back at Ronodin, who waited a couple of steps from the exit. Seth wondered how much he could rely on what Ronodin told him. Was it too late to run for it?\n\nWater fountained up and outward as a shape erupted from its surface, dousing Seth. Cressets still burning, Seth wiped saltwater from his eyes and beheld what looked like the gargantuan head of a lumpy, diseased lizard, infested with barnacles. Knobby horns projected in various shapes and directions, like the buds of malformed antlers, and the jaws gaped open to reveal a funnel of hooked teeth.\n\nSeth found he could no longer move. The sea dragon roared, and though Seth's bones seemed to quake and the entire cavern shuddered, he could not cover his ears. Supported by a purple, shell-encrusted neck, the head drew nearer, misshapen nostrils flexing as they sniffed.\n\n\"Speak to him,\" Ronodin encouraged.\n\nDespite his terror, Seth fought to move his lips. They refused to budge. Summoning all his strength, he tried to muster a hello, but his voice would not operate.\n\n\"This is no shadow charmer,\" the dragon said in a voice like the hulls of great ships grinding against rocky shores.\n\n\"His talents are still emerging,\" Ronodin said.\n\nYawning his jaws open, the dragon breathed out a warm, humid exhalation that reeked strongly of old fish. Seth stared past the uneven ranks of teeth to the leathery tunnel beyond, slimy brown surfaces wrinkled and clenching. He wanted to lean away from the stench but couldn't.\n\nThe dragon pulled back a bit, eyeing him with a flat gaze. \"He is crab food.\"\n\n\"Give the boy time,\" Ronodin said. \"Seth, now would be a good time to speak.\"\n\nSeth tried to turn his head to Ronodin. He wanted to clench his fists and yell in frustration. His body refused to respond.\n\n\"He makes kelp look clever,\" the dragon said. \"Why not present him as an offering?\"\n\n\"You don't need a boy,\" Ronodin said. \"You eat whales.\"\n\n\"I have not tasted human in a good while,\" the dragon said. \"Admittedly, they are bony and stringy, but they fear deliciously. The terror makes a fine seasoning.\"\n\nSeth wrestled with the accusation. He definitely didn't want to die, and the dragon scared him, but the fear was not smothering him. Ronodin had said he was safe within the triangle of cressets, and that appeared to be true. Although Seth was a helpless target, the dragon had not attacked. Seth wanted to show his courage, because his apparent fear seemed to make the dragon lose all respect for him. But his body would not comply.\n\n\"Please, Jibarro, give the boy a moment to compose himself,\" Ronodin said.\n\n\"Do not summon me for such dull entertainment,\" the dragon said; then his head plunged beneath the water, splashing Seth.\n\nA moment later, Seth could move. Ronodin approached briskly.\n\n\"What went wrong?\" Ronodin asked.\n\n\"That dragon could have eaten me,\" Seth said.\n\n\"Easily,\" Ronodin replied. \"You would have been mostly digested before reaching the stomach.\"\n\n\"Why couldn't I move?\" Seth asked.\n\n\"You tell me,\" Ronodin said.\n\n\"I tried to say hello,\" Seth said. \"I was keeping it simple.\"\n\n\"I sensed that your mind remained lucid,\" Ronodin said.\n\n\"Yes, I could think just fine,\" Seth said.\n\n\"Normally an inability to think is what stops a shadow charmer when faced by a dragon,\" Ronodin said.\n\n\"I could definitely think,\" Seth said. \"I just couldn't move.\"\n\n\"You have the power to overcome all magical fear,\" Ronodin said. \"You simply have to claim it.\"\n\n\"How?\" Seth asked.\n\nRonodin folded his arms. \"Evidently, when you are bombarded by extreme fear, the signals from your mind are not reaching your body. There is a disconnect between the mental and the physical.\"\n\n\"I knew what I wanted to say, where I wanted to move, but nothing would respond,\" Seth said.\n\n\"Look inside yourself, Seth,\" Ronodin said. \"Look deep. Can you find the darkness within you?\"\n\n\"I don't understand,\" Seth said.\n\n\"A demon called Graulas granted you a gift,\" Ronodin said. \"You have perceived the darkness and cold within a wraith. You have darkness in you as well. It establishes a degree of kinship with the undead. You can use it to hide, to extinguish fire, and to center yourself against fear.\"\n\nSeth tried to focus inward but wasn't sure what he was looking for. \"What should the darkness feel like?\"\n\nRonodin sighed. \"I'll give you a rare opportunity.\" He held out a hand. \"Take my hand and look into me. I was once a vessel of light. See what you find within me now.\"\n\nSeth took the offered hand, and his mind immediately perceived a void so startling, he worried he might lose himself in it. The vast darkness seemed sure and still. It took Seth a moment to remember to speak. \"The darkness is everywhere.\"\n\n\"Good,\" Ronodin said. \"You have similar power within you, though less developed. Close your eyes, and get curious about your center. Where is your essence housed? Behind your eyes? In your gut? Within your fingers?\"\n\n\"I don't get it,\" Seth said. \"I'm all me.\"\n\n\"Think harder,\" Ronodin challenged. \"Where is your core? Look for the part of you that feels and chooses and holds you together.\"\n\nSeth thought about it. \"Maybe in my chest? Maybe behind my eyes?\"\n\n\"What if it's an emergency, and you need to survive?\" Ronodin said. \"What takes over?\"\n\n\"It's in my chest,\" Seth said.\n\n\"There is a part of you that is in charge,\" Ronodin said. \"A part of you that insists no external influence is going to govern you.\"\n\n\"Yes, okay,\" Seth said. \"In my chest.\"\n\n\"Look for the darkness near there,\" Ronodin said. \"It is not yet throughout.\"\n\n\"I think I feel what you mean,\" Seth said. \"Almost like a cloud around my heart. Is that supposed to mean I'm evil?\"\n\n\"We're not talking about evil,\" Ronodin said. \"We're talking about darkness as a power source. Is day good and night evil? No. That distinction is ridiculous. Night is better for seeing the stars. Is not the cool of night preferable for traveling in the desert? Probably. Is sleeping in a dark room evil and sleeping in a bright one good? No, and I would argue that darkness is better suited to the purpose of sleep. The darkness you draw from is a resource. You can use it for good or evil.\"\n\n\"I guess so,\" Seth said.\n\n\"You sense the darkness within you?\" Ronodin asked. \"The gift you were given?\"\n\n\"I think so,\" Seth said.\n\n\"Good enough,\" Ronodin said. \"Tap into that darkness. Anchor yourself there. It can enable you to remain in charge of yourself, even against potent influences. The power there can be drawn upon to accomplish many things. Use it to extinguish this cresset.\"\n\nSeth opened his eyes to see Ronodin indicating one of the flaming cressets. Seth drew on the dark place Ronodin had helped him find and focused on the cresset, pushing coldness and darkness into it. The flames resisted.\n\n\"It's hot,\" Seth murmured.\n\n\"The heat is nothing compared to how cold and dark you can go,\" Ronodin said.\n\nSeth gritted his teeth and pushed with all that he had. The fire in the cresset shrank, dimmed, and went out with a puff of smoke. Seth found that he was panting, and his jaw hurt. Droplets of sweat beaded his brow.\n\n\"Very well done,\" Ronodin said richly, releasing Seth's hand. \"That cresset is magically waterproof. Not easy to put out.\" He used a torch to relight it.\n\n\"You can do this too?\" Seth asked.\n\n\"I draw on darkness in many ways,\" Ronodin said. \"Most unicorns are beings of light. It grew tedious to me. I traded my light to access darkness, making me unique. The dark unicorn.\"\n\n\"What does this have to do with talking to the dragon?\" Seth asked.\n\n\"Go to that same power source when Jibarro appears,\" Ronodin said. \"Draw on the strength there. Demand to stay in charge. Don't let any other stimulus suppress you.\"\n\nSeth nodded.\n\nRonodin walked over to the drum and picked up the thigh bone. \"Ready for another try?\"\n\n\"Shouldn't I rest?\" Seth asked. \"Train? What if I fail?\"\n\n\"Rules or no rules, this time he might eat you.\" Ronodin struck the drum three times.\n\nSeth wanted to protest. He didn't feel ready! He considered fleeing but decided it was too cowardly. At least he knew what to expect this time. Steeling himself, Seth turned and confronted the agitated water. The cavern shuddered. A geyser announced the arrival of Jibarro, his knobby horns grazing the top of the chamber.\n\n\"You again?\" the dragon asked, glaring. He drew near with a snarl. \"Don't tell me you suddenly grew a backbone?\"\n\nSeth could not move or reply. As he searched the dark place near his center, calm sureness spread through him.\n\n\"Hello,\" Seth said.\n\nHe suspected he could move, then gave a little wave to prove it.\n\nThe dragon looked to Ronodin. \"You're good.\"\n\n\"That's not news,\" Ronodin said. \"But I'm not doing this. The boy is talented.\"\n\nJibarro sniffed Seth, nostrils not quite touching his shirt. \"How is it done? Wires? A spell?\"\n\nSeth held his nose. \"What do you eat? Pollution?\"\n\nJibarro jerked away, eyes flashing. \"Was that disrespect?\"\n\n\"Independence,\" Ronodin said. \"Thank you, Jibarro. Your offering is already in place.\"\n\nThe head disappeared below the surface with a tremendous splash. Seth was absolutely soaked but didn't care. \"What offering?\"\n\n\"A large quantity of pickled squid,\" Ronodin said.\n\n\"Might explain the breath,\" Seth said, sitting down unsteadily on the rocky floor. He felt shaken but relieved. \"I did it!\"\n\n\"Lucky for you,\" Ronodin said. \"Otherwise you would not survive tomorrow night.\"\n\n\"What happens tomorrow?\"\n\nThe dark unicorn smiled. \"You have some captives to liberate.\"\n\nKendra did not realize she had fallen asleep until she was gently shaken awake. For a moment she felt disoriented. Then she recalled having come to Glory's stall to hide and think. Her favorite horse was gone, so the stall provided a cozy place where she could be alone.\n\nExcept she was no longer alone.\n\nAgad the wizard crouched before her. His smile looked strained. Seeing him brought Kendra fully awake.\n\n\"You're back!\" Kendra said. \"Any word about Seth?\"\n\nAgad started to rise, then, after a pause, went and sat on the opposite side of the stall from Kendra, back to the wall. \"Nothing definitive yet. I'm sorry. After the message from your grandparents, I went right to work. I'm still piecing it together.\"\n\n\"What are the pieces?\"\n\n\"I examined the barrel,\" Agad said. \"Its counterpart is now deep underwater.\"\n\n\"Where?\" Kendra asked.\n\n\"In the ocean, I believe,\" Agad said. \"Deep.\"\n\n\"What ocean?\"\n\nAgad shook his head sadly. \"I can't get a read on it. There is magical interference.\"\n\n\"Can we send somebody from our barrel to that one?\" Kendra asked.\n\nAgad shook his head. \"Not without someone to help them out on the other side. And I believe the barrel is extremely deep. I suspect the pressure alone would kill a person.\"\n\n\"They sank it instead of destroying it?\" Kendra asked.\n\n\"Meaning they may hope to use it again,\" Agad said. \"We should consider destroying the one we have.\"\n\n\"It might be our only link to Seth,\" Kendra said. \"Can't we lock it up? Make sure nobody can come through?\"\n\n\"We could,\" Agad said. \"You realize they may have scuttled the barrel someplace far from your brother. It may never serve as much of a clue.\"\n\n\"I know,\" Kendra said. She looked at the wizard. \"Do you have any idea who did this?\"\n\nAgad dropped his eyes. In that moment, Kendra realized how weary and haggard the old wizard looked, his skin tinged with gray that darkened around the eyes, his beard more matted and tangled than usual, his robes rumpled and stained.\n\n\"The Sphinx was involved,\" Agad whispered.\n\n\"You're sure?\" Kendra asked. Grandpa and Grandma Sorenson had suspected his participation.\n\nAgad nodded. \"Your grandfather used a hearing stone to contact me, and he told me what happened to Seth. I dropped everything and followed my intuition to Goblin Town.\"\n\n\"Is that a real place?\" Kendra asked.\n\n\"It's in Nevada,\" Agad said. \"Not a standard sanctuary, mind you. No caretaker. Without the milk, it looks to humans like a large ghost town. In reality, the ramshackle settlement is infested primarily with goblins, along with a roguish assortment of hobgoblins, kobolds, gremlins, and imps.\"\n\n\"Sounds lovely,\" Kendra said.\n\n\"Wonderful place to lose an eye,\" Agad said. \"Or a tooth. Or your life. Fortunately, goblins and their sort have a healthy respect for wizards.\"\n\n\"What did you find out?\" Kendra asked.\n\n\"I found Slaggo,\" Agad said.\n\n\"He got there quickly,\" Kendra said.\n\n\"I'm sure the Sphinx helped him,\" Agad said. \"The Sphinx didn't want anyone talking to Slaggo or Voorsh. It isn't his style to kill an accomplice. He wants those who aid him to be rewarded. I counted on the Sphinx deciding Slaggo and Voorsh could blend in better in Goblin Town than anywhere else.\"\n\n\"How did Slaggo escape Fablehaven?\" Kendra asked.\n\n\"I suspect the wizard Vernaz is helping the Sphinx,\" Agad said. \"Vernaz helped reconstruct Mendigo, which explains why the puppet was following secret instructions. Vernaz has the know-how and authority to take creatures across preserve boundaries, and also to transport individuals over great distances.\"\n\n\"Slaggo and Voorsh brought the barrel to the Sphinx?\" Kendra asked.\n\n\"That much we know,\" Agad said. \"When Knox inadvertently left them the keys, the goblins fled the dungeon with the barrel and exited Fablehaven through the front gate. Apparently some time ago the Sphinx gave Voorsh means to contact him. When the Sphinx heard about the barrel, he came immediately.\"\n\n\"The Sphinx has been to Fablehaven before,\" Kendra said. \"It never crossed my mind he might have made a deal with the goblins in the dungeon.\"\n\n\"He is a dangerous opportunist,\" Agad said. \"I fear it was a mistake to make him an Eternal.\"\n\n\"He has to die for anyone to open the new demon prison,\" Kendra said. \"He's good at staying alive. It makes sense for him to be part of the lock.\"\n\n\"And it preserves him,\" Agad said. \"Keeps him around, perhaps for centuries more, unless he is killed.\"\n\n\"Did you track him down?\" Kendra asked.\n\nAgad shook his head. \"I tried. I checked with insiders. But the Sphinx has disappeared. So has Vernaz.\"\n\n\"Did Slaggo see Vernaz?\" Kendra asked.\n\n\"A hooded figure was with the Sphinx,\" Agad said. \"There was no confirmation on the identity. But based on the description and the circumstances, it was very likely Vernaz.\"\n\nKendra knocked her head back against the wall of the stall a few times, hard enough to sting a little. \"What would the Sphinx want with Seth?\"\n\nAgad raised his eyebrows and spread his hands. \"I can only guess. Obviously, they are both shadow charmers. Or the Sphinx may have been working with somebody else who has an interest in your brother.\"\n\n\"Celebrant?\" Kendra asked. \"Ronodin?\"\n\n\"It's guesswork at this point,\" Agad said.\n\nKendra picked up a brittle piece of straw from the floor of the stall and twisted it between her fingers. \"The Sphinx originally allied himself with the demons. That door is closed, so now he might be scheming with the dragons?\"\n\n\"Maybe,\" Agad said.\n\n\"Or does he think he can train Seth as a sidekick?\" Kendra asked. \"Could he have known Seth lost his memories? He had barely lost them when he was kidnapped.\"\n\n\"I don't know,\" Agad said.\n\n\"That plan could work,\" Kendra said bleakly, straw crunching in her grasp. \"Whoever is behind it. Seth doesn't know who he is. He could be retrained from scratch. Reprogrammed by our enemies.\" Her eyes stung. \"Agad, we could end up fighting my little brother.\"\n\n\"We'll try to recover him before it comes to that,\" Agad said.\n\n\"But we don't know where to look,\" Kendra said.\n\n\"Not yet,\" Agad replied.\n\n\"Didn't you have a tracker on the Sphinx?\" Kendra asked.\n\n\"As part of becoming an Eternal, the Sphinx has the right to move and hide where he chooses,\" Agad said. \"He had two minders\u2014one we told him about and one we didn't. This is the first time we know of that he eluded both.\"\n\nKendra squeezed her eyes shut. This just kept getting worse. Would there ever be good news?\n\n\"I'm sorry that you kids were drawn into this crisis,\" Agad said.\n\n\"If this keeps up, the whole world will be drawn in,\" Kendra said.\n\n\"There is uncertainty and devastation all around,\" Agad said. \"All of Dragonwatch is working overtime on problems across the globe. We managed to win back a portion of Crescent Lagoon. It seemed the whole archipelago had fallen, but the caretaker reestablished a foothold on one island. Your friends Warren and Vanessa played a key role.\"\n\n\"Are they all right?\" Kendra asked.\n\n\"Last I heard they were safe and healthy,\" Agad said.\n\n\"I guess one less fallen sanctuary is a relief,\" Kendra said.\n\n\"It's a desperate fight,\" Agad said. \"We'll take what we can get.\"\n\n\"Are people looking for Bracken and Seth?\" Kendra asked.\n\n\"All of our agents are on the lookout,\" Agad said. \"And I am making it my personal mission to find your brother.\"\n\nThat brought Kendra some relief. At least Agad considered Seth a top priority.\n\n\"What about the sanctuary that completely fell?\" Kendra asked.\n\n\"Soaring Cliffs?\" Agad asked.\n\n\"Yes,\" Kendra said. \"Where Bracken was held for a time. Are those dragons rampaging? Destroying cities?\"\n\n\"Nothing like that yet,\" Agad said.\n\n\"Why not?\" Kendra asked.\n\n\"It's complicated,\" Agad said.\n\n\"Isn't that why we're fighting to keep these sanctuaries from falling? So dragons won't run wild?\"\n\nAgad furrowed his brow. \"I suppose as a caretaker you should know about our last lines of defense. Someday this knowledge may prove necessary. These secrets should not be repeated except to the most trusted ally at the greatest need.\"\n\n\"I know how to keep secrets,\" Kendra said.\n\n\"I would not share this otherwise,\" Agad said. \"Three of the seven dragon sanctuaries contain Dragon Temples. Each temple houses numerous priceless artifacts, along with one talisman of extreme importance. Here at Wyrmroost, that item is the Sage's Gauntlets. A wizard who masters the gauntlets can gain control over dragons.\"\n\n\"That would be amazing,\" Kendra said.\n\n\"For our side,\" Agad said. \"Do not forget that like all wizards, I was once a dragon. Using arts that are all but forgotten, I became mortal in exchange for greater magic. Though I have not been a dragon for a great while, I still remember how the existence of the Sage's Gauntlets offended me. All dragons feel likewise. There is also a shield that allows the bearer to repel dragons, and a harp that can soothe dragons to sleep. These talismans helped defeat the dragons long ago. As part of the treaty that brought the dragons to the sanctuaries, the dragons were allowed to guard these items in their temples. If we were to recover any of the three talismans, it would frighten and enrage the dragons, because the talismans make them vulnerable.\"\n\n\"Do we have any of the talismans?\" Kendra asked.\n\n\"No,\" Agad said. \"To pursue them would have been an act of war.\"\n\n\"But the dragons already declared war,\" Kendra said.\n\n\"Exactly,\" Agad said. \"So now, getting the talismans has become a priority.\"\n\n\"How does that explain why dragons are not attacking cities?\" Kendra asked.\n\n\"I'm getting there,\" Agad said. \"Another line of defense is the legendary Dragon Slayers.\"\n\n\"Like the Somber Knight?\" Kendra asked.\n\n\"Each sanctuary has a Dragon Slayer,\" Agad said. \"But there are five legendary Dragon Slayers abroad in the world. They are lost at present, but when Celebrant declared war, the door was opened for us to recruit them.\"\n\n\"Do you know how to find them?\" Kendra asked.\n\n\"We're working on it,\" Agad said. \"Finding them may not be easy after such a prolonged retirement. Aside from the legendary Dragon Slayers, we have a final line of defense between dragons and the mortal world.\"\n\n\"What?\" Kendra asked.\n\n\"The Sovereign Skull,\" Agad said. \"The fabled skull of Abraxas, the first dragon.\"\n\n\"It's really a skull?\" Kendra asked.\n\n\"Of the very first dragon,\" Agad said reverently. \"A pristine relic of crystalline beauty, more heavily enchanted and ensorcelled than any physical object in the world.\"\n\n\"What does it do?\" Kendra asked.\n\n\"Listen, the unbelief of humanity provides a degree of protection from the magical world,\" Agad said. \"The skull amplifies that effect, making human unbelief toxic to dragons. As long as the skull exists, mortals ignorant to the magical world are difficult for dragons to target, and unbelieving communities are impossible to invade.\"\n\n\"Where is it kept?\" Kendra asked.\n\n\"For the safety of the world, that must remain a secret,\" Agad said. \"The place was created by wizards, and only a few of us know the location.\"\n\n\"Are you one of them?\" Kendra asked.\n\n\"I should not say too much,\" Agad said.\n\n\"The dragons that escaped Soaring Cliffs can't attack the nonmagical world unless they destroy the skull,\" Kendra verified.\n\n\"Correct.\"\n\n\"But they can get up to other mischief?\"\n\n\"That is a definite concern,\" Agad said. \"The world has some defenses left as these preserves fall. But unless we defeat the dragons and get them back into sanctuaries, this will not end well. The longer the conflict rages, the more likely it becomes that the dragons will discover how to overthrow our last defenses.\"\n\n\"There is so much to do,\" Kendra said.\n\n\"We start by protecting the sanctuaries we still control,\" Agad said. \"You have done a remarkable job here, against terrible odds. I must say that you and your brother performed beyond all expectations.\"\n\n\"I just hope we can get him back,\" Kendra said.\n\n\"We need him back,\" Agad affirmed. \"Sentimentality aside, he is a caretaker of Wyrmroost. This sanctuary is safer with him here.\"\n\nThe long, low bellow of a horn reached Kendra's ears. \"The proudhorn,\" Kendra said.\n\nAgad stood. \"A dragon approaches.\"\n\n\"I hope it's Raxtus,\" Kendra said, getting up as well. \"Did you know he joined us?\"\n\n\"I heard,\" Agad said, exiting the stall. \"Not a moment too soon, from what I understand.\"\n\n\"He saved me,\" Kendra said.\n\nThey trotted out of the stable to see a gilded coach gliding through the air, pulled by a team of winged rams. Dragons flanked the coach to either side, flying a bit behind it.\n\n\"There is a sight Wyrmroost has not seen in centuries,\" Agad said.\n\n\"Flying rams?\" Kendra asked.\n\n\"The coach of Stormguard Castle,\" Agad said. \"After all this time, the curse is broken. Let's go see what they want.\"\n\nAgad and Kendra hurried to the courtyard where the coach landed. The handsome driver reined the six rams to a halt.\n\n\"They just flew over the wall,\" Kendra said. \"Did they have permission?\"\n\n\"The Blackwell Keep defenses don't repel the coach,\" Agad said. \"It's part of an ancient treaty.\"\n\nKendra noticed that the dragons tracking the coach had veered away. \"Dragons can't attack the coach?\" she asked.\n\n\"The coach of Stormguard Castle enjoys immunity on all diplomatic missions,\" Agad said. \"Even in wartime. The neutral status of the Fair Folk has some serious advantages. I'm sure the dragons would have attacked if it were possible.\"\n\nGrandpa and Grandma Sorenson were coming down stairs to the courtyard, as was Brunwin the minotaur. Marat and Henrick approached from a different direction.\n\nThe door to the coach opened and Tanu emerged, along with Knox and Lockland. Tanu and Knox looked tired but healthy, and no parts of them were silver or gold.\n\n\"Knox!\" Kendra called. \"Tanu! You're all right!\"\n\n\"Kendra!\" Knox called back, waving and hustling toward her. \"We wondered where you were! Did you get the prize?\"\n\n\"No,\" Kendra said.\n\n\"Did Tregain?\" Lockland asked.\n\nKendra winced. \"Tregain was killed while trying.\"\n\nLockland closed his eyes and pressed a fist to his forehead. Knox seemed to be resisting a grin.\n\n\"How did he fall?\" Lockland asked.\n\n\"He touched the stone and turned to ashes,\" Kendra said. \"Obregon too.\"\n\nLockland composed his features. \"My family and all of Stormguard Castle will rue these tidings. But casualties were to be expected in such a contest. What of your brother?\"\n\nKendra saw Knox and Tanu looking at her expectantly. She tried to tell them, but words wouldn't come. Only tears.\n\n\"Seth lost his memory and was kidnapped by the Sphinx,\" Agad said on her behalf. \"We are working to find him.\"\n\n\"Kidnapped?\" Knox asked.\n\n\"Taken away through the barrel,\" Grandpa Sorenson said, an edge to his voice.\n\nKnox blanched. \"Oh no.\"\n\n\"The goblins at Fablehaven gave the barrel to our enemies,\" Grandpa said, \"who used it to abduct Seth.\"\n\n\"No!\" Knox said, eyes staring wildly. \"No, no, no. I gave them the keys to the dungeon.\"\n\n\"It's not entirely your fault,\" Grandpa Sorenson said. \"You didn't take Seth, nor did you understand the risks of using the barrel.\"\n\n\"I'm an idiot,\" Knox said. \"I'm the worst.\"\n\n\"We all make mistakes,\" Grandpa said.\n\n\"Not that cause kidnappings,\" Knox said.\n\n\"True,\" Kendra mumbled. Then she raised her voice. \"I know you've been trying to help. Right now our focus is getting him back. Where are Lomo and Patton?\"\n\n\"Lomo remains at Stormguard Castle against his wishes,\" Lockland said. \"My father refused to let him come. I will return with the coach. As Fair Folk, we can't get directly involved in helping you.\"\n\n\"And Patton is helping the mounts return safely,\" Tanu said. \"Calvin is with him. But we have another problem.\"\n\n\"Fabulous,\" Kendra said. \"What?\"\n\n\"We stopped by Terrabelle in the coach to pick up Tess and the satyrs,\" Knox said. \"Tess was gone. The satyrs too. I guess Newel came here, but Doren, who was watching Tess, has disappeared. Along with Eve.\"\n\n\"Disappeared?\" Grandpa asked. \"Where did they go?\"\n\nKnox lowered his eyes. \"Nobody knows.\"\n\nHow could a dragon fit in there?\" Tess asked, pointing at what looked like a pair of wooden cellar doors on the ground within the sequoia grove. \"Is he tiny?\"\n\n\"He should be huge,\" Eve said, climbing down from her horse. \"I wonder if he has another way in.\"\n\n\"I guess one of us should stay with the animals,\" Doren said, swinging down from his gray mare.\n\n\"I thought you didn't like horses,\" Tess said.\n\nDoren held up a finger. \"I never bad-mouthed horses. I was a little hesitant to steal three of them.\"\n\n\"We borrowed them,\" Eve corrected.\n\n\"We borrowed them without permission,\" Doren amended. \"In order to sneak away from Terrabelle against the wishes of everyone in charge.\"\n\n\"You complained nonstop,\" Poza said.\n\n\"I mentioned I had never ridden a horse before,\" Doren said. \"And that I was worried I would feel like a centaur.\"\n\n\"Did you?\" Tess asked.\n\n\"A little,\" Doren admitted. \"Look, satyrs don't ride horses. I've never heard of it happening before. I broke protocol for you girls.\"\n\n\"Now you want to stay with the horses because you're afraid of the dragon,\" Eve said.\n\n\"I'm trying to be a good leader,\" Doren said. \"We need smart strategy. A peaceful dragon might be nice to two little girls. Kendra did fine with her visit. But the dragon might not like a satyr. Some creatures are biased against us.\"\n\n\"He makes a convincing argument,\" Poza said.\n\n\"Plus, somebody really should watch these borrowed horses,\" Doren said. \"It would be a long walk back. And it's getting dark.\"\n\n\"You'll be out here without the cloak of innocence,\" Eve told him.\n\nDoren shrugged. \"I'm still part of the group. It might still cover me.\"\n\n\"It might not,\" Eve said.\n\nDoren scratched his elbow and glanced around the grove. \"In an emergency, I could follow you in.\"\n\n\"And leave the horses?\" Tess cried.\n\n\"The horses could get away faster without me,\" Doren said.\n\n\"We do need to hurry,\" Eve said. \"And it probably makes sense to leave a guard for the horses.\"\n\n\"I brought you here,\" Poza said. \"May I excuse myself?\"\n\n\"What about finding our way back?\" Tess asked.\n\n\"Simply return the way we came,\" Poza said.\n\n\"But we came here without a path,\" Eve said. \"What if we get lost?\"\n\n\"You'll be fine,\" Poza said. \"That cloak works perfectly.\"\n\n\"Nothing would attack us, but we might starve,\" Tess said.\n\n\"And it will get dark,\" Eve added.\n\n\"Let the satyr find the way,\" Poza said. \"As part animal, he should have some instincts.\"\n\nDoren scratched his sideburn. \"I probably could retrace our route. I don't know, Poza. Would you trust your life to me?\"\n\n\"Absolutely not.\"\n\n\"What about the life of little Tess?\" Doren said.\n\nThe fairy glanced at Tess, then tossed up her arms in defeat. \"Fine, I'll stay. Don't take too long. I will not be assisting with the horses.\"\n\n\"Thank you,\" Tess said. She patted her horse, staring at the considerable drop to the ground. \"How do I get down?\"\n\nDoren helped Tess out of the saddle.\n\n\"Where do I tie up the horses?\" Doren asked.\n\n\"No need,\" Eve said. \"We train our horses well. Come help us open these doors.\"\n\nDoren walked over to the cellar doors. A pair of nearby stones had writing etched on them in both recognizable characters and mysterious glyphs. \"The languages I can read say 'Welcome,'\" he said. \"Anything else?\"\n\n\"I see the same,\" Eve said.\n\n\"What kind of dragon puts 'Welcome' on the door?\" Doren asked.\n\n\"A nice one?\" Tess asked.\n\n\"I get wary when something that could eat me wants company,\" Doren said. He bent down, grabbed a door handle, then paused. \"Do we know what's behind here?\"\n\n\"The way to the dragon,\" Tess said.\n\n\"Do you suppose this door is within his view?\" Doren asked.\n\n\"I've never been here before,\" Eve said.\n\nTaking a step back, Doren spat on his hands and rubbed them together. \"I guess we're about to find out.\" He bent forward and heaved open one of the doors. A long, straight stairway stretched down into darkness.\n\n\"No dragon yet,\" Tess said, sounding a bit disappointed.\n\n\"Foreboding nonetheless,\" Doren said. \"I wish you well. Keep that cloak handy.\"\n\n\"Come on,\" Eve said, after lighting a small lantern with a match. She took Tess by the hand and started down the stairs.\n\n\"See you soon, Poza,\" Tess said. \"You too, Doren.\"\n\nThe satyr gave her a nod and a wave. \"Maybe don't mention my name. I don't want to be overly known in dragon circles.\" The fairy flitted out of view as the door closed.\n\n\"I'm glad you brought a light,\" Tess said as they descended the stairs.\n\n\"Wouldn't be very smart to visit a dragon lair without one,\" Eve replied. \"This should have enough oil to last a couple of hours.\"\n\n\"The steps look like they go down forever,\" Tess said.\n\n\"It's the longest stairway I've seen,\" Eve said.\n\nGlancing back at intervals, Tess noticed when the doors to the surface receded into shadow. \"We're going deep underground.\"\n\n\"Are you getting nervous?\" Eve asked.\n\nTess looked ahead at the stairs descending into darkness beyond the lantern light. \"A little.\"\n\n\"We'll be fine,\" Eve said. \"I don't freeze in front of dragons. If you do, I can drag you out of there.\"\n\n\"I don't think I'll freeze up if it's a nice dragon,\" Tess said. \"My legs might get tired on the way, though.\"\n\n\"Just don't trip,\" Eve said. \"A fall down a stone staircase this long would break just about every bone you have.\"\n\nTess took extra care with the steps from that point downward. She was beginning to despair that the stairs would go on endlessly when a level place came into view.\n\n\"Finally,\" Eve said.\n\n\"I wonder how much farther to the dragon?\" Tess asked.\n\n\"Let's go find out.\"\n\nBeyond the stairs they followed a rounded tunnel to a heavy oaken door. \"Should we knock?\" Tess asked.\n\n\"We've come this far without permission,\" Eve said.\n\n\"It said 'Welcome' up top.\"\n\n\"True. Ready?\"\n\nTess felt nervous and excited. \"Do you think this is where the dragon is?\"\n\n\"Could be more stairs,\" Eve said. \"Or another hallway.\"\n\nEve opened the door, and they passed into a vast cavern. It took Tess a moment to spot the dragon's enormous shape, filling much of the room, but half-buried in stone rubble. For a moment she thought it might be a statue because the hide seemed so rocky. Then the craggy head raised up and swiveled in her direction, scattering masonry and flinging dust into the air.\n\n\"Two young girls,\" the dragon said in a startlingly genteel baritone. \"After years with no outside contact besides the ogres bringing food, I am getting the oddest visitors of late.\"\n\nTess gave a little curtsy. \"I'm Tess. This is Eve. I'm friends with the fairies. We heard you're a nice dragon.\"\n\nThe dragon shuddered and seemed to cough, making boulders shift and more dust take flight. \"Is that the new reputation for Dromadus the Terrible?\"\n\n\"Were you called that?\" Tess asked.\n\n\"I have been called many things,\" Dromadus said. \"Never nice.\"\n\n\"We heard you are a pacifist,\" Eve soothed.\n\n\"And who are you?\" Dromadus asked.\n\n\"Eve of the Fair Folk,\" Eve said.\n\n\"The other girl is mortal,\" Dromadus said.\n\nTess raised a hand. \"Tess.\"\n\n\"Tess is fairystruck but mortal,\" Dromadus said. \"And cloaked by innocence.\"\n\n\"I believe so,\" Eve said.\n\n\"We know you could probably still eat us if you wanted,\" Tess said.\n\n\"A prudent thing to know,\" Dromadus said. \"Yet do not underestimate the power of innocence. Many of the maladies that ruin the wise can be utterly confounded by innocence.\"\n\n\"Really?\" Tess asked.\n\n\"Innocence has a difficult time appreciating itself,\" Dromadus said. \"Though it has certain vulnerabilities, once lost, true innocence can never be regained. Anything both powerful and irrecoverable is most precious.\"\n\nTess gripped her cloak tightly. \"I'll be careful.\"\n\nThe dragon coughed lightly.\n\n\"Are you laughing?\" Eve asked.\n\n\"That would be a rarity,\" Dromadus said. \"Where did you hear I was a nice pacifist? Kendra and Seth, I suppose?\"\n\nWorried about getting her cousins in trouble, Tess looked to Eve.\n\n\"Seth told me a little about you,\" Eve said. \"He never called you nice. Everyone knows that no dragon is safe, even a peaceful one.\"\n\n\"Try not to spread any gossip about my gentle side,\" Dromadus said. \"Too many fools at Wyrmroost might take that for weakness. Some could get hurt.\"\n\n\"I won't spread it,\" Eve said.\n\n\"Me neither,\" Tess promised.\n\n\"Tell me, what brings you here today?\" Dromadus asked. \"Or did you simply hope to witness the peaceful dragon?\"\n\n\"That was part of it,\" Tess said.\n\n\"We're also hoping you can help us,\" Eve said. \"The dragons declared war.\"\n\n\"I'm aware,\" Dromadus said.\n\n\"It was unprovoked,\" Eve said.\n\n\"Many of the dragons in sanctuaries around the world consider themselves captives,\" Dromadus said.\n\n\"Dragons went to the sanctuaries by treaty,\" Eve said. \"It was part of the terms of their surrender.\"\n\n\"Many dragons born inside the sanctuaries never agreed to such terms,\" Dromadus replied. \"The war against the dragons occurred centuries ago. Defeated nations often regroup and fight again after time goes by. The dragons have old grievances to settle.\"\n\n\"Dragons have been treated well in the sanctuaries,\" Eve insisted.\n\nDromadus yawned. \"Dragons never belonged in close proximity to one another. By nature we are solitary.\"\n\n\"Then why do they have a king?\" Tess asked.\n\n\"Dragons have always established dominance,\" Dromadus said. \"But the king never really acted like a true leader until dragons began to be hunted and killed. As the fight against dragons swelled into a war, we needed to organize ourselves. A real king was required.\"\n\n\"Is a king needed now?\" Tess asked.\n\n\"To mount a war?\" Dromadus said. \"Yes, if that is what the dragons want. But dragons residing in a castle? That is a ridiculous imitation of humanity. We do not belong in colonies except in emergencies.\"\n\n\"The dragons are causing the emergency,\" Eve said. \"They have received humane treatment.\"\n\n\"You fail to understand dragons,\" Dromadus said. \"We have a strong need for independence.\"\n\n\"They have plenty of room at Wyrmroost to be independent,\" Eve said.\n\n\"Knowing there are boundaries decreases the feeling of independence,\" Dromadus said. \"Boundaries imposed by defeat are even worse.\"\n\n\"If the dragons wage another war, they could destroy themselves,\" Eve said.\n\n\"Or they will destroy everybody else,\" Tess said.\n\nDromadus sighed. \"That gets closer to the crux of the problem.\"\n\n\"Then you'll help us?\" Eve asked.\n\n\"I can't solve the problem,\" Dromadus said.\n\n\"You could try,\" Tess said. \"We're trying.\"\n\n\"She doesn't mean to be presumptuous,\" Eve said.\n\n\"I like her sincerity,\" Dromadus said. \"I no longer fear honesty. I have learned that anything important accomplished through dishonest means eventually unravels. Facing unpleasant truths protects us from building on faulty foundations.\"\n\n\"Can you help us?\" Eve asked.\n\nCraning his stony neck, the dragon looked around, sniffing the dusty air. \"I deliberately trapped myself in here, never to emerge, never to influence outside matters again. That intent has wavered of late. This is the danger of conversing with mortals.\"\n\n\"Danger?\" Tess asked. \"Are you afraid of us?\"\n\n\"Any wise creature of magic would be,\" Dromadus said.\n\n\"But you're so big,\" Tess said.\n\n\"I could try to rely on my size and strength,\" Dromadus said. \"I did for years. And I was a fool. I harmed and destroyed many I claimed to love. You are very dangerous, little one.\"\n\n\"Why?\" Tess asked.\n\n\"There is a divide between the magical world and the mortal world,\" Dromadus said.\n\n\"My brother couldn't see the fairies until he drank the milk,\" Tess said.\n\n\"Very good,\" Dromadus said. \"That is evidence of the divide. Magic is not for mortals. It is not inherent to their natures.\"\n\n\"But I can see fairies without milk,\" Tess said.\n\n\"Mortals can derive magic from magical sources. Some fairies took a liking to you and shared their powers. They changed you. But I wonder if they are aware that you also changed them?\"\n\n\"Did I change them?\"\n\n\"Just as magic is not for mortals, mortality is not for magical beings,\" Dromadus said. \"Those of an eternal nature do not change. We magical beings are nearly eternal, and so change comes slowly if at all. To many of us, change is a type of magic we do not understand.\"\n\n\"I change all the time,\" Tess said.\n\n\"I expect you do,\" Dromadus said. \"Such change feels as foreign to me as magic seems to you. Fairies do not typically appreciate mortals. They do not willingly reveal themselves to mortals. If fairies learned to like you, they have been changed in the process.\"\n\n\"That's a good change,\" Tess said.\n\n\"It depends on your perspective,\" Dromadus said. \"If those fairies get too friendly with mortals, they could get killed or trapped. They could get changed into imps. I noticed a satyr with you as you approached.\"\n\n\"How did you know?\" Tess asked.\n\n\"I have ways,\" Dromadus said. \"He was riding a horse.\"\n\n\"He is helping us,\" Tess said.\n\n\"But satyrs don't typically help anyone,\" Eve said with interest.\n\n\"How long have you known that satyr?\" Dromadus asked.\n\n\"Not long,\" Tess said. \"A few days.\"\n\n\"Does the satyr know other mortals?\" Dromadus asked.\n\n\"He is friends with Seth,\" Tess said.\n\n\"And that is how a satyr ends up riding a horse,\" Dromadus said. \"And helping humans, instead of indulging his own capricious interests.\"\n\n\"Because he is friends with Seth?\" Tess asked.\n\n\"Probably,\" Dromadus said. \"Any relationship with a mortal exerts influence. The divide becomes even more starkly evident with love. A romance between a magical being and a mortal is ultimately incompatible and ill-fated unless one joins the realm of the other.\"\n\n\"I have heard stories,\" Eve said.\n\n\"They tend to be tragic,\" Dromadus replied.\n\n\"You claimed Tess is dangerous to you,\" Eve said.\n\n\"Her cousins are too,\" Dromadus said. \"No matter how wise or powerful you are, no matter how fixed in your determinations, if you bring mortals around, you invite change. Think about yourself, Fair One. How long have you dreamed of an adventure like this?\"\n\n\"All my life,\" Eve said. \"That has been constant.\"\n\n\"Did you actually embark before or after interacting with mortals?\"\n\nEve paused. \"Wow. After.\"\n\n\"You begin to glimpse my peril,\" Dromadus said.\n\n\"You want to help us?\" Tess asked.\n\nThe dragon's head swooped in close, staring with black eyes. \"I never wanted to help mortals. I had vowed not to interact with the external world again. But when I met Seth and Kendra, the seeds of conflict were planted within me.\"\n\n\"What conflict?\" Eve asked.\n\n\"I do not agree with this war,\" Dromadus said quietly. \"I can see why the dragons feel it will benefit them. I can see why they suppose this war could right old wrongs and tear down boundaries and increase freedom. But I believe their aggression is wrong.\"\n\n\"Then you'll help us?\" Tess asked.\n\n\"I want to deny you,\" Dromadus said. \"I fear where this could end. But I will give you a bit of information that could help. If you appreciate the assistance, please do not make your source public.\"\n\n\"What do you know?\" Eve asked.\n\n\"You must hasten to Blackwell Keep,\" Dromadus said. \"Warn them that the dragons have an attack planned for tomorrow night. An attack they expect to succeed.\"\n\n\"How?\" Eve asked. \"Blackwell Keep is secure now.\"\n\n\"I truly have no more details,\" Dromadus said. \"But Celebrant is no fool.\"\n\n\"Can't you come with us?\" Tess asked.\n\n\"The aid I can offer is limited,\" Dromadus said. \"But I can watch you, and maybe share information again, if you trust me.\"\n\n\"How would you send information?\" Tess asked.\n\n\"Through trusted messengers, perhaps,\" Dromadus said. \"I make no promises.\"\n\n\"How will you watch us?\" Tess asked.\n\nThe head of the dragon swiveled away and rummaged in a rock pile. When the head returned, Dromadus opened his jaws and placed a jewel at her feet. \"Take the sapphire. If you keep it with you, I will be able to see you.\"\n\n\"Careful,\" Eve said. \"You'll be letting a dragon peer into Blackwell Keep.\"\n\nTess picked up the stone and held it up with the light of the lantern behind it, enjoying how the facets gleamed. \"He's a peaceful dragon. And he is helping us.\" She put the sapphire in her pocket.\n\n\"I hope so,\" Eve muttered.\n\n\"Hasten to the keep,\" Dromadus said. \"I do not know whether there is time to upset Celebrant's plan. The sooner you arrive, the better.\"\n\n\"We should go,\" Eve said.\n\n\"Wait, he gave me a present,\" Tess said. Facing the dragon, she dug into her pocket and pulled out a yo-yo. \"A satyr gave this to me. It's a yo-yo.\" She set it on the ground.\n\n\"Thank you, child,\" Dromadus said.\n\n\"Tess,\" she reminded the dragon, backing to the door.\n\n\"Thank you, Tess,\" Dromadus said. \"Travel swiftly and safely. Keep the cloak of innocence ever with you.\"\n\nEve took Tess by the hand and hurried her along.\n\n\"'Bye, Dromadus,\" Tess said. \"Thanks for the tip.\"\n\nOut behind the Blackwell Keep stable, Knox chucked a dirt clod at the wall, where it exploded in a puff of dust. He crouched to tear another clump from a chunky dirt pile, but the selected piece crumbled in his hand. Trying a different part of the pile, he dislodged a clod that held together.\n\nSeth had been taken by enemies and Knox could only blame himself. His insides constricted at the thought. He winged the dirt clod sidearm as hard as he could. The chunk burst into dust, leaving a little patch of dirt on the wall to commemorate the impact.\n\nKnox had enjoyed the feast at Stormguard Castle. Many official, fancy people had treated him like a celebrity. The food had been plentiful and delicious. When he had departed the castle, Knox had not worried that the praise might have been somewhat exaggerated. After all, he had taken a risk by going to Stormguard Castle, he had been part of the team that broke the curse, and if Seth and Kendra went off someplace else before the feast, then they chose to miss the victory party.\n\nKnox heaved a large dirt clump at the wall. It was one thing to accept a little more credit than he really deserved. It was another to learn he was the cause of Seth getting kidnapped. His cousin was in the hands of evil people because Knox had left the dungeon keys with the goblins back at Fablehaven.\n\nNow every word of praise came back to him like a hot knife. Every delicious bite of food seemed poisonous. Stan and some of the others had reminded Knox that his involvement had been indirect\u2014evil people had carried out the plan, not him. But at the very least Knox knew he was like a guard who had fallen asleep on duty and let the enemy harm his allies. No, worse: he was a clumsy outsider who had sidetracked the guards\u2014with disastrous consequences.\n\nKnox threw another hunk of dirt. And another. It made him sick to think that while he had feasted and laughed and bragged, his cousin who had actually saved the day was in the hands of the bad guys, probably getting tortured.\n\nIt almost made it worse that people had been mostly kind to him about his actions, as if they knew what he had done was so terrible it could mess him up for life. Maybe it should. Ruth had reminded him that he had not understood the consequences of what he was doing. Which was true, but Knox had known that it was wrong to steal the keys from Dale. He had known he wasn't supposed to sneak to Wyrmroost. What had felt like minor mischief had turned into a deadly mistake.\n\nAnother dirt clod smashed against the wall in a gritty burst. And another. He could not throw them hard enough. Part of him wished somebody was throwing the dirt clods at him. Or beating him, or locking him in jail. He had no means to express how remorseful he felt, no way to pay for what he had done. And no way to fix the mistake.\n\nKnox had not slept well the night before. This morning he had eaten a hasty breakfast and then slipped away to be alone. He crouched and tore off another hunk of dirt.\n\nBrunwin the minotaur came around the corner of the stable, and Knox dropped the dirt clod, suddenly worried he might have been violating rules. He tried to subtly brush the dirt from his hands.\n\n\"There you are,\" Brunwin said. \"Your sister has returned safely from Terrabelle.\"\n\n\"Good,\" Knox said. He didn't love showing too much interest in his sister, so he kept his reaction simple. Terrabelle was a safe place, but with all the trouble at the sanctuary, it was a relief to know Tess was all right.\n\n\"The Sorensons sent me to fetch you,\" Brunwin said.\n\n\"All right,\" Knox said, mildly relieved that the dirt clods did not appear to be an issue. \"Lead the way.\"\n\nRather than walk beside the shaggy minotaur, Knox followed. He began regretting the decision as the smell of the brute wafted back at him. He considered asking whether minotaurs ever bathed, but managed to hold his tongue.\n\nKnox was glad to know Tess was safe. He had assumed she would be. He could not imagine she had done anything truly daring with the cloak of innocence, but he could easily picture her creating commotion by wandering off.\n\nBrunwin escorted Knox to a room where Tess, Kendra, Newel, Doren, Marat, Tanu, and Agad sat alongside Stan and Ruth Sorenson at a long conference table. Patton leaned against a wall, arms folded. Stan invited Knox in and thanked Brunwin. The minotaur snorted and departed.\n\n\"Knox!\" Tess said. \"I saw a dragon. Did you see any?\"\n\n\"I saw some with you on the road,\" Knox said.\n\n\"I mean up close,\" Tess said. \"I talked to one. He used to be their king. He looked really rocky.\"\n\n\"I saw some dragons in human form,\" Knox said. \"And I got partway turned to gold.\" He looked to Patton. \"You made it.\"\n\n\"Arrived not long ago,\" Patton said. \"We took turns running along the road or flying just above it. The winged mounts made the trip much faster. Overall, smooth sailing. I'm not feeling so good, though. The life of a stingbulb is brief.\"\n\n\"I'm sorry,\" Knox said.\n\nPatton waved a hand and shook his head. \"Nothing to mourn. Plants bloom and then wilt. Just sorry I can't lend a hand much longer. I knew this was coming. I have maybe a day left.\"\n\n\"Calvin made it as well?\" Knox asked.\n\n\"I'm right here,\" the nipsie called from his place on the table. Knox had not yet seen him.\n\n\"Sorry about Seth,\" Knox said.\n\n\"We'll find him,\" Calvin replied. \"Just one more adventure to have.\"\n\nKnox glanced at his sister. \"Is Tess serious about seeing a dragon?\"\n\n\"Your sister took a terrible risk,\" Stan said. \"But it could work to our advantage.\"\n\nDoren raised a hand. \"And I rode a horse.\"\n\n\"I still can't believe it,\" Newel said in disgust.\n\n\"It wasn't as bad as you think,\" Doren said.\n\n\"Zero goat pride,\" Newel said.\n\n\"It was an emergency,\" Doren said. \"I had to supervise the child.\"\n\nStan cleared his throat.\n\n\"Right,\" Newel said. \"The task at hand. Stan has the floor.\"\n\nStan fixed his attention on Knox. \"Your sister and the daughter of Lord Dalgorel visited Dromadus, a former Dragon King who resides at this sanctuary.\"\n\n\"Did he know something about Seth?\" Knox asked hopefully.\n\n\"Not that he shared,\" Stan said. \"But apparently the dragons are planning an attack against Blackwell Keep. Tonight. They are confident of success.\"\n\n\"We rode through the night to get back to Terrabelle,\" Tess said. \"My friend Eve wanted to come here with me, but she got in trouble with her dad. I was so tired.\" She yawned. \"At least I got some sleep before Newel came.\"\n\n\"Newel and Henrick brought Tess and Doren here just now,\" Stan said. \"They came quickly after hearing the news. We couldn't find you so we sent Brunwin to retrieve you.\"\n\n\"I was behind the stables,\" Knox said. \"Thinking.\"\n\n\"Thinking?\" Tess asked skeptically.\n\n\"You should try it,\" Knox said. \"How are the dragons going to attack?\"\n\n\"Dromadus didn't know,\" Tess said. \"And don't tell anybody who told us about the dragons. It's a secret.\"\n\nKnox looked around the room. \"Your secret is spreading fast.\"\n\n\"Just so we can protect ourselves,\" Tess said. \"Dromadus wanted me to warn the caretaker.\"\n\nKnox glanced at Kendra, then looked to Stan. \"Any idea how they are going to attack?\"\n\n\"None so far,\" Stan said. \"We were just discussing the possibilities when you entered.\"\n\n\"The defenses of the keep are in perfect order,\" Agad said. \"Stronger than when I was caretaker here. With the scepter in our possession, the dragons should have no chance of breaching our defenses through force.\"\n\n\"And yet Celebrant is planning an attack that he expects to succeed,\" Ruth said.\n\n\"Perhaps Celebrant is letting anger cloud his judgment,\" Patton said.\n\n\"Or perhaps Dromadus is toying with us,\" Marat said. \"His assistance is most unexpected.\"\n\n\"Eve heard the dragon too,\" Tess said. \"She told Doren.\"\n\n\"Both of the girls seemed sure of the message,\" Doren confirmed.\n\n\"But the message itself could be false,\" Marat said. \"Dromadus is extremely old, and little is known about him since he shut himself away from the world. He could simply be eccentric. Or he could be scheming.\"\n\n\"We have to treat it like a real threat,\" Kendra said.\n\n\"I fully agree,\" Marat replied. \"Just be aware that all may not be as it seems with this warning.\"\n\n\"What would be the point of a fake warning?\" Knox asked.\n\n\"Could it be a prank?\" Tanu asked.\n\n\"Anything is possible,\" Agad said. \"Maybe Dromadus is working with Celebrant. Perhaps Celebrant wants to see how we react if we think an attack is imminent. He might hope we will show him a vulnerability as we scramble to prepare.\"\n\n\"Or the attack could be real,\" Marat said. \"There could be some defensive weakness we have overlooked. Or we might have a traitor among us.\"\n\nKnox considered the others. Surely nobody in this room would side with the dragons! Hadn't everyone present proven themselves? He wondered if any of them might think he was a traitor. After he had delivered the Fablehaven barrel to the enemy, they had a right to think anything.\n\n\"What about the barrel?\" Knox asked. \"Could that be the weakness?\"\n\n\"We should not ignore the possibility,\" Agad said. \"We placed the barrel in a cell in the dungeon with a trusted minotaur standing guard. With nobody to help an intruder out of the barrel, entering the keep in that way should be impossible.\"\n\n\"We could destroy the barrel,\" Tanu said.\n\n\"No,\" Kendra said. \"It's our only link to the people who took Seth.\"\n\n\"We can place an additional guard outside the cell with the barrel all night,\" Agad said. \"Just to be safe.\"\n\n\"Are there other vulnerabilities?\" Kendra asked.\n\n\"Betrayal might be the most viable,\" Stan said. \"A strong defense is best dismantled from the inside. Then again, maybe the dragons want to make us mistrust each other. Suspicion could do that.\" Stan turned toward Newel. \"You satyrs wanted to be in on this meeting.\"\n\n\"Seemed appropriate since Doren was involved directly,\" Newel said.\n\n\"What do you think about all of this?\" Stan asked.\n\n\"It's a long meeting,\" Newel said.\n\n\"About the defenses,\" Stan clarified.\n\n\"You've been warned, so be vigilant tonight,\" Newel said. \"Not much else to do. Can't run away.\"\n\n\"Some of you could flee,\" Kendra said. \"Tess has the cloak of innocence. I'm the only one who really has to stay.\"\n\n\"If you want to send away the satyrs or the younger children, that is one thing,\" Agad said. \"But Wyrmroost must not fall. The able defenders must stand and fight.\"\n\n\"We have to protect the keep, especially when the dragons might be bluffing,\" Patton said.\n\n\"Should we send Tess and Knox away from the sanctuary?\" Ruth asked.\n\n\"They could flee into the wilderness beyond the borders,\" Stan said. \"Wyrmroost is not in a convenient location. We would need to order a pickup to get them back to civilization.\"\n\n\"I'm not going anywhere,\" Knox said. \"Not until we find Seth.\"\n\n\"I'm not leaving either,\" Tess said.\n\n\"Sending anyone away from the safety of Blackwell Keep while the dragons are plotting may not be wise,\" Marat said.\n\n\"I've seen a lot of trouble come and go in my time,\" Patton said. \"I've made it through much of it by patiently staying in a secure place. Not letting myself get lured out. Choosing to face enemies on my terms.\"\n\nStan sighed. \"I've had similar experiences. Our safest bet is to keep everyone in our stronghold and to keep it secure.\"\n\n\"I will check all entrances and exits,\" Marat said. \"We'll post twice the normal watch.\"\n\n\"I will examine all magical defenses,\" Agad said.\n\n\"I'll brew up some special potions just in case,\" Tanu said.\n\n\"Let's take whatever reasonable precautions we can think of,\" Stan said. \"We have all afternoon to help ensure a quiet night.\"\n\n\"Might be prudent to have a good meal,\" Doren said. \"We don't want to face trouble on an empty stomach.\"\n\nRuth stood up. \"Hopefully, if we're vigilant, we'll end up having a calm rest tonight while the dragons snarl in frustration.\"\n\nThough Knox appreciated her attempt to be positive, he worried she wasn't giving the dragons enough credit. In his gut, he worried they were all missing something.\n\nAll right,\" Ronodin said, backing up to admire his handiwork. \"That suffices. Free yourself, if you can.\"\n\nSeth stood in a torchlit room, his back to a stone column, fastened in place by a chilly cocoon of chains and locks. Ronodin had just finished securing the final chain, but the weight and pressure of the bindings were already uncomfortable. \"Where do I start?\" Seth asked.\n\n\"The locks, of course,\" Ronodin said.\n\nSeth squirmed, but the chains hugged him maddeningly. He could barely move. Breathing felt laborious. Closing his eyes, he reached for the locks with his power but could only vaguely sense their locations.\n\n\"I can't connect my power to the locks,\" Seth said. \"I can barely feel them.\"\n\n\"They're not as easy to sense as the fire,\" Ronodin said.\n\n\"How am I supposed to pick them if I can't feel them?\" Seth asked.\n\n\"Your thinking is muddled,\" Ronodin said. \"You're not going to mechanically pick the locks any more than you physically doused the torch.\"\n\n\"Then how will they open?\" Seth asked. \"I don't see how cooling them will help.\"\n\n\"With the torch, you sought out the source of the flames,\" Ronodin said. \"You brought cold to the source of the heat. Don't seek out the physical mechanism of the lock so much as reach for what is binding you. The dark power inside you does not like to be bound. It can undo all but the most powerful confinements. Let the power do the work.\"\n\nSeth tried to focus on where he was bound. Weren't the chains all around him? He stared at one of the locks on the chains. He tried not to focus too much on the physical lock. It was just something binding him. Something his power could undo.\n\nSeth searched for the darkness near his center and found it. He tried to draw upon it as he had to quench the torch, but he wasn't sure what exact part of the darkness he was moving to the lock. Cold had seemed easier to isolate.\n\nIn frustration, he strained against the heavy links with his strength, but there was no give. He tried to use his power again, to no avail.\n\nA small bell rang in the distance.\n\n\"That is our signal,\" Ronodin said.\n\n\"What about the locks?\" Seth asked.\n\n\"Opening locks is more subtle than dousing fire,\" Ronodin said. \"It may take time.\" He snapped his fingers, and the chains fell away from Seth to pile around the base of the column. \"I was mostly distracting you from worrying about your upcoming mission.\"\n\nSeth stepped over the chains. \"This was misdirection?\"\n\nRonodin gave half a smile. \"And it was fun chaining you up.\"\n\n\"Wait,\" Seth said. \"The mission is now?\"\n\n\"It can be hard to judge time down here,\" Ronodin said.\n\n\"Where are we going?\"\n\n\"Follow me,\" Ronodin said, already walking.\n\n\"You're coming with me?\" Seth asked, hurrying to catch up.\n\n\"You'll go alone,\" Ronodin replied.\n\n\"Am I ready?\" Seth asked. \"What do I need to know?\"\n\n\"We'll judge your readiness by whether you survive,\" Ronodin said. \"You're about to learn all you need to know.\"\n\nSeth felt full of questions but didn't know where to start. They passed down a hall and into a dim room with only a wet barrel inside.\n\n\"Is that going to transport me?\" Seth asked.\n\n\"A barrel like this could lead almost anywhere,\" Ronodin said. \"One of my top operatives is on the other side to receive you. He will provide the help you need to succeed.\"\n\n\"Where exactly am I going?\" Seth asked. \"What will I have to do there?\"\n\n\"You are going to a prison,\" Ronodin said. \"You will free the captives there.\"\n\n\"Are they bad people?\" Seth asked.\n\n\"They are undead,\" Ronodin said. \"They were incarcerated because of their natures. They have been imprisoned for far too long.\"\n\n\"Will I have to open locks?\" Seth asked nervously.\n\n\"Not with your power,\" Ronodin said. \"The man who will receive you has the specifics. In short, you will go through the barrel and then do as you're told. Save your questions for him.\"\n\nSeth stared at the barrel. Was this the sort of activity he had done in his forgotten past? Maybe. Ronodin had certainly helped Seth regain access to dark powers, including conversing with wraiths. It didn't require much of a leap for Seth to imagine that he might have freed undead captives before. Left to himself, would he choose to do so right now? No way. But Ronodin was an expert at springing tasks on him without leaving much time to think.\n\n\"We've been underground since I came here,\" Seth said. \"At least I'll get to see the sky.\"\n\n\"Don't count on that,\" Ronodin said. \"Where you are going there may not be many windows.\"\n\n\"Do I have to do this?\" Seth asked.\n\n\"I suppose you could go tell the Underking you changed your mind,\" Ronodin said. \"I suspect the conversation would not end well.\"\n\nSeth glanced at the ephemeral manacle on his wrist. He wished he could make that binding come loose. For now, he was trapped. He had not chosen to come to this underworld, but to buy time he had made promises to the Underking. Right now, staying alive might be the best he could manage. If he survived long enough, maybe he could eventually take control of his life. What he wanted more than anything was some time alone to figure things out for himself. He would stay vigilant. Maybe going through the barrel would lead to a chance to escape.\n\n\"This will take me away from the domain of the Underking?\" Seth asked.\n\n\"Out of his domain, yes,\" Ronodin said. \"But make no mistake, Seth\u2014nowhere is out of his reach.\"\n\n\"I'll free his captives,\" Seth said.\n\nRonodin gave a nod. \"Into the barrel. Stay quiet. Follow the instructions you get on the other side. Perform as required. I hope to see you soon.\"\n\nSeth climbed into the barrel and faced Ronodin. \"Wish me luck.\"\n\nRonodin shook his head. \"Luck has no business where you are going. Your success will be decided by good planning and proper execution. Get it right. Crouch down.\"\n\nSeth hunched down inside the barrel. A strong hand grabbed his wrist and stood him up, then helped him out of the barrel.\n\nAn upturned flashlight balanced on the floor revealed that Seth was in a dungeon cell with the door ajar. The man with him had dark skin and short, beaded dreadlocks, and he wore black clothes. He had a handsome face and at first glance looked about thirty, but something in his eyes and his bearing suggested he might be quite a bit older.\n\nThe man smiled. \"Good to see you again, Seth,\" he whispered.\n\n\"Do I know you?\" Seth asked.\n\n\"Hard to know me when you hardly know yourself,\" the man said. \"We've had dealings in the past. I'm called the Sphinx.\"\n\n\"Half lion?\" Seth asked.\n\n\"Just a nickname,\" the Sphinx said. \"I'm a mortal like you. In fact, I'm more like you than you might expect.\"\n\n\"Did you lose your memory too?\"\n\n\"No, I'm a shadow charmer,\" the Sphinx said. \"I don't believe there are currently any others besides us.\"\n\n\"Only the two of us?\" Seth asked.\n\n\"It's a small club.\"\n\n\"If you're a shadow charmer, why do you need me?\"\n\n\"I was cursed by some wizards,\" the Sphinx said. \"As a result, my life was prolonged, but the tampering interfered with my powers. I can no longer communicate with the undead. I have only my defensive skills: hiding in dimness, dousing fire, picking locks.\"\n\n\"I don't know much about the mission,\" Seth whispered. \"We're freeing some of the undead?\"\n\n\"Ronodin can be so secretive,\" the Sphinx said. \"This shouldn't be too hard. This way. Don't trip on the minotaurs.\"\n\nThe Sphinx picked up the flashlight and covered the top with one hand so only a little light leaked out. He led Seth out of the cell. A pair of dead minotaurs were sprawled in the hallway near an ax and a mace.\n\n\"Did you do that?\" Seth asked.\n\n\"Self-defense,\" the Sphinx whispered. \"Minotaurs tend toward violence. It was my life or theirs. Come on.\"\n\nThe minotaurs were big, and their weapons looked impressive. Apparently the Sphinx knew how to fight.\n\n\"Where are we?\" Seth asked.\n\n\"The dungeon of a fort that holds many undead prisoners,\" the Sphinx whispered. \"Operated by cruel monsters. We don't want to get caught. Go silent for a few minutes. Follow the light.\"\n\nThe Sphinx faded out of view. Only the light seeping between his fingers remained visible. They passed down a hallway of rough stone with cell doors on either hand. Seth wondered what kind of prisoners might be hidden away down here. He smelled mildew and decay and heard the skittering of small creatures.\n\nThe flashlight paused at an iron door. Seth heard a click, and the door opened.\n\n\"I'm still trying to figure out locks,\" Seth whispered.\n\n\"I have my competencies,\" the Sphinx replied, coming back into view. \"This way.\"\n\nThey passed through the doorway, and the Sphinx closed the door behind them. After winding around and descending some stairs, they reached another iron door, this one mottled with corrosion and shedding brownish flakes.\n\n\"This door has been reinforced, physically and magically,\" the Sphinx said. \"Do you hear the voices?\"\n\n\"No,\" Seth said.\n\n\"Close your eyes,\" the Sphinx said. \"Concentrate.\"\n\nSeth complied, reaching out with his mind and his senses, listening. He noticed pressure, as if his ears needed to pop. The pressure diminished, and then it returned again. The sensation wavered, building and receding. \"I feel a throbbing. It's hard to explain.\"\n\n\"Perceptive,\" the Sphinx said. \"You sense voices and wills pushing again the magical barrier. The blunted impact of their efforts.\"\n\n\"The undead?\" Seth asked.\n\n\"Yes,\" the Sphinx said. \"The captives of the Blackwell.\"\n\n\"We're here to free them,\" Seth said.\n\n\"You're here to free them,\" the Sphinx said. \"I'm here to lend support and get you back to the Under Realm.\"\n\n\"You have more experience than me,\" Seth said. \"Can't you just do it?\"\n\n\"Not an option,\" the Sphinx said. \"I can no longer reach the undead. I'm able to sense them, but I cannot interact with them.\"\n\n\"How do I free them?\" Seth asked.\n\n\"Carefully,\" the Sphinx said. \"With conditions. They want out. When you enter the chamber, they will call to you, make promises, offer to serve you. Do not let them set the terms. You set the terms.\"\n\n\"How?\" Seth asked.\n\n\"Don't try to become their commander,\" the Sphinx said. \"There are too many undead in that well, and many are too powerful for any shadow charmer to control for any length of time. Have you any experience with revenants?\"\n\nThe word seemed familiar. He knew they were undead. \"Not that I remember. I've heard of them.\"\n\n\"What about liches?\" the Sphinx asked.\n\nSeth shook his head.\n\n\"Most of the undead have no mortal spark,\" the Sphinx said. \"A corpse may be involved, but the human lifespark has fled. What identity they have is alien. They are cursed to long for life without a chance to regain it. A revenant still has some of that lifespark trapped inside. And a lich deliberately retains some of its former human will. These creatures are more powerful and more pitiable because they have actually tasted life and are now imprisoned without it. I suspect most revenants and liches eventually long for release, though they have learned to cling to their contorted version of living. Until deliverance comes, they are most formidable.\"\n\n\"There are revenants in the well?\" Seth asked.\n\n\"And a lich,\" the Sphinx said. \"A former enchanter named Belrab. He transformed from dragon to wizard to lich. A horrible descent.\"\n\n\"He's powerful?\" Seth asked.\n\n\"You will address him first,\" the Sphinx said. \"And you will keep it brief.\"\n\n\"You can't help?\" Seth asked.\n\n\"Only with advice,\" the Sphinx said. \"I can hear them speaking, but it is gibberish to me. And I can't speak words that they can understand.\"\n\n\"But they will understand me?\" Seth checked. \"Even the revenants and the lich?\"\n\n\"It is a natural part of your gift,\" the Sphinx said, \"and a large portion of your power over them. Remember what they crave above all else?\"\n\n\"Life,\" Seth said.\n\n\"Your ability to understand them and respond to them is a connection to life,\" the Sphinx said. \"They are often feared, but seldom heard. If ever.\"\n\n\"They still might kill me,\" Seth said.\n\n\"If you get this wrong, we'll both die miserably,\" the Sphinx said. \"They like communication with the living but will soon set their sights on the light of life inside you. They will try to take it, and you will perish.\"\n\n\"Do they keep what they take?\" Seth asked.\n\n\"They only sample it for a moment,\" the Sphinx said. \"They leech the life from you, basking briefly in your waning vitality. An objectively small pleasure that feels large to them. You will die painfully.\"\n\n\"Why are we releasing them?\" Seth asked.\n\n\"Because they are unfairly trapped in ugly conditions,\" the Sphinx replied. \"And because the Underking desires it. And because we will flee.\"\n\nSeth bit his lip. Freeing such dangerous creatures seemed like it could have disastrous results. How had he gotten into this mess? Seth glanced at the spectral manacle on his arm. The Underking governed these creatures, and it made sense that he wanted them treated humanely. Unless Seth was ready to defy the Underking, he would have to support his causes for a time.\n\n\"Sounds like a terrible existence,\" Seth said.\n\n\"They are to be pitied,\" the Sphinx said. \"Sharks do not attack people out of cruelty. It is their nature to hunt. The undead are only following their natural inclinations. Locking them up is the cruelty, especially in a tiny space. You will not give these beings unlimited freedom, but you will grant them a wider area to roam. Care must be taken in how you release them.\"\n\n\"Tell me what to do,\" Seth said.\n\n\"Ignore their offers,\" the Sphinx said. \"The foolish will make absurd promises they can't really keep and the smart ones will deliberately attempt to entrap you.\"\n\n\"What offer do I make?\"\n\n\"There is a chain beside the well,\" the Sphinx said. \"Lowering the chain would allow some of the undead to exit. Others must be granted permission to use it. You must work out the details before dropping the chain.\"\n\n\"What terms do we want?\"\n\n\"Speak with Belrab first. Offer for him to be master of this castle if he agrees to remain in this castle and to keep the other undead here as well. We must make all the undead pledge to remain in this castle.\"\n\n\"Won't they still be trapped?\" Seth asked. \"Like they are already? If they're stuck in the castle? We're mainly trying to make them more comfortable?\"\n\n\"We'll also grant access to the connected roads,\" the Sphinx said. \"Limits must be set for the undead. We can't have them everywhere. Giving them a full castle to roam and some roads to haunt is a significant upgrade from being trapped at the bottom of a well.\"\n\n\"The castle is run by minotaurs?\" Seth asked.\n\n\"Along with an assortment of other monsters,\" the Sphinx said. \"They will flee if they have any sense. Before you lower the chain, you must obtain promises that we will be immune from the wrath of the undead who emerge.\"\n\n\"Should we make them promise not to hurt anyone?\" Seth asked.\n\n\"That would be like telling a shark not to hunt,\" the Sphinx said. \"It would be cruel, and probably not an agreement they could obey. We just need the ability to get away.\"\n\nSeth had no way to verify the truth of what the Sphinx was telling him, but the dead minotaurs supported the assertion that monsters controlled the castle. He didn't like the idea of loosing the undead in an occupied castle, but if the jailers were all monsters and could get away, he supposed they knew the risks associated with locking up powerful creatures. Did undead monsters deserve fewer rights than living monsters?\n\n\"You'll walk me through what to say?\" Seth asked.\n\n\"I'll stay at your side,\" the Sphinx said. \"I'll be able to understand the words you speak, but not what the undead say. We should hurry. If we're discovered, it will not go well for us.\"\n\n\"Lead the way,\" Seth said.\n\nThe Sphinx spread his palm against the corroded iron door. Multiple locks disengaged, the metallic clicks and clacks echoing in the stone hall. Hinges squealed as the Sphinx pushed the door open, making Seth cringe for fear of being heard.\n\nA jumbled flood of voices reached Seth, overlapping and full of desperate longing. The Sphinx looked at Seth. \"Hear them now?\"\n\n\"There are so many,\" Seth said, unable to make much sense of the chaotic gibbering.\n\n\"Hurry,\" the Sphinx said, leading Seth down a short corridor. \"The masters of this castle could discover us at any moment.\" They reached an oval chamber with a circular hole in the center. A thick chain snaked around one side of the room, each link with two holes\u2014one connecting to the previous link, and the other to the next.\n\nSeth peered over the brink of the circular pit. Roughly ten feet in diameter at the top, the bottom was out of view, lost in gloom. Boisterous pleas issued from the blackness, tangled cries for help mingling with inarticulate moans. Several voices mentioned the chain.\n\n\"There is little time,\" the Sphinx prompted. \"Ask for Belrab.\"\n\n\"I need to speak to Belrab,\" Seth said.\n\nThe frenzied appeals ended abruptly, and a slithery voice spoke out of the silence. \"You have my attention.\"\n\n\"I'll free you if you promise to remain in this castle and on the connected roads,\" Seth said, glancing at the Sphinx, who motioned for him to continue. \"And if you will keep the rest of the undead within those boundaries as well.\"\n\n\"I cannot speak for all,\" Belrab said. \"You seem to speak for others.\"\n\n\"Those are my terms,\" Seth said.\n\n\"You are with a cripple,\" Belrab said. \"A weakling shadow charmer putting words in your mouth that he has lost the competence to utter.\"\n\nSeth looked at the Sphinx. \"What is he saying?\" the Sphinx asked.\n\nSeth held up a finger for him to wait.\n\n\"You are young and powerful,\" Belrab said. \"With me at your side, those who seek to rule you will kneel in shame.\"\n\n\"I don't want a partner,\" Seth said. \"And you're going to miss your chance to get out.\"\n\nBelrab responded promptly. \"Very well. If you lower the chain and grant my freedom, I vow to remain in this castle and on the associated roads to rule over the rest of the undead here.\"\n\nThat was what Seth had been waiting to hear. At least with those limits, the undead couldn't just run wild.\n\n\"I need all of the undead to agree to remain in the castle and on the roads if I let them out of this pit,\" Seth said. \"If all do not agree, you'll stay down there.\"\n\nThe Sphinx nodded at Seth as many voices assented.\n\n\"Ask Belrab if all have agreed,\" the Sphinx said.\n\n\"Belrab, have all agreed?\" Seth asked.\n\n\"All save one wraith and two revenants,\" Belrab said.\n\n\"This is the last chance for that wraith and those revenants to agree,\" Seth said.\n\nThree gruesome voices assented to the terms.\n\n\"Was that them, Belrab?\" Seth asked.\n\n\"It was,\" the lich replied.\n\n\"Extra protection for us,\" the Sphinx prompted.\n\n\"You must all pledge to do me no harm,\" Seth said. \"And no harm to the crippled shadow charmer assisting me.\"\n\nHellish voices called out their agreement. Seth could feel their eagerness at the prospect of release.\n\n\"Crippled?\" the Sphinx asked.\n\n\"It's how Belrab described you,\" Seth muttered. Then he raised his voice. \"Belrab, have all agreed?\"\n\n\"All have agreed,\" Belrab responded. \"Send the chain.\"\n\n\"The chain, the chain,\" countless other voices cried.\n\nSeth glanced at the Sphinx. \"How many are down there?\"\n\n\"A lot,\" the Sphinx said. \"Uncounted.\"\n\n\"Before I send the chain, you must all pledge to always obey any commands I give,\" Seth said, \"along with promising to mean no harm to me and to do no harm to me.\"\n\nThere was silence for a moment.\n\n\"Hungry,\" one voice whispered.\n\n\"Cold,\" another murmured.\n\n\"And with these vows you willingly free us?\" Belrab verified.\n\n\"To remain in this castle, and on the connected roads, yes,\" Seth said.\n\n\"Very well,\" Belrab said. Other voices assented. \"We all agree. Send the chain.\"\n\n\"They agree,\" Seth said.\n\n\"Lower the chain,\" the Sphinx said. \"Quickly, I heard a door.\"\n\nSeth picked up the end of the chain, the links thick enough that he dragged it more than carried it to the edge of the well. The thought of the creatures behind all those sinister voices climbing toward him gave him a shiver as he hurled it over the edge.\n\nOnce the chain started falling, it had enough momentum to keep going, and the entire length unspooled downward with an uproarious clatter. After some time, an echoey clang rose from the depths. The chain jerked taut, the near end fastened to the wall.\n\n\"Time to go,\" the Sphinx said, hurrying out of the chamber to the short corridor.\n\nSeth followed, but they both stopped short as an aged man came through the iron door. His eyes went to Seth, then to the Sphinx. \"You,\" he accused.\n\n\"Hello, Agad,\" the Sphinx said. \"You have a containment problem on your hands.\"\n\nThe old man's eyes moved to Seth. \"What have you done?\"\n\nThe distressed sincerity of the question made Seth feel guilty. \"You'll want to get out of the castle,\" Seth said.\n\nThe old man's expression softened to one of concern. And then Seth felt rather than saw the dark presence behind him. The temperature of the air plunged. The old man looked past him and began to haltingly step forward, as if the movement challenged him.\n\n\"Come on,\" the Sphinx said, dodging past the old man and through the corroded iron door. Seth followed.\n\n\"You've been fooled, Seth,\" the old man called, voice strained.\n\n\"Hurry,\" the Sphinx said, running full speed. \"We need to get out of here.\"\n\n\"Who was that guy?\" Seth asked. \"He didn't look like a monster.\"\n\n\"One of the wizards who blocked my abilities,\" the Sphinx said. \"Don't let appearances deceive you. He's a ringleader of the creatures here.\"\n\nThe old wizard had looked devastated. He hadn't tried to attack them. Seth looked back, but the wizard was out of view. What might the wizard have told him if he'd had more time?\n\nThey ran in silence until they reached the dead minotaurs. The barrel awaited in the nearby cell. \"Well done, Seth,\" the Sphinx said. \"Mission accomplished. You first.\"\n\nSeth hopped into the barrel and crouched. Was he being fooled? It was entirely possible. As Ronodin's sure hands helped him out of the barrel, Seth tried to forget the look of grave disappointment on the old wizard's face.\n\nOn the long counter rested bottles and flasks of varying shape and size, some with fluid in them, some with sludge or dry matter, others empty. A cauldron bubbled on a fireplace in the corner, and several other pots and kettles were heating on makeshift burners, expelling pungent vapors. Other diverse containers held sundry ingredients.\n\n\"Kendra, could you pass me some bristle root?\" Tanu asked.\n\nShe grabbed the leather pouch and hurried over to Tanu. He opened the flap and dropped two pinches of what looked vaguely like red pencil shavings into a simmering pan. \"What are you working on there?\" she asked.\n\n\"A new gaseous potion,\" Tanu said. \"By the time we're done, I'll have a stockpile like never before.\"\n\nKendra looked at the potions on the counter. \"A lot of those are courage potions.\"\n\n\"If we face dragons this evening, those could be essential,\" Tanu said.\n\nKendra nodded. It felt good to be helping Tanu brew potions. Since the dragons were supposed to attack tonight, she wanted to stay busy. So far, the night had been quiet.\n\n\"You look tired,\" Calvin said from where he stood on the counter. \"Maybe you should sleep.\"\n\nKendra yawned. \"In a little while,\" she replied. \"I want to help finish this batch.\"\n\nMarat burst into the room, eyes wide. \"We have to go,\" he said urgently.\n\n\"What?\" Kendra asked, turning toward him in alarm.\n\n\"Someone has unleashed the entities from the Blackwell,\" Marat said. \"The attack is from within.\"\n\n\"Is Agad trying to put them back?\" Tanu asked, already gathering items.\n\n\"He did not seem hopeful,\" Marat said. \"It's either the road or the refuge.\"\n\nKendra could hardly believe what she was hearing. She had long known that if the undead emerged from the Blackwell, the whole sanctuary could be destroyed. Was that the threat they were suddenly facing?\n\nShe knew there was a fortified room at the heart of the keep called the refuge. Though the room was secure, there was no way out. But if they fled to the road, a lot would depend on whether the Fair Folk of Terrabelle would grant them sanctuary. If the Fair Folk strictly maintained their neutrality, the road would lead to a dead end. \"Can't we fight?\" Kendra asked.\n\n\"Not the denizens of the Blackwell,\" Marat said. \"We have to leave now or all avenues may be cut off.\"\n\n\"One moment,\" Tanu said as he raced to the courage potions, adding ingredients. \"These were tuned to dragons. We need them to protect against the undead as well. The dose might feel uncomfortable, but it beats getting petrified.\"\n\n\"How long?\" Marat asked.\n\n\"Quick and dirty, I can make them functional in less than a minute,\" Tanu said, shaking a potion with his thumb over the top, then adding a small spoonful of beige powder. He shook the bottle again and handed it to Kendra. \"Drink.\"\n\nShe took it from him, and he immediately began working on another. \"The undead will take over the castle, and the dragons are outside,\" she said.\n\n\"Road or refuge,\" Marat said simply.\n\nKendra had dealt with emergencies before. Seth had usually been with her, and little seemed to daunt him. When there was a battle to be fought, though scary, the prospect could be energizing. This was different. She felt so doomed that she was strangely calm. The only option was to go forward and try to weather what came.\n\nShe raised the bottle to her lips and took a sip. The fluid burned her tongue, but she swallowed anyhow.\n\n\"All of it,\" Tanu said, shaking up another bottle.\n\nKendra guzzled the rest in a few painful swallows. The searing liquid felt like it was stripping away the soft tissue in her mouth and throat. Tears streamed from her eyes, and she coughed after choking down the last mouthful.\n\n\"Sorry,\" Tanu said. \"No time to adjust for taste.\" He upended a bottle of his own and swallowed the contents.\n\nWarmth spread through Kendra's body, starting at her center and radiating out to her limbs. The doomed feeling melted away like a dispelled illusion. They still had a chance. This wasn't just up to fate. There was plenty to be done. \"We need to get moving,\" she said.\n\nTanu grimaced and exhaled noisily. \"That was fierce.\" He started shoving flasks and containers into his pack. \"I'll catch up. These potions could prove important before the night is through.\"\n\n\"I'll come with you, Kendra,\" Calvin called from the countertop.\n\nKendra picked him up and put him in her pocket. \"I'll keep you safe,\" she promised. She also collected her magical bow and her sack of gales.\n\n\"You must survive, Kendra,\" Marat said. \"You are the sole remaining caretaker of Wyrmroost. Road or refuge?\"\n\n\"What do you think?\" Kendra asked.\n\n\"The refuge is more accessible,\" Marat said. \"It also has stores of food and stronger defenses than the road. It is the fort within the fort.\"\n\n\"Let's go,\" Kendra said.\n\n\"I'll be right behind you,\" Tanu promised, still gathering ingredients and bottles.\n\nMarat ran from the room, and Kendra sprinted to follow. They raced down a corridor, turned a corner, then dashed down another hall.\n\n\"We have to cross this courtyard,\" Marat said.\n\n\"Okay,\" Kendra said.\n\n\"Brace yourself,\" Marat advised, opening a door.\n\nThey stepped out onto a walkway that overlooked the courtyard. A bright moon shed silver light onto the scene. The stone steps leading down to the courtyard were empty, but the courtyard was not.\n\nKendra had been feeling bold and hopeful in the corridor. Suddenly it was hard to breathe. Her throat was constricting, and her heart felt cold.\n\nShadowy figures prowled the courtyard below, shrouded in pockets of darkness. Indecipherable whispers burbled in her mind. The two slowest figures drew her gaze. One limped, somewhat dragging the weaker leg. The other moved more fluidly, but so slowly that Kendra could count to five between each footfall.\n\nMarat gave her a light slap on the cheek. \"Revenants,\" he said. \"Accompanied by a few wraiths. We move now or never. Others are spreading through the keep.\"\n\nThe fear Kendra felt was different from the paralysis inflicted by the dragons. This fear made her insides feel like they were turning to ice. \"Yes,\" she said with effort. Even with the help of the courage potion, movement required heavy concentration.\n\nMarat led the way down the stairs. As Kendra's eyes adjusted to the moonlight, she saw that the limping revenant was approaching a dwarf. The little fellow looked up at the oncoming threat, arms slack at his sides, his short sword loosely gripped in one hand. The dwarf was clearly immobilized. As the revenant reached toward him, Kendra held out her bow and pulled the string to her cheek.\n\n\"One,\" she murmured, and an arrow appeared.\n\nShe released, and the arrow streaked across the courtyard to lodge in the forehead of the revenant. The creature did not act bothered, and the revenant's skeletal hand covered the face of the dwarf. After violent convulsions, the small victim fell to the ground, hair, beard, and skin as white as snow.\n\n\"No conventional weapons,\" Marat scolded. \"You're trying to kill what is not alive. Hurry.\"\n\nKendra followed Marat to the bottom of the stairs, and they started across the courtyard. The viscosity of the air seemed to increase, as if she were pushing through water. And the temperature was plunging. Gritting her teeth, Kendra pressed forward.\n\n\"Do not stop,\" Marat advised. \"Stay with me.\"\n\n\"You can do it,\" Calvin said.\n\n\"It's cold,\" Kendra said.\n\n\"Those revenants are pointing at you,\" Calvin warned. \"Wraiths are coming.\"\n\nKendra didn't want to look. Marat was nearing a door. She kept her legs churning forward. Glancing back, she found the ominous forms of three wraiths closing on her, moving much faster than the revenants. And faster than her.\n\nKendra tried to speed up. Her muscles and joints refused to respond as desired. At least she was grinding forward. But the wraiths were gaining. Would it be like her dreams, where she couldn't outrun the monsters no matter how hard she tried?\n\nWith a snarl, the long body of a serpentine dragon coiled around her. The leonine head of the dragon roared at the wraiths, and they stopped advancing. Marat had reverted to his dragon shape\u2014Camarat.\n\n\"Go,\" Camarat said, lifting a segment of his long body so she could duck underneath. \"I can hold off the wraiths but not the revenants.\"\n\n\"You come too,\" Kendra said as she stumbled to the door.\n\n\"I will,\" Camarat said. \"You first.\"\n\nKendra pushed through the door into a corridor, and most of the iciness left her body. Marat joined her, with no sign he had just been a dragon.\n\n\"Thanks,\" Kendra said.\n\n\"Hurry to the refuge,\" Marat replied. \"Those were not the only undead on the prowl. Not nearly.\"\n\nThe hallways seemed foreign. Was it the darkness? She had not roamed much of the keep at night. Urgent whispers intruded on her thoughts, heard more with her mind than her ears. After passing through a gloomy room into another hall, Kendra saw Knox, Tess, and the satyrs up ahead. They were standing still, facing away from her. The cloak of innocence glowed with a silvery sheen.\n\n\"Trouble,\" Marat said, breaking into a full sprint.\n\nKendra was able to stay with him at first. But as she reached her cousins, she started to slow. The air grew chilly again.\n\nBeyond her cousins and the satyrs, coming toward them, was a gaunt woman dressed in the tattered remnants of what must once have been a beautiful gown. Her eyes were dark gray, without irises or pupils, and irregular open sores marred her skin. Her head tilted at an unnatural angle, and she was slowly jumping forward, feet bound together.\n\n\"Revenant,\" Marat said. \"A strong one.\"\n\nThe revenant was not coming quickly but scooting a bit closer with each jump. Kendra focused on her cousins. \"Knox, Tess, can you hear me?\" Neither looked her way or blinked. They seemed insensible. Kendra turned to Marat. \"Can I stop the revenant?\"\n\n\"Revenants are under enchantments,\" Marat said. \"I don't know how to break this one.\"\n\n\"Here,\" Tanu said, approaching from behind. He held up a bottle to Knox's lips and poured fluid into his mouth. \"Drink.\"\n\n\"You found us,\" Kendra said.\n\n\"Barely,\" Tanu said. \"It's like a festival night out there.\"\n\nKnox was drinking the potion, then spat some up.\n\n\"Swallow it all,\" Tanu demanded.\n\n\"It's like lava,\" Knox said.\n\n\"You can move now,\" Tanu said. \"And speak. So drink.\"\n\nMarat ran forward, dodging around the revenant, and soon passed out of view. With his eyes squeezed shut, Knox finished the little bottle. Tanu brought a potion to Tess's lips. The revenant drew nearer, her expression bland, her tongue shriveled and dry behind her remaining teeth.\n\nKendra wrenched her gaze from the revenant. \"We have to move,\" she said. \"What about the satyrs?\"\n\n\"These potions are for humans,\" Tanu said.\n\n\"Gross!\" Tess exclaimed, potion leaking down her chin. \"What is that?\"\n\n\"It'll let you run from the monsters,\" Tanu said.\n\n\"You can do it!\" Calvin cheered.\n\nCoughing and sputtering, Tess finished the potion.\n\nMarat returned, knifing past the revenant. \"The refuge has been sealed,\" Marat reported. \"Some of our people must have made it inside. The antechamber is swarming with the undead. We have no chance of getting in.\"\n\n\"The road?\" Kendra asked.\n\nMarat gave a nod.\n\nTanu hoisted Newel over one broad shoulder and Doren over the other. \"Show us the way.\"\n\n\"That lady looks like a corpse,\" Knox said, staring at the revenant.\n\n\"Welcome to our night,\" Kendra said, taking Tess by the hand. \"Stay together.\"\n\nMarat led them back toward the courtyard. \"Hopefully we'll find fewer undead upstairs.\" He opened the door to a stairway and led them up.\n\nAs they climbed, the satyrs began to squirm on Tanu's shoulders.\n\n\"Was it just me,\" Doren asked, \"or was she jumping?\"\n\n\"I was holding still on purpose,\" Newel said. \"Weighing my options.\"\n\nKendra rolled her eyes.\n\nTanu set the satyrs down.\n\nKendra could not help noticing the narrowness of the stairway, and she immediately imagined their peril if revenants pinned them in from above and below. She was relieved when they found no undead in the hall at the top of the stairs.\n\n\"What about Grandma and Grandpa?\" Kendra asked.\n\n\"I sent them and some others to the refuge before I went to fetch you and Tanu,\" Marat said. \"I didn't see them when I checked outside the refuge, so I assume they're inside.\"\n\n\"Then they're trapped,\" Kendra said.\n\n\"Cornered, for now,\" Marat said. \"But much safer than we are.\"\n\n\"What about the griffins?\" Kendra asked. \"And the horses?\"\n\n\"Patton went to free the mounts,\" Marat said.\n\n\"Could we fly away?\" Kendra asked.\n\n\"I expect many dragons would greet us in the sky,\" Marat said. \"The road is a better option if we can make it.\"\n\n\"Is there a chance your courage potion could work on us?\" Doren asked as they ran. \"I don't like becoming paralyzed.\"\n\n\"Give it a try,\" Tanu said, handing over a bottle.\n\nDoren took a sip and immediately began to cough wildly. With tears in his eyes, he passed the bottle back to Tanu. \"Pretty sure it's deadly to satyrs,\" he said.\n\n\"It felt gross and hot to me too,\" Tess said.\n\nMarat led them out onto a walkway atop one of the walls. Kendra felt her muscles seizing up again, and she saw dark entities patrolling the nearest courtyard. Marat stopped, one hand going to his temple.\n\n\"My brother Agad is reaching out to my mind,\" Marat said. \"The road is already taken by wraiths and revenants. We will not find safety there.\"\n\n\"What options do we have left?\" Tanu asked.\n\n\"We can try a little-used side door,\" Marat said. \"I'd rather not go over the wall in dragon form and attract attention. There will be no protection besides the cloak of innocence once we pass outside.\"\n\n\"Can it hold off Celebrant?\" Kendra asked.\n\n\"Possibly,\" Marat said. \"All items have limits. But we lack other options. The inhabitants of the Blackwell are claiming the keep and the road. At least dragons would provide a clean death.\"\n\n\"Are we going to die?\" Tess asked.\n\n\"Not if we hide behind your cloak,\" Doren said. \"Are you clear about who is under your protection?\"\n\n\"Kendra, Knox, the potion guy, you goat-men, the dragon person, and the little guy in Kendra's pocket,\" Tess said.\n\n\"There are wraiths ahead on the wall,\" Tanu warned.\n\n\"Then we must abandon subtlety,\" Marat said, his body elongating into a serpentine dragon. \"Take hold of me.\"\n\nKendra leaped onto the back of the dragon, straddling him and gripping as best she could. Tanu, Knox, and Tess climbed on as well. The satyrs stood petrified.\n\nCamarat shot into the air, body fluttering like a ribbon. Kendra hung on tightly. The dragon used several sets of his many legs to grab Newel and Doren and then flew low and quick, soaring over a building, his tale dislodging shingles, then down into a courtyard, where he skidded to a stop near a door in the wall. Kendra noticed half a dozen undead in the courtyard. A pair of emaciated men, one with a scraggly beard, the other wearing a helmet and dented breastplate, stood in front of the door.\n\n\"Revenants are barring the way out,\" Marat said after reverting to human shape.\n\n\"And wraiths are coming our way,\" Tanu said.\n\nFighting against the frozen feeling slowing her muscles, Kendra raised her bow. \"Maybe lots of arrows are better than one,\" Kendra said, hauling the bowstring to her cheek and aiming at the revenants blocking the door. \"Ninety.\"\n\nShe released the string, and ninety arrows sprang from her bow and turned the two revenants into pincushions, piercing flesh and armor. Many near misses thunked into the door as well. Neither revenant looked particularly bothered by the dozens of protruding arrows.\n\n\"No!\" Marat cried.\n\n\"What?\" Kendra asked.\n\n\"The cloak,\" Marat said, pointing.\n\nKendra saw that the cloak of innocence had lost its silvery glow. Her stomach dropped.\n\n\"None in the group can attack anything while under the protection of the cloak,\" Marat said, \"or the protection is lost to all.\"\n\nAs several wraiths, dark and almost translucent in the moonlight, drew nearer from across the courtyard, Kendra stared at Marat, her understanding catching up to his words.\n\nThey needed to leave the keep.\n\nShe had shot arrows at the revenants.\n\nArrows that had done no good.\n\nTheir only hope of survival when they left the keep was the cloak of innocence.\n\nBy shooting the revenants, she had negated that protection.\n\nWithout accomplishing anything, she had just ensured that she and all of those with her would die.\n\n\"I didn't understand,\" Kendra said numbly. \"Couldn't you have stopped me?\"\n\n\"I didn't make the connection until too late,\" Marat said. \"My first reaction was regret that the arrows would be futile. Only once you let them fly did I realize that, futile or not, it was an attack. None under the protection of the cloak can try to harm another without breaking the spell.\"\n\n\"We can all go gaseous and try to escape that way,\" Tanu said. \"At least the mortals.\"\n\nAfter glancing at Newel and Doren, who stood petrified, transfixed by the revenants, Kendra turned to Marat. \"Maybe you can fly the satyrs away to safety.\"\n\n\"If Celebrant is as ready as I expect, he will track and destroy you, gaseous or not,\" Marat said. \"And I cannot evade a host of dragons.\"\n\nA flaming figure burst from a door on the far side of the courtyard. Sheathed from head to foot in white fire, the burning humanoid raced toward a wraith and cut him in half at the waist with a fiery blade. The flaming newcomer continued toward Kendra and her allies.\n\n\"My brother,\" Marat said, directing the others to step aside.\n\nThe blazing wizard rushed the revenants blocking the door through the wall. They offered little resistance as he struck down one and then the other with consecutive swipes. After the revenants collapsed, Kendra felt the icy chill inside of her thaw, and the satyrs could move again.\n\nThe white flames fluttered out and Agad turned to face Kendra and the others, dropping to one knee. His hair and beard were singed and his robe was charred in places. One arm looked withered down to almost nothing, and part of his nose was missing.\n\nMarat ran to his brother and helped him up. The others gathered around them.\n\n\"I got there too late,\" Agad panted, his bloodshot eyes weary. \"There were too many of them, and Belrab was too powerful to challenge with so much support behind him. I tried to force them back down the well, and it nearly destroyed me.\" His eyes went to Kendra. \"It was Seth, assisted by the Sphinx.\"\n\n\"Seth was here?\" Kendra asked, hardly able to believe what she was hearing. \"With the Sphinx?\"\n\nAgad gave a nod. \"I saw them, Kendra. The Sphinx must have come through the barrel and hidden before Mendigo escaped with Seth. Mendigo would have brought him through while the barrel was missing. The Sphinx hid here until he killed the minotaurs, brought Seth through, and finally sent Seth back.\"\n\n\"Wraiths,\" Tanu warned.\n\nStaggering, Agad waved an arm, and a semicircular wall of white flame sprang up to block the wraiths from approaching the door. Marat supported his sagging brother.\n\n\"You're exerting yourself beyond your limits,\" Marat said.\n\n\"As is appropriate,\" Agad said. \"Ultimately, this sanctuary is my responsibility. If it falls, I should fall with it.\"\n\n\"We will need you in the greater conflict,\" Marat said.\n\nAgad frowned. \"I can't be everywhere, do everything. Instead I am doing what I can.\"\n\n\"Seth did this?\" Kendra asked, on the edge of hysteria, not wanting to believe it.\n\n\"He did not know me,\" Agad said. \"I doubt he understood what he was doing.\"\n\nKendra felt sick. Her worst fears were already playing out. Not only did their enemies have her brother, but he appeared to be siding with them.\n\n\"I suspect only Seth, with his caretaker status, could have opened the roads to the undead,\" Agad said. \"Who knows what fiction they told him to cajole his assistance? Seth did not fail us, Kendra. I failed to rescue him in time.\"\n\n\"You're sure Seth is gone?\" Kendra asked.\n\n\"I'm sure,\" Agad said. \"He and the Sphinx fled purposefully.\"\n\n\"We don't know where?\" Kendra asked. Even with danger surrounding them, more than anything, she wanted to find her brother.\n\n\"Only how,\" Agad said. \"Through the barrel, undoubtedly.\" The haggard wizard looked tiredly at the others. \"We have our own troubles now. We cannot stay here with the full wrath of the Blackwell unleashed. And unless we are extremely fortunate, we will not survive long beyond these walls.\"\n\n\"The cloak of innocence is undone,\" Marat said.\n\n\"I saw,\" Agad said. \"That was when I knew I must rush to your aid.\"\n\n\"Is there any chance left of getting away?\" Kendra asked.\n\n\"Not if the dragons are ready for us,\" Agad said. \"Which I expect they are.\"\n\n\"Should we try gaseous potions?\" Tanu asked.\n\n\"Keep them handy,\" Agad said. \"I would not use them yet. Flee as a gas and the dragons will destroy you with their breath weapons. We are defeated and trapped. The best chance we have left is to negotiate a surrender.\"\n\n\"We're giving up?\" Kendra asked.\n\n\"Not on the war,\" Agad said. \"But on Wyrmroost, perhaps. Without Blackwell Keep, we have lost. Simple as that.\"\n\nThe wall of white fire was beginning to dwindle. Wraiths were visible behind it, hesitant but near, their numbers growing.\n\n\"Maybe we can still sneak away,\" Newel said.\n\n\"I will cloak us in the best distracter spell I can muster,\" Agad said. \"Through the door on my count. Three, two, one.\"\n\nTess looked terrified. Kendra took her hand. Knox didn't look so well either. When he caught Kendra's gaze, he tried to smile. \"Are we going to be dragon food?\"\n\n\"I hope not,\" Kendra said.\n\nMarat held the door open as Agad and the satyrs exited. Kendra, Tanu, Knox, and Tess followed. Marat brought up the rear.\n\nKendra beheld an empty field, bordered on one side by the gray wall of the keep and on the others by trees, the stillness eerie after revenants and wraiths and fire. A soft breeze ruffled the tall grass. How did this quiet scenery exist so near the perils they had faced?\n\nFor a moment she felt hesitant relief.\n\nBut only for a moment.\n\nAs the door to Blackwell Keep shut, the colossal form of Celebrant landed in the field in front of Kendra, platinum scales glowing in the moonlight. Other dragons alighted all around them, quickly filling the field and cutting off any avenue of escape. The satyrs became immobilized with fear. The courage potions left Tanu, Knox, and Tess able to move.\n\n\"Pathetic concealment spell,\" Celebrant gloated in a voice like a boisterous choir of men speaking in unison. \"Look who we have here. It seems we have caught the crab outside of its shell.\"\n\nAgad stepped forward, stumbling a bit. \"We have come to negotiate surrender.\"\n\nCelebrant laughed darkly. \"You have come to negotiate, have you? I never received an invitation. It appears to me you are fleeing your fallen stronghold.\"\n\n\"Nevertheless, we are prepared to negotiate,\" Agad maintained.\n\n\"I must admit I enjoy this talk of surrender from the mighty Agad,\" Celebrant said. \"Are you still pleased to have traded your claws and fangs for robes?\"\n\n\"I stand by my decisions,\" Agad said. \"What are your terms?\"\n\n\"My first demand is that before negotiations begin, you must die,\" Celebrant said.\n\n\"Very well,\" Agad said, straightening. \"We will decide this by combat. You against me.\"\n\nKendra's eyes strayed to the damaged spot on Celebrant's chest where Madrigus had injured him in combat. Would the injury give Agad a chance?\n\n\"You misunderstand,\" Celebrant said. \"I was not proposing combat. This is a war. You have already lost this battle. You have no privileges. I need grant you no quarter.\"\n\n\"You are asking for my life,\" Agad said.\n\n\"Asking, telling, taking,\" Celebrant said. \"Take your pick.\"\n\n\"If I refuse?\" Agad said.\n\n\"No negotiations with the others,\" Celebrant said. \"And I take your life just the same.\"\n\n\"No,\" Marat said.\n\nAgad raised his good hand to halt his brother. \"I'm injured, Camarat. Deeply. Perhaps fatally already. I might have mustered one last grand attack. But this may accomplish more good.\"\n\n\"I grow impatient,\" Celebrant said.\n\n\"Very well,\" Agad said, striding unsteadily toward Celebrant. He fell to his knees before the dragon.\n\n\"After all of these years, it is pleasant to see you submit,\" Celebrant said. Jaws flexing open, his head snaked toward the old wizard.\n\nKendra turned her head and covered Tess's eyes. When she looked back, Celebrant jerked his head up, swallowing. Agad was gone.\n\n\"Subpar for a human,\" Celebrant said. \"Almost no decent meat on those old bones. Who speaks for you now? The caretaker, I presume?\"\n\n\"That's me,\" Kendra said, stepping forward.\n\n\"There you are,\" Celebrant said, eyes alight with triumph. \"For one so young, you have been a nuisance, but you are finally out of tricks, allies, and hiding places. Would you care to hear my terms?\"\n\n\"All right,\" Kendra said, trying to bury her grief for the moment. The others were counting on her. She glanced over at Knox and Tess. Hopefully the terms would allow some of them to survive.\n\n\"I invite you to formally surrender and trust my mercy,\" Celebrant said. \"Give me the caretaker's medallion and I will consider whether to let you and your comrades depart in peace.\"\n\nKendra looked back at Marat. \"Surrender and the sanctuary has officially fallen,\" he said.\n\n\"Or of course I could devour all of you now and claim the victory I have already won,\" Celebrant said.\n\nKendra wondered if there might be a reason he needed her official surrender. Would some barriers remain in place without it?\n\n\"I'll surrender if you promise to spare us,\" Kendra said.\n\n\"Were you absent for this discussion with Agad?\" Celebrant asked. \"You are in no position to haggle. I do not need your surrender. Make your choice or I destroy all of you this instant.\"\n\n\"Fine,\" Kendra said, angry as she took off the caretaker's medallion. She wished for the sword Vasilis. For a moment she considered her sack of gales, but she knew it was only effective against airborne dragons. \"I surrender.\" She threw the medallion toward the feet of the towering dragon.\n\n\"We have reached our destined ending,\" Celebrant said. \"I accept your surrender. And now all of you must die.\"\n\n\"What about mercy?\" Kendra cried, thinking first of Tess.\n\n\"Mercy is for those who have not insulted me,\" Celebrant said. \"Mercy is for those who surrender before they have been utterly abased. You were foolish to ever oppose me. I will delight in providing the exact ending you deserve.\"\n\n\"Enough!\" boomed a voice like thunder.\n\nFrom out of the sky, a man descended, landing beside Kendra. Clad in a dark robe, he had a short white beard and white hair. In one hand he carried a staff. Kendra had never seen him before.\n\nSnarling, Celebrant lunged at the newcomer but was repelled with a flash of red light when his face struck an invisible barrier. Celebrant opened his mouth and released a stream of white energy, but it splashed harmlessly against the barrier, brightening the night with a harsh glare.\n\nAt last Celebrant relented, glaring at the stranger. \"Who are you and where have you come from?\" the Dragon King asked.\n\n\"Had you simply shown mercy, you might never have met me,\" the newcomer said.\n\n\"Clearly you are a wizard, and not one that I know,\" Celebrant said.\n\n\"Right on the first count,\" the wizard said. \"Half right on the second.\"\n\nCelebrant's eyes widened. \"No.\"\n\n\"Yes,\" the wizard said.\n\n\"Dromadus,\" Celebrant said.\n\n\"Not anymore,\" the wizard said. \"I am Andromadus from now on.\"\n\n\"The art of becoming a wizard is lost,\" Celebrant declared.\n\n\"Not to me,\" Andromadus said. \"Archadius, the first of this order, taught me long ago. I never expected to apply the knowledge.\"\n\n\"You were already an embarrassment and an outcast,\" Celebrant said. \"You once wore my crown. Why would you further dishonor yourself and our kind?\"\n\n\"You are the embarrassment,\" Andromadus said sternly, in a voice that shook the ground. \"What honor is there in killing a young human girl who has surrendered? Such an execution is treatment for a worthy adversary. Do you consider her an enemy even now? A worthy opponent? Your better or your equal? Do you take pride in her defeat?\"\n\nThe questions elicited a murmur from some of the dragons.\n\n\"She is an insect to be squashed,\" Celebrant said, his head bumping up against the unseen barrier again.\n\n\"If so, you would have squashed her,\" Andromadus said. \"Instead, you had to drive her from Blackwell Keep with the aid of humans and unicorns. And then you solicit her surrender. Never have I begged an insect to submit.\"\n\nThe assertion was met with silence. Kendra noticed the other dragons staring at Celebrant intently. His status had been placed in question.\n\n\"I wanted to ensure no problems with the magical barriers of Wyrmroost,\" Celebrant said. \"The treaty is strong. With her surrender, our freedom is no longer in question. Dromadus, you have chosen an odd way to die.\"\n\n\"I will not die today,\" Andromadus said. \"Neither will Kendra or her friends. You have forgotten your true nature, Celebrant. Dragons are not humans.\"\n\n\"Dragons will rule over humans,\" Celebrant vowed.\n\n\"A real dragon would not care to rule over humans!\" Andromadus shouted. \"A real dragon would not gather his people to a castle like some human lordling. A dragon hunts. A dragon hoards. A dragon assumes authority.\"\n\n\"Our authority was taken!\" Celebrant cried.\n\n\"We lost our authority when we organized after human patterns and fought a war like mortals,\" Andromadus said. \"We lost that war. And, despite our defeat, those humans gave us space to live like dragons. We were in their power. They could have exterminated us.\"\n\n\"They will regret their failure,\" Celebrant said. \"We shall exterminate them.\"\n\n\"Where is the glory in that?\" Andromadus challenged. \"Some humans helped defeat us, alongside several of our kind who became wizards. But most humans do not even know there was a fight. You wanting to rule humans, to defeat humans, is like a human wanting to rule rabbits or defeat squirrels. It is not dignified. It runs against our very nature. It is not dragonlike.\"\n\n\"You are not dragonlike!\" Celebrant roared.\n\n\"I am true to my nature,\" Andromadus said. \"I side with the humans on this matter. It is time for my form to match my sympathies. This war is bad for dragons, bad for humans, and bad for the world. I now stand against you.\"\n\n\"The greatest coward of all dragons opposes me,\" Celebrant scoffed. \"The Dragon King who resigned!\"\n\n\"There is a kind of courage that looks to fulfill the expectations of others,\" Andromadus said. \"Some might call it duty. There is another kind that insists on being true to oneself.\"\n\n\"You cannot stand against us,\" Celebrant said harshly.\n\n\"I do not wish to fight you,\" Andromadus said. \"I wish for this war to end. I am still a friend to all dragons. I implore you to turn from this ludicrous path.\"\n\n\"You fear our victory,\" Celebrant exclaimed. \"You worry that after our triumph I would root out the decrepit old king and openly put him to shame.\"\n\n\"You could have attempted that long ago,\" Andromadus said.\n\n\"You have disgraced yourself more than I ever could,\" Celebrant said. \"You have become mortal.\"\n\n\"And you have disgraced the crown,\" Andromadus said. \"You are leading our kind toward folly and ruin.\"\n\n\"Not our kind anymore,\" Celebrant said. \"My kind. And my kind does not want to live like docile pets in a cage. My kind wants the dominance we deserve. This is the dawn of a new and endless age of dragons. Attack!\"\n\nThe dragons all around the field charged forward, slamming against the unseen barrier Andromadus had raised. The wizard held both hands high, trembling with effort. Judging from the way the dragons pressed against the barrier, Kendra could tell it was curved like a dome.\n\n\"You must depart,\" Andromadus said to her. \"Have I permission to send you?\"\n\n\"Where?\" Kendra asked.\n\n\"Away,\" Andromadus said. \"I will rejoin you when I can. I do not expect you will get a better offer.\"\n\n\"You're Dromadus?\" Tess asked.\n\n\"So I was,\" Andromadus said.\n\n\"You came!\" Tess cheered.\n\n\"So I did,\" Andromadus said with a small smile.\n\n\"Send us,\" Kendra said.\n\n\"As you wish,\" Andromadus said.\n\nKendra felt her body lurch. Everything temporarily went black, and then she dropped to her knees on a marble floor in a huge room decorated with absurdly oversized furniture. Knox, Tess, Tanu, Marat, and the satyrs were beside her, and directly before her loomed the giant Thronis.\n\nSeated alone at a stone table, Seth tried to spread butter across a hefty slice of wheat bread, but the room was too cold for the butter to smear well. Instead he did his best to shave the butter down to slivers and space them around the bread. He also had a broad bowl of various shellfish cooked in a tomato-based sauce, and a carafe of the kind of milk that allowed him to see magical creatures.\n\nRonodin and the Sphinx had started the meal with him, but they promptly departed together with a pair of wraiths toting the barrel, leaving Seth alone. The bread and butter tasted satisfying, and he generally liked most of the shellfish, though the black ones were too rubbery. Absently pushing the remaining tidbits around his bowl, Seth wondered what his favorite food might be. He could remember many different kinds of food, but he had no memory of how they tasted, or which he preferred. He had an abstract impression that many found pie delicious but could match it to no specific flavor.\n\nWater dripped from the ceiling in one corner of the room. He could sense the undead wandering nearby.\n\nMostly he felt haunted by the wizard at the Blackwell and his warning.\n\n\"You've been fooled, Seth.\"\n\nThough distressed by the problem of the emerging undead, the old man had seemed genuine. Seth had been told the castle was controlled by minotaurs and monsters, but the old wizard had seemed very human, and he had not attacked Seth or the Sphinx for what they had done.\n\nAnd the old man had known his name\u2014at least, the same name everyone else had been using.\n\nSeth frowned. How might he have been fooled?\n\nWhat if he had done something terrible? What if unleashing the undead from the well had brought horrors to a peaceful castle? He had seen dead minotaurs, but maybe monsters only worked down in the dungeon. He had granted the undead access to roads. What if those roads led to villages? Was unleashing the undead against anyone ever a fair option? Even if the only victims were monsters?\n\nThe old man hadn't looked like a monster. Could he have been the evil wizard who ruled the monsters? The Sphinx had said the old guy was one of the wizards who had stripped his shadow-charming powers. But if shadow charmers freed the undead to overrun inhabited castles, maybe the Sphinx had deserved some punishment.\n\nSeth had no proof the wizard was honest. Did the wizard really know him? Maybe the wizard knew that Seth had lost his memory and had made the comment to confuse him. But the wizard could as easily have been a dear friend as a dangerous enemy.\n\nFor all Seth knew, Ronodin could also be an enemy. Seth still had little context to build from. Ronodin had brought him to the Under Realm and gotten him indentured to the Underking. Seth was having to perform tasks just to stay alive. But Ronodin would argue it was for his good so he could relearn his abilities. Could anyone be trusted? What if it was him against everyone?\n\nRonodin was vague about so much. He refused to tell Seth exactly how he had lost his memory. He would reveal nothing about where this Under Realm was located. And he avoided specifics about Seth's history. That seemed suspect, though Ronodin assured him he would learn all in time.\n\nIt seemed reasonable to Seth that he may have once belonged in this Under Realm. After all, he could speak with the undead and had other dark powers. Might there be light in him as well? Seth knew he didn't want to cause harm to anyone. Even though he had dark abilities, he wanted to do good with them, not evil. Had he always felt that way? Even back before he could remember?\n\nHowever dark his history might be, did it have to define him? With his memory wiped clean, how much did it matter who he had been before? What if it turned out he didn't like who he had been? He wanted to know his history, but did he have to be a slave to it? Did he have to learn from somebody who kept secrets and acted like he owned him? In principle, Seth didn't mind freeing some of the undead, but not if people would get hurt.\n\nSeth ate more shellfish and munched on bread. The meal seemed like another hint that he was close to the ocean. What would he find if he got to the surface?\n\nHe thought about the girl Kendra, the first person he had met, who claimed to be his sister. She had run after him when the puppet Mendigo carried him away. Her desperation had seemed sincere. He wished he knew more about her. Was she really related to him? Or was it just a clever way to win over a person who had lost his memory? Had she been working with his enemies? Or was she on his side?\n\nThese questions had answers.\n\nHe needed them.\n\nRonodin had left him alone in an underground domain full of the undead. And Seth could communicate with the undead.\n\nHe pushed back his chair from the table, took one last sip of milk, dabbed his lips with the linen napkin, picked up the little lantern off the table, and walked into a neighboring chamber. Two wraiths stood there. Left to themselves, the wraiths did not do much besides sedately lurk. They were not like the wraiths in the well who had seemed so desperate for freedom, perhaps because these wraiths were already free.\n\n\"We need to talk,\" Seth said, pointing to one of the wraiths.\n\nAs you wish, the wraith responded, the chilling words flowing directly into Seth's mind.\n\n\"Can you speak out loud?\" Seth asked.\n\n\"If desired,\" the wraith replied in a slithery whisper.\n\n\"I like talking better,\" Seth said. \"Who am I?\"\n\n\"You are living,\" the wraith said.\n\n\"I mean specifically,\" Seth said.\n\nThe other wraith moved away. Seth saw no reason to stop it.\n\n\"You are untouchable,\" the wraith said.\n\nThe other wraith left the room. Seth wondered if it was deliberately avoiding the conversation.\n\n\"You can't touch me?\" Seth asked. \"Do you want to kill me?\"\n\n\"I serve the Underking,\" the wraith replied.\n\n\"Do you know my name?\" Seth asked.\n\n\"You are Seth.\"\n\n\"What else do you know about me?\" Seth asked.\n\n\"You are living. You are untouchable.\"\n\n\"Where are we?\" Seth asked.\n\n\"We are in the depths,\" the wraith whispered.\n\n\"If I went to the surface, where would we be?\" Seth asked.\n\n\"At the surface.\"\n\nSeth rolled his eyes. \"Where in the world would we be at the surface?\"\n\n\"The isle,\" the wraith said. \"But there are many ways.\"\n\n\"Many ways to the isle?\" Seth asked.\n\n\"Many ways to many places,\" the wraith said.\n\nSeth sighed. It was hard to get a straight answer. \"Does anyone here know more about who I am?\"\n\n\"The prisoner knows.\"\n\n\"What prisoner?\"\n\n\"You must not meet the prisoner,\" the wraith said.\n\n\"Why not?\"\n\n\"Orders from Ronodin.\"\n\n\"Is Ronodin in charge of you?\"\n\n\"I serve the Underking,\" the wraith said.\n\n\"But you follow orders from Ronodin?\"\n\n\"I heed orders authorized by the Underking.\"\n\n\"Do you follow orders from me?\" Seth asked.\n\n\"You are untouchable. I hear you. You hear me. I communicate.\"\n\n\"Where is the prison?\" Seth asked.\n\n\"You are to be kept from the prisoner,\" the wraith said.\n\n\"Orders from Ronodin?\"\n\n\"Orders,\" the wraith said.\n\n\"Okay,\" Seth said. \"Thanks.\"\n\nHe walked back to the room where he had eaten. The bread, seafood dish, and milk remained where he had left them. He knew his way to his quarters and went in that direction, but paused at his door and then kept going. He could sense more wraiths up ahead. Maybe he could get more out of a new wraith.\n\nSeth walked up to the next wraith he found. \"Will you take me to the prison?\"\n\nThe words of the wraith reached his mind. You are not to go there.\n\n\"Speak out loud.\" Seth wondered if he could use some cleverness to get an answer. \"What direction should I avoid to keep away from the prison?\"\n\n\"The wrong direction,\" the wraith said.\n\n\"Which way is that?\"\n\n\"The wrong way.\"\n\nSeth folded his arms. \"Who could tell me about the prison?\"\n\n\"Ezabar,\" the wraith said.\n\n\"Can you take me to Ezabar?\"\n\nThe wraith gave a curt bow and started walking. Seth followed, wondering if he should be worried or excited to speak with Ezabar. He had not yet dared to wander far unescorted by Ronodin, and he felt curious to see more of the Under Realm. Whether by slope or by stairs, the way descended downward, some portions of the journey like a cave, others enclosed by crude masonry. Seth frequently sensed undead presences, several of them unfamiliar and more disquieting than he had felt before, but he saw none of them.\n\nThe air grew colder, and the little lantern seemed to have more difficulty pushing back the darkness. The wraith stopped outside an elaborately carved doorway with bones fused into the stonework.\n\n\"Is this where I go?\" Seth asked.\n\n\"Ezabar,\" the wraith said.\n\n\"Are you coming in with me?\" Seth asked.\n\n\"No,\" the wraith said.\n\n\"Will you wait to take me back up?\" Seth asked.\n\n\"Yes,\" the wraith said.\n\nSeth took a step toward the door and then stopped. \"Is Ezabar dangerous?\"\n\n\"Yes.\"\n\nSeth walked up to the door, ancient wood bound in iron. If there was a prisoner he was not supposed to meet, Seth wanted to meet him. Or her. Or it. If talking to Ezabar was a loophole that could help the encounter happen, Seth figured it was worth the risk. He thought about knocking, then hauled the door open.\n\nColdness flowed from the open portal. In the tomblike room beyond, the lantern light penetrated the darkness to reveal a sinister figure at rest on a large marble slab. The apparent corpse wore a long chain-mail hauberk under a white tunic, hands folded on the hilt of the broadsword atop his chest. His faintly glowing skin had a blue tint and was translucent enough that Seth could make out the bones inside his hands and behind his face.\n\n\"Why do you disturb me?\" the supine figure asked in a deep, hollow voice.\n\n\"I'm looking for Ezabar,\" Seth said.\n\n\"You have found him,\" the figure said, rising from the slab to stand with his feet hovering a few inches above the ground, the sword clutched in one semitransparent hand. \"State your business.\"\n\nSeth worried he had disturbed a being he should have left alone. Goosebumps now textured both of his arms. He tried to act confident. \"I need to visit the prison.\"\n\n\"You are alive,\" Ezabar said.\n\n\"Don't hold that against me,\" Seth said. \"I'm doing my best with what I've got.\"\n\n\"Tell me, shadow charmer, what business have you in my prison?\" Ezabar asked.\n\n\"Your prison?\" Seth checked.\n\nEzabar rested the blade of his sword on one armored shoulder. \"I oversee all matters here.\"\n\n\"Are we at the prison?\" Seth asked.\n\n\"Nearly,\" Ezabar said.\n\n\"Could you give me a tour? I'm new.\"\n\n\"The living who enter my prison seldom return,\" Ezabar said.\n\n\"Are some in your prison living?\"\n\n\"All.\"\n\n\"No undead?\"\n\n\"None.\"\n\n\"Why don't I just go have a look?\"\n\n\"I must destroy any who enter the prison without permission,\" Ezabar said.\n\n\"But I'm untouchable,\" Seth said.\n\n\"No living being is untouchable here,\" Ezabar said.\n\n\"That's what I get for listening to wraiths,\" Seth muttered.\n\n\"Perhaps the wraiths were asked not to harm you,\" Ezabar allowed.\n\n\"Why not give me permission?\" Seth asked. \"Isn't my shadow charm working?\"\n\n\"You have yet to provide evidence to justify a visit,\" Ezabar said.\n\n\"What kind of undead are you?\" Seth asked. \"Is that too personal? You hold a better conversation than a wraith. And revenants don't seem to talk at all.\"\n\n\"Revenants are focused so deeply inward it is a wonder they can move,\" Ezabar said.\n\n\"You're a lich,\" Seth guessed.\n\n\"That is one label, I suppose,\" Ezabar said.\n\n\"What label do you use?\" Seth asked.\n\n\"You are either extraordinarily brave or exceedingly ignorant,\" Ezabar said. \"Either way, you assume too much familiarity. Either provide a reason for intruding on my rest or face the consequences.\"\n\n\"I heard there is a prisoner who knows me,\" Seth said, hoping to earn some points for honesty.\n\n\"I fail to see the relevance,\" Ezabar said.\n\n\"The Underking invited me to stay here,\" Seth said. \"I work with Ronodin. I just freed a bunch of undead from a well and gave them their own castle to haunt. But I've lost my memory. If one of your prisoners might know me, I want to have a talk.\"\n\n\"There is a prisoner who knows of you,\" Ezabar said. \"He would attempt to poison your mind if given the chance. Ronodin would not like that conversation to occur, though the Underking has not forbidden it.\"\n\n\"Wait, do you know me?\" Seth asked.\n\n\"I know enough,\" Ezabar said.\n\n\"If you lost your identity, wouldn't you want to gather information?\" Seth asked. \"Speak to anyone who knew you?\"\n\n\"I would never be so foolish as to lose something so precious,\" Ezabar said.\n\n\"I don't even know how I lost it,\" Seth said. \"Ronodin is teaching me. Why would he want to keep me away from a person from my past?\"\n\n\"This prisoner has harmed many of our kind,\" Ezabar said. \"He destroyed several who were close to me. He is a menace.\"\n\n\"I'm not going to blindly believe him,\" Seth said. \"But it's important I hear from multiple sources if I'm going to put together the pieces of my past. Unless Ronodin has something to hide . . .\"\n\n\"I do not care to guess at the mind of Ronodin,\" Ezabar said.\n\n\"Then why not let me interview the prisoner?\" Seth asked.\n\n\"I do not oversee a social parlor,\" Ezabar said. \"This is the prison of the Under Realm.\"\n\n\"I'm a guest here, and one of your prisoners could be useful to me,\" Seth said. \"Should I ask Ronodin? Is he who really controls access?\"\n\n\"Ronodin is a visitor here as well,\" Ezabar said. \"Albeit an honored one.\"\n\n\"How does it usually work to talk to a prisoner?\" Seth asked.\n\n\"Seldom is a prisoner interrogated,\" Ezabar said. \"Such rare circumstances are almost uniformly at the behest of the Underking.\"\n\n\"Then what do you manage here? Sounds like there isn't much to do.\"\n\n\"The prisoners are all living,\" Ezabar said.\n\n\"You already explained that,\" Seth said.\n\n\"I primarily keep the other inhabitants of the Under Realm from feeding on them.\"\n\n\"Gross,\" Seth said, wrinkling his nose. \"I guess that makes sense, though. Then why not just let me talk to the prisoner? I promise not to feed on him.\"\n\nEzabar gave a nod. \"It is amusing to hold a conversation with a live person. Especially one who is not imprisoned. What would you be willing to do for me if I grant the access you seek?\"\n\n\"I don't know,\" Seth said. \"You've got a tomb over your head, a sword to hold, a marble bed, cool armor. What do you get for the guy who has it all?\"\n\n\"What about a piece of information?\" Ezabar asked.\n\n\"You know I lost my memory,\" Seth said.\n\n\"This is information you can obtain,\" Ezabar said. \"Ages ago I dwelled in the country of Selona. Do you know it?\"\n\n\"I'm not sure,\" Seth said. \"I remember some places. I don't think I'm an expert with geography, and who knows how much I've forgotten. I'm not even sure where I'm from, though I would guess the United States, since a lot of the states and cities sound familiar.\"\n\n\"Selona is in Europe,\" Ezabar said. \"There was a boy called Toleron, son of the Duke of Hester. The child's mother was called Ingrid. It would interest me to know what became of him.\"\n\n\"Like what he did with his life?\" Seth asked.\n\n\"Share whatever you can learn,\" Ezabar said. \"The topic is of enough interest that if you vow to do this, I will grant the access you seek.\"\n\n\"Can I get to Selona?\" Seth asked.\n\n\"Not easily perhaps, or soon,\" Ezabar said. \"If you survive long enough, there are ways to get you close. Time is on my side.\"\n\n\"Okay,\" Seth said. \"I vow to do that.\"\n\n\"Very well,\" Ezabar said. \"Wait here.\" He gestured toward his marble slab. \"I must check on the prisoners before allowing you access. If you exit this chamber before I return, the offer is void.\"\n\n\"I'll wait right here,\" Seth said. \"Do you mind if I sit on your slab?\"\n\n\"I do not mind,\" Ezabar said. \"I shall return.\"\n\nThe lich left the room, closing the door. Seth sat down on the cold, hard marble, wondering how long he would have to wait.\n\nKendra knew Thronis only by reputation. Standing up tall, she barely came halfway up his hairy shins. He loomed before her, clad in sandals and a sky-blue toga. Liver spots stippled a bald head fringed by the short bristles of graying hair. Except for his enormous proportions, he looked like a regular man approaching the age of sixty, with a paunchy belly and inquisitive eyes.\n\n\"That guy is freaky,\" Knox murmured.\n\nThe enormous room of fine marble and granite was sparsely furnished with divans the size of sailing ships and ottomans bigger than gazebos. A bonfire roared inside a gargantuan hearth. A colossal mural on one wall depicted a giantess seated on an elaborate chair that had been carved into a mountainside. Numerous tiny humans groveled at her feet. On the far side of the room, doors stood open to a balcony that overlooked the night sky.\n\n\"Is this Stormcrag?\" Kendra called.\n\n\"I hear you just fine,\" the giant replied, his voice powerful but not deep. \"No need to shout. Welcome to my mountaintop. You must be Kendra.\"\n\n\"These are my cousins, Knox and Tess,\" Kendra said. \"And my friend, Tanu, along with the satyrs Newel and Doren.\"\n\n\"And I know Marat,\" Thronis said. \"You are welcome here. The new wizard known as Andromadus obtained my permission to send you.\"\n\n\"Where is he?\" Kendra asked, looking around.\n\n\"He pledged to return shortly,\" Thronis said. \"You have endured a difficult night. Can I provide any refreshments?\"\n\nKendra sagged. Were refreshments an option? It felt like a long time since she had found a chance to pause. Were they truly safe? Agad had just been devoured by Celebrant. It had looked like they would all die.\n\n\"I'm thirsty,\" Tess said.\n\n\"I wouldn't mind a drink,\" Doren said. \"If you're offering.\"\n\nThe giant walked away, footfalls sending vibrations through the furniture, and returned with a golden bowl, which he placed on the floor near Kendra. It came higher than her waist and was full of clear liquid.\n\n\"I have some cups for small folk,\" Thronis said, walking from the room.\n\n\"Are you sure he isn't going to eat us?\" Knox asked.\n\n\"Thronis always tells the truth,\" Kendra said.\n\n\"Not by rule anymore,\" Thronis said, returning to the room. \"The spell Agad cast on me died with him. But I do believe in honesty, and it has become quite a meticulous habit over the years.\"\n\nThe giant placed a large drawstring bag on the floor, and Tanu tugged it open. \"Plates and utensils too,\" Tanu announced as he withdrew several glass mugs and began handing them around.\n\nWhen Kendra dipped her mug in the fragrant liquid, she could feel it was warm. She tried a sip and tasted tepid water suffused with subtle herbal flavors, including mint. The aroma, the taste, and the temperature combined to make each sip relaxing.\n\n\"This looks like what a fancy guy would use to wash his hands,\" Knox said. \"But it tastes good.\"\n\n\"Thank you, Thronis,\" Kendra said.\n\n\"My pleasure,\" the giant replied. \"I very rarely get visitors.\"\n\nWith a red flash, Patton appeared, along with the winged horses Glory and Noble and many griffins from the stable, including Tempest. A moment later, Andromadus arrived with another burst of light.\n\n\"So many mounts,\" Thronis said.\n\n\"You have facilities for such?\" Andromadus asked.\n\nThronis gave a nod. \"I am fond of griffins and keep many of my own here. These are all welcome. You will reinforce the protections on my manse?\"\n\n\"Indeed,\" Andromadus said. Closing his eyes, the wizard raised his staff and began mumbling words, his free hand splayed. He lowered the staff. \"It is done. Your home is now a bastion of the founders.\"\n\nThronis smiled. \"The dragons would probably not try to attack us here, knowing the winds I could summon, but I don't mind my abode becoming more secure, especially now that the sanctuary has fallen and I have taken in fugitives.\"\n\nKendra turned to Andromadus. \"Did you see any sign of my grandma and grandpa?\"\n\n\"Your grandparents are safe inside the last refuge of the keep,\" Andromadus said. \"I had means to spy on the events of the evening. Henrick and the other survivors of the castle staff are in there as well. The spells protecting the refuge leave me no chance of gaining admittance. But the undead should fare no better. Neither should the dragons.\"\n\n\"How will they get out?\" Kendra asked.\n\n\"In time, a rescue could be planned,\" Andromadus said. \"That is a matter for another day.\"\n\n\"Thank you for saving us,\" Marat said. \"It was most unexpected.\"\n\n\"If you came earlier, you might have saved Agad,\" Knox said.\n\n\"I chose my moment deliberately,\" Andromadus said. \"I knew Celebrant would slay all of you. But knowing was different from hearing the sentence declared. I stood by for the end of Agad because he was an old and worthy opponent. Had Celebrant shown you mercy, Kendra, I would not have intervened.\"\n\n\"Are you on our side now?\" Kendra asked.\n\nAndromadus bowed his head. \"Against some of my better judgment, I consider myself your ally until the end of this war with the dragons.\"\n\n\"The dragons may rue this day,\" Thronis said. \"It has been many a year since the birth of a wizard. The arrival of a spellcaster of your stature is no small event.\"\n\n\"Time will reveal what impact I may have,\" Andromadus said.\n\n\"How could you hold off so many dragons?\" Knox asked.\n\n\"It helped that I had so recently converted into a wizard,\" Andromadus said. \"Archadius taught that my powers would spike for a time with my rebirth. He tried to coax me into this transformation long ago.\"\n\n\"You learned this technique from the first wizard?\" Kendra asked.\n\nAndromadus gave a nod.\n\n\"You were truly ancient,\" Thronis said. \"The power of the wizard is generally proportionate to the age and power of the transformed dragon.\"\n\n\"May that hold true,\" Andromadus said, \"for I am the oldest dragon to undergo this metamorphosis.\"\n\n\"Could you have taken out any of the dragons at the keep?\" Knox asked.\n\n\"Perhaps,\" Andromadus said. \"But I remain a pacifist. I do not intend to stop this war with violence.\"\n\nKendra felt her hopes sink a little. It would be more comforting if Andromadus was fully on their side and ready to fight. She sympathized with his disdain for war, but she felt certain fighting would be required if humanity was to survive.\n\n\"The preserve has fallen,\" Kendra said. \"I am no longer the caretaker?\"\n\n\"Correct,\" Andromadus said.\n\n\"Your obligation is fulfilled,\" Marat added. \"The sanctuary has fallen. You did very well under grueling conditions.\"\n\n\"Then I am free to search for my brother?\" Kendra asked.\n\n\"You are free to pursue whatever you wish,\" Marat said. \"Though finding Seth may be easier said than done.\"\n\n\"Tell me what happened to Seth,\" Thronis said. \"I found him both entertaining and useful when he fetched items from the Dragon Temple for me.\"\n\nResisting her sadness, Kendra summarized how Seth had lost his memory and was taken, along with how he had been used to open the Blackwell. Thronis listened with interest until she concluded.\n\n\"To think the Wizenstone was within the boundaries of this sanctuary,\" the giant mused.\n\n\"It is good that such unfathomable power has been sent away,\" Andromadus said. \"This war already has plenty of weapons.\"\n\n\"The Wizenstone could have ended the war,\" Marat said.\n\n\"Whoever won using the Wizenstone, the rest of us would lose,\" Andromadus said. \"The world would face a nearly invincible reign of terror.\"\n\nKendra hoped that wouldn't be the case if she had the Wizenstone, but she didn't want to argue the point. It was the sort of claim a person could prove only by doing.\n\n\"How can we escape Wyrmroost?\" Kendra asked. \"What about teleporting us out of here?\"\n\n\"Now that the preserve has fallen, I could send you a short distance beyond the boundaries,\" Andromadus said. \"At least for a time, before my power recedes to normal. But the dragons can also cross beyond the former boundaries of Wyrmroost. This mansion is the most fortified location within my reach. Anywhere else I send you could lead to a hasty demise.\"\n\n\"So we're trapped here?\" Kendra asked.\n\n\"I'll find a way out,\" Andromadus said. \"At present, I suggest you rest.\"\n\nThe wizard raised his staff and vanished with a red flash.\n\n\"Am I correct to understand that we are welcome guests here despite your enormity?\" Newel called with a hand beside his mouth.\n\n\"That is correct,\" Thronis said. \"And again, I have excellent hearing.\"\n\n\"Have you known many satyrs?\" Newel asked in a more regular tone.\n\n\"I hesitate to confess that your kind has stayed somewhat beneath my notice,\" Thronis said. \"I have heard of satyrs.\"\n\n\"Great,\" Doren murmured. \"He wasn't onto us. Keep drawing attention.\"\n\nNewel elbowed his friend. \"Satyrs are famously hungry.\"\n\nThronis nodded.\n\n\"I see now,\" Doren muttered. \"Clever stratagem.\"\n\n\"I have food in abundance for such tiny guests,\" Thronis said, striding out of the room.\n\nKendra turned to Tess. \"Are you all right?\"\n\n\"I don't like dragons anymore,\" Tess said. \"The only nice one became a wizard.\"\n\n\"There is another nice one,\" Kendra said. \"The rest are pretty dangerous. Worse now that we're at war with them.\"\n\n\"Can we go home?\" Tess asked.\n\n\"We can't yet,\" Kendra said. \"Soon, I hope.\"\n\nKendra walked over to Patton. \"How are you?\"\n\nHe was panting a little and looked flushed. \"I've been better. My body is failing. I managed to free the Luvians before the undead reached the stables. They should be running free. I took the flying mounts to the top of the highest tower. I hoped to swoop in and rescue you, but those dragons were everywhere. I was about to try when Andromadus showed up. Told me to lay low. He went and helped you, then came back for me, the griffins, and the winged horses.\"\n\n\"Thanks for all you've done,\" Kendra said.\n\n\"Wish I could do more,\" the stingbulb replied.\n\nThronis returned. He laid a platter on the floor with various offerings, including some green grapes the size of bowling balls, a wedge of cheese that would have barely fit in a bathtub, slices of salami larger than manhole covers, some olives that rivaled the grapes, and several slices of bread that Tess could almost have used as a bed.\n\n\"Where did you find such big food?\" Knox asked.\n\n\"Most of my food is magically augmented,\" Thronis said. \"Staying fed is one of the challenges of being a giant. I keep a greenhouse full of enhanced produce. You'll find the grapes are perfect right now. Enjoy.\"\n\n\"Where should we sleep?\" Kendra asked.\n\n\"I imagine the girls would prefer separate accommodations?\" Thronis asked.\n\n\"Sure,\" Kendra said, deciding some privacy for her and Tess would be nice.\n\n\"I can lead you there now if you wish,\" Thronis said.\n\n\"Thank you,\" Kendra said. She looked at Tess. \"You eat, and I'll be right back.\" Then Kendra jogged to follow the giant into a neighboring room.\n\nThronis laid out a comforter that looked big enough to cover a swimming pool, then folded it down to a smaller size and placed a sheet on top. \"Will this work?\"\n\n\"That will be great,\" Kendra said. Folded, the comforter was much thicker than any mattress she had ever rested on.\n\nThronis strolled out of the room, and Kendra was grateful for the moment alone. She would go eat in a moment, but for now she found herself holding back tears.\n\nEven without his memory, why would Seth have unleashed the undead from the Blackwell? Wyrmroost had fallen. Agad and probably some others of the castle staff had been killed.\n\n\"Are you all right?\" asked Calvin from her pocket.\n\n\"I almost forgot about you,\" Kendra said.\n\n\"I stayed quiet in case you didn't want the giant to know about me,\" he said.\n\n\"Probably smart,\" Kendra replied.\n\n\"Now you know a little more about how I feel,\" Calvin said. \"Being tiny in such a huge world.\"\n\n\"True,\" Kendra said. \"Thronis must look unbelievably big to you.\"\n\n\"Dragons are enormous,\" Calvin said. \"But I've never seen a person more than a fraction of his size. Are you holding up all right?\"\n\n\"I guess,\" Kendra said. \"Seth and I failed as caretakers.\"\n\n\"You did great,\" Calvin said.\n\n\"We barely lasted a week,\" Kendra said. \"And Seth played the key role in destroying Blackwell Keep.\"\n\n\"He didn't know what he was doing,\" Calvin said.\n\n\"At least now I can look for him,\" Kendra said. \"And Bracken. I guess that is the bright side.\"\n\n\"That's the spirit,\" Calvin said. \"You'll feel better after some food and rest.\"\n\n\"I feel terrible about Agad,\" Kendra said. \"He was so brave. It seems like everything is falling apart.\"\n\n\"Not everything,\" Calvin said. \"Just Wyrmroost. And maybe your family a little.\"\n\n\"I'm not sure if we can fix this,\" Kendra said. \"For now I'm going to focus on finding Seth. It may not be possible, and I might not be able to win him back to our side, but I'll give it all I have.\"\n\n\"It starts with getting some sleep,\" Calvin said.\n\nKendra sighed. \"I better grab a little food and bring Tess to bed.\"\n\n\"I'd like to try some of that cheese,\" Calvin said.\n\n\"I'm glad you're with me,\" Kendra said.\n\n\"I'm sworn to Seth,\" Calvin said. \"I'm with you until we find him.\"\n\n\"Let's hope he stops freeing the undead.\"\n\nSeth was making a shadow puppet on the wall when the temperature began to plunge. By the light of the single lantern, the edges of the shadow were crisp as it morphed between an alligator and a dragon depending on how flat he kept his hands. Curling one finger above the others produced an eye, and little changes in the size or shape of that small window of light significantly altered the personality of his creation.\n\nWithin a few minutes after the lich had departed, Seth had grown fidgety in the bleak room. He did not want to void their agreement by exiting the chamber, so he sat in sensory-deprived silence staring at bare stone surfaces while wishing he could bring himself to lie down and doze on the hard marble slab. It was just too creepy and uncomfortable for a nap.\n\nSeth did not realize how much colder the lich made the room until Ezabar returned, his grim skull visible behind the bluish skin of his face. Seth dropped his hands as the lich passed through the doorway, hoping the shadow puppetry hadn't made him look like a little kid.\n\n\"Come,\" Ezabar said.\n\nSeth picked up his lantern and followed the lich, clenching his jaw in an effort to prevent his teeth from chattering. Outside the room, Seth noticed the wraith waiting at a distance and waved. The wraith raised a hand in reply.\n\n\"Does the prisoner have a name?\" Seth asked.\n\n\"Any matters relating to the prisoner are yours to investigate,\" Ezabar said.\n\nThey reached an empty cylindrical cage fashioned from thick bands of intertwined iron. The sole occupant was a bedraggled cat that looked scrawny enough to slip between the bars. The cat's fur rose, back arching as it hissed.\n\n\"Is that a stray?\" Seth asked. \"Did the cat sneak in here?\"\n\n\"Nothing is in my prison by accident,\" Ezabar said.\n\n\"Why would you lock up a cat?\"\n\nThe lich gave no reply.\n\n\"Unless it isn't just a cat,\" Seth said.\n\n\"This way,\" Ezabar said.\n\nThe corridor expanded into a vast subterranean underscape. Dozens of iron cages stood among the rises and dips of the uneven floor. Many were square or rectangular, some cylindrical, a few almost spherical. The cages ranged from crude to ornate in design, and most looked empty. The few wretched figures Seth could see in cages looked human, but they were hunched on the floor with their backs to Ezabar, heads down, bodies still. Glancing at his lantern, Seth wondered how long it had been since any of the cowering inmates had seen light.\n\nOn the far side of a boulder, Ezabar gestured toward a broad cage with a circular base and a sloping, conical top. In the cage stood a young man with extremely fair skin and silver hair reaching his shoulders. His build was surprisingly lean and strong for a dungeon prisoner. Unlike the cringing figures Seth had seen, this prisoner regarded Ezabar defiantly. Then his gaze shifted to Seth.\n\n\"Seth?\" the prisoner called in a familiar tone. \"Could that really be you? Surely this is a ruse.\"\n\n\"This visitor wishes to speak with you,\" Ezabar said. \"He is favored by the Underking, and you will show him the respect he deserves.\"\n\nThe lich turned and walked away, his feet not quite touching the ground. As Ezabar moved away, Seth felt the chill in the air lessening, and some of the anxiety inside of him began to untangle.\n\nThe prisoner regarded Seth warily. Seth stepped closer to the bars of the cage, careful to stay out of reach.\n\n\"Who are you?\" the prisoner asked.\n\n\"That is the question,\" Seth said.\n\n\"The real Seth Sorenson would not be here,\" the prisoner said. \"He is fulfilling an assignment far away and has no connection to the Underking. Show me your true form.\"\n\n\"What kind of assignment?\" Seth asked.\n\n\"Are you an illusion?\" the prisoner asked. \"Or a replica, like a stingbulb? I can't get a read on you.\"\n\n\"I'm a real person,\" Seth said. \"I'm a kid with no memory of his childhood. I have questions about myself. I was told a prisoner here might know me.\"\n\n\"You can't be the person you look like,\" the prisoner said. \"How did you get here?\"\n\n\"A wooden puppet put me in a barrel,\" Seth said.\n\n\"Mendigo?\" the prisoner asked. \"How would one of the barrels have gotten here?\"\n\n\"You know about barrels that work like teleporters?\" Seth asked.\n\n\"Dual manifestations of the same barrel,\" the prisoner said. \"You can go into one and get pulled from the other.\"\n\n\"Sounds right,\" Seth said. \"I have almost no memories before coming here. I was in a castle with a girl who called herself Kendra.\"\n\n\"This is an act,\" the prisoner said, backing away. \"Are you telling me you forgot Kendra?\"\n\n\"Should I remember more about her?\" Seth asked. \"She claimed to be my sister.\"\n\nThe prisoner stepped forward, stretching one arm out through the bars. \"If you're serious, take my hand and grant me full permission to search your mind.\"\n\n\"That doesn't sound smart,\" Seth said. \"Why are you in here?\"\n\n\"You should know me well,\" the prisoner said.\n\n\"I can't remember anything,\" Seth said. \"I don't know who I can trust.\"\n\n\"I'm in here because my cousin has a score to settle with me,\" the prisoner said.\n\n\"Who is your cousin?\" Seth asked.\n\n\"Ronodin, the dark unicorn,\" the prisoner said.\n\n\"I know him,\" Seth said. \"He has been tutoring me.\"\n\n\"This is sick,\" the prisoner said.\n\n\"Are you a dark unicorn too?\" Seth asked.\n\n\"Ronodin is the only dark unicorn,\" the prisoner said. \"I am his cousin, Bracken, a true unicorn, and a friend to you and your sister.\"\n\n\"You mean Kendra?\" Seth asked.\n\n\"Yes, your only sister, Kendra,\" Bracken said. He stretched out his hand again. \"I'm a unicorn; you can trust me. We don't lie. I'm not trying to trick you or harm you. Let me see your mind. Maybe I can help you remember.\"\n\nThe possibility of this prisoner being able to help him was so enticing that it made Seth wary. \"How do I know you're really a unicorn?\"\n\n\"Did you notice how the presence of the lich did not affect me?\" Bracken asked.\n\n\"That might just mean you're powerful,\" Seth said. \"Turn into a horse and show me.\"\n\n\"I need my third horn for that,\" Bracken said.\n\n\"Unicorns only have one horn,\" Seth said.\n\n\"You really have forgotten a lot,\" Bracken said. \"We shed our first two horns, kind of like humans losing baby teeth. Kendra has my first horn. Ronodin overpowered me and took my second.\"\n\n\"Where is your third?\" Seth asked.\n\n\"It is part of a talisman called the Font of Immortality,\" Bracken said. \"Seth, you're searching right now. You came to me because this place doesn't feel right. Go with those instincts. Don't trust a dark unicorn. I'm not sure anyone in the world is less trustworthy than Ronodin.\"\n\n\"I have dark powers,\" Seth said.\n\n\"Of course he would use your powers as evidence,\" Bracken said. \"You're a shadow charmer. You speak arcane languages, you can hide in shadows, you once had an ally who was a demon. But you never chose spiritual darkness, Seth. You fought it. Let me see if I can help you.\"\n\nSeth looked at the offered hand. He had let Ronodin touch him. This guy seemed sincere. \"I still don't know if you're really a unicorn.\"\n\n\"You're choosing the wrong time to be cautious,\" Bracken said. \"I don't get visitors. You may not be allowed to stay long. Look around. Almost everyone in this realm wants to drain your life force. You may be able to communicate with the undead, but your best bet to find an actual friend here is among the prisoners.\"\n\n\"The prisoners are all safe?\" Seth asked.\n\n\"Absolutely not,\" Bracken said. \"But most of the poor individuals encaged down here are more likely to help you than liches or dark unicorns would be. I truly am a friend of yours, and I know many in your family\u2014unless you are deceiving me. I'll speak more freely when I know you really are who you appear to be.\"\n\n\"Fine,\" Seth said, taking the outstretched hand. \"You can look at my mind. But don't mess with it.\"\n\nBracken tightened his grip. His brow furrowed. \"You're almost blank. Seth, this really is you.\"\n\nSeth yanked his hand back. \"What can you see?\"\n\n\"You're not a phony,\" Bracken said. \"You have a lot of general knowledge. But your identity is gone, along with all of your specific memories from before your mind was wiped. It's like you were born this week.\"\n\n\"That's how it feels,\" Seth said. \"Who am I?\"\n\n\"You're Seth Sorenson,\" Bracken said. \"Co-caretaker of the dragon sanctuary Wyrmroost, alongside your sister, Kendra, and assisted by your grandparents.\"\n\n\"Caretaker of a dragon sanctuary?\" Seth asked.\n\n\"That's right,\" Bracken said. \"A weighty responsibility for one so young.\"\n\n\"What am I doing here?\" Seth asked.\n\n\"I can only assume you have been kidnapped by your enemies,\" Bracken said.\n\n\"Ronodin acts like he rescued me,\" Seth said.\n\n\"Look around,\" Bracken said. \"Do you feel rescued? Pay close attention. Do you really think he's your friend?\"\n\n\"I don't know,\" Seth said. \"Probably not a friend. Maybe a teacher. Or an ally. Either of you could be tricking me.\"\n\n\"It's funny,\" Bracken said, stepping back from the bars and stretching. \"You and I first met in a prison.\"\n\n\"Was I the prisoner?\" Seth asked.\n\n\"We both were,\" Bracken said. \"Of the Sphinx.\"\n\n\"Wait a minute,\" Seth said, retreating a couple of paces. \"You almost had me going, then you pushed it too far.\"\n\n\"What's so strange about meeting in a prison?\" Bracken asked.\n\n\"I've only really talked to three people since losing my memory,\" Seth said. \"You happen to bring up all of them? Kendra, Ronodin, and the Sphinx? She's good, they're bad?\"\n\n\"When did you see the Sphinx?\" Bracken asked.\n\n\"Don't play dumb,\" Seth said. \"You saw into my mind when I took your hand. That's why you worked the Sphinx into the conversation.\"\n\n\"I wasn't searching your recent memories,\" Bracken said. \"I don't see everything in an instant. I was checking to see whether it was really you. When did you run into the Sphinx?\"\n\n\"You're messing with me,\" Seth said.\n\n\"I'm not,\" Bracken said. \"He is a shadow charmer and one of the most dangerous men alive. We fought him together when he tried to open the demon prison, Zzyzx.\"\n\n\"Playing dumb makes you less believable,\" Seth said. \"You saw him helping me.\"\n\nBracken came back to the bars and held out a hand. \"Take my hand again and I'll look.\"\n\n\"No way,\" Seth said. \"I don't know what you might do to me this time, now that I'm onto you.\"\n\n\"How did the Sphinx help you?\" Bracken asked.\n\n\"We released some undead from a well,\" Seth said.\n\nBracken looked stunned. The reaction seemed authentic. \"Not the Blackwell.\"\n\n\"Yes,\" Seth said.\n\n\"Oh, Seth,\" Bracken said, both hands covering his mouth. \"What have you done?\"\n\n\"The undead were suffering down there,\" Seth said.\n\n\"Blackwell Keep is the central stronghold of Wyrmroost,\" Bracken said. \"The sanctuary you swore to protect. Did you give the undead unlimited access?\"\n\n\"Just the castle and the roads,\" Seth said.\n\nBracken flinched. \"The roads? Did you see the results?\"\n\n\"We left quickly,\" Seth said, feeling uncomfortable. \"An old wizard saw us as we left. He told me I had been fooled.\"\n\n\"That is an understatement,\" Bracken said. \"Seth, you cannot trust your captors. Who knows what else they may have you do? You have endangered your sister, your grandparents\u2014all the innocents at Wyrmroost.\"\n\nSeth hated the implications of what Bracken was telling him. He reminded himself that the supposed unicorn had lots of motivation to lie. His story fit the circumstances, but that would be easy to do after reading Seth's mind. \"I see why Ronodin told me you're dangerous.\"\n\n\"Excellent conclusion,\" Ronodin said, pacing into view from behind the nearby boulder.\n\n\"What have you done?\" Bracken asked with real anger in his voice.\n\n\"Don't be a sore loser, Bracky,\" Ronodin said. \"Seth, it was unwise to come here. You are talking to one of the deadliest con men in the world.\"\n\n\"That's what he said about you,\" Seth said.\n\n\"No surprise,\" Ronodin said with a smirk. \"Mister light-makes-right would purge the world of darkness if he could. He finds your powers disgusting. He'd crush us both like cockroaches, given the chance.\"\n\n\"Ronodin is the liar,\" Bracken said with intense calm.\n\nRonodin laughed. \"You would tell whatever story might free you in order to destroy me. I, on the other hand, have spared the life of this criminal, as you can see, Seth. Bound his wounds, locked him up safe and sound.\"\n\n\"You must find out who you are, Seth,\" Bracken said.\n\nRonodin raised his eyebrows. \"Many voices will try to persuade you of your identity. Your memory loss has been widely publicized. All of your worst enemies will try to win you to their side\u2014Kendra, Bracken, and the rest.\"\n\n\"You're not like Ronodin, Seth,\" Bracken said. \"Watch him control what you see and hear. Study his maneuverings. He's putting on a show, and you are in grave danger of hurting yourself and others.\"\n\n\"Enough of this abusive manipulation,\" Ronodin said. \"I'm not trying to blind or deafen Seth. To the contrary, I'm helping him hone his powers. I'm trying to protect him from liars while he reacquaints himself with his true nature. You are a shadow charmer, Seth. You are here among your kind. According to Bracken, if it waddles like a duck and quacks like a duck, it must be a horse.\"\n\n\"This is low, even for you, Ronodin,\" Bracken said, his tone furious.\n\n\"Few prisoners adore their captors,\" Ronodin said. \"I hold Bracken down here for the good of the world. I hope the day arrives when it will be safe to free him. Come, Seth. You have already spent too much time ingesting this poison.\"\n\n\"Stay sane down here,\" Seth said, backing away.\n\n\"Remember what I told you,\" Bracken said earnestly.\n\nSeth looked away and didn't look back. They walked around the boulder, and Bracken passed out of view.\n\n\"Am I in trouble for coming here?\" Seth asked Ronodin.\n\n\"I'd rather you hadn't,\" the dark unicorn said. \"I wanted to protect you from the confusion he creates. Bracken is very good at what he does. But you were inevitably going to encounter lies such as he is weaving. Perhaps it was useful to test your will against his make-believe.\"\n\n\"He told me I destroyed the dragon sanctuary that I was supposed to protect,\" Seth said, still unsettled.\n\nRonodin laughed. \"Forgive me, Seth, but the idea of you as the caretaker of a dragon sanctuary is absurd. You are talented, but you are a young teen. Such positions are given to beings with extensive experience. If Bracken were determined to lie, he could have come up with something less far-fetched. All you did tonight was liberate some captive zombies.\"\n\n\"Bracken said that you're his cousin,\" Seth said.\n\n\"Don't you find it odd that all of these people you don't remember claim to be related to us?\"\n\n\"Yes.\"\n\n\"I hope you didn't let him touch you,\" Ronodin said.\n\nSeth scrunched his face. \"I did.\"\n\n\"Then you gave him a wealth of information with which to manipulate you,\" Ronodin said. \"At least he's locked up. Steer clear of him in the future.\"\n\nSeth wasn't sure what to believe. What Bracken had shared sounded plausible. What Ronodin claimed could be true as well. For now, it was probably best to err on siding with the person who was in charge here.\n\n\"Avoiding Bracken sounds like good advice,\" Seth said.\n\nWhen Kendra came into the room where the boys had slept, she found Knox standing on the arm of a divan as if about to jump. Below him on the floor awaited a huge, folded quilt and a plump cushion that would be hard to miss. Newel and Doren stood off to one side watching.\n\n\"Knox, don't!\" Kendra called.\n\nHe looked at his cousin. \"I already did one from almost this high.\"\n\n\"That's like a three-story roof!\" Kendra called. \"Maybe four.\"\n\n\"With a giant fluffy pillow to catch me,\" Knox said, stepping off and tilting in the air to land flat on his back. The striped cushion enfolded him. After landing, he slid off the cushion wearing a huge smile.\n\n\"Excellent form!\" Doren called. He glanced at Kendra. \"Assuming you're going to foolishly ignore orders from the former caretaker.\"\n\n\"Knox, you're crazy,\" Kendra said, trying not to be won over by his smile. \"We can't afford for you to get hurt.\"\n\n\"I built up to it,\" Knox said. \"The jump might look dangerous, but it doesn't even hurt.\"\n\n\"What sports do you play?\" Kendra asked.\n\n\"Basketball and soccer, mostly,\" Knox said.\n\n\"Imagine I'm your coach,\" Kendra said folding her arms. She raised her voice. \"Knox, we can't afford sprained ankles at this point in the season. Cut out the dangerous recreation.\"\n\n\"I think my soccer coach would have jumped too,\" Knox said.\n\n\"Then I'm your basketball coach,\" Kendra said. \"More importantly, I'm your survival coach.\"\n\n\"This is a survival skill,\" Knox complained, gesturing at the satyrs. \"I'm bonding with the team.\"\n\n\"We're strictly bystanders,\" Newel said.\n\n\"We're staying near in case we need to put him back together again,\" Doren said.\n\n\"I'm sure you weren't encouraging him at all,\" Kendra said.\n\nThe satyrs glanced at each other.\n\n\"Can't blame a guy for watching a free show,\" Newel said.\n\n\"Hard to beat live entertainment,\" Doren said.\n\n\"Live unless he lands wrong,\" Kendra said.\n\n\"Point taken,\" Newel said. \"Game over.\"\n\n\"But I want to jump off the back!\" Knox complained. \"It's not much higher than the arm.\"\n\n\"No way,\" Kendra said. \"You're lucky you haven't gotten hurt already.\"\n\n\"As if you're in charge of me!\" Knox challenged.\n\nTanu walked into the large room, toting a generous chunk of cheese. Knox pointed at the potion master. \"Tanu didn't mind me jumping!\"\n\n\"Is there a problem?\" Tanu asked uncertainly.\n\nKendra took a deep breath. Was she just spoiling their fun? Seth would probably think so. Was she trying to force an issue that didn't matter? But the divan was so high! The back had to be at least the equivalent of a four-story building.\n\n\"He shouldn't jump from so high,\" Kendra said. \"It's too risky.\"\n\n\"It's a rookie mistake to get hurt outside of real trouble,\" Calvin called out.\n\n\"Listen to Calvin and Kendra,\" Tanu said. \"They have better sense than the rest of us.\"\n\n\"She's not in charge of me,\" Knox said.\n\nTanu's face grew grim. \"If you want to continue with us in peace, you will listen to her.\"\n\nKnox looked at Kendra awkwardly, then glanced at the divan. \"I guess I can go back to suffering.\"\n\n\"Good idea,\" Tanu said.\n\nWith a flash of red, Andromadus appeared. Calvin ducked into Kendra's pocket. Kendra could hardly believe who materialized along with the wizard.\n\n\"Warren?\" she cried in disbelief. \"Vanessa?\"\n\nWarren wore a bowler-style shirt, beige shorts with lots of pockets, and sturdy sandals. He looked tired and lightly sunburned. Vanessa had a deeper tan than usual and wore sleek sportswear on her lithe frame, her hair an enviable tumble of dark curls. Behind them, a long, painted canoe rested on the floor.\n\n\"Hey, Kendra,\" Warren said, hustling over to her and wrapping her up in a tight hug.\n\n\"Seth,\" Kendra said, her voice catching.\n\n\"I know,\" Warren replied.\n\nKendra felt some of her tensions relax. Somebody who cared about Seth almost as much as she did had arrived. Somebody with experience as an adventurer.\n\nWarren released Kendra, and Vanessa drew her near. Kendra leaned her head on Vanessa's shoulder.\n\n\"I'm sorry, Kendra,\" she said.\n\n\"You came,\" Kendra said.\n\n\"As quickly as we could,\" Vanessa assured her.\n\nKendra stepped away from the hug. Vanessa was a narcoblix, and she had once plotted against Kendra and her family. After biting people, she could control them in their sleep. At one point Vanessa had exercised control over Tanu, but Bracken had broken the connection.\n\n\"Bitten anyone lately?\" Kendra asked.\n\n\"Only those who deserved it,\" Vanessa said with an enigmatic smile.\n\n\"Your hair is curlier,\" Kendra said.\n\nVanessa pinched a few strands. \"Humidity does it every time.\"\n\n\"Tanu!\" Warren called.\n\nThe big Samoan shuffled over and embraced him. \"Good to see you're still in one piece,\" Tanu said.\n\n\"We've been through a lot,\" Warren admitted.\n\nTanu glanced at Vanessa. \"I was referring to your choice of partner.\"\n\n\"Warren will never be safer than with me,\" Vanessa said.\n\n\"Hello, Miss Santoro,\" Newel said, extending a hand stiffly.\n\n\"You're looking well,\" Doren added, taking his turn to shake her hand.\n\nKendra couldn't help smiling. She knew the satyrs had massive crushes on Vanessa and often spoke in hushed wonder about her intoxicating beauty.\n\n\"Who's the kid?\" Warren asked.\n\n\"My cousin Knox,\" Kendra said. \"He and his sister, Tess, are with us.\"\n\n\"That's a surprise,\" Warren said. \"Are they . . . seasoned?\"\n\n\"I'm learning fast,\" Knox said. \"I was partly turned to gold.\"\n\n\"Sounds like a good way to get your feet wet,\" Warren said.\n\n\"He was healed,\" Newel explained.\n\n\"We lost Agad,\" Kendra said. \"Wyrmroost has fallen.\"\n\n\"Andromadus gave us the headlines,\" Vanessa said.\n\n\"I didn't make the headlines?\" Knox complained.\n\nWarren scratched his cheek. \"It usually takes extreme tragedy. You healed.\"\n\n\"And I'm Calvin,\" came a voice from Kendra's pocket. \"I sometimes lie low, but it looks like we're all friends here.\"\n\nWarren stepped near and peered at Kendra's pocket. \"A nipsie?\"\n\n\"A giant nipsie, thanks to a spell,\" Calvin said. \"I'm sworn to Seth, but I'm traveling with Kendra until we find him.\"\n\n\"We'll take all the help we can get,\" Warren said.\n\n\"How did you get here?\" Kendra asked.\n\n\"Wasn't easy,\" Warren said. \"We came from Crescent Lagoon.\"\n\n\"The island dragon sanctuary?\" Kendra asked.\n\n\"It fell for a time,\" Vanessa said. \"We helped restore a safe haven for the caretaker.\"\n\n\"Most of the archipelago remains fallen,\" Warren said. \"There is still a lot of work to do.\"\n\n\"Can the dragons leave?\" Knox asked.\n\n\"They could have, for a time,\" Warren said. \"We're pretty sure none took advantage of the opportunity. When we reestablished a haven for the caretaker, the boundary barriers were restored.\"\n\n\"Will the protection hold?\" Kendra asked.\n\n\"Hopefully until we can find a more permanent solution,\" Warren said.\n\n\"The dragons can leave Wyrmroost?\" Knox asked.\n\n\"At their leisure,\" Andromadus said.\n\n\"Raxtus found us,\" Warren said. \"He told us about the trouble at Wyrmroost.\"\n\n\"Last night,\" Vanessa said, \"as your keep was falling.\"\n\n\"How did Raxtus get there?\" Knox asked. \"We're nowhere near the ocean.\"\n\n\"Raxtus was hatched by fairies,\" Warren said. \"He can travel between fairy shrines. The shrines at Wyrmroost and Crescent Lagoon were some of the first to be reconnected to the fairy realm.\"\n\n\"Raxtus carried us and the canoe through to Wyrmroost,\" Vanessa said. \"By the time we arrived, the keep had fallen. We hid at the fairy shrine until Andromadus found us.\"\n\n\"I was hoping to find a home for the refugees of Wyrmroost,\" Andromadus said. \"Somebody to take them away. Turned out I just needed a little patience. They came to me.\"\n\n\"Is the canoe like the barrels?\" Kendra asked.\n\n\"There is a matching canoe at Crescent Lagoon,\" Warren said. \"We can use this canoe to go straight there.\"\n\n\"We're going to an island?\" Kendra asked.\n\n\"Right now it's the only reasonable way out of here,\" Warren said.\n\n\"And we think you may be extra motivated to come,\" Vanessa said.\n\n\"Why?\" Kendra asked.\n\n\"We believe Ronodin has recently visited Crescent Lagoon,\" Warren said. \"The islands have ten sacred pools of pure water where pixies play and naiads dwell. Yesterday one was corrupted.\"\n\n\"And you think the most likely culprit is Ronodin,\" Tanu said.\n\n\"Why him?\" Knox asked.\n\n\"Just as a regular unicorn has the power to purify, Ronodin has the ability to corrupt,\" Warren said. \"Even though the preserve fell, the pools have power of their own to withstand tampering. There could be another explanation, but Ronodin is the top suspect.\"\n\n\"We know Ronodin has Bracken,\" Kendra said, \"and maybe could tell us something about Seth as well.\"\n\n\"Seems like our only lead, based on what Andromadus told us,\" Warren said.\n\n\"The Crescent Lagoon sanctuary almost fell,\" Kendra said.\n\n\"It did fall,\" Vanessa said. \"We reclaimed a small portion.\"\n\n\"How?\" Kendra asked.\n\n\"Sanctuaries are organized in different ways,\" Warren said. \"They were established by teams of wizards using various methods. The treaty at Crescent Lagoon is upheld by one hundred moai. Do you know that word?\"\n\nKendra shook her head.\n\n\"Easter Island heads,\" Tanu said.\n\n\"Yes,\" Warren said. \"Upright statues with huge noggins. On the island of Timbuli stands a ring of ten\u2014the Grand Moai. They surround a stone altar where the Sunset Pearl once rested in a basin. We're talking a magic pearl the size of a softball. Despite its being guarded by three cyclopses and a bunch of magical protections, the dragons stole the pearl, and all the guardian moai went dormant.\"\n\n\"Did anyone find the pearl?\" Kendra asked.\n\n\"I wish,\" Warren said. \"Most of the sanctuary remains undefended. Other than the ten Grand Moai, ninety other moai populate the islands.\"\n\n\"It's an archipelago?\" Kendra asked.\n\n\"What's that mean?\" Newel muttered.\n\n\"I think it's a musical term,\" Doren replied.\n\n\"It's a chain of islands,\" Calvin said.\n\n\"There are more than fifty,\" Warren said. \"Many of them are small, though several are quite large. The main stronghold was on the island Kusaka. I managed to reactivate a single moai on the island of Timbuli. It happens to be the moai overlooking Crescent Lagoon, for which the sanctuary is named.\"\n\n\"How did you reactivate the statue?\" Kendra asked.\n\n\"The moai require the sacrifice of your most precious physical possession,\" Warren said. \"A person or animal doesn't count. The offering doesn't need lots of intrinsic value, so long as it matters most to the giver.\"\n\n\"Some people don't have an item that clearly means more than all others,\" Vanessa said. \"I couldn't think of one.\"\n\nKnox faced Warren. \"What did you use?\"\n\n\"My Carson Sears rookie card,\" Warren said.\n\n\"Never heard of him,\" Knox said.\n\n\"He was my dad's friend,\" Warren said. \"We called him Uncle Carson. He only played for three innings in the major leagues. His one at bat was supposed to be a sacrifice bunt, but he beat the throw and got a single, so he has a perfect career batting average. My dad had the card printed. I kept it with me as a good luck charm.\"\n\n\"But you gave it to a statue,\" Kendra said.\n\n\"Put it right in the mouth,\" Warren said. \"The mouth closes if the offering is acceptable. Then it opens once the offering is consumed.\"\n\n\"So the preserve is okay now?\" Knox asked.\n\n\"Warren's offering was a short-term solution,\" Vanessa said. \"It temporarily appeased a single moai guarding a limited area. The Sunset Pearl is the object the moai consider most precious. When the pearl was on the altar, no other offerings were required.\"\n\n\"We have to find the pearl,\" Kendra said.\n\n\"We're working on it,\" Warren said. \"And we wouldn't mind help. But we also need to figure out why Ronodin corrupted one of the sacred pools.\"\n\n\"And learn if Bracken or Seth might be nearby,\" Kendra said.\n\n\"Or at least catch Ronodin and make him divulge their location,\" Vanessa said.\n\n\"I like it,\" Tanu said. \"When can we leave?\"\n\n\"With the canoe here, we can go whenever you want,\" Warren said. \"And if needed we can retreat back to Thronis.\"\n\n\"Great,\" Kendra said. \"My vote is we leave now.\"\n\n\"Leave where?\" Tess asked, coming into the room, her hair disheveled from sleeping. \"What's going on?\"\n\n\"We're going to the beach,\" Knox said.\n\nTess brightened. \"Will Mom and Dad be there?\"\n\n\"No,\" Knox said. \"Just more danger and dragons.\"\n\nTess shrugged. \"Still might be fun to make a sand castle.\"\n\nSeth jumped and turned when the door to his room opened without warning. Ronodin wore a gray cloak, tall boots, and heavy gloves. A second cloak was draped over one arm.\n\n\"Want to see some daylight?\" the dark unicorn asked.\n\n\"Yes,\" Seth said. \"The sun was beginning to feel like a fable.\"\n\n\"Don't get too excited,\" Ronodin said. \"It's gloomy out. But real daylight is filtering through the mist.\"\n\n\"Good enough,\" Seth said, catching the thrown cloak. He wrapped the heavy garment around his shoulders as he followed Ronodin from the room. \"I wondered if I'd ever see the surface here. I thought it might always be a mystery.\"\n\n\"The entrance to the Under Realm is a secret known only to a select few,\" Ronodin said.\n\n\"We're by the ocean, right?\"\n\n\"Wait and see,\" Ronodin said.\n\nTheir way wound upward until they reached a ladder of iron rungs that climbed a vertical shaft comparable to the inside of a chimney. Ronodin started upward and Seth came after, glancing up occasionally at the soles of his teacher's boots, but mostly keeping his face down to avoid the gritty particles the dark unicorn was dislodging.\n\nFifty rungs later, Ronodin paused and threw open a trapdoor overhead, admitting muted gray light. He pulled himself up, then turned and extended a hand to Seth, hauling him out of the murky shaft.\n\nSeth now stood high atop an ancient fort composed of rough, ill-fitted stone, next to a thick parapet overlooking a coastline. Slow mist curled around angular rocks and encompassed crumbling, bulky fortifications. With each inhalation, Seth smelled the salty scents of the sea mingling with the fresh moisture of the mist. The gray ocean surged and growled below, sinking and receding only to return with new vigor, attacking tide-sculpted formations with explosions of foamy spray.\n\n\"An ocean,\" Seth confirmed.\n\n\"The Pacific,\" Ronodin replied. \"This uncharted isle guards the entrance to a realm very few would deliberately seek.\"\n\n\"Lucky us,\" Seth said.\n\n\"Aren't we, though?\" Ronodin said. \"Striding with impunity where the brave dare not tread? This way.\"\n\nHe paced along the top of the fort and down some stone steps that led to a projection of rock overhanging the tumultuous waves. Further along the cliff, Seth spotted a couple of figures shuffling toward them through the mist, coming into clearer focus with every step.\n\n\"There are people here?\" Seth asked.\n\n\"Depends on your definition,\" Ronodin said. \"The people of the deep help guard this isle. They are the Underking's elite zombies, more powerful and deliberate than others of their kind.\"\n\nOne of the men coming toward them wore homespun clothes and was partially tangled in a fishing net. Another wore the tattered remnants of a sailor's uniform. Both appeared to have drowned long ago.\n\n\"Will they bother us?\" Seth asked.\n\n\"Not while we're guests of the Underking,\" Ronodin said. \"Unlike wraiths or revenants, they lack the faculties to really communicate with us. They primarily hear their king. Their own primitive thoughts are dominated by hunger.\"\n\nRonodin sat down, resting his hands on his knees, studying the restless water. After a final wary glance at the sea zombies, Seth sat beside the dark unicorn.\n\n\"How big is this island?\" Seth asked.\n\n\"Not very large, though you can seldom see all of it. The mist is a nearly permanent feature.\"\n\n\"Are you showing me I can't run away?\" Seth asked.\n\n\"You think this ocean is a problem?\" Ronodin exclaimed incredulously, waving a dismissive hand at the water. \"You can't escape because of the Underking.\"\n\nSeth glanced at the ghostly manacle on his wrist. \"Why are we here?\"\n\n\"That's a profound question. Some say to master our wills.\"\n\n\"On the surface,\" Seth clarified.\n\n\"I thought you could use a little time under the sky,\" Ronodin said. \"I don't mind the dark, but it gets tiresome. We're not dead. Variety helps.\"\n\n\"Why did you lock up your cousin?\" Seth asked. \"What did he do?\"\n\n\"Are you still dwelling on him?\" Ronodin asked. \"I tried to warn you that he is dangerous.\"\n\n\"He made some good arguments,\" Seth said.\n\n\"Many will have persuasive arguments,\" Ronodin said. \"Pay attention to who you are and who you want to become. If you let other people run your life, you'll never have more identity than a puppet.\"\n\n\"Why did you imprison him?\"\n\n\"Bracken is my enemy,\" Ronodin said. \"He lives to eradicate darkness. He has destroyed many of my friends and allies. He is a talented killer. He has captured and imprisoned others whom I respect. He estranged me from my family. He vilifies me wherever he goes. What is the ultimate triumph when you have an enemy? Capture him. I didn't maim him or kill him. I simply took Bracken out of circulation.\"\n\n\"He wanted me to help him get free,\" Seth said. \"He talked like he was an old friend of mine. Had an elaborate story.\"\n\n\"Bracken can be extremely convincing,\" Ronodin said. \"You chose to seek him out. Now you must sort through the information he planted in your mind.\"\n\n\"You plant information too,\" Seth said.\n\n\"All yours to weigh and categorize,\" Ronodin said. \"The Underking has a mission for us.\"\n\n\"More wraiths to release?\" Seth asked.\n\n\"This one is different,\" Ronodin said. \"We'll do it together. On a nearby island.\"\n\n\"How will we get there?\" Seth asked.\n\n\"By boat,\" Ronodin said.\n\n\"Do you know how to captain a boat?\" Seth asked.\n\n\"I know how to do almost everything,\" Ronodin said. \"I have been around for a very long time.\"\n\n\"Why are you helping the Underking?\" Seth asked.\n\nRonodin looked around, then held up a hand, palm outward. \"I'm shielding our conversation. Seth, I don't care one bit about the Underking. If I could dethrone him, I would in a heartbeat. I would laugh as I threw his crown in the ocean.\"\n\n\"Then why help him?\" Seth asked.\n\n\"Favors are the main currency in the magical world,\" Ronodin said. \"The Underking has been useful to me, so I owe him favors. If it ever became convenient for him to destroy me, or you, do you think he would hesitate? Not for a heartbeat.\"\n\n\"What are you really after?\" Seth asked. \"How is the Underking helping you?\"\n\n\"I don't like the idea of anyone ruling over somebody else,\" Ronodin said. \"No one should be oppressed.\"\n\n\"Aren't you oppressing Bracken?\" Seth asked.\n\n\"Bracken works for the oppressors,\" Ronodin said. \"He works to keep wide varieties of magical creatures imprisoned. I only have him locked up so I can free others.\"\n\n\"That's all you want?\" Seth asked.\n\n\"It's all I ask for myself,\" Ronodin said. \"Unconditional freedom to go where I want when I want and to do what I want. I use my freedom to grant freedom to others. Right now the long-oppressed dragons yearn for liberation, so I'm helping them. If they, in turn, oppress others, I will work against them.\"\n\n\"You're the friend of the underdog,\" Seth said. \"I like that.\"\n\n\"It never ends, Seth,\" Ronodin said. \"Most of the world craves power over others. One tyrant falls and ten clamor to replace him. Or her. Humans, dragons, demons, giants\u2014you name the person or the creature, they are almost always working to control somebody else. I fight against that trend.\"\n\n\"That's why you wanted to free the undead from the Blackwell?\" Seth asked.\n\n\"They had been pent up for centuries,\" Ronodin said. \"They deserved freedom. It indirectly also helped free some dragons.\"\n\n\"Was I at a dragon sanctuary?\" Seth asked.\n\n\"Depends on who you ask,\" Ronodin said. \"I would call it a dragon prison.\"\n\n\"Are any of Bracken's statements true?\" Seth asked. \"Was I supposed to protect that dragon prison? Did I have a role there? Is Kendra really my sister?\"\n\n\"You have a complex history,\" Ronodin said. \"You had been to that dragon prison before. There were times when it suited your purposes to work with Kendra and some of the humans. You also worked with the Sphinx and some demons, most notably Graulas, one of the most cunning demons in history.\"\n\n\"I partnered with a demon?\" Seth asked.\n\n\"Until you double-crossed and killed him,\" Ronodin said.\n\n\"Are you serious?\" Seth asked.\n\n\"Absolutely true,\" Ronodin said. \"The demon hadn't dealt fairly with you either. But when do they? I don't blame you for it.\"\n\n\"My history sounds confusing,\" Seth said.\n\n\"So is your present,\" Ronodin said. \"But your future doesn't have to be.\"\n\n\"What do you want from me?\" Seth asked.\n\n\"I want you to learn all you can,\" Ronodin said. \"Your powers mustn't go to waste. Or get muzzled by the light-makes-right brigade. I'd like to see you survive the Underking and earn your freedom. I'd appreciate your help with the coming adventures. But mostly I want you to develop to your highest potential.\"\n\n\"What's our mission on the island?\" Seth said.\n\n\"We're looking for a flower,\" Ronodin said.\n\n\"Really?\" Seth asked.\n\n\"Orders from our bloodless sovereign,\" Ronodin said. \"He has requested a certain flower from the island.\"\n\n\"Why?\"\n\nRonodin dropped his hand. \"It's known as the Everbloom. This flower never dies. The Underking wants to feed off its vitality.\"\n\n\"Are there many of these flowers?\" Seth asked.\n\n\"The Everbloom is singular,\" Ronodin said.\n\nSeth leaned in. \"When you lower your hand, can the Underking hear us?\"\n\n\"I doubt it,\" Ronodin said. \"He likely could if he cared to listen. I was just being cautious.\"\n\n\"How do we find the Everbloom?\"\n\n\"I have unearthed an old crypt on the island,\" Ronodin said. \"We need you to go inside, find a phantom called Dezia, and learn the location of the Everbloom. I tried to go in with the Sphinx but we couldn't get past the guards.\"\n\n\"You think I can?\" Seth asked.\n\n\"I hope so,\" Ronodin said. \"I'm not a shadow charmer, and the Sphinx has lost a lot of his power. I can only command the undead here on the isle because I have authority from the Underking. His power bridges the gap for me. Off this island, I can't communicate with the undead.\"\n\n\"This mission comes from the Underking?\" Seth asked.\n\n\"Yes,\" Ronodin said. \"In fact, he wanted me to once again offer you the chance to remain in the Under Realm. I know of no shadow charmer to have received that honor in more than a thousand years. It could be an opportunity to acquire tremendous power.\"\n\n\"I'm not sure I want the kind of power the Underking has to give,\" Seth said quietly. \"Did you know he offered me his crown?\"\n\nRonodin held up a hand, palm outward. \"Are you serious?\"\n\n\"Yeah.\"\n\n\"That is highly unusual,\" Ronodin said. \"Almost unbelievable.\"\n\n\"He told me I could outlast the stars,\" Seth said. \"But I think if I was like him, I would just want to die.\"\n\n\"You have touched on the problem with the undead,\" Ronodin said. \"They're so intent on persisting, on enduring, it seems they forget to consider whether theirs is an existence worth having.\"\n\n\"Right,\" Seth said.\n\n\"You wouldn't outlive the stars, because you wouldn't be alive,\" Ronodin said. \"Perhaps you could outlast them, and acquire staggering power. I believe those who are undead by choice are driven by fear. They are so afraid to stop existing that they cling to a nightmare.\"\n\n\"I'd rather take my chances with what might come next,\" Seth said.\n\n\"An offer to wear one of the five crowns would tempt anyone,\" Ronodin said.\n\n\"Not me,\" Seth replied.\n\n\"Probably wise,\" Ronodin said. He lowered his hand and stood. \"You'll join me on the mission?\"\n\nSeth looked up at the dark unicorn. Ronodin seemed to have good reasons for how he lived, and he had taught Seth a lot already. He had also brought Seth to an island where the Underking had fastened a ghostly chain to him. Was that the action of somebody who wanted everyone free? Until Seth was liberated from the Underking, he wasn't sure he had much choice but to play along. At least it sounded like this latest mission wouldn't endanger others.\n\n\"Do I have much choice?\" Seth asked.\n\n\"You could choose to die instead,\" Ronodin said. \"You could choose to participate halfheartedly. Your best chance to succeed and survive is to perform these tasks enthusiastically and well.\"\n\n\"I'll come,\" Seth said, standing up and brushing off his backside.\n\n\"I'll appreciate the company,\" Ronodin said. \"Ready to head back down? The mist is thickening.\"\n\n\"Aren't we going in a boat?\"\n\n\"Later.\"\n\nSeth looked around. The day was dimmer. He could see only half as far compared to earlier. Indistinct human shapes haunted the grayness at the limits of visibility.\n\n\"Let's go back under,\" Seth said with a last glance at the crashing waves. It might be dark underground, but at least he could look forward to a boat ride.\n\nStrong hands helped Kendra to her feet and supported her as she stepped out of the canoe. In an instant, after lying flat as if in a coffin, she had gone from a mountaintop at Wyrmroost to a beach curving around a placid lagoon protected by a long reef. It had recently rained, so the sand beneath her boots was damp, but the heavy clouds were clearing, unveiling a bright sun, and the fronds of the palm trees sparkled with droplets. The fresh, salt-tinged air was humid and warm.\n\nThe hands belonged to a shirtless redhead with freckly shoulders and a white coral necklace who looked like he should have a surfboard stashed nearby. Vanessa had already come through, and she greeted Kendra with a grin.\n\n\"This is Grady,\" Vanessa said.\n\nThe young man gave Kendra a smile and shook her hand with both of his. \"Welcome to Crescent Lagoon,\" he said. Then he turned and started hoisting Knox from the canoe.\n\n\"It's a paradise,\" Kendra said, her eyes sweeping the turquoise lagoon and the distant waves breaking against the reef.\n\n\"Don't let the beauty fool you,\" Vanessa said.\n\n\"Is that the moai?\" Knox asked, pointing.\n\nKendra turned to face the looming statue, a long, grim face of stone with a jutting brow, the head larger than the rest of the body. She had an eerie feeling that it was aware of her. From base to top it had to be almost three times her height.\n\n\"He's the last one awake on the island,\" Grady said. \"This lagoon is safe thanks to him.\"\n\n\"Just the lagoon?\" Kendra asked.\n\n\"Out to the reef,\" Grady said. \"Beyond that, you're on your own. Same if you head up the beach and around the point in either direction. Or if you trek inland more than a few hundred yards.\"\n\nGrady hauled Tess out of the canoe, holding her in the air with one hand, muscles clenching in his shoulder until he set her down on the sand. She spun around, eyes wide.\n\n\"Why didn't I bring my swimsuit?\" she cried.\n\n\"You need to be careful swimming here,\" Vanessa warned.\n\n\"She would be fine with me watching,\" Grady said. \"I've spent a lot of time in these waters.\" He reached into the canoe and helped Tanu out. \"I caught a big one.\"\n\n\"Thanks,\" Tanu said. \"I've always wanted to visit this sanctuary.\"\n\n\"Welcome,\" Grady said. He crouched and pulled out Newel. \"A satyr?\"\n\n\"This is nice,\" Newel said. \"Like a resort on television. Do you have any lounge chairs?\"\n\n\"We could probably find some,\" Grady said, giving Doren a hand out.\n\n\"Now we're talking,\" Doren said. \"Where can I get a pi\u00f1a colada?\"\n\n\"How many more satyrs are coming?\" Grady asked.\n\n\"Just us,\" Newel said.\n\n\"Let me guess,\" Doren said. \"Satyrs are not endemic to these islands.\"\n\n\"Correct,\" Grady said.\n\n\"We promise not to take over,\" Newel said.\n\n\"We'll leave plenty of space for the native species,\" Doren said.\n\n\"Especially if you keep us fed,\" Newel added.\n\nGrady pulled Warren from the canoe.\n\n\"I'm the caboose,\" Warren said.\n\n\"I wish Marat was coming,\" Knox said. \"Couldn't hurt to have at least one dragon on our side.\"\n\n\"He felt like he needed to watch over Wyrmroost,\" Kendra said. \"It's been his home for centuries. He feels allegiance to the protected territories that sided with us. I'm glad he'll be there to help them against the dragons.\"\n\n\"The borders of the territories are still strong,\" Tanu said.\n\n\"Yes, but now they're surrounded and unconnected,\" Kendra said. \"They need a strong ally who can bridge the gaps between them.\"\n\nWarren grabbed one end of the painted canoe, and Grady lifted the other. They started trudging inland over the sand.\n\n\"You're going to love the tree house,\" Vanessa said to Kendra.\n\n\"Tree house?\" Tess asked.\n\n\"Look,\" Vanessa said, pointing ahead.\n\nAs Kendra focused behind the moai, amid the treetops, she saw structures cunningly crafted to blend with the leaves and branches. She noted platforms with railings, some with roofs, and walkways between them, with rope ladders connecting different levels.\n\n\"It's big,\" Knox said. \"And hidden.\"\n\n\"You have no idea,\" Grady said. \"We call it the Monkey Maze. The tree house goes on and on. And there are no dead ends. Everything loops. If you think you reached a dead end, you're missing something.\"\n\n\"Wouldn't the structure get messed up as the trees grow?\" Knox asked.\n\n\"That's why we work with hamadryads,\" Grady said. \"Especially Azalea, the mistress of this grove. Over the years, the Monkey Maze has adapted to the shifting environment, endlessly refined by the menehune and hamadryads.\"\n\n\"Menehune?\" Kendra asked.\n\n\"Hairy little dwarfs,\" Tanu said. \"Surprisingly strong. Sort of the island version of brownies. They specialize in building.\"\n\n\"I want to go up there so badly,\" Tess said.\n\nThe closer they got to the trees, the more platforms Kendra could discern. Several appeared to be isolated, unconnected to the others, but Kendra suspected there must be a way to reach them, based on what Grady had explained.\n\nUnder the trees, droplets from the recent rainfall continued to percolate downward. Thanks to the warm air, Kendra found the cool drippings more refreshing than bothersome. She appreciated the lush green smell beneath the leaf shade. Having never been to a real jungle before, she found the smooth, wide trunks and fancifully sprawling limbs captivating, along with the dangling vines and intricate ferns. A glossy yellow frog with wavy black markings sprang from a tree trunk into a bush.\n\n\"Here we are,\" Grady said, pointing out a platform on the ground with a waist-high pole at each corner. Taut ropes connected the tops of the poles around the perimeter of the platform, apparently for safety, and other ropes stretched up into the leafy heights from each corner pole and from the center of the floor. Green, leaflike patterns disguised the ropes, and the wood of the platform matched the surrounding tree trunks. \"How about one adult and the three kids?\"\n\n\"Come on,\" Vanessa said, stepping over one of the guard ropes. Kendra ducked onto the platform. Knox placed a hand on a pole and, kicking his legs out sideways, jumped the rope.\n\n\"Is it like an elevator?\" Tess asked as she ducked the rope like Kendra.\n\n\"A primitive elevator,\" Grady said. \"You'll feel more secure if you hold the central line.\"\n\nThey all placed a hand on the rope rising from the center. Then Tess added her second hand. Kendra considered adding her free hand, but Vanessa's casual stance convinced her to refrain.\n\n\"Now think of your favorite treat,\" Grady said.\n\n\"Hot fudge ice cream cake,\" Tess said.\n\nGrady pulled a lever, and the platform started rising. After the initial lurch, Kendra was relieved to find the ascent smooth and steady. Soon the ground was only partially visible as limbs and leaves crowded them. As they got higher, Kendra noticed a rope moving through a series of pulleys off to one side, along with a metal counterweight that fell as they climbed. Kendra tried not to picture ropes snapping, and she squeezed the central line a little tighter.\n\nAfter decelerating, the simple elevator jerked to a stop, level with a much larger platform shaded by a thatched roof and enclosed by a chest-high guardrail. A bright yellow and blue parrot roosted on a perch, and Kendra noticed two large hammocks among the wicker furnishings. Four walkways branched out from the platform, two of them sturdy, with wooden guardrails, the others flimsier and held together by ropes.\n\nOnce they stepped off the elevator, it started back down. Knox stood by the edge, watching its descent.\n\n\"I get how a counterweight could make it rise,\" Knox said. \"But how is it going back down?\"\n\n\"By releasing the counterweight?\" Kendra guessed.\n\n\"Then they would need another counterweight to raise the next load,\" Knox said.\n\n\"Maybe they have one,\" Kendra said. \"Or a bunch.\"\n\n\"Where is everyone?\" Tess asked. \"I see lots of tree houses but no people.\"\n\nKendra looked around. At least ten other major platforms were in view, some higher than theirs, some lower, but she saw no other inhabitants.\n\n\"It's mostly deserted,\" Vanessa said. \"After the pearl was stolen, those living in Spring Fort, the Scarlet Tower, and the Monkey Maze fled to the villages.\"\n\n\"There are villages?\" Knox asked.\n\n\"Ten,\" Vanessa said. \"Though smaller, they operate like the protected territories at Wyrmroost.\"\n\n\"Hard to believe anything bad could happen here,\" Kendra said. \"It feels like paradise.\"\n\n\"When the sanctuary fell, the dragons could have torched these tree houses,\" Vanessa said. \"They seemed more interested in assembling. Many had been confined to separate islands. I think they wanted to size each other up and plot together. The majority gathered to Wanai, the northernmost isle.\"\n\nThe elevator reached the top again, carrying Newel and Doren. Both of the satyrs looked a bit pale and uncertain.\n\n\"Are you all right?\" Kendra asked.\n\n\"We do our best work down on the ground,\" Newel said, stepping off the elevator and moving toward the center of the platform, near the tree trunk.\n\n\"I've never minded a climb,\" Doren said, a step behind Newel. \"But living like a bird in a lofty nest seems a bit unnatural.\"\n\n\"Heights get to me, too,\" Kendra said.\n\nTess flopped into a hammock and giggled as it rocked. Kendra wanted to try the other one, but Knox beat her to it, so she wandered over to the parrot. She was tempted to pet its bright feathers, but the hooked beak looked thick and strong.\n\nThe elevator returned one last time bearing Warren and Tanu. Not long after they appeared, Grady vaulted the railing onto the tree-house floor.\n\n\"How did you get up?\" Knox asked.\n\n\"I have ways,\" Grady said. \"The caretaker is eager to meet you.\"\n\nKnox and Tess slid out of the hammocks, and the group followed Grady along walkways from one platform to another. He mostly used stable, level walkways, but one long rope bridge swayed and bounced beneath their feet enough that the satyrs wore grimaces the entire way.\n\nAt last they reached a conspicuously large tree with stairs winding up and around the mighty trunk. Kendra and the others spiraled higher until they came through a large hatch in a floor and entered the only completely enclosed platform Kendra had seen. Larger than most of the other living spaces, this one had shuttered windows, wooden furniture, and five lanterns.\n\nA detailed map hung on one wall, almost from the floor to the ceiling, depicting a curved chain of islands. In front of the map stood a Polynesian woman of medium height, her back to them, dressed in a simple floral wrap that showed off her toned shoulders and calves. She turned and smiled as they entered, the striking ferocity in her eyes lending an aggressive edge to her beautiful face, a ring of woven leaves surrounding her head.\n\n\"Welcome to Crescent Lagoon,\" she said, the words clearly enunciated in a way that made her English sound particularly well-learned. \"I am Savani. This is the hallowed island of Timbuli. Ordinarily I would have greeted you on Kusaka in the Spring Fort.\" She looked directly at Kendra. \"You must be the former caretaker of Wyrmroost.\"\n\n\"Yes,\" Kendra said. \"The sanctuary has fallen.\"\n\nSavani gave a small bow. \"This sanctuary had fallen until Warren and Vanessa awakened Taki, the moai who watches over our Monkey Maze and the Crescent Lagoon. As long as one moai remains active, the outer boundary of the sanctuary remains engaged. While Taki endures, no dragons can enter or exit our sanctuary.\"\n\n\"That should keep Celebrant away for now,\" Kendra said.\n\n\"You worry about the Dragon King,\" Savani said.\n\n\"He is waging a war,\" Kendra said. \"I know he wants to destroy me. Right now my priority is finding my brother and helping your sanctuary survive.\"\n\n\"Warren used a baseball card to appease the moai,\" Knox interjected.\n\n\"So I understand,\" Savani replied. \"The offering was consumed within hours, but it bought me time to prepare an offering of my own.\"\n\n\"What did you give?\" Kendra asked.\n\n\"The tablets containing the history of my people,\" Savani said.\n\n\"You lost your history?\" Kendra asked.\n\n\"The tablets have been copied many times,\" Savani said. \"Our history is not gone. But the original tablets were my most precious possession. My people entrusted the sacred records to my care.\"\n\n\"Who are your people?\" Tess asked.\n\n\"The firewalkers,\" Savani said. \"We appease Baga Loa, the great volcano of these islands. She is hidden by the clouds today. Perhaps she will emerge later. My people have served her for generations, helping anticipate and moderate her eruptions. Though the tablets represent our past, it was not worth clinging to them if doing so jeopardized our future.\"\n\n\"Will the tablets satisfy the moai for long?\" Kendra asked.\n\n\"I can read the moai to a degree,\" Savani said. \"Taki will be content with my offering for at least another couple of weeks. The moai were never meant to be kept awake by individual offerings. That aspect is for emergencies. The only way to truly reclaim this sanctuary will be to find the Sunset Pearl and appease all of the moai.\"\n\n\"Have you any idea where to find the pearl?\" Tanu asked.\n\n\"Our best efforts have yielded no leads,\" Savani said. \"I lost many of my most trusted companions when the sanctuary fell. Some perished in the chaos when the boundaries disappeared. Others have vanished since then while hunting the pearl. I hold out hope that some of my friends may have taken refuge in the villages or found other secure hiding places.\"\n\n\"We've tried to help in the pearl hunt,\" Warren said. \"So far we keep coming up empty.\"\n\n\"Are there fairies on the islands?\" Kendra asked.\n\n\"You can find a fairy shrine right here on Timbuli,\" Savani said. \"Many exotic varieties of sprites and fairies thrive in the tropics.\"\n\n\"Fairies listen to me,\" Kendra said. \"I could have them help in the search.\"\n\n\"I see that you are fairykind,\" Savani said.\n\n\"How?\" Kendra asked.\n\n\"Surely you have met others with eyes to recognize your splendor,\" Savani said. \"You shine so brightly. You carry more magical energy than any mortal I have met.\"\n\nKendra knew that Bracken talked about her that way. \"It sometimes surprises me to hear that. I can't see it.\"\n\n\"You are unique,\" Savani said. \"Your aid is most appreciated. The shrine and Pixie Falls are reliable places to find fairies on this island. The fairies have not approached any of the fairy feeders near the Monkey Maze since the sanctuary fell.\"\n\n\"Isn't it dangerous to go beyond the area protected by the moai?\" Knox asked. \"Can't a dragon attack us?\"\n\n\"Within the sanctuary, dragons are now free to roam everywhere except the little region Taki protects,\" Savani said. \"The triclops drove the dragons from this island and killed at least five of them. Now the triclops is the menace.\"\n\n\"Was the triclops freed when the boundaries came down?\" Kendra asked.\n\n\"Indirectly,\" Savani said. \"Mombatu the triclops was the designated Dragon Slayer of Crescent Lagoon. He had been in hibernation for so long that many of us questioned whether he still lived. But once the dragons ran wild, Mombatu returned.\"\n\n\"Doesn't he protect you?\" Kendra asked.\n\n\"He would attack a dragon before he would attack us,\" Savani said. \"But Mombatu is savage, and a danger to anyone who comes near him, including me. Legend tells that the original caretaker of Crescent Lagoon could calm Mombatu with a flute, but the method has been lost.\"\n\n\"Newel and Doren play the flute,\" Knox said.\n\n\"Don't mind the boy,\" Newel said.\n\n\"His brain has gone soft with all the danger,\" Doren maintained.\n\n\"But I heard you talking about\u2014\" Knox complained.\n\n\"Satyrs speak nonsense at all hours of the day,\" Newel interrupted smoothly.\n\n\"Our boasts should seldom be taken seriously,\" Doren said.\n\n\"Calming Mombatu requires a magical flute,\" Savani said.\n\n\"That makes our rudimentary piping skills even less relevant,\" Newel said.\n\n\"We'll try to stop interrupting the conversation,\" Doren said, giving Knox a glare.\n\n\"You satyrs are a welcome novelty on our island,\" Savani said. \"Tanu's reputation precedes him. And who is the young man?\"\n\n\"I'm Knox, and this is my sister, Tess. We just barely learned about magical stuff.\"\n\n\"Welcome,\" Savani said. \"Any other visitors I should know about?\"\n\n\"Just me,\" Calvin said from Kendra's pocket. \"Calvin the nipsie.\"\n\n\"A miniature surprise,\" Savani said. \"May the island treat you kindly.\"\n\n\"We understand a sacred pool was corrupted,\" Kendra said. \"Warren mentioned that Ronodin is a suspect.\"\n\n\"It appears to be his handiwork,\" Savani said. \"A menehune spotted him in the area shortly after the pool was spoiled.\"\n\n\"My brother was abducted by enemies of Dragonwatch,\" Kendra said. \"We know the Sphinx was involved, and Ronodin has worked with him in the past. Also, Ronodin was at Wyrmroost near the time my brother vanished. My brother had just lost his memories when he was taken. We want to find Ronodin to see what he knows.\"\n\n\"Few are more slippery than Ronodin,\" Savani said. \"We still don't know why the pool was corrupted. Hako the hunter plans to visit the site this afternoon to see what he can learn.\"\n\n\"Who remains of your staff?\" Tanu asked.\n\n\"You met Grady, and I just mentioned Hako, our gamekeeper,\" Savani said. \"Uma is frail and in a wheelchair, but she might interest you, potion master.\"\n\n\"Not Uma Stormbrewer,\" Tanu said.\n\n\"The same,\" Savani said. \"She joined our staff after her legs gave out. She is quite old but remains sharp.\"\n\n\"The thought of meeting her makes me shy,\" Tanu said. \"I'll need to brew myself a courage potion. She is one of the grand masters of potion making.\"\n\n\"There are also a handful of the menehune who interact with me,\" Savani said. \"It took long years to cultivate their trust.\"\n\n\"This is a big tree house for four people,\" Tess said. \"And a few dwarfs.\"\n\n\"More than a few dwarfs,\" Savani said. \"The menehune are not often seen, but they quietly keep all in order here. Still, the Monkey Maze was designed for many more occupants. Even at its peak, this tree community never approached full capacity, but now it feels abandoned.\"\n\n\"I want to go with Hako to check out the sacred pool,\" Kendra said.\n\n\"Are you sure?\" Vanessa asked.\n\n\"Definitely, if there is any chance it could help us find Seth,\" Kendra said.\n\n\"The dragons are gone from this island?\" Knox asked.\n\n\"I suppose they could return,\" Savani said. \"But the chances seem slight as long as the triclops is rampaging.\"\n\n\"The triclops must be really big,\" Tess said.\n\n\"He's almost a force of nature,\" Savani said. \"Mombatu may prove more hazardous than the dragons.\"\n\n\"I understand the dragons here tend to be smaller,\" Tanu said.\n\n\"It depends where you go,\" Grady said.\n\n\"The islands host many smaller cliff-dwelling and tree-dwelling dragons,\" Savani said. \"Some are small enough to hunt in groups. None of those rival the great dragons at a sanctuary like Wyrmroost. But do not forget the sea serpents and the sea dragons. Very few dragons can match some of those in scale.\"\n\n\"Sea serpents?\" Knox asked.\n\n\"Some are longer than any dragon I have heard about,\" Savani said. \"Over ninety-nine percent of this sanctuary is ocean. Our largest occupants are under the water, and those boundaries extend beyond sight of the islands.\"\n\n\"I don't want to swim anymore,\" Tess said.\n\n\"Crescent Lagoon is actually quite safe if you are supervised,\" Savani said. \"I don't mean the entire sanctuary\u2014I refer specifically to the lagoon for which the sanctuary is named. I learned to swim there, as have many others.\"\n\n\"I'll think about it,\" Tess said.\n\n\"Have you examined the site where the Sunset Pearl was kept?\" Tanu asked.\n\n\"Hako went there with Warren,\" Savani said.\n\n\"We didn't find much,\" Warren said. \"Ten really big moai in a ring facing inward. An altar of piled stones at the center with a bowl on top. Supposedly three guardians protected the pearl, but there was no sign of them. There were tracks from the triclops, though.\"\n\n\"Who were the guardians?\" Calvin asked. \"Could they have taken the pearl?\"\n\n\"They were secretive, with a reputation for being calm and wise,\" Savani said. \"Losing the pearl may have destroyed them.\"\n\n\"Could the triclops have taken it?\" Kendra asked.\n\nWarren shook his head. \"Mombatu didn't appear until after the pearl was long gone.\"\n\n\"Maybe we'll find some clues at the pool,\" Kendra said. \"And maybe I'll find a few fairies along the way.\"\n\nTanu stretched and groaned. \"I would like to consult with Uma before we go. I have dragon parts she may find interesting.\"\n\n\"I expect she will,\" Savani said.\n\n\"I'm coming too,\" Knox announced. \"To the pool.\"\n\n\"Me too,\" Tess said. \"I love pools.\"\n\n\"I think it would be safer if Tess stays here at the Monkey Maze,\" Savani said.\n\n\"We volunteer to watch her,\" Newell said, nudging Doren.\n\n\"We can play hide-and-seek,\" Doren added.\n\n\"That sounds fun,\" Tess clapped.\n\n\"Warren and I will join you as well, Kendra,\" Vanessa said. \"I wonder if somebody could create a diversion to distract the triclops? Lure him elsewhere on the island. Perhaps prevent him from falling upon us.\"\n\n\"I could try,\" Grady said.\n\n\"I wonder if two satyrs are required to watch one girl,\" Vanessa mused. \"Distracting the triclops would be very heroic.\"\n\nNewel swallowed loudly. \"I could come help, Grady. I have experience annoying lumbering brutes.\"\n\n\"That's true,\" Doren said.\n\n\"Don't underestimate Mombatu,\" Grady said. \"He is big, but he is also agile and impossibly strong. His hide can resist dragon fire and his hands can crush boulders.\"\n\n\"Sounds like my kind of party,\" Newel said feebly.\n\n\"I can watch Tess,\" Savani offered. \"I should remain here to make sure our last bastion on the islands remains secure.\"\n\n\"I guess that frees me up,\" Doren said without much enthusiasm.\n\n\"Let's get ready,\" Kendra said. \"I'm tired of being out of the hunt for my brother.\"\n\nThe dark craft bounded forward over the choppy swells, swooping down into troughs only to rise again. Water occasionally sloshed over the sides but somehow drained after reaching the bottom of the boat. Though there was room for at least six more people, Seth and Ronodin had no companions, and neither worked an oar. Seth wondered if Ronodin really had to keep his hand on the tiller, or if he just liked feeling in control. Since they had untethered the craft from a sheltered dock inside the sea cave, the boat had propelled itself.\n\nBeyond the mist surrounding the island, Seth had discovered a bright day, and he was enjoying the feeling of direct sunlight on his face for the first time in memory. Now the misty island was fading to a smudge behind them, and a new island was growing as they neared it. A towering volcano presided over several lesser mounts, slopes green with vegetation.\n\n\"Timbuli,\" Ronodin said. \"Soon you will glimpse other islands. And there are many more beyond our sight. This is the largest undiscovered archipelago in the ocean, a former dragon prison called Crescent Lagoon.\"\n\n\"Why hasn't it been discovered?\" Seth asked.\n\n\"These islands are shielded by distracter spells,\" Ronodin said. \"If a ship comes near, all aboard feel a subtle urge to steer away, along with an irresistible impulse not to notice what they avoided. Even a pair of eyes studying a satellite photo will feel an unnamed compulsion to ignore these islands.\"\n\n\"I don't feel that urge,\" Seth said.\n\n\"We're already within the prison,\" Ronodin said. \"From inside, the distractions lose their influence.\"\n\nSeth looked over the gunwale at the rolling sea. \"What's pushing the boat?\"\n\n\"Magic, of course,\" Ronodin said. \"It takes some magic to depart the Under Realm, and more to find it again. This vessel has enough to do both jobs.\"\n\n\"Can you do it without the boat?\" Seth asked.\n\n\"Naturally,\" Ronodin said with a smirk. \"But I'd rather work smart than hard.\"\n\n\"How does the water drain?\" Seth asked.\n\n\"A clever blend of engineering and magic,\" Ronodin said. \"You should ask about the bucket.\"\n\nSeth had hardly noticed the old bucket, even though it was the only gear in the boat besides a coiled line. \"Tell me.\"\n\n\"It's called a sweet bucket,\" Ronodin said. \"Not much to look at, but fill it with seawater, and the water turns fresh.\"\n\n\"Really?\" Seth exclaimed. \"Can I try?\"\n\n\"Be my guest.\"\n\nThough not terribly thirsty, Seth felt very curious. With the boat still bucking and diving, he leaned over the side and scooped some water into the bucket.\n\n\"No magic words?\" Seth asked.\n\nRonodin handed him a tin dipper. \"Try a sip.\"\n\nSeth scooped the dipper into the bucket. He tested the fluid with his tongue, then sipped some. Sure enough, the water was cool and fresh, without a trace of salt. \"I'm impressed.\"\n\n\"We all live to earn your approval,\" Ronodin said solemnly.\n\nSeth gazed ahead at the approaching island. \"We're going to the crypt during the day?\"\n\n\"Know your quarry,\" Ronodin said. \"This crypt is guarded by wraiths. And ultimately you will need to converse with a phantom. Shadow charmers sometimes prefer to operate at night because they hide so well in darkness, but daylight is no handicap to you. Wraiths and phantoms, however, are much less comfortable during the day. You do not want to confront them at the height of their powers.\"\n\n\"The wraiths will be hostile?\" Seth asked.\n\n\"They kept me out,\" Ronodin said. \"The Sphinx, too. Do you understand the nature of wraiths?\"\n\n\"I know they're cold,\" Seth said. \"And starving. It's nearly all they talk about unless you ask direct questions.\"\n\n\"Do you know how wraiths originate?\" Ronodin asked.\n\n\"No,\" Seth said.\n\n\"The creation of a wraith relies on a cruel trick,\" Ronodin said.\n\n\"Who would create a wraith?\"\n\n\"A powerful witch or wizard. A demon. A viviblix. Or one of the mightier undead, like a lich. Any of those prey upon individuals who want something desperately. The victims are eventually led to willingly cheat death until they receive what they desire.\"\n\n\"What happens if the people get what they want?\" Seth asked.\n\n\"Then they don't become wraiths,\" Ronodin said. \"The all-consuming lust for the unobtained prize is what fuels the transformation. If the desired reward is attained in life, the potential for a wraith is lost. The birth of a new wraith depends on the coveted prize remaining elusive. Once a being powerful in dark magic facilitates the change, the wraith can gain freedom only by receiving the desired object, which brings the postponed release of death.\"\n\n\"And if the person can't get what they want, they're trapped?\"\n\n\"Sometimes the originator of the deal will deliberately hide, withhold, or destroy the desired object to ensure an undead servant. At other times the targeted victims crave something they can never win, like the love of a certain woman, the respect of someone they admire, or to hold an important title or position. In any event, if wraiths ever obtain what they most want, they find release.\"\n\n\"Can I find out what the wraiths want?\"\n\n\"Easier said than done. By the time they are undead, wraiths are so miserable and dissatisfied that they avoid the topic. They rarely wish to discuss their agonizing disappointment. Wraiths seek distraction. They leech life and warmth. And if they find a master they can comprehend, wraiths like to surrender their wills by serving.\"\n\n\"The wraiths are serving the phantom?\" Seth asked.\n\n\"Or somebody who wanted the phantom protected,\" Ronodin said.\n\n\"Can I talk them out of guarding Dezia?\"\n\n\"You may fail, but you must try.\"\n\n\"Will I need to do favors for them?\" Seth asked, remembering the favor he already owed Ezabar.\n\n\"Wraiths are not terribly interested in favors,\" Ronodin said. \"Many wraiths think they want freedom, although their only real freedom would be finding the object of their desire. Their sole remaining pleasure is the fleeting rush of draining a life force. Most would be willing to serve a better master. If it were me, I would try to help them feel released from their duty of guarding the crypt. Offer them a better alternative. Uninfluenced by outside sources, wraiths routinely stay true to commands for decades or even centuries. But you can talk to them, meaning you have a chance to influence them.\"\n\n\"And if I make it to Dezia?\" Seth asked.\n\n\"Phantoms are less predictable,\" Ronodin said. \"I gave you a book on the subject.\"\n\n\"I may not have read every page,\" Seth said. \"I need the summary.\"\n\n\"Phantoms are ethereal,\" Ronodin said. \"They have no tangible body. Nobody knows exactly where they come from, but they are born out of deep tragedy.\"\n\n\"If a person lost everything, they could turn into a phantom?\"\n\n\"Such a person could cause the birth of a phantom,\" Ronodin said.\n\n\"Maybe I'm a phantom,\" Seth mused. \"I don't remember where I came from.\"\n\n\"You're very much a human boy,\" Ronodin said. \"The afflicted person does not turn into a phantom. The phantom is what fills the footprint of the tragedy. After coming into being, phantoms develop an independent identity. The fundamental nature of a phantom is difficult to discern.\"\n\n\"They can think and speak?\" Seth asked.\n\n\"You should be able to converse with the phantom,\" Ronodin said. \"How you get the information you need is up to you.\"\n\n\"Brute force?\"\n\nRonodin grinned. \"You could try.\"\n\n\"Favors?\" Seth asked.\n\n\"Whatever it takes,\" Ronodin said.\n\n\"This is starting to feel like quicksand,\" Seth said. \"I owe the Underking favors, and to do his favors, I'll need to owe other favors. I keep piling up new debts.\"\n\nRonodin laughed heartily. \"Welcome to the magical world!\"\n\n\"When does it end?\" Seth asked.\n\n\"Let me know if you figure that out,\" Ronodin said.\n\nSeth had plenty to think about as the island drew nearer, the prow of the craft aimed at a black beach flanked by low, rocky cliffs. How could he prepare for threats that remained largely unknown? He would have to rely on his ability to improvise as he discovered how the wraiths and the phantom interacted with him.\n\nAt length the boat moved into breakers and was propelled to shore at the fastest speed yet, waves frothing around them until the craft ran aground on fine black sand. Ronodin and Seth exited the front of the boat.\n\n\"Come,\" Ronodin said, as if talking to a dog. He snapped his fingers. \"Up. Up.\"\n\nThe boat surged forward in little scoots until it rested completely on dry sand. Ronodin scanned the waves. \"That will do.\"\n\n\"Obedient boat,\" Seth said.\n\n\"You should see it roll over,\" Ronodin remarked. \"This way.\"\n\nThey tromped off the sand and onto a path of hard mud that wound back into the trees. Ferns crowded the path, some lacy and delicate, others broad and shiny.\n\n\"Do you expect trouble?\" Seth asked.\n\n\"Why not?\" Ronodin said. \"We have a rampaging triclops on the prowl, the possibility of dragons, and the undead unwittingly awaiting our arrival.\"\n\n\"Aren't you on good terms with the dragons?\" Seth asked.\n\n\"I'm an expert at avoiding trouble,\" Ronodin said. \"I'm more concerned about you.\"\n\n\"So if dragons show up, I'm on my own?\"\n\n\"I want you safe. See the flowers over there?\"\n\nSeth looked where Ronodin was pointing and beheld huge blossoms the size of hula hoops, red and white petals pristine against the verdant backdrop, maybe fifteen paces off the path. \"The big ones?\"\n\n\"Yes,\" Ronodin said. \"They are responsible for the fragrance.\"\n\nSeth sniffed purposefully. A definite perfume mingled with the humid air. \"Smells tasty.\"\n\n\"Exactly,\" Ronodin said. \"Not just a pleasant aroma. The smell seems to promise you will be fed. See the green bulbs near the blossoms?\"\n\nThey were bigger than watermelons. Seth assumed they simply had not blossomed yet. \"Sure.\"\n\n\"They will hastily devour you if you come within range,\" Ronodin said.\n\n\"Carnivorous,\" Seth said, stepping away from the plant even though it was not near him.\n\n\"Aggressively carnivorous,\" Ronodin said. \"Strikes fast. Can bite through bone. Some birds can't resist the aroma. A meaty kill like you would satisfy the plant for months.\"\n\n\"Message received,\" Seth said. \"Don't stop and smell the roses.\"\n\n\"Not without expert guidance. If you want a rule of thumb, big and bright isn't right. Such species tend to be poisonous or carnivorous. Works in the jungle and in the sea as well. Some very nasty fish inhabit these waters.\"\n\n\"I want to see that flower eat something,\" Seth said, lingering as Ronodin moved on.\n\n\"Me too,\" Ronodin said. \"Sadly, we lack gullible children, and we have a schedule to keep.\"\n\nSeth hoped Ronodin was kidding.\n\nThey continued through the jungle. When they reached an open meadow, Seth noticed a moai on a nearby hilltop. \"Cool statue,\" he said.\n\n\"They once guarded this prison,\" Ronodin said. \"I prefer them asleep. It is why the Underking has chosen the present hour to send us on this mission.\"\n\nOn the far side of the meadow, the path continued. Several plants about Seth's height looked to have green dreadlocks instead of leaves. He amused himself by appreciating the variety of shrubs, flowers, and trees. Ronodin pointed out a tree that could kill by using its vines as nooses, and on the other side of the path, a spiky bush with exploding fruit.\n\nWhere a splashing stream crossed their path, Ronodin held a finger to his lips and crouched. Following his gaze, Seth looked upstream to where a red boar the size of a car was drinking. The creature abruptly raised its head, wide snout sniffing, tusks the size of baseball bats curving sharply.\n\nAs the beast turned toward Ronodin and Seth, the dark unicorn arose, one arm outstretched toward the boar, the other clutching what looked like a short, black unicorn horn. \"Depart in peace or meet your ruin,\" Ronodin said calmly.\n\nWith a squealing snort, the boar turned and ran, crashing through the foliage. Seth remained tense until it passed out of earshot.\n\n\"What if the pig had charged?\" he asked.\n\n\"Pork chops and bacon,\" Ronodin said, putting away the horn.\n\n\"Huge bacon,\" Seth said. \"Was that your horn?\"\n\n\"You saw nothing,\" Ronodin said.\n\n\"Okay,\" Seth said. \"Must have been my imagination.\"\n\n\"An attitude like that could take you far,\" Ronodin said.\n\n\"Should I have brought a weapon?\" Seth asked.\n\nRonodin stared at Seth. \"Would you have fought that boar with a sword?\"\n\nSeth smiled uncomfortably. \"That might not have gone well.\"\n\n\"Didn't you have some weapons in that satchel of yours?\" Ronodin asked. \"The one you carried when Mendigo brought you here?\"\n\n\"Random gear,\" Seth said. \"Like one glove. I had a knife in there. Should I have brought that?\"\n\n\"You used to always carry the satchel with you,\" Ronodin said. He produced a short sword. \"A weapon isn't a bad idea in case of emergency.\"\n\nSeth accepted the short sword. \"Thanks.\"\n\n\"It's yours,\" Ronodin said. \"Mendigo took it from its sheath when he grabbed you.\"\n\n\"It looks fancy,\" Seth said.\n\n\"I hope you don't have to use it,\" Ronodin said. \"Let's avoid trouble. Especially boars. Fighting is a last resort. Any clumsy oaf can fight.\"\n\nThey crossed the stream and continued along the path, moving generally uphill. After coming over a little rise, they saw a wide, muddy crater gaping before them. A few murky puddles sulked in the slick, black mud. On the far side of the crater, near the bottom, a smooth boulder rested beside a stone doorway.\n\n\"Is that the crypt?\" Seth asked, pointing at the open doorway with one hand while the other covered his nose.\n\n\"You guessed it,\" Ronodin said.\n\n\"Did you move the rock?\" Seth asked.\n\n\"No small feat,\" Ronodin said. \"It was enchanted. The Sphinx has a talent for opening things. The harder part was draining the pond.\"\n\n\"This was a pond?\" Seth asked. \"Of what? Sewage?\"\n\n\"Foul water,\" Ronodin said. \"This would give the sweet bucket a real test.\"\n\nSeth peered at the doorway. \"It looks like a cave. How far back does it go?\"\n\nRonodin slapped Seth on the back. \"That is for you to discover. We should hurry. If the triclops shows up, we might wish that boar had finished us.\"\n\n\"He's that tough?\"\n\n\"You won't believe how savage,\" Ronodin said. \"This way.\"\n\nRonodin led Seth down the least steep side of the empty pond. Tarlike mud sucked at each step, covering his shoes with squelchy gunk. Flies swirled above the squalid puddles. The stink was everywhere, but covering his nose and mouth reduced the impact.\n\nBefore long they stood before the stone frame of the doorway, gazing into the darkness beyond. \"Time to prove your mettle as a shadow charmer,\" Ronodin said. \"This will be a worthy test of your abilities.\"\n\n\"Pass-fail,\" Seth said.\n\n\"Exactly,\" Ronodin said. \"Sink or swim. Survive or perish. Off you go.\"\n\n\"You'll be here when I get back?\" Seth checked.\n\n\"Me or the triclops,\" Ronodin said. \"Or some other hideous creature. The faster you go, the better.\"\n\nSeth did not like the feel of the cool air wafting from the forbidding doorway. He had a hunch the cave stretched a long distance. \"Do you have a light?\"\n\nRonodin pulled out a holly wand. He snapped his fingers, and the tip lit up. He handed the wand to Seth.\n\n\"Smooth,\" Seth said, starting forward.\n\n\"Watch out for traps,\" Ronodin remarked.\n\nSeth paused. \"What kind of traps?\"\n\nRonodin shrugged. \"Anything you can imagine. It's a crypt that was hidden behind a boulder beneath a pond on an island full of monsters. There might be traps.\"\n\nSeth took a final glance at the blue sky and the tall trees surrounding the empty pond. Sword in one hand, glowing wand in the other, he stepped through the stone doorway into the chilly darkness beyond.\n\nSeth promptly realized that the cold inside the crypt came from more than just the absence of sunlight. The rough walls and ceiling of a natural cave led to a chamber lined with yellowed stone blocks. He could sense wraiths ahead in the dimness of the chamber, six of them, spread out in a loose semicircle.\n\nThe hostility from these wraiths was unmistakable, contrasting sharply with the neutral attention Seth received from the wraiths in the Under Realm. Seth paused at the entrance to the chamber. The darkness seemed thicker here, offering greater resistance to the modest light of his wand. Wraiths emerged from shallow niches in the walls around the room, slinking toward him. The malevolent cold emanating from the wraiths made the little hairs on his arms stand up. Seth felt like prey; it was time to act.\n\n\"Greetings,\" Seth said in the brightest voice he could muster. \"I brought a message.\"\n\nDezia needs no messages, one of the wraiths thought to him.\n\nYou do not belong here, another maintained.\n\nTake his warmth, a third expressed.\n\nHe is plump with life, another conveyed.\n\n\"Hold on,\" Seth said. \"My message is for you, not Dezia. Stop for a second. What's the big hurry?\"\n\nThe wraiths halted. Seth was poised to run away if necessary.\n\nHe speaks to us, one of the wraiths communicated.\n\n\"And I hear you,\" Seth said. \"You don't want my warmth. It would be a waste. I can help you.\"\n\nIt's cold here, one of the wraiths expressed.\n\nHe is alive, a different wraith spoke to his mind. He is warm.\n\n\"You can speak,\" Seth said. \"You don't have to think at me.\"\n\nWe commune as we choose, another wraith transmitted.\n\n\"Have you heard that this dragon prison fell?\" Seth asked.\n\nFor a time, one wraith expressed. Then one of the watchers revived.\n\n\"The hour of Crescent Lagoon is over,\" Seth said. \"I'm here to grant you freedom. You don't have to stay in a forgotten hole at a fallen prison. You can go where you want now.\"\n\nHe speaks pretty words, one of the wraiths put in.\n\nBeguiler, another wraith accused.\n\n\"That one watcher won't last long,\" Seth said. \"Besides, very few people even remember you're here. The world has moved on. You're guarding nothing. Nobody cares anymore.\"\n\nDrain him dry, one of the wraiths yearned, stepping forward.\n\nSeth held up a hand. \"I wouldn't do that. I'm the only friend you've got. The only voice of warning you're going to get.\"\n\nHear him, another wraith demanded.\n\nThe wraiths kept still.\n\nHungry, one of the wraiths expressed glumly.\n\n\"I come from the Underking,\" Seth said. \"I'm on an errand for him.\"\n\nThere is truth in his language, a wraith conveyed.\n\n\"I'm here to offer you a deal,\" Seth said. \"Are other wraiths guarding this crypt?\"\n\nWe six, one wraith expressed.\n\n\"If one of you agrees to serve me, the other five can go free,\" Seth said, hoping the deal sounded enticing. He had no real authority to allow them to leave, but he hoped that they remained there largely because nobody had given them permission to go. He also hoped that asking one wraith to serve him would make the others feel lucky about the bargain.\n\nServe you how? a wraith asked.\n\n\"Whatever I need until I leave the crypt,\" Seth said. \"The one who serves me will be released when I leave. You're never going to get what you really want by staying in this forsaken burrow. I have met tons of wraiths who live better than this. I freed some recently. Now I'm here to free you.\"\n\nWho will serve you? a wraith asked.\n\n\"The best one,\" Seth said.\n\n\"That would be me,\" one of the wraiths vocalized. No others challenged him. \"I accept your offer.\"\n\n\"Smart wraith,\" Seth said. \"What about the rest of you?\"\n\n\"Yes,\" a wraith said.\n\nYes, another conveyed.\n\nYes.\n\nI wish freedom, but to remain here for a time, a wraith expressed timidly.\n\nI wish the same, the final wraith added.\n\n\"If you promise not to harm me, you may stay here, free to leave when you choose,\" Seth said.\n\nI agree.\n\nAs do I.\n\n\"You who will serve me,\" Seth said. \"Do you promise me no harm during your service and afterward?\"\n\n\"You have my pledge,\" the wraith assured him.\n\n\"Congratulations,\" Seth said. \"The path to freedom begins now.\"\n\nMay we leave in the night? one of the wraiths asked.\n\n\"Sure,\" Seth said. \"As long as you don't trouble me.\"\n\nAll but one of the wraiths withdrew to their niches in the walls. Seth felt tension leaving his muscles. He had not been sure if the invitation would work.\n\n\"Ready for your first task?\" Seth asked the wraith who had agreed to serve him.\n\n\"Speak and I will serve,\" the wraith replied.\n\n\"Do you have a name?\" Seth wondered.\n\n\"Call me what you will,\" the wraith said.\n\n\"How about Midnight?\" Seth proposed.\n\n\"As you desire,\" Midnight replied.\n\n\"I need you to guide me safely to Dezia,\" Seth said.\n\n\"You claimed your message was for us,\" Midnight said.\n\n\"I have other business with her,\" Seth said.\n\n\"I shall escort you to her door,\" Midnight said. \"I cannot cross the threshold.\"\n\n\"Lead the way,\" Seth said.\n\nThe wraith strode away from the chamber down a winding corridor of natural stone. Seth followed, aware that the other wraiths were now behind him. He hoped their agreement not to harm him was binding.\n\nBefore the end of the corridor, Midnight stopped. \"If you look upon any of the idols in the next room, you will go blind. I can guide you.\"\n\nSeth tucked his hand inside his sleeve and extended his arm. \"All right.\"\n\nMidnight took his sleeve, and Seth closed his eyes. Icy cold spread from the place where the wraith touched his clothing. The wraith led him smoothly forward along a serpentine route. Eyes squeezed shut, Seth took halting steps, worried about running into obstacles, but the wraith led him true.\n\n\"You may look again,\" Midnight said, releasing his sleeve.\n\n\"We're through the room?\" Seth checked.\n\n\"Yes.\"\n\nSeth opened his eyes to find himself in another natural corridor of the cave. He realized that if there had been no room containing idols, and Midnight had been fooling him, there was no way for him to know. Seth started rubbing warmth back into his arm as the wraith led him forward.\n\nThe temperature began to rise, and Seth saw a red glow ahead. Around the next bend, they came into view of a circular room with a floor of hot coals, mostly black, but with irregular veins of fiery red. The air shimmered with heat.\n\n\"Dezia dwells beyond,\" Midnight said. \"I will wait here.\"\n\nSeth saw an open door on the far side of the room. It would take four or five consecutive jumps to get across. Even from down the hall, the heat was intense.\n\nThe coals were burning too uniformly and with too much heat to have been recently lit. He suspected some magic kept them eternally smoldering.\n\nHe edged forward, stopping about ten feet away from the room, when the heat became unbearable. His hair starting to singe, Seth took a few steps back to where the temperature was endurable.\n\n\"Can you cool it down?\" Seth asked Midnight.\n\n\"Alas, no. These coals mark the beginning of Dezia's dominion,\" the wraith replied.\n\nSeth flexed his fingers. How would it go if he tried to sprint across, maybe jumping as needed? He strongly suspected his carcass would get added to the coals.\n\nWhat if he cooled the coals? The room was orders of magnitude hotter than a torch. Seth closed his eyes and focused on the relentless heat. He turned his focus inward, searching for the darkness within and the associated reservoir of cold. He located the source and circled it, feeling for the coldest place from which to draw. Then he shifted his energy to project a steady flow toward the coals. The effort seemed futile against the intense heat, but as Seth poured out all the cold he could summon, he felt the temperature of the nearest coals wavering.\n\nGood, a whispery female voice spoke to his mind, startling him. Not the intended method of passage, but clever. Keep trying.\n\nSeth redoubled his efforts, trying to fully extinguish the coals the way he had doused the torch with Ronodin. The coals seemed to yield voluntarily to his power, and soon he had cooled a wide stripe of them across the center of the room. But the coals to either side continued to radiate significant heat.\n\nIn previous attempts, Seth had siphoned cold from himself to douse fire; now he wondered if perhaps he could call upon it to protect his body from heat. Drawing more gently from his power, Seth directed the cold to encompass his body, and the chill seeped over him until he was shivering.\n\nKeeping the coldness close, Seth raced onto the path of neutralized coals across the center of the room, kicking up ashes with each crunchy step. Intense heat encroached on the cold cocoon he had summoned, and he drew on the cold more powerfully until he dashed through the far doorway. As he staggered down a short hall beyond, the heat receded, and he disengaged from the darkness within.\n\nA few steps later, Seth found himself entering a domed cavern. Panting, he held up his holly wand. Crude pictograms decorated the smoother parts of the circular stone wall. Other than the archway he had used to enter, there appeared to be no other passages out of the room.\n\n\"Dezia!\" Seth called.\n\nI hear you, the phantom responded. It was the same susurrant voice that had aided him with the coals.\n\n\"Where are you?\" Seth asked.\n\nA short figure stepped into view through the far wall of the cavern, her dark form semitransparent, gauzy wrappings flowing as if in a strong breeze, although the air was still. The figure seemed to be wearing a wooden mask with crude holes cut for eyes, nose, and mouth, embellished by ornamental carvings. But when she spoke, the mouth of the mask moved.\n\n\"What business have you with me, shadow charmer?\" Dezia inquired, her voice matching the one Seth had heard in his mind.\n\n\"I'm looking for a flower,\" Seth said.\n\n\"Tell me more,\" the phantom said.\n\n\"The Everbloom,\" Seth explained.\n\n\"And why shouldn't I kill you for your impudence?\" Dezia asked. \"You discharged my wraiths.\"\n\n\"I'm on an errand for the Underking,\" Seth said.\n\nDezia waved a dismissive hand. \"What is that to me?\"\n\n\"Isn't he your king?\" Seth asked.\n\n\"The Underking is one of many powers in the darkness,\" Dezia said. \"I have no obligations to him.\"\n\nSeth had hoped his errand would carry more weight. Maybe she was posturing for bargaining power. \"I'm sorry about the wraiths. I needed to reach you.\"\n\n\"You're a shadow charmer,\" Dezia said. \"All you required was access. Dismissing them lacked elegance and courtesy. Had you left the wraiths in place, my secrets would have been better guarded from your rivals.\"\n\n\"I can ask if any of them want to stay,\" Seth said.\n\n\"Not likely after your invitation to freedom,\" Dezia said. \"Blunt but effective. I see some entities of note have offered to train you. I also observe that you have destroyed some powerful demons.\"\n\n\"How can you tell?\" Seth asked.\n\n\"The signs are there,\" Dezia said. \"The records of your interactions. It is evident that you stood in the presence of the Underking. No small feat for a mortal.\"\n\n\"I was telling you the truth,\" Seth said.\n\n\"You may know some true things,\" Dezia said. \"But no mortal possesses very much truth. A mortal telling the truth is like a blind woman painting a portrait.\"\n\n\"I know plenty of truth,\" Seth asserted.\n\n\"It can be difficult to perceive what one does not know,\" Dezia allowed. \"If I had a mind to teach you, there is much I could share.\"\n\n\"I'm looking for the Everbloom,\" Seth said.\n\n\"Merely the Everbloom,\" Dezia said. \"Do you understand what you are asking?\"\n\n\"It's a flower that never dies,\" Seth said.\n\n\"It's the supreme source of power on these islands,\" Dezia said.\n\n\"The Underking wants it.\"\n\n\"Everyone wants it. What if I refuse to help? Am I to be your next victim?\"\n\n\"I'm not looking for trouble,\" Seth said. \"Just information.\"\n\n\"You found trouble the moment you entered my crypt,\" Dezia said. \"You are either quite good at concealing your thoughts, or else your past is a fog.\"\n\n\"Do you know where I can find the bloom?\" Seth asked.\n\n\"That was your best question so far,\" Dezia said. \"There are limits to what I know about the location of the Everbloom.\"\n\n\"What can you tell me?\" Seth asked.\n\n\"I know generally where the Everbloom grows,\" Dezia said. \"Not specifically.\"\n\n\"Will you share what you know?\"\n\n\"Why should I?\"\n\n\"To help the Underking?\"\n\n\"Are you here because you love the Underking, or because you owe him?\"\n\nSeth could think of no clever response. \"I owe him.\"\n\n\"You want me to help you repay a debt to the Underking,\" Dezia said. \"What will you do for me?\"\n\n\"I'll let you teach me something,\" Seth tried.\n\n\"You'll let me do you a favor as payment for a favor?\" Dezia verified.\n\n\"How about we play hot or cold,\" Seth proposed.\n\n\"Hot, I'll wager,\" Dezia said.\n\n\"What is that supposed to mean?\" Seth asked. \"Don't I have to guess first?\"\n\n\"I won't help you that way,\" Dezia said.\n\n\"What do you want?\" Seth asked. \"What can I do for you?\"\n\n\"We're back to useful questions,\" Dezia said. \"I was losing hope. Do you know anything about me? Or about phantoms?\"\n\n\"You were created by a tragedy,\" Seth said.\n\n\"Are you deliberately explaining it poorly so I will correct you?\" Dezia asked. \"Not a bad tactic, but I will resist the urge.\"\n\n\"How were you created?\" Seth asked.\n\n\"Did you actually come here this unprepared?\"\n\n\"I rely on my charm,\" Seth said.\n\n\"Then you came to me unarmed. Many years ago, fifteen firewalkers lost their lives inside the volcano Baga Loa. They are buried here under those coals I helped you cross. Their demise opened my gateway to your world.\"\n\n\"Are firewalkers people?\" Seth asked.\n\n\"You are a stranger to the islands,\" Dezia said. \"As nereids are to the sea, and naiads are to fresh water, firewalkers are to volcanoes.\"\n\n\"Volcano nymphs?\" Seth asked.\n\n\"The true volcano nymphs are lost,\" Dezia said. \"The firewalkers are their descendants. They live more like mortals than proper nymphs. They wed and have offspring. And they tend to Baga Loa.\"\n\n\"Can they swim in lava?\"\n\n\"Almost,\" Dezia said. \"You asked what I want. Pledge to cast my token into the volcano, and I will reveal what I know about the Everbloom.\"\n\n\"What will the token do?\" Seth asked.\n\n\"Your throwing it into Baga Loa will grant me greater freedom to roam,\" Dezia said.\n\n\"What does that mean?\" Seth asked.\n\n\"If the sanctuary fully falls, I will be able to leave this place and find a new abode,\" Dezia said. \"If it stands, on festival nights I will be able to explore the island.\"\n\n\"Will you come bother me if you can roam?\" Seth asked.\n\n\"Only if you fail to cast the token into the volcano,\" Dezia said. \"Such failure would enable me to haunt you.\"\n\n\"How long do I have to get it there?\" Seth asked.\n\n\"One year from today,\" Dezia said.\n\n\"Where is the token?\" Seth asked.\n\nDezia drifted to a low stone altar on one side of the room and gestured at a primitive wooden doll. It had reedy hair made from dried grass, divots for eyes, and a simple smile painted onto the smooth, dark wood.\n\n\"A doll?\" Seth asked.\n\n\"My token,\" Dezia said. \"Will you deliver it to the volcano within one year if I tell you what I know about the location of the Everbloom?\"\n\nSeth doubted he would get a better offer. \"Sure.\"\n\n\"Then take custody of my token,\" Dezia said.\n\nSeth picked up the doll.\n\n\"The Everbloom grows somewhere within Baga Loa,\" Dezia said.\n\n\"Inside the volcano?\" Seth asked.\n\n\"A complex system of caves leaves portions of the mountain hollow,\" Dezia said. \"The firewalkers know those paths better than anyone. I am only sure that one firewalker knows the exact location of the Everbloom within those secret windings. Her name is Savani, and she is the caretaker of this sanctuary.\"\n\n\"That's all you know?\" Seth asked.\n\n\"Yes,\" Dezia said. \"I gave you the general location of the Everbloom, and I told you who can pinpoint the exact location. Now take my token and go. Hurry. With the pool gone, I mean to collapse the outer areas of this crypt to prevent easy entry.\"\n\n\"Now?\" Seth asked, looking toward the room full of hot coals.\n\n\"Summon your cold,\" Dezia said. \"I will aid you. Then run.\"\n\nSeth mustered all the cold he could and directed it toward the coals. Again, a central aisle of coals lost their heat. He wrapped cold around himself and ran, eyes squinting against the temperature and the particles in the air.\n\nAt the far side of the coals he found Midnight. \"We have to go,\" Seth said. \"This crypt is going to collapse.\"\n\n\"That poses no problem for me,\" Midnight said. \"But I will help you.\"\n\nSeth had almost forgotten about the room where he needed to close his eyes. \"Yes, please.\"\n\nHe hurried along the corridor, again tucking his hand into his sleeve and allowing Midnight to lead him. As Seth blindly followed the wraith, he listened for the rumblings of a cave-in. By the time he opened his eyes, he felt the first tremors.\n\nWhen Seth reached the chamber with the wraiths, the walls trembled, gently at first, then with greater vigor.\n\n\"Thank you,\" Seth called to Midnight, running for the entrance. \"Depart when you want!\"\n\nMidnight stayed close behind him. \"I am free when you emerge from the crypt?\"\n\nSeth lurched and stumbled as the quaking increased. Cracks were forming in the walls. Behind him, part of the ceiling collapsed.\n\n\"Yes,\" Seth said, pinballing off the walls of the corridor until he emerged from the crypt into the empty pond. Dust gushed from the opening behind him, accompanied by the thunder of falling rocks.\n\nFacing away from Seth, on the far side of the pond, stood a gigantic creature shaped like a man, but with a tough, gray hide like a rhinoceros and lumpy muscles across his back, legs, and arms. The huge brute clutched a wooden club that might once have been the trunk of a palm tree. Even at a distance, Seth could tell he came no higher than the creature's thigh.\n\nThe monstrosity turned, revealing a broad face with a flat nose and a huge forehead that accommodated three eyes spaced roughly in the shape of a triangle. As Seth had begun to expect, it was the triclops.\n\n\"Run!\" Ronodin called from somewhere beyond the triclops, and a rock struck the monster, turning its attention away.\n\nMore dust belched from the crypt, reminding Seth that the cave would offer no shelter, but also providing Seth with some cover. Keeping within the dust cloud as best he could, Seth raced up the side of the empty pond, away from the triclops.\n\nOnce out of the muddy crater, Seth crashed into the jungle, plowing through ferns and shrubs with reckless haste. By the time he looked back, vegetation impeded his view of the triclops.\n\nWhile glancing back, Seth blundered into something spongy and sticky, which to his disgust turned out to be the bell-shaped cap of a colossal yellow mushroom. Supported by a short, thick stalk, the bulbous cap went from Seth's knees to well above his head. To his horror, Seth found that every part of his body that had touched it could not pull away\u2014his right shoulder, right arm, right hand, right side, and right leg down to the knee were held fast as if by instant, perfect glue.\n\nHastily trying to wrench free, Seth made the mistake of pushing with his left palm, and that hand became immediately stuck to the adhesive surface. Seth struggled, hoping to tear off part of the mushroom or perhaps snap it free from the ground, but it was like trying to outpull a stubborn elephant.\n\nRather than succumb to his efforts, the mushroom, as if by some predatory instinct, leaned away from Seth just enough to lift him off the ground, causing more of his body to press against the sticky surface. Pathetically dangling from the tenacious mushroom, Seth kicked his free leg in frustration.\n\nBehind him, the triclops roared with a primal fury that triggered enough panic to halt Seth's struggles. Holding his breath, he prayed the behemoth would not venture his way.\n\nKendra crouched in the bushes near the drained pond, hands over her ears to dampen the roar of the triclops. After throwing a few rocks, Warren had hit the creature near one of the eyes, and now it was charging him. But Kendra's attention was focused on the place nearly halfway around the pond where her brother had vanished into the jungle. She had come to the corrupted pool hoping for a clue to finding Seth. The possibility that he would actually be there had not entered her mind.\n\nThe journey to the pool had been relatively calm until the last few minutes. Hako, a grim Polynesian man with a broad chest and thick limbs, had led them along quiet paths toward their destination, pointing out hazards along the way. As they traveled, Kendra had found two fairies, Nia and Beruni, who had joined her.\n\nThe trouble began as they drew near the pool. Hako had raised a fist, bringing the group to a halt, and after a few moments Kendra heard distant branches snapping in the jungle behind them.\n\n\"Mombatu,\" Hako had whispered, crouching low to place one hand on the ground. \"Scatter. Remember, he is drawn to motion. I will lure him past you.\"\n\nKendra had raced off the path beside Tanu, the two fairies flitting nearby. Warren, Vanessa, Doren, and Knox ran in the opposite direction. Hako had previously warned that if the triclops found them, their best chance for survival would be to separate and individually make their way back to the protected area around the Monkey Maze. When Kendra and Tanu reached some bushes near the edge of the vacant pond, Tanu had pointed to a fern that would conceal her and then scrambled off to find his own cover.\n\nShortly after Kendra took shelter beneath the lush fronds, she heard Hako taunting Mombatu. As Hako led the triclops from the jungle into the wide depression of the empty pond, to her surprise, Kendra saw Ronodin fleeing into the trees. Hako tumbled into the mud at the bottom of the empty pond and promptly disappeared. Whether the camouflage was through skill or magic, Kendra could not see him, and neither could Mombatu, who stomped around the muddy crater, brandishing his club in frustration.\n\nAfter that, the ground had begun rumbling, and seconds later, to her complete shock, Seth had emerged from the mouth of a cave in the waterless pond. Kendra had seen him only briefly before dust billowed from the opening and obscured her view. It was so unexpected that she hardly believed her eyes, but it was his face, his expression, his stance. Was Seth really alive? Was he free? Was he involved in some sort of adventure here on the island? Had she gotten lucky enough to encounter him? Somebody yelled for Seth to run, Warren started throwing rocks at Mombatu, and her brother dashed out of view.\n\nNow the triclops was pounding off into the jungle after Warren, and Hako emerged from the mud to chase Mombatu. According to the plan, if the triclops attacked, Kendra was supposed to withdraw immediately to the protections of Crescent Beach. But having glimpsed Seth, she knew retreat was no longer an option. Not until she caught up to him.\n\n\"Did you see the boy who came out of the cave?\" Kendra asked the fairies.\n\n\"Not as well as the triclops did,\" Nia said with a giggle. \"I only have two eyes.\"\n\n\"I saw him,\" Calvin said. \"It was Seth.\"\n\n\"I took some small notice of him,\" Beruni said.\n\n\"Can you fairies lead us to the boy?\" Kendra said. \"Going around the pond? To avoid being seen by Mombatu?\"\n\n\"This way,\" Nia said, fluttering ahead.\n\nAdvancing in a crouch, Kendra followed. Both fairies were hard to miss with their elaborate butterfly wings. She could hear Mombatu bulldozing through the jungle behind her, thankfully, not in her direction.\n\n\"Kendra!\" Tanu called in a stage whisper, appearing out of hiding. \"Where are you going? The lagoon is the other way.\"\n\n\"Seth is here,\" Kendra said. \"I saw him.\"\n\n\"He's here?\" Tanu exclaimed.\n\nKendra realized he must not have been in a position to view the empty pond when Seth had emerged. \"The fairies are leading me to him. I have to go.\"\n\n\"Be careful,\" Tanu urged. \"I'll check on Knox and the others. We'll draw the triclops away from you.\"\n\nKendra picked up her pace. If Seth had kept running, he had a big head start. Hopefully he was hiding. Either way, she wasn't going to give up until she found him. If he ran faster, she would run longer.\n\n\"Beruni,\" Kendra said, \"fly ahead and find Seth, then come report.\"\n\nBeruni darted out of view.\n\nKendra forced her way through the dense undergrowth, stumbling over creepers and roots, pushing aside fronds, rustling through ivy, trying to be swift and silent, with little success at either. Mombatu roared again, the naked rage of the bellow making Kendra fear for Knox and her friends.\n\n\"Nia, remember I can't fly,\" Kendra said. \"Lead me the best way for a clumsy human to go. And help me avoid anything dangerous.\"\n\n\"Don't drink from those,\" Nia replied, indicating a creeping vine with leaves shaped like cups. \"You'll hallucinate for hours.\"\n\nBeruni flew back into sight. \"He's not far ahead.\"\n\n\"Is he all right?\" Kendra asked.\n\n\"He won't be, without help,\" Beruni said.\n\nBreaking into a run, Kendra jumped a small trench, shoved through a curtain of vines, and wove among ferns until her brother came into view, his feet off the ground, most of his body affixed to the side of a huge mushroom the color of Dijon mustard. He was facing away from her as she approached, his free leg kicking halfheartedly. Part of her had worried she might never see him again, or that he would run from her, so finding him immobilized was a relief. But immediately she remembered the way he had looked at her like a stranger back at Stormguard Castle, and she wondered whether he would accept her help.\n\n\"Seth!\" Kendra ventured in a loud whisper. \"Are you okay?\"\n\n\"Do I look okay?\" he replied quietly. \"Who are you? I can't turn my head.\"\n\nKendra walked around to where Seth could see her. Even with one cheek mashed against the side of the mushroom, it was amazing to see his face. For a moment she was speechless. It was her brother, frustrated, dangling helplessly. She really had found him!\n\n\"Kendra?\" Seth asked.\n\n\"Do you remember me?\" Kendra asked.\n\n\"Sure, from the castle,\" Seth said. \"The first person I can remember. My sister, according to you. What are you doing here?\"\n\nKendra tried to resist feeling deflated by his continued lack of recognition. \"Looking for you,\" she said.\n\n\"Congrats,\" Seth said. \"I could use some help.\"\n\n\"A local warned me about these mushrooms,\" Kendra said. \"It will slowly absorb you.\"\n\n\"Unless the triclops decides I make a great topping for a fungal appetizer,\" Seth said.\n\n\"I can hardly believe it's really you,\" Kendra said.\n\n\"Please get me down,\" Seth said.\n\n\"Can you free him?\" Kendra asked Nia.\n\n\"No problem,\" Nia said. \"We have experience with death knells.\"\n\n\"Is that the name of this type of mushroom?\" Kendra asked.\n\n\"Because of the bell shape,\" Nia said. \"And the deadliness.\"\n\n\"Do fairies get stuck to them?\" Kendra asked.\n\n\"No self-respecting fairy,\" Beruni said with a sniff.\n\n\"Maybe in a windstorm,\" Nia said. \"Sometimes friendly animals get trapped. We help when we can. Should we free him?\"\n\nKendra considered the situation. Seth might do anything if he were free. He might turn on her. Physically, this was her brother, but mentally, who knew? Right now, the triclops was grunting and growling somewhere on the far side of the empty pond. If she wanted to convince Seth he was helping the wrong side, she needed him to listen. It might not be smart to immediately end his captive-audience status.\n\n\"Leave him for the moment,\" Kendra said, careful to speak in the language the fairies were using. \"Keep watch for the triclops. If Mombatu comes this way, free Seth immediately.\"\n\n\"Not you too!\" Seth complained. \"Not understanding the fairies was bad enough. What are you plotting?\"\n\n\"I wanted to make sure they understood me,\" Kendra said.\n\n\"Can they get me down?\" Seth asked.\n\n\"Probably,\" Kendra said.\n\n\"Then maybe we should hurry before I end up as a pizza topping.\"\n\n\"First I want to know where you have been.\"\n\nSeth sighed heavily. \"I get it. This is an interrogation.\"\n\n\"I want to help you,\" Kendra said. \"But you have been working against us.\"\n\n\"You shouldn't even be here,\" Seth said. \"Aren't you supposed to be watching over the dragon prison called Wyrmroost?\"\n\n\"I was the caretaker of that dragon sanctuary,\" Kendra said. \"So were you. Until you released the undead at our main fort and the sanctuary fell. Now I'm not caretaker of anything.\"\n\nMombatu roared loud enough to make Kendra flinch, though the triclops remained a fair distance away. Seth looked at her with startled eyes.\n\n\"Get me down,\" Seth said. \"My skin is tingling where it's touching the mushroom. I think it's already digesting me.\"\n\n\"Where have you been, Seth?\"\n\n\"You know.\"\n\n\"I really don't.\"\n\n\"The Under Realm.\"\n\n\"Where is that?\"\n\n\"It's the realm of the Underking.\"\n\n\"And now you're here?\" Kendra asked.\n\n\"For the moment,\" Seth said. \"We'll go back to the Under Realm.\"\n\n\"Is it nearby?\"\n\n\"It wouldn't be fair to say too much.\"\n\n\"Do you have to go back?\" Kendra asked. \"Could you stay? If you wanted?\"\n\n\"Not really,\" Seth said. \"I'm trying to earn my freedom from the Underking.\"\n\n\"Sounds like a great situation. Who took you there? The Sphinx?\"\n\n\"The guy from the castle. Well, a puppet he sent.\"\n\n\"Ronodin sent Mendigo? Seth, you were kidnapped.\"\n\n\"He called it a rescue. Meanwhile, you're keeping me stuck to a mushroom.\"\n\n\"I'm on your side,\" Kendra said. \"I'm your sister. You were taken from your family by an evil unicorn.\"\n\n\"And now you're risking me getting eaten by a triclops,\" Seth said.\n\n\"I'll get you down if the triclops comes this way,\" Kendra said.\n\n\"Are you daring me to call it?\" Seth asked.\n\n\"Stop being dumb,\" Kendra said.\n\n\"Stop holding me prisoner,\" Seth said.\n\n\"The dark unicorn is holding you prisoner,\" Kendra said.\n\n\"According to you and the other unicorn,\" Seth said.\n\n\"What other unicorn?\" Kendra asked urgently.\n\n\"That got your attention,\" Seth said. \"Help me down and I'll tell you.\"\n\n\"You're infuriating,\" Kendra said. \"We're on the same side. Stop making this hard.\"\n\n\"Everyone says I'm on their side,\" Seth said. \"Everyone wants things from me.\"\n\n\"What do you want?\" Kendra asked.\n\n\"Honestly? Some time on my own. Without anyone trying to control me.\"\n\n\"That could be good,\" Kendra said. \"Can you get time alone?\"\n\n\"Not yet,\" Seth said. \"Soon, I hope.\"\n\n\"Ronodin won't let you go,\" Kendra said. \"But if you're foolish enough to help him, he'll take advantage of you for as long as you allow it.\"\n\n\"If you're really my sister, get me down and let me work this out,\" Seth said.\n\n\"Of course I'm your sister,\" Kendra said. \"Don't you see the family resemblance? What do you want to know about yourself?\"\n\n\"Everything,\" Seth said. \"But I don't trust anyone enough to tell me. I don't know what stories to believe.\"\n\n\"Is Bracken in the Under Realm?\" Kendra asked.\n\n\"Good guess,\" Seth said. \"All caged up.\"\n\n\"He's all right?\" she asked, at first trying to hold back the tears, then almost glad for them, hoping they would show Seth her sincerity.\n\n\"He's not injured,\" Seth said. \"I mean, he's locked up in the Under Realm, so he could be better.\"\n\nKendra nodded, trying to find words. This whole situation was so twisted, having her own brother not trust her, hearing him speak glibly about Bracken's incarceration. But it wasn't Seth's fault. He was doing his best with limited information. She had to try to reach him.\n\nKendra took a step closer to him and whispered, \"You have to be curious about what happened. You have to wonder. You may not believe me, but here it is\u2014we were racing against dragons to get a powerful talisman called the Wizenstone. You and I were running from the dragons, trying to get into the room where the Wizenstone was kept, but to open the door, the price was your identity. You knew that and volunteered to do it. You were so brave. After you opened the door, you looked at me like a stranger.\"\n\n\"My first memory,\" Seth said.\n\n\"Then Ronodin showed up and started manipulating you,\" Kendra went on hurriedly. \"You saw what happened with Humbuggle and the stone. We got teleported away. And the Sphinx helped Ronodin kidnap you using the barrels. Before long he had you destroying the sanctuary you lost your memory to save.\"\n\n\"Sad story if it's true,\" Seth said. \"You've had time to work on it.\"\n\nKendra struggled to ignore the crushing disappointment of his disbelief. \"You're incredibly curious,\" she went on. \"You're not great at obeying rules. But you're not bad, Seth. You're good. You've fought some of the most dangerous villains in the magical world. You held strong and beat them. You have a true heart. That has to win out in the end.\"\n\n\"I talk to wraiths and phantoms,\" Seth said.\n\n\"You have dark powers,\" Kendra said. \"You never used them for evil. At least, not when I knew you.\"\n\n\"I'm not trying to do anything wrong,\" Seth said. \"If you're so good, why were you holding all of those undead creatures prisoners in a well?\"\n\n\"Because they're dangerous,\" Kendra said.\n\n\"You lock up anything you decide is dangerous?\" Seth asked. \"Dragons? What else?\"\n\n\"These preserves protect magical creatures from the outside world,\" Kendra said. \"They also protect the outside world from the creatures.\"\n\n\"Ronodin calls them prisons,\" Seth said.\n\n\"They may look that way,\" Kendra said. \"But the creatures aren't in cells. The dragons hunt. The satyrs play. They just have boundaries.\"\n\n\"Imposed by you?\" Seth asked.\n\n\"Imposed by treaties,\" Kendra said, \"after wars that nearly destroyed the world.\"\n\n\"The wraiths were in a cell,\" Seth said. \"A deep one.\"\n\n\"I guess some of the most dangerous creatures get treated like prisoners,\" Kendra said. \"I didn't set up the system. It might not be perfect. But it's definitely better than the end of humanity.\"\n\n\"If I tell you I believe you, will you let me down?\" Seth asked.\n\n\"I'm not trying to force you,\" Kendra said.\n\n\"Leaving me attached to a killer mushroom comes really close,\" Seth replied.\n\n\"You're different in some ways,\" Kendra said. \"You're you, but some of the feel is different. Where is your satchel?\"\n\n\"Back at the Under Realm,\" Seth said.\n\n\"You came without it?\" Kendra asked. \"That's hard to believe.\"\n\n\"I guess I always used to bring it,\" Seth said.\n\n\"It's full of your gear,\" Kendra said.\n\n\"Like a single glove?\"\n\n\"The glove makes you invisible when you hold still,\" Kendra said.\n\n\"That would have been good to know. What else?\"\n\n\"Lots of stuff,\" Kendra said. \"A knife. You always have some rope. You have a little statue of a leviathan that turns into a real one if you put it in the sea and say the right words.\"\n\n\"What words?\" Seth asked.\n\n\"I don't know,\" Kendra said. \"You probably wrote them down.\"\n\n\"I can check,\" Seth said. \"If I live.\"\n\n\"Would it convince you?\"\n\n\"An enemy might know the same specifics about me as a friend does,\" Seth said. \"In fact, an enemy might be more concerned with using little details to sway me to their side.\"\n\n\"A friend might do the same,\" Kendra said. \"Or a sister who loves you.\"\n\n\"You seem sincere,\" Seth said. \"I can tell you really care. That's kindness, at least. I haven't seen that out of Ronodin.\"\n\n\"I just want to help you,\" Kendra said. \"We grew up together.\"\n\n\"Can you understand how strange it is that you know me deeply, while for me, there is nothing familiar about you? My memory is fried. What you say could be true. If it is, I'm lucky to have such a kind sister.\"\n\n\"It's really frustrating that nothing seems to reach you.\"\n\n\"I hear your version of things,\" Seth said. \"If you're telling the truth, I'm so sorry. If you're lying, shame on you. Trust me to sort it out.\"\n\n\"Why are you here, Seth?\" Kendra said. \"What are you doing?\"\n\n\"I'm dangling from a mushroom,\" Seth said.\n\n\"Why did you come?\"\n\n\"Back to interrogation,\" Seth said.\n\n\"Ronodin corrupted that sacred pool,\" Kendra said. \"Why did you come here? What were you doing down there?\"\n\n\"Pushy,\" Seth said.\n\n\"I want to help,\" Kendra said.\n\n\"Then help me down,\" Seth said.\n\n\"You really don't have time for this,\" said a hushed voice from behind.\n\nKendra whirled to find Ronodin approaching. She reached for her bow and put a hand on the string.\n\n\"I'm not looking for a fight, unless you insist,\" Ronodin said. \"We have a bigger concern right now. With three eyes.\"\n\n\"I'll never forgive you for what you've done to my brother,\" Kendra said.\n\n\"As if I want or need your forgiveness,\" Ronodin said. \"You can be a convincing actress in a pinch\u2014I'll grant that you deliver your lines with feeling.\"\n\n\"What are you doing here?\" Kendra asked.\n\nRonodin went from whispering to yelling. \"I'm wondering when that brainless, blundering triclops will find us!\"\n\n\"Don't,\" Kendra whispered, wincing.\n\nRonodin lowered his voice again. \"You better take that speed potion Tanu prepared. I saw some of the others using it. If you don't lure that triclops away, we're all doomed.\" He lit a string of firecrackers and threw them at the base of a nearby tree. They began to explode with a sound like machine-gun fire. Kendra flinched at the racket.\n\n\"Run along,\" Ronodin encouraged.\n\nKendra moved her hand from the bowstring to a little flask. The dark unicorn was right about the potion. It was her insurance if Mombatu got close.\n\n\"Free him,\" Kendra said to Nia as she slung her bow over her shoulder.\n\n\"In exchange for your help, I offer you a hint,\" Ronodin said. \"Call it pity. You will simplify your problem if you lure Mombatu to the protected boundary.\"\n\nThe two fairies approached Seth. After several flashes of light, he dropped from the mushroom to the ground. Once the firecrackers stopped bursting, Kendra heard the rumble of Mombatu's approach.\n\n\"Will the boundary destroy him?\" she asked.\n\n\"In a manner of speaking,\" Ronodin said. \"Stop dawdling with the potion. I know how to avoid Mombatu. Your brother doesn't.\"\n\nKendra glanced at Seth. It hurt to think of losing him again. But Ronodin seemed willing to sacrifice him if Kendra didn't draw off the triclops. Seeing no alternative, she uncapped the flask and raised it to her lips. The fluid inside bubbled like soda and had a flavor like spicy grapes. She consumed it in three fizzy swallows.\n\nAnd everything slowed down.\n\nNia's wings swished back and forth lazily. Seth inched toward Ronodin. And Mombatu came into view, his mouth opening to roar as his three eyes surveyed the scene.\n\nKendra crouched and picked up a stone as the roar began. She flung it at his eyes, and the rock struck his cheek. Raising his club above his head, the triclops swung it down at her slowly enough for her to lunge out of the way. The blow flattened part of the sticky mushroom.\n\nAfter she hit Mombatu in the mouth with a second rock, he trained all three of his eyes on her. The snarling triclops ripped his club free and gave chase as she ran away. Behind Mombatu, Ronodin and Seth lay flat on the ground, silent and still, watching.\n\nDodging around trees and the larger plants, Kendra dashed toward the empty pond. Tanu had warned that the effects of the speed potion would not last very long and would leave her exhausted when it wore off, so she decided to sprint across open ground to maximize her temporary advantage. Also, she figured that crossing the empty pond would more efficiently get her to friends who could help distract Mombatu.\n\nIt agonized her to be running away from Seth. After all her worries, having found him, she was fleeing, leaving her brother with their enemy. But with the triclops in pursuit, what else could she do?\n\nMombatu raced after her, his ferocious roar spurring her on. She rushed down into the drained area, mud accumulating on her shoes and spraying into the air with every sloppy step. Nia and Beruni flanked her, despite their wings seeming to flap slower than normal.\n\nAs Kendra reached the far side of the pond, exhaustion began to take hold. Each breath became laborious, and the muscles in her legs started to burn. A glance back showed the triclops pounding through the mud at the middle of the empty pond, huge feet sinking to well above the ankles, his three-eyed gaze locked on her. After a couple more steps, the underbrush screened her view of her pursuer.\n\nFighting the fatigue in her limbs, Kendra charged forward, wading through ferns and knifing through gaps in heavier vegetation. While the potion lasted she could stretch her lead, but after it failed, Mombatu would have the advantage.\n\nMaintaining her top speed became increasingly hard. Breathing hurt, and no inhalation seemed to bring quite enough oxygen. Her legs began to feel wobbly, her head throbbed with each step, and her vision became hazy.\n\nBehind her, she heard branches cracking as the triclops reached the jungle. Despite her best efforts, Kendra had slowed, but she dug deep to resist the rubbery fatigue in her legs. And then the effects of the potion ended abruptly. The fairy wings beside her fluttered like normal. She staggered and had to brace herself against the side of a tree to keep from falling. Her strength had all but disappeared.\n\nThere was a triclops chasing her. She had to keep going. But she felt so light-headed that she wondered whether she would faint if she remained standing any longer, let alone if she tried to advance.\n\nStrong arms scooped her up and started running. Dreamily she realized that with somebody carrying her, she could relax for a moment. She glanced to see who held her and smiled. It was Warren.\n\n\"It's Vanessa's turn to distract Mombatu,\" Warren whispered. \"Sorry he came your way. We couldn't compete with those fireworks.\"\n\nWarren set her down and fell flat beside her.\n\n\"Stay low,\" he whispered. \"Hako was right that the triclops ignores stationary targets.\"\n\nHeart beating rapidly, Kendra rested her head against the forest floor. No bed ever felt so comfortable.\n\n\"The mental fog will clear in a moment,\" Warren said. \"Rest and let it pass.\"\n\nKendra heard Vanessa taunting the triclops amid raucous sounds of wood splintering. One good thing about Mombatu\u2014he was not subtle. His movements could be heard from a great distance.\n\nFootsteps approached; then Tanu crouched beside Kendra and waved something under her nose. The piquant tang of his custom smelling salts made her head snap up. Suddenly her mind was alert again, and she wondered how she could have paused to rest with a triclops in pursuit. Nostrils tingling, Kendra hopped to her feet.\n\n\"Quietly,\" Warren instructed.\n\nHe and Tanu led her through the lush foliage. The fairies orbited Kendra from a greater distance now that human companions had joined her. Kendra's body felt tired but functional. She was no longer near the verge of collapse.\n\n\"The speed potion wipes out your strength,\" Tanu said. \"I can't figure out how to get rid of that effect. But your vitality should return fairly quickly.\"\n\n\"The potion saved my life,\" Kendra whispered.\n\n\"That remains to be seen,\" Warren said. \"Hopefully we can lose Mombatu.\"\n\n\"Ronodin claimed that luring Mombatu to the border of the protected area would be our best move,\" Kendra said.\n\n\"I caught a glimpse of Ronodin,\" Tanu said. \"Why lead Mombatu to the border?\"\n\n\"He acted like it would neutralize Mombatu,\" Kendra said.\n\n\"How would a boundary kill him?\" Warren asked. \"It should only repel him.\"\n\n\"Ronodin didn't say it would kill him,\" Kendra said. \"He was being all smug, as if sharing a brilliant secret.\"\n\n\"Do you think he was telling the truth?\" Tanu asked.\n\n\"Ronodin has been an enemy to us,\" Kendra said. \"But the hint felt genuine to me.\"\n\n\"He may want Mombatu out of the way for his own reasons,\" Warren said. \"Ronodin is obviously up to mischief on the island.\"\n\n\"Do we think leading Mombatu to the border would work?\" Tanu asked.\n\nKendra considered it. \"I can't be certain, but I believe him.\"\n\n\"We have our own work to do on this island,\" Warren said. \"If taking the triclops to the border did neutralize him, it would free us up a lot.\"\n\n\"I'll catch up with Hako and talk to him,\" Tanu said, veering away from them.\n\n\"Where is Knox?\" Kendra asked.\n\n\"With Doren,\" Warren said. \"I sent them back. Knox used a speed potion. I used one also, and now Vanessa has too. Hako seems able to manage without one. Helps that he can disappear. He's like a chameleon.\"\n\n\"Is that a magic power?\" Kendra asked.\n\n\"I think so,\" Warren said. \"He tried to explain it to me. Something about becoming one with the island so he doesn't stand out. The reasoning sounded sketchy, but it seems to work.\"\n\nKendra noticed her strength returning. \"I think I can run again.\"\n\n\"Let's just walk quickly,\" Warren said. \"We have a lot of distance to cover, and moving quietly might be better right now than going fast.\"\n\nWalking behind Warren, Kendra trusted him to go the right way and avoid hazards. It was a relief to let somebody else worry about how to avoid getting eaten by a plant. \"Nia?\" she whispered.\n\nThe fairy drifted closer.\n\n\"Would you watch Mombatu?\" she asked. \"Warn us if he gets close.\"\n\nNia darted out of view.\n\n\"This is degrading work,\" Beruni said. \"Helping clumsy humans avoid getting crushed by a superior predator.\"\n\n\"It's my command that you help,\" Kendra said.\n\n\"Somehow you were authorized by the Fairy Queen,\" Beruni said. \"It must have been an act of pity. I have no choice but to do my part.\"\n\nKendra heard some rustling off to one side, and Vanessa emerged from tangled branches, her face scratched and damp with perspiration. She gave Kendra a weary nod.\n\n\"Any body parts missing?\" Warren asked.\n\n\"Hako diverted Mombatu after I ran out of steam,\" Vanessa said. \"Tanu revived me with the salts, though I'm still unsteady.\"\n\n\"Do you think if you bit Mombatu, you could control him in his sleep?\" Kendra asked.\n\n\"In theory,\" Vanessa said. \"His hide looks really thick, though.\"\n\n\"Mombatu would be a seriously powerful puppet to manage,\" Warren said.\n\n\"Would you still love me if I was ugly and over twenty feet tall?\" Vanessa asked.\n\n\"From a safe distance,\" Warren replied.\n\nA roar from Mombatu ended the chitchat and made them all flinch. Feathery fronds brushed by Kendra, rustling gently.\n\nTanu reappeared, flushed and gasping for air. \"Hako thought your plan was worth a try. We're going to lure Mombatu to Crescent Lagoon. Hako is getting tired, though. This will have to be a group effort.\"\n\n\"I'll take a turn baiting the triclops,\" Warren said. \"Want me to use another speed potion? We still have a couple.\"\n\n\"Maybe,\" Tanu said. \"Wait as long as you can.\" He pointed. \"Go that way.\"\n\n\"I'll just follow the sound of trees being uprooted,\" Warren said, hurrying away.\n\n\"I should do more cardio,\" Tanu panted, face shiny with perspiration.\n\n\"You're fast for a big man,\" Vanessa said.\n\n\"Not over long distances,\" Tanu said.\n\n\"I hope Newel and Grady are all right,\" Kendra said.\n\n\"I wish they could have occupied the triclops a little longer,\" Vanessa said.\n\n\"We suspected Mombatu would be a threat today,\" Tanu said. \"At least we didn't have to deal with him until we reached the sacred pool.\"\n\nKendra plodded forward, wondering how long this forced marathon would last. Tanu left to retrieve Warren. When they returned, Warren was leaning on the big potion master, looking pale and depleted. Mombatu continued to tirelessly trample through the jungle, roaring and growling. There seemed to be no end to his energy.\n\nVanessa went to distract the monster, and a short time later Hako caught up with the group. \"We're getting close,\" Hako reported. \"Mombatu is learning our tricks. Pausing more. Listening more. Hesitating before attacking the most obvious target. He isn't used to tracking small, evasive prey. He's more at home going head-to-head with dragons.\"\n\n\"Any bright ideas?\" Warren asked.\n\n\"Let's hope we can string him along a little while longer,\" Hako said.\n\n\"Look out!\" Vanessa yelled urgently from a distance.\n\nKendra heard the triclops plowing toward them.\n\n\"He didn't follow Vanessa,\" Hako said. \"He heard us. I'll try to redirect him.\" He pointed ahead. \"Hurry that way. About five hundred yards.\"\n\nWarren and Tanu broke into a full run, and Kendra followed. Hako raced off on a course to intercept Mombatu. The fairies zoomed up to Kendra and reported that Mombatu was coming.\n\n\"Slow him down,\" she said.\n\nThe fairies zipped away. Hako was shouting. Vanessa heckled Mombatu from farther away, but the triclops did not change course.\n\nA moment later the triclops shouldered aside a tree a little taller than himself and rumbled into view, eyes glaring, body lathered with foamy sweat. Bellowing in triumph, he charged.\n\n\"Split up,\" Tanu ordered, tossing Kendra a small flask.\n\nShe bobbled it but managed to hold on. \"Another one?\"\n\n\"Last one,\" Tanu said. \"Insurance if he gets too close. It should work again\u2014might wipe you out a little more than last time. Vanessa just took her second.\"\n\nWarren swerved left, Tanu cut to the right, and Kendra continued straight. She could run five hundred yards at high speed, couldn't she? Hadn't she run at least that far last time? Tanu had trusted her with the last potion. Shouldn't she help the others?\n\nKendra uncapped the flask and guzzled the fizzy contents. The world slowed down around her. Holding his club high, Mombatu pounded his blocky chest with his free hand as he ran. His lips peeled back, revealing blocky teeth designed for mashing.\n\nGlaring at Tanu, the triclops turned his way.\n\nKendra unslung her bow, pulled the string back, and muttered, \"Two hundred.\"\n\nAs she released the string, a swarm of arrows streaked toward the triclops. With shockingly quick reflexes, Mombatu covered his eyes with his free arm before the arrows arrived. They stuck better to his wooden club than to his thick hide, but the message was received. Mombatu flung his club at Kendra so hard that, even perceiving it at reduced speed, she barely managed to duck in time.\n\nWith a guttural grunt, Mombatu gave chase.\n\nThe race was on.\n\nKendra dashed ahead with everything she had. The exhaustion hadn't set in yet, and she added to her lead. A glance back showed the triclops running hard, arms and legs pumping, humps of muscle clenching and releasing, strings of saliva unfurling from his mouth.\n\nKendra focused on the route ahead. She couldn't waste time by tripping or getting caught in a thicket. She crashed through a stand of bamboo, poles rattling behind her. When she emerged from the bamboo, the exhaustion started. Muscling onward, she knew she had a little more time before she reached her limit again.\n\nScanning ahead for the Monkey Maze or the beach, Kendra saw no landmark to hint at the location of the border. She would just have to keep going until the border stopped the triclops. Or killed him. Or whatever was going to happen.\n\nHer breathing sped up and her lungs ached. The fire in her legs made her cry out. Her heart felt like it would burst. But her pace still resembled a sprint.\n\nBehind her, the stand of bamboo seemed to explode as Mombatu reached it. He wasn't brushing it aside\u2014he was blasting straight through. Glancing back, she saw that the triclops had gained on her. She didn't intend to run slower, but it hurt so much, and the thick vegetation was grasping at her.\n\nHer legs were becoming unreliable. Her vision was blurring.\n\nAnd then the effects of the potion ended.\n\nKendra stumbled forward, trying to maintain enough desperation to keep moving, willing herself not to pass out. She lumbered drunkenly ahead, the ground seeming to tilt back and forth beneath her feet.\n\nWhere were those smelling salts?\n\nShe heard Mombatu gaining quickly, roaring exultantly.\n\nShe reached a clearing. Up ahead, in the distance beyond the clearing, she saw some platforms in the trees.\n\n\"Send help,\" she mumbled incoherently, tottering out into the field.\n\nKendra didn't remember falling, but she noticed when she hit the ground, landing on her side. Woozy, gasping, she pushed up onto her hands and knees and crawled forward. She had to keep moving.\n\nHer arms gave out and she twisted onto her side, looking back as the colossal triclops stormed into the clearing. Raising his balled fists high, Mombatu closed in.\n\nBefore he reached her, with a dazzling flash, Mombatu split into three smaller creatures.\n\nKendra blinked her eyes and shook her head. Surely she was hallucinating. Was she seeing triple? The world was seesawing, and she was imagining the triclops had divided into smaller components.\n\nShe couldn't blink it away. The three beings remained separate. They gathered around her, looming over her, perhaps ten feet tall.\n\nKendra closed her eyes.\n\n\"Are you all right?\" a calm, deep voice asked.\n\n\"Tired,\" Kendra said. \"I took a speed potion.\"\n\n\"You're safe now,\" said another voice, deep and melodious. \"At least safe from us.\"\n\n\"I just need a minute,\" Kendra said, unsure if she could remain conscious. \"Who is talking? Did you really just split into three?\"\n\n\"Within areas protected by the watchers, we are the guardians of the Sunset Pearl,\" a third voice said, less deep than the other two, but perhaps the richest.\n\nKendra tried to sit up, but she slumped back down.\n\n\"Let yourself recover,\" the first voice said.\n\n\"I'm going to sleep well tonight,\" Kendra said, her mind cloudy. \"I got all my wiggles out.\"\n\nShe heard footsteps coming nearer, accompanied by ragged panting. Somebody dropped down beside her, and she smelled Tanu's salts.\n\n\"Yikes!\" she exclaimed, sitting up, eyes watering.\n\n\"I assume you're peaceful?\" Tanu checked with the guardians.\n\n\"We mean you no harm,\" the third voice answered.\n\nAfter rubbing her eyes clear, Kendra looked up. These three hulking humanoids had light gray bodies. Leaner than Mombatu, they had more defined muscles. Each was bald except for a long ponytail, and none of them had eyes\u2014just a big blank forehead above the nose.\n\n\"You must be Himalayan cyclopses,\" Tanu said.\n\n\"Correct,\" the second voice said.\n\n\"Your only eye is a third eye,\" Kendra said, remembering a conversation with Eve.\n\n\"We see you with the eyes of our minds,\" the third voice said.\n\n\"I'm Kendra. This is Tanu.\"\n\n\"I am Baroi,\" the third voice said.\n\n\"Hobar,\" the second voice said.\n\n\"Tal,\" the first voice said.\n\nKendra stood up, still tired but recovering. Tanu rose as well. Warren caught up to them.\n\n\"I thought I had seen everything,\" Warren said.\n\n\"Himalayan cyclopses,\" Kendra explained.\n\n\"Who merge into a triclops,\" Warren said. \"You boys almost killed us.\"\n\n\"Our memories of our other form are a blur of rage,\" Tal said calmly. \"We apologize for any harm done. That vicious state is a defense mechanism for the sanctuary. In times of peace, we are peaceful. When the protections go down, we merge into a form more appropriate for war.\"\n\n\"It was the design from the beginning,\" Hobar said.\n\nDespite the imposing size of the cyclopses, their polite demeanor and soothing voices set Kendra at ease. \"You were the guardians of the pearl.\"\n\nThey nodded.\n\nKendra considered them. \"If you walk away from this protected area, you will combine into the triclops again.\"\n\n\"We will not deliberately do so,\" Baroi said.\n\nKendra waved a hand experimentally. \"Do you see me doing that?\"\n\n\"I not only see you waving,\" Tal said, \"I saw you were going to do it.\"\n\n\"I can see you stopping,\" Hobar said.\n\n\"You see into the future?\" Kendra asked.\n\n\"The true eye of the mind has fewer limits than a sensory organ,\" Baroi said. \"We see in all directions. We see some things far off, across space or even time.\"\n\n\"Not all,\" Hobar added. \"Not nearly all.\"\n\n\"But we see much,\" Tal said. \"That which is nearby tends to be easiest to discern.\"\n\n\"Do you see the pearl?\" Kendra asked.\n\n\"We don't see it,\" Baroi said. \"But we can feel that it passed into the Under Realm.\"\n\n\"That's where my brother has been held,\" Kendra said. \"And my . . . friend.\"\n\n\"Do not be shy about how much you care for your friend,\" Hobar said.\n\n\"Can you help me get there?\" Kendra asked.\n\n\"We cannot go beyond the boundaries of this protected region without uniting into the Enforcer,\" Tal said. \"We certainly cannot go to the Under Realm.\"\n\n\"The Under Realm is near and far,\" Hobar said. \"The entrance shifts.\"\n\n\"To go there you must find the Phantom Isle,\" Baroi said. \"No small task.\"\n\n\"The Underking is there?\" Kendra asked.\n\n\"Ruling over a domain outside of this world,\" Tal said. \"The island is the gateway.\"\n\n\"The Phantom Isle appears and disappears,\" Hobar said. \"The location is ever changing.\"\n\nVanessa and Hako approached. Vanessa looked thrashed.\n\n\"I guess these guys no longer want to eat us,\" Warren said. \"They're bonding with Kendra.\"\n\n\"She radiates light,\" Baroi said. \"Perhaps she can retrieve the Sunset Pearl.\"\n\n\"But you don't know where to find the Phantom Isle,\" Kendra said.\n\n\"We do not know,\" Tal said.\n\n\"The isle can be located,\" Hobar said.\n\n\"You must recover the fool's treasure,\" Baroi said.\n\n\"At Desperation Beach?\" Hako asked.\n\n\"Sounds like a lousy place to vacation,\" Warren murmured.\n\n\"Obtaining the fool's treasure might be harder than finding the Phantom Isle,\" Hako said.\n\n\"Not harder,\" Baroi said. \"Necessary. The treasure begins the journey to the Phantom Isle.\"\n\n\"You must complete the journey, Kendra,\" Tal said.\n\n\"We foresee it,\" Hobar declared.\n\n\"Will I succeed?\" Kendra asked.\n\nThey remained silent, heads bowed.\n\n\"Unknowable,\" Baroi said finally.\n\n\"Is that the consensus?\" Warren asked.\n\n\"What one of us sees, all of us see,\" Tal said.\n\n\"With rare exceptions,\" Hobar said.\n\n\"We three are one,\" Baroi explained. \"Even when separate.\"\n\n\"Kendra, if you are willing to try, you might succeed,\" Tal said.\n\n\"And if you fail, others will try, building on what you left behind,\" Hobar said.\n\n\"Until there are no others willing to try, and darkness consumes the light,\" Baroi said.\n\n\"We're going to succeed,\" Warren said bracingly.\n\nKendra looked over her shoulder at Warren, then at Tanu, Vanessa, and Hako. \"I'll do my best,\" she said. \"But first, I'm really thirsty.\"\n\nSeth looked back at the island of Timbuli receding behind the boat and wondered if Kendra had survived the triclops. She had been moving like a blur when she raced out of sight. Seth hoped no harm had come to her\u2014especially because she might actually be his sister. It was annoying that she had left him stuck to the mushroom for so long, but in the end, she had distracted the triclops as if she really cared about him.\n\nThe stories Kendra and Bracken had told him combined to form a plausible scenario, except his dark powers didn't fit with their version very well. What role would his powers have had among people who fought and imprisoned dark creatures? Seth worried he might never sort out the truth about his past. How many different versions would he encounter?\n\nThe boat advanced purposefully over the swells, Ronodin at the rudder. Seth took a sip of water from the sweet bucket. The western horizon warmed to gold as the sun sank.\n\n\"We're leaving in a different direction than we came from,\" Seth noted.\n\n\"Nearly the opposite,\" Ronodin said.\n\n\"Are we going back to the Under Realm?\"\n\n\"Yes.\"\n\n\"By a different route?\"\n\n\"We're going directly back to the island we came from.\"\n\n\"How?\"\n\n\"Think,\" Ronodin said. \"If we're returning to the island by going in the opposite direction . . .\"\n\n\"The island moves?\"\n\nRonodin gave a small smile. \"It's why they call it the Phantom Isle.\"\n\n\"Also, a bunch of phantoms live there,\" Seth said. \"Wraiths, too. I got some information from Dezia.\"\n\n\"That's a relief,\" Ronodin said, \"since you collapsed the crypt and blocked access to her.\"\n\n\"I didn't collapse it.\"\n\n\"Was the cave-in a coincidence?\"\n\n\"Dezia collapsed the crypt as I was leaving.\"\n\n\"You learned where to find the Everbloom?\"\n\n\"Not exactly.\"\n\nRonodin furrowed his brow. \"That's distressing.\"\n\n\"I'm the one who should be frustrated,\" Seth said. \"I was the person risking my life! Going to Dezia wasn't my idea. She only knew the general location.\"\n\n\"What did you learn?\"\n\n\"The Everbloom is inside the volcano,\" Seth said. \"Baga Loa. In a cave. But there are lots of caves. It will be hard to find.\"\n\n\"Did Dezia share thoughts on how to pinpoint the exact location?\"\n\n\"She said a firewalker named Savani should know.\"\n\n\"I suspected she would know,\" Ronodin said. \"But Savani will not divulge the site.\"\n\n\"She is the caretaker of this dragon prison?\" Seth asked.\n\n\"Yes,\" Ronodin said. \"Seeking the location from her would be a dead end. Why did the phantom help you?\"\n\n\"I agreed to throw her token into the volcano,\" Seth said.\n\n\"Interesting,\" Ronodin said. \"What token?\"\n\n\"Some doll,\" Seth said.\n\n\"May I see it?\"\n\nSeth took the little doll from his pocket and handed it over. Ronodin released the tiller to examine the primitive doll with both hands. As Seth had suspected, the tiller seemed to manage just fine without Ronodin handling it.\n\nRonodin brought the doll to his face and smelled it deeply, then held the little figure of wood and woven grasses up against the sky, closing one eye to study it closely. \"Dezia wants this in the volcano?\"\n\n\"Yes, Dezia told me throwing the doll in there would allow her greater freedom to roam around on certain nights,\" Seth said. \"And allow her to find a new lair if the preserve falls.\"\n\nRonodin handed the doll back to Seth with a sigh. \"You did well enough.\"\n\n\"Thanks,\" Seth said, a little insulted by the paltry praise. \"I wonder if you could have left me stuck to that flesh-eating mushroom for a little longer?\"\n\n\"I had much to contend with,\" Ronodin said. \"The triclops arrived, along with Kendra and her friends.\"\n\n\"At least you came back for me,\" Seth said. \"I wondered if you would.\"\n\n\"Don't ever count on another,\" Ronodin said. \"We save ourselves or we perish.\" He put a hand back on the tiller and raised his voice. \"Take us to Bridge Cove.\"\n\nThe boat turned in a new direction.\n\n\"Change of plans?\" Seth asked.\n\n\"We can work with what you uncovered,\" Ronodin said. \"But we need an ointment brewed by the Pinaki people.\"\n\n\"Why?\"\n\n\"It will render you immune to heat,\" Ronodin said.\n\n\"Me?\"\n\n\"I'm not going into the volcano,\" Ronodin said.\n\n\"Wait, somebody has to go into the volcano?\" Seth asked. \"Is that even possible? That's going to take more than the right lotion.\"\n\n\"You won't go swimming in lava,\" Ronodin said. \"But you need to get close enough to accomplish your mission.\"\n\n\"Where will I throw the token?\" Seth asked. \"Does it have to be down the main shaft? Or would any pool of lava work?\"\n\n\"You won't toss the token into the volcano,\" Ronodin said.\n\n\"Then why do we need the ointment?\" Seth asked.\n\n\"So you can go get the Everbloom.\"\n\n\"I promised Dezia I would throw the token into the volcano,\" Seth said. \"She'll haunt me if I don't.\"\n\n\"Not if we haunt her first,\" Ronodin said. \"A token like she gave you shares a connection with her essence. We can exploit that connection.\"\n\n\"Do you mean we'll trick her?\" Seth asked. \"Is that fair?\"\n\n\"Was it fair of the phantom to ask you to perform a perilous task?\" Ronodin countered. \"Especially when she didn't provide the information you were seeking.\"\n\n\"She told me what she knew,\" Seth said.\n\n\"Did she?\" Ronodin said. \"Are you sure? The information she shared is of questionable value. I already assumed Savani knew the location of the Everbloom. And the volcano is a likely hiding place. In the absence of specifics, Dezia would be getting something for nothing. I can use the token to bind Dezia and ensure the phantom gave you the whole story.\"\n\n\"You're positive she won't end up haunting me?\" Seth asked.\n\n\"The phantom will learn that you are more formidable than you appear,\" Ronodin said. \"If she crosses you, she crosses your allies.\"\n\nSeth nodded, but inside he still felt like using the token to harass the phantom was a betrayal. How long before Ronodin did something similar to him?\n\nThe sun had fallen below the horizon when the little boat approached an island smaller than Timbuli, leaving the western clouds and sky smeared with pink, red, and purple. Seth watched the colors change and fade.\n\n\"Are sunsets typically this pretty?\" Seth asked.\n\n\"It's always a pleasure to watch light diminish,\" Ronodin said. \"Behold the island of Tiba Tiba. One of the lesser islands, but home to some interesting beasts and birds, along with the tribe we seek.\"\n\nBare projections of rock interrupted the jungle on this island. The low, jagged mountains complemented the rocky coast. The boat had to weave between inhospitable reefs to get in close.\n\n\"An arch,\" Seth pointed out, noticing a natural stone bridge ahead of them.\n\n\"The entrance to Bridge Cove,\" Ronodin said. \"I wouldn't recommend coming here without an enchanted vessel. The currents in the cove are merciless.\"\n\nAfter threading between contorted rocks, the boat passed between stone walls beneath the natural arch. Hemmed in by steep sides, the cove was not spacious. The water level rose and fell dramatically with the ebb and flow of the waves.\n\n\"How do we land the boat?\" Seth asked, looking up at the sheer walls of the cove.\n\n\"I'll stay with the boat,\" Ronodin said, indicating a wet, knotted rope hanging down the far wall of the cove. \"You're going to climb. Don't slip.\"\n\n\"I'm going alone?\" Seth asked as the boat bobbed toward the rope.\n\n\"I don't believe the Pinaki will deal with me,\" Ronodin said. \"As a mortal youth, you should be fine. Tell them of your need, and offer a royal kingfisher in the morning as payment.\"\n\n\"Is that a bird?\" Seth asked.\n\n\"A rare and magical bird,\" Ronodin said.\n\n\"Where am I going to get one?\"\n\n\"You won't,\" Ronodin said. \"It's what I'll do tonight as you enjoy their hospitality.\"\n\n\"Are they cannibals?\" Seth asked.\n\n\"If they ate you, it wouldn't be cannibalism,\" Ronodin said. \"You're a different species.\"\n\n\"What are they?\" Seth asked.\n\n\"You'll see,\" Ronodin said. \"Just be polite and explain your need for the ointment. Promise them the kingfisher.\"\n\n\"Are you sure you'll catch one?\"\n\nRonodin grinned. \"I'd bet your life on it.\"\n\n\"How will I get the bird from you?\" Seth asked.\n\n\"Come back to the cove in the morning,\" Ronodin said. \"You'll see it.\" He glanced upward. \"Use the rope.\"\n\nThe rope was now within reach, stretching down into the water near the bow. \"How will I find you again?\" Seth asked.\n\n\"Come back here,\" Ronodin said. \"The village isn't far.\"\n\n\"Where do I go after I climb the rope?\" Seth asked.\n\n\"They'll find you,\" Ronodin said. \"Off you go.\"\n\nSighing, Seth placed one foot on the gunwale, then reached out and caught hold of the rope. Leaving the boat behind, he hauled himself up one knot at a time, waiting until his feet were secure on a knot before reaching his hands higher.\n\nBy the time he reached the top, the little boat was out of the cove. In the dusk, Seth saw three little figures approaching, not much taller than his knee, two males and a female. They had big eyes, spindly limbs, and large, flat feet. Each carried a spear and moved over the rocks with agile ease, more leaping than walking.\n\n\"Are you the Pinaki?\" Seth inquired.\n\n\"What brings a human to our shore?\" said the little warrior in front, stouter than the other two.\n\n\"I need some fireproof ointment,\" Seth said.\n\n\"These are dangerous times to voyage,\" the stout one said. \"Your need must be great.\"\n\n\"Yes,\" Seth said.\n\n\"You came alone?\" the female asked.\n\n\"My friend dropped me off,\" Seth said. \"He left with the boat. He'll be back.\"\n\n\"Do you mean us any harm?\" the stout one asked.\n\n\"No,\" Seth said.\n\n\"We shall see,\" the stout one said. \"We will take you to the elder, and you will be tried.\"\n\n\"Is that how you always greet guests?\" Seth asked, not loving the idea of a trial.\n\n\"All strangers,\" the stout one affirmed.\n\n\"Follow us,\" the female said.\n\n\"I'm Seth, by the way.\"\n\n\"Call me Foamrider,\" the stout one said. \"That is Stoneturner, and she is Starbreaker.\"\n\nThey led him away from the cliffs over rocky terrain interspersed with sparse succulents. Ahead, at the edge of the jungle, Seth saw a wooden palisade just a little taller than he could reach. To the Pinaki people, the timber fencing must have seemed incredibly high.\n\n\"We have a human guest!\" Foamrider called.\n\nSmall doors opened at the base of the palisade. The Pinaki could comfortably stroll through with room to spare, but Seth had to get down and crawl. On the far side, Seth stood up in a bustling village crisscrossed by cords strung from house to house. Some cords continued to the top of the palisade in places, and up into the trees beyond the wooden wall. Many of the Pinaki traveled along the cords like tightrope walkers, their feet partly wrapping around the slender lines. Framed with bamboo, the walls and roofs of the huts were composed of tightly woven leaves. Shin-high torches had been lit against the deepening dusk.\n\nFoamrider guided Seth to a long hut on the left side of the village, forcing Seth to duck or step over a couple dozen lines to get there. It was by far the largest building in the community, the top of the roof almost reaching Seth's chin. The front doors opened, and an old man came out, his clothes brightened with shells and feathers.\n\n\"Elder!\" Foamrider called. \"A human guest is here to be tried.\"\n\nThe wizened elder looked up at Seth through squinted eyes framed by deep lines. \"Hello up there. You're young. What brings you here with the sanctuary fallen?\"\n\n\"Not all of it has fallen,\" Seth said.\n\n\"You should stay on protected ground,\" the elder said. \"These islands are not safe. Neither is the ocean.\"\n\n\"I need your heat-proof ointment,\" Seth said.\n\n\"You risked getting devoured by dragons for our renowned ointment?\" the elder asked. \"It will not stop dragon fire.\"\n\n\"I still need it,\" Seth said.\n\n\"So it seems,\" the elder said. \"Our village can shelter you for the night, but we cannot feed such a stomach as yours for long.\"\n\n\"Thank you,\" Seth said. \"I don't need to stay long. I just need the ointment.\"\n\n\"You would need much ointment for a body as large as yours,\" the elder said. \"Our supply is low, and lately the ingredients are scarce.\"\n\n\"I can give you a royal kingfisher for the ointment,\" Seth said.\n\nThe elder whistled. \"Where are you hiding the bird?\"\n\nSeveral of the villagers laughed.\n\n\"My friend will bring it in the morning,\" Seth said.\n\n\"Your friend has a royal kingfisher?\" the elder asked.\n\n\"He is retrieving it,\" Seth said. \"He will bring it in the morning.\"\n\n\"If you supply such a bird, we can spare enough ointment for you,\" the elder said. \"Come inside the assembly hall so I can try you.\"\n\nThe villagers murmured with interest.\n\nThe elder withdrew into the hall, and Seth dropped down and slithered through the doorway. Most of the inside was bare ground, the loose dirt swept away, so Seth had plenty of room to sit, if not a high enough roof to stand. Five colorful birds roosted in the rafters.\n\n\"Nice birds,\" Seth said.\n\n\"We would be glad to add a royal kingfisher,\" the elder said. \"We cannot renew our ointment supply until we have one.\"\n\n\"What do you use? The feathers?\"\n\n\"Our methods are private,\" the elder said. \"But the song of a royal kingfisher is essential. Place your hand on the heartstone.\" The elder tapped a large rock beside the hearth.\n\n\"Hearthstone?\" Seth asked, thinking he might have heard wrong.\n\n\"Heartstone,\" the elder said, tapping his chest.\n\n\"All right,\" Seth said. The surface felt a little warm from its proximity to the fire, but otherwise the stone seemed ordinary.\n\n\"Keep your hand upon the stone,\" the elder said. \"Do you intend to harm us?\"\n\n\"No,\" Seth said, glancing at the heartstone. Would it work like a lie detector?\n\n\"Are you a shadow charmer?\" the elder asked.\n\nSeth hesitated. Would admitting to that disqualify him? He decided getting caught in a lie would be much worse. \"Yes. How can you tell?\"\n\n\"Do you mean to cause mischief on our island with your shadow charming?\"\n\n\"No.\"\n\nThe elder gave a nod. \"You will pay for the ointment with a royal kingfisher?\"\n\n\"Yes,\" Seth said.\n\n\"Given the life you have lived up until this point, are you a trustworthy person?\" the elder asked.\n\nSeth considered Ronodin's plot to use the token against the phantom. It hadn't been his plan, and it hadn't happened yet. He thought about releasing the undead at Blackwell Keep. If he had betrayed people who trusted him, he hadn't done it deliberately.\n\n\"I'm not perfect,\" Seth said. \"But I do my best to be trustworthy.\"\n\nThe elder studied him for a moment. \"You can remove your hand,\" he finally said.\n\nSeth took his hand off the heartstone.\n\n\"You have a good heart,\" the elder said. \"Not perfectly pure, but a pristine heart is so rare, I have never seen it. You will receive your ointment in the morning. Tonight, this room is yours. It is the only space large enough to contain you. Are you hungry?\"\n\n\"Yeah,\" Seth admitted.\n\n\"Thirsty?\"\n\n\"A little,\" Seth said. He had sipped a lot from the sweet bucket.\n\n\"We will serve you food and drink,\" the elder said. \"I hope you like fruit and fish. Then you may rest here in peace. Our village remains secure.\"\n\n\"Thank you,\" Seth said.\n\n\"You realize the sanctuary is compromised? Nearly all protections and boundaries are down?\"\n\n\"Yes,\" Seth said.\n\n\"Enjoy our hospitality tonight. But tomorrow, we ask that you depart at first light.\"\n\nSeth nodded.\n\nThe elder smiled. \"Provide a royal kingfisher, and we will celebrate the trade.\"\n\nSeth hoped Ronodin wouldn't try to trick them. Was the heartstone why Ronodin had sent Seth in alone? Would the dark unicorn have failed the trial? Or did he already have a bad reputation among the Pinaki for some reason?\n\nThese little villagers seemed good and fair and hospitable. The evening meal they brought was fresh and flavorful even though the portions were small. After they left, Seth reclined on his back and gazed into the rafters, watching the movement of the birds.\n\nWhy had Ronodin avoided a village where the citizens seemed harmless and kind? Was breaking the deal with Dezia a smart idea? Ronodin had no problem with the betrayal. But it wasn't Ronodin's promise. Seth knew he was the person in danger if things went wrong, not the dark unicorn. Questions without clear answers kept Seth awake until long after the birds grew still.\n\nThe pale color in the sky hinted at the coming dawn. Kendra approached the north side of Crescent Lagoon, where the Himalayan cyclopses had gathered. The ocean was the calmest Kendra had seen it, the lagoon a silver pond in the predawn twilight.\n\nInside, Kendra was not calm. Last night, after planning a treasure hunt to Desperation Beach with Savani and Tanu, Kendra had realized that Calvin was no longer with her. She had searched her clothes, called his name, rummaged through her room, and backtracked down to the seashore. Warren had taken her to the clearing where she had first met the cyclopses, but the Tiny Hero was nowhere to be seen and did not answer her cries.\n\nWarren had sent her to bed with a promise that he would return to the corrupted pool. This morning he had informed her that he, Vanessa, and Hako had gone back there in the night, but the search yielded no success. He also told her that Savani had sent her trained parrot, Piharro, to look for him. Supposedly the bird was quite intelligent and had a knack for finding lost things. But Kendra still wanted to do more.\n\nThe cyclopses had wandered away from the Monkey Maze last night before the meeting with Savani. Kendra had looked for them in vain, hoping they could help her locate Calvin. This morning, as she and the others prepared to depart, she noticed them up on the high point to the north of the lagoon.\n\n\"Well met, Kendra,\" Tal said.\n\n\"Fair weather today,\" Hobar noted.\n\n\"You are glum,\" Baroi said.\n\n\"Hi, guys,\" Kendra said. \"I was carrying a tiny person with me yesterday. A nipsie. His name is Calvin. I somehow lost him.\"\n\n\"You had no tiny person with you when we met yesterday,\" Tal said.\n\n\"We would have noticed,\" Hobar said.\n\n\"Really?\" Kendra said. \"I guess that narrows things down. I must have lost him when we were traveling to the corrupted pool. Or while running from Mombatu. He was with me when we set out.\"\n\n\"Our apologies for any inconvenience we caused in our other form,\" Baroi said.\n\n\"He's so little,\" Kendra said. \"I have to find him. I looked for you three last night. Did you ever come inside the Monkey Maze? Hide out in one of the bigger tree houses?\"\n\n\"We do not go indoors voluntarily,\" Tal said.\n\n\"Confined spaces cloud our vision,\" Hobar said.\n\n\"From inside, how would we chronicle the seasons?\" Baroi asked.\n\n\"Or monitor the shapes of the clouds?\" Tal said.\n\n\"Or read the dance of the stars?\" Hobar said.\n\n\"We prefer the caress of the wind, the kisses of the rain, the reassurance of the earth,\" Baroi said.\n\n\"Here we stand now, to greet a new day dawning,\" Tal said.\n\n\"Can you see a nipsie anywhere?\" Kendra asked. \"With your special sight? He's a little bigger than normal. His people put a spell on him to make him tall. He's still miniature compared to us. About the size of my pinky.\"\n\n\"We're always looking,\" Hobar said. \"We do not see a nipsie.\"\n\n\"We will watch for him,\" Baroi promised.\n\n\"Thanks,\" Kendra said, still worried about Calvin. Was it possible she had squashed him during the chase yesterday? She had fallen several times. Calvin had always seemed surprisingly durable. Might he simply have fallen out and gotten lost? That sounded plausible.\n\n\"If you wish to cross to Shipwreck Key before low tide, the hour to depart is at hand,\" Tal said.\n\n\"Travel swiftly and safely,\" Hobar said.\n\n\"The fate of these islands rests upon your shoulders,\" Baroi said.\n\n\"No pressure, right?\" Kendra said.\n\n\"Vast amounts of pressure,\" Tal said.\n\n\"Like a congested volcano,\" Hobar added.\n\n\"Adapt and survive,\" Baroi said. \"Command the situation.\"\n\n\"Thanks,\" Kendra said, hurrying to the seaside trail where Tanu, Warren, Vanessa, Knox, Newel, Doren, and Hako awaited. Tess was staying behind again, with Grady and Savani.\n\n\"How did it go?\" Warren asked. \"Had they seen him? In a vision or anything?\"\n\n\"No,\" Kendra said. \"I'm really worried.\"\n\n\"He'll turn up,\" Tanu said. \"Calvin is very independent. He probably lassoed a bat and is touring the other islands.\"\n\n\"I bet those nonclopses are faking blindness,\" Knox said.\n\n\"Nonclopses?\" Doren asked.\n\n\"No eyes,\" Knox said. \"Why should imaginary eyes in their minds count?\"\n\n\"Because they can see with them,\" Kendra said.\n\n\"No way,\" Knox said. \"You know their eyes are hidden somewhere, sneaking peeks. Maybe on their palms. Or under their ponytails.\"\n\n\"They seem legitimate to me,\" Vanessa said.\n\n\"I bet they remember being Mombatu just fine,\" Knox said. \"They play dumb to seem innocent. Avoid consequences.\"\n\n\"Just because you would do that,\" Kendra said.\n\n\"What if they combine and come at us from behind?\" Knox asked.\n\n\"You have the wrong worries, young one,\" Hako said. \"Look ahead for today's trouble, not behind. The triclops is gone, but there are many dangers at this sanctuary, and away from the Monkey Maze we'll be in fallen areas, where all protections have collapsed.\"\n\n\"Other dangers don't erase the problem of a triclops sneak attack,\" Knox said. \"If it happens, I called it.\"\n\n\"We should hurry,\" Hako said. \"Reaching Desperation Beach precisely at low tide gives us the best chance to succeed.\"\n\nHe established a brisk pace. Kendra walked beside Warren. \"Should we have brought Knox?\" she whispered.\n\n\"It wasn't my first choice,\" Warren whispered back. \"He won't be denied. He wants to help find his cousin.\"\n\n\"We're now beyond the protection of the moai,\" Hako announced as the lagoon passed out of sight. \"Stay watchful. Everyone has a shovel?\"\n\nKendra's was more of a spade. Tanu wore his hefty shovel across his back. The satyrs used theirs like walking sticks.\n\n\"Yes,\" Warren replied for the group.\n\n\"I also brought chains and grapnels,\" Hako said. \"We're fishing for a rare prize today.\"\n\n\"We dig it up at low tide?\" Knox asked.\n\n\"In less than three hours,\" Hako said.\n\n\"We can get to the treasure that quickly?\" Knox asked.\n\n\"Proximity is not the problem,\" Hako said. \"Many have tried to retrieve this reward. Some claim to have seen the treasure box. But the tide always interferes before any can lay hold of it. And as the tide rises, the spider eels come.\"\n\nKendra took a little hop away from Hako, trying to banish the slithering, biting images that came to mind.\n\n\"Tell me these eels stay in the water,\" Newel said.\n\nHako shook his head. \"They venture onto the sand to pursue prey. The eels are voracious and run on a hundred spindly legs.\"\n\n\"Spiders have eight,\" Knox said.\n\n\"The number is off, but the legs resemble those of a spider,\" Hako said. \"We won't beat the eels in a fight. Once they show up, we leave. If we have the treasure, we run. If we don't, we also run.\"\n\n\"I have more speed potions,\" Tanu said. \"Some other potions as well. A few gaseous potions. Five doses of courage. I'm working on some new concoctions with Uma. That woman has forgotten more about brewing than I will ever learn.\"\n\n\"We should use some of the speed potions to dig,\" Vanessa said.\n\n\"The challenge hinges on getting to the treasure quickly,\" Hako said. \"So yes.\"\n\n\"I'm trying to remember the poem you told us last night,\" Doren said, twirling his shovel.\n\n\"Long ago, a wild boar was found with the map and the poem tattooed on it,\" Hako said. \"The islanders who copied it down introduced us to the fool's treasure. Digging at the spot revealed by the map has led to glimpses of the prize.\"\n\n\"Can we hear the poem again?\" Newel said.\n\nHako recited:\n\nSift deeply through the golden sand\n\nIf you by chance should come to stand\n\nBeside the wheel with bold command\n\nYou'll hold the treasure in your hand\n\n\"Has anyone seen a wheel while digging?\" Kendra asked.\n\n\"The traditions are old and confusing,\" Hako said. \"Some report to have seen a stone wheel near the treasure box. Others believe the wheel is in the box, or they claim to have glimpsed an insignia on the box. One treasure hunter theorized the wheel might refer to a charm on a necklace lost along the beach. The only aspect the stories agree upon is that the treasure always slips out of reach. Seasoned adventurers have gone mad with frustration.\"\n\n\"We need to do something others haven't,\" Kendra said.\n\n\"To get inside the box, we need to think outside the box,\" Knox said.\n\nKendra rolled her eyes at the triteness.\n\n\"Has anyone dug for the treasure using a speed potion?\" Warren asked.\n\n\"Not to my knowledge,\" Hako said.\n\n\"When the potion wears off, we might need stretchers to drag us away from the eels,\" Warren said.\n\n\"My salts will have to suffice,\" Tanu said.\n\n\"Have satyrs ever helped?\" Newel asked.\n\n\"I'm almost sure satyrs are a first,\" Hako said.\n\n\"Satyrs have noses for treasure,\" Doren said. \"Mark my words.\"\n\nThey mostly hiked in silence after that. The trail paralleled the sea, running alongside beaches, over rocky stretches, and across clifftops. Occasionally the path angled inland, through jungle or marshland, but it always returned to the seaside before long. The day brightened and warmed as the sun rose, reflecting off the water.\n\nAs they came to the end of a bluff, Hako pointed ahead to a narrow spit of sand, no wider than a sidewalk, extending from the coast to a low island perhaps a mile away. The relatively calm ocean lapped at the sandbar but never covered it.\n\n\"The crossing to Shipwreck Key will remain above water for less than two hours,\" Hako said.\n\n\"Twice a day?\" Kendra asked.\n\n\"Yes,\" Hako said. \"Low tide comes every twelve hours and twenty-five minutes.\"\n\n\"Lots of shipwrecks out there?\" Knox asked.\n\n\"Many in the water, and a strange amount on the beaches,\" Hako said. \"The formerly wooden vessels that ran aground are all petrified.\"\n\n\"Petrified ships?\" Warren asked.\n\n\"We have no explanation for it,\" Hako said. \"These islands harbor many mysteries.\"\n\n\"Hoofs are not designed for deep sand,\" Newel complained as they plodded out onto the strip flanked by the sea.\n\n\"Feet aren't perfect for it either,\" Knox said. \"Especially in shoes.\"\n\n\"What if we get stuck out there?\" Kendra asked.\n\n\"Shipwreck Key has a high point only thirty feet above sea level,\" Hako said. \"Nearly half of the islet disappears during high tide. If we get trapped, we'll wait it out. But with the protections dormant everywhere but Crescent Lagoon, let's not get stuck.\"\n\nKendra scanned the water as they walked. Scattered starfish inhabited the shallows, and an occasional ray glided by, broad fins rippling. The sandbar subtly widened as they went, until it joined the shore of Shipwreck Key. Tall palm trees populated the center of the islet, with broad, white beaches at either hand.\n\nWhere was Seth right now? Could he be on one of the islands? It had sounded like he spent most of his time in the Under Realm. It saddened her to think of Ronodin mentoring him. The dark unicorn could be very persuasive. It was hard to imagine what it might be like dealing with him without her memories. It was easy to picture how Seth might feel very confused and slow to trust anyone. Hopefully the present mission would bring her a step closer to rescuing him.\n\nHako turned right upon reaching the key. Fine white sand eventually gave way to a rocky section where Kendra stepped carefully. Crabs scuttled into hiding as the group approached, none bigger than a bar of soap. Some of them walked with upraised claws, as if signaling a touchdown.\n\nThe beach beyond the rocks glittered with golden sand. Three slanted masts of a large sailing ship projected from the sand about halfway down the beach, along with a large section of the bow, as if the stone ship had been frozen in an attempt to surge up out of the sand. No sails or rigging survived, but one of the masts had a crow's nest intact. A few smaller stony masts jutted from the sand in other places, along with one end of a petrified canoe.\n\n\"This way,\" Hako said, jogging swiftly over the sand.\n\nThe increased pace brought them quickly to the edge of the water, not far from the largest petrified ship. The waves lapped gently against the sand.\n\n\"Is it always this calm?\" Knox asked.\n\n\"Offshore reefs absorb many of the waves here,\" Hako said. \"But this is a very placid day. The spot where we want to dig is still too threatened by water. We'll start in about ten minutes.\"\n\n\"Did you ever think you would dig for buried treasure?\" Kendra asked Knox.\n\n\"I figured I would get around to it sooner or later,\" Knox said. \"The stone boats are weird.\"\n\n\"Yes,\" Kendra agreed.\n\n\"Think this beach has some sort of Medusa?\" he asked.\n\n\"Maybe,\" Kendra said. \"It is a magical preserve.\"\n\n\"A gorgon like Medusa wouldn't turn wood to stone,\" Warren said. \"At least, no gorgon I know about.\"\n\n\"It's a mystery,\" Hako said.\n\n\"At least I don't see any stone people,\" Knox said.\n\n\"The eels don't leave much behind,\" Hako said.\n\nGetting his chains and grapnels ready, Hako talked to Warren and Tanu about a strategy for pulling up the treasure box. Kendra practiced a little with her spade. The sand was fairly easy to scoop. She knelt down and pinched some damp sand between her fingers, feeling the fine grains. The sand was so golden that she wondered if it might be valuable.\n\n\"Is there real gold in the sand?\" Kendra asked.\n\n\"A little, I suspect,\" Hako said. \"The sand only looks golden here, though. Any you take away fades to a duller hue.\"\n\n\"Enchantment?\" Vanessa asked.\n\n\"Perhaps,\" Hako said, holding his shovel ready. \"Prepare your potions. We have only a limited time. Ready?\"\n\nKendra reached for her flask.\n\n\"Kendra and Knox, save your flasks to escape the eels,\" Tanu ordered. \"Warren and I will use ours first. Then Hako and Vanessa, if they choose. I have a few extra.\"\n\n\"You need to brew some for satyrs,\" Newel said.\n\n\"Someday, maybe,\" Tanu replied.\n\n\"Let's go,\" Hako said, jamming his shovel into the sand about ten feet from the nearest water. Kendra wondered why he had started so far from the water until she beheld how rapidly the hole widened.\n\nTanu and Warren began digging in fast motion. This was her first chance to calmly watch somebody else use a speed potion. Their incredible haste made her giggle. She and the others helped, and the hole grew quickly. By the time Tanu and Warren slowed, they were four feet below sea level, with heaps of moist sand surrounding the hole, forming a lumpy wall against the sea.\n\nHako, Newel, and Doren kept vigorously digging while Vanessa used smelling salts to revive Tanu and Warren. Kendra and Knox helped shape the barricade of sand around the hole, evening it out, making sure the side facing the ocean was piled highest.\n\nSoon Warren and Tanu were digging again. Kendra kept her eyes squinted to help protect them from the moist sand flying everywhere.\n\n\"Should I use the potion?\" Vanessa asked.\n\n\"Not yet,\" Hako said. \"Wait until we see the treasure box. We'll want a burst of speed right then.\"\n\nAs they reached six feet below sea level, Doren called out, \"I struck something!\"\n\nEveryone in the hole gathered to him, and together they scraped sand from the top of a petrified wooden box bound in iron. Hako crouched down, trying to fasten his grapnels to the box as the others dug deeper.\n\n\"Told you,\" Doren gloated. \"Satyrs have a nose for treasure!\"\n\n\"No!\" Warren cried.\n\n\"What?\" Kendra called. Too many bodies were crowded around the treasure box for her to see.\n\n\"It's sinking!\" Vanessa shouted.\n\n\"I can't see it anymore!\" Doren cried.\n\n\"Drink,\" Hako suggested, unstopping his potion.\n\nHe and Vanessa chugged the contents of their flasks and started digging frantically. The others moved back to give them room. The treasure box came back into view several times, but overall continued to sink. When the effects of the potion wore off, the deepest part of the hole was nearly nine feet below sea level. Hako and Vanessa slumped to the ground, and Tanu got out his smelling salts.\n\n\"This is like trying to dig a treasure out of quicksand!\" Warren complained.\n\nAs Newel, Doren, and Warren kept digging, Kendra noticed water leaking from the wall of the hole about five feet down. As she watched, the flow increased from several trickles to steady streams.\n\n\"Water,\" Kendra warned.\n\n\"I feel the box again,\" Warren called, shovel clanging.\n\nWater spouted into the hole from a dozen places. Hako and Vanessa struggled to stand. As they regained their balance, they found themselves up to their shins in seawater.\n\nWarren dropped to his knees, a grapnel in each hand. \"These won't hook on!\" he shouted in frustration. \"I can't get them to grip! No! It sank again!\"\n\n\"Keep trying,\" Hako called. \"The water is slowing down. I'll dig a channel to divert some of it.\"\n\n\"The tide isn't back in yet,\" Tanu complained. \"Why all the water?\"\n\n\"The fool's treasure is notoriously elusive,\" Hako said.\n\nFrustrated and feeling useless, Kenda stared at the sea. There was no sign of eels, but the water seemed slightly higher than earlier. As she peered up the beach at the huge ship projecting out of the sand, she noticed it was farther inland than seemed natural. Had it gotten stuck there at high tide?\n\nShe looked down into the hole at everyone splashing around, scooping up more water than sand at this point. They were going to fail. She knew it. This was what everyone in the past had tried. Hako's map had been used many times. People had dug at this spot. And everyone had always failed.\n\nGlancing back at the masts projecting from the sand, and thinking about the poem, Kendra had an idea. It might be ridiculous, but not much sillier than chasing a stone treasure box as it sank ever deeper into a watery pit.\n\n\"Guys,\" Kendra declared. \"I want to try something.\"\n\nThe ship has a wheel,\" Kendra said. \"What if the treasure is by the wheel of the petrified ship?\"\n\nStill digging, Hako grunted out the poem:\n\nSift deeply through the golden sand\n\nIf you by chance should come to stand\n\nBeside the wheel with bold command\n\nYou'll hold the treasure in your hand\n\n\"The map and the poem are as old as the treasure,\" Hako said. \"The map shows that this is the correct place to dig. The treasure has been seen here, including by us.\"\n\n\"It's called the fool's treasure for a reason,\" Kendra said. \"It has remained elusive because everyone has missed something. It's slipping away for a reason. The poem mentions a wheel. Have you seen one?\"\n\n\"No wheel,\" Hako said gruffly. \"Just a treasure box.\"\n\n\"I'll go investigate with Kendra,\" Vanessa volunteered. \"We have too many people down here anyhow. Help me up?\"\n\nNewel and Doren gave her a boost.\n\n\"We'll come supervise,\" Newel said. \"I know a cursed treasure when I see one.\"\n\n\"I'll watch for eels,\" Doren said.\n\nShovels in hand, Kendra and Vanessa ran toward the petrified ship. Charging across the deep, dry sand made Kendra feel slow and awkward.\n\n\"The wheel will be toward the stern,\" Vanessa panted. \"If I know my ships, it will be just behind the third mast.\"\n\nThree petrified masts jutted up from the sand, all at the same angle, the first almost uncovered toward the front of the stone ship. The bow of the ship faced seaward, as if the ship had backed in to the shore.\n\nThey reached the third mast. Kendra paused to marvel at how closely the stone mast mimicked the texture of the original wood.\n\n\"The sand looks pretty deep here,\" Kendra said.\n\n\"From the angle of the ship, it might be ten or twelve feet down to the deck,\" Vanessa said. \"To reach the wheel we'll need a hole deeper than the one by the sea. But the water shouldn't bother us here for some time.\"\n\n\"The eels might,\" Kendra said.\n\n\"We'll deal with them when they come,\" Vanessa said.\n\n\"Thanks for backing me up,\" Kendra said.\n\n\"It made more sense than the frenzy in that mudhole,\" Vanessa said.\n\nA couple of paces behind the mast, Vanessa started digging. Kendra began scooping sand as well. The satyrs caught up and lent their shovels to the effort. The hole did not take shape very well at first, where the sand was dry. But as they got deeper, the sand became damper and more manageable.\n\nNot long after, Knox joined them. \"Hako is really focused over there. I started to feel like I was getting in the way. I'm pretty sure he wants to take a bite out of the treasure box.\"\n\nA minute later Warren came along, soaked clothing dusted with sand, hair wet. \"The hole is totally flooded,\" Warren said. \"Hako is still catching glimpses of the box, diving for it. Tanu is there to help lift in case Hako can get those grapnels attached. I can't be over there any longer. Hako is like a captain going down with the ship.\"\n\nWarren drank his second speed potion and gave one to Vanessa as well. They worked furiously, causing a miniature sandstorm. The others turned away from them, still digging, backs getting steadily peppered with little clumps of sand. By the time Warren and Vanessa collapsed, they had the hole to almost six feet down.\n\nKnox ran back to borrow smelling salts from Tanu. Hako and Tanu returned with him, looking half drowned. Tanu used his salts to revive Warren and Vanessa.\n\n\"I wish you guys had remained at the treasure,\" Hako said. \"We could have used all hands there. I touched it several more times.\"\n\nKendra felt annoyed by his scolding. She didn't think her presence would have made any difference in reaching the petrified treasure box. They had all been there for several failed attempts before the hole got flooded.\n\n\"We're giving this a try,\" Vanessa said, picking up her shovel.\n\n\"Not for much longer,\" Hako said. \"The beach is quite level. When the tide comes in, the water rises fast. The sea swallows the entire beach at high tide, clear back to the trees. And don't forget the eels.\"\n\n\"Help dig for now,\" Vanessa said.\n\nKendra wanted to hug Vanessa. Hako had clearly become fixated on the treasure box and saw no value in her idea of digging to the wheel of the ship. But what if they found the treasure here? And even if they didn't, they could at least dismiss the idea of the ship wheel being relevant to the poem.\n\nTanu took another potion to help them dig faster, but Hako declined. With many shovels working, the hole widened and deepened. Kendra went to work at the top with Knox and Newel, moving displaced sand away from the hole.\n\n\"I think I've reached the spokes of the wheel!\" Warren called.\n\nKendra looked down as Warren dug away more sand, revealing the top of the stone steering wheel. Tanu sat off to the side as Vanessa revived him with smelling salts.\n\n\"Let's uncover it and get out of here,\" Hako said. \"If this leads to anything, I will eat my shirt.\"\n\n\"What's that?\" Knox called, pointing out to sea.\n\nKendra shifted her attention to the ocean. The sea had drawn closer since the last time she looked. The motion of something scuttling out of the water drew her eyes. The serpentine body slithered forward, assisted by dozens of legs that crooked up to the sides like those of a spider. The ugly head was larger than a basketball, the fierce mouth full of needle teeth.\n\n\"A spider eel!\" Kendra called. \"It's huge!\"\n\n\"Keep a low profile,\" Hako said, scrambling up out of the hole. \"It's a scout. When they spot us they will swarm.\"\n\nThe eel moved in bursts, first one way, then another, mouth opening wider in rhythmic pulses. After a few moments it returned to the sea. Farther down the beach, Kendra spotted another eel coming out of the water. Then two more.\n\nWhen Kendra checked on the progress in the hole, she found the steering wheel mostly uncovered. Knox shoveled furiously around it, having taken his speed potion. Kendra still had hers. Should she use it to dig, or save it to run?\n\nAn eel with a head the size of a sports bag came farther out of the water than the others had ventured, about a hundred yards to the left of the ship. Its body had to be forty feet long. Kendra kept a careful eye on it. How big did these things get? How many were out there? In hunter mode, Hako sank close to the sand and seemed to almost vanish. Kendra held very still until the big eel turned back to the water.\n\n\"It will get bad soon,\" Hako whispered loudly enough for all to hear. \"If we stay much longer, we will be devoured.\"\n\nKendra realized it was now or never, so she drank the potion and slid down the side of the hole. She started digging quickly while everyone else seemed to move in slow motion. As Tanu administered smelling salts to Knox, Kendra got down to the base of the steering wheel. Shovels scraping against the stone, she and the others began clearing the petrified deck.\n\nMost of the deck around the wheel looked like wooden planks turned to stone, but right in front of the wheel was a sandstone block, perhaps to let whoever was steering stand a little taller. Kendra saw no sign of a treasure.\n\nHer breathing was becoming labored and her vision swam. The shovelfuls of sand seemed to quadruple in weight. The muscles in her arms strained and burned. Kendra sat down as the others slowly expanded the clear area around the wheel. They were no longer digging a hole. They were excavating an archeological site.\n\n\"Widen the radius,\" Tanu said quietly.\n\nKendra's head was swimming. She wanted to lie down and sleep. The effects of the potion ended abruptly. Everyone seemed to be working faster.\n\n\"Help,\" Kendra murmured, her limbs exhausted. \"Salt.\"\n\nTanu put the salts under her nostrils, and alertness returned. They had cleared several feet in all directions around the ship's wheel without revealing anything of note.\n\nExcept for the oddly placed sandstone block. Could it secretly be a box? Was it there by accident? It looked very plain.\n\n\"Here come the eels in force,\" Hako said. \"Time's up. I'll run left, try to misdirect them. Give me a moment and then run right. It's now or never, people.\"\n\nHako moved out of view. Newel, Doren, and Knox started clambering out of the hole.\n\n\"If you by chance should come to stand,\" Kendra whispered, \"beside the wheel with bold command.\"\n\nShe went right up to the wheel and laid a hand on it.\n\n\"Show me the treasure,\" she whispered, hoping nobody heard her. It was a long shot but seemed worth trying.\n\nHer words had no discernible impact.\n\nThere had to be some trick to this. The poem asked for a wheel. They had found one. She was beside it. How was she supposed to make a bold command? According to the poem, that was the key to holding the treasure. Would a commander stand at the wheel? Or by the wheel? Where did she need to stand?\n\nVanessa and Warren climbed out of the hole.\n\n\"We're in trouble,\" Warren called down. \"Hako wasn't kidding. Hurry.\"\n\n\"We better go,\" Tanu said.\n\n\"Wait,\" Kendra said, stepping onto the sandstone block and placing both hands on the stone wheel.\n\nSomething was suddenly different.\n\nNo, everything was different.\n\nShe could feel the waves against the beach. Feel the ships buried in the sand.\n\nThe experience reminded her of the time she had touched the Oculus. She was perceiving more than her senses ordinarily allowed. Much more.\n\n\"No joke,\" Warren said. \"They're swarming. Hundreds of them.\"\n\nKendra could feel the eels slithering over the sand, their numberless feet scrambling.\n\n\"Come on,\" Tanu said, grabbing her arm.\n\nShe jerked out of his grasp. \"No. This is it. Something is happening.\"\n\n\"We're out of time,\" Tanu said.\n\n\"We can go gaseous,\" Kendra said.\n\nTanu held up two flasks. \"We might be cutting it close even for that.\"\n\nWhile talking to Tanu, Kendra continued her extrasensory experience. It was like the beach had become part of her. She could feel how deep the sand went, perceive the encroaching sea, distinguish where shells were buried. She could sense the hole filled with water left by their digging, the mounds of sand around it, the location of the petrified chest. She felt the footfalls of her fleeing friends. The hoofs of the satyrs.\n\nAnd the spider eels. So many spider eels.\n\nShe and her friends were in serious trouble.\n\nKendra brought her focus to the hole where she and Tanu were standing. She could feel the contours of the ship in the sand. Dozens of eels were squirming toward the ship. They would reach it momentarily. She reflexively wanted to raise the ship out of the way.\n\nAnd suddenly the entire ship was thrust up by the sand. Kendra not only felt the motion through her sand senses but perceived it happening around her. The sand shoved the ship upward, and suddenly she and Tanu were no longer in a hole. Tons of grainy particles cascaded off the deck to either side. Kendra helped will the sand off the deck, pushing with her mind, and the particles went flying.\n\nThe petrified ship now rested as high on the beach as it would upon water, leaving much of the vessel submerged in sand. The deck was now well beyond reach of the eels. Kendra could feel that if she used the sand to push the boat too high, it would tip.\n\n\"Did you do that?\" Tanu asked, wide-eyed.\n\n\"Yes,\" Kendra said. \"I can feel the sand. I can sense the whole beach. But I don't just feel it. I'm somehow connected to it. I can control the sand, almost like how I can control my hand.\"\n\n\"You're commanding it,\" Tanu said. \"You were right about the poem. Can you help the others?\"\n\nFrom her new, higher vantage point, Kendra could see her friends fleeing to the left and Hako running to the right, all retreating away from the water toward the palm trees. The seaward half of the beach now teemed with eels. Kendra saw Hako stab one in the eye with a knife after dodging its strike. He was about to be overwhelmed.\n\nBesides seeing Hako with her eyes, Kendra could feel his steps on the sand. She forced a wall of sand up around him, the beach immediately responding to her desire. She found that by compacting the sand, she could make it hold its shape nicely, protecting Hako within a tall, open-topped cylinder. Several eels flailed against the base of the cylinder, but none made a serious attempt to scale it.\n\nTurning her attention to Knox, Warren, Vanessa, Newel, and Doren, Kendra envisioned the beach forming an enormous sand castle around them. The creation was simple in shape but massive in scale, with an outer wall more than twenty feet above the beach and a main tower nearly twice that height.\n\nWith her friends safely behind sandy barriers, Kendra focused on the top layer of sand beneath the eels and curled it up and away toward the sea, catapulting hundreds of eels into the air over the water. As the layer of sand flung outward into a cloud, Kendra could still feel every grain, and she pulled them back in, returning them to the beach.\n\n\"Whoa,\" Tanu said. \"That was amazing.\"\n\n\"It isn't hard,\" Kendra said. The beach had somehow become an extension of herself. Controlling it seemed perfectly natural.\n\nA few eels wormed back onto the beach, and Kendra raised a twelve-foot wall just shy of where the sea had encroached, compacting the grains of sand so tightly that no water could leak through. Then she bowed the center of the wall outward, pushing back the tide until there was more beach exposed than when they had first arrived, leaving the ends of the wall curved inland so the water couldn't slip around the sides.\n\n\"I can feel the treasure box under the original hole we dug,\" Kendra said.\n\n\"Can you raise it to the surface?\" Tanu asked.\n\nKendra found it was a simple matter of pushing up the sand beneath the chest while gently squeezing the sand around it. A moment later the petrified treasure box erupted from the beach and rested in plain sight. Kendra reached out to the sand cylinder and created an archway for Hako to escape, then created a similar archway in the castle wall.\n\nHako emerged from his sandy confinement. A moment later the others exited the castle.\n\nTanu cupped his hands around his mouth. \"Kendra has control of the beach. The treasure box is on the surface. Get it before the tide rises.\"\n\nKendra was impressed by how loudly Tanu could shout when he needed to be heard. Everyone ran toward the treasure chest. Some eels started trying to slip around the ends of the seawall she had raised, but Kendra ripped up more sand and hurled them back into the water.\n\nKendra could feel her friends' footsteps on the sand. She considered moving the sand beneath them to hasten their progress, but she wasn't sure if she could do it with enough finesse to avoid harming them. She did move the treasure box toward them, but she left it still once they got close.\n\n\"I'll eat my shirt with mustard!\" Hako called out.\n\nThe group gathered around the petrified box. Warren and Tanu tried unsuccessfully to lift it.\n\n\"It's brutally heavy!\" Warren cried.\n\n\"I'll try to pick the lock!\" Hako called.\n\nEven from up on the ship, Kendra could see the big iron lock holding the lid closed. Hako got out some gear and knelt in front of the chest. Kendra repelled more eels trying to worm around the wall.\n\n\"Got it!\" Hako crowed.\n\nHe and Warren raised the lid together. Hako distributed the contents of the box to the others. Kendra could see gold and jewels.\n\n\"Mission accomplished!\" Hako called. \"Let's go!\"\n\nWarren closed the lid, and everyone started racing away from the water toward the palm trees. The satyrs flanked Knox.\n\n\"We should scram too,\" Tanu said.\n\n\"Let me finish up,\" Kendra said.\n\nShe let the petrified ship sink into the sand until the deck where she and Tanu stood was level with the beach. Kendra made sure the seawall was holding strong, flung away some encroaching eels one last time, and stepped off the sandstone block.\n\nThe instant Kendra moved off the sandstone, she lost all connection to the beach. Even after such a short time in sync with the sand, it felt a little like losing a limb. The sand was now separate and inert, no longer hers to command.\n\n\"Are you all right?\" Tanu asked.\n\nKendra realized she wasn't moving. \"I'm good. It was kind of a jolt losing the connection.\"\n\n\"We should hurry,\" Tanu said. \"Without you guarding the beach, who knows how soon it will be flooded with water and eels?\"\n\nThey leaped over the railing of the ship into the sand and followed the others into the palm trees. By the time Kendra reached the trees, she looked back to see water gushing around the ends of the seawall to fill the previously protected beach. Writhing eels came with the water.\n\n\"This way,\" Hako said, leading them deeper into the palms, angling back toward the crossing to the larger island. \"We may have been fast enough to return without waiting twelve hours.\"\n\n\"What was in the box?\" Kendra asked as they jogged.\n\nWarren held up a slim stone tablet. \"This has a map. And a lot of text. We'll sort through the rest later.\"\n\n\"The goal is to find the Phantom Isle,\" Hako said. \"The map could be the key.\"\n\n\"We found some gold and jewels too,\" Newel mentioned.\n\n\"Quite a good haul,\" Doren said.\n\n\"You are a genius, Kendra,\" Hako said. \"If we had relied on my instincts, that treasure would have gone unclaimed until the end of tides.\"\n\n\"That's why we're a team,\" Tanu said diplomatically. \"Thank you for leading us here, Hako.\"\n\n\"I just hope the treasure guides us where we want to go,\" Hako said.\n\nThey reached the white sand beach and found the strip of sand back to the island covered by an inch or two of water. Hako splashed out onto the slightly submerged sandbar.\n\n\"What about the eels?\" Knox cried.\n\n\"I never see them over here,\" Hako said. \"Hurry.\"\n\nKendra and the others splashed along behind Hako. At its deepest the water barely came above her ankles. When they arrived at the opposite beach, everyone stopped to catch their breath.\n\nKendra moved to where Warren was studying the stone tablet. The others gathered to him as well.\n\n\"The map has lots of words,\" Kendra said, trying to get a good look.\n\n\"We'll try to translate them when we get back to Crescent Lagoon,\" Warren said.\n\n\"I can read them,\" Kendra said.\n\n\"Being fairykind comes with some skills,\" Warren said, handing the tablet to her.\n\nKendra noticed that the islands were simply identified as \"land,\" but the ocean was full of details. She recognized the shape of Timbuli from the map in Savani's room. A trail from Crescent Lagoon ran down to an underwater settlement.\n\n\"This is a map to the merfolk village,\" Kendra said. \"It's under the sea.\"\n\n\"Scuba gear,\" Knox said.\n\n\"Way under the sea, by the look of it,\" Kendra said.\n\n\"Submarine?\" Knox asked.\n\n\"It says the demon Remulon holds the compass to the Phantom Isle,\" Kendra said. \"The merfolk protect the secret of his lair.\"\n\n\"The merfolk are not easy to deal with,\" Hako said.\n\n\"There is a formula for a potion,\" Kendra said. \"It's called the Elixir of Dry Depths.\"\n\n\"You're kidding,\" Tanu said. \"Does it list the ingredients?\"\n\n\"Looks like it,\" Kendra said.\n\n\"That potion is legendary,\" Tanu said. \"If we can brew it, we won't need scuba gear or submarines.\"\n\n\"What does it do?\" Knox asked.\n\n\"Under the influence of that potion, we could move through water as if it were air,\" Tanu said.\n\n\"Let me guess,\" Newel said. \"It won't work on satyrs.\"\n\n\"Probably not,\" Tanu said.\n\nDoren gave Newel a high five. \"Nap time.\"\n\nIn a quiet room in the Under Realm, Seth sat at a round, iron table with Ronodin and the Sphinx. The chairs were iron as well, cold and hard to the touch, and heavy enough that moving them was no small matter. On the table rested a lantern, a vial of ointment obtained from the Pinaki people, and Dezia's wooden token.\n\n\"May I?\" asked the Sphinx, reaching for the doll.\n\n\"Sure,\" Seth said.\n\nThe Sphinx picked up the little doll, hefted it, and began examining it from various angles, tapping it here and there with his forefinger. Ronodin watched intently.\n\nSeth glanced at the ointment. Earlier that day, he had found the royal kingfisher in a wooden cage at the appointed location and received the ointment as planned. Seth had met up with Ronodin at Bridge Cove without difficulty, and, after boating back to the Phantom Isle, Ronodin had gone off in private to summon the Sphinx.\n\n\"You want to use the phantom to help find the Everbloom?\" the Sphinx verified.\n\n\"Exactly,\" Ronodin said.\n\n\"I know just the person to do this for us,\" the Sphinx said. \"I'll need to keep the token.\"\n\n\"Can we destroy the token after we get the Everbloom?\" Seth asked.\n\nThe Sphinx shook his head. \"If you throw that token in the volcano after we use it to control the phantom, you invite retaliation. The token must be kept secure to hold Dezia bound.\"\n\nSeth frowned. \"I agreed to throw it in the volcano.\"\n\n\"The phantom took advantage of you,\" the Sphinx said. \"She asked too much from you for too little in return. Fortunately, she underestimated your friends.\"\n\n\"Outsmarting a phantom is standard practice, Seth,\" Ronodin said. \"If you're not in charge, you're getting played. The undead don't call the shots. Are you a shadow charmer or a shadow chump?\"\n\n\"Of course I don't want to be played,\" Seth said. \"I know I'm new at this. But I don't like going back on a promise.\"\n\n\"The phantom will not be pleased you broke the arrangement,\" the Sphinx said. \"But she will be powerless to retaliate. We can make her very useful. An incorporeal being will have a much easier time searching those caverns for the Everbloom than we would. The heat-repelling ointment you retrieved has limits. After Dezia finds a route, you'll have a direct path to the Everbloom instead of needing to negotiate a convoluted labyrinth where some paths lead to molten lava or volcanic creatures.\"\n\nSeth wasn't sure if he wanted to force Dezia to find a route. Yet both the Sphinx and Ronodin seemed to think this was the obvious move. Right now, they were his only allies. Did he want to risk throwing the token into the volcano even though they thought it would be foolish? Did he want to take sole responsibility for finding the Everbloom?\n\nRonodin folded his hands. \"Destroying the token in the volcano gives Dezia a very long leash. The phantom will have more power to haunt you if you follow her plan. Don't let the undead set the terms. The Sphinx's way allows Dezia less freedom to roam. And it gives you more information.\"\n\n\"All right,\" Seth finally said, feeling outnumbered and not seeing another option. \"Just try not to get me haunted.\"\n\n\"You'll see,\" the Sphinx assured him. \"It will be better this way.\"\n\n\"We have some experience in these matters,\" Ronodin said.\n\nThe Sphinx raised his head to look down one of the passages diverging from the room. \"Do you feel that?\"\n\nSeth reached out to mentally probe the darkness. A sinister presence was coming slowly toward them. \"Revenant,\" he said.\n\n\"Bardox,\" Ronodin specified.\n\n\"You know him by feel?\" Seth asked.\n\n\"He speaks for the Underking,\" Ronodin said.\n\n\"I cannot abide the company of a revenant,\" the Sphinx said. \"Those days have passed for me. Can I take the doll?\"\n\nRonodin looked to Seth.\n\nThe approach of the revenant was undeniable, but part of Seth wondered if they were using the situation as leverage to force him to quickly part with the phantom's token. There would be no going back after this. Was he really going to do this their way? How much did he know about phantoms anyway? \"Sure, go,\" Seth said.\n\nThe Sphinx grabbed the doll and marched away down a passage where Seth felt no dark presences. Ronodin put a finger against his lips and turned to watch the revenant approach.\n\n\"Remember how to interact with the Underking,\" Ronodin whispered. \"He is ancient and incredibly powerful. Be careful about sharing any desires. He'll exploit your wants and needs. Be careful what you agree to do for him.\"\n\n\"You make it sound like I'll be speaking directly to him,\" Seth whispered back.\n\n\"You will,\" Ronodin said. \"Bardox is just a mouthpiece. Almost like a telephone.\"\n\nA moment later, an emaciated ruin of a man hobbled into view, cradling a metal weight that was shackled to one ankle. He was hairless and pale, the tendons in his neck stood out, and his ribs jutted grotesquely. The thickest part of his arms were his elbows, and the thickest part of his legs were the knees.\n\n\"Bardox,\" Ronodin greeted. \"You're looking trim!\"\n\n\"I must confer with Seth alone,\" Bardox said, somehow shaping a strangled croak into words.\n\n\"That is my cue to exit,\" Ronodin said. He patted Seth's shoulder. \"Have a productive discussion. You can do this.\"\n\nSeth gave a nod.\n\nRonodin left.\n\nSeth waited.\n\nBardox stared emptily.\n\n\"I'm Seth.\"\n\nBardox offered no reply.\n\nAfter another moment, Bardox opened his mouth so wide it looked like he was screaming. The whispery voice of the Underking emerged, just as Seth remembered it.\n\n\"Have you given thought to my offer to remain here in the Under Realm?\" the Underking asked. Seth heard the words with his ears, but they also seemed to burrow into his mind. He felt a pull to obey whatever was asked and had to shake himself in order to resist.\n\n\"I don't want to stay here permanently,\" Seth said.\n\n\"You are in your element here,\" the Underking said. \"You could do very well.\"\n\n\"I'm content learning from Ronodin,\" Seth said.\n\n\"What do you really want?\" the Underking pressed, the whispery words squirming into Seth's psyche.\n\n\"I told you. I just want to study with Ronodin.\"\n\n\"Is that all? Look deeper and speak true. I have sensed questions growing within you. Dispense with false pretenses.\"\n\nSeth paused. Ronodin had warned him against sharing desires. But how much could it matter if the Underking already knew? Was there a chance the Underking could help?\n\n\"I want my memories back,\" Seth said.\n\n\"A mortal life is but a flicker,\" the Underking said.\n\n\"It's what I have,\" Seth said. \"I want to make sense of it.\"\n\n\"Very little lies beyond my reach,\" the Underking said. \"I could learn who took your identity and how it was accomplished. Pledge to serve me, and your memories will be restored.\"\n\n\"Aren't I already helping you?\" Seth asked.\n\n\"Thus far, you have proven useful,\" the Underking said. \"You would benefit infinitely more from a permanent arrangement. I do not make this offer casually or often. You have the potential to become as endless as night and shadow.\"\n\nThe suggestion tugged at him. The words felt compelling, as if only a fool would ignore such an opportunity. Seth felt as if he were treading water in a persistent current that was drawing him toward an undesired destination. If he drifted, if he stopped resisting, he would end up doing the will of the Underking. With effort, Seth held up the hand affixed to the ghostly manacle. \"I want this gone. I want to be free.\"\n\n\"Bring me the Everbloom,\" the Underking said.\n\n\"If I bring you the Everbloom, can I go free?\" Seth asked.\n\n\"After bringing me the Everbloom, if you still want freedom, I will grant it,\" the Underking said.\n\n\"What about my memories?\"\n\n\"Pledge to join me, and I will help you regain your memories,\" the Underking said.\n\n\"That's asking too much,\" Seth said.\n\n\"What do you expect these memories to do?\" the Underking said. \"Most mortals are lost and wandering. The more information they have, the more confused they become. Your memories are as likely to haunt and disappoint you as they are to provide comfort and clarity.\"\n\n\"They might do both,\" Seth said. \"But I deserve to know my past. I've heard conflicting versions. Is there a reason you don't want me to remember?\"\n\n\"Why should such a trifling matter concern me?\"\n\n\"Forget about helping me,\" Seth said. \"If it doesn't concern you, and since I've been helping you, can you allow me some time to investigate my past? As soon as possible? Ronodin comes and goes. Could I have some of that leniency?\"\n\n\"Ronodin has no obligation to me,\" the Underking said. \"He has worked off his debts. After you are free, you can go where you will.\"\n\n\"We may have the Everbloom before long,\" Seth said. \"Ronodin and the Sphinx came up with a trick that could work.\"\n\n\"Beware of your mentors,\" the Underking said. \"They are not wholly loyal to me.\"\n\n\"I'm not sure they're wholly loyal to anyone,\" Seth said.\n\n\"You have loyalty to give,\" the Underking said. \"They do not.\"\n\n\"They work for you, though,\" Seth said.\n\n\"Ronodin has his uses,\" the Underking said. \"But beware\u2014either the Sphinx or Ronodin may try to keep the Everbloom for himself.\"\n\n\"Why would they want it?\" Seth asked.\n\n\"The power, Seth,\" the Underking said, hunger in his tone. \"The Everbloom generates a vast amount of power. It is a nearly endless source of new life and magic. If your comrades are foolish enough to cross me, do not join them. Keep the Everbloom from them and bring it to me.\"\n\n\"Ronodin is my teacher,\" Seth said.\n\n\"And you have all agreed to fetch the Everbloom for me,\" the Underking said. \"Unless you would prefer to side against me?\"\n\nThe ghostly manacle burned coldly on Seth's wrist. He winced.\n\n\"I won't betray you,\" Seth said.\n\n\"Very wise,\" the Underking said. \"Pledge to resist Ronodin and the Sphinx if they try to claim the Everbloom.\"\n\n\"Aren't you demanding more than the original arrangement?\" Seth asked.\n\n\"This broadens our agreement,\" the Underking said.\n\n\"What do I get for promising more?\" Seth asked.\n\n\"Do not try my patience,\" the Underking said.\n\nSeth felt a tug to just agree. But his situation was bad enough already. He couldn't miss opportunities to maybe improve it. \"I should get something.\"\n\n\"Very well,\" the Underking said. \"The Hidden Sage resides upon the island of Omari, inside a small cave near the summit of Mount Dagaro. He has power to answer your questions regarding your memories.\"\n\n\"How will I get there?\" Seth asked.\n\n\"You may take my clever boat on this single excursion,\" the Underking said.\n\n\"The one I took with Ronodin?\" Seth checked.\n\n\"The same.\"\n\n\"When?\" Seth asked.\n\n\"Tomorrow, I will inform Ronodin that you have been commissioned to fulfill a private task for me, and that you have leave to take my clever boat for the afternoon. The rest is up to you.\"\n\n\"I just tell the boat I want to go to the island of Omari?\" Seth verified.\n\n\"Yes,\" the Underking said. \"After your excursion, regardless of the result, I expect you to retrieve the Everbloom for me.\"\n\n\"I'll do my best,\" Seth said.\n\n\"We have an accord,\" the Underking said. The revenant Bardox twitched, reminding Seth that he was speaking to the Underking through the mouth of another being. The decrepit figure turned and started trudging from the room, mouth still open in the shape of a scream. \"Remember our agreement,\" the Underking stressed. \"Stray at your peril.\"\n\nSeth cringed and drew air through his teeth as the manacle on his arm suddenly burned with coldness.\n\n\"Your debt is with me,\" the Underking said, the whispery voice slithering deep into Seth as Bardox withdrew. \"Your obligation is to me. Do not forget who is master of the Phantom Isle.\"\n\nWithered down to a frail, wrinkly frame and confined to a wheelchair, Uma Stormbrewer had reached an age at which time had robbed much from her physically, but her gnarled hands worked with steady precision on the cap she was knitting. The shirtless, barefoot menehune who had pushed her chair to the front of the room had ropy muscles and long, wild hair. He looked too fierce to attend an old woman, but he treated her with reverence and care.\n\n\"Kendra, Knox, may I present Uma Stormbrewer,\" Tanu said to an audience that included Warren, Vanessa, Grady, Savani, and Hako. \"She is a living legend among potion masters. Without her help, I would never have prepared the Elixir of Dry Depths so quickly. Her collection of ingredients was equally essential.\"\n\n\"Tanu is too modest,\" Uma said, knitting needles clicking. \"Some vital ingredients were among the dragon parts he brought here. I could not have concocted the potion without him.\"\n\nTanu looked pleased by the praise. \"You're very kind,\" he said. \"We produced enough of the elixir to allow six of us to make the journey.\"\n\n\"How long does it last?\" Kendra asked.\n\n\"Three days,\" Tanu said. \"Or until the potion is deactivated by the antidote. Uma will explain the dangers.\"\n\n\"Toby,\" Uma said, \"fetch my coconut milk.\"\n\nThe menehune raced over to a counter cluttered by bottles and jars containing any number of objects and ingredients. At three feet tall, he had to stretch to reach the glass, muscles writhing across his brown back. He snatched a straw and brought the glass to Uma, who took a sip and handed it back to him.\n\n\"His name is Toby?\" Knox murmured.\n\n\"His true name is a mouthful,\" Uma said. \"For me, he goes by Toby.\"\n\n\"Take a good look,\" Tanu said. \"You will seldom see a menehune interacting with humans.\"\n\n\"Now for the warnings,\" Uma said. \"You must not imbibe any other potions while under the influence of the elixir. At best, you would end the effects of the elixir. At worst, you could eradicate all water from your body, essentially mummifying yourself. To be extra safe, you may wish to avoid food and drinks as well. The water in anything you ingest will react uncomfortably with your body.\"\n\n\"We won't be able to drink water?\" Knox asked.\n\n\"Not without complications,\" Uma said.\n\n\"Explain how the elixir works,\" Tanu encouraged.\n\n\"Coconut milk,\" Uma said.\n\nThe menehune scurried to return the glass to Uma. She sipped again.\n\n\"It is called the Elixir of Dry Depths,\" Uma said. \"Given the name, how do you imagine the potion works?\"\n\n\"We won't get wet in the ocean?\" Knox blurted.\n\n\"We'll be able to breathe water?\" Kendra guessed.\n\n\"The effects are even more extreme than you describe,\" Uma said. \"You won't sense the ocean water any more than you would sense air. You'll breathe and move freely, even down to extreme depths.\"\n\n\"That's incredible,\" Kendra said.\n\n\"Will we get salt in our eyes?\" Knox asked.\n\n\"The magic will protect you from tiny particles in the water,\" Uma said. \"The elixir does carry inherent dangers. For example, after taking it, if you jump out of a boat, you will fall straight to the bottom of the sea.\"\n\n\"Whoa,\" Knox said.\n\n\"Off the coast of this island, towering drop-offs and deep trenches await,\" Uma said. \"While the elixir is active, great depths are as dangerous as great heights. If you do not take care, you will fall to your demise.\"\n\n\"I've never heard of a potion like this,\" Warren said.\n\n\"Until now, this elixir has not existed in the world for a long time,\" Uma said. \"The making of it has been lost for ages. I am honored to have helped bring this miracle back into the world.\"\n\n\"Tell them how to cancel the effects,\" Tanu said.\n\n\"All who use the potion must carry an antidote,\" Uma said. \"Make sure you don't cancel the effects while down deep. Not only would you get crushed by the pressure, but you could get trapped too far from the surface and drown.\"\n\n\"Could eating something cancel the effects of the elixir?\" Knox asked.\n\n\"Probably not,\" Uma said. \"But your body will not recognize the water in anything you consume. It will strain your system.\"\n\n\"What about the water in our bodies?\" Knox asked. \"We're largely made of water.\"\n\n\"The elixir will only change how you interact with outside sources of water,\" Uma said. \"In a pinch, the best way to introduce water to your body while under the elixir's influence would be to suck on ice. After partaking of the elixir, you will still interact with ice like any other solid.\"\n\n\"What about when the ice melts?\" Kendra asked.\n\n\"If you suck ice, some water will be assimilated before it moves out of phase with you,\" Uma said.\n\n\"What about our clothes?\" Knox asked. \"Will they get wet and all clingy?\"\n\n\"This one is sharp,\" Uma said. \"Coconut milk.\"\n\nToby brought her glass, and she took another long sip.\n\n\"The elixir is powerful enough to include your clothes as extensions of yourself,\" Uma said. \"But not items you carry or clothes you add after taking the potion. Let's imagine you put on a life jacket after imbibing the elixir. The flotation device would not be very effective because your body will have no buoyancy. After you take the elixir, it becomes difficult to make your body float.\"\n\n\"Remember, after the elixir, relative to you, a boat is essentially an aircraft,\" Tanu said. \"A life jacket capable of keeping you afloat would be like a jet pack, and without the ability to interact with the water, you would risk getting stranded as if in midair.\"\n\n\"Will there be side effects?\" Kendra asked.\n\n\"The map you found had no information about side effects,\" Uma said. \"This is an extraordinarily powerful potion. Elixirs of this caliber are often fine-tuned to minimize side effects. We'll find out if that holds true in this instance by trial and error.\"\n\n\"So if I encounter a shark, I'll be able to move as if on dry land,\" Warren said. \"But the shark will be in water, as if suspended in the air, and move accordingly.\"\n\n\"Yes,\" Uma said. \"Weapons or items you hold will react to the resistance of the water. But your body will feel no such constraints.\"\n\n\"Now the big question,\" Tanu said. \"Who should go? The elixir won't work on the satyrs. We have six doses.\"\n\n\"I'm coming,\" Kendra said.\n\n\"Me too,\" Knox said.\n\n\"Will it work on blixes?\" Vanessa asked.\n\n\"My other potions do,\" Tanu said. \"From a physiological standpoint, narcoblixes are basically mortal.\"\n\n\"Then Warren and I will join you,\" Vanessa said.\n\n\"I would love to come,\" Savani said. \"But our present situation requires a caretaker overseeing the sanctuary.\"\n\n\"I'll be your sixth man,\" Hako said.\n\nKendra glanced from Grady to Knox. \"Is Grady an option?\" she asked.\n\n\"The potion won't work on me,\" Grady said. \"I'm not exactly mortal.\"\n\nKendra wanted more of the story, but nobody offered an explanation. \"How friendly are the merfolk?\" she asked instead.\n\n\"They tend to distrust surface dwellers,\" Savani said. \"But the effects of the elixir will impress them. No mortal has walked freely in their sphere for many years. And I will provide you with an introduction\u2014the merfolk have respect for firewalkers, and for the position of caretaker. If I explain that your mission could help restore the sanctuary, I believe they will cooperate.\"\n\n\"How will you introduce us?\" Knox asked.\n\n\"We burn messages onto leather to send them underwater,\" Savani said. \"We made a full copy of the map from the fool's treasure using that technique so you won't risk losing the original.\"\n\n\"When should we go?\" Vanessa asked.\n\n\"We worked hard through the night to prepare the elixir by this morning,\" Tanu said. \"I propose we eat a hearty meal, drink a lot of water, and then meet down at the beach. We still have most of the day.\"\n\n\"I will come see you off,\" Uma said. \"It is a momentous hour.\"\n\nKnox leaned close to Kendra. \"Why do I suddenly feel like a guinea pig?\"\n\n\"You could stay above water,\" Kendra whispered. \"Nobody would blame you.\"\n\n\"What if you need somebody to stuff into a Quiet Box? Or to serve as eel bait?\"\n\n\"I can't argue with that.\"\n\n\"We have to get Seth back,\" Knox said. \"If that means being guinea pigs, oink oink.\"\n\n\"I don't think guinea pigs oink.\"\n\n\"Maybe the underwater kind do,\" Knox said. \"We'll know soon enough.\"\n\nKendra ate quickly, filled up on water, and got to the beach before the others. She found the cyclopses wading in the shallows of the lagoon.\n\n\"Any sign of Calvin?\" she asked.\n\n\"No nipsie sightings as of yet,\" Tal said.\n\n\"We are enjoying the lagoon until the rain comes,\" Hobar said.\n\n\"You are preparing to venture down to the depths,\" Baroi said. \"But you will not feel the water as we do.\"\n\n\"Is a storm coming?\" Kendra asked.\n\n\"A ten-minute downpour,\" Tal said. \"Then some minor showers off and on.\"\n\nKendra looked at the sky. The only clouds were to the north.\n\n\"Not for two hours,\" Hobar said.\n\n\"Will we find the compass down there?\" Kendra asked.\n\n\"The merfolk know how to guide you to the demon you seek,\" Baroi said. \"Be courteous. And helpful.\"\n\n\"If you will excuse us, we are removing invasive sea slugs from the shallows,\" Tal said.\n\n\"Sure,\" Kendra said, taking the hint to leave. She felt disappointed they still had no view of Calvin. Maybe Tanu was right, and Calvin had convinced a bird to take him to another island. He had a way with animals, and he liked to scout around. Maybe he had gone too far for the cyclopses to see him. Or maybe he was just too small.\n\nKnox came down to the beach and plopped on the sand beside Kendra. \"I keep trying to imagine this elixir working. It's hard.\"\n\n\"We'll see soon enough,\" Kendra said, staring at the ocean, wondering what lurked beneath the choppy surface.\n\n\"I've been wanting to ask you something,\" Knox said. \"I've been wondering this ever since I learned Fablehaven was magical.\"\n\n\"Okay,\" Kendra replied.\n\n\"Do you know what really happened to Grandma and Grandpa Larsen?\" Knox asked. \"When we all thought they died a few years back?\"\n\n\"Oh, wow,\" Kendra said. \"That's a good question. You wouldn't have known.\"\n\n\"The story we heard was almost unbelievable,\" Knox said. \"Supposedly they had to go into the witness protection program until some trial was over, and so their funeral was staged with wax figures. Even after they got back in touch, we weren't allowed to know any specifics\u2014for our protection, they said. At least they apologized for making us think they were dead.\"\n\n\"Their story isn't too far from the truth,\" Kendra said. \"The bodies were fakes. Stingbulbs, like Patton. Staging their deaths helped with the cover they needed to work as spies against an evil group called the Society of the Evening Star.\"\n\n\"So it was magical stuff,\" Knox said.\n\n\"Yes,\" Kendra said.\n\n\"I wonder what they'll tell our parents?\" Knox asked. \"To explain us being gone.\"\n\n\"My parents know the truth about Fablehaven,\" Kendra said. \"Maybe Grandma and Grandpa Larsen will have to let your parents in on the secret too. I guess it partly depends on how long we're gone.\"\n\n\"Also depends on if we survive,\" Knox said.\n\n\"I guess so,\" Kendra said. \"I hope I can give them better news about Seth than we have right now.\" She felt tears coming but held them back.\n\n\"Me too,\" Knox said.\n\nTanu and the others had begun gathering on the beach. Two menehune struggled to push Uma's wheelchair through the sand, the wheels digging furrows more than rotating. A third menehune shaded her with an umbrella.\n\nTess came up to Kendra and Knox. \"You're going in the water?\" she asked.\n\n\"Yes,\" Kendra said. \"Hiking, not swimming.\"\n\n\"I heard a rumor you'll see mermaids,\" Tess said.\n\n\"Yes,\" Kendra said. \"We'll be visiting the merfolk.\"\n\n\"If you see Gracie or Lila, tell them hello,\" Tess said.\n\n\"What?\" Kendra asked.\n\n\"They sometimes come up to those boulders by the shore,\" Tess said, pointing. \"They sing me songs.\"\n\nKendra realized Tess had been spending a fair amount of time around Crescent Lagoon on her own. \"Gracie and Lila?\" Kendra asked.\n\n\"Lila is the blonde one,\" Tess said. \"Be careful. Savani warned me not to let them touch me.\"\n\nTanu approached Kendra. He held a pouch on a belt. \"I thought you might want your unicorn horn. And probably the sack of gales, in case it could be useful.\"\n\n\"Would it work underwater?\" Kendra asked.\n\n\"It might,\" Tanu said. \"That would be a lot of bubbles. Of course, if it isn't designed to function underwater, it could send you flying all over the sea. But the sack of gales is small, and precious, so I thought you might want to keep it with you, along with the horn.\"\n\nKendra belted on the pouch. She placed the sack of gales and the unicorn horn inside.\n\nTanu was handing out elixirs. With each he included a small capsule.\n\n\"The antidote is in the capsule,\" Tanu announced. \"Only use it at the surface. Or very near. The capsule is waterproof. Carry it in a pocket.\"\n\nSavani was passing out simple necklaces with a crystal attached to each. \"These crystals draw light from sister crystals exposed to the sun. They will brighten your way in the deep.\"\n\n\"Should we drink?\" Knox asked.\n\n\"Down the hatch,\" Tanu said, tipping his vial back into his mouth.\n\nKendra studied the little vial. It held a smaller dose than most of Tanu's potions, the fluid a pale purple. She raised the vial to her lips and drank. The liquid tasted of tropical fruits with a citrus tang and was a little salty, with a slightly bitter aftertaste.\n\n\"Did it work?\" Knox asked.\n\n\"We'll know when we get in the water,\" Tanu said.\n\n\"I have some items for you,\" Savani said. \"First, a message of introduction to the merfolk, along with a copy of the map you discovered at Desperation Beach.\"\n\n\"Thank you,\" Tanu said, accepting them.\n\n\"For Hako, the loyal harpoon,\" Savani said.\n\nGrady stepped forward and presented Hako a seven-foot harpoon with a barbed tip. Hako accepted the weapon solemnly.\n\n\"We will also loan you Seaslayer,\" Savani said. \"The blade moves unhindered in the water and has an edge sharp enough for a clean shave. Your best swordsman should wield it.\"\n\n\"Warren,\" Tanu said without hesitation.\n\n\"It might be Vanessa,\" Warren said.\n\n\"You take it,\" Vanessa told him.\n\nWarren accepted the blade from Grady.\n\n\"We have extra harpoons for any who want them,\" Savani said.\n\nGrady helped Kendra choose a harpoon, slim and light, not quite as long as she was tall. Knox, Vanessa, and Tanu selected harpoons as well.\n\nWarren swished his gleaming sword in the air. \"This is a beautiful weapon,\" he said.\n\n\"May it serve you well,\" Savani said. \"Our hopes go with you.\"\n\n\"Time to get wet?\" Knox asked.\n\n\"Nope,\" Kendra said. \"Time to stay dry.\"\n\nHako gave a nod. \"I will take the lead. Tanu, perhaps you could stay near me with the map? Warren, I suggest you guard the rear. Keep your harpoons ready. Once we pass beyond the lagoon, we could encounter a wide variety of trouble. Remember, just as on land, the underwater boundaries are down. Provoke no creatures, but stay poised to defend yourselves.\"\n\nThe warnings helped remind Kendra that they were not just going into the ocean\u2014they were descending into the sea at a sanctuary for magical creatures. Predators much larger and more dangerous than sharks might prowl the depths below.\n\nHako and Tanu stepped into the lagoon. It was strange to see the two men wading deep into the water without causing a ripple.\n\nWhen Kendra entered the water, she felt nothing except the sandy bottom, as if the water were an illusion. That didn't change as she went deeper. In front of her, Tanu and Warren disappeared under the water, showing no propensity to float.\n\nAs the insubstantial water reached Kendra's neck, she stared across the surface from a low angle, feeling a natural inclination to hold her breath. Then she crouched down, ducking her head under the water. The coloring looked like she was underwater, though the water seemed unusually clear\u2014she could discern crisp outlines for a long distance in all directions. For a moment, all she could do was marvel at the panorama of sand, shells, kelp, coral reefs, sponges, sea fans, and brilliant tropical fish. It was weird not to feel like a clumsy intruder underwater. Though she could see water all around her, she felt no pressure from it, no hint of wetness. Breathing out produced no bubbles, and since exhaling felt normal, she took a small, experimental breath. Inhaling felt exactly like breathing air.\n\nClosing her eyes and taking another breath, Kendra felt like she could have been anywhere. Her last guess would have been underwater. The only hint of dampness was the gumminess of the sand beneath her feet.\n\n\"Isn't this great?\" Knox asked, grinning.\n\n\"Interesting,\" Kendra said. \"I hear you just fine. It feels perfectly normal to speak. It shouldn't, but it does.\"\n\nKnox ran over to a coral colony, scattering a school of yellow fish with dark blue stripes. He chased after some of the fish, and they darted in several directions to avoid him.\n\n\"If you want to indulge your curiosity, this is the place,\" Hako called back to the others. \"Once we leave the lagoon, we will have to be on guard.\"\n\nKendra appreciated the permission to explore. Wandering into deeper water, she marveled at the fish floating around and above her. Her freedom of movement created the illusion that the fish were flying. After stooping to pick up a shell, Kendra found it lighter than she expected, but she could feel the resistance of the water as she moved it, letting her indirectly perceive that she was, in fact, underwater. Kendra released the shell and watched it slowly drop to the seabed.\n\nUsing the butt of her harpoon like a walking stick, Kendra caught up to Hako and Tanu. Hako pointed out an iridescent crab the size of a toaster.\n\n\"Charm crab,\" Hako said. \"Don't stare too long or you might get mesmerized. Try to harm the crab and it will leave you stupefied.\"\n\n\"Stupified?\" Knox asked, joining them.\n\n\"In a state of deep confusion,\" Hako said. \"In this lagoon, it's hard to predict what might be magical. Treat everything you encounter with extra respect.\"\n\n\"This is the best playground ever,\" Knox said.\n\n\"It's a wonderland,\" Hako said. \"I've spent my life exploring these islands, but I've never experienced the water like this either. Unfortunately, we do have a mission.\"\n\nKendra nodded. First they needed to deal with the merfolk, then take a prize from a demon. All without getting devoured by sea dragons. \"Lead the way,\" she said.\n\nThe lagoon was deeper toward the middle, then got shallower again as they marched toward the open ocean. Hako led them around a jagged barricade of reefs until they stood at the brink of a high cliff. Kendra stayed a few steps back from the edge, staring in amazement at how far the ocean descended and how dark the depths became.\n\n\"That would be a serious fall,\" Knox murmured.\n\n\"Keep still,\" Hako whispered, putting a finger to his lips.\n\nBeyond the edge of the cliff, in the open ocean, a hammerhead shark the size of a school bus was swimming by, tail swishing smoothly. Kendra stared in awe as the tremendous shark traveled across her field of view, eventually swimming out of sight. She was relieved it had not turned toward them.\n\n\"That shark was unreal,\" Knox said. \"Like megalodon huge.\"\n\n\"We're sharing this ocean with all sorts of aquatic monsters,\" Hako said. \"The map indicates a trail this way.\" Hako started moving along the underwater cliff, parallel to its edge. \"Keep those harpoons handy.\"\n\n\"I have a feeling I'm going to miss the kiddie pool,\" Knox said, glancing back toward the lagoon.\n\n\"Let's just try not to get eaten,\" Kendra replied. \"Come on.\"\n\nThe longer Kendra walked, the more she appreciated that the island of Timbuli was merely the top of an underwater mountain, and that going deeper into the ocean meant climbing down it. Not all of the mountain was as steep as the cliffs just beyond the lagoon, and there really was something of a path that wound down the less precipitous slopes, occasionally twisting into switchbacks.\n\n\"Who makes a path underwater?\" Knox asked quietly. \"Fish with feet?\"\n\n\"We're at an enchanted sanctuary,\" Kendra said. \"Might be magic.\"\n\n\"Or left over from old times when our elixir was in greater use,\" Tanu said.\n\n\"Could also be the people of the deep,\" Hako said.\n\n\"Merfolk?\" Knox asked.\n\n\"No,\" Hako said. \"The people of the deep are the drowned undead who serve the Underking. I've explored underwater all around these islands, and I sometimes see paths. Occasionally on those paths I have spotted shambling wanderers.\"\n\n\"Like zombies?\" Kendra asked.\n\n\"More powerful and deliberate than typical zombies,\" Hako said. \"We'll keep watch for them\u2014and for a thousand other dangers.\"\n\nThe path they followed curved and dipped and occasionally rose but mostly descended. The light dimmed the deeper they hiked, making the crystals they wore shine more distinctly. Noon faded to twilight as the shimmer of the sunlit surface passed out of view.\n\nReefs and fish became scarce as they descended deeper. When they reached a flat, sandy shelf crowded with clams the size of trampolines, Hako carefully led them around the huge bivalves.\n\n\"Shouldn't we check for pearls?\" Knox asked.\n\n\"Never bother the grand clams,\" Hako said. \"They summon predators when disturbed.\"\n\nThey all trod more lightly after the warning. The mountainside became less steep for a time, allowing for easier passage. Hako had them drop and hold still, then pointed out a long sea serpent swooping and corkscrewing in the distance. Without ever coming near, it fluttered away like a ribbon in the wind.\n\n\"Are sea serpents dragons?\" Kendra asked.\n\n\"They are related to the serpentine dragons,\" Hako said. \"Sea serpents have no legs and no breath weapons, but the largest are the longest of all magical creatures.\"\n\nThe talk of dragons made Kendra think about Celebrant. Where was he now? What preserve was he trying to topple next? She hoped the members of Dragonwatch were finding ways to resist him. She really hoped they could find the Sunset Pearl. If she could help secure Crescent Lagoon from falling, it would be a measure of payback for losing Wyrmroost to him.\n\nJust as the underwater half-light was fading to black because of the depth, the path came to the brink of a steep drop-off. The trail continued down in a series of switchbacks until it reached an underwater valley. Several dozen enormous shells rested on the sand, shedding luminance. The diverse shapes and colors made Kendra recall seeing Las Vegas from a distance at night on a family road trip.\n\n\"Shield your lights,\" Hako said, placing a hand over his chest as they peered off the brink.\n\nKendra hastily covered the crystal she wore around her neck.\n\n\"What's making those lights down there?\" Knox asked quietly. \"Glowing crabs?\"\n\n\"Merfolk village,\" Hako said. \"Their structures are grown rather than built.\"\n\n\"But they glow,\" Knox said.\n\n\"Bioluminescent microbes inhabit them,\" Hako said. \"Enough to brighten the whole valley. If we dip much below the merfolk village, we enter the endless night of deep sea.\"\n\nKendra reconsidered the shells as buildings, impressed by their organic design. She could see little swimming figures silhouetted against some of the luminous structures. \"I see some merpeople,\" she said.\n\n\"You're looking at Alluvia Minor,\" Hako said. \"One of the last known merfolk villages.\"\n\n\"Are there many unknown ones?\" Knox asked.\n\n\"Hard to say for sure, if you consider the question,\" Warren said.\n\n\"We suspect that larger merfolk settlements survive in some unmapped depths,\" Hako said. \"Over the years, a few of the merfolk have alluded to secret cities.\"\n\n\"What are those people doing on the outskirts of the village?\" Vanessa asked.\n\n\"Sharp eyes,\" Hako said. \"You'll notice they are all over the valley floor.\"\n\nNow that Kendra focused her attention, she saw many figures milling about in the shadows beyond the vibrant illumination of the giant shells. \"Dozens,\" Kendra said. \"Maybe hundreds.\"\n\n\"People of the deep,\" Hako said. \"Hundreds of them. A full horde of the zombies. Magical barriers protect the village, but it is besieged.\"\n\n\"Can we get through the barriers?\" Knox asked.\n\n\"Mortals can cross any of the village barriers at this sanctuary,\" Hako said.\n\n\"Look at the whale,\" Tanu remarked, pointing.\n\nKendra followed his finger to see a half-devoured carcass of a whale in the shadows beyond the village, a large portion of the skeleton exposed. People of the deep swarmed the remains, tearing at the blubbery flesh with hands and teeth.\n\n\"How fast are these zombies?\" Warren asked.\n\n\"Faster than you might think,\" Hako said. \"They're more skilled and alert than most zombies.\"\n\n\"We didn't time this very well,\" Warren said. \"Is it zombie season?\"\n\n\"I wonder if the Underking is aware of our intent,\" Vanessa said.\n\n\"Who knows how long the siege has been going?\" Hako said. \"It may have started when the preserve fell.\"\n\n\"The village is surrounded,\" Tanu said. \"How do we get past them?\"\n\n\"It's too bad we can't swim,\" Hako said. \"The people of the deep move on the ocean floor. They don't float.\"\n\n\"We could swim right over them?\" Knox asked. \"Are we too deep to cancel the elixir?\"\n\n\"Almost certainly,\" Hako said. \"Especially since we would have no way to breathe.\"\n\n\"Do we try to wait them out?\" Vanessa asked. \"Or consider returning another time?\"\n\n\"The drowned ones could be here indefinitely,\" Hako said. \"At least until we renew the protections of the sanctuary. If we want to get into the village, now might be as good a time as later.\"\n\n\"And we can't afford to wait too long,\" Kendra said. \"The Underking has Bracken. And the Sunset Pearl. And Seth. Who knows what will happen to Bracken and Seth if we leave them there too long? And once we lose that last moai, the whole sanctuary falls.\"\n\n\"We have to succeed,\" Warren said. \"And nobody gets helped if we're dead.\"\n\nHako looked around. \"I don't see any of the people of the deep up here. If there were any behind us, traveling the switchbacks below us would leave us extremely vulnerable. Once we start down, we could get trapped.\"\n\n\"If those on the valley floor see us coming, we're in trouble too,\" Vanessa said. \"If they come up after us, our only option will be to retreat.\"\n\n\"And once we're on their level, they could swarm us,\" Tanu said.\n\n\"I think there is only one chance,\" Hako said.\n\n\"A sea serpent attacks them?\" Knox asked.\n\n\"Wouldn't that be nice?\" Hako said. \"We go down quietly. I will distract them. When they come after me, you run for the village.\"\n\n\"Is that your only move?\" Knox asked. \"Sacrificing yourself?\"\n\n\"I should be able to get away,\" Hako said. \"The loyal harpoon can carry a person through the water\u2014even a Dry Depths Elixir drinker like me, though it may be slow.\"\n\n\"Doesn't have to be fast if you can get out of reach,\" Warren said.\n\n\"Those are my thoughts,\" Hako said. \"It won't be easy. I may draw some of the zombies off, but you'll still have to fight your way through.\"\n\n\"Anyone have a better idea?\" Tanu asked.\n\nNobody volunteered a plan.\n\n\"I'll take the lead,\" Warren said. \"Vanessa will guard the rear. Tanu, cover Kendra and Knox.\"\n\n\"I wish I could use potions,\" Tanu said.\n\n\"I'd rather not be mummified,\" Knox said.\n\n\"So much depends on descending quietly and unseen,\" Hako whispered. \"Hold the crystals in your hands. Don't let light leak between your fingers. Stay low, tread carefully, and keep away from the edge of the path. I'll hang back on the last switchback. You'll know when to charge. If a way doesn't open to the village, remember you can backtrack to the switchbacks and flee.\"\n\nKendra remembered running from zombies at Obsidian Waste. She was worried about tangling with zombies again, especially if these were more capable.\n\nThey started down the trail. It was dim, but there was enough light to see where to step. Hairpin turn after hairpin turn, Kendra stayed low, without a view of the sandy valley below. She hoped that since she could not see the undead, they could not see her.\n\nAfter descending nine serpentine switchbacks, Hako hung back at the turn while the others descended the final stretch. Kendra kept as low as she could as the path joined the valley floor.\n\n\"If the zombies chase us up these switchbacks, I hope I'm not the slowest,\" Knox whispered.\n\nKendra glanced up. The top of the trail towered above them. She had to agree that an escape up the path was not appealing. What if she got eaten by zombies simply because she lacked better cardiovascular endurance?\n\nWarren held a finger to his lips and motioned for them to stay down and wait. Looking toward the glowing village, Kendra could see the silhouettes of many shuffling figures. The near side of the village was little more than a hundred yards away.\n\n\"Who wants fresh meat?\" Hako called to the multitude.\n\nKendra looked up the path to where he stood at the previous turn. Hako had uncovered his crystal and stood at the edge with his harpoon over his head. Leading with the harpoon, he leaped off the path at least sixty feet above the valley floor, but instead of falling, he was carried forward by the weapon, dangling from the shaft. He gradually lost altitude until he landed in the midst of the people of the deep. All of the sea zombies in view converged on Hako.\n\n\"Now,\" Warren said quietly, dashing forward.\n\nHolding her harpoon ready, Kendra raced behind, with Knox and Tanu beside her. The sandiness of the seafloor kept her from reaching her top speed, but not by much. The people of the deep still strode toward Hako, allowing Kendra and her friends to advance several steps unnoticed. Kendra and her group dashed through the spaces between the unaware zombies. Hako yelled and swiped with his harpoon as the zombies swarmed around him.\n\nThe zombies looked bloated and otherworldly, their hair and clothes billowing because they were underwater. A crusty old guy with a limp and a tricornered hat noticed Warren coming and turned to face him, raising a corroded cutlass. Warren hacked his head off cleanly before the man could swing, Seaslayer a blur. But the action caught the attention of other sea zombies.\n\nWaterlogged faces swung to face Kendra. Some of the zombies shifted course, leering and increasing their pace. Warren cut down another zombie, and Kendra felt more vacant eyes turning in their direction.\n\nA woman with her head tightly wrapped in a scarf came at Knox with an upraised knife, apron flowing, and he stabbed her with his harpoon before she got near enough to strike. Her body twisted, wrenching the harpoon from his grasp. She dropped her knife, then gripped the harpoon with both hands and started pulling it out of her torso. On the other side of Kendra, Tanu used his harpoon to deflect the thrust of a gaff. Warren lopped the arms off a weaponless zombie coming at him.\n\nKendra and her group were moving faster than the people of the deep. Hampered by the water, the zombies stepped slower, but their reflexes were quick. Kendra was less than halfway to the village when she noticed the zombies massing ahead of them, blocking their route. Warren cut sideways to go around the mob, but zombies encroached from all directions, and the gaps between them were getting scarce.\n\n\"We may not make it to the village,\" Tanu said.\n\n\"Fall back,\" Warren called.\n\nThough the zombies pressed from all sides, they were sparser behind the mortals. The group reversed direction, which put Vanessa in front. She turned her harpoon and used the blunt end to drive zombies back and deflect blows.\n\n\"We're in trouble,\" Knox said, dodging outstretched arms, weaponless.\n\nKendra handed him her harpoon, and he plunged it into a heavyset man reaching for them with bloated fingers, again losing his weapon as the man's hands closed around the shaft. Warren stepped up to fight beside Vanessa, carving space for them to move forward, but the way to the switchbacks closed off as more zombies crowded to block their retreat.\n\nKendra wondered if she should get out her sack of gales. If it didn't stabilize her underwater, she might end up like a failed rocket test. She also considered seeing if the unicorn horn would keep the zombies back, but she worried about losing it. She would save it to try as a last resort.\n\nSlack, discolored faces approached from all sides, some clutching weapons, most without. Their numbers made weapons irrelevant. Each time Warren dispatched one zombie, three pressed forward to take its place.\n\n\"Above you,\" called a voice.\n\nKendra looked up to see several mermen and a few mermaids swimming about ten feet above the swarming zombies. Their long fishtails with wide fins undulated gracefully. Both men and women had long hair, but the females wore pearls and combs in their tresses. Their wide shoulders and impressive musculature reminded Kendra of Olympic swimmers.\n\n\"Take the line,\" a mermaid urged.\n\nKendra realized they were dangling slender ropes. Kendra grabbed one with both hands, and a pair of mermaids lifted her off her feet. Tails flailed and muscles strained as the women elevated Kendra above the zombies. Her full weight was evidently a lot for them to handle.\n\nHer friends were being lifted above the zombies as well. Three powerful mermen were having a hard time with Tanu. The potion master kicked at people of the deep until a fourth merman joined and they hoisted him out of reach.\n\nKendra clung tightly to the line. The small knots in the slender rope helped Kendra just enough to support her weight. The people of the deep crowded beneath her, greedy hands reaching upward. The mermaids holding her line grimaced, tails sweeping vigorously, and began making slow progress toward the village.\n\n\"They feel full of iron,\" one of the mermen complained.\n\n\"These are depth walkers,\" another merman said. \"Their great weight is legendary.\"\n\n\"She is too heavy,\" one of the mermaids holding Kendra complained. \"Should we just feed her to the floorscrubbers?\" She glanced down at the zombies.\n\n\"Please don't,\" Kendra said. \"We were sent by the caretaker.\"\n\nTails still flailing, the mermaids looked down at Kendra in surprise.\n\n\"You speak Corilli?\" the one with the orange tail asked.\n\n\"I speak many languages,\" Kendra replied.\n\n\"We'll try to save you,\" the one with the blue tail said. She looked at her friend. \"I've never missed the golems more.\"\n\n\"Golems?\" Kendra asked.\n\n\"Four golems guarded our village until recently,\" the one with the orange tail explained, groaning, veins standing out in her neck.\n\n\"A lich came and drained their energy,\" the one with the blue tail said.\n\n\"Without the golems to ward them off, the floorscrubbers surrounded the village,\" the one with the orange tail said.\n\n\"Where are the golems now?\" Kendra asked.\n\n\"Too many questions,\" the one with the blue tail said, short of breath. \"Carrying you is harder than it looks.\"\n\n\"Try hauling the big one,\" one of the mermen holding Tanu grumbled.\n\n\"This is important,\" Kendra said. \"Are the golems still in the area?\"\n\n\"One is right over there,\" the mermaid with the orange tail said, jerking her head to one side. \"They lie where they fell.\"\n\nKendra saw a jumble of large sea rocks where the mermaid had indicated. \"What exactly was drained?\" Kendra asked. \"Magical energy?\"\n\n\"How should we know?\" the mermaid with the orange tail said.\n\n\"Magical energy,\" confirmed one of the mermen toting Tanu, muscles bulging, yellow tail swishing forcefully. \"These golems have turtle shells at their core that can store vast amounts of energy.\"\n\n\"Cyclonic turtles,\" another merman added.\n\n\"I think I can revive the golems,\" Kendra said. \"I'm fairykind. I'm a well of magical energy.\"\n\n\"How would you revive them?\" the mermaid with the blue tail asked.\n\n\"By touch,\" Kendra said. \"If you can get me over to one.\"\n\n\"You might survive,\" the mermaid with the orange tail said. \"The floorscrubbers don't climb on the golems.\"\n\n\"They might if they had a target,\" the mermaid with the blue tail said.\n\nKendra looked over at the heap of rubble that once was a golem. Would touching the stones be enough? Would direct contact with the shell be necessary? What if she was wrong? What if her energy didn't translate? If the golem didn't revive, she might be killed. Or zombified!\n\nThe mermaids jerked the line to the side and a harpoon glided past Kendra, thrown from below. Struggling, they lifted her higher. Kendra realized that if she fell, the impact alone would injure her.\n\nMore merfolk swam out from the village, several of them armed. Two mermen took Kendra's line from the mermaids. One of the mermen had red hair and a purple tail, the other, black hair and a green tail. Both bore spears, the heads shaped to imitate crab claws.\n\n\"I am Faro,\" the merman with the black hair said. \"This is Lars.\"\n\n\"I'm Kendra.\"\n\n\"What took you so long?\" the mermaid with the blue tail complained, rubbing the muscles in her arms.\n\n\"Word of this rescue is spreading,\" Lars said. \"Some are debating whether we should help.\"\n\n\"Meanwhile they would die,\" the mermaid with the blue tail said.\n\n\"It's why we skipped the discussion,\" Faro said.\n\n\"Kendra believes she can revive the golems,\" the mermaid with the orange tail said.\n\n\"She is fairykind,\" the other mermaid added.\n\n\"How would you do it?\" Faro asked.\n\n\"I just need to touch the golem,\" Kendra said. \"Or maybe the turtle shell.\"\n\n\"Getting to the golem is no problem,\" Lars said.\n\n\"Once you reach the seafloor, your safety could be compromised,\" Faro said.\n\nLooking down at the rioting zombies, Kendra wondered if she should just let the mermen carry her to the village. Getting dragged in by them would not be a very impressive arrival. However, if she could revive their golems, the merfolk might feel they owed her a favor. Plus, letting the golems clear out the people of the deep would make her departure a lot easier when the time came.\n\n\"I want to try,\" Kendra said.\n\nThe mermen veered away from the village toward the golem. They swam considerably faster than the mermaids with her weight, though they were straining as well. At least the golem was much closer to them than the village was.\n\nThey drew near the rock pile, sea zombies following below. \"Remember,\" Kendra said, \"I can't just drop. I fall like I'm in air, so lower me.\"\n\nFaro and Lars gently let her down atop the rock pile. Weedy vegetation grew among the rocks, and small shells pimpled the rough surfaces. Touching the rocks had no effect, so Kendra scrambled around searching for the turtle shell.\n\nAfter brief hesitation, one of the sea zombies stepped onto the rock pile, brandishing a large metal hook. Faro swooped down, stabbed him with his spear, and thrust him off the rocks. Lars joined the fight, skewering the next trespassing zombie.\n\nKendra moved from crack to crack, peering between the rocks. Deep inside one of the wider gaps, she caught sight of an iridescent turtle shell. Squirming in as far as she could, she stretched her arm forward until her fingertips brushed the shell. Upon contact, the shell blazed with light, and Kendra gasped as energy surged out of her. The rocks around her shifted, and she tumbled to the sandy seafloor.\n\nLooking up, Kendra found herself overshadowed by a nine-foot-tall figure made of stone and sand. The robust golem had a large head with concave hollows for eyes, a thick torso, and outsized hands and feet. With two big sweeps of his long arms, the golem cleared away at least a dozen sea zombies.\n\nFaro grabbed Kendra by one arm, Lars seized the other, and they swam her upward. The golem let them pass as they rose above the fight, then proceeded to punch and stomp sea zombies, crumpling bodies with every blow. For the first time, the people of the deep showed uncertainty, turning and trying to flee. Kendra heard shouts of relief from the merfolk carrying her friends, and cheering from the village.\n\n\"Are you all right, Kendra?\" Faro asked.\n\n\"I'm okay,\" Kendra said.\n\n\"Think you could do that three more times?\" Lars asked. \"We can use this golem to clear the way to the others. Our mystics can give it instructions.\"\n\n\"Sure,\" Kendra said, \"if you two come with me. You can help me find the shells and keep hold of me so I don't get dumped on the ground when the golems re-form.\"\n\n\"Consider it done,\" Lars said.\n\n\"We're in your debt,\" Faro said.\n\nKendra looked over at the rest of her group getting transported to the village, the golem hammering away at the zombies beneath them. Other merfolk were now streaming from the village to help.\n\n\"Let's wake up those sleeping golems,\" Kendra said.\n\nLord Quintus of the merfolk had silver hair and the upper body of a prizefighter. Rather than a throne, he rested on spongy cushions fitted inside of a giant clam shell, lying flat, like a bed. Kendra had seen no chairs in the large room, and, looking around, she realized that since the merfolk could float and had the lower bodies of fish, chairs might not even make sense for them.\n\nOutside the village, the four golems had the people of the deep on the run. With the help of Faro and Lars, reenergizing the golems had been relatively simple. Each golem they reanimated helped drive away the sea zombies so Kendra could revive the next. Once all four were in action, the sea zombies had no chance against the tireless onslaught.\n\nAfter revitalizing the golems, Kendra learned, to her relief, that Hako had made it to the village using his loyal harpoon. Now he, Tanu, Knox, Warren, and Vanessa stood with her before the leader of the merfolk.\n\n\"You have done us a great service,\" Lord Quintus said. \"Those floorscrubbers blocked access to many of the reefs and kelp forests where our foragers hunt and gather.\" He held up the document Tanu had given him. \"I see that Savani endorsed your visit. You are on an errand to recover the Sunset Pearl?\"\n\n\"We know the Phantom Isle is almost impossible to find,\" Kendra said. \"But we have to go there. A map told us a demon called Remulon has a compass that will help us.\"\n\n\"You are brave,\" Quintus said. \"The demon Remulon is more dangerous than almost any creature in the sea. I would never choose to visit his lair.\"\n\n\"We have to try,\" Kendra said.\n\n\"And the Phantom Isle is worse,\" Quintus said. \"If you can find it, the isle is guarded by the sea dragon Jibarro. Beyond lies the entrance to the Under Realm. Most would exert great efforts to avoid such a destination.\"\n\n\"The land of the dead,\" Kendra said.\n\n\"No, much worse,\" Quintus said. \"The domain of the neverdead.\"\n\nThe description made Kendra shiver. \"One problem at a time. Can you help us find Remulon?\"\n\n\"This demon is our neighbor,\" Quintus said. \"I cannot jeopardize the uneasy truce we've established, especially now, with most of the sanctuary's protections down. We will not engage directly with the demon. But if you insist on seeking him, we will show you the way.\" Quintus raised his voice. \"Who among our hunters will volunteer?\"\n\n\"I will lead them,\" Faro said, swimming forward, spear in hand.\n\n\"You could have no finer guide,\" Quintus said. \"Faro, show them to the lair of Remulon and then promptly return. How they fare with the demon is their concern.\"\n\n\"You know this demon,\" Kendra said. \"What's our best approach?\"\n\n\"Try not to let it come to a fight,\" Quintus said. \"I wish I could convince you to avoid the demon entirely.\"\n\n\"Does he have any known weaknesses?\" Kendra asked.\n\n\"He is frustratingly clever,\" Quintus said. \"He is fierce in combat. His lair is well protected. But he does have a passion for collecting novelties.\"\n\n\"So we might be able to work out a trade?\" Kendra asked.\n\n\"It is possible,\" Quintus said. \"Though I fear his primary interest will be in acquiring you. Thank you for your service to our village. You will be remembered as a friend of the merfolk.\"\n\n\"Thank you,\" Kendra said, relieved that at least they had a guide to the demon. She withdrew from the glowing assembly hall with her group.\n\nFaro swam up to her. \"Shall we depart?\"\n\n\"Sure,\" Kendra said.\n\nHe inclined his head. \"It is my honor to escort you.\"\n\n\"Before we go,\" Kendra said, \"do you know any mermaids named Gracie and Lila?\"\n\n\"Those two are always into mischief,\" Faro said.\n\n\"They know my cousin,\" Kendra said.\n\nHe nodded. \"That pair visits the surface more than most. Would you like me to look for them?\"\n\n\"If you don't mind,\" Kendra said, hoping their relationship with Tess might win her some special treatment or information.\n\nFaro swam away, and Knox approached her.\n\n\"We can be grateful for one thing,\" Knox said.\n\n\"What?\" Kendra asked.\n\n\"We're not supposed to eat. I saw their food.\"\n\n\"Gross?\"\n\n\"Squirmy,\" Knox said, pulling a disgusted face and shuddering.\n\n\"We get to negotiate with a demon instead,\" Kendra said.\n\n\"Think they would loan us some golems?\" Knox asked.\n\n\"I wish,\" Kendra said. \"They don't want trouble with the demon. Sending their golems to help us would probably be an act of war or something. We have a golem at Fablehaven\u2014Hugo. It would be a relief if he were here.\"\n\n\"How did you bring those golems back to life?\" Knox asked.\n\n\"They were out of magical energy,\" Kendra said. \"Ever since I became fairykind, I have been like a magical battery recharger.\"\n\n\"How did that fairykind thing happen?\" Knox asked.\n\n\"The power came from the Fairy Queen during an emergency at Fablehaven,\" Kendra said. \"It's a long story.\"\n\n\"Can you use that power against the demon?\" Knox asked.\n\n\"Not unless I have something to charge,\" Kendra said. \"Otherwise the ability is fairly useless.\"\n\nFaro returned with a pair of youthful mermaids, one with blonde hair, the other a brunette. \"Lila is the blonde?\" Kendra asked. \"And Gracie?\"\n\n\"Yes,\" Lila said. \"You're Kendra.\"\n\n\"I think you met my sister,\" Knox said. \"Tess.\"\n\n\"We did,\" Gracie said. \"Up at the surface. Cute little thing.\"\n\n\"She's my cousin, and she wanted me to tell you hello,\" Kendra said.\n\n\"Give her our regards,\" Lila said. \"We tried to coax her into the water, but she wouldn't come.\"\n\n\"Why invite her into the water?\" Knox asked.\n\nThe mermaids giggled. \"To bring her down deep with us,\" Lila said.\n\n\"You were going to drown her?\" Knox exclaimed.\n\n\"Sure,\" Gracie said.\n\n\"Didn't you like her?\" Knox asked.\n\n\"Of course,\" Lila said. \"We talked and sang songs.\"\n\n\"Then why drown her?\" Knox asked.\n\nGracie blinked as if surprised by the obvious question. \"The sea claims all the unwary.\"\n\n\"We weren't going out of our way,\" Lila said. \"No nets or traps.\"\n\n\"She was careful,\" Gracie said. \"Stayed well out of reach.\"\n\n\"She wouldn't come examine the shell I tried to show her,\" Lila said.\n\n\"As a favor to me, please let her be,\" Kendra said. \"She's part of our family.\"\n\nGracie shrugged. \"If she's careless, and we don't claim her, it'll be something else.\"\n\nLila elbowed Gracie. \"We're grateful to have the golems back,\" Lila said. \"We won't tempt your little cousin anymore.\"\n\n\"Shall we go?\" Faro asked.\n\n\"Sure,\" Kendra said. \"Nice to meet you two.\"\n\n\"Likewise,\" Lila said, swimming off with Gracie.\n\nKnox looked up at Faro. \"Why did you come help us against the undead? Seems like merfolk would rather see us drown.\"\n\n\"You were not merely careless humans tempting fate,\" Faro said. \"You were breathing water, as we do. It has been long years since we have seen humans walking the depths. There was some debate, but many of us wanted to meet you more than we wanted to let the floorscrubbers destroy you. I brought you gifts.\" He handed a slender spear to Kendra and another to Knox. \"To replace what you lost.\"\n\n\"Thanks,\" Kendra said. The spear was a little longer than the harpoon she had used, and it felt lighter but no less sturdy.\n\n\"Is it far to Remulon?\" Warren asked.\n\n\"Farther to walk than to swim,\" Faro said. \"We should arrive just after midday. He's closer to the surface than our village is, but we'll have to start by going deeper to get there. Come.\"\n\nAs Faro led them out of Alluvia Minor, they received a warm farewell, with many of the merfolk waving and calling out good-byes. Kendra waved back, but she had a hard time feeling close to creatures who were willing to drown children. Hopefully they were not all as indifferent to human life as Gracie and Lila. At least the merfolk had helped them with their quest, and for that she was grateful.\n\nHolding his spear in one hand and a lantern in the other, Faro led them across the sandy valley beyond the village, now clear of functioning sea zombies. The golems worked piling crushed corpses up against the valley wall and covering them with sand.\n\nFaro led them up to the outward edge of the valley, where they could see that the village and the valley were like a balcony on the mountainside, with the slope dropping off farther below. \"There are faster ways to go if you could swim,\" Faro said. \"This path will have to do.\"\n\nThe trail wound down until the only light came from their crystals and the lantern Faro held. As Kendra saw the light of her crystal flash upon new rocks and shells along the path, she felt like an intruder in these depths. Had light ever touched this landscape? Did it even belong here?\n\n\"Will our lights make us targets?\" Kendra asked.\n\n\"Most of the hunting down this deep is not done by sight,\" Faro said.\n\n\"Do you come down here a lot?\" Knox wondered.\n\n\"There is not much reason to go this deep,\" Faro said. \"We mostly forage higher than the village.\"\n\n\"Is that how you merfolk live?\" Knox asked. \"Foraging?\"\n\n\"The sea provides what we need,\" Faro said. \"We gather with gratitude.\"\n\n\"Have you ever been on land?\" Knox asked.\n\n\"Few merfolk venture into the dryness,\" Faro said. \"We leave the airy regions to your kind.\"\n\nAfter some time, the way started to trend up more than down. At one point, Faro had them pause and cover their lights. Several bioluminescent jellyfish drifted in a group, clear and bulbous with glassy tentacles, bodies laced with squiggly patterns that glowed a light blue. When Faro had them uncover their lights, the jellyfish seemed to vanish.\n\nAs they climbed, Kendra felt relief to see a haze of light above. She began to notice more fish. Her legs grew tired from the upward incline. The light from the surface gradually increased until their crystals were needed less. Marine life reappeared in greater abundance, including plants and sponges. They reached a large iron bell, incongruously resting on the seafloor, corroded and lumpy with barnacles.\n\n\"We're getting close,\" Faro said quietly. \"This bell marks the outskirts of the area claimed by Remulon, though he seldom leaves his cavernous lair. Does the water feel warmer to you?\"\n\n\"Under the effect of the elixir we don't feel the water,\" Hako said. \"But the temperature seems warmer.\"\n\n\"Thermal vents from the volcano heat certain pockets in this region,\" Faro said. \"Stay near me.\"\n\nThey advanced, passing an ancient cannon and three human statues that looked more suited for a museum than the bottom of the sea. Ahead, a sheer cliff loomed lofty, partially masked by a dense kelp bed. As they neared the base of the cliff, the way forward was barred by a high reef of black coral, overshadowed by towers of kelp.\n\n\"This razor coral forms a protective barrier around Remulon's lair,\" Faro whispered. \"Take great care\u2014the edges are so keen you could lose fingers without even feeling the cut. The electric kelp above it can shock you with more power than a lightning strike.\"\n\n\"That seaweed could fry us?\" Knox asked.\n\n\"Venture within range and it will electrocute you,\" Faro said. \"We have reached your destination. This is where I depart.\"\n\nKendra stared from the barrier to their guide. Was he really leaving now that they needed him most? \"Is there any way through?\" she asked.\n\nFaro jabbed at a low gap in the razor coral with his spear. \"From what I understand, you can get through here. But do not brush against the coral. Keep your wits about you. Farewell. I hope you find what you seek.\"\n\nAs Faro swam away, Kendra could not help feeling a little jealous. The hike to this point had been pleasant. Faro had the luxury of leaving before the scary part began.\n\n\"Whoever goes first will have to take it especially slow,\" Hako said. \"We don't know how the way might narrow or widen. Let's hope Faro was correct that it goes all the way through. Careful movements. Don't raise your head. Don't poke out your elbows. Stay aware of how much space you have on all sides. Watch out for unexpected protrusions. The good news is, the coral will remain stationary. Any damage we sustain will be our own fault.\"\n\n\"I'll go first,\" Warren said.\n\n\"Do we have a plan for dealing with the demon?\" Vanessa asked.\n\n\"What can we do besides improvise?\" Warren asked. \"Try to bargain.\"\n\n\"And fight if we must,\" Hako said.\n\n\"Entering the lair of a demon is unwise,\" Vanessa said. \"These creatures live to take advantage of others, to outsmart opponents. If not for our extreme need, I would not recommend going through with this. Don't let your guard down. Be wary of any agreement Remulon wants to make.\"\n\n\"Wish me luck,\" Warren said.\n\n\"Try to keep your fingers,\" Vanessa replied.\n\nWarren got down on his stomach and started scooting into the gap beneath the razor coral. Hako crouched behind him.\n\n\"Should we announce ourselves?\" Vanessa asked. \"It might be more polite.\"\n\nWarren stopped crawling. \"If we catch him asleep, that could work to our advantage.\"\n\n\"He probably knows we're here,\" Vanessa said. \"Anyone careful enough to have barriers like these will have systems in place to detect intruders. He might be listening to every word we say.\"\n\n\"Coming in!\" Warren called. \"Friends paying a neighborly call!\"\n\n\"Charming,\" Vanessa said.\n\nWarren wormed out of view, then Hako followed. Tanu gave a miserable sigh. \"I'm not built for this,\" he muttered before scooting after Hako in an army crawl, keeping his body as close to the ground as possible.\n\nKendra winced as she watched Tanu slither forward. He had a lot less margin for error than the others. She got down and squirmed in behind Tanu, figuring that if he could fit, she should have no problem. The close tunnel through the coral twisted and turned unpredictably. Dark spikes and blades jutted above her and to either side. Kendra took care to keep her head down and to stay centered in the gap. It was easy to forget she was underwater and to imagine herself crawling through a thorny briar. She kept envisioning her skin getting mangled by a casual brush against dozens of tiny blades. The grisly imagery helped her proceed with caution.\n\nAt last Tanu rose to his feet ahead of her, and Kendra reached the end of the spiny tunnel. She crawled out and gratefully accepted a hand up from Warren. She turned to congratulate Tanu for making it through and noticed some blood misting into the water from his shoulder, but froze when she realized he and Hako were facing the mountainside as if transfixed.\n\nA huge cavern hollowed out the base of the cliff, and in the opening awaited an octopus the size of a cement truck. The huge cephalopod was black with white suckers lining his tentacles, his intense red eyes fixed on them.\n\nKnox exited the razor coral, trailed by Vanessa. She inspected Tanu's shoulder and looked relieved. Once they all stood together, Hako stepped forward.\n\n\"We come seeking Remulon,\" Hako said.\n\nYou have found him, gamekeeper. The penetrating words came directly into Kendra's mind, the telepathic voice methodical and resonant. Step aside. My interest is in she who slayed our king.\n\nHako backed away and Kendra stepped forward, summoning her courage as she had when speaking to Celebrant. Showing fear would only hurt their chances. Tentacles flowed in rubbery tangles as the octopus shifted, oblong head pulsing. Kendra did not think it looked like a creature that should speak.\n\n\"I guess you mean me,\" she said.\n\nMy, how you shine, Remulon spoke to her mind. You come to me without Vasilis or any comparable armaments. Am I to understand you do not mean to destroy me?\n\n\"I've come hoping for help,\" Kendra said.\n\nYou bear the mark of Jubaya, Remulon communicated. I have a measure of respect for her, but she was no ally of mine.\n\nKendra had not been thinking about the mark. In the fairy realm, the demon Jubaya had marked her by taking her hand, and then instructed her to seek help from the demons against the dragons. \"We need your assistance,\" Kendra said.\n\nThe mark has never been used, Remulon conveyed. It was especially meant for three others. Now you seek help? And from me? How did you fare at Wyrmroost?\n\n\"The preserve has fallen,\" Kendra said.\n\nI am aware. And you did not solicit aid from Talizar? He might have rescued you.\n\nKendra remembered that Jubaya had mentioned three other demons who might help her. Talizar was one, but she had never tried to find any of them. She had never intended to seek help from the demons. Now seemed like a poor time to bring that up. \"It was a surprise attack,\" Kendra said.\n\nYou bring a peculiar test, Remulon mused. You have achieved much. You found favor with the Fairy Queen. You are a dragon slayer. You vanquished Gorgrog the Vile. And yet you come to me apparently defenseless, announced by the mark of Jubaya, groveling for assistance. What aid do you desire?\n\n\"We need to find the Phantom Isle,\" Kendra said.\n\nHis laughter filled her mind. I wondered if you meant to approach another monarch. But the Underking? Truly? In his domain? How long do you expect to survive?\n\n\"We have business there,\" Kendra said. \"Friends to help, and the Sunset Pearl to recover. Don't you demons want to see the dragons fail? Without the pearl, Crescent Lagoon will not hold the dragons in much longer. They will be free to go conquer the world.\"\n\nI know your methods are different from mine, Remulon expressed. Perhaps you see different ways to leverage power than I do. But I feel compelled to wonder . . . has your success been luck? What am I missing? It's true that you shine, but you and any who enter the Under Realm with you will perish. The darkness is overpowering. You would face hordes of revenants, legions of liches, and the Underking himself. To gaze upon him is to join the ranks of his undead.\n\nKendra ached to think of Seth and Bracken trapped in such a place. \"Let me worry about how we will do what we need to do. We need to know where to find the isle. Will you help us?\"\n\nI could, Remulon assured her. In my garden I have a nova song.\n\n\"I was told you have a compass,\" Kendra said.\n\nLess a compass, more a homing pigeon, Remulon explained. You are fairykind. Are you not familiar with the storied nova song?\n\nKendra looked back at Tanu. \"Can you hear him?\"\n\n\"I heard what he told Hako,\" Tanu said. \"I feel telepathy coming from him, but I can't catch the meaning.\"\n\n\"Have you heard of a nova song?\" Kendra asked.\n\n\"It's a legendary fairy,\" Tanu said. \"Maddox would trade all he has for one.\"\n\nA nova song is an amphibious fairy capable of extraordinary brightness, Remulon conveyed. They live to light the darkest spaces they can access.\n\n\"Your nova song can find the Phantom Isle?\" Kendra asked.\n\nIt is telling to realize the nova song could dive to the emptiest cave in the deepest trench at the bottom of the sea. Instead she would seek out the Phantom Isle. The darkness there resists and even extinguishes light. It is thicker, richer than any natural darkness. A real challenge for her. And part of the reason nova songs are so rare.\n\n\"You hold a nova song captive?\" Kendra asked. \"Why isn't she an imp?\"\n\nMy garden is protected within my caverns, but not behind any doors, Remulon explained. None of my trophies are indoors.\n\n\"Right,\" Kendra said. \"Fairies only fall if kept indoors overnight. Remulon, will you loan us the nova song? That's all we need.\"\n\nShe is my most prized possession. Yours is a doomed errand from which none will return. To lend her would be to lose her.\n\n\"You're underestimating me,\" Kendra said. \"I'll bring her back.\"\n\nImpossible! Remulon exclaimed. You would not make it past the sea dragon. If you did, you and anyone you bring would be overwhelmed by the undead. You will perish, paralyzed in the dark.\n\n\"I can speak in the presence of dragons,\" Kendra said.\n\nYou fail to understand. Even uninvited dragons would be paralyzed by the power and darkness of the Under Realm. Orogoro himself would fall. No potion or trinket would make a difference.\n\n\"I'll find a way,\" Kendra said.\n\nYour reckless optimism is instructive, Remulon expressed. If you are baiting me for some sort of trap, I sincerely applaud you. I am convinced this is exactly what it looks like.\n\n\"What do you mean?\" Kendra asked.\n\nDo you know who I am? Remulon asked.\n\n\"Kind of,\" Kendra said.\n\nThe demons of the deep are underappreciated by airbreathers, Remulon grumbled. But even on the surface my name is renowned. I have never been captured. I have never been bested. I am feared by the most formidable of my kind. Each of my tentacles has different abilities.\n\n\"You sound very impressive,\" Kendra said.\n\nYou came here unprepared. I believe you have a desperate need, and you visited me armed only with a mark from a demon and a vague hope I might help.\n\n\"We're willing to bargain,\" Kendra said.\n\nHow Gorgrog fell to you is incomprehensible. You have played this all wrong. What could you possibly offer me that would be more attractive than simply keeping you?\n\n\"Try me,\" Kendra said. \"What do you want?\"\n\nRemulon fell silent. A few of his tentacles stretched and curled. How much do your companions matter to you?\n\n\"Very much,\" Kendra said.\n\nUnconditionally bind yourself to me as my servant with an irrevocable oath. Fall on your knees and pledge to do my bidding for the rest of your days, and I will let you free your companions through your service, one by one.\n\nKendra stared at the octopus in disbelief. \"No way!\"\n\nYou solicited an offer, Remulon conveyed. You wish to reject the opportunity I present? Under the circumstances, it is very generous. I will not make another offer.\n\n\"You haven't captured us yet,\" Kendra said.\n\nRemulon laughed again.\n\n\"Is he threatening to capture us?\" Hako asked.\n\nYou are about to make the nova song my second most valued possession. The fairykind human who destroyed Gorgrog is an incomparable trophy. If your comrades posed a threat, I might consider slaying some of them. It will be a more pleasant novelty to keep all of you.\n\n\"He wants us,\" Kendra said, opening the small pouch on her waist and reaching for the unicorn horn.\n\nWith a yell, Hako flung the loyal harpoon at Remulon. It streaked through the water with supernatural speed, right on target, until a black tentacle lashed out, caught the harpoon, and snapped it like a toothpick.\n\n\"That was quick,\" Warren said in despair.\n\nThat tentacle is so quick I could kill you all before you understood what happened, Remulon conveyed. I could kill you in so many ways, but this tentacle will put you to sleep.\n\nKendra pulled out Bracken's first horn, accidentally dropping the sack of gales to the seafloor. Remulon raised a tentacle, showing the rows of white suckers on the underside. Dozens of dartlike spines fired from around the suckers, whizzing through the water toward Kendra and her friends. One pricked Kendra in the shoulder; another pierced her thigh.\n\nShe staggered, immediately drowsy. The unicorn horn fell from her hand, landing on the sack of gales. She pulled the slender spine from her thigh and stared at it blurrily. It was approximately the length of her hand.\n\nTurning, Kendra regarded her companions. Warren had dropped his sword and sunk to his knees. Hako was already flat on his back, and Tanu was sprawled nearby. Knox had fallen right by the tunnel through the razor coral. Vanessa swayed, took a few steps sideways, and collapsed.\n\nKendra hit the ground before she realized she was falling. As consciousness slipped away, she felt large suckers puckering against her skin.\n\nPuffy gray clouds shed a haze of rain to the west, and chaotic swells kept the boat rearing and plunging, but Seth was enjoying his time alone on the water. As he considered the solitude, Seth realized he had no memory of spending time on his own. Sure, he had been unchaperoned in the Under Realm before, but even in his room with the door closed, Seth could feel the undead around him and knew the Underking might be watching.\n\nThe clever boat maneuvered itself across the choppy water, leaving Seth free to ponder. Part of him wanted to tell the boat to take him to San Diego, but he had a hunch the Underking would override any command to flee. He rubbed at the ghostly manacle on his wrist, fingers passing through it. Was it absurd to think that if he delivered the Everbloom to the Underking, the manacle would come off and he could start living his own life? Was that too much to hope? Would there always be a catch? Another favor owed? More hoops to jump through?\n\nThe Sphinx and Ronodin had no intention of ever freeing the phantom Dezia. They would use her and then trap her as tightly as they could. Seth scowled. What would prevent them from doing the same to him?\n\nRonodin had brought Seth to the Under Realm knowing the Underking would strike a deal that bound him there. They might all be working together against him. Ronodin was teaching him the things he needed to know to serve as a useful tool. The dark unicorn spoke convincingly, but his actions didn't always add up. If he wanted to free everybody, why bind Dezia? Why lock up Bracken? Why force Seth into a situation in which the Underking could enslave him?\n\nSeth used the dipper to sip fresh water from the sweet bucket. Were liches and wraiths more worthy of freedom than he was? Should liches and wraiths be free to roam where they might encounter innocent people? What about the freedom of their victims?\n\nSeth knew he had dark powers. But did that mean he had to become like Ronodin? Or work for the Underking? The reality was, he didn't enjoy it, and his concerns about their practices and motives kept growing.\n\nHopefully the Hidden Sage could help him see more clearly. At minimum, Seth wanted to learn how to retrieve his past. He felt like he would do anything to get his memories back, no matter what they contained. He deserved to know who he had been. How much those memories influenced his future would be up to him to decide.\n\nThe tall, steep hills of Omari were rounded at the top, like colossal camel humps. Green with vegetation, they loomed larger as the clever boat neared the island, prompting Seth to wonder how such extreme formations took shape.\n\nSeth approached a beach with pebbly sand and a few boulders. He had asked the clever boat to land him near Mount Dagaro. Gazing up at the many hills towering over the beach, Seth wondered which was his desired destination. If Mount Dagaro was toward the interior of the island, it might be none of the hills he currently saw. Identifying the right one to climb felt daunting.\n\nAs the boat eased into the breakers, Seth saw a huge white gorilla lumbering from the jungle onto the beach, muscles rolling in its broad shoulders as it supported its steps with its knuckles. The beast was at least twice the height of a normal gorilla, and it gave Seth a hard stare, baring sharp teeth. Behind the ape, Seth saw at least two other oversized white gorillas partly hidden by vegetation.\n\n\"Turn around,\" Seth ordered the boat. \"We can't land here.\"\n\nThe boat halted, then backed away from the shore, angling expertly to avoid capsizing. The gorilla on the beach snarled and slapped its chest. Another hulking gorilla emerged from the foliage to join the first.\n\nThe Underking had not warned Seth about gorillas. What other dangers lurked on Omari? Though Seth had enjoyed the solitude in the boat, he now missed Ronodin's expertise and competence. Seth squeezed the gunwale. Was this why the Underking had let him venture to Omari? Did the Underking expect him to fail?\n\nBobbing on the swells, Seth tried to collect his thoughts. Those apes had guarded the beach as if they owned it, regarding him with open hostility. Even if he managed to get ashore, it would be foolish to wander a jungle patrolled by giant gorillas. They were way out of his weight class.\n\nWhat were his options? What could he use to his advantage?\n\n\"Circle the island,\" Seth said. \"Stay near the coast.\"\n\nThe boat complied.\n\nAs they made their way around the island, Seth reached out with his senses. If he could locate any undead, maybe he could gather information. Ideally he could charm some into becoming his bodyguards.\n\nSeth perceived no dark presences at first. But as they rounded a rocky headland, a new promontory came into view on the far side of a bay. A squat stone lighthouse stood on the point, emanating a familiar dark energy.\n\n\"Go to the lighthouse,\" Seth directed, scanning the coastline for danger. He didn't see any gorillas. The dilapidated remains of a dock sagged at one end of the bay, but he saw no buildings besides the lighthouse. A few exotic birds added splashes of color, and he noticed a lone fairy twinkling over some blossoms near the dock.\n\nThe boat took a circuitous route to the lighthouse, instinctively avoiding rocks and reefs, many of them lurking just beneath the surface of the water. When the boat came alongside the stony promontory, the lighthouse looked abandoned on the deserted shoreline. But Seth could sense two wraiths inside.\n\n\"Wait here,\" he told the boat, hopping onto the rocks and climbing up to a grassy area beside the lighthouse. Built of rough stone blocks, a low tower rose from a broad, hexagonal base. Seth wondered how long it had been since the lamp atop the structure had been lit.\n\nHe walked up to the heavy wooden door and found it locked. Remembering what Ronodin had taught him, Seth closed his eyes. Drawing upon the darkness inside of him, Seth projected his power at the door and commanded it to open. To his delight, he heard a click. When he tried the door again, it swung inward. The success marked the first time he had used his power to undo a lock.\n\nThere was no ignoring the cold presence of the wraiths. \"Hello?\" Seth called. \"Trick or treat. Anybody home?\"\n\nAfter stepping into a dusty room with webby furniture, Seth knew the lighthouse had a basement because he could sense the wraiths down there. They must have heard him enter\u2014he could tell they were in motion, coming up the stairs. The temperature in the room plunged, and Seth felt the hair on his arms standing up.\n\nIntruder, one wraith thought at him venomously.\n\n\"I can hear you,\" Seth said. \"I came to help.\"\n\nA door opened from the basement, and the wraiths entered the room. Seth stood firm against their icy malice.\n\nYou are young, one communicated.\n\nYou are warm, the other expressed.\n\n\"Tell me why you're here,\" Seth said.\n\nWe keep the lighthouse dark, one conveyed.\n\nShips crash on the reefs, the other volunteered.\n\n\"Seen many ships lately?\" Seth asked.\n\nThey offered no reply.\n\n\"This place is abandoned,\" Seth said. \"I bet the light stays out with or without your involvement. I've been helping wraiths around the islands, and now I need a hand. You know the boundaries are down?\"\n\nYes, one of the wraiths acknowledged.\n\n\"I'm doing errands for the Underking,\" Seth said. \"I need to climb Mount Dagaro. Do you know the way?\"\n\nYes, the other wraith confirmed.\n\n\"Who put you here to keep the lighthouse dark?\"\n\nNobody, one wraith expressed.\n\nWe found this purpose, the other conveyed.\n\n\"Want some variety?\" Seth asked. \"I need escorts.\"\n\nWhy should we aid you? one wraith asked.\n\n\"I'm a friend to wraiths,\" Seth said. \"I freed wraiths from a prison at Wyrmroost. I freed wraiths on Timbuli. Someday I'll help you.\"\n\nYou pledge to help us? the same wraith asked.\n\n\"If you promise to help me get to Mount Dagaro safely,\" Seth said. \"And then back to my boat. Someday, when you have a need, and I'm around, I'll help you. Maybe someday you'll want to get off this island. Maybe get recruited to the Phantom Isle, if that sounds good. Or else just find a really nice graveyard with some comfortable tombs. Until then, I can give you an exciting task, and I'm one of the few living people who can hear you. You've probably been cold? Hungry?\"\n\nI thirst, the other wraith communicated.\n\n\"I bet you thirst,\" Seth said. \"Can you drive away those big apes? Are they scared of you?\"\n\nAll animals flee us, one conveyed.\n\nThey flee or we bind them, the other expressed.\n\n\"That's what I need,\" Seth said. \"Some powerful wraiths to keep the gorillas away. How about it? Will you promise to take me safely to Mount Dagaro? And then back to my boat?\"\n\nYes.\n\nYes.\n\nSeth smiled. \"Do you want to start out traveling by boat? Or should we go over land?\"\n\nWe do not cross the water, one of the wraiths stressed.\n\n\"We'll hike, then,\" Seth said. \"I'll tell the boat to wait here. Should I give you names?\"\n\nIf you wish, the other wraith allowed.\n\n\"How about lighthouse names?\" Seth suggested. \"You can be Lampy. And we'll call you Salty. And if you don't mind, I like when you speak out loud.\"\n\n\"As you desire,\" Lampy said.\n\n\"Great,\" Seth said, looking from one expressionless wraith to the other. \"Let's go.\"\n\nThe hike to Mount Dagaro took more than two hours, and though Seth heard distant rustling on occasion, he saw no animals. The wraiths kept to the shadows when they could. They seemed taxed when crossing areas in direct sunlight, moving through the brightness with slower steps and bowed heads. Seth found their chilling presence both creepy and reassuring. If they kept the apes away and got him to the right mountain, he considered a little creepiness a small price to pay.\n\nMount Dagaro was one of the many soaring hills he had seen when approaching the island, taller than most. A faint, narrow trail spiraled up the mountain, sometimes steep enough that Seth had to use both hands and feet to climb. The sides of the hill were never quite straight down, but Seth suspected that if he started falling, he would not stop until he hit the bottom. The wraiths continued with him until the scant trail ended at a cave masked by a curtain of roots.\n\n\"We can proceed no farther,\" Salty said.\n\n\"We will await your return,\" Lampy said.\n\n\"Do you know anything about the Hidden Sage?\" Seth asked.\n\n\"We cannot see into the cave,\" Lampy said.\n\n\"Nor can we enter the cave,\" Salty said.\n\nSeth took their answers as a no. \"Wait here. I'll be back.\"\n\nSeth pushed through draperies of twisty roots that descended from the ceiling of the cave to the floor. It was hard to imagine a helpful sage living inside such a wild, secluded place. The cave looked more like a den for a monster. What if the Underking had sent him to his doom? Seth reminded himself that the Underking wanted the Everbloom. If he wanted Seth dead, why not have a bunch of liches gang up on him back in the Under Realm? There seemed to be no reason for the Underking to sabotage him, and that thought brought some consolation.\n\nBeyond the curtains of roots, the cave opened up into a single room. Small, slender stalactites decorated the ceiling, and low stalagmites pointed up from the floor. Water dripped from some of the stalactites into a shallow pool. Seth saw no branching corridors\u2014and no sign of life. He seemed to be alone at a dead end.\n\nHe was hoping to find the Hidden Sage. Maybe the sage was more hidden than just residing in an obscure cave on a mountaintop. Could there be a secret passage? Might the sage be invisible?\n\n\"I'm looking for the Hidden Sage,\" Seth tried. \"If you are here, I need help.\"\n\n\"And why should I aid a shadow charmer who brought wraiths to my threshold?\" answered a creaky little voice.\n\nThe words came from in front of him, but Seth couldn't see the speaker. \"I brought the wraiths to avoid getting pounded to death by giant apes. Are you invisible?\"\n\n\"To you I am, apparently,\" the voice answered, amused.\n\n\"Camouflage?\" Seth asked. He took a few steps in the direction of the voice. It seemed to come from down low, so he studied the ground. \"I lost my memory. I need advice.\"\n\n\"The world is in upheaval,\" the voice said. \"I save my counsel for those who deserve it.\"\n\n\"Please help him,\" answered a tiny voice that made Seth jump. It came from the pocket of his pants.\n\n\"What is going on?\" Seth asked, looking down.\n\n\"Sorry, Seth,\" the voice in his pocket apologized. \"I can't stay quiet any longer. Mighty sage, Seth is lost right now, but he is a warrior for good with a noble character.\"\n\nSeth saw a tiny head and a pair of arms poking out of his front pocket. They belonged to a handsome young man no bigger than his pinky finger. How long had he been in there? Was he a spy? Why did he sound so cheerful?\n\nSeth reached for the little person in his pocket. The miniature young man crawled willingly onto his palm. \"Amazing,\" Seth said. \"You're a miniature person.\"\n\n\"His endorsement carries weight with me,\" the creaky voice said.\n\nSeth glanced around. He still hadn't spotted the sage.\n\n\"When I say he's lost, I mean metaphorically,\" the little person said. \"He came here on purpose.\"\n\n\"I understood,\" the creaky voice replied.\n\nSeth finally identified the creaky speaker. Instead of rising to a point, one of the stalagmites ended with a little seat, and on it sat a small brown figure who, to a casual glance, looked like the top of the stalagmite.\n\nSeth knelt in front of the tiny being. \"Are you the Hidden Sage?\"\n\n\"I am Gabrinko,\" the little figure said. The texture of his skin made Seth wonder if the shriveled little person was made of wood. Standing up, he would be about the height of Seth's pointer finger, only a little taller than the miniature person in his hand. \"You may also refer to me as sage, or wise one.\"\n\nSeth returned his attention to the tiny young man in his palm. \"And who are you?\" Seth asked him.\n\n\"I'm Calvin, the Tiny Hero,\" the young man said brightly. \"I'm your only sworn vassal.\"\n\n\"My vassal?\"\n\n\"Keeper of your secrets, protector of your honor, servant to your interests, and defender of your causes.\"\n\n\"His support speaks well for you,\" Gabrinko said.\n\nSeth looked from Gabrinko to the Tiny Hero on his palm. \"How long have you been with me?\"\n\n\"I left Kendra and crossed to you the last time you talked to her,\" Calvin said. \"When you were stuck to the mushroom.\"\n\n\"I'm not familiar with that expression,\" Gabrinko said.\n\n\"I was actually stuck to a mushroom,\" Seth explained.\n\n\"I've stayed quiet since,\" Calvin said. \"A stowaway. A secret passenger. I heard you talking to Ronodin and to the Underking through that freaky undead puppet of his. Sorry to spy. I was worried that if I revealed myself, you might send me away.\"\n\n\"Or use you as fish bait,\" Gabrinko said.\n\n\"What kind of creature are you?\" Seth asked.\n\n\"A nipsie,\" Calvin said. \"Smallest of the fairy folk, though my people increased my size with a spell.\"\n\n\"And what are you?\" Seth asked Gabrinko.\n\n\"A wood sprite,\" Gabrinko said. \"We very rarely interact with big folk. My kind consider me an outcast, but the feeling is mutual.\"\n\n\"You live alone in a cave,\" Seth said.\n\n\"Keeps the rain off me,\" Gabrinko said. \"And no neighbors bother me here. Just the occasional adventurer.\"\n\nSeth returned his attention to Calvin. \"You're friends with Kendra?\" Seth asked.\n\n\"She's your sister,\" Calvin said. \"I can't believe you don't see that yet.\"\n\n\"I know she might be,\" Seth said. \"A lot of people tell me conflicting things.\"\n\n\"Some of those people kidnapped you to an island of the undead,\" Calvin said. \"Use your brain.\"\n\n\"My brain is the problem,\" Seth said. \"That's why I came here. I have a gaping hole where my memories should be. I can talk to the undead creatures on that island. My ability to recruit the help of two wraiths is the reason I made it to this cave alive. I can douse torches and hide in shadows. How can I be sure I don't belong there?\"\n\n\"I see how that could get confusing,\" Calvin admitted.\n\nSeth turned to Gabrinko. \"The Underking told me you could help me.\"\n\n\"I don't think the Underking expected me to help you,\" Gabrinko said.\n\n\"Why not?\" Seth asked.\n\n\"I suspect he did not believe you would reach me,\" Gabrinko said. \"He also knows I don't approve of the undead, and that I don't waste good advice on those who will not heed it.\"\n\n\"I came here with an open mind,\" Seth said.\n\n\"Give me your finger,\" Gabrinko said.\n\n\"Why?\" Seth asked.\n\n\"So I can read you,\" Gabrinko said. \"It must be voluntary.\"\n\nSeth held out his hand, pointer finger extended. Gabrinko laid a brown hand on his fingertip. Closing his eyes, the diminutive sage took some deep breaths.\n\n\"Calvin has done a service for you today,\" Gabrinko said. \"Based on what I see in your mind, I would have chosen not to help you.\"\n\n\"Why not?\" Seth asked.\n\n\"Your history is blank,\" Gabrinko said. \"Instantly suspicious. What might you be hiding? The memories I can see do you no favors. You deal with darkness. You ignored the desecration of a sacred pool. You participated in the ruination of a dragon sanctuary. You consort with wraiths and evildoers. You intend to rob this island of a glorious treasure to appease your dark master.\"\n\n\"The Everbloom?\" Seth asked.\n\n\"It imbues these islands with life and magic,\" Gabrinko said. \"Calvin, may I take your hand?\"\n\n\"Sure,\" Calvin said.\n\nSeth held out his palm to Gabrinko. Calvin stepped forward, and Gabrinko took his hand.\n\n\"This provides a clearer picture of you, Seth Sorenson,\" Gabrinko said. He released Calvin's hand. \"What do you wish to know?\"\n\n\"Is Kendra really my sister?\" Seth asked.\n\n\"Yes,\" Gabrinko said. \"And you were a co-caretaker of Wyrmroost. Until recently, you always tried to stand against evil, though, like all mortals, you made mistakes.\"\n\nSeth stared at Gabrinko. Did that mean Kendra was telling the truth about everything? What Bracken had told him matched Kendra's claims, so did that mean what Bracken said was true as well? The implications were too overwhelming to sort through right now. He needed to learn all he could from Gabrinko. \"How do I know you're telling me the truth?\"\n\n\"You don't,\" Gabrinko replied. \"But I just read your history as Calvin witnessed it and understands it. I have no reason to lie.\"\n\n\"The Underking recommended you,\" Seth said.\n\n\"There are always reasons to doubt,\" Gabrinko said.\n\n\"How can I get my memories back?\" Seth asked. \"I want to know my history. I'm tired of trying to piece together who I was based on what other people tell me.\"\n\n\"This is a worthy desire,\" Gabrinko said.\n\n\"Can you help me?\" Seth asked.\n\n\"You voluntarily surrendered your identity,\" Gabrinko said. \"It is the only way so much of you could have been taken. I believe your motives were noble.\"\n\n\"Can I get it back?\" Seth asked.\n\n\"Any magic that can be done can also be undone,\" Gabrinko said. \"Anything with a beginning must have an end. Curses can be lifted, spells can be broken, bindings can be loosed. It may not be easy, but it can be accomplished.\"\n\n\"I would do anything,\" Seth said.\n\n\"You have an essence, Seth,\" Gabrinko said. \"Call it a spirit, a soul, whatever you wish\u2014you have an eternal component, a core self, that cannot be changed without your willing permission. You can be influenced, Seth. You can be encouraged, helped, fooled, misled. For good or ill, your essence evolves. But never against your will. To truly change your essence, you must be a fully willing participant.\"\n\nSeth felt sick. \"Did they take my essence?\"\n\n\"No, Seth,\" Gabrinko said. \"If you had lost your essence, you would be gone. There would be nothing left to worry or hope or ask questions. You are the victim of a sophisticated curse. No curse is perfectly executed, but this came close. It took your memories and all knowledge related to your identity, but it left your general knowledge. You remember facts, but you have lost your opinions about those facts. You remember procedures, but not what those procedures mean to you. You don't have to relearn how to eat with a knife and fork. But you have no idea what you most enjoy eating. You retain familiarity with prevailing opinions on various topics, without an awareness of where you personally stand. You can't specifically recall how you relate to anything or anyone.\"\n\n\"That sounds right,\" Seth said.\n\n\"I can't directly perceive your essence,\" Gabrinko said. \"I suspect it remains largely unaltered. But it lacks context. You are trying to rebuild your convictions under the burden of considerable misinformation.\"\n\n\"What should I do?\" Seth asked.\n\n\"How you resolve this dilemma is yours to decide.\"\n\n\"Do you know who cursed me?\" Seth asked.\n\n\"Yes,\" Gabrinko said. \"And I know how he took your memories. I can probably discover where he went.\"\n\n\"Who?\" Calvin shouted.\n\n\"His name is Humbuggle,\" Gabrinko said. \"He is an ancient and powerful demon. Very cunning. He used the Wizenstone, a talisman of nearly incomprehensible power, to steal your identity.\"\n\n\"He was the little dwarf I saw?\" Seth asked.\n\n\"Yes,\" Gabrinko said. \"Near where your memories begin. The stone Kendra sent away was the Wizenstone.\"\n\n\"You can figure out where he went?\" Seth asked.\n\n\"I need one of your hands again,\" Gabrinko said. \"Palm up.\" Seth offered the hand without Calvin. Gabrinko sprang onto his hand and paced along some of the lines. Then he crouched and tapped Seth's wrist.\n\n\"Forgotten or not, your memories share a connection to you,\" Gabrinko said. He spat onto Seth's hand, crouched down, and rubbed it in. Eyes closed, one hand on Seth's palm, he raised the other. After a long moment, Gabrinko hopped to his feet.\n\n\"Any luck?\" Seth asked.\n\n\"I should have known,\" Gabrinko said. \"Humbuggle has gone to the Titan Valley dragon sanctuary.\"\n\n\"Where is Titan Valley?\" Seth asked.\n\n\"New Zealand,\" Gabrinko said. \"Near Australia.\"\n\nSeth held up the wrist with the ghostly manacle. \"Can you see this?\"\n\n\"I know what you see,\" Gabrinko said.\n\n\"Can you break it?\" Seth asked.\n\n\"I'm sorry, but I am not able,\" Gabrinko said. \"And you will not be going anywhere until you are free from your obligation to the Underking.\"\n\n\"So I have to bring him the Everbloom,\" Seth said.\n\n\"It would harm these islands,\" Gabrinko said. \"There would be irreversible damage to the ecosystem. Some magical life forms here would go extinct.\"\n\n\"If I don't take it, somebody else will,\" Seth said.\n\n\"Perhaps,\" Gabrinko replied.\n\n\"And I'll stay a slave to the king of the undead,\" Seth continued.\n\n\"Perhaps,\" Gabrinko said.\n\n\"At least I know where to go once I'm free,\" Seth said.\n\n\"I have words for Calvin,\" Gabrinko said.\n\n\"In private?\" Seth asked.\n\n\"Not necessary,\" Gabrinko said. He looked at Calvin. \"You want to break a curse.\"\n\n\"What curse?\" Seth asked.\n\n\"You don't remember,\" Calvin said. \"My people, the nipsies, were cursed by Graulas.\"\n\n\"If they made you big, how small are they?\" Seth asked.\n\n\"You would have a tough time seeing facial features with your naked eyes.\"\n\n\"Was I helping you try to break the curse?\"\n\n\"You killed the demon Graulas,\" Calvin said. \"Our curse came with a prophecy. All nipsies know it: 'The curse arose from the demon's blight; the lord who slays him will set it right. The slayer shall restore our pride, the Giant Hero at his side.'\"\n\n\"I promised to help?\" Seth asked.\n\n\"And I vowed to serve you,\" Calvin said solemnly.\n\n\"And I have spoken with Serena,\" Gabrinko said.\n\nCalvin whirled to face him. \"You have? She came here?\"\n\n\"I sent her to Titan Valley,\" Gabrinko said.\n\n\"Wait,\" Seth said. \"Is that where you send everyone?\"\n\n\"Depends on what the spit tells me,\" Gabrino said with a wink. \"Humbuggle helped Graulas create the nipsie curse. Humbuggle lived for many years at Titan Valley. I told Serena she might find clues in his old castle. I did not know Humbuggle was ever at Wyrmroost. But now he has returned to his former home.\"\n\n\"So Serena might still be at Titan Valley?\" Calvin asked.\n\n\"At present Humbuggle is there,\" Gabrinko said. \"I know Serena meant to go there.\"\n\n\"How long ago?\" Calvin asked.\n\n\"Less than a year,\" Gabrinko said.\n\n\"Who is Serena?\" Seth asked.\n\n\"My true love,\" Calvin said. \"She was selected to go study the curse before I was chosen. Some of my people fear she has perished. But I'm determined to find her.\"\n\nGabrinko stretched and yawned. \"And our interview is at an end. Please take your wraiths far from here. And if you want to be wise, protect the Everbloom at all cost.\"\n\n\"Thank you for the information,\" Seth said. \"Is there anything else you can tell us?\"\n\n\"Many things, but my mouth grows dry,\" Gabrinko said. \"You have your portion for now. Follow it, doubt it, ignore it\u2014whatever you choose. If you want my advice, keep Calvin close, Seth, and distance yourself from the Underking as you are able. Farewell.\"\n\n\"Do I keep you in my hand?\" Seth asked Calvin.\n\n\"That would get annoying for both of us,\" Calvin said. \"Your pocket is perfect.\"\n\nSeth pocketed Calvin and walked out of the cave.\n\nKnox awoke with his cheek pressed against the sand. His mind felt foggy, and he had no idea why he had gone to sleep on the beach. His weary muscles complained when he shifted, and his joints felt a little sore. He had a mild headache. Opening his eyes, he stared at a complex network of black needles and razors.\n\nA jolt of recognition went through him.\n\nHe remembered it all now\u2014the giant demon octopus, the razor coral, the spine that grazed his shoulder. Drugged and drowsy, he had dragged himself into the little tunnel before collapsing. Looking around, he saw that he had made it around a corner. From his current position he could not see the cliff or the area where Kendra and the others had fallen.\n\nNot wanting to turn around with razor coral on all sides, he carefully backed up until he could see out of the little tunnel. His limbs ached as if he had somehow slept wrong on all of them. The sandy expanse in front of the cavern was empty. Kendra's sack of gales still rested where it had fallen, the unicorn horn on top of it.\n\nThe others were gone. He was alone.\n\nThe light was dimmer than earlier, suggesting that somewhere above the surface of the ocean, the sun might be heading down. The light from the crystal he carried was more evident than it had been before.\n\nWhere was Kendra? Had the demon killed her? What about the others?\n\nThey were probably alive. Why would Remulon tranquilize them if he meant to kill them? It had sounded like the demon intended to capture them. Remulon had a reputation as a collector. Knox figured he had escaped the same fate as the others by getting far enough into the coral before passing out.\n\nSo what was he supposed to do?\n\nHe crawled deeper into the coral, around the corner, then a little beyond where he had awakened, just to be safe. If he was quick and quiet, he could probably slip away, go for help. And bring back who? The old lady in the wheelchair? The elixir wouldn't work on the satyrs, and who knew if there were ingredients to make more? At best, he might return with Savani and Grady. Could they somehow save the day? The merfolk had made it clear that they would not help.\n\nKnox stared at the coral. Was it really as sharp as everyone claimed? He reached in his pocket and found a playing card. He pulled it out and saw it was the joker. He had shown the satyrs a card trick the night before. Having made the joker disappear, he had never removed it from his pocket. Apparently the elixir had included the card as part of his clothes, because it wasn't soggy.\n\nSlowly, carefully, holding just the corner of the card, Knox brushed the joker against a dense snarl of keen edges. He felt no contact with the coral, not the faintest tug, but the card fell apart, shredded to ribbons that were diced smaller as they percolated through more coral to the sand.\n\nHe would look worse than that if he tried to take on Remulon. He knew it. He wouldn't last a second. The only way he would survive the encounter would be if the octopus chose to capture him. Did he want to spend the rest of his life as a prisoner?\n\nKnox knew he needed to make a choice. Was he going to run away and be safe or try to save Kendra and the others? Maybe Remulon was asleep? Or out hunting? Knox fidgeted, rubbing his thumb against his fingers. What if he could sneak into the cavern and untie the others? Was it possible they could all make it out of this alive?\n\nHe knew his hope of saving them was not very realistic. If he ran away, at least he could report what had happened. Maybe Savani could send for help. Bring people to the island through the canoe. Expert demon hunters. Dudes with big muscles and magical weapons.\n\nWhat if he fled and Remulon tracked him down before he could escape? Then he would be a dead coward instead of a dead hero. What if Remulon was just waiting for him to exit the coral? Those tentacles might be too big to reach into the tunnel. Maybe the instant he crawled out he would get snatched up.\n\nOr maybe Remulon had forgotten about him. What if he could walk to the surface without any problem? He didn't want Tess to lose her brother. It wasn't fair for his parents to lose their son. He had school and sports and friends waiting back in Texas. He had a whole life to live.\n\nWhy had he even come on this expedition? This predicament was unfair. Even if there wasn't help to send back, he could act like he went for help. His intentions would look good. Wouldn't people say he had done the smart thing?\n\nHe could leave. Nobody would blame him.\n\nThere was only one major problem.\n\nThe thought of running away made him sick.\n\nCould he live with leaving Kendra and the others behind? Without even checking if he could save them?\n\nNobody had made him come along.\n\nThey had invited him to stay behind.\n\nSo why had he come?\n\nBecause Seth had been captured. And it had been partly his fault. Largely his fault. Seth would not have been captured if Knox hadn't left the dungeon keys with the goblins back at Fablehaven. His cousin would still be a caretaker, and Wyrmroost would not have fallen.\n\nTrying to save the island sanctuary was a noble purpose. But Knox was mainly here to try to save Seth. And if Kendra was trapped in that cavern, not only Seth was in trouble, but another cousin as well.\n\nKnox grabbed a fistful of sand and squeezed it. Why had he come on this undersea adventure? He had thought maybe he could do something to help.\n\nEveryone had been captured but him. When was he going to get a better chance to help? To do something that mattered? To make up for the trouble he had caused?\n\nWas he really going to run away the first time nobody was watching? The first time everything depended on him? He remembered himself bragging about his bravery at the victory feast at Stormguard Castle. What if they could see him now?\n\nIt helped nobody for him to throw his life away. But he could at least check to see if there was a chance to free the others. If he freed them, he wouldn't be alone anymore. He would have their help. They might all escape.\n\nHe didn't have a weapon. He had dropped his spear when the spine pricked him. When he had looked a moment ago, the weapon was gone.\n\nAll the weapons were gone.\n\nUnless you counted the unicorn horn and the sack of gales. He remembered Kendra explaining that unicorn horns couldn't be stolen. That had to be why Remulon had left it.\n\nKnox wondered how it would work if he tried to pick it up. What if he wasn't trying to steal it? What if his intent was to return it to Kendra? The horn was the weapon Kendra had grabbed before Remulon attacked. Did she know it would hurt the demon?\n\nWith many glances over his shoulder, Knox started crawling backward. If things stayed quiet, he would at least take a look inside the cavern. If the effort got him captured or killed, at least he would have done his best. But he needed to think positively. If he was quiet and smart, maybe he could save the others.\n\nAt the end of the coral tunnel, Knox paused, listening intently. If Remulon was waiting for him to exit, this could be the end of life as he knew it. He backed out and looked around, but he saw no sign of the octopus.\n\nKnox rose cautiously to his feet. Stepping quietly, he walked over to the unicorn horn, crouched down, and gently touched the side of it, half expecting to get shocked. After he rubbed the pearly surface of the unicorn horn without any problem, he picked it up, along with the sack of gales beneath it.\n\nThe horn felt good in his hand, almost like a knife. If Remulon came out of the cavern at this moment, at least Knox had something to stab him with.\n\nExcept what if Remulon just shot more tentacle darts at him? There might not be a chance to poke the octopus. Kendra had carried the horn too, and it had done her no good.\n\nWhat about the sack of gales? Would it even bother Remulon? It was hard to predict how the sack would work underwater. Knox knew if he opened it, he might get blown all over the ocean. If he opened it right now, the rush of air might thrust him straight back into the razor coral.\n\nKnox walked into the wide mouth of the cavern. The cave didn't go far straight back from the entrance, but branched off to the left farther than he could see. The crystal he wore seemed brighter in the shade of the cavern. He covered it with his hand.\n\nProgressing deeper into the cave felt forbidding. Was it smart to blunder onward? What if he ran into Remulon? He might find his friends. But would Remulon have left them unguarded?\n\nInstead of turning left and going deeper into the cave, Knox tiptoed to the back wall of the entrance. There were enough large stones that he could easily hide. From where he stood, he could see out the mouth of the cavern to the razor coral and the electric kelp.\n\nIt might be best to meet Remulon on the most favorable ground he could find. With his back against the wall of the cave, maybe he could avoid getting blown around by the sack of gales. It would be risky, but so was any option at this point. If he found Kendra and the others and managed to free them, what were the chances of their escaping without the giant octopus hunting them down?\n\nMaybe his best option would be to surprise Remulon. Hold the showdown where he would have the best chance. He wanted to control how the encounter happened. Knox decided to yell something, then paused. He needed to call from outside the cavern or the acoustics might give him away.\n\nKnox ran out of the mouth of the cavern onto the sand. \"Kendra?\" he called loudly. \"Can you hear me? Where are you?\"\n\nAs he raced back to the rear wall of the cavern entrance, he heard a faint reply. \"Knox! Get out of here! Run!\" Her voice was faint, but he could tell she was yelling. She probably thought he was so stupid! Maybe she was right. . . .\n\nKnox reached the back wall and huddled down among the rocks, clutching his crystal tightly to conceal the light. Knox foresaw three possibilities: Remulon would either return from outside the cave, emerge from inside the cave, or not respond.\n\nKnox didn't have to wait long to find out. The enormous black octopus came gliding out from deeper inside the cave, tentacles winding and rippling. Remulon passed where Knox was hiding and drifted out the entrance.\n\nWhere are you, frail human? Remulon called out mentally. Deeper in the coral, perhaps. Imagining yourself secure? I see you took the horn. Do you think it frightens me? I am not diseased. It would be a minor irritant at worst.\n\nKnox tried to keep his cool. Was he really going to do this? He didn't have much choice left. By calling out, he had committed.\n\nThe giant black octopus glided closer to the coral. Why so silent? Will you cower in there until your elixir wears off? I have a safe air pocket where you can survive. It is your only hope. Do not vex me by remaining hidden.\n\nKnox stood up, braced himself against the back wall, and opened the sack of gales. Some little sound he made caused Remulon to look back at him at the last second, red eyes intense. A storm of bubbles gushed from the mouth of the sack, astonishing in volume and velocity. The geyser of air blasted out of the cave mouth and into Remulon.\n\nA single tentacle lashed out with blinding speed, clinging to a projection of stone at the mouth of the cave. The other tentacles flapped and flailed in the violent gust of bubbles, a couple getting sheared off as they hit the coral. The one tentacle anchoring Remulon looked tautly stretched to its limit.\n\nStop this at once! Remulon demanded, panic in his mentally projected words.\n\nKnox remembered that the wind inside the sack of gales was a limited resource. If it ran out with Remulon still alive, Knox knew he was doomed.\n\nStriding forward, Knox found he could walk while holding the sack. It fluttered in his grip, but it didn't blow him all over the place as he had worried! Keeping the airstream focused on the thrashing octopus, Knox approached the strained tentacle. With the sack of gales secured between one arm and his side, and one hand aiming the mouth of the sack, he raised the unicorn horn in his other hand.\n\nI will destroy all of you for this! Remulon threatened. I have never known defeat!\n\nKnox spoke through gritted teeth. \"You've never gone up against a Texan!\"\n\nHe brought the horn down, stabbing the tentacle as hard as he could. The meat of the tentacle sizzled and it released from the protuberance, wrenching the horn from Knox's hand. Driven by the underwater wind tunnel, Remulon flew helplessly into the coral.\n\nKnox closed the sack in time to see shreds and scraps of the octopus fanning out from where Remulon had blown through the barrier of razors. Above the mess, the torrent of bubbles rose out of sight. Only the top of the demon's head remained intact, but it was being dazzlingly electrocuted by the kelp. After several blinding flashes, the scorched remains of the head drifted down into the coral, where it was minced into uncountable particles.\n\nKnox sat down hard, tears streaming from his eyes as he laughed hysterically. It had actually worked! It had totally, perfectly worked! He wasn't going to die. The mighty aquatic demon had been grated into confetti!\n\nTears of shock and relief continued to fall. Knox felt the tears vanishing from his cheeks almost as soon as they were shed. His involuntary laughter finally started to subside.\n\nHe searched around on the floor of the cavern until he found the unicorn horn. He wondered if it might have octopus blood on it, but the horn looked pristine. Picking it up, he ventured deeper into the cave.\n\n\"Kendra?\" he shouted. \"Where are you?\"\n\n\"Knox,\" her reply came, faint because of the distance. \"Be quiet! Are you crazy?\"\n\n\"Remulon is gone!\" Knox called. \"Lead me to you!\"\n\n\"How is he gone?\"\n\n\"He's sushi! He's dead. Where are you?\"\n\nIt took some time calling back and forth, but Knox eventually heard her voice getting nearer. He was grateful for the light he wore as the cave got darker. After some time, he reached a large, fully lit chamber containing all sorts of strange items. The light came from bioluminescent fish and plants, no two of them alike. He paused to take in the bizarre variety of life and to admire the huge anchor, the suit of armor, the piles of gold and jewels, the statues, and the exotic shells.\n\n\"Knox?\" Kendra called, her voice near.\n\n\"I'm almost to you,\" Knox replied.\n\nHe followed her voice to a ramp on the far side of the room that led up to a chamber full of air. He could see where the water stopped, held back by the air pocket, but he felt no difference as he passed into the air. The room was very bright because of a brilliant white fairy seated on a perch. She had wings like a bat's, but scaly, and her slender tail ended in a silky tuft. Her body was in the form of a human but covered in white, reptilian scales.\n\n\"Knox,\" Kendra said with relief.\n\nShe was lying on the ground beside Warren, Vanessa, Tanu, and Hako. Their bodies were completely cocooned in gooey gray strands, with only their heads showing. All the prisoners except Kendra were unconscious.\n\n\"What did Remulon do to you?\" Knox asked.\n\n\"I'm not sure,\" Kendra said. \"I woke up like this. Is he really dead?\"\n\nKnox smiled. \"He finally met his match.\"\n\n\"You?\"\n\n\"Well, yeah. I'm from Texas.\"\n\nKendra rolled her eyes. \"I'm sure that was it.\"\n\n\"I attacked with your wind bag,\" Knox said. \"I don't think Remulon ever dealt with wind before.\"\n\n\"The sack of gales?\" Kendra asked. \"Really?\"\n\n\"I ambushed him,\" Knox said, unable to resist a big grin. \"And blew him into the razor coral.\"\n\nKendra's eyes widened. \"Knox, you're a genius!\"\n\n\"I'm just happy it worked!\"\n\n\"Knox, that was so clutch,\" Kendra said. \"I can't believe he's really gone.\"\n\n\"He's not completely gone,\" Knox said. \"He's in a million pieces. Part of him even got electrocuted.\"\n\n\"I was so worried about you,\" Kendra said.\n\n\"I dragged myself into the coral tunnel before I passed out,\" Knox said. \"He left me there.\"\n\n\"I should have been worried about Remulon,\" Kendra said. \"He had no idea who he was up against. I'm glad you got the horn.\"\n\n\"It came in handy,\" Knox said. \"While I was trying to blow him into the coral, his fast tentacle grabbed a sturdy rock. A stab from the horn made him let go.\"\n\n\"That's incredible,\" Kendra said. \"See if the horn works on this goo. I can hardly budge.\"\n\nKnox knelt beside Kendra and touched the point of the unicorn horn to the cocoon around her. The gray strands melted away wherever the horn made contact.\n\n\"It works,\" Knox said.\n\n\"Get me out,\" Kendra said.\n\nIt took only a minute before her gray bindings were melted away enough for Kendra to squirm free. Knox did some touching up, dissolving most of the gray matter that still clung to her.\n\n\"What about the others?\" Knox asked.\n\n\"I think they got hit with more octopus darts,\" Kendra said. \"Only two poked me.\"\n\n\"I only got scratched by one,\" Knox said. \"It was enough.\"\n\n\"My head hurts,\" Kendra said. \"But I'm so happy!\"\n\n\"Is that the fairy we want?\" Knox asked, motioning to the brilliant white creature.\n\n\"Yes,\" Kendra said. \"I've had a few words with the nova song. She speaks no English. She has been here for a long time. A little chain holds her to the perch.\"\n\n\"Will she fly away if we free her?\"\n\n\"I don't think so,\" Kendra said. \"I'll order her to stay with us. She technically has to obey me. But maybe we'll keep the chain attached just to be sure.\"\n\nWarren let out a groan. \"What is so bright?\" he complained groggily. \"Why can't I move?\"\n\n\"Remember the octopus?\" Kendra asked.\n\n\"Crud,\" Warren said. \"I do now. Are we in his belly?\"\n\n\"You're tied up in his lair,\" Kendra said. \"And you'll never believe what happened to Remulon.\"\n\nCraning his neck, Warren looked over at Kendra. \"What?\"\n\n\"Knox destroyed him,\" Kendra said.\n\n\"Wait, really?\" Warren asked. \"No offense, but did you say Knox destroyed him?\"\n\n\"None taken,\" Knox said. \"I showed him a little Texas hospitality.\"\n\n\"Meaning what?\" Warren asked.\n\n\"Sack of gales into the razor coral,\" Kendra said.\n\nWarren whistled.\n\n\"And we found the nova song,\" Knox said.\n\n\"I really wasn't looking forward to waking up,\" Warren said. \"I mean, even as I was fading out, I was kind of hoping to just stay asleep. This feels like a miracle.\"\n\n\"What now?\" Knox asked.\n\n\"You kids cut me loose,\" Warren said. \"Then we wake the others and get out of here.\"\n\n\"There is gold out in his garden,\" Knox said.\n\n\"Add grab some gold to the list,\" Warren said. \"Knox, we owe you, buddy. How did you keep it together and pull it off?\"\n\n\"Honestly,\" Knox said, looking from Kendra to Warren, \"I've never been more scared. I almost ran for it. But I couldn't leave without you guys.\" Tears sprang into his eyes again. \"I thought we were so dead. This is the best day ever. But now we need to rescue Seth.\"\n\nOn the floor of his private room within the Under Realm, Seth sat cross-legged with his satchel open, the contents spread in front of him. Having returned to the Phantom Isle just after nightfall, after dropping Lampy and Salty at their lighthouse and boating back from Omari, he had stolen quietly from the subterranean dock to his room, hoping to avoid interacting with Ronodin or the Sphinx.\n\n\"I was pretty smart,\" Seth said, approving of the contents laid out on the floor. He had first-aid equipment, string, duct tape, a compass, a matchbox, a magnifying glass, binoculars, a pocketknife, a dagger, a little mirror, a whistle, some beef jerky, hand sanitizer, vitamins, a flashlight, a small notepad, three pencils, a pen, some cash, a tightly folded rain slicker, a pouch of salt, several paper clips, a needle and thread, a pack of chewing gum, fingernail clippers, a bouncy ball, and a harmonica. An unlabeled bottle possibly held a potion.\n\nHe also had the single glove that could supposedly turn him invisible if he held still, as well as the leviathan carved from blue stone. He found a slip of paper with words to accompany the leviathan and also words for a tower. He didn't see a tower among the objects.\n\n\"I should have gone through this stuff earlier,\" Seth said. \"I'm definitely bringing the satchel next time.\"\n\nCalvin said nothing. He had suggested it would be best for him to remain silent while in the Under Realm, just in case the undead were listening.\n\nSeth picked up the harmonica and gave a little blow, generating a wheezy, discordant sound. He figured he had once known how to play it, and he wondered if he was any good. On a whim, he took a vitamin, then started packing the items back into the satchel.\n\nThe door opened without warning, and Ronodin entered. \"You're back late,\" he said. \"How did your errand go?\"\n\n\"Pretty well,\" Seth said.\n\nRonodin stared at him. \"What did you learn?\"\n\n\"What do you mean?\" Seth deflected.\n\n\"You choose an interesting juncture to start hiding information from me,\" Ronodin said.\n\n\"Are you saying you've been open with me?\" Seth asked.\n\n\"I've protected you from information that could confuse you,\" Ronodin said.\n\nSeth stared at him. He knew Ronodin was an enemy. But bringing that into the open right now would not be very strategic. \"I guess that is one way to look at it.\"\n\n\"You're different,\" Ronodin said.\n\n\"What do you mean?\"\n\n\"You look at me differently,\" Ronodin said. \"Who did you speak with?\"\n\n\"Don't I deserve a little private time? You take plenty.\"\n\n\"The Hidden Sage works with your enemies,\" Ronodin said.\n\n\"Who are you talking about?\"\n\n\"Your lies reveal more than the truth would,\" Ronodin said. \"You believe him. You've decided I'm your enemy and you must hide things from me.\"\n\n\"I'm confused,\" Seth said.\n\n\"At least you have that right,\" Ronodin said with an exasperated sigh. \"Your determination to dig into your past is poorly timed.\"\n\n\"Why?\"\n\n\"All is in place to retrieve the Everbloom,\" Ronodin said. \"The Sphinx has the location. The phantom will lead you to it.\"\n\n\"I'll be with Dezia?\" Seth asked.\n\n\"It's the most efficient way,\" Ronodin said. \"She'll be compliant and docile. The Sphinx made sure of that. We'll go fetch the Everbloom tomorrow. Unless you no longer have the stomach for it.\"\n\n\"What other choice do I have?\" Seth asked.\n\n\"I'd be fascinated to know,\" Ronodin said. \"The Everbloom is your ticket out of here. Hand that over to the Underking and you can go do whatever you want.\"\n\n\"The Underking would let me walk away,\" Seth said. \"Would you?\"\n\n\"Do you think the Underking is your friend?\" Ronodin asked. \"More than I am? Do you know what the Underking told me today?\"\n\nSeth wasn't sure he wanted to know. \"What?\"\n\n\"The Underking warned me to watch you,\" Ronodin said. \"He thinks you might try to keep the Everbloom for yourself.\"\n\n\"Really?\" Seth asked. \"Why would I do that?\"\n\n\"You tell me,\" Ronodin said. \"It seems preposterous. The price of your freedom has been set. You can't escape with the Everbloom unless the Underking frees you.\"\n\nSeth nodded.\n\n\"Unless you have some notion of being a hero,\" Ronodin said. \"Handing the Everbloom over to somebody else.\"\n\n\"Who?\" Seth asked.\n\n\"I don't know,\" Ronodin said. \"But the Underking is no fool. He has suspicions for a reason. It's just a flower, Seth. It's beautiful and mystical, but just a flower. If the lord of the undead wants a flower in order to let you go, bring him the flower and move on.\"\n\n\"That makes sense,\" Seth said.\n\n\"You haven't had a chance to live your life,\" Ronodin said. \"You could do a lot of good. But not bound here in the Under Realm. It isn't your fault that you're here. Pay the price and get free.\"\n\n\"I'm here because of you,\" Seth said. \"You brought me here.\"\n\n\"I brought you here so you could learn, Seth,\" Ronodin said. \"If not for me, your powers would have gone to waste. You think Kendra or any of those others would want you to regain dark abilities? They would be more than happy to leave you powerless, essentially lobotomized. I brought you where you could restore your potential.\"\n\n\"You brought me into slavery,\" Seth said.\n\n\"Not for long,\" Ronodin replied. \"Not if you play this smart. You could be free by this time tomorrow.\"\n\n\"And you'll let me go?\" Seth asked.\n\n\"Why would I want to keep you?\" Ronodin asked. \"I'd love to work with you in the future. But that is up to you. Can I get real with you?\"\n\n\"Sure.\"\n\n\"The Underking isn't on your side,\" Ronodin said. \"When he offered you his crown? That wasn't because he liked you.\"\n\n\"Why did he do it?\"\n\nRonodin looked at Seth seriously. \"To consume you. To utterly own you. Seth, you would have joined the undead not as their king, but as a puppet.\"\n\n\"I knew there had to be a catch,\" Seth said. \"I wasn't regretting my choice.\"\n\n\"So much depends upon tomorrow,\" Ronodin said.\n\n\"I know,\" Seth replied. \"We have the ointment to resist the heat. We know where to go. I'll do the job.\"\n\n\"You are saying words,\" Ronodin said. \"Telling me what you think I want to hear. I see the truth behind your assurances. You don't know what you're going to do.\"\n\n\"It makes sense to deliver the Everbloom and get free,\" Seth said.\n\n\"You're right about that. It would be the best choice. You didn't have to visit the Hidden Sage. You could have just come to me.\"\n\n\"What?\"\n\n\"You didn't have to go to some sage to learn where to get your memories,\" Ronodin said.\n\n\"You never answer my questions,\" Seth said.\n\n\"Your memories were taken by a demon that looks like a dwarf,\" Ronodin said. \"Humbuggle. You met him at the castle before he sent you away.\"\n\n\"Good to know,\" Seth said.\n\n\"He used the Wizenstone,\" Ronodin said. \"You would have to defeat Humbuggle to get your memories back.\"\n\n\"How can I do that?\" Seth asked.\n\n\"Nobody knows,\" Ronodin said. \"Nobody has done it.\"\n\n\"Where is he?\" Seth asked.\n\n\"Probably at a dragon sanctuary called Titan Valley.\"\n\nSeth considered Ronodin. Did he know what the Hidden Sage had told him? Was he sharing this information because he knew it no longer mattered? How did he know so much?\n\n\"I've been in this game for a long time,\" Ronodin said. \"I'm rarely a step behind. Don't try to outmaneuver me.\"\n\n\"Why are you doing all of this?\" Seth asked. \"What are you after?\"\n\n\"The world is changing,\" Ronodin said. \"The dragons are winning. There are going to be fewer prisons in the world. Magical sanctuaries are relics of the past. I celebrate this new era.\"\n\n\"What do you really want?\" Seth asked. \"It isn't just freedom for everybody.\"\n\n\"I want so many things,\" Ronodin said with relish. \"But tomorrow I will settle for delivering the Everbloom to the Underking. Sleep well. Be smart.\" Ronodin walked to the door. \"Goodnight, Seth.\"\n\n\"See you tomorrow.\"\n\nRonodin closed the door.\n\nSeth left everything where it was and climbed into bed. He didn't want to think. He deserved some rest. He wanted to sleep. And he knew it might take a while for him to drop off.\n\nRonodin had been completely right about one thing: Seth had no idea what he was going to do tomorrow.\n\nKendra snapped awake in the cool of the night, her hammock rocking as she sat up. White radiance shone through her window from the nova song chained to a perch outside the tree house.\n\nThe day before, Kendra, Knox, Warren, Vanessa, and Tanu had followed Hako on a long hike from the lair of Remulon out of the ocean, arriving at Crescent Lagoon just after dusk. They had taken their antidotes while standing in the shallows, and water became tangible to them again. Kendra had initially felt thirsty, noticing her hunger only after she had gotten enough to drink. They had eaten a big meal together before going to bed, too tired to plan for the next day.\n\nKendra had been full of worries as she climbed into her hammock. Remulon had warned that going to the Phantom Isle meant certain death. Lord Quintus had also stressed that people ran away from the Phantom Isle rather than toward it. Yet it was their next step. Seth and Bracken were there. There was no hiding from it.\n\nBut she felt different now than she had been feeling when she went to bed. A few hours of rest had simplified the problem in her head. Suddenly the way forward seemed clear.\n\nFeeling calm alertness, Kendra slid out of the hammock and got dressed. She quietly exited her tree house, then crossed a long rope bridge and a short stable one leading to the primitive elevator. There was no hint of dawn yet. After pulling a lever, she descended through the leafy canopy amid rustlings and dripping water until the elevator reached the jungle floor.\n\nKendra stepped off the little platform and walked to the lagoon. She gasped when she came through the trees to find the three eyeless cyclopses standing together on the sand, as if expecting her.\n\n\"You seek us,\" Tal said.\n\n\"You have awakened with a purpose,\" Hobar said.\n\n\"You wish to find the Fairy Queen's shrine,\" Baroi said.\n\n\"You guys are good,\" Kendra said.\n\n\"You sounded a clarion call,\" Tal said.\n\n\"We cannot escort you or we would revert to our more fearsome form,\" Hobar said.\n\n\"Instead we summoned a fairy to guide you,\" Baroi said.\n\n\"You can summon fairies?\" Kendra asked.\n\n\"Come out, Andressa,\" Tal said.\n\nA fairy bobbed out of hiding, tiger-striped butterfly wings sparkling. She fluttered near, allowing Kendra to appreciate her petite figure and playful elfin features.\n\n\"Most fairies do not perceive our summonings,\" Hobar said. \"Or else they choose not to heed them.\"\n\n\"Andressa hears well,\" Baroi said. \"We believe she is the most reliable fairy on the island.\"\n\n\"Pleased to meet you,\" Kendra said. \"I'm Kendra.\"\n\n\"The pleasure is mine,\" Andressa said. \"Are you certain you wish to visit the shrine? You shine brightly, but mortals who trespass there tend to die.\"\n\n\"I know it's never safe to visit a shrine of the Fairy Queen,\" Kendra said. \"But I have great need\u2014and a relationship with her majesty.\"\n\n\"I can show you the way and guide your return,\" Andressa said. \"You don't want to get lost with so many dangerous creatures roaming the night.\"\n\n\"Go,\" Baroi said. \"If you wish to return before dawn, you must hurry.\"\n\n\"You know what I intend?\" Kendra asked.\n\n\"We cannot see your mind,\" Hobar said.\n\n\"But in the quiet of the night, we could see you coming to us,\" Tal said. \"And we could see you on the trail to the shrine.\"\n\n\"Will I survive?\" Kendra asked.\n\n\"That depends on you,\" Baroi said.\n\n\"I'll be back,\" Kendra said. She had to try. Too many people were counting on her for her to back down now.\n\nThe fairy conducted Kendra to a path, lighting the way with her glow. Kendra didn't really need the illumination\u2014ever since she had become fairykind, no natural darkness was complete to her. Even the deepest shadows under the trees looked dim, not dark. Still, with branches overhead screening the moon, and her line of sight limited by dense vegetation, Kendra wondered if it was wise to strike off across the island without any of her companions. It seemed like something her brother would do.\n\nBut she had to go to the fairy shrine\u2014because to save Seth and Bracken, to restore the sanctuary, she and her companions had to go to the Under Realm. And if they went there without help, they were all going to die. The Fairy Queen was the most powerful being Kendra knew. Bracken was her son, and Kendra hoped a mission to rescue him would validate a claim to her assistance.\n\nShe would have been more comfortable if Warren or Vanessa were accompanying her. But none of her companions could risk approaching the Fairy Queen. Most mortals who dared invade the sacred terrain of the queen's shrine would be instantly struck down. Also, instinctively, Kendra knew she would better merit aid if she went alone.\n\n\"Keep up,\" Andressa said. \"An ocelot is stalking this way, but we won't cross paths if we hurry.\"\n\nKendra increased her pace to a jog, her eyes searching the dappled moonlight to either side of the trail. The prospect of unseen creatures prowling the undergrowth kept her anxiously alert.\n\n\"You can slow down now,\" Andressa said. \"You're much quieter at a walk.\"\n\nKendra settled into a comfortable walk. She decided that if she was going to invade the realm of the undead, she should try to enjoy the relative peace of strolling through a jungle teeming with dangerous magical creatures.\n\nAt least she knew where to find Bracken and her brother. That information had seemed unattainable a few days ago. She was closing in on her goal. There were still problems, though. Even if she managed to survive the Under Realm and find her brother, there was no guarantee Seth would come with her. That possibility made her ache inside.\n\nShe couldn't just force him to leave. That would do more harm than good. It would make her just like Ronodin. What she really hoped was that he would want to come with her.\n\nAs to surviving the Under Realm, Kendra needed the Fairy Queen to provide help. If not, she wasn't sure what to do.\n\nRemulon had been right when he scoffed that they had arrived at his lair with no plan. If it hadn't been for some luck and some shocking heroics from Knox, they would still be his prisoners. The Under Realm was sure to be much less forgiving if they just showed up expecting to prevail. This was no time to fall back on improvisation.\n\nKendra followed the fairy through a dense jungle, across brushy fields, along the outskirts of a bamboo forest, and up into some hills. Andressa flitted close to Kendra.\n\n\"We're almost there,\" the tiger-striped fairy said. \"I can't accompany you to the shrine or offer specific warnings. But be careful. Remember your true objective.\"\n\n\"Is it guarded by naiads?\" Kendra asked, thinking of the Fablehaven shrine.\n\n\"Not naiads,\" Andressa said. \"But the shrine is protected. If you are unworthy to be there, you will not survive.\"\n\nKendra had been to the Fairy Realm. But she had been escorted by Bracken, who was a rock star among the fairies. She felt sure her fairykind status rendered her worthy to visit the shrine, but she worried that its guardians might not agree.\n\n\"I'll await you here,\" Andressa said. \"This path ends at the shrine.\"\n\nKendra could hear water splashing. \"A waterfall?\"\n\n\"Follow that sound,\" Andressa said. \"I can say no more.\"\n\n\"Could I command you to tell me more?\" Kendra asked.\n\n\"Your authority to issue commands comes from the Fairy Queen,\" Andressa said. \"It was she who forbade us from revealing the secrets of this shrine.\"\n\n\"Fine,\" Kendra said. \"Thanks for leading me here. I'll see you soon.\"\n\nAlone, overshadowed by tall trees dripping with vines, Kendra proceeded along the path. She passed a few low, eroded ruins partially hidden by leafy creepers.\n\nThe steady churning of falling water grew in volume until Kendra entered a clearing open to the moon and stars. Ahead, water trickled and tumbled over the edge in several places along a low, curved wall of mossy rock, the largest waterfall at the center.\n\nThe incoming water spread over the clearing in a shallow pool. Carved tiki statues at least as tall as humans were spaced throughout the pond, heads disproportionately large, most submerged up to the shins. Little grassy islands poked above the water at random. Kendra could not see a stream flowing away from the pool, leaving her to wonder how water flowed in but not out. Perhaps it was escaping underground? Was the pool deeper in some places than it looked?\n\nIn Kendra's experience, though the area around a fairy shrine could be sizable, the shrine itself would be tiny. It was hard to guess where among the flooded statues she would locate the actual shrine.\n\nDespite the backdrop of falling water, the moonlit pool felt still and sacred. Kendra stepped hesitantly forward, wading into the ankle-deep water with her shoes on. Immediately she felt like a trespasser. Looking up, she found the nearest tiki statue gazing right at her. Had the head been facing her way before?\n\nAs she took another few steps forward, Kendra again felt unwelcome. Were more of the statues looking her way? It had to be a coincidence.\n\nKendra held still. Should she come back another time? Something felt off.\n\nNo! She had no other time. This had to happen tonight.\n\nKendra sloshed onward, searching the surroundings for the fairy shrine. There was no escaping the sense that she was unwanted and in danger. She felt the message as if it were being silently transmitted from every direction. She stopped again and found that all of the heads of the tiki statues were facing her. One had a blowgun raised to stone lips. Another looked ready to throw a hatchet.\n\nSomething supernatural was happening. The positions were definitely new. Were the statues about to attack?\n\nThe only sound was the splash of falling water.\n\n\"I belong here,\" Kendra called out. \"I'm fairykind, and I need help from the Fairy Queen. It's an emergency.\"\n\nThe impression that she was trespassing evaporated. Had the expressions of the tiki statues softened? She did not feel welcome, but she no longer felt rejected.\n\nKendra waded toward the main waterfall, the water never deeper than her shins. The tiki statues no longer seemed to follow her movements, and no weapons were aimed at her. As she neared the waterfall, Kendra noticed a golden necklace set with garnets arrayed on a stump. Her first impulse was to pick it up, but she realized it might be an offering to the Fairy Queen and decided to leave it alone. She also ignored an ivory scepter with a moonstone handle, which sat atop a little mound, and a silver javelin leaning against one of the tiki statues.\n\n\"I need to find the shrine,\" Kendra said, turning in a slow circle. \"Can anyone help me?\"\n\nAs she finished her rotation, she found one of the tiki statues pointing at the waterfall. She was almost positive the statue had not been positioned like that earlier.\n\nKendra approached the middle of the semicircular wall at the rear of the pool, where the water fell smooth and clear as glass in a curtain a little wider than her outstretched arms. She saw no fairy statue at the base of the waterfall, so she felt around underwater. After encountering nothing but slimy rocks, she looked up. Could the shrine be behind the falling water? It would be a shame to get wet for nothing.\n\nKendra scrutinized the area around the waterfall. She noted the occasional incongruous treasure, apparently unguarded, but still no fairy shrine. Three tiki statues now pointed at the waterfall.\n\n\"Thank you, I think,\" Kendra said. At least the night was warm; if she was going to get soaked, she probably wouldn't feel too chilled.\n\nShe walked up to the glassy cascade and tried to peer through it. She could see an alcove beyond. There was no space to go around. The waterfall shielded the alcove perfectly.\n\nReaching out a hand, she pushed a finger through the film of water and watched the curtain part a little beneath the intrusion. She liked how the water felt against her finger. It wasn't too cold.\n\nHolding her breath, Kendra stepped through the waterfall, getting thoroughly drenched in the process. The churning water sounded different inside the small, humid alcove. The floor was submerged, the water to her ankles.\n\nDelicately sculpted from white stone, a tiny statue of a fairy stood atop a small pedestal in a niche at the back of the alcove. A little bronze bowl rested near the statue. Kendra knelt in the water before the fairy figurine and pushed her rubbery wet hair away from her eyes.\n\n\"Hello, Fairy Queen,\" Kendra said. \"This is Kendra Sorenson. Thanks for letting me come this far. I'm here because I have an emergency. You've saved me before, and I've never needed your help more. Your son Bracken is imprisoned in the Under Realm. My brother, Seth, is stuck there too. And I need to retrieve the Sunset Pearl from there to restore this dragon sanctuary. I found a nova song who will lead me to the Phantom Isle. I can get there. But the island is guarded by a sea dragon, and I need a way to survive the Under Realm.\"\n\nThe air stirred, and Kendra smelled freshly tilled earth, ripe bananas, frosty pine needles, sea spray, cave minerals, mint leaves, and dozens of other aromas, some more familiar than others, simultaneously distinct and jumbled. The room brightened as the fairy statue glowed.\n\nKendra, I fear for my son, came the familiar voice of the Fairy Queen in her mind. Feelings of grief and worry washed over her.\n\n\"So do I,\" Kendra said, tears in her eyes.\n\nAlas, I cannot send anyone from my realm to help you, the Fairy Queen communicated. If my astrids, fairies, or any other minions were to enter the Under Realm on my orders, it would constitute an act of war. We simply cannot afford a war with the Underking.\n\n\"I'm willing to go,\" Kendra said. \"Can you help me succeed?\"\n\nThere is one way, the Fairy Queen expressed. It involves great risk for both of us, and it would not protect any of your companions. Success will be hard-won. It will depend on your determination and courage.\n\n\"I won't let you down,\" Kendra promised. \"I won't fail Seth or Bracken, either. I'll do whatever it takes.\"\n\nKendra felt the warmth of approval. Turn around and reach under the water.\n\nStill on her knees, Kendra turned and reached down, discovering that the water was inexplicably deep. She leaned forward until her face was just above the surface, both arms extended straight down without reaching the bottom. A hand brushed hers, passing her something metallic and circular.\n\nKendra lifted a beautiful circlet out of the water, wrought of a metal like radiant silver and inlaid with pristine gems, including a large white jewel in the front. The details were exquisite, the circlet formed by multiple strands in a perfect simulation of woven vines. Kendra knew instinctively that she had never held an object nearly as valuable.\n\nMy crown, the Fairy Queen explained.\n\nIn the emotion that accompanied the words, Kendra could feel the import of the gift. She got the feeling it was unprecedented. \"You're giving me your crown?\" Kendra asked.\n\nReturn it to me after you triumph, the Fairy Queen directed. The crown will offer protection when you enter the darkness. You shine so brightly, Kendra, yet many lack eyes to see your light. As a mortal with a wellspring of energy inside, you can light this crown as no other. Use it when you enter the Under Realm.\n\n\"Will it fit?\" Kendra asked.\n\nThe crown always fits perfectly, the Fairy Queen conveyed. The protection it provides depends upon you maintaining your courage. If the light within you dims, so will the crown.\n\n\"The crown won't protect others?\"\n\nAny who join you will face much greater jeopardy, with little chance to alter the outcome.\n\nKendra considered the implications. Could she really walk into the Under Realm alone? She supposed it would be better than endangering anyone else, especially if having backup wouldn't change the outcome.\n\n\"I'll return with Bracken,\" Kendra said.\n\nDo not lose my crown, the Fairy Queen admonished. You would leave my kingdom vulnerable, and I would not long remain queen. Take nothing that belongs to the Underking, only that which he has stolen from the world above. You must prevail.\n\nThe air stirred with uncountable aromas, and Kendra knew her connection to the Fairy Queen was severed. She stood up, holding the crown in one hand. She tested the depth where she had reached for the crown, only to discover the water was shallow again. After a final glance back at the inanimate figurine, she stepped out through the waterfall.\n\nInexplicably, passing back through the curtain of water dried her, except for her feet, which remained submerged. All of the tiki statues had their heads inclined toward her as if bowing. The moon and stars shone overhead. Kendra strode away from the pool determined to embark for the Phantom Isle before sunrise.\n\nKendra heard someone chopping wood as she approached the beach. Dawn was just beginning to color the sky, and birds had started to sing in the treetops. When Andressa led her into view of the lagoon, Kendra found six menehune hollowing out a tree trunk with axes and adzes. Two others were shaping outriggers.\n\nSavani stood on the beach to greet her, accompanied by the Himalayan cyclopses. Kendra changed the way she was holding the crown so her body shielded it from view. Was it smart to carry something so precious in the open like this?\n\nSavani approached Kendra and gave her a welcoming hug. Lips near her ear, the caretaker asked, \"Is that what I suspect it is?\"\n\n\"I can't believe she entrusted it to me,\" Kendra whispered back.\n\n\"Who are you?\" Savani asked, clearly impressed. \"Would you like a bag to conceal it?\"\n\n\"Yes, please,\" Kendra said.\n\n\"Are you exhausted?\" Savani asked.\n\n\"No,\" Kendra said. \"I woke up energized. It was dark but I wasn't sleepy.\"\n\n\"One moment,\" Savani said. She went and gave orders to a menehune, who scampered off. Then she waved for Kendra to join her by the cyclopses.\n\n\"Greetings, Kendra,\" Tal said.\n\n\"You had rare success at the fairy shrine,\" Hobar said.\n\n\"You gained the respect of its protectors,\" Baroi said.\n\n\"They pointed the way for me,\" Kendra said.\n\n\"Tell us your plans,\" Savani invited.\n\nKendra looked over to where the menehune were now vigorously shaping the exterior of the dugout canoe. \"Is that for me?\"\n\n\"That depends on your intentions,\" Savani said.\n\n\"I'm going to the Phantom Isle,\" Kendra said. \"Alone.\"\n\n\"You seek the Sunset Pearl,\" Tal said.\n\n\"You are prepared to venture into the dark,\" Hobar said.\n\n\"The way is fraught with peril,\" Baroi said. \"There will be opportunities to fall.\"\n\n\"The cyclopses woke me,\" Savani explained. \"They had a premonition of your intentions. You can journey into the Under Realm alone. But getting there is another matter. You will need assistance.\"\n\n\"I really don't want to drag others into this,\" Kendra said.\n\n\"I already have two volunteers,\" Savani said. \"They will help you get to the Phantom Isle. You'll be on your own for the Under Realm.\"\n\n\"Thank you,\" Kendra said. The thought of having others to help her navigate the ocean was a relief.\n\nRed hair tousled, Grady approached, holding the nova song by a slender chain with one hand and a burlap sack with the other. With the fairy above him straining against her tether, it looked like he was flying a small, beautiful kite. \"Will this do?\" Grady asked, holding out the sack.\n\nSavani accepted the sack and handed it to Kendra. \"Grady is one of the volunteers.\"\n\nHe smiled at Kendra. \"A final voyage for me.\"\n\n\"Grady is a selkie,\" Savani said. \"He was caught stealing from a village a couple of years ago and is working to earn his skin back.\"\n\n\"What does that mean?\" Kendra asked.\n\n\"I can turn into a seal,\" Grady said. \"But only if I have my skin.\"\n\n\"I'll entrust his sealskin to you, Kendra,\" Savani said. \"Once you arrive at the Phantom Isle, it will be his payment. Receiving his skin back should give him a good chance to escape.\"\n\n\"Don't give it to me early,\" Grady said. \"I won't be able to think straight. My instinct will be to go home to my people.\"\n\n\"You won't try to take it from me?\" Kendra asked.\n\n\"There have been occasions when I wanted to steal it,\" Grady said. \"Would have if I could have. But I believe in this mission to retrieve the Sunset Pearl. I vowed to Savani, and I vow to you, come what may, I won't take my skin until you offer it.\"\n\n\"All right,\" Kendra said. \"Who is the other volunteer?\"\n\n\"You'll join up with him after you embark,\" Savani said. \"When I first arose, I sounded the conch to solicit help from the merfolk. They sent envoys to the surface, and I explained our need. Before long they sent a hunter called Faro to escort you.\"\n\n\"I know him,\" Kendra said.\n\nSavani nodded. \"He and Grady will help you reach your destination. After that, all will depend on you. Do you wish to bid anyone farewell?\"\n\n\"I'd rather just slip away if I can,\" Kendra said. \"While I feel good about it.\"\n\nSavani issued a command in a language Kendra could not understand. A menehune approached and handed Savani a folded packet wrapped in banana leaves and tied with twine. She passed the bundle to Kendra.\n\n\"That contains the sealskin,\" Savani said. \"Keep it at the front of the canoe with you. Down by your feet is fine. But Kendra, if I were you, I would never take my hands off that sack. What it contains should never be set aside or entrusted to another.\"\n\n\"I agree,\" Kendra said, holding the crown close.\n\n\"Your craft is ready,\" Grady said.\n\nKendra turned to see that the menehune had gone and the two-person canoe was complete, with an outrigger on each side. Handing the nova song's chain to Savani, Grady kicked his sandals off and cast his shirt aside, then grabbed the front end of the canoe and dragged it into the lagoon. Kendra liked the smell of the freshly cut wood.\n\n\"They asked if we wanted it painted,\" Savani said, returning the bright fairy to Grady. \"I told them we had to hurry.\"\n\n\"They were fast,\" Kendra said.\n\n\"They placed their best enchantments on it,\" Savani said. \"It's watertight. Shouldn't be splintery. The canoe will eagerly go in the direction paddled. It will normally be ignored by dangerous creatures and will gently summon the kind of fish you might want to catch.\"\n\n\"Will we be fishing?\" Kendra asked.\n\n\"I hope not,\" Savani said. \"It depends how far you have to go to find the Phantom Isle.\"\n\nGrady stood in the water beside the dugout canoe. He had fastened the delicate chain of the nova song to the front. The scaly white fairy stood on the bow. Kendra realized it was time to depart.\n\n\"Tell my friends I love them and I'm sorry to sneak away,\" Kendra said. \"I'll be back.\"\n\n\"I will help them understand,\" Savani said.\n\n\"You can expect fair weather,\" Tal said.\n\n\"Hold fast to your mission,\" Hobar admonished.\n\n\"Return with honor,\" Baroi said. \"Many depend on you.\"\n\n\"I know,\" Kendra muttered. She wasn't sure how she could feel much more pressure.\n\nKendra waded out to the canoe, stashed the wrapped sealskin in the front, and climbed in. Grady handed her a paddle, then pushed the canoe forward and climbed in the back with a paddle of his own.\n\nLooking over her shoulder, Kendra waved. Savani and the cyclopses raised their hands in return. She helped Grady paddle out of the lagoon. What was she doing? Was she out of her mind? Was she really heading off to invade the Under Realm alone? Who did she think she was? Feeling the shape of the crown inside the sack on her lap gave her confidence.\n\nAs the canoe exited the lagoon, a familiar figure rose out of the water, clutching a spear that ended in the likeness of a crab claw. \"Welcome, Kendra,\" he said. \"Well met.\"\n\n\"Hi, Faro,\" Kendra said. \"You're going to see us safely to the island?\"\n\n\"When Savani sent her message, I knew I had to get involved,\" Faro said. \"Not only did you get rid of the people of the deep and restore our golems, you eliminated a demon who has threatened our safety for centuries. Now you are on a quest to rescue this sanctuary. You will need help getting past Jibarro. If you are willing to venture into the Under Realm alone, I am determined to help you evade the great sea dragon.\"\n\n\"Does Savani call the merfolk often?\" Kendra asked.\n\n\"Not often,\" Faro said. \"And, truthfully, we do not always respond. Let's just say that after all of your help, our ears were more attuned to the needs of those on the surface.\"\n\n\"I'm Grady of the selkie,\" Grady announced from the stern. \"We're grateful to have your assistance.\"\n\n\"It's my honor,\" Faro said. \"We need a heading.\"\n\nKendra looked at the nova song, still standing on the bow. \"Where do you want to go?\" Kendra asked her.\n\n\"You ordered me not to go toward the darkness,\" the fairy replied. \"I am repressing the urge.\"\n\n\"I promised you could follow your instincts before long,\" Kendra said. \"You can now.\"\n\n\"Thank you,\" the nova song said. \"And I promised that when you allowed me to obey my nature, I would share my name. Mizarine.\"\n\n\"Thanks, Mizarine,\" Kendra said.\n\n\"Thank you for not commanding me to share my name earlier than I preferred,\" Mizarine said. She flew until her chain pulled taut. \"The darkness is this way.\"\n\n\"We have a heading,\" Grady said, paddling. \"Ask her how far.\"\n\n\"Is it far?\" Kendra asked.\n\n\"For me, unchained, less than two hours,\" Mizarine said. \"At our present speed? Half a day, maybe more.\"\n\n\"Half a day will suffice,\" Grady said. \"I can go faster than this, and I don't tire easily.\"\n\n\"You're fortunate,\" Faro said. \"The Phantom Isle can range far and wide on the seas. It might have been many days away.\"\n\n\"What if it moves before we get there?\" Kendra asked.\n\n\"Possible,\" Faro said. \"Only one way to find out.\"\n\nKendra started paddling.\n\n\"Save your strength,\" Grady said. \"I have this.\"\n\n\"I'll go underneath the stern and push,\" Faro said. \"I can use the exercise.\"\n\nA moment after Faro disappeared beneath the surface, the canoe sped up significantly, gliding more rapidly up and down the swells. Kendra looked ahead at the nova song fluttering on her tether.\n\n\"Mizarine, you can rest on the canoe,\" Kendra said. \"Just correct us if we go off course.\"\n\n\"I have been bound to a perch for a long time,\" Mizarine said. \"I want to stretch my wings.\"\n\n\"If you should get free, I command you not to ditch me,\" Kendra said. \"Let me keep pace with you so we can find the Under Realm together.\"\n\n\"I will let you come to the darkness with me,\" Mizarine said, \"if that is your wish.\"\n\n\"The darkness of the Under Realm is dangerous,\" Kendra said. \"You don't have to go there.\"\n\n\"I must light the darkness,\" Mizarine said. \"For me there is no choice. I must fulfill the purpose of my existence.\"\n\nKendra watched as Timbuli diminished in the distance. She enjoyed moving over the ocean in the canoe. It seemed like an activity a tourist might pay to do. A glorious sunrise led to a mild day, with a light breeze and scattered white clouds. Kendra felt in no hurry to reach her destination. Whenever they got there would be soon enough.\n\nKendra knew the realm of the Underking would be full of horrors. She remembered the terrible undead creatures that had attacked Blackwell Keep, the revenants and wraiths. The Under Realm was an entire domain packed with those kinds of beings and worse. There were no guarantees the crown would fully protect her. This could be her last day of her life.\n\nBut she had faced days like this before. Kendra was willing to do anything to save Bracken and Seth, and for a long time she had known no way to help, so it felt good to be able to take action, even if it was risky. She had been warned to keep the light inside of her burning bright, and part of that meant remaining hopeful. Kendra felt sure the Fairy Queen would not have lent her the crown unless she believed she would get it back.\n\nKendra glanced down at the packet wrapped in leaves and twine. \"Grady, you have been a prisoner?\" she asked.\n\n\"More an indentured servant,\" Grady said. \"Working off my crimes.\"\n\n\"You were stealing?\" Kendra asked.\n\n\"Being a selkie means having a dual nature,\" Grady said. \"When we have spent enough time in the sea, we start longing for dry land. We always have the option of shedding our skin and walking about like a human. Occasionally we are coaxed out of the water by a pretty face. Other times we might crave a certain food or miss the experience of standing on a hilltop. Sometimes we get up to mischief.\"\n\n\"What were you trying to steal?\" Kendra asked.\n\n\"A jade figure from a memorial to the oceanids,\" Grady said. \"I took a fancy to it and decided it belonged in the water instead of on land. When we shed our skins, we take a risk. If somebody gets our sealskin, we can't return to the water except as a blundering human.\"\n\n\"Savani got hold of your sealskin,\" Kendra guessed.\n\n\"I have no hard feelings,\" Grady said. \"She was within her rights. It has happened to me before and will probably happen again. After I spend enough time as a human, it begins to feel like my native state. But the longing for the sea grows over time.\"\n\n\"Your mind-set alters when you become a seal?\" Kendra asked.\n\n\"Considerably, yes,\" Grady said. \"Becoming a seal represents a different form of existence, complete with new abilities and instincts. Once I'm in my seal form, my human inclinations melt away until the next time I come up on land.\"\n\n\"Thanks for helping me,\" Kendra said. \"I know this is dangerous.\"\n\n\"Whether I'm looking to my interests as a seal or a man, retrieving the Sunset Pearl would be good for the sanctuary,\" Grady said. \"Anyway, I'm not doing the hardest part. I'm in awe that you're willing to go into the Under Realm alone. My role is minor compared to yours. And besides, I get my skin back early this way.\"\n\nThe sun was climbing high in the sky when a smudge appeared on the horizon directly ahead of them. As they drew nearer, Kendra saw it was a patch of gray mist, like a low, murky cloud.\n\n\"We're almost there,\" Mizarine said.\n\n\"It's in the fog?\" Kendra asked.\n\n\"Yes,\" Mizarine said, batlike wings fluttering harder. \"Isn't it thrilling?\"\n\n\"We should pause and plan,\" Grady suggested, dragging his paddle to slow the canoe. Mizarine redoubled her efforts to pull them forward.\n\nFaro surfaced and rested a strong arm on one of the outriggers. \"Once we cross into the mist, Jibarro will hunt us.\"\n\n\"What's the best strategy?\" Grady asked.\n\n\"We should work together to propel the canoe to the island as fast as we can,\" Faro said. \"If Jibarro appears, I will antagonize him, draw him away.\"\n\n\"You'll be killed,\" Grady said.\n\n\"I'm very quick in the water and accustomed to confrontations with big game,\" Faro said. \"I'll make my escape. Just get Kendra to the shore without delay.\"\n\n\"We should free the nova song from the boat,\" Grady said. \"Kendra can hold the chain. We may need to move in great haste.\"\n\nGrady came forward, unfastened the slender chain from the front of the canoe, and handed it to Kendra. She noticed his eyes lingering for a moment on the wrapped sealskin. Then he returned to the stern and held his paddle ready.\n\n\"Are you going to push?\" Grady asked the merman.\n\nDucking his head, Faro vanished beneath the swells with a flick of his tail. Kendra hoped he could see farther underwater than they could see through the mist.\n\n\"Ready, Kendra?\" Grady asked.\n\nShe held her crown in one hand, the fairy chain in the other, and felt the sealskin in its leaf wrappings at her feet. She checked her sack of gales and the unicorn horn. The thought of perhaps presenting it to Bracken soon was encouraging. \"Sure.\"\n\nWith a sweep of Grady's paddle, the canoe surged forward much faster than ever, making Kendra rock back. She could only imagine how hard Faro was pushing under the water.\n\nCool mist breezed against Kendra's face as the canoe entered the grayness. Kendra peered ahead, hoping to see land before she saw a dragon. The visibility within the patchy fog varied, allowing her to see sometimes a hundred yards before them, sometimes fifty.\n\nWhen the island came into view up ahead, Kendra's relief was interrupted by a tremendous roar. Turning in the direction of the earsplitting bellow, Kendra saw the enormous head and neck of a dragon coming into view through the mist from off to one side.\n\n\"Get to shore,\" Faro said, surfacing and racing toward the dragon, spear upraised. \"I am Faro of the merfolk!\" he shouted in a surprisingly loud voice. \"I have come to slay the sea tyrant Jibarro! You must be his little niece. Fetch him for me!\"\n\nThe answering roar showed that the taunt had struck a nerve. Faro began to veer away from the island to lure the sea dragon away from the canoe. Grady paddled furiously.\n\n\"I always thought dragons belonged in the sky!\" Faro yelled. \"They look so majestic soaring against the clouds. Sea dragons are too fat and dull for such sport! They waddle like ducks and bob around like infants.\"\n\nThe island was coming into sharper focus. The ruins of stone fortifications mingled with jagged rocks. The forbidding coastline offered no obvious place for the canoe to land.\n\nJibarro had taken the bait. The huge sea dragon was closing in on Faro, who looked tiny by comparison. Kendra squeezed the crown through the sack. What spectacular dodge was going to save him? Faro looked doomed.\n\n\"Choke on me!\" Faro called, hurling his spear. It pinged harmlessly off the scales of Jibarro's neck. Jaws agape, the sea dragon launched forward.\n\nIn the last moment, Kendra saw Faro's eyes go to her. He seemed encouraging and at peace. Kendra realized the merman had never planned to avoid the sea dragon. Jibarro was too large and too fast. This had been a sacrifice from the start.\n\nHer heart squeezed at Faro's heroism as he disappeared into Jibarro's mouth, maybe fifty yards away. Why had he done it? Maybe he understood how badly the sanctuary needed the Sunset Pearl. Maybe he believed in her. She would never know for sure.\n\nSuddenly the canoe was catapulted into the air as an enormous tail rose up from beneath them. Flying out of her seat, Kendra let go of the nova song's chain but held tight to the crown. As Kendra soared through the air, she marveled that the dragon had kept track of the canoe even while moving away to devour Faro. She was also astounded at the size of the dragon\u2014that his head could be so far away and his tail could still reach her.\n\nKendra hit the water and sank amid the bubbles of her splash. Kicking her feet, feeling waterlogged in her clothes, Kendra rose to the surface. She saw Grady nearby.\n\nWithout the canoe they were mostly helpless. Neither she nor Grady had much chance of making it to shore ahead of the dragon. She saw no use in preventing Grady from becoming a seal. \"The sealskin is yours,\" she yelled.\n\nA thunderous roar announced that Jibarro was pivoting around to come her way. Grady dipped down under the water. Kendra started swimming toward the shore, hampered by her shoes and the sack in her hand. The unwelcoming shore wasn't far now, and Kendra realized she might get hurled against the rocks by the waves, but it seemed preferable to getting eaten.\n\nAnother roar told her that the sea dragon was closing in behind her. She was simply too slow. She would never reach the shore in time.\n\nSomething bulky nudged up against her under the water. After the initial surprise, Kendra recognized it was a chocolate brown seal. Assuming it had to be Grady, she threw her arms around the pinniped's neck and held her breath.\n\nThe large seal plunged down deep. Kendra had expected to get water in her face moving along the surface, but the seal had other ideas. Hoping the seal understood the limits of her lung capacity, she held on tightly.\n\nThey reached the underpinnings of the island and slipped into a little cave. The rocky walls brushed against Kendra as the seal darted forward, her clothes mostly insulating her against scrapes. Somewhere above and behind them, Jibarro was roaring. Kendra kept her eyes squeezed shut and held her breath.\n\nAs the seal swooped upward, Kendra fought the desperate urge to breathe. They finally broke through the surface of the water, and Kendra saw that the cave had widened as it curved upward into a large, landlocked pool, open to the sky. The water in the pool was turbulent, surging up and down, so she clung to the seal. The rocky walls of the pool extended at least twenty feet above the high point reached by the water, making Kendra feel like she was looking up from the bottom of a well. The sea dragon bellowed with a fury that made her ears buzz.\n\nThe seal towed Kendra near the wall of the pool and, when the surf was near the high point, helped boost her onto a shelf. Kendra released the seal and climbed to a higher shelf before the water surged up again. Panting, soaked, she covered her ears as the dragon roared again.\n\nIn that moment, with a hand over each ear, Kendra realized she no longer had the sack. She could hardly breathe as the implications sank in. She had deliberately kept hold of it when she fell into the ocean. She remembered feeling it in her grasp as she grabbed hold of the seal. That was her last sure memory of the sack in her hand.\n\nKendra watched the water in the salty pool rising and falling, almost reaching where she sat. The seal was gone.\n\nThe crown could be out in the ocean or in the cave. It might be directly beneath her. There was no way to be sure.\n\nKendra took a shuddering breath. Her insides felt hollowed out. Why had anyone trusted her? Why had she trusted herself? Bracken would stay captured. Seth would have no help. The Sunset Pearl would remain in the custody of the undead. Faro had died for nothing. The Fairy Queen had lost her crown. What would that mean for the Fairy Realm? And for fairies in general?\n\nJibarro roared again, sounding farther away. Part of Kendra wished the dragon had simply eaten her. It would have been a much cleaner death than what now awaited her.\n\nThe seal popped back into view, a sack in its mouth. Kendra felt an unexpected surge of hope. Could the crown still be in there? Was Grady really still helping her?\n\nThe next time the water surged high, the seal slid up onto the shelf where he had first deposited Kendra. She climbed down to that shelf and accepted the sack from the mouth of the seal. The crown was still inside.\n\nKendra flung her arms around the seal. \"Thank you, Grady. I was about to lose my mind. I won't let you down. I promise.\" She ended the hug and started wiping away tears of relief.\n\nWhen she looked back at the seal, he was Grady again, sealskin in his hands. \"Good job, Kendra,\" he said quietly. \"You can do this. Climb out of here and finish what you started.\"\n\n\"Okay,\" she said. \"Thank you so much. You saved me twice.\"\n\n\"You're trying to save so many of us,\" Grady said. He then pointed out the route she could follow to climb out. Kendra appreciated the advice. It was impressive that Grady had rescued her. She wondered how much he had needed to wrestle with his seal instincts to do so.\n\n\"Was it hard to help me?\" Kendra asked.\n\n\"A little,\" Grady said. \"My instincts were to flee. And I always feel a strong urge to shake off my land memories. That will happen soon enough. Go. I want to see you make it to the top.\"\n\n\"All right,\" Kendra said, putting the wet sack in her mouth. Stepping carefully, taking care to find good handholds and stable places to set her feet, Kendra made her way up. It was easier higher up, where the rocks were dry. After scrambling up from the highest ledge to the top, she took the sack from her mouth, then turned and looked back.\n\nGrady was a seal again. He barked once and then dove into the roiling water.\n\n\"You made it,\" a voice said behind Kendra.\n\nShe turned to find Mizarine fluttering nearby, shining white in the gray mist, the slender chain still dangling from her ankle. Kendra smiled. \"You waited for me.\"\n\n\"Following orders,\" the nova song said. \"The darkness is so close now.\"\n\n\"Let's go light it up.\"\n\nJust after midday, the keel of the clever boat scraped against sand, and Seth vaulted out onto the beach. He carried his satchel and had Calvin secreted in his pocket.\n\n\"Welcome back to Timbuli,\" Ronodin said, disembarking as well. \"Consider this your final exam. If you pass the test, you can leave school and start living life.\"\n\nSeth backed a few steps away as Ronodin called the boat up higher onto the beach. He still didn't have a clear plan, but he felt he should at least play along until he found the Everbloom. Otherwise Ronodin or the Sphinx might claim it without him, and he would lose the ability to influence the outcome.\n\n\"This way,\" Ronodin said. \"These caves make up a complex system with several entrances. The opening we're looking for isn't far.\"\n\nSeth followed Ronodin onto a trail between leafy philodendrons and fragrant anthuriums. The day was mild, less humid than normal, with a light breeze. It seemed like a strange day to confront the toughest choice he could remember facing.\n\nRonodin carried a duffel bag. Seth wondered what it might contain.\n\nThey branched left off the main path along a faint trail. After winding through trees and shrubs, they reached a grouping of boulders. In the ground between two of the largest boulders gaped a crack just large enough for a person to crawl down into.\n\n\"This isn't a cave,\" Seth complained.\n\n\"It leads into one of the largest cave systems on any island in the world,\" Ronodin said. He set down the duffel bag, unzipped it, and removed a garden trowel. \"Dig up the flower with this. The Everbloom is resilient, but try to get as many of the roots as you can.\" Then he held up a ceramic flowerpot. \"Keep the flower in here. Pack soil around it.\"\n\n\"Okay,\" Seth said.\n\n\"Time for your ointment,\" Ronodin said. \"I'll give you some privacy.\"\n\nAfter Ronodin walked out of view, Seth stripped down and took the broad stopper off the ointment jar. He gave a generous dollop of the ointment to Calvin, who went to work on himself inside a pocket.\n\n\"Need any help?\" Seth whispered.\n\n\"I'll be fine,\" Calvin said. \"You gave me enough to drown in. Your pocket might be a mess, though.\"\n\nSeth slathered bluish jelly all over his skin, leaving him greasy but hopefully impervious to the heat he was about to encounter. More than three-quarters of the jar was empty by the time he finished and put his pants on.\n\n\"Ronodin,\" Seth called. \"Can you get my back?\"\n\n\"Absolutely,\" his answer came from a distance.\n\n\"I could reach most of it,\" Seth said. \"I think I missed part of the middle, though.\"\n\nRonodin strode to him, picked up the jar, and began applying the ointment. \"Is it a comfort to know I have your back?\"\n\n\"At this moment?\" Seth asked. \"Sure. I'd rather not catch on fire. What about my clothes?\"\n\n\"The contact of the material against the ointment on your skin should grant heat resistance to your clothing. You can smear a little extra onto your clothes to be sure. Your back is done.\"\n\n\"Thanks,\" Seth said, putting his shirt back on.\n\n\"Isn't a situation like ours interesting once trust has disintegrated?\" Ronodin asked casually.\n\n\"Trust isn't gone,\" Seth said.\n\n\"No?\" Ronodin asked.\n\n\"I'm going to get the Everbloom. You're letting me.\"\n\n\"I trust that you have compelling reasons to act in your own interest,\" Ronodin said. \"I believe it is so sensible for you to fetch the flower and give it to the Underking that you might actually do it. And I'm also aware you might not.\"\n\n\"Giving the flower to the Underking is the key to my freedom,\" Seth said. \"You can trust I want this chain off me. But with you, I don't know what to trust.\"\n\n\"Probably wise,\" Ronodin replied. \"It's clumsy to trust anyone. People are uniformly imperfect and erratic. But if you have to guess, bet on them serving their own interests.\"\n\n\"Is that why you're unclear about what you want?\" Seth asked. \"To make it hard for people to guess what you'll do?\"\n\nRonodin grinned. \"A penetrating insight. You could have been a great pupil.\"\n\n\"Could have been?\"\n\n\"Had you really wanted to learn from me,\" Ronodin said.\n\n\"You're a very talented teacher,\" Seth said.\n\n\"Have you learned enough?\" Ronodin asked.\n\n\"What do you mean?\"\n\n\"Let's imagine everything you heard from the Hidden Sage is true,\" Ronodin said. \"Kendra and Bracken have shared their perspectives. Maybe others have too. Given all you have heard, how much do you suppose you can trust the light-makes-right brigade? What do you presume they want from you?\"\n\nSeth considered a careful answer. \"They act like they want to help me.\"\n\nRonodin laughed. \"They want to do what they do best, Seth. They want to lock you up. They'll lure you with honey if you respond. But make no mistake: you have broken their laws. They see you as a threat who must be neutralized.\"\n\n\"Maybe,\" Seth said. He hadn't considered this possibility, and he found the suggestion disconcerting.\n\n\"You really don't understand. They lock up dark creatures. They will imprison you simply for who and what you are, the same way they might imprison a wraith or a dragon. You have committed crimes against them, Seth, of your own free will. You destroyed an entire dragon sanctuary. You got some of the people protecting Wyrmroost killed\u2014I know this for a fact.\"\n\n\"You weren't honest about what I was doing,\" Seth said, distressed to hear that his actions had led to deaths. Hopefully Ronodin was lying about that.\n\n\"Really?\" Ronodin asked. \"You're going to tell them you released wraiths and revenants without knowing anyone might get hurt? Would you forgive a killer for not knowing how a gun functioned? How many would have to die before you stopped pardoning his ignorance?\"\n\n\"You tricked me,\" Seth said. \"I didn't understand what I was doing. I didn't have my memories. I still don't.\"\n\n\"You don't have to convince me,\" Ronodin said. \"I'm on your side. Bracken and his allies were holding creatures captive. The only crime those wraiths and revenants had committed was following their natures. Jailers assume the risk of being harmed by their prisoners. I think you are a hero, Seth, a liberator of the oppressed. The problem is how Kendra and Bracken view your actions. It's their perspective, their laws, that condemn you.\"\n\n\"Why are you telling me this?\"\n\n\"You might like to believe you once belonged among them,\" Ronodin said. \"Let's imagine you did. Even if that were true, somewhere in your hypothetical past, you don't belong there anymore. It's too late for you. In their eyes, your crimes are too great. In their eyes, your nature is too dark. They'll lure you in, take what you offer, then zip you into a straitjacket and leave you in a cell.\"\n\n\"Maybe I belong locked up,\" Seth said. \"Maybe you do too.\"\n\nRonodin gave Seth a sad smile. \"You need to believe that if you want to rejoin them. You need to accept that you are a villain, a fiend. Because that is the treatment you will receive. In their camp, you'll always be an outcast. With me, you're a prized pupil. You have a great deal to offer. You mean well. You're only beginning to realize your potential. You could make a positive difference in this world. Or you can let them lock you up and throw away the key.\"\n\n\"Thanks for the advice,\" Seth said.\n\nRonodin grew grave. \"I'm sure you're smart enough not to consider taking the Everbloom to them. I presume the sage advocated for you to leave the flower alone or to pass it to those who would guard it. Any who possess the Everbloom will use its power to serve their desires. Giving the flower to Kendra and her kind only increases their capacity to oppress you. Don't give them the rope they will use to hang you. And don't turn yourself into my enemy.\"\n\n\"I'd hate to do that,\" Seth said.\n\n\"So would I,\" Ronodin said. \"We have been allies up until now. You don't want to see what I do to my enemies, Seth. Not firsthand. And what the Underking would do to you is a hundred times worse. Don't ruin your life so you can give a flower to people who hate you. They want you in chains. Don't let somebody else collect the flower for the Underking. Use the flower to buy your life back. You deserve to seek out your memories.\"\n\nSeth glanced at the crack in the ground. \"Maybe I'll die in hot lava.\"\n\n\"Save the dramatics,\" Ronodin said. \"Bring the Everbloom to me. We'll give it to the Underking. Remember that I plan for all contingencies. Don't fight the inevitable.\"\n\n\"I hear you,\" Seth said. \"I just climb in?\"\n\n\"Chimney down using both sides,\" Ronodin said. He handed Seth a glowing holly wand. \"The descent isn't too far. You'll meet Dezia at the bottom. She will guide you to the bloom. Tread carefully\u2014Baga Loa is not a typical volcano. A cave system like this one would never form in the natural world. You may cross paths with some unusual creatures.\"\n\nNodding, Seth rubbed most of the remaining ointment onto his clothes and satchel. Then he squirmed down into the crack and started descending. Footholds abounded, and he reached the ground without much trouble. So far he noticed neither heat nor any glow of lava.\n\nDezia startled him when she came out of a wall, gauzy wrappings fluttering. \"I will guide you to the Everbloom,\" she said with little expression. Seth saw accusation in her eyes.\n\nShe turned and started gliding forward. Seth followed, feeling awkward. \"Sorry about the token,\" he said.\n\nThe phantom made no indication of having heard him.\n\n\"How are you?\" Calvin asked.\n\n\"Truthfully?\" Seth replied. \"Shaken up.\"\n\n\"Ronodin was trying to make you ashamed,\" Calvin said.\n\n\"I might deserve to be ashamed,\" Seth said. \"I've hurt people I should have protected. With or without my memories, there may be no way back to my old life.\"\n\n\"There never is,\" Calvin said. \"We're always moving forward. All you have is the present. Here we are. Just think about your next step. Forget your past for now. Start today. Get your next step right.\"\n\n\"It sounds pretty,\" Seth said. \"I'm not sure it fits what I've done.\"\n\n\"Kendra and the others know your memories were stripped,\" Calvin said. \"They just want you home safe.\"\n\nSeth stopped walking. \"Phantom, you can wait for me, or you can let me tell the Sphinx how you left me behind.\"\n\nThe phantom halted.\n\n\"Calvin, don't you get it?\" Seth said. \"If what you say is true, if Kendra and the good guys would forgive me, it almost makes it worse. I know what I did, Calvin. I may have been manipulated, but I wasn't forced. If I was a traitor, if I caused destruction by releasing evil, if I got people killed and made a sanctuary fall, I have to live with that. I'm not sure how I can.\"\n\n\"You've been in an impossible situation, Seth,\" Calvin said. \"You're trying to figure out who you are. I see it. And you'll make it. But not if you give up.\"\n\nSeth heaved a sigh. \"I might want to give up, but I won't. I worry, though: what if everything I do just causes more harm?\"\n\n\"Everyone causes harm,\" Calvin said. \"Sometimes we hurt others, and sometimes we hurt ourselves. If you've caused a problem, you learn and do better. You become smarter for next time. And you undo what harm you can.\"\n\n\"What about the Everbloom?\" Seth asked. \"What should I do?\"\n\n\"We've come this far,\" Calvin said. \"Let's have a look.\"\n\nSeth folded his arms and stared at the ground. What else was he supposed to do? Sit down and mope until Ronodin came looking for him? \"Go ahead, phantom,\" he said.\n\nDezia drifted forward again.\n\n\"Besides, you might be right,\" Calvin said. \"We may burn up in hot lava before we get there.\"\n\nSeth followed the phantom through the labyrinthine cave. After a number of turns, Seth wasn't sure if he could find his way back. The air started to get hotter. In one room, steam wafted up from little fissures in the floor. In another, a bubbling pool of hot water in one corner saturated the room with a steamy, eggy odor.\n\nSome passages narrowed so much, Seth had to turn sideways. Some wider chambers had room to spare, with stalagmites four or five times his height. The light from his wand did not always reach the ceiling, and it was hard to tell if some of the shadowy side passages went a long distance or ended after a few feet.\n\n\"Be extra quiet here,\" the phantom whispered.\n\nNoticing a red glow up ahead, Seth shielded the light of his wand with his hand and stepped carefully. In a neighboring chamber of the cave, Seth saw a blob of dark lava with molten redness glowing through the cracks. Fiery millipedes as long as Seth's arm skittered around the surface of the lava blob, some of them crawling into or out of the gaps to the red interior. He knew of no living creature that could handle such extreme temperature.\n\nThe air in the cave became increasingly hot and dry as Seth progressed. The stone around him smelled like it was baking. Seth crossed through one room where more than half the floor was a glowing pool of smoldering scarlet lava. He felt certain that without the ointment, his hair and clothes would burst into flame by proximity to such heat, but instead Seth discovered he could bear it. The air he breathed was somehow cooled by the ointment, and the ashy fumes did not clog his lungs.\n\nSeth caught more glimpses of lava as he continued to follow Dezia. In a vast room where a goopy flow of lava ran down part of one wall and hissing geysers erupted on the far side, the phantom pointed to an arching stone footbridge that led to an island in the middle of a lava lake. The bridge marked the first evidence of masonry Seth had seen since entering the cave.\n\n\"You all right, Calvin?\" Seth asked.\n\n\"I never imagined a place like this,\" the nipsie replied. \"I'm fascinated. Wouldn't turn away a cool drink, though.\"\n\nSeth walked over the bridge, feeling the heat of the stone through his shoes. In the center of the island, Seth spotted a single flower. He jogged to it and then stared in wonder.\n\nThe flower radiated light, almost like a stained-glass window with the sun behind it. Seth could not believe the complexity of the endless colors in the petals, tiny facets and slivers of every shade he knew and some he doubted he could ever describe. Like nature's finest kaleidoscope, the vibrant patterns shifted so that if he closed his eyes, he opened them to behold a new masterpiece.\n\n\"It's a miracle,\" Calvin said.\n\n\"It's beautiful,\" Seth said, kneeling down. The golden stem and leaves had a glow like heated metal. The blossom of the Everbloom was no bigger than Seth's fist, the stem barely a foot tall. Ronodin had done his homework. It would fit easily in the flowerpot Ronodin had prepared.\n\n\"We can't give this to the Underking,\" Seth said.\n\n\"I agree, boss,\" Calvin said.\n\n\"Which is a problem,\" Seth said, glancing at the ghostly manacle on his wrist. \"You gave me smart advice. I can start by making a good choice right now. This flower helps fuel these islands. It's exquisite. I don't want to harm it\u2014or the creatures that depend on it. I want to like myself, Calvin. I want to make choices I can live with.\"\n\n\"I like you already,\" Calvin said. \"You're doing great.\"\n\n\"I'm going to do everything I can to protect this flower,\" Seth said. \"That idea feels good. And maybe doing that will get me killed. But taking the flower to Ronodin might also lead to my death. I'd rather die liking myself than hating myself.\"\n\n\"I think the same way,\" Calvin said. \"And I'll do everything I can to keep you alive.\"\n\n\"To protect the flower, we have to start with the phantom,\" Seth said. \"Dezia, listen to me, I need to speak with you.\"\n\nThe phantom turned. \"I have led you to the Everbloom,\" she said tightly.\n\n\"I'm sorry I gave away your token,\" Seth said. \"I made a mistake.\"\n\nThe phantom did not answer.\n\n\"I want to fix this,\" Seth said. \"How can I free you?\"\n\nThe phantom stared at him. \"I would need a new token. You would have to bind me to the new one and break the connection with the old.\"\n\n\"Can I do that?\" Seth asked.\n\n\"As a shadow charmer, you have the power,\" Dezia said, a hint of hope in her voice.\n\n\"And then could I throw the new token into the lava?\" Seth asked. \"Keep my promise?\"\n\nDezia took a step toward him. \"You could, yes. Do you mean it?\"\n\n\"Would you still lead us out of here?\" Seth asked.\n\n\"Once my token was destroyed, you could no longer obligate me,\" Dezia said. \"But I pledge to lead you out of these caves if you do this service for me. And I will speak of you with admiration ever after.\"\n\n\"What would work as the new token?\" Seth asked.\n\n\"I must feel a connection with the object,\" Dezia said. \"Pluck a leaf from the Everbloom.\"\n\n\"Would that empower you too much?\" Seth asked.\n\n\"After being plucked, the leaf will lose the potential to greatly empower me,\" Dezia said. \"But it will possess some inherent power, simplifying your task of transferring the connection.\"\n\n\"The leaf will grow back?\" Seth asked.\n\n\"The Everbloom is alive,\" Dezia said. \"It sheds leaves and petals as all plants do.\"\n\nSeth crouched beside the Everbloom. One of the little leaves looked droopier than the others. \"Sorry,\" Seth told the flower. \"This is to protect you.\"\n\nWhen Seth touched the leaf, his mind suddenly became more alert, his body felt stronger, and he could perceive his power more distinctly. For a moment, his best hopes about himself seemed possible. His hunger and thirst disappeared. With a tug he plucked off the leaf, and the influx of energy ceased, though many of the effects lingered.\n\n\"That will do nicely,\" Dezia said. \"You know how to access your power?\"\n\n\"Yes.\"\n\n\"Can you sense me?\" Dezia asked.\n\n\"Yes,\" Seth said. Since touching the leaf of the Everbloom, Seth felt like his inner vision was clearer than ever. He could mentally sense the phantom in great detail.\n\n\"Can you feel my connection to the crypt?\" Dezia asked. \"And my connection to the token?\"\n\nSeth searched, studying the phantom with his mind. He closed his eyes and tried, then made another effort with them open. He discerned no connections.\n\n\"I'm sorry,\" Seth said.\n\n\"Keep trying,\" Dezia said. \"Unless you can sense my connections, you can't break them. You need to establish a link between me and the new token, allow the new token to share my connection to the crypt, and break the bond with my current token, the doll.\"\n\nSeth could not feel any hint of the connections she described. He had a sense of her presence, and that was all.\n\nHe glanced at the Everbloom. Holding the plucked leaf in one hand, Seth reached out with the other and rested a finger against the petals of the flower. Immediately he felt a powerful influx of energy, and his mind sharpened. He fixed his attention on the phantom. The dark umbilicus connecting her to the crypt was obvious, and so was the dark channel of power uniting her to the doll. The channel of power to the doll felt sullied. Somebody was tampering with it.\n\n\"I feel the connections,\" Seth said excitedly.\n\n\"Change them,\" Dezia said. \"Link me to the leaf. I am willing it. Help it to happen. Break the bond with my current token. Be quick about it. If they catch on to what we are doing, they will try to pull me away.\"\n\nSeth searched the power inside of him. The darkness he drew from to quench torches was enlivened by his contact with the Everbloom. Drawing on that power, Seth focused on creating a channel between Dezia and the leaf, but the connection to the current token blocked his effort.\n\nSeth concentrated on her bond with the doll. He could not sense the location of the doll, only that it was far away. He willed the phantom free of that bond, as he would when attempting to open a lock.\n\nThe bond with the current token resisted, gaining strength and trying to pull Dezia away. The phantom moaned. Forcing all of his effort into breaking the connection, Seth felt energy course through him from the Everbloom.\n\nThe bond with the doll broke. Suddenly it was easy to pour energy into the channel between Dezia and the leaf. When Seth felt that link solidify, he stopped touching the flower.\n\nThe instant his finger broke contact with the petals, Seth slumped forward, his body drenched in sweat. The struggle against the dark forces controlling the phantom had left him depleted. His heart hammered as if he had just run a mile at a brutal pace.\n\n\"Thank you, Seth,\" Dezia said. \"That was clumsy work, but you are very strong, at least with the help of the Everbloom. Now dispose of the leaf and I vow to lead you out of here.\"\n\n\"Can we go out a different way than we came in?\" Seth asked. \"I'm not sure Ronodin will be glad to see me.\"\n\n\"Whatever you wish,\" Dezia said. \"After a painful betrayal, you have done me an unexpected service.\"\n\n\"They have no hold on you now, right?\" Seth asked.\n\n\"Correct,\" Dezia said.\n\n\"And you won't help them find the Everbloom?\" he checked.\n\n\"Never,\" Dezia said. \"They abused me.\"\n\n\"Do you promise?\"\n\n\"I give my vow.\"\n\nSeth nodded. \"The flower is safer if I leave it here.\"\n\n\"What if Ronodin or the Sphinx get a new phantom to help them?\" Calvin asked.\n\n\"They need the ointment to come in here,\" Seth said. \"And they needed me to recruit the phantom. Neither of them can charm the undead.\"\n\n\"It would be a serious risk to remove the Everbloom,\" Calvin said. \"I bet it could destabilize the volcano.\"\n\nThe phantom laughed darkly. \"Any fool could see that much.\"\n\nSeth shook his head. \"Why am I not surprised they didn't warn me?\"\n\n\"Ronodin is usually so helpful,\" Calvin said.\n\nSeth walked over to the edge of the island, above the rich red lava. Even with the help of the ointment, the proximity made it almost unbearably hot. He could look at the lava directly for only a couple of seconds at a time.\n\n\"Is this as good a place as any to drop the token?\" Seth asked.\n\n\"Yes,\" Dezia replied.\n\nSeth released the leaf and watched it seesaw down onto the lava. A brief flame flared up, and it was gone.\n\n\"Thank you, Seth,\" Dezia said. \"Follow me.\"\n\nSeth gently patted the pocket with Calvin in it. \"Ready to get out of here?\"\n\n\"So soon?\" Calvin asked. \"I thought a sleepover could be fun. Another time, maybe. We can go . . . as long as you do the walking.\"\n\nThere are foul beings in the mist,\" Mizarine warned. \"I'll try to lead you around them.\"\n\nKendra glimpsed the fog-shrouded figures as she followed the nova song over and around decaying fortifications. She had taken the crown out of the sack and held it in her hand. She wondered if she should put it on. She decided she would if any of the zombies in the fog got too near.\n\n\"Is there a darkest place in the Under Realm?\" Kendra whispered.\n\n\"Yes,\" Mizarine answered.\n\n\"Will you stay with me until we get there?\" Kendra asked.\n\n\"I'll try,\" Mizarine said. \"You may want to hold my chain. The pull of the darkness gets stronger as we get closer.\"\n\nKendra took hold of the chain. She heard waves crashing against the rocks below. Off to one side, a peg-legged zombie in a long coat limped into view.\n\n\"Run,\" Mizarine said.\n\nKendra came as close as she could to running, trying to stay mindful of where she placed her feet on the uneven rocks. The nova song led her to the brink of a cliff above the sea. Kendra discovered stairs slanting diagonally down the face of the cliff, ending at a stone platform about halfway down to the turbulent water. A ladder descended from the platform to the surf below, and another set of stairs climbed to the clifftop in the opposite direction.\n\nA glance back revealed several zombies in pursuit. More climbed the ladder from the sea, and others rushed down the stairs from the opposite side of the clifftop, trying to beat Kendra to the platform.\n\n\"Hurry,\" the nova song urged.\n\nKendra raced down the stone stairs two at a time. The steps were damp from the mist, but there was no time for caution. \"The entrance is at the platform?\" she asked.\n\n\"Yes,\" the nova song said.\n\nOut of control, Kendra stumbled as she reached the platform several steps ahead of the oncoming zombies, but she managed to stay on her feet by catching herself against the cliff. At the rear of the platform yawned a large doorway framed in stone with no door. The dimness beyond was a shade most would see as black. With sea zombies on the stairs behind her, clambering up from the water and rapidly descending the opposite steps, Kendra had no time to hesitate.\n\nKendra followed Mizarine through the doorway, and the fairy began to sing. The high, trilling notes were calming and ethereal, music to inspire contemplation. As she sang, the fairy began to glow brighter, throwing light all over the wide hall extending from the doorway. The hall continued for a long way, with many passages branching off.\n\nGlancing back, Kendra found the sea zombies congregating at the entrance, but none crossed the threshold. Mizarine kept tugging Kendra forward.\n\nWraiths began to emerge from side passages, moving smoothly and silently, bringing a supernatural chill. A revenant, blindfolded and gagged, shuffled toward them from farther down the hall. The temperature plunged. Kendra rubbed the goosebumps on her arms. The wraiths stalked toward her. Another revenant hobbled into view, body partially decomposed, arms bound to her sides.\n\nKendra knew it was time. Part of her was scared to put on the crown. If it didn't work, her fate would be sealed. But there was no time for hesitation. The undead were seconds away from claiming her. The nearest wraith was barely ten steps away and closing.\n\nHolding Mizarine's chain in one hand, Kendra placed the Fairy Queen's circlet on her head. Immediately the crown flared to life, exponentially outshining the nova song. With her ears and mind, Kendra heard agonized shrieks as the wraiths fell back and the revenants halted. Looking at her hand, Kendra found her skin radiating light. Glancing back at the entryway, she saw the sea zombies recoiling, eyes averted.\n\nHope and confidence flooded into Kendra, more than a response to how brightly she gleamed and how the undead reacted. Resolve and strength flowed directly into her mind and heart from the crown. The crown glared brighter in response to her growing confidence. Tendrils of smoke wafted up from the retreating wraiths.\n\n\"I don't want to harm you,\" Kendra announced. \"You have taken things that don't belong here. I have come to retrieve them.\"\n\nThe nova song kept pulling Kendra forward, and she followed. The revenants turned and started moving sedately into side passages. Kendra walked straight down the hall. Wraiths fled the beacon she had become.\n\nThey were right to flee. Kendra knew she possessed in abundance something these shadow creatures lacked and feared. She was full of life and light. She could feel it coursing through her, radiating from her. She felt regal, like a queen. With the crown on, maybe she was a queen. No, not maybe. The truth of it became plain, and the realization freed her to burn even brighter.\n\nA door at the end of the hall opened, and an intimidating figure emerged. He was dressed in black armor, and his translucent face allowed Kendra to see his skull. A dark aura around him resisted her brightness.\n\nThe figure raised a hand toward her. Halt, intruder, he thundered in her mind. I am Stavius the Obscure, door warden of the Under Realm, a captain among liches. Depart or suffer my wrath.\n\nKendra imagined creatures like him tormenting Bracken. \"Step aside,\" she commanded.\n\nYou have no authority here, Stavius warned. This is the seat of our power. He pointed at her. Writhe.\n\nKendra's muscles suddenly felt sore, as if they were trying to cramp, and her vision became blurry. Gritting her teeth, she pictured Seth being indoctrinated by these foul beings. Power emanated from her crown, and Stavius staggered back. Strength flooded into her muscles and her vision sharpened. Looking down at herself, Kendra found her entire body shedding such glorious light that it almost looked like she was being consumed by white flames.\n\nImpossible, Stavius expressed in dismay. Come no closer. My touch will stop your heart.\n\nKendra sensed his uncertainty. \"You don't get it. Warn your friends. Unlike you, I have a heart. I'm alive. And I'm not afraid of you.\"\n\nBegone, Stavius demanded. I had thought to show you mercy. No longer. I will drain you to the brink of death and hold you there for ages beyond counting.\n\nKendra was drawing near. The lich was significantly taller than she was. In his heavy armor, normally he would have been very frightening. But under the bright light, Kendra could see right through him. He wasn't entirely tangible. The Fairy Queen had warned her to maintain her courage. If she backed down now, how would she save Bracken? How would she rescue Seth? How would she retrieve the Sunset Pearl? If this lich wanted to play chicken, he had picked the wrong girl on the wrong day.\n\n\"I wasn't half this strong when I killed Gorgrog,\" Kendra said. \"Stand down, or there will be nothing left of you to cast a shadow.\"\n\nThe lich took a step toward her. His next step was hesitant, and for the third, he looked as if he were leaning against a heavy wind. It ended up being only half a step.\n\nKendra was almost to the lich. Even bolstered by the power of the crown, Kendra felt hesitant to touch the creature. As she closed in, his aura of darkness was swept away, and he became even less substantial.\n\nStanding his ground, the lich stretched a trembling hand toward Kendra. She pitied him. He looked feeble, almost invisible.\n\nYou . . . will . . . perish.\n\nDid he believe that? Was he just being stubborn?\n\nKendra stopped. \"I didn't come here to harm you. I don't even know you. Move out of the way.\"\n\nYou . . . falter.\n\nKendra continued forward, barely able to see the dazzling flames consuming the lich because he had become so faint. But she heard the horrific scream in her mind. When she walked through the place where Stavius had stood, there was nothing left.\n\nPausing to look back, Kendra found the hallway bare. She wondered if the corridor had ever been so brightly lit.\n\n\"Hurry,\" Mizarine urged. \"Not much farther now.\"\n\nKendra went through the door the lich had guarded and down a long flight of stairs. Though the nova song tugged her forward, Kendra did not hurry. What if going to the darkest place in the Under Realm led her to the Underking? Was she ready for that?\n\nThen again, how was she supposed to find Seth, Bracken, and the Sunset Pearl without confronting the Underking? This was his domain. He would find her if he wanted. Judging by the splitting corridors and winding hallways, this place was a huge maze. Even if nobody attacked her, could she find what she was looking for amid all these twists and turns?\n\nAt least following the fairy to the darkest place in the Under Realm gave her a destination. She could make new plans from there. She found herself almost wanting to encounter the Underking. Confidence had worked against Stavius. She saw no advantage to fearfully avoiding the Underking.\n\nThe route wound onward, down inclines and stairs. At every junction, the nova song eagerly chose the next turn. Kendra was not simply lighting the halls\u2014she was scorching them. Her own light reflected harshly off the rough stone walls, and though none of the brightness bothered her eyes, she could tell it was intense.\n\nThe lower they traveled, the more cavelike the corridors became, though the floor remained smooth and comfortable for walking. The entire complex seemed abandoned. Kendra assumed the undead were staying out of her way.\n\nComing around a corner, Kendra unexpectedly found a familiar face awaiting her. The Sphinx stood in the middle of the hall, wearing a leather overcoat.\n\n\"Wow, Kendra,\" the Sphinx said, shielding his squinted eyes with one hand. \"Looking bright. Unbelievably bright.\"\n\n\"You betrayed us,\" Kendra accused, stopping. He wasn't undead, so she didn't know if her brightness would repel him. She grabbed the sack of gales just in case.\n\n\"You'll be glad to know Seth is alive and well,\" the Sphinx said.\n\n\"Where is he?\" Kendra asked.\n\n\"I honestly don't know,\" the Sphinx replied. \"I don't rank very high around here. Are you really wearing the crown of the Fairy Queen?\"\n\n\"How can I find Bracken and Seth?\"\n\n\"I see that you are,\" the Sphinx said. \"You and the crown really light each other up. I can't believe she parted with it, even to help her son. That was quite a risk.\"\n\n\"I'm also looking for the Sunset Pearl,\" Kendra said.\n\n\"Why not?\" the Sphinx said. \"Might as well. You want it all. The risk was in coming. Take what you can, if you can. Why don't I show you to the boss around here? The guy with the answers.\"\n\n\"The Underking?\" Kendra asked.\n\n\"He may not be happy to see you,\" the Sphinx said.\n\n\"I haven't seen anyone since leaving the entry hall,\" Kendra said.\n\n\"They're not used to light down here,\" the Sphinx said. \"That works in your favor. No missteps, though. They are all aware of you, and they will swarm if you stumble.\"\n\n\"Don't you try anything either,\" Kendra said.\n\n\"I'm not your enemy, Kendra,\" the Sphinx said. \"I never set out to hurt Seth. Just to help him reach his potential. He's already a much more powerful shadow charmer than before he forgot everything.\"\n\n\"What have you done to him?\"\n\n\"I didn't take his memory, Kendra,\" the Sphinx said. \"He made that choice. Ronodin and I have simply taught him about his powers.\"\n\n\"After you kidnapped him.\"\n\n\"I may have played a role,\" the Sphinx said. \"But you and I have different goals. You're working to save the sanctuaries. I'm working to save the world. Our methods might never match up.\"\n\n\"Was opening Zzyzx part of saving the world?\" Kendra challenged.\n\n\"In fairness, you never got to see how that would have played out,\" the Sphinx said. \"You interfered with the dragons, which is what led to them making their play for power.\"\n\n\"I don't have all day,\" Kendra said. \"Where is the Underking?\"\n\n\"You have a nova song,\" the Sphinx said. \"Good find. She can lead you to the Underking. I can lend some help with the door.\"\n\n\"Lead the way,\" Kendra said. \"Don't get close to me.\"\n\n\"Come,\" the Sphinx said.\n\nKendra followed the Sphinx, trying to restrain herself from blasting him with wind simply out of principle. He was as shameless as ever\u2014unwilling to admit his crimes, quick to offer explanations. She decided evil was most dangerous when operating with a smile. She preferred creatures like Stavius who looked and acted evil without apologies or excuses.\n\nThey did not go far before the Sphinx stopped at a large black door decorated with skulls that blazed white beneath the glare of Kendra and her crown. \"He's in there. He generates the darkness your nova song is seeking. Take care\u2014no darkness in the world can compete with his.\"\n\n\"How do I open the door?\" Kendra asked.\n\nThe Sphinx brushed a finger against a small, narrow skull with pointy teeth, and the door disappeared. Inky blackness awaited beyond the entryway. From where Kendra stood, her light did not seem to touch it.\n\n\"I heard the Underking can turn people undead,\" Kendra said.\n\n\"Merely looking upon him will cause that transformation,\" the Sphinx verified. \"Move cautiously. He has a lot of power here. You've been protected thus far. I think your only option is to rely on the crown.\"\n\nThe Fairy Queen had told her to be confident. So far that had gone well. The power from the crown filled her with surety, and her light grew brighter. Mizarine flapped her wings desperately, straining to pull Kendra forward.\n\nKendra stepped through the doorway into the darkness. The moment she entered, her light pressed the darkness back, forming a sphere around her. But the light did not extend to fill the room, and Kendra could feel the darkness squeezing the sphere of light smaller.\n\nBeyond the bright sphere, the darkness seemed heavy, like molten tar. As Kendra began to fear, she watched the sphere shrink inward.\n\nShe stamped her foot. No! Her light was brighter than a bunch of darkness. The sphere expanded with the thought, and Kendra felt power from the crown reinforcing the sentiment. The sphere of light around her grew bigger than it had been when she first entered. The nova song perched on Kendra's shoulder, singing softly and adding her light to the effort.\n\nWhispered words burrowed into her mind, quiet but penetrating. What feeble light intrudes upon my sanctum?\n\n\"Feeble?\" Kendra asked, insulted. As her light resisted the darkness, Kendra could feel the crown drawing upon her own energy. Focusing, Kendra forced more energy into the crown, and the sphere of light brightened and expanded.\n\nYou brought the crown of a rival into my presence, the voice observed. A foolhardy risk.\n\n\"We'll find out,\" Kendra said. \"Are you the Underking?\"\n\nI am master of this realm, the voice responded.\n\n\"I'm not here to invade,\" Kendra said. \"I'm here to retrieve some things you have that don't belong to you.\"\n\nEverything in the Under Realm belongs to me, the voice responded, incensed.\n\n\"I don't,\" Kendra said.\n\nThe Underking gave a slow, delighted laugh. You believe you can return to the surface?\n\n\"Yes,\" Kendra said, trying not to entertain any uncertainty.\n\nI lured you here. Removed all obstacles. Granted you access. Commanded the Sphinx to show you the way.\n\nKendra tried not to let the words frighten her, squeezing her eyes shut out of sheer anxiety. Should she keep them closed? If she saw the Underking would she become undead? Or would keeping them closed be more dangerous? When she opened her eyes, the sphere of light around her was smaller and still shrinking. \"Where is Seth? Where is my brother?\"\n\nWhat does it matter? Worry about yourself. The Fairy Queen wields power, especially in her realm. I would not venture there. You, my dear, are not the Fairy Queen. You have strayed far from where you belong and are in grave danger. Would you rather be a lich or a revenant? A lich boasts more power and autonomy. It depends upon your willingness.\n\nThe whispered words were calm and certain, but also unclean, as if they were leaving stains on her mind. The sphere of light contracted a bit more. Kendra couldn't think of a worse fate than joining the undead\u2014neither living nor able to rest, cursed to hunt and crave that which she could no longer enjoy herself. She pictured herself as a revenant, body degenerating as she listlessly wandered the world, forever unsatisfied.\n\nThe darkness constricted tighter. Seth needed her. Bracken needed her. If she failed, not only would she suffer a fate worse than death, but Crescent Lagoon would fall and the Fairy Queen would lose her crown. That would all contribute to the dragons winning their war against humanity. When the darkness overtook her, she would have failed everyone.\n\nIf you kneel and remove that crown, I will show you mercy, the Underking offered.\n\nThe proposal struck Kendra like a slap. She remembered with a sharp pang how Celebrant had encouraged her to surrender Wyrmroost so that she and her loved ones could obtain mercy. It had been a lie. Agad had died believing a similar lie. She and her group would have been murdered had Andromadus not intervened.\n\nMaybe the Underking could make her one of the undead. But if so, why talk about it? Why not take action? Confidence had brought her this far. Was she going to let him shake her? Why not doubt him instead of herself? If he was going to turn her into a revenant, he would have to fight for it. She wasn't going to surrender.\n\nResolve flared inside Kendra, and the light of her sphere swelled bigger and brighter than ever. Nobody would rescue her here. It was up to her to save Bracken and Seth. And to save herself. She already had all the help she was going to get. The Fairy Queen had given her the crown, which meant the Fairy Queen believed she could succeed.\n\n\"You have this all wrong,\" Kendra said, her light growing. \"I didn't come here to ask you for anything. I came here to tell you what I am taking. I'll destroy anyone who gets in my way, including you. Show yourself!\"\n\nNo answer came, and she started walking forward, her light growing. There were shallow pits in the black tile floor, and bones strewn off to the sides. She kicked a fragment of bone out of her path.\n\nYou are ensuring an eternity of suffering for yourself and those you love, the Underking threatened.\n\nThe voice seemed to be coming from up ahead. Kendra kept going. The darkness pressed more ferociously than ever, seeming to exert physical force against the sphere of brightness around her. Kendra had to lean forward to keep moving. Kendra pushed power into the crown and felt it feeding confidence back to her in return. After a moment of constricting, her sphere grew the largest it had been.\n\n\"You would know about an eternity of suffering,\" Kendra said boldly. \"You're living it. That's your best-case scenario. You need to do more than threaten. Where are you?\"\n\nNo answer came, but the darkness pressed frantically.\n\n\"Are you hiding from me?\" Kendra asked. \"Am I scaring you?\"\n\nVery well, the Underking answered coldly. You have been warned. Come to me, and we will finish this.\n\nThe darkness no longer pressed so hard against her light.\n\nFollow my voice, the Underking invited. Come and behold me.\n\nKendra felt a chill as she took another step. Being invited was much creepier than forcing her way forward. What if she was playing right into his hands? What if she transformed the instant she saw him? She felt tempted to close her eyes. But the Underking lived in darkness. He wanted to hide. Wouldn't closing her eyes help him? If she wanted to beat him, it would be with light. He might be encouraging her forward because all else had failed, and he hoped it would scare her. She had bet on confidence so far, and it had brought her to this point, including right through Stavius. If she was betting on confidence, she had to be brave. She kept walking.\n\n\"Where?\" Kendra asked.\n\nHere. Yes. Come meet your fate.\n\nHe sounded near. The front edge of her sphere would illuminate him at any moment.\n\nAnd then the darkness surged in as never before. Gasping, Kendra pushed back against it, but her sphere was shrinking fast. Soon it was scarcely lighting where she stood. Kendra resisted with all she had, barely holding the light steady.\n\nThe nova song sprang from her shoulder. With the sphere of light so small, there was enough slack on the chain for the fairy to cross into the darkness. Instantly the chain went limp, and the shriveled corpse of the fairy swung back into view, dangling lifelessly.\n\n\"No!\" Kendra cried, angry and horrified.\n\nYou're next, the Underking jeered.\n\nOutside the sphere of light, the nova song had instantly died. Kendra had to sustain her light. It was keeping her alive. And hopefully it could help her defeat the Underking. The chain dropped from her nervous fingers. Kendra took a step forward, walking over plain black tiles. One step. And then another.\n\nThe next step brought a skull into her circle of light. Browned with age, it was webbed with cracks and missing the jaw. Only two other bones rested beside it, also aged and discolored, an arm bone and a femur. A black crown rested on the brow of the skull. The crown reflected absolutely no light.\n\nSuddenly Kendra's sphere broadened to fill the room. It was a large chamber with many bones. But the bones before her were different somehow. They had presence. And the skull was wearing a crown. Kendra could hardly believe the implications.\n\n\"Is that you?\"\n\nLook into my eyes, the Underking invited.\n\n\"You don't have eyes,\" Kendra said. \"Just holes.\"\n\nYou're still alive, the Underking ranted. This is impossible.\n\n\"No wonder you prefer the dark,\" Kendra said.\n\nStop, the Underking demanded.\n\n\"No, you stop,\" Kendra said. \"You're just a broken old skull. Unless those bones are part of you?\"\n\nDo not gaze upon me, the Underking said.\n\n\"Too late,\" Kendra replied. \"Is that all you have left? A few corroded bones?\"\n\nI will never pardon this insolence, the Underking said.\n\n\"Why didn't you turn me into the undead?\" Kendra asked. \"Was that all just a lie?\"\n\nI have that power, the Underking said. But not while you are protected by the light.\n\n\"I get why you hide down here in the dark,\" Kendra said. \"This is where you pretend you still matter.\"\n\nI am appalled that you brought such resilient light into my sanctum, the Underking said. I am also impressed. I admire power. You have passed my trial and shown your strength. My crown is now yours.\n\nKendra looked at the crown. It looked like a void in the shape of a crown. The skull was pathetic. The crown was not.\n\nYour crown knows, the Underking whispered. Your crown recognizes the power it would wield united with mine. Light and shadow! You could rescue those you love. You could end the war with the dragons.\n\nKendra could sense the truth of his words. A vision came to her of a new infusion of power, like what she felt from the Fairy Queen's crown, but rich and dark, providing contrast. She could comprehend light and dark, life and death, above and below. Kings would kneel to her. Dragons would eat from her hand.\n\nThese opposing powers could only be united in a mortal, the Underking said. With these two crowns, you could go on to uncrown the other great monarchs. Nothing could escape your view. You would tower above all and penetrate the deep places. You would outlast this world. You would outlast the stars.\n\nKendra swayed, and she had to steady herself. This was no bluff. But she had not come here for a crown. She had come here to help Seth and Bracken. She had been warned to take only what the Underking had stolen.\n\n\"Why are you offering so much?\" Kendra asked.\n\nI need not offer, the Underking whispered to her mind. You have bested me. The crown is yours to claim.\n\n\"And I would keep it until I look like you?\" Kendra asked. \"No thanks.\"\n\nIt is your only hope, the Underking insisted.\n\n\"Enough of your lies,\" Kendra said. \"It would be an end to hope. I'm not interested. My fear is becoming like you. I'm not going to do that on purpose. I don't have to listen to you. I could completely shut you out of my mind.\"\n\nDo not close our dialog, the Underking conveyed.\n\n\"What's the point of sitting in a dark room as a skull?\" Kendra asked. \"Wouldn't you rather die than exist this way?\"\n\nNo, the Underking maintained.\n\n\"But you're not alive,\" Kendra said. \"Not in any of the ways that matter. There might be something better than this life. I think it's likely. Why choose to remain stuck here? Why not move on?\"\n\nYou do not understand, the Underking accused.\n\n\"I'm not going to wear your crown,\" Kendra said. \"But I might take it to the Fairy Realm. Maybe the Fairy Queen can figure out how to destroy it.\"\n\nMy crown is infused with pure will, the Underking explained. Will independent of life and death. Will cannot be taken or destroyed by another. It can only surrender. You cannot destroy my crown.\n\n\"Then I can take it to the Fairy Realm and lock it away,\" Kendra said.\n\nYou could, the Underking said, a smile in his voice. You would not be the first to attempt such a tactic. You may be resisting my crown now. Can you always? Can the Fairy Queen? Even when great need arises? What about her subjects? This was not always my domain. It once belonged to others. Any realm I inhabit eventually becomes the Under Realm. It only takes time. The Fairy Realm can become my new seat of power.\n\n\"But it would take time,\" Kendra said. \"Stopping you any amount of time is better than not stopping you at all.\"\n\nAnd there we have my problem, the Underking said. My crown would eventually corrupt the Fairy Queen or one of her minions. But it would take time, and I am comfortable here. I wonder if you have considered the chaos that would ensue were you to take my crown from this realm? No walls hold the undead here. They stay for me and for the darkness I provide. To remove my crown from here is to flood the world with darkness.\n\n\"I'll let the Fairy Queen figure out what to do with the crown,\" Kendra said.\n\nThat outcome is unacceptable to me, the Underking said. It is not worth the trouble. I prefer to remain where I am. I rarely make concessions, but today I am willing to bargain with you in exchange for your departure. What do you want?\n\n\"First tell me why you want me to claim your crown,\" Kendra said.\n\nWe are not merely this skull, the Underking said. We live on. We never end. There is surety in it. You could join us.\n\n\"We?\" Kendra asked.\n\nWe are every Underking. We endure. We increase. We never die.\n\n\"You want to add me,\" Kendra said. \"You want me to become part of you. It's all a horrible trick. You don't die because you're not really alive.\"\n\nYour life is a ridiculous spark, the Underking asserted. A random flash. But you could become so much more. You could see back before the beginning and on past the end. You could know all we know. And combined with the crown you now wear, there would be unimagined horizons to pursue.\n\n\"I like who I am,\" Kendra said. \"I don't want to be anything else.\"\n\nYou will soon be gone.\n\n\"I know I'm not permanent in this world,\" Kendra said. \"But guess what\u2014neither is the sun. I get to live my life. Someday death will free me and I will go someplace new. I don't want to get stuck here. What if this life is just the beginning?\"\n\nWhat if death means the end?\n\n\"I don't believe that,\" Kendra said. \"But I'd rather take that chance than become like you. Or like any of your servants. Being undead seems like a nightmare to me.\"\n\nWhat are your terms?\n\n\"I just want Bracken. And Seth. And the Sunset Pearl. Give me those things, and I'll leave you alone.\"\n\nThe skull remained as inert as ever. The Underking gave no response.\n\n\"You can keep your Under Realm,\" Kendra said. \"I don't want to stir up more trouble with you. I don't want to unleash the undead on the world. I'll leave, and I won't return unless you cross me again.\"\n\nMy identity cannot be known, the Underking finally expressed.\n\n\"I'll keep it a secret,\" Kendra said. \"I'll say we reached an arrangement. It isn't my business to reveal your identity. I can tell you think darkness is important. You can stay hidden.\"\n\nI cannot risk\u2014\n\n\"I give you permission to see my mind only to check if I am telling the truth,\" Kendra said. \"I will keep your secret. I mean it.\"\n\nInteresting. Yes, I see it is so.\n\n\"It will be the same as your darkness. The information will not spread. Give me what I want and I'll leave. And I'll leave you alone. If you leave me alone.\"\n\nYour light desecrates my sanctum, the Underking expressed. You have sufficient power to disturb my contemplations. I admit that you have placed me in a difficult situation. Existing for days without end includes learning certain lessons in practicality. What you ask is small compared to the relief of your absence. I grant your request.\n\n\"Thank you.\"\n\nYour brother is not here, the Underking conveyed. He has gone on an errand for me.\n\n\"Seth told me he is bound to you,\" Kendra said.\n\nTrue.\n\n\"Free him,\" Kendra said. \"And if he comes back, make sure he knows he is free, and send him away.\"\n\nDone, the Underking pronounced. What else?\n\n\"I need Bracken and the Sunset Pearl,\" Kendra said. \"And a way back to Timbuli.\"\n\nI will indulge your request, the Underking asserted. An assistant will take you to Bracken. He will be freed. The Sunset Pearl will be brought to you. None of my subjects will interfere with your departure. You will be escorted to my clever boat. It will take you to Timbuli.\n\n\"Bracken needs to leave with all of his possessions,\" Kendra said. \"Including his horn, if he had it.\"\n\nAgreed, the Underking confirmed. You have disturbed our environment long enough. You must promise to cause no harm to my subjects, to depart in peace, and never to return.\n\n\"As long as you don't cross me,\" Kendra said.\n\nAway with you, the Underking insisted. Your light is tiresome.\n\n\"You killed the nova song,\" Kendra accused, frustrated and sad.\n\nLight has no place here, the Underking warned.\n\n\"Can I take her body?\" Kendra asked, fighting the sting of tears. \"I don't want her to become something unnatural.\"\n\nFairies cannot join the undead, the Underking expressed. Leave her. She sought my darkness, and here she will remain. Depart.\n\n\"Keep your bargain,\" Kendra said.\n\nI hold to my vows.\n\nKendra turned and walked from the room.\n\nThe Sphinx was gone when Kendra exited the sanctum of the Underking. The door of skulls reappeared after she exited. Without Mizarine, Kendra was alone for the first time since embarking on this adventure.\n\nShe had survived her encounter with the Underking, and he had agreed to deliver what she had come for, but she would not be at peace until she was back on Timbuli with Bracken and the Sunset Pearl. She was disappointed by Seth's absence. Hopefully, wherever he was, he somehow knew he was free. Was there a chance he would come find her?\n\nA robed figure glided into view, feet not quite touching the floor. She had long, dark tresses and was slightly translucent, leaving her skeleton somewhat visible beneath her skin. \"Kendra Sorenson,\" she said.\n\nKendra found it unsettling that the lich knew her first and last name. \"Are you a lich?\"\n\n\"And your guide to the prison,\" the lich said. \"Follow me.\"\n\nKendra considered asking for her name, then decided she didn't care to know. She wondered how much the brightness of her crown mattered, since the Underking had promised that his subjects would not harm her. She put no extra effort into increasing the output, but even so, it glared more on the scale of a lighthouse than a lantern.\n\nThere was no keeping track of all the twists and turns, but Kendra noticed that overall their route continued to descend. She saw no undead besides the female lich until they found a new lich waiting for them near the top of a stairway. He was dressed like a Viking, with a horned helm, leather armor, and a braided beard. He refrained from speaking but set down a sack before walking away.\n\n\"You will find the possessions you requested in the bag,\" the female lich said.\n\nKendra opened the large sack and was surprised to discover two sizable unicorn horns, one slightly longer and thicker than the other. She had Bracken's first horn, so these had to be the second and the third. But wasn't his third horn part of the Font of Immortality? Not anymore, apparently.\n\nIn addition to the horns, at the bottom of the sack she found a pearl the size of a softball with a silky, variegated sheen. In the bright luminance emanating from Kendra, she could imagine numerous little sunsets reflected on its surface.\n\n\"Looks like it's all here,\" Kendra said.\n\nThey continued until they reached a cylindrical iron cage with a cat inside. Another lich awaited them\u2014a man wearing a white tunic and a chain-mail hauberk.\n\n\"I am Ezabar,\" he said. \"Keeper of this prison. We seldom receive guests, and we almost never release prisoners. Must you shine so brightly?\"\n\n\"It's who I am,\" Kendra said.\n\n\"Calumbra will await you here,\" Ezabar said. \"This way.\"\n\nThe female lich remained behind as Kendra followed Ezabar into a subterranean junkyard of cages. Spaced haphazardly across the rocky terrain, only some contained prisoners. The occupants looked miserable. All flinched away from Kendra and her light. She searched for Bracken but did not see him.\n\nAs she followed Ezabar past numerous cages of varied shape and size, she wished she had asked the Underking to release all of his prisoners rather than only the people she knew. It just hadn't occurred to her. And she was getting weary. So much time in the dark was making it harder to stay bright.\n\nAs she rounded a boulder, Bracken came into view inside a circular cage with a conical top. Unlike the other prisoners, he did not turn away as she approached. Despite his ragged surroundings, he looked healthy and fairly clean and didn't even squint. He seemed at home in her light. Leaving Ezabar behind, Kendra ran to the cage.\n\n\"Please let this be real,\" Bracken said.\n\n\"It's me,\" Kendra said. \"I have permission to let you out.\"\n\n\"What are you wearing?\" Bracken asked. \"Don't answer that. I'm fully aware, just astonished. Will you take my hand and give me permission to see your thoughts? This seems too good to be true.\"\n\n\"Sure,\" Kendra said. \"But you can't look at my negotiation with the Underking. It was part of the deal.\" Kendra held out her hand, and Bracken took it. The contact made him light up, taking on some of her shine.\n\n\"I don't believe it,\" Bracken said. \"How did you manage this?\"\n\n\"I can't say too much,\" Kendra said. \"But I was destroying liches with light, and they couldn't stop me.\"\n\nEzabar arrived at the cage with a key, unlocked the door, and opened it. He stepped aside.\n\n\"What's in the bag?\" Bracken asked as he exited.\n\n\"None of your business,\" said a voice from behind them. Kendra turned. It was Ronodin.\n\n\"We got these from the Underking,\" Kendra said. \"We have his permission to leave.\"\n\n\"Maybe,\" Ronodin said. \"But you don't have mine. I wish I could say I like your little hat, Kendra, but it brings back unfortunate memories.\"\n\n\"The Underking told his subjects to stand down and let us go,\" Kendra said.\n\n\"Fortunately, I am not his subject,\" Ronodin said. \"Those horns belong to me.\"\n\n\"Where is Seth?\" Kendra asked.\n\nRonodin smiled. \"Fulfilling his most treacherous mission thus far. I confess I left him behind when I got word the Under Realm had been invaded.\"\n\n\"Where?\" Kendra demanded, her brightness intensifying.\n\n\"Don't shine at me,\" Ronodin said. \"I may not like the light, but I also have no fear of it.\"\n\n\"I know what's in the bag,\" Bracken said. \"And I know what you meant to do.\"\n\n\"I had your mature horns,\" Ronodin said. \"I only lacked the first. Now they're all here.\"\n\n\"You should run away,\" Bracken said. He took the bag from Kendra and pulled out a horn.\n\n\"I came prepared,\" Ronodin said, raising a black horn into view.\n\n\"What?\" Bracken cried. \"Your horns were lost.\"\n\n\"I discarded them after they were corrupted,\" Ronodin said. \"I wanted nothing to do with my heritage. It was my biggest mistake. Others used them as power sources. It was messy work getting them back.\"\n\n\"All of them?\" Bracken asked.\n\n\"All three,\" Ronodin said with a grin. \"The Underking helped with two, and Celebrant assisted with the third. I have already paid in full.\"\n\n\"What was the payment?\" Kendra asked.\n\n\"You have no right to know,\" Ronodin said. \"A smart girl would guess.\"\n\n\"The Sunset Pearl?\" Kendra asked.\n\n\"That was one payment for the Underking,\" Ronodin said. \"All right. I'm bored. The other was releasing the undead from the Blackwell. Your brother performed perfectly. And the payment for Celebrant was the information about the Wizenstone.\"\n\n\"What about causing the fall of Blackwell Keep?\" Kendra asked.\n\nRonodin nodded. \"He still owes me for that one. It dovetailed so nicely with releasing the denizens of the Blackwell, I wanted to make sure I got credit for the consequences.\"\n\n\"You're a monster,\" Kendra said.\n\n\"I'm much more than that,\" Ronodin said.\n\n\"You don't want to fight me,\" Bracken warned.\n\n\"I really do,\" Ronodin said. \"Desperately. A duel might surpass what I had planned.\"\n\n\"He wanted to find out if he could taint me by corrupting my horns,\" Bracken said.\n\n\"Once I had all three of mine it might have worked,\" Ronodin said. \"If I had all three of yours. But this is better. More manly.\"\n\n\"Another time,\" Bracken said. \"Another place. Name it.\"\n\n\"Here,\" Ronodin demanded with more anger than Kendra had ever heard from him. \"Now. What's the matter? Worried Kendra could get hurt?\"\n\n\"I don't want to destroy you,\" Bracken said.\n\n\"Really?\" Ronodin asked. \"You could have fooled me.\"\n\n\"I discovered you were working with the demons,\" Bracken said. \"But you committed the crimes.\"\n\n\"You share the same narrow view as all our kind,\" Ronodin accused.\n\n\"Because of you, my father fell prey to the demons,\" Bracken said. \"He was trapped in Zzyzx for an unthinkable amount of time. Even after all that, unlike you, he never went dark.\"\n\n\"If he had, maybe he wouldn't have stayed chained up for so long,\" Ronodin said. \"En garde, Bracken.\"\n\nRonodin raised his horn, and in a swirling cloud of dark vapors, a transformation took place. In his place stood a powerful stallion, dark grey, with a fiery mane and tail. The orange horn glowed as if superheated. The dark unicorn immediately charged.\n\n\"He has learned to take his old shape,\" Bracken said, raising his horn. In a burst of light, Bracken was no longer a man, but a pure white unicorn. Muscles clenched as he bounded forward to meet Ronodin's charge.\n\nHorns lowered, the two unicorns converged as if jousting. Their horns clashed with a blazing flash, and the unicorns brushed past each other.\n\nKendra raced around to the far side of Bracken's cage, keeping it between herself and Ronodin. The dark unicorn showed no special interest in her, instead wheeling about to charge Bracken again. The white unicorn was doing the same.\n\nKendra tried to think of a way to help, but for the moment it seemed like staying out of the way might be the best idea. The unicorns galloped at each other again, their horns clashing brilliantly. After one pass, Bracked crashed against a tall, rusty cage, toppling it. After another, Ronodin glanced off a stone column and nearly fell.\n\nAs the fight progressed, they charged less and got in closer. Kendra soon realized the unicorns were using their horns less like lances and more like swords. Along with thrusting, they also slashed and parried. Both were giving as much attention to defending against the opponent's horn as to attacking.\n\nThe horns flashed each time they connected, highlighting the surreal landscape of cages with dazzling strobes. The unicorns circled in close, horns crossing and deflecting as each tried to gouge the other. Both unicorns reared, front hooves flailing, and Kendra saw a wound open on Bracken's shoulder, spilling silver blood. A new gash on Ronodin's neck bled reddish gold.\n\nYield, Ronodin commanded. This is my day.\n\nYou're flagging, Bracken replied. Not accustomed to this form, are you?\n\nYou haven't taken this shape in ages, Ronodin taunted.\n\nYou tarnished yourself, Bracken accused, lashing out with a strike that Ronodin deflected. Your horns are abominations.\n\nRonodin dove at Bracken with a lunging counterstrike. Yours are for show. They are the ornaments of mindless conformity.\n\nThey crossed horns, directly testing their strength. Neither unicorn moved to break the deadlock. Muscles straining, hooves shuffling, each strained to overpower the other.\n\nYou're weak, Ronodin mocked.\n\nYour coloring is absurd, Bracken countered. Gray and orange?\n\nNot as original as pearly white.\n\nYou look like a logo for a band.\n\nI could do this all day.\n\nHow about you push me over?\n\nKendra started sneaking toward Bracken from behind the cage. The horns quivered with the pressure of mutual exertion. She knew they could break the pose at any moment, so she rushed forward and rested a hand against Bracken's side and pushed all the energy she could muster into him.\n\nThe light of her crown intensified, Bracken's hair glowed, and his horn glared a blinding white. With a loud crack, the deadlock broke as Ronodin's horn snapped off.\n\nBack to human form, Ronodin stumbled and fell onto his side. Bracken touched his horn to the fallen one, and it turned an unblemished white.\n\n\"You'll pay for that,\" Ronodin muttered, upending a vial into his mouth.\n\nBreaking contact with Kendra, Bracken bounded toward his cousin, but Ronodin flew away in a flurry of black feathers. He had become an enormous raven. Without a backward glance, Ronodin flapped out of sight.\n\nLowering his head, Bracken shed his horn and resumed human shape. Blood stained his shirt at the shoulder.\n\n\"You're hurt,\" Kendra said.\n\n\"It's nothing,\" Bracken replied. \"I heal fast, and nothing gets infected. If I'm alive, I'll be fine.\"\n\n\"What happened with his horn?\"\n\nBracken put an arm around Kendra. \"You happened. That boost of energy you provided let me rip it from him. The horn really was flawed. Once it was severed from him, I was able to purify it.\"\n\nKendra looked at the white horn on the ground. \"His third horn is healed.\"\n\n\"As if it was never corrupted,\" Bracken said.\n\n\"Could that influence Ronodin?\" Kendra asked.\n\n\"He won't be able to transform into his unicorn shape anymore,\" Bracken said. \"The horns are connected to our identity, but they do not define who we are. I don't think I could heal him by purifying his horns any more than he could darken me by corrupting mine.\"\n\n\"His plan wouldn't have worked?\" Kendra asked.\n\n\"It would have weakened me,\" Bracken said. \"Made me less powerful. But my identity belongs to me. Only I can surrender that.\"\n\nKendra could not help thinking of her brother. \"Do you think Ronodin will go to Seth?\"\n\n\"Maybe,\" Bracken said. \"If Seth can be of use to him.\"\n\n\"That potion transformed him into a bird?\" Kendra asked.\n\n\"That magic was unfamiliar to me,\" Bracken said. \"But Ronodin deals with many whose company I find distasteful.\"\n\nEzabar approached them, his feet just a little above the ground. \"It appears your altercation with Ronodin has ended,\" he said.\n\n\"Yes,\" Bracken said. \"I'm tired of marinating in darkness. No matter your strength, it becomes exhausting.\"\n\n\"It's draining,\" Kendra agreed. \"Ezabar, get us out of here.\"\n\nWhen the ghostly manacle fell off with a sound like shattered glass, Seth was seated on a rock in the jungle near the mouth of a cave. \"Did you hear that?\" Seth asked.\n\n\"Hear what?\" Calvin replied.\n\n\"Like something breaking?\"\n\n\"No.\"\n\nSeth rubbed his wrist. The spectral manacle that had adorned his wrist since he had struck the deal with the Underking was gone.\n\n\"I don't get it,\" Seth said. \"I didn't deliver the Everbloom.\"\n\n\"What happened?\" Calvin asked.\n\n\"I think I'm free,\" Seth said. \"The ghost chain attached to my wrist disappeared.\"\n\n\"Could Dezia have done it?\" Calvin asked.\n\n\"I don't think so,\" Seth said. \"The Underking put it there.\"\n\nFollowing the phantom had allowed Seth to exit the caves without any major problems. As promised, she had taken him to a different opening from the one he had used to enter. After a final expression of gratitude, the phantom had departed.\n\n\"Maybe touching the Everbloom weakened it,\" Calvin theorized.\n\n\"Possibly,\" Seth said. \"That flower was incredibly powerful. I still feel energized.\"\n\n\"What now?\" Calvin asked.\n\n\"That's the question,\" Seth said. \"There's no reason to go back to the Phantom Isle. Eventually Ronodin will catch on that we're not returning and come looking for us.\"\n\n\"Do we go find Kendra?\" Calvin asked. \"I think I can lead us to the tree house.\"\n\n\"I'm not sure,\" Seth said.\n\n\"Why not? The tree house is still protected. They can help keep you safe from Ronodin and the Underking.\"\n\n\"I see why you like the option,\" Seth said. \"I only really want two things, though. I want some time alone, without people telling me who I'm supposed to be. And I want my memories back.\"\n\n\"The tree house won't be good for time alone,\" Calvin said.\n\n\"And what if Ronodin had a point?\" Seth asked. \"What if they want me locked up? Even if they try to do it humanely. What if they don't trust me enough to let me track down Humbuggle?\"\n\n\"You're thinking of going there on your own,\" Calvin surmised.\n\n\"Not completely on my own,\" Seth said. \"With my Tiny Hero.\"\n\n\"I'll admit we're a formidable team,\" Calvin said.\n\n\"I don't want other people to decide who I am,\" Seth said. \"I want to decide that for myself.\"\n\n\"You deserve to know your past,\" Calvin said.\n\n\"And after I know, I get to choose what comes next.\"\n\n\"How will we get there?\"\n\n\"What do you know about leviathans?\" Seth asked.\n\n\"Not a lot besides they're humongous,\" Calvin said.\n\n\"Some people carry snacks in their gear,\" Seth said. \"I carry humongous leviathans. If Kendra is right, I just put a little statue in the water and say some magic words.\"\n\n\"You didn't happen to trade a cow for this statue?\" Calvin asked.\n\n\"I'm not sure where I got it,\" Seth said.\n\n\"Let's say it works,\" Calvin said. \"We ride it like a horse? Will it obey you? What if it just swims off?\"\n\n\"We might end up needing a new plan,\" Seth admitted, rummaging in his satchel. He pulled out the little statue fashioned out of blue stone. \"But why not try this first?\"\n\n\"It looks like a whale,\" Calvin said.\n\n\"We'd definitely get style points,\" Seth said.\n\n\"I've never traveled by whale,\" Calvin said. \"Why not try?\"\n\n\"Think we can get to a beach without running into Ronodin?\" Seth asked. \"Or getting eaten by giant mushrooms?\"\n\n\"Worth a try,\" Calvin said.\n\nSeth stood up and followed the path leading away from the cave. It intersected another path, and Seth turned downhill. Before long they reached an overlook above a beach.\n\n\"This is the same beach where I arrived with Ronodin,\" Seth said.\n\n\"Are you sure?\" Calvin asked. \"I was mostly keeping my head down.\"\n\n\"I was just here,\" Seth said. \"See that rock shaped kind of like a mitten? And the palm trees on that little peninsula? I can even still see where the boat was on the sand.\"\n\n\"No boat?\"\n\n\"No boat.\"\n\nSeth stared up and down the coast and out to sea. The clever boat was nowhere in sight.\n\n\"What's Ronodin up to?\" Calvin asked.\n\n\"He might have hidden the boat to lure me down to the beach,\" Seth said. \"Make it look safe. But why would he expect me to go to the beach? If I was escaping with the Everbloom without a boat, I'd take off through the jungle. Maybe he left. But why would he leave without the Everbloom?\"\n\n\"You freed the phantom from her old token,\" Calvin said. \"Ronodin might have gotten word and figured you betrayed him.\"\n\n\"Wouldn't that make him hunt me down?\" Seth asked. \"The boat is gone. He might have really left.\"\n\n\"Maybe there was an emergency,\" Calvin guessed.\n\n\"I don't need the beach,\" Seth said. \"I want the tip of the peninsula.\"\n\n\"The beach might be too shallow for a humongous whale,\" Calvin said.\n\n\"We'll go carefully,\" Seth said. \"If Ronodin left, he may come back. If we can swing it, I want to be gone before he does.\"\n\nWith Calvin in his pocket, Seth continued down the trail to the beach. He walked quietly, watching and listening, but saw no sign of trouble. The beach lacked much cover, so once he reached it, Seth ran to the peninsula and out to the tip.\n\n\"Still no sign of Ronodin,\" Calvin said. \"There's a yellow fairy over by those rocks.\"\n\n\"I'll take her over a wild boar,\" Seth said. Looking out at the swells, he passed the stone whale back and forth between his hands. \"I'm supposed to toss this in the water and say the magic words.\" He fished the paper with the words out of his satchel. \"Do you think this could possibly work?\"\n\n\"I think we should find out,\" Calvin said.\n\nSeth threw the little stone whale out into the water. Then he read the words.\n\nWater exploded into the air as a gigantic whale expanded into existence. Seth ducked his head as water showered down, then marveled at the size of the beast before him. The water was clearly too shallow, because all but the belly was above it. The titanic creature had to be at least a hundred yards long.\n\nWhat is your request?\n\nThe words, ancient and slow, came into Seth's mind, surprising him. The leviathan did not seem distressed to be in shallow water. One eye the size of a hula hoop regarded him patiently.\n\n\"I can make requests?\" Seth asked.\n\nYou brought me back. I am yours to command.\n\nSeth laughed uncomfortably. \"Do you give rides?\"\n\nWhere would you care to go?\n\n\"New Zealand? As close to the Titan Valley dragon sanctuary as you can get me?\"\n\nMaking the beach rumble, the leviathan swiveled until its mouth was even with the tip of the peninsula. The mouth opened, and Seth felt like he was staring into a huge, fleshy cave fringed by teeth larger than traffic cones.\n\n\"Inside?\" Seth asked. \"Will I be able to breathe?\"\n\nThe forward void within me remains full of air. Do not proceed deeper than that.\n\n\"Should we go?\" Seth murmured to Calvin.\n\n\"Isn't this exactly what you wanted?\" Calvin asked.\n\n\"I pictured us riding up top,\" Seth said. \"But I guess this way we won't fall off.\"\n\n\"Should we leave a message for Kendra?\" Calvin asked.\n\n\"Ronodin will find it,\" Seth said.\n\n\"I suspect Ronodin already knows where you're going anyway,\" Calvin said. \"But my friends will be worried. They don't know where I am. Your sister could come in handy if we need help.\"\n\n\"Is there a good way to tell her?\" Seth asked.\n\n\"Let me think.\"\n\nCome. I desire deeper water.\n\n\"I'm coming,\" Seth said.\n\n\"Write it in the sand with a stick,\" Calvin said. \"At least it gives them a chance.\"\n\nSeth picked up a stick and made big block letters where the sand was a little wet:\n\nCALVIN LIVES\n\nTITAN VALLEY\n\n\"Better than nothing,\" Calvin said.\n\nCome, the leviathan insisted.\n\n\"Okay,\" Seth returned, stepping into the vast mouth. The whale kept still as Seth walked back through the moist, spongy surface of the throat into a large room with curved walls that moved as the leviathan breathed. The water in the room came up to Seth's knees. It smelled like a mix of wet blubber and old fish.\n\nIn the room there were a pair of battered rowboats, a desk, a table, and several crates and barrels. A squat, ugly creature sat in one of the rowboats. It had greenish skin, an oversized head, pointed ears, and a wide, lipless mouth.\n\n\"No!\" the creature cried. \"Not visitors.\"\n\nThe leviathan's mouth closed. A lantern glowed in each rowboat, so the interior of the leviathan did not become dark.\n\n\"This is my leviathan,\" Seth said.\n\nThe creature laughed. \"You no own leviathan. I be here when you dead.\"\n\n\"It's a hermit troll,\" Calvin whispered. \"They're squatters. They move from home to home.\"\n\n\"I've heard of them,\" Seth replied. He raised his voice to address the troll. \"We're here until we get to New Zealand. Can we share the leviathan for now?\"\n\nThe leviathan lurched, and Seth fell to his hands and knees in the slimy water. Everything kept shaking and jerking. He decided to stay down while the leviathan flopped toward deeper water.\n\nAre you at ease? came the leviathan's voice in his mind.\n\n\"I'm okay,\" Seth said. \"Just talking to a hermit troll.\"\n\n\"Me no hermit troll. Me fishy king.\"\n\nThe stores are meant to be shared. Let me know if you need a break on land.\n\n\"Those barrels have food?\" Seth asked. \"Thanks!\"\n\n\"Who you talk to?\" the hermit troll asked.\n\n\"The leviathan.\"\n\n\"You crazy.\"\n\n\"The leviathan said we have to share the stores.\"\n\nThe hermit troll shook his head emphatically. \"You get own stores.\"\n\nSeth stared. \"I'm pretty sure I could have the leviathan spit you out.\"\n\n\"We share,\" the hermit troll replied quickly.\n\n\"I'm Seth.\"\n\n\"Me Hermo.\"\n\n\"How long have you been in here, Hermo?\" Seth asked.\n\n\"Big long time.\"\n\n\"He was probably in stasis while the leviathan was a statue,\" Calvin said.\n\n\"Who that?\" Hermo asked.\n\n\"My secret friend,\" Seth said. \"We might as well be friends too, since we're sharing this space.\"\n\n\"No friend unless play game,\" Hermo said.\n\n\"What games do you like?\" Seth asked.\n\n\"You know Old Maid?\"\n\n\"Yeah. Do you have cards?\"\n\n\"No.\"\n\n\"Do you have any games?\"\n\n\"Mancala.\" The troll rummaged around in the boat, then held up a board full of divots and stones.\n\n\"I don't know how to play,\" Seth said, \"but I'll give it a try.\"\n\n\"I teach. You learn.\"\n\n\"Sure.\"\n\n\"Teacher always win.\"\n\nSeth sighed. \"I have a feeling this is going to be a long ride.\"\n\nWalking along the path to the fairy shrine was different under the light of day, with Bracken at her side. The cyclopses had replaced the Sunset Pearl the night before, which meant this path was perfectly protected all the way to the shrine. And with the Fairy Queen's son at her side, Kendra didn't worry about problems at the shrine, either.\n\n\"How is your shoulder?\" Kendra asked.\n\nBracken pulled back his loose shirt far enough for her to see the slightly raised scar.\n\n\"It's already a scar?\" Kendra asked. \"No scab?\"\n\n\"It will be faint by tomorrow morning,\" Bracken said. \"Gone by the afternoon.\"\n\nThe Underking's boat had left them at Crescent Lagoon yesterday evening, then had departed unmanned. Warren and Knox had complained the most about Kendra going to the Phantom Isle without them, but since she had returned with Bracken and the Sunset Pearl, they were left with little room for chastisement. The cyclopses had thanked her profusely and departed immediately to replace the pearl on the altar. The party at the tree house had lasted long into the night, but Kendra couldn't stop worrying about the unsolved mystery of Seth.\n\n\"Thinking about him again?\" Bracken asked.\n\n\"I don't know how to think about much else,\" Kendra said. \"It was the same with you.\"\n\n\"The cyclopses sensed the appearance of a leviathan here at Timbuli,\" Bracken said. \"That had to be Seth's doing.\"\n\n\"I'm sure,\" Kendra said. \"But where did he go?\"\n\n\"We'll find him,\" Bracken said.\n\n\"If he was on this island, why didn't he come find me?\"\n\n\"He was confused when I saw him,\" Bracken said. \"But he was still Seth. He'll figure it out. If he doesn't, we'll help him.\"\n\nKendra held up the crown. \"Think your mom will be happy to see this?\"\n\n\"I still can't believe she gave it to you,\" Bracken said. \"I don't even think it was the right choice.\"\n\n\"You thought I would blow it?\"\n\n\"No,\" Bracken said. \"But it was not a risk she should have taken for me or anyone.\"\n\n\"Even though it worked?\"\n\n\"Of course it seems brilliant now,\" Bracken said. \"But it embarrasses me that my mother took such a risk on my behalf. She endangered our entire realm.\"\n\n\"It wasn't just for you,\" Kendra said. \"We're fighting a war against the dragons. Celebrant is free now. Who knows what he is doing? She didn't want Crescent Lagoon to fall.\"\n\n\"I know her,\" Bracken said. \"It was mostly about me. Kendra, I can still hardly believe you came for me. I barely know what to say. I owe you my life.\"\n\nKendra smiled. \"Anytime.\"\n\nWhen they reached the pool with the tiki statues, Kendra was surprised to find Baroi waiting. He raised a hand in greeting. \"Hail, Kendra. Hail, Bracken.\"\n\n\"Hi, Baroi,\" Kendra said. \"I thought you three never left the Grand Moai.\"\n\n\"We are adjusting that stance,\" Baroi said. \"Our presence did not prevent the theft of the Sunset Pearl last time. We want to take more complete care of Timbuli.\"\n\n\"Sounds good to me,\" Kendra said.\n\n\"I came with a request for Bracken,\" Baroi said.\n\n\"What do you need?\"\n\n\"You are a unicorn.\"\n\n\"Yes,\" Bracken said.\n\n\"Ronodin corrupted one of the island's sacred pools,\" Baroi said. \"We hope you will purify it.\"\n\n\"That would be my honor,\" Bracken said.\n\n\"Thank you,\" Baroi said. \"And Kendra, we have good news for you.\"\n\n\"Tell me,\" Kendra said.\n\n\"The pearl augments our sight,\" Baroi said. \"We reached out to see if anyone had knowledge of whether Seth may have left with the leviathan. His departure is confirmed.\"\n\n\"That's a relief,\" Kendra said.\n\n\"A fairy witnessed the events,\" Baroi said. \"Seth had a tiny person in his pocket who matches the description of your missing nipsie.\"\n\nKendra bumped the heel of her hand against her forehead. \"Of course. Calvin must have gone to Seth when he was stuck to the mushroom! Why didn't I see that?\"\n\n\"The fairy also had a sense for where they might be going,\" Baroi said.\n\n\"Really?\" Kendra asked.\n\n\"Seth wrote four words in the sand before entering the leviathan,\" Baroi explained.\n\n\"Entering?\" Kendra asked.\n\n\"Leviathans are enormous,\" Bracken said. \"If a leviathan is friendly, large groups can travel comfortably inside of it.\"\n\n\"What were the words?\" Kendra asked.\n\n\"Calvin lives,\" Baroi said. \"Titan Valley.\"\n\n\"The dragon sanctuary?\" Kendra asked.\n\n\"Perhaps,\" Baroi said. \"Only those words were provided.\"\n\n\"Titan Valley is one of the big three,\" Bracken said. \"Like Wyrmroost, it has a Dragon Temple.\"\n\n\"I don't remember where it is,\" Kendra said.\n\n\"New Zealand,\" Bracken said. \"It is probably the most secure of all the dragon sanctuaries. Titan Valley has an extremely unconventional caretaker.\"\n\n\"Who?\" Kendra asked.\n\n\"The Giant Queen,\" Bracken said.\n\n\"She's one of the five monarchs,\" Kendra said.\n\n\"We cyclopses acknowledge her as our sovereign,\" Baroi said.\n\n\"And she is in charge?\" Kendra asked.\n\n\"It is the largest of the preserves,\" Bracken said, \"serving as a dragon sanctuary and as the homeland of the giants.\"\n\n\"That settles it,\" Kendra said. \"Let's get this crown back to your mother. Then we need to figure out how to beat a leviathan to New Zealand.\"\n\n\"That shouldn't be hard,\" Bracken said with a wink. \"Titan Valley has a fairy shrine.\"\n\n\"Another adventure,\" Kendra said, steadying herself. \"I can do this. I can't rest until Seth is safe.\"\n\n\"None of us will get much rest until we end this war with the dragons,\" Bracken said.\n\nKendra shuddered. It was daunting to consider what could be happening out in the world, beyond the boundaries of Crescent Lagoon. What progress might Celebrant be making in his war? \"We should consult with Andromadus before we go,\" Kendra said. \"And a member of Dragonwatch.\"\n\n\"We'll do all that,\" Bracken said. \"But it will take the leviathan time to reach New Zealand. You've been through so much. You've won some difficult victories. Crescent Lagoon is a fully functioning sanctuary, thanks to you. Seth is free from the Underking, and so am I. Today, we get to relax.\"\n\n\"Speak for yourself,\" Kendra said. \"I get nervous around your mom.\"\n\n\"Look what you're holding,\" Bracken said. \"I think you can expect the royal treatment.\"\nAcknowledgments\n\nI will never master writing a novel. Every new story presents unique problems to solve. This is my nineteenth novel, and, as usual, I faced several challenges I hadn't encountered, including trying to inhabit the point of view of a character who has lost his memories. With two books left to write, I have a grand plan for Dragonwatch, and discovering how to best tell this tale will be an adventure. I am grateful to those who helped me improve this book.\n\nMy agent, Simon Lipskar, lent keen insight, and the brilliant editors Chris Schoebinger, Liesa Abrams Mignogna, and Emily Watts helped with much of the heavy lifting. I had intensive assistance from Erlyn Madsen, who gave me the most thorough and useful editing I ever received from a friend. I also got great ideas and reactions from Cherie Mull, Jason and Natalie Conforto, and Pamela Mull. Even though he is battling cancer, my uncle Tucker Davis offered lots of thoughtful feedback. I failed to mention in the previous book that Tuck made up the Wizenstone when my siblings and I were kids. It was kind of him to let me use the name in the story.\n\nI owe a debt to Brandon Dorman for once again creating outstanding illustrations for the cover and interior, and to art director Richard Erickson from Shadow Mountain for working to get the look of everything just right. I also owe thanks to many others at Shadow Mountain for helping produce and market the book, including Ilise Levine, Rachael Ward, Troy Butcher, Callie Hansen, and Ali Nelson, and to the good team at Simon and Schuster, including Mara Anastas, Cassie Malmo, Caitlin Sweeny, Michelle Leo, Elizabeth Mims, and Alissa Nigro.\n\nI also need to thank my assistant Rene Lindsey for her invaluable support, and my kids, who are understanding when Daddy has deadlines and events. My friends and family make me a fortunate man.\n\nAnd, of course, I must thank you, the reader. When you purchase one of my books or recommend my stories to friends, teachers, and librarians, you help keep me employed and increase the chances for sequels to be written and for movies to be made. Also, when I meet you at events, it increases my motivation to share my stories. Thank you for your enthusiasm.\n\nConcurrent with Master of the Phantom Isle, I am releasing a picture book called Smarter Than a Monster: A Survival Guide for Young People, illustrated by the incredibly talented Mike Walton. The next novel I release will be Dragonwatch Book 4. If you are enjoying this series, I hope you already know that it is a sequel series to the Fablehaven books. Some of my other series would probably appeal to you as well, like Five Kingdoms and Candy Shop War.\n\nI'm happy to have completed three of the five Dragonwatch books. I've already started work on the fourth, and I can't wait to share the rest of the story!\nReading Guide\n\n1. At the start of this book, Seth has lost his identity. If you knew you would lose your identity in a week, how might you try to prepare?\n\n2. In this story, magical barrels are used to transport people instantly from one place to another. If you had a pair of magical barrels that could transport you back and forth between two locations, where would you place them? Why?\n\n3. Tess received protection from magical creatures with the cloak of innocence. Name some ways innocence can be a protection in real life.\n\n4. Why do you think Agad chose to surrender to Celebrant? Why did Kendra surrender? What else might they have done? Could any other option have yielded better results? Why or why not?\n\n5. Andromadus taught Tess and Eve that mortals could cause magical creatures to change. What examples from this series or the Fablehaven books show magical creatures changing because they interacted with mortals? Have you ever been changed because you interacted with somebody else? Share an example.\n\n6. After the Sunset Pearl was lost, the moai could be awakened by people offering the material possession they most treasured. What might you offer to revive a moai? Why do you treasure it?\n\n7. The cyclopses were known to glimpse the future. If you could have them look into your future and see only one of three options\u2014whom you will marry, how you will die, or who will win the next Super Bowl\u2014which would you choose? Why?\n\n8. What pressures kept Seth working with Ronodin? Why did Seth decide to rebel against the orders of the Underking? Could he have reached that conclusion earlier? If so, when?\n\n9. If you had the Elixir of Dry Depths, where would you explore? What would be the dangers?\n\n10. Why did Knox almost run away after he woke up in the razor coral? Why did he help Kendra instead? What would you have done in his circumstances? Explain.\n\n11. Kendra used light to fight the Underking. Why do you think the Underking preferred to remain in darkness? What would be the advantages of wearing the Underking's crown? What would be the disadvantages?\n\n12. Why do you think Seth decided to set off alone (with Calvin) at the end of the story? What else could he have done?\n\n13. Is this question unlucky to answer? Why or why not? Try this game of luck: Roll a die. Even number, your light beat the Underking. Odd number, the darkness overcame you. Now, try it five times in a row. How lucky or unlucky were you?\nAbout the Author\n\nBrandon Mull is the #1 New York Times best-selling author of the Fablehaven, Beyonders, and Five Kingdoms series. A kinetic thinker, Brandon enjoys bouncy balls, squeezable stress toys, and popping bubble wrap. He lives in Utah in a happy little valley near the mouth of a canyon with his four children and a dog named Buffy the Vampire Slayer. Brandon loves meeting his readers and hearing about their experiences with his books.\n\n## Contents\n\n  1. Under\n  2. Restless\n  3. Hero\n  4. Innocent\n  5. Jibarro\n  6. Defenses\n  7. Mortals and Magic\n  8. Threat\n  9. Liberation\n  10. Falling\n  11. Execution\n  12. Questions\n  13. Giant\n  14. Prisoner\n  15. Old Friends\n  16. Mission\n  17. Tree House\n  18. Landfall\n  19. Crypt\n  20. Triclops\n  21. Cyclopses\n  22. Heartstone\n  23. Shipwreck Key\n  24. Desperation Beach\n  25. Negotiations\n  26. Dry Depths\n  27. Merfolk\n  28. Remulon\n  29. Sage\n  30. Alone\n  31. Assignment\n  32. Wishing\n  33. Phantom Isle\n  34. Everbloom\n  35. Underking\n  36. Unicorns\n  37. Voyage\n  38. Victories\n  39. Acknowledgments\n  40. Reading Guide\n  41. About the Author\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# Think DSP\n\nDigital Signal Processing in Python\n\nAllen B. Downey\n\n# Think DSP\n\nby Allen  B.  Downey\n\nCopyright \u00a9 2016 Allen B. Downey. All rights reserved.\n\nPrinted in the United States of America.\n\nPublished by O'Reilly Media, Inc., 1005 Gravenstein Highway North, Sebastopol, CA 95472.\n\nO'Reilly books may be purchased for educational, business, or sales promotional use. Online editions are also available for most titles (<http:\/\/safaribooksonline.com>). For more information, contact our corporate\/institutional sales department: 800-998-9938 or _corporate@oreilly.com_.\n\n  * Editors: Nan Barber and Susan Conant\n  * Production Editor: Kristen Brown\n  * Copyeditor: Kim Cofer\n  * Proofreader: Rachel Head\n  * Indexer: Allen B. Downey\n  * Interior Designer: David Futato\n  * Cover Designer: Karen Montgomery\n  * Illustrator: Rebecca Demarest\n\n  * July 2016: First Edition\n\n# Revision History for the First Edition\n\n  * 2016-07-11: First Release\n\nSee <http:\/\/oreilly.com\/catalog\/errata.csp?isbn=9781491938454> for release details.\n\nThe O'Reilly logo is a registered trademark of O'Reilly Media, Inc. _Think DSP_ , the cover image of a smooth-billed ani, and related trade dress are trademarks of O'Reilly Media, Inc.\n\nWhile the publisher and the author have used good faith efforts to ensure that the information and instructions contained in this work are accurate, the publisher and the author disclaim all responsibility for errors or omissions, including without limitation responsibility for damages resulting from the use of or reliance on this work. Use of the information and instructions contained in this work is at your own risk. If any code samples or other technology this work contains or describes is subject to open source licenses or the intellectual property rights of others, it is your responsibility to ensure that your use thereof complies with such licenses and\/or rights.\n\n_Think DSP_ is available under the Creative Commons Attribution-NonCommercial 3.0 Unported License. The author maintains an online version at <http:\/\/greenteapress.com\/wp\/think-dsp\/>.\n\n978-1-491-93845-4\n\n[LSI]\n\n# Preface\n\nSignal processing is one of my favorite topics. It is useful in many areas of science and engineering, and if you understand the fundamental ideas, it provides insight into many things we see in the world, and especially the things we hear.\n\nBut unless you've studied electrical or mechanical engineering, you probably haven't had a chance to learn about signal processing. The problem is that most books (and the classes that use them) present the material bottom-up, starting with mathematical abstractions like phasors. And they tend to be theoretical, with few applications and little apparent relevance.\n\nThe premise of this book is that if you know how to program, you can use that skill to learn other things, and have fun doing it.\n\nWith a programming-based approach, I can present the most important ideas right away. By the end of the first chapter, you'll be able to analyze sound recordings and other signals, and generate new sounds. Each chapter introduces a new technique and an application you can apply to real signals. At each step you learn how to use a technique first, and then how it works.\n\nThis approach is more practical and, I hope you'll agree, more fun.\n\n# Who Is This Book For?\n\nThe examples and supporting code for this book are in Python. You should know core Python and you should be familiar with object-oriented features, at least using objects if not defining your own.\n\nIf you are not already familiar with Python, you might want to start with my other book, _Think Python_ , which is an introduction to Python for people who have never programmed, or Mark Lutz's _Learning Python_ , which might be better for people with programming experience.\n\nI use NumPy and SciPy extensively. If you are familiar with them already, that's great, but I will also explain the functions and data structures I use.\n\nI assume that the reader knows basic mathematics, including complex numbers. You don't need much calculus; if you understand the concepts of integration and differentiation, that will do. I use some linear algebra, but I will explain it as we go along.\n\n# Using the Code\n\nThe code and sound samples used in this book are available from this GitHub repository: _https:\/\/github.com\/AllenDowney\/ThinkDSP_. If you are not familiar with Git and GitHub, Git is a version control system that allows you to keep track of the files that make up a project. A collection of files under Git's control is called a \"repository\". GitHub is a hosting service that provides storage for Git repositories and a convenient web interface.\n\nThe GitHub home page for my repository provides several ways to work with the code:\n\n  * You can create a copy of my repository on GitHub by pressing the Fork button. If you don't already have a GitHub account, you'll need to create one. After forking, you'll have your own repository on GitHub that you can use to keep track of code you write while working on this book. Then you can clone the repository, which means that you copy the files to your computer.\n\n  * You can clone my repository. You don't need a GitHub account to do this, but you won't be able to write your changes back to GitHub.\n\n  * If you don't want to use Git at all, you can download the files in a ZIP file using the button in the lower-right corner of the GitHub page.\n\nAll of the code is written to work in both Python 2 and Python 3 with no translation.\n\nI developed this book using Anaconda from Continuum Analytics, which is a free Python distribution that includes all the packages you'll need to run the code (and lots more). I found Anaconda easy to install. By default it does a user-level installation, not system-level, so you don't need administrative privileges. And it supports both Python 2 and Python 3. You can download Anaconda from _http:\/\/continuum.io\/downloads_.\n\nIf you don't want to use Anaconda, you will need the following packages:\n\n  * NumPy for basic numerical computation ( _http:\/\/www.numpy.org_)\n\n  * SciPy for scientific computation ( _http:\/\/www.scipy.org_)\n\n  * matplotlib for visualization ( _http:\/\/matplotlib.org_)\n\nAlthough these are commonly used packages, they are not included with all Python installations, and they can be hard to install in some environments. If you have trouble installing them, I recommend using Anaconda or one of the other Python distributions that include these packages.\n\nMost exercises use Python scripts, but some also use Jupyter notebooks. If you have not used Jupyter before, you can read about it at _http:\/\/jupyter.org_.\n\nThere are three ways you can work with the Jupyter notebooks:\n\nRun Jupyter on your computer\n\nIf you installed Anaconda, you probably got Jupyter by default. To check, start the server from the command line, like this:\n\n    $ **jupyter notebook**\n\nIf it's not installed, you can install it in Anaconda like this:\n\n    $ **conda install jupyter**\n\nWhen you start the server, it should launch your default web browser or create a new tab in an open browser window.\n\nRun Jupyter on Binder\n\nBinder is a service that runs Jupyter in a virtual machine. If you follow the link _http:\/\/mybinder.org\/repo\/AllenDowney\/ThinkDSP_, you should get a Jupyter home page with the notebooks for this book and the supporting data and scripts.\n\nYou can run the scripts and modify them to run your own code, but the virtual machine you run in is temporary. Any changes you make will disappear, along with the virtual machine, if you leave it idle for more than about an hour.\n\nView notebooks on nbviewer\n\nWhen I refer to notebooks later in the book, I provide links to nbviewer, which provides a static view of the code and results. You can use these links to read the notebooks and listen to the examples, but you won't be able to modify or run the code, or use the interactive widgets.\n\nGood luck, and have fun!\n\n# Conventions Used in This Book\n\nThe following typographical conventions are used in this book:\n\n _Italic_\n\nIndicates emphasis, keystrokes, menu options, URLs, and email addresses.\n\n **Bold**\n\nUsed for new terms where they are defined.\n\n`Constant width`\n\nUsed for program listings, as well as within paragraphs to refer to filenames, file extensions, and program elements such as variable and function names, data types, statements, and keywords.\n\n **`Constant width bold`**\n\nShows commands or other text that should be typed literally by the user.\n\n# Safari\u00ae Books Online\n\n###### Note\n\nSafari Books Online ( _www.safaribooksonline.com_) is an on-demand digital library that delivers expert content in both book and video form from the world's leading authors in technology and business.\n\nTechnology professionals, software developers, web designers, and business and creative professionals use Safari Books Online as their primary resource for research, problem solving, learning, and certification training.\n\nSafari Books Online offers a range of plans and pricing for enterprise, government, education, and individuals.\n\nMembers have access to thousands of books, training videos, and prepublication manuscripts in one fully searchable database from publishers like O'Reilly Media, Prentice Hall Professional, Addison-Wesley Professional, Microsoft Press, Sams, Que, Peachpit Press, Focal Press, Cisco Press, John Wiley & Sons, Syngress, Morgan Kaufmann, IBM Redbooks, Packt, Adobe Press, FT Press, Apress, Manning, New Riders, McGraw-Hill, Jones & Bartlett, Course Technology, and hundreds more. 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You can access this page at _http:\/\/bit.ly\/think-dsp_.\n\nTo comment or ask technical questions about this book, send email to _bookquestions@oreilly.com_.\n\nFor more information about our books, courses, conferences, and news, see our website at _http:\/\/www.oreilly.com_.\n\nFind us on Facebook: _http:\/\/facebook.com\/oreilly_\n\nFollow us on Twitter: _http:\/\/twitter.com\/oreillymedia_\n\nWatch us on YouTube: _http:\/\/www.youtube.com\/oreillymedia_\n\n# Contributor List\n\nIf you have a suggestion or correction, please send email to _downey@allendowney.com_. If I make a change based on your feedback, I will add you to the contributor list (unless you ask to be omitted).\n\nIf you include at least part of the sentence the error appears in, that makes it easy for me to search. Page numbers and section titles are fine, too, but not as easy to work with. Thanks!\n\n  * Before I started writing, my thoughts about this book benefited from conversations with Boulos Harb at Google and Aurelio Ramos, formerly at Harmonix Music Systems.\n\n  * During the Fall 2013 semester, Nathan Lintz and Ian Daniher worked with me on an independent study project and helped me with the first draft of this book.\n\n  * On Reddit's DSP forum, the anonymous user RamjetSoundwave helped me fix a problem with my implementation of Brownian noise. And andodli found a typo.\n\n  * In Spring 2015 I had the pleasure of teaching this material along with Prof. Oscar Mur-Miranda and Prof. Siddhartan Govindasamy. Both made many suggestions and corrections.\n\n  * Silas Gyger corrected an arithmetic error.\n\n  * Giuseppe Masetti sent a number of very helpful suggestions.\n\nSpecial thanks to the technical reviewers, Eric Peters, Bruce Levens, and John Vincent, for many helpful suggestions, clarifications, and corrections.\n\nAlso thanks to Freesound, which is the source of many of the sound samples I use in this book, and to the Freesound users who contributed those samples. I include some of their wave files in the GitHub repository for this book, using the original filenames, so it should be easy to find their sources.\n\nUnfortunately, most Freesound users don't make their real names available, so I can only thank them by their usernames. Samples used in this book were contributed by Freesound users iluppai, wcfl10, thirsk, docquesting, kleeb, landup, zippi1, themusicalnomad, bcjordan, rockwehrmann, marcgascon7, and jcveliz. Thank you all! \n\n# Chapter 1. Sounds and Signals\n\nA **signal** represents a quantity that varies in time. That definition is pretty abstract, so let's start with a concrete example: sound. Sound is variation in air pressure. A sound signal represents variations in air pressure over time.\n\nA microphone is a device that measures these variations and generates an electrical signal that represents sound. A speaker is a device that takes an electrical signal and produces sound. Microphones and speakers are called **transducers** because they transduce, or convert, signals from one form to another.\n\nThis book is about signal processing, which includes processes for synthesizing, transforming, and analyzing signals. I will focus on sound signals, but the same methods apply to electronic signals, mechanical vibration, and signals in many other domains.\n\nThey also apply to signals that vary in space rather than time, like elevation along a hiking trail. And they apply to signals in more than one dimension, like an image, which you can think of as a signal that varies in two-dimensional space. Or a movie, which is a signal that varies in two-dimensional space _and_ time.\n\nBut we start with simple one-dimensional sound.\n\nThe code for this chapter is in `chap01.ipynb`, which is in the repository for this book (see \"Using the Code\"). You can also view it at _http:\/\/tinyurl.com\/thinkdsp01_.\n\n# Periodic Signals\n\nWe'll start with **periodic signals** , which are signals that repeat themselves after some period of time. For example, if you strike a bell, it vibrates and generates sound. If you record that sound and plot the transduced signal, it looks like Figure 1-1.\n\n###### Figure 1-1. Segment from a recording of a bell.\n\nThis signal resembles a **sinusoid** , which means it has the same shape as the trigonometric sine function.\n\nYou can see that this signal is periodic. I chose the duration to show three full repetitions, also known as **cycles**. The duration of each cycle, called the **period** , is about 2.3 ms.\n\nThe **frequency** of a signal is the number of cycles per second, which is the inverse of the period. The units of frequency are cycles per second, or **Hertz** , abbreviated \"Hz\". (Strictly speaking, the number of cycles is a dimensionless number, so a Hertz is really a \"per second\".)\n\nThe frequency of this signal is about 439 Hz, slightly lower than 440 Hz, which is the standard tuning pitch for orchestral music. The musical name of this note is A, or more specifically, A4. If you are not familiar with \"scientific pitch notation\", the numerical suffix indicates which octave the note is in. A4 is the A above middle C. A5 is one octave higher. See _http:\/\/en.wikipedia.org\/wiki\/Scientific_pitch_notation_.\n\nA tuning fork generates a sinusoid because the vibration of the tines is a form of simple harmonic motion. Most musical instruments produce periodic signals, but the shape of these signals is not sinusoidal. For example, Figure 1-2 shows a segment from a recording of a violin playing Boccherini's String Quintet No. 5 in E, 3rd movement.\n\n###### Figure 1-2. Segment from a recording of a violin.\n\nAgain we can see that the signal is periodic, but the shape of the signal is more complex. The shape of a periodic signal is called the **waveform**. Most musical instruments produce waveforms more complex than a sinusoid. The shape of the waveform determines the musical **timbre** , which is our perception of the quality of the sound. People usually perceive complex waveforms as rich, warm, and more interesting than sinusoids.\n\n# Spectral Decomposition\n\nThe most important topic in this book is **spectral decomposition** , which is the idea that any signal can be expressed as the sum of sinusoids with different frequencies.\n\nThe most important mathematical idea in this book is the **Discrete Fourier Transform** (DFT), which takes a signal and produces its **spectrum**. The spectrum is the set of sinusoids that add up to produce the signal.\n\nAnd the most important algorithm in this book is the **Fast Fourier Transform** (FFT), which is an efficient way to compute the DFT.\n\nFor example, Figure 1-3 shows the spectrum of the violin recording in Figure 1-2. The x-axis is the range of frequencies that make up the signal. The y-axis shows the strength or **amplitude** of each frequency component.\n\n###### Figure 1-3. Spectrum of a segment from the violin recording.\n\nThe lowest frequency component is called the **fundamental frequency**. The fundamental frequency of this signal is near 440 Hz (actually a little lower, or \"flat\").\n\nIn this signal the fundamental frequency has the largest amplitude, so it is also the **dominant frequency**. Normally the perceived pitch of a sound is determined by the fundamental frequency, even if it is not dominant.\n\nThe other spikes in the spectrum are at frequencies 880, 1320, 1760, and 2200, which are integer multiples of the fundamental. These components are called **harmonics** because they are musically harmonious with the fundamental:\n\n  * 880 is the frequency of A5, one octave higher than the fundamental. An **octave** is a doubling in frequency.\n\n  * 1320 is approximately E6, which is a perfect fifth above A5. If you are not familiar with musical intervals like \"perfect fifth\", see _https:\/\/en.wikipedia.org\/wiki\/Interval_(music)_).\n\n  * 1760 is A6, two octaves above the fundamental.\n\n  * 2200 is approximately C _\u266f_ 7, which is a major third above A6.\n\nThese harmonics make up the notes of an A major chord, although not all in the same octave. Some of them are only approximate because the notes that make up Western music have been adjusted for **equal temperament** (see _http:\/\/en.wikipedia.org\/wiki\/Equal_temperament_).\n\nGiven the harmonics and their amplitudes, you can reconstruct the signal by adding up sinusoids. Next we'll see how.\n\n# Signals\n\nI wrote a Python module called `thinkdsp.py` that contains classes and functions for working with signals and spectrums. You will find it in the repository for this book (see \"Using the Code\").\n\nTo represent signals, `thinkdsp` provides a class called `Signal`. This is the parent class for several signal types, including `Sinusoid`, which represents both sine and cosine signals.\n\n`thinkdsp` provides functions to create sine and cosine signals:\n\n    cos_sig = thinkdsp.CosSignal(freq=440, amp=1.0, offset=0)\n    sin_sig = thinkdsp.SinSignal(freq=880, amp=0.5, offset=0)\n\n`freq` is frequency in Hz. `amp` is amplitude in unspecified units, where 1.0 is defined as the largest amplitude we can record or play back.\n\n`offset` is a **phase offset** in radians. Phase offset determines where in the period the signal starts. For example, a sine signal with `offset=0` starts at  , which is 0. With `offset=pi\/2` it starts at  , which is 1.\n\nSignals have an `__add__` method, so you can use the `+` operator to add them:\n\n    mix = sin_sig + cos_sig\n\nThe result is a `SumSignal`, which represents the sum of two or more signals.\n\nA `Signal` is basically a Python representation of a mathematical function. Most signals are defined for all values of _t_ , from negative infinity to infinity.\n\nYou can't do much with a `Signal` until you evaluate it. In this context, \"evaluate\" means taking a sequence of points in time, `ts`, and computing the corresponding values of the signal, `ys`. I represent `ts` and `ys` using NumPy arrays and encapsulate them in an object called a `Wave`.\n\nA `Wave` represents a signal evaluated at a sequence of points in time. Each point in time is called a **frame** (a term borrowed from movies and video). The measurement itself is called a **sample** , although \"frame\" and \"sample\" are sometimes used interchangeably.\n\n`Signal` provides `make_wave`, which returns a new `Wave` object:\n\n    wave = mix.make_wave(duration=0.5, start=0, framerate=11025)\n\n`duration` is the length of the `Wave` in seconds. `start` is the start time, also in seconds. `framerate` is the (integer) number of frames per second, which is also the number of samples per second.\n\n11,025 frames per second is one of several frame rates commonly used in audio file formats, including Waveform Audio File (WAV) and MP3.\n\nThis example evaluates the signal from _t_ =0 to _t_ =0.5 at 5513 equally spaced frames (because 5513 is half of 11,025). The time between frames, or **timestep** , is 1\/1,1025 seconds, about 91 _\u03bc_ s.\n\n`Wave` provides a `plot` method that uses `pyplot`. You can plot the wave like this:\n\n    wave.plot()\n    pyplot.show()\n\n`pyplot` is part of matplotlib; it is included in many Python distributions, or you might have to install it.\n\nAt `freq=440` there are 220 periods in 0.5 seconds, so this plot would look like a solid block of color. To zoom in on a small number of periods, we can use `segment`, which copies a segment of a `Wave` and returns a new wave:\n\n    period = mix.period\n    segment = wave.segment(start=0, duration=period*3)\n\n`period` is a property of a `Signal`; it returns the period in seconds.\n\n`start` and `duration` are in seconds. This example copies the first three periods from `mix`. The result is a `Wave` object.\n\nIf we plot `segment`, it looks like Figure 1-4. This signal contains two frequency components, so it is more complicated than the signal from the tuning fork, but less complicated than the violin.\n\n###### Figure 1-4. Segment from a mixture of two sinusoid signals.\n\n# Reading and Writing Waves\n\n`thinkdsp` provides `read_wave`, which reads a WAV file and returns a `Wave`:\n\n    violin_wave = thinkdsp.read_wave('input.wav')\n\nAnd `Wave` provides `write`, which writes a WAV file:\n\n    wave.write(filename='output.wav')\n\nYou can listen to the `Wave` with any media player that plays WAV files. On Unix systems I use `aplay`, which is simple, robust, and included in many Linux distributions.\n\n`thinkdsp` also provides `play_wave`, which runs the media player as a subprocess:\n\n    thinkdsp.play_wave(filename='output.wav', player='aplay')\n\nIt uses `aplay` by default, but you can provide the name of another player.\n\n# Spectrums\n\n`Wave` provides `make_spectrum`, which returns a `Spectrum`:\n\n    spectrum = wave.make_spectrum()\n\nAnd `Spectrum` provides `plot`:\n\n    spectrum.plot()\n    thinkplot.show()\n\n`thinkplot` is a module I wrote to provide wrappers around some of the functions in `pyplot`. It is included in the Git repository for this book (see \"Using the Code\").\n\n`Spectrum` provides three methods that modify the spectrum:\n\n  * `low_pass` applies a low-pass filter, which means that components above a given cutoff frequency are **attenuated** (that is, reduced in magnitude) by a factor.\n\n  * `high_pass` applies a high-pass filter, which means that it attenuates components below the cutoff.\n\n  * `band_stop` attenuates components in the band of frequencies between two cutoffs.\n\nThis example attenuates all frequencies above 600 by 99%:\n\n    spectrum.low_pass(cutoff=600, factor=0.01)\n\nA low-pass filter removes bright, high-frequency sounds, so the result sounds muffled and darker. To hear what it sounds like, you can convert the `Spectrum` back to a `Wave`, and then play it:\n\n    wave = spectrum.make_wave()\n    wave.play('temp.wav')\n\nThe `play` method writes the wave to a file and then plays it. If you use Jupyter notebooks, you can use `make_audio`, which makes an Audio widget that plays the sound.\n\n# Wave Objects\n\nThere is nothing very complicated in `thinkdsp.py`. Most of the functions it provides are thin wrappers around functions from NumPy and SciPy.\n\nThe primary classes in `thinkdsp` are `Signal`, `Wave`, and `Spectrum`. Given a `Signal`, you can make a `Wave`. Given a `Wave`, you can make a `Spectrum`, and vice versa. These relationships are shown in Figure 1-5.\n\n###### Figure 1-5. Relationships among the classes in `thinkdsp`.\n\nA `Wave` object contains three attributes: `ys` is a NumPy array that contains the values in the signal; `ts` is an array of the times where the signal was evaluated or sampled; and `framerate` is the number of samples per unit of time. The unit of time is usually seconds, but it doesn't have to be. In one of my examples, it's days.\n\n`Wave` also provides three read-only properties: `start`, `end`, and `duration`. If you modify `ts`, these properties change accordingly.\n\nTo modify a wave, you can access the `ts` and `ys` directly. For example:\n\n    wave.ys *= 2\n    wave.ts += 1\n\nThe first line scales the wave by a factor of 2, making it louder. The second line shifts the wave in time, making it start 1 second later.\n\nBut `Wave` provides methods that perform many common operations. For example, the same two transformations could be written:\n\n    wave.scale(2)\n    wave.shift(1)\n\nYou can read the documentation of these methods and others at _http:\/\/greenteapress.com\/thinkdsp.html_.\n\n# Signal Objects\n\n`Signal` is a parent class that provides functions common to all kinds of signals, like `make_wave`. Child classes inherit these methods and provide `evaluate`, which evaluates the signal at a given sequence of times.\n\nFor example, `Sinusoid` is a child class of `Signal`, with this definition:\n\n    class Sinusoid(Signal):\n\n        def __init__(self, freq=440, amp=1.0, offset=0, func=np.sin):\n            Signal.__init__(self)\n            self.freq = freq\n            self.amp = amp\n            self.offset = offset\n            self.func = func\n\nThe parameters of `__init__` are:\n\n`freq`\n\nFrequency in cycles per second, or Hz.\n\n`amp`\n\nAmplitude. The units of amplitude are arbitrary, usually chosen so 1.0 corresponds to the maximum input from a microphone or maximum output to a speaker.\n\n`offset`\n\nIndicates where in its period the signal starts; `offset` is in units of radians.\n\n`func`\n\nA Python function used to evaluate the signal at a particular point in time. It is usually either `np.sin` or `np.cos`, yielding a sine or cosine signal.\n\nLike many init methods, this one just tucks the parameters away for future use.\n\n`Signal` provides `make_wave`, which looks like this:\n\n    def make_wave(self, duration=1, start=0, framerate=11025):\n        n = round(duration * framerate)\n        ts = start + np.arange(n) \/ framerate\n        ys = self.evaluate(ts)\n        return Wave(ys, ts, framerate=framerate)\n\n`start` and `duration` are the start time and duration in seconds. `framerate` is the number of frames (samples) per second.\n\n`n` is the number of samples, and `ts` is a NumPy array of sample times.\n\nTo compute the `ys`, `make_wave` invokes `evaluate`, which is provided by `Sinusoid`:\n\n    def evaluate(self, ts):\n        phases = PI2 * self.freq * ts + self.offset\n        ys = self.amp * self.func(phases)\n        return ys\n\nLet's unwind this function one step at a time:\n\n  1. `self.freq` is frequency in cycles per second, and each element of `ts` is a time in seconds, so their product is the number of cycles since the start time.\n\n  2. `PI2` is a constant that stores  . Multiplying by `PI2` converts from cycles to **phase**. You can think of phase as \"cycles since the start time\" expressed in radians. Each cycle is   radians.\n\n  3. `self.offset` is the phase when  . It has the effect of shifting the signal left or right in time.\n\n  4. If `self.func` is `np.sin` or `np.cos`, the result is a value between   and  .\n\n  5. Multiplying by `self.amp` yields a signal that ranges from `-self.amp` to `+self.amp`.\n\nIn math notation, `evaluate` is written like this:\n\nwhere _A_ is amplitude, _f_ is frequency, _t_ is time, and _\u03d5 0_ is the phase offset. It may seem like I wrote a lot of code to evaluate one simple expression, but as we'll see, this code provides a framework for dealing with all kinds of signals, not just sinusoids.\n\n# Exercises\n\nBefore you begin these exercises, you should download the code for this book, following the instructions in \"Using the Code\".\n\nSolutions to these exercises are in `chap01soln.ipynb`.\n\n##### Exercise 1-1.\n\nIf you have Jupyter, load `chap01.ipynb`, read through it, and run the examples. You can also view this notebook at _http:\/\/tinyurl.com\/thinkdsp01_.\n\n##### Exercise 1-2.\n\nGo to _http:\/\/freesound.org_ and download a sound sample that includes music, speech, or other sounds that have a well-defined pitch. Select a roughly half-second segment where the pitch is constant. Compute and plot the spectrum of the segment you selected. What connection can you make between the timbre of the sound and the harmonic structure you see in the spectrum?\n\nUse `high_pass`, `low_pass`, and `band_stop` to filter out some of the harmonics. Then convert the spectrum back to a wave and listen to it. How does the sound relate to the changes you made in the spectrum?\n\n##### Exercise 1-3.\n\nSynthesize a compound signal by creating `SinSignal` and `CosSignal` objects and adding them up. Evaluate the signal to get a `Wave`, and listen to it. Compute its `Spectrum` and plot it. What happens if you add frequency components that are not multiples of the fundamental?\n\n##### Exercise 1-4.\n\nWrite a function called `stretch` that takes a `Wave` and a stretch factor and speeds up or slows down the wave by modifying `ts` and `framerate`. Hint: it should only take two lines of code.\n\n The plural of \"spectrum\" is often written \"spectra\", but I prefer to use standard English plurals. If you are familiar with \"spectra\", I hope my choice doesn't sound too strange.\n\n# Chapter 2. Harmonics\n\nIn this chapter I present several new waveforms; we will look at their spectrums to understand their **harmonic structure** , which is the set of sinusoids they are made up of.\n\nI'll also introduce one of the most important phenomena in digital signal processing: aliasing. And I'll explain a little more about how the `Spectrum` class works.\n\nThe code for this chapter is in `chap02.ipynb`, which is in the repository for this book (see \"Using the Code\"). You can also view it at _http:\/\/tinyurl.com\/thinkdsp02_.\n\n# Triangle Waves\n\nA sinusoid contains only one frequency component, so its spectrum has only one peak. More complicated waveforms, like the violin recording in Figure 1-2, yield DFTs with many peaks. In this section we investigate the relationship between waveforms and their spectrums.\n\nI'll start with a triangle waveform, which is like a straight-line version of a sinusoid. Figure 2-1 shows a triangle waveform with frequency 200 Hz.\n\n###### Figure 2-1. Segment of a triangle signal at 200 Hz.\n\nTo generate a triangle wave, you can start with a `thinkdsp.TriangleSignal`:\n\n    class TriangleSignal(Sinusoid):\n\n        def evaluate(self, ts):\n            cycles = self.freq * ts + self.offset \/ PI2\n            frac, _ = np.modf(cycles)\n            ys = np.abs(frac - 0.5)\n            ys = normalize(unbias(ys), self.amp)\n            return ys\n\n`TriangleSignal` inherits `__init__` from `Sinusoid`, so it takes the same arguments: `freq`, `amp`, and `offset`.\n\nThe only difference is `evaluate`. As we saw before, `ts` is the sequence of sample times where we want to evaluate the signal.\n\nThere are many ways to generate a triangle wave. The details are not important, but here's how `evaluate` works:\n\n  1. `cycles` is the number of cycles since the start time. `np.modf` splits the number of cycles into the fraction part, stored in `frac`, and the integer part, which is ignored.\n\n  2. `frac` is a sequence that ramps from 0 to 1 with the given frequency. Subtracting 0.5 yields values between \u20130.5 and 0.5. Taking the absolute value yields a waveform that zigzags between 0.5 and 0.\n\n  3. `unbias` shifts the waveform down so it is centered at 0; then `normalize` scales it to the given amplitude, `amp`.\n\nHere's the code that generates Figure 2-1:\n\n    signal = thinkdsp.TriangleSignal(200)\n    signal.plot()\n\nNext we can use the `Signal` to make a `Wave`, and use the `Wave` to make a `Spectrum`:\n\n    wave = signal.make_wave(duration=0.5, framerate=10000)\n    spectrum = wave.make_spectrum()\n    spectrum.plot()\n\nFigure 2-2 shows two views of the result; the view on the right is scaled to show the harmonics more clearly. As expected, the highest peak is at the fundamental frequency, 200 Hz, and there are additional peaks at harmonic frequencies, which are integer multiples of 200.\n\n###### Figure 2-2. Spectrum of a triangle signal at 200 Hz, shown on two vertical scales. The version on the right cuts off the fundamental to show the harmonics more clearly.\n\nBut one surprise is that there are no peaks at the even multiples: 400, 800, etc. The harmonics of a triangle wave are all odd multiples of the fundamental frequency, in this example 600, 1000, 1400, etc.\n\nAnother feature of this spectrum is the relationship between the amplitude and frequency of the harmonics. Their amplitude drops off in proportion to frequency squared. For example, the frequency ratio of the first two harmonics (200 and 600 Hz) is 3, and the amplitude ratio is approximately 9. The frequency ratio of the next two harmonics (600 and 1000 Hz) is 1.7, and the amplitude ratio is approximately  . This relationship is called the **harmonic structure**.\n\n# Square Waves\n\n`thinkdsp` also provides `SquareSignal`, which represents a square signal. Here's the class definition:\n\n    class SquareSignal(Sinusoid):\n\n        def evaluate(self, ts):\n            cycles = self.freq * ts + self.offset \/ PI2\n            frac, _ = np.modf(cycles)\n            ys = self.amp * np.sign(unbias(frac))\n            return ys\n\nLike `TriangleSignal`, `SquareSignal` inherits `__init__` from `Sinusoid`, so it takes the same parameters.\n\nAnd the `evaluate` method is similar. Again, `cycles` is the number of cycles since the start time, and `frac` is the fractional part, which ramps from 0 to 1 each period.\n\n`unbias` shifts `frac` so it ramps from \u20130.5 to 0.5, then `np.sign` maps the negative values to \u20131 and the positive values to 1. Multiplying by `amp` yields a square wave that jumps between `-amp` and `amp`.\n\nFigure 2-3 shows three periods of a square wave with frequency 100 Hz, and Figure 2-4 shows its spectrum.\n\nLike a triangle wave, the square wave contains only odd harmonics, which is why there are peaks at 300, 500, and 700 Hz, etc. But the amplitude of the harmonics drops off more slowly. Specifically, amplitude drops in proportion to frequency (not frequency squared).\n\nThe exercises at the end of this chapter give you a chance to explore other waveforms and other harmonic structures.\n\n###### Figure 2-3. Segment of a square signal at 100 Hz.\n\n###### Figure 2-4. Spectrum of a square signal at 100 Hz.\n\n# Aliasing\n\nI have a confession. I chose the examples in the previous section carefully to avoid showing you something confusing. But now it's time to get confused.\n\nFigure 2-5 shows the spectrum of a triangle wave at 1100 Hz, sampled at 10,000 frames per second. Again, the view on the right is scaled to show the harmonics.\n\n###### Figure 2-5. Spectrum of a triangle signal at 1100 Hz sampled at 10,000 frames per second. The view on the right is scaled to show the harmonics.\n\nThe harmonics of this wave should be at 3300, 5500, 7700, and 9900 Hz. In the figure, there are peaks at 1100 and 3300 Hz, as expected, but the third peak is at 4500, not 5500 Hz. The fourth peak is at 2300, not 7700 Hz. And if you look closely, the peak that should be at 9900 is actually at 100 Hz. What's going on?\n\nThe problem is that when you evaluate the signal at discrete points in time, you lose information about what happened between samples. For low-frequency components, that's not a problem, because you have lots of samples per period.\n\nBut if you sample a signal at 5000 Hz with 10,000 frames per second, you only have two samples per period. That turns out to be enough, just barely, but if the frequency is higher, it's not.\n\nTo see why, let's generate cosine signals at 4500 and 5500 Hz, and sample them at 10,000 frames per second:\n\n    framerate = 10000\n\n    signal = thinkdsp.CosSignal(4500)\n    duration = signal.period*5\n    segment = signal.make_wave(duration, framerate=framerate)\n    segment.plot()\n\n    signal = thinkdsp.CosSignal(5500)\n    segment = signal.make_wave(duration, framerate=framerate)\n    segment.plot()\n\nFigure 2-6 shows the result. I plotted the `Signal`s with thin gray lines and the samples using vertical lines, to make it easier to compare the two `Wave`s. The problem should be clear: even though the `Signal`s are different, the `Wave`s are identical!\n\n###### Figure 2-6. Cosine signals at 4500 and 5500 Hz, sampled at 10,000 frames per second. The signals are different, but the samples are identical.\n\nWhen we sample a 5500 Hz signal at 10,000 frames per second, the result is indistinguishable from a 4500 Hz signal. For the same reason, a 7700 Hz signal is indistinguishable from 2300 Hz, and a 9900 Hz signal is indistinguishable from 100 Hz.\n\nThis effect is called **aliasing** because when the high-frequency signal is sampled, it appears to be a low-frequency signal.\n\nIn this example, the highest frequency we can measure is 5000 Hz, which is half the sampling rate. Frequencies above 5000 Hz are folded back below 5000 Hz, which is why this threshold is sometimes called the \"folding frequency\". It is sometimes also called the **Nyquist frequency**. See _http:\/\/en.wikipedia.org\/wiki\/Nyquist_frequency_.\n\nThe folding pattern continues if the aliased frequency goes below zero. For example, the fifth harmonic of the 1100 Hz triangle wave is at 12,100 Hz. Folded at 5000 Hz, it would appear at \u20132100 Hz, but it gets folded again at 0 Hz, back to 2100 Hz. In fact, you can see a small peak at 2100 Hz in Figure 2-4, and the next one at 4300 Hz.\n\n# Computing the Spectrum\n\nWe have seen the `Wave` method `make_spectrum` several times. Here is the implementation (leaving out some details we'll get to later):\n\n    from np.fft import rfft, rfftfreq\n\n    # class Wave:\n        def make_spectrum(self):\n            n = len(self.ys)\n            d = 1 \/ self.framerate\n\n            hs = rfft(self.ys)\n            fs = rfftfreq(n, d)\n\n            return Spectrum(hs, fs, self.framerate)\n\nThe parameter `self` is a `Wave` object. `n` is the number of samples in the wave, and `d` is the inverse of the frame rate, which is the time between samples.\n\n`np.fft` is the NumPy module that provides functions related to the Fast Fourier Transform (FFT), which is an efficient algorithm that computes the Discrete Fourier Transform (DFT).\n\n`make_spectrum` uses `rfft`, which stands for \"real FFT\", because the `Wave` contains real values, not complex. Later we'll see the full FFT, which can handle complex signals (see \"DFT of Real Signals\"). The result of `rfft`, which I call `hs`, is a NumPy array of complex numbers that represents the amplitude and phase offset of each frequency component in the wave.\n\nThe result of `rfftfreq`, which I call `fs`, is an array that contains frequencies corresponding to the `hs`.\n\nTo understand the values in `hs`, consider these two ways to think about complex numbers:\n\n  * A complex number is the sum of a real part and an imaginary part, often written  , where _i_ is the imaginary unit  . You can think of _x_ and _y_ as Cartesian coordinates.\n\n  * A complex number is also the product of a magnitude and a complex exponential,  , where _A_ is the **magnitude** and _\u03d5_ is the **angle** in radians, also called the \"argument\". You can think of _A_ and _\u03d5_ as polar coordinates.\n\nEach value in `hs` corresponds to a frequency component: its magnitude is proportional to the amplitude of the corresponding component; its angle is the phase offset.\n\nThe `Spectrum` class provides two read-only properties, `amps` and `angles`, which return NumPy arrays representing the magnitudes and angles of the `hs`. When we plot a `Spectrum` object, we usually plot `amps` versus `fs`. Sometimes it is also useful to plot `angles` versus `fs`.\n\nAlthough it might be tempting to look at the real and imaginary parts of `hs`, you will almost never need to. I encourage you to think of the DFT as a vector of amplitudes and phase offsets that happen to be encoded in the form of complex numbers.\n\nTo modify a `Spectrum`, you can access the `hs` directly. For example:\n\n    spectrum.hs *= 2\n    spectrum.hs[spectrum.fs > cutoff] = 0\n\nThe first line multiplies the elements of `hs` by 2, which doubles the amplitudes of all components. The second line sets to 0 only the elements of `hs` where the corresponding frequency exceeds some cutoff frequency.\n\nBut `Spectrum` also provides methods to perform these operations:\n\n    spectrum.scale(2)\n    spectrum.low_pass(cutoff)\n\nYou can read the documentation of these methods and others at _http:\/\/greenteapress.com\/thinkdsp.html_.\n\nAt this point you should have a better idea of how the `Signal`, `Wave`, and `Spectrum` classes work, but I have not explained how the Fast Fourier Transform works. That will take a few more chapters.\n\n# Exercises\n\nSolutions to these exercises are in `chap02soln.ipynb`.\n\n##### Exercise 2-1.\n\nIf you use Jupyter, load `chap02.ipynb` and try out the examples. You can also view the notebook at _http:\/\/tinyurl.com\/thinkdsp02_.\n\n##### Exercise 2-2.\n\nA sawtooth signal has a waveform that ramps up linearly from \u20131 to 1, then drops to \u20131 and repeats. See _http:\/\/en.wikipedia.org\/wiki\/Sawtooth_wave_.\n\nWrite a class called `SawtoothSignal` that extends `Signal` and provides `evaluate` to evaluate a sawtooth signal.\n\nCompute the spectrum of a sawtooth wave. How does the harmonic structure compare to triangle and square waves?\n\n##### Exercise 2-3.\n\nMake a square signal at 1100 Hz and make a `Wave` that samples it at 10,000 frames per second. If you plot the spectrum, you can see that most of the harmonics are aliased. When you listen to the wave, can you hear the aliased harmonics?\n\n##### Exercise 2-4.\n\nIf you have a spectrum object, `spectrum`, and print the first few values of `spectrum.fs`, you'll see that they start at zero. So `spectrum.hs[0]` is the magnitude of the component with frequency 0. But what does that mean?\n\nTry this experiment:\n\n  1. Make a triangle signal with frequency 440 and make a `Wave` with duration 0.01 seconds. Plot the waveform.\n\n  2. Make a `Spectrum` object and print `spectrum.hs[0]`. What is the amplitude and phase of this component?\n\n  3. Set `spectrum.hs[0] = 100`. What effect does this operation have on the waveform? Hint: `Spectrum` provides a method called `make_wave` that computes the `Wave` that corresponds to the `Spectrum`.\n\n##### Exercise 2-5.\n\nWrite a function that takes a `Spectrum` as a parameter and modifies it by dividing each element of `hs` by the corresponding frequency from `fs`. Hint: since division by zero is undefined, you might want to set `spectrum.hs[0] = 0`.\n\nTest your function using a square, triangle, or sawtooth wave:\n\n  1. Compute the `Spectrum` and plot it.\n\n  2. Modify the `Spectrum` using your function and plot it again.\n\n  3. Use `Spectrum.make_wave` to make a `Wave` from the modified `Spectrum`, and listen to it. What effect does this operation have on the signal?\n\n##### Exercise 2-6.\n\nTriangle and square waves have odd harmonics only; the sawtooth wave has both even and odd harmonics. The harmonics of the square and sawtooth waves drop off in proportion to  ; the harmonics of the triangle wave drop off like  . Can you find a waveform that has even and odd harmonics that drop off like  ?\n\nHint: there are two ways you could approach this. You could construct the signal you want by adding up sinusoids, or you could start with a signal that is similar to what you want and modify it.\n\n Using an underscore as a variable name is a convention that means, \"I don't intend to use this value.\"\n\n# Chapter 3. Non-Periodic Signals\n\nThe signals we have worked with so far are periodic, which means that they repeat forever. It also means that the frequency components they contain do not change over time. In this chapter, we consider non-periodic signals, whose frequency components _do_ change over time. In other words, pretty much all sound signals.\n\nThis chapter also presents spectrograms, a common way to visualize non-periodic signals.\n\nThe code for this chapter is in `chap03.ipynb`, which is in the repository for this book (see \"Using the Code\"). You can also view it at _http:\/\/tinyurl.com\/thinkdsp03_.\n\n# Linear Chirp\n\nWe'll start with a **chirp** , which is a signal with variable frequency. `thinkdsp` provides a `Signal` called `Chirp` that makes a sinusoid that sweeps linearly through a range of frequencies.\n\nHere's an example that sweeps from 220 to 880 Hz, which is two octaves from A3 to A5:\n\n    signal = thinkdsp.Chirp(start=220, end=880)\n    wave = signal.make_wave()\n\nFigure 3-1 shows segments of this wave near the beginning, middle, and end. It's clear that the frequency is increasing.\n\n###### Figure 3-1. Chirp waveform near the beginning, middle, and end.\n\nBefore we go on, let's see how `Chirp` is implemented. Here is the class definition:\n\n    class Chirp(Signal):\n\n        def __init__(self, start=440, end=880, amp=1.0):\n            self.start = start\n            self.end = end\n            self.amp = amp\n\n`start` and `end` are the frequencies, in Hz, at the start and end of the chirp. `amp` is amplitude.\n\nHere is the function that evaluates the signal:\n\n    def evaluate(self, ts):\n        freqs = np.linspace(self.start, self.end, len(ts)-1)\n        return self._evaluate(ts, freqs)\n\n`ts` is the sequence of points in time where the signal should be evaluated; to keep this function simple, I assume they are equally spaced.\n\nIf the length of `ts` is _n_ , you can think of it as a sequence of   intervals of time. To compute the frequency during each interval, I use `np.linspace`, which returns a NumPy array of   values between `start` and `end`.\n\n`_evaluate` is a private method that does the rest of the math:\n\n    def _evaluate(self, ts, freqs):\n        dts = np.diff(ts)\n        dphis = PI2 * freqs * dts\n        phases = np.cumsum(dphis)\n        phases = np.insert(phases, 0, 0)\n        ys = self.amp * np.cos(phases)\n        return ys\n\n`np.diff` computes the difference between adjacent elements of `ts`, returning the length of each interval in seconds. If the elements of `ts` are equally spaced, the `dts` values are all the same.\n\nThe next step is to figure out how much the phase changes during each interval. In \"Signal Objects\" we saw that when frequency is constant, the phase, _\u03d5_ , increases linearly over time:\n\nWhen frequency is a function of time, the _change_ in phase during a short time interval, _\u0394t_ , is:\n\nIn Python, since `freqs` contains   and `dts` contains the time intervals, we can write:\n\n    dphis = PI2 * freqs * dts\n\nNow, since `dphis` contains the changes in phase, we can get the total phase at each timestep by adding up the changes:\n\n    phases = np.cumsum(dphis)\n    phases = np.insert(phases, 0, 0)\n\n`np.cumsum` computes the cumulative sum, which is almost what we want, but it doesn't start at 0. So I use `np.insert` to add a 0 at the beginning.\n\nThe result is a NumPy array where the `i`th element contains the sum of the first `i` terms from `dphis`; that is, the total phase at the end of the `i`th interval. Finally, `np.cos` computes the amplitude of the wave as a function of phase (remember that phase is expressed in radians).\n\nIf you know calculus, you might notice that the limit as _\u0394t_ gets small is:\n\nDividing through by _dt_ yields:\n\nIn other words, frequency is the derivative of phase. Conversely, phase is the integral of frequency. When we used `cumsum` to go from frequency to phase, we were approximating integration.\n\n# Exponential Chirp\n\nWhen you listen to this chirp, you might notice that the pitch rises quickly at first and then slows down. The chirp spans two octaves, but it only takes 2\/3 s to span the first octave, and twice as long to span the second.\n\nThe reason is that our perception of pitch depends on the logarithm of frequency. As a result, the **interval** we hear between two notes depends on the _ratio_ of their frequencies, not the difference. \"Interval\" is the musical term for the perceived difference between two pitches.\n\nFor example, an octave is an interval where the ratio of two pitches is 2. So the interval from 220 to 440 Hz is one octave and the interval from 440 to 880 Hz is also one octave. The difference in frequency is bigger, but the ratio is the same.\n\nAs a result, if frequency increases linearly, as in a linear chirp, the perceived pitch increases logarithmically.\n\nIf you want the perceived pitch to increase linearly, the frequency has to increase exponentially. A signal with that shape is called an **exponential chirp**.\n\nHere's the `ExpoChirp` class definition:\n\n    class ExpoChirp(Chirp):\n\n        def evaluate(self, ts):\n            start, end = np.log10(self.start), np.log10(self.end)\n            freqs = np.logspace(start, end, len(ts)-1)\n            return self._evaluate(ts, freqs)\n\nInstead of `np.linspace`, this version of `evaluate` uses `np.logspace`, which creates a series of frequencies whose logarithms are equally spaced, which means that they increase exponentially.\n\nThat's it; everything else is the same as `Chirp`. Here's the code that makes one:\n\n    signal = thinkdsp.ExpoChirp(start=220, end=880)\n    wave = signal.make_wave(duration=1)\n\nYou can listen to the examples in `chap03.ipynb` and compare the linear and exponential chirps.\n\n# Spectrum of a Chirp\n\nWhat do you think happens if you compute the spectrum of a chirp? Here's an example that constructs a one-second, one-octave chirp and its spectrum:\n\n    signal = thinkdsp.Chirp(start=220, end=440)\n    wave = signal.make_wave(duration=1)\n    spectrum = wave.make_spectrum()\n\nFigure 3-2 shows the result. The spectrum has components at every frequency from 220 to 440 Hz, with variations that look a little like the Eye of Sauron (see _http:\/\/en.wikipedia.org\/wiki\/Sauron_).\n\n###### Figure 3-2. Spectrum of a one-second, one-octave chirp.\n\nThe spectrum is approximately flat between 220 and 440 Hz, which indicates that the signal spends equal time at each frequency in this range. Based on that observation, you should be able to guess what the spectrum of an exponential chirp looks like.\n\nThe spectrum gives hints about the structure of the signal, but it obscures the relationship between frequency and time. For example, we cannot tell by looking at this spectrum whether the frequency went up or down, or both.\n\n# Spectrogram\n\nTo recover the relationship between frequency and time, we can break the chirp into segments and plot the spectrum of each segment. The result is called a **Short-Time Fourier Transform** (STFT).\n\nThere are several ways to visualize an STFT, but the most common is a **spectrogram** , which shows time on the x-axis and frequency on the y-axis. Each column in the spectrogram shows the spectrum of a short segment, using color or grayscale to represent amplitude.\n\nAs an example, I'll compute the spectrogram of this chirp:\n\n    signal = thinkdsp.Chirp(start=220, end=440)\n    wave = signal.make_wave(duration=1, framerate=11025)\n\n`Wave` provides `make_spectrogram`, which returns a `Spectrogram` object:\n\n    spectrogram = wave.make_spectrogram(seg_length=512)\n    spectrogram.plot(high=700)\n\n`seg_length` is the number of samples in each segment. I chose 512 because the FFT is most efficient when the number of samples is a power of 2.\n\nFigure 3-3 shows the result. The x-axis shows time from 0 to 1 seconds. The y-axis shows frequency from 0 to 700 Hz. I cut off the top part of the spectrogram; the full range goes to 5512.5 Hz, which is half of the frame rate.\n\n###### Figure 3-3. Spectrogram of a one-second, one-octave chirp.\n\nThe spectrogram shows clearly that frequency increases linearly over time. However, notice that the peak in each column is blurred across 2\u20133 cells. This blurring reflects the limited resolution of the spectrogram.\n\n# The Gabor Limit\n\nThe **time resolution** of the spectrogram is the duration of the segments, which corresponds to the width of the cells in the spectrogram. Since each segment is 512 frames, and there are 11,025 frames per second, the duration of each segment is about 0.046 seconds.\n\nThe **frequency resolution** is the frequency range between elements in the spectrum, which corresponds to the height of the cells. With 512 frames, we get 256 frequency components over a range from 0 to 5512.5 Hz, so the range between components is 21.6 Hz.\n\nMore generally, if _n_ is the segment length, the spectrum contains   components. If the frame rate is _r_ , the maximum frequency in the spectrum is  . So the time resolution is   and the frequency resolution is:\n\nwhich is  .\n\nIdeally we would like time resolution to be small, so we can see rapid changes in frequency. And we would like frequency resolution to be small so we can see small changes in frequency. But you can't have both. Notice that time resolution,  , is the inverse of frequency resolution,  . So if one gets smaller, the other gets bigger.\n\nFor example, if you double the segment length, you cut frequency resolution in half (which is good), but you double time resolution (which is bad). Even increasing the frame rate doesn't help. You get more samples, but the range of frequencies increases at the same time.\n\nThis tradeoff is called the **Gabor limit** and it is a fundamental limitation of this kind of time\u2013frequency analysis.\n\n# Leakage\n\nIn order to explain how `make_spectrogram` works, I have to explain windowing; and in order to explain windowing, I have to show you the problem it is meant to address, which is leakage.\n\nThe Discrete Fourier Transform (DFT), which we use to compute `Spectrum`s, treats waves as if they are periodic; that is, it assumes that the finite segment it operates on is a complete period from an infinite signal that repeats over all time. In practice, this assumption is often false, which creates problems.\n\nOne common problem is discontinuity at the beginning and end of the segment. Because the DFT assumes that the signal is periodic, it implicitly connects the end of the segment back to the beginning to make a loop. If the end does not connect smoothly to the beginning, the discontinuity creates additional frequency components in the segment that are not in the signal.\n\nAs an example, let's start with a sine signal that contains only one frequency component at 440 Hz:\n\n    signal = thinkdsp.SinSignal(freq=440)\n\nIf we select a segment that happens to be an integer multiple of the period, the end of the segment connects smoothly with the beginning, and the DFT behaves well:\n\n    duration = signal.period * 30\n    wave = signal.make_wave(duration)\n    spectrum = wave.make_spectrum()\n\nFigure 3-4 (left) shows the result. As expected, there is a single peak at 440 Hz.\n\nBut if the duration is not a multiple of the period, bad things happen. With `duration = signal.period * 30.25`, the signal starts at 0 and ends at 1.\n\nFigure 3-4 (middle) shows the spectrum of this segment. Again, the peak is at 440 Hz, but now there are additional components spread out from 240 to 640 Hz. This spread is called **spectral leakage** , because some of the energy that is actually at the fundamental frequency leaks into other frequencies.\n\nFigure 3-4 (right) shows the spectrum of the windowed signal. Windowing has reduced leakage substantially, but not completely.\n\n###### Figure 3-4. Spectrums of a periodic segment of a sinusoid (left), a non-periodic segment (middle), and a windowed non-periodic segment (right).\n\nIn this example, leakage happens because we are using the DFT on a segment that becomes discontinuous when we treat it as periodic.\n\n# Windowing\n\nWe can reduce leakage by smoothing out the discontinuity between the beginning and end of the segment, and one way to do that is **windowing**.\n\nA \"window\" is a function designed to transform a non-periodic segment into something that can pass for periodic. Figure 3-5 (top) shows a segment where the end does not connect smoothly to the beginning.\n\nFigure 3-5 (middle) shows a \"Hamming window\", one of the more common window functions. No window function is perfect, but some can be shown to be optimal for different applications, and Hamming is a good, all-purpose window.\n\nFigure 3-5 (bottom) shows the result of multiplying the window by the original signal. Where the window is close to 1, the signal is unchanged. Where the window is close to 0, the signal is attenuated. Because the window tapers at both ends, the end of the segment connects smoothly to the beginning.\n\n###### Figure 3-5. Segment of a sinusoid (top), Hamming window (middle), and product of the segment and the window (bottom).\n\nHere's what the code looks like. `Wave` provides `window`, which applies a Hamming window:\n\n    #class Wave:\n        def window(self, window):\n            self.ys *= window\n\nAnd NumPy provides `hamming`, which computes a Hamming window with a given length:\n\n    window = np.hamming(len(wave))\n    wave.window(window)\n\nNumPy provides functions to compute other window functions, including `bartlett`, `blackman`, `hanning`, and `kaiser`. One of the exercises at the end of this chapter asks you to experiment with these other windows.\n\n# Implementing Spectrograms\n\nNow that we understand windowing, we can understand the implementation of `make_spectrogram`. Here is the `Wave` method that computes spectrograms:\n\n    #class Wave:\n        def make_spectrogram(self, seg_length):\n            window = np.hamming(seg_length)\n            i, j = 0, seg_length\n            step = seg_length \/ 2\n\n            spec_map = {}\n\n            while j < len(self.ys):\n                segment = self.slice(i, j)\n                segment.window(window)\n\n                t = (segment.start + segment.end) \/ 2\n                spec_map[t] = segment.make_spectrum()\n\n                i += step\n                j += step\n\n            return Spectrogram(spec_map, seg_length)\n\nThis is the longest function in the book, so if you can handle this, you can handle anything.\n\nThe parameter, `self`, is a `Wave` object. `seg_length` is the number of samples in each segment.\n\n`window` is a Hamming window with the same length as the segments.\n\n`i` and `j` are the slice indices that select segments from the wave. `step` is the offset between segments. Since `step` is half of `seg_length`, the segments overlap by half. Figure 3-6 shows what these overlapping windows look like.\n\n###### Figure 3-6. Overlapping Hamming windows.\n\n`spec_map` is a dictionary that maps from a timestamp to a `Spectrum`.\n\nInside the `while` loop, we select a slice from the wave and apply the window; then we construct a `Spectrum` object and add it to `spec_map`. The nominal time of each segment, `t`, is the midpoint.\n\nThen we advance `i` and `j`, and continue as long as `j` doesn't go past the end of the `Wave`.\n\nFinally, the method constructs and returns a `Spectrogram` object. Here is the definition of the class:\n\n    class Spectrogram(object):\n        def __init__(self, spec_map, seg_length):\n            self.spec_map = spec_map\n            self.seg_length = seg_length\n\nLike many init methods, this one just stores the parameters as attributes.\n\n`Spectrogram` provides `plot`, which generates a pseudocolor plot with time along the x-axis and frequency along the y-axis.\n\nAnd that's how `Spectrogram`s are implemented.\n\n# Exercises\n\nSolutions to these exercises are in `chap03soln.ipynb`.\n\n##### Exercise 3-1.\n\nRun and listen to the examples in `chap03.ipynb`, which is in the repository for this book, and also available at _http:\/\/tinyurl.com\/thinkdsp03_.\n\nIn the leakage example, try replacing the Hamming window with one of the other windows provided by NumPy, and see what effect they have on leakage. See _http:\/\/docs.scipy.org\/doc\/numpy\/reference\/routines.window.html_.\n\n##### Exercise 3-2.\n\nWrite a class called `SawtoothChirp` that extends `Chirp` and overrides `evaluate` to generate a sawtooth waveform with frequency that increases (or decreases) linearly.\n\nHint: combine the `evaluate` functions from `Chirp` and `SawtoothSignal`.\n\nDraw a sketch of what you think the spectrogram of this signal looks like, and then plot it. The effect of aliasing should be visually apparent, and if you listen carefully, you can hear it.\n\n##### Exercise 3-3.\n\nMake a sawtooth chirp that sweeps from 2500 to 3000 Hz, then use it to make a wave with duration 1 s and frame rate 20 kHz. Draw a sketch of what you think the spectrum will look like. Then plot the spectrum and see if you got it right.\n\n##### Exercise 3-4.\n\nIn musical terminology, a \"glissando\" is a note that slides from one pitch to another, so it is similar to a chirp.\n\nFind or make a recording of a glissando and plot a spectrogram of the first few seconds. One suggestion: George Gershwin's _Rhapsody in Blue_ , which you can download from _http:\/\/archive.org\/details\/rhapblue11924_, starts with a famous clarinet glissando.\n\n##### Exercise 3-5.\n\nA trombone player can play a glissando by extending the trombone slide while blowing continuously. As the slide extends, the total length of the tube gets longer, and the resulting pitch is inversely proportional to length.\n\nAssuming that the player moves the slide at a constant speed, how does frequency vary with time?\n\nWrite a class called `TromboneGliss` that extends `Chirp` and provides `evaluate`. Make a wave that simulates a trombone glissando from C3 up to F3 and back down to C3. C3 is 262 Hz; F3 is 349 Hz.\n\nPlot a spectrogram of the resulting wave. Is a trombone glissando more like a linear or an exponential chirp?\n\n##### Exercise 3-6.\n\nMake or find a recording of a series of vowel sounds and look at the spectrogram. Can you identify different vowels?\n\n Beginning a method name with an underscore makes it \"private\", indicating that it is not part of the API that should be used outside the class definition.\n\n# Chapter 4. Noise\n\nIn English, \"noise\" means an unwanted or unpleasant sound. In the context of signal processing, it has two different senses:\n\n  1. As in English, it can mean an unwanted signal of any kind. If two signals interfere with each other, each signal would consider the other to be noise.\n\n  2. \"Noise\" also refers to a signal that contains components at many frequencies, so it lacks the harmonic structure of the periodic signals we saw in previous chapters.\n\nThis chapter is about the second kind.\n\nThe code for this chapter is in `chap04.ipynb`, which is in the repository for this book (see \"Using the Code\"). You can also view it at _http:\/\/tinyurl.com\/thinkdsp04_.\n\n# Uncorrelated Noise\n\nThe simplest way to understand noise is to generate it, and the simplest kind to generate is **uncorrelated uniform noise** (UU noise). \"Uniform\" means the signal contains random values from a uniform distribution; that is, every value in the range is equally likely. \"Uncorrelated\" means that the values are independent; that is, knowing one value provides no information about the others.\n\nHere's a class that represents UU noise:\n\n    class UncorrelatedUniformNoise(_Noise):\n\n        def evaluate(self, ts):\n            ys = np.random.uniform(-self.amp, self.amp, len(ts))\n            return ys\n\n`UncorrelatedUniformNoise` inherits from `_Noise`, which inherits from `Signal`.\n\nAs usual, the evaluate function takes `ts`, the times when the signal should be evaluated. It uses `np.random.uniform`, which generates values from a uniform distribution. In this example, the values are in the range between `-amp` and `amp`.\n\nThe following example generates UU noise with duration 0.5 seconds at 11,025 samples per second:\n\n    signal = thinkdsp.UncorrelatedUniformNoise()\n    wave = signal.make_wave(duration=0.5, framerate=11025)\n\nIf you play this wave, it sounds like the static you hear if you tune a radio between channels. Figure 4-1 shows what the waveform looks like. As expected, it looks pretty random.\n\n###### Figure 4-1. Waveform of uncorrelated uniform noise.\n\nNow let's take a look at the spectrum:\n\n    spectrum = wave.make_spectrum()\n    spectrum.plot_power()\n\n`Spectrum.plot_power` is similar to `Spectrum.plot`, except that it plots power instead of amplitude. Power is the square of amplitude. I am switching from amplitude to power in this chapter because it is more conventional in the context of noise.\n\nFigure 4-2 shows the result. Like the signal, the spectrum looks pretty random. In fact, it _is_ random, but we have to be more precise about the word \"random\". There are at least three things we might like to know about a noise signal or its spectrum:\n\nDistribution\n\nThe distribution of a random signal is the set of possible values and their probabilities. For example, in the uniform noise signal, the set of values is the range from \u20131 to 1, and all values have the same probability. An alternative is **Gaussian noise** , where the set of values is the range from negative to positive infinity, but values near 0 are the most likely, with probability that drops off according to the Gaussian or \"bell\" curve.\n\nCorrelation\n\nIs each value in the signal independent from the others, or are there dependencies between them? In UU noise, the values are independent. An alternative is **Brownian noise** , where each value is the sum of the previous value and a random \"step\". So if the value of the signal is high at a particular point in time, we expect it to stay high, and if it is low, we expect it to stay low.\n\nRelationship between power and frequency\n\nIn the spectrum of UU noise, the power at all frequencies is drawn from the same distribution; that is, the average power is the same for all frequencies. An alternative is **pink noise** , where power is inversely related to frequency; that is, the power at frequency _f_ is drawn from a distribution whose mean is proportional to  .\n\n###### Figure 4-2. Power spectrum of uncorrelated uniform noise.\n\n# Integrated Spectrum\n\nFor UU noise we can see the relationship between power and frequency more clearly by looking at the **integrated spectrum** , which is a function of frequency, _f_ , that shows the cumulative power in the spectrum up to _f_.\n\n`Spectrum` provides a method that computes the `IntegratedSpectrum`:\n\n    def make_integrated_spectrum(self):\n        cs = np.cumsum(self.power)\n        cs \/= cs[-1]\n        return IntegratedSpectrum(cs, self.fs)\n\n`self.power` is a NumPy array containing the power for each frequency. `np.cumsum` computes the cumulative sum of the powers. Dividing through by the last element normalizes the integrated spectrum so it runs from 0 to 1.\n\nThe result is an `IntegratedSpectrum`. Here is the class definition:\n\n    class IntegratedSpectrum(object):    \n        def __init__(self, cs, fs):\n            self.cs = cs\n            self.fs = fs\n\nLike `Spectrum`, `IntegratedSpectrum` provides `plot_power`, so we can compute and plot the integrated spectrum like this:\n\n    integ = spectrum.make_integrated_spectrum()\n    integ.plot_power()\n\nThe result, shown in Figure 4-3, is a straight line, which indicates that power at all frequencies is constant, on average. Noise with equal power at all frequencies is called **white noise** by analogy with light, because an equal mixture of light at all visible frequencies is white.\n\n###### Figure 4-3. Integrated spectrum of uncorrelated uniform noise.\n\n# Brownian Noise\n\nUU noise is uncorrelated, which means that each value does not depend on the others. An alternative is Brownian noise, in which each value is the sum of the previous value and a random \"step\".\n\nIt is called \"Brownian\" by analogy with Brownian motion, in which a particle suspended in a fluid moves apparently at random, due to unseen interactions with the fluid. Brownian motion is often described using a **random walk** , which is a mathematical model of a path where the distance between steps is characterized by a random distribution.\n\nIn a one-dimensional random walk, the particle moves up or down by a random amount at each timestep. The location of the particle at any point in time is the sum of all previous steps.\n\nThis observation suggests a way to generate Brownian noise: generate uncorrelated random steps and then add them up. Here is a class definition that implements this algorithm:\n\n    class BrownianNoise(_Noise):\n\n        def evaluate(self, ts):\n            dys = np.random.uniform(-1, 1, len(ts))\n            ys = np.cumsum(dys)\n            ys = normalize(unbias(ys), self.amp)\n            return ys\n\n`evaluate` uses `np.random.uniform` to generate an uncorrelated signal and `np.cumsum` to compute their cumulative sum.\n\nSince the sum is likely to escape the range from \u20131 to 1, we have to use `unbias` to shift the mean to 0, and `normalize` to get the desired maximum amplitude.\n\nHere's the code that generates a `BrownianNoise` object and plots the waveform:\n\n    signal = thinkdsp.BrownianNoise()\n    wave = signal.make_wave(duration=0.5, framerate=11025)\n    wave.plot()\n\nFigure 4-4 shows the result. The waveform wanders up and down, but there is a clear correlation between successive values. When the amplitude is high, it tends to stay high, and vice versa.\n\n###### Figure 4-4. Waveform of Brownian noise.\n\nIf you plot the spectrum of Brownian noise on a linear scale, as in Figure 4-5 (left), it doesn't look like much. Nearly all of the power is at the lowest frequencies; the higher frequency components are not visible.\n\n###### Figure 4-5. Spectrum of Brownian noise on a linear scale (left) and log-log scale (right).\n\nTo see the shape of the spectrum more clearly, we can plot power and frequency on a log-log scale. Here's the code:\n\n    spectrum = wave.make_spectrum()\n    spectrum.plot_power(linewidth=1, alpha=0.5)\n    thinkplot.config(xscale='log', yscale='log')\n\nThe result is in Figure 4-5 (right). The relationship between power and frequency is noisy, but roughly linear.\n\n`Spectrum` provides `estimate_slope`, which uses SciPy to compute a least squares fit to the power spectrum:\n\n    #class Spectrum\n\n        def estimate_slope(self):\n            x = np.log(self.fs[1:])\n            y = np.log(self.power[1:])\n            t = scipy.stats.linregress(x,y)\n            return t\n\nIt discards the first component of the spectrum because this component corresponds to  , and   is undefined.\n\n`estimate_slope` returns the result from `scipy.stats.linregress`, which is an object that contains the estimated slope and intercept, coefficient of determination ( _R 2_), p-value, and standard error. For our purposes, we only need the slope.\n\nFor Brownian noise, the slope of the power spectrum is \u20132 (we'll see why in Chapter 9), so we can write this relationship:\n\nwhere _P_ is power, _f_ is frequency, and _k_ is the intercept of the line, which is not important for our purposes. Exponentiating both sides yields:\n\nwhere _K_ is _e k_, but still not important. More relevant is that power is proportional to  , which is characteristic of Brownian noise.\n\nBrownian noise is also called **red noise** , for the same reason that white noise is called \"white\". If you combine visible light with power proportional to  , most of the power will be at the low-frequency end of the spectrum, which is red. Brownian noise is also sometimes called \"brown noise\", but I think that's confusing, so I won't use it.\n\n# Pink Noise\n\nFor red noise, the relationship between frequency and power is:\n\nThere is nothing special about the exponent 2. More generally, we can synthesize noise with any exponent, _\u03b2_ :\n\nWhen  , power is constant at all frequencies, so the result is white noise. When   the result is red noise.\n\nWhen _\u03b2_ is between 0 and 2, the result is between white and red noise, so it is called **pink noise**.\n\nThere are several ways to generate pink noise. The simplest is to generate white noise and then apply a low-pass filter with the desired exponent. `thinkdsp` provides a class that represents a pink noise signal:\n\n    class PinkNoise(_Noise):\n\n        def __init__(self, amp=1.0, beta=1.0):\n            self.amp = amp\n            self.beta = beta\n\n`amp` is the desired amplitude of the signal. `beta` is the desired exponent. `PinkNoise` provides `make_wave`, which generates a `Wave`:\n\n    def make_wave(self, duration=1, start=0, framerate=11025):\n       signal = UncorrelatedUniformNoise()\n        wave = signal.make_wave(duration, start, framerate)\n        spectrum = wave.make_spectrum()\n\n        spectrum.pink_filter(beta=self.beta)\n\n        wave2 = spectrum.make_wave()\n        wave2.unbias()\n        wave2.normalize(self.amp)\n        return wave2\n\n`duration` is the length of the wave in seconds. `start` is the start time of the wave; it is included so that `make_wave` has the same interface for all types of signal, but for random noise, start time is irrelevant. `framerate` is the number of samples per second.\n\n`make_wave` creates a white noise wave, computes its spectrum, applies a filter with the desired exponent, and then converts the filtered spectrum back to a wave. Then it unbiases and normalizes the wave.\n\n`Spectrum` provides `pink_filter`:\n\n    def pink_filter(self, beta=1.0):\n        denom = self.fs ** (beta\/2.0)\n        denom[0] = 1\n        self.hs \/= denom\n\n`pink_filter` divides each element of the spectrum by  . Since power is the square of amplitude, this operation divides the power at each component by _f \u03b2_. It treats the component at   as a special case, partly to avoid dividing by 0 and partly because this element represents the bias of the signal, which we are going to set to 0 anyway.\n\nFigure 4-6 shows the resulting waveform. Like that of Brownian noise, it wanders up and down in a way that suggests correlation between successive values, but at least visually, it looks more random. In the next chapter we will come back to this observation and I will be more precise about what I mean by \"correlation\" and \"more random\".\n\n###### Figure 4-6. Waveform of pink noise with  .\n\nFinally, Figure 4-7 shows a spectrum for white, pink, and red noise on the same log-log scale. The relationship between the exponent, _\u03b2_ , and the slope of the spectrum is apparent in this figure.\n\n###### Figure 4-7. Spectrum of white, pink, and red noise on a log-log scale.\n\n# Gaussian Noise\n\nWe started with uncorrelated uniform (UU) noise and showed that, because its spectrum has equal power at all frequencies, on average, UU noise is white.\n\nBut when people talk about \"white noise\", they don't always mean UU noise. In fact, more often they mean uncorrelated Gaussian (UG) noise.\n\n`thinkdsp` provides an implementation of UG noise:\n\n    class UncorrelatedGaussianNoise(_Noise):\n\n        def evaluate(self, ts):\n            ys = np.random.normal(0, self.amp, len(ts))\n            return ys\n\n`np.random.normal` returns a NumPy array of values from a Gaussian distribution, in this case with mean 0 and standard deviation `self.amp`. In theory the range of values is from negative to positive infinity, but we expect about 99% of the values to be between \u20133 and 3.\n\nUG noise is similar in many ways to UU noise. The spectrum has equal power at all frequencies, on average, so UG is also white. And it has one other interesting property: the spectrum of UG noise is also UG noise. More precisely, the real and imaginary parts of the spectrum are uncorrelated Gaussian values.\n\nTo test that claim, we can generate the spectrum of UG noise and then generate a \"normal probability plot\", which is a graphical way to test whether a distribution is Gaussian:\n\n    signal = thinkdsp.UncorrelatedGaussianNoise()\n    wave = signal.make_wave(duration=0.5, framerate=11025)\n    spectrum = wave.make_spectrum()\n\n    thinkstats2.NormalProbabilityPlot(spectrum.real)\n    thinkstats2.NormalProbabilityPlot(spectrum.imag)\n\n`NormalProbabilityPlot` is provided by `thinkstats2`, which is included in the repository for this book. If you are not familiar with normal probability plots, you can read about them in Chapter 5 of _Think Stats_ at _http:\/\/thinkstats2.com_.\n\nFigure 4-8 shows the results. The gray lines show a linear model fit to the data; the dark lines show the data.\n\nA straight line on a normal probability plot indicates that the data come from a Gaussian distribution. Except for some random variation at the extremes, these lines are straight, which indicates that the spectrum of UG noise is UG noise.\n\nThe spectrum of UU noise is also UG noise, at least approximately. In fact, by the Central Limit Theorem, the spectrum of almost any uncorrelated noise is approximately Gaussian, as long as the distribution has finite mean and standard deviation, and the number of samples is large.\n\n###### Figure 4-8. Normal probability plot for the real and imaginary parts of the spectrum of Gaussian noise.\n\n# Exercises\n\nSolutions to these exercises are in `chap04soln.ipynb`.\n\n##### Exercise 4-1.\n\nA Soft Murmur is a website that plays a mixture of natural noise sources, including rain, waves, wind, etc. At _http:\/\/asoftmurmur.com\/about\/_ you can find their list of recordings, most of which are at _http:\/\/freesound.org_.\n\nDownload a few of these files and compute the spectrum of each signal. Does the power spectrum look like white noise, pink noise, or Brownian noise? How does the spectrum vary over time?\n\n##### Exercise 4-2.\n\nIn a noise signal, the mixture of frequencies changes over time. In the long run, we expect the power at all frequencies to be equal, but in any sample, the power at each frequency is random.\n\nTo estimate the long-term average power at each frequency, we can break a long signal into segments, compute the power spectrum for each segment, and then compute the average across the segments. You can read more about this algorithm at _http:\/\/en.wikipedia.org\/wiki\/Bartlett's_method_.\n\nImplement Bartlett's method and use it to estimate the power spectrum for a noise wave. Hint: look at the implementation of `make_spectrogram`.\n\n##### Exercise 4-3.\n\nAt _http:\/\/www.coindesk.com\/price_, you can download historical data on the daily price of a BitCoin as a CSV file. Read this file and compute the spectrum of BitCoin prices as a function of time. Does it resemble white, pink, or Brownian noise?\n\n##### Exercise 4-4.\n\nA Geiger counter is a device that detects radiation. When an ionizing particle strikes the detector, it outputs a surge of current. The total output at a point in time can be modeled as uncorrelated Poisson (UP) noise, where each sample is a random quantity from a Poisson distribution, which corresponds to the number of particles detected during an interval.\n\nWrite a class called `UncorrelatedPoissonNoise` that inherits from `thinkdsp._Noise` and provides `evaluate`. It should use `np.random.poisson` to generate random values from a Poisson distribution. The parameter of this function, `lam`, is the average number of particles during each interval. You can use the attribute `amp` to specify `lam`. For example, if the frame rate is 10 kHz and `amp` is 0.001, we expect about 10 \"clicks\" per second.\n\nGenerate about a second of UP noise and listen to it. For low values of `amp`, like 0.001, it should sound like a Geiger counter. For higher values it should sound like white noise. Compute and plot the power spectrum to see whether it looks like white noise.\n\n##### Exercise 4-5.\n\nThe algorithm in this chapter for generating pink noise is conceptually simple but computationally expensive. There are more efficient alternatives, like the Voss\u2013McCartney algorithm. Research this method, implement it, compute the spectrum of the result, and confirm that it has the desired relationship between power and frequency.\n\n# Chapter 5. Autocorrelation\n\nIn the previous chapter I characterized white noise as \"uncorrelated\", which means that each value is independent of the others, and Brownian noise as \"correlated\", because each value depends on the preceding value. In this chapter I define these terms more precisely and present the **autocorrelation function** , which is a useful tool for signal analysis.\n\nThe code for this chapter is in `chap05.ipynb`, which is in the repository for this book (see \"Using the Code\"). You can also view it at _http:\/\/tinyurl.com\/thinkdsp05_.\n\n# Correlation\n\nIn general, correlation between variables means that if you know the value of one, you have some information about the other. There are several ways to quantify correlation, but the most common is the Pearson product-moment correlation coefficient, usually denoted _\u03c1_. For two variables, _x_ and _y_ , that each contain _N_ values:\n\nwhere _\u03bc x_ and _\u03bc y_ are the means of _x_ and _y_ , and _\u03c3 x_ and _\u03c3 y_ are their standard deviations.\n\nPearson's correlation is always between \u20131 and +1 (including both). If _\u03c1_ is positive, we say that the correlation is positive, which means that when one variable is high, the other tends to be high. If _\u03c1_ is negative, the correlation is negative, so when one variable is high, the other tends to be low.\n\nThe magnitude of _\u03c1_ indicates the strength of the correlation. If _\u03c1_ is 1 or \u20131, the variables are perfectly correlated, which means that if you know one, you can make a perfect prediction about the other. If _\u03c1_ is near zero, the correlation is probably weak, so if you know one, it doesn't tell you much about the others.\n\nI say \"probably weak\" because it is also possible that there is a nonlinear relationship that is not captured by the coefficient of correlation. Nonlinear relationships are often important in statistics, but less often relevant for signal processing, so I won't say more about them here.\n\nPython provides several ways to compute correlations. `np.corrcoef` takes any number of variables and computes a **correlation matrix** that includes correlations between each pair of variables.\n\nI'll present an example with only two variables. First, I define a function that constructs sine waves with different phase offsets:\n\n    def make_sine(offset):\n        signal = thinkdsp.SinSignal(freq=440, offset=offset)\n        wave = signal.make_wave(duration=0.5, framerate=10000)\n        return wave\n\nNext I instantiate two waves with different offsets:\n\n    wave1 = make_sine(offset=0)\n    wave2 = make_sine(offset=1)\n\nFigure 5-1 shows what the first few periods of these waves look like. When one wave is high, the other is usually high, so we expect them to be correlated:\n\n    >>> **corr_matrix = np.corrcoef(wave1.ys, wave2.ys, ddof=0)**\n    [[ 1.    0.54]\n     [ 0.54  1.  ]]\n\nThe option `ddof=0` indicates that `corrcoef` should divide by _N_ , as in the equation above, rather than use the default,  .\n\nThe result is a correlation matrix. The first element is the correlation of `wave1` with itself, which is always 1. Similarly, the last element is the correlation of `wave2` with itself.\n\nThe off-diagonal elements contain the value we're interested in, the correlation of `wave1` and `wave2`. The value 0.54 indicates that the strength of the correlation is moderate.\n\n###### Figure 5-1. Two sine waves that differ by a phase offset of 1 radian; their coefficient of correlation is 0.54.\n\nAs the phase offset increases, this correlation decreases until the waves are 180 degrees out of phase, which yields correlation \u20131. Then it increases until the offset differs by 360 degrees. At that point we have come full circle and the correlation is 1.\n\nFigure 5-2 shows the relationship between correlation and phase offset for a sine wave. The shape of that curve should look familiar; it is a cosine.\n\n###### Figure 5-2. The correlation of two sine waves as a function of the phase offset between them. The result is a cosine.\n\n`thinkdsp` provides a simple interface for computing the correlation between waves:\n\n    >>> **wave1.corr(wave2)**\n    0.54\n\n# Serial Correlation\n\nSignals often represent measurements of quantities that vary in time. For example, the sound signals we've worked with represent measurements of voltage (or current), which correspond to the changes in air pressure we perceive as sound.\n\nMeasurements like these almost always have serial correlation, which is the correlation between each element and the next (or the previous). To compute serial correlation, we can shift a signal and then compute the correlation of the shifted version with the original:\n\n    def serial_corr(wave, lag=1):\n        n = len(wave)\n        y1 = wave.ys[lag:]\n        y2 = wave.ys[:n-lag]\n        corr = np.corrcoef(y1, y2, ddof=0)[0, 1]\n        return corr\n\n`serial_corr` takes a `Wave` object and `lag`, which is the integer number of places to shift the wave. It computes the correlation of the wave with a shifted version of itself.\n\nWe can test this function with the noise signals from the previous chapter. We expect UU noise to be uncorrelated, based on the way it's generated (not to mention the name):\n\n    signal = thinkdsp.UncorrelatedGaussianNoise()\n    wave = signal.make_wave(duration=0.5, framerate=11025)\n    serial_corr(wave)\n\nWhen I ran this example, I got 0.006, which indicates a very small serial correlation. You might get a different value when you run it, but it should be comparably small.\n\nIn a Brownian noise signal, each value is the sum of the previous value and a random \"step\", so we expect a strong serial correlation:\n\n    signal = thinkdsp.BrownianNoise()\n    wave = signal.make_wave(duration=0.5, framerate=11025)\n    serial_corr(wave)\n\nSure enough, the result I got is greater than 0.999.\n\nSince pink noise is in some sense between Brownian noise and UU noise, we might expect an intermediate correlation:\n\n    signal = thinkdsp.PinkNoise(beta=1)\n    wave = signal.make_wave(duration=0.5, framerate=11025)\n    serial_corr(wave)\n\nWith parameter  , I got a serial correlation of 0.851. As we vary the parameter from  , which is uncorrelated noise, to  , which is Brownian, serial correlation ranges from 0 to almost 1, as shown in Figure 5-3.\n\n###### Figure 5-3. Serial correlation for pink noise with a range of parameters.\n\n# Autocorrelation\n\nIn the previous section we computed the correlation between each value and the next, so we shifted the elements of the array by 1. But we can easily compute serial correlations with different lags.\n\nYou can think of `serial_corr` as a function that maps from each value of `lag` to the corresponding correlation, and we can evaluate that function by looping through values of `lag`:\n\n    def autocorr(wave):\n        lags = range(len(wave.ys)\/\/2)\n        corrs = [serial_corr(wave, lag) for lag in lags]\n        return lags, corrs\n\n`autocorr` takes a `Wave` object and returns the autocorrelation function as a pair of sequences: `lags` is a sequence of integers from 0 to half the length of the wave; `corrs` is the sequence of serial correlations for each lag.\n\nFigure 5-4 shows autocorrelation functions for pink noise with three values of _\u03b2_. For low values of _\u03b2_ , the signal is less correlated, and the autocorrelation function drops off to zero quickly. For larger values, serial correlation is stronger and drops off more slowly. With   serial correlation is strong even for long lags; this phenomenon is called **long-range dependence** , because it indicates that each value in the signal depends on many preceding values.\n\n###### Figure 5-4. Autocorrelation functions for pink noise with a range of parameters.\n\n# Autocorrelation of Periodic Signals\n\nThe autocorrelation of pink noise has interesting mathematical properties, but limited applications. The autocorrelation of periodic signals is more useful.\n\nAs an example, I downloaded from _https:\/\/freesound.org_ a recording of someone singing a chirp; the repository for this book includes the file ( _https:\/\/github.com\/AllenDowney\/ThinkDSP\/blob\/master\/code\/28042__bcjordan__voicedownbew.wav_). You can use the Jupyter notebook for this chapter, `chap05.ipynb`, to play it.\n\nFigure 5-5 shows the spectrogram of this wave. The fundamental frequency and some of the harmonics show up clearly. The chirp starts near 500 Hz and drops down to about 300 Hz, roughly from C5 to E4.\n\nTo estimate pitch at a particular point in time, we could use the spectrum, but it doesn't work very well. To see why not, I'll take a short segment from the wave and plot its spectrum:\n\n    duration = 0.01\n    segment = wave.segment(start=0.2, duration=duration)\n    spectrum = segment.make_spectrum()\n    spectrum.plot(high=1000)\n\n###### Figure 5-5. Spectrogram of a vocal chirp.\n\nThis segment starts at 0.2 seconds and lasts 0.01 seconds. Figure 5-6 shows its spectrum. There is a clear peak near 400 Hz, but it is hard to identify the pitch precisely. The length of the segment is 441 samples at a frame rate of 44,100 Hz, so the frequency resolution is 100 Hz (see \"The Gabor Limit\"). That means the estimated pitch might be off by 50 Hz; in musical terms, the range from 350 Hz to 450 Hz is about 5 semitones, which is a big difference!\n\n###### Figure 5-6. Spectrum of a segment from a vocal chirp.\n\nWe could get better frequency resolution by taking a longer segment, but since the pitch is changing over time, we would also get \"motion blur\"; that is, the peak would spread between the start and end pitch of the segment, as we saw in \"Spectrum of a Chirp\".\n\nWe can estimate pitch more precisely using autocorrelation. If a signal is periodic, we expect the autocorrelation to spike when the lag equals the period.\n\nTo show why that works, I'll plot two segments from the same recording:\n\n    def plot_shifted(wave, offset=0.001, start=0.2):\n        thinkplot.preplot(2)\n        segment1 = wave.segment(start=start, duration=0.01)\n        segment1.plot(linewidth=2, alpha=0.8)\n\n        segment2 = wave.segment(start=start-offset, duration=0.01)\n        segment2.shift(offset)\n        segment2.plot(linewidth=2, alpha=0.4)\n\n        corr = segment1.corr(segment2)\n        text = r'$\\rho =$ %.2g' % corr\n        thinkplot.text(segment1.start+0.0005, -0.8, text)\n        thinkplot.config(xlabel='Time (s)')\n\nOne segment starts at 0.2 seconds; the other starts 0.0023 seconds later. Figure 5-7 shows the result. The segments are similar, and their correlation is 0.99. This result suggests that the period is near 0.0023 seconds, which corresponds to a frequency of 435 Hz.\n\n###### Figure 5-7. Two segments from a chirp, one starting 0.0023 seconds after the other.\n\nFor this example, I estimated the period by trial and error. To automate the process, we can use the autocorrelation function:\n\n    lags, corrs = autocorr(segment)\n    thinkplot.plot(lags, corrs)\n\nFigure 5-8 shows the autocorrelation function for the segment starting at   seconds. The first peak occurs at `lag=101`. We can compute the frequency that corresponds to that period like this:\n\n    period = lag \/ segment.framerate\n    frequency = 1 \/ period\n\nThe estimated fundamental frequency is 437 Hz. To evaluate the precision of the estimate, we can run the same computation with lags 100 and 102, which correspond to frequencies 432 and 441 Hz. The frequency precision using autocorrelation is less than 10 Hz, compared with 100 Hz using the spectrum. In musical terms, the expected error is about 30 cents (a third of a semitone).\n\n###### Figure 5-8. Autocorrelation function for a segment from a chirp.\n\n# Correlation as Dot Product\n\nI started the chapter with this definition of Pearson's correlation coefficient:\n\nThen I used _\u03c1_ to define serial correlation and autocorrelation. That's consistent with how these terms are used in statistics, but in the context of signal processing, the definitions are a little different.\n\nIn signal processing, we are often working with unbiased signals, where the mean is 0, and normalized signals, where the standard deviation is 1. In that case, the definition of _\u03c1_ simplifies to:\n\nAnd it is common to simplify even further:\n\nThis definition of correlation is not \"standardized\", so it doesn't generally fall between \u20131 and 1. But it has other useful properties.\n\nIf you think of _x_ and _y_ as vectors, you might recognize this formula as the **dot product** ,  . See _http:\/\/en.wikipedia.org\/wiki\/Dot_product_.\n\nThe dot product indicates the degree to which the signals are similar. If they are normalized so their standard deviations are 1:\n\nwhere _\u03b8_ is the angle between the vectors. And that explains why Figure 5-2 is a cosine curve.\n\n# Using NumPy\n\nNumPy provides a function, `correlate`, that computes the correlation of two functions or the autocorrelation of one function. We can use it to compute the autocorrelation of the segment from the previous section:\n\n    corrs2 = np.correlate(segment.ys, segment.ys, mode='same')\n\nThe option `mode` tells `correlate` what range of `lag` to use. With the value `'same'`, the range is from   to  , where _N_ is the length of the wave array.\n\nFigure 5-9 shows the result. It is symmetric because the two signals are identical, so a negative lag on one has the same effect as a positive lag on the other. To compare with the results from `autocorr`, we can select the second half:\n\n    N = len(corrs2)\n    half = corrs2[N\/\/2:]\n\n###### Figure 5-9. Autocorrelation function computed with np.correlate.\n\nIf you compare Figure 5-9 to Figure 5-8, you'll notice that the correlations computed by `np.correlate` get smaller as the lags increase. That's because `np.correlate` uses the unstandardized definition of correlation; as the lag gets bigger, the number of points in the overlap between the two signals gets smaller, so the magnitude of the correlations decreases.\n\nWe can correct that by dividing through by the lengths:\n\n    lengths = range(N, N\/\/2, -1)\n    half \/= lengths\n\nFinally, we can normalize the results so the correlation with `lag=0` is 1:\n\n    half \/= half[0]\n\nWith these adjustments, the results computed by `autocorr` and `np.correlate` are nearly the same. They still differ by 1\u20132%. The reason is not important, but if you are curious: `autocorr` standardizes the correlations independently for each lag; for `np.correlate`, we standardized them all at the end.\n\nMore importantly, now you know what autocorrelation is, how to use it to estimate the fundamental period of a signal, and two ways to compute it.\n\n# Exercises\n\nSolutions to these exercises are in `chap05soln.ipynb`.\n\n##### Exercise 5-1.\n\nThe Jupyter notebook for this chapter, `chap05.ipynb`, includes an interaction that lets you compute autocorrelations for different lags. Use this interaction to estimate the pitch of the vocal chirp for a few different start times.\n\n##### Exercise 5-2.\n\nThe example code in `chap05.ipynb` shows how to use autocorrelation to estimate the fundamental frequency of a periodic signal. Encapsulate this code in a function called `estimate_fundamental`, and use it to track the pitch of a recorded sound.\n\nTo see how well it works, try superimposing your pitch estimates on a spectrogram of the recording.\n\n##### Exercise 5-3.\n\nIf you did the exercises in the previous chapter, you downloaded the historical prices of BitCoins and estimated the power spectrum of the price changes. Using the same data, compute the autocorrelation of BitCoin prices. Does the autocorrelation function drop off quickly? Is there evidence of periodic behavior?\n\n##### Exercise 5-4.\n\nIn the repository for this book you will find a Jupyter notebook called `saxophone.ipynb` that explores autocorrelation, pitch perception, and a phenomenon called the **missing fundamental**. Read through this notebook and run the examples. Try selecting a different segment of the recording and running the examples again.\n\nVi Hart has an excellent video called \"What is up with Noises? (The Science and Mathematics of Sound, Frequency, and Pitch)\"; it demonstrates the missing fundamental phenomenon and explains how pitch perception works (at least, to the degree that we know). Watch it at _https:\/\/www.youtube.com\/watch?v=i_0DXxNeaQ0_.\n\n# Chapter 6. Discrete Cosine Transform\n\nThe topic of this chapter is the **Discrete Cosine Transform** (DCT), which is used in MP3 and related formats for compressing music; JPEG and similar formats for images; and the MPEG family of formats for video.\n\nThe DCT is similar in many ways to the Discrete Fourier Transform (DFT), which we have been using for spectral analysis. Once we learn how the DCT works, it will be easier to explain the DFT.\n\nHere are the steps to get there:\n\n  1. We'll start with the synthesis problem: given a set of frequency components and their amplitudes, how can we construct a wave?\n\n  2. Next we'll rewrite the synthesis problem using NumPy arrays. This move is good for performance, and also provides insight for the next step.\n\n  3. We'll look at the analysis problem: given a signal and a set of frequencies, how can we find the amplitude of each frequency component? We'll start with a solution that is conceptually simple but slow.\n\n  4. Finally, we'll use some principles from linear algebra to find a more efficient algorithm. If you already know linear algebra, that's great, but I'll explain what you need as we go.\n\nThe code for this chapter is in `chap06.ipynb`, which is in the repository for this book (see \"Using the Code\"). You can also view it at _http:\/\/tinyurl.com\/thinkdsp06_.\n\n# Synthesis\n\nSuppose I give you a list of amplitudes and a list of frequencies, and ask you to construct a signal that is the sum of these frequency components. Using objects in the `thinkdsp` module, there is a simple way to perform this operation, which is called **synthesis** :\n\n    def synthesize1(amps, fs, ts):\n        components = [thinkdsp.CosSignal(freq, amp)\n                      for amp, freq in zip(amps, fs)]\n        signal = thinkdsp.SumSignal(*components)\n\n        ys = signal.evaluate(ts)\n        return ys\n\n`amps` is a list of amplitudes, `fs` is the list of frequencies, and `ts` is the sequence of times where the signal should be evaluated.\n\n`components` is a list of `CosSignal` objects, one for each amplitude\u2013frequency pair. `SumSignal` represents the sum of these frequency components.\n\nFinally, `evaluate` computes the value of the signal at each time in `ts`.\n\nWe can test this function like this:\n\n    amps = np.array([0.6, 0.25, 0.1, 0.05])\n    fs = [100, 200, 300, 400]\n    framerate = 11025\n\n    ts = np.linspace(0, 1, framerate)\n    ys = synthesize1(amps, fs, ts)\n    wave = thinkdsp.Wave(ys, framerate)\n\nThis example makes a signal that contains a fundamental frequency at 100 Hz and three harmonics (100 Hz is a sharp G2). It renders the signal for 1 second at 11,025 frames per second and puts the results into a `Wave` object.\n\nConceptually, synthesis is pretty simple. But in this form it doesn't help much with **analysis** , which is the inverse problem: given the wave, how do we identify the frequency components and their amplitudes?\n\n# Synthesis with Arrays\n\nHere's another way to write `synthesize`:\n\n    def synthesize2(amps, fs, ts):\n        args = np.outer(ts, fs)\n        M = np.cos(PI2 * args)\n        ys = np.dot(M, amps)\n        return ys\n\nThis function looks very different, but it does the same thing. Let's see how it works:\n\n  1. `np.outer` computes the **outer product** of `ts` and `fs`. The result is an array with one row for each element of `ts` and one column for each element of `fs`. Each element in the array is the product of a frequency and a time,  .\n\n  2. We multiply `args` by   and apply `cos`, so each element of the result is  . Since the `ts` runs down the columns, each column contains a cosine signal at a particular frequency, evaluated at a sequence of times.\n\n  3. `np.dot` multiplies each row of `M` by `amps`, elementwise, and then adds up the products. In terms of linear algebra, we are multiplying a matrix, `M`, by a vector, `amps`. In terms of signals, we are computing the weighted sum of frequency components.\n\nFigure 6-1 shows the structure of this computation. Each row of the matrix, `M`, corresponds to a time from 0.0 to 1.0 seconds; _t n_ is the time of the _n_ th row. Each column corresponds to a frequency from 100 to 400 Hz; _f k_ is the frequency of the _k_ th column.\n\n###### Figure 6-1. Synthesis with arrays.\n\nI labeled the _n_ th row with the letters _a_ through _d_ ; as an example, the value of _a_ is  .\n\nThe result of the dot product, `ys`, is a vector with one element for each row of `M`. The _n_ th element, labeled _e_ , is the sum of products:\n\nAnd likewise with the other elements of `ys`. So each element of `ys` is the sum of four frequency components, evaluated at a point in time and multiplied by the corresponding amplitudes. And that's exactly what we wanted.\n\nWe can use the code from the previous section to check that the two versions of `synthesize` produce the same results:\n\n    ys1 = synthesize1(amps, fs, ts)\n    ys2 = synthesize2(amps, fs, ts)\n    max(abs(ys1 - ys2))\n\nThe biggest difference between `ys1` and `ys2` is about `1e-13`, which is what we expect due to floating-point errors.\n\nWriting this computation in terms of linear algebra makes the code smaller and faster. Linear algebra provides concise notation for operations on matrices and vectors. For example, we could write `synthesize` like this:\n\nwhere _a_ is a vector of amplitudes, _t_ is a vector of times, _f_ is a vector of frequencies, and _\u2297_ is the symbol for the outer product of two vectors.\n\n# Analysis\n\nNow we are ready to solve the analysis problem. Suppose I give you a wave and tell you that it is the sum of cosines with a given set of frequencies. How would you find the amplitude for each frequency component? In other words, given `ys`, `ts`, and `fs`, can you recover `amps`?\n\nIn terms of linear algebra, the first step is the same as for synthesis: we compute  . Then we want to find _a_ so that  ; in other words, we want to solve a linear system. NumPy provides `linalg.solve`, which does exactly that.\n\nHere's what the code looks like:\n\n    def analyze1(ys, fs, ts):\n        args = np.outer(ts, fs)\n        M = np.cos(PI2 * args)\n        amps = np.linalg.solve(M, ys)\n        return amps\n\nThe first two lines use `ts` and `fs` to build the matrix, `M`. Then `np.linalg.solve` computes `amps`.\n\nBut there's a hitch. In general we can only solve a system of linear equations if the matrix is square; that is, if the number of equations (rows) is the same as the number of unknowns (columns).\n\nIn this example, we have only 4 frequencies, but we evaluated the signal at 11,025 times. So we have many more equations than unknowns.\n\nIn general if `ys` contains more than 4 elements, it is unlikely that we can analyze it using only 4 frequencies.\n\nBut in this case, we know that the `ys` values were actually generated by adding only 4 frequency components, so we can use any 4 values from the wave array to recover `amps`.\n\nFor simplicity, I'll use the first 4 samples from the signal. Using the values of `ys`, `fs`, and `ts` from the previous section, we can run `analyze1` like this:\n\n    n = len(fs)\n    amps2 = analyze1(ys[:n], fs, ts[:n])\n\nAnd sure enough, `amps2` is:\n\n    [ 0.6   0.25  0.1   0.05 ]\n\nThis algorithm works, but it is slow. Solving a linear system of equations takes time proportional to _n 3_, where _n_ is the number of columns in _M_. We can do better.\n\n# Orthogonal Matrices\n\nOne way to solve linear systems is by inverting matrices. The inverse of a square matrix _M_ is written  , and it has the property that  . _I_ is the identity matrix, which has the value 1 on all diagonal elements and 0 everywhere else.\n\nSo, to solve the equation  , we can multiply both sides by  , which yields:\n\nOn the right side, we can replace   with _I_ :\n\nIf we multiply _I_ by any vector _a_ , the result is _a_ , so:\n\nThis implies that if we can compute   efficiently, we can find _a_ with a simple matrix multiplication (using `np.dot`). That takes time proportional to _n 2_, which is better than _n 3_.\n\nInverting a matrix is slow, in general, but some special cases are faster. In particular, if _M_ is **orthogonal** , the inverse of _M_ is just the transpose of _M_ , written _M T_. In NumPy transposing an array is a constant-time operation. It doesn't actually move the elements of the array; instead, it creates a \"view\" that changes the way the elements are accessed.\n\nAgain, a matrix is orthogonal if its transpose is also its inverse; that is,  . That implies that  , which means we can check whether a matrix is orthogonal by computing _M TM_.\n\nSo let's see what the matrix looks like in `synthesize2`. In the previous example, _M_ has 11,025 rows, so it might be a good idea to work with a smaller example:\n\n    def test1():\n        amps = np.array([0.6, 0.25, 0.1, 0.05])\n        N = 4.0\n        time_unit = 0.001\n        ts = np.arange(N) \/ N * time_unit\n        max_freq = N \/ time_unit \/ 2\n        fs = np.arange(N) \/ N * max_freq\n        ys = synthesize2(amps, fs, ts)\n\n`amps` is the same vector of amplitudes we saw before. Since we have 4 frequency components, we'll sample the signal at 4 points in time. That way, _M_ is square.\n\n`ts` is a vector of equally spaced sample times in the range from 0 to 1 time unit. I chose the time unit to be 1 millisecond, but it is an arbitrary choice, and we will see in a minute that it drops out of the computation anyway.\n\nSince the frame rate is _N_ samples per time unit, the Nyquist frequency is `N \/ time_unit \/ 2`, which is 2000 Hz in this example. So `fs` is a vector of equally spaced frequencies between 0 and 2000 Hz.\n\nWith these values of `ts` and `fs`, the matrix, _M_ , is:\n\n    [[ 1.     1.     1.     1.   ]\n     [ 1.     0.707  0.    -0.707]\n     [ 1.     0.    -1.    -0.   ]\n     [ 1.    -0.707 -0.     0.707]]\n\nYou might recognize 0.707 as an approximation of  , which is  . You also might notice that this matrix is **symmetric** , which means that the element at   always equals the element at  . This implies that _M_ is its own transpose; that is,  .\n\nBut sadly, _M_ is not orthogonal. If we compute _M TM_, we get:\n\n    [[ 4.  1. -0.  1.]\n     [ 1.  2.  1. -0.]\n     [-0.  1.  2.  1.]\n     [ 1. -0.  1.  2.]]\n\nAnd that's not the identity matrix.\n\n# DCT-IV\n\nBut if we choose `ts` and `fs` carefully, we can make _M_ orthogonal. There are several ways to do it, which is why there are several versions of the Discrete Cosine Transform (DCT).\n\nOne simple option is to shift `ts` and `fs` by a half unit. This version is called DCT-IV, where \"IV\" is a roman numeral indicating that this is the fourth of eight versions of the DCT.\n\nHere's an updated version of `test1`:\n\n    def test2():\n        amps = np.array([0.6, 0.25, 0.1, 0.05])\n        N = 4.0\n        ts = (0.5 + np.arange(N)) \/ N\n        fs = (0.5 + np.arange(N)) \/ 2\n        ys = synthesize2(amps, fs, ts)\n\nIf you compare this to the previous version, you'll notice two changes. First, I added 0.5 to `ts` and `fs`. Second, I canceled out `time_units`, which simplifies the expression for `fs`.\n\nWith these values, _M_ is:\n\n    [[ 0.981  0.831  0.556  0.195]\n     [ 0.831 -0.195 -0.981 -0.556]\n     [ 0.556 -0.981  0.195  0.831]\n     [ 0.195 -0.556  0.831 -0.981]]\n\nAnd _M TM_ is:\n\n    [[ 2.  0.  0.  0.]\n     [ 0.  2. -0.  0.]\n     [ 0. -0.  2. -0.]\n     [ 0.  0. -0.  2.]]\n\nSome of the off-diagonal elements are displayed as \u20130, which means that the floating-point representation is a small negative number. This matrix is very close to _2I_ , which means _M_ is almost orthogonal; it's just off by a factor of 2. And for our purposes, that's good enough.\n\nBecause _M_ is symmetric and (almost) orthogonal, the inverse of _M_ is just  . Now we can write a more efficient version of `analyze`:\n\n    def analyze2(ys, fs, ts):\n        args = np.outer(ts, fs)\n        M = np.cos(PI2 * args)\n        amps = np.dot(M, ys) \/ 2\n        return amps\n\nInstead of using `np.linalg.solve`, we just multiply by  .\n\nCombining `test2` and `analyze2`, we can write an implementation of DCT-IV:\n\n    def dct_iv(ys):\n        N = len(ys)\n        ts = (0.5 + np.arange(N)) \/ N\n        fs = (0.5 + np.arange(N)) \/ 2\n        args = np.outer(ts, fs)\n        M = np.cos(PI2 * args)\n        amps = np.dot(M, ys) \/ 2\n        return amps\n\nAgain, `ys` is the wave array. We don't have to pass `ts` and `fs` as parameters; `dct_iv` can figure them out based on `N`, the length of `ys`.\n\nIf we've got it right, this function should solve the analysis problem; that is, given `ys` it should be able to recover `amps`. We can test it like this:\n\n    amps = np.array([0.6, 0.25, 0.1, 0.05])\n    N = 4.0\n    ts = (0.5 + np.arange(N)) \/ N\n    fs = (0.5 + np.arange(N)) \/ 2\n    ys = synthesize2(amps, fs, ts)\n    amps2 = dct_iv(ys)\n    max(abs(amps - amps2))\n\nStarting with `amps`, we synthesize a wave array, then use `dct_iv` to compute `amps2`. The biggest difference between `amps` and `amps2` is about `1e-16`, which is what we expect due to floating-point errors.\n\n# Inverse DCT\n\nFinally, notice that `analyze2` and `synthesize2` are almost identical. The only difference is that `analyze2` divides the result by 2. We can use this insight to compute the inverse DCT:\n\n    def inverse_dct_iv(amps):\n        return dct_iv(amps) * 2\n\n`inverse_dct_iv` solves the synthesis problem: it takes the vector of amplitudes and returns the wave array, `ys`. We can test it by starting with `amps`, applying `inverse_dct_iv` and `dct_iv`, and testing that we get back what we started with:\n\n    amps = [0.6, 0.25, 0.1, 0.05]\n    ys = inverse_dct_iv(amps)\n    amps2 = dct_iv(ys)\n    max(abs(amps - amps2))\n\nAgain, the biggest difference is about `1e-16`.\n\n# The Dct Class\n\n`thinkdsp` provides a `Dct` class that encapsulates the DCT in the same way the `Spectrum` class encapsulates the FFT. To make a `Dct` object, you can invoke `make_dct` on a `Wave`:\n\n    signal = thinkdsp.TriangleSignal(freq=400)\n    wave = signal.make_wave(duration=1.0, framerate=10000)\n    dct = wave.make_dct()\n    dct.plot()\n\nThe result is the DCT of a triangle wave at 400 Hz, shown in Figure 6-2. The values of the DCT can be positive or negative; a negative value in the DCT corresponds to a negated cosine or, equivalently, to a cosine shifted by 180 degrees.\n\n###### Figure 6-2. DCT of a triangle signal at 400 Hz, sampled at 10 kHz.\n\n`make_dct` uses DCT-II, which is the most common type of DCT, provided by `scipy.fftpack`:\n\n    import scipy.fftpack\n\n    # class Wave:\n        def make_dct(self):\n            N = len(self.ys)\n            hs = scipy.fftpack.dct(self.ys, type=2)\n            fs = (0.5 + np.arange(N)) \/ 2\n            return Dct(hs, fs, self.framerate)\n\nThe results from `dct` are stored in `hs`. The corresponding frequencies, computed as in \"DCT-IV\", are stored in `fs`. And then both are used to initialize the `Dct` object.\n\n`Dct` provides `make_wave`, which performs the inverse DCT. We can test it like this:\n\n    wave2 = dct.make_wave()\n    max(abs(wave.ys-wave2.ys))\n\nThe biggest difference between `ys1` and `ys2` is about `1e-16`, which is what we expect due to floating-point errors.\n\n`make_wave` uses `scipy.fftpack.idct`:\n\n    # class Dct\n        def make_wave(self):\n            n = len(self.hs)\n            ys = scipy.fftpack.idct(self.hs, type=2) \/ 2 \/ n\n            return Wave(ys, framerate=self.framerate)\n\nBy default, the inverse DCT doesn't normalize the result, so we have to divide through by _2N_.\n\n# Exercises\n\nFor the following exercises, I provide some starter code in `chap06starter.ipynb`. Solutions are in `chap06soln.ipynb`.\n\n##### Exercise 6-1.\n\nIn this chapter I claim that `analyze1` takes time proportional to _n 3_ and `analyze2` takes time proportional to _n 2_. To see if that's true, run them on a range of input sizes and time them. In Jupyter, you can use the \"magic command\" `%timeit`.\n\nIf you plot run time versus input size on a log-log scale, you should get a straight line with slope 3 for `analyze1` and slope 2 for `analyze2`.\n\nYou also might want to test `dct_iv` and `scipy.fftpack.dct`.\n\n##### Exercise 6-2.\n\nOne of the major applications of the DCT is compression for both sound and images. In its simplest form, DCT-based compression works like this:\n\n  1. Break a long signal into segments.\n\n  2. Compute the DCT of each segment.\n\n  3. Identify frequency components with amplitudes so low they are inaudible, and remove them. Store only the frequencies and amplitudes that remain.\n\n  4. To play back the signal, load the frequencies and amplitudes for each segment and apply the inverse DCT.\n\nImplement a version of this algorithm and apply it to a recording of music or speech. How many components can you eliminate before the difference is perceptible?\n\nIn order to make this method practical, you need some way to store a sparse array; that is, an array where most of the elements are zero. NumPy provides several implementations of sparse arrays, which you can read about at _http:\/\/docs.scipy.org\/doc\/scipy\/reference\/sparse.html_.\n\n##### Exercise 6-3.\n\nIn the repository for this book you will find a Jupyter notebook called `phase.ipynb` that explores the effect of phase on sound perception. Read through this notebook and run the examples. Choose another segment of sound and run the same experiments. Can you find any general relationships between the phase structure of a sound and how we perceive it?\n\n# Chapter 7. Discrete Fourier Transform\n\nWe've been using the Discrete Fourier Transform (DFT) since Chapter 1, but I haven't explained how it works. Now is the time.\n\nIf you understand the Discrete Cosine Transform (DCT), you will understand the DFT. The only difference is that instead of using the cosine function, we'll use the complex exponential function. I'll start by explaining complex exponentials, then we'll follow the same progression as in Chapter 6:\n\n  1. We'll start with the synthesis problem: given a set of frequency components and their amplitudes, how can we construct a signal? The synthesis problem is equivalent to the inverse DFT.\n\n  2. Then we'll rewrite the synthesis problem in the form of matrix multiplication using NumPy arrays.\n\n  3. Next we'll solve the analysis problem, which is equivalent to the DFT: given a signal, how do we find the amplitude and phase offset of its frequency components?\n\n  4. Finally, we'll use linear algebra to find a more efficient way to compute the DFT.\n\nThe code for this chapter is in `chap07.ipynb`, which is in the repository for this book (see \"Using the Code\"). You can also view it at _http:\/\/tinyurl.com\/thinkdsp07_.\n\n# Complex Exponentials\n\nOne of the more interesting moves in mathematics is the generalization of an operation from one type to another. For example, a factorial is a function that operates on integers; the natural definition for the factorial of _n_ is the product of all integers from 1 to _n_.\n\nIf you are of a certain inclination, you might wonder how to compute the factorial of a non-integer like 3.5. Since the natural definition doesn't apply, you might look for other ways to compute the factorial function, ways that would work with non-integers.\n\nIn 1730, Leonhard Euler found one, a generalization of the factorial function that we know as the gamma function (see _http:\/\/en.wikipedia.org\/wiki\/Gamma_function_).\n\nEuler also found one of the most useful generalizations in applied mathematics, the complex exponential function.\n\nThe natural definition of exponentiation is repeated multiplication; for example,  . But this definition doesn't apply to non-integer exponents.\n\nHowever, exponentiation can also be expressed as a power series:\n\nThis definition works with real numbers, with imaginary numbers and, by a simple extension, with complex numbers. Applying this definition to a pure imaginary number, _i\u03d5_ , we get:\n\nBy rearranging terms, we can show that this is equivalent to:\n\nYou can see the derivation at _http:\/\/en.wikipedia.org\/wiki\/Euler's_formula_.\n\nThis formula implies that _e i\u03d5_ is a complex number with magnitude 1; if you think of it as a point in the complex plane, it is always on the unit circle. And if you think of it as a vector, the angle in radians between the vector and the positive x-axis is the argument _\u03d5_.\n\nIn the case where the exponent is a complex number, we have:\n\nwhere _A_ is a real number that indicates amplitude and _e i\u03d5_ is a unit complex number that indicates angle.\n\nNumPy provides a version of `exp` that works with complex numbers:\n\n    >>> **phi = 1.5**\n    >>> **z = np.exp(1j * phi)**\n    >>> **z**\n    (0.0707+0.997j)\n\nPython uses `j` to represent the imaginary unit, rather than `i`. A number ending in `j` is considered imaginary, so `1j` is just _i_.\n\nWhen the argument to `np.exp` is imaginary or complex, the result is a complex number; specifically, an `np.complex128`, which is represented by two 64-bit floating-point numbers. In this example, the result is `0.0707+0.997j`.\n\nComplex numbers have attributes `real` and `imag`:\n\n    >>> **z.real**\n    0.0707\n    >>> **z.imag**\n    0.997\n\nTo get the magnitude, you can use the built-in function `abs` or `np.absolute`:\n\n    >>> **abs(z)**\n    1.0\n    >>> **np.absolute(z)**\n    1.0\n\nTo get the angle, you can use `np.angle`:\n\n    >>> **np.angle(z)**\n    1.5\n\nThis example confirms that   is a complex number with magnitude 1 and angle _\u03d5_ radians.\n\n# Complex Signals\n\nIf   is a function of time,   is also a function of time. Specifically:\n\nThis function describes a quantity that varies in time, so it is a signal. Specifically, it is a **complex exponential signal**.\n\nIn the special case where the frequency of the signal is constant,   is   and the result is a **complex sinusoid** :\n\nOr more generally, the signal might start at a phase offset _\u03d5 0_, yielding:\n\n`thinkdsp` provides an implementation of this signal, `ComplexSinusoid`:\n\n    class ComplexSinusoid(Sinusoid):\n\n       def evaluate(self, ts):\n            phases = PI2 * self.freq * ts + self.offset\n            ys = self.amp * np.exp(1j * phases)\n            return ys\n\n`ComplexSinusoid` inherits `__init__` from `Sinusoid`. It provides a version of `evaluate` that is almost identical to `Sinusoid.evaluate`; the only difference is that it uses `np.exp` instead of `np.sin`.\n\nThe result is a NumPy array of complex numbers:\n\n    >>> **signal = thinkdsp.ComplexSinusoid(freq=1, amp=0.6, offset=1)**\n    >>> **wave = signal.make_wave(duration=1, framerate=4)**\n    >>> **wave.ys**\n    [ 0.324+0.505j -0.505+0.324j -0.324-0.505j  0.505-0.324j]\n\nThe frequency of this signal is 1 cycle per second; the amplitude is 0.6 (in unspecified units); and the phase offset is 1 radian.\n\nThis example evaluates the signal at four places equally spaced between 0 and 1 second. The resulting samples are complex numbers.\n\n# The Synthesis Problem\n\nJust as we did with real sinusoids, we can create compound signals by adding up complex sinusoids with different frequencies. And that brings us to the complex version of the synthesis problem: given the frequency and amplitude of each complex component, how do we evaluate the signal?\n\nThe simplest solution is to create `ComplexSinusoid` objects and add them up:\n\n    def synthesize1(amps, fs, ts):\n        components = [thinkdsp.ComplexSinusoid(freq, amp)\n                      for amp, freq in zip(amps, fs)]\n        signal = thinkdsp.SumSignal(*components)\n        ys = signal.evaluate(ts)\n        return ys\n\nThis function is almost identical to `synthesize1` in \"Synthesis\"; the only difference is that I replaced `CosSignal` with `ComplexSinusoid`.\n\nHere's an example:\n\n    amps = np.array([0.6, 0.25, 0.1, 0.05])\n    fs = [100, 200, 300, 400]\n    framerate = 11025\n    ts = np.linspace(0, 1, framerate)\n    ys = synthesize1(amps, fs, ts)\n\nThe result is:\n\n    [ 1.000 +0.000e+00j  0.995 +9.093e-02j  0.979 +1.803e-01j ...,\n      0.979 -1.803e-01j  0.995 -9.093e-02j  1.000 -5.081e-15j]\n\nAt the lowest level, a complex signal is a sequence of complex numbers. But how should we interpret it? We have some intuition for real signals: they represent quantities that vary in time; for example, a sound signal represents changes in air pressure. But nothing we measure in the world yields complex numbers.\n\nSo what is a complex signal? I don't have a satisfying answer to this question. The best I can offer is two unsatisfying answers:\n\n  1. A complex signal is a mathematical abstraction that is useful for computation and analysis, but it does not correspond directly with anything in the real world.\n\n  2. If you like, you can think of a complex signal as a sequence of complex numbers that contains two signals as its real and imaginary parts.\n\nTaking the second point of view, we can split the previous signal into its real and imaginary parts:\n\n    n = 500\n    thinkplot.plot(ts[:n], ys[:n].real, label='real')\n    thinkplot.plot(ts[:n], ys[:n].imag, label='imag')\n\nFigure 7-1 shows a segment of the result. The real part is a sum of cosines; the imaginary part is a sum of sines. Although the waveforms look different, they contain the same frequency components in the same proportions. To our ears, they sound the same (in general, we don't hear phase offsets).\n\n###### Figure 7-1. Real and imaginary parts of a mixture of complex sinusoids.\n\n# Synthesis with Matrices\n\nAs we saw in \"Synthesis with Arrays\", we can also express the synthesis problem in terms of matrix multiplication:\n\n    PI2 = np.pi * 2\n\n    def synthesize2(amps, fs, ts):\n        args = np.outer(ts, fs)\n        M = np.exp(1j * PI2 * args)\n        ys = np.dot(M, amps)\n        return ys\n\nAgain, `amps` is a NumPy array that contains a sequence of amplitudes.\n\n`fs` is a sequence containing the frequencies of the components. `ts` contains the times where we will evaluate the signal.\n\n`args` contains the outer product of `ts` and `fs`, with the `ts` running down the rows and the `fs` running across the columns (you might want to refer back to Figure 6-1).\n\nEach column of matrix `M` contains a complex sinusoid with a particular frequency, evaluated at a sequence of `ts`.\n\nWhen we multiply `M` by the amplitudes, the result is a vector whose elements correspond to the `ts`; each element is the sum of several complex sinusoids, evaluated at a particular time.\n\nHere's the example from the previous section again:\n\n    >>> **ys = synthesize2(amps, fs, ts)**\n    >>> **ys**\n    [ 1.000 +0.000e+00j  0.995 +9.093e-02j  0.979 +1.803e-01j ...,\n      0.979 -1.803e-01j  0.995 -9.093e-02j  1.000 -5.081e-15j]\n\nThe result is the same.\n\nIn this example the amplitudes are real, but they could also be complex. What effect does a complex amplitude have on the result? Remember that we can think of a complex number in two ways: either the sum of a real and an imaginary part,  , or the product of a real amplitude and a complex exponential,  . Using the second interpretation, we can see what happens when we multiply a complex amplitude by a complex sinusoid. For each frequency, _f_ , we have:\n\nMultiplying by   multiplies the amplitude by _A_ and adds the phase offset _\u03d5 0_.\n\nWe can test that claim by running the previous example with complex amplitudes:\n\n    phi = 1.5\n    amps2 = amps * np.exp(1j * phi)\n    ys2 = synthesize2(amps2, fs, ts)\n\n    thinkplot.plot(ts[:n], ys.real[:n])\n    thinkplot.plot(ts[:n], ys2.real[:n])\n\nSince `amps` is an array of reals, multiplying by `np.exp(1j * phi)` yields an array of complex numbers with phase offset `phi` radians, and the same magnitudes as `amps`.\n\nFigure 7-2 shows waveforms with different phase offsets. With   each frequency component gets shifted by about a quarter of a cycle. But components with different frequencies have different cycles; as a result, each component is shifted by a different amount in time. When we add up the components, the resulting waveforms look different.\n\nNow that we have the more general solution to the synthesis problem\u2014one that handles complex amplitudes\u2014we are ready for the analysis problem.\n\n###### Figure 7-2. Real part of two complex signals that differ by a phase offset.\n\n# The Analysis Problem\n\nThe analysis problem is the inverse of the synthesis problem: given a sequence of samples, _y_ , and knowing the frequencies that make up the signal, can we compute the complex amplitudes of the components, _a_?\n\nAs we saw in \"Analysis\", we can solve this problem by forming the synthesis matrix, _M_ , and solving the system of linear equations,  , for _a_ :\n\n    def analyze1(ys, fs, ts):\n        args = np.outer(ts, fs)\n        M = np.exp(1j * PI2 * args)\n        amps = np.linalg.solve(M, ys)\n        return amps\n\n`analyze1` takes a (possibly complex) wave array, `ys`, a sequence of real frequencies, `fs`, and a sequence of real times, `ts`. It returns a sequence of complex amplitudes, `amps`.\n\nContinuing the previous example, we can confirm that `analyze1` recovers the amplitudes we started with. For the linear system solver to work, `M` has to be square, so we need `ys`, `fs` and `ts`, to have the same length. I'll ensure that by slicing `ys` and `ts` down to the length of `fs`:\n\n    >>> **n = len(fs)**\n    >>> **amps2 = analyze1(ys[:n], fs, ts[:n])**\n    >>> **amps2**\n    [ 0.60+0.j  0.25-0.j  0.10+0.j  0.05-0.j]\n\nThese are approximately the amplitudes we started with, although each component has a small imaginary part due to floating-point errors.\n\n# Efficient Analysis\n\nUnfortunately, solving a linear system of equations is slow. For the DCT, we were able to speed things up by choosing `fs` and `ts` so that `M` is orthogonal. That way, the inverse of `M` is the transpose of `M`, and we can compute both the DCT and inverse DCT by matrix multiplication.\n\nWe'll do the same thing for the DFT, with one small change. Since `M` is complex, we need it to be **unitary** , rather than orthogonal, which means that the inverse of `M` is the conjugate transpose of `M`, which we can compute by transposing the matrix and negating the imaginary part of each element. See _http:\/\/en.wikipedia.org\/wiki\/Unitary_matrix_.\n\nThe NumPy methods `conj` and `transpose` do what we want. Here's the code that computes `M` for   components:\n\n    N = 4\n    ts = np.arange(N) \/ N\n    fs = np.arange(N)\n    args = np.outer(ts, fs)\n    M = np.exp(1j * PI2 * args)\n\nIf _M_ is unitary,  , where   is the conjugate transpose of _M_ , and _I_ is the identity matrix. We can test whether _M_ is unitary like this:\n\n    MstarM = M.conj().transpose().dot(M)\n\nThe result, within the tolerance of floating-point error, is  , so _M_ is unitary except for an extra factor of _N_ , similar to the extra factor of 2 we found with the DCT.\n\nWe can use this result to write a faster version of `analyze1`:\n\n    def analyze2(ys, fs, ts):\n        args = np.outer(ts, fs)\n        M = np.exp(1j * PI2 * args)\n        amps = M.conj().transpose().dot(ys) \/ N\n        return amps\n\nAnd test it with appropriate values of `fs` and `ts`:\n\n    N = 4\n    amps = np.array([0.6, 0.25, 0.1, 0.05])\n    fs = np.arange(N)\n    ts = np.arange(N) \/ N\n    ys = synthesize2(amps, fs, ts)\n    amps3 = analyze2(ys, fs, ts)\n\nAgain, the result is correct within the tolerance of floating-point arithmetic:\n\n    [ 0.60+0.j  0.25+0.j  0.10-0.j  0.05-0.j]\n\n# DFT\n\nAs a function, `analyze2` would be hard to use because it only works if `fs` and `ts` are chosen correctly. Instead, I will rewrite it to take just `ys` and compute `freq` and `ts` itself.\n\nFirst, I'll make a function to compute the synthesis matrix, _M_ :\n\n    def synthesis_matrix(N):\n        ts = np.arange(N) \/ N\n        fs = np.arange(N)\n        args = np.outer(ts, fs)\n        M = np.exp(1j * PI2 * args)\n        return M\n\nThen I'll write the function that takes `ys` and returns `amps`:\n\n    def analyze3(ys):\n        N = len(ys)\n        M = synthesis_matrix(N)\n        amps = M.conj().transpose().dot(ys) \/ N\n        return amps\n\nWe are almost done; `analyze3` computes something very close to the DFT, with one difference. The conventional definition of the DFT does not divide by `N`:\n\n    def dft(ys):\n        N = len(ys)\n        M = synthesis_matrix(N)\n        amps = M.conj().transpose().dot(ys)\n        return amps\n\nNow we can confirm that my version yields the same result as `np.fft.fft`:\n\n    >>> **dft(ys)**\n    [ 2.4+0.j  1.0+0.j  0.4-0.j  0.2-0.j]\n\nThe result is close to `amps * N`. And here's the version in `np.fft`:\n\n    >>> **np.fft.fft(ys)**\n    [ 2.4+0.j  1.0+0.j  0.4-0.j  0.2-0.j]\n\nThey are the same, within floating-point error.\n\nThe inverse DFT is almost the same, except we don't have to transpose and conjugate `M`, and _now_ we have to divide through by `N`:\n\n    def idft(ys):\n        N = len(ys)\n        M = synthesis_matrix(N)\n        amps = M.dot(ys) \/ N\n        return amps\n\nFinally, we can confirm that `dft(idft(amps))` yields `amps`:\n\n    >>> **ys = idft(amps)**\n    >>> **dft(ys)**\n    [ 0.60+0.j  0.25+0.j  0.10-0.j  0.05-0.j]\n\nIf I could go back in time, I might change the definition of the DFT so it divides by _N_ and the inverse DFT doesn't. That would be more consistent with my presentation of the synthesis and analysis problems.\n\nOr I might change the definition so that both operations divide through by  . Then the DFT and inverse DFT would be more symmetric.\n\nBut I can't go back in time (yet!), so we're stuck with a slightly weird convention. For practical purposes it doesn't really matter.\n\n# The DFT Is Periodic\n\nIn this chapter I presented the DFT in the form of matrix multiplication. We compute the synthesis matrix, _M_ , and the analysis matrix,  . When we multiply   by the wave array, _y_ , each element of the result is the product of a row from   and _y_ , which we can write in the form of a summation:\n\nwhere _k_ is an index of frequency from 0 to   and _n_ is an index of time from 0 to  . So   is the _k_ th element of the DFT of _y_.\n\nNormally we evaluate this summation for _N_ values of _k_ , running from 0 to  . We _could_ evaluate it for other values of _k_ , but there is no point, because they start to repeat. That is, the value at _k_ is the same as the value at   or   or  , etc.\n\nWe can see that mathematically by plugging   into the summation:\n\nSince there is a sum in the exponent, we can break it into two parts:\n\nIn the second term, the exponent is always an integer multiple of  , so the result is always 1, and we can drop it:\n\nAnd we can see that this summation is equivalent to  . So the DFT is periodic, with period _N_. You will need this result for one of the exercises at the end of this chapter, which asks you to implement the Fast Fourier Transform (FFT).\n\nAs an aside, writing the DFT in the form of a summation provides an insight into how it works. If you review the diagram in \"Synthesis with Arrays\", you'll see that each column of the synthesis matrix is a signal evaluated at a sequence of times. The analysis matrix is the (conjugate) transpose of the synthesis matrix, so each _row_ is a signal evaluated at a sequence of times.\n\nTherefore, each summation is the correlation of _y_ with one of the signals in the array (see \"Correlation as Dot Product\"). That is, each element of the DFT is a correlation that quantifies the similarity of the wave array, _y_ , and a complex exponential at a particular frequency.\n\n# DFT of Real Signals\n\nThe `Spectrum` class in `thinkdsp` is based on `np.ftt.rfft`, which computes the \"real DFT\"; that is, it works with real signals. But the DFT as presented in this chapter is more general than that; it works with complex signals.\n\nSo what happens when we apply the \"full DFT\" to a real signal? Let's look at an example:\n\n    signal = thinkdsp.SawtoothSignal(freq=500)\n    wave = signal.make_wave(duration=0.1, framerate=10000)\n    hs = dft(wave.ys)\n    amps = np.absolute(hs)\n\nThis code makes a sawtooth wave with frequency 500 Hz, sampled at frame rate 10 kHz. `hs` contains the complex DFT of the wave; `amps` contains the amplitude at each frequency. But what frequency do these amplitudes correspond to? If we look at the body of `dft`, we see:\n\n    fs = np.arange(N)\n\nIt's tempting to think that these values are the right frequencies. The problem is that `dft` doesn't know the sampling rate. The DFT assumes that the duration of the wave is 1 time unit, so it thinks the sampling rate is _N_ per time unit. In order to interpret the frequencies, we have to convert from these arbitrary time units back to seconds, like this:\n\n    fs = np.arange(N) * framerate \/ N\n\nWith this change, the range of frequencies is from 0 to the actual frame rate, 10 kHz. Now we can plot the spectrum:\n\n    thinkplot.plot(fs, amps)\n    thinkplot.config(xlabel='frequency (Hz)', \n                     ylabel='amplitude')\n\nFigure 7-3 shows the amplitude of the signal for each frequency component from 0 to 10 kHz. The left half of the figure is what we should expect: the dominant frequency is at 500 Hz, with harmonics dropping off like  .\n\n###### Figure 7-3. DFT of a 500 Hz sawtooth signal sampled at 10 kHz.\n\nBut the right half of the figure is a surprise. Past 5000 Hz, the amplitude of the harmonics starts growing again, peaking at 9500 Hz. What's going on?\n\nThe answer: aliasing. Remember that with frame rate 10,000 Hz, the folding frequency is 5000 Hz. As we saw in \"Aliasing\", a component at 5500 Hz is indistinguishable from a component at 4500 Hz. When we evaluate the DFT at 5500 Hz, we get the same value as at 4500 Hz. Similarly, the value at 6000 Hz is the same as the one at 4000 Hz, and so on.\n\nThe DFT of a real signal is symmetric around the folding frequency. Since there is no additional information past this point, we can save time by evaluating only the first half of the DFT, and that's exactly what `np.fft.rfft` does.\n\n# Exercises\n\nSolutions to these exercises are in `chap07soln.ipynb`.\n\n##### Exercise 7-1.\n\nThe notebook for this chapter, `chap07.ipynb`, contains additional examples and explanations. Read through it and run the code.\n\n##### Exercise 7-2.\n\nIn this chapter, I showed how we can express the DFT and inverse DFT as matrix multiplications. These operations take time proportional to _N 2_, where _N_ is the length of the wave array. That is fast enough for many applications, but there is a faster algorithm, the Fast Fourier Transform (FFT), which takes time proportional to  .\n\nThe key to the FFT is the Danielson\u2013Lanczos lemma:\n\nwhere   is the _n_ th element of the DFT of _y_ , _e_ is a wave array containing the even elements of _y_ , and _o_ contains the odd elements of _y_.\n\nThis lemma suggests a recursive algorithm for the DFT:\n\n  1. Given a wave array, _y_ , split it into its even elements, _e_ , and its odd elements, _o_.\n\n  2. Compute the DFT of _e_ and _o_ by making recursive calls.\n\n  3. Compute   for each value of _n_ using the Danielson\u2013Lanczos lemma.\n\nFor the base case of this recursion, you could wait until the length of _y_ is 1. In that case,  . Or if the length of _y_ is sufficiently small, you could compute its DFT by matrix multiplication, possibly using a precomputed matrix.\n\nHint: I suggest you implement this algorithm incrementally by starting with a version that is not truly recursive. In Step 2, instead of making a recursive call, use `dft`, as defined in \"DFT\", or `np.fft.fft`. Get Step 3 working, and confirm that the results are consistent with the other implementations. Then add a base case and confirm that it works. Finally, replace Step 2 with recursive calls.\n\nOne more hint: remember that the DFT is periodic; you might find `np.tile` useful.\n\nYou can read more about the FFT at _https:\/\/en.wikipedia.org\/wiki\/Fast_Fourier_transform_.\n\n# Chapter 8. Filtering and Convolution\n\nIn this chapter I present one of the most important and useful ideas related to signal processing: the Convolution Theorem. But before we can understand the Convolution Theorem, we have to understand convolution. I'll start with a simple example, smoothing, and we'll go from there.\n\nThe code for this chapter is in `chap08.ipynb`, which is in the repository for this book (see \"Using the Code\"). You can also view it at _http:\/\/tinyurl.com\/thinkdsp08_.\n\n# Smoothing\n\nSmoothing is an operation that tries to remove short-term variations from a signal in order to reveal long-term trends. For example, if you were to plot daily changes in the price of a stock, it would look noisy; a smoothing operator might make it easier to see whether the price was generally going up or down over time.\n\nA common smoothing algorithm is a moving average, which computes the mean of the previous _n_ values, for some value of _n_.\n\nFor example, Figure 8-1 shows the daily closing price of Facebook stock from May 17, 2012 to December 8, 2015. The gray line is the raw data, and the darker line shows the 30-day moving average. Smoothing removes the most extreme changes and makes it easier to see long-term trends.\n\n###### Figure 8-1. Daily closing price of Facebook stock and a 30-day moving average.\n\nSmoothing operations also apply to sound signals. As an example, I'll start with a square wave at 440 Hz. As we saw in \"Square Waves\", the harmonics of a square wave drop off slowly, so it contains many high-frequency components.\n\nFirst I'll construct the signal and two waves:\n\n    signal = thinkdsp.SquareSignal(freq=440)\n    wave = signal.make_wave(duration=1, framerate=44100)\n    segment = wave.segment(duration=0.01)\n\n`wave` is a one-second slice of the signal; `segment` is a shorter slice I'll use for plotting.\n\nTo compute the moving average of this signal, I'll use a window similar to the ones in \"Windowing\". Previously we used a Hamming window to avoid spectral leakage caused by discontinuity at the beginning and end of a signal. More generally, we can use windows to compute the weighted sum of samples in a wave.\n\nFor example, to compute a moving average, I'll create a window with 11 elements and normalize it so the elements add up to 1:\n\n    window = np.ones(11)\n    window \/= sum(window)\n\nNow I can compute the average of the first 11 elements by multiplying the window by the wave array:\n\n    ys = segment.ys\n    N = len(ys)\n    padded = thinkdsp.zero_pad(window, N)\n    prod = padded * ys\n    sum(prod)\n\n`padded` is a version of the window with zeros added to the end so it is the same length as `segment.ys`. Adding zeros like this is called **padding**.\n\n`prod` is the product of the window and the wave array. The sum of the elementwise products is the average of the first 11 elements of the array. Since these elements are all \u20131, their average is \u20131.\n\nTo compute the next element of the moving average, we **roll** the window, which shifts the ones to the right and wraps one of the zeros from the end around to the beginning.\n\nWhen we multiply the rolled window and the wave array, we get the average of the next 11 elements of the wave array, starting with the second:\n\n    rolled = np.roll(rolled, 1)\n    prod = rolled * ys\n    sum(prod)\n\nThe result is \u20131 again.\n\nWe can compute the rest of the elements the same way. The following function wraps the code we have seen so far in a loop and stores the results in an array:\n\n    def smooth(ys, window):\n        N = len(ys)\n        smoothed = np.zeros(N)\n        padded = thinkdsp.zero_pad(window, N)\n        rolled = padded\n\n        for i in range(N):\n            smoothed[i] = sum(rolled * ys)\n            rolled = np.roll(rolled, 1)\n        return smoothed\n\n`smoothed` is the array that will contain the results; `padded` is an array that contains the window and enough zeros to have length `N`; and `rolled` is a copy of `padded` that gets shifted to the right by one element each time through the loop.\n\nInside the loop, we multiply `ys` by `rolled` to select 11 elements and add them up.\n\nFigure 8-2 shows the result for a square wave. The gray line is the original signal; the darker line is the smoothed signal. The smoothed signal starts to ramp up when the leading edge of the window reaches the first transition, and levels off when the window crosses the transition. As a result, the transitions are less abrupt, and the corners less sharp. If you listen to the smoothed signal, it sounds less buzzy and slightly muffled.\n\n###### Figure 8-2. A square signal at 400 Hz (gray) and an 11-element moving average.\n\n# Convolution\n\nThe operation we just performed\u2014applying a window function to each overlapping segment of a wave\u2014is called **convolution**.\n\nConvolution is such a common operation that NumPy provides an implementation that is simpler and faster than my version:\n\n    convolved = np.convolve(ys, window, mode='valid')\n    smooth2 = thinkdsp.Wave(convolved, framerate=wave.framerate)\n\n`np.convolve` computes the convolution of the wave array and the window. The mode flag `valid` indicates that it should only compute values when the window and the wave array overlap completely, so it stops when the right edge of the window reaches the end of the wave array. Other than that, the result is the same as in Figure 8-2.\n\nActually, there is one other difference. The loop in the previous section actually computes **cross-correlation** :\n\nwhere _f_ is a wave array with length _N_ , _g_ is the window, and \u2729 is the symbol for cross-correlation. To compute the _n_ th element of the result, we shift _g_ to the right, which is why the index is  .\n\nThe definition of convolution is slightly different:\n\nThe symbol _*_ represents convolution. The difference is in the index of _g_ : _m_ has been negated, so the summation iterates the elements of _g_ backward (assuming that negative indices wrap around to the end of the array).\n\nBecause the window we used in this example is symmetric, cross-correlation and convolution yield the same result. When we use other windows, we will have to be more careful.\n\nYou might wonder why convolution is defined like this, with the window applied in a way that seems backward. There are two reasons:\n\n  * This definition comes up naturally for several applications, especially analysis of signal-processing systems, which is the topic of Chapter 10.\n\n  * Also, this definition is the basis of the Convolution Theorem, coming up very soon.\n\nFinally, a note for people who know too much: in the presentation so far I have not distinguished between convolution and circular convolution. We'll get to it.\n\n# The Frequency Domain\n\nSmoothing makes the transitions in a square signal less abrupt, and makes the sound slightly muffled. Let's see what effect this operation has on the spectrum. First I'll plot the spectrum of the original wave:\n\n    spectrum = wave.make_spectrum()\n    spectrum.plot(color=GRAY)\n\nThen the smoothed wave:\n\n    convolved = np.convolve(wave.ys, window, mode='same')\n    smooth = thinkdsp.Wave(convolved, framerate=wave.framerate)\n    spectrum2 = smooth.make_spectrum()\n    spectrum2.plot()\n\nThe mode flag `same` indicates that the result should have the same length as the input. In this example, it will include a few values that \"wrap around\", but that's OK for now.\n\nFigure 8-3 shows the result. The fundamental frequency is almost unchanged; the first few harmonics are attenuated, and the higher harmonics are almost eliminated. So smoothing has the effect of a low-pass filter, which we saw in \"Spectrums\" and \"Pink Noise\".\n\n###### Figure 8-3. Spectrum of the square wave before and after smoothing.\n\nTo see how much each component has been attenuated, we can compute the ratio of the two spectrums:\n\n    amps = spectrum.amps\n    amps2 = spectrum2.amps\n    ratio = amps2 \/ amps    \n    ratio[amps<560] = 0\n    thinkplot.plot(ratio)\n\n`ratio` is the ratio of the amplitude before and after smoothing. When `amps` is small, this ratio can be big and noisy, so for simplicity I set the ratio to 0 except where the harmonics are.\n\nFigure 8-4 shows the result. As expected, the ratio is high for low frequencies and drops off at a cutoff frequency near 4000 Hz. But there is another feature we did not expect: above the cutoff, the ratio bounces around between 0 and 0.2. What's up with that?\n\n###### Figure 8-4. Ratio of spectrums for the square wave, before and after smoothing.\n\n# The Convolution Theorem\n\nThe answer is the Convolution Theorem. Stated mathematically:\n\nwhere _f_ is a wave array and _g_ is a window. In words, the Convolution Theorem says that if we convolve _f_ and _g_ , and then compute the DFT, we get the same answer as when computing the DFT of _f_ and _g_ , and then multiplying the results elementwise.\n\nWhen we apply an operation like convolution to a wave, we say we are working in the **time domain** , because the wave is a function of time. When we apply an operation like multiplication to the DFT, we are working in the **frequency domain** , because the DFT is a function of frequency.\n\nUsing these terms, we can state the Convolution Theorem more concisely:\n\n> Convolution in the time domain corresponds to multiplication in the frequency domain.\n\nAnd that explains Figure 8-4, because when we convolve a wave and a window, we multiply the spectrum of the wave with the spectrum of the window. To see how that works, we can compute the DFT of the window:\n\n    padded = zero_pad(window, N)\n    dft_window = np.fft.rfft(padded)\n    thinkplot.plot(abs(dft_window))\n\n`padded` contains the smoothing window, padded with zeros to be the same length as `wave`; `dft_window` contains the DFT of `padded`.\n\nFigure 8-5 shows the result, along with the ratios we computed in the previous section. The ratios are exactly the amplitudes in `dft_window`. Mathematically:\n\nIn this context, the DFT of a window is called a **filter**. For any convolution window in the time domain, there is a corresponding filter in the frequency domain. And for any filter that can be expressed by elementwise multiplication in the frequency domain, there is a corresponding window.\n\n###### Figure 8-5. Ratio of spectrums for the square wave, before and after smoothing, along with the DFT of the smoothing window.\n\n# Gaussian Filter\n\nThe moving average window we used in the previous section is a low-pass filter, but it is not a very good one. The DFT drops off steeply at first, but then it bounces around. Those bounces are called **sidelobes** , and they are there because the moving average window is like a square wave, so its spectrum contains high-frequency harmonics that drop off proportionally to  , which is relatively slow.\n\nWe can do better with a Gaussian window. SciPy provides functions that compute many common convolution windows, including `gaussian`:\n\n    gaussian = scipy.signal.gaussian(M=11, std=2)\n    gaussian \/= sum(gaussian)\n\n`M` is the number of elements in the window; `std` is the standard deviation of the Gaussian distribution used to compute it. Figure 8-6 shows the shape of the window. It is a discrete approximation of the Gaussian \"bell curve\". The figure also shows the moving average window from the previous example, which is sometimes called a **boxcar window** because it looks like a rectangular railway car.\n\n###### Figure 8-6. Boxcar and Gaussian windows.\n\nI ran the computations from the previous sections again with this window and generated Figure 8-7, which shows the ratio of the spectrums before and after smoothing, along with the DFT of the Gaussian window.\n\nAs a low-pass filter, Gaussian smoothing is better than a simple moving average. After the ratio drops off, it stays low, with almost none of the sidelobes we saw with the boxcar window. So it does a better job of cutting off the higher frequencies.\n\nThe reason it does so well is that the DFT of a Gaussian curve is also a Gaussian curve. So the ratio drops off in proportion to  , which is much faster than  .\n\n###### Figure 8-7. Ratio of spectrums before and after Gaussian smoothing, and the DFT of the window.\n\n# Efficient Convolution\n\nOne of the reasons the FFT is such an important algorithm is that, combined with the Convolution Theorem, it provides an efficient way to compute convolution, cross-correlation, and autocorrelation.\n\nAgain, the Convolution Theorem states:\n\nSo one way to compute a convolution is:\n\nwhere IDFT is the inverse DFT. A simple implementation of convolution takes time proportional to _N 2_; this algorithm, using the FFT, takes time proportional to  .\n\nWe can confirm that it works by computing the same convolution both ways. As an example, I'll apply it to the Facebook stock data shown in Figure 8-1:\n\n    import pandas as pd\n\n    names = ['date', 'open', 'high', 'low', 'close', 'volume']\n    df = pd.read_csv('fb.csv', header=0, names=names)\n    ys = df.close.values[::-1]\n\nThis example uses Pandas to read the data from the CSV file (included in the repository for this book). If you are not familiar with Pandas, don't worry: I'm not going to do much with it in this book. But if you're interested, you can learn more about it in _Think Stats_ at _http:\/\/thinkstats2.com_.\n\nThe result, `df`, is a `DataFrame`, one of the data structures provided by Pandas. `close` is a NumPy array that contains daily closing prices.\n\nNext I'll create a Gaussian window and convolve it with `close`:\n\n    window = scipy.signal.gaussian(M=30, std=6)\n    window \/= window.sum()\n    smoothed = np.convolve(ys, window, mode='valid')\n\n`fft_convolve` computes the same thing using the FFT:\n\n    from np.fft import fft, ifft\n\n    def fft_convolve(signal, window):\n        fft_signal = fft(signal)\n        fft_window = fft(window)\n        return ifft(fft_signal * fft_window)\n\nWe can test it by padding the window to the same length as `ys` and then computing the convolution:\n\n    padded = zero_pad(window, N)\n    smoothed2 = fft_convolve(ys, padded)\n\nThe result has   bogus values at the beginning, where _M_ is the length of the window. We can slice off the bogus values like this:\n\n    M = len(window)\n    smoothed2 = smoothed2[M-1:]\n\nThe result agrees with `fft_convolve` with about 12 digits of precision.\n\n# Efficient Autocorrelation\n\nIn \"Convolution\" I presented definitions of cross-correlation and convolution, and we saw that they are almost the same, except that in convolution the window is reversed.\n\nNow that we have an efficient algorithm for convolution, we can also use it to compute cross-correlations and autocorrelations. Using the data from the previous section, we can compute the autocorrelation of Facebook stock prices:\n\n    corrs = np.correlate(close, close, mode='same')\n\nWith `mode='same'`, the result has the same length as `close`, corresponding to lags from   to  . The gray line in Figure 8-8 shows the result. Except at `lag=0`, there are no peaks, so there is no apparent periodic behavior in this signal. However, the autocorrelation function drops off slowly, suggesting that this signal resembles pink noise, as we saw in \"Autocorrelation\".\n\n###### Figure 8-8. Autocorrelation functions computed by NumPy and fft_autocorr.\n\nTo compute autocorrelation using convolution, we have to zero-pad the signal to double the length. This trick is necessary because the FFT is based on the assumption that the signal is periodic; that is, that it wraps around from the end to the beginning. With time-series data like this, that assumption is invalid. Adding zeros, and then trimming the results, removes the bogus values.\n\nAlso, remember that convolution reverses the direction of the window. In order to cancel that effect, we reverse the direction of the window before calling `fft_convolve`, using `np.flipud`, which flips a NumPy array. The result is a view of the array, not a copy, so this operation is fast:\n\n    def fft_autocorr(signal):\n        N = len(signal)\n        signal = thinkdsp.zero_pad(signal, 2*N)\n        window = np.flipud(signal)\n\n        corrs = fft_convolve(signal, window)\n        corrs = np.roll(corrs, N\/\/2+1)[:N]\n        return corrs\n\nThe result from `fft_convolve` has length 2 _N_. Of those, the first and last   are valid; the rest are the result of zero-padding. To select the valid element, we roll the results and select the first _N_ , corresponding to lags from   to  .\n\nAs shown in Figure 8-8, the results from `fft_autocorr` and `np.correlate` are identical (with about 9 digits of precision).\n\nNotice that the correlations in Figure 8-8 are large numbers; we could normalize them (between \u20131 and 1) as shown in \"Using NumPy\".\n\nThe strategy we used here for autocorrelation also works for cross-correlation. Again, you have to prepare the signals by flipping one and padding both, and then you have to trim the invalid parts of the result. This padding and trimming is a nuisance, but that's why libraries like NumPy provide functions to do it for you.\n\n# Exercises\n\nSolutions to these exercises are in `chap08soln.ipynb`.\n\n##### Exercise 8-1.\n\nThe notebook for this chapter is `chap08.ipynb`. Read through it and run the code.\n\nIt contains an interactive widget that lets you experiment with the parameters of the Gaussian window to see what effect they have on the cutoff frequency.\n\nWhat goes wrong when you increase the width of the Gaussian window, `std`, without increasing the number of elements in the window, `M`?\n\n##### Exercise 8-2.\n\nIn this chapter I claimed that the Fourier Transform of a Gaussian curve is also a Gaussian curve. For Discrete Fourier Transforms, this relationship is approximately true.\n\nTry it out for a few examples. What happens to the Fourier Transform as you vary `std`?\n\n##### Exercise 8-3.\n\nIf you did the exercises in Chapter 3, you saw the effect of the Hamming window, and some of the other windows provided by NumPy, on spectral leakage. We can get some insight into the effects of these windows by looking at their DFTs.\n\nIn addition to the Gaussian window we used in this chapter, create a Hamming window with the same size. Zero-pad the windows and plot their DFTs. Which window acts as a better low-pass filter? You might find it useful to plot the DFTs on a log- _y_ scale.\n\nExperiment with a few different windows and a few different sizes.\n\n# Chapter 9. Differentiation and Integration\n\nThis chapter picks up where the previous chapter left off, looking at the relationship between windows in the time domain and filters in the frequency domain.\n\nIn particular, we'll look at the effects of a finite difference window, which approximates differentiation, and the cumulative sum operation, which approximates integration.\n\nThe code for this chapter is in `chap09.ipynb`, which is in the repository for this book (see \"Using the Code\"). You can also view it at _http:\/\/tinyurl.com\/thinkdsp09_.\n\n# Finite Differences\n\nIn \"Smoothing\", we applied a smoothing window to the daily stock price of Facebook and found that a smoothing window in the time domain corresponds to a low-pass filter in the frequency domain.\n\nIn this section, we'll look at daily price changes and see that computing the difference between successive elements, in the time domain, corresponds to a high-pass filter.\n\nHere's the code to read the data, store it as a wave, and compute its spectrum:\n\n    import pandas as pd\n\n    names = ['date', 'open', 'high', 'low', 'close', 'volume']\n    df = pd.read_csv('fb.csv', header=0, names=names)\n    ys = df.close.values[::-1]\n    close = thinkdsp.Wave(ys, framerate=1)\n    spectrum = wave.make_spectrum()\n\nThis example uses Pandas to read the CSV file; the result is a `DataFrame`, `df`, with columns for the opening price, closing price, and high and low prices. I select the closing prices and save them in a `Wave` object. The frame rate is 1 sample per day.\n\nFigure 9-1 shows this time series and its spectrum. Visually, the time series resembles Brownian noise (see \"Brownian Noise\"). And the spectrum looks like a straight line, albeit a noisy one. The estimated slope is \u20131.9, which is consistent with Brownian noise.\n\n###### Figure 9-1. Daily closing price of Facebook stock and the spectrum of this time series.\n\nNow let's compute the daily price change using `np.diff`:\n\n    diff = np.diff(ys)\n    change = thinkdsp.Wave(diff, framerate=1)\n    change_spectrum = change.make_spectrum()\n\nFigure 9-2 shows the resulting wave and its spectrum. The daily changes resemble white noise, and the estimated slope of the spectrum, \u20130.06, is near zero, which is what we expect for white noise.\n\n###### Figure 9-2. Daily price change of Facebook stock and the spectrum of this time series.\n\n# The Frequency Domain\n\nComputing the difference between successive elements is the same as convolution with the window `[1, -1]`. If the order of those elements seems backward, remember that convolution reverses the window before applying it to the signal.\n\nWe can see the effect of this operation in the frequency domain by computing the DFT of the window:\n\n    diff_window = np.array([1.0, -1.0])\n    padded = thinkdsp.zero_pad(diff_window, len(close))\n    diff_wave = thinkdsp.Wave(padded, framerate=close.framerate)\n    diff_filter = diff_wave.make_spectrum()\n\nFigure 9-3 shows the result. The finite difference window corresponds to a high-pass filter: its amplitude increases with frequency, linearly for low frequencies and then sublinearly after that. In the next section, we'll see why.\n\n###### Figure 9-3. Filters corresponding to the diff and differentiate operators (left) and integration operator (right, log-y scale).\n\n# Differentiation\n\nThe window we used in the previous section is a numerical approximation of the first derivative, so the filter approximates the effect of differentiation.\n\nDifferentiation in the time domain corresponds to a simple filter in the frequency domain; we can figure out what it is with a little math.\n\nSuppose we have a complex sinusoid with frequency _f_ :\n\nThe first derivative of _E f_ is:\n\nwhich we can rewrite as:\n\nIn other words, taking the derivative of _E f_ is the same as multiplying by  , which is a complex number with magnitude   and angle  .\n\nWe can compute the filter that corresponds to differentiation like this:\n\n    deriv_filter = close.make_spectrum()\n    deriv_filter.hs = PI2 * 1j * deriv_filter.fs\n\nI started with the spectrum of `close`, which has the right size and frame rate, then replaced the `hs` with  . Figure 9-3 (left) shows this filter; it is a straight line.\n\nAs we saw in \"Synthesis with Matrices\", multiplying a complex sinusoid by a complex number has two effects: it multiplies the amplitude, in this case by  , and shifts the phase offset, in this case by  .\n\nIf you are familiar with the language of operators and eigenfunctions, each _E f_ is an eigenfunction of the differentiation operator, with the corresponding eigenvalue  . See _http:\/\/en.wikipedia.org\/wiki\/Eigenfunction_.\n\nIf you are not familiar with that language, here's what it means:\n\n  * An operator is a function that takes a function and returns another function. For example, differentiation is an operator.\n\n  * A function, _g_ , is an eigenfunction of an operator,  , if applying   to _g_ has the effect of multiplying the function by a scalar. That is,  .\n\n  * In that case, the scalar _\u03bb_ is the eigenvalue that corresponds to the eigenfunction _g_.\n\n  * A given operator might have many eigenfunctions, each with a corresponding eigenvalue.\n\nBecause complex sinusoids are eigenfunctions of the differentiation operator, they are easy to differentiate. All we have to do is multiply by a complex scalar.\n\nFor signals with more than one component, the process is only slightly harder:\n\n  1. Express the signal as the sum of complex sinusoids.\n\n  2. Compute the derivative of each component by multiplication.\n\n  3. Add up the differentiated components.\n\nIf that process sounds familiar, that's because it is identical to the algorithm for convolution in \"Efficient Convolution\": compute the DFT, multiply by a filter, and compute the inverse DFT.\n\n`Spectrum` provides a method that applies the differentiation filter:\n\n    # class Spectrum:\n\n        def differentiate(self):\n            self.hs *= PI2 * 1j * self.fs\n\nWe can use it to compute the derivative of the Facebook time series:\n\n    deriv_spectrum = close.make_spectrum()\n    deriv_spectrum.differentiate()\n    deriv = deriv_spectrum.make_wave()\n\nFigure 9-4 compares the daily price changes computed by `np.diff` with the derivative we just computed. I selected the first 50 values in the time series so we can see the differences more clearly.\n\n###### Figure 9-4. Comparison of daily price changes computed by np.diff and by applying the differentiation filter.\n\nThe derivative is noisier, because it amplifies the high-frequency components more, as shown in Figure 9-3 (left). Also, the first few elements of the derivative are very noisy. The problem there is that the DFT-based derivative is based on the assumption that the signal is periodic. In effect, it connects the last element in the time series back to the first element, which creates artifacts at the boundaries.\n\nTo summarize, we have shown:\n\n  * Computing the difference between successive values in a signal can be expressed as convolution with a simple window. The result is an approximation of the first derivative.\n\n  * Differentiation in the time domain corresponds to a simple filter in the frequency domain. For periodic signals, the result is the first derivative, exactly. For some non-periodic signals, it can approximate the derivative.\n\nUsing the DFT to compute derivatives is the basis of **spectral methods** for solving differential equations (see _http:\/\/en.wikipedia.org\/wiki\/Spectral_method_).\n\nIn particular, it is useful for the analysis of linear, time-invariant systems, which is coming up in Chapter 10.\n\n# Integration\n\nIn the previous section, we showed that differentiation in the time domain corresponds to a simple filter in the frequency domain: it multiplies each component by  . Since integration is the inverse of differentiation, it also corresponds to a simple filter: it divides each component by  .\n\nWe can compute this filter like this:\n\n    integ_filter = close.make_spectrum()\n    integ_filter.hs = 1 \/ (PI2 * 1j * integ_filter.fs)\n\nFigure 9-3 (right) shows this filter on a log- _y_ scale, which makes it easier to see.\n\n`Spectrum` provides a method that applies the integration filter:\n\n    # class Spectrum:\n\n        def integrate(self):\n            self.hs \/= PI2 * 1j * self.fs\n\nWe can confirm that the integration filter is correct by applying it to the spectrum of the derivative we just computed:\n\n    integ_spectrum = deriv_spectrum.copy()\n    integ_spectrum.integrate()\n\nBut notice that at  , we are dividing by 0. The result in NumPy is NaN, which is a special floating-point value that represents \"not a number\". We can partially deal with this problem by setting this value to 0 before converting the spectrum back to a wave:\n\n    integ_spectrum.hs[0] = 0\n    integ_wave = integ_spectrum.make_wave()\n\nFigure 9-5 shows this integrated derivative along with the original time series. They are almost identical, but the integrated derivative has been shifted down. The problem is that when we clobbered the   component, we set the bias of the signal to 0. But that should not be surprising; in general, differentiation loses information about the bias, and integration can't recover it. In some sense, the NaN at   is telling us that this element is unknown.\n\n###### Figure 9-5. Comparison of the original time series and the integrated derivative.\n\nIf we provide this \"constant of integration\", the results are identical, which confirms that this integration filter is the correct inverse of the differentiation filter.\n\n# Cumulative Sum\n\nIn the same way that the `diff` operator approximates differentiation, the cumulative sum approximates integration. I'll demonstrate with a sawtooth signal:\n\n    signal = thinkdsp.SawtoothSignal(freq=50)\n    in_wave = signal.make_wave(duration=0.1, framerate=44100)\n\nFigure 9-6 shows this wave and its spectrum.\n\n`Wave` provides a method that computes the cumulative sum of a wave array and returns a new `Wave` object:\n\n    # class Wave:\n\n        def cumsum(self):\n            ys = np.cumsum(self.ys)\n            ts = self.ts.copy()\n            return Wave(ys, ts, self.framerate)\n\nWe can use it to compute the cumulative sum of `in_wave`:\n\n    out_wave = in_wave.cumsum()\n    out_wave.unbias()\n\nFigure 9-7 shows the resulting wave and its spectrum. If you did the exercises in Chapter 2, this waveform should look familiar: it's a parabolic signal.\n\n###### Figure 9-6. A sawtooth wave and its spectrum.\n\n###### Figure 9-7. A parabolic wave and its spectrum.\n\nComparing the two, we see that the amplitudes of the components drop off more quickly in the spectrum of the parabolic signal than in the spectrum of the sawtooth signal. In Chapter 2, we saw that the components of the sawtooth drop off in proportion to  . Since the cumulative sum approximates integration, and integration filters components in proportion to  , the components of the parabolic wave drop off in proportion to  .\n\nWe can see that graphically by computing the filter that corresponds to the cumulative sum:\n\n    cumsum_filter = diff_filter.copy()\n    cumsum_filter.hs = 1 \/ cumsum_filter.hs\n\nBecause `cumsum` is the inverse operation of `diff`, we start with a copy of `diff_filter`, which is the filter that corresponds to the `diff` operation, and then invert the `hs`.\n\nFigure 9-8 shows the filters corresponding to cumulative sum and integration. The cumulative sum is a good approximation of integration except at the highest frequencies, where it drops off a little faster.\n\n###### Figure 9-8. Filters corresponding to cumulative sum and integration.\n\nTo confirm that this is the correct filter for the cumulative sum, we can compare it to the ratio of the spectrum `out_wave` to the spectrum of `in_wave`:\n\n    in_spectrum = in_wave.make_spectrum()\n    out_spectrum = out_wave.make_spectrum()\n    ratio_spectrum = out_spectrum.ratio(in_spectrum, thresh=1)\n\nAnd here's the method that computes the ratios:\n\n    def ratio(self, denom, thresh=1):\n        ratio_spectrum = self.copy()\n        ratio_spectrum.hs \/= denom.hs\n        ratio_spectrum.hs[denom.amps < thresh] = np.nan\n        return ratio_spectrum\n\nWhen `denom.amps` is small, the resulting ratio is noisy, so I set those values to NaN.\n\nFigure 9-9 shows the ratios and the filter corresponding to the cumulative sum. They agree, which confirms that inverting the filter for `diff` yields the filter for `cumsum`.\n\n###### Figure 9-9. Filter corresponding to cumulative sum and actual ratios of the before-and-after spectrums.\n\nFinally, we can confirm that the Convolution Theorem applies by applying the `cumsum` filter in the frequency domain:\n\n    out_wave2 = (in_spectrum * cumsum_filter).make_wave()\n\nWithin the limits of floating-point error, `out_wave2` is identical to `out_wave`, which we computed using `cumsum`, so the Convolution Theorem works! But note that this demonstration only works with periodic signals.\n\n# Integrating Noise\n\nIn \"Brownian Noise\", we generated Brownian noise by computing the cumulative sum of white noise. Now that we understand the effect of `cumsum` in the frequency domain, we have some insight into the spectrum of Brownian noise.\n\nWhite noise has equal power at all frequencies, on average. When we compute the cumulative sum, the amplitude of each component is divided by _f_. Since power is the square of magnitude, the power of each component is divided by _f 2_. So on average, the power at frequency _f_ is proportional to  :\n\nwhere _K_ is a constant that's not important. Taking the log of both sides yields:\n\nAnd that's why, when we plot the spectrum of Brownian noise on a log-log scale, we expect to see a straight line with slope \u20132, at least approximately.\n\nIn \"Finite Differences\" we plotted the spectrum of closing prices for Facebook stock, and estimated that the slope is \u20131.9, which is consistent with Brownian noise. Many stock prices have similar spectrums.\n\nWhen we used the `diff` operator to compute daily changes, we multiplied the _amplitude_ of each component by a filter proportional to _f_ , which means we multiplied the _power_ of each component by _f 2_. On a log-log scale, this operation adds 2 to the slope of the power spectrum, which is why the estimated slope of the result is near 0.1 (but a little lower, because `diff` only approximates differentiation).\n\n# Exercises\n\nSolutions to these exercises are in `chap09soln.ipynb`.\n\n##### Exercise 9-1.\n\nThe notebook for this chapter is `chap09.ipynb`. Read through it and run the code.\n\nIn \"Cumulative Sum\", I mentioned that some of the examples don't work with non-periodic signals. Try replacing the sawtooth wave, which is periodic, with the Facebook data, which is not, and see what goes wrong.\n\n##### Exercise 9-2.\n\nThe goal of this exercise is to explore the effects of `diff` and `differentiate` on a signal. Create a triangle wave and plot it. Apply `diff` and plot the result. Compute the spectrum of the triangle wave, apply `differentiate`, and plot the result. Convert the spectrum back to a wave and plot it. Are there differences between the effects of `diff` and `differentiate` for this wave?\n\n##### Exercise 9-3.\n\nThe goal of this exercise is to explore the effects of `cumsum` and `integrate` on a signal. Create a square wave and plot it. Apply `cumsum` and plot the result. Compute the spectrum of the square wave, apply `integrate`, and plot the result. Convert the spectrum back to a wave and plot it. Are there differences between the effects of `cumsum` and `integrate` for this wave?\n\n##### Exercise 9-4.\n\nThe goal of this exercise is to explore the effect of integrating twice. Create a sawtooth wave, compute its spectrum, then apply `integrate` twice. Plot the resulting wave and its spectrum. What is the mathematical form of the wave? Why does it resemble a sinusoid?\n\n##### Exercise 9-5.\n\nThe goal of this exercise is to explore the effects of the second difference and second derivative. Create a `CubicSignal`, which is defined in `thinkdsp`. Compute the second difference by applying `diff` twice. What does the result look like? Compute the second derivative by applying `differentiate` to the spectrum twice. Does the result look the same?\n\nPlot the filters that correspond to the second difference and the second derivative and compare them. Hint: in order to get the filters on the same scale, use a wave with frame rate 1.\n\n# Chapter 10. LTI Systems\n\nThis chapter presents the theory of signals and systems, using musical acoustics as an example. It explains an important application of the Convolution Theorem, characterization of linear, time-invariant systems (which I'll define soon).\n\nThe code for this chapter is in `chap10.ipynb`, which is in the repository for this book (see \"Using the Code\"). You can also view it at _http:\/\/tinyurl.com\/thinkdsp10_.\n\n# Signals and Systems\n\nIn the context of signal processing, a **system** is an abstract representation of anything that takes a signal as input and produces a signal as output.\n\nFor example, an electronic amplifier is a circuit that takes an electrical signal as input and produces a (louder) signal as output.\n\nAs another example, when you listen to a musical performance, you can think of the room as a system that takes the sound of the performance at the location where it is generated and produces a somewhat different sound at the location where you hear it.\n\nA **linear, time-invariant system** is a system with these two properties:\n\nLinearity\n\nIf you put two inputs into the system at the same time, the result is the sum of their outputs. Mathematically, if an input _x 1_ produces output _y 1_ and another input _x 2_ produces _y 2_, then   produces  , where _a_ and _b_ are scalars.\n\nTime invariance\n\nThe effect of the system doesn't vary over time, or depend on the state of the system. So if inputs _x 1_ and _x 2_ differ only by a shift in time, their outputs, _y 1_ and _y 2_, differ by the same shift but are otherwise identical.\n\nMany physical systems have these properties, at least approximately:\n\n  * Circuits that contain only resistors, capacitors, and inductors are LTI, to the degree that the components behave like their idealized models.\n\n  * Mechanical systems that contain springs, masses, and dashpots are also LTI, assuming linear springs (force proportional to displacement) and dashpots (force proportional to velocity).\n\n  * Also, and most relevant to applications in this book, the media that transmit sound (including air, water, and solids) are well modeled by LTI systems.\n\nLTI systems are described by linear differential equations, and the solutions of those equations are complex sinusoids (see _http:\/\/en.wikipedia.org\/wiki\/Linear_differential_equation_).\n\nThis result provides an algorithm for computing the effect of an LTI system on an input signal:\n\n  1. Express the signal as the sum of complex sinusoid components.\n\n  2. For each input component, compute the corresponding output component.\n\n  3. Add up the output components.\n\nAt this point, I hope this algorithm sounds familiar. It's the same algorithm we used for convolution in \"Efficient Convolution\", and for differentiation in \"Differentiation\". This process is called **spectral decomposition** because we \"decompose\" the input signal into its spectral components.\n\nIn order to apply this process to an LTI system, we have to **characterize** the system by finding its effect on each component of the input signal. For mechanical systems, it turns out that there is a simple and efficient way to do that: you kick it and record the output.\n\nTechnically, the \"kick\" is called an **impulse** and the output is called the **impulse response**. You might wonder how a single impulse can completely characterize a system. You can see the answer by computing the DFT of an impulse. Here's a wave array with an impulse at  :\n\n    impulse = np.zeros(8)\n    impulse[0] = 1\n    impulse_spectrum = np.fft.fft(impulse)\n\nHere's the wave array:\n\n    [ 1.  0.  0.  0.  0.  0.  0.  0.]\n\nAnd here's its spectrum:\n\n    [ 1.+0.j  1.+0.j  1.+0.j  1.+0.j  1.+0.j  1.+0.j  1.+0.j  1.+0.j]\n\nThe spectrum is all ones; that is, an impulse is the sum of components with equal magnitudes at all frequencies. This spectrum should not be confused with white noise, which has the same _average_ power at all frequencies, but varies around that average.\n\nWhen you test a system by inputting an impulse, you are testing the response of the system at all frequencies. And you can test them all at the same time because the system is linear, so simultaneous tests don't interfere with each other.\n\n# Windows and Filters\n\nTo show why this kind of system characterization works, I will start with a simple example: a 2-element moving average. We can think of this operation as a system that takes a signal as an input and produces a slightly smoother signal as an output.\n\nIn this example we know what the window is, so we can compute the corresponding filter. But that's not usually the case; in the next section we'll look at an example where we don't know the window or the filter ahead of time.\n\nHere's a window that computes a 2-element moving average (see \"Smoothing\"):\n\n    window_array = np.array([0.5, 0.5, 0, 0, 0, 0, 0, 0,])\n    window = thinkdsp.Wave(window_array, framerate=8)\n\nWe can find the corresponding filter by computing the DFT of the window:\n\n    filtr = window.make_spectrum(full=True)\n\nFigure 10-1 shows the result. The filter that corresponds to a moving average window is a low-pass filter with the approximate shape of a Gaussian curve.\n\n###### Figure 10-1. DFT of a 2-element moving average window.\n\nNow imagine that we did not know the window or the corresponding filter, and we wanted to characterize this system. We would do that by inputting an impulse and measuring the impulse response.\n\nIn this example, we can compute the impulse response by multiplying the spectrum of the impulse and the filter, and then converting the result from a spectrum to a wave:\n\n    product = impulse_spectrum * filtr\n    filtered = product.make_wave()\n\nSince `impulse_spectrum` is all ones, the product is identical to the filter, and the filtered wave is identical to the window.\n\nThis example demonstrates two things:\n\n  * Because the spectrum of an impulse is all ones, the DFT of the impulse response is identical to the filter that characterizes the system.\n\n  * Therefore, the impulse response is identical to the convolution window that characterizes the system.\n\n# Acoustic Response\n\nTo characterize the acoustic response of a room or open space, a simple way to generate an impulse is to pop a balloon or fire a gun. The result is an input signal that approximates an impulse, so the sound you hear approximates the impulse response.\n\nAs an example, I'll use a recording of a gunshot to characterize the room where the gun was fired, then use the impulse response to simulate the effect of that room on a violin recording.\n\nThis example is in `chap10.ipynb`, which is in the repository for this book; you can also view it, and listen to the examples, at _http:\/\/tinyurl.com\/thinkdsp10_.\n\nHere's the gunshot:\n\n    response = thinkdsp.read_wave('180961__kleeb__gunshots.wav')\n    response = response.segment(start=0.26, duration=5.0)\n    response.normalize()\n    response.plot()\n\nI select a segment starting at 0.26 seconds to remove the silence before the gunshot. Figure 10-2 (left) shows the waveform of the gunshot. Next I compute the DFT of `response`:\n\n    transfer = response.make_spectrum()\n    transfer.plot()\n\n###### Figure 10-2. Waveform of a gunshot.\n\nFigure 10-2 (right) shows the result. This spectrum encodes the response of the room; for each frequency, the spectrum contains a complex number that represents an amplitude multiplier and a phase shift. This spectrum is called a **transfer function** because it contains information about how the system transfers the input to the output.\n\nNow we can simulate the effect this room would have on the sound of a violin. Here is the violin recording we used in \"Periodic Signals\":\n\n    violin = thinkdsp.read_wave('92002__jcveliz__violin-origional.wav')\n    violin.truncate(len(response))\n    violin.normalize()\n\nThe violin and gunshot waves were sampled at the same frame rate, 44,100 Hz. And coincidentally, the duration of both is about the same. I trimmed the violin wave to the same length as the gunshot.\n\nNext I compute the DFT of the violin wave:\n\n    spectrum = violin.make_spectrum()\n\nNow I know the magnitude and phase of each frequency component in the input, and I know the transfer function of the system. Their product is the DFT of the output, which we can use to compute the output wave:\n\n    output = (spectrum * transfer).make_wave()\n    output.normalize()\n    output.plot()\n\nFigure 10-3 shows the input (top) and output (bottom) of the system. They are substantially different, and the differences are clearly audible. Load `chap10.ipynb` and listen to them. One thing I find striking about this example is that you can get a sense of what the room is like; to me, it sounds like a long, narrow room with hard floors and ceilings. That is, like a firing range.\n\n###### Figure 10-3. The waveform of the violin recording before and after convolution.\n\nThere's one thing I glossed over in this example that I'll mention in case it bothers anyone. The violin recording I started with has already been transformed by one system: the room where it was recorded. So what I really computed in my example is the sound of the violin after two transformations. To properly simulate the sound of a violin in a different room, I should have characterized the room where the violin was recorded and applied the inverse of that transfer function first.\n\n# Systems and Convolution\n\nIf you think the previous example is black magic, you are not alone. I've been thinking about it for a while and it still makes my head hurt.\n\nIn the previous section, I suggested one way to think about it:\n\n  * An impulse is made up of components with amplitude 1 at all frequencies.\n\n  * The impulse response contains the sum of the responses of the system to all of these components.\n\n  * The transfer function, which is the DFT of the impulse response, encodes the effect of the system on each frequency component in the form of an amplitude multiplier and a phase shift.\n\n  * For any input, we can compute the response of the system by breaking the input into components, computing the response to each component, and adding them up.\n\nBut if you don't like that, there's another way to think about it altogether: convolution! By the Convolution Theorem, multiplication in the frequency domain corresponds to convolution in the time domain. In this example, the output of the system is the convolution of the input and the system response.\n\nHere are the keys to understanding why that works:\n\n  * You can think of the samples in the input wave as a sequence of impulses with varying amplitude.\n\n  * Each impulse in the input yields a copy of the impulse response, shifted in time (because the system is time-invariant) and scaled by the amplitude of the input.\n\n  * The output is the sum of the shifted, scaled copies of the impulse response. The copies add up because the system is linear.\n\nLet's work our way up gradually. Suppose that instead of firing one gun, we fire two: a big one with amplitude 1 at   and a smaller one with amplitude 0.5 at  .\n\nWe can compute the response of the system by adding up the original impulse response and a scaled, shifted copy of itself. Here's a function that makes a shifted, scaled copy of a wave:\n\n    def shifted_scaled(wave, shift, factor):\n        res = wave.copy()\n        res.shift(shift)\n        res.scale(factor)\n        return res\n\nThe parameter `shift` is a time shift in seconds; `factor` is a multiplicative factor.\n\nHere's how we use it to compute the response to a two-gun salute:\n\n    shift = 1\n    factor = 0.5\n    gun2 = response + shifted_scaled(response, shift, factor)\n\nFigure 10-4 shows the result. You can hear what it sounds like in `chap10.ipynb`. Not surprisingly, it sounds like two gunshots, the first one louder than the second.\n\n###### Figure 10-4. Sum of a wave and a shifted, scaled copy.\n\nNow suppose instead of 2 guns, you were to add up 100 guns fired at a rate of 441 shots per second. This loop computes the result:\n\n    dt = 1 \/ 441\n    total = 0\n    for k in range(100):\n        total += shifted_scaled(response, k*dt, 1.0)\n\nWith 441 shots per second, you don't hear the individual shots. Instead, it sounds like a periodic signal at 441 Hz. If you play this example, it sounds like a car horn in a garage.\n\nAnd that brings us to a key insight: you can think of any wave as a series of samples, where each sample is an impulse with a different amplitude.\n\nAs an example, I'll generate a sawtooth signal at 441 Hz:\n\n    signal = thinkdsp.SawtoothSignal(freq=441)\n    wave = signal.make_wave(duration=0.1,\n                            framerate=response.framerate)\n\nNow I'll loop through the series of impulses that make up the sawtooth, and add up the impulse responses:\n\n    total = 0\n    for t, y in zip(wave.ts, wave.ys):\n        total += shifted_scaled(response, t, y)\n\nThe result is what it would sound like to play a sawtooth wave in a firing range. Again, you can listen to it in `chap10.ipynb`.\n\nFigure 10-5 shows a diagram of this computation, where _f_ is the sawtooth, _g_ is the impulse response, and _h_ is the sum of the shifted, scaled copies of _g_.\n\n###### Figure 10-5. Diagram of the sum of scaled and shifted copies of g.\n\nFor the example shown:\n\nAnd more generally:\n\nYou might recognize this equation from \"Convolution\". It is the convolution of _f_ and _g_. This shows that if the input is _f_ and the impulse response of the system is _g_ , the output is the convolution of _f_ and _g_.\n\nIn summary, there are two ways to think about the effect of a system on a signal:\n\n  1. The input is a sequence of impulses, so the output is the sum of scaled, shifted copies of the impulse response; that sum is the convolution of the input and the impulse response.\n\n  2. The DFT of the impulse response is a transfer function that encodes the effect of the system on each frequency component as a magnitude and phase offset. The DFT of the input encodes the magnitude and phase offset of the frequency components it contains. Multiplying the DFT of the input by the transfer function yields the DFT of the output.\n\nThe equivalence of these descriptions should not be a surprise. It is basically a statement of the Convolution Theorem: convolution of _f_ and _g_ in the time domain corresponds to multiplication in the frequency domain.\n\nAnd if you wondered why convolution is defined as it is, which seemed backward when we talked about smoothing and difference windows, now you know the reason: the definition of convolution appears naturally in the response of an LTI system to a signal.\n\n# Proof of the Convolution Theorem\n\nWell, I've put it off long enough. It's time to prove the Convolution Theorem (CT), which states:\n\nwhere _f_ and _g_ are vectors with the same length, _N_.\n\nI'll proceed in two steps:\n\n  1. I'll show that in the special case where _f_ is a complex exponential, convolution with _g_ has the effect of multiplying _f_ by a scalar.\n\n  2. In the more general case where _f_ is not a complex exponential, we can use the DFT to express it as a sum of exponential components, compute the convolution of each component (by multiplication), and then add up the results.\n\nTogether these steps prove the Convolution Theorem. But first, let's assemble the pieces we'll need. The DFT of _g_ , which I'll call _G_ , is:\n\nwhere _k_ is an index of frequency from 0 to   and _n_ is an index of time from 0 to  . The result is a vector of _N_ complex numbers.\n\nThe inverse DFT of _F_ , which I'll call _f_ , is:\n\nHere's the definition of convolution:\n\nwhere _m_ is another index of time from 0 to  . Convolution is commutative, so I could equivalently write:\n\nNow let's consider the special case where _f_ is a complex exponential with frequency _k_ , which I'll call _e k_:\n\nwhere _k_ is an index of frequency and _n_ is an index of time.\n\nPlugging _e k_ into the second definition of convolution yields:\n\nWe can split the first term into a product:\n\nThe first half does not depend on _m_ , so we can pull it out of the summation:\n\nNow we recognize that the first term is _e k_, and the summation is   (using _m_ as the index of time). So we can write:\n\nwhich shows that for each complex exponential, _e k_, convolution with _g_ has the effect of multiplying _e k_ by  . In mathematical terms, each _e k_ is an eigenvector of this operation, and   is the corresponding eigenvalue (see \"Differentiation\").\n\nNow for the second part of the proof. If the input signal, _f_ , doesn't happen to be a complex exponential, we can express it as a sum of complex exponentials by computing its DFT, _F_. For each value of _k_ from 0 to  ,   is the complex magnitude of the component with frequency _k_.\n\nEach input component is a complex exponential with magnitude  , so each output component is a complex exponential with magnitude  , based on the first part of the proof.\n\nBecause the system is linear, the output is just the sum of the output components:\n\nPlugging in the definition of _e k_ yields:\n\nThe righthand side is the inverse DFT of the product  . Thus:\n\nSubstituting   and  :\n\nFinally, taking the DFT of both sides yields the Convolution Theorem:\n\nQED.\n\n# Exercises\n\nSolutions to these exercises are in `chap10soln.ipynb`.\n\n##### Exercise 10-1.\n\nIn \"Systems and Convolution\" I describe convolution as the sum of shifted, scaled copies of a signal.\n\nBut in \"Acoustic Response\", when we multiply the DFT of the signal by the transfer function, that operation corresponds to **circular convolution** , which assumes that the signal is periodic. As a result, you might notice that the output contains an extra note at the beginning, which wraps around from the end.\n\nFortunately, there is a standard solution to this problem. If you add enough zeros to the end of the signal before computing the DFT, you can avoid the wraparound effect.\n\nModify the example in `chap10.ipynb` and confirm that zero-padding eliminates the extra note at the beginning of the output.\n\n##### Exercise 10-2.\n\nThe Open AIR library provides a \"centralized... on-line resource for anyone interested in auralization and acoustical impulse response data\" ( _http:\/\/www.openairlib.net_). Browse its collection of impulse response data and download one that sounds interesting. Find a short recording that has the same sample rate as the impulse response you downloaded.\n\nSimulate the sound of your recording in the space where the impulse response was measured, computed two ways: by convolving the recording with the impulse response and by computing the filter that corresponds to the impulse response and multiplying by the DFT of the recording.\n\n My presentation here follows _http:\/\/en.wikipedia.org\/wiki\/LTI_system_theory_.\n\n# Chapter 11. Modulation and Sampling\n\nIn \"Aliasing\" we saw that when a signal is sampled at 10,000 Hz, a component at 5500 Hz is indistinguishable from a component at 4500 Hz. In this example, the folding frequency, 5000 Hz, is half of the sampling rate. But I didn't explain why.\n\nThis chapter explores the effect of sampling and presents the Sampling Theorem, which explains aliasing and the folding frequency.\n\nI'll start by exploring the effect of convolution with impulses; I'll use that effect to explain amplitude modulation (AM), which turns out to be useful for understanding the Sampling Theorem.\n\nThe code for this chapter is in `chap11.ipynb`, which is in the repository for this book (see \"Using the Code\"). You can also view it at _http:\/\/tinyurl.com\/thinkdsp-ch11_.\n\n# Convolution with Impulses\n\nAs we saw in \"Systems and Convolution\", convolution of a signal with a series of impulses has the effect of adding up shifted, scaled copies of the signal.\n\nAs an example, I'll read a signal that sounds like a beep:\n\n    filename = '253887__themusicalnomad__positive-beeps.wav'\n    wave = thinkdsp.read_wave(filename)\n    wave.normalize()\n\nAnd I'll construct a wave with four impulses:\n\n    imp_sig = thinkdsp.Impulses([0.005, 0.3, 0.6,  0.9], \n                           amps=[1,     0.5, 0.25, 0.1])\n    impulses = imp_sig.make_wave(start=0, duration=1.0, \n                                 framerate=wave.framerate)\n\nand then convolve them:\n\n    convolved = wave.convolve(impulses)\n\nFigure 11-1 shows the results, with the signal in the top left, the impulses in the lower left, and the result on the right.\n\n###### Figure 11-1. The effect of convolving a signal (top left) with a series of impulses (bottom left). The result (right) is the sum of shifted, scaled copies of the signal.\n\nYou can hear the result in `chap11.ipynb`; it sounds like a series of four beeps with decreasing loudness.\n\nThe point of this example is just to demonstrate that convolution with impulses makes shifted, scaled copies. This result will be useful in the next section.\n\n# Amplitude Modulation\n\nAmplitude modulation (AM) is used to broadcast AM radio, among other applications. At the transmitter, the signal (which might contain speech, music, etc.) is \"modulated\" by multiplying it with a cosine signal that acts as a \"carrier wave\". The result is a high-frequency wave that is suitable for broadcast by radio. Typical frequencies for AM radio in the United States are 500\u20131600 kHz (see _https:\/\/en.wikipedia.org\/wiki\/AM_broadcasting_).\n\nAt the receiving end, the broadcast signal is \"demodulated\" to recover the original signal. Surprisingly, demodulation works by multiplying the broadcast signal, again, by the same carrier wave.\n\nTo see how that works, I'll modulate a signal with a carrier wave at 10 kHz. Here's the signal:\n\n    filename = '105977__wcfl10__favorite-station.wav'\n    wave = thinkdsp.read_wave(filename)\n    wave.unbias()\n    wave.normalize()\n\nAnd here's the carrier:\n\n    carrier_sig = thinkdsp.CosSignal(freq=10000)\n    carrier_wave = carrier_sig.make_wave(duration=wave.duration, \n                                         framerate=wave.framerate)\n\nWe can multiply them using the `*` operator, which multiplies the wave arrays elementwise:\n\n    modulated = wave * carrier_wave\n\nThe result sounds pretty bad. You can hear it in `chap11.ipynb`.\n\nFigure 11-2 shows what's happening in the frequency domain. The top row is the spectrum of the original signal. The next row is the spectrum of the modulated signal, after multiplying by the carrier. It contains two copies of the original spectrum, shifted by plus and minus 10 kHz.\n\nTo understand why, recall that convolution in the time domain corresponds to multiplication in the frequency domain. Conversely, multiplication in the time domain corresponds to convolution in the frequency domain. When we multiply the signal by the carrier, we are convolving its spectrum with the DFT of the carrier.\n\nSince the carrier is a simple cosine wave, its DFT is two impulses, at plus and minus 10 kHz. Convolution with these impulses makes shifted, scaled copies of the spectrum. Notice that the amplitude of the spectrum is smaller after modulation. The energy from the original signal is split between the copies.\n\nWe demodulate the signal by multiplying by the carrier wave again:\n\n    demodulated = modulated * carrier_wave\n\nThe third row of Figure 11-2 shows the result. Again, multiplication in the time domain corresponds to convolution in the frequency domain, which makes shifted, scaled copies of the spectrum.\n\n###### Figure 11-2. Demonstration of amplitude modulation. The top row is the spectrum of the signal; the next row is the spectrum after modulation; the next row is the spectrum after demodulation; the last row is the demodulated signal after low-pass filtering.\n\nSince the modulated spectrum contains two peaks, each peak gets split in half and shifted by plus and minus 20 kHz. Two of the copies meet at 0 kHz and get added together; the other two copies end up centered at plus and minus 20 kHz.\n\nIf you listen to the demodulated signal, it sounds pretty good. The extra copies of the spectrum add high-frequency components that were not in the original signal. These are so high that most speakers can't play them and most people can't hear them, but if you have good speakers and good ears, you might.\n\nIn that case, you can get rid of the extra components by applying a low-pass filter:\n\n    demodulated_spectrum = demodulated.make_spectrum(full=True)\n    demodulated_spectrum.low_pass(10000)\n    filtered = demodulated_spectrum.make_wave()\n\nThe result is quite close to the original wave, although about half of the power is lost after demodulating and filtering. That's not a problem in practice, because much more of the power is lost in transmitting and receiving the broadcast signal. Since we have to amplify the result anyway, another factor of 2 is not an issue.\n\n# Sampling\n\nI explained amplitude modulation in part because it is interesting, but mostly because it will help us understand sampling. **Sampling** is the process of measuring an analog signal at a series of points in time, usually with equal spacing.\n\nFor example, the WAV files we have used as examples were recorded by sampling the output of a microphone using an analog-to-digital converter (ADC). The sampling rate for most of them is 44.1 kHz, which is the standard rate for \"CD-quality\" sound, or 48 kHz, which is the standard for DVD sound.\n\nAt 48 kHz, the folding frequency is 24 kHz, which is higher than most people can hear (see _https:\/\/en.wikipedia.org\/wiki\/Hearing_range_).\n\nIn most of these waves, each sample has 16 bits, so there are 216 distinct levels. This \"bit depth\" turns out to be enough that adding more bits does not improve the sound quality noticeably (see _https:\/\/en.wikipedia.org\/wiki\/Digital_audio_).\n\nOf course, applications other than audio signals might require higher sampling rates in order to capture higher frequencies, or higher bit depth in order to reproduce waveforms with more fidelity.\n\nTo demonstrate the effect of the sampling process, I am going to start with a wave sampled at 44.1 kHz and select samples from it at about 11 kHz. This is not exactly the same as sampling from an analog signal, but the effect is the same.\n\nFirst I'll load a recording of a drum solo:\n\n    filename = '263868__kevcio__amen-break-a-160-bpm.wav'\n    wave = thinkdsp.read_wave(filename)\n    wave.normalize()\n\nFigure 11-3 (top) shows the spectrum of this wave. Now here's the function that samples from the wave:\n\n    def sample(wave, factor=4):\n        ys = np.zeros(len(wave))\n        ys[::factor] = wave.ys[::factor]\n        return thinkdsp.Wave(ys, framerate=wave.framerate)\n\nI'll use it to select every fourth element:\n\n    sampled = sample(wave, 4)\n\nThe result has the same frame rate as the original, but most of the elements are zero. If you play the sampled wave, it doesn't sound very good. The sampling process introduces high-frequency components that were not in the original.\n\nFigure 11-3 (bottom) shows the spectrum of the sampled wave. It contains four copies of the original spectrum (it looks like five copies because one is split between the highest and lowest frequencies).\n\n###### Figure 11-3. Spectrum of a signal before (top) and after (bottom) sampling.\n\nTo understand where these copies come from, we can think of the sampling process as multiplication with a series of impulses. Instead of using `sample` to select every fourth element, we could use this function to make a series of impulses, sometimes called an **impulse train** :\n\n    def make_impulses(wave, factor):\n        ys = np.zeros(len(wave))\n        ys[::factor] = 1\n        ts = np.arange(len(wave)) \/ wave.framerate\n        return thinkdsp.Wave(ys, ts, wave.framerate)\n\nAnd then multiply the original wave by the impulse train:\n\n    impulses = make_impulses(wave, 4)\n    sampled = wave * impulses\n\nThe result is the same; it still doesn't sound very good, but now we understand why. Multiplication in the time domain corresponds to convolution in the frequency domain. When we multiply by an impulse train, we are convolving with the DFT of an impulse train. As it turns out, the DFT of an impulse train is also an impulse train.\n\nFigure 11-4 shows two examples. The top row is the impulse train in the example, with frequency 11,025 Hz. The DFT is a train of four impulses, which is why we get four copies of the spectrum. The bottom row shows an impulse train with a lower frequency, about 5512 Hz. Its DFT is a train of eight impulses. In general, more impulses in the time domain correspond to fewer impulses in the frequency domain.\n\nIn summary:\n\n  * We can think of sampling as multiplication by an impulse train.\n\n  * Multiplying by an impulse train corresponds to convolution with an impulse train in the frequency domain.\n\n  * Convolution with an impulse train makes multiple copies of the signal's spectrum.\n\n###### Figure 11-4. The DFT of an impulse train is also an impulse train.\n\n# Aliasing\n\nIn \"Amplitude Modulation\", after demodulating an AM signal we got rid of the extra copies of the spectrum by applying a low-pass filter. We can do the same thing after sampling, but it turns out not to be a perfect solution.\n\nFigure 11-5 shows why not. The top row is the spectrum of the drum solo. It contains high-frequency components that extend past 10 kHz. When we sample this wave, we convolve the spectrum with the impulse train (second row), which makes copies of the spectrum (third row). The bottom row shows the result after applying a low-pass filter with a cutoff at the folding frequency, 5512 Hz.\n\n###### Figure 11-5. Spectrums of the drum solo (top), the impulse train (second row), the sampled wave (third row), and the result after low-pass filtering (bottom).\n\nIf we convert the result back to a wave, it is similar to the original wave, but there are two problems:\n\n  * Because of the low-pass filter, the components above 5500 Hz have been lost, so the result sounds muted.\n\n  * Even the components below 5500 Hz are not quite right, because they include contributions from the spectral copies we tried to filter out.\n\nIf the spectral copies overlap after sampling, we lose information about the spectrum and we won't be able to recover it.\n\nBut if the copies don't overlap, things work out pretty well. As a second example, I loaded a recording of a bass guitar solo.\n\nFigure 11-6 shows its spectrum (top row), which contains no visible energy above 5512 Hz. The second row shows the spectrum of the sampled wave, and the third row shows the spectrum after the low-pass filter. The amplitude is lower because we've filtered out some of the energy, but the shape of the spectrum is almost exactly what we started with. And if we convert back to a wave, it sounds the same.\n\n###### Figure 11-6. Spectrum of a bass guitar solo (top), after sampling (middle), and after filtering (bottom).\n\n# Interpolation\n\nThe low-pass filter I used in the last step is a so-called **brick wall filter** ; frequencies above the cutoff are removed completely, as if they hit a brick wall.\n\nFigure 11-7 (right) shows what this filter looks like. Of course, multiplication by this filter in the frequency domain corresponds to convolution with a window in the time domain. We can find out what that window is by computing the inverse DFT of the filter, which is shown in Figure 11-7 (left).\n\n###### Figure 11-7. A brick wall low-pass filter (right) and the corresponding convolution window (left).\n\nThat function has a name\u2014it is the normalized **sinc function** , or at least a discrete approximation of it (see _https:\/\/en.wikipedia.org\/wiki\/Sinc_function_):\n\nWhen we apply the low-pass filter, we are convolving with a sinc function. We can think of this convolution as the sum of shifted, scaled copies of the sinc function.\n\nThe value of sinc is 1 at 0 and 0 at every other integer value of _x_. When we shift the sinc function, we move the zero point. When we scale it, we change the height at the zero point. So when we add up the shifted, scaled copies, they interpolate between the sampled points.\n\nFigure 11-8 shows how that works using a short segment of the bass guitar solo. The line across the top is the original wave. The vertical gray lines show the sampled values. The thin curves are the shifted, scaled copies of the sinc function. The sum of these sinc functions is identical to the original wave.\n\n###### Figure 11-8. Close-up view of a sequence of samples (vertical gray lines), interpolating sinc functions (thin curves), and the original wave (thicker line across the top).\n\nI'll say that again, because it is surprising and important:\n\n> The sum of these sinc functions is identical to the original wave.\n\nBecause we started with a signal that contained no energy above 5512 Hz, and we sampled at 11,025 Hz, we were able to recover the original spectrum exactly. And if we have the original spectrum, exactly, we can recover the original wave exactly.\n\nIn this example, I started with a wave that had already been sampled at 44,100 Hz, and I resampled it at 11,025 Hz. After resampling, the gap between the spectral copies is 11.025 kHz.\n\nIf the original wave contains no energy above 5512 Hz, the spectral copies don't overlap, we don't lose information, and we can recover the original signal exactly.\n\nThis result is known as the Nyquist\u2013Shannon Sampling Theorem (see _https:\/\/en.wikipedia.org\/wiki\/Nyquist-Shannon_sampling_theorem_).\n\nThis example does not prove the Sampling Theorem, but I hope it helps you understand what it says and why it works.\n\nNotice that the argument I made does not depend on the original sampling rate, 44.1 kHz. The result would be the same if the original had been sampled at a higher frequency, or even if the original had been a continuous analog signal: if we sample at frame rate _f_ , we can recover the original signal exactly, as long as it contains no energy at frequencies above  . A signal like that is called **bandwidth limited**.\n\n# Summary\n\nCongratulations! You have reached the end of the book (well, except for a few more exercises). Before you close the book, I want to review how we got here:\n\n  * We started with periodic signals and their spectrums, and I introduced the key objects in the `thinkdsp` library: `Signal`, `Wave`, and `Spectrum`.\n\n  * We looked at the harmonic structure of simple waveforms and recordings of musical instruments, and we saw the effect of aliasing.\n\n  * Using spectrograms, we explored chirps and other sounds whose spectrum changes over time.\n\n  * We generated and analyzed noise signals, and characterized natural sources of noise.\n\n  * We used the autocorrelation function for pitch estimation and additional characterization of noise.\n\n  * We learned about the Discrete Cosine Transform (DCT), which is useful for compression and also a step toward understanding the Discrete Fourier Transform (DFT).\n\n  * We used complex exponentials to synthesize complex signals, then we inverted the process to develop the DFT. If you did the exercises at the end of Chapter 7, you implemented the Fast Fourier Transform (FFT), one of the most important algorithms of the 20th century.\n\n  * Starting with smoothing, I presented the definition of convolution and stated the Convolution Theorem, which relates operations like smoothing in the time domain to filters in the frequency domain.\n\n  * We explored differentiation and integration as linear filters, which is the basis of spectral methods for solving differential equations. It also explains some of the effects we saw in previous chapters, like the relationship between white noise and Brownian noise.\n\n  * We learned about LTI system theory and used the Convolution Theorem to characterize LTI systems by their impulse response.\n\n  * I presented amplitude modulation (AM), which is important in radio communication and also a step toward understanding the Sampling Theorem, a surprising result that is critical for digital signal processing.\n\nIf you got this far, you should have a good balance of practical knowledge (how to work with signals and spectrums using computational tools) and theory (an understanding of how and why sampling and filtering work).\n\nI hope you had some fun along the way. Thank you!\n\n# Exercises\n\nSolutions to these exercises are in `chap11soln.ipynb`.\n\n##### Exercise 11-1.\n\nThe code in this chapter is in `chap11.ipynb`. Read through it and listen to the examples.\n\n##### Exercise 11-2.\n\nChris \"Monty\" Montgomery has an excellent video called \"D\/A and A\/D | Digital Show and Tell\"; it demonstrates the Sampling Theorem in action, and presents lots of other excellent information about sampling. Watch it at _https:\/\/www.youtube.com\/watch?v=cIQ9IXSUzuM_.\n\n##### Exercise 11-3.\n\nAs we have seen, if you sample a signal at too low a frame rate, frequencies above the folding frequency get aliased. Once that happens, it is no longer possible to filter out these components, because they are indistinguishable from lower frequencies.\n\nIt is a good idea to filter out these frequencies _before_ sampling; a low-pass filter used for this purpose is called an **anti-aliasing filter**.\n\nReturning to the drum solo example, apply a low-pass filter before sampling, then apply the low-pass filter again to remove the spectral copies introduced by sampling. The result should be identical to the filtered signal. \n\n# Index\n\n### A\n\n  * abs, Complex Exponentials\n  * acoustic response, Acoustic Response\n  * aliasing, Harmonics, Aliasing, Exercises, DFT of Real Signals, Modulation and Sampling, Exercises\n  * AM, Amplitude Modulation\n  * amplifier, Signals and Systems\n  * amplitude, Spectral Decomposition, Signal Objects, Computing the Spectrum, Complex Exponentials\n  * amplitude modulation, Amplitude Modulation\n  * Anaconda, Using the Code\n  * analog-to-digital converter, Sampling\n  * analysis, Discrete Cosine Transform, Synthesis, The Analysis Problem\n  * angle, Computing the Spectrum, Complex Exponentials, Complex Exponentials\n  * anti-aliasing filter, Exercises\n  * aplay, Reading and Writing Waves\n  * Audio widget, Spectrums\n  * autocorrelation, Autocorrelation, Autocorrelation, Efficient Autocorrelation\n  * autocorrelation function, Autocorrelation of Periodic Signals\n\n### B\n\n  * band stop filter, Spectrums\n  * Bartlett's method, Exercises\n  * bias, Pink Noise, Integration\n  * Binder, Using the Code\n  * bit depth, Sampling\n  * BitCoin, Exercises, Exercises\n  * Boccherini, Periodic Signals\n  * boxcar window, Gaussian Filter\n  * brick wall filter, Interpolation\n  * Brownian noise, Uncorrelated Noise, Brownian Noise, Serial Correlation, Finite Differences, Integrating Noise\n\n### C\n\n  * carrier wave, Amplitude Modulation\n  * Cartesian coordinate, Computing the Spectrum\n  * characterizing a system, Signals and Systems\n  * chirp, Linear Chirp, Exercises\n  * circuits, Signals and Systems\n  * circular convolution, Convolution, Exercises\n  * cloning the repo, Using the Code\n  * complex amplitude, Synthesis with Matrices\n  * complex conjugate, Efficient Analysis\n  * complex exponential, Complex Exponentials, Complex Signals, Proof of the Convolution Theorem\n  * complex number, Computing the Spectrum, Complex Exponentials\n  * complex plane, Complex Exponentials\n  * complex signal, The Synthesis Problem\n  * complex sinusoid, Complex Signals, Differentiation\n  * component, Signals, Proof of the Convolution Theorem\n  * compression, Exercises\n  * contributors, Contributor List\n  * convolution, Filtering and Convolution, Convolution, Systems and Convolution, Convolution with Impulses, Amplitude Modulation\n  * Convolution Theorem, Filtering and Convolution, The Convolution Theorem, Cumulative Sum, Systems and Convolution, Proof of the Convolution Theorem\n  * correlation, Brownian Noise, Pink Noise, Correlation, Correlation as Dot Product, The DFT Is Periodic\n  * correlation matrix, Correlation\n  * cosine, Signals\n  * CosSignal, Signals\n  * cross-correlation, Convolution, Efficient Autocorrelation\n  * cubic waveform, Exercises\n  * cumsum, Linear Chirp, Exercises\n  * cumulative sum, Linear Chirp, Cumulative Sum, Integrating Noise\n  * cycle, Periodic Signals, Triangle Waves\n\n### D\n\n  * Danielson\u2013Lanczos lemma, Exercises\n  * DCT, Discrete Cosine Transform\n  * Dct, The Dct Class\n  * DCT-IV, DCT-IV\n  * demodulation, Amplitude Modulation\n  * derivative, Linear Chirp, Differentiation\n  * DFT, Spectral Decomposition, Triangle Waves, Discrete Cosine Transform, Discrete Fourier Transform, DFT, Exercises, Proof of the Convolution Theorem\n  * diff, Exercises\n  * differential equations, Signals and Systems\n  * differentiate, Exercises\n  * differentiation, Differentiation and Integration, Differentiation\n  * discontinuity, Leakage\n  * Discrete Cosine Transform, Discrete Cosine Transform\n  * Discrete Fourier Transform (see DFT)\n  * dominant frequency, Spectral Decomposition\n  * dot product, Correlation as Dot Product, Synthesis with Arrays\n\n### E\n\n  * eigenfunction, Differentiation, Proof of the Convolution Theorem\n  * eigenvalue, Differentiation, Proof of the Convolution Theorem\n  * equal temperament, Spectral Decomposition\n  * Euler, Complex Exponentials\n  * Euler's formula, Complex Exponentials\n  * exponential chirp, Exponential Chirp\n  * Eye of Sauron, Spectrum of a Chirp\n\n### F\n\n  * Facebook stock price, Smoothing, Efficient Convolution\n  * Fast Fourier Transform (see FFT)\n  * FFT, Spectral Decomposition, Computing the Spectrum, Exercises, Efficient Convolution\n  * fft, Computing the Spectrum\n  * filter, Pink Noise, The Convolution Theorem, The Frequency Domain, Differentiation, Integration, Windows and Filters\n  * finite difference, The Frequency Domain\n  * floating-point, DCT-IV\n  * folding frequency, Aliasing, DFT of Real Signals\n  * fork, Using the Code\n  * frame, Signals\n  * Freesound, Contributor List, Exercises, Exercises, Autocorrelation of Periodic Signals\n  * frequency, Periodic Signals, Signal Objects, Linear Chirp\n  * frequency component, Signals, Synthesis, Analysis\n  * frequency domain, The Convolution Theorem, The Frequency Domain, Amplitude Modulation\n  * frequency resolution, The Gabor Limit, Autocorrelation of Periodic Signals\n  * full DFT, DFT of Real Signals\n  * fundamental frequency, Spectral Decomposition, Triangle Waves\n\n### G\n\n  * Gabor limit, The Gabor Limit, Autocorrelation of Periodic Signals\n  * gamma function, Complex Exponentials\n  * Gaussian filter, Gaussian Filter\n  * Gaussian noise, Uncorrelated Noise, Gaussian Noise\n  * Gaussian signal, Exercises\n  * Gaussian window, Exercises, Exercises\n  * Geiger counter, Exercises\n  * Git, Using the Code\n  * GitHub, Using the Code\n  * glissando, Exercises\n  * gunshot, Acoustic Response\n\n### H\n\n  * Hamming window, Windowing, Exercises\n  * harmonic structure, Harmonics, Triangle Waves, Square Waves, Exercises\n  * harmonics, Spectral Decomposition, Triangle Waves, DFT of Real Signals\n  * Hart, Vi, Exercises\n  * Hertz, Periodic Signals\n  * high-pass filter, Spectrums, Finite Differences\n\n### I\n\n  * identity matrix, Orthogonal Matrices, Efficient Analysis\n  * imaginary number, Complex Exponentials\n  * impulse, Signals and Systems, Systems and Convolution, Systems and Convolution, Convolution with Impulses, Amplitude Modulation\n  * impulse response, Signals and Systems, Exercises\n  * impulse train, Sampling\n  * input, Signals and Systems\n  * installation, Using the Code\n  * integral, Linear Chirp\n  * integrate, Exercises\n  * integrated spectrum, Integrated Spectrum\n  * integration, Differentiation and Integration, Integration, Cumulative Sum\n  * interpolation, Interpolation\n  * interval, Exponential Chirp\n  * inverse DCT, Inverse DCT\n  * inverse DFT, DFT\n\n### J\n\n  * Jupyter, Using the Code, Spectrums\n\n### L\n\n  * lag, Serial Correlation, Autocorrelation\n  * leakage, Leakage, Smoothing\n  * linear algebra, Synthesis with Arrays\n  * linear system, Signals and Systems\n  * log-log scale, Brownian Noise, Integrating Noise\n  * logspace, Exponential Chirp\n  * long-range dependence, Autocorrelation\n  * low-pass filter, Spectrums, Pink Noise, The Frequency Domain, Gaussian Filter, Windows and Filters, Amplitude Modulation\n  * LTI system, Signals and Systems\n\n### M\n\n  * magnitude, Computing the Spectrum, Complex Exponentials\n  * major third, Spectral Decomposition\n  * matplotlib, Using the Code\n  * matrix, Synthesis with Arrays\n  * matrix inverse, Orthogonal Matrices\n  * matrix multiplication, Synthesis with Matrices\n  * mechanical systems, Signals and Systems\n  * missing fundamental, Exercises\n  * modf, Triangle Waves\n  * modulation, Amplitude Modulation\n  * Montgomery, Chris, Exercises\n  * moving average, Smoothing\n\n### N\n\n  * NaN, Integration\n  * nbviewer, Using the Code\n  * noise, Noise, Exercises\n  * non-periodic signal, Non-Periodic Signals, Exercises\n  * normal probability plot, Gaussian Noise\n  * normalize, The Dct Class, Efficient Autocorrelation\n  * NumPy, Using the Code, Using NumPy, Convolution\n  * Nyquist, Interpolation\n  * Nyquist frequency, Aliasing\n\n### O\n\n  * octave, Spectral Decomposition, Exponential Chirp\n  * operator, Differentiation\n  * orthogonal matrix, Orthogonal Matrices, DCT-IV, Efficient Analysis\n  * outer product, Synthesis with Arrays, Synthesis with Matrices\n  * output, Signals and Systems\n\n### P\n\n  * padding, Smoothing, Efficient Autocorrelation, Exercises\n  * Pandas, Efficient Convolution, Finite Differences\n  * parabola, Cumulative Sum\n  * Pearson's correlation, Correlation\n  * perfect fifth, Spectral Decomposition\n  * period, Periodic Signals, Square Waves\n  * periodic, Leakage\n  * periodic DFT, The DFT Is Periodic\n  * periodic signal, Periodic Signals, Autocorrelation of Periodic Signals\n  * phase, Linear Chirp\n  * phase offset, Signals, Signal Objects, Computing the Spectrum, Correlation, Exercises, Complex Signals\n  * pink noise, Uncorrelated Noise, Pink Noise, Exercises, Serial Correlation, Efficient Autocorrelation\n  * pitch, Periodic Signals\n  * pitch estimation, Autocorrelation of Periodic Signals\n  * pitch perception, Exercises\n  * pitch tracking, Exercises\n  * Poisson noise, Exercises\n  * polar coordinate, Computing the Spectrum\n  * power, Uncorrelated Noise, Brownian Noise, Exercises\n  * power series, Complex Exponentials\n  * power spectrum, Brownian Noise\n  * proof, Proof of the Convolution Theorem\n  * Python 2, Using the Code\n  * Python 3, Using the Code\n\n### R\n\n  * random, Uncorrelated Noise, Pink Noise\n  * random walk, Brownian Noise\n  * real DFT, DFT of Real Signals\n  * real FFT, Computing the Spectrum\n  * red noise, Brownian Noise\n  * repository, Using the Code\n  * resolution, The Gabor Limit\n  * response, Systems and Convolution\n\n### S\n\n  * sample, Signals, Aliasing, Systems and Convolution\n  * sampling, Modulation and Sampling, Sampling\n  * Sampling Theorem, Modulation and Sampling, Interpolation\n  * sawtooth chirp, Exercises\n  * sawtooth waveform, Exercises, Exercises, Systems and Convolution\n  * scale, Wave Objects, Systems and Convolution, Exercises, Convolution with Impulses\n  * scientific pitch notation, Periodic Signals\n  * SciPy, Using the Code, Exercises\n  * segment, Signals\n  * serial correlation, Serial Correlation\n  * Shannon, Interpolation\n  * shift, Wave Objects, Serial Correlation, Smoothing, Systems and Convolution, Exercises, Convolution with Impulses\n  * Short-Time Fourier Transform, Spectrogram\n  * sidelobe, Gaussian Filter\n  * signal, Sounds and Signals\n  * Signal, Signals, Signal Objects\n  * signal processing, Preface\n  * sinc function, Interpolation\n  * sine, Signals\n  * SinSignal, Signals\n  * sinusoid, Periodic Signals\n  * Sinusoid, Signals, Signal Objects\n  * smoothing, Smoothing, The Convolution Theorem\n  * Soft Murmur, Exercises\n  * sound, Sounds and Signals\n  * sparse array, Exercises\n  * spectral analysis, Discrete Cosine Transform\n  * spectral decomposition, Spectral Decomposition, Differentiation, Signals and Systems\n  * spectral leakage, Leakage, Smoothing\n  * spectral method, Differentiation\n  * spectrogram, Non-Periodic Signals, Spectrogram, Implementing Spectrograms, Exercises\n  * spectrum, Signals, Triangle Waves, Spectrum of a Chirp, Uncorrelated Noise, The Frequency Domain, Sampling\n  * Spectrum, Spectrums\n  * square matrix, Analysis\n  * square waveform, Square Waves, Exercises, Smoothing\n  * standard deviation, Correlation\n  * standardizing, Using NumPy\n  * static, Uncorrelated Noise\n  * STFT, Spectrogram\n  * stock prices, Integrating Noise\n  * stretch function, Exercises\n  * SumSignal, Signals\n  * symmetric matrix, Orthogonal Matrices\n  * synthesis, Exercises, Discrete Cosine Transform, Synthesis, The Synthesis Problem\n  * system, Signals and Systems\n\n### T\n\n  * temperament, Spectral Decomposition\n  * timbre, Periodic Signals\n  * time domain, The Convolution Theorem, Finite Differences\n  * time resolution, The Gabor Limit\n  * time-invariant system, Signals and Systems\n  * timestep, Signals\n  * transducer, Sounds and Signals\n  * transfer function, Acoustic Response, Systems and Convolution, Systems and Convolution\n  * transpose, Orthogonal Matrices, Efficient Analysis\n  * triangle waveform, Triangle Waves, Exercises, The Dct Class\n  * trombone, Exercises\n  * tuning fork, Periodic Signals\n\n### U\n\n  * UG noise, Gaussian Noise\n  * unbias, Triangle Waves\n  * uncorrelated noise, Uncorrelated Noise\n  * uniform noise, Uncorrelated Noise\n  * unitary matrix, Efficient Analysis\n  * UU noise, Uncorrelated Noise, Serial Correlation\n\n### V\n\n  * vector, Orthogonal Matrices\n  * video, Exercises\n  * violin, Spectral Decomposition, Acoustic Response\n  * virtual machine, Using the Code\n  * Voss\u2013McCartney algorithm, Exercises\n  * vowel, Exercises\n\n### W\n\n  * WAV file, Reading and Writing Waves, Sampling\n  * Wave, Signals, Wave Objects\n  * waveform, Periodic Signals\n  * white noise, Integrated Spectrum, Finite Differences, Signals and Systems\n  * window, Leakage, Windowing, Smoothing, Exercises, The Frequency Domain, Windows and Filters\n  * windowing, Windowing\n\n# About the Author\n\n**Allen B. Downey** is a Professor of Computer Science at Olin College of Engineering. He has taught at Wellesley College, Colby College and U.C. Berkeley. He has a Ph.D. in Computer Science from U.C. Berkeley and Master's and Bachelor's degrees from MIT.\n\n# Colophon\n\nThe animal on the cover of _Think DSP_ is a smooth-billed ani ( _Crotophaga ani_ ), a large bird that is part of the cuckoo family. It is found in Florida, the Bahamas, Caribbean islands, and parts of Central and South America.\n\nSmooth-billed anis have black plumage, long tails, and large ridged beaks. They feed on the ground, with a diet made up of termites, insects, and even small lizards and frogs. The birds prefer semi-open habitats with a mix of fields and brushy thickets. As human settlements and deforestation have affected their territory, anis have adapted by frequenting farm pastures and eating the insects flushed out by livestock.\n\nThis species is very social and is always found in noisy groups. Mating pairs nest communally with several other couples, taking turns to construct a bowl-shaped nest high in a tree, incubate eggs, and feed the chicks. Each female lays 4\u20137 eggs, but nests have been found with up to 29 eggs.\n\nMany of the animals on O'Reilly covers are endangered; all of them are important to the world. To learn more about how you can help, go to _animals.oreilly.com_.\n\nThe cover image is from the _Braukhaus Lexicon_. The cover fonts are URW Typewriter and Guardian Sans. The text font is Adobe Minion Pro; the heading font is Adobe Myriad Condensed; and the code font is Dalton Maag's Ubuntu Mono.\n  1. Preface\n    1. Who Is This Book For?\n    2. Using the Code\n    3. Conventions Used in This Book\n    4. Safari\u00ae Books Online\n    5. How to Contact Us\n    6. Contributor List\n  2. 1. Sounds and Signals\n    1. Periodic Signals\n    2. Spectral Decomposition\n    3. Signals\n    4. Reading and Writing Waves\n    5. Spectrums\n    6. Wave Objects\n    7. Signal Objects\n    8. Exercises\n  3. 2. Harmonics\n    1. Triangle Waves\n    2. Square Waves\n    3. Aliasing\n    4. Computing the Spectrum\n    5. Exercises\n  4. 3. Non-Periodic Signals\n    1. Linear Chirp\n    2. Exponential Chirp\n    3. Spectrum of a Chirp\n    4. Spectrogram\n    5. The Gabor Limit\n    6. Leakage\n    7. Windowing\n    8. Implementing Spectrograms\n    9. Exercises\n  5. 4. Noise\n    1. Uncorrelated Noise\n    2. Integrated Spectrum\n    3. Brownian Noise\n    4. Pink Noise\n    5. Gaussian Noise\n    6. Exercises\n  6. 5. Autocorrelation\n    1. Correlation\n    2. Serial Correlation\n    3. Autocorrelation\n    4. Autocorrelation of Periodic Signals\n    5. Correlation as Dot Product\n    6. Using NumPy\n    7. Exercises\n  7. 6. Discrete Cosine Transform\n    1. Synthesis\n    2. Synthesis with Arrays\n    3. Analysis\n    4. Orthogonal Matrices\n    5. DCT-IV\n    6. Inverse DCT\n    7. The Dct Class\n    8. Exercises\n  8. 7. Discrete Fourier Transform\n    1. Complex Exponentials\n    2. Complex Signals\n    3. The Synthesis Problem\n    4. Synthesis with Matrices\n    5. The Analysis Problem\n    6. Efficient Analysis\n    7. DFT\n    8. The DFT Is Periodic\n    9. DFT of Real Signals\n    10. Exercises\n  9. 8. Filtering and Convolution\n    1. Smoothing\n    2. Convolution\n    3. The Frequency Domain\n    4. The Convolution Theorem\n    5. Gaussian Filter\n    6. Efficient Convolution\n    7. Efficient Autocorrelation\n    8. Exercises\n  10. 9. Differentiation and Integration\n    1. Finite Differences\n    2. The Frequency Domain\n    3. Differentiation\n    4. Integration\n    5. Cumulative Sum\n    6. Integrating Noise\n    7. Exercises\n  11. 10. LTI Systems\n    1. Signals and Systems\n    2. Windows and Filters\n    3. Acoustic Response\n    4. Systems and Convolution\n    5. Proof of the Convolution Theorem\n    6. Exercises\n  12. 11. Modulation and Sampling\n    1. Convolution with Impulses\n    2. Amplitude Modulation\n    3. Sampling\n    4. Aliasing\n    5. Interpolation\n    6. Summary\n    7. Exercises\n  13. Index\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nNew York City\nBig City Food Biographies Series\n\nSeries Editor\n\nKen Albala, University of the Pacific, kalbala@pacific.edu\n\nFood helps define the cultural identity of cities in much the same way as the distinctive architecture and famous personalities. Great cities have one-of-a-kind food cultures, offering the essence of the multitudes who have immigrated there and shaped foodways through time. The Big City Food Biographies series focuses on those metropolises celebrated as culinary destinations, with their iconic dishes, ethnic neighborhoods, markets, restaurants, and chefs. Guidebooks to cities abound, but these are real biographies that will satisfy readers' desire to know the full food culture of a city. Each narrative volume, devoted to a different city, explains the history, the natural resources, and the people that make that city's food culture unique. Each biography also looks at the markets, historic restaurants, signature dishes, and great cookbooks that are part of the city's gastronomic makeup.\n\nBooks in the Series\n\nNew Orleans: A Food Biography, by Elizabeth M. Williams\n\nSan Francisco: A Food Biography, by Erica J. Peters\nNew York City\n\nA Food Biography\n\nAndrew F. Smith\n\nRowman & Littlefield\n\nLanham \u2022 Boulder \u2022 New York \u2022 Toronto \u2022 Plymouth, UK\nThe publisher has done its best to ensure that the instructions and\/or recipes in the book are correct. However, users, especially parents and teachers working with young people, should apply judgment and experience when preparing recipes. The publisher accepts no responsibility for the outcome of any recipe included in this volume.\n\nPublished by Rowman & Littlefield\n\n4501 Forbes Boulevard, Suite 200, Lanham, Maryland 20706\n\nwww.rowman.com\n\n10 Thornbury Road, Plymouth PL6 7PP, United Kingdom\n\nCopyright \u00a9 2014 by Rowman & Littlefield\n\nAll rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without written permission from the publisher, except by a reviewer who may quote passages in a review.\n\nBritish Library Cataloguing in Publication Information Available\n\nLibrary of Congress Cataloging-in-Publication Data\n\nSmith, Andrew F., 1946\u2013\n\nNew York City : a food biography \/ Andrew F. Smith.\n\npages cm \u2014 (Big city food biographies series)\n\nIncludes bibliographical references and index.\n\nISBN 978-1-4422-2712-5 (cloth : alk. paper) \u2014 ISBN 978-1-4422-2713-2 (electronic : alk. paper) 1. Food\u2014New York (State) \u2014New York\u2014History. 2. Food habits\u2014New York (State) \u2014New York\u2014History. 3. Food industry and trade\u2014New York (State) \u2014New York\u2014History. 4. Cooking, International\u2014History. 5. New York (N.Y.) \u2014History. I. Title.\n\nTX360.U63N498 2014\n\n641.59747\u2014dc23 2013032098\n\n \u2122 The paper used in this publication meets the minimum requirements of American National Standard for Information Sciences\u2014Permanence of Paper for Printed Library Materials, ANSI\/NISO Z39.48-1992.\n\nPrinted in the United States of America\n\nContents\n\nContents\n\nBig City Food Biographies, Series Foreword\n\nChronology\n\nChapter 1: The Material Resources\n\nChapter 2: From Colonization to the Present\n\nChapter 3: A Culinary Stew\n\nChapter 4: From Garden to Grocery\n\nChapter 5: Drinking in the City\n\nChapter 6: Eateries\n\nChapter 7: Historic Cookbooks, Dishes, and Recipes\n\nBibliography\n\nAbout the Author\n\nBig City Food Biographies, Series Foreword\n\nCities are rather like living organisms. There are nerve centers, circulatory systems, structures that hold them together, and of course conduits through which food enters and waste leaves the city. Each city also has its own unique personality, based mostly on the people who live there but also on the physical layout, the habits of interaction, and the places where people meet to eat and drink. More than any other factor, it seems that food is used to define the identity of so many cities. Simply say any of the following words and a particular place immediately leaps to mind: bagel, cheesesteak, muffuletta, \"chowda,\" and cioppino. Natives, of course, have many more associations\u2014their favorite restaurants and markets, bakeries and donut shops, pizza parlors, and hot dog stands. Even the restaurants seem to have their own unique vibe wherever you go. Some cities boast great steakhouses or barbecue pits; others, their ethnic enclaves and more elusive specialties like Frito pie in Santa Fe, Cincinnati chili, and the Chicago deep dish pizza. Tourists might find snippets of information about such hidden gems in guidebooks; the inveterate flaneur naturally seeks them out personally. For the rest of us, this is practically unchartered territory.\n\nThese urban food biographies are meant to be not guidebooks but rather real biographies, explaining the urban infrastructure, the natural resources that make each city unique, and most importantly the history, people, and neighborhoods. Each volume is meant to introduce you to the city or reacquaint you with an old friend in ways you may never have considered. Each biography also looks at the historic restaurants, signature dishes, and great cookbooks that reflect each city's unique gastronomic makeup.\n\nThese food biographies also come at a crucial juncture in our culinary history as a people. Not only do chain restaurants and fast food threaten the existence of our gastronomic heritage, but also we are increasingly mobile as a people, losing our deep connections to place and the cooking that happens in cities over the generations with a rooted population. Moreover, signature dishes associated with individual cities become popularized and bastardized and are often in danger of becoming caricatures of themselves. Ersatz versions of so many classics, catering to the lowest common denominator of taste, are now available throughout the country. Our gastronomic sensibilities are in danger of becoming entirely homogenized. The intent here is not, however, to simply stop the clock or make museum pieces of regional cuisines. Cooking must and will evolve, but understanding the history of each city's food will help us make better choices, will make us more discerning customers, and perhaps will make us more respectful of the wonderful variety that exists across our great nation.\n\nKen Albala University of the Pacific\n\nChronology\n\n1626 | New Amsterdam is founded as a fur-acquiring operation by the Dutch West India Company. Old World plants and animals are introduced into what will become New York.\n\n---|---\n\n1664 | English warships capture New Amsterdam and change the city's name to New York. English immigration.\n\n1762 | Samuel Fraunces opens a tavern, first named the Queens Arms, later named for its owner. Today, the building is the oldest surviving structure in New York City.\n\n1763 | The Seven Years' War ends; the British acquire Canada and all of North America east of the Appalachian Mountains to the Mississippi River. American merchants thrive during the war, supplying the British army and navy with food and other necessities.\n\n1764 | To help pay for the war, the British parliament begins to pass bills taxing products imported into North America to help cover the costs of British troops that are stationed in Canada. New Yorkers form Sons of Liberty in New York and launch resistance to the British laws, particularly to the Sugar Act and a series of other revenue measures imposed by parliament.\n\n1773 | British parliament lowers the tax on tea; the British East Indies Company rushes ships with tea to America assuming that colonials will buy the lower-costing tea. The ships containing tea are turned back; tea is thrown into New York Harbor, just as it had been previously in Boston.\n\n1775 | The War for American Independence begins; British forces occupy New York City the following year and continue to do so for more than seven years. Surrounding farmers refuse to send food into the city, and the British garrison and the loyalists living in the city depend on shipments of food from Ireland to survive.\n\n1783 | Peace treaty is signed granting independence to the United States; British forces leave New York. General George Washington celebrates by dining at Fraunces Tavern in lower Manhattan.\n\n1784 | The sailing ship Empress of China sailed out of New York Harbor headed for China and returns with a load of tea; it is the first American vessel to trade with China.\n\n1789 | George Washington is inaugurated first president of the United States in New York City; he hires Samuel Fraunces, proprietor of a city tavern, to cater his food during his tenure in New York.\n\n1807 | William Havemeyer opens a sugar refinery in New York; the company he founded will become one of the most important sugar companies in nineteenth-century America.\n\n1812 | War of 1812 with Britain begins; international trade is greatly reduced until its end in 1815.\n\n1819 | Canning of salmon, lobster, and oysters begins in New York City.\n\n1825 | Erie Canal is completed; grains and agricultural goods from upstate New York and the Midwest flow into New York. The city increases financial, commercial, and manufacturing operations, especially for flour and sugar refining.\n\n1827 | The Delmonico brothers, immigrants from the Italian region of Switzerland, open a caf\u00e9 in New York; it will blossom into one of the city's most famous upper-class restaurants, which survives until 1923.\n\n1830s | Large numbers of German immigrants begin arriving in New York\u2014they will continue to arrive until World War I. They establish larger beer breweries, delicatessens, grocery stores, street vendors\u2014one of whom will be credited with inventing the American hot dog\u2014and restaurants, which serve hamburger steak.\n\n1832 | Vegetarian and health reformer Sylvester Graham lectures in New York, arguing for temperance and vegetarianism; supporters open Graham Boardinghouse in New York, which followed his advice about food and beverages.\n\n1840s | Irish immigrants begin flooding into New York; they open saloons and grocery stores, and work in boardinghouses. The \"swill milk war\" begins with newspapers and magazines publishing expos\u00e9s about the city's milk, beer, and distilling industries.\n\n1860s | Italian immigrants begin to flood into the city, creating Little Italy. The Great Atlantic & Pacific Tea Company (soon shortened to A&P grocery store) chain opens in New York.\n\n1861 | The Civil War begins; New Yorkers join regiments to fight for the Union; the war ends four years later. New York restaurants thrive during the war.\n\n1863 | Riots in New York against the draft require U.S. military to help control parts of the city.\n\n1864 | Union League Club launches effort to give Union soldiers and sailors a turkey dinner on Thanksgiving.\n\n1867 | Charles Feltman, a pushcart vendor on Coney Island, New York, is credited with selling sausages on white rolls. These are considered the first hot dogs.\n\n1869 | The transcontinental railroad completed; New Yorkers can travel by train to San Francisco, and food can be shipped across the continent. The grocery store chain that will eventually become A&P establishes stores outside of Manhattan and will eventually become the largest chain in America.\n\n1870s | Chinese immigrants begin to move into the city, eventually creating Chinatown.\n\n1880s | Bagels are sold by street vendors.\n\n1890 | Jacob Riis publishes the expos\u00e9 How the Other Half Lives: Studies among the Tenements of New York, reporting that more than half a million people living in the tenements were either begging for food or subsisting on charity.\n\n1894 | Charles Ranhofer, the former chef at Delmonico's, publishes The Epicurean, with the recipes that made the restaurant famous.\n\n1898 | Five boroughs\u2014Manhattan, Brooklyn, Staten Island, Queens, and the Bronx\u2014merge to form Greater New York.\n\n1902 | New York's first Automat opens. They survive until 1991.\n\n1903 | America's first pizzeria opens in New York's Little Italy.\n\n1916 | Nathan Handwerker, a Polish immigrant, opens a hot dog stand at Coney Island, New York; it's soon known as \"Nathan's Famous.\"\n\n1917 | United States enters the war; Congress passes restriction on making beer and spirits as a way to help conserve grain for use by the military and to send to allies. New York distilleries and breweries decrease or halt production of beer and spirits. \"Wheatless, Meatless, and Porkless\" days are declared by the mayor to support the war effort.\n\n1919 | Eighteenth Amendment to U.S. Constitution is approved, outlawing the manufacture, sale, importation, and transportation of alcoholic beverages in America; the following year, Prohibition begins. Saloons, bars, breweries, distilleries, and many restaurants close in New York; speakeasies open, and bootleggers flourish.\n\n1930 | Michael J. Cullen opens one of the nation's first supermarkets\u2014King Kullen\u2014in Jamaica, New York.\n\n1933 | Prohibition ends; bars and a few breweries reopen.\n\n1939 | New York's World Fair opens in Flushing Meadow, Queens. It has more than eighty restaurants that introduce New York fairgoers to a wide variety of foreign foods. Two restaurants emerge from the fair and will influence the city's future restaurants for decades.\n\n1941 | The United States enters World War II. Gourmet magazine launched, and Henri Soul\u00e9 opens Le Pavillon restaurant in New York's plush Upper East Side. Both will reintroduce haute cuisine into New York.\n\n1942 | Rationing goes into effect and will not completely end until 1947; New Yorkers plant Victory Gardens to help the war effort.\n\n1945 | World War II ends. Veterans return with an experience of and taste for European and Asian food cultures.\n\n1957 | Craig Claiborne is hired as a food columnist by New York Times; his cookbook, The New York Times Cook Book (1961), sold more than three million copies; it familiarized New Yorkers with regional American, European, and some Asian recipes.\n\n1959 | Four Seasons restaurant created by restaurant impresario Joe Baum opens.\n\n1965 | Immigration Act makes it possible for immigrants from all over the world to bring their food cultures to America; New York restaurant kitchens benefit from the influx of culinary professionals from around the world.\n\n1968 | Gael Greene is hired as the restaurant critic and later food columnist for New York Magazine.\n\n1973 | Michael and Ariane Batterberry publish On the Town in New York: From 1776 to the Present, New York's first food history.\n\n1975 | Mimi Sheraton is hired as the New York Times restaurant critic and transforms the genre of newspapers reviewing restaurants. She describes her techniques in her memoir, Eating My Words: An Appetite for Life (2004).\n\n1976 | Windows on the World, designed by the team led by restaurant impresario Joe Baum, opens in the World Trade Center. New York's first Greenmarket opens; today, fifty-four Greenmarkets are open in all boroughs of the city.\n\n1990 | Molly O'Neill is hired as a reporter for the New York Times and later becomes a weekly food columnist for the paper's Sunday magazine. She publishes three cookbooks in the 1990s, including New York Cookbook (1992).\n\n1993 | Food Network is launched in New York; its programs\u2014as well as those of other networks\u2014popularize food preparation and other culinary matters.\n\n1994 | Saveur, a food magazine, is started in New York.\n\n1997 | Amanda Hesser becomes a food reporter for the New York Times, and later the food editor for the newspaper's Sunday magazine. Her cookbook, The Essential New York Times Cook Book, was published in 2010.\n\n2001 | The World Trade Center is destroyed; New Yorkers pull together to move beyond the tragedy. Construction begins on a new World Trade Center, which is completed in 2013.\n\n2009 | William Grimes publishes Appetite City: A Culinary History of New York, the city's second book-length culinary history.\n\n2010 | Gourmet magazine closes, ending an era of upscale food publishing. Lidia Bastianich, her son Joe Bastianich, and Mario Batali extend the reach of their interpretations of Italian cuisine with Eataly, a fifty-thousand-square-foot emporium that is devoted to Italian food and culinary traditions.\nChapter One\n\nThe Material Resources\n\nLand, Water, and Air\n\nTwenty thousand years ago there wasn't much to see\u2014or eat\u2014where the culinary capital that is New York City now rises. There was no Manhattan Island, Hudson River, or Long Island Sound, and no forests, fish, animals, or ocean; the Atlantic, at that time, was 130 miles to the east. There was nothing but a wintry wasteland of snow and chunks of ice, for this was the southeastern boundary of the Laurentide ice sheet. When the ice sheet retreated, it created the physical characteristics of the future city\u2014its islands and hills, and its rivers and shoreline. Tundra and grasslands expanded into the void left by the glacier, and big game\u2014woolly mammoths, mastodons, bison, bears, sloths, giant beavers, caribous, and saber-toothed tigers\u2014roamed into the area. Small bands of Paleo-Indians followed them. Little is known about these hunters other than they were highly mobile and their principal weapon was a small spear propelled by a spear thrower. The small spear points, made from antler or stone, easily penetrated the hide of their prey but also slipped out easily, permitting their reuse. The big game disappeared\u2014likely due to the expert hunting skills of the Paleo-Indians and climatic fluctuations.\n\nThe climate and geographical features stabilized about eight thousand years ago. Forests grew, and dozens of different habitats provided homes for a complex array of trees, plants, animals, fish, shellfish, and birds. The most comprehensive early account of the natural features of the New York area was recorded by Adriaen van der Donck, a Dutch lawyer and colonist who arrived in New Amsterdam in 1641. He was later given an estate north of Manhattan that would later become a city named after him, Yonkers. Van der Donck carefully studied the natural flora and fauna of the colony and published his observations in A Description of the New Netherlands (1655). The winters were colder in New Netherland than in Holland at the time, he reported. This was likely true, as the Little Ice Age that had arrived a few hundred years before greatly lowered temperatures in eastern North America. It would continue to affect the climate until the early nineteenth century. He also reported that the summers were warmer and the soil was perfect for farming, especially on Long Island and Staten Island.\n\nThe food plants native to the area, Van der Donck recounted, included \"several kinds of plums, Wild or small cherries, juniper, small kinds of apples, many hazel-nuts, black currants, gooseberries, blue India figs, and strawberries in abundance all over the country, some of which ripen at half May, and we have them until July; blueberries, raspberries, black-caps, &c., with artichokes, ground-acorns, ground beans, wild onions, and leeks like ours, with several other kinds of roots and fruits, known to the Indians.\" There were also ample fowl, particularly wild turkeys, which were easy to hunt year round, and migratory birds, such as \"geese, ducks, widgeons, teal, brant and many more,\" which were available much of the year.\n\nThe Hudson estuary, much of which was below sea level, was a tidal oasis for fish, shellfish, and sea mammals. Anadromous fish (those that spawn in freshwater rivers and annually migrate to and from the ocean)\u2014shad, herring, sturgeon, striped bass, salmon, sea trout, and others\u2014frequented the Hudson River and migrated annually through the bay to the ocean. In the surrounding bay and waters, there was plenty of herring, mackerel, halibut, scoll, sheepshead, \"carp, snook, forrels, pike, trout, suckers, thickheads, flounders, eels, palings, brickens and lampreys,\" noted Van der Donck. They became a major food source for the area, as did turtles, crabs, lobster, mussels, scallops, eels, and clams. Some lobsters, claimed Van der Donck, were \"five to six feet in length; others again are from a foot to a foot and a half long, which are the best for the table.\" The most common shellfish were the oysters; according to later estimates, the New York Bay and Long Island Sound had the most prolific oyster beds in the world. The bay was also frequented by marine mammals\u2014seals, porpoises, and the occasional whale\u2014which were occasionally stranded on beaches. Just offshore in the Atlantic was yet another abundance of oceanic fish and seafood. All in all, the New York City area was a natural culinary wonderland.\n\nThe Lenape\n\nAbout 6,500 years ago, the Munsee\u2014a loose collection of groups of Delaware Indians who settled the area from eastern Connecticut to central New Jersey and eastern Pennsylvania\u2014arrived in the New York area. They spoke dialects of Algonquian, as did hundreds of other indigenous groups living in eastern North America. In New York, the Munsee called themselves Lenape, or \"real people.\" Archaeologists have identified about eighty Lenape habitation sites in the five boroughs of New York, but few remains have been located. How many Lenape lived in the area when the Europeans arrived in the early seventeenth century is unknown. Estimates vary from a low of 3,800 to a high of 15,000. Based on the number of archaeological sites, Staten Island likely had the largest Indian population.\n\nThe Lenape operated in small autonomous bands that varied in size from a few dozen to hundreds of members. Each band maintained fishing and hunting \"rights\" within the particular territory where they lived. Their villages were typically situated on hills near freshwater sources; larger villages were surrounded by palisades. Their wigwams\u2014small round structures\u2014and longhouses\u2014larger multifamily buildings\u2014were constructed of grass, wood, and bark. The hearth was located in the center of their abodes, and a hole in the roof permitted the smoke to escape. The Lenape's only domesticated animal was the American dog used for hunting and, occasionally, as a pack animal. It disappeared shortly after Europeans introduced Old World dogs into the area.\n\nNative peoples enjoyed a rich and diverse diet. With spears, stone axes, and traps, men hunted birds and animals. They acquired bows and arrows about 2,800 years ago, and this greatly enhanced their hunting efficiency. They also collected eggs from wild birds and gathered a wide variety of readily available plant foods: grapes, gooseberries, raspberries, strawberries, fruit, chestnuts, hickory nuts, acorns, roots, barks, and leaves. They caught shad, herring, eels, alewife, turtles, and other sea animals with spears or bag-shaped nets. They gathered shellfish, particularly clams, oysters, lobsters, and mussels. From white or purple seashells found at Long Island Sound, the Lenape made wampum, small beads pierced with a central hole so that they could be strung together. Wampum was used for ritual and ornamental purposes. The Lenape strung them together to make belts and ornaments. When the Dutch arrived, wampum became a medium of exchange, and eventually it was recognized as legal tender throughout the Dutch colony of New Netherland.\n\nThe Lenape developed a network of trails throughout the area that gave them easy access to wooded areas for hunting and coastal areas for fishing and gathering shellfish. They made pitch-sealed bark buckets and fired clay pots. The pot walls tapered to a thin rim, making them lighter and easier to move from place to place, if relatively fragile. Still, if a pot broke in transit, it was easy enough to make a new one, which they found preferable to carrying heavier, sturdier vessels. The pots and buckets were used for cooking by filling them with water and dropping in heated stones, which quickly brought the water to a boil.\n\nLenape bands did not settle into towns, at least as Europeans understood the term. They were seminomadic; they moved occasionally depending on soil productivity and game availability. They employed the slash-and-burn method of agriculture, creating fields by felling big trees and burning small trees and underbrush. In these open areas, they planted corn, beans, and squash, which had been domesticated in Mesoamerica but had been disseminated into the New York area about a thousand years ago.\n\nIn the fall, when the harvest was done, the Lenape burned their fields, as well as nearby woods and meadows. In the spring, they burned off any vegetation that had grown on the fields before they planted their crops. Burning the woods cleared the area of entangling underbrush and improved the productivity of the land. The fires did not usually harm large trees, which only lost their outer bark and lower branches, thus ensuring a supply of nuts or fruits. The clearings also created a habitat for animals, which could then be hunted more easily. (This practice was so useful that Dutch colonists would later do the same thing.) They remained in a specific place for several years, until the productivity of the land diminished; then they moved to another location and started over.\n\nLenape women prepared fields for planting by making small molehills about two and a half feet apart. In April, they planted five or six grains of corn in each hill. When the plants were two or three feet high, the women sowed beans under them, and the bean vines climbed the cornstalks like trellises. Occasionally, squash seeds were planted between the mounds and grew along the ground: they help to keep the soil moist between showers. The beans, corn, and squash were harvested in the fall; some of the produce was eaten fresh, and some was dried for use during the winter and spring. Dried food was stored in clay pots, pitch-sealed bark buckets, woven baskets, or bags made of rushes or wild hemp. These were buried in pits dug into the house floors or stashed beneath sleeping platforms. For sustaining winter meals, they boiled fresh meat with dried corn or beans until soft enough to eat. They also boiled beans with corn to make a type of succotash.\n\nThe Lenape may have also planted Jerusalem artichokes (Helianthus tuberosus) and sunflowers (Helianthus annuus), which grew wild in the area. The sunflower had been domesticated by North American Indians about five thousand years ago. The seeds were dried, shelled, and eaten; they were combined with corn and beans to make succotash. The sunflower plant was closely related to the Jerusalem artichoke, a somewhat knobby, thin-skinned underground tuber that was usually boiled or roasted. The dried tubers and dried sunflower seeds were also ground into flour. These were the only two plants that were domesticated in eastern North America and widely disseminated in pre-Columbian times.\n\nDuring the summer months, women weeded the fields and piled more dirt on the mounds. In August, they removed barren cornstalks and sucked out the sweet sap, which was one of the few natural sweeteners available to them. In the fall, the adult men and some women left the village to hunt and fish, leaving other women, the elderly, and children in charge of the fields to keep deer and other animals from gobbling up the fields. The hunters were absent for months at a time. They ate as much as they could, becoming fat, which helped them survive the long winter and spring. They also dried meat and fish and brought them back to the settlement.\n\nIn the spring, men fished with dragnets that they had knitted from wild hemp fibers. They mainly caught shad and white salmon, which, according to an early European account, was \"of very good flavor . . . it has white scales; the heads are so full of fat that in some there are two or three spoonfuls, so that there is good eating for one who is fond of picking heads.\" In the summer, the men hunted \"deer, fawns, hares, and foxes and all such\" with bows and arrows reported another European. They commonly boiled their meat and fish, and also dried both for later consumption. When needed, the leathery dried fish or meat was pounded into powder and boiled with ground cornmeal. This was a common dish in late winter and early spring, when other provisions had been exhausted. If the supply of dried food gave out, the Lenape went out hunting again or headed for the bay or sound, where shellfish were usually abundant.\n\nDried corn was the Lenape's most important staple, particularly during the winter and spring when other food sources were at a premium. To render it edible, they boiled it and then pulverized it with a stone on a block or with a wooden mortar, usually a hollowed-out tree stump. With the coarser cornmeal, the Lenape made porridge, which they called sappaen; this was a mainstay of their winter diet. They sifted out the fine flour, mixed it with lukewarm water, and kneaded it into dough. This was molded into round, flat cakes, which were heated in the hot ashes of a fire. When the Europeans arrived, some were unimpressed with the Indian cornbread, which did not taste or look like the wheat bread they expected. As one later explained, \"They gave us a small piece when we entered, and although the grains were not ripe, and it was half baked and coarse grains, we nevertheless had to eat it.\" Despite bad reviews, early European colonists adopted it as a staple of their diets.\n\nThe Lenape also made cakes and porridge out of acorns. As most acorns contain a toxin (tannic acid), this was a complicated process. The nuts either were subjected to long soaking or were repeatedly boiled and rinsed. When waterlogged, the shells were cracked open and the nutmeats removed. The acorns were pulverized with a mortar and pestle and the resulting mush soaked additional times to leach out the tannin. Then it was dried in the sun and finally pounded into a powder. They used acorn flour as they did cornmeal, to make porridge or cakes.\n\nThe Indians had no set schedule for meals; each family ate when they were hungry. They shared all of their meals with anyone who wanted to eat, with no expectation of compensation, reported early European settlers. The exceptions were ceremonial dinners, where each family in a community contributed delicacies. When the Lenape entertained, their special foods were beavers' tails, bass heads, or very fat meat stewed with parched cornmeal and bruised shelled chestnuts, reported a Dutch colonist. When the Lenape traveled and had no expectation of finding or hunting for food along the way, they carried small bags of parched cornmeal, which took up little space. When water was added to the meal, the parched corn expanded and provided a filling meal. According to the Dutch, the Lenape could subsist on a quarter of a pound of this mixture per day for several days, if necessary.\n\nThe Lenape did not eat excessively, even on their feast days. The quality and quantity of food were the same for all; higher status did not entitle one to more or better food or the special culinary luxuries that Europeans expected. No food was wasted.\n\nTheir main beverage was water, but when grapes were in season, the Lenape occasionally crushed them and drank the juice. They also drank the water in which their food was cooked. They consumed no alcoholic beverages until the arrival of the Europeans. Once exposed to alcohol, some drank to excess, becoming malicious, \"insolent and troublesome,\" reported Dutch observers. Once Indians became accustomed to alcohol, however, Van der Donck reported, they bore \"it as well as our own people do.\" Most refused to drink alcohol, and they called those Indians who drank to excess fools. The Dutch government forbade the sale of alcoholic beverages to Indians as did other colonial governments. Despite the laws, Europeans regularly traded or gave alcohol to them, and alcoholism contributed to their dissolution of families and their communities.\n\nWhen the Dutch West India Company made the decision to establish a settlement on Manhattan, Peter Minuit, the director of the colony of New Netherland, bought the island in April 1626 for about sixty guilders worth of goods (estimated today to be worth about four thousand dollars) from some Lenape, who, according to the Dutch, were empowered to speak for all Lenape on Manhattan. The Lenape had agreed, claimed the Dutch, to move north of the island to what is today the Bronx. How many Lenape lived on Manhattan at the time is variously estimated to have been as few as 300 or as many as 1,200, and they lived in several different communities. Later, the Dutch made similar accommodations with other groups on Long Island and Staten Island. Europeans identified these groups with the places where they lived, such as the Jamaica (beaver place), Canarsies (grassy place), and the Rockaways (sandy place).\n\nSome Lenape never left the area, and others returned to the places where they had once lived. It was estimated in 1628 that three hundred Lenape still inhabited Manhattan. Relations between them and the colonists deteriorated. A Dutch observer complained that the Indians were \"a bad race of savages, who have always been very obstinate and unfriendly towards our countrymen.\" The friction was abetted by conflicting systems of agriculture. Dutch introduced Old World animals that roamed freely, eating Lenape crops, and the Dutch destroyed their hunting grounds. Conflicts erupted over stolen pigs, cattle, and even peaches. These escalated into wars that ended with the Lenape's defeat.\n\nWar, the disappearance of game, the growth of alcoholism, and the introduction of viral diseases by the Dutch all contributed to the destruction of the Lenape. They disappeared from Manhattan by the end of the seventeenth century, but small groups survived in Brooklyn, Staten Island, and the Bronx until the nineteenth century.\n\nEuropean colonists altered the area's natural environment, fashioning it into a traditional European landscape. They chopped down forests to make way for buildings, roads, and farms, which quickly dominated Manhattan and the other boroughs. Hills were leveled; valleys were filled in; and ponds, swamps, springs, and streams were covered over. Wharves, built to enable ships to sail right up to the island and easily load and unload, destroyed sandy beaches and wildlife habitats. Rocks, debris, garbage, trash, and construction materials were dumped into the surrounding waterways, which expanded the size of Manhattan, Long Island, and other islands in the harbor. The Hudson River, the East River, and the harbor became polluted; the aquatic and avian diversity that once thrived and fed the city's inhabitants disappeared.\n\nThese natural losses were noted and mourned by the mid-nineteenth century. New Yorkers preserved some natural habitats by creating parks, and the results are some of the nation's most beautiful urban parks, sanctuaries, and botanic gardens, such as Central Park in Manhattan, Prospect Park and the Brooklyn Botanic Garden in Brooklyn, Van Cortlandt Park and the New York Botanic Garden in the Bronx, and the Jamaica Bay Wildlife Sanctuary in Queens. Only during the last forty years have New Yorkers become particularly concerned with water and soil pollution. Good progress has been made: wetlands have been preserved; water pollution has declined; and some native fish, shellfish, and birds have returned. Oyster beds have been planted in the harbor to help filter the pollution out of the water and improve biodiversity. These were important steps, but they have not re-created the area's abundant natural environment that existed before the arrival of large numbers of Europeans in the seventeenth century.\nChapter Two\n\nFrom Colonization to the Present\n\nHenry Hudson, an English explorer, was engaged by the Dutch East India Company to find a northern route to the Pacific coast of Asia. If a northern passage could be found, it would shorten the distance and time to reach the wealth of India and China. Hudson sailed northward along the eastern coast of North America when, in September 1609, he sighted a large bay. Hudson sailed his ship, de Halve Maen (the Half Moon), through those narrows and into the bay, and he traveled up the estuary called the Great River of Mountains. He turned back near where the city of Albany is today. Along the river he traded with native peoples, acquiring corn, oysters, and fruit. One sailor proclaimed, \"They haue great store of Maiz or Indian Wheate, whereof they make good Bread.\" A Dutch officer on board reported that the Indians possessed \"an abundance of provisions, skins, and furs, of martens and foxes, and many other commodities, as birds and fruit, even white and red grapes, and they traded amicably with the people.\" The sailors themselves netted \"ten great Mullets\" and a ray so large that four men had difficulty wrestling it onboard the ship.\n\nBack in Holland, the Dutch were impressed with the furs that Hudson brought back from his voyage. Other Dutch ships soon arrived in the bay and further explored the river, which they named the North River. (It wouldn't be called the Hudson River until 1909, when New Yorkers celebrated the three hundredth anniversary of Hudson's voyage.) The Dutch commenced a fur-trading enterprise with the Indians, which proved so lucrative that in 1614, the Dutch established a small outpost at Fort Orange, today Albany. Early Dutch traders found an abundance of game, migratory fowl, fresh and saltwater fish, shellfish, turtles, and even buffalo. They were solely focused on acquiring furs (beavers, otters, minks, and other animals). From local Indian tribes they acquired maize and other foodstuffs. They neglected to establish the gardens, farms, and orchards necessary to sustain their growing community.\n\nA growing food shortage was not the worst problem they faced, however: the English claimed all of eastern North America and had complained to the Dutch about their fur-trading activities along the Hudson River. The English were serious about their claim: in 1620, English colonists, many of whom ironically lived in Holland, were granted authority to establish a community at the mouth of the Hudson River. For reasons that remain unclear, these English colonists, later called Pilgrims, settled at Plimoth Plantation, in Massachusetts. Had this group settled in New York, the Dutch trading center upriver would have been cut off and New York's culinary history would have been very different.\n\nThe Dutch created the West India Company in 1621, and the fur-trading venture in North America became one of its responsibilities. In 1625, the company ordered the director general of its North American trading operations to concentrate colonists in a single settlement. The southern tip of Manhattan Island was selected, and construction was begun on a small fort to protect the colony. More colonists arrived from Europe, bringing cattle, dogs, sheep, hogs, horses, and agricultural implements. By November 1626, 270 colonists were living in New Amsterdam, as the settlement was called. They established farms, planted crops, and raised animals, but their main activity was fur trading, and food shortages emerged. In 1628, the Reverend Jonas Micha\u00eblius, the first minister of the Dutch Reformed Church in the settlement, reported that the colonists failed to \"obtain proper sustenance for want of bread and other necessaries.\" Butter and milk, he wrote, \"cannot here be obtained\" except at \"very high price.\" He urged the West India Company to send more farmers and dairymen. This complaint worked. Dairy farmers were recruited to settle in New Amsterdam, and milk, butter, and cheese became staples, as they were in Holland.\n\nColonists planted wheat, but the early varieties did not fare well. Having learned from the Lenape how to plant corn, the Dutch ground the kernels and made cakes by mixing cornmeal with water and baking the dough in hot ashes\u2014just as the local Indians had taught them. Dutch colonists also ate the Indian corn dishes samp (a type of hominy) and suppawn (a thick cornmeal porridge), but they mixed these with butter, a European ingredient unknown to the Lenape. Colonists planted other Old World grains, such as barley, and a wide variety of fruits (apples, berries, cherries, and grapes), nuts (almonds and hazelnuts), and vegetables (onions, peas, and turnips).\n\nUnlike Boston, New Amsterdam never developed a large deep-sea fishing fleet; Boston was closer to the Grand Banks, where cod, the premier fish of the day, thrived, so that city had a geographical advantage. But there was a more important reason. The streams, rivers, lakes, bays, Long Island Sound, and the nearby Atlantic offered such abundance and diversity of fish, shellfish, and seafood that fishing hundreds of miles offshore was simply unnecessary. According to Adriaen van der Donck, \"All the waters of the New-Netherlands are rich with fishes.\" In early June, sturgeon were extremely plentiful in the rivers, but they were not preferred and were not eaten at all when large. In fact, he reported that no one took the trouble to catch them for profit, and among those who caught sturgeon for sport, the \"roes from which the costly caviaer was made\" were discarded. This resource would later be recognized as a valuable commodity, and in the nineteenth century New York City became the world's largest caviar exporter until the sturgeon disappeared due to overfishing and pollution.\n\nOysters were so abundant in the bays and harbors around Manhattan and Long Island that they were considered a cheap everyday food rather than a luxury, but Van der Donck complained that they didn't have any valuable pearls in them. Oysters from the bay and East River were highly praised. Two visitors in 1669 exclaimed that the oysters \"are large and full, some of them not less than a foot long, and they grow sometimes ten, twelve and sixteen together, and are then like a piece of rock. Others are young and small. In consequence of the great quantities of them, everybody keeps the shells for the purpose of burning them into lime.\"\n\nIt took time for Old World wheat varieties to acclimatize to the New World, but by the 1640s wheat grew abundantly in the Hudson Valley and on Long Island; it was shipped to New Amsterdam, where flour was milled and made into wheat bread and bakeries were in operation. After annexation by the English, New York became the world's largest milling center. Grain, flour, bread, and biscuits became major export items, and New York, along with New Jersey and Pennsylvania, would later be called the \"bread colonies.\" Ships leaving New York Harbor stocked up on bread, hardtack, and flour before venturing out to sea. By the 1760s, 75 percent of New York's exports were agricultural products in the form of flour or biscuits. During the following decades, the city became the world's largest merchant-milling center.\n\nFew natural sweeteners were available in the New Netherland colony. American Indians did draw sap from several trees, the most productive of which was the sugar maple, but sugar from the maple was not manufactured extensively until the following century. Honey would have been another possibility, but American bees produced little of it. About 1638 the Dutch introduced the Old World bee (Apis mellifica) into New Netherland, and this honey-producing strain quickly established itself throughout colonial America. The Indians called the bees \"white men's flies.\"\n\nThe rapid expansion of sugarcane planting in the Dutch (and other) colonies in the Caribbean and South America gave New Amsterdamers easy access to refined sugar, although it was expensive and only the wealthy could afford to buy a lot of it. Sugar was occasionally granulated but more often came in clumps or in seven- to ten-pound \"loaves,\" which were large, solid, white cones. Shears, mini-axes and hammers were used to snip off pieces of the sugar cone, which would then be placed in water to dissolve. A much cheaper sweetener was molasses, a by-product of sugar refining. Molasses was the sweetener used by the poor, but it would later find its way into a wide variety of foods, including cookies, cakes, and porridge, and was also used to flavor stewed vegetables and meats.\n\nRather than expend time and effort raising and growing all of their own food, New Amsterdamers acquired game from the Indians; imported spices, coffee, tea, and wine from Europe; and bought beef, pork, wheat, butter, lamb, fish, butter, malt, salt, fruit, and cider apples from Long Island, New England, and Virginia. As the profitability of fur trapping dwindled, the West India Company faced financial ruin. The Dutch never emigrated in large numbers to New Amsterdam. Officials permitted immigrants and refugees with diverse national, religious, ethnic, and linguistic backgrounds\u2014including peoples from Germany, Italy, Spain, and England\u2014to settle in New Amsterdam.\n\nThe colony had another problem: it was geographically between two rapidly expanding English colonies, New England to the north and Virginia to the south. In an attempt to provoke a war with the Netherlands, the English sent four warships into New Amsterdam's harbor, and the Dutch colony surrendered without firing a shot in 1664. The Second Anglo-Dutch War broke out several months later, and the Dutch occupied English sugar plantations in what is today Surinam, a country in northern South America that few New Yorkers have ever heard of. In the seventeenth century, sugar was extremely valuable, so when the Dutch came out on top during the war, they kept the lucrative sugar plantations, and the English were stuck with the floundering fur-trading colony, which they renamed New York, in honor of James Stuart, Duke of York, the future King James II.\n\nAt the time, the city had a population of about 1,500, and only about half were of Dutch\/Walloon heritage. Despite this small population, decisions made and actions taken during the Dutch period greatly affected the city's culinary future. These included the introduction of a wide range of Old World plants and animals along with the establishment of dairying and baking. These set the city and its environs on the path to becoming an agricultural powerhouse that survived for two centuries. Just as significant was the willingness of the Dutch to allow many different nationalities and religious groups to settle in the region, a tradition that continued when the colony was turned over to the English. Each new wave of immigrants enriched the city with their own foodways, a precedent that would create an incredible culinary mosaic that is enjoyed by New Yorkers to this day. Yet another significant contribution of the Dutch was their view that New Amsterdam was an economic operation; trade and commerce have continued to shape the city and the food produced and consumed in it.\n\nColonial New York\n\nUnder English administration, New York left its origins as a small, unprofitable fur-trapping colony behind and became a major Atlantic trading center. The Hudson River was navigable father than any other river on the eastern seaboard. The Hudson valley was a rich agricultural area, especially for growing wheat, which was shipped down to New York City; the city became an important milling center and a \"bread basket\" for other colonies and for stocking ships. Other commodities, such as livestock, timber, and apples, were also shipped down the Hudson to New York, where they were converted into salable products such as flour, bread, wooden casks, and hard cider. These were shipped to other American colonies but mostly to the West Indies, where sugarcane was the dominant cash crop. Molasses, sent back from the islands, came to be the most widely used sweetener in America. More important, molasses could be converted into rum, which quickly became New York's spirit of choice.\n\nThe West Indies also shipped raw sugar to New York; these partly refined cane sugar syrups were sweeter and more expensive than molasses, but they could be refined into pure sugar. When the first New York sugar refinery was established is a matter of debate. Historian Mary L. Booth claimed that the city's first sugar refinery opened in 1689 on Liberty Street. Sugar historian Noel Deerr deemed this too early, placing the date in 1725, when the New York legislature granted a monopoly to Robert Hooper to create a refinery. Others soon followed. By 1728 another refinery, run by Samuel Bayard, was operating on Wall Street; it sold refined loaf sugar, powder sugars, and candy. In 1754, the Livingston family constructed a six-story refinery on Crown Street, and John van Cortlandt established one on Liberty Street. William Rhinelander opened the Rhinelander Sugar House at the corner of Rose and Duane Streets in 1763, and later the family acquired another refinery. Refineries were the largest buildings in the city at the time, and sugar refining was one of the city's most important industries.\n\nRevolutionary New York\n\nDuring the Seven Years' War (1756\u20131763), Great Britain defeated the French army in North America, and as a result of the treaty, it acquired Canada along with a vast territory west of the Allegheny Mountains that extended to the Mississippi River. The war had been costly, and many members of the British parliament concluded that the American colonists, who had gained much from the war, should pay a portion of the debt and the cost of upkeep of the British troops stationed in the newly acquired territories.\n\nThe British colonies had indeed reaped tremendous benefits from the war. Merchants had made profits supplying the British army and navy with food and equipment, and more important, the greatest and only real military threat to the colonies\u2014the French in Canada\u2014had been removed. In 1764, the British parliament passed the Sugar Act, which levied a tax on American imports of five products, the most important of which was molasses. A British tax on molasses imported from the French, Spanish, and Dutch colonies in the Caribbean had already been on the books for thirty years, but colonists had simply subverted the law through smuggling and bribing customs officials. The Sugar Act actually lowered the existing import duties from six pence to three pence, but it also had stiff enforcement provisions, which, if carried out, would harm the American sugar-refining and whiskey-distilling industries. Many colonists also viewed the tax as a restriction of their liberties.\n\nThe Sugar Act was vigorously opposed by New Yorkers. Some opponents engaged in public demonstrations against it in front of the Merchants' Coffee House, a central location in the city. Others protestors were more destructive. An angry mob trashed Vauxhall, a tavern and pleasure garden (a small private park), after its owner, a former British army officer, came out in favor of the act. Still other opponents created the Sons of Liberty, a semi-secret organization that met at taverns and would soon organize resistance to other acts passed by the British parliament and the royal governor's appointed by the Crown.\n\nIt was no surprise that New York taverns played such an important role in revolutionary activity that led to the American Revolution. Taverns were more than just places to drink, eat, and occasionally stay the night: In colonial America, taverns functioned as community centers. Their owners were generally well informed about current happenings, and they shared what they knew. Meetings of groups such as militiamen were frequently convened at taverns and public houses. Men congregated at taverns, passing on information and discussing current events. Many taverns provided newspapers and magazines for customers to peruse.\n\nThe protests worked. The British parliament rescinded the Sugar Act but continued to pass a series of acts intended to reaffirm its right to levy a tax on the colonies. These acts generated even bigger protests, culminating in the outbreak of the Revolutionary War in Massachusetts in 1775.\n\nA few months after the British evacuated Boston in March of 1776, British forces landed on Staten Island and, a few weeks later, on Long Island. Within a matter of days they had captured New York City and more than three thousand American soldiers. Supporters of the Revolution fled the city, loyalists returned, and the city was occupied for the next seven years\u2014longer than any other American city. Thousands of British troops and German mercenaries were garrisoned in the city. In addition, thousands of loyalists and escaping slaves from the surrounding areas and colonies flocked to New York. The city's population swelled from seventeen to thirty-four thousand during the war.\n\nSeveral thousand American soldiers captured by the British ended up imprisoned in abandoned churches and sugar refineries. The unlucky ones ended up in the holds of decommissioned British ships anchored off the coast of Brooklyn. These facilities were overcrowded, and thousands died from the lack of food and clean water, as well as from typhus and other diseases, for which no medical assistance was provided.\n\nThroughout out the Revolutionary War, British and American military forces engaged in guerilla warfare in the hinterland around New York. Cattle rustling and crop burnings were common. Farmers on Long Island and New Jersey eventually supported the War for Independence and refused to sell their products to the British occupiers of New York City. The British army had to import commodities by ship, and the cost of food in British-occupied New York jumped 800 percent during the war. When food shipments failed to arrive, the British considered abandoning the city. Only the timely arrival of a victualing fleet from Ireland prevented mass starvation.\n\nPeace negotiations began in April 1782; in mid-August 1783, the British governor of New York informed General George Washington that his forces were withdrawing from the city. During the next three months, twenty-nine thousand British military personnel, loyalists, and freed slaves left the city, while those who had fled in 1776 trickled back to reclaim what was left of their property. New York was in a shambles. Fires had swept through the city in 1776 and 1778, and an explosion on an ammunition ship in the East River had damaged the docks and other port facilities. When the war ended, more than one-third of the city's buildings had been destroyed, replaced by shanties that housed penniless loyalists and military personnel.\n\nOne tavern that survived was owned by Samuel Fraunces, who was born in the West Indies. Like most tavern owners, Fraunces offered a set dinner as well as alcoholic beverages. He prepared meals in the English style, and his desserts were particularly popular. Fraunces left New York City when the British took over. He later returned and resumed running his tavern. Fraunces aided American prisoners of war during the British occupation, for which a grateful Continental Congress awarded him two hundred pounds in 1782. Many British soldiers drank and ate in his tavern, and he reportedly passed on information gleaned from his guests to Washington's army, which was encamped outside the city.\n\nThe Treaty of Paris was signed on September 3, 1783, ending the Revolutionary War, and the British military authorities officially turned over New York City on November 25, 1783. Called \"Evacuation Day,\" New Yorkers would celebrate it with revelry and alcoholic beverages for the next sixty years. George Washington and the Continental Army marched down Broadway to cheering crowds. That night, a public dinner was held for Washington at Fraunces Tavern, and there, nine days later, Washington gave his farewell address to the officers of the Continental Army.\n\nRevitalizing a City\n\nThe city slowly reinvented itself, an undertaking it would repeat many times in the centuries that followed. As New York reestablished its trade and financial center, the city took a commercial lead over its rivals, Boston and Philadelphia\u2014a margin that continued to increase during the following years. After the U.S. Constitution was approved in 1788, New York became the nation's temporary capital. When George Washington was inaugurated president at Federal Hall in 1789, he selected Samuel Fraunces as his head steward. While in New York, Washington hosted dinners and receptions lavish enough to shock some members of his cabinet and Congress. An observer described one such dinner in August 1789: \"First was the soup; fish roasted and boiled; meats, gammon, fowls, etc.\" It was followed by dessert: \"apple pies, pudding, etc.; then iced creams, jellies, etc.; then water-melons, musk-melons, apples, peaches, and nuts.\"\n\nThe city's population almost doubled\u2014from thirty-three thousand in 1790 to sixty thousand by the end of the decade\u2014and swelled even more during the early nineteenth century. This created a large concentration of potential customers that made commercial production of bakery products, refined sugar, and canned goods financially viable.\n\nBakeries\n\nBakeries had been in operation in the city since early colonial days, but during the early nineteenth century city dwellers opted to pay for the ease and convenience of buying bakers' bread, and the housewife's chore of baking bread shifted to the task of buying it. Bakers sold their goods retail or wholesaled their products to grocery stores, restaurants, and street vendors. Commercial bakeries were subject to market forces: local laws controlled the price and characteristics of bakers' bread, making it difficult for bakers to make a profit. Bakers responded by reducing costs: they adulterated their products with powdered chalk or plaster to make the bread whiter and heavier, or at least so claimed Sylvester Graham, a Presbyterian minister, temperance advocate, and outspoken advocate for vegetarianism. Graham frequently spoke to large crowds in New York and elsewhere, inveighing against bakers and their pallid, impoverished bread, as well as butchers. He believed that women should bake bread at home, eschewing refined white flour and using coarse, unbolted (whole grain) wheat flour instead. His views caught on in New York City, and beginning in 1833, Graham boardinghouses, which institutionalized his views, sprang up in the city to enable his adherents to follow Graham's health food regimen.\n\nGraham's views toward professional bakers were not, however, shared by most New Yorkers, who increasingly bought commercially baked loaves. The city's baking industry was regularly renewed by an influx of professional bakers from abroad. By the early nineteenth century, Scots-Irish bakers dominated the industry. When German and Irish immigrants began to arrive in the 1830s, many went into baking. By 1855, however, Germans, many of whom were Jewish, dominated the city's commercial bakeries. As other immigrant groups entered the city, they, too, opened bakeries that offered goods particularly associated with those groups. Both the quality and diversity of bread and other baked goods improved throughout the nineteenth century. By the early twentieth century, few New Yorkers made bread at home.\n\nLarge commercial bakeries and hundreds of small artisanal bakeries\u2014many owned by immigrants\u2014sustained city dwellers in the late nineteenth century. In 1889, six bakeries united to create the New York Biscuit Company. Other bakeries joined the organization, and in 1898 the company was renamed the National Biscuit Company, which in 1971 officially shortened its name to \"Nabisco.\" It operated a manufacturing facility in New York until after World War II.\n\nNabisco created many of America's favorite cookies and crackers, including Uneeda Biscuits, Triscuits, Barnum's Animal Crackers, Oreos, Nilla Wafers, and Ritz Crackers. It has grown into a huge corporation by acquiring other companies, including those that produced Shredded Wheat, Planters Nuts, and LifeSavers. Its most iconic New York brand was Mallomars\u2014cookies with a round biscuit base (similar to a Graham cracker), topped with a dome of marshmallow and coated with dark chocolate. The former Nabisco plant in Manhattan, at Ninth Avenue and Fifteenth Street, is now home to a very successful and upscale \"mini-mall\" of food purveyors called Chelsea Market. The Food Network's headquarters and studios, and several high-end restaurants, are in the same building.\n\nBird's eye view of Manhattan. Courtesy of the Library of Congress. Reproduction Number: LC-DIG-pga-03183\n\nSugar Refining\n\nNew York's sugar-refining industry quickly revived after the Revolutionary War. A German immigrant, William Havemeyer, arrived in New York in 1799. He had been superintendent of a sugar refinery in London. In New York, he ran Edmund Seaman & Company's sugar house on Pine Street and then opened his own refinery in Manhattan in 1807. His son, William F. Havemeyer, joined the family business, later going into partnership with a cousin, Frederick C. Havemeyer.\n\nNew York was an ideal location for the sugar-refining industry. Its port facilities were the best on the East Coast, and it was easy to transport raw sugar into the city from the Caribbean. The city had a large population that wanted the sweetener, and the city's transportation grid meant that the refined product could be easily transported to other regions of the country. By 1830, the city had eleven sugar-refining companies. Thirty years later, this had increased to twenty-six, which were collectively valued at thirty-five million dollars (today somewhere between one and five billion dollars). New York City dominated sugar refining in America, and this dominance increased over the years. Eventually, as manufacturing methods improved and production rose dramatically, sugar prices took a steep drop in the nineteenth century. Several companies went out of business, but the surviving firms expanded their dominance of the industry. In 1864, the Havemeyers constructed the largest and most technologically advanced sugar refineries in the world in Williamsburg.\n\nBy 1872, New York processed 59 percent of the sugar imported into America; fifteen years later, this had risen to 68 percent. As a result of mass production, the price of sugar continued to decline. In 1887, the Havemeyers and seven other sugar industry leaders formed the Sugar Trust. Its intent was to lower production in order to raise prices and profits for all the companies. The trust was taken to court, and in 1891 it was allowed to reorganize. It acquired additional companies, and the resulting conglomerate was named the American Sugar Refining Company. Inefficient plants were closed while others were combined, and American Sugar Refining unofficially, but effectively, fixed the price of refined sugar. In 1900, the company created a subsidiary, Domino Sugar, to market the sugar that the parent company refined. By 1907, the American Sugar Refining Company controlled 97 percent of all production of refined sugar in America. Although this percentage declined after 1921, when the company lost a major antitrust case, it continued to dominate American sugar refining throughout the twentieth century. New York's refineries were slowly closed as more efficient\u2014and less costly\u2014factories were built elsewhere. The large Domino sugar refinery in Williamsburg, the last sugar refinery in New York, closed in 2004. The property is being redeveloped; preservationists hope that at least one building, emblazoned with \"Domino Sugar\" in big yellow letters resembling a box of sugar, will be landmarked.\n\nSugar was used in many ways, but one of the most important was confectionery, and industries making candy emerged in New York City. Tootsie Rolls, a perennial American favorite, were created by Leo Hirschfield, an immigrant from Austria. In 1896, he began turning out small, log-shaped, chewy chocolate caramels in his candy store. In 1905, he named the candy after his daughter, Clara, whose nickname was \"Tootsie,\" and began manufacturing them. Tootsie Rolls were the first penny candy to be individually wrapped. Hirschfield named his business the Sweets Company of America, and in 1917 he began advertising nationally. In 1931, the company introduced the Tootsie Pop, which was a hard candy lollipop with a Tootsie Roll center. During World War II, Tootsie Rolls were placed in soldier's ration kits, mainly because they could survive extremes of temperature and humidity. After the war, the company targeted its advertising on youth by sponsoring popular children's television shows. The company was renamed Tootsie Roll Industries, Inc., and today it is headquartered in Chicago.\n\nSamuel Born, a Russian Jew who immigrated to New York in 1910, invented the \"Born Sucker Machine,\" which automatically inserted sticks into lollipops, thereby revolutionizing the lollipop trade. In 1917, he opened a retail candy store in Brooklyn. In 1923, Born started his own manufacturing company: Just Born. In 1932, the company relocated to Bethlehem, Pennsylvania. Its most famous product is Peeps, marshmallow chicks in bright Easter colors.\n\nYet another candy company was launched by Edward Noble, owner of the Mint Products Company of New York. In 1913, he acquired the rights to manufacture LifeSavers, ring-shaped mint candies that had been invented the previous year in Ohio. Noble packaged stacks of the candies in foil and sold them for a nickel a roll. They were distributed to bars, where displays were installed next to cash registers. The idea was that mint candy served as a \"lifesaver\" for drinkers, disguising the smell of alcohol on their breath. In 1915, the United Cigar Store chain placed the displays in their many stores, and Noble also marketed LifeSavers to drugstores, grocery stores, and other retailers. LifeSavers became one of the most successful hard candy lines in America. The company was eventually sold to Wrigley Company, best known for their chewing gum.\n\nMany food and beverage corporations were headquartered in New York, and many operated factories, but beginning in the mid-twentieth century most manufacturing operations left the city. During the early twenty-first century, the New York City Economic Development Corporation encouraged food startups, particularly in Brooklyn and Queens. Today, dozens of small companies have been launched, including Kombucha Brooklyn (tea-based beverages), Empire Mayonnaise, McClure's Pickles, Brooklyn Soda Works, Steve's Ice Cream, Mad\u00e9casse (chocolate bars), People's Pops (ice pops), Cuties Pies, King County Jerky Co., ten breweries, and one distillery.\n\nFresh, Pickled, Canned, and Bottled\n\nFrom colonial times, raw oysters were sold on the streets and later in oyster houses, saloons, cellars, and restaurants. Oysters were also pickled, packed in small casks, and sent to other colonies and the Caribbean. In 1748, Swedish-Finnish botanist Peter Kalm described the elaborate process in which New Yorkers pickled them. The oysters were first shelled, boiled in a pot, removed, and dried:\n\nThen some nutmeg allspice and black pepper are added, and as much vinegar as is thought sufficient to give a sourish taste. All this is mixed with half the liquor in which the oysters are boiled, and put over the fire again. . . . At last the whole pickling liquid is poured into a glass or earthen vessel, the oysters are put into it and the vessel is well stopped to keep out the air. In this manner, oysters will keep for years, and may be sent to the most distant parts of the world.\n\nLater, pickled oysters were shipped throughout the Midwest and even to California and England. When railroads were constructed, fresh oysters were sent to restaurants and grocery stores throughout the East Coast.\n\nIt was logical that oysters were among the first foods canned in New York. The sealable tin can was invented in England in the early nineteenth century. English immigrant Thomas Kensett arrived in New York about 1806; within a few years he had begun a canning business with his father-in-law, Ezra Dagget, also an English immigrant. They were among the first in the United States to hermetically seal tins of food\u2014salmon, lobsters, oysters, meats, fruits, and vegetables. They were awarded the first U.S. patent for preserving food in 1825.\n\nBy the 1850s, many canning factories operated in New York City and in Astoria on Long Island. One firm, Wells, Miller & Provost, packed tomatoes in the city. The editors of the American Agriculturist claimed that canned tomatoes put up by this firm tasted as \"nice sweet and fresh as if just picked from the vines.\" Canned goods were still expensive, and only the well-to-do could afford them. High-end restaurants adopted canned fruits and vegetables in order to serve their delighted customers out-of-season delicacies in the winter. The price of canned goods dropped dramatically after the Civil War, and by the 1870s, they were affordable to most New Yorkers and routinely sold in city grocery stores.\n\nFor the home canners, John L. Mason, owner of a metalworking shop on Canal Street, invented a self-sealing zinc lid and glass jar in 1858. The screw-on lid greatly simplified the canning process and made the jars genuinely reusable. It revolutionized fruit and vegetable preservation in the home. As the jars were easy to use and comparatively inexpensive to produce, their popularity soared. By 1860, Mason jars were shipped throughout the United States.\n\nFood Transportation\n\nDuring the colonial era, overland transportation to and from New York by wagon over dirt roads was slow at the best of times, and often impossible in bad weather. After the American Revolution, surveys were conducted to evaluate the possibility of constructing a canal that would link the Hudson River with the Great Lakes. The War of 1812 was a decisive factor in the final decision to construct the canal. It became apparent during the war that there was no easy way to move troops and supplies across the state to support military efforts. After the war, pressure to build a canal mounted, and construction finally began on July 4, 1817. Those who proposed and approved the Erie Canal's construction\u2014and those who bought bonds to finance it\u2014hoped that the canal would encourage settlement around the new waterway and that farms along the route would ensure an adequate food supply for New York's rapidly growing population.\n\nAs sections of the canal were completed, agricultural commodities, particularly wheat, began to travel southward with unprecedented efficiency. This was the beginning of a flood of grain from upstate New York that poured into the city, where it was milled into flour and then loaded onto oceangoing vessels for shipment to coastal cities in the eastern and southern United States, the Caribbean, and Europe. Likewise, most goods that were shipped westward through the canal came through New York City. The canal was completed in 1825, and by the late 1820s, hundreds of sailing vessels and fifty steamboats traveled up and down the river to and from New York City, while more than 1,400 ships arrived annually in New York Harbor from foreign ports. Within fifteen years of the canal's opening, more tonnage moved through the Port of New York than the combined tonnage of the nation's three other largest ports\u2014Boston, Baltimore, and New Orleans.\n\nThe commercial success of the Erie Canal spurred on the construction of railroads. Tracks were first laid northward from New York City. This meant that dairy and produce farmers who lived north of the metropolis, but not on a navigable river, could ship fresh milk and produce into the city daily.\n\nBy the 1850s, railroads connected New York and Chicago, which greatly enhanced the transportation of agricultural goods to New York from the Midwest during the Civil War. Live cattle were soon shipped from the Midwest to New Jersey. They were shipped across the Hudson and herded daily through city streets to the Meatpacking District, situated in the area from West Sixteenth Street south to Gansevoort Street, and from the Hudson River east to Hudson Street. By 1900, more than 250 packing plants were operating in the district, but that was the industry's peak.\n\nAfter the Civil War, the invention of the refrigerated railroad car in the 1870s permitted livestock to be slaughtered in Midwestern feedlots and then transported to New York. It was no longer necessary for live cattle to be shipped to New York and through its streets. The shipment of cheap dressed beef from the Midwest reduced the need for local butchers. In 1860, New York City had more than two thousand; only a few hundred survived in 1900. Butchers hoped that their customers' appreciation of the quality and flavor of their meats would breed loyalty, but shoppers readily passed up locally butchered beef and pork for cheaper cuts shipped in from the Midwest.\n\nAfter the completion of the transcontinental railroad in 1869, agricultural goods\u2014particularly fruit and vegetables\u2014could be shipped from San Francisco to New York. Other railroads connected the city with the South. By the 1880s, truck gardeners in Virginia and North Carolina were shipping produce to New York; by 1899, Florida growers were shipping oranges and other citrus to the city by train.\n\nThe twentieth century saw the construction of all-weather roads and, eventually, the interstate system. The advent of long-haul trucks with roof-mounted refrigeration systems, in 1948, made it possible for refrigerated and frozen foods to be easily transported from even the most isolated communities to New York. The invention of shipping containers and the construction of container ships in 1956 made it easier for many foods\u2014particularly fruit, vegetables, grains, coffee, chocolate, and meat\u2014to be shipped from Central and Latin America, and in winter, shipments are particularly heavy.\n\nHolidays and Celebrations\n\nFrom its earliest days, New Yorkers enjoyed a range of secular, religious, and unusual holidays and relished the attendant food and beverage traditions associated with them. In colonial times, at least some residents celebrated the feast of Bacchus (March 16), Shrovetide or Carnival (typically forty days before Easter), Easter, Pinxter (Whitsuntide), May Day (May 1), Ascension Day, Christmas, New Year's, Twelfth Night\/Day (January 5\u20136), and Voorbereiding (\"preparation for the Lord's Supper,\" observed quarterly). Holidays were typically celebrated with revelers firing guns into the air, beating drums, and drinking excessive amounts of liquor. City officials passed regulations attempting to prevent these activities, but they failed.\n\nSinterklaas, or St. Nicholas Day (December 5 or 6), was also celebrated in New York, at least until the 1740s, when it was officially dropped from New York City's list of official holidays. It may have been dropped, but it was not forgotten. Washington Irving popularized the Dutch celebration of a completely fictional Saint Nicholas in his History of New-York (1809), which just happened to be published on St. Nicholas Day. He described the arrival of the saint on horseback on the Eve of Saint Nicholas, who was now identified as the patron saint of New York. John Pintard, a descendent of a Huguenot family that arrived in 1690, paid for the publication of a broadside with an illustration of a stern Saint Nicholas. Over the fireplace hung two stockings\u2014one filled with goodies (perhaps toys, oranges, sugar plums, and \"oley cooks\"), the other, a stick.\n\nChristmas was celebrated in the city in colonial times, but like other holidays, it was a time to fire guns into the air and drink. Drunkenness and rowdiness prevailed on the streets. Some of the shooting was purposeful. Historian Mary Booth, writing in 1860, reported that in colonial times, the day started with an exchange of presents; then, \"the young men repaired to the 'commons' or 'Beekman's swamp' to shoot at turkeys which were set up for a target. Each man paid a few stuyvers for a shot, and he who succeeded in hitting the bird took it off as a prize.\" For those who celebrated Christmas (some Protestant denominations considered it to be a \"Papist holiday\"), the turkey was a dinner favorite.\n\nFrom the early nineteenth century, efforts were underway to tame Christmas. The name \"Santa Claus,\" a derivative of Saint Nicholas and Sinter-klaas, first appeared in print in New York in the 1770s and has been associated with Christmas ever since. The association of Christmas with a cozy family celebration complete with a large dinner was popularized by Washington Irving in his Sketch Book (1819). An acquaintance of John Pintard and Washington Irving was Clement Clarke Moore, an Episcopalian minister of Walloon ancestry. He wrote \"An Account of a Visit from St. Nicholas\" (1823), commonly known as \"'Twas the Night before Christmas.\" He intentionally set the arrival of Santa Claus on Christmas Eve (although some early versions say New Year's Eve, not Christmas). The shift from Christmas to Christmas Eve avoided potential religious issues. Illustrator Thomas Nast, a German immigrant who arrived in New York in 1850, created the modern image of jolly old St. Nick, which first appeared in Harper's Weekly in 1862. By this time, many New Yorkers celebrated a less rowdy Christmas complete with presents for the children and a family dinner.\n\nNew Year's Day was the main secular holiday during the city's early years. Both the Dutch and English exchanged gifts. The holiday was also celebrated by visits to the homes of friends and acquaintances, where refreshments were served. Some celebrated New Year's by \"going from House to House with Guns and other Fire Arms, and being often intoxicated with Liquor\"\u2014a practice that had existed at least since 1675. Laws were passed to prevent such displays, and the city police doubled their forces on these holidays, but the celebratory shooting continued until the 1840s. The custom of New Year's Day visits continued well into the nineteenth century. Here, from the 1830s, is an account of a round of such visits:\n\nWe began at nine in the morning, and at five in the afternoon we ceased, having visited sixty-seven houses. In some cases, in consequence of the great number of \"callers\" in a house, we merely walked in and said \"Happy New Year,\" or \"Compliments of the Season,\" \"Thank you, we dare not indulge,\" \"Good-morning.\" At other houses, when the young ladies were especially interesting, a few minutes' conversation and a sip of cherry bounce or coffee, \"Good-morning,\" and off to another house. Such was the routine of the young men, while the elder, having fewer visits to make, remained longer at their calls and indulged in the table, lavishly spread with crullers, doughnuts, cookies (New Year's cakes), pickled and stewed oysters, chicken, turkey, mince-pies, jellies, etc., and with wines and liqueurs.\n\nOn New Year's 1837, New York's mayor opened his house to the public. Gentlemen came, politely tendered their greetings, and left; a few drank \"a single glass of wine or cherry bounce, and a morsel of pound-cake or New Year's cookies,\" reported diarist Philip Hone. Then the rabble showed up demanding refreshments. They devoured the beef and turkey, wiped their greasy fingers on the curtains, drank the mayor's liquor, and discharged \"riotous shouts of 'Huzza for our Mayor!'\" The police had to be called in to restore order. For many, New Year's was simply a day away from working and a day for drinking; by evening, the streets were frequently littered with casualties of the festivities.\n\nFor the upper classes, hostesses offered their visitors beverages, such as rare wines, cognac, bourbon, and, occasionally, champagne. An English visitor to New York City on New Year's in 1870 found that eggnog was \"the peculiar beverage of New Year's day,\" but also served were apple toddy, milk punch, brandy smash, and mixed drinks, such as the popular Tom and Jerry, an eggnog-like mixed drink with eggs, sugar, rum, and brandy served hot. By the end of the nineteenth century, however, the temperance movement had followers, at least among some well-to-do New Yorkers. When the English writer George Sala visited the city on New Year's 1880, he was mainly offered coffee, bouillon, and chocolate, but no alcohol.\n\nOver time, the celebration shifted to New Year's Eve, and the midnight glass (or more) of champagne became obligatory along with singing \"Auld Lang Syne.\" Public celebration of the stroke of midnight were celebrated across the city, but beginning in 1904 these celebrations began to concentrate at Times Square; watching the ball drop at 11:59 p.m. on December 31 became de rigueur in 1907. The Times Square celebration became a national event with the advent of radio and a global event watched by millions after the arrival of television.\n\nNew holidays were added to the city's calendar after the American Revolution. After 1784, the Fourth of July, Independence Day, was celebrated annually in the city with fireworks, parades, entertainment, drinking, and food. In 1797, Joseph Delacroix, owner of the Vauxhall Garden, offered celebrants entertainment plus ice cream, punch, or lemonade (but no alcohol) for six shillings. In 1838, an English visitor named Frederick Marryat described the city's Fourth of July celebration:\n\nOn each side of the whole length of Broadway, were ranged booths and stands, similar to those at an English fair, and on which were displayed small plates of oysters, with a fork stuck in the board opposite to each plate; clams sweltering in the hot sun; pineapples, boiled hams, pies, puddings, barley-sugar, and many other indescribables. But what was remarkable, Broadway being three miles long, and the booths lining each side of it, in every booth there was a roast pig, large or small, as the centre attraction. Six miles of roast pig! and that in New York City alone; and roast pig in every other city, town, hamlet, and village in the Union. What association can there be between roast pig and independence? Let it not be supposed that there was any deficiency in the very necessary articles of potation on this auspicious day: no! the booths were loaded with porter, ale, cider, mead, brandy, wine, ginger-beer, mint juleps, besides many other compounds.\n\nYet another new holiday was Thanksgiving. It had been celebrated in New England since the late eighteenth century. Many New Englanders moved to the city, and they introduced New Yorkers to all the trappings of their celebration. In 1830, New York was the first state outside of New England to declare Thanksgiving a state holiday. As one New Yorker wrote in the 1830s, \"The Yankee Thanksgiving, with its turkey, cranberry sauce, mince, pumpkin apple pie and cider, found favor with the dames of knickerbocker proclivities, who not to be outdone, had added the indigestible doughnut and cruller to the dyspepsia provoking list.\"\n\nAn unusual Thanksgiving celebration took place during the Civil War. When President Abraham Lincoln issued a proclamation declaring that Thanksgiving would be celebrated on November 24, 1864, George W. Blunt, a member of New York's Union League Club, proposed to lift the spirits of Union soldiers and sailors in Virginia by supplying them \"with poultry and pies, or puddings, all cooked, ready for use.\" The Union League Club appealed to hoteliers, restaurateurs, bakers, and private individuals (with the means to do so) to roast twenty or more turkeys and chickens and send them to a central location so they could be shipped south.\n\nDelmonico's, the city's most fashionable restaurant, contributed the services of its chefs to stuff and roast thousands of turkeys. These delicacies were packed up and sent by train, ship, and wagon all the way to Virginia. In just three weeks, the Union League Club collected more than $56,500 toward the purchase of 146,586 pounds of poultry. New York's Fulton Market supplied much of that and donated a $3,386 profit back to the fund. An additional 225,000 pounds of poultry was received as contributions, along with enormous quantities of other Thanksgiving dinner ingredients. Northern newspapers crowed about the feast. The New York Herald proudly announced, \"A people in the midst of a bloody war, having tens of thousands of soldiers in the field, and war ships studding every sea, yet every day expanding into greater commercial importance, and celebrates its national elections, feasts, and holidays with peace and harmony.\"\n\nNew Yorkers prospered during the Civil War as government contracts poured in for manufacturing military hardware and constructing and resupplying ships. Jobs were readily available, and wages were high. Food was plentiful in the city throughout the war, and the city's wealth meant that its restaurants thrived. Women joined sanitary commissions, which helped wounded soldiers and their families. The well-to-do created the Union League Club in 1863 to support the war effort. As at the Century Club, the city's first, formed in 1847, the Union League's members were men. Clubs made it possible for members to drink, dine, and socialize among themselves without having to resort to the city's public restaurants and bars.\n\nThanksgiving and Christmas were connected by an invention of Macy's department store employees: the Thanksgiving Day Parade. In 1924, Macy's employees organized a procession that started on 145th Street in Harlem and ended in front of the Macy's store, on Herald Square in midtown. Traditionally, the last float of the parade carries Santa Claus, who officially kicks off the Christmas shopping season. Today, millions of visitors and television viewers watch the trademark giant balloons as they're marched down the avenues. Virtually all the restaurants along the parade route that are open are filled, but most New Yorkers view the parade on TV and eat Thanksgiving dinner at home with their families and friends.\n\nCulinary Education and Organizations\n\nBy the mid-nineteenth century, increasing wealth in New York City meant that more families were able to hire domestics. The problem with many immigrant household servants, especially those from Ireland, was that they were unable to prepare the food that employers wanted. One solution was developed by Pierre Blot, a Frenchman who came to New York about 1855. He lectured on the culinary arts, and in 1863 published his first book, What to Eat, and How to Cook It. Two years later, he launched a cooking school called the Culinary School of Design and styled himself the \"professor of gastronomy.\" His second book, Hand-Book of Practical Cookery, was published in 1867, and he followed it with a series of articles on culinary topics in The Galaxy, a monthly magazine. These gave him great visibility. With financial assistance from a daughter of the railroad magnate Commodore Cornelius Vanderbilt, Blot opened the New York Cooking School, America's first French cooking school. It mainly catered to wealthy women, who brought their cooks with them for training, and it lasted only a few years, but the idea for cooking schools reemerged after the Civil War, as more immigrants streamed into New York and the need for domestics and restaurant cooks swelled.\n\nDuring the 1860s, a German immigrant, Gesine Lemcke, opened the Greater New York Cooking School on East Forty-second Street and another branch in Brooklyn. Her aim was to train cooks, butlers, and waitresses. When Lemcke died in 1904, her daughter became the principal, and the school continued to operate until 1934.\n\nIn 1873, the nation went into an economic recession that continued for several years. Many immigrants lost their jobs, and new employment was hard to find. Juliet Corson, a thirty-two-year-old librarian at the Working Women's Library in New York, became concerned about the plight of poor women in the city. Corson volunteered as secretary to the Women's Educational and Industrial Society of New York, whose purpose was to offer free vocational training for unemployed working-class women. She was asked to teach cookery, even though she had little experience or culinary knowledge; the hope was that her students might find jobs as domestics. Corson studied German and French cookbooks and began offering classes in her home. In 1874, the society moved into a building that housed classrooms as well as free dormitories for unemployed women taking courses at the school. The following year, Juliet Corson began submitting articles on cookery for newspapers and magazines, and lecturing at various venues around the city.\n\nCorson's cooking courses attracted the attention of middle- and upper-class women, and she decided to launch her own cooking school. In November 1876, she opened the New York Cooking School, with a sliding-scale tuition that made it affordable for women from every walk of life. The school offered a series of twelve lessons. Based on her lectures and experiences, Corson wrote The Cooking Manual of Practical Directions for Economical Every-Day Cookery, which presented her culinary philosophy\u2014to make \"the most wholesome and palatable dishes at the least possible cost.\" Corson's cooking school and lectures were so successful that she received inquiries from many cities, including Philadelphia and Boston, on how to start and manage a cooking school. She published Cooking School Text Book and Housekeepers' Guide (1879), which became the textbook for many schools in other cities, such as Boston and Philadelphia. It included a section on \"Dietary for Schools,\" which delineated the foods and beverages students needed and made suggestions for breakfasts and dinners. Copies of the \"Dietary\" were reprinted by the secretary of the Interior and sent to schools around the country.\n\nHarper's magazine proclaimed Corson to be \"the benefactor of the working classes, for she teaches them how to make two dishes where formerly they made but one; and the friend of women, for she has shown them the way to a useful and honorable profession.\" Her efforts\u2014and those of home economics professionals\u2014led to the teaching of cookery in New York City's public schools.\n\nDuring the twentieth century, other culinary schools opened in New York. One was launched by Dione Lucas, an Englishwoman born in Italy and brought up in France. She attended Le Cordon Bleu cooking school in Paris and opened a restaurant in London just before World War II. When the war broke out, she immigrated to New York where she opened the Cordon Bleu Cooking School at her restaurant in Manhattan. It offered lessons for \"five dollars apiece\" and \"a course of forty-eight lessons for $192.\" The school's name was later changed to the Gourmet Cooking School, which she continued to run out of her various restaurants. Another culinary school was opened by cookbook author James Beard in his townhouse on West Tenth Street and later on West Twelfth Street.\n\nSome older schools still in operation are Peter Kump's New York Cooking School, founded in 1975 and renamed the Institute for Culinary Education in 2001; the Natural Gourmet Cooking School, founded by Annemarie Colbin in 1977; De Gustibus Cooking School, launched at Macy's in 1980 by Arlene Feltman Sailhac; and the French Culinary Institute in Soho, founded by Dorothy Cann Hamilton in 1984 and later renamed the International Culinary Center.\n\nIn addition to culinary schools, city universities have developed academic food studies programs. New York University, for instance, launched the Department of Nutrition, Food Studies, and Public Health in the 1990s. It offers undergraduate, master's, and doctoral programs in food studies, culture, and food systems. The New School also has recently started an undergraduate food studies department, and other city universities have offered numerous academic courses of various aspects of food.\n\nMany culinary organizations have been launched in New York. The Soci\u00e9t\u00e9 Culinaire et Philanthropique was founded in 1865. It is the oldest culinary association in America; its membership includes chefs, cooks, pastry chefs, bakers, and butchers. It is headquartered in New York, and its \"Salon of Culinary Art\" is presented each year during the International Hotel\/Motel & Restaurant Show at the Jacob Javits Convention Center. Les Dames d'Escoffier was launched in 1973 by Carol Brock, then Sunday food editor at the New York Daily News. An organization of professional women in the food and wine industry, the New York Women's Culinary Alliance, was launched in 1981 by Sara Moulton and Maria Reuge. The Culinary Historians of New York was founded in 1985 to stimulate and share knowledge about food history. Slow Food USA was launched in Brooklyn in 2000.\n\nWhen cookbook author and cooking teacher James Beard died, Peter Kump, a former student of Beard's, raised funds to buy Beard's Greenwich Village townhouse. Along with others, including Julia Child and Jacques P\u00e9pin, Kump helped launch the James Beard Foundation, which is housed in Beard's former home. Since 1990, the James Beard Foundation has given awards for culinary excellence to chefs, journalists, and other culinary professionals.\n\nWorld War II\n\nThe United States entered World War II in December 1941. Like many other Americans, New Yorkers with lawns and backyards planted \"Victory Gardens,\" raising a large amount of vegetables. Tomatoes were planted in Rockefeller Plaza and penthouses often sported beans. New York families changed their eating habits as some traditional foods became unavailable and domestic servants left to join the army or take war-related jobs during war.\n\nDuring the war, many New Yorkers volunteered or were drafted in the armed services, and many other soldiers and sailors came through the city on their way overseas. Canteens were set up for military personnel, and women volunteers prepared cakes and sandwiches that were consumed with coffee, milk, and fruit when available. One canteen in Brooklyn served fourteen thousand servicemen in a single month in 1942.\n\nRationing went into effect in 1942 and continued until 1947, and it became difficult for restaurateurs and homemakers to acquire all the beef, wheat, sugar, imported spices, wines, and other products that they wanted. Another serious problem was the shortage of workers. Women entered the workforce and soon dominated sales forces in grocery stores and food production industries and as waiters in restaurants, diners, and other eateries. Many employees worked overtime, and as there were few consumer goods on the market, they pocketed their salaries or dined in restaurants. The war years were particularly good ones for the city's restaurants. From 1939 to 1946, restaurant sales in New York quadrupled. When the war ended, restaurant profits soared even more: New Yorkers wanted to compensate for the hardships during the war, and many had money that could not be spent during the war.\n\nFood Journalism\n\nFood journalism was one reason the New York culinary scene took off after World War II. Previously, food articles in magazines and newspapers had supplied recipes and helpful hints and were often thinly veiled (or overt) promotions for new manufactured food products and the latest kitchen equipment. But as the twentieth century rolled on, some newspaper food writers became extremely popular and influential. Clementine Paddleford, a Kansas native with a degree in industrial journalism, became food editor at the New York Herald Tribune in 1936, having previously written for the New York Sun and New York Telegram. By the early 1950s, her columns reached twelve million readers, and in 1953 Time magazine named her \"America's best known food editor.\" She was also a frequent contributor to Gourmet. Clementine Paddleford's book How America Eats (1960) is a collection of her columns on regional food.\n\nJust before World War II, American food journalism went upscale when authors began writing about more sophisticated culinary matters. Some prominent U.S. writers traveled to and lived in France between World War I and World War II, and this period nurtured some classic reportage and books about food. A. J. Liebling, an alumnus of the Columbia School of Journalism, wrote a series of articles for the New Yorker, which he later compiled into his influential work Between Meals: An Appetite for Paris (1962), published a year before his death.\n\nCzech-born Joseph Wechsberg emigrated to the United States in 1938. Having taught himself English, he talked his way into a job with the New Yorker just before shipping out to Europe with the U.S. Army. Wechsberg often profiled great French chefs Fernand Point and Alexandre Dumaine. Wechsberg also wrote the gastronomic memoir Blue Trout and Black Truffles: The Peregrinations of an Epicure (1953) and Dining at the Pavillon (1962), the story of New York's premiere French restaurant.\n\nFood journalism took another step forward when Earle MacAusland launched Gourmet in January 1941. He knew little about culinary matters, so he handpicked an editorial staff of experts. They included Pearl Metzelthin, author of The World Wide Cook Book: Menus and Recipes of 75 Nations (1939); and Louis de Gouy, a New York chef, restaurateur, and cookbook author who became the magazine's \"Gourmet Chef.\" MacAusland also hired James Beard, who had just published his first cookbook, Hors D'oeuvre and Canap\u00e9s (1940). The distinguished food writers M. F. K. Fisher, Jane Grigson, and Joseph Wechsberg, among others, were early contributors to Gourmet. At first, the magazine's main focus was on French food, but the articles and recipes covered cuisines from all over the world. In addition to Gourmet, the upscale, glossy, New York\u2013based food magazines include Bon App\u00e9tit, Saveur, and Food and Wine, which was launched by Michael and Ariane Batterberry. The Batterberrys later became publishers of Food Arts, a magazine for professionals. They also wrote On the Town in New York from 1776 to the Present (1973), the first attempt at a culinary history of New York City.\n\nNonfood magazines also began publishing serious articles and columns about food. Vogue, for instance, hired Jeffrey Steingarten as a columnist in 1989. His books include The Man Who Ate Everything (1996) and It Must've Been Something I Ate (2002). The New Yorker has continued to offer frequent food articles and publishes an annual Food Issue each November. Calvin Trillin, who joined the New Yorker staff in 1963, traveled around the country writing the \"U.S. Journal\" column, which was often about food. He has published a number of food-related books, including American Fried: Adventures of a Happy Eater (1974); Alice Let's Eat: Further Adventures of a Happy Eater (1978); Third Helpings (1983); and Feeding a Yen (2003).\n\nJohn McPhee began writing for the New Yorker in 1965. His meticulously detailed prose lends itself to long, in-depth pieces. For the article \"Giving Good Weight,\" about the early years of New York City's Greenmarkets, McPhee spent time working on a farm and manning a farmer's stand in the city. McPhee's books include Oranges (1967) and The Founding Fish (2002), an examination of the natural and culinary history of shad.\n\nWhen New York magazine was launched in 1968, Gael Greene joined the staff, serving as a restaurant critic until 2000 and as a columnist until 2008. She documented the city's expanding interest in food and the rise of its restaurant culture. Her book Bite: A New York Restaurant Strategy (1972) is a collection of articles about the city's most notable eating places; she later recounted experiences culinary and otherwise in her memoir Insatiable: Tales from a Life of Delicious Excess (2006).\n\nRaymond Sokolov, a Paris-based correspondent for Newsweek, became the New York Times restaurant critic and food editor from 1973 to 1975. For more than twenty years, Sokolov wrote the column \"A Matter of Taste\" for Natural History magazine: he crisscrossed the country trying to capture traditional American foodways before they were lost forever. Some of these columns were later collected in Fading Feast: A Compendium of Disappearing American Regional Foods (1981). In 2006, Sokolov became founding editor of the \"Leisure and Arts\" section of the Wall Street Journal's weekend edition and also served as its restaurant critic.\n\nNew York Times\n\nNo source influenced what New Yorker's ate more than did the New York Times. For more than seventy years, it has hired professionals to write about food. Professional food journalism took off when, in 1957, the New York Times hired its first food columnist, the Mississippi-born food critic and writer Craig Claiborne, who had trained at L'Ecole H\u00f4teli\u00e8re in Lausanne, Switzerland. Claiborne was among the first restaurant critics to dine at restaurants incognito, and he was the first newspaper writer to review restaurants using the star system. Claiborne continued to write for the New York Times on and off for the next thirty-five years. He would write or coauthor seventeen food books, including The New York Times Cookbook and his autobiography A Feast Made for Laughter (1982).\n\nFlorence Fabricant began writing a food column, \"In Seasons,\" for the East Hampton Star in 1972 and earned praise from Craig Claiborne. By December 1972, she was writing for the New York Times, where she has remained a columnist ever since. She has written or edited ten cookbooks, including her latest, The New York Restaurant Cookbook: Recipes from the City's Best Chefs (2009). Another New York food columnist, Marian Burros, expanded the food-writing brand at the New York Times to include consumer subjects like food safety and government-related food programs. She has written nine cookbooks including (with Lois Levine) Elegant but Easy Cookbook (1984).\n\nAs culinary journalism, and especially restaurant criticism, became a mainstay of local and national newspapers, attempts were made to put some distance between the writers and food companies, restaurants, and advertisers. Some food journalists were no longer allowed to go on press trips or to accept free meals or merchandise. Newspaper writers had been reviewing restaurants for years, but the professional standards for reviewing were set by Mimi Sheraton of the New York Times. She describes her techniques in her memoir, Eating My Words: An Appetite for Life (2004).\n\nSome of America's top culinary writers have been attracted to the New York Times. William Grimes arrived at the New York Times in 1989 and since then has written extensively about food, including restaurant reviewing. He is the author of five books, including Straight Up or On the Rocks: The Story of the American Cocktail (2002) and Appetite City: A Culinary History of New York (2009). Molly O'Neill, who had written for the Boston Globe and New York Newsday, began a ten-year stint in 1989 as a reporter for the New York Times and subsequently became the weekly food columnist for the paper's Sunday magazine. In 1995, she was the first food writer nominated for a Pulitzer. She is the author of four cookbooks, the most recent being One Big Table (2010). Her landmark article \"Food Porn,\" in the Columbia Journalism Review in 2003, took a critical look at food journalism, and she edited the comprehensive tome American Food Writing: An Anthology with Classic Recipes (2007). Amanda Hesser began working as a reporter for the New York Times in 1997; she wrote restaurant reviews and also edited the Sunday magazine. Hesser left the Times in 2008 but has continued writing for magazines and for her website, food52.com. For her books The Cook and the Gardener (1999) and Cooking for Mr. Latte (2009), Amanda Hesser won Literary Food Writing Awards from the International Association for Culinary Professionals. Kim Severson wrote about food for the San Francisco Chronicle for six years before arriving in 2004 at the New York Times, where she covered everything from Girl Scout cookies and Jamaican beef patties to the fate of New Orleans restaurants after Hurricane Katrina. Severson has won four James Beard Awards for journalism. She also has published a number of books, including The Trans Fat Solution: Cooking and Shopping to Eliminate the Deadliest Fat from Your Diet (2004), The New Alaska Cookbook: Recipes from the Last Frontier's Best Chefs (2009), and a memoir, Spoon Fed: How Eight Cooks Saved My Life (2010). Severson is now the Atlanta bureau chief for the Times.\n\nIn September 1997, the Times revamped its \"Living\" section into a \"Dining\" section that appeared on Wednesdays. It included recipes, restaurant reviews (previously published on Fridays), reports on new food products, calendars of culinary events, and articles of local interest as well as wider-ranging pieces on food issues and happenings around the United States and the world.\n\nOther New York newspapers, such as the New York Daily News, the Village Voice, the New York Post, and the Wall Street Journal, have also covered the city's food scene, reviewing restaurants and publishing articles about food trends.\n\nTelevision Food\n\nTelevision stations experimented with cooking shows during the early 1940s, but none stirred up much public interest. James Beard, a trained actor and opera singer, had hosted a series of fifteen-minute programs called I Love to Cook! for NBC in 1946; they aired after a variety show, Elsie Presents, introduced by a puppet version of Borden's Elsie the Cow. None of Beard's programs have survived, but Beard was reportedly awkward and ill at ease in front of the camera. Regardless of Beard's lack of telegenic skills, television programmers liked the idea, and other shows soon followed. Dione Lucas hosted the first national thirty-minute cooking show on CBS in 1948. Her programs, To the Queen's Taste, followed by Dione Lucas's Cooking Show, focused on French food, but her stern and somewhat forbidding presentation made French cookery seem complex and difficult. It was Julia Child's French Chef that launched food television in 1962.\n\nFollowing Child's lead, other chefs, restaurateurs, and food personalities took to the airwaves, and cooking shows appeared on both commercial and the newly emerging public television stations. Their perennial popularity culminated in the creation of the Television Food Network in 1993 in New York City. At first it focused on instructional cooking shows\u2014colloquially referred to as \"dump and stir\" programs. They were relatively inexpensive to make and formulaic in their construction; many such programs had already aired on other television stations. Several new programs were developed, featuring chefs and personalities such as New Yorker's Sara Moulton, Mario Batali, and Rachael Ray\u2014all of whom went on to host their own programs. By 2001, the Food Network was reaching seventy-five million households, with some of the most affluent TV viewers in America. By 2013, the network was being watched in ninety-six million homes\u2014virtually every American home that had a cable or satellite TV hookup\u2014and its website had seven million users monthly. While the network's initial audience was primarily female, increasing numbers of men began watching its programs. Today, almost half the viewers of the Food Network are male.\n\nThe Scripps Networks Interactive and the Hearst Corporation launched the Food Network Magazine in 2008. This monthly magazine promotes and builds on the chefs and programs on the Food Network and Cooking Channel. The magazine had 1.1 million subscribers by the end of its first year and today reaches an estimated 5.4 million readers. The Food Network was so successful that in 2010 Scripps Networks Interactive, the owner of the Food Network, launched a second cable television channel, the Cooking Channel, which focuses on instructional programs.\n\nFood Problems and Solutions\n\nThe New York food scene is not just about feeding the wealthy. It has also been about feeding those in need. The Humane Society of New York City began giving food to families of debtors in 1787. In 1791, it began offering food assistance to the poor. It officially launched a soup kitchen in 1802. The Humane Society printed up tickets that it sold to individuals and organizations, which, in turn, gave the tickets to the poor. When Congress passed the Non-Importation Act in 1808, trade with Great Britain was embargoed, and many New Yorkers became unemployed. Thousands sought help in the Humane Society's soup kitchen. This was only a small number compared with those who needed food assistance during the War of 1812. By 1815, their soup kitchen served seventy thousand New Yorkers.\n\nMany societies were formed in the nineteenth century to assist immigrants or the poor in New York. The French Benevolent Society offered food assistance to indigent Frenchmen. The Irish Emigration Society, formed in 1841, offered assistance to Irish immigrants. The St. Andrew's Society of New York, formed in 1826, offered food assistance to immigrants from Scotland. When New York went into a major financial crisis during the 1870s, the society opened five stands to serve the poor. In the first three weeks of operation, they served 1,500 people. It became the model for subsequent soup kitchens in the city.\n\nFree coffee at Bowery Mission for unemployed circa 1880. Courtesy of the Library of Congress. Reproduction Number: LC-DIG-pga-03183\n\nOn October 24, 1929, the New York Stock Market crashed, beginning the Great Depression. During the next eleven years, hundreds of thousands of New Yorkers were out of work. By 1931, eighty-four bread lines operating in New York served eighty-five thousand people daily. Soup kitchens and missions fed tens of thousands more. Shantyvilles became common throughout the city. Many out-of-work New Yorkers became beggars; others tried to sell apples on the street. When President Herbert Hoover was asked about the massive number of apple vendors in New York, he replied, \"Many people have left their jobs for the more profitable one of selling apples.\" In 1932, he reported that \"the hoboes . . . are actually better fed than they have ever been. One hobo in New York got ten meals in one day.\" In the same year, thirty-two New Yorkers died of starvation and 110 from malnutrition, an estimated twenty thousand school children were reported to be malnourished, and tens of thousands more suffered from nutrition-related diseases.\n\nBernarr Macfadden, the vegetarian health guru, established a five-million-dollar foundation to launch penny restaurants. His first opened in December 1931. It was a cafeteria-style operation that served \"cracked wheat, Scotch oatmeal, lima bean soup, green pea soup, soaked prunes, seeded raisins, whole wheat bread, butter, raisin coffee and cereal coffee.\" Each dish cost one cent. Macfadden said he charged for the food so that unemployed workers would not feel that they were on charity. His penny restaurants served 1,500 to 3,000 customers per day. There were no chairs, just counters where customers stood while eating. At the height, they served an estimate eleven thousand New Yorkers. Others opened similar operations. The Salvation Army offered meals for a nickel. Diners were given trays filled with food, such as corned beef and cabbage, potatoes, coffee, stewed prunes, and bread. Unlike Macfadden's restaurants, this one had tables and chairs. It served men and women of all ages, regardless of \"race, creed or color.\"\n\nBeginning in 1853, the New York's Children Aid Society began offering lunches for some students in vocational schools. In 1908, the New York City Board of Education instituted three-cent lunches for students. The Federal School Lunch Act, passed in 1946, established federally subsidized lunches for the nation's school children.\n\nThe Food Stamp Act, passed in 1964, offered federal assistance to qualifying low- or no-income people. Funded through the U.S. Department of Agriculture, it provided qualified individuals and families with stamps they could redeem in grocery stores and supermarkets. In 2008, the program was renamed the Supplemental Nutrition Assistance Program (SNAP), and the system has shifted from stamps to Electronic Benefits Transfer (EBT) cards that operate like credit or debit cards. In 2001, eight hundred thousand New Yorkers qualified for food stamps. Ten years later, this had reached 1.6 million people, or about one-quarter of all adult New Yorkers. An estimated 750,000 residents live in what are called \"food deserts,\" areas where there are no grocery stores or supermarkets that sell fresh fruits and vegetables. Recently, large chains, such as Whole Foods, have agreed to open outlets in food deserts that will offer better quality foods at reasonable prices. Also, it is now possible to use EBT cards at most of the city's Greenmarkets.\n\nToday, more than 1.4 million city residents and the homeless are food insecure. Many organizations in the city have focused on helping New Yorkers who do not have enough food. The New York City Coalition against Hunger was formed in 1983 by nonprofit soup kitchens and food pantries to coordinate the activities of the emergency food providers. Today, the coalition represents more than 1,100 hunger-relief groups working in the city. Programs are available for the homeless as well as the elderly who are unable to purchase or prepare their own meals.\n\nAt the other end of the spectrum are those New Yorkers who are overweight. In 2011, it was estimated that 25 percent of New Yorkers were obese, and this is projected to hit 51 percent by 2030. Recently, the city's health department with the encouragement and support of Mayor Michael Bloomberg, has launched a number of efforts to curb obesity. In 2006, New York City required chain restaurants to post calories of the foods that they served; this went into effect three years before this became a national requirement. In January 2012, the city health department launched a program, \"Cut Your Portions, Cut Your Risk,\" to encourage overweight New Yorkers to eat less at meals. The city health department has also attempted to reduce the consumption of sugary sodas as a means of reducing obesity.\nChapter Three\n\nA Culinary Stew\n\nImmigrants and Their Food\n\nFrom the beginning of European colonization, New York's population embraced myriad groups that differed culturally, linguistically, religiously, and racially. In addition to American Indians, there were British, Dutch, Swedish, German, Irish, and French settlers; enslaved peoples were shipped in chains from Africa and the Caribbean. After the colonial period, immigration from abroad rapidly increased, as did migration from other regions of America. These groups have shaped the city\u2014and its foods and beverages\u2014for four hundred years.\n\nWhile New Amsterdam was established by the Dutch, the majority of early European settlers were Walloons from what is today Germany, northern France, and Belgium. Peter Minuit, the first director of the West India Company's operation in New Amsterdam, was a Protestant refugee from what is today North Rhine\u2013Westphalia in Germany. Walloons also settled at the Waelenbogt, or Walloon Bay, which is today in Brooklyn. Most New Amsterdam colonists understood Dutch, but Rev. Jonas Micha\u00eblius occasionally preached in French to reach those whose primary language was not Dutch. In addition, slaves were brought to New Amsterdam within months of the community's founding, and they continued to be imported throughout the colonial period.\n\nThe Dutch population in New Amsterdam was never very large. When the fur trade petered out, immigration languished, and other Europeans were permitted to settle in the colony. In 1643, the director general of New Amsterdam estimated the total male population of the city and its surrounding area at between four and five hundred. New Amsterdamers spoke eighteen different languages and included a number of English settlers; some freed slaves; Jewish refugees from Pernambuco, the former Dutch colony in Brazil; and some Swedes, Danes, French, and at least one Italian. By the time English military forces captured the city in 1664, only about half of those in the colony were of Dutch lineage.\n\nAs part of the agreement that surrendered New Netherland to England, Peter Stuyvesant, the Dutch director general, convinced the English to grant the Dutch colonists ownership of their estates, which were especially vast along the Hudson Valley. As the English and others immigrated to New York, the Dutch sold small parcels of their land for high prices, thus making themselves wealthy. Their financial position helped create an aristocracy that continued to influence New York for more than two hundred years.\n\nPrecisely what early Dutch colonists in New Amsterdam ate is unclear. No descriptions of meals have been uncovered. Archaeological evidence indicates that New Amsterdamers ate bread (made from wheat or rye), milk, cheese, game, apples, and some beef and pork, as well as maize, squash, and beans. Corn was converted into porridge and mush. Pumpkins were commonly consumed as fruit.\n\nSeveral later descriptions of upstate Dutch fare have survived. Peter Kalm, a Swedish-Finnish scientist who visited New York in 1749, reported that the Dutch along the Hudson typically breakfasted on bread, butter, cheese, and milk, and that dinner, usually served at midday, consisted of one main dish accompanied by a \"great salad\" dressed with vinegar.\n\nManuscript cookbooks authored by New Yorkers of Dutch heritage date to the late eighteenth and early nineteenth centuries but do not likely reflect what Dutch colonists ate before 1664. However, unlike other immigrant groups who wanted to assimilate into their adopted land, the Dutch in New York maintained many of their traditions, including their language, well into the nineteenth century. This was partly due to their common membership in the Dutch Reformed Church, which kept many traditions alive.\n\nDutch settlers continued to arrive in New York after 1664. These immigrants likely introduced culinary traditions that were emerging or popular in Holland, and they were likely the source for some \"Dutch\" culinary traditions popularized by nineteenth-century writers. Waffles (from the Dutch wafel), for instance, were first recorded in New York City by 1744, and they were such a hit that \"waffle frolics\" and \"waffle parties\" were enjoyed in the city for decades afterward. Coleslaw (koolsla, a cabbage salad) was first cited in 1794; doughnuts, which may have been of Dutch, English, or German derivation, were sold in a public market by a Dutch woman by 1796; cookies (koekjes, \"small cakes\" that were, in fact, very large) were mentioned in New York by 1808; and crullers (krullen, \"curls\") were consumed by 1820 and were frequently cited throughout the nineteenth century. Pretzels had been made in Europe by the Dutch and Germans by the late seventeenth century, but the word does not appear in print in New York until the 1820s. Many other Dutch favorites\u2014hutspot (a dish of potatoes and vegetables, usually served with meat), olykoek (a deep-fried cake), and oliebollen (a spherical doughnut often studded with dried fruit and typically served on New Year's Day)\u2014were likely introduced into New York at an early time but were not recorded until decades\u2014or centuries\u2014later.\n\nNew Yorkers of Dutch heritage influenced the city's culinary life throughout the nineteenth century. A mid-nineteenth century writer for Harper's reported that the gardens and farms of \"Dutch Fraus\" in Brooklyn provided cabbage, potatoes, turnips, and other vegetables to Manhattan's public markets. But Dutch traditions were fading. In 1881, Gertrude Lefferts Vanderbilt complained that Dutch Americans had given up their traditions, including the foods of their forefathers.\n\nLefferts's House, Prospect Park, Brooklyn. Courtesy of the Library of Congress. Reproduction Number: HABS NY,24-BROK,2-3\n\nDuring the colonial period, each immigrant group contributed to the city's foods and beverages, but English settlers put down the deepest roots. Not only were English settlers most numerous, but also they continued to immigrate to New York in large numbers throughout the colonial period and long afterward.\n\nEnglish immigrants brought cookery books and recipe collections with them; English culinary styles came to dominate colonial New York's foodways and continued to do so well after the American Revolution. English cookbooks were imported into New York, and beginning in 1790, they were published in the city. Even after cookbooks authored by Americans began to be published in New York, English traditions continued to thrive, influencing what New Yorkers ate and drank. English-style pancakes, sweet and savory pies, tarts, sausages, puddings, rusks, roast beef, pastries, ketchups, dumplings, stews, soups, and condiments dominated New Yorkers' tables. These were accompanied by English-style beer, sweet wines from the Atlantic islands, and mixed drinks, such as possets and punches, that were popular in England.\n\nEnglish immigrants opened boarding houses, restaurants, and saloons, and established food businesses. English-born Richard Davies, who had founded the Canton Tea Company in New York, invented a popular mixture of teas that was marketed as \"English Breakfast.\" Another English immigrant, Samuel Bath Thomas, opened a bakery called the \"Muffin House\" in New York in 1880. When his baked goods became popular, he formed the S. B. Thomas Company and began manufacturing \"Thomas's English Muffins,\" a product that remains part of American breakfasts today.\n\nAnother English immigrant was William Loft, the son of a candy maker. In 1860, he opened a candy store in Manhattan, which developed into a chain that was a part of New York City life for 130 years. The company he and his son created eventually manufactured and sold more than 350 types of candies and was one of the largest candy store chains in America.\n\nAn English culinary import that became a perennial New York City favorite was the sandwich. Sandwiches first appeared in England in the 1760s. The sandwich concept migrated to New York, and recipes for them were published in the city by 1816. Virtually all early nineteenth-century American sandwich recipes were taken almost verbatim from British cookbooks. But Americans made the sandwich their own, and special sandwiches have become some of New York's defining foods\u2014from hotdogs and hamburgers to the Reuben sandwich and pastrami on rye.\n\nThe British culinary influence may have predominated in early America, but other nationalities made contributions. Under British colonial control, immigrants from other countries were also permitted to settle in the city. More Dutch, Scandinavian, French, and German immigrants arrived. When the first census was taken in 1790, 33,470 people lived in Manhattan. Of these 2,250 were new immigrants from Germany, and 5,000 were (mostly Protestant) immigrants from Ireland.\n\nImmigrants from abroad continued to move into the city. By 1825, Manhattan's population had reached 166,000. Immigration patterns changed in the 1830s as industrialization, revolutionary activity, crop failures, poverty, and religious persecution abroad drove waves of Europeans from their home countries. The first new wave of immigrants came from Germany.\n\nGerman Immigrants\n\nGermans had trickled into New York since the 1600s, but immigration picked up after the American Revolution. William Havemeyer, born in what is today Lower Saxony, started a sugar refinery in New York City. His descendants grew the firm into America's largest sugar-refining business, which would become known as Domino Foods. Another German immigrant, John Jacob Astor, came from a village near Heidelberg. He first went into the fur trade, where he made his first fortune. When he saw the potential profits in the tea trade with China, Astor jumped in, shipping furs to China in exchange for tea. This trade, begun in 1800, earned Astor a second fortune. He invested in Manhattan real estate, reaping huge profits when he sold the latter to a rapidly expanding city and made yet another fortune. He purchased property around his home on Broadway in downtown Manhattan, and here he built a block-square hotel called the Astor House. When it opened in 1836, it was the city's largest and most luxurious hotel, and its dining room served some of the city's finest French cuisine.\n\nDuring the 1830s, the pace of German immigration increased due to a series of unsuccessful revolutions and agriculture failures in the southern and western parts of the country. Those immigrants who had already arrived in New York wrote home telling family and friends about life in America, thus creating a chain migration. By 1840, twenty-four thousand Germans lived in Manhattan; others crossed the East River and moved to Brooklyn Village and Williamsburg, where, by 1847, they comprised two-thirds of the entire population. In the 1840s, bad weather again destroyed crops in southern Germany, and a steep rise in food and beer prices led to widespread riots there that continued for months. Southern Germany, especially Bavaria, suffered yet another severe blow in 1848, when a revolution was brutally suppressed. More Germans fled to New York, and they continued to do so for decades afterward.\n\nBy 1860, the city was home to two hundred thousand Germans, and many lived east of Bowery and north of Division Street in Manhattan\u2014an area that came to be called Kleindeutschland (\"little Germany\"), which was New York's first ethnic ghetto. As the immigrants became more affluent, they moved northward to the city's more fashionable neighborhoods, such as Yorkville on Manhattan's Upper East Side, Ridgewood in Queens, and Bushwick and Williamsburg in Brooklyn. By 1880, the number of Germans in New York had doubled again, forming almost one-third of the city's entire population. New York had more German-speaking people than any other city in the world except Berlin and Vienna.\n\nNewcomers went into food-related businesses of one kind or another. Some opened hotels, boardinghouses, inns, and saloons. In 1854, a German immigrant set up a coffee and chocolate factory that also produced \"German-style mustard.\" By 1855, two-thirds of the city's butchers were German, and German bakers outnumbered the Irish or Scots bakers who had previously dominated the profession. German immigrants ran the majority of the city's grocery stores from 1855 on; 75 percent of the grocers were from the Lower Rhine, particularly Hanover. By 1867, Germans had a near monopoly on the market-garden business supplying New York City.\n\nBeer was the German national beverage, but the heavy, dark, British-style brews made in New York did not appeal to the immigrants. Brewing was an obvious career choice for the new arrivals, many of whom had followed this trade back home. German-owned breweries sprang up around the city to make the lighter lagers favored in the old country. Saloons owned or operated by Germans served the lager, and there German immigrants (only the men, of course) could socialize with their compatriots, eat a meal, and seek help finding work and housing.\n\nSingle German women had few employment opportunities; many became domestics or worked in boardinghouses. Domestic employees at least had the security of a roof over their heads and food to eat; they earned only about four to six dollars per month. The alternative was the factories, where working conditions could be horrendous. Domestics could also easily move from one place of employment to another as the job market was rapidly expanding. By 1855, about 15 percent of all domestics in New York City were German immigrants. Since these women cooked some, if not all, of the family's meals, their culinary traditions had some influence on their employers' way of eating.\n\nGerman immigrants also found work in New York's rapidly growing restaurant scene. Some were cooks or chefs, while others worked as waiters in the city's best restaurants. Others opened small restaurants of their own, which were known for serving generous portions of food at low prices. In 1867, these were described as\n\nqueer, dingy, rattle-trap dining-houses in which families of the Teuton race\u2014men, women, and children\u2014appear to spend a great deal of their time. The bar was piled with joints and manufactured meat adapted to the strong German stomach; enormous fat hams, not thoroughly boiled, for the German prefers his pig underdone; rounds of cold corned beef, jostled by cold roast legs and loins of veal; pyramids of sausages of every known size and shape, and several cognate articles of manufactured swine-meat. . . . [There were also] baskets full of those queer, twisted, briny cakes which go variously, I believe, by the names of Pretzel and Wunder; sardine-boxes piled upon each other . . . huge glass jars of pickled oysters, flanked by huge earthen jars of caviare. Raw onions in heaps give a tone to the combined odors of all these; and through this confusion of smells come powerful whiffs of the Limburger and Sweitzer cheeses, without which the menu of no German restaurant would be considered complete. . . . Conspicuously posted upon the walls are the Weinlisten, from which documents you gather that white wine is to be had at from one dollar and a quarter to three dollars per bottle, and red wine at from one dollar to four. The inevitable keg of lager-bier lies upon its slanting trestles, behind one end of the counter.\n\nAuguste Ermisch, an immigrant from Mecklenburg-Schwerin, opened a restaurant at Nassau and John Streets in Manhattan. In 1873, his menu featured German bread, beef with macaroni, Vienna sausage with mashed potatoes and sauerkraut, wienerschnitzel, calf's tongue, lentil soup with bologna sausage, fish balls and red cabbage, dumplings, and even pumpkin pie. One of the more interesting items on the menu was \"Hamburger steak,\" described in an article in the New York Times as \"a beefsteak redeemed from its original toughness by being mashed into mincemeat and then formed into a conglomerated mass. This is very appetizing, but conscience compels us to state that it is inferior to the genuine article, which can also be had here in a very satisfactory condition of tenderness.\" Hamburg, or hamburger steak, was an inexpensive dish made by grinding scraps of beef left after butchering choicer cuts, such as Porterhouse steak or sirloin. This dish would become commonplace throughout the country within ten years; by the 1890s, hamburgers were sold by street vendors, who served them in buns for a convenient stand-up meal, thus creating the hamburger sandwich.\n\nIn Williamsburg, German immigrants Carl and Peter Luger opened a caf\u00e9, bowling alley, and billiards hall in 1887. After Carl died, the bowling alley and billiards hall were jettisoned and the establishment was renamed Peter Luger Steak House. In Manhattan, August Guido L\u00fcchow, an \u00e9migr\u00e9 from Hannover, went to work in a beer hall on Fourteenth Street in 1877 and, by 1879, had saved up enough money to buy the place. William Steinway, manufacturer of pianos, was a fan and frequent patron of the establishment, which became a favorite among New York's musical elite. As L\u00fcchow's became popular it grew from a beer garden into one of New York's most fashionable restaurants.\n\nGerman immigrants and their progeny, such as Charles Gulden, William Entenmann, and Richard Hellmann, opened food-related businesses. Charles Gulden, the son of a German immigrant, worked at his uncle's company, Union Mustard Mills and then in 1867 launched his own mustard company on Elizabeth Street in New York. By 1883, Gulden's product line included many different mustard varieties and other products. His spicy brown mustard has survived, although the company is now owned by ConAgra. William Entenmann, an immigrant from Stuttgart, Germany, opened a small bakery selling small cakes, breads, and rolls on Rogers Avenue in Brooklyn in 1898. His business thrived, and he expanded his product line. During the 1920s, the company moved to Bay Shore, Long Island. During the 1950s, it expanded sales throughout the Northeast and later throughout America; today, its brands are among the largest-selling pastries and donuts in America. Richard Hellmann, who arrived in New York City in October 1903, opened a deli on Manhattan's Upper East Side in 1905. On September 1, 1912, he began selling his store-made \"Blue Ribbon Mayonnaise.\" Within fifteen years, his company was the largest mayonnaise manufacturer in the world. In 1927, Richard Hellmann sold his company; Hellmann's mayonnaise eventually became the most profitable brand in the product line of Best Foods, Inc., which today is owned by Unilever USA, headquartered in New Jersey.\n\nThe two world wars hurt German restaurants and many German Americans left the city. Most German restaurants closed. L\u00fcchow's, one of New York's last prominent German restaurants, survived for decades before finally closing in 1986. The steakhouse that Carl and Peter Luger launched in Williamsburg, however, has survived, and other restaurants serving German-style food have opened around the city.\n\nIrish Immigrants\n\nGermans were only the first of several waves of immigrants to hit New York. The second was the Irish. Irish immigrants had been coming to the city since colonial times. Many came as indentured servants and some as convicts; still others were merchants, artisans, and shopkeepers who paid their own way. Most early Irish immigrants were Protestants.\n\nEarly Irish immigrants went into food-related businesses in New York. One successful Irish entrepreneur was John Hill, who opened the New Beef Steak and Oyster House in 1774. Another was William Niblo, who arrived after the American Revolution. He apprenticed at a coffeehouse and married the owner's daughter. In 1814, he opened the Bank Coffee House. In 1823, he purchased the grounds of a small circus, called \"The Stadium,\" at the corner of Broadway and Prince Street. There he opened Niblo's Garden, the city's most fashionable resort, complete with a restaurant. In 1834, Niblo put up a theater, where plays and other entertainments were performed throughout the mid-nineteenth century.\n\nBeginning in the early 1830s, Irish Catholics began to immigrate to New York. Most were poor, from rural areas; unskilled and uneducated, they were not prepared for the demands of the urban workplace. They were also Catholic in a mainly Protestant city, and they would face prejudice, poverty, and hunger for decades to come.\n\nSome Irish immigrants opened eating establishments. Daniel Sweeney arrived in New York during the 1820s. He made money as a water vendor while learning the restaurant trade. In 1836, he opened a restaurant to serve middle-class workers in downtown Manhattan, offering oysters, roast beef, mutton, beans, pies, and desserts. It was an immediate success, and Sweeney later opened a hotel, which became a center for Irish immigrants during the following decades. Patrick Dolan, an Irishman who arrived in New York in 1846 and worked at Sweeney's restaurant, opened his own very successful restaurant in downtown Manhattan. Other Irish immigrants became importers. One of New York's largest importers of beer and wine was Thomas McMullen, who in 1852, published Hand-Book of Wines, Practical, Theoretical, and Historical; with a Description of Foreign Spirits and Liquors, one of the first books published in the United States on wine and how to make it.\n\nWhen the Potato Famine hit Ireland in 1846, Irish immigration swelled to flood stage. In 1847, an estimated eighty-five thousand Irish men, women, and children departed for North America, many bound for New York. Decades after the Potato Famine subsided, the Irish continued to immigrate. By 1860, there were more than two hundred thousand of them in New York, forming about one-fourth of the city's total population. By 1890, more than four hundred thousand people of Irish heritage lived in Manhattan (and thousands more in Brooklyn and Queens). There were more Irish in New York than Dublin.\n\nMany single women went into domestic service. In 1846, two-thirds of the domestic servants in New York City were Irish. Within a decade, 80 percent of the thirty-five thousand foreign-born domestics and waiters in Manhattan were Irish. Coming as they did from a famine-stricken land, these women were not necessarily able cooks, and they were unfamiliar with American food. Some, according to their employers, were unable to cook even potatoes properly by American standards: they \"only half boil their potatoes, leaving the centre so hard, that it is called the bone of the potato.\" And they persisted \"in bringing half-boiled potatoes to the table, notwithstanding our repeated orders to the contrary.\" But Irish kitchen help gradually became familiar with American food and its preparation, and they brought these skills and habits home to their families when they married.\n\nOther Irish women became fruit and vegetable peddlers. They bought oranges, apples, and other groceries from middlemen and sold them on the streets. Those who were successful rented stalls in one of the city's markets. A few immigrants opened small hotels, and those who could afford to buy houses frequently turned them into boardinghouses.\n\nStill others went into the grocery business, making their greatest profits in liquor sales. In a short period, Irish dominated New York grocery stores. Some gained economic prominence. Perhaps the most successful Irish immigrant grocer was Thomas Butler, who arrived in America in 1871. After working in hotels, he and a partner started a grocery store in 1882. They painted the shop front green and emblazoned their names on it in gold lettering. They sold their goods at low prices, and the public thronged to the store. Butler bought out his partner the following year and began to open more stores. By 1914, there were 235 Butler Grocery Stores. The chain continued to grow, reaching 1,100 stores in the New York area by the time Thomas Butler died, in 1934. Despite its Irish American ownership, and the fact that many of its customers were of Irish heritage, the chain did not sell any particularly \"Irish\" food products.\n\nIn Ireland, the diet of the rural poor was largely based on potatoes, oatmeal, buttermilk, some vegetables, and very little\u2014if any\u2014meat, which was simply beyond their means. The same was true of whiskey. In New York, food prices and alcoholic beverages declined throughout the nineteenth century. Irish immigrants, poor as they were, ate and drank well compared to how they had fared in Ireland. An observer noted in 1864 that Irish immigrants in New York bought cheap whiskey and cheap provisions, especially meat that \"they have been accustomed to consider the luxury of the rich, and they go in for it accordingly. They eat meat three times a day, rudely cooked, and in large quantities. Whisky of an execrable quality, is plentiful and cheap.\"\n\nSome Irish newcomers became waiters at New York City restaurants, while others opened working-class restaurants, especially oyster cellars. Many immigrants opened or worked in saloons. In 1851, Seamus O'Daoir, owner of a \"porter house\" on New York's Duane Street, wrote a poem for the Irish American promoting his drinking establishment and extolling the connection between the Irish and alcohol. Another enterprising immigrant, John McSorley, opened an alehouse for workingmen on East Seventh Street in 1854. McSorley's Old Ale House, New York's oldest continuously operating saloon, still serves ale, porter, and their famous liverwurst-and-onion sandwiches. Other Irish saloons served potatoes to go along with the alcohol.\n\nMcSorley's Old Ale House, which claims to be New York's oldest saloon, 2013. Courtesy of Kelly Fitzsimmons\n\nBars and saloons served as places of employment for Irish immigrants. Bartenders helped men find jobs and get a meal when necessary. Irish saloons were also places where men socialized and talked politics. In Hungering for America (2001), Hasia Diner observed that\n\nalcohol linked the Irish in America to the emotionally satisfying world of past memory and ushered them into a comfortable world of friendship with others like themselves. Above all, it heightened their Irish identity. In saloons, under the influence of alcohol, they declared their unswerving loyalty to Ireland, the place they had left but claimed still to serve, while they articulated a deep American patriotism.\n\nIrish saloons became the core of Irish American political control over Tammany Hall, which dominated New York City politics until the Depression.\n\nIrish immigrant Patrick J. Clarke became a bartender at a dingy saloon on Fifty-fifth Street and Third Avenue in 1902. Ten years later, he had saved enough money to buy the place. It was just another little neighborhood bar until 1945, when it was selected as the location for the Billy Wilder film The Lost Weekend, based on a book by Charles R. Jackson, who had done much of his writing in the bar. Celebrities discovered P. J. Clarke's, and it morphed into a restaurant, serving hamburgers, pies, and other dishes. Clarke died in 1948, but P. J. Clarke's survived and expanded to other locations in New York and in other cities.\n\nIrish immigrants formed regiments in the New York militia, such as the Sixty-ninth, also called the \"Irish Brigade.\" When the Civil War broke out in 1861, it was among the first units to respond to President Lincoln's call for volunteers to put down the rebellion. The \"Fighting Sixty-ninth\" served with great distinction during the war, and afterward it traditionally marched at the head of the city's St. Patrick's Day Parade, widely celebrated in New York since the nineteenth century. The day featured serious drinking and lavish meals featuring corned beef and cabbage, Irish stew, Irish soda bread, and colcannon (mashed potatoes with kale or cabbage).\n\nJewish Immigrants\n\nThe third wave of immigrants were Jews, initially from Germany and later from Eastern Europe. In 1654, when the Portuguese expelled the Dutch from Pernambuco, which is today in Brazil, some Dutch refugees, of whom twenty-seven were Jews, arrived in New Amsterdam. Eventually, after considerable debate among city leaders, Jews won the right to own property. Few Jews immigrated subsequently, but this changed suddenly with the advent of the first big wave of German immigration: an estimated eighty thousand of the Germans who began arriving in the 1830s were Jewish. Many settled permanently in New York\u2014they were urbanites who had little difficulty adjusting to city life. Many ended up as butchers and bakers. For Passover in 1859, nine Jewish bakeries in New York used 2,200 barrels of flour making matzah, most of which was shipped to other communities.\n\nDuring the 1880s, a new group of Jewish immigrants poured into New York City, fleeing political oppression, pogroms in Russia, crushing poverty in Eastern Europe, and the turmoil caused by the breakup of the Ottoman Empire. Unlike the earlier Jewish immigrants, the newcomers were largely poor, from rural villages; thus, many were unprepared for city life. They moved into tenement buildings in Manhattan's Lower East Side, which had previously been home to German immigrants. Many of these Eastern Europeans were Orthodox Jews, and they strictly followed the laws of kashruth (Kosher dietary laws). Meats and some other foods had to be prepared according to these rules. To meet these needs, immigrants opened butcher shops, bakeries, restaurants, caf\u00e9s, and cafeterias.\n\nThe meals prepared in these establishments (or at home) were simple fare. In addition to basic soups, stews, grain dishes, and the like, they included gefilte fish, pumpernickel and rye breads, challah, kreplach, latkes (potato pancakes), and blintzes (cheese-filled pancakes). Some Jews moved into commercial production of foods, such as bagels, which were sold by vendors on the streets of New York by the early twentieth century.\n\nMany Jewish-owned businesses advertised that they were kosher, selling only kosher foods, wines, and liquors. But were they truly kosher? This issue perplexed New York Jews for decades. In 1887, the Orthodox synagogues in New York and other cities imported a rabbi from Vilna, Russia (now Lithuania), in part to regulate the kosher meat business. His efforts\u2014and those of others\u2014failed. In 1923, the Union of Orthodox Jewish Congregations launched a supervision and certification program for processed foods. It wasn't until New York State passed and enforced a Kosher Law in 1934 that this matter was, for the most part, resolved.\n\nDuring the late nineteenth and early twentieth centuries, the vast majority of the city's pushcart vendors and door-to-door salesmen were Jewish; many moved up the retail ladder to open their own shops. By 1899, these included \"140 groceries, 131 kosher butchers, 36 bakeries, 9 bread stands, 14 butter and egg stores, 24 candy stores, 7 coffee shops, 10 delicatessens, 9 fish stores, 7 fruit stands, 2 meat markets, 10 sausage stores, 13 wine shops, 13 grape wine shops, and 10 confectioners.\" Jewish grocers thrived throughout the early twentieth century. By 1938, there were almost ten thousand grocery stores that catered largely to the Jewish population in New York. In addition, about five hundred \"appetizing stores,\" such as Russ & Daughters and Zabar's, specialized in prepared foods: pickled, smoked, and salted fish, such as lox, herring, and white fish, as well as dried fruit, nuts, pickles, breads, candy, and imported delicacies.\n\nDiversity of sausages and other products for sale in a New York deli, 2013. Courtesy of Kelly Fitzsimmons\n\nBy comparison to other immigrant groups, most immigrant Jews drank little alcohol. In Jewish neighborhoods, saloons languished. As an alternative, two companies began manufacturing soda, which was well received in part due to its purported healthfulness. When the Sugar Trust's activities increased the price of sugar, Jewish soda manufacturers shifted to make seltzer (plain soda water), referred to as \"Jewish champagne\" and labeled the \"staple beverage of Yiddish New York.\" By 1907, there were more than one hundred Jewish seltzer manufacturers in New York. A Jewish candy store owner in Brooklyn, Louis Auster, has been credited with inventing one of the city's most iconic beverages, the egg cream. Despite its name, this classic fountain drink contains only soda water, milk, and chocolate syrup.\n\nJewish immigrants found employment in New York City, and many thrived economically. Those who had often gone hungry in their homelands could now enjoy foods that had been the province of the upper classes there. As Hasia Diner notes, \"The formerly poor started to eat blintzes, kreplach, kasha-varnitchkes, strudel, noodles, knishes and more importantly, meat every day.\"\n\nJewish restaurants opened throughout the city. Ratner's, in Manhattan, was founded by Jacob Harmatz and Alex Ratner in 1905. It was a kosher dairy restaurant, where no meat was served. At the height of its fame, Ratner's would serve ten thousand meals per week. Other early and notable kosher restaurants included Steinberg's on the Upper West Side of Manhattan and Garfield's in Brooklyn. Neighborhood caf\u00e9s and coffeehouses flourished on the Lower East Side, some of them catering to intellectuals\u2014writers, poets, and artists. The Garden Cafeteria, a dairy restaurant on bustling East Broadway, was the most famous of these.\n\nLike their non-Jewish German counterparts, Jewish immigrants also opened delicatessens, which eventually became sit-down restaurants. Following the laws of kashruth, these offered only meat, fish, and pareve (neutral) foods; a typical selection featured chicken soup, corned beef, gefilte fish, lox, knishes, pastrami, chopped liver, tongue, and garlic pickles. Some delis made their own beef sausages\u2014salami, bologna, and frankfurters\u2014but most found it cheaper and easier to buy from Isaac Gellis, a Jewish immigrant from Berlin, who opened a kosher sausage company on Essex Street shortly after the Civil War. Other companies, such as the Williamsburg Genuine Kosher Meat Products Company in Brooklyn and the Hebrew National Kosher Sausage Factory on the Lower East Side, later provisioned Jewish delis. Jewish delis also popularized old-country breads, notably rye and pumpernickel.\n\nGradually, as secular Jews and non-Jewish diners became steady customers, Jewish delicatessens expanded their menus to include cream cheese and farmer cheese to spread on bagels and bialys, herring in cream sauce, blintzes, and cheesecake. Of course, these establishments lost their kashruth-observing patrons in the process.\n\nSome Jewish delis became iconic New York establishments. Joel Russ, a Jewish immigrant from what is today Poland, began his career in 1907 selling Polish mushrooms, which he carried on his shoulders. He saved up enough money to buy a pushcart and then a horse and wagon. In 1914, he opened his first store on Orchard Street on the Lower East Side and expanded his offerings to include salmon and herring. His business would eventually become Russ & Daughters, which is still one of New York's premier appetizing stores.\n\nRuss & Daughters appetizing store, Manhattan, 2013. Courtesy of Kelly Fitzsimmons\n\nLouis Zabar, a Jewish refugee from Ukraine, started selling smoked fish in Brooklyn in 1934; he later moved the business to the Upper West Side of Manhattan. Still family owned and run, Zabar's now offers a full range of gourmet foods and kitchenware at its single vast store on Broadway at Eightieth Street, and smoked fish is still a huge part of the business.\n\nDuring the Depression, deli owners had a difficult time selling their relatively expensive meats and imported products. Mogen Dovid Delicatessen Corporation, a trade association, encouraged Jewish deli owners to appeal to a wider audience and to keep their prices moderate; they published a periodical for deli owners that included a list of suggested prices for standard deli items. Many Jewish delis sell kosher meats but do not otherwise adhere to the laws of kashruth, including closing the store for the Sabbath. Some notable Manhattan delis past and present are the Carnegie, the Second Avenue Deli, the Stage, and Katz's.\n\nKatz's Deli, on Houston Street, Manhattan, 2013. Courtesy of Kelly Fitzsimmons\n\nFood played an important part in the Jewish communities in New York. As Hasia Diner wrote, \"Food drew Jewish men and women to the streets. In the immigrant and first American-born generations, Jews lived in relatively compact Jewish neighborhoods. The provision of food to Jews by Jews enhanced the sense of community.\"\n\nBy 1910, more than 1.2 million Jews lived in New York, and this number continued to increase; by the 1920s, the Jewish population of the city was nearly two million. This immigration stopped in 1924 but picked up again in the 1970s, when the city became the destination of choice for thousands of Soviet Jews fleeing relentless oppression. Many moved to Brighton Beach, Brooklyn, now dubbed \"Little Odessa.\" Today, New York is home to about one hundred thousand Russian Jews, more than any other place in the world. Russian restaurants, caf\u00e9s, clubs, and bars have proliferated in the area, and neighborhood shops supply imported specialty items from Russia. One immigrant from Odessa, Tatiana Varzar, opened a small restaurant on the boardwalk in Brighton Beach that grew into Tatiana Restaurant, one of the most popular eating establishments and nightclubs in the neighborhood.\n\nItalian Immigrants\n\nAlthough Italians lived in New York since colonial times, they were few in number. Some Italian immigrants became prominent in the early nineteenth century. Ferdinand Palmo, who arrived in New York in 1808, opened Palmo's Garden on Broadway near Duane Street. Its restaurant, the Caf\u00e9 de Mille Colonnes, was filled with gilded columns and huge mirrors. Palmo hired Italian musicians for the band at the restaurant, which quickly became one of the city's hot spots. In 1844, he opened an opera house that specialized in Italian opera.\n\nAccording to the 1850 census, fewer than four thousand people of Italian heritage lived in the United States. In the 1850s, Italian immigration picked up momentum. By the 1860s, Italian foods such as spaghetti, macaroni, and vermicelli were manufactured in the city, and merchants imported Chianti and other Italian wines.\n\nItalian immigration surged after the Civil War. Most of the newcomers were from poverty-stricken areas of southern Italy, and they came to New York seeking economic opportunities. They became street vendors and gardeners. Italian commission merchants established themselves in the peanut business, hiring their fellow Italian immigrants as pushcart peddlers to sell the nuts. By 1870, there were several hundred Italian peanut vendors on the streets of New York. Some moved up the retail chain by buying carts and hiring others to operate them, or by setting up stationary stands. Some also returned to Italy and recruited others to come to America, guaranteeing them jobs as peanut vendors. Some Italians hired in this way returned home after they had made enough money; most, however, encouraged their families to immigrate to America, thus creating another chain migration.\n\nImmigrants mainly lived in tenements south of Greenwich Village in what would become Little Italy. Where they could, they kept chickens, goats, and on occasion pigs in their tenement basements; for those lucky enough to live in a house, backyard gardens were the norm. Others lived or worked on small farms around the city where they grew fruits, vegetables, and herbs commonly used in Italian cooking: these included certain types of lettuce, Savoy cabbage, tomatoes, eggplant, onions, asparagus, beans, potatoes, broccoli, celery, peas, thyme, parsley, mint, basil, garlic, and oregano. These were marketed only to Italian immigrants at first, but eventually these items caught on with New Yorkers of other ethnic groups.\n\nSome Italian food products were unavailable in New York, so Italian-owned businesses imported olive oil, Parmesan and other cheeses, sausages, anchovies, pastas, and coffees. These were supplied to Italian American grocers, who introduced them to other New Yorkers. By the 1930s, Italians ran 10,000 grocery stores, 875 butcher shops, and 1,150 restaurants in the city. Many of the city's other restaurants hired Italian chefs, waiters, and kitchen help. Italian cookbooks were published in New York beginning in 1911. Italian foods were popularized through festivals, such as the Feast of San Gennaro, which has been celebrated since 1926 along Mulberry Street in Little Italy. New dishes based on Italian heritage emerged; for example, spaghetti and meatballs, a combination not traditionally served in Italy, became an American mainstay.\n\nItalian immigrants opened fish markets and introduced New Yorkers to types of seafood they hadn't appreciated before. Few New Yorkers, for instance, ate tuna\u2014it was considered too oily, its flesh too dark. In the autumn of 1897, the New York Times published an astounding saga of an Italian fishmonger who had bought a nine-hundred-pound tuna for nine dollars at Fulton Market. \"About his store that night a changing group made up of old women, old men, young girls, and small boys with bulging eyes worshiped at the shrine. The old people spoke of fish in their day in far-off Italy, and revealed the secret of the wonderful sauce. To this last the young girls paid careful attention.\" Within two decades, New Yorkers began to eat canned tuna, which by the 1940s, became the nation's most consumed fish.\n\nItalians also opened ice-cream parlors, quick-lunch establishments, and restaurants. Many were highly rated, such as Morelli's in Little Italy. Others were identified as \"spaghetti joints\"\u2014usually very small establishments in cellars. One was described in 1891 as \"lit by a smoky kerosene lamp. A little bar is in one corner, and narrow, wooden benches, black with use, run around the walls and are fastened to them. Here five cents will buy a plate of maccaroni [sic], a bit of toast, and a cup of coffee.\"\n\nBy the 1930s, Manhattan had about six hundred Italian restaurants, while Brooklyn had three hundred and the Bronx another two hundred and fifty. Most remained small \"hole-in-the-wall\" joints with sawdust on the floor and few refinements of d\u00e9cor or service. They served big bowls of macaroni and spaghetti, and pizza with tomato sauce and cheese. At the other end of the Italian restaurant spectrum were restaurants such as Monetta's on Mulberry Street, Barbetta's on West Forty-sixth Street, and Sardi's near Times Square. Sardi's served both Italian and French cuisine, and was (and remains) a popular theater crowd hangout in the Theater District. Whether large or small, these restaurants popularized common Italian foods.\n\nItalian immigrants also started factories that produced Italian foods. Macaroni had been known in America since the late eighteenth century, but it was considered an exotic luxury. The Atlantic Macaroni Company began operating in 1897 and La Rosa Macaroni Company in 1914. Emanuelle Ronzoni formed the Ronzoni Macaroni Company in 1915. It specialized in farina macaroni, farina spaghetti, and pasta all'uovo (egg noodles). Along with La Rosa, Ronzoni became one of the largest pasta makers in America. Yet another business was launched by Joseph Kresevich and his wife Angela, immigrants from Trieste, Italy, who started the Stella D'Oro Biscuit Company in 1932.\n\nIn time, Italian foods and dishes reached all levels of New York society. By the late ninetieth century, spaghetti had become a mainstream dish. Pizza was sold in Little Italy by 1902. Lombardi's Pizzeria, which opened in 1905, still stands on the corner of Spring and Mott Streets. Pizza was eaten mainly by Italian Americans until after World War II, when returning GIs who had served in Italy sought out dishes they had enjoyed overseas. Pizza was on its way to becoming one of New York's (and America's) favorite foods. Still, as late as 1956, the New York Times had to define pizza as \"a circular mixture of dough, cheese, sauce and Italian lore.\" New York\u2013style pizza\u2014a thin crust with a thin layer of sauce\u2014remains the city's favorite pizza.\n\nItalians had a greater influence on New York culinary scene than did many other immigrant groups. As historian Richard Hooker said, their cookery \"was strange enough to interest Americans but not so different as to be unacceptable or difficult.\" Italian food has continued to thrive in New York, especially as chefs and restaurateurs such as Lidia Bastianich and Mario Batali have created Italian food empires. In 2010, Lidia Bastianich, her son Joe Bastianich, and Mario Batali opened Eataly, a fifty-thousand-square-foot emporium devoted to Italian food and drink. New Yorkers have accepted Italian food with open arms, whether it be homemade lasagna layered with locally made mozzarella and ricotta; \"a slice\" from a neighborhood pizza joint, folded in two and dripping with oil; or an elegant repast fragrant with truffles and complemented with fine Tuscan wines.\n\nChinese Immigrants\n\nChinese immigrants arrived in New York City beginning in 1855, when a census included thirty-eight Chinese men. When construction began on the transcontinental railroad in 1863, an estimated ten thousand Chinese laborers helped build the western portion, and Chinese cooks came along to prepare their meals. With the railroad's completion in 1869, the Chinese population\u2014and the delights of their culinary repertoire\u2014spread eastward across America. By 1873, more than five hundred Chinese lived in New York City, in the area that would become Chinatown. Even after Congress passed the Chinese Exclusion Acts in 1882, Chinese continued to migrate to New York from the West Coast, and illegal immigration continued through Canada. New York's Chinese population steadily increased.\n\nRacial discrimination prevented Chinese immigrants from employment in many professions; restaurants were an exception. Chinese restaurants opened in Chinatown, and initially they mainly served the immigrant population. Like many ethnic restaurants, they served more than cheap comfort food; they also helped immigrants to assimilate, became gathering places for new arrivals, and occasionally functioned as aid societies, lending money when necessary and helping people find work.\n\nPort Author Restaurant, Chinatown, during Chinese New Year's celebration circa 1890s. Courtesy of the Library of Congress. Reproduction Number: LC-DIG-ggbain-00025\n\nFew New Yorkers ventured into Chinese restaurants except out of curiosity. One brave soul reported in 1891 that a Chinese restaurant on Mott Street served rice, stewed pork, pig's feet, duck, chicken, boiled cabbage, onions, bamboo shoots, celery, beans, soy sauce, and tea. Special party menus included pigeon, sea worms, bird's nest soup, muscles, and rice wine. The observer of this restaurant may well have seen chop suey, which was described by the Brooklyn Eagle as a combination of \"pork, bacon, chickens, mushroom, bamboo shoots, onion, pepper.\" Others reported that chop suey contained mushrooms, parsley, cornstarch, water chestnuts, bean sprouts, seaweed, and many other ingredients. There was no agreed-upon definition, and each establishment made it with whatever ingredients were available. The only thing clear about chop suey was that it originated in America, although it was based on Chinese culinary traditions.\n\nBy the late 1880s, chop suey joints were popular in New York. A writer in Leslie's Illustrated warned in 1896 that \"an American who once falls under the spell of chop suey may forget all about things Chinese for a while, and suddenly a strange craving that almost defies will power arises\" and \"he finds that his feet are carrying him to Mott street.\" A 1915 description of these chop suey establishments noted that non-Chinese diners found the food \"really toothsome and gratifying.\" But the author, who had traveled extensively in China, also pointed out that the food had little in common with the food actually served in China. Chinese Americans rarely ate anything like it, either. One wrote in 1902 that \"there is a great deal of pork in it, and it is too greasy. It is made more to suit American tastes.\" Other Chinese Americans regarded chop suey as \"a culinary joke at the expense of the foreigner.\"\n\nBy 1900, New York's Chinatown had more than one hundred restaurants. Their basic menu was fairly set: chop suey, chow mein, egg foo young, and a noodle soup called yat gaw mein, or \"yakaman.\" Egg rolls probably didn't appear until the 1920s and fortune cookies, an American invention, not until the 1940s. These dishes, although based on home-style village food, had been relentlessly adapted to American tastes. By the 1930s, Chinese restaurants, such as the Bamboo Forest on MacDougal Street and the Chinese Delmonico's on Pell Street became popular.\n\nThe Chinese immigrants themselves, whether at home or in restaurants, clung steadfastly to the traditional fare from whatever region or country they had come from. They imported products from China and created their own market gardens, which supplied home kitchens and Chinese restaurants. They ate mainly rice, vegetables, fish (frequently salted), shrimp, other seafood, fowl, special herbs, and pork. They drank tea, eschewed dairy products, and rarely consumed sugar. They stir-fried, braised, or steamed their food in woks, and they ate with chopsticks.\n\nChinese restaurateurs intentionally moved out of Chinatown, where competition was severe and few non-Chinese were willing to eat. By 1910, more than half of the Chinese-owned restaurants in New York were located outside Chinatown. After World War I, Chinese restaurants expanded in size and number as their foods became more acceptable to their non-Asian neighbors. The Chinese immigrant population of New York increased again after World War II. In 1958, the city had more than three hundred Chinese restaurants, of which thirty were in the Times Square area alone. By 1960, only fifty of the nearly six hundred Chinese restaurants in New York were located in Chinatown.\n\nBy 1988, the 781 Chinese restaurants in New York City employed more than fifteen thousand workers. New immigrants brought other culinary traditions with them. Unlike earlier arrivals, who were mainly Cantonese, the new immigrants came from Sichuan, Hunan, and other regions. Some had lived in other countries before arriving in New York. Restaurants specializing in Chinese regional cuisines from Peking, Sichuan, and Hunan opened around the city. By the twenty-first century, the city had an estimated half a million residents of Chinese heritage. The Chinese restaurant business continued to grow and prosper, in part because of a demand for inexpensive Chinese food and the continued inflow of Chinese immigrants willing to work for low wages. At the other end of the economic spectrum are trendy upper-class Chinese restaurants offering traditional and innovative Chinese fare. New Yorkers seem to have unlimited appetites for the foods of China.\n\nGreek Immigrants\n\nWhen Greek immigrants arrived in America in the late nineteenth century, they wanted the foods that they remembered from back home. Many immigrants went into the food service business. Beginning in the 1890s, Greek immigrants in New York established stores that imported and sold Greek wines, cheese, olive oil, black olives, dried fish, figs, and many other foods. These operations subsequently expanded to Chicago, Boston, and other large northern cities. Other Greek immigrants opened candy stores, fruit stands, ice-cream parlors, bakeries, and coffeehouses.\n\nMany Greek immigrants became street food vendors, selling whatever potential customers would buy. When zoning and licensing laws made it more difficult to operate pushcarts, Greek vendors bought coffee shops and small restaurants. By 1913, at least two hundred of these \"third-class\" restaurants served customers on Seventh Avenue alone. A few upscale Greek restaurants opened on Sixth Avenue. There, according to one observer, \"soup, roast lamb, potatoes, salad, Greek pudding and bread may be secured for thirty-five or forty cents.\" Greek immigrants soon dominated the city's corner diners. Usually their food was very similar to that of other neighborhood restaurants; one distinctive characteristic of the Greek diners was their use of paper cups that featured the Parthenon. Greece is a coffee-loving country, and many Greek immigrants opened coffeehouses. After World War II, especially during the 1960s and 1970s, more Greeks immigrated to New York, particularly to Astoria in Queens, Bay Ridge in Brooklyn, and Washington Heights in Manhattan. The city became home to more than two hundred thousand people of Greek heritage. Greek businessmen purchased coffee shops and hired the new immigrants to operate them. Others launched restaurants, many of which served Greek foods, while others opened pizza parlors.\n\nTom's Greek diner in Brooklyn, 2013. Courtesy of Kelly Fitzsimmons\n\nSlaves from Africa and Migrants from the South\n\nEnslaved Africans had been brought to New York since colonial days. By 1700, slaves and freed slaves comprised an estimated 14 to 21 percent of the city's entire population. Most slaves lived with families, and many ended up as cooks. Freed slaves established small communities that were later incorporated into New York. When the British army left New York at the end of the American Revolution, three thousand freed slaves left with them. When the first census was taken in 1790, out of a total population of 33,470 people living in Manhattan, 2,369 were slaves.\n\nSlavery was outlawed in the state in 1837, and many African Americans ended up as street vendors. Others worked as cooks and waiters in the city's restaurants. When the Civil War ended in 1865, African Americans from the South began to migrate into the city; some took jobs as cooks and servants in private homes, schools, and hotels. Many worked as cooks in the city's booming restaurants, and their cooking abilities were highly praised by observers.\n\nA wave of African Americans into the city increased after 1903 and reached its peak in the 1920s, when tens of thousands of African Americans moved in, seeking jobs in factories. Many ended up in Harlem, which at the time was occupied by Irish and Jewish immigrants and squatters. During the 1920s and 1930s, West Harlem became the center of the \"Harlem Renaissance.\" Nightclubs such as the Cotton Club featured the greatest African American entertainers and supplied their patrons with beer and other alcoholic beverages even during Prohibition. The Cotton Club menu included Chinese dishes, such as \"Chinese soup,\" moo goo gai pan, and egg foo young; other items included baked oysters Cotton Club, filet mignon, broiled live lobster, steak sandwiches, crabmeat cocktails, and scrambled eggs and sausage. Although the Cotton Club was segregated\u2014only whites were admitted as customers\u2014other Harlem clubs were integrated, providing an opportunity for African Americans and whites to socialize.\n\nAfrican Americans brought their culinary traditions with them when they came north. Edna Lewis, who was born in Virginia, arrived in New York during the Depression. In 1949, she became the cook at Caf\u00e9 Nicholson, a restaurant opened by a friend on New York's well-to-do Upper East Side. Caf\u00e9 Nicholson became a gathering place for literary greats including Tennessee Williams, William Faulkner, and Truman Capote, as well as other New Yorkers in the arts. Lewis became known for her perfectly roasted chickens and her lofty souffl\u00e9s. In 1957, Lewis left the restaurant and worked at various food-related jobs, including catering. Although Lewis wrote notable cookbooks, such as The Edna Lewis Cookbook (1972) and The Taste of Country Cooking (1976), she continued to work in New York restaurants, most notably as chef at the venerable Brooklyn restaurant Gage and Tollner. In 1995, Edna Lewis received the first Living Legend Award from the James Beard Foundation.\n\nIn the 1960s, some African American culinary traditions were popularized under the rubric of \"soul food,\" and by 1964 Harlem became known as \"Soul City.\" One African American, Sylvia Woods, became particularly associated with \"soul food\" in New York. Woods, who was born in Hemingway, South Carolina, moved to New York in 1941 and married Herbert Woods. In 1962, the couple bought a luncheonette, which they named \"Sylvia's.\" They served hamburgers and French fries as well as foods traditional in the South, such as fried chicken, ribs, hot cakes, corn bread, collard greens, black-eyed peas, candied yams, and rich desserts. Six years later, they moved to a larger restaurant on 126th Street. Sylvia's became a tourist attraction, drawing visitors from all over the world. In the 1990s, Woods launched Sylvia Woods Enterprises, supplying canned soul food to supermarkets. A consultant recommended that \"Sylvia's of Harlem\u2014Queen of Soul\" be placed on the product labels, and thereafter Sylvia Woods was called \"the Queen of Soul.\" She wrote two cookbooks: Sylvia's Soul Food: Recipes from Harlem's World Famous Restaurant (1992) and Sylvia's Family Soul Food Cookbook (1999).\n\nYet other African American culinary traditions were espoused by the Nation of Islam, which forbade many of the traditional African American foods, especially pork, but retained a fondness for sugary desserts and well-cooked vegetables. Its culinary hallmark became the bean pie, akin to pumpkin pie but filled with sweetened, spiced, mashed kidney beans.\n\nHispanic Immigrants\n\nPuerto Rico was acquired by the United States through conquest during the Spanish-American War (1898). During World War I, America needed soldiers, and to acquire them, Congress passed the Second Organic Act in 1917, granting Puerto Ricans U.S. citizenship, which gave them freedom to travel freely between the island and the United States. This plus Congress's limitation of immigration from other countries, encouraged Puerto Ricans to migrate to New York, particularly East Harlem in Manhattan, south of the Gowanus Canal in Brooklyn. By the late 1920s, there were an estimated 150,000 to 200,000 Puerto Ricans living in New York, and they were served by more than 125 restaurants and 200 bodegas and colmados (neighborhood grocery and convenience stores).\n\nPuerto Ricans continued to migrate to New York during the Depression. This turned to a flood after World War II and reached a peak in 1953 when seventy-five thousand people left the island. The mostly rural, unskilled immigrants settled predominantly in the northeastern part of Manhattan, which is known as El Barrio or Spanish Harlem. This migration continued, and by 1964, more than 10 percent of New York's total population was of Puerto Rican heritage. Puerto Ricans organized social, cultural, and sports clubs that often sponsored public festivals. The Puerto Rican club Los j\u00edbaros (the Puerto Rican peasants) organized festivals that highlighted the preparation of foods like chicharrones (fried pork rinds), tostones (twice-fried green plantains), and arroz con dulce (coconut milk rice pudding).\n\nIn El Barrio, Puerto Ricans strived to maintain their food habits. Carnicer\u00edas (butcher shops), cuchifritos (snack stands), and bodegas brought Puerto Rican flavors to the neighborhood. In the 1950s, Puerto Ricans owned and supported food stalls in the covered food market that they baptized La Marqueta. This market became like a piece of the Caribbean in Manhattan, where they sold yuca and other root vegetables, yaut\u00eda (underground stem of a plant of the genus Xanthosoma), plantains, green bananas, culantro (long-leaf cilantro), fruits, gandules (pigeon peas), avocados, and mav\u00ed (bark used to make a slightly alcoholic drink).\n\nPrudencio Unanue, a Basque who moved to Puerto Rico when he was seventeen, immigrated to New York and, in 1928, opened a small firm in New York to import products like olives, olive oil, and sardines from Spain. In 1936, Unanue founded Goya Foods, which sold rice and beans in bulk in La Marqueta and supplied bodegas with a wide array of foods for the Puerto Rican table: dried codfish, tropical fruit syrups, papaya and guava preserves, chorizo sausages, anchovies, sardines, spices, and cans of prepared foods like pasteles (savory plantain and green banana cakes) and mondongo (tripe soup). Goya has expanded and diversified according to the changes in migration trends and as the new generations lack the skills to prepare traditional foods. Now Goya is a large food importer and processing company that is the leader in its market. In spite of the large size of the company, family members still serve in key positions.\n\nHispanics are currently one of New York's fastest-growing immigrant groups. Dominicans mainly settled in Washington Heights in Manhattan or in Queens. Many have gone into operating bodegas and small restaurants catering to the city's Hispanic population. Few Mexicans lived in New York prior to 1980, but since then, the Mexican population has increased several fold. More than six hundred Mexican-style restaurants have opened, and some offer specific Mexican regional styles. In addition, taco trucks prowl the streets.\n\nJapanese Immigrants\n\nFew Japanese immigrants arrived in New York prior to the 1950s. In 1882, Congress passed the Chinese Exclusion Acts, which also restricted immigration from Japan. In 1908, the United States and Japan concluded a gentleman's agreement in which the United States would not officially halt Japanese immigration, but the Japanese government agreed to deny visas to laborers who wanted to emigrate to America. The few Japanese restaurants established in New York prior to World War II catered almost exclusively to Japanese ex-pats and visiting tourists from Japan, although there were exceptions such as the Miyako on West Fifty-eighth Street and the Daruma on Sixth Avenue. To attract a wider clientele, they offered sukiyaki and other Americanized dishes.\n\nDuring the early 1950s, this began to change as Japanese officials connected with the United Nations arrived in the city. Japanese markets in New York began to sell raw fish for sashimi and sushi to resident Japanese nationals, but raw fish was also bought by some \"Caucasians who have tasted and liked this speciality,\" reported a Japanese market owner in 1954.\n\nBy the late 1950s, Japan had recovered from the devastation of World War II, and its economy was booming. At the time, many American corporations were headquartered in New York City, and Japanese businessmen flocked there, opening offices managed by Japanese nationals. These professionals demanded authentic, high-quality Japanese food, not the made-up American dishes identified as \"Japanese,\" and they could afford it. Japanese chefs and sushi masters were imported to feed their compatriots. New restaurants opened, manned by sushi professionals, and their Japanese customers invited New York counterparts to share in Japanese food. Japanese cuisine is complex and subtle, but what received the most attention from their colleagues were sashimi\u2014sliced filets of raw fish\u2014and sushi\u2014vinegared rice topped with raw fish (among other things). Before then New Yorkers did not eat raw fish, and it seemed unlikely that most would ever sit down to a meal of raw seafood.\n\nThis began to change in the 1960s. Craig Claiborne, the New York Times' food editor, discovered Japanese food in 1961. Two years later, he cautioned that raw fish was probably still \"a trifle too 'far out' for many American palates.\" By 1965, he described New York City as \"a metropolis with a growing public enthusiasm for the Japanese raw fish specialties, sashimi and sushi.\" Two years later, he noted that \"gastronomically there has been no phenomenon in recent years to equal the proliferation of Japanese restaurants, East Side, West Side and up and down the town.\"\n\nRecent Immigrants\n\nThe Immigration Act of 1965 liberalized U.S. immigration law, making it easier for people from other countries to move to America. Jet air service made travel easier, swifter, and more affordable. New York became a major destination city for immigrants. In 1970, 18 percent of the city's population was foreign born; by 2010, it was 40 percent. Prior to 1970, most immigrants hailed from Europe; after that date, they were mainly from the Caribbean, Latin America, South and East Asia, the Middle East, and Africa. White flight began, and the city's racial composition changed. By 1990, Asians made up 7 percent of the city's total population, Hispanics about 25 percent, and African Americans slightly more, but of the last group, five hundred thousand were from the Caribbean, particularly Haiti and Jamaica.\n\nAsian immigrants have included Koreans, Filipinos, and Indians. Korean immigrants settled in enclaves in Flushing and Elmhurst, Queens. Beginning in the 1970s, Korean immigrants began to retail produce, groceries, and fish. They soon operated 90 percent of the city's produce stores. By 2000, the metropolitan area was home to more than two thousand Korean-owned produce retail stores. Most have been very successful, but in 1988 Korean greengrocers in African American neighborhoods have run into ethnic conflicts and boycotts. Recently, food trucks selling Korean tacos have become popular on New York streets. Korean restaurants line West Thirty-first to West Thirty-third Streets, which is now called Koreatown, and Chef David Chang, the son of Korean immigrants, has achieved culinary stardom with his Momofuko restaurant group.\n\nFilipinos began to immigrate to New York after 1970. Filipino restaurants have opened particularly in Queens and Brooklyn, and in the East Village in Manhattan. They serve traditional Filipino fare, such as bangus (a boneless baby milkfish), adobo (the Filipino national dish), arroz caldo (rice porridge), and Spam fried rice. Some opened specialty food stores in Manhattan. Many have settled in Flushing, Queens, where a Filipino fast-food chain, Jolly Bee, just opened a franchise.\n\nSmall numbers of Indians lived in New York prior to World War II. Indian migration picked up after the war. They were diverse: they adhered to different religions (Hindu, Muslim, Christian, Buddhist, Sikh, and smaller groups such as Biharis and Jains), castes, and classes. Many were vegetarian. Indian immigrants came from different regions of India and had different culinary traditions. Some opened spice stores and restaurants on Lexington Avenue between Twenty-sixth and Twenty-ninth Streets, which was soon called \"Little India.\" Others opened grocery stores in those neighborhoods where Indian immigrants lived. These sold a variety of spices, basmati rice, flours, cans and jars of spicy pickles, sweets, snacks, and fresh vegetables. In the 1990s, Indian restaurants began to open in Flushing, Long Island City, and Jackson Heights, Queens, where many Indian immigrants lived. Some restaurants focused on specific Indian cuisines, such as those from Punjab, Gujarat, Bengal, or South India. Most served regional variations of dishes, such as rice, curries, Indian breads, Indian lassi (a yogurt-based drink), roti, samosas, tamarind sauce, and tandoori chicken. Some Indian restaurants were run by Jews and served kosher Indian food. Others were fast-food operations, called \"curry in a hurry\" outlets. Still other Indians opened pizzerias and other non-Indian fast-food operations. Smaller numbers of Pakistanis and Bangladeshis have followed similar patterns.\n\nThe first significant number of early Arab immigrants began arriving in the 1880s. They hailed initially from Lebanon and Syria, and most were Christian. After 1970, immigration from Arab countries picked up. This second wave of Arab immigrants included many Muslims. In 2010, there were an estimated 370,000 Americans of Arab heritage living in the New York area. The center of the Arab community in the city is a section of Atlantic Avenue in Brooklyn, where Arab restaurants, grocery stores, bakeries, and other shops are concentrated.\n\nArab Americans (along with Jewish Americans) popularized the falafel, which is usually made from fried ground chickpeas, fava beans, or the combination of the two, and other vegetables. These combinations are usually stuffed into pita bread. Falafels and stuffed pita bread have been sold by street vendors in New York for years and have recently become a favorite of many of the city's food trucks.\n\nOne unusual group has been Afghan refugees who began arriving in New York after the Soviet occupation of Afghanistan in 1979. Many have become food vendors; others have opened ethnic restaurants; and still others have opened up fast-food chicken establishments. Kentucky Fried Chicken chain was probably founded by an Afghan immigrant, and many Afghans are owners of these outlets today.\n\nEthiopians and Eritreans arrived in New York beginning in the 1960s, and some opened Ethiopian restaurants. Ethiopians introduced New Yorkers to wat, a thick stew, and injera, a large flatbread made from teff. Eritreans introduced New Yorkers to similar dishes (typically spicy hot) as well as their interpretations of Italian dishes. During the 1980s, Senegalese began immigrating to New York. They were followed in the 1990s by Guineans, Malians, Ghanaians, Nigerians, Ivory Coasters, and peoples from other African countries. Many were street vendors, but others opened grocery stores in Harlem, Clinton Hill, University Heights, Flatbush, and the Bronx. These sold specialty foods imported from West Africa. During the early twenty-first century, entrepreneurs began opening restaurants. Many serve traditional foods of their home countries, such as cheb, mafe, and yassa. Others have gone into the food delivery services, which today is dominated by West Africans.\n\nPeoples from the West Indies (mainly Jamaica, Barbados, Trinidad, and Guyana) arrived in New York after 1964. Many ended up working in restaurants or as street vendors. Their culinary traditions have just begun to influence mainstream New York. Common Caribbean dishes include callaloo (edible leaves of various plants used as greens or in making thick soups), chicken, beef or goat patties, curried goat, fungi, Jamaican patty (pastry that contains various fillings and spices and is often sold, surprisingly, in pizza joints), peas and rice (peas or beans and rice), and roti (a round flat bread); patties and stews are now common foods sold by street vendors and food trucks. Many Caribbean foods are sold in Brooklyn on Labor Day at the Caribbean Day Parade, which claims to be the largest parade in the world with an estimated two to three million people participating and watching.\n\nJamaican-born Cheryl Smith, chef and co-owner of the Global Soul Restaurant in Brooklyn. Courtesy of Kelly Fitzsimmons\n\nSome West Indians, especially Jamaicans, are Rastafarian, members of a quasi-religious group whose dietary precepts incline them to vegetarianism (some Rastas eat fish but not shellfish). The approved foods, called \"ital,\" are pure, natural, and unprocessed; they are supposed to enhance \"Livity,\" or life energy. Consumption of alcohol is strongly discouraged. Rastas have opened restaurants in New York neighborhoods where large numbers of them have settled, including Crown Heights and Bedford-Stuyvesant in Brooklyn, the north Bronx, and around Laurelton and Cambria Heights, Queens.\n\nDuring the 1960s throughout the 1970s, many Haitians emigrated to the United States. An estimated four hundred thousand people of Haitian ancestry now live in New York, especially in Flatbush and East Flatbush in Brooklyn and Springfield Gardens in Queens. Many Haitian Americans work in New York grocery stores, restaurants, bakeries, and bars. Haitian Americans have begun to influence mainstream American foodways through street vendors and small restaurants, mainly in areas where large numbers have migrated.\n\nIt isn't just that immigrant groups have moved into New York and started selling traditional or modified culinary treats from their homelands. The city creates opportunities for groups to interact in ways that they might never have in their points of origin. Groups and individuals have learned from others, and what's frequently offered on restaurant menus or on the family table show signs of these interconnections. Although many restaurants purport to offer \"authentic\" foods and beverages (and some do), it is more common that immigrants have added nontraditional styles and dishes to their culinary repertoires.\nChapter Four\n\nFrom Garden to Grocery\n\nA Tour of Markets, Vendors, and Other Food Sellers\n\nEarly European colonists in New York grew, raised, and prepared much of their own food in their yards and farms. Houses had kitchen gardens and chicken coops. Cows and pigs were often allowed to roam freely. As the city grew, urbanization decreased room for gardening and farming at the tip of Manhattan; farmers and dairymen moved northward on the island or to Long Island or New Jersey, where they supplied the city's expanding population. Days were set for farmers to bring in their goods to sell to city residents. Others imported and sold specialty items, such as spices, sugar, tea, chocolate, coffee, wine, and spirits from Europe and the Caribbean, and other foodstuffs from other North American colonies. Still others, such as butchers and bakers, retailed food directly to city residents. Street vendors, farmers' markets, and grocery stores provided a complex distribution system for a wide variety of foodstuffs that New Yorkers enjoyed\u2014and continue to enjoy today.\n\nStreet Food\n\nStarting during the colonial days, vendors hawked their wares on city streets. They sold an incredible array of products, including many foods and beverages: milk, buns, clams, fish, strawberries, radishes, potatoes, sweet potatoes, onions, muffins, gingerbread, corn, oysters, soda, baked pears, chestnuts, peanuts, tea rusks, and even horseradish, to name just a few. Potatoes were grown in western Connecticut and were brought by sloop to New York City, where they were sold by men pushing wheel barrels for twenty-five to fifty cents per bushel.\n\nAfrican American women and children sold baked pears from glazed earthenware dishes that were filled with homemade syrup. They also vended steaming-hot \"hot corn\" from pails. They were sold in its green husk, which provided a no-expense container. According to a writer in 1808, corn was sold abundantly in the fall. Vendors provided salt to customers, and the hot corn made for \"very pleasant eating.\" To attract buyers, some vendors shouted out, \"Hot corn, hot corn, here's your lily white hot corn; hot corn, all hot; just come out of the boiling pot!\" Others called out,\n\nHot corn, hot corn\u2014\u2014\n\nHere's your lily-white corn;\n\nAll you that's got money\n\n(Poor me that's got none)\n\nCome buy my lily hot corn,\n\nAnd let me go home.\n\nOn Saturdays, vendors flocked to Canal Street west of Broadway, where they sold fish, oysters, and clams, which they had caught or gathered themselves. The \"clam man\" distinctive cry was,\n\nHere's clams, here's clams, here's clams to-day,\n\nThey lately came from Rockaway;\n\nThey're good to roast, they're good to fry,\n\nThey're good to make a clam pot-pie.\n\nHere they go!\n\nWater was crucial for everyday life in the city. Vendors acquired water from springs on the island and brought it twice daily via cart for those in the city. The quality of water was particularly important for brewing tea, so New York City erected a tea-water pump over the spring at Chattam and Roosevelt Streets. Vendors acquired water from this (and later additional) pumps and sold it on the streets, yelling out, \"Tea water! Tea water! Come out and get your tea water!\" To go with the tea were rusks (dried biscuits) that were sold by bakers' boys who yelled out, \"Tea ruk, ruk, ruk, tea ruk.\"\n\nFruit and vegetables were seasonal treats. Cherries were favorites in June and July. Apples were sold in the fall and winter, and as many of the sellers were Irishwomen, they soon acquired the sobriquet \"Apple Marys.\" Peaches sold in the summer. Each vendor had their own cries: \"Penny apiece! bananas and peaches!\" \"Pe-e-e-eaches! Pe-eaches! Fi-i-i-I've cents a quart!\" \"'E-e-ere you are! Delaware peaches, nice 'n' ripe! o-o-o-only five cents a bushel!\"\n\nFarmers who kept cows on the outskirts of the city drove around with milk carts and sold their beverage at from six to ten cents a quart. Others carried pails filled with milk, scooped it out of their large pails, and poured it into containers provided by customers. The pails themselves were open at the top and were exposed to dust and sweat. As immigrants flooded into the city during the nineteenth century, the demand for milk increased. Distributors carried several gallons at a time by employing a wooden yoke \"three feet long chiseled out and smoothed to fit over the shoulders and the back of the neck, with nicely rounded arms extending over the shoulders. A light chain or a rope was suspended from each arm with a hook at the end. With this yolk across his shoulders the carrier stood between two pails or other containers, and by stooping forward attached the hooks to the vessels; then straightening up, the weight of the vessels rested on his shoulders.\" As they walked along the streets, they shouted, \"Here's Milk, Ho!\"\n\nOther vending businesses were based on nuts. Vendors selling chestnuts was a common sight in New York until the native American chestnut tree was wiped out by a fungal disease in the late nineteenth century. Old World chestnuts have been introduced into America, and roasting chestnuts were common on city streets, particularly about Christmas time.\n\nMany sold peanuts, which were low cost, and all sellers had to do was roast them. By 1787, a small trade in peanuts was carried on between Charleston and New York. The unshelled peanuts were sold by a few vendors, who roasted them at their stalls and sold them to passersby. Englishman Henry Wansey enjoyed roasted peanuts on his tour of the United States in 1794. He found them unusual enough to take a supply home to England, where he served them to guests. On January 23, 1803, a letter written by \"Jonathan Oldstyle,\" the pen name of Washington Irving, appeared in New York's Morning Chronicle complaining about the \"crackling of nuts\" at the Theater on Park Row, then New York City's only theater. Irving later reported, presumably humorously, that college students enjoyed \"pea-nuts and beer, after the fatiguing morning studies.\" Peanut shells continued to be a nuisance throughout the nineteenth century.\n\nThe peanut vending business in New York escalated into a big business after the Civil War. In 1867, Thomas Rowland of Norfolk, Virginia, bought peanuts from farmers and shipped a small batch to an Italian merchant, also called a \"pardone,\" who hired Italian immigrants to sell the consignment. The pardone equipped the vendors with a pushcart and a bag of peanuts on credit. Later, this model was expanded into a wide variety of goods. The peddlers were required to sell a certain quantity of goods within a given period of time or their carts were taken away and they were refused further supplies. Vendors returned 25 percent of their sales to the pardone and pocketed the remaining 75 percent. By 1870, several hundred vendors sold food through this system. Jewish and Greek immigrants followed a similar pattern, selling other merchandise, and those three large immigrant groups controlled the vending trade until the 1920s.\n\nA similar vending operation was launched by pecan growers. During the Civil War, many Union soldiers became familiar with this kind of nut while in Louisiana, Texas, Mississippi, and other areas where they grew. After the war, the pecan trade from the South to New York picked up. By 1867, pecans were sold in New York markets from December through April. R. C. Koerber began shipping pecans from Texas to New York City on a large scale in 1884. Three years later, he sold twenty thousand pounds, and in 1890 more than one hundred thousand pounds. Not all were sold by street vendors\u2014they were also supplied to restaurants, hotels, and groceries.\n\nIn 1870, an author claimed in a Harper's magazine article that vendors generated only \"a few shillings' worth\" of sales. However, vending could be a lucrative business. In 1887, a house on Mulberry Street in New York City was sold at auction to an Italian who bid more than twenty-four thousand dollars. Much to the surprise of those at the auction, the buyer was a street vendor who operated a peanut stand on Park Avenue. He paid cash on the spot. Another vendor, John Courtsounis, a Greek immigrant who came to the United States in 1901, sold roasted peanuts on the street outside Columbia University. He made enough money to pay for his sisters and brothers to come to America and to open a restaurant.\n\nThe majority of vendors during the second half of the nineteenth century were Irish, Italian, and Jewish immigrants, but there were also African Americans and even a few Chinese (who sold pineapple candy and cigars). A writer for the New York Tribune reported in 1869 that \"they furnish impromptu dinners and breakfasts for a shilling; prepare oyster-stews while you take out your pocket-book, and bake waffles while you determine the time of day. They dispose of frozen custards and sour milk, sweetened, for ice cream; soda-water without gas; lemonade without lemons.\" A reporter writing for Scribner's Monthly reported in 1870 that street vendors\n\nmainly sold fruit, apples, peaches, chestnuts, candy, cakes and pies, soggy yellow buns, and frightful red tarts, sweetmeat-stand, with its jars and boxes and heaps of many-colored confectionery, sticks and caramels and lozenges, and round, rough cakes, red, white, or yellow, of a cocoa-nut sweet-stuff peculiarly characteristic of New York candy-stands, and \"ice-water, one cent a glass.\" Here is an old man calmly compounding red lemonade in a huge tin pan; and yonder is a soda-water stand with its green marble urn, its company of inverted glasses, and its array of richly colored syrups in bottles generously large. Here is a basket of rough-coated pine-apples, and a burly woman with gold ear-rings converting the imperfect ones into piles of juicy slices, which a bit of ice keeps cool and fresh. There is a long stand with a green heap of uncut water-melons and a tempting array of red-fleshed, black-seeded halves and quarters.\n\nSome vendors targeted juveniles. They dispensed mead, spruce beer, cakes, and ginger pop. Others catered to the needs of first-generation immigrants. For some groups, street food became an important way of retaining their ethnic identity. Pushcart peddlers selling old-country foods played a particularly important role for many immigrants in the late nineteenth century. Still others revised foods from the old country to create popular foods that attracted a large number of New Yorkers. Although there were few Mexican Americans in New York at the time, three foods\u2014tamales, frijoles, and chili con carne\u2014became well known. Chili con carne\u2014a Tex-Mex invention\u2014became a popular street food in New York in the 1880s and was sold in restaurants by the following decade. The popularity of the tamale on the streets of New York was spurred on by the Mexican Food Corporation, which employed white-clad street vendors selling tamales wrapped in corn husks and shouting, \"We are the boys of the hot tamale.\" The success of this tactic was attributed to the exotic nature of the tamale at that time. The upper class considered tamales a \"Bohemian eatable\"; one tea party host invited \"ten Hot Tamale men\" to serve tamales (along with another such Bohemian treat, the Frankfurt sausage) to his guests. Tamales and chili con carne were sold on the street until the early 1920s, when they were no longer exotic, and cheap restaurants began selling them, such as the Chili Villa on West Forty-ninth Street.\n\nFor immigrants, who were familiar with European street vendors, street sales were a logical venture. They did not need to be highly skilled or fluent in English; many immigrant vendors operated in areas dominated by their own language group, spoke the language of their customers, and sold foods that immigrants preferred. There were plenty of opportunities\u2014legal and otherwise\u2014to sell food on the streets of New York. More important, the city's traditional system of supplying residents with food\u2014that is, retail grocery stores and public markets\u2014were hard pressed to meet the challenge of feeding the city's rapidly growing population, which increased from 391,000 in 1840 to 2.5 million in 1890.\n\nNot everyone was impressed with the vendors or their cries. A reporter in 1855 referred to them as\n\nruffians who, in the most dirty way, peddle oysters from an old wagon at one cent each. Their furniture consists of stentorian lungs, from which the most ear-splitting cries disturb the peace of every street and the temper of all the denizens, a pail of nasty water, a soda-water bottle of vinegar and another of a ferocious compound called pepper-sauce, and a box of salt, pepper, and street-dust mixed. Buying and selling only the cheapest oysters in the dirtiest way, they offer many spoiled ones\u2014very likely to be productive of disease, and otherwise engender and minister to ill-health.\n\nCity officials tried to regulate street vendors beginning in 1691 when a city ordinance forbade the vendors from operating until two hours after the opening of the city's public markets. In 1707, the city tried to completely ban all vendors, but it didn't work. This wasn't much of a problem until the mid-nineteenth century when immigrants began to flood into the city. In the late nineteenth century, New York City officials instituted a law requiring pushcart vendors to relocate every thirty minutes. Most vendors ignored the law except when a policeman was sighted, in which case they moved their pushcarts in a circle and repositioned them within a few feet of where they had been. The law was difficult to enforce in the crowded neighborhoods filled with recently arrived immigrants, such as the Lower East Side. The law was rescinded in 1915.\n\nAnother solution was to establish open-air pushcart markets, which forced vendors into specific areas. The first was established on Hester Street in 1886. The most famous and popular open-air pushcart venue was Paddy's Market. It stretched on both sides of Ninth Avenue between Thirty-fifth and Forty-second Streets with sometimes as many as two hundred pushcarts. The Irish, Italian, French, Middle Eastern, and Greek vendors sold produce, breads, meat, fish, poultry, spices, coffees, and teas. Many more open-air markets were established, but they did not prevent other vendors from operating on busy streets.\n\nYet another solution was licensing vendors. The City Bureau of Licenses was created, and vendors were required to obtain licenses. This was strongly supported by food retailers, such as grocery stores and restaurants, who believed that street vendors took business away from them and that streets and sidewalks filled with vendors discouraged their customers. Rather than complain about the competition, owners argued that the congestion caused by street vendors made it difficult for emergency vehicles, such as fire engines, to navigate the city.\n\nExpos\u00e9s about the food vendors sold frequently appeared in newspapers and magazines. Vendors acquired fruit and vegetables, claimed one, from the refuse from docks, railroad yards, auctions, and wholesale markets, where broken boxes and rotten food were discarded. The fruit was small and often defective, and the food was sold in unsanitary conditions, at least so said those writing expos\u00e9s.\n\nThe continuous clamor generated pressure for the city to study vendor problems. A Pushcart Commission was appointed in 1906. They found that Jewish, Italians, and Greek immigrants made up 97 percent of more than four thousand legal pushcart vendors. The Italians sold fruit and vegetables; the Greeks, fruit and ice cream; and the Jews, the largest group of vendors, sold a much larger, more diverse array of foods and beverages. These vendors mainly operated on the streets and sidewalks of Manhattan and in the Jewish neighborhoods of Brooklyn, such as Brownsville. The commission found that many pushcart vendors often had to bribe police, other city officials, and store owners in order to operate. If payoffs were not made, arrests often followed. In 1904 alone, more than five thousand vendors were arrested for obstructing sidewalks and streets and for failure to move their stands every thirty minutes as then required by city ordinance. Some were unusual arrests\u2014like the one where it was said that a whistle on a peanut cart bothered a local church. Little evidence was found of health hazards with the food sold by the vendors\u2014in fact, the commission found fewer problems with street vendors than with grocery stores: vendors had no ability to store food and therefore had to buy small quantities of fruit and vegetables daily, while grocery stores usually bought large quantities and kept them for longer periods of time. Contrary to what merchants claimed, the commission found that open-air vendor markets increased business for local stores.\n\nThe commission offered few solutions to the problem of congestion on the streets and sidewalks of New York, and during the next few years, the number of vendors increased dramatically. In 1913, the city designated vending areas, such as streets with access to the Manhattan, Williamsburg, and Brooklyn Bridges in hopes of reducing the number of vendors in Manhattan.\n\nThe U.S. Department of Agriculture (USDA) examined New York vendors in 1925. It found that the ethnic composition of the vendors had changed: 63 percent were Jewish, mainly immigrants from Eastern Europe; 32 percent were Italian; and the remaining were German, Irish, African Americans, and others. An estimated one million Manhattanites were dependent on these vendors for all or part of their food. The Italian vendors usually sold peppers, cheese, garlic, nuts, and olives; Jewish vendors typically sold fish, onions, potatoes, cabbage, carrots, pickles, soda, halvah, knishes, chestnuts, pretzels, and ice cream. The report also found few sanitary problems with New York City street vendors\u2014in fact, vendors received fewer violations from the Health Department than did grocery stores and stalls at large farmer's markets. The main exception, according to the report, was the fish sold by street vendors, which was, according to the report, a \"menace to health.\" The report recommended that sanitary conditions be strictly enforced to protect food from flies, dust, and insects. This was especially a problem for those foods that were typically not washed after purchase, such as candy, bread stuffs, pretzels, dried figs, and unshelled nuts.\n\nA more serious problem for vendors was sparked by the arrival of automobiles in the city. Vendors who operated in the streets caused traffic delays, especially during rush hours. In 1918, the city declined to issue more licenses, and it began to strictly enforce existing laws; many vendors went into other occupations or moved to other cities. The number of legal street vendors dropped from almost 6,000 to 2,383 in 1920. The city established the Department of Public Markets to give more supervision to vendors. During the 1920s, reporters lamented the loss of the city's iconic street vendors, such as those selling hot tamales, apples, pretzels, hot dogs, and hot corn. Enforcement was one reason for the decline; a sharp decline in immigration was another: Congress passed laws restricting immigration, and in 1924, it virtually ended all legal immigration.\n\nOne of the new hot spots in the street food world in the 1920s was orange juice. Orange growers in California and later Florida launched national campaigns to encourage Americans to drink orange juice. They targeted one campaign directly at children, whom they encouraged to open juice stands. Orange juice took off when vitamin C was popularized, and orange growers convinced Americans that they needed orange juice every day. On New York streets, vendors acquired oranges and juiced them while customers waited for the fresh drink. New Yorkers Robert T. Neely and Orville A. Dickinson thought that selling orange juice was a great idea. They formed Nedick's Orange Juice Company in 1913. Named for the first three syllables of their last names, it later expanded into a fast chain that sold a wide variety of foods and beverages along with their iconic orange drink. The Nedicks chain folded in the 1980s, but vendors continued to sell fresh squeezed orange juice until the price of oranges escalated to the point where a small cup of fresh squeezed orange juice cost more than customers were willing to pay for a street beverage.\n\nDespite the laws and complaints, the number of street vendors exploded when the Depression hit. In 1930, it was estimated that there were 7,900 street vendors on city streets, and as the Depression deepened, many more prowled the streets. Vendors, many of whom were not licensed, clogged the streets and often sold defective products and preyed on unsuspecting tourists, giving the city a bad name. When New York City was selected as the site for the 1939 World's Fair, Mayor Fiorello LaGuardia decided that the streets needed to be cleaned up. He created indoor markets where vendors were located. This worked for a while\u2014until vendors took once again to the streets.\n\nChanges in immigration patterns again altered the ethnic composition of street vendors, their offerings, and the types of foods that New Yorkers eat. The Immigration and Nationality Act amendments passed Congress in 1965, and the Immigration and Reform Act, passed in 1986, lessened restrictions on immigration from Asia, particularly China, the Caribbean, Central America, and Africa.\n\nSucceeding mayors tried to restrict them by making vendors acquire a permit from the health department and pay retail tax. In their application they had to provide a list of food to be served and agree to pay retail tax. As the number of permit seekers exceeded the number of available permits, they were distributed via a lottery, a practice that continues today. During the 1990s, Mayor Rudy Giuliani tried to close more than one hundred streets to vendors and tried, unsuccessfully, to force vendors into pushcart markets. Other restrictions include that carts must be at least ten feet from the crosswalk and twenty feet from any doorways.\n\nPushcart vendors provided the less affluent the opportunity to sample different ethnic foods, and to this day immigrant vendors still offer foods like \"coco helado\" (water ices) and button-sized Hong Kong\u2013style teacakes on the streets of New York.\n\nMotorized vending vehicles plied New York City streets as early as the 1920s. Good Humor ice-cream trucks lured children with their sleigh bells, and beginning in the 1950s, soft-serve ice-cream vans, such as Mister Softee, rolled along or parked near schools and playgrounds, their siren song a maddeningly repetitious recorded jingle. Beginning in the late 1990s, food trucks began to proliferate in the city. Some of the pioneers of this resurgence sold coffee, desserts, or both, but they were soon joined by young entrepreneurs offering an array of ethnic specialties: Chinese dumplings, Belgian waffles, wienerschnitzel, fish tacos, and Margherita pizza. Coming full circle, artisan ice-cream makers now park their sleek vans in upscale neighborhoods and sell their premium treats in respectful silence.\n\nStreet Vendors in Manhattan, 2013. Courtesy of Kelly Fitzsimmons\n\nDuring the last decade, food trucks have become fixtures on New York City streets. There are even some areas designated for food trucks. In 2010, when Central Park's Tavern on the Green closed, selected food trucks were permitted to lease space behind the building; they were even granted liquor licenses. With just a fraction of the overhead costs required by a brick-and-mortar restaurant, a food truck also has the advantage of mobility. The operator can go where people are known to gather, putting word out via Twitter and Facebook to inform customers of that day's location. Some city restaurateurs, though, feel that food trucks are taking business away from them and have tried to restrict their growth through increased regulation. The New York City Food Truck Association (NYCFTA) was formed in January 2011 to represent food truck operators. In the same year, Zagat's New York Restaurant Guide began to rate food trucks.\n\nFood truck in Manhattan, 2013. Courtesy of Kelly Fitzsimmons\n\nPublic Markets\n\nFor more than two hundred years, public markets were the primary retail and wholesale food sources for most New Yorkers. Financed and regulated by the city, public markets were originally scheduled events held in fields. Market days made it possible for city residents to buy goods carted in by local farmers. Hogs and cattle were so numerous in New Amsterdam by 1641 that city officials set specific days for livestock to be driven into the city and sold to other farmers or city residents.\n\nThe first public market was established in 1648, when Peter Stuyvesant created an outdoor market between Whitehall and Broad Streets on what was then the East River. It was initially scheduled for Mondays, but in 1656, it was changed to Saturday, when \"the people from the country bring various wares, such as meat, bacon, butter, cheese, turnips, roots, straw, and other products of the farm to this City for sale.\" Two years later, the first meat market was opened; rather than selling live animals, it sold butchered meat.\n\nThe first market house was constructed in 1676. It was located in what was then a shallow valley at Maiden Lane and the East River. As it was located in a wetland, the Dutch called it Vlaie, meaning swamp. The name was later Anglicized, and the place became known as the \"Fly market.\" By 1680, sloops transported food down the Hudson as well as across the East River from Long Island and across the Hudson from New Jersey.\n\nGoods were also sold through public auctions, or \"vendues.\" As an English visitor, Sarah Knight, reported in 1704, \"They have Vendues very frequently, and make their Earnings very well by them, for they treat with good Liquor Liberally, and the Customers Drink as Liberally and generally pay for't as well, by paying for that which they Bidd up Briskly for, after the sack has gone plentifully about, tho' sometimes good penny worths are got there.\"\n\nDuring the early eighteenth century, New York sported five markets along the East River at the ends of streets. These were originally open-air markets, but large buildings were eventually constructed. The market hall on Broad Street housed vendors as well as serving as a location for balls, dinners, and meetings. The markets were officially open two days a week, although some vendors in them sold goods every day except Sunday.\n\nDepending on the season, poultry and game, including wild geese, ducks, and other waterfowl, were plentiful. In the spring and autumn, pigeons were so abundant that, in 1754, one penny would purchase six. All meat, fish, and fruit was sold through public markets until 1763, when a poulterer set up shop near City Hall. Milk was the exception: it was sold almost exclusively by street vendors. Markets were well supplied, mainly by farmers in New Jersey and on Long Island. City authorities inspected the markets, and if they found vendors selling underweight products or impure food, the vendors were fined and food was occasionally destroyed. On the eve of the American Revolution, an English visitor reported that the city's four markets were well provisioned.\n\nTwo markets were abandoned after the American Revolution, but three new ones were constructed along the East River. Farmers brought their products to these markets on specified days, and their customers\u2014housewives, employees, servants, or slaves\u2014picked up what was needed. By 1807, the city's markets were closed on Sundays but open the other six days. The markets were abundantly supplied in season with an incredible diversity of game, meat, and seafood. They sold eight game animals (bear, deer, raccoon, groundhog, opossum, squirrel, rabbit, and hare)\u2014some dead, others alive; five amphibious creatures, such as turtles; fourteen different shellfish; fifty-one different species of birds; and sixty-two different types of fish. Hucksters in the market sold vegetables and fruit. The novelist James Fenimore Cooper accurately proclaimed in 1828 that \"it is difficult to name fish, fowl, or beast that is not, either in its proper person, or in some species nearly allied to it, to be obtained in the markets of New-York.\"\n\nUnusual game remained an important part of the city's public markets for decades. Thomas DeVoe, a butcher at the Jefferson Market, reported in his book The Market Assistant (1867), that \"every item of human food [is] sold in the Public Markets,\" including venison, elk, caribou, reindeer, black-tailed deer, mule deer, moose, antelope, pronghorn antelope, bighorn sheep, rocky mountain goat, hares, rabbits, guinea pigs, squirrels, black bears, raccoons, lynx, opossums, groundhogs, porcupines, skunks, beavers, otters, badgers, muskrats, and many, many more.\n\nVendors and wagons typically arrived at the markets at 3:00 a.m. on weekdays but 2:00 a.m. on Saturday, the busiest market day. In some cases, the farmers had begun their journeys the previous afternoon, traveling up to thirty miles from Long Island or New Jersey and then waiting for the ferry across the East River or the Hudson. By 7:00 a.m., the markets were sufficiently stocked, but supplies continued to arrive for the next few hours, creating massive traffic jams as wagons waited to unload their produce and customers fought to enter, shop, and then leave with their purchases.\n\nThe earliest customers at the markets were those who shopped for hotels and restaurants, such as the Fifth Avenue, St. Nicholas, Brevoort, Metropolitan, Astor, Hoffman, St. James, and Delmonico's. They selected the \"choicest beef, the fattest mutton, the freshest cutlets, the earliest fruits and vegetables; for the reputation and patronage of their houses depend upon the excellence of their table,\" proclaimed a New York reporter in 1869. After these buyers had made their rounds, prices dropped somewhat. Then came shoppers from well-to-do and middle-class households, who bought the meat, fish, fruits, and vegetables required for the coming week. After 2:00 p.m., prices dropped steeply as stall owners tried to clear out their inventory; the customers at this hour were the proprietors of boardinghouses, street vendors, and others more focused on price than on quality.\n\nAs the city's population grew, more public markets were built. By 1867, there were eleven in the city; the two largest and most important were Washington and Fulton Markets. Washington Market was opened in 1812 on the west side of lower Manhattan, on Washington Street between Fulton and Vesey Streets, and later expanded to the surrounding area. In 1867, it consisted of a series of rectangular buildings constructed around a circular building with a small cupola on top. The market, which included many stalls, stores, and extensive warehouses, was touted as the largest market in the United States\u2014some said the largest in the world. And it was indeed massive, with more than five hundred stalls and three thousand vendors. About thirty thousand people were estimated to earn their livelihood there. In a time before refrigeration, New Yorkers had to buy their food frequently. The Washington Market alone served about one hundred thousand customers daily during a busy season, such as Christmas. Customers came from all over the region to purchase their groceries. It was estimated that Washington Market provided for at least two million people. In addition, foods were packaged at Washington Market and transported to other states, including California.\n\nGeneral view of Washington Market, 1877. From \"How New York is Fed,\" Scribner's Monthly 14 (October 1877): 729. Courtesy of Kelly Fitzsimmons\n\nWashington Market sold a wide variety of produce. Early in the season\u2014in April and May\u2014fruits and vegetables came from Bermuda and the Carolinas. In the high season for local produce\u2014late summer and early fall\u2014more than six hundred wagons from New Jersey, Long Island, and western New York came to the market daily with fresh farm products. The most important fruit sold in the markets was the apple, which arrived from upstate New York at the beginning of October and disappeared by the end of November. During these two months, New Yorkers purchased half a million barrels (each containing about 225 apples), and most of these were sold through the public markets or by street vendors. The second most important fruit was the peach, which came in from New Jersey, Delaware, and Maryland. During their two-month season, an estimated twenty thousand bushels of peaches were sold daily. Other fruit with significant sales were pears, grapes, and berries.\n\nWashington Market was known for its meat, poultry, and game, and observers marveled at the variety and quantity. In 1828, an observer reported seeing canvasback ducks from the Susquehanna; venison from New Jersey, Long Island, and the Catskill mountains; grouse from Hempstead, on Long Island; snipe from the Newark meadows; and partridges from Bull Hill along the Hudson River. An 1839 visitor to the market found \"three hundred head of deer, with quantities of bear, rackoons [sic], wild turkies [sic], geese, ducks, and every variety of bird in countless profusion.\" Once railroads connected New York with the Midwest, in the 1850s, meat could be transported from as far away as Kansas and Illinois. \"The best beef, mutton, veal and lamb the country affords are displayed upon the stalls,\" wrote a marketgoer in 1869. \"Those roasts and steaks, those hind-quarters, those cutlets, those breasts with luscious sweetbreads, would make an Englishman hungry as he rose from the table.\"\n\nFulton Market, on the east side of Manhattan, was established in 1820. Initially, it sold a wide variety of goods. As one observer reported, it was\n\na butcher's store, a fruiterer's stall, an oyster-counter, a coffee shop, a poultry-yard, and a fish-monger's establishment. It is every thing in one\u2014a magnum not in parvo, but a magnum in magna. It is one vast repository for the sale of every article of diet you could fancy from a lamb-chop up to a \"steak for two,\" from a shrimp up to a lobster, from a cup of coffee up to the largest table d'hote fare you could pick out.\n\nBut Fulton Market was best known for its fish and seafood. An 1828 visitor found a profusion of cod from Newfoundland; lobster, catfish, flounder, eel, mussels, and clams from the Long Island Sound; and softshell crabs from Baltimore. The oysters came from Blue Point, Long Island; the York River and Chincoteague Island in Virginia; and Chingarora in Raritan Bay, New Jersey. By 1853, it was estimated that there were more than a thousand vessels engaged in bringing in oysters to the New York markets.\n\nExterior of Fulton Fish Market. Courtesy of the Library of Congress. Reproduction Number: LC-DIG-ppmsca-12798\n\nA separate fish market was constructed at Fulton Market in the 1831; right on the docks, it could receive fresh seafood straight off the boats. From the Fulton Fish Market, fish and seafood were packed and shipped out via railroads and ferries to a radius of four hundred miles. By the 1860s, two hundred schooners and smacks plus smaller oyster boats supplied the Fulton Market with \"cod, halibut, haddock, herring, mackerel, blackfish, bluefish, smelt, weak fish, and white fish, eels, porgees, sea bass, striped bass, trout, sturgeon, sheep's head, flounders, and many others too numerous to mention.\" Fish and seafood were brought in from a five-hundred-mile radius around the city. In addition, freshwater fish from the Great Lakes were shipped to the city via the Erie Canal and railroads. During the \"R\" months, some 250,000 oysters were eaten every day in New York, and about 25,000 daily during the remaining months of the year. Most came through the Fulton Market.\n\nThe Fulton Fish Market thrived both as a wholesale outlet serving restaurants, hotels, and fishmongers and as a retail operation, selling directly to New Yorkers. Some stalls sold seafood to be eaten on the spot; Dorlon's oyster bar, one of the more popular, served patrons well into the night. It was started in the 1840s by Alfred P. Dorlon, who worked at his father's stall wholesaling oysters. Dorlon and his brother later opened an oyster stall of their own and placed a few tables out front, where people could eat oysters. Over the next ten years, Dorlon's expanded to become one of New York's most fashionable oyster bars. It was described in the late 1860s as plainly furnished, without tablecloths or carpets. As to the patrons, \"No opera, soiree, fashionable palace, can boast of a more fashionably dressed and distinguished company. Fastidious ladies, who at home dwell in splendid boudoirs and sit in perfumed chambers, take Dorlan's [sic] on their way from the opera, for a stew or a saddle-rock roast.\" The \"men of wealth, and women of society; fastidious scholars, and authors of renown\" who ate at Dorlon's had to stand in line waiting their turn. The business survived until 1888, when the Fulton Fish Market became seedy and well-to-do customers took their business elsewhere. Also, the oysters in New York Harbor and Long Island Sound were overharvested, and the price of oysters skyrocketed.\n\nAs early as 1857, New York City newspaper editorials advocated closing downtown markets. Their arguments initially focused on congestion: if the markets closed, thousands of carts and wagons would be removed from the streets, permitting other businesses to move in. During the following decade, more newspapers and magazines criticized the chaos, disorder, shabbiness, and filthiness of the city's markets. The buildings were in disrepair, and customers complained that vendors were not only rude but also guilty of fraudulent practices and that other crime was rampant in the markets.\n\nThomas DeVoe became the city's superintendent of markets in 1871. He was well aware of the problems confronting public markets, and he energetically launched reforms. New buildings were constructed at some markets, while others were renovated. But it wasn't enough. Demographic trends were against the existing public markets. Most were located at the southern end of Manhattan, but during the mid-nineteenth century, prosperous New Yorkers moved northward on the island, as immigrants and the poor moved into lower Manhattan. The immigrants were better served by pushcart vendors who charged lower prices than market stallholders. The well-to-do continued to send their servants down to fetch food from the markets, but as grocery stores, butcher shops, and specialty stores moved uptown, it became easier to shop locally.\n\nThere were other reasons for the decline of particular public markets. One reason for the success of the Fulton Market had been its proximity to the Brooklyn ferry terminal. Brooklynites who worked or shopped in Manhattan passed by or through the market. After the completion of the Brooklyn Bridge, in 1883, foot traffic dropped off because the bridge exit was blocks away from the market. Shops and grocery stores opened around the access roads leading to the bridge. The Manhattan and Williamsburg Bridges took even more customers away from Fulton Street. By 1912, the only market surviving on the site was the wholesale fish market.\n\nPublic markets stopped generating profits for the city and gradually became a liability. In 1899, Deputy Controller Edgar J. Levey said that it was time for the city to \"go out of the public market business.\" Several markets were closed, but the Washington Market survived; it was rebuilt in 1884 and renovated twice during the early twentieth century. Its retail building was finally closed in 1956, and the wholesale operation survived a few more years. The block on which the market stood was eventually incorporated into the World Trade Center.\n\nInterior of Public Market New York City, 1948. Courtesy of the Library of Congress. Reproduction Number: LC-DIG-ppmsca-12798\n\nAs public markets closed, the city needed a wholesale market to supply restaurants, hotels, and grocery stores with fresh produce, meat, and other products. In 1962, ground was broken for the New York City Terminal Market at Hunt's Point, in the Bronx. The site offered easy access to rail, truck, and ship transport without disrupting traffic and was easy for buyers to reach as well.\n\nThe Fulton Fish Market survived through the twentieth century. Additional buildings were constructed in 1936, and one building was replaced in 1939. But it was downhill from there. In the 1970s, fishing boats ceased landing their catches at the market, refrigerated trucks becoming the best mode of delivery. In 2005, however, the Fulton Fish Market was relocated to Hunt's Point.\n\nGreenmarkets\n\nDuring his tenure as mayor in the 1950s and early 1960s, Robert F. Wagner supported a farmer's market at City Hall that featured the eight \"Miracle Gardens,\" which were developed on formerly litter-strewn vacant plots around the city. In 1970, the privately funded Council on the Environment of New York City was created as a policy organization to write reports about environmental issues in the city. It evolved into an agency that actively supported worthwhile efforts to improve the environmental quality of the city.\n\nIn October 1973, John L. Hess, a columnist for the New York Times, commended the city of Syracuse for permitting farmers to sell their goods on one city street every Tuesday. This article was read by Barry Benepe, a forty-eight-year-old urban planner, who convinced one of his colleagues, Bob Lewis, that New York City should have a farmers' market as well. With support from the Council on the Environment of New York City, they worked for three years, acquiring the necessary permits, locating a site, and convincing farmers to give the new market a try.\n\nIn July 1976, their efforts bore fruit when the Greenmarket opened on a dirt lot surrounded by a fence on Second Avenue and Fifty-ninth Street. Seven plant, vegetable, and fruit growers trucked in their farm produce from Long Island, New Jersey, and the Hudson Valley on Saturdays. Television news picked up the story and gave the Greenmarket visibility. A few months after it opened, the city convinced Benepe and Lewis to move the operation downtown to Union Square. At the time, the parking lot surrounding Union Square Park was a hotbed of drug dealing. Around the square, businesses were faltering and stores were closing. With the city's support, the Greenmarket thrived, and the neighborhood followed suit, becoming a center for upscale restaurants and other businesses.\n\nThe city's second Greenmarket opened in Brooklyn, at the intersection of Atlantic and Flatbush Avenues; others followed elsewhere in the city. Today, there are fifty-four farmers' markets serving all five boroughs. Some are seasonal, open only during the growing season, while others operate all year round. Some markets are open two days a week, others just one; the Union Square Greenmarket operates four days a week throughout the year.\n\nThe Council on the Environment of New York City changed its name to GrowNYC in 2007. It supports the city's Greenmarkets, builds community gardens, and encourages recycling. In 2011, GrowNYC initiated a new program, placing receptacles at Greenmarkets to collect produce scraps for composting. The compost is used on urban farms and garden projects throughout the city.\n\nGrowNYC, a recycling program for vegetable matter, at the Union Square Greenmarket, 2013. Courtesy of Kelly Fitzsimmons\n\nRetailing\n\nRetailing of food developed slowly in the city over the course of the seventeenth century. During the Dutch period, the West India Company maintained a store that sold goods, including imported food and beverages, to the colonists. Ships entering the harbor often sold their goods right off the wharf or moved their goods to warehouses, where they were sold directly to customers.\n\nBreweries and bakeries wholesaled their products to taverns but also sold directly to customers. From the earliest days, city government tried to regulate these artisans and require that they sell their goods at reasonable prices. Bread was regulated after 1649, and at least some bakers were charged with violating regulations. The content and weight of breadstuffs were monitored, as were the prices that bakers could charge. Since the cost of ingredients and the mandated minimum weights of certain items meant that sometimes the baker could not make a profit, so bakers figured out ways to trim their costs by using cheap grain and fillers and selling bread that was under the legal weight. In 1681, baker David Provoost was brought up on charges for \"having bad breed not fitt for Sale.\" He was let off with a warning, but other prosecutions followed.\n\nNew York law required butchers to operate through public markets, but in the 1840s butchers began to open freestanding stores. As many New Yorkers preferred buying their meat and poultry from a shop rather than trying to navigate the large, overcrowded public markets, the law was changed. Shops specializing in fish and seafood soon opened in the city as well.\n\nFrom colonial times, small stores sold specialty goods, such as spices, tea, chocolate, or coffee, that were not typically sold by street vendors or in public markets. Small stores emerged in the mid-eighteenth century, selling imported products\u2014spices, coffee, tea, chocolate, pickles, fresh fruit (oranges, lemons, and mangoes), figs, cooking oil, olives, ketchup, capers, herrings, and walnuts, as well as pickled anchovies, lobsters, and oysters. Confectioners sold sugar plumbs, sugar candy, \"Jarr Raisins and Cask ditto; Currants, Figgs and Prunes; Almonds in the Shell; Cloves, Mace, Cinnamon and Nutmeg; Ginger, Black Pepper and Allspice; Dry Citron by the Box or smaller quantity; West India Sweetmeats of all Sorts; Preserves of all Sorts, such as Currants, Jellys, Quinces, Grapes, Strawberries, Raspberries, Damsons, Peaches, Plumbs and sundry other sorts,\" noted an eighteenth century advertisement.\n\nGrocery stores selling both perishable and nonperishable foods appeared in the late eighteenth century. By 1817, the city had an estimated 1,500 family-owned grocery stores, many owned by Irish immigrants. Most sold wine or spirits, particularly rum, which is where they made their profits. They were often located in the basements or attics of boardinghouses or tenement buildings, and they served as gathering places for immigrants to meet and socialize. By the mid-nineteenth century, family-owned grocery stores far outnumbered the stalls in public markets.\n\nSpecialty stores focused on particular products, such as imported tea. The Great Atlantic Tea Company was founded by George F. Gilman and George H. Hartford around 1860. It was similar to other retail stores of the time. Gilman, the owner of a leather company, also dabbled in shipping interests. He imported small consignments of tea, which he sold to wholesalers. Hartford was his assistant. Hartford convinced Gilman that they would make more money if they cut out middlemen and retailed the tea directly to the public. Gilman and Hartford opened a retail tea shop in New York City and also began selling tea by mail; they eventually named their business the Great American Tea Company. Sales were good enough that Gilman and Hartford started buying tea in larger quantities, which enabled them to cut their expenses and set their retail prices even lower, underselling other grocers. They expanded their line to include coffee and other luxury products.\n\nBy 1865, the partners had five small stores in New York City, thus creating America's first grocery chain. The completion of the transcontinental railroad in 1869 made it possible for the company to receive shipments of tea and other specialty goods from Asia via San Francisco and then ship them by train throughout the country. Hartford and Gilman began to open stores across the nation, and to reflect the newly bicoastal operation, they changed their firm's name to the Great Atlantic & Pacific Tea Company, subsequently shortened to A&P. Over the next few decades, the A&P stores gradually augmented their inventory to include a full range of groceries.\n\nThe early success of the A&P was condemned by independent grocers, who attacked the company for its predatory pricing and the resulting destruction of small family-owned businesses. Despite such criticism\u2014or perhaps because of it\u2014A&P built more stores and expanded their product lines. It quickly became America's largest grocery chain as it opened stores throughout the United States. By 1925, A&P had more than fourteen thousand stores across the United States.\n\nOther entrepreneurs, impressed by A&P's early success, started their own chains. The Jones Brothers Tea Company was founded in Scranton, Pennsylvania, in 1872 by Cyrus, Frank, and Charles Jones. When the Jones brothers expanded their chain to New York, they changed its name to Grand Union Tea Company, soon shortened to \"Grand Union.\" In 1883, Bernard H. Kroger began a chain-store operation in Cincinnati. Two German immigrants, brothers Charles and Diedrich Gristede, opened their first grocery store in New York in 1891; Gristede's grew into an upscale chain with stores in New York's wealthier neighborhoods.\n\nItalian immigrant Luigi Balduchi sold bananas in Greenpoint from 1915 until 1925 when he and his wife Maria opened a produce stand there. Twenty years later, they moved to Greenwich Avenue in Manhattan and eventually sold upscale meat, fish, bakery goods, and other products. By the late twentieth century, it had expanded into a small chain that proclaimed itself to be the premier destination for food lovers.\n\nAnother Italian immigrant Nicholas D'Agostino and his brother launched a market by Italian immigrants in 1932 on Eighty-third Street and Lexington Avenue. They opened additional stores, and today the company operates thirteen supermarkets in New York City. Nathan Glickberg opened a small fruit and vegetable stand on West Seventy-fourth Street in Manhattan in 1933. In 1954, he moved his operation to Broadway and renamed it Fairway Market. The company expanded during the 1980s but ran into stiff competition. Today, it is a small chain with five stores in the New York metropolitan area that concentrate on specialty items. One of Fairway's competitors was started by Mike Citarella as a fish market in Harlem in 1912. In 1941, he moved the store to Columbus Avenue. In the 1980s, the company expanded its offerings to include specialty meat, prepared foods, pastry and produce, and other gourmet food. Another upscale specialty food retailer was launched by Giorgio DeLuca and Joel Dean in Soho in 1977. Nine years later, Dean & DeLuca moved to Broadway and Prince Street. It has opened additional stores in New York and other sites.\n\nSupermarkets\n\nIn 1930, Michael J. Cullen, the son of Irish immigrants, opened a grocery store he named \"King Kullen\" on Jamaica Avenue in Queens. He stocked more than a thousand basic food products at prices that undercut those at local markets. There was also free parking in lots adjacent to the store, drawing in customers in cars. Encouraged by the success of the first location, Cullen quickly opened more stores, and by 1936 fifteen King Kullen stores served Queens, the Bronx, and Long Island. King Kullen undercut the prices at traditional chain and independent grocers by at least 10 to 15 percent, making up the lost profit through high sales volume. In the New York metropolitan area, supermarkets generated annual sales in excess of fifteen million dollars during the Depression.\n\nThe Depression hit just as Cullen debuted his first King Kullen store, and it helped make his business model a success. Price became the number 1 factor for consumers in deciding where to shop. Beginning in the mid-1930s, many grocery chains closed their smaller stores and opened supermarkets. Independent grocers were unable to follow suit, however, and thousands went out of business in New York City during the late 1930s and early 1940s. World War II slowed the growth of supermarkets, as wartime restrictions halted the construction of stores, foods were rationed, and many food processors concentrated on military contracts. Before the war, almost all grocery store clerks had been young men; as they enlisted and left their jobs, women filled their places. When the war ended, women continued to work in supermarkets and made up a majority of their employees.\n\nSupermarkets thrived in New York during the 1950s and 1960s. A&P remained America's largest food retailer until 1975, when it ran into financial reversals, as did other large chains in the city. This decline was largely due to increased competition in the city. Small grocery stores, bodegas, and delis cut into supermarket business, as did larger supermarkets outside of the city that were accessible by car.\n\nDuring the early 1970s, alternative food-distribution systems emerged, such as food coops and community-supported agriculture (CSA). In 1976, the first Greenmarket was established in Union Square, selling foods grown, raised, and prepared locally, which was defined as within two hundred miles of the city. Eventually some 120 more Greenmarkets opened around the city, and these, too, reduced sales at city supermarkets. Specialty grocery store chains, such as the Gourmet Garage, the Garden of Eden, and Union Market, have attracted a wide following in the city. Fresh Direct, an online grocery service, opened in 2002. In 2005, Whole Foods, a supermarket chain that emphasizes \"natural and organic products,\" opened a store on Union Square and subsequently other stores elsewhere the city.\n\nDelicatessens\n\nYet another retail operation is the delicatessen. German immigrants imported European specialty foods, such as European wines and brandies, Westphalia ham, Magdeburg sauerkraut, Gruy\u00e8re cheese, German poppy-seed oil, pasta from Italy, and fish from the Netherlands. These items were sold directly to restaurants and delicatessens. According to Artemas Ward, chronicler of the nineteenth-century grocery trade, the first delicatessen opened on Grand Street in New York City around 1868. Delicatessens were patronized by German Americans, of course, but they also attracted passersby who were enticed by the window displays of foreign and \"fancy\" foods. Delicatessens sold cooked, ready-to-eat meats, sausages, poultry, and fish, as well as cheeses, teas, mushrooms, caviar, olive oil, pickles, and other imported goods.\n\nDelicatessens filled the gap between butcher shops, which mainly sold uncooked meats, and general grocery stores, which offered bulk and packaged goods and liquor. Originally, deli customers purchased the prepared food products and took them elsewhere to eat, but as delicatessens thrived and expanded, some made a place for customers to sit down and eat in the store, particularly at lunchtime.\n\nDelicatessens remain an important cultural component of New York, but the term is used much more loosely today. As other new groups of immigrants\u2014Italians, Greeks, Puerto Ricans, West Indians, Chinese, and Koreans\u2014moved into the city, many found that \"deli\" ownership proved an excellent entry point. While some delis today still specialize in cured meats, pickled fish, and other Old World specialties, most sell a general line of packaged and prepared foods. Deli owners from newer immigrant groups, though, often include their native favorites; for instance, Korean American\u2013owned delis frequently sell kimchi and a few other Asian delicacies, while West Indian proprietors offer savory turnovers called \"patties.\"\n\nToday, New York's marketers and food retailers offer home cooks and restaurateurs a wide selection of local and imported produce, meats, fish, seafood, spices, poultry, grains, and prepared foods. Most are available year round, although the quality often depends on the season and the food miles that the products must travel to reach the city. Likewise, New Yorkers have a vast choice when it comes to beverages to which we now turn.\nChapter Five\n\nDrinking in the City\n\nDuring New York's colonial period, most people quenched their thirst with plain water. Clean water was plentiful in springs, lakes, and rivers throughout much of Manhattan, Staten Island, Long Island, and the Bronx, but in New Amsterdam, at the tip of Manhattan, the fresh water supply was limited. The Dutch colonists collected rainwater in cisterns and shallow wells, but there wasn't enough for their needs. Deeper wells were dug, but there still was not sufficient fresh water for drinking, cooking, brewing, and washing, or for watering the ever-growing herds of livestock. Residents who were able to traveled north, beyond the city's walls, to collect fresh water from springs.\n\nUnder English colonial rule, wells were systematically dug, and in 1688 a public well system was established. Still, problems persisted as the population grew: wells became polluted not only with seepage from privies, garbage heaps, and cemeteries but also with effluent from commercial operations such as hatters, dyers, and starch makers. During the next century, several solutions were tried to solve the water problem. For cooking, New Yorkers boiled or filtered their water before they used it. Some believed that adding alcohol to water would purify it. Those who could afford the cost paid carters who brought water into the city\u2014presumably from rural springs, but buyers could never be sure where the water came from.\n\nYet another solution was to buy bottled spring water from upstate. In 1820, Dr. Darius Griswold began bottling water at Saratoga's Congress Springs. He stuck with the business for just a few years before leasing the bottling rights to Dr. John Clarke and Thomas Lynch in 1823. Saratoga Mineral Water, the first commercially successful bottled water in the United States, was shipped to New York City and then distributed to other eastern cities. Other bottled waters soon followed.\n\nDespite these efforts, yellow fever, cholera, and other waterborne diseases ravaged the city again and again. The well-to-do usually headed out of the city during summer, when such epidemics typically struck. It was the poor who suffered the most. Medical opinion of the day placed the blame on the poor themselves\u2014for eating fresh fruit, especially watermelons, and drinking bad water.\n\nHealth and economic concerns finally convinced the city government to embark on a massive project that would bring fresh water to Manhattan via an aqueduct. The Croton River north of the city was dammed, reservoirs were built, and a forty-two-mile aqueduct was constructed. The project began in 1837; when it was completed five years later, New Yorkers celebrated with parades and fireworks. Private houses were connected directly to the system, so the homes of the wealthy had running water and indoor privies for the first time. The less privileged continued to draw water from public taps for decades to come, but at least it was safe to drink. During this same period, the city began to construct a sewage system that piped raw sewage into the bay, which at least helped protect some of Manhattan's fresh water.\n\nThe Founding Beverages\n\nIt was not by choice that water was the most common drink in colonial New York; even water acknowledged as pure was considered a distant second best to alcoholic beverages. Alcoholic beverages were thought to be more nutritious, and there was a class stigma associated with drinking water\u2014after all, only the poorest of the poor had no alternative. The Dutch and English preferred beer, and breweries were established within a decade of the city's founding. Colonists also liked their wine and brandy.\n\nEven before the arrival of hard liquor in the colony, alarms were raised regarding the amount of alcohol consumed by city residents. William Kieft, who became the director general of the colony in 1638, believed that \"much mischief and perversity is daily occasioned by immoderate Drinking.\" One-quarter of all the buildings in New Amsterdam, he claimed, were \"grog-shops or houses where nothing is to be got but tobacco and beer.\" He passed a law preventing the sale of beer during religious services, although he encouraged the sale of beer through the Dutch West India Company's store, the profits of which helped fund the colony.\n\nConcluding that the production of alcohol could be used to generate income for the city (and himself), Kieft established the colony's first distillery on Staten Island. Then he concluded that the production of alcohol could be used to generate income for the city. When he made a disastrous decision to go to war against the Indians in 1643, money was needed to pay for the war. Betting that an excise tax on beer would be a good revenue source, Kieft imposed the tax of $1.20 per cask. Tapsters (tavern owners) who refused to pay the tax were shut down and their beer confiscated and given to soldiers. This tax, however, did not raise enough revenue to cover the colony's war expenses, and Kieft was recalled home to Holland.\n\nPeter Stuyvesant, Kieft's successor, maintained the beer tax and imposed a fee of two cents on each wine jar sold. These taxes did not affect home brewers, merchants, or really anyone else. They did not apply to the state-owned brewhouse, nor to the sale of wine and brandy by the colonial government. These generated more revenue for the Dutch West India Company than did anything other than the fur trade.\n\nStuyvesant forbade excessive drinking at night and any drinking at all on Sunday, but his edict went \"unobserved.\" In 1648, he passed another ordinance to regulate and license \"Taverns, Alehouses and Tippling places\" within the colony. The result was that by 1654 a large number of illegal establishments were serving beer and wine. Shortly thereafter, rum began to be imported into New Amsterdam; its consumption increased after the city's conquest by the English. Rum was distilled in New York by the end of the seventeenth century, and the industry grew quickly. By 1768, New York had seventeen distilleries producing half a million gallons of rum annually.\n\nNew York's upper classes preferred mixed drinks, which appeared in many guises. Flips (beer sweetened with sugar, molasses, dried pumpkin, or honey and strengthened with rum) had become popular by 1690. Possets, made from spiced hot milk and eggs combined with ale or beer, evolved into eggnog and other beverages. Syllabub\u2014spiced milk or cream whipped to a froth with sweet wine or cider and sugar\u2014was a spirituous drink for festive occasions. Shrubs, composed of citrus juice from imported oranges, lemons, and limes mixed with various spirits, were popular drinks before the War of Independence, as were hot toddies (made of liquor, water, sugar and spices) and cherry bounce (made from cherry juice and rum). The most popular mixed drink was punch, usually composed of rum, citrus juice, sugar, and water, with myriad variations. Milk punches, made with egg yolks, sugar, rum, and grated nutmeg, were also common for parties and balls. There were iced punches for summer and hot punches for winter. Sangaree, a mixture of wine, water, sugar, and spices, was an Anglo-American version of sangria, a common drink in Spain.\n\nAlcoholic beverages were often mentioned in newspapers of the day, and they even occasionally inspired poetry of sorts. In February 13, 1744, the New York Gazette published \"A Receipt for All Young Ladies That Are Going to Be Married, to Make a Sack Posset.\" The recipe likely originated in England, but it was widely reprinted in America:\n\nFrom famed Barbados, on the Western Main,\n\nFetch sugar half a pound: fetch sack from Spain,\n\nA pint; and from the Eastern Indian coast\n\nNutmeg, the glory of our Northern toast;\n\nO'er flaming coals together let them heat,\n\nTill the all-conquering sack dissolves the sweet;\n\nO'er such another fire let eggs, twice ten,\n\nNew born from foot of cock and rump of hen;\n\nStir them with steady hand, and conscience pricking\n\nTo see the untimely fate of twenty chicken;\n\nFrom shining shelf take down your brazen skillet;\n\nA quart of milk from gentle cow will fill it;\n\nWhen boiled and cooled, put milk and sack to egg,\n\nUnite them firmly, like the triple League;\n\nThen covered close, together let them dwell\n\nTill Miss twice sings, You must not kiss and tell.\n\nEach lad and lass snatch up their murdering spoon,\n\nAnd fall on fiercely, like a starved dragoon.\n\nTo support New York's fondness for alcoholic beverages, the number of taverns and other places serving and selling them increased rapidly. By 1773, the city had \"396 licensed taverns and many illegal groggeries that sold alcohol.\" Rum and whiskey were the beverages of choice for the lower classes\u2014they were cheap, and they had more of a kick than the more expensive beer and wine. By the mid-1820s, New York State had 1,129 distilleries, and New York City alone had more than 1,600 \"spirit sellers,\" who sold whiskey for thirty-eight cents a gallon.\n\nAccording to one British observer, this abundance of sellers and low price meant that \"the people indulge themselves to excess, and run into all the extravagancies of inebriety.\" He concluded that \"the friends of humanity would rejoice in the establishment of an adequate check to an evil of such dangerous consequence in a country which promises to become eminent in greatness and power.\" Despite concerns, New York City headed in precisely the opposite direction. Many immigrants, who began arriving in the 1830s, went into brewing and distilling. By 1855, more than five hundred German and Irish immigrants were brewers and distillers. Many more immigrants went into selling beer and spirits. By the middle of the nineteenth century, Manhattan alone had an estimated five thousand liquor stores and taverns.\n\nHot Beverages\n\nThe Dutch were importing tea to New York by about the middle of the seventeenth century, and it soon became the city's hot beverage of choice. Throughout the eighteenth century, tea consumption steadily increased, at least among the ladies of society. Good tea required good water, and in colonial times a thriving business was launched selling \"tea water\" from particular springs. Garardus Comfort, a cooper, owned a house with a well. He began distributing Comfort's Tea Water some time before 1741. Special tea water pumps were identified, and vendors began selling tea water from pumps and from vendors on the streets. By 1774, it was estimated that three thousand families got their household drinking water from the \"tea water men.\"\n\nSome took the tea-drinking habit to extremes. In 1734, a New Yorker wrote, \"I am credibly informed that tea and china ware cost the province, yearly, near the sum of \u00a310,000; and people that are least able to go to the expence, must have their tea tho' their families want bread. Nay, I am told, often pawn their rings and plate to gratifie themselves in that piece of extravagance.\" In 1762, William Smith, writing in his history of New York, noted bitterly that \"our people both in town and country are shamefully gone into the habit of tea-drinking.\" Indeed, New Yorkers loved their tea. It was sometimes served at meals but especially in \"the afternoon, as in England,\" as a coffeehouse owner reported. Tea gardens, also called pleasure gardens, were also launched in New York by the mid-eighteenth century.\n\nIn 1773, the British parliament imposed a tax on the sale of tea in the colonies. Every schoolchild knows how Bostonians responded with the Boston Tea Party on December 16, 1773. Less well known is the similar drama that played out in New York. In April 1774, two ships\u2014the Nancy and the London\u2014entered New York Harbor at about the same time, both carrying tea. The tea consignee to the Nancy sent a note to the captain, reporting that the city's populace was \"violently opposed\" to the landing or vending of the tea and \"that any attempts in us, either to effect one or the other would not only be fruitless, but expose so considerable a property to inevitable destruction.\" The Nancy returned to England without unloading its tea. The captain of the London claimed that there was no tea onboard, but when the ship was searched, it was found to contain eighteen chests of tea. The Sons of Liberty, disguised as Indians, threw the tea into the harbor and threatened the captain, who set sail to England at his earliest opportunity. Among those Americans, tea drinking stopped during the War for Independence in part to show support for the revolution and also because it was difficult to import due to the British blockade. The exception was New York loyalists, who consumed it when it could be obtained.\n\nAfter the war, tea drinking again became fashionable among the city's well-to-do, and tea-drinking occasions regained their former glory. Now tea could be acquired directly from China rather than through the British or Dutch. Robert Morris, a Philadelphia merchant and a signer of the Declaration of Independence, saw a great opportunity. It was time to cut out the middlemen in Europe and trade directly with China. Morris joined with others to outfit a ship, the Empress of China, which sailed out of New York on February 22, 1784, and returned fifteen months later with a cargo mainly of tea. When this tea was sold, each investor netted thirty thousand dollars (equivalent to about 1.6 million dollars today). Other ships made the journey to China as well, and soon a thriving business between New York and China was underway and fortunes were made in the trade.\n\nAfternoon tea was served at most restaurants and boardinghouses, and tea leaves were sold by specialty shops. By 1860, there were sixty-five retail tea shops in the city. James Geery at 119 Mulberry Street, at the corner of Bayard Street, was widely known for the excellence of his teas. J. Stiner & Company had a thriving business in importing and selling tea; they opened additional shops, thus becoming one of America's first chain-store retailers. By 1867, it operated ten stores. Another tea shop was launched by George F. Gilman and George H. Hartford, who imported teas, coffees, and other products. Gilman and Hartford opened a retail tea store in New York City and also began selling tea through the mail. Their business would eventually evolve into the A&P grocery store chain.\n\nTearooms became common in New York during the late nineteenth century. These were largely owned, operated, and frequented by women, who saw these establishments as good alternatives to the saloons where men congregated. While a tearoom did not necessarily revolve around afternoon tea\u2014most were ladies' lunchrooms, homey or elegant\u2014tea was an important item on their menus. Tearooms became particularly popular during Prohibition as alternatives to the speakeasies, and they went out of fashion after Repeal in 1933.\n\nCoffee arrived in New York not long after tea. In 1696, Lieutenant John Hutchins opened a coffeehouse called the King's Arms; it became the unofficial headquarters of English \u00e9migr\u00e9s in New York. The first floor of the two-story wooden structure had a coffee bar and booths; the second floor contained rooms for public and private meetings. For years, it was the city's only coffeehouse. The Exchange Coffee House, probably located at the foot of Broad Street, opened by 1730. The Merchants Coffee House, at the end of Wall Street, opened in 1731. It promptly became the premier coffeehouse in New York; during the British occupation of the city during the Revolutionary War, it was the venue for auctioning off captured American vessels. In 1784, the Bank of New York was launched in the Merchants Coffee House, and it continued to be the place where public meetings were held and business deals were consummated until it burned down in 1804.\n\nSeveral other coffeehouses were opened, the most important being the Tontine Coffee House at the corner of Wall and Water Streets, which opened in 1793. After the Merchants Coffee House burned down, the Tontine Coffee House became the most prominent in the city. Typically, coffeehouses sold hot chocolate and food as well as coffee and tea; in 1808, some added a new beverage\u2014soda water.\n\nCompared with making tea by simply pouring water over tea leaves, brewing coffee was a complicated process, which was one reason it took a while for coffee to catch on in New York. At first, unbranded, green coffee beans were sold from open barrels, scooped out by the customer or grocer and then weighed and priced. One New Yorker observed that exposed to air, odors, and dust, the beans, \"from long transit and resting in the small stores, became tough and tasted as much of codfish and oil as [they] did of coffee.\" New Yorkers roasted the coffee beans at home and ground them as needed. Coffee roasting was an art, and roasting beans in a pan over a fireplace or even with a coffee roaster was unpredictable. For those who were skilled roasters, the coffee was flavorful and aromatic. For those less skilled, beans were scorched or underroasted and did not develop their full flavor. Home roasting of the beans often produced an acrid brew that had to be camouflaged with milk, cream, and sugar.\n\nTea, however, was more expensive, and coffee had a greater caffeine payload. In addition, tea had a lingering feminine association, while coffee quickly became the more \"manly\" choice. New Yorkers began to consume ever-increasing quantities of coffee, and the city became the center of America's coffee trade. By the early nineteenth century, supply could not keep up with demand. Coffee prices jumped from 42 cents per pound in 1810 to $1.08 in 1812. At the time, coffee was mainly grown on the Arabian Peninsula and on a few Caribbean islands, but as fortunes were made in the coffee trade, more plantations were established in Brazil and the Dutch colonies of Java and Sumatra. As they began to be imported into the city, coffee prices started to fall\u2014to thirty cents per pound in 1823 and bottomed out at eight cents seven years later. While this decline in price hurt coffee producers, it was a boon to consumers. New Yorkers became hooked on coffee. When the federal government abolished import taxes on coffee in 1832, its consumption jumped even more. New York averaged three pounds of coffee per person in 1830; by 1840, this reached more than five pounds; and nine years later, it topped out at eight pounds. Coffee had surpassed tea as the city's hot beverage of choice.\n\nRoasted coffee was also sold at retail in New York, but if it sat in the store for very long, it lost its flavor. New York inventor and manufacturer Jabez Burns addressed a key problem in coffee roasting\u2014the fact that the beans rested directly on a hot surface, where they could quickly scorch. In 1864, Burns devised a screwlike device to keep the beans moving as they roasted. The equipment was too bulky and expensive for home use, but it was ideal for large-scale coffee-roasting operations.\n\nCounterfeit and adulterated coffee was a common problem throughout the nineteenth century. In one 1881 study, Francis Thurber, a New York City grocery distributor, deplored \"the adulteration of coffee and the vast scale on which it is practiced\" and noted that despite what was being sold in many grocery stores, \"a primary requisite for making a good cup of coffee is, of course, coffee.\" Thurber's efforts and those of others enticed newspapers to examine coffee adulteration; newspapers such as the New York Times alerted readers to these problems by publishing articles with headlines such as \"Poison in Every Cup of Coffee.\"\n\nDespite initial concerns with the stimulating qualities of coffee and tea, temperance advocates eventually concluded that these beverages were at least good alternatives to alcohol. Temperance coffeehouses, or \"temperance drinking saloons,\" were opened to counter the ubiquitous bars and taverns. The coffeehouse offered a place where people could socialize without being tempted by strong drink. As one observer commented, \"A coffee-house is a liquor saloon without liquor. It is a place where a workingman can get a well-cooked meal at a cheap rate, where he can read his newspaper and enjoy his game of chess or draughts without being preached at. It is the saloon, less the saloon's evil concomitants.\"\n\nGrowth of Bars and Grogshops\n\nThe precise origin of the word \"bar,\" meaning a place where liquor is served, is obscure. Some historians have suggested that it emerged from the grogshops commonly found in seaports in the eighteenth century, or in taverns. Others believe that bars started in various kinds of retail stores, such as groceries, barber shops, and bakeries, where liquor was sold for off-premise consumption as a side business. Over time, the retail businesses declined and liquor became the main attraction. Spirits were stored in barred cages to prevent unauthorized access. When customers arrived, the proprietor opened a small window in the bars above the counter and poured drinks from a small selection of bottles. Early establishments had only a few common mugs used by all customers. Counters initially were just small wooden planks, but as time went on, the counters were enlarged to accommodate more drinkers, and the counters were renamed \"bars.\" From their humble beginnings bars increased in size and attractiveness, the range of drinks they served diversified, and their accouterments became more refined.\n\nNew York's City Hotel, built in 1794, had a bar room off the lobby with a \"liquor cage\" so that the alcohol could be locked up when the bar was closed. The early nineteenth century saw a proliferation of great hotels in New York and such bars were de rigueur. Over time, bars were upgraded into much larger rooms with long hardwood (mahogany or oak) counters often adorned with intricate carving. Elaborate mirrors, more woodwork, and an impressive display of liquor bottles formed the \"back bar\" facing the drinkers. Counters sported brass foot rails and sometimes handrails, designed to accommodate standup drinking.\n\nAs bars became popular, a new profession emerged\u2014the bartender. The first publicly acclaimed bartender was Orsamus Willard of the City Hotel. According to a British naval officer who stayed at the hotel in 1826,\n\nThe entrance to the house is constantly obstructed by crowds of people passing to and from the bar-room, where a person presides at a buffet formed upon the plan of a cage. This individual is engaged, \"from morn to dewy eve,\" in preparing and issuing forth punch and spirits to strange-looking men, who come to the house to read the newspapers and talk politics.\n\nWillard, who lived at the hotel, was identified by another British visitor as a man \"who never forgets the face of a customer.\" Willard served traditional wines, brandies, and spirits\u2014but also concocted inventive assortments of mixed drinks. His most famous and popular beverage was the mint julep, which consisted of \"four or five stalks of unbruised mint\" combined with ice, brandy, water, and sugar. For his celebrated apple toddy, Willard rolled apples in wet brown paper and buried them \"in live embers till they are thoroughly roasted and quite soft; then a fourth part of apples, a fourth part of brandy, a fourth part of water, a lump of ice, and the whole to be rich with a fourth part of sugar, makes the agreeable compound.\" His slings had a ratio of one-third rum, gin, or brandy to two-thirds water, sweetened with sugar and spiced with nutmeg. When Willard added bitters to the sling, it became a cocktail, as described by another English visitor.\n\nThe Metropolitan Hotel, which opened on Broadway in 1852, also had a fashionable bar. Its most prominent bartender was a man named Jerry Thomas, who about 1851 settled in New York, where he operated the first of his four bars, all of which received wide acclaim. In 1858, he tended bar at the Metropolitan Hotel, where he was particularly known for his flamboyant manner of mixing drinks, a style that would continue to be the model for future bartenders. Then Thomas did what no other bartender had done: he started writing down his recipes for publication. Commissioned by the New York publisher Dick & Fitzgerald, he assembled a collection that would be published in 1862 as The Bar-Tender's Guide; or How to Mix Drinks. For sophisticated New Yorkers the cocktail soon achieved stardom.\n\nSwill Milk\n\nAnother important beverage in New York was milk. Shortly after New Amsterdam was founded, the Dutch imported cows, and dairy farming became an important occupation on the island of Manhattan. Unfortunately, the growth of the human population outpaced that of their dairy herds, leading Reverend Jonas Micha\u00eblius to complain that milk \"cannot here be obtained\" except at \"very high price.\" He urged the Dutch West India Company to send more farmers and dairymen to the colony. The Dutch complied, and the colonists were again able to enjoy an abundance of milk and other dairy products, as they had in the old country. Herds of dairy cattle flourished in New Amsterdam and in the Dutch settlements that grew up in Brooklyn.\n\nIn colonial days, many New York families kept their own cows. Some enterprising individuals tied cows to stakes in city streets and fed them garbage and then drank\u2014or sold\u2014the milk. Property owners charged for the privilege of herding cows on their streets. According to dairy historian John Dillon, \"The disposition of the manure was a provision of the lease contract,\" and at the time there was an unlimited demand for manure to fertilize Manhattan's farms and gardens.\n\nAs the city's population grew, the demand for milk increased, and the business of distributing it became more profitable; dealers multiplied and competition became intense. In the 1820s, some ambitious New York City dairymen adopted a British system that exploited another popular beverage\u2014beer\u2014to support milk production. Eager to maximize profits from their operations, brewers built cow barns adjacent to their breweries. Mash, or slop\u2014the watery by-product of brewing with grain\u2014was siphoned from the brewery directly to cows' mangers through wooden chutes. By the 1830s, an estimated eighteen thousand cows in New York City and Brooklyn were being fed almost exclusively on brewery mash. Brewers saved money by not having to cart away the waste, and they enjoyed substantial profits from the sale of milk. They made additional profits by selling the cow manure to farmers, and when the cows died, they sold the carcasses to butchers.\n\nRobert M. Hartley, the founder of the New York Temperance Society in 1829, investigated the city's brewing and distilling industries and concluded that milk production was one of their main profit centers. In fact, he concluded that the profits generated by the dairy operations were so substantial that, if they were eliminated, New York's alcohol industry would collapse. Hartley's investigation of the dairies generated some shocking discoveries: the dairies were poorly managed and badly kept, and the cows were frequently diseased\u2014some to the point that they could not stand and had to be winched up for milking. The milk from brewery and distillery dairies was low in butterfat and extremely thin, with a pale bluish cast. Brewers and distillers (as well as other milk distributors) were known to adulterate milk to make it appear wholesome. They stretched it with water; brightened it up with powdered chalk, carrot juice, and annatto; broke in raw eggs for creaminess; and even sweetened the milk with molasses so children would ask for it.\n\nHartley suspected that poor quality and adulterated milk was the primary reason for New York City's steep increase in infant mortality during the 1830s. He launched a campaign against \"swill milk\" or \"slop milk,\" publishing a series of pamphlets and articles opposing its sale. In 1842, he compiled his views in America's first book about milk, An Historical, Scientific, and Practical Essay on Milk, as an Article of Human Sustenance.\n\nOn the take from brewery and distillery owners, city officials ignored his campaign, and the number of breweries exploded in New York during the 1840s and 1850s, and so did the number of dairies. But Hartley's efforts had raised an alarm and caught the interest of newspapers, which began investigations of their own into the atrocious conditions in the dairies. In 1847, the New York Tribune reported on \"one of the largest and filthiest swill milk stables and urged the city to act.\" Yet another series of expos\u00e9s ran in the New York Evening Post during the early 1850s; newspaperman John Mullaly collected these articles in a book, which was published in 1853. The newly launched New York Times began featuring articles on swill milk in 1854. In 1858, Frank Leslie's Illustrated Newspaper \"declared war\" on the brewery dairies of Manhattan and Brooklyn. The first article, \"Startling Exposure of Milk Trade of New York and Brooklyn!\" reported that the stables surrounding the distilleries were \"disgusting,\" \"dilapidated and wretchedly filthy.\" The controversy heated up and turned violent when one of the newspaper's reporters was murdered while developing a story.\n\nDespite the barrage of expos\u00e9s, lectures, pamphlets, books, and newspaper and magazine articles, the sale of swill milk continued unabated. The brewing and distilling industries were just too ingrained in the municipal political system to permit anything more than superficial modifications to be mandated. It was not until 1896 that the city required milk purveyors to be licensed, established standards for milk, and hired inspectors to examine milk to ensure that dealers complied with the law. Large milk companies could more easily comply with these requirements, while many of their smaller competitors were driven out of business by the burden of the new regulations. But the end result was a relatively safe milk supply for the city.\n\nThe campaign against swill milk encouraged pure food laws in New York City and State. For almost three decades, national legislation was blocked by liquor industry lobbyists and those politicians on the take from the industry. This changed in 1906 when Upton Sinclair, a relatively unknown novelist, published The Jungle, an expos\u00e9 of the meat industry in Chicago. Within months, the Pure Food and Drug Act and the Meat Inspection Act passed Congress.\n\nLager Beer\n\nNew Amsterdamers planted barley and wheat and brewed their own beer at home. The Dutch writer David Pietersz de Vries reported that \"our Netherlanders raise good wheat, rye, barley, oats, and peas, and can brew as good beer here as in our Fatherland, for good hops grow in the woods; and they who make it their business can produce enough of those things, as everything can be grown which grows in Holland, England, or France.\" The first brewery opened in 1633. Oloff Stevensen van Cortlandt, progenitor of one of the city's most prominent and wealthy families, later acquired the brewery and enlarged it. Additional other breweries were established by other families as the city's population increased. In addition, large casks of beer, such as Bass ale, were imported.\n\nDuring the early nineteenth century, beer production in New York declined in part because of the deteriorating quality of the city's water. German immigrants had opened new breweries in New York\u2014the first as early as 1834. City brewers, who generated almost five hundred thousand dollars (more than one billion dollars today) in annual sales at the time, complained to the city government about the inadequate water supply. This concern, and others, encouraged the city to embark on a massive project to bring water to Manhattan from upstate via the Croton Aqueduct.\n\nGeorge Gillig had served apprenticeships in breweries in Bavaria before he immigrated to New York in 1836. Four years later, he opened a brewery on Lexington Avenue in 1840; he was reportedly the first New Yorker to brew lager beer, which was lighter bodied, less bitter, weaker, and fizzier than other beers and ales traditionally consumed in the city. Still, lager beer was popular among German immigrants from southern Germany, and Gillig opened two more breweries, one in Williamsburg, the other in Staten Island.\n\nOther early German brewers included Joseph Doegler and Frederick and Maximilian Schaefer. Doegler, a Bavarian, arrived in New York in 1843 and began working in a brewery. He purchased Gillig's brewery in 1853. The Schaefers purchased the Sebastian Sommers brewery in New York City in 1842; six years later, they began marketing a lager beer under their own name. Franz Ruppert, who had come to New York from G\u00f6llheim, Germany, sometime before 1842, launched what would eventually become the Jacob Ruppert Brewing Company with his brother in Turtle Bay (in Manhattan on the East River) in 1851.\n\nBy 1854, New York had \"twenty-seven breweries, and many of them, such as Turtle Bay, Gilley's, and Schaefer's, brew more than ten thousand barrels, of thirty gallons each, of lager beer in the course of the year,\" reported the New York Tribune. These breweries produced eighty-five thousand barrels, equivalent to thirty million gallons, which were distributed by two thousand retail outlets, independent of large hotels and restaurants. Other German breweries opened in Williamsburg in Brooklyn. By 1872, there were seventy lager breweries in New York.\n\nThe main outlet for lager beer was saloons owned or frequented by German immigrants. They also served German-style food, and some establishments provided a free lunch to those who bought beer. Lunches at German-owned saloons included foods like blutwurst, sausage, sardines, herring, ham, pig's feet, rye bread, hamburgers, roast beef, corned beef, onions, pumpernickel bread, salads, cheeses, crackers, bologna, and pretzels. At first it was just those saloons that catered to immigrants that served lager beer, and only Germans who drank it, but this soon changed. As lager beer rolled out of the local breweries, New Yorkers began flocking to places that served it. Other city saloons and bars soon jumped on the lager wagon. In September 1855, the New York Times reported that \"a very large portion of the inhabitants of the City of New York, who a few months since, were in the habit of drinking one or more glasses of rum, gin or brandy, every day, are now using about the same number of glasses of lagerbier instead.\" Lager beer was an improvement, according to a reporter for the New York Herald who believed that the \"increase in the use of lager beer will do more to overthrow drunkenness and rowdyism, and to establish good order on their ruins than all the labors and the efforts\" of temperance advocates. By the 1860s, an estimated three to four thousand establishments sold lager beer in Manhattan alone, with many more in surrounding communities, and it was estimated that more than half of their patrons were not German immigrants. As one observer reported in the 1870s, \"New Yorkers ran mad after it, and nothing was spoken of or drunk but Lager, or Logger beer, as it was more usually called.\"\n\nIn addition to serving beer in bars and saloons, many brewers set up their own beer gardens\u2014large halls planted with trees and flowers; sometimes part of the ceiling or a wall could be opened to let in the sun in good weather. These beer gardens could hold as many as three to four thousand people at a time. Some were family oriented; others catered to the less-affluent immigrants. The largest beer garden in New York was the Atlantic Garden, reported a popular writer in 1872, an \"immense room with a lofty curved ceiling, handsomely frescoed, and lighted by numerous chandeliers and by brackets along the walls. It is lighted during the day from the roof. At one side is an open space planted with trees and flowers, the only mark of a garden visible.\" The Atlantic Garden contained a restaurant, \"several bars, a shooting gallery, billiard tables, bowling alleys, and an orchestra.\" Another observer wrote that there were \"dense clouds of tobacco-smoke, and hurry of waiters, and banging of glasses, and calling for beer, but no rowdyism.\"\n\nBeer gardens served a variety of functions other than drinking. They were social centers, concert halls, eating establishments, and athletic fields; civic, religious, and organizational meetings were held in them as well. Beer gardens were not just for German Americans; mainstream New Yorkers and immigrants from elsewhere frequented them as well.\n\nWine\n\nColonial New Yorkers imported a variety of wines, such as canary, port, sherry, and muscadine. Sack (fortified white wines from Spain and the Canary Islands) and hock or Hockheimer (German white wines) were also common by the late eighteenth century. Frequent hostilities between England and France during the colonial period made it difficult for New Yorkers to import French wines, but after the Revolutionary War, French wines were sold in the city. Particular favorites included claret, a dark French Bordeaux, Burgundy, and Champagne. Specialty wine stores opened. A Swiss-born sea captain named Giovanni Del-Monico sold his ship in 1824 and with the proceeds opened a wine shop in the city. His customers were mostly Frenchmen and other European immigrants or foreign visitors. To maximize profits, Del-Monico imported casks of wine and did the bottling himself, but he wasn't satisfied with the business. He sold his wine shop in 1826 and returned to Switzerland to visit his family. There, he convinced his brother Pietro, a pastry chef, to return with him to New York and open a caf\u00e9 and pastry shop. He sold his business and later with his brother opened Delmonico's restaurant, which stocked French and other wines. By the 1830s, all high-end restaurants in the city served a wide range of imported wines.\n\nSpecialty stores, such as Acker Merrall & Condit that opened on Chambers Street in 1856, sold a wide range of imported wines to restaurants and well-to-do New Yorkers. (Acker Merrall & Condit has survived and claims to be the oldest wine store in America.) German immigrants sold wines from Germany in saloons, bars, beer gardens, and restaurants, and Italian immigrants imported and sold wines from Italy in grocery stores and Italian restaurants. Jewish immigrants produced kosher wines that were sold in New York by the 1880s. California wines were sold in New York by the late nineteenth century, and New York State wineries began selling wine in the city shortly thereafter. Both tended to be cheap wines that targeted immigrant communities.\n\nWines continued to be imported, but an English visitor to America in 1878 reported that it was \"very expensive, and it is only once in a while that it appears on the tables even of the rich\u2014so seldom that it cannot be called a custom. Sherry, port, claret, madeira, and champagne are rare luxuries, and only appear on festive occasions, or in the houses of epicures and wealthy foreigners.\" Five years later, another British visitor, George Sala, was assured by Americans that \"the custom of wine drinking would speedily fade out altogether from good society in America\" due to the temperance movement.\n\nThe Temperance Movement\n\nThe temperance movement began in the late eighteenth century and slowly picked up steam in New York State during the early nineteenth century. The movement's popularity was understandable. According to one British observer, the easy availability and low price of alcohol meant that \"the people indulge themselves to excess, and run into all the extravagancies of inebriety.\" Temperance crusaders estimated that three-quarters of the crime and pauperism in New York City was related to liquor consumption.\n\nThe temperance movement in New York took off in the 1830s, when advocates joined forces. They faced one very serious problem. If New Yorkers were not to drink alcoholic beverages, what would they drink? Public water fountains were constructed after the completion of the Croton Aqueduct in 1842; later, temperance and other groups funded the construction and maintenance of \"ice-free\" water fountains\u2014particularly in public parks\u2014where people could drink water without expense. Temperance advocates also encouraged restaurants to offer glasses of water to patrons when they sat down.\n\nYet another solution popped up from scientists. In the 1700s, they discovered that carbon dioxide was the source of the bubbles in natural springs, as well as the \"fizz\" in beer and champagne. By 1800, beverage manufacturers had discovered that they could produce their own fizzy water by adding a solution of sodium bicarbonate, thus creating carbonated water. However, the carbonation process was more commonly accomplished under high pressure using sulfuric acid, and manufacturing carbonated water this way was more difficult and dangerous. Workers could be seriously burned by the acid, and containers of charged soda water sometimes exploded. In 1809, Benjamin Silliman, a Yale chemistry professor, and a partner purchased costly soda-making equipment and opened two retail soda-water outlets, one at the City Hotel and the other at the Tontine Coffee House. He sold Ballston water (from an upstate spa), soda, or seltzer (plain carbonated water), which was made on the premises. An Irishman, George Usher, may have beaten them to it.\n\nSoda water continued to be part of the New York beverage scene, especially after easier and safer ways of producing it were invented. By the mid-nineteenth century, vendors sold seltzer and ginger pop, spruce beer, mead, and other goodies. One New Yorker wrote about the soda vendors in the mid-nineteenth century:\n\nTheir locale was almost universally designated by a sign, on which, for the mead and beer, was delineated a bottle with a stream of liquor pouring from it into a tumbler at its side, with a uniformity of outline and of curve that would have done credit to a geometrical draughtsman. The ginger pop was designated in a very different manner, being totally devoid of any illustration of convenience or economy; inasmuch as two men were portrayed as fulfilling the regulations of a duel, being placed opposite to each other and each extending a bottle, from which emanated a stream of liquor propelling a cork, which was as the bullet of a pistol.\n\nDuring the late nineteenth century, soda fountains emerged. The temperance movement supported them as an alternative to bars. Plain soda was cheap at two cents per glass; with flavorings it cost a nickel, which virtually everyone could afford. On Wall Street stood Delatour's soda-water stand, and during the summers Delatour dispensed drinks to long lines of customers. Soda fountains competed to offer the most flavor selections. One offered thirty-two different syrups, eight kinds of mineral water, and soda water. By 1891, New York had more soda fountains than it had bars.\n\nA large variety of flavorings and ingredients were combined with soda. During the 1880s, a popular specialty was made with chocolate syrup, cream, and raw eggs mixed into soda water. In poorer neighborhoods, a less expensive version of this treat was created called the Egg Cream (made without the eggs or cream), which remains an iconic New York beverage.\n\nMost temperance advocates concluded that these worthwhile alternatives to alcoholic beverages were not enough to rid the city of excessive drinking. They turned to stronger measures.\n\nProhibition\n\nDespite widespread support for Prohibition in upstate New York, most New Yorkers strongly opposed it. Liquor was an important business in the city: an 1897 survey of the borough of Manhattan found a ratio of one liquor distributor to every 208 residents. Saloons were also an important base of support for Tammany Hall's political machine, which had controlled the city government since 1854.\n\nDuring and after the Civil War, even more saloons and bars opened, and there was no shortage of customers. One of the few victories of the New York temperance movement was achieved in 1866 when, prompted by temperance advocates, the city banned the sale of liquor on Sunday. This caused Germans immigrants to rally, proclaiming, \"Liberty and Lagerbier!\" Others just crossed the Hudson River to New Jersey, where there were plenty of breweries, beer gardens, and saloons open on Sundays. Still others went to saloons and bars that continued to serve alcohol on Sunday despite the new law. Ignoring this law set a precedent and served as a model for responding to legislation restricting alcoholic consumption that continued right up to\u2014and through\u2014Prohibition in the 1920s.\n\nBy 1870, Manhattan had 7,071 licensed suppliers of liquor and probably an even greater number of illegal establishments. The free lunches offered on bar counters were a strong draw for workingmen\u2014all they had to do was buy a drink, and they could eat their fill. Hot lunches were served from 11:00 a.m. to 3:00 p.m. A choice of \"pea soup, chowder, bean soup, lamb stew, beef stew, pork and beans, etc.\" was offered by one New York saloonkeeper. \"There were some ten plates of cold lunch besides, consisting of bologna, liver sausage, spiced fish, pickled herring, smoked or shredded fish, sliced cabbage, onion, bread pretzels, potato salad, radishes, etc.\" Free lunches ensured that customers acquired the habit of drinking liquor at noon, and this helped to develop a regular clientele for saloons and bars. Moreover, the food served at saloons was often salty, which increased the desire to drink more alcohol, which, of course, was where the bars made their profits.\n\nThe New York State legislature tried to restrict drinking in 1896, when it passed the Raines Law, banning the sale of liquor on Sunday. But the law had a good-sized loophole: alcohol could be sold with meals in hotels with more than ten rooms. Within months, hundreds of \"Raines hotels\" were in operation, and many of these also served as brothels. Other drinking establishments began reopening on Sundays, but the authorities usually ignored them provided they kept a low profile. A survey of Manhattan and Bronx saloons in 1908 found that 5,000 of the 5,820 legal saloons served alcohol on Sunday. This should have given Prohibition advocates an inkling of problems to come, but little attention was paid to enforcement at the time\u2014or later.\n\nThe Anti-Saloon League, the main national lobbying organization for Prohibition, considered New York City the \"liquor center of America,\" and their literature cited statistics proclaiming that every week people in the city consumed 75,000 quarts of gin, 76,000 quarts of brandy, 100,000 quarts of champagne and wine, 498,000 quarts of whiskey, 33 million quarts of domestic beer and ale, and 300,000 quarts of assorted other alcoholic beverages. If accurate, New Yorkers\u2014and visitors\u2014were downing an awful lot of alcohol.\n\nDespite strong pro-drinking forces in New York City, the Eighteenth Amendment, which outlawed the manufacture, sale, transportation, and importation of alcohol in the United States, became the law of the land at the stroke of midnight on January 16, 1920. The National Prohibition Act, commonly known as the Volstead Act, had an immediate effect: the consumption of liquor dropped off immediately. Saloons disappeared, never to reopen. Public drunkenness sharply declined, and hospitals dealt with far fewer of the medical problems associated with drinking and alcoholism. Still, it remained fairly easy to acquire alcohol all through Prohibition in New York. The main alcohol distributor became the speakeasy, a shady operation usually located in a dark and poorly ventilated cellar or a hidden back room. It supposedly acquired its name because customers spoke quietly about such places in public and when inside so as not to alert the authorities. Customers achieved admission only after proper scrutiny by guards who checked out all comers by eyeing them through a peephole in the door. The liquor at speakeasies, sold at inflated prices, was of dubious origin and notoriously harsh, foul tasting, and sickening.\n\nPrior to Prohibition, saloons, bars, and other drinking establishments were stocked with the products manufactured by commercial breweries, distilleries, and vintners, whose practices were regulated. The alcohol served in speakeasies, however, was made by amateurs; \"bathtub gin\" was not just a figure of speech. Spirits were sometimes made with poisonous wood alcohol. Imported liquor was available but only at extremely high prices, and a lot of what was sold as imported spirits was actually domestic bootleg liquor poured into fancy bottles with fake labels. Customers never knew where their drinks came from or what they were made from. In one year alone, the deaths of 625 New Yorkers were directly attributable to ingesting poisoned alcohol and another 1,200 to alcohol-related causes.\n\nNew York City Deputy Police Commissioner watching agents pour liquor into sewer following a raid during the height of Prohibition. Courtesy of the Library of Congress. Reproduction Number: LC-USZ62-123257\n\nThe major flaw in Prohibition was enforcement. The federal government allocated only 4.75 million dollars for enforcement throughout the United States, and only 129 federal enforcement agents were assigned to New York City. The framers of the Eighteenth Amendment assumed that local and state police would also uphold the law. But many state and local agencies lacked the trained manpower to do so; nor did they have the ability to deal with the wave of organized crime that Prohibition engendered.\n\nNew York City police were particularly lax in enforcing the liquor ban. The New York State legislature passed the Mullan-Gage Act on April 5, 1921, which made violations of the Volstead Act also violations of state law, requiring state and local police to enforce federal law. According to Samuel Hopkins Adams, a writer for Collier's, \"Restaurants, private houses, blind tigers, and the hip pocket of the casual wayfarer were raided with a disconcerting impartiality. A baby carriage, even, was held up and found to harbor a bottle which did not contain milk.\" During the first two months after the passage of the Mullan-Gage Act, four thousand New Yorkers were arrested for violating it; however, less than five hundred were indicted, only six were convicted, and not one received a prison term. When Al Smith, an ardent opponent of Prohibition, became governor of New York in 1923, the bill was repealed.\n\nFrom that time on, the city's police generally declined to arrest anyone as long as liquor distributors were unobtrusive and bribed them and city officials on time. The lack of effective enforcement gave speakeasies free rein, and by 1922 there were an estimated five thousand illegal booze joints in the city. When organized crime began centralizing the illegal manufacture, distribution, and retail sales of liquor, money and power flowed to the crime syndicates. As New Yorkers increasingly flouted Prohibition, a generation came of age with little respect for the law.\n\nWealthier New Yorkers never felt that the Volstead Act was intended for them anyway; people with plenty of cash had no problem securing bootlegged wines, liquors, brandies, or any other alcoholic beverages throughout the Prohibition era. They socialized with their peers at private cocktail parties throughout Prohibition. The urban middle class also had access to alcohol through speakeasies, and if they chose, they could (and did) experiment with apparatuses required for making bathtub gin. It was the working class and the poor who suffered most during Prohibition. The favored drink of less-privileged Americans\u2014beer\u2014was almost unobtainable: the raw ingredients were bulky, brewing required a lot more space than distilling or concocting spirits, and the finished product came in big wooden barrels, much harder to conceal than glass bottles. In rural areas, the time-honored art of moonshining went into high gear, with tens of thousands of illegal stills turning out \"white lightning.\" Many minority groups in New York City, such as Italian and Jewish immigrants, did not believe that drinking was immoral, and they did not believe it should be illegal. They also knew that fortunes could be made providing liquor, and many bootleggers and underworld bosses came from these groups.\n\nIn 1929, the New York City police commissioner estimated that the city had thirty-two thousand speakeasies\u2014double the number of all establishments serving alcohol (legally or otherwise) prior to Prohibition. Others said that the commissioner's estimate was far off\u2014 there were really one hundred thousand speakeasies in the city. Mabel Willebrandt, the U.S. assistant attorney general responsible for prosecuting violations of the Volstead Act, commented on the lawlessness in New York's speakeasies: \"It can not truthfully be said that prohibition enforcement has failed in New York. It has not yet been attempted.\"\n\nAs the violence of organized crime increased, newspapers, magazines, radio programs, and movie-theater newsreels kept Americans, even in rural areas, apprised of the latest murders, shootings, and arrests associated with illegal alcohol. Prohibition agents earned $1,800 per year, and New York City police officers even less. Both could supplement their income substantially by accepting bribes, and many did. As enforcement efforts faltered, Congress stepped in and passed the Jones Act, which greatly increased the penalties for violating the Volstead Act. The first conviction was a felony, with a maximum penalty of five years imprisonment and a ten-thousand-dollar fine. Even with such harsh provisions in place, enforcement couldn't keep a lid on the liquor trade. The Jones Act did alienate many prominent Americans, and opposition to Prohibition strengthened. In 1929, Pauline Sabin, a wealthy and politically well-connected New York socialite who occasionally sipped alcoholic beverages, formed the Women's Organization for National Prohibition Reform. In less than a year it counted more than one hundred thousand members; by 1931, its membership had reached three hundred thousand; and by the time of the presidential election in November 1932, it had more than 1.1 million members.\n\nDespite growing opposition to the Volstead Act, widespread disregard for its provisions, and the associated violent crime and corruption, the Eighteenth Amendment might not have been repealed had it not been for the Depression. Republicans had strongly associated themselves with Prohibition, while many Democrats, especially Catholics, Jews, and those of Irish, Italian, and German origin, strongly opposed it. When the New York stock market crashed in 1929 and the economic system collapsed, the Republicans, who had controlled Congress since 1919 and the presidency since 1921, were blamed. In November 1932, President Herbert Hoover lost his reelection bid and the Democrats swept into power with large majorities in both houses of Congress. President Franklin Delano Roosevelt, former governor of New York, pledged to get the economy moving again, and repeal of Prohibition was part of his plan. Roosevelt and his allies presented repeal as a means of generating revenue. Reinstating the excise tax on the sale of liquor was thus one argument for ending Prohibition. Others argued that repeal would relaunch industries that had once employed hundreds of thousands of Americans. Still others believed that repeal would help put an end to the burgeoning crime generated by the illegal production and sale of alcohol. On February 20, 1933, the U.S. Congress approved the Twenty-first Amendment repealing Prohibition. The Amendment was ratified by enough states, and on December 5, 1933, the \"noble experiment,\" as it was called, ended. Throughout the city, countless toasts were raised to repeal.\n\nBeverage Diversity\n\nSince Prohibition, New Yorkers have enjoyed a diversity of beverages. New York coffeehouses thrived during the 1960s, when they were frequent gathering places for poets, writers, beatniks, and counterculture, but later they began to disappear. They were not resurrected in New York until the arrival of Starbucks, Stumptown Coffee, and other premium coffee vendors.\n\nThe commercial production of wine began in New York State in the nineteenth century, and California wineries exported wine to New York beginning by the early twentieth century; these wines tended to be generic, cheap, and low quality and thus were the perfect beverage for the poor and alcoholics. They found few customers among the middle- and upper-class New Yorkers, who, if they drank wine, continued to consume imported wine from Europe.\n\nIronically, legal wine consumption skyrocketed during Prohibition. Wineries legally supplied wine to Catholics and Jews who consumed it for religious reasons. For everyone else, wineries manufactured nonalcoholic grape juice with instructions on how to convert it into wine. It was legal to make wine at home, and many New Yorkers gave it a try. The homemade stuff filled a void during Prohibition, but with repeal, wine consumption plummeted as other alcoholic beverages became available again.\n\nAs fine French restaurants flourished in New York City, so did the sale of imported European wines. Specialty wine shops, such as Acker Merrall & Condit (founded in 1856), Sherry-Lehmann Wine & Spirits (founded in 1934), and Astor Wines & Spirits (founded in 1946), broadened their selections and increased their supplies to meet the needs of restaurants and well-to-do New Yorkers. During the 1960s, California wines came of age, equaling and sometimes surpassing French vintages. New York State wines also improved, and both are now regular offerings in restaurants and liquor stores.\n\nGrowing interest in wine led the New York Times to assign a weekly wine columnist, Frank Prial, in 1972. In 1976, Kevin Zraly first offered his comprehensive wine course at Windows on the World, the restaurant on the 107th floor of the World Trade Center; the course is still offered, now at a midtown hotel. Today, wines are imported into New York from around the world and around the country, and New Yorkers are enthusiastic and well-informed wine drinkers.\n\nNew York's brewing industry did not return to business as usual after Prohibition's end. Many New York breweries did not reopen. Still, as late as 1962, New York City produced 10 percent of the beer consumed in America, but brewing in the city was a dying industry. The Jacob Ruppert Brewing Company survived until 1965. In 1969, Schlitz closed its Brooklyn plant, and Schaeffer's, the city's last large brewery, closed in 1976. But during the following decade, brewing revived. In 1984, the Manhattan Brewing Company opened in a Soho restaurant. Three years later, the New Amsterdam Brewing Company began producing beer in Manhattan. Although they did not survive for long, they revived New Yorkers' interest in locally made beer. The Manhattan Brewery in Soho opened in 1984. Today Manhattan sports two breweries (Chelsea Brewing Company, at Chelsea Piers, and 508 Greenwich, a restaurant where the proprietor makes his own beer); Brooklyn, three (Brooklyn, Sixpoint, and KelSo); and Queens, one (SingleCut). The Greenpoint Beer Works in Brooklyn's Clinton Hill brews beer at Heartland Brewery, a chain of seven brewpubs in Manhattan launched in 1995. The Brooklyn Brewing Company brewed its first beer in 1996 and today is the city's largest brewery.\n\nHeartland Brewery in Union Square, Manhattan, 2013. Courtesy of Kelly Fitzsimmons\n\nSoda continued to be bottled in New York. The Pepsi-Cola Bottling Company in College Point, Queens, continues to bottle soda, and its bottles have \"New York\" printed on top. Soda fountains closed after World War II, but several have opened recently in all five boroughs.\n\nNot everyone was happy with sugary soda. In 2013, the city forbade the sale of sugary drinks in containers larger than sixteen ounces in restaurants, movie theaters, and other eating places. This has been challenged in court and may not take effect, but it has made New Yorkers aware that sugary sodas contain a large amount of calories. As of this writing, the decision has been appealed and it is still in the courts.\n\nSome New Yorkers developed new, nonalcoholic beverages. In 1972, three entrepreneurs launched a very small company, called the Unadulterated Food Products, Inc., in Brooklyn. They sold fruit juices, sodas, and seltzers in wide-mouth bottles. The products were sold in health food stores, promoted with the slogan \"Made from the best stuff on Earth.\" This \"new age\" marketing approach was highly successful, and the company marketed other beverages, such as lemonade. In 1988, Unadulterated Food Products launched its first flavored tea. In 1993, the company changed its name to the Snapple Beverage Company. Now part of the large multinational Dr. Pepper Snapple Group, it has high international sales\u2014and burgeoning competition.\n\nStill other New Yorkers have opened juice bars. In 1996, Doug Green opened Liquiteria in the East Village. He served raw, cold-press juices, and he promptly became the guru of the juice movement. Others, such as Denise Mari and Doug Evans of Organic Avenue, jumped on the juice wagon. National chains, such as Jamba Juice and Red Mango, have opened outlets in the city. Today, hundreds of establishments offer juice and smoothies to New Yorkers.\nChapter Six\n\nEateries\n\nTo a stranger, New-York must seem to be perpetually engaged in eating. Go where you will between the hours of 8 in the morning and 6 in the evening, and you are reminded that man is a cooking animal. Tables are always spread; knives and forks are always rattling against dishes; the odors of the kitchen are always rising. Is the appetite of the Metropolis ever appeased?\n\nThis was how the reporter and author Junius Henri Browne introduced his chapter on restaurants in his Great Metropolis: A Mirror of New York (1869). The only difference between then and now is that in the city that never sleeps, it also never stops eating. It didn't start out this way.\n\nTaverns and Gardens\n\nThe city's first commercial eating establishments were taverns. Drinking, of course, was their primary function\u2014eating was an afterthought. The sale of alcoholic beverages was where taverns made their main profits\u2014just as it is in most New York restaurants today. The City Tavern may have been New Amsterdam's first tavern. Constructed in 1643, it was a two-story building with a basement. It could house several travelers and occasionally prisoners. It was also a place of business where public and private deals were transacted. In addition to selling beer, wine, and brandy, it also sold food, although precisely what kind of food has not survived. Other taverns were soon constructed, one by a Frenchman who named it The White Horse, a name that will be commonly employed by subsequent taverns. Another was opened by a Metje Wessels, whose \"eating house\" served beer and wine in 1656, and it became the place for dinners and festivities. Precisely what was served was not recorded. The exception was the terrapin, which was noted as one of the most popular dishes. Perhaps the Dutchman Adriaen van der Donck was referring to this when he wrote in 1655, \"Some persons prepare delicious dishes from the water terrapin which is luscious food.\"\n\nTraditionally, American taverns were licensed to sell wine for consumption on the premises; inns were licensed to lodge travelers and to sell beer and ale. Over time, these distinctions broke down\u2014taverns also lodged travelers, and both taverns and inns sold beer, ale, wine, and cider. By the early eighteenth century, rum was the prevailing beverage served in taverns. Higher-class establishments also offered coffee, tea, and chocolate. Taverns also served food. Unlike restaurants, which arrived in America in the early nineteenth century and provided menus (or bills of fare, or whatever they were), taverns served whatever was available, and diners usually had little or no choice as to their victuals.\n\nDutch taverns survived the English takeover of the city, but English taverns soon outnumbered them. Some, such as those operated by Roger Baker, Michael Howden, and John Parmyter, served the gentry. Elias Chardavoine's victualing house served French food, something that was quite unusual at the time. France, of course, was known for its haute cuisine, which was greatly appreciated by England's wealthy, but most Americans considered it effete and a waste of money.\n\nBy the 1730s, the social life of the city revolved around taverns, one of which invited diners to sample \"the most Dishes of Meat\" in the \"best Order,\" and \"drink the richest Wine,\" according to a newspaper advertisement. In 1775, Edward Bardin's tavern served \"Roast Beef, Veal, Mutton, Lamb, Ducks and Chickens, Gammon, Lobsters, Pickled Oysters, Custards, and Tarts of Different Kinds. Chicken Pies ready for Supper every night. Tea and Coffee every afternoon.\" The number of taverns and other alcohol sellers increased rapidly as the population grew.\n\nThe upper classes, particularly those engaged in businesses, frequented coffeehouses. In 1696, Lieutenant John Hutchins opened a coffeehouse called the King's Arms; it became the unofficial headquarters of English \u00e9migr\u00e9s in New York. Many more coffeehouses were opened in New York City during the early eighteenth century, serving good food as well as beverages.\n\nSome taverns sported \"gardens,\" offering summertime entertainment\u2014fireworks, concerts, plays, and outdoor dining. A 1763 advertisement for the \"Spring Gardens\" offered the \"best of green tea &c. Hot French rolls will be provided. N. B. Pies and tarts will be drawn from 7 in the evening till 9, where gentlemen and ladies may depend on good attendance; the best of Madeira, mead, cakes, &c.\" The most popular tavern garden was Vauxhall, named for the garden in London. It offered \"Reception of Ladies, Gentlemen, etc., and will be illuminated every evening in the Week; Coffee, Tea, and Hot Rolls at any hour in the day, neat Wines and other Liquors, with Cakes.\"\n\nThe city's most famous early garden was constructed by William Niblo, an Irish immigrant, who, in 1823, purchased the grounds of a small circus, called \"The Stadium,\" at the corner of Broadway and Prince Street. There he opened Niblo's Garden, featuring a restaurant and concerts. It quickly became one of the most important eating establishments. Particularly popular was its ice cream and its sherry cobblers, which were sipped throughout the gardens. The British aristocrat and philosopher Thomas Hamilton dined at Niblo's in 1833 and described his meal of \"oyster soup, shad, venison, partridges, grouse, wild-ducks of different varieties, and several other dishes.\"\n\nTaverns thrived well into the nineteenth century, but their functions\u2014drinking, eating, and sleeping\u2014were increasingly replaced by hotels, bars, saloons, boardinghouses, and independent restaurants. Taverns survived well into the twentieth century, but by then, they were little different than saloons and bars of the time. Some taverns became quite popular and famous: the White Horse Tavern in Greenwich Village was frequented by writers, actors, poets, counterculture leaders, and political activists in the 1950s and 1960s; and the Stonewall Inn tavern, also in Greenwich Village, is credited with launching the gay rights movement in America in 1969.\n\nHotel Dining\n\nThe nation's first luxury hotel was New York's City Hotel, built in 1794. For almost half a century the four-story, block-long hostelry on lower Broadway was New York's premier inn, the place where wealthy and celebrated visitors stayed when they visited the city. The hotel operated on the American plan, where guests paid both for rooms and all meals (in contrast to the \"European plan,\" which included breakfast only). Manhattan businessmen ate dinner or had tea in the hotel's dining room. Dinners often included choices of twelve to sixteen meat, poultry, or seafood dishes along with vegetables and fruit.\n\nAstor House, Broadway and Vesey Streets in New York City, 1907. Courtesy of the Library of Congress. Reproduction Number: LC-DIG-ds-01872\n\nBy the mid-1830s many New York hotel dining rooms were serving la haute cuisine fran\u00e7aise. At the Globe Hotel at 66 Broadway, customers enjoyed \"excellent French cookery,\" according to an English visitor. He was even more taken with the food at the palatial Astor House, which employed French chefs to run its dining rooms. Menus were incredible as illustrated by the Astor's \"Table-d' H\u00f4tel\" for Wednesday, March 21,1838:\n\nVermicelli Soup\n\nBoiled Cod Fish and Oysters\n\nBoiled Corn'd Beef\n\nBoiled Ham\n\nBoiled Tongue\n\nBoiled Turkey and Oysters\n\nBoiled Chickens and Pork\n\nBoiled Leg of Mutton Oyster Pie\n\nCuisse de Poulet Sauce Tomate\n\nPoitrine de Veau au Blanc\n\nSalade de Volaille\n\nBallon de Mouton au Tomate\n\nT\u00e9te de Veau au Marinade\n\nCasserolle de Fomme de Terre garnie\n\nCompote de Pigeon\n\nRolleau de Veau \u00e0 la Jardiniere\n\nC\u00f4telettes de Veau Saut\u00e9\n\nFilet de Mounton Piqu\u00e9 aux Ognons\n\nRonde de B\u0153uf\n\nFricandeau de Veau aux Epinards\n\nC\u00f4telettes de Mouton Pan\u00e9e\n\nMacaroni au Parmesan\n\nRoast Beef\n\nRoast Pig\n\nRoast Veal\n\nRoast Leg of Mutton Roast Goose\n\nRoast Turkey\n\nRoast Chickens\n\nRoast Wild Ducks\n\nRoast Wild Goose\n\nRoast Guinea Fowl\n\nRoast Brandt\n\nQueen Pudding\n\nMince Pie\n\nCream Puffs\n\nDessert.\n\nWhen the Astor House opened, guests paid three dollars per week for room and full board. Its bar was one of the largest in the city, and it offered a prodigious free lunch for all customers who bought drinks. Not surprisingly, the prices escalated rapidly as well-to-do foreign visitors discovered the hotel.\n\nThe success of the Astor House encouraged others to build luxury hotels. At the former site of Niblo's Garden on Broadway at Prince Street, Stephen Van Rensselaer built the Metropolitan Hotel in 1852. Its restaurant became one of the most important in the city. The St. Nicholas Hotel was built on Broadway at Broome Street and cost the unheard of price of one million dollars. It had two dining rooms and two taprooms. In the dining rooms, one English visitor in 1856 was amazed at the number and variety of dishes and beverages from which diners could choose:\n\nTwo soups, two kinds of fish, ten boiled dishes, nine roast dishes, six relishes, seventeen entries, three cold dishes, five varieties of game, thirteen varieties of vegetables, seven kinds of pastry, and seven fruits, with ice-cream and coffee. The wines numbered eight brands of Madeira, seventeen of sherry, eighteen of champagne, six of port, four of Burgundy, twenty of hock, sixteen of claret, six sauternes, nine varieties of brandy, three liqueurs, and Scotch ale, India pale-ale, and London porter.\n\nMenu from the Fifth Avenue Hotel, 1903. Courtesy of the Library of Congress. Digital id: rbpe 1310100a <http:\/\/hdl.loc.gov\/loc.rbc\/rbpe.1310100a>\n\nThe Fifth Avenue Hotel facing Madison Square surpassed all hotels in New York when it was completed in 1859, and its menu at its restaurant was impressive. Its culinary delights, however, were topped with the completion of New York's most famous hotel and restaurants at the Waldorf Astoria Hotel, which started as two separate hotels. William Astor, the grandson of John Jacob Astor, built the Waldorf Hotel in 1893 on Fifth Avenue and Thirty-fourth Street. Four years later, his cousin, John Jacob Astor III, built the Astoria Hotel on a lot adjacent to the Waldorf. A corridor between the two hotels was constructed, and the complex became known as the Waldorf Astoria; it is generally referred to, however, as the Waldorf.\n\nThe Waldorf was particularly famous for its dining rooms. One reason was a Swiss immigrant, Oscar Tschirky, who arrived in New York in 1883 and worked as a busboy at Hoffman House and later waited tables at Delmonico's. Before long he became head of the restaurant's catering department, where he became popular with some of New York's most famous people. Tschirky joined the staff at the Waldorf two months before it opened. His management skills helped launch the hotel's restaurants. He was put in charge of its private dining rooms, which were frequented by the city's most famous people, such as the financier J. P. Morgan, the actress Lillian Russell, and the wealthy, overweight businessman and gourmet Diamond Jim Brady. Tschirky made a point of remembering what important guests preferred to eat and drink. His success in the dining rooms elevated him to the position of hotel steward responsible for supplying and preparing all the restaurants with provisions, and he managed all of the restaurant staff, including thirty-five chefs and hundreds of employees. The restaurants served French cuisine, and Oscar, as he was called (because most Americans were unable to pronounce his last name, he asserted in his autobiography), proclaimed that eating at the Waldorf was just as good as eating in a top Parisian restaurant.\n\nPalm Court dining room at the Waldorf Astoria Hotel, circa 1902. Courtesy of the Library of Congress. Reproduction Number: LC-DIG-ds-02798\n\nTschirky instituted a number of changes at the hotel that were later adopted by other restaurants. Traditionally, upper-class menus were written in French; the Waldorf's menus were written in English, although a bilingual menu was available on request. It was not the first restaurant in America to do so, but other city restaurants soon followed its lead. Another change was in the number of items listed on the menu. Some haute cuisine restaurant menus had reached gargantuan proportions, weighing in at a thousand items. Tschirky maintained the quality of the French food served at the hotel but simplified the menu and decreased the number of courses served. Other city restaurants soon followed his lead with this innovation as well.\n\nIn January 1907, the Waldorf Astoria made a bold move\u2014it put up a sign on the hotel's bulletin board announcing that \"ladies without escort will be served in the restaurants at any hour.\" This broke an unwritten rule of ignoring single women who sat down at its restaurants. Upper-class restaurants had \"discouraged\" unescorted women diners. Waiters ignored them and refused to take their orders. This policy, of course, did not apply to wealthy women known at the restaurants. Women could dine in private meeting rooms at hotels and restaurants. On April 20, 1868, the city's prominent women intentionally met at Delmonico's large meeting room on its second floor and formed the Sorosis Club, the nation's first club for professional women. It sent a shock wave through the city, but the club continued to meet, usually for lunch at meeting rooms of high-end restaurants, such as the Waldorf Astoria.\n\nThe Waldorf restaurants were the first upper-class eateries in the city to announce that single women could dine alone, and strong public pressure forced them to reverse this policy shortly after they made it, but it did begin public discussion and some eateries changed. At the time, single women could dine at tea rooms and at the Women's Lunch Club, where exotic salads and \"ice cream overpoured with maple syrup and walnuts, and other innovations of endless charm\" were served, reported a writer. In 1909, women shoppers were served lunches \"with expedition and sometimes with courtesy\" on the top floors of department stores. When the Nineteenth Amendment to the Constitution granted women the right to vote in 1920, single women demonstrated their freedom by visiting speakeasies, nightclubs, and restaurants throughout the city, including the Waldorf.\n\nThe Waldorf influenced what Americans ate. The hotel restaurants popularized chafing dishes, which they used to prepare two of their iconic dishes: chicken \u00e0 la king and lobster Newburg. Neither of these dishes originated at the Waldorf, but it did originate the \"Waldorf salad,\" consisting of chopped celery, apples, and mayonnaise, artistically arranged on a bed of lettuce. Walnuts and other items were added later. It quickly became popular throughout the nation, as did \"Oscar\" sauce, a sauce that the hotel bottled and sold.\n\nProhibition closed most of New York's haute cuisine restaurants; the Waldorf restaurants did survive, but they did not thrive. Tschirky believed that \"Prohibition ruined fine cooking and robbed America of some of its most distinctive native dishes,\" such as terrapin stew and lobster Newburg, for the well-to-do who could afford these dishes and were not interested in them without accompanying alcohol.\n\nThe owners of Waldorf and Astoria agreed in 1929 to move their hotels to make way for the Empire State Building and to relocate to Park Avenue, where the now merged Waldorf Astoria Hotel was even grander than their previous incarnations. It had three dining rooms, one of which was the Palm Court, where Tschirky served as ma\u00eetre d'h\u00f4tel until he retired in 1943. His influence survived as other restaurateurs studied and copied his mode of operating.\n\nOyster Saloons\n\nOysters were one of New York's most prominent and cherished foods. From the earliest colonial days, vendors sold them on the streets. Small wagons provided New Yorkers with oysters, \"biscuits, pepper, and ginger beer; in short, for a few pence, the carter or mechanic has a whet which might satisfy even a gourmand.\" Others sold oysters flavored by ketchup for a penny apiece from \"rude huts, paralytic shanties\" along the East River. Those who frequented \"these al fresco oyster-houses are longshoremen, truckmen, stevedores, sailors, and others of that ilk, and a very large bowl of oyster soup, not stew, can be obtained for 5 cents.\" Others sold oysters from \"floating-houses,\" which preserved live oysters for several days by placing them in baskets in the cellar and attic of the oyster boat.\n\nA step up from the floating houses were a large number of \"oyster saloons,\" \"oyster and coffee saloons,\" and \"oyster and lager beer saloons.\" They were mainly constructed in cellars and typically identified themselves with balloons made of \"bright red muslin stretched over a globular frame of rattan or wire and was illuminated at night by a candle placed within.\" Most oysters were served raw, but some were cooked in various ways. Many establishments were on the \"canal street plan,\" which meant that you could eat as many oysters as you liked for twelve cents. By 1835, an estimated five thousand New Yorkers were engaged in retailing oysters.\n\nRumors abounded that bartenders gave spoiled oysters to those who ate too many.\n\nCharles Dickens, visiting New York in 1842, wrote that he considered such places \"pleasant retreats\" with \"wonderful cookery of oysters, pretty nigh as large as cheese plates.\" Another British visitor reported that \"there is scarcely a square without several oyster-saloons; they are aboveground and underground, in shanties and palaces. They are served in every imaginable style escolloped, steamed, stewed, roasted, 'on the half shell,' eaten raw with pepper and salt, devilled, baked in crumbs, cooked in pates, put in delicious sauces on fish and boiled mutton.\"\n\nAccording to Scottish poet and journalist Charles Mackay who visited New York in 1859, the oysters were\n\ncooked in twenty, or, perhaps in forty or a hundred, different ways. Oysters pickled, stewed, baked, roasted, fried, and scolloped; oysters made into soups, patties, and puddings; oysters with condiments and without condiments; oysters for breakfast, dinner, and supper; oysters without stint or limit\u2014fresh as the fresh air, and almost as abundant\u2014are daily offered to the palates of the Manhattanese, and appreciated with all the gratitude which such a bounty of nature ought to inspire.\n\nOyster cellars catered to the well-to-do as well as the less affluent. Upper-class oyster bars were filled with mirrors, pictures, and other artistic enhancements. Tables could be closed off with curtains, which provided customers with privacy. Less ornate cellars had a counter at one side. Behind it, oyster shuckers, frequently blacks, quickly opened oysters \"for their customers, who swallow them with astonishing relish and rapidity.\" Good shuckers could open 3,500 oysters a day\u2014but the average was about 2,500.\n\nThe greatest concentration of oyster houses was found along Canal Street, but again, according to an English visitor in the 1860s, the oyster should be called \"the national dish; it is at least the great dish of the Atlantic States.\" He continued,\n\nThe oyster is the sine qua non of all dinner parties and picnics, of all night revels and festive banquets. For tenpence you may have a large dish of them, done in any style you will, and as many as you can consume. The restaurants ostentatious and humble are in the season, crowded with oyster lovers: ladies and gentlemen, workmen and seamstresses, resort to them in multitudes, and for a trifle may have a right royal feast.\n\nOysters were so plentiful that everyone thought the supply was inexhaustible. But there was a limit. The beds were overharvested, and production rapidly declined in the 1840s. To meet demand, oysters were shipped in from Long Island Sound and then from the Chesapeake Bay. Estimates varied as to how many vessels were engaged in the oyster trade: some said 1,500; others said ten times that. But it was a dying occupation. Pollution and untreated human waste that drained into the harbor destroyed oyster beds. The beds in Harlem River were closed in the 1870s. Health scares associated with them frightened New Yorkers away from the dives and street vendors. Many oyster saloons became seedy, and upper-class New Yorkers migrated to other eateries. When oysters were thought to have caused a typhus outbreak in the 1890s, more beds were closed by health officials. By 1916, most oyster beds in New York Bay were closed or destroyed by pollution. The most iconic oyster restaurant in the city today is the Grand Central Oyster Bar, which opened in 1913. It imports all of its oysters from Long Island and other areas where oysters thrive. Its signature dishes include oyster pan roast and oyster stew.\n\nOyster Bar Restaurant, Grand Central Terminal, Manhattan, 2013. Courtesy of Kelly Fitzsimmons\n\nBoardinghouse Fare\n\nA British actor who immigrated to America in 1797 reported that New York was more a village and a collection of inns than a real city. But he was delighted with the culinary choices at one upscale boardinghouse: \"[It served] fish, ham, beef, boiled fowls, eggs, pigeons, pumpkin pies, lobsters, vegetables, tea, coffee, cider, sangaree, and cherry-brandy!\" A later British visitor to New York also thought well of the fashionable boardinghouse where he stayed:\n\nIn the first-rate boarding-houses of the American commercial capital, there is an entire absence of all vulgarity. The attendance and culinary arrangements are after the models of some of the best English private establishments, and the general tone of the society refined enough for the most fastidious; nor are the expenses so exorbitant as in many a noisy, disagreeable hotel.\n\nBoardinghouses had been common in New York since colonial days. As merchants, clerks, salesmen, seamstresses, mechanics, and immigrants streamed into the city in the early nineteenth century, lower Manhattan became crowded. The island's well-to-do sold their three- and four-story houses and moved uptown. Entrepreneurs purchased the old houses, subdivided the rooms, and converted them into boardinghouses. As landlords made more money selling liquor and other provisions than they did by renting out the rooms, they often converted the parlors or cellars into saloons, or made them into grocery stores that sold liquor. There were not enough old houses to meet the needs of incoming immigrant families, so entrepreneurs began purchasing old nonresidential buildings and converting them into boardinghouses.\n\nSome boardinghouses catered to specific groups, such as Germans, Irish, or Chinese. Others catered to specific professions, such as actors. They were less expensive than hotels, and guests often remained in residence for long periods of time. Guests were typically served three meals a day. In 1842, Walt Whitman, then a twenty-two-year-old New York journalist, proclaimed that the \"universal Yankee nation\" was a \"boarding people.\" But sixteen years later he grossly exaggerated in proclaiming that three-quarters of the adults in New York City lived in boardinghouses.\n\nBoardinghouse fare depended on the proprietors, and many residents were unhappy with what they were served. Asa Greene, author of the novel The Perils of Pearl Street (1834), recorded his impressions of the food served by a boardinghouse landlady he called \"Mrs. Conniption.\" Breakfast consisted of salted shad or mackerel, dried beef steak, and occasionally stale sausages, or heavily salted pork, and bread that was \"heavy as a grindstone.\" Water \"drawn from the Manhattan hydrant or the pump\" was converted into a hot drink by the addition of \"a small quantity of damaged coffee, burnt crust, or roasted rye, well pulverised.\" Dinner included cheap meat (mutton, beef, or pork), which \"was dry as a chip, and totally destitute of any inviting qualities.\" Vegetables consisted of \"watery potatoes, sliced beats, boiled cabbage, and so forth.\" Sour apple dumplings, rice pudding with molasses, tart apple pies \"with the crust as strong as sole-leather,\" or \"shrivelled peaches\" were served for dessert. The tea table consisted of dry bread with \"extra-salted butter,\" and weak tea that was made with a \"thimbleful of tea . . . put into a quart, a gallon, or some other assignable quantity of water.\" This was flavored with small amounts of brown sugar and watered-down milk.\n\nIn the same spirit, Thomas Butler Gunn, author of The Physiology of New York Boarding-Houses (1857), wrote that \"meals were uniformly served up 'neither cold, nor hot.'\" The soups \"might have been improved by a less liberal allowance of grease and unground pepper, of which latter there always remained a deep sediment\u2014as of small shot\u2014in each plate.\"\n\nVegetarian Eateries\n\nOther boardinghouses were established by vegetarians and temperance advocates. These were promoted or inspired by Sylvester Graham. In 1833, Asenath Nicholson launched a temperance boardinghouse. She teamed up with Graham to institutionalize his dietary principles in her establishment, and she later published the first \"Graham\" recipes devised in accordance with his precepts. Other Graham boardinghouses opened, and many survived until late in the nineteenth century. Noted guests at such places included New York newspapermen Horace Greeley and William Lloyd Garrison. The aforementioned Thomas Butler Gunn was surprised and pleased with the offerings at one of the Graham houses:\n\nWe had no meats, no fish, no gravy-soups. Tea and coffee were also rejected, as stimulants. But every variety of vegetable appeared at our table, as also fruit and pastry. (No butter entered into the composition of the latter, that being a tabooed article.) Bananas, melons, peaches, grapes, oranges, cherries, pine-apples; all the daintier forms of Vegetarian fare were provided with a liberal hand. The display, indeed, exceeded our expectations.\n\nThe New York Vegetarian Society was organized in 1852; it organized dinners. One in 1894 held at the St. Denis Hotel in 1894 offered an impressive menu:\n\nSoup.\n\nCream of celery.\n\nRelishes.\n\nOlives. Tomatoes. Cucumbers. Salted Almonds. Pickled Walnuts. Haricot beans on bread with curry sauce.\n\nRemoves.\n\nBraised lettuce with mushroom sauce and celery croquettes.\n\nStewed oyster plant with rissole and sweet potatoes Lyonnaise.\n\nBrussels sprouts, cream sauce.\n\nLemon ice.\n\nFrench peas, country style.\n\nBaked stuffed tomatoes with spaghetti a la Milannaise.\n\nFried squash, Creole style, with corn fritters.\n\nMixed salad with toasted crackers.\n\nSweets.\n\nRice and apricots. Croute of mixed fruits. Orange salad. Nesselrode pudding.\n\nDessert.\n\nStilton, Roquefort, and Camembert cheese. Fruits of the season. Nuts. Raisins. Cakes. Tea. Coffee. Chocolate.\n\nVegetarian boardinghouses went out of business by 1900, but vegetarian restaurants, such as The White Rose or The Laurel, were operating in the city. These sold vegetarian steaks, cutlets, filets, and duck. According to one author in 1904, their vegetables were \"amazingly\" well cooked; he also reported that their mushroom and nut dishes were outstanding, a black cream of mushroom soup was \"worthy of a plutocrat\" and the coffee made from peanuts was delicious. The publication of Upton Sinclair's Jungle (1906) and other exposes about the meatpacking industry encouraged many New Yorkers to become vegetarians. Vegetarian restaurants survived and today thrive as do vegan and raw food restaurants.\n\nTenements\n\nBoardinghouses were so profitable that entrepreneurs began converting large warehouses and other buildings into \"tenant houses,\" later called tenements. A large brewery erected in 1792 in the Five Points neighborhood, for instance, was in 1837 converted into a dwelling that eventually housed more than seven hundred Irish and African Americans. Unlike boardinghouses, tenements did not provide food for those renting rooms and often did not have kitchens.\n\nAs more immigrants flooded into the city, entrepreneurs converted larger old buildings for multiple dwellings. The rooms were tiny\u2014often just big enough for a bed\u2014and most did not have running water, heat, ventilation, or indoor plumbing. Instead, there were outdoor privies with inadequate sewage systems. Water had to be carried in buckets from wells or pumps on the street. New York City and State passed laws to improve conditions in tenements, but the laws often went unenforced. As more immigrants moved into the tenements, they became dangerously overcrowded. According to the New York Tribune, garbage was set in boxes on the streets: the waste was \"composed of potato-peelings, oyster-shells, night-soil, rancid butter, dead dogs and cats, and ordinary black street mud.\" The garbage boxes formed \"one festering, rotting, loathsome, hellish mass of air poisoning, death-breeding filth, reeking in the fierce sunshine, which gloats yellowly over it like the glare of a devil whom Satan has kicked from his councils in virtuous disgust.\"\n\nAs more immigrants streamed into the city, more tenements were created to house them. In 1900, there were 42,700 tenement buildings in New York, housing almost 1.6 million people. Jacob Riis's expos\u00e9 How the Other Half Lives: Studies among the Tenements of New York (1890) concluded that more than half a million people living in the tenements were either begging for food or subsisting on charity.\n\nOne building that was, surprisingly, converted into a tenement was Fraunces Tavern. Originally built as a family home, it was converted into a tavern in 1762. In the early nineteenth century, it was converted back into a house, with street-level shops added later, before it ended up as a tenement for immigrants. In 1904, the building was bought by the New York chapter of the Sons of the American Revolution, who tried to restore the building to its original eighteenth-century condition. Today, it claims to be the oldest surviving building in Manhattan; it functions as a museum and a restaurant serving food and beverages loosely based on colonial fare.\n\nAnother tenement building, at 97 Orchard Street on the Lower East Side, was built in 1864. It was designed to have twenty three-room apartments. Each apartment\u2014about 325 square feet\u2014would typically house a family of seven or eight. There was no running water and no toilets, although the apartments did have small kitchens. Residents were able to prepare simple meals. In 1988, this became the Tenement Museum.\n\nRoom in The Tenement Museum , 97 Orchard Street. Courtesy of the Library of Congress. Reproduction Number: LC-DIG-highsm-14234\n\nForeign Food\n\nBy the mid-nineteenth century, restaurants, bistros, and cafes thrived in New York, as did more affordable oyster houses and other inexpensive eating establishments. Manhattan alone had an estimated five or six thousand restaurants and eating houses. They ranged from the elegant and expensive Delmonico's to \"the subterranean sties where men are fed like swine, and dirt is served gratis in unhomoeopathic doses\" with \"broken earthen-ware, soiled table-cloths, and coarse dishes,\" as a New York reporter wrote in 1869.\n\nMany restaurants served \"foreign\" food, and some were widely frequented. \"Gosling's French and American Dining Saloons\" on Nassau served a thousand customers a day at lunch. It was good and cheap. In 1843, three customers ate \"roast veal farci, with two slices of bread, potatoes, a pickle, and a 'smaller' of beer,\" and the total cost for all three diners was about thirty cents. By comparison, a meal for three at Delmonico's cost more than fifteen dollars. Henri Mouquin's opened in 1857 on Fulton and Nassau. Their \"diner du jour\" consisted \"of soup, one entree, and cut of roast meat with one vegetable, and cheese with bread at discretion.\" It cost \"25 cents, with a pint of red wine at 12 cents extra,\" according to a magazine writer in 1866. Mouquin later moved his restaurant uptown to Twenty-eighth Street and Sixth Avenue where he upped the prices and became a \"rendezvous for epicures.\"\n\nChinese, Italian, and German restaurants thrived in New York by the late nineteenth century. Chop suey joints became common in Five Points (an area later called Chinatown) beginning in the late nineteenth century. By 1898, eleven Chinese restaurants could be found on Mott and Pell Streets. By the 1930s, Chinese restaurants were common even on Times Square. Italian restaurants opened as immigrants from Italy flooded into the city, but in the 1890s, they rapidly increased as New Yorkers fell in love with spaghetti joints where meals were accompanied by a bottle of red wine. Others became very popular, such as the one opened by Stefano Moretti on Fourteenth Street in the 1860s. German restaurants were common in New York by the 1870s. They were located off streetcar lines and sold German foods and wines. Some served the city's well-to-do, such as L\u00fcchow's in Manhattan and the Peter Luger Steak House in Williamsburg.\n\nChophouses, \"adaptations\" of English grillrooms, were also common. In the 1860s, they were clustered around Houston and Bleecker Streets. They served a wide range of foods, such as served \"stewed tripe, liver and bacon, mutton-chops, porter-house steak, and cuts from 'joints.' The bill of fare is a written one hung up at the bar. The prices are moderate, and the food better cooked than in almost any of the other eating-houses.\" The De Soto, the leading chophouse in 1866, was run by an English immigrant named William H. Garrard. It was frequented by actors and \"dramatic critics\" who ate \"broiled kidneys and Welch rarebits\" and drank ale and cocktails. Forty years later, among the popular chophouses were Farrish's on John Street, Engel's near Herald Square, and Browne's on Broadway near Fortieth Street. These were \"much frequented\" by \"actors and singers and have fascinating collections of old prints and quaint photographs.\" They were inexpensive. For \"a chop, baked potato, a bit of water-cress, plenty of good bread, and English pickles,\" the cost was forty to eighty cents. They also served \"broiled kidneys, porterhouse steaks, imported Bass, porter, or stout, Scotch ale, or half and half (properly pronounced \"'arf 'n' 'arf\"), served in the pewter and drawn from the wood, are also to be had at their best at these places, and at a reasonable price.\"\n\nBy the time that the Depression hit, foreign food was readily available, as a writer noted in 1931:\n\nWithin a few square miles, you can sample the foods of India, Syria, Japan, and Normandy; you can eat the foods of the French, the German and the Irish; the specialties of the Italians and the Swedes and the Russians and the Danes. You can drink Turkish, European and Florentine coffee; rose-water or Danish beer. You can revel in Smorgasbord, Hors d'oeuvres, or Antipasto; and buy a meal for 50 cents or 50 dollars.\n\nThere were also Jewish restaurants. The most famous was the Caf\u00e9 Royal, on Second Avenue, then the heart of New York's Yiddish rialto. It opened in 1908 and quickly became the place where socialists, artists, writers, and Yiddish performers mingled. In his book The Strangest Places (1939), Russian immigrant Leo Calvin Rosten proclaimed the Caf\u00e9 Royal to be \"the Delmonico's, the Simpson's and the Fouquet's of Second Avenue, all in one.\" He continued, \"Everybody who is anybody in the creative Jewish world turns up at the Caf\u00e9 Royal at least one night a week. To be seen there is a social duty, a mark of distinction, and an investment in prestige.\"\n\nForeign and ethnic food was and is a defining element in the city's foodscape, and today restaurants offer foods and beverages from hundreds of different national, regional, and ethnic groups.\n\nSpeed Dining\n\nUntil the mid-nineteenth century, New Yorkers engaged in business went home for afternoon dinner. As the city grew and residences were separated from places of employment, lunchrooms emerged. They typically offered fast service and cheap food. Most customers ate quickly and returned to work. An English visitor in 1828 described the \"Plate House,\" one of the city's early quick-service lunch establishments, as a long, narrow, dark room with two rows of boxes \"just large enough to hold four persons.\" Attendants glided \"up and down, and across the passage, inclining their heads for an instant first to one box, then to another, and receiving the whispered wishes of the company, which they straightway bawled out in a loud voice, to give notice of what fare was wanted.\" As customers ordered their food, attendants yelled them out: \"Three beef, 8 !\" \"Half plate beef, 4!\" \"One potato, 5!\" \"Two apple pie, one plum pudding, 8!\" The numbers at the end identified the table. Other colloquial names for dishes were \"a plate of Siamese Twins\" (fish balls); \"woodcock\" (pork and beans); \"boned turkey,\" \"corduroy,\" and \"West Broadway\" (hash); \"Irish goose\" (codfish, baked or boiled); and \"a plate of Tennessee, and be quick about it\" (hot corn bread). Within seconds the food was delivered. Customers ate in silence, ate quickly, and left.\n\nTime was money, whether at a restaurant or in a boardinghouse. Speed of delivery and speed of eating was everything. In the 1830s, an Englishman described breakfast at his boardinghouse:\n\nHere was no loitering nor lounging; no dipping into newspapers; no apparent lassitude of appetite; no intervals of repose in mastication; but all was hurry, bustle, clamour, and voracity, and the business of repletion went forward, with a rapidity altogether unexampled. The strenuous efforts of the company were, of course, soon rewarded with success. Departures, which had begun even before I took my place at the table, became every instant more numerous, and in a few minutes the apartment had become . . . \"a banquet-hall deserted.\"\n\nAnother reported that New Yorkers spent \"from five to ten minutes for breakfast, fifteen to twenty for dinner, and ten for supper. . . . Each person, as soon as satisfied leaves the table without regard to his neighbors; no social conversation follows.\"\n\nLater in the century, a French visitor observed, \"In five minutes they manage to gulp down a certain amount of food, pay and go. An intelligent barkeeper attracts customers by the facilities he can offer for rapid feeding. 'Try our quick lunch' is stuck up in the streets.\"\n\nAnd there was noise. As an American reporter noted in 1869,\n\nFrom 12 o'clock to 3 of the afternoon, the down-town eating-houses are in one continuous roar. The clatter of plates and knives, the slamming of doors, the talking and giving of orders by the customers, the bellowing of waiters, are mingled in a wild chaos. The sole wonder is how any one gets anything; how the waiters understand anything; how anything is paid for, or expected to be paid for. Everybody talks at once; everybody orders at once; everybody eats at once; and everybody seems anxious to pay at once.\n\nMost lunchrooms were mom-and-pop shops that sold sandwiches, soup, and few other items. By the 1920s, lunchroom chains began to develop in New York. Most chains had only two or three establishments. These shared a name and were advertised as a system. Large chains might also have had central food distribution and preparation. One such chain was W. F. Schrafft's, launched by Frank Shattuck in 1898 as a candy store in Boston. By 1915, Schrafft's had extended its operation to New York City. By 1922, the chain had twenty-two stores.\n\nFaster and more efficient ways of feeding workers continued to emerge in the city. One solution that developed was the Exchange Buffet, which opened in 1885 across the street from the New York Stock Exchange. It catered only to males, who proceeded down a long buffet containing items such as sandwiches, salads, and cakes to eat and tea, coffee, or milk to drink. They picked the foods that they wanted and juggled them in their hands to tall counters where they ate standing up. They bused their own dishes, tallied their own bill, and then told the cashier what he owed. This self-service operation based on the honor system was surprisingly successful, and the company opened thirty-five more establishments in Manhattan and Brooklyn during the next few years.\n\nOther businessmen saw crowds surging about the Exchange Buffet and developed their own model. In 1889, William and Samuel Child opened their first lunchroom on the main floor of the Merchants Hotel. Cleanliness was an important part of the operation, and they dressed waitresses in white aprons and white caps. Unlike the Exchange Buffet, they catered mainly to women. They are also credited with introducing the tray, so that customers didn't have to juggle their dishes. This was so successful that the brothers opened more cafeterias, more than one hundred by 1925.\n\nCafeterias particularly thrived in large institutions\u2014corporate dining rooms; government offices; museums; military installations, such as Governor's Island; and prisons, such as Riker's Island. New York YMCAs opened their first cafeteria in 1916. New York schools soon opened cafeterias for their students. By 1929, the city had 786 cafeterias.\n\nA restaurant chain that thrived during the Depression was Chock full o'Nuts, which was launched by a Russian immigrant, William Black, who had opened a nut stand in a basement in Times Square in 1926. The chain was successful, and Black opened eighteen additional shops around the city. When the Depression hit, he converted his nut shops into budget luncheonettes, charging a nickel each for coffee and a sandwich consisting of cream cheese, chopped nuts, and lightly toasted date-nut bread. He later added soup and pie to the menu. By the 1950s, he owned twenty-five outlets in Manhattan and two in Brooklyn. In 1953, he came out with his own brand of coffee, also called Chock full o'Nuts. It later expanded to a chain with 125 restaurants. Although the chain did not survive, its brand of coffee did.\n\nYet another type of self-service operation that thrived in New York was the coin-operated vending machine. They were developed in the United Kingdom in the 1880s. The Thomas Adams Gum Company began selling gum in vending machines on New York's elevated train platforms in 1888. Coin-operated machines offered the means to sell food in many places without the need for a sales force. They took off nationally in 1901, when F. W. and H. S. Mills debuted penny-in-the-slot machines, which became ubiquitous across America shortly thereafter selling gum and candy.\n\nEuropeans began selling a wide variety of foods through vending machines, especially in transportation hubs. Philadelphians Joseph Horn and Frank Hardart ordered the coin-operated machines from a German firm and, in 1902, opened America's first Automat. These were self-service operations that dispensed food through coin-operated machines with glass windows. Food, such as sandwiches, pies, coffee cake, cookies, pudding, hot dogs, vegetables, and side dishes, was prepared in advance and placed in glass compartments so that customers could see what they were buying. Heated or refrigerated compartments provided hot and cold food. Empty compartments were filled by workers who prepared food behind the row of machines.\n\nIn December 1902, James Harcombe, a New York restaurateur, opened the first Automat in New York. It was fifty feet long and twenty-five feet wide, but it had \"an attractive front and a handsome interior,\" reported the New York Times, and it should have, for Harcombe spent seventy-five thousand dollars in lavishly decorating the restaurant. Scientific American stated that, when compared with the average caf\u00e9, the Automat was \"illuminated with extravagant splendor but is a dismal place compared with it.\" He advertised it as \"Europe's Unique Electric Self-serving Device for Lunches and Beverages. No Waiting. No Tipping. Open Evenings until Midnight.\" Prices ranged from a nickel to a quarter. To compete with high-end restaurants, the Automat offered a wild assortment of lavish dishes, including \"lobster \u00e0 la Newberg.\" In addition to food and coffee, it also offered beer, wine, and cocktails. Despite widespread promotion, the restaurant did not thrive, and it closed about 1907.\n\nHorn & Hardart expanded its operation to New York City in 1912, and within a few years, the company operated fifteen Automats in the city. In September 1922, the company opened a combination Automat cafeteria, which they claimed was the largest restaurant in the city with the capacity to feed ten thousand New Yorkers daily. (But one of the most poignant images of the Automat, painted by Edward Hopper in 1927, was a woman alone drinking a cup of coffee in an Automat.) In order to maintain the quality of its operations, the company opened a central commissary that supplied all of its operations in New York. Horn & Hardart did well during the 1920s and thrived during the Depression. In 1933, the company did something unusual\u2014it hired Francis Bourdon, a classically trained chef who had worked at top restaurants in Europe and America, to run its commissary, a position he held for the next thirty-five years. He developed new recipes, gave the chain some prestige, and guaranteed that the quality of food remained the same in every outlet.\n\nAfter World War II, Horn & Hardart expanded to fifty outlets in the city, and by 1950 most had converted into cafeterias. Then decline set in. It was a combination of issues\u2014New York City began charging tax on food, and it was impossible to put pennies into the nickel slots. The company lost money on its popular high-quality coffee, so it upped the price to a dime. Business declined when they began to water down their coffee and decrease the quality of their food. Finally, fast-food establishments took away even more business. Within four decades, most automats were closed, many having been converted into Burger King outlets. The last Automat in New York closed in April 1991. But this was not the end of the concept. Many businesses and colleges have cafeterias, where workers and students can acquire food items through coin-operated machines, and of course, coin-operated candy, snack, and soda machines are ubiquitous throughout New York.\n\nCheap food was available from other eating establishments. By the 1880s, cheap hash houses were common. In these, customers acquired bread and butter, potatoes, and pickles for just fifteen cents. Hamburger steak was available for just eight cents. Even cheaper food could be had at the Italian basement eating houses, which served macaroni, toast, and coffee for a nickel. The winter of 1902 was extremely cold, and many New Yorkers were out of work. Bernarr Macfadden opened a cafeteria for the unemployed at City Hall Park. It charged a penny per course, and when it closed in the spring, it actually made a profit. When the Depression hit in 1929, Macfadden duplicated and expanded his concept. Yet another response to the Depression, at least for the customers who had some money, was the \"all you can eat\" restaurant. These were restaurants with chairs, tables, and waiters. Customers were charged an entrance fee\u2014sixty cents or a dollar\u2014and then they could eat all they wanted on any item on the menus.\n\nHigh-End Restaurants\n\nThe French and Swiss never immigrated in large numbers to New York, but they had a disproportionate influence on New York food, for many were cooks, confectioners, chocolatiers, bakers, restaurateurs, and caf\u00e9 owners. After the French Revolution in 1789, many French refugees came to the United States, where they shared the legacy of French cuisine. One of those who arrived on American shores was Captain Joseph Collet, who opened an ice-cream parlor. The French writer Jean Anthelme Brillat-Savarin wrote admiringly that Collet had \"earned a great deal of money in New York in 1794 and 1795, by making ices and sherbets for the inhabitants of that commercial town.\" Collet opened the \"Commercial Hotel\"\u2014a fancy name for a boardinghouse with a coffeehouse on the lower floor.\n\nFran\u00e7ois Guerin opened a confectionery and a small lunchroom on lower Broadway opposite the City Hotel in 1815. He sold \"imported and domestic confectionery, inviting specimens of pastry and cake, bottles of choice French cordials, fancy boxes filled with Parisian bon-bons interspersed with the fruits then in market.\" He also served soup and sandwiches and sardines. Louis Curtillet, an immigrant from Provence, sold \"meat pies, plum puddings, confections de cr\u00e8me glac\u00e9es, and punch \u00e0 la Romaine.\" In the 1840s, John Pinteux's Caf\u00e9 de Mille Colonnes offered coffee, juleps, and ice cream. It was patronized by the \"gayer bloods, who loved glare and glitter and the noise of Broadway.\"\n\nIn December 1827, Swiss immigrants John and Peter Delmonico opened a little caf\u00e9 on William Street, near the harbor. It had just six tables; there, patrons could enjoy European-style pastries, ices, bonbons, cakes, and coffee. In March 1830, the caf\u00e9 was expanded into a fine dining establishment that was listed in the city directory as \"Delmonico & Brother, Restaurant Fran\u00e7ais.\" After a fire swept lower Manhattan and their caf\u00e9 was destroyed, the Delmonico brothers bought Collet's caf\u00e9 and boardinghouse, while they began to construct a new restaurant.\n\nAn enlarged Delmonico's on South William Street opened in 1837. It offered salads and other novel French dishes, and the restaurant became very popular. Its eleven-page menu listed a full selection of French dishes (with English translations opposite), such as Potage aux huitres, Salade de chicor\u00e9e, Blanquette de Veau \u00e0 la Perigueux, Meringues \u00e0 la cr\u00e8me, and an impressive French wine list with Bordeaux, vintage 1825. Delmonico's quickly became the city's premier restaurant where the upper classes and visiting dignitaries dined. While some meals could be had for less than fifteen dollars for three, other meals weighed in at more than eighty dollars per person, a very costly meal at the time. But the food was good. Frederick Marryat, an English naval officer, traveler, and novelist, visited Delmonico's and other French restaurants in the 1830s and reported that they served \"excellent\" French food. A visiting Frenchman, Georges Savuvin, went even further, when he reported in 1893 that the food served at Delmonico's was \"better and more sumptuous\" than that served \"in the best restaurant in Paris.\"\n\nNewspapers and magazines reported on the lavish dinners for the city's upper crust, as well as visiting dignitaries and royalty. One dinner and ball attended by eight hundred of New York's social elite cost $350,000, an exorbitant amount in 1897. Another dinner for forty guests cost $10,000, or $250 per dinner, again an unheard of amount at the time.\n\nAnnual Banquet of the Sons of the Revolution in the State of New York, Delmonico's restaurant, February 22, 1906. Courtesy of the Library of Congress. Reproduction Number: LC-USZ62-139580\n\nDelmonico's success encouraged the establishment of other French restaurants. By the mid-1830s, many New York hotels served French food in their restaurants. The Astor House employed French chefs to run its dining rooms, and a number of these men became quite famous. The French influence was clearly visible on the menus of these restaurants as well as the way the food was served via service \u00e0 la fran\u00e7aise (service in the French style), meaning that all food was served at the same time. Carl Schurz, a German visitor to the city in the 1850s, described this experience at the Union Square Hotel restaurant:\n\nSome fifteen or twenty negroes, clad in white jackets, white aprons, and white cotton gloves, stood ready to conduct the guests to their seats, which they did with broad smiles and curiously elaborate bows and foot scrapings. A portly colored head-waiter in a dress coat and white necktie, whose manners were strikingly grand and patronizing, directed their movements. When the guests were seated, the head-waiter struck a loud bell; then the negroes rapidly filed out and soon reappeared carrying large soup tureens covered with bright silver covers. They planted themselves along the table at certain intervals, standing for a second motionless. At another clang of their commander's bell they lifted their tureens high up and then deposited them upon the table with a bump that made the chandeliers tremble and came near terrifying the ladies. But this was not the end of the ceremony. The negroes held fast with their right hands to the handles of the silver covers until another stroke of the bell resounded. Then they jerked off the covers, swung them high over their heads, and thus marched off as if carrying away their booty in triumph. So the dinner went on, with several repetitions of such proceedings, the negroes getting all the while more and more enthusiastic and bizarre in their performances.\n\nThe French themselves shifted to service \u00e0 la russe (service in the Russian style) during the mid-nineteenth century. This meant that food was served in courses\u2014typically appetizers, salad, soup, and entr\u00e9es followed by desert. American restaurants did not convert to service \u00e0 la russe until later in the nineteenth century.\n\nFrench cookery in New York reached a peak at the beginning of the twentieth century, when hotel restaurants run by French or French-trained chefs\u2014at the Waldorf Astoria, Essex House, and Ritz-Carlton, among others\u2014were among the nation's premiere eating places. Wealthy Americans also hired French chefs to preside over their household kitchens.\n\nNew forms of restaurants emerged during the late nineteenth century. Louis Sherry opened an elegant restaurant at the corner of Fifth Avenue and Thirty-seventh Street. The restaurant's decor and its French chefs soon attracted New York's social elite as well as those who loved good food. To prepare the food, the restaurant had sixteen chefs and two hundred kitchen employees. Sherry's was soon followed by others. Thomas Shanley, an Irish immigrant, opened a small restaurant on Twenty-third Street in 1891. It was in the old theater district. He drew theatrical celebrities from the theaters, and his business grew. The theater district moved to Times Square, and in 1910 Shanley moved his restaurant to Broadway between Forty-second and Forty-third Streets. He offered \"light entertainment\" in addition to the food and liquor. It had an opulent interior, and its food was good and costly but of secondary importance to customers who came for the entertainment, for it had a full orchestra, the first restaurant in the city to have one. Shanley is credited with being the first in New York to launch what would later be called a lobster palace. It was an overnight sensation.\n\nOther lobster palaces emerged about the same time. These included those owned by George Rector and Captain James Churchill, and J. B. Martin's Caf\u00e9 Martin, Maxim's, John Murray's Roman Gardens, Reisenweber's, Bustanoby's, and many more. Typically, they were built on or near Longacre Square (later renamed Times Square), the city's theater district. Lobster palaces were expensive, elegantly decorated, and sported live entertainment, such as orchestras, dancing, and floor shows. They served a variety of dishes, but lobster was their signature dish. They also possessed liquor licenses permitting them to serve alcohol throughout the night, and in many ways they were the city's first nightclubs. Selling liquor was where they made their profits.\n\nHigh-end restaurants also made much of their profits from the sale of liquor that preceded, accompanied, and followed a meal. French restaurants, for instance, particularly featured expensive imported wines as part of their dining experience, and the sale of alcoholic beverages played an important role in their profits. When Prohibition went into effect in 1920, restaurants, particularly lobster palaces and haute cuisine establishments, were hit hard. Louis Sherry saw the writing on the balance sheet and closed his restaurant a few months before Prohibition went into effect. Shanley's closed in March 1923. After nearly a century of glorious history, Delmonico's did the same in May 1923.\n\nThere were notable exceptions. The Russian Tea Room opened on Fifty-seventh Street in 1927. Initially, it served as a gathering place for those fleeing the Russian Revolution, but it quickly attracted a wide audience by offering Russian and Continental cuisine. Sardi's restaurant, opened in the theater district by Italian immigrant Melchiore Pio Vincenzo Sardi (aka Vincent Sardi) in 1927, served Italian and French fare; its most popular dishes in 1930 were Moules marini\u00e9re, the classic Normandy mussel dish, bouillabaisse, and filet of sole. Other popular exceptions included the \"21\" Club, which sold liquor throughout Prohibition, and The Colony, which was reborn in the midst of Prohibition. Before World War I, The Colony was a somewhat shady bistro with a gambling den upstairs. Gene Cavallero became headwaiter in December 1919, and in 1921 he teamed up with two other waiters to buy the restaurant. They remodeled the premises in ten days and reopened with a menu of fine French cuisine. The Colony was almost immediately discovered by society matron Anne Vanderbilt (Mrs. W. K. Vanderbilt Jr.), who was followed by other members of New York's fashionable set. It was a very expensive restaurant, and its specialties were chafing dishes, game, and its hors d'oeuvres. By late 1920s, The Colony was so much the favored haunt of New York's wealthy that the restaurant closed during the summer, its clientele having decamped for Europe.\n\nHaute Cuisine Revival\n\nProhibition ended in December 1933, but by this time, the country was in the depths of the Depression and few restaurants served haute cuisine. This changed after New York hosted the 1939 World's Fair. Located in Flushing Meadow in Queens, the fair had an optimistic theme, \"Dawn of a New Day,\" where visitors could glimpse \"the World of Tomorrow.\" The Food Zone included 150 cows sponsored by Borden and a wheat field\u2014the city's first in sixty-eight years\u2014sponsored by Continental Baking, makers of Wonder Bread. The Beach Nut exhibit illustrated how coffee was grown, harvested, and roasted. Standard Brands exhibited Fleischman's Yeast, Royal Desserts, and Chase & Sanborn Coffee.\n\nMore than eighty restaurants were located around the fairgrounds. The Japanese restaurant served sukiyaki; the Italian, spaghetti; the Swedish, a smorgasbord; and the Mexican, chili con carne. Two restaurants emerged from the fair to become New York culinary landmarks. One was opened by Dario Toffenetti, an Italian immigrant and Chicago restaurateur. He acquired the concession for a restaurant at the fair and decided to open a restaurant in New York. At Forty-third Street and Broadway, he built a large modern restaurant complete with an escalator and massive chandeliers. It could seat a thousand people, and the day it opened in August 1940, it served 8,500 customers. It offered simple, low-cost food, such as \"Spaghetti a la Toffenetti with Fresh Meat Sauce,\" supposedly derived from an obscure manuscript that Mrs. Toffenetti had located in Bologna. Toffenetti's restaurant was open twenty-four hours a day, and for almost thirty years it was one of New York's most popular restaurants, serving thousands of customers daily.\n\nA very different type of restaurant was launched by Henri Soul\u00e9, the ma\u00eetre d'h\u00f4tel at the restaurant at the French Pavillon. The restaurant, which the New York Times called \"an epicure's delight,\" served classical French food. Wealthy New Yorkers were familiar with the French cuisine; during Prohibition and the Depression, they had traveled to France, where they dined at the world's finest restaurants. The French restaurant at the World's Fair, however, attracted a wide audience. Most fairgoers had never before sampled French food; they did so at the Pavillon, and they liked what they tasted.\n\nHenri Soul\u00e9 remained in New York when the fair ended and opened a restaurant on New York's fashionable Upper East Side in 1941. Hoping to trade on the restaurant's popularity at the fair, he named the establishment Le Pavillon. Soul\u00e9 was joined by many of his former colleagues, such as Pierre Franey, who had been the poisson commis (assistant fish cook). Soul\u00e9 attracted the rich and famous to dine. Le Pavillon was a solid success from the beginning as New York's social and economic elites\u2014Astors, Cabots, Kennedys, Rockefellers, and Vanderbilts\u2014dined regularly at the restaurant. It survived World War II and thrived during the 1950s. Soul\u00e9 made a virtue of snobbery. Presumably to protect the sensibilities of his refined customers, he refused entrance to those who did not meet his standards. Soul\u00e9 is also said to have coined the term \"Siberia,\" referring to the tables closest to the kitchen (considered the worst in any restaurant).\n\nWhen Henri Soul\u00e9 died in 1966, the New York Times restaurant critic Craig Claiborne eulogized him as the \"Michelangelo, the Mozart and Leonardo of the French restaurant in America.\" Soul\u00e9's greatest achievement, according to Claiborne, was his nurturing of restaurateurs and other influential individuals on America's food scene. Pierre Belin and Paul Arepejou, who started at Le Pavillon, opened La Poini\u00e8re in 1954. Roger Fessaguet, who had moved from Le Pavillon to La Caravelle, made La Caravelle the \"incubator for some of New York's better-known chefs.\" Michael Romano worked with Fessaguet until restaurateur Danny Meyer tapped him to become the chef at the Union Square Caf\u00e9. David Ruggerio went from La Caravelle to Le Chantilly, and Cyril Renaud became the chef and owner of Fleur de Sel.\n\nLe Pavillon survived only a few years after Soul\u00e9's death, but French cuisine continued to thrive in New York. A host of restaurants serving a variety of regional French cuisines as well as Escoffier's classics appeared in New York. Lut\u00e8ce opened in 1961, with Chef Andr\u00e9 Soltner at the helm. For more than three decades, Lut\u00e8ce was considered one of the finest restaurants in America. Le P\u00e9rigord, opened by Georges Briguet in 1964, still upholds the tradition of fine French cuisine in New York. Sirio Maccioni, who came to work at The Colony in 1956, opened Le Cirque with Jean Vergnes in 1974. It has moved premises and transformed itself over the years, but Le Cirque continues to employ notable chefs, and alumni of its kitchens include Alain Sailhac, Jacques Torres, Daniel Boulud, David Bouley, and Michael Lomonaco.\n\nAmong New York City's longest-tenured French restaurants are La Grenouille and La Mangeoire, serving classic and country cuisines, respectively; of more recent vintage but highly regarded are Restaurant Daniel, Jean Georges, and Le Bernardin.\n\nManagement Groups\n\nJoe Baum, a graduate of the Cornell's hotel management school, was hired by Rikers Restaurant Associates (later renamed Restaurant Associates) in 1953. It operated twenty-four restaurants in the New York City area. They had just acquired the Newarker, a restaurant at Newark Airport, then a small regional airfield. Baum was placed in charge of developing the Newarker.\n\nBaum hired Albert Stockli, a classically trained Swiss chef, to run the kitchen. Together, they devised an unusual menu with showy, attention-grabbing features\u2014large portions, sparklers in birthday cakes, and lots of dishes that were flamb\u00e9ed tableside. The restaurant lost twenty-five thousand dollars in that first year, but it soon became a \"destination\" eating place. Despite its location in the airport, the Newarker served a thousand customers a day\u2014most of them not airline passengers\u2014and grossed twenty-five million dollars a year.\n\nBaum was placed in charge of the Restaurant Associates' specialty restaurants in 1955. He hired big-name consultants, such as James Beard, Julia Child, Jacques P\u00e9pin, Barbara Kafka, and Rozanne Gold, to help him think through restaurant concepts and menus. Then he recruited top talent to design and run the restaurants. He is credited with popularizing themed restaurants, such as the Aurora (French), the Hawaiian Room at the Lexington Hotel (Polynesian, complete with hula dancers), La Fonda Del Sol (South American), Zum-Zum (German sausage), Quo Vadis (French and Italian), the Forum of the Twelve Caesars at Rockefeller Center (Roman), and many others, such as the Brasserie, Hudson River Club, Caf\u00e9 Greco, Tavern on the Green in Central Park, and The Four Seasons. The expenses were enormous, and Restaurant Associates began to lose money. Baum was let go in 1970.\n\nA few weeks after leaving the Restaurant Associates, Baum was hired by the New York Transit Authority to design a world-class restaurant at the top of the brand-new World Trade Center and to consult on other eateries in the buildings. He developed the Market Bar and Dining Rooms on the ground-floor concourse, Windows on the World at the top of the North Tower, and twenty other eating places for the massive new office complex. Windows on the World opened in April 1976, quickly becoming a landmark and a destination restaurant for tourists and New Yorkers alike. From then on, Baum continued to design restaurants. He took over and redesigned the Rainbow Room in Rockefeller Center, which reopened in 1988. When Joe Baum died, in 1998, he and Restaurant Associates had created 167 restaurants.\n\nWith some notable exceptions, such as The Four Seasons, most of Baum's restaurants have not survived, but his approach has influenced many who came after him: the chefs and restaurateurs Drew Nieporent, Daniel Boulud, Danny Meyer, Jean-Georges Vongerichten, Stephen Hanson, and David Chang have followed in Baum's footsteps, designing new restaurant concepts in New York and around the world. Drew Nieporent, a Cornell hotel management graduate, worked at Tavern on The Green, before he formed the Myriad Restaurant Group, which operates Tribeca Grill, Nobu New York City, The Daily Burger at Madison Square Garden, and thirty-two other restaurants. Danny Meyer created the Union Square Hospitality Group, including the Union Square Caf\u00e9, Gramercy Tavern, Shake Shack, The Modern, Caf\u00e9 2, Terrace 5 and others. Korean American David Chang opened his first restaurant, Momofuku Noodle Bar, in the East Village in 2004. He then opened other restaurants in New York and other cities and created the Momofuku restaurant group to manage them.\n\nRestaurant Workers\n\nThe first modern fast-food chain to hit New York was White Castle, which arrived during the Depression. Initially, it sold nickel square hamburgers, soda, coffee, and pie. Hamburgers were prepared on an assembly-line format to fulfill orders as quickly as possible. Outlets stayed open late at night and were often located near businesses, such as newspaper plants, that operated twenty-four hours a day. It served as a model for other fast-food chains that moved into the city after World War II. These included McDonald's, Burger King, Wendy's, Taco Bell, Au Bon Pain, Chipotle, Starbucks, Jamba Juice, and many more.\n\nToday, fast-food chains employ fifty thousand workers in the city. Most are paid minimum wage (currently $7.25 an hour), and most work only part time. Few receive medical insurance, paid sick days, or vacation days. In November 2012 and again in April 2013, workers at McDonald's, Burger King, Wendy's, Taco Bell, and other fast-food chains went on one-day strikes asking for a living wage.\n\nImmigrants form the main workforce behind the food service industry and their suppliers, such as meatpackers, and farm laborers, such as those who pick tomatoes. According to the Restaurant Opportunities Center of New York, 47.5 percent of the restaurant workers in the city were foreign born in 1980; by 2000, this had reached 64.5 percent; today the number is much higher. More than 125,000 immigrants\u2014many undocumented\u2014are employed in the city's restaurants. Most are paid minimum wage\u2014and some even less. The number of undocumented workers in restaurants is likely to increase in future years. As a Manhattan chef and restaurateur explained in 2011, \"We always, always hire the undocumented workers. . . . It's not just me, it's everybody in the industry. First, they are willing to do the work. Second, they are willing to learn. Third, they are not paid as well. It's an economic decision. It's less expensive to hire an undocumented person.\"\n\nIn 2011, 69 percent of all small restaurant and other food service business owners in New York City were foreign born. In other culinary sectors, immigrants make up an even larger percentage: 84 percent of small grocery store owners were foreign born, as are most of those employed in other restaurants and grocery stores. A 2007 report, \"Unregulated Work in the Global City,\" conducted by the Brennan Center for Justice, stated that in the food retail industry, \"the workforce is almost exclusively immigrant, from Mexico, Central America, Korea, Africa, the Caribbean, and South Asia. Delivery workers are mostly African immigrants.\"\nChapter Seven\n\nHistoric Cookbooks, Dishes, and Recipes\n\nDuring the last two hundred years, the majority of America's important cookbooks have been brought into being by New York City publishers. Although both Boston and Philadelphia were (and are) home to some major publishing houses, including Little, Brown & Co., Houghton Mifflin, and J. W. Lippincott, New York was and is the capital of the American publishing industry. Many cookbook authors and editors live in the city, and New York's intense restaurant culture gives rise to an entire genre of cookbooks.\n\nBefore American authors began churning out culinary works, British authors were dominant. English colonists brought cookery manuscripts and cookbooks with them. British cookery books, such as Martha Bradley's British Housewife, E. Smith's Compleat Housewife, and Penelope Bradshaw's Family Jewel, among others, were sold by 1761 at the Bible and Crown, Hugh Gaine's bookshop on Hanover Square. Beginning in 1790, New York publishers issued reprints of British cookery books. A popular title was Susannah Carter's Frugal Housewife, originally published in London in 1772, which was first printed in New York in 1792. The 1803 edition included \"an appendix containing several new receipts adapted to the American mode of cooking.\" It is unknown who wrote the twenty-nine recipes, but eighteen clearly reflect American traditions, such as recipes for making Indian pudding (corn pudding), pumpkin pie, cranberry tarts, and several recipes that are sweetened with maple syrup.\n\nBritish cookbooks naturally influenced the earliest American cookery writers. Amelia Simmons's American Cookery (1796), the first cookbook written by an American, was not unlike the British works that preceded it. American Cookery, however, did call for many New World ingredients such as corn and cranberries. It was the first book to offer distinctively American recipes, such as \"Indian Slapjack\" made from cornmeal. Originally published in Hartford, Connecticut, and reprinted many times, American Cookery was not printed in New York until 1822.\n\nThe first \"New York\" cookbook, The Universal Receipt Book or Complete Family Directory by a Society of Gentlemen in New York (1814), is attributed to Richard Alsop, a wealthy poet and satirist, who, in fact, resided in Hartford, Connecticut. The Society of Gentlemen had existed off and on since 1749. It consisted mainly of physicians, but by 1814 it had expanded to include a range of well-to-do \"gentlemen.\" For at least a decade, Alsop collected recipes from his friends and acquaintances. Alsop's brother-in-law published it. The culinary recipes were generally reflective of the American diet at the time: some notable exceptions featured ketchup and tomatoes, and there is even a recipe for pineapple ice cream.\n\nAs New York became America's financial capital, publishers flocked to the city. Many published cookbooks, although most of their authors were not New Yorkers; they lived elsewhere in the country, or overseas, notably in the United Kingdom and France. Among the more interesting works published in New York include two homemakers encyclopedias. Writer Elizabeth Ellet, who was born upstate but lived much of her life in New York City, wrote a \"Domestic Encyclopedia\" titled The Practical Housekeeper (1857). Her tome, which weighed in at 599 pages, was intended to be a \"complete system of Domestic Economy.\" It was broken down into three sections: \"housekeeping, cooking, and drugstore-type concerns (soaps, perfumes, medical guide).\" It was revised and reprinted throughout the nineteenth century. A similar work, The Housekeeper's Encyclopedia (1861) was written by a \"Mrs. E. F. Haskell,\" and it too went through several printings during the nineteenth century and has recently been reprinted in facsimile. Similar works continued to be published well into the twentieth century. Isabel Ely Lord's Everybody's Cook Book (1924), developed at the Pratt Institute in Brooklyn, contained more than 3,400 recipes. America's Cook Book (1937), compiled by the Home Institute of the New York Herald Tribune, included more than three thousand recipes.\n\nImmigrant and International Cookbooks\n\nCookbooks by and for immigrants began to appear in the late 1850s. Anna May wrote Die kleine New Yorker k\u00f6chin (The little New York cook). Its recipes were mainly written in German, and the book's intended audience was the new German immigrant population who wished to find work as domestics. First published in 1859, it was reprinted several times. A more influential cookbook was written by a French immigrant, Pierre Blot, who came to the United States around 1855. He founded the New York Cooking Academy, America's first French cooking school, which didn't last long. With financial assistance from the daughter of Commodore Cornelius Vanderbilt, Blot subsequently opened the New York Cooking School, which mainly catered to the wealthy women and their cooks who were interested in studying a simplified version of the classic techniques of French cookery. He wrote articles and two cookbooks, What to Eat, and How to Cook It (1863) and Hand-Book of Practical Cookery (1867), which went through several printings.\n\nImmigrants and members of ethnic and religious groups have continued to publish cookbooks in New York. Gesine Lemcke, a German immigrant who launched a cooking school in Union Square, published Desserts and Salads (1901) and European and American Cuisine (1902). In 1918, Mrs. Maria Matilda Ericsson Hammond, who ran a cooking school on East Forty-sixth Street, self-published The Swedish, French, American Cook Book; she brought out a Swedish-language edition, Mrs. Ericsson Hammonds Svensk-Amerikanska Kokbok, in 1926. Florence Greenbaum, an instructor in cooking and domestic science at the Young Women's Hebrew Association, wrote The International Jewish Cook Book: A Modern \"Kosher\" Cook Book (1918). The book went through four editions; later editions included \"1600 recipes according to the Jewish dietary laws with the rules for kashering: the favorite recipes of America, Austria, Germany, Russia, France, Poland, Roumania.\"\n\nImmigrants who arrived in the United States after World War II have also become part of the New York cookbook landscape. Italian American cookbook authors are legion, and most of them settled in New York. Pino and Fedora Bontempi were pioneers in television cooking history and started a long-running program, \"Cooking with the Bontempis,\" in 1949. They published The Bontempi Cookbook in 1965, and it became a standard work. Marcella Hazan, who came to New York from Emilia-Romagna in 1955, opened the School of Classic Italian Cooking in 1969. Her first book, The Classic Italian Cook Book: The Art of Italian Cooking and the Italian Art of Eating, was published in 1973; this book and its sequel, More Classic Italian Cooking, have become mainstays of the American cookbook shelf. A native of Abruzzo who first came to New York in 1959, Anna Teresa Callen also became a cooking teacher and published several books, including Food and Memories of Abruzzo (1998). Lidia Bastianich arrived in New York from Trieste as a young refugee in 1958. In 1971, she and her husband, Felice, opened a restaurant in Forest Hills, Queens. Their Manhattan restaurant, Felidia, opened in 1981 to great acclaim. Lidia Bastianich's first cookbook, La Cucina Di Lidia: Distinctive Regional Cuisine from the North of Italy, was published in 1990. She has since written twelve more books and has been a star of the PBS cooking-show roster since 1998.\n\nCookbooks for the Working Class\n\nA different direction was set by Juliet Corson who opened the New York Cooking School in 1876. She targeted her teaching at not only middle-class women and their children but also poor and working-class women. Based on her lectures, Corson penned the Cooking School Text Book and Housekeepers' Guide, which simplified cooking and promised that anyone who could follow simple directions could become a cook in \"twelve easy lessons.\" At this time, economic woes gripped the country; unemployment was at 14 percent. Wages were cut, leading railroad workers to strike in 1877 and paralyzing the nation. Corson saw that struggling families needed help in selecting and preparing cheap, nutritious meals. At her own expense, she printed fifty thousand copies of a pamphlet, Fifteen Cent Dinners for Workingmen's Families (1877). The pamphlet received widespread acclaim, and the recipes were republished in newspapers and magazines across the country. She subsequently published Twenty-five Cent Dinners for Families of Six (1878), Meals for the Million (1882), and Family Living on $500 a Year (1888)\u2014all of which were widely distributed\u2014and the simple and inexpensive recipes reflected what many working-class New Yorkers ate.\n\nHaute Cuisine Cookbooks\n\nCookery at the other end of the economic spectrum was reflected in the cookbooks authored by New York City restaurant chefs. During the second half of the nineteenth century, the Metropolitan Hotel's restaurant on Broadway was among the city's more fashionable eateries. In 1864, a small softcover book, 250 Rare Receipts, including the Celebrated 100 Metropolitan Hotel Recipes, and over 150 other Valuable Receipts, was published. Although no author was identified, it was reprinted several times, setting the stage for hundreds of future cookbooks written by chefs who gained fame at New York City's hotels, restaurants, and clubs.\n\nSoon French chefs who worked in New York City restaurants began to write cookbooks. In 1884, Felix D\u00e9li\u00e9e, successively chef at the New York Club, the Union Club, and Manhattan Club, published a 620-page tome The Franco-American Cookery Book, and it was regularly reprinted until 1910. Alessandro Filippini was a Swiss \u00e9migr\u00e9 who started out as a cook at Delmonico's in 1848, rose to the post of chef de cuisine, and finally became manager of one of the Delmonico's restaurants. In 1888, when he retired, Filippini began writing cookbooks. The first of these, The Table (1889), presented simplified versions of recipes he prepared at the restaurant. One of his later works was The International Cook Book (1906). Other Delmonico chefs followed in his footsteps: Charles Ranhofer, a French-born chef who replaced Filippini when he retired, produced his masterwork The Epicurean (1894). It was the most comprehensive French cookbook published in the United States up to that time. Unlike Filippini's simplified version of Delmonico's cookery, Ranhofer's imposing tome was extremely detailed and featured menus for every imaginable event and occasion. His work included many of the recipes that made Delmonico's famous, including lobster \u00e0 la Newberg and baked Alaska. When The Epicurean was published, Leopold Rimmer, then in charge of one of Delmonico's dining rooms, bitterly complained that Ranhofer had given \"away all the secrets of the house.\"\n\nOscar Tschirky, ma\u00eetre d'h\u00f4tel at the Waldorf, collected recipes from the chefs at the hotel's restaurants and in 1896 published The Cook Book by \"Oscar\" of the Waldorf. It includes many recipes that the Waldorf was known for, such as the Waldorf salad. Three years later, Annie Clarke published The New Waldorf Cook Book, containing more than a thousand recipes for cooking and household management. After World War II, French cookbooks were regularly released by New York publishers. Dione Lucas, a British immigrant, opened a restaurant and cooking school in New York; in 1947, she published her recipe collection The Cordon Bleu Cook Book. Until the advent of Julia Child, it was considered the standard work on the subject. Joseph Donon, a chef to New York's wealthy, wrote The Classic French Cuisine, published in 1959.\n\nMagazines and Cookbooks\n\nWorld War I, followed by Prohibition and then by World War II, put a damper on restaurants and cookbooks published by their chefs. But after World War II, haute cuisine restaurants reemerged, and chefs began to publish cookbooks. Chef Louis P. De Gouy, who had studied with Escoffier, worked in many European and American restaurant kitchens, including the Waldorf's. In 1931, De Gouy opened the Institute of Modern and Practical Cooking in New York City. His Gold Cookbook (1947) was published the year he died, and his wife and daughter compiled and published several additional cookbooks posthumously. Before his death, he was identified as the \"Gourmet chef\" for Gourmet, a magazine launched in New York by Earle MacAusland in 1941.\n\nMacAusland sought out the best food writers of the day to contribute to his magazine. He was foresighted enough to hire James Beard, who had just published his first cookbook, Hors D'oeuvre and Canap\u00e9s, with a Key to the Cocktail Party; M. F. K. Fisher, whose first book, Serve It Forth, had appeared in 1937; Pearl V. Metzelthin, author of the World Wide Cook Book (1939); and chef Louis Diat, whose book Cooking \u00e0 la Ritz (1941) included the first published recipe for vichyssoise. Other contributors to Gourmet were popular writers such as Joseph Wechsberg, who later wrote Dining at the Pavillon (1962), among other culinary books, and the British cookbook writers Jane Grigson and Elizabeth David. Recipes from Gourmet were collected and published in cookbooks beginning in 1951.\n\nOther national food magazines got their start in New York. Food and Wine was first published in 1978. The brainchild of Michael Batterberry, his wife, Ariane, and three others, Michael Batterberry, who served as the magazine's editor in chief, described the magazine as taking \"food seriously but not pompously.\" The Batterberrys coauthored On the Town in New York, from 1776 to the Present (1973), the first \"food biography\" of New York. In 1988, they launched a trade publication, Food Arts magazine.\n\nSaveur was launched in New York in 1994 \"to satisfy the hunger for genuine information about food in all its contexts . . . heritage and tradition, home cooking and real food, evoking flavors from around the world (including forgotten pockets of culinary excellence in the United States).\" Saveur has also published many cookbooks based on recipes that first appeared in the magazine.\n\nNewspapers and Cookbooks\n\nNew York newspapers got in on the act as well as they hired food columnists. Ida C. Bailey Allen, author of Mrs. Allen's Cook Book (1917), was hired as the food editor of the New York American in 1924. She continued to write cookbooks, including Mrs. Allen on Cooking, Menus, Service (1924)\u2014reissued eight years later as Ida Bailey Allen's Modern Cook Book. Allen hosted a variety of radio cookery programs and eventually wrote more than fifty cookbooks and cook booklets, including When You Entertain\u2014What to Do, and How (1932), which promoted Coca-Cola; 375,000 copies were distributed.\n\nClementine Paddleford served as a food columnist for several New York publications, including the New York Herald Tribune, the New York Sun, and the New York Telegram. Paddleford's weekly readership topped twelve million during the 1950s and 1960s. In 1953, Time magazine named her America's \"best-known food editor.\" Her cookbooks included Recipes from Antoine's Kitchen (1948), How America Eats (1960), Clementine Paddleford's Cook Young Cookbook (1966), and her posthumously published Best in American Cooking: Recipes Collected by Clementine Paddleford (1970).\n\nPaddleford was soon eclipsed by Craig Claiborne, a Southerner, who was hired by the New York Times in 1959 as its first food columnist. He authored The New York Times Cookbook in 1961 and revised it in 1980. It sold more than three million copies. Claiborne invited some of America's top French chefs and cookbook writers, including future cookbook writers Pierre Franey, Ren\u00e9 Verdon, Jean Vergnes, and Jacques P\u00e9pin, to cook in his home kitchen. Claiborne noted down their recipes and comments as they prepared a meal and presented the recipes in his newspaper columns. He was later hailed as the man who \"demystified haute cuisine, and even made it look like fun.\" Pierre Franey and Craig Claiborne became friends, and they collaborated on five cookbooks, including Classic French Cookery (1970). Claiborne went on to author or coauthor twenty cookbooks, including The New York Times International Cookbook (1971), four volumes of Craig Claiborne's Favorites from the New York Times (1975\u20131978), Craig Claiborne's the New York Times Food Encyclopedia (1985), and The Best of Craig Claiborne: 1,000 Recipes from His New York Times Food Columns and Four of His Classic Cookbooks (1999).\n\nAfter Claiborne left the New York Times, some of his successors also became cookbook authors, such as Molly O'Neill and Amanda Hesser. Other New York Times writers who have published cookbooks include Kim Severson and Florence Fabricant, who has written or edited ten cookbooks, including her latest, The New York Restaurant Cookbook: Recipes from the City's Best Chefs (2009). Ruth Reichl, who served as the restaurant critic for the Times before becoming the last editor of Gourmet, published several cookbooks, including The Gourmet Cookbook: More than 1000 Recipes (2006); Gourmet Today: More than 1000 All-New Recipes for the Contemporary Kitchen (2009), and memoirs, which include recipes.\n\nCelebrity Chefs' Cookbooks\n\nJacques P\u00e9pin immigrated to the United States in 1959 and began working at New York's famous French restaurant Le Pavillon. He left the restaurant after eight months and went to work for the Howard Johnson's restaurant chain. P\u00e9pin left Howard Johnson's in 1969 to open La Potagerie, a soup restaurant in New York. A publisher approached him about writing a soup cookbook, which P\u00e9pin declined to do, but this discussion launched his career as an author. Among his books are A French Chef Cooks at Home (1975); La Technique (1976), which stayed in print for the next twenty-two years; its companion volume, La Methode (1979; both of these are virtual bibles to many chefs), and Jacques Pepin's The Art of Cooking (1987\u20131988).\n\nIn 1949, Edna Lewis, the granddaughter of Virginia sharecroppers, came to cook at Caf\u00e9 Nicholson, a restaurant on New York's well-to-do Upper East Side. Her first book, The Edna Lewis Cookbook (1972), focused on her experiences at Caf\u00e9 Nicholson and her subsequent catering business. After Lewis met Knopf editor Judith Jones, who specialized in cookbooks, she wrote a second book, The Taste of Country Cooking (1976), which focused on the culinary experiences of her youth. Lewis continues to work in restaurants, most notably the venerable Gage and Tollner in Brooklyn. She also wrote In Pursuit of Flavor (1988) and coauthored, with Scott Peacock, The Gift of Southern Cooking (2003).\n\nNew York celebrities, chefs, and restaurateurs have continued to publish a wide range of cookbooks. New York theatrical producer Crosby Gaige procured recipes from home economists in each state and compiled them in New York World's Fair Cook Book: The American Kitchen (1939). He went on to publish The Standard Cocktail Guide: A Manual of Mixed Drinks Written for the American Host (1944).\n\nFrank Case's Feeding the Lions: An Algonquin Cookbook (1942) built on the popularity of the Algonquin \"Round Table,\" where New York City's literary and journalistic lights gathered for meals and witty conversation. It included comments from the famous\u2014Dorothy Parker, Robert Benchley, and the like\u2014on their favorite dishes. Some of the other eateries that spearheaded the trend in publishing cookbooks or restaurant \"biographies\" were The Colony (1945), El Borracho (1950), L\u00fcchow's (1952), The House of Chan (1952), Sardi's (1953), Longchamps (1954), The Lobster (1958), Romeo Salta (1962), and Leone's (1967).\n\nItalian immigrant and restaurateur Sirio Maccioni opened Le Cirque in 1974. He hired some of New York's top chefs, including Alain Sailhac and Daniel Boulud. Along with coauthor Pamela Fiori and Alain Ducasse, he published A Table at Le Cirque: Stories and Recipes from New York's Most Legendary Restaurant (2012). French restaurateur Daniel Boulud moved to New York in 1982 and opened the Polo Lounge at the Westbury Hotel. In 1986, he became the executive chef at Le Cirque. He opened his own restaurant, Daniel, in 1993. He published his first cookbook, Cooking with Daniel Boulud, the same year. His latest cookbook is Cocktails and Amuse-Bouches (2011), coauthored with mixologist Xavier Herit.\n\nAlsatian Andr\u00e9 Soltner became chef-owner of Manhattan's Lut\u00e8ce in 1961, a career that lasted thirty-four years. His Lut\u00e8ce Cookbook (1995) published many of the recipes that made the restaurant a success. Danny Meyer, owner of the Union Square Caf\u00e9, cowrote with Chef Michael Romano The Union Square Caf\u00e9 Cookbook: 160 Favorite Recipes from New York's Acclaimed Restaurant (1994). A sequel, Second Helpings from Union Square Caf\u00e9, was published in 2001.\n\nSmaller but still iconic New York restaurants, many of them ethnic and specialty eateries, also published cookbooks. These include The Little Italian Cookbook by Christiana Lindsay and Alfred Lepore (1968), from Caff\u00e9 Ferrara in Little Italy; Princess Pamela's Soul Food Cookbook (1969); Cooking Creatively with Natural Foods (1972), by Edith and Sam Brown of Brownie's, a beloved mostly vegetarian restaurant; Ratner's Meatless Cookbook (1975) by Judith Gethers and Elizabeth Lefft, from a Jewish dairy restaurant on Delancey Street; Pat Miller's Serendipity Cookbook (1999) from the fanciful Upper East Side dessert shop\/celebrity clubhouse (Andy Warhol, Marilyn Monroe, and Truman Capote were regulars); and three books published in 1999: Sharon Lebewohl's Second Avenue Deli Cookbook, from her family's archetypal Jewish deli; Sylvia Woods and Melissa Clark's Sylvia's Family Soul Food Cookbook, from the renowned Harlem restaurant; and Welcome to Junior's, by Marvin and Walter Rosen, whose father founded the Brooklyn cheesecake mecca.\n\nCheesecakes at Junior's Restaurant, Brooklyn, 2013. Courtesy of Kelly Fitzsimmons\n\nNew York Dishes\n\nNew York City cooks, chefs, bartenders, and homemakers have created untold numbers of original dishes and drinks. Many groundbreaking recipes were first published in New York. Dishes, recipes, and beverages with the words \"Manhattan,\" \"New York,\" \"Coney Island,\" or \"Waldorf\" in their names, however, did not necessarily originate in the city, although many have been popularized by New York restaurateurs, cookbook authors, newspaper columnists, pushcart vendors, deli owners, and bartenders. Here are a few that are often associated with New York over the years.\n\nManhattan Clam Chowder\n\nThe first recipe for a clam chowder containing tomatoes\u2014what is today typically called \"Manhattan clam chowder\"\u2014appeared in Jessup Whitehead's Cooking for Profit (1893). Whitehead described it as popular in restaurants as a lunch dish, and he subtitled the recipe \"Coney Island Style.\" Coney Island (now, thanks to landfill, no longer an island) was a Brooklyn beach resort where a popular amusement park was established in the 1840s. Joseph Vachon's Book of Economical Soups and Entr\u00e9es (1903) used the name \"Coney Island clam chowder\" for a soup made with tomato ketchup rather than milk. A recipe for \"Fulton Market clam chowder\" appeared in The Grand Union Cook Book (1902) by Margaret Compton. The 1931 Pictorial Review Standard Cook Book called the soup \"New York clam chowder,\" and three years later, Virginia Elliot and Robert Jones renamed it \"Manhattan clam chowder\" for their book Soups and Sauces (1934). The name caught on, but the idea of replacing milk with tomatoes in clam chowder remains a point of contention between New Yorkers, New Englanders, and their respective partisans. Chef Louis P. De Gouy, author of The Soup Book (1949), concluded that it was one of those subjects, like politics and religion, that could never be discussed lightly. In his book, he offered ninety-eight chowder recipes, including two for \"Manhattan-manner\" clam chowder. Leaving the subject more blurred than clarified, De Gouy claimed that these actually originated in New England.\n\nRegardless of its real origin or ingredients, Manhattan clam chowder never became as popular as the creamy New England version in the city. Culinary impresario James Beard described it as a horrendous soup \"which resembles a vegetable soup that accidentally had some clams dumped into it.\" Despite such disparagement, Manhattan clam chowder recipes still appear in cookbooks, and the soup is still found on city restaurant menus.\n\nLobster \u00e0 la Newberg\n\nChefs at fashionable restaurants often flattered the egos of their wealthy patrons by naming dishes after them. As the story goes, in 1876 Ben Wenberg, a wealthy shipowner and a frequent customer at Delmonico's, showed Charles Delmonico an extremely luxurious way of preparing lobster in a chafing dish. Delmonico put the dish on the menu as \"Lobster \u00e0 la Wenberg.\" For unclear reasons\u2014possibly a falling out between the two men\u2014the name was later changed to lobster \u00e0 la Newberg. Alessandro Filippini and Charles Ranhofer, both Delmonico's chefs, as well as other nineteenth-century cookbook writers, included recipes for it in their books.\n\nLobster \u00e0 la Newberg: Split two good-sized, fine, freshly boiled lobsters. Pick all the meat out from the shells, then cut it into one-inch-length equal pieces. Place it in a saucepan on the hot range with one ounce of very good, fresh butter. Season with one pinch of salt and half a saltspoonful of red pepper, adding two medium-sized, sound truffles cut into small dice-shaped pieces. Cook for five minutes; then add a wine-glassful of good Madeira wine. Reduce to one-half, which will take three minutes. Have three egg yolks in a bowl with half a pint of sweet cream, beat well together, and add it to the lobster. Gently shuffle for two minutes longer, or until it thickens well. Pour it into a hot tureen, and serve hot. Source: Alessandro Filippini, One Hundred Ways of Cooking Fish (New York: Charles L. Webster, 1892), 66.\n\nLobster \u00e0 la Newberg or Delmonico: Cook six lobsters each weighing about two pounds in boiling salted water for twenty-five minutes. Twelve pounds of live lobster when cooked yields from two to two and a half pounds of meat with three to four ounces of coral. When cold detach the bodies from the tails and cut the latter into slices, put them into a sautoir, each piece lying flat, and add hot clarified butter; season with salt and fry lightly on both sides without coloring; moisten to their height with good raw cream; reduce quickly to half; and then add two or three spoonfuls of Madeira wine; boil the liquid once more only, then remove and thicken with a thickening of egg-yolks and raw cream. Cook without boiling, incorporating a little cayenne and butter; warm it up again without boiling, tossing the lobster lightly, then arrange the pieces in a vegetable dish and pour the sauce over. Source: Charles Ranhofer, The Epicurean (New York: R. Ranhofer, 1894), 411.\n\nBaked Alaska\n\nThe United States purchased Alaska from Russia in 1867, and there was a flurry of interest about the new territory. Delmonico's began serving a dish called \"Alaska Florida.\" It was described by English writer George Sala, in December 1879 as\n\nan entremet called an \"Alaska.\" The \"Alaska\" is a baked ice. A beau mentir qui vient de loin; but this is no traveller's tale. The nucleus or core of the entremet is an ice cream. This is surrounded by an envelope of carefully whipped cream, which, just before the dainty dish is served, is popped into the oven, or is brought under the scorching influence of a red hot salamander; so that its surface is covered with a light brown crust. So you go on discussing the warm cream souffl\u00e9 till you come, with somewhat painful suddenness, on the row of ice.\n\nSala was mistaken about one crucial point: the topping is not whipped cream but meringue, which is a poor conductor of heat. This keeps the ice cream from melting while the meringue browns under the salamander (a heated iron).\n\nSimilar recipes soon appeared in other cookbooks and cookery magazines under various names. They were usually simplified versions of Ranhofer's extravaganza, and the name that stuck was \"baked Alaska.\"\n\nCocktails\n\nSome of America's most celebrated cocktails likely originated in New York or were named after boroughs. The Manhattan\u2014typically made with whiskey, sweet vermouth, and bitters\u2014may have been invented at New York's Manhattan Club during the 1870s. It first appeared in a bartender's guide in 1884. The second edition of Jerry Thomas's Bar-Tenders Guide (1887) includes a formula consisting of vermouth, rye whiskey, bitters, two dashes of Cura\u00e7ao or maraschino, and ice. Today a Manhattan is frequently stirred with ice and then strained into a cocktail glass. (A bartender at the Manhattan Club may also have created the first tomato \"cocktail,\" a mixture of oyster juice, tomato juice, Tabasco sauce, chili pepper sauce, ketchup, and lemon juice.)\n\nNew York City's best-known borough was not the only one to have cocktails named for it. The Brooklyn cocktail is credited to one Maurice Hegeman, who moved to Brooklyn from Cincinnati and worked in Schmidt's Caf\u00e9. Schmidt's was the first saloon one encountered after crossing the Brooklyn Bridge from Manhattan. In 1910, Hegeman gave the borough's name to a combination of hard cider, absinthe, and ginger ale. Over the years, the ingredients changed (or perhaps the name was appropriated), and today's Brooklyn cocktail is typically made with whiskey, dry vermouth, and maraschino liqueur. Some credit the Bronx cocktail to Joseph S. Sormani, a Bronx restaurateur who began serving the drink in 1905: it was a blend of gin, orange juice, and vermouth. According to legend, Sormani acquired the recipe in Philadelphia. Others credit the Bronx cocktail to Johnnie Solon, a bartender at the Waldorf Astoria Hotel. The somewhat obscure Queens cocktail consists of gin, pineapple juice, and both dry and sweet vermouth. There's a Staten Island martini\u2014obviously a recent invention\u2014made with coffee vodka, dry vermouth, and lime juice; a cocktail called the Staten Island Ferry is a blend of rum and pineapple juice.\n\nThere's no definitive origin story for the martini. Recipes for a drink called the \"Martinez\" appeared beginning in 1884. The second edition of Jerry Thomas's Bar-Tenders Guide (1887) includes one made with gin, vermouth, bitters, two dashes of maraschino liquor, and ice. The following year, Harry Johnson's New and Improved Illustrated Bartenders' Manual offered a martini cocktail recipe (gin, vermouth, bitters, Cura\u00e7ao, gum syrup, and ice) and also illustrations of two different \"Martine\" glasses. Who precisely invented the martini\u2014and how to concoct a perfect one\u2014have been topics of discussion ever since.\n\nThe martini was served in New York from the late nineteenth century onward. A 1930 recipe published during Prohibition included an orange peel, olive, pecan, or walnut in addition to Gordon's gin and vermouth. It was shaken, not stirred. The martini became specifically associated with New York in the 1960s, when businessmen famously gathered for the \"three-martini lunch.\" It became infamous in 1977, when President Jimmy Carter complained bitterly about the \"three-martini lunch,\" which he believed was an inappropriate business expense that was tax deductible. He called for the Internal Revenue Service to remove this and other loopholes. One participant fondly remembered them in a New York Times letter to the editor in 1989. They consisted of six-ounce martinis made up of six ounces of gin, a drop of vermouth, and a thin strip of lemon peel floating on the top, surrounded by a handful of silvery slivers of ice. \"Straight up\" was the way most people drank in the 1960s; ordering \"on the rocks\" was seen as a sign of weakness, as was the substitution of vodka. Olives displaced too much gin, and the only people who drank Gibsons were the effete heads of publishing companies. The three-martini lunch has disappeared, but the martini continues to thrive in New York.\n\nWaldorf Salad\n\nWaldorf salad is a combination of chopped tart apples, celery, and mayonnaise. The recipe was first published in The Cook Book by \"Oscar\" of the Waldorf (1896). Author Oscar Tschirky was in charge of the dining rooms at the Waldorf Hotel. According to Tschirky, Waldorf salad was served as a side dish, not as a salad course or main course. The Waldorf salad became an American favorite, and variations emerged. Beginning in 1898, it was usually served on a bed of lettuce. Chopped walnuts were added during the first decade of the twentieth century, and they have become part of the standard recipe. Grapes and raisins are other common additions. At some point the Waldorf salad came to be served as an entr\u00e9e; a popular variation includes cubes of roast chicken to make it a more substantial dish.\n\nWaldorf Salad: Peel two raw apples and cut them into small pieces, say about half an inch square, also cut some celery the same way, and mix it with the apple. Be very careful not to let any seeds of the apples be mixed with it. The salad must be dressed with a good mayonnaise. Source: Oscar Tschirky, The Cook Book by \"Oscar\" of the Waldorf (Chicago: Werner, 1896).\n\nSteaks, Potatoes, and Eggs\n\nConfusion surrounds the term \"New York steak.\" It likely did not originate in New York, but the term has been adopted in many local restaurants. A New York steak is variously defined as a porterhouse steak with the filet removed; a boneless top sirloin; or a strip steak. To make matters more confusing, it can also be called New York strip, New York cut, shell steak, New York sirloin, or sirloin strip. Outside of New York, the term may be used to refer to a completely different cut. (One steak that may have originated in New York is the porterhouse.)\n\nToday the term \"Delmonico steak,\" or \"New York steak,\" refers to two rounds of boneless rib eye tied together with butcher's twine. Charles Ranhofer includes three steak recipes in The Epicurean that are named for the restaurant. Here's one:\n\nDelmonico Sirloin Steak of Twenty Ounces, Plain (Bifteck de Contrefilet Delmonico de Vingt Onces, Nature): Cut from a sirloin slices two inches in thickness; beat them to flatten them to an inch and a half thick, trim nicely; they should now weigh twenty ounces each; salt them on both sides, baste them over with oil or melted butter, and broil them on a moderate fire for fourteen minutes if desired very rare; eighteen to be done properly, and twenty-two to be well done. Set them on a hot dish with a little clear gravy . . . or ma\u00eetre d'h\u00f4tel butter. Source: Charles Ranhofer, The Epicurean (New York: R. Ranhofer, 1894), 487.\n\nDelmonico's steak is often served with Delmonico potatoes, which are rich mashed potatoes served gratin\u00e9e\u2014with a browned topping of grated cheese (and sometimes breadcrumbs). Recipes for them were published beginning in 1889. Alessandro Filippini, a former Delmonico's chef, published his own in 1906.\n\nDelmonico Potatoes: Place four good-sized boiled and finely hashed potatoes in a frying pan with one and a half gills cold milk, half gill cream, two saltspoons salt, one saltspoon white pepper and a saltspoon grated nutmeg; mix well and cook on the range for ten minutes, lightly mixing occasionally. Then add one tablespoon grated Parmesan cheese, lightly mix again. Transfer the potatoes into a gratin dish, sprinkle another light tablespoon grated Parmesan cheese over and set in the oven to bake for six minutes, or until they have obtained a good golden colour; remove and serve. Source: Alessandro Filippini, The International Cook Book (New York: Doubleday, Page, 1906), 204.\n\nMany stories surround the origins of eggs Benedict, but they all agree that the dish was invented in the 1890s in a New York restaurant (usually identified as Delmonico's or one of the restaurants in the Waldorf Astoria hotels). The recipe evolved over the years before being codified as a slice of ham and a poached egg atop an English muffin, lavished with Hollandaise sauce. A \"Talk of the Town\" item published in the New Yorker in 1942 gives credit to Lemuel Benedict, a retired stockbroker who was a veteran of New York's glamorous Gilded Age restaurant scene: he had dined with Diamond Jim Brady at Delmonico's, and Caruso had been a dinner guest in his home. Benedict claimed that forty-eight years earlier (which would have been 1894), in the throes of a hangover, he had ordered breakfast at the Waldorf \u2014\"some buttered toast, crisp bacon, two poached eggs, and a hooker of hollandaise sauce\"\u2014with which, continues the writer, \"he then and there proceeded to put together the dish that has, ever since, borne his name.\" Oscar Tschirky, author of The Cook Book by \"Oscar\" of the Waldorf (1896), includes a similar recipe, substituting muffins for the toast and sliced chicken for the bacon, but he calls the dish \"Philadelphia eggs.\" Did Lemuel Benedict have a faulty memory or an inflated ego? Or was Oscar Tschirky loath to credit a customer with the creation of such a successful dish? Regardless of its actual origin, eggs Benedict is still a popular brunch choice in the city's restaurants.\n\nPhiladelphia Eggs: Split some fresh muffins in two, toast them to a fine color and lay them on a dish. Cut cooked chicken white meat the same size as the half muffin, lay them on top, then a poached egg over, and cover with Hollandaise sauce, made as follows: Melt a quarter of a pound of fresh butter and when quite hot add two raw egg yolks and the juice of half a lemon; whip well till it becomes creamy and consistent, then use. Source: Oscar Tschirky, The Cook Book by \"Oscar\" of the Waldorf (Chicago: Werner, 1896), 588.\n\nConey Island Hot Dogs\n\nNew Yorker Charles Feltman has often been credited with selling America's first hot dog\u2014a sausage served in a bun. In 1871, Feltman, an immigrant from Hanover, Germany, is reported to have operated a pushcart near the Coney Island amusement park in Brooklyn, selling sausages on rolls with sauerkraut. In Germany, such sausages are called, among other names, Frankfurters (i.e., from Frankfurt) or Wieners (i.e., from Wien, [Vienna]). By the 1890s, Feltman's ten-cent specialty had come to be called a \"hot dog,\" and his business had grown from a pushcart into a large restaurant with table service that sold a variety of foods including sausages from individual serving stands.\n\nThe genius behind the most famous hot dog stand in New York\u2014and America\u2014is Nathan Handwerker, a Polish immigrant who worked at Feltman's restaurant. Handwerker stayed at Feltman's for a year and then took three hundred he had saved from his wages and opened a hot dog stand a few blocks away. Handwerker cooked his hot dogs on a twelve-foot grill and sold them for a nickel apiece, thus undercutting his former boss. To keep the customers coming, Handwerker installed signs with horns that sounded like fire-engine sirens. When the subway arrived at Coney Island in 1920, people from all over the city could visit Coney Island for a nickel, and a hot dog from \"Nathan's Famous Frankfurters\" (or simply \"Nathan's Famous\") was a must for beachgoers. Nathan's began marketing retail packages of its beef frankfurters, based on Ida Handwerker's original recipe, in grocery stores, and they are now sold nationwide. There are also hundreds of Nathan's franchises around the country. Every summer, on the Fourth of July, Nathan's sponsors a highly publicized hot dog\u2013eating contest on the boardwalk at Coney Island. Contestants down as many hot dogs (and buns) as they can in ten minutes. The current records are sixty-five for a male contestant and forty for a woman.\n\nThe Original Nathan's Hot Dog stand on Coney Island, 2013. Courtesy of Kelly Fitzsimmons\n\nHamburgers\n\nHamburgers\u2014a cooked ground beef patty in a bun\u2014had been sold in New York since the late nineteenth century. Their main advantage was that they were cheap and easy to make. Street vendors with a grill could sell them as could diners, caf\u00e9s, stands, and restaurants. During the Depression, so many hamburger places opened up in Times Square that it became known as \"Hamburger Row.\" Another type of hamburger operation entered the city in 1933\u2014White Castle, a chain launched more than a decade before in Wichita, Kansas. Within a few years, fifteen White Castle outlets were established in the city offering cheap hamburgers, coffee, soda, and apple pie.\n\nHamburgers remained one of New York's favorite foods. It was served in virtually every coffee shop, restaurant, and caf\u00e9. The Hamburger Heaven, a chain launched during the Depression, thrived after the Second World War. It served relatively expensive burgers, cakes, and coffee. Another chain, Burger Heaven, which today has three outlets, opened in the 1940s. Other restaurateurs made their own burgers and gave them exotic names\u2014the Prima Burger, the Plush Burger, and Phoebe's Whamburger. The P. J. Clarke chain was famous for its bacon cheeseburger, which was proclaimed \"the Cadillac of burgers\" by Nat King Cole. Beginning in the 1970s, national hamburger chains McDonald's and Burger King emerged. There were also local chains, such as the Shake Shack, which was launched in Madison Square Park in 2004.\n\nThe hamburger was also a menu item on upper-class restaurants. In 1975, New York's famous \"21\" Club introduced the \"21 Burger,\" priced at $21. It was served \"nude\"\u2014without a bun. Burger connoisseurs pronounced it \"the best burger in the world,\" and it was undoubtedly the most expensive in the world at that time. In New York City in 2001, four-star chef Daniel Boulud opened db Bistro Moderne, a more casual cousin to his elegant Restaurant Daniel. On the menu was the \"db Hamburger,\" a ground sirloin patty stuffed with the rich, tender meat of beef short ribs as well as foie gras and truffles. At twenty-nine dollars, it quickly became a smash hit, and the burger was featured in the New York Times and culinary magazines. It was as if the gauntlet had been thrown down, and other restaurateurs began to vie for the title of \"best burger\"\u2014and highest price.\n\nMarc Sherry, owner of New York's venerable Old Homestead Steakhouse, offered a forty-one-dollar hamburger, with Kobe beef, lobster, mushrooms, and microgreens on a Parmesan twist roll. Daniel Boulud retaliated with the \"db Hamburger Royale\" with double truffles for nine-nine dollars. This culinary creation, served only during truffle season, won the Guinness Book of Records' certificate for the most expensive commercial hamburger in the world at the time.\n\nMany New York restaurants have their special hamburgers.\n\nBagels, Cream Cheese, and Lox\n\nThe point of origin for the bagel, a small doughnut-shaped bread, has been disputed, although it seems to have Eastern European roots. Bagels are unusual in that they are boiled before being baked, giving them a glossy finish. Jewish bakers in New York were making bagels, and street vendors were selling them, as early as the 1880s. During the early twentieth century, bagel factories opened, and bagels were sold in delicatessens. These small, plain, ring-shaped breads were little known outside Jewish neighborhoods until Polish-born Harry Lender brought the bagel to a wider American audience: by 1955, his baking plant was packaging bagels, six to a plastic bag, and these became a standard item in supermarkets nationwide. As bagels became popular, they lost their original characteristics to become bigger, softer, puffier, and sweeter. Embellished and flavored bagels became the norm, where once the only choices were \"plain\" and \"egg.\" American bagels were topped with poppy seeds, sesame seeds, onions, garlic, or coarse salt, or swirled with cinnamon and raisins. A wide variety of frozen and fresh bagels are sold in supermarkets and are served in coffee shops, diners, and fast-food chains.\n\nThe combination of bagels and cream cheese likely originated in New York. Though the Jewish immigrants from Eastern Europe were familiar with soft, unripened cheeses, such as cottage cheese, milk was relatively scarce in their rural villages. Only the very rich would have had the luxury of adding cream to the pressed, soft curds, which is how cream cheese is made. Rather, the immigrants first encountered cream cheese upon their arrival in the United States. It had been made as early as 1841; an upstate New York farmer, William Lawrence, began mass producing it in 1877 under the name \"Philadelphia Cream Cheese.\" By the early 1880s, there were several cream cheese brands to be found in New York City markets. Cream cheese, however, was an expensive \"fancy cheese,\" still beyond the budgets of the newly arrived immigrant families. By the early 1920s, however, the children of these immigrants had climbed the rungs of social mobility and were able to afford such luxuries. In addition, the mass production of cream cheese by corporations such as Phenix, beginning in 1903, had lowered the price of the cheese. Finally, the taste of cream cheese underwent a transformation with the introduction of a new high-fat cream cheese by Breakstone Brothers Dairy in 1923.\n\nIn the early 1940s, newspapers reported that the combination of bagels and lox was sold by New York delis as a Sunday morning treat. Historically, lox was salmon cured in a salt brine; today, it usually refers to smoked salmon. After World War II, virtually everyone, even Bob Hope, was eating bagels, cream cheese, and lox. In 1951, a show called Bagels and Yox, billed as the \"Yiddish American Revue,\" played on Broadway. The following year, the New York Times identified the combination of bagels, cream cheese, and lox as \"an old New York Sunday morning tradition,\" and reported that \"together they're magnificent; apart they're only wonderful, according to fanciers.\" For many New Yorkers, bagels and cream cheese are typical breakfast or brunch treats.\n\nBialys\u2014chewy, flat rolls\u2014have also become an iconic New York food. They are similar to bagels but are not boiled before baking; rather than having a glossy, hard exterior, they are flour dusted and yielding to the bite. Instead of a hole, the bialy has a depression in the middle which is filled with onions, garlic, or other seasoning. Unlike the bagel, which is now commonly sold throughout America, the bialy remains very much of New York\u2014especially when eaten fresh from the oven at Kossar's bakery on the Lower East Side.\n\nCorned Beef, Pastrami, and Reuben Sandwiches\n\nIn the late nineteenth century, New York delicatessens sold a variety of cured, smoked, salted, and brined meats, including corned beef and pastrami. Developed long before artificial refrigeration, \"corning\" is a way of preserving meat by coating it heavily with salt. Corned beef is simply salted beef, which was part of British and Irish culinary traditions that came to America in colonial times. It is usually made from a brisket, a fatty, flavorful cut. Recipes for preparing and cooking corned beef appear in many nineteenth-century American cookbooks. Corned beef is one of the few traditional Irish foods to be recognized as such in the United States; the combination of corned beef and cabbage is a must for St. Patrick's Day celebrations in New York. Here's one recipe from New York cookbook teacher and author Juliet Corson:\n\nCorned Beef Brisket: Choose about seven pounds of the navel end of the brisket of corned beef; wash it well in plenty of cold water; put it over the fire in a large pot containing sufficient cold water to cover the beef, and let it slowly approach the boiling point, removing all scum as it rises. The beef can either be left flat, or the bones be taken out and a compact roll be made of it, secured by a string. Boil the beef steadily and gently for four hours after it begins to boil; it may then be served hot, or may be pressed until cold before it is used.\n\nCabbage is usually served with corned beef; cooked according to directions given among the vegetable recipes, it is excellent; or it may be boiled with the beef, only it should not be boiled longer than just time enough to make it tender, which is less than half an hour usually. Source: Juliet Corson, Miss Corson's Practical American Cookery (New York: Dodd, Mead, 1886), 338.\n\nBy the late nineteenth century, corned beef became a staple item in New York delicatessens. Today corned beef sandwiches are still a top-of-the-menu item at the city's kosher and Jewish-style delis.\n\nThe making of pastrami is a complicated process involving brining, drying, spicing, smoking, and then steaming a piece of meat. The seasonings may include salt, peppercorns, garlic, paprika, ginger, and other spices. The method probably evolved in Asia Minor, possibly in Turkey, and eventually took hold in Romania (once ruled by the Turks), where goose breast was a favorite meat for making pastrami.\n\nThe technique of pastrami making was brought to New York by Romanian Jews in the late nineteenth century. There are two conflicting (and undocumented) stories about the first New York deli to serve pastrami. One claim is made for Sussman Volk, a Lithuanian immigrant who opened a butcher shop on Delancey Street in New York in 1888. In a 1953 article in Collier's magazine, Jacob Altman, president of the Manhattan Provision Company, claimed that Volk had acquired the recipe for making pastrami from a Romanian immigrant. Once Volk began offering pastrami in his butcher shop around 1903, according to Altman, there was such a demand for it that he turned the shop into a restaurant specializing in pastrami sandwiches. Others credit Willy Katz, an immigrant, who, in 1903, entered into a partnership in the Icelandic Brothers Delicatessen on Ludlow Street on the Lower East Side. Katz's Deli now stands at the corner of Houston and Ludlow, a mecca for pastrami lovers and a rare survivor of the immigrant era in a neighborhood that has become hip, upscale, and polyglot.\n\nThe Reuben sandwich\u2014layers of corned beef, Swiss cheese, sauerkraut, and Russian or Thousand Island dressing on rye or pumpernickel\u2014was invented in New York by Arnold Reuben, a German immigrant who opened a small deli on the Upper West Side about 1908. He later moved to Fifty-eighth Street and expanded his business into a restaurant that specialized in sandwiches. The first located print reference to a \"Reuben sandwich\" is in a 1927 newspaper article that described special sandwiches Reuben devised for celebrities who frequented his restaurant late at night, after the theater. Around 1914, he came up with the \"Annette Seelos Special\" for one of Charlie Chaplin's leading ladies. It consisted of \"ham, cheese, turkey, cole slaw and dressing.\" While not quite today's Reuben, the Annette Seelos Special was its clear antecedent.\n\nReuben sandwich at Katz's Deli, Manhattan, 2013. Courtesy of Kelly Fitzsimmons\n\nReuben's restaurant attracted celebrities even during the Depression, when Arnold Reuben spent one hundred thousand dollars \"flossing up his two sandwich shops,\" which had grown, according to a 1933 newspaper article, \"to a national institution.\" British dignitaries visiting the city in 1937 visited one of these restaurants to dine with celebrities and enjoy \"a Reuben Sandwich,\" which was a generic term for the fifty or so different sandwiches that Arnold Reuben served. One called the Col. Jay Flippen's sandwich consisted of \"corned beef on toasted rye bread with a thin slice of Swiss cheese melted over the top, mustard and cole slaw,\" which was very close to today's Reuben.\n\nJoseph Oliver Dahl, a New York restaurant expert and prolific author, changed the name for this combination in his Menu Making for Professionals in Quantity Cookery (1939). His \"Ruben\" sandwich consisted of \"Rye Bread, Switzerland Cheese, Sliced Corn Beef, Sauerkraut, Dressing.\" The book was intended for use in \"Hotels, Restaurants, Clubs, Schools, Fountains, Tearooms, Resorts, Camps, Cafeterias, Hospitals and Institutions.\" This book was widely distributed, and it went through several reprintings through 1950. It standardized the formula, although when it hit menus, the spelling was changed to \"Reuben.\" It became a staple in New York in many of the city's \"Jewish-style\" delis even though the combination of corned beef and Swiss isn't kosher (meat and cheese may not be eaten together). Many other sandwiches, such as pastrami and corned beef on rye, were also popularized by New York's delis and restaurants.\n\nNew York Cakes\n\nOne of the most iconic New York foods that emerged into the mainstream is Jewish-style cheesecake. New Yorkers consumed cheesecake since colonial times, and it was not much different from that of the cheesecakes that other Americans (and Europeans) consumed. These were made from ricotta cheese, cottage cheese, curd cheese, sour cream, or farmer's cheese, and the crust was typically made of pastry. During the early twentieth century, a very different type of cheesecake, made from cream cheese, emerged in New York delis and restaurants. It was smoother than other cheesecakes. Arnold Reuben claimed to have originated this recipe in 1928. During the following decade, his cheesecake became popular in New York, and he began to ship it to other places. An article in the Los Angeles Times in 1939 reported that he shipped it \"anywhere from New Orleans, to London to Paris.\" The shipping charges were often four times more than the cost of the cheesecake, but people were willing to pay for it. He later complained that his recipe was stolen by others, specifically Leo Lindermann.\n\nLindermann was a German-Jewish immigrant who opened Lindy's, a restaurant near Times Square in 1921. By 1948, he was selling a velvety cream-cheese-based cheesecake, plain or with sweet fruit toppings. It had a cookie-like crust, and the cake was flavored with citrus zest. Another cheesecake seller was Junior's in Brooklyn. It had been established in 1950 by Harry Rosen, who was born on the Lower East Side, sits on a thin layer of sponge cake, and is flavored only with vanilla. Sylvia Balser Hirsch, a Texan transplanted to New York City, provided New York's upscale restaurants and gourmet shops with a distinctive vanilla cheesecake (and a range of other desserts) in the 1970s and 1980s under the name \"Miss Grimble.\" Her preference was for cookie-crumb crusts.\n\nOther New York cakes were also popular\u2014at least for a time. Ebinger's Baking Company was a small family-owned chain in Brooklyn that began on Flatbush Avenue in 1898. It was known for its classic lemon cupcakes, \"crumb buns,\" and its Brooklyn Blackout Cake, a very dark devil's food layer cake with custardy chocolate filling and frosting; the whole cake was coated with a layer of chocolate cake crumbs. The chain folded in 1972, but memories of its Blackout Cake remain alive for those lucky enough to have sampled the original. Many bakers and cookbook writers have tried to replicate the original recipe, which has not survived.\n\nYet another iconic New York dessert cake is the Charlotte Russe. In the classic French recipe for this dessert (a \"Russian\" charlotte), a mold is lined with ladyfingers or slices of cake and then filled with whipped cream or custard and chilled so that it can be sliced. The New York City version is made as an individual serving: a disk of yellow sponge cake is placed in the bottom of a cylindrical paper case, and whipped cream is piped in until it mounds on top. The confection is topped with a maraschino cherry (and sometimes with chocolate sprinkles). Sometimes the paper case was made in two parts, so that the bottom can be pushed up as the cream was consumed. As the Charlotte Russe was mostly whipped cream, it was a cool-weather specialty, sold only in the autumn and winter. These iconic treats were typically sold in bakeries and candy stores from the 1930s to the 1960s; there is hardly any place in the city that still makes them.\n\nEgg Creams\n\nNonalcoholic drinks made with raw eggs, cream, and fizzy water were popular soda-fountain fare in the late nineteenth century; chocolate and vanilla syrups were the most common flavorings. To make a more affordable treat, the egg and cream were sometimes eliminated, and the simplified \"egg cream\" was a mixture of milk and chocolate syrup, frothed with carbonated water (usually seltzer, dispensed from a blue- or green-glass bottle topped with a silvery syphon). Several Brooklynites have been credited with its invention, including Louis Auster. Typically made with Fox's U-Bet Chocolate Syrup (invented by Brooklynites Herman and Ida Fox in 1895), egg creams were standard fare in soda fountains and \"candy stores\" (corner grocery\/paper\/sweet shops) throughout New York by the 1920s. Vanilla egg creams arrived in the 1950s, as did coffee-flavored ones. The egg cream largely disappeared in the late twentieth century, although it can still be found in some quintessential New York City restaurants, such as Junior's in Brooklyn, and in surviving \"candy stores,\" mostly in the outer boroughs.\n\nCookies\n\nSeveral cookies originated in New York City; the best known is the Oreo, a round cookie with vanilla cream sandwiched between two dark chocolate wafers. The National Biscuit Company (later Nabisco), which was based in the city until the 1950s, introduced Oreos in 1912 to compete with Sunshine Bakeries' Hydrox Biscuit Bonbons, which had come on the market two years earlier. Oreos were produced in the National Biscuit Company bakeries, on Ninth Avenue between Fifteen and Sixteenth Streets\u2014a block where the street sign now reads \"Oreo Way.\" Oreos are arguably the most famous commercial cookies in American history.\n\nA New York favorite, but less successful elsewhere in the country, is the Mallomar, a round biscuit that tastes something like a Graham cracker, topped with a dome of marshmallow and coated in dark chocolate. These were first made by Nabisco in 1913, at the same bakeries where Oreos were produced. Because the chocolate coating melts in warm temperatures, Mallomars are made and sold only in the autumn and winter; 70 percent of the total production is snapped up by New Yorkers.\n\nEven more iconic to New York is the black-and-white cookie, which is typically sold in bakeries and delis. It is a large, somewhat soft, chewy cookie frosted half and half with chocolate and vanilla icing.\n\nPretzels\n\nPretzels, which were made in similar ways to bagels, were sold in New York by the 1860s. Street vendors carried them on long poles, a custom that would survive for decades. Today, big, somewhat soft pretzels are sold from the same carts that offer hot dogs and, in the winter, roasted chestnuts. Even in the early days they were usually made by professional bakers, but some old recipes for home-baked pretzels have survived.\n\nPretzels: Sift together two cupfuls of flour, one teaspoonful of salt and a pinch of ginger; add one egg beaten, one-third of a cupful of butter, one-half of a yeast cake dissolved, and stir in enough sweet milk to make a very stiff dough; beat with biscuit roller until smooth, and let rise; cut off small pieces and roll them between the hands into strips, form small rings, pinch the ends of the dough together, and let rise; put them a few at a time in salted boiling water, let cook until they begin to come to the top of the water, take out quickly, sprinkle with salt, put in a greased pan, and bake a light brown. Source: Miss Sigel, \"Pretzels,\" Table Talk 27 (February 1912): 102.\n\nThe New York City food scene has come a long way since, where the city now stands, Lenape Indians hunted game; gathered nuts, berries, and oysters; caught shad and sturgeon; and grew maize, squash, and beans using slash and burn agricultural methods. Almost four hundred years ago European colonists introduced Old World foods and beverages into the area, and over the succeeding centuries, immigrants\u2014first from Western Europe and later from the rest of the world\u2014streamed into the city, bringing their own food culture. These contributions are all ingredients in the simmering, ever-changing culinary stew that is New York City food.\n\nTo meet the culinary needs of the city's 8.3 million residents and its fifty-two million annual visitors, New York has developed one of the most complex food systems in the world: there are more than fifty thousand restaurants; some five thousand street food vendors and food truck operators; tens of thousands of supermarkets, corner stores, bodegas, delis, greenmarkets, meat markets, wine shops, and bakeries; and hundreds of soup kitchens and food pantries. If past is prologue, New York City's unique, quixotic culinary life will continue to sustain, surprise, confuse, and amuse anyone lucky enough to eat there.\n\nBibliography\n\nNew York Histories\n\nAllen, Irving L. City in Slang: New York Life and Popular Speech. New York: Oxford University Press, 1993.\n\nAnbinder, Tyler. Five Points: The 19th Century New York City Neighborhood That Invented Tap Dance, Stole Elections, and Became the World's Most Notorious Slum. New York: Free Press, 2001.\n\n[Beard, Frank]. The Night Side of New York: A Picture of the Great Metropolis after Nightfall. New York: J. C. Haney, 1866.\n\nBernard, John. Retrospections of America, 1797\u20131811. Edited by Laurence Hutton and Brander Matthews. New York: Harper & Brothers, 1887.\n\nBooth, Mary L. The History of New York City. New York: W. R. C. Clark, 1860.\n\nBrowne, Junius Henri. The Great Metropolis: A Mirror of New York. Hartford, CT: American Publishing, 1869.\n\nBurrows, Edwin G., and Mike Wallace. Gotham: A History of New York City to 1898. New York: Oxford University Press, 1999.\n\nCampbell, Helen. Darkness and Daylight: Lights and Shadows of New York Life. Hartford, CT: A. D. Worthington, 1891.\n\nCantwell, Anne-Marie E., and Diana diZerega Wall. Unearthing Gotham: The Archaeology of New York City. 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The Immigrant Scene: Ethnic Amusements in New York, 1880\u20131920. Minneapolis: University of Minnesota Press, 2008.\n\nHarlow, Alvin F. Old Bowery Days: The Chronicles of a Famous Street. New York: D. Appleton, 1931.\n\nHarris, Charles Townsend. Memories of Manhattan in the Sixties and Seventies. New York: Derrydale Press, 1928.\n\nHasswell, Charles H. Reminiscences of an Octogenarian of the City of New York. New York: Harper & Brothers, 1897.\n\nHughes, Rupert. The Real New York. New York: Smart Set, 1904.\n\nJacobs, Jaap. The Colony of New Netherland: A Dutch Settlement in Seventeenth-Century America. Ithaca, NY: Cornell University Press, 2009.\n\n\u2014\u2014\u2014. \"'It Has Pleased the Lord That We Must Learn English': Dutch New York after 1664.\" In Dutch New York between East and West: The World of Margrieta van Varick, edited by Deborah L. Krohn and Peter Miller, 55\u201365. New Haven, CT: Yale University Press, 2009.\n\nJameson, J. Franklin, ed. Narratives of New Netherland, 1609\u20131664. New York: Charles Scribner's Sons, 1909.\n\nJanowitz, Meta F., and Diane Dallal, eds. Tales of Gotham, Historical Archaeology, Ethnohistory and Microhistory of New York City. New York: Springer, 2013.\n\nKernan, Frank J. Reminiscences of the Old Fire Laddies and Volunteer Fire Departments of New York and Brooklyn. New York: M. Crane, 1885.\n\nKrohn, Deborah L., and Peter Miller, eds. Dutch New York between East and West: The World of Margrieta van Varick. New Haven, CT: Yale University Press, 2009.\n\nLevinton, Jeffrey S., and John R. Waldman, eds. The Hudson River Estuary. Cambridge: Cambridge University Press, 2006.\n\nMackeever, Samuel Anderson. Glimpses of Gotham, and City Characters. New York: National Police Gazette Office, [1881].\n\nMayer, Grace M. Once upon a City: New York from 1890 to 1910. New York: Macmillan, 1958.\n\nMcKay, Ernest A. The Civil War and New York City. Syracuse, NY: Syracuse University Press, 1990.\n\nMensch, Barbara. South Street. New York: Columbia University Press, 2007.\n\nMorris, Lloyd R. Incredible New York. New York: Random House, 1951.\n\nPeterson, Arthur Everett, and George William Edwards. New York as an Eighteenth Century Municipality. New York: Columbia University Press, 1917.\n\nReitano, Joanne R. The Restless City: A Short History of New York from Colonial Times to the Present. New York: Routledge, 2006.\n\n\u2014\u2014\u2014. The Restless City Reader: A New York City Sourcebook. New York: Routledge, 2010.\n\nRiis, Jacob August. How the Other Half Lives: Studies among the Tenements of New York. New York: Charles Scribner's Sons, 1890.\n\nSandler, Corey. Henry Hudson: Dreams and Obsession. New York: Citadel Press, 2007.\n\nShepp, James W., and Daniel B. Shepp. Shepp's New York City Illustrated: Scene and Story in the Metropolis of the Western World. Chicago: Globe Bible, 1894.\n\nShorto, Russell. The Island at the Center of the World: The Epic Story of Dutch Manhattan and the Forgotten Colony That Shaped America. Prince Frederick, MD: Recorded Books, 2004.\n\nSingleton, Esther. Social New York under the Georges, 1714\u20131776. New York: D. Appleton, 1902.\n\nSmith, Matthew Hale. Sunshine and Shadow in New York. Hartford, CT: J. B. Burr, 1868.\n\nSpann, Edward K. Gotham at War: New York City, 1860\u20131865. Wilmington, DE: Scholarly Resources, 2002.\n\nStill, Bayrd. Mirror for Gotham: New York as Seen by Contemporaries from Dutch Days to the Present. New York: New York University Press, 1956.\n\nVanderbilt, Gertrude Lefferts. The Social History of Flatbush, and Manners and Customs of the Dutch Settlers of Kings County. New York: D. Appleton, 1881.\n\nvan der Donck, Adriaen. A Description of the New Netherlands. Edited by Thomas F. O'Donnell. Syracuse, NY: Syracuse University Press, 1968.\n\nVan Dyke, John C. The New New York. New York: Macmillan, 1909.\n\nWilson, James Grant, ed. The Memorial History of the City of New York, from the First Settlement to the Year 1892. 4 vols. [New York]: New-York History, 1892\u20131893.\n\nCity Food Histories\n\nBatterberry, Michael, and Ariane Batterberry. On the Town in New York: The Landmark History of Eating, Drinking, and Entertainments from the American Revolution to the Food Revolution. New York: Routledge, 1999.\n\nGrimes, William. Appetite City: A Culinary History of New York. New York: North Point Press, 2009. Appetite City television series: <http:\/\/a002-vod.nyc.gov\/html\/appetite_city.php?id=1221.>\n\nLawson, Annie H., and Jon Deutsch, eds. Gastropolis: Food and New York City. New York: Columbia University Press, 2008.\n\nSchwartz, Arthur. Arthur Schwartz's New York City Food: An Opinionated History and More Than 100 Legendary Recipes. New York: Stewart, Tabori & Chang, 2004.\n\nShulman, Robin. Eat the City: A Tale of the Fishers, Trappers, Hunters, Foragers, Slaughterers, Butchers, Farmers, Poultry Minders, Sugar Refiners, Cane Cutters, Beekeepers, Winemakers, and Brewers Who Built New York. New York: Crown, 2012.\n\nChapter 1. The Material Resources: Land, Water, and Air\n\nGrumet, Robert S. First Manhattans: A History of the Indians of Greater New York. Norman: University of Oklahoma Press, 2011.\n\nHomberger, Eric. The Historical Atlas of New York City: A Visual Celebration of 400 Years of New York City's History. New York: Henry Holt, 1994.\n\nKoeppel, Gerard T. Water for Gotham: A History. Princeton, NJ: Princeton University Press, 2000.\n\nLewis, Tom. The Hudson: A History. New Haven, CT: Yale University Press, 2005.\n\nOtto, Paul. \"Intercultural Relations between Native Americans and Europeans in New Netherland and New York.\" In Four Centuries of Dutch-American Relations, edited by Hans Krabbendam, Cornelis A. van Minnen, and Giles Scott-Smith, 178\u201391. Albany: State University of New York Press, 2009.\n\nPritchard, Evan T. Native New Yorkers: The Legacy of the Algonquin People of New York. San Francisco: Council Oak Books, 2002.\n\nSanderson, Eric W. Mannahatta: A Natural History of New York City. New York: Abrams, 2009.\n\nChapter 2. From Colonization to the Present\n\nBluestone, Daniel. \"'The Pushcart Evil.'\" In Landscape of Modernity, edited by David Ward and Oliver Zunz, 287\u2013312. New York: Russell Sage Foundation, 1992.\n\nBradley, Elizabeth L. Knickerbocker: The Myth behind New York. New Brunswick, NJ: Rivergate Books, 2009.\n\nCahn, William. Out of the Cracker Barrel: The Nabisco Story, from Animal Crackers to Zuzus. New York: Simon & Schuster, 1969.\n\nFederal Writers Project. The Italians of New York. New York: Random House, 1938.\n\nFrench, Earl R. Push Cart Markets in New York City. Washington, DC: Agricultural Economic Bureau, USDA, 1925. <http:\/\/naldc.nal.usda.gov\/download\/CAT10675911\/PDF>.\n\nKellar, Jane Carpenter, comp. and ed. On the Score of Hospitality: Selected Receipts of a Van Rensselaer Family, Albany, New York, 1785\u20131835; A Historic Cherry Hill Recipe Collection. Albany, NY: Historic Cherry Hill, 1986.\n\nLoukaitou-Sideris, Anastasia, and Renia Ehrenfeucht. Sidewalks: Conflict and Negotiation over Public Space. Cambridge, MA: MIT Press, 2009.\n\nRay, Krishnendu. \"Exotic Restaurants and Expatriate Home Cooking: Indian Food in Manhattan.\" In The Globalization of Food, edited by David Inglis and Debra Gimlin, 213\u201326. Oxford: Berg, 2009.\n\nRestad, Penne L. Christmas in America: A History. New York: Oxford University Press, 1995.\n\nServenti, Silvano, and Fran\u00e7oise Sabban. Pasta: The Story of a Universal Food. Translated by Antony Shugaar. New York: Columbia University Press, 2002.\n\nSijs, Nicoline van der. Cookies, Coleslaw, and Stoops: The Influence of Dutch on the North American Languages. [Amsterdam]: Amsterdam University Press, 2009.\n\nTaylor, Dennis S., et al. \"Street Foods in America: A True Melting Pot.\" World Review of Nutrition and Dietetics 86 (2000): 25\u201344.\n\nChapter 3. A Culinary Stew: Immigrants and Their Food\n\nBalinska, Maria. The Bagel: The Surprising History of a Modest Bread. New Haven: Yale University Press, 2008.\n\nBayor, Ronald H., and Timothy Meagher, eds. The New York Irish. Baltimore: Johns Hopkins University Press, 1996.\n\nBercovici, Konrad. Around the World in New York. New York: Century Co., [1924].\n\n\u2014\u2014\u2014. \"The Greatest Jewish City in the World.\" Nation, September 12, 1923, 259\u201361.\n\nBinder, Frederick M., and David M. Reimers. All the Nations under Heaven: An Ethnic and Racial History of New York City. New York: Columbia University Press, 1995.\n\nBurgess, Thomas. Greeks in America. Boston: Sherman, French, 1913.\n\nDiner, Hasia R. Hungering for America: Italian, Irish, and Jewish Foodways in the Age of Migration. Cambridge, MA: Harvard University Press, 2001.\n\nErnst, Robert. Immigrant Life in New York City, 1825\u20131865. New York: Kings Crown Press, 1949.\n\nFoner, Nancy, ed. New Immigrants in New York. New York: Columbia University Press, 2001.\n\nGabaccia, Donna R. We Are What We Eat: Ethnic Food and the Making of Americans. Cambridge, MA: Harvard University Press, 1998.\n\nGolden, Harry. The Greatest Jewish City in the World. New York: Doubleday, 1972.\n\nHendricks, Glenn L. The Dominican Diaspora: From the Dominican Republic to New York City; Villagers in Transition. New York: Teachers College Press, Columbia University, [1974].\n\nKhandelwal, Madhulika Shankar. Becoming American, Being Indian: An Immigrant Community in New York City. Ithaca, NY: Cornell University Press, 2002.\n\nKorrol, Virginia S\u00e1nchez. From Colonia to Community: The History of Puerto Ricans in New York City. Berkeley: University of California Press, 1994.\n\nLaguerre, Michel S. American Odyssey: Haitians in New York City. Ithaca, NY: Cornell University Press, 1984.\n\nMarx, Jeff. \"The Days Had Come of Curds and Cream: The Origins and Development of Cream Cheese in America, 1870\u20131880.\" Food, Culture and Society 15 (2012): 177\u201395.\n\nMin, Pyong Gap. Caught in the Middle: Korean Communities in New York and Los Angeles. Berkeley: University of California Press, 1996.\n\nPozzetta, George E. \"The Italians of New York City, 1890\u20131914.\" PhD diss., University of North Carolina at Chapel Hill, 1971.\n\nRischin, Moses. The Promised City: New York's Jews, 1870\u20131914. Cambridge, MA: Harvard University Press, 1962.\n\nS\u00e1nchez Korrol, Virginia. From Colonia to Community: The History of Puerto Ricans in New York City. Berkeley: University of California Press, 1994.\n\nWang, Xinyang. Surviving the City: The Chinese Immigrant Experience in New York City, 1890\u20131970. Lanham, MD: Rowman & Littlefield, 2001.\n\nZiegelman, Jane. 97 Orchard: An Edible History of Five Immigrant Families in One New York Tenement. New York: Smithsonian Books\/HarperCollins, 2010.\n\nChapter 4. From Garden to Grocery: A Tour of Markets, Vendors, and Other Food Sellers\n\nBenepe, Barry, and John L. Hess. Greenmarket: The Rebirth of Farmers Markets in New York City. New York City: Council on the Environment of New York City, [1978].\n\nCaldwell, Alison. \"Will Tweet for Food: The Impact of Twitter and New York City Food Trucks, Online, Offline, and Inline.\" Appetite 56, no. 2 (April 2011): 522.\n\nThe Cries of New-York. New-York: Printed and sold by S. Wood, Juvenile Book-store, 1808.\n\nDeVoe, Thomas F. The Market Assistant. New York: Hurd and Houghton, 1867.\n\n\u2014\u2014\u2014. The Market Book Containing a Historical Account of the Public Markets. Vol. 1. New York: Printed for the author, 1862.\n\nGreat Atlantic & Pacific Tea Company. Colonel Goodbody's Red Setter Menus. Montvale, NJ: Author, n.d.\n\n\u2014\u2014\u2014. Grocery Catalogue 1912. Glen Cove, NY: Author, 1912.\n\nHutcheson, John C. \"The Markets of New York.\" Harper's, July 1867, 229\u201336.\n\nLevinson, Marc. The Great A&P and the Struggle for Small Business in America. New York: Hill and Wang, 2011.\n\nMendelson, Anne. \"To Market, to Market: A Profile of Thomas F. De Voe.\" NYFoodstory 1 (Fall 2012): 3, 14\u201316, 18.\n\nSeiger, Karen E. Markets of New York City: A Guide to the Best Artisan, Farmer, Food, and Flea Markets. New York: Little Bookroom, 2010.\n\nSlantez, Priscilla Jennings. \"A History of the Fulton Fish Market.\" Log of the Mystic Seaport 36 (Spring 1986): 14\u201325.\n\nTangires, Helen. Public Markets. New York: W. W. Norton, Library of Congress, 2008.\n\n\u2014\u2014\u2014. Public Markets and Civic Culture in Nineteenth-Century America. Baltimore: Johns Hopkins University Press, 2003.\n\nWalsh, William I. The Rise and Decline of the Great Atlantic & Pacific Tea Company. Secaucus, NJ: Lyle Stuart, 1986.\n\nWasserman, Suzanne. \"Hawkers and Gawkers: Peddling and Markets in New York City.\" In Gastropolis: Food and New York City, 153\u201373. New York: Columbia University Press, 2009.\n\nWhite, Charles Henry. \"The Fulton Street Market.\" Harper's, September 1905, 616\u201323.\n\nChapter 5: Drinking in the City\n\nAnderson, Will. The Breweries of Brooklyn: An Informal History of a Great Industry in a Great City. Croton Falls, NY: Anderson, 1976.\n\nBryson, Lew. New York Breweries. Mechanicsburg, PA: Stackpole Books, 2003.\n\nByles, W. Harrison. Old Taverns of New York. New York: Frank Allaben Genealogical Company, 1915.\n\nDeGennaro, Jeremiah J. \"From Civic to Social: New York's Taverns, Inside and Outside the Political Life of the United States after the Revolution.\" Master's thesis, University of North Carolina at Greensboro, 2008.\n\nMitchell, Joseph. McSorley's Wonderful Saloon. New York: Pantheon, 1992.\n\n\"Old New York Coffee-Houses.\" Harper's, March 1882, 481\u201399.\n\nSmith, Andrew F. Drinking History: Fifteen Turning Points in the Making of American Drink. New York: Columbia University Press, 2012.\n\nChapter 6: Eateries\n\nBeriss, David, and David Sutton, eds. The Restaurants Book: Ethnographies of Where We Eat. Oxford: Berg, 2007.\n\nBerry, Rynn, and Chris Abreu-Suzuki, with Barry Litsky. The Vegan Guide to New York City. 17th ed. New York: Ethical Living, 2010.\n\nBradley, Jimmy, Andrew Friedman, and David Sawyer. The Red Cat Cookbook: 125 Recipes from New York City's Favorite Neighborhood Restaurant. New York: Clarkson Potter, 2006.\n\nBrody, Iles. The Colony. New York: Greenberg, 1945.\n\nBromell, Nicolas. \"The Automat: Preparing the Way for Fast Food.\" New York History 81, no. 3 (July 2000): 300\u2013312.\n\nChappell, George Shepard. The Restaurants of New York. New York: Greenberg, 1925.\n\nDearing, Albin Pasteur. The Elegant Inn: The Waldorf-Astoria Hotel, 1893\u20131929. Secaucus, NJ: Lyle Stuart, 1986.\n\nDiehl, Lorraine B., and Marrianne Hardart. Automat: The History, Recipes, and Allure of Horn & Hardart' s Masterpiece. New York: Clarkson Potter, 2002.\n\nErenberg, Lewis A. Steppin' Out: New York Nightlife and the Transformation of American Culture, 1890\u20131930. Westport, CT: Greenwood, 1981.\n\nEvans, Meryle R. \"Knickerbocker Hotels and Restaurants 1800\u20131850.\" New-York Historical Society Quarterly 36 (1952): 377\u2013410.\n\nFairchild, Henry Pratt. Greek Immigration to the United States. New Haven, CT: Yale University Press, 1911.\n\nFederman, Mark Russ. Russ & Daughters: Reflections and Recipes from the House That Herring Built. New York: Schocken, 2013.\n\nFreeland, David. Automats, Taxi Dances, and Vaudeville: Excavating Manhattan's Lost Places of Leisure. New York: New York University Press, 2009.\n\nGamber, Wendy. The Boardinghouse in Nineteenth-Century America. Baltimore: Johns Hopkins University Press, 2007.\n\nGarrett, Thomas Myers. \"A History of Pleasure Gardens in New York City, 1700\u20131865.\" PhD diss., New York University, 1978.\n\nHaley, Andrew P. Turning the Tables: Restaurants and the Rise of the American Middle Class, 1880\u20131920. Chapel Hill: University of North Carolina Press, 2011.\n\nJames, Rian. Dining in New York. New York: John Day, 1930.\n\nLebewohl, Sharon. The Second Avenue Deli Cookbook: Recipes and Memories from Abe Lebewohl's Legendary Kitchen. New York: Villard, 1999.\n\nMaccioni, Sirio, and Peter Elliot. Sirio: The Story of My Life and Le Cirque. Hoboken, NJ: Wiley, 2004.\n\nManzo, Joseph T. \"From Pushcart to Modular Restaurant.\" Journal of American Culture 13, no. 3 (Fall 1990): 13\u201321.\n\nMariani, John, with Alex Von Bidder. The Four Seasons: A History of America's Premier Restaurant. New York: Crown, 1994.\n\nMeyer, Danny, and Michael Romano. The Union Square Caf\u00e9 Cookbook: 160 Favorite Recipes from New York's Acclaimed Restaurant. New York: HarperCollins, 1994.\n\nRimmer, Leopold. A History of Old New York Life and the House of the Delmonicos. New York, 1898.\n\nSardi, Vincent, and Richard Gehman. Sardi's: The Story of a Famous Restaurant. New York: Henry Holt, [1953].\n\nSchriftgiesser, Karl. Oscar of the Waldorf. New York: E. P. Dutton, 1943.\n\nSegrave, Kerry. Vending Machines: An American Social History. Jefferson, NC: McFarland, 2002.\n\nShuldiner, Alec Tristin. \"Trapped behind the Automat: Technological Systems and the American Restaurant, l902\u2013l99l.\" PhD diss., Cornell University, 200l.\n\nSlomanson, Joan Kanel. When Everybody Ate at Schrafft's: Memories, Pictures, and Recipes from a Very Special Restaurant Empire. Fort Lee, NJ: Barricade Books, 2006.\n\nThomas, Lately [pseudonym for Robert V. P. Steele?]. Delmonico's: A Century of Splendor. Boston: Houghton Mifflin, 1967.\n\nWechsberg, Joseph. Dining at the Pavillon. Boston: Little, Brown, 1962.\n\nWhere and How to Dine in New York: The Principal Hotels, Restaurants and Caf\u00e9s of Various Kinds and Nationalities Which Have Added to the Gastronomic Fame of New York and Its Suburbs. New York: Lewis, Scribner, 1903.\n\nWilliams, Ellen, and Steve Radlauer. The Historic Shops and Restaurants of New York. New York: Little Bookroom, 2002.\n\nZagat Survey. New York City Restaurants 2009. New York: Author, 2008.\n\nChapter 7: Historic Cookbooks, Dishes, and Recipes\n\n[Alsop, Richard.] The Universal Receipt Book or Complete Family Directory by a Society of Gentlemen in New York. New York: I. Riley, 1814.\n\nAppel, Jennifer, and Allysa Torey. The Magnolia Bakery Cookbook: Old-fashioned Recipes from New York's Sweetest Bakery. New York: Simon & Schuster, 1999.\n\nAstor, Mrs. Jane. The New York Cook-Book. New York: G. W. Carleton, 1880.\n\nBastianich, Lidia. Lidia's Italian Table. New York: William Morrow, 1998.\n\nBlot, Pierre. What to Eat, and How to Cook It. New York: D. Appleton, 1863.\n\nBradley, Jimmy, Andrew Friedman, and David Sawyer. The Red Cat Cookbook: 125 Recipes from New York City's Favorite Neighborhood Restaurant. New York: Clarkson Potter, 2006.\n\n[Centaur]. New York Receipt Book. New York: Centaur, 1881.\n\nClaiborne, Craig. The New York Times Cook Book. New York: Harper & Row, 1961.\n\nClark, Annie. The New Waldorf Cook Book. New York: F. Tennyson Neely, 1899.\n\nCorson, Juliet. Miss Corson's Practical American Cookery. New York: Dodd, Mead, 1886.\n\nFilippini, Alessandro. One Hundred Ways of Cooking Eggs. New York: Charles L. Webster, 1892.\n\n\u2014\u2014\u2014. One Hundred Ways of Cooking Fish. New York: Charles L. Webster, 1892.\n\n\u2014\u2014\u2014. The International Cook Book. New York: Doubleday, Page, 1906.\n\n\u2014\u2014\u2014. The Table: How to Buy Food, How to Cook it, and How to Serve It. New York: Charles L. Webster, 1889.\n\nGreenbaum, Florence Kreisler. The International Jewish Cook Book: A Modern \"Kosher\" Cook Book. New York: Bloch, 1918.\n\nHesser, Amanda. The Essential New York Times Cook Book: Classic Recipes for a New Century. New York: W. W. Norton, 2010.\n\nLahman, DeDe, Neil Kleinberg, and Michael Harlan Turkell. Clinton St. Baking Company Cookbook: Breakfast, Brunch and Beyond from New York's Favorite Neighborhood Restaurant. New York: Little, Brown, 2010.\n\nLewis, Edna. The Taste of Country Cooking. New York: Alfred A. Knopf, 1976.\n\nLewis, Edna, and Evangeline Peterson. The Edna Lewis Cookbook. Indianapolis: Bobbs-Merrill, 1972.\n\nMay, Anna. Die kleine New Yorker k\u00f6chin [The little New York cook). New York: Verlag von Steiger, 1859.\n\nMayor Mitchel's Food Supply Committee. Hints to Housewives. New York: Author, 1917.\n\nO'Neill, Molly. New York Cookbook. New York: Workman, 1992.\n\nP\u00e9pin, Jacques. The Apprentice: My Life in the Kitchen. Boston: Houghton Mifflin, 2003.\n\nRanhofer, Charles. The Epicurean. New York: R. Ranhofer, 1894.\n\nRose, Peter G., trans. and ed. The Sensible Cook: Dutch Foodways in the Old and the New World. Syracuse, NY: Syracuse University Press, 1989.\n\n[Sauer Company]. Fifty Recipes on Flavoring by a Famous New York Chef. Richmond, VA: C. F. Sauer, [1920s].\n\nSchwartz, Arthur. Arthur Schwartz's New York City Food: An Opinionated History and More than 100 Legendary Recipes. New York: Stewart, Tabori & Chang, 2004.\n\nSigel, Miss. \"Pretzels.\" Table Talk 27 (February 1912): 102.\n\nTschirky, Oscar. The Cook Book by \"Oscar\" of the Waldorf. Chicago: Werner, 1896.\n\nAbout the Author\n\nAndrew F. Smith has taught culinary history at the New School in New York since 1995. He is the author or editor of twenty-five books, including American Tuna: The Rise and Fall of an Improbable Food (2012), Drinking History: 15 Turning Points in the Making of American Beverages (2012), and Food and Drink in American History: A \"Full Course\" Encyclopedia (2013). He serves as the editor in chief of the Oxford Encyclopedia on Food and Drink in America (2013) and series editor for the Edible series. He has written more than three hundred articles in academic journals, popular magazines, and newspapers, and has served as historical consultant to several television series. For more about him, visit his website: www.andrewfsmith.com\/.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\n_For my marvellous pal Nicola,  \nbecause of the seal bit_\nContents\n\nLast Time in Mermaid World...\n\n1. Meri Pebble, the Spy Mermaid\n\n2. Zelda on Thick Ice\n\n3. Blubble the Ice-skating Seal\n\n4. The Magic of Mermaid Rockits\n\n5. Slushville Is So Slush\n\n6. Simoseal\n\n7. Meri Does Some Midnight Spying\n\n8. Underwater Igloo Avenue, Flurry Falls\n\n9. Meri Screams I AM MERI, a Lot\n\n10. Maritza Mist\n\n11. Multistorey Clam-Car Car Park (Say It Fast)\n\n12. Wartina's Necklace\n\n13. Gronnyupple in Trouble\n\n14. The Power of Paris\n\n15. Jelly and the Whale Bus\n\n16. Realm Reach\n\n17. The Glorious Pinkness of Pinkly\n\n18. How Do You Get an Axolotl Out of a Vending Machine, and Other Important Questions\n\n19. Meri and the Burguin Sale\n\n20. Vampire Rocks and the Ice Leech\n\n21. Long-Lost Steve\n\n22. Nightmare Dolphin\n\n23. Meri Makes Some Friends\n\n24. Slippery Octopolli\n\n25. Beak de Beak\n\n26. Maritza Sends a Shark\n\n27. The Right Ticket\n\n28. Paris Rants and Raves about Gulper and Rose\n\n29. Steve and the Dolphin Balloon\n\n30. Jawella's to the Rescue (Sort of)\n\n31. Steve Arrives at Rainbow Landing\n\n32. So Many Mermaid Towns, So Little Time\n\n33. Seahorse Life\n\n34. To Mume\n\n35. Wearing a Vending Machine\n\n36. Back on the Dolphin Balloon\n\n37. Goblin Sharks GALORE and an Unexpected Face\n\n38. Sabrina Explains Herself\n\n39. Showdown\n\n40. Goodbye, Maritza Mist\n\n41. Back to Underwater Igloo Avenue\n\n42. Just Mermaids\n\nWitch Wars\n\n1. Down the Plughole\n\nLast Time in Mermaid World...\n\nThe gang grew bigger! At first it was just Beattie and her twin best friends, Mimi and Zelda \u2013 and, of course, Steve, the talking seahorse. But now they have Paris too. She's a human who, with the help of a magic necklace, can morph into a mermaid and various other sea creatures. And in the Crocodile Kingdom, they befriended Gronnyupple, the Seahorse Surprise-guzzling water witch, who told Beattie she's a water witch too! And they also figured out Mimi is a fishtalker. So that's the gang \u2013 two water witches, one fishtalker, a human who can morph into a mermaid, a talking seahorse and a... Zelda.\n\nWhen we left them, they were on their way to Frostopia in a clam car. Gronnyupple is convinced Maritza Mist is in trouble. Maritza is also a water witch and she owns the Maritza Mist's Water Witch Catalogue, where all water witches order their magic. Gronnyupple didn't receive her latest catalogue order, which made her suspect something had happened to Maritza \u2013 and when they heard that two immortal mermaids who had used Maritza's magic potions had escaped from Viperview Prison, they started to worry. So they set off to find her.\n\nBut getting into Frostopia is a challenge because the kingdom is closed to mermaids from other realms. Beattie is hoping they'll be able to glide in undetected, which is wishful thinking because nothing ever seems to go to plan...\n\nFirst, though, it's time to check in on a mermaid called Meri, in the far-off \u2013 and much warmer \u2013 underwater city of Fortress Bay. Because that's where this story really begins...\n1\n\nMeri Pebble, the Spy Mermaid\n\nMeri Pebble had been a Fortress Bay spy for as long as she could swish her tail, and her morning routine was _always_ the same.\n\nShe swam out of bed, a fish brushed her teeth, and she pulled her long black hair into a slick ponytail. Then she ate some of her favourite Sandcrackle cereal.\n\nAfter breakfast, Meri collected her jellyfish assistant, Lady Wriggles, and together they swam to the northerly tower of Fortress Bay headquarters to do their morning rounds.\n\nOnly on this particular morning, she was running very late.\n\n'Excuse me, coming through,' Meri said as she wriggled her way past all the spy mermaids crowding the corridors. Meri was Fortress Bay's most promising young spy, so she had some special extra duties that none of the other mermaids knew about.\n\nAs she swam along, she swished her tail from side to side, making images flash across it. Her spy-mermaid tail looked plain and ordinary, but she could see pictures from all around the world on it. Spy mermaids kept an eye on former criminal mermaids, human divers who needed distracting and whales assisting with spy missions. Meri loved watching the mermaids of the underwater kingdoms, all free and having fun. They had no idea what dangers she was protecting them from.\n\nSometimes she wished she could be like them, but she couldn't leave Fortress Bay until her training was complete. And spy training took _years_.\n\nIn a quiet corridor she pressed her tail against a metal panel. Lady Wriggles did the same, only with her little jellyfish face. The panel glowed and clicked back to reveal a room beyond \u2013 a room filled with frost and freezing water.\n\n'Do you want to see who can create the biggest nose icicles today?' Meri asked Lady Wriggles.\n\nLady Wriggles rearranged her tentacles into a NO. She never wanted to play that game.\n\n'Suit yourself,' Meri said with a smile as the door closed behind them. Frost formed on her jet-black hair as she swam deeper inside.\n\n'Spy MP 241 reporting from Tower Five, secret lock four,' she said, slotting her tail into a gap in the floor.\n\nThere was another click and Meri rose out of the ground on a platform. In front of her, another piece of the floor rose too \u2013 a thick tube, dark and frozen.\n\nLady Wriggles began punching buttons on a control panel.\n\nMeri gave her a nod. 'Spy MP 241 illuminating the ice narwhal.'\n\nA light burst from the tube.\n\n'Spy MP 241 speaking. I can confirm that the ice narwhal is\u2014'\n\nHer eyes grew wide.\n\n'Gone!'\n\nLady Wriggles looked like she'd been electrified.\n\n'And the list?' Meri said, her eyes darting to the back of the door, where a very important list was kept.\n\nThe list had vanished.\n\n'Whoever took the ice narwhal also took the list,' Meri said. 'Oh this is bad. Quick, Lady Wriggles. _The telephone_.'\n\nMeri paused.\n\n'We can't mention this to anyone, Lady Wriggles. We'll be in so much trouble. We've got to get it back before anyone finds out.'\n\nLady Wriggles rearranged her tentacles into an Mmmmm.\n\n'Please,' Meri begged.\n\n' _Swim Together, Sink Together_ ' was the mermaid spy and sea-creature assistant pledge \u2013 Lady Wriggles had very little choice in the matter.\n\n'Oh, and one more thing,' Meri went on. 'Please could you make a fake replacement ice narwhal? Just until I find the real one.'\n\nLady Wriggles reluctantly scooped up some ice from the floor and squished it into shape.\n\nMeri placed it in the tube. 'Good narwhal sculpting! Now, the telephone.'\n\nThe jellyfish swam off and returned with a human phone.\n\nA fish popped out of the receiver to take the message, because that's how mermaids make phone calls.\n\n'CODE BLUE, CODE BLUE. Urgent message to be delivered to Maritza Mist of Realm Nine, Frostopia. The ice narwhal is GONE.'\n\n2\n\nZelda on Thick Ice\n\nBeneath the freezing waters of Antarctica, Frostopia's perfect ice walls rose from the depths. It was a mermaid kingdom that didn't let outsiders in. A familiar clam car floated nearby, but only Beattie was in it.\n\n' _Oh I do like to be in Frostopiaaaaa! Oh I do like to be stuck to a waaaalllll!_ ' Zelda sang \u2013 from where she was frozen, upside down on the city's outer wall.\n\nZelda was impulsive. She liked taking risks, playing shockey \u2013 which she was excellent at \u2013 and annoying Steve, Beattie's talking seahorse.\n\n'MY EARS ARE SCREAMING \u2013 you're _so_ out of tune!' Steve said as Paris and Mimi tried to pull Zelda free.\n\n'Only because I'm upside down,' Zelda shot back.\n\n'Stop wriggling,' Mimi and Paris said at once, as they tried in vain to peel Zelda off the wall.\n\nMimi, Zelda's twin, was acquainted with her antics. Paris, on the other hand, was still getting used to her.\n\n'Why is Zelda's tail making such a squeaking noise?' Beattie whispered. 'We'll be heard for miles around!'\n\n'Oh calm down,' Zelda said. 'Look, there's no one here.'\n\nBeattie didn't know why she bothered trying to keep them all out of trouble. She was almost always unsuccessful.\n\n'What if we slide her down the wall by yanking her hair?' Paris suggested.\n\nZelda scowled. 'No one touches my hair.'\n\n'I know!' Gronnyupple said as she rifled around in her backpack and pulled out a tiny potion bottle.\n\nShe lobbed the bottle at Zelda. It exploded, spreading a strange green gloop everywhere. Zelda's perfectly flicked green hair began flashing.\n\n'Hmm...' Gronnyupple said. 'That was meant to free you, but instead it's turned your hair... into a shining beacon.'\n\n'Great,' Beattie groaned, hitting her head repeatedly off the clam-car steering wheel.\n\n'Beattie...' Gronnyupple said. 'I see Frostopia mermaids...'\n\nZelda tried to turn. 'I can't see them! I can just see... bright-green light.'\n\nSteve dived into his false teeth, making them snap shut.\n\nGronnyupple was right. In the distance, Beattie could make out two mermaids, their tails decorated with icicles and their hair an electric shade of blue. They rode on the backs of killer whales with SECURITY written in thick frost across their noses.\n\n'Hurry!' Beattie hissed at the others, pointing madly at the mermaids.\n\n'Uh-oh,' Mimi said.\n\nGronnyupple shook her head in disbelief. ' _How_ did they see us?'\n\nBeattie stared at Zelda's flashing hair. 'I suppose it will always be a mystery.'\n\n'I hope we don't get eaten,' Gronnyupple said quietly as she shoved some Seahorse Surprise sweets in her mouth.\n\n'Eaten?' Beattie gulped.\n\nGronnyupple shrugged and pointed at the whale on the left. 'I know a hungry face when I see one.'\n\nBeattie gave the Frostopia mermaids a shy wave, but they glided straight past her and halted next to Zelda.\n\n'It's not what you think,' Zelda said quickly. Her face had gone purple from being upside down for too long, and her hair was still flashing.\n\nThey floated in awkward silence until one of the mermaids steered his whale towards the clam car and leaned inside.\n\n'I've never seen a clam car like this before,' he said.\n\nThe other mermaid swam down and peered in the passenger window. 'Aha!'\n\nBeattie froze.\n\n'I KNOW WHO YOU ARE,' the mermaid said.\n\nBeattie's tail began to shake. The false teeth next to her started to chatter \u2013 which meant Steve was scared too.\n\n'You're Gillica and the team!' the mermaid said, flashing her a smile. 'We've been expecting you.'\n\n'I'm not Gill\u2014' Beattie began, but Zelda cut her off.\n\n'That's right... I'm Gillica! Can Gillica get unstuck from the wall now, please?'\n\n'Excellent,' the one on the left said, swivelling her whale around and shouting 'RELEASE!' Zelda slipped from the wall and dived into the clam car.\n\nSteve tutted at her.\n\nThey watched as the security mermaids swam off and began rearranging icicles on the wall.\n\n'Cool,' Paris said. 'It must be a secret pattern code to get in!'\n\n'Zelda,' Beattie hissed. 'How are we supposed to get out of this one?'\n\n'I think what you meant to say, Beattie, is thank you _so much_ , Zelda, for saving us from being eaten by a security whale.'\n\nThe whale looked offended.\n\n'He says that's a really offensive stereotype,' Mimi said. Mimi was a fishtalker and she could understand sea creatures.\n\n'The sea must be so loud for you,' Paris said, but Mimi was concentrating on the whale.\n\n'And anyway,' she continued, 'he had a big breakfast, so at most he'd only be eating two of us.'\n\nThe whale shifted its gaze to Steve.\n\n'And probably Steve,' Mimi added.\n\n'Eat me?' Steve scoffed. 'But I'm wearing _COUTURE_.'\n\nMimi stared at the whale for a second and nodded. 'The whale says he doesn't know what that is.'\n\n'Get ready, great and honourable guests!' the security mermaid shouted, turning her whale to face the wall. 'Frostopia awaits!'\n\nThe icicles began to glow and the wall started to crack, revealing an icy tunnel that dipped down into the darkness below.\n\n'WOAH,' everyone said at once (apart from Steve, who said 'CRIPES!' \u2013 which was a word he was trying to bring back).\n\n'Line up your clam car and keep the engine turned off, please,' the security mermaid said. 'Let the ice tunnel do the work... AND MAY I BE THE FIRST TO SAY WE CANNOT _WAIT_ FOR YOUR SHOW!'\n\n'Show?' Paris whispered. 'What show?'\n\n'And Gillica, who is me, cannot wait either,' Zelda said awkwardly, giving Beattie a wink.\n\nBeattie gritted her teeth and reluctantly drove the clam car towards the mouth of the tunnel. As soon as the car made contact, the water got even colder and she began to shiver.\n\nSteve pointed his tail at his fur cone top. 'Always dress for the occasion,' he said. 'And preferably in couture. You'll have to go shopping for a new outfit once we get to Frosto\u2014'\n\nThe clam car dropped, hurtling faster and faster until they were pinned to the ceiling of the clam car.\n\n'PPPPPPPP-IIIIIIIIIIIIIII-AAAAAAAAAA!' Steve finished.\n\nMiraculously, they had found a way into a mermaid kingdom that had not been seen for years. A kingdom of myths, legends and magic!\n\nAnd hopefully a shop that sold warm stuff.\n\n'ARE WE NEARLY THERE?' Paris shouted, her hair and eyelashes frosting over.\n\nThere was a little click and they slipped out of the tunnel. Icicles exploded around them like fireworks, and curtains of ice fish parted as their speed reduced. And that's when Beattie saw it \u2013 the icy realm of Frostopia, frozen beneath them. It was bigger than the Hidden Lagoon and the Crocodile Kingdom combined!\n\n'Maritza Mist is down there somewhere,' Gronnyupple said as she peeled a bag of Seahorse Surprise off her face. 'And we're going to find her!'\n\nIt wasn't going to be easy, and Frostopia's magic meant the realm stretched further than anyone could see...\n3\n\nBlubble the Ice-skating Seal\n\nFrostopia was unlike anything Beattie could have imagined. Huge frozen towers rose up from the depths, connected by icicle-covered archways. Killer whales gathered in clumps, and thousands of mermaids swam the streets as loud clangs and beeps and voices flooded the water around them.\n\n'IT'S LOUD, ISN'T IT?!' Gronnyupple yelled into a killer whale's face.\n\nOver near the parked clam car, there was a problem.\n\n'So... it turns out,' Zelda said slowly, 'Gillica is a world-famous ice-skating mermaid who dresses as a seal called Blubble.'\n\nThe guards edged forward for her autograph.\n\n'Here's your costume!' a security mermaid said, placing a rubbery fake-seal costume in Zelda's arms. 'Thank you for sending it ahead of your arrival. It's all security checked and ready to go.'\n\nZelda squeezed it and it began to play a little jingle.\n\n_She skates, she glides, she moves from side to side._\n\n_She's big and she's trouble,_\n\n_Yeah... she's Blubble._\n\nZelda forced a smile.\n\nBeattie couldn't help but laugh. But it came out in slow motion, because her lips were almost frozen shut.\n\n'Now,' the other guard said, 'you'll want to see the stage we've set up for the Blubble show.'\n\nZelda sneakily pulled a map from the guard's pocket and handed it to Beattie.\n\n'We'll have to escort you to the stage, of course,' the guard went on. 'You aren't allowed out of the Glisten Quarter.'\n\n'We need to lose them,' Zelda whispered to Beattie.\n\n'I beg your pardon?' the guards said at the same time.\n\n'Your beg is pardoned,' Mimi answered, bowing.\n\nThe guards exchanged confused looks. 'Out-of-towners,' one mumbled to the other. 'Stay there, and we'll load up the whales with the extra kit we need for the stage. I hear you're going to be jump-skating through hoops.' The guard pointed over to a shark wearing a wig. 'And what is the wigged shark going to do?'\n\nZelda stared blankly at the shark.\n\n'That would be... telling,' she said.\n\nAs soon as the guards swam off, everyone huddled around the map.\n\n'Where to?' Zelda said. 'We need to decide quickly.'\n\nFrostopia was laid out like a lopsided snowflake split into four uneven quarters \u2013 Slushville, Flurry Falls, Floe and the Glisten Quarter. But Beattie saw there was something strange about the map \u2013 Slushville and Flurry Falls were separated from Floe and the Glisten Quarter by a huge underwater waterfall. One side said South Pole, the other said North Pole.\n\n'Woah,' Beattie whispered. 'Half of it is on the other side of the world! The waterfall must be a magical portal!'\n\n'We expanded to the North Pole thousands of years ago,' a mermaid said as she swam past. 'You can only get into Frostopia from the South Pole waters, but once inside, you can use the underwater waterfall to take you to the north.'\n\nThey stared eagerly at the map.\n\n'I like the look of Flurry Falls,' Paris whispered.\n\n'Slushville,' Zelda said with an approving nod. 'Then we'd get to go through the waterfall.'\n\n' _Floe!_ ' Mimi sang as she swam around their heads and slapped the top left of the map.\n\n'Slushville is a paradise,' Steve read aloud, moving his snout along the text descriptions on the map. 'Full of shops and fun. Floe is a little village, but it's got the funniest mermaids in town. Flurry Falls is a bit mysterious \u2013 it just says \"Strange things happen in Flurry Falls.\"'\n\nSteve spun round. 'Slushville. We're going shopping in Slushville.'\n\n'But we have to stay inside the Glisten Quarter,' Beattie whispered back.\n\n'Only if you're Gillica,' Steve pointed out. 'But we're not Gillica. We're not even meant to be here at all.'\n\n'There's no time for shopping,' Gronnyupple said. 'We need to find Maritza Mist before we do anything. She's in danger, I just know it. She didn't deliver my catalogue order, and those water witches who escaped from Viperview Prison have probably got her by now!'\n\n'We don't know that the two are even linked yet,' Beattie said, trying to comfort her.\n\n'And how do you know Maritza Mist isn't in Slushville? In a shop?' Steve said.\n\nGronnyupple stopped chewing her Seahorse Surprise. 'I don't, actually. Let's go to a shop in Slushville!'\n\n'Can we go quickly?' Zelda groaned, her teeth chattering. Frost had spread across her tail and was mushrooming from her hair.\n\nBeattie snapped an icicle off her nose. 'Steve's right \u2013 we can't save Maritza Mist if we're frozen. A quick stop to pick up something warm to wear \u2013 and then we go and find her.'\n\n'Where will we get the money to pay for new clothes?' Paris asked as the guard returned for another of Gillica's trunks.\n\n'Oh, and here's your fee,' the guard said. He handed Zelda a frozen box full of Frosties, Frostopian money.\n\n'Well, that was easy,' Beattie mumbled as Zelda filled her waistcoat pockets.\n\nThey waved as the guard swam off again.\n\n'WE'LL BE WAITING RIGHT HERE,' Zelda lied. 'WE'RE NOT GOING _ANYWHERE_.'\n4\n\nThe Magic of Mermaid Rockits\n\nSabrina the seahorse taught most of Meri's spy-school lessons in Fortress Bay. She was loud, despite her size, and had once famously wrestled a whale and won.\n\n'ROCKITS!' Sabrina roared as sparkly rocks covered in mermaid scales landed in front of each of the students. 'Who can tell me what they do? Meri?'\n\nThe Teenies rolled their eyes in unison. The Teenies were triplet mermaids and Meri's sunken bunkmates. They said the same thing and did the same thing at the same time, always.\n\nMeri cleared her throat and tried to steady her voice. It was difficult to act normal with the ice narwhal missing, especially as it was Sabrina who had secretly tasked her with looking after it. 'Rockits are the latest in virtual spy technology. The rocklike device links up with your tail, so not only can you see scenes from around the world, you can swim _within_ them, as if you're really there.'\n\n'Your voice is weird today,' Sabrina said, swimming closer.\n\n'Your... face is weird today,' Meri joked awkwardly.\n\nSabrina hovered so close Meri went cross-eyed.\n\n'You're wrong about my face, but you're completely correct about the Rockits,' Sabrina said. 'Now, let's see what you can all do with one, shall we?'\n\nAs soon as Meri clasped the cold rock in her hand, she knew where she wanted to go. She swished her tail until it showed the route of her telephone fish. Then she held the Rockit out in front of her and \u2013 _swoosh_.\n\nEverything looked a little see-through and fake, but she was there, swimming towards the ice walls of Frostopia next to the little fish. It morphed as it got closer, switching colour and size to resemble a Frostopia fish, and wriggled through a small crack in the wall. Meri, being in virtual reality, glided right through beside it, but something made her stop.\n\nA couple of mermaids were tunnelling through the wall using what she knew to be a rare water-witch Melt potion. They looked familiar.\n\nThree coughs hit her ear, along with a prod. She dropped the rock and was back in the classroom, swimming against the ceiling, her tail flipping at speed.\n\n'Show-off,' the Teenies all said at once.\n\n'What did you see?' Sabrina asked eagerly. 'You looked like you were surprised.'\n\nMeri swam back to her workstation. 'Nothing,' she said quickly. 'My Rockit didn't work.'\n\n'Why is she even in spy school?' the Teenies chorused, circling around Meri with matching smirks.\n\n'WHY ARE _YOU THREE_ HERE?!' Meri shouted back. 'YOU'RE... THREE TIMES MORE LIKELY TO GET CAUGHT SPYING!'\n\n'Good comeback,' the Teenies said sarcastically. 'Triplet jokes \u2013 so creative.'\n\n'Meri was destined to come here because she was born with the spy tail,' Sabrina butted in, prodding each of them with her nose. 'And you triplets might prove useful. It's handy to have three identical mermaids.'\n\n'BUT WE'RE SO DIFFERENT,' the Teenies said.\n\nSabrina sighed. 'We'll leave it here for today. Take your Rockits and practise. We'll try again tomorrow.'\n\nThe trainee spies swam out into the corridor, where spy whales and students floated before their next class.\n\nMeri hung back until the coast was clear, then she fired up the Rockit again.\n\nThe mermaids were still there, tunnelling through the wall. She swam closer to get a better look at their faces, and saw something that made her drop the Rockit in shock.\n\n'The ice narwhal,' she whispered to Lady Wriggles. 'They've got the ice narwhal!'\n5\n\nSlushville Is So Slush\n\nBack in Frostopia, Beattie parked the clam car and dived through the icy underwater waterfall to the North Pole. They landed in Slushville, where everything was slushy, and the reason it was slushy was because it was home to _a lot_ of seals. Mermaids with frozen tails and fluffy hats barged past Beattie and disappeared down the sloppy streets.\n\n'Shopping time!' Steve cheered.\n\n'Look at everyone's tails,' Paris said as they weaved their way through the slush and the crowds. 'They're almost all the same \u2013 black with splodges of white, just like the killer whales!'\n\nBeattie worried their tails would be an instant giveaway that they weren't from Frostopia, but every so often a mermaid with a blue tail or a pink or purple one would swim past too.\n\nZelda dodged seals like the professional shockey mermaid she was. She stopped and leaned arrogantly against one of them. 'Hurry up, slowmaids!'\n\n'Um, Zelda,' Mimi said, 'that seal says, and I quote, \"If you continue to touch me, I will kill you.\"'\n\nZelda shot the seal a nervous look before backing off. 'It's a good thing you're a fishtalker, Mimi.'\n\n'I can see more seals than sea!' Paris cried. She shook her necklace and morphed into a jellyfish, hoping that being significantly smaller would help with the navigating, but she seized up and her little jellyfish stingers froze.\n\n'Ah,' Mimi said, picking her up before she sank. 'Paris is telling me in Jellyfish language that she's too frozen to make her necklace work.' She gave it a shake and Paris, with a _FLLLISSSSPIP_ , morphed into a shark.\n\nThe seals scattered.\n\n'Perfect,' Beattie said as Paris cleared a path for them.\n\nThey glided through town, past a mermaid with a tall fur hat who called, 'Welcome to Slushville! Head to Polar Peaks for half-price Icetipple!'\n\nAbove them, Beattie could see lots of holes in the ice ceiling.\n\nAll around them, mermaids flopped into gondolas pulled by seals and then shot up through the holes. Beattie could see that each of the holes had a dancing hologram next to it \u2013 there was a polar bear hologram holding a sign that read _POLAR PEAKS_ , and a smiling mermaid hologram with a T-shirt that said _GLAMBEY'S FROZEN FASHION_.\n\n' _That's_ where we need to go,' Steve said, spotting the word 'fashion'.\n\nA seal pulling a large gondola lolloped up to Beattie. They all got in, apart from Paris, in shark form, who put a fin in the boat. The seal barked. Everyone stopped dead in their tracks and stared.\n\nAnother seal nodded madly at a sign: _WE'LL TRANSPORT ANYTHING THAT DOESN'T EAT US_.\n\n'I won't eat you!' Paris cried, but it came out as 'Gnash, gnash, gnash!'\n\nShe shook the necklace with her teeth and morphed into a mermaid. 'Better?' she asked.\n\nThe seal nodded.\n\n'To Glambey's Frozen Fashion!' Steve shouted, and the seal glided off towards the hole in the ice above them.\n\n'Welcome to GLAMBEY'S FROZEN FASHION!' the hologram greeted them as they shot through and skidded into an icy cave.\n\n'We're above water!' Zelda yelled, punching the air. 'We're in human territory. Wait \u2013 is this dangerous? Because danger is my middle name!'\n\n'No it's not, it's Pamela,' Mimi reminded her.\n\n'It's very safe,' a mermaid said as she slid next to them in another gondola. 'Humans don't come this deep into the ice. You're at the top of the world!'\n\nBeattie excitedly leaned forward in the gondola as they careered towards ice rails filled with furry tops and glistening tail embellishments. Up ahead, mermaids pulled by seals plucked items off the shelves \u2013 hats and gloves and little icy bags. Over at the changing rooms, a mermaid wearing a sparkling waistcoat slipped out from behind the frost-covered curtains.\n\n'I want something like that!' Zelda said. 'I love a waistcoat.'\n\n'We're here to get _warm_ things,' Steve protested, dipping his head in a puddle at the bottom of the gondola. Unlike mermaids and seals, Steve couldn't breathe out of water.\n\nZelda stuck her tongue out at him. 'I'm getting the waistcoat.'\n\nBeattie shivered and pulled a coat with a fluffy trim from the racks. Frostopia was _cool_.\n\nMimi chose a T-shirt with a half penguin, half hamburger on it.\n\n'That's my favourite cartoon!' a young mermaid called as she shot past.\n\n'That's weird \u2013 I instantly feel much warmer,' Mimi said.\n\n'Everything in this shop will keep you warm,' the shop owner said as she slid past on a seal.\n\n'You need to get this, Paris,' Steve said, holding up a furry crown.\n\n'Do you think it might be a bit much?' Paris asked, reluctantly putting it on.\n\n'Absolutely not,' Steve said. 'If anything, it's not enough. Now we just need something for Gronnyupple.'\n\nBeattie spotted her in the corner of the shop, belly up, putting the Blubble seal costume on a passing seal \u2013 and laughing hysterically.\n\n'Gronnyupple!' Zelda shouted. 'WHAT ARE YOU DOING?'\n\nThe seal got a fright and flopped out of the shop, still wearing the costume.\n\nBeattie shuffled over and peered down through the hole. 'If the guards find us outside of the Glisten Quarter _and_ we've lost the Blubble costume... Oh cods, this isn't good.'\n\n'I love Blubble!' an ancient-looking mermaid said as she swam past with a gondola full of hats.\n\n'It's fine!' Gronnyupple said. 'We'll easily find it. It's a seal dressed up as a... Ah, I see what I've done.'\n\nSteve shook his head. 'This is not turning out to be the successful shopping trip I wanted!'\n\nGronnyupple bit her lip. 'Sorry, I just thought the seal would look great dressed like\u2014'\n\n'A seal,' Steve finished.\n\n'We could say we misplaced the costume?' Gronnyupple suggested. 'Or we could say a killer whale ate it?'\n\nA killer whale popped its head through the hole.\n\n'He says you really need to stop with that stereotype,' Mimi said. 'Killer whales don't eat everything. They don't even like being called killer whales. They actually prefer \"deluxe dolphins\".'\n\n' _Wait_ ,' Gronnyupple said, sniffing the air. 'I smell a water witch.'\n\n'Is it Beattie?' Zelda said flatly.\n\nBeattie's eyes widened. She could smell it too! It smelled like the candyfloss humans ate, only a bit burned and soggy.\n\nGronnyupple pulled at Zelda's waistcoat, sniffing it excitedly.\n\n'Get off!' Zelda cried. But Gronnyupple was already peeling the waistcoat from her back.\n\nShe held it up and pointed at the label.\n\n'Made by MM,' Beattie read.\n\n'Maritza Mist!' Gronnyupple cried. 'It's got to be her! How many people have the initials MM?'\n\n'Marjorie Mussels, Max Mako, Martina Mackerel, Moya von Manatee,' Steve listed. 'Molly Marine, Mairie Mul\u2014'\n\n' _Exactly_ ,' Gronnyupple interrupted. 'Almost _no one_. I bet it's her, and that the shop owner knows where we can find her!'\n\nThey all cheered! And then a goblin shark came crashing into the shop and grabbed Gronnyupple's tail in its teeth.\n6\n\nSimoseal\n\nThe goblin shark refused to let Gronnyupple go. It was a strange-looking shark, covered in amber jewels, with twinkling amber eyes \u2013 certainly not the kind of shark that looked as though it belonged in Frostopia.\n\n'Looks like you're the Seahorse Surprise sweet now,' Steve joked.\n\n'STEVE!' Beattie shouted. 'So inappropriate.'\n\n'LET HER GO,' Paris demanded, but the goblin shark shook its head.\n\n'Seriously,' Zelda said, buttoning up her waistcoat in anticipation of a chase. 'LET THE MERMAID GO.'\n\nThe shark took one look at Zelda and dived through the icy floor, shattering it into tiny shards!\n\nBeattie tumbled back into the water, surrounded by clumps of clothes and screaming mermaids.\n\n'GRONNYUPPLE!' she cried. 'GRONNYUPPLE!'\n\n'You find Maritza... I'll handle this,' Paris said, and with a _FLIIISSSIP_ she morphed into a shark and tore after Gronnyupple.\n\n'Excuse me,' came Steve's voice from where the shop owner had landed, face first on the back of a polar bear. 'Not a good time to ask, I realise \u2013 but do you happen to know where the designer of these clothes lives?'\n\n'Mee met mem melivered my meal,' the mermaid mumbled through the polar bear's fluff.\n\nSteve turned to the others. 'THEY GET THEM DELIVERED BY SEAL! Mimi, work your magic!'\n\n'Thank you for the clothes,' Beattie said, placing some Frosties sheepishly next to the shop owner's smooshed face.\n\nMimi swam from seal to seal, asking if they could help.\n\n'THIS ONE SAYS HER NAME IS SUSASEAL, BUT I NEED TO FIND SIMOSEAL. HE'S THE ONE WE NEED.'\n\nThey looked around at the hundreds of seals pulling their gondolas.\n\n'SIMOSEAL?' Beattie shouted hopefully.\n\nA seal came lolloping over. Mimi did the thumbs up.\n\n'Simoseal says he knows where Maritza Mist lives \u2013 at 000 Underwater Igloo Avenue, Flurry Falls.'\n\n'You know,' Zelda said, 'I'm jealous of all your powers \u2013 Paris can morph and Beattie and Gronnyupple are water witches, but actually, Mimi, I think your fishtalking might be the best!'\n\n'I CAN'T HEAR YOU!' Mimi said to Zelda as a particularly chatty shoal of fish swam past.\n\nThey dived into Simoseal's gondola.\n\n'PLEASE TAKE US TO MARITZA MIST!' Zelda shouted.\n\nThe seal drop-dived, pinning them all to the back of the gondola.\n\n'I hope we don't lose Paris!' Beattie yelled as her cheeks wobbled madly as they tore through the sea.\n\nWhen she dared to look down, she saw little igloos being battered by thundering underwater waterfalls. A frozen sign cobbled together from sunken treasure spelled out _FLURRY FALLS_.\n\n'Don't worry, Beattie,' Zelda said. 'Paris will find Gronnyupple and be back with us in no time. When she's a shark, she has an incredibly powerful sense of smell. She'll pick us up.'\n\n'I hope you're right,' Beattie said with a gulp. 'We can't lose Paris as well.'\n7\n\nMeri Does Some Midnight Spying\n\nMeri swam back to her room after a long day. She had struggled to concentrate in her afternoon spy classes. And she barely touched her dinner, even though it was her _favourite_ chomp chops, secretly sourced from Jawella's, the shark restaurant in Hammerhead Heights in the Hidden Lagoon.\n\n'Not like you to not eat your chomp chops,' the Teenies said.\n\n'I wasn't hungry,' Meri said, flopping on to her top bunk.\n\nThe Teenies floated up beside her.\n\nMeri rolled over, pulled out her seaweed notebook from under her seaweed pillow and began scribbling in it, keeping one eye on her tail.\n\n'You're looking at your tail, but you're not meant to spy out of hours,' the Teenies said. 'Who are you spying on?'\n\n'No one,' Meri said. 'I was just admiring how slimy my scales look today.'\n\nThe Teenies retreated to their bunk \u2013 an extra wide one so they could sleep side by side. But Meri could tell they weren't convinced.\n\nShe watched scene after scene of Frostopia swimways and houses and shops and igloos, but she couldn't see the mermaids with the ice narwhal.\n\n'Where are you?' she whispered to her tail.\n\nShe waited until she heard the Teenies snoring in unison, and then pulled out her clamshell compact. On the shell's hidden screen, she scrolled through pictures of mermaids on the most-wanted list, until she found them.\n\n'The escaped Viperview mermaids,' she muttered to herself, writing their names in her notebook: _Gulper and Rose Rockweed_.\n\nShe read the report that accompanied their pictures. A human called Susan Silkensocks had damaged Viperview Prison with a ship and they had escaped with a mermaid called Ommy, who was captured instantly when he swam into a rock and knocked himself out.\n\nMeri gasped and shook her head when she read the next line: ' _Gulper and Rose Rockweed found and returned to Viperview. Case closed._ '\n\n'But they aren't in Viperview,' Meri said, floating off her bunk and hitting the ceiling. 'They're breaking into Frostopia!'\n\n'MERI! GO TO SLEEP!' the Teenies shouted, before immediately snoring again.\n\nMeri sneak-swam to the door. She needed to tell Sabrina the mermaids hadn't been caught \u2013 they were free and dangerous. But she wouldn't mention the ice narwhal thing. Not yet. Maybe she could convince Sabrina to let her go on her first real mission to capture the Viperview mermaids. And then she could sneakily get the ice narwhal and put it back before anyone noticed.\n\n'GO TO SLEEP, MERI!' Sabrina shouted. 'The Rockweeds have been caught. Everything is under control.'\n\n'But I saw them,' Meri said, swimming around the outside of Sabrina's tiny bejewelled house.\n\n'Impossible,' Sabrina said. 'I'll show you on your tail \u2013 look, that's them in Viperview Prison.'\n\nMeri stared at her tail in disbelief. It was true, they were in Viperview. 'But... I'm _certain_ it was them.'\n\nA roll of seaweed floated towards her.\n\n'The capture notes,' Sabrina said. 'We interviewed them after they were caught. As you can see, nothing out of the ordinary.'\n\n**SPY TT 202:** Gulper. Rose. What's your favourite thing to do?\n\n**Rose:** Knitting.\n\n**Gulper:** Plotting evil things.\n\n**SPY TT 202:** Fascinating! Do you knit with sunken wool, and what kind of evil things do you plot?\n\n**Rose:** Nope. Hair.\n\n**Gulper:** Destroy the world. FISH EYE! A delightful interview! Thank you to Gulper and Rose. CONFIRM: It is Gulper and Rose. CASE CLOSED.\n\n'What does Fish Eye mean?' Meri asked. 'Gulper said FISH EYE.'\n\n'It's the things that fish have on their faces,' Sabrina explained. 'Along with lips and... Do they have noses? Look, Meri, if this is a silly attempt to get out of Fortress Bay and go on a mission, it is definitely not working.'\n\nMeri slumped. 'I just thought I could\u2014'\n\n'GO TO BED, MERI,' Sabrina said firmly. 'That's enough now. Don't make me call Lady Wriggles.'\n\nLady Wriggles was both an assistant and a nightmare if you interrupted her beauty sleep. The last thing Meri needed was an angry jellyfish stinging her. _Swim Together, Sink Together_. If Meri was in trouble with a teacher, then so was Lady Wriggles.\n\n'Fine,' Meri huffed. 'But you're wrong, Sabrina! I know what I saw!' She swam off towards the dorms, watching her tail as she went. She could see a bunch of guards swimming furiously through the streets of Frostopia, waving a CATCH GILLICA THE ICE-SKATING SEAL poster. The imagery switched to Flurry Falls just as Meri reached the empty canteen. She grabbed a chomp chop and munched on it, trying to think.\n\nA mermaid with a flick of green hair slid into view on her tail. She was in a gondola with two other mermaids. Meri sighed at the sight of them \u2013 'Oh to have friends!'\n\nThey didn't look like they were having fun as they were pulled through the icy waters by a seal at breakneck speed, but still.\n\nMeri wiped the chomp chop juice from her lips and pulled out her clamshell compact. She scanned it against the mermaids on her tail.\n\nMERMAIDS FROM THE HIDDEN LAGOON.\n\nPOTENTIALLY BAD MERMAIDS IN FORBIDDEN TERRITORY.\n\nENTRY OCCURRED THIS MORNING.\n\nALERT OUT IN FROSTOPIA FOR THEIR CAPTURE \u2013 SEAL COSTUME INCIDENT.\n\nUNDER CONTROL, NO FORTRESS BAY INVOLVEMENT NEEDED.\n\n'Why are you so far from home?' Meri muttered.\n\nA guilty-looking whale emerged from the kitchen and swam away sheepishly. He was clearly having a midnight snack.\n\n'I didn't mean you!' Meri called after him. She chewed on her chomp chop again as she read through the profiles. She stopped when she got to Mimi's.\n\n'A fishtalker, eh?' An idea came to her. 'Well, if I can't get out of here, maybe Mimi can help me get the ice narwhal back...'\n\nShe pressed the FISHTALKER TALK button on her clamshell compact and took a deep breath.\n8\n\nUnderwater Igloo Avenue, Flurry Falls\n\nHundreds of icicle-covered arches glistened for as far as Beattie could see. Fast-flowing waterfalls broke through from the surface and landed in icy pools, where hundreds of mermaids lazed and drank Icetipple. Killer whales floated around, and above them Beattie could see baby mermaids taking swimming lessons with polar bear cubs.\n\nMimi looked around in wide-eyed wonder as they soared past in the gondola.\n\n'Beautiful, isn't it?' Beattie said wistfully.\n\n'Every single sea creature is shouting that they're merry,' Mimi chirped. 'Is that nice or weird?'\n\n'Merry?' Zelda said.\n\n'I AM MERI! CAN YOU HEAR ME?' a group of polar bear cubs shouted.\n\n'I AM MERI!' a whale bellowed.\n\nA fish turned to her and screamed, 'I AM MERI!'\n\nMimi turned round in the gondola, her mouth hanging open.\n\n'Mimi?' Zelda said, nudging her. 'Everything all right?'\n\n'Yes...' she answered finally. 'This place is wonderful, every creature is so MERRY!'\n\nFar away in Fortress Bay, Meri Pebble hit her head against a desk.\n\nThe seal weaved through the frozen arches before coming to a halt outside a small igloo. The initials MM were carved next to the frozen door.\n\n'Maritza Mist's house,' Beattie said.\n\nShe reached out to knock, then turned to the others. 'What if she's... dangerous?'\n\nZelda rolled her eyes. 'She's some old mermaid who makes magic for a catalogue. I think we'll live.'\n\nShe swam past Beattie and knocked for her.\n\nThe door creaked open.\n\n'Hello?' Beattie whispered as they swam inside. It was a tiny igloo with only a couple of doors leading to other rooms and some sunken human paintings hanging lop-sided on the wall.\n\nThey heard a creak and a swish of a tail coming from the next room. Something clattered and the door flew open!\n\n'Maritza?' Beattie said hopefully.\n\nIt was indeed Maritza Mist. But she was not what they were expecting. Not what they were expecting _at all_.\n9\n\nMeri Screams I AM MERI, a Lot\n\n'How many times do I have to scream \"I AM MERI\" before this Mimi mermaid realises that all the sea creatures aren't _merry_? I'm trying to tell her I AM MERI and I AM TRYING TO CONTACT HER! What's the point of this _use any creature to deliver a message to a fishtalker_ technology if the mermaid DOESN'T GET IT?!'\n\nShe took a deep breath as she watched her tail \u2013 a fish swam past Mimi's nose. 'I AM MERI!' she shouted again.\n\n'We _know_ ,' the Teenies said.\n\nMeri froze and tried to act casual as they floated past with their canteen trays. She wasn't sure how long they'd been there. The canteen was full and she hadn't even noticed \u2013 it was Sandcrackle cereal time already.\n\n'I AM MERI!' Meri tried again.\n\nLady Wriggles thrust a piece of seaweed in her face.\n\n_TRY CHANGING YOUR NAME TO GORDON._\n\n'Lady Wriggles,' Meri said. 'You're a _genius_.'\n10\n\nMaritza Mist\n\nBeattie floated in silence, staring at the mermaid Gronnyupple believed to be the most powerful water witch in the world.\n\nShe had icy hair pulled into three neat buns, and a small black and white splodged tail. She was wearing a Burguin T-shirt, just like Mimi.\n\n'She's about seven years old,' Zelda whispered to Beattie out of the corner of her mouth. She turned to Mimi and mouthed, 'SEVEN.'\n\nMaritza Mist's ear grew bigger. 'Listening spell, dears,' she said, pointing at it. 'It's a new creation of mine. Oh, and I am actually exceedingly old. But I smeared my tail in immortality paste when I was seven, so now I'm immortal and stuck looking like this forever. It's why I sound like a granny, dears, but I look like you.'\n\nBeattie and the others rolled back on their tails in fits of relieved giggles.\n\n'Why did you cover your tail in immortality paste?' Beattie asked when she'd stopped laughing.\n\n'Because I have to try out my spells first, dear. I can't just sell any old thing! Now, answer me this \u2013 why are you here?'\n\n'Well, at first we wanted to come because you didn't deliver a catalogue order to the Crocodile Kingdom,' Beattie said.\n\n'That's a long way to come to complain about tardy deliveries.'\n\n'No, it's not a complaint \u2013 Gronnyupple thought something had happened to you,' Zelda explained. 'But you're _immortal_ and indestructible! She'll be so relieved.'\n\n'Oh poor Gronnyupple. I was just a little delayed because I wanted to throw in a free listening spell for her as a surprise! Why isn't she here?'\n\nBeattie tried to think of the best way to phrase the goblin shark situation.\n\n'Goblin shark,' Mimi chirped, 'which is not ideal, but she'll be happy to know you're safe \u2013 she was convinced the immortal mermaids who escaped from Viperview had got you and that's why you didn't deliver the catalogue order.'\n\nMaritza's face fell. 'They escaped from Viperview?'\n\nBefore Mimi could reply, the door flew open. They ducked for cover as frozen splinters rained down on them.\n\nTwo mermaids floated in with orange and black-scaled tails and smug grins.\n\n'Maritza?' Beattie asked. 'Who are these mermaids?' She could tell by the way they were all staring at each other that they weren't friends.\n\n'Why, it's Gulper and Rose Rockweed. How did you escape from Viperview?' Maritza asked.\n\n' _WELL_ \u2014' Rose Rockweed began, but a fish interrupted her.\n\nIt was tiny and exhausted and swam through the door, spewing bubbles that rearranged themselves into letters.\n\n_CODE BLUE... CODE BL... U... E. Urgent... message... to... be... delivered... to... Maritza... Mist... of... Realm... Nine... Frostopia... The... ice... narwhal... is GONE._\n\nRose Rockweed threw her arms in the air. 'Well, that ruined my big speech \u2013 I've been practising it all the way here! It's true, we have the ice narwhal!'\n\nBeattie could have sworn Maritza Mist's eyes flashed with panic, but she floated like a mermaid who had everything under control.\n\nRose shoved a small icy sculpture in Maritza's face. ' _And_ we got the list of where you hid the rest of them. They just leave these things lying around in Fortress Bay,' she laughed. 'It's too easy.'\n\n'What's an ice narwhal?' Beattie asked, but Maritza Mist didn't answer.\n\n'Very good, Rockweeds,' she said. 'But why are you here?'\n\nGulper and Rose began swimming in threatening circles. 'To stock up on magic, of course! We have a lot to do and we need some of your potions to help us.'\n\n'It's time for you to go, dears,' Maritza said quickly to the others, as a bunch of seals appeared with a _pop_ and began shoving them out of the doorway.\n\n'Why are the seals shoving us outside?' Beattie asked, trying to fight her way back in.\n\n'SEAL POP,' Maritza said knowingly, waving an empty bottle. 'Tell Gronnyupple I'll put one in her next order. Now, off you go.'\n\n'Oh,' Mimi said. 'They're all called Gordon! And one is called Lady Wriggles It's Not Working. That's a long name.'\n\nThe seals lined up outside like blubbery bouncers.\n\nThere was no way in.\n\nThe four mermaids and Steve swam around the side of the igloo and crammed together at the window, watching helplessly as Gulper and Rose began ransacking the place.\n\n'GIVE US YOUR POTIONS!' they yelled.\n\nSpare seals went flying. Ice shattered. Maritza Mist began throwing potions willy-nilly.\n\nOne landed on Rose's head and made it swell to the size of a polar bear.\n\n'She's going to pop!' Zelda cried, just as Rose's head deflated like a balloon and sagged on to her shoulders.\n\n'Is she dead?' Mimi whispered.\n\n'They're immortal,' Beattie said. 'This could go on for a while.'\n\nGulper grabbed Maritza Mist by the tail and flung her across the room. 'YOU CAN'T STOP US!' he yelled.\n\nMaritza landed on a freeze potion. She tried to flick her tail, but the end was frozen solid and she began to sink. 'Drats, that's going to take a few minutes to wear off.'\n\n'MOVE!' Rose shouted to Gulper as she threw another potion at Maritza. But he was too busy laughing.\n\nThe potion hit him square in the face and his ears began to droop until his earlobes were as long as his body.\n\n'WHY DOES EVERYTHING SOUND LOOOONG?' he cried.\n\nMaritza Mist cackled as Rose sneaked up behind her and began smearing a strange magical paste over her arms.\n\n'No!' Maritza cried. 'Not\u2014'\n\nAll the objects in the room came flying towards her.\n\n'The magnet potion,' she finished with a sigh. She tried to swim, but she was well and truly sunk.\n\n'Let's go to Pinkly Lagoon and get the next ice creature,' Rose said as Gulper grabbed fistfuls of potions and headed for the door. 'SOON EVERY MERMAID KINGDOM IN THE WORLD WILL BE OURS!'\n\nHe flicked a potion at the wall.\n\n'Duck!' Beattie cried as the wall exploded.\n\nThey watched as Gulper and Rose danced off into the distance.\n\n'Maritza,' Beattie said, hurrying inside. 'Are you all right?'\n\nMaritza Mist cackled. 'Of course I am! That's the problem. I'm immortal, and unfortunately so are Gulper and Rose. And that makes them unstoppable. The mermaid world is about to fall into very bad hands and I fear we are doomed.'\n\n'Doomed?' Beattie whispered nervously.\n\nMaritza Mist smeared a potion on her lips and they magically extended into a megaphone. 'DOOMED, DEARS.'\n11\n\nMultistorey Clam-Car Car Park (Say It Fast)\n\nBack in Fortress Bay, Meri had decided that there was only one thing for it \u2013 if she was going to fix this mess and get the ice narwhal back, she was going to have to do it herself. Changing her name to Gordon was not going to cut it.\n\n'And that means WE'RE SNEAKING OUT OF HERE!' she said to a wildly unenthusiastic Lady Wriggles.\n\nMeri swam through the multistorey clam car car park and chose a ride. She picked the clam car's lock and the roof flipped open.\n\n'Easy,' she said, while Lady Wriggles hid her eyes with her stinger.\n\nThe place was completely empty, apart from the Teenies, whose job it was to guard the cars. But, like always, they were huddled in the far corner, reading identical copies of _SPY TWEEN_ magazine.\n\nMeri began punching buttons on the gem-studded control panel above her head. She'd chosen the clam car with special spy features, including camouflage mode.\n\n'I know what you're thinking,' Meri said, as Lady Wriggles looked at her with her unchanging jellyfish face. 'How are we going to get past the Teenies?'\n\nLady Wriggles wasn't thinking that \u2013 even three small rocks would be harder to get past than the Teenies.\n\n'But I have a plan!' Meri said proudly as she steered the car towards the exit. 'Engage camouflage mode!'\n\nLady Wriggles could hear Meri's heart beating like a manic octopus. After all, Meri had only seen the outside world on her tail \u2013 she'd never actually left Fortress Bay before.\n\n'I hope this works,' she whispered as the clam car inched slowly past the Teenies.\n\nAlmost immediately, they snapped their heads up.\n\n'WAIT A _MINUTE_ ,' the three of them said.\n\nMeri held her breath.\n\n'ACCORDING TO _SPY TWEEN_ , ROCKITS ARE THE NEW COOL SPY GADGET THIS SEASON. CAN YOU BELIEVE SABRINA MIGHT ACTUALLY BE COOL? THAT'S BLOWN MY TINY FISHY MIND!'\n\nMeri continued to hold her breath as they drove slowly through the sand security perimeter and out of Fortress Bay.\n\n'We did it!' Meri squealed as soon as they were out of earshot. 'Maybe I'll meet those mermaids from the Hidden Lagoon in Frostopia. It would be nice to make some friends.'\n\nLady Wriggles turned on the radio and the Squid Sisters started blaring from the speakers.\n\n'You're right,' Meri said with a sad smile. 'Spy mermaids can't have friends.'\n12\n\nWartina's Necklace\n\n'Now, I'm very sorry, dears, but I have a few things to do,' Maritza said as she swam towards the door at the back of her igloo.\n\n'Things to do with the ice narwhal?' Beattie guessed.\n\nMaritza Mist stopped.\n\n'It's safer if you don't get involved.'\n\n'We could help,' Beattie said. 'We're excellent at stopping disasters.'\n\n'And at causing them,' Zelda whispered to Mimi.\n\n'Are you good or bad mermaids?' Maritza asked.\n\nBeattie beamed. 'Good ones. Definitely.'\n\n'Depends who you ask,' Zelda mumbled.\n\nMaritza made her way through the door and into a cluttered room beyond.\n\nThey followed.\n\nPotion bottles floated about their heads, while a polar bear in the corner folded catalogues by sitting on them.\n\n'I created one ice creature for every mermaid kingdom around the world, and Frostopia's was the ice narwhal. Each one contained a little sprinkle of good potion to encourage unity \u2013 mermaids of the world united! But I had a friend named Wartina Brittle who sneakily switched the potions, making the ice creatures _very bad_ \u2013 so the potion within them caused division and destruction instead. And what made it worse was that when combined they'd bring together the badness of every mermaid kingdom in the world! If a mermaid got hold of every single one and attached them to their tail, they'd be the most powerful and destructive mermaid we've ever seen.'\n\n'Did Wartina use them?' Beattie asked. 'Did she become all-powerful?'\n\n'Almost,' Maritza Mist explained, 'but I managed to stop her. The problem was I couldn't do anything about the magic within the ice creatures. All I could do was hide them far away from each other to limit their power. Though the damage had been done \u2013 they divided and destroyed the bonds between many mermaids. And as the years passed, mermaids in different realms no longer spoke to one another. Some still speak to or visit one another, but it's not like it used to be. They built ice walls and hidden cities, until realms like your Hidden Lagoon forgot about their other mermaid friends and believed places like the Crocodile Kingdom to be nothing but a myth.'\n\n'Wartina doesn't sound like a very nice friend,' Mimi said. 'But I _really_ like her name.'\n\n'So Gulper and Rose want that magic now. They want to become all-powerful,' Beattie said. 'We can't let that happen!'\n\n'What about trying to remove the immortality-paste spell? Then Gulper and Rose would be gone forever,' Zelda suggested.\n\nMaritza Mist shook her head. 'Reversing or removing a spell can only be done with Razzle Dazzle kelp, and it would be a _miracle_ if we found any.'\n\n'How can you tell if it's Razzle Dazzle kelp?' Beattie asked.\n\nMaritza Mist smiled. 'It's always showing off.'\n\n'SOUNDS LIKE MY KIND OF KELP!' Steve joked.\n\n'It's impossible to find,' Maritza Mist said sadly. 'Believe me, I've looked. It would be nice to undo this immortality spell on me too.'\n\n'So Gulper and Rose are going to try to collect all the ice creatures,' Beattie said.\n\nMaritza Mist nodded gravely. 'They already have the ice creature from Rainbow Landing \u2013 a seahorse. They stole it hundreds of years ago, after they used the immortality paste. I knew then exactly what they were up to, so I framed them for a crime and got them locked away in Viperview.'\n\n'Let's frame them again!' Zelda said. 'I'll commit a little crime, and then we'll tell everyone it was them.'\n\n'Zelda,' Beattie said, sounding shocked. 'You can't commit a _crime_.'\n\n'You haven't even heard my crime ideas yet,' Zelda said with a pout.\n\nThe polar bear in the corner groaned and pointed towards the door \u2013 there, floating sadly on the spot, was a familiar figure.\n\n'Paris!' Beattie cried, racing over and hugging her. 'You found us!'\n\n'You used your extra-sensitive shark nose, didn't you?' Zelda said. 'I told everyone that's what you'd do!'\n\n'No,' Paris said. 'I just asked the mermaid in the shop where you were \u2013 apparently Mimi shouted out the address really loudly before you left.'\n\nMimi bowed.\n\n'I lost Gronnyupple,' Paris said with a tear in her eye. 'The goblin shark was too fast. I don't know where she's gone.'\n\n'Oh no,' Beattie said. 'Maybe Maritza can help us find her. She's got a slight problem on her hands with some immortal mermaids who ransacked her igloo and stole potions to help them take over the world, but\u2014'\n\nThere was a crash as Maritza tumbled over the polar bear. She was wide-eyed and impossibly pale, like she'd seen a ghost jellyfish.\n\n'Maritza?' Beattie whispered.\n\n'That necklace,' Maritza said, choking out the words as if they were frozen in her throat. 'Where did you get it?'\n\nParis looked down at her necklace. 'It was a present from Arabella Cod, the mermaid queen of the Hidden Lagoon. She gave it to me for saving her life.'\n\n'It's magic,' Zelda said. 'Paris is a human, but with the necklace she can morph into all sorts of sea creatures.'\n\n'There's something I didn't tell you about Wartina Brittle,' Maritza said. 'She was a human... and that was her necklace.'\n13\n\nGronnyupple in Trouble\n\nFar away in Amberberg, where mermaid palaces made of amber wound around the rocks, Gronnyupple was in trouble.\n\nMermaids and codfish clad in amber jewels floated in the room where the goblin shark had dropped off Gronnyupple.\n\n'SEND HER TO VIPERVIEW!' they shouted. 'FOR ONE HUNDRED YEARS!'\n\n'Can I have some Seahorse Surprise?' Gronnyupple asked. 'Or my backpack full of spells... I mean SPOILS... wait, no, that sounds like I've stolen something.'\n\nAn old mermaid judge with a luminous orange tail read the seaweed notes in front of him.\n\n'Gronnyupple Wondra Crab, you have been brought here by goblin shark because you are accused of robbing Amberberg Central Bank and stealing over one million Ambug coins. You also stand accused of robbing the city's crown jewel \u2013 the Amber Flubby \u2013 and of looting various shops around the city. How do you plead?'\n\nGronnyupple thought for a second. 'Um, well... it would be something like _pleeeeease! Pleeeeeease!_ ' She clasped her hands together. 'And my hands would be like this, for extra effect.'\n\nThe judge stared down his nose at her.\n\nAnother mermaid, with orange hair twisted into a cone shape, swam around Gronnyupple, holding up a packet of Seahorse Surprise.\n\n'This is your calling card. A half-eaten packet of Seahorse Surprise \u2013 you leave one at the locations of all your crimes.'\n\n'But lots of mermaids eat Seahorse Surprise.'\n\nThe mermaid laughed. 'Indeed, but you _love_ Seahorse Surprise, don't you?'\n\n'YES!' Gronnyupple cried, punching the water in triumph. 'But I'd _never_ leave them half-eaten. I _always_ finish the packet. I hope when you find the real thief you'll punish them for wasting good sweets.'\n\n'You must prove to us that you are not responsible for these crimes,' the judge said. 'If we don't believe you, then you will be sent to VIPERVIEW PRISON!'\n\n'Do they have Seahorse Surprise there?' Gronnyupple asked hopefully.\n14\n\nThe Power of Paris\n\nParis held the crystal pendant of her necklace firmly in her hand. 'Should I, um, take it off?'\n\n'NO!' Beattie, Mimi, Zelda and Maritza all shouted.\n\n'You're a human, remember?' Beattie said. 'If you take the necklace off you'll morph back\u2014'\n\n'And we're miles under the ice!' Zelda finished.\n\n'And humans swim slowly,' Mimi added.\n\n'You must not do that, dear,' Maritza said, swimming around Paris. 'The necklace is yours now. I made it myself!' She stopped and stared, her eyes growing wide. 'You know, Paris might be just what we need to defeat Gulper and Rose forever.'\n\nParis looked concerned.\n\n'Water-witch magic gets stronger in the hands of a human,' Maritza explained. 'I tell all the water witches who get my catalogue about the bad human who nearly destroyed us all. Gronnyupple knows this story, or at least the short version of it. I designed the necklace as a fishtalker device \u2013 the mermaids who wear it can hear the sea creatures. But a human who wears it can _become_ those creatures. When Wartina Brittle collected and then tried to use the ice creatures, there was a snag in her plan \u2013 the ice creatures cracked when she held them. Being a human made the magic too strong for the ice to contain it. And, of course, if an ice creature broke\u2014'\n\n'The magic would lose its power, it would be useless,' Beattie said.\n\n'Exactly,' Maritza said. 'So she went off to find a mermaid who would help her \u2013 someone who could hold the magic. She ended up convincing Mary Ruster, who wasn't a bad mermaid at all, just clumsy and very silly with her magic.'\n\n'Her ancient sunken ship is in our lagoon!' Beattie cried. 'Did she succeed in helping Wartina?'\n\n'No,' Maritza said. 'I stopped them with the help of the Fortress Bay mermaids. They gave Wartina an ultimatum \u2013 be locked away in Fortress Bay or leave for land and never return. So of course she chose land. She threw that necklace away and I never saw her again. It became known as the morphing necklace among the very few who knew of its existence. And every couple of hundred years a mermaid might hear whispers about it and ask me if I'd made it.\n\nShe produced some seaweed letters.\n\n_DEAR MARITZA,_\n\n_DID YOU MAKE THE MORPHING NECKLACE?_\n\n_FROM KRILKY DRAGONHOLM_\n\nI pretend I know nothing. I tell them it's either an ancient magic or made up, and hope they'll forget about it. So anyway, back to the story. After Wartina was banished, I hid the ice creatures to minimise their power and stop any mermaid trying to use them, and then I sent the ice narwhal and the list of locations to Fortress Bay, and that's where they've been for hundreds of years. Until now.'\n\n'So what you're saying,' Beattie said slowly, 'is that because Wartina was a human, the ice creatures were so powerful they cracked in her hands. So if Paris can get hold of an ice creature\u2014'\n\n'There is a strong possibility she could destroy it,' Maritza Mist finished.\n\nBeattie flicked her tail excitedly. 'And in order for the ice creatures to work, Gulper and Rose need _all_ of them. But if we destroy one\u2014'\n\n'You win,' Maritza said.\n\n'Where are they hidden?' Beattie asked eagerly.\n\n'Each mermaid realm has one, though I've forgotten most of the locations now \u2013 and Gulper and Rose have the list. Frostopia's is the narwhal, which Gulper and Rose already stole from Fortress Bay. They also have the ice seahorse from Rainbow Landing \u2013 they got that one hundreds of years before they were sent to Viperview. And now they're on their way to Pinkly Lagoon, where I hid an ice leech in the old town of Vampire Rocks.'\n\n'But they have a head start,' Zelda groaned. 'It'll take us ages to get to another kingdom.'\n\n'Well, for that I have one more trick up my sleeve,' Maritza Mist said as she threw on a fur-lined cape. 'Come with me.'\n15\n\nJelly and the Whale Bus\n\nMeri put the clam car on auto-swim and curled up in the back seat for a nap. Lady Wriggles was already asleep, wrapped neatly around the steering wheel. They were in crystal-clear waters, filled with slow-swimming fish.\n\n' _AHOOOHGAH!_ '\n\nMeri snapped awake.\n\n' _AHOOOHGAH!_ '\n\n'Lady Wriggles,' she whispered. 'What creature makes that noise? I haven't been taught that one.'\n\n' _AHOOOHGAH!_ '\n\nLady Wriggles covered her eyes with her tentacles. She knew what sea creature made that noise.\n\n'Seriously,' Meri said, desperately scanning the data on her tail. 'What makes that\u2014'\n\n'JELLY HERE! YOU ARE DRIVING IN FORBIDDEN HUMAN TERRITORY. PLEASE BOARD THE WHALE BUS.'\n\nMeri flung herself at the control panel and punched a button. The clam car nestled into the seabed and turned the colour of sand.\n\n'NICE TRY, BUT MY OLD EYES CAN SEE EVERYTHING,' came the voice, as a whale with a sunken human bus on its back floated past the window. A jellyfish with a horn strapped to its head tapped on the windscreen, making Meri and Lady Wriggles jump.\n\n'THIS IS QUENTIN, MY JELLYFISH. HE'LL SHOW YOU WHERE TO PARK YOUR CLAM CAR. YOU'LL GET IT BACK WHEN WE RETURN TO THE LOWER REALMS \u2013 SAFE MERMAID TERRITORY. IF YOU RESIST, I WILL SET THE WHALE ON YOU.'\n\n'Great,' Meri groaned.\n\nAfter she'd parked the clam car on the roof of the bus, Meri wrapped Lady Wriggles in her hair and swam aboard. She was greeted by a jolly old mermaid with a shrivelled tail.\n\n'How could you see us?' Meri demanded.\n\nJelly smiled a toothless smile. 'These old eyes see everything... And you forgot to turn your headlights off.'\n\n'Codfuffle,' Meri muttered in frustration.\n\n'Where are you from?' Jelly asked.\n\n'I can't say,' Meri replied quickly.\n\n'Well, you _have_ to say,' Jelly said, her smile growing. 'Because now that you're on the bus, I have to take you to safe mermaid territory.'\n\n'Frostopia then,' Meri said.\n\n'You sneaking in?' Jelly asked hopefully.\n\nMeri shook her head.\n\n'That's a shame,' Jelly said. 'I like rebels.'\n\nMeri looked into the almost empty bus. There was only an old man and old lady mermaid on board. And a shopping trolley.\n\n'They're always trying to go for holidays on land with the humans,' Jelly whispered, rolling her eyes. 'In a shopping trolley.'\n\nThe mermaids were wearing matching T-shirts that read, _I WENT ON A HUMAN ROLLERCOASTER IN A SHOPPING TROLLEY AND ALL I GOT WAS THIS EXCELLENT T-SHIRT (and a fine)_.\n\n'This is going to be a long ride,' Meri whispered to Lady Wriggles.\n16\n\nRealm Reach\n\nBeattie and the others followed Maritza Mist through the icicle-covered alleyways of Flurry Falls, and came to a stop at a towering ice wall.\n\n'Dead end,' Zelda said, turning to leave.\n\n'No,' Beattie whispered, running her hand over the ice. 'There's something in there.'\n\nThey looked up and gasped \u2013 frozen within the wall were a dozen colourful sea creatures.\n\n'THERE'S A SEAHORSE!' Steve wailed, fake-fainting on to Beattie's palm.\n\nMaritza took a deep breath and held her hands up to the wall. She turned to Beattie. 'Do the same \u2013 you can make magic, can't you?'\n\nBeattie nearly tumbled back on her tail. She had no idea how Maritza could possibly know that. When Beattie had first met Gronnyupple, she had used water-witch potions from Maritza's catalogue. But as time went on, Beattie realised she could make the magic herself, though she wasn't very good at it yet.\n\n'You know, dear,' Maritza said, steadying her splodged tail. 'There's usually only one water witch in the world who can make magic. She or he creates the potions for the catalogue and sends them out to all the water witches.'\n\n'But you can also make magic,' Beattie said slowly. 'So that means there's two of us.'\n\nMaritza smiled. 'I know, dear \u2013 isn't it strange! There's only meant to be one. They always have and always will run the catalogue \u2013 it's their _destiny_. But now there's two of us. It's probably because I'm immortal and not meant to be here. Anyway, what's important now is that you help me reactivate this wall.'\n\nZelda backed away, pulling Mimi and Paris with her.\n\nSteve looked wide-eyed from Beattie to the wall.\n\nThe water was eerily quiet and Beattie could hear what sounded like a dripping tap. She raised her hands, and glittering water began gushing down the frozen wall, sweeping her friends sideways.\n\n'KEEP GOING!' Maritza shouted as the creatures in the wall began to emerge \u2013 sculpted replicas made of colourful ice.\n\n'Oh good,' Steve said. 'They're not real creatures \u2013 they'd be so cold!'\n\n'Perfect,' Maritza said, clapping her hands. 'Not bad, Beattie, not bad at all.'\n\n'What is this?' Zelda said, swimming up to the icy inky-coloured octopus on the wall. She reached out a hand.\n\n'CAREFUL!' Maritza shouted, but it was too late. The octopus began swirling around and around until it wasn't an octopus any more but a dark watery opening in the wall.\n\n'Cool!' Zelda cried, pushing her whole head in. She reeled back in shock. 'It's Octopolli in there! The realm where the mermaids have octopus tentacles for tails!'\n\n'Oh calm down,' Steve said, prodding the colourful seahorse with his nose. It spun just as the octopus had and revealed a multicoloured watery portal.\n\nBeattie pulled him back before he could disappear inside. She didn't want Steve getting lost.\n\nMeanwhile, Maritza was busy transforming the other sea creatures into swirling portals.\n\n'A long time ago, when water witches were everywhere and Frostopia ruled the mermaid kingdoms,' Maritza explained, 'this magic wall was built to connect all the realms.'\n\nShe touched the final creature \u2013 a pink dolphin.\n\n'It's called Realm Reach, and it will take you to any mermaid realm in the world in a single swish of the tail.'\n\nBeattie looked at Paris, whose mouth was hanging open so wide a fish swam inside.\n\nMaritza rushed back to them, her tail flicking fast. 'Frostopia has been largely cut off from the rest of the world for many years.'\n\n'So Realm Reach was forgotten,' Beattie said.\n\nMaritza nodded sadly. 'We seemed more interested in arguing, or ignoring each other completely. I have kept watch over the wall in the hope that one day the world would be a better place and mermaids would use it again. Gulper and Rose showed up long after Realm Reach was forgotten, so they don't know it exists.'\n\nBeattie swam up to the Pinkly Lagoon portal, where Gulper and Rose had said they were going. It was dark inside, but Beattie could feel the water was warm and it smelled of sun-soaked sand.\n\nZelda began swishing her tail excitedly. 'With this wall we can reach the other realms before Gulper and Rose! They don't stand a chance!'\n\n'What's the ice creature in Pinkly Lagoon?' Paris asked.\n\n'A leech,' Maritza Mist said, making Paris tense up. 'Pinkly Lagoon is famous for its leeches.'\n\nMaritza swam over to a block of ice not much taller than herself and raised her hands again. The ice exploded, revealing a battered old vending machine hidden within.\n\n'An axolotl vending machine!' Zelda cried. 'We have one of those in the Hidden Lagoon. It contains small creatures for carrying big things.'\n\n'I invented them!' Maritza Mist said, banging the vending machine with her tail. 'They put one here for mermaids going on shopping trips to other realms. They need those axolotls to carry their shopping.'\n\nAn axolotl dropped out of the slot and rolled into her hand.\n\n'Hello, dear Grubs, it's been a long time,' she said, stroking the axolotl affectionately. She placed him in Mimi's hand. 'He will help you carry the ice creatures \u2013 they can be heavy, even though they're small. He's my most trusty sea-creature friend. He's been living here for thousands of years waiting for mermaids to return to the wall. I tried to get him to move, but he wouldn't.'\n\nMimi stroked Grubs as he nuzzled into her hand.\n\n'Take care of him,' Maritza said, gazing at him fondly. 'He is very special to me.'\n\n'He says you're very special to him too,' Mimi said.\n\n'Aren't you coming?' Beattie asked.\n\n'No,' Maritza said quietly.\n\n'You'd better have a good reason,' Zelda said, folding her arms crossly. She didn't like anyone who wasn't a team player. It wasn't the shockey way.\n\n'Actually, I have three reasons,' Maritza said. 'Firstly, Gulper and Rose might try to steal more magic from my igloo \u2013 that's why they raided the place \u2013 so I need to get back and make sure my potions are well protected and hidden.'\n\n'And secondly?' Zelda said, thawing slightly.\n\n'Secondly, I need to find out where that goblin shark was taking Gronnyupple. Can you tell me anything about it?'\n\n'It had amber eyes and was covered in amber jewels,' Beattie said.\n\n'Amberberg then,' Maritza said.\n\n'Pardon?' Beattie, Mimi, Zelda and Paris all said at once.\n\nMaritza pointed to the amber-coloured swirling portal. 'A city where the palaces are carved in amber. Plus they use goblin sharks to collect criminals.'\n\n'Criminals?' Beattie choked.\n\n'Yes, criminals, dear. I can't leave the igloo, but I'll find out as much as I can, so that once we've defeated Gulper and Rose we can _all_ go and get Gronnyupple.'\n\nBeattie reluctantly nodded. It seemed mean not to rescue Gronnyupple now that they knew where she was. But they didn't have much choice.\n\n'And thirdly, the Burguin shop in the Glisten Quarter is having a major sale today.'\n\n'Oooh,' Mimi said, looking down at her Burguin T-shirt. 'Can you get me something?'\n\nMaritza Mist winked. 'Remember, head for Vampire Rocks \u2013 you'll find the leech in the old part of town, down the darkest alleyway.'\n\n'LET'S DO THIS!' Zelda roared. She and Paris dived excitedly through the portal to Pinkly Lagoon.\n\n'This was meant to be a little detour to check Maritza Mist was OK,' Beattie mumbled to Steve. 'And now we're saving the world.'\n\n'You _hope_ ,' Steve said.\n17\n\nThe Glorious Pinkness of Pinkly\n\nSteve floated in silence in Flicko City, the capital of Pinkly Lagoon. If he'd had more of a mouth, his jaw would have been on the floor.\n\nThe portal had taken them right to the middle of town, where chattering seahorses whizzed past every few seconds.\n\n'I thought I was the only talking seahorse in the world,' Steve said glumly. 'I'm not a miracle and it's been proven otherwise... I'm _normal_. THIS IS HORRIFIC!'\n\nMermaids floated past, unaware that Beattie and the others had just arrived via portal from another realm. Most of them had pink tails \u2013 the older the mermaid the more neon-pink the tail.\n\n'Afternoon,' a really ancient mermaid croaked, nearly blinding them.\n\nMimi put on her sunglasses.\n\nDolphins somersaulted around them, and multicoloured fish weaved in and out of pastel-pink mermaid townhouses nestled in the coral reef.\n\nParis shook her necklace and morphed into a dolphin.\n\nBeattie spotted a group of cool-looking mermaids floating nearby \u2013 one had a pink top with shell sleeves and another had a gold dolphin choker.\n\nFeeling inadequately dressed, Beattie swam down a road called Skerry Street and stopped next to a clump of singing seahorses.\n\n'Excuse me, which way to Vampire Rocks?'\n\n'Leeeeeffffttt,' the seahorses chimed in sync.\n\nAnd off Beattie and her friends went, towards a highway filled with dolphins pulling mermaids to their destinations.\n\n'I'M NOT A TAXI,' Paris shouted as a group of mermaids grabbed hold of her fin, but it came out as 'Eeeeee ee EE'.\n\n'Steady up!' one of them yelled, shaking Paris and making her morph back into a mermaid.\n\nThe mermaids holding her fin screamed and shot off in all directions. Zelda laughed as they dipped and dived to avoid the traffic.\n\nIt wasn't long before the highway trickled out into a glistening pink lagoon. Mermaids leapt out of the shallow water on to lilos that floated on the surface.\n\nBeattie and the others nervously raised their heads out of the water. They were in a human cove, but there wasn't a single human in sight, just thousands of mermaids floating on lilos.\n\n'This is so dangerous,' Beattie said, looking around her. 'Guys... ?'\n\nBut they'd gone.\n\nBeattie craned her neck to get a better look and spotted them lazing on the lilos!\n\nSteve bounced back into the water, gasping for breath. 'Too... much... air.'\n\n'Get back here,' Beattie ordered as Zelda leapt on to Mimi's lilo, making her topple over with a splash.\n\n'HUMAN!' someone suddenly roared.\n\nThousands of mermaids dived back into the water as a troop of dolphins rose up and crowded the surface to hide them.\n\n'LOOK AT THE DOLPHINS!' a human kid cried from a boat sailing along the coastline.\n\nBeattie's heart was beating in her mouth. She'd never been so close to humans in her mermaid form.\n\nZelda popped her head above the water right next to the boat.\n\n'A mermaid!' the little kid cried, but the adult humans on the boat weren't listening.\n\n'Why aren't you combing your hair, like a proper mermaid?' the kid whispered.\n\nZelda rose up out of the water until she was nose to nose with him. 'I'm not here to comb my hair. I'm here to _save the world_.'\n\n'Zelda!' Beattie snapped, pulling her back under. 'Stop talking to the human.'\n\nThey floated silently under the dolphins until the boat with the humans sailed on.\n\n'BACK TO LAZING AROUND!' a mermaid cried, and they burst out of the water once again.\n\n'And they do this for a _holiday_?' Beattie said, practically hyperventilating.\n\n'Oh come on,' Zelda said, heaving Beattie on to a lilo. 'It's fun!'\n\nBeattie looked over at Mimi, who was applying sun cream to Paris's face.\n\n'We don't have time for this,' Beattie groaned. _'_ And where's Grubs?'\n\nSteve's head popped out of the water. 'I have some bad news.'\n18\n\nHow Do You Get an Axolotl Out of a Vending Machine, and Other Important Questions\n\nThe problem with the axolotl vending machines was that they were very popular \u2013 in every mermaid kingdom.\n\n'The mermaid took it,' Steve said, swishing his tail. 'And put it in...' He prodded the vending machine with his nose.\n\n'How do we know which one is ours?' Beattie asked. 'Oh, it could take all day for it to spit out the right axolotl!'\n\nMimi put an ear to the vending machine. 'I'll listen for his voice.'\n\nBut it was soundproof. She couldn't hear a thing.\n\nZelda charged and hit it with a _clang_.\n\n'Argh!' she cried, when she didn't even dent it.\n\nParis scooped some stray Pinkly pounds from the ground and slotted them into the vending machine.\n\nAn axolotl fell out.\n\n'No,' Mimi said. 'That one's got a Pinkly accent.'\n\nParis tried again.\n\n'That one is just RUDE,' Mimi said.\n\nParis slotted in the last coin.\n\nA fat fish flopped out.\n\n'That's a fish,' Mimi said helpfully.\n\n'We could come back for Grubs later?' Zelda sug\u00ad gested. 'He's used to being inside vending machines!'\n\n'Grubs is Maritza's special axolotl!' Beattie cried. 'We can't leave him \u2013 what if another mermaid gets him and he's lost forever? Oh WHY DOES THIS STUFF ALWAYS HAPPEN TO ME?!'\n\n'This has never happened to you before,' Zelda said. 'You've never had an axolotl called Grubs who got stuck in a vending machine.'\n\n'I meant STUFF _LIKE_ THIS!' Beattie clarified.\n\nA queue of mermaids began to form behind them, all eager to use the vending machine.\n\n'There's only one solution, but it will involve swimming very fast afterwards,' Zelda said, moving towards Paris.\n\n'What?' Beattie asked nervously.\n\nZelda whispered in Paris's ear, then grabbed her necklace and shook it. With a _snap_ , she morphed into a huge shark, took a bite out of the top of the vending machine, and axolotls spewed out everywhere!\n\n'Grubs!' Mimi cried, scooping up a small axolotl.\n\n'FREE AXOLOTLS!' the mermaids in the queue cried, grabbing fistfuls of them and kissing their little faces.\n\nA siren sounded. Mermaids with dolphin tails and dolphin-shaped sunglasses appeared. 'STOP RIGHT THERE, BAD MERMAIDS!'\n\n'This is the swimming fast bit,' Zelda said, grabbing Beattie's arm and pulling her through the winding coral paths towards a sign that read _VAMPIRE ROCKS_.\n19\n\nMeri and the Burguin Sale\n\nMeri smiled as she detached her clam car from the roof of the bus and climbed into the driver's seat. 'I hope we meet again one day, Jelly!' she shouted out of the window.\n\n'I ONLY MEET MERMAIDS IN TROUBLE, SO I HOPE NOT!' Jelly shouted back, and with a honk of Quentin the jellyfish's horn the whale bus took off.\n\n'Right,' Meri said, punching the controls and steering the clam car towards Frostopia's ice wall. There was a whirring sound, and in one swift twist the clam car burrowed into the ice.\n\nMeri checked the controls. 'Lady Wriggles, do you know where we will emerge once we get to the other side of the wall?'\n\nLady Wriggles consulted the screens and punched some buttons.\n\n_BURGUIN SALE_ popped up.\n\n'What's... Burguin sale?' Meri whispered.\n\nAfter about five minutes, Meri and Lady Wriggles found themselves in the midst of what looked like a riot.\n\n'I WANT PARTY-EDITION BURGUIN!'\n\n'GIVE ME THAT GREEN HAMBURGER! I NEED GREEN TO COMPLETE THE SET!'\n\n'WILL YOU STOP SLAPPING LIMITED-EDITION GOLD BURGUIN IN THE FACE! HIS BEAK IS _FRAGILE!_ '\n\n'Excuse me,' Meri said quietly to a mermaid wearing a stack of caps with a sort of burger-penguin cartoon on them. 'Are you at war?'\n\n'WE'RE SALE SHOPPING!' the mermaid roared, grabbing a tiny multicoloured burger and flipping it open. A little model of the burger-penguin floated out.\n\n'OH, YOU GOT BIKINI BURGUIN!' a mermaid shouted as she floated past. 'THAT'S SO UNFAIR! I'VE BEEN TRYING TO FIND THAT ONE FOR YEARS!'\n\nMeri grabbed a burger. 'Burguin,' she said, reading the label, 'is half burger, half penguin. The most popular cartoon character in Frostopia. Find the ice Burguin and you get an eternal supply of Burguin merchandise and a chance to visit the Burguin Studios in Floe.'\n\nMeri tossed the burger over her shoulder. 'So it's not a war,' she said to Lady Wriggles, 'it's _shopping_. They seem almost the same thing\u2014' Meri stopped. Across the frozen room of hysterical mermaids was a young-looking mermaid with three icy buns on her head.\n\n'Maritza Mist!' she shouted, before realising how not-at-all stealth she was being.\n\nThe mermaid with the three buns spun round in shock.\n\n'You made me drop my Burguin burgers,' Maritza Mist said.\n\nMeri straightened her tail and floated perfectly upright.\n\n'Please confirm the status of your health,' she said robotically.\n\n'Still immortal,' Maritza Mist said, trying not to smile. 'You must be from Fortress Bay.'\n\n'And,' Meri coughed, 'please advise the status of the ice narwhal, if known.'\n\n'On its way to Pinkly Lagoon,' Maritza Mist said.\n\n'PINKLY?!' Meri roared, making all the mermaids drop their Burguins. 'Well then, I have to go there too.'\n\n'I know a shortcut,' Maritza Mist said.\n\nTHE FREEZY\n\nFrostopia's Coolest Magazine!\n\nBURGUIN INTERVIEW!\n\nToday we interviewed Izzy Iceberger of Flurry Falls, who has found the very rare Bikini Burguin! Bikini Burguin is like Original Burguin, but he's wearing a bikini.\n\n**_Reporter:_** Izzy, how did you find Bikini Burguin?\n\n**_Izzy:_** I bought him in a shop.\n\n**_Reporter:_** That must've been expensive, Bikini Burguin is very rare.\n\n**_Izzy:_** Yes, I sold my house.\n\n**_Reporter:_** Doesn't your house belong to your parents?\n\n**_Izzy:_** Yes, but now it also belongs to Mrs Melterson, who bought it in exchange for the money I needed to buy Bikini Burguin.\n\n**_Reporter:_** Aren't your parents... angry?\n\n**_Izzy:_** They don't know yet.\n\n**_Reporter:_** But Mrs Melterson will be in your house, with all her stuff. What if she redecorates the place \u2013 how will you explain that to your parents?\n\n**_Izzy:_** I'll tell them Mrs Melterson is an interior designer who gave the house a makeover.\n\n**_Reporter:_** And what if Mrs Melterson eats food from the fridge?\n\n**_Izzy:_** I'll say she's a robber.\n\n**_Reporter:_** Thank you, Izzy Iceberger!\n20\n\nVampire Rocks and the Ice Leech\n\nVampire Rocks stretched out in front of Beattie \u2013 a lumpy expanse covered in mermaids covered in leeches.\n\n'Welcome to Vampire Rocks Leech Spa. How many, please?'\n\n'NONE,' Zelda said, disgusted.\n\nBeattie nudged her with her tail. 'Zelda, the ice creature is in there.'\n\n'Yeah,' Zelda said. 'But Maritza failed to tell us the old part of town is now a LEECH SPA.'\n\n'We're not here for the spa,' Paris said. 'We just want to swim around.'\n\n'That's nice,' the mermaid said, sounding like she didn't think it was nice at all. 'But no one's allowed past me without a spa ticket.'\n\n'Four and a seahorse, then,' Zelda said reluctantly.\n\nA shower of leeches rained down on them.\n\n'MERS MON IN MY MOUF!' Beattie cried, spitting in disgust.\n\n'Spa rules! You must stay still when in the spa and remain in your allocated spa bed. We hope you have a relaxing experience,' the mermaid oozed. 'Would anyone like a glass of Gloopy?'\n\nMoving, let alone swimming, was nearly impossible in the leech spa \u2013 they were covered from head to tail in leeches, and the glasses of Gloopy made them sleepy.\n\nBeattie's tail felt heavy and there was a leech sucking on her eyebrow. A clock floated in front of her, with a little rainbowfish moving the hands.\n\n'YOU'LL BE GORGEOUS AT HALF PAST THE RAINBOWFISH!' the mermaid in charge of the spa shouted as she swam past.\n\n'I'll be gorgeous when we get the ice leech,' Zelda said. 'Gorgeously happy anyway.'\n\nMimi peeled the leeches off her face and rolled on to her belly. 'The rainbowfish says the old town streets run under the spa. We can reach them from those caves over there behind the rocks.'\n\n'Let's go before the owner comes back,' Zelda hissed, swimming fast towards the rocks. Mimi followed her.\n\n'Beattie,' Paris whispered. 'Where's Steve?'\n\nBeattie looked around. She could only see mermaids covered in leeches and clocks with rainbowfish.\n\n'YOU'LL BE GORGEOUS AT HALF PAST THE RAINBOWFISH!'\n\n'Steve?' Beattie said, scanning the water for him.\n\n'He'll be fine,' Zelda whispered. 'We'll be back in no time. Now, come on!'\n\nThey dived into a crack in the rocks, slithering down until they hit sand and the rocks opened into a tunnel.\n\nBeattie had a horrible feeling, and it wasn't because the leech was still on her eyebrow.\n21\n\nLong-Lost Steve\n\nSteve floated in the seahorse section of the Vampire Rocks spa. It was different from the mermaid section, with bubbling baths of leech juice for them to stew in, and tiny cocktails of Gloopy.\n\nSteve had gone a bit cross-eyed from drinking too much of it.\n\n'You visiting for long?' the seahorse bubbling next to him asked.\n\n'No, just a quick holiday,' Steve lied. 'Do you live here?'\n\n'Live here?' the seahorse laughed. 'Seahorses don't _live_ here, silly! Have you been drinking that Gloopy through your nose?! We're _all_ from Rainbow Landing \u2013 we just holiday here, like you. My name's Stella.'\n\n'Steve,' he replied.\n\nShe gasped and slipped under the leech juice. He knew he was an impressive seahorse, even if he was no longer a miracle, but he'd never got that kind of reaction for just saying his name.\n\n'Hello?' Steve called.\n\nThe leech juice bubbled.\n\nStella burst back up, coughing and spluttering. 'You're Steve?'\n\n'Yes,' Steve replied.\n\n'Long-lost Steve!'\n\n'Well, actually I usually go by Steve and at the very most _Miracle_ , because where I live I'm the only talking seahorse.'\n\n'So you're _not_ from Rainbow Landing. You _are_ Long-lost Steve!'\n\n'I'm pretty sure I'm just Steve,' Steve said. 'I was found next to a sunken ship, yelling at some tuna. The mermaid who found me gave me to her daughter as a pet of sorts, and I've been part of their family ever since. Most would argue the best part of their family. And by most, I mean me.'\n\n'YOU ARE LONG-LOST STEVE! ARROGANT LONG-LOST STEVE!' Stella cried. 'EVERYONE!' she yelled across to the thousands of other seahorses stewing in leech-juice pools. 'IT'S LONG-LOST STEVE!'\n\nA ripple of excitement spread through the spa.\n\n'Who is Long-lost Steve?' Steve whispered.\n\n'He's the missing seahorse. The son of rock stars Stevie and Steven!'\n\n'Rock stars?' Steve said, his eyes lighting up.\n\nStella nodded. 'I'm heading back to Rainbow Landing today \u2013 come with me! I can reunite you.'\n\n'But... my family. My Beattie, she would never \u2013' Steve looked over the rocks and down to the level below, where Beattie had been lying covered in leeches \u2013 'leave me,' he finished in surprise when he realised she was _gone_.\n22\n\nNightmare Dolphin\n\n'So this was the Pinkly Lagoon of the olden days,' Beattie said as they passed ancient caves with seaweed-lined windows. There were no Burguin cartoons, swimways or clam cars.\n\nMimi, who had swum ahead, was charging back towards them in a burst of speed.\n\n'The good news is, I've found the ice leech,' she said.\n\n'YES!' Beattie roared, swimming faster.\n\n'The bad news is that a dolphin is swimming away with it.'\n\nThe ground began to rumble and the rock walls started to crumble as two pink dolphins came into view.\n\n'Eeee E eeeee EeeeEEEEeee,' the bigger dolphin squeaked.\n\n'It's telling the other dolphin that she should've only drunk one of the potions, like the label said,' Mimi translated for the others.\n\n'Potions? Wait a minute!' Beattie cried. 'They're not dolphins, they're Gulper and Rose in disguise!'\n\nRose was growing bigger and bigger \u2013 she could barely fit in the cramped tunnel \u2013 and at the same time her face was becoming more and more mermaid-like. It wasn't long before she looked like a dolphin with a mermaid nose and eyebrows.\n\n'IT'S LIKE A NIGHTMARE!' Zelda screamed, trying to squeeze past.\n\n'SHE'S GOING TO BURST!' Paris cried, just as the rock walls gave way, releasing Rose the gigantic dolphin on to the unsuspecting spa mermaids above. Beattie shielded her face as rocks and leeches rained down on her like the worst shower in the world.\n\n'GIANT DOLPHIN ATTACK!' a mermaid screeched overhead. They swam in all directions, smacking into each other and shrieking.\n\nRose and Gulper shot off.\n\n'Cods,' Beattie said as she sat down on a rock next to Zelda and surveyed the scene. Leeches went flying past their noses as mermaids flopped about in the wreckage.\n\n'And this,' Zelda said, gesticulating with a leech, 'is why you'll never find me at a spa.'\n\n'We were too late,' Beattie grumbled as they swam back towards the spa reception.\n\n'Well, technically it wasn't our fault,' Zelda pointed out. 'How were we supposed to know someone would scoop up Grubs and put him in a vending machine?'\n\n'If you hadn't been playing on the lilos, it would never have happened,' Beattie said, crossing her arms angrily.\n\n'But you said you LOVED the lilos!' Zelda said.\n\n'No I didn't,' Beattie protested.\n\n'Oh yeah, that was Steve. Never mind.'\n\n'Steve!' Beattie said, looking around. 'We have to get him before we go.'\n\n'We hope you enjoyed your leechy, beachy spa experience,' the spa mermaid said, her hair pasted to her face and a leech up each nostril. 'We're sorry the spa collapsed and a nightmare dolphin tried to kill us all. To apologise for this inconvenience, your spa treatment today is free. We hope to see you soon, so you can become gorgeous at half past the rainbowfish!'\n\n'We've got to pick up Steve, our seahorse,' Beattie said. 'He's still inside somewhere.'\n\n'Oh, Steve,' the mermaid said. 'He left.'\n\n'Left?' Beattie said.\n\n'Jumped on the dolphin balloon to Rainbow Landing, just before the place crumbled.'\n\n'Dolphin balloon?' Beattie said in disbelief.\n\n'It's like a hot-air balloon,' the mermaid said flatly. 'We stole the idea from the humans. It's attached to a dolphin, then the basket is filled with seahorses, and the dolphin swims to Rainbow Landing, diving every so often so the seahorses can be dunked in water to breathe.'\n\n'Does it travel fast?' Beattie asked.\n\n'Oh yes,' the mermaid said. 'He's well on his way to Rainbow Landing.'\n\nTHE FREEZY\n\nFrostopia's Coolest Magazine!\n\nBURGUIN INTERVIEW!\n\nToday we interviewed Aimi McWhale of Floe, who has found an Original Burguin! Original Burguin looks like Original Burguin, because they are the same thing and almost everyone in Frostopia has one.\n\n**_Reporter:_** Aimi, you have an Original Burguin.\n\n**_Aimi:_** Is it a rare one?\n\n**_Reporter:_** No.\n\n**_Aimi:_** I need to find a rare one then. Maybe I'll get my pet seal on the job. She's called Slippy and she likes to slime mermaids.\n\n**_Reporter:_** Is that why I'm covered in slime?\n\n**_Aimi:_** Maaaaayyyybeeee.\n\n**_Reporter:_** Thank you, Aimi McWhale!\n23\n\nMeri Makes Some Friends\n\nBeattie dived through the portal to Realm Reach in shocked silence. Steve was gone.\n\n'Well, they may have the ice leech, but we have Realm Reach!' Zelda said, trying to cheer everyone up.\n\n'And now so do we,' came a whisper as Gulper and Rose squeezed through the portal. 'Thanks for the tip. This will be much quicker than swimming \u2013 even with speedy potions, it still takes a _loooong_ time!'\n\nThey disappeared through the portal to Octopolli.\n\n'Did that just happen?' Zelda said, rubbing her eyes. 'Did Gulper and Rose just follow us here?'\n\n'They've gone to Octopolli!' Beattie cried. 'Ugh, we're so _bad_ at this!'\n\n'So we'll go somewhere else,' Paris said. 'Let them get the ice creature in Octopolli. We'll go to\u2014'\n\n'The Kingdom of Mume,' Zelda finished.\n\n'No,' Beattie said firmly. 'We need to follow them. You saw what havoc they caused at the spa \u2013 they might _hurt_ someone in Octopolli.'\n\n'But, Beattie,' Zelda protested, 'we only need one ice creature to stop them \u2013 what's the point in chasing them?'\n\nBeattie thought for a moment. She pictured all the mermaids lazing at the leech spa fleeing in a leech-smeared panic. 'No. We know where they're going, so we have a responsibility to protect other mermaids.'\n\n'Fine,' Zelda said. 'But for the record, my idea is better.'\n\nGrubs began wriggling as Maritza Mist floated around the corner. She stopped when she saw their faces. 'No luck, dears?'\n\nBeattie looked down at her tail. Maritza had trusted them and they'd messed it up.\n\n'Well, I have someone to help you,' Maritza said as a mermaid with jet-black hair and a determined smile appeared beside her.\n\n'I'm Meri,' she said shyly. 'I've been _waaaaatching_ you.'\n\n'Tone it down,' Maritza said. 'That sounded creepy.'\n\n'Ah,' Mimi said. 'Merry! Just like all the sea creatures in Frostopia. What's your name?'\n\nMeri paused. 'No, um, my name is Meri. With an i.'\n\nMimi looked confused.\n\n'She's from Fortress Bay, the spy school in the Arabian Sea,' Maritza explained.\n\nMeri shifted awkwardly. 'I'm, um, sort of meant to be undercover.'\n\n'She's undercover, dears.'\n\n'I'm here to help,' Meri said as Maritza nudged her forward. 'I know the locations of the ice creatures. It was my job to protect the list and the ice narwhal. I know that list off by heart!'\n\n'ISN'T THAT JUST WHAT WE NEEDED?!' Maritza said cheerily. 'I'd forgotten where I'd hidden most of them, so you have helped hugely already. YOU'RE THE BEST SPY MERMAID IN THE WORLD, DEAR!'\n\n'I'm actually pretty average,' Meri mumbled.\n\n'She has a jellyfish in her hair,' Paris whispered to Zelda.\n\n'I also have news of Gronnyupple,' Maritza said. 'Meri and I have established that she is alive and well in Amberberg, where they think she's a criminal mastermind.'\n\n'But she's not a criminal mastermind!' Zelda scoffed. 'Wait... Maybe she was just pretending to be goofy and weird!'\n\n'She's completely innocent,' Meri said. 'It looks like she's been framed, but I don't know why. Maritza is going to look into it while we go after Gulper and Rose. She'll try to break her out, but it won't be easy. Amberberg has the toughest goblin sharks in the world.'\n\n'But who would frame Gronnyupple?' Beattie asked. 'Who even knows Gronnyupple? She's a loner. She lives in the Crocodile Kingdom and hangs out in an abandoned launderette, where she practises spells.'\n\n'It's very strange,' Meri said. 'I wonder if Gulper and Rose could've had a hand in it.'\n\n'But why Gronnyupple?' Paris asked. 'She wouldn't hurt a fish. She's hardly a threat to them.'\n\n'It might be part of a bigger plan,' Meri said. 'It's too soon to tell. But the fact that she's a water witch might be significant. Now, let's get on with the mission.'\n\n'Steve would love to meet a spy,' Beattie said quietly as she stared at the empty false teeth.\n\n'I've been following your progress so far,' Meri said, pointing at her tail. 'Spy mermaids can see anywhere in the world on their tails.'\n\n'Did you see where Steve went?' Beattie asked hopefully.\n\nMeri looked at her blankly. 'Who?'\n\nBeattie sighed. 'Never mind.' She looked up at the multicoloured Rainbow Landing portal. Her chances of getting there, locating Steve and getting back before Gulper and Rose found the ice creature in Octopolli seemed small.\n\n'Beattie,' Zelda said, nudging her.\n\n'I don't understand why he would leave me,' Beattie said as her eyes welled with tears.\n\nMimi sidled up beside her. 'It might be an insensitive moment to mention it, but this rainbowfish is wondering if Steve's false teeth bedroom is available to rent? He'd very much like to live in them.'\n\nBeattie snapped the false teeth closed and dived through the portal to Octopolli.\n\n'We'll find him, Beattie!' Zelda cried as she raced after her.\n\n'After we get the ice creature from Octopolli!' Paris added as she dived through.\n\nMimi stared at the rainbowfish. 'I'm afraid it's a firm no on the rental.'\n24\n\nSlippery Octopolli\n\nThe Realm Reach portal took them to Inko, the biggest city in Octopolli.\n\nIt was dark, with murky grey water and ink-black buildings. Mermaids with octopus tentacles slunk past suspiciously.\n\n'Would you rather have an octopus tail that constantly picked your nose, or an octopus head?' Zelda asked.\n\nMimi and Paris seriously considered the question.\n\n'Would the tentacles hurt when picking my nose?' Mimi asked.\n\nParis tapped her chin. 'Would I be allowed to wear earrings if I had an octopus head?'\n\n'I'm so sorry about them,' Beattie whispered to Meri.\n\n'It's not a problem,' Meri said. 'I share a room with the Teenies \u2013 triplet mermaids \u2013 at spy school and they're far worse. This is actually fun!'\n\nBeattie wondered what it was like at Fortress Bay, and what spy school involved.\n\n'The ice octopus is hidden in the Beak de Beak area of Octopolli,' Meri went on.\n\n'Perfect \u2013 let's figure out how to get there,' Beattie said as Zelda and Paris helped Mimi put an octopus tentacle up her nose.\n\n'It does hurt!'\n\nTHE FREEZY\n\nFrostopia's Coolest Magazine!\n\nBURGUIN INTERVIEW!\n\nToday we interviewed Ralph Snowland of the Glisten Quarter, who has found the Seahorse Surprise Burguin! Seahorse Surprise Burguin looks like Original Burguin, but he's hiding Seahorse Surprise sweets in his burger bun!\n\n**_Reporter:_** Ralph, what did you do when you found Seahorse Surprise Burguin?\n\n**_Ralph:_** I screamed!\n\n**_Reporter:_** Because you were so delighted, I'm sure!\n\n**_Ralph:_** I had just found out that Gillica, who plays Blubble the Ice-Skating Seal, was CANCELLING her shows because some other mermaids pretended to be her and then ran away, so she wasn't allowed into Frostopia and went home \u2013 and that is why I screamed.\n\n**_Reporter:_** Weren't you happy to find the Seahorse Surprise Burguin though? They are pretty rare!\n\n**_Ralph:_** I don't really know what Burguin is. I screamed my way into the Burguin shop and got it by accident.\n\n**_Reporter:_** How is that even possible?\n\n**_Ralph:_** They threw it at me to get me to stop screaming, but it landed in my mouth so I got to keep it.\n\n**_Reporter:_** Thank you, Ralph!\n25\n\nBeak de Beak\n\nA long cage weaved around Inko city like a sinister caterpillar. Inside, Beattie could see mermaids riding huge gold tentacles, slithering fast to their destinations.\n\n'The Inko Express will take us to Beak de Beak,' Meri said, pointing at the cage. Mermaids were lining up in their hundreds along its edges. 'Inko Express is its official name, but the mermaids of Octopolli call it the Inko Slinko.'\n\n'How come you know so much about other mermaid places?' Zelda asked suspiciously.\n\n'I've studied them all. It's how you become a good spy. For example, I know that in your Hidden Lagoon there's an excellent restaurant called Jawella's inside a shark. I get their chomp chops delivered to the canteen in Fortress Bay by special request.'\n\n'No way!' Zelda said, sounding impressed.\n\nBeattie swam around the Inko Slinko map. Inko was one of seven cities in Octopolli, and Beak de Beak was at the end of the line.\n\n'Is this thing safe?' Paris asked, just before a tentacle carriage slammed against the cage. 'Never mind,' she whimpered.\n\nThey waited for an empty one before they opened the cage door and dived on. Beattie tried to get a grip but it was almost impossible, the whole thing was as slippery as \u2013 well \u2013 an octopus.\n\nThey slithered off at breakneck speed.\n\n'IT JUST SEEMS TO BE GOING IN CIRCLES AROUND INKO!' Beattie shouted to Meri, who was balancing side-saddle on the tentacle carriage with ease.\n\n'Beattie, _look_!' Paris yelled as the seabed beneath them opened up to reveal a pit of glowing jellyfish.\n\n'OH COD!' Beattie cried, grabbing on to Paris and Mimi just as the carriage plunged into the pit.\n\n'I HAVE A JELLYFISH STUCK TO MY FAAAACCEEEE!' Zelda roared, peeling it off, just as another one attached itself to her hair.\n\nThey hurtled through a dark passageway, where eels popped out of caves holding little seaweed flyers.\n\n_VISIT THE POD BOX IN PODLEY, the best gallery for sunken human paintings! We have a new drawing by the incredibly popular human artist Teddyage4. Plus some fascinating doodles by Homework._\n\nThe Inko Slinko whipped on through another clump of jellyfish, only this time they extended their stingers as the carriage passed through.\n\n'The stingers are slowing us down,' Beattie said in amazement as they reached a caged area.\n\n'SLIPPERY ROCK,' came a crackle. 'PLEASE SLINK OFF HERE FOR SLIPPERY ROCK.'\n\nZelda pointed enthusiastically at a poster. 'Look, Beatts! Slippery Rock is home to the best swimming complex in Octopolli, with over a thousand whirlpools and a never-ending flume. I want to go on the never-ending flume!'\n\n'Read the rest,' Beattie said. 'It says only one mermaid has ever got on the never-ending flume, and she's still on it, _ninety-five_ years later.'\n\n'What a way to spend ninety-five years,' Zelda said dreamily.\n\nThey shot off again, passing Squiddy Circus, home to a circus run by a squid, then around a bend full of the largest sunken ships Beattie had ever seen, before stopping to refuel at Ringsworth, the location of the world's only finishing school for octopuses.\n\nBeattie was getting impatient.\n\n'Look!' Paris shouted. 'UP AHEAD! It's Gulper and Rose!'\n\n'Where?' Meri said, staring at her tail, trying to get a better view.\n\n'About nine carriages ahead,' Paris said as she tried to flop off the tentacle carriage. Its tip smacked her, pushing her back on.\n\nMeri tapped her tail. 'Good spot. They'll reach Beak de Beak five minutes before us. We have to slow them down.'\n\n'How?' Beattie said, just as Mimi fell forward, clutching her ears. Meri did the same.\n\nZelda grabbed her twin's shoulders and shook them. 'Mimi! What is it?'\n\n'MERI PEBBLE! MERI PEBBLE! MERI PEBBLE!' boomed all the passing eels and octopuses.\n\n'MERI,' Mimi said loudly. 'SOMEONE'S CALLING YOU.'\n\n'Oh no,' Meri said, curling into a ball. 'It's Sabrina. She's discovered I left Fortress Bay. I really thought that whale I dressed up in my clothes would've fooled her...'\n\n'MERI PEBBLE! RETURN TO BASE AT ONCE!'\n\n'LOOK AHEAD ON THE TENTACLE CARRIAGES!' Meri shouted to Sabrina. 'THE VIPERVIEW PRISON ESCAPEES ARE RIDING THE INKO SLINKO!'\n\n'MERI,' an eel said sternly, while Mimi translated for the non-fishtalkers. 'THIS IS BEYOND RIDICULOUS NOW. YOU STOLE THE MOST EXPENSIVE CLAM CAR TOO.'\n\nLady Wriggles burrowed deeper into Meri's hair.\n\n'I CAN SEE YOU, LADY WRIGGLES!' Sabrina's voice boomed.\n\n'I'll be back soon!' Meri said, defiantly pushing the eel away.\n\n'You left Fortress Bay without permission?' Zelda said.\n\n'Sabrina didn't believe me about Gulper and Rose,' Meri said. 'She showed me footage of them in Viperview \u2013 they look exactly the same, but they can't be in two places at once.'\n\n'Wait,' Beattie said. 'Did either of them say FISH EYE?'\n\n'Yes,' Meri said, sounding shocked. 'How do you know that?'\n\n'Beattie once used one of Maritza Mist's spells on us,' Zelda explained. 'It's a potion that doubles you, but a side effect is your double sometimes shouts FISH EYE!'\n\n'So they used a doubling potion!' Meri said, scrunching up her fist. 'Those cunning mermaids.'\n\n'MERI!' Sabrina yelled. 'IS THIS THING ON? ARE THE SEA CREATURES SHOUTING MY MESSAGES? HELLO?!'\n\nMimi clutched her ears as the sound increased. She needed to tell Sabrina about the potion!\n\n'MERI!' came Sabrina's roar. But this time it was too loud for all the sea creatures that could hear it. The eels reared up and began to scatter, the octopuses formed what Beattie could only describe as a wriggling panic, and the tentacle carriages began to jump and dive as if they'd forgotten where they were going.\n\n'This is our chance,' Zelda said, grabbing Beattie's arm and pulling her off the tentacle. They swam as fast as they could towards Gulper and Rose. 'It's like I always say, a little chaos can save the day.'\n26\n\nMaritza Sends a Shark\n\nMaritza Mist uncorked another bottle of SEAL POP and peered out of the icy window of her igloo. Gulper and Rose were nowhere to be seen, but she'd hidden the potions under her Burguin collection, just to be safe.\n\n'Now,' she said, flipping open the clamshell compact Meri had left for her, 'let's see if we can save Gronnyupple.' She prodded the compact with her finger until a menu came up.\n\nSABRINA'S PORTAL\n\nMERMAID SEARCH\n\nSPY CASE DATABASE\n\nREALM ASSISTANCE\n\nMaritza pressed Realm Assistance and another menu popped up. She sighed \u2013 she wasn't a fan of technology.\n\nJELLY PICK-UP\n\nUNDERCOVER MERMAIDS ON LAND\n\nWHALE DIVERSIONS\n\nFISHTALKER TALK\n\nSHARK FRIENDS\n\n'Hmm,' she mumbled, before randomly prodding the screen.\n\nAnother menu popped up, this time with a selection of Shark Friends photos.\n\nMaritza groaned and clicked the first one.\n\nJAWELLA'S SHARK, LOCATED IN THE HAMMERHEAD\n\nHEIGHTS REGION OF THE HIDDEN LAGOON, CURRENTLY\n\nA POPULAR RESTAURANT.\n\nVARIETY: MEGALODON.\n\nSPEED: FAST.\n\nBITE SCOPE: IMPRESSIVE.\n\nMOOD: JOYFUL.\n\nSPECIAL TALENTS: IS ALSO A RESTAURANT.\n\n'Perfect,' Maritza said, clicking on it. A live picture flashed up of the Jawella's shark swimming slowly between the tall rock towers of Hammerhead Heights. It was covered in fairy lights and had a sign that flashed _JAWELLA'S_. Maritza smiled and pressed the 'CALL SHARK' button.\n\n'Rescued by a restaurant,' she said with a nod. 'Gronnyupple will love that.'\n27\n\nThe Right Ticket\n\n'CARNIVAL!' mermaids roared as they swam up and over the grand rubbery gates. Clouds of ink wafted past like mist on a spooky day as the bright lights of the Beak de Beak carnival danced across the murky waters.\n\n'TICKETS!' came a shout through the deafening sound of drumming octopuses. Up ahead, Beattie could see a mermaid in a huge headdress supervising a line of jellyfish stamping tickets. And at the very front, holding out a ticket smugly, was Rose Rockweed.\n\n'Unbelievable,' Beattie said.\n\n'You can use your magic to make us some tickets,' Mimi said to her. Then she turned to Meri. 'Beattie is a water witch, but a special one like Maritza \u2013 they can make magic.'\n\n'Wow, that's incredibly cool, Beattie,' Meri said. 'We need to get inside and fast. Can you do it?'\n\nThey floated in front of Beattie, staring. She wished Steve was here \u2013 he would explain how rubbish she was at magic.\n\n'You can do it,' Paris said kindly.\n\nBeattie squeezed her eyes shut and imagined some tickets for the Beak de Beak carnival. She held her hand out and...\n\n'Beatts, these are tickets to a Best Beak contest,' Zelda said, inspecting them. 'In Rainbow Landing.'\n\n'Sorry,' Beattie said. 'I was thinking about Steve. I'll try again.'\n\nShe felt the soft squidginess of seaweed in her hand.\n\n'Nope,' Meri said. 'These tickets are for a clam-car and beak-cleaning service in Podley.'\n\n'FISHELBOWS!' Beattie yelled (it was a very rude word).\n\n'Wait, I have an idea,' Paris said, plucking the useless tickets from Beattie's grasp and handing them one each. 'Make sure you let me stamp them.' She shook her necklace and morphed into a jellyfish.\n\n'GET BACK TO WORK!' the mermaid with the huge headdress roared at her.\n\nThey all wiggled with excitement as Paris the jellyfish wafted off towards the ticket line.\n\nAs they got close to the front Beattie could hear her heart pounding in her chest \u2013 or perhaps it was the octopus drummers.\n\nParis took one look at their tickets and zapped them with her stinger.\n\n'Genius,' Beattie whispered as they swam in.\n\n'Now what?' Zelda said, resisting the urge to jump on one of the rides.\n\nMeri pointed at an ornate carousel. It had all sorts of sea creatures whirling around at speed as mermaids cackled and cheered and practically fell off it.\n\n'The ice creature is in the carousel,' Meri said with an awkward cough. 'In the golden dolphin specifically.'\n\n'Great,' Zelda said. 'We'll break it open.'\n\n'The carousel's golden dolphin seat always has a rider on it \u2013 it's what mermaids come to this carnival for. Dolphins don't swim in these waters, so none of them have seen one. That's why it's the most popular thing in the whole place!'\n\n'Why didn't Maritza just hide them at the back of her wardrobe or in a trusty fish, like a normal mermaid?' Beattie said as Paris the jellyfish joined them.\n\n'BACK TO WORK!' a mermaid cried, grabbing Paris the jellyfish and flinging her into an arcade stall.\n\n'THROW THE SHELL HAT ON THE JELLYFISH AND WIN BOTH THE SHELL AND THE JELLYFISH!' the mermaid at the stall shouted.\n\nA mesmerised kid mermaid chucked a giant shell, which landed squarely on Paris's jellyfish head.\n\n'Right,' Beattie said, 'everyone remain calm. Zelda, go and get Paris back. Mimi, Meri and I will get the ice creature.'\n\n' _Mimi, Meri and IIIII_ ,' Mimi sang unhelpfully.\n\nThey waded through the forest of tentacles as the crowds of mermaids grew thicker and thicker.\n\nBeattie got as close to the carousel as she could. The golden dolphin seat sailed past her eyeline, taunting her.\n\n'Mimi,' she said firmly. 'Fin-fu it.'\n\n'But I'm only supposed to use fin-fu for good.'\n\n'THIS _IS_ FOR GOOD!' Beattie said.\n\n'But I'll be destroying a special carousel,' Mimi said, clearly in turmoil.\n\n'They can fix it quickly,' Meri said encouragingly. 'Have you seen how many tentacles the mermaids have around here?! It'll take them no time at all!'\n\nMimi slowly closed her eyes and flicked her finger. The beautiful golden dolphin seat crumbled, pinging the mermaid riding it all the way out of the Beak de Beak carnival.\n\n'He'll be _fine_ ,' Beattie said. She looked into the crowds, wondering where Gulper and Rose were.\n\n'Beattie,' Meri said, 'the ice creature... it's\u2014'\n\nBeattie scrunched up her fists, expecting the news to be bad.\n\n'HERE!' Meri cried.\n\nBeattie stared into the crumpled seat \u2013 curled up in the middle was a tiny ice octopus.\n\nMermaids began to crowd around, whispering.\n\n'NOTHING TO SEE HERE!' Beattie called out. 'JUST A STANDARD MALFUNCTION. VERY TYPICAL OF DOLPHIN-SHAPED THINGS. IT'LL BE FIXED SOON.'\n\nGrubs leapt from Mimi's shoulder and wriggled under the ice creature until it was on his back.\n\n'Did we just do what I think we did?!' Beattie said, spotting Gulper and Rose, who were still waiting to get on the ride.\n\n'WE DID IT!' Meri roared. She composed herself quickly. If spy school had taught her anything it was that if you celebrate a successful mission, the bad guys will take that moment to swoop in and...\n\n'LEAVE US ALONE!' Beattie cried as Gulper and Rose appeared above them.\n\nMimi fin-fu chopped them and sent them flying into the carousel.\n\nBeattie felt a strange tingling in her tail as all around them the water began to bubble.\n\n'Ooh, the water witch is getting angry,' Gulper laughed. 'Look at her hair!'\n\n'It's turned red,' Mimi said with a smile.\n\n'What?' Beattie looked around for a mirror.\n\n'And now it's going a bit yellow,' Meri said.\n\n'Oh, it's just hair,' Beattie said. She raised Grubs in the air. 'Who cares when we have Grubs and the ice\u2014'\n\n' _She skates, she glides, she moves from side to side. She's big and she's trouble_ ,' a voice sang _._\n\nA seal lolloped past and knocked Grubs out of Beattie's hand.\n\n' _Yeah... she's Blubble._ '\n\nBeattie stared in disbelief as Grubs landed in Rose Rockweed's hand. She ripped the ice creature off the poor axolotl and threw him back to Beattie.\n\n'HOW DID THAT SEAL GET IN HERE?!' Beattie roared. 'This is all Zelda's fault for pretending to be Gillica in the first place!'\n\n'Got her!' Zelda said cheerily as she emerged from the crowd with Paris the jellyfish firmly in her grasp. 'Wait \u2013 what did I miss?'\n28\n\nParis Rants and Raves about Gulper and Rose\n\n* _JELLYFISH SILENCE_ *\n29\n\nSteve and the Dolphin Balloon\n\nThe rainbow-patterned dolphin balloon glided through the air off the south coast of Australia.\n\nThe hot sun was beating down on it, casting a rainbow glow across the water.\n\nSteve had been vomiting the whole way there.\n\nThe dolphin made to dive again and...\n\n_BLEURGH!_\n\nThey hit the water. When the balloon and basket rose back up into the air, fifty-seven seahorses were covered from head to tail in vomit \u2013 and SCREAMING.\n\n'STOP TIMING IT SO YOU THROW UP RIGHT BEFORE WE GO UNDER!' Stella the seahorse shouted at Steve.\n\nThe dolphin began to dive again.\n\n'NO! STEVE... DON'T!'\n\n_BLEURGH!_\n\n' _HOW_ MUCH LONGER?' one of the seahorses wailed.\n\nThe dolphin dived again.\n\n_BLEURGH!_\n\nSteve wiped his nose on the edge of the basket and slumped over, exhausted. That's when he saw it, just up ahead \u2013 island upon island covered in rainbow flags.\n\nAnd suddenly he felt much better.\n30\n\nJawella's to the Rescue (Sort of)\n\nThe Amberberg mermaids heard the Jawella's shark before they saw it.\n\n'Do you hear that swishing? And clinking?'\n\nMermaids drinking foam shakes in delicate amber teacups stopped and looked around.\n\n'It sounds like plates rattling around in...'\n\n'A SHARK!'\n\nThe Jawella's shark came speeding into view and lurched forward, mouth open and \u2013\n\n_Ping_\n\n\u2013 bounced delicately off the building it had been instructed to free Gronnyupple from. It floated on the spot, mouth wide and awaiting further instructions.\n\n'Oooh,' the Amberberg mermaids said, swimming inside the shark's mouth and sitting down for lunch.\n\n'This is nice,' one of them said, picking up a menu.\n\n'What are chomp chops?'\n\nBack in Frostopia, Maritza Mist emerged from her ice mist shower with an icicle towel wrapped around her head. She stared into the clamshell compact and frowned. 'The shark didn't complete its mission? Now, how did that happen?'\n\nShe closed the compact. She'd have to think of something else.\n31\n\nSteve Arrives at Rainbow Landing\n\nThe dolphin dived deeper and deeper until the basket of seahorses spied a glistening coral reef, which rearranged itself in a perfect rainbow as they passed. The purple section of it flipped up and the dolphin swam through.\n\n'Welcome!' boomed a cheery voice. 'TO RAINBOW LANDING!'\n\nSteve spun around, taking it all in. Everything sparkled and mermaids with shiny rainbow tails stopped and waved from their colourful gardens as the dolphin balloon glided by.\n\nOn a twinkling bridge separating one rainbow-coloured seahorse house from another, an angelfish played the violin.\n\nThey pulled into a rocky area full of resting balloon dolphins. A family with eight tiny seahorse babies floated into the basket next to them.\n\n'TO PINKLY FOR HOLIDAY!' the littlest one cheered.\n\n'Steve?' came a voice. 'Long-lost Steve!'\n\nBefore Steve knew what was happening, two seahorses were hugging him and crying.\n\n'It's us!' the one that was doing the most crying said. 'It's Stevie and Steven \u2013 Mum and Dad! We've found you, after all these years!'\n\nSteve was overwhelmed. It was more of a shock than the time Zelda had convinced him to hug an electric eel. He missed them all a little bit, even Zelda, but he pushed those thoughts aside. After all, they had left him alone in the Vampire Rocks spa.\n\nHe was with his real family now.\n\nThe seahorses guided Steve in a daze towards a spectacular house covered in candy-cane striped turrets with rainbows beaming from the windows.\n\n'I'm home,' Steve said with a smile. And he meant it.\n32\n\nSo Many Mermaid Towns, So Little Time\n\nEvery time Beattie returned to Realm Reach she liked it less and less.\n\nMaritza greeted them with a big smile. 'Any luck, dears?'\n\nBeattie stared down at her tail.\n\n'We got it,' Meri said. 'But then we lost it.'\n\n'Well, I have news on Gronnyupple,' Maritza said.\n\nBeattie looked up excitedly. 'She's back?'\n\n'No,' Maritza said, still smiling. 'I sent a shark to rescue her, but it didn't work like it was supposed to.'\n\nMeri took the clamshell compact and reviewed the footage. 'It pinged off the building? A megalodon shark? And then it just floated there, being a _restaurant_?'\n\nMaritza nodded.\n\nMeri looked worried.\n\n'What is it?' Beattie asked.\n\n'It's just... strange,' Meri said. 'The shark didn't complete its mission \u2013 it was meant to bite into the building so Gronnyupple could swim out and into its mouth. And then she'd be carried home.'\n\n'Maybe the shark wasn't hungry,' Mimi said.\n\n'No,' Meri said quietly. 'It would take some serious magic to stop a spy-mermaid shark attack. And it wasn't Gulper and Rose because they were in Octopolli, not Amberberg. Someone else must've done it... maybe there's a water witch in Amberberg that we don't know about, who is more powerful than we could ever imagine. But that's almost impossible!'\n\n'Well, that's a mystery for another time,' Maritza said. 'Gulper and Rose just dived through the portal to Jewelport \u2013 and you're running out of time.'\n\nThe chase was on.\n\nThe mermaids headed to Jewelport, a trading town filled with gems and trunks of treasure.\n\n'I'VE GOT GEMS IN MY EYES!' Zelda shouted, as Gulper and Rose burst from a pile of jewels with an ice turtle.\n\nParis tried to whack them with a treasure chest, but it just flattened Rose's face.\n\n'LOO LAN'T LOP US,' she said with a squint smirk as she squashed her face back together.\n\n'I don't think I can swim much further,' Beattie wheezed.\n\n'Come on, Beattie!' Zelda roared. 'Back to Realm Reach. Pick up the pace!'\n\nEvery time they passed the multicoloured portal to Rainbow Landing, Beattie wanted nothing more than to dive inside and find Steve.\n\n'There's no ice creature in Rainbow Landing, Beattie,' Zelda would remind her. 'Gulper and Rose got that one years ago.'\n\nThey tore through the different realms, losing with ease. They swam fast in the Hidden Lagoon, in case their friends from home spotted them, and chomped their way through Jellywiches and another defeat in the Crocodile Kingdom.\n\nThey dived through the Hermit Grove portal and into darkness. It was night-time there, but Gulper and Rose emerged from the gloom with grim smiles pasted on their faces.\n\nGulper was holding an ice crab.\n\n'Ugh,' Meri said as a family of hermit crabs attached themselves to her ears.\n\nNext was the only portal as icy cold as Frostopia \u2013 Beluga Town!\n\n'WELL, HELLO THERE, FRIENDS. WELCOME TO BELUGA TOWN!' chirped a mermaid called Moira Wet. 'The exclamation mark is actually in the town's name, because we're so excited you're here!'\n\nThey'd arrived at the Shrimpol station on the edge of town. A huge swirl of ice stretched beneath them as far as the eye could see. Mermaids proceeded through the swirl, some free-swimming, others on sledges pulled by beluga whales. Moira Wet, Beluga Town!'s most loved mermaid, was guiding beluga whales into bays and sticking Shrimpol pumps in their mouths.\n\n'Right,' Beattie said, hurrying towards the swirl, 'let's finally get an ice creature!'\n\nMeri stopped in front of a whale so ancient it was barely moving.\n\n'It's not\u2014' Beattie said quietly.\n\nThe whale did an almighty burp and spat out Gulper and Rose.\n\n'HIDDEN IN THE ANCIENT WHALE!' Rose bellowed, holding up a tiny ice whale.\n\n'OH COME _ON_!' Beattie cried.\n\n'See you in Amberberg,' Gulper said with a smirk.\n\nZelda turned to Beattie. 'We're going to Mume \u2013 it's time we tried my plan. There are only two ice creatures left \u2013 in Amberberg and Mume. This is our last chance to outsmart them.'\n\nTHE FREEZY\n\nFrostopia's Coolest Magazine!\n\nBURGUIN INTERVIEW!\n\nToday we interviewed Patricia von Polar of Floe, who has found the rarest Burguin of all \u2013 the Ice Burguin! Ice Burguin looks like Original Burguin but is made of ice.\n\n**_Reporter:_** Patricia, you found the rarest Burguin of them all \u2013 how do you feel?\n\n**_Patricia:_** I LOST IT!\n\n**_Reporter:_** That's so great \u2013 wait, what?\n\n**_Patricia:_** It slipped out of my hand when a seal slimed me \u2013 almost certainly on purpose!\n\n**_Reporter:_** If you could send a message to the seal who did this, what would you say?\n\n**_Patricia:_** I don't speak seal.\n\n**_Reporter:_** Thank you, Patricia von Polar!\n33\n\nSeahorse Life\n\nSteve swam down Rainbow Landing's main street, with its multicoloured shops and fabulously dressed mermaids. He bought a tiny Sparkle Shake in a colourful cup and a rainbow-patterned cone top. And he paid for them in smiles.\n\n'Look!' seahorses cried as he passed. 'It's Long-lost Steve!'\n\nHis parents zoomed by, wearing their fancy shell jetpacks, which made them go extremely fast. 'Don't be home too late,' they called. 'We're having a party for you!'\n\nHe picked up a copy of _Seahorse Sunrise_ , the daily read, and swam off into a gorgeous kelp forest to find some peace and quiet.\n\nLONG-LOST STEVE NOW JUST STEVE!  \nSteve, the son of rock stars Stevie and Steven, has been found! There's a party tonight at their mansion to celebrate his return. The dress code is FABULOUSLY FOUND: wear fabulous things you've found while being fabulous.  \nEvery guest is asked to bring a smile and dye their hair their favourite colour of the rainbow!  \nWe love you, Steve!\n\nSteve rocked back and forth on a kelp strand like he was in a hammock.\n\n'Who needs mermaids who leave you alone in leech spas when you can have rainbows and lots of seahorses with names really similar to yours in your life,' he said to himself.\n\nThings were great.\n\nBeattie had probably saved the world by now.\n\nEven if she hadn't, someone else probably had.\n\nA sparkly piece of kelp floated down to his eye level and began tap-dancing.\n\nHe smiled and carried on reading. But then something occurred to him. Slowly, he lowered the magazine.\n\n'You're not... Razzle Dazzle kelp, are you?'\n\nThe sparkly kelp bowed and two other sparkly bits of kelp joined in the dancing.\n\nSteve remembered the conversation they'd had with Maritza Mist about reversing magic. He could see Beattie, nodding sadly. And he could see Maritza explaining it all.\n\n' _How can you tell if it's Razzle Dazzle kelp?_ ' came the voice from his memory in a long drawn-out groan.\n\n'Because it's always showing off!' Steve cried, staring at the dancing kelp in awe. 'This is a MIRACLE! I'M A MIRACLE! Did you decide to come out of hiding for me because we're both miracles? I _AM_ A MIRACLE! I can't wait to tell Zelda...'\n34\n\nTo Mume\n\nThe mermaid Kingdom of Mume, hidden near the shores of Japan, was the most beautiful in the world.\n\nBeattie and the others arrived at its centre, next to a peaceful garden of tiny coral clumps pruned to perfection. In the distance they could see bright lights and billboards.\n\n'Look at the billboards,' Mimi said. 'It's Clippee!'\n\nClippee was Mimi's favourite cartoon character \u2013 a lobster in a dress, with an evil arch nemesis called O.\n\nIt made Beattie miss Steve even more \u2013 Clippee was his favourite too.\n\nMermaids with exquisitely painted tails glided past coral houses with seaweed walls. They smiled politely at the strange mermaids as they went by.\n\nBeattie's hair turned blue.\n\n'Your hair is doing that thing again,' Paris said, giving her a nudge.\n\nBeattie pulled at her plait.\n\n'No time to worry about your hair,' Zelda said. 'Meri, lead us to the ice creature!' The Mume ice creature was in a vending machine, in the area of Mume known as Vending Machine City.\n\n'You're joking,' Zelda said as they floated at the edge of a sprawling mass of machines. Things pinged and jingled, neon lights flashed and mermaids swam from place to place, putting money in slots and pulling strange items out.\n\n'Here's one for merkitten accessories,' Beattie said.\n\nMimi fin-fued it slyly and a large shell fell out. Beattie cracked it open and found a tiny merkitten hat in the style of a narwhal.\n\n'So this is the vending machine to use if you want to dress up your merkitten as a narwhal,' Zelda said. 'That's not weird.'\n\n'IT'S BRILLIANT!' Paris said, swimming somersaults down the street over mermaids dressed in the most fabulous costumes. Beattie swam past a mermaid wearing strings of glowing jellyfish-shaped earrings teamed with a glowing Clippee T-shirt.\n\n'How are we ever going to find _the_ vending machine?' Beattie despaired. 'There are thousands \u2013 maybe even millions of them.'\n\n'Well, Vending Machine City is always growing,' Meri said. 'And every year they add more and more. The one Maritza hid the ice creature in is the first axolotl vending machine she ever created. But she made sure this one was broken \u2013 so no one could get the ice creature. It would sit among the millions of other vending machines, hidden forever.'\n\n'It's actually quite a good hiding place,' Zelda said. 'And at least it's fun \u2013 Gulper and Rose are probably having to squeeze through a shark's bottom to get their ice creature.'\n\n'Found it,' Mimi said casually.\n\nBeattie raced over and inspected what Mimi was pointing at. It was an axolotl vending machine with a little _MM_ scratched on the back. The sides were decorated with pictures of mermaids with crazy curly hairstyles and sea-sprayed fringes holding up cute axolotls in tiny costumes \u2013 fish, turtle, shark, dolphin \u2013 you name it, the axolotls were wearing it. Behind the glass only the costumes remained; the axolotls were long gone.\n\n'According to the list, the ice creature is a blobfish,' Meri said, trying to stick her head inside the flap at the bottom. 'It's probably hidden at the back, behind the costumes. Any idea how we get it out?'\n\nZelda swam up and plucked Grubs from Mimi's hand. 'Go get 'em, Grubs,' she said, feeding him into the flap.\n\nThey waited.\n\nThere was a rattling noise as Grubs disappeared through the costumes.\n\nBeattie and Meri exchanged hopeful smiles. Mimi tapped her tail.\n\n_FOUR HOURS LATER_\n\nZelda smooshed her face against the glass. 'He probably thinks we've put him away...'\n\nMeri fed Lady Wriggles into the flap to fetch him, and they waited again.\n\n'Oh no,' she said, slapping a hand to her forehead. 'Lady Wriggles has one weakness\u2014'\n\n'She's a fragile jellyfish?' Zelda guessed.\n\n'No,' Meri said. 'She's got a costume OBSESSION.'\n\nThey turned to see Lady Wriggles wriggling into a lobster costume.\n\n'Mimi,' Beattie said. 'Fin-fu it open.'\n\nMimi flexed her tail and slapped it hard against the vending machine. It rolled over, but it didn't break.\n\n'It's indestructible,' she said.\n\nIt was Paris's turn next. She shook her necklace and morphed into a shark.\n\nParis opened her jaws wide and charged, but her teeth barely made a dent.\n\n'What did she make this out of?!' Meri cried, flicking it with her tail. 'Ow. Something solid.'\n\nBeattie scrunched up her fist and closed her eyes.\n\n'Oooh,' Zelda said, nudging Paris the shark. 'Beattie's going to do magic.'\n\nBeattie knew she could make things happen if she tried really hard. She rolled up her tail and dug her nails into her palms. 'Come on, come on, come on,' she muttered.\n\nThere was a _CLANG_ and a _crunch_. Something was happening!\n\nBeattie opened her eyes and grinned \u2013 little axolotl costumes floated around her head. 'I did it!' she shouted. 'I broke the vending machine open!' She tried to raise her arms in triumph, but found she couldn't.\n\n'Beattie!' Zelda cried. 'You're _in_ the vending machine!'\n35\n\nWearing a Vending Machine\n\nBeattie moved at a sea-slug pace, the very tip of her tail poking out of the bottom of the vending machine. Grubs sat inside with her, on her head, and Lady Wriggles wrapped herself around her arm. Mermaids turned and stared as Beattie inched past them, making her way back to the portal. If they'd looked closely, they'd have seen she was clutching an ice blobfish in her hand.\n\n'Almost there, Beatts,' Zelda said encouragingly as the others helped lift the vending machine from behind to stop it sinking.\n\n'Are you holding the vending machine up to help me?' Beattie shouted, making bubbles form on the glass.\n\n'No, no, you're doing it _all_ yourself, you powerful water witch, you,' Meri lied, giving Zelda a wink.\n\nThey wriggled their way through the crowds of mermaids. Beattie watched as they pulled out seahorse-shaped hair accessories and cool T-shirts from various vending machines.\n\n'Look,' Mimi said, hitting a machine with her tail as she passed. 'This one has Burguin T-shirts, and _Chops & Slinky_, the detective croc and eel duo cartoon from the Crocodile Kingdom. They have _everything_.'\n\n'But _you_ have the one thing _we_ want,' came a familiar voice.\n\nBeattie tried to spin round. 'Did I just hear\u2014?'\n\n'ROSE!' Rose screeched as her face popped up in front of the glass, making Beattie scream.\n\nZelda launched into attack mode, but Rose was too powerful. Her entire fist inflated and she knocked Zelda into a vending machine filled with dancing lobsters.\n\nMeri was sent flying into an arguably better vending machine filled with delicious frozen Icetipple, and Paris the shark was slapped away as if she were a tiny fish.\n\nMimi held her finger up in a threatening fin-fu pose, but soon found herself tumbling on top of her twin.\n\n'BYE-BYE,' said Gulper with a smirk as Rose advanced on Beattie.\n\n'GIVE ME THE BLOBFISH!' she yelled.\n\nGulper held the vending machine still as Rose reached in and wrenched the ice blobfish from Beattie's grasp.\n\n'DO SOMETHING, GRUBS!' Beattie cried, so Grubs had a nap.\n\nBeattie tried to swim, but without the others holding up the vending machine she hit the seabed with a _clang_.\n\n'Now all we have to do is get the Amberberg ice creature,' Rose said.\n\n'WHAT?' Beattie said. 'But we came here because you were going to Amberberg!' She almost felt better \u2013 if they hadn't got the ice creature there, then they still had one more chance to stop them.\n\nRose and Gulper exchanged looks and sniggered. 'We just said that, you silly mermaid. We're immortal. We have _forever_ to find the ice creatures, so the only thing we have to do right now is stop you from getting any of them and slowing us down. You're not chasing us, Beattie Shelton. _We're_ chasing _you_.'\n\nBeattie stared wide-eyed as they disappeared into the distance.\n\nThe others came racing over, with various vending machine accessories in their hair. Meri crunched on an Icetipple icicle.\n\n'It's over,' Beattie said from inside the vending machine.\n\nParis put both palms on the glass and fixed Beattie with a sincere stare. 'Come on, Beattie. It's not over until it's over.'\n\nThe vending machine made a whirring noise and a tiny dolphin outfit for an axolotl floated out.\n\nThey stared at it in silence.\n\n'I suppose we might as well try,' Beattie finally said.\n\nParis was right. And she knew if Steve was still with them, he'd say the same thing.\n\nLittle did she know, Steve was headed her way.\n36\n\nBack on the Dolphin Balloon\n\n'BLEURGGGGHHH' went Steve as the dolphin balloon dived towards the water once more.\n\nOnly this time it was a swimming pool, because Steve had convinced the dolphin to take a shortcut across Australia.\n\n'THERE'S SEAHORSE VOMIT IN THE SWIMMING POOL!' someone shouted as the dolphin flopped on, leaping from pool to pool.\n\n'CRIPES!' Steve cried as they leapt off a cliff and hit the ocean, charging full pelt towards the icy waters of Frostopia.\n\n'BE THERE SOON TO SAVE THE DAY, BEATTIE!' he called out as the Razzle Dazzle kelp jiggled about in the dolphin's mouth.\n37\n\nGoblin Sharks GALORE and an Unexpected Face\n\nThe whole city of Amberberg glowed in gorgeous hues of orange and gold. It was a mismatched place, filled with spectacular amber palaces and lumpy goblin sharks.\n\n'It's this way,' Meri said. 'She hid the last ice creature \u2013 a starfish \u2013 in the old palace vaults.'\n\n' _They're_ in a rush,' Zelda said as they watched hundreds of mermaids trying to squeeze through the grand doors of the largest building in town \u2013 the old palace.\n\n'I hear they have a star witness who can prove Gronnyupple stole the Amber Flubby,' a mermaid said to his friend as they swam past. 'I hope she goes to Viperview forever.'\n\n'Gronnyupple!' Beattie said, spinning round, still in her vending machine.\n\nZelda turned to Meri and mouthed, 'Amber Flubby?'\n\n'It's a jewel,' Meri said. 'The mermaids of Amberberg believe it's very powerful. It's not. But it is very expensive.'\n\n'But what do we do?' Beattie asked. 'We can't have Gronnyupple sent to Viperview. And we can't let Gulper and Rose get the final ice creature.'\n\n'We're in luck,' Meri said with a smile. 'Because Gronnyupple and the final ice creature are in the same building.'\n\nInside the old palace's courtroom, everyone was trying to swim closer to the front, hoping to get a look at the devious Gronnyupple in the flesh.\n\n'She's a _criminal mastermind_ ,' one mermaid said.\n\n'No she's not!' Zelda laughed.\n\nParis watched mermaids putting their handbags in the mouths of goblin sharks.\n\n'What are they doing?'\n\n'That's where they leave their bags. They collect them afterwards,' Meri explained.\n\n'Mermaids are so weird,' Paris said.\n\n'Humans are weirder,' Zelda said. 'I've seen a toaster.'\n\nThe crowd fell silent, apart from the mermaids next to Beattie's vending machine, who were giggling at her.\n\n'It's fashion,' Beattie said. 'You'll all be wearing it soon.'\n\n'Can I go now?' came a familiar voice from the front of the courtroom.\n\nZelda, Mimi and Paris craned their necks and spotted Gronnyupple. She looked smaller in the big watery room with the big important mermaids.\n\nZelda scrunched up her fists. 'We're about to see who framed Gronnyupple.'\n\n'We call to the stand our STAR WITNESS, who saw Gronnyupple hide the Amber Flubby. This hero not only saw the criminal and reported her to us, but they also found the Amber Flubby and returned it to our city.' The judge mermaid held up a glob of amber.\n\nThe crowd went wild.\n\n'Silence, please. Allow our star witness to take the stand.'\n\nA seahorse glided into the room.\n\nMeri's mouth fell open.\n\nBeattie and the others stared at her.\n\n'What is it, Meri?' Zelda whispered.\n\n'I know that seahorse,' Meri said in disbelief. 'It's Sabrina.'\n38\n\nSabrina Explains Herself\n\nThe trial played out in slow motion for Meri as she tried to understand why Sabrina would have framed Gronnyupple.\n\n'A seahorse runs Fortress Bay spy school,' Zelda said. It was the third time she'd said it, but she just couldn't quite get over it.\n\n'Just my year,' Meri managed to say. She went to pull out her spy clamshell compact, but then realised she'd left it with Maritza to arrange Gronnyupple's rescue.\n\nMeri gasped again.\n\n'The shark,' she said. 'Remember when Maritza sent the Jawella's shark to save Gronnyupple and it failed? I thought someone had done some meddling magic, but I never considered the other way a mission could be stopped \u2013 and that's if someone at Fortress Bay called it off at the last minute. The only one who has access to my clamshell compact and the authority to call things off is Sabrina.'\n\nThey all looked at the tiny seahorse.\n\n'But why would she frame Gronnyupple?' Zelda said. 'And then stop her from being rescued?'\n\n'Maybe she wanted to get her out of the way,' Meri said. 'Gronnyupple is a powerful water witch, after all.'\n\n'But technically not as powerful as Beattie, who can _make_ magic,' Zelda said. 'Why didn't they frame her too?'\n\n'If there's no good reason why Sabrina would frame Gronnyupple, we must be missing something,' Beattie said.\n\n'MERI!' came a tiny seahorse shout. 'What are you doing in Amberberg?'\n\nMeri tried to hide under the crowds, but it was no good \u2013 they parted, revealing her lying face down in the sand.\n\nShe looked up at Sabrina.\n\n'Oh... hi,' she said sheepishly.\n\n'Meri,' Sabrina hissed. 'Get out of here now \u2013 and take your friends and the... vending machine... with you.'\n\nBeattie growled at Sabrina.\n\n'Get them out of here. I've got this under control,' Sabrina snapped.\n\n'Oh I bet you do, traitor!' Meri cried. 'You framed Gronnyupple, and she's my friends' friend.'\n\n'You're a spy, Meri,' Sabrina whispered. 'You can't have friends.'\n\n'I'm her friend,' Mimi said, putting her arm around Meri.\n\nZelda slunk off undetected to the vaults to get the ice creature.\n\nBeattie swivelled in her vending machine and spotted two mermaids in large amber hats heading that way too.\n\n'IT'S GULPER AND ROSE!' Beattie cried, trying to point but hitting the vending machine glass instead. 'IN BIG AMBER HATS!'\n\nParis looked around. Almost everyone was wearing big amber hats.\n\n'They're closing in on Zelda,' Beattie said. 'Paris, go! Get the ice creature and stop them.'\n\nParis nodded and patted the vending machine, making more axolotl costumes fall out.\n\n'I'm on it!'\n\nShe shook the necklace and morphed into a jellyfish. Beattie watched helplessly from her vending machine as her friend wafted through the crowds.\n\n'GRONNYUPPLE WONDRA CRAB,' the judge said.\n\n'OH LOOK!' Gronnyupple cheered, spotting Paris the jellyfish. 'It's my friend Paris!'\n\n'That's a jellyfish,' the judge said flatly. 'And I'm afraid, Gronnyupple, we have decided you are GUILTY.'\n\n'What?!' Beattie shouted.\n\nSabrina shot off.\n\n'GET HER!' Beattie yelled. Mimi and Meri lifted Beattie and wobbled outside \u2013 and suddenly saw a familiar restaurant.\n\n'The Jawella's shark,' Mimi said, bowing grandly. 'Good day, sir.'\n\nSabrina stared at Mimi.\n\n'She's the weird one,' Meri whispered. 'Incredible fin-fu skills though.'\n\nThe thick amber door behind them flew open, revealing Gronnyupple.\n\n'Beattie!' Gronnyupple cheered, staring at the vending machine. 'You've got a new look! It's very... rectangular!'\n\nThe Jawella's shark opened its mouth wide.\n\nGronnyupple looked at Beattie and Mimi and then Meri.\n\n'Oh hello,' she said. 'You're a new one!'\n\nSabrina began to guide Gronnyupple into the shark.\n\n'YOU CAN'T DO THIS!' Meri cried. 'SHE'S MY FRIENDS' FRIEND!'\n\n'Meri,' Sabrina hissed. 'This isn't real \u2013 it's a set-up. It was all planned. I've been trying to get hold of a water witch to help out with lessons at the spy school, but they're so difficult to find. So I framed this one for a fake crime, and now I'm going to tell her she can either go to Viperview or teach a few classes.'\n\n'That's so sneaky,' Meri said.\n\nSabrina shrugged. 'I never said I was nice. This is how spies work.'\n\n'You could've just asked her,' Meri said. 'Gronnyupple is really kind, and if you got to know her, she'd definitely help you.'\n\n'No time for that emotional stuff,' Sabrina scoffed. 'This was more effective.'\n\n'Oooh,' Gronnyupple said from inside Jawella's. 'It's a restaurant in here!'\n\nThe shark took off, soaring high over the Amberberg buildings.\n\n'Meri, some things are beyond your spy grade,' Sabrina said. 'You need to stop meddling. It's time to come back to spy school with me.'\n\n'But we're not finished yet,' Beattie said, as Mimi took Meri's hand and held it tightly.\n\n'We still have to stop Gulper and Rose,' Mimi said. 'And Meri is helping us.'\n\nSabrina rolled her eyes. 'They. Are. In. Viperview. Locked away! Meri, stop spreading such nonsense!'\n\n'But Gulper and Rose never went back to Viperview!' Meri cried. 'They used a doubling potion to double themselves \u2013 that's why Gulper said \"FISH EYE!\" It's a side effect of the potion!'\n\n'It's true,' Beattie said. 'I've used that potion myself.'\n\n'Gulper and Rose are collecting ice creatures,' Meri blurted out, scared to say the next bit. 'And they have the ice narwhal, Sabrina. I was late checking it one day, and when I got there, Gulper and Rose had stolen it. I lost the ice narwhal.'\n\nMeri's tail suddenly felt lighter \u2013 it was such a relief to admit her big mistake. She wished she'd done it sooner, maybe then things wouldn't be so bad now.\n\n'But I saw the ice narwhal after you left. It was there, safe in the lock-up,' Sabrina said.\n\nMeri shifted awkwardly.\n\n'I made Lady Wriggles make a fake one. She was just being loyal to me.'\n\nSabrina floated on the spot, her mouth hanging open. 'Well, this is a disaster.'\n\n'It's all under control,' Beattie said. 'We have a plan.'\n\n'I don't know how to say this without sounding mean,' Sabrina said, 'but you're telling me everything is under control while trapped inside a vending machine.'\n\n'I GOT IT!' Zelda roared, swimming straight into the vending machine and knocking herself out. A tiny ice starfish floated down and landed on the bit of Beattie's tail hanging out of the machine.\n\n'WE DID IT!' Beattie cheered, but Gulper and Rose weren't finished with them yet.\n\nTHE FREEZY\n\nFrostopia's Coolest Magazine!\n\nBURGUIN INTERVIEW!\n\nToday we interviewed Kiko Frostwater of Slushville, who has found the rare and elusive Upside-Down Burguin! Upside-Down Burguin looks like Original Burguin, but due to a factory malfunction, his head is sticking out of the bottom of the burger bun and his flippers are sticking out of the top.\n\n**_Reporter:_** Kiko, you found Upside-Down Burguin \u2013 what is the first thing you did when you realised it was a special collector's item?\n\n**_Kiko:_** Took it apart and stuck it back together the right way.\n\n**_Reporter:_** So... now it's not the rare and special Upside-Down Burguin, it's just an Original Burguin?\n\n**_Kiko:_** I think so!\n\n**_Reporter:_** But you can get Original Burguins everywhere, there was only one Upside-Down Burguin in the whole world!\n\n**_Kiko:_** And I fixed him.\n\n**_Reporter:_** Thank you, Kiko...\n39\n\nShowdown\n\nSabrina screamed as Gulper and Rose tried to grab the ice starfish.\n\n'STOP THAT!' Beattie cried, curling her tail around the tiny creature.\n\nRose rolled her eyes and flung the vending machine up high, where it smashed into the amber spires, splitting in two.\n\n'I'M FREE!' Beattie bellowed as she tumbled out with Grubs and Lady Wriggles.\n\nShe tore the ice creature from Rose's grasp and chucked it at Zelda, who caught it and dived through the portal to Realm Reach.\n\nEveryone followed, fighting to squeeze through the portal first.\n\n'Sorry I doubted you!' Sabrina called faintly to Meri.\n\nMeri smiled and dived into the portal.\n\n'You look weird without the vending machine,' Mimi said to Beattie as they also tumbled back through to icy waters. 'I'd got used to you wearing it.'\n\n'GIVE US THE ICE CREATURE!' Gulper yelled.\n\nZelda threw it to Paris.\n\n'NEVER!' Paris cried, as it began to crack in her hand.\n\n'Keep hold of it!' Beattie shouted. 'Don't let go!'\n\nRose growled and hurtled towards Paris, who held the ice starfish behind her back. But Rose wasn't after the ice creature. Before anyone could stop her, she ripped the necklace from Paris's neck and smashed it against the ice. She cackled madly as what remained of it sank to the bottom of the sea.\n\n'NO!' Beattie screamed. 'She's morphing back into a human!'\n\nParis's tail stretched and began forming into legs and feet. She puffed out her cheeks and looked up in panic \u2013 there was nothing but solid ice above her.\n\n'DON'T WORRY, A MIRACLE IS COMING!' came a familiar voice as a seahorse whizzed past Beattie's nose and halted in front of Paris.\n\n'Steve's back?' Beattie said in astonishment.\n\nParis smiled, her face pale and her eyes slowly beginning to close.\n\n'I'LL SAVE HER!' Steve shouted, blowing urgently into her ear.\n\n'That's not her _mouth_ , Steve,' Zelda said, hurrying over.\n\n'Oh,' Steve said. 'Is that not how you do it?'\n\nBeattie grabbed the necklace from the seabed. She could feel her tail getting hot. Her hair began to change colour, from blue to pink to green to orange to black to red. She summoned her powers, holding the necklace up as high as she could, thinking only of Paris. She couldn't stop her hands from shaking.\n\n'How's it going, dears?' Maritza Mist said as she arrived on the scene. 'Oh, not well.'\n\nThe tiny sea creatures that had fallen off the necklace reappeared around Beattie's colour-changing hair.\n\n'Wow,' Maritza whispered to no one. 'She's good.'\n\nBeattie held on tightly to the necklace and focused all her energy. Her tail felt like it was on fire. Her fingers tingled as if there was magic trapped inside them.\n\n'I'LL FIX IT!' she screamed as the necklace re-formed itself. She flopped over, exhausted.\n\n'Paris!' Meri cried, grabbing the necklace and securing it around her neck.\n\nShe held Paris's unmoving body in her arms.\n\n'Paris?' she whispered.\n\n'NOOOOO!' Steve wailed. 'NOT PARIS!'\n\nBut then Paris's feet began to wriggle.\n\nThe knee-high socks she was wearing started to unravel.\n\n'IT'S WORKING!' Meri cried. 'BEATTIE, YOU DID IT!'\n\nParis's eyes snapped open and she took a deep watery breath. She was a mermaid again!\n\n'Nice tail,' Rose said mockingly. 'But I have something nicer. I have the ice starfish, and that makes ALL OF THEM.'\n\nShe started placing them on her tail, one by one. With each ice creature she added, she grew frostier and frostier.\n\n'It would be funny if she turned into one of those human snowmen, wouldn't it?' Zelda joked.\n\nBeattie shot her a look.\n\n'OK. Not the right time for snowman jokes,' Zelda said.\n\n'We always knew we'd get out of Viperview and finish what we started,' Rose boasted to Maritza. 'Did you think you could stop us with some amateur water witches and _normal_ mermaids?'\n\nParis charged at Rose, but she batted her away like she was a shrimp.\n\nBeattie tried too, but found herself crumpled in a heap next to Paris.\n\n'HELLO THERE!' Gronnyupple's voice called over the confusion. Up above, a shark circled. 'I got the Jawella's shark to drop me off here! I'm going to teach a class at Fortress Bay Spy Sch\u2014'\n\nShe stopped when she saw Gulper and Rose.\n\n' _She skates, she slides_ ,' came a voice from nearby.\n\nGronnyupple grinned as the Blubble seal swam past. 'I found the seal! What a day!'\n\n'Come _here_ ,' Zelda hissed at Gronnyupple.\n\nTogether, Beattie, Mimi, Zelda, Maritza Mist, Paris, Meri and Gronnyupple lined up to face Rose.\n\n'Now what?' Meri asked.\n\n'We start a band?' Zelda joked.\n\nSteve interrupted, the stream of Razzle Dazzle kelp curled around his tail. 'Beattie, I have so many questions about what your hair is doing, but no time right now. I found this \u2013 it's the Razzle Dazzle kelp Maritza said can reverse spells. You know, make Gulper and Rose _not_ immortal any more.'\n\n'Are you sure it's Razzle Dazzle kelp?' Beattie whispered excitedly.\n\nRose laughed. 'WE'RE GOING TO RULE THE WORLD!' She fixed her eyes on Beattie. 'And no magic mermaid will survive. Well, we'll keep Maritza as a servant, so she can make potions for us. But we'll get rid of the rest.'\n\nMaritza sighed. 'I never should have made that immortality paste.'\n\nRose's face was frosty now, and she had only a few ice creatures left to add before her tail was complete. 'I'LL PROBABLY MAKE THEM STAMP PICTURES OF MY FACE ON ALL CLAM CARS TOO. OR MAYBE I'LL CRUSH ALL THE CLAM CARS WITH MY ALL-POWERFUL FISTS!'\n\nMaritza spotted the Razzle Dazzle kelp and her eyes lit up. 'Gulper and Rose need to hold it,' she mouthed. 'And then they need to say the magic word \u2013 _cigamdiamrem_.'\n\n'What?!' Beattie mouthed back.\n\n'It's \"mermaid magic\" backwards,' Maritza whispered.\n\nShe watched as Rose continued to attach the ice creatures to her tail. The others looked to Beattie for directions, but Beattie didn't know what to do.\n\n'I'm willing to sing,' Steve whispered. 'Would opera be helpful?'\n\n'Leave this to me,' Zelda said, pulling at her waistcoat. 'Hey, Rose!'\n\n'Yes, Zelda,' Rose said with a sigh.\n\n'Can you remind me what happens when you've added all those little ice creatures to your tail?'\n\n'THEN I BECOME ALL-POWERFUL AND THE MERMAID WORLD IS MINE!'\n\n'So what about you?' Zelda said to Gulper. 'You get nothing? You did half the work!'\n\nGulper looked down at the ice creatures he was handing Rose. 'Wait, why don't I get to be all-powerful?'\n\n' _Because_ ,' Rose spat, 'only one mermaid can wear all the ice creatures. So only _I_ can become all powerful. You get to hang out with me, that's all you get.'\n\n'But _I_ could wear all the ice creatures,' Gulper said.\n\nRose laughed. 'Don't be RIDICULOUS!'\n\nZelda flashed Beattie a smug smile.\n\n'What's so ridiculous about it?' Gulper said, his voice sharp.\n\n'Oh come on, Gulper,' Rose said. 'It's _you_.'\n\n'She may not be magic,' Mimi said, 'but my twin is _excellent_ at causing trouble.'\n\nBeattie's hair turned pink.\n\n'Your hair is doing that thing again,' Mimi said.\n\n'Now what?' Meri asked.\n\n'I WOULD BE SO MUCH BETTER AT RULING THE WORLD!' Gulper roared at Rose.\n\n'NO YOU WOULDN'T!' Rose scoffed. 'AND ANYWAY, MY FACE WILL LOOK SO MUCH BETTER ON ALL THE SEAWEED FLAGS I'M GOING TO MAKE \u2013 FOR EVERY MERMAID CITY IN THE WORLD!'\n\n'WELL AT LEAST THAT'LL BE _TERRIFYING_!' Gulper said.\n\n'YEAH!' Rose cheered before stopping. 'Wait,' she said. 'DID YOU JUST MAKE FUN OF MY FACE?'\n\n'At least they don't have that new magical kelp that would allow them _both_ to become all-powerful,' Beattie said loudly.\n\nGulper swam fast towards her and snatched it.\n\n'Oh no,' Beattie said, winking excitedly at Zelda. 'What a disaster.'\n\n'Aha!' Gulper said, swimming eagerly over to Rose. 'This will split the magic.'\n\n'But I don't want to split the magic!' Rose said, grabbing the other end of the kelp.\n\nThey pulled and pulled until it snapped in two.\n\n'It smells weird, what did you say it was?' Rose asked, sniffing it.\n\n'Cigamdiamrem,' Beattie said with a smile.\n\n'Cigamdiamrem?' Gulper and Rose both said at once.\n\n'Is that even a word?' Rose said as she tried to throw it away. But she couldn't \u2013 it was stuck to her fingers.\n\n'NO!' Gulper cried, realising his mistake. He shook his hand furiously, but the kelp wouldn't budge.\n\nIn a snap, the pair grew older and older, morphing and twisting until they disappeared with a _pop_.\n\n'IT WORKED!' Beattie cried.\n\nParis picked up the ice narwhal and held it firmly in her hands. It began to crack.\n\n'Come on,' Beattie whispered, willing it to work.\n\nIt continued to crack, shaking in Paris's hand, until it made a popping sound and exploded in a puff of snowflakes.\n\nThey all watched as the sparkle of magic within the other ice creatures faded. The magic was finally broken.\n\n'NO WAY! WE DID IT!' Beattie roared as Meri, Zelda, Mimi and Gronnyupple swam in victory circles around Paris, cheering and slapping their tails together.\n\nSteve and Sabrina eyed each other suspiciously.\n\n'Are you Long-lost Steve?' she asked.\n\n'Sort of,' Steve said.\n\n'I'm your sister,' Sabrina said. 'But I'm a really important spy, so I won't be home much.'\n\n'Steve!' Beattie said, grinning madly. 'You still have a little bit of Razzle Dazzle kelp stuck in your tail.' She held it up.\n\nThere was a cough. She looked over to see Maritza floating quietly.\n\n'I was wondering,' Maritza Mist said, eyeing the kelp in Beattie's hand, 'if we could do one final reverse spell.'\n\nBeattie smiled. 'Sure! On what?'\n\n'On me, dear,' Maritza said. 'It's time for me to go.'\n40\n\nGoodbye, Maritza Mist\n\nBeattie's face fell. 'You want to be gone forever?'\n\nThe seal in the Blubble costume floated behind Beattie, playing the jingle,\n\n_She skates, she glides, she moves from side to side._\n\n_She's big and she's trouble,_\n\n_Yeah... she's Blubble._\n\n'Unbelievable,' Beattie muttered as the seal disappeared into the Mume portal.\n\n'I've been here too long,' Maritza said. 'No one wants to live forever.'\n\n'But you're SEVEN,' Zelda said.\n\nMaritza Mist laughed. 'I only _look_ seven. I'm actually four thousand and eight years old. And four days, if you're counting, which I have been, believe me.'\n\n'Aw no,' Mimi said. 'We just missed your birthday!'\n\n'You can free me, Beattie, after all these years!' Maritza said.\n\n'But I don't want you to go,' Beattie said sadly. 'We've only just got to know you. And who is going to send out your magic catalogue orders?'\n\n'Why, Gronnyupple, of course!'\n\n'Seriously?' Zelda whispered. 'Have you seen her use magic?!'\n\nMaritza Mist smiled. 'There's no one who loves magic more than Gronnyupple, plus she wouldn't be alone. She'd have Beattie to make the magic for her.'\n\n'Me?' Beattie spluttered.\n\n'Dear, you are much better at making magic than I ever was. And look, your hair is changing with your moods, like the true chosen one.'\n\n'I thought she might have a disease,' Mimi confided to Maritza.\n\n'That always happens when you start making magic,' Maritza said. 'It calms down after a while.'\n\n'Wait,' Zelda said. 'Beattie's hair has gone a deep blue. What does that mean?'\n\n'Sad,' Beattie said as a tear rolled down her face. She took the last remaining bit of kelp and held it in her hand.\n\nGronnyupple hugged Maritza Mist tightly. 'Thank you for the wonderful water-witch potions, and for always delivering my catalogue orders on time, apart from that last one.'\n\nMaritza beamed at her. 'And thank you for being my favourite customer, and for knowing there was trouble ahead and acting so quickly. Your instincts are good, Gronnyupple. Look after Grubs for me.'\n\nSteve, Mimi, Zelda and Paris took it in turns to hug Maritza goodbye.\n\n'And goodbye, Beattie,' she said. 'Don't be doing any more spells near vending machines.'\n\n'It really has been one of the greatest things in my life to have met you,' Beattie said, slowly handing Maritza Mist the kelp.\n\nMaritza took the kelp and held it tightly. 'Goodbye, dears.'\n\nBeattie closed her eyes as Maritza whispered the magic word.\n\nIt all happened so fast. First, Maritza morphed into a teenager!\n\n'Ooooh!' Steve said.\n\nThen a mermaid in her thirties.\n\n'Aaaaah!' Steve said.\n\nThen a mermaid in her sixties, then eighties, then nineties.\n\n'Yeulch,' Steve said.\n\n'Oh stop being such an ageist little seahorse,' Zelda said.\n\nOld Maritza Mist smiled and waved one final time.\n\nAnd then she was gone.\n41\n\nBack to Underwater Igloo Avenue\n\nThe mermaids floated sadly back to Maritza Mist's house.\n\n'I think I'll come here and run the catalogue from Maritza's house when I'm not visiting Fortress Bay,' Gronnyupple said. 'I'm a big fan of the frost. And the fashion. And as long as Realm Reach is there, I can easily get back to the Crocodile Kingdom to stock up on Seahorse Surprise! Hey, Beattie,' she rambled on excitedly. 'Maybe you could create a potion that makes Seahorse Surprise.'\n\n'Mmm, maybe,' Beattie said as she swam on ahead, though she wasn't really listening.\n\n'Oh, Beattie,' Steve said, opening up the false teeth. 'Thank you for getting me this wonderful rainbow-coloured pillow!'\n\nBeattie peered in to see the rainbowfish lying flat in the false teeth.\n\n'That's not a pillow,' Zelda said, relishing the opportunity to annoy him. 'That's your new ROOMMATE.'\n42\n\nJust Mermaids\n\nBeattie, Mimi and Zelda were back in the twins' bedroom in Periwinkle Palace in the Hidden Lagoon. Zelda was practising her shockey moves and Mimi was slumped on the bed with a foam shake as she caught up on all the episodes of _Clippee_ that she'd missed.\n\nBeattie rested her elbows on the window ledge and looked out on the Swirlyshell streets below. Mermaids with all kinds of tails floated past: black and white whale ones from Frostopia, crocodile prints from the Crocodile Kingdom, Mume tails and Jewelport and Beluga Town! ones.\n\nThe whole mermaid world was talking about Beattie and her friends who had taken on Gulper and Rose and won. News about Realm Reach had spread, and now the realms were united once more. Mermaids could travel the world, seeing the sights and making new friends.\n\n'No point in being ice-olated,' Zelda had joked.\n\nThe little clamshell compact on the twins' desk began buzzing. Zelda picked it up and threw it across the room. 'Beatts, it's for you!'\n\nThere on the shell screen was Gronnyupple's face. Grubs was asleep on her head, snoring loudly.\n\n'How are the new spells coming along, Beattie? I'm getting ready to send the next catalogue out.'\n\n'Almost there!' Beattie said. 'It's going to be the best one yet.'\n\n'Have you made the spell that makes you melt?'\n\nBeattie shook her head. 'Too dangerous.'\n\n'I wouldn't like to be slushy forever, that's true,' Gronnyupple said.\n\nThe clamshell compact began buzzing again. 'Oh, I've got another call.'\n\nParis appeared on the screen next to Gronnyupple.\n\n'It's Paris!' Beattie called over to Zelda and Mimi.\n\n'How is it being a human again?' Zelda asked.\n\nParis rolled her eyes. 'It's hideous! I have so many toes! AND KNEES!'\n\nThey all burst out laughing.\n\nThe clam compact jiggled and Meri popped up on the screen too.\n\n'I have five minutes between spy classes,' she said as Lady Wriggles floated past in the background. 'Sabrina said my method of spying is so good I'm going to be given my first official mission!'\n\n'Cool!' Zelda said.\n\n'We miss you!' Beattie added.\n\n'I'm going to come and see you soon, if I can sneak away,' Paris said. 'Is everything going well with the catalogue?'\n\n'Oh yes!' Gronnyupple said. 'Tell her about your new thing, Beattie. The special one.'\n\nBeattie held up some rings with different-shaped gems \u2013 a dolphin, a shell, a pufferfish... 'Well, I took the magic that makes my hair change colour and I put it into these rings. I call them _mood rings_. When you wear them, they'll show your mood in a colour.'\n\n'ISN'T IT AMAZING?!' Gronnyupple roared, spitting Seahorse Surprise all over the screen.\n\nZelda put hers on and it flashed all the colours of the rainbow. Mimi's did too.\n\n'What does that mean?' Zelda asked.\n\n'Happy,' Beattie said with a smile.\n\nSteve sailed in through the window, carrying a multicoloured seahorse mansion. Zelda's ring instantly flicked to yellow.\n\n'What's yellow?' Gronnyupple shouted through the screen.\n\n'Annoyed,' Beattie whispered.\n\n'Oh, thank you _so much_ , Mimi, for saying we could keep our mansion here until we find a permanent location,' Steve said, dumping it on top of Zelda's shockey gear.\n\nZelda growled.\n\nStevie and Steven, Steve's parents, came swishing through the window, blowing into shells and making an irritating honking noise.\n\nZelda covered her ears.\n\n'They're rock stars,' Steve said proudly.\n\n'So you're moving here for good?' Mimi asked Stevie and Steven.\n\n'For half the year,' Steven said. 'We'll do six months in Rainbow Landing and the other half here \u2013 bringing our mansion with us!'\n\nAnother little seahorse swam into the room. 'And this,' Steve said, 'is my big sister, Sabrina.'\n\n'Hi, Sabrina,' everyone chimed.\n\nThey knew exactly who Sabrina was, but they couldn't let Steve's parents know, because of the top-secret spy thing.\n\n'You really should visit Rainbow Landing,' Stevie said, rubbing her nose against Mimi's. 'They'd love your rainbow tail!'\n\nBeattie's mum, the travel writer Belinda Shelton, sailed past the window with her notepad. 'Mimi, are you ready?'\n\n'Oh yes,' Mimi said, switching off the _Clippee_ cartoon and swimming out of the window.\n\n'We're off to Hammerhead Heights to interview the Jawella's shark for _Clamzine_. Mimi is going to translate for me.'\n\n'Ugh, I wish I was there!' Paris cried from the clamshell compact.\n\n'Where are you right now, Paris? We could pick you up!' Beattie's mum offered. 'It's so easy now with Realm Reach!'\n\nBeattie snapped the compact closed. 'She's really far away.'\n\n'Let's all meet at Jawella's for dinner tonight then!' Beattie's mum called back.\n\nBeattie opened the clamshell compact to a smiling Paris, Gronnyupple and Meri.\n\n'Why don't you tell your mum that Paris is a human on land?' Gronnyupple whispered. 'Your mum is cool, and she's not anti-human.'\n\n'Because I don't want to listen to all the questions she'll have about toes!' Beattie said. 'I'll send you each a mood ring. Look out for a crabagram!'\n\nAnd with that they said goodbye.\n\nZelda flopped on to her bed. 'Beattie, given everything we've achieved, I think it's officially official.'\n\n'What is?' Beattie said.\n\nZelda grinned. 'We are _bad_.'\nThere is a world where witches live, deep down below the sink pipes...\n\nRead on for a sneak peek at the first book in Sib\u00e9al Pounder's Witch Wars series\n\nAVAILABLE NOW!\n\nDown the Plughole\n\nIt would have been very difficult to spot Fran the fairy on the day this story begins. Her dress may have been puffy, her hair may have been huge, but she was barely the size of a small potato.\n\nFran was slowly sidestepping across a garden lawn, holding a large, limp leaf in front of her. She didn't want the owner of the garden to see her because Miss Heks was a terrible old woman with a grim face and size eleven shoes. If she had seen Fran she would've squashed her immediately.\n\nFran and her leaf were on a mission. There was something very important in the shed at the bottom of Miss Heks's garden. That something was a girl called Tiga Whicabim.\n\n'You!' Tiga said, pointing at a slug that was sliding its way across an old stone sink. 'You will be the star of my show! You will play the role of Beryl, an ambitious dancer with severe hiccups.'\n\nTiga had been in the shed for hours. The evil Miss Heks had been her guardian for as long as Tiga could remember and she had quickly learned to keep out of her way. If she didn't, the old bat would make her sew up the holes in her disgusting, scratchy dresses. Or she would force Tiga to run up and down the garden in her gigantic, ugly shoes, bellowing things like 'FASTER!' and 'OH, DID YOU TRIP?' from the kitchen window.\n\nTiga shone a torch on the slug.\n\n'You are going to be the best actor the world has ever seen!' she cried.\n\nFran sighed when she saw that.\n\nNot because she'd finally found Tiga, after a long and perilous journey that had almost ended with her being eaten by a dog.\n\nNo, the reason Fran sighed was because she loved a bit of acting!\n\nDespite her small size, Fran was a big deal in the world of show business. Everyone called her Fran the Fabulous Fairy (a name she had made up for herself). She had hosted many award-winning TV shows like _Cooking for Tiny People_ and _The Squashed and the Swatted_ and she'd played the lead role in _Glittery Sue_ \u2013 a tragic drama about a small lady called Sue who got some glitter in her hair and couldn't get it out again.\n\n'An actor you say!' Fran said, making Tiga jump.\n\nTiga stared, mouth open, at the small person that marched across the shed and \u2013 very ungracefully, and with much grunting \u2013 climbed up the leg of her trusty old rocking chair.\n\nFran stretched out a hand.\n\n'Very delighted to meet you, Tiga! Now, it's pronounced _Teega_ , isn't it? That's what I thought! I'm very good at names and absolutely everything else. I'm Fran the Fabulous Fairy. But you can call me Fran. Or Fabulous. BUT NEVER JUST FAIRY. I hate that.'\n\nTiga, understandably, assumed she had gone mad. Or at the very least fallen asleep.\n\nShe squinted at the little thing with big hair and then looked to the slug for reassurance, but it was sliding its way across the floor as if it knew exactly who Fran was, and was trying to escape.\n\n'I don't think,' Fran said, pointing at the slug, 'that she should be acting in the lead role. She is slimy and not paying much attention.'\n\nFran wiggled a foot and a beehive of hair just like her own appeared on top of the slug's head.\n\n'Much, much, _much_ better,' she said.\n\nTiga panicked \u2013 the slug had _hair_! Not any old hair, a beehive of perfectly groomed hair! It was a split-second reaction, but with a flick of her hand she batted the fairy clean off the rocking chair.\n\nFran wobbled from left to right and tried to steady herself.\n\n'Did you just _swat_ me?' she snapped. 'The ultimate insult!'\n\nTiga tried to avoid eye contact and instead looked at the slug. She couldn't be sure, but it looked a lot like it was shaking its head at her.\n\n'WITCHES ARE NOT ALLOWED TO SWAT FAIRIES. IT IS THE LAW,' Fran ranted.\n\n'I'm sorry!' Tiga cried. 'I didn't think you were real \u2013 I thought you were just my imagination! You don't need to call me a witch.'\n\n'Yes I do,' said Fran, floating in front of Tiga with her hands on her hips. 'Because you are one.'\n\n'I am one what?' Tiga asked.\n\n'One witch,' said Fran as she twirled in the air, got her puffy dress caught in her wings and crash-landed on the floor.\n\n'BRAAAAT!' came a bellow from across the garden. 'Time to leave the shed. Your dinner is ready!'\n\nTiga glanced nervously out of the window. 'If you are real, although I'm still not convinced you are, you'd better leave now. Miss Heks is a terrible old woman and she will do horrible, nasty, ear-pinching things to you.'\n\nFran ignored her and went back to twirling in the air. 'What are you having for dinner?'\n\n'Cheese water,' Tiga said with a sigh. 'It's only ever cheese water.'\n\nFran thought about this for a moment. 'And how do you make this cheese water?'\n\n'You find a bit of mouldy old cheese and you put it in some boiling water,' said Tiga, looking ill.\n\nFran swooped down lower and landed on the sink. 'Well, I'm afraid we don't have cheese water in Ritzy City \u2013 it's mostly cakes.'\n\nTiga stared at the fairy. 'Ritzy where?'\n\n' _Riiiitzzzzzy Ciiiiity!_ ' Fran cheered, waving her hands in the air.\n\nTiga shrugged. 'Never heard of it.'\n\n'But you're a witch,' said Fran.\n\n'I am not a witch!' Tiga cried.\n\n'You SO are!'\n\n'I am not!'\n\n'Definitely are,' said Fran, nodding her head. 'Even your name says so.'\n\nAnd with that she flicked her tiny finger, sending a burst of glittery dust sailing across the room.\n\n_TIGA WHICABIM_, the dust read.\n\nThen it began to wobble and rearrange itself into something new.\n\n_I AM A BIG WITCH_.\n\n'You've cheated somehow,' Tiga mumbled, moving the dust letters about in the air. Most people would've believed Fran by this point, but Tiga wasn't used to magic and fun and insane fairies. So, despite this very convincing evidence that she might just be a witch, Tiga still walked towards the door. Towards the cheese water.\n\n'TIGA!' bellowed Miss Heks. 'YOUR CHEESE WATER HAS REACHED BOILING POINT.'\n\n'Cheese water,' Fran chuckled. 'Wait! Where are you going, Tiga?'\n\n'To eat dinner,' said Tiga. 'Bye, Fabulous Fairy Fran. It was lovely to meet you.'\n\nFran raised a hand in the air. 'Wait! _What?_ You're not coming with me to Ritzy City, a place of wonder and absolutely no cheese?'\n\nTiga paused. Even if it was a mad dream, it was better than cheese water. She turned on her heel and walked back towards Fran.\n\nFran squealed and squeaked and did somersaults in the air.\n\n'WHAT'S GOING ON IN THERE? I KNOW YOU CAN HEAR ME, YOU LITTLE MAGGOT!' Miss Heks shouted.\n\nTiga could see Miss Heks stomping her way towards the shed.\n\n'Quick!' Fran cried. 'We must go to Ritzy City right now!'\n\n' _How?_ ' Tiga cried, frantically looking around the shed for an escape route.\n\n'Down the sink pipes, of course,' Fran said as she shot through the air and straight down the plughole.\n\n'Come on, Tiga!' her shrill little voice echoed from somewhere inside the sink.\n\nTiga leaned over the stone sink and stared down the plughole.\n\nThere was nothing down there. No light. And certainly no city, that was for sure.\n\nThe door to the shed flew open and splinters of old wood went soaring through the air.\n\n'WHAT IS GOING ON?' Miss Heks bellowed.\n\n'NOW!' Fran yelled.\n\nTiga wiggled a finger in the plughole.\n\n_This is nonsense_ , she thought, just as she disappeared.\nFind out where it all began\n\n'Dive into fin-a tastic, madcap adventure that's funnier than being tickled giant by a octopus!'\n\nAward-winning author David Solomons on Bad Mermaids\n\nAVAILABLE NOW!\n\nBeattie, Zelda, Mimi and Steve the talking seahorse will return in the next fabulously fishy book in the series.\n\nCOMING SOON!\nBLOOMSBURY CHILDREN'S BOOKS\n\nBloomsbury Publishing Plc\n\n50 Bedford Square, London WC1B 3DP, UK\n\nThis electronic edition published in 2019 by Bloomsbury Publishing Plc\n\nBLOOMSBURY, BLOOMSBURY CHILDREN'S BOOKS and the Diana logo are trademarks of Bloomsbury Publishing Plc\n\nFirst published in Great Britain in 2019 by Bloomsbury Publishing Plc\n\nText copyright \u00a9 Sib\u00e9al Pounder, 2019\n\nIllustrations copyright \u00a9 Jason Cockcroft, 2019\n\nSib\u00e9al Pounder and Jason Cockcroft have asserted their rights under the Copyright, Designs and Patents Act, 1988, to be identified as Author and Illustrator of this work\n\nAll rights reserved   \nYou may not copy, distribute, transmit, reproduce or otherwise make available this publication (or any part of it) in any form, or by any means (including without limitation electronic, digital, optical, mechanical, photocopying, printing, recording or otherwise), without the prior written permission of the publisher. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages.\n\nA catalogue record for this book is available from the British Library\n\nISBN: 978-1-4088-7716-6 (PB)   \nISBN: 978-1-4088-7717-3 (eBook)\n\nTo find out more about our authors and their books please visit www.bloomsbury.com where you will find extracts, author interviews and details of forthcoming events, and to be the first to hear about latest releases and special offers, sign up for our newsletters.\n\n#\n\n  1. Cover\n  2. Title Page\n  3. Dedication\n  4. Contents\n  5. Last Time in Mermaid World...\n  6. 1. Meri Pebble, the Spy Mermaid\n  7. 2. Zelda on Thick Ice\n  8. 3. Blubble the Ice-skating Seal\n  9. 4. The Magic of Mermaid Rockits\n  10. 5. Slushville Is So Slush\n  11. 6. Simoseal\n  12. 7. Meri Does Some Midnight Spying\n  13. 8. Underwater Igloo Avenue, Flurry Falls\n  14. 9. Meri Screams I AM MERI, a Lot\n  15. 10. Maritza Mist\n  16. 11. Multistorey Clam-Car Car Park (Say It Fast)\n  17. 12. Wartina's Necklace\n  18. 13. Gronnyupple in Trouble\n  19. 14. The Power of Paris\n  20. 15. Jelly and the Whale Bus\n  21. 16. Realm Reach\n  22. 17. The Glorious Pinkness of Pinkly\n  23. 18. How Do You Get an Axolotl Out of a Vending Machine, and Other Important Questions\n  24. 19. Meri and the Burguin Sale\n  25. 20. Vampire Rocks and the Ice Leech\n  26. 21. Long-Lost Steve\n  27. 22. Nightmare Dolphin\n  28. 23. Meri Makes Some Friends\n  29. 24. Slippery Octopolli\n  30. 25. Beak de Beak\n  31. 26. Maritza Sends a Shark\n  32. 27. The Right Ticket\n  33. 28. Paris Rants and Raves about Gulper and Rose\n  34. 29. Steve and the Dolphin Balloon\n  35. 30. Jawella's to the Rescue (Sort of)\n  36. 31. Steve Arrives at Rainbow Landing\n  37. 32. So Many Mermaid Towns, So Little Time\n  38. 33. Seahorse Life\n  39. 34. To Mume\n  40. 35. Wearing a Vending Machine\n  41. 36. Back on the Dolphin Balloon\n  42. 37. Goblin Sharks GALORE and an Unexpected Face\n  43. 38. Sabrina Explains Herself\n  44. 39. Showdown\n  45. 40. Goodbye, Maritza Mist\n  46. 41. Back to Underwater Igloo Avenue\n  47. 42. Just Mermaids\n  48. Witch Wars\n  49. 1. Down the Plughole\n  50. eCopyright\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n**Russia, Freaks and Foreigners**\n\n**Three Performance Texts**\n\n# Russia, Freaks and Foreigners\n\n**Three Performance Texts**\n\nJames MacDonald\n\n**_To Mother, Pop, Susan and Inna  \nFour of God's truly elect_**\n\nFirst Published in the UK in 2008 by  \nIntellect Books, The Mill, Parnall Road, Fishponds, Bristol, BS16 3JG, UK\n\nFirst published in the USA in 2008 by  \nIntellect Books, The University of Chicago Press, 1427 E. 60th Street, Chicago,  \nIL 60637, USA\n\nCopyright \u00a9 2008 Intellect Ltd\n\nAll rights reserved. No part of this publication may be reproduced,  \nstored in a retrieval system, or transmitted, in any form or by any means,  \nelectronic, mechanical, photocopying, recording, or otherwise, without  \nwritten permission.\n\nA catalogue record for this book is available from the British Library.\n\nCover Design: Gabriel Solomons  \nCopy Editor: Holly Spradling  \nTypesetting: Mac Style, Nafferton, E. Yorkshire\n\nISBN 978-1-84150-186-4\/EISBN 978-1-84150-229-8\n\nPrinted and bound by Gutenberg Press, Malta.\n\n## CONTENTS\n\nIntroduction\n\n**Part One: Texts**\n\nBread and Circus Freaks\n\nThe Sweetheart Zone\n\nEmigr\u00e9s\n\n**Part Two: Essays**\n\nGetting to Know James MacDonald\n\n_Peter Thomson_\n\nDirector's Notes\n\n_Martin Harvey_\n\nFreaks, Food and Fairy Tales: Confronting the Limits of Disability in _Bread and Circus Freaks_\n\n_Thomas Fahy_\n\nEpilogue\n\n_Su Elliott_\n\nContributors\n\n## INTRODUCTION\n\n_Russia, Freaks and Foreigners: Three Performance Texts_ does not constitute a trilogy in the formal sense, but several things serve to unite the texts and to justify their collection as a unified book. They are all texts whose first performances were directed by one man. Two of the three were written as part of an undergraduate module in interpretive acting, led by the same man. All three share thematic concerns with various forms of difference. Consummately, I hope, they provide the theatre reader with a dynamic experience of this complex and far-reaching subject \u2013 personally, socially, culturally.\n\nPerformance texts are primarily for performance, and their dissemination in book form necessitates a discussion of their performance characteristics. This is especially true of new texts (whose performance history is extremely short). I have been ideally served in this respect by having accompanying essays by colleagues of long standing who can discuss the texts in ways that newly published texts are not often discussed in print. Martin Harvey, who directed the performances, discusses the theatrical qualities of the texts; Peter Thomson has known my work for thirty years and gives a historical perspective on its genesis; Thomas Fahy has done extensive work on the literary\/dramatic nature of difference and brings this to bear in a detailed analysis of _Bread and Circus Freaks_ ; Su Elliott closes the volume with her reflections on what it was like to perform one of these texts. Their views were expressed independently, and it may well be that they even contradict one another in certain respects. But this is the essence of critical discourse, and any author is privileged indeed to have his\/her work focused in this way. Whatever they say, each contributor is a MacDonald expert as well as a distinguished practitioner, and their comments can only enhance the reader's isolated perusal of these texts.\n\nAs for my own view (for what it is worth), I have always sought to employ disability much as Dostoyevsky uses epilepsy, to comment on the human condition as a whole. Many writers have used deformity in this way (Rabelais, Gogol, Nathanael West, Carson McCullers), and I feel in kindred spirit with them. _Bread and Circus Freaks_ , for example, was intended to be a contemporary re-working of the Russian vaudeville, a comic play often centred on mistaken identity. As Peter Thomson has noted, questions of identity have rarely been absent from all my work, and Martin Harvey has given a partial glimpse into the reasons why. Transforming life into artefact, with regard to _Circus Freaks_ , I meant the title to refer to all the characters, as Thomas Fahy notes, and for each character to do a 'turn' in presenting her\/his take on society. Each is somehow an outsider, and each has something distinctive to say about society. Without their separateness, they would not be outsiders.\n\nI meant the play to be funny, and audiences have definite views not only on the subjects they accept as comic but also on the ways this comedy can be purveyed. Peter Thomson talks about this in relation to a 1984 play he helped produce in London. Peter is quite right that the agent who was helping me at the time had serious commercial misgivings about disability as a subject. She said she had had a terrible time trying to market _A Day in the Death of Joe Egg_ and that the characters in my play were even more depressing to watch. A _Times_ reviewer seemed to bear this out when he averred that _Whose Life Is It Anyway?_ offers a far better view of disability than my play. But inasmuch as Brian Clarke's play is really about euthanasia, its disability statement could almost be seen as fascistic. And I ear-wigged on two patrons in wheelchairs who said of my play's verisimilitude, 'He'd have to be disabled (in order) to know that.'\n\nPerhaps Peter is correct when he suggests that it takes direct knowledge of what he calls 'the other side' to fully appreciate depictions of it. But shying away from a subject (for whatever reason) is hardly the best way to respond to it, and theatre has always been in the vanguard when it comes to tackling difficult subjects. Tom Fahy has done splendid work in examining the whole history of the freak show. But outlawing such exhibitions may simply result in removing 'deformity' from the social agenda. One aim of the publication of all three texts is to place issues of deformity and dislocation squarely before the public in such a way that they must respond beyond polite detachment, and unrestrained laughter is a clear sign of honest engagement. In this way the performance of all three texts was an attempt to push back two boundaries \u2013 presenting the subject at all, and then challenging audiences to find the displays comic. Neither aim was carried out in a spirit of hostility toward the audience. I was not asking anyone to confront suppressed prejudices. But equally, failure to acknowledge a subject can be as cruel as actively promulgating it. For all the brutality of some of the characters' behaviour, they do not flinch from the subjects of deformity and dislocation, and their candour was what we attempted to bring before audiences.\n\nThis was certainly so in the case of Martin Harvey's production of _Bread and Circus Freaks_ , and so it has remained in our work with students on the other two texts in this volume, all of which has summoned a tremendous amount of courageous commitment from the performers. This may even exceed one author's fondest dream(s). Performers in the first play were willing to risk unpopularity on the progressive fringe to achieve something of devastating novelty, and then student performers have been eager to commit themselves to work that goes beyond progressive curriculum, involving workshop engagement with issues that have remained hidden for far too long. I cannot exaggerate the extent of this courage; indeed, I have no wish to in paying the fullest tribute to all the contributors to this volume (among whom I am happy to include Sam King, Roberta Mock and everyone else at Intellect). What you have done here is genuinely remarkable, as the life of this volume will clearly bear out. For this reason alone, I have high hopes (and unstinting gratitude) for it.\n\nJames MacDonald\n\n## PART ONE: TEXTS\n### BREAD AND CIRCUS FREAKS\n\n#### A One Act Vaudeville\n\nFirst performed at the Finborough Theatre, London, on 6 March 2002, under the direction of Martin Harvey and with the following cast:\n\nPANIA ANDREYEVNA | Su Elliott\n\n---|---\n\nINNA IGOROVNA | Leah Fells\n\nMARIANNA SELIGMAN | Leslie O'Hara\n\nOSIP PISHCHIK | Michael Bottle\n\nVOLKOV | Stephen Harvey\n\nDesign | Kamal Desei\n\nRussia Consultant | Inna Rodina\n\nIn the countryside surrounding Petersburg, winter.\n\nMarianna (Leslie O'Hara) and Pania (Su Elliott) struggle for control of Inna (Leah Fells). _Bread and Circus Freaks_ , Finborough Theatre, March 2002. Photo: Marilyn Kingwill.\n\n**Settlement No. 7, some 70 kilometres from Petersburg, a village impacted by snow and by circumstance \u2013 the remnants of a collective farm. A bread shop immediately after the morning delivery. PANIA ANDREYEVNA, 42 but looking 15 years older and very thin, supports INNA IGOROVNA through the back door. INNA is nearly twenty but looks five years younger. Most distinctively, though, she is heavily spastic from cerebral palsy. PANIA stations INNA on a chair and begins lifting the bread trays from the floor to the counter**.\n\nINNA _(After watching her for several seconds.)_ I wish I could help you.\n\nPANIA: You what? _(Preoccupied.)_ No...you can't do it.\n\nINNA: If we could maybe carry the trays \u2013\n\nPANIA: ...I know. We'd feel like millionaires. I'd feel like a millionaire anyway. I'd feel like I didn't need to work...and then I wouldn't employ you...and then you wouldn't work...You'd be an outcast, for people to take pity on or worse...because pity soon turns to contempt...and then you might starve...they'd look for your footprints in the snow...and then maybe some night \u2013 or maybe first light \u2013 someone would discover the corpse of an under-fed girl...half-eaten by wolves...and then they'd say, 'Wasn't that the waif that used to work for Pania Andreyevna? I'm sure of it.' And then they'd come looking for me...and charge me with murder after the fact...and all because I let you help me lift the trays. I wouldn't feel much like a millionaire then, would I? A million's not much good against a murder rap, is it? Especially if it's all an illusion. You can't pay a murder judge with it. 'If she felt she had a million, she could have showed more compassion to poor Inna Igorovna.' That's what they'd say...that's the verdict they'd bring in on me. It's worth a little inconvenience to be spared a fate like that, now wouldn't you say?\n\n**She's been working the while**.\n\nINNA: I wish I could help you.\n\nPANIA: Oh-ho, you're helping me. You think I'm running a charity ward? You're serving our hungry horde while I put my feet up in back. Maybe I'll even fall into the third degree of sleep... so that when you have a riot here \u2013 when demand far exceeds our supply \u2013 you won't be able to wake me. I won't know anything about it. What do you reckon to that? I'm throwing you in at the deep end, my Inka. I don't believe in charity.\n\nINNA: We don't have wolves here, do we?\n\nPANIA: I didn't say I was throwing you to the wolves \u2013 I said \u2013\n\nINNA: ...half-eaten by wolves...if you let me go. Don't you remember? Were you just trying to scare me?\n\nPANIA: I was trying to show you the nonsense you're talking \u2013 you could help me lift the trays. I can't do it \u2013 I don't see how on earth you think you can. _(Calculating.)_ One, two, three...eight, nine...nine. There isn't going to be enough brown. I told them last time to make it four dozen. They can't hear straight \u2013 they've given me four dozen white. Deaf bastards! Well, I thought I was doing you a favour, my darling, but now it looks like \u2013\n\nINNA: You've done me lots of favours, Pania Andreyevna. You've given me this job, for a start...\n\nPANIA: I mean I put up a notice the delivery was going to be late...just to give us time to ourselves.\n\nINNA: Do people really read notices? I don't think so.\n\nPANIA: ...well, with four dozen white loaves, and next-to-no brown...you'd better believe they'll do something constructive. Otherwise, they'll be taking our blood before midday.\n\nINNA: You're not serious.\n\nPANIA: How long have I been selling bread? You'd better believe they'll be asking for bread... and then taking our blood just as soon as we've run out. _(A beat.)_ Can you use a hunter?\n\nINNA: What's a hunter?\n\nPANIA: ...or maybe it'd be all right if you just point.\n\nINNA: I'm sure I can point, but what's \u2013 ?\n\nPANIA: A hunting rifle, of course, are you that naive?\n\nINNA: You mean you...fire on people?\n\nPANIA: Yeah, well, mainly I just point.\n\nINNA: And does it work?\n\nPANIA: If it didn't, I'd be shooting at them, wouldn't I? It works, all right. People round here are such cowards.\n\nINNA: So they wouldn't really take our blood. You were just pretend-\n\nPANIA: What are you talking about? They'd certainly try if the delivery was like it was today, and if I didn't have my hunter.\n\nINNA: My God...\n\nPANIA _(Overlapping.)_ : What do you think I got it for \u2013 effect? They'll try it with you \u2013 you'll see.\n\nINNA: I don't want to fire on people!\n\nPANIA: I'd better stay awake then.\n\nINNA: You never told me it was going to be dangerous.\n\nPANIA: What'd you think they were, friends? Furry bunny rabbits?\n\nINNA: All right, I admit, I thought one or two thieves, maybe.\n\nPANIA: Oh, you did, did you? There wouldn't be thieves round here, darling. We've grown up together, you know? We're just crazy, that's all. Especially when we're forced to eat white when they haven't brought brown. Then we're apt to grow a little irate, you know? And then the only thing that can calm us is a hunter, even if it's only pointed at us.\n\nINNA: My mother heard...I guess she saw it on television...The President made a big speech.\n\nPANIA: It couldn't have been on television.\n\nINNA: Oh, you know the one I mean?\n\nPANIA: One what?\n\nINNA: ...speech \u2013 the one he just made.\n\nPANIA: It wouldn't have made any difference, if he made it on television. Your mother couldn't have seen it.\n\nINNA: What makes you so sure?\n\nPANIA: ...we only have radio here...and the radio said the television tower burnt down.\n\nINNA: I don't think you \u2013\n\nPANIA: Are you simple as well as naive? I'm telling you straight \u2013 the whole place is in cinders. That's what the radio said.\n\nINNA: Well, yes, I know that...\n\nPANIA: Well, then, what are you \u2013 ?\n\nINNA: ...but only one tower. Not enough to stop broadcasting.\n\nPANIA: Your mother must have heard it on the radio, like everybody else.\n\n**INNA is vexed**.\n\nINNA: Anyway, the President issued the warning that Russia is steadily dying off. What do you say, d'you think it's true?\n\nPANIA: I know it \u2013 there's not enough brown.\n\nINNA: He means, she said, because older people are dying off faster than babies are being born.\n\nPANIA: Older people, I tell you, aren't the problem \u2013 it's that people producing the babies are dying off. Older people don't produce, do they. They're better dead ... in the ground...as manure.\n\nINNA: But who would you say are the others \u2013 soldiers and people like that?\n\nPANIA: Russians aren't getting their fibre... _(Shouting.)_...when all they'll deliver is mouldy white bread.\n\nINNA: It's not mouldy, is it?\n\nPANIA: It gets mouldy \u2013 nobody eats it. _(A beat.)_ You like soldiers, do you? You'd like to be raped by a soldier?\n\nINNA: No, you said...I thought soldiers must be who you meant...they'd all died.\n\nPANIA: Not in my lifetime. There's not enough of 'em. In the history books, maybe. Yes, in the history books about wars. Is that your schoolgirl fantasy, is it? Foreign wars with Russian soldiers raping the flesh off you? Sex isn't that good.\n\nINNA: That's not my fantasy, Madame Pania.\n\nPANIA: Ha \u2013 says you.\n\nINNA: It must be yours.\n\nPANIA: It's nobody's, that's the whole point.\n\nINNA: That's what the President says.\n\nPANIA: What does the President know that I don't...except where to find some brown bread? About sex, though, he can't tell me nothing. And what is this 'Madame Pania'? We've established I'm no millionaire.\n\nINNA: But I wanted to show I respect you.\n\nPANIA: Be a good girl then. Don't say 'Madame' just so you can give me an argument. I'd rather you were rude and then listened to some good advice.\n\nINNA: I listen.\n\nPANIA: You can't talk and listen.\n\nINNA: That's called 'discussion'.\n\nPANIA: It's called showing me no respect.\n\nINNA: All right. I'm sorry.\n\nPANIA: So you should be. _(A beat.)_ I don't hold discussions with people. It's just the way I'm made. God said at my birth, 'This one won't discuss anything with anyone. We can make sure she has a pretty face, strong hands. We'll give her other things to get by with in life. But when other people hold their discussions, it'll be like she's not in the room.' So \u2013 that's my personal message from God, that's my fate. I'm healthy enough, good and strong.\n\nINNA: You can produce babies.\n\nPANIA: Oh, yes, my insides still work. My mother had me when she was 47. There's only one thing wrong with birth in this village. You discuss things. Do you know what that is? Shall I tell you? The men who are still living are all past it. I'm wasting away.\n\nINNA: ...so the President's right.\n\n**There is a thumping against the door**.\n\nPANIA: Unless I want to produce with a goat.\n\nINNA: What was that?\n\nPANIA: The billygoat mounting the nanny.\n\nINNA: Oh, you're joking.\n\n**INNA crosses the space to get a better look at the goats from one of the two windows. Her walk is asymmetrical and slow**.\n\nSomebody should take a photograph \u2013 that's hilarious.\n\nPANIA: Maybe you should still be at school.\n\nINNA _(Still amused.)_ : Why's that? It's true \u2013 they're really making love.\n\nPANIA: You've seen dogs do it.\n\nINNA: Not that often...and somehow it's not so funny.\n\nPANIA: You should maybe study mathematics.\n\nINNA: Why? I know what's what \u2013 I'm not that innocent.\n\nPANIA: With mathematics on your mind, the sight of the goats might not be so funny.\n\nINNA: What's wrong with having a laugh?\n\nPANIA: There's no men, just like there's not enough brown. So what's the point of liking sex?\n\nINNA: No, you can't call that sex \u2013 it's just funny.\n\nPANIA _(A beat.)_ : Maybe you should study biology.\n\n**A pause**.\n\nINNA _(Trying to peer through the window.)_ : Who'd have thought they could do it in this cold?\n\nPANIA: They were forced to push the train for the final kilometre.\n\nINNA: Eh? What was that, Mad \u2013 Pania Andreyevna?\n\nPANIA: It doesn't matter in the least. Go on with your dreaming.\n\nINNA _(Turns back to PANIA.)_ : No, I'm sorry. What did you say?\n\nPANIA: Don't apologize, Inna. It gets on my nerves.\n\nINNA: But you said something I didn't hear. I want to know what it was. Unless you don't want to tell me. Did God say, 'You mustn't repeat yourself'?\n\nPANIA: Let's leave God up in heaven. I was talking about the bread train.\n\nINNA: What, and there being no brown?\n\nPANIA: Not enough brown, I said.\n\nINNA: Well, other than pointing your hunter \u2013\n\nPANIA: ...and the fact that you take no interest at all. I was only trying to tell you how the bread train had to be pushed the last bit.\n\nINNA: What do you mean, pushed? That's terrible...isn't it?\n\nPANIA: It's bloody hard work.\n\nINNA: What \u2013 you mean you did it, too?\n\nPANIA: All the women helped.\n\nINNA: What did the men do, stand and watch?\n\nPANIA: There were no men.\n\nINNA: So who did the driving?\n\nPANIA: What the hell does that matter?\n\nINNA: Well, if it was a man, it matters quite a lot, I'd have thought.\n\nPANIA: Who else would they get to drive the damn train? But where were the others to push it? That's what I'm saying. It took fifteen or twenty of us women.\n\nINNA: Well, yes, it would, fifteen or twenty at least. And I'll bet it took you an age and three quarters.\n\nPANIA: It was on a downward incline.\n\nINNA: Yes, all right, but even so...\n\nPANIA: You see what I'm trying to tell you now.\n\nINNA: ...and I see what the President meant.\n\nPANIA: And he doesn't help us. _(Sighs.)_ I'm sure he's a great man...he has a great plan for Russia and all that...\n\nINNA: You can't expect him to help you push the train!\n\nPANIA: Fool. I'm not talking about that.\n\nINNA _(A beat.)_ : He once lived in Petersburg, you know.\n\nPANIA: Well, I never met him.\n\nINNA: I didn't say you did.\n\nPANIA: Well, then, why bring it up? He once lived in Peters, so what? He should have helped push?\n\nINNA: Now you're making fun of me.\n\nPANIA: Then you can say I don't understand today's youth...that I must be past it. I'm good only for giving out not enough loaves. Our Lord's grace is no longer in me. All right. Let the hungry go on to the city. I want a good long rest.\n\nINNA: ...and let me look after the shop for you.\n\nPANIA: ...when there aren't any people.\n\nINNA: But there will be. You said so yourself.\n\nPANIA: Yeah, too many for one. They'll come in, see you just about standing there...defenceless... and they'll reckon they can exploit the situation. The pushier ones will take immediate advantage... they'll grab the bread without paying...\n\nINNA: But you said they're not thieves.\n\nPANIA: Today's youth have never seen really desperate people. They think they can cope with anything.\n\nINNA: Where'd you get the idea I ever said that?\n\nPANIA: What makes you think you can cope on your own? You're nothing but a kid, a crippled kid at that...living in fantasy land. That may be all right for kids...\n\nINNA: I wish you'd stop saying that.\n\nPANIA: ...but I have to cope in the real world, where trains have to be pushed and then the bread don't get delivered...\n\nINNA: Those sailors who were drowned weren't living in fantasy land...or the soldiers dying in Chechnya.\n\nPANIA: What are you bringing them up for?\n\nINNA: Aren't they just as much youth of today?\n\nPANIA: All right, but you're not dying in Chechnya, are you. What makes you think you can speak for them? Are you a sailor's widow?\n\nINNA: I was just saying what the President said.\n\nPANIA: ...you said he could help us.\n\nINNA: No. I just thought it was interesting he comes from \u2013\n\nPANIA: He's not there now, so what's 'interesting' about it? He's in Moscow sorting out other things. I wouldn't expect him to help us.\n\nINNA: Nor would I.\n\nPANIA: He's got to try to console the poor mothers, poor man. What can he do? He's not God.\n\nINNA: He says he'll look after the children.\n\nPANIA: All their lives? My father was a disabled war veteran they also promised to 'look after'. A blanket, two towels and the message, 'You're lucky you'll die in bed.' 'Look after.'\n\nINNA: And that was in Soviet times. They're saying it's that much worse now.\n\nPANIA: Who are you quoting now...who's 'they'?\n\nINNA: Newspapers, I guess, I don't \u2013\n\nPANIA: They're blasphemers, whoever they are. I don't want to hear you blaspheme. Your mother specifically asked me, she did: 'Look after my Inna, Pania Andreyevna. Make her useful for something. I have to find work in the city.' She's one of my oldest friends.\n\nINNA: Were you Pioneers together?\n\nPANIA: I never made friends in the Pioneers...bunch of hooligans.\n\nINNA: Who, the Pioneers?\n\nPANIA: Of course. _(A beat.)_ You were never a Pioneer, were you?\n\nINNA: They wouldn't let me join.\n\nPANIA: You had a lucky escape, let me tell you.\n\nINNA: I'd have been asked to perform feats of skill, which I couldn't...or go rock climbing.\n\nPANIA: They'd have asked all cadets to burn holes in your hands \u2013 your torture as their rite of passage.\n\nINNA: The Pioneers?\n\nPANIA: Believe me. Organized criminals, like everybody. Only smaller. Don't look at me like that \u2013 I know.\n\nINNA: You mean they tortured you?\n\nPANIA: How could they torture me? I'm not 'unclean'. I could always look after myself. Nobody picked fights with Pania Andreyevna. I'd have bloodied my knuckles on anyone that tried.\n\nINNA: Were you a bit of a bully?\n\nPANIA: What are you talking about, 'bully'? Crazy, are you?\n\nINNA: No, I'm not saying now.\n\nPANIA: Now or ever...is that what you think? You'd characterize me as a bully, would you?\n\nINNA: No, believe me, I \u2013\n\nPANIA: Perhaps next you'll be telling all your crippled friends, 'Pania Andreyevna stuck pins in me, I swear...she stuck lighted fags up my arse.' The Pioneers would do that. I quit the Pioneers when I was younger than you are...and I don't allow smoking in here.\n\nINNA: I was only joking.\n\nPANIA: I'm splitting my sides. You've a warped sense of I-don't-know-what \u2013 it isn't humour. Just like the Pioneers.\n\nINNA: Isn't it dangerous to speak out against them?\n\nPANIA: Can anyone hear me? They can't. They've disbanded, like everything else. It's all for nothing. Sailors go down with their ship; towers get razed to the ground. Fucking Chechens won't die \u2013 they seem to come out of the ground.\n\nINNA _(Sighing.)_ : ...and there's no bread.\n\nPANIA: Not enough of it anyway. _(Crossing to a bread tray and selecting a brown loaf for her.)_ Here.\n\nINNA: What? No, I don't want \u2013\n\nPANIA: What, you're not like the rest of us? You live on thin air? Are you a sprite?\n\nINNA: No, of course I eat bread, but I have no objection to the white loaf.\n\nPANIA: Unlike the rest of us. Aren't you human? Look here... _(Showing her.)_ It's mouldy. There's gobs of mould here and here.\n\nINNA: Well, then no one should eat it...they shouldn't have sent it.\n\nPANIA: No, that's right \u2013 you go and complain. You make your way through the snow and the shit, falling down every third step, and you tell them, 'These loaves are mouldy' \u2013 and you know what they'll say?\n\nINNA: All right, it's just my opinion.\n\nPANIA: 'You're mouldy yourself, slut' \u2013 that's what they'll say. That's their opinion.\n\nINNA _(Taking the white loaf.)_ : It's not that bad. Look, I can cut the bits off. ( _Inspects_.) I think you must be exaggerating. I don't see any \u2013\n\nPANIA: I think you must be dim-witted or something, not to want brown and then not to see any fault in the white.\n\nINNA: Well, take mine, if you're \u2013\n\nPANIA: I've one of my own. Don't be so noble, your first day of work. That's why I say you're dim-witted. They won't bless you for it \u2013 they'll just rob me blind. That's not what I'm paying you for, got it straight?\n\nINNA: I'm not talking of anyone else. I want you to have two \u2013 you deserve it.\n\nPANIA: 'Deserve' \u2013 I deserve to retire. But I'm saying I can't eat two brown before one of them goes mouldy. I eat for balance, you know. Bread and cabbage. Potatoes. And then eggs. We've got dozens of eggs, but no one will touch them. It's brown bread or nothing at all. They'll turn vicious. If you let me have this, back it goes on the pile.\n\nINNA: But your husband may want it instead.\n\nPANIA: What do you know about husbands? He's already got what he wants \u2013 his freedom from me. 'They're taking on men at the ironworks' \u2013 and him a tractor mechanic. He hit the trail three months back....Bryansk Region, somewhere like that.\n\nINNA: But I'm sure he'll return, like my mother.\n\nPANIA: And I'm sure you're reading your storybooks. I've changed the locks.\n\nINNA: I'm sorry.\n\nPANIA: You think I ought to be? He'd come back crawling from drink, like a mongrel. I wouldn't give him the barn. If your mother comes back, she'll say I'm not feeding you. _(Holding out a brown loaf.)_ So I'm supposed to lie?\n\nINNA: She didn't ask you to look after me.\n\nPANIA: You weren't there at the time.\n\nINNA: But I'm nineteen, nearly twenty years old.\n\nPANIA: Nearly twenty and gormless.\n\nINNA _(Fed up, snatching the brown.)_ : Gimme the brown.\n\nPANIA _(Holding onto the loaf.)_ : I'll put it under the counter, next to mine. _(As she does so.)_ They'll do all they can to get them off us, of course. But I've got the hunter good and ready.\n\nINNA: Well, you said so yourself \u2013 they're welcome to mine.\n\nPANIA: Let's give them the muzzle instead. _(As she goes back to shifting the loaves.)_ Who's looking after you, my dear?\n\nINNA: I thought you said you are.\n\nPANIA: ...I mean with your mother away. Have you gone back to the home?\n\nINNA: Which home \u2013 the institution? I haven't been there since I was twelve. I heard they closed it at last, thank the Lord.\n\nPANIA: ...like I said, everything's breaking down.\n\nINNA: I'd call that definite progress. I'm far better off.\n\nPANIA: But the others? They might not have caring mothers with kind employers for friends. What happened to your schoolgirl friends?\n\nINNA: We lost touch.\n\nPANIA: I expect a good many \u2013 maybe most \u2013 simply died...starved to death. Maybe their carcasses went into animal feed.\n\nINNA: What! Now you're trying to make me feel guilty \u2013 I didn't leave them to starve. You talk like \u2013\n\nPANIA: You can be sure it wasn't me.\n\nINNA: It's only your idea they died anyway. Maybe they're \u2013\n\nPANIA: What, only my idea? Another would be that they went into business, emigrated to America, maybe. Yes, that's a definite possibility.\n\n**A pause**.\n\nINNA: Who was the worst Pioneer you remember?\n\nPANIA: I was.\n\nINNA: No, the meanest...the absolute bully, bordering on sadist. Can you remember?\n\nPANIA: Why do you want to know?\n\nINNA: Every one of the guardians we had was worse. We were simply left there if we so much as soiled our beds or our clothing. I'm talking for days at a time. My memory of a good meal was rice porridge, not even semolina. There was vermin of every description...and the rooms were left untidy and unclean. They'd killed off a lot of my friends through neglect.\n\nPANIA: You're saying you've already been raped.\n\nINNA: That's not exactly what I'm saying.\n\nPANIA: You'll have to tell me 'exactly'.\n\n**Somebody tries the door**.\n\nINNA: I'm saying it wasn't a home. I remember one time \u2013\n\nPANIA _(Crossing to the window.)_ : Oh, my God...more goats.\n\nINNA: Maybe it's our first customer.\n\n**INNA crosses to the window, too**.\n\nPANIA: We're not ready to open yet. _(Calling.)_ Can you hear me all right?\n\nINNA: I told you nobody reads notices. Shall I let them in?\n\nPANIA: You do, and you're going to have to wait on them.\n\nMARIANNA _(From outside.)_ : It's cold out here.\n\nPANIA: It's the middle of winter...what does she expect? _(Looking at INNA.)_ What's the matter with you?\n\nINNA _(Agitated.)_ : I can't possibly wait on her.\n\nPANIA: Ridiculous \u2013 why not?\n\nINNA: You do it, please.\n\nPANIA: I told you \u2013 it's too early.\n\nINNA: Well, all right, we'll just leave her. _(Whispers.)_ It's someone I don't want to see.\n\nPANIA: Oh, for God's sake. _(As INNA starts out the other way.)_ Hey, come back here.\n\nINNA: Can't you say I've gone to the toilet or something? Believe me, she won't go away.\n\nPANIA: Yeah, well, we're in business, in case you didn't know it. Are you going to do this with every customer? Fine help I hired. You might as well go home. Don't come back. Stay at home with your Barbie Doll.\n\nINNA _(Part concealed.)_ : No, it's just her \u2013 I promise.\n\nPANIA: You mean she's a troublemaker? I'll get rid of her then.\n\n**They cross \u2013 PANIA to the front door, INNA goes somewhere out of sight**.\n\nINNA: She's fine...you probably know her, in fact...it's just \u2013\n\nMARIANNA: It's getting colder, if you have any heart...\n\nPANIA: Just give me a fraction of a minute... _(Then to INNA.)_ Don't bring your personal problems in here. I don't want you shitting on my doorstep. _(Muttering.)_ Her own is too precious, I suppose. _(Calling.)_ We don't open till 10 o'clock...like it says on the notice.\n\nMARIANNA _(Still her voice.)_ : But it's perishing out here.\n\nPANIA: The notice...you know?\n\n**Gesturing to no avail, PANIA now admits MARIANNA SELIGMAN, same age and figure as PANIA but dressed somewhat garishly in a brightly coloured dress and coat and headscarf. Her appearance suggests a gypsy fortune teller. She still tries to get in the door**.\n\nMARIANNA: It's bread day \u2013 I knew I had to come early.\n\nPANIA: Too early \u2013 two hours to go.\n\nMARIANNA: You're usually open by this time.\n\nPANIA: Stock-taking; not for the public.\n\nMARIANNA: But I saw the train come...saw them unloading.\n\nPANIA: There's still the notice...and you could try other places.\n\nMARIANNA: But there are none...you know that.\n\nPANIA: You can't hang that one on me. It's my shop \u2013 I suppose I can open when I like. I could even close down the business.\n\nMARIANNA: Don't be silly, Pania Andreyevna.\n\n**There's a bit of a game between them with the front door**.\n\nPANIA: What if I turned my hunter on you?\n\nMARIANNA: You don't want to go against the law.\n\nPANIA: What would you call what you're doing...trying to break in? (Giving up.) Criminal trespass, they call it.\n\nMARIANNA: You wouldn't really shoot me.\n\nPANIA: Don't press your luck. I was thinking mainly of the door anyway. You're dead meat... but I'm left with a door to fix. It's too cold.\n\nMARIANNA: That's just what I said.\n\nPANIA _(Territorial.)_ : There's no fire in here. _(Hurriedly sitting on the chair.)_ Nowhere to sit neither.\n\nMARIANNA: That's all right \u2013 I only want bread.\n\nPANIA: I'm afraid I can't allow you to stockpile.\n\nMARIANNA: Would you prefer me to wait?\n\nPANIA: That's right \u2013 out there.\n\nMARIANNA: Can I help you at all? What if we moved all the trays?\n\nPANIA: What if we left them alone? You're not going to make out you work here...get round me like that.\n\nMARIANNA: I was just thinking it was easier, that's all.\n\nPANIA: I'll tell you what's 'easier'...\n\nMARIANNA: I told you I'd wait...I don't mind standing...you're kind enough to let me \u2013\n\nPANIA: The quicker I serve you, the sooner you'll go. _(Giving her a white loaf.)_ One to a customer.\n\nMARIANNA: You don't mean that's all they delivered!\n\nPANIA: ...and it might be that's all I ordered. R2.38. And don't even think about asking for brown.\n\nMARIANNA: I'm delighted with this. If you haven't got change yet...\n\nPANIA: Don't you worry about me. I've got change. _(Fairly snatching her money and going to the cashbox.)_ R2.38 from R5...\n\nMARIANNA: I may want something else if it's all right.\n\nPANIA: Tell me now, quick. Eggs?\n\nMARIANNA: We have our own.\n\nPANIA: Aren't you the lucky ones. You never asked me to sell any. If you did, I might be able to let you have brown...today even.\n\nMARIANNA: We're in business with someone in the city.\n\nPANIA: You should support local industry.\n\nMARIANNA: Most would tell you the city is local. Fifteen dozen in one week, sometimes 30.\n\nPANIA: You're making it up! I can't clear that much in six months.\n\nMARIANNA: Well, the village isn't big enough, is it?\n\nPANIA: Getting smaller by the day. Why don't you move to the city? _(A beat.)_ Cabbage or beetroot? Don't tell me you grow that.\n\nMARIANNA: All right, I'll have a cabbage. My daughter goes off to the city this week. That's why I want the bread.\n\nPANIA: With some beets, it'll come to a straight R5.\n\nMARIANNA: Splendid \u2013 I can make borsht.\n\nPANIA: We've no sour cream.\n\nMARIANNA: That's all right \u2013 Zhenka Rodin supplies me.\n\nPANIA: I'd watch that if I was you \u2013 Zhenka Rodin's an alcoholic. His cream is probably tainted with bugs. They put rats in the sausage as well. Rodin's too drunk to know what he's doing. Relieves himself where he stands.\n\nMARIANNA: My family was raised on his produce.\n\nPANIA: And you're still standing? Miraculous.\n\nMARIANNA: And what's this you say about sausages? He doesn't sell any, does he?\n\nPANIA: He's not a villain...no, I never said that.\n\nMARIANNA: It wouldn't matter if he was, if his milk is all right.\n\nPANIA: All I ever said was he drinks too much. And you'd never say 'wouldn't matter' if you died from contaminated milk.\n\nMARIANNA: You're not very cheerful this morning.\n\nPANIA: I'm just speaking the truth. Just trying to warn you, my darling. My bread, for example, is always fresh.\n\nMARIANNA: I can see it is.\n\nPANIA: ...won't have any trouble digesting that loaf.\n\nMARIANNA: I'm going to make a currant pudding out of it.\n\nPANIA: What are you talking about, pudding? From that fresh loaf?\n\nMARIANNA: That's right \u2013 for Shura's farewell dinner.\n\nPANIA: It's a thousand times better than that.\n\nMARIANNA: Better than what, farewell dinners?\n\nPANIA: Stodgy puddings. I only have regular customers, you know.\n\nMARIANNA: All right, I won't come again.\n\nPANIA: I can name a clear dozen that would live off that loaf for a week. They wouldn't throw it away in a pudding.\n\nMARIANNA: Well, I can't give it back to you.\n\nPANIA: Don't worry. I can't have it back...after you've pawed it.\n\nMARIANNA: They probably do put it in puddings, you know.\n\nPANIA: Don't you try telling me about my dear customers. Dear to me, they are. Precious. Why, we've known each other for years. We don't do business with Zhenka Rodin's like.\n\nMARIANNA: You're insulting me, aren't you?\n\nPANIA: You're insulting yourself \u2013 Zhenka Rodin. Why should I want to insult you?\n\nMARIANNA: That's just what I'm \u2013\n\nPANIA: I let you in, didn't I? You couldn't be bothered to read the notice, but I let you in anyway. The train broke down, so we had to push it. Then I discovered the delivery was short.\n\nMARIANNA: I thought it must be.\n\nPANIA: But none of that mattered. There you were. 'Here's a woman in trouble', I told myself... '...a woman that's cold, desperate for bread'.\n\nMARIANNA: And I'm very grateful.\n\nPANIA: What more do you want me to do?\n\nMARIANNA: You don't mean that you actually pushed the train?\n\nPANIA: It ran out of diesel \u2013 what else could we do?\n\nMARIANNA: No wonder you wanted to open late.\n\nPANIA: You don't have to worry about me. I'm a strong, healthy woman \u2013 I'll never go off to the city.\n\n**A pause**.\n\nMARIANNA: Doesn't little Inna Igorovna work here?\n\nPANIA: What about it?\n\nMARIANNA: I just wondered, that's all.\n\nPANIA: And what about if she works here...what's it to you?\n\nMARIANNA: Is she late?\n\nPANIA: What business is it of yours? I'm warning you \u2013 if you've come here to make trouble for Inna...if the two of you are in some kind of feud...\n\nMARIANNA: I'm not out to make any trouble...\n\nPANIA: ...you'll be doing it all on my time. That makes it my business, I suppose. Doesn't it?\n\nMARIANNA: Look, let me...I'm Inna's teacher, all right?\n\nPANIA: She told me she left school.\n\nMARIANNA: Well, that's right. She said she was going to be working here...she recommended you.\n\nPANIA: No, she didn't.\n\nMARIANNA: All right, believe what you want. You will anyway.\n\nPANIA: I believe what Inna herself told me. Why shouldn't I? I'm her mother's best friend.\n\nMARIANNA: How is Zoya Igorovna?\n\nPANIA: Don't pretend like you know her, too.\n\nMARIANNA: I don't at all, but you see I believe you.\n\nPANIA: Oh, you do? Inna tells me she's terrified of you. What do you think of that?\n\nMARIANNA: I don't...I think she didn't.\n\nPANIA: Are you calling me a liar or are you just shocked by the truth? _(Running on.)_ And before you say anything else, I don't want to know anything about it.\n\nMARIANNA _(A beat.)_ : What time is she coming in?\n\nPANIA: That doesn't matter \u2013 I'm not going to allow you to wait. You got your bread?\n\nMARIANNA: ...and you got your money.\n\nPANIA _(Trying to push her out.)_ : ....concluding the business between us...Goodbye.\n\nMARIANNA: Won't you at least...give her a message from me?\n\nPANIA: I'm not getting involved in your squabbles.\n\nMARIANNA: But look...I'm sorry...this is absolutely fantastic to me.\n\nPANIA: I don't give that for what you're telling me. _(She makes a Russian gesture signifying 'nothing' \u2013 her thumbs between index and middle fingers pointed at the person.)_\n\nMARIANNA: But we've had no squabble. It's almost as though you're talking about other people. I'm devoted to Inna Igorovna. I taught her for years. I wanted to see where she works.\n\nPANIA: Why?\n\nMARIANNA: I'm interested in her welfare...we both are.\n\nPANIA: I'm not a charity warden.\n\nMARIANNA: Well, of course not.\n\nPANIA: I'm going to put her to work...she's going to have to work hard.\n\nMARIANNA: I know that \u2013 you're giving her a chance. That's why I wanted to meet you.\n\nPANIA: She ought to be working \u2013 nearly twenty years old. Does she think she's going to be kept all her life, like a Number One cripple? She's not crippled at all, some would say.\n\nMARIANNA: It's good to hear you talk that way. That's just what she needs.\n\nPANIA: Oh, so you're her employer, are you? Well, she's working for me now \u2013 you had your chance.\n\nMARIANNA: Wonderful. I said I'm her teacher.\n\nPANIA: She's unskilled. What on earth could you have taught her, for God's sake \u2013 mathematics?\n\nMARIANNA: Art.\n\nPANIA: What \u2013 'art'? Oh-ho-ho. That's not a trade.\n\nMARIANNA: I didn't say it was.\n\nPANIA: So you trained her for nothing. I see.\n\nMARIANNA: I'm not going to argue with you. We'd never get anywhere with it. I just wanted to say \u2013\n\nPANIA: She told me about you and all.\n\nMARIANNA: I'm sure she did. That's where we have common ground.\n\nPANIA: You were the worst form of bully, she said.\n\nMARIANNA: I...beg your pardon?\n\nPANIA: '...bordering on the sadistic...'\n\nMARIANNA _(A beat.)_ : I think we must be talking about different people.\n\nPANIA: Those were her very words. Society judged her to be unclean, and you made sure she stayed that way.\n\nMARIANNA: I'd prefer you not to say any more.\n\nPANIA: It's the same girl, though, isn't it. She gave me a graphic account.\n\nMARIANNA: Please don't say any more! I'm sorry that I bothered you. _(Hurt, she starts toward the door.)_\n\nPANIA _(Following after her.)_ : In Soviet days, we could send you to prison.\n\n**INNA comes on from the back**.\n\nINNA: But I never meant Madam Seligman!\n\nMARIANNA: Inna?\n\nPANIA: Eh? You stood where you're now standing and wept it out. Like you're weeping now.\n\nMARIANNA: You make me want to weep.\n\nINNA: But I was talking about the home guardians...not...for a minute...not you.\n\nPANIA: ...and is that why you were terrified when you saw her just now? _(To MARIANNA.)_ She ran and hid from you. Now I know why. _(To INNA.)_ Now I understand perfectly, my darling.\n\nMARIANNA: Well, I'm afraid I don't. _(To INNA.)_ How did I terrify you exactly? Please tell me.\n\nINNA: But you didn't. It's all a \u2013\n\nMARIANNA: Why were you hiding from me? Did I beat you? _(To PANIA.)_ I critiqued her artwork \u2013 that was my job. _(To INNA.)_ Are you upset that I didn't proclaim you a genius? Another Frida Kahlo?\n\nINNA: I'm not upset!\n\nMARIANNA: Go to art school, test yourself out, if you want to find out what you're really worth.\n\nINNA: No...wait a minute...that's what upsets me.\n\nMARIANNA: You confuse interest with instant acclaim.\n\nINNA: No, I don't! Really.\n\nMARIANNA: Kahlo had to wait...years. She lived a virtual lifetime in her husband's shadow. That's the whole point about Kahlo \u2013 I told you.\n\nPANIA: Who's this Kahlo you're talking about?\n\nMARIANNA: Woman still-life artist...Mexican...disabled.\n\nPANIA: And that's why she's terrified...another cripple? She's not even Russian. _(To INNA.)_ You'll never meet her.\n\nMARIANNA: She died in the 1950s.\n\nPANIA: A dead cripple? _(To MARIANNA.)_ I know what this is \u2013 guilty conscience. _(Back to INNA.)_ You're thinking about your dead comrades, aren't you? _(Then back to MARIANNA.)_ She was talking about it today.\n\nMARIANNA: I don't think you quite understand.\n\nINNA: No, she doesn't.\n\nPANIA _(To MARIANNA.)_ : How do you know? You weren't here. _(To INNA.)_ If that's all you've got to worry about... _(To MARIANNA.)_ You know what they'd call it in Soviet times, don't you? Morbid self-interest. She'd be judged a State enemy...a pariah.\n\nMARIANNA: She was born a pariah.\n\nPANIA: What do you want me to do about it? In Soviet times, I'd be put under surveillance for even employing such a person, and you're both pointing the finger at me.\n\nMARIANNA: ...because these are different times. And who's pointing the finger? You are!\n\nINNA: ...and you were calling the Pioneers criminals. You didn't like Soviet life.\n\nPANIA: I was talking about bad Pioneers. The Pioneer ideal was beautiful. It's how we lived. And you were excluded \u2013 enemy!\n\nINNA: ...because I couldn't walk.\n\nPANIA: So? Complain to your paediatrician \u2013 I can't help you.\n\nMARIANNA _(Seizing INNA's arm.)_ : Come on, darling. I've got the funding to send you to art school.\n\nINNA: No!\n\nMARIANNA: But this is no place at all for you. You heard her just now. I'm not at all sure the misunderstanding wasn't the real truth. _(To PANIA.)_ You're the sadist, my dear. That much is perfectly obvious. How could she join the Pioneers?\n\nPANIA: She said it \u2013 she didn't.\n\nMARIANNA: So you wanted perfect bodies, like the Germans?\n\nPANIA: They'd have called you a traitor for saying that. You'd have been shot the same day.\n\nMARIANNA: You're living the glorious Soviet myth.\n\nPANIA: Hey, you can't take her out of here.\n\nMARIANNA: I'll pay you her wage, don't you worry.\n\nPANIA: And just who do I get to replace her at this notice? Any minute now they'll be breaking down the door.\n\nMARIANNA: And just how do you think she can help you? You need a strong, healthy girl for work of that kind. Could she push the bread train? You know she couldn't.\n\nINNA: No, all right, but I wanted to.\n\nMARIANNA: Now you're just being ridiculous. You couldn't even lift the bread tray.\n\nINNA: Not the train, I know, or the tray. But the job, yes, the ordinary job.\n\nMARIANNA: ...but not this job.\n\nINNA: Believe it or not, I thought this one, yes. That's why I was frightened to see you, you see \u2013 you disapprove.\n\nMARIANNA: It's your life.\n\nPANIA: ...and my shop, if you're sure you don't mind.\n\nMARIANNA: I'm not sure you'll really be happy.\n\nPANIA: Another bourgeois conceit....\n\nINNA: I'm not here to be happy.\n\nPANIA: That's right \u2013 she's here to work.\n\nMARIANNA: But you can do so much better.\n\nPANIA: At what \u2013 her 'artwork'?\n\nMARIANNA: I can get you a place. _(To PANIA.)_ Let her do what she can, yah?\n\nPANIA: She can help me.\n\nMARIANNA: A fit girl would be better...you know it.\n\nPANIA: She'll do for now.\n\nMARIANNA _(To INNA.)_ : '...for now', you see? She's no intention whatever of making you permanent.\n\nINNA: And I've no idea I'll be any good.\n\nPANIA: It's time you learned, eh? I was moving heavy machinery when I was half her age.\n\nMARIANNA: That's pure fabrication \u2013 we never had child labour.\n\nPANIA: She's bone idle at twenty years old!\n\nMARIANNA: But she's not \u2013 that's my point.\n\nPANIA: Oh, no, I forgot \u2013 she produces 'art'.\n\nMARIANNA: Others do as well. Why are you so inflexible?\n\nPANIA: I'm a worker.\n\nMARIANNA: And that makes the rest of us what, parasites? I teach. You went to school, didn't you?\n\nPANIA: Not your sort of school.\n\nMARIANNA: That's pure envy.\n\nPANIA: Bourgeois pig. While Russia is starving herself, you produce what \u2013 fine _objets d'art_ , still-lives of what, exactly? And just what does she produce that you're so anxious to bring forth to the world?\n\n**At some point here, the two women tug for INNA**.\n\nINNA: Oh, no, please. Both of you.\n\nPANIA: No, really, I'm genuinely interested. I didn't know a Kandinsky walked into my life. Is my shop here all part of your study? You want to do a still-life...an in-depth study of the rat? I'll just try to see where he's gone. I suppose you don't mind \u2013 you might even prefer \u2013 that he's dead. Yes, it might give your statement more meaning.\n\nMARIANNA: You had industrial art in the thirties.\n\nPANIA: We've no industry! You want to announce to us all life is shitty? We know that already \u2013 we can smell our own shit. Are you about to tell us what to do with it to make our life better, is that it? Oh, well, thank you.\n\nMARIANNA: I'm not prepared to have this discussion.\n\nPANIA: I don't discuss things \u2013 I told her that.\n\nINNA: Just before you came in.\n\nMARIANNA: That's not the point anyway \u2013 it's what Inna herself wants to do.\n\nPANIA: You're wrong \u2013 it's what I want in my shop. You came in here wanting my bread.\n\nMARIANNA: I came because of Inna.\n\nPANIA: She's not for sale.\n\nMARIANNA: ...and she's not a commodity.\n\nPANIA: ...that's right \u2013 she's pretty well useless.\n\nMARIANNA: Then why don't you just let her go? You're not bothered about keeping her, admit it. You just detest giving in. I've met women like you before. Object on principle.\n\nPANIA: I made a promise to her mother.\n\nMARIANNA: Yah.\n\nPANIA: 'Make her useful, Pania Andreyevna', that's exactly what she said.\n\nMARIANNA: As if either one of you cares either way. Do you know where her mother is now?\n\nPANIA: Finding a job in the city.\n\nMARIANNA: Or she could be halfway to Bryansk. And when is she going to return? Do we know that? If she cared, they'd have gone off together. 'Making her useful' would have sent Inna out to beg. That mother of hers didn't even care enough to exploit her. And you're going to stand there pretending there was ever anything stronger between them than the biological fact?\n\nINNA: Can't you stop this please?\n\nPANIA: There you go upsetting the poor girl \u2013 talking that way about her mother.\n\nINNA: Supposing she didn't really care...supposing she's not coming back, for all anyone knows.\n\nPANIA: You let this old witch upset you too much. _(To MARIANNA.)_ It's you that wants to exploit her, dear.\n\nINNA: All right, my mother didn't care for me all that much. I know that. It might have been law, but she needn't have sent me away...even if I'm Number One disabled, I'm not helpless. She didn't have to...\n\nPANIA: Don't allow yourself to be deluded \u2013 she's a good woman, Zoya Igorovna. She wanted the best for you.\n\nINNA: Do you really know her? Because I never did. My Auntie Yulia paid me more visits than anyone else in the family. It was always my dream to leave to look after her when I was old enough. But then she died. It was only after I left that I discovered the details. She had consumption. No one bothered to tell me. I understand they never gave me a thought. I was fourteen and a half...when they closed down the house, and I went to look for my mother. It wasn't straightforward \u2013 I only had one address, I don't remember being there. 'Somewhere outside of Petersburg', someone directed me. I'd never even taken the train. You could say I was like a young colt...my first steps of freedom. I saw a moving staircase, going down. I'd never been on one, and I was plainly terrified...All those people...I thought I'd be trampled to death once I tried. And suddenly I wished I was back at the home. I started to cry, and I heard people round me, shouting, 'Who is she?' 'What's she done?' 'Somebody had better get hold of a policeman'...as though I'd committed a crime!\n\nPANIA: There \u2013 you see? She'd be lost by herself in the city. You heard her say it \u2013 trampled to death.\n\n**Gradually, for this account, INNA becomes 'able-bodied' \u2013 to the point of rebounding from side to side like a rubber ball**.\n\nINNA: ...somebody suddenly took hold of my arm...I thought at first a policeman...and that I'd be taken off, thrown in detention somewhere...or worse...down the moving staircase. But no, he was moving us to the open air, doing it in such a way I found...gentle. He asked me where I was headed, and I heard myself saying automatically, 'The train for Settlement Number 7.' So we got in a taxi, then into a train...I fell asleep from the motion...only to find when I woke up \u2013 he'd delivered me to my mother's door. I think before I fell asleep he told me his name was Pavel Zakharin. Did I dream it? He volunteered that his father had been a general in the Patriotic War. I think I went to sleep believing, 'We're not likely to get lost then.' And...oh, yes, I think he made me eat a supper of brown bread, buttermilk and radishes...She wasn't unhappy to see me, my mother. We embraced...then she put me to bed. And I know I was dreaming this time...about a moving staircase...and the dormitory at the home. But in this dream the general's son turned out to be Yekaterina Olmyoskaia, the last matron we had at the home. It was her face I saw when I woke up next morning, though of course it was my mother's voice. 'It's 7.00, and I've made rice porridge if you're hungry, and then you'd better get ready for school.' That's when I knew that my new life had begun, and that I was bound to survive.\n\nPANIA ( _To MARIANNA.)_ : And you were her teacher, I suppose.\n\nMARIANNA: Only one of them, though I think I was probably the one that most cared.\n\nINNA: I felt I was always your favourite...I always wanted to ask why.\n\nPANIA: Jews always want to mother everybody. _(To MARIANNA.)_ Was your daughter at home at the time?\n\nMARIANNA _(To INNA.)_ : You never reminded me of Shura, but I thought of Kahlo, I admit, straightaway. And that painting by Velasquez \u2013 the group portrait with the little girl who's adored by the dwarf.\n\nINNA: You've shown me a photograph of it.\n\nMARIANNA: The aristocracy always looked after their lame ones...like beloved family pets. It wasn't looked on as misfortune to be lame.\n\nPANIA: Here, she's an act at the circus, so what?\n\nMARIANNA: I want to look after you, darling.\n\nINNA: I know that, Marianna.\n\nPANIA: And did you happen to hear what else she said? She doesn't want that...she's afraid of the city, for one thing. And, for another, she's no liking for being a 'family pet'.\n\nINNA: That's not it either.\n\nPANIA: Why is it you're always arguing with me?\n\nMARIANNA: You're speaking for her...\n\nPANIA: ...and you're sending her back where she came. Where's the sense in that?\n\nMARIANNA _(To INNA.)_ : You know that's not true. _(A beat.)_ I'm giving her the chance of a career.\n\nINNA: Perhaps next year.\n\nMARIANNA: They won't hold the place. And how do you know you wouldn't be able to cope on your own? You spent an afternoon, when you'd just been released. What kind of test is that?\n\nINNA: You expect me to go from that straight to living on my own.\n\nMARIANNA: It's six years later, and you're not on your own. You'd be living with Shura.\n\nINNA: But she doesn't know me. You expect her to look after someone she doesn't even know...a \u2013\n\nMARIANNA: Don't you dare use the word 'cripple'.\n\nINNA: ... 'total stranger' I was going to say. She'd expect me at least to contribute.\n\nMARIANNA: And so you would...why not?\n\nINNA: You don't know that, Marianna.\n\nMARIANNA: Better than you do yourself.\n\nPANIA: There, you see? You want to do her living for her.\n\nINNA: It's not that I don't think I would contribute or make my own friends...\n\nMARIANNA: Shura's a good girl...responsible.\n\nINNA: No, but she didn't adopt me, did she. _(A beat.)_ Can't we stay as we are?\n\nMARIANNA: And how's that, exactly? You're not still at school.\n\nINNA: We can still see each other.\n\nMARIANNA _(Disappointed.)_ : Yah, sure.\n\nINNA: It's better this way...you'll see. _(Moving toward her.)_ I know my limitations, better than anyone else, which is natural.\n\nMARIANNA: So you want to be a lone wolf, a pariah, as our friend said.\n\nINNA: I only wish I could make you understand.\n\nMARIANNA: You wish to persuade me. I understand very well.\n\nINNA: It's only because of my condition.\n\nMARIANNA: It's the condition of Russia, my dear. People become friends with their condition... they depend on it, finally become enslaved by it. Like the people in this village.\n\nPANIA: I'll give you 'enslaved', all right.\n\nMARIANNA: You do nothing, quite frankly, to help yourselves.\n\nPANIA: No, of course. We just moved a whole bloody train. Of course, that's no match at all for talking about art and culture and civilisation. Move mountains you do with your talk, I forgot.\n\nMARIANNA: You make me sound like a Pre-Revolutionary.\n\nPANIA: An Anti-Revolutionary, sure.\n\nMARIANNA: Stop living in the past. _(A beat.)_ I don't blame you entirely, either of you. My daughter insists she knows better than I do.\n\nINNA: But you want her to go to the city...don't you?\n\nMARIANNA: I only want her to live, to act right. She convinced me to send her to Germany, to study architecture. What happens? She comes back a year later, no qualification, no job, of course. And she asked for the place for herself.\n\nINNA: She's got to have it, of course \u2013 she's your birth daughter. I'm glad you told me.\n\nMARIANNA: But she doesn't deserve it...as you do, if only you've sense enough.\n\nINNA: She's your daughter...you're both artists. You can't ask me to replace her in your life.\n\nMARIANNA: I never said anything like that.\n\nINNA: You did without realising it, through your actions. That's not fair to any of us, dear.\n\nMARIANNA: What can you possibly know about it?\n\nINNA: Not a thing, in specifics. We've never met. But I'd want my own mother to care. _(Moving toward her.)_ Make it up with her, Marianna, before you lose her for good. She's asking that of you. I know it.\n\nPANIA: I know it and all. She has to want more than a mouldy bread pudding.\n\nMARIANNA: And there's nothing I can do to persuade you...you're determined to fight your own battles.\n\nINNA: I'll be thinking of you, though, more than you could know. _(Choking back the tears.)_ I'm ever so glad we had this chance to...\n\nMARIANNA: That's precisely what disturbs me. _(As she goes out.)_ Make sure you don't turn her into the family cat \u2013 you'll skin her and eat her alive.\n\n**She goes off**.\n\nPANIA: That's a marvellous thought, isn't it?\n\nINNA: She's really a generous person.\n\nPANIA: That daughter of hers is a whore.\n\nINNA: What?\n\nPANIA: That's what she's gone to the city to do. I've seen her round the place, looking for clients.\n\nINNA: I'm sure you're mistaken.\n\nPANIA: That shitheel Zhenka Rodin \u2013 he's the one. Mafia agent, he arranged for her transfer. That's what you were headed for, my deluded darling. Tonight, as you play with your finger paints, please reflect on your avoided fate.\n\nINNA: I'm sure that's not true. Marianna would have to know.\n\nPANIA: I could tell at once she's a fool. You did exactly the right thing to send her packing. I can just hear her moaning into the night, 'Oh, why have I been so forsaken', because she's such an idiot. That's exactly what she deserves. _(She surveys the state of her shop \u2013 loaves still on the trays.)_ And here, look. We haven't managed to do a thing with these...all through entertaining your friends. That's where indulging the arts gets you.\n\n**Someone else \u2013 this time a man \u2013 is outside the door**.\n\nOSIP: Madame Pania Andreyevna!\n\nINNA: Shall I open up for him?\n\nPANIA: In your babyfat dreams \u2013 we're through entertaining for one day!\n\nOSIP: I've got something I need to discuss with you.\n\nPANIA: That did it \u2013 discussion is out, just like he is...and he's going to stay that way.\n\nINNA: He could be a customer.\n\nPANIA: All right, you tell him to fuck a donkey \u2013 let's see what you're made of.\n\nINNA: Well, I'm hardly likely, am I...to tell him that. _(Crosses to the door and calls out.)_ We're not open yet...Can you come back in...half an hour?\n\nOSIP: I'm not a customer, tell her.\n\nINNA: Oh, dear. Aren't you?\n\nPANIA _(Through the door.)_ : Then don't bother returning at all! _(To INNA.)_ Did you happen to tell all your friends?\n\nINNA: I don't know who it is.\n\nOSIP: I'm here, as you might say, 'officially'. That is, on business...urgent business. Well, I'm here anyway.\n\n**This compels PANIA to peek through a window to see who it is**.\n\nPANIA: Oh, God...Osip Pishchik.\n\nINNA: Is he a regular?\n\nPANIA: Regular pain in the arse, yeah. I forgot to warn you about him.\n\nINNA: Warn me \u2013 what about?\n\nPANIA _(Through the door.)_ : No delivery today, sweetheart....back to beddy-byes with you.OSIP: Is that you, Pania dear?\n\nPANIA: Can you remember where your bed is?\n\nOSIP: I'm not here about bread!\n\nPANIA: Not 'bread', dear, I said bed \u2013 your bed. You shouldn't have left it so early.\n\nOSIP: Of course, I'll buy bread, if that's what you want. You're in business yourself, I know. You can't afford to let a working moment go by without profit. I'll add to that profit, if I can.\n\nPANIA:Can you hear me at all? There's no bread today. The bastards forgot to deliver.\n\nOSIP: ...but I still need to speak to you.\n\nPANIA: Shall I send for your friend Volkov...the Inspector? Would that fill you with comfort and joy?\n\nOSIP: Volkov, you say? That's very funny.\n\nPANIA: We can ask him to put you to bed, if you want.\n\nOSIP: No, I don't think we want Volkov just yet. Of course, he'll have to be in on it at some time or other.\n\nPANIA: Then I'd leave, if I was you, before he appears on the scene.\n\nOSIP: No, I've got to see you first, dear...for your own good.\n\nPANIA _(To INNA.)_ : You'd be having to deal with this sooner or later. Just as well you see it now.\n\nINNA: Osip Pishchik? But I do know him. He doesn't bother you, does he?\n\nPANIA: Pain in the arse.\n\nINNA: That surprises me, because I always thought he was \u2013\n\nPANIA: He's one of the ones they ought to clear out. They would do it, no problem, in Soviet times...\n\nINNA: If you mean what I think \u2013\n\nPANIA: I mean 'imprison' or 'hospitalize'...get rid of anyway.\n\nINNA: Then I'm afraid I can't agree.\n\nPANIA: You've only just come out yourself.\n\nINNA: ...from the home, not from prison...and it was four or five years ago. It was a kind of a prison.\n\nPANIA: I know what I meant. Too sheltered to know that people like Pishchik exist, and you've got to deal with them the best way you can.\n\nINNA: What do you mean? I've known people like Pishchik...I've known Pishchik. I think he's nice...I think you should let him in.\n\nPANIA: And I think you're crazy.\n\nINNA: Well, as you say \u2013 it's your shop. I'm only the shopgirl...and this is only my first day.\n\nPANIA: And we've had one interruption already.\n\nINNA: My friend Marianna. Yes, that's very true.\n\nPANIA: So would you please let me deal with this?\n\nINNA: I just think you're being unfair...all right \u2013 fire me.\n\nPANIA: I just want you to shut up a minute.\n\nINNA: If I could perhaps talk with him? There's no need at all for police.\n\n**She starts to unlock the door**.\n\nPANIA: Once he gets in, we'll never get rid of him...ourselves.\n\nINNA: Oh, we won't need 'to get rid of him' at all. He came to the home all the time. He did magic tricks for us. Then he...went away again.\n\nPANIA: Wait a minute. _(Hand against the door.)_ What do you mean he did tricks for you?\n\nINNA: Magic tricks. He made us all laugh. Of course, we were only about eight or nine...they probably wouldn't amuse me now.\n\nPANIA: (He's worse than I thought.) What else did he do?\n\nINNA: Told us stories? I forget. That's probably because he wasn't that good at storytelling.\n\nPANIA: Did he touch you at all? Did those freaks let him handle you?\n\nINNA: What 'freaks'? What are you on about?\n\nPANIA: Those guardians you say were like sadists...that made a habit of handling you...did they let people like Pishchik \u2013\n\nINNA: Why do you say, 'people like Pishchik'? No one did anything to us!\n\nPANIA: So what were you talking about earlier?\n\nINNA: Neglect, I was saying \u2013 not that.\n\nPANIA: Do you understand \u2013 ?\n\nINNA: I know. Chikatilo. I'm not stupid.\n\nPANIA: But you could have forgotten...They say \u2013\n\nINNA: Well, they're wrong. It was only ten years ago, in my very young \u2013 so you're telling me \u2013 life. I forgot nothing at all about their care...Pishchik did nothing like what you're talking about. I'd have known, wouldn't I?\n\nPANIA: What were they doing letting him in?\n\nINNA: I told you \u2013 he did tricks. Maybe they thought they wouldn't have to work the time he was there. I dunno. But I can tell you how it happened. He'd bribe them. That's clear enough, isn't it? Nobody anywhere has enough money. He paid them to let him entertain us. I remember, because the first time he said he came from my village...and I thought maybe he was related to me in some way...maybe married to Auntie Yulia. He didn't come very often anyway...not more than two or three times...\n\nPANIA: All right, if he didn't do anything, who else \u2013 ?\n\nINNA: I don't want to talk about...it's rubbish, okay? No one did anything to us. We lived a fairy tale.\n\nOSIP _(Singing it.)_ : I'm still wait-ing\/Pania Andry-ov-na! _(Speaking it.)_ I'm still here.\n\nINNA: I thought he must have died or moved away.\n\nPANIA: And aren't you thrilled now he's never done either?\n\n**OSIP now launches into the beginning of 'March of the Aviators'**.\n\nOSIP: 'We were born to make fairy tales come true...'\n\nPANIA _(To INNA.)_ : Did he ever do that at the home?\n\n**She now moves to open the door**.\n\nOSIP: '...We have steel wings for arms!' _(Almost falling through the door.)_ Hello, my dear. _(Seeing INNA.)_ And who's this...? No! It can't be...it's little Inyechka!\n\nINNA _(Tentatively.)_ : Hello, Osip Pishchik.\n\nOSIP: You're a butterfly now \u2013 a speckled yellow for sure.\n\n**We now see he's a shambling 50-year-old**.\n\nINNA: I wasn't sure you'd remember me...once Pania Andreyevna said \u2013\n\nOSIP: Why, it's only been a few years. And you're a living portrait now, aren't you...a tiny masterpiece.\n\nPANIA: Let's leave it before you get overwhelmed by the artwork. We had one exhibition already.\n\nINNA: I hope you're all right.\n\nOSIP: Me? Oh, yes, fine, should I say indestructible, why? Did you imagine I'd died?\n\nINNA: I've thought of you very often, in fact...I missed you.\n\nOSIP: Oh, don't say that.\n\nINNA: But it's quite true.\n\nOSIP: That breaks my heart, you know? Because I never went back to see you. I did try one or two years ago. But then they said the home had been closed down.\n\nINNA: It was more than two years ago....more like five.\n\nOSIP: Is it as long as that? Those poor little kiddykins.\n\nPANIA: All right, stop it. She works here now. What do you want, bread?\n\nOSIP: Do you want me to have it?\n\nPANIA: I want you to say what you want and then go...I don't want you hanging round. You can see Inna later \u2013 another time and a very large distance from here, please.\n\nOSIP: I love meeting Inna again, of course \u2013 instantly I'm a young man. But it's you, my dear Pania Andreyevna, that I've come to see...on a personal matter, you understand? A bit delicate, a bit personal.\n\nINNA: I'll leave the room.\n\nPANIA: You stay right there. He doesn't know me 'personally'.\n\nOSIP: What, not know you? I've known you since schooldays.\n\nPANIA: Oh, yeah?\n\nOSIP: It's perfectly true we've not seen so much of each other in recent years. But I assure you, I remember the day you organized the cleaning when the janitors were snowbound after that job at Settlement 32. _(For INNA's benefit.)_ Our school had been closed the previous month, and, somehow, it hadn't been cleaned. The teachers \u2013 well, you can imagine \u2013 were beside themselves \u2013 there was no ground staff, and the state of the toilets alone left us shocked. But your fearless employer there... _(To PANIA.)_ ...you took it all on yourself, didn't you, dear. You looked Principal Comrade Luzhkov straight in the eye, and you said, 'Don't worry, Comrade Principal, we'll have it spotless overnight'! And you did, didn't you, working away through the night. It was as though we'd never been closed. You saved the ground staff their jobs, didn't you? I remember thinking at the time, 'This one is destined for leadership, no doubt.'\n\nPANIA: Yeah? And now look where I am.\n\nOSIP: And the school numbered two or three hundred...it wasn't small. It must have taken most of the night, yes, to clean.\n\nINNA: This morning she helped push the bread train.\n\nOSIP: Who could doubt it? Wings of steel, you see? Even now. You're in the most capable hands you could find. She won't fail you. Overnight, the entire school. Four whole storeys.\n\nPANIA: I'm not impressed.\n\nINNA: You're too modest.\n\nPANIA: You want something, Osip Yegorovich. You didn't come here to reflect on my glory.\n\nOSIP: I came to warn you.\n\nPANIA: ...of what?\n\nINNA: Something to do with the shortage, I bet.\n\nPANIA: I know all about that...see? That's all they sent. Do you want a white loaf? _(Proffering loaf.)_ Here, take it...and go. Get out, go. _(Tries to push him out the door.)_\n\nOSIP: I discovered a body...late last night.\n\nPANIA: Yes, I know \u2013 the body of your great grandfather...He appeared to you out of the mists of time.\n\nOSIP: No, I think it's your husband, in fact.\n\nPANIA: Fuck off.\n\nINNA: You really shouldn't joke like that, Mr Osip.\n\nOSIP: Dear me, did you think I was joking?\n\nINNA: Well, I hoped you were.\n\nPANIA: I think he's plain wrong \u2013 drunk, deluded, missing a few of your marbles. I told her to be on her guard.\n\nOSIP: I don't doubt it sounds strange to you \u2013 unexpected and therefore a shock. You're probably wondering if you heard right...and once it hits you, no doubt, there will be grief. Had you known each other very long, say, from schooldays?\n\nPANIA: You found no body!\n\nINNA: Denial, that is. Marianna told me. It's what they now call being in denial. She says I'm in denial about my condition, but that that's a good thing. Usually it's said of something...not so good.\n\nOSIP: Of course, I can't be sure it's his body or not. But you should take a look.\n\nPANIA _(Physical contact.)_ : Can you see me, old man? Are you listening to me? My husband's not dead, and there's no body no place. It's just a figment of your deluded...your warped mind...and I've got a day's work ahead of me. I was out there pushing a bloody bread train at 4.00 in the morning.\n\nOSIP: You should have a little lie-down first, of course.\n\nPANIA: I'll lie down, all right. I'll lie you down, lay you out, I'll flatten you blin-style, how's that? Idiot.\n\nOSIP: I'm not for a moment suggesting you murdered him.\n\nPANIA: Next time I'll murder you, no problem.\n\nOSIP: But you've got to admit to yourself the possibility, the mere possibility \u2013\n\nPANIA: Will you kindly shut up?\n\nOSIP:...like, for instance, he's not at home, and he hasn't been home for a matter of \u2013\n\nINNA:...three or four months. Isn't that what you told me?\n\nPANIA: You can shut up and all. Whose side are you on?\n\nINNA: Oh, I'm not suggesting it either. But I don't think Mr Osip would lie or make a mistake like that. I definitely think there's been \u2013\n\nPANIA: You definitely nothing-on-earth. You can't move to use public transport. I told you he went to the city. If you don't believe me \u2013\n\nINNA: My memory is that he could be anywhere. Bryansk was one place you mentioned.\n\nPANIA: It doesn't matter, all right? Because I know where he is and the whore he's shacked up with, all right? Got the picture?\n\nINNA _(A beat.)_ : You still ought to see the body.\n\nOSIP: I think Pania Andreyevna wants to give herself time to adjust.\n\nINNA: When, exactly, did you discover \u2013 ?\n\nOSIP: Late last night, early this morning.\n\nINNA: And, exactly, what state was it in?\n\nOSIP: Well, I'm hardly what you'd call expert. I mean an expert could determine everything, possibly immediately. All I saw was a male \u2013 snow-covered, of course, in weather like this...\n\nINNA: And half-eaten by wolves?\n\nOSIP: What?\n\nINNA: I mean...could he have starved, possibly?\n\nOSIP: Quite possibly, yes, though I wouldn't like to venture offhand. But wolves? Where'd you get that idea?\n\nPANIA: From the storybook, like you did.\n\nOSIP: People have died through the ages in such circumstances. You can't deny that, dear.\n\nPANIA: They've also been shot in battle...they die of plague, rare diseases and heart failure. How does that affect me?\n\nINNA: Did he meet with a violent death, do we know? Did you see blood?\n\nPANIA: Perhaps he was still breathing...gasping for life...and you walked away from the scene. Perhaps we can hold you responsible.\n\nOSIP: I'm fully prepared to tell what I know...and leave my judgement to God.\n\nPANIA: Oh, that's that then \u2013 you'll burn in hell. _(Sighs.)_ That's a load off my mind anyway.\n\nINNA: Why so reluctant to believe, Pania Andreyevna? Where does it profit Mr Osip?\n\nOSIP: It's true, you know \u2013 I came here as an article of faith, of friendship, though we've always been a bit distant. I always thought you disapproved of me, for some reason I never knew.\n\nPANIA: No reason, eh?\n\nOSIP: No, I never knew it, that's all. It's been a complete mystery to me ever since \u2013\n\nPANIA: Because we never knew each other at school or anywhere else. There you are \u2013 mystery solved.\n\nOSIP: Of course we did. There's only one school in the village. It's perfectly true we weren't in the same classroom together...I'm a year or two older than you are.\n\nPANIA: Ten years and you're just barely literate. _(To INNA.)_ That janitor nonsense he vomited up? Pure invention. You can add 'fraud' to his list of unforgettable attributes.\n\nOSIP: I respect and admire you deeply.\n\nPANIA: You accuse me of murder, that's what.\n\nOSIP: I only talked about a body in the snow.\n\nPANIA:...and there's bodies of chickens and dogs as well as, occasionally, human remains. What gave you the notion I'd be interested? Am I a...what is it now...taxiderm?\n\nOSIP: It's a short distance only from here. We'd definitely have to say circumstantial.\n\nPANIA: Circumstantial of what?\n\nINNA: Maybe someone's heart gave out when you were moving the train.\n\nPANIA: We'd have recovered it...sent for the doctor. _(To OSIP.)_ Well? I'm waiting.\n\nOSIP: I'll tell you...everything I know. He was already dead...dunno how many hours...but half-buried beneath the latest drift.\n\nPANIA: How enlightening! And what does Volkov say about the matter?\n\nOSIP: Volkov?\n\nPANIA: Village constable, yes. _(To INNA.)_ Watch closely now. _(To OSIP.)_ You naturally first thought of contacting him, the man in charge.\n\nOSIP: No, frankly, I didn't.\n\nPANIA _(To INNA.)_ : Shit-laden scared of him, see? _(To OSIP.)_ You didn't? Wasn't that naughty of you?\n\nOSIP: I thought first of the Mafia.\n\nPANIA: Oh, I see, yes, the \u2013\n\nOSIP:...Mafia, yes, of a Mafia hit. Things have gone seriously wrong in recent times in this village. _(To INNA.)_ I tremble for you when I think of the world you're being reborn into. Though things couldn't have been all that pleasant in your place of confinement, the world we've come to know now is not the same one we've known for years. Cases of perversion were once isolated, perhaps even removed altogether. Harder, yes, there was a finality. But absolute outcomes served absolute values as well \u2013 absolutely. Now what there is is a corruption they're calling 'a necessary'. Necessary what, of course, no one says. _(To PANIA.)_ I half-thought that Volkov was mixed up in it, see?\n\nPANIA: Oh, but surely you trust Volkov, don't you? You must have known him at school.\n\nOSIP _(Whispers.)_ : You can keep a secret?\n\nPANIA _(Playing along.)_ : Yes, what is it?\n\nOSIP: I will tell him, yes, but in my own time. _(Whispers.)_ It's all part of my plan to trap him.\n\nPANIA: Congratulations.\n\nOSIP: I'll know how far he's involved by his reactions, you see. I'll assume his complete innocence straight away. Then I'll assume that he's squarely involved \u2013 his own man did it. And I'll determine the truth from his reaction to my forthright declaration: 'I found a body in the snow!'\n\nPANIA: Lunatic.\n\nINNA: I think that's very sound.\n\nPANIA: You're both lunatics. _(Yawns.)_ It's not yet 8.00...and I've been up for a day and a half... another full day ahead. You want nourishment? Here, take it in both hands. _(Pushing bread in OSIP'S direction.)_ Somebody died, and there's not enough bread. Early winter \u2013 there's not enough fuel. Nobody has enough money, so they barter their lives away...or they thieve. Twenty-five possible reasons why people die before their time. Don't come to me with your revelations \u2013 you don't even have a decent roof. So now tell me \u2013 did you do it yourself for the money?\n\nOSIP: You...what?\n\nPANIA: I'm not thinking Mafia money \u2013 you're just broke. Haven't eaten for two or three days, for one thing.\n\nINNA: Wait a minute. First you didn't believe him...\n\nPANIA: I'm just showing how easy it is. _(To OSIP.)_ Why come to me...why not the authorities?\n\nOSIP: I told you \u2013 he seemed to resemble your husband.\n\nPANIA: What does that mean? You were a friend of his...you were anxious to see justice done?\n\nOSIP: Well, no, as a matter of fact, I heard you'd hired a disabled girl...and I wanted to see if it was the girl that I knew from long ago.\n\nPANIA: And what's that to do with the body?\n\nOSIP: Body?\n\nPANIA: That you say you found, that's right.\n\nOSIP: Oh, well, I thought that his face looked familiar....\n\nPANIA: We're back to him now, are we?\n\nOSIP:...and I knew that you keep a shotgun in back.\n\nPANIA: Sounds a situation ripe for blackmail. You may remember, I've been asking what it is you want.\n\nOSIP:...all right, now, I've been asking you for nothing at all, except maybe your trust. That hunter of yours has tongues wagging...people are saying it's extreme...woman in your situation, a village fixture.\n\nPANIA: The extremity is the bread shortage \u2013 people shortly become desperate.\n\nINNA: I'm sure you just mean it as a precaution \u2013 you wouldn't actually assault anyone.\n\nPANIA: I'm a woman alone in this 'extreme situation'.\n\nOSIP: Yes, and people are wondering aloud about that. They wonder all sorts of things.....they always have. 'Why is she alone? She needn't be...she wasn't, we thought. What, exactly, is going on? Where is her husband? And what about this shotgun of hers...?'\n\nINNA:...that she calls her hunter.\n\nOSIP: People are apt to be inventive when they don't really know for sure.\n\nPANIA: It's none of people's fucking business...or yours. Do you hear me? You think you're all spies...that we're still in the Soviet era...you inform at will and gain preferment. All that's finished, or hadn't you heard? There's nothing to gain from it now \u2013 Russia has nothing whatever to give. You want bread \u2013 here it is. Take what you can of it. _(She begins tossing loaves.)_ You think you smell blood somewhere? So did I \u2013 across the tracks...early morning \u2013 the shocking appearance of a dead elephant! Its neck was broken, its head was severed from its body, like a broken toy...all except for the fresh elephant blood that came gushing out like a fountain!\n\nINNA: Oh, please, don't go on about it. _(Turning her head away.)_ Oy.\n\nPANIA: Don't be squeamish, dear \u2013 it's an excellent basis for your still-life examination of the elementary. You can't afford to turn away. A mere hunter can't account for such carnage. Besides, a hunter \u2013 both weapon and man \u2013 always makes such a clean kill. This was mutilation in the extreme...and she missed it \u2013 compounded tragedy.\n\nOSIP: She lives with her own mutilation, poor darling. You can't blame her.\n\nPANIA: What you mean is she's always been protected from real life. Her friend said as much \u2013 she's like a pet, a child's toy. If she's broken by everyday circumstances, she gets thrown away. That's the way it works now. If you can't afford a replacement, you steal or you do completely without. Children outgrow their toys anyway. Only idiots keep them as artefacts. The rest of us settle for real life. _(To INNA.)_ You want to go with him to see the body, dear?\n\nINNA: I...only suggested that you might.\n\nPANIA: What on earth for? I've just got back from an elephant shoot! That's enough for one morning in peacetime.\n\nINNA: What's the point of seeing anyway? I never knew your husband.\n\nPANIA:...whereas I did, you're saying... naturally enough.\n\nINNA: Well, you could identify the body if necessary.\n\nPANIA: What would I need to identify him for? That's not the murder in question. And I wouldn't be guilty if it was. I didn't need to kill him, can you understand that? No? He was dead to me already...through his child.\n\nINNA: What are you talking about, Pania?\n\nPANIA: Oh, for God's sake now, what do you think? And you try to convince me you're not Barbie Doll innocent. You'll have to try harder, that's all. _(To OSIP.)_ Hey, you, granddad \u2013 this is my confession, don't you want it?\n\nOSIP: No, you misunderstand \u2013 I never said you did it.\n\nPANIA: Congratulations. I didn't need to \u2013 I'd murdered his child.\n\nOSIP: I supported you right the way along.\n\nINNA _(Relieved.)_ : For a minute there I...didn't know what to think.\n\nPANIA: All it was, if you really want it, was that I found out the night after he left that I'd be having his kid if I kept it to term. Well, I also found out that I really wasn't bothered that he left...so what would I want with a daily reminder of him for the next twenty years. There's your murder.\n\nINNA _(Impulsively.)_ : Oh, dear Pania Andreyevna!\n\n**She embraces and kisses her repeatedly**.\n\nPANIA: What? We've got all this work to do, what? You want us to play with your dolls?\n\n**INNA goes down on her knees to recover the strewn loaves**.\n\nINNA: Oh, I'll work for you, don't worry. I'll work hard. Every day I'll get up and think, 'I'm going off to work now...and isn't it fantastic?'\n\nPANIA: All right, you don't have to get on the floor.\n\nINNA: Oh, but it needs to be done...it's soon time to open. And I can, you see? I'm perfectly supple. I'm that much younger than you are, and now I can see that you need me. Oh, how you've suffered in life. I can see what they mean now \u2013 Pania Andreyevna, the Bread Lady. What a glorious thing to be needed for life!\n\nPANIA: All right, you'll be sounding like him in a minute...and then I really will start to worry.\n\nOSIP: I need you, too, Inna Igorovna.\n\nINNA: You...what?\n\nOSIP: I love you, my little Inyechka. You're my ideal, you are!\n\nINNA: Oh, no, for...I'm sure you're mistaken, Mr Osip.\n\n**The revelation upsets INNA so that her spasticity becomes particularly uncoordinated. Her movements are not unlike those of a trapped butterfly**.\n\nOSIP: Oh, now, you've taken a spill. _(Crosses to her.)_ This is my fault...let me help you up. You've suffered far more than the rest of us.\n\nINNA: Oh, no, I haven't at all. I'm not at all what you think. Pania Andreyevna, please make it clear to him...I can get up...I'm far better doing it by myself.\n\n**There's some physical awkwardness between the two of them before PANIA crosses to sort out the tangle**.\n\nPANIA: One's as helpless as the other... and you're both making a pigsty of my shop. Out of the way, old man. Go on. Get on out of it. You think she wants to hear you drool? And what would people think if they came in right now? 'She doesn't know the first thing about keeping a shop.' They'd think I was giving looters an absolute invitation...and then I'd lose all my stock.\n\nOSIP: No, I don't think you understand.\n\nPANIA: I understand people, all right. The basic ingredients don't change all that much \u2013 what's there to understand?\n\nOSIP: But I tell you...I love her...I want us to marry.\n\nINNA: Oh, my God.\n\nOSIP: Yes, I'm sorry, my darling, for not telling you first...for telling her before I told you. There it is, though...it can't be avoided.\n\nINNA: But I don't love you...I mean, I don't know you. I haven't seen you since childhood, have I...you're something sweet from my childhood.\n\nPANIA: You mean he's ridiculous...a clown.\n\nOSIP: You don't know me at all. _(Of INNA.)_ She knows.\n\nINNA: Yes, I'm sure you're a very good person.\n\nOSIP: I've kept myself pure.\n\nPANIA: Ha!\n\nINNA: I'm sure of it. I can see it in your eyes. You can always read a person from their eyes... and your character...your smile begins in your eyes...that's a beautiful sign.\n\nOSIP: I knew him, you see...the...the general's son.\n\nINNA: What?\n\nOSIP: The man that brought you here from the home...reunite you with your mother...I knew him personally. He was a personal friend.\n\nINNA: Pavel Zakharin?\n\nOSIP: That's it \u2013 Zakharin. He was your guardian angel that day.\n\nINNA: I feel now that you must have known him. I'm glad you told me...especially glad now that we've met again.\n\nOSIP: Fate \u2013 that's what it is. God's will. And I'll be your guardian angel hereafter.\n\nPANIA: I've had enough of this.\n\n**She moves to force him out the front door, but OSIP suddenly discovers his reserve and stands his ground**.\n\nOSIP: You leave me alone...you hear me? We're conversing in unfathomed depth.\n\nINNA: I'm still your friend...I look forward to knowing you again. But it's like I tried to tell my friend Marianna. I need to move on my own first...learn who I am in the world. I've been sheltered all my life, really sheltered.\n\nOSIP: She's an angel.\n\nINNA: I'm nothing like that. I couldn't get on the metro myself at the age when Pania Andreyevna was tractor driving. I'll never drive a tractor, all right. But I am good for something in the world. I must be. God's given me the chance that He didn't give all the others. I've got to try it for them as well...for us all...for the future of us all.\n\nOSIP _(Focused on INNA but addressing PANIA.)_ : All that...and you think she's 'unclean'!\n\nPANIA: I never said that...I said you were...\n\nOSIP: What do you know about it anyway? You haven't understood a word of what she's saying.\n\nPANIA: I understand she needs to work.\n\nOSIP: She needs to be recognized for what she is.\n\nPANIA _(To INNA.)_ : Oh, God, now he's saying you're divine.\n\nOSIP: That's right...she is \u2013 a message from God for the nation.\n\nPANIA _(To INNA.)_ : And you think this is what the President meant in his speech? That's where sex leads you, my darling, right there, to him.\n\nINNA: I'm sure everything with him is as it should be.\n\nPANIA: What's that? Do you have any idea? Is it icon worship he's after? He might have you believing...you're for all the ages...that you're older than he is \u2013 the dirty young man.\n\nINNA: I think you're being hugely unfair.\n\nPANIA: And I say he's hugely indecent. Has he exposed himself yet? Do we have to wait till he starts exposing himself? He's been stalking you for years...magic roundabouts.\n\nOSIP: You want to leave what doesn't concern you alone.\n\nPANIA: I'm well inside the law here \u2013 sexual assault. I could fire on your assets and no one would say I'd done wrong.\n\nINNA: Oh, no, I beseech you both...anything at all except that. _(Tearfully, she gets hold of PANIA and wants OSIP to leave for his own sake.)_ You've got to believe that she means it.\n\nOSIP _(Joyous.)_ : Let her, let her...let her draw a bead. You and me are indestructible!\n\nINNA: Oh, my God, how did this happen?\n\nOSIP _(Starting off.)_ : Never give it another moment's worry \u2013 I'll see you tonight after work.\n\n**VOLKOV comes on, blocking OSIP'S retreat. He is a typical officer \u2013 ex-military, sadistic, here dressed in a formal overcoat and fur hat**.\n\nVOLKOV: You're out very early this morning, Pishchik.\n\nPANIA: Oh, I was beginning to think you'd never get here.\n\nVOLKOV: Don't worry \u2013 I've been trailing him for hours. _(To OSIP.)_ Why? What have you been up to?\n\nOSIP: What? It's bread day...wanted to get here early.\n\nVOLKOV: Wanted to be the first one, eh?\n\nPANIA: That's a lie for a start \u2013 he said he didn't want bread.\n\nVOLKOV: And it's not bread day on the slag heap, is it.\n\nOSIP: I watched the train coming in...watched them pushing the train. You ask her...ask Pania Andreyevna. There were fifteen of you, weren't there, dear.\n\nVOLKOV: And you'd have us believe you were that \u2013 what, hungry? \u2013 that you stood watching them all that time.\n\nPANIA: You didn't offer to help, did you? Bastard!\n\nVOLKOV: Oh, well, I expect he had other things on his mind. More important things.\n\nPANIA: Fifteen fair to ageing and infirm women. Lazy bastard.\n\nVOLKOV: Why don't you tell us what those important things were?\n\nPANIA: He came in here all fired up about a body. Lunatic bastard.\n\nVOLKOV: A body, eh?\n\nPANIA:...specifically, the dead body of a certain man.\n\nVOLKOV: Are you talking about a human corpse?\n\nOSIP: I found it...just lying there.\n\nVOLKOV: And so you came in here and frightened these women about it?\n\nOSIP _(To PANIA.)_ : I didn't frighten you, did I? Tell him the truth.\n\nPANIA: Of course, I was frightened. He came in here babbling about \u2013\n\nINNA: I wasn't the least bit frightened, and I'm sure he meant no harm. I knew him from childhood, for one thing.\n\nVOLKOV: But how is it he made his way in here? That in itself is suspicious... _(To OSIP.)_...isn't it? You seek the police if you find a body...that's anyone's first thought, their first impulse almost, in such a situation. To seek an alternative, well, that needs looking into.\n\nOSIP: All right, don't worry \u2013 I'll show you the body. You can't intimidate me.\n\nVOLKOV: I'm not trying to intimidate anybody \u2013 I'm just after the truth.\n\nOSIP: We're all after the truth...in our own way.\n\nVOLKOV: And you were thinking, were you, of maybe taking over my job?\n\nOSIP: Why did you follow me? Intimidation.\n\nVOLKOV: Well...you wander the streets at night...up on slag heaps, with corpses. You think that's normal behaviour?\n\nOSIP: You've never caught me at anything, have you?\n\nVOLKOV: There's always a first time...Go home to bed. That's where normal people are at this hour. _(Restraining hand.)_ Not now. You've got some explaining to do.\n\nPANIA: And there's a little matter of sexual habits.\n\nINNA: No, there isn't.\n\nPANIA: You stay out of it this time, dear. We've known this species a lot longer than you.\n\nINNA: But you don't know that about him \u2013 I refuse to believe...\n\nPANIA: That's right \u2013 he didn't declare himself wildly just now...propose 'marriage' to you? I must have misheard.\n\nINNA: You misunderstood anyway. He didn't mean marriage.\n\nOSIP: What else do you think I meant?\n\nINNA: Oh, my God, don't say that. _(To VOLKOV.)_ I knew him when I was a child, you see, sir.\n\nVOLKOV: What's he been doing with children?\n\nPANIA: There \u2013 you see? What did I tell you?\n\nINNA: He can't tell anything from that. _(To VOLKOV.)_ He did magic tricks for us.\n\nPANIA:...in a cripples' home he bribed the guardians to exploit with his 'magic tricks'.\n\nINNA: I tell you he never touched us!\n\nPANIA:...and I say he's got her spooked.\n\nOSIP: And I've loved you ever since.\n\nINNA: Oh, no!\n\nVOLKOV: That's good enough for me.\n\n**VOLKOV now moves on OSIP, while INNA still tries to reason**.\n\nINNA: You mustn't say things like that, Mr Osip \u2013 it isn't....people don't understand.\n\nVOLKOV: You hear what she said, 'Mr Osip'? It isn't nice...it just isn't done. You shouldn't say things like that...and you certainly can't be allowed to do anything. We've got to see to it that you don't.\n\n**With each stressed word, VOLKOV squeezes OSIP'S testicles harder, and INNA'S sobbing increases**.\n\nINNA: Let him go...can't you see...oh, please stop it, he didn't do anything. _(To PANIA.)_ My God, can't you see what you started? You've got to tell him to stop! _(Hanging onto VOLKOV.)_ You've got to stop this!\n\nVOLKOV _(Breathless.)_ : All right...I've stopped. I've stopped him and all. He won't do anything to anyone from now on. He's no danger to anyone.\n\nINNA _(Sobbing.)_ : He never was.\n\nVOLKOV: You really know nothing about him, dear. He's a menace. They all are, people like him.\n\nINNA: I don't care. I don't care what you say I don't know. I want you to leave him alone, all right? Just leave him.\n\nVOLKOV: I can't do that, dear. He's got to come with me...for his own protection if nothing else.\n\nINNA: Yes, I've heard what you do for people's 'own protection'.\n\nPANIA: Don't be a little fool...you want to anger him against you?\n\nVOLKOV: It's all right. _(To INNA.)_ I can't ignore the charges against him, sweetheart.\n\nINNA: Charges? I'm not making any charges. She's not either, are you?\n\nVOLKOV: I'm talking about the vagrancy...I mean, look at him, dear. He's disgusting. Do you really think we can have that disturbing people's \u2013 ?\n\nINNA: All right, I'm just like him \u2013 a menace \u2013 you'd better arrest me as well.\n\nPANIA: I told you to keep your mouth shut.\n\nINNA: Ask Pania Andreyevna what I'm talking about. She knows I was judged a pariah.....unclean. They locked me up for years, too.\n\nVOLKOV: I know that in Soviet times...\n\nINNA: What makes these times any better? Turning on him? What's he ever done that's so terribly different from anyone else? He struggles like the rest of us...he hasn't enough...of everything. How is that different from Pania Andreyevna? Because he wasn't fool enough to push a bloody train? Why did it need to be pushed anyway?\n\nPANIA: You couldn't have done it.\n\nINNA: But why should you? Is that post-Soviet freedom, is it? The train doesn't work, and there's not enough brown.\n\nPANIA: There \u2013 she's solved the mystery of the ages. We can all go home to bed.\n\nINNA: All right, laugh. But if Pishchik goes with him, I'm no longer working for you.\n\nVOLKOV: It's out of her hands, dear \u2013 he's broken the law.\n\nINNA: I don't care what he's done \u2013 that's what I'm prepared to do.\n\nPANIA: And after that, what \u2013 your exceptional art? _(To VOLKOV.)_ Perhaps she'll wait for him through prison.\n\nINNA: Is he going to release him?\n\nPANIA _(A beat.)_ : To hell with both of you...I don't care what you do. _(As INNA shuffles past her.)_ Hey, what do you think you're doing? I said I didn't care. You're not going to make me either. You can just put that back.\n\n**INNA has taken a brown loaf from a tray**.\n\nINNA: What if you were to release him in my care?\n\nPANIA: She doesn't know what she's saying.\n\nINNA _(Giving VOLKOV the brown loaf.)_ : But you know that's not true, don't you?\n\nPANIA: Congratulations.\n\nVOLKOV _(To PANIA.)_ : You've got to admit...he's not nearly as bad as some that are still roaming wild.\n\nPANIA: You release him so they can play doll's house together. Doesn't that make you uplifted?\n\nVOLKOV: Don't worry \u2013 I'll still keep my eye out for him.\n\nINNA: I promise you \u2013 he'll be all right.\n\nPANIA: They're going to 'complete each other's destinies' \u2013 how fulfilling can life ever be?\n\nVOLKOV: Well, I'll tell you \u2013 if there's nothing else I can do for you...\n\nPANIA: Don't let on that any of this happened under my roof. I'd never live it down.\n\nVOLKOV _(As he leaves.)_ : I'm not even sure what time it is.\n\n**A pause**.\n\nPANIA _(To INNA.)_ : If you're serious, you'll get him out of here fast. Look at the damage he's caused.\n\nINNA: He's got a cottage nearby? _(As PANIA crosses the space out of view.)_ Don't worry, I'll come right back.\n\nPANIA _(Slightly off.)_ : I mean it \u2013 I don't care what you do. You can go off to live in a cave, for all I care. I'm still looking for the third degree of sleep.\n\n**INNA is now alone with OSIP**.\n\nINNA _(Bending over him.)_ : Mr Osip...are you conscious?\n\nOSIP _(Suddenly very much alive.)_ : So you've considered my proposal...and it's yes?\n\nINNA: I can't marry you, my dear. It wouldn't be decent, like they said. I'm...I'm young enough to be your daughter. But I do want to thank you most sincerely, and to greet you with all of my heart. I never dreamed about anything like this.\n\nOSIP: I'm very young, they tell me, in my heart.\n\nINNA: Yes, in a most particular way \u2013 I can see it \u2013 you've no age at all. _(Kisses him.)_\n\nOSIP: That means \u2013\n\nINNA: I know very well what it means, my dear.\n\nOSIP:...no! That I'm not going to die. Isn't that...!\n\nINNA: Are you sure you can walk all right?\n\nOSIP: Walk? Wings of steel, I have. I can just about fly.\n\nINNA _(Cheerful.)_ : You'll have to wait for me then...I'm slow.\n\n**They leave the space supporting each other as the lights fade**.\n\n### THE SWEETHEART ZONE\n\n#### A Play\n\nPerformed as part of a second-year module by Drama students of the University of Exeter, under the direction of Martin Harvey, on 23 January 2007.\n\nMISHA SHATKOVSKY | Nick Cosslet\n\n---|---\n\nNINA | Kate Kalvinos\n\nZHENIA | Katie Brayne\n\nNADIA | Harri Dobby\n\nLUDKA | Nikki Valvis\n\nGRANNY | Lucy Holland\n\n'LOLA' | Alice Beaumont\n\nSHURA | Leah Peregrine-Lewis\n\nYULIA | Ruth Ley\n\nJACQUI | Chloe Nicholson\n\nENGLISH TANYA | Lucy Hill\n\nPOLYA | Emma Tomlinson\n\nRAISA | Lucy Drewett\n\nAMIRA | Kelly Ward\n\nJADVIGA | Esther Finney\n\n'EVA BRAUN' | Annabel Norbury\n\nDR. ORLOVA | Helen Bryer\n\nKOMANDIR | Catherine Downes\n\nASKA | Victoria Horn\n\nVALODIA | Simon Cox\n\nThe women's zone of a labour camp on the Volga in post-Soviet Russia.\n\n##### Character descriptions\n\nMISHA SHATKOVSKY Career secret policeman, forties, with a family and a reasonably sized flat. Against stereotype, Misha is ordinary and honest.\n\nNINA One of the zone's leaders by virtue of her personal history. In her late thirties. Father blinded in combat, so Nina looked after him until his death. Her crime was to lead a protest against punitive taxes on street merchants, incurring the hostility of local Duma member, a politician with dubious business interests.\n\nZHENIA Grade One (most severely) disabled, she was imprisoned probably for nothing more serious than alleged vagrancy. In and out of various homes for most of her 32 years.\n\nNADIA Another of the zone's leaders. Alleged to have carried out a mercy killing, she probably did nothing more than watch helplessly as her daughter died of a miscarriage. In her mid-40s.\n\nLUDKA Severely mentally challenged and imprisoned for it, she had a job painting railway sidings and such. Fellow workers on one job got drunk, fell asleep and died from the paint fumes. Ludka was blamed. In her forties. She has a strong sense of being beset by evil demons, everywhere.\n\nGRANNY Zone matriarch, close to 60, she conceals a dark secret about why she is here.\n\n'LOLA' Now 30, she married a teacher in her teens, and he made life miserable for her with his jealousy. She murdered him as a result.\n\nSHURA Early thirties, vampish, the boss's favourite until a new woman came on the scene, then Shura went berserk.\n\nYULIA Early thirties, in prison for a trifle. Wants to be popular.\n\nJACQUI Late twenties. Street-wise by virtue of an unconventional lifestyle. Served short sentence for drugs, worked as a maid for a Mayfair prostitute.\n\nENGLISH TANYA Late twenties. First extended stay abroad. Works in a travel agency.\n\nPOLYA and RAISA Late teens, transfers from juvenile camp. Arrested for vagrancy, they have lived so far by various dubious means short of prostitution.\n\nAMIRA Mid-30s. Jewish in a widely anti-semitic Russia, she faces immediate prejudice and plays up to it by currying favour with the prison authorities.\n\nJADVIGA Early thirties. A Balt in post-Soviet Russia, she instantly arouses the antagonism of anyone aware of her origins. Probably imprisoned for letting her temporary residence permit lapse.\n\n'EVA BRAUN' Early thirties. Relatively new prison warder, specially recruited for this assignment.\n\nDR. ORLOVA Early forties. Her namesake was Soviet Russia's first mega film star, and Dr. Orlova benefits from this to the extent of inspiring maternal trust among the inmates. She deserves it, though her loyalty, in the end, is toward the authorities.\n\nKOMANDIR Early forties. A Soviet-era 'aparatchik' with little patience for western-style humanism. These women, to her, are scum.\n\nASKA Late twenties. Imprisoned for a trifling offence, she has been put in solitary because it is known that she is the sister of a secret policeman who may be involved in counter-government activity.\n\nVALODIA Mid-20s. New to the prison service, ex-military. Probably served in Chechnya.\n\n**The audience doubles as a group of official observers, presumably watching the prisoners from behind wire netting. A lone man, MISHA SHATKOVSKY, stands out. The women periodically address their remarks to him. A searchlight combs the compound. Then the prisoners are marched out in three groups of five. They are in blacks with white headscarves round their heads and grey identity triangles on their chests. The compound consists of the yard, a large work shed, the dormitory and 'infirmary'\/psycho unit. Two of the groups are made up of seasoned inmates. The third, of new arrivals, is desultory. They huddle together in twos, with a lone inmate hovering round them. She is a Latvian named JADVIGA**.\n\nTANNOY: All right then, inmates. Nice and tidy and wide awake!\n\nSHURA _(Second group.)_ : What the hell for?\n\nNADIA _(First group.)_ : They don't want to torture us in our sleep.\n\nYULIA _(Second group.)_ : Afraid we won't feel the full effect.\n\nNINA _(First group, shielding her eyes from the searchlight.)_ : I don't like the way the sun's glaring at me.\n\nLOLA _(Separating herself from the first group; directly addressing the audience.)_ : What the fuck do you think you're doing?\n\nMISHA: We're only here to observe you. Try not to notice us.\n\nGRANNY _(First group, older.)_ : You don't want your name taken, do you, my dear?\n\nLOLA: What do you think we are, monkeys...at the fucking zoo? _(Making a face at MISHA.)_ That's right. I'm talking to you.\n\nYULIA: Don't wet yourself over it.\n\nLOLA: Yeah? And what do you know about anything? _(To MISHA.)_ Is she your girlfriend or something?\n\nMISHA: Can't you stand friendly interest?\n\nLOLA: From a sex maniac?\n\nNINA _(Crossing toward the third group.)_ : Granny's right, though. If you start anything, they'll take you down.\n\nMISHA: Take you down where? Come on, girls. I'm on your side.\n\nLOLA: That's a laugh and three quarters! _(To NINA.)_ D'you see them?\n\nNINA: I see them all right \u2013 I'm not sightless, am I? _(To MISHA.)_ Try not to take such obvious pleasure. We're just like everybody else, even with our clothes off.\n\nSHURA: Maybe he finds us attractive...irresistible even.\n\nLOLA: I'll give them all irresistible...rend the atmosphere thick with words they hope they've forgotten.\n\nMISHA: We're not here to piss you off, I promise you.\n\nYULIA: I heard them say he's a KGB spy.\n\nNINA _(To the third group.)_ : So you girls have only just arrived.\n\nTANNOY: Get back into line there.\n\nLOLA: Why don't you come here and lick my arse!\n\nNINA: At least you can turn off the searchlight \u2013 it's daybreak.\n\nTANNOY: Go back to your own line, Petrova.\n\nNINA: I'll stay right here with the rookies.\n\nJADVIGA: We don't want troubles for you.\n\nNINA: You're not Russian?\n\nJADVIGA: None of us. From Latvia, me.\n\nPOLYA _(Whispering, to RAISA.)_ : What's a fucking Balt doing here?\n\nNINA _(To POLYA and RAISA.)_ : You're Russian, aren't you? Sparrows from juve camp.\n\nRAISA _(To POLYA.)_ : Don't tell her anything.\n\nNINA: You're going to get awfully fed up...nobody else to talk to.\n\nJADVIGA _(Of the remaining two.)_ : From USA.\n\nNINA: Is that right, Americans?\n\nJACQUI: English. _(To MISHA.)_ Do you understand, mister? Sir? They locked us up here for no reason.\n\n**MISHA smiles vaguely**.\n\nTANYA: He thinks you're flirting with him. _(To MISHA.)_ But we're not on the game, honest.\n\nJACQUI: What makes you think he understands you any better?\n\nTANYA _(Whispers.)_ : I don't give a flying (fart).\n\nMISHA _(Whispers.)_ : I wish I could help you.\n\nNINA _(Smiling, with gestures.)_ : My name...Nina.\n\nJACQUI: Oh. Right. My name's Jacqui...and this is Tanya.\n\nNINA: Tanya? You? But then you're Russian.\n\nJACQUI: No, no, no, no...Tanya is English as well. We are English.\n\nTANYA: I'm going to be mental, at this rate.\n\nNINA: He is...KGB. You know, secret policeman?\n\nTANYA: Oh, God. That's all we need.\n\nTANNOY: All right, you toe-suckers. Back into line.\n\nSHURA: Where do you think we are?\n\nLOLA: You tell us, please, what we're doing here.\n\nAMIRA: I can tell you...I just heard all about it.\n\n**AMIRA'S entrance causes a small stir. No one speaks for a moment. POLYA has her eyes fixed on ZHENIA, a disabled woman in the first group. Her mate, NADIA, becomes gradually upset**.\n\nSHURA: All right, nark, tell us.\n\nAMIRA: I keep telling you, I'm not a nark. I'm just back from 'psycho', aren't I?\n\nYULIA: So you keep telling us. _(To her group.)_ Something very dodgy about a person that has to announce her good intentions all the time. Don't you agree with me, ladies?\n\n**General hubbub**.\n\nAMIRA:...and something even worse about a shit-stirrer.\n\nSHURA: I couldn't agree more. You're that as well...which makes you even \u2013\n\n**General hubbub**.\n\nGRANNY _(To AMIRA.)_ : Just tell us your news, lovie. Are they going to reduce our rations again?\n\nAMIRA: They've appointed a new 'executive officer' to oversee us... 'Exec'...get it?\n\nNINA _(To JADVIGA and the others.)_ : A 'zek' is the slang term for 'inmate'.\n\nYULIA: Are you trying to say he's an ex-convict? _(General: 'Oh, charming...' 'That means he'll be even worse.')_\n\nAMIRA: I'm trying to say it's a woman...what you asked for, after all.\n\nLOLA _(To the audience.)_ : Did you lot have anything to do with it?\n\nAMIRA: Aren't you always sending protest letters to Moscow? Talk about shit-stirrers. _(To MISHA.)_ She's the one that brought you here, right?\n\nMISHA: We're not here on anyone's say-so. But can't you tell me, please. What did she mean by 'psycho'?\n\nSHURA _(To AMIRA.)_ : We remember how you didn't sign.\n\nMISHA: What did they do to you there?\n\nAMIRA _(To MISHA.)_ : I can't talk to you. _(To NINA.)_ I'm telling you now how they've gone and answered you. I just heard about it.\n\nYULIA: You didn't catch sight of her, did you?\n\nAMIRA: No, but I heard her referred to as a terror, nicknamed Eva Braun, to prove it.\n\nSHURA _(Derisively.)_ : Eva Braun!\n\nAMIRA _(Pointing.)_ : And there she is!\n\n**'EVA BRAUN' appears above, imperious, like a Soviet commissar**.\n\nEVA BRAUN: You were told to get into line...like you do every morning. Don't for a second imagine my regime's going to be any different, except, very possibly, harder.\n\n**General grumbling**.\n\nAMIRA: See what I mean?\n\nEVA BRAUN: Well, isn't this just what you asked for \u2013 a woman exec?\n\nYULIA: Why don't you meet us, then, on our level, face-to-face?\n\nLOLA: And are you responsible for that lot out there? Is he also part of your 'regime'?\n\nEVA BRAUN _(Descending.)_ : You'll find it easier to pretend they're not there.\n\nNADIA: What, like the bugging devices planted by our beds?\n\nGENERAL: 'Don't say a word'; 'Can't even talk out here now' etc.\n\nEVA BRAUN: Well, you could try confiding your troubles to him.\n\nNADIA: One Day in the Life of Masha Denisova.\n\nJACQUI _(Nudging TANYA.)_ : I think they're talking about those people out there.\n\nTANYA: Tourists at Auschwitz.\n\nEVA BRAUN: Well! People are very interested. Aren't you meant to be the world-famous 'sweetheart zone'? You'll probably be asked for your autographs.\n\nLUDKA _(First group, addressing audience.)_ : Can you really help us? What's your reason for being here? Someone cast a spell...didn't they?\n\nNADIA: It's an obvious trap. Can't you see? That's the face of a government thug.\n\nEVA BRAUN: All right, then, keep stum. You'll be making my job that much easier.\n\nLOLA: I think she's a regular cream puff. _(To EVA BRAUN.)_ That right? Your name's loaded with irony, isn't it?\n\nYULIA: What, like you're loaded with horse shit?\n\nNADIA: What are you getting on Lola's case for?\n\nNINA: Of course, we're sweet.\n\nGRANNY: How else could we look after each other?\n\nNADIA: Well said, Granny dear.\n\nEVA BRAUN: Then it's obvious we're a match. My name... _(Jeers.)_ For those who don't know it, my name is Volkova...Marina?\n\nLOLA _(Derisively.)_ : Say that again, please. Vulcan?\n\n**General laughter**.\n\nEVA BRAUN: I heard your last chief was called Lionheart.\n\nLOLA: Lionheart?\n\nSHURA _(Laughter.)_ : And Littlebrain!\n\nYULIA: Liverlips!\n\nEVA BRAUN: And yet you complained about my predecessor! _(Noise.)_ Why else do you think I'm here?\n\nLOLA: She'll tell us next she's our union rep!\n\nEVA BRAUN: Okay, we've had our friendly joke. _(Noise.)_ The men's camp is only over that wall.\n\nLOLA _(To MISHA.)_ : Is that where you're from?\n\nYULIA: Maybe they're really lovers. _(To MISHA.)_ That right, lover? _(Makes a lascivious gesture toward him.)_\n\nMISHA: I'm really trying to help you. _(Makes a farting noise.)_\n\nEVA BRAUN: I'll send you there one at a time if we don't get some order here in the next twenty seconds! _(Points gun. Hush.)_ That's more like it. I'll be calling the roll here, and with each name your individual task for the morning.\n\nNADIA: What do you mean, task?\n\nSHURA: We all know what we do here.\n\nNINA: All except the rookies.\n\nSHURA: They'll find out.\n\nEVA BRAUN: That's why we're having a general review. Besides, I mean to try something new my first day. You're in favour of that, aren't you?\n\nNINA: Don't make out you're giving us options.\n\nGENERAL: 'You tell her, Ninochka.'\n\nNINA: We sew gloves and very little else...sometimes mailbags.\n\nLOLA: The occasional woollen knicker, eh, Granny?\n\nEVA BRAUN: You're still going to do that, don't worry. But this compound is more like a pig pen, come on.\n\nSHURA: It's more like a prison, what do you want?\n\nEVA BRAUN: A garden of tulips! _(General hooting.)_ Isn't that just what you asked for...what you want? _(Above the din.)_ What's the point of having a woman in charge?\n\nLOLA: To practise homosex without getting caught!\n\nNINA: Hang on, ladies! _(To EVA BRAUN.)_ Are you serious, an actual garden?\n\nYULIA: To grow carrots and beetroot and spuds?\n\nEVA BRAUN: Oh, no, I said flowers \u2013 tulips, or daffs, if you want.\n\nYULIA: What's the point of that? You can't eat them.\n\nSHURA: What do you bet?\n\nLOLA _(To the audience.)_ : I bet it's on account of you lot.\n\nSHURA: I'll swallow anything that doesn't taste of slurry. Even tulip bulbs.\n\nEVA BRAUN: You lot are just wasting time.\n\nSHURA: You eat sunflower seeds, don't you? Russian chocolates?\n\nYULIA: You don't have any, do you?\n\nEVA BRAUN: I have the rota all worked out.\n\nNADIA: Rota? I thought she said 'roll'.\n\nSHURA _(To EVA BRAUN.)_ : You don't have to worry, dear. There's no place for us to escape to.\n\nLOLA: I think we ought to discuss this first...maybe put it to a voice vote.\n\nEVA BRAUN: Belova, north garden.\n\nNADIA: 'Belova, north garden'...what does that mean?\n\nEVA BRAUN _(Pointing.)_ : Over there.\n\nNINA _(Whispering, to LOLA.)_ : I don't think we're given a choice.\n\nEVA BRAUN: Budina, north garden...Zhukova, north garden...Larina, north... You'd better hurry.\n\n**NADIA, POLYA, AMIRA and SHURA have separated from their respective groups but do not form a group of their own**.\n\nNADIA: What is this, Get-Acquainted Day?\n\nSHURA: Let's suppose for a moment I don't want her _(AMIRA)_ near me.\n\nPOLYA: Raisa and I are the new ones. Are you separating us just for spite?\n\nNINA: I'll go with you, whatever happens.\n\nPOLYA: Well, thanks, but I'd just as soon...\n\nNADIA: Wait a minute. Look. We're already in groups. Why don't we just take the north side here, and then they can \u2013\n\nEVA BRAUN: Not all of you have been allocated gardening duties.\n\n**General hum**.\n\nNADIA: I don't understand. I thought this was gardening detail.\n\nEVA BRAUN: Some of you will have to continue with sewing. The quota, I understand, is 50 pair per person per day.\n\nLOLA: She probably wants them to make up the shortfall. _(To EVA BRAUN.)_ Now I see how you came by your nickname.\n\nNINA: All right, how many and who, exactly?\n\nJACQUI: I haven't a clue what's going on.\n\nTANYA: I think she's checking names.\n\nJACQUI: Then maybe they've missed us.\n\nJADVIGA: She give us all our job.\n\nJACQUI: You speak English!\n\nJADVIGA: Little only.\n\nEVA BRAUN: Everybody's on gardening duty except...\n\nLOLA:...except...?\n\nJADVIGA _(Whispering.)_ : They ask for us dig.\n\nJACQUI: Dig? Dig what?\n\nTANYA: Our own graves, it looks like.\n\nEVA BRAUN:...Fedina and Kuzina.\n\nGENERAL: 'What!' 'Why only two?' etc.\n\nEVA BRAUN: You two will carry on with the sewing. _(Consulting her paper.)_ Mironova, south garden....\n\nYULIA: There's not going to be enough garden.\n\nNADIA: I think there's been some mistake.\n\nLOLA: There's no mistake, sweetheart. They planned it this way.\n\nSHURA: Yeah...why only two?\n\nNADIA: You've chosen the two who never do sewing. Why?\n\nEVA BRAUN: Exactly. It's time they started.\n\nLOLA: I told you.\n\nPOLYA _(To RAISA, though her eyes have been on ZHENIA.)_ : I was afraid they'd put her with us.\n\nNADIA: Back me up, will you, Nina. _(To EVA BRAUN.)_ But they can't sew. Zhen, in particular. By rights, she shouldn't be here.\n\nEVA BRAUN: Correct me if I'm mistaken. She is here...and this is a labour camp, not a Black Sea resort?\n\nNADIA: But...just look at her.\n\nEVA BRAUN: I can't take my eyes off her \u2013 so what?\n\nNINA: She's crippled, Chief.\n\nNADIA: Any fool can see that.\n\nNINA: She's never gone near a machine.\n\nEVA BRAUN: What is she then, the camp mascot?\n\nZHENIA: Don't make excuses for me, girls. _(To EVA.)_ I can do the work.\n\nEVA BRAUN: You hear that? She's itching to start.\n\nNADIA: All right, let me go in her place. She'll dig a wonderful garden for you. That's more or less what she does. She's the zone's Cinderella. That's \u2013\n\nEVA BRAUN: I know what that is. Every camp has one.\n\nNADIA: Well, she's ours.\n\nNINA: And Ludka doesn't work a machine either \u2013 she's \u2013\n\nEVA BRAUN: Cinder's assistant, I suppose.\n\nNINA: Ludka, dear. This is the new chief's first day on the job. She'd be grateful if you could give her your name.\n\nLUDKA: Last night I assumed the form of the Grand Duchess Anastasya. _(Looking at MISHA.)_ Rudolpho asked me to the ball. I don't like to torment a heart.\n\nNADIA: Satisfied? It's for real, you know.\n\nMISHA: I don't know anything about it.\n\nEVA BRAUN: Don't encourage them then. They said you're just here to observe.\n\nNINA: She'd be likely to sew all the gloves without fingers. That's the point.\n\nEVA BRAUN: You're saying she belongs in the psycho ward.\n\nNINA: All we're saying, in all honesty \u2013\n\nEVA BRAUN: That's the only other place for either of them. Now you'll find the spades and other gardening implements...\n\n**General din**.\n\nEVA BRAUN _(Voice raised.)_ : This isn't a matter for general debate.\n\nLOLA _(To the audience.)_ : What do you think? Is this a proper way to run a prison? It's not true there are no political prisoners any more. Tell the media, will you? We're all political.\n\nJACQUI _(To JADVIGA.)_ : Can you get someone to get in touch with our Embassy?\n\nJADVIGA _(With gestures.)_ : Cannot hear you.\n\nJACQUI: The British Embassy. We're the victims of false arrest. It's some scuzz-ball pimp that's the real culprit!\n\nTANYA: That's going to make them drop everything.\n\nEVA BRAUN: Shut the fuck up, the whole fucking lot of you! _(Beat.)_ Where's that Balt? _(Spotting JADVIGA.)_ You.\n\nJADVIGA: I'm from Riga, Madame. Jadviga Glazevsky.\n\nEVA BRAUN: Thrilled to bits that you told me that, sweetheart. _(To the group.)_ You all see her? She's the new Cinderella. Come with me.\n\nJADVIGA: But I have to translate....tell them\n\n**They go off and there is a momentary hush**.\n\nNADIA _(Breaking the silence.)_ : 'Eva Braun' was far too demure, if you ask me. She's more like the Beast of Belsen \u2013 what's her name? Ilse...Ilse Krupp.\n\nYULIA: Don't tempt fate.\n\nPOLYA: Where's she taken her, the psycho ward? What's that?\n\nYULIA: Punishment wing...they keep you on half rations...chained to the wall.\n\nAMIRA _(Going from group to group.)_ : She really belongs in the sick bay...on her way to the cemetery. She's the last one we want as our Cinderella. No doubt that's the idea...to kill us off painfully...one or two at a time.\n\nNINA _(Explaining to POLYA and RAISA_.): She's the one we elect to test things out for us.\n\nAMIRA: Yes, that's it \u2013 fetch and carry and cook and clean. Haven't you noticed the sores on her arms and forehead? I got close enough.\n\nNINA: She's trying to say she's syphilitic, like every other foreigner who finds her way in here. _(To AMIRA.)_ That's right, isn't it?\n\nAMIRA: Not just syphilis, my dears...speed.\n\nNINA: One day you'll go too far.\n\nAMIRA: Threaten me all you like. I saw her lesions.\n\nYULIA: Then maybe you've got it.\n\nJACQUI _(To TANYA.)_ : Do you have a clue what they're talking about?\n\nTANYA: They took her off, for some reason.\n\nAMIRA: Sick. She has... _(Gestures.)_...you know?\n\nJACQUI: I don't like the sound of that. You suppose she means Aids?\n\nTANYA: Why should that matter to you?\n\nJACQUI: I'm not HIV.\n\nTANYA: We both are, thanks to your friends.\n\nJACQUI: Then you've nothing to worry about either.\n\nMISHA: That is not what the guard said.\n\nJACQUI: You speak English!\n\nTANYA: Can you get us out of here? They said you're...\n\nMISHA: They are fools only. I will can translate. The guards asks her to carry. _(Of AMIRA.)_ This one...make scandal. Must be what they call parrot.\n\nJACQUI: We had them in England.\n\nTANYA: That's what you were, eh?\n\nMISHA: Possible we help each other.\n\n**A solitary cell in the psycho ward. ASKA is trying to carry the slop bucket across the space to the back, where she will be permitted to deposit the waste. But the bucket is heavy and she falters. There is the sound of clanking keys. ASKA cowers. VALODIA comes on**.\n\nVALODIA: Jesus Christ! _(Crosses to her. Helps her to her feet.)_ You didn't try to lift that yourself, did you. That's a two-zek job.\n\nASKA: And I'm in solitary.\n\nVALODIA: Yeah, I know, but the least they could do is help you with it...if they want to keep you alive.\n\nASKA: Maybe that's it. Maybe they just want me dead.\n\nVALODIA: No, they want you alive, take my word for it.\n\nASKA: Why should I? Why the fuck \u2013 ?\n\nVALODIA: 'cause I wouldn't be here otherwise. _(Helps her sit down on a bench.)_ Are you all right now? And don't worry no more about the bucket. I'll empty it now and tell them they've got to help you from now on. You're not on hunger strike, are you? That's just stupid. It's so cold in here, you risk not being able to have babies anyway. If you starve yourself as well...\n\nASKA: I'm not on strike.\n\nVALODIA: Yeah, well, that's a good thing. For you, I mean. They couldn't give a...a Chechen's balls. They'd just take this long plastic tube through your nose...and you wouldn't be able to breathe...\n\n**ASKA breaks down**.\n\nNo, it's all right. You say you're not on strike, and you eat regular. Not a banquet, I know...\n\nASKA _(Wailing.)_ : I don't know why I'm here!\n\nVALODIA: Yeah, well, I know. You're here for the same reason I'm here. To find out exactly what it is you...Don't you really know, honest? Jeez, that's going to land me right in the... _(Pointing to the contents of the bucket.)_ in here. _(A beat.)_ They couldn't have made a mistake.\n\nASKA: Oh, yes, I'm sure that's what they've done. It's all a mistake. It has to be. I've done nothing the other girls haven't done. In fact they've done even worse. Murder, one or two of them did.\n\nVALODIA: Believe me, you're not like the others. By their lights at least.\n\nASKA: But just tell me. What have I done? I took a handbag. That's what I was accused of anyway. And then when I was interviewed by the police, I wouldn't pay over a bribe.\n\nVALODIA: That's just your cover story.\n\nASKA: It's what happened.\n\nVALODIA: They think there has to be more.\n\nASKA: But can't you tell me...what they think it is.\n\nVALODIA _(A beat.)_ : All right. I'm going to lay it on the line. They don't think you're just like the others...the minnows. Even the nymphet that bumped off her old man. The ones you hear three times a week over Prison Radio, including the pissheads. 'I drank to get over my sad love affair...and then I stole in order to drink...and then my lover took over my flat, with his family.' God in heaven, you're not one of those. Are you?\n\nASKA: I didn't even take the bag.\n\nVALODIA: I believe you. So do they. About the bag anyway. But what I don't get \u2013 and what you're going to help me with \u2013 is why they should call you 'political'.\n\nASKA: But I'm not. I swear to you...cold sober. What's 'political' anyway?\n\nVALODIA: They're convinced you can tell me.\n\nASKA: And who are you? You're not another bent policeman, are you? You offered to help me...you said someone would help me...with the bucket. You remember? There's no such thing as political any more. You know that.\n\nVALODIA: And you never once carried a message for someone?\n\nASKA: I mean...look at me. I'm shitting myself!\n\nVALODIA: Not even for a member of your family?\n\n**Scene shift**.\n\nNADIA _(Dovetailing.)_ : What about it, ladies? Are we going to let Zhenia do the sewing on her own? _(To the audience.)_ Any doubt in your mind that it was deliberate? I'm telling you. She has no business being here. It's not like she committed a crime, except being born like she is.\n\nYULIA: You want us to withdraw our labour...maybe even a hunger strike. And for what, exactly?\n\nNADIA: What do you mean, for what? For Zhenia!\n\nPOLYA: The last thing I want is a hunger strike.\n\nSHURA: That's right. I want to know when we're going to eat.\n\nYULIA: It's normally right after roll-call.\n\nSHURA: And it's normally no more than gruel.\n\nRAISA: It's not so much hungry as thirsty...my God!\n\nSHURA: Don't tell me. They gave you fish when they met you off the Moscow train and drove you here.\n\nYULIA: A special sort of herring \u2013 added salt to make you crazy with thirst.\n\nRAISA: Have you got anything...?\n\nYULIA:...and then force you to drink your own piss. _(Starting to move.)_ I keep my jar right next to Shura's.\n\nRAISA: Keep your jar...of your own urine?\n\nYULIA: That way you never run short.\n\nRAISA: No, don't...show me.\n\nYULIA: I was going to offer you some.\n\n**General laughter**.\n\nPOLYA: Can't you tell when your leg's being pulled?\n\nNINA: I don't know. The way the tea's brewed, it does often taste like a form of waste. _(Passes her a mug of tea.)_ This was fresh this morning.\n\nRAISA: What is it?\n\nNINA: Bless us and keep us...tea.\n\nRAISA: What else is in it?\n\nNINA: I think Granny added a strawberry leaf for flavour.\n\nRAISA: Th-that's all right.\n\nNINA: I know it's all right...it's freshly brewed tea!\n\nPOLYA: She's never going to believe you.\n\nNINA: She'll die of thirst then. _(She downs the tea in one.)_ Care to smell my breath?\n\nLOLA: Any more where that came from? I mean tea.\n\n**The women begin to move toward the area where the first group stood. ZHENIA sets about serving them**.\n\nNADIA: What are we going to do about Zhenia?\n\nZHENIA: And Ludka, too, don't forget.\n\nNADIA: Why don't we take it in turns to do her shift?\n\nZHENIA: You can't ask them to do that.\n\nNADIA: You keep out of it. Hang on. Ladies? If I go with her to sew the first ten...who's going second? You'll help us, won't you, Neen?\n\nNINA: If that's what we decide, sure.\n\nNADIA: Who's got a better idea?\n\nZHENIA: I can do what I can by myself. I'm not helpless.\n\nNADIA: Sure, you are. She's setting you up, like Lola said.\n\nLOLA: Lola said what? Who's quoting my words in vain?\n\nNADIA _(To MISHA.)_ : You're authorized to report this, right? It's deliberate provocation. You can tell them so.\n\nMISHA: I can tell them anything you like, if only you'll tell me. Can you tell me about the punishments?\n\nLOLA: In your dreams, nark.\n\nNINA: Whether it was or it wasn't provocation, what's helping Zhenia going to achieve?\n\nNADIA: Are you serious? I can't believe you said that, Nina.\n\nNINA: I mean we can refuse altogether. _(To POLYA.)_ What? _(POLYA whispers something.)_ And I don't think we should ask the new girls, anyway.\n\nNADIA: All right, but there's nearly a dozen of us.\n\nPOLYA: I meant she makes me nervous.\n\nRAISA: She's afraid to be around people like her.\n\nNADIA: People like who?\n\nRAISA: Ask her about it. I'm going to get some of that... _(Pushes past the women near her to the area where tea is being served.)_\n\nNADIA _(To NINA.)_ : Did you hear that? _(To POLYA.)_ Someone just might take a sudden and violent dislike to you.\n\nGRANNY: I think I have an answer for what we might do.\n\nAMIRA _(To LOLA.)_ : That Balt has been gone an awfully long time.\n\nLOLA: Why don't you ring her up?\n\nAMIRA: I tell you they've taken her...\n\nGRANNY _(Overlapping.)_ : We resume our practice of knitting woollens, those of us that feel the cold.\n\nYULIA: You're talking about all of us.\n\nNADIA: That's what I said \u2013 we take it in turns.\n\nGRANNY: No, no. With this, none of us go anywhere near the machines.\n\nNADIA: Including Zhenia.\n\nGRANNY: Especially Zhenia. _(To ZHENIA.)_ We must do what we can to protect you, my dear.\n\nZHENIA: I can't let you take the risk, Granny. Ludka and I'll just have to do what we can. We'll cope, eh, Lud?\n\nLUDKA: Anyone with black eyes has the Devil himself for a near relation. _(To MISHA.)_ Not a brother-in-law ten times removed. I mean blood close. _(She whimpers.)_\n\nMISHA: What's the matter with her? _(To LUDKA.)_ Were you beaten?\n\nNINA _(Crossing to comfort her.)_ : You frightened her, what do you think? _(To LUDKA.)_ Just look into our eyes, Lud. They're only blue.\n\nLOLA: I'll give anybody five buckshee to spit in his eye.\n\nNINA: All right, Lo.\n\nLOLA: You'd do it for big American bucks, wouldn't you, Lud?\n\nNINA: I said, all right.\n\nLUDKA _(Tearfully.)_ : Why were you reading my letters?\n\nMISHA: She's crazy. I'm not here to hurt anyone.\n\nNINA: And just how does that excuse you? You're a government nark.\n\nMISHA: Maybe I'm more of a friend than you think. But I just asked what's wrong with her.\n\nLOLA: She's bewitched.\n\nGRANNY: Would somebody pass me that bag from over there? Don't you do it, Zhenia. _(To LOLA.)_ You're full of beans this morning, Larissa Osipovna.\n\nLOLA: I didn't know you knew my real name.\n\nGRANNY: Right over there behind the sluice buckets.\n\nLOLA _(Crossing to the buckets.)_ : Did you all hear that? Larissa, not a tramp.\n\nNINA _(To RAISA.)_ : We called her Lolita when she told us she married a teacher when young. _(To LOLA.)_ We called you that out of affection.\n\nLOLA: You were jealous as hell.\n\nNINA: Well, that in itself is a tribute.\n\nLOLA _(Of the bag's contents.)_ : What the hell is this!\n\nGRANNY: Bring it here and I'll show you.\n\nLOLA: What did you do, rob a warehouse? _(General reaction.)_ Did you know this was here, girls? Reams of it! Finest cotton! How in the world did you ever get...! _(Showing NINA.)_ It's the real thing, eh, Nina?\n\nNINA: I'm sure it is.\n\n**The women gravitate toward that area**.\n\nSHURA: Who did you find to bribe?\n\nGRANNY _(Conscious of MISHA.)_ : Don't show it to everybody.\n\nNINA: This looks like surgical...It's from the infirmary!\n\nGRANNY _(To MISHA.)_ : They'd sell us anything for the right price.\n\n**LOLA collects the bits the women are inspecting and takes the bag over to GRANNY. A number of them follow her**.\n\nSHURA: I'll bet most of those were taken off corpses.\n\nNINA: They're not going to be missed then.\n\nGRANNY: Most of it, really, was confiscated.\n\nSHURA: You mean it's our own stuff.\n\nGENERAL: 'I had a silk shimmy they were pleased to take off me'\/ 'I had some silk smalls'... etc.\n\nNINA: How did you manage to buy it all back? With what?\n\nGRANNY: You want me to tell you my secrets? _(As she sorts through the bits.)_ I think someone should tell our foreign guests about the zone's underwear policy.\n\nAMIRA _(With gestures.)_ : You must...you know...strip!\n\nTANYA: Fuck off.\n\nJACQUI: Why? Is it some lesbian ritual?\n\nMISHA: It's a joke, huh?\n\nGRANNY: No, no, no, we don't ask you. But first step punishment...strip to your bras and \u2013\n\nJACQUI: \u2013 knickers, I got you. Punishment for what?\n\nGRANNY: Anything they want. Zhenia, the cripple, was ordered to work on machines. If she does not, she will need to strip and stand out in the cold.\n\nJACQUI: Jesus, for how long?\n\nMISHA: Maybe when she will dead.\n\nAMIRA: That's right.\n\nGRANNY _(In translation.)_ : But you did not tell them about their culottes. If you have anythings like French underwears, they will either be stolen or you will die of the cold.\n\nTANYA _(To JACQUI.)_ : I could kill you right now, myself...save them the trouble.\n\nJACQUI: Oh, grow up. These ladies are coping, aren't they? And I made it through two years in Holloway.\n\nTANYA: Jailbirds!\n\nJACQUI: No, it's like travelling in India. The quickest way to get sick as a budgerigar is to order their English cuisine everywhere you go. If you eat what the natives eat \u2013\n\nTANYA: You can spare me your eastern philosophy. _(To MISHA.)_ But you're an official of some kind, aren't you? You've got to speak up for us.\n\nMISHA: No, no, I am no things. If they catched me speak with you even \u2013\n\nJACQUI: You must have contacts at our embassy...You've probably been posted to London in your time?\n\nTANYA _(Frustrated.)_ : Anybody out there been raped and speaks English?\n\nJACQUI: Oh, eff off!\n\nMISHA: I do not understand. You want to know about \u2013\n\nJACQUI: She's having a laugh. That's what journalists ask when they arrive in a foreign city. She must have read that somewhere.\n\nMISHA: English journalists.\n\nJACQUI: Of course. It's a joke, really.\n\nMISHA: Different cultures. First question in Russia is who you must pay.\n\nTANYA: I'm not arsed. _(To MISHA.)_ What'll you take to get us out of here? How much?\n\nMISHA: I coulds do little for you, really.\n\nTANYA _(To AMIRA.)_ : All right...who do you know?\n\nJACQUI _(As though AMIRA is fluent.)_ : You see, Tanya's pissed off 'cause she was arrested for being a prostitute. But...I mean she's not...We were both staying at this tourist hotel.\n\nTANYA: Can't you see you're wasting your \u2013\n\nMISHA: Intourist? They arrested you there?\n\nJACQUI: I guess that's where it was. And it was a mistake, of course. I mean...it's ridiculous.\n\nTANYA: Will you shut the...up? _(To audience.)_ Don't pay any attention.\n\nJACQUI: But I've worked for a prostitute back in London...in Soho, although she moved up market to somewhere in Mayfair.\n\nTANYA: Great. They'll think you're my lesbian pimp.\n\nJACQUI: Oh, no! I was Gillian's maid. Gillian, the woman I worked for. And I clean up for her... answer the door to clients. It was all very...nice really. I know that must be hard to imagine.\n\nMISHA: I do not think any imagine it.\n\nJACQUI: Well, I know. That's what I'm \u2013\n\nMISHA: I do not know that I image it. You were in prison for cleaning house?\n\nJACQUI: That was something else entirely.\n\nTANYA:...for being a druggie.\n\nJACQUI: For having a partner who was a druggie. He dealt drugs, that was the point.\n\nTANYA: The point is, he left you holding the stash...and she's right. Nobody knows what you're talking about or could give a fuck.\n\nJACQUI: I don't care. I'm as narked off about it as you are, and I needed to tell somebody. So. Thanks for listening.\n\nTANYA: They had no choice.\n\nJACQUI: Yes, they did. Oh, yes. I almost forgot. I'm more used to this than she is. That was my real \u2013\n\nTANYA _(Dovetailing.)_ : If there's anybody out there who knows any English...\n\nJACQUI: Being raped is optional, I guess.\n\n**Overlapping this, the other inmates are seen taking an interest in GRANNY'S project, so AMIRA makes her way over there**.\n\nSHURA: Do you know what? This isn't too bad. As ideas go.\n\nGRANNY: It'll save many a kidney. _(To POLYA and RAISA.)_ That's what I can't understand about young girls. You want to risk your lives every time you go out \u2013 half naked, for fashion.\n\nPOLYA: You got me wrong, grandma. I'm not stripping on no one's account.\n\nRAISA: That's true. The bathhouse might be another town, for all the notice she takes. _(Drains her mug.)_ Great pee, by the way. That strawberry leaf is like sugar.\n\nPOLYA: Why don't I give you real pee? You're almost stupid enough to drink it.\n\nNINA _(Suddenly bursting out laughing.)_ : I remember once riding in a bus, this drunk got up and all the way down the aisle he kept muttering, 'Right side, idiots; left side, whores.' And finally, when he got within two seats of me, the woman stood up and announced: 'You're quite wrong. I am not what you called me. I never had a lover in my life!' To which he replied, calm as Christmas, 'In that case, you sit over there!' _(Laughs.)_\n\nAMIRA: And which side of the aisle were you?\n\nSHURA _(Overlapping.)_ : How is it you never gave these to us before?\n\nYULIA: Before what? It's been thirty degrees for the last umpteen weeks. You want to have fried kidneys?\n\nNADIA: This would do for you, Ludka darling. _(Holds out her hand.)_ Come sit by me. That's right. Now what we want you to do is take hold of the stitching in each piece of cloth and undo it, so that we then have many more pieces of cloth.\n\nYULIA: That brings up an interesting problem, though, doesn't it?\n\nNADIA: What do you mean?\n\nYULIA: Which side of the bus would Lud have been on?\n\nSHURA: She'd have had to stand in the middle!\n\nPOLYA _(Mocking.)_ : And what did you have for your breakfast?\n\nRAISA _(Giggles.)_ : Fried kidneys and piss!\n\nNADIA: You see that? They're only encouraged.\n\nYULIA: Oh, for God's sake.\n\nSHURA: No, no. You're quite right. Sorry, Nadia. I'm sorry, Lud. Seriously. Here, look. I'll get you started. Come on.\n\nNADIA: Just leave well enough alone.\n\nNINA: How can you wonder they treat her as strange when you protect her the whole time?\n\nNADIA: Is that what you really think?\n\nGRANNY: She thinks everyone is the devil.\n\nSHURA: You trust us, don't you, Ludka? My eyes are azure blue...and I haven't got a tail.\n\nNADIA: Shu-ra.\n\nSHURA _(Showing her the material.)_ : You take hold of it like this...and if we were doing it professionally, we'd have scissors, of course. But that could be classed as a weapon.\n\nGRANNY: You want this? _(Holds out a small implement.)_\n\nSHURA: You must be a sorceress \u2013 yes! It's tiny but...this is exactly what it was made for.\n\nPOLYA: Got any more?\n\nSHURA: It's Luda's job.\n\nLUDKA: If it's mine, then let me have it.\n\nSHURA: Certainly, my dear.\n\nNADIA: She's not going to hurt herself, is she?\n\nSHURA: She can't even hurt the stitch. _(To LUDKA.)_ Silly Nadia.\n\nLUDKA: Let me...have it! _(She seizes hold of the material and tries to rip it apart with her hands, merely growing in frustration.)_\n\nNADIA: What are you forcing on her?\n\nSHURA: You heard what your friends said, didn't you? Butt out of it. Liuda's just fine.\n\nYULIA: Can you remember the paint work you did, Lud?\n\nLUDKA _(Continuing to struggle with the material_.): I painted sidings.\n\nYULIA: I know you did. You painted the football stadium, too.\n\nLUDKA: Not the stadium...are you crazy? _(Laughs.)_ I painted the seats.\n\nYULIA: I beg your pardon, my dear.\n\nSHURA: And then they asked you to paint the changing rooms, didn't they.\n\nLUDKA: Don't remember.\n\nSHURA: I think it's important that you try to, my lovely. Can you repeat after me? 'I...was... raped.'\n\nNADIA: All right, get out of it...both of you.\n\nSHURA: That's what happened, though, isn't it?\n\nNADIA: Nina, help me with these maniacs, for God's sake. Have Zhenia prepare her some breakfast.\n\nNINA _(Arm round LUDKA.)_ : Sure, but we can't watch over them every second. _(To SHURA.)_ What did you do anyway?\n\nSHURA: I was helping her with the stitching. Nadia's the one that said she should try.\n\nNADIA: I'm not the one that upset her.\n\nSHURA: Who said she's upset?\n\nNADIA: 'Repeat after me!'\n\nLUDKA: I remember being locked in a room in the cellar of...what was it?\n\nYULIA: Stadium.\n\nLUDKA: That's right. You asked me about that, didn't you? I was there with two men, and they told us to paint the whole room. But the room went dark, and the men fell asleep and the paint smelled just terrible. I climbed out through a window. And next day they told me the men didn't wake up.\n\nNINA _(To NADIA.)_ : Is that what actually happened?\n\nNADIA: I don't know.\n\nNINA: Where did the rape come in?\n\nSHURA: She must have been handled at some point.\n\nNINA: Listen to me, Lud, very carefully. Okay? Did those men ever touch you?\n\nLUDKA _(Angry and laughing.)_ : I told you \u2013 they went to sleep!\n\n**Perhaps a collective sigh**.\n\nYULIA: Did any man ever touch you?\n\nNINA: No, I'm sorry. That's too much for her before breakfast.\n\nZHENIA _(Partially in view.)_ : I've been waiting since you asked me...how long is it now?\n\nAMIRA: We were told not to eat until Speedy came back with the tools.\n\n**They ignore her, but only SHURA and YULIA follow LUDKA into the dining tent**.\n\nRAISA: Aren't you eating?\n\nPOLYA: What do you think?\n\nNADIA: Both of you don't mind starving to death. Very interesting.\n\nPOLYA: That one's plain loco...and the crip makes me cringe.\n\nNADIA: I warned you about that already.\n\nPOLYA: I just want to know where I am. They told me prison, but now I'm not sure. And what are you all doing humouring them? Is that your punishment? The place I was at before, they'd have left it to us to get rid of the likes of her.\n\nGRANNY: And where was that, my dear?\n\nPOLYA: Does it matter? That's another thing. We were always told not to reveal why we were inside. It wasn't the done thing to ask.\n\nNADIA: And why was that, do you think \u2013 it was unspeakably bad?\n\nNINA: Our pasts are what draw us together. They inspire us.\n\nGRANNY: We confess our sins. Isn't that right, Polya?\n\n**LOLA has been helping GRANNY sort through the old clothing. She addresses GRANNY but intends the other women to overhear**.\n\nLOLA: Sure, I'll bite. I'm not ashamed of killing my husband. _(To the audience.)_ You got that, folks? And I was unfairly arrested for it, too.\n\nGRANNY: He was well connected.\n\nLOLA: He was a positive shit...I mean positively, he was \u2013\n\nSHURA: He caught you cheating on him, admit it.\n\nLOLA: He saw me sharing my lunch on the bench with a schoolmate! We were 16 years old. You want to know what peeping is? Let me tell you. Boris would even hide out in the bushes.\n\nNINA: Waiting to catch you making love?\n\nLOLA: Watching...what...I did...in class.\n\nSHURA: With your schoolmate...\n\nLOLA: With the textbook! Lenin's additions to dialectical materialism, and there is my devoted spouse slobbering away like a diseased dog. Homeless, even. They're going to round him up, gas him and flog his fur for carpets.\n\nSHURA: Except that you got there first.\n\nLOLA: I smothered him with a cushion, yes. One of mother's finest heirlooms. Would you like me to give you the context, Inspector? He locked me out of the kitchen. Clever, eh? The noted professor who's so distinguished, he won't even allow his toy wife to have lunch...or breakfast or dinner...or anything at all. 'This is my food', he calmly announces one day before lunchtime. 'If you're hungry, fine, go earn your own.' And the law backed him up in this. Support a nymphet? She deserves to starve. Of course, I killed him. He could have been eaten to death.\n\nGRANNY _(To ZHENIA.)_ : Fetch her some porridge, love.\n\nLOLA: No, I don't mean I'm hungry now.\n\nGRANNY: Of course, you're hungry. Every one of us is, even poor Liuda Antipova...transformed into phantoms of our real selves.\n\n**ZHENIA has gone toward the dining area**.\n\nLOLA: That's really what happened to me.\n\nNINA: And I think that's saying too much for Ludka.\n\nGRANNY: And when we talk of personal histories, of course, yours is the one we all look to.\n\nNINA: I'll help with the sewing, Granny, gladly. But if you don't mind...\n\nGRANNY: She had a clothing stall in the local street market \u2013\n\nNINA: I really wish you wouldn't.\n\nGRANNY: Better tell it yourself then, quick. The mayor wanted you out...\n\nNINA: He wanted to shut us down, yes. My mother's protest was much more heroic.\n\nGRANNY: But yours brought them out in the streets. 'People's enemy' was invoked in your honour.\n\nNINA: It's true that we're all political prisoners.\n\nGRANNY: Some more than others, my dear.\n\nNINA: They raised the licensing tax...three times more than I made. They wanted us 'off the street'. _(Tearfully.)_ Poor Galusik was killed for her efforts. I was merely...chased out of town. _(Changing the subject quickly.)_ My mother was nearly arrested for talking to a neighbour in front of the Orthodox Church. They let her go when she told them my father fought at Stalingrad. He did fight at Stalingrad. He was blinded there. Had his fingers blown off as well.\n\nAMIRA: And I tell you again and again I was arrested after the bastard I married threw my son into the fire. I called the police on him, so they arrested me.\n\nNADIA: Have you no particle of shame? These women have risked their lives many times, and you mock them.\n\nAMIRA: I'm a Christian! Like she is.\n\nNADIA: And tell us next how you took Calvary's burden on your back...up the steps to Gethsemane.\n\nAMIRA: Fine. You tell me how I come to be here.\n\nSHURA: We all think you're a nark.\n\nNADIA: I don't know. You probably kicked a halfwit to death.\n\nAMIRA: But they're doing away with such people...so they'd thank me.\n\nNADIA: Or maybe it was a policeman.\n\nAMIRA: Well, then you should.\n\nNADIA: I don't know. All any of us knows is...you're no hero.\n\nGRANNY _(After a moment.)_ : Most of you must think I bribed someone a pretty penny for these fineries...the same as I bribed the city officials when I made and sold my moonshine for an unheard of profit. And you could have carried on with the town's blessing, Nina, if only you'd given the right person the right price.\n\nNINA: I had nothing to give.\n\nGRANNY: There's always something that someone wants badly enough.\n\nNINA: It's true. I have a beautiful, talented niece that the mayor saw. She sang in the choir of the Baptist Church, and he asked if she'd sing at a friend's wedding. I wouldn't let her. She was the age Lola was when she married, and the mayor was scarcely proposing. So. I said she couldn't and paid for my niece's salvation with my own. No doubt Lola would say it was worth it.\n\nGRANNY: But I paid with actual money, didn't I? The same as I did for these clothes. How is it that I, too, am in here with Lola and Nina and the rest of you sinners?\n\nNINA: Does it matter?\n\nNADIA: You were double-crossed.\n\nGRANNY: It might matter, yes, to our sisters that think secrets are better kept secret. _(Beat.)_ Our neighbour Petrovich was the best DIY specialist in our area. Larger. His renown was county wide. Friends would ask him to lay a forecourt and he'd do it himself, single-handed, within a day. And all I needed him for was a fence...repair job. Not even a new fence...which he could have erected, no problem, in almost the same amount of time. Why belittle a man for doing precisely what you ask...and for free. He didn't want payment, even knowing my moonshine, itself, was renowned. I did give him lunch, of course. He came promptly at eight and finished just before midday...and it would have taken a sizable tremor to uproot that fence now. So a decent lunch and a bottle or two of my finest would have been the least he should have expected. And I daresay that's what he'd have got if things had taken their natural course. Well, they didn't, needless to say. They couldn't have gone more disastrously, for both of us, though it was the last thing you might have thought. Petrovich became suddenly and violently ill, during lunch, and it was while wondering what to do that I suddenly realized what I'd done. It wasn't an attack of food poisoning or a bottle I'd got badly wrong. It wasn't the moonshine at all, as it happened. I'd given him sulphuric acid instead.\n\nNINA: Oh, my God.\n\nLOLA: Is there any reason you're doing this to yourself?\n\nGRANNY: I can't have you thinking I'm heroic. There are causes, yes...then there are crimes.\n\nNADIA: But it was a mistake, a genuine \u2013\n\nGRANNY: Oh, utterly. On his part as well. Believing it to be vodka, he downed it in one. It was only with the reaction that we realized. I did, at any rate. And then, well, you can imagine. Paralytic with horror...with fear.\n\nNADIA: But...you did manage to save him. He recovered.\n\nGRANNY: I need you to hear this...all of you.\n\nNADIA: Just tell me the worst didn't happen.\n\nGRANNY: It did. You've heard about those people who, having witnessed a murder, go into agonising detail about how badly they felt. The child-killer that weeps at the thought of his victim's suffering. I couldn't call for help.\n\nNINA: But there was no telephone...anywhere in your street. Your neighbours were all out at work.\n\nGRANNY: I was afraid of them laughing at me.\n\nNADIA: Laughing at you? He was fighting for life!\n\nGRANNY: Don't I know it! All the way to the street...clutching his throat. Then he lay there... and I watched him. _(Sighs.)_ Have you ever seen a poisoned rat die? It takes hours.\n\nNADIA: This is becoming unbearable.\n\nGRANNY: Eventually, Petrovich's son found him outside their front door...still alive. And they got him to hospital quick. But by then, of course, the poison was well in his bloodstream...and he...Like that.\n\nNADIA: I hope your neighbours held you in ridicule.\n\nGRANNY: They did, my dear, don't worry. Petrovich, I told you, was renowned. And even though the court acquitted, my neighbours condemned me outright. I'm in prison escaping their sentence.\n\nLOLA: But that doesn't make sense. 'Officially' \u2013\n\nNADIA: Does that matter?\n\nYULIA: And I thought I was awful...for stealing a goat! Honestly, that's all I did. It was for the kids.\n\nSHURA: My crime was even less...let me tell you. _(Beat.)_ What the hell am I doing this for? _(She tosses down the material.)_\n\nYULIA: She'll only sell it back to the screws for a huge profit...which we won't know about. I'd like to have her kidneys...boiled.\n\nSHURA _(Rising and moving toward the dining hut.)_ : Right, you gammy bitch. When are we having this porridge you been boasting about?\n\nAMIRA _(Following.)_ : I'll sort us out something better, don't worry. In a place like this, you need to know who to trust...and that's never everyone. They've been falsely suspicious of me since I came.\n\nPOLYA _(Getting up as well.)_ : It's all one happy sluice bucket, you ask me.\n\nNADIA _(To NINA.)_ : I can't ask for their help now.\n\nNINA: I think you were pushing your luck from the start. Optional hunger strikes are one thing. The food's repulsive anyway. Asking someone to do someone else's job...I'm sure some would rather put their daughters on the streets.\n\nAMIRA _(Her voice.)_ : Get out of the way, girl. You've been told what to do, and it certainly isn't this.\n\nNADIA _(Starting off.)_ : I'd better go see...\n\nNINA: Don't provoke anybody.\n\n**JACQUI and TANYA cross to the others**.\n\nJACQUI: What do you suppose is going on over there?\n\nMISHA: Are you getting ready to eat?\n\nNINA: We're getting ready for a fight.\n\nMISHA: But then I really don't understand, unless it's a joke.\n\nJACQUI: Why should you? You just got here, like we did.\n\n**The sound of smashing crockery**.\n\nTANYA: No, nothing at all...just a prison riot.\n\nZHENIA _(On the verge of tears.)_ : She's crazy...all I did was pour tea.\n\nAMIRA _(Her voice.)_ : She's useless!\n\nLOLA _(Her voice.)_ : Well, then, let her be useless. What difference does it make?\n\nZHENIA: But I wasn't. She's just...she wants to do \u2013\n\nNINA: I think we all know what she's like.\n\nAMIRA: Go on and laugh. I was just trying to help her.\n\nJACQUI: Then why is she crying?\n\n**JADVIGA returns with a barrow of gardening implements**.\n\nJADVIGA: Well! Here they are. Anyone want to help me unload?\n\n**The women come out to see what has happened. They cross to JADVIGA with increasing interest. General: 'She was serious!'\/ 'See if there's any veg.'**\n\nAMIRA: What do you think I am, your skivvy? What's going on?\n\nGENERAL: 'What kind of seeds are they?'\/ 'Does anybody remember what side they're on?'\/ 'You mean idiots or whores?'\/ 'What difference does it make? Just grab a spade and dig.'\n\nAMIRA: You can't do it like that. Let me through.\n\nYULIA: Oh, lighten up, you old sore spot.\n\nSHURA: She probably thinks it's a trick.\n\nAMIRA: I don't say it's anything like that, but \u2013\n\nLOLA: \u2013 you hate the thought of not being in charge.\n\n**The women begin selecting implements and going off to different areas. Their conversation, too, generally overlaps**.\n\nAMIRA _(To JADVIGA.)_ : You were a long time coming. Why was that?\n\nLOLA: She had a lot to carry...for God's sake.\n\nJADVIGA: I had to go for a physical.\n\nAMIRA: Something's wrong with you, right?\n\nSHURA: All Cinderellas submit to a physical.\n\nAMIRA: But something's wrong with her, she said.\n\nYULIA: We've heard you say it.\n\nJADVIGA: Something is wrong with me.\n\nAMIRA: You hear? You hear? _(To JADVIGA.)_ Go on...tell us.\n\nJADVIGA: They say I have gall stones. _(Hoots of laughter greet this.)_ It's not funny.\n\nYULIA _(Continuing to laugh.)_ : No, of course it's not, my dear. _(Perhaps an arm round her.)_ Only she told us you had...\n\nJADVIGA: Had what? Please?\n\nAMIRA: Go on \u2013 tell them.\n\nJADVIGA: How would you know anyway? We've never spoken.\n\nYULIA _(To AMIRA.)_ : Your turn \u2013 speak up. _(Beat. To the others.)_ She doesn't know a fucking thing!\n\nAMIRA: I know who's next for the psycho ward.\n\nSHURA: Only because you're a nark. You tell them who's next.\n\nAMIRA: You do it yourselves, by lodging protests.\n\nJADVIGA: Excuse me, I don't want to be the cause of any fighting between you.\n\nSHURA: You won't be the cause, believe me.\n\nJADVIGA: No, you don't understand. It's not right for me to have enemies when I've just come.\n\nAMIRA: I'm not the one to make enemies.\n\nSHURA: Ah. So you're not the one who made a point of telling us...a point of it, mind you \u2013\n\nYULIA: Leave it out, Shur. _(To JADVIGA.)_ I think you're right.\n\nJADVIGA: I'm going to need everyone's help with my diet.\n\nYULIA: You're the right one to be Cinderella then. It's your job to taste the food. If anything seems too salty for you, tell us. We'll do something about it quick.\n\nAMIRA: You might even succeed in getting veg...to plant, I mean.\n\nJADVIGA: Oh, yes. _(Announcing it.)_ She told me everybody's got to begin working their patch of soil and then, when it's good and soft, to start laying the flower seeds in very neat rows.\n\nLOLA _(From her patch.)_ : She talks as though she's never had a garden! _(To the audience.)_ Anybody here with a camera? This is probably what they mean by a photo op.\n\nJADVIGA _(To her group.)_ : It feels strange giving the orders my first day.\n\nYULIA: No doubt intended to keep you from being too friendly with the inmates.\n\nJADVIGA: No, I told you \u2013 I want no enemies. I'm only relaying their message. _(To JACQUI and TANYA.)_ Do you understand, please?\n\n**AMIRA, JACQUI and TANYA have crossed to these women**.\n\nJACQUI: But we need to ask you something.\n\nJADVIGA: They want you also to plant...flowers, see?\n\nJACQUI: No, that's not it.\n\nTANYA _(Overlapping.)_ : Who is it you saw, Jadviga? It could be very important to us.\n\nJACQUI _(Overlapping.)_ : She's convinced that they made a mistake.\n\nJADVIGA: Please, girls. My English, you know, very bad.\n\nAMIRA: They're trying to say that they're here 'by mistake'. We all by mistake, darlings. _(With a nod toward MISHA.)_ They think he's here 'to help them'.\n\nJADVIGA: Oh, yes, thanks for reminding me. Somebody told me they want to interview the two English girls.\n\nAMIRA: That's a fucking lie!\n\nJADVIGA: No, that's what she told me. _(To JACQUI.)_ You must go when they call you.\n\nJACQUI: Who, that guard that was here?\n\nJADVIGA: No, it was the woman that saw me, I think it was Dr Orlova. She was a doctor anyway.\n\nAMIRA: Yeah, her name's Orlova. _(Beat.)_ Maybe they want to give you physicals.\n\nJADVIGA: She said they'd be here some time today.\n\nAMIRA: They're...coming here? _(General reaction.)_ It's a trick, I told you. She's probably a plant.\n\nJADVIGA: No, I give you my Bible oath \u2013 that was the mess\u2013\n\nMISHA _(To JACQUI and TANYA.)_ : You understand? They want to talk to you here...I want to talk to you, too.\n\nJADVIGA: He speaks English?\n\nAMIRA: Fucking KGB fuck.\n\nTANYA: And I don't really give a fuck. I'll talk to anyone... _(Audience.)_... even you lot. _(Raised voice.)_ You hear me? I am in-no-cent.\n\nJADVIGA: Why you don't dig while you wait?\n\nTANYA: Eh? What's the point? I'm getting out of here.\n\nJACQUI: Why not plant a tree in your honour? Give them something to remember you by?\n\n**The two of them start to handle the soil. MISHA signals to JADVIGA**.\n\nJADVIGA: I've just come here. I don't want any trouble.\n\nMISHA: And I'm head of the official delegation from Moscow!\n\nJADVIGA _(Crossing over to him, timidly.)_ : What is it you want, please?\n\nMISHA _(Softening.)_ : When you went downstairs...to the psycho ward...\n\nJADVIGA: The where?\n\nMISHA: So they refer to it as the hospital. Never mind. Did you hear anybody being tortured? Anybody? Anything?\n\nJADVIGA: You're mistaken, sir.\n\nMISHA: I'm aware how they threaten you.\n\nJADVIGA: Don't say I was threatened...I wasn't. I just went in a shed for the tools. I heard nothing at all.\n\n**A stifled cry returns us to the cell where EVA BRAUN stands over a terrified ASKA. VALODIA stands by. EVA has a weapon**.\n\nEVA: Hit her again.\n\nVALODIA: I'm sure she's telling the truth.\n\nEVA: I tell you to do it. They say her husband has links with the vermin that murdered those children. You want them on your conscience?\n\nASKA: I haven't got a husband!\n\n**VALODIA slaps her**.\n\nEVA: But whoever did it is well-placed to murder the government. Isn't that right?\n\nASKA: I don't know who you... _(VALODIA makes as if to slap her.)_ I have a brother who's in the KGB. The FSB as it is now.\n\nEVA: There it is _(To VALODIA.)_ , you see?\n\nASKA: But he's nobody really. And he wouldn't tell me anyway. They don't even say who they are.\n\nVALODIA: That's generally true. I had a schoolfriend...and for years afterwards. Nobody ever told me.\n\nEVA: Shut up. You want us to think you're a spy?\n\nVALODIA: I don't see the point in beating her to death. She can't tell you anything.\n\nASKA: Valodia's right. I haven't seen or heard from my brother in years. They're told to lose touch with their families. Mine was.\n\nEVA: Valodia's right? _(To VALODIA.)_ You were ordered to get the truth out...and you fucked her instead? I could shoot you both for high treason.\n\nVALODIA: I'd take the gun off you. _(To ASKA.)_ She's not going to shoot, sweetheart.\n\nASKA _(With the gun at her head.)_ : I'll tell you whatever you want to know!\n\n**VALODIA grabs EVA from behind and chokes her**.\n\nASKA: She's not dead.\n\nVALODIA: I think she is.\n\nASKA: Tell me she's only unconscious.\n\n**VALODIA checks to see that EVA is dead. Then he begins dragging her off**.\n\nASKA: Please don't treat her like that. _(He laughs.)_ I'm serious. Show some respect.\n\nVALODIA: She wouldn't have thought twice about killing you. _(Stops dragging her.)_ She only just came. She won't even be missed. _(Smiles.)_ You want to pray over her?\n\nASKA _(Sobbing.)_ : I shouldn't be here!\n\nVALODIA: That's for fucking sure.\n\nKOMANDIR _(Her voice.)_ : Has she talked yet?\n\n**VALODIA and ASKA exchange anxious looks**.\n\nVALODIA: I don't think she knows all that much, Komandir.\n\nKOMANDIR _(Coming on.)_ : That means you're afraid to find out. _(Seeing EVA sprawled out.)_ What's that?\n\nVALODIA: I thought she was one of the zeks, to be honest. She came at me with a knife.\n\nKOMANDIR: She came at you? You're a liar. She happens to be a new guard.\n\nASKA: That's right, he's lying. She went crazy...came at us both with a gun. He had to do it. I'll tell them.\n\nKOMANDIR: Shut up. There's a delegation from Moscow, out front. I'm not going to tell them one fucking thing. Neither are you, sweetheart. You're going to behave like the rest of them out there, and someone nicknamed Eva Braun, she never existed.\n\nASKA: She's a Jew who liked going with Germans. We...we sent her over the wall!\n\nVALODIA: Don't worry \u2013 they'll never ask.\n\n**In the area, nearest the food hut, NADIA and ZHENIA join NINA and a gnomic GRANNY. NINA continues to work away at the material**.\n\nNADIA: Are you going to help me with the gloves, Neen? The two of us could polish off both women's quotas.\n\nNINA: I tell you, we're better off doing no work...and Granny's right about these thermals.\n\nZHENIA: And I think you're right. _(To NADIA.)_ There's no reason for you to do my work for me.\n\nNINA: No sense in it either.\n\nNADIA: You both make me sick. No, that's wrong. Forgive me. Saddened is what I feel about you. This isn't the Soviet era, all right? There's no need for quotas or time sheets or... 'doing our share'. _(Pointedly at GRANNY.)_ We can afford to help our neighbour. _(To ZHENIA.)_ You're right, my dear. I'm not doing it for you; I'm doing it for my own daughter.\n\nZHENIA: Oh. Is she...?\n\nNADIA: If friends don't exist, there's only families...and I'm doing what I have to to look after mine. Oksana's daughter came down with meningitis and her no-good-of-a-husband left them flat. 'I got no money for wooden legs', he announced, and she brought the girl home to me. Only I couldn't do very much for the mite. So Oksana started drinking...then and more. I'd have told her to go back to Moscow if it hadn't been for Katia. Turns out that's just what I should have done. She strangled the girl by the river that Thursday, then disappeared. I haven't seen her since. My neighbours noticed both of them gone, and contacted the police. Katia's body was found. I confessed...came here. _(Starting toward the shed.)_ Tell me how I'm not responsible.\n\n**A pause**.\n\nZHENIA: I've got to go to her.\n\nNINA: Stay here.\n\nZHENIA: Don't be silly. She's in there for my \u2013\n\nNINA: And the last thing she wants is your damned interference. Did you leave a mother at home?\n\nZHENIA: What does that \u2013 ?\n\nNINA: Just answer my question.\n\nZHENIA: No, she died. I was placed in a hostel...the way most of us are.\n\nNINA: But you had a mother, right?\n\nZHENIA: What sort of a question is that \u2013 had a mother?\n\nNINA: When you were growing up.\n\nZHENIA: I don't remember. The one story they told me was that I was placed before an open window, hours after my birth. A nurse rescued me and had me placed in a hostel. Another nurse there took constant care of me. Rachel. She was Jewish. She was like Nadia...looked after me. She even tried to adopt me, take me to Israel with her. I don't know why they didn't agree to it. I left the hostel at eighteen, fit for nothing. I was due to be sent to an adult one, if I'd got a job. Somewhere I heard that Rachel was killed in an explosion...not where you'd think. On holiday, in Egypt. Lots of Russians go there these days. They're richer than Rachel could ever have been.\n\nNINA: Want to do something for all of us do you, dear? Pretend that the stories you heard aren't true. Rachel's just the other side of the barrier...and she's planning to take you home with her...to the Promised Land.\n\nZHENIA: What good would that \u2013 ?\n\nNINA: All right, insist on the truth and break a mother's grieving heart. Did your own mother die of a stroke?\n\nZHENIA: Stop it!\n\nNINA: I'm fed up with the whole business. _(Rises.)_ Where's the other slacker?\n\nZHENIA: Ludka? She's sleeping, I think.\n\nNINA: Too degrading for you to do the same, I suppose. Well, why don't you dig in the garden, like Nadia wants?\n\nZHENIA: Couldn't I stay here with you?\n\nNINA: I told you I'm tired of looking after you. She's right \u2013 you've less call to be here than a pregnant woman, and a letter in the right quarter would ensure that you never came back. It's a pity you never learned to write.\n\nZHENIA: I can write.\n\nNINA: That's what I mean \u2013 it's a pity. _(Starts to walk with a limp.)_ I've got this damned spur in my heel. Do you suppose that makes me disabled?\n\nGRANNY: You must beat your heel hard...if you want to lose the spur.\n\nNINA: I think I'll hang onto it...for bargaining purposes. _(Calls.)_ Hey, Cinders! One of your duties is to sluice out for us!\n\n**She disappears behind the partition designated as a toilet**.\n\nJADVIGA _(To LOLA.)_ : Excuse me...does that mean...?\n\nLOLA: Go in there as soon as she's finished...and see what needs seeing to. Then you take it out back and er...do what you think should be done with it. You'll get the idea.\n\nJADVIGA: Believe me, I do not mind...but that it should be so near the kitchens area!\n\nLOLA: That, believe me, is the whole idea.\n\nSHURA:...and why we so often confuse or daily bread with our daily...you-know. (I'm watching my language for her benefit.)\n\nJADVIGA: Why? You think Baltic women never say shit?\n\nLOLA: She wasn't aware that you carried it, that's all.\n\nJADVIGA: It's true. Baltic facilities are, on the whole, much better. We are in Europe now. _(She drifts off toward the toilet.)_\n\nYULIA _(Giggles.)_ : She seems really anxious to get stuck in.\n\nLOLA: Well, there's too many of us here for one plot.\n\nYULIA: She said groups of three.\n\nLOLA: I count four.\n\nYULIA: But she's gone off...Oh, you mean...\n\nLOLA: Zhenia's here.\n\nYULIA: I know. I just thought...she'd be watching.\n\nSHURA _(Quickly in.)_ : Three's a daft number anyway. Why don't we simply pair off? There's no target, is there? All right, what's the difference?\n\n**The gardening detail is now SHURA and YULIA, LOLA and ZHENIA, JADVIGA, JACQUI and TANYA, POLYA, RAISA and AMIRA. The routine is innocuous: mulching and spreading of seeds. General: 'How far down do we go?' 'Till you find soft earth.' 'Anyone find any carrots? 'No, nothing but daffs.'**\n\nSHURA: Did they find her baby among the reeds?\n\nYULIA: What?\n\nSHURA: Nadia's baby.\n\nYULIA: Her daughter's \u2013\n\nSHURA: Don't be stupid. She killed it. Don't ask me why, and I don't really care whose it is. But she's in here, isn't she? There's a reason, all right. Nobody's that noble or unlucky.\n\nYULIA: I stole a goat...how unlucky is that?\n\nSHURA _(Laughs.)_ : It wouldn't surprise me if she killed and cooked it for food.\n\nYULIA: No, it was only the milk for my kids.\n\nSHURA: What? What have your kids to do with Nadia?\n\nYULIA: I'm talking about why I'm in here.\n\nSHURA: Who said anything about that? I was trying to tell you how I don't believe Nadia's story. There's got to be more to it.\n\nYULIA: But eating the baby?\n\nSHURA: Look around you. Anyone here could be capable.\n\nYULIA: After being in here \u2013\n\nSHURA: I mean before they came.\n\nYULIA _(After looking.)_ : Zhenia?\n\nSHURA: Among the most likely.\n\nYULIA: Come on.\n\nSHURA: I'm not arguing with you. _(Beat.)_ I'm among the least likely, if it comes to that. _(YULIA laughs.)_ All right, laugh. But the reason I'm in here is even less harmful, and all true. _(Beat.)_ I spat in someone's face. That's all.\n\nYULIA: You want to say you're like Zhukova or Nina protesting against the authorities.\n\nSHURA: Less harmful, I said. Want to hear about it...or are you too much in love with your goats?\n\nYULIA: That's \u2013\n\nSHURA: There was this girl that worked in our office...and she had a baby. She had a flat as well. Now that I think of it, her name was Nadia. Oh, she got on my tits something terrible. She thought she was Pavlova of the typing pool, and she hated me because I knew her for what she really was. Her eyes used to narrow like a Chink's when she passed my desk. So finally one day, I took her on. 'You're very lucky, my dear', I said, 'I often wonder if you know just how lucky you are.' 'I'm not sure I know what you mean, Aleksandra Ivanovna.' You see how she was? Baiting me, the whole time. And the whole office gawking at us...Pavlova, you see? She knew she had her audience, her adored public. So, I said, 'I just mean, my darling, that in my township, the last thing they'd reward you with is a flat. Most people have to wait twenty years...and here you are. You'd be the absolute laughing stock, I tell you. You might even find that baby of yours lying face down among the reeds. Why, they'd label you \"whore\" and simply spit in your face...' And \u2013 to demonstrate, yeah? \u2013 I... _(Mimes act of spitting in Nadia's face.)_ Well, that made her simply white with rage...and she stormed off into the boss's office. He must have been knobbing her himself. Maybe he only wanted to knob her. He called me in anyway, and as a result...\n\nYULIA: As a result, what?\n\nSHURA: I lose my job. You see?\n\nYULIA: Not really. Who was she, this Nadia? An unmarried mother. That's nobody.\n\nSHURA: Common as dogshit, uh-huh. That's my point.\n\nYULIA: But you didn't kill her. Or did you? Is that why you're \u2013 ?\n\nSHURA: Let's just drop it, all right? You obviously don't under \u2013\n\nYULIA: Well, no, I don't, because people aren't put into prison for \u2013\n\nSHURA: Oh, aren't they? Well, then, tell me my crime. Go ahead, tell me.\n\nYULIA: I'm just saying \u2013\n\nSHURA: You were just milking a goat, you said.\n\nYULIA: That's right.\n\nSHURA: So what, did you shoot the herder that owned the goat? You must have done. That's what you're telling me.\n\nYULIA: I did nothing!\n\nSHURA: Well, neither did I!\n\n**A pause**.\n\nMy back is killing me. Do you suppose they want us to do this all day?\n\nYULIA: I read somewhere that if you sleep curled up, like you do as a kid, it saves your spine from hurting.\n\nSHURA: That's very useful advice.\n\nYULIA: I read it somewhere. Or maybe I heard it over Prison Radio.\n\nSHURA _(Overlapping.)_ : When are we ever going to get to sleep?\n\nYULIA: What?\n\nSHURA: We'll probably be crippled for life before then.\n\nYULIA _(Nudging SHURA.)_ : Hey...look.\n\n**YULIA directs SHURA'S attention to the plot where ZHENIA and LOLA are digging. What they are unaware of is the interest POLYA and RAISA are taking in them \u2013 preparing to play a trick, presumably on ZHENIA. Simultaneously AMIRA is going from group to group**.\n\nAMIRA: Care to sign the petition, my dears?\n\nPOLYA: Why don't you injure yourself?\n\nRAISA _(Giggles.)_ : We never learned how to read.\n\nAMIRA: Oh, but that isn't necessary. It's about the food.\n\nPOLYA: What time is dinner?\n\nRAISA: I never got any breakfast.\n\nAMIRA: Consider yourselves rescued, dear. She's in there this minute preparing the next fatal meal. Fatal, I tell you...she's diseased. Now I can ensure that what food we get is the best possible quality. _(Whispering.)_ I have influence. That's what they say about Granny, but it's just not true. She admitted it.\n\nRAISA: You want to move?\n\nAMIRA: What?\n\nRAISA: You're right where the begonias are going!\n\nPOLYA: They're daffs.\n\nAMIRA: No, you don't seem to understand.\n\nPOLYA: Will you use your influence to tell the cook we didn't get any breakfast. Do they want us to eat the soil?\n\n**ZHENIA and LOLA are farther along, though ZHENIA is the worker. LOLA might almost be indulging her**.\n\nZHENIA: I know it isn't the right thing to say, but I'm almost enjoying myself. It's just like working an allotment.\n\nLOLA: They're right, you know. You shouldn't be here.\n\nZHENIA: Don't remind me. If someone stands lookout, I can get to the shed before they find out.\n\nLOLA: No. You shouldn't be here. _(Of the audience.)_ Ask them. Pregnant women are automatically discharged. Or those with chronic illnesses.\n\nZHENIA: That explains it. I was stricken before I came.\n\nLOLA: They had no business sending you here! _(Beat.)_ What did you do anyway? It can't have been serious.\n\nZHENIA: I stole from a shop.\n\nLOLA: I don't believe you. You're not quick enough.\n\nZHENIA: I wasn't...I was caught.\n\nLOLA: You were homeless, was that it?\n\nZHENIA:...and I went into a shop.\n\nLOLA: Guns blazing, I suppose. Have it your way. But in your shoes, I'd protest to the rafters. _(This last line is directly to the audience.)_ I'd get them to bring in the TV cameras so quick, they'd think it was Beslan mark two.\n\n**AMIRA has drifted off to another area while LOLA is addressing the audience. RAISA and POLYA call to ZHENIA**.\n\nRAISA: You must have done this work before.\n\nZHENIA: What?\n\nRAISA: The way it's all laid out in rows like that, neat. I'd say you've been doing it most of your life.\n\nZHENIA: No, not really. I'm not very far with this. I've just \u2013\n\nRAISA: Didn't we hear you say you had...an allotment?\n\nZHENIA: No, I said this was like an allotment...and we only had a small garden. It wasn't mine.\n\nRAISA: Go on, your whole arm's as green as a meadow.\n\nPOLYA: I sure wish somebody'd help me with this.\n\nRAISA: Help you? I've been squatting here doing everything you say!\n\nPOLYA: I mean an expert. Me and Rai are just inner-city brats...no better than gypsies really.\n\nZHENIA: Oh, no, it's easy. _(She starts to cross over.)_\n\nLOLA: Hey, hold on there.\n\nRAISA: We're only asking her advice.\n\nLOLA: You two are doing just fine. Leave her alone.\n\nPOLYA: What're you getting so antsy about? It's not a Soviet competition, is it? Aren't we sisters?\n\nRAISA: You keep telling her she can't do anything. I'd say that's your problem. She's great.\n\nLOLA: I know she's \u2013\n\nPOLYA: You don't. You keep saying she's rubbish. We heard you. 'You shouldn't be here.'\n\nLOLA: Mind your own business.\n\nZHENIA: It's all right, Lo. Don't row.\n\nLOLA: I'm not rowing.\n\nAMIRA: I'll report how you been fighting.\n\nLOLA: You keep your mouth shut.\n\nAMIRA: Sign my petition then.\n\nLOLA: Petition for what?\n\nAMIRA: We need a new Cinderella. The Polack's diseased, and they're planning to use that to get us all transferred to the psycho ward.\n\nLOLA: Oh, I bet. All 58 of us.\n\nAMIRA _(Pointing to GRANNY.)_ : She knows. She had a look at the medicine they're planning to give us when she bartered for that cloth! They're going to lace our food with this new form of drug. The Polack's doing it right now.\n\nLOLA: She's Lithuanian!\n\nGRANNY: From Latvia. I wouldn't discount what she's saying, just because it's her that's saying it.\n\nAMIRA: I got you!\n\nLOLA: It's Amira we're talking about.\n\nGRANNY: Exactly.\n\n**Simultaneously (and unnoticed) ZHENIA has crossed over to POLYA's and RAISA's plot**.\n\nZHENIA: What is it you're finding so difficult? You just work the soil into a nice soft texture.\n\nPOLYA: We had a business helping people like you.\n\nZHENIA _(Beat.)_ : Then you divide it into rows.\n\nPOLYA: We may even have helped some of your friends.\n\nZHENIA: You have to make sure that the soil is deep enough.\n\nRAISA: She doesn't seem to be listening.\n\nZHENIA: I thought you wanted help.\n\nPOLYA: Didn't you live in a hostel...for cripples?\n\nZHENIA: But...we're not from the same city.\n\nRAISA: No, we advertised everywhere, didn't we?\n\nPOLYA: That's right \u2013 all over Russia.\n\nZHENIA: What do you mean, advertised what?\n\nRAISA: Our company.\n\nZHENIA: You had a company?\n\nPOLYA: For helping cripples, yeah.\n\nRAISA: She doesn't believe us.\n\nZHENIA: It's not that.\n\nRAISA: She doesn't believe us.\n\nPOLYA: It's like this, see. We put the adverts in various papers: 'We find suitable work for crippled people...any age, any degree of crippleness.'\n\nZHENIA: I wish you'd stop saying that word.\n\nRAISA: What, you think they weren't crippled?\n\nZHENIA: It's not a nice word.\n\nPOLYA: They weren't in the greatest condition, let me tell you. You're an Olympian gymnast, let me tell you, next to most of the ones we had. No arms or legs...the worst of them were. I don't know how they got there.\n\nZHENIA: And what work could you find them to do?\n\nRAISA: Sounds difficult, doesn't it. But we managed it.\n\nPOLYA: We set them to work raising money.\n\nZHENIA: For charity?\n\nPOLYA: Uh-huh.\n\nRAISA: That's right. Only some of them wouldn't cooperate, see. So we had to \u2013\n\nPOLYA: \u2013 penalize them a bit.\n\nRAISA: That's right, make them less greedy.\n\nZHENIA: I don't think I want to hear any more.\n\nPOLYA: Oh, we wouldn't have trouble with you.\n\nRAISA: We were a business.\n\nPOLYA: How'd you like to be cheated by a bunch of \u2013\n\nRAISA: \u2013 armless bandits. _(Giggles.)_\n\nPOLYA: Bet you'd wish you were back in here...\n\nRAISA: Somebody willing to do your shift.\n\nPOLYA: We had nothing like that on the outside.\n\nRAISA: No fear. They'd take our share of the money without a qualm.\n\nPOLYA: Damned lucky they couldn't get very far.\n\nRAISA: That's right. We'd be looking to you to make up the difference on their behalf.\n\nPOLYA: And that wouldn't have been very fair, would it.\n\nRAISA: Lucky for you that we killed them.\n\nPOLYA: Yeah, damned lucky.\n\n**As part of the game they have blocked ZHENIA from leaving, but now NADIA has noticed and crosses over to them**.\n\nNADIA: What the hell's going on here?\n\nRAISA: Not a thing \u2013 why?\n\nPOLYA: We was just getting acquainted.\n\nRAISA: She was just helping us with our garden.\n\nPOLYA: Allotment expert.\n\nNADIA _(To ZHENIA.)_ : Are you all right?\n\nZHENIA: Yes.\n\nNADIA: You look like you're ready to cry.\n\nZHENIA _(On the verge.)_ : I'm fine.\n\nRAISA: What, do you think we threatened her, with a knife, maybe?\n\nPOLYA: Women prisoners don't do that kind of thing.\n\nRAISA: She thinks we're different.\n\nPOLYA: Even where we come from.\n\nNADIA: You need to bear that in mind.\n\n**Perhaps simultaneously AMIRA approaches NINA as she returns to the compound**.\n\nNINA: You know I'm not going to sign anything, so you might just as well...\n\nAMIRA: You'd support a hunger strike, wouldn't you? You did it when they refused Zhenia visits.\n\nNINA: Who've you got to visit you?\n\nAMIRA: Don't say it like that. I might have a husband like everyone else, except Zhenia.\n\nNINA: Fine, you show me your written request and their refusal, and I'll consider \u2013\n\nAMIRA: That's not what I meant. Don't support me to be Cinderella, I can bear it. But don't eat the food that they cook \u2013 not after she's handled it. _(NINA sighs.)_ It's a plot, I promise you. She's infected...and so will we be, one by one.\n\nNINA: Why don't you invent some incurable disease for yourself so they'll release you? I'll back you up then. Or...I know. We can all arrange to help you over the wall to the men's zone... then you can get preggers and get released that way.\n\nAMIRA: You don't have to tell me that all of you hate me; I felt it from the first. And I done less than any of you. I suppose that's why.\n\nNINA: Don't be silly. We'd all like to leave here. We just want to help you.\n\nLOLA _(To MISHA.)_ : You clocked that, did you? You've got our names? Our families can expect a knock at the door in the small hours, can they? Kiss my arse!\n\nMISHA _(Addressing LOLA.)_ : Hey, Killer! What are you in for? What more can I do to you? _(Motions to silence her.)_ You're all gob and no breakfast. Hear me out first, eh? We know zeks are dying off like flies, by the hour. But now someone tells us they're girls...and I mean your average Russian, not a Chechen, two eyes and a bomb up her hijaab. So, of course, now everybody's concerned. Maybe there's Chechens in charge. But I need to know all about this psycho ward. Who they have there...how long for...and how many, exactly, have never come back. _(Beat.)_ What is it? Don't look at me like some western pervert or something...like we're all out here freezing to see how you girls have a slash.\n\n**DR ORLOVA, in glasses and a white coat, comes on, wheeling a trolley with an assortment of medicines. Her entrance is greeted by a frisson of interest**.\n\nNADIA: Now what do you suppose she's doing here with that?\n\nAMIRA: I tell you because of the food...it's been tainted.\n\nDR ORLOVA: Where can I find the new women?\n\nAMIRA: What did I tell you? She's after the Balt.\n\nPOLYA _(Overlapping.)_ : They told me they test you for crabs.\n\nRAISA: Shout it out, why don't you.\n\nPOLYA: You'll see.\n\nDR ORLOVA _(Consulting a clipboard.)_ : Apparently two Americans.\n\nJACQUI: That must mean us.\n\nTANYA: We're English.\n\nDR ORLOVA: Prostitutes.\n\nJACQUI: That must have been why we're arrested. _(To JADVIGA.)_ It couldn't possibly be true, tell her.\n\nTANYA: Kidnapped by these two gorillas....\n\nJADVIGA: They don't speak a word of Russian, and they're terrified.\n\nTANYA:...and I'm not having anyone poking me...scrutinizing my discharge.\n\nJADVIGA: She's afraid to be examined.\n\nDR ORLOVA: Tell her this isn't a medical.\n\nJADVIGA _(Mainly to TANYA.)_ : No, no, this is safe place. This woman is Dr Orlova...she is good person. I saw her in clinic with patient, very kind.\n\nJACQUI _(To TANYA.)_ : You're always so arsed about everything.\n\nTANYA: And what do you think all this is, a field trip? It's not some mistake in our reservations. We're in prison here. They sever limbs before you're released.\n\nDR ORLOVA _(To JADVIGA.)_ : Tell her to shut up.\n\nJADVIGA: You must, you know, keep mouth closed.\n\nDR ORLOVA: They have to answer some questions. If they answer correctly, they may be released.\n\nJADVIGA: You will need answer questions.\n\nTANYA: My mouth's closed.\n\nJACQUI: We'll do whatever we can.\n\nTANYA: And we demand to see some official. You got that? _(Over-emphasising, to DR ORLOVA.)_ English embassy. We want.\n\nDR ORLOVA _(Pigeon English.)_ : English whore.\n\nJACQUI: But that's it. This is all a mistake.\n\nDR ORLOVA _(Of MISHA.)_ : And who the hell is this?\n\nAMIRA: Someone said he's a KGB spy.\n\nMISHA: I'm the head of a delegation from Moscow. _(To JACQUI and TANYA.)_ And I speak little English.\n\nJACQUI: I understood the word 'Moscow'.\n\nTANYA: Then he's some kind of official. _(To JADVIGA.)_ We want him to represent us, get it?\n\nMISHA: I speak English, that is all.\n\nDR ORLOVA: I want to question them in camera, away from Moscow's prying eyes.\n\nJADVIGA: You have go with her...without him.\n\nTANYA: We're not going anywhere without him.\n\nMISHA: She does not want me.\n\nTANYA: Then we're saying nothing. _(To JADVIGA.)_ Tell her.\n\nJADVIGA: But this is one way you can leave.\n\nDR ORLOVA _(Showing JADVIGA photo.)_ : Were they taken to this hotel?\n\nMISHA _(Stepping in.)_ : This your hotel?\n\nJACQUI: It might have been.\n\nTANYA: Yes, that's the one.\n\nDR ORLOVA: And they saw this man meet with this woman?\n\nMISHA: You saw these peoples?\n\nTANYA: Yup. Definitely.\n\nJACQUI: You can't be sure.\n\nTANYA: And you were out of your face. _(To MISHA.)_ She was drugged at the time.\n\nJACQUI: You know that's not true. _(Slightly hushed.)_ I'm not going to lie to get out of here.\n\nTANYA: And you're effing daft.\n\nJADVIGA: Both are very nervous.\n\nDR ORLOVA: And one of them's lying. _(Distinctly.)_ Hotel Peking.\n\nJACQUI and TANYA: No!\/Yes. That's the one...and those are people we saw, absolutely. Who is he?\n\nMISHA: He must be oligarch. _(To DR ORLOVA.)_ Is that right?\n\nJADVIGA: And they hold his girlfriend in 'psycho'. If you can know her...\n\nJACQUI: But we don't!\n\nTANYA: Shut up!\n\nMISHA: What does she look like? Maybe I know her.\n\nDR ORLOVA: That's not possible.\n\nMISHA: But you won't object if I go with them to see her.\n\nDR ORLOVA: They're not going. I was asked to see if the foreigners knew her, and they obviously don't.\n\nMISHA: But you're holding someone. Why can't I see her?\n\nDR ORLOVA _(Intimately.)_ : You don't get it. It's a trick, see? They say yes, and we can hold them indefinitely.\n\nTANYA: What's going on here?\n\nMISHA: I try to help you.\n\nDR ORLOVA: Look, I just told you, there's nobody.\n\nMISHA: You just told me there was.\n\nDR ORLOVA: But not the one you want. Forget about it.\n\nTANYA: Will somebody tell us what's going on?\n\nJADVIGA: I think maybe she is playing game.\n\nDR ORLOVA: Cinderella, I want you to help with these medicines.\n\nAMIRA: She can't do anything for you, my dear. Didn't they tell you she's a pox-ridden Balt?\n\nTANYA _(Overlapping, to MISHA.)_ : Game or not, you're going to help us.\n\nMISHA: I try to get to psycho.\n\nTANYA: That's not good enough. We want a message delivered to the British Embassy.\n\n**For the moment, MISHA stays with the two British women while DR ORLOVA sets about distributing medicines to various members of the zone**.\n\nPOLYA _(To LOLA.)_ : What does this mean, she tools them up?\n\nLOLA: I wouldn't count on it, if I were you.\n\nRAISA: We're just trying to figure out what's what.\n\nDR ORLOVA _(Answering AMIRA.)_ : You think that's the same as a terrorist, do you? They give Russia maximum points during Eurovision. And I've tested her for everything going, even jaundice. She's antiseptically clean.\n\nAMIRA: You don't understand. _(Gesturing.)_ She's a plant, to inform to him. I heard them conferring earlier.\n\nJADVIGA: That's not true, doctor. No more than my disease was.\n\nDR ORLOVA: And what, exactly, would they be conferring about?\n\nAMIRA: How you help certain girls with addictions, for instance.\n\nDR ORLOVA _(Loud on purpose.)_ : And what about your trick of inhaling bags of sugar to make it look like a shadow on your lung?\n\nAMIRA: What are you talking about?\n\nDR ORLOVA: And you never complained of an ulcerous stomach?\n\nAMIRA: And which of us hasn't?\n\nDR ORLOVA: Do you like being put on the spot?\n\nAMIRA: That's what I am, all right, all the time. You all know it. And I've never understood why. I'm as frightened of them as you are.\n\nDR ORLOVA: Here's something to calm you down then. _(Administers a quick injection.)_\n\nAMIRA: I didn't mean it like that.\n\nDR ORLOVA: Anybody know where Fedina's got to?\n\nAMIRA: You ought to give her something.\n\nDR ORLOVA _(To JADVIGA.)_ : The backward girl...helps out in the kitchen sometimes.\n\nJADVIGA: I've met her this morning, yes.\n\nAMIRA: That's where she is, of course. Curled up, asleep.\n\nDR ORLOVA: I don't think she drinks, does she, girls?\n\nAMIRA: You mistake me. I never said 'drunk'.\n\nLOLA: No, you just knocked her unconscious.\n\nAMIRA: You see there? That's just what I mean. For no reason at all.\n\nDR ORLOVA _(To JADVIGA.)_ : Do you want to fetch her in here?\n\n**On cue, LUDKA materializes**.\n\nTo help us settle a long-standing dispute, Fedina. Did this woman assault you physically?\n\nLOLA: We're all behind you, Lud. There's no need for you to be frightened.\n\nDR ORLOVA: Shut up. _(To LUDKA.)_ Did she hit you?\n\nLUDKA: Hit me? No, she wouldn't dare.\n\nAMIRA: From her own lips, you all heard it.\n\nLUDKA: She just fixed me with her evil eye.\n\nAMIRA: That's too silly. We all know she's mad (as a hatter).\n\nLUDKA: Her eye, as everyone knows, is as black as coal, and her heart's like burnt cork. Of course, she fixed me. I've been under her spell for more than a week.\n\nDR ORLOVA: And what would you like to have done to her? Burning?\n\nLUDKA: Remove the spell.\n\nAMIRA: I'm not going to stay here to listen...\n\nLUDKA _(Close to manic.)_ : Remove the spell and beg my forgiveness.\n\nLOLA _(And others.)_ : We'd all like to hear that.\n\nLUDKA: All right, beg forgiveness from all my sisters. Go on. Let's hear you sing it, like a prayer! Make her do it, doctor.\n\nDR ORLOVA: She'd better start saying her prayers, full stop.\n\nAMIRA _(Trying to push past.)_ : Oh, go to...\n\nDR ORLOVA: That solution I gave you a minute ago. Sodium Omeprazol. It has a strangely debilitating effect on the nervous system.\n\nAMIRA: You said it was to calm me down.\n\nDR ORLOVA: Mm-hm. In a very short while you'll begin to experience decline in your motor skills.\n\nAMIRA _(Rushing out.)_ : Baastaards!\n\n**The compound hoots with laughter**.\n\nDR ORLOVA: That should keep her out of the way for a good while.\n\nJACQUI: What the hell's going on here? She's going to inject every one of us?\n\nJADVIGA: No, no...only her...who says I carry sexual disease.\n\nDR ORLOVA _(To one or two of the other women.)_ : Only flour paste and salt.\n\nTANNOY: What are you doing there, doctor?\n\nDR ORLOVA: Making checks. Some may be seriously \u2013\n\nTANNOY: They go to 'psycho' if they're not well. Your job was to find out about the drugs. Did you find out?\n\nDR ORLOVA: The English girls know nothing.\n\nTANYA: Wait. Just a minute there.\n\nTANNOY: And what are you doing with Moscow at your throat?\n\nDR ORLOVA: You're the one that arranged that...aren't you?\n\n**Everyone looks up. KOMANDIR is the closest to a Soviet-era official to exist in post-Soviet Russia. She is severe in appearance and in manner. Like 'EVA BRAUN', she comes on from a height and descends to ground level. As with DR ORLOVA, there is a frisson at her entrance**.\n\nJADVIGA _(To the English.)_ : She is real boss of prison. I saw her before.\n\nKOMANDIR: I heard joyous laughter a moment ago. You think you're on holiday in Yalta? _(Someone starts to speak, but she speaks through her to TANYA.)_ And you want very much to go to 'psycho', is that right? Very well!\n\nMISHA: No! You just heard the good doctor. They don't know a thing. I'm the one you need to interview. Take me now.\n\nKOMANDIR: Oh, yes, that's been arranged. But now what about the rest of you? You've got this looking like your private allotments.\n\nLOLA: This is what we were ordered to do...by the new commander...Eva Braun.\n\nKOMANDIR: I'm the only commander here. _(To DR ORLOVA.)_ Is that right?\n\nDR ORLOVA: Marina Volkanova, that's right.\n\nLOLA: That's just what she said her name was.\n\nKOMANDIR: Then one of us must have been lying. Care to guess which one?\n\nLOLA: Then you don't want us for gardening detail?\n\nKOMANDIR: I don't want you out here to shit. _(Faint mumbling.)_ Any questions? I can always get the good doctor to inject the nervous among you with the same thing you pretended to give that other bitch. _(To DR ORLOVA.)_ You have it for real?\n\nDR ORLOVA _(Showing.)_ : Right here, my dears.\n\nKOMANDIR: Which one of you is Kuzina?\n\nZHENIA _(Inches forward.)_ : Here, Komandir.\n\nKOMANDIR: Let's see your hands.\n\nZHENIA: My...?\n\nKOMANDIR _(Grabbing her hands.)_ : These are lady's hands...artist's...musician's. What do you think you're showing me, art work? I want to see ulcers, like theirs! Let's see how you thread a needle. Here. _(She holds one out. ZHENIA drops it.)_ Butterfingers!\n\nNADIA: That's not fair. _(To the others.)_ I don't care who she says she is.\n\nKOMANDIR: You're not going to say it's too hard for her?\n\nNADIA: You know the answer to that without asking.\n\nDR ORLOVA: They call her the camp Cinderella, Komandir. She and Fedina cook and clean what they can...and take it in turn to taste the food.\n\nKOMANDIR: But who was it who told me the new one was doing that \u2013 the diseased Balt.\n\n**This causes something of a stir among the women**.\n\nJADVIGA _(To DR ORLOVA.):_ Please tell her I'm not diseased!\n\nNADIA _(Above the din.)_ : What the fuck does it matter? We all know which of us shouldn't be here.\n\nKOMANDIR: Excuse me! You all are here, yes? And that should mean you ought to do something. Every one of you.\n\nNADIA: All right, but not these two. _(To the others.)_ Agreed?\n\nZHENIA _(Speaking up and thus stopping the answer.)_ : I can sew if you reduce the quota.\n\nKOMANDIR: But why should we want to do that? _(Small murmur.)_ Your crime's as severe as the others. Are you saying your sentence should be less? And how do the rest of you feel about that?\n\nPOLYA: Would that make our punishment less?\n\nKOMANDIR: That's what she's saying.\n\nNADIA: No, I'm not! The age of the quota is over.\n\nKOMANDIR: And the age of the free market is here.\n\nNADIA: And what about your observers from Moscow? _(To the audience.)_ You have a view on what you've seen, haven't you?\n\nGRANNY: But that doesn't mean they'll release her, Nadinka.\n\nMISHA: But don't certain conditions guarantee release?\n\nKOMANDIR: Ah. You're trying to tell us she's pregnant.\n\nMISHA: She's Grade One Disabled. _(To DR ORLOVA.)_ Isn't that right?\n\nDR ORLOVA: I'd be very surprised if she ever worked.\n\nKOMANDIR: So you're trying to force us to set her free?\n\nZHENIA: But I don't want to leave.\n\nKOMANDIR: All right, she's free.\n\nMISHA: But you're not serious.\n\nKOMANDIR: What do you want, a sworn affidavit? _(Of LUDKA.)_ What about her?\n\nLUDKA: Why is she pointing at me? _(To KOMANDIR.)_ You have raven's eyes!\n\nLOLA _(Stepping in.)_ : Shh, shh, shh...It's all right, Lud. She's entitled to do that. She's the boss.\n\nLUDKA _(To DR ORLOVA.)_ : Can you give her something to make her stop?\n\nDR ORLOVA: That's right. I'll give her something, don't worry. Sleeve? _(Prepares to inject LUDKA.)_\n\nLUDKA: No, not me. Her.\n\nDR ORLOVA _(To KOMANDIR.)_ : She'll go to sleep in a minute or two.\n\nLOLA: For how long? _(Looking round.)_ I hope all of you noticed what just happened here. _(To MISHA.)_ Did you?\n\nMISHA: I saw it, all right. _(To KOMANDIR.)_ Where does she go now, the 'psycho' ward?\n\nDR ORLOVA: She goes back where she goes every night...and half the days. _(To LUDKA.)_ Go on now, dear. You're all right.\n\nLUDKA _(Going out.)_ : And you'll make sure that nobody follows me?\n\nDR ORLOVA: Perfectly sure, dear. I'll be standing guard.\n\nLUDKA _(Going off.)_ : There's a dagger of vengeance right by my side.\n\n**Everyone watches**.\n\nMISHA _(To DR ORLOVA.)_ : And what happens next? One day you'll give her enough so she won't wake up?\n\nNADIA _(To ZHENIA.)_ : You see why we're anxious to get you out of here?\n\nZHENIA: I do, but... _(She looks at KOMANDIR.)_\n\nNADIA: She's afraid of the same thing happening to her. _(To ZHENIA.)_ Is that right?\n\nKOMANDIR: What, with Moscow staring down our gullet? _(Looking round.)_ I don't want her here! The other one, all right. You can put her to sleep. But you're a pain in the butt cracks. Create tensions. Everybody taking sides...before even nine in the morning. Oh, get her out of here, please.\n\nZHENIA: But where would I go?\n\nDR ORLOVA: With me, to sign the release forms.\n\nZHENIA: No, you don't understand.\n\nKOMANDIR: If she stays here, she works the full quota. There's no other way.\n\nZHENIA: All right, then, I'd just as soon die here, among the people I've known. You're as much family as I've ever known. If I leave, I'll be homeless. And then, back inside, somewhere else.\n\nPOLYA: If we were out, we could find her a place to stay.\n\nRAISA: That's right...and a job to do.\n\nZHENIA: You see? I'll end up like...their friends.\n\nRAISA: What's the matter with that? They're at peace now.\n\nPOLYA: One room to a customer.\n\nNADIA: And that's not going to happen to you. _(To POLYA and RAISA.)_ You're in here for a long time to come.\n\nPOLYA: Oh, we have loads of friends on the outside.\n\nNADIA: So's Nina. _(NINA perks up at the mention of her name.)_ You're from Zhenia's home town, aren't you, Neen? Close enough anyway.\n\nNINA: Yeah, sure. All my nieces and nephews...the brothers that are still alive.\n\nNADIA: And surely somebody be happy to take you in.\n\nNINA: Oh, yes, I can give you my Tanya's address right now.\n\nKOMANDIR: There's time enough for that later on. Are we all right now? I want this sorted and everybody back to the jobs they were meant to be doing before they got out here this morning.\n\nLOLA: And what happened to Eva?\n\nJADVIGA: What do you want me to do?\n\nKOMANDIR: Which one are you, the drug dealer's whore? Oh, no, you're the Balt whore.\n\nJADVIGA: I'm from Latvia.\n\nDR ORLOVA _(Reaching out to ZHENIA.)_ : And you come with me, my dear.\n\nMISHA: Hang on just a second. _(To KOMANDIR.)_ She's taking her off...to the 'psycho' ward?\n\nKOMANDIR: You mean to say Moscow disapproves?\n\nDR ORLOVA: I already told you, there's no oligarch.\n\nMISHA: That's right. You said it's a trick.\n\nNADIA: No, she's being released. You helped to secure it.\n\nMISHA: I just want to know how many others you're holding.\n\nKOMANDIR: How many others we've tortured to death? Feast your eyes!\n\n**Tableau of VALODIA and ASKA together. During what follows, ZHENIA goes off with DR ORLOVA**.\n\nASKA: Misha!\n\nMISHA: It is you. _(Of VALODIA.)_ Who's this? _(To VALODIA.)_ If you laid one fingertip on her, you bastard, I'll \u2013\n\nASKA: No, no, you don't understand. Valodik has been my protector.\n\nMISHA: Protector, my arse. He's a prison thug.\n\nVALODIA: And she's in here, I guess, only because of you.\n\nMISHA: I'll kill (you).\n\nVALODIA: It's okay by me. I was only asked to find out what she knew.\n\nASKA: And I could tell them next to nothing...\n\nKOMANDIR: So now we're planning to release her, too, like the other one, unscathed. Without a blemish, in fact.\n\nMISHA: Because you figure you got me. Your quack called it a trap.\n\nKOMANDIR: In front of how many witnesses? You didn't ingest anything, did you? Orlova never injected you.\n\nVALODIA _(Laughing.)_ : And the last thing I want is to wrestle you to the ground. I think you owe her some kind of line, even if you do it in private.\n\nASKA: I don't want you to tell me anything, Misha. I don't know anything, as I told them. And I want to keep it that way. I want you to tell me if you're all right, only. It's even all right if you lie. I'll still know.\n\nMISHA: I'm not all right, no.\n\nASKA: What?\n\nMISHA: I could scarcely be worse and still be alive. _(Addressing the audience.)_ All my colleagues out there think I'm spying on them \u2013 I'm the one under surveillance. Ask them! I must know when and where the next Chechen attack comes? You're asking the wrong man!\n\nASKA: Oh, I'm so relieved. _(Matching his hands on the fence.)_ Oh, Mishenka!\n\nMISHA: What's the difference? They still think I know.\n\nVALODIA: You might just as well finger me. I was at Beslan and the theatre siege. Chechen lady had her bullet thrower dead on me. I thought I'd be killed because she didn't shoot.\n\nASKA: I don't care...I'm not bothered about anything now. You're alive...you're not injured... and they can keep me in here for however long \u2013\n\n**Suddenly over the tannoy there is the piercing sound of someone in pain. It freezes the action for a long moment**.\n\nLOLA: What the fuck was that? There's nobody out there.\n\nASKA: You're quite wrong.\n\nLOLA: Eva Braun?\n\nASKA: She died an hour ago. He killed her. And then she _(Of KOMANDIR.)_ told him to hush it up.\n\nVALODIA: But I did that to stop her from killing you.\n\n**The scream is repeated**.\n\nNADIA: You bastards! _(She moves toward the KOMANDIR, but VALODIA blocks her way.)_\n\nKOMANDIR: That's right. Keep her there. _(Calling out.)_ Orlova!\n\nNINA: I think somebody ought to tell us what's going on.\n\nASKA: I tell you they're torturing someone.\n\nKOMANDIR: Not you. _(To DR ORLOVA.)_ Can you tell them, no one's been tortured?\n\nDR ORLOVA: It's all right. I promise you, everything's...\n\nNINA: What was that scream then, rusty pipes?\n\nDR ORLOVA: I give you my word \u2013 your friend's perfectly safe.\n\nNADIA: Bring her here then.\n\nDR ORLOVA: I can't...at the moment.\n\nNADIA: Why not, if you say she's all right? Bring her to us!\n\nVALODIA: You know I don't want to hurt anyone.\n\nNINA: Nadia, dear, everything's for the best...whatever happens.\n\nPOLYA: I don't see what you're worried about. We all heard her say that she'd sooner die here.\n\n**Suddenly NADIA lunges at POLYA with a sharp gardening implement. The move catches everyone off-guard. Then several move in to pull NADIA off POLYA. Detaching herself from the action, LOLA addresses the audience**.\n\nLOLA: Well! What does that make it now? Three killings inside the women's zone? That's unheard of! And what's Moscow going to say?\n\n**Instant blackout**.\n\n##### Author's note\n\nThere is a veritable library of documentation about Russian prison life. I am particularly indebted to Irina Ratushinskaya and her memoir _Grey is the Colour of Hope_ (Hodder and Stoughton: 1986) for the general ambience of prison life in this play. The characters, their stories and individual behaviour are, creatively, mine.\n\n### EMIGR\u00c9S\n\n#### A Grotesque Comedy\n\nWorkshopped as part of a second-year module by Drama students of the University of Exeter in January 2005, under the direction of Martin Harvey and with the following distribution of parts for the various scenes:\n\nIMMIGRATION OFFICER | Matthew Lockwood\n\n---|---\n\nTANYA | Laura Sibbick\/Ginny Henning\/Megan Atkin\/Jenny Barbrook\/Liz Norwalk\n\nINDIAN | Laura Kerr\n\nMAKSIM | James Louis\/Tim Schofield\/Mark Evans\/Tom Armstrong\/Peter Lawrence\n\nROY | Patrick Godin\n\nANNA | Nancy Scott\n\nINNA | Eeva Hautala\n\nBRANGWEN | Megan Atkin\n\nBEZNIK | David Dimitriou\n\nBETTY | Nancy Scott\n\nOLIVER | Jamie Williams\n\nMISS HIGHLIGHTS\/FRANKLYN | David Dimitriou\n\nMcBRIDE | Matthew Blackhurst\n\nNATALYA | Laura Kerr\n\nPOLBOI | James Anniballi\n\nALENKA | Carys Stephen\n\nMARTINS | Matthew Lockwood\n\nSCOOBY | Martin Harvey\n\nVarious places in the West Country (Act One) and in Scotland (Act Two).\n\n##### ACT ONE\n\n**Unit set representing various locations mainly in the West Country and in Scotland. The fluid nature of the scene changes should have some suggestion of dreams. MAKSIM is dark-haired and thin, TANYA is blonde and tiny. Their first encounter is with an Immigration officer. In general, when foreign characters speak in accented, broken English, it's assumed that they are trying to speak English. When their speech is normal, it's to be assumed that they are speaking in their own language**.\n\nIMMIGRATION: You and your...\n\nTANYA: He's looking at me funny. Can you get him to stop it?\n\nIMMIGRATION: Is something the matter?\n\nMAKSIM:...wife. She's my wife. There's something the matter with her eyes.\n\nIMMIGRATION: We call that shifty. _(Looks down at documents.)_ You came from the Ukraine.\n\nMAKSIM: She...she is Ukrainian. We came from Kaliningrad.\n\nIMMIGRATION: Isn't that part of a Baltic republic? It's almost Euroland. How can you say you've been persecuted? Are you on the run? _(He consults a file.)_\n\nMAKSIM: I'm a trained barrister. _(A beat.)_ You can tell by my knowledge of English...can't you? My wife's a registered nurse, back in Russia.\n\nIMMIGRATION _(To TANYA.)_ : How is your English, dear?\n\nTANYA: I little speak. Learn. Learn. Blakha-mukha. _(To MAKSIM.)_ Tell him you're teaching me.\n\nIMMIGRATION: Try telling her not to profane.\n\nMAKSIM: She didn't say...\n\nIMMIGRATION: 'Blacka...' something.\n\nMAKSIM: It means nothing. Gibberish.\n\nIMMIGRATION: So you say. She could be laughing at me while I grin like a Butlins rep. _(A beat.)_ Before too much longer, you characters will have to apply for asylum before leaving.\n\nMAKSIM: What does that mean...that you're sending us back?\n\nIMMIGRATION: I may have a love child in Bucharest...where she's likely to grow up under-advantaged. _(Stamps document hard.)_ And I'm a permanent civil servant. Don't you think I'm entitled to special consideration? _(Handing him documents.)_ Report to the detention centre at Harrow.\n\nMAKSIM: What about our passports?\n\nIMMIGRATION: We keep them safe. _(Holds out vouchers.)_ Give you these vouchers instead. _(Beat.)_ You think you're entitled to anything better? If you'd have said you lost these, we would have sent you back...within the hour.\n\n**Indian checkout worker (woman) confronts TANYA about her purchases**.\n\nWOMAN: Tights and sanitary towels aren't on the list.\n\nTANYA: I sorry. Little spik English.\n\nWOMAN: You cannot get tights and sanitary towels with your vouchers \u2013 _(As if to a deaf person.)_ It's not allowed.\n\n**MAKSIM approaches them**.\n\nTANYA _(To MAKSIM.)_ : Did you hear that? She's not English. _(To the WOMAN.)_ Don't they have feminine hygiene in your country?\n\nWOMAN: She can blaspheme as much as she likes, tell her.\n\nMAKSIM: She's not blaspheming.\n\nTANYA _(Almost in tears to the WOMAN.)_ : Nids this panties, understand? I nids.\n\nWOMAN; They're not pants, for a \u2013 _(To MAKSIM.)_ Will you tell her they're not underwear?\n\nMAKSIM: She knows what they are; she just doesn't know the English.\n\nWOMAN: I'm not having this conversation, right? Vouchers are only for the essentials. A s.t.'s is not an essential.\n\nMAKSIM: I agree with Tanya \u2013 you don't have periods in this country?\n\nWOMAN: I'll call the manager \u2013 say you've been sexually harassing me.\n\nMAKSIM: Call him, yes, fine. Then we can get what we want.\n\nWOMAN: Nobody ast you to come here, did they. Shouldna come if you don't wanna follow the rules.\n\nTANYA: I'll put them back where I found them. _(To WOMAN.)_ Tek back, you see? _(Under her breath.)_ Blakha-mukha.\n\nWOMAN: Give them back. I don't trust you. _(Beat.)_ My family came in the fifties. You think we had it right? Filled my knickers with dogshit my first week at school...and we was jolly-just-come, just like you. _(Beat.)_ Tell her to use toilet paper.\n\n**ROY WINTERS taps TANYA on the shoulder. She gives a start**.\n\nROY: Can I have a word with your husband? We have various folk to look after you.\n\nTANYA: Excuse, please. I little spik.\n\nROY: Oh, God. Are you both in the same boat? Kosovan? We have three of you staying here at the moment.\n\nTANYA: Rossian. From Rossia.\n\nROY: That's wonderful. There's nobody here who speaks...\n\nMAKSIM: I speak English.\n\nROY: Are you sure? Wonderful, right. That's a miracle. Well. You won't need the vouchers in Devon. The Government pays us for each of you, bed and board. You're free to look for whatever work you can find. But that may be no better than the poultry farm outside Newton Abbot. Or if you and your wife want to stay together, there's always the meal shifts here. My name's Roy Winters, by the way. Like the season, and with an 's'. Celestine, the guy doing it now, is a gift from the gods, but he's also from darkest Africa, and the Kosovans won't touch a thing he cooks. He scrubs up better than they do, twice a day. But I keep thinking they'll hack him like they hack the heads off their 600 daily cocks and hens. You might help to prevent an axe murder, fancy that.\n\n**ANNA buttonholes TANYA, her speech overlapping with ROY'S**.\n\nANNA: Is he off again, like he's trying to get some chamber of something-or-other to leave their fortunes with us for Whitsun week. I be his second wife, Anna, and his first Brangwen was Welsh, if you can picture the gabby pair they made.\n\nROY:...and she doesn't speak any English.\n\nANNA:...and so you're telling me that he does? It's that rare when I has a woman to talk to. _(To TANYA.)_ We do get the odd married couple, like yourselves. But mostly they're daughters of Allah, which is very odd, indeed. You never see hide nor hair even when they're right there in front of you...and as for airing their thoughts \u2013\n\nROY: Have you even asked her name?\n\nANNA: A simple smile communicates volumes. _(To TANYA.)_ He's referring to the fact that this used to be an old people's home, and I never talked to any of the old dears. _(To ROY.)_ This one's as young and as bright as somebody's little sister...\n\nMAKSIM: My wife's name is Tanya.\n\nANNA: What did I tell you?\n\nROY: They want to be left alone.\n\n**The couple start out, but their progress is blocked by BRANGWEN with another resident, INNA**.\n\nBRANGWEN _(Accented and in mid flow.)_ :...expect you could do with a long sit-you-down.\n\nANNA: Brangwen!\n\nBRANGWEN: And you must be wondering what I think I'm doing.\n\nANNA: You was here once upon a winters. Now I seem to recall it's somewhere else, in town.\n\nBRANGWEN: The man at the station didn't know that, you see. When this woman here struggled to say Winters Rest, he got hold of me on the run. On my way to Whipton Village, I was, only what could I do? She needs help.\n\nROY _(To MAKSIM.)_ : She didn't train in with you?\n\nMAKSIM: We never saw her before.\n\nBRANGWEN: From one platform to the next and the one after that, she's been missing her connections all day long. Right the way down to Cornwall, she could have been. And her tears alone told me that. She can't speak a word of good English.\n\nANNA: One more waif and stray.\n\nBRANGWEN: It's my idea she's some sort of Russian.\n\nTANYA _(Lighting up, to INNA.)_ : Russia? You're from Russia?\n\nINNA _(Lighting up.)_ : Thank you, God. Someone who speaks my language.\n\nTANYA: Of course, yes, we're from Zhitomir.\n\nINNA: I'm sure He's been watching over me.\n\nMAKSIM: We're not really Ukrainian.\n\nINNA:...I have in-laws in Uter-Mylovsk'.\n\nBRANGWEN: I've a notion they're known to each other.\n\nROY: Are you going to let us get to bed?\n\nANNA: They'll need towels and a fresh set of bedding...each.\n\nBRANGWEN: And don't forget the food now.\n\nROY: Betty's gone home for the evening.\n\nBRANGWEN: But I'm yere.\n\nANNA: She's getting her feet back under the table.\n\nBRANGWEN _(Moving toward the kitchen.)_ : Just tell me what there is to eat.\n\nANNA: I'll do whatever wants doing.\n\nROY: Why can't we all just be friends?\n\nANNA: The Mormons is settled in Utah, my own love. Your decree puts you firmly with me, alone, under the one roof.\n\nBRANGWEN: Of course, yes, I'm on my way home.\n\nANNA: Has she pitched tent out in the garden or something?\n\nBRANGWEN: I don't want the least little fuss now.\n\nROY: She didn't ask to come this evening.\n\nANNA: Is it all right? I'm asking her to leave.\n\n**Overlapping with this**.\n\nMAKSIM: We actually came from Kaliningrad.\n\nTANYA: His sister lives there.\n\nINNA: I've never lived anywhere else but Bryansk.\n\nMAKSIM: We went through there the year before last.\n\nINNA: How long have you lived in this country?\n\nMAKSIM: Forty hours.\n\nTANYA: They kept us in Harrow last night.\n\nMAKSIM: Did you fly to Heathrow from Moscow?\n\nINNA: It seems like a month in the train. I changed at Harrow and nearly caused a panic when I crossed the tracks on foot.\n\nTANYA: God in heaven.\n\nMAKSIM: The rail's electrified.\n\nTANYA: She wouldn't know that.\n\nINNA: Of course not, in Russia it's dead.\n\nMAKSIM: They must have thought they were watching a suicide.\n\nINNA: With my luggage?\n\nMAKSIM: Minor detail.\n\nTANYA: It's very strange.\n\nINNA: At Harrow in someplace called Tesco's \u2013\n\nTANYA:...you used vouchers.\n\nINNA: I hit a man. He started putting my things in his shopping bag, so I got him to stop. How could I know he was trying to help me?\n\nMAKSIM: How could he know she was Russian?\n\nINNA: His face like this... _(Shows horror.)_...and the woman behind grabbed my arm. Someone else was beginning to ring the police, but then the first man explained.\n\nTANYA: Embarrassed to tell you what happened to us.\n\n**Suddenly someone outside throws a brick through the window. The alarm goes off and doors open upstairs. Several male shouts further orchestrate the disturbance**.\n\nBEZNIK _(Off, heavily accented.)_ : Somebody trying to kill us!\n\nROY _(Calling up the stairwell.)_ : It's all right. It's only the local booze brigade saying goodnight on the way home. Happens two or three times a month, and it's nothing personal.\n\nANNA: Should you be putting it to them precisely in those terms?\n\nROY: What, you want to start a vendetta?\n\nBRANGWEN: Would you consider seeing me home?\n\nINNA _(To TANYA.)_ : Ground-floor windows in Russia are fitted with iron bars.\n\nTANYA _(To MAKSIM.)_ : Are we going to stay here?\n\nMAKSIM: You think we should go back to Harrow...maybe Kaliningrad?\n\nROY: I tell you everything's all right. You're perfectly safe.\n\n**Police siren sounds**.\n\nMAKSIM: Excuse me, please. But this is the kind of thing we thought we were escaping by coming to Britain.\n\nROY: Well, what do you want me to do about it?\n\nTANYA _(To INNA.)_ : My husband's a lawyer.\n\nROY: It's my broken window, not yours.\n\n**Lighting shift**.\n\n**Kitchen. BETTY, wizened and 60ish, is talking to INNA as they both prepare food. BEZNIK, Kosovan, is trying to chat up TANYA, peeling spuds**.\n\nBETTY: You must stop me if I'm speaking too quickly or...or quickly, that's right.\n\nINNA: I little understand.\n\nBETTY: That's a problem for most of us, in a place such as this. And I was here in the days when they looked after old folks.\n\nINNA: Sorry.\n\nBETTY _(Loudly.)_ : I'm saying it used to be a Home.\n\nINNA: This is not home? I don't understand.\n\nBETTY: You don't get it, my dear. Not a home like in 'house', but like in old people's home...for people that can't look after theirselves.\n\nINNA: Hospital?\n\nBETTY: No! I mean, something like. Only not the same thing...you know \u2013 different.\n\nINNA: Difficult, English language. Many meanings same word.\n\nBETTY: Well, there's not many as would confuse house for a nursing home. Or a hospital.\n\nINNA: Hospital have nurses in England...or no.\n\nBETTY: You're not just being clever, are you? _(Ire rising.)_ I said an old people's home!\n\nINNA: What means 'clever'?\n\nBETTY: Smart-arse...\n\nINNA:...'arse'?\n\nBETTY: Ask your English teacher, dear. I ent no genius. Only don't ask him that, for God's sake. He'll think you been mixing with the wrong sort of people.\n\nINNA: Russia language one word for one thing.\n\nBETTY: That's all right \u2013 spare me the Russian for...It's your reverse side, you know \u2013 this. _(Turns and touches her bottom.)_\n\nINNA: Russia people show to people they don't like.\n\nBETTY: They don't! Amazing, that is. It's almost the meaning of 'smart-arse', and we've barely a sentence to say to each other! When's your birthday? I can make you a chocolate cake.\n\n**Overlapping**.\n\nBEZNIK: I try spikking English for you. My name Beznik, you know? Bez-nik? From Kosovo. Parents kill by Serbia bastard. Sister rape. You like the whiskey? Johnnie Walker red.\n\nTANYA _(Low tone in 'her own tongue'.)_ : You couldn't be from Kosovo, no. I don't believe it. Your eyes are blue. _(To BEZNIK, with gestures.)_ Your eye...very! _(Pointing to her own eye.)_ Very.\n\nBEZNIK _(Grinning.)_ : My eye, yes, very!\n\nTANYA: Very! I mean...how?\n\nBEZNIK: Your name?\n\nTANYA: My eye is blue, of course! _(Lower tone.)_ What does he think? I'm a natural blonde.\n\nBEZNIK _(Idiot slowly.)_ : What is your name, please?\n\nTANYA: Tanya. Tatiyana.\n\nBEZNIK: That is 'very'.\n\nTANYA _(Responding in kind.)_ : Most women in Russia.\n\nBEZNIK: What? _(Grinning.)_ Your name? Fantastic. Russia drink vodka. We drink Russia vodka to Russia women \u2013 okay? Russia vodka...Stolichnaya. Okay?\n\nTANYA: Russia vodka...okay.\n\nBEZNIK: You, me and...your boy together drink.\n\nTANYA: 'Boy'? You mean my husband? _(Deliberately.)_ You know the meaning of 'husband'? Married...boy...man and me.\n\nBEZNIK: I go finding wife very day.\n\nTANYA _(Low tone.)_ : You think you're going find her through me? _(To him.)_ All right...I go now... Bye-bye. _(Starting off.)_\n\nBEZNIK: You want drink now, upstairs?\n\nTANYA: I'm off to get my husband, you pervert. You keep out of my way. _(Fending his arm away.)_ No, no...no touch me \u2013 I'm married, see? _(Shows her ring finger.)_ Innochka, quick. Over here.\n\nINNA: What is it?\n\nTANYA: Safety in numbers.\n\n**BETTY watches them and addresses BEZNIK**.\n\nBETTY: We meant to warn you about this one. _(Sternly.)_ Beznik. You pack it in, you hear me? Mr Roy speaks louder than I do. Do you want to hear it from him?\n\nBEZNIK _(Corresponding tone.)_ : I don't want to marry you, old lady. No England passport, no, thanks.\n\nBETTY: I never asked you to marry me, you perfect loon. I'm talking about these here girls. You're not to speak to them, hark to me? Or smile, or just about nod. They're off limits to you in any dimension, and I'm not bothered if you can't catch my every word. You get my meaning, I'm sure, very clearly, and if you don't, we can circulate Mr Roy.\n\nTANYA _(To INNA.)_ : You don't smell something burning, do you?\n\nINNA: Burning?\n\nTANYA _(Rushing to the big oven.)_ : My God \u2013 it's the whole roast. _(Angrily, to BETTY.)_ You kept us talking so much, it's all burnt.\n\nINNA: How could it be Betty's \u2013 ?\n\n**The smoke alarm goes off**.\n\nTANYA _(Showing food.)_ : You see it? There's nothing left....and Roy will ask us to pay for it. Maybe dock us some wages or...withdraw our food. You're my witnesses, eh? _(To BETTY.)_ I'm going to tell him it's all your fault.\n\nBETTY: I don't get a word what you're saying, dear. I know you're very angry about something...\n\nTANYA _(To BETTY.)_ : Crazy...you are... _(Touches her temple.)_\n\nBETTY: Because I tried to protect yer? I thought you needed help.\n\nINNA: You can't talk to someone like that \u2013 it's very rude.\n\nBETTY: He was going to merlest yer.\n\nTANYA _(Thrusting the tray of food at BETTY.)_ : You want the bitch? All right then, have it. Go on. Eat it yourself...off the floor. Senile...somebody ought to put the bitch down.\n\n**The food falls to the floor. INNA bends down to pick it up**.\n\nLet her lick it up. It's what she wants. ( _To BETTY, pointing_.) You go...like dog.\n\nINNA: Your language gives me a headache. Please stop.\n\nBETTY _(Tearfully.)_ : I was only trying to help yer.\n\nINNA: Beznik, help, please.\n\nBEZNIK _(Eagerly, coming forward.)_ : Yah, what, darling lovely, I help.\n\n**INNA goes for a mop to try to shut off the alarm with the mop handle. BEZNIK eagerly grabs hold of her waist**.\n\nTANYA: Why don't you bring the bucket for the floor?\n\nBETTY: I'd do anything for you to forgive me. Only please tell me what I done wrong.\n\nTANYA _(Waving her hand as though she's trying to wave away an insect.)_ : Blakha-mukha... can't she get the point?\n\nINNA _(Trying to show him.)_ : You must...like this...do.\n\nBEZNIK _(Laughing, holding tighter.)_ : Like this?\n\nINNA _(Flaring.)_ : I am not lovely darling for you...I am witches! I keel you. Understand?\n\nBEZNIK _(Angry.)_ : Fine, okay, fuck you very much. Fuck you.\n\n**Smoke alarm upstairs goes off. Voices off seem to echo the level of disturbance here. Doors open noisily. ANNA remonstrates before coming on with MAKSIM**.\n\nANNA: You think you can do what you like in this house?\n\nMAKSIM: When the food is as awful as this is, I do, yes. It becomes a necessity.\n\nROY: What the hell's going on?\n\nANNA: There's a thing here called fire regulation \u2013 with a \u00a31,000 fine attached.\n\n**INNA manages to stop the alarm in the kitchen when ROY, MAKSIM and ANNA come on. The upstairs alarm stops shortly afterwards, followed by a slamming bedroom door**.\n\nROY: So you're saying it wasn't by mistake?\n\nANNA: What wasn't? A bloody barbecue in the bathroom? He has to be some sort of deranged.\n\nMAKSIM: It's raining outside.\n\nANNA: And if it wasn't?\n\nMAKSIM: We'd have it outside.\n\nROY: W-wait a minute. _(To MAKSIM.)_ She's serious \u2013 a barbecue?\n\nBETTY: That's exactly what we had in here.\n\nROY: Yeah, but I mean like...how?\n\nINNA _(To TANYA.)_ : You don't know what they're talking about, do you?\n\nTANYA: It must be the fire alarm.\n\nINNA: But we'd have become mad. Are they saying I did wrong?\n\nTANYA: Not this one, Innochka...something to do with Maksim.\n\nMAKSIM _(Overlapping, to ROY.)_ : Three days ago in Tesco. It would have gone off another day.\n\nROY: But how in the loo did you cook it? And why?\n\nMAKSIM: The standard of hygiene is very poor.\n\nANNA: Pardon?\n\nMAKSIM: A Primus stove.\n\nTANYA: Do you want to tell us what's going on?\n\nINNA: Just tell me I'm not to blame.\n\nROY: Overlooking the obvious insult, my wife is right \u2013 it's against regulations.\n\nMAKSIM: Nothing happened, did it?\n\nBETTY: They're saying I burnt the roast.\n\nMAKSIM _(After glancing at BETTY.)_ : That's exactly why I did it.\n\nANNA: We could take you to court.\n\nMAKSIM: Fine, you do that...I'll take you on.\n\nROY: I've had dealings with barrack-room lawyers before. In the Army. That's what we called you geezers that thought you knew everything and sounded off whenever you could. Always threatening to take us to court, you were.\n\nMAKSIM: I never threatened \u2013 you asked me why.\n\nROY: Well, now I'm asking you to \u2013\n\nMAKSIM: You also asked us to help you avoid a blood bath. At \u00a31.50 an hour, \u00a35 daily for Tanya? That's you...avoiding the minimum wage. We had wages like that back in Russia.\n\nANNA: So what are they doing in this country? Ask him.\n\nROY: There's always the poultry farm.\n\nMAKSIM: You think we wouldn't consider it? You've got us working three shifts. How much are you getting for each of us?\n\nROY: Less than I'm paying you.\n\nMAKSIM: How do we know? And what, exactly, are you giving us for it? _(Looking at BETTY.)_ Food we can't eat...by a woman who must have been one of the inmates before. You can't even give us a double room, and the beds, with their plastic sheets, must have been used by the inmates as well. They smell like it. Tanya looked after people like her back in Russia. Now you're asking us to work with her? It cannot be done.\n\nBETTY: He's not talking about me, is he?\n\nROY: You don't hear anyone else sounding off.\n\nMAKSIM: No, of course. To slaughter poor chickens is easier.\n\nBETTY: Does he know what his own wife was doing before \u2013 with that one? _(Points to Beznik.)_ And look what he's doing now to Inna.\n\nBEZNIK _(Grins.)_ : We good friends...Russia\u2013Kosovo.\n\n**INNA removes his hands from her waist**.\n\nTANYA: I wish I understood what you're telling them.\n\nMAKSIM _(To ROY.)_ : I could report you to the Home Office on several counts.\n\nROY: And Strasbourg...could you?\n\nMAKSIM: Of course, but London's enough.\n\nROY: All right then, try it.\n\nMAKSIM: Exploitation...sub-standard living conditions.\n\nROY: Yeah, you told me. Have a go.\n\nINNA _(Hushed, to TANYA.)_ : I don't like it. They both look...\n\nTANYA: You're not going to hit him, are you dear?\n\nINNA: Roy could call the police...and he's bigger and stronger anyway.\n\nTANYA _(Moving to stop him.)_ : Please don't, Maksim.\n\nMAKSIM: I'm not going to touch him, don't worry.\n\nTANYA: Well, then come back upstairs. _(Trying to lead him off.)_ Come on.\n\nROY: I figured you'd bottle out in the end.\n\nMAKSIM: Will you give us a room of our own anyway?\n\nROY _(Barks laughter.)_ : Why should I?\n\nMAKSIM:...two single rooms and we're married.\n\nROY: No, I mean where's your threat? How much teeth have you really got? Strasbourg! You're two trouble-makers, that's all. Pains in the arse. I'd rid myself really if you weren't around. And where else, exactly, are you going to go? _(Starting off.)_ If there's nothing else you want to whinge...\n\nMAKSIM: We could leave here tomorrow.\n\nROY: And I'd be straight on the phone. Obliged to report it, aren't I? Violation of terms. They'd get after me for that, sure. I could go to gaol for not reporting you...I'd almost certainly get the chop.\n\nANNA: What on earth are you saying \u2013 who's going to gaol?\n\nROY: Our legal beagle reckons he's nailed me.\n\nANNA: I want you both to stop this. It's not at all nice.\n\nROY: It's not at all anything, my love. That's why I can't be arsed. But if you think you got hold of something, as I say...Give us a laugh.\n\nBEZNIK: I want us all drink.\n\n**Lights begin to fade except on TANYA and on INNA, picked out by spotlight. The others are frozen in shadow**.\n\nTANYA: Maksim explained it to me upstairs. He's really angry with Roy, and the way he explained it...there's no going back.\n\nINNA: What do you mean? Betty's really a good person. You made her cry after you left.\n\nTANYA: I can't talk about that now \u2013 we're leaving.\n\nINNA: You can't do that.\n\nTANYA: Maksim says we can...we must, in a way.\n\nINNA: What 'must'?\n\nTANYA: What all that was about today. If no one knows where we're going, Maksim says, what can they do? Maksim says he's in touch with a man who can give us a job and a resident permit.\n\nINNA: I don't believe it.\n\nTANYA: I can't argue with you. _(A beat.)_ They turned down our application.\n\nINNA: But you can appeal.\n\nMAKSIM _(Shouting to ROY.)_ : I'm not going to take any more of it!\n\nROY: You're barking.\n\nTANYA: On what grounds? This is a genuine offer, you see? A job and a residence permit. Maxim's a lawyer, Inna. Who's going to cheat us? He's worked for oligarchs.\n\nINNA: I don't want to know any more. Don't.\n\nTANYA: I can't say any more. I don't know it. Except...I'll keep in touch.\n\n**Lighting shift. Everyone but TANYA and MAKSIM leave the space and OLIVER takes ROY'S place when the lights come up again**.\n\nOLIVER: We cater for a kind of ex-collusive clientele. I don't mean they're funny peculiar. But we've only the twelve _en suite chambres_ , you see? And we date back to Tudor times\/look at the beams. Do you know about 'Murder Weekends'? Themes like that for the paying guests. Well, we don't go into any of that, thank you. I'm not averse to hearing we have the odd ghost, that can add to the overall _\u00e9lan_ as long as it's not taken to extremes\/you said you were what nationality?\n\nMAKSIM: Russian.\n\nOLIVER: Does your little friend speak? _(To TANYA.) Bonjour, ma petite_.\n\nMAKSIM: Does she speak what, English, you mean?\n\nOLIVER: Well, yes. But if she doesn't, it's a fair bet she's not going to say much at all. At least nothing the rest of us can understand. What's she saying now, for example? (Point to me, I think.)\n\nMAKSIM: Tanya and I are married, by the way.\n\nOLIVER: That's exactly our sort of guest \u2013 marrieds. Young or old, but they come to us attached, if you catch my deeper meaning. One word says it all, like the most succinct holiday brochure. One look and you know we don't readily welcome pets or large family or anything one might describe as brash. So if you think you'd be happy to join us...\n\nMAKSIM: You talked of a resident permit.\n\nOLIVER: Oh, yes, you said Russian, didn't you. We've had an assortment of Asians working for us, Orientals...\n\nMAKSIM: You're not racist, are you?\n\nTANYA _(Hushed voice.)_ : He looks to me like a blue boy.\n\nOLIVER: Is that what it sounds like, Russian? It must be colourful.\n\nTANYA: I've heard it's the English disease. I think we should go somewhere else.\n\nOLIVER: Wait a moment \u2013 we had a Yusipov staying here at one time. I think it might be an omen, don't you? We can try it out anyway, all of us, and if we find we're not a match...\n\nMAKSIM: What about salary?\n\nOLIVER: Well, now, which do you want answered first? 'Am I racist?' If they've worked here, how can I be? You want a resident-something or other. Why? Is England a police state?\n\nMAKSIM: We came in as asylum seekers! We need permission in order to stay!\n\nOLIVER: I hear perfectly out of both ears.\n\nMAKSIM: I'm sorry. Chinese and Asians have worked here. Were they deported?\n\nOLIVER: They moved on by their own choice, as I remember. Does \u00a35 an hour appeal to you? That was your third point.\n\nMAKSIM: Is that for each of us?\n\nOLIVER: You like? I told you the kind of people we have here. Nothing about us is cut-rate.\n\nMAKSIM _(To TANYA.)_ : He said we can have \u00a35 each an hour... _(To OLIVER.)_...for how many hours?\n\nOLIVER: To start with? Seven-to-ten thirty, midday-to-six.\n\nMAKSIM _(To TANYA.)_ : Eight and a half hours, finish at six.\n\nTANYA: I don't think that's possible. Hotel work isn't like that. Think about it. We're servants. We work different hours to other people.\n\nOLIVER: What's your name, dear?\n\nTANYA _(To MAKSIM.)_ : What's \u2013 ? _(To OLIVER.)_ My name?\n\nOLIVER: Ah, she understands that much. Ex-cellent.\n\nTANYA: Tanya.\n\nOLIVER: Ex-quisite.\n\nTANYA _(To MAKSIM.)_ : I want to leave.\n\nMAKSIM: No, let's try it for a while anyway. _(A beat.)_ Where else are we going to go? Tonight?\n\nTANYA: Has he talked about what kind of rooms they're giving us?\n\nMAKSIM: One room, I think it is. _(To OLIVER.)_ And we'd be living where?\n\nOLIVER: A studio. Downstairs.\n\nMAKSIM _(To TANYA.)_ : That's right...in the basement.\n\nOLIVER: There was a Tanya in _Eugene Onegin_ , wasn't there?\n\nTANYA _(To MAKSIM.)_ : Why can't we live outside?\n\nMAKSIM: Outside where? We're at the heart of the Devon Moors.\n\nTANYA: Yes, exactly, trapped.\n\nOLIVER: Can I show it to you, the room?\n\nMAKSIM _(To TANYA.)_ : He's asking to take us down now.\n\nTANYA:...and what, rob and kill us?\n\nOLIVER: What about something to drink?\n\nMAKSIM: It's something we never got at Winter's.\n\nTANYA looks from one to the other.\n\n**Lighting shift**.\n\n**The sound of doors slamming in regular succession accompanies MAKSIM and TANYA moving in. The light intensifies on the room, where 'MISS HIGHLIGHTS' is seen. This boy is androgynous rather than camp. He flinches when FRANKLYN speaks**.\n\nMAKSIM: We could save as much as \u00a310,000 and then go to Canada.\n\nTANYA: As long as we don't get deported.\n\nMAKSIM: I'm referring to when we're deported, say, in two years' time. Trust me, these things take that long.\n\nFRANKLYN _(Mock chanting, off.)_ : Bless and keep us, Miss Highlights. Miss H lets her countenance shine upon us and give us peace. Blessed be the name of Miss H!\n\nTANYA: Alcoholics are living here, too?\n\nMAKSIM _(Crossing to the 'door'.)_ : I don't know how many others.\n\nTANYA: Other women would be nice.\n\n**MAKSIM looks out. MISS HIGHLIGHTS signals him from inside the room**.\n\nMISS HIGH: Don't provoke him.\n\nMAKSIM: Huh? Who?\n\nMISS HIGH: Franklyn Gerontius \u2013 Oliver's nephew. Are you the new ones? We need to become acquainted.\n\nMAKSIM _(Squinting.)_ : Where are you?\n\nMISS HIGH _(Into the light.)_ : In here.\n\nFRANKLYN: I have a feel-ing...you're not alone!\n\nMISS HIGH: Can you come inside if you're coming? He'll attack if he sees you.\n\nMAKSIM: We're trying to find our room.\n\nMISS HIGH: Yes, in here. _(Crossing to them.)_ He's come back from the pub in the village. _(Moving MAKSIM inside.)_...and he's wild.\n\n**They are 'in the room'**.\n\nMAKSIM: Tanyush?\n\nTANYA: We've got to get to bed...if we start work at seven.\n\nMISS HIGH: I work in the kitchen...my name's Dorian. But he calls me something provocative. Don't ask me to tell you what. It's based on the fact that I colour my hair, though. I detest him.\n\nMAKSIM: How many others are working here?\n\nTANYA: Maksim.\n\nMAKSIM: Wait a minute. _(To MISS HIGHLIGHTS.)_ Any other women on the staff?\n\nMISS HIGH: You're not frightened of us, are you?\n\nMAKSIM: I was thinking of my wife.\n\nMISS HIGH _(To TANYA.)_ : Oh, yes, hello. I can see what you mean. Well, let's see. We did have a Japanese girl...Yuni. Then there's Georgina, looking after the rooms. I think you'll be working with...Why are you looking so oddly at me? _(To MAKSIM.)_ Please tell her to stop it.\n\nMAKSIM: She doesn't speak English.\n\nMISS HIGH: Yes, well, I'm asking you. I can't help how I look.\n\nMAKSIM: Her problem is she doesn't understand. _(To TANYA.)_ He thinks you're being rude.\n\nTANYA: I'm just trying to get him to leave.\n\nMAKSIM _(To MISS HIGHLIGHTS.)_ : She means nothing by it.\n\nTANYA _(To MISS HIGHLIGHTS.)_ : You...go...away!\n\nMAKSIM: He says there's another girl...working with you.\n\nTANYA: I don't give a fuck \u2013 get him out of here.\n\nMISS HIGH: What's that, Russian you're speaking?\n\nMAKSIM: We're very tired, if you wouldn't mind...\n\nMISS HIGH: I am, myself. So let me tell you about the sleeping arrangements. The sofa lets out into a bed.\n\nMAKSIM: We'll discover all that for ourselves.\n\nMISS HIGH: No, you see, it's going to be my bed...where I sleep. Unless you have other ideas.\n\nMAKSIM: But this is our room.\n\nMISS HIGH: Yes...and the bedroom. And mine. Though I understand, you want privacy.\n\nMAKSIM: We want this room to ourselves!\n\nTANYA: What's going on here? Isn't this ours?\n\nMAKSIM _(To TANYA.)_ : Just a minute here.\n\nMISS HIGH: Well, where am I going to sleep? Franklyn's liable to do anything after his drinking...and to me especially...you don't know what he's capable of.\n\nMAKSIM: That's nothing to do with us.\n\nMISS HIGH: It has to be if you're staying here \u2013 you can't escape it.\n\nMAKSIM _(Starting out.)_ : I'm going to go see the manager.\n\nMISS HIGH: Oliver? He's the owner, and he'll be asleep.\n\nTANYA: Where are you going? Don't leave me with him.\n\nMAKSIM: I'll only be a minute.\n\nMISS HIGH: I warned you, remember. About Franklyn, too. Please don't tell him I'm here.\n\nFRANKLYN _(Shadowy.)_ : You enjoy coming out, don't you, Dor-ian? Miss Highlights? So why are you hiding from me?\n\nMISS HIGH: Don't go out yet, I beg you.\n\nTANYA: This is crazy. Let's get out of here. You hear me, Maksim? Telephone a taxi, back to the station. Winter's Rest is far better than this. _(To MISS HIGHLIGHTS.)_ You must go. We must to sleep. Go. Go. To sleep.\n\nMISS HIGH: If only I could, love.\n\n**MAKSIM has been on his mobile**.\n\nMAKSIM: Mr... _(Looking at MISS HIGHLIGHTS.)_\n\nMISS HIGH:...Oliver.\n\nMAKSIM: There seems to be a confusion, can you come down? _(A beat.)_ Maksim Pavlov.... _(A beat.)_ All right, but I tell you, we can't stay under these conditions. _(A beat.)_ I'm sorry.\n\nMISS HIGH: Sorry about what? You never got him to come down?\n\nTANYA: What the fuck's happening?\n\n**OLIVER appears out of the shadows**.\n\nOLIVER: I never gave it another thought, to be honest. Aren't you Russkies used to communal living?\n\nMAKSIM: You didn't know we were Russian when I rang up.\n\nOLIVER: I'd no idea there'd be two of you. Dorrie used to share with a boy named Jan. Call it foolish, but I assumed you'd be more like him...Well, alone, anyway.\n\nMAKSIM: But I told you on the phone there were two of us.\n\nOLIVER: I know you did \u2013 it slipped my mind. I must have been thinking about something else. Or I thought about what I expected to happen. Have you ever experienced that?\n\nMAKSIM: Are you going to suggest a solution? We want to get quickly to bed.\n\nOLIVER: Well, now, Georgina's room is just down the hall. They'll be working together \u2013 why don't she and Tanya share?\n\nMAKSIM: Because Tanya and I are married. Do you understand?\n\nOLIVER: I'm familiar with the practice. There's no need to be rude.\n\nMAKSIM: Two men and a woman can't live together. Can't you see?\n\nOLIVER: For the sake of the sofa, I'm sure Dorrie wouldn't mind my saying \u2013 he's hardly a \u2013\n\nMAKSIM: I don't care what he is, we're not going to stay here if he does.\n\nOLIVER: All right. I'll sort something out by the morning.\n\nMAKSIM: My wife wants it sorted out now.\n\nOLIVER: Quite a terrier, is she? Is it all right he leaves his things tonight?\n\nMISS HIGH: Are you asking me to leave?\n\nOLIVER: The Russians are doing the asking.\n\nMISS HIGH: But you agree to it? I'm afraid to go out that door.\n\nOLIVER: Nonsense, I'm going out with you.\n\nMISS HIGH: Franklyn's back...he'll see where we've gone.\n\nOLIVER: Is that so difficult?\n\nMISS HIGH: Your nephew, I'm sorry to say, is very strange. You must know of his attitude toward me.\n\nOLIVER: But it's not personal, dear. He's a drinker, and drinkers, you know, hate the world. I hope that doesn't sound sanctimonious, when hospitality's our metier...\n\nMAKSIM: Excuse me, please.\n\nTANYA: You want me to do it myself?\n\n**Lighting shift**.\n\n**FRANKLYN comes out of the shadow, with a bottle or two in tow**.\n\nFRANKLYN: Don't think I'm intruding \u2013 I just brought you these.\n\nMAKSIM _(Trying to move him out.)_ : And then you're leaving?\n\nFRANKLYN: Sorry?\n\nMAKSIM: We've been working since dawn.\n\nFRANKLYN: There's the weekend ahead.\n\nMAKSIM: Yeah, we're working then, too, double shift.\n\n**TANYA is slightly groggy from trying to rest**.\n\nTANYA: No..no...He brought us drink? Let him stay. _(To FRANKLYN.) Spasobichki_. Sank you very much.\n\nFRANKLYN _(Moving in.)_ : You know what they say about wetting the baby's head? Well, I wet the weekend ahead...even when it's Tuesday. You got any glasses?\n\nTANYA _(Up on her feet with alacrity.)_ : No, no...I do. You zits down.\n\nMAKSIM _(Correcting her.)_ : 'Sit' down.\n\nTANYA: 'Zit.'\n\nMAKSIM: 'Sit'...Try saying it with 'ts'.\n\nFRANKLYN: Try shutting up about it...What does it matter? I'm with the wogs in this \u2013 I sits where I pisses.\n\nMAKSIM: She has to start learning good English.\n\nFRANKLYN: Yeah, but you know what, for your guests? It's bloody boring.\n\n**A pause. TANYA, gingerly, gives FRANKLYN a glass**.\n\nMAKSIM: You work...is that right...in the kitchen?\n\nFRANKLYN: So you got Wanda Wonderbra out of here? _(Lifts his glass.)_ To you! _(To TANYA.)_ Spastic \u2013 whatever. Like to cut out its heart...fry it up in a mixed grill including its goolies. He'd probably be grateful for that. And I don't want him anywhere near me.\n\nMAKSIM: In Russia, until very recently, people like him were thought criminal.\n\nFRANKLYN: Wait a minute. What do you know about Russia?\n\nMAKSIM: We're Russian.\n\nFRANKLYN: Oh, yeah, like I believe that. You believe I'm related to Fu Manchu? _(Slants his eyes.)_\n\nMAKSIM: If you said you were, why not?\n\nFRANKLYN: You're fucking whacked. Ought to cut out your heart and all.\n\nTANYA _(Hushed.)_ : I don't think he's very nice.\n\nMAKSIM: Yeah, all right.\n\nFRANKLYN: What language is that, Serbo-Croat? _(Laughs.)_ So when, exactly, were you in the joint?\n\nMAKSIM: I'm sorry?\n\nFRANKLYN: I thought you said you knew creatures like Ladybird here back in prison.\n\nMAKSIM: I think you misunderstood.\n\nFRANKLYN: I think you're some kind of Porky-pier. If he turned up in any of the places I've been in, he'd be brown bread after the first day. _(Looks at him.)_ Oh, Jeez, are you thick or what? Dead, you know? Ker-plunk. Pasted.\n\nMAKSIM: I think that's what I was saying.\n\nFRANKLYN: You never said that at all. What do you take me for?\n\nTANYA: Just leave him alone.\n\nFRANKLYN: What's she saying about me now? _(To TANYA.)_ Making jokes about my equipment, were you? _(Hands to his flies.)_ I got to prove you wrong?\n\nMAKSIM: She asks...can you leave us alone...please?\n\nFRANKLYN _(In his face.)_ : Please...Who wants to fuck with you anyway? _(Gets up to leave. Without the bottles.)_ It's bloody bor-ing.\n\n**Lighting shift**.\n\n**TANYA and MAKSIM cross to meet INNA. In the course of the scene \u2013 taking place in a pub \u2013 OLIVER comes out of the shadows to talk to MAKSIM \u2013 presumably back at the hotel. The scenes should dovetail. The first starts with TANYA trying to get INNA to have another drink**.\n\nINNA: No, please, no more for me.\n\nTANYA: Really, it's all right...because we work here, you see?\n\nMAKSIM: This bar is attached to the hotel.\n\nTANYA: 'One get...one free'.\n\nINNA _(Giggles.)_ : 'One get...one free'...\n\n**While they're pouring another round of drinks, for INNA as well \u2013 doubles**.\n\nMAKSIM: It's actually, 'Buy one...get one free'.\n\nINNA _(Not hearing him.)_ : I'm so embarrassed to go out of the shop with 25 packets of Kit Kat...five tubs margarine\n\nTANYA: We have to take what we can get in this job.\n\nMAKSIM: It's not that bad.\n\nTANYA: It's only because the boss fancies you. _(To INNA.)_ It's true, you know. Every one in the place is some kind of blue boy.\n\nINNA: It's all over England.\n\nTANYA: I'm beginning to think you're right.\n\nINNA: On the television, late last night, there was a programme like a training course...They were out in the park, you know, hunting, like wolves. _(Pulls a predatory face.)_ And then the camera followed them into the public convenience...my God. It was awful...and so funny, believe me. One of them... _(Laughs.)_....put his hose in a hole...and another one, in the booth next him, takes the hose and he...I-don't-know-what, you know? And there it was, right there, on the television. England, believe me, strange country.\n\nTANYA: Where was Roy while you watched this? Was Celestine there?\n\nMAKSIM: Inna's not there any more. Don't you remember?\n\nTANYA _(to INNA)_ : You never told us.\n\nINNA: I married.\n\nTANYA: Inna!\n\nMAKSIM: Yes, I knew.\n\n**TANYA looks from one to the other**.\n\nMAKSIM: I told you she was moving out.\n\nINNA: I told you myself, one week after you rang me you'd found the job here.\n\nTANYA: Well, yes, of course I remember that part. I thought you...I was getting ready to ask if you wanted a job here. We could both do with the company, and I'm sure Oliver wouldn't mind. He'll do whatever Maksim wants, very nearly. That's the reason we're here.\n\nINNA: I can't work. I'm no longer after asylum.\n\nTANYA: What do you mean? We can't work. _(To MAKSIM.)_ Get him over here. She can start this week.\n\nINNA: No, you don't understand. Explain it, Maksim.\n\nTANYA: You can explain it. I'll follow.\n\nINNA: According to English law \u2013\n\nMAKSIM: You and I aren't after anything, except money.\n\nTANYA _(waving him away)_ : I know that much. _(To INNA.)_ According to law, what?\n\nINNA: If I apply as the wife of an English man \u2013\n\nTANYA: He is English, your husband?\n\nINNA: Yes, he is...and disabled, in fact.\n\nTANYA: Well, what does that have to do with anything?\n\nINNA: It had to do with a lot, once I got into Court. Home Office said I married the only one I could find.\n\nTANYA: Well, that's true, isn't it?\n\nINNA: But I never did anything like Beznik or Abdullah. Finding a girl in a pub, sleeping with her, and saying they're going to get married the next day.\n\nMAKSIM: You don't drink, for a start.\n\nTANYA: Fine, but who's going to know that?\n\nINNA: Anyway, while they're deciding my case, I can't work.\n\nTANYA: But if he's disabled, neither does he. So you're both homeless.\n\nINNA: No, it's not like it is back in Russia...thank you, God.\n\nOLIVER _(loudly, emerging out of the shadows)_ : And if you'd had faith in me from the start....\n\n**Something of a transition. TANYA and INNA recede in shadow while OLIVER takes MAKSIM into the spotlight**.\n\nOLIVER: I understand it's been difficult for you.\n\nMAKSIM: Your nephew's mad...dangerous.\n\nOLIVER: But he doesn't work here.\n\nMAKSIM: He lives here.\n\nOLIVER: Not all the time.\n\nMAKSIM:...a criminal record and Tanya's plain frightened. And who can protect her from that?\n\nOLIVER: Police can...the laws of this country. There's a restraining order forbidding him any contact with ordinary folk. I wouldn't be surprised if he was already on remand. Maybe goes back at the weekends.\n\nMAKSIM: Your business, not ours.\n\nOLIVER: Well, it's not really \u2013 how can it be \u2013 my...\n\nMAKSIM: It's not meant to be mine and Tanya's \u2013 that's what I'm trying to tell you.\n\nOLIVER: Don't you like it with me?\n\nMAKSIM: What do you mean...?\n\nOLIVER: The money I give you, it's not enough?\n\nMAKSIM: Tanya's been working twelve hours sometimes.\n\nOLIVER: I'm glad we're talking about this.\n\n**A pause**.\n\nYou know, now that Christmas is coming...\n\nMAKSIM: What, you want us to work even more?\n\nOLIVER: You're not giving me a chance. Now admit it. You ask for my help, yet you won't hear what I have to offer. Does that even make sense? No, it doesn't. Admit it.\n\nMAKSIM: What is it you want?\n\nOLIVER: We've received several compliments about you from the guests. We must treat you as featured attractions, it seems. And I know why they say it. You're like a pair of walking...what are they called...Russian dolls?\n\nMAKSIM: Matrioshki?\n\nOLIVER: Yes, exactly, that's right.\n\nMAKSIM: And then at Christmas, Father Christmas appears with the Snow Queen...each other's costume.\n\nOLIVER: Well, now, isn't that fun. It chimes perfectly, in fact, with what I had in mind. Would you consider showing us?\n\nMAKSIM: Well, no, the costumes are back home in Russia.\n\nOLIVER: Oh, really? I had in mind, anyway, a sort of family gathering...minus Franklyn, you may be sure. But we'd be on our own, with maybe one or two welcome additions...\n\nMAKSIM: You're giving us the time off?\n\nOLIVER: Oh, yes, as much as you want...afterwards.\n\nMAKSIM: 'After'?\n\nOLIVER: It's the slack time, January. And doesn't that anyway coincide? Don't Russians celebrate later than we do? It's perfect timing, in fact.\n\nMAKSIM: And you could guarantee us...what? How much extra?\n\nOLIVER: So you'll do it then...\n\nTANYA _(Long and plaintively.)_ : No-oh-oh!\n\n**A mini shift has TANYA arguing with MAKSIM. Party laughter is heard, off. Toward the end of their discussion, the lights rise on 'a bedroom' where BRANGWEN is laughing to the brink of her control**.\n\nMAKSIM: Hasn't he always paid what he promised?\n\nTANYA: And I want some fucking time off. You hear that \u2013 fucking? I'm serious.\n\nMAKSIM: I never thought you weren't. I told Oliver \u2013\n\nTANYA: 'You told Oliver' \u2013 what? Your wife wants him to lay off her husband? Did you? He should hire Inna to work with your overworked wife.\n\nMAKSIM: That's up to Inna.\n\nTANYA: He can make her an offer, can't he? To satisfy me? Doesn't he think I want some laughter?\n\nMAKSIM: That's what I thought he was offering us.\n\nTANYA: In other words, no, I'm his servant to swab out his messes. That's very nice. _(Trailing off.)_ The stuff my doll's dreams are about. Fuck him into an early grave.\n\n**She tosses a glass down in anger, and this coincides with a bottle breaking in BRANGWEN'S area, bringing a shift in lighting. BRANGWEN addresses someone momentarily out of view**.\n\nBRANGWEN _(Happily)_ : We'll have to pay extra for that, you know. To say nothing of other replacements. And then it's Christmastime. What if they run out? I hope this isn't a sample of our married life to come.\n\n**TANYA comes on with cleaning material. They don't recognize each other for the moment**.\n\nTANYA: Something's break here?\n\nBRANGWEN: Well, yes, we're sorry. A bottle...\n\nTANYA: Bottle. All bottle. Champagne.\n\nBRANGWEN: Well, yes, I know, it's a frightful waste.\n\nTANYA: Fucking...mess.\n\nBRANGWEN: I...pardon? I can't believe what I just heard. Were you swearing at me?\n\nTANYA: I must pay...fucking mess. Clean up. After pay.\n\nBRANGWEN: That's not really the \u2013\n\nTANYA:...fucking.\n\nBRANGWEN: Stop saying that to me.\n\n**A pause. TANYA begins to clean up. BRANGWEN is tearful**.\n\nI suppose it doesn't matter to you at all you've completely spoiled my honeymoon. This dream of a holiday hotel...for which we've paid an undisclosed fortune has now turned into a nightmare for us. And very much you care. Yes, thank you, very much.\n\n**BEZNIK now comes into view and immediately recognizes TANYA**.\n\nBEZNIK: Darling Tanya. Khalo!\n\nTANYA: What are you talking about?\n\nBRANGWEN: Why are you calling...why 'darling'? This woman used language to me.\n\nBEZNIK: You don't know...little Tanya?\n\nBRANGWEN: Well, I never thought...\n\nBEZNIK: Brangwen and me...married now.\n\nBRANGWEN:...and are you working here? you and...erm.\n\nBEZNIK: Wonder...yes...we drink champagne!\n\nTANYA: You could if it wasn't all over the floor...for me to clean up. And how did you happen to come here anyway? Of all places?\n\nBRANGWEN: It's probably my own inner failing. But you know I can't understand a syllable of what you're saying.\n\nTANYA: Did you tail after Inna, is that how you did it?\n\nBEZNIK: Inna? Inna marry, too...American English man. To America, bye-bye.\n\nTANYA: Did the Home Office offer you money for turning us in? Passport?\n\nBEZNIK: Passport? I get passport, sure.\n\nBRANGWEN: I'm sure it would be lovely to stay here and natter, if we only spoke each other's... It's hard enough knowing what he says.\n\nTANYA: I just want to tell you...it won't work. _(Trying to speak English.)_ Okay? Fucking off.\n\nBRANGWEN _(Tearful.)_ : That's the language she used in the first place! And she can't tell me what I did wrong.\n\n**Dovetailing into this scene, the lights come up sharply on FRANKLYN smashing the glass of the fire extinguisher**.\n\nBEZNIK: Fucking hell.\n\nBRANGWEN: Does that mean we're now on fire!\n\n**OLIVER challenges FRANKLYN**.\n\nOLIVER: Are you completely out of your senses?\n\nFRANKLYN _(Axe in hand.)_ : I'm warning you, coz. Stay back.\n\nOLIVER: I suppose it's utterly slipped your notice or concern that you've also triggered the police alarm.\n\n**Doors heard opening**.\n\nThere's no need whatever to panic. Believe me. It's a false alarm! Somebody's seasonal prank... out of hand. Happy hols one and all. _(Back to FRANKLYN.)_ They're going to be here any minute, don't worry.\n\nFRANKLYN: Fucking...animal.\n\nOLIVER: That's it, yes. Run for your life. _(He's referring to the fact that FRANKLYN himself looks like a hunted animal. He lets out a primal scream as he runs off.)_\n\nVOICE: Is somebody going to shut that damn thing off?\n\nOLIVER: I've just had a thought about something else. There is no need to panic. But regulations, I'm afraid. We'll all have to vacate the building. I do apologize.\n\n**At another part of the stage, MAKSIM and TANYA find each other**.\n\nMAKSIM: Are you all right?\n\nTANYA: Depends what you mean by \u2013\n\nMAKSIM: But you're not hurt or anything. And there's no fire?\n\nTANYA: There's a mess in the bedroom. Are you sure there isn't?\n\nMAKSIM: Only that lunatic Franklyn...and they know it. It's nothing to do with me this time.\n\nTANYA: We're still going to have to leave.\n\nMAKSIM: I've sorted it out.\n\nTANYA: I said 'had to', you idiot. Our cover's blown. The Home Office are onto us.\n\nMAKSIM: My God...was it Inna.\n\nTANYA: No, of course it wasn't Inna, my idiot darling.\n\n**The fire alarm stops**.\n\nMAKSIM: My God. Who?\n\n**Instant fade**.\n\n##### ACT TWO\n\n**Scotland**.\n\n**MAKSIM and TANYA face each other centre stage as though they are about to dance. Down centre facing the audience, hotel manager McBrodie speaks, heavily accented**.\n\nMcBRODIE: I've already told you I like what you have to offer. On that I trust we've no argument. It's your wee wife we're concerned with exclusively. If she'll no learn English, how can you continue to work together? I mean, you...well, the ceiling's the limit, as far goes the hotel trade. And I'll give it to you \u2013 she'd be a positive peach in reception. Without the language, however? Can you honestly see her in anathing other than cleaning up? She couldna even wait table. Best advised she attend evening classes.\n\nTANYA: Scotland's wonderful.\n\nMAKSIM: We're going to take breathtaking trips.\n\nTANYA: Without worrying about closing hours.\n\nMAKSIM: He told you to take an evening class.\n\nTANYA: That won't interfere with my drinking.\n\nMcBRODIE _(Approaching him.)_ : You take, for example, a professional-amateur golf tournament. Organizers have offered this as the main catering venue. What if I was to arrange it that you'd be in overall charge? I don't want to hear that your wife has qualms. Her opinion hasn't been sought. You see what I'm saying to you. Every man in the room has a wife which he talks to in exactly the same way. It's understood.\n\nTANYA _(An English-lesson-sentence, heavily accented.)_ : I have my bread in the hope that it does not fall butter-side down. This is what is known as a homily.\n\n**She takes a drink. NATALYA taps her on the shoulder. McBRODIE stands over her**.\n\nNATALYA: Excuse me, please. You work at the hotel? You are Russian?\n\nTANYA _(Looks at her.)_ : You're a guest?\n\nNATALYA _(Warmly.)_ : I work there, too, from today. _(Holding her hand out.)_ Natalya Rospova... Tasha. What's the matter?\n\nMcBRODIE _(Overlapping.)_ : The verb 'to conjugate' is the one you must treat as the key. It does not mean sleep with your boyfriend, rather, 'Which tense must I use when?' and how. That is its true definition. All right?\n\n**TANYA, meanwhile, has attempted to walk past NATALYA**.\n\nNATALYA: I'm sure you're the right girl. I've seen you there talking with somebody else...Is that your husband?\n\nMcBRODIE: Example: the auxiliary verb 'must', auxiliary meaning, 'to help'. One verb helping another, like one person helping another.\n\nTANYA _(Cod 'English' to NATALYA.)_ : I must to learn the English.\n\nMcBRODIE: Exactly. 'I must do what I'm paid for', precise meaning, 'I have no choice.' Got the idea? 'Must' is essential vocab for you...more important than 'Give me a drink.' 'Give me money.' You must, before you can get, work. 'Get one free' doesn't work...at least for me.\n\nNATALYA: Why are you being so difficult?\n\nTANYA: Can you speak it? If we spend all our time together, we'll both get sacked.\n\nNATALYA: I don't know anybody else.\n\nTANYA: Then you'll learn that much quicker than I do.\n\n**McBRODIE now approaches NATALYA**.\n\nMcBRODIE _(Slowly.)_ : Do you know any English language?\n\nNATALYA: I try...try to learn...to study.\n\nMcBRODIE: That's a good girl. And are you going to college? Classes? In the evening?\n\nNATALYA: Night?\n\nMcBRODIE: The night time is only for study and sleep. Never ever for drinking and...other things.\n\n**MAKSIM approaches TANYA**.\n\nMAKSIM: If you tried hard, I think you could work waiting tables. There's a restaurant attached. I think you'd better. I had to turn him down.\n\nTANYA: Why? What do you mean?\n\nMAKSIM: The conference he wanted me to work at would have meant leaving you. And I told him no. So I think he'll be watching us closely from now on.\n\n**TANYA is a bit groggy**.\n\nCan you at least try to get out of bed?\n\nMcBRODIE: There's an opening for someone with your kind of promise. I had hopes for another girl, but I've been let down.\n\n**NATALYA smiles obligingly without really comprehending**.\n\nTANYA: Couldn't we go somewhere else...if he let us go?\n\nMAKSIM: You said you liked Scotland so much.\n\nTANYA: Somewhere else in Scotland.\n\nMAKSIM: Wouldn't we have the same problems? You can't speak.\n\nNATALYA _(To McBRODIE.)_ : I want to learn...all I can.\n\nMcBRODIE: And I want to help you, my dear. Okay, that's splendid.\n\n**MAKSIM tries to rouse TANYA**.\n\nMAKSIM: How much did you have to drink?\n\nTANYA: It's not fair. You never have anything like the amount that I do, so you're always all right. And you speak English. You see? It's just not fair.\n\nMAKSIM: So you're saying it's my fault for speaking English?\n\nTANYA: It's plainly my fault that I don't.\n\nMAKSIM: We couldn't have come here if I hadn't.\n\nTANYA: Exactly. You did it to suit yourself.\n\n**McBRODIE and NATALYA separate. He crosses to talk to MAKSIM, she crosses over to TANYA**.\n\nTANYA: No, you go off to the conference \u2013 what's stopping you? I'll go alone to the pub.\n\nMAKSIM: You're going to have more to drink?\n\nTANYA:...after cleaning the fucking loos, sure, why not?\n\nMcBRODIE _(To MAKSIM.)_ : I'm afeared I need this quiet word with you.\n\nMAKSIM:...about working the conference, yes? I'm available for it.\n\nMcBRODIE: You may be available, but I've had to look elsewhere, I'm sorry. And someone's had a word in my ear you've borrowed one of our space heaters. Now when I said you were hotel material, I didna expect you'd be taking it all for granted.\n\nMAKSIM: The heater's mine... Do you want me to show the receipt?\n\nMcBRODIE: I'm everything you could want for an employer, but \u2013\n\nMAKSIM: The heater's mine!\n\nMcBRODIE: You're not serious.\n\nMAKSIM: I bought it from Curry's two weeks ago. _(Showing receipt.)_\n\nMcBRODIE _(After looking at it.)_ : Someone's engaging in tittle-tattle.\n\nMAKSIM: Who?\n\nMcBRODIE: Ah, no. There's no room in the hotel for secret vendettas. We'll draw a line under this like we've forgotten your mistake with the sporting tournament.\n\nMAKSIM: What do you mean my mistake?\n\nMcBRODIE: I tell you it's forgotten.\n\nMAKSIM: But I never knew you were ordering me!\n\nMcBRODIE: You're a clever boy. I've already told you. Only you want to make sure you don't miss your chance. Like arguing with your employers whenever the fancy takes you. Or letting the envious ones hold you back.\n\nMAKSIM: I don't think I know what you mean.\n\nMcBRODIE: Course you do \u2013 you're a clever macelvoy. And fond of you as I am, I cannot afford to make a place for that wife of yours merely because she's your wife. Who are you, when the dust settles? And who the hell-and-gone is she? One look and the world knows you married beneath yourself. That may be your business, but you offered me something different in a worker. Something better. You'll no tell me I'm wrong about this.\n\nMAKSIM: She works the same hours I do.\n\nMcBRODIE: Doing what? I've got her cleaning the toilets because she's your wife. But I have to be honest with you \u2013 we've our own untrained lassies for that kind of work. And you led me to think I'd be buying a pair.\n\nMAKSIM: Tanya's family is starving in the Ukraine.\n\nMcBRODIE: That cannot concern me for its own sake. Someone's told me she's fond of her tipple.\n\nMAKSIM: What does that mean? My knowledge of English does not include jargon.\n\nMcBRODIE: I'm trying to tell you she's not good enough. _(A beat.)_ With a higher position, you could afford to keep her at home.\n\nMAKSIM: At home where? We'll lose the room here if we don't both have jobs. _(McBRODIE sighs.)_ How much longer can you keep her on? Do you want me to train her? What do you want her to do...and when? For God's sake, speak.\n\n**Overlapping**.\n\n**TANYA appears with a bucket and mop, busy 'cleaning the toilet'. NATALYA coughs**.\n\nNATALYA: The manager seems very friendly to me. Is he that way to you?\n\nTANYA _(After a beat.)_ : No. He's a swine.\n\nNATALYA: Is that because you don't know English very well? He tried teaching me English, I think.\n\nTANYA: He wants your bottom.\n\nNATALYA: My bottom? For hard work? I think he wants me to work very hard.\n\nTANYA: He wants sex.\n\nNATALYA: That's...not possible, is it?\n\nTANYA: That's up to you.\n\nNATALYA: No, I mean \u2013\n\nTANYA: It's not possible with me \u2013 I'm married. And he knows it. So I'm cleaning his toilets. Give him your arse \u2013 for sex \u2013 and maybe you'll get a bouquet. If you don't, I'll probably fight you for this job. _(Making an appropriate sound.)_ What a prospect.\n\nNATALYA: You see, I came alone...Yaroslavl.\n\nTANYA: I couldn't give a damn.\n\nNATALYA: My mother left us \u2013 my brother and me \u2013 for a whole summer when I was twelve. Daniel was nine. Our father left us for good just after I was born...my mother says. That's why she needs to go out and find work. And there was nothing in our flat for those three months but dried pasta and tea. And I haven't been able to eat pasta since.\n\nTANYA: If you're twelve, open up. Let me see your baby teeth.\n\nNATALYA: You haven't been listen \u2013\n\nTANYA: That's right. And if you're only twelve, then you're too young to be working here. Don't look to me to mother you.\n\nNATALYA: She came back...by the end of the summer!\n\nTANYA: There's no fluff on you, that's for damn certain.\n\nNATALYA: What, you think I should feel ashamed? What for? We're two women trying to have a conversation.\n\nTANYA: I am one woman trying to clean out the shit from this bowl...because my parents haven't had light for two years.\n\nNATALYA: Haven't had light? Why? Are they blind? I'm so \u2013\n\nTANYA: No, Natalya Only-Pasta-For-Three-Months. They've been without electricity.\n\nNATALYA: That's not poss \u2013\n\nTANYA: I don't give a damn.\n\nNATALYA: But where in Russia is anything that bad?\n\nTANYA: Outer Siberia. _(A beat.)_ They're in the Ukraine.\n\nNATALYA: My brother's become alcoholic.\n\nTANYA: My dog has fleas. My brother skins dogs for a living. Eats the flesh, sells the fur. The neighbours below on the balcony shout, 'Hey, Sos! The dogs are dripping fat...and they stink.'\n\nNATALYA: He doesn't still stay with your parents, does he?\n\nTANYA: God...She must be twelve. We've all had a hard time in Russia \u2013 that's why we left. The older the person, the harder the circumstance, in a way. There's been more to overcome. I don't want to know you as friend. I'm not lonely...I'm not especially kind.\n\nNATALYA: Someone said you're a nurse.\n\nTANYA: Someone said it. Not someone. My magpie husband. So you've met him. How thrilling for you. _(A beat.)_ You want some vodka...all right?\n\nNATALYA: No, I told you \u2013 my brother's an alcoholic. _(Suddenly she starts to stumble and cough.)_\n\nTANYA: I'm not so sure that you're not.\n\n**McBRODIE now separates from MAKSIM**.\n\nNATALYA: I sometimes...well, lately I've been having dizzy spells. But not from drink. Oh, no. We learned not to have it in the house. _(A beat.)_ I've always been worried it was my fault...the poor care I took of Dan when \u2013\n\nTANYA: Christ in Heaven, no more of your stories.\n\nMcBRODIE _(Calling back.)_ : I'll be ready with another log on the fire!\n\nNATALYA: He wouldn't really want me for \u2013\n\nTANYA: How on earth would I know?\n\nNATALYA _(Calling after him; heavily accented.)_ : Mr McBrodie!\n\nTANYA: Good God. She's gone to find out.\n\n**NATALYA'S gone after McBRODIE. MAKSIM moves to join TANYA, who 'finishes up'. Their scene together lapses into a local pub. ARABELLA, a local, hovers**.\n\nMAKSIM: Hallo.\n\n**TANYA greets this much as she greeted NATALYA'S overture \u2013 preoccupied with work**.\n\nTANYA: Have you gone and come back or not bothered to go?\n\nMAKSIM: He asked someone else, in the end.\n\nTANYA: You can't blame me for that!\n\nMAKSIM: I'm not going to.\n\nTANYA: You owe me a drink for it, in fact. I just about ordered you to go. So staying was your decision.\n\nMAKSIM: I said it was. Let's go without talking about it.\n\n**This is probably the shift to the pub**.\n\nTANYA: Another Russian's now working at the hotel.\n\nMAKSIM: I know \u2013 McBrodie told me.\n\nTANYA: But you don't know who it is.\n\nMAKSIM: Is she all right? _(He pours her a drink.)_\n\nTANYA: What do you mean? You think I met her? Exchanged Moscow addresses...swapped anecdotes?\n\nMAKSIM: There must have been something between you. McBrodie couldn't have told you, too.\n\nTANYA: And when did you meet? She said you'd been talking together.\n\nMAKSIM: She didn't tell you? Then it couldn't have been very good.\n\nTANYA: Let's stop this, all right? I'm not jealous, if that's what you mean. I think she's young, silly, she's nothing in common with us, and I'm going to steer clear. Please get me another drink.\n\nMAKSIM: You're drinking too quickly.\n\nTANYA: And she made me think too much about home.\n\n**MAKSIM rises to fetch more drink. ARABELLA seizes her chance. Unless otherwise noted, TANYA is assumed to be speaking Russian**.\n\nARABELLA: Men are such boors when you need them to be something else. _(Sitting.)_ They force you to take extreme and unladylike measures.\n\nTANYA _(Deadpan.)_ : My husband's coming right back.\n\nARABELLA: Is that Russian you're speaking? Dosvidaniya is about all I know.\n\nTANYA: That's just what I want to say to you \u2013 goodbye.\n\nARABELLA: Oh, right. That's how you pronounce it. _(Rising.)_ If I got us both White Russians, d'ye reckon we'd begin to forget our unfortunate histories? The unfortunate parts of our pasts?\n\nTANYA: She's actually taking off!\n\n**ARABELLA has gone only as far as the table to get her own drink**.\n\nARABELLA _(Sitting down.)_ : We'll make it the next round \u2013 I see we're both plentifully supplied. _(Looking at her.)_ You're an angel in miniature, aren't you? You're exquisite. You must have been told that before.\n\nTANYA _(Now in English.)_ : My husband...come back...quickly. Husband...Man.\n\nARABELLA: And then you're so soft to the touch. _(Begins running her fingers over TANYA's face.)_\n\nTANYA: Stop.\n\nARABELLA: You mean it's exciting you? It is me...I don't mind admitting. I wish I knew the Russian for 'turned on'. _(More emphatically.)_ The word for 'beautiful'... you?\n\n**MAKSIM returns**.\n\nMAKSIM: Excuse me, we're together.\n\nARABELLA _(Trying it on.)_ : I don't think so, McTavish \u2013 she and I are. Find yourself something to fiddle with.\n\nTANYA _(Beseechingly.)_ : What's she trying to do?\n\nMAKSIM: Why are you so surprised? You called it the English Disease.\n\nTANYA: Here in Scotland?\n\nARABELLA: Hey...hey. Would you mind not talking beyond me? It's indescribably rude.\n\nMAKSIM: We're married...all right? Now why don't you leave us alone?\n\nARABELLA: Does your brother think you're not entitled to any fun of your own? _(To MAKSIM.)_ I'll wager you've no idea how hard this woman works.\n\nMAKSIM: And you'd be precisely and utterly wrong.\n\nARABELLA: 'Precisely and utterly', is it? I say she's a poppet. _(To TANYA.)_ And you want to come home with me.\n\nTANYA: At least make her stop touching me.\n\nMAKSIM _(Standing up, taking ARABELLA'S hand away.)_ : She doesn't want you to touch her.\n\nARABELLA: I bet I could make you care. _(A beat.)_ Can't we even exchange mobile numbers? Come on.\n\nTANYA: I can't believe this just happened to me.\n\nMAKSIM: Yeah, whatever you do, don't smile.\n\n**He begins to lead TANYA away. ARABELLA tries to follow, calling after them**.\n\nARABELLA: My name really is Arabella! I love you! Hey...hey, Golden Balls. I bet she doesn't give it to you tonight!\n\n**Lighting shift**.\n\n**As they move back 'to the hotel', NATALYA is featured stumbling and coughing, out of their view. During their next exchange, she makes her way 'to the toilet'**.\n\nTANYA: I don't know what's wrong with this country. _(Leans on MAKSIM for support.)_ 'Great Britain'. Sexual maniacs. After you down there so we have to go here. And here they go after me.\n\nMAKSIM: It might not have happened if...\n\nTANYA: What 'might'? I can understand men...you know. There's Beznik. But her...? And those maniacs that were after you. Oliver...and 'Miss...' What was her name?\n\nMAKSIM: His...Are you all right?\n\nTANYA: Yes, I'm all right. I'm something like angry...Sad...for her.\n\nMAKSIM: Sad for who \u2013 Natalya?\n\nTANYA: 'Who?' That woman just now. Who's Natalya?\n\nMAKSIM: From Yaroslavl.\n\nTANYA: Oh. _(A beat.)_ She told you where she was from? You chatted that much? She's a magpie like you.\n\nMAKSIM: She lived with her mother and \u2013\n\nTANYA: Yeah, don't bother. She also told me.\n\n**NATALYA is heard throwing up**.\n\nThere she is now. My God. Whatever you say about me, I'm not that bad.\n\nMAKSIM: What, you think that's from drink?\n\nTANYA: She says she doesn't...\n\nMAKSIM: I think she must really be ill. _(Crosses to the toilet.)_ Are you all right...? Natalya?\n\nTANYA _(Following him.)_ : That's the one I just finished cleaning.\n\nMAKSIM _(Over her.)_ : Shut up. _(Back to NATALYA.)_ Can you speak? Can you open the door?\n\nNATALYA _(Struggling.)_ : Who's that?\n\nMAKSIM: Maksim Pavlov. We met this afternoon. _(A beat.)_ Do you know where you are?\n\nNATALYA _(A beat.)_ : Is...Tanya with you?\n\nMAKSIM _(Somewhat relieved.)_ : Yes, she's here. _(Guiding TANYA forward.)_ Talk to her.\n\nTANYA _(Hushed.)_ : And what do I say? _(To NATALYA.)_ Tasha? It's me. Can you come out? What can we do for you? What's wrong?\n\nMAKSIM: She'll need a doctor.\n\nTANYA: Shut up. _(To NATALYA.)_ Tasha, please let us help you. Open the door. _(To MAKSIM.)_ She said she suffers from dizziness.\n\n**NATALYA comes out of the darkness looking wretched. She needs help in order to stand, and her words are partially indistinct. She's also crying profusely. MAKSIM and TANYA both support her**.\n\nNATALYA:...can't....s-see...c-can't...s-spea. _(Recovering somewhat.)_ Tan-tan-yush? I'm sorry. I've been sick.\n\nTANYA: What's wrong with you?\n\nMAKSIM: I'm going to phone the Emergencies. _(As he leaves them.)_ Can you manage?\n\nTANYA: Don't leave us. Let's get her to bed first. _(To NATALYA.)_ All right?\n\n**The trio cross the stage**.\n\nNATALYA: You're both...very kind.\n\nTANYA _(Past NATALYA, in hushed voice.)_ : We're not, you know. I was horrible to her. _(To NATALYA.)_ I didn't know you were ill.\n\nNATALYA: I want to sleep...very...\n\nMAKSIM: I'm phoning an ambulance.\n\nTANYA _(To NATALYA.)_ : No, you shouldn't sleep yet. Maksim is getting you a doctor. Can you stay awake? It's free in this country. Are you still awake? Did we tell you my family's from Zhitomir?\n\n**Lighting and scene shift as MAKSIM begins to ring through. Funereal atmosphere as McBRODIE and TANYA stand apart, each delivering a 'eulogy'**.\n\nMcBRODIE: Nabody toold me, ya see? Here am I expecting to see her at work...she's not there. And next week she's away. I'm beginning to be a wee bit angry, thinking she's done a runner. She did, too, it turns out. The ultimate runner, aye.\n\nTANYA: And what I tell her? 'We've all had it harder than you'. Shows what I know. We're all still alive. Twelve years old \u2013 she was trying to tell me. She's like those kids in Beslan. I hope she's meeting them now. I'm sure she likes it.\n\n**POLBOI, in black leather, approaches MAKSIM**.\n\nPOLBOI: Who's it for?\n\nMAKSIM: Russian girl. Natalya. _(Realizes he's speaking to another Russian.)_ Hallo?\n\nPOLBOI _(Shaking hands.)_ : Polboi...Viktor...Vitia.\n\nMAKSIM: Pavlov. _(A beat.)_ Are you family?\n\nPOLBOI: Let's go for a drink.\n\nMAKSIM: I'm meant to be at work. _(Smiles.)_ This isn't Russia.\n\nPOLBOI: I'd like to know what happened.\n\nMAKSIM: Well...Damn it, yes, why not? You've come all this distance.\n\nPOLBOI:...and she was alone. Was she?\n\nMAKSIM _(Looking over at TANYA.)_ : I'll just get my wife.\n\nPOLBOI: Can't we talk about this by ourselves?\n\nMAKSIM: Yes, of course. Only Tanya was with her as well.\n\nPOLBOI: I'd feel easier.\n\nMAKSIM _(Trying to catch TANYA's eye.)_ : Tanya?\n\nPOLBOI _(Booming voice.)_ : Tanya!\n\n**TANYA sees them**.\n\nTANYA: What's happening? _(Starts across.)_\n\nMAKSIM _(Calling.)_ : No, no \u2013 you stay there. I'm just going...\n\n**POLBOI has MAKSIM in a bear hug**.\n\nMAKSIM: A relative of Tasha's!\n\nTANYA: I don't think so! Her brother is younger...And she didn't have any other males! Leave him alone.\n\nMAKSIM: I'll be all right!\n\nPOLBOI: Worries about you, your missus? That's great.\n\nMAKSIM _(Hurting from his grasp.)_ : Where do you want to go?\n\nPOLBOI: Are we in private? This is good enough. _(He relaxes his grip and takes out a bottle of vodka.)_ You got glasses?\n\nMAKSIM: Once again, it's not like in Russia. Here they have pubs. I'm not sure it isn't actually illegal to drink in the street.\n\nPOLBOI: But you said we were private, yeah?\n\nMAKSIM: You're not her brother, are you?\n\nPOLBOI: Whose? Oh, that girl's?\n\nMAKSIM: You didn't actually know her, did you? Who are you really?\n\nPOLBOI: Don't piss yourself. I knew her. Natalya, right? How'd she die again?\n\nMAKSIM _(A beat.)_ : She had a brain tumour.\n\nPOLBOI: Shit.\n\nMAKSIM: And she lay in a coma for three days. Tanya and I rushed her in...operated on the next day. Woke up once after surgery...and then \u2013\n\nPOLBOI: That's all right, don't tell me any more. _(A beat.)_ It's a bitch, isn't it. _(Drinks.)_\n\nMAKSIM: What makes it worse is that we didn't really know her ourselves, except for about three days. She'd just started at the hotel where we work...\n\nPOLBOI: And where is that again?\n\nMAKSIM: In this city. _(A beat.)_ 'Again'?\n\nPOLBOI: Drink up...come on. _(Proffers the bottle.)_\n\nMAKSIM: I don't want any more.\n\nPOLBOI: 'Any more'? You haven't had anything yet. Now come on. We got to drink to Natalya. Don't we?\n\nMAKSIM: I've got to get back.\n\nPOLBOI: Not really your line of work, is it? _(Gripping his shoulder.)_ What are you, a waiter? Boot black? _(Short laugh.)_\n\nMAKSIM: I asked you \u2013 who the hell are you?\n\nPOLBOI: Just a lawyer...like you.\n\n**Somewhere round this point ALENKA approaches TANYA. The two duologues now overlap**.\n\nALENKA: Hallo, Tanya.\n\nTANYA _(Eyes fixed on MAKSIM.)_ : Don't disturb me... _(Realizes who she is talking to_. _Stunned.)_ Alenka! _(Moves a step closer.)_ Am I dreaming?\n\nALENKA _(Moving next to her.)_ : No, it's me. _(Kisses her.)_ See? Care to touch me? _(Trying to guide her hand.)_\n\nTANYA _(Withdrawing her hand.)_ : No, I believe you...now. How are you? Are you well? Are you safe?\n\nALENKA: No, no, don't worry, I'm all right.\n\nTANYA: I'll go get Maksim; he's \u2013\n\nALENKA: Yeah, I know. Leave him for a bit.\n\nTANYA: So you know who he's talking to? What's it about?\n\nALENKA: So. How are the two of you \u2013 are you well and safe?\n\nTANYA: Well, yeah. We've just been attending a funeral, which isn't very \u2013 Russian girl we worked with. Suddenly...you know.\n\nALENKA _(Distracted.)_ : A tragedy then.\n\nTANYA: Yah. _(A beat.)_ So what are you doing here? Are you staying? What's happened?\n\nALENKA: You mean at home in Kaliningrad? Or Moscow?\n\nTANYA: Just...anything. Don't make me fish.\n\n**Overlapping**.\n\nPOLBOI: The fact is I'd like to offer you another job. Better, cleaner. Much better paid.\n\nMAKSIM: I can't go back to Russia.\n\nPOLBOI: No, I know that. I'm talking about here.\n\nMAKSIM: In Scotland?\n\nPOLBOI: Back in England. Outside London. Surrey \u2013 the gold brick belt.\n\nMAKSIM: We're afraid to go back \u2013\n\nPOLBOI: Yeah, I know all about that. It's all right. We give you insurance.\n\nMAKSIM: What kind of \u2013 ?\n\nPOLBOI _(Laughing.)_ : Diplomatic immunity!\n\nMAKSIM: If you're a lawyer, can't you do the job?\n\nPOLBOI: Uh-uh. The client wants someone with your expertise. In fact, he specifically asked for you. _(Laughs.)_ What an accolade, eh? Nobody ever asked for me personally. It'd be a moment in my career, if they did. So I guess this is your moment.\n\nMAKSIM: One of them.\n\nPOLBOI: One of them, eh? Bastard. _(Grins.)_\n\nMAKSIM _(A beat.)_ : How did you know where to find me?\n\nPOLBOI: You're in hiding, eh? It wasn't too difficult. _(Pours a drink.)_ We'll toast your success, all right? Then I'll tell you. Okay? _(Raises his glass.)_ Success?\n\nMAKSIM: Thanks. _(Drinks.)_\n\nPOLBOI: So. _(Pointing.)_ You see those two ladies over there? Recognize them?\n\nMAKSIM: Well, my wife and...\n\nPOLBOI:...and...?\n\nMAKSIM:...my sister.\n\n**Overlapping**.\n\nALENKA: I have to tell you this, late as it is. I'm sorry for the way I treated you, Tanya. You've been a good wife to Maksim, I understand and respect that. And it isn't your fault you're in the spot you're both in.\n\nTANYA: What 'spot', exactly?\n\nALENKA: Well, you know...hiding out...here. If Maksim hadn't done what he did, you'd both be working back home...in your chosen professions. And I'm sure you advised him against it, didn't you. As I did myself.\n\nTANYA: We're supporting my parents in Zhitomir.\n\nALENKA: Oh, I know. The poverty there is appalling. And the meagre amount you earn here.\n\nTANYA: It's enough. They're very grateful.\n\nALENKA: All right, but it's nothing like what you could send if you still lived in Kaliningrad. Where we'd all live together. Come on. And it's all Maksim's fault \u2013 that we don't, that you have to live doing terrible jobs.\n\nTANYA: Why are you saying this? Now? I asked you \u2013 what's going on?\n\nALENKA: All right. You want to know what they're talking about? We'll ask them.\n\n**The quartet come together, up centre**.\n\nPOLBOI: This is some kind of reunion, I guess.\n\nMAKSIM: And I guess you don't want to leave us.\n\nPOLBOI: Oh, it's not allowed. I'm obliged to stay with you at all times.\n\nMAKSIM: It's worse than being in prison then. Thanks, Alenka.\n\nALENKA: What are you talking about? Nikolka is in prison...\n\nMAKSIM:...thanks to me.\n\nALENKA: All right then, yes, thanks to you.\n\nMAKSIM: That's what I mean. _(Mockingly.)_ Poor Nikolka.\n\nALENKA: We gave you a great job. What'd you have to go sticking your nose into things that were none of your business.\n\nMAKSIM: Because that's what my business was beginning to be, and I didn't like it.\n\nALENKA: Shit \u2013 why do you have to be such a crusader? And married to the Red Cross. You think it does anything else but make trouble?\n\nMAKSIM: His trouble was trouble for me. Didn't he tell you? Okay, make your mind up who to believe. My side, he was all but ordering me to make a hit on the rival ship owner. Who might well have had me marked anyway. I'd have been shot if I'd gone through with it. I always thought he knew that...and wanted to get rid of me.\n\nALENKA: Rid of you? You were his brother-in-law...knew all his business. You were even responsible for much of it.\n\nMAKSIM: And he sent you here...to remind me.\n\nALENKA: No.\n\nMAKSIM: I don't believe you...Well, do you believe me? You've made your choice, Lyen. It's obvious.\n\nALENKA: No, it's not. Even if you don't give a damn about me any more, there's your own circumstances to think of. You care about those or you don't really care about anything. That's why you'll do what he wants. _(Referring to POLBOI.)_\n\nPOLBOI: It's not a difficult job we're asking. Deliver and explain a set of contracts, that's all. What else do lawyers do, after all?\n\nMAKSIM: It's the penalty for not doing it I'm thinking about.\n\nPOLBOI: Oh, sure.\n\nMAKSIM: There's security, being a boot black...or washing toilets, what she does. You sleep soundly at night, even when it's freezing cold. _(To TANYA.)_ Tell them...\n\nTANYA: Tell them...what?\n\nMAKSIM: How happy you are. The pubs stay open.\n\nTANYA: Right, there are no closing hours.\n\nMAKSIM: And the toilets are...\n\nTANYA:...even the British toilets are clean.\n\nMAKSIM: A tulip garden.\n\nTANYA: Well, I wouldn't go that far.\n\nMAKSIM: People use them for making love.\n\nTANYA _(giggling)_ : Oh, yes, that's true. Apparently. I've never seen it myself. Well, I mean, why should I? Maksim and I have our own room. All right, it's not the same as we had in Kaliningrad. But it's bigger than most rooms in Moscow, and we're well fed.\n\nALENKA: And these people know your future is limited...that you broke your conditions of entry...once you're found, you'll be sent back. Not to a country of your choosing, but to Russia. They know that. One phone call is all it takes.\n\nMAKSIM: They know it now anyway. As I say, thanks.\n\nPOLBOI: Why have you all got long faces? 'One phone call.' One errand is all we want. It's the same as a long weekend break to the Lake District. That's where it is, in fact. Yes. Believe me. Surrey...London...Lake District. The only addresses I know. And after that, your problems are over.\n\nALENKA: You've got to do more than think about it. But that's my advice, and you've always only gone your own way.\n\n**Light fades on POLBOI and ALENKA and leaves MAKSIM and TANYA alone. Two more shadowy figures begin to appear. But for the moment MAKSIM is busy packing and explaining to TANYA why**.\n\nMAKSIM: Take the job and life would be over. You've got to help me decide.\n\nTANYA: What? Your sister said you always did only what you wanted.\n\nMAKSIM: And you take her side? She put us here. What does she know? Do you want to go over our past mistakes? Your family needed money.\n\nTANYA: I know.\n\nMAKSIM: What could I make as a Law lecturer? Less than you as a nurse.\n\nTANYA: I know all that. So what? She said you never took my advice; I didn't. I know what you know...that I never gave it. Maksim? What you've done has always been right.\n\n**A pause**.\n\nMAKSIM: Well, I'm asking for your advice now. _(She nestles close to him. He stops packing.)_ We have to go to Canada now, I think, that's certain. The question is how do we do it? Do we assume to take what Polboi asks...maybe even do the job...I don't know. As soon as we seem to have done it, then we get on the first available flight. That's the first \u2013 and the logical \u2013 option, I think. The other is simply to lean on McBrodie \u2013 ask his contacts to shield us out of here, within the week. It's possible, I think. We'd leave from some Highland airstrip. They wouldn't know where it was. He's even got contacts once we land...if we agree to continue to work in hotels...in Nova Scotia, for instance. That wouldn't be bad to begin with, and they probably don't have as many perverts anyway.\n\nTANYA: But...can I tell you what I think?\n\nMAKSIM: That's why I asked.\n\nTANYA: What if they have this place watched? That man and his friends.\n\nMAKSIM: That's what makes me want to go straight for the first option. Why not begin to do what they want? I know things...that they want to know.\n\n**D.I. MARTINS comes out of the shadows, calling out MAKSIM'S name as he approaches**.\n\nMARTINS: Maksim Pavlov?\n\nMAKSIM: Yes?\n\nMARTINS: Your current employer told me where to find you and...is this your...wife?\n\nMAKSIM _(An echo of the first scene.)_ : We're married, yes.\n\nMARTINS: And is it correct that you left Winters Asylum Centre to work at the Crestview Hotel outside Chagford?\n\nMAKSIM: You're from the Home Office.\n\nMARTINS: Devon and Cornwall Police. _(Flashes his badge.)_\n\nTANYA _(Under her breath.)_ : My God...\n\nMARTINS: I'd like if I may to ask you some questions concerning events that took place there on the evening of 5 December. You were present?\n\nMAKSIM: We both were.\n\nMARTINS: And you both are or were applying for asylum...Application which has subsequently been refused. _(Consulting notebook. Continuing to look down as he speaks.)_ We're not concerned with that as such but as I say with what was alleged to have taken place on the evening or the night-time in question.\n\nMAKSIM: You're...not here about our asylum application?\n\nMARTINS: Why? You've violated conditions of entry, is that right? _(Looks at them now.)_ You've failed to report to the police ever since you left Winters Asylum Centre.\n\nMAKSIM: We're continuing with our appeal. Our solicitor is \u2013\n\nMARTINS: I'm investigating another matter.\n\nTANYA _(In the silence.)_ : Maksim? What's going on?\n\nMAKSIM: Quiet. _(To MARTINS.)_ What matter is \u2013 ?\n\nMARTINS: Are you acquainted with Franklyn Gerontius, nephew of the owner of the Crest \u2013?\n\nMAKSIM: Yes, we both are...or were.\n\nMARTINS: You have a clear picture of who I mean.\n\nMAKSIM: Absolutely.\n\nMARTINS: Do you both speak English, by the way? I understand you're Russian?\n\nMAKSIM: I'm a solicitor. Yes, I do. Tanya's English is a little \u2013\n\nTANYA _(To MARTINS.)_ : Little...yes...Better.\n\nMARTINS: But you both remember seeing him that night.\n\nMAKSIM: I think so, yes. That was the evening he set off the fire alarm.\n\nMARTINS: You're sure it was him?\n\nMAKSIM: He boasted he'd done it.\n\nMARTINS: And what sort of man was he? Did you have much contact?\n\nMAKSIM: Oh, yes. Not that very evening, but \u2013\n\nMARTINS: So you were aware of his habits.\n\nMAKSIM: No, only when he was at the hotel.\n\nTANYA: Awful man, Franklyn. Awful...dirty...maniac. I hate him.\n\nMARTINS _(After a moment.)_ : He was found with his throat cut at 2.34 a.m. on 5 December last.\n\nMAKSIM: So you're saying we're under suspicion.\n\nMARTINS: Suspicion of what? I'm saying nothing for the moment. I'm trying to form a picture of who he was.\n\nMAKSIM: Didn't he have a police record?\n\nMARTINS: How do you know that? Did he tell you himself?\n\nMAKSIM: I can't remember. Yes, I'm sure he did. He boasted...as I say. He would boast about many things.\n\nMARTINS: Go on.\n\nMAKSIM: Well, that was his character. He didn't work at the hotel, as you probably know. His uncle, Oliver, kept a room for him.\n\nMARTINS: And how did you happen to know him?\n\nMAKSIM: Well...he'd bother the hotel staff. It was his character, his sport.\n\nTANYA: Tell him how he'd speak to us.\n\nMARTINS: And what about the others?\n\nMAKSIM: What?\n\nMARTINS: Was his behaviour with them the same?\n\nMAKSIM: I don't know what you mean.\n\nTANYA: I am sure he wanted rape me! I keel him...you know?\n\nMAKSIM: Tanya!\n\nTANYA: Why not? It's the truth.\n\nMAKSIM: Because he's saying he was murdered. Now he'll think that we had something to do with it.\n\n**A pause**.\n\n_(To MARTINS.)_ When Tanya said 'kill', I'm sure she didn't mean...I'm sure you must have heard that before. It means nothing, doesn't it? And when she said rape.\n\nMARTINS: We've already got someone in custody. We're just trying to build up a case.\n\nMAKSIM: Oh, well, of course. _(A beat.)_ You couldn't say who it is?\n\nMARTINS: Of course not.\n\nMAKSIM: Of course not...sorry.\n\nMARTINS: But in fact you saw nothing of Gerontius that evening or night?\n\nMAKSIM: Heard only...shouting and swearing. Yes, that's right.\n\nMARTINS: All right. Thank you very much. _(A beat.)_ Would you be available for further questioning in the event?\n\nMAKSIM: In the event of what?\n\nMARTINS: You could be called as material witnesses. That's all.\n\nMAKSIM: When this case goes to trial?\n\nMARTINS _(Starting to go.)_ : Yes, that's right. _(Turns back.)_ I think I can tell you this much. From everything we've been able to piece together, it was a murder made to happen. In other words, he's not likely to be missed. Several people, at least, openly echoed your wife's sentiments. Goodnight.\n\n**A pause**.\n\nTANYA: Maksim?\n\nMAKSIM: Just a minute.\n\nTANYA: Did I understand? We're not going to be able to leave?\n\nMAKSIM: And all this time...we've been worried about being deported...and the police have known about us...all this time. And, in the end, when they finally catch up with us, they're worried about someone else? An English maniac? Is that how you see it, too?\n\nTANYA: I'm not sure. I didn't really understand what he was saying.\n\nMAKSIM: But about Franklyn \u2013 you understood that.\n\nTANYA: Well, he's dead. Yes. Good.\n\nMAKSIM: That's right! You even made what they call 'incriminating statements', and he couldn't have cared less. You were expressing the general opinion!\n\nTANYA: Well, all right...\n\nMAKSIM: No, it's better than that, don't you see? We could stay here forever... without the asylum. And they wouldn't care!\n\nTANYA: Are we going to do that?\n\nMAKSIM: Well, the pressure is off us at least. _(A beat.)_ Why were you so sure that our cover had been blown?\n\nTANYA: That night? I saw Beznik with Brangwen.\n\nMAKSIM: What do you mean?\n\nTANYA: They were staying there. She said they were married.\n\nMAKSIM: You're crazy.\n\nTANYA: That's what I mean. They'd tracked us down to the hotel, and were waiting to inform on us.\n\nMAKSIM: But even if they did, it didn't matter.\n\nTANYA: Didn't it?\n\nMAKSIM: Of course not. What did he say?\n\nTANYA: You speak English.\n\nMAKSIM: They've known about us from the beginning. And yet...we're still here. _(A beat.)_ You want to go for a drink...to celebrate?\n\nTANYA: I don't want to see that crazy girl.\n\nMAKSIM: We'll go somewhere else. Or I'll be with you the whole time. Nobody's going to bother you.\n\n**They start out**.\n\nTANYA: I keep thinking it would have been nice to take Tasha with us....to have her as a friend.\n\nMAKSIM: You know, I completely forgot about her? It's a shame.\n\nTANYA: It's worse than that. For her, of course. But also for us. We're alone here except for each other. Her friendship would have been \u2013\n\nMAKSIM: That's why I still think Canada's our future.\n\nTANYA: You mean it's possible?\n\nMAKSIM: I think it's essential. You said it exactly. We're alone here except for maniacs, hotel managers...\n\nTANYA:...dirty toilets.\n\nMAKSIM: It's better than Russia.\n\nTANYA: It's a toilet. What's better about that?\n\nMAKSIM: Canada!\n\n**SCOOBY steps out of the shadows**.\n\nSCOOBY: Hey, Pavlov!\n\nMAKSIM: What? Are you one of Polboi's goons?\n\nSCOOBY: My name's Scooby.\n\nMAKSIM: If you're also a friend of my sister...\n\nSCOOBY: What?\n\nMAKSIM: Tell her she can go to hell.\n\n**SCOOBY draws a pistol and shoots MAKSIM. He falls. It's too sudden and quick for TANYA to take it in immediately**.\n\nSCOOBY: Yeah, I'll tell her, all right. _(To TANYA.)_ What's her name?\n\n**TANYA lets out a primal scream**.\n\n**Instant fade**.\n\n## PART TWO: ESSAYS\n### GETTING TO KNOW JAMES MACDONALD\n\n_Peter Thomson_\n\nIt is a fact, recorded in one of my notebooks, that I first read a play by James MacDonald in February 1977. It was called _Assumptions About Evelyn Ackroyd_ , and written for radio. It had signs, my notebook tells me, of being 'the work of an efficient and experienced writer'. Evelyn Ackroyd is a thirty-three-year-old spinster, but what else is she? That depends on whom you trust. If there is such a thing as an essential self, the young MacDonald seems to be asking, how does it square with other people's views of that self? Who has the better claim to be 'right', Evelyn Ackroyd or the world that summarises her? The discomforting possibility that personal identity is nothing other than a compound of personal appearance and personal circumstance is rarely absent from any of the thirty or so MacDonald plays I've read or seen since 1977: the 'self' as contingent. In November 1977 I read _Zapped_ , a highly charged play in which reasons for caring for the disabled are brought into question. The ultra-realistic presentation of the dialogue \u2013 clipped and approximate, as in life, and with constant overlapping of speeches \u2013 was a feature. MacDonald's achievement was to create complex characters \u2013 two young women, both with lesbian yearnings perhaps, and a disabled man. This play would resurface, much changed, in 1985 as _Balance Is Stillness_ , and then be again rewritten for publication in _Peering Behind the Curtain_ (2002), a volume on the 'extraordinary body', edited by Thomas Fahy and Kimball King. Together with _Crippled Ruth_ (1993, published 1996), it represents MacDonald's published 'disability' plays. ( _Bread and Circus Freaks_ , which appears here, belongs more properly to his 'Russian' plays.)\n\nIt's clear from my notebooks that I was wondering what made the author of these two very different plays ( _Assumptions_ and _Zapped_ ) tick, and _His Daughters Bury Her_ , which I read in December 1978, kept me guessing. There was a _tour de force_ of a row between the three daughters in Act Two, and the same scrupulous concern with characterisation. I was evidently reminded, at the time, of Arthur Miller's _The Price_. It is certainly a 'family' drama, and one which shares Miller's preoccupation with the Ibsenite past. If I was wondering what made MacDonald tick, he was evidently wondering the same about his characters. That was about as much as I could say about him. We hadn't, after all, met (so why was I reading his plays?), but we soon would; and the circumstance of our meeting is interesting enough to be worth recording.\n\nSince the year 1975, I had been the first 'reader' of plays submitted through Clive Wolfe to the International Student Playscript Competition, and 1978 was a bumper year. Not only were there sixty-eight entries, but also several of high quality. I despatched, as always, the short-listed texts to Clive, for subsequent semi-final readings (perhaps already by Stephen Jeffreys) preliminary to the final judgement of Alan Ayckbourn, and then got on with my life. But the mills of God grind on, and when the miller is as industrious in the cause of student drama as Clive Wolfe the product is unpredictable. He broached me ('broached' is just the right verb to describe Clive's approach in such matters) with the idea that we might mount rehearsed readings of the short-listed plays at the Northcott Theatre in Exeter. And so \u2013 at some time in 1979 \u2013 we did. There was another phone-call between the broaching and the event. The relevant bit went something like this:\n\nCLIVE: Have you met James MacDonald?\n\nME: No.\n\nCLIVE: I'll be interested to know what you make of him.\n\nME: Why?\n\nCLIVE: Well... put it this way. You may be surprised.\n\nAnd I suppose I was, though I've known him for so long that it's difficult to recall the first time.\n\n_His Daughters Bury Her_ (I've often been unconvinced by James's titles) was one of the plays given a rehearsed reading. I may have forgotten some of the others. Barry Sutcliffe's _Gerontius_ was an obliquely quizzical view of Elgar and his 'public'. Sutcliffe went on to edit a volume of plays by George Colman the Younger and Thomas Morton before setting himself up as a publisher. Now contemplating retirement, he 'blogs' in a website called 'Complete Pants'. Paul Unwin's _Pavilion_ was a witty one-act piece in which two very English 'chaps' exposed their prejudices while watching their cricket team being thrashed by a black touring side. Unwin went on to be joint-creator of _Casualty_ for the BBC, but would probably rather be remembered for his artistic directorship of the Bristol Old Vic and his superb short film, _Syrup_ (1994). And then there was the extraordinary Elizabeth Gowans. She had submitted five plays, all of them worth staging ( _Casino_ was produced at the Soho Poly in 1992). _The Twain_ , which was the one we read, was set in a West Australian Ore Processing Plant, and showed just what a gifted writer could do by interleaving sex and Australian ideals of mateship. Gowans is, I believe, a major writer lost. I 'googled' her, and found only that she had scripted a television mini-series called _Heart of the High Country_ (1985). I can't remember whether it was before or after that that she wrote to me from the Raffles Hotel in Singapore, saying that she wasn't planning to write any more. And I met James MacDonald.\n\nTo an unpredictable extent, we are what we look like. For all the difference it makes, the abstemious James MacDonald, who is refused service in a pub because he looks drunk to the landlord, is drunk. Very probably, the fact that he walks unsteadily was the first thing I noticed. But it wasn't only appearances that Clive Wolfe was alluding to. When you are confronted with a person whose real life is in his or her writing, the encounter can be disconcerting. It didn't take me long to realize that James (the transition from MacDonald to James begins here) is not a congenitally disabled writer. He is a writer who is congenitally disabled, and that's a different thing. Disability may have knowingly coloured some of his writing, but writing uncoloured by personal experience is as rare as it's boring. We are determined by gender and parentage, by appearance and circumstance (I recently found out who Paris Hilton is), and by what we have so far made of ourselves. By the time I met him, James was decisively a playwright. Peggy Ramsay had noticed that, and signed him up as part of her distinguished list. But a playwright needs confirmation through performance, and Ramsay was not selling James's plays to theatre managements. He registered for a PhD and moved from London (where he'd been seriously mugged) to Exeter (where his only mugging was comparatively frivolous). The move involved his abandoning a thesis on Henry James (Henry James!) and starting one on the little-known Northern Irish playwright, George Shiels (1886\u20131949). It so happens that Shiels spent most of his adult life in a wheelchair (as the result of an accident, not congenital incapacity), but even if James were to tell me that that was why he was drawn to him, I would have my doubts. Shiels is a mordant comic realist, with an uncultivated eye for the grotesque, and the writing of the thesis coincided with a shift in James's own work, away from realism towards the grotesque. (In a recent letter to me, he has referred to Rabelais and Gogol as 'the patron saints of comic deformity', and he has opened himself to their influence.) This was already evident in the next play I read (in January 1980), _This England_ , subsequently rewritten as _Too Many Monkeys_ and performed at the Northcott Theatre in November 1983. The family is still a central issue here, but the image is distorted: an unloved and unlovely mother with an epileptic daughter and a disabled son (an expert on the planes of World War Two, who needs his nappy changed). There is a particularly disturbing episode in which the daughter's boyfriend mocks the boy while flirting with his mother: Rabelaisian, James would say, viewing disability from the other side.\n\nFrom around this time, I date a restlessness in James's writing. _A Nuclear Family_ (read in January 1981) is a play of dialogue wars that reach no resolution. How is a playwright to keep on writing without the promise of performance? _Too Many Monkeys_ became his favourite, but was that because it was at least staged? The decision to book the Bloomsbury Theatre in London for a short run of _Caliper_ in the late summer of 1984 was a bid to bring James to public notice. It didn't work, for any of a number of reasons. There was a student cast (a competent one), and an abrasive central performance from John Hilton (next heard of as a milkman in Lancashire). But this was \u2013 like many of James's best plays \u2013 a comedy without a distinctive 'feel-good factor'. Peggy Ramsay attended, with a thumbs-down face. Her calculation, I think, was that the punters had no fondness for pieces that knew (and showed) that disabled people can be just as nasty as the rest of us. After that \u2013 or so it seemed to me \u2013 she abandoned any attempt to sell MacDonald to the professional theatre. Not that he was likely to stop writing (as I've said, that's what he is \u2013 a writer). _Self-Service_ (read in June 1986) is a complex 'comedy' about graft and corruption among architects and the government in a not-far-distant Britain. _Heavy Petting_ (read in February 1987) is a radio play which, like _Caliper_ , recognizes, and to some extent relishes, the selfish sexuality of the disabled while at the same time exposing the harmful naivety of the able-bodied (fancy someone in a wheelchair wanting sex!). _Opportunity Knocks_ (read in May 1989) is a beautifully crafted one-act piece which focuses on a rent-boy trading in a motorway caf\u00e9. All of these are insistently plays for performance \u2013 unperformed. Little wonder that James had accepted a commission to write a small-scale community play for the district of Wonford (Exeter) in 1988. _Women and Children First_ deals with Exeter during World War Two, including the impact of Hitler's 1942 Baedeker blitz of the city. Bread and butter work for a frustrated playwright.\n\nThrough the 1980s and on into the 1990s, long conversations with James were a constant stimulus to my interest in plays and playwrights. Like all real writers, he lives in the past of his art as well as in the present. It was in December 1990 that I read _The History of Our Dreams_ , on the face of it a huge departure from his previous work. The subject here is Stalin, and the treatment owes something to Brechtian epic \u2013 one scene after another rather than one scene because of another. In retrospect, I can see the significance of Brecht's probing question \u2013 'who notices what he takes for granted?' \u2013 in all the early plays which I have referred to so far. Physical disability had been the vantage-point from which James's mental agility allowed him insights into the unnoticed, but he was uneasy about being labelled as the dramatic spokesman for a misunderstood underclass. (I remember his reluctance to have _Crippled Ruth_ staged in the inaugural Platform Four season of 1993, and his response to the success of this exquisite one-act piece included an element of squirming.) The Stalin play, I now believe, marks a point of disjunction in James MacDonald's _oeuvre_. Increasingly since then, he has sited his sense of otherness (he would, I think, prefer difference) in Russia \u2013 particularly in the largely unnoticed (by the western democracies) post-Cold-War Russia. This volume contains three of James's most recent 'Russian' plays. Two earlier ones, _Left-Handed Enterprise_ and _Kleptocrats_ , were published, under the cover title _As Russia Goes_ , in 2000. The first of them had received a staged reading (by the Soho Theatre Company) at the Cockpit in London in 1995. In sum (and the sum is incomplete) these Russian plays constitute a quite extraordinary body of work. I know of nothing like it. Hardy's Wessex? Trollope's Barsetshire? But these are novels \u2013 and written from inside their country of origin. James's Russian project is fired, in part, by a wish to impart urgent (before it is too late) information. He has relocated his imagination.\n\nIt is possible that the process of relocation began in 1985\u20136. I had the job of selecting a play for performance by a group of students, six women and six men. There had to be serious acting jobs for all of them, and it was no easy thing to find a text that provided them. Scouting around, I looked at Gorky's last play, _Vassa Zheleznova_. 'Some meaty parts for women \u2013 very little for men', I noted in November 1985: 'Could James write a scene for e.g. Polya and Krotkikh: Melnikov and his son: Krotkikh and the Captain: Polya and Yevgeny\/Melnikov: the Captain and Ludmila? Or a whole new Act plus an extension of the Captain?' Over the Christmas vacation he did something like that, not according to my prescriptions but growing from them, and we presented Gorky (with MacDonald) in the spring of 1986. It was a much stronger play than Gorky's, and the cast was well above student average (Naomi Cooke, who played Vassa, and Kate Hale, who played her alarmingly powerful secretary, went on to found Foursight Theatre). But the point of significance here is that James had found clues in the text that were invisible to me (and, I suspect, to Gorky, whom he flattered with Gogolian eccentricities). My suspicion is that, between _Vassa Zheleznova_ and _The History of Our Dreams_ , James's reading took on a decidedly Russian tinge. I remember \u2013 but I don't remember when \u2013 a conversation about Sukhovo-Kobylin's bizarre trilogy, and I think we both read Erdman's _The Suicide_ at much the same time. Certainly he was pressing me into reading Soviet texts which did more for him than they did for me. It was several years later that he made a punishing trip to St. Petersburg, and a little after that that he married a Russian asylum seeker (Inna is now both a source and an addressee of his plays: she has become, in a sense, his muse). His dramatic project has merged with his life.\n\nTeaching in the Exeter Drama Department remains part of that life, too. _The Sweetheart Zone_ and _Emigr\u00e9s_ are intriguing products of that. They link back to projects he and I shared: not only _Vassa Zheleznova_ but also a dramatisation of _A Tale of Two Cities_ from the perspective of the revolutionaries (rather than that of Dickens's middle- and upper-class victims) and the writing of up-to-the-minute Living Newspapers for a pre-determined number (and gender) of actors. These are plays that would not have been written if they were not about to be performed. It's a very obvious point that this harks back to the origins of drama, as well as echoing the practice of contemporary 'fringe' companies. Aeschylus didn't write the _Oresteia_ speculatively \u2013 and if Shakespeare and Moli\u00e8re had had to have their plays vetted by a selection committee, most of them would have been lost to history. James's early plays, the ones I have briefly described above, are probably, in this sense, 'lost'. The 'wrighting' (in brief contrast to 'writing') of a text-for-performance is a distinct craft, more humble, perhaps, than the art of the dramatist, but not easier. It requires an intuitive relationship with the actor for whom the role is designed. Would Shakespeare have been able to create Hamlet had there been no Burbage? Fighting my way, at a reading, through the individual densities of _The Sweetheart Zone_ , I found myself yearning for the clarity of a performance (which, alas, I did not see). It is a text which vibrates with creatable idiosyncrasies, but I find them hard to create for myself in the loneliness of an armchair. _Emigr\u00e9s_ (which I did not see either) is more readily permeable \u2013 Russia relocated in a Britain which doesn't know how to assimilate it. But my first encounter with both of these texts served as a Brechtian reminder to take nothing for granted. Am I still getting to know James MacDonald?\n\nI end with another kind of reminiscence. It's March 2002, and I'm in search of the Finborough Theatre in London. It's a city that keeps springing new bits on you, and the new bits blur in with the old. I think I'm right that the theatre is above a pub that stands at the unlikely pinnacle of a triangle of roads. I know that the spectators were seated on two sides of a sparsely furnished room, and that _Bread and Circus Freaks_ shared the booking with Christopher Dunkley's _Mirita_. What I seem to remember is that the audience response was respectful, but not fully engaged. Respect is the kind of homage that polite British people pay to disability, and there was Leah Fells (an ex-student James had worked with) embodying disability. I sit in the audience wondering whether we're being really respectful or really patronising (am I remembering the unanticipated surge of fondness I felt for James's wayward first partner when he told me that she'd declined to have his baby because he'd drop it?), and anyway the play isn't about disability \u2013 it's about Russia. And it's funny! Not particularly 'feel-good', but definitely funny. So why don't we laugh? There's an ethos of uncertainty about just what the play is doing. (The same thing happened as long ago as _Caliper_.) Whoever this 'MacDonald' is, he doesn't write according to a discernible emotional formula. I leave feeling vaguely dissatisfied, not with the actors' performance so much as with my own. Only later do I learn that the last-night audience found its way to relax into laughter, thereby releasing the play. Martin Harvey has suggested to me that the laughter was led by his old mentor, the inspirational Hungarian director George Roman, and by George's Hungarian wife. Maybe you have to know something about 'the other side' \u2013 feel something from the other side \u2013 to appreciate fully any of James's plays. Or maybe you have to act in them.\n\n### DIRECTOR'S NOTES\n\n_Martin Harvey_\n\nIn the early 'nineties I was working on a production of _Of Mice and Men_ for the Belgrade Theatre in Coventry. I had asked James to come and speak to the cast, to give them a literary, political and historical context to the piece. James spoke at the read-through to an enthusiastic and talented group of actors, who were eager to pick his brains further in the pub that evening. The Belgrade had booked James into a hotel in the centre of town; and so, after one of those great evenings in which it is possible to see a company forming in front of your eyes, we put James into a taxi to take him to his hotel.\n\nThe next morning James relayed a story which will remain with me to the end of my days: the night manager refused him entrance to the hotel until he had taken him to an ATM to withdraw the cash for the night's accommodation (even though this had been booked and payment agreed by the theatre). If James is on his own, he is often mistaken for a drunk (even though he very rarely takes alcohol), a tramp, an idiot, an untouchable; the night manager probably saw him as all of these. This is how James had to pay for his determination to remain independent. Needless to say, the theatre took the hotel to task and, as a reinforcement of his independence, James insisted on staying there again on a subsequent visit to the cast. This time he was given red carpet treatment, with which he was clearly unimpressed.\n\nI relate this story because I think it gives insight into James's world and more specifically into where his writing is coming from and the characters, situations and settings that interest him. These plays are peopled by outsiders, misfits, those who are often regarded as non-persons; the wrongly labelled who have to fight in order just to validate their existence, and ask for little else but to be regarded as genuinely useful contributors to the communities in which they find themselves and from which they are often, in one way or another, excluded.\n\nOsip (Michael Bottle) and Inna (Leah Fells) after a fall. _Bread and Circus Freaks_ , Finborough Theatre, March 2002. Photo: Marilyn Kingwill\n\nGiven the central motif linking the plays which appear in this volume (and others which do not), one might expect these 'outsider _s_ ' to be treated with kid gloves, ennobled, even sanitized. Far from it, no holds are barred; these characters inhabit a world which is grotesque, where body-image and bodily functions are foregrounded, heightened and celebrated. At the climax of _Bread and Circus Freaks_ Osip Pischik and Inna thrash around on the floor together in a comically grotesque dance, which is not intended to elicit pity but rather freedom through laughter.\n\nIn another play of James's, _The Monkey Sanctuary_ , this grotesque celebration of the disabled body reaches its peak in a scene where a disabled man, shut away from society, is given a very rough but highly sexually charged bath by the woman of his dreams. She is Russian and she deals with him and his genitals without a trace of coyness or circumspection. It is this frankness which holds the key to the 'Russia' question. I too have asked James on numerous occasions, 'Why another play set in Russia?' (He even wrote a play set in Chicago to shut me up.) Since directing _Bread and Circus Freaks_ , I have come to a much clearer understanding of the driving force behind this issue.\n\nJames does not labour under some delusion that disabled people and other marginals are treated more fairly, with more kindness, or are given more value in Russia than anywhere else; in fact, from his own experiences, he may well believe the reverse to be true. What he has been attracted by is a culture, which he views as robust, containing prejudice which is expressed openly and not sanitized by English niceties or American correctness; emotions which are fully expressed and which can swing violently from tears or anger to laughter in a moment; and where the taboos surrounding the body are non-existent. He has found here a world where natural reality can be exaggerated, distorted and ultimately exploded, so as to be re-assembled into a different kind of truth, fuelled by imagination.\n\nThis place may not be _Russia_ at all, in the sense of a country we can objectively find, but more a place which is created from stories; stories from literature or true stories orally related, distorted and coloured in the telling. In short, it is a place of the imagination, based on observable truths, then taken beyond them into a kind of dreamscape. The physical settings in which these _dreams_ take place are outside the mainstream of life, in Russia or elsewhere: a cut-off village; the nebulous world of hostels and shady hotels, inhabited by those without the correct papers; a work camp for women, where the very rules of existence can change in the blink of an eye.\n\nIt is not my intention here to indulge in detailed literary criticism of these plays (my contact with them has been far too visceral for that), more to give a context through which to understand their style, which has proved to be increasingly important for me in their production. My principal concerns are with approaches to acting, from preparation, through rehearsal, to performance, and with the relationship between actors and audience.\n\nAll characters are real. That is to say, they are based on a perceived reality. Many of them have, at their core, true stories made up from observed incidents, related histories, documentary evidence. The basis for the setting of _Bread and Circus Freaks_ comes from a documentary film about the bread train and the irascible woman who sells the bread, which has to be pushed into the village. The actor playing Pania starts from there and from the invented fragments of the character's related history, her time in the pioneers, her marriage, her abortion, her schooldays etc., and weaves these together into a sense of character. This, of course, conforms to the first principles of Stanislavskian work, namely to look for the _given circumstances_ and, because the plays are made up from many stories, there is a wealth of potential _given circumstance_. Problems, however, may arise if those circumstances are not treated with some latitude, and the actor should be aware that a slavish attempt to weave them together into a linear structure or _through-line_ may lead to frustration. Stories may only be half true, they may be exaggerated, they may be a deliberate fabrication, so looking for cut and dried consistency is dangerous. But half-truth, exaggeration and fabrication only deal with truth as fact; behind each one of them is another truth, the truth of imagination.\n\nIt may be that this nether world, somewhere between the real and the imagined, is a place where greater truths can be explored, away from the crippling exactitudes of bare fact and recordable history. In _Bread and Circus Freaks_ , Inna breaks out of her physical disability as she describes her journey in search of her mother. Here truth, desire, fear and resolution are manufactured by Inna as she seeks to create her own space, her own voice, independent of those who are fighting for the right to speak for her. When Osip proposes to Inna and calls her his ideal, he is reaching for a world in his imagination, where he is no longer marginalized but can be validated through caring and being cared for. The actor must find the comfort in the desire for sanctuary and not in some physical or sexual imperative. If the others misread him, Inna understands.\n\nIn order for the actors to be able to play all this, it is necessary to search for a coherent style beyond naturalism. These are not characters, separated from the audience, going about their lives with consistency, unaware of their theatrical and imaginative context. There is a playfulness and improvisational quality which requires a conscious contact. Neither is the work confrontational, requiring that kind of head-on contact which pushes an audience back on its heels, shaming them into complicity or creating brutal alienation. This is more by way of an invitation, a suggested complicity. I imagine the audience as another character, as if on stage, with whom all is shared, sometimes directly \u2013 never as an aside, everybody in the space, audience included, shares in everything. (In _The Sweetheart Zone_ , James takes this a step further and gives the audience a defined character, that of a body of observers, constantly present and visible to the characters on stage.)\n\nThis requires a strong sense of shared space and even, to some extent, shared light. When we produced _Bread and Circus Freaks_ at the Finborough, we worked in traverse, audience facing each other across the playing space. Rather than the usual black-box studio, we painted the whole space blue and, although concentrating light on the playing space, allowed a diffuse light to illuminate the audience and the wall behind. The intention was to give a sense of one space, without division, into which we had all been invited.\n\nThe consequent required style of playing can be tricky for the actors to find, particularly for those used to hiding behind character or metaphorical footlights. An easy acceptance of the presence of the audience and an easy communication with them, once found, allows actors and audience alike to enter, enjoy, and reflect upon the imaginative world of the play. When hit upon well, this style works as a release mechanism on actor and audience alike, and we found that, as we got more and more at home with it during the run of performances and more and more relaxed in the relationship with the audience, the more the grotesque nature of the piece could be released, without jarring, and the funnier it became.\n\nIf this style can be tricky for accomplished professionals to find (and I don't think we could have wished for a better cast than played _Bread and Circus Freaks_ at the Finborough) it is quite a demand to make of students. Nevertheless we were keen to pursue the idea in _\u00c9migr\u00e9s_ and _The Sweetheart Zone_ , both of which formed the culminating project for the _Interpretative Acting_ module which James and I have been teaching together for the past few years at Exeter University. Both plays were, in fact, written specifically for the students on the module and aimed (amongst other things) to bring together workshops undertaken throughout the module.\n\nThe module, which aims to teach approaches to acting with text, begins with naturalism and with _An Actor Prepares_ by Konstantin Stanislavski. The students grapple with _given circumstances, the magic if, emotion memory_ etc., endeavouring to work the theory into acting naturalistic texts ranging from Chekhov to Wesker. It is not an intensive acting course (nor could it ever be, lasting as it does, over one university semester), but rather seeks to introduce the students to a range of ideas, which are worked on practically, and from which they can begin to assemble a _toolbox_ with which to approach their work on text. After work on Stanislavski, there are workshops on Michael Chekhov's techniques, Shakespeare and verse handling, and Bertolt Brecht (on which subject they mainly come to university armed with misunderstandings given to them at school).\n\nThe work on a play by James follows on from this, giving the students a chance to gain a sense of company, ensemble and indeed ownership of the work. In feedback, which we ask for at the end of the module, this project always rates highly with them, both for the challenges it lays down and for the excitement it generates. Each year, it seems, the project has become more ambitious, starting out three years ago with _\u00c9migr\u00e9s_ , which was presented, in edited form, as a script-in-hand rehearsed reading, to _The Sweetheart Zone_ which, in early 2007, was presented as an off-text studio production.\n\nThe plays present a number of challenges for the students beyond acting style. Firstly, the subject matter: the students have already, within workshops on Brecht, been asked to consider the political content of a piece (usually a short scene from, say, _The Caucasian Chalk Circle_ ) and to come to a collective view of what that content is and how to present it; now they are asked to consider the entire content of a new play in what is now a large group or company. For _\u00c9migr\u00e9s_ we had a sometimes heated debate on the subjects of immigration and asylumseekers; for _The Sweetheart Zone_ , the debate ranged from the nature of power and its abuses, to the problems of the outsider and bullying.\n\nSecondly, the very act of working with a new text brings with it disciplines for the students to master (disciplines which many professionals have problems with), namely coping with a changing text, understanding the actor's role in shaping that text, and simply playing a role that has no history of being played before. This last carries with it a responsibility to the play and the text which they may not have encountered before. Often these days, and very often within the university context, texts are viewed as merely starting points from which to create something new; but in the first outing of a new text, the job is to expose the play. Read any review of a modern production of a Shakespeare play and it will talk about the production, the performances, and the interpretation. A review of a new work deals first and foremost with that work, as there is no history of productions against which to measure interpretation.\n\nThirdly, because of the size of the groups that take this module, and because of James's refusal to be beaten by cast size and gender split, the students are asked to work in a large ensemble, a task which taxes the powers of concentration, generosity and listening. The fact that the students have managed this task so well is not only a tribute to their own ability to find generous ways of working, but also to James's generosity of spirit, in the writing, in rehearsal, in the re-writing and as an audience of his own work, which has now become the work of many.\n\nJames moreover uses the creation of this text to bring together the different strands followed throughout the course. So the actors have to develop a sense of their character's history; many of them have stories to tell, which may, as in _Bread and Circus Freaks_ be embellished fact (but are mainly based on real-life stories that have been related to James). They have to understand how their character and their stories fit the project of the play and what we, as a company, have collectively agreed we are trying to say with it. They have to communicate that in an easy but direct relationship with an audience which must be included. Above all, in such a large ensemble, each must take responsibility to keep the focus clear and moving towards the climactic moment. This moment, when Zhenia, in _The Sweetheart Zone_ , is forced to cross the stage, unaided and with all eyes on her, brings together in a single _gestus_ the cruelty of the bully to the outsider. One might even say that, distilled in this moment, are all the moments in James's life which resemble the moment in Coventry when he was marched to the ATM.\n\nProfessionals and students alike who have encountered these plays have found them very rewarding to work on. They have depth, humour, complexity of style and the kind of language that actors like, namely the kind of language they love to speak but for the most part could not write themselves. As a director, I can say that they have all presented me with something to learn and something to say. I commend them to you.\n\n### FREAKS, FOOD AND FAIRY TALES: CONFRONTING THE LIMITS OF DISABILITY IN _B READ AND CIRCUS FREAKS_\n\n_Thomas Fahy_\n\nTapping into the tradition of the carnival and vaudeville, James MacDonald's _Bread and Circus Freaks_ offers a powerful critique of the social prejudices that limit, restrict, and interpret the disabled body. The entire work is set in Pania Andreyevna's bread shop. Pania claims that she was a close friend of Inna's mother, and for this reason, she has agreed to give Inna, a twenty-year-old girl with cerebral palsy, a job there. With Inna's disability as the centrepiece, the play depicts four people who enter the shop in succession and ultimately define themselves in relation to Inna and her body \u2013 Pania (her aggressive employer whose verbal cruelty seems to be fuelled by the psychological burden of her abortion), Marianna Seligman (a former art teacher who tries to compensate for her failures as a mother by pressuring Inna to go to art school), Osip Pishchik (a vagrant whose romantic obsession with Inna began when she was institutionalized as a young girl), and Volkov (an ex-military constable who represents the government's oppressive authority). MacDonald uses each character to depict the ways in which many people \u2013 even when they believe themselves to be well intentioned \u2013 tend to equate disability with helplessness, dependence, and even personal failure.\n\nHis critique of this equation announces itself most clearly through the play's ironic humour, much of which comes from the fact that the true 'freaks' on stage are the able-bodied characters, whose attitudes about society and difference feed into self-destructive delusions. In fact, the word 'freak' only appears once in the play itself, and it is applied to the people who run government institutions for the disabled: 'Did those freaks let him handle you? [...] Those guardians you say were like sadists' (47). Freakishness, MacDonald suggests through this pointed use of the word, is about treating difference inhumanely, not about the body itself.\n\nMikhail Bakhtin's oft-discussed notion of the carnivalesque can provide an effective starting point for understanding MacDonald's use of freakishness here. As Bakhtin explains, the carnival provides an accepted outlet for the lower classes to satirize the wealthy and privileged. It inverts power structures that reinforce social hierarchies \u2013 allowing participants to parody the excesses of political power, affluence, and physical difference. In fact, Bakhtin's _Rabelais and His World_ reminds us that the carnivalesque contains an important 'bodily element' (19). The masks and disguises of the carnival typically play into anxieties about the body \u2013 degradation, irregularity, carnality, defecation and mortality. But these images are only meant to be temporary. Costumes can be taken off and put away. According to Bakhtin, 'carnival is not spectacle seen by the people; they live it, and everyone participates because its very idea embraces all the people. While carnival lasts, there is no other life outside it' (7). Outside the performance and laughter of the carnival, therefore, the body is expected to be contained and controlled. But the visibly disabled body does not allow this containment; it operates as an unwanted reminder of bodily degradation and irregularity. And this is one of the concerns taken up in MacDonald's play, whose protagonist, Inna, has been institutionalized for much of her childhood because of her disability.\n\nMacDonald also explores this concern through his use of traditions associated with vaudeville \u2013 a popular form of entertainment in the nineteenth and early twentieth centuries that was comprised of various performance styles, such as burlesque, freak shows, minstrelsy and musical numbers. As part of the theatrical productions of the American Museum, for example, P. T. Barnum crafted _entr'acte_ vaudeville that featured singers, blackface, jugglers, and some of his more famous freak performers, including Henry Johnson ('What Is It?') and Tom Thumb, who was billed as the smallest man in the world. The incorporation of freak shows into vaudeville further reinforced the ways in which the extraordinary body was viewed as a form of entertainment. Scholars such as Leslie Fiedler, Robert Bogdan, Rosemarie Garland Thomson, Rachel Adams and others have shown that the freak represented the Other for the audience. The performer's body became a visible marker for exclusion, and it often reinforced the audience's own sense of belonging to a group different from the one onstage \u2013 a more 'perfect' group. At the same time, Thomson argues that the audience both despised and identified with the Other: 'the spectator enthusiastically invested his dime in the freak show not only to confirm his own superiority, but also to safely focus an identificatory longing upon these creatures who embodies freedom's elusive and threatening promise of not being like everybody else' ( _Extraordinary Bodies_ 69). In other words, the viewer's longing for freedom from the quotidian as well as from social norms was an integral part of this exchange.\n\nCertainly, the title of MacDonald's play invites these historical comparisons. It not only alludes to the Latin phrase _panem et circenses_ (bread and circuses) from Juvenal's _Satire X_ \u2013 which criticizes both the government's use of entertainment and food to distract the public from more pressing socio-political concerns and the public's complacent acceptance of this \u2013 but the addition of the word 'freaks' also announces the focus of MacDonald's social critique. His work addresses the problems of equating disability with spectacle. More specifically, through a vaudevillian sequence of unrelated interactions and a thematic link between food and community, _Bread and Circus Freaks_ provides a dramatic mosaic that challenges the audience to confront extreme versions of their own biases and to reject the historical association between disability and freakishness.\n\n#### Not enough brown: the hunger for acceptance and belonging\n\nThroughout the play, MacDonald uses food as a central metaphor for the often unacknowledged needs and desires of the disabled. All of the characters hunger for something \u2013 a job, a cause, independence, love, justice \u2013 to give their lives meaning. Yet in a work whose title and setting suggest the importance of literal food, it is notable that no one eats. The bread shop merely distributes food. Nothing is baked or made there. And if we are to believe Pania's complaints about the shortage of brown bread and the mouldy condition of the white, this bread shop also fails to provide sustenance for the community.\n\nBread doesn't offer nourishment here, and it also doesn't forge the kind of social bonds that are traditionally associated with the sharing of food. As E. N. Anderson explains in _Everybody Eats: Understanding Food and Culture_ , 'food is used in every society to communicate messages. Pre-eminent among these are messages of group solidarity; food sharing is literally sacred in almost all religions and takes on a near-sacred quality in many (most?) families around the world' (7). He goes on to argue that:\n\none main message of food [...] is solidarity. Eating together means sharing and participating. [...] The other main message is separation. Food marks social class, ethnicity, and so on. Food transactions define families, networks, friendship groups, religions, and virtually every other socially institutionalized group. Naturally, one group can try to use food to separate itself, while another is trying to use food to eliminate that separation. (125)\n\nThis tension between solidarity and separation is certainly evident in MacDonald's play \u2013 where the characters desire community but largely seem isolated from one another. And the lack of food sharing and consumption reinforces this.\n\nMacDonald specifically links literal\/metaphoric hunger with loneliness in order to highlight the importance of a more inclusive community \u2013 one that acknowledges and treats disability with open-mindedness and compassion. At the outset of the play, Pania reminds Inna that having a job prevents her from being 'an outcast, for people to take pity on or worse... because pity soon turns to contempt... and then you might starve' (13). Being an outcast leads to social and literal starvation here. Yet Pania, who later reveals that she had an abortion after her husband left her, is arguably a marginalized figure as well. The lack of intimacy in her life is highlighted humorously by her reaction to the goats having sex outside the bread shop: 'There's no men, just like there's not enough brown. So what's the point of liking sex?' (18). Food and sex are in short supply in Pania's world, and this prevents her from finding pleasure in life. It also keeps her emotionally distant from the community she lives in and serves. Not surprisingly, she has internalized an adversarial relationship with her customers \u2013 relying on a rifle ('hunter') to quell potentially violent crowds. These qualities make her far less connected to the community than Inna, who has several visitors throughout the play. Nevertheless, Pania throws out the words 'cripple', 'outcast' and 'pariah' like punches to remind Inna of her othered status \u2013 a status that is inescapable because of her body.\n\nThis solidarity\/separation dichotomy gets reinforced through the image of bread as well, which tends to foster a violent competitiveness among people, not solidarity ('They'll be asking for bread... and then taking our blood just as soon as we've run out' [14]). Pania, for example, is a seller of bread, a provider, and she views this profession as having an important social value. She repeatedly tells stories of her own physical strength (pushing the bread train and lifting the heavy trays) to remind Inna of her limitations as an individual and her failure to contribute something meaningful to society. It is not surprising, therefore, that Pania refuses to let Inna do any physical work. She needs Inna to remain a foil for her own sense of productivity and usefulness. At one point, she even offers Inna some brown bread in a gesture that amounts to a handout \u2013 negating Inna's desire to work.\n\nINNA: No, I don't want \u2013 [...] No, of course I eat bread, but I have no objection to the white loaf.\n\nPANIA: Unlike the rest of us. Aren't you human? Look here... _(Showing her.)_ It's mouldy. There's gobs of mould here and here. [...]\n\nINNA _(Taking the white loaf.):_ It's not that bad. Look, I can cut the bits off. _(Inspects.)_ I think you must be exaggerating. I don't see any\u2013\n\nPANIA: I think you must be dim-witted or something, not to want brown and then not to see any fault in the white. (23)\n\nFor Pania, difference exists only in extremes (brown\/white, right\/wrong, human\/disabled). It requires the absolute rejection of one thing over the other \u2013 so that one cannot refuse to eat brown bread without seeing the white as flawed. Likewise, one cannot accept difference (in terms of disability, sexuality, ethnicity, etc.) without calling into question accepted notions of normality. Hence, Inna must stay an isolated figure. To preserve Pania's sense of social importance\/relevance, Inna must be seen as a 'cripple' and by implication subhuman.\n\nMacDonald continues to explore this tension between separation and solidarity through two figures (the artist and the lover) that reject food in order to follow their passions. Of course, _Bread and Circus Freaks_ is not _commedia dell'arte_ , and MacDonald offers inventive variations on these types to highlight age-old prejudices that still limit the disabled. First of all, Marianna, Inna's former art teacher, tries to direct young people to pursue the arts. Here, she purchases bread as a pretence for speaking with Inna \u2013 in an attempt to nurture\/nourish her. Marianna has secured a scholarship for Inna, and, presenting Frida Kahlo as a model, she urges Inna to develop her art. 'Go to art school, test yourself, if you want to find out what you're really worth' (34). On the surface, Marianna seems to have Inna's best interests in mind, but her own attitudes about disability are not so different from Pania's. She fundamentally believes that Inna's 'worth' is in question, yet she hopes that art will remedy this to some extent. Art can give Inna a place in society by providing an outlet to transform her disability into something aesthetic, as Kahlo did. In other words, art can provide a way for Inna to define herself completely by her disability. At one point, Marianna snaps at Pania: 'And just how do you think she can help you? You need a strong, healthy girl for work of that kind. Could she push the bread train? You know she couldn't' (36). Inna replies that she 'wanted to' and that she yearns for an 'ordinary job', but Marianna can't hear this. For her, Inna doesn't need a practical existence, as represented by jobs and bread; she needs the protection of educators and social workers. She needs an outlet \u2013 art \u2013 to define herself entirely by her own body.\n\nMarianna's failure stems, in part, from the fact that she already sees Inna as an art object, and MacDonald presents this desire to read\/interpret disability (as opposed to seeing the individual) as one of the impediments that prevent the disabled from achieving a broader social acceptance. Marianna's initial interest in Inna, for example, comes from the way her body reminds Marianna of Diego Vel\u00e1zquez's _Las Meninas_ : 'And that painting by Vel\u00e1zquez \u2013 the group portrait with the little girl who's adored by the dwarf. [...] The aristocracy always looked after their lame ones... like beloved family pets. It wasn't looked on as misfortune to be lame. [...] I want to look after you, darling' (40). For Marianna, Inna's body represents need and dependency. It relegates her to the realm of the metaphoric. On one hand, Marianna wants to recast the Vel\u00e1zquez with Inna as family pet and herself as the beneficent aristocrat. On the other hand, Marianna reads Inna's body as an image for the failures of Russia more broadly. 'It's the condition of Russia, my dear. People become friends with their condition... They depend on it, finally become enslaved by it. Like the people in this village. [...] You do nothing, quite frankly, to help yourselves' (42). Both of these interpretations suggest that disability is either a condition that requires the magnanimous assistance of the able-bodied or something that must be overcome. It never occurs to Marianna to accept Inna (or her own daughter) for who she is \u2013 an individual with her own desires and goals. Instead, she warns Inna that to accept her disability is to be trapped by it.\n\nMacDonald presents the lover figure, Osip, as another example of how disability is often read in aesthetic and metaphoric terms. Even though Osip is a vagrant, he isn't interested in eating bread. Instead, he hungers for intimacy and connection, just as he longs to relive the time when he performed magic tricks for institutionalized children. That was a period in his life when he felt as if he were making a meaningful contribution to others. His love for Inna is not only an attempt to compensate for this loneliness (as well as his social status as a pariah), but it also comes from his idealization of her. He calls her 'a living portrait, [...] a tiny masterpiece' (48), and 'an angel' (58). The latter invokes another clich\u00e9d view of disability: reading the extraordinary body in terms of the divine. Ancient Greek and Roman society, for example, interpreted natural phenomena as the result of cosmic or divine forces: fires, epidemics, the appearance of a comet or eclipse and the birth of an extraordinary body were believed to presage the doom of an empire and the breakdown of social order. Similarly in the Middle Ages, Christians viewed disabled bodies as divine warnings against the dangers of pride, disobedience and waning faith. Osip parrots these interpretations explicitly: 'She needs to be recognized for what she is. [...] That's right... she is \u2013 a message from God for the nation' (59). The problem here (other than the fact that he is thirty years her senior and that they first met when she was a child) is that Osip views her disability as a kind of panacea for his own suffering. Like Pania and Marianna, he insists that Inna's body proves a level of suffering worse than his own. Her body, therefore, evinces her need for help: 'I love you, my little Inyechka. You're my ideal, you are! [...] Let me help you. You've suffered more than the rest of us' (57). Throughout _Bread and Circus Freaks_ , it is this kind of selfish need to feel better off than someone else that drives the characters' obsession with disability.\n\nAll of the able-bodied characters fail to see Inna's basic hunger \u2013 her desire to be needed. At one point, she tells Pania: 'Now I can see that you need me. Oh, how you've suffered in life. I can see what they mean now \u2013 Pania Andreyevna, the Bread Lady. What a glorious thing to be needed for life!' (57). Pania (a name which may be an allusion to the Pania in Maori mythology who was betrayed by her husband when he placed food in her mouth to prevent her from returning to the sea) is the 'bread lady'. Her place in this village is clearly established, and Inna desires exactly this \u2013 a meaningful, recognized role in society that has nothing to do with her body. At the end of the play, she does find an outlet for this \u2013 Osip. When Volkov, the constable, threatens to arrest him, Inna offers to be his caretaker. 'He's a menace', Volkov explains. 'They all are, people like him [...] I mean, look at him, dear. He's disgusting' (63). Inna immediately recognizes that the language Volkov applies to Osip has also been used to describe her: 'I was judged a pariah... unclean. They locked me up for years, too' (63). Inna has successfully fought this kind of prejudice throughout her life. She has, in effect, forged a community of her own. She has found a job, turned away the assistance of Marianna, and become the caretaker\u2013lover of Osip. None of these things has changed society's view of her, however. Pania still sees her as a cripple, and her decision to help Osip does seem to run counter to her earlier desire 'to move on my own first... learn who I am in the world. I've been sheltered all my life, really sheltered' (59). Perhaps, this is MacDonald's point. She may not be sheltered any longer, but she will always be subject to the Panias, Mariannas, and Volkovs of the world. Nevertheless, these prejudices don't have to limit Inna, MacDonald suggests. Her choice to be with Osip offers a hopeful alternative. In spite of the limitations placed on her, she still has the ability to be compassionate, to help ease the suffering of those around her.\n\nMacDonald's introduction of Volkov broadens the critique of his play, moving it from the problems with individual prejudices against disability (seeing the extraordinary body in terms of spectacle) to an indictment of the systemic practices that attempt to render disability invisible. As an ex-military figure and village constable, Volkov represents the law in its strictest, most unyielding form. Like the banking uniform that Gregor Samsa's father wears in Kafka's _Metamorphosis_ , Volkov's uniform \u2013 along with his hateful words and abusive actions \u2013 makes him a symbol for legally sanctioned oppression and violence. His desire to remove 'people like' Osip (the homeless, the sick, the disabled) 'for [their] own protection' (63) is not only arrogant and spiteful, but it is also callous. He judges Osip on appearance alone, just as Pania judges Inna's job at the bakery as an act in the circus (40). MacDonald clearly crafts _Bread and Circus Freaks_ to attack this kind of politics of visibility. It was through a similar legerdemain of costuming, rehearsed behaviour, spiel and true-life pamphlets that freak show entrepreneurs like Barnum created 'freaks' \u2013 dressing up African Americans as Fiji cannibals and using padded clothing to make a performer the fattest woman in the world, for example. This link between Inna and a circus act, therefore, illustrates the terrible legacy of the freak show and how it continues to shape social attitudes about disability.\n\nVolkov's (and by extension the government's) main response to difference, however, is to render it invisible. As a society, we put criminals behind bars not only as a means of punishment but also as an act of erasure \u2013 removing them from sight. In _Crime and Punishment in America_ , for example, Elliot Currie discusses the effects of this in the United States:\n\nOur growing reliance on incarceration helps us avoid confronting a host of deep and stubborn social problems. [...] A swollen correctional system allows us to sweep these problems under the rug, but it does not make them go away, and indeed makes them worse. (191)\n\nMacDonald's play offers a similar warning about the institutionalization of the disabled in the post-Soviet era. It only heightens the challenges facing this community because it provides an official stamp for labelling anyone who is different as an enemy of the state and of the social order.\n\n#### Disability and rewriting fairy tales\n\nYou don't have to read more than a few pages of Grimm's fairy tales to find acts of violence, cruelty, and vengeance associated with food. The queen in 'Snow White' \u2013 who uses an apple to poison Snow White \u2013 is eventually tortured to death with red-hot, iron shoes. Hansel and Gretel, whose parents have abandoned them in the woods because they are too poor to provide them with bread \u2013 ultimately kill the witch who plans to eat them. And the stepsisters in 'Cinderella' throw peas and lentils into the ashes for Cinderella to pick up. At the end of the story, they mutilate their feet to trick the prince into marrying them and are blinded for their wickedness. MacDonald taps into this tradition throughout _Bread and Circus Freaks_. Food often represents a longing for happiness in fairy tales, but the dangers of hunger are never far off. MacDonald uses the threat of danger\/violence in this play to rewrite the fairy tale in terms of disability. Inna, a poor girl who works in a bread shop, is the princess\/maiden of this play. Her parents are absent, and she hungers for something more in life. But it is not external beauty that defines her; it is her disability. MacDonald asks us to consider the implications of this inversion. Snow White, for example, is spared her life and helped time and time again because of her stunning beauty. Inna, on the other hand, is continually restricted by her physical appearance. She remains a perpetual outcast and circus act. She is not locked in a tower but an institution \u2013 a period in her life that she ironically calls 'a fairy tale' (48). Nor does a prince rescue her. Instead she must rescue the homeless, emasculated Osip.\n\nThe version of romance that MacDonald presents here reinforces the idea that traditional fairytale narratives are not available to the disabled. Inna may dream of princes on horseback, but Pania repeatedly transforms fantasies of romanticized love into stories of rape.\n\nPANIA: You like soldiers, do you? You'd like to be raped by a soldier? [...] Is that your schoolgirl fantasy, is it? Foreign war with Russian soldiers raping the flesh off you? Sex isn't that good.\n\nINNA: That's not my fantasy, Madame Pania.\n\nPANIA: Ha \u2013 says you.\n\nINNA: It must be yours.\n\nPANIA: It's nobody's, that's the whole point. (16)\n\nPania's fantasy rejects the romance of fairy tales for all women, but she suggests that the impediments to love are greater for the disabled \u2013 always alluding to the dangerous wolves that lurk outside the bread shop. Little Red Riding Hood may have been tricked by a wolf to stray from the path to grandmother's house, but this beautiful girl is still able to escape his treachery with the help of a huntsman. From Pania's perspective, Inna cannot stray from the path of her condition and expect to be saved. The wolves (outside world) will quickly tear her to pieces if she no longer has a job (the protection of the able-bodied). MacDonald's rewriting of such fairytale narratives and images suggests how far the disabled experience is removed from mainstream culture and society.\n\nIn a 2002 interview, James MacDonald lamented the failures of British society to deal with difference and the need more broadly for disability to be recognized and dealt with humanely. 'If people can't even acknowledge it, how can they engage with it to overcome the problem? We are still dealing with the difference thrown up by class barriers, so others \u2013 sexual, ethnic, and physical \u2013 must be some way off being addressed. Things have improved. Disabled people are now generally visible at least. But being accepted remains a future stage of progress; how far in the future I don't know' (Fahy and King, _Peering Behind the Curtain_ , 112). His call for acceptance drives _Bread and Circus Freaks_. This powerful play challenges audiences to see the ways in which prejudices and misguided altruism limit the disabled. His use of humour also reminds us that it's okay to laugh. Ultimately, laughter might be the best tool for helping us recognize the absurdity of our own biases and to do something about it.\n\n#### Works Consulted\n\nAdams, Rachel (2001), _Sideshow U.S.A.: Freaks and the American Cultural Imagination_ , Chicago: University of Chicago Press.\n\nAllen, Robert C. (1991), _Horrible Prettiness: Burlesque and American Culture_ , Chapel Hill: University of North Carolina Press.\n\nAnderson, E. N. (2005), _Everybody Eats: Understanding Food and Culture_ , New York: New York University Press.\n\nBakhtin, Mikhail (1984), _Rabelais and His World_ , (trans. H\u00e9l\u00e8ne Iswolsky), Bloomington: Indiana University Press.\n\nBogdan, Robert (1988), _Freak Show: Presenting Human Oddities for Amusement and Profit_ , Chicago: University of Chicago Press.\n\nCurrie, Elliot (1998), _Crime and Punishment in America_ , New York: Metropolitan Books.\n\nFahy, Thomas (2006), _Freak Shows and the Modern American Imagination: Constructing the Damaged Body from Willa Cather to Truman Capote_ , New York: Palgrave Macmillan.\n\nFahy, Thomas and Kimball King (eds.) (2002), _Peering Behind the Curtain: Disability, Illness, and the Extraordinary Body in Contemporary Theater_ , New York: Routledge.\n\nFiedler, Leslie (1978), _Freaks: Myths and Images of the Secret Self_ , New York: Simon and Schuster.\n\nHarris, Neil (1973), _Humbug: The Art of P. T. Barnum_ , Boston: Little, Brown.\n\nLott, Eric (1993), _Love and Theft: Blackface Minstrelsy and the American Working Class_ , New York: Oxford University Press.\n\nReiss, Benjamin (2001), _The Showman and the Slave: Race, Death, and Memory in Barnum's America_ , Cambridge, MA: Harvard University Press.\n\nThomson, Rosemarie Garland (1997), _Extraordinary Bodies: Figuring Physical Disability in American Culture and Literature_ , New York: Columbia University Press.\n\n\u2014 (1996), 'Introduction: From Wonder to Error \u2013 A Genealogy of Freak Discourse in Modernity', in Rosemarie Garland Thomson (ed.), Freakery: Cultural Spectacles of the Extraordinary Body, New York: New York University Press, 1 \u2013 19.\n\nWilson, Dudley (1993), _Signs and Portents: Monstrous Births from the Middle Ages to the Enlightenment_ , New York: Routledge.\n\n## EPILOGUE\n\n_Su Elliott_\n\nI read _Bread and Circus Freaks_ again the other day to refresh my memory of Pania Andreyevna and the play. I'd forgotten (how could I?) how very, very funny it is and what a complex and interesting character _she_ is.\n\nI don't know how I 'approach' a part. I like to read the play myself and then enjoy the actual first-day play-reading with the rest of the cast, and then sort of get stuck in. But there are two really vital elements that make your life oh-so-much easier, and these are a good director who isn't going to flaunt his\/her ego and who has plenty of ideas _and_ a good script. Well, we actors were lucky in _Bread and Circus Freaks_ as we had both in spades.\n\nPeople aren't Mr Men. You can't say someone is just nasty. They may be, in certain circumstances, and your subjective view of them may not change from that. But in your heart you know that there are people who find the person fun or helpful or warm. It's the same in theatre; but with James the complexities of the characters are already written in, and that's so much fun to work on. Though as with any good piece of writing, there's plenty to discover along the way that isn't immediately obvious, and of course the audience will find their own insights.\n\nI loved the monumental selfishness of Pania. If she thinks she's not getting enough attention, she practically says, 'Well, that's enough about me, now what about _ME_?' She can be cruel. She _wants_ to be cruel, and she's very hard on Inna, but then again, it's a hard life and a hard place to live, and she knows that. It's 'tough love', as we would call it now, but as you work into the play, you can see that there is kindness there, and there are absolutely no pretensions whatsoever. The exchanges with Marianna were particularly fun to do. Pania certainly knows more than she lets on \u2013 after all, she makes reference to Kandinsky. But of course it doesn't suit her to be on the same intellectual footing with Marianna: she'd lose the argument then, and she needs to have the upper hand at all times. When Pania's logic is sliding all over the place as she duets and dives in this rapid-fire argument, Marianna says, 'You just detest giving in' \u2013 an understatement if ever there was one. I loved the lack of relaxation in Pania, the competitiveness, and how anyone encountering her has to be on their toes. Nothing came easy to this woman, so why should it come easy to anyone else? She's like a recalcitrant teenager. You can hear her saying, 'Yeah, yeah, whatever', and having an answer for everything.\n\nWhen she's with Pishchik, she matches his slyness; but it's then in the script that James gives the information that her husband has left her and that there is a deep hurt there. But James isn't finished, because _then_ you realize that she's had an abortion, which reverts to her line early on, 'Oh, yes, my insides still work', and is one of the keys to the way she reacts, and to her own pain.\n\nI can't remember any difficulties working on and performing this piece, apart from the fact that we could have done with a bit more time. The style of acting suited me very well. I'm rather partial to confronting an audience head-on \u2013 as Pania does. It calls for a mixture of confrontation and charm; otherwise they'll be put off. The humour (here) helps, and it's such fun to do. I loved the line about Pioneers being Organized Criminals, like everybody...( _Ghoulish pause_.)...only smaller. There are loads like that \u2013 too many to quote here. I do know that when you work on a play, you're all in it together \u2013 the crew, the cast, the writer and the director, and when it works (together with the audience), it becomes more than the sum of its parts, and that's when it takes off and you know it was a Good Job Done. But it's a hell of a lot easier when you've got a good script to start with. So...thanks, James!\n\n## CONTRIBUTORS\n\nPeter Thomson, Professor Emeritus of Drama at Exeter University, is the author of a score of books on Shakespeare, Brecht and theatre, including _Shakespeare's Theatre_ , _Shakespeare's Professional Career_ , _Brecht's Mother Courage and Her Children_ and _English Drama: 1660\u20131900_. For Intellect he has edited the quarterly journal _Studies in Theatre and Performance_.\n\nMartin Harvey, former Associate Director of the Northcott Theatre, Exeter, has had a varied and acclaimed theatre career as actor, director and writer, with such notable productions as _King Lear_ , _Of Mice and Men_ and the one-act opera _Isabella and the Pot of Basil_. He is currently Teaching Fellow in Drama at the University of Exeter.\n\nThomas Fahy, Director of American Studies at Long Island University, New York, is the author and editor of several notable books, including _Freak Shows and the Modern American Imagination_ , _Considering Alan Ball_ , _Considering Aaron Sorkin_ and _Considering David Chase_. He has also published numerous critical articles and two novels, _Night Visions_ and _The Unspoken_.\n\nSu Elliott was born in Newcastle upon Tyne and trained at the Guildhall School of Music and Drama in preparation for an acting career that has embraced every part of the industry. Her West End performances include _Can't Pay_ , _Won't Pay_ and _Adrian Mole_ ; for Mike Leigh she appeared in _Secrets and Lies_ and, a featured role, _Home Sweet Home_. Her extensive television work includes _Auf Wiedersehen_ , _Pet_ , _Coronation Street_ and _Holby City_. Su wrote and performed in a one-woman adaptation of fairy tales to the delight of Fortnam and Mason customers young and old throughout the holiday period, 2007.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nH. G. Wells\n\npar\n\nLaura El Makki\n\nGallimard\nLaura El Makki est l'auteur, avec Marie Berthoumieu, de _Henry David Thoreau_ (Folio Biographies no 115, 2014). Elle est productrice, sur France Inter, des \u00e9missions \u00ab Un \u00e9t\u00e9 avec Proust \u00bb et \u00ab Un \u00e9t\u00e9 avec Victor Hugo \u00bb (publi\u00e9es en co\u00e9dition France Inter \/ \u00c9quateurs). Elle est \u00e9galement sc\u00e9nariste de fictions radiophoniques.\n\nAutre publication : _\u00c7a peut pas faire de mal_ , Proust, Hugo et Mme de Lafayette lus et comment\u00e9s par Guillaume Gallienne (co\u00e9dition Gallimard \/ France Inter, 2014).\nJe crois que je ne regrette rien.\n\nH. G. Wells\n\n_Tono Bungay_\nPrologue\n\n\u00ab Il faut que tu m'aimes. \u00bb Puis elle a ajout\u00e9 : \u00ab Sinon je me tue. J'ai du poison. J'ai un rasoir 1 *1. \u00bb Il avait tout pr\u00e9vu, sauf \u00e7a.\n\nCe soir-l\u00e0, Wells avait enfil\u00e9 un smoking impeccable, nou\u00e9 sa cravate, mis ses belles chaussures noires, et peign\u00e9 ses cheveux en les plaquant bien derri\u00e8re l'oreille gauche. Il ne lui manquait plus que son chapeau, pos\u00e9 dans le couloir. On l'attendait : un d\u00eener tr\u00e8s important \u00e0 l'autre bout de la ville avec le secr\u00e9taire d'\u00c9tat de l'Inde, Edwin Samuel Montagu. Il \u00e9tait d\u00e9j\u00e0 en retard, son taxi gar\u00e9 en bas. Mais en sortant du dressing pour se diriger vers la porte d'entr\u00e9e, un bruit avait retenu son attention. Des pas provenant de son bureau. Cela ne pouvait pas \u00eatre Jane, elle \u00e9tait \u00e0 la campagne avec les enfants. Sa femme de chambre ? Non plus, c'\u00e9tait son jour de repos. Il avait pouss\u00e9 la porte, balay\u00e9 la pi\u00e8ce du regard et l'avait aper\u00e7ue, allong\u00e9e langoureusement sur le tapis pr\u00e8s de la chemin\u00e9e, v\u00eatue d'un long imperm\u00e9able qui d\u00e9couvrait son corps enti\u00e8rement nu. Elle portait des bas noirs, des escarpins, du rouge aux l\u00e8vres, et maintenait d'une seule main ses cheveux au-dessus de sa nuque. C'\u00e9tait elle, Hedwig Gatternigg, la jeune Autrichienne, celle qui ressemblait tant \u00e0 Mona Lisa. Il y avait plusieurs mois, elle lui avait \u00e9crit : elle voulait traduire en allemand la biographie qu'il avait consacr\u00e9e au professeur Frederick William Sanderson 2, et il lui avait donn\u00e9 son accord. Depuis, les lettres ne cessaient de s'amonceler devant sa porte, des d\u00e9clarations d'amour ampoul\u00e9es, des po\u00e8mes soigneusement recopi\u00e9s. Il se souvenait de s'\u00eatre senti flatt\u00e9 et de lui avoir fait l'amour un apr\u00e8s-midi de printemps. Cela ne devait pas durer, cela ne _pouvait_ pas durer : il y avait Jane, et puis surtout Rebecca \u2014 les choses \u00e9taient devenues difficiles avec elle, mais il l'aimait toujours, enfin, il croyait l'aimer, dix ans bient\u00f4t qu'ils \u00e9taient ensemble. Oui, cette aventure avec Hedwig devait cesser, sa vie \u00e9tait d\u00e9j\u00e0 assez compliqu\u00e9e. D'ailleurs, il avait pris un plaisir particulier \u00e0 ne pas r\u00e9pondre \u00e0 ses courriers et \u00e0 ignorer ses appels quotidiens, pensant na\u00efvement qu'elle se lasserait. Rien \u00e0 faire : la voil\u00e0 en face de lui, belle et ridicule \u00e0 la fois. Entre ses doigts s'agite effectivement une lame de rasoir qui brille \u00e0 la lumi\u00e8re du feu.\n\n\u00ab Il faut que tu m'aimes. \u00bb Il n'a pas r\u00eav\u00e9, elle vient de prononcer ces mots. Que faire ? Aller vers elle ? Lui prendre le rasoir des mains ? Trop dangereux. Dans ce genre de situation, il faut des t\u00e9moins, on ne sait jamais. Tr\u00e8s doucement, il se tourne vers la porte, h\u00e8le la domestique et lui demande de t\u00e9l\u00e9phoner au concierge en bas. C'est un homme de confiance, costaud qui plus est. Mais \u00e0 peine a-t-il d\u00e9tourn\u00e9 la t\u00eate \u2014 quelques secondes, pas plus \u2014 que l'Autrichienne s'est d\u00e9j\u00e0 tranch\u00e9 les poignets. Le sang coule maintenant, et la vague rouge\u00e2tre s'\u00e9tale sur le tapis. Il se pr\u00e9cipite vers elle, appuie sur les plaies comme il peut. \u00ab Laisse-moi mourir 3 \u00bb, dit-elle en se d\u00e9battant. Il la tient fermement, hurle \u00e0 l'aide. Le concierge arrive en courant, accompagn\u00e9 de deux policiers. Tous les trois la soul\u00e8vent et l'emm\u00e8nent, tandis qu'elle articule des \u00ab Je l'aime \u00bb imperceptibles \u00e0 ses sauveteurs. Et puis, plus rien. Le silence profond lui fait entendre les battements de son propre c\u0153ur. Il se rel\u00e8ve et constate que son costume n'en est plus un. Si quelqu'un entrait \u00e0 l'improviste dans la pi\u00e8ce, il ne saurait comment expliquer que, malgr\u00e9 les apparences, il n'a tu\u00e9 personne. Sa montre indique une heure d\u00e9cid\u00e9ment bien tardive. Il d\u00e9croche le t\u00e9l\u00e9phone, annule le d\u00eener et s'enfonce dans le fauteuil de son bureau. Bient\u00f4t, la presse sera l\u00e0, les reporters demanderont \u00e0 lui parler, il devra expliquer l'inexplicable \u2014 une jeune femme s'est introduite chez lui, nue et arm\u00e9e d'un rasoir, pour lui demander de l'aimer. Qui pourra croire cela ? Il imagine d\u00e9j\u00e0 les gros titres \u00e0 la une des journaux le lendemain : \u00ab Nuit tragique dans l'appartement d'un c\u00e9l\u00e8bre auteur 4. \u00bb C'est une catastrophe \u2014 pour lui, pour sa famille, pour Rebecca. Il a cinquante-sept ans et la d\u00e9sagr\u00e9able impression que sa vie lui \u00e9chappe. Rien de tout cela n'aurait d\u00fb arriver. Bien s\u00fbr, cette notori\u00e9t\u00e9, il l'a cherch\u00e9e, l'amour des femmes aussi (constamment, d\u00e9sesp\u00e9r\u00e9ment), mais il n'a jamais convoit\u00e9 la rubrique des faits-divers. Tout ce qu'il voulait, lui, c'\u00e9tait \u00eatre \u00e9crivain.\n\n*1. Les notes bibliographiques sont regroup\u00e9es en fin de volume, p. 224.\nEnfant des \u00e9toiles\n\nOn a install\u00e9 un pupitre en bois. Dessus, on a pos\u00e9 un livre ouvert. On lui demande de s'asseoir et de faire comme s'il lisait, tout en regardant l'objectif. La position n'est pas du tout naturelle. Sa m\u00e8re, Sarah, lui fait signe de se redresser puis s'approche une derni\u00e8re fois pour le peigner \u00e9nergiquement, rabattant les cheveux sur le c\u00f4t\u00e9 gauche, comme toujours, derri\u00e8re l'oreille. C'est un jour important : celui du portrait photographique. Wells a dix ans, il porte son plus beau gilet, boutonn\u00e9 jusqu'au cou, un pantalon en laine, des chaussettes propres et ses souliers habituels qu'on a vernis pour l'occasion. Bon \u00e9l\u00e8ve, il s'ex\u00e9cute, croise les bras devant le livre, esquisse le d\u00e9but d'un sourire et ne bouge plus. Voil\u00e0, c'est fait. Sarah remercie le photographe, prend la main de son fils, ils repartent d'un pas press\u00e9. Il faut rentrer et rouvrir la boutique ; d'habitude, elle n'est jamais ferm\u00e9e \u00e0 cette heure-ci, les clients vont se poser des questions. En arrivant, ils constatent que cela ne servait \u00e0 rien de courir. Il n'y a personne, ni devant le magasin ni aux alentours. La rue est vide, la vaisselle prend la poussi\u00e8re dans la vitrine. On enl\u00e8ve le beau gilet au profit d'une chemise en lin un peu froiss\u00e9e. Il n'est pas encore midi, le gar\u00e7on doit aller chercher de l'eau \u00e0 la pompe et rapporter de la cave un ou deux sacs de charbon. La nuit derni\u00e8re a \u00e9t\u00e9 fra\u00eeche.\n\nEn traversant la cour, il croise son p\u00e8re, Joseph, mais ne lui adresse pas la parole \u2014 il conna\u00eet trop ses humeurs. Quand il est occup\u00e9 \u00e0 \u00e9laguer son fameux pied de vigne, mieux vaut ne pas le d\u00e9ranger. Cette petite plante maigrelette qui tente de pousser contre le mur en pierre, c'est la sienne, personne n'a le droit d'y toucher. Le cricket et le jardinage, voil\u00e0 les deux seules choses qui int\u00e9ressent cet homme las et grincheux. Cela fait des ann\u00e9es qu'il a d\u00e9sert\u00e9 le commerce familial. \u00c0 l'origine, pourtant, il avait convaincu sa femme d'investir toutes leurs \u00e9conomies dans ce vieux magasin d'objets d'occasion qui appartenait \u00e0 l'\u00e9poque \u00e0 un cousin lointain. Il disait que c'\u00e9tait une bonne affaire, qu'ils prendraient un nouveau d\u00e9part, qu'ils seraient heureux. Ils s'\u00e9taient rencontr\u00e9s trois ans plus t\u00f4t \u00e0 Up Park, la propri\u00e9t\u00e9 d'une riche aristocrate (Miss Featherstonhaugh). Joseph \u00e9tait jardinier, Sarah, femme de chambre. Lui, vigoureux, sanguin, n'ayant que faire de la religion et des conventions. Elle, les yeux bleus, l'habitude de respecter les r\u00e8gles, la foi chr\u00e9tienne chevill\u00e9e au c\u0153ur. Le coup de foudre est imm\u00e9diat. Ils se marient. Peu apr\u00e8s, Joseph perd son emploi pour d\u00e9sob\u00e9issance \u00e0 son sup\u00e9rieur. Sarah, elle, perd ses parents, et la jeune fille h\u00e9rite ainsi d'une lourde dette. N'y connaissant rien en affaires, ils n'h\u00e9sitent pourtant pas \u00e0 aller s'installer \u00e0 Bromley, dans le Kent, persuad\u00e9s d'y trouver leur salut. Ils sont jeunes, enthousiastes, et depuis peu parents d'une petite Fanny : ils y arriveront. La boutique se trouve au rez-de-chauss\u00e9e d'une maison en pierre. Eux habitent \u00e0 l'\u00e9tage. Leur voisin, le boucher du village, tue ses moutons la nuit. En face, se dresse le clocher de l'\u00e9glise. Dans leur entr\u00e9e, une vieille statue, repr\u00e9sentant le dieu Atlas, accueille les visiteurs.\n\nMais les clients se font rares, la fa\u00efence prend racine sur les \u00e9tag\u00e8res, et Sarah s'inqui\u00e8te d\u00e9sormais quotidiennement. \u00ab Je crains que nous n'ayons fait une b\u00eatise 1 \u00bb, \u00e9crit-elle dans son journal intime. Quelle b\u00eatise, oui ! Jamais, cependant, elle ne se plaint. Si elle ne tient pas bon, qui le fera ? Joseph est de plus en plus absent, pr\u00e9f\u00e9rant s'adonner \u00e0 sa passion du cricket qui rapporte un peu d'argent au foyer. Il ne lui parle plus, ne la regarde presque pas, et la touche tr\u00e8s occasionnellement. Sarah s'occupe seule de la maison et s'isole parfois dans la journ\u00e9e pour noircir ses carnets, consignant les \u00ab jours malheureux 2 \u00bb, les all\u00e9es et venues \u00e0 l'\u00c9glise, l'amour qu'elle porte, malgr\u00e9 tout, \u00e0 son \u00ab Joe ch\u00e9ri 3 \u00bb, et les naissances de ses enfants \u2014 uniques \u00e9v\u00e9nements de sa vie.\n\n1857, naissance du premier fils, Frank. 1862, Frederik. Deux ans plus tard, l'a\u00een\u00e9e tombe malade. Fanny meurt brutalement \u00e0 l'\u00e2ge de neuf ans. Sarah est inconsolable. Les mois passent, et elle accouche d'un troisi\u00e8me fils. C'est lui, Herbert George \u2014 deux pr\u00e9noms vite oubli\u00e9s au profit d'un diminutif taquin. \u00ab Bert \u00bb (et parfois \u00ab Bertie \u00bb) est un miracle vivant : cadeau du ciel, espoir de la famille, futur mod\u00e8le d'\u00e9ducation et d'ob\u00e9issance. Mais le \u00ab petit gar\u00e7on trapu au nez retrouss\u00e9 4 \u00bb et \u00e0 la \u00ab t\u00eate couleur de lin 5 \u00bb ne se laisse pas dompter. La cuill\u00e8re d'huile de foie de morue s'avale tous les matins sans trop de difficult\u00e9s, mais pour la bagarre avec les enfants du village, c'est lui qui d\u00e9cide. Une t\u00eate br\u00fbl\u00e9e, voil\u00e0 ce qu'il est au fond. Qu'on essaie un peu de lui dicter sa conduite.\n\n\u00c0 la maison, il s'ennuie ferme. Il d\u00e9teste cet \u00ab int\u00e9rieur mesquin et besogneux 6 \u00bb, ce \u00ab trou triste 7 \u00bb qui leur sert de foyer et d'o\u00f9 ils ne sortent quasiment jamais. \u00c0 une douzaine de miles de l\u00e0 grouille l'\u00e9nergie londonienne, facilement accessible par le train. Mais l'obscurit\u00e9 du 47 High Street ne laisse pas le temps de r\u00eaver \u00e0 ces choses-l\u00e0. Il y a de la boue dans la cour et des punaises dans les lits. C'est l\u00e0, dans une puanteur et une humidit\u00e9 que la m\u00e9moire ne peut effacer, que Wells a vu le jour le 21 septembre 1866. Les murs sont sordides, le sol crasseux, les pi\u00e8ces insignifiantes. Le salon n'en est pas un, et ce qui pourrait faire office de biblioth\u00e8que n'est qu'une construction \u00ab d\u00e9labr\u00e9e 8 \u00bb \u2014 mais l'une de celles qu'on a envie tout de m\u00eame d'escalader, quand les parents ont le dos tourn\u00e9.\n\nDans la cour d\u00e9saffect\u00e9e, le gar\u00e7on r\u00eave de grandes batailles, mais n'a rien pour jouer aux soldats. Souvent, il descend au sous-sol, dans la cuisine, pr\u00e8s de l'unique fen\u00eatre qui lui permet d'observer les pieds des passants. C'est l\u00e0 certainement qu'il apprend \u00e0 aimer les femmes \u2014 fantasmant des heures enti\u00e8res sur leurs jupes, imaginant le galbe de leurs jambes, la douceur de leur peau, l'odeur de leur cou \u2014 et qu'il fait ses premiers pas en lecture. Contre l'un des murs de la pi\u00e8ce est affich\u00e9e une grande feuille sur laquelle sont \u00e9crits des lettres majuscules et des chiffres. Tous les jours, il les r\u00e9p\u00e8te, consciencieusement, \u00e0 haute voix, avec sa m\u00e8re. Les journ\u00e9es passent lentement, ponctu\u00e9es par des repas qui interdisent toute discussion. Une fois sortis de table, le soir, Joseph fume sa pipe en silence, et Sarah s'assoit pr\u00e8s du feu pour repriser les vieux v\u00eatements. Elle raccommode tout, rapi\u00e8ce, coupe et recoud les endroits ab\u00eem\u00e9s des pantalons et des chaussettes. Si seulement elle pouvait ne plus s'acharner \u00e0 rendre les choses moins laides, cesser de s'\u00e9puiser \u00e0 sauver les apparences. Elle r\u00eave tant de l'aisance bourgeoise, de l'insouciance de ceux qui ont de l'argent. La pauvret\u00e9 lui est insupportable, et encore davantage si elle devient visible aux autres. Alors il faut faire comme si elle n'existait pas. Ce qui se passe \u00e0 l'int\u00e9rieur des murs de la maison de l'Atlas doit rester secret : la pestilence de la pi\u00e8ce \u00e0 vivre comme le manque de pain. Bert et ses fr\u00e8res ne doivent rien laisser entendre de leur quotidien et ont l'interdiction formelle de jouer avec les enfants des ouvriers : ne pas dire qu'ils sont pauvres, et surtout ne pas donner l'occasion aux voisins de s'en douter.\n\nSur la table de nuit de Sarah est pos\u00e9 le livre de Miss Strickland, _Les Reines d'Angleterre_ \u2014 son ouvrage pr\u00e9f\u00e9r\u00e9. La grande Victoria, sur le tr\u00f4ne depuis plus de vingt ans, l'int\u00e9resse par-dessus tout. Parfois, lorsque le convoi royal passe par chance \u00e0 Chislehurst, tout pr\u00e8s de Bromley, Sarah se fait belle et emm\u00e8ne ses fils. Au milieu de la foule venue comme elle admirer les fastes de la cal\u00e8che, elle sourit et tape des mains, pressant Bert d'enlever son chapeau. Lui ne s'explique pas les origines de cette admiration, ni sa propre hostilit\u00e9 envers ce que sa m\u00e8re lui impose d'aimer. Mais il ne s'en plaint pas, trop \u00e9tonn\u00e9 de voir cette \u00ab femme fatigu\u00e9e 9 \u00bb, abattue par les d\u00e9ceptions quotidiennes et vivant dans l'espoir impossible d'un jour meilleur, soudainement heureuse.\n\nL'autre raison de vivre de Sarah est Dieu. Bert ne comprend pas pourquoi il faut le prier matin et soir, et lui ob\u00e9ir le reste du temps. \u00ab J'ai senti le manque de fondement de ces choses avant que j'aie os\u00e9 le penser 10 \u00bb, \u00e9crit-il bien plus tard dans son autobiographie. L'omnipotence pr\u00e9sum\u00e9e de cet \u00eatre invisible lui donne tr\u00e8s t\u00f4t le sens du doute. Peu \u00e0 peu, il entrevoit ce qui s'apparente pour lui \u00e0 une imposture. Prier un dieu qui n'emp\u00eache pas les guerres de se faire, les in\u00e9galit\u00e9s de se produire et les enfants de mourir, \u00e0 quoi cela pourrait-il servir ? Dieu n'est rien d'autre qu'un \u00ab mouchard 11 \u00bb, un \u00ab \u00e9pouvantail 12 \u00bb, \u00ab une image d'amabilit\u00e9 incongrue, arrang\u00e9e, un m\u00e9lange monstrueux d'homme et d'infini, qui dupe les \u00e2mes simples par de vagues promesses de miracles 13 \u00bb. Quant au diable, il l'aper\u00e7oit un jour en ouvrant au hasard un vieux livre. \u00c0 l'int\u00e9rieur, deux pages sont coll\u00e9es ensemble. Intrigu\u00e9, le gar\u00e7on approche l'ouvrage de la bougie, et voit au travers une illustration \u00e9tonnante : des hommes sont jet\u00e9s au feu par une immense cr\u00e9ature qui tient une fourche, et tentent de se d\u00e9battre. Il comprend que sa m\u00e8re a elle-m\u00eame coll\u00e9 ces pages, craignant que son petit dernier n'apprenne l'existence de l'enfer. Trop tard.\n\nBert est sceptique, mais il ne croit pas en rien pour autant. Son imagination, cet espace de \u00ab pure fantaisie 14 \u00bb, est sa seule religion. Dans cette sph\u00e8re bien d\u00e9limit\u00e9e, il aime exercer son esprit \u00e0 l'inenvisageable. \u00ab Ni impossible, ni incroyable 15 \u00bb : le credo de Joseph Davis \u2014 le h\u00e9ros d' _Enfants des \u00e9toiles_ , l'un des romans les moins connus de l'auteur \u2014 est celui de Wells petit, qui a la conviction farouche que le monde ne se limite pas \u00e0 la perception qu'on lui en donne ; qu'autre chose existe probablement, quelque part. La machine \u00e0 explorer le temps ne lui est pas encore apparue, ni la potentialit\u00e9 d'une quatri\u00e8me dimension, mais la n\u00e9cessit\u00e9 du r\u00eave l'habite d\u00e9j\u00e0. Il se repr\u00e9sente l'espace comme un grand n\u00e9ant sombre, il s'interroge sur l'hypoth\u00e8se d'un gigantesque tremblement de terre \u00e0 l'origine du monde et se demande constamment si la Terre n'est vraiment \u00ab qu'une pointe d'\u00e9pingle 16 \u00bb dans tout ce noir. L'enfant qui r\u00eave d'explorations n'accorde vraiment aucune importance \u00e0 la lumi\u00e8re divine : il aime d\u00e9j\u00e0 celle des \u00e9toiles.\nCoup de chance\n\nLa place pr\u00e9f\u00e9r\u00e9e de Bert est le canap\u00e9 du salon. Depuis peu, il y a exceptionnellement \u00e9lu domicile \u00e0 la suite d'un accident qui allait bouleverser son existence. Tout a commenc\u00e9 quand Sutton, le fils d'un commer\u00e7ant de Bromley, l'a jaug\u00e9 du regard, sans raison. L'apr\u00e8s-midi se d\u00e9roulait tranquillement, tout le monde assistait \u00e0 une partie de cricket, il faisait beau, on s'amusait. \u00ab T'as un probl\u00e8me, Sutton ? \u00bb Non, il veut juste lui montrer qui est le plus fort. Cinq minutes plus tard, Bert se retrouvait presque empal\u00e9 sur le pieu d'une tente, le tibia en sang. Mais la souffrance fut \u00e9ph\u00e9m\u00e8re. Au fond de lui, il jubilait. Il savait que cette blessure lui assurait d'\u00eatre dorlot\u00e9 par sa m\u00e8re. Il pourrait ainsi r\u00e9clamer tous ses mets pr\u00e9f\u00e9r\u00e9s \u2014 gel\u00e9e, fruits, p\u00e2t\u00e9 de cochon et poulet. Le voil\u00e0 donc install\u00e9 \u00ab princi\u00e8rement 1 \u00bb, et cela pour plusieurs semaines, dans la pi\u00e8ce \u00e0 vivre du foyer, la t\u00eate appuy\u00e9e contre un gros coussin et les yeux dirig\u00e9s avec fiert\u00e9 sur sa cicatrice au mollet. Tous les jours, il re\u00e7oit de la visite, la m\u00e8re de Sutton, notamment, qui culpabilise \u00e0 la place de son fils et redouble d'attention pour lui, arrivant les bras charg\u00e9s de jouets et de livres. \u00c9tat de gr\u00e2ce. Cet \u00ab art d'abandonner ...] un corps recroquevill\u00e9 et insensible[ 2 \u00bb au confort d'un divan lui convient \u00e0 merveille. Surtout, les plaisirs de la lecture s'offrent \u00e0 lui sans qu'il les ait vraiment recherch\u00e9s, et lui procurent une joie immense. Ces livres dans ses mains, quotidiennement, pendant des heures : il ne le sait pas encore, mais c'est le \u00ab coup de chance 3 \u00bb de sa vie.\n\nSon p\u00e8re prend le relais de Miss Sutton et se rend chaque jour \u00e0 Market Square pour emprunter un ou deux bouquins. Le go\u00fbt de son fils pour la litt\u00e9rature l'inqui\u00e8te, il n'y voit aucune utilit\u00e9. Mais la gu\u00e9rison de sa jambe s'\u00e9ternisant, il veut lui faire plaisir. Bert ne retient ni le titre ni le nom de l'auteur des ouvrages qu'il lit. L'histoire seule l'int\u00e9resse, la vie des personnages, et le cong\u00e9 que son esprit prend, instantan\u00e9ment, de la r\u00e9alit\u00e9. Avant de dormir, il visite le Tibet ou le Br\u00e9sil dans un recueil illustr\u00e9 de gravures sur bois ; \u00e0 son r\u00e9veil, il d\u00e9couvre la parent\u00e9 entre les chats, les tigres et les lions et la nuit d'apr\u00e8s, l'existence du gorille, dans _Illustrated_ _Natural History_ (\u00ab L'Histoire naturelle illustr\u00e9e \u00bb), la r\u00e9f\u00e9rence litt\u00e9raire du Britannique John George Wood. L'image de ce dernier animal le poursuit longtemps dans sa jeunesse, quand il doit monter se coucher le soir et qu'arriv\u00e9 en haut des escaliers, il imagine que la b\u00eate le guette sur le palier sombre. Il n'est pas rare qu'il se mette \u00e0 courir \u00e0 ce moment-l\u00e0, se jetant sur la poign\u00e9e de la porte de sa chambre, fuyant un danger imaginaire. La m\u00eame sc\u00e8ne terrifiante est reproduite, des ann\u00e9es plus tard, dans _L'\u00cele du docteur Moreau_ , lorsque Edward Prendick, rescap\u00e9 d'un naufrage et recueilli par deux savants marginaux sur une \u00eele d\u00e9serte, court dans la jungle pour \u00e9chapper aux hommes-singes et autres cr\u00e9atures immondes, trafiqu\u00e9es par la folie scientifique. C'est une peur r\u00e9elle mais jouissive, l'une de celles qui font \u00ab compl\u00e8tement perdre la t\u00eate 4 \u00bb :\n\nTant que je vivrai, je me rappellerai la terreur de cette poursuite. Je courais au bord des flots et j'entendais de temps en temps le clapotis des pas qui gagnaient sur moi. Au loin, d\u00e9sesp\u00e9r\u00e9ment loin, brillait faiblement la lueur jaune. La nuit, tout autour de nous, \u00e9tait noire et muette. Plaff ! Plaff ! faisaient continuellement les pieds de mon ennemi. ...] Nous courions ainsi sous les \u00e9toiles tranquilles[ 5.\n\n\u00c0 cette \u00e9poque, Bert aime les sensations fortes et la vie des grands hommes. Il emploie d'ailleurs chacune de ses journ\u00e9es \u00e0 se familiariser avec les \u00e9v\u00e9nements de l'Histoire. La guerre de S\u00e9cession n'a bient\u00f4t plus de secrets pour lui, tout comme les travaux de Washington Irving, les aventures de Christophe Colomb et celles du duc de Wellington. Mais aucun de ces livres n'a le prestige de la collection \u00ab Punch \u00bb \u2014 des p\u00e9riodiques augment\u00e9s de dessins politiques et de caricatures qui permettent au gar\u00e7on de d\u00e9velopper son sens critique et son go\u00fbt pour le dessin. Peu \u00e0 peu, son id\u00e9e du monde et des interactions entre les hommes se pr\u00e9cise : les empereurs fran\u00e7ais l'inspirent, le tigre du Bengale l'interroge, l'oncle Sam l'amuse.\n\nC'est aussi l'\u00e9poque des premiers fantasmes. Dans ces journaux, les pays apparaissent sous la forme de magnifiques d\u00e9esses \u2014 Britannia, Colombia ou la France sont des femmes avant d'\u00eatre des territoires, \u00ab bras nus, cous nus, montrant de beaux seins nus, laissant voir des cuisses brillantes et portant des v\u00eatements qui \u00e9taient une r\u00e9v\u00e9lation \u00e0 l'\u00e9poque des volants et des crinolines 6 \u00bb. Wells tient entre ses mains des corps sublimes et g\u00e9n\u00e9reux. La femme est l\u00e0, tout pr\u00e8s de lui, et l'accompagne en r\u00eave, lui faisant vivre d'\u00e9tranges \u00ab \u00e9treintes ensommeill\u00e9es 7 \u00bb. Le coup de chance est donc double, car la d\u00e9couverte de la lecture l'am\u00e8ne \u00e0 celle du sexe : dans cette \u00ab liaison unilat\u00e9rale avec son matelas 8 \u00bb, le gar\u00e7on explore clandestinement les d\u00e9bordements de son d\u00e9sir, avec cette heureuse conviction \u2014 qui ne le quittera plus \u2014 de ne pouvoir trouver le repos que dans les bras voluptueux d'une belle cr\u00e9ature. Le plaisir est d'autant plus intense qu'il l'\u00e9prouve contre le Saint-Esprit maternel qui loge dans la chambre \u00e0 c\u00f4t\u00e9. Sa jambe enfin gu\u00e9rie, Bert est n\u00e9anmoins press\u00e9 par ses parents d'abandonner ses lectures pour d'autres occupations. Mais le ver est dans le fruit. Jamais plus l'ancien bless\u00e9 ne passera un jour sans tourner les pages d'un livre.\n\nD'autres lectures obligatoires le freinent dans cette ascension des monts litt\u00e9raires. Son professeur, Thomas Morley, l'oriente un temps vers l'histoire de l'\u00c9gypte ancienne. Cet \u00c9cossais sans dipl\u00f4me a cr\u00e9\u00e9 sa propre \u00e9cole au coin de la South Street, \u00e0 quelques m\u00e8tres de la maison des Wells. Sa classe compte une petite trentaine d'\u00e9l\u00e8ves, fils de bourgeois d\u00e9sargent\u00e9s ou de fermiers imp\u00e9cunieux, ne pouvant pr\u00e9tendre \u00e0 un meilleur enseignement. L'institution est v\u00e9tuste, et fonctionne selon le \u00ab vieux syst\u00e8me 9 \u00bb, contrairement au d\u00e9veloppement de nouvelles \u00e9coles d'\u00c9tat (les \u00ab British Schools \u00bb ou \u00ab Natural Schools \u00bb) qui b\u00e9n\u00e9ficient d'une approche in\u00e9dite, dict\u00e9e par la loi de 1871 sur l'\u00e9ducation primaire. Mais Sarah croit fermement que l'Acad\u00e9mie de Bromley peut encore former les esprits. Elle y inscrit donc ses fils surnomm\u00e9s par les enfants des autres \u00e9coles \u2014 plus prestigieuses \u2014 \u00ab les bouledogues de Morley 10 \u00bb. Bert serre les dents, dissimulant tant qu'il peut son \u00ab c\u0153ur rancunier 11 \u00bb. Un jour, le chien pourra montrer les crocs parce qu'il fera partie des meilleurs.\n\nMorley est col\u00e9rique mais bienveillant. Et il est dr\u00f4le : c'est un homme qui alterne constamment entre une rigueur excessive et un laxisme d\u00e9raisonnable. Son invariabilit\u00e9 vestimentaire fait jaser ses \u00e9l\u00e8ves mais sa canne en bois fait peur. L'\u00e9ventualit\u00e9 d'une frappe sur les doigts n'est pourtant pas la punition la plus redout\u00e9e. La gageure, pour ces jeunes \u00e9coliers, est d'\u00e9viter la souffrance de se tenir en \u00e9quilibre, debout sur un banc, les bras hauts, en soutenant une pile de livres, sans jamais montrer un signe d'\u00e9puisement. Et puis, il y a des moments plus sereins, quand les siestes incongrues du professeur permettent aux \u00e9coliers de jouer aux quilles. Au creux de ces minutes de pure libert\u00e9, la gloire revient au plus audacieux, \u00e0 celui qui ose faire quelques pas de danse ou aller dessiner sur le tableau noir. T\u00e9m\u00e9raire, Bert tente \u00e0 chaque fois l'aventure.\n\nMorley n'a qu'un but : faire de ses \u00e9l\u00e8ves de futurs employ\u00e9s de bureau \u2014 des comptables, si possible. Mais il a d'autres projets pour Bert, le seul de la classe qui trouve gr\u00e2ce \u00e0 ses yeux. Il d\u00e9c\u00e8le chez lui une curiosit\u00e9 inhabituelle, un d\u00e9sir insatiable d'\u00e9vasion, une pr\u00e9disposition pour les sciences aussi. Bert excelle en math\u00e9matiques, en trigonom\u00e9trie : il a \u00ab un cerveau bon pour les grandes lignes 12 \u00bb. Morley tente de lui apprendre le fran\u00e7ais, sans succ\u00e8s. Les r\u00e8gles grammaticales et la conjugaison des verbes d\u00e9couragent l'\u00e9l\u00e8ve qui d\u00e9veloppe \u00e0 cet instant une peur noire \u00ab des voyelles dans toutes les autres langues 13 \u00bb. Quand il ne travaille pas, il dessine avec fr\u00e9n\u00e9sie sur ses cahiers : des formes nettes, des visages expressifs, des constructions d\u00e9taill\u00e9es. Cette pratique se d\u00e9ploie dans une d\u00e9sob\u00e9issance toute particuli\u00e8re : Bert peut enfin tenir sa plume comme il l'entend sans se soucier de son professeur qui l'oblige, en classe, \u00e0 l'immobiliser sous une certaine inclinaison. Jusqu'alors, l'acte d'\u00e9crire \u00e9tait \u00e9trangement \u2014 physiquement \u2014 douloureux pour l'enfant. D\u00e9sormais, la main court avec aisance sur le papier.\nL'esprit guerrier\n\nDans la cour de l'\u00e9cole, il y a un endroit o\u00f9 Bert joue au cricket apr\u00e8s la classe, avec son camarade Sydney Bowkett. Ils peuvent passer des heures tous les deux, absorb\u00e9s par les tactiques de chacun, attentifs au moindre mouvement, enterrant par la m\u00eame occasion le souvenir d'une dispute qui les avait s\u00e9par\u00e9s pendant plusieurs mois. Une broutille que le temps a effac\u00e9e. Maintenant, ils sont redevenus \u00ab copains 1 \u00bb, rien ne peut dissoudre leur amiti\u00e9, ils se ressemblent tellement et surtout sont persuad\u00e9s d'\u00eatre tous les deux \u00ab dou\u00e9s de qualit\u00e9s exceptionnelles 2 \u00bb. Dans leur esprit se forment les m\u00eames r\u00eaves de grandeur. Ils br\u00fblent les heures \u00e0 inventer des histoires, \u00e0 imaginer des d\u00e9cors en plein c\u0153ur de l'Himalaya ou de l'Afrique centrale, et s'y mettent en sc\u00e8ne, cr\u00e9ant des personnages qui portent leur nom. Une fois revenus \u00e0 la r\u00e9alit\u00e9, ils courent dans les bois, explorent la nature, capturent des animaux, ou attendent la tomb\u00e9e de la nuit afin de sauter les barri\u00e8res pour p\u00e9n\u00e9trer dans les jardins du voisinage, \u00ab pour voir un peu 3 \u00bb. Mais quand il s'agit de parler aux filles, les deux amis sont moins loquaces. Ils regardent l'amour avec suspicion et le baiser avec d\u00e9go\u00fbt. Non seulement cela ne sert \u00e0 rien, mais ce n'est pas en courant apr\u00e8s les demoiselles qu'on mesure le courage d'un homme. Le vrai d\u00e9fi consiste \u00e0 immobiliser son doigt dans une flamme pendant une trentaine de secondes. La peau br\u00fble, mais les esprits se gonflent de fiert\u00e9. S'ils peuvent supporter cette douleur, ils sont pr\u00eats \u00e0 entreprendre de grandes choses.\n\nMis \u00e0 part Sydney, il n'y a personne d'autre, surtout pas ses fr\u00e8res. L'a\u00een\u00e9, Frank, ne s'int\u00e9resse qu'au bricolage et ne se laisse pas approcher. Freddy, le deuxi\u00e8me fils, est moins redoutable, mais n'\u00e9tudie pas dans la m\u00eame \u00e9cole que Bert. Le d\u00eener est le seul moment de la journ\u00e9e o\u00f9 les trois gar\u00e7ons sont r\u00e9unis, sans pouvoir cependant dire quoi que ce soit : ils ont l'interdiction formelle de poser une question, et m\u00eame de parler du beau temps. Chacun mange dans son coin, en silence, puis se l\u00e8ve de table et s'isole. Provocateur, turbulent, Bert s'occupe tout seul et ne partage pas ses jouets avec ses fr\u00e8res.\n\nSarah laisse faire, comme toujours. Son astuce pour calmer les esprits, c'est d'emmener Bert, d\u00e8s qu'elle le peut, en dehors de la maison. Ils vont au Crystal Palace, un b\u00e2timent construit pour l'Exposition universelle de 1851, qui tr\u00f4ne au milieu de Hyde Park, \u00e0 Londres. Dans les jardins qui entourent la b\u00e2tisse sont expos\u00e9s des moulages de pl\u00e2tre repr\u00e9sentant des dinosaures et toutes sortes de cr\u00e9atures disparues qui attisent la curiosit\u00e9 du gar\u00e7on. Sa m\u00e8re lui explique qu'il s'agit l\u00e0 d'animaux \u00ab ant\u00e9diluviens 4 __ \u00bb. Bert s'interroge. Comment cet ichtyosaure de dix m\u00e8tres de long, sorte de grand dauphin aux nageoires dynamiques, aurait-il pu se noyer lors du D\u00e9luge ? Soit sa m\u00e8re lui ment, soit l'animal en question ne devait pas \u00eatre si fort que \u00e7a. Le gar\u00e7on se procure alors le volume du _Cosmos_ d'Alexander von Humboldt *1, s'informe sur toutes les esp\u00e8ces animales et les \u00e8res g\u00e9ologiques. D'autres questions le taraudent dont celle, incessante, de l'origine du monde. Sa m\u00e8re, ne sachant que lui dire, se d\u00e9barrasse comme elle peut de la r\u00e9ponse. \u00ab Un jour, on saura tout cela 5 \u00bb, lui dit-elle sans trop y croire.\n\nMais Bert est impatient, maladivement curieux et surtout, il pense \u00eatre immortel. Rien ni personne ne peut l'atteindre : il est fort et invuln\u00e9rable, comme son pays, l'Angleterre, le territoire de la R\u00e9forme \u00e0 l'origine d'un immense empire \u00e0 travers le monde. Lui aussi veut participer au cours de l'Histoire, prendre part au destin de l'humanit\u00e9. Il r\u00eave de \u00ab devenir un grand dictateur militaire comme Cromwell, un grand r\u00e9publicain comme George Washington, ou comme Napol\u00e9on dans sa premi\u00e8re phase 6 \u00bb. Et lorsque son camarade Sydney ne peut pas jouer avec lui certains apr\u00e8s-midi, Bert arpente seul les rues de Bromley, parlant \u00e0 voix haute sans s'en rendre compte, livrant bataille, donnant l'ordre \u00e0 ses soldats de lancer les canons. Il se cache derri\u00e8re les arbres, met sa main en visi\u00e8re et regarde, au loin, l'objet fantasm\u00e9 de sa conqu\u00eate. Les habitants du village observent avec perplexit\u00e9 ce jeune adolescent marchant au pas militaire, agitant ses bras dans le vide et chuchotant des phrases incompr\u00e9hensibles. Dans sa t\u00eate se d\u00e9roule un combat plus r\u00e9el que jamais :\n\nLes ennemis \u00e9taient fauch\u00e9s par milliers. Ils se dispersaient sur les pentes raides. Et venait la contre-attaque ; ma cavalerie et moi nous balayions les masses rompues vers Croydon ; nous les poursuivions sans merci \u00e0 travers une nuit sanglante, et enfin leurs restes vaincus se rendaient piteusement vers l'aube pr\u00e8s du Keston Fish Pond. Et j'entrais \u00e0 cheval dans les villes conquises ou sauv\u00e9es \u00e0 la t\u00eate de mes troupes, et mes cousins et mes camarades, se penchant aux fen\u00eatres, me reconnaissaient avec surprise. Et les rois et les pr\u00e9sidents et les puissants de cette terre venaient saluer en moi le sauveur. J'\u00e9tais simple m\u00eame dans la victoire. Je prenais des d\u00e9cisions sages et fermes au sujet de la morale et des m\u0153urs et surtout de ces grands magasins qui avaient men\u00e9 mon p\u00e8re \u00e0 la faillite. Envers des ennemis inv\u00e9t\u00e9r\u00e9s, les monarchistes, les catholiques, les non-Aryens et d'autres, je me montrais juste et s\u00e9v\u00e8re. C'\u00e9tait un travail r\u00e9barbatif \u2014 mais c'\u00e9tait mon devoir **** 7...\n\nL'effusion de sang, la proximit\u00e9 de l'adversaire, le risque incessant de la mort et, bien s\u00fbr, le plaisir de la gloire, l'ivresse de la reconnaissance : la guerre est un jeu, mais surtout une joie dont Bert d\u00e9finit les r\u00e8gles pr\u00e9cises dans un petit cahier qu'il tient secret.\n\nJoseph ferme les yeux sur ces gamineries. Il est l'exact contraire de ce que son fils r\u00eave de devenir. Pas d'h\u00e9ro\u00efsme ni de courage chez lui : un peu de l\u00e2chet\u00e9, et beaucoup de paresse. C'est un homme plut\u00f4t rustre et solitaire, ne cessant d'\u00e9laborer de nouveaux projets mais manquant cruellement d'ambition. Sa vie passe, emp\u00eatr\u00e9e dans un perp\u00e9tuel inach\u00e8vement. Pour Wells, c'est \u00ab un rat\u00e9 8 \u00bb, rien d'autre. Le p\u00e8re et le fils ne partagent rien d'intime, leur entente est tout juste cordiale. Un \u00e9v\u00e9nement les rapproche. Un dimanche matin, Joseph \u00e9lague comme \u00e0 son habitude le fameux pied de vigne dans la cour de la maison, et tombe. L'accident est grave, Joseph ne peut plus marcher normalement, ce qui compromet sa seule activit\u00e9 \u2014 le cricket, la principale source de revenu de la famille. Sa longue convalescence et les d\u00e9penses colossales li\u00e9es aux soins m\u00e9dicaux appauvrissent le foyer et conduisent Sarah \u00e0 prendre une d\u00e9cision : les enfants doivent commencer \u00e0 travailler. Frank est d\u00e9j\u00e0 plac\u00e9 en tant qu'apprenti chez un drapier. Freddy suit le m\u00eame chemin, et bient\u00f4t Bert. Les \u00ab trois t\u00eatards sortis de l'eau avant que leurs poumons et leurs pattes puissent fonctionner convenablement 9 \u00bb sont pr\u00e9cipit\u00e9s dans la vie active pour quelques grappes de raisin.\n\nL'existence telle que Bert l'envisage \u2014 celle qui rime avec lectures, r\u00e9flexions et cr\u00e9ation \u2014 s'\u00e9loigne peu \u00e0 peu de son esprit. Sa m\u00e8re ne lui laisse pas m\u00eame le temps d'y r\u00eaver en le pla\u00e7ant, d\u00e8s 1880, chez un drapier de Windsor, \u00ab Rodgers and Denyer \u00bb, en tant qu'apprenti. Lui qui ne vit que pour la po\u00e9sie de Walter Scott et les _Voyages_ du capitaine Cook se retrouve derri\u00e8re la caisse d'une boutique d'\u00e9toffes, debout, les mains qui s'emm\u00ealent, les pieds qui tiraillent, attendant qu'un client passe la porte. Dix heures par jour, il plie les tissus, rend la monnaie, nettoie les vitres. Le soir, il rejoint un dortoir qui abrite tous les autres commis. Une dizaine de lits align\u00e9s, \u00e9troits, grin\u00e7ants, conduisent \u00e0 quatre lavabos dans un coin. Extinction des feux \u00e0 dix heures et demie. Le matin, il se lave \u00e0 l'eau froide et ne mange pas \u00e0 sa faim, mais ne proteste jamais. Il accomplit toutes les t\u00e2ches qui lui incombent et r\u00e9cup\u00e8re son maigre salaire \u00e0 la fin de la semaine \u2014 quelques pi\u00e8ces qui ne lui appartiennent pas, qu'il doit envoyer \u00e0 sa m\u00e8re pour les besoins du foyer. Bert, \u00e0 quatorze ans, ressemble d\u00e9j\u00e0 \u00e0 un \u00ab marxiste int\u00e9gral 10 \u00bb. Dans un essai publi\u00e9 en 1920, dans lequel il affirme toute sa haine du penseur allemand dont il esp\u00e8re un jour \u00ab raser 11 \u00bb la barbe, il reconna\u00eet avoir ressenti une indignation proche des disciples de l'auteur du _Capital_ :\n\nJ'avais \u00e9t\u00e9 soudain arrach\u00e9 \u00e0 l'\u00e9cole, enferm\u00e9 dans une abominable boutique, et l'on me dressait \u00e0 une vie de labeur mesquin et fastidieux. On m'imposait un travail si dur, et de si longues heures, que toute id\u00e9e de progr\u00e8s personnel paraissait d\u00e9sesp\u00e9r\u00e9ment hors de port\u00e9e. J'aurais mis le feu \u00e0 la boutique si je n'avais \u00e9t\u00e9 convaincu qu'elle \u00e9tait assur\u00e9e au-del\u00e0 de sa valeur 12.\n\nCette d\u00e9sertion forc\u00e9e du cadre scolaire va hanter son \u0153uvre litt\u00e9raire qui se dresse tout enti\u00e8re contre ce qui emp\u00eache l'esprit de se former. Wells a vu l'enseignement lui \u00e9chapper, il a \u00e9t\u00e9 cet assoiff\u00e9 de connaissances, cet enfant perdu dans la nuit \u00e0 la recherche de \u00ab la flamme immortelle \u00bb. C'est le titre qu'il donnera \u00e0 l'un des plus beaux plaidoyers pour l'\u00e9ducation qu'il d\u00e9die \u00e0 \u00ab tous les ma\u00eetres et ma\u00eetresses d'\u00e9cole \u00bb, en 1919 :\n\nQuelle est ici-bas ...] la t\u00e2che de celui qui enseigne ? C'est la plus haute de toutes les t\u00e2ches humaines. C'est d'assurer la croissance de l'Homme, de l'Homme divin, dans les \u00e2mes humaines. Qu'est-ce qu'un homme, sans instruction ? Il na\u00eet comme naissent les b\u00eates, compos\u00e9 d'\u00e9go\u00efsme avide et d'\u00e2pre d\u00e9sir, cr\u00e9ature de passions et de craintes. Il ne sait consid\u00e9rer toutes choses que dans leurs relations avec lui-m\u00eame. L'amour m\u00eame n'est pour lui qu'un march\u00e9. Son maximum d'effort n'est que vanit\u00e9, puisqu'il doit mourir. Et il n'y a que nous, les \u00e9ducateurs, qui puissions l'\u00e9lever au-dessus de cette pr\u00e9occupation de lui-m\u00eame. Nous, les \u00e9ducateurs... Nous pouvons le lib\u00e9rer et l'introduire dans un cercle d'id\u00e9es plus large, qui le d\u00e9passe et dans lequel il puisse enfin s'oublier compl\u00e8tement, lui et ses mis\u00e9rables fins personnelles. Nous pouvons lui ouvrir les yeux sur le pass\u00e9 et l'avenir, et l'immortelle vie de l'Humanit\u00e9. Et ainsi, gr\u00e2ce \u00e0 notre intervention, il \u00e9chappera \u00e0 la mort et \u00e0 la futilit\u00e9. Un homme sans instruction n'est que lui tout seul, aussi isol\u00e9 dans ses fins et sa destin\u00e9e que n'importe quelle b\u00eate ; un homme initi\u00e9 est un homme lib\u00e9r\u00e9 de cette \u00e9troite prison du moi et participant d\u00e9sormais \u00e0 une vie immortelle qui a commenc\u00e9 nous ne savons quand, et qui se d\u00e9veloppe au-dessus et au-del\u00e0 de l'immensit\u00e9 des \u00e9toiles[ 13...\n\nLa prison, la futilit\u00e9, la mort. Elles sont l\u00e0, autour de lui, mis\u00e9rable drapier post\u00e9 au milieu des tissus. Au fond de son ventre, la r\u00e9bellion gronde, comme chez son personnage Alfred Polly qui envisage aussi de tout faire br\u00fbler : ces \u00e9toffes qu'il n'aime pas vendre, ces outils qu'il d\u00e9teste manipuler, cette vie qu'il n'a pas choisie. L'histoire de cet homme en pleine crise existentielle, trop l\u00e2che pour s'ouvrir la gorge mais juste assez d\u00e9sesp\u00e9r\u00e9 pour prendre la route, constitue l'un des romans 14 les plus bouleversants de l'auteur \u2014 l'un de ses pr\u00e9f\u00e9r\u00e9s aussi. \u00c0 l'instar de ce h\u00e9ros en qu\u00eate de r\u00e9ponses, Wells a toujours cherch\u00e9 un moyen de se lib\u00e9rer du cadre qu'on lui imposait. La clef demeure, \u00e0 ses yeux, le savoir, qu'il souhaite inoculer, tel un \u00ab virus 15 \u00bb, \u00e0 toute l'humanit\u00e9. Mais l'adolescent n'a pas encore en sa possession cette seringue magique. Dans son morne quotidien, l'unique \u00e9chappatoire est le r\u00eave et sa routine ne l'emp\u00eache pas de nourrir secr\u00e8tement sa \u00ab vie int\u00e9rieure 16 \u00bb. Derri\u00e8re son comptoir, Bert est la copie conforme \u2014 l\u00e0 aussi \u2014 de Hoopdriver, le h\u00e9ros du roman _La Burlesque \u00c9quip\u00e9e du cycliste_ , drapier de son \u00e9tat : un homme qui porte une poign\u00e9e d'\u00e9pingles sur sa veste mais dont les jambes discr\u00e8tement muscl\u00e9es indiquent qu'il n'est sur Terre que pour p\u00e9daler. La vie de cet aventurier contrari\u00e9 est une perp\u00e9tuelle attente du jour de cong\u00e9, qui lui permettra de d\u00e9valer les collines des environs sur sa \u00ab machine 17 \u00bb d'acier.\n\nLe r\u00eave de Bert n'a pas la forme d'une bicyclette : il est plus rectangulaire. Les semaines passant, l'apprenti trouve une cachette au sous-sol pour ses livres, et quand la boutique se vide, il descend, s'assoit derri\u00e8re les piles de tissus et s'abandonne \u00e0 tous ces mondes parall\u00e8les. La litt\u00e9rature est son rem\u00e8de \u00e0 l'ennui. Elle est \u00ab une r\u00e9v\u00e9lation 18 \u00bb, elle indique les autres possibilit\u00e9s d'existence, l'espace pr\u00e9serv\u00e9 derri\u00e8re les \u00ab apparences superficielles 19 \u00bb. Et s'il ne parvient pas \u00e0 finir un livre dans son terrier, il le glisse dans sa poche et l'emm\u00e8ne, le dimanche, chez son oncle Pennicott, propri\u00e9taire d'une petite auberge \u00e0 Surly Hall, tout pr\u00e8s de son lieu de travail. L\u00e0-bas, l'adolescent apprend \u00e0 godiller, joue avec ses cousines (Kate et Clara) et tombe pour la premi\u00e8re fois sous le charme d'une femme \u2014 une c\u00e9l\u00e8bre actrice, Ellen Terry, qu'il invite \u00e0 faire un tour de barque en lui offrant des fleurs. Le retour \u00e0 Windsor est souvent difficile, surtout lorsqu'un lundi matin, son patron l'accuse de voler dans la caisse. Bert d\u00e9ment, explique que l'argent manquant n'est que le r\u00e9sultat de ses fautes d'inattention, et se fait renvoyer.\n\nSarah est tr\u00e8s inqui\u00e8te. Portant \u00e0 bout de bras le foyer, elle pensait ne plus avoir \u00e0 se soucier du sort de Bert et pouvoir compter sur lui pour renflouer les comptes. Elle-m\u00eame a d\u00e9laiss\u00e9 la boutique de vaisselle, v\u00e9ritable gouffre financier, et s'est fait engager dans la propri\u00e9t\u00e9 de son ancienne ma\u00eetresse \u00e0 Up Park, r\u00e9cup\u00e9r\u00e9e depuis peu par Miss Frances Bullock. La voici de retour dans cet immense domaine, pr\u00e8s de vingt ans apr\u00e8s l'avoir quitt\u00e9, le m\u00eame bonnet en dentelle viss\u00e9 sur la t\u00eate, habill\u00e9e d'une robe de soie noire et d'un tablier blanc. Son mari et ses enfants viennent parfois lui rendre visite le dimanche. Joseph vit maintenant tout seul dans la maison de l'Atlas.\n\nBert aime se rendre \u00e0 Up Park pendant les vacances. Il conna\u00eet tous les recoins de cette fabuleuse b\u00e2tisse qu'il d\u00e9crit, plus tard, dans son roman _Tono Bungay_. La grande maison \u00ab grise 20 \u00bb aux cent dix-sept fen\u00eatres, situ\u00e9e pr\u00e8s de South Haring, est entour\u00e9e d'un parc et fascine le gar\u00e7on qui en imprime dans sa m\u00e9moire les moindres contours. Il arpente des heures enti\u00e8res les couloirs tentaculaires du rez-de-chauss\u00e9e, ceux qui relient la cuisine \u00e0 l'escalier de service, et qui m\u00e8nent plus loin au grand salon. Traverser cette longue pi\u00e8ce, bord\u00e9e de lourds rideaux de soie, est une exp\u00e9rience unique. Plus on avance, mieux on distingue la sculpture pos\u00e9e sur l'immense chemin\u00e9e tout au fond : Romulus et Remus aux pieds de la louve. Ce d\u00e9dale d'alc\u00f4ves, de boudoirs et de corridors annonce d\u00e9j\u00e0 le \u00ab monde souterrain 21 \u00bb et labyrinthique des Morlocks, ces cr\u00e9atures visqueuses et carnivores que l'Explorateur du Temps, \u00e0 la recherche de sa machine perdue, rencontre dans les t\u00e9n\u00e8bres.\n\nMais l'architecture d'Up Park lui donne surtout un aper\u00e7u des relations entre les hommes. Ce domaine est un microcosme de soci\u00e9t\u00e9, une construction parfaitement r\u00e9fl\u00e9chie dans laquelle chaque individu a une place bien d\u00e9finie \u2014 un rang qu'il doit tenir et qui d\u00e9termine son identit\u00e9. Les domestiques vivent en bas et ne montent dans les \u00e9tages que lorsqu'ils y sont invit\u00e9s. Bert, fid\u00e8le \u00e0 sa vocation de pionnier, se fond dans toutes les strates de l'\u00e9difice, nouant des amiti\u00e9s avec les servantes et regardant les privil\u00e9gi\u00e9s des hauteurs avec un m\u00e9lange de suspicion et d'envie. En 1939, il met en mots une obsession qui est d\u00e9j\u00e0 la sienne en cette ann\u00e9e 1881 :\n\nIl y avait les classes sup\u00e9rieures que l'on respectait, et les classes inf\u00e9rieures que l'on ne respectait pas, et les pauvres \u00e9taient oblig\u00e9s de travailler car c'est ainsi qu'allait le monde. Plus je pourrais m'approcher des classes sup\u00e9rieures, mieux cela vaudrait pour moi 22.\n\nCurieux, toujours un mot \u00e0 la bouche, jouant de son l\u00e9ger accent de la campagne, il se faufile dans chaque pi\u00e8ce. Il aime particuli\u00e8rement l'impressionnante biblioth\u00e8que de Sir Harry Feathertonhaugh dans laquelle il d\u00e9couvre les ouvrages de Thomas Paine, Jonathan Swift, Plutarque, Platon et Voltaire. Les chambres des gouvernantes sont aussi son terrain de jeu : il y organise souvent des repr\u00e9sentations th\u00e9\u00e2trales clandestines (jouant \u00e0 lui seul tous les r\u00f4les d'une histoire invent\u00e9e pour l'occasion). Un jour, une temp\u00eate de neige s'abat sur la r\u00e9gion, l'obligeant \u00e0 rester confin\u00e9 dans la chambre de sa m\u00e8re d\u00e9sesp\u00e9r\u00e9e de le voir d\u00e9ambuler sur son lieu de travail. Alors, pour passer le temps, il cr\u00e9e un journal, \u00ab The Up Park Alarmist \u00bb, qu'il distribue \u00e0 tout le personnel du domaine. Tout pr\u00e8s de cette m\u00eame pi\u00e8ce, dans un coin inhabit\u00e9 de la maison o\u00f9 s'entassent de vieux meubles, Bert trouve un t\u00e9lescope pos\u00e9 sur son tr\u00e9pied qu'il rapporte dans sa chambre et installe pr\u00e8s de la fen\u00eatre. Le soir suivant, l'\u0153il coll\u00e9 au viseur, il regarde la Lune dans la nuit glaciale. Que distingue-t-il \u00e0 travers la lunette ? Wells n'a laiss\u00e9 aucune trace de cette premi\u00e8re rencontre avec l'espace. Mais il est ais\u00e9 d'imaginer son excitation, sa perplexit\u00e9 aussi, face \u00e0 cette chose indescriptible. La beaut\u00e9 de l'astre a d\u00fb lui appara\u00eetre furtivement \u2014 la m\u00eame beaut\u00e9 qui fut d\u00e9taill\u00e9e des ann\u00e9es plus t\u00f4t par Victor Hugo, autre observateur nocturne. \u00ab C'\u00e9tait trouble, fugace, impalpable \u00e0 l'\u0153il, pour ainsi parler. Si rien avait une forme, ce serait cela 23. \u00bb Ces deux-l\u00e0 ont ressenti le m\u00eame vertige, ont \u00e9t\u00e9 envahis par la m\u00eame \u00ab perte du r\u00e9el 24 \u00bb, ont pens\u00e9 la m\u00eame chose, \u00e0 plusieurs dizaines d'ann\u00e9es d'intervalles :\n\nCe songe \u00e9tait une terre. Probablement, on \u2014 qui ? \u2014 marchait dessus, on allait et venait dans cette chim\u00e8re ; ce centre conjectural d'une cr\u00e9ation diff\u00e9rente de la n\u00f4tre \u00e9tait un r\u00e9cipient de vie ; on y naissait, on y mourait peut-\u00eatre 25...\n\nPuis la Lune se voile, le songe se referme, et Bert est envoy\u00e9 \u00e0 Wookey, dans le Somerset, en tant qu'\u00e9l\u00e8ve-ma\u00eetre dans une \u00e9cole primaire. Un proche de sa m\u00e8re, Alfred Williams \u2014 l'\u00e9poux de la s\u0153ur de Pennicott \u2014, enseigne \u00e0 la National School. Quand Bert le rencontre pour la premi\u00e8re fois, il est marqu\u00e9 par son nez pointu, le crochet qui lui fait office de bras, et la \u00ab __ quantit\u00e9 extraordinaire de cheveux qui pousse dans ses oreilles 26 \u00bb. Cet homme lui pla\u00eet. Il est farouchement oppos\u00e9 \u00e0 toute forme de religion et a un sens de l'humour tr\u00e8s d\u00e9velopp\u00e9. Jusqu'alors, l'enfance de Bert s'\u00e9tait d\u00e9roul\u00e9e dans une sorte de pesanteur, les jours tombaient comme des pierres, lourds et bruyants. Soudain, le voil\u00e0 \u00e0 l'int\u00e9rieur d'une \u00e9cole \u2014 un lieu qu'il avait renonc\u00e9 \u00e0 r\u00e9int\u00e9grer un jour \u2014, il apprend le m\u00e9tier de professeur et est enfin consid\u00e9r\u00e9 comme \u00ab une grande personne 27 \u00bb.\n\nIl loge chez Williams, avec l'une de ses cousines, de deux ans son a\u00een\u00e9e. La jeune fille n'a peur de rien et parle de tout, sans aucune censure. Bien d\u00e9cid\u00e9e \u00e0 initier l'adolescent \u00e0 la sexualit\u00e9, elle l'emm\u00e8ne souvent dans les bois, ils discutent des heures durant des diff\u00e9rentes mani\u00e8res d'embrasser ou de faire l'amour. Ces conversations sont tr\u00e8s \u00ab instructives 28 \u00bb pour lui et aboutissent \u00e0 des d\u00e9buts de mise en pratique. Bert n'y prend aucun plaisir. \u00ab La chose r\u00e9elle, comme elle me fut alors pr\u00e9sent\u00e9e, me parut une affaire inconfortable, qui donnait chaud, et un peu honteuse 29. \u00bb L'exp\u00e9rience tourne court car Williams, son logeur, se fait vite \u00e9carter de l'enseignement pour le motif qu'il ne poss\u00e8de pas de titre pour exercer. Bert doit partir. Il rassemble donc ses affaires et rejoint, de nouveau, sa m\u00e8re \u00e0 Up Park. Avant m\u00eame qu'il ne pose ses valises, Sarah lui a d\u00e9j\u00e0 trouv\u00e9 une place en tant qu'apprenti pharmacien, dans la commune de Midhurst. La pension est ch\u00e8re, Wells ne comprend pas le latin \u2014 indispensable pour exercer ce m\u00e9tier \u2014 et il d\u00e9cide donc de d\u00e9missionner, un mois seulement apr\u00e8s avoir pris son poste.\n\nLe temps de trouver une autre place, Sarah l'inscrit \u00e0 la _grammar school_ de Midhurst, un coll\u00e8ge dirig\u00e9 par Horace Byatt qui prend sous son aile le jeune Wells. Ce professeur dublinois a une dizaine d'\u00e9l\u00e8ves dans sa classe. Il leur apprend le latin et plusieurs disciplines scientifiques. Bert retrouve le sourire, travaille tant qu'il peut, apprend ses le\u00e7ons avec acharnement, noircit des cahiers entiers, \u00e9crit m\u00eame une histoire intitul\u00e9e \u00ab La bataille de Bungeldum \u00bb, que Byatt expose sur un chevalet face aux \u00e9l\u00e8ves, tant il est impressionn\u00e9 par les capacit\u00e9s du nouveau venu. Enfin, Bert a trouv\u00e9 sa place, il \u00e9tudie comme les autres gar\u00e7ons de son \u00e2ge, il a acc\u00e8s \u00e0 tout ce qui peut calmer ses \u00ab yeux pleins d'une interrogation farouche 30 \u00bb. Indiff\u00e9rente \u00e0 ses succ\u00e8s, Sarah a toujours en t\u00eate de lui faire embrasser, d\u00e9finitivement, la profession de drapier, et parvient \u00e0 le placer \u00e0 Southsea, dans l'atelier de M. Edwin Hyde. Apr\u00e8s seulement quelques mois, Bert doit quitter l'\u00e9cole qu'il avait eu tant de mal \u00e0 retrouver et d\u00e9m\u00e9nage, le c\u0153ur lourd. De toute fa\u00e7on, il doit se rendre \u00e0 l'\u00e9vidence : lui et ses fr\u00e8res sont \u00ab pris comme des rats 31 \u00bb.\n\nIl y a de la d\u00e9tresse dans ce d\u00e9part, l'impression d'entrer dans une prison dont il ne pourra jamais sortir. De la fatigue aussi : toujours partir, \u00eatre en mouvement, ne jamais s'attacher \u00e0 un endroit ni \u00e0 personne. Accoud\u00e9 \u00e0 la fen\u00eatre du dortoir qu'il partage avec ses nouveaux coll\u00e8gues, il observe, dehors, la vie suivant son cours et ceux qui ont la libert\u00e9 de d\u00e9cider : \u00ab Ma derni\u00e8re chance s'\u00e9tait \u00e9vanouie. \u00c0 ce moment-l\u00e0 je ne pouvais d\u00e9couvrir ni en moi-m\u00eame, ni dans le monde, tel qu'il se pr\u00e9sentait dans cette rue \u00e9troite que je regardais ...] la plus faible possibilit\u00e9 d'une nouvelle fuite[ 32. \u00bb Il ne peut pas se plaindre : les conditions de travail sont plut\u00f4t bonnes, il est bien nourri, et gagne de l'argent. Mais quand en aura-t-il termin\u00e9 avec ces pi\u00e8ces de satinette qui ne cessent d'aller de travers ? Avec celles en cretonne qui ne se laissent pas mater ? Quand est-ce que son patron, le redoutable John Key, arr\u00eatera-t-il de le r\u00e9primander inutilement ? Cela aura-t-il m\u00eame une fin ? \u00c9tait-ce cela la vie ? \u00ab Quitte ce m\u00e9tier, avant qu'il ne soit trop tard, quitte-le \u00e0 tout prix 33 \u00bb, voil\u00e0 ce qu'il se r\u00e9p\u00e8te chaque soir avant de s'endormir.\n\n\u00ab Changer la vie. \u00bb Wells n'a pas encore lu Rimbaud \u2014 _Une saison en enfer_ a paru outre-Manche dix ans plus t\u00f4t, en 1873 \u2014, mais il prononce sans le savoir le m\u00eame credo po\u00e9tique. \u00ab Si tu veux quelque chose, et si tu la veux vraiment, prends-la, et fiche-toi des cons\u00e9quences. ...] Si ta vie n'est pas satisfaisante, change-la ; ne supporte jamais une vie terne et triste, car la pire des choses qui peut t'arriver, si tu luttes et si tu continues \u00e0 lutter pour t'\u00e9chapper, c'est de subir une d\u00e9faite ; et cela n'est jamais tout \u00e0 fait certain jusqu'\u00e0 la fin, qui est la mort, et la fin de tout[ 34 _._ \u00bb Rien d'impossible donc. Le fils \u00e9crit \u00e0 sa m\u00e8re, tentant de la persuader que le m\u00e9tier de drapier ne lui pla\u00eet pas, qu'il veut retourner \u00e0 l'\u00e9cole, que son avenir, apr\u00e8s tout, lui appartient. Sarah ne veut rien entendre, trop occup\u00e9e \u00e0 pr\u00e9parer la \u00ab confirmation \u00bb de son fils comme membre de l'\u00c9glise anglicane. Bert courbe le dos une fois de plus et ob\u00e9it \u00e0 l'injonction maternelle. Mais s'il n'a pas encore le courage de rompre avec sa m\u00e8re, il a celui de provoquer le vicaire de Portsmouth. Face \u00e0 lui, le jeune homme s'avance avec de gros sabots en affirmant croire \u00e0 la th\u00e9orie darwinienne de l'\u00e9volution, et avouant s'interroger s\u00e9rieusement quant \u00e0 la possibilit\u00e9 du p\u00e9ch\u00e9 originel. L'effet attendu se produit. Le vicaire manque de s'\u00e9touffer. Bert repart satisfait et avec la ferme intention de se vanter de ses exploits. De retour au dortoir, il anime le d\u00e9bat aupr\u00e8s de ses camarades de chambr\u00e9e, agite ses bras, mart\u00e8le que l'enfer n'est qu'une pure construction, un mensonge invent\u00e9 \u00ab pour effrayer les gosses de dix ans 35 \u00bb, et que le besoin de Dieu se manifeste chez tous ceux qui n'ont pas assez de courage pour affronter la vie. Les pas du patron se font entendre dans l'escalier de service. Il faut dormir, et bient\u00f4t, les lumi\u00e8res s'\u00e9teignent. Mais la \u00ab r\u00e9volte de son] esprit[ 36 \u00bb est d\u00e9j\u00e0 en marche et ne s'arr\u00eatera plus. Un jour, \u00e0 l'occasion du sermon d'un pr\u00e9dicateur qu'il juge particuli\u00e8rement mauvais, il constate, depuis le fond de l'\u00e9glise o\u00f9 il peut tout observer : \u00ab J'avais honte d'\u00eatre assis l\u00e0, dans ce bain de cr\u00e9dulit\u00e9 37. \u00bb\n\nUn temps, il pense prendre la fuite. Il en discute avec son fr\u00e8re Frank qui lui-m\u00eame n'en peut plus de cette existence sans perspective. D'un caract\u00e8re moins affirm\u00e9 que son benjamin, Frank d\u00e9cidera de tout quitter, mais quinze ans plus tard. Bert, lui, est press\u00e9, d\u00e9sesp\u00e9r\u00e9. Il pense m\u00eame au suicide \u2014 il en \u00e9bauchera l'\u00ab anatomie \u00bb, bien apr\u00e8s, dans un essai encore in\u00e9dit en fran\u00e7ais 38. La rimbaldie n'est pas loin, cette fougue d\u00e9sesp\u00e9r\u00e9e de l'esprit qui ne veut pas se contenter de la normalit\u00e9 ou du convenable. S'il ne vit pas _vraiment_ , \u00ab \u00e0 quoi bon la vie 39 ? \u00bb. Bert, pr\u00eat \u00e0 mettre son existence dans la balance, d\u00e9cide d'aller trouver sa m\u00e8re.\n\nUn dimanche de l'hiver 1882, \u00e0 l'aube, il se faufile hors du dortoir, passe la grille et parcourt \u00e0 pied dans la brume les dix-sept miles qui le s\u00e9parent d'Up Park. Le ventre vide, il tient son col de veste qui le prot\u00e8ge du froid. Il risque une s\u00e9v\u00e8re correction \u00e0 son retour. Mais la d\u00e9sob\u00e9issance est jouissive et cette escapade matinale lui donne de l'\u00e9nergie. En arrivant \u00e0 destination, il pose un ultimatum \u00e0 sa m\u00e8re : soit elle le laisse faire ce qu'il veut, soit il se tue. C'est la premi\u00e8re fois qu'il s'oppose \u00e0 elle, qu'il exprime une quelconque volont\u00e9. Sarah, effray\u00e9e, finit par c\u00e9der, mais lui fait comprendre qu'\u00e0 son sens, il commet une grave erreur. C'est gagn\u00e9. Bert reprend le train direction Midhurst. Il est heureux, pense \u00e0 la t\u00eate que fera le terrible John Key lorsqu'il va lui annoncer sa d\u00e9mission, et fredonne sur un air joyeux ces quelques phrases improvis\u00e9es :\n\nRousp\u00e8te, rousp\u00e8te, vieux J. K., vieux J. K., vieux J. K.\n\nLe sacr\u00e9 gosse _est_ parti, _est_ parti, _est_ parti.\n\nLe sacr\u00e9 gosse est parti, est parti, et pour toujours 40.\n\n*1. Explorateur et g\u00e9ographe allemand.\n\u00c0 l'\u00e9cole\n\nFin de l'ann\u00e9e 1882. Robert Louis Stevenson vient de publier _L'\u00cele au tr\u00e9sor_. James Joyce et Virginia Woolf sont n\u00e9s. Charles Darwin est mort. Et Wells, seize ans tout juste, a l'\u00e2ge d'endosser le titre d' _assistant master_ \u00e0 Midhurst, l'\u00e9cole dirig\u00e9e par Byatt.\n\nLog\u00e9 chez une vendeuse de confiserie, il est appliqu\u00e9, ne manquant aucun cours, travaillant d'arrache-pied sur ses le\u00e7ons et ne s'accordant de r\u00e9pit que pour dormir. Dans sa chambre \u00e0 l'\u00e9tage, tout n'est que livres et feuilles volantes. Au-dessus de sa table de travail, sur son mur, il a accroch\u00e9 un tableau dessin\u00e9 de sa main pour organiser au mieux son temps. Il craint l'oisivet\u00e9, la procrastination, et n'a qu'un seul but en t\u00eate : se \u00ab bourrer de connaissances 1 \u00bb, rattraper le temps perdu. Quand il n'\u00e9tudie pas, il sort marcher avec Harris, le jeune homme qui partage sa chambre. La cadence est sportive : quatre miles par heure minimum, pour exercer le souffle et muscler les jambes. Toujours un livre en poche, il lui arrive de s'allonger en pleine nature, quelques instants, et de d\u00e9vorer un chapitre. C'est ainsi qu'il lit _La R\u00e9publique_ de Platon. Il aime ce philosophe comme \u00ab un fr\u00e8re 2 \u00bb. C'est lui qui l'initie \u00e0 ses premi\u00e8res id\u00e9es politiques. Bert est notamment s\u00e9duit par la perspective d'une soci\u00e9t\u00e9 construite sur l'int\u00e9r\u00eat commun (une d\u00e9couverte qu'il d\u00e9signe comme sa \u00ab premi\u00e8re rencontre avec l'id\u00e9e communiste 3 \u00bb), mais aussi par la fonction des gardiens, en charge de la cit\u00e9, et par la place donn\u00e9e aux esprits \u00e9clair\u00e9s (les philosophes), \u00ab \u00e9pris de ...] science[ 4 \u00bb, \u00ab totalement d\u00e9pourvus de cupidit\u00e9 5 \u00bb et dot\u00e9s d'une \u00ab excellente m\u00e9moire 6 \u00bb. Il y a aussi l'ouvrage d'Henry George, _Progr\u00e8s et pauvret\u00e9_ , gr\u00e2ce auquel il se familiarise avec la critique du capitalisme. En revanche, il n'a pas l'intention de lire Marx, \u00ab ce th\u00e9oricien renferm\u00e9, \u00e9gocentrique et malveillant 7 \u00bb \u00e0 la \u00ab philosophie pr\u00e9tentieuse 8 \u00bb. Dans son esprit se forment les contours d'un socialisme qu'il souhaite salvateur, qui rendrait v\u00e9ritablement les hommes heureux.\n\n\u00c0 dix-huit ans, ses interrogations sur l'Univers n'ont pas faibli, ni celles concernant la religion. Mais le jeune homme a pass\u00e9 sa confirmation \u00ab en toute soumission 9 \u00bb, pour les besoins de son travail \u2014 l'ensemble du personnel enseignant de Midhurst devait faire partie de l'\u00c9glise d'Angleterre. Son ath\u00e9isme ne diminue pas, bien au contraire, et il ne cesse de penser \u00e0 la sexualit\u00e9, c\u00f4toyant r\u00e9guli\u00e8rement les jeunes filles des alentours \u2014 vendeuses ou jeunes apprenties du village. Leurs accoutrements lui d\u00e9plaisent et l'emp\u00eachent de voir ne serait-ce qu'un bout de peau \u2014 un bras ou une jambe. Et puis ces femmes ne se laissent pas approcher facilement \u2014 ou bien c'est lui qui, trop intimid\u00e9, n'ose pas faire le premier pas. Bert pr\u00e9f\u00e8re parler aux prostitu\u00e9es, mais n'a pas les moyens de s'offrir un moment avec elles : \u00ab un shilling par semaine comme argent de poche ne laisse pas de marge pour l'amour v\u00e9nal 10 \u00bb.\n\nTous les matins, il porte son unique cravate rouge sur des chemises us\u00e9es et des pantalons trop courts. Puis il se rend dans sa classe, un groupe de petits gar\u00e7ons qu'il \u00e9duque comme il peut. Mais son int\u00e9r\u00eat n'est pas l\u00e0 : Byatt veut qu'il travaille la g\u00e9ologie, la chimie \u00e9l\u00e9mentaire, les math\u00e9matiques, toutes ces mati\u00e8res qui lui permettront de passer les examens de fin d'ann\u00e9e et qui, s'il les r\u00e9ussit, rapporteront au directeur de belles sommes d'argent. Bert est re\u00e7u \u00e0 toutes les \u00e9preuves d'\u00e9valuation, et se voit proposer de postuler pour une place d'\u00e9tudiant boursier \u00e0 l'\u00c9cole normale des sciences de South Kensington. Byatt ne l'y encourage pas, car il veut absolument le garder \u00e0 ses c\u00f4t\u00e9s. Bert remplit donc le formulaire bleu sans en parler \u00e0 personne, l'envoie. La r\u00e9ponse arrive un matin : il est choisi pour assister aux cours du c\u00e9l\u00e8bre professeur Huxley (le grand-p\u00e8re de l'auteur du _Meilleur des mondes_ ). Byatt d\u00e9sapprouve ce d\u00e9part, soutenu par Sarah, \u00e9videmment, qui voit en Huxley un dangereux ath\u00e9e. Bert ne les \u00e9coute pas et fait ses valises.\n\nAvant le d\u00e9but des cours, il a quelques semaines devant lui et en profite pour rentrer \u00e0 la maison de l'Atlas. Il n'a pas vu son p\u00e8re depuis longtemps. Il trouve un homme \u00e9puis\u00e9, difficilement soutenu par une canne, vivant dans une maison n\u00e9glig\u00e9e depuis que Sarah ne s'en occupe plus. Bert arrive un matin, au milieu du salon en d\u00e9sordre, et salue son p\u00e8re d'un sourire. La douceur de ces retrouvailles est inhabituelle. Il a d\u00e9sormais de la tendresse pour cet homme triste qui n'a rien accompli dans sa vie. Pour la premi\u00e8re fois, le p\u00e8re f\u00e9licite le fils. Ils apprennent \u00e0 se conna\u00eetre. Ces jours pass\u00e9s ensemble sont d'un grand r\u00e9confort pour Bert. Ils font de longues promenades, jouent aux \u00e9checs et parlent des femmes. Une certaine Mary, fille de cuisine d'Up Park, commence \u00e0 hanter les nuits de Wells. Elle est d'une \u00ab beaut\u00e9 indescriptible 11 \u00bb et bondit sur lui un soir d'hiver, l'embrassant dans les couloirs de l'immense propri\u00e9t\u00e9, alors qu'il rend visite \u00e0 sa m\u00e8re pour No\u00ebl. Bert, plaqu\u00e9 contre le mur, trop surpris pour faire le moindre mouvement, ne parvient pas \u00e0 la retenir. Il ne l'a jamais revue, mais garde intact le souvenir de cette \u00e9treinte, du c\u0153ur de la jeune fille qui bat contre sa poitrine, de \u00ab son corps souple dans sa l\u00e9g\u00e8re robe jaune 12 \u00bb. Il ne peut plus se passer de ce nouveau besoin charnel, ne sachant comment vivre avec une telle tension en lui. Il ne r\u00eave pas de mariage, ni d'enfant, mais d'une femme \u00e9nigmatique, libre et puissante, qui ressemblerait en tout point \u00e0 celle des Samoura\u00efs \u2014 cet ordre de l'excellence d\u00e9fini en 1905 dans son inqui\u00e9tant roman _Une utopie moderne_.\n\nCe fantasme le poursuit lorsqu'il s'installe \u00e0 South Kensington. Il est d'abord log\u00e9 chez une amie de sa m\u00e8re. Wells sympathise avec cette nouvelle famille, et noue des liens avec la s\u0153ur du mari de sa logeuse. Cette fois-ci, pas de g\u00eane, les mains se baladent sous les v\u00eatements, mais la jeune fille r\u00e9siste, emp\u00eachant l'exp\u00e9rience de se poursuivre. Puis il part s'installer chez sa tante et son oncle au 181 Euston Road, une maison peu confortable, d\u00e9pourvue de salle de bains, mais au c\u0153ur de laquelle il va avoir la chance de rencontrer le premier amour de sa vie : Isabel Mary Wells, sa cousine. Elle a le m\u00eame \u00e2ge que lui, \u00ab un visage grave et charmant, des traits fermement model\u00e9s, un front large, une bouche particuli\u00e8rement ravissante, un joli cou, et un joli menton 13 \u00bb. Tous les jours, il se rend \u00e0 pied \u00e0 South Kensington en sa compagnie \u2014 elle travaille comme retoucheuse \u00e0 deux rues de l\u00e0.\n\n\u00c0 l'\u00c9cole normale des sciences, l'unique b\u00e2tisse de brique rouge d'Exhibition Road, Bert s'inscrit sur le registre, au milieu d'autres \u00e9tudiants, et se dirige vers le laboratoire de biologie, son terrain de jeu pour les prochaines ann\u00e9es \u00e0 venir. C'est un jour important pour lui, l'\u00ab un des grands jours de s]a vie[ 14 \u00bb, dont il retrace les lignes principales en 1911 dans _L'Amour et M. Lewisham_ \u2014 l'histoire partiellement autobiographique d'un \u00e9tudiant qui int\u00e8gre cette prestigieuse universit\u00e9 et qui d\u00e9couvre, dans le m\u00eame temps, l'amour... Voici donc Bert entour\u00e9 de microscopes, de mat\u00e9riel de dissection et de nombreux tiroirs qui renferment divers sp\u00e9cimens et comptes rendus d'exp\u00e9riences. Et chose miraculeuse : il va rencontrer Thomas Henry Huxley, l'un des plus c\u00e9l\u00e8bres professeurs du Royaume-Uni.\n\nN\u00e9 en 1825, ce biologiste de renom est un ami de Darwin. Plus : il est son avocat le plus fervent. C'est lui qui a pris la d\u00e9fense du naturaliste lors du d\u00e9bat qui eut lieu \u00e0 Oxford suite \u00e0 la publication de _De l'origine des esp\u00e8ces_. Face \u00e0 l'\u00e9v\u00eaque Samuel Wilberforce, Huxley se vante de descendre du singe et hurle sa foi profonde en la th\u00e9orie de l'\u00e9volution. Wells sait tout cela. Il n'a jamais connu Darwin (d\u00e9c\u00e9d\u00e9 l'ann\u00e9e o\u00f9 lui-m\u00eame arrive \u00e0 South Kensington), mais il a lu ses ouvrages et Huxley lui en parle beaucoup. \u00ab Tous les deux \u00e9taient de tr\u00e8s grands hommes, \u00e9crit Wells, ils pensaient d'une fa\u00e7on courageuse, fouill\u00e9e et simple, ils parlaient et \u00e9crivaient sans r\u00e9serve et d'une mani\u00e8re compr\u00e9hensible ; ils menaient une vie modeste et d\u00e9cente ; ils \u00e9taient tous les deux de puissants lib\u00e9rateurs de l'esprit ....] De petits bonshommes se hisseront sur les \u00e9paules des g\u00e9ants, jusqu'\u00e0 la fin des temps, et de petits oiseaux souilleront les nids o\u00f9 ils ont \u00e9t\u00e9 couv\u00e9s[ 15. \u00bb\n\nL'enseignement d\u00e9livr\u00e9 par Huxley \u2014 et en creux, l'\u00e9tude approfondie des th\u00e8ses de Darwin \u2014 ouvre \u00e0 Wells tout le champ des possibles. Il d\u00e9borde d'enthousiasme, alors m\u00eame que son \u00e9poque est encore frileuse quant \u00e0 l'id\u00e9e que l'homme descendrait du singe... L'Univers lui appara\u00eet tout d'un coup comme un vaste syst\u00e8me biologique. S'il comprend si bien ces deux ma\u00eetres de pens\u00e9e, c'est parce qu'ils le confortent dans l'id\u00e9e d'une \u00e9volution naturelle de l'humanit\u00e9, dissoci\u00e9e des notions religieuses. Ces derniers ont cru au progr\u00e8s. Il fera de m\u00eame. En lui germent d\u00e9j\u00e0 les extravagances scientifiques du docteur Moreau. Wells travaille intens\u00e9ment, passe des heures dans ce laboratoire, absorb\u00e9 par une dissection d'animal, le nettoyage d'un fossile, ou le d\u00e9chiffrage de diverses notes. Aux c\u00f4t\u00e9s de Howes \u2014 le pr\u00e9parateur hyperactif qui l'assiste dans sa t\u00e2che \u2014, il envisage toutes sortes d'exp\u00e9riences qui pourraient attirer l'attention de son mentor, ce \u00ab vieillard au visage carr\u00e9, au teint jauni et aux] yeux bruns et vifs[ 16 \u00bb, celui qu'il regarde comme l'homme qu'il voudrait \u00eatre.\n\nWells passe ses examens avec succ\u00e8s, obtenant la mention de \u00ab premi\u00e8re classe \u00bb. Il partage cette note avec un camarade nomm\u00e9 Jennings, fils d'un instituteur qui devient vite son ami. Tous les deux travaillent des nuits enti\u00e8res au laboratoire, d\u00e9battant jusqu'au petit matin sur des questions qui les divisent. Wells a trouv\u00e9 quelqu'un \u00e0 sa mesure \u2014 ou plut\u00f4t \u00e0 sa d\u00e9mesure : ambitieux et aussi passionn\u00e9 que lui, pr\u00eat \u00e0 tout pour percer le myst\u00e8re de l'univers. Leurs discussions sont anim\u00e9es, ils sont souvent en d\u00e9saccord, mais Jennings r\u00e9ussit, parfois, \u00e0 le convaincre. Il lui est \u00e9galement loyal. Wells habite d\u00e9sormais dans une chambre spartiate et, manquant souvent d'argent, il lui arrive de se priver de manger. L'\u00e9cole sert un go\u00fbter en fin d'apr\u00e8s-midi, il se serre le ventre en attendant. Mais le jour o\u00f9 Jennings d\u00e9couvre cette situation, il part frapper \u00e0 sa porte, d\u00e9croche son manteau et l'emm\u00e8ne dans un petit restaurant au coin de la rue. Un professeur qu'il admire, un v\u00e9ritable ami sur qui il peut compter : cette premi\u00e8re ann\u00e9e ne pouvait mieux se passer.\n\nDans les couloirs aussi, la camaraderie est l\u00e0. On \u00e9change des livres, on d\u00e9bat des le\u00e7ons, on s'exerce \u00e0 l'\u00e9loquence. Wells est de toutes les discussions. Au milieu de ses nouveaux amis \u2014 Arthur Morley Davies, William Burton, Elizabeth Healey, Richard Gregory \u2014, il s'enflamme pour les questions d'\u00e9volution et de religion, ou pour la n\u00e9cessit\u00e9 du socialisme. John Ruskin, William Morris, Goethe et bien d'autres sont au centre de leurs r\u00e9flexions qui vont bient\u00f4t trouver un autre th\u00e9\u00e2tre. Dans une salle faiblement \u00e9clair\u00e9e, au sous-sol de l'universit\u00e9, la Soci\u00e9t\u00e9 des d\u00e9bats voit le jour. Il y a du mat\u00e9riel d'\u00e9tude entass\u00e9 \u00e7\u00e0 et l\u00e0, des chaises cass\u00e9es, des vieux bureaux, et puis l'estrade, petite, bancale, mais sur laquelle les membres du cercle montent \u00e0 tour de r\u00f4le pour prendre la parole. Il faut s'imposer, interrompre les camarades, parler plus fort que l'autre. Les questions li\u00e9es \u00e0 la religion et \u00e0 la politique sont proscrites mais Wells refuse de se soumettre \u00e0 cette r\u00e8gle, car ne pas parler de ces sujets revient pour lui \u00e0 ne pas regarder l'existence en face. Alors tout devient pr\u00e9texte \u00e0 montrer son \u00ab ind\u00e9pendance irr\u00e9v\u00e9rencieuse 17 \u00bb. L'assembl\u00e9e crie au scandale, Wells se met \u00e0 dos ses meilleurs amis, mais tout cela l'amuse. On le fait sortir de force. Quatre \u00e9tudiants l'attrapent par les bras et les jambes, Bert se d\u00e9bat et se retrouve violemment jet\u00e9 au sol. \u00ab D\u00e9sagr\u00e9ment passager 18. \u00bb\n\nLes esprits \u00e9chauff\u00e9s se calment, et la troupe cr\u00e9e le _Science Schools Journal_ , destin\u00e9 \u00e0 tous les \u00e9tudiants de South Kensington. Wells en est le r\u00e9dacteur en chef et le principal contributeur, \u00e9crivant sous de nombreux pseudonymes \u2014 Walter Glockenhammer, Septimus Browne, ou encore Sosthenes Smith 19. Les cours ne comptent plus, les devoirs de sciences s'amoncellent sur son bureau, les bonnes r\u00e9solutions de rentr\u00e9e sont oubli\u00e9es. Et la deuxi\u00e8me ann\u00e9e commence sous des auspices plus fragiles.\n\nL'\u00e9l\u00e8ve brillant, assidu, voire obsessionnel, devient \u00ab le rebelle m\u00e9content, fac\u00e9tieux, inquiet et aga\u00e7ant 20 \u00bb. N'ayant plus Huxley comme professeur, il d\u00e9sapprouve fortement les programmes propos\u00e9s par l'\u00e9cole, et toutes les m\u00e9thodes d'enseignement. Chaque jour est une d\u00e9ception, il est en col\u00e8re contre ce syst\u00e8me p\u00e9dagogique bancal. Sa confiance en lui ne diminue pas, bien au contraire : il persiste dans la voie de ses r\u00eaves, mais se rend chaque matin en cours avec la sensation de perdre son temps. Guthrie a remplac\u00e9 Huxley. La cinquantaine bien avanc\u00e9e, il est \u00ab lourd et lent 21 \u00bb, et n'a aucun talent oratoire. Wells reste dans son coin, r\u00e9pond pr\u00e9sent en cours mais n'\u00e9coute pas. Les bras sont crois\u00e9s, le regard est noir, et la voix haut port\u00e9e. Ne penser qu'\u00e0 s'amuser. Applaudir pour d\u00e9stabiliser un professeur, faire du bruit en soufflant dans un chalumeau en caoutchouc : Bert devient un \u00ab blanc-bec absolument d\u00e9testable 22 \u00bb. Il attend la fin des cours avec impatience pour poser devant l'objectif d'un camarade aux c\u00f4t\u00e9s d'un squelette de singe, le bras pos\u00e9 sur les \u00e9paules du primate, et l'autre main soutenant fi\u00e8rement un cr\u00e2ne d'homme. Au bout de quelques semaines, il gagne sa r\u00e9putation d'agitateur. Il est dans sa bulle, sans volont\u00e9 de s'en extraire, regardant le monde avec distance et amusement :\n\nL'existence est entour\u00e9e au nord et au sud, \u00e0 l'est et \u00e0 l'ouest, en haut et en bas, par le myst\u00e8re. Dans ce cadre se trouve comme une petite maison dans un paysage \u00e9trange, vaste, froid et magnifique, la vie sur notre plan\u00e8te, une vie dont je suis un petit point temporaire, une impression. Il y a des sujets d'int\u00e9r\u00eat sans limites, \u00e0 l'int\u00e9rieur de cette maison, je veux en profiter de toutes mes forces. Quand m\u00eame, on \u00e9prouve de temps en temps le besoin d'en sortir et de scruter ces immensit\u00e9s \u00e9nigmatiques. Mais pour une petite chose comme moi, il n'y a rien \u00e0 faire l\u00e0-dehors. \u00c0 la fin, des apparences m\u00e9taphysiques lointaines peuvent vouloir dire quelque chose, mais aussi loin que vont ma volont\u00e9 pr\u00e9sente et mon activit\u00e9, elles ne veulent rien dire du tout. La physique se r\u00e9tr\u00e9cit jusqu'\u00e0 l'infinit\u00e9simal dans une \u00e9tincelle vacillante, \u00e0 l'int\u00e9rieur d'une ampoule de verre, \u2014 ou elle tourbillonne \u00e0 travers les profondeurs de l'espace avec les n\u00e9buleuses en dehors de la Voie lact\u00e9e ; et apr\u00e8s un temps je cesse de plonger mon regard \u00e0 l'int\u00e9rieur du verre et de scruter les \u00e9toiles, et je rentre \u00e0 l'int\u00e9rieur de la maison 23.\n\nBiologie, stratigraphie, min\u00e9ralogie, p\u00e9trologie, cristallographie. \u00ab Il fallait apprendre tout cela par c\u0153ur 24. \u00bb Judd est le nouveau Guthrie \u2014 monotone, ennuyeux. Huxley est d\u00e9cid\u00e9ment incomparable. Heureusement que les roches, inspect\u00e9es au microscope, \u00e9merveillent les pupilles. Elles sont belles, d'une \u00ab beaut\u00e9 significative 25 \u00bb, mais \u00e0 peine son \u0153il d\u00e9c\u00e8le-t-il leurs courbes et leurs couleurs qu'il faut s'occuper d'autre chose. Manque de temps, travail b\u00e2cl\u00e9, enthousiasme tari. Il y a pire que la d\u00e9ception : il y a le d\u00e9sint\u00e9r\u00eat, le moment pr\u00e9cis o\u00f9 plus rien n'a d'importance. Wells avait pass\u00e9 les grilles de la prestigieuse \u00e9cole avec le r\u00eave qu'il \u00ab apprendrait] tout[ 26 \u00bb, il se rend compte qu'on ne lui enseigne rien. \u00c0 force de dilettantisme et d'exasp\u00e9ration, l'\u00e9tudiant \u00e9choue \u00e0 l'examen final de g\u00e9ologie. L'annonce des r\u00e9sultats lui fait l'effet d'un \u00ab ballon piqu\u00e9 d'une \u00e9pingle 27 \u00bb. Il n'a pas de ressources ni de relations, et ne sait comment annoncer la nouvelle \u00e0 ses parents qui ont tant dout\u00e9 de sa r\u00e9ussite universitaire. Mais s'il a \u00e9chou\u00e9 en sciences, il r\u00e9ussira en litt\u00e9rature. Il a la conviction secr\u00e8te, profonde, inexplicable, qu'il a \u00ab l'\u00e9toffe d'un homme de lettres 28 \u00bb. Il quitte donc l'\u00c9cole normale des sciences sans avoir excell\u00e9 nulle part, forc\u00e9 de renoncer \u00e0 une carri\u00e8re scientifique, et cherchant un moyen rapide de gagner sa vie. Il trouve finalement une place de professeur \u00e0 la Holt Academy, \u00e0 Wrexham.\n\nQuitter South Kensington, c'est aussi s'\u00e9loigner d'Isabel. D\u00e8s lors que Wells s'\u00e9tait install\u00e9 sur le campus de l'universit\u00e9, il voyait sa cousine tous les dimanches, l'embrassant en cachette, lui murmurant mille choses : pourquoi il est important de douter de Dieu, de croire en la puissance de l'art et en la possibilit\u00e9 d'une r\u00e9volution sociale. Habill\u00e9 d'une jaquette et d'un chapeau haut de forme, Wells soutient le regard d'Isabel, lui expliquant les interrogations qui le tiennent en vie. Plus ils \u00e9changent, plus il se rend compte que la jeune fille n'a pas la m\u00eame passion que lui pour la lecture. Et quand il veut l'initier aux ouvrages de Ruskin, elle d\u00e9cline poliment. Wells n'insiste pas, mais il esp\u00e8re : \u00ab un jour, d'une fa\u00e7on ou d'une autre, je ferais quelque chose de tr\u00e8s beau, qui aurait un grand succ\u00e8s, et le monde serait \u00e0 mes pieds ; ses restrictions mentales s'\u00e9vanouiraient, et elle se rendrait compte que dans tout ce que je disais, dans tout ce que je faisais et voulais, j'avais eu raison 29 \u00bb.\n\n\u00c0 ce moment, Wells est plus amoureux de l'image qu'il veut renvoyer \u00e0 Isabel que d'Isabel elle-m\u00eame. Il faut absolument que cette femme l'aime \u2014 qu'elle aime ce \u00ab corps scandaleusement maigre 30 \u00bb, ce m\u00e8tre soixante-cinq qui ne p\u00e8se pas plus de cinquante-neuf kilos tout mouill\u00e9, et cet accoutrement de vagabond frott\u00e9 occasionnellement avec une \u00e9ponge. Assez du r\u00eave, de ce d\u00e9sir brid\u00e9, de ces beaut\u00e9s qu'il croise au coin des rues sans imaginer les toucher. Le corps est peut-\u00eatre insignifiant, mais la t\u00eate est bien pleine. La carrure d'un gentleman ne vaut rien face \u00ab \u00e0 l'ambition et au d\u00e9sir d'un bon d\u00e9veloppement intellectuel 31 \u00bb. Oui, il en est certain : un jour, il aura sinon le monde, du moins toutes les femmes \u00e0 ses pieds.\nInvincible\n\nWells \u00e9crit beaucoup, tout ce qui lui passe par la t\u00eate. Le style est peu assur\u00e9 mais les intrigues se tiennent, et son petit public d'intimes lit ses quelques nouvelles avec enthousiasme. S'il osait vraiment, il enverrait un texte au _Family Herald_ qui publie des fictions. Il ose, on le paie trois sous, et ses initiales sont imprim\u00e9es au bas de la page. Sa d\u00e9cision est prise : il sera \u00e9crivain. En attendant, il doit travailler, gagner de l'argent. \u00c0 Wrexham, il donne quelques cours la journ\u00e9e et \u00e9crit la nuit, dans une chambre qu'il partage avec deux pr\u00eatres calvinistes. Quand ses colocataires font leur pri\u00e8re, lui tente d'oublier Isabel (qui ne lui donne aucune nouvelle), et pense \u00e0 la jeune Annie Meredith, la fille du pasteur de la paroisse voisine, avec laquelle il a entam\u00e9 quelques jours auparavant un flirt \u00ab tr\u00e8s vif et anim\u00e9 1 \u00bb. Elle vient le voir jouer au football. Sur le banc, elle l'observe et lui sourit. Bert fait du mieux qu'il peut pour renvoyer le ballon, craignant qu'elle ne se rende compte de la supercherie \u2014 sur le terrain, il est tout sauf dou\u00e9.\n\nCe jour-l\u00e0, la pelouse est humide. La partie a commenc\u00e9 depuis quelques minutes, et Wells tombe d'un coup, percut\u00e9 par un autre joueur. Il se rel\u00e8ve, se pensant juste un peu sali par la boue, et reprend le jeu. Mais la t\u00eate lui tourne, une douleur intense lui engourdit le bas du dos, et il rentre chez lui en boitant. Le m\u00e9decin vient le lendemain et lui annonce que son rein gauche est litt\u00e9ralement \u00e9cras\u00e9. L'histoire se r\u00e9p\u00e8te : plus d'une d\u00e9cennie apr\u00e8s sa jambe cass\u00e9e, le voil\u00e0 de nouveau alit\u00e9, immobile, lisant toute la journ\u00e9e _La Foire aux vanit\u00e9s_ de William Thackeray. Les h\u00e9morragies sont constantes et son mal de dos se double d'une tuberculose chronique. On lui dit que ses jours sont compt\u00e9s. Il y croit, redoute ses derni\u00e8res heures, et regrette d\u00e9j\u00e0 tout ce qu'il n'a pas v\u00e9cu :\n\nJe d\u00e9testais l'id\u00e9e de la mort jusque dans mes os, je la d\u00e9testais furieusement et agressivement. J'\u00e9tais exasp\u00e9r\u00e9 de ne pas pouvoir devenir c\u00e9l\u00e8bre, de ne pas avoir vu le monde. J'\u00e9tais encore plus exasp\u00e9r\u00e9 contre ce reste de retenue en moi, qui mena\u00e7ait de me laisser mourir vierge. Je passais par une r\u00e9surrection enrag\u00e9e du d\u00e9sir sexuel. Un curieux ressentiment s'accumula en moi contre ma cousine Isabel, parce qu'elle n'avait pas d'amour pour moi. J'avais envie de sortir et de poursuivre des femmes inconnues. Je me reprochais mes scrupules, qui m'avaient emp\u00each\u00e9 d'aborder les femmes qui faisaient le trottoir \u00e0 Londres 2.\n\nEt puis, il y a la peur, r\u00e9currente, provoqu\u00e9e par son sang dans sa bouche, l'impression d'\u00ab \u00eatre \u00e9trangl\u00e9, glac\u00e9 et enferm\u00e9 3 \u00bb. La vie s'arr\u00eaterait donc l\u00e0 ? Pas d'exploits ni de notori\u00e9t\u00e9, d'amour ni de sexe ? Aucun \u00ab couronnement 4 \u00bb ? Les semaines passent, et il est toujours en vie. Alors, il fait ses valises et part pour Up Park rejoindre sa m\u00e8re. Son p\u00e8re est depuis peu install\u00e9 dans les environs, pr\u00e8s de Rogate Station. La maison de l'Atlas a \u00e9t\u00e9 vendue, la boutique aussi et No\u00ebl arrive. Le r\u00e9veillon se f\u00eate avec les domestiques, autour de l'immense table de la cuisine. Malgr\u00e9 sa sant\u00e9 fragile, Bert est heureux et vit ces instants comme s'ils \u00e9taient les derniers.\n\nImpossible de savoir combien de temps il lui reste \u00e0 vivre. Une chose est s\u00fbre, il faut combattre le sentiment d'abattement, la m\u00e9lancolie, utiliser ce temps \u00e0 bon escient ; lire et \u00e9crire encore davantage. Shelley, Keats, Heine, Whitman, Hawthorne : aupr\u00e8s de ces po\u00e8tes il comprend que l'\u00e9criture s'apprend. \u00c9merveill\u00e9 par leurs \u0153uvres, il veut les imiter. Mais sous sa plume, les sonnets grincent : \u00e0 peine \u00e9crits, ils sont jet\u00e9s au feu. La forme ne convient pas \u00e0 Bert. Le fond non plus d'ailleurs : lui s'int\u00e9resse surtout aux questions politiques. Depuis son lit, il veut d\u00e9truire ce monde qui ne lui laisse aucune chance. C'est ce qu'il \u00e9crit \u00e0 son amie Elizabeth Healey au d\u00e9but de l'ann\u00e9e 1888 :\n\nLes choses m'ont \u00e9t\u00e9 donn\u00e9es pour que je leur lance des pierres, parfois je m'imagine que c'est ce qui me donne le plaisir le plus profond : jeter des objets contre des objets pour les briser 5.\n\nWells a des r\u00eaves de puissance. La plupart des personnages de ses romans auront les m\u00eames. Hoopdriver (dans _La Burlesque \u00c9quip\u00e9e du cycliste_ ), Kipps (dans le roman \u00e9ponyme), ou encore Alfred Polly (dans _L'Histoire de M. Polly_ ) apparaissent au d\u00e9part tous d\u00e9sarm\u00e9s, vuln\u00e9rables, mais d\u00e9ploieront une force rare, une capacit\u00e9 \u00e0 se battre, quoi qu'il leur en co\u00fbte, contre l'hostilit\u00e9 du monde. Le journaliste britannique Ivor Brown, auteur d'une biographie de H. G. Wells, ne se trompait pas en soulignant l'importance de la rencontre, au d\u00e9but des ann\u00e9es 1920, entre l'\u00e9crivain et Charlie Chaplin :\n\nCe n'est pas un simple caprice qui amena Wells \u00e0 inviter Charlie Chaplin dans sa maison de l'Essex. Les deux hommes ont en commun un sens du comique, un g\u00e9nie comique. La tragi-com\u00e9die du petit homme aux prises avec un grand univers qui le d\u00e9concerte, \u00e0 l'esprit toujours quelque peu embrum\u00e9, et qui est pourtant fondamentalement sup\u00e9rieur \u00e0 toutes les forces m\u00e9caniques, est un th\u00e8me qui leur est cher \u00e0 tous les deux 6.\n\nLe printemps arrive avec ses nouvelles r\u00e9solutions. Wells sort un apr\u00e8s-midi, et part seul marcher dans les bois. Pr\u00e8s d'un arbre, il s'allonge et r\u00e9fl\u00e9chit. \u00ab J'en ai assez de mourir 7. \u00bb C'est l\u00e0, du haut de ses vingt et un ans, au beau milieu des jacinthes sauvages, qu'il revient \u00e0 la vie sans aucune intention d'y renoncer. Vite, rentrer, d\u00e9marcher toutes les \u00e9coles de Londres et des environs. Prendre un nouveau d\u00e9part. En attendant les r\u00e9ponses, Wells \u00e9crit un texte dont il est pleinement satisfait et qu'il s'empresse d'envoyer \u00e0 ses amis de South Kensington, qui ont repris la direction du _Science Schools Journal_. Le r\u00e9cit est rapidement publi\u00e9 en trois \u00e9pisodes. Il a pour titre \u00ab The Chronic Argonauts \u00bb (\u00ab Les Argonautes \u00e0 la conqu\u00eate du temps \u00bb), premi\u00e8re matrice d'une machine qui promet d'explorer le temps.\n\nWells re\u00e7oit une r\u00e9ponse positive de l'universit\u00e9 de Londres, qui l'autorise \u00e0 compl\u00e9ter ses dipl\u00f4mes. Gare de Saint-Pancras, Theobalds Road, et le voici dans une mansarde entre un vieux lavabo et un lit rouill\u00e9. La cloison du mur est mince, il entend tous les soirs le m\u00eame couple faire l'amour. La frustration est \u00e0 son comble, et la concentration difficile. Alors, il trouve la parade : en toussant ou en faisant grincer sa chaise, les \u00e9bats s'arr\u00eatent. Il peut travailler tranquillement et passer ses examens avec succ\u00e8s \u2014 ceux de zoologie et de g\u00e9ologie qui lui manquaient. Le poste d' _assistant master_ qu'on lui propose de prendre quelques semaines plus tard \u00e0 la Henley House School, dans le quartier londonien de Kilburn, tombe \u00e0 point nomm\u00e9, et lui permet de continuer \u00e0 \u00e9crire son premier essai, _The Rediscovery of The Unique_ _(La Red\u00e9couverte de l'unique)_ , une r\u00e9flexion sur l'\u00e9volution de l'humanit\u00e9 qu'il parvient \u00e0 faire publier, en 1891, dans _The Fortnightly Review._\n\nPour la premi\u00e8re fois, Wells gagne assez d'argent pour envisager de s'installer dans un endroit d\u00e9cent. Il veut aussi \u00e9pouser Isabel qu'il n'a pas cess\u00e9 de convoiter. Il l'aime \u2014 ou croit l'aimer \u2014, mais sait que la r\u00e9ciproque n'est pas certaine. Comme le narrateur de _Tono Bungay_ , cr\u00e9\u00e9 des ann\u00e9es plus tard, il se prend parfois \u00e0 r\u00eaver qu'il passe son bras autour de la taille de l'\u00e9lue de son c\u0153ur \u2014 Marion \u2014, un soir dans le m\u00e9tro, et qu'il se penche tout pr\u00e8s de son oreille :\n\n\u00ab Je vous aime \u00bb, murmurai-je, puis, mon c\u0153ur battant \u00e0 se rompre, je l'attirai, vivante image de la beaut\u00e9, vers moi, et je posai ma bouche sur ses l\u00e8vres froides et passives.\n\n\u00ab Vous m'aimez ? dit-elle, en s'arrachant \u00e0 moi. C'est mal ! \u00bb Puis, comme le train entrait en gare : \u00ab Il ne faut le dire \u00e0 personne... Je ne sais pas... Vous n'auriez pas d\u00fb... \u00bb\n\nDeux voyageurs mont\u00e8rent et mirent fin \u00e0 mes d\u00e9clarations.\n\n[...] Quand nous nous rev\u00eemes, elle me d\u00e9clara que je ne devrais jamais refaire \u00ab cela \u00bb.\n\nBaiser ses l\u00e8vres m'\u00e9tait d'abord apparu comme le comble du bonheur. Et pourtant ce ne devait \u00eatre que le commencement de mes d\u00e9sirs. Je lui dis que ma seule ambition \u00e9tait de l'\u00e9pouser.\n\n\u00ab Mais, objecta-t-elle, vous n'avez pas encore de situation \u2013 \u00e0 quoi bon parler de cela ? \u00bb\n\n[...] \u00ab Je le veux, dis-je. [...]\n\nJe n'\u00e9tais qu'\u00e0 quelques centim\u00e8tres des l\u00e8vres que j'avais bais\u00e9es ; [...] je voyais s'ouvrir entre nous un ab\u00eeme d'ann\u00e9es, de labeur, d'attente, de d\u00e9ceptions et d'incertitudes.\n\n\u00ab Je vous aime, dis-je. Ne m'aimez-vous pas ? \u00bb\n\nElle me fixa de ses yeux graves o\u00f9 ne se lisait aucune \u00e9motion.\n\n...] J'aurais d\u00fb m'apercevoir d\u00e8s ce moment que la flamme qui br\u00fblait en moi ne se communiquait pas \u00e0 son \u00e2me. Mais comment aurais-je su ? [...] Je la voulais, stupidement et par pur instinct ****[ 8... \u00bb\n\nIsabel, comme Marion, se laisse finalement aller \u00e0 cet amour. L'engagement est conclu et les fian\u00e7ailles annonc\u00e9es, malgr\u00e9 les caract\u00e8res incompatibles, les passions et les ambitions divergentes. Mais il y a plus grave : Isabel est tr\u00e8s fuyante lorsqu'il s'agit de rapprocher son corps du sien, tandis que Wells ne supporte plus sa virginit\u00e9, et franchit le pas en se mettant \u00e0 aller r\u00e9guli\u00e8rement voir des prostitu\u00e9es. Cela ne change cependant rien \u00e0 ses projets : il est, pour une raison bien obscure, obs\u00e9d\u00e9 par l'id\u00e9e d'\u00eatre mari\u00e9. La c\u00e9r\u00e9monie a lieu \u00e0 l'automne 1891, dans une \u00e9glise de Wandsworth, au sud de la capitale. Wells serre les dents. Pour l'ath\u00e9e qu'il est, se tenir dans la maison d'un dieu auquel il ne croit pas est douloureux. L'un de ses fr\u00e8res, Frank, a fait le d\u00e9placement, et pleure \u00e0 chaudes larmes \u2014 premi\u00e8re et unique d\u00e9monstration d'un attachement fraternel. La nuit de noces est une catastrophe. Wells d\u00e9borde de d\u00e9sir pour sa femme qui se montre tr\u00e8s effray\u00e9e. Elle ne dit rien et pleure. De l'autre c\u00f4t\u00e9 du lit, lui n'arrive pas \u00e0 dormir et s'en veut d'avoir \u00e9t\u00e9 un peu brutal ou pas assez patient. Il faut penser au lendemain, aux copies \u00e0 corriger et aux grenouilles \u00e0 diss\u00e9quer \u2014 tout pour ne pas affronter le d\u00e9sastre de sa vie.\n\nDepuis quelque temps, il enseigne la biologie \u00e0 l'University Correspondence College de Londres, il \u00e9crit dans plusieurs journaux, et r\u00e9fl\u00e9chit sans cesse \u00e0 de nouvelles histoires \u00e0 raconter. Il envoie \u00e0 Harris, le r\u00e9dacteur en chef de la _Fortnightly Review_ , un texte in\u00e9dit, \u00ab L'Univers rigide \u00bb, dans lequel il envisage un monde en quatre dimensions. Sans le lire en entier, Harris l'imprime, se rend compte qu'il est all\u00e9 un peu vite, et convoque le jeune professeur dans son bureau. Wells, s'imaginant que son travail a \u00e9t\u00e9 appr\u00e9ci\u00e9, arrive dans son plus beau costume, d\u00e9goulinant de sueur apr\u00e8s avoir couru, encombr\u00e9 d'un parapluie inutile qu'Isabel lui a dit de prendre avant de partir. Harris le regarde avec s\u00e9v\u00e9rit\u00e9, lui lance son article \u00e0 la figure et lui dit :\n\n\u00ab Vous m'avez envoy\u00e9 cet \"Univers rrr...rrigide\" [...] Mon Dieu ! Je n'en comprends pas le premier mot. Qu'avez-vous voulu dire ? Pour l'amour de Dieu, dites-moi de quoi il s'agit. Quel est le sens de tout cela ? Qu'est-ce que vous avez eu dans la t\u00eate ? [...]\n\n\u2014 Eh bien ! Vous voyez ! dis-je.\n\n\u2014 Je ne vois pas ! dit Harris. C'est justement ce \u00e0 quoi je n'arrive pas.\n\n\u2014 L'id\u00e9e, dis-je. L'id\u00e9e... [...] Si vous consid\u00e9rez que le temps est comme l'espace... alors, \u2014... Je veux dire que si vous le traitez comme une quatri\u00e8me dimension, alors vous voyez...\n\n\u2014 Bon Dieu ! de quelle fa\u00e7on j'ai \u00e9t\u00e9 roul\u00e9 ! interjeta Harris dans un apart\u00e9 avec le bon Dieu. Je ne peux pas l'utiliser. ...] Il faut d\u00e9truire la composition[ 9... \u00bb\n\nWells, pris pour un fou, se fait cong\u00e9dier sans d\u00e9lai. Il rentre chez lui, dans la nouvelle maison qu'il occupe avec Isabel \u00e0 Fitzroy Road, s'enferme dans sa chambre, et d\u00e9cide de ne plus rien \u00e9crire pendant plusieurs mois. Ses amis l'encouragent \u00e0 reprendre la plume, \u00e0 croire en ses id\u00e9es, m\u00eame saugrenues. Wells s'y remet, et re\u00e7oit bien plus tard une lettre de ce m\u00eame Harris lui proposant une collaboration dans son nouveau journal. Entre-temps, Oscar Wilde avait vant\u00e9 les m\u00e9rites de cet esprit certainement en avance sur son \u00e9poque. Wells d\u00e9cline poliment. D'autres projets ont \u00e9merg\u00e9 depuis. Il a publi\u00e9 son premier livre, _Manuel de biologie_ , avec une pr\u00e9face de G. B. Howes, l'assistant de Huxley \u00e0 South Kensington. Et sa vie a chang\u00e9. Sans surprise, Isabel et lui se sont \u00e9loign\u00e9s. Ils ne se parlent presque plus et cohabitent avec une politesse forc\u00e9e, se demandant chacun comment ils ont pu croire \u00e0 la possibilit\u00e9 de leur couple. Inutile de chercher les raisons de ce g\u00e2chis : leur mariage \u00e9tait vou\u00e9 \u00e0 l'\u00e9chec depuis le d\u00e9part.\n\nIl existe une disposition traditionnelle \u00e0 distribuer le bl\u00e2me ou la sympathie \u00e0 propos de chaque rupture entre un homme et une femme. Les gens qui vous racontent leur histoire vous disent qu'il lui \u00e9tait infid\u00e8le, ou qu'elle n'\u00e9tait pas digne de lui, ou qu'elle n'a fait aucun effort et ainsi de suite. Mais dans la plupart des ruptures entre hommes et femmes, le manque d'harmonie existait depuis le commencement ...]. Ma cousine et moi avions \u00e9t\u00e9 pouss\u00e9s l'un vers l'autre par les accidents de la vie[ 10.\n\n1893, la d\u00e9cision est prise. Reste \u00e0 la mettre en pratique. Wells veut quitter Isabel pour les beaux yeux noirs d'une autre jeune femme. Elle s'appelle Amy Catherine Robbins \u2014 \u00ab Jane \u00bb \u2014, c'est l'une de ses \u00e9tudiantes, blonde, affranchie de toute religion, et plut\u00f4t bavarde. Un jour, \u00e0 la fin d'un cours, elle vient lui parler. Les semaines passent, ils d\u00e9battent pendant des heures, \u00e9changent des livres et boivent le th\u00e9. C'est avec elle qu'il veut d\u00e9sormais vivre. Il faut tout avouer \u00e0 Isabel, mettre fin \u00e0 une union qui \u00e9tait vou\u00e9e d\u00e8s le d\u00e9part \u00e0 l'\u00e9chec ; avoir le courage de faire face \u00e0 ses d\u00e9sirs. La t\u00eate est volontaire, r\u00e9solue, mais le corps d\u00e9cline, une fois de plus. Depuis plusieurs jours, Wells est affaibli par une \u00e9ni\u00e8me crise pulmonaire, une mauvaise bronchite qui l'oblige \u00e0 calmer ses ardeurs. Un chemin s'ouvre \u00e0 lui pourtant. Son \u00ab aimable Destin\u00e9e 11 \u00bb lui fait signe. Tandis qu'il esp\u00e8re, sans trop y croire, s'installer avec Jane, un certain W. W. Astor, millionnaire am\u00e9ricain, rach\u00e8te la _Pall Mall Gazette_ , et place le brillant Harry Cust \u00e0 la t\u00eate de la r\u00e9daction. Cust cherche de nouvelles plumes jeunes, audacieuses. Wells tente sa chance, et envoie un texte inspir\u00e9 d'une de ses r\u00e9centes lectures : une nouvelle de l'\u00c9cossais J. M. Barrie (le futur cr\u00e9ateur de _Peter Pan_ ), trouv\u00e9e par hasard dans une librairie. Le personnage de Barrie est justement un journaliste qui puise les sujets de ses articles non pas dans l'actualit\u00e9 br\u00fblante qui l'entoure mais dans les menus d\u00e9tails du quotidien. Wells, qui s'est octroy\u00e9 quelques jours de repos \u00e0 la plage, d\u00e9cide de suivre cet exemple et r\u00e9dige un article au titre amusant : \u00ab Sur le s\u00e9jour au bord de la mer \u00bb. C'est dr\u00f4le, simple, bien \u00e9crit : cela pla\u00eet imm\u00e9diatement. Wells jubile, et reprend des forces.\n\nToutes ces ann\u00e9es pass\u00e9es \u00e0 publier dans des revues \u00e9tudiantes avaient pr\u00e9par\u00e9 le jeune homme \u00e0 un avenir journalistique insoup\u00e7onn\u00e9. Son nouvel employeur ne cesse de le solliciter, Wells est d\u00e9bord\u00e9, il ressort ses vieux travaux datant du _Science Schools Journal_ , des brouillons, des notes jamais utilis\u00e9es. Puisqu'il a carte blanche, il en profite. En moins d'un an, il \u00e9crit une quinzaine d'articles pour la _Pall Mall Gazette_ , et l'argent s'amoncelle. Des sommes importantes qui l'autorisent \u00e0 faire des projets. Sa premi\u00e8re d\u00e9pense n'est pas pour lui, ni pour Jane. Une chose lui tient \u00e0 c\u0153ur : le sort de ses proches. Sa m\u00e8re venant de se faire licencier, tout comme son fr\u00e8re, il les installe avec son p\u00e8re dans une maison qu'il loue en bordure de Londres. Il est heureux de pourvoir \u00e0 leurs besoins, de leur offrir un confort qui leur a toujours \u00e9chapp\u00e9. Ce n'est pas le grand luxe, mais c'est d\u00e9j\u00e0 \u00e7a. Wells donne \u00e0 sa famille ce qu'elle avait il y a si longtemps perdu : le repos et la tranquillit\u00e9. Lui est encore loin de ses r\u00eaves de romancier, mais il s'est fait un nom et s'est gorg\u00e9 d'une confiance in\u00e9branlable qui ne le quittera plus.\n\nToutes les r\u00e9dactions de la capitale cherchent \u00e0 le faire travailler \u2014 la _Saturday Review_ et la _New Review_ , notamment, pour lesquelles il r\u00e9dige des comptes rendus de romans et des critiques dramatiques. Ses textes sont incisifs et son existence se colore bient\u00f4t d'une intransigeance inaccoutum\u00e9e. Il faut saisir le moment et partir, tout de suite. Wells mesure la souffrance qu'il inflige \u00e0 Isabel. Cela fait des mois qu'elle est au courant de sa liaison avec Jane. Elle sait que sa rivale est plus jeune et surtout plus intellectuelle qu'elle ne le sera jamais. Elle a toujours senti que quelque chose la maintenait \u00e0 l'\u00e9cart de son mari, qu'elle avait une lecture et une compr\u00e9hension du monde tout \u00e0 fait diff\u00e9rentes de la sienne. Elle laisse donc la place \u00e0 Jane sans m\u00eame r\u00e9sister.\n\n\u00c0 l'\u00e9poque, le divorce est rare, tr\u00e8s mal vu, et ne peut \u00eatre prononc\u00e9 que si un adult\u00e8re est commis et prouv\u00e9. Il est fortement probable que Wells et Jane avaient consomm\u00e9 leur idylle depuis longtemps, mais il est hors de question pour l'\u00e9crivain de rendre tout cela public. Il en va de sa r\u00e9putation. Sa carri\u00e8re litt\u00e9raire d\u00e9colle tout juste, l'enjeu est trop important pour risquer de tout faire voler en \u00e9clats. Il doit trouver un moyen de concilier son d\u00e9but de gloire avec ce bouillonnement sentimental. Les deux lui sont vitaux.\n\nSans surprise, Joe et Sarah d\u00e9sapprouvent cette dangereuse s\u00e9paration qui expose leur fils aux regards malveillants. Ils ne manquent pas non plus d'accuser Isabel de n'avoir pas \u00e9t\u00e9 \u00e0 la hauteur de ses devoirs d'\u00e9pouse. Peu importe. Wells fait ses valises, embrasse une derni\u00e8re fois Isabel, lui promettant de continuer \u00e0 subvenir \u00e0 ses besoins, et part. Jane l'attend. Ils ont trouv\u00e9 un petit appartement \u00e0 Mornington Place, le quartier londonien de ses ann\u00e9es estudiantines. Il s'en va sans regret, avec la certitude d'avoir enfin trouv\u00e9 sa partenaire de vie.\nLa machine se met en route\n\nAvec Jane, la vie est douce, ils sont \u00ab les meilleurs camarades du monde 1 \u00bb, mais la cristallisation n'a pas lieu. Aucun doute qu'elle est une partenaire de choix, une parfaite ma\u00eetresse de maison ainsi qu'une secr\u00e9taire fiable (elle l'encourage \u00e0 \u00e9crire et recopie ses manuscrits). Wells a d\u00e9j\u00e0 d\u00e9tourn\u00e9 les yeux d'elle, tout en lui vouant une profonde affection. Leur tendre compagnonnage se refl\u00e8te dans les nombreux dessins qu'il \u00e9bauche \u00e0 cette \u00e9poque, des \u00ab picshua \u00bb comme il les appelle. Jane y est toujours repr\u00e9sent\u00e9e comme celle qui d\u00e9tient le savoir et la v\u00e9rit\u00e9. Wells endosse plus souvent le r\u00f4le de l'\u00e9crivain stupide, grincheux et maladroit, attendant les jambes crois\u00e9es le verdict de son juge. Une s\u00e9rie de croquis les montre tous les deux dans des situations burlesques : Jane tapant sur la t\u00eate de Wells pour le punir d'avoir mal \u00e9crit ; Jane mettant sous clef ses manuscrits afin qu'il ne les modifie pas ; Jane, encore, tenant son mari en laisse, et le promenant comme un chien docile qui a besoin d'exercice...\n\nLes propositions de collaboration abondent. Au d\u00e9but de l'ann\u00e9e 1894, Lewis Hind, r\u00e9dacteur en chef du _Pall Mall Budget_ , propose \u00e0 Wells de devenir un collaborateur r\u00e9gulier de son journal. Cust lui a parl\u00e9 de lui, de ses comp\u00e9tences en \u00e9criture et de ses connaissances en sciences. Sa plume l'int\u00e9resse. \u00c0 l'heure o\u00f9 les nouveaut\u00e9s techniques sont en marche, Hind demande \u00e0 sa jeune recrue d'y consacrer une rubrique fictionnelle \u2014 un espace dans lequel il pourra donner libre cours \u00e0 son imaginaire. Wells s'empresse d'accepter. \u00catre pay\u00e9 pour ce qu'il sait le mieux faire \u2014 \u00e9crire \u2014 avec pour sujets ses th\u00e9matiques favorites, il ne pouvait r\u00eaver situation plus id\u00e9ale. Il se met au travail et r\u00e9dige une \u00e9tonnante histoire, \u00ab The Stolen Bacillus \u00bb (\u00ab Le Vol du microbe \u00bb), les aventures d'un homme qui s'injecte le virus du chol\u00e9ra dans le but de contaminer les autres. C'est la premi\u00e8re nouvelle d'une longue s\u00e9rie que Wells va \u00e9crire jusqu'en 1895. Cust, qui a eu vent du succ\u00e8s de ses publications, le recommande sans plus tarder \u00e0 l'un de ses amis, \u00e9crivain, journaliste et directeur du magazine _The Observer_ , William Ernest Henley. Ce dernier propose \u00e0 Wells de travailler pour lui. Nouvelle opportunit\u00e9 tomb\u00e9e du ciel qui enthousiasme l'\u00e9crivain en devenir, assoiff\u00e9 de reconnaissance. Enferm\u00e9 dans son bureau, il ouvre son tiroir \u00e0 vieilles id\u00e9es, ressort de l'ab\u00eeme \u00ab The Chronic Argonauts \u00bb, timidement publi\u00e9 des ann\u00e9es auparavant, et retravaille son manuscrit. Wells retrouve le h\u00e9ros qu'il avait alors cr\u00e9\u00e9 \u2014 le savant Nebogipfel \u2014 mais d\u00e9cide de l'appeler d\u00e9sormais \u00ab l'Explorateur du Temps \u00bb. Cet homme a une \u00e9trange lueur dans les yeux, car il a fait une d\u00e9couverte, l'une de celles que la majorit\u00e9 des individus ne croirait pas : \u00ab le Temps n'est qu'une sorte d'Espace 2 \u00bb dans lequel les hommes peuvent aussi se mouvoir, seulement, ils ne le savent pas. Voyager dans le temps, abolir les ann\u00e9es, enjamber les si\u00e8cles : tout cela est possible si l'on embarque dans une fabuleuse machine. Pas de place pour les incr\u00e9dules. Le r\u00e9cit que l'Explorateur du Temps fait \u00e0 ses convives n\u00e9glige les limites du possible. \u00ab Dites que j'ai fait un r\u00eave dans mon laboratoire 3 \u00bb, lance-t-il \u00e0 ceux qui mettraient en doute la v\u00e9racit\u00e9 de son propos. Lui sait bien ce qu'il a v\u00e9cu. Il se souvient d'avoir cru mourir quand le voyage a commenc\u00e9, passant de la lumi\u00e8re aux t\u00e9n\u00e8bres, d\u00e9passant la Lune et c\u00f4toyant le Soleil, glissant sur le Temps comme on d\u00e9vale une montagne. Dans un avenir lointain, en 802 701, il d\u00e9couvre la Terre d\u00e9barrass\u00e9e du commerce et des maladies, ressemblant \u00e0 un immense jardin, et habit\u00e9e par des \u00eatres doux et joueurs \u2014 les \u00c9lo\u00efs \u2014, descendants des hommes. Mais ce paradis n'est pas exempt d'obscurit\u00e9 et de crainte, aliment\u00e9es par les Morlocks qui vivent dans des souterrains et se nourrissent de leurs semblables. Le voyage dans le temps montre la d\u00e9g\u00e9n\u00e9rescence d'une humanit\u00e9 o\u00f9 subsiste l'opposition fondamentale des privil\u00e9gi\u00e9s et des plus faibles. Wells a l'ambition de faire r\u00e9fl\u00e9chir ses lecteurs, de les d\u00e9ranger aussi. Ce que la vie propose \u2014 les rapports de force, la violence, le chagrin \u2014, le roman doit le montrer, l'interroger.\n\nCette \u00ab machine \u00bb donne une vision jusque-l\u00e0 in\u00e9dite du futur et de l'\u00e9volution de l'homme, Wells allant jusqu'\u00e0 \u00e9voquer, sans la nommer, la fin du monde, d\u00e9crite comme une \u00e9clipse :\n\nL'obscurit\u00e9 croissait rapidement. Un vent froid commen\u00e7a \u00e0 souffler de l'est par rafales fra\u00eechissantes, et le vol des flocons s'\u00e9paissit. Du lointain de la mer s'approch\u00e8rent une ride l\u00e9g\u00e8re et un murmure. Hors ces sons inanim\u00e9s, le monde \u00e9tait plein de silence. Du silence ? Il est bien difficile d'exprimer ce calme qui pesait sur lui. Tous les bruits de l'humanit\u00e9, le b\u00ealement des troupeaux, le chant des oiseaux, le bourdonnement des insectes, toute l'agitation qui fait l'arri\u00e8re-plan de nos vies, tout cela n'existait plus. Comme les t\u00e9n\u00e8bres s'\u00e9paississaient, les flocons, tourbillonnant et dansant devant mes yeux, devinrent plus abondants et le froid de l'air devint plus intense. \u00c0 la fin, un par un, les sommets blancs des collines lointaines s'\u00e9vanouirent dans l'obscurit\u00e9. La brise se changea en un vent g\u00e9missant. Je vis l'ombre centrale de l'\u00e9clipse s'\u00e9tendre vers moi. En un autre instant, seules les p\u00e2les \u00e9toiles furent visibles. Tout le reste fut plong\u00e9 dans la plus grande obscurit\u00e9. Le ciel devint absolument noir 4.\n\nLe texte terrifie et fascine Henley qui en publie le premier \u00e9pisode le 17 mars 1894, et d\u00e9cide d'\u00e9diter la version d\u00e9finitive dans _The New Review,_ \u00e0 partir de 1895. Cette premi\u00e8re _scientific romance_ s'intitule _Time Machine_ \u2014 litt\u00e9ralement \u00ab La Machine du Temps \u00bb. Wells y a mis toute son \u00e9nergie : il esp\u00e8re ainsi toucher un large public, r\u00eavant m\u00eame d'un succ\u00e8s comme il n'en a encore jamais connu. Et bient\u00f4t, le mot est \u00e9crit l\u00e0, sous ses yeux, dans le journal. Wells boit son caf\u00e9 et lit l'adjectif accol\u00e9 \u00e0 son nom : g\u00e9nie. Il aurait donc r\u00e9ussi, lui, le provincial, l'\u00e9tudiant contrari\u00e9 ? Finie, l'amertume des jeunes ann\u00e9es ! Il a eu raison d'y croire, de pers\u00e9v\u00e9rer contre l'avis de ses parents. Tout le monde allait d\u00e9sormais entendre parler de lui. L'\u00e9diteur Heinemann publie le livre dans son int\u00e9gralit\u00e9. Wells a vingt-huit ans, et il entre dans la cour d'Oscar Wilde, Thomas Hardy et Rudyard Kipling \u2014 tous les c\u00e9l\u00e8bres \u00e9crivains britanniques de son \u00e9poque. Il rencontre Bernard Shaw et Henry James. Ce dernier est tr\u00e8s int\u00e9ress\u00e9 par le travail de Wells, mais la r\u00e9ciproque n'est pas aussi \u00e9vidente. James a certes une profonde connaissance de la nature humaine et des interactions sociales, mais sa syntaxe est lourde : \u00ab Ses paragraphe me font penser \u00e0 un \u00e9cheveau de laine avec lequel un chat aurait jou\u00e9 5. \u00bb\n\nSollicit\u00e9 par les grands du monde des lettres, enivr\u00e9 de fiert\u00e9, Wells envoie _La Machine \u00e0 explorer le temps_ \u00e0 une personne dont il attend encore l'approbation. Sur le dos du paquet ficel\u00e9, il \u00e9crit le nom de son cher professeur d'universit\u00e9, T. H. Huxley, impatient de recevoir un signe de lui.\n\nPapier, plume, encre. Lumi\u00e8re. Wells \u00e9crit avec endurance, toute la journ\u00e9e, des dizaines de nouvelles r\u00e9unies aujourd'hui dans plusieurs recueils. _Select Conversations With An Uncle_ (\u00ab Conversations choisies avec un oncle \u00bb), _The Wonderful Visit_ (\u00ab La Merveilleuse Visite \u00bb), _The Stolen Bacillus and Other Incidents_ (\u00ab Le Vol du microbe \u00bb). Le rythme est effr\u00e9n\u00e9, l'auteur infatigable. Wells sait qu'il lui reste encore beaucoup de chemin \u00e0 parcourir, qu'un succ\u00e8s ne veut rien dire ; qu'il joue aussi avec le feu en abandonnant son m\u00e9tier de professeur et la s\u00e9curit\u00e9 de revenus fixes ; qu'il doit payer une pension \u00e0 Isabel (dont il a finalement divorc\u00e9) et pourvoir aux besoins de Jane (qu'il a officiellement \u00e9pous\u00e9e), tout en finan\u00e7ant la construction de sa nouvelle maison \u00e0 Sandgate, face \u00e0 la mer. La nuit, il dort peu, inquiet \u00e0 l'id\u00e9e de ne pas y arriver, ind\u00e9cis sur le chemin qu'il doit prendre. Il aime autant Shelley que Darwin. Comment concilier ses diff\u00e9rentes aspirations ? Et puis il y a cette terrible nouvelle, la mort d'Huxley, son ma\u00eetre, celui qui lui a donn\u00e9 l'envie d'apprendre et d'interroger la r\u00e9alit\u00e9. Wells ne sait pas s'il a eu le temps de lire _La Machine \u00e0 explorer le temps_ , et s'il a m\u00eame re\u00e7u le paquet qu'il lui avait envoy\u00e9. Le vieil homme ne lui a jamais r\u00e9pondu. Dans sa t\u00eate se bousculent les souvenirs de dissection, les heures pass\u00e9es au laboratoire \u00e0 choisir le bon instrument, \u00e0 \u00e9mettre des hypoth\u00e8ses. Tout cela lui donne envie de poursuivre un manuscrit laiss\u00e9 inachev\u00e9, un texte de jeunesse peupl\u00e9 d'hommes-b\u00eates effrayants qui terrorisent la face du monde.\n\nWells cr\u00e9e un monstre. _L'\u00cele du docteur Moreau_ para\u00eet en 1896. Quelques jours avant sa sortie, le _Times_ pr\u00e9vient ses lecteurs :\n\nNous nous sentons dans l'obligation de nous \u00e9lever contre certains exc\u00e8s qui peuvent mener nous ne savons o\u00f9 et d'adresser quelques paroles de mise en garde au lecteur qui ne se doute de rien et d\u00e9sire \u00e9chapper au d\u00e9go\u00fbtant et au r\u00e9pugnant... Ce livre doit \u00eatre tenu hors de port\u00e9e des jeunes, et tous ceux qui poss\u00e8dent bon go\u00fbt, fine sensibilit\u00e9 ou des nerfs peu solides doivent \u00e9viter de le lire. C'est tout simplement un sacril\u00e8ge de faire baigner la belle nature dans le sang et dans l'antiseptique de l'amphith\u00e9\u00e2tre de vivisection 6.\n\nUn sacril\u00e8ge, ou une prouesse, c'est selon. Les aventures de Prendick, Montgomery et Moreau sont d'une violence inou\u00efe, mais aussi d'une rare lucidit\u00e9 sur les dangers des pouvoirs de l'homme. Wells pousse le darwinisme \u00e0 son extr\u00eame et imagine le pire : l'humanisation des animaux, leur remodelage en cr\u00e9atures r\u00e9pugnantes, fa\u00e7onn\u00e9es par une parodie de Cr\u00e9ateur \u2014 le docteur Moreau \u2014, proscrit de la soci\u00e9t\u00e9 que rien n'arr\u00eate. Le narrateur de l'histoire (Prendick, donc) a fait naufrage dans le Pacifique, quelque part entre le P\u00e9rou et Hawa\u00ef. C'est un nouveau Cruso\u00e9, un autre Gulliver. Il est recueilli par un navire transportant une \u00e9trange cargaison, et conduit par Montgomery qui l'emm\u00e8ne sur une \u00ab \u00eele infernale 7 \u00bb. L\u00e0-bas, il d\u00e9couvre l'horreur : des cochons, des singes, des loups, des l\u00e9opards, et tant d'autres races modifi\u00e9es, trafiqu\u00e9es entre les mains de Moreau. Cet homme pratique l'ablation, la greffe, et m\u00eame la lobotomie. Dans la nuit noire, les cris des animaux tortur\u00e9s retentissent, le repos des sens est impossible. Il y a du Frankenstein dans ce scientifique fou, un homme non seulement passionn\u00e9 mais d\u00e9barrass\u00e9 de tout questionnement \u00e9thique, courant apr\u00e8s \u00ab la limite extr\u00eame de la plasticit\u00e9 dans une forme vivante 8 \u00bb. Wells explore le registre fantastique, mais plus encore la robinsonnade. Moreau serait-il le descendant n\u00e9vros\u00e9 de Thomas More (l'auteur de _L'Utopie_ ) __ et du comte Arthur de Gobineau (l'inventeur de la th\u00e9orie de l'in\u00e9galit\u00e9 des races) ? La combinaison des deux noms fonctionnerait, et il n'est pas impossible que Wells ait lu les ouvrages respectifs de ces \u00e9crivains pour cr\u00e9er cet \u00eatre hybride, tout aussi id\u00e9aliste qu'id\u00e9ologue. Moreau est l'arch\u00e9type du m\u00e9chant, celui qui aime faire souffrir et n'a que faire de la morale. \u00c0 travers lui, Wells pose la question de la cruaut\u00e9 : qui est, ici, le plus d\u00e9nu\u00e9 d'humanit\u00e9 ? Est-il possible que les hommes ne soient, peut-\u00eatre, que des b\u00eates ? \u00c0 la fin du r\u00e9cit, Prendick, de retour \u00e0 Londres, ne peut s'\u00f4ter cette id\u00e9e de l'esprit :\n\nJe n'ose esp\u00e9rer que la terreur de cette \u00eele me quittera jamais enti\u00e8rement, encore que la plupart du temps elle ne soit, tout au fond de mon esprit, rien qu'un nuage \u00e9loign\u00e9, un souvenir, un timide soup\u00e7on ; mais il est des moments o\u00f9 ce petit nuage s'\u00e9tend et grandit jusqu'\u00e0 obscurcir tout le ciel. Si, alors, je regarde mes semblables autour de moi, mes craintes me reprennent. Je vois des faces \u00e2pres et anim\u00e9es, d'autres ternes et dangereuses, d'autres fuyantes et menteuses, sans qu'aucune poss\u00e8de la calme autorit\u00e9 d'une \u00e2me raisonnable. J'ai l'impression que l'animal va repara\u00eetre tout \u00e0 coup sous ces visages ...][ 9.\n\nWells, la plume encore ensanglant\u00e9e de son \u00e9pouvante bestiale, a d\u00e9j\u00e0 mille autres id\u00e9es en t\u00eate. Il est lanc\u00e9 \u00e0 toute vitesse, sans intention de ralentir. La pratique de la bicyclette lui est \u00e0 cette \u00e9poque quasi quotidienne. C'est en p\u00e9dalant \u2014 seul ou en tandem avec Jane \u2014 qu'il r\u00e9fl\u00e9chit \u00e0 ses prochains livres, qu'il imagine de possibles d\u00e9cors. En passant un jour pr\u00e8s d'une plaine, il se dit qu'elle serait le parfait th\u00e9\u00e2tre pour faire atterrir une soucoupe volante. Reste \u00e0 d\u00e9cider de l'aspect des extraterrestres. Petits, visqueux, avec une grosse t\u00eate ? On verra. Pour l'instant, il pense \u00e0 un personnage dr\u00f4le et attachant. Hoopdriver devient le protagoniste d'une incroyable \u00e9pop\u00e9e cycliste qui tranche r\u00e9solument avec l'atmosph\u00e8re oppressante de son dernier roman. Et puis il songe \u00e0 un th\u00e8me in\u00e9dit : un scientifique un peu solitaire trouverait le moyen de dispara\u00eetre physiquement de la Terre, tout en restant vivant... Griffin, le personnage de cette nouvelle histoire, devient l'\u00ab Homme invisible \u00bb, supr\u00eamement intelligent, excessif et, l\u00e0 encore, amoral.\n\nEn moins de deux ans, trois romans, trois succ\u00e8s \u00e9ditoriaux. Wells s'entoure d'agents litt\u00e9raires qui l'aident \u00e0 n\u00e9gocier ses contrats et \u00e0 gagner des sommes importantes. Il ach\u00e8te une maison \u00e0 ses parents et, chose rare pour le fils d'une femme de chambre, s'offre les services d'une domestique. Tr\u00e8s sollicit\u00e9 par la presse, il re\u00e7oit les journalistes dans son bureau, d\u00e9cor\u00e9 \u00e0 la derni\u00e8re mode, entour\u00e9 de beaux livres et d'objets soigneusement choisis. Il est fier de cette r\u00e9ussite, il veut en profiter. Il a des envies de voyages, part d\u00e9couvrir Rome et la c\u00f4te amalfitaine avec Jane et leur ami Gissing, ce \u00ab pauvre esprit tracass\u00e9 10 \u00bb rencontr\u00e9 lors d'un d\u00eener en ville. Lui aussi est \u00e9crivain, r\u00e9put\u00e9 pour son temp\u00e9rament et sa plume taciturnes, mais Wells a le don de le faire rire. L'amiti\u00e9 se noue autour de points communs fondamentaux. M\u00eame haine des conventions et de la foi chr\u00e9tienne, m\u00eame condition physique d\u00e9sastreuse \u2014 lui aussi souffre des reins \u2014, et surtout, m\u00eame amour des femmes. Ces quelques jours pass\u00e9s en Italie avec la nouvelle vedette de la litt\u00e9rature anglaise lui sont tr\u00e8s agr\u00e9ables, mais le retour \u00e0 la r\u00e9alit\u00e9 est difficile. Gissing se laisse mourir lentement, et Wells le veille jusqu'\u00e0 son dernier souffle, observant cet homme qui \u00ab gaspille] son grand et beau cerveau \u00e0 d\u00e9pr\u00e9cier la vie, car il ne voulait pas [...] la regarder droit dans les yeux[ 11 \u00bb.\n\nLui fait tout pour profiter de chaque instant. Cette notori\u00e9t\u00e9 soudaine, ces invitations qui se multiplient, ces personnes qui veulent absolument lui serrer la main : Wells entre dans la danse sans conna\u00eetre les pas. Une nouvelle vie mondaine commence pour lui, mais il n'a pas de smoking, ne sait pas comment on mange le caviar, et ne conna\u00eet pas la moiti\u00e9 des boissons qu'on lui sert. Un peu pataud, encore timide ; il est comme un homme qui d\u00e9barque sur Terre apr\u00e8s un long voyage depuis une autre plan\u00e8te : perdu. Son costume tout neuf qu'il a fait faire sur mesure lui bride la poitrine, et ses chaussures vernies lui font mal aux pieds. Alors il ne s'attarde jamais tr\u00e8s longtemps, serre quelques mains et rentre chez lui. Il a du mal \u00e0 croire \u00e0 tout cela. Une heureuse surprise le ram\u00e8ne \u00e0 la r\u00e9alit\u00e9. En feuilletant le journal, il tombe par hasard sur un nom familier. Sydney Bowkett. Son ami d'enfance, perdu de vue depuis des ann\u00e9es, est auteur dramatique. L'article qui lui est consacr\u00e9 est \u00e9logieux. \u00c9mu, Wells lui \u00e9crit une lettre, une foule de souvenirs lui reviennent en m\u00e9moire : ces apr\u00e8s-midi pass\u00e9s \u00e0 jouer aux billes, \u00e0 imaginer l'avenir, \u00e0 se faire des promesses. La vie a s\u00e9par\u00e9 les deux camarades, d'un coup. Wells \u00e9tait, \u00e0 l'\u00e9poque, parti si vite de Bromley pour commencer \u00e0 travailler qu'il a l'impression d'avoir laiss\u00e9 cette amiti\u00e9 en suspens, puis de l'avoir oubli\u00e9e. Sydney lui r\u00e9pond et lui propose qu'ils se revoient. Il n'a pas chang\u00e9, ou si peu. Lui aussi est mari\u00e9. L'amour des femmes est ce qui rapproche d\u00e9sormais les anciens complices.\n\nUn matin, Wells re\u00e7oit une lettre du r\u00e9dacteur en chef du magazine _Woman_ , Arnold Bennett, qui lui \u00e9crit pour lui dire tout le bien qu'il pense de son \u0153uvre. Hasard ou co\u00efncidence, dans son enfance, Bennett a eu pour professeur Horace Byatt. Il n'en fallait pas plus pour que naisse une correspondance entre les deux hommes. Joseph Conrad agrandit bient\u00f4t ce nouveau cercle de connaissances : celui qui est en train d'\u00e9crire _Lord Jim_ est s\u00e9duit par une critique de son camarade d'\u00e9criture qu'il consid\u00e8re comme un auteur \u00e0 part. Wells est tr\u00e8s impressionn\u00e9 par sa prestance, bien qu'il soit \u00ab plut\u00f4t trapu, avec un dos vo\u00fbt\u00e9 et une t\u00eate comme enfonc\u00e9e dans son corps 12 \u00bb. Quelque chose le trouble : il ne voit pas ses yeux, opaques, t\u00e9n\u00e9breux, et qui semblent fixes. Il n'y a que ses l\u00e8vres qui bougent, lentement, racontant des souvenirs d'aventures et des nuits pass\u00e9es dans la jungle. Mais plus ils parlent, plus ils se rendent compte qu'ils n'ont pas la m\u00eame vision des choses. Ce bateau, par exemple, qui s'appr\u00eate \u00e0 accoster et que Conrad pointe du doigt un apr\u00e8s-midi sur la plage, Wells ne l'a m\u00eame pas remarqu\u00e9. Son ami lui demande de d\u00e9crire ce qu'il voit. Wells r\u00e9pond que cela ne sert \u00e0 rien, qu'il ne peut pas d\u00e9crire un objet qu'il n'a m\u00eame pas vu de lui-m\u00eame. Il en a trop dit, ou pas assez. Conrad hausse le ton : comment peut-il ignorer ce qu'il y a sous les yeux ? La r\u00e9alit\u00e9 est l\u00e0, \u00e0 port\u00e9e de regard, et il y aurait tant de choses \u00e0 dire sur cette forme lointaine, \u00e0 peine discernable mais quand bien m\u00eame tangible ! Peu importe, Wells ne l'a pas vu, ce bateau n'est pas digne d'int\u00e9r\u00eat. Selon lui, il ne suffit pas de voir pour \u00e9crire. L'histoire seule compte, pas l'objet. En quelques secondes, deux visions de la litt\u00e9rature ont \u00e9merg\u00e9 pour ne jamais cesser de s'opposer. Conrad comprend que son ami n'est pas aussi _r\u00e9aliste_ que lui, ou plut\u00f4t qu'il est un \u00ab r\u00e9aliste du fantastique 13 \u00bb \u2014 hypoth\u00e8se qui se v\u00e9rifie l'ann\u00e9e suivante, avec le choc de _La Guerre des mondes_.\n\nD'un c\u00f4t\u00e9, des hommes heureux, insouciants, et qui se croient invuln\u00e9rables. De l'autre, un puissant projectile venant de la plan\u00e8te Mars, et lanc\u00e9 \u00e0 vive allure contre la Terre. Bient\u00f4t, le \u00ab cataclysme 14 \u00bb, l'incompr\u00e9hension, l'exode, et la lutte pour la survie. \u00c0 quoi Wells r\u00eave-t-il donc pour \u00e9crire de telles histoires ? Que se passe-t-il dans sa t\u00eate \u00e0 ce moment ? D'o\u00f9 lui viennent ces fantasmes d'invasion, ces cauchemars de fin du monde ? Comment cet homme, vivant tranquillement \u00e0 la campagne avec sa femme, et ne sortant que pour son tour quotidien de bicyclette, imagine-t-il m\u00eame un jour l'apparence d'un extraterrestre ?\n\nCeux qui n'ont jamais vu de Martiens vivants peuvent difficilement s'imaginer l'horreur \u00e9trange de leur aspect, leur bouche singuli\u00e8re en forme de V et la l\u00e8vre sup\u00e9rieure pointue, le manque de front, l'absence de menton au-dessous de la l\u00e8vre inf\u00e9rieure en coin, le remuement incessant de cette bouche, le groupe gorgonesque des tentacules, la respiration tumultueuse des poumons dans une atmosph\u00e8re diff\u00e9rente, leurs mouvements lourds et p\u00e9nibles, \u00e0 cause de l'\u00e9nergie plus grande de la pesanteur sur la Terre et par-dessus tout l'extraordinaire intensit\u00e9 de leurs yeux \u00e9normes \u2014 tout cela me produisit un effet qui tenait de la naus\u00e9e. Il y avait quelque chose de fongueux dans la peau brune huileuse, quelque chose d'inexprimablement terrible dans la maladroite assurance de leurs lents mouvements 15.\n\nLes histoires d'invasion prosp\u00e8rent \u00e0 cette \u00e9poque. Les lecteurs se r\u00e9jouissent des romans qui mettent en sc\u00e8ne une arm\u00e9e \u00e9trang\u00e8re attaquant la Grande-Bretagne. Mais Wells veut aggraver cette peur, tenter d'exprimer ses propres inqui\u00e9tudes, ses angoisses fondamentales. C'est en se promenant avec son fr\u00e8re Frank, venu lui rendre visite, que l'arriv\u00e9e des Martiens sur Terre lui appara\u00eet comme un formidable d\u00e9but d'intrigue. La vanit\u00e9 des hommes serait terrass\u00e9e par une \u00e9nergie destructrice sans pareille. Ce serait un bain de sang, un bal d'atrocit\u00e9s. Enfin quelque chose enl\u00e8verait \u00ab cette sereine confiance en l'avenir, qui est la plus f\u00e9conde source de la d\u00e9cadence 16 \u00bb. Le fantasme de la destruction devient une id\u00e9e fixe. Peut-\u00eatre est-ce une trace de cette attirance pour l'enfer dont la simple pens\u00e9e lui \u00e9tait interdite quand il \u00e9tait enfant. Peut-\u00eatre aussi cela vient-il de lui-m\u00eame, de son corps malade, ch\u00e9tif, qui ne cesse de c\u00f4toyer la menace de la mort au quotidien. Une \u00e9ni\u00e8me rechute le cloue au lit plusieurs semaines. Mais le nouveau si\u00e8cle approche. Il s'ouvrira au soleil, \u00e0 Sandgate, en bord de mer, dans la nouvelle maison que Wells et Jane occupent d\u00e8s le d\u00e9but de l'ann\u00e9e 1901.\nAnticipations\n\nC'est un beau soir d'\u00e9t\u00e9. Wells est assis sur le sable. Sa maison est juste l\u00e0, derri\u00e8re lui : grande, un peu sur\u00e9lev\u00e9e, id\u00e9alement expos\u00e9e sur cette baie du Kent qu'il aime tant. Au-dessus : la lune, ronde, brillante, promontoire de tous les songes. Un jour, des hommes pourront cesser de l'observer au t\u00e9lescope, et y aller vraiment. Ils d\u00e9couvriront que les plantes y poussent, que des centaines de cavernes sont creus\u00e9es dans le sol, o\u00f9 des souterrains sont irrigu\u00e9s d'eau et enduits d'or... Le roman permet d'imaginer ces choses-l\u00e0, de leur donner vie. La recette est simple : mettre deux scientifiques (Bedford et Cavour) dans une sph\u00e8re et les lancer tr\u00e8s loin dans le ciel ; puis les faire sortir de la capsule, marcher en bondissant jusqu'\u00e0 ce qu'ils se perdent et soient faits prisonniers par une colonie d'insectes g\u00e9ants \u2014 des bergers qui gardent leurs troupeaux de veaux lunaires... Les aventures des _Premiers hommes dans la Lune_ 1 sont une anticipation \u00e9tonnante. Wells s'inscrit ici dans le sillon de Cyrano de Bergerac ( _Histoire comique des \u00c9tats et Empires de la Lune,_ en 1657) _,_ Murtagh McDermot ( _A Trip To The Moon_ , __ \u00ab Un voyage vers la Lune \u00bb), en 1728) et bien s\u00fbr de Jules Verne, qui, dans son c\u00e9l\u00e8bre diptyque, _De la Terre \u00e0 la Lune_ et _Autour de la Lune,_ avait envisag\u00e9 la satellisation d'une capsule autour de l'astre. C'\u00e9tait en 1870.\n\nComme eux, Wells r\u00eave de la Lune et y met les deux pieds, au risque de contrarier son pr\u00e9d\u00e9cesseur fran\u00e7ais qui ne manque pas de se moquer de son succ\u00e8s :\n\nOn m'a envoy\u00e9 ses livres et je les ai lus. C'est tr\u00e8s curieux et, ajouterai-je, tr\u00e8s anglais. Mais je ne vois pas de possibilit\u00e9 de comparer son \u0153uvre et la mienne. Nous ne proc\u00e9dons pas de la m\u00eame mani\u00e8re. Il me semble que ses histoires ne reposent pas sur des bases tr\u00e8s scientifiques. Non, il n'y a pas de rapport entre son \u0153uvre et la mienne. Je vais sur la Lune dans un boulet de canon lanc\u00e9 par un canon. Ce n'est pas une invention. Lui va dans Mars avec un a\u00e9ronef qu'il construit dans un m\u00e9tal qui supprime la loi de la gravitation. \u00c7a, c'est tr\u00e8s joli, ...] mais montrez-moi ce m\u00e9tal. Qu'il nous le fabrique[ 2 !\n\nNon seulement Verne ne cache pas son exasp\u00e9ration, mais il confond Mars avec la Lune ! Ce qu'il ne supporte pas, c'est la manie qu'a Wells de m\u00ealer ses fantasmes \u00e0 la science. Et ce qu'il n'a pas pu voir (Verne meurt en 1905), c'est que son homologue britannique n'avait pas compl\u00e8tement tort : ses hypoth\u00e8ses \u2014 sa folie douce \u2014 ont trouv\u00e9 un \u00e9cho inattendu dans l'histoire du XXe si\u00e8cle.\n\nCar Wells est un anticipateur. Son intranquillit\u00e9 chronique le m\u00e8ne sans cesse \u00e0 r\u00e9fl\u00e9chir au futur. Il a beau \u00eatre arriv\u00e9 au sommet, avoir balay\u00e9 les doutes, les chambres miteuses et les femmes qui ne veulent pas de lui ; \u00eatre avec Jane, qui vient de mettre au monde leur premier fils, George Philip ; avoir ses livres traduits dans le monde entier, et de l'argent plein les poches ; \u00e0 cet instant pr\u00e9cis, tandis qu'il observe le ciel en \u00e9coutant le clapotis des vagues, il lui manque quelque chose \u2014 l'essentiel \u2014, mais il ne sait pas quoi. Cette existence bien rang\u00e9e, il a tout fait pour l'avoir, sans jamais la souhaiter vraiment. Pourquoi s'est-il fix\u00e9 \u00e0 un endroit, \u00e0 une personne ? Tout l'ennuie : cette vie qui s'assoupit, cette \u00e9poque qui expire, cette litt\u00e9rature qui stagne. Il faut penser l'avenir. Mieux : le pr\u00e9voir et l'\u00e9crire.\n\nIl vient de publier _L'Amour et M. Lewisham_ , r\u00e9cit d'une aventure amoureuse pour lequel il a fait appel \u00e0 ses souvenirs d'\u00e9tudiant londonien. C'est un livre dont la tonalit\u00e9 tranche r\u00e9solument avec celle des pr\u00e9c\u00e9dents. Bennett le lui fait remarquer. Selon lui, Wells ne devrait \u00e9crire que des ouvrages de science-fiction, et laisser de c\u00f4t\u00e9 le reste \u2014 les romans plus r\u00e9alistes comme celui-ci. Mais Wells a besoin d'\u00e9crire les deux, d'\u00eatre \u00e0 la fois dans l'imagination pure et dans le v\u00e9cu. La r\u00e9daction de _La Machine \u00e0 explorer le temps_ rel\u00e8ve pour lui de l'intuition, tandis qu'un livre comme _L'Amour et M. Lewisham_ est le fruit d'un travail plus laborieux \u2014 presque plus _s\u00e9rieux_ \u00e0 ses yeux. Les deux sont en tout cas \u00ab une distillation m\u00e9thodique et prudente de ses] pens\u00e9es, [ses] sentiments, [ses] exp\u00e9riences et [ses] impressions[ 3 \u00bb. Infatigable, son esprit est d\u00e9j\u00e0 lanc\u00e9 ailleurs. Wells r\u00e9fl\u00e9chit vaguement \u00e0 l'histoire d'une sir\u00e8ne nageant au large des c\u00f4tes de Sandgate. Le titre est tout trouv\u00e9 \u2014 _Miss Waters \u2014_ et le ton \u2014 quelque part entre le conte fantastique et la tragi-com\u00e9die. Son dernier roman, _Les Premiers Hommes dans la Lune,_ commence \u00e0 bien se vendre. Et puis un certain Robert W. Paul, un ing\u00e9nieur talentueux, pionnier dans l'industrie cin\u00e9matographique britannique, lui a \u00e9crit une lettre de louanges. Il a ador\u00e9 _La Machine \u00e0 explorer le temps_ , il veut en faire un spectacle o\u00f9 hommes et femmes seraient install\u00e9s dans une sorte de plateforme ferm\u00e9e, entour\u00e9s de toiles gigantesques, de bruits et d'effets sp\u00e9ciaux en tout genre \u2014 spectacle qui leur donnerait la sensation v\u00e9ritable d'un voyage dans l'espace et le temps. Ce qu'il a en t\u00eate ressemble \u00e0 s'y m\u00e9prendre \u00e0 certaines de nos attractions modernes : un \u00ab grand-huit \u00bb avant l'heure. L'id\u00e9e est novatrice, elle enchante Wells qui donne son accord de principe. L'\u00e9crivain n'en revient pas que sa machine puisse prendre forme, que la fiction puisse peut-\u00eatre devenir une exp\u00e9rience r\u00e9elle hors du commun. Pour diverses raisons (surtout financi\u00e8res), le projet n'aboutit pas. Wells n'en prend pas ombrage. Le cin\u00e9ma n'a pas fini de s'int\u00e9resser \u00e0 lui.\n\nMalgr\u00e9 ces bonnes nouvelles, il n'a pas la t\u00eate \u00e0 \u00e9crire d'autres intrigues. Face \u00e0 ce nouveau si\u00e8cle, dans une Angleterre qui vient de perdre sa reine (Victoria est morte le 22 janvier 1901), les jeux de l'imagination font place \u00e0 une ambition qui doit se conjuguer au futur. Wells veut \u00e9crire sur \u00ab la fa\u00e7on dont iront les choses de ce monde au XXe si\u00e8cle 4 \u00bb. Pr\u00e9dire, avec assurance, ce que l'on ne peut que supposer. Endosser, pour ne plus jamais le quitter, le costume de proph\u00e8te. Le recueil d'articles r\u00e9unis sous le titre _Anticipations_ est une autre incroyable machine \u00e0 explorer le futur. Elle donne \u00e0 l'avenir ses contours et ses couleurs. Elle vise loin, et souvent juste. Le livre, v\u00e9ritable best-seller \u00e0 sa sortie, est une \u00ab clef de vo\u00fbte 5 \u00bb dans l'\u0153uvre de l'\u00e9crivain qui l'a en t\u00eate depuis des ann\u00e9es, revenant subtilement \u00e0 ses premi\u00e8res amours : l'\u00e9tude de l'homme et de sa place dans l'Univers.\n\nWells a toujours eu l'esprit constructif. Il fait partie de ces \u00eatres qui voient \u00ab le monde comme un immense chantier 6 \u00bb \u00e0 b\u00e2tir et \u00e0 consolider. Tous les livres qu'il a jusqu'\u00e0 pr\u00e9sent \u00e9crits sont tourn\u00e9s vers l'apr\u00e8s \u2014 jamais vers le pass\u00e9 :\n\nC'est dans le futur que nous allons, demain est la chose importante et f\u00e9conde pour nous. C'est demain qui renferme tout ce qui reste \u00e0 sentir pour nous, nos enfants et tous ceux qui nous sont chers. ...] Beaucoup de gens croient [...] qu'il ne peut y avoir aucune esp\u00e8ce de certitude touchant l'avenir. Vous ne pouvez pas plus savoir quelque chose de l'avenir, m'assurait r\u00e9cemment un ami, que vous ne savez \u00e0 quel endroit un chat va sauter. Tous ceux qui ont cette opinion, tous ceux qui consid\u00e8rent l'avenir comme une source perp\u00e9tuelle de surprises convulsives, comme une imp\u00e9n\u00e9trable, incurable, \u00e9ternelle t\u00e9n\u00e8bre, ont certes raison de tirer des \u00e9v\u00e9nements qui selon eux sont arriv\u00e9s certainement les valeurs qu'il est n\u00e9cessaire d'attacher aux choses. C'est notre ignorance de l'avenir et notre persuasion que cette ignorance est absolument incurable qui donnent seules au pass\u00e9 son \u00e9norme pr\u00e9dominance dans nos pens\u00e9es[ 7.\n\nAlors, il lance les d\u00e9s, pour voir. Dans le futur selon Wells, les hommes ne subissent plus la distance. Plus ils pourront se d\u00e9placer facilement, mieux ils vivront. La locomotive fut une formidable invention, mais l'automobile la d\u00e9passera (plus \u00ab silencieuse, l\u00e9g\u00e8re, \u00e9l\u00e9gante et puissante 8 \u00bb), sans parler du camion \u00e0 moteur qui transportera les marchandises sur de longs trajets. L'homme sera un voyageur libre, insouciant, il \u00ab s'arr\u00eatera s'il lui pla\u00eet en tel ou tel endroit pour y cueillir des fleurs, et s'il \u00e9prouve, certain matin, le d\u00e9sir de s'attarder au lit d'auberge, il fera prier le chauffeur d'attendre 9 \u00bb. Plus d'embouteillages de carrioles ou de diligences. Wells pr\u00e9voit l'\u00e9mergence de l'\u00e9changeur, afin que les v\u00e9hicules se croisent sans se ralentir. C'est une refonte de l'organisation urbaine qu'il envisage, une ville mouvante, technique, audacieuse et qui abriterait \u2014 pourquoi pas \u2014 quarante millions de citadins. Le proph\u00e8te a les yeux plus gros que le ventre, mais il n'est pas fou. Il pense \u00e0 l'am\u00e9lioration du quotidien de chacun par quelques menus progr\u00e8s techniques. Les maisons seront pens\u00e9es et construites avec un chauffage centralis\u00e9 ; le t\u00e9l\u00e9phone deviendra un objet indispensable et \u00ab perfectionn\u00e9 10 \u00bb dont chaque individu pourra disposer personnellement ; \u00ab la force d'invention 11 \u00bb de l'homme le conduira aussi \u00e0 optimiser les guerres, \u00e0 les pr\u00e9parer avec discipline. Fusils et ba\u00efonnettes seront remplac\u00e9s par une \u00ab artillerie et des engins de destruction et d'intimidation dont nous ne r\u00eavons pas encore 12 \u00bb, et par d'\u00e9tonnantes \u00ab machines volantes \u00bb qui situeront d\u00e9finitivement les combats dans les airs.\n\nLa Premi\u00e8re Guerre mondiale n'a pas encore eu lieu, et Wells voit d\u00e9j\u00e0 les mitrailleuses, les tranch\u00e9es, les blind\u00e9s, et derri\u00e8re toute cette ing\u00e9nierie, une menace constante, un conflit in\u00e9vitable et g\u00e9n\u00e9ralis\u00e9, un monde sans aucune s\u00e9curit\u00e9 \u00e0 la recherche d'un syst\u00e8me politique plus fiable que ce \u00ab leurre verbal 13 \u00bb qu'est la d\u00e9mocratie :\n\nIl n'existe pas, dans nos \u00c9tats modernes, de gouvernement d\u00e9mocratique \u00e9lectif qui, en cinq minutes, ne puisse \u00eatre d\u00e9montr\u00e9 absurde. ...] Ni les hommes ni leurs droits ne sont identiquement \u00e9gaux, car ces droits varient avec chaque individu, et il est indiscutable que le minimum ou le maximum de bonheur g\u00e9n\u00e9ral est loin de d\u00e9pendre directement du contr\u00f4le public [...][ 14.\n\nIl faut r\u00e9fl\u00e9chir \u00e0 un r\u00e9gime nouveau, \u00e9difier un \u00ab \u00c9tat mondial 15 \u00bb qui permette aux individus de mieux vivre ensemble. Dans ce monde vivront des hommes \u00ab capables 16 \u00bb, forts car intelligents, et ayant une id\u00e9e tr\u00e8s claire de l'ordre social. Balayer l'ignorance, la col\u00e8re et les r\u00e9volutions gr\u00e2ce \u00e0 une \u00e9lite de techniciens visionnaires qui n'aura qu'une seule pr\u00e9occupation : l'\u00e9ducation. Au sein de cette \u00ab R\u00e9publique nouvelle 17 \u00bb \u2014 la r\u00e9f\u00e9rence \u00e0 Platon est \u00e0 peine dissimul\u00e9e \u2014, l'\u00e9cole aura pour mission de former les esprits au raisonnement et \u00e0 la critique, tous les \u00e9crivains seront financi\u00e8rement entretenus par les citoyens, et les dialectes dispara\u00eetront au profit d'une langue unique (l'anglais) qui facilitera la communication mondiale. Wells d\u00e9veloppe le fil de sa pens\u00e9e sans aucune retenue, allant m\u00eame jusqu'\u00e0 \u00e9mettre des d\u00e9clarations r\u00e9pugnantes pour nos sensibilit\u00e9s modernes. Son opinion sur \u00ab ces multitudes fourmillantes de demi-Blancs, de demi-Noirs, de n\u00e8gres et de Jaunes 18 \u00bb peut effectivement choquer, surtout lorsqu'il pr\u00f4ne leur expulsion de sa soci\u00e9t\u00e9 id\u00e9ale : \u00ab ...] le monde est le monde, non pas une institution charitable, et je vois qu'il leur faudra dispara\u00eetre. L'ensemble du plan et du sens universels tend vers leur disparition. S'ils ne r\u00e9ussissent pas \u00e0 former des personnalit\u00e9s saines, vigoureuses, distinctes dans le monde spacieux de l'avenir, il leur faudra succomber et s'\u00e9teindre[ 19 \u00bb. La violence du discours nous conduit aujourd'hui \u00e0 nous interroger sur la coh\u00e9rence du nouveau syst\u00e8me politique pr\u00f4n\u00e9 par Wells, et sur le paradoxe qui s'en d\u00e9gage. Le progr\u00e8s dont il r\u00eavait pour les hommes \u00e9tait bel et bien d\u00e9nu\u00e9 de toute humanit\u00e9. Lui qui souhaitait tant porter les esprits vers les hauteurs du savoir et de l'\u00e9ducation avait embourb\u00e9 le sien dans le racisme le plus ignoble. Ces propos, h\u00e9las, \u00e9taient admis \u00e0 l'\u00e9poque dans certains cercles comme celui des Fabiens qui voyaient dans l'eug\u00e9nisme une mani\u00e8re comme une autre d'apporter une solution concr\u00e8te aux probl\u00e8mes sociaux.\n\nCes progressistes en vogue lisent d'ailleurs avec passion l'ouvrage de Wells et lui transmettent leur envie de travailler avec lui. Sa rencontre en 1903 avec Beatrice et Sydney Webb \u2014 deux t\u00e9nors de ce groupe de r\u00e9flexion \u2014 lui donne de l'espoir. Wells se rend bien compte qu'il r\u00eave seul, qu'il parie sur l'avenir sans savoir \u00ab combien de temps il faudra pour que tout cela se r\u00e9alise 20 \u00bb. Avec les Webb et la soci\u00e9t\u00e9 fabienne \u2014 fond\u00e9e en 1884, et r\u00e9unissant des socialistes d\u00e9sireux de r\u00e9former la Grande-Bretagne \u2014, une porte s'ouvre. Voil\u00e0 l'ancien \u00e9l\u00e8ve dissident aux c\u00f4t\u00e9s de Bernard Shaw, Graham Walls, Annie Besant, Sydney Olivier, Hubert Bland : tous les phares de l'intelligentsia de l'\u00e9poque qui ont pour ambition de transformer le pays en profondeur. Wells est plut\u00f4t enthousiaste, il pense que ses projets ont une chance de voir le jour, et il est encore tr\u00e8s habit\u00e9 par son r\u00e9cent voyage aux \u00c9tats-Unis ; un s\u00e9jour vivifiant, \u00e9nerg\u00e9tique, qu'il effectue pour le compte du journal londonien _The Tribune_. Il a pass\u00e9 un peu plus d'un mois \u00e0 d\u00e9couvrir la c\u00f4te Est, \u00e0 se nourrir de toute une litt\u00e9rature prometteuse, et \u00e0 rencontrer des personnalit\u00e9s passionnantes. Maxime Gorki est l'une d'elles. Les deux hommes sont faits pour s'entendre : tous deux autodidactes, volontaires, d\u00e9sireux d'embellir l'humanit\u00e9 par la voie de l'\u00e9ducation. Chacun conna\u00eet l'\u0153uvre de l'autre. \u00c0 cette \u00e9poque, l'\u00e9crivain russe est en exil et vit de ses maigres ressources sur l'\u00eele de Staten Island, pr\u00e8s de New York. Proche de L\u00e9nine et des progressistes qui veulent r\u00e9former le pays, Gorki s'est fix\u00e9 provisoirement en Am\u00e9rique avec pour objectif de recueillir des soutiens et de l'argent pour le Parti. Mais on a r\u00e9serv\u00e9 un accueil glacial \u00e0 cet \u00e9tranger qui voyage avec une femme (russe, elle aussi) hors des liens du mariage. Il trouve ainsi aupr\u00e8s de Wells un confident inattendu, un ami avec lequel il peut partager les valeurs qui lui sont ch\u00e8res. Transport\u00e9 par l'assurance discr\u00e8te de cet homme au regard sombre, Wells se met \u00e0 r\u00eaver de la Russie. Un jour, peut-\u00eatre, il ira.\n\nUne autre entrevue, d'une importance capitale, se d\u00e9roule lors de ce s\u00e9jour : la rencontre avec le pr\u00e9sident des \u00c9tats-Unis, Theodore Roosevelt. Wells mesure sa chance d'\u00eatre re\u00e7u par un tel homme, celui qui a apport\u00e9 ce vent de \u00ab grande lib\u00e9ration 21 \u00bb dans le pays. Invit\u00e9 \u00e0 la Maison-Blanche, il passe avec lui une partie de l'apr\u00e8s-midi \u00e0 marcher dans les jardins de la demeure pr\u00e9sidentielle en \u00e9voquant l'avenir du monde. Il d\u00e9couvre un esprit vif, enthousiaste, qui d\u00e9taille les \u00e9ventualit\u00e9s du futur avec lucidit\u00e9. L'Europe le laisse perplexe \u2014 il ne sait pas de quoi ce continent est capable \u2014, l'Asie ne l'int\u00e9resse pas, et la Russie l'indiff\u00e8re. Wells d\u00e9fend son point de vue, \u00e9num\u00e8re ses arguments en faveur d'un monde unifi\u00e9 (le probl\u00e8me, selon lui, se trouve dans la multiplicit\u00e9 des religions qui, toutes, tendent \u00e0 l'universalit\u00e9 sans y parvenir), et d'un possible syst\u00e8me socialiste capable de porter l'humanit\u00e9 (l'id\u00e9e est nouvelle \u00e0 cette \u00e9poque, l'Am\u00e9rique ne vivant que pour et par le capitalisme). Roosevelt \u00e9coute, attentif, mais il est davantage int\u00e9ress\u00e9 par le romancier que par l'utopiste. Il a lu avec beaucoup d'int\u00e9r\u00eat les livres de Wells, et garde un souvenir imp\u00e9rissable de _La Machine \u00e0 explorer le temps_ , un r\u00e9cit empreint \u00e0 ses yeux de pessimisme. Assis au milieu de la grande pelouse de la Maison-Blanche dans un fauteuil en osier, Roosevelt revient sur l'id\u00e9e, souffl\u00e9e par Wells dans le livre, d'une d\u00e9cadence prochaine de l'humanit\u00e9, et lui dit : \u00ab En supposant qu'apr\u00e8s tout, ...] ceci soit vrai, que tout doive aboutir \u00e0 vos papillons et vos gnomes, cela n'a aucune importance pour le pr\u00e9sent. Notre effort est r\u00e9el. Cela vaut la peine de continuer. Cela vaut la peine de continuer... m\u00eame dans ces conditions-l\u00e0[ 22 ! \u00bb Roosevelt parle en s'agrippant \u00e0 l'anse du fauteuil, l'autre poing serr\u00e9 vers le ciel. Wells regarde cet homme si puissant, si convaincant, lui dire qu'il faut croire non seulement \u00e0 l'avenir, mais aussi aux actions de tous les individus qui se battent, au jour le jour, pour la justice et la libert\u00e9. Le pr\u00e9sent, le _moment_ pr\u00e9sent. Wells, la main en visi\u00e8re et les yeux constamment fix\u00e9s \u00e0 l'horizon, l'avait presque oubli\u00e9.\n\nIl quitte ce grand pays dynamique rempli d'espoir, avec l'intention d'\u00e9crire un essai sur ce formidable mod\u00e8le pour l'Europe *1. Quelque chose lui pla\u00eet en Am\u00e9rique, une ferveur, un enthousiasme serein, l'impression que tout est possible. Ce n'est pas le cas en Angleterre. Non seulement il n'a jamais \u00e9t\u00e9 re\u00e7u \u00e0 Buckingham, mais surtout les Fabiens sont plus frileux qu'il ne le pensait, et ne pr\u00eatent aucune attention \u00e0 ses dol\u00e9ances. Il rentre donc en Angleterre revigor\u00e9, retrouvant les bras de la belle Rosamund Bland, qui a la moiti\u00e9 de son \u00e2ge et la f\u00e2cheuse manie de lui souffler dans le cou d\u00e8s qu'ils sont seuls. C'est la fille adoptive d'Edith et Hubert Bland, un couple d'amis fabiens qui ont une maison de vacances pr\u00e8s de Dymchurch, dans les environs de Sandgate. Il a rencontr\u00e9 Rosamund chez eux un dimanche apr\u00e8s-midi. Elle lui a tout de suite plu. Elle aussi fait partie des Fabiens, mais s'occupe de la Pouponni\u00e8re, l'antichambre du groupe qui r\u00e9unit les jeunes adh\u00e9rents. Depuis peu, ils se voient en cachette, et projettent de passer un week-end \u00e0 Paris, rien que tous les deux. Le jour de leur d\u00e9part clandestin, Hubert Bland les intercepte \u00e0 la gare de Paddington. Il avait bien remarqu\u00e9 des regards \u00e9chang\u00e9s et une complicit\u00e9 douteuse. Et puis il avait entendu Rosamund sortir plus t\u00f4t que pr\u00e9vu de la maison. La confrontation est terrible : Hubert, visiblement irrit\u00e9, son monocle se balan\u00e7ant au bout de son ruban, s'approche avec d\u00e9termination, frappe Wells d'un coup de poing au visage, et prend sa fille par le bras avant de s'en aller. Wells, \u00e0 terre, n'a encore une fois rien vu venir. Depuis son altercation avec Sutton, dans sa prime jeunesse, il s'\u00e9tait pourtant jur\u00e9 que plus jamais il ne se ferait battre par un homme. En vain.\n\nIl a une marque rouge sur la joue et d\u00e9cide de se faire discret pendant quelques jours. Laisser l'orage passer, et surtout \u00e9viter de croiser Hubert. Une fois r\u00e9tabli, il d\u00e9cide de marquer son retour et de formuler toutes les propositions qu'il a en t\u00eate pour am\u00e9liorer le fonctionnement interne des Fabiens. Aucun membre du groupe n'a encore os\u00e9 remettre en cause la Soci\u00e9t\u00e9. Wells y voit une opportunit\u00e9, d\u00e9cide d'affronter ses pairs et de leur lire un texte critique, dont il est l'auteur, au titre sans \u00e9quivoque : \u00ab Faults of the Fabian \u00bb (\u00ab Les fautes des Fabiens \u00bb). Dans cette allocution, il presse ses camarades de mener une propagande socialiste aupr\u00e8s de tous les intellectuels du pays afin de pr\u00f4ner le progr\u00e8s. Il veut surtout que les Fabiens s'allient au parti travailliste, et qu'ils trouvent ainsi un engagement politique concret. Mais Wells est mauvais orateur, voire \u00ab remarquablement absurde et incapable 23 \u00bb selon ses propres termes. Il ne se sent pas \u00e0 l'aise dans ce r\u00f4le de d\u00e9tracteur, et \u00e9prouve sa maladresse au moment de prendre la parole. Les plus jeunes l'\u00e9coutent avec int\u00e9r\u00eat, les vieux chefs sont sceptiques et refusent majoritairement toute id\u00e9e de changement. Wells tente alors sa chance aupr\u00e8s des antennes locales, plus petites, plus ind\u00e9pendantes, mais se rend compte de la difficult\u00e9 d'imposer un quelconque \u00e9lan. Les Fabiens ne le rejettent pas, mais lui font comprendre qu'il ne sert \u00e0 rien de s'agiter. Wells, lui, n'a qu'une envie : secouer le monde, le r\u00e9veiller de son sommeil scl\u00e9rosant. L'agitateur tire sa r\u00e9v\u00e9rence, envoie sa lettre de d\u00e9mission et retourne \u00e0 ses livres, l\u00e0 o\u00f9 il peut concentrer toute son \u00e9nergie. Il quitte les Fabiens dans l'amertume, avec le sentiment de n'avoir pu changer les choses et, coll\u00e9e \u00e0 la peau, une encombrante \u00e9tiquette de \u00ab proph\u00e8te \u00bb.\n\nSon d\u00e9part se teinte d'un \u00e9ni\u00e8me parfum de scandale. Celui qui s'est permis de critiquer les grands pontes de l'organisation a aussi nou\u00e9 une relation des plus passionn\u00e9es avec l'une des recrues du groupe, jolie et brillante, fille de politicien : Amber Reeves, vingt ans \u00e0 peine. Son visage, son allure, servent de mod\u00e8le romanesque au portrait de la nouvelle h\u00e9ro\u00efne de Wells, qui a pour nom Ann Veronica. La voici donc, \u00ab les cheveux noirs, de beaux sourcils et le teint clair ; et les forces qui avaient model\u00e9 les traits avaient pris go\u00fbt \u00e0 leur ouvrage et s'y \u00e9taient attard\u00e9es pour les rendre fins et d\u00e9licats. Elle \u00e9tait svelte ; tant\u00f4t elle donnait l'impression d'\u00eatre grande : sa d\u00e9marche et son port avaient une l\u00e9g\u00e8ret\u00e9 all\u00e8gre \u00e9voquant une personne qui se sent d'ordinaire et de fa\u00e7on habituelle bien dans sa peau ; et tant\u00f4t elle se vo\u00fbtait l\u00e9g\u00e8rement sous le poids de la pens\u00e9e. Ses l\u00e8vres se rejoignaient avec une expression \u00e0 mi-chemin du contentement et d'un sourire esquiss\u00e9 ; son abord \u00e9tait empreint d'une r\u00e9serve pudique et, \u00e0 l'abri de ce masque, elle se sentait furieusement insatisfaite, aspirant \u00e0 la libert\u00e9 et \u00e0 la vie 24 \u00bb.\n\nComme avec Rosamund, il n'a eu pas besoin de la s\u00e9duire. La jeune femme lui a tr\u00e8s t\u00f4t manifest\u00e9 publiquement son int\u00e9r\u00eat. Elle conna\u00eet par c\u0153ur certains de ses ouvrages et adh\u00e8re sans r\u00e9serve \u00e0 toutes ses id\u00e9es. Maud, la m\u00e8re d'Amber \u2014 elle aussi membre des Fabiens \u2014, assiste \u00e0 ce coup de foudre intellectuel et encourage sa prog\u00e9niture \u00e0 se rapprocher de Wells, qu'elle appr\u00e9cie pour son charisme et son temp\u00e9rament de feu. Mais elle est loin de se douter que les chastes promenades et les longues discussions philosophiques vont bient\u00f4t faire place \u00e0 des liens plus intimes. Amber s'adresse \u00e0 Wells en le nommant \u00ab Ma\u00eetre 25 \u00bb ; lui l'appelle \u00ab Dusa \u00bb \u2014 diminutif de \u00ab Medusa \u00bb dont la t\u00eate coup\u00e9e, tenue par Pers\u00e9e et sculpt\u00e9e par l'Italien Cellini, est depuis longtemps admir\u00e9e par la jeune fille. Un jour, n'y tenant plus, elle se jette \u00e0 l'eau et lui avoue qu'elle est amoureuse. \u00ab De qui 26 ? \u00bb, demande subtilement l'int\u00e9ress\u00e9. Pour toute r\u00e9ponse, Amber l'embrasse fougueusement. Nous sommes en 1908. C'est le d\u00e9but d'une liaison charnelle immortalis\u00e9e dans le roman _Ann Veronica_ , qui cr\u00e9era lors de sa publication une immense pol\u00e9mique.\n\nWells et Amber vivent leur amour en cachette \u00e0 Londres, o\u00f9 elle poursuit ses \u00e9tudes et lui continue de creuser son sillon litt\u00e9raire. Mais l'idylle clandestine est \u00e9ph\u00e9m\u00e8re, Amber ne r\u00e9sistant pas \u00e0 l'envie de se confier \u00e0 sa m\u00e8re (inqui\u00e8te) et \u00e0 d'autres amis (bavards). Wells se rend bient\u00f4t compte que la moiti\u00e9 de l'universit\u00e9 est au courant \u2014 des \u00e9l\u00e8ves mais surtout des professeurs qu'il conna\u00eet bien. Elle en a aussi parl\u00e9 \u00e0 ce jeune homme, Rivers Blanco White, \u00e9tudiant en droit, qui nourrit depuis longtemps pour elle des sentiments amoureux. Il lui a d\u00e9j\u00e0 propos\u00e9 de l'\u00e9pouser, mais elle a d\u00e9clin\u00e9, ne pouvant se r\u00e9soudre \u00e0 se s\u00e9parer de Wells. Est-ce Rivers qui, pour se venger, a d\u00e9voil\u00e9 le secret de leur liaison au p\u00e8re de la jeune fille (lequel, de mani\u00e8re \u00e9tonnante, est rest\u00e9 imperm\u00e9able \u00e0 la scandaleuse nouvelle) ? Difficile \u00e0 savoir. Au sein des Fabiens, on ne parle que de cela. Le p\u00e8re d'Amber tente de trouver une issue aux rumeurs. Il convoque sa fille un soir et l'oblige \u00e0 accepter la main de Rivers. Prise au pi\u00e8ge, Amber supplie Wells de lui faire un enfant, pensant qu'une grossesse pourrait compromettre ce mariage forc\u00e9. Une fois enceinte, elle annonce la nouvelle \u00e0 son fianc\u00e9 qui reste sur ses positions. Alors, il faut partir. Apr\u00e8s en avoir averti Jane qui n'est dupe de rien et ferme les yeux sur cette situation d\u00e9sormais publique, Wells emm\u00e8ne Amber \u00e0 Victoria Station. Ensemble, ils partent pour Le Touquet, en France, dans un chalet que l'\u00e9crivain a lou\u00e9 pour quelques semaines. L\u00e0-bas, enfin seuls, ils r\u00e9fl\u00e9chissent \u00e0 ce qu'ils peuvent faire. Fuir, dispara\u00eetre ? Non. Divorcer de Jane ? Non plus. Wells ne veut pas la quitter, il aime son \u00e9pouse et ses enfants. S'il faut choisir, il choisit sa famille. Face aux menaces constantes des parents d'Amber qui entendent la d\u00e9sh\u00e9riter si elle ne revient pas \u00e0 la raison, les fugitifs prennent donc une d\u00e9cision : renoncer \u00e0 leur amour. Wells raccompagne sa jeune prot\u00e9g\u00e9e, un matin, sur le quai de Boulogne et la regarde s'\u00e9loigner en bateau. Peu de temps apr\u00e8s, Amber \u00e9pouse Rivers.\n\nD\u00e9vast\u00e9, Wells n'a pas la force de rentrer chez lui. Jane, avertie de toute l'affaire, le rejoint avec les enfants. Elle n'exprime aucun ressentiment, aucune ranc\u0153ur. \u00ab _Wonderful_ 27 \u00bb (\u00ab merveilleux \u00bb) est le mot employ\u00e9 par l'auteur des ann\u00e9es plus tard pour d\u00e9crire son \u00e9pouse \u00e0 cet instant. \u00c0 ses c\u00f4t\u00e9s, il se remet \u00e0 \u00e9crire, d\u00e9cide de tourner la page en vendant Spade House, lieu de l'\u00e9chec du triangle amoureux, \u00ab cette maison qui] menac[e] de devenir le d\u00e9cor d\u00e9finitif de [s]a vie[ 28 \u00bb. Il part s'installer dans la capitale, \u00e0 Hampstead, ouvrant ainsi un nouveau chapitre. Mais tr\u00e8s vite, il re\u00e7oit des nouvelles d'Amber. Son union avec Rivers n'a rien d'\u00e9panouissant. Elle a quitt\u00e9 leur maison et cherche un endroit o\u00f9 vivre et mener sa grossesse \u00e0 terme. Wells prend les choses en main, et l'installe dans la demeure d'une amie, \u00e0 Blyth, dans le comt\u00e9 de Northumberland, pr\u00e8s de la fronti\u00e8re \u00e9cossaise. L\u00e0-bas, il lui rend visite r\u00e9guli\u00e8rement, et prend soin d'elle.\n\n_Ann Veronica_ est publi\u00e9. Cette h\u00e9ro\u00efne vierge qui tombe amoureuse d'un homme mari\u00e9 \u2014 son professeur de biologie, Capes \u2014 sans craindre de le lui dire est le parfait miroir de ce qu'ils ont v\u00e9cu. Amber craint que tout cela n'envenime encore davantage un climat d\u00e9j\u00e0 tourment\u00e9. Mais il est trop tard pour penser \u00e0 cela. Le livre est en vente partout, et suscite un \u00e9clat dans le cercle des Fabiens, choqu\u00e9s \u00e0 l'id\u00e9e qu'une jeune fille (on soup\u00e7onne \u00e9videmment qu'il s'agit d'Amber) puisse montrer son attirance pour l'autre sexe. \u00ab Je vous d\u00e9sire. Je veux que vous soyez mon amant. Je veux me donner \u00e0 vous. Je veux \u00eatre tout ce que je peux pour vous 29. \u00bb Ces quelques mots sont r\u00e9volutionnaires, tout comme la formulation de la n\u00e9gligence assum\u00e9e des conventions :\n\nVous n'\u00eates pas pour moi un homme parmi d'autres : je veux dire, un repr\u00e9sentant de votre sexe. Vous \u00eates simplement un \u00eatre particulier qui \u00e9chappe \u00e0 toutes les classifications du monde. Vous \u00eates n\u00e9cessaire \u00e0 ma vie. Je n'ai jamais rencontr\u00e9 quelqu'un comme vous. Que vous m'apparteniez compte plus que tout. Rien d'autre ne vient en balance. La moralit\u00e9 passe apr\u00e8s cela. Peu importe que nous ne puissions jamais nous marier. Je n'ai absolument peur de rien : du scandale, des difficult\u00e9s, du combat... J'ai plut\u00f4t envie d'eux. J'en ai vraiment envie 30.\n\nWells cr\u00e9e un personnage f\u00e9minin qui n'a pas honte de ses actions, et qui agit sans que soit prononc\u00e9e contre lui de sentence. Non seulement Ann Veronica est audacieuse, mais son audace n'est ni condamn\u00e9e ni punie par le romancier. Wells le sait, il va trop loin. Il se f\u00e9licite de mettre en mots ce que les femmes pensent tout bas :\n\nMon livre fut \u00e9crit essentiellement pour exprimer le ressentiment et la d\u00e9tresse qu'\u00e9prouvent aujourd'hui beaucoup de femmes devant le fait qu'elles d\u00e9pendent pratiquement et in\u00e9vitablement de quelque individu masculin qu'elles n'ont point d\u00e9lib\u00e9r\u00e9ment choisi, et en accord complet avec l'id\u00e9e naturelle, mais peut-\u00eatre anarchique et antisociale, qu'il est intol\u00e9rable pour une femme d'avoir des rapports sexuels avec un homme qu'elle n'aime pas, et naturel, d\u00e9sirable et admirable qu'elle d\u00e9sire en avoir avec un homme qu'elle a elle-m\u00eame choisi et, encore plus, qu'elle souhaite en avoir des enfants 31.\n\nCe n'est pas la premi\u00e8re fois que Wells se place du c\u00f4t\u00e9 de la femme. D\u00e9j\u00e0 en 1905, dans _Une utopie moderne_ , il avait \u00e9nonc\u00e9 plusieurs mesures visant \u00e0 mettre en valeur son r\u00f4le dans la soci\u00e9t\u00e9. Constatant que les femmes \u00e9taient libres \u00ab en th\u00e9orie et aucunement en pratique 32 \u00bb \u2014 notamment sur le plan \u00e9conomique \u2014, il avait imagin\u00e9 un service d'allocations qui permettrait de r\u00e9mun\u00e9rer celles qui mettaient au monde un enfant. Selon lui, devenir m\u00e8re \u00e9tait, au m\u00eame titre que les m\u00e9tiers exerc\u00e9s par les hommes (et aussi glacial que cela puisse para\u00eetre), \u00ab un service rendu \u00e0 l'\u00c9tat 33 \u00bb, et devait en cons\u00e9quence \u00eatre pris en compte.\n\nPeu apr\u00e8s sa publication, _Ann Veronica_ est interdit en biblioth\u00e8que, s\u00e9v\u00e8rement critiqu\u00e9 par l'\u00c9glise 34 et dans la presse \u2014 ce qui favorise, \u00e9videmment, les ventes. Wells tente de calmer les esprits, et modifie la fin de son texte en d\u00e9crivant le mariage d'Ann Veronica et du divorc\u00e9 Capes. Il le regrette vite. Non seulement il a l'impression de donner raison aux moralistes, mais surtout, rien n'arr\u00eate le \u00ab torrent d'injures, de fausset\u00e9s, de condamnations h\u00e2tives 35 \u00bb. Un critique, en particulier (John St Loe Strachey, r\u00e9dacteur en chef du _Spectator_ et contradicteur coutumier de Wells), trouve un malin plaisir \u00e0 clouer au pilori cette histoire d'amour moderne :\n\nLe d\u00e9go\u00fbt et l'indignation que nous inspire ce livre tiennent \u00e0 ce qu'il risque fort d'\u00e9mousser chez l'individu le sens de la continence qui est essentiel \u00e0 un \u00c9tat solide et sain. Il nous apprend en v\u00e9rit\u00e9 qu'il n'est rien de tel que l'honneur d'une femme, ou s'il existe, il ne saurait \u00eatre rien d'autre qu'un rempart contre la faiblesse de la tentation... si une envie animale ou un d\u00e9sir est suffisamment absorbant, alors il faut lui ob\u00e9ir. L'abn\u00e9gation est un r\u00eave et la retenue une illusion. De telles choses n'ont pas leur place dans le monde fangeux des fantasmes de Mr Wells. Son univers est fait d'hermines et de furets, qui ignorent ce que peuvent \u00eatre le devoir et l'abn\u00e9gation 36.\n\nWells riposte, s'offusque que son livre ne puisse pas circuler entre toutes les mains et revendique ce \u00ab monde fangeux \u00bb qui est le sien. Le roman anglais est \u00e9touff\u00e9 par la tradition, Wells veut l'encanailler, le m\u00ealer aux interdits et surtout, qu'il embrasse \u00ab l'amour libre \u00bb. Cette notion appara\u00eet dans _Une utopie moderne_ , l'essai romanc\u00e9 publi\u00e9 quelques ann\u00e9es plus t\u00f4t. Toujours tr\u00e8s influenc\u00e9 par la pens\u00e9e platonicienne, Wells imagine un monde id\u00e9al, \u00ab sain et heureux 37 \u00bb, tout entier d\u00e9vou\u00e9 \u00e0 l'\u00e9ducation des individus, et dirig\u00e9 par une \u00e9lite \u2014 l'ordre des Samoura\u00efs \u2014 travaillant \u00e0 la construction d'un v\u00e9ritable syst\u00e8me socialiste \u2014 un monde qui ressemble davantage, dans les faits, \u00e0 un \u00c9tat policier. Pas de places pour les faibles, il faut que \u00ab les inf\u00e9rieurs soient mis hors d'\u00e9tat de cro\u00eetre et de multiplier 38 \u00bb. Wells ne veut pas des idiots et des fous, des pervers et des incapables, de ceux aussi qui peuvent \u00eatre \u00ab atteints de certaines maladies r\u00e9pugnantes et transmissibles 39 \u00bb. Il a non seulement le go\u00fbt de l'excellence, mais surtout celui de l'ordre et de l'autorit\u00e9, refusant d'envisager un monde dirig\u00e9 par \u00ab n'importe qui 40 \u00bb. L'\u00e9lite qu'il esp\u00e8re voir \u00e9merger, il l'imagine scrupuleusement s\u00e9lectionn\u00e9e \u2014 d\u00e8s la naissance, gr\u00e2ce \u00e0 la pratique r\u00e9pandue de l'eug\u00e9nisme, puis plus tard, au sein des classes sociales sup\u00e9rieures (les \u00ab Kin\u00e9tiques \u00bb et les \u00ab Po\u00e9tiques \u00bb). Hommes ou femmes, ils ont l'interdiction formelle de boire, de fumer, de chanter, ou m\u00eame de parier de l'argent, leur vie \u00e9tant consacr\u00e9e \u00e0 l'instruction et \u00e0 l'exemplarit\u00e9. Ils suivent un r\u00e9gime alimentaire strict, exercent r\u00e9guli\u00e8rement leur esprit \u00e0 la m\u00e9ditation et pratiquent entre eux ce fameux \u00ab amour libre \u00bb. L'auteur envisage ainsi comme principe fondamental un \u00e9tat d'affranchissement total concernant les relations sentimentales ou charnelles, consid\u00e9rant que \u00ab la morale priv\u00e9e d'un citoyen adulte ne regarde en rien l'\u00c9tat 41 \u00bb. L'infid\u00e9lit\u00e9 n'est donc pas une faute, elle est admise et fait partie d'un processus d'\u00e9quilibre entre les individus.\n\nWells, lui-m\u00eame, met un soin particulier \u00e0 explorer cette conception depuis de nombreuses ann\u00e9es. Ses frasques donjuanesques sont connues de tous. On parle de May Nisbet, fille d'un journaliste du _Times_ , quinze ans \u00e0 peine, qui aurait surgi un jour dans son salon en l\u00e9g\u00e8re robe de plage ; de Violet Hunt, une proche de Bennett, romanci\u00e8re franco-britannique avec laquelle il \u00e9cume les restaurants de Londres ; de Dorothy Richardson, une amie de Jane, blonde \u00e0 la plume incendiaire en qu\u00eate d'un \u00e9quilibre spirituel, qui aime faire l'amour dans les foug\u00e8res ; de cette prostitu\u00e9e aussi, rencontr\u00e9e \u00e0 Washington, qui serait tomb\u00e9e amoureuse de lui \u00e0 la suite d'un t\u00eate-\u00e0-t\u00eate en maison close ; ou encore de cette Australienne qui a lu _Kipps_ et qui \u00e9crit au romancier pour lui dire tout le bien qu'elle pense de ses livres, avant de le retrouver dans une chambre d'h\u00f4tel. Dans les faits, donc, Wells applique au jour le jour sa th\u00e9orie, mais tr\u00e8s vite, elle n'est plus assum\u00e9e, voire se trouve ni\u00e9e par l'\u00e9crivain qui comprend que cela peut nuire \u00e0 sa r\u00e9putation. Wells en fait l'aveu dans une interview 42, des ann\u00e9es plus tard, expliquant ce revirement d'opinion par une prudence \u00e9conomique. Il ne pouvait pr\u00f4ner quelque chose qui le conduirait \u00e0 se mettre \u00e0 dos des lecteurs, et l'emp\u00eacherait probablement de vendre beaucoup de livres.\n\nLes mois passent, et Rivers veut r\u00e9cup\u00e9rer son \u00e9pouse. Amber et Wells d\u00e9cident d'un commun accord de se s\u00e9parer pour de bon. Les f\u00eates de No\u00ebl sont c\u00e9l\u00e9br\u00e9es, et la fugitive met au monde une petite fille, Anna Jane. De nombreuses ann\u00e9es s'\u00e9coulent avant que Wells ne se rende compte de sa contradiction. Lui qui encourageait l'amour libre refusait secr\u00e8tement \u00e0 la fois de \u00ab lib\u00e9rer \u00bb Jane et de partager Amber :\n\nJe th\u00e9orisais \u00e0 propos de l'amour libre, mais je gardais Amber pour moi seul. J'essayais de maintenir une configuration triangulaire. Jane demeurait invinciblement ma femme, et Amber la jeune ma\u00eetresse ; nous nous comprenions, nous nous affirmions, magnifiquement 43.\n\nIl revoit son ancienne ma\u00eetresse apr\u00e8s la guerre et suit de loin la croissance de sa fille qui excelle dans ses \u00e9tudes. Les derni\u00e8res nouvelles qu'il re\u00e7oit d'Amber lui parviennent dans une lettre datant de l'\u00e9t\u00e9 1939 : elle le remercie sinc\u00e8rement pour ces ann\u00e9es pass\u00e9es \u00e0 l'aimer, et lui assure ne rien regretter de leur histoire :\n\nNous sommes rentr\u00e9s hier de Wales et avons re\u00e7u ton livre \u2014 qui occupera nos pens\u00e9es \u2014 comme une b\u00e9n\u00e9diction. \u00c0 un moment pareil, quand la vie telle que nous la connaissons semble se terminer pour nous tous, on r\u00e9fl\u00e9chit \u00e0 ce qu'on a v\u00e9cu, et m\u00eame s'il n'y avait pas ce livre je pense que je t'aurais \u00e9crit pour te remercier. Ce que tu m'as donn\u00e9 tout au long de ces ann\u00e9es \u2014 un amour qui m'a sembl\u00e9 parfait, l'influence de ton esprit, et notre fille \u2014 ne m'a jamais quitt\u00e9 depuis. Je n'ai jamais pens\u00e9 un instant que tout cela n'en valait pas la peine 44.\n\n*1. _The Future in America : A Search After Realities_ (1906).\nLa poursuite de l'amour\n\nAmber. Corps sublime, \u00e2me tendre, fantasme absolu. En \u00e9tait-il pour autant vraiment amoureux ? Il voudrait dissoudre toutes ces interrogations qui lui renvoient le poids des ann\u00e9es \u2014 ce qu'il a \u00e9t\u00e9, ce qu'il a endur\u00e9 et r\u00e9ussi, ce qu'il aurait pu faire, ce qu'il aimait ou croyait aimer. Sur ce dernier point, il ne sera jamais s\u00fbr. L'amour demeure myst\u00e9rieux. Il est une chim\u00e8re qu'il voudrait attraper, qui semble \u00e0 port\u00e9e de main mais se d\u00e9robe sans cesse \u00e0 lui : l'Ombre-Amoureuse (\u00ab _Lover-Shadow_ \u00bb). Et pourtant, depuis toujours, il porte en lui l'esp\u00e9rance d'une possibilit\u00e9, l'id\u00e9e que, peut-\u00eatre, chaque individu en ce monde a une \u00e2me s\u0153ur quelque part :\n\nJe pense que dans l'esprit de chacun, possiblement depuis l'\u00e2ge le plus tendre, il existe un complexe d'attente et d'espoir \u2014 toujours grandissant, toujours plus subtil ; une agr\u00e9gation de pens\u00e9es aimables et excitantes ; des concepts de rencontre et de r\u00e9action tir\u00e9s de l'observation, des descriptions, et du drame ; des r\u00eaveries de d\u00e9lices sensuelles et d'extases ; des r\u00eaveries de compr\u00e9hension et de r\u00e9ciprocit\u00e9 ; ce que j'appellerai \u00ab _Lover-Shadow_ \u00bb et que je pense aussi essentiel \u00e0 nos vies que la conscience de soi. C'est une conscience autre. Aucun \u00eatre humain n'affronte le monde conscient de sa compl\u00e8te solitude ; aucun \u00eatre humain, je crois, ne vit ou ne peut vivre sans cette pr\u00e9sence qui pour \u00eatre vague, prot\u00e9iforme et changeante n'en est pas moins tr\u00e8s r\u00e9elle, \u00e0 c\u00f4t\u00e9 de la persona. Voil\u00e0 ce que je veux dire par _Lover-Shadow_. Il s'agit du corr\u00e9lat ins\u00e9parable de la persona, lorsqu'il est question de diriger nos vies. M\u00eame si on lui refuse toute reconnaissance ; m\u00eame si on lui d\u00e9nie toute existence ; le _Lover-Shadow_ est bien l\u00e0, pr\u00e9sidant \u00e0 l'\u00e9criture des livres, des po\u00e8mes, des images 1.\n\nLui aussi, au fond, \u00e9crit des livres pour cela, m\u00eame lorsqu'il \u00e9voque la guerre \u2014 c'est juste une autre mani\u00e8re de parler d'amour. Inscrire sur le papier ce qu'il ne peut, dans la vraie vie, saisir, tenter de comprendre ce qu'est ce sentiment fugitif : voil\u00e0 la v\u00e9ritable \u00ab anticipation \u00bb, la seule vraie raison d'\u00e9crire, ce qui justifie m\u00eame le risque de devenir \u00e9crivain. Il sait trop ce qu'est le d\u00e9sir, il en a connu, des femmes, \u00e9treint, des corps \u2014 certains d\u00e9sormais oubli\u00e9s, d'autres dont il a encore la marque. \u00ab In\u00e9branlable 2 \u00bb, dirait Shakespeare :\n\nAu mariage de deux esprits s'ils sont sinc\u00e8res\n\nJe ne con\u00e7ois d'obstacle. Car l'amour\n\nN'est pas l'amour, si l'un change quand l'autre change,\n\nOui si, trahi, il d\u00e9sire trahir.\n\nAh non ! Lui, c'est le phare en son point fixe,\n\nIn\u00e9branlable au fort de la temp\u00eate,\n\nC'est l'\u00e9toile des barques d\u00e9sempar\u00e9es,\n\nDont calculable est le ciel mais inestimable le don.\n\n[...]\n\nL'amour ne souffre pas en ces heures br\u00e8ves,\n\nIl va droit jusqu'\u00e0 celle du Jugement.\n\nEt si c'est faux, cela, et qu'on me le prouve,\n\nAu diable mes \u00e9crits ! Nul n'aura donc aim\u00e9 3.\n\nLa vague le submerge d'un seul coup, \u00e7a le prend \u00e0 la gorge, \u00e7a lui pique les yeux. Ce c\u0153ur, l\u00e0, dans sa poitrine, n'aurait peut-\u00eatre jamais aim\u00e9. Il a toujours remarqu\u00e9 qu'il battait lentement. Jamais il ne s'emballe. M\u00eame \u00e0 l'enterrement de sa m\u00e8re, son rythme cardiaque ne s'est pas acc\u00e9l\u00e9r\u00e9. Il se souvient du malaise qu'il a ressenti ce jour-l\u00e0. C'\u00e9tait en 1905, juste avant l'\u00e9t\u00e9. Elle venait de mourir et lui se demandait que penser de cette femme qui \u00e9tait partie dans l'air moite de sa chambre, sans dire un mot, sans jamais se plaindre, sans rien demander ; que penser de son comportement \u00e0 lui, son fils, qui ne lui avait jamais exprim\u00e9 sa tendresse, sa reconnaissance, et qui la voyait assez peu ces derni\u00e8res ann\u00e9es. Elle \u00e9tait venue, pourtant, passer quelques jours \u00e0 Spade House. Comme elle avait \u00e9t\u00e9 heureuse ! Elle se mettait au soleil d\u00e8s la fin d'apr\u00e8s-midi, sur une chaise en osier face \u00e0 la mer. Elle \u00e9tait habill\u00e9e de noir, comme \u00e0 son habitude. Elle souriait.\n\nLa veille de l'enterrement, il avait retrouv\u00e9 un papier dans une corbeille pr\u00e8s de son lit. Sarah y avait gribouill\u00e9 quelque chose, d'une \u00e9criture mal assur\u00e9e, et il avait reconnu le mot inscrit sur la tombe d'Huxley, qui \u00e9voquait non pas l'immortalit\u00e9, mais le doux repos \u00e9ternel. Sa pieuse m\u00e8re, qui avait tant d\u00e9nigr\u00e9 ce professeur ath\u00e9e et critiqu\u00e9 le choix de son fils de suivre ses cours, avait scrupuleusement recopi\u00e9 ces lignes avant de mourir. Cela le touche de penser qu'elle ait pu \u00eatre \u00e9mue par les mots de son mentor, qu'elle ait m\u00eame pu les \u00e9crire en pensant \u00e0 lui. Ce m\u00eame jour, en mettant de l'ordre dans ses affaires, il \u00e9tait aussi tomb\u00e9 par hasard sur son journal intime 4, et il s'\u00e9tait empress\u00e9 de le lire. Il n'aurait pas d\u00fb, car ce qu'il a d\u00e9couvert dans ce cahier interdit l'emp\u00eache maintenant de dormir : page apr\u00e8s page, sa m\u00e8re \u00e9crit sa frustration de vivre et ce chagrin insondable, jamais gu\u00e9ri, de la perte de sa fille a\u00een\u00e9e. C'est inscrit noir sur blanc : si elle a voulu \u00e0 tout prix retomber enceinte apr\u00e8s la mort de Fanny, c'\u00e9tait pour la remplacer. Elle d\u00e9sirait avoir une fille. Au lieu de cela, Herbert est venu au monde.\n\nLui se trouvait l\u00e0, en face du cercueil qui s'enfon\u00e7ait dans le sol, observant la couche de terre rabattue sur le corps de celle qui lui a donn\u00e9 la vie sans le vouloir vraiment. Il se souvient d'avoir eu honte au point de vouloir dispara\u00eetre : il avait fallu que sa m\u00e8re meure pour qu'il sache non seulement qui elle \u00e9tait, mais surtout pour qu'il se rende compte qu'il ne l'avait jamais aim\u00e9e \u2014 qu'il l'aimait d\u00e9sormais, \u00e9trangement, et qu'il \u00e9tait trop tard pour le lui dire 5. L'a-t-il assez dit \u00e0 Jane ? Non, il le devrait. Cependant, ils ont un accord tous les deux : leur libert\u00e9 avant tout. Leur mariage est r\u00e9el, leur union est ouverte, chacun ayant le droit de vivre ce qu'il veut tant que cela ne menace pas l'\u00e9quilibre du foyer familial. Seul Wells en a pour l'instant profit\u00e9. Guid\u00e9 par son d\u00e9sir, amen\u00e9 naturellement \u00e0 rencontrer de nouvelles personnes, sans cesse courtis\u00e9 par de jeunes lectrices, il est de ces hommes qui ne savent pas dire non. Lui pr\u00e9f\u00e8re dire : \u00ab Pourquoi pas 6 ? \u00bb Michael Sherborne, auteur d'une biographie de H. G. Wells, in\u00e9dite en fran\u00e7ais, a retrouv\u00e9, dans un num\u00e9ro de la _Pall Mall Gazette_ datant de mai 1894, un po\u00e8me intitul\u00e9 \u00ab \u00c9pisode \u00bb, sans nom d'auteur, mais dont tout laisse \u00e0 penser qu'il a \u00e9t\u00e9 \u00e9crit par Wells. Dans ces quelques phrases se refl\u00e8te la double vie du romancier, partag\u00e9 entre ses idylles d'un jour et son calme quotidien de mari :\n\nUne rencontre sous un arbre verdoyant\n\nSous une lumi\u00e8re douce traversant les feuilles\n\nUne rencontre oui, et une passion \u00e0 conna\u00eetre\n\nSi je lis bien tes yeux.\n\nUne s\u00e9paration sous l'arbre vert,\n\nUne passion d\u00e9licate de la douleur\n\n(Et sobrement je retrouve ma\n\nJane, mature et \u00e9l\u00e9gante) 7.\n\nJane n'a de son c\u00f4t\u00e9 jamais d'aventures. Soit elle ne le veut pas, soit elle n'en a pas l'occasion. En tout cas, elle n'ignore rien des \u00e9carts de son mari, et les supporte en silence.\n\nElle s'occupe de tout, recopiant les manuscrits de Bert, g\u00e9rant ses contrats, sa comptabilit\u00e9, organisant les soir\u00e9es, et le soutenant moralement en toutes circonstances. Elle seule peut \u00e9mettre un jugement sur les \u00e9crits de son mari que ce dernier prend vraiment au s\u00e9rieux. Elle est son critique le plus redoutable, le plus juste. Wells ne cesse de lui demander son avis, d'attendre ses remarques. Tout pour lui, peu pour elle. Ses go\u00fbts \u2014 pour la plume de Marcel Proust, la d\u00e9licatesse de Katherine Mansfield, la peinture figurative et les charades qu'elle invente par dizaines pour amuser les invit\u00e9s \u2014 comme ses ambitions (foisonnantes), elle les garde secrets. \u00c0 bien y r\u00e9fl\u00e9chir, elle se sent bien telle qu'elle est, dans cette grande maison avec ses enfants, \u00e0 s'acheter les robes qu'elle d\u00e9sire, \u00e0 se rendre aux d\u00eeners mondains, \u00e0 partir en voyage quand elle le souhaite. Les pr\u00e9tentions litt\u00e9raires qu'elle peut avoir sont r\u00e9elles, mais elles doivent rester secondaires. Un plaisir solitaire. \u00c0 la fin de la journ\u00e9e, lorsqu'elle s'est occup\u00e9e de toutes les affaires de Wells, elle se met \u00e0 son bureau et \u00e9crit : des nouvelles, des po\u00e8mes, des textes \u00e9pars qu'elle range pr\u00e9cieusement dans une chemise en carton, gliss\u00e9e dans le tiroir de la commode. Personne d'autre qu'elle ne les lit, elle n'envisage m\u00eame pas de les pr\u00e9senter \u00e0 un \u00e9diteur, que penserait-il d'elle, l'\u00e9pouse et secr\u00e9taire de l'\u00e9crivain le plus connu d'Angleterre ? Elle a bien song\u00e9 \u00e0 en envoyer quelques-uns, mais y a renonc\u00e9 assez vite.\n\nElle aime son mari, elle ferait tout pour lui \u2014 elle vit _pour_ lui. Elle l'aime quand il tourne en rond dans le jardin en fumant sa pipe, quand ils font du tandem le dimanche dans la campagne, quand elle le regarde, assis dans un fauteuil du salon, \u00e9couter de la musique sur son gramophone. Elle aime sa fantaisie, cette bulle dans laquelle ils vivent. Il peut bien se perdre dans les bras d'autres femmes, rien ne pourra alt\u00e9rer leur amour. Ils sont \u00e9poux mais ils sont aussi partenaires. C'est la seule chose qui compte. Pour le reste, elle ferme les yeux, n'\u00e9tant dupe de rien. \u00ab J'ai fait tout ce que j'ai voulu faire, de sorte que chacune de mes pulsions sexuelles a pu s'exprimer 8. \u00bb Rien d'impossible pour Wells, jamais. L'esprit travaille et le corps exulte, sans frustration. Et cela, en fait, n'a rien \u00e0 voir avec l'amour, l'attachement absolu, avec ce r\u00eave d'une \u00ab pr\u00e9sence 9 \u00bb fondamentale qui lui \u00f4terait tout souci d'exister. Il n'est pas comme \u00ab la moiti\u00e9 des gens dou\u00e9s d'imagination qui] succombent \u00e0 cette conviction aussi naturellement que les canetons vont \u00e0 l'eau[ 10 \u00bb. Lui avance dans un d\u00e9sert, tels les \u00ab P\u00e8lerins de l'Amour 11 \u00bb, en sachant qu'il aura toujours soif. Aimer les femmes est une n\u00e9cessit\u00e9 vitale, incontr\u00f4lable.\n\nAmber. Il faut \u00e9touffer ce chagrin. Le rem\u00e8de est irlandais et s'appelle Elizabeth von Arnim \u2014 \u00ab Little E \u00bb. S\u00e9par\u00e9e de son mari (un comte allemand) et install\u00e9e en Angleterre avec ses cinq enfants, cette aristocrate s'adonne occasionnellement \u00e0 l'\u00e9criture tout en privil\u00e9giant une vie tr\u00e8s spontan\u00e9e. C'est une bonne vivante qui s'ennuie lorsqu'il s'agit de parler politique ou philosophie. Cela pla\u00eet beaucoup \u00e0 Wells qui vit \u00e0 ses c\u00f4t\u00e9s des moments de pure extase. Ils s'amusent ensemble, voyagent dans le nord de l'Europe et quelquefois en Italie. Tandis qu'avec son mari le sexe \u00e9tait une \u00ab affaire d\u00e9sagr\u00e9able 12 \u00bb, avec Wells Elizabeth vit une aventure de tous les instants \u2014 \u00e0 leur actif, des lits cass\u00e9s dans les h\u00f4tels, des parenth\u00e8ses \u00e9rotiques dans les endroits publics. Comme cet apr\u00e8s-midi qu'ils ont pass\u00e9 \u00e0 lire le _Times_ dans les bois, assis contre un arbre. Un article d\u00e9non\u00e7ait la tendance moraliste de la jeune g\u00e9n\u00e9ration dont une partie s'offusquait de la libert\u00e9 de ton d'une certaine Rebecca West... Wells ne conna\u00eet pas cet \u00e9crivain dont tout le monde parle, et ne s'en inqui\u00e8te pas davantage. Ce qu'il veut faire en revanche, c'est montrer \u00e0 cette journaliste qu'elle a tort, que certains n'ont que faire de la morale. En moins de deux minutes, ils se d\u00e9shabillent et font l'amour l\u00e0, sous les arbres, litt\u00e9ralement _sur_ le texte de Mrs Humphrey Ward. R\u00e9pondre \u00e0 une id\u00e9e par un acte, et br\u00fbler le journal tout de suite apr\u00e8s.\n\nLes choses deviennent plus difficiles d\u00e8s lors que \u00ab Little E \u00bb commence \u00e0 nourrir une inexplicable jalousie envers Jane. Elle ne cesse de la critiquer devant Wells, de la d\u00e9pr\u00e9cier et de vouloir lui en donner une mauvaise image. Parfois, quand Wells \u00e9crit une lettre \u00e0 son \u00e9pouse alors qu'il est en d\u00e9placement avec Elizabeth, celle-ci se met dans des col\u00e8res noires. Elle veut davantage. Plus que le simple amusement et que ces escapades au soleil. Elle veut vivre avec Wells et le lui fait violemment savoir. \u00c0 l'automne 1912, il fait finalement ses valises, agac\u00e9 par ces remontrances qu'il pense ill\u00e9gitimes, et rejoint en h\u00e2te son foyer. \u00ab Little E \u00bb rentre seule \u00e0 Londres, et ne cesse d'\u00e9crire \u00e0 son amant pour lui demander pardon. Il accepte finalement de la revoir, sonnant un jour \u00e0 sa porte de mani\u00e8re impromptue. Les semaines ont pass\u00e9, et Wells a fait d'autres conqu\u00eates qui ne sont pas pass\u00e9es inaper\u00e7ues. Les reproches fusent, Elizabeth l'accusant de toutes les fautes dont celle de ne l'avoir jamais aim\u00e9e. Wells r\u00e9plique : si elle se permet de lui dire cela, c'est elle qui ne l'aime pas. \u00ab Tu n'as \u00e9t\u00e9 que la moiti\u00e9 d'un amant 13 \u00bb, lui lance-t-elle pour finir. Ils se s\u00e9parent sur ces mots, Wells claquant la porte derri\u00e8re lui.\n\nComme \u00e0 chaque bouleversement \u2014 qu'il soit professionnel ou personnel \u2014, il en profite pour donner une nouvelle impulsion \u00e0 sa vie. Cette rupture correspond donc \u00e0 son installation dans l'Essex, pr\u00e8s de la ville de Dunmow, dans une maison de grand style surplombant le parc de la comtesse de Warwick. Cela lui rappelle son enfance, et Up Park. La demeure, entour\u00e9e de c\u00e8dres, est vaste et luxueuse, mais s'appelle pourtant \u00ab Little Easton Rectory \u00bb. Jane et les enfants s'y sentent tr\u00e8s vite chez eux ; Wells aussi, m\u00eame s'il garde un pied-\u00e0-terre dans la capitale, \u00e0 St James Court \u2014 son havre de paix et de libert\u00e9.\n\nD\u00e9cid\u00e9ment, il ne comprend rien \u00e0 l'amour. Il en est presque d\u00e9go\u00fbt\u00e9. Il a trop vu quels pouvaient en \u00eatre les lourdeurs, les d\u00e9bordements et les dangers (l'emprise, la possession, la jalousie, tout ce qu'il d\u00e9teste). Il en fait le sujet de son nouveau roman, _Mariage_ , l'histoire d'un jeune couple issu de la classe moyenne \u2014 un professeur en sciences et sa jolie femme \u2014 confront\u00e9 aux al\u00e9as de l'existence. L'essor de la vie professionnelle du mari (Trafford) tranche avec la tranquillit\u00e9 d'une union qui devient peu \u00e0 peu morose. Malgr\u00e9 l'\u00e9loignement et les crises, sa femme, Marjorie, demeure aux c\u00f4t\u00e9s de son \u00e9poux, l'aidant dans son travail et d\u00e9ployant toute son \u00e9nergie pour que son couple ne sombre pas. Toute ressemblance avec des personnes r\u00e9elles est, \u00e9videmment, totalement fortuite. Le roman est salu\u00e9 par la critique pour son aspect moral. Le v\u00e9ritable amour, ce serait donc cela : traverser le meilleur comme le pire, mais rester ensemble, quoi qu'il advienne. Cette id\u00e9e met tous les lecteurs et lectrices d'accord, tous sauf une, Rebecca West, qui critique s\u00e9v\u00e8rement la vision que Wells donne de la femme \u2014 faible, conciliante, acceptant un r\u00f4le inf\u00e9rieur \u00e0 celui de son mari. Son opinion est d\u00e9taill\u00e9e dans un magazine au titre r\u00e9v\u00e9lateur, _The Free Woman_. Il fallait oser s'attaquer au romancier publiquement.\n\nVolontaire, t\u00e9m\u00e9raire, r\u00e9volt\u00e9e : cette jeune fille a tout d'une h\u00e9ro\u00efne dramatique. C'est d'ailleurs en d\u00e9couvrant une pi\u00e8ce d'Henrik Ibsen ( _Rosmersholm_ ) dans laquelle appara\u00eet la fictionnelle Rebekka West, que la journaliste d\u00e9cide d'en faire sa signature. Avec ces deux initiales, elle se sent pr\u00eate pour affronter le monde, d\u00e9jouer les r\u00e8gles qu'on lui impose, crier haut et fort ses id\u00e9es \u00e0 qui veut les entendre. Disparue la petite Cicely Isabel Fairfield, n\u00e9e d'un p\u00e8re journaliste irlandais et d'une m\u00e8re \u00e9cossaise morte trop t\u00f4t. Fascin\u00e9e par la sc\u00e8ne, grande admiratrice de Sarah Bernhardt, la future Rebecca suit des cours \u00e0 l'Acad\u00e9mie d'art dramatique de Londres, \u00e9crit dans des journaux d'avant-garde des articles engag\u00e9s sur le droit des femmes, la discrimination raciale, l'avenir de la classe moyenne, ou encore le mouvement des suffragettes, jusqu'\u00e0 ce jour du 19 septembre 1912 o\u00f9 elle consacre ce fameux article au nouveau livre du plus populaire des \u00e9crivains britanniques. Et comme s'il ne lui suffisait pas de tourner en ridicule cette Marjorie bien peu cr\u00e9dible \u00e0 ses yeux, elle d\u00e9signe son auteur comme \u00ab la vieille fille des romanciers \u00bb \u2014 \u00ab _he is the old maid among novelists_ 14 \u00bb, \u00e9crit-elle en conclusion de son article.\n\nTr\u00e8s vite inform\u00e9 de l'affaire, Wells est moins contrari\u00e9 qu'intrigu\u00e9, et il d\u00e9cide d'inviter cette \u00ab femme libre \u00bb \u00e0 d\u00e9jeuner chez lui. Il s'attend \u00e0 rencontrer une journaliste acari\u00e2tre et frustr\u00e9e. Rien de cela. Quand il ouvre la porte, il se retrouve face \u00e0 une femme extr\u00eamement attirante, vingt-deux ans \u00e0 peine, de grands yeux noirs, une bouche pulpeuse, et un franc-parler d\u00e9concertant. D\u00e8s qu'ils se mettent \u00e0 table, le ton monte et aucun des deux ne touche \u00e0 son assiette. Ils sont trop occup\u00e9s \u00e0 d\u00e9battre de tel livre ou de telle question philosophique. Rebecca ne ressemble \u00e0 aucune autre. Elle enchante Wells qui ne manque pas de remarquer son \u00ab esprit perturb\u00e9 et magnifique 15 \u00bb. \u00c0 la fin du d\u00e9jeuner, ils conviennent de se revoir. Leur deuxi\u00e8me rendez-vous se d\u00e9roule dans la biblioth\u00e8que de Wells, autour de livres ouverts dont ils discutent \u00e0 voix basse, avant de s'embrasser. D\u00e8s lors, il est son \u00ab jaguar \u00bb, elle, sa \u00ab panth\u00e8re \u00bb 16.\n\nCe devait \u00eatre la notori\u00e9t\u00e9, l'assurance qui \u00e9manait de lui \u2014 ces yeux fixes qui jamais ne tremblent \u2014 qui ont plu \u00e0 Rebecca. Elle sait qu'elle n'est pas la seule, qu'il y a Jane et toutes les autres. Aussi, elle n'exige rien. Les deux amants se fr\u00e9quentent en cachette, \u00e0 l'abri du regard envahissant de la famille West, qui veut pr\u00e9server Rebecca du scandale. Cette derni\u00e8re n'\u00e9coute rien ni personne. Elle est heureuse et profond\u00e9ment amoureuse, ce qui ne l'emp\u00eache pas de nourrir des points de discorde avec son amant. L'\u00e9criture en est un. Ils ont tous les deux une approche tr\u00e8s particuli\u00e8re du roman. Les mod\u00e8les de Rebecca s'appellent James Joyce, Henry James et D. H. Lawrence. Ceux de Wells sont bien diff\u00e9rents : Charles Dickens, Jonathan Swift et Laurence Sterne, entre autres. Le point litigieux se noue surtout autour de l'\u0153uvre de James. Au d\u00e9but de leur relation, Rebecca a d\u00e9but\u00e9 une \u00e9tude tr\u00e8s \u00e9logieuse de l'auteur du _Tour d'\u00e9crou_. Au m\u00eame moment, Wells a \u00e9t\u00e9 approch\u00e9 par l'Academic Committee of the Royal Society of Literature, un cercle tr\u00e8s ferm\u00e9 r\u00e9unissant tous les grands \u00e9crivains de l'\u00e9poque. J. M. Barrie, Thomas Hardy, Bernard Shaw et bien s\u00fbr Henry James en font partie. Wells, qui garde un mauvais souvenir des assembl\u00e9es collectives des Fabiens, d\u00e9cline poliment l'invitation. Quand il est question d'art, il vaut mieux faire cavalier seul. Mais un matin, on lui apporte une lettre de James en personne qui tente de le faire changer d'avis. Rien \u00e0 faire : Wells pr\u00e9f\u00e8re \u00ab l'anarchie 17 \u00bb en litt\u00e9rature, tout plut\u00f4t que ces organisations acad\u00e9miques qui gomment les individualit\u00e9s.\n\nAvec James, les liens sont de plus en plus tendus, le premier n'ayant pas dig\u00e9r\u00e9 le refus du second \u2014 un refus qu'il prend pour de l'arrogance. De son c\u00f4t\u00e9, Wells s'amuse. Il \u00e9crit _Boon_ , un roman, ou plut\u00f4t une satire, secr\u00e8tement dirig\u00e9e contre ce James si susceptible, dont il d\u00e9sapprouve totalement la d\u00e9marche fictionnelle. Le livre est \u00e9crit comme une conversation entre George Boon \u2014 romancier c\u00e9l\u00e8bre \u2014 et ses amis qui sont r\u00e9unis lors d'un colloque, Boon s'\u00e9vertuant \u00e0 d\u00e9fendre sa vision de la litt\u00e9rature comme une chose sacr\u00e9e qui doit se fondre dans une structure parfaite. Wells va contre cette opinion. Le roman doit \u00eatre pour lui mouvant, foisonnant, il doit permettre la digression, la r\u00e9flexion, l'inattendu \u2014 tout ce que Rebecca, elle aussi, refuse. \u00c0 la sortie du livre, \u00e0 l'\u00e9t\u00e9 1915, Wells, un brin provocateur, envoie un exemplaire \u00e0 James qui lit l'ouvrage et le d\u00e9teste, choqu\u00e9 de voir son nom et sa r\u00e9putation moqu\u00e9s de la sorte. Wells sourit et r\u00e9pond \u00e0 l'int\u00e9ress\u00e9 que tout cela est de bonne guerre : apr\u00e8s tout, c'est James qui, le premier, l'avait publiquement attaqu\u00e9 dans un journal en critiquant son \u0153uvre : _Le Nouveau Machiavel_ et _Mariage_ , notamment. Les voil\u00e0 d\u00e9sormais \u00e0 \u00e9galit\u00e9. Mais James est en col\u00e8re, il \u00e9crit \u00e0 son adversaire pour lui dire tout le mal qu'il pense de son \u0153uvre et pour rompre tout lien avec lui. Wells r\u00e9plique, tentant d'apaiser les choses : \u00ab Pour vous, la litt\u00e9rature comme la peinture est une fin ; pour moi, la litt\u00e9rature comme l'architecture est un moyen, elle a une utilit\u00e9 18. \u00bb La discorde se cl\u00f4t sur cette distinction de la fonction des arts. James meurt quelques ann\u00e9es plus tard, sans avoir r\u00e9pondu \u00e0 aucune des lettres que Wells continuait pourtant \u00e0 lui \u00e9crire.\n\nAvec Rebecca, la relation est parfois houleuse, notamment \u00e0 cause de ces diff\u00e9rences d'opinions sur la th\u00e9orisation litt\u00e9raire. Plusieurs fois, Wells pense la quitter, mais ils se retrouvent toujours, et s'aiment avant de se d\u00e9chirer \u00e0 nouveau. Tr\u00e8s vite, Rebecca tombe enceinte. Ils ne s'y attendent pas, Rebecca vient juste de se voir proposer un poste \u00e0 responsabilit\u00e9 au _New Freewoman_ , Wells n'a de temps que pour ses livres. Mais la r\u00e9alit\u00e9 les rattrape. Leur fils, Anthony, na\u00eet le 4 ao\u00fbt 1914 \u2014 date \u00ab m\u00e9morable 19 \u00bb, dira l'\u00e9crivain, puisqu'il s'agit de l'entr\u00e9e en guerre du pays. Une nouvelle (une autre) vie de famille commence alors pour lui, sans qu'il l'ait d\u00e9sir\u00e9e ; les deux amants se connaissent finalement tr\u00e8s peu. Jane apprend la nouvelle sans heurt et accepte que son mari partage son temps entre ses deux maisons \u2014 Rebecca et Anthony sont install\u00e9s \u00e0 quelques kilom\u00e8tres de la propri\u00e9t\u00e9 d'Easton. La jeune m\u00e8re est seule, d\u00e9laiss\u00e9e par sa propre famille, qui d\u00e9sapprouve cette union clandestine, constamment jug\u00e9e par les domestiques embauch\u00e9s par Wells, qui observent d'un mauvais \u0153il cette situation. Pour \u00e9viter les scandales en public, le petit gar\u00e7on est \u00e9duqu\u00e9 dans l'ignorance de ses origines \u2014 il appelle sa m\u00e8re \u00ab Tata Panth\u00e8re 20 \u00bb. La famille est souvent amen\u00e9e \u00e0 faire ses valises et \u00e0 changer de maison, deux, trois, quatre fois, jusqu'\u00e0 ce que Rebecca n'en puisse plus et demande \u00e0 Wells de choisir. Jane ou elle. Comme \u00e0 son habitude, Wells est cat\u00e9gorique, jamais il ne brisera son mariage. Rebecca hurle de col\u00e8re. Shakespeare ne disait-il pas que l'amour \u00e9tait \u00ab in\u00e9branlable au fort de la temp\u00eate 21 \u00bb ? L'amant r\u00e9solu d\u00e9cide de le croire, et de laisser passer la bourrasque.\nFace \u00e0 la guerre\n\nUne autre temp\u00eate, plus dangereuse, se pr\u00e9pare en Europe. D'un c\u00f4t\u00e9, des d\u00e9mocraties \u2014 la France, l'Italie et la Grande-Bretagne, premi\u00e8re puissance \u00e9conomique. De l'autre, des monarchies plus ou moins autoritaires \u2014 l'Autriche-Hongrie, la Russie et bien s\u00fbr l'Allemagne. Mais sur l'\u00e9chiquier, les alliances font se m\u00e9langer les r\u00e9gimes qui promettent de se soutenir les uns les autres si un conflit \u00e9clate. Il y a celle qui porte l'empreinte du chancelier Bismarck (la Triple-Alliance r\u00e9unissant les Empires allemand, austro-hongrois et le royaume italien), et l'autre que l'on nomme Triple-Entente (France, Grande-Bretagne et Russie). Les deux font s'opposer les \u00e9ternels ennemis, anciens adversaires de la guerre de 1870 : la France et l'Allemagne. Tout cela tient pour l'instant, mais le fil risque de se rompre face \u00e0 la mont\u00e9e des nationalismes.\n\nWells observe l'Europe depuis son bureau d'Easton Glebe, et r\u00eave secr\u00e8tement de partir \u2014 loin. Un pays l'attire principalement : la Russie. Il ne saurait dire pourquoi. Apr\u00e8s tout, ses romanciers l'ont toujours profond\u00e9ment ennuy\u00e9, il d\u00e9teste le froid et il n'a pas de passeport \u2014 ce curieux objet exig\u00e9 depuis peu \u00e0 la fronti\u00e8re. Mais l'ambassadeur de Russie \u00e0 Londres insiste pour qu'il aille visiter sa belle contr\u00e9e, et l'invite officiellement \u00e0 faire le d\u00e9placement en janvier 1914. Wells fait donc ses valises et se retrouve \u00e0 Saint-P\u00e9tersbourg (bient\u00f4t rebaptis\u00e9e Petrograd) en plein hiver. \u00c0 son arriv\u00e9e, ses guides l'attendent : le diplomate et journaliste Maurice Baring, le fondateur du parti constitutionnel d\u00e9mocrate russe, principale force d'opposition politique, Vladimir Dmitrievitch Nabokov (le p\u00e8re de l'auteur de _Lolita_ ) et la publiciste Ariadna Vladimirovna Tyrkova, l'\u00e9pouse d'un proche de Wells (Harold Williams, journaliste du _Times_ ).\n\nUn pied dehors et d\u00e9j\u00e0 son regard se perd dans l'immensit\u00e9 blanche. De la neige partout, des champs enti\u00e8rement recouverts et des maisons \u00e0 peine visibles. Surr\u00e9aliste. Cette marche dans la ville de Pierre le Grand est relat\u00e9e plus tard dans un roman, _Jeanne et Pierre_ , \u00e9voquant ces \u00ab boutiques noir et or 1 \u00bb, cette \u00ab foule nocturne 2 \u00bb s'engouffrant dans le Palais du Peuple, et la superbe Neva gel\u00e9e qui, pour des yeux habitu\u00e9s \u00e0 la Tamise, semblait scintiller sous les \u00e9toiles. L'\u00e9crivain veut voir le pays, conna\u00eetre son peuple, comprendre le tsarisme. Il veut se fondre dans cette poudre laiteuse et observer la vie qui s'y d\u00e9roule. Apr\u00e8s avoir fait un crochet par les terres recul\u00e9es de la campagne, visit\u00e9 avec d\u00e9lectation les isbas (maisons traditionnelles russes, enti\u00e8rement construites en bois) et les \u00e9coles de plusieurs villages, Wells se dirige vers Moscou. Le rouge du Kremlin appara\u00eet aux yeux de l'\u00e9tranger comme le phare au navigateur. Il est charm\u00e9 par \u00ab l'\u00e9trange cath\u00e9drale Saint-Basile et la magnificence du monast\u00e8re de Troitsky 3 \u00bb, par les d\u00f4mes des \u00e9glises et leurs croix qui s'\u00e9rigent dans le ciel blafard, mais aussi par ces nombreux \u00ab pr\u00eatres barbus 4 \u00bb et ces citadins recouverts de fourrure. Si la ville ne pr\u00eate pas autant \u00e0 r\u00eaver que ses alentours, le voyageur appr\u00e9cie malgr\u00e9 tout ses sorties nocturnes, notamment au th\u00e9\u00e2tre. Il garde un souvenir m\u00e9morable d'une repr\u00e9sentation des _Trois s\u0153urs_ d'Anton Tchekhov, \u00e9bloui par les fastes du d\u00e9cor et la ferveur des spectateurs. Ce pays tremble d'une \u00e9nergie inhabituelle, \u00e0 l'oppos\u00e9 de son Angleterre natale \u2014 grise et monotone. Il le quitte apr\u00e8s une douzaine de jours d'excursions merveilleuses, avec l'impression d'avoir touch\u00e9 \u00ab les limites de l'humanit\u00e9 5 \u00bb, et en sachant qu'il reviendra.\n\nDe retour \u00e0 Easton Glebe, il se met \u00e0 \u00e9crire. Au-dessus de lui, le ciel se couvre. Depuis sa table de travail, Wells ne veut pas croire au conflit mondial qui pourtant se pr\u00e9pare. Il pr\u00e9f\u00e8re rester dans la th\u00e9orie, la pr\u00e9vision, continuer \u00e0 penser que la paix est possible, avertir les hommes que la guerre peut r\u00e9ellement les d\u00e9truire. Alors, il joue au m\u00e9t\u00e9orologue, fait s'obscurcir les nuages et jette, non pas un \u00e9clair, mais une \u00e9paisse fum\u00e9e \u00e0 l'horizon. Ce n'est pas du bois qui flambe, c'est l'humanit\u00e9 qui part en poussi\u00e8re :\n\nLe monde n'\u00e9tait plus le m\u00eame. On apercevait une esp\u00e8ce de vibration. ...] C'\u00e9tait comme si toutes les conduites d'eau, toutes les machines enfouies dans le sol et tous les c\u00e2bles des rues s'\u00e9taient mis \u00e0 battre \u2014 tel un c\u0153ur qui bat. Et tout autour d'elle soufflait comme un vent \u2014 le vent du d\u00e9sarroi. [...] Plus rien n'existait que cet \u00e9clair pourpre et ce vacarme, assourdissant et continu, embrasant l'univers[ 6.\n\nLe d\u00e9sastre a d\u00e9sormais un nom : la fission nucl\u00e9aire. Tout commence par cette conf\u00e9rence \u00e0 laquelle assiste un jeune chimiste ambitieux. Le scientifique sur l'estrade montre \u00e0 tous les \u00e9tudiants une bouteille contenant de l'uranium, c'est-\u00e0-dire \u00ab au moins autant d'\u00e9nergie que celle que nous obtiendrons en br\u00fblant cent soixante tonnes de charbon \u00bb. \u00ab Si d'un mot, continue le professeur, en un instant, je parvenais \u00e0 lib\u00e9rer brutalement cette \u00e9nergie, ici et maintenant, elle nous exploserait au visage et nous r\u00e9duirait en lambeaux 7. \u00bb L'homme assis dans les gradins, et qui \u00e9coute religieusement la d\u00e9monstration, est l'un des personnages qui ouvrent le nouveau roman de Wells : _La Destruction lib\u00e9ratrice_. \u00c9crit \u00e0 la fin de l'ann\u00e9e 1913, publi\u00e9 quelques semaines avant le d\u00e9clenchement du conflit mondial, le livre se veut une mise en garde contre cette formidable d\u00e9couverte de la puissance du radium. Wells est tr\u00e8s au fait des travaux de Rutherford et Soddy, deux scientifiques (l'un n\u00e9o-z\u00e9landais, l'autre britannique) qui analysent la transformation radioactive d\u00e8s 1903, et observeront quinze ans plus tard la premi\u00e8re r\u00e9action nucl\u00e9aire.\n\nLa possibilit\u00e9 d'une explosion est encore hypoth\u00e9tique, mais Wells, depuis sa table de travail champ\u00eatre, y croit fermement et choisit la France comme th\u00e9\u00e2tre de l'apocalypse. Sous sa plume, Paris prend des couleurs brumeuses, \u00e9trangement rouge\u00e2tres. Ce n'est pas son sixi\u00e8me sens qui parle. L\u00e0 encore, le romancier est simplement attentif \u00e0 tout ce qui se passe autour de lui, \u00e0 l'\u00e9volution de la soci\u00e9t\u00e9, et \u00e0 l'escalade du progr\u00e8s technique dont il est le premier admirateur. Dans ce livre, il retrace avec pr\u00e9cision toute l'histoire de l'homme et de ses inventions, depuis la caverne pr\u00e9historique jusqu'\u00e0 l'\u00e9closion de la pens\u00e9e et, surtout, le moment o\u00f9 l'individu a pris conscience de sa force \u00ab illimit\u00e9e 8 \u00bb. Apr\u00e8s avoir apprivois\u00e9 les animaux, ma\u00eetris\u00e9 le feu, sculpt\u00e9 le bois et trouv\u00e9 comment fr\u00e9quenter la mer, l'homme \u2014 et \u00ab le singe qui sommeille 9 \u00bb en lui \u2014 d\u00e9couvre la bombe atomique. Wells pr\u00e9voit son explosion pour l'ann\u00e9e 1956, pas avant. Selon lui, l'\u00eatre humain n'est pas encore pr\u00eat \u00e0 une telle avanc\u00e9e scientifique, une quarantaine d'ann\u00e9es suppl\u00e9mentaires seraient bien n\u00e9cessaires \u00e0 l'\u00e9laboration de cette arme. Wells vise presque juste, devan\u00e7ant Robert Oppenheimer (souvent pr\u00e9sent\u00e9 comme le p\u00e8re de l'arme nucl\u00e9aire), se trompant d'\u00e0 peine une d\u00e9cennie (Hiroshima est pulv\u00e9ris\u00e9e en 1945) et anticipant dans le m\u00eame temps une seconde guerre mondiale que personne n'a encore os\u00e9 imaginer.\n\nAu c\u0153ur de ce chaos \u00e9merge un homme, un Fran\u00e7ais : Leblanc, ambassadeur \u00e0 Washington. Il va tout faire pour mettre un terme au conflit. Le choix du nom n'est pas anodin. Dans le roman, Leblanc est celui qui apporte une lueur d'espoir au c\u0153ur d'un monde qui a perdu toute clart\u00e9. De lui vient le sursaut salvateur. \u00ab La question de savoir si un Leblanc peut encore exister, la question de savoir s'il est toujours possible de provoquer un acc\u00e8s de clairvoyance cr\u00e9atrice dans l'esprit de l'homme, d'enrayer ce glissement continuel vers la destruction, est pour l'heure l'une des questions les plus cruciales \u00e0 laquelle le monde va devoir r\u00e9pondre 10 \u00bb, \u00e9crit-il dans sa pr\u00e9face au roman, publi\u00e9e en 1921. S'il y a des hommes assez stupides pour se faire la guerre, il y en aura d'autres plus intelligents pour l'arr\u00eater : \u00ab Il est \u00e9vident que l'auteur est, par nature, enclin \u00e0 esp\u00e9rer qu'il subsiste pareille \u00e9ventualit\u00e9 11. \u00bb\n\nMais le r\u00e9veil est douloureux. En ce mois de juin 1914, l'archiduc Fran\u00e7ois-Ferdinand s'est fait tuer \u00e0 Sarajevo par un nationaliste serbe, et la guerre est l\u00e0, sous les yeux de Wells, plus r\u00e9elle que jamais \u2014 il ne voulait pas y croire. Il est d\u00e9sempar\u00e9 de voir les hommes engag\u00e9s dans l'horreur, honteux de s'\u00eatre tromp\u00e9 d'\u00e9poque. Il pensait que ce serait pour plus tard. Il sait qu'il a perdu en cr\u00e9dibilit\u00e9, que ses lecteurs qui lui faisaient jadis confiance ne le prennent plus au s\u00e9rieux. Certains rient de cette pr\u00e9diction atomique fantasque. Pour la premi\u00e8re fois, le masque du proph\u00e8te glisse de son visage.\n\nComment a-t-il pu se tromper \u00e0 ce point ? Cela fait vingt ans au moins qu'il ne fait qu'\u00e9crire sur d'hypoth\u00e9tiques guerres, qu'il en r\u00eave la nuit. Il y a encore peu (1908), il imaginait une bataille a\u00e9rienne qu'il offrait \u00e0 ses lecteurs comme une gourmandise empoisonn\u00e9e. Regardez ce que la guerre peut faire \u00e0 l'homme, semblait-il dire, lui, le passionn\u00e9 d'aviation qui n'avait pas encore effectu\u00e9 son premier vol mais qui d\u00e9crivait d\u00e9j\u00e0 un monde vu d'en haut, et attaqu\u00e9 de toutes parts, peupl\u00e9 d'hommes inquiets et vivant dans un climat g\u00e9opolitique tendu. Dans _La Guerre dans les airs_ , les grandes puissances mondiales se jaugent et craignent l'\u00e9mergence de nouvelles nations telles que l'\u00c9gypte ou l'Inde ; l'Allemagne, pays le plus puissant du monde, est dirig\u00e9e par le prince Karl Albert qui a pour seul objectif de conqu\u00e9rir la Terre enti\u00e8re. La ville de New York, que tout le monde croyait invuln\u00e9rable, est attaqu\u00e9e. La riposte am\u00e9ricaine est redoutable, un bombardement a\u00e9rien g\u00e9n\u00e9ralis\u00e9, tout s'\u00e9croule. Le monde n'est plus qu'un tas de miettes.\n\nLa fiction n'est pas tr\u00e8s \u00e9loign\u00e9e de la r\u00e9alit\u00e9. La Premi\u00e8re Guerre mondiale arrive avec son cort\u00e8ge d'engins in\u00e9dits, tous pens\u00e9s pour \u00ab bombarder \u00bb l'adversaire. Le mot est nouveau, la m\u00e9thode encore h\u00e9sitante, et les fameuses bombes, rudimentaires. Mais les cons\u00e9quences n'en sont pas moins impressionnantes, surtout lorsque les Allemands utilisent leurs r\u00e9cents dirigeables (les zeppelins). Les premiers missiles s'abattent sur Londres en 1916, et touchent des civils. Au mois de mai, l'arm\u00e9e britannique organise sa d\u00e9fense. Elle croit \u00e0 une guerre d'usure et s'est dot\u00e9e d'un attirail qu'elle pense infaillible. La flotte l\u00e8ve donc l'ancre et se dirige vers Jutland, au nord du Danemark. La bataille a lieu de nuit et voit se confronter pr\u00e8s de deux cent cinquante navires. Les Anglais perdent, leur trentaine de dreadnoughts (navires au blindage m\u00e9tallique, construits pour \u00eatre insubmersibles) est coul\u00e9e, avec six mille hommes \u00e0 bord. Wells serre les poings. Il avait pr\u00e9venu que la menace allemande viendrait de ses sous-marins, et il a toujours su que ces dreadnoughts n'\u00e9taient pas fiables \u2014 il y avait m\u00eame consacr\u00e9 un article quelques ann\u00e9es plus t\u00f4t. Personne n'y avait pr\u00eat\u00e9 attention.\n\nIl se sent inutile avec son papier et sa plume. Il lui faut aller voir la guerre de plus pr\u00e8s, savoir \u00e0 quoi ressemble une tranch\u00e9e. Et voil\u00e0 Wells qui pose un pied en France, \u00e0 Arras, en compagnie de l'arch\u00e9ologue Osbert Guy Stanhope Crawford, engag\u00e9 sur le front. Ils marchent \u00e0 hauteur de la terre d\u00e9vast\u00e9e, \u00ab dans un long et haut couloir aux murs de boue 12 \u00bb, sous le soleil qui tape, par une journ\u00e9e de tr\u00eave. Au loin, on entend des obus exploser, mais cela ne les inqui\u00e8te pas. On leur a donn\u00e9 \u00e0 chacun un casque d'acier qu'ils gardent sous le bras, au cas o\u00f9. Crawford conna\u00eet bien les environs et demande au visiteur de rester courb\u00e9, de ne surtout pas laisser d\u00e9passer la t\u00eate en dehors de la tranch\u00e9e. Wells, pour le plaisir de la d\u00e9sob\u00e9issance, fait le contraire, laisse glisser furtivement son regard et ne voit qu'une \u00e9tendue d\u00e9serte. Aucun Allemand \u00e0 l'horizon, mais une plaine fumante, trou\u00e9e \u00e7\u00e0 et l\u00e0. Ce terrain chahut\u00e9 par la folie des hommes ne devait plus jamais quitter son esprit.\n\nCela le hante : le bruit, l'odeur de la guerre, sa violence. Une nuit, dans sa chambre d'Easton Glebe, la pluie tombe \u00e0 torrents contre la vitre et le sort brusquement d'un mauvais r\u00eave. Dans son sommeil, il a vu des soldats luttant dans les tranch\u00e9es, il les a entendus hurler, aval\u00e9s par une bouche d'ombre. C'est une image insupportable qui l'am\u00e8ne \u00e0 r\u00e9fl\u00e9chir. Il se l\u00e8ve, allume une bougie, prend une feuille dans le tiroir de la table et commence \u00e0 dessiner. Son id\u00e9e est simple : trouver un moyen d'am\u00e9liorer le quotidien de ces hommes qui manquent mat\u00e9riellement de tout. Alors il esquisse le projet d'un t\u00e9l\u00e9ph\u00e9rique mobile qui, gr\u00e2ce \u00e0 un syst\u00e8me de c\u00e2bles et de traction, servirait \u00e0 approvisionner tous les soldats sur les lignes de combat. Les premi\u00e8res lueurs du jour apparaissent, Wells est satisfait de sa trouvaille. Dans la foul\u00e9e, il \u00e9crit \u00e0 Winston Churchill. L'ancien officier revenu des Indes et d'Afrique a rejoint depuis peu le parti lib\u00e9ral, et a \u00e9t\u00e9 nomm\u00e9 Premier Lord de l'Amiraut\u00e9 (soit ministre de la Marine). Wells sait combien cet homme d'\u00c9tat s'int\u00e9resse aux innovations \u2014 il a notamment \u00e9t\u00e9 sensible \u00e0 l'id\u00e9e du romancier de cr\u00e9er des chars de combat (les futurs \u00ab Churchill \u00bb, qui verront le jour \u00e0 l'occasion de la Seconde Guerre mondiale). Et surtout, ils ont d\u00e9j\u00e0 deux points communs : une personnalit\u00e9 fougueuse et la passion immod\u00e9r\u00e9e des avions.\n\nLa rencontre s'effectue dans l'atelier de Clare Sheridan, une journaliste et sculptrice en vogue qui est aussi la cousine de Churchill. Ce dernier est s\u00e9duit par l'id\u00e9e de Wells et fait tout pour l'aider \u00e0 la mettre en \u0153uvre. Wells, qui n'a jamais aim\u00e9 les uniformes, se retrouve \u00e0 travailler en lien direct avec l'arm\u00e9e, entour\u00e9 d'esprits qui ne comprennent pas sa vision des choses. Le t\u00e9l\u00e9ph\u00e9rique se construit tant bien que mal, sans pour autant apaiser l'angoisse fondamentale qui monte en lui. Ce premier pas dans la lutte contre les Allemands en appelle d'autres, et Wells n'a aucune envie de reculer. Ses amis ne comprennent pas cette alliance soudaine avec l'ordre militaire, lui qui a toujours revendiqu\u00e9 son pacifisme et qui n'a cess\u00e9 de d\u00e9crire la guerre dans ses livres afin, justement, d'\u00e9viter qu'elle ne se produise dans la r\u00e9alit\u00e9. Une seule obsession l'habite d\u00e9sormais : abattre l'ennemi. Et comme beaucoup d'intellectuels de l'\u00e9poque, son arme demeure l'\u00e9criture. Contact\u00e9 par Lord Northcliffe (grand patron de presse qui a lanc\u00e9 en 1896 le _Daily Mail_ , un quotidien qui s'arrache \u00e0 la vente), il se rend \u00e0 Crewe House, \u00e0 Londres, un b\u00e2timent d'o\u00f9 s'organise dans le plus grand secret la propagande britannique contre l'Allemagne. Northcliffe, qui conna\u00eet le travail de Wells et lui a d\u00e9j\u00e0 confi\u00e9 par le pass\u00e9 une enqu\u00eate sur la classe ouvri\u00e8re du pays, lui propose de devenir le responsable de la pr\u00e9paration de la propagande. C'est un honneur pour l'\u00e9crivain qui impressionne les troupes par sa verve et son intransigeance. Il veut donner une ligne directrice \u00e0 cet appareil de guerre, et l'orienter tant qu'il peut non seulement vers la victoire, mais vers la paix durable :\n\nUne paix n\u00e9gative n'est pas notre but ! C'est naturellement d\u00e9j\u00e0 quelque chose qu'un r\u00e9pit dans la souffrance qu'on subit, et dans celle qu'on inflige ; mais c'est une chose encore plus grande d'\u00eatre affranchis : et la paix affranchit les peuples. Elle leur donne la libert\u00e9 de penser, de vivre, d'accomplir un travail qui vaut la peine d'\u00eatre accompli, de b\u00e2tir au lieu de d\u00e9truire, de se consacrer \u00e0 la poursuite ou \u00e0 la cr\u00e9ation des choses qui leur semblent supr\u00eames, au lieu de passer leur temps \u00e0 \u00e9viter d'\u00eatre tu\u00e9s. La paix est une coupe vide, que nous pouvons remplir \u00e0 notre guise, \u2014 c'est une opportunit\u00e9 que nous pouvons saisir, ou n\u00e9gliger 13.\n\nLe message est clair, et il circule dans toutes les instances de l'organisation afin d'enrayer les possibles erreurs. Wells sait qu'il a affaire \u00e0 des minist\u00e8res qui n'envisagent que la violence et qui ont parfois des \u00ab intentions mesquines 14 \u00bb. Alors il mart\u00e8le partout sa conviction qu'une sortie du conflit est possible. De jour comme de nuit, il distribue des brochures aux agents secrets, fabrique de faux journaux et des tracts \u00e0 charger dans la cale des avions. Ses livres aussi, comme une sorte de m\u00e9lodie lancinante, \u00e9voquent tous la guerre *1.\n\nAu c\u0153ur de ces sombres ann\u00e9es, un roman marque profond\u00e9ment les consciences : _M. Britling commence \u00e0 y voir clair_. Wells y d\u00e9crit le quotidien d'un \u00e9crivain britannique pendant le conflit mondial. Dans ce personnage typiquement anglais, aur\u00e9ol\u00e9 d'un succ\u00e8s important mais inexplicablement attach\u00e9 \u00e0 ses origines modestes, vivant dans la campagne bucolique, il est ais\u00e9 de reconna\u00eetre l'autoportrait du romancier. M. Britling \u00ab figurait dans l'annuaire litt\u00e9raire de deux continents ; les derni\u00e8res ann\u00e9es l'avaient vu se hausser jusqu'au rang d'\u00e9crivain consacr\u00e9 et f\u00eat\u00e9 par le public am\u00e9ricain cultiv\u00e9, voire m\u00eame connu d'un cercle choisi de lecteurs anglais... ...] Il parlait de tout ; il avait des id\u00e9es sur tout ; il lui \u00e9tait aussi naturel d'avoir des id\u00e9es sur tout qu'il l'est \u00e0 un chien de vous flairer les talons. Lui flairait les talons de la r\u00e9alit\u00e9[ 15 \u00bb. Et l'odeur hum\u00e9e est celle de la guerre. Tout y est peint avec pr\u00e9cision dans le but de montrer l'absurdit\u00e9 du combat : la surprise des premi\u00e8res attaques, d'abord, \u00e0 laquelle M. Britling refusait de croire (\u00ab Les grandes puissances d'Europe peuvent se chicaner et se menacer entre elles, n'emp\u00eache qu'elles sauront d\u00e9sormais r\u00e9gler leurs conflits autrement que par les armes 16 \u00bb) ; puis les peurs li\u00e9es \u00e0 \u00ab la catastrophe en marche 17 \u00bb, \u00e0 l'entr\u00e9e de l'Angleterre dans le conflit, \u00e0 l'avenir du monde d\u00e9cid\u00e9ment incertain ; enfin, le quotidien des soldats dans les tranch\u00e9es, d\u00e9crit dans le roman avec une pr\u00e9cision glaciale, comme cet \u00e9pisode relat\u00e9 par Hugh, le fils de M. Britling engag\u00e9 sur le front, qui raconte dans une lettre la mort au front d'Ortheris, l'un de ses camarades :\n\n\u00ab Un obus arrive, me renverse, sans me blesser. Tout \u00e9tourdi, couvert de boue, je me redresse sur les genoux, et je [le] vois [...] \u00e0 six m\u00e8tres de l\u00e0, les jambes broy\u00e9es. Une bouillie rouge...\n\nIl avait l'air stup\u00e9fait. \u00ab Foutu, fit-il. Foutu. \u00bb\n\nIl fixait les yeux sur moi ; et soudain, il eut un rire grimac\u00e9. [...] \u00ab Je croyais que tu y \u00e9tais, reprit-il, et c'est moi. J'suis propre ! Pire qu'un cochon saign\u00e9. Foutue salet\u00e9 ! Foutu, c'est le cas de le dire, h\u00e9 ? \u00bb [...]\n\nConstern\u00e9, je le contemplais, ne sachant que r\u00e9pondre. [...]\n\nBrusquement, pendant que je d\u00e9bouchais mon bison, Ortheris, abandonnant son r\u00f4le de guerrier sto\u00efque, eut un instant de r\u00e9volte. Il fit une moue d'enfant capricieux qui refuserait d'aller se coucher et grommela d'un ton rageur : \u00ab J'voulais pas partir encore, et, pourtant, m'v'l\u00e0 flamb\u00e9 ! \u00bb\n\nCe furent ses derni\u00e8res paroles. Il tourna la t\u00eate un peu de c\u00f4t\u00e9, comme d\u00e9go\u00fbt\u00e9 de tout... s'\u00e9vanouit \u2014 et mourut.\n\nUn bon moment, je restai l\u00e0, essayant de le faire boire.\n\n...] soudain, je me mis \u00e0 pleurer... comme un gosse. Moi non plus, je ne voulais pas qu'il parte ! Je voulais garder mon petit voyou faubourien. Je le voulais de toutes mes forces[ 18... \u00bb\n\nIl ne s'agit plus pour le romancier de s'attarder sur les questions politiques \u2014 la guerre comme enjeu entre les nations mondiales \u2014, mais de donner une id\u00e9e de la mort, de rendre compte du choc des pertes humaines, colossales. Ce livre touche en plein c\u0153ur Maxime Gorki qui entreprend d'en superviser la traduction en russe. La lettre qu'il adresse \u00e0 Wells quelques semaines plus tard est une v\u00e9ritable d\u00e9claration d'amiti\u00e9 :\n\nCher ami,\n\nJe viens de terminer la correction de la traduction russe de votre dernier livre, _M. Britling_ , et je d\u00e9sire vous exprimer mon admiration, car vous avez \u00e9crit un livre magnifique. Sans aucun doute, c'est le meilleur livre, le plus hardi, le plus v\u00e9ridique et humain \u00e9crit en Europe pendant cette guerre maudite ! Je suis certain que par la suite, quand nous serons redevenus plus humains, l'Angleterre sera fi\u00e8re que la premi\u00e8re voix de la protestation, et une protestation si v\u00e9h\u00e9mente, se soit \u00e9lev\u00e9e en Angleterre contre les atrocit\u00e9s de la guerre, et tous les hommes sinc\u00e8res et intelligents prononceront votre nom avec reconnaissance. Votre livre est de ceux qui vivront longtemps. Vous \u00eates un homme grand et excellent, Wells, et je suis si heureux de vous avoir vu, de pouvoir me souvenir de votre visage, de vos yeux magnifiques. J'exprime peut-\u00eatre cela d'une fa\u00e7on quelque peu primitive, mais je veux tout simplement vous dire qu'en ces temps de cruaut\u00e9 et de barbarie universelle, votre livre est un grand livre 19.\n\nDans ce livre, Wells place son h\u00e9ros face au d\u00e9c\u00e8s de son fils, ce Hugh si enthousiaste et volontaire qui voulait faire comme tous les autres, \u00eatre un homme quel qu'en soit le p\u00e9ril. Et voil\u00e0 Britling \u2014 et derri\u00e8re lui, Wells \u2014 faisant appel \u00e0 Dieu. Deuxi\u00e8me choc. L'incroyant absolu, l'ath\u00e9e de naissance, se tourne vers le ciel comme ultime recours, \u00e0 la recherche d'un \u00ab capitaine de la R\u00e9publique mondiale 20 \u00bb capable de faire advenir la paix.\n\nCe virage spirituel \u00e9tonne beaucoup. Les lecteurs ne sont pas habitu\u00e9s \u00e0 voir leur auteur favori, l'enfant terrible de la litt\u00e9rature britannique, s'\u00e9pancher sur cette foi nouvelle, presque artificielle, qui rayonne aussi dans un essai au titre sans \u00e9quivoque : _Dieu, l'invisible roi_. L'\u00e9lan mystique est de courte dur\u00e9e (rapidement, l'auteur embrasse de nouveau ses premi\u00e8res convictions), mais il t\u00e9moigne bien, \u00e0 cet instant, du d\u00e9sarroi de Wells que la guerre d\u00e9sesp\u00e8re. Jadis, pourtant, il n'h\u00e9sitait pas \u00e0 critiquer les hommes, ces \u00eatres faibles, qui s'attachaient \u00e0 une force sup\u00e9rieure. Croire en Dieu signifiait qu'on n'\u00e9tait pas arm\u00e9 pour la vie. C'est maintenant \u00e0 son tour d'\u00eatre pleinement _d\u00e9sarm\u00e9_. Wells cherche une issue pour sortir le monde \u2014 et lui-m\u00eame \u2014 de ce d\u00e9sastre.\n\nLe 11 novembre 1918, l'armistice est sign\u00e9, officialisant la victoire des Alli\u00e9s face \u00e0 l'Allemagne. C'est un jour particulier en Europe, chaque pays f\u00eatant \u00e0 sa mani\u00e8re la fin des combats. \u00c0 Londres, la foule se presse au Victoria Memorial, brandissant des drapeaux, chantant, hurlant, tapant des mains. Wells est l\u00e0 aussi, ballott\u00e9 par le flot des corps et des voix, tentant de se frayer un chemin au milieu de ce d\u00e9lire. La guerre est finie, mais il n'a pas le c\u0153ur \u00e0 sourire. Il est envahi par un profond sentiment d'\u00e9chec. Des centaines de milliers de victimes, peut-\u00eatre un million : ce sont en tout cas les chiffres provisoires avanc\u00e9s par les gouvernements europ\u00e9ens. Il n'a jamais autant ressenti la fragilit\u00e9 du monde et la n\u00e9cessit\u00e9 d'imaginer un syst\u00e8me qui permette de cr\u00e9er de nouvelles bases communes.\n\nDepuis plusieurs mois, il est devenu membre de la Ligue de la Soci\u00e9t\u00e9 des Nations \u00e0 Londres, un groupe d'intellectuels qui r\u00e9fl\u00e9chit \u00e0 l'avenir mondial. Au sein de cet organisme \u00e9merge une id\u00e9e, plus vaste et ambitieuse : cr\u00e9er une sorte de comit\u00e9 qui r\u00e9unirait les grandes puissances, ferait respecter le droit international, et garantirait non seulement la paix mais aussi la s\u00e9curit\u00e9 collective. L'id\u00e9e devient un projet qui s\u00e9duit le pr\u00e9sident am\u00e9ricain Wilson et qui prendra le nom de Soci\u00e9t\u00e9 des Nations *2 en janvier 1919, juste avant la signature du trait\u00e9 de Versailles. Wells croit profond\u00e9ment en la puissance am\u00e9ricaine. Selon lui, le Nouveau Monde peut et doit conduire les autres nations hors du gouffre de la guerre. Et derri\u00e8re ce r\u00eave d'un \u00ab ordre nouveau 21 \u00bb r\u00e9siste une foi in\u00e9branlable en l'homme et en sa capacit\u00e9 \u00e0 rebondir :\n\nTous les hommes ont dans leur nature des instincts m\u00e9lang\u00e9s. Il n'y en a pas un seul exempt d'avidit\u00e9 ou de bassesse, ou incapable de d\u00e9sir de vengeance. Apr\u00e8s la d\u00e9tresse et les pertes d'une lutte comme celle o\u00f9 nous sommes, il n'est que trop \u00ab naturel \u00bb d'\u00eatre pr\u00eat \u00e0 empoigner et \u00e0 punir, d\u00e8s que la balance de la victoire penche en notre faveur. Il ne faut pas attacher une trop grande importance au patriotisme agressif de la presse dans les pays bellig\u00e9rants. Il nous faut garder un peu d'humour dans notre interpr\u00e9tation des motifs de l'ennemi, et nous rappeler que, bien que l'homme ait encore assez du singe dans son temp\u00e9rament, cela ne fait pas de lui un diable incurable. Le m\u00eame Allemand qui exultera en lisant qu'un bateau avec des passagers britanniques a \u00e9t\u00e9 torpill\u00e9 par un sous-marin, ou qui, les yeux riv\u00e9s sur une carte, r\u00eavera d'une Allemagne gigantesque, qui s'\u00e9tendrait d'Anvers jusqu'\u00e0 Constantinople, fond\u00e9e sur le sang et l'horreur, et dominant le monde, \u2014 le m\u00eame sera, dans ses moments lucides, tout pr\u00eat \u00e0 se soumettre \u00e0 un plan de bonne volont\u00e9 g\u00e9n\u00e9rale, pourvu que ce plan assure, \u00e0 lui et aux siens, un degr\u00e9 tol\u00e9rable de prosp\u00e9rit\u00e9 et de bonheur. L'\u00e9l\u00e9ment belliqueux se trouve chez tout homme, mais il n'est pas incurable chez la plupart d'entre eux 22.\n\nMise en application d\u00e8s l'ann\u00e9e suivante, et s'appuyant sur une base de cinq membres permanents, la Soci\u00e9t\u00e9 des Nations (S.D.N.) pi\u00e9tine et ne convainc pas. Elle est rapidement d\u00e9sert\u00e9e par les Am\u00e9ricains qui ont peur que leurs int\u00e9r\u00eats ne soient pas sauvegard\u00e9s et, plus que tout, elle d\u00e9\u00e7oit Wells qui r\u00eave toujours, en silence, de son \u00c9tat mondial. Le moment est peut-\u00eatre venu de tout refonder, de b\u00e2tir un monde meilleur, \u00e0 sa mani\u00e8re, sans rien attendre ni des Am\u00e9ricains ni de quiconque. Ce qui n'a pas fonctionn\u00e9, c'est l'\u00e9ducation, il en est s\u00fbr. C'est le probl\u00e8me fondamental de toute soci\u00e9t\u00e9 en crise. L'unit\u00e9 doit donc passer par le savoir, seul capable d'enrayer le chaos.\n\n*1. _La guerre qui tuera la guerre_ (1914), _La Guerre de l'avenir_ (1917), _Jeanne et Pierre_ (1918).\n\n*2. Anc\u00eatre malheureux de l'Organisation des Nations unies, cr\u00e9\u00e9e sur ses ruines \u00e0 l'issue de la Seconde Guerre mondiale.\nUn monde meilleur\n\nLa paix est sign\u00e9e, et Wells commence l'\u00e9criture d'un ouvrage qui sera l'un de ses plus importants : _The Outline of History_ ( _Esquisse de l'histoire universelle_ ), une grande fresque qui retrace l'\u00e9volution de la civilisation. Le romancier laisse place au professeur qui \u00e9nonce un cours magistral, ambitieux, une le\u00e7on d\u00e9livr\u00e9e au plus grand nombre qui promet de lui apporter un regain de notori\u00e9t\u00e9, de confiance (il en a besoin) \u2014 et beaucoup d'argent. D'abord, ses \u00e9diteurs l'ont mis en garde, l'avertissant que ses lecteurs attendaient probablement de lui une nouvelle fiction, de l'aventure, du suspense, tout ce qu'il sait si bien raconter. Mais Wells a la conviction profonde qu'il fait ce qu'il doit faire : proposer \u00e0 tous les hommes \u00ab une base commune d'id\u00e9es g\u00e9n\u00e9rales 1 \u00bb. Tant pis si les lecteurs ne le suivent pas.\n\nIl se met au travail, et effectue des recherches colossales pour lesquelles Jane l'aide quotidiennement. C'est un dur labeur \u2014 une ann\u00e9e enti\u00e8re de r\u00e9daction \u2014, l'ouvrage le plus complexe qu'il ait entrepris jusqu'alors. L'\u00e9crivain y apporte d'incessantes corrections, n'\u00e9tant jamais pleinement satisfait du r\u00e9sultat. Jane tape le manuscrit et v\u00e9rifie les r\u00e9f\u00e9rences de certaines informations. Le travail se termine bient\u00f4t et le livre arrive en librairie. Wells esp\u00e8re que l'\u00e9tudiant comme l'ouvrier l'ach\u00e8teront. Il n'a pas pens\u00e9 \u00e0 l'universitaire ou au sp\u00e9cialiste pendant la r\u00e9daction, mais au lecteur lambda, \u00e0 celui qui est m\u00fb par la curiosit\u00e9 et l'envie. Derri\u00e8re l'\u00e9criture de la grande Histoire se loge la sienne, celle d'un fils de commer\u00e7ant qui \u00e9tait destin\u00e9 \u00e0 tout sauf \u00e0 devenir \u00e9crivain, mais qui s'est accroch\u00e9 \u00e0 l'\u00e9cole comme \u00e0 un rocher. Il n'a pas oubli\u00e9 les livres qu'il cachait sous ses blouses d'apprenti drapier, les \u00e9coliers qu'il voyait passer avec envie dans la rue, lui qui devait garder la caisse de la boutique. Il n'oubliera jamais le jour o\u00f9 il en a eu assez, ce matin d'hiver o\u00f9 plus rien n'avait d'importance que ce feu int\u00e9rieur, cette envie irr\u00e9pressible de dire \u00ab non \u00bb \u00e0 sa m\u00e8re, quitte \u00e0 en mourir. La th\u00e8se qu'il d\u00e9veloppe dans son _Histoire universelle_ suit subtilement la trajectoire de sa propre vie. L'\u00e9ducation l'a sauv\u00e9, il en est l'exemple parfait. Apprendre lui a donn\u00e9 l'envie d'\u00e9crire et de transmettre son savoir aux autres. Juste avant le d\u00e9clenchement de la guerre, il avait d'ailleurs trouv\u00e9 l'\u00e9ducateur parfait en la personne de Frederick William Sanderson, un professeur aux m\u00e9thodes avant-gardistes \u00e0 qui il d\u00e9cide de consacrer une biographie *1. L'homme dirige l'\u00e9cole d'Oundle, o\u00f9 sont inscrits les fils de Wells. Il est d\u00e9crit comme \u00ab original et vigoureux \u00bb, partisan d'une \u00ab \u00e9ducation modernis\u00e9e \u00bb 2. L'une de ses id\u00e9es \u00e9tait par exemple de b\u00e2tir une \u00ab Maison de la Vision \u00bb, \u00e0 c\u00f4t\u00e9 du b\u00e2timent scolaire, afin que chaque \u00e9l\u00e8ve puisse y aller pour penser, au calme, entour\u00e9 d'une collection de livres et d'archives, n\u00e9cessaires pour d\u00e9velopper des facult\u00e9s de r\u00e9flexion. Sanderson est mort avant d'avoir pu concr\u00e9tiser ce projet. Wells l'a toujours eu en t\u00eate et l'a port\u00e9, d'une autre fa\u00e7on, dans ses livres. L'intelligence, il en est s\u00fbr, doit se d\u00e9ployer \u00e0 grande \u00e9chelle. Le savoir doit devenir la \u00ab clef \u00bb de toute existence :\n\nNous \u2014 moi et les gens de mon esp\u00e8ce \u2014 avons d\u00e9termin\u00e9 exactement comment est faite cette clef d'une plus parfaite vie humaine. Par un labeur incessant : \u00e9tudes, propagande, \u00e9ducation, par des suggestions cr\u00e9atrices, par des sacrifices s'il en faut, en luttant courageusement et en nous mesurant intr\u00e9pidement avec la stupidit\u00e9, l'hypocrisie et la m\u00e9chancet\u00e9, nous pouvons encore d\u00e9couper, lisser et polir et ins\u00e9rer dans la serrure et tourner cette clef d'une communaut\u00e9 heureuse du monde, avant qu'il ne soit trop tard. Le royaume des cieux est mat\u00e9riellement \u00e0 notre port\u00e9e 3.\n\n\u00ab Le royaume des cieux \u00bb devient, effectivement, une possibilit\u00e9 r\u00e9elle pour au moins cent mille lecteurs (c'est le nombre d'exemplaires vendus du livre, lors de sa publication en 1920), et plus tard pr\u00e8s de deux millions (lorsque l'ouvrage est traduit et vendu \u00e0 travers le monde). Wells ne touche plus terre, et se targue d'\u00eatre l'auteur d'ouvrages plus populaires que la Bible. Des \u00e9diteurs \u00e9trangers le contactent pour acheter les droits. Les pays francophones notamment, comme la Belgique, saluent son courage et son intelligence. Le proph\u00e8te est de retour, il a endoss\u00e9 son costume favori (celui de l'\u00e9ducateur) et veut d\u00e9sormais se consacrer exclusivement \u00e0 l'\u00e9criture d'essais. Jane l'encourage. Rebecca, au contraire, voit tout cela d'un tr\u00e8s mauvais \u0153il. Selon elle, il se g\u00e2che, il gaspille du temps et de l'\u00e9nergie pour des livres qui ne servent \u00e0 rien, alors qu'il devrait continuer \u00e0 \u00e9crire des romans, inscrire sa marque dans l'histoire litt\u00e9raire. Mais il a d\u00e9j\u00e0 en t\u00eate la suite, deux volumes pour compl\u00e9ter une s\u00e9rie qu'il envisageait d\u00e8s le d\u00e9part comme un triptyque : _La Science de la vie_ , publi\u00e9 en 1930, et _Le Travail, la Richesse et le Bonheur de l'humanit\u00e9_ , publi\u00e9 en 1931.\n\nLe succ\u00e8s est grisant et conduit Wells \u00e0 entreprendre plusieurs voyages. Il est notamment invit\u00e9 par le bolchevique L. B. Kamenev \u00e0 se rendre en Russie (\u00e0 peine remise des bouleversements r\u00e9volutionnaires) pour rencontrer L\u00e9nine, le nouveau chef du pays \u00e0 la t\u00eate du Conseil des commissaires du peuple. L'\u00e9crivain jouit aupr\u00e8s du parti vainqueur d'une grande aura. Les bolcheviques esp\u00e8rent qu'ils pourront, gr\u00e2ce \u00e0 lui, renouer le dialogue \u2014 totalement rompu depuis qu'ils sont au pouvoir \u2014 avec l'Angleterre. Apr\u00e8s Stockholm et Tallinn, Wells arrive donc \u00e0 Petrograd, sinon en dernier espoir, du moins en interm\u00e9diaire de choix, le 26 septembre 1920, en compagnie de son fils George Philip (\u00ab Gip \u00bb). Il sait qu'il est attendu, qu'il a une certaine renomm\u00e9e dans le pays (plusieurs de ses livres sont traduits en russe), et qu'il est probable qu'il trouve ici un homme politique capable d'entendre ses id\u00e9es. D\u00e9\u00e7u par la S.D.N., il n'a pas abandonn\u00e9 son projet d'\u00c9tat mondial, et compte bien profiter de ce s\u00e9jour pour exposer ce qu'il a en t\u00eate.\n\nDepuis 1914, date de son dernier voyage l\u00e0-bas, la Russie a bien chang\u00e9. L'archa\u00efsme du syst\u00e8me politique, les d\u00e9faites de l'arm\u00e9e russe durant le premier conflit mondial et les famines \u00e0 r\u00e9p\u00e9tition ont conduit \u00e0 la R\u00e9volution de f\u00e9vrier 1917 qui a provoqu\u00e9 le renversement du r\u00e9gime tsariste. Mais l'all\u00e9gresse populaire a bient\u00f4t fait place au d\u00e9sordre, et le gouvernement provisoire n'a pas r\u00e9sist\u00e9. Peu \u00e0 peu, le parti bolchevique a affirm\u00e9 sa l\u00e9gitimit\u00e9 et sa volont\u00e9 de prendre le pouvoir. Au mois d'octobre de la m\u00eame ann\u00e9e, une nouvelle r\u00e9volution a \u00e9clat\u00e9 avec en t\u00eate de cort\u00e8ge L\u00e9nine, ancien avocat et homme politique fougueux qui a d\u00e9j\u00e0 v\u00e9cu la d\u00e9portation en Sib\u00e9rie, et qui a de grands espoirs pour son pays. Aux c\u00f4t\u00e9s du militant marxiste L\u00e9on Trotski, L\u00e9nine installe, pour de bon, le r\u00e9gime bolchevique. Pour les grandes puissances du monde, il s'agit d'un coup d'\u00c9tat, pis : du d\u00e9but d'une dictature. Pour Wells en revanche (qui, pourtant, soutenait le tsarisme il y a encore quelques ann\u00e9es), ce soul\u00e8vement est v\u00e9cu comme une joie, un envol d\u00e9mocratique. Certes, les bolcheviques sont des \u00ab socialistes marxistes 4 \u00bb. Certes, Wells n'a jamais admir\u00e9 l'auteur du _Capital_ , ce \u00ab raseur de la pire esp\u00e8ce 5 \u00bb. Mais il sait aussi qu'il ne faut pas s'arr\u00eater aux d\u00e9tails, perdre de vue l'essentiel. Tout ce qui assure la paix et la possibilit\u00e9 d'un syst\u00e8me universel est bon, \u00e0 ses yeux. Sans m\u00eame les conna\u00eetre, Wells d\u00e9cide donc de croire en l'honn\u00eatet\u00e9 de ces nouveaux dirigeants, attendant avec impatience de les rencontrer.\n\nWells est log\u00e9 chez son ami Gorki, \u00e0 Petrograd. Tr\u00e8s heureux de le revoir, il le trouve plut\u00f4t en bonne sant\u00e9, malgr\u00e9 les r\u00e9centes mauvaises nouvelles sur son \u00e9tat. Lorsqu'il arrive dans son appartement, une femme est dans le salon. Elle s'appelle Moura, elle a vingt-sept ans, porte une robe noire et une veste militaire kaki. Belle, presque arrogante avec une main dans sa poche et l'autre qui tient une cigarette. Wells la salue et se pr\u00e9sente. Elle lui dit qu'ils se connaissent d\u00e9j\u00e0, qu'ils s'\u00e9taient crois\u00e9s bri\u00e8vement quelques ann\u00e9es plus t\u00f4t, en 1914, lors d'un d\u00eener ici m\u00eame, \u00e0 Petrograd. Comment peut-il ne pas s'en souvenir ? Elle ne lui en tient pas rigueur, l'invite \u00e0 s'asseoir, et lui annonce qu'elle sera son interpr\u00e8te pour toute la dur\u00e9e de son s\u00e9jour. Wells tente de mieux la conna\u00eetre, et demande \u00e0 Gorki quelques renseignements. Moura est n\u00e9e en 1892. Elle est la fille d'un fonctionnaire russe de lign\u00e9e ukrainienne, Ignace Platonovitch Zakrovski. Arriv\u00e9e en Angleterre \u00e0 l'\u00e2ge de onze ans, sous la coupe d'un demi-fr\u00e8re conseiller de cour \u00e0 l'ambassade russe de Londres, la jeune femme parle couramment l'anglais. Apr\u00e8s un premier mariage en 1911, elle s'engage dans les services secrets britanniques pendant les quatre ann\u00e9es du conflit mondial, et fait la connaissance de Robert Bruce Lockhart, espion lui aussi, qui devient son amant. Cet homme dressera d'elle un portrait, l'un des rares connus \u00e0 ce jour :\n\nElle consid\u00e9rait de haut toutes les petitesses de l'existence et prenait toutes choses avec un courage qui ne se d\u00e9mentait jamais. Sa vitalit\u00e9, due vraisemblablement \u00e0 une constitution de fer, \u00e9tait incomparable et r\u00e9confortait tous ceux qui la voyaient. L\u00e0 o\u00f9 elle aimait, l\u00e0 \u00e9tait son univers, et sa philosophie de la vie lui en faisait accepter toutes les cons\u00e9quences. C'\u00e9tait une aristocrate ; elle aurait pu \u00eatre communiste, bourgeoise jamais 6.\n\nGorki, qui rencontre Moura en 1919, a d\u00fb percevoir tout cela. La jeune femme, m\u00e8re de deux enfants \u2014 Tania et Paul \u2014, est devenue sa secr\u00e9taire et confidente. Wells la soup\u00e7onne d'\u00eatre sa ma\u00eetresse.\n\nC'est souvent en sa compagnie que l'\u00e9crivain sort se promener. Dehors, Petrograd s'offre \u00e0 lui d'une mani\u00e8re bien \u00e9trange \u2014 cette ville qu'il avait trouv\u00e9e si belle la premi\u00e8re fois, et qu'il d\u00e9couvre aujourd'hui d\u00e9serte, comme \u00e9teinte. Partout, des rues vides, des boutiques ferm\u00e9es \u2014 sauf peut-\u00eatre un ou deux fleuristes qui survivent inexplicablement. Il n'y a rien \u00e0 manger, le peuple meurt de faim, mais si l'on veut, on peut acheter des chrysanth\u00e8mes. Curieux, Wells se risque \u00e0 aborder les habitants, \u00e0 leur poser des questions sur leur quotidien, leurs peurs et leurs espoirs. Ils lui racontent la r\u00e9volution, l'effondrement de la monarchie, et surtout la faim et le froid, tenaces, omnipr\u00e9sents. L'\u00e9crivain est en col\u00e8re. Ce pays se meurt, ces gens n'ont plus rien pour se nourrir, se chauffer ni se soigner, cela se passe sous les yeux du monde entier, et personne ne r\u00e9agit \u2014 surtout pas l'Angleterre. Le gouvernement sovi\u00e9tique a organis\u00e9 un syst\u00e8me de rationnement, mais ce n'est pas assez pour remplir les ventres. Les commerces clandestins se propagent \u00e0 chaque coin de rue, quelques pommes de terre se vendent sur des \u00e9tablis de fortune \u00e7\u00e0 et l\u00e0, et des cuisines de quartier sont ouvertes jusque tard dans la nuit. Wells voit tout cela, mais ne veut pas croire que les bolcheviques en sont responsables : \u00ab ...] cette d\u00e9solation de la Russie ne r\u00e9sulte pas des attaques subies par le syst\u00e8me, qui serait d\u00e9truit par une force maligne. C'est le syst\u00e8me lui-m\u00eame qui, atteint de corruption, s'est \u00e9puis\u00e9 et s'est effondr\u00e9. Ce n'est pas le communisme qui a b\u00e2ti ces villes immenses et impossibles, c'est le capitalisme. Ce n'est pas le communisme qui a plong\u00e9 dans une guerre \u00e9puisante de six ans ce colossal empire insolvable et grin\u00e7ant ; c'est le capitalisme europ\u00e9en[ 7 \u00bb.\n\nAvec Gorki, Wells assiste \u00e0 de nombreuses r\u00e9unions, des mondanit\u00e9s plut\u00f4t d\u00e9plaisantes mais n\u00e9cessaires. Les maisons d'ouvriers, les prisons : il voit le tragique du quotidien du peuple, sa pauvret\u00e9, sa mis\u00e8re. Les mots qu'il choisit pour ses articles sont sans \u00e9quivoque : \u00ab effondrement \u00bb, \u00ab d\u00e9b\u00e2cle \u00bb, \u00ab chaos \u00bb 8. Par chance, il pousse la porte d'un vieux laboratoire toujours en exercice, peupl\u00e9 de scientifiques de renom (Oldenbourg, Karpinsli, Pavlov, Radlov et bien d'autres) qui poursuivent tant bien que mal leurs travaux de recherche. Wells veut les aider, et fait son possible pour leur faire envoyer d'Angleterre des livres et du mat\u00e9riel dont ils ont besoin. Il visite aussi des foyers pour enfants abandonn\u00e9s et des \u00e9coles : dans l'une d'elles, il se rend compte que son arriv\u00e9e a \u00e9t\u00e9 savamment pr\u00e9par\u00e9e, les \u00e9l\u00e8ves l'accueillant en \u00e9non\u00e7ant par c\u0153ur les titres des ouvrages dont il est l'auteur. Contrari\u00e9, Wells ne s'attarde pas, mais reconna\u00eet que le syst\u00e8me \u00e9ducatif russe est bon, qu'on y apprend des choses. D\u00e8s qu'il le peut, il s'\u00e9chappe de la ville et part dans les provinces \u00e9loign\u00e9es. L\u00e0-bas, il observe \u00e0 sa grande surprise que les paysans sont plut\u00f4t bien lotis et parviennent \u00e0 vivre de leur r\u00e9colte, contrairement aux autres classes de la population. L\u00e0 encore, son appr\u00e9ciation est \u00e0 l'oppos\u00e9 du s\u00e9jour de 1914 : probablement influenc\u00e9 par Gorki qui n'h\u00e9site pas \u00e0 critiquer ces \u00ab moujiks \u00bb que Tolsto\u00ef porte aux nues, Wells regarde les habitants de la campagne comme une menace.\n\nLa distraction arrive le soir, dans les salles obscures des th\u00e9\u00e2tres et des op\u00e9ras qui tiennent encore debout. Depuis la R\u00e9volution, les arts sont en difficult\u00e9 sauf la sc\u00e8ne, qui survit gr\u00e2ce aux g\u00e9n\u00e9reuses subventions de l'\u00c9tat. \u00c0 la nuit tomb\u00e9e, la foule, argent\u00e9e ou non (l'entr\u00e9e est gratuite), rejoint les fauteuils d'orchestre et oublie la rudesse de ces temps troubl\u00e9s en \u00e9coutant _Le Barbier de S\u00e9ville_. Et soudain, \u00ab on e\u00fbt dit que rien n'avait chang\u00e9 en Russie 9 \u00bb.\n\nWells ne perd pas de vue son objectif : rencontrer le nouvel homme fort du pays, L\u00e9nine. Apr\u00e8s plusieurs d\u00e9marches fastidieuses, il est introduit dans la Maison des Invit\u00e9s de Moscou puis dans les appartements du \u00ab r\u00eaveur du Kremlin 10 \u00bb, accompagn\u00e9 par M. Rothstein, interpr\u00e8te, et par un photographe am\u00e9ricain, Wigfrid Humphries, venu pour immortaliser ce moment politique important. En entrant dans son bureau, Wells est frapp\u00e9 de trouver un petit homme assis derri\u00e8re une grande table, au milieu d'un beau d\u00e9sordre, \u00ab le visage plut\u00f4t basan\u00e9, les traits agr\u00e9ables et mobiles, un sourire enjou\u00e9 et l'habitude ...] de contracter un \u0153il quand il cesse de parler[ 11 \u00bb. La discussion s'engage rapidement, l'Anglais demandant \u00e0 son h\u00f4te ce qu'il compte faire pour reb\u00e2tir cette Russie en ruine, L\u00e9nine s'\u00e9tonnant que l'Angleterre n'ait toujours pas entam\u00e9 sa r\u00e9volution sociale. Il se r\u00e9jouit de cette entrevue \u2014 il avait \u00e9t\u00e9 auparavant d\u00e9\u00e7u de ne pouvoir se faire entendre face \u00e0 une d\u00e9l\u00e9gation de d\u00e9put\u00e9s travaillistes. Il expose \u00e0 Wells tous ses projets, parfois utopiques (notamment l'\u00e9lectrification \u00e9tendue \u00e0 toutes les provinces du pays, qui aboutirait \u00e0 l'av\u00e8nement de l'\u00e8re industrielle, et \u00e0 la disparition de la paysannerie). Wells sourit. Il \u00e9coute cet homme r\u00eaver \u00e0 voix haute. Le pays est trop atteint, trop affaibli pour mettre en place ce type de bouleversements. Mais L\u00e9nine, confiant, le rassure : \u00ab Revenez dans dix ans voir ce que nous aurons accompli ...][ 12. \u00bb Wells repart ind\u00e9cis, troubl\u00e9, mais avec la certitude qu'il n'est plus le seul \u00e0 d\u00e9sirer un monde meilleur.\n\nIl ne lui reste que quelques jours avant de rejoindre l'Angleterre. Il les passe \u00e0 Petrograd avec Moura, qui l'emm\u00e8ne dans des endroits insolites et lui fait part de sa vision de l'avenir pour la Russie. Mais Wells ne veut pas parler politique avec elle. Un soir, de retour dans l'appartement de Gorki apr\u00e8s une nuit en ville, ils partent chacun dormir dans leur chambre. En pleine nuit, Moura se r\u00e9veille avec l'impression que quelqu'un se tient pr\u00e8s d'elle. En effet, Wells est l\u00e0, dans l'obscurit\u00e9, au pied de son lit. Ils s'embrassent et font l'amour juste \u00e0 c\u00f4t\u00e9 de la chambre de Gorki qui dort paisiblement. Wells est amoureux comme il ne l'a jamais \u00e9t\u00e9 13. Le lendemain, il lui faut partir. Elle l'accompagne sur le quai de la gare, pense qu'ils ne se reverront jamais, et lui dit juste : _\u00ab I will never forget you_ 14. \u00bb (\u00ab Je ne t'oublierai jamais. \u00bb) Dans le train, il regarde Moura s'\u00e9loigner, la Russie d\u00e9filer sous ses yeux, et il pense \u00e0 Rebecca, \u00e0 leur amour contrari\u00e9, \u00e0 cette relation qui le gangr\u00e8ne. Cela doit cesser, une fois pour toutes.\n\nDe retour en Angleterre, il raconte son r\u00e9cent voyage dans une s\u00e9rie d'articles publi\u00e9s dans le _Sunday Express_ entre le 31 octobre et le 28 novembre 1920, et r\u00e9unis dans un recueil intitul\u00e9 _La Russie dans l'ombre_. C'est un travail documentaire pr\u00e9cis et critique qui tente d'analyser la situation politique et sociale du pays. Surtout, Wells veut convaincre l'Occident que le bolchevisme est la seule et unique solution pour le peuple russe. L'ouvrage provoque l'ire de certains politiques et intellectuels : plusieurs journalistes lui reprochent sa connivence avec celui qui est consid\u00e9r\u00e9 comme un ennemi de la d\u00e9mocratie (L\u00e9nine) ; d'autres, des expatri\u00e9s russes vivant en Angleterre, pointent du doigt son caract\u00e8re versatile. Pour beaucoup de monde, Wells s'est mis du mauvais c\u00f4t\u00e9 de l'\u00e9chiquier. Mais l'int\u00e9ress\u00e9 ne sourcille pas. Il demande m\u00eame une entrevue \u00e0 Lord Curzon, le ministre britannique des Affaires \u00e9trang\u00e8res. L'homme est plut\u00f4t aust\u00e8re, conservateur, et refuse d'entendre ce que son invit\u00e9 veut lui dire. Wells a beau d\u00e9ployer l'argument humain \u2014 la n\u00e9cessit\u00e9 de venir en aide non pas \u00e0 des f\u00e9rus de marxisme mais \u00e0 des hommes qui se meurent, avant tout \u2014, Curzon reste flegmatique. D'un ton cinglant, il r\u00e9plique. Non, il ne d\u00e9bloquera pas de moyens financiers pour aider la Russie \u00e0 sortir du gouffre de l'apr\u00e8s-R\u00e9volution. Et non, jamais il n'approuvera le gouvernement bolchevique, ce syst\u00e8me politique qui ose critiquer la Grande-Bretagne \u00e0 l'\u00e9tranger. Wells n'insiste pas, le salue, et franchit la porte avec le sentiment d'avoir \u00e9chou\u00e9. Le pas est lourd, la t\u00eate baiss\u00e9e. \u00c0 cet instant, il est persuad\u00e9 que l'Angleterre a manqu\u00e9 une marche de son destin : en refusant de regarder la Russie en face, elle souille son id\u00e9al d\u00e9mocratique. Il est d\u00e9\u00e7u, mais surtout en col\u00e8re de n'avoir pour interlocuteurs que ce genre de \u00ab d\u00e9ficients infantiles, qui devraient \u00eatre ou renvoy\u00e9s \u00e0 l'\u00e9cole pour \u00e9tudier les \u00e9l\u00e9ments de la sociologie, ou \u00eatre reconnus comme des esprits endommag\u00e9s, et mis de c\u00f4t\u00e9 comme incapables de diriger les affaires publiques 15 \u00bb. Lui n'a pas l'intention de passer \u00e0 c\u00f4t\u00e9 de la construction du monde et veut s'en donner les moyens. Son projet de r\u00e9former la soci\u00e9t\u00e9 en profondeur, de cr\u00e9er une humanit\u00e9 unie et solide, ne peut aboutir que s'il en a le pouvoir concret. Alors, comme il se jetterait d'une falaise dans le vide, il se pr\u00e9sente aux \u00e9lections l\u00e9gislatives du c\u00f4t\u00e9 des travaillistes. La t\u00e2che est ardue et p\u00e9nible. Mener campagne l'\u00e9puise : il court les r\u00e9unions, les comit\u00e9s, donne des interviews, tente de faire entendre sa voix. Mais il doit faire face aux vieux repr\u00e9sentants du Labour Party qui n'appr\u00e9cient gu\u00e8re son volontarisme. Les \u00e9lections sont un \u00e9chec. Wells est moins embarrass\u00e9 que perplexe. Il a le vertige.\n\n*1. _Un grand \u00e9ducateur moderne. Sanderson, directeur du coll\u00e8ge d'Oundle_ (1924).\nD\u00e9grisement\n\n1923, Londres : cet \u00ab ...] amas chaotique de rues, de b\u00e2timents, [et] de gens courant sans raison \u00e7\u00e0 et l\u00e0[ 1 \u00bb. Et lui, fig\u00e9 dans son appartement de Whitehall Court, toujours enfonc\u00e9 dans le fauteuil de son bureau, \u00e0 laisser filer les heures, \u00e0 se demander ce qu'il fait l\u00e0. Depuis l'incident avec Hedwig, il n'a pas boug\u00e9. On vient de l'appeler. Son admiratrice suicidaire a \u00e9t\u00e9 transf\u00e9r\u00e9e au Charing Cross Hospital, elle va bien et passe la nuit en observation. Lui est toujours couvert de sang, incapable de faire le moindre mouvement. Il faudrait se lever peut-\u00eatre, changer de chemise, se mettre au lit. Le soleil a d\u00e9clin\u00e9, il ne s'est m\u00eame pas rendu compte que son bureau \u00e9tait plong\u00e9 dans le noir. Dehors, il entend les voitures se garer : les journalistes, sans aucun doute. La domestique entre, lui dit que plusieurs personnes veulent lui parler. D'un geste de la t\u00eate, il fait \u00ab non \u00bb et ferme les yeux. Il est tellement fatigu\u00e9, il aimerait dormir, oublier toutes ces femmes, ces visages et ces corps qui l'emp\u00eachent de se reposer, d'\u00e9crire \u2014 de vivre. Revoil\u00e0 la famili\u00e8re d\u00e9tresse, comme une vieille rengaine qui vient oppresser son c\u0153ur. Il aimerait tellement s'en d\u00e9faire. Parfois, il a peur de finir comme les h\u00e9ros de ses romans, tel celui des _Coins secrets du c\u0153ur_ , le dernier en date, qui met en sc\u00e8ne un homme dans une impasse existentielle. Ne sachant comment changer de vie, il d\u00e9cide d'attendre qu'elle passe. Wells se demande ce qu'il lui a pris d'\u00e9crire une telle histoire. Lui n'a jamais fait cela. \u00c0 choisir, il ferait comme le protagoniste de _M. Barnstaple chez les hommes-dieux_. Dans cette fantaisie romanesque, un homme en pleine crise quitte femme et enfant pour des vacances solitaires et arrive finalement en Utopie. Oui, cela lui conviendrait mieux. Changer d'horizon, d'univers. Il a besoin de partir. \u00ab Si ta vie n'est pas satisfaisante, change-la \u00bb : il n'a pas oubli\u00e9 le credo de ses jeunes ann\u00e9es. Jusqu'ici, il l'a sans cesse appliqu\u00e9, fuyant toujours l'inertie, la s\u00e9curit\u00e9 et m\u00eame le confort. Mais aujourd'hui, la fatigue lui p\u00e8se. Il faut qu'il se mette debout. Demain matin, il a rendez-vous avec Rebecca. Il ne sait pas ce qu'il va lui dire. Qu'il a couch\u00e9 avec Moura ? Qu'une femme enti\u00e8rement nue s'est ouvert les veines chez lui ? Qu'il ne l'aime plus ? Tout cela, oui, entre autres. Wells se l\u00e8ve, se dirige vers la porte, et dispara\u00eet dans la p\u00e9nombre de son appartement.\n\nC'est une matin\u00e9e agr\u00e9able. Assis sur un banc de Kensington Garden avec Rebecca, il savoure la douceur de l'\u00e9t\u00e9 qui commence tout juste. Sa voix est sereine, elle lui sourit, lui annonce qu'elle se rend avec une amie \u00e0 Marienbad, qu'il est le bienvenu s'il veut se joindre \u00e0 elles. La proposition ne peut pas se refuser. Et puis, cela fait longtemps qu'ils ne sont pas partis avec leur fils Anthony, en famille. Jane comprendra \u2014 elle comprend toujours. Les vacances sont agr\u00e9ables bien que parfois houleuses, certaines soir\u00e9es se terminant dans les cris et les larmes, Wells se rendant compte (pour la \u00e9ni\u00e8me fois) qu'ils ne sont pas faits pour \u00eatre ensemble 2. \u00c0 sa grande surprise, c'est Rebecca qui d\u00e9cide de rompre. Elle doit bient\u00f4t se rendre aux \u00c9tats-Unis pour une s\u00e9rie de conf\u00e9rences. Peut-\u00eatre est-ce le moment qu'ils se s\u00e9parent pour de bon. Alors, ils se font leurs adieux. Rebecca s'en va au mois d'octobre ; Jane, elle, a emmen\u00e9 les enfants dans les Alpes. Wells se retrouve seul \u00e0 Londres, triste, et encombr\u00e9 d'une mauvaise bronchite. Un peu de soleil lui ferait du bien, celui de Lisbonne par exemple. Il part et s'installe \u00e0 l'h\u00f4tel Mirimar de la capitale portugaise pour plusieurs mois, profitant de cette nouvelle libert\u00e9 pour s\u00e9duire une jolie rousse, Dorothy Petrie, avec laquelle il se rend peu apr\u00e8s au pays basque. Une rousse, c'est bien la premi\u00e8re fois que cela lui arrive.\n\nSes doigts caressent la longue chevelure de feu, mais son c\u0153ur est ailleurs. Cela fait des mois qu'il ne pense qu'\u00e0 Moura, qu'il lui \u00e9crit et esp\u00e8re la voir. Il sait depuis peu qu'elle a r\u00e9ussi \u00e0 s'enfuir de Russie et qu'elle a \u00e9pous\u00e9 Nicola\u00ef Budberg, un baron dont elle a pay\u00e9 les dettes de jeu en \u00e9change d'un passeport estonien. Tallinn est sa nouvelle patrie. La vie \u00e0 Petrograd devenait impossible pour elle. Gorki aussi a d\u00fb partir, \u00e9chappant \u00e0 de r\u00e9currentes menaces de mort. Il s'\u00e9tait permis de critiquer le parti bolchevique, et de pointer du doigt l'attitude de L\u00e9nine. Il savait pourtant qu'il n'avait pas le droit de critiquer le chef. Atteint d'une tuberculose chronique, le vieil ami de Wells a pr\u00e9f\u00e9r\u00e9 s'exiler. Il s'est install\u00e9 sur la c\u00f4te amalfitaine, \u00e0 Sorrente, o\u00f9 il coule des jours paisibles. Il se souvient de leur dernier d\u00eener \u00e0 tous les trois, \u00e0 Petrograd, la veille de son d\u00e9part pour l'Angleterre. C'\u00e9tait une belle soir\u00e9e. Il avait pu toucher Moura, il avait senti qu'il \u00e9tait amoureux d'elle. Maintenant, il en \u00e9tait s\u00fbr. Parfois, quand il se prom\u00e8ne dans les rues de Londres, il a l'impression de la voir, de reconna\u00eetre son allure, son parfum. Mais non, c'est juste une brune qui passe \u2014 il y en a d\u00e9cid\u00e9ment beaucoup trop en Angleterre.\n\nDans sa bo\u00eete aux lettres, il trouve un courrier de Rebecca qui est de retour des \u00c9tats-Unis et demande \u00e0 le voir. Si Moura est insaisissable, Rebecca, elle, demeure omnipr\u00e9sente. Crispation. Il pensait pourtant cette affaire r\u00e9gl\u00e9e. Il n'a plus rien \u00e0 lui dire, plus rien \u00e0 vivre avec elle. Il a toujours eu le sentiment de _presque_ l'aimer, mais de ne jamais tout \u00e0 fait y parvenir 3. Rebecca appartient au pass\u00e9, elle fait d\u00e9j\u00e0 partie de ses livres (elle est Amanda dans _La Recherche magnifique,_ et sera Helen dans _Le Monde de William Clissold_ ), c'est dire s'il a tourn\u00e9 la page. Une autre lettre le sort bient\u00f4t de sa torpeur : c'est Amber qui lui \u00e9crit pour lui donner des nouvelles de leur fille, Anna Jane. La petite a bient\u00f4t quatorze ans, elle est, selon les dires de sa m\u00e8re, une jeune \u00e9l\u00e8ve brillante avec de grandes ambitions \u2014 comme son p\u00e8re. Parfois, c'est \u00e9trange, il oublie qu'il a aussi une fille et que cette derni\u00e8re n'a aucune id\u00e9e que son g\u00e9niteur s'appelle Herbert George Wells. Peut-\u00eatre est-ce mieux comme cela. Peut-\u00eatre que les choses ne doivent pas tout le temps \u00eatre dites. Sur ces sages r\u00e9flexions, il ouvre une troisi\u00e8me enveloppe : une invitation pour la r\u00e9union de la Soci\u00e9t\u00e9 des Nations \u00e0 Gen\u00e8ve, au mois de septembre prochain \u2014 dans quelques semaines, donc. Cela lui \u00e9tait totalement sorti de l'esprit.\n\nLe voil\u00e0 en Suisse, dans sa chambre d'h\u00f4tel, assis sur le rebord du lit, et entour\u00e9 de ses valises. Le t\u00e9l\u00e9phone sonne. Il h\u00e9site \u00e0 d\u00e9crocher \u2014 \u00e0 peine arriv\u00e9, on le d\u00e9range d\u00e9j\u00e0. Il saisit finalement le combin\u00e9, articule un son et \u00e9coute. C'est une femme. Mieux : c'est Odette Keun. Elle est dans le m\u00eame h\u00f4tel que lui, \u00e0 quelques portes de sa chambre, et lui demande de venir la rejoindre. Il ne l'a encore jamais vue, mais la conna\u00eet depuis au moins un an. Elle est \u00e9crivain, elle aussi, et fille de diplomate. Elle a du sang italien dans les veines et la r\u00e9putation de toujours arriver \u00e0 ses fins. Au d\u00e9but de l'ann\u00e9e 1923, l'un de ses ouvrages \u00e9crits en fran\u00e7ais ( _Sous L\u00e9nine_ ) est traduit en anglais, et le r\u00e9dacteur en chef du magazine _Adelphi_ , Middleton Murry, demande \u00e0 Wells de r\u00e9diger une notice pour signaler la parution du livre. Wells s'ex\u00e9cute, un peu machinalement, sans savoir \u00e0 quoi il s'expose. Odette est l'une de ses plus ferventes lectrices, cela fait des ann\u00e9es qu'elle lui voue une admiration sans faille et qu'elle ne part pas en voyage sans un de ses romans dans ses bagages. Quand elle apprend que son \u00e9crivain pr\u00e9f\u00e9r\u00e9 lui a consacr\u00e9 un article, elle saisit l'opportunit\u00e9 pour lui \u00e9crire et lui faire une proposition os\u00e9e : qu'il vienne la voir chez elle \u00e0 Paris, et qu'ils deviennent amants. Amus\u00e9, Wells avait, \u00e0 l'\u00e9poque, d\u00e9clin\u00e9 poliment. En a-t-il envie \u00e0 cet instant ? Apr\u00e8s tout, il n'y a plus Rebecca, il n'y a pas Jane, et pas encore Moura. Alors il accepte, raccroche, r\u00e9ajuste sa veste et sort. La chambre d'Odette se trouve au bout du couloir, \u00e0 quelques m\u00e8tres.\n\nIl est en face de la porte : c'est le bon num\u00e9ro. Il frappe, per\u00e7oit un mouvement et soudain se retrouve devant une femme qui lui dit d'entrer. \u00c0 l'int\u00e9rieur, il fait noir, les rideaux ont \u00e9t\u00e9 tir\u00e9s, et \u00e7a sent le jasmin. Odette le d\u00e9shabille, lui dit qu'elle est heureuse qu'il soit l\u00e0, et qu'elle l'aime depuis toujours 4. Cette femme le fascine : elle a grandi \u00e0 Constantinople, a pass\u00e9 plusieurs ann\u00e9es dans un couvent dont elle s'est fait renvoyer le jour o\u00f9 son confesseur lui a fait des avances. Arriv\u00e9e \u00e0 Paris, elle devient la ma\u00eetresse d'un professeur de sociologie, un certain Bernard Lavergne, commence \u00e0 \u00e9crire un essai intitul\u00e9 _Une femme moderne_ et d\u00e9dicac\u00e9... \u00e0 Wells 5. Odette est excentrique, dr\u00f4le, il a besoin d'elle, de sa bonne humeur, mais n'en est pas amoureux. Cela ne l'emp\u00eache pas de l'emmener dans le sud de la France, pr\u00e8s de Grasse, dans une maison qu'il loue pour l'hiver. \u00ab Lou Bastidon \u00bb (c'est le nom de la propri\u00e9t\u00e9) cache leurs amours pendant plusieurs mois, \u00e0 l'abri des regards indiscrets, et loin de la vie londonienne dont Wells s'est lass\u00e9. Jane a tout accept\u00e9. Elle conna\u00eet bien son \u00e9poux, et sait qu'il a besoin de temps pour lui seul. Depuis des ann\u00e9es, ils vivent de mani\u00e8re tr\u00e8s ind\u00e9pendante : elle, \u00e0 Easton Glebe et lui, \u00e0 Whitehall Court. Il lui rend souvent visite le week-end, et elle passe lui dire bonjour lorsqu'elle est de passage dans la capitale (elle aussi poss\u00e8de un petit appartement, pr\u00e8s de Bloomsbury, au c\u0153ur duquel elle peut \u00e9crire \u00e0 sa guise). Cette libert\u00e9 leur convient bien \u00e0 tous les deux.\n\nLe soleil de Grasse est curatif, tout comme cette terre jonch\u00e9e d'oliviers. Wells s'y sent bien. Ses probl\u00e8mes comme ses frustrations s'\u00e9loignent. Mais \u00e0 presque soixante ans, l'heure de la retraite n'a pas encore sonn\u00e9. Son esprit fourmille de mille id\u00e9es, il n'a pas abandonn\u00e9 le projet de sauver le monde de sa propre perte, et il est pr\u00eat \u00e0 \u00e9crire de nouveaux romans. _Entre-temps_ , d'abord, un r\u00e9cit social tr\u00e8s document\u00e9 sur le monde ouvrier en Angleterre. Puis _Le Monde de William Clissold_ , d\u00e9dicac\u00e9 \u00e0 Odette, publi\u00e9 en 1926 et d'embl\u00e9e annonc\u00e9 comme le grand retour de l'\u00e9crivain. Ces derni\u00e8res ann\u00e9es, la science-fiction n'est plus \u00e0 la mode. La litt\u00e9rature est devenue introspective : ce n'est pas le futur qui int\u00e9resse les lecteurs, ce sont l'intimit\u00e9, les \u00e9motions, les m\u00e9andres du moi. Le groupe de Bloomsbury est d\u00e9sormais incontournable, Virginia Woolf s'est fait remarquer avec _La_ _Chambre de Jacob_ et _Mrs Dalloway_ , tout comme James Joyce qui a publi\u00e9 _Gens de Dublin_ et surtout _Ulysse._ Pour Wells, la gloire est pass\u00e9e, on l'observe comme un essayiste, un conf\u00e9rencier, plus du tout comme un romancier. Peu importe, ses ambitions sont intactes, et tant qu'il peut \u00e9crire, tant qu'il peut se faire entendre, il continue. Il r\u00eave d'une entente secr\u00e8te entre les hommes, non pas d'un complot mais d'une \u00ab conspiration \u00bb qui s'accomplirait dans la lumi\u00e8re \u2014 au grand jour :\n\nCe n'est qu'\u00e0 travers une coop\u00e9ration consciente, universelle des hommes de la science, des travailleurs scientifiques, des hommes habitu\u00e9s \u00e0 diriger l'industrie productive, des hommes capables de contr\u00f4ler les art\u00e8res de l'afflux du cr\u00e9dit, des hommes qui peuvent avoir le contr\u00f4le des journaux et des politiciens, \u2014 que le grand syst\u00e8me d'\u00e9changes, qui a \u00e9t\u00e9 mis en marche presque par inadvertance, peut \u00eatre d\u00e9velopp\u00e9 d'une fa\u00e7on ordonn\u00e9e, et telle qu'elle donne des espoirs pour l'avenir.\n\nLes hommes de cette sorte, qu'ils le veuillent ou non, sont les vrais r\u00e9volutionnaires de notre monde 6...\n\nChercher les prochains \u00ab conspirateur[s] au grand jour \u00bb et \u00e9laborer une v\u00e9ritable solidarit\u00e9 intellectuelle deviennent les nouvelles obsessions de l'\u00e9crivain. L'expression va bient\u00f4t devenir un livre.\n\nLa douceur du sud de la France d\u00e9cide Wells \u00e0 investir dans une nouvelle maison. Il en fait construire une, selon ses d\u00e9sirs, dans cette belle r\u00e9gion qu'il aime tant, sur \u00ab un terrain avec de jolis rochers, des vignes, du jasmin, et un cours d'eau tout pr\u00e8s 7 \u00bb, et la baptise \u00ab Lou Pidou \u00bb \u2014 contraction de \u00ab le petit dieu \u00bb, surnom fran\u00e7ais qu'Odette aimait donner \u00e0 Wells. Mais la somptueuse propri\u00e9t\u00e9, les fr\u00e9quentes vir\u00e9es sur la c\u00f4te, et les amis de l'\u00e9crivain qui passent r\u00e9guli\u00e8rement, ne masquent pas les tensions qui \u00e9mergent au sein du couple. Charles Chaplin en fait l'exp\u00e9rience. Le com\u00e9dien arrive \u00e0 Lou Pidou un matin et s'extasie devant la beaut\u00e9 du lieu. Il remarque notamment l'inscription grav\u00e9e au-dessus de la chemin\u00e9e (\u00ab _Two lovers built this house_ \u00bb, \u00ab Deux amants ont construit cette maison \u00bb) et se tourne vers Wells en souriant, heureux de voir son ami amoureux. Mais son h\u00f4te lui explique sans attendre que tout cela n'est pas aussi simple qu'il y para\u00eet : qu'un ma\u00e7on est venu r\u00e9cemment enlever ce bout de pierre car il s'\u00e9tait disput\u00e9 avec Odette et pensait la quitter. Ils s'\u00e9taient finalement rabiboch\u00e9s et Wells avait donc demand\u00e9 au ma\u00e7on de revenir pour restaurer l'inscription. Aujourd'hui, leur amour \u00e9tait au plus mal, et il pensait tr\u00e8s s\u00e9rieusement faire de nouveau appel \u00e0 cet homme patient et d\u00e9vou\u00e9. Quelques secondes plus tard, Chaplin et lui ouvraient une bonne bouteille.\n\nWells a tout pour \u00eatre heureux. Il a deux femmes et deux maisons, dans deux pays diff\u00e9rents, que demander de plus ? Mais Odette est maladivement jalouse de Jane, ouvrant toutes les lettres que l'\u00e9pouse envoie \u00e0 son mari \u2014 devenant de plus en plus insupportable. Face \u00e0 ses reproches, \u00e0 ses cris, et malgr\u00e9 tout l'argent qu'il lui donne chaque jour depuis trois ans, Wells mesure pour la premi\u00e8re fois \u00ab le tracas de vivre 8 \u00bb avec elle. \u00ab Ce sont les femmes ...] qui nous obligent \u00e0 travailler et nous imposent tous les tracas. [...] Somme toute, je ne fais pas l\u00e0 une d\u00e9couverte si originale : relisez donc le troisi\u00e8me chapitre de la Gen\u00e8se[ 9. \u00bb Ce que pense le narrateur d' _Effrois et fantasmagories_ , un recueil de nouvelles cocasses, __ est \u00e0 peu pr\u00e8s identique \u00e0 ce que ressent Wells \u00e0 cet instant. Et puis la maison lui cause beaucoup de soucis \u2014 le jardinage, l'entretien de la voiture, des all\u00e9es, et cet anonymat qui ne joue plus en sa faveur. Comme souvent dans ce genre de situations, il laisse passer l'orage.\n\nDe l'autre c\u00f4t\u00e9 de la Manche, Jane la discr\u00e8te pr\u00e9pare ses valises. Elle doit rejoindre son mari \u00e0 Paris pour une conf\u00e9rence qu'il donne \u00e0 la Sorbonne dans quelques jours. Wells fait de m\u00eame et quitte Odette, devenue en quelques semaines \u00ab mauvaise et d\u00e9testable 10 \u00bb, pour sa tendre et fid\u00e8le \u00e9pouse. Dans le train, il peaufine son discours. Cette invitation fran\u00e7aise est tr\u00e8s importante, c'est un honneur d'\u00eatre accueilli l\u00e0-bas. Jane l'attend \u00e0 la gare, cela fait des semaines qu'ils ne se sont pas vus, Wells est heureux. La conf\u00e9rence a lieu l'apr\u00e8s-midi, ils n'ont pas beaucoup de temps. Arriv\u00e9 sur place, on le dirige vers le grand amphith\u00e9\u00e2tre. La salle est comble, il se hisse derri\u00e8re son pupitre et annonce le sujet du jour : \u00ab La d\u00e9mocratie soumise \u00e0 r\u00e9vision \u00bb. Depuis son exp\u00e9rience (malheureuse) avec la Soci\u00e9t\u00e9 fabienne, il n'a pas eu l'occasion de s'exprimer devant un tel public. Certes, il a \u00e9nonc\u00e9 ses id\u00e9es dans ses livres, dans les journaux et face \u00e0 de grandes personnalit\u00e9s politiques, mais aujourd'hui ses propos r\u00e9sonnent comme jamais dans le pays qui fut le berceau des Lumi\u00e8res.\n\nL'auditoire \u2014 plut\u00f4t jeune \u2014 est attentif, et Wells f\u00e9licit\u00e9 pour son intervention. Comme d'habitude, il est observ\u00e9 en anticipateur un peu fantaisiste, en romancier qui ne peut s'arr\u00eater de r\u00eaver. Pour beaucoup, ce qu'il explique ce jour-l\u00e0 n'est rien d'autre qu'un remaniement de ses th\u00e8ses pass\u00e9es, \u00e0 savoir le projet d'un syst\u00e8me politique et social qui ressemble davantage \u00e0 une utopie qu'\u00e0 une r\u00e9alit\u00e9 potentielle. On le salue, on le complimente, lui ne sait comment exprimer cette inqui\u00e9tude qui l'habite, ce pressentiment qu'il faut agir avant que le monde ne bascule de nouveau. Nous sommes en 1927, la Seconde Guerre mondiale n'est m\u00eame pas encore un fantasme, mais lui sait qu'il y a urgence, qu'il faut trouver une alliance, former une communaut\u00e9 d'esprits. Les sourires sont de rigueur, les mains se serrent fermement, et Wells ravale son amertume qui risque de le faire flancher. Jane est \u00e0 ses c\u00f4t\u00e9s, souriante et vigoureuse. Cela fait des mois qu'elle apprend le fran\u00e7ais en pr\u00e9paration de cet \u00e9v\u00e9nement. Parmi la foule de personnalit\u00e9s venues \u00e0 leur rencontre, la plus grande partie est fran\u00e7aise, et Jane parle \u00e0 tout le monde avec aisance. Elle irradie d'intelligence et impressionne Marie Curie qui a fait le d\u00e9placement. La scientifique d\u00e9sirait depuis longtemps rencontrer le fameux couple d'Anglais.\n\nElle irradie et, pourtant, elle sait qu'elle est gravement malade. Une douleur abdominale l'affaiblit depuis des semaines. Les m\u00e9decins sont cat\u00e9goriques : c'est un cancer, trop avanc\u00e9 pour \u00eatre gu\u00e9ri. Wells l'ignore encore, Jane ne veut pas l'inqui\u00e9ter. Ils se quittent apr\u00e8s la conf\u00e9rence, lui redescend dans le Sud, elle rentre \u00e0 Londres. Mais son \u00e9tat s'aggrave, et leur second fils, Frank, d\u00e9cide de prendre les devants. Jane est trop triste pour annoncer \u00e0 son mari la mauvaise nouvelle \u2014 il ne lui reste plus que quelques mois \u00e0 vivre. Frank \u00e9crit alors \u00e0 son p\u00e8re et Wells, sous le choc, envoie imm\u00e9diatement une lettre \u00e0 Jane, l'assurant de son retour prochain, de son amour profond, et lui promettant qu'il va s'occuper d'elle.\n\nCh\u00e8re m\u00e8re, ma tendre \u00e9pouse, j'ai re\u00e7u aujourd'hui une lettre de Frank m'apprenant que tu as \u00e9t\u00e9 gravement malade et qu'il est fort possible que cela continue. Ma ch\u00e9rie, je t'aime comme je n'ai jamais aim\u00e9 personne au monde, et j'arrive pour m'occuper de toi et tout faire pour te rendre heureuse.\n\nTon Bins qui t'aime 11.\n\nIl arrive en Angleterre quelques jours plus tard, se rendant directement \u00e0 Easton Glebe. Il trouve Jane amaigrie mais souriante, refusant de se plaindre. Elle rassure tout le monde, pr\u00e9pare son d\u00e9part, console ses enfants et leur demande de ne pas porter le deuil quand elle sera morte. Wells voit leurs trente ans de mariage d\u00e9filer dans sa t\u00eate. Les premi\u00e8res ann\u00e9es \u00e0 Mornington Place, l'excitation de ses premiers succ\u00e8s litt\u00e9raires, puis le soutien ind\u00e9fectible de Jane, leur travail commun, son sens de la critique, son indulgence, surtout. Il veut profiter du peu de temps qu'ils ont encore \u00e0 vivre ensemble et fait tout pour lui faciliter le quotidien. La journ\u00e9e, ils passent des heures \u00e0 \u00e9couter ses musiques pr\u00e9f\u00e9r\u00e9es (Bach, Mozart, Purcell), \u00e0 parler, se rem\u00e9morer leurs bons souvenirs, se relire leurs premi\u00e8res lettres d'amour. Quand elle s'endort, il se met \u00e0 son bureau et travaille \u00e0 son prochain ouvrage, _La Science de la vie_. Pour ce nouveau livre, il travaille conjointement avec son fils Gip mais aussi avec Julian (le petit-fils de T. H. Huxley et fr\u00e8re d'Aldous), grand biologiste \u2014 et futur premier directeur de l'Unesco.\n\nLe mois d'octobre arrive. Jane meurt la veille du mariage de Frank. La cr\u00e9mation a lieu dans le quartier londonien de Golders Green. Tous leurs amis sont l\u00e0 : le couple Shaw notamment, le fid\u00e8le Arnold Bennett, et m\u00eame Virginia Woolf qui ne manquera pas de relater cet \u00e9v\u00e9nement dans son _Journal_. De retour \u00e0 Easton Glebe, Wells ne sait pas ce qu'il va faire. Jane \u00e9tait la seule qui apportait un \u00e9quilibre \u00e0 sa vie \u2014 un \u00e9quilibre et une \u00ab dignit\u00e9 12 \u00bb. Au fond, il a toujours cru qu'elle lui survivrait, c'est lui qui est malade depuis tant d'ann\u00e9es, ce corps l'a tellement trahi qu'il aurait d\u00fb mourir dix fois avant elle. Cette libert\u00e9 tant d\u00e9sir\u00e9e lui devient soudain r\u00e9elle, pesante. Il est seul, enfin, il n'est plus oblig\u00e9 de rendre de comptes \u00e0 personne, mais il est triste et abhorre d\u00e9sormais ce qu'il a si longtemps convoit\u00e9. Toutes les femmes du monde peuvent sonner \u00e0 sa porte, il ne leur ouvrira pas. La maison lui para\u00eet bien vide, trop grande pour lui \u2014 Gip est parti aux \u00c9tats-Unis et Frank aussi doit bient\u00f4t y aller (il \u00e9crit des sc\u00e9narios pour le th\u00e9\u00e2tre et le cin\u00e9ma). Wells range les affaires de Jane, tous ses cahiers, ses manuscrits qu'elle gardait pr\u00e9cieusement dans son secr\u00e9taire. Il savait qu'elle \u00e9crivait, mais n'avait jamais pris le temps d'ouvrir un seul de ses manuscrits. Le voil\u00e0, assis dans son bureau, \u00e0 tout lire, \u00e0 comprendre qu'il a pu devenir ce qu'il est justement parce que Jane avait accept\u00e9 de rel\u00e9guer son ambition dans les tiroirs. Il met tout en ordre, classe les diff\u00e9rents textes et en fait un livre ( _Le Livre de Catherine Wells)_ , publi\u00e9 en 1928.\n\nEaston Glebe est mise en vente. Avec l'argent de la transaction, il finit de payer Lou Pidou, ce gouffre financier et sentimental dont les couloirs sont toujours hant\u00e9s par l'envahissante Odette. Il ne sait plus comment s'en d\u00e9barrasser. De toute \u00e9vidence, le couple qu'il formait avec Jane ne ressemblait \u00e0 aucun autre, jamais plus il ne retrouvera un tel \u00e9quilibre, un tel bonheur \u2014 une telle femme. Odette, qui esp\u00e8re d\u00e9sormais r\u00e9cup\u00e9rer Wells pour elle toute seule, l'accueille avec sa moue de reproche. Elle veut habiter \u00e0 Paris avec lui, quitter Lou Pidou qui devient une prison. Wells refuse, il n'a aucune envie d'aller jouer les amants passionn\u00e9s dans la capitale, et ne veut surtout pas envisager un quotidien avec elle. Elle est son amante, elle le restera. La maison r\u00e9sonne souvent de leurs cris, et Wells prend l'habitude de s'isoler dans la chambre. Seul, il se demande comment continuer ainsi.\nLe conspirateur amoureux\n\n\u00ab Mais que faites-vous donc 1 ? \u00bb La question lui est pos\u00e9e sans qu'il sache d'o\u00f9 elle vient. En fait, il imagine qu'on la lui pose, \u00e0 lui, l'\u00e9crivain accompli, le c\u00e9l\u00e8bre romancier qui rencontre les grands de ce monde, habite \u00e0 Londres, Paris, Moscou, et tente tant bien que mal d'attacher son c\u0153ur \u00e0 celui d'une femme. Si jamais quelqu'un lui demande ce qu'il fait, il doit \u00eatre pr\u00e9par\u00e9 \u00e0 r\u00e9pondre, il doit surtout ne pas h\u00e9siter et s'inclure dans la grande ronde des penseurs de son temps. \u00ab Nous, travailleurs intellectuels \u00e0 facult\u00e9s cr\u00e9atrices, nous transformons les bases m\u00eames de la vie humaine 2. \u00bb Il sait bien ce que les gens s'imaginent : qu'il m\u00e8ne une vie parall\u00e8le, privil\u00e9gi\u00e9e, en dehors des contingences de la r\u00e9alit\u00e9. Ces derni\u00e8res ann\u00e9es, il a gagn\u00e9 des sommes importantes : sa famille et lui sont \u00e0 l'abri du besoin. Rien de comparable, donc, au quotidien de l'ouvrier ou du commer\u00e7ant. Ses jeunes ann\u00e9es br\u00fbl\u00e9es \u00e0 attendre le client derri\u00e8re la caisse l'ont persuad\u00e9 d'une chose : cette vie de labeur pass\u00e9e aux d\u00e9pens d'une autre \u2014 plus po\u00e9tique \u2014 n'\u00e9tait pas faite pour lui. Il admet avoir plong\u00e9 \u00ab dans l'eau 3 \u00bb, \u00eatre all\u00e9 se perdre dans les profondeurs de son imagination, et remonter de temps en temps \u00e0 la surface pour inspirer. Selon lui, les hommes de plume ont deux solutions : \u00ab Il nous faut respirer ou mourir 4. \u00bb Lui choisit de respirer. S'il continue \u00e0 vivre, malgr\u00e9 la mort de Jane, malgr\u00e9 les soucis que lui cause Odette, ce n'est pas pour le plaisir : il a une \u00ab t\u00e2che particuli\u00e8re 5 \u00bb \u00e0 accomplir.\n\nUne seule chose le fait tenir : l'\u00e9criture. _La Conspiration au grand jour_ est devenu un livre, concis, ramass\u00e9, limpide, l'un de ceux dont Wells est le plus fier et qui r\u00e9unit les points fondamentaux de son projet. Dans cet essai, il invite les hommes \u00e0 engager une r\u00e9volution mondiale, \u00e0 ouvrir la br\u00e8che d'un mouvement r\u00e9conciliateur, \u00e0 former une fraternit\u00e9 sereine. Wells n'a jamais cru en Dieu \u2014 ou si peu, et si vite \u2014, mais il a toujours cru en l'homme. C'est une foi tenace, indestructible. Il cherche des alli\u00e9s, il esp\u00e8re que cet opuscule soul\u00e8vera les foules, mais la parution du livre ne provoque que de timides vagues. Il r\u00e9cidive avec un petit texte interrogatif : _Que devons-nous faire de nos vies ?_ La question est lanc\u00e9e comme une bouteille \u00e0 la mer. Toujours rien. Wells doit bien se rendre \u00e0 l'\u00e9vidence : le proph\u00e8te est inaudible, il \u00ab conspire \u00bb tout seul.\n\nC'est \u00e0 ce moment-l\u00e0 qu'il re\u00e7oit un appel de Berlin. Wells est invit\u00e9 \u00e0 donner une conf\u00e9rence devant les d\u00e9put\u00e9s allemands et un parterre de repr\u00e9sentants \u00e9trangers en tout genre. Paul Loebe, le pr\u00e9sident du Reischtag, l'accueille en grande pompe. Et tandis qu'il marche vers la tribune, Wells entrevoit une possibilit\u00e9, un espoir \u2014 toujours ce regain d'optimisme qui le caract\u00e9rise. Depuis un an, les sociaux-d\u00e9mocrates sont au pouvoir en Allemagne, et aujourd'hui ils sont l\u00e0, face \u00e0 lui, tout comme, probablement, les futurs nazis qui l'\u00e9coutent sans rien dire. Chacune de ses paroles est traduite en allemand. Wells pr\u00eache pour une paix durable et concr\u00e8te, pour la cr\u00e9ation d'un \u00c9tat f\u00e9d\u00e9ral mondial. Il met en garde l'auditoire contre la guerre qui, selon lui, est imminente. \u00ab En d\u00e9pit de tous les pactes et de toutes les promesses, le monde marche indiscutablement vers des guerres nouvelles 6 \u00bb, dit-il gravement. Personne ne r\u00e9agit. Il continue : \u00ab Bient\u00f4t les canons partiront d'eux-m\u00eames, la psychose de la guerre nous coupera la parole et le caporal nous poussera de nouveau dans la cour d'une caserne 7. \u00bb Il ne croit pas si bien dire. Hitler, qui est entr\u00e9 depuis peu en politique, est justement un caporal ambitieux qui a v\u00e9cu les tranch\u00e9es et qui a de grands projets pour son pays. Mais \u00e0 ce moment, c'est tout juste si Wells est entendu et applaudi.\n\nEn sortant de la salle, et apr\u00e8s quelques salutations de rigueur, son regard s'arr\u00eate sur un visage. La personne s'approche de lui. \u00ab C'est toi 8 \u00bb, dit soudain la voix. En face de lui se tient Moura. Elle \u00e9tait dans la salle, elle a entendu son discours. Huit ans qu'il ne l'a pas vue, qu'il a esp\u00e9r\u00e9 la revoir et la croiser par hasard. C'est bien elle : ces cheveux noirs, ce nez un peu cass\u00e9, cette peau blanche, si fine, qu'il aper\u00e7oit dans son d\u00e9collet\u00e9. Elle a trente-six ans, et s'appelle d\u00e9sormais Moura Budberg. Ils se parlent quelques instants, bouscul\u00e9s par la foule des convives, et se faufilent rapidement vers la sortie. \u00c0 cette \u00e9poque, Moura est suspect\u00e9e d'\u00eatre un agent double (au service des Russes et des Allemands), et certains historiens ont m\u00eame \u00e9tabli l'hypoth\u00e8se qu'elle \u00e9tait all\u00e9e trouver Wells ce jour-l\u00e0 dans l'intention de l'utiliser pour un rapprochement avec les pays de l'Ouest 9. Il devait aussi y avoir un peu d'envie de revoir un homme qui avait sans aucun doute marqu\u00e9 son esprit. Attabl\u00e9s au restaurant, Moura lui raconte sa nouvelle vie en Estonie, ses enfants, son amiti\u00e9 avec Gorki qui n'a pas faibli. Wells lui demande si elle a \u00e9t\u00e9 sa ma\u00eetresse \u2014 si elle l'est, tout court \u2014, elle nie en bloc, jure qu'elle n'a eu que six amants dans sa vie, que Gorki n'en fait pas partie. La discussion se prolonge, ils boivent \u00e9norm\u00e9ment, Wells est impressionn\u00e9 par la quantit\u00e9 de vodka qu'elle peut ingurgiter. Son charme est inchang\u00e9, magnifi\u00e9 par les ann\u00e9es. Il avait bien retrouv\u00e9 une photographie d'elle quelque temps auparavant, mais le clich\u00e9 ne lui \u00e9tait pas favorable. Moura est ce genre de beaut\u00e9-l\u00e0 : celle qui ne supporte pas la fixation.\n\nMoura exerce un tel \u00ab magn\u00e9tisme 10 \u00bb sur lui qu'il pense quitter Odette d\u00e9finitivement. S'il lui avoue son amour, il la pr\u00e9vient aussi : il ne peut lui promettre de lui \u00eatre fid\u00e8le \u2014 il ne l'a jamais \u00e9t\u00e9, avec aucune femme. \u00ab _Very well my dear_ 11 \u00bb (\u00ab Tr\u00e8s bien mon ch\u00e9ri \u00bb), lui r\u00e9pond Moura. Elle ne lui demande rien, elle veut juste avoir l'occasion de le voir de temps en temps. Cela aurait d\u00fb satisfaire Wells. Il n'en est rien. Ses inqui\u00e9tudes, au contraire, augmentent. Il sait que Moura pla\u00eet aux hommes, aux femmes. Il sait qu'il risque de ne pas \u00eatre le seul amour de sa vie. Cette situation est nouvelle pour lui, \u00eatre avec une femme qui n'exige rien de lui. Leur idylle d\u00e9bute donc dans une clandestinit\u00e9 d\u00e9licieuse et se poursuit en Angleterre, \u00e0 l'abri du regard d'Odette \u00e0 qui Wells ne laisse pas le choix : il faut qu'elle parte en vacances, voici justement des billets pour l'\u00c9gypte.\n\n\u00c0 son retour, Wells lui donne rendez-vous dans un parc de Londres pour mettre un terme \u00e0 leur relation. Odette arrive, elle lui raconte qu'elle est plong\u00e9e dans l'\u00e9criture de son nouveau livre, _I Discover the English_ ( _Je d\u00e9couvre les Anglais_ ), et qu'elle est tr\u00e8s excit\u00e9e \u00e0 l'id\u00e9e d'\u00eatre prochainement publi\u00e9e en Angleterre. Gr\u00e2ce \u00e0 lui, elle a pu rencontrer des personnes importantes, \u00eatre int\u00e9gr\u00e9e dans les cercles intellectuels de la ville. Elle est persuad\u00e9e de toucher \u00e0 la gloire. Manque d'occasion ou de courage ? Wells ne lui dit rien et ils repartent par des chemins s\u00e9par\u00e9s. Sur les routes de Suisse, Odette lui \u00e9crit constamment, Wells ne r\u00e9pond pas. Irrit\u00e9e, elle d\u00e9cide d'en faire part aux plus proches amis de son amant pour les avertir de la situation et leur demander d'intervenir. Toujours rien. Ultime menace : elle annonce par lettre \u00e0 Wells qu'elle va \u00e9crire un livre et tout r\u00e9v\u00e9ler de lui \u2014 ce que les gens ne savent pas, la partie sombre de sa personnalit\u00e9, ses secrets, ses enfants ill\u00e9gitimes, ses obsessions. Elle le menace aussi de publier toutes les lettres qu'il lui a \u00e9crites, de rendre leur intimit\u00e9 publique. Wells sourit \u00e0 de telles d\u00e9marches et l'encourage \u00e0 se d\u00e9p\u00eacher car il pr\u00e9voit lui aussi d'\u00e9crire son autobiographie, comptant bien rendre sa vie priv\u00e9e _publique_ 12.\n\nUn temps, il a mis sa fatigue physique et morale sur le compte de cette histoire de c\u0153ur. S'il dormait peu, s'il avait mal \u00e0 la t\u00eate et au ventre, c'\u00e9tait \u00e0 cause d'Odette. Mais le docteur Norman lui annonce une mauvaise nouvelle. Wells est diab\u00e9tique. Il faut qu'il se m\u00e9nage, qu'il prenne soin de lui, le m\u00e9decin est formel, avec sa sant\u00e9 fragile, il devrait faire attention. \u00c0 peine quelques semaines plus tard, il apprend la mort de son vieil ami, Arnold Bennett, qui a succomb\u00e9 \u00e0 la typho\u00efde. Depuis leur premi\u00e8re lettre \u00e9chang\u00e9e en 1897, ils ne se sont jamais perdus de vue, m\u00eame si, ces derniers temps, ils s'\u00e9taient \u00e9loign\u00e9s l'un de l'autre. Tous les deux se ressemblaient tellement : curieux, francs, autodidactes. Conrad est mort aussi. De leur petit groupe de camarades \u00e9crivains, il ne reste que lui et Bernard Shaw. Rapidement, un autre d\u00e9c\u00e8s vient assombrir cette ann\u00e9e 1931 : celui de sa premi\u00e8re femme, Isabel. Malgr\u00e9 leur divorce, ils continuaient de s'\u00e9crire, il lui envoyait r\u00e9guli\u00e8rement de l'argent. Elle meurt et Wells reste hant\u00e9 par l'inach\u00e8vement. Il aura \u00e9t\u00e9 \u00e0 la fois si amoureux d'elle, et si incapable de l'aimer. L'id\u00e9e m\u00eame de ce paradoxe le plonge dans une profonde d\u00e9solation.\n\nLe spleen est enray\u00e9 gr\u00e2ce \u00e0 l'\u00e9criture de deux romans \u2014 _M. Blettsworthy sur l'\u00eele Rampole_ et _La Dictature de M. Parham_ \u2014 ainsi qu'\u00e0 la publication du troisi\u00e8me tome de son histoire universelle \u2014 _Le Travail, la Richesse et le Bonheur de l'humanit\u00e9_. Mais ses livres ne se vendent pas \u2014 plus. Depuis une dizaine d'ann\u00e9es, son nom est associ\u00e9 \u00e0 celui d'un conf\u00e9rencier, d'un \u00e9ducateur, mais pas d'un \u00e9crivain. La gloire est venue si vite \u2014 \u00e0 trente ans, il publiait cons\u00e9cutivement _La Machine \u00e0 explorer le temps, La Guerre des mondes_ et _L'\u00cele du docteur Moreau_ \u2014 qu'il a oubli\u00e9 qu'elle pouvait \u00eatre rare, et fugace. Ses romans ne plaisent plus autant d\u00e9sormais, bien qu'il ait toujours son cercle de fid\u00e8les lecteurs. Trois d'entre eux ont d'ailleurs cr\u00e9\u00e9 un groupe de r\u00e9flexion en son honneur : The H. G. Wells Society. L'\u00e9crivain est flatt\u00e9, mais demande \u00e0 changer le titre. _The Open Conspiracy_ (\u00ab La Conspiration ouverte \u00bb) lui convient mieux. Et puis la B.B.C. l'a contact\u00e9. Les dirigeants de cette radio veulent l'inviter \u00e0 exposer ses id\u00e9es \u00e0 l'antenne. Jusqu'\u00e0 pr\u00e9sent, il a toujours d\u00e9clin\u00e9, craignant de ne pouvoir \u00eatre totalement libre de dire ce qu'il veut. Mais la cha\u00eene insiste, lui assure qu'il n'y aura pas de censure, et fait transmettre le message par une s\u00e9duisante jeune femme. Wells peut difficilement refuser, et signe donc un contrat pour une s\u00e9rie d'\u00e9missions bien r\u00e9mun\u00e9r\u00e9es. L'exp\u00e9rience est int\u00e9ressante mais laborieuse, Wells n'\u00e9tant pas familier de l'exercice. Apr\u00e8s chaque \u00e9mission \u2014 diffus\u00e9e en direct et \u00e9cout\u00e9e par un tr\u00e8s grand nombre d'Anglais \u2014, Wells salue les auditeurs, \u00e9puis\u00e9, n'ayant qu'un seul but en t\u00eate : boire un peu de cette bouteille de whisky qu'on a mise \u00e0 sa disposition sur la table, au cas o\u00f9 il en aurait envie.\n\nSur le chemin du retour, il pense \u00e0 ce qu'il a et \u00e0 ce qui lui manque. Peut-\u00eatre est-il temps de vivre, enfin, l'amour apr\u00e8s lequel il court depuis si longtemps ? Au mois d'avril 1932, il se lance et demande \u00e0 Moura de devenir sa femme. Elle refuse. Pas pour des raisons sentimentales : elle est follement amoureuse de lui, elle confie m\u00eame \u00e0 son ami l'\u00e9crivain Somerset Maugham (qui ne comprend pas comment elle peut s'attacher \u00e0 cet homme plus vieux qu'elle et un peu gros) qu'elle ne peut se passer de lui car \u00ab il sent le miel 13 \u00bb. En fait, elle n'a jamais compris ce que le mariage venait faire entre deux personnes qui s'aimaient. \u00ab _Let us go on as we are_ 14 \u00bb (\u00ab Continuons ainsi \u00bb), lui r\u00e9pond-elle. Pourquoi ? Et Moura l'embrasse pour toute r\u00e9ponse. Ils partent ensemble \u00e0 Salzbourg et Vienne, puis dans les Alpes. Ce s\u00e9jour dans les montagnes est lumineux, passionn\u00e9 \u2014 parfait. Wells, qui n'en d\u00e9mord pas, lui renouvelle sa demande. Encore non. Elle lui promet de toujours \u00eatre l\u00e0 pour lui. Elle veut juste savoir qu'elle peut \u00eatre libre. Wells se rend compte avec accablement de la dure r\u00e9alit\u00e9 : la femme qu'il aime est peut-\u00eatre plus \u00ab wellsienne \u00bb qu'il ne l'a jamais \u00e9t\u00e9. Il ne comprend pas, insiste, hausse le ton et parvient \u00e0 lui soutirer des explications suppl\u00e9mentaires. Le couperet tombe. Cela fait des semaines qu'ils s'\u00e9crivent de nouveau, Gorki et elle. Il vient de perdre son fils, il est tr\u00e8s malade et la r\u00e9clame \u00e0 ses c\u00f4t\u00e9s, en Russie. Wells continue de lui d\u00e9clarer son amour, l'emm\u00e8ne partout, la pr\u00e9sente \u00e0 tous ses amis. C'est la premi\u00e8re fois qu'il a envie de montrer son amour pour une femme. Moura se laisse porter, tout en gardant ses distances.\n\nDe l'autre c\u00f4t\u00e9 du Rhin, quelque chose a insidieusement grandi. Le syst\u00e8me politique allemand a subi une impulsion glaciale. \u00c0 l'hiver 1933, Hitler est nomm\u00e9 chancelier de la R\u00e9publique de Weimar. Chef du Parti national-socialiste des travailleurs allemands depuis 1921, il a mont\u00e9 les marches du pouvoir une \u00e0 une, b\u00e2tissant autour de lui un culte m\u00e9galomane et installant progressivement une chasse \u00e0 tous les opposants. Les nazis ont pris le pouvoir, et la pers\u00e9cution des intellectuels commence. Au m\u00eame moment, Wells est d\u00e9sign\u00e9 pour remplacer John Galsworthy (d\u00e9c\u00e9d\u00e9 r\u00e9cemment) \u00e0 la direction du P.E.N. Club, une association internationale d'\u00e9crivains cr\u00e9\u00e9e afin de r\u00e9unir les plumes du monde entier et de permettre le dialogue entre elles. \u00ab PEN \u00bb comme \u00ab stylo \u00bb, bien s\u00fbr, mais aussi comme ce qui rassemble en trois lettres l'ensemble des gens de lettres : les po\u00e8tes (\u00ab P \u00bb), essayistes et \u00e9diteurs (\u00ab E \u00bb), et les romanciers (\u00ab N \u00bb comme \u00ab novelists \u00bb). Wells se retrouve propuls\u00e9 \u00e0 la t\u00eate de cette soci\u00e9t\u00e9 litt\u00e9raire, d'embl\u00e9e confront\u00e9e \u00e0 l'arriv\u00e9e du F\u00fchrer qui terrorise l'Europe. La priorit\u00e9 est de prot\u00e9ger les intellectuels allemands, les \u00e9crivains juifs et ceux qui appartiennent \u00e0 la gauche, dont certains se sont d\u00e9j\u00e0 fait arr\u00eater et emprisonner. Il tente de convaincre ses troupes, participe \u00e0 des rencontres et se rend \u00e0 Raguse et \u00c9dimbourg afin de trouver des alli\u00e9s. La r\u00e9sistance s'organise aussi \u00e0 Rome, Vienne et Berlin o\u00f9 les P.E.N. Clubs respectifs assurent une petite propagande locale qui porte haut la valeur de libert\u00e9 d'expression. Wells fait tout pour prot\u00e9ger ses coll\u00e8gues et, derri\u00e8re eux, la litt\u00e9rature. Il sent quelque chose approcher.\n\nSes inqui\u00e9tudes nourrissent un roman. Dans _The Shape of Things to Come_ (litt\u00e9ralement \u00ab La forme des choses \u00e0 venir \u00bb, encore in\u00e9dit en fran\u00e7ais), une guerre mondiale d\u00e9bute en 1940 par l'embrasement du couloir polonais... Pr\u00e8s de sept ann\u00e9es avant le d\u00e9clenchement r\u00e9el du conflit, Wells \u00e9crit approximativement la date et le lieu. Proph\u00e8te, toujours. La d\u00e9b\u00e2cle qu'il \u00e9bauche est sans commune mesure avec ce qu'il a pu imaginer auparavant. Le combat qui oppose principalement l'Allemagne et l'Italie \u00e0 la Pologne et \u00e0 la France est d'une violence inou\u00efe. Le romancier a renonc\u00e9 \u00e0 l'arme atomique (il a toujours en t\u00eate le souvenir de la parution de _La Destruction lib\u00e9ratrice_ , qui lui a caus\u00e9 beaucoup de tort), mais a dot\u00e9 ses personnages d'une arme tout aussi dangereuse \u2014 le gaz \u2014 qui d\u00e9cime des milliers d'hommes sur son passage. Tout s'\u00e9croule : les syst\u00e8mes comme les b\u00e2timents, les individus et les r\u00eaves. Au c\u0153ur de cette nouvelle fin du monde, une conspiration \u00e9merge du brouillard et des hommes \u00e9clair\u00e9s reprennent en main l'humanit\u00e9. Qu'il ait raison ou tort, il poursuit sa route r\u00eav\u00e9e : celle d'un \u00ab _happy ending_ \u00bb, d'un sursaut ultime qui pr\u00e9parerait le monde de demain.\n\nLe cin\u00e9ma le convoite de plus en plus. Paramount ach\u00e8te les droits de _L'\u00cele du docteur Moreau_ et en fait un film d'horreur ( _Island of Lost Souls_ ) en 1933, avec Charles Laughton et Bela Lugosi (premier interpr\u00e8te de Dracula sur grand \u00e9cran) dans les r\u00f4les principaux. Le septi\u00e8me art lui pla\u00eet, il r\u00e9fl\u00e9chit d'ailleurs \u00e0 y mettre un pied en tant que sc\u00e9nariste, mais attend l'opportunit\u00e9. \u00c9crire des fantaisies scientifiques ne l'int\u00e9resse plus. Tout ce qui pouvait l'amuser il y a trente ans \u2014 ces \u00ab exercices de l'imagination 15 \u00bb qui ne visaient qu'\u00e0 divertir le lecteur, qu'\u00e0 le mener, par l'illusion de l'\u00e9criture, \u00e0 l'id\u00e9e de l'impossible \u2014 lui semble aujourd'hui obsol\u00e8te. \u00ab Je crois que je m'emploie mieux maintenant en \u00e9tant plus pr\u00e8s de la r\u00e9alit\u00e9, \u00e0 essayer de faire une analyse pratique des perplexit\u00e9s sociales croissantes 16 \u00bb, \u00e9crit-il dans une pr\u00e9face \u00e0 l'occasion de la r\u00e9\u00e9dition en un seul volume de ses quatre plus grands romans scientifiques ( _La Machine \u00e0 explorer le temps, La Guerre des mondes, L'Homme invisible_ et _L'\u00cele du docteur Moreau_ ). Il ajoute :\n\nFace aux cataclysmes r\u00e9els, le monde n'a pas besoin de nouveaux cataclysmes fantastiques. Ce jeu est termin\u00e9. Qui a besoin des humeurs invent\u00e9es de M. Parham \u00e0 Whitehall, quand nous pouvons observer jour apr\u00e8s jour M. Hitler en Allemagne ? Quelle invention humaine peut se camper face aux farces fantastiques du destin ? J'ai tort d'en vouloir aux critiques. La r\u00e9alit\u00e9 a pris une feuille \u00e0 mes livres et s'est mise en passe de me remplacer 17.\n\nCe jour-l\u00e0, \u00e0 la terrasse du restaurant The Queen's, dans le quartier de Sloane Square, Wells prend le soleil, les yeux pliss\u00e9s \u00e0 cause de la lumi\u00e8re. Il est l'heure de d\u00e9jeuner, les Londoniens s'affairent dans tous les sens mais son corps \u00e0 lui est comme fix\u00e9 dans le temps et l'espace. Immobile. Il a l'impression d'\u00eatre dans un univers sans limites, que cette ville qu'il conna\u00eet si bien \u00e9gale l'infini. Il est assis derri\u00e8re sa table, en terrasse, mais c'est comme s'il \u00e9prouvait l'apesanteur, ou voguait sur l'eau. Londres, apr\u00e8s tout, est \u00ab une mer o\u00f9 il faut nager 18 \u00bb. Depuis peu, il a recouvr\u00e9 la vue, les choses lui apparaissent clairement. Il ne peut pas pr\u00e9tendre vouloir un monde meilleur si lui-m\u00eame ne tente pas, aussi, d'am\u00e9liorer son existence. Jane est morte, Isabel aussi. Amber est heureuse avec leur fille, Anna Jane. La petite a grandi, elle \u00e9tudie \u00e0 la prestigieuse London School of Economics, et est tomb\u00e9e amoureuse d'un certain Eric Davis. Wells a pris l'habitude de la voir souvent, jouant le r\u00f4le de l'ami, du confident. Un jour, tandis qu'ils d\u00e9jeunent ensemble, elle lui annonce qu'elle pr\u00e9voit de partir s'installer en Afrique du Sud avec Eric, l'universit\u00e9 du Cap lui offre un poste de conf\u00e9renci\u00e8re dans sa discipline. Wells, follement jaloux que sa fille unique lui \u00e9chappe pour ce jeune homme qu'il n'appr\u00e9cie gu\u00e8re, veut l'en emp\u00eacher. Il lui \u00e9crit une lettre, lui avoue ce qu'elle a s\u00fbrement d\u00e9j\u00e0 devin\u00e9 : qu'il est son p\u00e8re, qu'\u00e0 ce titre il s'oppose non seulement \u00e0 son d\u00e9part mais \u00e0 son union avec Eric. Une semaine passe sans qu'il re\u00e7oive de nouvelles d'Anna Jane. Il lui \u00e9crit de nouveau. La jeune fille lui r\u00e9pond qu'elle vivra comme elle l'entend, que cela lui plaise ou non.\n\nRebecca, elle, va bient\u00f4t se marier. Elle lui a annonc\u00e9 la nouvelle l'autre jour, entre un th\u00e9 et des g\u00e2teaux secs. Elle a beaucoup chang\u00e9. Sa fougue d'antan a laiss\u00e9 place \u00e0 un certain conservatisme, mais elle a l'air heureux \u2014 m\u00eame si leur fils Anthony lui cause du souci. Wells aussi aimerait \u00eatre heureux. Cela ne tient pas \u00e0 grand-chose : juste \u00e0 un poids dont il doit se d\u00e9lester.\n\nUne ombre vient le priver du soleil qui chauffait sa peau. Quelques secondes s'\u00e9coulent avant qu'il ne reconnaisse Odette. Elle est en retard, mais elle est l\u00e0, c'est tout ce qui compte. Ils commandent leur repas, Wells mange tandis qu'elle d\u00e9blat\u00e8re sur son incons\u00e9quence, sa m\u00e9chancet\u00e9, et qu'elle renouvelle ses menaces \u00e0 son encontre. \u00ab _Go ahead with it_ 19. \u00bb Vas-y, lui dit-il, publie ce que tu veux sur mon compte, cela n'a pas d'importance. Puis il lui annonce qu'il la quitte, qu'\u00e0 bient\u00f4t soixante-sept ans, il veut continuer \u00e0 vivre sans elle, qu'il a rencontr\u00e9 une femme formidable (Moura) et qu'il est \u00e9perdument amoureux d'elle. Il la rassure aussi, elle peut rester \u00e0 Lou Pidou, il continuera \u00e0 s'occuper de sa s\u0153ur Maggie (qui vit seule avec trois enfants, sans argent, et que Wells entretient depuis des ann\u00e9es). Odette se l\u00e8ve brusquement et se dirige vers les toilettes du restaurant. Quand elle revient, Wells a pay\u00e9 la note et l'attend debout, \u00e0 l'entr\u00e9e. Il h\u00e8le un taxi, ouvre la porti\u00e8re, elle s'assoit, le supplie encore un peu et il claque la porte. Enfin libre. Toute cette histoire fera, il en est s\u00fbr, un tr\u00e8s bon livre *1.\n\n*1. _Dolor\u00e8s_ (1938).\nUn cerveau pas si ordinaire\n\nNeuf heures du matin, face au miroir. Impossible d'\u00e9viter son reflet. \u00c0 vingt ans d\u00e9j\u00e0, il n'\u00e9prouvait que du d\u00e9go\u00fbt en se regardant, gamin ch\u00e9tif qu'on bousculait sur les terrains de football et qui n'avait pas de quoi se nourrir. Aujourd'hui, il s'est rattrap\u00e9, il est riche, aime la bonne ch\u00e8re et a de l'embonpoint. Mais il ne ressemble toujours \u00e0 rien. Il aurait aim\u00e9 \u00eatre bel homme, avoir \u00ab un beau corps 1 \u00bb et s'en servir pour l'amour. Au lieu de cela, il est laid et les yeux encore endormis de la veille ne l'emp\u00eachent pas de voir avec acuit\u00e9 sa silhouette dans la glace. Ce n'est plus du d\u00e9go\u00fbt, c'est de l'aversion. Des cheveux en moins et des rides, \u00e7\u00e0 et l\u00e0, par centaines. Cela pla\u00eet aux femmes apr\u00e8s tout, c'est l\u00e0 sa chance. En revanche ce grand front, trop grand pour une si \u00ab petite 2 \u00bb t\u00eate... L'inverse serait plus gracieux \u2014 quoique. Si seulement il pouvait savoir ce qu'il y a dans ce caillou. S'il pouvait assister, par miracle, \u00e0 sa propre autopsie. Contempler son enc\u00e9phale perc\u00e9, enfin accessible... Oui, imaginer la taille de cette chose, ses circonvolutions, ses beaut\u00e9s, ses d\u00e9fauts : le spectacle lui plairait bien. Il sait pr\u00e9cis\u00e9ment ce qu'il dirait au moment o\u00f9 les m\u00e9decins d\u00e9couvriraient l'int\u00e9rieur de son cerveau : \u00ab Je vous l'avais dit. Bande de cons 3. \u00bb Cette phrase est parfaite. Il faudra l'inscrire sur sa tombe \u2014 ne pas oublier d'en parler aux enfants.\n\nUn cerveau correct, donc. Pis : ordinaire. Pas celui d'Albert Einstein ou de David Lloyd George. Non, son cerveau malheureusement n'a l'envergure ni de celui d'un g\u00e9nie ni de celui d'un Premier ministre. Il ne lui a jamais permis d'exceller en calcul mental, lui a souvent fait d\u00e9faut lorsqu'il s'agissait de se rappeler le nom d'un ami, et le ralentit parfois au bridge 4. Le probl\u00e8me vient peut-\u00eatre des vaisseaux sanguins : \u00ab Je soup\u00e7onne mon art\u00e8re carotide de ne pas irriguer ma mati\u00e8re grise aussi librement et g\u00e9n\u00e9reusement qu'elle devrait le faire 5. \u00bb De l\u00e0, sans doute, son cr\u00e2ne \u00e9troit, inhabituellement \u00e9triqu\u00e9 par rapport au reste de son corps, et qui amuse les foules. C'est toujours la m\u00eame chose, d\u00e8s qu'il \u00e9change son chapeau contre celui d'un autre, il se retrouve syst\u00e9matiquement dans le noir complet. Alors il ravale sa col\u00e8re, rit \u00e0 grands \u00e9clats \u2014 trop pr\u00e9occup\u00e9 par \u00ab cette terreur nationale qu'a tout bon Anglais d'\u00eatre surpris dans l'embarras 6 \u00bb \u2014 et s'en sort par une pirouette. Rien de grave, sinon que sa t\u00eate n'est plus seulement _petite_. Elle l'est publiquement.\n\nDans ses mains pourtant, elle lui para\u00eet lourde, bien remplie. De cette caboche sont sorties les histoires les plus fantasques (un homme qui \u00e9chappe aux cr\u00e9anciers en devenant invisible 7 ; un autre qui sacrifie sa jeunesse pour un h\u00e9ritage 8, un autre encore, en vacances, qui rencontre une sir\u00e8ne 9). Petite, malgr\u00e9 tout. Il faut se rendre \u00e0 l'\u00e9vidence, ne pas embellir le tableau, d\u00e9crire l'exacte image que ce miroir lui renvoie en cette matin\u00e9e ensoleill\u00e9e : un homme de taille moyenne, que la jeunesse a quitt\u00e9, \u00e9puis\u00e9 par des nuits trop courtes, accabl\u00e9 par des tourments consid\u00e9rables (l'\u00e9criture, les femmes, sa r\u00e9flexion sur l'avenir du monde), inquiet de ne peut-\u00eatre jamais atteindre une pleine satisfaction existentielle ; persuad\u00e9, aussi, qu'une catastrophe le guette. Cela n'a jamais \u00e9t\u00e9 autrement. Il a d\u00e9j\u00e0 fr\u00f4l\u00e9 la mort deux fois, son rein gauche le fait atrocement souffrir, des difficult\u00e9s respiratoires l'emp\u00eachent de vivre normalement depuis des dizaines d'ann\u00e9es, et comme si cela ne suffisait pas, il est diab\u00e9tique. \u00c0 vrai dire, il n'a pas peur du noir. \u00ab La mort ne me concerne pas 10. \u00bb Il aimerait juste que tout cela continue encore un peu : l'\u00e9criture, les femmes, et sa r\u00e9flexion sur l'avenir du monde. Il a soixante-huit ans, et n'a pas termin\u00e9 sa t\u00e2che. Il pourrait encore d\u00e9montrer tant de choses, publier tant de livres, et aimer tant de fois.\n\nIl est temps de s'y mettre. Assez du miroir qui brille et du temps qui passe. Il se met \u00e0 sa table de travail et \u00e9crit les premi\u00e8res lignes de son autobiographie. Il vise la transparence totale, mais la prudence lui souffle un mot qui deviendra son titre et le d\u00e9douanera de toute erreur potentielle : \u00ab _experiment_ \u00bb (essai, tentative). Il n'a jamais eu de difficult\u00e9 particuli\u00e8re \u00e0 trouver l'inspiration. Ce jour-l\u00e0, sa main est fi\u00e9vreuse et in\u00e9puisable. Il \u00e9crit vite, comme \u00e7a lui vient, des souvenirs, des images, des phrases entendues ou prononc\u00e9es. Il a un si grand besoin de \u00ab soulager s]on esprit[ 11 \u00bb et de \u00ab bavarder][ 12 \u00bb avec son lecteur, d'appara\u00eetre tel qu'il est : un \u00ab vagabond fatigu\u00e9 13 \u00bb qui tente de s'extraire de la m\u00e9lancolie. Il n'est pas mieux qu'un autre, ni plus fort ni plus intelligent. Il conna\u00eet ses failles : le manque de s\u00e9r\u00e9nit\u00e9, sa facult\u00e9 \u00e0 se laisser envahir par le doute, son volontarisme aussi et son ambition, qui l'ont amen\u00e9 \u00e0 convoiter la gloire, et \u00e0 ne plus pouvoir s'en passer. Mais la vie, en fin de compte, l'a g\u00e2t\u00e9, m\u00eame si aujourd'hui ses convictions demeurent inaudibles :\n\nC'est le destin de tous les cr\u00e9ateurs de n'aboutir qu'\u00e0 des choses imparfaites et incompl\u00e8tes. Nous transigeons tous. Nous subissons tous des \u00e9checs. L'histoire de la vie de n'importe quel travailleur aux facult\u00e9s cr\u00e9atrices \u2014 de sa nature m\u00eame, par la divergence de ses objets, par ses succ\u00e8s, jamais complets, par ce continuel et grotesque passage du sublime aux basses n\u00e9cessit\u00e9s de la vie, avec sa tendance \u00e0 l'\u00e9vasion, \u2014 est une pure com\u00e9die 14 _._\n\nUne com\u00e9die, une farce. Mais il ne d\u00e9sesp\u00e8re pas de changer le monde, de le rendre meilleur. C'est peut-\u00eatre son tourment fondamental, ce qui le fait avancer, ce qui l'use en m\u00eame temps : \u00ab le d\u00e9sir d'accomplir quelque chose qui d\u00e9passerait s]a capacit\u00e9 ant\u00e9rieure[ 15 __ \u00bb. Tous les efforts qu'il d\u00e9ploie depuis tant d'ann\u00e9es sont dirig\u00e9s vers ce but impossible, vers ce futur imperceptible et fuyant. Wells aime la vitesse, il n'est pas sportif mais a toujours aim\u00e9 courir, aller voir ce qui se passe plus loin, avant les autres. Il y est parvenu, parfois, mais il sait aussi qu'il se trompe, souvent. Il y a une chose, n\u00e9anmoins, dont il est persuad\u00e9 : \u00ab Nous ne r\u00e9alisons jamais les splendeurs secr\u00e8tes de nos intuitions ; cependant quelques-uns parmi nous arrivent \u00e0 faire des choses qui semblent valoir la peine qu'ils se sont donn\u00e9e 16. \u00bb Wells esp\u00e8re faire partie de ceux-l\u00e0.\n\nIl doit bient\u00f4t partir aux \u00c9tats-Unis (pour rencontrer le pr\u00e9sident Roosevelt) et en Russie (pour interviewer Staline, officiellement au pouvoir depuis 1929). Il demande \u00e0 Moura de l'accompagner \u2014 surtout en URSS, il a besoin de son regard sur le pays, de son aide aussi pour communiquer. Partir sans elle serait sans int\u00e9r\u00eat. C'est non. Elle a des choses \u00e0 faire, des gens \u00e0 voir en Estonie. Ils se retrouveront \u00e0 son retour.\n\nWells est sur le pont du _Washington_ , le bateau qui l'emm\u00e8ne de l'autre c\u00f4t\u00e9 de l'Atlantique. Rien devant lui, ni derri\u00e8re. Juste l'Oc\u00e9an. Il r\u00e9fl\u00e9chit \u00e0 ce qu'il va dire au Pr\u00e9sident, il pense surtout lui parler d'\u00e9ducation \u2014 la colonne vert\u00e9brale de son propos. Il se rend compte d'une chose : lui et sa petite t\u00eate quittent un pays o\u00f9 l'instruction se meurt pour un autre, en face, qui subit la m\u00eame situation. Comment deux nations aussi \u00e9volu\u00e9es que la Grande-Bretagne et les \u00c9tats-Unis peuvent-elles \u00eatre en retard sur ces questions ? Son r\u00eave \u00e0 lui n'a pas chang\u00e9, il n'a pas oubli\u00e9 ses lectures de jeunesse. Comme Shelley, il attend un nouveau printemps :\n\nLe grand \u00e2ge du monde rena\u00eet ; les ann\u00e9es d'or reviennent ; la terre, comme un serpent, renouvelle ses v\u00eatements us\u00e9s de l'hiver ! Le Ciel sourit ; fois et empires ne jettent plus qu'une faible lueur, semblables aux d\u00e9bris d'un songe \u00e9vanoui 17...\n\nLe bateau accoste \u00e0 New York dans la brume du matin. L'arriv\u00e9e au port s'est faite au milieu des sir\u00e8nes \u2014 le _Washington_ a manqu\u00e9 de toucher un navire allemand qui passait juste \u00e0 c\u00f4t\u00e9. Wells profite de quelques jours dans la ville avant de se rendre \u00e0 la Maison-Blanche. Il aime New York, sa foule vibrante, son intranquillit\u00e9. Assis au restaurant ou d\u00e9ambulant dans les salles des mus\u00e9es, il laisse tra\u00eener ses oreilles, rencontre des Am\u00e9ricains, s'impr\u00e8gne de l'atmosph\u00e8re et pr\u00e9pare son entrevue. Washington est une ville qu'il conna\u00eet bien, cela fait plusieurs fois qu'il s'y rend (le premier Roosevelt, Wilson et Harding l'avaient d\u00e9j\u00e0 re\u00e7u). Il arrive dans l'apr\u00e8s-midi, accueilli par le brain-trust du Pr\u00e9sident \u2014 un groupe d'\u00e9conomistes et d'universitaires, experts r\u00e9put\u00e9s, qui conseillent le chef de l'\u00c9tat. Wells serre toutes les mains. Felix Frankfurter, Rexford Tugwell, Hugh S. Johnson : les \u00ab cerveaux \u00bb de Roosevelt lui posent quelques questions et Wells ne sait que penser de cet amas d'intellectualit\u00e9s disparates qui l'entourent comme des abeilles autour du miel. Ce sont eux qui ont mis en place la politique du New Deal pour faire face \u00e0 la Grande D\u00e9pression. Wells est sceptique jusqu'\u00e0 ce qu'il soit introduit dans un grand bureau. Il d\u00e9couvre alors un pr\u00e9sident infirme, le torse attach\u00e9 au dos d'une chaise roulante, les jambes dissimul\u00e9es sous une couverture. Mais d\u00e8s que Roosevelt lui adresse la parole, son handicap dispara\u00eet, il est d'une \u00e9loquence et d'un humour rares. Son \u00e9pouse est \u00e0 ses c\u00f4t\u00e9s. Ce couple est beau \u00e0 voir, ils ne cherchent pas \u00e0 plaire, ils _se_ plaisent. Wells leur parle librement de son r\u00eave d'un monde unifi\u00e9, de l'Est et de l'Ouest qu'il imagine un jour avancer main dans la main. Roosevelt acquiesce, il voudrait surtout que les deux grandes communaut\u00e9s anglophones (les \u00c9tats-Unis et l'Angleterre) se rapprochent davantage. Wells repart sans certitudes r\u00e9elles, mais avec le sentiment d'avoir un alli\u00e9. En Russie, Staline l'attend. Il veut voir aussi Gorki. Lui seul peut l'aider \u00e0 convaincre le gouvernement russe de s'allier aux Am\u00e9ricains.\n\nJuillet 1934. Wells, accompagn\u00e9 de son fils Gip (d\u00e9sormais biologiste et parlant bien le russe), prend l'avion \u00e0 Croydon, au sud de Londres, direction Moscou. L'orage gronde dans le ciel, ils sont oblig\u00e9s d'atterrir \u00e0 Berlin, et de passer la nuit dans cette ville d\u00e9sormais nazie. Ils repartent t\u00f4t le lendemain, tandis que le soleil se l\u00e8ve. Wells, qui survole l'Europe dans ce ciel dor\u00e9, regarde les nuages et n'en revient pas des avanc\u00e9es techniques de ces derni\u00e8res ann\u00e9es. Lorsqu'il d\u00e9crivait, il y a plus de trente ans, les trajets a\u00e9riens dans ses romans, il ne s'imaginait pas qu'ils deviendraient si courants. Tout le monde, il en est s\u00fbr, pourrait bient\u00f4t voyager dans les airs, se rendre d'un point \u00e0 un autre du globe en quelques heures, plus rapidement que par aucun autre moyen de transport. Bient\u00f4t, la distance n'aurait plus d'importance.\n\nVoil\u00e0 enfin Moscou. Wells est frapp\u00e9 par la vue qu'il a du pays depuis le ciel : des chantiers partout, des usines, qui montrent bien que l'Union des r\u00e9publiques socialistes sovi\u00e9tiques a pris le virage de l'industrialisation. La premi\u00e8re chose qu'il fait est de se rendre sur le mausol\u00e9e de L\u00e9nine. Le \u00ab petit homme \u00bb est mort en janvier 1924 et repose sur la place Rouge, tout pr\u00e8s du Kremlin. \u00ab Son visage \u00e9tait cireux et p\u00e2le, et ses mains inqui\u00e8tes \u00e9taient au repos. Sa barbe me semblait plus rouge que je ne l'avais dans mon souvenir. Son expression est tr\u00e8s digne et simple, un peu path\u00e9tique. Il a du courage et quelque chose d'enfantin : les supr\u00eames qualit\u00e9s humaines. Et il dort, trop t\u00f4t pour la Russie 18. \u00bb Autour de lui, des femmes prient. Dehors est inscrite cette formule en gros caract\u00e8res : \u00ab La religion est l'opium du peuple. \u00bb Tout cela appartient d\u00e9sormais au pass\u00e9.\n\nLe nouveau p\u00e8re des peuples s'appelle Joseph Staline. Apr\u00e8s avoir dirig\u00e9 le comit\u00e9 central (plus haute instance du Parti communiste de l'Union sovi\u00e9tique) et \u00e9loign\u00e9 Trotski du pouvoir, il est devenu l'un des dirigeants les plus redout\u00e9s d'Europe, mettant en place la collectivisation des terres et faisant r\u00e9gner autour de lui une forte terreur. Wells s'attend \u00e0 rencontrer un individu rustre, ferm\u00e9 au dialogue. Mais au lieu du \u00ab montagnard g\u00e9orgien 19 \u00bb il trouve un homme souriant, presque doux. Staline le re\u00e7oit v\u00eatu d'une vulgaire chemise blanche et d'un pantalon noir, dans une pi\u00e8ce quasiment vide de meubles. Wells a quarante minutes \u2014 pas plus \u2014 pour le convaincre de la n\u00e9cessit\u00e9 d'un \u00c9tat mondial organis\u00e9 et unifi\u00e9. Il veut combattre avec lui la b\u00eatise du Bureau anglais des affaires \u00e9trang\u00e8res et du Commonwealth, l'\u00e9go\u00efsme de Mussolini et le d\u00e9lire allemand qui s'\u00e9tale en la personne de Hitler. Il lui dit qu'il revient d'Am\u00e9rique, qu'il aimerait que Roosevelt et lui puissent se parler. Staline l'\u00e9coute et se dit pr\u00eat \u00e0 travailler \u00e0 ses c\u00f4t\u00e9s m\u00eame s'il lui semble que leur conviction profonde diff\u00e8re. Pour Wells, les hommes sont tous bons et perfectibles, tandis que Staline est un homme de doutes, qui n'accorde pas sa confiance facilement. Wells veut lui dire \u00e0 quel point l'\u00e9ducation est importante pour b\u00e2tir une soci\u00e9t\u00e9 solide, libre en actes et en paroles. Il lui demande s'il se sent pr\u00eat pour la libert\u00e9. Il n'aura pas de r\u00e9ponse claire. Selon le dirigeant russe, la critique est possible, mais elle doit se faire \u00e0 l'int\u00e9rieur du Parti. Wells regrette de ne pas parler le russe, de ne pas pouvoir communiquer plus directement avec son interlocuteur. La discussion dure finalement trois heures. Wells repart satisfait et avec la conviction \u00e9trange de n'avoir \u00ab jamais rencontr\u00e9 un homme plus sinc\u00e8re, plus loyal et plus honn\u00eate 20 \u00bb. Deux mois plus tard, leur conversation est officiellement publi\u00e9e dans le journal _The New Statesman_.\n\nApr\u00e8s cette rencontre, il a pr\u00e9vu de voir quelques amis : les \u00e9crivains Alekse\u00ef Tolsto\u00ef et Alexandre Belia\u00efev, auteurs de romans de science-fiction, mais aussi le physiologiste Ivan Pavlov, et bien s\u00fbr Gorki. Il le retrouve tr\u00e8s chang\u00e9, habitant une demeure luxueuse, son sourire habituel masqu\u00e9 par une m\u00e9fiance de rigueur. Son engouement pour l'\u0153uvre de l'\u00e9crivain britannique est retomb\u00e9. Wells l'ennuie, tandis que Knut Hamsun \u2014 l'auteur de _La Faim_ \u2014 le passionne. D'autre part, l'ami russe devenu stalinien n'appr\u00e9cie gu\u00e8re les all\u00e9es et venues de son camarade britannique entre les diff\u00e9rents bureaux politiques du monde. \u00c0 ses yeux, l'Am\u00e9rique est l'ennemi, elle ne sera jamais un alli\u00e9 du syst\u00e8me sovi\u00e9tique. Wells lui explique qu'il vient pour d\u00e9fendre l'int\u00e9r\u00eat des individus libres et surtout celui des \u00e9crivains, menac\u00e9s de toutes parts. En tant que pr\u00e9sident du P.E.N. Club, et face \u00e0 la mont\u00e9e du nazisme, il veut promouvoir la libert\u00e9 d'expression. Gorki, lui, veut le contraire : d\u00e9sormais prot\u00e9g\u00e9 par le r\u00e9gime stalinien, il consid\u00e8re que tout le monde ne peut pas exprimer ce qu'il veut. Il est impossible pour Wells d'entendre ce genre de d\u00e9claration, impossible de \u00ab s\u00e9questrer l'imagination \u00bb, de dire : \u00ab Vous pouvez imaginer jusque-l\u00e0, mais pas plus loin 21. \u00bb Son id\u00e9e de cr\u00e9er un P.E.N. Club russe tombe \u00e0 l'eau. Pour d\u00e9tendre l'atmosph\u00e8re, Gorki lui montre ce sur quoi il travaille, pensant que cela le mettra en joie : la construction d'un gigantesque et luxueux palais d\u00e9di\u00e9 \u00e0 la biologie. Pas de quoi lui remonter le moral. En jetant un \u0153il \u00e0 ces plans d'architecte tapageurs, Wells se demande \u00e0 quoi pense son ami, lui qui vit dans un pays o\u00f9 le papier est encore une denr\u00e9e rare, o\u00f9 les rues sont un chaos de voitures et de charrettes, et les ventres toujours vides. La magie a op\u00e9r\u00e9 un temps, mais il doit bien se rendre \u00e0 l'\u00e9vidence : les r\u00eaves qu'il projetait pour la belle contr\u00e9e blanche ne sont pas r\u00e9alisables. Il constate son impuissance \u00e0 toucher sinon le peuple, du moins son camarade : \u00ab Je peux leur parler, les \u00e9branler m\u00eame, mais je ne peux pas les obliger \u00e0 voir 22. \u00bb\n\nGorki et lui ont toujours besoin d'un interpr\u00e8te, m\u00eame si le visiteur commence \u00e0 \u00eatre familier de la langue russe. Andreychin \u2014 c'est le nom du traducteur \u2014 demande \u00e0 Wells de mani\u00e8re tr\u00e8s informelle quels sont ses projets apr\u00e8s la Russie. Il lui dit qu'il doit rejoindre une amie en Estonie, Moura Budberg. Andreychin l\u00e8ve soudain les yeux et s'exclame qu'il la conna\u00eet bien. Elle \u00e9tait justement l\u00e0, la semaine pr\u00e9c\u00e9dente. Wells ne comprend pas et lui assure que non, la semaine pass\u00e9e, Moura \u00e9tait en Estonie et lui aux \u00c9tats-Unis, c'\u00e9tait convenu comme cela, il ne pouvait en \u00eatre autrement. Et de rench\u00e9rir en disant \u00e0 l'interpr\u00e8te qu'il avait re\u00e7u il y avait trois jours une lettre d'elle post\u00e9e de Tallinn \u2014 impossible, donc. G\u00ean\u00e9, Andreychin acquiesce et avoue s'\u00eatre peut-\u00eatre tromp\u00e9. Wells commence \u00e0 r\u00e9fl\u00e9chir, \u00e0 compter, \u00e0 se rem\u00e9morer les dates. Le calendrier d\u00e9file dans sa t\u00eate, et il se souvient surtout des r\u00e9ticences de Moura qui ne voulait pas l'accompagner. Deux fois, il le lui a demand\u00e9. Deux fois, elle a refus\u00e9. Pour en avoir le c\u0153ur net, il d\u00e9cide d'interroger Gorki et d'amener subtilement le sujet dans la conversation \u2014 il ne sait rien de leur liaison, Moura ne voulait pas qu'il sache. Il va donc vers lui et, l'air de rien, lui dit qu'il avait beaucoup appr\u00e9ci\u00e9, lors de son dernier s\u00e9jour, la pr\u00e9sence de son ancienne secr\u00e9taire, Moura, et qu'il regrette de ne pas la voir ce jour-l\u00e0. Andreychin, le visage bl\u00eame, traduit. Wells les regarde \u00e9changer longuement, sans rien comprendre. Puis l'interpr\u00e8te se tourne vers lui, lui transmet la r\u00e9ponse de Gorki : Moura est effectivement une femme formidable, elle est venue lui rendre visite trois fois cette ann\u00e9e-l\u00e0, et elle \u00e9tait encore l\u00e0 une semaine auparavant. Wells cache p\u00e9niblement sa surprise, salue son ami et s'en va. Avant de reprendre l'avion, il envoie une carte postale \u00e0 Moura lui demandant des explications.\n\nIl quitte l'URSS, cette terre de \u00ab litt\u00e9rature surveill\u00e9e 23 \u00bb et de chagrin de l'\u00e2me, ne sachant comment faire face \u00e0 la femme qu'il aime. Elle l'attend \u00e0 l'a\u00e9roport de Tallinn, lui demande si tout va bien, et ne fait aucune mention de la carte. Wells ne dit rien. Ils se dirigent vers un restaurant du centre-ville puis, apr\u00e8s un d\u00e9jeuner silencieux, prennent le train pour une petite ville de province o\u00f9 elle s\u00e9journe. L\u00e0, Wells peut l'interroger. Elle se justifie comme elle peut, lui assure que son dernier s\u00e9jour russe n'\u00e9tait pas pr\u00e9vu, qu'elle voulait revoir Gorki, que Moscou lui manquait. Wells lui r\u00e9torque que, d'apr\u00e8s Andreychin, elle y est all\u00e9e trois fois en un an, et qu'elle ne lui a rien dit. Elle nie tout, arguant que Gorki a d\u00fb commettre une erreur, ou qu'Andreychin a mal traduit ses propos. Wells ne la croit pas, se convainc qu'elle l'a certainement tromp\u00e9 avec leur ami commun, qu'il ne pourra pas lui pardonner. Le soir m\u00eame, il fait un cauchemar. Il marche, en pleine nuit, dans une sorte de taudis. Il ne voit rien puis aper\u00e7oit soudain Moura qui tient dans ses bras un grand sac. Il lui demande ce que ce sac contient et celui-ci explose en une multitude de vieux journaux. Il lui demande avec qui elle se trouvait en Russie, et Moura tombe litt\u00e9ralement en pi\u00e8ces, comme un pantin de bois qui se briserait sous l'impulsion d'un mouvement. Sa t\u00eate roule \u00e0 ses pieds, il la soul\u00e8ve et d\u00e9couvre qu'elle est vide 24. Dans la r\u00e9alit\u00e9, cette vacuit\u00e9 s'\u00e9tend \u00e0 leur couple, Wells \u00e9tant incapable de lui pardonner. L'amour n'a plus lieu d'\u00eatre \u00ab libre \u00bb pour lui \u00e0 cet instant. L'amour est plut\u00f4t une ombre \u00e9paisse, et persistante.\n\nAvidit\u00e9, sottise et grandes esp\u00e9rances : voil\u00e0 \u00e0 quoi se r\u00e9sume son histoire avec les femmes 25. Avidit\u00e9 d'aimer, sottise de conqu\u00e9rir, espoir que cela sera davantage qu'une illusion. Wells n'a jamais renonc\u00e9 \u00e0 ce r\u00eave d'une entente parfaite, d'une femme avec laquelle il pourrait partager, selon ses dires, une maison et un jardin, et qui serait \u00e0 la fois compr\u00e9hensive, passionn\u00e9e et travailleuse. La liste est longue, il l'\u00e9num\u00e8re en d\u00e9tail dans la suite de son autobiographie r\u00e9dig\u00e9e en cette fin d'ann\u00e9e 1934 \u2014 _H. G. Wells in Love_ (\u00ab La Vie amoureuse de H. G. Wells \u00bb) \u2014 dont le manuscrit fut tenu confidentiel jusqu'\u00e0 sa mort, et finalement publi\u00e9 en 1984 par son fils Anthony.\nEntre-temps\n\nUne histoire de plus qu'on aura du mal \u00e0 croire. C'est Amber qui la lui raconte au t\u00e9l\u00e9phone, encore sous le choc de ce \u00e0 quoi elle vient d'assister. Odette est venue frapper \u00e0 sa porte, furieuse, et lui a propos\u00e9 d'aller saccager la nouvelle maison de Wells, Hanover Terrace, dans le quartier de Regent's Park. Comme pour l'assurer de sa d\u00e9termination, Odette lui a montr\u00e9 un revolver qu'elle cachait sous son manteau. S'il ne voulait pas d'elle et de son amour, il allait le regretter, et puisque Amber aussi avait souffert \u00e0 cause de cet homme, il fallait qu'elles y aillent toutes les deux. D'apr\u00e8s le r\u00e9cit d'Anthony West (rapport\u00e9 par l'un des biographes d'H. G. Wells, Michael Sherborne), Amber aurait r\u00e9ussi \u00e0 s'emparer de l'arme, \u00e0 la restituer \u00e0 la police de son quartier, et \u00e0 convaincre Odette de rentrer chez elle se reposer.\n\n_The world is a world of disorder and menace_ 1. \u00bb (\u00ab Le monde est un monde de d\u00e9sordre et de menace. \u00bb) Les m\u00eames qualificatifs (d\u00e9sordre, menace) pourraient \u00eatre employ\u00e9s, \u00e0 ce moment pr\u00e9cis, pour la vie de Wells. Mais heureusement, de m\u00eame qu'il a un plan de sauvetage pour le monde, il en a aussi un pour lui-m\u00eame : d'abord, ne plus donner de nouvelles \u00e0 Odette (cela lui demande peu d'efforts), ensuite, ne pas quitter Moura (il en est bien incapable) et, enfin, c\u00f4toyer d'autres femmes (beaucoup d'autres), autant que possible, pour consoler son c\u0153ur de l'impossibilit\u00e9 fondamentale de l'amour. L'Italie et la Riviera lui ont ouvert de nouveaux horizons. Il y a pass\u00e9 quelques semaines \u2014 rendant visite \u00e0 son ancienne compagne \u00ab Little E \u00bb, elle aussi install\u00e9e dans le Sud \u2014 avant de s\u00e9journer \u00e0 Lou Pidou. Cela fait longtemps qu'il n'\u00e9tait pas all\u00e9 dans sa demeure fran\u00e7aise, \u00e9vitant \u00e0 tout prix de croiser Odette, gardienne des lieux. Moura, qui l'accompagnait, est rentr\u00e9e un peu plus t\u00f4t. Ces derniers temps, elle \u00e9tait tr\u00e8s pr\u00e9occup\u00e9e, toujours inqui\u00e8te pour ses amis \u00e0 l'\u00e9tranger, et plut\u00f4t distante. C'est \u00e9trange comme \u00e0 cet instant il la d\u00e9teste autant qu'il l'aime, les deux sentiments s'\u00e9quilibrant parfaitement : \u00ab Moura est Moura, comme toujours : humaine, fautive, sage, idiote, et je l'aime 2. \u00bb\n\nLa m\u00e9lancolie s'est peu \u00e0 peu effac\u00e9e au contact d'une certaine Constance Coolidge \u2014 belle, passionn\u00e9e et am\u00e9ricaine \u2014, avant-go\u00fbt romantique de son voyage outre-Atlantique qui l'a amen\u00e9 \u00e0 voir, de nouveau, Roosevelt et \u00e0 d\u00e9couvrir, davantage, cette nation (les \u00c9tats-Unis ont fait le pari \u2014 dangereux, selon lui \u2014 du New Deal), et \u00e0 rencontrer une blonde charismatique. Elle s'appelle Martha Gellhorn, elle est journaliste, elle a vingt-six ans. Quarante-deux ans d'\u00e9cart, c'est beaucoup, il en convient. Leur liaison fait du bruit : on taxe la jeune fille (qui a des vell\u00e9it\u00e9s litt\u00e9raires) d'opportunisme, on regarde Wells avec suspicion. Mais que peut-il face \u00e0 une jolie femme ? Pas grand-chose. Avec Martha, la vie est simple. Les paysages rocheux du Connecticut ont abrit\u00e9 un temps leurs amours, Wells quittant cette nature sauvage pour Los Angeles o\u00f9 l'attendait son ami Chaplin, avec Paulette Goddard. Le com\u00e9dien et sa compagne, en plein tournage de leur film _Les Temps Modernes_ , \u00e9taient tr\u00e8s int\u00e9ress\u00e9s par l'un des romans de Wells, _L'Histoire de M_. _Polly_ , dont ils ont achet\u00e9 les droits. Le film sera finalement r\u00e9alis\u00e9 en Angleterre, quinze ans plus tard.\n\nDe retour \u00e0 Londres, Wells a compris qu'il entrait dans la derni\u00e8re \u00ab phase 3 \u00bb de sa vie. Il a pris une d\u00e9cision soudaine mais r\u00e9fl\u00e9chie : acheter cette superbe maison \u00e0 Regent's Park, le quartier de ses jeunes ann\u00e9es, celui o\u00f9 chaque coin de rue est associ\u00e9 \u00e0 des souvenirs d'Isabel ou de Jane. La maison \u2014 Hanover Terrace \u2014 est spacieuse et lui donne l'impression d'\u00eatre son ch\u00e2teau de Ferney \u00e0 lui. La premi\u00e8re chose qu'il a faite en investissant les lieux a \u00e9t\u00e9 de placer un buste de Voltaire dans la salle \u00e0 manger. Comme le philosophe fran\u00e7ais (dont il lisait, adolescent, les contes, allong\u00e9 dans la biblioth\u00e8que d'Up Park), il souhaite depuis ce domaine prot\u00e9g\u00e9 continuer \u00e0 observer le monde \u2014 et surtout ne jamais le perdre de vue. Moura, tel un fant\u00f4me, erre de temps en temps dans les couloirs de la demeure sans jamais s'y installer vraiment. Forment-ils toujours un couple ? Ils s'interrogent tous les deux. Un jour, Moura demande \u00e0 Wells de r\u00e9unir leurs amis pour un banquet de noces. La romanci\u00e8re et dramaturge britannique Enid Bagnold, qui faisait partie des convives, a relat\u00e9 l'\u00e9pisode dans ses _M\u00e9moires_. La soir\u00e9e s'est d\u00e9roul\u00e9e dans un restaurant londonien de Soho :\n\nUne longue table \u00e9tait d\u00e9j\u00e0 mise, o\u00f9 nous pr\u00eemes place. [...] Je m'approchai de Moura pour la f\u00e9liciter. Elle me sourit calmement : \u00ab Je ne l'\u00e9pouse pas. Il se l'imagine. Je ne suis pas si b\u00eate ! \u00bb\n\nQuand tout le monde fut l\u00e0, ils entr\u00e8rent en se tenant le bras. Ce fut l'all\u00e9gresse g\u00e9n\u00e9rale, les congratulations, le champagne. Mais, au milieu du repas, Moura demanda la parole et se leva :\n\n\u2014 Tout ceci n'est qu'une plaisanterie, dit-elle. Nous vous avons jou\u00e9 un tour. Nous ne nous sommes pas mari\u00e9s aujourd'hui et nous n'avons pas l'intention de le faire \u00e0 l'avenir 4.\n\nWells, \u00e0 l'autre bout de la table, sourit comme il peut.\n\nDepuis des mois, Wells pr\u00e9pare l'adaptation de son roman _The Shape of Things to Come_. Le film ( _Things to Come_ ) (\u00ab Des choses \u00e0 venir \u00bb) a \u00e9t\u00e9 r\u00e9alis\u00e9 sans qu'il ait pu v\u00e9ritablement donner son avis sur le projet, par un homme qu'il juge \u00ab incomp\u00e9tent 5 \u00bb (William Cameron Menzies), dont le manque suppos\u00e9 d'imagination aurait mis l'auteur en furie. Il assiste, un peu \u00e0 contrec\u0153ur, \u00e0 la premi\u00e8re projection au Leicester Square. Moura, elle, se trouve en Russie, au chevet d'un Gorki mourant. Elle a voulu partir seule, Wells ne s'y est pas oppos\u00e9, m\u00eame s'il soup\u00e7onnait son ami de vouloir r\u00e9cup\u00e9rer son ancienne ma\u00eetresse. La r\u00e9alit\u00e9, malheureusement, lui a donn\u00e9 tort : il apprend, peu de jours apr\u00e8s, le d\u00e9c\u00e8s de son vieux camarade.\n\nIl y a du monde, ce soir du 19 octobre 1936, au Savoy Hotel de Londres. Le P.E.N. Club a invit\u00e9 plus d'une centaine de personnes, tri\u00e9es sur le volet, \u00e0 venir c\u00e9l\u00e9brer le soixante-dixi\u00e8me anniversaire de Wells. Moura est \u00e0 ses c\u00f4t\u00e9s, tout le monde l'observe. L'\u00e9crivain est heureux, ne sent pas le temps passer sur lui, et en fait part \u00e0 l'assembl\u00e9e dans un discours amus\u00e9 qui lui ressemble : \u00ab Ce soir, je suis tout \u00e0 fait dans la situation d'un petit gar\u00e7on qui, lors d'une magnifique r\u00e9ception, a re\u00e7u une quantit\u00e9 de beaux jouets et a juste commenc\u00e9 \u00e0 les \u00e9taler sur le plancher. Tout d'un coup, sa nurse arrive : \"Il se fait tard, Bertie, lui dit-elle, c'est l'heure de ranger vos jouets.\" Mais je n'ai pas la moindre envie de ranger mes jouets, je d\u00e9teste l'id\u00e9e de m'en aller 6. \u00bb Des jouets, justement, il en manipule de nouveaux chaque jour en r\u00e9fl\u00e9chissant, avec toute la t\u00e9nacit\u00e9 qui le caract\u00e9rise, \u00e0 r\u00e9aliser son prochain r\u00eave : cr\u00e9er une encyclop\u00e9die moderne, quelque chose de grand et d'in\u00e9dit, qui toucherait toute l'humanit\u00e9 ; un savoir d\u00e9ploy\u00e9 \u00e0 l'infini, \u00e9clair\u00e9, puissant, qui n'aurait pas la forme d'un livre, mais plut\u00f4t celle d'un dispositif.\n\nIl a bien r\u00e9fl\u00e9chi. Le savoir ne peut pas se contenter du papier, il doit pouvoir \u00eatre accessible \u00e0 tous sous une forme moins lourde et plus fonctionnelle : le savoir a besoin du microfilm. Ce proc\u00e9d\u00e9 existe d\u00e9j\u00e0 depuis la fin des ann\u00e9es 1860, mais il est encore tr\u00e8s confidentiel. Pourtant, Wells en est certain, la reproduction analogique est la condition de la diffusion des connaissances \u2014 il en fait mention lors de la premi\u00e8re conf\u00e9rence qu'il donne \u00e0 ce sujet, en novembre 1936. Diderot et d'Alembert ne sont pas all\u00e9s assez loin dans leur d\u00e9marche. Il faut aller au bout de l'id\u00e9e, permettre aux hommes non seulement d'acqu\u00e9rir le savoir, mais d'y acc\u00e9der de la mani\u00e8re la plus simple et permanente qui soit. L'id\u00e9e n'est plus de figer l'\u00e9poque dans des livres, mais de la voir se refl\u00e9ter sur un support qui la d\u00e9multiplierait. Wells \u2014 dont l'enfance fut tant priv\u00e9e de cette curiosit\u00e9 intellectuelle qui fait d'un homme un esprit en marche \u2014 formule un souhait : celui du \u00ab _world brain_ \u00bb (cerveau mondial). Sa d\u00e9marche est ambitieuse. Il se sent \u00e0 la hauteur du d\u00e9fi et l'\u00e9taye dans un ouvrage \u00e9ponyme, publi\u00e9 en 1938.\n\n_Brynhild, Enfants des \u00e9toiles, Dolor\u00e8s_ et _World Brain_ : en moins de deux ans, Wells a publi\u00e9 quatre livres, et re\u00e7u le titre honorifique de \u00ab docteur en litt\u00e9rature \u00bb, d\u00e9livr\u00e9 par l'universit\u00e9 de Londres. La sant\u00e9 est bonne et l'esprit vaillant, comme toujours, malgr\u00e9 la mauvaise nouvelle de la conf\u00e9rence de Munich dont il apprend le triste d\u00e9nouement. Depuis son arriv\u00e9e au pouvoir en Allemagne, Hitler et ses ambitions terrorisent l'Europe. Il a notamment exig\u00e9 l'annexion de la r\u00e9gion des Sud\u00e8tes (en Tch\u00e9coslovaquie) peupl\u00e9e de germanophones. Apr\u00e8s plusieurs semaines d'h\u00e9sitation, Mussolini prend les devants et organise une conf\u00e9rence afin de r\u00e9gler ce qui s'apparente \u00e0 une \u00ab crise \u00bb. Le Premier ministre britannique, Neville Chamberlain, ainsi que le pr\u00e9sident du Conseil fran\u00e7ais, \u00c9douard Daladier, rejoignent donc le Duce et le F\u00fchrer \u00e0 Munich, le 28 septembre 1938, pour convenir ensemble d'un accord. La d\u00e9cision prise donne satisfaction au chancelier. Edvard Bene\u0161, le pr\u00e9sident tch\u00e9coslovaque \u00e9cart\u00e9 de cette r\u00e9union, est contraint de quitter son propre pays. Wells, qui le conna\u00eet bien pour l'avoir d\u00e9j\u00e0 rencontr\u00e9 lors de sa tourn\u00e9e europ\u00e9enne pour le P.E.N. Club, lui \u00e9crit une lettre chaleureuse, condamnant fermement la signature de ces accords, et assistant avec horreur aux d\u00e9buts de l'invasion nazie.\n\nUne autre invasion se pr\u00e9pare outre-Atlantique \u2014 une invasion \u00e0 laquelle personne n'\u00e9tait pr\u00e9par\u00e9. La veille de Halloween, le 30 octobre 1938, peu apr\u00e8s 20 heures, un bulletin sp\u00e9cial d'informations interrompt un programme musical de la cha\u00eene CBS (principale radio du pays) annon\u00e7ant l'arriv\u00e9e imminente des Martiens en Am\u00e9rique. Un certain professeur Farrel, de l'observatoire du mont Jennings \u00e0 Chicago, aurait observ\u00e9 sur la plan\u00e8te Mars une s\u00e9rie d'explosions de gaz se dirigeant vers la Terre. Quelques minutes plus tard, un autre bulletin d'informations, plus inqui\u00e9tant, fait \u00e9tat d'un objet en flammes venu du ciel, qui se serait \u00e9cras\u00e9 pr\u00e8s d'une ferme de Grover's Hill, dans le New Jersey. Un reporter est sur place et rend compte, en direct, de ce qu'il voit, d\u00e9crivant la terreur des habitants alentour et la pr\u00e9sence de la police qui tente de contenir la foule. Soudain, le reporter d\u00e9crit une cr\u00e9ature \u00e9trange, ressemblant \u00e0 un serpent et tenant une arme qu'elle projette contre les hommes. Dans sa voix, on peut entendre la peur. Il se met \u00e0 crier, la liaison se brouille puis plus rien \u2014 le silence, abyssal, a envahi les ondes.\n\nDans les studios new-yorkais de la CBS, au vingti\u00e8me \u00e9tage d'une grande tour, un homme fige ses mains en l'air, tel un chef d'orchestre. Il porte une chemise blanche \u00e0 manches courtes, des bretelles noires et une cravate nou\u00e9e n\u00e9gligemment. Sur ses oreilles, un casque. Devant lui, un pupitre et un micro. Tout autour, vingt-sept musiciens et autant de com\u00e9diens, qui le regardent, attendent son signal. Orson Welles a vingt-trois ans, il est aux commandes d'une \u00e9mission hebdomadaire de fictions radiophoniques, et il n'a pas encore conscience du vent de panique qu'il vient de d\u00e9clencher dans pr\u00e8s d'un million de foyers. Passionn\u00e9 de th\u00e9\u00e2tre et de litt\u00e9rature, l'homonyme du romancier a d\u00e9j\u00e0 foul\u00e9 les planches et mis en sc\u00e8ne une adaptation tr\u00e8s remarqu\u00e9e du _Macbeth_ de Shakespeare. Depuis peu, il est \u00e0 la t\u00eate de sa propre troupe et s'amuse, tous les lundis soir, \u00e0 mettre en ondes ses romans pr\u00e9f\u00e9r\u00e9s. Ceux de H. G. Wells sont parmi ses favoris et, la soir\u00e9e de Halloween approchant, il a l'id\u00e9e d'adapter _La Guerre des mondes_ en transposant l'action qui se d\u00e9roule en Angleterre sur le territoire am\u00e9ricain. Cela devait \u00eatre un divertissement : ce fut une panique. Une grande partie des auditeurs n'a pas entendu le g\u00e9n\u00e9rique de l'\u00e9mission (qui pr\u00e9cise bien qu'il s'agit d'une \u00ab pi\u00e8ce radiophonique \u00bb), et se retrouve perplexe \u00e0 l'\u00e9coute de ces cris, de ce silence qui dure maintenant depuis plusieurs secondes. Puis une voix (celle d'Orson Welles) reprend l'antenne et annonce l'invasion martienne. Dans un pays non seulement affaibli par la crise \u00e9conomique de 1929 mais dans l'appr\u00e9hension constante d'un nouveau conflit mondial, cette annonce (d'un r\u00e9alisme total) est v\u00e9cue comme un v\u00e9ritable drame. Dans les maisons, on s'inqui\u00e8te, on se r\u00e9fugie chez le voisin, on pense \u00e0 partir. La proph\u00e9tie de la premi\u00e8re page de _La Guerre des mondes_ se serait donc r\u00e9alis\u00e9e, punissant ces hommes arrogants qui, \u00ab avec une suffisance infinie, ...] allaient de-ci, de-l\u00e0 par le monde, vaquant \u00e0 leurs petites affaires, dans la sereine s\u00e9curit\u00e9 de leur empire sur la mati\u00e8re[ 7 \u00bb.\n\nFace \u00e0 la vague de peur provoqu\u00e9e par l'\u00e9mission, Orson Welles et son \u00e9quipe sont forc\u00e9s d'arr\u00eater le programme et de s'expliquer aupr\u00e8s des dirigeants de la CBS, qui commencent \u00e0 recevoir des plaintes. Une conf\u00e9rence de presse est organis\u00e9e le lendemain matin et Welles (secr\u00e8tement amus\u00e9 qu'on l'ait pris au s\u00e9rieux) est amen\u00e9, face \u00e0 une foule de journalistes, \u00e0 pr\u00e9senter ses excuses. Le futur r\u00e9alisateur de _Citizen Kane_ est d\u00e9sormais connu de tous, son visage, serein et mal ras\u00e9, fait le tour du monde. Et tandis qu'Hollywood le convoite d\u00e9j\u00e0, Wells (l'autre) le d\u00e9couvre un matin dans le journal, alors qu'il est confortablement assis dans son canap\u00e9. Tout ce tapage n'amuse gu\u00e8re l'\u00e9crivain : il ne supporte pas l'id\u00e9e que son nom soit associ\u00e9 \u00e0 ce genre de plaisanterie. Sans perdre de temps, il contacte ses avocats, pensant engager une action en justice. Mais il se rend vite compte que cet \u00e9v\u00e9nement lui sert de publicit\u00e9 et qu'il c\u00e9l\u00e8bre, paradoxalement, toute la puissance de sa pens\u00e9e. Ces milliers de gens, quels qu'ils soient, ont cru l'espace de quelques secondes ce que lui-m\u00eame a \u00e0 peine os\u00e9 r\u00eaver.\nC'est la fin\n\nLa Terre vue du ciel. Wells ne s'en lassera jamais. C'est dans les airs qu'il se sent le mieux \u2014 le plus \u00e0 sa place. Cette fois-ci, il traverse litt\u00e9ralement le globe en direction de l'Australie et de la Nouvelle-Z\u00e9lande, pour une s\u00e9rie de r\u00e9unions et de conf\u00e9rences. Son corps le fait souffrir, il se sent lentement vieillir mais n'a pas l'intention de changer sa fa\u00e7on de vivre. Depuis l'\u00e9tranger, il fustige la position politique australienne et britannique face \u00e0 la situation allemande qui promet de d\u00e9g\u00e9n\u00e9rer, renvoyant dos \u00e0 dos les deux Premiers ministres de ces pays (Chamberlain et Lyons) qu'il qualifie de \u00ab conciliateurs \u00bb. Il quitte l'oc\u00e9an Indien apr\u00e8s y avoir sem\u00e9 la temp\u00eate, rejoignant Moura pour quelques jours en Gr\u00e8ce.\n\nWells est tr\u00e8s inquiet pour l'avenir du monde. Il ne peut supporter d'assister \u00e0 la pers\u00e9cution des intellectuels et \u00e0 celle des Juifs. Il fait son possible pour sauver le traducteur tch\u00e8que Pavel Eisner de l'arrestation, mais sauve, \u00e0 son insu, son homonyme, l'autre Eisner \u00e9tant donc d\u00e9port\u00e9 \u00e0 Terez\u00edn sans pour autant y mourir puisqu'il ira s'installer, apr\u00e8s la Lib\u00e9ration, aux \u00c9tats-Unis 1. Le sort de son ami Sigmund Freud le pr\u00e9occupe \u00e9galement. Le psychanalyste, plong\u00e9 dans la r\u00e9daction d'un ouvrage sur Mo\u00efse, re\u00e7oit l'\u00e9crivain \u00e0 l'automne 1938. L'entrevue est br\u00e8ve mais a le don de divertir Wells dont l'esprit est accabl\u00e9 de contrari\u00e9t\u00e9s en tout genre \u2014 notamment financi\u00e8res, l'\u00e9crivain \u00e9tant depuis peu rappel\u00e9 \u00e0 l'ordre par le fisc pour de l'argent qu'il aurait oubli\u00e9 de verser. Et puis la guerre le saisit dans son sommeil. Septembre 1939 : l'Allemagne a attaqu\u00e9 la Pologne, chaque pays alentour r\u00e9fl\u00e9chissant \u00e0 la marche \u00e0 suivre. Apr\u00e8s une vague d'ultimatums, l'Angleterre, au m\u00eame titre que la France, d\u00e9clare la guerre au IIIe Reich. Wells se trouve au m\u00eame moment \u00e0 Stockholm, \u00e9crivant un \u00e9ni\u00e8me essai sur la condition de l'humanit\u00e9 ( _The Fate of Homo Sapiens_ , \u00ab Le Destin de l'Homo Sapiens \u00bb), et faisant son possible pour rentrer \u00e0 Londres.\n\nToujours actif sur les fronts litt\u00e9raire et journalistique, il se sent pourtant impuissant face \u00e0 ce conflit tant redout\u00e9. Le d\u00e9part de Chamberlain \u2014 pouss\u00e9 vers la sortie au d\u00e9but de l'ann\u00e9e 1940 \u2014 le satisfait, mais il ignore encore si son rempla\u00e7ant, Churchill, sera \u00e0 la hauteur de la t\u00e2che. Le nouveau Premier ministre promet de ne pas plier l'\u00e9chine face au F\u00fchrer, tandis que, de l'autre c\u00f4t\u00e9 de la Manche, le mar\u00e9chal P\u00e9tain a appel\u00e9 \u00e0 un cessez-le-feu. Wells, lui, n'a pas non plus l'intention d'abandonner son pays, et veut continuer \u00e0 diffuser ses id\u00e9es \u00e0 travers le monde. En ces temps troubles pour la nature humaine, il se sent l'\u00e2me d'un pr\u00e9dicateur et s'embarque pour un \u00e9ni\u00e8me voyage en Am\u00e9rique \u2014 qui est devenue pour lui une seconde patrie. Ses proches le pr\u00e9viennent du danger qu'il court en prenant la mer dans un climat international si tendu et incertain. Mais lui n'\u00e9coute que ses envies. Il a d\u00e9j\u00e0 parcouru des milliers de kilom\u00e8tres dans les airs, il ne peut rien lui arriver sur l'eau.\n\nArriv\u00e9 \u00e0 New York, il assiste \u00e0 la premi\u00e8re du _Dictateur_ de Chaplin. L'irr\u00e9v\u00e9rence de son ami, incarnant une r\u00e9plique \u00e0 peine dissimul\u00e9e de Hitler, lui donne le sentiment que rien n'est perdu. Il aimerait voir la \u00ab reconstruction 2 \u00bb dont il r\u00eave depuis tant d'ann\u00e9es. Il a l'occasion d'en parler avec Orson Welles, le nouveau prodige du cin\u00e9ma am\u00e9ricain, qu'il rencontre pour la premi\u00e8re fois \u00e0 la radio. Juste apr\u00e8s l'attend Martha, d\u00e9sormais mari\u00e9e \u00e0 Ernest Hemingway, qui l'a convi\u00e9 \u00e0 un d\u00e9jeuner en ville. Il est heureux de la revoir, elle lui parle de son m\u00e9tier \u2014 le reportage de guerre \u2014 et lui de ses projets \u2014 un essai contre l'\u00c9glise catholique ( _Crux ansata_ , \u00ab Croix ans\u00e9e \u00bb) et tr\u00e8s certainement une courte halte aux Bermudes (il a besoin de prendre le soleil).\n\nIl rentre \u00e0 Londres, en janvier 1941, \u00e9puis\u00e9 par ce tour am\u00e9ricain. Chez lui l'attendent Moura et plusieurs manuscrits laiss\u00e9s inachev\u00e9s. Son entourage s'\u00e9tonne de sa vigueur \u00e0 toujours tenir la plume. Mais, au fond, l'\u00e9criture n'a jamais \u00e9t\u00e9 pour lui une question de dynamisme :\n\nIl y a une sorte de l\u00e9gende journalistique d'apr\u00e8s laquelle j'aurais une facult\u00e9 d'enthousiasme et une \u00e9nergie sans limites. Rien n'est plus loin de la r\u00e9alit\u00e9. ...] \u00c9crire une quantit\u00e9 de livres et d'articles n'est pas un t\u00e9moignage d'\u00e9nergie, mais d'habitudes s\u00e9dentaires. Les gens qui ont un v\u00e9ritable exc\u00e8s quantitatif d'\u00e9nergie deviennent des Mussolini, des Hitler, des Staline, des Gladstone, des Beaverbrook, des Northcliffe, des Napol\u00e9on. Il faut des g\u00e9n\u00e9rations pour d\u00e9blayer le terrain apr\u00e8s leur d\u00e9part. Mais ce qui restera apr\u00e8s moi n'aura pas besoin d'\u00eatre d\u00e9blay\u00e9. Justement \u00e0 cause de mon apathie constitutionnelle, ce que je laisserai sera caract\u00e9ristiquement libre de tout fatras individuel, restera disponible, et durera autant qu'on en aura besoin[ 3.\n\nL'apathie. Wells la fait sienne pour le plaisir de la formule, mais elle est difficilement cr\u00e9dible le concernant. Le corps n'est plus autant en mouvement, mais l'esprit est encore vif, lucide. Il sursaute m\u00eame, lorsqu'un certain Eric Arthur Blair (plus connu sous le nom de George Orwell) se risque \u00e0 critiquer son \u0153uvre \u00e0 la B.B.C. : selon lui, le c\u00e9l\u00e8bre anticipateur s'est fourvoy\u00e9 en croyant que le salut du monde se trouvait dans la science, alors m\u00eame que c'est elle qui semble aujourd'hui l'an\u00e9antir. Le futur auteur de _1984_ (grand admirateur, pourtant, de son a\u00een\u00e9, et certainement l'un de ses plus proches h\u00e9ritiers litt\u00e9raires) ne craint pas la foudre. La r\u00e9ponse de l'int\u00e9ress\u00e9, qui a d\u00e9j\u00e0 rencontr\u00e9 Orwell dans des circonstances plus sereines, n'a pas tard\u00e9 : \u00ab _I don't say that at all. Read my early works, you shit_ 4. \u00bb (\u00ab Je ne dis pas du tout cela. Lisez mes premiers travaux, petit merdeux. \u00bb) D'apathie, il n'en est donc, pour l'instant, pas du tout question. En revanche, l'affaiblissement physique se confirme. Wells dort de plus en plus, et re\u00e7oit de fr\u00e9quentes visites du m\u00e9decin. Son fils, Anthony, qui s'est install\u00e9 par pr\u00e9caution dans la ferme voisine de Hanover Terrace, sait que son p\u00e8re est malade, qu'il est atteint d'un cancer du foie, mais il n'ose pas lui annoncer la nouvelle. Il a peur de le voir abandonner la vie. C'est finalement Rebecca qui le lui apprend. Au d\u00e9but, l'\u00e9crivain n'y croit pas, pense qu'il y a une erreur de diagnostic, qu'il a encore de longues ann\u00e9es devant lui. Il n'y a m\u00eame pas deux ans de cela, il publiait un \u00e9ni\u00e8me roman ( _Un homme averti en vaut deux_ ). Ce livre ne pouvait pas \u00eatre son dernier.\n\nMais le d\u00e9sespoir commence \u00e0 l'envahir. Son \u00e9tat se d\u00e9grade au fil des semaines et son impatience \u00e0 voir le monde sortir de sa torpeur se mue en lente r\u00e9signation. La guerre se rapproche de lui. Des amis proposent de l'h\u00e9berger loin de Londres, notamment aux \u00c9tats-Unis, mais il ne veut pas quitter sa maison, m\u00eame si elle doit devenir son tombeau. Il s'est d\u00e9j\u00e0 s\u00e9par\u00e9 de Lou Pidou (Odette ayant finalement d\u00e9m\u00e9nag\u00e9), abandonnant ce paradis d'orangers \u00e0 un \u00ab myst\u00e9rieux chat noir \u00bb, il ne peut pas partir d'ici maintenant. Peu \u00e0 peu, ses voisins s'en vont, ses sorties se font plus rares, les fen\u00eatres de sa grande demeure sont condamn\u00e9es, et lui, impassible, reste clo\u00eetr\u00e9 \u00e0 l'int\u00e9rieur sous le bruit des missiles qui s'\u00e9crasent tout autour. Le palais de Buckingham et la cath\u00e9drale Saint-Paul ont d\u00e9j\u00e0 \u00e9t\u00e9 touch\u00e9s lors du Blitz, en 1940. D\u00e9sormais, ce sont de nouveaux missiles qui tombent sur l'Angleterre, des puissants projectiles nomm\u00e9s V1 et V2 qui d\u00e9truisent son jardin et font trembler les vitres. Sa maison s'effrite comme sa vie, la r\u00e9alit\u00e9 semblant \u00e9trangement rejoindre ses plus folles pr\u00e9dictions.\n\nPlus rien n'a d'importance, surtout pas cette s\u00e9ance photographique pr\u00e9vue depuis des semaines pour le _Countryman,_ et qu'il n'est plus possible d'annuler. Wells est debout, sous un arbre de son jardin qui n'est pas encore tomb\u00e9, face \u00e0 sa maison d\u00e9labr\u00e9e. Le photographe lui demande de prendre la pose entre deux toussotements. La m\u00eame sc\u00e8ne s'est d\u00e9roul\u00e9e il y a exactement soixante-dix ans. Lui, tir\u00e9 \u00e0 quatre \u00e9pingles, les cheveux plaqu\u00e9s derri\u00e8re l'oreille, \u00e9coutant sa m\u00e8re lui dire de bien regarder l'objectif, souriant dans le vide. Il n'\u00e9tait alors qu'un petit gar\u00e7on qui voulait accomplir de grandes choses. Y est-il parvenu ? Il a fait son possible, il s'est accroch\u00e9 \u00e0 la vie et a travers\u00e9 le monde avec une seule obsession : le fa\u00e7onner selon son d\u00e9sir, montrer aux hommes le chemin du progr\u00e8s \u2014 un chemin sinueux, et par endroits contestable. Tel Atlas, dont la petite statue dominait l'entr\u00e9e de sa maison d'enfance, il a voulu porter ce m\u00eame monde sur ses \u00e9paules \u2014 quitte \u00e0 en souffrir.\n\nCe monde est finalement d\u00e9livr\u00e9. Le 30 avril 1945, Hitler se donne la mort, une semaine plus tard, l'Allemagne capitule. Mais les valeurs semblent invers\u00e9es, et la plus grande crainte du vieil \u00e9crivain se voit confirm\u00e9e sans qu'il s'y attende, un matin d'\u00e9t\u00e9. Le 6 ao\u00fbt 1945, la r\u00e9alit\u00e9 d\u00e9passe la fiction et prend le nom d'Hiroshima. La B.B.C. contacte Wells et lui demande s'il serait d'accord pour intervenir sur les ondes afin d'\u00e9voquer ce d\u00e9sastre nucl\u00e9aire. Mais il ne peut pas, il souffre trop \u2014 du corps mais surtout du c\u0153ur. \u00ab Qu'allons-nous devenir ? \u00bb, demandait le vicaire au narrateur de _La Guerre des mondes_ , \u00e0 quoi ce dernier r\u00e9pondait : \u00ab Il ne faut pas perdre la t\u00eate. Il y a encore de l'espoir. ...] beaucoup d'espoir \u2014 malgr\u00e9 tous ces ravages[ 5 ! \u00bb\n\nMais \u00e0 cet instant, point d'espoir. Wells pense \u00e0 Stefan Zweig. L'\u00e9crivain a fui son pays, l'Autriche, pour Londres, avant de s'exiler en Am\u00e9rique du Sud. Le 22 f\u00e9vrier 1942, \u00e0 l'\u00e2ge de soixante ans, ne supportant plus de voir le monde courir \u00e0 sa perte, il a pr\u00e9f\u00e9r\u00e9 une fiole de poison au poison de vivre. Wells comprend cela, l'id\u00e9e du suicide lui a d\u00e9j\u00e0 travers\u00e9 l'esprit \u2014 quand il \u00e9tait jeune et malheureux dans l'atelier de draperie, et aujourd'hui, tandis que la maladie le consume lentement. Depuis plusieurs semaines, il fait face au m\u00eame dilemme : le chagrin ou le n\u00e9ant. Mais comme le h\u00e9ros de Faulkner dans _Si je t'oublie, J\u00e9rusalem_ , il choisit le chagrin 6. Termin\u00e9, l'optimisme coutumier, celui qui le portait sans cesse vers le futur, qui l'amenait \u00e0 voir plus loin que tous les autres. L'avenir est impossible, le pr\u00e9sent se meurt : l'existence est arriv\u00e9e \u00e0 bout de ses ressources \u2014 \u00ab au bout de son rouleau 7 \u00bb. Il faut imaginer cet homme, qui n'a cess\u00e9 de parcourir le temps et l'espace, d\u00e9sormais incapable d'envisager une quelconque issue, aveugl\u00e9 par la confusion de son \u00e9poque, irr\u00e9m\u00e9diablement bloqu\u00e9 dans l'ici et maintenant, convaincu \u00ab qu'il n'y a pas \u00e0 sortir de l'impasse, \u00e0 la contourner ou \u00e0 la traverser 8 \u00bb, puisque \u00ab c'est la fin 9 \u00bb. Il est difficile d'expliquer ce pessimisme, de trouver son origine. Anthony n'en cherche pas la raison : il consid\u00e8re que son p\u00e8re l'a toujours eu en lui :\n\nCe qui se passa \u00e0 mesure que ses facult\u00e9s d\u00e9clin\u00e8rent, \u00e0 partir de 1940, est qu'il fit retour vers ses croyances premi\u00e8res et profond\u00e9ment ancr\u00e9es en lui, au sujet des r\u00e9alit\u00e9s de la condition humaine. C'\u00e9tait par nature un pessimiste et il avait fait violence \u00e0 ses intuitions et \u00e0 ses perceptions rationnelles tout \u00e0 la fois, en affirmant, vers le milieu de sa carri\u00e8re, que l'esp\u00e8ce humaine \u00e9tait capable de cr\u00e9er un monde meilleur pour elle-m\u00eame par un effort de volont\u00e9 10.\n\nTandis que l'horloge tourne et lui rappelle le poids des heures, Wells voit sa vie d\u00e9filer sur une toile blanche, comme un film qui serait projet\u00e9 sur un \u00e9cran de cin\u00e9ma. Des livres et des femmes, par centaines. Mais cela lui semble n'avoir jamais exist\u00e9. \u00ab Nos amours, nos haines, nos guerres et nos batailles, elles-m\u00eames aussi d\u00e9nu\u00e9es de substance qu'un r\u00eave, ne sont rien qu'une fantasmagorie qui danse sur ce tissu 11. \u00bb Il n'y a plus rien \u00e0 attendre des hommes, la faillite du monde est sous ses yeux, tous les jours dans les journaux, sur les ondes. Que lui reste-t-il ? La certitude qu'\u00ab il a v\u00e9cu gaiement et abondamment, qu']il a r\u00e9chauff\u00e9 ses deux mains au feu de la vie[ 12 \u00bb et il serait bon d'ajouter \u00e0 celui des femmes. La derni\u00e8re d'entre elles, Moura, est pr\u00e9sente quotidiennement \u00e0 ses c\u00f4t\u00e9s. Elle lui fait la lecture, envoie des lettres qu'il lui dicte. Certains jours, il ne la reconna\u00eet pas. Ces crises de d\u00e9lire sont passag\u00e8res. D'autres, il avoue n'\u00eatre arriv\u00e9 \u00e0 rien. Le d\u00e9sespoir nie l'\u0153uvre radicalement. M\u00eame le lointain souvenir des premiers succ\u00e8s n'existe plus. Le chagrin passe. Gip, Anthony et Moura le calment, et Wells se rendort paisiblement.\n\nIl aimerait savoir comment la mort le saisira. Il se souvient de celle de son p\u00e8re : un matin, apr\u00e8s avoir lu le journal en d\u00e9gustant un pudding, il s'est lev\u00e9 de son lit et s'est effondr\u00e9 sur le sol. M\u00eame chose pour son grand-p\u00e8re, sauf que le vieil homme s'est entour\u00e9 d'un beau d\u00e9cor : il est tomb\u00e9 dehors, face au coucher du soleil. Voil\u00e0 une \u00ab fa\u00e7on magnifique 13 \u00bb de partir. Comme il aimerait s'affaisser d'un coup, quitter la vie aussi pr\u00e9cipitamment qu'il y avait go\u00fbt\u00e9. Il se demande bien ce qu'il y aura apr\u00e8s \u2014 et s'il y a m\u00eame quelque chose. Peut-\u00eatre devra-t-il assister au Jugement dernier, assis, comme il l'avait imagin\u00e9 dans l'une de ses nouvelles 14, entre Darwin et Henri VIII ? Peut-\u00eatre sera-t-il condamn\u00e9 \u00e0 partir sur une lointaine plan\u00e8te, peupl\u00e9e de \u00ab corps agr\u00e9ables \u00e0 voir 15 \u00bb. Ce serait plaisant.\n\nLe souhait s'accomplit dans son lit, le 13 ao\u00fbt 1946, apr\u00e8s avoir rempli une grille de mots crois\u00e9s dans le _Times_. \u00ab _Go away, I'm all right_ \u00bb (\u00ab Allez-vous-en. Tout va bien \u00bb) sont les derni\u00e8res paroles qu'il aurait adress\u00e9es \u00e0 son infirmi\u00e8re. Quelques jours plus tard, Gip et Anthony prennent le large avec l'urne qui contient ses cendres. Ils tiennent \u00e0 ce qu'elles rejoignent l'\u00e9tendue bleue qu'il aimait tant contempler et traverser. N'avait-il pas \u00e9crit dans _Tono Bungay_ : \u00ab Tous nous jouons un r\u00f4le et nous disparaissons, luttant pour accomplir une mission dont le sens nous reste cach\u00e9, en route vers la haute mer 16 \u00bb ? Les deux fils savent qu'ils sont au bon endroit, et regardent leur p\u00e8re s'envoler au large de Studland.\nANNEXES\nREP\u00c8RES CHRONOLOGIQUES\n\n1866. |  _21 septembre_ : naissance de Herbert George Wells \u00e0 Bromley, dans le Kent.  \n---|---  \n1874. |  Int\u00e8gre l'\u00e9cole de Thomas Morley.  \n1880. |  D\u00e9bute son apprentissage en atelier de draperie.  \n1883. |  Devient r\u00e9p\u00e9titeur \u00e0 la Midhurst Grammar School et obtient une bourse d'\u00e9tudes pour la Normal School of Science, \u00e0 South Kensington.  \n1884. |  \u00c9tudie \u00e0 la Normal School of Science sous la direction du professeur et grand biologiste T. H. Huxley. Il cr\u00e9e avec quelques camarades le _Science Schools Journal_ , et tombe amoureux de sa cousine, Isabel Mary Wells.  \n1887. |  Enseigne \u00e0 la Holt Academy de Wrexham. Il se blesse gravement lors d'une partie de football.  \n1888. |  Publie les premiers \u00e9pisodes du r\u00e9cit \u00ab The Chronic Argonauts \u00bb (premi\u00e8re matrice de _La Machine \u00e0 explorer le temps_ ) dans le _Science Schools Journal_.  \n1890. |  Obtient son dipl\u00f4me de biologie.  \n1891. |  Enseigne \u00e0 l'University Correspondence College. \u00c9pouse sa cousine, Isabel. Publie son premier essai, _La Red\u00e9couverte de l'Unique_ dans la _Fortnightly Review_.  \n1893. |  D\u00e9cide de devenir \u00e9crivain. Publie _Text-Book of Biology_ ( _Manuel de biologie_ ), son premier livre.  \n1894. |  Divorce d'Isabel pour Amy Catherine Robbins, dite Jane. Le couple s'installe \u00e0 Londres, \u00e0 Mornington Place. Publie _La Machine \u00e0 explorer le temps_ et _La Merveilleuse Visite_.  \n1896. |  Publie _La Burlesque \u00c9quip\u00e9e du cycliste_ et _L'\u00cele du docteur Moreau_.  \n1897. |  Publie _L'Homme invisible_.  \n1898. |  Publie _La Guerre des mondes_.  \n1900. |  S'installe \u00e0 Sandgate, dans le Kent. Publie _L'Amour et M. Lewisham_.  \n1901. |  Naissance de son premier fils, George Philip Wells dit Gip. R\u00e9fl\u00e9chit \u00e0 la cr\u00e9ation d'un \u00c9tat mondial. Publie _Anticipations_ et _Les Premiers Hommes dans la Lune._  \n1902. |  Publie _Miss Waters_.  \n1903. |  Naissance de son deuxi\u00e8me fils, Frank. Rencontre Beatrice et Sydney Webb et adh\u00e8re \u00e0 la Soci\u00e9t\u00e9 des Fabiens. Publie _Mankind in the Making_.  \n1905. |  Publie _Une utopie moderne_ et _Kipps_. Mort de sa m\u00e8re, Sarah Wells.  \n1906. |  Voyage pour la premi\u00e8re fois aux \u00c9tats-Unis. Rencontre Maxime Gorki \u00e0 New York et d\u00e9jeune avec le pr\u00e9sident Theodore Roosevelt \u00e0 la Maison-Blanche.  \n1908. |  Naissance d'Anna Jane Blanco White, fille qu'il a avec sa ma\u00eetresse Amber Reeves. S'installe \u00e0 Hampstead. Publie _La Guerre dans les airs_.  \n1909. |  Publie _Tono Bungay_ et _Ann Veronica_. Rencontre Elizabeth von Arnim dite Little E.  \n1910. |  S'installe \u00e0 Easton Glebe. Publie _L'Histoire de M. Polly_.  \n1912. |  Publie _Mariage_. Rencontre Rebecca West, journaliste pour _The Freewoman_.  \n1914. |  Voyage pour la premi\u00e8re fois en Russie. Naissance d'Anthony, fils qu'il a avec Rebecca West. Publie _La Destruction lib\u00e9ratrice_ , roman dans lequel il met en garde les hommes contre les dangers de l'arme nucl\u00e9aire.  \n1915. |  Publie _La Recherche magnifique_. Se dispute avec Henry James suite \u00e0 la publication de _Boon_.  \n1916. |  Visite les fronts fran\u00e7ais et italien. Publie _M. Britling commence \u00e0 y voir clair_. Br\u00e8ve parenth\u00e8se religieuse.  \n1917. |  Publie _La Guerre de l'avenir_ , _Dieu, l'invisible roi._ Participe \u00e0 la propagande anti allemande.  \n1919. |  Publie _La Flamme immortelle_ , un plaidoyer en faveur de l'\u00e9ducation. R\u00e9fl\u00e9chit \u00e0 la cr\u00e9ation d'une Soci\u00e9t\u00e9 des Nations.  \n1920. |  Voyage en Russie, rend visite \u00e0 Gorki, rencontre L\u00e9nine et Moura Budberg. Publie _Esquisse de l'histoire universelle,_ grand succ\u00e8s \u00e9ditorial, et _La Russie dans l'ombre_.  \n1922. |  Se pr\u00e9sente comme candidat aux \u00e9lections l\u00e9gislatives, du c\u00f4t\u00e9 du Parti travailliste, mais n'est pas \u00e9lu. Publie _Les Coins secrets du c\u0153ur_.  \n1923. |  Hedwig Gatternigg, une admiratrice, tente de se suicider chez lui, \u00e0 Londres. Publie _M. Barnstaple chez les hommes-dieux_.  \n1924. |  Publie _Un grand \u00e9ducateur moderne_ , essai sur le professeur Sanderson. Rencontre Odette Keun. S'installe \u00e0 Grasse dans une grande maison, Lou Bastidon.  \n1926. |  Publie _Le Monde de William Clissold_.  \n1927. |  Fait construire la villa Lou Pidou \u00e0 Grasse. Conf\u00e9rence \u00e0 la Sorbonne. Mort de Jane.  \n1928. |  Tente d'insuffler un nouvel \u00e9lan pour unifier le monde en publiant _La Conspiration au grand jour_. Vend sa maison d'Easton Glebe.  \n1929. |  Donne une conf\u00e9rence au Reischtag, \u00e0 Berlin. Revoit Moura Budberg. Ils deviennent amants.  \n1930. |  S'installe de nouveau \u00e0 Londres. Publie _La Dictature de M. Parham_ et _La Science de la vie_ , \u00e9crit en collaboration avec Julian Huxley et G. P. Wells.  \n1933. |  Publie _The Shape of Things to Come_. Prend la succession de John Galsworthy \u00e0 la direction du P.E.N. Club.  \n1934. |  Voyage aux \u00c9tats-Unis et en U.R.S.S., rencontre F. D. Roosevelt et Staline. Publie _Une tentative d'autobiographie_.  \n1935. |  Travaille sur le sc\u00e9nario de _Things to Come_ , adaptation de son roman \u00e9ponyme.  \n1936. |  F\u00eate son 70e anniversaire au Savoy Hotel de Londres. Nomm\u00e9 docteur \u00e8s lettres _honoris causa_ par l'universit\u00e9 de Londres. Publie _The Anatomy of Frustration_.  \n1937. |  Publie _Enfants des \u00e9toiles_ , _Brynhild._  \n1938. |  Voyage en Australie. Publie _Dolor\u00e8s_. R\u00e9fl\u00e9chit \u00e0 un syst\u00e8me de diffusion du savoir sur microfilm dans son essai _World Brain_. Orson Welles adapte _La Guerre des mondes_ sur les ondes am\u00e9ricaines.  \n1939. |  Voyage en Su\u00e8de au moment o\u00f9 l'Angleterre d\u00e9clare la guerre \u00e0 l'Allemagne. Publie _The Fate of Homo Sapiens_.  \n1940. |  \u00c0 Londres pendant le Blitz. Donne plusieurs conf\u00e9rences aux \u00c9tats-Unis.  \n1941. |  Publie son dernier roman, _Un homme averti en vaut deux_.  \n1942. |  Pol\u00e9mique avec George Orwell.  \n1945. |  Publie _\u00c0 fin de course_ , son dernier livre, le plus pessimiste de tous.  \n1946. |  _13 ao\u00fbt :_ il meurt \u00e0 Londres dans sa maison de Hanover Terrace.\nBIBLIOGRAPHIE S\u00c9LECTIVE\n\n\u0152UVRES DE H. G. WELLS TRADUITES EN FRAN\u00c7AIS\n\nRomans\n\n_La Machine \u00e0 explorer le temps_ (1895), Mercure de France (1959), Gallimard, coll. Folio, traduit de l'anglais par Henry-D. Davray, 1975.\n\n_La Merveilleuse Visite_ (1895), Mercure de France, traduit de l'anglais par Louis Barron, 1903.\n\n_L'\u00cele du docteur Moreau_ (1896), Mercure de France (1901), Gallimard, coll. Folio, traduit de l'anglais par Henry-D. Davray, 1975.\n\n_La Burlesque \u00c9quip\u00e9e du cycliste_ (1896), Mercure de France (1906), Gallimard, coll. Folio, traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, 1984.\n\n_L'Homme invisible_ (1897), Soci\u00e9t\u00e9 d'\u00e9ditions litt\u00e9raires et artistiques, Librairie Paul Ollendorff (1901), Albin Michel (1958), Le Livre de Poche, traduit de l'anglais par Achille Laurent, 2012.\n\n_La Guerre des mondes_ (1898), Mercure de France (1900), Gallimard, coll. Folio, traduit de l'anglais par Henry-D. Davray, 2005.\n\n_Quand le dormeur s'\u00e9veillera_ (1899), Mercure de France, traduit de l'anglais par Henry-D. Davray, 1904.\n\n_L'Amour et M. Lewisham_ (1900), Mercure de France (1903), Gallimard, coll. Folio, traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, 1984.\n\n_Les Premiers Hommes dans la Lune_ (1901), Mercure de France (1901), Gallimard, coll. Folio, traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, 1984.\n\n_Miss Waters_ (1902), Mercure de France (1906), Gallimard, coll. Folio, traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, 1984.\n\n_Place aux g\u00e9ants_ (1904), Mercure de France, traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, 1904.\n\n_Kipps_ (1905), Payot, traduit de l'anglais par L. Wolff, 1922.\n\n_Au temps de la com\u00e8te_ (1906), Mercure de France (1910), Gallimard, coll. Folio, traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, 1984.\n\n_La Guerre dans les airs_ (1908), Mercure de France (1910), Gallimard, coll. Folio, traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, 1984.\n\n_Tono Bungay_ (1909), Payot, traduit de l'anglais par \u00c9douard Guyot, 1929.\n\n_Ann Veronica_ (1909), Mercure de France (1911), Le Rocher, traduit de l'anglais par Joseph Dobrinsky, 1991.\n\n_L'Histoire de M. Polly_ (1910), Mercure de France (1911), Gallimard, coll. Folio traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, 1978.\n\n_Le Nouveau Machiavel_ (1911), Albin Michel, traduit de l'anglais par Madeleine Rolland, 1924.\n\n_Mariage_ (1912), Payot, traduit de l'anglais par P. Hollard et M. Lefebvre, 1926.\n\n_La Destruction lib\u00e9ratrice_ (1914), Bruxelles, Grama, traduit de l'anglais par Patrick Delperdange, 1995.\n\n_M. Britling commence \u00e0 y voir clair_ (1916), Librairie Payot et Cie, sans mention de traducteur, 1918.\n\n_Jeanne et Pierre_ (1918), Payot et Cie, sans mention de traducteur, 1922.\n\n_La Flamme immortelle_ (1919), Payot et Cie, traduit de l'anglais par M. Butts, 1920.\n\n_Les Coins secrets du c\u0153ur_ (1922), Payot, traduit de l'anglais par M. Butts, 1924.\n\n_M. Barnstaple chez les hommes-dieux_ (1923), Albin Michel, traduit par Louis Labat, 1926.\n\n_Enfants des \u00e9toiles_ (1937), Gallimard (1939), coll. Folio, traduit de l'anglais par Armand Pierhal, 1984.\n\n_Brynhild_ (1937), La Table ronde, traduit de l'anglais par Louis Chantem\u00e8le, 1946.\n\n_Dolor\u00e8s_ (1938), Les Deux Rives, traduit de l'anglais par H. F. Merville, 1946.\n\n_Un homme averti en vaut deux_ (1941), Les \u00c9ditions universelles, traduit de l'anglais par Louis Chantem\u00e8le, 1946.\n\nContes et nouvelles\n\n_Une histoire des temps \u00e0 venir,_ Mercure de France, traduit de l'anglais par Henry-D. Davray, 1901.\n\n_L'Empire des fourmis et autres nouvelles_ , Le Livre de Poche, traduit de l'anglais par Joseph Dobrinsky, 1990.\n\n_Effrois et fantasmagories_ , Mercure de France (1911), Gallimard, coll. L'Imaginaire, traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, 1984.\n\n_Le Joueur de croquet_ , Gallimard (1938), coll. Folio, traduit de l'anglais par Marie Tadi\u00e9, 1988.\n\nEssais\n\n_Anticipations, ou De l'influence du progr\u00e8s m\u00e9canique et technique sur la vie et la pens\u00e9e humaines_ (1901), Mercure de France, traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, 1904.\n\n_Une utopie moderne_ (1905), Mercure de France, traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, 1907.\n\n_La guerre qui tuera la guerre_ (1914), \u00c9ditions et Librairie, traduit de l'anglais par Cecil Georges-Bazile, 1915.\n\n_La Guerre et l'Avenir_ (1917), Albin Michel, traduit de l'anglais par Cecil Georges-Bazile, 1917.\n\n_Dieu, l'invisible roi_ (1917), Payot et Cie, traduit de l'anglais par M. Butts, 1917.\n\n_La Russie dans l'ombre_ (1920), A. M. M\u00e9taili\u00e9, traduit de l'anglais par Bernard Loing, 1985.\n\n_Esquisse de l'histoire universelle_ (1920), Payot, traduit de l'anglais par \u00c9douard Guyot, 1925.\n\n_Un grand \u00e9ducateur moderne, Sanderson, directeur du coll\u00e8ge d'Oundle_ (1924), Librairie F\u00e9lix Alcan, traduit de l'anglais par M. Butts, 1929.\n\n_La Conspiration au grand jour_ (1928), Aubier, traduit de l'anglais par Odette Keun et J. Fournet, 1929.\n\n_Une tentative d'autobiographie, d\u00e9couvertes et conclusions d'un cerveau tr\u00e8s ordinaire_ (1934), Gallimard, traduit de l'anglais par Antonina Vallentin, 1936.\n\n_\u00c0 fin de course_ (1945), La Table ronde, traduit de l'anglais par Louis Chantem\u00e8le, 1946.\n\n\u0152UVRES DE H. G. WELLS NON TRADUITES EN FRAN\u00c7AIS\n\nEssais\n\n_Mankind in the Making,_ Londres, Chapman and Hall, 1903.\n\n_The Future in America : A Search After Realities_ , Londres, Chapman and Hall, 1906.\n\n_What Are We To Do With Our Lives ?_ , Londres, Heinemann, 1931.\n\n_The Anatomy of Frustration,_ Londres, The Cresset Press, 1936.\n\n_World Brain_ , Londres, Methuen and Co., 1938.\n\n_The Fate of Homo Sapiens : An Unemotional Statement of The Things That Are Happening To Him Now And of The Immediate Possibilities Confronting Him_ , Londres, Secker and Warburg, 1939.\n\n_Arnold Bennett and H. G. Wells_ , Harris Wilson, \u00e9d., Londres, Hart-Davis, 1960 ; Urbana, University of Illinois Press, 1960.\n\n_H. G. Wells in Love, Postscript of An Experiment in Autobiography_ (1984), G. P. Wells \u00e9d., Faber and Faber, 2008.\n\nRomans\n\n_Boon_ , Londres, T. Fisher Unwin, 1915.\n\n_The World of William Clissold : A Novel at A New Angle_ , Londres, Ernest Benn, 1926.\n\n_The Shape of Things to Come_ , Londres, Hutchinson, 1933 ; New York, Macmillan, 1933.\n\nOUVRAGES SUR H. G. WELLS\n\nEn fran\u00e7ais\n\nJoseph ALTAIRAC, _H. G. Wells, parcours d'une \u0153uvre_ , Encrage, coll. R\u00e9f\u00e9rences, 1998.\n\nDavid LODGE, _Des vies \u00e0 \u00e9crire_ , traduit de l'anglais par Martine Aubert, Rivages, 2014.\n\nDavid LODGE, _Un homme de temp\u00e9rament_ , traduit de l'anglais par Martine Aubert, Rivages, 2012.\n\nAntonina VALLENTIN, _H. G. Wells ou La Conspiration au grand jour_ , Stock, 1952.\n\nJean-Pierre VERNIER, _H. G. Wells et son temps_ , Publications de l'universit\u00e9 de Rouen, coll. \u00c9tudes anglaises, 1971.\n\nEn anglais\n\nJohn R. HAMMOND, _H. G. Wells and Rebecca West_ , Londres et New York, Harvester Wheatsheaf, 1991.\n\nNorman et Jeanne MACKENZIE, _The Time Traveller_ , Londres, Weidenfeld and Nicolson, 1973 ; New York, Simon and Schuster, 1973 ; \u00e9d. r\u00e9vis\u00e9e Londres, Hogarth Press, 1987.\n\nMichael SHERBORNE, _H. G. Wells, Another Kind of Life_ , Londres et Chicago, Peter Owen, 2012.\n\nAnthony WEST, _H. G. Wells, Aspects of A Life_ , New York, Random House, 1984 ; Londres, Hutchinson, 1984.\n\nGeoffrey WEST, _H. G. Wells : A Sketch For A Portrait_ , Londres, Howe, 1930 ; New York, Norton, 1930.\n\nAUTRES\n\nEnid BAGNOLD, _Autobiography_ , Little, Brown, 1970.\n\nNina BERBEROVA, _Histoire de la baronne Boudberg_ , traduit du russe par Michel Niqueux, Actes Sud, 1988.\n\nVictor HUGO, _Le Promontoire du songe_ (1834), Gallimard, 2012.\n\nR. H. Bruce LOCKHART, _M\u00e9moires d'un agent secret britannique en Russie_ _(1912-1918)_ , traduit de l'anglais par Lucien Thomas, Payot, 1933.\n\nWilliam SHAKESPEARE, _Les Sonnets_ , traduits de l'anglais par Yves Bonnefoy, \u00ab Po\u00e9sie\/Gallimard \u00bb, 2007, sonnet 116.\n\nAnthony WEST, _Principles and Persuasion_ , New York, Harcourt, Brace and Company, 1957.\nNotes\n\nPROLOGUE\n\n. Cit\u00e9 dans Herbert George Wells, _H. G. Wells in Love, Postscript to An Experiment in Autobiography_ , G. P. Wells \u00e9d., Faber and Faber, 1984, p. 105. L'ensemble des extraits de cet ouvrage est traduit par l'auteur.\n\n. H. G. Wells, _Un grand \u00e9ducateur moderne_. _Sanderson, directeur du coll\u00e8ge d'Oundle_ (1924), traduit de l'anglais par M. Butts, Librairie F\u00e9lix Alcan, 1929.\n\n. H. G. Wells, _H. G. Wells in Love_ , _op. cit._ , p. 105.\n\n. _Ibid_., p. 106.\n\nENFANT DES \u00c9TOILES\n\n. H. G. Wells, _Une tentative d'autobiographie, d\u00e9couvertes et conclusions d'un cerveau tr\u00e8s ordinaire_ (1934), traduit de l'anglais par Antonina Vallentin, Gallimard, 1936, p. 38.\n\n. _Ibid_., p. 39.\n\n. _Ibid_., p. 36.\n\n. _Ibid_., p. 23.\n\n. _Ibid_.\n\n. _Ibid_.\n\n. _Ibid_., p. 37.\n\n. _Ibid_., p. 26.\n\n. _Ibid_., p. 40.\n\n. _Ibid_., p. 42.\n\n. _Ibid_.\n\n. _Ibid_., p. 43.\n\n. _Ibid_.\n\n. H. G. Wells, _Enfants des \u00e9toiles_ (1937), traduit de l'anglais par Armand Pierhal, Gallimard, coll. Folio, 1984, p. 55.\n\n. _Ibid_., p. 54.\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 63.\n\nCOUP DE CHANCE\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 49.\n\n. _Ibid_.\n\n. _Ibid_., p. 71.\n\n. H. G. Wells, _L'\u00cele du docteur Moreau_ (1896), traduit de l'anglais par Henry-D. Davray, Gallimard, coll. Folio, 2014, p. 69.\n\n. _Ibid_.\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , pp. 50-51.\n\n. _Ibid_., p. 53.\n\n. _Ibid_.\n\n. _Ibid_., p. 54.\n\n. _Ibid_., p. 60.\n\n. _Ibid_., p. 61.\n\n. _Ibid_., p. 18.\n\n. _Ibid_., p. 59.\n\nL'ESPRIT GUERRIER\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit_., p. 69.\n\n. _Ibid_.\n\n. _Ibid_., p. 70.\n\n. _Ibid_., p. 64.\n\n. _Ibid._\n\n. _Ibid_., p. 65-66.\n\n. _Ibid_., p. 66.\n\n. _Ibid_., p. 46.\n\n. _Ibid_., p. 86.\n\n. H. G. Wells, _La Russie dans l'ombre_ (1920), traduit de l'anglais par Bernard Loing, A. M. M\u00e9taili\u00e9, 1985, chap. III, p. 71.\n\n. _Ibid_., p. 70.\n\n. _Ibid_., p. 71.\n\n. H. G. Wells, _La Flamme immortelle_ (1919), traduit de l'anglais par M. Butts, Payot et Cie, 1920, pp. 70-71.\n\n. H. G. Wells, _L'Histoire de M. Polly_ (1910), traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, Gallimard, coll. Folio, 1978.\n\n. H. G. Wells, _Une tentative d'autobiographie, op. cit.,_ p. 493.\n\n. _Ibid_., p. 78.\n\n. _La Burlesque \u00c9quip\u00e9e du cycliste_ (1896), traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, Gallimard, coll. Folio, 1984, p. 19.\n\n. _Ibid_., p. 9.\n\n. _Ibid_.\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 32.\n\n. H. G. Wells, _La Machine \u00e0 explorer le temps_ (1895), traduit de l'anglais par Henry-D. Davray, Gallimard, coll. Folio, 1975, p. 98.\n\n. H. G. Wells, _The Fate of Homo Sapiens : An Unemotional Statement of The Things That Are Happening To Him Now And of The Immediate Possibilities Confronting Him,_ Secker and Warburg, 1939, p. 5, cit\u00e9 par Jean-Pierre Vernier, _H. G. Wells et son temps_ , Publications de l'universit\u00e9 de Rouen, coll. \u00c9tudes anglaises, 1971, p. 37.\n\n. Victor Hugo, _Le Promontoire du songe_ , Gallimard, coll. L'Imaginaire, 2012, p. 20.\n\n. _Ibid_.\n\n. _Ibid_., p. 21.\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 82.\n\n. _Ibid_.\n\n. _Ibid_., p. 83.\n\n. _Ibid_.\n\n. _Ibid_., p. 94.\n\n. _Ibid_., p. 95.\n\n. _Ibid_.\n\n. _Ibid_., p. 101.\n\n. _Ibid_., p. 102.\n\n_Ibid_., p. 102.\n\n. _Ibid_., p. 106.\n\n. _Ibid_., p. 107.\n\n. _Ibid_., p. 108.\n\n. H. G. Wells, _The_ _Anatomy of Frustration_ , The Cresset Press, 1936.\n\n. H. G. Wells, _Une tentative d'autobiographie, op. cit_., p. 102.\n\n. _Ibid_., p. 103.\n\n\u00c0 L'\u00c9COLE\n\n. H. G. Wells, _Une tentative d'autobiographie,_ _op. cit._ , p. 114.\n\n. _Ibid_.\n\n. _Ibid_., p. 115.\n\n. Platon, _La R\u00e9publique_ , traduit par Georges Leroux, GF, 2002, Livre VI, p. 316.\n\n. _Ibid_., p. 317.\n\n. _Ibid._ , p. 318.\n\n. H. G. Wells, _Une tentative d'autobiographie,_ _op. cit._ , p. 116.\n\n. _Ibid_.\n\n. _Ibid_., p. 121.\n\n. _Ibid_., p. 118.\n\n. _Ibid_.\n\n. _Ibid_.\n\n. _Ibid_., p. 182.\n\n. _Ibid_., p. 129.\n\n. _Ibid_., p. 131.\n\n. _Ibid_.\n\n. _Ibid_., p. 154.\n\n. _Ibid_.\n\n. Jean-Pierre Vernier, _H. G. Wells et son temps_ , _op. cit._ , p. 85.\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 134.\n\n. _Ibid_., p. 136.\n\n. _Ibid_., p. 154.\n\n. _Ibid_., p. 147.\n\n. _Ibid_., p. 149.\n\n. _Ibid_., p. 150.\n\n. _Ibid_., p. 151.\n\n. _Ibid_., p. 190.\n\n. _Ibid_., p. 191.\n\n. _Ibid_., p. 189.\n\n. _Ibid_., p. 184.\n\n. _Ibid_., p. 185.\n\nINVINCIBLE\n\n. H. G. Wells, _Une tentative d'autobiographie,_ _op. cit._ , p. 194.\n\n. _Ibid_., p. 197.\n\n. _Ibid_.\n\n. _Ibid_.\n\n. Cit\u00e9 par Geoffrey West, _H. G. Wells : A Sketch for A Portrait_ , Londres, Howe, 1930 ; New York, Norton, 1930, p. 52, repris par Jean-Pierre Vernier, _H. G. Wells et son temps_ , _op. cit._ , p. 64.\n\n. Ivor Brown, _H. G. Wells_ , Nisbet, 1923, p. 35, cit\u00e9 par Jean-Pierre Vernier, _H. G. Wells et son temps_ , _op. cit._ , p. 145.\n\n. _Une tentative d'autobiographie_ , _op. cit._ , p. 204.\n\n. H. G. Wells, _Tono Bungay_ (1909), traduit de l'anglais par \u00c9douard Guyot, Payot, 1929, Livre II, chap. V, pp. 179-181.\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 237.\n\n. _Ibid_., pp. 237-238.\n\n. _Ibid_., p. 246.\n\nLA MACHINE SE MET EN ROUTE\n\n. H. G. Wells, _Une tentative d'autobiographie,_ _op. cit._ , p. 282.\n\n. H. G. Wells, _La Machine \u00e0 explorer le temps_ , _op. cit._ , p. 15.\n\n. _Ibid_., p. 158.\n\n. _Ibid_., pp. 152-153.\n\n. H. G. Wells, _Literary Criticism_ , Patrick Parrinder et Robert Philmus, \u00e9d., Brighton, Harvester Press, 1980, Totowa, NJ, Barnes and Noble, 1980, pp. 190-191. (Traduction de l'auteur.)\n\n. Extrait du _Times_ , 17 juin 1896, cit\u00e9 par Geoffrey H. Wells, _The Work of H. G. Wells_ , p. 7, repris par Jean-Pierre Vernier, _H. G. Wells et son temps_ , _op. cit._ , p. 130.\n\n. H. G. Wells, _L'\u00cele du docteur Moreau_ , _op. cit._ , p. 21.\n\n. _Ibid_., p. 115.\n\n. _Ibid_., pp. 209-210.\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 354.\n\n. _Ibid_., p. 358.\n\n. _Ibid_., p. 375.\n\n. Michael Sherborne, _H. G. Wells, Another Kind of Life_ , Peter Owen, 2012, chap. 7, p. 125. (Traduction de l'auteur.)\n\n. H. G. Wells, _La Guerre des mondes_ (1898), traduit de l'anglais par Henry-D. Davray, Gallimard, coll. Folio, 2005, p. 27.\n\n. _Ibid_., p. 48.\n\n. _Ibid_., p. 308.\n\nANTICIPATIONS\n\n. H. G. Wells, _Les Premiers Hommes dans la Lune_ , traduit de l'anglais par Henry-D. Davray, Mercure de France, 1901.\n\n. Propos recueillis dans _T.P.'s Weekly_ , 9 octobre 1903, par Ribert H. Sherard, dans _Textes oubli\u00e9s de Jules Verne_ , \u00e9dit\u00e9s par F. Lacassin, 10-18, 1977 ; dans _Les Chefs-d'\u0153uvre de H. G. Wells_ , pr\u00e9face de Francis Lacassin, Omnibus \/ SF, 2007.\n\n. Lettre du 15 juin 1890, adress\u00e9e \u00e0 Arnold Bennett, dans _Arnold Bennet and H. G. Wells_ , Harris Wilson, \u00e9d., Londres, Hart Davis, 1960 ; Urbana, University of Illinois Press, 1960, p. 45. (Traduction de l'auteur.)\n\n. H. G. Wells, _Anticipations_ , _ou De l'influence du progr\u00e8s m\u00e9canique et technique sur la vie et la pens\u00e9e humaines_ , traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, Mercure de France, 1901, chap. I, p. 5.\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 392.\n\n. H. G. Wells, \u00ab La d\u00e9couverte de l'avenir \u00bb, _Anticipations_ , _op. cit._ , p. 6.\n\n. _Ibid_., pp. 20-24.\n\n. _Ibid_., p. 19.\n\n. _Ibid_., p. 21.\n\n. _Ibid.,_ p. 69.\n\n. _Ibid_., p. 242.\n\n. _Ibid_., p. 199.\n\n. _Ibid_., p. 171.\n\n. _Ibid_.\n\n. _Ibid_., p. 280.\n\n. _Ibid_., p. 190.\n\n. _Ibid_., p. 308.\n\n. _Ibid_., p. 362.\n\n. _Ibid_., pp. 362-363.\n\n. _Ibid_., p. 130.\n\n. H. G. Wells, _Une tentative d'autobiographie,_ _op. cit._ , p. 472.\n\n. _Ibid._ , p. 475.\n\n. _Ibid._ , p. 404.\n\n. H. G. Wells, _Ann Veronica_ (1909), traduit de l'anglais par Joseph Dobrinsky, Le Rocher, Jean-Paul Bertrand \u00c9diteur, coll. Alph\u00e9e, 1991, chap. I, p. 13.\n\n. H. G. Wells, _H. G. Wells in Love_ , _op. cit._ , p. 74.\n\n. _Ibid._ , p. 75.\n\n. _Ibid._ , p. 81.\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 290.\n\n. H. G. Wells, _Ann Veronica_ , _op. cit._ , p. 419.\n\n. _Ibid_., p. 425.\n\n. Geoffrey West, _H. G. Wells : A Sketch for A Portrait_ , _op. cit._ , p. 184, cit\u00e9 par Jean-Pierre Vernier, _H. G. Wells et son temps_ , _op. cit._ , p. 306.\n\n. H. G. Wells, _Une utopie moderne_ (1905), traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, Mercure de France, 1907, p. 207.\n\n. _Ibid._ , pp. 208-209.\n\n. David Lodge relate cet \u00e9pisode dans son roman _Un homme de temp\u00e9rament_ (traduit de l'anglais par Martine Aubert, Rivages, 2012), et cite la phrase d'un chanoine de l'\u00e9poque d\u00e9clarant : \u00ab Je pr\u00e9f\u00e9rerais envoyer ma fille dans une maison contamin\u00e9e par la dipht\u00e9rie ou la typho\u00efde que mettre ce livre entre ses mains \u00bb, p. 424.\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 292.\n\n. Cit\u00e9 par David Lodge dans _Un homme de temp\u00e9rament_ , _op. cit._ , p. 426.\n\n. H. G. Wells, _Une utopie moderne_ , _op. cit._ , p. 199.\n\n. _Ibid_., p. 202.\n\n. _Ibid_., p. 158.\n\n. _Ibid_., p. 403.\n\n. _Ibid_., p. 225.\n\n. Cette interview est cit\u00e9e par Bertrand Russell dans _Portraits from Memory_ , James Press, 2007, pp. 77-78, et reprise par Jean-Pierre Vernier dans _H. G. Wells et son temps_ , _op. cit._ , p. 303 : \u00ab Wells fut attaqu\u00e9 dans la presse non pas \u00e0 cause de son pacifisme, mais parce qu'il s'\u00e9tait fait le champion de l'amour libre. Il r\u00e9pondit avec quelque humeur qu'il ne s'\u00e9tait point fait le champion de l'amour libre, mais avait simplement imagin\u00e9 les effets possibles de nouveaux \u00e9l\u00e9ments dans l'atmosph\u00e8re, sans pr\u00e9ciser s'il consid\u00e9rait ces effets comme bons ou mauvais. Ceci me semblait malhonn\u00eate, et je lui demandai : \"Pourquoi vous \u00eates-vous d'abord fait le champion de l'amour libre et l'avez-vous ni\u00e9 ensuite ?\" Il r\u00e9pliqua qu'il n'avait pas encore mis assez d'argent de c\u00f4t\u00e9 sur ses droits d'auteur pour vivre des int\u00e9r\u00eats et qu'il n'avait point l'intention de se faire ouvertement le champion de l'amour libre avant d'avoir atteint ce but. \u00bb\n\n. H. G. Wells, _H. G. Wells in Love, op. cit._ , p. 77.\n\n. _Ibid._ , lettre du 25 ao\u00fbt 1939, p. 86.\n\nLA POURSUITE DE L'AMOUR\n\n. H. G. Wells, _H. G. Wells in Love_ , pp. 53-54.\n\n. William Shakespeare, _Les_ _Sonnets_ , sonnet 116, traduit de l'anglais par Yves Bonnefoy, \u00ab Po\u00e9sie \/ Gallimard \u00bb, 2007, p. 274.\n\n. _Ibid_.\n\n. David Lodge rapporte cette anecdote extraite de _Aspects of A Life_ , d'Anthony West (1984), dans son essai _Des vies \u00e0 \u00e9crire_ , traduit de l'anglais par Martine Aubert, Payot et Rivages, 2014, p. 236.\n\n. H. G. Wells \u00e9voque la disparition de sa m\u00e8re en ces termes dans _Tono Bungay_ , _op. cit._ , p. 93.\n\n. H. G. Wells, _H. G. Wells in Love,_ _op. cit._ , p. 53.\n\n. Cit\u00e9 par Michael Sherborne, _H. G. Wells, Another Kind of Life_ , _op. cit._ , pp. 110-111. (Traduction de l'auteur.)\n\n. H. G. Wells, _H. G. Wells in Love_ , _op. cit._ , p. 53.\n\n. H. G. Wells, _M. Britling commence \u00e0 y voir clair_ (1916), sans mention de traducteur, Librairie Payot et Cie, 1918, p. 116.\n\n. _Ibid._\n\n. _Ibid_., p. 117.\n\n. H. G. Wells, _H. G. Wells in Love_ , _op. cit._ , p. 87.\n\n. _Ibid_., pp. 90-91.\n\n. Extrait de l'article de Rebecca West, publi\u00e9 dans _The Freewoman_ le 19 septembre 1912, cit\u00e9 par J. R. Hammond dans _H. G. Wells and Rebecca West_ , Londres et New York, Harvester Wheatsheaf, 1991, p. 50.\n\n. H. G. Wells, _H. G. Wells in Love_ , _op. cit._ , p. 95.\n\n. J. R. Hammond, _H. G. Wells and Rebecca West_ , _op. cit._ , p. 51. (Traduction de l'auteur.)\n\n. \u00ab J'ai une objection insurmontable concernant toute acad\u00e9mie militaire ou artistique, toute hi\u00e9rarchie, toute suggestion de contr\u00f4les ou de crit\u00e8res fixes pour ces choses... Notre monde, je veux dire le monde du travail de la cr\u00e9ation et de la repr\u00e9sentation que nous accomplissons, se porte mieux, j'en suis convaincu, dans l'anarchie \u00bb, H. G. Wells cit\u00e9 par J. R. Hammond, _H. G. Wells and Rebecca West_ , _op. cit._ , p. 53. (Traduction de l'auteur.)\n\n. Citation extraite de l'ouvrage de J. R. Hammond, _H. G. Wells and Rebecca West_ , _op. cit._ , p. 58 ; traduit de l'anglais par Martine Aubert dans le roman de David Lodge, _Un homme de temp\u00e9rament_ , _op. cit._ , p. 556.\n\n. H. G. Wells, _H. G. Wells in Love, op. cit._ , p. 96.\n\n. J. R. Hammond, _H. G. Wells and Rebecca West_ , _op. cit._ , p. 92. (Traduction de l'auteur.)\n\n. William Shakespeare, _Les Sonnets_ , sonnet 116, _op. cit._ , p. 274.\n\nFACE \u00c0 LA GUERRE\n\n. H. G. Wells, _Jeanne et Pierre_ (1918), sans mention de traducteur, Payot et Cie, 1922, chap. II, t. II, pp. 152-153.\n\n. _Ibid._\n\n. _Ibid._\n\n. _Ibid._\n\n. _Ibid_.\n\n. H. G. Wells, _La Destruction lib\u00e9ratrice_ (1914), traduit de l'anglais par Patrick Delperdange, Grama, Bruxelles, 1995, p. 91.\n\n. _Ibid_., p. 35.\n\n. _Ibid._ , pp. 19-20.\n\n. _Ibid._ , p. 20.\n\n. _Ibid._ , pr\u00e9face, 1921, p. 14.\n\n. _Ibid._\n\n. H. G. Wells, _M. Britling commence \u00e0 y voir clair_ , _op. cit._ , p. 364.\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 437.\n\n. _Ibid_., p. 433.\n\n. H. G. Wells, _M. Britling commence \u00e0 y voir clair_ , _op. cit._ , pp. 16-17.\n\n. _Ibid_., p. 97.\n\n. _Ibid_., p. 140.\n\n. _Ibid_., pp. 389-390.\n\n. Lettre dat\u00e9e de d\u00e9cembre 1916, adress\u00e9e \u00e0 H. G. Wells, cit\u00e9e par Nina Berberova dans _Histoire de la baronne Boudberg_ , traduit du russe par Michel Niqueux, Actes Sud, 1988, pp. 133-134.\n\n. H. G. Wells, _M. Britling commence \u00e0 y voir clair_ , _op. cit._ , p. 467.\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 439.\n\n. _Ibid_., pp. 440-441.\n\nUN MONDE MEILLEUR\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 449.\n\n. _Ibid._ , p. 452.\n\n. _Ibid._ , p. 16.\n\n. H. G. Wells, _La Russie dans l'ombre_ , _op. cit._ , p. 69.\n\n. _Ibid_.\n\n. R. H. Bruce Lockhart, _M\u00e9moires d'un agent secret britannique en Russie_ (1912-1918), traduit de l'anglais par L. Thomas, Payot, Paris, 1933, p. 256.\n\n. H. G. Wells, _La Russie dans l'ombre_ , _op. cit._ , p. 48.\n\n. _Ibid_., p. 40.\n\n. _Ibid_., p. 53.\n\n. _Ibid_., p. 95.\n\n. _Ibid._ , p. 99.\n\n. _Ibid_., p. 102.\n\n. \u00ab J'\u00e9tais tomb\u00e9 amoureux, plus compl\u00e8tement et sinc\u00e8rement que je ne l'avais \u00e9t\u00e9 auparavant, en Russie, en 1920 \u00bb, H. G. Wells, _H. G. Wells in Love, op. cit._ , p. 109.\n\n. _Ibid.,_ p. 166.\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 484.\n\nD\u00c9GRISEMENT\n\n. H. G. Wells, _Tono Bungay_ , _op. cit._ , p. 144.\n\n. \u00ab Nous sommes fondamentalement incompatibles \u00bb, H. G. Wells, _H. G. Wells in Love, op. cit._ , p. 108.\n\n. \u00ab J'\u00e9tais toujours proche d'aimer Rebecca pendant notre liaison, tandis qu'elle m'a souvent beaucoup aim\u00e9 \u00bb, _ibid._ , p. 161.\n\n. \u00ab Elle voulait me donner toute sa vie. Elle ne voulait rien d'autre qu'\u00eatre \u00e0 mon service \u00bb, _ibid._ , __ p. 125.\n\n. \u00ab \u00c0 H. G. Wells, tu nous as impos\u00e9 tes songes \u00bb, _ibid._ , __ p. 123.\n\n. H. G. Wells, _Le Monde de William_ _Clissold_ , dans H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , pp. 464-465.\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 463.\n\n. Titre de la neuvi\u00e8me nouvelle du recueil _Effrois et fantasmagories,_ traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, Gallimard, coll. L'Imaginaire, 1984.\n\n. _Ibid_., p. 205.\n\n. Odette est d\u00e9crite comme suit : \u00ab une personne mauvaise et d\u00e9testable ; vaniteuse, bruyante et faiblement scandaleuse \u00bb, H. G. Wells, _H. G. Wells in Love_ , _op. cit._ , p. 116.\n\n. Lettre cit\u00e9e par J. R. Hammond, _H. G. Wells and Rebecca West_ , _op. cit._ , p. 62. (Traduction de l'auteur.)\n\n. \u00ab Elle a stabilis\u00e9 ma vie. Elle lui a donn\u00e9 une maison et une dignit\u00e9 \u00bb, M. Sherborne, _H. G. Wells, Another Kind of Life_ , _op. cit._ , p. 280. (Traduction de l'auteur.)\n\nLE CONSPIRATEUR AMOUREUX\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 8.\n\n. _Ibid_., pp. 8-9.\n\n. _Ibid_., p. 9.\n\n. _Ibid_.\n\n. _Ibid_.\n\n. Extrait du discours de H. G. Wells au Reischtag, cit\u00e9 par Antonina Vallentin, _H. G. Wells, ou La Conspiration au grand jour_ , Stock, 1952, p. 296.\n\n. _Ibid_.\n\n. H. G. Wells, _H. G. Wells in Love_ , _op. cit._ , p. 167.\n\n. Anecdote relat\u00e9e par M. Sherborne, _H. G. Wells, Another Kind of Life_ , _op. cit._ , p. 284.\n\n. H. G. Wells, _H. G. Wells in Love_ , _op. cit._ , p. 169.\n\n. _Ibid_., p. 143.\n\n. \u00ab Je lui ai dit de se d\u00e9p\u00eacher parce que j'\u00e9tais en train d'\u00e9crire mon autobiographie et que je n'avais pas l'intention de laisser grand-chose d'intime dans l'ombre \u00bb, H. G. Wells, _H. G. Wells in Love_ , _op. cit._ , p. 157.\n\n. Ted Morgan, _Maugham, A Biography_ , New York, 1980, p. 382. (Traduction de l'auteur.)\n\n. H. G. Wells, _H. G. Wells in Love_ , _op. cit._ , p. 170.\n\n. H. G. Wells, \u00ab Pr\u00e9face aux romans scientifiques \u00bb, traduit par Nelly St\u00e9phane, _Europe_ , 64e ann\u00e9e, no 681-682, janvier-f\u00e9vrier 1986, p. 44.\n\n. _Ibid_., p. 48.\n\n. _Ibid_.\n\n. H. G. Wells, _Tono Bungay_ , _op. cit._ , p. 162.\n\n. H. G. Wells, _H. G. Wells in Love_ , _op. cit._ , p. 158.\n\nUN CERVEAU PAS SI ORDINAIRE\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 185.\n\n. _Ibid_., p. 19.\n\n. H. G. Wells, pr\u00e9face \u00e0 _La Guerre dans les airs_ (1908), traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, Gallimard, coll. Folio, 1984, cit\u00e9 par David Lodge dans _Un homme de temp\u00e9rament_ , _op. cit._ , p. 17.\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 17.\n\n. _Ibid_., p. 19.\n\n. H. G. Wells, _Miss Waters_ (1902), traduit de l'anglais par Henry-D. Davray et B. Kozakiewicz, Gallimard, coll. Folio, 1984, p. 19.\n\n. H. G. Wells, _L'Homme invisible_ (1897), traduit de l'anglais par Achille Laurent, Le Livre de Poche, 2012.\n\n. H. G. Wells, \u00ab L'histoire de feu M. Elvesham \u00bb, _Effrois et fantasmagories, op. cit._\n\n. H. G. Wells, _Miss Waters, op. cit._\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 197.\n\n. _Ibid_., p. 12.\n\n. _Ibid_., p. 13.\n\n. _Ibid_.\n\n. _Ibid_., p. 12.\n\n. _Ibid_., p. 10.\n\n. _Ibid._ , p. 11.\n\n. Percy Bysshe Shelley, _Hymne pour un monde nouveau_ , _\u0152uvres po\u00e9tiques compl\u00e8tes_ , traduit de l'anglais par F\u00e9lix Rabbe, Sandre, t. II, _Hellas_ (1821), p. 328.\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 490.\n\n. _Ibid_., p. 504.\n\n. H. G. Wells, _La Russie dans l'ombre_ , _op. cit._ , p. 172.\n\n. _Ibid.,_ p. 178.\n\n. _Ibid_., p. 187.\n\n. _Ibid_., p. 181.\n\n. Le r\u00eave est d\u00e9crit avec pr\u00e9cision dans _H. G. Wells in Love, op. cit._ , p. 184.\n\n. _Ibid_., p. 197.\n\nENTRE-TEMPS\n\n. _H. G. Wells in Love_ , _op. cit._ , p. 219.\n\n. _Ibid._\n\n. _Ibid._ , p. 211.\n\n. Cit\u00e9 par Nina Berberova, _Histoire de la baronne Boudberg_ , _op. cit._ , p. 261.\n\n. M. Sherborne, _H. G. Wells, Another Kind of Life_ , _op. cit._ , p. 211. (Traduction de l'auteur.)\n\n. Cit\u00e9 par Antonina Vallentin, _H. G. Wells ou La Conspiration au grand jour_ , _op. cit._ , p. 335.\n\n. H. G. Wells, _La Guerre des mondes_ , _op. cit._ , p. 23.\n\nC'EST LA FIN\n\n. Cette anecdote est rapport\u00e9e par M. Sherborne, _H. G. Wells, Another Kind of Life_ , _op. cit._ , p. 323.\n\n. \u00ab Je pense que si j'\u00e9cris encore je devrais \u00e9crire un livre intitul\u00e9 Reconstruction, une sorte d'assemblage et de critique des espoirs et des projets. Il faut que j'y pense \u00bb, H. G. Wells, _H. G. Wells in Love_ , _op. cit._ , p. 226.\n\n. H. G. Wells, _Une tentative d'autobiographie,_ _op. cit._ , pp. 22-23.\n\n. Cit\u00e9 par Norman et Jeanne MacKenzie, _H. G. Wells, The Time Traveller_ , Littlehampton Book Services Ltd, 1973, p. 431 ; cit\u00e9 par M. Sherborne, _H. G. Wells, Another Kind of Life_ , _op. cit._ , p. 333.\n\n. H. G. Wells, _La_ _Guerre des mondes_ , _op. cit._ , p. 128.\n\n. \u00ab Entre le chagrin et le n\u00e9ant, je choisis le chagrin \u00bb, W. Faulkner, _Si je t'oublie, J\u00e9rusalem_ , traduit de l'am\u00e9ricain par Maurice-Edgar Coindreau, Gallimard, coll. L'Imaginaire, 2001, p. 337.\n\n. H. G. Wells, _\u00c0 fin de course_ (1945), traduit de l'anglais par Louis Chantem\u00e8le, La Table ronde, 1946, p. 16.\n\n. _Ibid_., p. 23.\n\n. _Ibid_.\n\n. Anthony West, _Principles and Persuasions_ , New York, Harcourt, Brace and Company, 1957, p. 6. (Traduit par Jean-Pierre Vernier dans _H. G. Wells et son temps_ , _op. cit._ , p. 518.)\n\n. _Ibid_., p. 33.\n\n. _Ibid_., p. 80.\n\n. H. G. Wells, _Une tentative d'autobiographie_ , _op. cit._ , p. 128.\n\n. H. G. Wells, \u00ab Une vision du Jugement dernier \u00bb, _Effrois et fantasmagories_ , _op. cit._ , p. 227.\n\n. _Ibid._ , p. 240.\n\n. H. G. Wells, _Tono Bungay_ , _op. cit._ , p. 581.\nREMERCIEMENTS\n\nJe remercie, pour leur soutien et leur \u00e9ternelle bienveillance, Marie, Romane, Anne-Sophie, Ollivier, Martine, Ali et V\u00e9ronique.\nCouverture : H. G. Wells en 1915. Photo \u00a9 George C. Beresford \/ Beresford \/ Getty Images (d\u00e9tail).   \nExtrait de Things to Come, r\u00e9alis\u00e9 par William Cameron Menzies, 1936, inspir\u00e9 de l'ouvrage de Wells The Shape of Things to Come.\n\n\u00a9 \u00c9ditions Gallimard, 2016.\n\n\u00c9ditions Gallimard  \n5 rue Gaston-Gallimard  \n75328 Paris  \n<http:\/\/www.gallimard.fr>  \n _\u00a9 \u00c9ditions Gallimard, 2016._\nH. G. Wells\n\npar Laura El Makki\n\n\u25a0 \u00ab Sans r\u00eaves, la vie humaine serait insupportable pour la plus grande multitude des \u00eatres humains. \u00bb\n\nHerbert George Wells (1866-1946) est un homme press\u00e9 de vivre et d'aimer. Lecteur de Platon et de Darwin, ce fils de commer\u00e7ants anglais, d'abord professeur de biologie, abandonne vite la dissection des grenouilles pour laisser libre cours \u00e0 son imagination d\u00e9brid\u00e9e. Tandis que l'\u00e8re victorienne agonise, il donne naissance \u00e0 une quatri\u00e8me dimension ( _La Machine \u00e0 explorer le temps_ ), fait d\u00e9barquer les Martiens sur Terre ( _La Guerre des mondes_ ) et envisage les cons\u00e9quences de la manipulation g\u00e9n\u00e9tique ( _L'\u00cele du docteur Moreau_ ). \u00c9crivain prolifique, courtis\u00e9 par les hommes politiques de son temps et par les femmes qu'il collectionne sans compter, Wells agace Jules Verne quand il se dit \u00ab proph\u00e8te \u00bb. C'est pourtant lui qui, avant m\u00eame le premier conflit mondial, anticipe l'arme atomique, ou encore, bien avant Internet, songe \u00e0 la construction d'un syst\u00e8me global de connaissances, accessible \u00e0 toute l'humanit\u00e9.\nCette \u00e9dition \u00e9lectronique du livre _H. G. Wells_ de Laura El Makki  \na \u00e9t\u00e9 r\u00e9alis\u00e9e le 26 janvier 2016 par les \u00c9ditions Gallimard.\n\nElle repose sur l'\u00e9dition papier du m\u00eame ouvrage  \n(ISBN : 9782070462308 - Num\u00e9ro d'\u00e9dition : 274316).\n\nCode Sodis : N66435 - ISBN : 9782072574474.\n\nNum\u00e9ro d'\u00e9dition : 274317.\n\n_Ce document num\u00e9rique a \u00e9t\u00e9 r\u00e9alis\u00e9 parAps-Chromostyle_\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Nick Hodson of London, England\n\n\n\n\nReal Gold, by George Manville Fenn.\n\n________________________________________________________________________\n\n________________________________________________________________________\nREAL GOLD, BY GEORGE MANVILLE FENN.\n\n\n\nCHAPTER ONE.\n\nA CHAT IN A BOAT.\n\n\"Bother the old fish!\"\n\n\"Yes; they won't bite.\"\n\n\"It's no good, Perry; they are having their siesta.  Let's get in the\nshade and have one too.\"\n\n\"What! in the middle of the day--go to sleep?  No, thank you.  I'm not a\nforeigner.\"\n\n\"More am I; but you come and live out here for a bit, and you'll be\nready enough to do as the Romans--I mean the Spaniards--do.\"\n\n\"Not I, Cyril, and I don't believe fish do go to sleep.\"\n\n\"What?  Why, I've seen them lie in shoals here, perfectly still; basking\nin the hot sunshine, fast asleep.\"\n\n\"With their eyes shut?\"\n\n\"Gammon!  Fish can't shut their eyes.\"\n\n\"Then they can't go to sleep.--My! it is hot.  I shan't fish any more.\"\n\nTwo boys sitting in a boat half a mile from the shore, and sheltered by\na ridge of rocks from the tremendous swell of the vast Pacific Ocean,\nwhich to north and south curled over in great glistening billows upon\nthe sand--in the former instance, to scoop it out, carry it back, and\nthen throw it up farther away; in the latter, to strike upon sheer rocks\nand fly up in silver spray with a low deep sound as of muttered thunder.\nAway to the west there was the great plain of smooth damasked silver,\nlost at last in a faint haze, and all so bright that the eyes ached and\nwere dazzled by its sheen.  To the east, the bright-looking port of San\nGeronimo, with a few ships, and half-a-dozen long, black, red-funnelled\nscrew-steamers at anchor; beyond them wharves and warehouses, and again\nbeyond these the houses of the little town, with a few scattered white\nvillas rising high on terrace and shelf of the steep cliffs.  The place\nlooked bright and attractive seen from the distance, but dry and barren.\nNothing green rested and refreshed the eye.  No trees, no verdant <DW72>\nof lawn or field; nothing but sand in front, glittering rock behind.\nEverything suggested its being a region where no rain fell.\n\nBut, all the same, it had its beauty.  More, its grandeur, for\napparently close at hand, though miles away in the clear distance, rose\nthe great Sierra--the mighty range of mountains, next to the Himalayas\nthe highest in the world--and seeming to rise suddenly like a gigantic\nwall right up into the deep blue sky, cloudless, and dazzling with the\nice and snow.\n\nThe two boys, both of them, though fair by nature, tanned now of a warm\nreddish brown, were of about the same age, and nearly the same physique;\nand as now they twisted the stout lines they had been holding round the\nthole pins of the boat, which softly rose and fell with a pleasant\nlulling motion, the first who had spoken unfastened the neck-button of\nhis shirt.\n\n\"Hullo!  Going to bathe?\"\n\n\"Bathe!  No, thankye.  I should wake up the sharks: they'd bite then.\"\n\n\"Ugh!\"\n\n\"Yes, you may shudder.  They grow fine about here.  Why, before I'd made\na dozen strokes, you'd hear me squeak, and see me go down and never come\nup again.\"\n\n\"How horrid!  You don't mean it, though, do you?\"\n\n\"Yes, it's true enough.  I'm going to have a nap.\"\n\nAs the boy spoke, he lay back in the stern of the boat, and placed his\nbroad Panama hat over his face.\n\n\"I say, Perry, old chap!\" he continued, with his voice sounding whistly\nthrough the closely-woven hat.\n\n\"What?\"\n\n\"If you smell me burning, wake me up.\"\n\n\"All right,\" said the lad addressed as Perry; and resting his elbows on\nhis knees, he sat gazing up at the huge towering mountain nearest at\nhand for a few minutes, then:\n\n\"Cil!\"\n\n\"Hullo!\" drowsily.\n\n\"Don't go to sleep, old chap; I want to talk to you.\"\n\n\"I can't go to sleep if you talk.  What is it?\"\n\n\"I say, how rum it seems for it to be boiling hot down here, and all\nthat ice and snow to be up there.  Look.\"\n\n\"Yes,\" said Cyril, \"'tis its nature to.  I don't want to look.  Seen it\nbefore.\"\n\n\"But how far is it up to where the snow is--a thousand feet?\"\n\n\"What?\" cried Cyril, starting up into a sitting position, with his hat\nfalling off.\n\n\"I said how far is it up to where the snow is?\"\n\n\"I know you did,\" cried the boy, laughing, \"and you said, was it a\nthousand feet?\"\n\n\"Yes, and it was stupid of me.  It must be twice as high.\"\n\n\"Perry Campion, you are a greenhorn.  I say: no offence meant; but my\ndear, fresh, innocent, young friend, that snow is three miles high.\"\n\n\"Well, I know that, of course.  It must be much more to where it is.\"\n\n\"Sixty or seventy,\" said Cyril, whose drowsiness had departed, and who\nwas now all life and eagerness.  \"The air's so clear here that it's\nhorribly deceiving.  But I didn't mean that: I meant that the snow's\nquite three miles straight up perpendicular in the air.\"\n\n\"Nonsense!\"\n\n\"But I tell you it is.  If you were to rise straight up in a balloon\nfrom here, you'd have to go up three miles to get on a level with the\nsnow.\"\n\nPerry Campion looked fixedly at his companion, but there was no\nflinching.\n\n\"I'm not gammoning you,\" said Cyril earnestly.  \"Things are so much\nbigger out here than they look.\"\n\n\"Then how big--how high is that mountain?\" said Perry.\n\n\"Nearly four miles.\"\n\n\"But it seems to be impossible.\"\n\n\"It isn't, though,\" said Cyril.  \"That one's over twenty thousand feet\nhigh, and father has seen much bigger ones up to the north.  I say,\nsquire, you've got some climbing to do.  You won't hop over those hills\nvery easily.\"\n\n\"No,\" said Perry thoughtfully.  \"It will be a climb.\"\n\n\"I say: whereabouts are you going?\"\n\n\"I don't know.  Right up in the mountains somewhere.\"\n\n\"But what are you going for?\"\n\n\"I don't know that either.  To travel, I suppose.\"\n\n\"Oh, but the colonel must be going for something,\" cried Cyril.  \"I\nbelieve I know.\"\n\n\"Do you?  What?\"\n\n\"Well, you don't want me to tell you.  I suppose the colonel has told\nyou not to tell anybody.\"\n\n\"No,\" said Perry quickly.  \"He has not told me.  Why do you think he's\ngoing?\"\n\n\"Prospecting.  To search out a good place for a mine.\"\n\nPerry looked at him eagerly.\n\n\"The Andes are full of places where there might be mines.  There's gold,\nand silver, and quicksilver, and precious stones.  Lots of treasures\nnever been found yet.\"\n\n\"Yes, I've heard that there are plenty of minerals,\" said Perry\nthoughtfully.\n\n\"And besides,\" said Cyril, grinning, \"there's all the gold and silver\nthat belonged to the Incas.  The Indians buried it, and they have handed\ndown the secret of the different places to their children.\"\n\n\"Who have dug it up and spent it,\" said Perry.\n\n\"No.  They're too religious.  They dare not.  They keep the secret of\nthe places till the Incas come again to claim their country, and then it\nwill all be dug up, golden wheels, and suns, and flowers, and cups, and\nthings that the Spaniards never found.  That's it; your father's going\nafter the treasures.  But if he is, you'd better look out.\"\n\n\"Why?\"\n\n\"Because if the Indians thought you were after that, they'd kill you in\nno time.\"\n\nPerry looked at him searchingly.\n\n\"Oh, I mean it,\" said Cyril.  \"Father has often talked about it, and he\nsays that the Indians consider it a religious duty to protect the\nhiding-places of these treasures.  There was a man took a party with him\nup into the mountains on purpose to search for them.\"\n\n\"Well?  Did he find anything?\"\n\n\"Don't know.  Nobody ever did know.\"\n\n\"How was that?\"\n\n\"He never came back.  Nor any of his people.\"\n\n\"Why?  What became of them?\"\n\n\"I tell you they went up into the mountains and never came back.  The\nIndians know what became of them.\"\n\n\"But was no search made for them--no examination made of the Indians?\"\ncried Perry, looking aghast.\n\n\"Search!  Where?  Indians!  What Indians?\" said Cyril sharply.  \"You\nforget how big the place is, and what great forests and wilds there are\nover the other side.\"\n\n\"But it sounds so horrible for a party like that to disappear, and no\nmore to be heard of them,\" said Perry.\n\n\"Yes, but the Indians are savages, and, as father said, they think they\nare doing their duty against people who have no right in the country, so\nyour father will have to look out.  I wish I were going with you, all\nthe same.\"\n\n\"You're safer in San Geronimo, if it's as bad as you say,\" cried Perry.\n\n\"Oh, it's bad enough, but I shouldn't mind.\"\n\nThere was silence for a few minutes, during which time both lads sat\ngazing dreamily up at the vast range of mountains before them, with its\nglittering peaks, dark cavernous valleys, and mysterious shades, towards\nwhere the high tablelands lay which had been the seat and home of the\nbarbaric civilisation of the Incas, before ruin and destruction came in\nthe train of the Spanish adventurers who swept the land in search for El\nDorado, the City of Gold.\n\nPerry Campion was the first to break the silence.\n\n\"How long have you been out here, Cyril?--Cil, I say, I shall call you\nCil.\"\n\n\"All right, I don't mind, only it won't be for long.  You go next week,\ndon't you?\"\n\n\"Yes, I suppose so,\" said Perry, glancing again at the mountains.\n\n\"Wish I were going with you.  What did you say?--how long have I been\nout here?  Nearly four years.  Father sent me over to England to be\neducated when I was six, and I was at a big school at Worksop till I was\ntwelve, and then he sent for me to come out here again.\"\n\n\"Weren't you glad?\"\n\n\"Of course.  It was very jolly at school; but school isn't home, is it?\"\n\n\"Of course not.\"\n\n\"Father said I could go on reading with him, and it would brush up his\nclassics, which had grown rusty since he turned merchant.\"\n\n\"Wasn't he always a merchant, then?\"\n\n\"My father?\" cried Cyril.  \"No, he was a captain in the army, and had to\ngive up on account of his health.  The doctors said he was dying.  That\nwas twelve years ago; but he doesn't look like dying now, does he?\"\n\n\"No, he looks wonderfully strong and well.\"\n\n\"Yes.  This place suited him and mother because it was so dry.\"\n\n\"And then he took to being a merchant?\"\n\n\"Yes; and ships off drugs, and minerals, and guano, and bark.\"\n\n\"What! for tanning?\"\n\n\"Tanning!  Ha! ha!  No, no; Peruvian bark, that they make quinine of.\nPhysic for fevers.\"\n\n\"Oh!  I see.\"\n\n\"It's very jolly, and he makes plenty of money; but I do get so tired\nsometimes.  I should like to go to sea, or to travel, or something.  I\nhate being always either at studies or keeping accounts.  I wish I were\ngoing along with you.\"\n\n\"To be killed by the Indians,\" said Perry drily.\n\n\"I should like to catch 'em at it,\" cried Cyril.  \"But I'd risk it.\nWhat an adventure, to go with your father to hunt out the places where\nthe Indians buried the Incas' gold!\"\n\n\"My father did not say he was going in search of that,\" said Perry.\n\n\"No; he's too close.  But that's it, safe enough; you see if it isn't.\nOnly think of it--right up in the grand valleys, where it's almost dark\nat mid-day, and you walk along shelves over the torrents where there\nisn't room for two mules to pass, and there are storms that are quite\nawful sometimes.  I say, I'd give anything to go.\"\n\n\"I wish you were going, Cil.\"\n\n\"You do?\" cried the boy excitedly.  \"I say: do you mean that?\"\n\n\"Of course I do,\" said Perry, looking amused at his companion's\neagerness.  \"We've got on right enough together since we have been\nstaying at your house.\"\n\n\"Got on?  I should think we have,\" cried Cyril.  \"Why, it has been no\nend of a treat to me for you to be at our place.  I can't get on very\nwell with the half-Spanish chaps about here.  They're gentlemen, of\ncourse, with tremendously grand descents from Don this and Don that; but\nthey're not English boys, and you can't make English boys of them.\"\n\n\"Of course not.\"\n\n\"Ah, you may laugh,\" continued Cyril, \"but would you believe it?  I\ntried to get up a cricket club, and took no end of pains to show them\nthe game, and they all laughed at it, and said I must be half mad.\nThat's being Spanish, that is!  It's no wonder their country's left all\nbehind.\"\n\n\"Then the cricket was a failure?\" said Perry.\n\n\"Failure?  It ended in a fight, and I went home and burned the stumps,\nbats, and balls.\"\n\n\"What a pity!\" cried Perry.\n\n\"That's what father said, and it did seem too bad, after he'd had the\ntackle brought out from England on purpose.  I was sorry afterwards; but\nI was so jolly wild then, I couldn't help it.\"\n\n\"How came there to be a fight?\" said Perry after a pause, during which\nhe watched the frank, handsome face of his companion, who was looking at\nthe great peak again.\n\n\"Oh, it was all about nothing.  These Spanish chaps are so cocky and\nbumptious, and ready to take everything as being meant as an insult.\nLittle stupid things, too, which an English boy wouldn't notice.  I was\nbowling one evening, and young Mariniaz was batting.  Of course he'd got\nhis bat and his wits, and he ought to have taken care of himself.  I\nnever thought of hitting him, but I sent in a shooter that would have\ntaken off the bail on his side, and instead of blocking it, he stepped\nright before the wicket.\"\n\n\"What for?\" said Perry.\n\n\"Ah, that's more than I know,\" said Cyril; \"and the next moment he\ncaught it right in the centre of his--er--middle.\"\n\n\"Ha! ha!\" laughed Perry merrily.\n\n\"It knocked all the wind out of him for a minute, and then, as soon as\nhe could speak, he was furious, and said I did it on purpose--in\nSpanish--and I said it was an accident that all people were liable to in\ncricket, and that they ought to be able to defend themselves.  Then he\nsaid he was able to defend himself.\"\n\n\"That meant fighting,\" cried Perry, growing more interested.\n\n\"Of course it did, but I wasn't going to notice it, for the mater said I\nwas to be very careful not to get into any quarrel with the Spanish\nfellows, because they are none too friendly about my father being here.\nThey're jealous because he's a foreigner, when all the time there isn't\na more splendid fellow living than my father,\" cried the boy warmly.\n\"You don't half know him yet.\"\n\n\"Well, what happened then?\" said Perry, as he noted the warm glow in the\nboy's cheeks and the flash of his eyes.\n\n\"Oh, Mariniaz appealed to three or four of the others, and they sided\nwith him, and said that they saw me take a long breath and gather myself\nup and take a deadly aim at his chest, and then hurl the ball with all\nmy might, as if I meant to kill him.\"\n\n\"What rubbish!\" cried Perry.\n\n\"Wasn't it?  You couldn't teach chaps like that to play cricket, could\nyou?\"\n\n\"Of course not.  They didn't want to learn.\"\n\n\"That was it; and they egged Mariniaz on till he called me an English\nbeast, and that upset me and made my tongue loose.\"\n\n\"Well?\"\n\n\"He said he knew from the first I had a spite against him, and had been\ntrying to knock him over with the ball; and, feeling what a lie it was,\nI grew pepper, and told him it wasn't the first time an English ball had\nknocked over a Spaniard, for I got thinking about our old chaps playing\nbowls when the news came about the Armada.\"\n\n\"Yes?\" cried Perry, for Cyril had stopped.\n\n\"Well, then, he turned more yellow than usual, and he gave me a\nbackhanded smack across the face.\"\n\n\"And what did you do?\" cried Perry hotly, for the boy once more stopped.\n\n\"Oh, I went mad for a bit.\"\n\n\"You--went mad?\"\n\n\"I suppose so.  My mother said I must have been mad, so I expect I was.\"\n\n\"But you don't tell me,\" cried Perry impatiently.  \"What did you do?\"\n\n\"I don't know.\"\n\n\"Yes, you do: tell me.\"\n\n\"I can't recollect, and I never could.  I only know I turned very hot\nand saw sparks, and that there was a regular banging about, and\nsometimes I was up and sometimes I was down; and then all at once I was\nstanding there, with Mariniaz lying on the ground crying, and with his\nnose bleeding.  Another chap was sitting holding his handkerchief to one\neye, and two more were being held up by some of the players, who were\ngiving one of them some water to drink, while the other was showing them\na tooth which he held in his fingers.\"\n\n\"Then you'd whacked four of them?\" cried Perry excitedly.\n\n\"I don't know,\" said Cyril, with his face screwed up.  \"I suppose I had\nbeen knocking them about a bit, and they wouldn't fight any more.  They\nall said I was an English savage, and that I ought to be sent out of the\nplace; and then I began to get a bit cooler, and felt sorry I had\nknocked them all about so much.\"\n\n\"I don't see why you should,\" cried Perry.\n\n\"But I did.  It made such an upset.  There was no end of a bother.  My\nmother cried about it when I went home, and said I should never look\nmyself again; and when my father came home and saw me with bits of\nsticking plaster all over my face and knuckles, he was in a regular\npassion, for he had been hearing about it in the town, and had words\nwith the other boys' fathers.  Then he made me tell him all about it\nfrom the beginning, sitting back, looking as fierce and stern as could\nbe, till I had done; and I finished off by saying, `What would you have\ndone if you had been me?'\n\n\"`Just the same as you did, Cil, my boy,' he cried, shaking hands; and\nthen my mother looked astonished, and he sat back in his chair and\nlaughed till he cried.  `Why, mother,' he said, `they tell us that the\nEnglish stock is falling off.  Not very much, eh?  One English to four\nSpanish.'\n\n\"`But it's so terrible,' my mother said.  `Yes,' said my father,\n`fighting is very disgraceful.  No more of it, Cil, my lad; but I've\nmade a mistake: I ought to have made a soldier of you, after all.'  I\nsay, though, Perry, I do wish I were going with you, all the same.\"\n\n\"I tell you what,\" cried Perry; \"I'll ask my father to ask yours to let\nyou go with us.\"\n\n\"You will?\" cried Cyril, making a rush.\n\n\"Mind! we shall have the boat over.\"\n\nIt was a narrow escape, but by sitting down they made the boat right\nitself.\n\n\"Yes, I'll ask him to.  I say, though, it isn't so dangerous as you say,\nis it?\"\n\n\"They say it is, particularly if you are going to hunt for the gold the\nIndians have buried.\"\n\n\"But I don't know that we are.  Would you go, even if it is so\ndangerous.\"\n\n\"Of course I would,\" cried Cyril excitedly.  \"I do so want a change.\nAhoy!  Hurray!  Dinner!\"\n\n\"Eh?  Where?\" cried Perry.\n\n\"Look.  Father's hoisting the flag.\"\n\nHe pointed in the direction of one of the white villas up on the high\ncliff <DW72>, where a union jack was being run up a tall signal staff by\na figure in white, clearly seen in the bright sunshine, while another\nfigure was evidently using a telescope.\n\n\"There's my father watching us,\" said Perry, shading his eyes.\n\n\"Lend a hand here and help to haul up this stone,\" cried Cyril, and\ntogether the boys hauled up the heavy block which served for an anchor.\n\nFive minutes after, they were rowing steadily for the wharf--Incas'\ntreasure, perils from Indians, fights with Spanish boys, and heights of\nsnow peaks forgotten in the one important of all questions to a hungry\nyouth--_Dinner_.\n\n\n\nCHAPTER TWO.\n\nA FAILURE.\n\nDinner was over at Captain Norton's.  Mrs Norton had left the\ndining-room, after begging her son and his visitor not to go out in the\nbroiling heat.  The boy had promised that he would not, and after he had\nsat listening to Colonel Campion's--a keen grey-haired man, thin, wiry\nin the extreme, and giving promise of being extremely active--talk to\nhis father about the preparations for his trip up into the mountains,\nCyril gave Perry a kick under the table, and rose.\n\nTaking the sharp jar upon his shin to mean telegraphy and the sign,\n\"Come on,\" Perry rose as well, and the two boys, forgetful of all\nadvice, went and sat in the dry garden, where every shrub and plant\nseemed to be crying out for water, and looked as if it were being\nprepared for a _hortus siccus_ beloved of botanists, and where the sun\ncame down almost hot enough to fry.\n\nHere the boys had a long discussion about the promise Perry had made in\nthe boat; after which they waited for an opportunity.\n\nMeanwhile, as the two gentlemen sat chatting over their cigarettes,\nCaptain Norton, a frank, genial, soldierly-looking man, said:\n\n\"So you mean to take all the risks?\"\n\n\"Risks!\" said the colonel, turning his keen eyes upon the speaker, as he\nlet the smoke from his cigarette curl up toward the ceiling.  \"You an\nold soldier, and ask that?\"\n\n\"Yes,\" said Captain Norton.  \"I have been here a long time now, and know\nsomething of the country.\"\n\n\"Are the risks so very great, then?\"\n\n\"To an ordinary traveller--no: to a man going with some special object\nor search--yes.\"\n\n\"I did nut say that I was going on a special search,\" said Colonel\nCampion quickly.\n\n\"No, but everything points to it; and as you came to me with letters of\nintroduction from an old friend and brother-officer, I receive you as my\nfriend, and treat you as I would a brother.\"\n\n\"And as the man whom you treat as a brother, I am very reticent, eh?\"\n\n\"Very,\" said Cyril Norton's father; \"and if I try to know why you are\ngoing upon so perilous a journey, it is not from curiosity, but because\nI am eager to save you from running into danger.\"\n\nColonel Campion held out his hand, which was taken, and the two men sat\nfor a few moments gazing in each other's eyes.\n\n\"If I spoke out, Norton, you would immediately do everything you could\nto prevent me from going, instead of helping me; so I am silent, for I\nhave made up my mind to go, and no persuasion would stop me.\"\n\n\"Then you are going on an insane quest of the treasures of gold said to\nhave been buried by the Incas' followers to preserve them from the\nSpaniards.\"\n\n\"Am I?\" said the colonel quietly.\n\n\"I take it for granted that you are; so now, listen.  It will be a very\ndangerous search.  That the gold exists, I do not doubt; and I feel\npretty sure that the Indians have had it handed down from father to son.\nWhere this gold is hidden in the mountains is a sacred trust, which\nthey in their superstitious natures dare not betray.  It means death to\nany one who discovers one of these hoards.\"\n\n\"If found out,\" said the colonel, smoking, with his eyes half shut.\n\n\"He would certainly be found out,\" said the captain, \"and if you persist\nin going, you must run the risk; but I beg of you not to take that boy\nPerry with you, to expose him to these dangers.\"\n\n\"What am I to do with him, then?\"\n\n\"Leave him with us.  He will be happy enough with my boy Cyril; and my\nwife and I will take every care of him.\"\n\n\"Thank you, Norton,\" cried the colonel warmly; \"I am most grateful.  But\nyou are wrong: he would not be happy if he stayed here and I went alone;\nI believe he would prefer running all risks with me.  How odd!\" added\nthe colonel, smiling; \"here he is, to speak for himself.\"\n\nFor at that moment the door was softly opened, and Perry stood there,\nlooking startled.\n\n\"Come in, boy, come in,\" cried the colonel.\n\n\"I--I beg; our pardon; I thought Captain Norton had gone.\"\n\n\"No, and we were just talking about you.\"\n\n\"About me, father?\"\n\n\"Yes; Captain Norton thinks it would be too risky and arduous a journey\nfor you up into the mountains, and he says you are to stay here and make\nyourself happy with Cyril till I come back.\"\n\nThe lad looked delighted.\n\n\"Oh father!\" he cried.  Then, quick as thought, his manner changed.\n\n\"It is very good of Captain Norton,\" he said gravely, \"but I could not\nstop here and let you go alone.\"\n\n\"Don't be hasty, Perry, lad,\" said the captain kindly.  \"There, I'm\ngoing down to the wharf; you and your father chat it over, and we'll\ntalk about it when I return.\"\n\nHe left the room, passing out through the veranda.\n\n\"Well,\" said the colonel, looking away at the window, \"I think he's\nright, and you had better stay, Perry.\"\n\n\"I don't think you do, father,\" replied the boy.  \"Besides, you promised\nto take me.\"\n\n\"Um!  Yes, I did, my lad; but circumstances have altered since then.\nThey say it's dangerous up there among the Indians.\"\n\n\"Then you had better not go, father,\" said Perry quickly.\n\n\"I have undertaken to go, and I am going,\" said the colonel firmly.  \"I\ngave my word.\"\n\n\"And you can't break it, father?\"\n\n\"No, my boy, not honourably.\"\n\nPerry laughed softly.\n\n\"Hullo!  What does that mean, sir?\" cried the colonel.  \"Glad I am going\ninto danger?\"\n\n\"Of course not, father,\" said Perry.  \"I was only laughing because you\npromised to take me, and you can't break your word.\"\n\nThe colonel leaned back and laughed.\n\n\"And I've come with a petition, father,\" said Perry.\n\n\"Petition?\"\n\n\"Yes; you said that it would be nice for me to be with Cyril Norton.\"\n\n\"Yes, I rather like the lad.  He's a rackety, wild young dog, but\nthere's a good deal of the gentleman about him.  But what do you mean!\nYou said you did not want to stay here.\"\n\n\"Yes, father, but he wants to stay with us.\"\n\n\"Stay with us?  We're not going to stay here.\"\n\n\"I mean, go with us.  He is wild to go.  Take him with us, father.  I\nshould like it so much.\"\n\n\"Why, Perry, my boy, you're mad,\" said the colonel.  \"If the journey is\nso risky that Captain Norton wishes me to leave you here, do you think\nit likely that he will let his son go?\"\n\n\"Perhaps he would with you, father.  He trusts you.\"\n\n\"Not to that extent.\"\n\n\"Try him, father.  It would be so nice to have Cil with us.\"\n\n\"Nice for you, sir--double responsibility for me.\"\n\n\"You wouldn't mind that, father, and we would help you so.\"\n\n\"Yes, nice lot of help I should get from you.\"\n\n\"You don't know, father; but, I say, you will ask him?\"\n\n\"Ask him yourself, sir,\" said the colonel firmly; \"here he is.\"\n\nFor at that moment steps were heard in the veranda, and Captain Norton\nappeared.\n\n\"Don't let me disturb you,\" he said; \"I came back for some bills of\nlading.--Well, Perry, you're going to stop and keep Cil company, eh?\nI'll have the big boat out and newly rigged for you boys.  You can fish,\nand sail, and--\"\n\n\"But I'm not going to stay, sir,\" said Perry quietly.\n\n\"Not going to stay!  I'm very sorry.  But you must think better of it.\nSleep on it, my lad.  That journey in the mountains will be too arduous\nfor a lad like you.\"\n\n\"Oh no, sir.  I'm light and strong, and--\"\n\n\"Yes?  And what?  You are afraid of outstaying your welcome?  Nonsense,\nboy; you'll be conferring a favour upon us.  I shall be glad for Cil to\nhave your company.  He likes you.\"\n\nPerry exchanged glances with his father, who nodded, and his eyes seemed\nto say, \"Now's your time.\"\n\n\"Yes, sir, and I like Cil.  We get on together, and--and he wants to go\nwith us!\"\n\nPerry uttered the last words hurriedly, and then wished that he had not\nsaid them, for the captain looked at him quite fiercely.\n\n\"What!\" he exclaimed.\n\n\"Cil said he would give anything to go with us, sir, and I promised to\nask my father if he would take him.\"\n\n\"Well,\" said Captain Norton sternly, \"and have you asked him?\"\n\n\"Yes, sir.\"\n\n\"What does he say?\"\n\n\"He says no,\" said the colonel firmly.  \"There is no doubt, I suppose,\nthat I am going to run some risks, and I begin to feel now that I am\nhardly warranted in exposing my own son to these dangers.  I should\ncertainly not be right in exposing the son of a friend to them, even if\nthat friend consented, which he would not.  Am I right, Norton?\"\n\n\"Quite right,\" said the gentleman addressed.\n\n\"Then we need say no more about it,\" cried the colonel.  \"Pray, my boy,\nhelp us by dissuading your new friend from thinking about so mad a\nproject.  We must not make Captain and Mrs Norton regret their kindness\nto us.\"\n\n\"No, father.  I understand,\" said Perry.\n\n\"Then there is an end of the matter,\" said the colonel.\n\n\"Not quite,\" said their host, smiling, \"I am still hoping that you will\nstay with us, Perry.\"\n\n\"No, sir,\" said the boy, very firmly now, \"I am going with my father.  I\nwish, though, you would let Cil come too.\"\n\n\"Impossible, my lad,\" said the captain.\n\n\"Then now let's change the subject,\" said the colonel.  \"I do not start\nyet for a week, and plenty of things may occur to alter all our opinions\nand determinations.\"\n\n\"They will not alter mine,\" said the captain firmly.  \"If you both alter\nyours, I shall be very glad.  There, I must go now.\"\n\nCaptain Norton gave Perry a friendly nod, and left them once more.\n\n\"There, Perry, you hear?\"\n\n\"Yes, father, but he may alter his mind.\"\n\n\"Don't expect it, my lad; Captain Norton is firm as a rock in all he\ndecides upon.\"\n\n\"So is Cyril, father.\"\n\n\"Not quite,\" said the colonel, smiling; \"the stuff is soft yet, and will\nhave to yield.  There, go and tell him you have failed.\"\n\n\"Yes, father,\" said Perry sadly.\n\n\"And you mean to go with me?\"\n\n\"Of course, father.\"\n\n\"Very well,\" said the colonel, and Perry left the room.\n\n\n\nCHAPTER THREE.\n\nPREPARING TO START.\n\n\"Well, did you ask him?\" cried Cyril eagerly, as Perry went out into the\nparched garden, the boy pouncing out upon him from behind a patch of\ndry-looking shrubs.\n\n\"Yes, I asked him, and then your father came in.\"\n\n\"Yes,\" said Cyril eagerly, \"I saw him, and kept in hiding, because I\nthought it best to leave it for you to do.  Well, what did your father\nsay?\"\n\n\"He as good as said no.\"\n\n\"Yes, at first,\" cried Cyril.  \"I knew he would.  But he came round.\"\n\n\"And then your father came in.\"\n\n\"Yes?\"\n\n\"And my father made me ask him what he had to say about it.\"\n\n\"Yes?  Do go on, old chap.  You are so slow.\"\n\n\"The captain was quite angry, and wouldn't listen to the idea for a\nmoment.\"\n\n\"That was because he had made his plans for you to stay with me.  But he\ncame round, didn't he?\"\n\n\"No,\" said Perry sadly.  \"He was firm as a rock, and they are both dead\nagainst it.  I should have liked for you to come, Cil.\"\n\nThere was a dead silence; and as Perry looked at his companion, he saw\nthat his brow was full of deep lines, and that the boy's face looked\nhard and set, the eyes fixed, and the lips tightened together into quite\na hard crease.\n\nPerry looked at him for a few moments, feeling pained to see the way in\nwhich the lad took his disappointment.\n\n\"I'm so sorry, Cil,\" he said at last.\n\nCyril did not seem to have heard him, and after a pause Perry spoke\nagain.\n\n\"Perhaps your father will give way before we go.\"\n\n\"What?\"\n\nPerry started, the word sounded so sharp and harsh.\n\n\"I say perhaps he'll give way before we go.\"\n\n\"No, he won't.  He never does.  Father says a thing, and means it.\"\n\n\"It's very disappointing,\" said Perry, \"but it's of no use to fret.\"\n\nCyril laughed bitterly.\n\n\"You're going,\" he said sharply.  \"It can't disappoint you.\"\n\n\"Yes, it can.  I am disappointed.  I don't care about going so much now\nwithout you.\"\n\n\"Then stop here with me,\" cried Cyril sharply.\n\n\"I can't,\" was the reply.  \"You wouldn't give up going if you were me.\nDon't let's think any more about it now, but go and do something.\"\n\nCyril made no reply, but walked straight away out of the garden and then\ndown towards the harbour, while Perry watched him for a few minutes\nsadly, and then followed slowly, missed sight of him, and after quite a\nlong search found him sitting on the edge of his wharf, where the sun\nbeat down most fiercely, and staring straight out to sea.  \"Cil!\" said\nPerry, after going close up, but without exciting the slightest notice\nof his presence.\n\nThere was no reply.\n\n\"Cil--don't be sulky with me.\"\n\n\"Not sulky,\" came with quite a snap.\n\n\"Well, angry then.  It isn't my fault.  I wish you could come.\"\n\n\"Didn't say it was your fault.\"\n\n\"Then why do you take it like that?\"\n\nCyril turned upon him quite fiercely.\n\n\"What's the good of talking?\" he cried.  \"You can't understand.  You go\nsailing about with your father and seeing things everywhere.  I never go\neven into the forest.  It's horrible always shut up here with\nbook-keeping and classics.  I wish sometimes I was only one of the\nIndians, like that one yonder.\"\n\nPerry felt disposed to say, which one? for there was a second Indian\nclose by; but wishing to brighten his companion, and turn the current of\nhis thoughts, he merely said:\n\n\"Well, I shouldn't wish to be a she Indian.\"\n\n\"Those are not shes--they're both men,\" said Cyril sharply.\n\nPerry looked at the pair incredulously, for they certainly had a most\nfeminine aspect, being broad of figure and face, plump-cheeked, and with\nthick long hair cut square across the forehead and allowed to hang down\nbehind.  Their eyes were dreamy-looking and oblique, their faces\nperfectly devoid of hair, and to add to their womanish look, they wore a\nloose kind of cotton garment, which hung down from their shoulders\nnearly to their ankles.\n\n\"I say, what are they doing?\" said Perry, as he stared at the pair.\n\n\"Taking snuff.  That's their way.  They carry some in a little bag, and\nwhen they want to take any, they put the powder in that little\nsiphon-like pipe, and hold it to their nose, and another one blows it\nup.  That one sitting down's the guide father is getting for you.--Here,\nhi!\"\n\nThe Indians looked round, nodded, finished the snuff-taking business,\nand then came deliberately toward the boys.\n\n\"They're Antis,\" said Cyril, as Perry watched the two sleepy-looking\nIndians curiously, and noted that they were both about his own height.\n\nThe men came close up, and stood there smiling, waiting to be spoken to;\nand as Perry had hoped, their presence took Cyril out of himself for the\ntime.\n\n\"Been to see my father?\" said Cyril in a mongrel kind of Spanish.\n\nOne of the Indians nodded.\n\n\"And his father too?\"\n\nThe man replied that he was going now.  So Cyril interpreted the few\nwords.\n\n\"That's the worst of them; and it's so hard to make them understand\nexactly what you mean.  He didn't know what I meant, and had not been--\nWhat say?\"  For the Indian had muttered something which he repeated.\n\n\"Wants to know if I'm going too,\" said Cyril bitterly; and he shook his\nhead at the Indian, when both smiled and looked pleased.\n\nCyril gave his teeth a grind.  \"You beggars,\" he cried in English,\n\"looking glad because I'm disappointed.--And I've given that first chap\nmany a good tuck out, and lots of tobacco dust for snuff, and paid him\nno end of times for birds he has shot with his blowpipe, besides buying\nbutterflies and eggs he has brought down out of the mountains.  All\nright, though; I'll serve them out.--I say,\" cried the boy, and a\ncomplete change came over him, \"can you speak Spanish?\"\n\n\"I?  No, not a word.\"\n\n\"That's a pity.  You'll have to learn a few words, so as to be able to\ntalk to these chaps.  But you'll soon pick them up--some Indian, some\nSpanish, and some half-and-half.  Wait a moment; I want to talk to this\nchap about--about your going.\"\n\nHe began to speak to the man in a low voice, and then grew more and more\neager, while the Indian began by smiling and looking amused, but,\ndirectly after, shook his head, and seemed to be refusing something\nwhich Cyril was asking.  Then Perry saw the lad put his hand in his\npocket and give the Indian a good two-bladed pocket-knife, whose\nkeenness he demonstrated to the great interest of the Indian, who tried\nit on one of the heavy posts by the wharf, and then transferred it to\nhis pocket with a smile of satisfaction, nodding his head now to\neverything Cyril said.\n\nTheir conversation lasted for some time, and Perry began to grow\nimpatient after he had satisfied his scrutiny of the two Indians'\nappearance, and wondered why they should disfigure themselves by\npainting horizontal lines from their noses across their cheeks.\n\n\"There,\" cried Cyril, speaking rather excitedly, \"it's all right now.\nHe says he'll take great care of you, and wait upon you as if you were\nhis father, and always find the best places for sleeping, and mind you\ndon't tumble down into any of the great gaps.  But, I say, Perry, old\nchap, you do wish I was going, don't you?\"\n\n\"Of course I do.\"\n\n\"Ah, well, I suppose I must give in and make the best of it, mustn't I?\"\n\n\"I'm afraid so.\"\n\n\"And you can't write to me and tell me how you are getting on.  There\nare no post-offices up there.\"\n\n\"No, I suppose not.\"\n\n\"You suppose not!\" cried Cyril, laughing, and looking as if his bitter\nfit had quite passed away.\n\n\"Why, you're going where you'll hardly see a soul, unless you meet a\nparty coming down from the mines, or bringing bales of bark.  There, I'm\nnot going to look grumpy any more, but I did feel savage for a bit.\"\n\n\"That's right.  Let's make the best of it while we're together, and do\nsome more fishing, or have a mule ride or two.\"\n\n\"No,\" said Cyril decisively, \"that's all over now.  Father told me this\nmorning that I should have to work and help you make all your\npreparations, for there would be no end to do.  Come along.  They're\ngoing up to see your father now.\"\n\nThe two Indians were both moving off, and the boys followed to the\nhouse, where they were witnesses to the meeting, Captain Norton having\nfollowed shortly, and acting as interpreter between the parties.\n\n\"It is rather awkward,\" he said, \"but I daresay you will soon pick up\nenough of their jargon to make them understand.\"\n\n\"Oh yes,\" said the colonel.  \"I could gather the man's meaning from the\nSpanish words he used.\"\n\n\"Then you will soon manage.  Of course, if you had been a Spaniard, it\nwould have been easy enough.\"\n\n\"I shall not worry about that part of the business,\" said the colonel,\n\"so long as the man is willing, and will do his best.  But we shall want\ntwo others to attend to the mules.\"\n\n\"He understands that.  He is going to bring another trustworthy fellow.\nHe proposed doing so himself.\"\n\n\"And they can manage the mules?\"\n\n\"Oh yes, you may trust them.  This man, Diego, as we call him, has been\nin the habit of coming down from the mountains for years to trade and\nsell.  I consider that I was very lucky in getting him for you.  When\nwill you start, shall I tell him?\"\n\n\"On the sixth day from now.\"\n\n\"That is soon, is it not?\"\n\n\"No; that ought to be time enough to get our mule-packs ready, and a\nsufficiency of stores.  I have everything else.\"\n\n\"Don't hurry,\" said Captain Norton.  \"You are very welcome here, and I\nshall regret your going.\"\n\n\"I know that,\" said the colonel warmly, \"but I am eager to begin, and\nshall be restless till I start.\"\n\nThe captain nodded, and said a few words to the Indians, who replied,\nand then took their departure, it being fully understood that they would\nbe there, ready, on the sixth morning.\n\n\"Ha!\" exclaimed the colonel, \"that is satisfactory.--Now then, Perry, my\nboy, call up John Manning to unpack the luggage, and we'll make our\nselection of what we mean to take.  Captain Norton will keep in store\nfor us all that we decide to leave, and he will help us with his\nexperience in making our selection.--And you will help too, Cyril, will\nyou not?\"\n\n\"Of course, sir.\"\n\n\"Thanks.  Sorry I can't take you, my lad, but your father is right.\"\n\nThose next five days passed almost like magic.  Six highly-bred mules\nwere selected by Captain Norton's help, and furnished with packages and\nhide ropes, besides more for riding purposes.\n\n\"But we shan't be able to manage so many, sir,\" said John Manning, a\nlithe, dry-looking man of about forty, who had been the colonel's\nservant when he was in the army, and had stayed with him ever since, to\nPerry's great disgust; for the lad declared that he was the most\ndisagreeable fellow under the sun, since he was always grumbling.\n\nIt was quite true, for he found fault with everything to the two boys;\nthough silent, as if he were still in the ranks, in the presence of the\ncolonel.  But he quite won Cyril's heart in one of his grumbles, and\nalways after, during their preparations, the boy declared that he was\ncapital fun, and that he liked him.\n\n\"There, young gentlemen,\" said John, \"that's as much toggery as I can\nget in the colonel's soft portmanter, and you'll have to make shift,\nMaster Perry, if you want any more flannels and things.\"\n\n\"Oh, there'll be enough, John,\" said Perry.  \"A fellow don't want\ncollars and cuffs up in the mountains.\"\n\n\"But there ain't enough, sir.  The man must ha' been a hijot as made\nthat portmanter.  If it had been six inches longer, it would have held\never so much more.\"\n\n\"Why, of course it would,\" said Cyril contemptuously.\n\n\"It ain't my business,\" continued the man; \"I'm only a servant.  But\nwhat ought to ha' been done was to have had Mr Cyril here with us, and\nfilled a portmanter up with his things.  Then they'd ha' balanced quite\neasy on the mule's back.\"\n\n\"Yes, that's what ought to have been done,\" said Cyril excitedly.\n\n\"I wish you'd hold your tongue, John,\" cried Perry angrily.\n\n\"All right, sir.  Cut it out, if you like.  We're in savage lands, and\nthere's no magistrates to stop it, for all I know.  But there, sir,\nthat's all I can do as I see.\"\n\n\"How are you getting on?\" cried the colonel, joining them.  \"All packed\nnow?\"\n\n\"Yes, sir,\" said John Manning, drawing himself up stiffly.\n\n\"Did you oil the rifles and pistols?\"\n\n\"Oh yes, sir; I went all over the armoury.  Everything's in perfect\norder.\"\n\n\"And the cartridges?\"\n\n\"Some in every package, sir; so that you can always get a few.\"\n\n\"That's right.\"\n\nBy this time the captain had had an abundance of the most portable and\nuseful provisions packed, simplicity having been especially studied; and\non the evening of that fifth day, it was felt that nothing more could be\ndone.\n\n\"I can think of nothing else to help you, Campion,\" said Captain Norton.\n\n\"No, you have done wonders for me.  There's only one thing I wish.\"\n\n\"What is it?\"\n\n\"That you were coming too.\"\n\n\"Colonel Campion!\" cried Mrs Norton, as the boys exchanged glances.\n\n\"I beg your pardon, madam,\" said the colonel.  \"I will not be so\nselfish.  No, I do not wish that.--Come, boys, make the most of your\nlast hours together.  Shall you be up to see us off in the morning,\nCyril?\"\n\n\"Of course,\" said the boy with a sigh.\n\n\"To be sure,\" said the captain; \"and we'll ride a few miles with you--\neh, Cil?\"\n\n\"No, thank you father, I'd rather not,\" said the lad dolefully.  \"I'll\nbid them good-bye here.--Coming out, Perry?\"\n\n\"Yes,\" said the latter.\n\n\"Don't be long, my lad,\" said the colonel.  \"I want you to get to bed in\ngood time.  You must be up by four.\"\n\n\"Breakfast will be ready by then,\" said Mrs Norton.\n\n\"All right, father,\" said Perry, and the two lads went out into the soft\nmoonlight, to be accosted directly by John Manning.\n\n\"I was looking for you, Master Perry, sir,\" he said.  \"I've been\na-making of my will, and want you to see me sign it, and witness it.\"\n\n\"You want to sign your will?\" cried Perry, laughing.\n\n\"Yes, sir; this here's going to be my last journey, I'm afraid, for one\no' them mules has marked me down.  He means to kick me over the first\npressy pass we comes to.\"\n\n\"Don't let him,\" cried Cyril.  \"If he's going to, shove him over\ninstead.\"\n\nJohn Manning stared.\n\n\"Thankye, sir, I will.  Now, do you know, I never thought o' that.\"\n\n\"Come along, Cil,\" said Perry, laying his hand upon his companion's\nshoulder, and they strolled along to where they could look over the\nsparkling lights of the town, away across the glittering ocean, with its\nbroad path of silver, and then back up to the huge mountain, whose icy\ntop flashed in the brilliant moonbeams, while every here and there the\ndeep ravines marked the sides with an intense black.\n\nThey neither of them spoke, both feeling too sad at heart, but stood\nthere, rapt in thought about the coming morrow, till they were\ninterrupted by the coming of John Manning.\n\n\"Colonel says it's lights out, young gentlemen,\" he said respectfully.\n\"There's allers something wrong in this world.--You ought to ha' been\nwith us, Master Cyril, sir, in this forlorn-hope job.  But, I suppose,\nwe must make the best of it.\"\n\n\"Yes,\" said Cyril bitterly.  \"I suppose we must.\"\n\nA quarter of an hour later the lads were in their bedrooms, listening to\nthe hum of the mosquitoes, and feeling weary, but restless in the heat.\nCyril felt as if he could not sleep for thinking of the coming day, but\nall the same, he went off soundly in spite of his depressing thoughts,\nand woke up with a start, to find that his father was standing by his\nbedside.\n\n\"Half-past three, my lad,\" he cried.  \"Up with you, and act like a man.\nShow our visitors that you can be unselfish, and let's start them\nhappily upon their expedition.\"\n\nCyril tried to say, \"Yes, father,\" cheerfully, but not a word would\ncome.\n\n\"Sulky?\" said Captain Norton rather sternly.  \"I'm sorry that you turn\nlike that.  I'll talk to you this evening, Cyril, my boy.\"\n\nThe boy drew his breath hard, but he said no word, only began hurriedly\nto dress, as his father left the room.\n\n\n\nCHAPTER FOUR.\n\nTHREE SHADOWS.\n\n\"Hallo, sir,\" cried Captain Norton, as they stood outside in the\nenclosure where the mules were being loaded, \"where's the a other man?\"\n\nThe Indian guide looked a little troubled, but spoke out quickly in his\nhalf-Indian, half-Spanish jargon.\n\n\"He will come.  He will meet us soon in the mountains.\"\n\n\"Is that to be depended upon?\" said the colonel harshly; for the absence\nof one man of his force jarred upon his military precision.\n\n\"Yes.  I have always found the Antis trustworthy.\"\n\n\"But we shall be a man short for the mule-driving.\"\n\n\"No,\" said Cyril quickly.  \"They want no driving.  All you have to do is\nto start the leading mule, and the others will follow right enough.\"\n\n\"One more thing,\" said the colonel, who had had many a weary march\nacross the hot dusty plains of India.  \"Ought we not to take water?\"\n\n\"No; the Indians will take you from spring to spring.  They know all the\nstreams and falls in the mountains.\"\n\nThe mules were laden after a good deal of squealing and kicking, and,\nduring the process, John Manning shook his head, and confided to Perry\nthat the big leading mule with the bells had squinted round and shaken\none hind-leg at him.\n\n\"He means me, Master Perry, sir.  I ought to have got that will done.\"\n\n\"Nonsense! it's all right,\" cried the boy; and soon after, an\naffectionate farewell was taken of the Nortons, it being decided, at the\nlast moment, that the captain should not accompany them.  Then the\nlittle mule train started in the darkness up the bridle road leading\nstraight away for the mountains, Cyril sending a cooee-like call after\nthem as they reached the first turn of the zigzag road, and, ten minutes\nafter, they were slowly rising above the town, which still lay in the\ndarkness below.\n\nThe guide went first quite out of sight with the leading baggage mule,\nthe others following; then the colonel walked next, beside his mule,\nwith John Manning behind him; lastly, followed Perry with his mule, and\nthe second Indian came last of all.\n\nThe road was fairly wide at first, giving room for three mules to have\nwalked abreast, but their habit was to keep in single file, and, in\nspite of all efforts on Perry's part, his animal followed the example of\nothers, and walked close to the edge.\n\nAs the day broke, John Manning noticed the trouble his young master was\ntaking, and he shook his head.\n\n\"'Tain't no good, sir; I've been a-trying as hard as a man can try to\nget the crittur to walk like a Christian, and he won't.  One of 'em 'll\ngo over the edge directly, and kill hisself, and serve him right.\"\n\nBut the mule team plodded on, in their slow patient way, higher and\nhigher, while from time to time the travellers stopped to gaze back away\nover the town, at the glittering, far-spreading sea, till all at once,\nafter surmounting the last zigzag up the side of the mountain, the\nleading mule turned a sharp corner and disappeared from Perry's view,\nthe others following, just as if they had entered a door in the side of\nthe mountain.  But, upon leaching the spot, Perry found that they had\nentered a chasm in the <DW72>--a huge rift, not twenty feet wide, and\nmade quite dim by the distance to where it opened upon the sky; while\nbelow, it rapidly ran together, and closed some forty feet beneath the\nledge along which the path ran, and with a swift gurgling stream\nhurrying downward to the shore.\n\nIt was Perry's first sight of a mountain stream whose waters came direct\nfrom the melting snow of the heights above, where winter always reigned,\nbut he could see little but an occasional flash as the mules plodded on\nclose to the edge of the path, which, as it rose, grew narrower and more\nrugged.  And, as they still ascended, and the walls on either side of\nthe gorge shut out the light, the boy shuddered, and wondered whether\nthe way would become more dangerous, for, if so, he felt that he dared\nnot mount and ride where a false step on the part of the mule would send\nhim down headlong from the shelf-like track, twenty--forty--why, it must\nbe a hundred feet down to the stream!\n\n\"Two, I should say, boy,\" said the colonel, for Perry had involuntarily\nspoken aloud.  \"Don't take any notice of the depth; you'll soon get used\nto it.  Look at the mules, how they keep to the very edge.\"\n\n\"Yes, it's horrible, father.  The guides ought to train them to keep\nclose to the wall.\"\n\n\"The mules know best, boy.  They are used to carry loads which spread\nout on either side, and they avoid the wall because it is as dangerous.\nThey might catch their burden against it, and be jerked off.\"\n\n\"I don't think I shall ever get used to such paths as this.\"\n\nThe colonel laughed.\n\n\"Not in half a day,\" he said.  \"In a short space of time you will run\nalong them as fearlessly as if you were on an English road.\"\n\n\"But are there many like this?\"\n\n\"Pooh, this is nothing, Perry.  You are going up into a land of wonders,\nwhere everything is so vast and grand that you will have no time to feel\nnervous.\"\n\n\"But what are we going for?\" asked Perry.\n\nThe colonel turned and looked his son full in the eye.  Then, smiling:\n\n\"Wait,\" he said.  \"You will know in good time.\"\n\nPerry felt abashed, and wished that he had not asked, mentally\ndetermining not to question his father again, while, as he recalled his\nconversations with Cyril, he began to feel that his new friend's ideas\nmust be right.  Directly after, he felt sure that they were, for John\nManning edged up to him, where the path was a little wider, and said in\na whisper:\n\n\"Master won't tell you, then, Master Perry?\"\n\n\"Won't tell me what?\" said Perry rather shortly.\n\n\"What we're going after.  Strikes me as we're going treasure-hunting,\nand we shall get into one of them wonderful valleys you read of in the\n_'Rabian Nights_, where the precious stones lies about so thick, you can\nscoop 'em up.\"\n\n\"Oh, nonsense!\"\n\n\"Do you know what the next country is to this?\"\n\n\"Well, I suppose, if you went far enough over the mountains, you'd come\nto Brazil.\"\n\n\"Zackley, sir; and what comes from Brazil?\"\n\n\"Nuts,\" said Perry, laughing.  \"Hard-shelled, oily nuts, that are\nhorribly nasty to eat.\"\n\n\"Yes, sir, and di'monds.  So don't you say it ain't likely that we may\ncome to a valley of precious stones, because it strikes me that's what\nit means.\"\n\nOnward and upward, along paths partly natural, partly cut in the sides\nof the gorge where the stream ran, and about mid-day Perry began to\nrealise how high the mountains were, for, upon reaching an opening where\nhe could look up and down, he saw that they had been climbing up and up\nfor about seven hours, and were able to look down at a wonderful\npanorama of mountain-side and valley; but upon looking upward, the great\nsnowy peak appeared to be as far away as when they started.\n\nJust then the guide spoke to the leading mule, his voice echoing back\nfrom across the gorge, for they had reached a <DW72> where the sun shone,\nand there were patches of grass and green shrub which promised pasture\nfor the animals.  They all stopped at once, waited patiently to be\nrelieved of their burdens; and then, when the packs were neatly arranged\nin a circle, the patient beasts threw themselves down, had a good roll,\ntossing their legs high in the air, so as to balance themselves for a\nfew moments upon their spines, and then rose again, to begin nibbling at\nsuch herbage as they could find.\n\nJohn Manning busied himself at once and started a fire, while, taking a\ntin, the second Indian went down the steep <DW72> to the bottom of the\ngorge, and toiled up again with his load of clear icily-cold water, into\nwhich, when it boiled, a small handful of tea was thrown, the tin\nremoved from the fire, and the provisions the colonel's servant had\ntaken from a basket were served out.\n\nThe Indians took what was given to them, and sat down by themselves,\nwhile the others partook of their portions with great gusto.\n\nThen, upon looking round, Perry found that the Indians were fast asleep,\nand asked his father whether he ought not to go and wake them up.\n\n\"No, boy; they'll take their mid-day nap and wake up soon.\"\n\nAnd so it proved, for at the end of a couple of hours, the two men\nsuddenly sprang up, caught the leading mule and led him back to the\npath, the others following and standing patiently to be laden.\n\nThen onward again till dark, when the guides halted at a spot like the\nlast, the fire was lit, the evening meal prepared, and, well tired out,\nPerry lay down to pass the first night in the awful solitude to which\nthey had climbed, and gazed up at the brilliant stars seen between large\nwalls of rock.  He wondered what Cyril was doing; felt that it would be\nimpossible to sleep cushioned on that hard rock, and fell asleep\ndirectly, as a matter of course.\n\nThe night was cold up there beneath the glittering stars, but when Perry\nwoke up, warmly rolled in his blanket, there was a sight before him that\nwas as new to him as it was grand.\n\nRight away, apparently at the head of a long narrow valley, and high up\ntoward the heavens, there was a huge peak that might have been the mass\nof glittering rock from which broke away the diamonds, emeralds,\nsapphires, rubies, opals, and amethysts, which went to form the valleys\nof precious stones of which John Manning had spoken.  For it was all\ndark below, but up there one of the gigantic Andean peaks was bathed in\nthe full blaze of the rising sun.\n\nThe boy lay gazing up, enraptured, thinking of the delight of climbing\nup into such a world of glory, and then rolling himself out of his\nblanket, he leaped up, with the stiffness and uneasiness of the past\nnight quite gone, for the colonel's voice reached him:\n\n\"Now, my lad, jump up; breakfast.\"\n\nAt the same moment he heard the crackling sound of burning wood, and in\na nook of the great wall of rock, where otherwise it would have been\nquite dark, the glow of a bright fire shone upon the intent, hard face\nof John Manning, who was baking a bread cake upon a disc of iron, while\nthe two plump, effeminate-looking Indians watched him complacently.\n\nJust below, the mules were cropping the green herbage, and from below\nthem came the rush, roar, and splash of falling water.\n\n\"That's right,\" cried the colonel, holding out his hand.  \"Slept well?\nFind your bedroom draughty?\"\n\n\"I don't know, father,\" said Perry.  \"I was looking at the stars one\nminute, and the next I was staring at that glorious peak.\"\n\n\"Glorious indeed, my boy.  Hah!  There's nothing like a tramp in the\nmountains, and a night's rest in the beautiful, crisp, fresh air.  Come\nalong down to the dressing-room.\"\n\n\"Where?\" said Perry, staring.\n\n\"Down yonder for a wash,\" said the colonel merrily; and, leading the\nway, they descended the precipitous <DW72> to where the stream ran\nthundering by, reaching first a place where it was not above a couple of\nyards across.\n\n\"Why, I thought it would be bigger than this,\" said Perry, \"from the\nnoise it makes.\"\n\n\"Bigger than you think, my lad,\" said the colonel.  \"There is an\nenormous amount of water going by here.  I daresay that crack is a\nhundred feet deep.  Look at the speed at which it runs.\"\n\n\"Yes, it seems to run fast.\"\n\n\"Seems!\" cried the colonel.  \"Here, give me your hand.  Don't be afraid.\nStop a moment; roll up your sleeve above your elbow.  That's it.  Set\nyour feet fast, and trust to me.\"\n\nThe boy obeyed, and after making sure of his own footing, the colonel\nlet his son sink down sidewise till he was nearly horizontal, and could\nplunge his arm right into the stream above the elbow.\n\nIt was a strange sensation for the boy to be sinking lower and lower,\ngazing in the gloom at that rushing, glassy water, which, as it darted\nalong, carried with it another stream--one of air, which blew his hair\nabout and felt icily-cold, but nothing to compare with the water into\nwhich he plunged his arm.\n\nThe shock was electric.  It was as if he received a blow from a mass of\nice which numbed him, and gave his limb a sudden snatch and drag to draw\nit from the socket.\n\nPerry gave a gasp, and pulled his arm out of the torrent.\n\n\"Ugh!\" he ejaculated.  \"It's freezing.\"\n\n\"Yes; would you like a plunge in?\"\n\n\"What! there, father?  It would sweep me away.\"\n\n\"Yes, if you were a thousand times as strong, my lad.  The force is\ntremendous.  Come along here.\"\n\nHe led the way upwards to where there was a fall of some few feet, and\nat the side a shallow pool of the water, wonderfully round, and forming\na basin, giving them ample room for their ablutions; after which, fresh\nand glowing, they climbed up past the mules to where the breakfast was\nwaiting, the hot coffee, bread, and frizzled charqui, or dried beef,\nbeing partaken of with an appetite Perry had never felt before.\n\nThen the remains were packed up, the squealing mules loaded, and they\nstarted once more; now rising a thousand feet, now descending, but\nalways following the stream deeper and deeper into the mountains, till\nthe grandeur and weird sternness of the gorge's defiles through which\nthey passed grew monotonous, so that at the end of two days Perry began\nto long for some change and the open sunshine, away from the tremendous\nprecipices which closed them in, and, in spite of the elasticity of the\nair, had sometimes a strangely depressing effect.\n\nJohn Manning felt it, evidently, and sought every opportunity of keeping\nPerry by his side, so as to have a good grumble about the colonel.\n\n\"I don't know what he could be thinking about, Master Perry, to come to\nsuch a place as this.  It's the world's end, I say.  We shan't have a\nbit o' shoe to our foot when we've gone a bit farther.\"\n\n\"Why don't you ride more, then?\" said Perry.  \"You've got a mule on\npurpose.\"\n\n\"What!\" said John Manning, turning sharply round, \"ride that mule?  No,\nthankye, sir.  I've seen him kick.  I'm not going to give him a chance\nto send me over his head down into one of them cracks.  I believe some\nof them go right through the world.  Look at this one now.  I can't see\nno bottom to it--can you?\"\n\nHe pointed down into the deep chasm along one of whose sides the rough\npath led.\n\n\"No, not from here,\" said Perry, glancing down, and wondering at the\nabsence of giddiness.\n\n\"Nor from anywhere else, sir,\" grumbled the man.  \"Why, if any one told\nme that if you dropped down there, you'd come out somewhere by Simla, I\nshould believe him, for I know they go right through.\"\n\n\"Nonsense!\" said Perry, laughing.  \"There, father's beckoning to me.\"\n\nIt was evening once more, and they were coming again to an opening among\nthe lower mountains, where they would halt for the night.  In fact, half\nan hour later the leading Indian checked his mule in a bowl-shaped\nhollow, where there was a dense little wood of goodly-sized trees, and a\nthread-like fall of water came curving down into a mossy basin, while\nthe whole place was brightened by the reflection from the mountains,\nmade dazzling now by the setting sun.\n\nThe preparations were made for the evening meal with quite military\nprecision; the arms were placed near the fire, and, as if in imitation,\nthe two Indians placed together their long stave-like blowpipes and bows\nand arrows, before helping to unload the mules, and then sitting down\npatiently to administer snuff to each other, and wait to be asked to\njoin the meal.\n\n\"It's very awkward, Perry, my lad,\" said the colonel suddenly.  \"We\nought to have had a guide who could understand us more easily.\"\n\n\"It _is_ awkward, father,\" said Perry.\n\n\"Come and help me now, and between us we may make the man comprehend.\"\n\nPerry followed to where the Indians were squatting down in their loose\ncotton smock-frock-like garments, and at a sign the leader rose.\n\n\"The other man--where is the other man?\" said the colonel in Spanish;\nbut the Indian gazed at him vacantly, till in a fit of inspiration Perry\nrepeated his father's words as nearly as he could, and then began to\ncount, laughing as he said in Spanish, \"One--two\"--and gave an inquiring\ntone to the word \"three,\" as if asking for it.\n\nThe man smiled and pointed to the ground as he answered, and then closed\nhis eyes and let his head fall over upon one hand.\n\n\"What does he mean by that, father?\" said Perry.\n\n\"I can't tell, my boy, unless he wants to show us that the other man is\ncoming while we sleep.\"\n\nThey tried again, but could get no farther.  The Indian grew excited at\nlast and voluble, making gestures too, pointing forward and then at the\nground, ending by pointing at them in turn, throwing himself down and\npretending to sleep.\n\n\"I give it up,\" said the colonel, turning away towards the fire; \"but we\nmust try to learn their language as fast as we can, or we shall never\nget through our journey.\"\n\nA good meal near the fire, whose glow was comfortable enough up at the\nheight they now were, and then father and son strolled a little way\nabout their camp, the wood proving very attractive; but the darkness\nsoon closed in, and they began to return toward the fire, which glowed\nbrightly and cast the shadows of the Indians against the rock-wall as\nthey sat prolonging their meal.\n\n\"What is John Manning doing eating with them?\" said the colonel\nsuddenly.  \"I told him he was to keep those men in their places.  They\nare my paid servants while--\"\n\n\"He's over yonder,\" said Perry, interrupting his father, \"stacking guns\ntogether, I think.\"\n\n\"Nonsense!\" said the colonel; \"there he is, sitting with the Indians,\neating.\"\n\n\"He can't be, father; I can see him over there.\"\n\n\"But look, boy, there are three shadows against the cliff.\"\n\n\"One--two--yes, there are three shadows,\" said Perry dubiously; \"but it\nis something to do with the fire.\"\n\n\"Absurd, my lad; there are three shadows cast by three men sitting\nthere.\"\n\n\"I know,\" cried Perry excitedly; \"that is what the Indian meant--that\nthe third fellow would be here to-night to sleep, or while we slept.\"\n\n\"To be sure,\" cried the colonel, in a tone full of satisfaction.  \"I'm\nglad of it, not but what we could have got on without him, for the mules\ngo well enough, but because it proves the guide to be trusty and a man\nof his word.\"\n\n\n\nCHAPTER FIVE.\n\nPERRY IS STARTLED.\n\nThe guide came to the colonel smiling as soon as he saw him seated, and\npointed to; the other side of the fire, as he spoke words which\nevidently announced the coming of the promised assistant.\n\nThe colonel replied in Spanish, and the Indian went back to his\ncompanions.  Soon after, the smell from John Manning's pipe rose on the\ncool night-air, and Perry sat talking to his father in a questioning\nmood.\n\n\"When are we going over the top of one of the snow-mountains, father?\"\nhe said.\n\n\"I have no intention of going over the top of either of the mountains,\"\nreplied the colonel.  \"We have nothing to gain but hard labour up there.\nWe want to get through the first ridge, and on to the rich tablelands,\nor among the beautiful valleys.\"\n\nPerry said \"Oh!\" in a tone of voice which suggested \"Do we?  I did not\nknow.\"  Then aloud: \"How high up are we now?\"\n\n\"About eight thousand feet, I should say; perhaps a little more, for it\nis rather cold.  There, let's get to sleep; I want to start early and be\nwell on our way soon after sunrise.\"\n\nThe colonel had his desire, for, long before the lower part of the ridge\nwas quite light, the mules were all loaded, and the party made their\nstart, with Diego the Indian leading, the new arrival second, and the\nother man right in the rear as before.\n\nPerry had one glance at the new-comer, and made out that he was a more\nstunted fellow than the others.  In other respects he seemed to be\nsimilar in aspect, but wore a good deal of radiating paint upon his\ncheeks, from which it was drawn along in lines right up to his brows,\nand downward toward the jaws.  He wore the same loose, many-folded gown,\nreaching just to his knees, and carried a bow, arrows, and a long\nblowpipe, but he was wanting in his friends' plumpness and breadth of\nshoulder.\n\n\"Looking at the new mule-driver, Perry?\" said the colonel.  \"Yes?  Seems\nto be quite a stripling.  But so long as he does his work well enough,\nit does not matter.\"\n\nHe did do his work and well, as it proved, trudging along by the mules,\nhelping to unload and load again, managing those under his charge\nadmirably, and proving to be most industrious in fetching water.  But he\nwas timid and distant to a degree, shrinking away when either of the\nEnglish party approached him, and on one occasion showing so evident an\nintention to hurry away into the mountains, that the colonel checked his\nson when next he saw him making for the Indian lad.\n\n\"Let him be,\" said the colonel; \"he's wild as a hawk, and he doesn't\nlook particularly clean.\"\n\n\"No,\" said Perry, laughing, \"he is a grub.  Those fellows don't wash, I\nsuppose, for fear of spoiling their paint.\"\n\nThey ceased then to take much notice of their fresh follower all through\nthat day and the next; and the Indian trudged on beside the luggage\nmules, with his shoulders up and his head bent, as if he were carefully\nwatching where he should next place his feet, speaking to nobody but\nDiego, when the guide left the leading mule for a few minutes to stop\nand look right along the line, inspecting the loads as the mules passed\nhim, smiling at the colonel and Perry, and exchanging rather a fierce\nlook with John Manning; for, somehow, these two did not seem to be the\nbest of friends.  Then he would let his companion who guarded the rear\ncome right up, walk beside him, talking for a few minutes, and then\nstart forward again at a trot, passing them once more till he had\nreached the leading mule.\n\nThere was little change that day, always a constant succession of\nprecipitous walls to right and left, their way being along a narrow\nshelf, with the stream they followed thundering beneath them, sometimes\na hundred feet beneath, at others perhaps a thousand, and quite\ninvisible, but making itself evident by the echoing roar of the rushing\nwaters.\n\nThey passed nobody, neither did they see a single animal to tempt them\nto use rifle or gun that hung by its sling across their backs, till late\nin the afternoon, when, just as they turned the corner of a great\nbuttress of rock, a huge bird suddenly swept by, gazing wonderingly at\nthem.  By one consent, father and son paused to watch the ease with\nwhich the great-winged creature glided along the gorge, half-way between\nthe top and the stream below, turned suddenly and came back, as if to\nrenew their acquaintance, and then curved round again, sweeping along\nfor a short distance, and again wheeling round, not in a series of\ncircles, but ellipses, each turn sending it almost without effort higher\nand higher, till it had reached a sufficient elevation, when it passed\nout of sight over the wall on their left.\n\n\"Eagle?\" said Perry.\n\n\"Vulture,\" replied the colonel.  \"There you have seen one of the biggest\nbirds that fly.  Didn't you notice its naked head?\"\n\n\"Yes; and it had quite a comb over the top, and a ruff round its neck.\nI thought it was an eagle from its great hooked beak.\"\n\n\"The featherless head is a general mark of the vultures,\" said the\ncolonel.  \"I wish I had had a shot at it; but I don't know: I don't want\nto be burdened with bird-skins, especially of such a size as that.\"\n\n\"What a monster to skin!\" said Perry thoughtfully.  \"Why, its wings must\nhave been six or seven feet from point to point.\"\n\n\"Double the length--say fourteen or fifteen, my boy,\" replied the\ncolonel.  \"It must have been that.  Old travellers used to make them out\nto be twenty-five or thirty feet from wing-tip to wing-tip; but they do\nreach the size I say.  Hallo! what are we stopping for?\"\n\n\"Why, there's a bridge,\" cried Perry; \"and the path goes along on the\nother side of the gorge.\"\n\n\"And what a bridge,\" muttered the colonel.\n\nHe might well exclaim, for it was formed in the narrowest part of the\ngloomy gorge, and though not more than five-and-thirty feet in length,\nit looked perilous in the extreme, being formed merely of a couple of\nthick ropes of twisted fibre, secured at either side round masses of\nrock, and with a roadway made by rough pieces of wood laid across and\nfirmly bound to the ropes.\n\n\"A suspension bridge with a vengeance,\" continued the colonel.  \"We\nshall never get the mules to cross that.\"\n\nAnd he had perfect warrant for his words.  For some forty yards below,\nthe water foamed along in a perfect torrent, falling heavily from a\nshelf above, and sending up quite a thick mist, which magnified the\nsurrounding objects and added to the gloom of the place.\n\nPerry felt appalled, but the halt was of short duration, for after\nturning to them and shouting something which was almost inaudible in the\nroar of the torrent, the Indian stepped on to the bridge, and walked\ncoolly across, half hidden by the mist; while the mule which played the\npart of leader bent its head, sniffed at the stout boards which formed\nthe flooring, stepped on and walked carefully across, with the bridge\nswaying heavily beneath its weight.\n\n\"Not so bad as it looked, my lad,\" said the colonel, as the next mule\nfollowed without hesitation.  Then, after a pause, their new Indian\ncrossed, followed with the mule by which he had walked, and then the\nrest, including those from which the travellers had dismounted, for no\none thought of venturing to ride across the chasm.\n\n\"Our turns now, Perry,\" said the colonel.  \"How do you feel?\"\n\n\"Don't like it,\" said Perry huskily.\n\n\"Summon up your nerve, my lad; forget that there is any torrent beneath\nyou, and walk boldly across.  Here, I'll go first.\"\n\n\"No, no, please don't,\" cried Perry, setting his teeth.  \"I'll go.\"\n\n\"Go on, then,\" said the colonel.\n\nThe boy descended from the few yards of loose stony way to where the wet\nrough-hewn boards began, drew a deep breath, and stepped on to the\nbridge, conscious that the guide was looking back, and that the new\nIndian was at the other end, watching him earnestly, with his lips\nslightly parted and his teeth bared.\n\nTo Perry it was a sign that their attendant felt the danger of the\nplace, and was watching to see him fall.  And if he did, he felt nothing\ncould save him, for he would be swept away in an instant down that\nnarrow chasm full of rushing water, where it was impossible for any one\nto climb down and stretch out a helping hand.\n\nOne step, two steps, three steps, all descending, for the middle of the\nbridge hung far lower than the ends, and Perry could feel it vibrato\nbeneath him, and his nervous dread increased.  And yet it was so short a\ndistance to where the Indians were waiting, as he stepped cautiously on\ntill he was well past the middle, when all at once the sky above him\nseemed to be darkened over his head, there was a peculiar, whistling,\nrushing sound, and looking up sharply, Perry saw that the huge bird\nwhich had passed out of sight had wheeled round and was flying so close\nabove him, that it seemed as if its object were to strike at him with\nits powerful talons.\n\nAs a matter of fact, the bird swept by five-and-twenty feet above his\nhead, but it was near enough to destroy the lad's balance as he started\nand bent down to avoid the fancied blow.  The colonel uttered a loud cry\nof warning, and Perry made an effort to recover himself, but this\nstagger caused the bridge to sway, and in another moment or two he would\nhave been over into the torrent had not the bridge vibrated more heavily\nas a guttural voice whispered to him:\n\n\"Quick! _mano_--hand!\"\n\nIt was accompanied by a sharp drag as his own was seized, and,\nrecovering his balance, he half ran--was half pulled--up the <DW72> into\nsafety on the other side.\n\nPerry felt giddy and dazed as the Indian loosed his hold and hurried\naway among the mules, while before he had half recovered himself, his\nfather had crossed and was at his side.\n\n\"Perry, my lad, you sent my heart into my mouth.\"\n\n\"Yes,\" faltered the boy.  \"It was very horrid.  That bird.\"\n\n\"It was startling, my lad, but you ought to be able to walk boldly\nacross a place like that.\"\n\n\"Ahoy! colonel!\" came from the other side, as John Manning hailed them.\n\n\"What is it?\" shouted back the colonel.\n\n\"Hadn't I better go back, sir?\"\n\n\"Back?  No.  Come over!\"\n\nJohn Manning took off his hat and scratched his head, looking down at\nthe hanging bridge and then up at his master.\n\nJust then there was a shout from Diego and some words in the Indian\ntongue, which resulted in the other Indian offering his hand to the\ncolonel's servant, who resented it directly.\n\n\"No,\" he growled; \"I'll do it alone.  One must be safer by one's self;\"\nand stretching out his arms like a tight-rope dancer, he came down\ncautiously, stepped on to the bridge and slowly walked across, the\nIndian following at a trot, as if astonished at any body finding so good\na pathway difficult.\n\n\"I hope there ain't many more o' them spring playthings, sir,\" said John\nManning gruffly.  \"I thought Master Perry was gone.\"\n\n\"Nonsense!\" said the colonel shortly.  \"That great bird startled him.\nForward again; the men are going on.--Perry, my boy, you must give that\nIndian lad a knife, or something as a present: he saved your life.\"\n\n\"Yes, father,\" said the boy, looking dazed and strange.  \"I--I'm better\nnow.\"\n\n\"Yes, of course you are.  Pish! we mustn't dwell upon every slip we\nhave.  There, think no more about it,\" he continued, as he noticed the\nboy's blank, pale face.  \"Go on, and mount your mule.\"\n\n\"I think I would rather walk,\" said Perry.\n\n\"Walk, then,\" said the colonel shortly, and he went on and mounted his\nmule.\n\n\"Quick! _mano_--hand!\" buzzed in Perry's ear, and at the same time he\nseemed to hear the booming roar of the torrent beneath his feet, and the\nrush of the huge bird's wings just above his head--\"Quick! _mano_--\nhand!\"\n\n\"I say, Master Perry, sir, don't look that how,\" said John Manning in a\nlow voice; \"you're as white as taller candle.  You're all right now.\"\n\n\"Yes,\" said Perry, trying hard to recover his natural balance.  \"I'm all\nright now.\"\n\n\"You've made the colonel look as black as thunder, and it wasn't our\nfault.  They've no business to have such bridges in a Christian country.\nBut it was enough to scare any one, my lad.  I thought that there bird\nmeant to have you.\"\n\n\"That was fancy,\" said Perry hastily.  \"I ought to have known better.\"\n\n\"No, it wasn't fancy, my lad.  I think he'd have had you, only seeing us\nall about made him give you up.  But it's all right.\"\n\n\"All right?\"\n\n\"Yes, sir, we're on the c'rect track.\"\n\n\"Of course we are,\" said Perry, as they marched on once more behind the\nmules, followed by the Indian.\n\n\"You dunno what I mean, sir,\" said John Manning testily.  \"I meant on\nthe track for one o' them di'mond valleys.  Know what that bird was?\"\n\n\"Yes; a condor.\"\n\n\"Con grandmother, sir.  It was a roc, one o' them birds as carried\nSindbad out o' the valley.  This was only a chicken, I should say; but\nit was a roc, all the same.\"\n\n\"What nonsense!\" said Perry.  \"That was all fancy tale and romance.\"\n\n\"Not it, sir.  I might have thought so once, but I don't now.  Let me\nask you this, sir,\" said Manning: \"suppose there was no way out or no\nway into the valleys we've come along, could you climb up the sides?\"\n\n\"No, of course not.\"\n\n\"And if you'd heard tell of birds with wings thirty foot across before\nyou'd seen 'em, would you have believed in them?\"\n\n\"No, and I don't now.\"\n\n\"What! after one of 'em come down to attack you, and we scared it away.\"\n\n\"That was only about half the size.\"\n\n\"Oh, come, Master Perry, sir, don't get a haggling about trifles; there\nain't much difference between fifteen foot and thirty.  You mark my\nwords, sir, the colonel's been studying up his _'Rabian Nights_, and\nhe's on the right track now for one of them valleys, and we shall go\nback to San-what's-its-name with these ugly-looking donkey mules loaded\nup with all kinds of precious stones.  You're a lucky one, Master Perry,\nsir, and your fortune's about made.\"\n\n\"Think so?\" said Perry, for the sake of speaking, for he was very\nthoughtful.\n\n\"Yes, sir, I just do; and as for me, I hope it's going to be my luck to\nget just a few nubbly bits for myself, so as I can buy myself a cottage\nand a bit o' garden, and keep a pig, so as to live retired.  You'll come\nand see me, Master Perry, then, won't you?\"\n\n\"Of course,\" said the boy, and then, making a trivial excuse to get\naway, he hurried along the line of slow-going mules to see that his\nfather was right in front before their guide, who walked by the first\nmule; then there were three more plodding along, just far enough behind\neach other to be safe from any playful kick.  By the head of the third\nmule their new Indian driver was walking with his bow over his shoulder,\na handful of long arrows tucked under his arm, and his head bent down\nwatching his footsteps.\n\nPerry kept behind at some distance, watching the Indian's every gesture,\ntill he saw his father returning, for the track had become wider, and\nthe boy watched intently; for he saw the colonel bend down from his mule\nand tap the Indian on the shoulders as he said a few words in Spanish.\nBut what they were Perry was too far off to hear, the mules too making a\ngood deal of clattering on the rocky track, which noise was echoed all\naround in a wonderful way.\n\n\"It must have been my fancy, but I could have been sure he said\nsomething to me in English,\" muttered Perry.  \"I was so excited, I\nsuppose.\"\n\n\n\nCHAPTER SIX.\n\nA NIGHT ALARM.\n\n\"Did you give the Indian lad the knife?\" said the colonel as they came\nabreast.\n\n\"No, father.\"\n\n\"Go and do it at once, and mind how you give it; the fellow's as wild as\na hawk.  I thought he was going to spring over the precipice as soon as\nI touched him.\"\n\nPerry took out the pocket-knife he had with him, and stepped forward;\nbut a word from his father checked him.\n\n\"I don't want to make too much fuss over this, Perry, my lad,\" he said,\n\"but you displayed a great want of nerve.  You did not act like a\nhealthy, sturdy, English boy, and but for that Indian's quick decision,\nyou would have lost your life.\"\n\n\"Yes, father, I'm afraid so.\"\n\n\"Then, for goodness' sake, my lad, try to shake off this girlish\ncowardice, or you'll make me regret bringing you.\"\n\n\"I'll try, father,\" said the boy, his face flushing hotly.\n\n\"That's right.  I'm sure Captain Norton's son would have cut a better\nfigure.\"\n\nPerry's face grew hotter, and he felt a bitter feeling of annoyance at\nbeing compared so unfavourably with the lad who had been his companion.\n\nThe feeling was only momentary, though, and he went on and overtook the\nIndian, with the knife in his hand.\n\nHe was going to give it without a word, but the idea that, perhaps,\nafter all, the half-savage being might understand a few words of\nEnglish, flashed into his mind, and he said:\n\n\"This is not worth much, but I hope you'll keep it in memory of my\ngratitude for your bravery to-day.\"\n\nTo his disgust, the Indian paid not the slightest attention, but trudged\non barefooted beside the mule, as if perfectly unconscious of any one\nbeside him, and Perry's nerves being all on the jar, he felt irritated\nat giving, un-noticed, a pretty speech.\n\n\"Here, catch hold,\" he said.  \"This is for you.\"\n\nHe thrust the knife into the Indian's grimy hand as he spoke, and then\nwalked on to where Diego received him with a smile of welcome, and began\ntalking directly in his mongrel tongue, perfectly content if the boy\nseemed to understand a word here and there, when he pointed to\ncavernous-looking holes in the cliff face opposite to him, to some\nbrighter and greener spot in the gorge, or to some distant fall which\nglittered in the sunshine which came obliquely down into the narrow\nvale.\n\nAll at once there was a beating of wings, and one of the huge condors,\nstartled from the eyrie it occupied high up above their heads, suddenly\nthrew itself off, and began to fly round, rising higher and higher,\nwhile the Indian rapidly fitted one of the long feathered arrows be\ncarried to the string of his bow, waited till the great bird was gliding\nby, and then loosed the shaft.  The arrow struck the condor in the wing,\nand made the huge bird give itself an angry jerk, as if it were disposed\nto turn upon its aggressor; but as Perry watched, the bird gave a few\nrapid beats with its pinions, shooting upwards rapidly, and though it\nwas some distance away, the air was so clear that Perry distinctly saw\nthe long feathered arrow shaken out of the condor's white wing, and fall\nslowly down into the depths of the gorge, while the great bird literally\nshot up for some distance, and then glided over a shoulder of the\nmountain they were flanking, and disappeared.\n\nThe Indian looked at Perry and shook his head, as he muttered some words\nwhich were easily interpreted.\n\n\"Lost my arrow, and did not get my bird.\"\n\n\"And a good thing too,\" said Perry.  \"It would have been of no use, and\nonly wanton destruction.\"\n\nThe man nodded and smiled as if Perry's words were full of sympathy for\nhis loss.  But they fell upon other ears as well, for the colonel was\nclose behind.\n\n\"Rather misdirected sympathy, I'm afraid, Perry, my lad,\" he said.  \"The\nbird would have been no use to us, but I dare say its death would have\nsaved the lives of a good many young vicunas and llamas.\"\n\nPerry stared for a moment or two, and then: \"Oh yes, I know.  Do they\nlive up in these mountains?\"\n\n\"Yes, you'll see plenty by-and-by.\"\n\n\"Sort of goats, aren't they, father?\"\n\n\"Well, my boy, they partake more of the nature of a camel or sheep, as\nyou'll say whenever you see the long-necked, flat-backed creatures.  But\nit's getting time for camping.  The mules are growing sluggish, and\nsniffing about for food.\"\n\n\"I hope we shan't camp here,\" said Perry with a shiver.\n\n\"Not an attractive place, but I daresay Diego has some spot marked out\nin his eye, for he has evidently been along here a good many times\nbefore.\"\n\nTen minutes later, as the snowy peaks which came into view began to grow\nof a bright orange in the western sunshine, one of the mules in front\nuttered a whinnying squeal, and the rest pricked up their ears and\nincreased their pace.\n\n\"Steady there!  Wo-ho!\" shouted John Manning.  \"Hadn't we better sound a\nhalt, sir, or some of 'em 'll be over the side of the path.\"\n\n\"I think we may trust them; they smell grass or something ahead, and\nknow it is their halting-place.\"\n\n\"But look at that brown 'un, sir; he's walking right out from under his\nload.\"\n\nA few hitches, though, and a tightening of the hide ropes, kept the\nloosened pack in its place; and soon after, to Perry's great delight,\nthe gorge opened out into a bright green valley, where, a snug,\nwell-sheltered nook being selected, the mules were once more unloaded,\nand a fire lit.  Then, thanks to John Manning's campaigning cleverness,\nbefore the light on the mountain tops quite died out, they were seated\nat a comfortable meal, with a good fire crackling and burning between\nthem and the Indians, wood for once in a way being fairly plentiful,\nthere being a little forest of dense scrubby trees low down by the\nstream which coursed through the bottom of the valley.\n\n\"Not quite such a savage-looking place, Master Perry,\" said John\nManning, when the colonel had taken his gun and gone for a final look\nround before they retired to their blankets on the hard ground.\n\n\"Savage!  Why, it's beautiful,\" cried Perry, who had been watching the\ncolours die out on one snowy peak.\n\n\"Yes, sir, I suppose it is,\" said the man, shaking his head; \"but we\ndidn't take all the trouble to see things look beautiful.  We can do\nthat at home.  What I'm thinking is that the place don't look healthy.\"\n\n\"Not healthy?  Up here in the mountains?\"\n\n\"Tchah!  I don't mean that way, sir; I mean healthy for your pocket.\nThis looks like a place where you might have a farm and gardens, and\nkeep sheep.  You'd never come here to search for di'monds, and\nsapphires, and things.\"\n\n\"N-no,\" assented Perry.\n\n\"O' course not.  We want good wild broken stone muddle over rocky\nplaces, where you have to let yourselves down with ropes.\"\n\n\"Or ride down on rocs' backs, eh, John?\"\n\n\"Yes, sir, that's your sort.  We've passed several good\nwholesome-looking places that I should have liked to have hunted over;\nbut of course the colonel knows best, and he is leading us somewhere for\nus to have a regular good haul.  Tired, sir?\"\n\n\"Yes, pretty well, but one feels as if one could go on walking a long\nway up in these mountains.\"\n\n\"Well, sir, we've got every chance, and I'd just as soon walk as get\nacross one of these mules, with your legs swinging, and the thin,\nwiry-boned crittur wriggling about under you.  I always feel as if my\none was groaning to himself, and looking out for a good place where he\ncould thrust his hind-legs up and send me flying over his head into the\nair, where he could watch me turn somersaults till I got to the bottom.\"\n\n\"Oh, they're quiet enough,\" said Perry.\n\n\"Oh, are they, sir?  Don't you tell me.  My one never misses a chance of\nrubbing my leg up against a corner, and when he has done there, he goes\nto the other extreme and walks right along the edge, so that my other\nleg is hanging over the side; and if I look down, I get giddy, and\nexpect that every moment over we shall both go.\"\n\n\"I tell you, they don't mean anything,\" cried Perry.\n\n\"Then why does my one, as soon as he knows he has frightened me, begin\nto show his teeth, and laugh and wriggle his ears about, as if he were\nenjoying himself right down to the roots.  I don't believe these mules\nare any good, Master Perry, that I don't, and as aforesaid, I always\nfeel as if I'd rather walk.\"\n\nFurther conversation was put an end to by the return of the colonel, and\nsoon after, leaving the Indians crouching near the fire, which they\nseemed reluctant to leave, the English party sought the corner which had\nbeen selected for their sleeping-place, rolled themselves in their\nblankets, and with valises for pillows, and their stores piled up for a\nshelter from the wind, they were not long in dropping off to sleep.\n\nPerry's was sound enough at first, but after a time he began to dream\nand go through the troubles connected with crossing the swinging bridge\nagain.  He found himself half-way across, and then he could go no\nfarther in spite of all his efforts, till, just as the condor was about\nto take advantage of his helplessness, and descend to fix its talons in\nthe sides of his head and pick out his eyes, the Indian made a snatch at\nhim, and dragged him across for him to awake with a start.\n\nIt was all so real that his brow was wet with perspiration, but he\nsettled what was the cause, and changed his position peevishly.\n\n\"That comes of eating charqui late at night, and then lying on one's\nback,\" he muttered, and dropped off to sleep again directly.\n\nBut only to begin dreaming again of the condor, which was floating\noverhead, spreading its wings quite thirty feet now; and there was the\nscene of the day repeated with exaggerations.  For the Indian guide bent\nan immense bow, and sent an arrow as big as a spear whizzing through the\nair, to strike the huge bird, which swooped down close by, and looked at\nhim reproachfully, as it said in a whisper: \"I only came to bring back\nyour knife.\"\n\nPerry lay bound in the fetters of sleep, but all the same, his ears\nseemed to be open to outer impressions, for the words were repeated\nclose to him, and he started up on to his elbow.\n\n\"Who's there? who spoke?\" said a low firm voice close to him.  \"That\nyou, Perry?\"\n\n\"Yes, father,\" replied the boy, as he heard the ominous _click-click_ of\nthe double gun that lay by the colonel's side.\n\n\"What were you doing?\"\n\n\"Nothing, father.  I just woke up and fancied I heard some one speak.\"\n\n\"There was a whisper, and some one brushed against me just before.  Did\nyou move from your place?\"\n\n\"No, father,\" said Perry, feeling startled now.\n\n\"Manning!\"\n\n\"Sir!\"\n\n\"Have you been moving?\"\n\n\"No, sir; fast asleep till you woke me, talking.\"\n\n\"Then some one has been visiting us,\" whispered the colonel.  \"Hah!\nwhat's that?\"\n\n\"Something rustling along yonder, sir.\"\n\n_Bang! bang_!  Both barrels were discharged with a noise which seemed to\nhave awakened all the sleeping echoes of the mountains around their\ncamp.\n\nThen, as the colonel hastily reloaded his piece, Perry and John Manning\nsprang up, each seizing his gun, and waited.\n\n\"I missed him; but, whoever it is, he won't come prowling about again.\nFollow me quickly.  Stoop.\"\n\nBending down, they hurried across the few yards which intervened between\nthem and the smouldering ashes of the fire, which, fanned now and then\nby the breeze sweeping along the valley, gave forth a faint\nphosphorescent-looking light, by which they could just make out the\nfigures of the three Indians standing with their bows and arrows ready,\nas if about to shoot.\n\n\"Which of you came over to us?\" said the colonel in Spanish; but there\nwas no reply, and the speaker stamped his foot in anger.  \"What folly,\"\nhe cried, \"not to be able to communicate with one's guide!\"\n\n\"Could it have been some one from the valley lower down?\" whispered\nPerry, who then felt a curious startled sensation, for he recalled\nperfectly the words he had heard while asleep, or nearly so: \"I only\ncame to bring back your knife.\"\n\n\"Then it must have been the little Indian, and he could speak English\nafter all.\"\n\nAccusatory words rose to Perry's lips, but he did not speak them.  A\nstrange reluctance came over him, and he shrank from getting the poor\nfellow into trouble, knowing, as he did, that his father would be very\nsevere on the intruder upon their little camp.  For it was a fact that\nthe little Indian had crept up to where they slept and spoken to him.\nThe excitement had prevented him from noticing it before, but he held in\nhis hand the proof of the visit, tightly, nervously clutched: the knife\nwas in his left hand, just as it had been thrust there while he slept.\n\n\"Attend here,\" said the colonel.  Then very sternly: \"You cannot\nunderstand my words, perhaps, but you know what I mean by my actions.\nOne of you came for some dishonest purpose to where we lay sleeping, and\nI wonder I did not hit whoever it was as he ran.--Give me your hand,\nsir,\" he cried; and he seized and held Diego's right hand for a few\nmoments.\n\nThen dropping it, he held out his hand to the other Indian, who eagerly\nplaced his in the colonel's palm.\n\n\"An outside enemy, I'm afraid,\" muttered the colonel; \"they are both\nperfectly calm.--Now you, sir,\" he continued, turning to the last comer,\nwho hesitated for a moment, and then held out his hand.\n\nThis was all in the dim starlight, the figures of the men being made\nplainer from time to time by the faint glow from the fire; but their\nfaces were quite in the shade as the colonel took the last comer's hand\nand grasped it tightly, while Perry's heart began to beat, for he felt\nthat the discovery was coming; and hence he was not surprised at the\ncolonel's fierce and decided action.\n\n\"Your pulse galloping,\" he cried angrily, as he dragged the dimly-seen\nfigure forward.  \"Perry, Manning, cover those two men, and if they make\na gesture to draw their bows, fire at once.--Now, you scoundrel, it was\nyou, and you had come to steal.\"\n\n\"No, he had not, father; he came to give me back my knife.\"\n\n\"What!\" cried the colonel angrily.\n\n\"It's a fact; he put it in my hand while I slept; and here it is.\"\n\n\"Then--\"\n\n\"It's quite true, sir, and no good to keep it up any longer.\"\n\n\"Cil!\" cried Perry in astonishment.\n\n\"Yes.  Don't be very angry with me, Colonel Campion.  I felt obliged to\ncome; I couldn't stop away.\"\n\n\"Why, you treacherous young rascal,\" cried the colonel, shaking him\nviolently.\n\n\"Don't, sir, please; you hurt!\" cried Cyril half angrily.\n\n\"How dare you mutiny against your father's commands, and come after us\nlike--?\"\n\n\"I dunno,\" said Cyril mournfully.  \"I felt obliged; I wanted to be with\nPerry there.\"\n\n\"But to come masquerading like this, sir!  How dare you?\"\n\n\"I dunno, I tell you,\" said the boy petulantly.  \"It isn't so very nice\nto come over the stones without shoes or stockings, and only in this\nthing.  It's as cold as cold, besides being painted and dirtied up as I\nam.  My feet are as sore as sore.\"\n\n\"And serve you right, you young dog.  What will your father say?\"\n\n\"I don't know what he'd have said if you'd shot me,\" grumbled Cyril.\n\nThe colonel coughed.\n\n\"You precious nearly did, you know,\" continued Cyril querulously.  \"I\nheard the shots go crashing in among the bushes as I ran.\"\n\n\"Then you shouldn't have come prowling about the camp in the middle of\nthe night,\" cried the colonel.  \"Of course, sir, I took you for some\nwild beast or marauding Indian.\"\n\n\"Well,\" said Cyril, \"now you know, sir, and I suppose I can go back and\ntry to sleep.\"\n\n\"Go back?  Yes, sir, first thing--to your father,\" cried the colonel\nfiercely.  \"I suppose he does not know you have come?\"\n\n\"No, sir.\"\n\n\"Of course not.  A pretty disgraceful escapade, upon my word, sir!  I\nonly wish I were back in my regiment, and you were one of my subalterns.\nI'd punish you pretty severely for this, I promise you.\"\n\n\"Would you, sir?\" said Cyril drearily.  \"I thought I was getting\npunished enough.  I'm sorry I disturbed you, sir; I only wanted to get\nclose up, and touch Perry's hand.\"\n\n\"Bah!\" cried the colonel.  \"Why did you want to touch Perry's hand?\"\n\n\"Because I was so lonely and miserable, lying there with my feet sore.\nI couldn't sleep, sir.  The stones have cut them, and I was afraid to\nwash them, for fear you should see how white my legs were.\"\n\nThe colonel coughed.\n\n\"Here; stop a moment, sir,\" he said, in rather a different tone.  \"You\nsee, I might have shot you.\"\n\n\"Yes, sir,\" said Cyril dolefully.  \"And it did seem hard to be shot at,\nbecause I felt glad the poor fellow didn't go off the bridge.\"\n\nThe colonel coughed again.\n\n\"Hum, ha, yes,\" he said, a little huskily.  \"It was a very narrow\nescape, of course, and you behaved very well.  You--er--yes, of course,\nyou quite saved his life.  But I shall say no more about that now.--\nHere, Manning, get Mr Cyril Norton a couple of blankets.--And you'll\ncome and lie down by us, sir; and mind this: no more evasions, no\nattempts to escape.\"\n\n\"I shan't try to escape in the dark,\" said Cyril drearily.  \"Where\nshould I escape to, sir?\"\n\n\"Ah! of course.  Where to, indeed!  So recollect you are a prisoner,\ntill I place you back safely in your father's hands.--Stop!  Halt!  What\nare you doing, Perry?\"\n\n\"Only shaking hands with him, father,\" said the lad.\n\n\"Then don't shake hands with him, sir.  Shake hands with gentlemen, and\nnot with lads who disgrace themselves by disobeying their father's\norders, and satisfying their own selfishness by causing others intense\nanxiety.\"\n\nPerry drew in a long, deep breath, which did not go down into his lungs\nproperly, but seemed to catch here and there.\n\n\"One moment,\" said the colonel; \"can you make that man Diego\nunderstand?\"\n\n\"Yes, sir.\"\n\n\"Then tell him and his companion to go to sleep again.\"\n\nCyril said a few words to the guide, and the two Indians dropped down at\nonce, close to the warm ashes.\n\n\"I suppose, then, he knew all about your escapade, sir, eh?\" cried the\ncolonel.  \"Of course, he must have got you the Indian clothes and\npaint.\"\n\n\"It was all my fault, sir; don't blame him,\" said Cyril humbly.  \"I'm\nvery sorry I did it now.  It seemed--\"\n\n\"Seemed?  Well, what did it seem, eh?  There, hold your tongue now, and\ngo and lie down by Perry.  Recollect you are in an old soldier's camp,\nand I forbid all talking now.  Stop!--er--are you hungry?\"\n\n\"No, sir; I can't eat,\" said Cyril bitterly.\n\n\"Humph!  There, go and lie down, both of you, and get to sleep.--Once\nmore, no talking, Perry; not till to-morrow morning.--Good-night, both\nof you.\"\n\nBy this time John Manning had taken two soft blankets out of one of the\npacks, and handed them to the prisoner with a very unmilitary whisper.\n\n\"My!\" he said, \"what a game, Mr Cyril.\"\n\nBut neither of the boys smiled.  They lay down in Perry's old place, and\nCyril uttered a sigh of content, and then a stifled sob, as he felt\nPerry's hand seeking for his to hold it tightly.\n\n\"Good-night,\" he whispered, as Perry bent over him, and then there was\nanother whisper.\n\n\"Can father send you back, Cil?\" and the answer came:\n\n\"It's too late now.  No.\"\n\nJust then the colonel lay down again in his old place, and another\nrustling told that John Manning was curling up in his.\n\n\"Good-night, Perry, my boy,\" said the colonel.\n\n\"Good-night, father,\" replied Perry, and then to himself, \"Oh, I hope\nhe'll say good-night to poor old Cil.\"\n\nHe had his wish.\n\n\"Good-night, Cyril,\" came rather huskily.\n\n\"Good-night, sir,\" said the boy, in a voice he could hardly keep steady.\n\nAnd then came:\n\n\"Thank God I did not hit you when I fired, my lad.\"\n\nThen there was nothing heard but the whispering of the wind below them\namong the trees.\n\n\n\nCHAPTER SEVEN.\n\nTHE SLIPPERY WAY.\n\n\"Awake, Cil?\" whispered Perry, just as daylight was making its way down\ninto the depths of the valley, and a faint glow became visible on one of\nthe snow peaks.\n\n\"Yes,\" was whispered back, \"these two hours.\"\n\n\"Couldn't you sleep?\"\n\n\"No; not for thinking.  It's all very well for you, but I've got to hear\nwhat your father _says_ this morning.\"\n\nThis was unanswerable, and Perry remained silent for a few minutes,\nwondering what he had better say next.\n\nThen the inspiration came.\n\n\"Look here, Cil,\" he said; \"you won't get on any the better for having a\npainted and dirty face.  I'll get a bit of soap, and we'll go down and\nhave a good wash.\"\n\n\"What's the good?\" said Cyril.  \"Dirty painted face goes best with\nthings like this.\"\n\n\"Yes, but you're not going like this,\" said Perry.  \"You must put on\ndecent clothes.\"\n\n\"Haven't got any,\" said Cyril sourly.\n\n\"No, but I have--two spare suits, and you shall have one.\"\n\nCyril gave a start.\n\n\"I say, Per,\" he whispered excitedly, \"do you mean that?\"\n\n\"Of course I do.  My things will fit you, and you can have a regular\nrig-out.\"\n\n\"Oh!\" ejaculated Cyril.  \"Come on then, quick.\"\n\nThey stole out of their corner to the baggage pile, where Perry pointed\nto the portmanteau containing his kit, signing to Cyril to take one end\nand help him to bear it a dozen yards away to where a huge mass of rock\nhad fallen from above.\n\n\"Here we are,\" cried Perry, dragging out one of the suits that had been\nmade expressly for the journey.  \"They'll fit you, I know.\"\n\n\"Fit!\" cried Cyril excitedly; \"of course they will.  Once get myself\ndecent, I shan't so much mind what the colonel says--I mean, I can bear\nit better.  I did feel such a poor miserable wretch when he was talking\nto me in the night.  It all seemed so easy just to dress like one of the\nIndians; but as soon as I was in that long shirt thing, with my bare\nlegs and feet, I felt as if I'd suddenly turned into a savage, and\ndaren't look any one in the face.\"\n\n\"And I don't wonder at it,\" growled a deep voice.  \"Here, what game's\nthis, young gents?\"\n\nThe boys looked up to see that John Manning was peering over the rock,\nand they were so startled for a few moments that neither spoke.\n\n\"Going off again, and you with him, Master Perry?  Well, you don't do\nthat while I'm here.\"\n\n\"Don't be so stupid, John,\" cried Perry, recovering himself.  \"Can't you\nsee what we're doing?\"\n\n\"Yes, that's what I can see, making of yourselves a little kit apiece,\nready to desert, both of you.\"\n\n\"Rubbish!\" cried Perry.--\"That's all, isn't it, Cyril?\"\n\n\"Boots!\" said Cyril dolefully; \"but I don't know how I am going to get\nthem on.\"\n\n\"Oh, a good bathing will do that.  Here you are.--Now, John Manning,\nfasten this up again, and take it back.\"\n\n\"Honour, Master Perry?\"\n\n\"Honour what?\"\n\n\"You're not going to desert?\"\n\n\"You go and light a good fire and get breakfast ready; we're going down\nto have a bathe.\"\n\n\"Oh, that's it, is it?\" said the old soldier, chuckling.  \"Well, a bath\nwould improve Master Cyril.  Shall I bring you down a tin of hot water,\ngentlemen.\"\n\n\"You be off, and hold your tongue.  I don't want my father to know until\nwe get back.\"\n\n\"All right, gentlemen,\" said John Manning, grinning; \"but I say, Master\nCyril, there'll be court-martial on you arter breakfast.\"\n\n\"Come along, and don't mind him,\" whispered Perry, and they hurried down\nto the side of the torrent, where they had to spend some time before a\nsuitable place was found where they could bathe without being washed\naway, for the water ran with tremendous force.  But at length a safe\nspot was hit upon, where the stream eddied round and round; and here\nPerry's tin of soap was brought into play with plenty of vigour, there\nbeing no temptation to prolong their stay in water which had come\nfreshly down from the snow, and which turned their skins of a bluish\nscarlet by the time they were dressed.\n\n\"Shall I pitch this smock-frock thing into the stream?\" said Perry, with\na look of satisfaction at his companion.\n\n\"Throw it away?  No.  Perhaps your father will order me to keep it to\nwear, and make me give back your clothes.\"\n\n\"I know my father better than that,\" cried Perry warmly.\n\n\"But see how he went on at me last night, and how he'll go on at me\nagain to-day.  I wish I hadn't done it.\"\n\n\"I'm glad you are come, Cil,\" said Perry; \"but it does seem a pity.\nWhatever made you do it?\"\n\n\"I hardly know,\" said the boy sadly.  \"I was so down in the dumps\nbecause I couldn't come with you, and I did so long, for it seemed as if\nyou were going to have all the fun, and I was to be left drudging away\nat home, where it was going to be as dull as dull without you.  And then\nI got talking to Diego, and when he heard that I was not coming too, he\nsaid he should give it up.  He wasn't coming with three strangers, he\nsaid, for how did he know how people with plenty of guns and powder and\nshot would behave to him.\"\n\n\"He said that?\" cried Perry.\n\n\"Yes, and a lot more about it, and he wanted me to ask father again to\nlet me come.\"\n\n\"And did you?\"\n\n\"No; where would have been the use?  When father says a thing, he means\nit.  Then Diego turned quite sulky, and I thought he was going to give\nup altogether.  That was two days before you were going to start, and I\nbegged him not to throw you over, and he said he wouldn't if I came too;\nand when I told him my father wouldn't let me, he said why not come\nwithout leave?  And after a great deal of talking, in which he always\nhad the best of me, because I wanted to do as he proposed, at last I\nsaid I would, and he got me the Indian dress and the bow and arrows.\"\n\n\"And when did you start?\"\n\n\"That same night, after they'd gone to bed at home.  I'd got the things\nall ready, and I soon dressed and locked up the clothes I took off in a\ndrawer they weren't likely to look into, so that they might keep on\nexpecting to see me back, thinking I'd gone out next morning early, and\nthat would give me a start of all that night and all next day.\"\n\n\"What a thing to do!\" said Perry.\n\n\"Yes; wasn't it?  Didn't seem so bad in the hurry and worry of getting\noff I didn't think about anything but hurrying on after you, and then I\ngot very tired and hot, and that kept me too from thinking about\nanything but catching up to you.\"\n\n\"But how did you know the way?\" said Perry.\n\n\"Oh, that was easy enough.  Diego told me which road he should take, and\nI'd been along there before as far as the place where he said he would\nwait for me.\"\n\n\"Yes, he said when you would come.\"\n\n\"And when at last I was getting nearer to you, I began to lose heart\naltogether, and I'd eaten all the food I brought with me; and I'd had so\nlittle sleep, because I was obliged to overtake you before you started.\nIf I had not--\"\n\nHe stopped short, and Perry stared at him.\n\n\"Go on,\" he said at last.  \"If you hadn't what?\"\n\n\"If I hadn't caught up to you, it would have been all over.\"\n\n\"Nonsense!  Why?  You'd have gone back.\"\n\n\"No.  I'd been one whole day without anything to eat, and I couldn't\nhave got back, tired as I was, in less than four days.  I should have\nlain down and died.\"\n\n\"But you'd have met somebody,\" said Perry.\n\n\"Up here?  No.  There's a caravan of llamas comes down about twice a\nyear, and now and then a traveller comes along, but very seldom.  How\nmany people did you meet?\"\n\n\"Not one.\"\n\n\"No, and you were not likely to.  I knew this, and it made me keep on\nwalking to overtake you, for it was my only chance.\"\n\n\"But did you think about what a risky thing you were going to do before\nyou started?\"\n\n\"No,\" said Cyril sadly; \"all that came after, and there was no going\nback.\"\n\n\"But what a way your father and mother must be in.  What will they\nthink?\"\n\n\"Oh, don't, don't, don't!\" groaned Cyril.  \"Think I haven't gone over it\nall, times enough?  I never thought how much there was in it, or what\ntrouble it would make till it was too late.  Do you think I'd have come\nto be near you for a minute last night, if I'd known that the colonel\nwas going to shoot at me?\"\n\n\"Of course not.\"\n\n\"And that's the way with lots of things: one don't think about them till\nit's too late.  Hush, here he comes.\"\n\nFor while the boys were busy talking, they had climbed up the side of\nthe valley, and come close up to the fire before they were aware of it.\n\n\"Humph!\" ejaculated the colonel sternly.  \"So you've given up being a\nsavage then, young fellow, eh?\"\n\n\"Yes, sir,\" said Cyril humbly.\n\n\"You'll join us at breakfast, then, eh?\"\n\n\"I don't feel as if I could eat anything, thank you, sir.\"\n\n\"No, I shouldn't think you did; I don't think I should have much of an\nappetite if I had behaved to my father and mother as you have behaved to\nyours.  But there, you are my friend's son, and I must be hospitable, I\nsuppose.  Come and have breakfast, and then the sooner you are off back,\nthe better.\"\n\nPerry stared at his father so hard that the colonel noticed it.\n\n\"Well, boy,\" he said, \"what is it?\"\n\n\"I was thinking about what you said, father.\"\n\n\"About his going back?  Well, what about it?\"\n\n\"How is he to go all the way back by himself?\"\n\n\"The same way as he came, sir, of course.\"\n\n\"He couldn't do it, father.  His feet are sore, and he'd have to carry\nall the provisions he'd want on the way.\"\n\n\"Provisions!  To carry?  Why, he hasn't got any.--Have you, sir?\"  Cyril\nshook his head.  \"Then how do you expect to get back?\"\n\n\"I don't know,\" said the boy sadly.  \"No!\" thundered the colonel.  \"Of\ncourse you don't know.  Nice sort of a young scoundrel you've proved\nyourself.  Scoundrel?  No: lunatic.  You can't go on with us, because,\nout of respect for your father, I won't have you; and you can't go back\nalone, because you have no stores.  What do you mean to do--lie down and\ndie?\"\n\n\"Perhaps I'd better,\" said Cyril bitterly; \"there seems to be nothing\nelse I can do.\"\n\n\"Well, don't lie down and die anywhere near where I'm camping, sir,\nbecause it would be very unpleasant, and spoil my journey.  What time do\nyou start back, now you can go decently?\"\n\n\"Now, sir,\" said Cyril, and he turned sharply and took a step to go, but\nthe colonel caught him by the shoulder.\n\n\"Come and have your breakfast first, sir.  If you can behave badly to\nyour father and mother, I cannot, by ill-treating their son.  No\nnonsense: come and sit down, and I'm very glad to see that you are\nbeginning to realise what a mad trick it is of which you have been\nguilty.--Ready, Manning?\"\n\n\"Yes, sir,\" came back from the fire, and a minute later they were all\nseated in silence, partaking of the hot coffee and fried bacon made\nready for them by Manning, who gave Cyril a bit of a grin as he saw the\nchange in his appearance.\n\nThe colonel ate heartily, but Perry's appetite was very poor; and Cyril\ncould hardly master a morsel, in spite of the colonel's manner becoming\nless harsh.\n\n\"Come, boy,\" he said, \"eat.  You've a long journey back, and you'd\nbetter make much of the provisions, now you have a chance.  I'll send\nyour father a line in pencil for you to bear, and to exonerate me from\ncausing him so much uneasiness.  By the way, how many days do you think\nit will take you to get back?\"\n\nCyril tried to answer indignantly, but the words seemed to stick in his\nthroat; and Perry's face grew red at what he considered to be his\nfather's harsh treatment of the lad whom he looked upon as his friend.\nThere was a painful silence, then, for some minutes, during which the\ncolonel went on with his breakfast, and Perry sat with his eyes dropped,\nunable to get any farther.\n\nAll at once, Cyril spoke out in a half-suffocated voice, as he looked up\nindignantly at the colonel.  \"Isn't it too hard upon me, sir,\" he cried,\n\"to keep on punishing me like this?  You know I cannot go back, or I\nshould have gone long ago.\"\n\n\"I want to punish you, sir, because I want to make you feel what a mad\nthing you have done, and how bitterly cruel you have been to a father\nwho trusted in your honour as a gentleman, and a mother whose affection\nfor you was without bounds.\"\n\n\"But, don't I know all that?\" cried Cyril, springing up and speaking\npassionately now.  \"Hasn't it been torturing me for days past; and\nwouldn't I have gone back if I could, and owned how wrong I had been?\"\n\n\"Only you had found that, when once you had foolishly put your foot on\nthe slippery decline, you could not get back to the starting-point, and\nhave gone on gliding down ever since,\" said the colonel, speaking\nquietly.  \"Yes, my lad, I believe you have been bitterly sorry for your\nfoolish escapade since you started, and you have been severely punished.\nThere, I will say no more about it.\"\n\n\"And you will help me to get back, sir?\"\n\n\"If an opportunity occurs.  As soon as we meet an Indian who can be\ntrusted, you can take two of the mules, and a sufficiency of provisions\nto last till you get back.  I am a man short now, or one of these should\nreturn with you at once.  I am sorry for your people, but I cannot turn\nback now, and I'm sure your father would not ask it of me.\"\n\n\"Thank you, sir,\" said Cyril humbly.\n\n\"There,\" cried the colonel, \"I have done my duty by you, boy.  You have\nhad your punishment, and you have taken it bravely.  I have no more to\nsay, especially as you are not yet out of the wood, but have your father\nto meet.\"\n\n\"Yes, sir, I have my father to meet,\" said Cyril.\n\n\"Then, now eat your breakfast, and let's get on again.  Take off that\nmiserable face, for I shall not refer to the trouble again.\"\n\nHe held out his hand.  Something very like a sob escaped from Cyril's\nlips, as the boy made a quick snatch at his hand, and held it in his for\na moment or two.\n\nThen the breakfast went on in silence, and Perry's appetite suddenly\nreturned; while Cyril did not do so very badly after all.\n\n\n\nCHAPTER EIGHT.\n\nSIGNS OF SUSPICION.\n\nHalf an hour later, the little caravan was in motion, and, for the first\ntime the preparations were delightfully easy.  Eager to be of some\nservice, and to try to make up for what he had done, Cyril began to help\nto load the mules, and above all, helped the colonel.\n\nFor the latter was trying hard to make the guide understand that he\nwould like to pass through the patch of forest below them, before they\nascended the mountain path visible away to their left; and the man\nstared at him in the most blank way possible, and then kept on pointing\nto a couple of great fagots which lay tightly bound upon one of the\nmules' backs.\n\n\"It's all right, sir; let me speak to him,\" cried Cyril eagerly.  \"He\nthinks you keep on telling him you want wood for the next fire we make,\nand he says he has got plenty.\"  Then, turning to the guide, he rapidly\nsaid a few words in the rough dialect of Indian and Spanish, with the\nresult that the man gave the colonel a sharp look, and then nodded his\nhead, and went off with the leading mule.\n\nPerry gave his father an eager look, and the colonel, who was smiling\nwith satisfaction at the ease with which a difficulty had been smoothed\naway, frowned.\n\n\"Oh yes, it's very nice,\" he said; \"but I cannot afford to have an\nintelligent interpreter on such terms as these, Master Perry.  There,\nget on; I said I would not refer to the trouble any more.--Hi!  Cyril,\nmy lad, you'd better ride that black mule.\"\n\n\"Ride--the mule, sir?\" said the boy hesitatingly.\n\n\"Yes; your feet are cut and sore.  Rest till they are better.\"\n\n\"Hurrah!\" whispered Perry.  \"Jump up, old chap.  Here, I'll give you a\nleg.  I shall ride, too, to-day.\"\n\nThe next minute, both boys were mounted, and following the last mule\nwith the second Indian.\n\nThat patch of scrubby forest looked to be close at hand, but it took\nthem nearly an hour to reach it, everything being on so grand a scale\namong the mountains; but at last they began to thread their way through,\nwith the colonel eagerly examining the different trees, the Indians\nnoting his actions curiously, but always hanging their heads again if\nthey thought that they were observed.\n\nThe colonel kept up his examination, but did not seem very well\nsatisfied; and soon after, the bushy trees with their shining green\nleaves were left behind, and they journeyed on through what had looked\nat a distance like fields of buttercups, but which proved to be a large\ntract covered with golden calceolaria, whose rounded turban-like flowers\nglistened in the sun.  This looked the more beautiful from the abundance\nof grass, at which the mules sniffed carelessly, for they had passed the\nnight eating.\n\nThen before starting upward, there was the rapid stream to cross at a\nspot where the rocks had fallen in a perfect chaos from the\nmountain-side, completely filling up the chasm along which the water\nran; and here they could hear it rushing, gurgling, and trickling down a\nhundred channels far below, in and out amongst the rugged masses of rock\nwhich dammed it back.\n\nThe mules made no difficulty about going over here, merely lowering\ntheir muzzles, and sniffing at the cracks and holes as they felt about\nwith their forefeet, and climbed more than walked across to the solid\nrock and the bare, very faintly marked, stony track, which led up and up\nto a narrow gap in the mountains, evidently a pass.\n\nSteeper and steeper grew the way, now zigzagging along a stiff <DW72>,\nnow making a bold dash at the mountain-side, over loose stones which\nwent rolling down, setting others in motion till regular avalanches\nrolled down into the valley hundreds of feet beneath.\n\n\"Have you ever been here before, Cil?\" said Perry, who now rode close\nbehind his friend.\n\n\"No.  Never any farther than the place where I overtook you.\"\n\n\"Isn't this very dangerous?\" continued Perry, as the mules climbed up,\nsending the loose stones rattling down to their right.\n\n\"Eh?  Dangerous?  I don't know.  I was wondering what they are thinking\nat home.  Yes, I suppose it is dangerous.\"\n\n\"Then hadn't we better get down and walk?\"\n\n\"What for?  We couldn't walk up so well as the mules.  They've got four\nlegs to our two.  They're a deal more clever and sure-footed than we\nshould be.\"\n\nPerry kept his seat, fully expecting to have the mule make a slip, and\nthen for them to go rolling down hundreds of feet into the valley; but\nin due time the gap-like opening was reached, and through this place,\nwith the walls on either side so steep that they looked an if they had\nbeen cut, they passed into a narrow valley, or rather chasm, looking as\nif the mountains had been split down to their roots by some earthquake;\nand a chill of horror ran through Perry, as he checked his mule where\nthe rest were panting and recovering their breath.\n\n\"Not a very cheerful-looking place, boys,\" said the colonel, as he\nsurveyed the great chasm, running apparently for miles through the\nmountains, zigzagging, returning upon itself, and always dark and\nprofound in its lower part; so deep, in fact, that from where they stood\nit might have gone right down to the centre of the earth, while upward\nthe sides rose, wall-like, toward three huge peaks, which looked\ndazzlingly white.\n\nAll at once Perry started, and it seemed as if an electric shock had\npassed through the mules.  For there was a tremendous booming roar some\ndistance away, followed by peal after peal, as if of thunder running for\nmiles amongst the mountains, and not dying away till quite a couple of\nminutes had elapsed.\n\n\"Thunder,\" whispered Perry.\n\n\"No, I think not,\" said Cyril below his breath.--\"What was that, Diego?\"\nhe said in the man's tongue.\n\nThe answer was laconic, and accompanied by a smile.\n\n\"He says some of the snow fell over yonder, out of sight.\"\n\n_Crash_!\n\nThere was another roar, followed by its echoes.\n\n\"Look! look!\" cried Cyril excitedly.  \"There, just below that place\nwhere the sun shines on the ice.\"\n\n\"Yes, I see it,\" said Perry; \"a waterfall.\"  And he shaded his eyes to\ngaze at the glittering appearance of a cascade pouring over a shelf of\nice into the depths below.\n\n\"Waterfall!\" said the colonel, smiling.  \"There is no water up there to\nfall.  It is a cataract of pieces of ice and solidified snow, thousands\nof tons of it broken away through the weight and the mass being loosened\nby the heat of the sun.\"\n\n\"Gone!\" cried Cyril.\n\n\"To appear again, lower down,\" said the colonel, and they watched the\nglittering curve of dazzling ice as it reappeared and made another leap,\nand again another and another, lower down, till it finally disappeared\nby falling into some chasm behind a fold of the mountain.  But the roar\nof the ice was continued like distant thunder, telling how enormous the\nfall must have been, though dwarfed by the distance into a size that\nappeared trifling.\n\nThen the boys sat gazing at the black gulf before them, with its huge\nwalls, which were nearly perpendicular in places.\n\n\"I say, of course, we're not going along that way?\" said Perry\nnervously.\n\n\"I don't know,\" replied Cyril; \"the tracks generally do go along the\nworst-looking places.\"\n\n\"But how can they have been so stupid as to pick those?\" said Perry\npetulantly.\n\n\"They don't pick them,\" replied Cyril.  \"Only they are obliged to go\nalong any places there are.  Yes, we shall have to go along yonder.\"\n\n\"Impossible.\"\n\n\"How would you go, then?\" said Cyril.  \"We're not flies; we can't climb\nup those walls; and you couldn't go over the mountains if you wished,\nbecause of the ice and snow.  You must go in and out round them where\nthe valleys are open, and this is open enough.  There is no other way.\"\n\n\"But, I say, shan't you be--er--just a little afraid to go down there?\"\n\n\"No,\" said Cyril quietly.  \"I don't feel afraid a bit.  There's only one\nthing I feel afraid of now.\"\n\n\"What's that?  Falling off one of the precipices?\"\n\n\"No,\" said Cyril sadly.  \"Meeting my father.\"\n\nPerry was silent, and his friend turned to Diego, who was going from\nmule to mule, examining the knots in the hide ropes by which the baggage\nwas secured to the pack-saddles.\n\n\"Which way does the road go now?\" he asked.\n\nThe man pointed straight along the black chasm running from below them\naway into the distance.\n\n\"Along there?\" whispered Perry, as he comprehended the gesture.\n\n\"Yes, I thought so,\" said Cyril coolly.  \"There can be no other way.\"\n\n\"But what else did he say?\" asked Perry breathlessly.\n\n\"He said, did your father want to go on any more.\"\n\n\"What's that?\" cried the colonel.\n\nCyril repeated the man's remark.\n\n\"Tell him of course, till I wish him to stop.\"\n\nCyril delivered the message, and the man spoke again, gesticulating and\npointing along the deep valley.\n\n\"He says, sir, that there is no place farther on where you will get a\nbigger valley, and that there are plenty of snow-mountains farther\nback.\"\n\nThe colonel made a gesture full of impatience.\n\n\"What does he mean, Cyril?  Doesn't he want to go any farther?\"\n\n\"I think that's it, sir.  I'll ask him what he means.\"\n\nCyril turned to the guide again, and there was a short, eager\nconversation, carried on for a minute or so.\n\n\"He says, sir, that the way along the track is very dangerous.  It goes\nalong that side, to the left, and the path is very narrow.  If any one\nslipped, he would fall right to the bottom.\"\n\n\"It must be the regular way across the mountains, where mules are\naccustomed to go, and he undertook to guide me; so tell him I go on.\"\n\nCyril conveyed the colonel's words to the man, who looked annoyed, and\nglanced suspiciously at the colonel as he said a few words, to which the\nboy replied angrily.\n\n\"What's that? what's that?\" cried the colonel.\n\nCyril hesitated.\n\n\"Speak out, sir; what is it?  Why don't you speak?\"\n\n\"He said he wanted to know where you wanted to go, and what for?\" said\nCyril, watching the colonel rather anxiously.\n\n\"Tell him as far as I please, and where I please,\" said the colonel\nsternly.  \"Now then, at once; and tell him I should advise him not to\nask me any more questions.  Forward!\"\n\nCyril interpreted the words, and the Indian looked sharply at his\nemployer, to see in his eyes the glances of a man accustomed to command,\nand without a word he took the rein of the leading mule, and went away\nto the left, seeming to Perry as if he were passing over the edge of a\nprecipice, so suddenly the descent began, a dozen yards away.\n\nBut, as is often the case among the mountains, that which had looked so\nterrible at a little distance, last its dangerous aspect when boldly\napproached, for, following closely upon the luggage mules, Perry reached\nthe edge of that which he had supposed to be a precipice, and found that\nit was only a <DW72>, going downward; but it was quite steep enough to\nrequire great care in crossing it, and the mules showed their\ncomprehension of the fact that it must not be attacked lightly, by the\nway in which they walked, slowly and carefully, making sure of every\nstep they took, till they were well across the green <DW72>, and on to\nsolid rock once more.\n\nAnd now it was plain that the man had not exaggerated, for their path\nlay along what is known to geologists as a fault in the rock of which\nthe side of the valley was composed--that is to say, the upper part of\nthe huge mass appeared to have slipped sidewise, leaving four or five\nfeet of the lower part of the valley wall like a shelf, and along this\nthe mules began to walk cautiously, taking the greatest care that their\nloads did not touch the side of the rock, and consequently walking as\nclose to the edge as possible.\n\nThe man had not exaggerated in the least.  The shelf-like paths they had\npreviously traversed were in places perilous enough, but here the bottom\nof the chasm-like valley was quite hidden from the travellers, and\nimagination added largely to the depth whenever either of the boys stole\na glance downward.\n\nNo one spoke, but they rode on in single line, feeling appalled by the\nawful nature of the place, hour after hour, for the path wound and\nzigzagged, and seemed without end.  At every slip of a mule's hoof, at\nevery kick against a loose stone, Cyril felt his pulses leap, and Perry\nturned cold with apprehension; while, whenever Cyril turned to look\nround at his friend, each saw in the other's face a hard set look, and a\nstrange, almost despairing stare in his eyes.\n\nThey were conscious of there being a rushing torrent somewhere far\nbelow, but it was down in the region of gloom, and they went on for\nhours without once catching even a gleam of the water, which at times\nsent up a dull thunderous roar, at others died away into a faint\nmurmuring vibration, as if it were making for itself a subterranean\nchannel through the bottom of the chasm.  But little attention was paid\nto that, each of the travellers keeping his eyes fixed upon the narrow\npath in front, and rarely glancing up at the rocky wall on their left,\nor down into the profound gulf upon their right.\n\nIt was well on in the afternoon when, in turning an angle where the path\nshot off suddenly to their left, they came upon a wide opening lit up by\nthe sun; but, saving that it was light, it was more repellent to the eye\nthan the path along which they had come.  For it was one wild chaos of\ntumbled-together rocks, looking as if, by some convulsion of nature, the\nwhole of that portion of the valley side had been shattered and tumbled\ndown from the shoulder of a huge mountain, destroying the pathway, and\nleaving in its place a broad stretch of masses of rocks, from pieces\nhundreds of tons in weight, to fragments not larger than a man's head.\n\nProgress across this appeared impossible, but the guide went on for a\nfew minutes and then stopped; for rugged as the place was, it possessed\nthe quality of being level enough to enable them to make a halt for\nrefreshments, without being on a narrow shelf where there was not room\nfor a mule to be turned.\n\nHideous as the place was, every face brightened, for the strain of\nfeeling in great peril was for the time removed, and even the mules\nshowed their satisfaction by whinnying to each other, and giving\nthemselves a shake, as they began to sniff about and browse upon the dry\nvegetation which grew amongst the fallen stones.\n\n\"Hah!\" ejaculated the colonel, as he got off his mule, and looked round\nand above at the pure blue sky.  \"One feels as if one could breathe and\nmove now.\"\n\n\"Yes,\" said Perry, with a shudder; \"it was horrible.\"\n\n\"Nonsense, boy,\" cried the colonel.  \"It was not a place one would\nselect for a nice walk, but I should not have liked to miss such a\njourney.  People at home do not know there are such wildly-grand places\nin the world--eh, Cyril?\"\n\n\"No, sir,\" replied the latter eagerly, for a pleasant word or two from\nthe colonel was like a gleam of sunshine in his breast; \"but it was\ndangerous.  I should not have liked to get off my mule on that shelf.\"\n\n\"Not on the precipice side, certainly,\" said the colonel.\n\n\"Why, there wasn't any room on the other,\" cried Perry; \"and if one had\nturned giddy, one would have gone down, down--ugh!\"\n\n\"Yes, the place did look deep,\" said the colonel, \"but no one did turn\ngiddy, and the mules went along as steadily as if they had been on a\nturnpike road.--Well, Manning, what's the matter?\"\n\n\"I was thinking about our having to go back along that there path, sir.\"\n\n\"Well, I daresay we shall,\" replied the colonel, \"but you don't mind.\"\n\n\"Not mind, sir?\" cried the old soldier gloomily.\n\n\"Not you, my man.  I grant it is a little dangerous, but not so bad as\nwalking along a shelf in the Nagari pass, with a Belooch behind every\nstone, taking aim at one with his long matchlock.\"\n\nJohn Manning grinned, took off his hat, and scratched his head.\n\n\"You did not complain about the danger then,\" continued the colonel.\n\n\"No, sir, I didn't, did I!\" said the man, wrinkling up his face a little\nmore; \"and I ain't going to grumble about this neither.  I'll go\nwherever you lead, colonel, like a soldier should.\"\n\n\"Yes; I knew that when I chose you to come with us, Manning,\" said the\ncolonel quietly.  \"Well, what about dinner?  We had better have it upon\nthat flat-topped stone.\"\n\n\"I shan't be five minutes, sir; but I was hesitating about that stone.\nIt's just in the hot sunshine, and if there are any snakes about here,\nthat seems a likely place.\"\n\n\"Any snakes about here, Diego?\" asked Cyril, and the man shook his head,\nand replied that it was too cold.\n\nA few minutes later they were enjoying a hearty meal, and the mules were\nrevelling in their freedom from their loads, while the two Indians sat\nmunching their sun-dried strips of meat, and talking together in a low\nvoice.\n\n\"All these stones and rocks tumbled down from above, I suppose, sir?\"\nsaid Cyril, after a prolonged look upward at the peak which rose high\nabove them, with its smooth sides glittering with snow, and a thin,\nwhite, gauzy cloud just hiding the extreme point.\n\n\"Yes, my lad,\" said the colonel, shading his eyes, and looking up.  \"The\nsnow hides the old scar, but I should say that during some eruption the\nwhole side of the crater fell outward, and crumbled down to here, as you\nsay.\"\n\n\"Crater?\" cried Cyril.\n\n\"Yes; don't you see that it is a volcano?\"\n\n\"I did not, sir.  Then those clouds up there are smoke?\"\n\n\"More likely steam.\"\n\n\"Steam?  Those clouds?\" cried Perry, gazing up.  \"And is this a burning\nmountain?\"\n\n\"Yes.  You will be able to say you have been on the side of a volcano,\"\nsaid the colonel quietly.  \"Look at all this broken stone about; how\nglistening a great deal is, as if it had been molten.  That piece, too,\nlooks like scoria.\"\n\n\"Then hadn't we better go on at once?\" cried Perry, getting up from the\nstone on which he was seated.\n\n\"What for?  Are you afraid of an eruption?\" said the colonel, with a\nshade of contempt in his voice.--\"Feel that stone where he was sitting,\nCyril; perhaps it is warm.\"\n\n\"Yes, it was quite warm when I sat down upon it,\" said Perry hastily.\n\"All the stones about here are nearly hot.\"\n\n\"Of course they are, sir,\" cried his father.  \"Have they not been baking\nin this hot sunshine?  There, sit down and finish your dinner.\nMountains don't break out into eruption without giving some warning.\"\n\n\"But this must have been quite lately, sir,\" said Cyril, to turn the\ncolonel's fire.\n\n\"Geologically lately, my lad,\" he said, picking up and examining a\nstone, \"but not in our time, nor our grandfathers'.  In all probability\nthese stones came crumbling down some hundreds of years ago.\"\n\n\"Then you think there is no fear of another eruption, father?\"\n\n\"If I did think there was, do you think I should be sitting here so\ncalmly?\" replied the colonel.\n\nPerry had nothing to say to this, and he soon after became interested in\na conversation which took place between Cyril and the guide, waiting\nimpatiently until it was at an end.\n\n\"What does he say?\" asked Perry, as Cyril turned away.\n\n\"That as soon as we've passed this rough place there's another path,\nlike the one we've come by, and he wants to know if your father means to\nrisk it.\"\n\nPerry felt a shrinking sensation, but he said nothing, knowing how\ndetermined his father was when he had set his mind upon a thing.\n\n\"I told him we were going, of course.  But, I say, Perry,\" whispered\nCyril, \"how far does he mean to go?\"\n\nPerry shook his head.\n\n\"Is it any use to ask him where he means to stop?\" whispered Cyril.\n\n\"No; not a bit.\"\n\n\"Hallo!  Look here!\" cried Cyril, and Perry snatched up his piece from\nwhere it lay.\n\n\"Look out, father!\" he cried, as one by one, with solemn, slow stride,\nsome half-dozen peculiar-looking, flat-backed, long-necked animals came\ninto sight round an angle of the valley at the far side of the chaos of\nstones amongst which they had made their halt.\n\n\"Put down that gun.  Don't be stupid,\" cried Cyril.  \"Can't you see they\nare llamas?\"\n\n\"What if they are?  I suppose they are good to eat.\"\n\n\"I shouldn't like to try one,\" cried Cyril, laughing.\n\nThe colonel had now caught sight of the animals, which kept on coming\nround the corner in regular file, with their long necks held up stiffly.\n\n\"Quite a caravan,\" the colonel said.  \"Ask Diego what they are\ncarrying.\"\n\n\"I know, without asking, sir,\" said Cyril eagerly.  \"They're bringing\ndown Quinquina--kina, as they call it.  You know, sir--bark.\"\n\n\"Hah!\" ejaculated the colonel eagerly, and he took out the little double\nglass he carried to examine the train of animals, which had evidently\ncome from the track that they were to pursue after their halt.\n\n\"You're wrong, I think, my lad,\" said the colonel, after a long\nexamination through his glass.  \"They have all got bales of something on\ntheir backs, and, judging from the outside, I think they are skins or\nhides.\"\n\n\"Yes, sir, that's right,\" cried Cyril, \"but it is bark inside.  They\nmake the bark up into bales, and cover them with hides before binding\nthem up.  I know; I've seen them before.\"\n\nThe colonel continued his inspection, and Cyril hurriedly questioned the\nguide before speaking to the former again.\n\n\"He says they are taking the kina down to the port, and that they will\nhalt here to rest.\"\n\n\"Then we'll stay a little longer and see them,\" said the colonel,\nclosing his glass after seeing several armed men turn the corner and\nbegin to climb beside the llamas over the rugged stones.\n\n\n\nCHAPTER NINE.\n\nCYRIL SCENTS DANGER.\n\nAs the men in charge of the llamas came in sight of the colonel and his\nparty, they waited for more and more to join them, and it soon became\nplain that they expected or meditated an attack; but a peaceful message\nsent on by the colonel gave them confidence, and the swarthy men, for\nthe most part armed, came on, followed now by their charge, till the\ngreat opening in the rock-wall was filled by the drove of rough,\nwoolly-looking animals; there being over five hundred in the caravan,\nand each bearing about a hundredweight of the precious fever-averting\nbark.\n\nDiego and Cyril's powers were soon brought into requisition for\ninterpreting; the strangers willingly stating where they were going, but\nproving themselves as eager to know the colonel's business as he showed\nhimself about the bark bales, before the mules were once more loaded,\nand the English party started again, so as to get to the end of the\nvalley before dark.\n\nThe coming of the caravan had given the boys encouragement, for, as\nCyril argued to Perry, the track could not be so very bad if that drove\nof animals bearing loads could come along it in safety.\n\n\"I don't know about that,\" replied Perry.  \"I had a good look at them.\nShort-legged, broad things like these, with soft spongy feet like\ncamels, seem made for walking up here among the rocks; while the mules,\nwith their long legs and hard hoofs, look as if they might slip and go\nover at any time.\"\n\nThis was just after they had started, and found, as soon as they had\ncleared the rocky chaos, that the shelf path was so wide that the lads\nwere able to ride abreast; and as the colonel had gone right in front\nwith the guide, the boys began talking about the men with the llamas.\n\n\"Any one would think your father wanted to go into the kina trade,\" said\nCyril, who was rapidly recovering his spirits.  \"Did you notice how the\nIndian frowned when Diego kept on talking to him, and I asked all those\nquestions for your father?\"\n\n\"I thought he seemed impatient and tired, and as if he wanted to sit\ndown and rest.\"\n\n\"Oh, it wasn't that,\" said Cyril quietly; \"it's because they want to\nkeep all about the bark trees very secret, so that no one else shall be\nable to grow it and supply it for sale.  You heard my father say how the\npeople who went in search of the trees never came back again.  Father\nfeels sure that they were murdered.\"\n\n\"No; that was the people who went after the treasures.\"\n\n\"Oh, was it?  I forget.  Perhaps it was both,\" said Cyril.  \"My head got\nin such a muddle over my coming after you, that things are mixed.  I\nsuppose it was because Colonel Campion asked so much about the kina.\"\n\n\"Father takes a great interest in everything; that's why he travels and\nhas come here,\" said Perry.  \"Look, there goes a condor.\"\n\n\"Well, let him go,\" said Cyril.  \"He isn't good to eat, and you've got\nplenty of provisions to last till you get to some village on the other\nside of the mountains.  But, I say, it does seem strange that you people\nshould come here of all places in the world.\"\n\n\"I don't see it,\" replied Perry.  \"It's a very wonderful place to come\nto, but I wish it wasn't quite so dangerous.  I keep feeling afraid of\nturning giddy.\"\n\n\"Yes, it's a wonderful place to come to, and I had no idea that the\nvalleys were so awful and deep; but I should enjoy it if it wasn't for\nthinking of them at home.  I hope they believe I've come after you.\nWish I'd left a line to say where I had gone.\"\n\n\"It's too late to wish that now,\" said Perry.\n\n\"Yes, but one can't help wishing it all the same.  I wish I knew why\nyour father has come up here.\"\n\nAt that moment there was a warning shout from forward, and another from\nJohn Manning in the rear, for the boys had been so wrapped in their\nthoughts that they had not noticed how rapidly the path was narrowing.\nThey had, however, another hint, and that was from Cyril's mule, which,\nfrom long training on similar paths, knew exactly what to do, and went\non ahead, while Perry's stopped short on the narrowing shelf which\nfollowed all the windings and angles of the rocky wall, and had become\nso strait that Perry shrank from watching the laden mules, whose loads\nevery now and then brushed against the stones, and one completely caught\nagainst a rough projection, making the intelligent animal that bore it\nstop and ease away a little, leaning more and more over the precipice\ntill Perry's hands turned cold and wet, and he held his breath.  Just,\nthough, as he was about to close his eyes, so as not to see the poor\nbrute plunge headlong down to where it would certainly be dashed to\npieces, the load escaped from the awkward corner, and the mule trudged\non just as before, while Perry heard a deeply-drawn sigh just behind\nhim.\n\n\"I thought he'd have gone, Master Perry,\" said John Manning.  \"Mules\nain't got no nerves, that's for certain, and if ever you hear any one\nsay in the future as a donkey's a stupid animal, you tell him he don't\nknow what he's talking about.\"\n\n\"That mule's sensible enough, at all events,\" said Perry, without\nventuring to turn his head, lest he should have to look down into the\ngulf.\n\n\"Sensible, sir?  Why, he acted just as a human being would.  I call it\nwonderful.  I say, Master Perry, though.\"\n\n\"Yes?  But I wish you wouldn't talk to me so, while we are going along a\nplace like this.\"\n\n\"Don't say so, Master Perry, because I want to talk.  It keeps one from\nfeeling a bit skeary, because this is a place, sir, really.\"\n\n\"Well, what do you want to say?  Speak loud, for I can't turn round to\nlisten.\"\n\n\"But if I speak loud, the colonel will hear me, sir, and I want to talk\nabout him.\"\n\n\"Well, go on then; what is it?\"\n\n\"Can't you tell me, sir, where we're going to, and what we're going\nfor?\"\n\n\"We're going over the mountains, John.\"\n\n\"Well, sir, I know that; but what are we going for?\"\n\n\"To find the valley of diamonds, and throw down lumps of meat for the\nrocs to fetch out.\"\n\n\"No, no, sir, that won't do,\" said John Manning, shaking his head.  \"As\nyou said to me the other day, that's only a story out of the _'Rabian\nNights_, and not real truth, though these places might just as well be\nsomething of the kind, from the looks of them.  But, I say, sir, you do\nknow where we're going, and what for, don't you?\"\n\n\"No, I have not the slightest idea.  Ask my father yourself.\"\n\n\"What, sir!  Me ask the colonel about the plan of his campaign?  Why, I\nshould as soon have thought of asking the Dook o' Wellington.\"\n\n\"We shall know in good time, I daresay,\" said Perry; and then a slip on\nthe part of one of the mules ahead made them turn cold once more.\n\nBut the clever animal recovered itself on the instant, and for hours\nthey kept on along this path, till the boys despaired of reaching its\nend, and began to calculate on the possibility of having to encamp on a\nplace like that for the night.\n\nBut it is a long lane that has no turning, and just when there was a\nsensible deepening of the gloom, and the peeps they had of the sky\noverhead were of a golden amber, they turned an angle and became aware\nof an increase in the murmuring sound of water, which thenceforth grew\nlouder and louder, till it was evident that they were approaching some\nextensive fall.\n\nAn hour later they were in full sight of where it came thundering down\nhundreds of feet, spouting forth from a gap, and plunging down on to a\nhuge buttress of rock, which shot it off again far into the air,\ndistributing it so that it went on down into the valley like a misty\nrain, and without a sound arising from below.\n\nThe fall was magnificent, for, as they approached, the upper part was\nturned to gold by the setting sun, and to add to the beauty of the\nscene, there was a patch of forest on either side, and the narrow shelf\nwas broadening out to where it ran into a side valley, all golden green\nand darkened shadow.  For they had reached the end of the terrific\ngorge, and there were scores of places just in front ready for the\nformation of ideal camps, without the risk of an incautious step sending\nits unhappy author thousands of feet down into the depths below.\n\nIn another half-hour they were in a place which, by comparison with the\nsterile defile of darkness and depression, seemed to the lads beautiful\nin the extreme; and after a hearty meal, while the colonel was looking\nround the camp, as he called it, and having a farewell glance for the\nnight at the mules, which were thoroughly enjoying the abundance of\ngrass, Cyril sat looking very thoughtful and depressed.\n\n\"He's thinking of home and his people,\" said Perry to himself, and then,\non the impulse of the moment:\n\n\"I say,\" he cried, \"why didn't my father send you back along with the\nllama train?  I never thought of that before.\"\n\n\"Are you in such a hurry to get rid of me?\" said Cyril bitterly.\n\n\"No, of course not; but as he said he should send you home by the first,\nI thought it strange that he had not done so.\"\n\n\"Because they were not going to San Geronimo,\" said Cyril quietly.\n\"They would turn off to the north, just where I first joined you, and I\nsuppose he thought, after what I had suffered, it would be too cruel to\nsend me to find a great deal of my way back with people like that.\"\n\nThey relapsed into silence for a time, during which period John Manning\ncleared away and washed up as methodically as if he were at home, while\nthe two Indians sat by the fire munching away at the supply of biscuit\ngiven to them.\n\n\"What are you thinking about, Perry?\" said Cyril at last.\n\n\"The stars.  How big and bright they are up here.  What were you\nthinking about?\"\n\n\"Diego, our guide.\"\n\n\"What about him?--that he ought to be fonder of water, even if it is\nicily-cold?\"\n\n\"No,\" said Cyril seriously.  \"I want to know why he has turned so quiet\nand serious, and why he seems to be always watching your father in such\na peculiar way.\"\n\n\"Father was sharp with him, and ordered him to go on, when he seemed to\nwant to go back.\"\n\n\"Yes, and I suppose he did not quite like it; but that isn't all.\"\n\n\"What is all, then?\" said Perry.\n\n\"Ah, that's what I want to find out.  He puzzles me.  He's thinking\nabout something, and I shouldn't wonder if he has taken it into his head\nthat your father has come up here to look for the Incas' treasures.\"\n\n\"Pooh!  Why should he think that?\" returned Perry.\n\n\"Because these Indian chaps are horribly suspicious as well as\nsuperstitious.  They would think it a horrible sin to touch the gold if\nthere is any; and if it is found, they would be ready to defend it.\"\n\n\"What with?  Bows and arrows?\" cried Perry, laughing.\n\n\"Yes, and blowpipes.\"\n\n\"Why don't you introduce pop-guns as well?\"\n\n\"Because they are toys,\" said Cyril seriously, \"and blowpipes are not.\nDon't you know the tiny darts they send out are poisoned, and that one\nwill kill anything it hits?\"\n\n\"Is that true?\" said Perry, whose eyes dilated at the idea.\n\n\"Quite true.  I saw a man kill several birds with the darts.  They died\nalmost directly they were struck, and I have been told by father that he\nhas seen small animals die in a few minutes after being scratched.\"\n\n\"But do you think--Oh, what nonsense!  You have got your head crammed\nwith that idea about the gold.\"\n\n\"Perhaps so,\" said Cyril thoughtfully, \"and maybe I'm wrong.  But I\ndon't like to see old Diego turn so gruff and distant, and it seemed\nstrange for him to go and talk for a long time with the Indians in\ncharge of the llamas.  I saw them look very strangely and suspiciously\nat your father afterwards.\"\n\n\"Those Indians?  Why, what could it be to them?  Ah, the Peruvian\nIndians are said to be joined together to protect everything belonging\nto the old days when they were a great nation, and keep it for the time\nwhen the Incas come back to rule over them again.\"\n\n\"Say, Master Perry,\" said John Manning in a low voice, \"your eyes are\nyounger than mine.  Just cast 'em along the rock path we come to-day.\"\n\n\"Yes, what for?\"\n\n\"Are you looking straight along, sir?\"\n\n\"Yes.\"\n\n\"Well, what do you see?\"\n\n\"Nothing at all.\"\n\n\"Try again, sir.\"\n\nBoth Perry and Cyril looked along the path, tracing it faintly in the\ncoming night for some distance along, beyond where the great fall came\nthundering down.\n\n\"I can't see anything,\" said Perry.\n\n\"Nor I,\" said Cyril.  \"Yes, I can.  There's something that looks like\nshadows moving.\"\n\n\"Steady, sir; don't seem as if you were noticing it, but notice it all\nthe same.  It struck me as strange ten minutes ago, but I thought it was\nfancy.  But you see it, sir, and it must be right.  Now then, sir, what\ndo you make that to be?\"\n\n\"Indians,\" said Cyril promptly.\n\n\"That's right, sir--what I thought; and they're watching us, and after\nno good.\"\n\n\"What!  Do you think they are hanging round the camp to try to steal?\"\n\n\"Don't know, sir,\" said John Manning gruffly.  \"I hope that's the\nworst.\"\n\n\n\nCHAPTER TEN.\n\nJOHN MANNING THINKS.\n\nJohn Manning's curious remark sent a thrill through Cyril, and, trying\nhard not to appear as if watching, he strained his eyes in the direction\nindicated, but the gloom had increased, and neither he nor Perry could\nmake out anything more.\n\n\"What do you mean by `you hope that's the worst?'\" said Perry.\n\n\"Well, sir, I hardly know how to tell you.\"\n\n\"Speak out,\" said Cyril rather huskily.  \"If you think there's danger,\ntell us, so that we can tell the colonel, and put him on his guard.\"\n\n\"Well, young gents, I did give him a hint once, but he nearly jumped\ndown my throat,\" said John Manning.\n\n\"What!\" cried the boys in a breath.\n\n\"Well, sir, that's what you clever folk call methy-physical.  I told him\nI didn't think the Indians was to be trusted, and that I fancied they\nwere keeping an eye upon everything he did, and he insulted me, sir.\"\n\n\"Nonsense, John,\" said Perry.  \"My father wouldn't insult you.\"\n\n\"O' course you stick up for your dad, Master Perry, as is quite right\nnatural, and your duty to.  But I put it to you, Master Cyril: he's a\nsoldier, and I'm a soldier, and if one soldier calls another a stoopid\nold woman, with no more pluck than a quill pen, isn't that an insult?\"\n\n\"But Colonel Campion did not mean it, I'm sure,\" said Cyril impatiently.\n\"Now then, don't waste time.  What is it you think?\"\n\n\"Well, sir, I think our Indians said something to those Indians who were\nwith the llamas, and three or four turned back and followed after us.\"\n\n\"Are you sure?\" said Cyril anxiously.\n\n\"Well, sir, I'm sure I saw some of them dodging us and following.  I\nwasn't very sure at first, for I thought p'raps the colonel was right,\nand I was a bit of an old woman growing scared at shadows; but I feel\npretty sure now.\"\n\n\"But why should they follow us?\" asked Cyril tentatively.  \"You have\nsome idea in your head.\"\n\n\"Well, sir, I have; and whether it's right or wrong I can't say, but it\nseems to me as these people are all in league together, and they don't\nwant anybody to come up in the mountains.  They want to know what we're\nabout.\"\n\n\"But don't you fancy that, because it is what you have been thinking,\nJohn,\" said Perry.  \"You have been wonderfully anxious to know where we\nwere going, and what for.\"\n\n\"That's a true word, sir,\" replied the man, \"but I think they want to\nknow too.  It seems to me they're afraid we want to take something out\nof their country.\"\n\n\"Nonsense,\" said Perry.\n\n\"Nonsense!\" said Cyril sharply.  \"He's right, Perry.  It's just what I\ntold you, and--\"\n\n\"Now, quick, Mr Cyril!\" whispered John.  \"Turn towards me, as if you\nwere going to speak, and look toward the path we came by.\"\n\nCyril responded quickly, and saw by the light of the fire, which had\njust then blazed up brightly, a dark face peering at them over a great\npiece of rock.  He even saw the flash of the fire in the watcher's eyes,\nand then, as he pretended to hand something to Manning, his look was\naverted for a moment, and when he glanced again in the same direction,\nthe face was gone.\n\nCyril responded quickly, and saw a dark face peering at them over a\ngreat piece of rock.\n\n\"Now, Master Cyril, what do you say?\" whispered John Manning.\n\n\"I say it may only be curiosity,\" replied Cyril, \"but certainly we are\nbeing watched, and the colonel ought to know directly.\"\n\n\"Hah!\" ejaculated the old soldier, with a sigh of satisfaction, \"this\nputs one in mind of old times up in the hill-country, with the <DW65>s\nwaiting to go at you with matchlock or knife.  I didn't think I was ever\ngoing to have the luck to see a bit of fighting again.\"\n\nPerry started, and Cyril's face looked in the firelight as if it was\nflushed.\n\n\"Where's the colonel?\" he said quickly.  \"Be steady, Perry, old chap.\nThere's nothing to be frightened about.  Don't look as if anything was\nthe matter.  Come and find your father, and let's go and speak to Diego\nand the other man.\"\n\n\"But I can't speak to them,\" said Perry excitedly.\n\n\"Never mind, I will.  Come along.\"\n\n\"And suppose they shoot at us,\" whispered Perry, \"with an arrow or\nblowpipe?\"\n\n\"They'd better!\" said Cyril grimly.  \"But they won't do that.  Come on.\"\n\nHe walked on toward the fire, behind which the two Indians were\ncrouched, apparently enjoying the warmth and the charqui they were\nmunching; but they gazed furtively up at the two boys as they came up,\nand one of them started slightly as Cyril made a sudden stoop, but\nbecame impassive directly when the boy picked up two or three\nhalf-burned brands and threw them into the middle of the fire before\nholding his hands out to the flame.\n\n\"The waterfall makes it feel cold up here, and damp,\" he said to the\nguide in his patois, and the man smiled as he spoke, and then pointed up\na defile away above them as he replied.\n\n\"What does he say?\" asked Perry.\n\n\"That the wind comes down that narrow rift from the snow, and it is that\nwhich makes it cold.  I only half understand him.\"\n\nHe turned laughingly to the guide, and said a few words to that effect,\nand the man laughed and nodded as he replied.\n\n\"Oh, what a big fib!\" said Cyril merrily.  \"He says I speak his tongue\nbeautifully.--Oh, there's the colonel looking round at the mules.\nThey're having a beautiful feed here.  Plenty of grass for the mules,\"\nhe said to the Indian, and the man nodded again, and said it was good.\n\nThe colonel said something very similar, as the boys strolled carelessly\nup, at a time when Perry felt as if he must run to his father, shouting:\n\"Look out!  Danger!\"\n\n\"We must stay here two or three days, boys,\" the colonel said.  \"The\nmules will revel in this grass and fresh water, and make up for their\nfasting lately.\"\n\n\"I think not, sir,\" said Cyril, speaking carelessly, and making believe\nto pat one of the mules, which turned sharply round and showed him its\nheels.\n\n\"What do you mean, sir?\"\n\nCyril told him quickly; and as he spoke, the colonel's hand twitched,\nand went involuntarily to his side, as if he were seeking a sword.\n\n\"Humph!\" he ejaculated.  Then quietly, and looking at the mules: \"That's\nright, lads; don't make a sign.  I daresay John Manning is right.  He\nhas eyes like a hawk, and he is true as steel.  Well, I'm not surprised.\nI half expected it, though not quite so soon.\"\n\n\"What shall we do then, father?\" said Perry anxiously.  \"Go back?\"\n\n\"Englishmen don't go back, Perry,\" said his father gravely.  \"They would\nnot have colonised the whole world if they did.  No, boy, we are going\non, and I don't think there is anything to fear.  These people are all\njoined together to watch every stranger who comes into their country, in\ndread lest they should be in search of the Incas' treasures, and they\nwould be ready to fight in defence.\"\n\n\"And kill us, father,\" said Perry, with his lips paling in the\nfirelight.\n\n\"If we let them, boy.  But we are well-armed, John Manning and I, and\nknow how to use our weapons if it should come to a struggle, which I\ndoubt.\"\n\n\"Then you have come in search of something, sir!  I knew it,\" cried\nCyril.\n\n\"Yes, I have come in search of something, boy, and I mean to find it and\ntake it away out of the country in spite of all their watchfulness and\ncare.  Now, then, what do you say to that?  Are you afraid, and do you\nwant to get back?\"\n\n\"I don't know, sir,\" said Cyril quietly.  \"Yes, I do.  I can't help\nfeeling a bit frightened like.  I don't want to, but I do.\"\n\n\"And you wish to go back?  For I warn you I am going on in spite of all\nobstacles.\"\n\n\"No, I don't,\" said Cyril quietly.  \"I shall go with you.  I'm not going\nto leave Perry in the lurch.\"\n\n\"There's a coward for you, Perry, my boy,\" said the colonel, laughing.\n\"You must be a very good sort of a fellow to have made a friend like\nthat; one who risks his father's anger to come with you, and who is now\nready to run more risks for your sake.\"\n\n\"I'm afraid it isn't that, sir,\" said Cyril frankly.  \"I wanted to come\nbecause I thought it was going to be a great treat.\"\n\n\"There, say no more now.  Listen to me.  I shall take it for granted\nthat we have spies in the camp, and that, consequent upon their\ncommunication to the men of the llama caravan, some of that party are\nfollowing us.  Of course the poor fellows consider that they are\nperforming a religious duty, so I shall not charge them with their\naction.  They will go on watching us till they find I have done\nsomething which calls for immediate action.  Till then we are safe.\"\n\n\"Then you will not do anything, sir?\" said Cyril, looking quite aghast.\n\n\"Oh yes, I shall be upon my guard.  From now there will be watch set\nevery night in camp, and we shall sleep with our arms charged and ready\nfor action at a moment's notice.\"\n\n\"Yes,\" said Cyril, with a sigh of satisfaction.\n\n\"You can handle a gun, Cyril?\"\n\n\"Yes, sir, after a fashion.  I have often been up in the hills with my\nfather, shooting.\"\n\n\"That will do,\" said the colonel.  \"Now let's go and have a look at the\nfalls before setting watch and going to our blankets.  Tell the men to\nkeep up a pretty good fire, Cyril.\"\n\nHe led the way to where the Indians were seated as he spoke, and nodded\nto them smilingly as Cyril gave his orders; and then, as the men quickly\nobeyed them, the colonel led the way to the edge of a cliff!  From here\nthey could see the large body of water come gliding down in a curve from\nfar away up in the darkness, to gleam in the firelight as it passed\nthem, and then dive down into the deeper darkness below.\n\n\"An awful-looking place, boys, in the darkness,\" said the colonel\nquietly.  \"There now, we'll seek our blankets--at least you shall, for I\nshall take the first watch; John Manning will take the second.\"\n\n\"Shall I sit up with you, father?\" said Perry.\n\n\"No, my lad, we must husband our resources.  Your turn will come\nto-morrow night.  Remember what I said about the guns.  Make no show,\nbut have your ammunition ready for use at a moment's notice.  The\nIndians will see that, you may depend upon it, and act accordingly.\"\n\nHalf an hour later the two boys were lying inside a little shelter\nformed of the mules' packs and a wall-like mass of rock, listening to\nthe roar of the falls, and watching the figure of the colonel standing\ngazing out into the night, as he rested his chin upon the barrel of his\npiece.\n\n\"I shan't go to sleep to-night,\" said Perry in a whisper.\n\n\"Oh yes, you will.  I shall,\" replied Cyril.\n\nJust then John Manning came close up, with his gun in his hand.\n\n\"Good-night, gentlemen,\" he said.  \"Colonel says I'm to come and lie in\nthe shelter here.  Don't kick in the night, please, because I'm going to\nbe at your feet.  I had a messmate once out in India, who, when we were\nin barracks, used to sleep like a lamb, but so sure as we were on the\nmarch and had to share a tent, which meant he slept in his boots, you\nmight just as well have gone to sleep with a pack of commissariat mules,\nfor the way in which he'd let go with his heels was a wonder.\nGood-night, gentlemen, good-night.\"\n\n\n\nCHAPTER ELEVEN.\n\nTHE PERIL THICKENS.\n\nThere must have been something wonderfully lulling in the roar of that\nfall, and a feeling of great confidence in the fact that the colonel\nwould keep watch over them half the night, and John Manning, stern,\ntried, old soldier that he was, for the second half; for, though the\nboys lay there, fully convinced that they would not be able to sleep,\nand had visions of knife-armed Indians creeping toward them through the\ndarkness, they soon dropped off, and rested uninterruptedly for eight\nhours, when they sprang up at a touch from John Manning.\n\n\"If you gentlemen will relieve guard,\" he said quietly, \"I'll see about\nbreakfast.\"\n\nSoon after, as if nothing whatever had happened, they all sat down to a\nhearty meal, and that over, once more started upon their journey through\nthe mountains; the Indians seeming more willing, and at the suggestion\nthat the mules should rest for a day or two in that luxuriant pasturage,\neagerly assuring the colonel, through Cyril, that for days to come they\nwould find plenty, and that the road would be easier.\n\nOn hearing this, the colonel decided to go on, and soon found that the\nguide's words were correct; for, during the next six days, they\ntraversed smiling valleys, with grass and trees in abundance.\nSnowcapped hills rose high above them; but where they journeyed, they\nwere in a beautiful temperate climate, with rich soil and abundance of\nflowers.\n\nThis part of their journey was delightful; for the way along the passes\nwas easy, and the colonel, who was a dead shot, several times over added\nto their larder with his gun.\n\nBut they went on in no false security; for several times over they\npassed Indians, and were made fully aware of the fact that every mile\nthey took was carefully watched, and that the leader of the expedition\ninspected no mountain shelf, cave, or patch of dwarf forest, without his\nacts being duly noted, though in no observant way.\n\nDiego proved to be a perfect guide; and, making no objections now, he\nled them steadily on in a way which would have disarmed suspicion with\nsome people; but the colonel was quietly on the alert, and went on\nexamining plant, flower, and tree, at one time with all the patient care\nof a botanist; and at another time, when they were climbing some rugged\nshelf in a ravine, letting no mineral escape his observation.\n\nAnd all the while the little party, though they made no sign, were\nperfectly well aware that they were being watched.\n\n\"Strikes me that when we've got it, they won't let us take a simple lump\nof gold out of the place, Master Cyril.\"\n\n\"No,\" said the boy drily.\n\n\"But I mean some of the precious stones, Master Perry.  I shall have\nthem.\"\n\n\"How?\"\n\n\"Swaller 'em, sir, if I can manage it without being seen.  Why, do you\nknow I went down by that bit o' stream, last night, to bathe my feet,\nand before I got there, I stopped short and sneezed, and before I had\ntime to say, `Bless me!' there was an Indian's head popped up over a\nbush, and another from behind a stone, to see what was the matter.\"\n\n\"Yes; I've noticed something of the kind,\" said Cyril thoughtfully.\n\"But I shouldn't advise you to swallow any stones you find.\"\n\n\"Why, sir?\"\n\n\"Because they won't agree with you.\"\n\n\"They agree with chickens,\" said Manning, grinning, \"and make their hard\nfood digest, so I don't see why they shouldn't agree with me, sir.  But,\nI say, Master Perry, let it out now; I'm sure you'd feel a deal happier\nif you told us what the colonel's hunting for.\"\n\n\"I shall not tell you, because I don't know.  My father knows best about\nwhat he's doing, I daresay.  We thought, the other day, that we were in\ngreat danger; but you saw how quietly he took it, and how it all came to\nnothing.\"\n\n\"Perhaps the time has not come yet,\" said Cyril rather seriously; \"don't\nlet's talk too soon.\"\n\nNo more was said then; but a few days later, the others thought of how\nprophetic the boy's words had proved.\n\nBut it was not until another fortnight had passed, and a day had arrived\nwhen, after journeying through a deep defile of a similar character to\nthat which they had threaded upon the day when they met the llama\ncaravan, they reached a point upon the <DW72> of a huge mountain, from\nwhich they looked down over a glorious picture of hill and dale, verdant\nforest and wide-reaching plain, with, in two places, thin serpentine\nthreads of water glistening in the sun.\n\n\"At last,\" said the colonel gravely.  \"It has been a long journey, boys,\nbut we have reached the point I sought.\"\n\nCyril looked at him inquiringly; and Perry, who felt that he was\nexpected to speak, said: \"Yes; it's very grand.  How different to being\nin amongst the mountains!\"\n\n\"Yes, boy; we can breathe out here.  Did you notice the water in the\nlast two streams we passed?\"\n\n\"Yes; very beautiful with the overhanging trees, father.\"\n\n\"Yes; but the way they ran?\"\n\n\"No,\" said Perry.\n\n\"Look yonder, then,\" said the colonel, pointing to a little rivulet\nwhich leaped out from between two masses of rock.  \"Where is that\ngoing?\"\n\n\"Into another stream, I suppose,\" said Perry, \"and that will run into\nanother, and so on, till they all together form a big river, and run\ninto the ocean.\"\n\n\"Yes; but what ocean, my boy?  Don't you see that we have crossed the\nwatershed?  Till the last day or two, all the streams we passed have\nbeen going constantly west into the Pacific.  Now we have passed through\nthe mountains, and found the eastern <DW72>, where all run down to make\nthe vast rivers which flow into the Atlantic.\"\n\n\"I should not have known,\" said Perry.\n\n\"Nor I,\" said Cyril; \"but its much fresher out here.\"\n\n\"Yes, we have left the dry region behind, to get into the land of rains\nand many waters.  We saw no such forests as those which are spread\nbefore us even at this height.\"\n\n\"Is this high, sir?\" asked Cyril.\n\n\"Yes, my lad, about nine thousand feet.\"\n\n\"And shall we go back the same way?\" asked Perry.\n\n\"Possibly, my lad, but more probably not.  It depends upon the way the\nIndians treat us.\"\n\n\"But we can never find our way back any other way,\" cried Cyril.  \"Don't\nthink about it.  We should be lost up here in these mountains.\"\n\n\"No, because we have a guide with us, my boy, and if I can help it, he\nwill not leave us till he has seen us safely back.\"\n\nCyril said nothing, but the thought occurred to him:\n\n\"Suppose we wake some fine morning, and the guides are gone.\"\n\nThey camped that night on the <DW72> of the hill, and till it was growing\ndark, the colonel busied himself with his glass, carefully, as it seemed\nto the boys, inspecting the forest in every direction, and ending by\nclosing the telescope with a satisfied smile, which was not lost upon\nCyril.\n\n\"He has found it,\" he said to Perry, as soon as they were alone.\n\n\"Found what?\"\n\n\"What he has come after.\"\n\nPerry looked at him wonderingly.\n\n\"You have found out?\" he said.\n\n\"No, I wish I had; but didn't you see how pleased he seemed when he came\nback to supper, and said that we should camp here for a few days?\"\n\n\"Yes, I noticed that.\"\n\n\"Well, doesn't it mean that we have got to the spot at last that he was\nin search of?\"\n\nPerry shook his head.\n\n\"Well, you see if we don't find out something to-morrow.\"\n\nPerry had almost forgotten his companion's words at breakfast-time the\nnext morning, but they came to his memory as soon as they had done, for\nthe colonel said:\n\n\"Now, boys, we'll make a little expedition along the edge of the forest\nhere this morning.  Cyril, tell the men to mind the mules don't stray\ntoo far, and keep up the fire.\"\n\nJohn Manning looked sharply at the colonel, as much as to say: \"Then you\nmean me to come also, sir?\"\n\n\"Yes, I want you to carry spare ammunition and the game-bag.  I hope we\nshall have some sport along here,\" said the colonel, who had caught the\nold soldier's inquiring gaze.\n\nHalf an hour later, they were tramping along the mountain-<DW72>, through\nopen woods that were quite park-like, and gave them glimpses of the\nfar-spreading region below, all vested in a delicate bluish mist, while\nwhere they journeyed all was brilliant sunshine.  There was a delicious\nfeeling of spring in the air, for though the sun was hot, the air was\ncrisp and cool, making the task of walking easy, and giving the\ntravellers a feeling of elasticity, wanting when passing through the\ngloomy gorges of the huge mountain-chain.\n\nThe colonel led off as if he were quite accustomed to the place, though\nthere was no sign of a track, and before they had gone far, Perry\nwhispered a hope that they would not lose their way.\n\n\"No fear,\" said Cyril.  \"We are keeping the mountains on our left, and\nwe must keep them on our right as we go back.  We have only to keep\nalong till we strike the stream, and follow it up or down till we reach\nour fire.  I daresay we shall see it long before we are near, by the\nsmoke.\"\n\nThey had gone quite a couple of miles without seeing any trace of game,\nthe woods being wonderfully silent.  The colonel was on in front, and\nthe two boys about twenty yards behind, each bearing a gun, when Perry\nsuddenly paused.\n\n\"Where's John Manning?\" he said.  \"I thought he was close up.\"\n\nThey waited, and then whistled several times, but there was no response,\nand then Cyril ran back to where the land was more open, but still there\nwas no sign; and he was about to run forward again, and signal to the\ncolonel to stop, when the missing man suddenly appeared with his piece\nat the trail, running hard, but keeping himself bent down, to avoid\nbeing seen.\n\n\"What's the matter?\" said Cyril, as the old soldier came up.  \"Seen a\ndeer?\"\n\n\"No, sir; I only had a suspicion.\"\n\n\"What of?\"\n\n\"Struck me that Master Diego would come after us to see which way we\nwent.\"\n\n\"Well?\"\n\n\"Yes, I hung back to watch, and he's half a mile behind, tracking us by\nour footmarks, with his head down, or else he'd have seen me.\"\n\n\"Come on, and tell the colonel.\"\n\nThey hurried forward, and joined Perry, waiting for them anxiously.\n\n\"At last,\" he said excitedly.  \"Did you see?\"\n\n\"See what?\"\n\n\"Those Indians.\"\n\n\"No.  Where?\" said Cyril.\n\n\"On both sides, among the trees.  They are watching us.  What ought we\nto do?\"\n\n\"Rearguard closes up on the main body,\" said John Manning quickly.\n\"Single file, and at the double.  Now, sir, you head the advance.\nMarch.\"\n\nCyril sprang forward to overtake the colonel, looking down as he trotted\nforward in search of trampled-down grass and broken twigs; but from the\nfirst he saw nothing, neither could he hear a sound, and after some\nminutes' progress, he pulled up short, and breathing hard.\n\n\"We had better spread out now,\" he said, \"or we shall overlook him.\"\n\n\"Didn't you see which way he went?\" said John Manning.\n\n\"Out this way somewhere,\" said Perry.  \"He can't be far away.\"\n\n\"Enemy closing in,\" said the old soldier in a low voice.  \"Forward, my\nlads.  We must find him now.\"\n\nThose last words sent a chill through Cyril, who sprang forward again,\nand then nearly uttered a shout; for, about fifty yards in front, he\ncaught sight of the colonel standing half hidden by the thick growth at\nthe edge of a clearing, where some dozen or so of men were busy\napparently cutting wood.  Beyond them were two rough huts thatched with\nboughs, and piled up in little stacks were fagots of the wood which the\nmen had cut down.\n\nThey were so busy over their task, that they had not noticed the\npresence of the colonel, neither did they hear the approaching footsteps\nas they worked on.  But the colonel did, and turned and hurried back to\nmeet the boys, holding up his hand to command silence.  His eyes were\nflashing with satisfaction as they came up, and he had a branch of one\nof the trees about them in his hand.\n\n\"We thought we had lost you,\" said Perry excitedly.  \"What's that?\"\n\n\"The magic tree I have come all these thousands of miles to seek, boy,\nand now--\"\n\n\"Will you cast your eye this way, sir?\" said John Manning, in a hoarse\nlow voice of warning.  \"Indians all around.  Do it mean mischief, or are\nthey only friends?\"\n\nHe gave his head a backward wag as he spoke, and as Cyril looked\nexcitedly in the indicated direction, he saw that which made him thrust\nhis hand into his pouch to count the cartridges; for if the coming\nIndians meant offence, they as travellers were in mortal peril of losing\ntheir lives.\n\n\n\nCHAPTER TWELVE.\n\nAT BAY.\n\nTo have attempted to escape, the colonel said, would be madness, for it\nwould have suggested fear of the approaching Indians, and made them\nthink at once that the visit to their secluded haunts meant no good to\nthem; so throwing his piece into the hollow of his left arm, and bidding\nthe others do as he did, Colonel Campion took a few steps forward to\nmeet the Indians, and held out his hand.\n\nThis had the effect of making them halt a few yards from them, and\nkeeping their faces fixed upon the English party, they talked rapidly\namong themselves.\n\nAt that moment Cyril caught sight of Diego hanging back among the men in\nthe rear.\n\n\"There's our guide, sir,\" he said hurriedly.  \"Shall I call him?\"\n\n\"He there?\" said the colonel sharply.  \"Yes, call him.  No: go through\nthem, and fetch him, boy.\"\n\nCyril hesitated for a moment or two, and his heart beat high; but the\norder had been given in true military style, and it had its influence.\nThe boy felt that he would be backed up by the colonel in all he did,\nand throwing his gun over his right shoulder, he stepped boldly forward,\nfinding that the white was master even here; for the Indians, taken by\nsurprise at his firmness, parted at once to let him pass, and then\nCyril's pulses beat a little more rapidly, for the men closed up again,\nshutting him off from his friends.\n\nThe boy felt this, but he knew that he must not show fear, and without a\nmoment's hesitation, he walked on up to where Diego stood half hidden\nbehind a couple of the Indians, and clapping his hand upon the man's\nshoulder, \"Come,\" he said, \"the colonel wants you.\"\n\nThe guide shrank at Cyril's touch, and looked at his fellows for\nsupport, but no one stirred, and uttering a low sigh, the man allowed\nhimself to be marched away to where the colonel stood, the Indians\ngiving way on either side, and then closing up again in silence, and\nwithout the slightest show of menace.\n\nFor to them it was as if a superior being had calmly passed among them\nand fetched one, each man feeling relieved that he was not the one\nselected, and that, had he been, he would have felt compelled to go.\n\n\"Well done, British boy,\" said the colonel to Cyril, as he stopped\nbefore him with the guide, who looked of a curious dusky colour now; his\neyes showing the whites around the iris, and his lips seeming parched as\nhe moistened them hastily with his tongue from time to time.\n\n\"Now then,\" continued the colonel sharply, as if he were addressing a\ndelegate from a mutinous company of his old regiment, \"why have you\nbrought all these men after me, sir?--Interpret quickly, Cyril.\"\n\nThis was done, and the man's voice trembled as he answered.\n\n\"He says they made him come, sir,\" said Cyril.\n\n\"Which is a lie,\" cried John Manning; \"for he has been dodging us all\nthe time.\"\n\n\"Silence there.  'Tention!\" cried the colonel harshly, and the old\nsoldier drew himself up smartly, lowered and then shouldered arms, just\nas if he had been on parade.\n\nIt was a trifle, but it had its effect upon the Indians, giving them a\ngreat idea of the importance of the colonel, who stood there, erect and\nstern, issuing his orders; and in their eyes he was a great white chief,\nif not a king.\n\n\"Now,\" he said sharply, \"let that boy ask him what these people want.\"\n\nCyril interpreted and obtained his answer, the peril of their position\nsharpening the boy's faculties, and making him snatch at words of which\nhe was in doubt.\n\n\"They have come,\" said Cyril, \"to see why you are here.  They say you\nhave no right to come amongst the kina gatherers, and that you must go\nback to the coast at once.\"\n\n\"Indeed!\" said the colonel haughtily.  \"We shall see about that.  Tell\nthem, boy, that I am the English chief of a great white queen; that I\nhave come into this country to examine it and its products, and that I\nwill shoot dead with this piece the first man who dares to interfere\nwith me and mine.\"\n\n\"Hear, hear!\" growled John Manning.\n\n\"Silence in the ranks,\" cried the colonel sharply; while, gaining\nconfidence, Cyril's voice partook somewhat of his leader's imperious\ncommand, as he repeated the words as loudly as he could, so that all\nmight hear.\n\nThere was a low fierce murmur from the little crowd, which was now\naugmented by the bark peelers, who closed the English party up from the\nrear.\n\n\"What do they say?\" cried the colonel, taking a step forward, and\ncocking his piece at the same moment.\n\n\"That they will make us prisoners, sir,\" said Cyril.\n\n\"Who dared say that?\" roared the colonel, and taking another step\nforward, he looked fiercely round, with the result that to a man the\nIndians bent their heads before him, and not one dared look him in the\nface.\n\n\"Hah!\" he ejaculated, \"that is better.  Now tell them I wish to see the\nkina gathered and prepared.\"\n\nCyril gave the interpretation of his words, and Diego and an old Indian\ncame humbly forward and laid down their bows and arrows at his feet.\n\nThe colonel took a step and planted his foot upon the weapons.  Then\ndrawing back, he pointed down.\n\n\"Pick them up!\" he said sternly in English, and repeated the words in\nSpanish, when a low murmur of satisfaction arose, and the men stooped,\nlifted their weapons, and then making deprecating signs, they led the\nway into the clearing where the cinchona trees had been cut down, and\nthe people had been busy collecting and drying the bark.\n\nThe colonel went on first, and Cyril and John Manning next, followed by\nPerry and Diego.\n\n\"It does one good, Master Cyril,\" whispered John Manning, \"it does one\ngood again, my lad.  That's the sort of man the colonel is.  Fit for a\nking, every inch of him.  There ain't many men as would have faced a\nbody of savage Indians with their bows and arrows like that.  He's the\nright sort of stuff, ain't he? and yet they let him leave the army and\ngo on half-pay.\"\n\n\"Yes, but do you think there will be any treachery?\" replied Cyril.\n\n\"No, sir, I don't, so long as we show 'em we mean to keep the upper hand\nof 'em.  They daren't.  They know the colonel meant what he said, and\nfelt that every word he said was true, and that a big chief had come\namong 'em.\"\n\n\"Yes, I could see that,\" said Cyril.\n\n\"My word, he was like a lion among a lot o' them big, long-necked sheep,\nsir; and you did your part of it splendidly.\"\n\n\"I did?\" said Cyril, looking at the man in wonder.\n\n\"Yes, you, sir.  I only wish our Master Perry would speak up as bold.\"\n\n\"Why, John Manning,\" said Cyril, half laughing, \"if you only knew how I\nfelt.\"\n\n\"I do, sir.\"\n\n\"Not you, or you would not talk like that.\"\n\n\"I tell you I do, sir.  You felt just as I did first time I went into\naction, and heard the bullets go whizzing by like bees in the air, and\nsaw some of them sting the poor fellows, who kept on dropping here and\nthere, many of 'em never to get up again.  I thought I was in a terrible\nfright, and that I was such a miserable coward I ought to be drummed out\nof the regiment; but it couldn't have been fright, only not being used\nto it; and I couldn't have been a coward, because I was in the front\nrank all the time, close alongside of your father; and when we'd charged\nand driven the enemy flying, the colonel clapped me on the shoulder and\nsaid he'd never seen a braver bit of work in his life, and of course he\nought to know.\"\n\n\"I did feel horribly frightened, though,\" said Cyril.\n\n\"Thought you did, sir, that was all.  You couldn't have done it better.\"\n\n\"I don't know,\" said the boy, smiling.  \"Suppose the Indians had found\nme out?\"\n\n\"Found you out, sir?  Bah!  If it comes to the worst, they'll find out\nyou can fight as well as talk.  Now, just look here, sir; didn't you\never have a set to at school, when you were at home in England?\"\n\n\"Yes, two or three.\"\n\n\"And didn't you feel shimmery-whimmery before you began?\"\n\n\"Yes.\"\n\n\"And as soon as you were hurt, forgot all that, and went in and\nwhipped.\"\n\n\"Well, yes, I suppose so.\"\n\n\"Of course you did, sir.  That's human nature, that is.  But, I say, Mr\nCyril, sir, what does it all mean?  Why has the colonel come out here?\nHe can't have come just to see people cut down a few trees and peel off\nthe bark.\"\n\n\"I begin to think he has.\"\n\n\"But I could have taken him down in Surrey, sir, and showed him into\nwoods where they were doing all that to the oak trees, without coming\nout here, or running any risks of getting an arrow sent through you,\njust as if you was a chicken got ready to roast.\"\n\n\"I don't quite understand it yet,\" said Cyril; \"but don't talk any more\nnow.  Look, look! what is he going to do?\"\n\n\n\nCHAPTER THIRTEEN.\n\nIN TREASURE LAND.\n\nThey were by this time close up in front of the huts of the bark\ngatherers, when all at once one of the huge condors came swooping along\noverhead, looking gigantic up against the sky.  And then it was as if a\nsudden idea had struck the colonel, who raised his piece, took aim,\nlowered it, and hesitated; for the huge bird was at a great distance,\nand the people looked at him wonderingly.  The next moment his rifle was\nat his shoulder again, there was the flash and puff of white smoke, the\nsharp crack, and the rumbling echoing roar in the mountains, as the\ncondor was seen to swerve and then dart straight upward.\n\n\"Missed!\" muttered John Manning, \"but he felt the bullet.\"\n\n\"Hit!\" cried Cyril excitedly, for all at once the bird's wings closed,\nand it fell over and over and then dropped like a stone, crashing in\namong the trees about a hundred yards away.\n\nThe Indians had looked on at first incredulously, and several of them\nexchanged glances as the condor shot upward as if to escape unharmed;\nbut the moment it turned over and began to fall, they set up a loud\nshout and rushed off to pick up the fallen bird, the whole crowd making\nfor the dense patch of forest, and then walking back steadily, bearing\nthe bird in triumph.\n\n\"Rather a risky thing to do, boys,\" said the colonel, reloading as he\nspoke.  \"If I had missed, I should have done harm to the position we\nhave made in these people's estimation.  But I felt that I could hit the\nbird, and now they will believe that I may prove a terrible enemy in\nanger.\"\n\n\"Do it?  Of course he could,\" whispered John Manning.  \"I've known him\ntake a rifle from one of our men lots of times, and pick off one of the\nBeloochees who was doing no end of mischief in our ranks up in the\nmountains.\"\n\nBy this time the Indians were back, looking full of excitement, and\nready almost to worship the white chief who had come amongst them, with\nsuch power of life and death in his hands--powers beside which their\nbows and arrows and poison-dealing blowpipes seemed to them to be\npitiful in the extreme.  They laid the body of the great bird, which was\nstone-dead, at his feet, and then looked at him wonderingly, as if to\nsay, \"What next?\"\n\nThat shot had the effect which the colonel had intended to produce, for\nto a man the Indians felt the terrible power their white visitor held in\nhis hand, and each felt that he might be the object of his vengeance if\nany attack was made.\n\nBut Colonel Campion felt that the effect was only likely to be\ntemporary, and that he must gain the object for which he had made his\nperilous journey as quickly as possible, and begin to return before the\nimpression had worn off.\n\nBidding Cyril then tell their guide that he should camp there for a few\ndays, he sent the two men back for the mules, giving orders that they\nshould take a couple of the Indians who had followed them to help.\n\nHis manner carried the day, and the party of four departed.\n\n\"I suppose it's all right, Master Cyril,\" whispered John Manning; \"but I\nshould have thought we'd ha' done better by fortifying our own camp, and\nnot running our heads right into the lion's mouth; but the colonel knows\nbest, and we've only got to obey orders.\"\n\nCertainly that seemed to be the safest course to pursue--a bold one; so\nin this spirit, and as if the colonel felt that there was nothing\nwhatever to be feared from the people, the mules and packages were\nbrought up.  A snugly-sheltered spot was selected, close to a spring\nwhich came gushing from the rock, and a fresh camp made; the party going\nand coming among the cinchona gatherers as if they were invited\nvisitors; while the Indians themselves looked puzzled, and watched every\naction from a distance.\n\nThat night, beside the fire, surrounded by the dense growth of the\nlife-preserving trees he had sought, the colonel became more\ncommunicative.\n\n\"You boys have, I daresay, canvassed why I undertook this expedition,\"\nhe said, \"and, I suppose, took it for granted that I came in search of\nthe gold supposed to be hidden by the Peruvians, to save it from the\nrapacity of the Spaniards.\"\n\n\"Yes, sir; that's what I thought,\" said Cyril.\n\n\"Or else to find one of the di'mond walleys,\" growled John Manning.\n\n\"This is not the right direction for them, my man,\" said the colonel,\nsmiling.  \"You have to seek for them between the leaves of books.  No,\nboys; I came to seek something of far greater value to my\nfellow-creatures than a buried store of yellow metal, which may or may\nnot exist.  It is possible that a number of the sacred vessels from some\nof the old temples may have been hidden by the priests, who, at their\ndeath, handed down the secret to their successors; but I think it is far\nmore likely to be a fable.  Still, the Indians believe in it, and if\nthey knew that a discovery had been made, they would destroy the lives\nof the finders, sooner than that the gold should be taken out of the\ncountry.\"\n\n\"Then you have not come to find the gold, sir?\" said Cyril; while Perry\nlay there upon his chest, resting his chin upon his hands, and elbows on\nthe earth, gazing up in his father's face.\n\n\"No, boy; I have come, and I am running some risks, I know, to drag out\ninto the light of day the wondrous medicine which has saved the lives of\nhundreds of thousands, and made it possible for men to exist in the\nfever-haunted countries spread around the globe.\"\n\n\"You mean quinine,\" said Cyril.  \"Father always keeps a bottle in his\ndesk.\"\n\n\"Yes, I mean quinine, the beautiful crystals obtained from the bark of\nthese trees, boy; the medicine kept so jealously guarded here, the only\nplace where it is produced, high up on the eastern <DW72> of these\nmountains.  I have come to seek it, and have found it far more easily\nthan I expected: we are sitting and lying here right in the middle of\none of the cinchona groves.\"\n\n\"But we can't take away much, father, even if they will let us,\" said\nPerry.\n\n\"Wrong, boy.  I hope that we shall be able to bear away, unseen, enough\nto stock the world, and to make the drug, which is a blessing to\nhumanity, plentiful, instead of civilised Europe having to depend upon\nthe supply from here--from this carefully-guarded place.\"\n\n\"You mean to take away some young trees,\" said Cyril excitedly.\n\n\"I should like to do so, but that is a doubtful way, my boy.  The young\ntrees would be awkward to carry, and transplanting trees often means\nkilling them.  We must try something better than that, though.  I shall\nsee what I can do in making one bundle, with the roots carefully bound\nup in damp moss.\"\n\n\"Yes, we might do that,\" assented Cyril, \"but we didn't bring a spade.\"\n\n\"Let us find some tiny trees, and we'll do without a spade,\" said the\ncolonel quietly.  \"But I am in this position, boys.  I know very little\nabout the trees we see around us.  That they are the right ones there\ncan be no doubt, for the Indians are camped here, cutting them down, and\npeeling off and drying the bark.  There are several kinds which produce\ninferior kinds of quinine; but these laurel-like evergreen trees produce\nthe true, the best Peruvian bark; and it is to take away the means of\npropagating these trees in suitable hot mountainous colonies of our own,\nthat we are here.  Now, how is it to be done?\"\n\n\"Indians won't let it be done, sir,\" said Manning.  \"Here, I know lots\no' places up Simla way where it would grow fine.  Up there, north o'\nCalcutta, sir.\"\n\n\"Yes; there are spots there where it might be grown, or in the mountains\nof Ceylon,\" said the colonel; \"but we have to get it there.\"\n\n\"I know,\" said Cyril.  \"Let's get heaps of seed.  Why, we might till our\npockets that way.\"\n\n\"Yes; that is my great hope, boys; so, whenever you see seed-pods or\nberries nearly ripe, secure them.  But we are surrounded by\ndifficulties.  We may be here at the wrong time of year, though I\ncalculated that as well as I could; and now that we are here, I have\nbeen terribly disappointed, for so far, instead of seeing seed, I have\nnoted nothing but the blossoms.  It is as if we are too early, though I\nhope these are only a second crop of flowers, and that we may find seed\nafter all.\"\n\n\"But these sweet-smelling flowers, something like small lilac, are not\nthe blossoms of the trees, are they?\" said Perry.\n\n\"Yes, those are they,\" said the colonel.  \"Now my secret is out, and you\nknow what we have to do.--Well, Manning, what is it?\"\n\n\"My old father had a garden, sir, and he used to grow little shrubs by\ncutting up roots in little bits, which were often dry as a bone when he\nput them in, but they used to grow.\"\n\n\"Yes,\" said the colonel.  \"Quite right; and now we are here, in spite of\nall opposition, we must take away with us seeds, cuttings of twigs, and\nroots, and if possible, and we can find them, a number of the tiny\nseedlings which spring up beneath the old trees from the scattered seed.\nThere, that is our work, and all must help.--Do you hear, Manning?\"\n\n\"Oh yes, sir, I hear, and if you show me exactly what you want, I'll do\nmy best; but, begging your pardon, sir, ain't it taking a deal o'\ntrouble for very small gains?\"\n\n\"No, my man, the reward will be incalculable.\"\n\n\"All right, sir, you know best.  I'll do what you tell me, and when\nwe've got what we want, I'll fight for it.  That's more in my way.  But,\nbegging your pardon once more, wouldn't it be better for you to go to\nthe head-man, and say, through Master Cyril here: `Look here, young\nfellow, we've come a long journey to get some seed and young plants of\nthis stuff; can't you make a sort of trade of it, and sell us a few\npen'orth civilly.'\"\n\nThe colonel laughed.\n\n\"No.  They will not let us take a seed out of the country if they can\nprevent it.  I will tell you all the worst at once.  They will make a\nbold effort to master the dread with which I have succeeded in inspiring\nthem, and fight desperately to stop us when we get our little store.\"\n\n\"Then, begging your pardon again, colonel, wouldn't it ha' been better\nto have come with a couple of companies of foot, and marched up with\nfixed bayonets, and told him that you didn't mean to stand any nonsense,\nbut were going to take as much seed as you liked?\"\n\n\"Invited the rulers of the country to send a little army after us?\"\n\n\"Yes, of course, sir; but they've got no soldiers out here as could face\nBritish Grenadiers.\"\n\nThe colonel was ready to listen to every opinion that night, and he\nreplied quietly:\n\n\"I thought it all out before I started, and this was the only way--to\ncome up into the mountains as simple travellers, reach the hot <DW72>s\nand valley regions where the cinchona grows, and then trust to our good\nfortune to get a good supply of the seed.  But, even now, from our start\nfrom San Geronimo we have been watched.  You have noticed it too, boys.\nEven the guide we took has arrayed himself against us from the first,\nand, while seeming to obey my orders, has taken care to communicate with\nevery one we passed that he was suspicious of my motives.  Every mile we\nhave come through the mountain-range has been noted, and will be noted,\ntill we get back.\"\n\n\"Why not go back, then, some other way, sir?\"\n\n\"Because we cannot cross the mountains where we please.  The road we\nfollowed is one which, no doubt, dates from the days when the Incas\nruled, and there are others here and there at intervals, but they will\nbe of no use to us.  Somehow or other, we must go back by the way we\ncame, and I hope to take at least one mule-load with us to get safely to\nEngland.  There, that is enough for to-night.  Now for a good rest and\nwe shall see what to-morrow brings forth.  Cyril and Perry, you will be\non sentry till as near midnight as you can guess, and then rouse me.\nI'm going now to take a look round at the mules, and then I shall lie\ndown.\"\n\nHe rose and walked away to where the mules were cropping the grass,\nwhich grew abundantly in the open places, and as soon as he was out of\nhearing, John Manning began to growl.\n\n\"All right, young gentlemen,\" he said, \"I'm ready for anything; but, of\nall the wild scarum-harum games I was ever in, this is about the\nwildest.  Come up here to steal stuff! for that's what it is, and you\ncan't call it anything else.  I've know'd people steal every mortal\nthing nearly, from a horse down to a pocket-knife.  I've been where the\n<DW65>s tickled you when you was asleep and made you roll over, so that\nthey could steal the blanket you lay upon.  I've seen the crows in Indy\nsteal the food out of the dogs' mouths; but this beats everything.\"\n\n\"Why?\" said Perry shortly.\n\n\"Why, sir?  Because physic's a thing as everybody's willing enough to\ngive to someone else; I didn't think it was a thing as anybody would\never dream o' stealing.  As you may say, it's a thing as couldn't be\nstole.\"\n\n\"Father knows what he is about,\" said Perry shortly.\n\n\"Course he does, sir.  Nobody denies that.  We've got to begin taking\nphysic with a vengeance.  All right: I'm ready.  And I was thinking all\nthe time as we should bring back those four-legged jackasses loaded with\ngold and precious stones.  All right, gentlemen.  As I said before, I'm\nready; and it's a good beginning for me, for I shall get a long night's\nrest; so here goes.\"\n\nHe rolled himself in his blanket, then lay down with his feet near the\nfire, and began to breathe the heavy breath of a sleeper the next\nminute.\n\n\"Well, Cil,\" said Perry, \"what do you think of it?\"\n\n\"Don't know,\" said Cyril.  \"Yes, I do.  They're wonderfully watchful\nover the bark, and as soon as they know what we are after, they'll stop\nus.\"\n\n\"Then we must not let them see what we are after, my lad,\" said the\ncolonel, who had returned unseen.  \"We must collect plants and flowers\nof all kinds, and load a couple of the mules.  That will help to disarm\nsuspicion.--Pieces loaded?\"\n\n\"Yes, sir.\"\n\n\"That's right.  We must keep military watch now regularly; but there\nwill be nothing to fear to-night.\"\n\n\n\nCHAPTER FOURTEEN.\n\nTHE NIGHT-WATCH.\n\nThose were very encouraging words, and they seemed to tingle in the\nboys' ears as the colonel followed his servant's example, rolled a\nblanket about his shoulders, and lay down with his head resting on one\nof the mules' loads; but the impression soon died away, leaving the lads\nclose together, with their guns resting on the grass, listening in the\ndeep silence of the starlit night, and for some time without speaking a\nword.\n\n\"Come a little farther away,\" whispered Perry at last.  \"I want to\ntalk.\"\n\nThey moved a few yards away from the sleepers, and stopped beneath a\ngreat spreading tree at about equal distance from the colonel and the\nfire, which glowed faintly, but gave sufficient light for them to see\nDiego and the other Indian squatted down, making tents of their long\ngarments, and with their chins bent down upon their breasts; but whether\nasleep, or waking and watchful, it was impossible to say.\n\n\"Well?\" said Perry at last, after they had been straining their ears to\ncatch different sounds, now the trickling murmur of falling water, now\nsome strange cry from far away in the woods, or the whisper of a breeze\nwhich came down from the mountains to pass away among the trees.\n\n\"Well?\" said Cyril.\n\n\"Isn't it awfully quiet?\"\n\n\"Yes.\"\n\n\"Look over there, just to the left of the fire.  Isn't that some one\nwatching us?\"\n\n\"Tree trunk,\" said Cyril laconically.\n\nThere was a pause, and then Perry whispered again.\n\n\"I say, I don't want to be cowardly, but there's some one coming slowly\nthrough the trees.  I caught a glimpse of his back.  He's stooping\ndown--there, between those two big trunks, where it's open.  Don't you\nsee--stooping?\"\n\n\"Yes, I see, and nibbling the grass as he comes.  One of the mules.\"\n\nPerry shaded his eyes--needlessly, for there was no glare to shut out--\nand he soon convinced himself that his companion was right.\n\nBut he felt annoyed, and said testily:\n\n\"I wish you wouldn't be so ready to contradict everything I say.\"\n\nCyril laughed softly.\n\n\"Why, you didn't want it to be an enemy, did you?\"\n\nPerry made no reply, and they stood for some time together in silence,\nlistening to the _crop, crop_ sound made by the mules, and the\nwhispering sighs of the wind, which came down sharp and chill from the\nmountains.  At last Cyril spoke again.\n\n\"Let's walk round the camp.\"\n\n\"You can't for the trees.\"\n\n\"Oh yes, we can.  It's cold standing here.  We'll work in and out of the\ntrees, and make a regular path round.  It will be better than standing\nstill.\"\n\n\"Very well,\" said Perry shortly.  \"Go on first.\"\n\nCyril shouldered his piece and stepped off cautiously for a couple of\ndozen yards, and then struck off to the left, meaning to make the fire\nact as a centre round which they could walk, keeping guard and\nthemselves warm; but before he had gone many steps he stopped short.\n\n\"Look here,\" he whispered, \"you are a soldier's son, and ought to teach\nme what to do in keeping guard.\"\n\n\"There's nothing to teach,\" said Perry.  \"All you've got to do is to\nkeep a sharp lookout.\"\n\n\"Yes, there is.  If we keep together like this, we leave a lot of the\ncamp exposed.  What we ought to do is for one to go one way, and one the\nother; then meet, cross, and go on again.  It would be far better.\"\n\n\"But then we should be alone so long.  We had better keep together.\"\n\n\"Very well,\" said Cyril shortly; but he owned to himself that he felt\nbetter satisfied, for it was lonely, depressing work there in the\ndarkness.\n\nCyril stepped forward again, going slowly and carefully through the\nthick growth, making as little noise as possible, and trying to keep as\nnearly as possible to the same distance from the fire--no easy task, by\nthe way--but he had not gone far before he stopped short and started\naside, bringing his gun down to the present.  For, all at once, from out\nof the darkness, some one seemed to strike at him, the blow cutting\nthrough the twigs and leaves by which he was surrounded with a loud\nwhistling noise, while the stroke was so near, that he felt the air move\nclose to his face.\n\n\"Fire--fire!\" whispered Perry excitedly.\n\n\"What at?  I can't see any one,\" replied Cyril, as he stood with his\nfinger on the trigger.\n\nHe felt his heart beat with a heavy throb, and his hands grew moist, as\nhe tried hard to pierce the darkness, and fix his eyes upon the enemy\nwho had made so cowardly a blow at him; but the thick branches shut out\nevery ray of light, and the silence was now painful in the extreme.  The\nposition was the more startling from the fact that neither could tell\nfrom which side the next blow would come.\n\nBut still that blow did not fall, and it seemed to Cyril, as he stood\nthere holding his breath, that the Indian who had struck at him so\ntreacherously was waiting until he moved, so as to make sure before\nstriking again.  At last the painful tension came to an end, for\nsuddenly, from just in front, there was a heavy sigh, and _crop, crop,\ncrop_, followed by a burst of laughter from the boy.\n\n\"Oh, I say, Perry,\" he cried, \"what a game!  Fancy being scared like\nthat by a mule.\"\n\n\"Then it was one of the mules?\"\n\n\"Of course; we frightened the poor thing, and it kicked out at us.  Come\nalong.\"\n\nHe bore off a little to one side, and they passed the browsing animal,\nand, though describing rather an irregular circle, made their way round\nthe fire, getting back pretty exactly to the place from which they\nstarted.\n\nThis was repeated several times, and then, for a change, Cyril proposed\nthat they should strike off a little, straight away from the camp.\n\nPerry was willing, and they put their plan in operation, for no special\nreason other than that of seeing the ground was clear in different\ndirections, and to relieve the monotony of the watch.\n\n\"You lead now,\" said Cyril, in a low voice, so as not to disturb the\nothers, who, in thorough confidence that a good watch would be kept, and\nthat there was no fear of any danger, were sound asleep.\n\nPerry led on, finding the way more open a short distance from the camp,\nbut he had not led thirty yards when he stopped short.\n\n\"Hallo! another mule?\" said Cyril.\n\n\"Indian!\" said Perry huskily; and, as Cyril pressed forward to his\ncompanion's side, there, hard to define, but plain at last, stood one of\nthe Indians, who raised his arm and pointed back, uttering two or three\nwords in a guttural tone.\n\n\"What does he say?\"\n\n\"That we must go back to the fire.  Perhaps we had better,\" said Cyril.\n\"I don't like his being there, though.  Look here,\" he said quickly;\n\"let's make haste back, and go right out the other way.\"\n\n\"What for?\" said Perry, following his companion.\n\n\"I'll tell you directly.\"\n\nFive minutes later they were checked just on the other side by another\nIndian who started up right in their path.\n\n\"Come and warn my father,\" said Perry excitedly.  \"They're going to\nattack us.\"\n\n\"No; I think not,\" replied Cyril decisively.  \"They're sentries.  Come\nand try another way.\"\n\nHe led off again, after they had returned to the fire, finding that they\nwere not followed, and that all was still; and again they were stopped\nby an Indian starting up and ordering them back.\n\n\"That's it,\" said Cyril quietly; \"they've surrounded us with sentries.\"\n\n\"To attack us?\"\n\n\"No; to see that we don't escape; and while we were walking round and\nround, they were within a few yards of us, listening to all our\nmovements.\"\n\n\"But they couldn't have been there then, or they would have started up\nas they did just now.\"\n\n\"No; we weren't doing anything they minded; but as soon as we tried to\ngo straight away, they stopped us.  Let's try once more.\"\n\nHe led off quickly again, with the same result; and then Perry turned\nback to where his father lay asleep.\n\n\"What are you going to do?\" whispered Cyril.\n\n\"Wake up my father, of course.  We are attacked.\"\n\n\"Don't do that,\" said Cyril decisively.  \"We are not attacked, or they\nwould have seized us at once.  I'm sure they are only guarding us, to\nmake sure that we don't try to escape.  It's of no use to wake him till\nthe proper time.\"\n\nPerry hesitated.\n\n\"But we are in danger.\"\n\n\"No; I don't think we are.  They are watching us, but they don't mean to\nattack us, or they would do so.  You'll see now.  We've come among them,\nand they'll keep us under their eye, and perhaps will not let us go\nagain.  Look here: let's go and speak to Diego.\"\n\nPerry was easily led, and yielding to his companion's decisive manner,\nhe followed to the fire and then round to the other side, where the\nIndian guide and his companion were squatted down with their chins\nresting upon their chests.\n\nThey made no sign as the boys came silently up, and appeared to be fast\nasleep; but Cyril knew better, for he saw in the dim glow shed by the\nfire, a slight tightening of the man's hand upon his bow.\n\n\"They're asleep,\" whispered Perry.  \"Better come to my father.\"\n\n\"Asleep with one eye open, and on the watch,\" said Cyril quietly, and he\nbent down and whispered a few words.\n\nThey were electric in their effect, for both men raised their heads, and\ntheir eyes glittered in the faint light from the fire.\n\n\"Didn't take much waking,\" said Cyril, with a little laugh.  Then\nturning to Diego, he said, in the man's half-Spanish jargon:\n\n\"Why are the Indians on the watch all round here?\"\n\nThe man looked at the speaker intently.\n\n\"Are the Indians watching all round?\" he said quietly.\n\n\"You know they are.  Why is it?  To keep us from going away?\"\n\nThe man looked at him intently, and then nodded his head.\n\n\"And suppose we try to go away, what then?  Would they fight?\"\n\n\"Yes,\" said the guide gravely.\n\n\"And try to kill us?\"\n\n\"Yes, they would kill you.\"\n\n\"Try to, you mean.\"\n\n\"No,\" said the man gravely.  \"Kill you.  You are few, they are many.\"\n\n\"Stop a moment,\" said Cyril, as the man turned his head aside wearily.\n\"Will they try to kill us if we stay?\"\n\n\"No.\"\n\nCyril tried to get more information from the man, but he shook his head,\nand made a pretence of being so lazy and unable to comprehend the boy's\nwords, that Cyril gave up in disgust, and turned impatiently away.\n\n\"It's of no good to-night,\" he said.  \"We heard all that he is likely to\nknow.  Let's walk round again.\"\n\n\"But they may strike at us in the dark.\"\n\n\"No, they will not do that.  I'm not afraid.  Let's go through with our\nwatching, till we think it's midnight, and then wake up the colonel.\"\n\n\"We'd better call him now.\"\n\n\"No; if we did, it would only be giving a false alarm, when we know that\nthere is no danger.  Come along.\"\n\nThe weaker mind yielded to the stronger, and the march round was begun\nagain, one which required no little courage, knowing, as the boys did,\nthat there must be quite a dozen Indians within striking distance, and\nevery rustle they heard, made probably by one of the grazing mules,\nmight be caused by an enemy creeping forward to strike a blow.\n\nAt last, when they felt that it must be getting toward midnight, Cyril\nproposed that they should go back close to where the colonel lay asleep,\nand they had not been standing near him ten minutes, hesitating to call\nhim for fear he should be awakened too soon, when he suddenly made a\nhasty movement, opened his eyes, looked round, and sprang to his feet.\n\n\"Midnight, boys,\" he said, \"is it not?\"\n\n\"We don't know, father, and did not like to call you too soon.\"\n\n\"Yes, it must be about midnight,\" he said decisively, \"or I should not\nhave woke up.  Well, is all right?\"\n\n\"No, father,\" whispered Perry.\n\n\"Oh yes; there's nothing to mind,\" said Cyril hastily.  \"We only found\nthat there are a lot of Indians round about the camp.\"\n\n\"You saw them?\"\n\n\"Yes, sir.  So soon as we moved a little way, a man rose up and stopped\nus.\"\n\n\"On one side?\" said the colonel.\n\n\"All round, sir.\"\n\n\"On guard, then, in case we wished to escape.  We're prisoners, my lad,\nfor the present.  However, they will not venture to hurt us, unless we\ngive them good reason, by loading up the mules to take away something\nthey consider ought to be kept here, and that we shall not be ready to\ndo for some days to come.\"\n\n\"That's what I wanted Perry to feel sir,\" said Cyril, \"but he would have\nit that they were going to attack us to-night.\"\n\n\"There is no fear of that, my boy,\" said the colonel firmly.  \"There,\nlie down, and sleep till breakfast-time; there is nothing to fear.\"\n\n\"But are you going to watch alone, sir?\"\n\n\"Yes, quite alone, my lad,\" said the colonel, smiling.  \"There, take my\nplace; I'm rested now, and you have nothing to mind.  Don't meet perils\nhalf-way; its bad enough when they come.  Till they do, it is our duty\nto be patient and watch.  Afterwards we must fight--if it is necessary.\nNow--to bed.\"\n\nThe boys obeyed, and the colonel commenced his solitary watch.\n\n\n\nCHAPTER FIFTEEN.\n\nCOLLECTING THE GOLD.\n\n\"Ever see 'em ketch eels at home, Master Cyril?\" said John Manning one\nmorning.\n\n\"We used to set night lines in the lake at school,\" said Cyril.  \"We\nthrew the bait out ever so far, and tied the other end to a brick sunk\nin the water.\"\n\n\"Oh yes: but I don't mean that way, where every twopenny eel spoils four\npen'orth o' good line and hooks.  I mean with an eel-trap, one of those\nmade of osiers, so that it's very easy to get in, but very hard to get\nout.\"\n\n\"Yes; I saw some of those once,\" cried Perry, \"up by a weir.  But why?\nThere are no eels here.\"\n\nJohn Manning chuckled, and shook all over, as if he enjoyed what he was\nsaying.\n\n\"Not many, sir, but quite enough.  We're the eels, and we've wriggled\nourselves right into a trap, and there's no getting out again.\"\n\n\"It doesn't seem as if there were,\" said Cyril thoughtfully; \"but we're\ngetting what the colonel wanted, and I don't think the Indians have\nnoticed it yet.\"\n\n\"'Tain't for want of looking, sir,\" said the old soldier.  \"I go for a\nbit of a walk in one direction, and begin picking something, and feel a\ntickling about the back.  `Some one's eyes on me,' I says to myself, and\nI go a bit farther, and feel the same tickling in front.  Then one side,\nthen t'other, and it's always eyes watching.\"\n\n\"Yes,\" said Perry.  \"We've been a week here, and I get so sick of it: I\nnever move without there being some one after me; and the worst of it\nis, you don't see him coming, but find him watching you from behind a\nrock, or out of a bush.\"\n\n\"Yes,\" said Cyril, \"it isn't nice.  They crawl about like snakes, and\nalmost as quietly.\"\n\n\"Don't matter,\" said John Manning, with another chuckle.  \"We can be as\ncunning as they.  How have you young gents got on since the colonel give\nhis orders?\"\n\n\"Pretty well,\" said Cyril.  \"Of course it's of no use to try and get\nroots or cuttings, they look too sharp after us; but I've found some\nseed, and he has got more than I have.\"\n\n\"How much have you got, both of you together?\" asked the old soldier,\nwith his eyes twinkling.\n\n\"Nearly a handful, I should say,\" replied Cyril.\n\n\"A handful, sir!  Why, what's that?  I've got quite half a gallon.\"\n\n\"You have?\" cried Perry.  \"Father will be so pleased.\"\n\n\"Course he will, sir,\" said John Manning, with a self-satisfied smile.\n\"`Get every seed you can,' he says, `and they'll hardly notice you.'\n\n\"`Right, sir,' I says, and I set to work quietly, going a bit here, and\na bit there, in among the trees, making believe I was making for them\ncocoa-nut leaves as the Indians chew; and whenever I caught one of the\nInjuns watching me, I picked a leaf, and began to chew it, and nodded at\nhim, and said _bono, bono_.  You should have seen how he grinned and\nshowed his teeth at me, Master Cyril, and I could see he was thinking\nwhat a fool this Englishman was.  But I wasn't quite so stupid as he\nthought, eh?\"\n\n\"But that's not cocoa-nut leaf,\" said Cyril, \"but the leaf of the coca.\"\n\n\"Well, sir, that's what I say.  I know it isn't the nuts but the leaves\nthey chew.\"\n\n\"But the coca leaf's a different thing.\"\n\n\"Course it is, sir; one's a leaf and t'other's a nut.\"\n\n\"But, don't you see, cocoa-nut leaf and coca leaf are different things?\"\n\n\"No, sir; but it don't matter.  They think I'm hunting for them leaves\nto chew, and they laugh at me, and all the time I'm getting a good heap\nof the seeds the colonel wants.  'Tain't the first time he's sent me to\nforage.\"\n\n\"But where are the seeds?\" said Cyril.\n\n\"All right, sir,\" said John Manning, with a look full of cunning.\n\"Never you put all your eggs in one basket, sir.\"\n\n\"Of course not; but I hope you've put them in a dry place.  Seeds are no\nuse if they're not kept dry.\"\n\n\"They're all right, sir.  I've got some in each of my pockets, and some\nalong with my cartridges in my satchel, and some inside the lining of my\ncoat, and a lot more round my waist.\"\n\n\"Round your waist?\" cried Cyril.  \"You can't wear seeds round your\nwaist.\"\n\nJohn Manning chuckled once more.\n\n\"Can, if you put 'em in an old stocking first, sir,\" he said.  \"But look\nhere, young gents, as I'm so much more lucky than you are, and know\nbetter where to go for 'em, you'd better take part o' mine, and leave me\nfree to fill up again.\"\n\n\"Yes, that will be best,\" assented Perry.  \"I can take a lot in my\npockets.\"\n\n\"Any one looking, sir?\"\n\n\"Very likely; but I shall take no notice.  They won't know what we're\nchanging from one pocket to the other, so let them watch.\"\n\n\"All right, sir; then here goes,\" said the old soldier, thrusting a hand\ndeep down into his trousers pocket, and drawing out a quantity of seed.\n\"Here you are, sir; and I'd make believe to eat a bit in case any one is\nwatching.\"\n\nBut as they were seated out of the sun, in the shade of the rough hut\nthat had originally been put up for drying the kina bark, they were\npretty well hidden from watchers, and able to carry on the transfer in\ncomparative secrecy.\n\n\"But this isn't seed of the cinchona tree,\" cried Cyril excitedly.\n\n\"What!\" said the old soldier sharply, and as if startled.  Then altering\nhis tone to one of easy confidence, with a dash of the supercilious.\n\"Don't you talk about what you can't understand, sir.  These here are\nwhat the colonel showed me, and told me to pick for him.\"\n\n\"They're not the same as my father told me to pick,\" cried Perry.\n\n\"Well, seeing as you're young gents, and I'm only a sarvant,\" grumbled\nthe man, \"it ain't for me to contradict, and I won't; but I will say\nthem's the seeds the colonel told me to pick, and there they are, and\nyou'd better put 'em away.\"\n\n\"I'm not going to put these in my pocket,\" said Cyril, \"for I know\nthey're wrong.\"\n\n\"And I certainly shan't put them in mine,\" said Perry.\n\n\"Look here, young gents, ain't this a bit mutinous?\" said John Manning.\n\"Colonel's orders were that we should collect them seeds, and if you'd\ngot the best lot, I should have helped you; but as you haven't got the\nbest lot, and I have, ain't it your duty to help me?\"\n\n\"Yes; and so we should, if you hadn't made a blunder.\"\n\n\"But I ain't, young gents; these here are right.\"\n\n\"No,\" said Perry.  \"These are right,\" and he took a few seeds from his\npocket.\n\n\"And these,\" said Cyril, following his companion's example.\n\n\"Not they,\" cried John Manning warmly.  \"They ain't a bit like mine.\"\n\n\"No, not a bit,\" said Cyril triumphantly.\n\n\"No, nor his ain't like yours, Master Perry.\"\n\nThe boys stared, for this was a new phase of the question, and they\neagerly inspected the treasures.\n\n\"I'm sure I'm right,\" said Perry confidently.\n\n\"And I'm sure I'm right,\" cried Cyril.\n\nJohn Manning put his arms round his knees, as he sat on the ground, and\nrocked himself to and fro, chuckling softly.\n\nAt that point the colonel came up, and looked round wonderingly.\n\n\"You're just in time, father,\" cried Perry.  \"Look at this seed John\nManning has collected.--Show him, John.\"\n\nThe old soldier triumphantly pulled out a handful, and held it under the\ncolonel's nose.\n\n\"What's that?\" said his master.\n\n\"The seed you told me to forage for, sir.\"\n\n\"Absurd!  There: throw it away.\"\n\n\"Throw it away, sir?\"\n\n\"Of course.  It is not what I told you.  There, take and throw it away,\nwhere the Indians see you do it, and they will pay less attention next\ntime they see you collecting.\"\n\nJohn Manning said nothing then, but went out of the slight hut frowning,\ncame back, and the colonel turned to the boys.\n\n\"Well,\" he said, \"what have you got?\"\n\nThey both eagerly showed a little of the seed, and the colonel uttered\nan ejaculation full of impatience.\n\n\"No, no,\" he said; \"pray be careful.  That is not the same as you got\nfor me the day before yesterday.\"\n\n\"Not mine?\" cried Perry.\n\n\"No, sir; nor yours either, Cyril.  They are both cinchona, but of the\ninferior, comparatively useless kinds.\"\n\nJohn Manning chuckled.\n\n\"But the seeds are so much alike, sir,\" said Cyril.\n\n\"Yes, but the broken capsules with them are not, boy.  The good splits\ndown one way, the inferior the other.  There, I suppose I must give you\nall another lesson.  Come and have a walk at once.\"\n\nHe led the way out, all taking their guns, in the hope of getting a\nlittle fresh provision, as well as to throw off the attention of the\nIndians, who smiled at them pleasantly enough, as they looked up from\ntheir tasks of cutting and peeling the bark from the trunks and\nbranches, most of the men with their jaws working, as they chewed away\nat the coca leaf, which every one seemed to carry in a little pouch\nattached to the waist.\n\nNo one seemed to pay further heed to them, but they were soon conscious\nthat they were being watched, for an Indian was visible, when they went\npast the spot where their two guides were watching the browsing mules;\nand then, as they plunged into the forest, from time to time there was\nan indication that they were being well guarded, and that any attempt at\nevasion would result in an alarm being spread at once.\n\nOnce well out among the trees, the colonel began picking leaf and flower\nindiscriminately, to take off the watcher's attention; but he contrived,\nat the same time, to rivet the boys' attention upon the flower and seed\nof the most valuable of the cinchona trees, indicating the colour of the\nblossom, and the peculiarities of the seed-vessels, till even John\nManning declared himself perfect.\n\n\"Seeds only,\" said the colonel.  \"I give up all thought of trying to\ntake plants.  We must depend upon the seeds alone, and we ought to get a\ngood collection before we have done.\"\n\n\"And then, father?\" asked Perry.\n\n\"Then we go back as fast as we can, if--\"\n\n\"If what?\" asked Perry.\n\n\"The Indians will let us depart.\"\n\n\"That's it, sir,\" put in John Manning.  \"What I was saying to the young\ngentleman this morning.  They don't mean to let us go.  We've regularly\nwalked into a trap.\"\n\nThere was silence for a few moments, the colonel frowning, as if\nresenting the interference of his servant, but directly after he said\nquietly:\n\n\"I'm afraid you are right, John Manning, but we must set our wits\nagainst theirs.  In another week we shall have quite sufficient of the\ntreasured seed to satisfy me--that is, if you three are more careful--\nthen we must start back, before our stores begin to fail.\"\n\n\"What about the guides, sir?\" said Cyril.  \"They will not help us.\"\n\n\"No,\" said the colonel.  \"Not the Indian guides, but I have a little\nEnglish guide here, upon which we shall have to depend.  There must be\nother passes through the mountains, and we know that our course is due\nwest.  We shall have to trust to this.\"\n\nHe held out a little pocket-compass as he spoke, and then, after they\nhad added somewhat to the store of seed already collected, both boys\nthis time making the proper selection of tree from which to gather the\nreproductive seeds, they walked slowly back toward the camp.\n\nBut not alone: the Indians who had followed them outward, returning\nslowly behind them, carefully keeping far in the background, and trying\nto conceal the fact that they were on the watch; but it was only too\nplain to all that it would require a great deal of ingenuity to escape\nnotice and get a fair start when the time came for making their escape.\n\n\n\nCHAPTER SIXTEEN.\n\nPREPARING FOR FLIGHT.\n\n\"I say, Cil, I don't quite know what to make of it,\" said Perry, a few\ndays later.  \"These people are as civil and amiable as can be; they\nsurely won't try to stop us when we want to go?\"\n\n\"You wait and see,\" was the reply.  \"They will.  I know them better than\nyou do.\"\n\n\"But they don't think we have got anything to take away.\"\n\n\"Perhaps not; but they will think that as soon as we are out of their\nsight we shall be searching for and taking something away that they want\nto preserve, and if we do get away unseen, they will be after us\ndirectly.\"\n\n\"Well, we shall soon see,\" said Perry rather gloomily, as he sat gazing\ndown into a deep valley running due south, in whose depths a bright\ngleam here and there told of the presence of water.\n\n\"Yes, we shall soon know now.  Your father and John Manning have been\ncarefully examining the mules, and going over the stores and packages.\"\n\n\"Have they?  I didn't know.\"\n\n\"I did, and then they came out here and sat for some time over their\nguns.\"\n\n\"On the lookout for birds?\"\n\n\"On the lookout to see if this way would do for us to escape.\"\n\nPerry whistled.\n\n\"Did they tell you so?\"\n\n\"No; but I put that and that together.\"\n\n\"Put why go this way?  This does not lead over the mountains.\"\n\n\"Because the Indians will not think we should choose this route.\"\n\n\"But we couldn't get over the mountains from down there.\"\n\n\"We must,\" said Cyril quietly.\n\n\"But,\" said Perry, \"we can't get the mules and their loads away without\nDiego knowing.\"\n\n\"Must again,\" replied Cyril.  \"We can't escape without a supply of food,\nand we must have the mules to carry it, for we may be weeks wandering\nabout in the gorges of the mountains.  So it's must, must, must, my lad.\nWe've got it to do, and we're going to do it.\"\n\n\"I say.\"\n\n\"Well--what?\"\n\n\"Do you think it will come to a fight?\"\n\n\"Not if your father can help it; but if it does, we shall have to do\nsome shooting.\"\n\nPerry drew his breath hard.\n\n\"Don't stare down the valley any more,\" said Cyril, after a pause.\n\n\"Why?  It's very beautiful.\"\n\n\"Because you're watched.  We're watched always, sleeping or waking.\"\n\n\"Then we shall never be able to get away,\" said Perry despondently.\n\n\"Must, my lad.  Why, we're not going to let a pack of half-savage\nIndians prove too clever for us.  What are you thinking about?  There,\nlet's get back at once, or they'll be thinking we mean something by\nsitting here.\"\n\nPerry rose and followed his companion, who made several halts in the\nforest before they reached the shelter-hut, to find the colonel and John\nManning away; but they returned soon after, each carrying a couple of\ngood-sized birds, which gave a colour to their morning's walk.\n\nThis game John Manning bore off to prepare by the fire which Diego and\nhis companion kept going night and day; and as soon as he had gone, the\ncolonel seated himself, and looked curiously from one boy to the other.\n\n\"Well Cyril,\" he said sharply, \"ready to go home and meet your father?\"\n\n\"Yes, sir,\" replied the boy promptly.  \"I want to get it over.\"\n\n\"And you, Perry, ready to go back to where you can sleep in a decent bed\nagain?\"\n\n\"Yes, father,\" replied Perry; but there was a dubious tone to his words.\n\n\"That's right.  Listen, then, both of you.  I trust to you to make no\nsign whatever, but to go on precisely the same as usual, so as to keep\nthe Indians in ignorance of our intentions.\"\n\n\"Then you are going to make a start, sir?\" said Cyril eagerly.\n\n\"All being well, very soon, my lad.\"\n\n\"But the mules, sir?\"\n\n\"Ah, we shall see about that,\" said the colonel.  \"I have now got\ntogether quite as much of the seed as I dared to hope for, and it would\nbe foolish to delay longer.  These Indian labourers are only working for\nsomebody of importance, and if whoever he may be comes and finds us\nhere, our position may be made very unpleasant, so I have decided for us\nto start at dark, to-morrow evening.\"\n\nThis announcement caused a peculiar fluttering in the breasts of both\nlads, for they felt that they would not be able to get away without a\nstruggle, since that they were detained here until some one in authority\narrived, seemed certain; and they well understood how necessary it was\nfor them to get away if possible.\n\nThe rest of the day passed like a feverish dream to Cyril, whose\nthoughts were of a very mingled nature.  On the one hand, there was the\nrisk to be run in making their escape, and the long perilous journey\nbefore them; on the other hand, there was home at San Geronimo, and his\nfather's stern face rose before him, full of reproach for his conduct;\nand now, more than ever, he asked himself how he could have been so mad\nand so cruel to those who loved him, as to leave in the way he had.\n\nToo late for repentance then, as he knew, and he had to face the\ninevitable, and take the punishment he deserved as patiently as he\ncould.\n\nToward dark the boys found themselves alone with John Manning, who\nwhispered: \"Been over the arms and ammunition, gentlemen, and they're in\nsplendid order.  Bit touched with rust, but that won't interfere with\ntheir shooting.\"\n\n\"Don't talk about it,\" said Perry petulantly.\n\n\"Can't help it, sir.  We're off to-morrow night, and some of us may have\nto cover the retreat.  You can't do that sort o' work without tools.\"\n\n\"Look here,\" said Cyril eagerly.  \"How about the mules?\"\n\n\"I don't know, sir,\" replied the old soldier.  \"That's the puzzle of it.\nBut the colonel knows what he means to do, of course.  I've been with\nhim before, when he was going to make an advance.\"\n\n\"But this is a retreat,\" said Cyril sharply.\n\n\"What, sir?  Retreat?  British soldiers don't retreat.  Of course they\nhave to make an advance the other way on sometimes.  You can't always be\ngoing in one direction; but they don't retreat.  It'll be all right,\nthough, sir.  You'll see: for following orders, I've got all the packs\nready to stow on the saddles at a moment's notice, and we shan't leave\nnothing behind.\"\n\nThey had a hint soon after of there being a plan all ready, for the\ncolonel came and hunted Cyril out to act as interpreter, and walked down\nwith him to where Diego and his companion were seated, while the mules\nwere browsing here and there, some fifty yards away.\n\n\"Now, interpret as well as you can,\" said the colonel.  \"Tell him that I\nam very angry about the state of the mules, which look half-starved.\nThe feed about here is disgraceful, and all the time there is a splendid\nsupply on the other side of the clearing, beyond where the Indians are\ncutting and stacking the bark.\"\n\nCyril's voice shook a little from anxiety as he began his\ninterpretation, but it soon grew stronger, and he gave the colonel's\nwishes with so much energy that the guide looked terribly disturbed as\nhe replied.\n\n\"What does he say?\" cried the colonel angrily.\n\n\"That the head-man of the kina gatherers gave orders that they were to\nbe pastured here.\"\n\n\"Then tell him to go to the head-man, and say I order them to be moved\nat once over to the other side of the huts, ready for me when I wish to\ngo on.\"\n\nDiego started off at once, and returned soon after with the head-man and\nabout a dozen of the Indians, to whom the colonel's wishes were\nrepeated; and then came quite a deprecating reply that it was\nimpossible, for the woodcutters were going in that direction the very\nnext day, and the mules would be disturbed again.\n\n\"Tell him my mules are of more consequence than his bark gatherers,\"\nsaid the colonel, \"and that I insist upon the mules being moved.\"\n\nThere was a laboured interpretation, a short buzz of conversation, and\nthen a reply came through Diego that the head-man would obey the white\nchief's orders, and remove the mules to better pasture; but it could not\nbe there, in the place he wished.\n\n\"Tell him anywhere, so long as the poor beasts are properly fed.\"\n\nThe colonel stalked away, with his rifle in the hollow of his arm, the\nIndians giving place obsequiously; but he turned back to Cyril.  \"Tell\nJohn Manning to stop and see where they are driven, and then come and\nreport to me.--You two follow.\"\n\nCyril gave the colonel's orders, and then went after him to the hut,\nwhere they sat waiting for nearly an hour before Manning arrived.\n\n\"Well, where are the mules?\"\n\n\"They've driven 'em out of the bit of forest, sir, and down on the other\nside toward the <DW72> of that big valley.\"\n\n\"Hah!\" ejaculated the colonel; and then, after a pause, \"The very spot.\"\n\n\"But you said the other side,\" said Cyril; \"at the back of their huts.\"\n\n\"Where I knew they would not have them,\" said the colonel.  \"It looked\nto them, in their childish cunning, like an attempt on my part to get\nthe animals down toward the point from which we came; and, of course,\nthey would not do that.  I hardly expected such good fortune, boys; but\nthe mules are in the very place I wish.  Now we have to devise a means\nof getting those mules loaded unseen, and then starting off down the\nvalley as soon after dark to-morrow night as possible.\"\n\nA long conversation followed as to those best means, and the colonel\nheard each one's proposal impatiently.\n\nPerry said it was impossible, and that they must all take as much\nprovision as they could carry, and leave the mules behind.\n\nJohn Manning said there was only one way of doing it, and that was for\nhim to take the stock off one of the guns, and as soon as it was dusk\ncreep round the camp, and catch every one of the sentries by surprise,\nand then club him, and bind his hands and feet.\n\n\"I could stun 'em, sir, and then they couldn't give no alarm.\"\n\n\"You mean, murder the poor wretches,\" said the colonel quietly.\n\n\"No, no, sir; not so bad as that,\" grumbled the man.  \"These Injuns have\ngot heads as thick as rams.  More likely to spoil the gun.\"\n\n\"Now you, Cyril,\" said the colonel quietly.\n\n\"I can't propose anything, sir,\" said Cyril frankly.  \"It seems to me\nthat we might pass one or two of the Indians, but the others would see\nor hear the mules.\"\n\n\"And you can propose nothing else?\"\n\nCyril shook his head, and the colonel got up and went out of the hut, to\ngo and walk up and down where the Indians were busy, giving first one a\nfriendly nod, and then another, evidently to their great satisfaction.\n\nThe party in the hut watched him for a few moments, and then John\nManning said:\n\n\"There aren't no better way, gentlemen, than mine.  I don't want to kill\nnone of 'em, so long as they don't try to kill me, or any of you.  If\nthey do, why, of course, it makes me feel nasty, and as if I could do\nanything to stop 'em.\"\n\n\"It's too horrid and butcher like,\" said Cyril firmly.\n\n\"Yes,\" assented Perry.\n\n\"Very well, then, gentlemen, suppose you propose a better way.  It's of\nno use to go an' say, `Please we're tired of staying here, and want to\ngo,' because that only would be waste of breath.\"\n\n\"Yes,\" said Perry sadly.  \"We shall never get away till they give us\nleave.\"\n\n\"Hear that, Mr Cyril, sir; that's my young master, and the son of a\nstout soldier as never turned his back on an enemy in his life.  Don't\nsay you're going to give up like that, sir.\"\n\n\"No,\" said Cyril, setting his teeth.  \"I'm not going to give up, and he\nis not going to give up either.  We'll get away somehow, though we can't\nsee the way just now.\"\n\n\"That we will, sir,\" cried John Manning excitedly.  \"Bri'sh wits again'\nInjun wits.  Bah! who says we can't beat them?  It's all right,\ngentlemen.  I know the colonel, and have known him since he was a slip\nof an ensign, and I was not much more than a raw Johnny of a boy fresh\nfrom the awkward squad.  I say I know the colonel, and he's only been\nleading us on.  Wait till to-morrow night.  He's got some dodge or\nanother ready to fire off, and this time two days we shall be on our way\nback, and the Injuns'll be howling like mad, because they can't make out\nwhich way we've gone.\"\n\n\n\nCHAPTER SEVENTEEN.\n\nPERRY'S HORROR.\n\n\"How do you feel, Cil?\" said Perry, in the middle of that same night.\n\n\"Horrible.  Can't sleep.  I am hot and itchy, and all of a fidget about\nthings.\"\n\n\"Father said we were to take things coolly, when he said good-night.\"\n\n\"Didn't say how, did he?\" whispered Cyril.  \"I shall be so glad when we\nbegin doing something.  Anything's better than this waiting to begin.  I\nsay--\"\n\n\"Yes.\"\n\n\"Isn't it near morning?\"\n\n\"No, it ain't,\" said a gruff voice in an ill-used tone.  \"How's a man to\nget a good sleep before he relieves the colonel, if you two young gents\nkeep on twisting about and talking?\"\n\n\"All right, we'll be quiet now.\"\n\n\"Ay, do, my lads.  Get a good sleep, and have a nap or two to-morrow,\nfor we shall be travelling all night.\"\n\nThere was silence for the rest of the time in the little camp, broken\nonly by a weary sigh or two, for no sleep came to the restless lads; and\nthe next morning found them red-eyed and feverish in spite of the bathe\nthey had in the intensely cold water of the neighbouring mountain rill.\n\nAnd all that day they were on the strain, and constantly on the watch\nfor the colonel, hoping that he would become communicative.  But he was\nvery quiet, and spent the greater part of the day either sleeping or\npretending, and lounging about watching the Indians busy cutting down\ntrees, or peeling the boughs and twigs.\n\nJohn Manning, too, looked wonderfully lazy, and avoided the boys, who at\nlast began to look at each other in despair.\n\n\"I can't make it out,\" said Perry at last.  \"We are not going to-night,\nor father would have said something--don't you think so?\"\n\n\"Don't know.\"\n\n\"But you don't think we can be going?\"\n\n\"I think we are,\" replied Cyril, \"and they are doing all this to throw\nthe Indians off their guard.\"\n\nDinner-time came, for which meal John Manning had prepared a very\nsatisfactory dish from some charqui flavoured with fruit and vegetables,\nand the boys anxiously waited again for some communication from the\ncolonel.  But he was still reticent, and after the meal was over, Diego\nand his companion were summoned and left to clear the tin bowl which did\nduty as a dish, a duty they always carried out to perfection, never\nleaving it so long as there was a scrap to finish.\n\nThen came the long, hot, weary afternoon, which the two boys started to\npass under the boughs of a sturdy tree, both feeling their irritability\nincrease as they listened to the crackling and breaking of wood near at\nhand, and the murmur of voices from the Indians, who kept on busily with\ntheir work.\n\nThere was a humming noise in the air, as the insects darted here and\nthere in the hot afternoon sunshine; and from where the two lads lay,\nthey could see the mountains <DW72> down rapidly into the long deep\nvalley, filled now with a soft golden haze, while the air was delicious\nwith the aromatic perfume shed by the trees around.\n\nCyril felt hot, feverish, and weary still, but at the same time, as he\nlay there, it seemed as if that valley at his feet was very beautiful\nwith the sun lighting it up from end to end, and that it would be a pity\nto start that night, before he had had a good restful sleep, and then--\ndirectly after it seemed to him--he felt vexed with Perry for worrying\nand shaking him.  The next moment he started up to find that the valley\nbelow looked dark, and the sun was on the other side of the mountains,\nwhile the colonel was standing over him, smiling.\n\n\"That's better, boys,\" he said.  \"I'm glad that you have both had a good\nrest.  You will be all the fresher for your walk.\"\n\n\"Then you are going to-night, sir?\"\n\n\"Hush!  Yes; of course.--Perry.\"\n\n\"Yes, father.\"\n\n\"Don't go away, either of you, and you must not look excited.  Come and\nhave supper--it is ready--and then wait about by the hut while the\nguides have theirs.  You will take no notice of anything, but loiter\nabout outside while John Manning and I act.  But be ready to help, if I\ncall upon you.\"\n\n\"We'll do all you wish, sir,\" said Cyril excitedly.\n\n\"Then do it calmly,\" said the colonel.  \"Mind this, the Indians must not\nhave a suggestion that shall make them suspicious.  To them everything\nmust seem as if we were patiently submitting to our rather easy\ncaptivity.  Come.\"\n\nThe colonel led the way back to the fire, close to which their meal was\nspread by John Manning, and as the boys drew nearer, they saw that Diego\nand his companion were hanging about as if wondering why they had not\nbeen summoned sooner to partake of the meal.\n\n\"Yes, we're late,\" said the colonel aloud, and setting the example, he\ntook his place and began to eat as calmly as if nothing important was on\nthe way.\n\n\"Come, boys,\" he said quietly, \"make a good meal, and don't look\nanxious; there are some of the Indians coming up.  Recollect what I\nsaid.\"\n\nCyril tried to act his part, and said something in a laughing way to\nPerry, but it fell very flat.  Still, there was nothing in the scene to\nattract attention, and though they were all aware that work for the day\nhad long ceased near the huts, and the Indians who were not partaking of\ntheir simple meal, were strolling about, and many of them keenly\nwatching the white party, no head was turned.  At last the colonel asked\nif all were done, and then rose and signed to Diego and the other man to\ncome and take their places.\n\nThis they did eagerly, and from where Cyril stood now in the\nsemi-darkness, he could see the men's faces by the light of the fire,\nand that they were eating hungrily.\n\n\"Did you look to the mules?\" said the colonel in a low voice.\n\n\"Yes, sir, all ready.\"\n\n\"That's right.  Now, boys, the Indians have strolled back, and I don't\nthink they have set any watch yet.  Keep on walking to and fro as you do\nsometimes, with your arms on each other's shoulders.  Keep between the\nfire and the Indians' clearing, and take no notice of anything you see.\nWe shall not leave you behind.\"\n\nCyril's heart beat violently, and he heard Perry utter a low sigh as he\nthrew his arm over his companion's shoulder and they began walking to\nand fro about twenty yards from their fire, while the low hum of many\nvoices came from the clearing where the Indians were talking together\nbefore settling themselves for the night.\n\nMeanwhile Diego and his companion were eating away as if they had\nsuffered a three days' fast, and showed no sign of leaving off, till all\nat once, just as the boys turned, they became aware of the fact that the\ncolonel had gone from the spot where they had seen him last, and that he\nand John Manning had suddenly appeared in front of the guides, where\nthey were eating.  By the light of the fire they saw that guns were\npresented at the men's heads, with the effect of making them throw out\ntheir arms to seize their weapons, but before they could effect anything\nfor their defence, they were thrust backward, and Cyril at the same\nmoment saw by the firelight Diego lying upon his back, with the\ncolonel's foot upon his chest, and the other man in a similar position,\nheld down by John Manning.\n\n\"Keep on walking,\" Cyril said aloud to Perry, for the latter had\nstopped, panting and startled, and Cyril felt him quiver as he\nhalf-forced him along.\n\n\"What are they going to do?  Kill them?\" whispered Perry.\n\n\"They're going to master them,\" replied Cyril.  \"Don't speak like that.\nRecollect our orders.  It is to save them from being seen.\"\n\nThe boys kept on their walk, watching the proceedings by the fire as\nmuch as they could, but in less than five minutes there was nothing to\nsee, for both the guides were bound with a hide rope from the mules'\npackages; and urged onward by threats from the colonel's and John\nManning's pieces, they had passed out of sight among the bushes in an\nenforced stooping position, a faint crackling telling of the direction\nin which they had gone, while a louder crackling and snapping told, with\nthe accompanying blaze, that something had been thrown upon the fire.\n\n\"The bows and arrows,\" whispered Perry, and they kept up their\nmonotonous tramp to and fro.\n\n\"What are they doing now?\" said Perry suddenly, and then he started, for\nCyril burst out into a merry laugh, and gave him a sharp slap on the\nback, so suddenly, and with such force, that Perry stumbled forward, and\nnearly fell.\n\n\"Are you mad?\" cried the boy furiously.\n\n\"Not quite,\" said Cyril merrily.  \"Here, give us your hand, old chap:\nI'll haul up.  That's your sort.  Ahoy!  There you are again.\"\n\nHe said all this boisterously, and then in a low whisper:\n\n\"Keep it up.  Hit me, or do something.  Two Indians have come up close\nto watch.\"\n\nPerry trembled violently, but he tried to follow out his companion's\nplan, and turning upon him, engaged in a mock struggle, each making\nbelieve to throw the other for a minute or two, and then laughingly\nresuming their walk to and fro.\n\nThose laughs were very hysterical, though, and Perry's next words came\nwith gasps as he said:\n\n\"See the Indians now?\"\n\n\"No; they're either gone back or they're hiding.\"\n\n\"Which?  Let's go and see.\"\n\n\"We can't,\" replied Cyril.  \"Our orders are to walk up and down here, as\nif nothing were wrong.  Can't you see it will make them believe we are\ngoing on as usual?\"\n\n\"Yes,\" said Perry huskily; \"but I wish my father would come now.\"\n\n\"So do I.\"\n\n\"Those two may have got the better of them.\"\n\n\"Not they,\" said Cyril stoutly.  \"It would take three Indians to get rid\nof your John Manning.  Your father will take care they do nothing.\nDon't take any notice.  Hear that?\"\n\n\"Yes, some one going away through the bushes.  Those two hadn't gone,\nand they were hiding.\"\n\n\"Yes.\"\n\n\"But are they both gone now?\"\n\n\"I only heard one,\" said Cyril, beginning to whistle a merry tune, but\nbefore he had got through the first strain, there was another faint\nrustling among the trees.\n\n\"There goes the other,\" said Cyril quietly, and then he broke into a\nloud yawn.  \"Heigh--he--ha--hum,\" he said.  \"How dark it has grown.\"\n\n\"Listen,\" whispered Perry.\n\n\"I heard it,\" said Cyril.  \"One of the mules squeaking.\"\n\n\"No, it was a horrible cry.  Some one has been killed.\"\n\n\"There goes another then,\" said Cyril, as a peculiar sound came from the\nforest.\n\n\"Yes, they are killing the guides.\"\n\n\"I tell you, it was the squeaking of the mules.  I know the sound well\nenough.\"\n\n\"I'm sure you're wrong,\" protested Perry.\n\n\"And I'm sure _you_ are.  If it was the cry of some one being killed,\nwouldn't there be a rush of the Indians, to see what was the matter?\"\n\n\"If they heard it.\"\n\n\"And they would.  Trust them for that.  The mules are excited and\ncalling to one another.  I believe they are being loaded.\"\n\n\"Oh, how can you take it all so coolly?\" groaned Perry.  \"My heart beats\nas if it would break, and I feel a curious choking sensation at the\nthroat, and all the time you take it as if there was nothing the\nmatter.\"\n\n\"Do I?  You don't know,\" said Cyril.  \"I believe I'm worse than you are;\nbut never mind, try to laugh.\"\n\n\"Laugh,\" said Perry piteously.  \"I feel as if I could sit down and cry.\"\n\n\"Leave that to the girls, lad.  We've got something else to do.  Don't\nstop.  We must keep on, so as to keep the Indians from thinking there's\nanything wrong.  There, cheer up.  Can you sing any thing?\"\n\n\"Sing!\" cried Perry, in a voice full of reproach.\n\n\"Very well, then, I must whistle softly.\"\n\nHe commenced a tune, and got through a few bare.  Then he ceased as\nsuddenly as he had begun, and began talking.\n\n\"I say it was very plucky of your father, wasn't it?  The boldness of\nthe plan has made it do.  The Indians could not even think we should\nmake such an attempt.\"\n\nFor a full hour the boys kept up that painful tramp up and down, Perry\ngrowing more and more silent, and Cyril bursting out from time to time\nwith a little peal of forced laughter.  Twice over, they were conscious\nof the presence of the watchful Indians creeping furtively among the\ntrees; but the actions of the boys allayed their suspicions, and they\nwent back as softly as they came.\n\n\"Was it never to end?\" the lads asked themselves, and though neither\nmade any allusion to their thoughts, they were tortured by fancies of\nwhat might have happened, till at last Perry was certain that, instead\nof the colonel and John Manning killing the two guides, these two men\nhad turned upon them and stabbed them to the heart.\n\nAt last the boy could bear this thought no longer.  He fought hard to\nkeep it to himself, but it would have vent finally, and as they turned\nto continue their weary tramp, he suddenly caught Cyril fiercely by the\narm.\n\n\"They won't come back to us,\" he whispered.  \"They cannot.  Diego and\nthe other man turned upon them, killed them, and those were their cries\nwe heard.  They're both dead, Cil--they're both dead.\"\n\n\"And your father has come to tell us he has been killed,\" said Cyril,\nwith a forced laugh, which was more like a hoarse cry of agony.  \"At\nlast,\" he groaned: \"I don't think I could have borne it any longer.\"\n\n\"What do you mean?\" said Perry.\n\n\"There--by the fire.  Here they come.\"\n\nPerry looked sharply round in the direction pointed out by his\ncompanion, and then the pulses of both seemed to stand still, for they\nheard the approach of Indians from the direction of the clearing.\nAlmost at the same moment, they could plainly see by the faint light of\nthe fire, not the colonel and John Manning coming to fetch them at last,\nbut the figures of the guides bending down, and then beginning to\napproach, in the soft furtive manner of a couple of wild beasts about to\nmake their fatal spring.\n\n\n\nCHAPTER EIGHTEEN.\n\nADVENTURES OF A NIGHT.\n\n\"He was right,\" muttered Cyril, as the blood rushed to his head and made\nhim feel giddy; \"and now they mean to have us, but--\"\n\nHe stopped short, and his teeth made a grating sound as he seized Perry\nby the shoulder.  \"Can you fight?\" he whispered.  \"I--I don't half\nknow,\" groaned Perry.  \"I'll try.\"\n\n\"That's right.  We must,\" the boy continued.  \"They shall find we're\nEnglish after all.\"\n\n\"What are you going to do?\" said Perry, holding on by his companion's\narm.\n\n\"Get our guns.  They're close by the fire there.  What are those two\ndoing?\"\n\n\"I don't know,\" was the reply, and Perry gazed hard at the two guides,\nwho were stooping about the fire.  \"Yes, I do; they're putting on more\nwood.\"\n\n\"Then, as soon as they come toward us, we must run round and try to get\nour guns.\"\n\nThey stood in the darkness watching for some moments, while the guides\nstill busied themselves about the fire, wandering here and there, as if\nbusy about something; though, after seeing the flames rise, on the first\nportion of wood being added, their object appeared vague.\n\nAll at once the rustling toward the clearing recommenced, and the boys\nlooked sharply in that direction, fully expecting that the first attack\nwould come from there; but the sound grew fainter, and they knew that\nthe Indians must be going back, apparently satisfied with their\nscrutiny.  This meant the danger lessened for the moment by one half;\nand Cyril now gripped his companion's shoulder more tightly.\n\n\"Now, then,\" he said, \"let's get round by the trees to the other side.\"\n\n\"Too late,\" said Perry; \"they are coming here.\"\n\nCyril glanced toward the fire, but no one was visible.  In the brief\nmoments during which their backs were turned, the guides had\ndisappeared, and all was silent; not a sound suggested the spot from\nwhich the enemy would advance.\n\n\"We must chance it,\" whispered Cyril.  \"Quick; come along this way.\nQuiet.\"\n\nThey started away to their right, so as to get round to the back of the\nfire; but as fate had it, they went right into the arms of those whom\nthey were seeking to avoid.  Not forty steps had been taken cautiously\nthrough the dark shadows beneath the trees, before Perry uttered a cry\nas the two guides sprang up in their path.\n\n\"This way, Cil; run,\" he whispered.\n\n\"Hush!  Silence!\" came in a familiar voice.  \"Don't you know us, boys?\"\n\nBoth Cyril and Perry were speechless, so great was the emotion caused by\nthe surprise, and they stared at the dimly-seen, bare-headed figures\nwearing the Indians' long, loose garments.\n\n\"Now, quick,\" said the colonel, stripping off the Indian frock, \"off\nwith yours, too, Manning.\"\n\nThe man obeyed with all a well-drilled soldier's celerity and silence,\nand, stooping down, the colonel was about to thrust the cotton garments\nin amongst the undergrowth, when Cyril, who had now recovered himself,\nwhispered a few words to the colonel.\n\n\"Good!  Capital!\" he said.  \"Only quick, and we'll wait here.\"\n\nCyril snatched at the two frocks, and, stooping down, laid them, well\nstretched out, at a short distance from the fire, where, in the dim\nlight, they gave a rough idea of covering a couple of Indians stretched\nout in sleep.\n\nIt was only the work of a minute, and then Cyril was back to where Perry\nstood excited and nervous, for the feeling was strong upon him that,\nafter all, his father and Manning had slain the two guides.\n\n\"Where are the mules?\" said Cyril to the colonel.\n\n\"Silence!  Follow.  Stoop till we are well beyond the fire.\"\n\n\"But our guns, sir?\" said Cyril.\n\n\"I said silence, boy!\" replied the colonel, and they went off in single\nfile for about a couple of hundred yards in and out among the trees,\ntill the colonel stopped short, and the boys made out that they were\nstanding by the mules, which were waiting, all ready laden, and with\nhanging heads, ready to proceed on their journey.  Then, without another\nword, the colonel took the rein of the old leader, started off, and\nsteadily and quietly the others followed, the unladen last, while John\nManning and the two boys followed for some time.\n\n\"Here, take your fireworks, my lads,\" whispered John Manning at last.\n\"Pouches are fastened to 'em, and well filled with ammunition.  I'll\nhelp you to put 'em on as we go.\"\n\nAll this in a whisper, and then Perry said: \"You thought of our wanting\nthem, then?\"\n\n\"Rum sort of soldier if I hadn't, my lad,\" growled the man.  \"Steady.\nKeep on walking.  Under your right arm, my lad.  That's it.--Now you,\nMr Cyril.\"\n\n\"Mine's on all right,\" was the reply; and then it was always onward and\ndownward, in and out among the trees, with all around so dark beneath\nbranches, that, but for the steady, slow pace of the mules, which never\nhesitated for a moment, the journey would have been next to impossible.\nAnd all the time, as the rustling, soft, trampling noise made by the\nanimals' hoofs went on, very few words were spoken, for every ear was\nattent and strained to catch the first announcement of the pursuit\nhaving begun.\n\nThe two boys felt no inclination to converse, but tramped on silent\nenough, while, when anything was said, John Manning was the speaker.  He\nwould begin by enjoining silence in the ranks, and the minute after,\nfind he had something he must say.\n\n\"Don't think they've took the alarm yet, gentlemen,\" he said, after a\nlong time.  \"That dodge o' yourn with the Injuns' frocks was splendid.\nWhen they do come, take your word from me, as I command the rearguard;\nand fire low, for we must give them a volley.\"\n\nPerry shrank from their old servant involuntarily, for it seemed to him\nhorrible that John Manning should speak in so cheery a tone from time to\ntime, when, only a short time back, he had imbrued his hand in the blood\nof their two guides.  But at last he felt constrained to speak, the\nwords coming forth unbidden.\n\n\"Those two guides,\" he said huskily.\n\n\"Ay, poor chaps, it seemed hard, sir,\" replied the old soldier; \"but it\nwas us or them, and, of course, it had to be them.  We was obliged to do\nit, or else how was I to get the mules loaded?\"\n\n\"But it seems so horrible,\" said Cyril.\n\n\"Oh, I don't know, sir.  Sort o' tit for tat.  They wouldn't ha' been\nvery particular about us, and it was, as you may say, in self-defence.\nBut, I say, Mr Cyril, don't you think I got all those packs down to the\nmules pretty quick, and the beasts laden?\"\n\n\"Wonderfully quickly,\" said Cyril.\n\n\"It was, sir, though I say it as shouldn't say it.  I did get warm over\nthe job.  Thought I should have had no end o' trouble with 'em, but they\ntook it as quietly as lambs; and as soon as they found out what was\ngoing on, the pack-mules all hung together and waited their turns, while\nthe saddle mules seemed to be looking on.\"\n\n\"Of course that was after the--after Diego and the other man--\"\n\n\"Of course, sir.  There'd ha' been no mule packing if we'd left those\ntwo chaps to lift up their lovely voices, and shout to their friends for\nhelp.  That would not have done, eh, Mr Cyril?\"\n\n\"No; I suppose not, if we were to escape.\"\n\n\"And that's what we had to do, sir; for, as the colonel said to me more\nthan once, `We're not safe, John Manning, for sooner or later they'll\nfind out why I have come, and then I would not answer for our lives.'\nBut we're off now in spite of 'em, and well provisioned too.  My word, I\ndid get a warming over those mules; but the colonel's wonderful handy at\nthe loading, and helped me well.  You see, he superintended a lot out in\nIndia, when we had mules and camels to carry our baggage.  And we did it\nall fine.  Listen.\"\n\nThey paused, but the faint pattering of the mules' hoofs was the only\nsound; and they followed on again, John Manning keeping silence for a\ntime, and then bursting out with a chuckle.\n\n\"I told you so yes'day, young gentlemen.  The colonel 'll have some\ndodge to get us off, and there you are!  He led, and it was grand the\nway in which he had worked it out.  He didn't tell me till to-night, and\nwhen he had done, I laughed out.  `Think it will do, John Manning?' he\nsaid.  `Do, sir?'  I says.  `Of course it'll do;' and it's done.  Don't\nsuppose those two liked it much, poor fellows, but they had to put up\nwith it.\"\n\n\"Oh, John Manning,\" cried Perry excitedly, unable to bear it any longer,\n\"how can you treat it so lightly?  If you had tied and bound the poor\nwretches, it would have been different, but to drag them away and kill\nthem in cold blood!  It is horrible.\"\n\n\"Well I _am_ blessed!\" exclaimed the old soldier, in a tone and with an\nemphasis that showed how he was startled.\n\n\"And I'll never believe that my father meant it to be so.\"\n\nJohn Manning gave Cyril a dig with his elbow, and he winked one eye, but\nthe act was invisible in the darkness.\n\n\"Why, it was him as 'vented the plan, sir.  I only helped carry it out.\"\n\n\"Oh!\" ejaculated Perry.\n\n\"Hadn't we got to escape, sir?\"\n\n\"But in such a way!\"\n\n\"Why, it was a splendid way, Master Perry.  But I say I am ashamed of\nyou to go private court-martialling your own father in that way, and\nfind such fault with him for helping you to get off!\"\n\n\"I'm not going to judge him,\" said Perry.  \"I only say it was horrible.\"\n\n\"Well, yes, sir, it was, and is,\" said the old soldier, giving Cyril\nanother dig.  \"Can't say as I should like to lie all night on my back\nwith my hands tied behind me to a big pole, and my ankles and knees\nserved the same, just as if I was going to be roasted for a cannibal's\ndinner, and to make it worse, an old worsted stocking rammed into my\nmouth, and a cloth tied over it and behind my neck, to make sure I\ndidn't get it out.\"\n\n\"What!\" cried Perry.\n\n\"I said a stocking rammed into my month, sir, so as I shouldn't holler,\nonly breathe.  It is hard on a man, but what was you to do?\"\n\n\"Then you didn't kill them,\" cried Perry joyfully.\n\n\"Kill 'em,\" said John Manning, in a tone full of disgust.  \"Did you ever\nknow a British soldier, as was a soldier, go killing folk in that way,\nsir, when they'd been made prisoners?  Master Perry, sir, I'm ashamed o'\nyou for thinking such a thing o' your father, as is as fine an officer\nas ever stepped.\"\n\n\"Not so much ashamed of me as I am of myself,\" said Perry huskily.\n\"Then Diego and the other man are all right?\"\n\n\"They don't think so,\" said the old soldier with a chuckle.  \"They're\nprecious uncomfortable by this time, for I rammed the stockings pretty\nfar, and I tied them knots with those new hide ropes as tight as they'd\ndraw.\"\n\n\"Quiet there, quiet,\" said the colonel sternly, for he had stopped and\nlet the mules pass him.  \"No more talking for the present.  Can you hear\nanything?\"\n\n\"No, sir, not a sound,\" said John Manning.  But even as he spoke there\nwas a faint cry borne on the night wind from high up the valley, and\nsituated as they were, that sound could only have one meaning--pursuit.\n\n\n\nCHAPTER NINETEEN.\n\nTHE DARK WAY.\n\n\"They've missed us,\" said Cyril excitedly.  \"Shall I run to the leader,\nsir, and hurry him on?\"\n\n\"No, my lad,\" said the colonel, \"we shall do nothing by hurrying.  Our\nretreat must be carried out slowly.  We can get on no faster than the\nmules will walk.  Keep on as we are.\"\n\nHe left them after listening for a few minutes, and hurried forward to\nreach his place again by the leading mule, for the sagacious beast had\ngone steadily on, followed by the others, acting as if it knew its duty\nas well as a human being--that duty being to follow the easiest course\noffered by the valley, which ran parallel with one of the outer ranges\nof foot-hills, there being no track whatever to act as guide.\n\n\"Sounds quite reviving,\" said John Manning in a whisper.  \"We've had so\nmuch dull do-nothing times, that it quite freshens one up.\"\n\n\"How long will it be before they overtake us?\" said Perry anxiously.\n\n\"How long have we been coming here, sir?\" replied the old soldier.\n\n\"I don't know--an hour, I suppose.\"\n\n\"Yes, sir, an hour.  Well, if they knew the way we came and followed on,\nit would take them hours more than it has taken us.\"\n\n\"Why?\" said Cyril sharply.\n\n\"Why, sir? because,\" said John Manning, with one of his dry chuckles,\n\"they'll have to come along very slowly, searching among the trees as\nthey come, for fear of overrunning the scent; for as it's dark, they've\ngot nothing to guide 'em, and I hope they won't find much when it's\nlight, for the sun will soon dry up the dew which shows the marks made\nby brushing it off.  We're all right till they hit the track we've come,\nand that won't be till some time to-morrow, if they hit it then.\"\n\n\"Oh, they'll know the way we've come,\" said Perry, who was breathing\nhard from excitement.\n\n\"They must be very clever then, sir,\" said John Manning drily.  \"I\nshould say they'll think we've made for the way we came.\"\n\n\"Speak lower,\" said Cyril.  \"Why?\"\n\n\"Because, says they, these white fellows haven't got any guides now, and\nthey only know one road, so they're sure to take it.\"\n\n\"Yes, that sounds likely,\" said Perry sharply; \"but how was it we could\nhear them shouting?\"\n\n\"I know that,\" said Cyril.  \"The air is so clear right up here in the\nmountains, and the wind is this way.  It's like seeing.  You know how\nclose the peaks seem when they're twenty miles away.\"\n\n\"Yes, sir, and sounds run along a hollow like this wonderfully.  Why, I\nremember in one of the passes up in India, we in the rearguard could\nhear the men talking right away in the front as easily as if we were\nclose to them.\"\n\n\"But look here,\" said Cyril.  \"Diego or the other fellow must have seen\nwhich way we came.\"\n\n\"They must have been very sharp then, sir, for I took care to tie a\nlittle biscuit bag over each of their heads, only I left holes for their\nnoses to come out and breathe.  Don't you fret, young gentlemen; we've\ngot the start, and I don't believe the fight 'll begin 'fore to-morrow\nevening, if it do then.\"\n\n\"You know, then, that it will come to a fight,\" said Perry.\n\n\"Well, say a skirmish, sir.  We in the rearguard 'll have to be divided\ninto three companies, and keep on retiring one after the other, and\ntaking up fresh ground to protect the baggage-train.  It's all right,\ngentlemen, and it'll be quite a new experience for you both.  You'll\nlike it as soon as the excitement begins.\"\n\n\"Excitement?\" cried Perry.  \"Suppose one of us is shot.\"\n\n\"Ah, we don't think of that, sir, in the army,\" said John Manning.  \"We\nthink of the enemy getting that.  But, if one of us is so unlucky, why,\nthen, he'll be clapped on a mule's back and go on with the\nbaggage-train.\"\n\nThe two boys stopped then to listen, but all was silent save the faint\nrustling made by the mules in front as they went steadily onward in\ntheir leader's track.  The night was dark, but the stars glittered\nbrilliantly overhead in a broad strip which showed how deep down the\nvalley had grown, and how wall-like the sides rose in their blackness.\n\n\"I say,\" whispered Perry, stopping short.  \"Doesn't it make you feel\nshivery?\"\n\n\"No,\" said Cyril.  \"Shuddery.  We seem to be going on, down and down, as\nif this were a <DW72> leading right underground.  I shall be glad when\nthe daylight comes, so that we can see where we are going.--Hear any one\ncoming?\"\n\n\"No, but let's go on, or we may be left behind.\"\n\n\"Well, we are left behind now.\"\n\n\"But suppose we missed the others.  It would be horrible.\"\n\n\"No fear,\" said Cyril; \"the valley's getting narrower and narrower, and\nif we keep on, we're sure to overtake the mules.\"\n\nCyril was right, for in a few minutes they heard the faint patter of the\nhoofs again, and were glad to keep close in the rear, for instinctively\nthe patient beasts picked out the easiest way.  And now from being a\nsmooth, grassy, park-like, open valley, the route they followed began to\ncontract into a gorge, from whose wall-like sides masses of stone had\nbeen tumbled down in the course of ages, till the bottom was growing\nmore difficult to traverse every mile they passed; while, for aught they\nknew in the darkness, they might be skirting precipice and pitfall of\nthe most dangerous kind, depending, as they were, entirely upon the\nmules.\n\nThey had suggestions of there being unknown depths around, for to their\nleft there was the gurgling, rushing sound of water, apparently deep\ndown beneath the fallen stones, sometimes louder, sometimes dying away\ninto a murmur; till all at once, as they turned a corner into sudden,\ncomplete darkness--for the long band of starry light overhead was now\nshut out--they were startled by a deep echoing, booming roar, and a\nchilling damp air smote them in the face as it came down, evidently from\nsome gorge to their right, which joined the one along which they had\ntravelled.\n\nIt needed no explanation.  Light failed, but they knew as well as if\nthey were in broad sunshine that they were face to face with a huge\ncascade which came gliding down from far on high into some terrific\nchasm far below, while the change from the calm silence of the valley\nthey had traversed to the deafening sound which rose from below, was\nconfusing and strange to such a degree, that they came to a stand.\n\nIt was not that the noise was so great, as that it seemed, paradoxical\nas it may sound, so huge and soft, and to pervade all space, to the\nexclusion of everything else.  As Cyril said afterwards, it was a noise\nthat did not pierce and ring in your ears, but stopped them up and\nsmothered all speech; while the darkness was so deep, that no one felt\nthe slightest desire to take a step forward.\n\nPerry was the first to make any move, for all at once he felt for Cyril,\nplaced his lips close to his ear, and said excitedly:\n\n\"My father: can you hear him?\"\n\n\"No,\" replied his companion, after a pause.  \"I can only hear the\nwater.\"\n\n\"Then he must have fallen in.--Here, John Manning.  Where is the\nlantern?\"\n\n\"Tied to the first mule's pack, sir.\"\n\n\"Oh!\" exclaimed Perry excitedly, and then he shouted \"Father!\" as loudly\nas he could, but the cry seemed to be driven back in his face.\n\n\"I'll light a match, sir,\" cried Manning, and after a few moments there\nwas a flash, the gleam of a light, and the shape of the old soldier's\nhands, with the tiny flame gleaming ruddily between his fingers; but,\nsave that the boys saw the familiar rugged features of the man's face\nfor a few moments, they saw nothing more, and the darkness grew painful\nas the match went out.\n\nJohn Manning struck another light, got the splint well in a blaze, and\ntossed it from him; but there was nothing to be seen but mist.  The boys\nnow shouted together, but without result, and a chilling sensation of\ndread came over them as they grasped each other's wet cold hand, not\ndaring to stir, and with the horrible feeling increasing upon them that\nsome terrible tragedy must have happened to their leader.\n\nJust when the sensation of horror was at its height, John Manning's\nvoice was heard.\n\n\"What had we best do, gentlemen--go forward or go back?\"\n\n\"We ought to go forward,\" said Cyril.\n\n\"Yes, that's what I feel, sir,\" shouted the man; \"but next step may be\ndown into the pit.\"\n\n\"We must go on,\" said Perry excitedly; \"my father wants help.  He's in\ndanger, I'm sure, or he would have made some sign.\"\n\nAs he spoke, he snatched his hand from Cyril's grasp, and took a step or\ntwo forward into the black darkness.\n\n\"Perry!\" shouted Cyril, in a voice which sounded like a faint whisper,\nas he felt himself seized by the shoulder, John Manning's great hand\nclosing upon it like a vice, and holding it firmly.\n\n\"Where's Master Perry?\"\n\nNo answer escaped Cyril's lips for a minute.  He felt suffocated, and it\nwas not until John Manning had shaken him violently and repeated his\nquestion twice, that he panted out the single word, \"Gone.\"\n\n\"Can you see where--has he fallen in?\" was panted in his ear.\n\n\"No; he stepped from me to help the colonel, and then he was gone.\"\n\nJohn Manning groaned, and Cyril felt the strong man's hand trembling,\nand the vibration thrilled through the boy's frame until every nerve\nquivered with the horrible dread which assailed him.\n\nAll at once he felt the lips at his ear again.\n\n\"Let's shout together, sir,\" was whispered, and they tried hard to make\ntheir voices heard, calling together with all their strength, but they\ndid not seem to be able to pierce the roar which pressed, as it were,\nupon them; and though they repeated the cry at intervals and listened\nfor a reply, none came.\n\n\"It's no good, Mr Cyril, sir,\" groaned John Manning.  \"I'm ready, sir,\nto do anything to try and save my poor colonel and Master Perry; what\ncan I do?  It's like chucking away my life and yours, sir, to stir a\nstep.\"\n\n\"Yes, and I'd help you,\" said Cyril despairingly; \"but we dare not move\nin this terrible darkness.\"\n\n\"Shall we try to go back, sir?\"\n\n\"No,\" shouted Cyril firmly.  \"We must not do that.\"\n\n\"What then, sir?  What can we do?\"\n\n\"Wait for daylight,\" Cyril shouted back in the man's ear.  Then softly\nto himself: \"And pray.\"\n\n\n\nCHAPTER TWENTY.\n\nWAITING FOR DAYLIGHT.\n\nAs John Manning afterwards said, those were hours to make a man's hair\nturn grey, and to Cyril every minute seemed to be indefinitely\nprolonged, as he stood till he felt his knees begin to give way beneath\nhim, and finally sank cautiously down upon them--John Manning imitating\nhis movement--till they both rested upon wet, slippery rock.\n\nThere they crouched with strained ears, waiting for the light which\nseemed as if it would never come, while the noise was crushing them\nback, as it were, upon themselves, and dulling their brains till all was\nto Cyril like some terrible dream.  There were moments when he felt as\nif his senses were leaving him, and the sensation was almost welcome,\nfor the agony at last grew greater than he could bear.\n\nHe had reached this pitch as he crouched there with his arm drawn\ntightly through John Manning's, when he felt the man's grasp upon him\nloosened, and the next moment he felt a thrust.\n\nHe knew directly what it meant.  Following the movement, he became\nconscious of some pale, bluish-looking smoke on his left, and as this\ngrew clearer, he realised that it was not smoke, but a thick mist\nbetween him and the coming light of day; but for a few minutes there was\nnothing more.\n\nThen by slow degrees this dim, grey appearance grew and expanded, till\nthe boy made out that the mist rose out of the depths before them, and\nat last that he and John Manning were crouching upon a ledge of rock on\none side of a great gulf, down into which the waters thundered from\ntheir right, while overhead the wall of rock rose up nearly straight,\nthe light of day being shut out by the dense mist which rose from below.\n\nThis light increased rapidly now in pale gleams from the left, and a\nfaint, soft diffusion from above, showing that they were where a vast\nrift in the mountain joined at right angles the valley they had\ndescended, while the rocky sides were so close that they nearly met\noverhead.  But some time elapsed before they could make out more, the\nsteamy mist obscuring everything, and preventing them from seeing\nanything of Perry or the colonel.\n\nThey had both risen to their feet, and clasping hands, began, as soon as\nit was possible to see a step or two, to try to penetrate farther in;\nbut before they had gone half-a-dozen steps, John Manning, who looked\nmisty and unsubstantial to Cyril, stopped short and pointed downward in\nfront of him to where the rock looked slippery as glass.\n\n\"He went down there, sir,\" he shouted, and loosening his grasp, threw\nhimself down upon his chest, and wormed himself forward, so as to get\nhis head over the gulf and look down.\n\nCyril watched the man in agony, fully expecting to see him glide forward\nout of sight; but in a few minutes he worked himself back, rose, and\nplaced his lips to the boy's ear again.\n\n\"Can't see.  All one thick cloud of spray.\"\n\nCyril gave a great start, for at that moment, from out of the misty\ngloom, the colonel strode forward to meet them.\n\n\"Thank goodness,\" he shouted.  \"I was very nervous about--Where's\nPerry?\"\n\nCyril and John Manning, whose faces had lit up with pleasure, now gave\nhim a despairing look, which made him seize Cyril by both arms.\n\n\"My boy!\" he gasped.  \"Where's my boy?\"\n\nThere was no reply.  There was none needed, for the colonel read in\ntheir faces what was wrong.  He had seen them, too, trying to look down\ninto the misty gulf below, and there was a horrible look of despair in\nhis countenance as he pointed mutely down into the terrible-looking\ngloom.\n\nThen going right to the edge, he tried to look over, but drew back a\nlittle and stretched out his hand to John Manning, hooking his fingers\nthe while.\n\nThe old soldier stepped forward.  Long discipline and training had made\nhim ready to grasp his master's wishes, and planting his right foot\nagainst a projecting piece of the rock, he hooked his fingers in the\ncolonel's, and then hung slightly back, giving a little and a little\nmore, till the latter was able to lean right out and gaze down.\n\nIt was by this time far lighter, and the mist was here and there\ntransparent, as it came eddying up more and more like the clouds of\nsmoke from a fire, but there was no piercing even the lightest parts;\nand giving this up in despair, Colonel Campion rose up, made a sign to\nthem to stand firm, and then stepped rapidly in the direction from which\nthey had seen him come.\n\nOne minute they saw his figure growing fainter along by the side of the\nrock-wall, the next he had disappeared in the gloom and mist.\n\n\"Let's follow,\" said Cyril, with his lips to John Manning's ear.\n\nThe man shook his head.\n\n\"Soldier never leaves his post without orders,\" he replied.  \"Better\nstay, sir.\"\n\nCyril hesitated, but stayed; now watching the spot where the colonel had\ndisappeared, now letting his eyes wander round the place, which, as the\ngrowing light of day penetrated it more and more, was still awful\nenough, with its whirling mist, gloom, and deafening roar of invisible\nwater falling behind the pearly veil, but far from being as terrible as\nwhen it was all shrouded in deep obscurity.\n\nFor the light came down softly from high above their heads, showing that\nthough the rocky walls nearly approached, there was a firmly-defined\nband that would probably be bright and golden when the sun rose, but\nJohn Manning's words were justified as he suddenly leaned forward and\nsaid:\n\n\"What a place, sir!  It's a wonder there ain't four of us gone for\ngood.\"\n\nJust then the colonel reappeared with half-a-dozen of the raw hide ropes\nused about the mules for lassoes, tethering, and binding on their loads.\n\nThese he threw down, and John Manning followed his example as he began\nto knot them together.\n\n\"Bear me?\" shouted the colonel to the old soldier.\n\n\"Two of you, sir,\" said the latter; \"but you lower, I'll go.\"\n\nThe colonel shook his head angrily--the task of speaking was too much in\nhis state of anguish--and he went on trying the knots he made, while\nCyril picked up one end and examined a couple of the knots before making\na strong loop, and passing it over his head and shoulders.\n\nHis action passed un-noticed for a few moments, for he had drawn back;\nbut when the last rope was joined to the others, the colonel turned and\ngrasped the boy's intention.\n\n\"God bless you, my lad,\" he cried, \"but I cannot let you go.\"\n\nCyril hardly heard a word in the midst of that deep-toned, booming\nthunder, but he grasped their import, and stood firm.\n\n\"Yes,\" he shouted.  \"I'm light.  Lower me down.\"\n\nA curious sensation attacked him as he spoke, and he knew that he was\nturning pale, but he faced in the direction of the gulf, and tried hard\nto pull himself together.\n\n\"Perry would have gone down after me,\" he said to himself, \"and it isn't\nso very dangerous after all.\"\n\nBut all the while he knew that it was, and also that it was a task\ncalling for nerve, determination, and strength, all three of which he\nseemed to be wanting in when face to face with the dense, wreathing mist\nof that terrible gulf.\n\n\"I don't care.  I'm afraid, horribly afraid,\" he muttered between his\nteeth.  \"But I'll go.  I'd go if it was twice as dangerous, if it's only\nto let father know I'm not all bad.\"\n\nMeanwhile, a short discussion, painfully hard, went on between the\ncolonel and John Manning, the former hesitating, the latter insisting.\n\n\"He's light, and can do it better than you.  Perhaps we couldn't pull\nyou up, nor you me.\"\n\nThen the colonel held out his hand to Cyril, who grasped it eagerly, but\nin an instant the colonel's face began to work, and he drew the lad to\nhis breast, held him there for a brief moment, and then released him.\n\n\"I'm not afraid now,\" shouted Cyril, and he stepped at once to the edge,\nand, as the line was tightened, went down on his face, passed his legs\nover, and, grasping the line with both hands, glided down; seeing the\nfaces of the two men who held the rope disappear, then the shelf; and\nthe next minute, as he was lowered, he saw nothing but the light mist\nwhich closed him in, and struck dank and chilly to his face and hands.\n\nHe had expected to swing to and fro in the air, and had prepared himself\nto grasp at the rock, and try to prevent himself from turning round and\nround; but to his surprise he found that he was on a sharp incline, down\nwhich he was sliding easily, for the rock was covered with a slippery\nmossy growth, over which his hands glided whenever he tried to check his\ncourse; for, in spite of his determination, the desire to do this\nmastered him.  Anything to stop himself from going down into that awful\nplace at some terrible depth below, where the water was churning round\nand round, and tossing up this mist of spray.  To go down into that must\nmean instant death; and after all, what good was he going to do?  Poor\nPerry had slipped, gone over the edge, and then not fallen headlong, but\nglided down at a terrible rate, with no power to arrest his course; and,\nif he were not down there below, he must have been swept out by the\nstream, and be far away down the river by then.\n\nThese thoughts came quickly as he slipped gently down, keeping his face\ntoward the roaring water and churning mist, but seeing nothing; for the\ndarkness now, as he was lowered more, began to increase.\n\nDown, down, down!  Was there no end to the rope?  How long it seemed\nbefore it was checked.  Still Cyril tried hard to make out something of\nthe whereabouts of his friend.  But no; if he turned to the right,\ntoward where there was the hissing noise of the falling water, all was\nblack, as black as it was below in the fearful hollow into which it\nplunged, to send up that deafening, reverberating thunder.  At last to\nthe left there, where he knew the chasm must open into the valley by\nwhich they came, he could see a faint suggestion of light, such light as\none sees when looking towards a candle with the eyes tightly closed, and\nwhen trying to peer through the veined lids.\n\nThen, to his horror, he was being lowered again, for he had believed\nthat the end of the hide rope was reached.\n\nIt seemed a great depth down before there was another check, though\nprobably it was not more than a dozen or twenty feet; and once more, as\nhe tried to grasp the slimy rock behind him, he peered about vainly,\nknowing that if poor Perry had once begun to glide down that horrible\n<DW72>, he must have gone right on down to the bottom.\n\nThen there was a heavier strain upon his chest, and to his intense\nrelief, now that he felt how vain his effort had been, he turned his\nface toward the rocks, and tried to help by climbing, as he was being\ndrawn up.\n\nVain effort.  Hands and feet glided over the slippery moss, and he soon\nsubsided, and waited in increasing agony, while he was steadily hauled\nup.  For, in descending, his senses were hard at work, and he was\nmomentarily hoping to rest upon some shelf where he might come upon\nPerry.  But now he had nothing to do but think of himself and his risks,\nand, in spite of the effort to be brave, he could not keep his mind from\ndwelling upon the knots of the several ropes, and wondering whether\nthose John Manning tied were as firm as the colonel's, and whether the\nrope itself might not have been frayed by passing over the rocks, and\ngive way just before he reached the shelf.\n\nAt last, with head burning, hands and feet like ice, and clothes\ndrenched with the spray, he felt himself seized by John Manning's strong\nfingers and lifted into safety.\n\nIt had now become light enough for him to see well around; the mist on\nhigh was turning roseate and warm by reflection, for the sun was rising;\nand the colonel turned from him with a look of agony, and stood with his\nback to them, while John Manning unloosed the rope.\n\n\"Nobody could come out of such a place as that, my lad,\" he said,\n\"alive.\"\n\n\n\nCHAPTER TWENTY ONE.\n\nTHE PURSUIT.\n\n\"I'll go down again, sir,\" said Cyril, when the colonel had turned back,\nand he had tried to make him understand the nature of the place, as far\nas he had been able to make out.\n\nBut the colonel shook his head.\n\n\"We must go back, and try to reach the stream where it flows out, my\nboy,\" he said.  \"We can do no good here.--Come, Manning, and fetch the\nmules.\"\n\nJohn Manning stared, and seemed as if he could not understand.\n\n\"The mules, sir--go back and find the stream?  What about the Indians,\nif they are coming on?\"  The colonel had forgotten their pursuers.  \"The\nmules,\" he said then; and he led the way on into the mist, Cyril\nfollowing him wonderingly along the continuation of the rocky shelf for\nabout a hundred yards, and glancing back from time to time to see that\nJohn Manning was close behind, untying the knots of the hide ropes as he\ncame.\n\nEvery step took them nearer to the great waterfall, and in the dim light\nCyril now made out that the path was wider; but all at once it seemed to\nend in front of a gleaming sheet of water reaching from the thick mist\nbelow right up to where the rock-walls appeared to give place to the\nspray-clouded sky.  And there, just before them, all huddled together,\nstood the mules, ready to turn toward them as they approached.\n\n\"They brought me as far as this last night,\" said the colonel, \"and then\nstopped.  No wonder, poor brutes, they would go no farther; and I was\nlost in the darkness, and dared not turn back.  I stood with them till\ndaybreak, hoping you all were safe, and then--\"\n\nCyril uttered a wild cry of joy, one which made itself heard by all, for\na bare-headed misty figure, whose presence they had not been aware of as\nit followed them, suddenly caught the colonel's arm, placed its lips to\nhis ear, and cried:\n\n\"Quick, father--the Indians; they're coming down the valley fast.\"\n\nIn the face of such news as Perry bore, there was no time to ask\nquestions about his escape, but as the colonel grasped the boy's arm,\ntrembling the while with excitement, his heart throbbing with joy, he\ncried:\n\n\"How far away?\"\n\n\"Not half a mile.  I could see them coming down the valley.\"\n\n\"This way,\" said the colonel promptly, and he supplemented his words\nwith gestures, as, still holding his son's arm tightly, he led them on\nthrough the mist of fine spray inward toward where the mules were\nstanding together.  And now as they approached the fall, a great deal of\nthe horror caused by the darkness and noise passed away, for the mist\ngrew opalescent from the sunshine far above, and though progress looked\nterribly perilous, they could see the extent of their danger, and there\nwas no mystery of hidden peril, no constant dread of unknown chasms\nwaiting to engulf them at their next stride.\n\nFor they knew now that they were in one of Nature's wildest and grandest\nrifts, where a goodly-sized river, after tearing its way along the\nprofound depths of a narrow gorge, had reached a spot where by some\nearthquake convulsion this gorge had suddenly, as it were, broken in\ntwo.  One part had dropped several hundred feet, forming a profound\nchasm into which the water from above leaped in one great glistening\nwave, smooth as so much gleaming glass, to be broken up into spray as it\nreached the jagged rocks below, and there eddy and foam in what was\nundoubtedly a huge basin, from which the mist arose, while the broken\nwater swept on down into the valley to join the little stream by whose\nside they had come.\n\nThe leading mule threw up its head as the colonel approached, and its\nparted teeth and drawn-back lips suggested that it was whinnying a\nwelcome or a demand for food.  But the great fall before them, and the\nknowledge that at any time the Indians might appear from out of the\ndense mist and commence their attack, gave the colonel eyes for only one\nthing, and that a way out of what seemed to be a perfect _cul de sac_.\n\nThe deafening roar, of course, prevented all consultation, and the mist\nadded to the confusion; but these had their advantages for the\nfugitives, veiling their actions from their pursuers, and preventing any\nsound made by the mules from being heard.\n\nAnd as Cyril watched their leader's actions, and then caught an\nencouraging look from John Manning, who gave his head a jerk in the\ncolonel's direction, as if to say: \"It's all right, he'll find his way\nout,\" the boy felt in better spirits.  The terrors of the night were\ngone; they were all there safe, and there was the possibility of the\nIndians feeling as much in awe of the terrible chasm as they had\nthemselves, and hence shrinking from making their way through the mist,\nand giving them the credit of going on down the valley by the greater\nstream which issued from beneath the falls.\n\nCyril's thoughts were many, and in the reaction from the terrible\ndespair from which he had suffered, he was ready to accept anything\nshort of the marvellous; and consequently he was in nowise surprised on\nseeing their leader go right on into the darkness, peering here and\nthere, and the leading mule follow him and Perry, the rest getting in\nmotion directly, and going on into the mist till the last had\ndisappeared.\n\nJust then John Manning, who had turned to look back, wiping the moisture\nfrom his face, clapped Cyril on the shoulder, and placed his lips close\nto the boy's ear.\n\n\"Can't see 'em coming.  This'll scare 'em from following.  They'll think\nnobody but mad folk would ever come along here.  I say, he's found a way\nbehind the fall.\"\n\nBut John Manning was wrong.\n\nThey followed the direction taken by the last mule, together stepping\ncautiously onward through the mist, for the rugged shelf they were on\nwas dripping with moisture, and felt slippery beneath their feet, while\nto their left there was the huge body of water always gliding down into\nthe spray which eddied up to meet it.  Then, to their intense\nastonishment, they stepped right out of the dense, clinging mist, which\nhid everything, into a clear atmosphere.  It was quite in twilight that\nthey stood, but the falling water brought with it a cool current of air;\nand as they both stopped for a moment to gaze and wonder, there to their\nleft was the great fall rushing down clear of the rock behind, and\nleaving plenty of room for any one to pass through to the other side,\nbeneath the water, had the shelf been continued there; but it passed\nround to their right, as if Nature had made a natural staircase,\nzigzagging up the side of the gorge; and there, some distance above\nthem, were the colonel and Perry, mounting slowly after the leading\nmule, which showed no hesitation about proceeding now that it was day.\n\nJohn Manning nodded, and they followed up and up the giddy path, now\nleaving the fall some distance behind, now approaching it again, but\nalways near enough to be terribly impressed by the vast curve of\ngleaming black water, which, as they rose higher, could be plainly seen\nplunging down into what appeared to them as a dark grey cloud.\n\nFrom time to time the colonel looked back and waved his hand, stopping\nat last at a spot where the natural track curved suddenly round a sharp\npoint of rock.  The mules followed one by one, their heads right down,\nand their feet carefully planted at every step, till the last had gone\nround; and then in turn Cyril and John Manning climbed up, and before\npassing the sharp rock, stopped to gaze down into the vast rift up whose\nside they had mounted so far.\n\nFrom this point the whole of the wild zigzag was visible right to where\nthe grey veil of mist shut off the level shelf where they had passed the\nnight, and John Manning's lips had just parted to utter some words about\nthe horrible nature of the place, when Cyril started back and jerked his\ngarment, to make him follow suit.\n\nThe old soldier was keenly alive to danger, and dropping upon his knees,\nhe joined Cyril in cautiously looking over the edge of the rocks they\nhad just ascended, softly bringing the muzzle of his piece to bear upon\nwhat he saw.\n\nFor, as he gazed down, there in the gloom, not two hundred yards away as\nan arrow would fly, but at a distance which it had taken them nearly\nhalf an hour to climb by the gradual ascent, was the figure of an Indian\nstanding out just clear of the mist, and peering cautiously about, as if\nsearching every rock and crevice around.\n\nThe next minute another had joined him, coming out of the mist\ncautiously, and with the tentative motion of one who was on strange\nground.\n\nThen came another and another, with their figures looking huge and\ngrotesque as they stood in the mist, and then suddenly shrinking into\nthe stature as of dwarfs, as soon as they were clear.\n\nOne by one they came on, till there were at least thirty collected\ntogether, and all gazing about cautiously, as if in dread.\n\nAs Cyril knew from his own experience, they could only converse with\ndifficulty, so that he was not surprised to see that one of them, who\nappeared to be the leader, was gesticulating and pointing here and\nthere, and finally upward toward where the two fugitives were watching\nevery act.\n\nBut, as the boy watched the Indians keenly, it was very evident that\nthey were far from confident, and he soon decided that they were as much\npanic-stricken by the horror of the place as he and his friends had been\novernight.  At last, though after a great deal of pointing upward and\nhesitation, it seemed as if they were all reluctantly about to continue\nthe pursuit, for their leader took a few steps forward and waved them\non.\n\nBut they did not stir, save to crowd together a little more and press\ntoward the wall of rock, away from the fall.\n\n\"They don't like it,\" whispered John Manning, for it was becoming\npossible, where they lay, to make a few words audible without shouting.\n\"Strikes me they're so scared, that if we were to send one of these big\npieces of rock rolling down, they'd beat a retreat.\"\n\n\"Look, look!\" whispered Cyril.\n\n\"I am,\" said John Manning, for all at once a couple more of the Indians\nsuddenly appeared from out of the mist, in whom they recognised Diego\nand his fellow-guide, the former holding something in his hand which he\nwas showing to all in turn with a great deal of gesticulation,\naccompanied by eager pointings down into the depths below the fall, and\nback through the mist.\n\n\"What's he got there?\" whispered John Manning.  \"Something to eat?  He\nwants them to go back.\"\n\n\"I know,\" said Cyril so loudly that his companion caught his arm.  \"It's\nPerry's cap.\"\n\n\"What!\" cried the old soldier.  \"I know how it is.  They've found it\nsomewhere down the stream, where it had been washed, and he's saying\nthat we must all have tumbled in there and been swept away.\"\n\nThis appeared to be a very likely interpretation, for, with a great\ndisplay of eagerness, the men hurried back through the mist till all\nwere gone.\n\n\"Let's make haste on and overtake them,\" said Cyril eagerly.  \"I want to\nask Perry where he left his cap.\"\n\n\"And he'll tell you, sir, that he didn't leave it anywhere, but had it\ntook away by the water.\"\n\n\"Are they in sight?\" said the colonel, bending down over them.  \"You\nwere quite right.  This is an excellent place to keep them back.  Yes,\"\nhe continued, on hearing the surmises of the two watchers, \"that must be\nit, and they have gone back to follow the stream.\"\n\nHe led the way again, and they followed to where Perry was anxiously\nlooking back, as the mules steadily went on higher and higher up the\ngloomy gorge, where the great stream was hurrying and foaming along\ntoward where it would make its plunge; while the thunderous roar of the\nfall was rapidly dying away, shut out, as it now was, by curve after\ncurve of the valley.\n\nThe place was black and forbidding enough, but as they got on another\nmile or two, their journey was brightened by the glow upon the ridges\nand <DW72>s on high where the sun reached, and the grassy sides of the\nlower mountains looked delightful after their long experience of black,\ndripping stone.\n\nMany a look back was given as they went on higher and higher, every step\ntaking them more into the mighty range, and fortunately due west; and,\nweary as they all were, intense was the longing to hurry their steps.\nBut that last was impossible.  They were dependent upon the mules for\ntheir supply of food, and the cautious animals only had one pace, and\nthis regulated their masters'.\n\nAt last, when utterly exhausted, a halt was called just at a sharp turn\nin the gorge, where water could be reached, and the rocks sheltered them\nand the mules from pursuers; while they gave them the opportunity of\nscanning the narrow way for nearly a mile, so that if a watch was kept,\nit was impossible for them to be taken by surprise.\n\nThere was some stunted herbage too, here, upon which, as soon as they\nhad drunk, the mules began to browse.  But no load was removed, arms\nwere ready for an attack, and the only mule that was lightened was the\none that bore the provisions.\n\nAnd now Perry was questioned more closely about his escape, and Cyril\nheard it from his lips for the first time.\n\n\n\nCHAPTER TWENTY TWO.\n\nPERRY'S PERIL.\n\n\"There isn't much to tell,\" said the boy with a shiver.\n\n\"Never mind; tell me: I want to know.  What's the matter--cold?\"\n\n\"No, I'm warm enough now,\" said Perry, \"for my clothes have got dry; but\nit makes me shiver as soon as I think about it, and I feel as if I\nalways shall.  It's a thing I shall dream about of a night, and wake up\nfeeling the water strangling me.\"\n\nCyril looked at him in wonder, and the boy tried to smile, but it was a\nvery pitiful attempt, and he went on hurriedly.\n\n\"You know how horrible all that was when I felt sure that my father had\ngone down somewhere, and something forced me to go and try to find him.\nAnd then, as I went on through the mist, I only took three or four steps\nbefore my feet gave way, and I was sliding at a terrible rate down, down\nto where the water was thundering and roaring.\"\n\n\"Was it very deep?\" said Cyril, for his companion paused.\n\n\"I don't know; I seemed to be sliding along very fast, and then I was\nfighting for breath, and being dashed here and there, and I suppose I\nwas carried along by the water almost as swiftly as I slid down that\ndreadful <DW72>.  Then, after fighting for my breath, all was confusion\nand darkness, and I can't remember any more till I found myself lying\namong some rocks.  The water was rushing and foaming over my legs, and\nevery now and then rushing up over my chest, and making me feel so in\nfear of being drowned that I climbed a little, and then a little more,\ntill I was out of the water, but afraid to move in the darkness in case\nI should fall in again.\"\n\n\"Where were you?\" said Cyril.\n\n\"I didn't know then, but lay aching with the cold, and listening to the\nrushing water; while it was so dark, that I felt sure that I must have\nbeen washed into some great hole underground, where I should lie till I\nwas dead.\"\n\n\"We felt all kinds of horrors about you,\" said Cyril, \"but you seem to\nhave suffered more than we did.\"\n\n\"I don't know,\" said Perry plaintively.  \"It was very bad, though, and\nif I hadn't fallen at last into a sort of stupor, I've thought since\nthat I should have gone mad.\"\n\n\"Stupor!\" said Cyril, smiling.  \"You mean you went to sleep.\"\n\nPerry looked at him so reproachfully that Cyril felt the blood flush\ninto his cheeks, and the colour deepened as his companion said: \"How\ncould a fellow go to sleep when he believes his father has been killed,\nand he has himself just escaped from a horrible death?\"\n\n\"Don't take any notice of what I said,\" cried Cyril hurriedly; \"I did\nnot mean it.\"\n\n\"I know you did not.  I suppose it was from being so exhausted.  I felt\nas if I had been stunned, and could neither think nor stir, and then\nthis curious feeling came over me, and everything passed away.  It was\nnot sleep.\"\n\n\"No, no; don't say that again,\" cried Cyril apologetically.  \"How long\nwere you like that?\"\n\n\"I don't know, only that it was still dark when I came to, and sat\nwondering where I was, and whether I should ever see the light again, so\nmiserable and desolate you cannot think.\"\n\n\"Yes, I can,\" said Cyril warmly; \"I felt bad, too, when I thought you\nwere drowned, and went down to try to find you.\"\n\n\"What!\" cried Perry excitedly.  \"You went down to try to find me?\"\n\n\"Oh yes,\" said Cyril coolly.  \"Didn't you know?  They put a rope round\nme and let me down.\"\n\n\"Cil!\"\n\n\"Well, don't make a fuss about it,\" said Cyril, laughing.  \"They had\nhold of the rope.\"\n\n\"But the place was so awful.  Didn't you feel frightened?\"\n\n\"Horribly, of course, and it was ever so much worse when I'd got to the\nend of the rope, and felt that you must be gone.  But never mind that.\nGo on.  You were saying how miserable you were.\"\n\n\"Yes,\" said Perry thoughtfully, \"till all at once I caught sight of\nsomething high up, just as if it was a point of light coming through a\ncrack in the roof of the cavern into which I had been washed.\"\n\n\"And was it?\"\n\n\"No,\" said the boy, with his eyes brightening, \"it was the first light\nof morning shining miles up on the ice of one of the great peaks, and as\nI watched it, I saw it get brighter and then begin to glow as if it were\na precious stone.  The light gradually stole down lower and lower, till\nit seemed to come right into my heart; and from that moment I began to\ngrow strong and hopeful, and something seemed to tell me that I should\nsee you all again.\"\n\n\"Hah!\" ejaculated Cyril, as he watched his friend's countenance; \"I wish\nsomething of that kind had come to me when I was feeling worst.\"\n\n\"You weren't alone,\" said Perry, smiling.  \"Well, as soon as I found\nthat I was just at the edge of a rushing torrent, I knew that if I\nfollowed it up, I should come to the mouth of the gorge where you must\nbe, and I began to climb along the side, getting warmer every minute;\nand I felt more hopeful too, for I began to think how clever my father\nwas, and that he would have been able to save himself, or have been\nsaved, just as I was.\"\n\n\"And then you soon found the mouth of the gorge where the water came\nout?\"\n\n\"Yes, and the place where we turned in last night, instead of going\nright on down the main valley.  It was quite a climb up to the path, but\nI dragged myself up; and just then I happened to turn my eyes along the\nway we came just as I was warmest, and then I turned cold again.\"\n\n\"Because you saw the Indians?\"\n\nPerry nodded, and the boys sat in silence for a few minutes, looking up\nat the sunlit sky, which appeared like a broad jagged path running along\nhigh above their heads.\n\n\"What are you thinking about?\" said Perry suddenly, as he noted the\nthoughtful, deeply-lined brow of his companion.\n\n\"Eh?  Oh, nothing much,\" replied Cyril.  \"Only that when I knew you were\ncoming up into the mountains, I felt so jealous of you, and I fancied\nthat you were coming to see all kinds of wonders and make great\ndiscoveries, and that it would be one grand holiday, day after day, and\ninstead of that--I say, we haven't had so very much fun yet, have we?\"\n\n\"Plenty of adventures,\" replied Perry thoughtfully.\n\n\"Yes, plenty of adventures.\"\n\n\"It's been so hard upon you, though, from the first.  You were so upset\nwhen you joined us.\"\n\n\"And serve me right,\" cried Cyril angrily.  \"I'd no business to do it; I\nbelieve they think at home that I'm dead.  Nothing's too bad to happen\nto me.\"\n\n\"Then you're sorry you came?\"\n\n\"Yes; horribly.  I don't mind all we've gone through, because it has\nseemed to stir me up so, and made me feel as if I'd got more stuff in\nme; and it ought to, for sometimes I've felt, since we came, that I\nbehaved like a miserable, thoughtless coward.\"\n\n\"No one could call you a coward,\" said Perry firmly.\n\n\"Oh yes, they could--a miserable, selfish coward.\"\n\n\"I should just like to hear any one call you one,\" said Perry viciously,\nand with a hard, fierce look in his countenance.\n\n\"Then you soon shall,\" said Cyril.  \"I call myself one a dozen times a\nday.  There, I'm a coward.\"\n\n\"But I meant some one else.\"\n\n\"You wait long enough, and you'll hear my father call me one.\"\n\n\"You're not.\"\n\n\"Yes, I am, and I shall deserve all he says--that is, if we ever get\nback to San Geronimo.\"\n\n\"Don't talk like that,\" said Perry.  \"What's to prevent us?\"\n\n\"Indians,\" said Cyril mournfully.\n\n\"But we've left them behind.\"\n\n\"For a bit.  They'll hunt us out again somewhere.  They've got all the\nadvantage of us.  I daresay there are thirty or forty of them hunting\nus, and what one doesn't know of the country, another does; and as they\nspread out, they'll warn every Indian they meet, so as to run us down,\nfor they're sure to feel now that we're after the buried treasures, and\nthey'll give us credit for having found them.\"\n\n\"Why?\"\n\n\"Because we have escaped.  Every pass will be guarded, and every valley\nsearched, so that they are sure to come across us at last.--Look,\nthey're going to start.  Come along.\"  And picking up their guns, the\nboys joined the colonel and John Manning, who were tightening up the\nropes round two of the loads.\n\n\"Better trust the leader, Manning,\" said the colonel.\n\n\"Yes, sir.  He seems as good as a guide; and if you set his head\nstraight, he'll take us somewhere; and where he goes, the others'll\nfollow.  Rum thing, too, sir.\"\n\n\"Oh, I don't know,\" said the colonel; \"these animals have passed their\nlives in the mountains.\"\n\n\"Of course, sir, but I didn't mean that.  I meant it was a rum thing for\nthem to follow their leader in this way, for they all hate him like\npoison, and kick at him whenever they have a chance; and as for the way\nhe kicks at them, I wonder sometimes he doesn't get his heels stuck in\ntheir ribs, so that he can't get out no more.  'Tis their natur' to, eh,\nMaster Cyril, sir?--Ah, would yer!\"\n\nThis to one of the mules, whose heels must have itched, for it was\nsoftly turning itself round as if seeking somewhere to administer a good\nround kick.\n\nThen all was ready for a start; but first the colonel mounted the side\namong the rocks, to search the valley with his glass.\n\nHe was soon satisfied that the Indians were nowhere within sight, and\ntaking advantage of the high position he occupied, he turned the glass\nin the other direction, to scan the way they were about to go.\n\nAll there was utterly silent and desolate.  There were the rocks\neverywhere, hardly relieved by a patch of green, and he was about to\ndescend and start the mules, when he caught sight of Cyril hurrying back\ntoward him, and signing to him to stay where he was.\n\n\"What is it?\" he cried, as he saw the boy's anxious face.\n\n\"Look up to your left, sir, just above where that big rock sticks out\njust as if it must fall.\"\n\n\"Yes, I see,\" said the colonel; \"with another just above.\"\n\n\"That's it, sir.  Look just between those two blocks.\"\n\n\"Yes, I have the place.\"\n\n\"Well, sir, there are two Indians there watching us.\"\n\n\"No, my lad, there are no Indians there.  Take the glass and look for\nyourself.\"\n\nCyril snatched the glass, directed it to the steep, precipitous side of\nthe gorge, and then uttered an ejaculation full of annoyance.\n\n\"They're gone, sir, but I'm sure there were two men there.\"\n\n\"Then if so, they must be close to the same spot now.  I hope you are\nwrong, but of course you may be right.  Let's go on, and if they are\nthere, we shall be sure to catch sight of them, for they must go forward\nor backward.\"\n\n\"Would you go on?\" said Cyril dubiously.\n\n\"At any cost, boy.  We cannot go back to that awful chasm to pass\nanother night.  There, back with you, but keep your eyes on the position\nin which you saw the men.\"\n\nCyril was silenced, and half ready to suppose that in his anxiety he had\ndeceived himself; and in a few minutes he was back with the colonel,\nbeside Perry and the mules, but without seeing anything in the direction\nhe had pointed out.\n\n\"Ready?\"\n\n\"Yes, sir, but my eyes are not quite so good as they were, sir, and I\nfancied I saw some one creeping along the side of the rock, up yonder to\nthe right.\"\n\n\"Left, John Manning,\" cried Cyril, \"and I saw it too.\"\n\n\"You saw something on your left, sir?  Then I am right, and my eyes are\ntrue.  There's Injuns watching us, sir, and if we don't look out, we\nshall have arrows sticking in our skins.\"\n\n\n\nCHAPTER TWENTY THREE.\n\nAT THE BIVOUAC.\n\nThe boys were heartily glad when, just before nightfall--night, which\nfell much sooner, shut in there in the deep valleys of the Andes--the\ncolonel snatched at a suggestion made by John Manning.\n\n\"Water, sir, coming out of that slit in the rock, plenty o' breastwork,\nand a bit of green stuff for the mules.\"\n\n\"Yes, we'll halt here.  We are not likely to find a better place,\" said\nthe colonel.\n\nSo instead of tramping on for another hour, a halt was called early, the\npackages formed into a shelter in front of the \"slit\" in the rock, as\nJohn Manning called it, a place which suggested its being a way into a\ngood-sized cavern, and then a fire was lit, and they prepared their\nmeal.\n\nFor no more had been seen of the Indians, and though the colonel had a\nshrewd suspicion that they might still be in chase of them, those which\nhad been seen in the valley were, he concluded, only wanderers, whom\nthey had startled while on some hunting expedition, and whom they would\nprobably see no more.\n\nThe fire was only used to heat the water for their coffee, and as soon\nas this was made, carefully extinguished by John Manning, so as not to\nattract attention if any one was still about; and then they sat, glad of\nthe rest, eating biscuit and charqui, and sipping coffee from the tin.\n\nOver the meal, John Manning made a report respecting what he called the\ncommissariat department.\n\n\"Stores getting low, sir,\" he said.\n\n\"Yes, I must supplement them with one of the guns,\" said the colonel.\n\"I have been so much taken up with getting the cinchona seed, that I\nhave hardly thought of anything else.\"\n\nVery little was said then for some time, the weariness mentally felt by\nall making them ill disposed for conversation; but just before dark the\ncolonel carefully inspected their surroundings, and with John Manning's\nhelp, made a few arrangements for their defence.\n\n\"I don't think they would dare to attack us if they found where we are,\"\nsaid the colonel; \"but we must be prepared.\"\n\n\"Is it worth all this trouble and risk, father?\" said Perry, who was, in\naddition to being weary and low-spirited, stiff, and a good deal\nbruised.\n\n\"What! to get the seed, boy?\"\n\nPerry nodded.\n\n\"Lie down and rest, and wait till the knowledge comes to you, boy.\nThere, I'll speak out and ask you a question.  Do you think it is good\nfor humanity at large for one of the greatest blessings discovered by\nthem, for the prevention and cure of a terrible ill, to be solely under\nthe control of one petty, narrow-minded government, who dole it out to\nthe world just as they please, and at what price they like?  Why, such a\nblessing as quinine ought to be easily accessible all the world round,\nand if I can succeed in getting our precious little store safely to\nEngland, it will be the beginning of a very great work.  Worth the\ntrouble?  Why, the tenth part of what I have obtained of full ripe seed,\nof what is undoubtedly the finest white-flowered kind, would be worth a\nhundred times the labour and risk we have gone through--worth even\ngiving up life, my lad, so that others might benefit by what I have\ndone.\"\n\n\"But suppose, when we get it to England, it won't grow,\" said Perry.\n\n\"Why, you doleful young croaker!\" cried the colonel merrily, \"I don't\nexpect it to grow in England.  Tropic plants do not flourish in our\nlittle, cool, damp isle.  There are plenty of places, though, where it\nwould grow, if we get it safely home.\"\n\n\"Getting it wet isn't good for it, is it?\" said Perry sleepily.\n\n\"You are thinking of what you have in your pockets,\" said the colonel.\n\"That will not have hurt, for it would dry again pretty soon.--You have\nyours safe, Cyril?\"\n\n\"Yes, sir, there's about three pounds in my pockets.\"\n\n\"I have as much, and John Manning a little more, while I have a small\npacket in each of the mules' loads.\"\n\n\"So as to make sure of saving some of it?\" said Cyril eagerly.\n\n\"Yes, that is the idea, my lad,\" said the colonel.  \"Now, boys, Manning\nand I will take it in turns to watch.  There, get a good rest, and don't\nthink that I should have gone through all this labour, risk, and\nexcitement unless I had felt that I was doing something well worthy of\nthe trouble; so make up your minds to get it safely to San Geronimo.\"\n\nHe left them, as usual, to see where the mules were grazing, and Cyril\nsat gazing down before him.\n\n\"What's the matter?\" said Perry.\n\n\"I was thinking that it's all very well for you people to get back home,\nonly it isn't so pleasant for me.\"\n\n\"Father will speak to Captain Norton for you,\" said Perry.\n\n\"No: I don't want him to.  I shall speak myself.  I wouldn't have my\nfather see me sneak in behind yours in that cowardly way.  Oh dear, I\nwish it was over!\"\n\n\"Mules feeding well and all quiet, boys,\" said the colonel; \"and to all\nappearance there isn't a soul near us for miles.--By the way, Manning,\ndid you go into the cave?\"\n\n\"No, sir.  Did you tell me?  Seemed too damp to use for sleeping.\"\n\n\"No, I did not tell you; but get the lantern and let's look inside.  We\ndon't want to be disturbed by some animal coming out in the night.\"\n\nManning took the battered lantern, and led the way to where the spring\ncame gushing out of what at a distance looked like a long, narrow,\nsloping crack, but which proved to be, on closer acquaintance, large\nenough for a man to walk in upright by stepping from stone to stone,\nround about which the water came gurgling and bubbling out.\n\nIt was about a dozen yards from where their fire had been lit, amongst\nthe stones fallen at different times from the heights above; and as they\napproached, a low musical rippling greeted their ear in a pleasant\nmurmur, suggesting that the spring must come for some distance through a\nlow, natural passage, whose stony walls caused the echoings of silvery\nsplashings, which now grew louder and more strange.\n\n\"Yes, too damp-looking for a resting-place,\" said the colonel; \"and it\ndoes not look like the lair of any dangerous beast, but we may as well\nexamine it, and we ought to have done so before.  Why, boys, it would\nmake quite a fortress if we had to defend ourselves.  Plenty of\nwater-supply, and ample room to drive in the mules.\"\n\nJohn Manning had gone inside at once, and as soon as he was a short\ndistance from the narrow entrance, he struck a light and applied it to\nthe candle within the lantern, holding it above his head, and then\ncautiously picking his steps along from stone to stone in the bed of the\nstream.\n\nWhish, whirr, came a peculiar sound, and, as if moved by one impulse,\nthe two boys rushed out, startled, to stand looking back, wondering why\nthe colonel had not followed.\n\n\"What was that?\" cried Cyril.\n\n\"I don't know.  Something rushed by my head,\" said Perry excitedly, as\nhe looked vainly round in the dim light, and then back at the faintly\nlit-up entrance to the cave, where the lantern, now invisible behind a\ncurve, shone upon the moist stone wall.\n\n\"Come along back,\" cried Cyril; \"what cowards they will think us.  It\nmust have been birds.  Ah! yes; look, dozens of them,\" he cried,\npointing to where what seemed to be faint shadows kept gliding out and\nshooting upward over the face of the rock, to disappear at once in the\nevening gloom.\n\n\"Think they are birds?\" said Perry, in an awe-stricken voice.\n\n\"Birds or bats,\" said Cyril.  \"How stupid to be startled like that!\nCome along.\"\n\nHe sturdily led the way back, ashamed of the sudden access of fear which\nhad come upon him; though entering so strangely weird-looking a place by\nthe feeble light of a lantern, and when unnerved by long toil and the\ndangers they had lately passed through, it was not surprising, and\nstronger folk might easily have been scared.\n\nHe had hardly got well inside again before his face was brushed by a\nsoft wing, and he felt ready to run back once more, but this time he\nmastered the dread, and felt that Perry's hand was laid upon his arm\njust as the colonel's voice, which sounded hollow, echoing, and strange,\nsaid softly: \"Goes in, perhaps, for miles.--Look, boys.\"\n\nThe voice sounded close to his ear; but to his surprise he found that\nthe lantern was quite a hundred yards in, and the light glimmering from\nthe surface of the tiny stream, while there was plenty of room on either\nside for them to walk.\n\n\"Where are you, boys?\" said the colonel, more loudly.\n\n\"Here, sir; coming,\" cried Cyril, who grasped the fact, now, that their\nsudden rush out had not been noticed.\n\n\"It's all safe so far; no crevices or chasms,\" said the colonel; and as\nthe two lads approached, \"Did you see the birds?  They are flying about\noverhead in flocks.  Hark at the rush of their wings!\"\n\nAs he ceased speaking, and his voice was no longer reverberating and\nwhispering about overhead, a peculiar fluttering, whirring sound, as of\nmany wheels in rapid motion, struck upon the boys' ears, a sound which\nadded strangely to the mysterious air of the place.  It was evident,\ntoo, that the roof was now far above their heads, giving room for the\nstrange dwellers in darkness to wheel and swoop about, often so close\nthat the wind raised by their pinions beat upon the explorers' cheeks.\n\n\"Lucky I'd got the lantern door shut,\" said John Manning, in a strange\nwhisper, \"or they'd have blown it out a dozen times over.--Shall I go\nany farther, sir?\"\n\n\"No; it is of no use.  But what a hiding-place!  There's room, Manning,\nfor quite a brigade.--What's that?\"\n\nA sharp crash fell upon their ears, as of a stone dislodged somewhere\nhigh up in the distance; and this was evidently the case, for they heard\nit rattle down, loosening others, and sending a reverberating echo along\nthe cavern, which told of its vastness being greater than they had\nbefore imagined.\n\n\"One of the birds loosened a stone, sir,\" said John Manning.  \"Look out:\nhere they come.\"\n\nFor, evidently alarmed by the falling stones, there was now the rush as\nof a mighty wind, and the little party could feel that a great flock of\nbirds was passing overhead toward the entrance, hurriedly making their\nescape out into the open air.\n\n\"Let's follow their example,\" said the colonel; \"we are only wasting\ntime.  But this would make a capital retreat if we were attacked; and we\ncould defend it against hundreds.\"\n\n\"Till we were starved, or burned out,\" grumbled John Manning.\n\n\"It would take a forest on fire to burn us out of this, sir,\" said the\ncolonel.  \"What! make difficulties?  We have plenty to encounter\nwithout.  Now then, forward with the light.\"\n\nJohn Manning faced round, and led on at once, while, as he held up the\nlantern, the dark mass of birds in a regular train could be seen passing\non toward the entrance, which was reached directly after, both boys\nuttering a sigh of relief on finding themselves once more in the outer\ndarkness, where they could breathe freely, and feel as if a great danger\nhad been escaped.\n\n\n\nCHAPTER TWENTY FOUR.\n\nTHE CAVE'S MOUTH.\n\nAs soon as they were outside, Cyril looked round for the birds,\nexpecting to see them swooping about in all directions, but there was\nnothing visible between him and the stars; and with the peculiar nervous\nfeeling which he had felt in the cavern assailing him again, he turned\nto the colonel, who laughed.\n\n\"Well,\" he said, \"did you think it was something of what the Scotch call\n`no canny,' my lad?\"\n\nCyril felt more uncomfortable still.\n\n\"Do you think they really were birds?\" he said.\n\n\"Of course; the South American cave-bird.  A regular nocturnal\ncreature.\"\n\n\"What! a sort of owl, sir?\"\n\n\"No.  Perry here has seen their relatives at home.\"\n\n\"I?  No, father,\" said the boy wonderingly.\n\n\"Nonsense.  What about the nightjars you have seen hawking round the oak\ntrees in Surrey, after sunset?\"\n\n\"Oh yes, I remember them,\" cried Perry.\n\n\"Well, these are, I fancy, birds of a similar kind, but instead of\nfrequenting trees, they live in flocks in these dark caverns, and go out\nof a night to feed.  Our light startled them just as they were about to\ntake flight.  This must be one of their great breeding-places.--But no\nmore chatter.  Sleep, and get a good night's rest.\"\n\nEasier said than done.  The boys lay down in company with John Manning,\nbut it was long enough before either Cyril or Perry could drop off!\nThey would close their eyes, but only by an effort, for they were always\nready to start open again at some sound high up on one or the other side\nof the narrow winding valley.  It was cold too, in spite of the\nblankets, and when Cyril did at last slumber, he felt that he could\nhardly have been asleep an hour, as he started up into wakefulness\nagain.\n\nSomething was wrong he was sure, and he stretched out his hand to touch\nJohn Manning, who awoke instantly and sat up.\n\n\"All right,\" he said, in a low voice.\n\n\"No, no, don't move,\" whispered Cyril, grasping his arm.  \"I fancied I\nheard something.\"\n\n\"Eh?  Fancied?  Perhaps it was fancy, sir.  I'll ask the colonel.\"\n\n\"Listen first.\"\n\nThey knelt there in the darkness, attent for some minutes.\n\n\"Don't hear anything, sir.  I'll go and speak to the colonel.  What did\nyou fancy?\"\n\n\"I--I don't know,\" faltered Cyril.  \"It must have been while I was\nasleep.  Yes,\" he whispered excitedly, \"that was it.\"\n\n\"The mules!\" said the old soldier.  \"What are they doing here in camp?\"\n\nFor there came plainly now the soft pattering of hoofs on the stony\nground, and directly after a tall figure loomed up out of the darkness.\n\n\"Want me, sir?\" said John Manning, in a quick whisper.\n\n\"As you are awake, yes.  There is something stirring close at hand,\nwhether wild beast or Indian I can't say.  Keep watch, and cover us\nwhile I get the mules into that cave.\"\n\nJohn Manning's double gun was already in his hand, and he stood fast\nwhile the colonel went by with the leading mule, the others following.\nThen directly after the soft pattering ceased, and the watchers knew\nthat the patient animals had been led right into the cave.\n\n\"Hear anything, Master Cyril?\" whispered John Manning.\n\n\"No.\"\n\n\"And one can't see down in this dark gash,\" grumbled the man.  \"We\nhumans are worse off than any of the animals.  We can't see so well, nor\nhear so well, nor smell so well, nor run, nor fly.  Lucky for us, we've\ngot gumption enough to make telescopes and steam-engines and ships, or I\ndon't know what we should do.\"\n\n\"Who's that?\" said the colonel, returning.  \"Cyril?\"\n\n\"Yes, sir.\"\n\n\"Go and stand at the mouth of the cave, and mind that the mules don't\ncome out.\"\n\nCyril obeyed, and took up his position on a stone in the gurgling\nstream, to stand listening to the soft patter of the mules within, and\nto the faint whispers which came time after time from where he had left\nthe colonel and John Manning.\n\nHe had been at his position for some few minutes, turning from time to\ntime in the darkness to cast a furtive look back into the entrance of\nthe cavern, hardly able to restrain a shudder, as he thought of its\nunknown depths and the strange sound they had heard of the stone\nfalling, and he could not help wishing that Perry was with him for\ncompany's sake.\n\nFor there was a terrible feeling of lonesomeness there in the darkness,\nespecially at a time like that, when he had just been roused from an\nuneasy sleep by something unexplained at which the colonel had taken\nalarm.\n\n\"He said either Indian or wild beast,\" mused the boy.  \"What wild beast\ncould there be?\"  There were, he knew, the wild varieties of the llamas,\nguanacoes, and the like, but they were timid, sheep-like creatures; and\nthere were, he knew, pumas, the South American lions, as they were\ncalled, and perhaps jaguars--both these latter cat-like, nocturnal\ncreatures; but they were animals of the forests, and not of these\nsterile, rocky valleys.  Still, there might be other dangerous beasts in\nplenty, and his eyes wandered here and there, and he held his gun ready,\nthough in that deep gloom he felt that he would be quite at the mercy of\nanything which attacked.\n\nHe had just reached this point, when his thoughts took a fresh\ndirection--suppose some savage creature should be in the cave, and\nsuddenly spring upon him from behind.\n\nHe turned cold with horror, and tried to call for help, but his mouth\nand throat were dry from the nervous trepidation he suffered; for he had\nsuddenly been touched just below the shoulders, something big having\ngiven him a rude thrust.  This was followed by another, which nearly\nsent him down into the water from the stone.\n\nBut he recovered himself, turned sharply, and struck out with his right\nhand--a quick angry blow, while he felt as angry with himself for his\nabsurd cowardice, the second thrust having awakened him to the fact that\nhe had received a heavy push from the head of one of the mules, which\nhad come silently close up, and was desirous of getting out again into\nthe open air.\n\nCyril's blow drove the animal hastily back, and as he stood listening,\nhe heard the effect of his sharp action, for there was a good deal of\npattering about when the mule turned sharply to its companions, driving\nthem farther in.  Then there was silence once more.\n\n\"How easy it is to let one's self be frightened,\" thought Cyril.  \"I\nwish I were braver, and more like a man.\"\n\nThen he wondered why the colonel and John Manning did not come to him,\nand whether they were searching about for the cause of alarm.  All was\nvery still now, and it was some time since he had heard a whisper.\n\n\"Very likely I shall hear a shot fired,\" he thought, and making up his\nmind not to be startled if he did, for that it would be a good sign and\na proof that the cause of their night alarm had either been killed or\nfrightened away, he stood gazing out into the darkness in all\ndirections, and then smiled and complimented himself on his firmness.\n\n\"Not going to be scared at that,\" he muttered, for there had been a\nsudden clattering of hoofs among the stones inside the cavern--just such\na sound as would be made if one of the mules had kicked out at its\ncompanions, and made them start.\n\nAll was silent again for a minute, and then there was a faint splash.\n\n\"One of them gone down to drink,\" said Cyril to himself, and he turned\nnow and looked inward along the narrow opening, and could see faintly\none of the stars reflected in the black water, now twinkling, now\nburning brightly.  Then it disappeared, as if a cloud had passed across\nthe heavens, though that could not be, for another star gleamed closer\nto him, but that was blotted out too.\n\n\"One of the mules coming out,\" he said, starting and raising his hand,\nwhen there was a sudden bound made by something which had been crawling\nslowly out of the cave's mouth; and as the boy struck at it wildly, his\nfist touched something warm and soft, and the object, whatever it was,\nmade a stone or two rattle where it alighted, and then was gone.\n\nCyril raised his gun, but he did not draw trigger, for it was folly to\nfire quite at random, and he was leaning forward, peering into the\ndarkness, when a faint click made him turn again toward the mouth of the\ncave, just in time to be driven backward and lose his feet as another of\nthe creatures leaped out and dashed away into the darkness.\n\nTwo, and they were not mules, though evidently four-footed creatures.\nBut what could they be? he asked himself, as he recovered his feet and\nstood with presented piece, his heart throbbing, and his finger on the\ntrigger, ready to fire at the next movement from the cave.  They could\nnot be pumas, for the touch he had of the first one's body was not\nfurry; neither could they be large monkeys, for they would not have\nsmooth bodies, and besides, these creatures were too large.\n\nHe was still in doubt, when there was a sound behind him, and as he\nturned sharply, a husky whisper:\n\n\"Don't fire, my lad.  What was that?\"\n\n\"Did you hear it, Manning?\"\n\n\"Yes, and had a glint of some one running by me.\"\n\n\"Some one?\"\n\n\"Yes.  Indian, I think; did you see him?\"\n\nCyril told him of what he had seen, and was just finishing, when there\nwas a faint whisper and a movement of a stone or two as some one hurried\nup.\n\n\"Manning--Cyril--\"\n\n\"Yes, sir.\"--\"Yes, sir.\"\n\n\"Look out.  Some one passed me just now.  Who's this--Perry?\"\n\n\"Yes, father,\" came in excited tones from out of the darkness.  \"Was it\nyou who fell over me?\"\n\n\"No: when? where?\"\n\n\"Just now.  Then it must have been Cyril.  He went down heavily, but\njumped up and ran away.\"\n\n\"Indians, sir,\" said John Manning, in a low angry growl.  \"They passed\nthe line of sentries, and must have been trying to spot the camp.\"\n\n\"Absurd.\"\n\n\"Fact, sir.  Ask Mr Cyril here.\"\n\n\"Yes, sir; two Indians--I thought they were wild beasts--came crawling\nout of the cave and jumped by me.\"\n\n\"You saw them?\"\n\n\"Oh no, sir: it was so dark; but I hit at one of them and felt him.\"\n\n\"Came out of the cavern?\"\n\n\"Yes, sir.\"\n\n\"But are you sure?\"\n\n\"Quite, sir; I heard them frighten the mules, which began trampling, and\nthen one of the men sprang out.\"\n\n\"Shall I light the lantern, sir, and go in and see if there's any more?\"\ngrowled John Manning.\n\n\"No, my man; in all probability there were only the two, and perhaps\nthey were not enemies to be feared.  Possibly we have scared them more\nthan they scared us.\"\n\nCyril mentally demurred to that, but said nothing, and the colonel went\non:\n\n\"I'm afraid our night's rest has gone,\" he said, \"for it is impossible\nto lie down with the knowledge that Indians who may be enemies are\nabout.--Did you see anything as you made your round, John Manning?\"\n\n\"No, sir; but I heard something twice.  It may have been only an animal,\nbut something moved a few little stones up to the left.  When I went\ncautiously up, whatever it was had gone.  Did you see or hear anything,\nsir?\"\n\n\"I thought I heard a whisper a short distance away, but I could not be\nsure.  I am sure, though, that some one glided by me, and Perry here had\nthe best of evidence that one of the Indians fell over him.\"\n\n\"Unless it was Cyril; he did lie down to sleep by me, father.\"\n\n\"I've been on guard here by the cave's mouth for ever so long,\" said\nCyril sharply, as if resenting the fact that his companion should have\nbeen sleeping while he watched.\n\n\"Then it was an Indian,\" said the boy sharply.\n\n\"We have the mules safe, Manning,\" said the colonel, \"and now we must\nmake sure of the baggage.  Stand together, boys, facing two ways, while\nManning and I get the packs into the cave.\"\n\n\"But there may be more Indians in there, sir,\" said Cyril.\n\n\"If there are, we must drive them out.  That must be our fort for the\npresent.\"\n\nAt that moment there was a faint whistle from a distance, and it was\nanswered from somewhere high up on the mountain-side.\n\n\n\nCHAPTER TWENTY FIVE.\n\nBETWEEN TWO STOOLS.\n\nNo time was lost.  The two boys were posted at the cavern entrance, one\nto try to check any advance from the valley, the other to guard against\nthe escape of the mules, and stay by his presence any Indian who might\nstill be in hiding.\n\nThis latter was Cyril's duty, and this time he set his teeth and stepped\nright within the opening, encouraged by the fact that he had the colonel\nand John Manning constantly coming and going with the mule-packs, till\nonly two were left to bring in.\n\n\"I can hear people coming nearer,\" whispered Perry just then, and the\ncolonel threw his load down at his son's feet.\n\n\"You lads carry that in,\" he said.--\"Manning, quick, we must get those\nother packs.  They're coming on.\"\n\nJohn Manning, who was walking back from the cave, hastened his pace, and\nthe two men hurried through the darkness to where the last two packages\nlay.\n\n\"You keep watch,\" said Cyril.  \"I'm the stronger, and will get the pack\ninside.\"\n\nPerry said nothing, but felt glad and yet sorry, for he dreaded to\nre-enter the cave alone, and at the same time was ashamed to relegate\nthe task to his companion.\n\nBut there was no time for hesitation.  Something serious was evidently\ngoing on by the spot where the packages had been stacked, for there were\nshouts and cries, and Cyril stooped to lift the pack, meaning to hurry\ninto the cave and then return to pick up the gun he left with Perry, and\nstand ready to support their companions, in case he could do any good.\n\nThe package was heavy, but he hoisted it on to one shoulder, and was\nabout to bear it into the cave, when he was driven backward, and fell\nheavily, to be trampled under foot by a couple of men who charged out,\nplainly showing that there were others inside.\n\nIt was almost momentary work.  The men were there and then gone in the\ndarkness, and, sore and angry, Cyril struggled to his feet.\n\n\"Why didn't you fire?\" he cried.\n\n\"What at?  I might have hit you, or perhaps my father,\" protested Perry.\n\n\"Trampling on a fellow like that,\" grumbled Cyril, rehoisting his load.\n\n\"Yes; they had each got a pack.\"\n\n\"What! our packs?\" cried Cyril excitedly.\n\n\"Yes; I could just make that out,\" said Perry.\n\n\"Oh!\" ejaculated Cyril, stepping close in, and throwing down his load so\nas to regain his gun, \"what will the colonel say?\"\n\nNot what the boy expected, for just then he came panting up with John\nManning, carrying a pack between them; while the rattling of the loose\nstones told that they were being pursued.\n\n\"Quick, both of you,\" cried the colonel, \"fire in the direction of the\nnoise.\"\n\nCyril's gun spoke out with both barrels rapidly, one after the other,\nthe flashes cutting through the darkness, and the reports being followed\na few seconds later by quite a volley of echoes, which ran reverberating\nalong the gorge, to die away slowly in the distance; but before they had\nceased, the little party was well inside the very doubtful shelter they\nhad chosen, and John Manning posted at the entrance with his loaded\npiece.\n\n\"Why didn't you fire?\" whispered Cyril.\n\n\"I did.\"\n\n\"That you didn't.  I did twice.\"\n\n\"I mean,\" said Perry, \"I pulled the trigger, but the thing wouldn't go\noff.--Oh!\"\n\n\"What's the matter?\" said Cyril eagerly, as he reloaded his piece.\n\n\"Don't say anything,\" whispered Perry.  \"I forgot to cock it.\"\n\n\"A narrow escape, Manning,\" said the colonel just then.\n\n\"Tidy, sir,\" replied the old soldier; \"but I don't like losing that\npack.  Shall I make a charge and fetch it in?\"\n\n\"Madness, man,\" said the colonel.  \"Let it go.  We've got all the others\nsafe.\"\n\n\"No, father,\" cried Perry excitedly; \"two Indians rushed out of this\nplace while you were gone, and each man had one of the packs.\"\n\n\"What!\" cried the colonel in a despairing tone; \"three of my precious\npackages of seed--gone?\"\n\nNo one spoke; but from out of the darkness came the peculiar sound of\none grinding his teeth, and a pang of misery and disappointment shot\nthrough Cyril as the colonel said bitterly:\n\n\"Two of you with guns, and you could not check those brutes.\"\n\n\"No, sir,\" growled John Manning; \"how could they without bay'nets?\n'tain't to be done.\"\n\n\"It was all so sudden, father,\" put in Perry, his words saving John\nManning from a stern reproof.  \"Cyril was knocked down, and there was\nnot time to fire.\"\n\n\"And if there had been,\" whispered Cyril maliciously, \"your gun would\nnot have gone off.\"\n\n\"Beg pardon, sir,\" said John Manning, \"I daresay we can get back the\nseed in the morning: they'll keep the good things, and throw what they\nthink is rubbish away.\"\n\n\"No,\" said the colonel, speaking sternly, \"the three bags in those\npackages are gone.  It is the main object of these men to keep the seed\nfrom being taken out of the country.  Where is the lantern, Manning?\"\n\n\"Somewhere along with the packages, sir.  I think we brought it in with\nthe second lot.\"\n\n\"You keep guard, while we search the cave.  A sharp lookout, mind.--\nPerry, come with me.--You stay with Manning, Cyril, till I return.\"\n\n\"Sharp lookout, sir!\" growled the old soldier.  \"Who's to keep a sharp\nlookout in the dark, and how's a man to guard the inside and outside\ntogether?--Say, Master Cyril, we're in a pretty tidy hole here, and\nit'll take all we know to get out of it again.\"\n\n\"Oh, we shall manage,\" said Cyril sharply; \"but three packs gone.\nThat's terrible!\"\n\n\"'Tis, sir, and they'd all got in what's of more consequence to us now\nthan seed--a whole bag each of rice and meal, without counting\ndelicacies in the shape o' pepper and mustard.\"\n\nJust then there was the crackling of a match, followed by a faint glow,\nand the lantern shed its light around, gleaming from the running water,\nand showing dimly the mules standing in a group with their heads\ntogether.  Then as Cyril stood waiting and watchful, he saw the lantern\ngo on and on as if the colonel were zigzagging about to and fro, now\napproaching the little stream, now going right away.  Sometimes the\nlight passed beyond intervening rocks, and disappeared for a minute,\nthen came into sight again; but there was no sign of other occupant in\nthe great cave, whose extent was evidently vast.\n\n\"Don't see no more o' they bat birds buzzing about,\" said Manning\nsuddenly.  \"I hope they'll come back.\"\n\n\"Why?\" said Cyril.\n\n\"Foodling,\" growled the old soldier.  \"We may have to stand a siege, and\nit ain't bad to know you've got plenty of meat and water on the spot.\"\n\n\"What's that noise?\" whispered Cyril.\n\n\"Some on 'em crawling about on the stones outside yonder.  I heered 'em,\nand if they don't keep off--I don't want to shoot no one, had enough of\nit when I was out in Indy, sir; but duty's duty, and if they won't leave\nus alone, they must be taught how.  See anything o' the lantern now?\"\n\n\"No; it has gone out of sight some time.\"\n\n\"Humph!  I hope they won't go too far and lose theirselves, sir, because\nthey can't be spared.  I knowed of a man losing himself in a stone\nquarry once under ground, but they found him afterwards.\"\n\n\"Half-starved?\" said Cyril eagerly.\n\n\"Quite, sir.  It was a year after he went down.  I don't like work under\nground.  It's only fit for rats or worms.  See the light now?\"\n\n\"No: what's that?\"\n\n\"Something moving inside, sir.\"\n\n\"The mules?\"\n\n\"No, sir; their hoofs are not so soft as that.  Sounds to me as if some\nof 'em was going to make a rush, and we haven't a bay'net to bless\nourselves with.  You fire, sir, at once before they come on.\"\n\nCyril did not hesitate, but without shouldering his piece, he drew\ntrigger with the result that they heard, mingled with the reverberations\nof the report, a faint pattering noise as of retreating feet.\n\n\"Well done, sir.  Reload quickly.  They were going to rush us, and\nthat's taught 'em we were on the kwy wyve as the Frenchies call it.\"\n\n\"Keep a sharp lookout your way,\" said Cyril as he hurriedly reloaded,\nhis fingers trembling from his excitement.\n\n\"That's what I'm doing, sir, with my ears.  I've been on sentry before\nwith different kind of Indians on the lookout to bring you down with\nbullets.  I shall hear 'em, I dessay.\"\n\n\"But look here, John Manning, we've stopped those men from coming, and\ndriven them back on the colonel.\"\n\n\"Yes, sir, and all the worse for them, for he's sure to hear them and be\non the lookout.  Strikes me that the cave swarms with Indians, and that\nour first job ought to be to clear the place.  But look out, and don't\nbe in too great a hurry to shoot now, sir, because your shot 'll bring\nour friends back to us.  Perhaps it came in quite right, for they may\nhave lost their way.\"\n\nThen some minutes passed, and a noise was heard which made Cyril lower\nhis gun again, but a voice warned him that he must not fire.\n\n\"Where are you?\" cried the colonel.\n\n\"Here, sir.\"\n\n\"Thank goodness.  We had an accident, fell over a stone, and put out the\nlight.  This place is tremendous, and we should have hardly found our\nway out of it, had it not been for your shot.  Did you mean it as a\nrecall?\"\n\nCyril explained, and the colonel came to the conclusion that it was\nuseless to explore farther, for there was room for a hundred of the\nenemy to hide and elude them, so vast was the number of huge blocks\nlying about, masses which had fallen from the roof during some\nconvulsion of nature.\n\n\"We must wait for daylight,\" he said at last.  \"It is impossible to make\nany plans till then.\"\n\nBut all the same the colonel arranged his little force so that it might\ntell to the best advantage; he and Perry securing themselves behind a\nblock of stone to guard from an attack within, while Cyril had to join\nJohn Manning in guarding the entrance from an attack from outside, where\nthey had the satisfaction soon after of seeing one of the mountain peaks\nappear, pale and ghastly looking, over the other side of the gorge,\nwhile all below was intensely black.\n\nOnce they heard a peculiar cry which might have come from Indians or\nsome wild creature, quadruped or night-bird; but otherwise all was still\nin the gorge, as they strained their eyes in their endeavours to pierce\nthe darkness in search of danger.\n\nAt last weariness began to tell upon Cyril, and his head nodded gently,\nthen went down so suddenly that he started up, angry, and in dread lest\nManning should have been aware of his lapse.  For it was horrible at a\ntime like that, when perhaps the lives of all depended upon his\nwatchfulness.\n\n\"It was too dark, and he did not notice it,\" thought Cyril, with a glow\nof satisfaction pervading his breast.\n\n\"Yes, it's hard work, as I well know, sir,\" said Manning quietly.  \"When\nI was a soldier first, I used to think it killing work to keep on sentry\nwhen one would have given anything to have a good sleep.\"\n\n\"You noticed it, then,\" said Cyril.\n\n\"Noticed it, my lad? why, of course.  Seeing how dark it is, you might\nhave had a doze and me not known anything but there you were, very\nquiet; but when you says to me, as plain as a young man can speak, `I'm\ntired out, and my eyes won't keep open any longer,' why, of course, I\nknow you're off.\"\n\n\"But did I say that?\"\n\n\"Not exactly, sir, but you said `gug,' and I heered your teeth chop\ntogether when your chin went down upon your chest.\"\n\n\"Oh!\" ejaculated Cyril bitterly, \"and I did try so hard.\"\n\n\"Course you did, sir, but human nature's the nat'ralist thing there is,\nand it will have its own way.  I'd say have a snooze, but orders were\nthat you was to watch, and watch you must.\"\n\n\"Yes,\" said Cyril firmly, \"and I will keep awake now.\"\n\nHe kept his word for fully ten minutes, and then his efforts were vain.\nIf the peril had been ten times greater, he would have dropped off all\nthe same; but he had not slept a minute before there was the sharp\nreport of a gun which came bellowing out of the cave's mouth, and the\nboy started up once more as if it were he who had been shot; while from\nclose at hand there was a rush of feet, and John Manning fired at once\ninto the darkness, with the result that there was another rush from\nCyril's right.\n\n\n\nCHAPTER TWENTY SIX.\n\nIN THE GORGE.\n\n\"Well,\" cried the colonel, as the echoing died away, \"are they coming\non?\"\n\n\"They were, sir, without us knowing it,\" said Cyril.  \"Your shot\nfrightened them, and then Manning fired and startled some more.\"\n\n\"The mules warned me,\" said the colonel, \"as they did you, but a shot\nsent the rascals back.--Hah! the light coming at last.\"\n\nFor the mountain peaks were beginning to glow, and the clouds which hung\nround the highest showed tints that were quite crimson, while the light\nwas now slowly stealing down into the deep gorge, bringing with it\nrelief from the terrible anxiety of the night.\n\nThen, as it grew brighter, it became evident that the Indians had drawn\noff for a time, not a sign being visible of their presence anywhere in\nthe deep valley, while inside the cavern all was so still that it was\nalmost impossible to believe that any danger could be lurking there.\n\nBut the danger was ever present, and it was not until John Manning had\nbeen posted well inside the cavern, ready to fire in the event of any\nfresh advance, that preparations were made for a very necessary meal;\nafter which it became requisite to hold a council of war, when it was\ndecided that to stay where they were would be madness, and that nothing\nremained for them but to show a bold front and push on at once.\n\nPerry looked so dubious that his father smiled.\n\n\"Don't you see, my boy, how dependent we are upon the mules?  Well, the\nmules must be turned out to graze, and we shall be as safe journeying on\nas posting ourselves to guard them.  Besides, if we stop here, the\nIndians will conclude that we are afraid to go on, and this will give\nthem courage; whereas, if we advance boldly, they will give us the\ncredit of being braver than we are.\"\n\n\"And if we shut ourselves up in that cave, how long will the provisions\nhold, sir, if I may make so bold?\" said John Manning.\n\n\"Quite right,\" said the colonel nodding.  \"Don't you know, Perry, that a\nwise man once said that an army does not gallop along, but crawls upon\nits stomach?\"\n\n\"Crawls?\" said Perry.\n\n\"You don't understand, boy.  He meant that an army can only move as fast\nas provisions can be supplied to it.  That is our case.  If we take the\nrisk of shutting ourselves up here--a dreary, depressing plan, by the\nway--we can only hold out till our provisions come to an end.  Better\nfar make a bold dash onward toward the other side of the mountains.\nEvery step we take will be toward civilisation and safety, while every\nstep the Indians take in pursuit will be toward land where they know\nthat they will be at a disadvantage.  There, I do not see why I should\nexplain all this to you, but I want you to have confidence in me.  And\nyou too, sir.\"\n\n\"Oh, I have, father,\" cried Perry.\n\n\"And I'm sure I have, sir,\" said Cyril warmly, \"only I can't help\nfeeling that we shall be safer in the cavern than out there in the\nnarrow valley, where these people can shoot us down when they like.\"\n\n\"Of course you feel that, my lad,\" said the colonel, \"but I am under the\nimpression that they will hesitate about shooting at us.  I fancy that\nthey will strain every nerve to master us and capture all our stores, in\nthe full belief that we are taking out of the country valuables that\ntheir traditions and the teachings of their rulers bind them to defend.\nIf they had liked, I feel convinced that some one of us, perhaps two,\nwould be wounded and helpless by now.--What do you say, John Manning?\"\n\n\"Well, sir, I'm 'bout divided in my opinions.  One time I think they\nmust be such bad shots, they're afraid to show it, and another I get\nthinking that they've got an idea of your being an indefatigable sort of\na gentleman.\"\n\n\"Well, I am, John Manning,\" replied the colonel smiling; \"and you know\nit too.\"\n\n\"Course I do, sir, but that isn't what I mean, sir: 'tain't\nindefatigable; it's a word that means something to do with armour, and\nthe more you shoot at any one, the more you won't hurt him.\"\n\n\"Invulnerable,\" said Cyril.\n\n\"That's it, sir,\" cried the old soldier, slapping his leg.  \"Lor', what\na fine thing it is to have been brought up a scholar.\"\n\n\"You are right, Manning.  I impressed them, I suppose, by my shooting,\nand they have evidently some idea of that kind in their ignorance.\nWe'll take advantage of it and start at once.--Very tired, boys?\"\n\nThey were silent.\n\n\"Shall we start now, or try to get some rest, and then start at night?\"\n\n\"Let's start now, sir,\" said Cyril firmly.  \"The darkness makes cowards\nof us--I mean, makes one of me, for I'm always fancying dangers all\naround.\"\n\n\"Are you ready, Perry?\"\n\n\"Yes, father.  Let's go at once.\"\n\n\"We will,\" said the colonel, \"for I'm afraid that we should make very\npoor progress after dark.  In with you then, and let's load up the\nmules; they must take a mouthful of grass wherever there is any as we\ngo.\"\n\nThe very thought of getting on chased away a great deal of the\nweariness, and the little party were soon hard at work in the\nsemi-darkness, just inside the cave, fastening on the packs.  Then all\nmounted the riding mules, and without a moment's hesitation rode out,\nthe colonel with the leading mule turning up the gorge, which ran pretty\nnearly due west and rising higher at every step, while John Manning and\nthe two lads formed the rearguard.\n\n\"Ten times better than being shut up there in the dark, my lads,\" said\nthe old soldier, sitting up erect in his saddle, with the butt of his\npiece resting upon his thigh.  \"It's like being in the cavalry.--See any\nof 'em, Master Cyril, sir?\"\n\n\"No, not a soul in sight.  Have they gone right away?\"\n\nJohn Manning chuckled.\n\n\"Just far enough to keep a sharp eye on us, sir.  They're hiding\nsomewhere behind the stones.\"\n\nBut for the space of an hour, as they rode on in the shade of the early\nmorning, there was no sign of an enemy either to front or rear, and\ninspirited by the crisp mountain air, the boys felt their spirits rise,\nand were ready to banter John Manning about what they looked upon as his\nmistake.\n\n\"Depend upon it,\" said Perry, \"they drew right back to go and camp for\nthe day, and rest, before coming to attack us again.\"\n\n\"They were soon rested then, sir,\" said the old soldier drily, \"for\nthere's two of 'em up yonder behind those pieces of rock.\"\n\n\"Where?  Nonsense.  Birds: condors, perhaps, on a shelf.\"\n\n\"Perhaps so, sir,\" said John Manning; \"but they're birds that can make\nsignals, and your father sees them too.\"  For just then the colonel drew\nhis mule aside, and let the rest pass on, while he waited for the\nrearguard to come up.\n\n\"Be on the alert,\" he said as they came up; \"the Indians are high above\nus on the mountain-side, and they are making signs to others right up\nthe gorge.  Close up.\"\n\nThen going nearer to his old servant, he whispered:\n\n\"Keep a sharp eye up to right and left, and if they open the ball, jump\ndown, and don't hesitate about taking good aim at the first who tries to\nstop us.\"\n\n\"Right, sir.  But how do you think they will open the ball, sir?\nArrows?\"\n\n\"No: as our old friends in the North-west did, John Manning.  Ah, look,\nthey have begun.\"\n\n\"Yes, sir; I expected it,\" cried John Manning, as the mules all stopped\nshort, their leader having suddenly swerved aside to avoid a little\navalanche of stones and masses of rocks which came tearing down from far\non high, right across their course.\n\nIt began by the dislodging of one great mass, which was forced over from\na rocky shelf, and before this had rolled half-way, it had started\nhundreds more, the attempt being so well contrived that the pieces of\nrock, which came leaping and bounding down with a clashing sound like\nthunder, would certainly have crushed one or more of the mules, but for\ntheir quick appreciation of the danger.\n\nTwo sharp replies to this attack were given from the loaded pieces, and\nthe Indians disappeared; but when, after a great deal of coaxing, the\nmules were got into motion once more, the colonel urging the leader\nround beyond where the stones had fallen, the boys uttered a warning\ncry, for another mass of rock was started from high up on the other\nside, and with such good effect that the rush of stones it started\ncaught the tiny caravan half-way, and to the misery and despair of all,\none of the best mules lay with its pack half buried, and the poor\ncreature's head crushed almost flat.\n\nThis time the boys fired up at a single man perched fully twelve hundred\nfeet above the narrow bottom of the gorge, and he dropped out of sight,\nwhile the colonel and John Manning leaped down again, and rapidly\nremoved the stones which impeded them in their efforts to loosen the\npack-ropes and remove the load to another mule.\n\nThis was a hindrance of a good half-hour, and the colonel looked very\nstern as he gave the order to advance again, when they gained about a\nmile, the gorge opening a little.  Then the huge towering walls\ncontracted once more, and a black-looking prospect opened before them,\nfor there, so narrow that there was barely room for the mules to go\nsingly, was their way, through a black-looking rift, above which the\nmountains on either side rose in shelves admirably adapted for the\nenemy's defence, and promising so ill for the little party, that the\ncolonel hesitated for a minute while he used his glass.  Then, as all\nwas still, no sign of an enemy visible, he gave the order to advance, in\nthe hope that the place was too precipitous for the Indians to occupy.\n\nVain hope!  Ten minutes later the mule he rode lay quivering on the\nground; the colonel having the narrowest of narrow escapes from a\nterrible death.\n\n\"Forward!\" he said firmly, but as he spoke, another avalanche and\nanother came thundering down, and seeing the madness of attempting to\nproceed, he gave the word to retreat.\n\nIt was needless, for the mules had already sprung round, and were\nhurrying back at a rate that was faster than anything they had yet\nshown.\n\nThere was no yell of triumph from on high, and no sign of enemy on\neither hand as they rode back, face to face with the fact that exit from\nthe gorge in that direction was impossible, and that unless they could\nfind some side ravine leading in the direction they wished to pursue,\nthere was nothing for it but to retrace their steps right to the\ncinchona cutters' camp, and from thence make their way home by the road\nthey came.\n\n\"It does seem so strange,\" said Perry, as they rode back; \"they don't\npursue us, but let us ride quietly on.  Are they satisfied with the\nmischief they have done?\"\n\nCyril made no answer, for he was watching the colonel, where he rode by\nhimself, grave and stern, impressed as he was by the feeling that all\nhis efforts would be brought to nought, if he could not devise some\nmeans of reaching San Geronimo.  At that moment it looked hopeless, and\nas if he would be thwarted at every step, for though the enemy were\ninvisible, he had had bitter experience of the fact that their knowledge\nof the mountain paths placed them at a tremendous advantage, and he felt\nthat in all probability they were even then where they could watch every\nmovement, and had perhaps got well ahead, ready to stop them by another\nattack upon the mules.\n\nIt was past mid-day now, and the hope of getting well onward toward\nsafety was completely crushed; the sun was now shining fiercely down\ninto the gorge, heating the rocks so, that the reflection was almost\nworse than the scorching rays from above.  The mules were panting and\nthirsty, and the exertions of the day, coming upon such a broken night,\nwere telling terribly upon all, so that reluctantly, and as their only\nresource, the colonel allowed the old leading mule to make straight for\nthe stream which came gurgling out of the cave they had left that\nmorning.  Here, to the surprise of all, the intelligent beast, instead\nof stooping to drink at once, stepped carefully among the stones right\ninto the cool shade within, the rest following, and five minutes later\nPerry and Cyril were seated inside upon one of the packs.\n\n\"And all that day's work thrown away,\" said Cyril sadly.\n\n\"And two mules dead,\" whispered Perry, so that his father should not\nhear.\n\n\"But we saved their loads,\" replied Cyril.\n\n\"What for?\" said Perry, in the same low voice.  \"It's of no use: we\nshall never get away alive.\"\n\n\n\nCHAPTER TWENTY SEVEN.\n\nREADY FOR THE WORST.\n\nNature must be satisfied at whatever cost, and, as soon as possible, the\nprovisions were unpacked, while a fire, which had been lit with the\nscraps of wood collected as they returned, burned brightly, the smoke\ndrawing farther into the cavern, and being rapidly dispersed.  Then, as\nsoon as the kettle, suspended by an iron rod over the flames, began to\nboil, a very small handful of tea was thrown in, and the tin lifted off\nand laid aside.\n\nOnly some very wooden cake bread, and some very dry tough beef, with a\nstrong flavour of being imperfectly sun-cured; but how delicious it was\nwhen washed down by the warm, unsweetened, milkless tea!\n\nThey sat on the stones a little way inside the entrance, where the cool\ndampness of the cave was relieved by the soft, warm, sunny air which\nfloated in from the gorge; and a sharp lookout was kept for the Indians,\nbut not a sign betrayed their presence.  A short exploration of the\ninterior, too, was made, but there was not a sound to be heard, save an\nimpatient stamp or two from the mules; and, concluding that the enemy\nhad all left the place, the colonel returned to where he had left the\nboys on guard, but only to find them both fast asleep, and John Manning\nwalking up and down slowly by the stream, with his gun over his\nshoulder.\n\n\"Don't wake 'em, sir,\" said the man appealingly: \"they tried very hard\nto hold up, but it warn't to be done; and if I hadn't got up and taken\nto marching here on sentry-go, I should have been fast asleep too.\"\n\n\"Poor fellows,\" said the colonel sadly, \"it was too much to expect after\nwhat they have gone through.  There, lie down for a couple of hours, and\nthen I'll wake you to relieve me.\"\n\n\"Begging your pardon, sir, if you'd take first turn,\" began the old\nsoldier, but he was cut short directly by a sharp gesture, and, without\na word, he lay down with his piece in his hand, and was asleep directly.\nThe colonel took his place, and then began to walk slowly to and fro,\nnow right out where the sinking sun glistened upon the surface of the\nstream, now back inward, with the walk growing darker, till it was quite\nblack.\n\nAnd as he marched to and fro, the colonel reflected upon the\nhopelessness of their position.  As he approached the open sunshine, he\nfelt lighter-hearted; but as he turned and walked inward toward the cold\nand darkness, his spirits sank once more, and he saw no way out of their\ndifficulties save the giving up of that for which he had come all those\nmany thousand miles.  Then he stopped short, for Cyril had suddenly\nsprung to his feet, looking horribly guilty and ashamed.\n\n\"I am so sorry, sir,\" he faltered.  \"I did not mean to go to sleep.\"\n\n\"You could not help it, my lad,\" said the colonel, laying his hand upon\nthe boy's shoulder.  \"You had reached the limit of human endurance.\nBut, Cyril, my lad, you are being sharply punished for your mad\nescapade.\"\n\n\"Yes, sir,\" said the boy sadly, \"and I suppose I deserve it.  I should\nmind all this much less if we had arranged with my father that I should\ncome.\"\n\n\"Well done, young philosopher,\" said the colonel, as he kept his hand on\nthe lad's shoulder, and marched him to and fro.  \"Come, as you can think\nso well, give me your advice.  You know why I have come--to obtain this\nseed to place in the hands of those who will cultivate it, and make the\nworld independent of the one source of supply.\"\n\n\"Yes, sir,\" said Cyril wonderingly.\n\n\"And you see the position to which I am reduced.\"\n\n\"Yes, sir.  Will the Indians kill us?\"\n\n\"Not without paying dearly for it,\" said the colonel sternly.  Then\nchanging his manner: \"No, my lad.  These people are only half-savage,\nand look upon what they are doing as a duty.  I do not think they will\nkill us if they can get possession of all our baggage.  They might keep\nus in captivity until means are taken to free us; but I don't know--I\nhardly feel that our lives are safe.\"\n\n\"Not very, sir,\" said Cyril grimly, \"if they rolled big stones.  It\nmight have been us instead of the mules this morning.\"\n\n\"Yes, we have had some narrow escapes; but what shall I do now--give up\nand own that I am beaten?\"\n\n\"And let them have the kina seed, sir?  No, that I wouldn't; I'd fight\nfor it first,\" cried Cyril excitedly.\n\n\"Do you know what you are talking about?\" said the colonel excitedly.\n\n\"Yes, sir,\" cried Cyril, \"I was very much down a while ago; but I've had\nsomething to eat now, and a sleep and--What! give up to a pack of\nsavages, and let them rob you of all we have worked so hard to get?\nThat I wouldn't while there was a charge of powder left.\"\n\n\"Humph!\" ejaculated the colonel, looking at the boy searchingly.\n\n\"That's what my father would say if he were here.  I wish he were.\"\n\n\"To thrash you for leaving home in that cowardly way?\"\n\n\"No, sir,\" said Cyril quietly.  \"My father never thrashed me, and he\nnever would.  He always said it was degrading a boy to beat him, and\nthat he was a poor parent who could not rule his children without blows.\nHe told me he thought he could hurt me a good deal more by his words,\nand so he always could.\"\n\n\"Perhaps so, sir,\" said the colonel sternly; \"but see what a mistake he\nhas made, and what a miserable young dog you have turned out.\"\n\nCyril was silent for a few moments.\n\n\"I hope I'm not all bad, sir,\" he said.  \"I'm sure I've bitterly\nrepented what I have done.\"\n\n\"And been severely punished, too, my lad,\" said the colonel kindly.\n\"Your father is quite right, and when I tell him how you have behaved--\nas, please God, I hope I shall--if he is the man I believe him to be, he\nwill shake hands with you as I do now, and say, `Let's forget the\npast!'\"\n\n\"Colonel Campion!\" cried Cyril, snatching at the hand extended to him.\n\nHe could say no more, for something seemed to rise in his throat and\nchoke him, while the colonel rested his arm affectionately upon his\nshoulder once more, and walked him up and down toward the light and back\nagain.\n\n\"So you'd fight for it and stand out, eh?\" said the colonel at last.\n\n\"Yes, sir, that I would,\" cried Cyril excitedly, \"and I'm sure Perry\nwill say the same.\"\n\nThe colonel was silent for a few moments, gazing straight before him\ninto the darkness, and he asked himself whether he would be justified in\nrunning all these risks.  He shook his head, and they turned and marched\ndown again toward the light, where the rippling stream looked rosy in\nthe evening, and the rocks on the other side of the gorge were all\naglow.  And there was so much in the brightness and hope and prospect of\nsuccess that the feelings of dread, the shrinking from a terrible\nordeal, and all hesitation fled away.\n\n\"Yes,\" he said firmly as he stopped short; \"a civilian might hesitate\nand give up, but I cannot, my lad.  I determined to carry out this task,\nbelieving that by it I should vastly benefit suffering humanity at\nlarge, and Heaven helping me, I will achieve my aims.\"\n\n\"And you will fight it out, sir?\" cried Cyril.\n\n\"I will, my boy, to the bitter end.  I made the Indians fear me, and if\nthey attack us now, they shall fear me more, for I have our lives to\nsave as well.  Now, go and lie down.\"\n\n\"No, sir, I don't feel sleepy; let me watch with you.\"\n\n\"You are a soldier for the time being, my lad, and if we are to succeed,\ndiscipline must be observed.  In an hour's time I rouse up John Manning\nto relieve me, and toward morning, when you are rested and refreshed,\nyou shall be called to take your watch.\"\n\nCyril lay down without a word, meaning to think out all that had been\nsaid, and feeling happier than he had since he left home.  But he did\nnot think, for in a few seconds after he had stretched his weary limbs\nupon the rocky floor, all the corners and points of his bed became soft\nas eider-down, and he dropped into a deep dreamless sleep, leaving the\ncolonel seated on a rock, leaning forward with his gun between his legs,\nand his form looking black against the soft glowing light at the\nentrance of the cave.\n\n\n\nCHAPTER TWENTY EIGHT.\n\nNATURE IS MISTRESS.\n\n\"I haven't slept long,\" thought Cyril as he woke with a start to see the\ncolonel seated as he had left him when sleep came.  He did not appear to\nhave moved, and all was perfectly still.  One thing, however, puzzled\nhim, and that was the light.  It was soft and warm and glowing then; now\nit was clear and bright.\n\nAll at once he saw something come into sight from the side and stand\nlooking in.  It was the old leading mule, and it stood there munching\naway at a tuft of green stuff which it held in its teeth, and then\ndropped, and stooped its head to take a long drink of water.\n\n\"Have the Indians gone?\" thought Cyril, \"and has the colonel let the\nmules out to feed?\"\n\nHe sat up for a moment or two, and then lay down again.\n\n\"I may as well sleep till I am called,\" he thought, and he lay listening\nto the heavy breathing of John Manning and Perry, whom he could see\nlying in precisely the same position as when he lay down a few minutes\nbefore.\n\nBut was it a few minutes before?  Cyril asked himself as he saw the mule\nlift its head with the water dripping from its lips, and then pick up\nthe tuft of green twigs, and go on munching again.  It could not have\nbeen longer, for the colonel had not been relieved yet; but the light\nwas so strange.  Last time he looked, the opposite side of the gorge was\nglowing with the sunset rays; now it was in shadow, and the sun was\nshining just inside the mouth of the cave, and making the water flash\nlike molten silver.\n\nThe mule stared in toward him, and Cyril made a bull, as a sudden\nthought flashed through his mind.\n\n\"Why, it's to-morrow morning!\" he mentally exclaimed as he started up,\n\"and I've been asleep all this time.\"\n\nHe went to where the colonel sat, and though the stones rattled a little\nbeneath his feet, the watcher did not stir, but sat as if gazing\nstraight out past the mule, when Cyril said reproachfully:\n\n\"You shouldn't have let me sleep so long, sir.\"\n\nBut there was no answer, and at that moment there was a dull sound, a\nwhinnying sigh, as the mule gave a violent start, throwing out its legs\nin all directions, and scattering the stones before gathering itself\ntogether, and making a tremendous leap to go galloping away, not so\nquickly, though, but that Cyril saw it had a long arrow sticking in its\nback.\n\n\"What's that?  Who?  You, Cyril?\" cried the colonel, springing to his\nfeet.  \"What!  Have I been to sleep?\"\n\n\"I don't know, sir; I only just woke up.\"\n\n\"Why, it's morning,\" cried the colonel angrily.  \"What was that noise?\"\n\n\"The old mule.  Some one shot an arrow, and stuck it in her back.\"\n\n\"But how came it there?  I--I--Ah! and I blamed you, boy, for falling\nasleep at your post.\"\n\n\"Anything the matter, sir?\" said John Manning, coming forward, followed\nhastily by Berry.\n\n\"Matter, man; yes.  I have kept watch by sleeping all night.\"\n\n\"Well, sir, I don't wonder,\" said the man.  \"I have too, for you didn't\ncall me to go on duty.\"\n\n\"Oh, shame!\" exclaimed the colonel.  \"And at a time like this!\"\n\n\"I don't see no shame in it, sir,\" growled John Manning bluntly.  \"You\nwanted sleep like the rest of us, and you couldn't fight against it.\nYou ain't an iron man, sir.\"\n\n\"Silence, sir!\" cried the colonel angrily, and he hurried right out of\nthe cave, closely followed by the boys, in time to see the mule\ndisappear low down the gorge, galloping madly among the stones.\n\n\"Another--our most valuable pack mule gone,\" cried the colonel, in a\nvoice full of the anger and annoyance he felt.  \"The poor brute must\nhave strayed out to graze.\"\n\n\"Yes, sir; I saw it eating when I awoke, and then directly after the\narrow struck it.\"\n\n\"Then the Indians must be close at hand.  Come back and help to secure\nthe rest of the mules, or they may be straying out after their leader.\"\n\nCyril looked in the colonel's face, and then pointed down the valley at\nsomething moving, plainly seen now in the clear morning atmosphere.\n\n\"Yes; I see it.  One moment till I focus the glass.\"\n\nThe colonel held his little double glass to his eye, and then let his\nhand fall to his side in dismay on learning that the mischief was worse\nthan he had anticipated, for there was another mule on the far <DW72>;\nand just then John Manning came up with Perry from an expedition within\nthe cave.\n\n\"Yes: what is it?\" cried the colonel sharply.\n\n\"Sorry to have to report it, sir,\" said Manning, in a low deep growl,\n\"but the mules have strayed out in the night.\"\n\n\"All?\" said the colonel hoarsely.  \"Is there not one left?\"\n\n\"No, sir; as far as I can make out, not one.\"\n\nThe colonel's brows knit, and he stood motionless for a few moments.\nThen without a word he walked right in to where it was quite dark.\n\n\"Light the lantern,\" he said sharply, and in a few moments a match\ngleamed out, and the candle was soon after shedding its rays in all\ndirections.  But hardly had John Manning given a glance round, than he\nuttered a sharp ejaculation.\n\n\"What is it?\" cried the colonel.  \"Can you see the rest of the mules?\"\n\n\"No, sir.\"\n\n\"Well, what is it?\" cried the colonel impatiently.\n\n\"Don't you see, father?\" cried Perry excitedly; \"the Indians must have\ncrept in while we slept.  They have carried everything away.\"\n\n\"What?\" raged out the colonel as he looked wildly round where his\nservant was holding up the lantern, and then he uttered a groan.\n\nFor it was too true.  Every pack had gone, and with them the possibility\nof holding out against the cunning enemy who had been in their midst.\n\nThe same thoughts came to each of the boys--thoughts of all they had\nheard before setting out, of those who ventured into the mountains in\nsearch of the Incas' treasures never being heard of more, and a curious\nchilling feeling of despair came over them.\n\nWould they ever see home again?\n\nBut the colonel was not the man to give way to despair.  The position\nwas terrible--right out there amid the gigantic mountains, with the only\nroads through them those naturally formed by the torrents in the wild\ndeep gorges, shut in by precipices of the most stupendous nature, with\nno other guide than their compass, and surrounded by enemies who might\nat any moment make an attack; while, so far from being able to make a\nprolonged defence now, the Indians had robbed them of the means.\n\nThis was the position as it struck them all at first, but the colonel\ngave it a different aspect directly.\n\n\"Council of war,\" he said cheerily, as he led the way out into the\nsunshine, and sat down upon a block of stone.  \"Come, Perry lad, a\nsoldier's son must not look like that as soon as he is face to face with\ndifficulties.  John Manning will tell you that he and I have been in\nworse straits than this up in the hill-country.--Eh, Manning?\"\n\n\"Of course, sir.  This is nothing.  Such a fine morning, too.  Why, if\nthe snow lay twenty foot deep, as we've seen it, and we didn't know\nwhether we had any fingers or toes, we might begin to holler.\"\n\n\"Of course,\" said the colonel.  \"So let's see what is to be done.\"\n\nThe two boys exchanged glances, but said nothing.\n\n\"Now, then,\" cried the colonel, \"let's look the worst straight in the\nface, and then we will calculate our advantages.  First of all, the\nenemy.\"\n\n\"None in sight, sir,\" said John Manning.\n\n\"No; they have gone.  They have achieved their aim in getting possession\nof our baggage.\"\n\n\"Don't you think they'll attack us, sir?\" said Cyril.\n\n\"No, certainly not.  They have been here, and found us asleep.  Had they\nwished to slay us, there we were at their mercy, and not a hair of our\nheads has been touched.  Next trouble.\"\n\n\"Walked off with all the wittle, sir,\" growled John Manning.\n\n\"Yes, that's bad; but we have plenty of water, and shall not fail in our\njourneyings for that.  Bad thing to be journeying through a wild country\nwith not a drop of water, eh, Manning?  We have done that.\"\n\n\"Don't talk about it, sir.  And the sun all the time drying the very\nmarrow out of your bones.\"\n\n\"Yes, those were terrible times, my man.  We shall not suffer that way,\nand as to food, we have our guns.  What about ammunition?\"\n\n\"All got plenty, sir,\" growled John Manning.  \"I've seen every morning\nthat our pouches were full.\"\n\n\"That's right, then.  With a little hunting, we must find some game in\nthese valleys, scarce as it has been.  Then there is the coca leaf,\nwhose virtues we must try, failing anything better.  Oh, come, we are\nnot so very badly off.\"\n\n\"Then you will try at once to make for one of the tracks through the\nmountains, sir?\" said Cyril.\n\n\"And live by hunting, father?\" cried Perry.\n\n\"And give up, after trying so long, and being so near success, my boy?\"\nsaid the colonel.  \"Come, come: Britons would not have made their name\nknown all over the world if they had gone on like that.\"\n\n\"Look here,\" cried Cyril excitedly; and then he stopped and looked\ndoubtingly at the colonel.\n\n\"Well, I'm waiting, my lad.  Every suggestion has its value at a time\nlike this.  Speak out.\"\n\n\"I'm afraid you'll think the idea too wild, sir.\"\n\n\"I will say so if it is, my boy.  Let's have it.\"\n\n\"I thought this, sir,\" said Cyril hesitatingly: \"we're well-armed; the\nIndians are afraid of you, and they are most likely retreating back to\ntheir camp with our mules and baggage, along with the seed we worked so\nhard to get.\"\n\nHe stopped again, and looked from one to the other, very red in the face\nall the time.\n\n\"Well, go on,\" said the colonel encouragingly.\n\n\"Why not show them we're not a bit afraid, and go on in pursuit of them\nat once?  I don't believe they'd fight, and if they did--well, we should\nhave to do the same.\"\n\n\"Hooray!\" shouted John Manning, throwing up his cap, catching it again,\nand then drawing himself up stiffly as he glanced at the colonel: \"Beg\npardon, sir.\"\n\nThe colonel merely nodded, but said to himself: \"I wish Percy had spoken\nlike that.\"  Then turning to Cyril: \"You said, why not go in pursuit?\"\n\n\"Yes, sir,\" said Cyril, throwing off his hesitation, and speaking now\nwith his eyes sparkling, and cheeks flushed with excitement.  \"Why not?\"\n\n\"Exactly, my boy, why not?\" said the colonel.  \"We were caught unawares,\nand I have blamed myself, an old soldier, severely for the greatest\nlapse of which an officer can be guilty--eh, John Manning?--sleeping on\nduty in face of the enemy.\"\n\n\"Awful bad, sir, in time of war.\"\n\n\"Yes, but there are bounds to human nature's endurance, John Manning;\nand though I would not own it to myself, I was utterly exhausted.\"\n\n\"All was, sir.\"\n\n\"Then now we must make up for it.--Cyril, my lad, you have proposed\nexactly what I intended to do.  Fortunately, we made a good meal last\nnight.  To-day we must feast again when we have retaken the baggage.--\nAll ready?  Fall in.\"\n\nThe boys followed the colonel's example, and leaped to their feet.\n\n\"Light marching order,\" said the colonel, \"so we ought to get along\nfast.  That mule we saw, Cyril, shows that the others have been taken\ndown the valley toward the great fall.  There is no choice of road here,\nso I take it that the Indians are making their way straight back to\ntheir camp.  Now, one word more.  See that your weapons are ready for\nimmediate use; no talking, but keep all your energies devoted to making\nobservations in every direction.  No rift or ravine likely to hide the\nenemy must be passed, if it is one possible for mules to climb.  Now,\nforward.\"\n\nThen with a feeling of exhilaration that the boys could hardly\ncomprehend, the little party started off with the colonel leading, and\nJohn Manning with his gun over his shoulder marching last, with a look\nin his face that suggested his feeling that he was guarding the rear of\na column of advance once again.\n\n\n\nCHAPTER TWENTY NINE.\n\nA MILITARY MOVEMENT.\n\nAt the first spot which gave him an opportunity to examine the gorge,\nthe colonel mounted to a narrow shelf and made good use of his glass,\ndescending at last to say: \"They have got a good start of us, but there\nis something about a quarter of a mile on that I can't make out.\nForward cautiously.\"\n\nFive minutes later the colonel halted again and sent Perry to the left,\nclose up under that side of the gorge; Cyril to the right, with orders\nto advance in a line with him, and be ready to fire if there were any\nneed.\n\nIt was quite a military movement, and the boys' hearts beat heavily at\nwhat seemed like the first initiation in real warfare; but before they\nhad gone far Cyril uttered a shout, and pointed forward to something now\nhidden from the colonel by a cluster of rocks in the lowest part of the\nravine, close to where the river ran with a deep-toned roar far below.\n\nA minute later they were alongside the object, which proved to be the\nleader, lying as near as it could get to the deep gash in which the\nwater was foaming.  The poor wounded beast had in its struggles broken\nthe long arrow nearly level with its skin, and in its agony of thirst it\nhad been trying to reach the water, but fallen upon its side.\n\nAs Cyril came close up, the mule raised its head and uttered a piteous\nwhinnying noise, looking up in the face of one who had many a time\nbroken off some green spray of juicy growth to feed it as it trudged\nalong with its load; but its eyes were already glazing, and it was the\npoor creature's last effort, for the head fell back heavily: there was a\ncurious quivering of the legs, which struck out once as if their owner\nwere galloping, and then all was still.\n\n\"Poor brute,\" said the colonel.  \"We cannot bury it, for there is no\nsoil here, even if we had tools.  Forward, my lads.\"\n\nHe led on, and the boys followed, feeling low-spirited; but they soon\nhad something else to think about, for just as they were approaching one\nof the narrowest parts of the gorge, Perry stopped short.\n\n\"Where's John Manning?\" he said.\n\nCyril followed suit, but no sign of the old soldier met his eyes, and\nthey communicated with the colonel, who looked very anxious and much\ndisturbed.\n\n\"We must return,\" he said.  \"Why, boys, you ought to have kept in touch\nwith him.  Double.\"\n\nThey all started back, but before they had gone a hundred yards the\ncolonel cried \"Halt;\" for there in the distance was the missing man\ncoming on at a rate which meant that he would soon overtake them.\n\n\"Did you see what he has been doing?\" said Cyril, as they were once more\non the march.\n\n\"Been stopping to get something,\" said Perry, \"but I could not see what.\nCould you?\"\n\n\"Ugh!  Yes,\" said Cyril, with a shudder of disgust.  \"He doesn't want\nfor us to be starved, but who's going to eat mule?\"\n\nThe tramp was long and tedious, but being no longer controlled by the\npace of the baggage animals, the little party made far better progress\nthan when they were making their way up the valley; yet the distance\nthey had come was far greater than they had anticipated, and for long\nenough there was no sign of the Indians having passed that way.  But\nthey kept on, the colonel feeling convinced that they had passed no side\nravine up which the mules could have been driven; while, having these\nanimals at their command, the colonel felt certain that the Indians\nwould not carry the loads.\n\nAt last, during the hottest part of the afternoon, a halt was called,\nand they made for a huge rock which overhung on one side, offering a\ntempting shade from the burning sun; but before they reached it Cyril\nuttered an eager cry.\n\n\"Look! look!\" he said excitedly, and he pointed to where there were\nmarks about a patch of herbage where the mules had been cropping the\ncoarse stuff, as well as browsing upon some tufts of bushes, whose green\ntwigs were bitten and broken off, and here and there leaves which had\nbeen dropped were still so fresh that it was evident that they could not\nlong have been left.\n\nThis discovery, and a faint trace or two of the Indians having been with\nthe mules, had a better effect upon the party than hours of rest.  For\nthey knew now that the treasured packs, containing not only\nthe necessities upon which they depended for life, but the\ncarefully-collected seed, were only a short distance ahead, and that if\nthey pushed on with energy they ought to overtake them.\n\nThe rest depended upon the strong arms of the two men.\n\nThey went on then at once, but no fresh sign encouraged them, and at\nlast the closing in of the ravine and the piled-up mountain in front\nwarned them that they were approaching the gloomy chasm into which the\nriver plunged.  In fact, half an hour after, the deep booming roar of\nthe fall began to be audible, as if coming from somewhere high up on the\nmountain-side.\n\n\"If it's coming to a fight,\" said Perry, \"I hope it will not be in that\ndeep cavernous place near the fall.  I feel as if I hardly dare go down\nto it after what happened.\"\n\n\"Nonsense; you'll feel plucky enough when it comes to the point.  Come\nalong.\"\n\n\"Steady, young gentleman.  Halt,\" growled John Manning.  \"You should\nkeep your eyes open for what's going on in front.  Look at the colonel.\"\n\nThey gazed forward, and saw that the last-named gentleman was signing to\nthem to stop and follow his example of crouching down; and directly\nafter they saw him creeping on toward the spot, from which so short a\ntime before they had seen the Indians come out from behind the veil of\nmist, and amidst the roar of the falls point upward, making signs to\neach other, and then disappear.\n\nIt was now so gloomy, that it was hard to make out their leader's\nmovements, for the light only reached them from the narrow opening high\noverhead, and where the little river raced onward toward the fall it was\nrapidly growing black; but in a few minutes the colonel signed to them\nto come on, and at a word from John Manning they advanced quickly,\nstooping in obedience to a sign from the colonel's hand, and reached him\nat last where he crouched behind a stone.\n\nHe did not speak, but pointed, and first John Manning, then Perry, and\nlastly Cyril peered cautiously over the stone, the latter being only in\ntime to see that they were quite right in their surmises, for there\nbelow was a party of about thirty well-armed Indians, slowly making\ntheir way down the last of the many zigzags of the path toward where the\nmist rose like a dark veil, the wind which blew down the gorge keeping\nit, as it rose from the great gulf, from spreading in their direction,\nbut beating it back into a dense cloud, to float slowly out into the\nvalley beyond.\n\nThere they were walking very cautiously, some ten in front, and next, to\nCyril's great joy, the remainder of their train of pack-mules, well\nloaded with the treasures taken from the cave.  The rest of the party\nfollowed about a dozen yards behind.\n\nCyril remained watching till the head of the little column readied the\nveil of mist, went on, and was completely blotted out the next minute,\none by one, the Indians being visible in the gloom, and then gone.\n\nNext it was the turn of the animals, and as Cyril watched, it was very\ncurious to note how a mule would be visible for a time, then its head\nand shoulders would disappear, and lastly it would be entirely swallowed\nup.  The remainder of the Indians followed, one by one, till the last\nman, who seemed to be their chief, was alone, and he turned back to gaze\nupward, narrowly scrutinising the zigzag path by which his party had\ndescended for a few minutes before he followed the rest.  Then the\ngloomy place was utterly deserted, and momentarily growing blacker, so\nthat the way down could not be seen.\n\nThere was no fear now of their words being heard above the booming roar\nwhich came up out of the chasm, and the colonel explained his\nintentions.\n\n\"I am going to call upon you all to show your nerve,\" he said, \"for we\nmust go down, and pass along by the fall through the darkness and mist.\"\n\n\"But is it safe, father?\" said Perry anxiously.\n\n\"Safe or no, my lad, it is a chance I cannot let slip,\" replied his\nfather.  \"You saw their leader looking back, Cyril?  He was evidently\nsatisfied that there was no pursuit, and he and his people will conclude\nthat once they get through the mist they will be safe, and no doubt camp\njust on the other side for the night.--Manning, we must steal through\nthe mist, creep up, and surprise them.  Two or three of our little\nvolleys will, I believe, put them to flight if we come upon them\nsuddenly, and then we must return this way, and show our teeth fiercely\nif they pursue.\"\n\n\"Right, sir; I'm ready,\" said John Manning.  \"You lead, and we'll\nfollow.\"\n\n\"Let's rest, then, till it is quite dark.  I believe I can lead you\nright through to the other side without much risk, for after our\nexperience we know the worst.  Besides, where they can go, we can go.\nThere, wait a bit, boys, and we shall have our own again, and the\nwherewithal to make a good meal.  No, no,\" the colonel added as John\nManning raised his head.  \"Only in case of utter starvation, my man.  We\ncan hold out a little longer, and I hope we shall not have to come to\neating mule.  Now, lie down, all of you, and rest.  In a couple of\nhours' time I shall start.\"\n\n\n\nCHAPTER THIRTY.\n\nTRAPPED.\n\nHungry and faint, it was a dreary time passed during that halt; but in\nspite of all, it was restful, though the stones were hard, and there\nwere moments when Cyril felt as if he could go off fast asleep, and\ndream of banquets, as hungry people are said to do.\n\nBut there was no sleeping, and, as nearly as could be guessed, at the\nend of two hours the colonel rose, and gave the word \"Forward.\"\n\nThen began the journey down the long zigzag, every turn bringing them\nnearer to the spot where the river took its great plunge into the gulf.\nThe roar grew deeper and louder, though still smothered by the dense\nmist, and as they drew nearer, there was the damp odour of water,\nbreathed in the smallest of vesicles, as it was churned and then whirled\nupward to battle with the breeze descending the gorge.\n\nThe last turn of the descending path at last, and then the colonel\nstopped, for they were face to face with the great black veil of mist.\n\n\"Now,\" he said, with his lips close to each one's ear in turn, \"you\nfirst, Perry, take a grip of the stock of my gun, and pass your own\nbackward as I do mine for Cyril to take hold.  Then Cyril will do the\nsame for John Manning to grasp, and we shall be linked together and well\nin touch.  I shall lead, of course.  Courage, boys, and no hesitation.\nWe shall soon be through.  Now, forward.\"\n\nIn half-a-dozen steps the darkness, which had been relieved by the faint\ngleaming of the stars peering down into the gorge, became intense, for\nthey were once more in the mist, and guided only by the gentle drag upon\nthe guns, as without hesitation the colonel led on, keeping close to the\nwall upon his left.\n\nThe noise of the water thundering down was more confusing than ever, the\nmist more stifling; but the boys gathered confidence as they went on,\nand Perry was too much occupied in following his father's steps, to\nthink much about the horrible slip into the gulf below; while Cyril, as\nhe stepped on manfully, kept trying to recall how far the way was\nthrough, and calculated that they must be fully half-way.\n\nHe had just arrived at this conclusion, when he turned angrily to resent\nwhat he looked upon as absurd behaviour on the part of John Manning, who\nsuddenly grasped him tightly, pinning his arms down to his sides, and\nflinging him up against the rock-wall as far as possible from the edge\nof the gulf.\n\n\"How stupid!\" he cried aloud, though not a word was heard.  \"I'm all\nright.  Now you've broken the chain.\"\n\nHe had arrived at this point, when he felt a rope passed rapidly round\nhim, binding his arms to his sides.  Then he was thrown upon his back,\nand in spite of his struggles, his legs were treated in the same way,\nafter which a cloth was bound over his face, so tightly as to be almost\nsuffocating.  Lastly, he felt himself lifted head and heels, and borne\nforward, dizzy, confused, and wondering what had happened to his\ncompanions, and finally bound to conclude that they must have been\ntreated precisely in the same way.  He felt that this must be so, and\nthat the Indian cunning had been too much for the colonel's strategy, a\nparty having remained in waiting in full knowledge that they were\npursued, and ready to pounce upon them, just in a spot where an attack\nwould be least expected and surest of success.\n\nAll at once, as the boy was borne along, feeling satisfied that it was\nuseless to struggle and folly to exert himself and shout, it occurred to\nhim that his bearers were going closer to the edge of the gulf, for the\nroar of the water seemed to be more deafening.  There could only be one\nreason for this, he argued--it was his turn to be thrown in, and the\nothers must be gone.\n\nThe horrible thought made him begin to struggle with all his might, but\nat the first writhe a strong additional arm was passed over his body,\ngripping him tightly to its owner's side, and in this fashion he became\nhelpless, and was carried forward, to grow calmer, for he awoke to the\nfact that his life was certainly for the present safe.\n\nThen a curious feeling of faintness came over him, the heat of the cloth\nover his mouth was suffocating, bright specks of light danced before his\neyes, there was a singing in his ears, and then everything seemed to be\nat an end, till the stars were looking down at him from far on high, and\nabove the low distant booming of the fall he could hear the pleasant\nsilvery gurgle of water, and the heavy breathing of sleepers close at\nhand.\n\nBy degrees the boy's head grew clearer, but at the expense of his body,\nfor as the power of thinking brightened, his limbs grew heavier, numb,\nand helpless, and the effort he made to turn over upon one side proved\nto be in vain.  He felt that the cloth which bound his lips was gone,\nbut there was no inclination to cry for help, and he lay perfectly\nstill, wondering whether his companions were near, and then utterly\nexhausted, all passed away again, but this time he slept.\n\nIt must have been near morning once more, when Cyril awoke with a\nfeeling of something warm touching his ear, and a voice whispered:\n\n\"Careful, my lad.  I've cut you free, and I'm going to cut the colonel\nand Master Perry clear.  Now try and rub your legs gently.  We must make\na dash for it, as soon as you're ready.  Don't speak.\"\n\nThe lips were removed from his ear, and there was a faint rustling, that\nwas all.\n\nHe tried to obey the orders he had received, but for some minutes there\nwas scarcely any sense of feeling in his hand, or in the part he\ntouched, but he worked on, feeling hopeful now.  John Manning was\nfighting for their freedom, and the others must be close at hand, but he\nfelt that if they were as helpless as he, they would not be of much use\nin an attempt to escape from their captors.\n\nAnd as Cyril went on softly rubbing circulation into his numbed and\nswollen legs once more, a faint point of light high up in the clouds,\nwhere an ice peak was catching the first rays of the coming morn, shone\nout like a hopeful sign to tell him that all was not yet lost.\n\nFor quite half an hour he kept up the gentle friction, bringing back\ncirculation, but with it intense pain.  Then his heart bounded, and he\nforgot his agony, for John Manning crept close to him again.\n\n\"Been rubbing?\" he whispered.\n\n\"Yes.\"\n\n\"Can you fight?\"\n\n\"I'll try.\"\n\n\"All right then, boy; the others are ready, and I think the Indians are\nasleep.  We must make a dash for it now, before they make up their minds\nto put us out of our misery, for I am afraid it's that they mean.\"\n\n\"What do we do first?\" whispered Cyril, who felt the power rapidly\ncoming back into his legs.\n\n\"Wait till the colonel joins us with Master Perry.  They're coming as\nsoon as they feel it safe, and then we dash back for the falls, and\nretreat up the gorge.  When we jump up, keep together and run.  Hit out,\nlad, at anybody who tries to stop you.  They're only cowards after all,\nbut the colonel's coming.  Now get up softly.  Ready?\"\n\n\"Yes.\"\n\nAt that moment there was a fierce yell, and Cyril was dashed back upon\nthe ground, three Indians flinging themselves upon him; there was a dull\nblow, a groan, and John Manning cried aloud:\n\n\"They've done for me, lad; run for it, if you can get free.  Tell the\ncolonel I did my duty to the last.\"\n\nAlmost at the same moment Cyril, as he fiercely struggled, heard a\nshrill cry of agony from Perry, a shout from the colonel, and the\nreports of half-a-dozen guns fired in rapid succession.\n\nThen all was blank, for a heavy blow on the side of the head made the\nlad insensible to what was passing around.\n\n\n\nCHAPTER THIRTY ONE.\n\nFATHER AND SON.\n\nWhen Cyril opened his eyes and began to look about, his head was aching\nviolently, and a swimming sensation made everything near him look misty\nand indistinct.  But he was conscious that the sun was shining brightly\nall around, and that he was lying in the shade cast by a tree, whose\nfoliage was so familiar that he closed his eyes again to think and\nwonder whether he was dreaming.\n\nFor that was unmistakably a cinchona tree, one of those he had thought\nabout so much of late.\n\nHe opened his eyes again, and looked round to see that there were\nseveral mules about grazing on the rich grass, and there was a peculiar\nodour in the air which he knew to be caused by burning wood.\n\nA low buzz of conversation was going on, too, somewhere close behind\nhim, and he tried to look round, but the movement gave him so much pain\nthat he let his head sink down, uttering a weary sigh, which was\nevidently heard, for there was a rustling sound behind him, and some one\ncame and bent down and took his hand, at the same time laying another\nupon his forehead and gazing into his eyes.\n\nFor some moments nothing was said; Cyril, with his heart beating\nheavily, gazing up into the eyes that looked down into his, while he\nwondered more than ever what it all meant.\n\n\"Don't you know me, my boy?\" was said at last, and a half-hysterical cry\nescaped the lad's lips as he clung to the hand which grasped his.\n\n\"Yes, father!  But--but what does it all mean?\"\n\n\"That you must lie still and rest for a bit.  You have had a nasty blow\non the head, but you will soon be better.\"\n\n\"But--where are we?--where is Perry, and where is the colonel?  I can't\nthink, but I don't understand why you are here.\"\n\n\"You can ask yourself that last question by-and-by, my lad.\"\n\nCyril shrank a little, for those words were more potent than any\nreproach, and Captain Norton went on:\n\n\"You were asking about your friends.  They are all here, but have been\nhurt more or less.  We only came up just in time.\"\n\n\"You came up--just in time?  Oh, I remember now.  We were fighting and\ntrying to escape, and somebody fired.  Was it you, father?\"\n\n\"Yes, my lad, my friends and I.  If we had not arrived as we did, I'm\nafraid that there would have been a tragedy here in this valley, for the\nIndians were roused, and I believe that you would none of you have lived\nto see another day.\"\n\n\"And the Indians: where are they now?\"\n\n\"Far away, my lad.  They will not face firearms.\"\n\n\"But you came, father--after me?\"\n\n\"Of course, as soon as I grasped the fact that you had followed Colonel\nCampion.  At first I would not think it possible that my son could treat\nus at home as you had; but when, from a man who had come over the\nmountains with a llama train, I learned that he had seen you, I did what\nI felt it to be my duty to do for your mother's sake.\"\n\nCyril's hands went up to his face for a few moments, and then they were\ngently pressed aside.\n\n\"This is no time for blaming you, Cyril,\" said the captain gently; \"you\nare injured.  Get well, my boy.  But you asked me how I came here.  As\nsoon as I knew that you were with Colonel Campion, I got the help of two\nor three friends, and our servants, and we obtained mules and came on in\nsearch of you.  I did so, for, in addition to my duty to you, I repented\nletting a brother-officer come upon what I felt more and more was an\nexceedingly risky expedition.  It has proved so, has it not?\"\n\n\"I'm afraid so, father,\" sighed Cyril.  \"Would the Indians have killed\nus?\"\n\n\"It seems so.  You were utterly outnumbered, and from what I can gather,\nI suppose they believe you were hunting for and had found some of the\nold treasures buried here in the mountains.\"\n\n\"Oh no,\" cried Cyril; \"they were quite wrong.\"\n\nAnd he explained the object of the colonel's mission.\n\n\"They would not believe that, my boy, though they would have been just\nas ready to stop anything of the kind.  I found, on tracing you to their\ncamp, that you had come down in this direction, and the man who acted as\nour guide gathered that there was some trouble on the way, and thus made\nme hurry on after you.  I should have come up with your party sooner,\nonly three times over we were tricked into following another track, our\nguide proving perfectly untrustworthy directly after he had been in\ncommunication with the people at the back camp.  However, I came up with\nyou in time, just as a fierce fight was going on, and your party were\nbeing worsted.  A few shots drove the Indians off, and for the present\nwe are safe.\"\n\n\"And the mules and their loads?\"\n\n\"There are our mules,\" said the captain quietly.\n\n\"No, no; I mean ours,\" cried Cyril.\n\n\"I have seen no others.  There are none here.\"\n\n\"But they've taken the kinia seed that the colonel came to collect.  We\nmust go and attack them at once.\"\n\n\"We must get from here on to the regular track through the mountains as\nsoon as we can, my boy,\" said the captain sternly.  \"We do not know\nwhether we may not ourselves be attacked by a strong body of the\nIndians.  I cannot do as I like, for I must study my friends; but if I\ncould, I would not run any risk in the face of such odds: so if Colonel\nCampion can by any possibility sit a mule, we shall begin our retreat at\nonce.  What?  Can you stand?\"\n\n\"Yes, father.  Only a little giddy; and I want to see the colonel and\nJohn Manning.\"\n\nFor Cyril had raised himself to his feet, and his father led him at once\nto where his companions lay close by, where their rescuers had formed\ntheir temporary camp, and were now making a hearty meal.\n\nPerry was lying back with his head bandaged, John Manning was suffering\nfrom a severe knife wound, and the colonel lay looking very hollow of\ncheek, for he also in the fight had received a bad knife thrust, and to\nCyril it seemed that it would be impossible for the party to begin their\nretreat for some days to come.\n\nBut as soon as he awoke, the colonel declared himself able to sit a\nmule, and John Manning insisted upon the hurt he had received being\nmerely a scratch; so, as the case was urgent, a start was made that same\nafternoon, and a few miles made before they were overtaken by night, and\nencamped, setting a careful watch in case of attack.\n\nBut none came, the lesson given by Captain Norton quelling all present\ndesire for a closer acquaintance with the firearms; and soon after\ndaybreak they were once more in motion, the leader retracing the way\ntaken by his friends in their attempted escape till they were close up\nto the cinchona camp, which they found deserted.\n\nA long halt was necessary here on account of the injured party, but two\ndays later they were on their way again, after a long consultation\nbetween Colonel Campion and their friends.\n\n\"Did you hear what was said?\" asked Perry, as he and Cyril rode side by\nside wherever the track would allow.\n\n\"Yes, everything; your father wanted to stay here for a bit and make an\nexpedition or two in search of the Indians, so as to try and recover the\nbaggage and mules.\"\n\n\"Of course,\" said Perry.  \"It's horrible to go back like this, regularly\nbeaten.  But they wouldn't?\"\n\n\"No: my father said he was willing, but the rest would not.  They said\nthey had come to help to save all our lives, and bring me back, but they\nwere not going to risk their own any more to satisfy--\"\n\n\"Well, satisfy what?\" said Perry, for his companion checked himself.\n\n\"Like to know?\"\n\n\"Of course.\"\n\n\"Satisfy your father's mad-brained ideas.\"\n\n\"Mad-brained indeed!\" cried Perry indignantly.  \"And didn't father say\nthey must go?\"\n\n\"No,\" replied Cyril, laughing, \"because he had no authority, and he was\nperfectly helpless.  You see he couldn't go himself.\"\n\n\"I only wish he was strong enough,\" cried Perry.  \"He would soon show\nsome of them.\"\n\n\"Hasn't he shown them enough?  My father's right.\"\n\n\"What, in giving up?\" cried Perry indignantly.\n\n\"No, in behaving like a good soldier, and drawing off his forces when he\nis beaten.  Father told him that it was folly to go on now in his\nhelpless state.  That, injured as he was, he would kill himself and you\nand your man too, for you had neither mules, provisions, nor weapons,\nand that the only thing to do was to go back.\"\n\n\"And what did my father say?\" cried Perry hotly.\n\n\"Nothing.  He only held out his hand without speaking, and they stood\nfor half a minute.\"\n\n\"But it's horrid to be beaten and go back like this, robbed of all our\nbelongings, and just too when we had succeeded so well.  The cowards!\nAll that party against us.  I feel as if I couldn't go back to San\nGeronimo.\"\n\n\"So do I,\" said Cyril dolefully.\n\n\"You?  What have you got to mind?\"\n\n\"What have I got to mind?  All that my father will say when we get back,\nthough I don't worry about that so much.\"\n\n\"What, then?\"\n\n\"I've got to meet my mother.\"\n\n\"Well, but she won't say anything unkind to you.\"\n\n\"No,\" said Cyril sadly, \"not a word; but she'll look at me as I often\nseem to see her looking at me now, and asking me how I could behave so\ncruelly to her.  It half killed her, father says, for my boat was\nmissing for a fortnight.  One of the fishermen had taken it away, and\nshe thought I had gone out in her, and was drowned.\"\n\nPerry was silent, and soon after the boys had to separate, and ride in\nsingle file about the middle of the little line, Captain Norton and two\nof his friends forming the rearguard, in case of attack.\n\nBut though the return journey was very slow, on account of the weakness\nof the injured part of the little caravan, and there was every\nopportunity for the Indians to fall upon them had they been so disposed,\nthey went on, day after day, unmolested, and their nights were\nundisturbed.\n\nThose long narrow shelves of rock at the sides of the defiles seemed as\nif they would never end, but the clear crisp mountain air was wonderful\nin its curative effects; and while Perry was quite well again, and Cyril\nhad about forgotten his injury, Colonel Campion and John Manning, though\nboth thin of face, and generally a good deal pulled down, were strong\nenough to walk down--at the close of the last day's journey--the long\n<DW72> which led to Captain Norton's house on its platform high above the\nsea.\n\n\"Where's Cyril?\" said Perry suddenly to Captain Norton.  \"I haven't seen\nhim these two hours.\"\n\nCaptain Norton stopped at the edge of the narrow path, and pointed down\nto the dry-looking garden at the back of his house, where the tall,\ntapering flagstaff stood up, with the British colours fluttering in the\nsea-breeze.\n\nPerry shaded his eyes, and through the clear evening air he could\ndistinctly see his companion standing by a lady, and looking up at the\nlittle mule train filing down the <DW72>.\n\n\"Why, he has run on home!\"\n\n\"Yes,\" said the captain.  \"I sent him on to meet his mother alone.\nPerry, my lad, for the sake of all who hold you dear, never be guilty of\nsuch a selfish, thoughtless act as his.\"\n\n\"I'll try not,\" replied the boy thoughtfully; and then in an animated\nway: \"But, I say, Captain Norton, if it had not been for his thoughtless\nact, where would we three have been now?\"\n\nThe captain smiled and looked at the colonel, who had heard all that had\nbeen said.\n\n\"That's a question I would rather not try to answer, my lad.  There, no\nmore: I've promised Cyril to bury the past.\"\n\nWeak as he still was from his injuries, and smarting from the bitter\ndisappointment of his failure, Colonel Campion seized the first\nopportunity which occurred of getting a passage up to Panama, the two\nboys parting with many promises of keeping up a correspondence, which\nwere none too faithfully fulfilled.  Perry wrote from Panama, and again\nfrom Barbadoes on the way home.  Then three years elapsed before Cyril\nhad a letter, though Captain Norton had heard again and again from his\nfriend the colonel.\n\nHere is a portion of the letter Cyril received:\n\n\"I don't suppose they will do it, but I think they ought to make my\nfather F.L.S. and F.R.S. and F.G.S., and all the rest of it, besides\nknighting him.  For only think, in spite of all the disappointment of\nlosing the packages of seed we so carefully made up, the little lots we\nhad in our pockets, including those you gave me at San Geronimo out of\nyours--I mean that day on board the packet, when you said, `You may as\nwell take these, for they're no use to me--' I say, all these were\ndistributed and set, and with the exception of one lot, pretty well all\ngrew, and they have made small plantations in Java, Ceylon, India, and\none or two other places, so that in the course of time there'll be\nquinine in plenty in hot places all over the world.  Which lot do you\nthink it was failed?  You, in your modesty, will say your own.  Not it,\nbut mine; and I'll tell you how it was--through my fall down into that\nhorrid place.  The seed was of course soaked, and it went off mouldy, I\nsuppose.  At all events, none of it grew.\"\n\n\"Hah!\" exclaimed Captain Norton as he heard the letter read.  \"It was a\ndaring thing to do--a brave soldier's deed.  How many poor wretches in\nthe future who struggle back from some deadly fever will ever hear of or\nbless his name?  Hardly one.\"\n\n\"But we shall have the satisfaction, father, of knowing that we helped\nto save them all the same.\"\n\n\"Right, boy,\" cried the captain, bringing his hand heavily down upon his\nson's shoulder.  \"You did your share, and it would be a poor world\nindeed if we did all our good actions for the sake of the reward.\"\n\n\"But mine was not a good action, father,\" said Cyril gravely.\n\n\"Ah, well,\" said his father, \"it is a matter of the past.  I made you a\npromise then, and we will not argue that.\"\n\nThe End.\n\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Real Gold, by George Manville Fenn\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nDedicated to JD for his inspiration  \nand to Rob, my son, to give him inspiration.\n\n## ACKNOWLEDGEMENTS\n\nI would like to give my thanks to all the crew who willingly volunteered their stories about their careers on Concorde. Certain episodes will always remain strongly etched on the mind and it is those episodes of an extraordinary life that I have happily recalled.\n\nIn particular, I would like to thank cabin service manager Dick Bell, Scarlett Geen, Louise Brown, Dee Bull, Jill Channon, Annie Carter, Carol Cornwell, Steve Brennan, Sue Drayton, Bernadette Forrest, Jeannette Hartley, Robert Bailey, Maggie Sinclair, Maggie Coles, Captain John Hutchinson, Captain Jock Lowe, Chief Pilot Mike Bannister and, last but not least, Purser Julia van den Bosch, who has flown more supersonic miles than anyone on this planet!\n\nI would also like to thank all the crews who flew with me for all the special memories we shared. Thanks to Amy Rigg and all at The History Press for their help and support. To special friends Jilly Cooper and Fred Finn and to Richard Noble for their kind contributions to the book.\n\nFinally, this book is not intended in any way as a technical narrative on Concorde so any omissions on its performance, handling and statistics will, I hope, be forgiven by the readers of this book.\n\nFront cover photos \u2013 thanks to Mike Bannister and Frederic Carmel.\n\n## CONTENTS\n\n  1. Title\n  2. Dedication\n  3. Acknowledgements\n  4. Foreword by Richard Noble\n  5. Foreword by Fred Finn\n  6. Introduction\n  7. The End of an Era \u2013 Tears and Cheers: London Heathrow Airport\n  8. Concorde Beginnings and Design Issues\n  9. Now, What shall We Call it? Naming Issues\n  10. The Stage is Set and a New Icon is Born\n  11. The American Response\n  12. The Air Stewardess: How it all Began\n  13. A Flying Start: the 747 Fleet\n  14. Subsonic to Supersonic\n  15. The Supersonic Experience \u2013 'Flying Hotel' to 'Pocket Rocket'\n  16. The Working Day: Behind the Scenes\n  17. The Flying Wardrobe\n  18. Flight 001 is Now Ready for Boarding\n  19. Life at the Sharp End\n  20. Into the Stratosphere, the Curvature of the Earth and to New York\n  21. A Well-stocked Wine Cellar and Food Fit for a Prince\n  22. The New York Stopover and Dinner with a Singing Legend\n  23. Concorde Charters and a Good Excuse to Display Concorde's Flying Abilities\n  24. The Vienna Ball\n  25. Round-the-World Charters and One for Opera Lovers\n  26. Promoting Concorde: Unique Times and a Trip to Detroit\n  27. Barbados Bound\n  28. Life with the Superstars, Rock Stars and Mega Stars\n  29. Stars of Stage and Screen\n  30. Rock Stars on Board\n  31. The Supersonic Commuters\n  32. The 11-Mile-High Club \u2013 the Shortest Chapter\n  33. Breaking Records: Washington to Nice, 11 September 1984, and a Kind-hearted Bellboy\n  34. Life Doesn't Always Go According to Plan\n  35. Do Not Upset the Captain!\n  36. All in a Day's Work\n  37. The Royals and Other Famous Flights\n  38. Crew Shenanigans: You Would Expect it, Wouldn't You?\n  39. The Concorde Crash\n  40. The Aftermath\n  41. And Finally...\n  42. The Future of Supersonic Flying\n  43. Postscript: Crossing the Atlantic \u2013 3 Hours and 30 Minutes to Seventeen Days\n  44. Life after Concorde\n  45. Concorde Timeline\n  46. Concorde's Vital Statistics\n  47. Plates\n  48. Copyright\n\n## FOREWORD\n\n## BY RICHARD NOBLE\n\nIt's June 2009 and I am standing under the wings of Concorde Delta Golf at Brooklands. There are crowds of people around and we are here to launch the cockpit simulator which had been rescued from Filton, where it had been irreverently trashed when Concorde flying ceased. There are quite a few children present and it suddenly occurs to me that they will never see or experience Concorde fly. They must be wondering what all this is about and why they have been let down by technology and our current less-than-amazing culture.\n\nI'm a lucky one. I was asked by British Airways (BA) to set a record for the fastest ever triple crossing of the Atlantic to celebrate ten years of Concorde transatlantic flying. We were taking G-BOAA to Kennedy (John F. Kennedy Airport, New York), and it was the same aircraft and crew who made the inaugural flight on 22 November 1977.\n\nWe are BA 001 and we leave at 10.30 a.m. to the second. I change Concordes at Kennedy, running across the tarmac to G-BOAG which is waiting for me, and by 1830 hours I am in the cockpit and on finals to Heathrow, Runway 27, when a message comes through for me, 'Richard please pick up the BA chairman's baggage on your return flight \u2013 it got left behind!'\n\nArmed with the chairman's cases I am on my way back to Kennedy for the second time that day \u2013 and on arrival we all have a humongous party at Mortimer's in New York. The next morning I am on a Concorde flight back to London and then I am immediately off to work in our office, selling our production light aircraft to Brazil. I have absolutely no jet lag. Including stops, the three trips took 11 hours and 22 minutes. The average speed, including stops, was 864mph.\n\n'Do anything special over the weekend?' asks our chief designer, Bruce Giddings.\n\n'Well, yes, I can claim to be the only person ever to have crossed the Atlantic three times in 12 hours...'\n\nSally Armstrong's book tells you all about the very special Concorde people who kept the service running to the very high standard that BA demands. It was a lifetime's experience to fly with them.\n\nBut there is another, even more important story. Concorde was a dramatic symbol and a source of huge pride to the British. I remember standing outside our dentist's surgery in Teddington, Middlesex, and seeing Concorde make its standard low level U-turn overhead to set off west for Kennedy. The whole street had stopped and everyone was looking up. Of course, this happened every day \u2013 and every day the Brits took enormous pride in their truly beautiful aircraft.\n\nWhat Concorde did was inspire and encourage a vast number of people to become engineers, scientists and flight crew. It was a symbol of outstanding achievement by an extraordinary generation of very capable and focused people \u2013 and now their children will never see it fly. Even worse, the reality of Concorde has been replaced by singing, cooking and dancing on television and fakery in computer gaming. Concorde and its service was real. No wonder that today there is a critical shortage of engineers and scientists in our country. Could we ever do this again? I doubt it.\n\nSo, Concorde was a very special programme in a very special age. Just how special? Well, read Sally Armstrong's book \u2013 and you'll find out. I promise you won't be disappointed!\n\nRichard Noble\n\nKingston upon Thames\n\nLand-speed record holder, 1983\u201397\n\n## FOREWORD\n\n## BY FRED FINN\n\nFred Finn, the world's most travelled man.\n\nWhen I was asked to write the foreword to this wonderful book, Vintage Champagne on the Edge of Space, I was thrilled to accept. I flew with Sally Armstrong on the 'Queen of the Skies' many times.\n\nI was privileged to belong to the Concorde group of friends. With 718 flights in Concorde, I got to know the wonderful cabin crew very well during a thirty-year love affair with this extraordinary, beautiful and graceful lady of the skies. As a result we became good friends \u2013 there was always a chilled bottle of DP [Dom Perignon] shampoo under the seat in front when I boarded into my seat, 9a.\n\nThere were occasions when I was asked to give my seat to someone they didn't want to leave behind. On those occasions I used to sit in the jump seat behind the captain, most of whom I got to know socially, having played cricket for the Concorde cricket team in the annual match in Allworth village at the local pub. It was the home of the first captain, Brian Calvert, who organised the very first Concorde charter with the gamekeeper and the poacher on board.\n\nConcorde was the only aircraft that had this friendly, fun and yet extremely professional atmosphere, where it appeared that everyone knew the regulars and, in my case, I was known as Fred. I used to look forward to being on board and, of course, it was quicker than a rifle bullet. I got so used to the atmosphere on Concorde flights that I began to think the flights were too quick. I didn't ever want this part of my life and world ever to stop.\n\nThank you to my many Concorde crew friends, some of whom I still see and continue to enjoy that special relationship that only Concorde could have created, and to Sally for this real account of life on Concorde.\n\nWhere else could you have John Denver playing 'Country Road', Paul McCartney drawing happy faces, and the 'boss' Bruce Springsteen asking 'how many flights for Fred now?' to the crew?\n\nConcorde, and my friends on Concorde, I miss you \u2013 it was a disaster that it was pulled out of the skies so prematurely, nothing else comes close. It was so much, much more than just an aircraft. The aircraft lived and breathed \u2013 Concorde was alive.\n\nThe world has gone back to those ubiquitous bi-bodied impersonal sons of Boeing and Airbus, a retroactive step. Nothing will ever be the same again in the world of aviation.\n\nFred Finn\n\n718 flights in Concorde, 16 million miles \u2013 Guinness World Records held since 1983\n\nDirector of Livingstone's Travel World\n\nCo-founder of 'Quicket', the mobile travel app\n\nPresident of Save Concorde Group\n\nSally Armstrong.\n\n## INTRODUCTION\n\nShe's fast, she's slinky and the captain loves her.\n\nSo ran the headline in a newspaper. This was not some story about a captain's love affair with a stewardess, this was not some insider's 'tell all' story about a romance in the flying world \u2013 this was the description of a unique aircraft. This was about Concorde.\n\nThere are not many aircraft in aviation history that can match the description, or earn the love and devotion from the many hundreds who worked on her, both on the ground and in the air. Concorde, with her unique elegant silhouette, was impossible to ignore as she streamed across the skies. To compare her to a very beautiful woman is easy. She was the ultimate head turner. She had a beautifully designed body, she was elegant and, like a beautiful woman, never failed to cause a stir of attention.\n\nFor the captains and flight crew 'going to work' was always a joy. Flying your own 'dream machine' that had cost zillions was not a bad way to earn a living and most pilots will confess to that. For the cabin crew it was the ultimate in flying as a career.\n\nFor the passengers it invariably stirred a childhood excitement. Flying on the outer edge of space never failed to impress even the most regular passengers. It was exhilarating. It was unlike any other passenger aircraft in aviation history. Even after its twenty-seven years' service it was still an icon. Here is my story of Concorde, the ultimate flying machine.\n\nSally Armstrong\n\n## THE END OF AN ERA \u2013 TEARS AND CHEERS:\n\n## LONDON HEATHROW AIRPORT\n\nOn 24 October 2003, the world's press, TV camera crews and many thousands of spectators gathered at London's Heathrow Airport. Every available spot around the perimeter roads was crammed with devoted fans. Police cordons were set up to ensure the spectators kept to their side of the barriers. A grandstand had been specially erected to hold 1,000 people.\n\nContrary to most VIP arrivals it wasn't the guest list on board the incoming aircraft that was the subject of their attention. It wasn't the arrival of a mega pop star who fans waited excitedly to see. It was the actual aircraft that held their focus of attention. But this was no ordinary arrival and no ordinary landing. A momentous event was about to happen. The landing of this iconic aircraft would herald the end of an era of supersonic flying.\n\nIt was Concorde the world waited for. This aviation phenomenon of the twentieth and twenty-first centuries was making its final commercial flight into London's Heathrow Airport. At the exact appointed time and, like a diva making her appearance right on cue on the world's stage, the distinctive shape of Concorde appeared through the clouds, its pointed drooping nose leading the swanlike contours of its wing structure. Elegant and graceful, its four Rolls-Royce Olympus 593 engines throttled back as it made its final approach.\n\nThe spectators held their breath, marvelling for the last time at Concorde's timeless elegance. History was being made, and this was one piece of history the public wanted to be a part of. Concorde was very close to their hearts and, in a time of economic uncertainty and upheavals in politics, it gave them a reason to be proud of being British. Concorde fans had come in their thousands to pay their last respects. Many millions more around the world watched on TV.\n\nAs Concorde approached Heathrow Airport for the last time in its twenty-seven years of commercial history, few could take in this moment. It was incomprehensible that the familiar shape that passed overhead on its 'regular as clockwork' arrival time into Heathrow would be, from this day, missing from our skies \u2013 that we would never again witness its feminine, graceful contours as it accelerated away to its supersonic altitude.\n\nConcorde was to be consigned to the history books. Could we believe that the fastest passenger aircraft ever, able to cross the Atlantic in the same time it takes a train to get from London to Manchester, would now be ignominiously fated to sit in a hangar, never to fly the skies again?\n\nThe world was witnessing the end of an era, and the British public along with all the workers who been a part of its history were mourning its fate. Concorde had been the miracle of the twentieth and twenty-first centuries.\n\nThe first Concorde (G-BOAE) landed at 4.01 p.m. from Edinburgh. This flight had left London in the morning. Many of its passengers for the return flight were British Airways (BA) staff who had been lucky enough to win standby tickets in a prize draw. They had flown from London to Edinburgh subsonic. In the Executive Lounge at Edinburgh Airport they had enjoyed a champagne style reception as they watched Concorde land to the sound of bagpipes and cheering crowds. Amongst the passengers was a group of British Airways cabin crew. They felt very sorry for the working crew who, as one said, 'were very emotional. The crew hold Concorde very dear to their hearts.' As they took off it wasn't only the crew who were feeling the weight of the moment. For the passengers on board, the emotionally charged atmosphere must have been an ironic mix of a party and a funeral wake.\n\nThe second Concorde (G-BOAF), having completed a supersonic flight around the Bay of Biscay, followed at 4.03 p.m. Three minutes later, the final Concorde (G-BOAG), flight BA 002, landed from New York. This Concorde had departed from London the evening before. Its passengers, who were avid enthusiasts, had paid \u00a39,000 for a ticket.\n\nConcorde had experienced a wonderful send-off in New York, its departure commemorated by a water canon spray of blue, red and white to evoke the colours of Great Britain, France and America. As flight BA 002 made its final approach into Heathrow with all the grace and finesse of the great ballerina Darcy Bussell (who was herself on that flight), the event was, without doubt, a tear-jerking moment.\n\nFew who witnessed the sight of Concorde's final landing will forget the poignancy as Captain Mike Bannister taxied the aircraft and dipped Concorde's nose in salute to the crowds, waving the Union Jack from out of the cockpit window. The air traffic controllers responded with the message, 'The eagles have landed \u2013 welcome home.'\n\nChief Pilot Captain Bannister, who had joined the Concorde fleet in 1977 as its youngest pilot, commented that they had tried to make the day of Concorde's retirement a celebration that both the public and airline could look back on with pride.\n\nFlight 002 was a celebrity packed flight. Luminaries such as the BA chairman, Lord Marshall, Sir David Frost, the Duke of Kent and Sir Andrew Lloyd Webber were on board. Sir Terence Conran, who two years earlier had redesigned the Concorde interior, was a guest along with Julien Macdonald, designer of the new BA uniforms. Actress Joan Collins and her husband, Percy Gibson, Jeremy Clarkson, politician Tony Benn and Bernie Ecclestone were all on the VIP passenger list.\n\nPurser Julia van den Bosch is the longest serving Concorde stewardess. She joined it in 1976 and was on this last flight from New York. She shares her memories of the last flight:\n\nWe were all very aware that we were making history. When we arrived in New York the day before our last flight home, we sat down in the hotel and did a lot of signing of memorabilia and autographs and gave interviews. I had also given a lot of interviews in London before Concorde's departure as the interest was global \u2013 journalists had arrived from all over the world. Before I left London, friends and neighbours were coming to my door asking me if I would take something belonging to them on the flight with me just so they could say it had been on Concorde's last flight. They felt that strongly about it.\n\nIn New York we signed first day covers, flight certificates and gave each other mementos. We had the British Airways wine and food supervisors come to tell us exactly what we would be serving for the flight home and we went through the meal schedule. The evening concluded with a wonderful last night dinner at La Cirque.\n\nThe really strange thing was that British Airways would not allow the occasion to be written up as 'sad' and kept insisting that the last flight was a 'celebration' of Concorde, and we were all briefed accordingly. Of course the press spent the whole time trying to catch photos of us crying to prove the airline wrong!\n\nNone of us could feel that it was a celebration. It was an incredibly sad occasion, but equally we wanted to do our best. In one interview with the BBC I said that I wanted to say thank you to Concorde, because it had been quite some journey and I had done such wonderful things in my time with her. Likewise with the passengers, I think they all were very aware of the historical significance of the flight \u2013 especially as we had people on board who had been hugely connected with the aircraft right from the beginning.\n\nThe flight itself was chaotic \u2013 our cabin had all the press, two of whom, Mary Nightingale and Jeremy Bowen, were broadcasting live, and those who weren't broadcasting live were trying to interview the celebs, so it was 'Oh I'll eat later', or 'leave it over there would you and I'll get round to it' \u2013 so I said to the crew, 'We have to say no. There is no later, it's now or never! And on this flight, we really mean it's now or never!'\n\nFor all the crews who had worked on Concorde, the engineers, ground crew, pilots and cabin crew, it was the break-up of a close-knit family. The Concorde crews and staff were an elite group who never lost sight of the privilege of working on such a special aircraft. To fly Concorde was an unrivalled aviation experience.\n\nTo work on it was to be a part of a special team and at the forefront of the British Airways flagship. For the flight crew it meant keeping to the tight arrival and departure schedules. For the engineers it meant keeping Concorde up to its mechanical best and for the ground crew, to ensure the passengers check-in ran like clockwork. The cabin crew had to be at their most professional, with a service that was second to none, and the five-course meal served during the constrained flight time had to be as near perfect as possible, given the size of the supersonic restaurant and its 100 guests.\n\nIt must have been a supremely difficult decision for British Airways management to pull the plug on their flagship aircraft and many believe it was indeed a step backwards for mankind. Who can honestly say they never stopped to stand and stare as it flew overhead? Concorde never became mundane. Its familiarity in the skies never bred indifference. If it was still flying today I'm sure the same would be true. As it cruised over London just after 1800 hours, at the start of its Atlantic journey, and then the return flight from New York coming over just before 2200 hours, its unique engine noise never let one forget that overhead was an exotic and glamorous icon.\n\nAnother great British icon was Frankel, the nation's favourite racehorse. He had bowed out of his spectacular racing career and experienced a similar exit on his retirement in 2012. Thousands of fans and ardent racegoers turned out to watch him at Ascot in his last ever race. It made headlines across the world. It was a sad day to see such a fine horse being retired to a stud farm. Frankel was a true thoroughbred and unbeaten in his race history. In the case of Concorde, however, there was no future generation to carry on the marvels of supersonic flight.\n\nWith her exceptional flying ability, Concorde was often described as a thoroughbred by the men and women who flew her. Jock Lowe, the longest serving pilot on the Concorde fleet and former president of the Royal Aeronautical Society, elucidates on this. He describes her as being more advanced than the Apollo 11 that put the first men on the moon and, in terms of manoeuvrability and power, she was on a par with or even better than any military aircraft of the time.\n\nAs a member of the Concorde cabin crew and flying supersonic at 60,000ft, it was for me another day of 'going to the office', and in this case going to the office was always a consummate adventure.\n\n## [CONCORDE BEGINNINGS  \nAND DESIGN ISSUES](contents.htm#a6)\n\nThe development of Concorde is a wonderfully intriguing story in itself. It was as early as 1956 that a British project was started with the view to developing a supersonic passenger aircraft, and two years later it had been agreed by the Supersonic Transport Advisory Committee that it was indeed a feasible project.\n\nIn 1962 the British and French governments signed an agreement for joint design, development and manufacture of a supersonic airliner. A workforce of over 200,000 were engaged on the project, which incidentally was on the same scale as the American space programme to the moon. Ten years later, British Overseas Airways Corporation (BOAC) ordered five Concorde into production and Air France ordered four.\n\nTo get her to the stage of commercial operation there was, naturally, a huge amount of costly development and many obstacles to overcome and there is no question that flying ordinary men at an altitude of 60,000ft without space suits was one of the greatest technological achievements of the age. The intricacies of design and development were such amazing achievements for the time that it could be described as the new Elizabethan age in aircraft development. They are deserving of a book in themselves, of which there have been many superb ones written by the pilots.\n\nHowever, some of the design development cannot be passed by without a mention. The simple, yet elegant shape of Concorde with the swept-back delta wing was unique to a passenger aircraft. Although efficient in supersonic flight, it was not brilliant at giving the aircraft the lift needed to propel it into the skies. Simple paper aeroplanes were used to test the wing shapes. This simple form of testing was revealed years later when papier m\u00e2ch\u00e9 prototypes were uncovered in a dusty warehouse. These simple experiments graduated to wood then metal versions.\n\nThe Concorde wing has the appearance of total simplicity, but this was one area on the development side that took more attention than any other \u2013 5,000 hours of testing was conducted in the wind tunnel. Looking at the delta-shaped wing head-on to the aircraft, the wing can be seen to twist and droop. As aircraft speeds have increased over time, the amount of 'sweepback' that can be seen on the wings has also increased. On traditional aircraft, the vortex (swirling air needed to give the aircraft lift) is formed on the wing tip. These can be seen on a damp day as an aircraft comes into land or take off. Trails of vapour are seen on the wing tips.\n\nHowever, on Concorde's delta wing, because of the higher angle of attack on take-off (most subsonic jets leave the runway at an angle of 3\u20134 degrees and Concorde's angle of lift was 10\u201311 degrees), the vortices are formed along the entire wing tip. Thus, the end result of all the development was that the supersonic aircraft could fly at a complete range of speed, from the take-off speed, to fly gracefully at Mach 2 and then slowing down on her approach for landing.\n\nAnother massive challenge facing the designers of this innovative passenger aircraft was the issue over the air intakes. No jet engine can accept air into its compressors at supersonic speed. Necessarily the air had to be slowed by 1,000mph, from Mach 2 to Mach 0.5, in order to enter the engines. The British engineers achieved a feat that had never been done before with computers that technically were not in common use at the time. These computers controlled the flow of air into the engine via hydraulically powered ramps that moved up and down to control the airflow and reduce its speed.\n\nA view inside Concorde's cockpit. (Courtesy of Louise Brown)\n\nTo power her 180 tons into the air were four Rolls-Royce Olympus engines with massive thrust \u2013 38,000lb, or in layman's speak, the equivalent of 3,000 family saloon cars. The power was accelerated by reheats or afterburners, a device used by fighter jets. Seen as a spectacular red glow of flame on take-off they also came with a bucketful of decibels. Take-off, from inside Concorde and watching from the ground, was nothing if not spectacular.\n\nThe take-off speed for most passenger jets varies between 160 and 180mph (260 and 290km\/h). Concorde's was much faster at 225mph (360km\/h) and from standstill to full take-off speed took 30 seconds. And that was 30 seconds of thrill seeking for those on board. Not only because of the speed but also the angle of climb.\n\nAnother design feature that had to be addressed was the visibility issue in the cockpit during take-off and landing. This led to the drooping nose design. By lowering it 5 degrees during taxi and take-off, and 12 degrees during landing, the pilot could have a clear view of the runway. After landing this was raised to prevent damage to the nose. After take-off the nose, along with a moving visor, was raised to improve its aerodynamic performance.\n\nConcorde had carbon fibre brakes which back in the 1960s was a technological marvel, along with the fly-by-wire system which belonged to a new age of flying. This system replaced the heavy manual flight controls with an electronic interface so that commands from computers stabilised the aircraft.\n\nOnce the cruising speed of 1,350mph was reached, there was an issue with the outside temperature of the fuselage. Concorde's speed created friction that, in turn, created heat. The nose at this stage was hot: 127\u00b0C (261\u00b0F) to be precise. The rear of the aircraft was slightly cooler at 100\u00b0C (212\u00b0F). This was another developmental problem that set it apart from the subsonic commercial aircraft. A special aluminium alloy had to be developed to withstand these temperatures. Also, the visor was designed to be used as a heat shield and to deflect the heat from the cockpit. Because of this heating issue with the outside skin it also meant that the British Airways logo had to be minimal and the majority of the skin painted in highly reflective white paint to deflect the heat.\n\nConcorde grew 8\u201310in while in flight and the flight engineer's cap that was placed there during the last flight is now a permanent fixture. (Courtesy of Mike Bannister)\n\nThis friction and heat also caused the aircraft to expand during supersonic cruise. Concorde could grow in length by up to 300mm (1ft). This expansion was evident in the flight deck next to the engineer's console and the bulkhead adjoining it. During flight the gap made a good perch for the engineer's cap. But, of course, he had to remember to remove it before the aircraft reduced height and speed, otherwise it would have been its compulsory resting place until the aircraft flew supersonic again!\n\n## [NOW, WHAT SHALL WE  \nCALL IT? NAMING ISSUES](contents.htm#a7)\n\nFinally, there was an issue with the naming of this technological marvel. This, you would have thought, would have been the easiest part of the whole story. Compare it to building a house. After years and years of planning, design and building, then comes the easy bit \u2013 giving it a name. The naming of the house is the final piece of tinsel on the Christmas tree. Deciding on a name for Concorde after years of hardcore development should have been easy, but when it came down to a French\u2013English co-production sadly this was not so. It created tension between the French and British governments at the time.\n\nThe word 'concorde' comes from the French word meaning harmony\/agreement\/union. In English we have the same word but without the 'e'. As it turned out, none of those meanings applied to the final naming of the aircraft. There was disharmony, there was disagreement, there was division and bickering that only politicians could create when it came to Anglo-French relations.\n\nInitially, it was agreed Concorde was to be spelt with an 'e'. Then, Prime Minister Harold Macmillan decided to drop the 'e'. However, in 1967 when Concorde was finally presented to the world in Toulouse, Tony Benn, Minister for Technology, suddenly announced that Concorde would have the 'e', thereby avoiding another fracas with the French.\n\nOne would have thought that would have been the end of the 'e' story, but no \u2013 apparently there was a nationalist uproar, which only died down when Benn calmed the protesting masses by saying that the 'e' on the end of Concorde stood for 'excellence', 'England', 'Europe' and 'entente (cordiale)'!\n\nAs if this wasn't enough, some Scotsman with a bee in his bonnet wrote to Mr Benn and informed him that part of the aircraft had been made in Scotland (the nose cone to be exact). Another civil war was averted when Mr Benn explained that 'e' (well-used by now) also stood for 'Ecosse' (the French word for Scotland).\n\nAnd the final 'e' story. In ecclesiastical terms, there is the definitive statement that even God loved Concorde. The prayer goes, 'O God who art the author of peace and lover of concord' (and not an 'e' in sight!).\n\nAs Tony Benn pointed out, Concorde came into being at a very bad time for the economy when oil prices were high and there were moves afoot to cancel Concorde production. However, with 200,000 jobs now at stake in its construction, it would have been an unwise manoeuvre to follow the Treasury's policy to scrap the project.\n\nTony Benn recalls one of the Concorde test flights where he sat behind the pilot wearing a parachute. Having been an RAF pilot he was quite happy to do this, although he never quite understood what use it would have been.\n\nAfter the French President George Pompidou had agreed to keep Concorde, Tony Benn invited all the employees on a flight around the Bay of Biscay in the summer of 1974:\n\nMany of them had been working on aircraft for twenty to thirty years but many of them had never flown, and so a few were quite nervous. We went all over the Bay of Biscay, and we'd all brought sandwiches and Thermos flasks. It was like a charabanc trip to the seaside.\n\n(By kind permission of the Guardian newspaper,  \n17 October 2003)\n\n## [THE STAGE IS SET AND  \nA NEW ICON IS BORN](contents.htm#a8)\n\nWhen Concorde was revealed to the world in Toulouse in 1967, Brian Calvert, a very influential figure in the development of Concorde and one of a small group of BOAC captains involved in the task of preparing Concorde for service, describes vividly the day he saw the prototype for the first time. In his book Flying Concorde he says:\n\nIt was a bitterly cold December day. A tiered stand with a thousand guests faced the hangar whose closed doors hid the first prototype. The hangar doors rolled back accompanied by the usual warning whistles... and there she was, sideways on, long sharp and sleek. The shuffling and foot stamping died out, and a long silence followed as the crowd took it all in. As Concorde was towed out slowly in a wide arc we saw for the first time how the extraordinary shape seemed to change as different angles were presented. There had been photographs of course but this, the first sight of a real Concorde, stopped the breath.\n\n(By kind permission of Airlife Publishing, an imprint of The Crowood Press Ltd)\n\nTwo years later, Concorde's chief test pilot, Brian Trubshaw, took Concorde 002 on its maiden flight from Filton. Concorde prototype 001 flew supersonic for the first time on 1 October 1969 for nine minutes.\n\nBrian Trubshaw and Brian Calvert, along with many others who were involved with the early proving flights, were indeed brave men. They were testing the unknown. Only military jets and a select group of young, fit, military pilots in pressure suits had done anything similar. Never before had men flown so close to space in an aircraft designed for commercial use. (It is interesting to note that during the test flights the pilots wore parachutes and an escape hatch had been built in the floor of the prototype Concorde in case of a potential disaster.)\n\nIt was not until 1972, five years after its initial unveiling in Toulouse, that the first orders for Concorde were taken. At the time it appeared that the whole of the aviation industry wanted to buy into supersonic flight. Options for orders were placed by many of the American airlines, notably TWA and Pan American. Iran Air, Air India, China, Qantas, Air Canada, Japanese Airlines (JAL), Lufthansa and Sabena all showed interest. A total of sixteen airlines placed options for seventy-four Concordes.\n\nIn a televised speech, President John F. Kennedy had spoken proudly of his dream of America having its own supersonic passenger aircraft, and a project \u2013 the Boeing SST \u2013 was started. The dream, however, was shelved when the true costs of development were revealed. The financial clout of Wall Street preferred money-making schemes to the massive development costs on a project that was still in doubt. They opted instead for the development of the jumbo jet. And so, in 1973, on the day the options were due to run out, both TWA and Pan American pulled the plug on Concorde. The other countries followed suit like a falling pack of cards.\n\nRussia had been keen to join the 'supersonic club' with its own transport. It developed the prototype TU144, dubbed 'Concordski' by the British press, but its development was seriously jeopardised when it crashed spectacularly at the Paris Air Show in 1973.\n\nIt was left to the diminutive island of Great Britain and its neighbour, France, to pursue the global dream despite the escalating costs that were, by 1973, breathtakingly over budget. Had both of these governments decided to abandon the supersonic project, it is a daunting thought that Concorde may never have taken to the skies as a commercial concern and the world would have lost one of the greatest achievements ever.\n\nIn 1962 the total bill was expected to be between \u00a3150 million and \u00a3170 million. In 1974 this was revised to a cost of \u00a3974 million to France and Britain. The three main factors given for this sixfold increase were inflation, currency devaluation and design changes. However, as it was pointed out to the disgruntled taxpayer who effectively owned the aircraft, to have got the sums right would have been a superhuman achievement. The designers and engineers on the project were doing something that had never been done before with a commercial aircraft. They could not have known what frontiers they were pushing in this breakthrough for the aerospace industry.\n\nAnd the spin-off to all this development and cost? The advancement in so many sectors of industry has been hugely influenced, from the use of computers in design and product applications, to new types of machine tools and the technology of paints, glass, plastics and non-ferrous material \u2013 all because of Concorde. Great Britain and France led the world in this new technology. Looking back to those nerve-racking days of its early development, there cannot be many who are not justly proud that this scheme evolved into making Britain a nation proud of its achievements.\n\n## THE AMERICAN RESPONSE\n\nIt was on 21 January 1976 that Concorde BA 300 began its first commercial service from Heathrow. Captain Brian Calvert, Captain Norman Todd and Engineer John Liddiard had the honour of flying this inaugural flight to Bahrain. The planning and preparation had taken many months and the world eagerly anticipated this new dawn of a supersonic era.\n\nAmong the passengers were a mixture of fare payers. Lord and Lady Leathers had taken the precaution of booking twelve years earlier. Invited guests included the Duke of Kent and senior politicians. The world's press were also on board to film and broadcast the flight.\n\nSimultaneously, at the Charles de Gaulle Airport in Paris, an Air France Concorde was also preparing for its inaugural flight to Rio de Janeiro via Dakar. It was another landmark in Concorde's history when the two aircraft took off at precisely the same time, and it was estimated that worldwide 250 million people watched this moment.\n\nHowever, Concorde's position as 'Queen of the Skies' was not unanimously received around the world. In Sydney it had arrived to an unwelcome reception by anti-noise and environmental lobbyists. In America, there were plenty of objectors, including an environmentalist group called 'the Citizens League Against the Sonic Boom'. In the New York Times a full-page advert appeared:\n\nSST (Super Sonic Transport) AIRPLANE OF TOMORROW\n\nBREAKS WINDOWS, CRACKS WALLS, STAMPEDES CATTLE AND WILL HASTEN THE END OF THE AMERICAN WILDERNESS.\n\nNew York fought hard to prevent the arrival of the new noisy aircraft. Some regarded this as a sulky gesture from the USA who led the aviation industry but had not achieved their own supersonic transport. On 2 December 1970 the US Senate banned Concorde on environmental grounds.\n\nSlowly, acceptance of Concorde was won in spite of its American antagonists. Its first appearance in the United States was to Dallas\/Fort Worth Airport in September 1973 to celebrate the dedication of the new airport. It was greeted enthusiastically. By the end of 1974 Concorde had made appearances at Boston, Miami, Los Angeles, San Francisco and Anchorage, and its arrival proved to be an overall success. In Los Angeles, an estimated 250,000 turned out to see it.\n\nSpecial permission by the US Secretary of Transportation was finally given for a regular service into Washington's Dulles International Airport and on 24 May 1976 both Air France and BOAC commenced their services. Before landing, both aircraft simultaneously flew over the US capital and then made parallel approaches to Dulles Airport. They touched down together and landed to a rapturous welcome. The two Concordes parked nose to nose in front of the control tower. This became a regular route and between 1984 and 1991 the route was extended to Miami via Washington three times a week.\n\nHowever, Concorde's viability depended on the New York to London route. In October 1977 two Concordes (one Air France) flew to New York for a test flight to monitor the engine noise on take-off. Captain Walpole and his crew were met by a hostile press reception but he and his team had worked long and hard to find a way to reduce the noise levels. As they took off with a 25\u00b0 angle of bank and a throttling back of the engines, the noise monitors hardly registered. Concorde's future was saved. Its route to New York was passed.\n\nIn less than a month its first commercial service landed at New York's John F. Kennedy Airport. However, the objectors persisted in making their views known and the crew were a target. As they left the terminal heading for the city, the protesters would bang on their coach and were vociferous in their anti-Concorde remarks. It must have felt quite ironic to the crews who were hailed as pioneers of twentieth-century travel and lauded in their own and other countries, yet were made to feel like criminals by a minority in a country that was so advanced compared to the rest of the world.\n\nBut enough of the technical metamorphosis of this aircraft. It's time to switch to the more human side of the story and how the author came to fly on Concorde.\n\n## [THE AIR STEWARDESS:  \nHOW IT ALL BEGAN](contents.htm#a10)\n\nMy first memories of aeroplanes were as a child lying on the grass in my garden at home and watching the occasional jet airliner pass high overhead, its vapour trails striating a blue sky in dramatic white pastels. Like many children with time on their hands and left to their own dreams and imaginations, I would reflect on the unobtainable. In my childhood, flying was not the commonplace means of transport for taking the family to foreign destinations on holiday. Holidays were spent at home or, for the more adventurous, via an interminably long car journey to distant places like Devon or Cornwall.\n\nFlying was what dreams were made of and as I gazed up at this world so very far from my own it left me wondering. How many passengers were inside the tiny metal tube racing across the sky? Where were they going? What was happening inside the aircraft at that moment? And how lucky were those glamorous stewardesses to be up there flying to exotic places! They were handpicked, and the consummate image of glamour and mystique. The perception of a stewardess as a tall, thin and beautiful girl was a reality then and not just in the imagination of a public relations executive. Some air hostesses, as they were then called, came from the cream of society \u2013 ex-debutantes with a spare year on their hands joined an airline to see the world. Flying was a prestigious job and with it came a suitcase full of glamour. It wasn't until Concorde landed on our horizons that this old-fashioned concept was to return.\n\nThe exotic dream of getting airborne, however, was different in reality, with my first passenger flight at the age of 17. It was not filled with the wonderment I had anticipated. Flying for the first time and alone, it was a case of nerves supplanting any idea of romance. But it did enable me to leave behind a rain-soaked English summer and transported me to the south of France to start my first foreign holiday job teaching sailing to French holidaymakers.\n\nSeveral years later, having chosen my vocation as a teacher, I was flat sharing with a BOAC stewardess in Richmond, Surrey. Dutifully, and happily I might add, I would catch the bus to East Sheen Primary School every day and life became a satisfactory routine of teaching small children, amongst other things, the necessary social skills of toilet training, keeping lunch where it was designed to be once eaten (in the stomach and not thrown up in the classroom) and advising them on the uses of a handkerchief \u2013 green runny noses were not an attractive sight to anyone, least of all their squeamish teacher.\n\nReturning home on a dank and dark winter's afternoon, having been enclosed in a germ-breeding capsule for the common cold with thirty-two infants, I was greeted by the most enormous bowl of exotic fruits on the living room coffee table. Perched mountainously high were avocados, pineapples, mangoes and peaches, all freshly flown in from Nairobi market that morning. In a vase was a towering arrangement of orange and crimson Bird of Paradise flowers. A winter's living room had, in an instant, been transformed into a luxurious haven that smelt of a tropical paradise. To bring further exotica into the picture, my very beautiful flat mate, a public relations man's dream, wafted into the room looking exceptionally healthy and with a suntan the colour of burnt toffee. The last straw was seeing her line of pretty bikinis drying above the bathroom radiator. Her 'work' for that week had been a three-day trip to Nairobi staying at the Hilton Hotel, sunbathing by the pool along with shopping trips to the market and eating very healthily. And she got handsomely paid for all this.\n\nAs much as I loved teaching I was easily persuaded that, at the age of 22, the world was my oyster and that there were more exciting ways of earning a living, together with acquiring a midwinter suntan and the opportunity for foreign travel, such as had been the dreams of my childhood.\n\nThe prerequisites for a stewardess were slightly different then to what they are now. In the sixties and seventies, the stewardesses stayed in the job for probably no more than a year to eighteen months. It was a way to see the world, but not a chosen career as such. Many of the privately educated girls used flying as a sort of finishing school. Working as a stewardess was then still in its glamour phase and there was an aura of mystery surrounding the life. None of the girls were allowed to be married, so it was a tough choice \u2013 marry the man or lose the job. There were several girls, however, who would slip the ring off their finger before a flight hoping to get away with their 'crime' of being married. The company obviously deemed that by the age of 35 all stewardesses were past their 'fly by date'. That was the retirement age, or after ten years of service, whichever came first. Emancipation of women still had a long way to go even then...\n\nPrevious careers in teaching or nursing helped enormously when trying to get through the interview stage. A good pair of legs was also a bonus and a slim build and attractive looks were no doubt taken into account. And of course there was the smile. This was the one essential that came with the job and was used to great advantage in times of stress situations on the aircraft and for calming awkward passengers. Also required was a coolness of character in times of the odd emergency and potentially catastrophic situation (an engine fire and no landing gear could not be ruled out).\n\nDespite the tension and nerve-racking wait for the job confirmation letter to drop through the letter box, my fairy godmother came to my assistance with my new career plans. Later that year, after two interviews and six weeks of training for 747s, I too was flying the skies proudly in my BOAC uniform.\n\n## A FLYING START \u2013 THE 747 FLEET\n\nThe 747 fleet had been in operation for four years when I joined the airline and in this year, 1974, BEA (British European Airways or 'Back Every Afternoon') had merged with BOAC (long-haul routes) to form British Airways. The first 747 commercial flight had taken place from New York to London in January 1970, but what was so remarkable was that this innovative and huge engineering feat had taken a mere twenty-eight months to design.\n\nLike most sceptics of the time and before I started flying, I doubted that a plane of that size could actually get airborne. It appeared to defy the laws of gravity as it lumbered down the runway for what seemed like forever. Just as the last landing strip of lights appeared, the nose would lift and slowly the big beast would start to climb. There was always that doubt in my mind that it would not quite make it.\n\nThis was brought home to me quite vividly when, in the early days of my flying career on the 747, we left New York one wild, stormy night with a fifth engine on the wing. This had to be repositioned back to London and obviously the aircraft was designed to carry the extra load. However, we all doubted the idea at the time. We had a full aircraft of 430 passengers and it was a nail-biting moment on take-off. The 747 seemed to take forever to leave the runway ('Are we going by road?' one passenger asked). The captain and his flight crew made no secret of the fact that they had white knuckles during the take-off as the conditions were not ideal to get the heavyweight into the air.\n\nThe ease of working on such a spacious aircraft was wonderful. The wide aisles and galleys allowed the crew plenty of working space. With a cabin crew of thirteen plus three flight crew, the longer trips that could extend to twenty-one days were epic. Trips down through the Middle East, India, Singapore, Hong Kong, Melbourne, Sydney and Auckland were interesting in many respects. Sightseeing was always part of the itinerary if we had the time during a stopover.\n\nInvariably the crew got on well, and more often than not the trips were a non-stop party. The job description of cabin crew then was often recalled as 'flying from party to party in an aluminium tube'. The crew 'post flight' party was virtually compulsory whatever time of night or day we landed. Even if we arrived at some unearthly early hour of the morning it was expected that all crew would meet for drinks in someone's room. It was a tough call at times, when all I wanted was for my head to hit the pillow, but, arriving at the designated room, an alcoholic drink would be put in my hand 'to help the jet lag', and a casual 'debrief' of the flight followed. Plans for the remaining stopover time were also made.\n\nSome of these parties would warrant several chapters on their own, especially if the crew got on well and were 'party animals'. Affairs between the crew happened whichever location we happened to be in. Beautiful places like Mauritius and the Caribbean naturally lent themselves more for these to develop, with night beach parties making it hard sometimes to resist 'the romance of the moment'.\n\nHowever, most crew, particularly the married ones, were very philosophical about these arrangements and the 'affairs' generally finished once the crew landed back at Heathrow. In those days the reporting office where we checked out after a flight was named 221. Aptly, there were many '221 divorces'. It was at times very difficult to adjust to coming home to the realities of life after an intense few weeks away, working hard and partying hard and in great company every day and night. There are few jobs where the workplace is so varied or the work colleagues spend so much time together. We were spending most of our time working and relaxing together and great friendships were formed.\n\nWe were entertained royally by generous hosts in the countries we visited. There were the infamous sheik's parties in Bahrain hosted by the ruler himself at his beach house, there were High Commission lunches in Delhi, party invitations to the Hong Kong Yacht Club, invitations to go to cocktail parties on board cruise liners in Bermuda, opportunities to go sailing in Sydney, Barbados and many other exotic locations. It was a privileged world.\n\nI understand life with the airline is different now. The crews lead a far healthier lifestyle, drinking less and not socialising as much. The crew allowances, which for us were more than generous, meant that we wined and dined in style whilst away. With the downturn in the airline economy, the basic pay for cabin crew has been much reduced and so, wisely enough, the crews save their allowances and probably have much healthier livers for it!\n\nAfter forty-four years the 747s have not yet been retired and are still the reliable 'workhorse' for many airlines. At any one time there are 100,000 people flying in 747s somewhere in the skies.\n\n## SUBSONIC TO SUPERSONIC\n\nAfter seven years of flying the world on 747s and the body clock getting more confused by the day with jet lag, I decided it was time for a change. Flying to different parts of the globe was fine, but when the crew schedule takes you westwards to Anchorage and Osaka, Japan, and then east via India to Hong Kong, followed by yet another swing of the body clock pendulum to Los Angeles, there comes a time when the sleep routine becomes totally disrupted. The time changes were enormous.\n\nAdded to that and not helping the jet lag was my 'other' life of flying to the Grand Prix circuits in various parts of the globe with a Formula One team. I had also suffered health problems having succumbed to two bouts of salmonella in two years. The immune system had worn down somewhat and my stamina was not what it used to be. All these factors were pushing me to a more routine lifestyle \u2013 if 'flying' can in any way be mentioned in the same breath as 'routine'. My next month's schedule was to include a trip to Bombay, interesting in its myriad of cultures and lifestyles, Johannesburg, a city living on its nerves, and finally Anchorage and Osaka.\n\nAlaska and Japan were both favourite destinations for the crew despite the huge time change. Anchorage, one of the most northern cities on the planet, was in those days little more than a one-horse town where reindeer roamed the garden suburbs, and crew trips to the spectacular mountain glaciers were always a bonus. The country around was wild and beautiful and earthquake tremors were commonplace. Biplanes taking off and landing on the great stretches of water around Anchorage were common forms of transport for the locals.\n\nThe next sector to Osaka was, again, a very extraordinary culture and it never failed to fascinate. Eating out at night was a case of choosing our dishes from the plastic replicas in the windows of the restaurants. Shopping was a new experience, too. As the stores opened their doors at exactly the appointed hour, the staff would line up inside the foyer entrance to greet their customers. A Japanese member of staff stood at the bottom of every escalator with a cloth in her gloved hand holding the handrail to ensure its cleanliness. On other occasions we flew into Tokyo, and whatever else was on the agenda, no visit was complete without a ride on the bullet train. One memorable trip was to the city of Kyoto where a walk through the Imperial Palace gardens was both moving and restorative to the weariness of the previous long flight. However, my body clock was telling me it was time for a change. And that is exactly what happened...\n\nHaving spotted a management notice in the corridors of Crew Reporting that they were looking for Concorde crew, I wasted no time in arranging a meeting with my cabin crew manager to see if I passed the grades to join the 'Elite Fleet'. Although I would miss the glamorous locations that the 747 flew to, I was not going to miss the brain-numbing time changes.\n\nAfter an interview in which the annual reports on my standard of work had been scrutinised and a file of complimentary letters read through, I was accepted. There followed a week's Concorde familiarisation in the training centre near Heathrow Airport and our class of thirteen 'graduated' and were ready to fly the 'pocket rocket'. The instruction for this new work environment had included learning the fine wines and food served during the flight and, more importantly, the strict routines of cabin service. Even though I was used to working in first class on the 747 there were still big changes ahead \u2013 mainly the passenger number being up to 100 instead of twenty-four. There were also the safety procedures to learn and exams to pass.\n\nThe last day of Concorde training group in 1981, receiving our Concorde wings.\n\nAs with every aircraft, one could not take to the skies without precise knowledge of the stowage of safety equipment and the evacuation drills. Concorde equipment was, as you would imagine, the very latest in technology. One of the situations that we had to be trained for was the evacuation of the aircraft on water. The theory was that if the aircraft had to land on water, the passengers would be directed out of the aircraft doors with their lifejackets already on but not inflated. (Lifejackets would be inflated once on the wing.) The plus side to all this was that the wings were so large that a full complement of 100 passengers could assemble on them and wait their turn to get into the life rafts that the crew would have dutifully inflated.\n\nSo in theory, we as crew would be faced with traumatised passengers tottering out onto the wings all dressed in the BA fashion statement of yellow lifejackets, distraught at having to leave behind their Manolo Blahnik shoes, laptops and Gucci handbags on an aircraft that they were now abandoning for a wet rubber inflatable and a ration of Mars Bars. It didn't bear thinking about!\n\nThe downside to this very hypothetical dire situation was that if Concorde had ever to land in rough seas in an emergency, it would have more than likely gone straight down to the bottom of the ocean. When I asked the flight crew about Concorde's abilities to land on water they were very cautious about the idea. They did admit that although the beautiful bird was built for speed at high altitude, its aquatic abilities were extremely limited and they would not have wanted to put them to the test. Whether Concorde could have remained afloat for very long, we shall thankfully never know.\n\nWhen Concorde moved to short-haul operations in 1981, for reasons I will explain later, we also had to train to fly on the BAC 1-11, an aircraft that has now retired to sit in the Museum of the History of Flight (there is also one at Brooklands Museum in Surrey). I will always remember the day aboard a mock-up of this plane in the Safety Training Centre, when we were being given instruction on how to deploy the slides. I'm not sure what my face was registering \u2013 incredulity that such a bizarre method would pass aviation regulations or the giggle that I was suppressing at the thought of having to evacuate, slowly, a full load of passengers. The words in my head, to quote John McEnroe, were, 'You cannot be serious'.\n\nI hope I will be forgiven by any devotees of the 1-11 who will probably be very upset that I could besmirch this aircraft. They will probably quote to me on the number of times its safety record passed with flying colours and that the procedures worked perfectly. The drill, however, was quite archaic after the Concorde and 747s. Speedy it was not. There were stairs for evacuation at the rear of the aircraft, but if the engines were on fire then you would have to exit at the front of the aircraft, which is where the slides were located in a hatch above the two exit doors. First, these had to be opened and lowered. Second, two very tall, very strong crew members (I was not in this category) had to lower a heavy canvas package onto the floor and uncover the slide. Third, two able-bodied volunteers (passengers, not crew \u2013 would they be entitled to a refund?) had to then scale down the side of the aircraft (rope provided) and hold out the slide for the passengers to flee the fire or whatever hazard had forced us to abandon the plane. This means of escape was never going to happen quickly and there was something quite genteel about the whole set-up. One can imagine the crew saying, 'One at a time please, and, sir\/madam, we will be serving you tea on the tarmac shortly.'\n\n## THE SUPERSONIC EXPERIENCE \u2013 'FLYING HOTEL' TO 'POCKET ROCKET'\n\nConcorde was an entirely new working environment. The familiar spacious cabins of the 747s were replaced with a narrow, low-ceilinged interior (the cabin height being 6ft 5in (1.96m)) and grey leather seats that exuded exclusivity. This was more private jet than flying machine for the masses. With it came a new reverence for the job. Despite my seven years' flying experience I still remember being extremely nervous on my first Concorde flight.\n\nUp until 1981 the Concorde crews led a pretty easy life. They worked, on average, a 7-hour week. The pay was comparatively low due to fewer overseas allowances, but the glamour stakes were high. When Concorde first started its commercial routes in 1976, the crew were regarded as pioneers. Julia van den Bosch, one of the original stewardesses who stayed on it until its retirement in 2003, recalls a publicity reception at the Dorchester Hotel in London. One of the guests, the Duchess of Argyll, came up to Julia and shook her by the hand. 'So brave,' she commented. 'You are so brave to fly on an aircraft to the edge of space.'\n\nThe British Airways long-haul flight rules (which included any flights outside of Europe) made provision that the cabin crews were entitled to four days off after a flight back to their London base. In the early days, Concorde came under this umbrella. This, of course, provided an envious lifestyle and one Cockney steward on short haul was known to refer to the Concorde crew as 'those from 'arrods whilst the rest of us were from Woolies'.\n\nIn the seventies and early eighties, the schedules to Washington included a five-day stopover. There were idyllic days in the summer of long lunches, picnics and rowing on the Potomac River. To while away the time the crews also got into a routine of making up afternoon tea for the incoming crew.\n\nIn New York there was a similar relaxed lifestyle. A leisurely crew breakfast in one of the local coffee shops was followed by shopping at Bloomingdales, Macy's and Korvettes, a visit to the Guggenheim and other galleries, long drawn out lunches at Tavern on the Green in Central Park and the lure of theatre, shows and opera in the evening. The Rainbow Room, Studio 54 and other famous hotspots were also on the crew agenda. In winter, the stewardesses' luggage contained exotic fur coats for the luncheon parties that followed their arrival into New York. On the fourth day they would fly home in 3 hours and 30 minutes, or less, to spend another four days unpacking their shopping and enjoying a pretty civilised home life.\n\nThe first stopover of a Concorde route to south-east Asia was the inaugural flight between London Heathrow (LHR) and Bahrain (BAH) on 21 January 1976. The route was mainly overland, forcing the Concorde to fly at subsonic speeds on large portions of the London\u2013Bahrain sector. But the aircraft still managed to save around 2 hours and 30 minutes over the regular flights with a Mach 0.95 cruise speed. In December 1977, three flights a week to Singapore commenced but were stopped after just three return flights following complaints from the Malaysian Government regarding the noise over the Straits of Malacca.\n\nThe Bahrain to Singapore trip for flight and cabin crew was nine days and run jointly with Singapore Airlines. The flight crew and operational crew were British Airways whilst the cabin crew was 50 per cent Singaporean girls who, incidentally, never failed to win over the flight crew and no doubt added extra exotica to their flying schedule. The aircraft used for this route (G-BOAD) was painted with the Singapore Airlines livery on one side and British Airways on the other.\n\nIn 1979 these services were resumed, this time avoiding Malaysia, but sadly the plug was pulled on this service in November 1980 as the revenue figures showed the route had low passenger numbers and was losing \u00a32 million a year. It was rumoured that after the showing of the controversial film Death of a Princess on British TV, the Saudi Government declined to have Concorde and this service was also halted.\n\nThe flights to Dallas\/Fort Worth in 1979 became a loss-making route and were cancelled in 1980. Concorde was now left with destinations to New York, Washington, Miami and various charters. Air France discontinued their service to Washington in 1982; from then on New York became its only scheduled destination.\n\nBy 1981 the future of Concorde looked bleak. The government had lost money on Concorde operations annually and it was mooted that it would cease to fly. The government gave the airline two years to turn the loss around. The chairman of British Airways, Sir John King, remained convinced that Concorde was a premier product and underpriced. Market research revealed that many high-flying businessmen did not actually buy their own tickets, this task was delegated to their PA or secretary. Therefore, many did not know the cost and, when asked, thought the fare was much higher than it actually was. Ticket prices were then raised to match these perceptions. Sir John persuaded the government to sell Concorde outright to his airline for a mere \u00a316.5 million plus the first year's profits. In two years Concorde changed its fortunes \u2013 it started making big profits.\n\nTiming, as they say, is everything in life and it soon became apparent that the exclusive Concorde crew lifestyle was neither viable for the company nor realistically feasible to run conjointly with the other hard-working fleets of BA. When I joined the Concorde fleet in 1981 it had been operating for five years and a cost-saving exercise was about to be implemented. British Airways crew management, in their wisdom, decided to change the flying regulations and bring Concorde working hours in line with the rest of the BA fleets (an 'arrod's\u2013Woolies merger, perhaps?).\n\nI had six weeks of the 'old style life' before the merger. From then on Concorde was switched to the short-haul fleet under the auspices of shorter flying hours compared with those of the long-haul flights. With the 3-hour 30-minute flying time across the Atlantic, the cabin crews were now able to work the next day, forgoing their four days off previously guaranteed under long-haul rules. They could now fly six consecutive days, and this could include another Concorde flight across the Atlantic, or a BAC 1-11 or 737 flights into Europe. This now became a matter of flying hours and not distance and the six-day short-haul roster came into force, allowing three days off in the UK after this work period. And we were effectively trained to fly on three aircraft.\n\nIn the early days of the changeover, I was one of the few left who had flown on the aircraft whilst under the long-haul agreement. Many of the original Concorde stewardesses, having rightly guessed that this was the end of a very comfortable era, had left under the new rules. Their elitist lifestyle had become part of Concorde history, and so a new intake of short-haul crew was trained for the Concorde operation. The more experienced of us were called upon by the office hierarchy to initiate the new recruits and I found myself flying across the Atlantic on six consecutive days. Six crossings in six days was a tall order and I was completely exhausted at the end of my 18,000-mile commuting week. Fortunately this situation did not last for long.\n\nJulia van den Bosch, Concorde's longest serving stewardess, sitting in the engine housing. (Courtesy of Julia van den Bosch)\n\nLouise Brown sitting with a colleague in the engine housing. (Courtesy of Louise Brown)\n\nThe stress and workload for the new crews and experienced hands that ensued in those initial weeks were not ideal. The short-haul crew, in fairness, had not been trained to do a first-class service before their Concorde training so their learning curves were steeper than Concorde's climb rate after take-off with an empty load. They very quickly learnt that speed was of the essence and landing with meal trays still in their hands was not the best way to arrive at their destination.\n\nDespite our reputation for superlative professionalism there was inevitably the odd calamity, and not only during the change over from long-haul crew to short-haul crew. We can all recall the landings when the reverse thrust of the engines brought the aircraft to a shuddering halt on the runway. During this procedure on a number of occasions one of the trolleys would jump out of its restraining straps and trundle off on its own journey down the aisle gaining momentum with the speed of the aircraft's braking. One trolley, on such an occasion, even managed to reach the flight deck door. Strapped in as we were, the crew could do little to halt is wayward path. I often wondered what the passengers made of this.\n\n## [THE WORKING DAY:  \nBEHIND THE SCENES](contents.htm#a14)\n\nA cabin crew day started with checking in at the British Airways Crew Reporting Office based on the perimeter road at Heathrow. In the seventies and eighties this was a utilitarian building that resembled a cross between an enlarged army Nissen hut and a small industrial unit. British Airways finally upped their image in the nineties by replacing it with a huge, super smart, state-of-the-art building with blue glass windows called the Compass Centre. The flight crews before this had their own check-in office at Terminal 1.\n\nLiving in Kew, my trip to the airport was a short drive on the M4. However, there were crew members who travelled far greater distances to operate their flights. Some commuted from Manchester, Jersey, Scotland and even France.\n\nGetting to the airport was not without its tense moments. If you were just a few minutes late without informing the check-in staff (and there were no mobile phones then) you were automatically stood down from the flight and replaced by a standby crew member at Crew Reporting. This had its advantages on some of the long, unpopular long-haul night flights, but not on Concorde.\n\nHaving checked in at the reporting desk and collected the internal mail that included future flying schedules or rosters, the crew assembled in one of the allocated briefing rooms for a 20-minute briefing by the cabin service director (or CSD as we called him). He would cast a beady eye around to make sure everyone was properly attired and smartly dressed. The Concorde uniforms were the same as the other fleets but the hats, which sometimes rested in the hat rack for embarkation of passengers on other flights, were obligatory on Concorde, as were the white gloves. Personal appearance was absolutely of key importance and special care was taken with this. No scrappy hairdos or overdue haircuts were permitted.\n\nIn general, the more 'easy-going' the cabin service director was the more we relaxed and committed to enjoying our flight. Those who 'nit-picked' and took the job too seriously were often an adverse influence on the crew. If we were relaxed it would reflect on our initial meeting with the passengers. And, being the consummate professionals that we were, the relaxed attitude in no way reflected on our prompt, efficient service.\n\nWe would be assigned or allowed to choose our working positions in the cabin. These would be either in the front or rear cabin or the galleys, one at the front and one at the rear. Although the CSD was ultimately responsible for the whole of the aircraft and its passengers, he worked in the front cabin and the purser was in charge of the rear backed up by a team of two other crew in each cabin.\n\nDuring the briefing the CSD would give us the flight time, the name of the captain, first officer and engineer and the type of meal to be served. There were invariably special requests for different dietary foods and these were pointed out.\n\nWorking in the galley meant responsibility for the timings and cooking of the meals but did not require the smiling, friendly persona presented to the passengers for the whole flight. We all had an off day now and again where we would prefer to do galley duties, unless of course the movie or rock star of our dreams was on board!\n\nI have been asked on many occasions if there were two classes on the aircraft. The front cabin seated forty passengers and the rear sixty, but both were exactly the same in standard and type of service. There was, however, a snob value to sitting in seat 1A or 1D at the very front, which the regular clientele competed for. The royals always sat there as it was a quieter seat and provided the most privacy.\n\nDuring the crew briefing famous names, politicians, top executives and regular clientele who were travelling with us were also brought to our attention. It always added an extra buzz to the working day if there was a well-known face on board. If it was a movie or rock star we were always curious to meet them in the flesh. Names such as Barry Gibb, Robert de Niro, Tom Cruise or Paul McCartney invariably brought a smile to the female faces at the briefing. One had to compare this to my previous occupation of teaching \u2013 where would one meet such luminaries in the everyday workplace? Could one go home and say 'Well guess who came into my \"office\" this morning?' Henry Kissinger, Michael Jackson or Elizabeth Taylor?\n\nAfter the briefing, a crew bus would transport us across to Terminal 3 where Concorde was being prepared for its flight. A bevy of busy ground engineers, catering, cargo and fuel trucks were all part of the activity that surrounded the 'tethered bird' waiting patiently for its release into supersonic flight.\n\nOnce aboard we introduced ourselves to the three flight crew, who by this time were well into their preflight checks. In the galleys, hot drinks were prepared for them and the menus were unwrapped, along with newspapers and magazines which were prepared for presentation to the passengers. The drinks trolleys and meals were checked, and plastic covers lifted off the meal trays and canap\u00e9s. The toilets and cabins were carefully scrutinised for tidiness and cleanliness. The CSD would perform a thorough check of safety equipment before the passengers' coats arrived.\n\n## THE FLYING WARDROBE\n\nIn the seventies and eighties when consummate wealth reached its highest amongst the Concorde jet set, their opulent fur coats and designer outdoor wear would be wheeled onto the aircraft on a hanging rail. These would be carefully stowed into a compact wardrobe area. I emphasise 'carefully' because some of the animal pelts were worth a year's salary to some of us lesser mortals. Whether flying to or leaving New York or Washington, where the temperatures in winter could drop to well below zero, the minks were a pretty common item. There was more often than not a quick appraisal of those we thought the most chic. Sometimes we went a step further and modelled them, much to the amusement of the male crew.\n\nThe men's coats and jackets were equally appealing and made of the softest wool or cashmere. Designer names such as Burberry, Ralph Lauren and Yves St Laurent were all familiar labels. For a first-time crew on Concorde this was a taste of who and what was to come, and the passengers invariably oozed wealth. This was no ordinary air travel. The clientele dressed for the flight in exclusive designer clothes and jewels that would collectively warrant a Securicor van at their destination. Designer handbags and briefcases were of the finest leather or crocodile. Not only were our guests well dressed but the cabin air would be scented with the most exclusive aftershaves and perfumes. If money had a smell, this was it.\n\nAn interesting change was noted by one of my colleagues, who commented that after the champagne-drinking, mink-wearing hedonism of the eighties there followed a period of more austere outdoor wear. The ostentatious minks were replaced by regular coats and instead of the champagne quaffing high-flyers, a more demure breed of Perrier drinkers took their place.\n\nHowever, during my time in the eighties, there was one flight when a female passenger came aboard wearing a three-quarter length mink coat. This was unusual, as most were glad to part company with their coats before boarding the aircraft. When I asked her if she would like me to hang up her coat she politely refused and, to the crews' incredulity, sat through the whole flight wearing it. We could only assume that it was so expensive she didn't want to lose sight of it.\n\nWhen we landed in London, it was not unusual for customs to bring sniffer dogs onto the aircraft to check for drugs. On this occasion the dog grew very excited at the seat at the rear of the aircraft where this particular female passenger had been sitting. The customs were pretty convinced this was a 'drug seat'. We never did find out whether HM Customs & Excise caught up with the passenger in the customs hall to check the exact contents of her coat.\n\nAnother incident regarding fur coats happened whilst I was working on the 747 fleet and before I transferred to Concorde. We were travelling from New York to London and again a lady decided to wear her thick fur coat all the way to London. As I was passing her seat during the meal service I happened to notice her 'feeding' her coat a morsel of steak!\n\nUnsure how to ask her why she was feeding her coat and taken aback at what I had just seen, I relayed what I had seen to the purser in charge of the cabin. Before congratulating myself on my astute powers of observation, I accompanied the purser to the seat. He grew suspicious when he detected a movement from within the coat. 'Are you concealing something in your coat?' he politely asked. The sweet old lady timidly admitted her cover was blown and showed us a miniature dog curled up inside a large inside pocket. This, of course, was strictly illegal. The captain almost blew a gasket upon being told he had prohibited items in the cabin and instructed the dog be removed immediately and isolated from its owner. One can only assume that this incident would involve a lot of paperwork for a tired flight crew arriving into London, hence his irascibility.\n\nFive minutes later this poor pooch was transferred to an upturned wire drinks basket and sat bewildered and shivering in its new 'home' by a drafty aircraft door. As a fervent animal lover I was extremely upset by his poor treatment. Not only had the dog been removed from the warmth and comfort of its owner's fur coat, but it was confused by its noisy environment so, seeking to give it some solace, I put my hand through the cage to stroke it. My kind gesture was rebuffed with a nasty bite to the finger!\n\nLittle did I know that informing the purser of the dog's response would land me in a lot of trouble. Having relayed the incident to the captain, the purser told me to report to the flight deck immediately. I was summarily interrogated by a very cross captain. 'What were you doing going near that dog?' He fumed. 'Why did you touch it?' Innocently I defended my case by saying the animal looked lonely and frightened. Then, seeing this was going to land me in trouble, I mentioned that it was only a little nip. 'Lucky for you then,' he replied tetchily, 'because if it had drawn blood, you, along with the dog, would both be turned over to the quarantine authorities.' How unfair, I thought, as I returned to my work, that a little kindness had landed me in such trouble.\n\n## [FLIGHT 001 IS NOW  \nREADY FOR BOARDING](contents.htm#a16)\n\nConcorde had its own check-in desk at Heathrow and this was where the exclusive experience for the supersonic passenger began. It was quick, it was smart and it was the queue (if indeed there ever was one) that every airline passenger wanted to join. No queuing at McDonald's or Costa Coffee for these fortunates. The Concorde boarding pass opened up the glitz of exclusive airline comfort. The luxury of the departure lounge offered newspapers, telephones, fax machines (before laptop and mobile phone days), television and secretarial services. An abundance of drinks, champagne, canap\u00e9s, snacks, fruit and coffee were also available.\n\nTwenty minutes before the flight's departure, the passengers would be escorted onto Concorde. The crew's first job was to give them a warm welcome and if we had flown with them before we always greeted them personally by name. Concorde had, as David Frost said, 'an exclusive private club atmosphere'. It was small, it was friendly and whenever you could you addressed the passenger by their name.\n\nStorage space was minimal so it was often quite a feat to stow the carry-on luggage in the hat racks. One of the most memorable items I have offered to assist with was the Stradivarius violin belonging to Sir Yehudi Menuhin. Sir Yehudi informed me very politely that he could stow this without any help. He and his wife were the most charming, unpretentious couple, and I still have his signed menu.\n\nOn another flight, Sir Yehudi accepted the offer for the violin to be stowed in the rear wardrobe. Julia van den Bosch, who was the purser at the time, recalls how she took the priceless instrument wrapped in a cloth cover and stowed it very carefully in the rear wardrobe. This was checked several times during the flight and it appeared that no harm could possibly come to it. Towards the end of the flight, however, a passenger sitting in the rear seat in front of the wardrobe was in a hurry to go to the bathroom and, forgetting the tray that was on his lap, leapt up and the glass of wine that was on it became airborne... and flew into the wardrobe. Julia was horrified to discover the cloth covering one of the most expensive string instruments in the world had not escaped the deluge of flying wine.\n\nSummoning up all her reserves, she broke the news to the 'maestro'. Sir Yehudi wasted no time in getting out of his seat and hotfooting it down the aisle to inspect the damage. To his and everyone else's huge relief the violin inside the case was untouched. Perhaps that is why, on the flight on which I met him, he decided the best place for his instrument was in the overhead locker right above his head and away from any possible calamities.\n\nOn a flight to Washington I was once handed three bottles of wine by two Arab passengers and expressly asked to take great care of them. These bottles weren't ordinary duty-free. The Arabs informed me of their cost \u2013 \u00a31,000 apiece, which in today's money equivalent would probably be nearer \u00a33,000. They were duly stowed away with immense care.\n\nOn one occasion, after the passengers had boarded the aircraft, an aircraft dispatcher (or 'red cap', as we called them) came on with a package under his arm wrapped in brown paper. He casually he told me this was the cargo the captain was expecting on the flight deck. This being an unusual procedure, I asked the red cap what was in the brown paper.\n\n'A Cezanne,' he said with a wink.\n\nSure enough, when I went onto the flight deck the captain asked me to remove the brown paper and there it was. The first and last time I ever held a genuine Cezanne.\n\nJulia van den Bosch recalls an Arab charter where a live falcon came on board and was taken up to the flight deck for the duration of the flight. She reminisced, thinking that if the aircraft had gone down, one of the last things that may have been recorded in the black box would have been the panicky squawk of a bird!\n\nOnce settled in their seats, the passengers would be offered vintage champagne and soft drinks, hot towels, menus and newspapers. An order would be taken for drinks and cocktails to be served straight after take-off. Even for these wealthy and widely travelled people, the beginning of the supersonic experience was a moment of excitement. Despite their feigned nonchalance as they tried to read their newspapers, few passengers could disguise their pleasure and anticipation as the seat belts were buckled up, followed by the safety demonstration and the preflight welcome, giving the flying speed of 1,650mph and a flying time to New York of 3 hours and the last numerical was given in exact minutes (normal range between 20 and 24 minutes). This preflight routine could never compare to flying in other aircraft. The suspense was palpable as Concorde pushed back from its stand and taxied down passed the stands of its bigger brothers.\n\nThe first flight (or supernumerary) for any cabin crew on Concorde was always special. There was the excitement of flying supersonic for the first time, plus the prestige of working amongst a very exclusive clientele. The experience of my first flight more than exceeded my expectations. Suddenly one was entering a new working environment. With it came a new reverence for the job. It was 'back to school' again, and the crew I flew with on my first flight took on the role of school prefects who I revered and adhered to their every word. First-time flight nerves also kicked in.\n\nThe captains were very kind to the new recruits and invariably offered them a seat on the flight deck for take-off and landing, which again was a pretty special way to start one's new working environment. At the end of the flight, the new crew were given a certificate with all the crew names signed on it.\n\nMy supernumerary had a special significance as I was eased in very gently to the whole routine. There were only a few passengers on board, but sitting at the rear of the aircraft was Mick Jagger, along with ten other passengers at most. The purser mentioned to him that it was my first flight and Mick kindly volunteered to have his photo taken with me and signed a menu. I met him several times on flights after that, and he was often accompanied by Gerry Hall.\n\n## LIFE AT THE SHARP END\n\nIt is a little known fact that there are more US astronauts on the planet than there are Concorde pilots...\n\nI would often ask to go up on the flight deck for take-off and landing and the flight crew never seemed to mind. The supernumerary seat was next to the engineer and directly behind the captain. It was folded up in a convoluted manner that required a knack to unfold it, which I never seemed to master. However, once seated the engineer would ensure that I was properly strapped in. We shared many giggles as one of the lap straps had to go from the base of the seat between my legs and up to the locking system at chest height. This was fine if one was wearing trousers, as we could with one of our uniform designs, but with a skirt, dignity was cast aside.\n\nThe engineers invariably enjoyed the struggle of fixing it into position. The undignified crotch strap was attached to a five-point harness with two shoulder straps and a waist strap. This was quite a procedure and I did wonder how the female royals who visited the flight deck coped with this. I later found out that Princess Margaret was given a tea towel to place over the hitched up area of her dress, thereby preserving her modesty.\n\nWhilst I enjoyed the view from the cockpit window, the flight engineer would undertake another thirty checks, including the transfer of fuel from a rear trim tank for use before take-off. The captain would be steering the aircraft by means of a small handle on the side panel. No mean feat, as the nose wheel was 38ft behind him and the main wheels were 97ft behind. Unlike any other aircraft, Concorde had a springy feel to it in the cockpit during taxiing as the long narrow fuselage flexed over any bump on the taxi, or runway.\n\nSo, for 10 minutes of my working day I would sometimes swap the back end of the 'office' for the front end, and this was without doubt the most exhilarating one. For those watching take-off from the ground it was a unique experience, especially at night. The red glow of the afterburners shedding 30ft of flames from the four engines was an unforgettable sight and a reminder of Concorde's similarity to the space shuttle blast from Cape Canaveral.\n\nOf the many take-offs I experienced from the flight deck, leaving Miami one beautiful morning was a sublime experience. In the cramped space surrounded by dials and three busy flight crew at work, one could fully appreciate the technology of Concorde. As we edged back from the stand many cameras and binoculars were aimed at the sleek fuselage and as we taxied to the runway a Pan Am captain was heard to request a delay in take-off so he could watch us depart.\n\nPre-take-off checks complete, speed and heading confirmed to the control tower, engines to be throttled back 48 seconds after take-off. All was ready for our departure to Washington. The first officer even had time to comment on a battered old aircraft poking out of a hangar. 'That would make a good investment for you, Sally,' he remarked.\n\nLining up on the runway with only twenty-four passengers aboard, we reached 225mph in 18 seconds and lifted off gracefully, climbing high and fast into a dazzling aquamarine sky. Below us, the freeway was busy with morning traffic, the waterways had a steady flow of cruisers heading towards the ocean. Empty beaches appeared before us and then the ocean, glinting like a sea of diamonds. We must have looked a beautiful sight as we climbed away from the city.\n\nThe Washington to Miami service was stopped in 1991 due to low passenger numbers and, from a crew point of view, it was missed. Miami gave us that glimpse of winter sunshine on our few hours off. The Washington service finished in 1994, also due to poor passenger loads.\n\nMach 2.\n\n## INTO THE STRATOSPHERE, THE CURVATURE OF THE EARTH AND TO NEW YORK\n\nDuring the taxi to its departure point on the runway Concorde would burn almost 1.2 tons of fuel, or 2 per cent of its total capacity. I remember on one occasion, when flying out of New York, Concorde was sitting on the tarmac for 20 minutes. The fuel tanks by this time were registering a drop significant enough for us to return to the stand so that we could be topped up to ensure a safe passage home. Concorde had to have fuel reserves of 45 minutes \u2013 either for a holding pattern or a diversion to another airport. In spite of the amount of fuel the aircraft burnt even whilst on the runway, air traffic control did not give Concorde priority for take-off. The New York authorities, however, did allow Concorde to queue jump when it came to landing at John F. Kennedy Airport.\n\nOne Concorde captain, who was not so cautious about the fuel reserves after leaving New York, ignored the advice of the flight engineer to divert to Shannon to refuel and pressed on to London. Punctuality was more of an issue than 'gas in the tanks' on this occasion. The Concorde did land safely at Heathrow but ran out of fuel after leaving the runway. That was a close shave for the company. Perhaps lawsuits would have been issued faster than a Concorde take-off had the truth come to light.\n\nListening to the conversations between the captain and control tower through a pair of headphones was also part of the take-off perk and it was fascinating to listen in to other aircraft and the directives of the air traffic controller. The captain would wait for our call sign and permission to take off. The calm resonant voice of the air traffic controller directing the take-off procedure would come through the headset: 'Okay Speedbird 193 cleared for take-off.'\n\nYou could almost envisage the other aircraft waiting for their turn, bowing in deference to their faster cousin. It is also pretty hard to put into words that moment when take-off clearance was granted. The captain would engage the throttles and as they went fully forward, there would be a slight nudge in the back as acceleration gathered pace down the runway. Again another surge of excitement and thrill that in 30 seconds one would become airborne and soar into the skies. No lumbering down the runway 747-style for what seemed like forever, to Staines and beyond. Concorde was like a caged bird. As soon as she saw the runway she couldn't wait for her release to be airborne.\n\nThe take-off speed was 225mph (223km\/h) faster than any other commercial aircraft. Concorde burnt 25,629 litres of fuel per hour.\n\nLeaving London's Heathrow Airport the captain would, by law, have to throttle back for noise abatement. Even with the afterburners switched off, the climb would be 1,000ft per minute. Gaining height steadily, the nose and visor were raised. Life in the cockpit became quieter. Climbing over Reading at 8,000ft the full power of the four Rolls-Royce Olympus engines would be initiated again and the climb speed at that point would be 3,000ft per minute. At 28,000ft the passengers were glued to the Mach meters at the front of each cabin. At this height we would be cruising at Mach 0.95, just below the speed of sound, and flying over the Bristol Channel.\n\nOnce clear of the Welsh coastline, the reheat switches would be flicked on by the engineer, two at a time. The afterburners would kick in and the Mach meters in the cabin would clock up Mach 1. The pre-lunch cocktails would have been served and for first-time supersonic flyers they would invariably toast the 'Mach 1 Experience' with a glass of champagne. A mere 34 minutes after take-off, Concorde had reached Mach 2. We were now streaking across the Atlantic at 1,350mph \u2013 a mile every 2.7 seconds \u2013 and at 58,000ft in our 'pocket rocket'.\n\nDuring the winter evening service to New York, we would take off in the dark and, heading west, we would experience a spectacular golden glow 'sunrise'. The aircraft flying faster than the rotation of the earth would catch up the sun and we could watch it 'rising' and then setting again before we landed. This was a completely unique experience. Before the supersonic experience no man had seen the sun 'rise' in the west.\n\nTurbulence, fortunately, was rare at this height. We were above the jet stream and thunderstorms. There was no sense of movement. If there was a time and place where man felt small this would be it. The curvature of the earth in its midnight blue mantle could be seen quite clearly and if one had time to pause and reflect it would be to wonder at the beauty and magnificence of it all. We were on the edge of space, leaving behind the busy life of one continent to arrive at another. In the meantime, there was 3,400 miles of ocean below, the sky around us and a small aircraft catching up the rotation of the earth.\n\nDid we ever see anything below us at that altitude? We passed over the occasional transatlantic jet which appeared to be going backwards in comparison to our speed (800mph faster) and our height (20,000ft higher). The Atlantic Ocean always appeared a smooth shade of blue-grey. No waves were visible from this height, but I do remember a brief glimpse of the Queen Elizabeth 2 (QE2). The flight crew had been in radio communication with her and as we passed over her at Mach 2 we were able to see her very briefly, a tiny sliver of the hull 11 miles below us. I believe on this occasion the captains discussed what was on their lunch menus \u2013 the QE2's side of the conversation being somewhat lengthier than that of the captain on our flight.\n\n## A WELL-STOCKED WINE CELLAR AND FOOD FIT FOR A PRINCE\n\nOn the 747 first-class service to New York, a five-course meal was served to a maximum of twenty-four passengers at a leisurely pace. Time was not an issue as we had 7 hours and 30 minutes to feed and look after our passengers.\n\nNow, multiply the seating in Concorde's flying restaurant by four, divide the time by two, cut the working space by four, and the maths work out as pretty much impossible on paper. There must have been a lot of head-scratching in the logistics department when the meal plans in this five-star 'space rocket' were devised.\n\nService would begin immediately after the seatbelt sign was switched off, the drinks order having been taken on a notepad prior to take-off. The cocktails, expertly prepared by the galley steward, would be served to passengers on a silver tray. These were accompanied by prettily decorated bite-size canap\u00e9s of caviar, goose galantine, prawns and smoked salmon.\n\nOur repertoire of cocktails sounds somewhat dated now compared to the kaleidoscopic lethal concoctions found in bars today. Our cocktail choice of a Manhattan, Old-Fashioned and Highball, Americano and Negroni belonged to the seventies 'James Bond' era and the genteel habitu\u00e9s of the Waldorf Astoria and the Rainbow Room. Uncomplicated cocktails they were, but no less effective. They were not for the faint-hearted, especially at high altitude. A champagne cocktail, a mix of champagne, brandy, sugar and Angostura bitters, could have warranted a 'proceed with caution' notice. More popular were the standard no-frills Bloody Mary or dry Martini.\n\nFor the older English generation on charters, who made Concorde their once-in-a-lifetime experience, the tipple for the demure old ladies was often a port and lemonade, sherry or sweet Martini, no doubt a reflection back to the high day and holiday mood when those drinks were a 'push the boat out' sort of moment. For the men, they were happy with cans of beer after the celebratory glass of champagne.\n\nThe wine and drinks menu was, however, a huge temptation for the wine connoisseur, regular commuter or first-timer to the Concorde experience. It was a supremely difficult task to resist such expensive vintage wines, champagnes and top label spirits. However, most of the regulars were well aware of the dangers of mixing flying at high altitude and the high alcohol content of cocktails. They preferred to sip at the expensive wines and champagne. Invariably a neatly manicured finger would be pointed to the glass for the exact amount to top up and one dare not exceed the limit of the finger pointing!\n\nDuring this part of the service, Concorde held the reputation for being the fastest, highest and most expensive cocktail bar in the world. The view out of the windows was pretty spectacular too. The earth's curvature could be quite clearly seen and at 11 miles high the sky is a deep indigo.\n\nAfter a second drinks round, the glasses would be cleared away. The meal set-up was preceded by laying a Damask tablecloth on the fold-up tables in front of the passengers followed by a tray with cutlery, a miniature Royal Doulton condiment set and linen napkins. The cold starter course was usually seafood \u2013 crab legs, lobster or salmon. One such starter was described as 'Fresh Maine lobster and barquette of Sevruga caviar garnished with artichoke, lemon and mayonnaise', and always popular. Iranian caviar also came on the menu at regular intervals and was served in small sealed pots along with a garnish of onion and chopped egg white and yolk as an accompaniment. (Nothing could beat the sheer hedonism of getting home after a flight and spooning up the contents of one of those jars, that is, if one was lucky enough to find one unopened after the meal service.)\n\nThis would be followed by a main course presented on Royal Doulton china and delivered to the passenger on a silver tray. There was no handling of plates by hand. The plate was transferred from tray to table using a 'spong' (a name and method peculiar to airlines, I believe). A spong is a neat, long-handled serving tool with a lip at the end that holds the dish in place whilst it is carefully being transferred to the passenger tray. In my experience it was foolproof and I never heard of any crew member dropping a plate using this sturdy, reliable piece of equipment.\n\nThe main courses were a gourmet's delight. The menu out of London might include an 'English Game Pie prepared from marinated venison, pheasant and morel mushrooms'. Turbot was often a favourite on the menu. The American pronunciation was often to leave off the last syllable. Turbot would invariably be pronounced 'turbo' and a fillet steak would be 'fillee' (cue smile from the stewardess). I did have one American passenger who pronounced pheasant with a hard 'p' (as in the poor, working on the land variety). I was tempted ask her how she would like her country bumpkin served up. But at times like these diplomacy prevails. I don't think America have 'pheasants' (the people variety) in their countryside. I cannot remember an instance of the American species. Did they have the dubious promotion of 'farm workers'? If not, I stand corrected.\n\nThe menu choice of steak would include dishes such as Tournedos P\u00e9rigord, 'a fillet of prime beef grilled and coated with a rich dark sauce prepared with truffles from P\u00e9rigord accompanied by buttered leaf spinach, artichokes, tomato and wild rice pilaf'. A spring menu might include lamb, under the grand title of 'Noisettes d'agneau Prince Orloff: small steaklets of lamb seared on a hot griddle [naturally not on the aircraft] topped with a mixture of shredded ox tongue, pimentos, onions and cream served with Madeira sauce'.\n\nGame was frequently included in the menu. Noisettes of venison with chestnuts came in a rich game sauce and was garnished with Morello cherries. Pheasant, grouse, duck and roast quail also made appearances on the winter menu. Although quail stuffed with foie gras had a novelty value, this was not a particular favourite with the passengers as it was small and boney. However the jus, a wine sauce again prepared with truffles from P\u00e9rigord, was delicious.\n\nOn one occasion when this was included on the menu we had a steward on the crew who was of half Greek origin. The Greeks are well known for their consumption of anything small, airborne and with two wings. After the meal service was finished our steward who, I would add, had a great sense of humour, explained to us that in Greece the bird was eaten whole without removing any of the bones. The comment was greeted by a cynical 'Oh yes' from the three stewardesses in the galley.\n\n'Okay I'll show you,' he replied. Removing one of the remaining birds from the oven he chomped his way casually through it and swallowed it, bones and all. Our eyes were popping and we were left in awe and admiration for his sturdy Greek digestive system!\n\nOn the 10.30 a.m. flight from London, and on the morning departure from New York, a brunch was served. This could include kippers coated in breadcrumbs and a pan-fried, fluffy omelette, or the 'Concorde Brunch' of pork loin, pork sausages accompanied by button mushrooms and Anna potatoes. And by the time the passengers arrived in New York at 9.30 a.m. local time, they were in time for another late breakfast.\n\nThere was always a variety of seasonal salads to accompany the main course, aptly called on some flights 'a millionaire's salad' and served with a choice of dressing. A selection of French and English cheeses followed the main course in true French tradition. These could include cheeses such as Tomme de Savoie, English Stilton or Cheddar served with vintage port. The meal finished with a delicate and refreshing dessert. This usually included a fresh fruit ingredient such as fresh pineapple and figs served in a brandy and passion fruit syrup set on a base of macaroon, or a seasonal fruit salad served with kirsch.\n\nHaving cleared the remains of the meal service, the crew would commence a tea and coffee service accompanied by friandises \u2013 rich chocolate petits fours and mints \u2013 as the finale to this fine-dining experience.\n\nThe team of chefs who would prepare the menus, which changed weekly, would be highly distinguished. Michel and Albert Roux, whose restaurants included Le Gavroche in London and the Waterside Inn at Bray, were two of the chefs who designed menus for British Airways First Class and Concorde. Anton Mossimann, Shaun Hill, Richard Corrigan, Burton Race and Mark Edwards, Liam Tomlin and Claudia Fleming also had input over the years.\n\nA selection of menus.\n\nMenu centre-spread.\n\nOn one occasion, Albert and Michel Roux were passengers on one of my flights to the USA. I felt extremely apprehensive as I served them one of their creations! Would it be the right temperature, would it be cooked to the perfection that they demanded in their kitchens? However, they were both very complimentary in their comments and smiled appreciatively. They were happy that we had done a good job.\n\nThe wines were of a standard that would make the most avid wine buff salivate in wine heaven. In the 1980s the champagne offered on the flights would be a vintage Cuv\u00e9e de Rene Lalou Mill\u00e9sim\u00e9, described on the menu as a 'prince amongst champagnes \u2013 a privilege to drink and a rare gift to receive'. Another was Champagne Louis Heidsieck Dry Monopole 1975. One champagne of particular interest was Laurent-Perrier Grand Si\u00e8cle in a uniquely shaped green bottle. This shape dated back to one used in the time of Louis XIV and the champagne was a blend of three vintages. In the early days of Concorde, Dom Perignon was served and, no doubt, in a class of its own.\n\nThe wines were pretty exceptional too and all of superior vintage. The red wine could be a Ch\u00e2teau Latour, one of the great wines of the Bordeaux region. The Les Forts de Latour 1981 is a Cabernet Sauvignon Merlot from the Pauillac region of France (the average price is now between \u00a377 and \u00a3100 a bottle). Another excellent red wine was a Ch\u00e2teau Brane-Cantenac Margaux 1979, a wine from the Haut Medoc region of Bordeaux. A Ch\u00e2teau Cantemerle 1974, from the same region, also featured on the menu.\n\nThe white wines that would accompany such delights as a fricassee of fruits de mer or a lobster Newburg could be a 1982 Meursault or a Labour\u00e9-Roi Meursault 1985. Another favourite with passengers was a Puligny-Montrachet 1980 from the village famous for producing two of the world's greatest wines, Le Montrachet and Chevalier-Montrachet. A Chablis Premier Cru 'La Fourchaume' 1983, a classical Pinot Chardonnay grape from one of the four best premier cru vineyards, was also served.\n\nThe wines, of which there would only be one of each type on board (i.e. one red, one white and one champagne), were presented to the passenger from a wine basket. It must have been a difficult choice for the executive commuters flying to New York to attend high-powered meetings to resist the wines on offer.\n\nThe saddest part about these expensive and classic vintage wines was the wastage. During the approach to our destination, any remaining wine had to be tipped down the sink. Even a bottle opened for half a glass of wine had to be disposed of. No, we didn't cry, but it almost brought tears to the eyes as the vintage champagne and wines disappeared down the sink.\n\nThere was a good selection of liqueurs and a Remy Martin Napoleon brandy offered with coffee. Before the 'no smoking' rule on aircraft came into effect, cigars were also offered with the coffee service. These would be Jamaican Macanudos and, whilst many of the male passengers enjoyed them, it wasn't so pleasant for the crew struggling through a fog of heavy aromatic cigar smoke. Thankfully, times have changed in the aircraft environment and the crews are no longer subjected to passive smoking hazards.\n\nIf the transatlantic flights were 100 per cent full, which they rarely were, the cabin crew were under a lot of pressure to get the service completed in time. Serving a gourmet dining experience to 100 passengers, where the service had to appear unhurried and calm, was achievable, but there were times when the last trays were being stowed as the landing lights of the runway came into view. The pulse was racing and the adrenalin pumping and often we would have wished to go and ask the captain to do another circuit to allow us to finish clearing away!\n\nHowever, we took great delight in announcing the flight time to the last minute, and depending on the winds, this would vary on a regular basis between 3 hours 20 minutes and 3 hours 24 minutes. The captain was always very pleased to announce if we had beaten the projected flight time. Each crew member had a personal best for the Atlantic crossing and I remember mine from New York to London was 2 hours and 56 minutes.\n\nUnlike my own record-beater where I omitted to record the second hand, the fastest time across the Atlantic still stands at 2 hours, 52 minutes and 59 seconds in 1996 by Captain Leslie Scott. This was preceded by a New York to London flight captained by Norman Britton who, in April 1990, managed 2 hours, 54 minutes and 30 seconds. These world records were helped by the hand of nature, namely the strength of the Atlantic tailwinds which assisted considerably in shrinking the flight time from a west to east direction.\n\nConcorde also once flew from Miami to London in 3 hours and 50 minutes but with only two passengers aboard (a saving of over 5 hours and 15 minutes on the subsonic route). When Black Monday hit the financial world in 1987 the aircraft flew back shortly afterwards from New York. There was one solitary passenger aboard (the others no doubt counting their financial losses) and on arrival his luggage had been lost. That took some explaining away by the BA ground staff, but no doubt the passenger had felt very privileged to have the aircraft to himself and the undivided attention of six cabin crew.\n\nWorking to such a high standard, not to mention altitude, had its rewards. When we finally landed in New York with a full complement of passengers we had all made a strong crew bond. There was relief that the service had gone smoothly, as it invariably did, and pride that each and every one of us had worked hard to make a success of the service. We were undoubtedly a small close-knit family.\n\n## THE NEW YORK STOPOVER AND DINNER WITH A SINGING LEGEND\n\nThe ground crew who met the Concorde flight in New York were the nicest and friendliest bunch of Americans you could meet. They were always so genuinely pleased to see us, conscientious in their work and nothing was ever too much trouble for them.\n\nHaving cleared customs in John F. Kennedy Airport and collected our luggage, a crew coach would be waiting for us to take us into the city. Occasionally when we were feeling reckless we took the short helicopter flight from the airport into midtown Manhattan. That was always a fun way to enter the bustling city \u2013 and the quickest! On the coach we would relieve the boredom of the 'stop\u2013start' drive in heavy traffic by guessing the time it would take for us to pass through the Lincoln Tunnel. Bets were placed, money parted with, and it was always amusing if the cabin crew beat the flight crew who regarded themselves as experts in this game.\n\nThe discussion then inevitably moved on to the rest of the day's arrangements. Where were the girls shopping? Where was the venue for lunch? Which bar were we heading for in the evening? You could always guarantee to meet the male crew members in the evening at some bar or other. A favourite was an Irish bar, Rosie O'Grady's, or, for the flight crew, the Triple Inn. On smarter occasions it would be the Rainbow Room, the Russian Tea Rooms or one of the many chic restaurants New York had to offer. There was always a member of the crew keen to go to the theatre, which we did quite often.\n\nHaving checked into the hotel in Manhattan and after a quick change, the girls would invariably meet up and shop for 4 or 5 hours. (I often wonder how we did it for that long, having just walked the Atlantic!) The city had so many attractions to visit. Shopping would be interspersed with a trip to an art gallery, a walk in Central Park and plenty of breaks for coffee in the numerous bars.\n\nIf the cruise ship the QE2 was in port we invariably received an invitation via the flight crew to go aboard for drinks and lunch. Concorde did many charters with the QE2. Passengers would fly out to New York on Concorde and join the cruise ship for a five-day trip back to Southampton. I remember one female passenger with her husband who had been more than pleased and grateful to fly back. They had had one of the suites on the QE2 on the higher deck. During a particularly bad crossing on this occasion the ship had been extremely lively in the Atlantic seas, with the result that she had spent the entire time in her cabin. The supersonic flight was a very welcome alternative as a means to return home.\n\nBefore the 10.30 a.m. flight back to London we would meet up for breakfast at the local coffee shop round the corner. It was a 'home from home' experience. The waitresses knew us all by name. They were typical flamboyant New Yorkers and many a time they would regale us with their hilarious stories about their love lives. Similarly, they would know quite a bit about the domestic lives of the crew members. I'm sure they must have guessed about the various crew affairs as there were the regular couples who always flew together and that was surely no coincidence.\n\nThe nightlife in New York started early, which suited us well having flown to a 4-hour time change. Some of the bars in town would be heaving by 6.30 in the evening, discos and dancing in full flow. It was an eye-opening insight into New York City life. The city workers would flock to the bars, enjoy partying and be out in time for their commuter trains to take them home at a decent hour. It epitomised the Americans' way of life \u2013 they are known for working hard and partying hard, and we certainly took full advantage of it.\n\nOne of my most unforgettable evenings in New York was spent with the singer John Denver. Returning to New York on Concorde after a concert in London, he asked to visit the flight deck and he was up there for quite some time. As he was missing the dinner service I popped my head into the flight deck to see if he would like to eat. He was sitting in the left-hand seat, no doubt having a crash course in the myriad dials in front of him. John was a qualified pilot and so was totally absorbed and intrigued by the technology in front of him. I confess that in the dimmed light of the flight deck, our eyes met. It was one of those moments when the heart strings pull and you just know the chemistry is there. Later on he signed a menu for me and wrote, 'Sally, you have a lovely smile. Peace, John Denver'.\n\nLater we got chatting and he asked me if I flew into New York very often. 'About once a week on average,' I replied. 'Sometimes twice.' He gave me one of his charismatic smiles and asked when I was going to be over next. Quite by chance our dates for being in New York coincided and, to my surprise, he asked if I would have dinner with him. The reply was naturally affirmative.\n\nTwo weeks later, I was bound for New York to meet up with my special date. Nature, however, was to play a bad card by producing a very stormy night and things were not about to run smoothly. We approached New York in the middle of a mighty rainstorm that made landing impossible. We were diverted to Newark. This had never happened to me before, nor would it ever again, but on this night of all nights fate stepped in.\n\nConcorde made a bumpy approach through heavy rain clouds into Newark. We disembarked the passengers and waited. The captain, aware of my date as were all the crew, disappeared off to the operations room to sort out transport into the city and also to ask if John Denver's private jet had landed. He returned to the waiting crew who were as anxious as I was to know the outcome. Yes, his plane had landed at 4 p.m. that afternoon before the rainstorm. The captain was wonderfully sympathetic and asked for a coach to be sent as soon as possible. However, the journey still took 2\u00bd hours. The New York skyline looked bleak and the coach trip seemed to take forever. We munched disconsolately on cheese rolls and drank brandy to warm ourselves up. It was around 9 p.m. when we finally checked into the hotel.\n\nI had by this time given up all hope of seeing John for dinner. At the hotel check-in desk there were two messages to say Mr Denver had called. The crew were full of sympathy for my missed opportunity. Feeling pretty low, I went up to my room and resolutely took off my make-up, got undressed and slipped into bed. By this time it was 9.30 p.m. (1.30 a.m. UK time) and any chance of another call from John seemed out of the question. Superstars are probably not used to being stood up!\n\nHaving made myself comfortable between the sheets I was startled when the phone rang. True to his word, John was determined to keep the date. He told me that he and some friends were in a Japanese restaurant on the Upper West Side and would I be up to joining them? If there was a record time for a female getting showered, dressed and make-up on, I would most certainly have made the record book! Half an hour later, outwardly calm and composed but adrenalin bursting in my chest, I walked into the restaurant where John, with four of his male friends, gave me a very warm welcome. We drank sake and ate sushi and the weariness of the flight and delay evaporated as I got to know more about this amazing man.\n\nWe carried on drinking in a bar across the road. So absorbed were we in each other we hardly noticed the others leaving. There were crayons and paper covering the tables. John drew a big smiley face and wrote my name underneath it. We talked about his farm, his love of the wild open spaces, the Rockies being his favourite wilderness, and I became totally absorbed by him. He spoke passionately of wanting to be one of the first 'ordinary' men in space. He was firmly committed to the idea. He spoke vehemently about hunger in the world and was resolute about finding solutions to this. We discussed his concerts and his songwriting.\n\n'How do feel when you walk out on stage in front of a vast audience like the Albert Hall and there's just you and your guitar?' I asked.\n\nHe smiled one of his enigmatic smiles. 'Just great, Sally,' he said. 'I love singing to the world whether it's two or 2,000.'\n\nOne subject that we didn't broach was his love life. He had divorced the year before. He was, however, very enthusiastic about his young daughter Jesse Belle, whom he had come to visit in New York. 'You must come and meet her,' he said 'Come over for a weekend.'\n\nHe was, without doubt, the most unpretentious star I have ever met. His million megawatt smile and absolute enthusiasm for life would 'fill a room', as they say. It was infectious and despite all his stardom he made you feel very special. Leaving the bar we stepped into his long, black stretch limo and headed off to the appropriately named Sally's Bar opposite the crew hotel. By this time it was late and the bar was quiet. He dismissed his driver and we sat drinking coffee and reflecting on life. Finally we walked back to my hotel and, stepping into the lift, we wondered when we would see each other again. The electricity between us would have driven that lift up to the twenty-third floor on its own. I knew I was going to find it hard to say goodnight, and as we stood outside the door to my room he asked if he could come in.\n\nAs tempted as I was, the reality of getting up in a few hours to work back to London was very much in the back of my mind, and I felt pretty sure that after the fabulous evening we would see each other again. I often reflect on this decision and still have this image in my mind of John walking back to his hotel in Central Park in the rain, at 3 in the morning. I fell asleep wondering about what might have been...\n\nThe phone rang the next morning as I was getting ready for crew pick-up. It was John to say how much he enjoyed the evening and he hoped we might see each other again. We must not let distance come between us, he insisted. He gave me his address and phone number in Colorado. In the hotel lobby the crew were waiting for me and desperate to hear of the night's developments. I think I could have flown home under my own steam that day I was so elated.\n\nJohn died in 1997 at the age of 53 as the result of an air crash. The small plane he was flying solo was new to him and the official report was that the fatal crash off the coast at Monterey, California, was pilot error. He was, however, an extremely experienced pilot. In his huge contribution to music he was the fourth highest selling American solo artist of all time, the others being Elvis Presley, Michael Jackson and Frank Sinatra. He left a legacy of hauntingly beautiful music, magical words and, for one lady, the memories of a beautiful night in rainy New York.\n\nNaturally, as Concorde cabin crew we met many handsome, wealthy and unavailable men. They wore Saville Row suits and becoming smiles. There was always the odd one or two who were persistent in asking for our phone numbers. One of my regular passengers was a very highly educated Indian shipping magnate. He was always very charming and he invited me for dinner in a top London Indian restaurant. I remember quite clearly the drinks order consisted of three different types of bottled water. I don't remember being asked if I would like a glass of wine!\n\nThen there was Tom, a very suave New Yorker with Paul Newman eyes. We had dinner in New York on several occasions. He knew all the best restaurants and treated me royally. One evening we ended up at a small caf\u00e9 \u2013 the Lone Star Caf\u00e9. I was thrilled when we walked in to find one of my favourite singers, Don McLean, was playing. We stayed to hear him play his memorable numbers 'Vincent' and 'Starry, Starry Night'.\n\nThere were other stewardesses who dated famous people. A stewardess friend of mine was elated when she had met Eric Clapton on a flight and he asked for her phone number. Was this to be another 'nearly, not quite' moment, as happened so many times in our job? We all wanted to be the first to be updated on the progress report of Mr Clapton's dating. Meeting up with Annie over the next few days, the burning question was 'had she gone out on date with him?' She was mortified that he had rung twice when she had been out and left two messages. If mobile phone development had progressed a few years ahead of time who knows what might have happened?\n\nOne of my flying girlfriends befriended a passenger who always took her to Flushing Meadow to watch the international tennis tournaments. He was also very generous when he came to London and lent her his Rolls-Royce with a chauffeur. We enjoyed a very hedonistic shopping trip in London with our obliging chauffeur. There were, however, very few stewardesses who met the man of their dreams on these flights and managed to marry him! Mostly we were all dating boyfriends at home and as mine was involved with the Grand Prix world that again was another glamorous lifestyle.\n\n## CONCORDE CHARTERS AND A GOOD EXCUSE TO DISPLAY CONCORDE'S FLYING ABILITIES\n\nLong-serving stewardess Jeannette Hartley became the first person to charter Concorde for a group of friends in 1981. Jeannette, who joined Concorde in 1977 and has an impressive record of 744 Atlantic crossings in twenty years, is a member of the Naval Reserve. Her colleagues there had often spoken of the desire to fly supersonic and Jeannette approached Concorde charters and a price was agreed to charter the aircraft. For 1 hour and 15 minutes' flight around the Bay of Biscay, British Airways agreed to charge \u00a320,000. However, to Jeannette's disappointment, only eleven people came forward to take the flight even though the cost had been reduced to \u00a317,500. At the time she was being interviewed by a woman's magazine and British Airways public relations department decided to seize on this to market her idea and the following day the Daily Mail published a picture of her standing in front of Concorde promoting her charter idea.\n\nIn three weeks Jeannette had filled not only one but two Concordes for her charter flight. Readers who had seen the opportunity of flying supersonic for \u00a3175 wasted no time in getting in touch with her. Jeannette told me, 'I was inundated with cheques. Letters would come through the post addressed to Ms Jeannette Hartley, Concorde Stewardess, Wexham, Bucks.' They all found their way to her house and Jeannette admits she could have filled another Concorde. After that, the round the Bay of Biscay charters really took off.\n\nJeannette Hartley's first charter with British Airways in 1981 around the Bay of Biscay. (Courtesy of Jeannette Hartley)\n\nOver the years we flew many charters. These charters were the saviours of Concorde's accounts and became very popular with Concorde fans due to their affordability. Some were brief 2-hour flights over the Bay of Biscay, giving our guests the experience of flying supersonic with a glass of champagne in hand. Others were of a longer duration. The first-timers on the charters were always so enthusiastic, as were the children who were lucky enough to be included. It was a pleasure to see their faces as they first stepped aboard, many speechless with excitement. During these short trips the emphasis was more on the performance of the aircraft than the in-flight service.\n\nAfter serving lunch to our charter passengers we regarded ourselves lucky if we got any of the cutlery, china or glassware back. Everything that bore the Concorde logo was regarded as fair game by the passengers and some would take as much as they thought they could get away with as a memento of their flight. Most charter passengers had saved for a long time for their flight of a lifetime and they made sure they got their money's worth. Certificates signed by the crew and captain were also part of the package. An even greater delight was the invitation by the crew to visit the flight deck that, understandably, no passenger refused.\n\nOne of the memorable charters was to the RAF base at Mildenhall in Suffolk. We landed to see thousands of spectators lining the fencing alongside the runway to watch our arrival. However, Concorde was not the only highlight of the day. The Red Arrows display team was also there along with the American aerobatics team, the Vulcan bomber and the sinister Blackbird (also capable of flying supersonic, but at Mach 3 and at an altitude of 85,000ft).\n\nThe Red Arrows display from our vantage point was pretty unique \u2013 standing on the wings of the parked Concorde. We had removed our shoes and stepped out onto the long curved wing from one of the aircraft doors. Undoubtedly we had a good view, as the men in their magnificent flying machines never failed to impress. It was quite a bizarre experience too, seeing them flying in the Concorde formation from our unique position.\n\nAfter watching this gravity defying display, we went to meet the pilots as they parked their nine aircraft close to ours. I was naively taken aback that as they stepped out of their cockpits they were as cool and relaxed as if they had just stepped off a bus! Not only were we in total awe of the pilots' flying prowess but it did not go unnoticed by the girls that they were all exceptionally good-looking, charming and very easy to talk to.\n\nIt was our turn next to impress the crowds. The 100 passengers who had paid for an hour's charter eagerly embarked Concorde for a 'spin round the bay'. Now, whether the 'jump jockeys' on the flight deck were trying to emulate the Red Arrows is debatable. Whether it was the fact that the aircraft was very light, having no baggage in the hold, but the 'boys at the front' were more than ready to show off their flying skills as well as the performance of Concorde. Our guests for the trip, grannies and youngsters alike, were excitedly awaiting the take-off. Concorde raced down the runway and headed up into the skies. There was huge G-force and the captain proudly announced that we had just climbed 11,000ft per minute. My stomach certainly felt like it had done just that.\n\nFurther aerobatics were in store. We headed out over the North Sea and then turned to approach the runway at the airbase. Beautiful to watch from the ground, no doubt, the captain did a superb approach, wheels skimmed the runaway with great precision and then we were off again climbing as fast as we had done the first time. We banked over and headed out to reach Mach 1. I have never been a lover of fairground rides and this was one I definitely wanted to get off.\n\nSome of the passengers did not respond well either and I'm sure they had not been expecting to be part of such a heart stopping air display. Many delivered their champagne and snacks back into sick bags. Others were probably of the opinion the aircraft performed in this way all the time. I was moved to comment to the flight crew that the severity of the climb was indeed a first for me, and being an experienced flyer even I had found it a nauseating one. Sitting in their flight deck seats, beaming from ear to ear and no doubt immensely proud of themselves, the two pilots proudly announced they had given a superb display and it wasn't often they had the chance to fly such a lightweight Concorde and therefore show off its true performance levels... Mischievous boys let loose in a fairground!\n\nOn another occasion Concorde was flying in formation with another two Concordes at an air show. There were no passengers on this particular flight so the two stewardesses were standing idly in the front galley next to the flight deck, the door of which happened to be open. Now one of the stewardesses was well known as being extraordinarily verbose and was never short of words. The two girls were carrying on an avid conversation obviously unconcerned about the pretty crucial flying going on a few feet away from them. From the flight deck came a sharp retort from the captain. 'Will you two parakeets shut up? I'm trying to fly in formation!'\n\nConcorde had flown several times with the Red Arrows at air displays. On one occasion Julia van den Bosch, our longest-serving supersonic stewardess, recalled the event:\n\nThe captain wanted to stay as close to the Red Arrows as aviation law would permit. He also wanted to show Concorde off to the crowds below. Not to be left out of the Red Arrows consummate display he in turn performed a few aerobatics as part of the display. Unfortunately for the passengers this meant fighting off queasy stomachs and the laws of gravity.\n\nJulia recalls that the view from the flight deck was amazing as they followed in the wake of such a highly experienced flying team, and she comments that this was a typical example of how working on Concorde made it so unlike any other aircraft. 'On which aircraft,' she commented, 'would your day's work include flying in formation with the Red Arrows in front of thousands of avid and enthralled fans?'\n\nIn 1986 Concorde was specially chartered to a group of Americans from New York's John F. Kennedy Airport. A hundred Americans had paid for the privilege of getting as close to one of Halley's Comet's rare appearances as humanly possible. It was, at the time, passing close to the earth and had not been seen since 1901. After several glasses of champagne the Americans were really getting excited at seeing this very rare spectacle in the sky.\n\nFinally the captain announced that we would be passing the comet on the left-hand side but that he would be turning the aircraft so that the passengers on the right could have an equally good view. However, overexcited at glimpsing the comet there was a mad rush of passengers from the right-hand side to get a look at it. We had to announce that it was best if everyone remained in their seats. Champagne-fuelled as they were, there was a distinct reluctance to comply with this. The scientists had reported that this was a very bad year for seeing the comet and that it could only be seen with a telescope. Whether we saw it for sure is still open for debate, but our enthusiastic guests had a great time anyway.\n\nThe Red Arrows through the Concorde cockpit window. (Courtesy of Mike Bannister)\n\n## THE VIENNA BALL\n\nOne of the more glamorous annual charters was to fly 100 passengers to Vienna for the New Year's Eve Ball at the Hofburg Imperial Palace. This event was the grand successor to the Imperial Ball and the palace was home to the great Hapsburg family. It was undeniably a great sense of occasion and a very glittering way to see in the New Year.\n\nBall gowns and dinner suits were packed as most of the crew had agreed to attend the ball. Having flown into Vienna the day before we had a wonderful day exploring the city and its wine bars. I had by this time met my husband who worked on Concorde as a cabin service director and we were lucky enough to have been rostered for this trip together. However, our waltzing was twenty years out of practise so my husband and I spent some time in the afternoon in our hotel room going through the steps of the Viennese waltz.\n\nFeeling very privileged to be at such an event, we drew up in taxis in front of the palace feeling very much the part in our long ball gowns and the men in their dinner suits. Our first sight of the Hofburg Palace was memorable and as we entered the interior we were even more taken aback at the grandeur. At the top of the ornately decorated sweeping staircase were two enormous arrangements of pink roses. Next to these were two very large busts on plinths of two of the male Hapsburg dynasty. To our amazement, one of the Hapsburg family members was there to greet us. Wearing a long white dress, this vision of beauty looked like a princess \u2013 as she probably was. The night was magical and we took to the floor to dance to the magnificent orchestra playing the Viennese waltzes.\n\nWe weren't the only novice dancers on the dance floor that night, but it was such a grand setting few noticed the occasional wrong-footed step. As we returned to the hotel just after midnight, the sky was ablaze with fireworks that lit up the city. In the park opposite, the revellers went on until the early hours of the morning. The Viennese certainly know how to bring in the New Year.\n\nThe next evening we were scheduled to fly back to London. The passengers had experienced a wonderful New Year and were now quite weary after all the celebrations. No sooner had we taken off than the crew were aware that something was amiss. One of the stewardesses was hastily summoned to the flight deck. As she came back into the cabin, her face, as pale as an ostrich egg, registered shock. A light on the engineer's panel had shown that there was smoke in the aircraft hold and the possibility of a fire.\n\nBarrie, the cabin service director who had been at the rear of the aircraft at the time, was also summoned to the captain and was told that we would have to return to Vienna with the possibility of a full passenger evacuation. In the meantime, the stewardess, who was still looking like she had seen the end of the world coming, and not wanting to let the standards slip even in an emergency, immediately went to the hat rack, found her hat and put it on.\n\n'Why are you doing that?' questioned one of her colleagues.\n\n'Well, if we have to evacuate the aircraft in a hurry at least I am correctly dressed for it.' Even with the prospect of an emergency situation our colleague was more than ready to look every inch the Concorde stewardess as she took on her role directing passengers down the slide and off the aircraft.\n\nFollowing a jettisoning of fuel to lighten the landing weight, our Concorde was met by a fleet of fire engines and emergency vehicles, their lights pulsating in the darkness. It was soon established that this warning light had in fact had a faulty reading, but nonetheless it had to be rectified before we could leave again for London. Our departure was thus delayed by a couple of hours until the problem was fixed. It was a salutary reminder that our much loved aircraft was not infallible and even she would pick her untimely moments to have the odd malfunction.\n\nThe Concorde charter to the New Year Vienna Ball at the Hofburg Palace; guests being greeted by a member of the Hapsburg family.\n\nThe grand entrance to the Hofburg Palace.\n\nAnother potential safety concern occurred to this same stewardess when she was working in the front galley on another flight. As you can imagine, one of the responsibilities of a stewardess is to have a heightened awareness of any unfamiliar noise that might be heard whilst in the cabin. My colleague was instantly alerted when she heard a buzzing noise emanating from the flight deck. This was not a sound she could identify and her first thoughts were that it was an emergency signal emanating from one of the instruments. Apprehensively, she ventured into the flight deck not sure what to expect. However, to her surprise the sound was no more than an electric shaver and she was greeted by the captain having a shave. So much for strange noises!\n\n## ROUND-THE-WORLD CHARTERS AND ONE FOR OPERA LOVERS\n\nThe 'Round-the-World' charters became a successful money earner for Concorde and must have surely been the ultimate experience for the flying enthusiast.\n\nIn 1988 Imperial Tobacco sponsored a thirteen-day trip for competition winners of their John Player Special brand of cigarette. This was quite a unique charter as some of the passengers had never flown before. Finding themselves with a round-the-world ticket flying supersonic must have been mind-blowing and, in some cases, quite a daunting prospect for these first-timers. These were ordinary working folk who suddenly found themselves on a holiday of a lifetime and about to embark on a trip they would only have previously dreamt about.\n\nThe flight left London via New York. Their next stopover was San Francisco for a couple of nights, then on to Honolulu where everyone was met with the traditional welcome of a colourful flower lei. Even Concorde was included and had one draped across her nose. Guam, an island in the Pacific, was next on the stopover. Not since Captain Cook landed in 1778 had an arrival caused such a stir as thousands turned out to watch Concorde's arrival. From Guam on to Hong Kong, then to Bali, Colombo and Bahrain; it was a trip of a lifetime for those lucky enough to be on it and for the fortunate crew.\n\nIn 1989 another charter took place with an American company. As the passengers were elderly there was also a doctor on board. The passengers were all from the southern states of America, very much 'old money' and very charming and gracious. Dee Bull, a fellow stewardess, takes up the story:\n\nOriginally we were to position to Hong Kong and work back to Delhi the next day. However, it was decided the crew should be there for the handing over of the departing crew so we had in fact arrived a day early to attend a cocktail party. We then had five days off whilst our passengers were flown up to Beijing with a Chinese carrier [Concorde was not allowed to go there].\n\nSo we started the trip well rested (although as ex-short-haul crew we rushed around Hong Kong trying to fit in all the sightseeing and shopping, including a boat trip to Macau for the day).\n\nThe catering and wines on Concorde were superb. The only instruction we were given when we left London was that at the end of each sector, all wine was to be 'disposed of' as we would have new provisioning for the next sector \u2013 it was difficult to know how to dispose of it. However, an economic solution from the crew was reached!\n\nWherever we arrived we were met with chaotic scenes as the station managers were eager to impress friends and guests with a tour of Concorde. As soon as all our passengers had disembarked our wonderful captain, Chris McMahon, gave the order that no one was to be allowed onto the aircraft for 15 minutes. The flights were pretty hard work, so it was wonderful to just sit and relax with a little of what had to be 'disposed of'!\n\nIn Delhi we flew up to the Taj Mahal with our guests on Air India. They were staying overnight in order to see the breathtaking sight of the Taj at sunset and dawn. We, however, had to get back for the next leg of the trip the following morning so we had to take our life in our hands and endure the death-defying taxi ride back to Delhi.\n\nWhilst at the Taj Mahal, one of the stewardesses, Margaret, slipped and sprained her ankle. The doctor strapped it up but she was very anxious not to be left behind. Help was at hand in the guise of the 'PR' pilot, Dick, who had worked as cabin crew when the company had decommissioned Trident aircraft and were overstaffed with first officers. Dick volunteered to work in the cabin. The passengers were told of Margaret's accident and Dick, the 'new recruit', became a firm favourite in the cabin.\n\nWe flew on to Cairo and our first day included a visit to the museums and 'Son et Lumiere' at the pyramids. Our final evening was one of mystery and exotic Arabian hospitality at its best. Wearing the traditional dress of 'dish dashes' we were taken out to the pyramids. Imagine the scene amongst the backdrop of one of the greatest wonders of the world. As we approached a huge Bedouin tent, we were greeted by riders on magnificent dancing horses carrying large flares to guide us in. It was a magical evening.\n\nThe world's best belly dancer was there with swirling dervishes and a magnificent meal was laid on for us. (There was even a whole roast lamb on a spit outside, and I remember going out into the dark night and seeing only the outline of the pyramids in front of me, and having to pinch myself that I was actually getting paid for this!)\n\nOne of the ladies on board I particularly recall was elegant and must have been in her eighties. She had porcelain skin and in her gloved hands she smoked a cigarette with a cigarette holder. She had to have a wheelchair. Whilst her husband was disembarking with the hand baggage, I heard her say to him in a strong, deep southern drawl, 'You be careful there, honey'.\n\n'Don't you worry about me' he replied.\n\nTo which she replied, 'I ain't worried about you, I'm worried about your pacemaker!'\n\nSadly when we got back, despite taking all precautions I went down with salmonella, I guess it was a hazard of the job. No one else did, so it was just bad luck on my part.\n\nIn the mid-eighties Concorde was chartered to fly to Luxor in Egypt. A hundred opera fans had booked to experience Verdi's famous opera Aida in the unique setting of the 3,500-year-old Luxor Temple. Placido Domingo topped the bill with a cast of 1,500 singers, dancers, musicians and extras and even a lion. It was estimated that 30,000 people in total attended the opera, including sixty European princes and princesses.\n\nRegrettably, although we were not far from the venue, we failed to get tickets for the opera. Our accommodation was a Nile riverboat with a view across to the Valley of the Kings and this made an interesting interlude from the normal five-star city hotels we stayed in. The next morning, determined to enjoy the Nile experience, a group of us visited the Karnak Temple, the most visited site after the pyramids at Giza. Its size and preservation was quite awe inspiring.\n\nA trip to Egypt would not have been true to form if someone had not gone down with a serious stomach bug. Sadly, on our return to London, it was one of our female passengers who had to be stretchered off.\n\nAnother incident aboard involved one high-ranking, opera-loving cabinet minister. Prior to landing we were giving out boxed models of Concorde as a keepsake of the flight. Most passengers were very appreciative of their 'gift'. Not so the minister, who abruptly told the stewardess, 'Why would I want one of these?' and handed it back to her. She was visibly upset by his unwarranted rudeness. The watchword in situations like these was to 'keep smiling'. Sometimes it wasn't as easy as you imagine.\n\nIt was always company policy that each Concorde passenger received a gift at the end of the flight. These varied greatly from grey leather bridge sets to leather hip flasks, silver drinks labels, Concorde leather gift tags, pens, diaries, notebooks, writing pads, scarves and ties, to name but a few. No doubt these are now amongst some of the passengers' prized possessions.\n\nThere was always one who wanted to take an extra souvenir home with them. I think the most audacious request I recall was when a passenger very seriously asked me if he could have a dozen of the miniature salt and pepper sets for his dining table at home. 'They would be quite unique wouldn't they?' he suggested. I couldn't disagree with him, could I?\n\n## [PROMOTING CONCORDE:  \nUNIQUE TIMES AND A TRIP TO DETROIT](contents.htm#a24)\n\nOne of the tasks of the Concorde fleet, as flagship to British Airways, was to be involved in promotional work. Detroit was one of the destinations, no doubt to impress our American cousins who had slowly come round to the 'noisy' supersonic aircraft.\n\nIt was a memorable trip, not least because we flew an empty Concorde over there. To sit in one of the passenger seats with no work to do gave the flight an added exclusivity and made a pleasant change from working across the Atlantic. One of the stewards served us lunch and that even further enhanced the hedonistic experience. It was the only time I ever flew passenger on Concorde and got paid for it.\n\nOur first stopover was New York. From there two Concordes were to fly to Detroit and land on parallel runways. Captain John Hutchinson was flying one and the other, which I was on, was captained by Colin Morris. It had been decided that John's Concorde would taxi out first at JFK followed by ours. Now, whether there was a competitive spirit between these two captains, or Colin Morris just wanted to have a joke with John, I'm not sure, but it was very amusing when Colin jumped the queue and we arrived at the departure runway first. There was a great deal of friendly bickering over the airwaves. Colin wanted to be the first to arrive in the Detroit area.\n\nAs it happened, both aircraft arrived there at the same time. One by one we did a planned overshoot at 200ft and then came around to land simultaneously on parallel runways. It was an unforgettable flying moment as both captains did a superb job and the two aircraft landed perfectly together. It was certainly a novel experience to look across at the other runway and see another Concorde landing at the same time.\n\nA crowd of 20,000 spectators had gathered to watch the display. The control tower relayed to the flight crew that they had never seen such spectacular flying. In the airport manager's lounge there was quite a reception waiting for us as TV and press were there, as well as VIPs and members of the airline's executive management. We felt quite the celebrities and after that spectacular arrival we were as jubilant as the crowds who had watched our arrival.\n\nCaptain Morris volunteered me to be interviewed by one of the TV stations. Thus the job of stewardess widened its horizons and I found my 15 minutes of fame on live TV. One of the questions I was asked was if one experienced jet lag on the Atlantic crossings. Momentarily I hesitated. I think I might have been in trouble with the company if I had said yes. The theory behind such short flying times is that there is no jet lag and certainly from the passengers' point of view I had never heard of them suffering from it. Even 'three times across the Atlantic in one day' Richard Noble, in his record-breaking event, commented that he had no jet lag.\n\nThe curious observation about working on Concorde was that for the first six months all the crews were absolutely fine and the newcomers who had left long haul were delighted that this was one of the bonuses of flying supersonic. There was no great mix of time changes. On long haul one could be in Tokyo one week, San Francisco the next and Sydney the following week. The body clock could become very scrambled. However, on Concorde, large numbers of us agreed that after approximately six months of supersonic flying a form of tiredness equivalent to jet lag hit our systems. The flight deck never had a problem but for those of us making two trips to New York a week, it most definitely had an effect.\n\nAfter the American press had taken photographs for the local newspapers and the TV stations had had their fill of interviews we left the welcoming reception to watch the two Concordes depart for London, again with a spectacular display. Taking off pretty much one after the other, they climbed into the skies then returned, banking overhead before starting their supersonic climb across the Atlantic. We all cheered and waved and our hearts swelled with pride at such superlative flying. Along with the Red Arrows, Concorde was an icon of aviation and made us all justly proud to be British.\n\nThat night there were two Concorde crews remaining and we were all in full party mode. The airport authorities had laid on a fantastic party for us and we all felt very fortunate to have been part of a landmark in history-making. One of the comments from the airport managers was that we all seemed to be a very close and friendly group. And that's how it was. We felt immensely privileged to work on such a fleet.\n\nWith all the official duties over and enjoying the ample American hospitality we did let our hair down a little. One of the sillier moments of the evening was my introduction to a game called 'Dead Ants'. There were seven of us, flight crew included, in the lift descending to the ground floor for the party. When one of the crew called 'dead ants' we all had to fall onto the floor and act dead. Unfortunately at this moment the lift arrived at its destination and to the amazement of the couple standing at the open doors there were seven of us giggling and sprawled on the floor. Thank goodness they hadn't recognised the captain and first officer out of uniform.\n\nThe trip finished in suitable style when we flew back to London first class on a British Airways 747. 'A long way to go for a party', I wrote in my diary.\n\n## BARBADOS BOUND\n\nThe first scheduled Concorde flights to Barbados began in February 1984 and the incoming crew on this inaugural flight were treated royally and invited to a champagne reception at the governor general's residence.\n\nI was scheduled to fly the return flight to London. We had flown out on Concorde to New York the previous day and were booked on a passenger flight to Barbados on British West Indian Airlines. Leaving a cold, stony-grey New York and finding ourselves in the warmth of the Caribbean was indeed a winter's tonic and a wonderful break from working out of London.\n\nDuring my years flying on 747s I had spent many holidays visiting Barbados. Being such a regular it was like a homecoming to be staying on the island again where I had many friends. Our schedule included a day off and the next day proved to be a lovely break from work as the whole crew chartered a catamaran and we sailed along the beautiful coastline, sampling the rum punch that was liberally served to us by our Caribbean friends of long standing.\n\nSunset is always a magical time in the Caribbean and it is the one place in the world I have visited where people gather on the beach to witness the ending of the day, locals and tourists alike. It is truly a moment of tranquility as the sun slips behind the skyline and it gives one a moment for reflection and gratitude at being in such a heavenly place. And there were many happy memories of being there.\n\nLuciano Pavarotti loved to holiday in Barbados and in the nineties he flew out on Concorde and stayed at the Sandy Lane Hotel. On one occasion, my husband and I were there with our 4-year-old son. It was a glorious morning and we were taking a very early stroll along the Sandy Lane beach (my young son liked to rise early, whichever time zone he was in). Standing in the sea was this larger than life figure, Pavarotti, with his Italian secretary\/girlfriend, Nicoletta, whom he later married in 2003. As my son looked across to him, Pavarotti began to sing in his wonderfully sonorous voice, 'Good Morning Good Morning'. I don't think my son was fully aware that he had just been serenaded by one of the world's greatest living legends in opera.\n\nWith the commencement of the Concorde service, the morning after our arrival was business as usual. We enjoyed a relaxed hotel breakfast in the sunshine and it was a very pleasant thought that all the crew seated around the table were really looking forward to going to work.\n\nAt 10 a.m. precisely we joined officials and VIPs at the Grantley Adams Airport waiting for Concorde's final approach into Barbados. There was a wonderful atmosphere. Crowds had gathered on the viewing galleries and rooftops of the terminal. A steel band dressed in green and yellow was playing at full volume as a warm breeze wafted over us. We spotted the first glimpse of the silver-winged bird as it approached from out of a clear blue sky. Glinting in the sunlight, it throttled back to pass overhead, and turned up the coastline before finally coming into land. With its familiar poise for landing, stick legs protruding down and droop nose, it was like a much-loved family pet that, as soon as one saw it, one's heart lurched.\n\nOfficials stepped onto the tarmac to greet the passengers and crew. There were smiles all round, especially from the passengers who probably were still in shock that the 'bird' they were emerging from had delivered them from London in 3 hours and 30 minutes, less than half the flying time of the subsonic jets. Breakfast before leaving home and an afternoon dip in the Caribbean Sea was quite a good way to start a holiday!\n\nAs soon as we were allowed onto the aircraft to prepare for the return journey, we realised that patience was the order of the day. Stowing food and beverages on Concorde was a first for the Barbados catering crew. What they lacked in knowledge of the new stowing procedures they made up for in enthusiasm. It was a case of dealing with the unfamiliar, and teething problems were bound to happen. Despite the delay of getting everything on board, the local catering team was immensely proud to be a part of the supersonic experience.\n\nConcorde operated a twice weekly service to Barbados throughout the winter and it was invariably full. It also carried many of the stars and celebrities who visited the island.\n\n## LIFE WITH THE SUPERSTARS, ROCK STARS AND MEGA STARS\n\nIf I had to name all the famous people I had met during my time on Concorde it would be an extremely long list of both household names and icons. They could be divided into categories of movie stars, rock stars, opera stars and sports stars, with a subcategory for rising stars, fading stars and ever-shining stars. But enough of the galaxy.\n\nThere were also thespians, movie producers, film directors, as well as bankers and business executives who were hot names in the finance world and who were regular commuters. There were chairmen of global companies, media barons, authors and no doubt mafia men and the odd criminal from the underworld.\n\nWhatever their trade, whatever their talent, whatever their notoriety, good or bad, they were all treated the same. If you had a boarding pass for the pointy-nosed aircraft, you were treated like a star whether you were a pensioner who had spent your life savings on an hour round the bay, or a first-time flyer with no inkling of the finesse of the machine you were about to fly on.\n\nOn a few unscheduled occasions, British Airways management decided that instead of using the usual 737 for the early flight to Glasgow, they would lay on a Concorde, unknown to the passengers. This was a superb piece of advertising as the unexpected arrival of Concorde on a commuter run to Glasgow was quickly snatched up by the media and attracted a lot of TV coverage.\n\nMost passengers were absolutely thrilled to have this rare experience thrust upon them. However, there were one or two, and I hope you know who you are out there, who walked on completely nonchalant and whose only concern was to know if the flight would arrive on time! I guess the equivalent would be if someone was catching a Number 9 bus to work and along came a stretch limo and chauffeur to take him to his destination. Wouldn't there be some acknowledgement of such a pleasant surprise? I can't imagine anyone not being in awe of the upgrade.\n\nAt Glasgow Airport we were greeted in true Scottish style by some extremely tall bagpipe players who, positioned at the bottom of the aircraft steps, played the passengers off the aircraft and across the tarmac \u2013 not a bad way to start a morning's work.\n\nIn the icon category of famous passengers, the American movie producer of the James Bond films, Cubby Broccoli, is one I remember well. He had enormous presence. Albert 'Cubby' Broccoli was born in Queens, New York City and in 1962 produced the first Bond movie, Dr No, with Harry Saltzman. They went on to make more successful Bond movies until they parted company in 1976. Cubby's last movie, GoldenEye, came out in 1995 and he died a year later. He was indeed an imposing character and I guess many a female 'wannabe' actor in my situation would have been in awe of meeting this famous person. They might even have grabbed the moment to try to get an introduction into the movie world.\n\nHowever, I had no such aspirations as, several years earlier, I had been given a taster of movie making. I had flown into Toronto on a 747 trip and on our day off, myself and another stewardess went to a prestigious lakeside country club to relax and sunbathe. I guess this was a perk of the job that we had access to such establishments which cost the members dearly in annual fees. Having settled ourselves on sunbeds we did notice that there was unusual activity around us in the form of cameras, lighting and film crew. However, our attention was focused on chilling out and getting sun onto our pale winterised bodies. A while later, I was aware of a shadow standing over me and looked up to see one of the film crew looking at me. 'How would you like to be in a movie?' he asked. This was the director speaking.\n\nMy immediate answer was, 'I will have to call my agent first' (I did think he was joking). He smiled at my reply then asked, if I agreed to do it would I go to make-up and did I have any other clothes with me? As these offers don't come along every day, I disengaged myself from the comfortable lounger and followed the director to the film crew. Suddenly I was being treated quite royally. The guys did a lighting test on me, they patiently explained what I had to do (NOT look at the cameras) and after an hour of lights and cameras being positioned correctly we were ready to film. Then the male lead arrived whom I was to play opposite. I can't remember him engaging me in much conversation and it was down to work straight away.\n\nI have to add at this point that I was a stand-in for the famous star, Kim Novak. Apparently I had the same length blonde hair and, as I was to be slightly off the main shot and it was only a back view, it didn't matter too much about the looks! The scene took a brief 5 minutes in which the male lead answered the phone at a table on the terrace overlooking the lake with me beside him. Was I paid for my time? No. Did I ever get to see the movie on screen? No. And probably, like all good pieces of film, my part ended up on the cutting room floor.\n\n## STARS OF STAGE AND SCREEN\n\nIn the movie mogul world, those who could be counted as 'regulars' were actor and film producer Sir Richard Attenborough and Michael Winner. They often sat together. A quiet, unassuming character was Jim Henson, creator of the Muppets, who died in 1990. The legendary director Steven Spielberg also flew on many occasions.\n\nBarbra Streisand, who flew on it several times, was remembered for her comment that she believed the nose of Concorde was inspired by her own nose shape and therefore the aircraft should have been called 'the Streisand'. Lauren Bacall travelled with us and, despite having a reputation of being a tough cookie, Purser Julia van den Bosch recalls having a long heart to heart with her. Julia had just broken up with her boyfriend and Miss Bacall spent some time with her listening sympathetically. 'Keep busy' were her words of advice, which was not difficult to do in our job.\n\nIn the A-list film star category, Liz Taylor had to rank amongst the highest, along with Robert de Niro, Tom Cruise, Claudette Colbert, Ben Kingsley, Richard Dreyfuss, Peter Sellers and Michael Douglas, to name but a few. Sadly, some of those are no longer with us. Peter Sellers was accompanied by his actress wife, Lynne Frederick. After Peter Sellers' death in 1980 she went on to marry David Frost. This was dissolved a year later and she died at the age of 39 of substance abuse.\n\nSir Richard Harris, also now passed to the great heavenly Concorde 'check-in', was a regular and it was ironic that his normal persona of being a 'hell-raiser' and alcoholic was very much absent. As the flight progressed I found out why. He had given up drinking and turned vegetarian. Known for his famous role in Camelot and his award nomination for his role in The Field, he was also a poet, singer and performed in theatre. Latterly, his best remembered role was that of Dumbledore in the first two Harry Potter films. Sadly, this once flamboyant character died at the age of 72 in 2002 from Hodgkin's disease.\n\nThe late Bob Hope, comedian of great style, made the infamous comment when seeing the size of the Concorde toilet that it was no larger than a broom cupboard. 'Hey guys, you really have to decide what you're gonna do before going into one of these. Either reverse in, or forward for us guys.'\n\nThere was the elusive Rudolph Nureyev who confessed to the crew he was petrified of flying. He remained anonymous, wrapped up in a scarf and balaclava and immersed in his own fears for the whole flight. He died from AIDS in January 1993 at the young age of 54.\n\nAnother famous face who flew on Concorde was Grace Jones. On two occasions she boarded the aircraft wearing a floor length dark brown cloak with a hood (very Scottish Widows advert) that covered her face completely. She kept her face covered for the entire flight, neither eating nor drinking anything.\n\nJohn F. Kennedy junior, the son of President John F. Kennedy and Jackie Kennedy, was on a Concorde flight a short while before he died in 1999. Julia van den Bosch, who was purser aboard the flight to New York, commented that not only was he incredibly good-looking but he oozed charisma and charm. He was 38 years old at the time of his death. He had been piloting a light aircraft off the coast of Martha's Vineyard, Massachusetts, when it crashed into the Atlantic Ocean. His wife, Carolyn Bessette, and sister-in-law, Lauren, were also on the flight. All three were killed.\n\nOther famous celebrities who were pilots themselves always received an invitation to the flight deck. At the higher end of the pilots was Buzz Aldrin, one of the astronauts on the Apollo 11 mission to the moon in 1969. He was the second person to walk on the moon. He was made extremely welcome by the flight crew and I'm sure they had many stories to exchange about flying in space or on the edge of it. It is worthy of mention that there are more astronauts than Concorde pilots \u2013 a rare breed indeed.\n\nElizabeth Taylor was an icon I had always wanted to meet, and on the occasion she travelled with us it was undoubtedly a very special experience. She was between husbands at the time, having divorced American politician John Warner in 1982 and was yet to marry husband number eight, Larry Fortensky, in 1991. She boarded the aircraft with a male companion and sat at the rear of the front cabin. In the row across the aisle from her was Michael Douglas accompanied by a starlet.\n\nAs we taxied along the runway Miss Taylor was not in her seat. We found her holding the toilet door open and washing a cosmetic bag in the sink. When asked if she needed assistance she replied, 'No, its fine. I just spilt some make-up in my bag here.' And she obliged us by returning to her seat.\n\nI was working in the rear cabin for the flight but was keen to have another look at the movie legend I had always admired. My girlfriend Bernadette, working in the front cabin, suggested I take the drink Miss Taylor had ordered to her. 'And just take a look at those violet eyes,' she added. It was one of those moments in one's career one never forgets \u2013 the first sight of those amazing eyes. And the beauty that had decorated the film screens during my impressionable years. Never as a teenager would I have imagined meeting her. Drink delivered, I returned to continue the drinks service.\n\nOn my way up to the front cabin later in the flight I passed Bernadette chatting to Miss Taylor. I paused, curiosity getting the better of me. 'May I say, Miss Taylor, your ring is absolutely stunning,' Bernadette was saying. I paused some more. Bernadette was no stranger to expensive jewellery; in fact, along with clothes and shoes, it could be counted as one of her passions in life. Miss Taylor appeared flattered.\n\n'Well try it on, honey,' she said giving us both one of those intriguing smiles as she slipped the ring off her finger and handed it to Bernadette. This was the ring she wore almost daily for forty years. It was 33.19 carats and bought for her by her husband, Richard Burton. She removed it as casually as if it were a piece of insignificant costume jewellery.\n\n'May we?' Bernadette asked pointing to the toilets.\n\n'Sure, go take a look,' Miss Taylor replied, probably amused at the incredulity on our faces. Bernadette and I, ensconced in the privacy of the toilet, both tried on the ring, admiring its beauty in the mirror. What a moment to have such a beautiful diamond on one's finger, and belonging to such a movie legend. It was a pity that mobile phones with cameras were not on the market then. Having gloated over this enormous diamond we returned it to its owner. On Miss Taylor's death in 2011 the ring sold at auction for \u00a35.6 million, surely the most expensive piece of jewellery that will ever sit on my finger!\n\nAs a mention, Miss Taylor was not averse to removing her rings to show them to other people. There was an even bigger 69-carat ring that Richard Burton had bought for her 40th birthday. Sitting next to Princess Margaret at a dinner party the princess commented that the ring was 'vulgar'. Miss Taylor offered it to her to try on, which the princess did. 'Not so vulgar now it's on your finger, is it Ma'am?' was her brave comment.\n\nThe iconic film star Claudette Colbert shared common ground with Elizabeth Taylor in that they both played Cleopatra. Miss Colbert's performance was in 1934, almost thirty years earlier than Miss Taylor's. When Concorde started its winter schedules to Barbados, Miss Colbert used this service to fly to her home on the west coast of the island. In the late thirties she was the highest paid Hollywood actress and co-starred with names such as Edward G. Robinson, John Wayne and Clark Gable. When I met her at the age of 82 she looked incredibly youthful. She told me she loved Barbados and owed her good health to the climate there. 'And this,' she said, pointing to her topped up glass of champagne. 'These two things keep me in good health.'\n\nAnother Concorde passenger and film star of the old movie era was Lillian Gish. She earned the title the 'First Lady' of American cinema and had a film career spanning seventy-five years. Her role in silent cinema is legendary and her last film was at the age of 93 in which she appeared with Bette Davis. She died in 1993, aged 99. For such a lady the experience of flying Concorde must have been truly memorable. As she told my flying colleague, Louise Brown, when she was born flying hadn't even been invented. She was 10 years old when the American Wright brothers took to the air in their flying machine.\n\nHigh-profile film star Liza Minnelli had just married her agent, David Gest, when she flew on Concorde. She had, by all accounts, been very successful at a dramatic weight loss. During the breakfast service, she was about to accept a croissant from the bread basket. However, a halting hand from her husband reached across to stop her. Miss Minnelli did not argue the point and refused the temptation. A while later, however, she passed by the galley and asked for a croissant, which she ate with great relish. As a tribute to her attention to detail she did ask the crew member if there were any telltale crumbs around her mouth before returning to her seat and her watchful husband.\n\nConcorde was, more than any other aircraft, a magnet for the rich and famous and it was interesting to see that even well-known stars were excited about flying on it. In the fashion world, Ralph Lauren and Gloria Vanderbilt commuted from New York to London. Est\u00e9e Lauder, who co-founded the American Est\u00e9e Lauder cosmetics company with her husband, was a pleasure to meet. She was famous for her quote, 'There are no ugly women, only women who don't care or believe they're not attractive.' She was, without doubt, an amazing example of the excellence of her products. Her complexion was beautiful. I took it as a very Est\u00e9e(med) compliment when she commented that I had a good complexion.\n\nMick Jagger\n\nJohn Denver (courtesy of the author)\n\nRalph Lauren (courtesy of the author)\n\nMargaret Thatcher\n\nPaul McCartney\n\nLillian Gish\n\nRobert de Niro (courtesy of the author)\n\nHenry Kissinger (Unless indicated, all courtesy of Louise Brown)\n\n## ROCK STARS ON BOARD\n\nOver the years, Concorde carried a fair selection of the world's top musicians \u2013 rock stars and classical stars alike. In the operatic world Placido Domingo, Luciano Pavarotti and Kiri Te Kanawa were all celebrities we had the pleasure of meeting (as they were regular passengers).\n\nBoy George, the pop icon of the seventies and early eighties, was a colourful, friendly guy who was always expertly made up. When I met him he was travelling with his partner, male singer Marilyn. Before they boarded the flight from New York they had had a disagreement over something. This resulted in Marilyn sitting at the rear of the aircraft and Boy George in the front cabin. I was allotted to play messenger girl between the two in an attempt to sort out their differences. I obviously failed, as they got off in London still not talking!\n\nDiana Ross travelled solo on occasions. Her huge wide tresses of hair were amazing. I was on one flight working with my husband who had flown with her quite recently. In fact, when she boarded the aircraft she addressed him by his first name \u2013 that really impressed the crew. Sadly, there was an incident prior to leaving London when she was taken off the aircraft by police for insulting a security officer. However, she still flew Concorde back to New York the following day.\n\nUnlike today's high security where the flight crew are locked in the flight deck and have very little interaction with the passengers, in the 1980s any passenger who asked to see the flight deck was automatically invited up. When a name like Dolly Parton appeared on the passenger manifest the invitation from the captain was issued very smartly. I clearly remember the captain's beaming expression when we told him Dolly Parton was on board. 'Bring her up to see us,' he said enthusiastically.\n\nThe engineer and first officer were also very keen to meet her. After finishing her meal Dolly, looking every inch the glamorous star and as beautiful in the flesh as she appears in films and on TV with her waspish waist, stunning figure and a mass of blonde hair (a wig which she owns up to), sauntered up to see the 'boys in charge'. She was up there quite a while and I'm sure the conversation didn't exclusively revolve around the technicalities of flying. When Dolly emerged from the flight deck she gave us one of her radiant flirtatious smiles. 'Hey, I sure left those guys feeling horny,' she remarked, with a wink and in her deep southern drawl.\n\nRobin Gibb first came on one of my flights with his personal assistant, Dwina. Robin was a strict vegetarian (like Sir Richard Harris) and special food was always put on for him. The second time I met them, Dwina was blossoming with pregnancy and the third time, Dwina was married and now to be addressed as Mrs Robin Gibb. It was interesting to note the different changes in circumstances with some of our regulars.\n\nLike Robin, Barry Gibb flew regularly to Miami where both brothers had homes. Other singers who travelled with us were Ringo Starr with his wife Barbara Bach, and Paul McCartney. It was not unknown for Paul McCartney to pick up his guitar and sing in the aisles when he was married to Linda. On one occasion when steward Steve Whitehead was working on board, it was his birthday. Elton John was also on the aircraft and both he and Paul McCartney sang a duet to him.\n\nMichael Jackson, unlike his public persona of being a very private individual, was very affable to the crew and passengers. Purser Julia van den Bosch recalls being nervous about asking for his autograph, but he was more than happy to sign menus for her and other fans on the flight. He was surprisingly interactive and he was a listening person, Julia recalls, a rare attribute amongst some of the superstars. Jackson met his untimely death in 2009, aged 50.\n\nWhitney Houston's father, John Houston, travelled with us from New York to London at a time when his daughter was at the height of her career. He was her business manager and Whitney was on tour in Europe. Chatting to him after the meal service, I commented to him on what a talented daughter he had and how I loved her music. He thanked me and asked if I would like to see her in concert in London. Two tickets were arranged for collection at Wembley Stadium and my girlfriend and I had superb seats to watch the talented performer.\n\nMy husband, however, was unable to go as he was away at the time. Two weeks later, John Houston was on a flight returning from London to New York. He remembered me and asked if I enjoyed the concert. I enthused about it and thanked him profusely, adding that my husband was quite envious that he had missed it owing to his work schedule. Without a second's hesitation Mr Houston suggested, 'Why don't you take him to the next concert?' Whitney was performing in Germany and then returning to Wembley for the last part of the tour. Sure enough, John arranged tickets from the New York office and my husband and I were able to see her. It showed the caring and kindness of Whitney's gentle-natured father.\n\nThere were other occasions of generosity shown by the passengers. When the renowned chef Delia Smith flew to Barbados one Christmas she invited all the crew to lunch at the Royal Pavilion on Boxing Day. Similarly, Burton Race extended an invitation to all the crew to his Michelin-star restaurant in Berkshire.\n\nFlying Concorde opened up so many opportunities, and in some places even just the word 'Concorde' worked the magic. Steward Steve Whitehead had flown in with the crew to New York for an overnight stopover. Learning that Frank Sinatra was appearing at Carnegie Hall that night he wasted no time in going down to the box office, a few blocks away from the hotel, to see if there were any tickets available.\n\nNot unexpectedly, Steve was told the concert was a complete sell-out. Looking suitably disappointed, Steve said he had flown over on Concorde in the hope of seeing Frank Sinatra in concert (not entirely untrue) and whether there was any chance there might be a spare ticket. The guy in the box office, obviously impressed that someone had taken Concorde for the event, told him to come back at 5.30 that evening. Dressed smartly in a suit and looking 'the business', Steve was escorted through the hall and to a box 50ft from the stage. As the evening got under way he was joined by some people in the box next to him. Once Lionel Hampton had done the warm-up, Sinatra came on stage. After the interval, he announced that there was one of the greatest American singers in the audience. The spotlights zoomed on to the person sitting next to Steve, and Sinatra introduced Tony Bennett. 'A memorable evening', Steve recalled...\n\nOne of the events that caught the interest of the world's media was the Concorde flight on 13 July 1985 with Phil Collins aboard as part of the Live Aid concert. This monumental event, organised by Bob Geldof and Midge Ure in aid of Ethiopia's starving population, was televised around the world. Taking place at Wembley Stadium in London and JFK Stadium in Philadelphia it was described as the biggest rock concert ever, with an amazing line-up of talent. Princess Diana and Prince Charles attended the concert in London.\n\nPhil Collins took to the stage, playing his numbers and a duet with Sting before leaving by helicopter to Heathrow where he caught the evening Concorde flight to John F. Kennedy Airport. It was reported that this was watched by 1 billion TV viewers. In a moving gesture, Concorde flew low over the stadium and as it did so dipped its wings to the roar of approval from the crowds. It was, as a stewardess friend of mine who was there said, 'a very special moment in a very special day'. When the flight landed in New York, Phil Collins was taken to Philadelphia by helicopter where he performed on stage as part of the Live Aid concert there.\n\n## THE SUPERSONIC COMMUTERS\n\nOf the regular commuters on Concorde no one stood out more than Fred Finn. He flew Concorde as often as we did! He knew us well, called us by our Christian names and was always very relaxed on board. He flew a remarkable 718 flights on Concorde and was named as the world's most travelled Concorde passenger, having clocked up 3 million miles, the equivalent of flying to the moon and back six times! Later, Guinness World Records awarded him the accolade of the world's most travelled man, having clocked up 15 million miles in total. Fred Finn is now president of the 'Save Concorde Group'. This team leads an enthusiastic membership aimed at getting Concorde into the skies again, albeit in a heritage capacity.\n\nAnother familiar face was David Springbett. David was a caring man who was always ready to listen to one's problems and give good advice, especially about career moves. If the cabin service had finished, he would invite you to sit down and for a few minutes we would put the world to rights. He was a great adventurer and had he been born a century or two before, I'm sure would have been one of the world's great explorers. He had huge belief that everything was achievable.\n\nA few years previously he had set the world speed record for flying round the world and was entered in the Guinness Book of Records. In 1980 he had flown subsonic from Los Angeles to London, then on to Singapore via Concorde. The next stops were Bangkok, Manila, Tokyo and Honolulu, arriving back in Los Angeles on a 747. The trip had taken 44 hours and 6 minutes. He had also ensured that this trip would not cost him a penny. He had bet $500 at 20\u20131 that he could beat the existing record, and thereby raised enough money to cover the cost of his challenge.\n\nA year later he beat the world record for getting from London's financial district to Wall Street, New York. Naturally this included a transatlantic flight on Concorde. With helicopters standing by at both ends, the one in London landing on a runway 20m from Concorde, he beat the record by 11 minutes, in a time of 3 hours, 59 minutes and 44 seconds. He also beat the record going from west to east.\n\nHe was yet another personality whose enthusiasm for life was infectious and it was stimulating to come across these characters on a regular basis. David kindly gave me a signed copy of the book Skiing Down Everest and the Guinness Book of Records, in which his records are documented.\n\nAnother speed record-breaker was Richard Noble. Delightfully unassuming, he was nonetheless fascinating and spoke quite humbly of his exploits. In 1983 he had helped break the world speed record in the jet-powered Thrust 2 that reached speeds of 633.46mph in Nevada's Black Rock Desert. In 1997 he was project director of the team that built Thrust SSC, which was to break both the record and the sound barrier on the desert sands of Nevada with a speed of 764.03 miles an hour (Mach 1.02).\n\nRichard also went into the aviation record books. In November 1987 he completed three Atlantic crossings on Concorde in one day (not something I would have chosen to do, even as a passenger). His new project, Bloodhound SSC, is the ultimate 1,000mph jet and rocket-powered car. It will be capable of achieving Mach 1.4, and is to be driven by Andy Green. This is, first and foremost, an international education initiative designed to further careers in science, technology, engineering and mathematics by showcasing these subjects in the most exciting way possible. This fascinating project will be tested in 2015 and 2016 in South Africa.\n\nIn the sporting world there were the extremes of emotion on board Concorde with the golfing elite. When the Brits brought home the Ryder Cup there were huge celebrations on board. A not-so-jubilant group were the American team after a 'drubbing' at the Belfry, recalls flying colleague Dick Bell. 'The mood on board was none too pleasant and the team proceeded to brood all the way home. Mind you,' says Dick, 'the crew were all golfers and smirking for most of the flight!'\n\nIn the Formula One motor-racing world, Bernie Ecclestone, F1 supremo, was a regular and liked to sit in seat 1A. Having known F1 team owner Frank Williams through my F1 days, it was a pleasure to meet him again on a couple of flights. He is a true gentleman and has managed extraordinarily well to lead a normal life despite the permanent injuries from his tragic accident.\n\nAnother passenger it was a pleasure to meet was none other than our boss, chairman of British Airways, Lord King. He was flying to New York with the American ambassador and he was extremely charming and approachable. A very likeable boss! He was responsible for transforming the loss-making airline into one of the most profitable in the world. In 1981, he purchased Concorde directly from the government and within a year was turning this 'white elephant' into a profit. Advertising experts were called in to change the brand image of the company and, although 23,000 jobs were shed, Lord King managed to boost staff morale when it was at a low ebb and modernise operations. Media reports suggest that the London\u2013New York service made an annual \u00a320 million operating profit by the early 2000s. Lord King passed away in 2005.\n\nNo chapter on the famous and regular commuters would be complete without mention of Sir David Frost. Sir David was a very regular flyer, commuting up to twenty times a year across the Atlantic and making 400 flights in twenty years. Known for his untidiness with the newspapers and his love of red wine, he was nonetheless a colourful character whom it was always a pleasure to serve. On the last ever commercial flight on which he was a guest he was asked what made Concorde special, and he summed it up beautifully by saying it had a 'friendly club atmosphere'.\n\nIf Sir Richard Attenborough was on the flight they often sat together, and if awards could be given for the most charming, affable and indeed lovable character it would have to be Sir Richard. Impeccably mannered, he was always most endearing with his 'darling' prefix when addressing us.\n\nA very close second would be John Travolta. Known for his acting roles in Saturday Night Fever and, later, Pulp Fiction, John is an accomplished pilot, flying his own 707. He also owned three Gulfstreams and a Learjet. No doubt John enjoyed his supersonic flying experience, and despite his stardom and fame was one of the nicest and most unassuming of the A-listers.\n\n## [THE 11-MILE-HIGH CLUB \u2013   \nTHE SHORTEST CHAPTER](contents.htm#a30)\n\nUndoubtedly any discussion about Concorde from those curious about the supersonic lifestyle would have to include a subtle enquiry about any shenanigans that took place in flight.\n\nI am sorry to disappoint my readers but there was very little that went on, to my knowledge, and the lack of space was more than likely the reason. The toilets were more cramped than other aircraft and although I know of one famous American movie star who disappeared in there with his travelling companion, I doubt that whatever they got up to was very comfortable! These events were far more common on 747s due to the length of the flight, the easy access to alcohol and no doubt the boredom factor.\n\nThe elite members who joined the 11-Mile-High Club were few and far between, and they ran the gauntlet of being found out \u2013 which they invariably were. One couple who had managed to sneak into the toilet together were obviously getting carried away as the door was taking a good pounding. One of the stewards was concerned that this might disturb the passengers and informed the flight crew of these antics. There followed an announcement from the flight deck that went something like this: 'Ladies and gentlemen, we are now flying at Mach 2, twice the speed of sound and at a height of 58,000ft and for the two passengers who have just joined the 11-Mile-High Club \u2013 congratulations!'\n\n'The male in question returned to his seat as cool as a cucumber, the female emerged with a face as pink as rhubarb,' commented my flying colleague, Bernadette. On their seat was a certificate signed by the crew. It read: 'We hope the Concorde was the climax of your trip.' Another crew member had written, 'So pleased you could come on Concorde today.'\n\nOn another occasion, a couple sitting up at the front of the aircraft were getting amorous and it was noted that they moved to the rear of the aircraft, which was pretty empty. Still puffing on his cigar the male passenger put his other hand to good use with his partner, and her moaning could be heard by the crew in the galley. The three aft cabin crew concealed behind the galley curtain peered out to see what happened next. Like the other passengers they were also handed a certificate with appropriate comments written on it. Part of our training was observation \u2013 and there was little that escaped our notice...\n\n## [BREAKING RECORDS: WASHINGTON TO NICE, 11 SEPTEMBER 1984, AND A  \nKIND-HEARTED BELLBOY](contents.htm#a31)\n\nIt would be fair to say that there were not many people in the world who could afford to charter Concorde in the eighties and indeed I don't know of anyone who did, apart from the trip I am about to describe. The mother of the king of one the Arab royal families had been in Los Angeles for an operation. She and her entourage had flown from Los Angeles to Washington, where it had been arranged for a Concorde to fly her to Nice.\n\nAs I was selected for this VIP trip, the schedule was to fly out to Washington, spend the night in a hotel there and the following morning to fly our guests to Nice. At a special briefing in London we were given specific instructions on how to serve food and beverages in the Arab custom \u2013 only with the left hand. No alcoholic beverages would be offered in accordance with their drinking laws. Arabic coffee and pots would also be part of the catering along with Arabic food. And finally it was compulsory to wear white gloves to 'meet and greet' the royal family.\n\nOur Concorde flight out to Washington was very relaxed as our twelve passengers consisted of BA staff who were part of the organisational team for this exclusive charter. On our arrival at Washington Dulles Airport we were soon aware of the hospitality of our Arab hosts who had spared no expense in looking after us. We were met by three extremely long, black stretch limos complete with tinted glass, TVs and a bar. There was one for the flight crew and two for the cabin crew of six (they obviously didn't want to overcrowd the mobile party room!). No sooner had we closed the doors than the sound of corks could be heard popping and glasses of champagne raised to such a pleasant day's work. It was another memorable moment to add to the long list of everyday life with the supersonic fleet.\n\nIn the limousine, royal Arab charter.\n\nThere were ways and means of acquiring the white gloves! Royal Arab charter.\n\nOur accommodation for the night was a superior five-star hotel and much grander than any we normally stayed in. If this was how the rich and famous lived then I guess we were fortunate to enjoy their lifestyle for a short time. However, the euphoria of champagne and luxury soon dissipated when I unpacked my suitcase to find that my white gloves that were compulsory for this trip were missing. Since this small detail had been emphasised at the briefing and we had all nodded a positive 'yes' when asked if we had all our gloves with us, I was thrown into a mild panic.\n\n'Where could I buy a pair of white gloves?' I asked the hotel receptionist, swallowing hard and very aware of the late hour. This would not look good on my records if, having been selected to do this exclusive charter, I turned up barehanded. The receptionist glanced at the clock. It was now six in the evening and she was sorry to tell me but all the stores would be closed by now. In another panic I phoned all the girl crew and asked if they had packed a spare pair. The replies were all negative. Then I had a flash of inspiration \u2013 had I not seen the bellboy unloading our luggage wearing white gloves?\n\nImmediately I hotfooted it down to the lobby and, yes, there he was still wearing white gloves. But how could I persuade him to part with them? Asking nicely was my only tactic, I decided, unless I was to take him hostage in a corridor to save my own skin. The bellboy looked at me doubtfully. They were his only pair and he too needed them for his job. If anyone in the immediate vicinity had overheard me pleading with the bellboy to part with his gloves I'm sure they would have been greatly amused. Finally, with the promise of a leather Concorde luggage tag, the deal was done. The gloves were removed, the tag handed over and I was saved. However, the gloves were a tad too large and that evening a lot of time was devoted to shrinking them in hot water and restoring them to an acceptable standard of white.\n\nThe royal Arab charter \u2013 the crew at Nice Airport before flying back to London.\n\nThe certificate we received from Captain Cook on the royal Arab charter. We made history with the longest ever Concorde flight, 4,566 miles (642 subsonic) in 4 hours 7 minutes.\n\nIn Nice after the royal Arab charter.\n\nAfter a leisurely breakfast and feeling every inch superstars, we waited as the enormous black stretch limos pulled up outside the hotel to take us to the airport. The Concorde ground staff were already hard at work. Seats from the front few rows had been removed to allow more space for the royal party. The special catering, of which there was a consummate amount and enough to furnish a banquet, had to be stowed so that we knew where everything was.\n\nWhen all was ready, we waited for the party who, not unexpectedly, were late. The captain no doubt had a slot of take-off times on his mind but when you charter the world's most expensive aircraft I guess it is the guests' prerogative to turn up when it suits them.\n\nI was more than ready to display my newly acquired gloves and congratulated myself on the deal that had been brokered with the bellboy. After what seemed an interminably long time the royal party arrived. The royals, which included the mother of the king and four princes and princesses, were seated in the front section, whilst the entourage of forty-eight were seated in the rear. Suffice it to say the royal party itself was very relaxed, whilst we found the entourage the most demanding of passengers.\n\nIt was dark when we landed at Nice, but as we turned off the runway into our specially allotted parking area we were left in no doubt of the prestige of our passengers. Waiting for the doors to open and welcome the royals were a body of smartly suited minders. Behind them sat a queue of seventeen Mercedes limousines waiting to ferry the passengers to their destination.\n\nOur working day finished at a hotel in Nice, and over drinks and a debrief Captain Cook informed us that we had made history that day. The flight from Washington to Nice was the longest recorded, a distance of 4,566 miles in 4 hours and 7 minutes, and the flight crew had signed certificates for us to record this superlative day.\n\nThe only remaining task for this charter was to reposition Concorde back to London. Captain Cook, being one of the most amiable of captains, asked us, probably tongue in cheek, what time we would like him to schedule our departure (that was definitely a first... and last in my flying career). We all agreed that after breakfast would be perfect and so at 10 a.m. precisely we left to return the aircraft to London. The captain performed a splendid take-off and fly-past over Nice Airport. 'Just to show the French we know how to fly these aircraft', he was heard to comment.\n\nWe were soon headed back to the real world and the routine of normal flights, but not without a handsome tip for all of us from our Arab guests.\n\n## [LIFE DOESN'T ALWAYS  \nGO ACCORDING TO PLAN](contents.htm#a32)\n\nThere were many tales about the seating arrangements on the aircraft. One financial advisor to an Arab royal family booked a seat purely for his briefcase, which no doubt contained slightly more money than the cost of the fare...\n\nAnother time, a high-ranking minister came onto the aircraft and was very insistent that he didn't want anyone to be seated next to him and asked specifically not to be interrupted during the flight to New York as he had a considerable amount of work to do. It was not unusual for many of the commuter executives to go straight to meetings after their arrival into JFK. The flying time was therefore used for work preparation, and it was not uncommon for some of them to attend their meetings and fly back to London that same day on the evening flight.\n\nOn this particular flight there was a very attractive female with long, dark hair, equally dark eyes and slim figure. She spoke with a South American accent and was extremely dressed up for the occasion. Wearing a short cream silk dress (had she just come from a party the previous night?) and jewellery that would fill a Boodles shop window, she did stand out amongst the mostly male commuter flight to New York.\n\nDespite the lack of space in the aisles, she left her seat and strolled slowly up and down the cabin casually making note of the passengers. I watched her with interest (again, careful observation is part of our job) as it was unusual for a passenger to be so restless. When I next spotted her she was bending over talking to the minister who was supposedly hard at work. 'Is everything alright, sir?' I asked, discreetly glancing towards the South American beauty who had by now seated herself next to him.\n\nHe smiled as if he had just won the jackpot. 'Everything's fine,' he replied, looking at me guiltily.\n\nI hoped ministers weren't in a habit of changing their minds so readily about government decisions as this one was about his seating arrangements that morning. As I served the meal to them, there were no signs of the minister's copious amounts of paperwork. The notes had been put away and the two were heavily engaged in conversation. All was obviously going well between them until, during the coffee service, one of the stewards came blustering into the galley. 'Mon Dieu!' he exclaimed. 'I have just spilt a glass of port over a lady's cream dress. Well, it wasn't my fault exactly. She knocked my hand.'\n\nWe both grabbed cloths and soda water and rushed up to where the accident had happened. This, I hasten to add, was a very rare occurrence and it would reflect badly on our standards of exemplary service. It was the pretty South American girl who, very embarrassingly, had a very large red stain in the crotch area of her dress. To our surprise she refused our help to remove it. 'Eet is okay,' she said. 'I will do it mayself.'\n\nThe steward was profusely apologetic and suggested that maybe she go into the toilets to use the soda that was very good at removing stains, he assured her. Showing not the slightest sign of distress at the ruined silk dress, she stood up and went towards the toilets in the mid-section of the aircraft. Alongside these are the crew take-off seats and a small area that can be curtained off. When I next passed, there was a very cosy, intimate picture behind the curtains of the minister kneeling with his hand halfway up her dress attempting to remove the stain with a cloth. I did wonder, as they departed the aircraft, whether it was his intention to buy her a new dress or just remove the one she was wearing in one of New York's hotel rooms...\n\nIt was very rare in all the years of Concorde service that our food made bad tabloid news. However, there was one serious incident when some of the food was contaminated with the food poisoning bug salmonella. Passengers and cabin crew were taken ill, some of them seriously. The source was traced to aspic on the canap\u00e9s.\n\nThe whole episode proved to be very expensive for BA, both from a cost point of view and the public relations angle. All claims were met very generously. However, there were some who were taking no chances. Businessman Peter de Savary, who had been hospitalised after this outbreak, was in no mood to jeopardise his health on his next Concorde flight, and who could blame him? Greeting him on the flight I was handed a Fortnum & Mason's hamper to be placed in the hat rack. At the start of the meal service I guessed what was coming, but the question had to be asked \u2013 was he dining with us? He replied no and looked taken aback, as if I had just asked him if he wanted another salmonella sandwich. The hamper was removed from the hat rack and he proceeded to eat a luxurious picnic with enthusiasm.\n\nAnother story about quirky eating habits is recalled by a colleague of mine, cabin service director and cabin crew manager Dick Bell:\n\nI remember once we had the actress Dyan Cannon as a passenger. She boarded with a large hatbox and did not want to give it up, so as the seat next to her was empty we strapped the hatbox into it. Did she want a drink? the crew asked. 'No', was the reply. Was she going to eat on the flight? Again, the answer was 'No'. After take-off she opened the lid of the hatbox and it was full to the brim with popcorn. She ate the entire contents of the box during the flight with no liquids with which to wash down the popcorn, not even water.\n\n## DO NOT UPSET THE CAPTAIN!\n\nThere is probably only one occasion where I have witnessed the captain having a difficult time with a Concorde passenger. It was an evening flight to New York and we were all anxious to get away as the weather was deteriorating. However, we had a tall American who swaggered on board and promptly lit up a cigar. Politely, I asked him to extinguish this as it was against regulations, for the simple reason that the aircraft had just been refuelled (we had about 95 tons of it around us, to be exact).\n\nThe passenger carried on smoking as if he hadn't heard a word. Getting somewhat concerned at this flagrant breach of safety I repeated my request more urgently. This was having absolutely no effect, so the cabin service director stepped in and still there was no response.\n\nFinally the captain walked smartly down the aisle, hat on, jacket buttoned and looking every inch the commander. I have to say that he was not one of our tallest captains, rather short in fact, so when this very tall American stood up there was a considerable height difference. The American was not budging on extinguishing the cigar and one can only assume that his bravado was the result of too many pre-departure drinks in the lounge. It was a battle of wills, and to make matters even worse the American blew smoke into the captain's face. At this flagrant insouciance, the captain's face turned rouge. This was undoubtedly, I'm sure, the first time anyone had ever questioned his authority in uniform. It was the spark that lit the touch paper.\n\nAt the sight of two uniformed airport police approaching his seat, the American finally realised that he was in a bit of trouble. Hastily extinguishing his cigar, he was ignominiously escorted off the aircraft whilst pleading that he had to get home for an important dinner that night. Shamefully for him, he missed it and spent a night in jail before receiving a heavy fine and leaving the next day.\n\n## ALL IN A DAY'S WORK\n\nIn every strata of society there are always the prima donnas and quite unsurprisingly there was the odd one on Concorde flights. This particular lady was married to a media baron. She was indeed one of the world's lovable eccentrics.\n\nThe aircraft doors had already closed, but were hastily reopened for her late arrival. She was travelling with her son to Miami. However, it transpired that he had lost his passport and therefore had to stay behind. Our middle-aged passenger was dressed in a black taffeta mini skirt, a creased, cream, flouncy silk shirt complete with coffee stains and flat brown lace-up shoes. As she bent over in the middle of the aisle, and in full view of the passengers, she made no attempt to hide an ample display of leg covered in black tights and a crepe bandage over one knee.\n\nPrior to the meal service, she had been drinking copious amounts of champagne interspersed with Panadol and smelling salts. On the presentation of her meal of guineafowl she immediately asked for more potatoes and added that if she didn't like the fowl she would change it for veal. The afternoon tea service was also met with a demand for more sandwiches.\n\nOn arrival in Washington, usually an hour's stopover before the sector to Miami, 'her ladyship' insisted that a message be sent to her husband to order some tranquilisers that she had left at home in London. Just as the doors were closing for departure she again insisted that another message be sent to her maid in London, as she had lost her American Express card and she thought it could be in her bed!\n\nAt this point, our normally very relaxed cabin service director came to the rear of the aircraft to tell us this story and he was, by this time, pretty wound up. Still drinking champagne, our passenger decided to order a wheelchair for her arrival, not a bad idea considering the amount of alcohol she had drunk.\n\nWe later heard from the ground staff that she had plagued them for 4 hours after her arrival. We last saw 'her ladyship' checking in for her flight to the Caribbean arguing the case against paying excess luggage. It turned out that she had twelve excess bags and this was going to cost her $80 apiece. 'But my maid packed for me,' she protested. Still, the check-in girl was unrelenting.\n\nThen our harassed passenger tried the well-used celebrity tactic, 'Do you know who I am?' The calm response came that even if she were the President of the United States she would still have to pay for excess luggage. So we left her ladyship flying off to an island in the Caribbean, her son left behind in England, her American Express card somewhere amongst the bed clothes and her tranquiliser prescription somewhere in the making. Fortunately, not all our passengers had such domestic problems.\n\n## [THE ROYALS AND OTHER  \nFAMOUS FLIGHTS](contents.htm#a35)\n\nIn February 1977 the Queen and Prince Philip flew Concorde for the first time from Barbados to London at the end of her Silver Jubilee tour.\n\nThen, in February 1979, they flew to Kuwait at the start of a Middle East tour. Cabin crew member Jeannette Hartley, who was on the tour, remembers it vividly. The Queen, Prince Philip and their entourage disembarked in Kuwait whilst the Concorde crew flew on to Bahrain and had four days there to await the royal party who were arriving on the Royal Yacht Britannia. The crew were hoping to visit the royal yacht the following afternoon but were turned away by armed guards. However, once they had explained in greater detail that they were Her Majesty's Concorde crew they were let through the gates and enjoyed a wonderful visit aboard the yacht. Other places on the tour were Riyadh and Dhahran, where the Queen met the crew and shook hands with them.\n\nIn 1985, the Queen Mother flew supersonic on her 85th birthday. This was a birthday present to her from British Airways. She was accompanied by Viscount Linley, Lady Sarah Armstrong Jones and Chairman of BA, Lord King. The Queen Mother had mentioned to Captain John Hutchinson that every time she heard Concorde coming over Clarence House she would go out onto the balcony and wave. John responded by saying that, in future, when he flew Concorde over London he would flash the aircraft headlights in salute. It became a tradition for all Concorde captains to do this thereafter.\n\nAnother royal who had flown Concorde on several occasions was HRH the Duke of York. He had a very relaxed persona. One time, on his return from Shannon, the flight was quite empty and some of the crew were taking a break and having a meal at the rear of the aircraft. Imagine their embarrassment when they saw the prince walking down the aisle as they were tucking into a steak. He quickly put them at their ease and chatted to them as casually as if they were all old friends.\n\nA colleague of mine, Dick Bell, recounts his experiences of flying with the royals:\n\nIn 1984 I became a cabin crew fleet director and with two other colleagues managed all of the Concorde cabin crew. Along with my day-to-day duties I was also given the responsibility of managing flights for the royal family and the prime minister of the day and any other special flights such as air shows and Ryder Cup flights. These flights were planned down to the finest detail. Wherever you went you knew there would be thousands of people watching your arrival, such was the appeal of the aircraft, so everything had to be just right. It really was a flying superstar.\n\nTo fly in formation with the Red Arrows was fantastic but not as nerve-racking as the first time I was involved in the arrangement and operating of a royal flight for Her Majesty the Queen. Many other royal flights followed, but the first is always the one you remember the most. It was a twelve-day tour of the United States and, like the Queen Mother, the Queen is a joy to look after and to be around. She puts you at ease the first time you meet her.\n\nThe crew underwent several briefings during the build-up to the flight, usually starting a month before. The Air Commodore of the Royal Flight was in charge of the itinerary and the crew worked closely with him on protocol matters. His advice was to 'be natural'. The briefings included everything from the Queen's likes and dislikes, to menu choices and protocol, and how she liked her drinks. Two favourite drinks of Her Majesty are a very dry Martini and a gin and Italian vermouth, the old 'gin and it', as we knew it. The only difference is that Her Majesty likes it two-thirds gin and one-third Italian, instead of the other way round (nice, I tried it). Prince Philip had a preference for Watney's Red Barrel that, somehow, was still available.\n\nBecause these flights were regarded as private flights there were no announcements and no safety demonstrations. Her Majesty made herself at home and had photographs of the family around her in the cabin that was furnished with a dining table and two chairs. She is a small eater and, although I have mentioned her favourite drinks, her consumption of alcohol was minimal.\n\nThe 1991 US tour included stops at Washington, Miami, Tampa and Austin, Houston and Kentucky. The Queen travelled with her entourage including her surgeon, ladies-in-waiting, a butler, maids and hairdresser and many others. During the flight, the Queen would invite prominent members of her entourage \u2013 maybe her surgeon and the foreign minister \u2013 to dine with her. It was interesting to note that the age-old royal protocol still firmly applies that when the Queen finishes eating, so does everyone else around the table.\n\nAn interesting part of any royal flight takes place after landing. Before landing, the Queen's dressers (she has two) laid out on her bed the outfit she was to wear on arrival along with the jewellery, gloves, hat and handbag. As soon as the aircraft pulled off the runway and whilst taxiing, the Queen went into her 'bedroom' to change, so when she emerged from the aircraft she had only been wearing the outfit for no more than 2 minutes. And the aircraft had to arrive exactly on time \u2013 to the minute.\n\nOne incident I will always remember happened just prior to landing in Washington. The Queen was finishing her lunch and she knocked a glass of red wine directly into her handbag, which was beside her on the floor. I took away the handbag and cleaned it up.\n\nOn another occasion on a 767 flight to South Africa the aircraft landed to a very windy day. The doors were opened and as the Queen stepped out in her somewhat large hat she saw the press standing behind the barriers waiting to photograph her. Behind clenched teeth and looking directly ahead she commented, 'I think this is the wrong hat!'\n\nBeing the consummate professional that she is, she managed to battle with the wind and keep her hat firmly in place.\n\nJulia van den Bosch recalls her experiences of royal flights where the ladies-in-waiting had to change in the toilet, not an easy task in such a confined space, whilst the accompanying male members changed into their ceremonial dress in the cabin. During the tours to the USA they landed at air force bases and not the hugely organised commercial airports. Here, there were no catering trucks driving up alongside to replace the used cutlery and plates. Julia recalls that whilst they were lavishly entertained at embassy parties on the tour they still had the job of washing up in the small galley sink, a far cry from the glamour of the tour they were on.\n\nFlying colleague Sue Drayton hosted Princess Diana on Concorde to the annual Council of Design awards at the Lincoln Center in 1986. (Princess Diana, at that time, had dramatically changed hairstyle to having it gelled back for the occasion.) Sue recalls that her amazing navy dress hung on the back of the flight deck door and Sue was most concerned that it didn't arrive smelling of food from the galley, which was just behind the entrance to the flight deck.\n\nAnother of the concerns when royals are flying is that the toilets are always kept pristine, and checks are made constantly. However, just before landing it was not unknown for excess rubbish to be bagged up and stored in the toilet for landing. Sue had just done this when Princess Diana made a last-minute visit to the toilet. On opening the door and seeing the confined space loaded up with rubbish bags, she smiled at Sue saying, 'Sorry, a girl's gotta do what a girl's gotta to do', and waited patiently whilst Sue hastily and, no doubt somewhat embarrassed, removed the offending bags.\n\n## CREW SHENANIGANS: YOU WOULD EXPECT IT, WOULDN'T YOU?\n\nThe flight crew were always good fun to be with as they loved their job and were always enthusiastic about their work. There were twenty-four captains and the same number of co-pilots and flight engineers. Some had an interesting history. For instance, one had won the King's Cup Air Race, another had been the UK aerobatic champion and another as a young fleet air arm pilot had been involved in a mission to sink the Torrey Canyon, a tanker that had run aground on the Cornish coast in 1967. Suffice it to say, he missed his target and his first drop incinerated the cliff top! This became a bit of a joke amongst the flying crew. In fact, on one Concorde flight the first officer calmly announced that the captain flying the aircraft had been assigned to bomb the Torrey Canyon and missed. There was nervous sniggering amongst the passengers...\n\nIt was inevitable that members of a small select fleet leading a pretty exclusive lifestyle and enjoying nights away in romantic places would form close friendships. It happened on long-haul flights. It also happened on Concorde.\n\nSometimes these 'friendships' developed into something more, and inevitably relationships were sometimes formed between the flight crew and stewardesses. The flight crew also had a bid line so it was simple to arrange flying with a woman of their choice! There was the heady cocktail of attractive, single (and sometimes not so single) females and the silver braid of the flight crew flying a prestigious plane and, hey bingo, it's a mix as lethal as a cigarette and a tanker of fuel. We all knew who was seeing whom. Some made little secret of the fact. If a couple did manage to keep their relationship low profile, it didn't last very long. Everyone who knew went about their own business and passed little comment, and they did not play 'judge and jury' at what went on.\n\nI do recall one amusing evening, but perhaps not so amusing for the couples involved. British Airways held a reception for Concorde flight crew and cabin crew and their wives and partners at the Concorde Centre at Heathrow Airport. I remember it well as I arrived on crutches having just torn my knee ligaments on a skiing holiday. However, I'm sure the pain in my knee was mild in comparison to the discomfort of some of the flight crew who arrived with their wives.\n\nIt was very unusual for any of us to come face to face with the 'anonymous spouses'. I think it was generally a case of 'out of sight, out of mind' for some of the crew. And I don't remember many incidences where the pilots took their wives away on trips with them \u2013 for the simple reason that the New York trips were so short, it was hardly worth all the effort of flying 3,500 miles for one night on the town.\n\nSo, at this party there was a highly charged atmosphere of girlfriends mingling together, while avoiding their flying 'other halves'. It really was a case of keeping an eye on who was in the room and sidestepping around any would-be hazard in the form of a wife. Because of this, the atmosphere was quite strained and I did wonder how many present actually enjoyed the evening.\n\n## THE CONCORDE CRASH\n\nOne of the questions I am most frequently asked when Concorde comes up in conversation is why did the world's only supersonic aircraft have to stop flying and was it the Air France Concorde Paris air crash that sealed its fate?\n\nOn 25 July 2000 the unimaginable happened. An Air France Concorde with a full complement of 100 passengers and nine crew took off from Charles de Gaulle Airport. As flight AF4590 gathered speed down the runway and lifted into the air, the controllers in the control tower, who had just given the aircraft clearance for take-off, were horrified to see flames pouring from its left-hand side wing. They wasted no time in informing the captain (the flight crew would have been unable to see the catastrophic fire that was about to change the destiny of that flight and all those aboard).\n\nAt less than 200ft, the aircraft reared up, its damaged wing now engulfed in fire. It dipped to one side and fell back to earth. The doomed aircraft had made just 90 seconds of its flight. The captain's intention had been to take it to Le Bourget Airport, a few kilometres away. As it was, the aircraft could not make that distance and came down in the suburb of Gonesse, hitting a small hotel and killing four people and all passengers and crew. The intensity of the fire was so great that the first firefighters on the scene could not approach the crash site.\n\nFlight AF4590 had been carrying German tourists, including three children, bound for New York. The passengers were to join a cruise to Ecuador, South America, via the Panama Canal. It was to have been a holiday of a lifetime.\n\nThe Concorde experience had not started well for them. The aircraft had been delayed by 2 hours after it was found that there was a problem with Number 2 engine and its reverse thrust. The captain had waited until the fault had been rectified. Captain Marty was 54 years of age and had 317 hours of experience on the aircraft. He was a daredevil skier and had gained fame by windsurfing across the Atlantic some years before.\n\nThe problem fixed, the passengers boarded the aircraft, excitement mounting now that they were finally on their way to enjoy the supersonic experience to New York. However, as the Concorde gathered speed down the runway, a fire started and a sequence of events unfolded that made this incredibly safe and beautiful aircraft doomed to a tragic end.\n\nHow had the fire started? One of our senior and most likeable Concorde captains, John Hutchinson, flew Concorde for fifteen years. He is an aviation consultant, speaker and broadcaster and his views are much respected in the airline industry. In 1999 he was installed as Master of the Guild of Air Pilots and Navigators. He commented on the accident (I quote), 'the fire on its own should have been eminently survivable, the pilot should have been able to fly his way out of trouble'. Captain Hutchinson believes that there were a number of factors \u2013 a combination of operational error by the flight crew and 'negligence' by the Air France maintenance department that led to the crash.\n\nA Continental Airlines DC10 bound for Houston had taken off minutes earlier, losing a 44cm piece of metal from its undercarriage. It was lying on the runway and in the path of Concorde flight AF4590 about to take off. As the fully loaded Concorde gathered speed it hit the metal strip with such force that it caused one of its tyres to burst. Even though Concorde tyres had been developed to withstand impact they had not been designed to withstand hitting such a large piece of metal at such speed.\n\nWith the impact of the tyre bursting, a large piece of rubber from one of the tyres flew up and with the force of a missile hit the fuel tank. The fuel tanks were so full that the impact of the rubber hitting the tank caused a shock wave that had nowhere to dissipate, causing it to rupture. Fuel escaped from the hole and ignited by a spark, probably from an electrical wire. There was now a dramatic fire situation.\n\nCaptain Marty and his crew were at that time beyond the point of return. They were committed to take off. The vivid images we have are of a rooster tail fireball spewing out from the Number 2 engine and Concorde's final moments before it plunged back to earth.\n\nOne of the disturbing facts that came out of the accident report was that as the aircraft gathered speed, it was skewing down the runway towards the grass verge. As a result it hit a yellow landing light that disintegrated into engine Number 1, thereby causing it to fail. It would appear that Captain Marty had no choice but to take to the air. The aircraft was about to leave the runway and at that speed there would have been a major incident and fire on its own.\n\nTo add to the potential of a further catastrophe, the Concorde would have been heading straight towards a 747 that had just landed from Japan and was waiting to cross the runway from which the Concorde was taking off. On this aircraft were President Jacques Chirac and the First Lady of France, returning from a G7 summit in Tokyo. It would have been a 'no choice' decision. If the captain of the Concorde had continued off the runway and onto the grass there would have been a cataclysmic accident and many, many fatalities \u2013 and France could have lost its president.\n\nThe take-off speed of flight AF4590 was 188 knots, 11 knots below the minimum recommended speed. Under normal circumstances the captain still had enough speed to climb away, but no longer enough power. Number 1 engine had failed and Number 2 engine had been shut down by the flight engineer because of the fire. This was at a crucial time when the remaining thrust from that engine was vital for its survival. But what had caused the aircraft to yaw to the left as it sped along the runway?\n\nThis remains a controversial subject. Two leading Air France Concorde pilots prepared a sixty-page report of the crash which they submitted to the investigating judge. They maintain it was the nose wheel that had been improperly repaired causing the skewing motion. A week before the crash, the doomed aircraft had been in the hangar for maintenance. With both the British Airways and Air France Concordes, they were subject to strict procedures of inspection. Various load-bearing components such as the undercarriage bogeys had to be replaced at regular intervals and after a few hundred flying hours.\n\nThe stress on the wheels is more than any other aircraft, the reason being that the delta shape wings provide hardly any lift on the ground. On Concorde's take-off, lift is achieved when the captain pulls up the nose and pitches the aircraft to an angle of 11 degrees. Up until then the wheels and bogeys bear the full 185 tons of Concorde weight.\n\nIn the case of the doomed Concorde, it left the hangar missing a spacer. This is a 304.8mm-long and 127mm-diameter piece of anodised aluminium \u2013 a vital component of the landing gear that keeps the wheels in alignment. Although it had flown to New York twice in that time, the condition deteriorated to the extent that when flight AF4590 lined up on the runway there was nothing to keep the front wheels in line with the back. It was similar to a supermarket trolley that veers to one side if the wheels are not aligned properly.\n\nThe BEA (French accident investigation bureau) maintain that the aircraft's yaw to one side was not caused by the missing spacer but by the fact that the aircraft had lost the power of both engines on the left side. However, the Concorde captains disagree. In the training simulator they are trained to experience an engine loss shortly before take-off. In John Hutchinson's words, 'It's no big deal at all. The yaw is totally containable.'\n\nFor a few agonising seconds after take-off the captain got the aircraft up to 210 knots, only for Number 1 engine, which had begun to recover, to fail again. But what makes this story even more cataclysmic in its unfolding is the other factors that came to light after the investigation.\n\nIt wasn't only mechanical errors that led to the crash. The aircraft was overweight. When the captain lined up on the runway his instruments would have showed him that he had 1.2 tons of extra fuel on board. This should have been burnt off during the taxi. Further, there were nineteen extra bags that were not included in the manifest (the aircraft's documentation). These had been loaded at the last minute and weighed another 500kg. This brought the aircraft to within an estimated 6 tons of its allowable weight given the wind conditions at the time.\n\nIn the interval between the Concorde leaving the terminal and reaching the runway, another circumstance occurred. When the flight left the terminal there was no wind. In between leaving the stand and reaching the take-off point the wind had increased to 8 knots, but it was a tailwind. An aircraft will always take off into the wind, the reason being that this gives it extra lift. John Hutchinson comments, 'The captain should have insisted on taxiing back to the other end of the runway, as most Concorde pilots have done in the past, and taken off into the wind. It appeared that the captain and his crew did not react to this information at all.' They failed to respond to the fact that the aircraft was now several tons overweight and beyond its aft centre of gravity limit, given the conditions. A senior industry source stated that, 'Even in a take-off with all four engines working normally you are well beyond the point where the test pilot has been prepared to tread.'\n\nAnother factor in the fate of this flight was that had the aircraft not been overweight, it would have been airborne before it hit the metal strip that caused the fire, and the experts say the captain would have got away with it. But, according to Captain Hutchinson, his options were severely compromised and the captain found himself, 'trying to save a one-time thoroughbred which was responding like a flying pancake'.\n\nIn the concluding chapter of this tragic unfolding of events, flight AF4590's fate was sealed with one final act \u2013 25ft off the ground, the ailing Number 2 engine which had ingested some of the runway light was shut down by the flight engineer. This breached the set procedures that require that no engine is shut down until the flight is stable at 400ft, and then only on a set of commands from the captain. The flight was doomed, and the great white bird so graceful in flight fell to an untimely and horrific end, along with its precious cargo of passengers.\n\nThe crash was a horrifying reminder that Concorde was not the invincible piece of supersonic machinery that we had all assumed after its twenty-four years. For myself and other crew members, the image of that crash remains as indelible as any other heart-wrenching events of our generation, ranking alongside such historical moments as the assassination of John F. Kennedy and the collapse of the New York Twin Towers on 11 September 2001. It was impossible to comprehend.\n\n## THE AFTERMATH\n\nAfter the disaster, ironically enough BA did not ground their Concorde fleet. Normal flights were resumed the following day, much to the anger of the Air France Concorde management, who believed that all the Concordes should have been grounded after such a disaster. It was three weeks later on 15 August that an announcement was issued informing that the certificate of airworthiness for the fleet was to be withdrawn. At the time, a British Airways Concorde was taxiing onto the runway bound for New York. It was ordered to return to the stand and all flights were suspended.\n\nAlthough Concorde had the highest safety record of any commercial airliner, like any long-serving aircraft faults were beginning to appear. In 1989, fourteen years before it came out of service, defects were beginning to manifest themselves. An American commentator and author, William F. Buckley Jnr, had chartered Concorde for a round-the-world trip in a bid to beat the speed record. With 100 Americans on board, the rudder, a 9ft piece of tail, disintegrated whilst flying at 40,000ft. The captain was unaware of this until the control tower at Sydney International Airport noticed a sizeable chunk missing from the Concorde's tail.\n\nDespite Concorde's reputation as being the best maintained aircraft in the world, this did little to reassure some of the American passengers who flew home from Sydney by subsonic means. It must have been a huge shock to the flight crew to emerge from the front door of the aircraft and look down the fuselage to find the huge gap where the rudder had been.\n\nDuring a period of thirteen years Concorde suffered another four problems with its rudder sections falling off. Moves were made to strengthen their design. Failing rudder sections were not the only incidents. There were another three in six weeks on one occasion. On a flight to New York, a Concorde had to return to London after an engine was shut down. Later, three aircraft windows shattered as cracks were found in one of the outer layers of the one of the windows.\n\nThe irony of these incidents was that, after the Paris crash in 2000, the remaining British Airways and Air France Concordes were given a \u00a330 million refit. The fuel tanks were strengthened and made safer with Kevlar, designed to be bulletproof. The tyres were also strengthened. An internal refurbishment was carried out by Sir Terence Conran and designer Adam White at a cost of millions of pounds. Thirteen months after the Air France crash, the British Airways fleet had their certificate of airworthiness restored. Concorde was ready to take to the skies again and loyal Concorde passengers were royally entertained by the Chairman of BA with caviar and champagne in one of the hangars. BA was desperate to woo them back to their favourite aircraft.\n\nBefore the scheduled services were put into action, Captain Mike Bannister had arranged to fly Concorde halfway across the Atlantic as a familiarisation for the pilots, crew and ground staff. On its return, everything having gone so well, the pilots, the cabin crew and all those involved in Concorde's re-entry into service were very upbeat. Concorde was to be restored to her natural place in the skies again.\n\nHowever, on their return to London they were met with devastating news. An aircraft had flown into one of the Twin Towers in New York. Further news came through that another plane had deliberately crashed into the second tower. All at British Airways knew what this meant for Concorde. In the 9\/11 disaster Concorde lost forty of its most frequent commuters \u2013 bankers, executives and heads of industry. Concorde's revenue would be hugely hit.\n\nBut the writing was already on the wall for the supersonic era. Despite its good safety record, and it was the best maintained aircraft out of any other fleet, management were deliberating its future. Air France, having had another incident with one of its aircraft, was also deliberating whether to keep its Concordes in service. Bookings were down and these were sadly exacerbated after the disaster of 9\/11.\n\nBritish Airways also had to accept the economic realities of the Concorde operation. Passenger numbers were falling, yet the fuel costs for supersonic flying were astronomical. During the grounding of the Concorde after the Air France crash it was also noted that the airlines made more money flying their first-class passengers subsonically than on the fuel-guzzling Concorde.\n\nAnother factor was Airbus. The aircraft's manufacturer had detailed discussions with British Airways confirming the need for an enhanced maintenance programme in the coming years. Maintenance charges would go up by \u00a340 million. British Airways decided that such a huge investment could not be justified at a time when there was a global downturn in all forms of premium travel and, more relevantly, on Concorde.\n\nProlonging Concorde's life was becoming a losing battle and, as BA pointed out, by the year 2009 Concorde would be 40 years old. Concorde, however, never made its 40th birthday. It was to retire after twenty-seven years of commercial operations. In the case of Concorde, age was purely relative. As a BA spokesman was quick to point out, the aircraft did fewer landings and take-offs than any other fleet. It was in the air for considerably less time than the subsonic aircraft, and was therefore in much better shape.\n\nBut the few problems that occurred attracted more attention from the press, for the sole reason that it was such a high-profile aircraft, and it was to survive barely two years after the Paris crash. However, despite the 9\/11 disaster, services resumed in November 2001.\n\nTwo years later the crowds were gathering at Heathrow to watch its final exit from the skies. After the last flight had landed at Heathrow and the press, the TV crews and public had disbanded after what had been a momentous day, the poignancy of the event was recalled by Chief Pilot Mike Bannister, who had flown the last commercial flight in from New York a few hours earlier. As he left the hangar he recalls it was a misty night, and on the tarmac neon lights were shining on five perfectly serviceable Concordes that he knew would never fly commercially again. It must have been a heart-wrenching moment.\n\nBritish Airways now had the dilemma of what to do with these iconic aircraft. It was decided that they should be shared around the globe. In October 2003 G-BOAC flew to Manchester, where it is now a fixture at the Runway Visitor Park. This was followed five days later by G-BOAG, which flew to Seattle via New York to the Museum of Flight. The final resting place for G-BOAD was New York and the Intrepid Sea Air Space Museum. Concorde G-BOAA, which never flew again after the grounding in 2000, is at the National Museum of Flight in Scotland and G-BOAB, retired in 2000, now resides somewhere at Heathrow Airport.\n\nA popular destination for Concorde during the winter months was Barbados. Alongside Barbados' Grantley Adams Airport is a hangar that now houses Concorde G-BOAE. Here, visitors can enjoy the 'Concorde Experience' and even try a flight simulator.\n\nAt Brooklands Museum in Surrey is the production aircraft G-BBDG that was used extensively in the test programme in the early seventies. On 13 February 1974 it was first flown by Peter Baker and Brian Trubshaw from Filton to Fairford, and it was the first aircraft to carry 100 passengers at Mach 2. At Duxford Imperial War Museum is the pre-production Concorde (GAXDN) that was used for testing.\n\nThe final flight ever of a Concorde took place on the historic day of 26 November 2003 when Concorde G-BOAF took off for Filton, near Bristol. It was no coincidence that in 1969 the first British Concorde had taken off from here. The public was again out in vast numbers \u2013 1,000 spectators watched her fly from Heathrow. The sight of crowds lining the Clifton Suspension Bridge as she passed over will remain etched on my memory as I watched it on TV.\n\nAt Filton, amongst the 20,000 crowd to see G-BOAF land, was HRH the Duke of York, who received Concorde's last flight log from Chief Pilot Mike Bannister. His words gave a fitting end to the Concorde chapter when he described Concorde in his speech as the 'icon of the century'.\n\nOn board this flight were crew members Louise Brown and Julia van den Bosch, who between them had over fifty years' supersonic flying experience. Louise recalls the poignancy of taking leave of the aircraft for the final time. Both she and Julia were very subdued. Louise got off first. She left Julia, the longest ever serving Concorde stewardess, to say her own personal goodbyes to the ghosts of the past and the aircraft to which she had devoted twenty-seven years of her career.\n\n## AND FINALLY...\n\nConcorde has been hailed many times as the greatest engineering achievement of the twentieth century and there will not be many who would wish to contradict this statement. In every way Concorde was way ahead of its time when its design and development began in the sixties.\n\nIt left us with a wonderful legacy that, although at the time of its inception appeared to be a massive drain on the country's finances, gave the British public an icon to be proud of. It gave the future generation a hope that our small nation can still be a world leader in new technology. Although only a small number of the population ever flew on the British Airways Concorde (over 2 million), those who did will never forget the experience. So many times I have listened to someone's account of their flight on Concorde. Their eyes light up, their voice becomes animated and they recall every detail indelibly embedded in their memories. Then their ebullience fades as they mourn its demise.\n\nAnd one final story about our beloved aircraft, as told by Captain Jock Lowe. It is as amusing as it is revealing of Concorde's prowess:\n\nAmerican spy planes had been sitting over Cuba for years. The pilots of these aircraft were at 60,000ft and sitting in spacesuits. One day the spy plane was asked to move 30 miles north as an airliner was passing. And there went Concorde... 100 passengers in shirt sleeves drinking vintage champagne.\n\n## THE FUTURE OF SUPERSONIC FLYING\n\nSo what is the answer to the question so many Concorde fans would like to know? What are the chances of Concorde ever flying again? Ex-operations manager at Brooklands Museum, Jan Knott, who was in charge of Concorde charters for twenty years, has this to say: 'There's no one in the world who'd love to see that happen more than I would. But the costs of getting it up and running again would just be phenomenal.'\n\nBritish Airways removed any chance of its reappearance into the skies, even in a heritage role, when it disabled its electrical and hydraulic systems and, according to one expert, to reinstate these would cost hundreds of millions of pounds. Air France Concorde enthusiasts are keeping their options open and hoping to maintain the engines in an airworthy condition. However, the Save Concorde Group, of which there are 3,000 members and with Fred Finn as president, is still holding on to the belief that it can be done in a heritage role.\n\nAnd as for the future of supersonic transport? The buzz word is hypersonic travel and speeds of up to Mach 6 at 250,000ft \u2013 London to New York in 1 hour, London to Tokyo in 2 hours... and the development of a spy plane that has neither the beauty nor elegance of Concorde. Pictures of this American spy plane, Aurora, came to light when mysterious loud bangs were heard the length and breadth of Great Britain and in New York in December 2014. A theory was put forward that Aurora was flying across the Atlantic!\n\nAgain, the cost and design issues of developing something similar into a passenger plane are being mooted as astronomical. And so, whilst we send missions to Mars and find out more about our universe with techniques that scientists could only dream about several decades ago, we have no immediate replacement for supersonic travel \u2013 and probably not in my lifetime.\n\nConcorde will be confined to the history books and supersonic passenger travel will be a part of it too. For now...\n\nConcorde at the Intrepid Museum of New York. (Courtesy of Frederic Carmel, freeimages.com)\n\n## [POSTSCRIPT:  \nCROSSING THE ATLANTIC \u2013 3 HOURS AND 30 MINUTES TO SEVENTEEN DAYS](contents.htm#a41)\n\nFlying across the Atlantic on Concorde on a regular basis brought home to me how technology in such a short time had shrunk the world. Three and a half thousand miles of ocean was reduced in time to a mere lunch service. The early explorers could have had no comprehension that one day man would achieve this. It was as hypothetical as putting a man on the moon.\n\nThe time between the first powered aeroplane by brothers Wilbur and Orville Wright in 1903 and the rolling out of the first prototype Concorde from a Toulouse hangar was a mere sixty-four years. It was a remarkable achievement in the timescale of history.\n\nHaving flown across the Atlantic many times in a subsonic aircraft in a time of 7 hours 30 minutes and crossed it the fastest possible way, supersonically, it occurred to me that to experience the full enormity of what Concorde could achieve, it was only right that this ocean should be crossed by slower methods.\n\nIn 1996 my family and I booked a cabin on the QE2 and made the crossing in four and a half days. This was not without incident. Once out in the Atlantic we experienced the full force of a gale and it was interesting to watch the height of the waves as we sat enjoying dinner in an almost deserted restaurant. Cabin portholes were sealed off and we were not allowed up on deck to take a breath of air with a 60mph wind ready to whisk us off the decks as the ship ploughed on towards Newfoundland. As we approached the coast we ran into a blanket of fog and were diverted around the Long Island crash site of the TWA 747 that had gone down a few weeks earlier. Speculation was rife, even then, that it had either been a terrorist attack or an accident by the American Navy during missile practise.\n\nDuring this eventful voyage, and in between perusing the five-course menus and wave watching from the comfort of the ship, it set me wondering about what it would be like sailing in those massive breaking waves on a small yacht. From the safety of the massive tonnage of the QE2, it seemed inconceivable that a yacht could survive in that vast boiling morass of breaking foam \u2013 and so another challenge was born.\n\nTwo years later I had signed up to do an Atlantic crossing on an 80ft round-the-world Whitbread yacht called Creightons Naturally. Exactly how old and tired this boat was and its standard of seaworthiness was only made clear to me when we set out from Las Palmas in the Canaries bound for St Lucia. Three hundred miles into the Atlantic Ocean, life aboard was full of drama. Having left the calms of the Canary Islands we headed into strong winds. We all had to take a turn at helming and this was not easy in the cross-beam breaking waves. It took a considerable amount of concentration.\n\nIn one day we managed to damage two spinnakers (the large curved sail at the front), destroy one large grinder on deck (a hefty steel rounded instrument for 'grinding' in the sails) and the glass cover for the instrument panel. The cabin below was a chaotic mess of a spinnaker graveyard and a chandler's workshop.\n\nAside from crewing duties we all had to take turns at preparing meals. The galley was small and antiquated and cooking three meals a day for eighteen people was not easy when the yacht was being tossed around by the Atlantic swells. It made Concorde's Atlantic crossing seem extremely civilised in comparison. To add to our galley duty day we were also assigned to cleaning the floors and heads (toilets) after finishing the cooking duties.\n\nFour days into the trip we were allowed our first outdoor shower using sea water, with a quick hosing off with a litre of fresh water. It felt wonderful. Over the next few days we had problems with the rigging and almost keelhauled a spinnaker after the spinnaker halyard (the rope that holds the spinnaker up to the top of the mast) snapped. One of the crew climbed the mast to retrieve it and found another of the mast's support ropes had frayed. Racing was now abandoned as crucially we had to sail, or rather nurse, the boat across the Atlantic to prevent the mast falling down. All this drama, and 1,400 miles to the nearest landfall.\n\nDuring that time the wind dropped, the Atlantic swell lessened to a more or less flat pond and we were becalmed. Boredom now became an issue. Luckily, a yacht on its way to Barbados passed close by. It was motoring and diverted course to come and see us. Seizing the opportunity to relieve the tedium, our male crew dressed up as pirates and swam the 100 yards to 'board' the Finnish boat and drink vodka with its crew. We also gave them some fuel from our well-stocked tanks. This small interlude greatly enhanced the routine of wind watching and listening to the sound of sails lurching idly in the 'doldrums'. The girls on the crew also took a dip in the Atlantic. It made us feel very insignificant when one stopped to consider the vast ocean below and around us.\n\nBy this time we were drinking putrid-tasting water that was rationed. We did not have the luxury of a water maker that other transatlantic yachts took with them by way of necessity. We had also finished the last green vegetable in the form of tinned peas. I developed a sympathy with the sailors of old who grew sick and went down with scurvy. Would we be in similar condition when we arrived in St Lucia? At least those sailors had enjoyed a daily rum ration \u2013 that was not on our ship's agenda!\n\nOn day ten the navigator shouted to everyone, 'We've only got 1,000 miles to go!' Now, for someone like myself who had only cruised the Solent or done a sprint across 90 miles of the North Sea, this sounded like a lot of miles, but we were by now on the homeward leg.\n\nBy day seventeen we spotted landfall. This was a defining moment in the trip. I was roused from my bunk wearing only a T-shirt and underwear and was promptly handed the helm by the skipper. After weeks of seeing nothing but ocean this dot of land became the focus of our attention. This was one of my most exhilarating moments of the trip as I helmed the boat towards our destination which, until now, had been an unseen island existing only on the chart plotter. My state of undress was only a minor matter.\n\nAs the island of St Lucia grew larger we revelled in our achievement. We had finally made it across that large ocean despite all the challenges that nature had thrown at us. We had experienced a tropical storm with a deluge of non-stop rain lasting 24 hours, complete with a water spout, wild seas, flat calms and beautiful sunsets and nights lit by a million stars.\n\nWe had also shared many ridiculously funny moments. Even during the many crises when we lost faith in the seaworthiness of the yacht we had bonded together. Someone amongst the crew had always managed to find the funny side of things and bring a smile to our faces. During the crossing we had spotted flying fish, dolphins and a spouting whale. The joy of being so close to nature and the ocean was something one could never ever experience flying on a supersonic jet across the Atlantic.\n\n## LIFE AFTER CONCORDE\n\nThe supersonic crews are now part of a small elite club consigned to history. The beautiful aircraft we worked on are now sadly sitting idly in museums. Whatever the future holds for supersonic flying, we all have wonderful memories and I hope this book will give an insight into the lifestyles of the Concorde crews for Concorde lovers to enjoy.\n\nWhat did we do after we left British Airways? Obviously there are too many to mention here, but here are a few of my old friends:\n\nJeannette Hartley is a guide at Brooklands on Concorde.\n\nJulia van den Bosh leads a very busy life as a textile designer and exhibits nationally.\n\nLouise Brown lives in the Cotswolds and works part time as an estate agent.\n\nBernadette Forrest is now married and still leads an enviable life travelling to exotic holiday destinations.\n\nJill Channon is married to Mike Channon, the racehorse trainer.\n\nScarlett Geen is married to a wine importer and is a very keen tennis player.\n\nSeveral of the BA Concorde captains, including Mike Bannister, John Hutchinson, Brian Walpole and Jock Lowe, are sought after as guest speakers at various society dinners and functions, as well as giving lectures aboard cruise ships.\n\nAnd finally myself? I am a very keen sailor living by the coast, and a member of three yacht clubs. I also work as a yachting photographer: visit my website at www.solentpics.com. When I'm not enjoying myself on the water, I have found time to pen several novels.\n\nConcorde profile. (Shutterstock, Image ID: 2922877. Copyright: Kevin)\n\n## CONCORDE TIMELINE\n\n1959: Government committee recommends that the UK builds a supersonic passenger jet.\n\n29 November 1962: Anglo-French treaty signed. A wide range of airlines begin to make non-binding orders.\n\n2 March 1969: First flight.\n\n1 October 1969: First supersonic flight.\n\n21 January 1976: Concorde enters service with BA300, flying from Heathrow to Bahrain.\n\n24 May 1976: Concorde flies to Washington DC.\n\n22 November 1977: Concorde starts flying to New York, JFK.\n\n27 March 1984: Concorde begins flights to Miami via Washington.\n\n13 July 1985: Phil Collins performs in both Live Aid concerts in London and Philadelphia, courtesy of Concorde.\n\n25 July 2000: Air France Concorde crashes outside Paris with 113 killed.\n\n15 August 2000: BA flight to New York returns to gate when cause of Paris crash emerges. Concorde grounded.\n\n11 September 2001: First post-crash proving flight with passengers lands at Heathrow just after the attacks on the World Trade Center.\n\n10 April 2003: BA and Air France announce the end of Concorde.\n\n24 October 2003: Concorde's last passenger flight, BA 002 New York to Heathrow.\n\n26 November 2003: Last flight, to Filton.\n\n## CONCORDE'S VITAL STATISTICS\n\nLENGTH: 203ft 9in (62.1m)\n\nWING SPAN: 83ft 8in (25.5m)\n\nFUSILAGE WIDTH: 9ft 6in (2.9m)\n\nHEIGHT: 37ft 1in (11.3m)\n\nINTERNAL HEIGHT: 6ft 5in (1.96m)\n\nFUEL CAPACITY: 26,286 imperial gallons (119,500 litres)\n\nRANGE: 4,143 miles (6,667km)\n\nTAKE-OFF SPEED: 250mph (400km\/h)\n\nCRUISING SPEED: Mach 2 (1,350mph\/2,160km\/h) up to 60,000ft (18,288m)\n\nLANDING SPEED: 187mph (300km\/h)\n\n## PLATES\n\nConcorde taking off. (Courtesy of Peter March)\n\nAnother distinctive shot of Concorde taking off, showing the vortex effect on the wing. (Courtesy of Peter March)\n\nConcorde climbing, showing the afterburners. (Courtesy of Peter March)\n\nConcorde nose. (Shutterstock, Image ID: 2922876. Copyright: Kevin)\n\nThe distinctive nose cone of the supersonic aircraft. Concorde shot from beneath. (Shutterstock, Image ID: 224094943. Copyright: af8images)\n\nConcorde at Singapore. (Courtesy of Mike Bannister)\n\nThe leather seating inside Concorde. (Shutterstock, Image ID: 162863810. Copyright: Marbury)\n\nThe curvature of the earth. (Courtesy of Mike Bannister)\n\nThe captain's view. (Courtesy of Mike Bannister)\n\nThe pilots' panel. (Courtesy of Mike Bannister)\n\nSally and crew on location.\n\nThe Mildenhall Air Show, Suffolk, with the crew standing on Concorde's wing watching the Red Arrows in Concorde formation in 1983.\n\nThe crew, including senior pilot Jock Lowe and Dick Bell (cabin service director), at Mildenhall Air Show with the Red Arrows.\n\nA Concorde unscheduled service to Glasgow. We were met with the red carpet and bagpipes, a true Scottish welcome!\n\nA round-the-world charter in Honolulu.\n\nFred Finn in front of his beloved Concorde.\n\nConcorde presentation, 10 March 2001, after refurbishment of aircraft prior to its re-entry into service. (Courtesy of Louise Brown)\n\nConcorde in her final livery. (Courtesy of Mike Bannister)\n\nThe last flight was an emotional experience. (Courtesy of Julia van den Bosch)\n\nAny opinions expressed in this material are those of the author and BA employees the author interviewed, and do not reflect the view of British Airways.\n\nCover illustrations. Front: top: \u00a9 Zimmytws\/Dreamstime.com; bottom: curve of the Earth \u00a9 Mike Bannister; Concorde \u00a9 Frederic Carmel.\n\n## COPYRIGHT\n\nFirst published in 2015\n\nThe History Press\n\nThe Mill, Brimscombe Port\n\nStroud, Gloucestershire, GL5 2QG\n\nwww.thehistorypress.co.uk\n\nThis ebook edition first published in 2015\n\nAll rights reserved\n\n\u00a9 Sally Armstrong, 2015\n\nThe right of Sally Armstrong to be identified as the Author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.\n\nThis ebook is copyright material and must not be copied, reproduced, transferred, distributed, leased, licensed or publicly performed or used in any way except as specifically permitted in writing by the publishers, as allowed under the terms and conditions under which it was purchased or as strictly permitted by applicable copyright law. Any unauthorised distribution or use of this text may be a direct infringement of the author's and publisher's rights, and those responsible may be liable in law accordingly.\n\nEPUB ISBN 978 0 7509 6656 6\n\nOriginal typesetting by The History Press\n\nEbook compilation by RefineCatch Limited, Bungay, Suffolk\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\n\n\nProduced by Wayne Hammond, Malcolm Farmer and the Online\nDistributed Proofreading Team at http:\/\/www.pgdp.net\n\n\n\n\n\n\nPUNCH, OR THE LONDON CHARIVARI.\n\nVOL. 107.\n\nAUGUST 18, 1894.\n\n\n\n\nMORE ORNAMENTAL THAN USEFUL.\n\n(_A Legend of the Results of the School Board._)\n\nThe Committee sat waiting patiently for candidates. Although the papers\nhad been full of advertisements describing the appointments the\n_reclames_ had had no effect. There were certainly a number of persons\nin the waiting-room, but the usher had declared that they did not\npossess the elementary qualifications for the post that the Committee\nwere seeking to fill with a suitable official.\n\n\"Usher,\" cried the Chairman at length with some impatience; \"I am sure\nyou must be wrong. Let us see some of the occupants of the adjoining\noffice.\"\n\nThe usher bowed with a grace that had been acquired by several years\nstudy in deportment in the Board School, and replied that he fancied\nthat most of the applicants were too highly educated for the coveted\nposition.\n\n\"Too highly educated!\" exclaimed the representative of municipal\nprogress. \"It is impossible to be too highly educated! You don't know\nwhat you're talking about!\"\n\n\"Pardon me, Sir,\" returned the Usher, with another graceful inclination\nof the head, \"but would not 'imperfectly acquainted with the subject of\nyour discourse' be more polished? But, with your permission, I will obey\nyou.\"\n\nAnd then the official returned to usher in an aged man wearing\nspectacles. The veteran immediately fell upon his knees and began to\nimplore the Committee to appoint him to the vacant post.\n\n\"I can assure you, Gentlemen, that, thanks to the School Board, I am a\nfirst-rate Latin and Greek scholar. I am intimately acquainted with the\nHebrew language, and have the greatest possible respect for the Union\nJack. I know all that can be known about mathematics, and can play\nseveral musical instruments. I am also an accomplished waltzer; I know\nthe use of the globes, and can play the overture to _Zampa_ on the\nmusical-glasses. I know the works of SHAKSPEARE backwards, and----\"\n\n\"Stop, stop!\" interrupted the Chairman. \"You may do all this, and more;\nbut have you any knowledge of the _modus operandi_ of the labour\nrequired of you?\"\n\n\"Alas, no!\" returned the applicant; \"but if a man of education----\"\n\n\"Remove him, Usher!\" cried the Chairman; and the veteran was removed in\ntears.\n\nA second, a third, and a fourth made their appearance, and disappeared,\nand none of them would do. They were all singularly accomplished.\n\nAt length a rough man, who had been lounging down the street, walked\ninto the Council-chamber.\n\n\"What may you want, Sir?\" asked the Chairman, indignantly.\n\n\"What's that to you?\" was the prompt reply. \"I ain't a going to tell\neveryone my business--not me--you bet!\"\n\n\"Ungrammatical!\" said Committee Man No. One. \"Very promising.\"\n\n\"Uncouth and vulgar!\" murmured Committee Man No. Two.\n\n\"Where were you educated?\" queried the Chairman.\n\n\"Nowheres in particular. I was brought up in the wilds of Canada.\nThere's not much book learning over there,\" and the rough fellow\nindulged in a loud hoarse laugh.\n\n\"Ah! that accounts for your not having enjoyed the great advantages of\nthe School Board. Have you seen the circular--have you read the details\nof the proposed appointment?\"\n\n\"Me read!\" cried the uncouth one; \"oh, that is a game! Why I can't read\nnor yet write!\"\n\n\"Better and better,\" said Committee Man No. One.\n\n\"First rate,\" murmured Committee Man No. Two. \"I think we have at length\nfound our ideal.\"\n\nThen the usher read the advertisement.\n\n\"What! shake the hall mat!\" cried the candidate. \"Why I could do that\nlittle job on my head!\"\n\nSo there being no other applicant for the post, the backwoods' ignoramus\nwas appointed office-sweeper at a couple of hundred pounds a year.\n\n\"Rather high wages,\" said the Chairman to himself, as he went home on\nthe top of an omnibus; \"but what can one expect when we educate all the\nchildren at the cost of the rates. Last year there was an additional\nfarthing; this year we have to pay five shillings, and goodness only\nknows how much it will be hereafter!\"\n\nAnd as he thought this, the Chairman (in the names of the rest of the\nratepayers) heartily cursed the School Board.\n\n       *       *       *       *       *\n\n[Illustration: RETURNED EMPTY.\n\n_Old Mayfly_ (_who had dropped his Flask further down stream, and has\njust had it returned to him by Honest Rustic_). \"DEAR ME! THANK YOU!\nTHANK YOU!\" (_Gives him a Shilling._) \"DON'T KNOW WHAT I SHOULD HA' DONE\nWITHOUT IT!\" (_Begins to unscrew top._) \"MAY I OFFER YOU A----\"\n\n_Honest Rustic._ \"WELL, THANK Y', SIR, BUT ME AND MY MATE, NOT SEEIN' A\nHOWNER ABOUT, WE'VE TA'EN WHAT THERE WERE INSIDE.\"]\n\n       *       *       *       *       *\n\nRE-DRESS REQUIRED.\n\n    [A writer in the _Lancet_ draws attention to the fact that the\n    regular hospital nurse's uniform is now worn as ordinary ladies'\n    attire.]\n\nThere's no doubt my new costume is _very_ becoming. I like the idea or\nthe cape, and the apron is just perfect, while the little bonnet suits\nme to a T. Met cousin FRED, who said it was \"fetching,\" and that \"they\nwanted some of my sort at the hospitals.\" I said I thought the patients\nhad good enough nurses at present; he replied \"he didn't mean the\npatients--he meant the doctors.\" Of course I couldn't stand the drudgery\nof a nurse's life; but that's no reason why I shouldn't appropriate the\nuniform, is it?\n\nWalking down street. Met another nurse--a real one, I suppose. She\nstared, turned red, and then looked horribly offended. I believe she\nmust have made some sign to me that I didn't understand. Are Nurses\nFreemasons, I wonder? Quite a secret society, it seems. Really that sort\nof thing oughtn't to be allowed. It makes things so awkward for the\nimpost--the imitators, I mean.\n\nJust got home after _dreadful_ incident! I was in a Bayswater Square,\nwhen suddenly a man driving round a corner in a cart got upset, and was\npitched on to the road close to me. A small crowd gathered immediately,\nand evidently expected _me_ to help. One man shouted \"Hi! Come and bind\nup his head, Miss!\" And his head was actually bleeding! I couldn't do\nanything, except feel awfully inclined to faint, and then the mob began\nto hiss and jeer! Somebody said I must know how to render \"first aid to\nthe injured,\" and if I didn't come quick the man would bleed to death. I\nwas so frightened I ran away, and the mob ran after me, and I had to\ntake shelter in a shop, and ask the shopman to explain to the crowd that\nI was not really a nurse at all. Then they used dreadful expressions,\nand I had to be got out by a back way. I don't think the costume is half\nas becoming as it seemed this morning; I'm going to sell it as a\n\"cast-off garment.\" Lucky for me it wasn't a torn-off garment!\n\n       *       *       *       *       *\n\nScott on the New Woman.\n\n(_As the Wizard of the North would have written now._)\n\n  New Woman! in our hours of ease\n  A smoking rival hard to please,\n  Wishing to put Man in the shade,\n  Collar his togs and take his trade;\n  When pain and anguish wring the brow,\n  A swaggering, \"spanking\" _Pipchin_ thou!\n\n       *       *       *       *       *\n\n[Illustration: \"THE COW WAS THE STAMP TO IMPRESS SUPERIOR BUTTER.\"\n\n\"ARF A POUND ER MARGARINE, PLEASE; AN' MOTHER SAYS WILL YER PUT THE COW ON\nIT, 'COS SHE'S GOT COMPANY!\"]\n\n       *       *       *       *       *\n\nHINT FOR THE ALPINE SEASON.\n\n(_Adapted freely from the Old Royal Repartee._)\n\n_Middle-aged would-be Mountaineer (loq.)._\n\n  Fain would I climb, but,--well, my belt's too small.\n\n            _Mr. Punch (in reply)._\n\n  If your girth grows, Sir, _do not climb at all_!\n  Your Alpen-stock put by, ere the world mock.\n  And you become an (Alpine) Laughing-stock.\n  Though Alps on Alps arise _you_ stop in bed,\n  And let a younger man yon glaciers tread.\n  The dangers of steep slides and deep crevasses\n  Are not for elderly donkeys, but young asses.\n  The Himalayas woo you still to pant on?\n  Well, treat 'em as you would an arch young wanton,\n  Think of your legs, the boys, the girls, the Missus,\n  And do not play the elderly Narcissus.\n  To witch the world with noble \"Icemanship\"\n  Is tempting, yes, but if you chance to slip,\n  Your bones a fathomless abyss may strew,\n  An Alpine death,--and they'll all pine for you.\n  Man after fifty fits not the sublime,\n  So stay at home nor seek a foreign climb.\n  The plague of guide, and chum, and wife and daughter,\n  Is Senex who will climb and didn't oughter.\n  Stick to your Alpine Club, but like old foodles,\n  Pay, stop at home, and play at whist at Boodles'.\n  Decline with the old mania to be bitten,\n  And you will own this tip is diamond-written\n  (Like good Queen Bess's repartee on glass),\n  And that you're saved from being an old ass!\n\n       *       *       *       *       *\n\nLINES IN PLEASANT PLACES.\n\nVI.--KEW GARDENS.\n\n         In the gardens at Kew\n         It were certainly sweet\n         To be wand'ring with you,\n         Far from city and street;\n  'Twere the one thing, dear NELLIE, my joy and content to complete\n         In the gardens at Kew.\n\n         In the gardens at Kew,\n         If my way I might take\n         By the water with you,\n         Oh! how merry we'd make,--\n  I am sure you would dote on the dear little ducks in the lake\n         In the gardens at Kew.\n\n         In the gardens at Kew,\n         Having tea _a la fraises_,\n         We would cheerfully stew\n         'Neath the fierce solar rays,\n  And in \"eloquent silence\" you'd meet my affectionate gaze\n         In the gardens at Kew.\n\n         In the gardens at Kew\n         We would sit in the shade\n         For an hour or two,\n         Without chaperone's aid,\n  And your head on my shoulder (who knows?) might be lovingly laid\n         In the gardens at Kew.\n\n         In the gardens at Kew,\n         Far away from the crowd,\n         Though I'm longing for you,\n         To stern Fate I have bowed:\n  For it grieves me, dear NELLIE, to tell you, \"_No dogs are allowed_\"\n         In the gardens at Kew!\n\n       *       *       *       *       *\n\n_NOT_ MASTER OF HIMSELF THOUGH CHINA FALL.\n\n    [\"The Emperor (of China) is still cursed with the violent temper of\n    his adolescence, and \"breaks things.\"--_\"Times\" Correspondent at\n    Pekin._]\n\n Oh! is this announcement plain truth?\n   Or is it mere genial mockery?\n And _what_ does this choleric youth\n   Of China thus break--is it crockery?\n It does seem unfitting, you know--\n   At least as we Westerners see things--\n That the lord of Souchong and Pekoe\n   Should be guilty of smashing up tea-things!\n Of course, if he had an idea\n   Of breaking the Japanese bondage,\n Or breaking their hold on Korea,--\n   Well, youth is a fiery and fond age,\n And old age _might_ find an excuse\n   For breaking the peace; but kind wishes\n Can hardly invent an excuse\n   For breaking the plates and the dishes.\n He is youthful, like little AH SID,\n   It would be very mean to malign a\n Mere boy; yet a true Chinese kid\n   Should not start with the smashing of China!\n\n       *       *       *       *       *\n\nThe Cry of the (Literary) Croakers.\n\n Batrachians may doubt if King Stork or King Log\n   Be the Frog-pond's most suitable lord and controller;\n But Grub Street's unfortunate _un_lauded frog\n   Loathes the rule of the new King Log-Roller!\n\n       *       *       *       *       *\n\nMEM. BY AN OVERWORKED ONE.\n\n With \"brain-fag\" our swift, feverish age is rife,\n And death is oft the mere \"fag-end\" of life.\n\n       *       *       *       *       *\n\nSOMETHING LIKE A \"PACKED MEETING.\"--The meeting of the various Arctic\nExpeditions in the Polar Ice Pack.\n\n       *       *       *       *       *\n\n[Illustration: \"TO BE WELL SHAKEN BEFORE TAKEN!\"]\n\n       *       *       *       *       *\n\n\"JUSTICE AS SHE IS SPOKEN IN FRANCE.\"\n\nDEAR MR. PUNCH,--Now that we are close upon the silly season, when it is\nmost difficult to get interesting \"copy\" for the columns of the daily\npapers, may I be permitted to make a suggestion? No doubt you have seen\nan account of the examination of CASERIO SANTO by the President of the\nCourt on the occasion of his trial. Could not the idea be naturalised in\nLondon by the Metropolitan Police Magistrates? I would not, of course,\npropose to apply the method in cases of a serious character, but used in\nwhat are known as \"the night charges,\" the practice would become very\ninteresting. To better explain my meaning. I will imagine that a\nprisoner who has been arrested on a charge of being \"drunk and\nincapable\" is standing in front of his worship.\n\n_Magistrate_ (_with sarcasm_). You are sober now.\n\n_Prisoner_ (_in the same tone_). As a judge.\n\n_Magistrate_ (_indignantly_). Judges are always sober.\n\n_Prisoner_ (_with a laugh_). How should _you_ know?--you, who are only a\nmagistrate!\n\n    [_Murmurs._\n\n_Magistrate._ You insult me! But that will not serve you. Drink is the\ncurse of the country!\n\n_Prisoner._ You have tried it? It has been a curse to you!\n\n    [_Cries of disapproval._\n\n_Magistrate._ You are young to bandy words with one old enough to be\nyour father!\n\n_Prisoner._ My father! You my father! What an honour!\n\n_Magistrate._ I do not envy him! Nor your mother!\n\n_Prisoner_ (_excitedly_). You shall not speak of my mother. My mother is\nsacred. She shall not be referred to in the tainted atmosphere of a\nCourt of Justice.\n\n    [_Applause._\n\n_Magistrate._ This hypocrisy shall not serve you. You never loved your\nmother!\n\n    [_Prolonged sensation._\n\n_Prisoner._ Your worship, you are a liar!\n\n    [_Loud cheers._\n\n_Magistrate._ This to the Bench from the gutter! For you know\nyou were found drunk and incapable in the gutter. What were you\ndoing there?\n\n_Prisoner_ (_tearfully_). I was dreaming of my mother, my loved\nmother.\n\n    [_Sympathetic applause._\n\n_Magistrate._ You do not deserve to have a mother!\n\n    [_Prolonged sensation._\n\n_Prisoner_ (_scornfully_). Only a magistrate could make such a\ncold-blooded observation!\n\n    [_Cheers._\n\n_Magistrate._ For all that you are fined five shillings and costs!\nRemove the wretched prisoner!\n\n    [_The accused was then removed amidst expressions of sympathy from\n    the body of the Court._\n\nThere, Sir, would not that be far better reading than paragraphs\nabout gigantic gooseberries and leaders upon the sea serpent?\nPerhaps my suggestion may be adopted in the proper quarter.\nHoping that this may be the case, the police case,\n\n I remain,\n\n Yours respectfully,\n\n THE MAN IN THE REPORTER'S BOX.\n\n       *       *       *       *       *\n\nNOBLESSE OBLIGE.\n\n(_New Version._)\n\n \"Let Art and Commerce, Laws and Learning die,\n But leave us still our Old Nobility!\"\n Without them, in our democratic day,\n Who will the part of princely patriot play?\n Who else will keep a splendid Family Seat,\n And claim--for its defence--a mighty Fleet?\n Who else will make Bank Holidays a joy\n To wandering workman and to wondering boy?\n Who else will rear big fortunes upon Rent,\n Or palaces on Unearned Increment?\n Monopolise art's treasures and life's pleasures,\n And throw out dangerous democratic measures?\n Who else will keep up England's glorious name?\n Who else preserve her prestige--and her game?\n Who else will wear the purple and the ermine,\n And proudly stamp out Socialistic vermin?\n Who else in one grand field-day, 'midst the Peers,\n Undo the labours of _ig_noble years?\n Who else in solemn ranks, like three-tailed Turks,\n Defend the power of Privilege and Perks?\n And 'tis these most magnanimous Mamelukes,\n Our patriot Earls and foe-defying Dukes,\n A traitorous Chancellor would dare to--_Tax_!!!\n Ah! where's the dungeon, and oh! where's the axe?\n _Noblesse oblige!_ But sure the obligation\n Cannot involve that horror, Graduation!\n Is't not enough to rule, and guide, and bless,\n And soar as shining samples of Success?\n While with our Nobles England's glory waxes,\n The Proletariat's proud to--pay the Taxes!\n\n       *       *       *       *       *\n\nLYRE AND LANCET.\n\n(_A Story in Scenes._)\n\nPART VII.--IGNOTUM PRO MIRIFICO.\n\nSCENE XII.--_The Amber Boudoir at Wyvern--immediately after_ Lady\nCANTIRE _and her daughter have entered._\n\n_Lady Cantire_ (_in reply to_ Lady CULVERIN). Tea? oh yes, my dear;\nanything _warm_! I'm positively perished--that tedious cold journey and\nthe long drive afterwards! I always tell RUPERT he would see me _far_\noftener at Wyvern if he would only get the Company to bring the line\nround close to the Park Gates, but it has _no_ effect upon him! (_As_\nTREDWELL _announces_ SPURRELL, _who enters in trepidation_.) Mr. JAMES\nSPURRELL! Who's Mr.----? Oh, to be sure; _that_'s the name of my\ninteresting young poet--_Andromeda_, you know, my dear! Go and be\npleasant to him, ALBINIA, he wants reassuring.\n\n_Lady Culverin (a trifle nervous)._ How do you do, Mr.--ah--SPURRELL?\n(_To herself._) I _said_ he ended in \"'ell\"! (_Aloud._) So pleased to\nsee you! We think so much of your _Andromeda_ here, you know. Quite\ndelightful of you to find time to run down!\n\n_Spurrell (to himself)._ Why _she_'s chummy, too! Old Drummy pulls me\nthrough everything! (_Aloud._) Don't name it, my la--hum--Lady CULVERIN.\nNo trouble at all; only too proud to get your summons!\n\n_Lady Culv. (to herself)._ He doesn't seem very revolutionary!\n(_Aloud._) That's so sweet of you; when so many must be absolutely\nfighting to get you!\n\n_Spurr._ Oh, as for that, there _is_ rather a run on me just now, but I\nput everything else aside for _you_, of course!\n\n_Lady Culv. (to herself)._ He's soon _reassured_. (_Aloud, with a touch\nof frost._) I am sure we must consider ourselves most fortunate.\n(_Turning to the_ Countess.) You _did_ say cream, ROHESIA? Sugar, MAISIE\ndearest?\n\n_Spurr. (to himself)._ I'm all right up to now! I suppose I'd better say\nnothing about the horse till _they_ do. I feel rather out of it among\nthese nobs, though. I'll try and chum on to little Lady MAISIE again;\nshe may have got over her temper by this time, and she's the only one I\nknow. (_He approaches her._) Well, Lady MAISIE, here I _am_, you see.\nI'd really no idea your aunt would be so friendly! I say, you know, you\ndon't mind _speaking_ to a fellow, do you? I've no one else I can go\nto--and--and it's a bit strange at first, you know!\n\n_Lady Maisie (coloured with mingled apprehension, vexation, and pity)._\nIf I can be of any help to you, Mr. SPURRELL----!\n\n_Spurr._ Well, if you'd only tell me what I ought to _do_!\n\n[Illustration: \"My keys! Why, what do you want _them_ for?\"]\n\n_Lady Maisie._ Surely that's very simple; do _nothing_; just take\neverything quietly as it comes, and you _can't_ make any mistakes.\n\n_Spurr. (anxiously)._ And you don't think anybody'll see anything odd in\nmy being here like this?\n\n_Lady Maisie (to herself)._ I'm only too afraid they _will_! (_Aloud._)\nYou really _must_ have a little self-confidence. Just remember that no\none here could produce anything a millionth part as splendid as your\n_Andromeda_! It's _too_ distressing to see you so _appallingly_ humble!\n(_To herself._) There's Captain THICKNESSE over there--he MIGHT come and\nrescue me; but he doesn't seem to care to!\n\n_Spurr._ Well, you _do_ put some heart into me, Lady MAISIE. I feel\nequal to the lot of 'em now!\n\n_Pilliner_ (_to_ Miss SPELWANE). Is _that_ the Poet? Why, but I\nsay--he's a _fraud_! Where's his matted head? He's not a bit ragged, or\nrusty either. And why don't he dabble? Don't seem to know what to do\nwith his hands quite, though, _does_ he?\n\n_Miss Spelwane (coldly)._ He knows how to do some very exquisite poetry\nwith _one_ of them, at all events. I've been reading it, and _I_ think\nit perfectly marvellous!\n\n_Pill._ I see what it is, you're preparing to turn his matted head for\nhim? I warn you you'll only waste your sweetness. That pretty little\nLady MAISIE'S annexed _him_. Can't you content yourself with _one_\nvictim?\n\n_Miss Spelw._ Don't be so utterly idiotic! (_To herself._) If MAISIE\nimagines she's to be allowed to monopolise the only man in the room\nworth talking to!----\n\n_Captain Thicknesse_ (_to himself, as he watches_ Lady MAISIE). She is\nlookin' prettier than ever! Forgotten me. Used to be friendly enough\nonce, though, till her mother warned me off. Seems to have a good deal\nto say to that Poet fellow; saw her colour up from here the moment he\ncame near; he's _begun_ Petrarchin', hang him! I'd cross over and speak\nto her if I could catch her eye. Don't know, though; what's the use? She\nwouldn't thank me for interruptin'. She likes these clever chaps;\ndon't signify to her if they _are_ bounders, I suppose. _I_'m not\nintellectual. Gad, I wish I'd gone back to Aldershot!\n\n_Lady Cant. (by the tea-table)._ Why don't you make that woman of yours\nsend you up decent cakes, my dear? These are cinders. I'm afraid you let\nher have too much of her own way. Now, tell me--who are your party?\nVIVIEN SPELWANE! Never have that girl to meet me again, I can't _endure_\nher; and that affected little ape of a Mr. PILLINER--h'm! Do I see\nCaptain THICKNESSE? Now, I don't object to _him_. MAISIE and he used to\nbe great friends.... Ah, how do you _do_, Captain THICKNESSE? Quite\npleasant finding you here; such ages since we saw anything of you! Why\nhaven't you been near us all this time?... Oh, I may have been out once\nor twice when you called; but you might have tried again, _mightn't_\nyou? There, _I_ forgive you; you had better go and see if you can make\nyour peace with MAISIE!\n\n_Capt. Thick. (to himself, as he obeys)._ Doosid odd, the Countess\ncomin' round like this. Wish she'd thought of it before.\n\n_Lady Cant. (in a whisper)._ He's always been such a favourite of mine.\nThey tell me his uncle, poor dear Lord DUNDERHEAD, is _so_ ill--felt the\nloss of his only son so terribly. Of course it will make a great\ndifference--in many ways.\n\n_Capt. Thick._ (_constrainedly to_ Lady MAISIE). How do you do? Afraid\nyou've forgotten me.\n\n_Lady Maisie._ Oh no, indeed! (_Hurriedly._) You--you don't know Mr.\nSPURRELL, I think? (_Introducing them._) Captain THICKNESSE.\n\n_Capt. Thick._ How are you? Been hearin' a lot about you lately.\n_Andromeda_, don't you know; and that kind of thing.\n\n_Spurr._ It's wonderful what a hit she seems to have made--not that I'm\n_surprised_ at it, either; I always knew----\n\n_Lady Maisie (hastily)._ Oh, Mr. SPURRELL, you haven't had any tea! _Do_\ngo and get some before it's taken away.\n\n    [SPURRELL _goes_.\n\n_Capt. Thick._ Been tryin' to get you to notice me ever since you came;\nbut you were so awfully absorbed, you know!\n\n_Lady Maisie._ Was I? So absorbed as all that! What with?\n\n_Capt. Thick._ Well, it looked like it--with talkin' to your poetical\nfriend.\n\n_Lady Maisie (flushing)._ He is not _my_ friend in particular; I--I\nadmire his poetry, of course.\n\n_Capt. Thick. (to himself)._ Can't even speak of him without a change of\ncolour. Bad sign that! (_Aloud._) You always _were_ keen about poetry\nand literature and that in the old days, weren't you? Used to rag me for\nnot readin' enough. But I do now. I was readin' a book only last week.\nI'll tell you the name if you give me a minute to think--book\neverybody's readin' just now--no end of a clever book.\n\n    [Miss SPELWANE _rushes across to_ Lady MAISIE.\n\n_Miss Spelw._ MAISIE, dear, how are you? You look _so_ tired!\nThat's the journey, I suppose. (_Whispering._) Do tell me--is that\nreally the author of _Andromeda_ drinking tea close by? You're a\n_great_ friend of his, I know. Do be a dear, and introduce him to me!\nI declare the dogs have made friends with him already. Poets have\nsuch a wonderful attraction for animals, haven't they?\n\n    [Lady MAISIE _has to bring_ SPURRELL _up and introduce him:_ Captain\n    THICKNESSE _chooses to consider himself dismissed_.\n\n_Miss Spelw. (with shy adoration)._ Oh, Mr. SPURRELL, I feel as if I\n_must_ talk to you about _Andromeda_. I _did_ so admire it!\n\n_Spurr. (to himself)._ Another of 'em! They seem uncommonly sweet on\n\"bulls\" in this house! (_Aloud._) Very glad to hear you say so, I'm\nsure. I've seen nothing to touch her myself. I don't know if you noticed\nall her points----?\n\n_Miss Spelw._ Indeed, I believe none of them were lost upon me; but my\npoor little praise must seem so worthless and ignorant!\n\n_Spurr. (indulgently)._ Oh, I wouldn't say _that_. I find some ladies\nvery knowing about these things. I'm having a picture done of her.\n\n_Miss Spelw._ Are you really? _How_ delightful! As a frontispiece?\n\n_Spurr._ Eh? Oh no--full length, and sideways--so as to show her legs,\nyou know.\n\n_Miss Spelw._ Her legs? Oh, of _course_--with \"her roseal toes cramped.\"\nI thought that such a _wonderful_ touch!\n\n_Spurr._ They're not more cramped than they ought to be; she never\nturned them _in_, you know!\n\n_Miss Spelw. (mystified)._ I didn't mean that. And now tell me--if it's\nnot an indiscreet question--when do you expect there'll be another\nedition?\n\n_Spurr. (to himself)._ Another addition! _She_'s cadging for a pup now!\n(_Aloud._) Oh--er--really--couldn't say.\n\n_Miss Spelw._ I'm sure the first must be disposed of by this time. I\nshall look out for the next _so_ eagerly!\n\n_Spurr. (to himself)._ Time I \"off\"ed it. (_Aloud._) Afraid I can't say\nanything definite--and, excuse me leaving you, but I think Lady CULVERIN\nis looking my way.\n\n_Miss Spelw._ Oh, by all _means_! (_To herself._) I might as well praise\na pillar-post! And after spending quite half an hour reading him up,\ntoo! I wonder if BERTIE PILLINER was right; but I shall have him all to\nmyself at dinner.\n\n_Lady Cant._ And where is _Rupert_? too busy of _course_ to come and\nsay a word! Well, some day he may understand what a sister is--when it's\ntoo late. Ah, here's our nice unassuming young poet coming up to talk to\nyou. Don't _repel_ him, my dear!\n\n_Spurr. (to himself)._ Better give her the chance of telling me what's\nwrong with the horse, I suppose. (_Aloud._) Er--nice old-fashioned sort\nof house this, Lady CULVERIN. (_To himself._) I'll work round to the\nstabling presently.\n\n_Lady Culv. (coldly)._ I believe it dates from the Tudors--if that is\nwhat you mean.\n\n_Lady Cant._ My dear ALBINIA, I _quite_ understand him; \"old-fashioned\"\nis _exactly_ the epithet. And I was born and brought up here, so perhaps\nI should know.\n\n    [_A footman enters, and comes up to_ SPURRELL _mysteriously._\n\n_Footman._ Will you let me have your keys, if you please, Sir?\n\n_Spurr. (in some alarm)._ My keys! (_Suspiciously._) Why, what do you\nwant _them_ for?\n\n_Lady Cant. (in a whisper)._ Isn't he _deliciously_ unsophisticated?\nQuite a child of nature! (_Aloud._) My dear Mr. SPURRELL, he wants your\nkeys to unlock your portmanteau and put out your things; you'll be able\nto dress for dinner all the quicker.\n\n_Spurr._ Do you mean--am I to have the honour of sitting down with all\nof _you_?\n\n_Lady Culv. (to herself)._ Oh, my goodness, what _will_ RUPERT say?\n(_Aloud._) Why, of course, Mr. SPURRELL; how can you ask?\n\n_Spurr. (feebly)._ I--I didn't know, that was all. (_To_ Footman). Here\nyou are, then. (_To himself._) Put out my things? he'll find nothing to\nput out except a nightgown, sponge bag, and a couple of brushes! If I'd\nonly known I should be let in for this, I'd have brought dress-clothes.\nBut how _could_ I? I--I wonder if it would be any good telling 'em\nquietly how it is. I shouldn't like 'em to think I hadn't got any. (_He\nlooks at_ Lady CANTIRE _and her sister-in-law, who are talking in an\nundertone._) No, perhaps I'd better let it alone. I--I can allude to it\nin a joky sort of way when I come down!\n\n       *       *       *       *       *\n\nTO MY BEEF TEA.\n\n(_By Our Dyspeptic Poet._)\n\n When the doctor's stern decree\n Rings the knell of libertee,\n   And dismisses from my sight\n   All the dishes that delight;\n When my temperature is high--\n When to pastry and to pie\n   Duty bids me say farewell,\n   Then I hail thy fragrant smell!\n\n When the doctor shakes his head,\n Banning wine or white or red,\n   And at all my well-loved joints\n   Disapproving finger points;\n When my poultry too he stops,\n Then, reduced to taking \"slops,\"\n   I, for solace and relief,\n   Fly to thee, O Tea of Beef!\n\n But--if simple truth I tell--\n I can brook thee none too well;\n   Thy delights, O Bovine Tea,\n   Have no special charm for me!\n Though thou comest piping hot,\n Oh, believe I love thee not!\n   Weary of thy gentle reign--\n   Give me oysters and champagne!\n\n       *       *       *       *       *\n\n\"CLUBS! CLUBS!\"\n\n    [\"FRY of Wadham,\" illustrious all-round athlete of Oxford, holds\n    that Golf is no better than \"glorified Croquet.\"]\n\n Oh, FRY of Wadham, you've opened your mouth,\n And \"put your foot in it!\" Here in the South,\n Talked to death by wild golfers, we're likely to cry\n Hooray, to see Link-lovers roasted by FRY.\n Golf-glorification's a terrible tax on\n The muscular Cricketing, Footballing Saxon,\n To whom the game seems just a little bit pokey.\n But FRY of Wadham, Sir, \"glorified Croquet\"!\n Champion of Champions, you're going to catch it!\n Each man loves his sport, swears no other can match it\n _Chacun a son gout!_ And he's rather to blame\n Who's prompt to make game of another man's Game!\n\n       *       *       *       *       *\n\n\"TO BE TAKEN AS READ.\"\n\nDEAR MR. PUNCH,--Thanks to the action of the Circulating Libraries, it\nseems that the old-fashioned three-volume novel is doomed to become a\nwork of the past. Most of the popular writers have abandoned it, and now\nthe publishers are beginning to fight shy of it. The principal argument,\nI believe, in favour of its retention is that it gives a chance to \"the\nlittle read.\" The Circulating Libraries are called upon to fill boxes\nintended for the edification of subscribers in the country, and in these\nreceptacles of light literature I believe the unpopular authors have\ntheir greatest chance. But as a matter of fact, although a romance may\nbe sent to a peruser, it is not within the scope of civilisation to\ncause that romance to be read. According to statistics I believe about\nsixty per cent of the second and third rate is only sampled by the\nrecipients of the aforesaid boxes. The last couple of pages of the third\nvolume are largely read, whilst the remainder of the work is saved from\nthe labours of the paper-knife. As this is so, would it not be as well\nto give a \"common form\" _finale_ to serve as a model for novels _in\nextremis_? To make my meaning plainer I will give an example.\n\n[Illustration]\n\nLet me suppose that the country subscriber has received a novel per\nparcels post called _The Deed in Drab_. Instead of having to cut some\nnine hundred pages, he finds gummed to the inside of the cover what I\nmay call\n\nTHE LAST CHAPTER.\n\nAnd so amidst the joy bells of the old church and the songs of the\nnightingales, and the pleasant laughter of the little children, EDWIN\nand ANGELINA were married. As they passed under the oaken porch the Duke\ngave them his blessing. Need it be said they lived happily--like a\nprince and a princess in fairy tale--for ever after?\n\nCaptain MONTMORENCY GUILT, kicked out of his club and warned off the\nTurf at Newmarket, left England with his ill-gotten gains for Cairo.\nArrived in Egypt, he disappeared into the Soudan. Those of the Arabs who\ncame from the desert declare that there is a white ruler in Khartoum.\nWhether it be he, who knows? Still, the stories of cruelty brought back\nby the swarthy traders are not unsuggestive of the man who brought poor\nPAULINE to her grave and broke the Bank at Monte Carlo.\n\nEDWARD WATTS _did_ marry MARY BEETLES, and they are now doing well at\nLittle Pannington. The village all-sorts shop has grown into a \"Stores,\"\nand those who are in the know say that at a near date it will be\nconverted into a \"Company, Limited.\" Be this as it may, EDWARD and MARY\ndrive to chapel in their own gig.\n\nAnd what became of PAUL PETERSON? Overwhelmed with the secret sorrow\nthat could never be shared by another, he went his way to the wilds of\nAustralia. And there, under the starlight influence of the Southern\nCross, and amidst the glorious glaciers of the Boomerang Mountains, he\ntries to forget the terrible and half-forgiven details of the \"Deed in\nDrab.\"\n\nTHE END.\n\nThere, Sir, you have the ending of ninety-nine novels out of\na possible hundred. In the hands of an experienced writer the\nsentences might be so adapted as to meet the requirements of the\nbook completing the century. Surely the suggestion is worthy of\nthe attention of a MUDIE, and the consideration of a W. H. SMITH.\n\n Yours faithfully,\n\n MULTUM IN PARVO.\n\n       *       *       *       *       *\n\n[Illustration: SUPPRESSIO VERI.\n\n_Mr._ \"AND HOW OLD ARE YOU, DEAR CHILD?\"\n\n_Little Miss._ \"I SHOULD LIKE TO SAY I'M EIGHT--BUT MAMMA WON'T LET\nME!\"]\n\n       *       *       *       *       *\n\nYE GENTLEMEN OF HOLLAND.\n\nAN ODE TO THE DUTCH CRICKETERS.\n\nAIR--\"_Ye Mariners of England._\"\n\nI.\n\n Ye Gentlemen of Holland\n   That guard your native stumps,\n Ye come to bat on wickets damp,\n   And block the ball that bumps.\n The \"glorious game\" you play amain,\n   And may you match the foe;\n And smite left and right,\n   While the balls for \"boundaries\" go;\n While your batsmen run 'em fast and long,\n   And the balls for \"boundaries\" go!\n\nII.\n\n The spirits of your fathers\n   Should watch you from the wave!--\n The brine, it was their field of fame;\n   On turf you're just as brave.\n As VAN TROMP'S and DE RUYTER'S did\n   Your manly breasts must glow\n As you smite left and right,\n   While the balls for \"boundaries\" go;\n Whilst the batsmen run 'em fast and long,\n   And the balls for \"boundaries\" go!\n\nIII.\n\n BRITANNIA loves to encounter\n   Her ancient foes--in peace.\n Our march is to the wickets green,\n   Our home is at the crease.\n With volleys from her native wood\n   She meets the friendly foe,\n As they smite left and right,\n   And the balls for \"boundaries\" go;\n While the batsmen run 'em fast and long,\n   And the balls for \"boundaries\" go!\n\nIV.\n\n The willows of old England,\n   Dutch willows shall not spurn!\n Your team we'll cheer when they depart,\n   We'll welcome their return!\n Then, then ye willow-warriors,\n   Our song and feast shall flow\n To the fame of your name,\n   When to Holland back ye go;\n When the shout \"How's that?\" is heard no more,\n   And to Dutchland back ye go!\n\n       *       *       *       *       *\n\nPUTTING HIS FOOT IN IT;\n\n_Or, The Wilful Markee._\n\n    [\"The House of Lords, for some reason, always assumes special care\n    of Ireland, a fact which may account for a few of the curiosities of\n    Irish political and domestic economy.\"--_Mr. Punch's Essence of\n    Parliament, June 3, 1861._]\n\nAIR--\"_Widow Machree.\"_\n\n Wilful Markee, it's loike thunder ye frown,\n     _Ochone! Wilful Markee!_\n Faith ye'd plase yer proud Parthy by kicking _me_ down,\n     _Ochone! Wilful Markee!_\n How haughty your air,\n As you kick me down-stair!\n Faix, I wondher ye dare\n   In this oisle of the free!\n Och, ye autocrat churl,\n Me poor head's in a whirl.\n     _Ochone! Wilful Markee!_\n\n Wilful Markee, Oireland's chance is now come,\n     _Ochone! Wilful Markee!_\n Whin everything smoiles must the Tories look glum?\n     _Ochone! Wilful Markee!_\n Sure the Commons, wid prayers,\n Have sint me upstairs;\n Who is it that dares\n   Wid me form disagree?\n _Don't_ haughtily pish\n At ould Oireland's last wish!\n     _Ochone! Wilful Markee!_\n\n Wilful Markee, whin a Bill enters in.\n     _Ochone! Wilful Markee!_\n To be kicking it out in this stoyle is a sin.\n     _Ochone! Wilful Markee!_\n Surely hammer and tongs\n To bad ould days belongs;\n Far betther sing songs\n   Full of family glee.\n Oireland's bad bitter cup\n Do not harshly fill up,\n     _Ochone! Wilful Markee!_\n\n And do ye not know wid yer bearing so bould,--\n     _Ochone! Wilful Markee!_\n How ye're kaping the poor tinants out in the could?\n     _Ochone! Wilful Markee!_\n Wid such sins on your head,\n Sure your peace will be fled;\n Could you slape in your bed\n   Widout thinking to see\n My ghost or my sprite\n That will wake ye each night\n     Groaning _Ochone! Wilful Markee!_\n\n Then take my advice haughty Wilful Markee,\n     _Ochone! Wilful Markee!_\n And loike \"Compensation Bill\" do not trate _me_!\n     _Ochone! Wilful Markee!_\n Of stroife we all tire,\n Then why stir the ould fire?\n Sure hope is no liar\n   In whisperin' to me,\n Hate's ould ghost will depart\n When you win Oireland's heart!\n     _Ochone! Wilful Markee!_\n       *       *       *       *       *\n\n[Illustration: \"PUTTING HIS FOOT IN IT.\"]\n\n       *       *       *       *       *\n\n[Illustration: STROLL IN THE ZOO. SUNDAY AFTERNOON.]\n\n       *       *       *       *       *\n\nTHE MESSAGE FROM MARS.\n\n(_Per favour of Mr. Punch._)\n\n_Mr. Punch._ So you've not been signalling to Mother Earth, after\nall, my noble Warrior?\n\n_Mars (with a wink)._ What do _you_ think? Why should I dig\ncanals 100 miles wide, and 2,000 miles long, or build bonfires as big\nas Scotland, when I can always communicate what I may have to say\nthrough you?\n\n Because Mars looks spotty or misty,\n Some dreamers, with intellects twisty,\n    Imagine, old horse,\n    Mars is playing at Morse!\n All bosh! You ask DYSON or CHRISTIE.\n\n_Mr. Punch._ Mr. MAUNDER \"has you under his special charge,\" hasn't he?\n\n_Mars._ Much obliged to Mr. MAUNDER, I'm sure! Wants to take my photo,\ndoesn't he? As if I were a mere politician, a popular comedian, or\n'ARRIET at the seaside on a Bank Holiday!\n\n_Mr. Punch._ Have you any Bank Holidays in your planet?\n\n_Mars._ Thank Sol, _Mr. Punch_, we have outlived the epoch of taking our\npleasure in spasms, like your cockney victims of the vulgar voluptuary's\nSt. Vitus's dance!\n\n_Mr. Punch._ Don't be uppish, old man! 'Tis an ill-bred age of Kodaks,\nand Interviews, and other phases of popular Paul Pryism. But you've had\nyour ignominious moments, Mars. If a \"snap-shot\" could have been taken\nat you when held prostrate, chained, and captive, at the feet of Otus\nand Ephialtes, or, still worse, when caught with Venus in the iron net\nof Vulcan:--\n\n All heaven beholds, imprison'd as they lie,\n And unextinguish'd laughter shakes the sky.\n\n_Mars._ Spare me, excellent _Punch_. Eugh! Thank heaven Olympus knew no\nKodaks then, or \"the gay Apollo\" would yet longer have had the laugh of\nme.\n\n_Mr. Punch._ Pardon me for awaking unpleasant memories! But even gods\nshould not be bumptious, especially when, like the _Second Mrs.\nTanqueray_, they \"have a past.\"\n\n_Mars._ Well, anyhow I've been able to baffle the camera-wielders up to\nnow. My ruddy countenance and \"bluish radiance\" have beaten Greenwich,\nand even licked the Lick! As they themselves admit, \"Mars up to the\npresent has defied cameral detection.\"\n\n_Mr. Punch._ But what about those \"bright spots\"?\n\n_Mars._ Have you no \"bright spots\" even on your dull and foggy old\nplanet? I have often noticed one at 85, Fleet Street. In June and\nDecember it emits thousands of brilliant sparks of a \"bluish radiance,\"\ntoo. But I don't jump to the conclusion that you are \"signalling\" to me.\nLook, the naked eye can see the Punchian \"_projection lumineuse_\" even\nfrom here!\n\n_Mr. Punch._ I do not have to \"signal\" my messages to \"Hellas\" or\n\"LOCKYER'S Land\" by canals or \"ten million arc lights of 100,000\ncandle-power apiece.\" Like the Sun, I am self-luminous, and do not, like\nthe finest planets, shine by reflected light.\n\n_Mars._ True for you. And from your own intellectual observatory, like\nTEUFELSDROECKH \"alone with the stars,\" you ofttimes scan the heavens\nwhen, as LONGFELLOW says:--\n\n \"----the first watch of night is given\n To the red planet Mars.\"\n\n_Mr. Punch._ Precisely!\n\n    [_Murmurs musingly._\n\n And earnest thoughts within me rise\n   When I behold afar,\n Suspended in the evening skies\n   The shield of that red star.\n\n A star of strength! I see thee stand\n   And smile upon my pain;\n Thou beckonest with thy mailed hand,\n   And I am strong again.\n\n The star of the unconquered will\n   He rises in my breast.\n Serene, and resolute and still,\n   And calm, and self-possessed.\n\n_Mars._ Ah yes! that's all very pretty and poetical, and I'm much\nobliged to HENRY WADSWORTH and the other bards who have lyrically\nglorified me. But _Punch_, old man, _you and I know better_! Mother\nEarth has ever paid, and payeth still, far too much worship to Mars--the\nMars of her own militant fancy. To tell you the truth, _Punch_, I'm sick\nof my old _metier_, especially since Science stepped in and bedevilled\nit past bearing with her big guns, and dynamite-bombs, and treacherous\ntorpedoes; weapons more fit for grubby Vulcan's subterranean Cyclops\nthan a god, a gentleman and a soldier like me.\n\n_Mr. Punch._ Hoho! That's the way the (LOCKYER'S) land lies, eh?\n\n_Mars._ Exactly, _I_ wasn't signalling to your stupid, conservative,\nbellicose old world, which, like the Bourbons, learns nothing and\nforgets nothing. Could I write in plain Titanic capitals across a\nthousand square miles of my smoothest surface Mars's Straight Tip to\nMother Earth, viz.:--\n\n FIGHTING'S AN EXPENSIVE BORE,\n SO DISARM AND WAR NO MORE!\n\nwhat effect would it have on any of you, from civilised England, with\nyou to enlighten it, to the furious fighting dragons who are tearing\neach other in the eastern seas? None! But if any of your quidnuncs\nreally want to know what I _would_ say if I _did_ signal, tell them old\nMars, grown wiser, has turned up War; has nailed his raven to a\nbarn-door as a warning; has made a pet of Peace's soft-plumed dove; and\nstrongly advises the belligerent boobies on earth who take his old name\nin vain, and play his abandoned game still, to--_go and do likewise_!!!\n\n_Mr. Punch._ By the cestus of Venus, and so I will!!!\n\n       *       *       *       *       *\n\nODE TO IXION.\n\n(_By a Sympathetic, but Superficial Observer._)\n\n Oh! the hardest of hearts some compassion must feel\n For that modern Ixion, the Man on the Wheel!\n See him scouring the roads on his spindly-spoked spider,\n Dust-hid till you scarce tell the \"bike\" from its rider;\n His abdomen shrunken, his shoulders up-humped,\n With the gaping parched lips of one awfully pumped.\n _Could_ a camel condemned to the treadmill look worse?\n Sure those lips, could he close them, would shape to a curse\n On his horrible doom! As I gaze and stand by,\n With a pang at my heart, and a tear in my eye,\n I think of Ixion, the Wandering Jew,\n That Cork-legged Dutchman--the Flying One, too,\n And other poor victims of pitiless speed;\n And I own, while _their_ cases were frightful indeed,\n The Bicyclist's fate is the worser by far.\n Poor soul!!! The small \"pub,\" and a \"pull\" at the \"bar,\"\n Appear your best comfort. Imagine the cheer\n Of a slave of the \"bike\" whose sole solace is beer!\n You can't see the prospect; your eyes are cast down\n Like BUNYAN'S _Muck-raker_; your brows in a frown\n Of purposeless effort are woefully knit;\n Of Nature's best charms you perceive not a bit.\n The hedge your horizon, the long, dusty road\n Is your sole point of sight. Wretched victim, what goad\n Of Fate, or sheer folly, thus urges you on?\n Old torments--like poor Io's gadfly--are gone,\n And yet, like Orestes, the Fury-whipped, you\n Wheel on, as some comet wheels on through the blue\n In billion-leagued cycles less dreary than is\n The cycle on which round the wide world you whiz!\n Eh? _Cutting a record?_ You _like_ it? The goose!!!\n A task without pleasure, a toil without use!\n Poor soul! You are worse than Ixion, I feel,\n For _he_ was not tied _by himself_ to the wheel!\n\n       *       *       *       *       *\n\n[Illustration: CONJUGAL EGOTISM.\n\n\"WHAT A STUPID PAPER THIS IS, ROBERT! NOT A WORD ABOUT _YOU_ IN IT!\"]\n\n       *       *       *       *       *\n\nThe Plaint of the Unwilling Peer.\n\n From my M.P.'s seat I--oh, the pity!--must move.\n   I am one of Rank's sorrowful heirs;\n For the Commons Fate bids me dissemble my love,\n   But _why_ did she kick me upstairs?\n\n       *       *       *       *       *\n\nON TICK.--The Modern Novel is a blend of the Erotic, the Neurotic, and\nthe Tommy-rotic.\n\n       *       *       *       *       *\n\nWHERE TO GO.\n\n _Antwerp_--if you are not tired of Exhibitions.\n _Boulogne_--if you don't mind the mud of the port.\n _Cologne_--if you are not particular about the comfort of your nose.\n _Dieppe_--if you like bathing in the foreign fashion.\n _Etretat_--if solitude has commanding charms.\n _Florence_--if you are partial to 100 deg. in the shade.\n _Genoa_--if you have no objection to mosquitoes.\n _Heidelberg_--if you are not tired of the everlasting castle.\n _Interlacken_--if the Jungfrau has the advantage of novelty.\n _Java_--if you wish to eat its jelly on the spot.\n _Kandahar_--if you are not afraid of Afghan treachery.\n _Lyons_--if you are fond of riots and _emeutes_.\n _Marseilles_--if you are determined to do the Chateau D'If.\n _Naples_--if you are anxious to perform an ante-mortem duty.\n _Ouchy_--if you like it better than Lausanne.\n _Paris_--if you have not been there for at least a fortnight.\n _Quebec_--if you are qualifying for admission to a lunatic asylum.\n _Rome_--if you have never had the local fever and want to try it.\n _Strasbourg_--if you are hard up for an appropriate destination.\n _Turin_--if it is the only town you have not seen in Italy.\n _Uig_--if you affect the Isle of Skye in a thunder-storm.\n _Venice_--if you scorn stings and evil odours.\n _Wiesbaden_--if you can enjoy scenery minus gambling.\n _Yokohama_--if you are willing to risk assault and battery.\n _Zurich_--if you can think of no other place to visit.\n\n[Illustration]\n\nN.B.--The above places are where to go on the keep-moving-tourist plan.\nBut when you want to know \"WHERE TO STAY,\"--we reply, \"AT HOME.\"\n\n       *       *       *       *       *\n\nTHE INCONVENIENCED TRAVELLER'S PHRASE-BOOK.\n\n(_To be Translated as Required._)\n\nWhy have you thrown my boxes down with such violence that their contents\nhave become distributed on the platform?\n\nWhy is it necessary to strike me on the head with a stick because I am\ntaking my proper place at the ticket-office?\n\nWhy have you refused to give me change for a sovereign, minus the\neighteenpence you have the right to charge for my fare?\n\nWhy do you close the door of communication when I offer a remonstrance?\n\nWhy can I not obtain redress upon complaint to the station-master?\n\nWhy am I chased off the premises by a private policeman when I am\nanxious to catch the next train?\n\nWhy is my luggage being placarded with places that certainly do not\ncorrespond with my desired destination?\n\nWhy can I not have my tea cool enough to drink? and why I am hurried out\nof the refreshment-room before I can discuss my bread and butter?\n\n[Illustration]\n\nWhy must I pay half-a-crown for comestibles valued on the card at less\nthan a shilling?\n\nWhy am I forced into a carriage already overcrowded with aged females,\nsickly children, and snarling spaniels?\n\nWhy can I not have a seat, considering I have paid the full fare, and\namply tipped the guard?\n\nWhy can I not have a window open, considering that the glass stands at\nninety in the shade?\n\nWhy can I not smoke, having chosen a smoking carriage?\n\nWhy should I be dictated to by a disagreeable and elderly stranger, who\nsnores half the journey, and helps herself to ardent spirits in the\ntunnels?\n\nWhy should I be threatened with imprisonment, and be only pardoned by\nrepaying my fare because I have lost my ticket?\n\nAnd, lastly (for the present), why have I been carried to Little\nPeddlington-on-the-Ditch when I desired to reach the British Coast _en\nroute_ for Paris?\n\n       *       *       *       *       *\n\nAIRS RESUMPTIVE.\n\nIII.--THE RIME OF THE ANCIENT SAWBATH-BREAKER.\n\n(_Being a Record of the 12th._)\n\n It was an ancient poacher-man,\n   Bronzed as a penny-bun;--\n \"By thy beady eye, now tell me why,\n   Thou offspring of a gun,\n\n O tell me why beneath thy\n   Exceeding hoary tuft\n Precisely half a brace of grouse chin's\n   Hangs, admirably stuffed?\"\n\n He blinked his beady eye; his voice\n   Was singularly clear;\n And as I listened to his tale\n   I could not choose but hear.\n\n \"Mon, ye mun ken I have not aye\n   Been sec a feckless loon;\n In me behold the wreck of what\n   Was once The MCAROON.\n\n Oft have I made a merrie bag\n   Across my native heath;\n Shot o'er my ain ancestral dawgs\n   Or aiblins underneath.\n\n Until lang syne, a monie year--\n   Ye couldna weel be born--\n The blessed twalfth of August fell\n   Upon a Sawbath morn.\n\n Braw were the birds, my gun was braw,\n   My bluid was pipin' hot;\n I thocht it crime to gie 'em time--\n   Allowance like a yacht.\n\n Scarce had I bagged but ane wee bird,\n   There was the de'il to pay:\n It's unco deadly skaith wi' Scots\n   To break the Sawbath day.\n\n The billies wha the nicht before\n   Were fou at my expense,\n They deaved the meenister aboot\n   My verra bad offence.\n\n An' a' the Kirk declared the work\n   Was perfect deevilrie,\n An' hung the bird by this absurd\n   Arrangement whilk ye see.\n\n Twal' month an' mair my shame I bear\n   Beneath the curse o' noon,\n A paltry wraith of what was once\n   The Laird o' MCAROON.\n\n An' aye when fa's the blessed twalfth\n   Upo' the Sawbath day,\n I bear the bird in this absurd\n   An' aggravatin' way.\"\n\n The ancient ceased his sorry tale,\n   And craved a trifling boon,\n To wet the whistle of what was once\n   The Laird o' MCAROON.\n\n       *       *       *       *       *\n\n[Illustration: THE OBSTINACY OF THE PARENT.\n\n_Emily Jane._ \"YES, I'M ALWAYS A-SAYIN' TO FATHER AS 'E OUGHTER RETIRE\nFROM THE CROSSIN', BUT KEEP AT IT 'E WILL, THOUGH IT AIN'T JUST NO MORE\n'N THE BROOM AS 'OLDS 'IM UP!\"]\n\n       *       *       *       *       *\n\nDitto to Mr. Courtney.\n\n As after jackdaw chatter and owl-hooting,\n Gratefully follows Philomel's dulcet fluting;\n So, after HANBURY'S gibes and HEALY'S jeers,\n COURTNEY'S cool reason gladdens patriot ears.\n _O, si sic omnes!_ But though his sole voice\n Sound \"in the wilderness,\" yet _some_ rejoice\n To hear, 'midst blare of venomed wrath and vanity,\n The moving tones of brave, sound-hearted sanity.\n\n       *       *       *       *       *\n\nTHE FLY ROUTE TO CASTLES IN THE AIR.\n\n(_By Our Imaginary Interviewer._)\n\nI found The great man surrounded by plans and models of any number of\nwonderful inventions. Here was a clever scheme for spending a week's\nholiday in the Mountains of the Moon, there a recipe for removing the\nspots from the face of the sun. It would take too long to give an\ninventory of all the marvels. Enough to say their name was legion.\n\n\"And so you have discovered the secret of aerial navigation?\" I asked,\nafter I was comfortably seated.\n\nThe great man smiled. He evidently had solved the difficult problem.\n\n\"I suppose that now you and all will be able to do without ships and\nrailways? I presume we shall be independent of cabs and omnibuses?\"\n\n[Illustration]\n\nOnce more there was a smile. I was answered. \"Of course,\" I continued,\n\"you will be able to take your aerial contrivances to all the countries\nof the earth? What is there to prevent you from starting flying-machines\nfrom London to Paris, or Berlin, or even Timbuctoo?\" Again there was a\npleasant smile. Evidently my guess was a good one.\n\n\"You will be able to travel thousands of miles without the assistance of\nrails? You will dispense with land and water? All you will require will\nbe the atmosphere, and that is always with us--always at our service.\"\n\nAgain my suggestions remained uncontradicted.\n\n\"It is truly marvellous,\" I remarked; \"truly marvellous! And you have\ncommenced? You have been able to float through the air for a dozen, a\nhundred feet?\" There was a smile once again.\n\n\"And yet, perhaps, as railways and steamships are still 'firm' on the\nStock Exchange, it may be just as well to allow our holdings in those\nsecurities to remain undisturbed? What do you think? It is scarcely time\nto speculate for a fall?\" Once more he smiled, and as smiling is\ninfectious, I joined him in his merriment.\n\n       *       *       *       *       *\n\nTO A VETERAN CHAMPION.\n\n    [At Clifton, on Aug. 9, in Gloucestershire _v._ Middlesex, Dr. W. G.\n    GRACE completed his 1000 runs in first-class matches this summer.\n    The other players who share this distinction are ABEL, ALBERT WARD,\n    and BROCKWELL.]\n\n Well hit! _Mr. Punch_ chalks it up once more--\n   Your ten-hundredth run between the \"creases\"!\n   Why, this (at twenty-two yards apiece) is\n Twelve-miles-and-a half for this season's score!\n\n But stay! we've no business to \"notch\" each mile!\n   With your cuts and draws, and your drives and trick hits,\n   You've only to stand still before the wickets,\n And straight to the boundary \"fours\" compile!\n\n With ABEL, WARD, BROCKWELL, you hold your own,\n   As '94 cricket now nears its finish;\n   We'll hope your four figures will ne'er diminish--\n As \"Grand Old Bat\" you shall e'er be known!\n\n       *       *       *       *       *\n\nQUEER QUERIES.--THE LAW AND THE LADY.--Can it really\nbe true that at a place called Onehunga, in New Zealand, they have\na lady as Mayor? Surely this is altogether \"_ultra vires_,\" as well\nas being ultra-virile! My legal knowledge--which is considerable--convinces\nme that there is a fatal flaw in the so-called election of a\nwoman to the chief post in a municipality, even in New Sheland--I\nmean New Zealand. It's quite settled law that a _femme sole_ cannot\nbe a Corporation; then how, I should like to know, can she preside\nover a Corporation? Possibly some legal readers will say if their\nopinion coincides with mine.\n\n BARRISTER (UNCALLED FOR).\n\n       *       *       *       *       *\n\nESSENCE OF PARLIAMENT.\n\nEXTRACTED FROM THE DIARY OF TOBY, M.P.\n\n_House of Lords, Monday Night, August 6._--MARKISS expected to continue\nto-night that speech around the Budget he didn't commence on second\nreading of the Bill. Sat mysteriously quiet on that occasion.\nUnexpectedly broke out at following sitting, wanting to know what\nHERSCHELL meant by saying Judicial Committee of Privy Council had\narrived at conclusion that Lords had no power to amend a money bill.\n\"Where's your report?\" he asked. \"Produce it.\"\n\nLORD CHANCELLOR didn't happen to have it in his waistcoat pocket or\nsecreted in wig. MARKISS gave notice that he would to-night formally\nmove for production of report. Flutter of interest in House. Commons\nflocked in prepared for some fresh \"blazing indiscretion.\" Found the\nMARKISS sitting on woolsack chatting with LORD CHANCELLOR. Held book\nbetween them, as young persons about to marry are wont to do when\nattending morning or evening service. Vague idea that presently they\nwould rise and sing a hymn. LORD CHANCELLOR quite equal to it, being a\nbig gun at the Bar Musical Society and very fond of the Opera. Nothing\nhowever came of it, at least, not in that direction. When hour for\npublic business arrived MARKISS left woolsack carrying the tune book\nwith him. His motion for report of Judicial Committee stood half way\ndown Orders of the Day. When it was reached MARKISS said nothing.\nNaturally other peers were silent, and whilst commoners accustomed to\nother ways of transacting business were marvelling as to what had\nhappened, and what would follow, House adjourned, practically for a\nweek.\n\n\"Well,\" said SARK for once nonplussed; \"certainly if there is a place in\nthe world where 'e don't know where 'e are, it's the House of Lords.\nWhen a peer is expected to speak he sits dumb. When arrangements have\nbeen made for a quiet sitting, the MARKISS or some other big gun is sure\nto go off unexpectedly with alarming consequences.\"\n\n_Business done._--Irish Evicted Tenants Bill passed Report Stage in\nCommons.\n\n_Tuesday._--It is the unexpected that happens in the House of Commons.\nBefel to-night with dramatic suddenness. Third reading of Evicted\nTenants Bill moved. At eleven o'clock JOSEPH resumed his seat with\npleased consciousness of having cast some balm, in the shape of vitriol,\nover Irish Question. House crowded; DEVONSHIRE, in depression and dinner\ndress, looked down from Peers' Gallery. Over the clock sat SANDHURST,\npresently to move first reading of Bill in House of Lords. Arranged Bill\nshould finally leave Commons to-night. Only one hour in which PRINCE\nARTHUR might speak, and JOHN MORLEY reply. JOSEPH having despatched his\nfinal arrow at his old friends the Irish Members, the shaft being barbed\nwith points composing pleasing legend, \"Violence, Agitation,\nDishonesty,\" PRINCE ARTHUR rose, with evident intent of showing, as has\nhappened several times this Session, how the same sort of thing may be\nsaid with better effect in quite another way.\n\n[Illustration: The Macgregor proposes to \"toss the Caber\"--next\nSession!]\n\nSimultaneously from below gangway uprose the tall figure of JOHN DILLON.\nOpposition roared with despairing indignation. Everything settled, to\nlast button on the gaiter; JOSEPH had had his half-hour; Prince ARTHUR\nwould take his, honourably leaving JOHN MORLEY his thirty minutes. Then\nDivision called; Bill read third time; sent on to Lords; Commons\ncomfortably home by half-past twelve. And here was JOHN DILLON claiming\nthe right to reply to attacks and inuendos of the genial JOSEPH!\n\nTumult rose; DILLON folded his arms and faced it. A bad sign\nthat gesture. Remember it in years gone by, when all things were\ntopsy-turvey; when FORSTER was Chief Secretary, and, next to PARNELL,\nthe hope of the Irish Members fighting for Home Rule was JOSEPH\nCHAMBERLAIN.\n\nDILLON in that attitude evidently immoveable; various suggestions\noffered. Evade the Twelve o'Clock Rule, and sit till all was over;\nadjourn the Debate. Finally agreed that Debate should be adjourned till\nto-morrow--to-morrow, the day on which, at end of last real fight of\nSession, most Members were off on the delayed holiday.\n\nOut of this dilemma PRINCE ARTHUR delivered a grateful House. Had\nprepared his speech through long sitting; doubtless had many bright\nthings to say; but what was one speech among so many? Perish his speech,\nrather than the whole arrangements of Parliamentary week be upset. So\ngracefully stood aside; DILLON took his half hour; JOHN MORLEY followed\nin vigorous fighting form, marking fresh step in steady improvement as\nParliamentary debater; and before midnight all was over.\n\n_Business done._--Evicted Tenants Bill read third time by 199\nvotes against 167.\n\n_Wednesday._--M. de Londres--the Hangman, as blunt Britons put\nit--called to-day. House engaged on Committee of Equalisation of Rates\nBill; seat found for Monsieur under Gallery, where private secretaries\nof ministers and heads of public offices sit when Bills affecting their\ndepartments are under discussion.\n\n\"Monsieur has something to do with the Home Office, _n'est ce pas?_\" I\nasked SARK. \"Looked in, I suppose, to help ASQUITH?\"\n\n\"No,\" said the Member for SARK. \"It's not that. He's heard House intends\nto suspend the Standing Orders. Wants to see how _we_ go to work. Not\nabove taking a wrinkle even from amateurs.\"\n\n\"Ah,\" said W. P. JACKSON, throwing up his hands with gesture of despair.\n\"Knew it would come to this under present Government. First the\nguillotine, then the gallows.\"\n\n_Business done._--Quite a lot.\n\n_Thursday._--Southerners long heard of pleasurable hours spent in\nCommittee-room upstairs, where Scotch Members been engaged for weeks in\nGrand Committee on their Local Government Bill. Such badinage! such\npersiflage! not omitting refreshing influences of another kind familiar\nin _Noctes Ambrosianae_. 'Tis said, when conversation flagged quite usual\nthing for J. B. BALFOUR and CHARLES PEARSON to strip off coats and\nwaistcoats, place two umbrellas crosswise on floor, and go through\nsword-dance, TREVELYAN in the chair leading off colourable imitation of\nbagpipe accompaniment, in which Committee joined in mad chorus.\n\nNot sure about that. Absolutely no doubt that on last day of meeting all\nthe members stood on chairs, with one foot on the table, and, holding\nhands, sang \"_Auld Lang Syne_.\" Bound to say they seem to have exhausted\nall their hilarity in Committee-room. PARKER SMITH still a good deal to\nsay; HOZIER not uncommunicative; and WALTER M'LAREN enjoys keen\nsatisfaction of insisting on Division that presents smallest minority of\nthe series. But, on the whole, House seems filled with what SARK tell me\nEdinburgh, occasionally suffering from the visitation, calls \"an\neasterly haar.\"\n\nThrough the cold, wet, white fog, comes one gleam of light. JOHN MORLEY\nbrings in a Bill making further provision with respect to Irish\nCongested Districts Board. SPEAKER puts customary question, \"Who is\nprepared to bring in this Bill?\" \"Mr. ARTHUR BALFOUR and myself,\"\nresponds the CHIEF SECRETARY; and the House gratefully goes off into a\nfit of laughter.\n\n\"Lovely in life,\" exclaims DAVID PLUNKET, looking with almost equal\naffection on his two right hon. friends, \"on the Congested Districts\nBoard (Ireland) Bill they are not divided.\"\n\n_Business done._--Scotch Local Government Bill.\n\n_Friday._--Another \"Nicht wi' BURNS.\" Sadder even than the last. But\nsooner over. By eleven o'clock report stage agreed to. \"Shall we take\nthird reading now, or would you like a third night with the Bill?\" asked\nTREVELYAN.\n\nA shudder ran through the House; when it was over Bill hurried past\nfinal stage.\n\n_Business done._--Winding-up rapidly.\n\n       *       *       *       *       *\n\nTHE NEW NEWNESS.\n\n  \"There is nothing new under the sun.\"\n    So said the proverbial preacher.\n  But surely 'twas only his fun!\n    A modern and up-to-date teacher\n  Would tell him that Humour, and Art,\n    And Daughters, and Wives, and Morality,\n  All aim to make a fresh start\n    In novel (and nauseous) reality;\n  And the wail of the Wise Man will be, pretty soon,\n    \"There is nothing _old_ under the sun--or the moon!\"\n\n[Transcriber's Note:\n\nAlternative spellings retained.\n\nPunctuation normalized without comment.]\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Punch, or the London Charivari, Volume\n107, August 18, 1894, by Various\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n# Copyright\n\nCopyright \u00a9 2018 by Franchesca Ramsey\n\nCover design by Brian Lemus\n\nCover photography \u00a9 Erin Patrice OBrien\n\nCover copyright \u00a9 2018 by Hachette Book Group, Inc.\n\nHachette Book Group supports the right to free expression and the value of copyright. The purpose of copyright is to encourage writers and artists to produce the creative works that enrich our culture.\n\nThe scanning, uploading, and distribution of this book without permission is a theft of the author's intellectual property. If you would like permission to use material from the book (other than for review purposes), please contact permissions@hbgusa.com. Thank you for your support of the author's rights.\n\nGrand Central Publishing\n\nHachette Book Group\n\n1290 Avenue of the Americas\n\nNew York, NY 10104\n\ngrandcentralpublishing.com\n\ntwitter.com\/grandcentralpub\n\nFirst ebook edition: May 2018\n\nGrand Central Publishing is a division of Hachette Book Group Inc. The Grand Central Publishing name and logo is a trademark of Hachette Book Group, Inc.\n\nThe Hachette Speakers Bureau provides a wide range of authors for speaking events. To find out more, go to www.hachettespeakersbureau.com or call (866) 376-6591.\n\nThe publisher is not responsible for websites (or their content) that are not owned by the publisher.\n\nISBN 978-1-538-76104-5\n\nE3-20180404-JV-NF\n\n# CONTENTS\n\n  1. Cover\n  2. Title Page\n  3. Copyright\n  4. Dedication\n  5. **Introduction:** Girl Walks into the Comments Section\n  6. 1 Famous in Four Hours\n  7. 2 Black-lash\n  8. 3 My Reign as YouTube's Callout Queen\n  9. 4 Between a Loc and a Hard Place: The Chescalocs Story\n  10. 5 Lessons from Going Public, Part 1: If You Can't Stand the Heat, Get Out of the Spotlight\n  11. 6 Lessons from Going Public, Part 2: Objects on Social Media Are Not as Close as They Appear\n  12. 7 Guess Who's Coming to Dinner? (It's Lena Dunham.)\n  13. 8 Stop Hating and Start Studying\n  14. 9 Activist Lent\n  15. 10 Unfriended\n  16. 11 Last Name Basis\n  17. 12 Eulogies for Cringeworthy Comments\n  18. 13 Self-Care Is Not Selling Out (Unless It Is)\n  19. **Conclusion:** Activism Is Like Long Division\u2014You Have to Show Your Work\n  20. Franchesca's Simple Explanations of Not-So-Simple Concepts\n  21. Acknowledgments\n  22. Newsletters\n\n# Navigation\n\n  1. Begin Reading\n  2. Table of Contents\n\n_For my mother, the best friend, best blessing, and best role model a girl could ask for._\n\n# INTRODUCTION\n\n## GIRL WALKS INTO THE COMMENTS SECTION\n\nI know the exact date I went from being a nobody, minding my own business in my corporate retail job, to being \"internet famous\"\u2014and inadvertently making a lot of girls cry.\n\nI have a long and complicated history with the internet. I basically grew up online. I built my first website in middle school after spending the summer at computer camp learning how to code. My first boyfriend was a kid I met in an AOL chatroom. Smartphones and digital cameras didn't exist back then, so our late-'90s version of sexting was me taking a Polaroid of my nipple and scanning it. When I'm super famous and my long-lost internet boyfriend inevitably comes out of the woodwork and releases that photo to the press, I have a statement ready: \"That low-res mess of pixels is not recognizable as a human breast.\"\n\nIn high school I bought my own domain name, franchesca.net, and started blogging about my life before it was actually called blogging. I kept that up through college before making the leap to video in 2006, one year after YouTube was founded. I spent the next six years making YouTube videos in my spare time, just for fun; the topics spanned everything from hairstyle tutorials to informational discussions about safe sex to original songs about student loan debt. (A sample: \"Went off to school to get my education \/ Little did I know debt was part of the equation\"\u2014I know, I'm good.)\n\nI'd spend hours each week filming and editing videos after work\u2014and sometimes _during_ work, when no one was paying attention\u2014but I never had a very big audience. My comment section generally broke down into three categories:\n\n1. \"LOL\"\n\n2. \"Kill yourself.\"\n\n3. My mom scolding me about typos in graphics and inappropriate jokes\n\nThen, one day, it actually happened\u2014every YouTuber's not-so-secret dream: One of my videos went viral. And I don't mean _Huffington Post_ viral. I'm talking supermassive, mainstream-news viral\u2014an unstoppable contagion, if contagions also had some good side effects. It launched my career.\n\nYou could say it all happened because my high school's alumni Christmas party left me sick of white people's shit. I was frustrated by the same tired conversations I kept having with friends and acquaintances I had known for decades, the head patting, the hair yanking, and the gently racist observations that seem to just roll off the tongue after a few drinks. So I did the only thing I knew how to do moderately well: I put my frustrations into a video.\n\nAnd before I knew it, my life got turned upside down, _Fresh Prince_ style.\n\nThat video was \"Shit White Girls Say... to Black Girls,\" and it was right on brand for the kind of social commentary I'd been begging people to watch on my YouTube channel and would soon be known for. Within a couple of hours of uploading \"SWGSTBG,\" I was officially the new (sh)it girl. I soon quit my job to pursue acting and internetting full-time.\n\nIn the years since, a lot has changed. These days, most people know me as a fiercely passionate, outspoken social justice advocate, laying down truth on my MTV show _Decoded_ or popping up as a commentator on cable news. But while I am extremely proud of the conversations my work has sparked around the world, I'm still embarrassed to admit that none of this was expected. Before my career exploded, I was pretty comfortable pushing pixels at my desk job as a graphic designer.\n\nIn other words, I didn't set out to be an activist, and I've made a lot of mistakes along the way. And precisely because of that, I think I have a lot to teach folks who find themselves on this same journey, struggling to find their voice and stand up for what they believe in without screaming at some guy who calls himself LethalDUMPS22 on Twitter that he doesn't know your life! Of course he doesn't know your life\u2014he has chosen to go by the name LethalDUMPS22.\n\nWhile I've never rocked a war bonnet at Coachella, I have put my foot in my mouth more than a few times. Social media means we're all living our lives in public now, and many of us are learning how to be advocates, allies, and activists in public, too. And while there may be nothing that can prepare you for the perks and pitfalls of being elected the \"racism referee\" among your friends and family, there are some proven methods for coping when someone writes a 1,500-word Tumblr post about how you are obviously oblivious to your self-hating racism. (The social justice glossary at the end of this book\u2014\"Franchesca's Simple Explanations of Not-So-Simple Concepts\"\u2014may also come in handy.) I've been on the receiving end of more than a few of those missives\u2014and, I hate to admit it, have dished them out, too.\n\nAs the conversation about social justice broadens, I wish we could be more understanding of those who may be coming to it later than others. I was at a speaking gig at a university recently, and toward the end of the talk a girl in the audience asked me a question that made me really sad. She began by sharing that she felt guilty about how \"ignorant\" she had been in high school, where she'd been the only Asian student but had never thought about it much. Once she got to college, she started watching my videos, and now she couldn't get over all the times she hadn't objected to people's offensive comments, and times when she'd said disrespectful things herself. She ended by saying that she wanted to go back to her hometown and raise awareness at her high school, but she worried she would be hypocritical for doing so.\n\nIt can be really scary to admit that there are a lot of things you don't know. We live in a world where people are quick to pounce on you if you express confusion or ask a question, and many online activists aren't honest about the fact that they didn't always know what the gender binary was, either. Too often, people climb the ladder and say, \"Hell yeah, I climbed the ladder! And I beat you to the top!\" The thing about the ladder is that you never stop climbing, and if you think you have, you have a lot more work to do.\n\nI told the \"guilty\" college student standing in front of me that I didn't even start thinking about this stuff until way after college, when a mob of angry Tumblr users descended on my social media accounts to explain to me exactly why I was a hopeless person who had no idea what she was talking about. I may not have always had the vocabulary to explain what I was talking about, but I was certainly not hopeless. My wake-up call didn't come until my twenties, and I'm still learning, too.\n\nIt can be hard to remember this today, when the stakes for these kinds of conversations are higher than ever. Because much of this work is happening online, especially for young people, the crucial nuances of face-to-face interaction are almost nonexistent. What's more, we now have a record of everything. It used to be that if you made a crass joke, someone might call you out on it, you could learn, and you could move on. But now these offhand, often accidental comments can be used as an indictment of someone's character in perpetuity. There are experts in the field of Problematic Archaeology who will spend hours combing through your tweets and preteen blogging efforts to uncover the most offensive artifacts of your past. (I've been those people\u2014and I found what I was looking for.) Often, disagreements and misunderstandings escalate so quickly that there isn't time to reflect, understand where you went wrong (if you actually went wrong at all), and figure out how to fix it.\n\nThis book is an attempt to show you that mistakes are inevitable, and that what's actually important is how we use them to make a better world. For all the hate and abuse I get from all points on the political spectrum, I've been fortunate that many people have been compassionate about showing me where I messed up and helping me get back on track. I wanted to pay that generosity forward somehow, and I'm doing it the best way I can: by pulling my own receipts and dragging my former self.\n\n# CHAPTER ONE\n\n# FAMOUS IN FOUR HOURS\n\nIf I could do it all over again, I would have bought a better wig. The hair I wore to parody white girls was platinum blond, of course, and it hung a little past my shoulders in the kind of long layers beloved by women who star in reality TV shows set in California. In that way, it was perfect. But it also didn't really fit my head.\n\nThe morning before my life changed was like any other, except that I spent most of it looking at endless footage of myself in this bad wig. I'd set \"Shit White Girls Say... to Black Girls\" to upload to YouTube the night before; high-speed internet was a luxury I couldn't yet afford, and my connection was slow as hell. When I woke up, I saw that the upload had failed. Before I left for my job as a graphic designer at Ann Taylor, I started the upload again, thinking my hour-long commute would give it plenty of uninterrupted time to transfer and that I could finish the posting process from work. I looked at the still I'd chosen for the video: My eyebrows were halfway up my forehead, and my smile was cheesy. A few blond strands were caught on my hand, which was paused in the middle of raising the roof\u2014part of my imitation of a white girl singing Nicki Minaj's \"Super Bass.\" I didn't know it at the time, but millions of people were about to meet my freeze-frame face rocking this look.\n\nWhen I got to work, I saw that my janky internet had pulled through. Our office had an open floor plan, which, while gorgeous and modern, was not architecturally convenient for maintaining a side hustle. As I edited my video's description box, I switched back and forth between YouTube and the image of Demi Moore in business-casual that I was Photoshopping for our new ad campaign, constantly looking over my shoulder to watch out for someone about to catch me in the act. \"Ms. Ramsey... have you been producing humorous videos for the internet on company time?\" (I have since accepted that procrastination is a fundamental part of any job requiring a computer.) I created a prewritten \"Click to tweet\" link, added all my social media links, and made sure to note the royalty-free music site I used for the end credits per instructions from Patrick, my then-aspiring-lawyer boyfriend. (Getting sued is one of my major career phobias.)\n\nWith the description box finished, \"Shit White Girls Say... to Black Girls\" was ready for her close-up. I changed the setting from private to public, and then, a full hour after arriving at work, finally decided to give Ann Taylor my full attention. I spent the morning listening to my _Hunger Games_ audiobook and wielding my Magic Wand and Healing Brush on ladies in tasteful cardigans.\n\nBefore \"SWGSTBG,\" I was convinced I was going to go viral for a parody video I'd made a few months before, \"Student Loan Countdown.\" It was based on the criminally underrated Beyonc\u00e9 bop \"Countdown,\" and the idea for the parody came to me as soon as the original music video dropped. I worked around the clock to post mine in time to take advantage of the press already surrounding Bey. I stayed up all night writing lyrics; I went to American Apparel as soon as it opened to buy turtlenecks to match the ones she wears in the video; I spent hours tinkering with GarageBand to get my knockoff track to sound passable; and I exhausted myself learning Beyonc\u00e9's choreography and shooting multiple takes to get it exactly right. I got my parody up in twenty-four hours, which is some Beyonc\u00e9-level dedication, if I do say so myself. I'd worked hard writing and editing \"SWGSTBG,\" but it was nothing like what I did for \"Student Loan Countdown,\" which generated about 100,000 views and was my most popular video ever. So I'd lowered my expectations. I thought \"SWGSTBG\" would do well, but for me that meant a couple hundred thousand views, maybe.\n\nWhen lunch rolled around and I was able to pull myself away from the dual sagas of Katniss Everdeen and pleated slacks, I noticed my phone was buzzing a bit more than usual. My Gchats were blinking. My inbox was full of unread emails\u2014sprinkled in with the usual YouTube comments were messages from Fox, the _Village Voice_ , MSNBC, and the _Huffington Post_. Normally, when I made a video, I would spend an hour emailing blogs and media asking them to feature it. But I was at work, so I hadn't done any of that. Why were these people emailing me?\n\nI typed \"Shit White Girls Say to Black Girls\" into Google, and there, at the top of the search results, was me, cheesing in that platinum-blond wig, next to one million views. I took a screenshot and pasted it into an email to my mom, my best friend De'Lon, and Patrick. Then I burst into tears. This was not a poignant, cinematic, twinkle-in-my-eye moment, but a sloppy, snotty, _my first boyfriend dumped me and we were SUPPOSED TO GET MARRIED_ kind of cry. I don't know if it was the shock of it, or feeling overwhelmed because I had gone from no one watching my videos to, suddenly, everyone watching. It only took six long years and four incredibly fast hours. I was clutching my face, rocking back and forth, and mumbling, \"Oh my God, oh my God,\" when a coworker came over and put her hand on my shoulder. \"Is everything okay?\"\n\nI looked up at her. Where could I begin?\n\nWhen \"SWGSTBG\" hit, I had no idea what going viral would entail, which I guess is why I burst into tears. Though I was working full-time, I barely made enough money to cover rent, bills, and my student loans. Patrick and I were living above an out-of-work opera singer whose erratic yet rigorous practice schedule meant my filming was often interrupted midmonologue by a burst of muffled eighteenth-century baritone. Once, I had to go down there at three a.m. to ask if he could please stop working on his Italian arias. He responded, \"How did it sound, though?\" While I've always been more of a lover than a fighter, it took everything in me to not slap him across the face and say, \"It sounded like you need to shut the hell up.\" If he wasn't singing, he was fighting with his wife, who was upset that he was out of work. It was like my life was a sitcom about an aspiring actress in New York.\n\nBy that time, I had been making videos and posting them on YouTube for about six years. My first was a hair tutorial in which I explained how I put my hair up without a hair tie. The \"secret\" was taking two sections of locs and tying them into a bow. Very high-level stuff. The video was, to put it lightly, awful. To start, I was backlit, which is a total no-no. Always know where your light is, folks: in front of you. Then the tutorial hit a few snags when my narcissism got the best of me\u2014\"I'm trying to look at myself in the camera and I'm realizing it looks weird,\" I noted, aloud, about halfway through. But I persevered. I followed up that video with a blurry fifty-six-second clip of my dog Kaya jumping around on my bed to a heavy-metal soundtrack (chosen by Patrick). Soon I expanded my repertoire to making things people might actually want to watch, developed a little audience, and began devoting most of my free time (and much of my time on the clock at my day job) to YouTube.\n\nAlthough I never had a specific goal, exactly, I knew I wanted to \"work in entertainment\" in some way. I'd racked up a nice chunk of student loan debt for half of an acting degree from the University of Michigan; after a few semesters taking classes on clowning, stage combat, and rolling around on the ground like a baby, I still had no idea how to break into the business, so I transferred to the graphic design program at Miami International University of Art & Design.\n\nAfter graduating, I didn't necessarily think YouTube was going to get me into show business. Back then, there was no formula for YouTube success because there was no such thing as YouTube success, no webcam-to-riches tales of network executives discovering sensations and offering them development deals. In 2008 YouTube launched its Partner Program, which gave people the opportunity to make money off the ads that played before their videos. After being denied twice, I was finally approved to participate and managed to make a little money from ads\u2014I got a check for a hundred dollars every few months, if that. Though that's not to say I didn't see the potential. In 2008, right before I moved to New York in order to be close to auditions and agents, I won a contest to interview celebrities on the red carpet for the Emmys; this was the first of many times I was sure I was going to become famous and subsequently did not become famous. But really, YouTube was just a place to mess around, make stuff exactly how I wanted to make it, and meet people.\n\nEspecially after I moved to New York, posting videos on YouTube was a way to scratch my performer's itch without having to claw my way through the city's stand-up scene. I didn't like to go out or drink; open mics were always late at night, and they usually paid in alcohol, if they paid at all. Because I lived in Queens, far away from the clubs downtown, I'd end up getting home at one or two in the morning before having to wake up at seven thirty for work the next day. Unlike most of the other comics I met, I couldn't work as a bartender or waiter because I can't even carry a single glass of wine across my living room without becoming a danger to myself and others. So I took odd jobs, like handing out club flyers and stuffing envelopes, in between working retail and graphic design temp jobs. Nevertheless, it was cool to be able to reach thousands of people on YouTube, rather than the same seven white dudes who frequented comedy clubs, and I liked being able to film multiple takes, especially when I had to pause for my neighbor's nightly opera performances.\n\nIn 2011 I thought for sure I'd gotten my big entertainment break when I entered YouTube's NextUp contest and, after two excruciating rounds of voting, managed to win. Along with twenty-four other YouTubers from around the country, I spent a week at the Google offices in New York City learning the ins and outs of YouTube and being mentored by some of the platform's top creators. We also got grants to invest in our channels. I spent most of the money on video equipment, including a new camera, computer, editing software, and lights, before putting the rest into savings and toward my hefty student loans. I decided this was my chance to get serious and make something out of my channel.\n\nBut even as I got more and more invested in my fans and my videos, in the real world I kept quiet about my YouTube jobby.  In those days, most people thought of YouTube as a place to watch videos, not make videos. Whenever I told people I had been making hairstyle tutorials and comedy videos for years, they would look at me as if I had just said something ludicrous, like \"I dunno, I just don't think Lupita Nyong'o is THAT pretty.\" ( _Who_ would say this?!) YouTube was so new that many people didn't see the point in investing money and hours into a platform that was for movie trailers and cat videos. My friends from my performing arts high school didn't consider what I did \"real acting,\" and although my family supported me, they didn't get it, either. When I told my grandmother, who doesn't have internet, that I made videos in my bedroom and posted them online\u2014and that sometimes my boyfriend was in them\u2014her response was \"Do you have clothes on, Frannie?\"\n\nNevertheless, by the time the \"Shit Girls Say\" phenomenon hit in late 2011, I'd developed a voice and a dedicated audience of about ten thousand subscribers. If you somehow managed to not be one of the forty million people who watched the four-part \"Shit...\" series, I'll bring you up to speed. \"Shit Girls Say\" starred comedian Kyle Humphrey in drag, going through a series of quick scenes depicting stereotypical things \"girls\" say. When delivered in Humphrey's over-the-top upspeak and mashed together in a two-minute supercut, ordinary lines like \"Do you know anything about computers?\" and \"Go into my purse...\" and \"Shut _UP_ \" took on a universal relatability. It went viral thanks to a perfect combination of \"funny 'cause it's true,\" \"dude in a wig,\" and \"ha ha, girls are dumb.\"\n\nAs I watched the \"Shit...\" parodies pop up around the internet, I started to think about making my own. Then, about a week after the original video appeared, I got my inspiration from a video posted by stand-up comedian Billy Sorrells: \"Shit Black Girls Say.\"\n\nWell, sort of. As soon as the video began to play, Sorrells's portrayal of a stereotypical black girl named \"Peaches\" who liked blingy sunglasses and _Basketball Wives_ made me feel weird. While the idea of a man in drag imitating the harmless things women and girls say has undertones of misogyny, in general \"Shit Girls Say\" felt lighthearted, all in good fun. I'm sure every person on earth, regardless of gender, has said, \"Could you do me a favor?\" at one time or another, and that's partially what made it funny. But the Peaches character felt different. Sorrells's dramatic, obnoxious black woman character seemed more like the butt of the joke than someone who was laughing along with it. There were also a few casual quips about domestic violence, as if that could ever possibly be funny.\n\nToday, I know the vocabulary word to explain exactly why \"Shit Black Girls Say\" made me uncomfortable: _misogynoir_. The term, coined by activist Moya Bailey, describes the unique interplay of racism and sexism that black women face. Sorrells was drawing on an all-too-familiar trope: Black male comedians like Tyler Perry and Eddie Murphy donning drag and regurgitating the same racist stereotypes that white supremacy uses to oppress black women. These portrayals paint us as loud, angry, aggressive, hysterical, overly sexual, neck-swerving, gum-popping clich\u00e9s who scream, \"Oh, no, he didn't!\" on loop. While these tropes persist all over modern media like Instagram, YouTube, Vine (RIP), and film and television, they're not too different from the mammy and Jezebel stereotypes promoted by Jim Crow\u2013era advertisements and cartoons.\n\nBut at the time, I couldn't have explained any of this to you; all I knew was that I didn't relate. Like everyone else, I saw \"Shit Black Girls Say\" all over Facebook, and even a few of my coworkers were sending it around, laughing hysterically and exchanging _omg so true!_ 's. (As the only black woman on my team, I found this particularly alienating.) The whole point of the \"Shit...\" meme was for audiences to see themselves in the character. But that wasn't my version of blackness; Peaches was like a cross between my high school bullies and Martin Lawrence's Sheneneh character. The more I thought about it, the more angry I got that this video might make my white coworkers and friends believe stereotyping blackness\u2014and especially black womanhood\u2014in this way was acceptable. As long as it's done by a black person, with a dose of humor!\n\n_If that's not me_ , I thought, _then what \"shit\" do I say?_ I started to brainstorm ideas for my own take on the meme. I considered \"Shit Black Suburban Girls Say,\" since I grew up in the suburbs, but that was too clunky. I toyed with \"Shit Oreos Say,\" since that's what people used to say I was\u2014an Oreo.  But my fear of litigation stopped me there. What if Nabisco tried to come after me for trademark infringement? After racking my brain for days, I decided it was probably too late to make a spinoff of my own anyway.\n\nThat changed around Christmas. While home in Florida I went to my high school's annual holiday reunion. That year's party wasn't any different from those of years past, except that night I was driving, which meant I wasn't drinking. As the night went on, things got awkward, and I guess this was the first time I was sober enough to recognize just how much I put up with people treating me differently. I had shown up to the party with my locs (duh\u2014they're attached to my head), which I hadn't had in high school, and as everyone around me got drunker, they became bolder. The usual questions about my hair, including \"Is it real?\" and especially \"Can I touch it?\" started to come out. And like most white people I'd ever talked to about my hair, these sloppy drunks already had their hands out, en route to groping my head, when they \"asked.\"\n\nI'm embarrassed to say that I didn't exactly stand up for myself when my friends started acting this way. I did my best to bob and weave as hands flew at me left and right, but I didn't have the courage to say outright, \"Please keep your grubby fingers to yourself,\" or \"Notice I'm not asking if your breasts are real because that's none of my business.\" Dealing with white people faux pas as a black woman is tricky: If you get upset, you can quickly be labeled the \"angry black girl\"; if you're too passive, it seems like you're giving permission, or letting racism slide. I had always been the token black girl in the group, so I knew this struggle well. Before I went natural,  I hardly knew anyone in real life with natural hair, let alone with locs, so I can see how my hair seemed new and strange to some of my friends. But curiosity doesn't give you free rein to treat me like a baby goat at the petting zoo. I know I'm just as cute as a baby goat, but back off.\n\nThe combination of this party and \"Shit Black Girls Say\" made something click. While part of me cringed, another part knew that this could be the makings of an incredible video. I whipped out my phone and started making a list of all the \"Can I touch your hair?\"\u2013type comments I could think of. _You're not the same as other black people. He's so cute for a black guy! You can say the N-word, but I can't? You guys can do_ so much _with your hair. It kind of feels like Cheetos! Is that racist? Is_ that _racist? That's_ NOT _racist!_ An old friend, Megan, who was at the alumni party, provided the perfect inspiration for the nasal voice and the lazy way I waved that cigarette around in some of the scenes. As soon as I got back to New York, I called my friend with a fancy camera, Eric Walter, and got to work on what would become \"Shit White Girls Say... to Black Girls.\"\n\nThough many articles explaining \"Here's What It's Like to Be a Viral Video Star\" have been published over the last few years, in 2012 there was no handbook for what to do when you suddenly find yourself thrust into the spotlight. Sitting there in my cubicle, bawling my eyes out in front of my coworkers, I felt like the entire world was watching to see what I would do next. It was amazing to go viral for something I'd written and produced\u2014a friend of mine had his first viral hit with a video of his mom sleepwalking, which is funny but not exactly what you want to be known for as a comedian. Still, all the attention was overwhelming.\n\nThe first of many unsettling realizations was that I still had to finish the workday. Sadly, YouTube does not show up at your office with one of those giant Publishers Clearing House checks, and though you may feel pretty proud of yourself, you are not immediately infused with the confidence to give your boss the finger and leave the office in a blaze of upwardly mobile glory. I wouldn't have done that anyway\u2014I liked my job. Instead, I had to find some way to explain to the growing crowd of concerned coworkers gathering around me that I had been making YouTube videos for years, and finally one had gone viral. I felt like I was about to tell my spouse that I had a secret family, or confess that I liked to dress up in furry costumes and play with kitty litter in the bedroom.  My private hobby was about to become super public.\n\nI knew my coworkers had seen \"Shit Girls Say\" and \"Shit Black Girls Say,\" and although virality and internet fame were not the daily occurrences then that they are today, they were still concepts most people understood. _What's the worst that can happen?_ I thought. I hadn't studied the Ann Taylor code of conduct nearly as closely as I should have (to say I skimmed it would be an overstatement), but I was pretty sure satire wasn't a firing offense.\n\nI took a deep breath and said, \"I made a response to that 'Shit Girls Say' video, and now it's going viral.\"\n\nKate, the most sophisticated of the bunch, was visibly surprised. \" _You_ make YouTube videos?\"\n\nThen, to my own surprise, Richard chimed in with, \"What's that supposed to mean? I've made a few videos.\"\n\nJenna asked, \"Well, can we watch it?\"\n\nYou know how embarrassing it is when you show someone a video you think is really funny and they don't laugh? And you have to sit there as they watch the video on your phone, not laughing, waiting for the tense period of not-laughing to end?\n\nAs my coworkers crowded around my desk to watch \"SWGSTBG,\" I envisioned their faces falling one by one as they tried to assure me, \"Oh, yeah, it was really... something.\" After making hundreds of videos, I'd gotten over the discomfort and nervousness that came along with venturing into the comments, but seeing people I actually knew watch my work, in real time, was super uncomfortable.\n\nThe first thirty seconds felt like an eternity. Did they think it was funny? Did they think it was offensive? Jenna was a blonde\u2014did she think this was about her? None of them had ever said any of the phrases in the video to me, but I couldn't be certain they hadn't said those things to anyone else. Maybe this wasn't a good idea after all. Was I ready to talk about race with my coworkers?\n\nFinally, they started laughing. They got it! Maybe some of them would share it! (I stopped myself from chiming in with \"Don't forget to subscribe!\" when it was over\u2014too thirsty.) Throughout the day they morphed into my own personal cheerleading section, yelling out to the whole office when one of their friends posted the video on Facebook. When it was finally late enough in the day that I could go home, Richard hugged me and said, \"I can't wait to tell people we work together! Girlfriend!\" He'd never called me \"girlfriend\" before. Had I just unwittingly opened an annoying door I could never close, or was it a harmless callback? I decided it was the latter and sank into the hug. This felt good. Maybe \"Do I still have a job?\" was written all over my face as I packed up, or perhaps the universe was just telling me to calm down, but either way, my supervisor, Jonathan, leaned over my desk and cooed, \"Who knew we had a celebrity on our team?\"\n\nWhen I got home, my dogs, Fil (short for Filthy McNasty) and Kaya, hadn't gotten the memo that I was a star\u2014they just wanted to go on a walk. I tried to use it as an opportunity to relax, but I couldn't stop scrolling through my phone. My inbox was flooded, my Twitter mentions were a mess, and I may have paused mid\u2013poop scoop to check the view count on the video. It was a wonder we made it to the park and back without getting hit by a car. I can hear the news story now: \"Viral star struck and killed in Queens intersection while responding to 'y u so rassit?' tweet. More at eleven.\"\n\nBack in my apartment I finally had a moment to reflect. On one hand, I felt like I'd finished a marathon\u2014all the time I spent writing and editing my videos, tweaking them to perfection, staying up late, and waking up early was finally paying off in a way I could never have anticipated. On the other, it was like being bestowed a gift from the internet gods\u2014my career was about to change in a major way, and the opportunity seemed like it had dropped out of the sky. I knew the old saying \"There's no such thing as an overnight sensation,\" and I knew I had been building the foundation for this success for years. But like so many other viral videos, \"SWGSTBG\" went viral because of a precise combination of timing, hard work, and luck, and the luck part was trippy.\n\nBefore I could tackle my raging inbox, I had to take care of some family business. Though I still couldn't bring myself to say \"shit\" in front of her, I called my mom to fill her in on all the exciting emails and opportunities that were pouring in. She christened me \"Miss Viral.\" Then, Patrick's mom, Nancy, called to congratulate me\u2014and apologize.\n\nThe first time I met Nancy\u2014who is, like Patrick, white\u2014I had just spent the night with Patrick and was slipping out of the apartment they lived in together the next morning. Suddenly, I heard the front door open, and there was my boyfriend's mother, walking in from her night shift working as a nurse. Apparently Patrick had told her about me, because she said, \"You must be Fran!\" I'd recently added fire-engine red to my locs, which I thought was daring and cute but in retrospect probably made my hair actually look like Cheetos. After I'd sheepishly introduced myself, Nancy added, \"Oh, I love your hair!\" and extended her hand to give my locs a stroke. I was very clearly still wearing last night's makeup and rocking severe bed head  so there was no way to mistake what I'd been up to with her son the night before. I decided to not make the situation any worse, so I said thank you and made my exit.\n\nAfter praising my performance in \"SWGSTBG\" on the phone, Nancy got serious. \"Oh my God, Fran\u2014I feel so terrible about touching your hair.\" I hadn't made the video to call her out, but I'd inadvertently done just that. I told her not to worry about it, but the moment made me realize how often I didn't tell people how I was actually feeling, especially if there was a power dynamic at play. I'd been dodging head pats and holding my tongue around coworkers, friends, bosses, and teachers for my whole life. Instead of saying something about how uncomfortable their comments or behavior made me, I'd channeled those feelings into a parody video. \"SWGSTBG\" brought to the surface how silent I had been\u2014as a teenager growing up in West Palm Beach, as a young woman at college, and all throughout my early twenties.\n\nEven before I fully grasped what the success of the video would mean for my life and my career, I realized it wasn't just about my hair. It was about growing up in Florida, where tanning is practically an Olympic sport, and having white girlfriends joke, \"I'm blacker than you.\" It was about fielding comments that a certain man was \"pretty hot for a black guy.\" It was about realizing that \"I'm not into _______\" is really code for \"I see all _______ as the same.\" It was about not being \"like other black girls\" because I was \"so nice\" or \"one of the good ones.\" It was about all the seemingly benign comments and expressions of disregard that added up to a constant reminder that I was different from other people, and that my difference was, for some reason, important. In many ways, being called \"so nice\" or held up as \"one of the good ones\" becomes a kind of muzzle. Though I didn't know it at first, \"SWGSTBG\" was me taking off the muzzle. I'd found my voice.\n\nAs I scrolled through my inbox and social media accounts that night, it was really weird to see people from high school posting my video\u2014did they not remember saying those things to me? Maybe they did, and they felt ashamed, or maybe they didn't and just wanted to get credit for knowing me. Producing a viral video has the strange power of elevating acquaintances to the level of \"oldest and best of friends\"\u2014anyone who had direct access to me began reaching out. I got so many texts from random numbers saying, \"Everyone on my Facebook is sharing your video!\" that I was tempted to respond, \"Same phone. Who dis?\"\n\nMeanwhile, every white girl I'd ever met decided to reach out to ask which lines were about her. I've never felt closer to Carly Simon in my life. (\"You're so vain, you probably think this song is about you.\") When a girl I'd worked with in Miami, who eventually did become a friend, texted to ask, \"Tell me the truth. How many of those comments were from me?\" I was completely honest: \"I had to cut yours because it was so stupid I didn't think anyone would believe it.\" She responded with a string of cry-laughing emojis. The incident I was thinking of was the time she gave me a black shirt to wear to our company golf tournament because, \"You know, you're black.\" And now you see why her comment landed on the cutting-room floor.\n\nMore importantly, I knew that my message was resonating with people who'd had to deal with the same awkward, inappropriate situations I did. The positive comments and emails and friend requests I received the day I went viral came from girls, boys, women, men, and everyone in between who had grown up just like me. They were the only person of color in all their high school and company photos. Like me, they'd been treated to bizarre throwaway comments for their whole lives. And like me, they hadn't known how to respond, so they didn't.\n\nAs much as I wanted to reply to each and every one of my new supporters, I had some media requests to answer. In addition to the _Huffington Post_ , MSNBC, Fox, and the _Village Voice_ , I got messages from _Saturday Night Live_ and the team that would eventually become my talent agents. Producers from places like MTV, NPR, and _Anderson Live_ were asking if I would do interviews or appear on their shows.\n\nAnd they were asking if I would bring the wig. The principle of the wig still stood: It was an immediate signal about the theme of the video and the message I was trying to get across. Nevertheless, I always told them no. Though I still have that thing tangled up in a box in my closet somewhere, I knew I was destined to be more than the girl with the wig.\n\n# CHAPTER TWO\n\n# BLACK-LASH\n\nThe biggest audition I'd snagged in the first four years I'd been in New York was for the reality TV competition show _The Glee Project_ , where twelve singers competed for a chance at a guest-starring role on the Fox musical comedy show _Glee_. Very highbrow stuff. Needless to say, I did not book the job. Despite occasionally getting carded at bars and taking karaoke way too seriously, I guess I wasn't believable as a teenage singing sensation on a should've-been-canceled-after-one-season Fox show.\n\nBut following \"SWGSTBG,\" suddenly everyone wanted a piece of me\u2014after years of celebrating anytime a casting breakdown said \"open to all ethnicities,\" I had more interview requests, meetings with agents, and audition opportunities than I knew what to do with.\n\nI tried to respond to as many as possible, which translated to leaving early for lunch to take calls away from my desk. At one point I volunteered to \"help\" one of my coworkers by accompanying her on an office coffee run just so I could take a call with my soon-to-be agent in the Starbucks restroom.\n\nWhile radio and magazine interviews were exciting, nothing could prepare me for a message from _Anderson Live_ , Anderson Cooper's daytime talk show, where he covered trending stories that didn't quite fit into his news beat at CNN. A producer from the show emailed me a few days after \"SWGSTBG\" blew up. My cue to \"head to lunch.\"\n\nI'd spent the past few days reading tons of articles about the \"Shit Girls Say\" meme, so I felt more than prepared to handle the producer's pre-interview spiel. Besides your usual \"Where are you from? What inspired the video?\" type of questions, we talked at length about the negative response to my jokes. Was I surprised people thought the video was \"racist toward white people\"?\n\nI told her I wasn't surprised that people were upset or confused, but it was pretty revealing that no one had reacted negatively to the original video, \"Shit Girls Say,\" which portrayed women as obnoxious, dumb, and preoccupied with trivial things. Or to \"Shit Black Girls Say,\" which was basically just another rehash of the sassy black woman stereotype. Sure, they were supposed to be a bit over the top, but they didn't exactly present women, or more specifically black women, in a positive light. Apparently, I said, they had gotten a boys'-club pass. I'd seen little to no backlash against those videos. They certainly weren't being invited on national television to discuss the complicated politics of their two-minute satirical riffs on everyday interactions. I didn't quite have the language to express it at the time, but in retrospect I can see that the negative response to my video was a clear symptom of a racist and sexist culture that was viscerally offended by the idea of a black woman talking about her lived experience.\n\nThe _Anderson_ interview looked like it was going to be a go. Then, right before we got off the phone, the producer asked if I'd invite the old friend on whom I'd based my \"SWGSTBG\" voice and mannerisms, Megan, to come on the show. I told her I'd give it a shot, but realistically I wasn't so sure it was an option. There's just no easy way to say, \"Hey, would you be interested in going on national television to talk about all the dumb things you've said to me throughout our fifteen years of friendship?\" But I must have done an okay job of it, because I called her up and she agreed\u2014I'm sure the free flight from LA to New York, combined with Megan being an aspiring actress looking for screen time, didn't hurt. The night before the taping, I braided my hair, laid out the dress I planned to wear (it was from Target), and went to bed early. I was nervous but excited\u2014I saw _Anderson_ as a chance to put myself on the map.\n\nMy segment was only twelve minutes long, but the taping lasted more than half an hour. Anderson was warm, charming, and even more handsome in person. (For those wondering, yes, his eyes really are that blue. \"Silver fox\" is a total misnomer\u2014dude straight-up looks like a wolf.) He joked with me between takes and complimented my poise and professionalism, which made me blush. Did I already mention how handsome he was? As I studied the way he read from the teleprompter and bantered with the audience, I knew I wanted to be where he was someday.\n\nAside from one strained exchange with an audience member, the experience went well. Even Megan seemed to be having a good time when Anderson pulled her out of the audience and we all laughed together about her nasal voice. I felt like I'd nailed it\u2014which was good, because I had to run from the taping to a meeting uptown, where I officially snagged an agent.\n\nIn the week between taping _Anderson_ and the segment going up, I had also been invited to audition for _Saturday Night Live_. This was nearly a year before the conversation about why there wasn't a black woman on _SNL_ hit a fever pitch, but still: I knew there wasn't a black woman in the cast at the time, and I couldn't help thinking that maybe _I_ could be the black woman on _SNL_. The audition invite didn't provide many details\u2014just that those auditioning should prepare a character showcase\u2014and my new agent, Scott, told me to \"do four characters I loved.\"\n\nEager to support their brand-new client, Scott and a group of people from his team said they would come cheer me on. I had been watching the show for my entire life\u2014you can't claim to want to be a comedian without spending some amount of time proclaiming, \"Live from New York, it's Saturday night!\" alone to your bathroom mirror. The opportunity was a dream come true.\n\nMore precisely, it was one of those dreams where you're onstage, you suddenly realize you're naked, and everyone starts laughing at you. But in my case, not many people laughed.\n\nAlmost as soon as I walked into the club, I knew I was going to bomb. The audition invitation had been vague because everyone else trying out was a UCB -trained comic who had studied _SNL_ auditions, which require a very specific kind of comedy, for years. They didn't need instructions. It had never even occurred to me to ask my agent for sample audition tapes or to look for audition videos on YouTube\u2014you know, the very platform that had earned me an audition in the first place. (PSA: YouTube has tons of _SNL_ audition tapes and screen tests from talented comedians who did and didn't make the cut.) To top it all off, I'd been working on my characters for about a week, and instead of trying out my routine beforehand in front of an audience at a comedy club, I opted for the acclaimed \"Above the Opera Singer Theater,\" aka my living room. Patrick gave me some feedback, but getting one person's perspective is not the same as having an audience, especially when that one person is like Patrick and doesn't really know much about pop culture. As much as I love him, he once asked me if Selena was still alive, and I couldn't help but wonder, aloud, \"Who the hell did I marry?\" I hadn't performed stand-up since moving to New York, and YouTube comedy is completely different\u2014you can edit the hell out of yourself to make sure all your jokes hit.\n\nFor my character showcase, I had come up with the idea that I'd be at an awards show and play both the presenter and the winners: Tyra Banks was hosting, and Lil Wayne, Britney Spears, my grandma, and Megan from \"SWGSTBG\" would be accepting awards. Not the most original pitch by any stretch of the imagination, but I figured it would get the job done. This was nothing like what the other comics had prepared. Instead of a generic, predictable location for their impressions, they'd introduce each character with an oddly specific, bizarre scenario\u2014kind of like the premise of an SNL sketch: \"This is Tom Hanks, who has stubbed his toe because he's been helping someone move,\" or \"Here's Sof\u00eda Vergara as a real estate agent in New York.\" I can do a great Britney Spears impression, but I had no context for it. (Example: \"Here's Britney Spears babysitting for a Russian diplomat!\")\n\nOut of twelve _SNL_ hopefuls that night (including Sasheer Zamata, who did eventually join the cast), I was scheduled to go on eleventh, which made the whole thing more excruciating. Over and over, I watched people who were so prepared, so funny, and so on point that I wondered if my agent was going to drop me when I finally wandered onstage. Or if I should just get it over with and drop myself.\n\nWhen my turn came around, I decided I needed to get through my performance as fast as possible. I introduced each character as Tyra Banks, praying the audience would get my basic _America's Next Top Model_ references. \"I only have one photo in my hand, and it's of Franchesca's grandma\"; \"I was rooting for you, Britney Spears! We were all rooting for you!\" It was exactly as bad as I expected it would be. My performance was less professional comedy routine and more Twenty-Fifth Annual Dunder Mifflin Paper Company Talent Show. While I don't think I'll ever be right for _SNL_ , I would totally place at DMPC. I swear my Britney is that good.\n\nAfter the final comic finished, Scott and the rest of the group from the agency walked me to the subway. I was trying very hard to hold it together, but I was humiliated\u2014not just because of my performance, but because I worried they thought I was lazy and unprepared. I hadn't even met some of them before, and I hated that this was the first thing they'd seen me do. Somehow, they didn't fire me, but as I made the hour-long trek back up to Queens, humbled and embarrassed, I realized I had let all the attention from \"SWGSTBG\" go to my head. I had worked for years honing my specific skill set, but that didn't mean I was an expert.\n\nThough the _SNL_ audition was a bruise to my ego, I was able to bounce back for the airing of my _Anderson_ interview. \"SWGSTBG\" was well past six million views, and despite a Lil Wayne impression that consisted of nothing more than an aluminum foil grill, shades, and a backwards hat, I still officially had an agent. I was getting called for auditions left and right, and my very first national TV interview was about to debut. I was glowing.\n\nI didn't have access to a TV at work, so I was stuck reading tweets about the show as it aired instead of watching it live. I didn't have cable or DVR at home, either, so my friend Dominic recorded the episode and a group of us planned to watch it after work. I spent the entire day refreshing my timeline instead of Photoshopping statement blazers, and at five p.m. I sped out of the office like I had just been told there was an athleisure sample sale in the lobby. (What can I say? I love high-end sweatpants at almost affordable prices.)\n\nThat evening, I sat in Dom's tiny living room with Pat and our friends, drinking and eating snacks as if we were about to watch the Oscars. After Dom's grand introduction\u2014\"Tonight we're gathered to celebrate Fran because we love her and because we're all jealous she got to meet Anderson fucking Cooper!\"\u2014the segment began to play. My heart was pounding as suddenly I was on-screen in my platinum-blond wig, saying:\n\nNot to sound racist, but you can say the N-word, but I can't?\n\nMy best friend was black\u2014well, she's still black, but we're not really friends anymore.\n\nOh my God, I'm practically black! Twinsies!\n\nThis is so ghetto.\n\nSo cute for a black guy, right?\n\nCan I touch it? Okay. I'm already touching it a little. Is this real? Wait, it's not real? It is. So nappy.\n\nI think what I like the most about them is they're not, like, stereotypical, like, black people. You know what I mean?\n\nIt's almost like you're not black.\n\nWhen the clip from \"SWGSTBG\" was over, we were transported back to the studio, where Anderson and I were sitting down for a one-on-one. While I certainly remembered being there, I almost couldn't believe how comfortable and professional I looked. My little black dress looked damn good, my hair was piled high in my expertly braided updo, and my eyes sparkled like your grandmother's favorite holiday shawl. I heard myself explaining how I thought the \"Shit Girls Say\" videos were funny, but I couldn't really relate to them. \"Being a YouTuber,\" I was saying, \"I thought, _How can I jump on this and get tons of views?_ Because I want to get views, too\u2014I'm going to be honest.\"\n\nThe audience laughed, and so did our living room viewing party. Seeing myself on television instead of on my computer screen was surreal. And I was pleasantly surprised to find that all the talk about the camera adding ten pounds was clearly bullshit.\n\nThen Anderson got into the drama portion of the interview\u2014the backlash to my video. Slowly, I began to look less poised. \"Have you been surprised at the reaction?\" he asked, in that Scholarly Dad voice he has. \"Some of the comments online have said it's racist, what you've done.\"\n\nI knew this was coming, and I had practiced my answer before the taping. \"Well, I think to be fair, my intention was not to label anyone as a racist, because in all honesty, none of those comments are actually racist.\"\n\nIf you're thinking: _Wait a second, 2012 Franchesca. Those comments_ AREN'T _racist? They're as racist as a \"12 Reasons Obummer Is a Kenyan Muslim\" novelty calendar!_ \u2014well, so were a lot of people. I'll get to that in a second.\n\nI went on to explain that curiosity was totally understandable, but treating black women like they were animals at a petting zoo was not. \"I think a lot of times people are trying to relate to me and show me that we're cool and we can get along,\" I said. \"They're trying to be nice, but then the wrong thing slips out.\" It's almost funny to look back at how conciliatory and nonconfrontational I was being. How could anyone be offended by that?\n\nAnderson took a few comments from black women in the audience, and they said the video was a great icebreaker for racial dialogue with their friends and coworkers. They found it relatable and funny and were happy to share it with their friends. Awesome. But then, Anderson approached an older white woman in the audience. Back at Dom's apartment, I rolled my eyes\u2014I already knew how this exchange played out.\n\n**Salty Lady:** Hi. I was just wondering, if you say it had no racism or connotations, why do you use white and black girls? Why didn't you just say, you know, \"Funny Things Stupid People Say to Me\" rather than using the black and the white?\n\n**Me [visibly annoyed, to some laughter in the audience]:** Okay. Once again, the original video...\n\nAs I watched my eyebrows rise higher and higher into my perfectly sculpted updo, I felt like I was about to crawl out of my skin. _Let me not be the \"angry black girl\" on national TV!_ I thought. Even though I had every right to find this woman's comment infuriating\u2014had I not just explained the concept minutes before?\u2014I was projecting the stereotype onto myself.\n\nShe kept going:\n\n**Salty Lady:** Saying black and white is what I'm getting at.\n\n**Me:** I know. I'm going to answer your question. The original video that this is based off of is \"Stuff Girls Say.\" Then the most popular parody of that video was \"Stuff Black Girls Say.\" So my idea...\n\nAnderson cut in to ask if I had been offended by the previous \"Shit Girls Say\" videos. I assured him that no, I hadn't, and was about to explain further when my number one fan cut in again.\n\n**Salty Lady:** I feel like white people are the only people that can be racist.\n\n**Me [obviously exasperated]:** Nothing said in the video is racist. Honestly. I mean, asking about my hair, saying that you think someone is attractive despite being black\u2014I don't think any of those statements are racist. I do think that they are ignorant. I mean, I really am sorry if people are offended or they are upset about it, but I think that if you don't see anything of yourself in it, then I don't think it should be offensive.\n\nThough I did do an admirable job of replacing every single \"Shit\" with \"Stuff\" for daytime audiences, I was starting to feel bad about how I looked on TV. On top of the tense back-and-forth, the way the segment had been edited made me look bored and nervous as I sat listening to the audience's questions and comments. I'd been on TV before, but only local news (\"Florida girl wins contest to go to the Emmys\"), and the _Anderson_ segment involved more production than I could have expected. Now, having worked on TV, I don't think it was malicious\u2014you've gotta get that B-roll.  I learned an important show-biz lesson that day: As dubious as the name may be, \"resting bitch face\" is real, and it's the stuff of TV magic. But at the time, I felt a little manipulated. Patrick shot me a confused, angry look. Was this about to go south?\n\n\"Ugh!\" Dominic said. \"That lady was so obnoxious. More power to you\u2014I would've told her off. Seriously, sit down!\"\n\nI tried to go back to the segment, which had moved on to man-on-the-street interviews about the video. Most reactions were positive, with lots of black women agreeing they'd heard the statements before, and even a few white women agreeing that the comments were commonplace. Then the segment reached the last young woman.\n\n**Carnival Pretzel\u2013Level Salty Girl:** By doing it, she's being racist to me.\n\nIt was like I had been transported back to that exact moment and was reliving it in real time. How could she think that me talking about my (very benign) experiences as a black woman was racist? And how did I not scream in the studio the minute she pulled the reverse racism card?\n\n**Anderson:** It's interesting. Franchesca, you heard that last young woman saying that you were being racist to her.\n\n**Me:** I don't think that talking about ignorance is racist. Like I said, I'm not labeling anyone racist, because that would imply that the statements were saying that someone is better than another race, and that's not what any of the statements are doing.\n\nAfter the commercial break, we got to the portion with Megan, and my friends began laughing it off and chatting among themselves. But I couldn't stop going over my responses to Anderson's questions. Actually, maybe saying someone is \"so cute for a black guy\" _does_ imply racial superiority of some kind, right? Using \"It's almost like you're not black\" as a compliment lumps all black people together, and it suggests they're generally worse than white people. Working through the whole thing felt too complicated to explore in a twelve-minute TV segment, so I told myself to let it go. Even though I can see now where I messed up, I had done pretty well for a girl who'd had no publicist, no agent, no manager, and only a pep talk from her boyfriend and dogs to prepare.\n\nBut not everyone was willing to give me so much credit. Or any credit. At all.\n\nIf I could go back in time, I would say, with confidence, \"My video is not racist because I genuinely do not believe that racism against white people exists. Nothing I say or do could ever oppress white people, and my video certainly has not led to the oppression or mistreatment of white people. It might hurt some feelings, but this video isn't about hating white people. It's not about hating anyone. It's shining a light on the experiences black women and people of color face every single day in a world that says we're less than or 'weird' because we aren't white. And if that makes you uncomfortable, maybe you should consider whether the problem lies with you, not me.\"\n\nInstead of a time machine, I had Black Tumblr.\n\nIf I thought my inbox was bad when I first went viral, it was even worse after my _Anderson_ interview aired. A whole new wave of emails, messages, and comments crashed into every social media platform I was on. (Basically all of them.) As the days went on, the comments got more and more vitriolic\u2014it was like my haters were taking time to build the perfect nasty insult. People of all races\u2014or who claimed they were of all races\u2014were calling me names: the N-word, Uncle Tom, slur after slur after slur. I was devastated, and I couldn't understand it.\n\nBefore \"SWGSTBG\" went viral, I prided myself on being super engaged with my modest fan base, so my YouTube account was set to forward any comments on my videos to my email inbox; I wanted to be able to respond to comments in real time. Now that I was the new face of \"reverse racism,\" thousands of hateful comments came straight to me, unfiltered. Although I eventually turned off my email notifications, for months I answered angry comments regularly. I was just so surprised by the backlash that I thought I could reason with people.\n\nFrom one side, there was the conservative white supremacy stuff: I was a racist who hated white people and I should go back to Africa. Having grown up in a fairly diverse and liberal community, I'd never been called a racist before, and I was not prepared. I opted for the _kill them with kindness_ approach: \"I don't hate anyone, but I'd love to visit Africa someday! Thanks for the encouragement! ;)\"\n\nEventually I forced myself to ignore the barrage of racist ramblings\u2014a lesson I would return to over and over again in my career. But the criticism from the other side\u2014from the left\u2014was harder to disregard.\n\nI'd been on the microblogging site Tumblr for a few years, steadily building a loyal audience by posting my hair tutorials and comedy videos and by reblogging viral content and occasionally adding personal commentary or jokes. After keeping a journal throughout high school and college, and abandoning LiveJournal when it was taken over by tweens, moving to Tumblr was a logical next step for me. The site served as a collaborative digital scrapbook\u2014the community was just as much of a draw as the ease of publishing. It took all the work out of finding an audience of like-minded people with similar interests, and the site allowed microcommunities, ranging from fan fiction writers to conspiracy theorists to natural-hair bloggers, to proliferate.\n\nAmong these groups, Tumblr is probably best known for its active social justice community; it's a particularly good platform for marginalized folks to organize, vent, and boost news stories about race, gender, sexuality, mental health, and disability that would otherwise be ignored. When \"SWGSTBG\" went viral, I unwittingly attracted the attention of this community. They rallied behind me when Perez Hilton reuploaded the video without permission on his site (which lost me views, and money). They also used the video as a jumping-off point for lengthy discussions about the dangers of unchecked white privilege, power dynamics in majority-white spaces, respectability politics, and microaggressions. But as Tumblr users unpacked the issues, I was nowhere to be found\u2014I was busy imagining my life as a celebrity, complete with a late-night comedy show gig and an extensive new wardrobe.\n\nWhile I was riding high from my _Anderson_ segment, my newfound Tumblr audience wasn't impressed. Why hadn't I shut down the woman who accused me of being racist toward her? Why hadn't I explained that racism against white people doesn't exist because there's no social structure oppressing white people? Why didn't I explain the concept of white privilege? Why had I said none of the comments in my video were racist when they clearly were? Why had I not used my new national TV platform to define the term \"microaggression\" for the masses? How did I not know about microaggressions? How dare I express sympathy with some of the people who'd been hurt by my video? How dare I have a white boyfriend? How dare I be so stupid?\n\nI was used to white people saying they were upset by or didn't understand my video, but these messages were coming from black people, some of them even from academia. I was transported back to my middle and high school days, when my black peers would tell me I wasn't being black in the right way, that I \"sounded like a white girl\" or wished I were white. To hear Black Tumblr tell it, I had bombed _Anderson_ , and no amount of praise from my agents or parents\u2014or especially from my white boyfriend\u2014was going to change their minds. I knew I'd started a conversation, but only then did I realize I'd suddenly been handed a gigantic platform to talk about race. It was becoming painfully obvious that I was far from prepared. I hadn't understood how much \"SWGSTBG\" meant to the people who felt they needed it most, and to them, I had fumbled the opportunity to turn it into something bigger than just a viral video.\n\nThe backlash, or _black-lash_ , as I started to call it, came to a head when I posted a video sent to me by a white fan. Titled something like \"Why 'Shit White Girls Say to Black Girls' Isn't Racist,\" the clip consists of a twenty-something white guy explaining power dynamics, institutional racism, microaggressions, and the fallacy of \"reverse racism\"\u2014basically everything Black Tumblr had wanted me to say on my _Anderson_ appearance. I thought it was a perfect breakdown that, in a tight five minutes, gave voice to things I'd captured in the video but hadn't been able to quite put into words. So I posted the video on Tumblr, adding a \"disclaimer\": The video repeated things black people have been saying for ages, but the sad truth was that some people would only listen if they were coming from a white person.\n\nWow. This, too, was the wrong thing to do. More than a few people responded that a white \"translator\" shouldn't be necessary for people to understand the issue. I agreed, but did so while holding my ground. I wrote a response on my Tumblr: \"This shouldn't be an 'us vs. them' situation. The fact is that some people will refuse to digest the original message simply because it's coming from a person of color... Call me an idiot if you'd like, but I don't see how supporting someone (regardless of race) whose goal is to educate others deserves a side eye.\"\n\nI shared this and sat back to wait for cookies and agreement for my excellent points. Instead, I got called out, in posts like \"Chescaleigh, sit the fucking fuck down\":\n\nThis here is when I lay a big ass smack down on white-identifying wannabe cracker motherfucking ass... First, you ain't white. I know you wanna be chile, but you're not... You think whitey gonna see any different? HELL to the no. They see you as a N. I. G. G. E. R. PERIOD. Don't for a minute doubt that white people don't see your black ass as a straight up niggerrrrr... It's not \"unfortunate\" that some white people will refuse to take an anti-racist message from a person of color. IT'S RACIST, you racist apologist shitstain on the pants of fucking life... Seriously, what the fuck do you think happened on the Anderson show? White people set you up to be browbeat and shit on. And while I had no problem with you taking it because hey, it's a show, it's something, you actually ate their shit. In a fucking shit pie, served to you, and now you're vomiting it all over everyone, including your now EX-supporters.\n\nI was absolutely crushed. I cried for what felt like an eternity. But instead of logging off, I went down the rabbit hole, hoping to find a way out. As I read every message, comment, and reblog trying to figure out where I'd gone wrong, I felt like my computer was going to burst into flames. I'd fallen into \"callout culture\"\u2014as quickly as I'd become internet famous, I was in danger of losing the trust of the audience I'd built with those two minutes and thirty seconds of satire. I was ripped to shreds because I didn't know what I didn't know during my _Anderson_ appearance, and now I was being ripped to shreds because I didn't know how to respond to being ripped to shreds. I'd gone from confident viral-video-star Franchesca to the high school version of myself\u2014insecure, uncertain, and trying to balance my true self with the woman the world, and especially other black people, wanted me to be. I had just started to shake off the silly \"racist\" label from trolls and confused white people. But how could I shake off being a \"coon\" to the very black people who had, just days before, championed me for telling their stories?\n\nI thought I was hot shit for making a viral video, but it was becoming clear that I had a lot to learn if I wanted to get to Anderson Cooper status someday. This is not an arena where you want to wing it, but I was not ready to be thrust into the role of activist and spokesperson. Like at my _SNL_ audition, I hadn't even realized there was something to wing.\n\nThough many of the commenters were totally off base, some of them had a point. There was no reason to prioritize a white voice in a conversation about black experiences. And while sarcasm and passive aggression made me feel superior to the trolls who slung personal insults and racial slurs, there was no place for it when interacting with people who called themselves fans.\n\nOne of the only reasons I didn't collapse into a puddle of hopelessness after the one-two punch of my failed _SNL_ audition and incurring the wrath of Black Tumblr was that not everyone was calling for me to be permabanned from the cookout. A few black women reached out with some constructive criticism. They sent me reading lists and invites to private Facebook groups, and suddenly my Tumblr dashboard was filled with blogs like _Fuck-Yeah-Feminists!_ , _POC Creators_ , _Colorlines_ , _Racialicious_ , and _RacismSchool_ , which is basically exactly what it sounds like\u2014a primer on all the concepts I'd experienced all the time but had never heard of. My eyes were being opened to new ways to talk about race, the media, and politics. I was starting to feel like I was back in school again, but for a different reason: I was cramming. I wanted to learn as much as I could as fast as possible, not only for myself but because all of a sudden people were coming to me for advice, and asking me for answers I didn't have. Soon I realized that my great talent lay in what I'd been doing, accidentally, in \"SWGSTBG\" and for years on YouTube: I had a knack for explaining things with jokes and pop culture.\n\nSomehow, my agency didn't drop me after I bombed _SNL_ , and it was really cool to realize that the team believed in me and saw my potential. (It helped to remember that Scott had no interaction with Black Tumblr, so he had no idea there was a corner of the internet that believed I should be banned from speaking publicly ever again.) And while it was becoming painfully obvious that I needed to proceed with caution, I now had the support of new online friends and mentors I could vent to and ask for support.\n\nMeanwhile, at my day job, I was sneaking around so much that I felt like I was trying to hide an affair. Thankfully, my manager didn't really care that I was balancing my new life as an activist\/actress with my full-time job; I always managed to get my work done on time and had no problem staying late to finish projects. Nevertheless, I quickly realized that if I wanted to take advantage of all the opportunities in front of me, I would have to quit. My boss didn't seem surprised when I told him I needed to resign, but I couldn't help but cry\u2014I really liked the job, and I had just discovered the beauty of my employee discount. I'm not sure why, but people really sleep on Ann Taylor and their sister store, Loft. I'll admit I wasn't hip to it until I worked there myself, but now let me pass this totally not-sponsored good word on to you: Both brands have cute work and casual wear and bomb accessories, their shoes go up to size 11, and while most people know them as petite-friendly brands, they also carry tall sizes and go up to women's size 18. Now do you understand why I cried so hard?\n\nWhile I didn't have a job to fall back on, I was able to quit because, soon after this whirlwind of a month, YouTube sent me a huge check\u2014about $30,000 initially, and then $15,000 more after the popularity of \"SWGSTBG\" sent traffic to some of my other videos. I had no idea people were making that kind of money on YouTube\u2014and neither did my bank teller.\n\nWhen I went to deposit the first payment at my local branch, a little giddy, a little terrified that the check would somehow disintegrate in my hand between the front door and the counter, the man working asked, \"Whoa. Why is Google giving _you_ this much money?\" It didn't register at the time that I probably should have interpreted this as a backhanded compliment, if not outright racist\u2014I was wondering the same thing myself. Instead of asking to speak to the manager, or just getting a new teller, I asked him if he'd seen the viral YouTube video \"Shit White Girls Say... to Black Girls,\" and explained that my check was payment for the ad views my video had racked up. He admitted to somehow missing the \"Shit Girls Say\" craze. I whipped out my phone so we could watch right there at the counter.\n\nThankfully, he laughed. Although it was a small thing, it gave me some much-needed perspective. Since the nesting dolls of controversy had taken up residence on my desk, my life had felt like it was taking place in the computer, in angry blog posts and furious messages. Getting a chuckle out of this random guy reminded me that my audience wasn't just made up of the most vocal or most online people. Reaching local bank employees was just as important as reaching Tumblr influencers, and slowly but surely, I was developing the vocabulary and confidence to have important conversations with both.\n\n# CHAPTER THREE\n\n# MY REIGN AS YOUTUBE'S CALLOUT QUEEN\n\nIt's impossible to be part of the YouTube community and not be at least familiar with Jenna Mourey, the vlogger better known as Jenna Marbles. The OG \"queen of YouTube,\" she has more than seventeen million subscribers and a life-size wax figure at Madame Tussauds. One of her first videos, \"How to Trick People into Thinking You're Good Looking,\" got over five million views in the first week she posted it; you might also know \"How to Avoid Talking to People You Don't Want to Talk To\" or \"Drunk Makeup Tutorial.\" Her sense of humor is self-deprecating and punny, but she often mixes in a somewhat feminist message. (\"Trick People\" was a pretty biting\u2014if occasionally casually offensive\u2014commentary on the ridiculous beauty standards society holds for women.)\n\n\"Things I Don't Understand About Girls Part 2: Slut Edition,\" which Jenna posted in December 2012, was different. Off the bat, the tone was a little more bitter, a little more \"I'm about to _say some things_.\" Filming in her dining room, with her familiar Spider-Man poster behind her, she begins with a disclosure: The video \"isn't going to be that funny.\" Instead, she says, it's going to be \"questions that I have for sluts.\"\n\nShe tells her viewers they should feel free to disagree with her and sound off in the comments. Then, she proceeds to aggressively attack women who have \"a lot of casual sex.\" She goes on and on about one-night stands, which she does not understand because they put women at risk for getting \"murdered and herpes.\" She criticizes a subcategory of sluts that she calls \"stupid sluts\" and suggests that women who have a lot of casual sex don't deserve respect, making a particular note that sluts who fuck other people's boyfriends are just \"asking for it.\" Etc. It's almost ten minutes long.\n\nObviously, there are a lot of problems with this video. The whole premise, for starters. And are women in committed relationships somehow immune from violent crime or sexually transmitted infections? Several feminist YouTubers took Jenna to task for her textbook sex-negative slut-shaming. They did an awesome job of explaining the nuances of how the language Jenna used to denounce \"sluts\" was exactly the same as the language people use to silence sexual assault survivors and promote the centuries-old sexual double standard we all know and hate. They never attacked her, or said she was a despicable person for making the video. They just calmly explained where she'd gone wrong, why it was wrong, and what she could have done better.\n\nWhen I watched the video, I was disappointed, and I couldn't shake the overwhelming sense of dread that one of her very first comments\u2014about how engaging in \"risky behavior\" like one-night stands can lead to getting raped or murdered\u2014had made me feel. But when I saw what Jenna's followers were commenting on the video, my shock and sadness turned to anger: People were agreeing with her, and many of them seemed young or impressionable. They wrote that they knew women who had been raped and were indeed huge \"sluts.\" Others commented that they had been assaulted, and that this video had opened their eyes to what they should have done differently.\n\nI contemplated diving into the comments, but instead I decided to make a response video because I thought I could contribute something unique to the conversation\u2014a personal experience. In \"How Slut Shaming Becomes Victim Blaming,\" I told half a million people the story of how I \"lost\" my virginity. In reality, I did not misplace my virginity\u2014I know where it went. A girlfriend got me concert tickets for my eighteenth birthday, and I spent the day leading up to the show drinking, feeling pressured by the group of older guys we were with. By the time the concert was over, I was wasted, and I passed out. It wasn't until the next morning that I realized I'd been, in the words of my giggling girlfriend, \"soooo bad\"\u2014I'd \"had sex\" with one of the guys we were with, her boyfriend's older roommate. I remembered none of it, and I couldn't stop feeling guilty about how I could \"let\" myself be date-raped. I begged my friend, who was also my coworker at Blockbuster Video at the time, not to tell anyone we worked with about it. She immediately told everyone, including my manager, and they didn't handle it with discretion at all\u2014they let me know they thought I was a huge \"slut.\" On top of trying to deal with the trauma of being raped, I was totally humiliated. I couldn't afford to quit, so I asked to be moved to a different store and tried to put the whole thing behind me.\n\nI never thought I'd tell that story on the internet. For one thing, I had only ever told my husband, Patrick\u2014I had to call my mom before I posted the video to prepare her for it. Plus, I was still struggling to brand myself as an actress and comedian, and there is nothing funny about this story. I'd quit my graphic design job to focus on acting and writing full-time, but I was still living off my \"SWGSTBG\" money, since I wasn't booking many gigs. I worried about what my new agent would think.\n\nBut I knew that coming forward would be worth it if my story helped just one person. By the time Jenna released her video, I'd gone to therapy, worked through what happened to me, and finally recognized that it was not my fault, but the process had taken a long time. If I had seen Jenna's video soon after my rape, I don't know how I would have dealt with it. For many young women, Jenna is the pinnacle of coolness and success\u2014something to aspire to\u2014and I believed that if her thoughts on \"sluts\" were allowed to float around unchecked, they would cause harm.\n\nAny nerves I felt about calling out a YouTube celebrity\u2014the most famous woman on YouTube, really\u2014dissipated when I thought about all the women who might see my response and feel less alone. Though I was nowhere near Jenna's level of fame, people were starting to see me as someone who had it all figured out. And I was careful not to attack or blame her\u2014I just thought that talking about my experience could show people what it actually means to be a survivor of sexual assault. The truth is, it can happen to anyone.\n\nIn turn, I learned what it actually means to come out as a survivor of sexual assault. When \"Victim Blaming\" went viral, I got a lot of positive responses, and I never doubted my decision to tell my story\u2014people sent messages saying the video had helped them gather the courage to press charges against their rapist or talk to their children about consent. But I also saw firsthand how our culture shames survivors who tell their stories. Jenna's fans were brutal\u2014they said that I was just jealous of her and that I was racist (because of \"SWGSTBG\"), racial slurs were sprinkled throughout my comments, and folks argued that I had somehow ruined the life of the man I was \"accusing of rape\" even though I hadn't named him. (I don't even remember his name.)\n\nI'd hoped that after I spilled my guts, Jenna would see the light and apologize. Maybe my video\u2014combined with the other responses\u2014could be a teachable moment for her. Maybe she'd respond to the criticism with a \"Heal the World\"\u2013style musical apology starring various YouTube celebrities and donate the iTunes proceeds to RAINN.\n\nShe didn't do any of that. In the immediate aftermath, she silently removed the video from her channel and posted a condescending Facebook status about how people needed to learn to take a joke. She'd opened her video by encouraging differing opinions, but it was clear she wasn't interested in hearing anything other than \"lolol sluts.\" In a _New York Times_ interview about a year later, she said she'd been \"crucified\" for it.\n\nThat sucked, but whatever. I hadn't made my video for Jenna, anyway; I'd made it for her young fans. Though I had to endure a racist, sexist backlash (which, as you can tell, quickly became way too familiar to me), I knew I'd contributed something positive to the conversation. And it was a reminder that doing the right thing is hard and sometimes doesn't have the outcome you expect, or want.\n\nI thought that was that, and I moved on. Then, three years later, I got a suspicious email through the contact form on my website. I thought it was someone trolling me, which happens a lot, especially when it comes to my more controversial videos. But I asked around and figured out the email really did come from who the sender said she was. It was Jenna Marbles, writing to apologize for the way she handled the situation, and to thank me for having the courage to call her out and tell my story.\n\nI'd had a lot of experience with what we now know as callout culture after the _Anderson Live_ appearance. But the situation with Jenna Marbles was the first time I got attention for being the caller and not the unsuspecting person picking up the phone. Though strangers who can't spell send you vile, insulting messages no matter whose side you're on, it was obviously a very different experience to be the referee.\n\nInstead of shying away, I decided to dive in. Because I wasn't working regularly, I spent a lot of time on Twitter, Facebook, Tumblr, and YouTube, and although I wouldn't say I went looking for stuff to get mad about, I was more likely to chime in on trending stories or wade into comment sections to tell someone why what they said was racist, sexist, ableist, homophobic, transphobic, classist, or otherwise absolutely not okay. Eventually people started telling me I should comment on certain issues, sending me bigoted or prejudiced posts and asking if I'd heard about some awful thing so-and-so had said.\n\nWhenever I got involved\u2014which was often\u2014I felt like I had to. Because no one else was, or because no one with an audience was, and I wanted to use my newish success for good. It felt like an uphill battle, but one I genuinely thought was worth it. I developed such a reputation as a callout queen that people began acting like it was my job\u2014I became the go-to girl for defending marginalized people from bigotry, the final leg in the internet's racism relay. Posted a \"comedy\" sketch about slave rape? I was coming for you. An eight-minute video complaining about diversity and HIV testing? Please hold while I charge my camera batteries. In the habit of dropping the N-word and donning blackface and never seeing repercussions? Until now! I didn't care how many subscribers you had or how beloved you were, I was happy to put my SEO  skills to work and dig up your oldest, most racist videos, screencap them, and set them free like an injured bird I'd nursed back to health that was now ready to fly. It was cool to have an online identity and to be praised for it, and although the onslaught of abuse from my adversaries' fans often left me shaking and crying in front of my computer, I would always get an adrenaline rush out of it. I felt like I was changing the world\u2014or at least the internet\u2014for the better.\n\nThe first time I realized that I might have been misplacing my newfound social media power, if not using it for outright evil, came a few months after the Jenna Marbles incident. A man claiming to be named Andrew Moskowitz posted a Facebook comment on a _New York Times_ article about racist hiring practices; claiming to be a hiring manager, Moskowitz wrote that he threw away a r\u00e9sum\u00e9 if it had a name like Tamisha on it. On Twitter and Tumblr, people were enraged, and they took it upon themselves to get him fired from his job. But first, they had to find out where he worked.\n\nSomeone on Twitter started saying he worked at the Monroe Cotton Mills in Monroe, Georgia. The evidence? Well, Sherlock, he'd checked into the company on his Facebook page, so clearly they were mailing out his W-2. Not to mention that the idea of a racist working at a former cotton mill was almost poetic. With the company name in hand, everyone began bombarding the Monroe Cotton Mills with emails demanding action against Racist Andrew. Meanwhile, other users had confused the Monroe Cotton Mills with the nearby Cotton Warehouse and began sending _them_ similar messages, as well as posting negative reviews on the business's Yelp page. The managers of both companies were perplexed.\n\n\"I have NEVER heard that name until 11 AM this morning, when I started getting emails sent to me during church asking why I hired a racist,\" one manager told the _Daily Dot_. \"I have no idea where this guy works,\" the other manager said.\n\nThe Mills\/Warehouse confusion was quickly quashed, but no one was willing to believe that the vigilante anti-racism force we'd assembled was mistaken and Moskowitz didn't work at either place. I, too, thought something fishy was going on, so I called the Monroe Cotton Mills and, in my best Southern belle accent, asked to speak to Mr. Moskowitz. The receptionist told me that although he wasn't there when we called, Andrew Moskowitz did work for the company and would be in on Monday. Got him!\n\nWhen people suggested that maybe we should lay off until we had more information, I confidently replied that I had called the company and confirmed Moskowitz did work there. I continued: The manager of the Cotton Mills was probably just attempting to cover up their racist mistake by saying Moskowitz didn't work there, so we should call back on Monday, when he'd be in the office. After who knows how many people called his office and screamed at this receptionist for her duplicity, someone finally figured out that she worked for an answering service, not the Cotton Mills, and was required to answer any inquiry with \"He's not in right now, but he can get back to you some other day.\" We had wasted not only our time and energy, but the time and energy of a bunch of innocent small-business employees.\n\nI was embarrassed, but the drama was addicting. Even when it backfired, I thought it ultimately generated more awareness of important issues. At the end of the day, wasn't I doing this for the issues?\n\nIn retrospect, I'm pretty sure some of the people I spent hours calling out on Twitter were baiting me. Many YouTubers and \"internet personalities\" have made careers out of getting their audiences to join out-of-proportion controversy, and I fell for it more times than I'd like to admit. Last year, I found a post on the _Daily Stormer_ , the white supremacist message board that was instrumental in organizing the Charlottesville Unite the Right rally, that detailed how easy it was to troll me. There were screenshots featuring me going back and forth with racists, trying to reason with them and getting madder and madder. It was a wake-up call. There are people whose Twitter feeds consist entirely of antagonistic @-replies that they send to random people until someone bites. This is a game to them. They want you to waste your time on them instead of having productive conversations with people who matter.\n\nI was always calm and reasonable when I presented my points, but it didn't matter\u2014each time, I was inundated with abuse, slurs, and death threats. People contacted my agent and suggested I should be dropped because of my \"anti-white\" tendencies. The trolls' responses were always totally disproportionate to the critique\u2014I'd argue it would be pretty cool if police could stop killing unarmed black people, and someone would respond \"I hope you become the next hashtag\"\u2014but it doesn't matter. Even if you know you're right, eventually you start to get tired.\n\nSocial media has always thrived on the kind of drama fueled by callout campaigns\u2014all you have to do is write a single tweet about why what Katy Perry did was #NotOK, and suddenly you've mobilized tens of thousands of people to... do what, exactly? The sense of deep moral righteousness each party feels in the heat of the moment, furiously typing out clapbacks and subtweets before members of the opposing #TeamWhoever can get their barbs in, gives the whole thing the illusion of importance when it's almost always infighting, or attention seeking. I used a track-and-field analogy earlier, but sometimes it feels more like the WWE\u2014laying the smack down with the perfect GIF.\n\nAfter a while, I still felt like I was changing the internet, but I wondered if it might be for the worse. I was letting these attacks consume my time, and more and more it started to feel like my career depended on other people saying stupid shit instead of me saying good shit. I was giving so much of my attention to people who were intentionally posting hateful things online instead of focusing on my own career and my own audience.\n\nIt all came to a head in early 2016, around the time I was hired to work on _The Nightly Show with Larry Wilmore_. I was invited to the Sundance Film Festival to participate in a program the festival was cosponsoring with YouTube. It was my first time at Sundance, and I was excited to attend the premiere of _The Skinny_ , a new web series that was going to run on Refinery29. It was being billed as an unapologetically feminist project, and Jill Soloway,  the creator of _Transparent_ , was executive-producing it.\n\nSo I was pretty surprised when the series began to roll and it was racist. Not, like, a little bit racist, but like _Hot Asian women keep \"stealing\" men from the white protagonist and it's NOT FAIR_ racist. _Black guys are being objectified, but also portrayed as dangerous rapists_ racist. _A joke about police brutality in the first episode_ racist.\n\nI sat through all six episodes, feeling more and more defeated, and when the lights came on and the panel discussion began, I assumed someone was going to say something. But the praise was rapturous. _The Skinny_ was a feminist masterpiece! Its body-positivity was transformative! Jill Soloway and their team were really \"standing up to patriarchy\"!\n\nWhen the Q&A period rolled around, I stood up to ask a question. I didn't want to offend Jill, or make a scene\u2014I was invited to Sundance as an up-and-coming voice, I had just gotten my first job writing for TV, and honestly, the altitude was fucking with my head\u2014so I tried to be positive. \"There are lot of funny elements in this,\" I said, \"and I want it to succeed. But although Jill mentioned intersectionality, the only women of color in the series were being mocked and resented for their sexuality. The series also fetishized black men while implying that they're dangerous. Why did you go in that direction? Have you considered how to rectify the issue for the next season?\"\n\nJill and the moderator didn't know what to do. Jill went on the defensive; the moderator tried to talk her way out of it. \"How are black men being fetishized?\" the moderator asked. \"It's true! Black men love me, and I'm fat!\"\n\nIt was too much. After the panel, I just wanted to get out of there, but as soon as the crowd was dismissed, Jill Soloway and the moderator came over and asked to talk. I didn't really want to, and I told them so, but a crowd was forming, and from a career standpoint, I knew it probably wasn't wise to snub everyone's new favorite television director. But then Jill said something that made me downright sick to my stomach. \"You're right,\" they said. \"We need more women of color who write. Where are they?\"\n\nFor a second, I couldn't say anything. Where are they? I was a black woman who had been invited to Sundance because I was a promising TV writer. I was standing right in front of Jill, and it was like they couldn't see me. Or didn't want to.\n\nI began to feel backed into a corner, literally and figuratively. I heard Jill say, \"Next time you're in LA you need to come over.\" I told them I didn't want to come over and didn't want to be their friend, and that if they wanted more women of color to participate in their projects they would have to work to find them. Maybe start by looking at Sundance? They went back to saying it was actually not racist to produce a scene in which a white woman asks a bunch of black men if they would fuck her and then wonders if black men have lower standards than white guys.\n\nWhen I finally got back to my hotel room, I was nervous and fuming, and I took out my phone. This is probably not the best way to handle your anger, but it was what I did: I recorded a series of videos, talking about how fucking frustrating the whole thing was, even crying a little at the end, and posted it to my Snapchat.\n\nThe video was resonating with other women of color who'd had to deal with white-feminist doublespeak\u2014the \"I'm on your side\" that becomes \"I'm actually more on your side than you are.\" My fans started encouraging me to post the video on my blog so more people would be able to see it, since Snapchat videos disappear after twenty-four hours. I was still mad, so I slapped on a title\u2014\"Here's What Happens When You Call Out White Feminists\"\u2014and up it went.\n\nI should have known the post would take off\u2014I was cursing and crying and dropping pop-culture keywords like \"Jill Soloway\" and \"white feminists.\" I made it in the heat of the moment and didn't take enough time to process what had happened, which made it ideal content for the internet. Soon enough, bloggers began to blog, and angry commenters began to harass Jill, leaving comments on their Instagram calling them a racist bitch and saying the world would be better off if they killed themselves. Jill emailed me to let me know they were still on my side, not my enemy, and to again extend the invitation to come to their house. I told them not to email me again.\n\nWhen the furor died down a few days later, I felt gross on multiple levels. I still had a bad taste in my mouth from the original incident, but I also felt responsible for my fans, who had taken their harassment campaign to inappropriate levels. I worried the frenzy would jeopardize my job on _The Nightly Show_. Even though I'd never directly sic my followers on someone, I was starting to realize how much influence I had. I was not just a girl from South Florida making videos in her bedroom\u2014I had a following of all sorts of random people who could and would do things I don't agree with on my behalf.\n\nThere wasn't much I could do at that point\u2014the damage was done. But I was so embarrassed by the hysteria. A couple of weeks later, when Jill announced they would be directing a new Amazon series, I saw a comment from someone saying they weren't going to watch the show because of what Jill \"did to Franchesca Ramsey.\" Though it was by no means the strongest language ever used by an internet commenter, it was clear that I'd had a lasting impact on this person's reputation. My story had reached corners of the internet I hadn't anticipated, and I wasn't sure how to feel about it.\n\nI made a vague tweet and Tumblr post reminding my audience that my speaking publicly about an issue or person wasn't a call to arms. From then on, I knew I had to take this stuff more seriously\u2014for my own mental health, for my career, and in order to create the kind of online community I didn't dread participating in.\n\nThat's not to say I stopped getting into Twitter arguments, because I didn't. A queen doesn't give up her crown so easily. During my first year on TV, as my visibility increased, I couldn't help but try to defend myself against all the new people, both anonymous and public, coming for me. But I did it the old-fashioned way\u2014passive-aggressive shade. No naming names, and I always shared articles from trusted sources instead of random people whose worldview happened to align perfectly with mine. I realized that if I wanted to effect any actual change, my commentary had to focus on the systems, not individual scapegoats.\n\nI learned the hard way that callout culture is much more complicated than it seems\u2014it's not as simple as \"If you see something, say something.\" I used to treat social media like a tool; in reality, it's more like a double-edged sword. Just as I've developed a career and an audience on YouTube, Facebook, Twitter, Instagram, and Snapchat, I've seen how easy it is for information, whether it's true or false, to spread quickly on these platforms, and I know it's almost impossible to take it back once it's out there. Motives are questionable and drama is real.\n\n_Okay_ , you're probably thinking, _but the internet is full of prejudice and bigotry. What should I do? How do I know if I should call someone out or take the conversation offline? And what am I even supposed to say? You know I'm addicted to Facebook, so don't even think about telling me to delete my account and spend more time with my family._\n\nI would never tell you to delete your account. (Though you probably should spend more time with your family.) Instead, I made a guide.\n\n# CALLING IN VS. CALLING OUT: THE GUIDE\n\n**CALL OUT (VERB): TO BRING ATTENTION PUBLICLY TO ANOTHER PERSON'S BIGOTED SPEECH, BEHAVIOR, SOUND BITE, JOKE, LYRIC, ARTICLE, FACEBOOK POST, TWEET, INSTAGRAM STORY, SNAPCHAT STORY, ROLE IN A TELEVISION SHOW OR FILM, OR PERFORMANCE, ESPECIALLY ON _SATURDAY NIGHT LIVE_ OR AT THE MTV VIDEO MUSIC AWARDS**\n\n**GOAL: TO MAKE THE BIGOTED PERSON AWARE OF THEIR MISTAKE AND\/OR TO RAISE AWARENESS ABOUT A GIVEN ISSUE**\n\nCalling out shady behavior online has become many people's first response when they encounter prejudice, bigotry, or anything they think might possibly indicate prejudice or bigotry if you squint real hard and write a two-thousand-word essay about it. But in most cases, it should be reserved for taking large, powerful entities\u2014celebrities, brands, and media\u2014to task when they screw up. Which they inevitably do. This is because (1) they should know better, and (2) taking Emma \"Definitely White\" Stone out to coffee to discuss the history of whitewashing in Hollywood wasn't really an option for most of us after she was cast as a mixed-race tour guide in the film _Aloha_. (She was supposed to be a quarter Chinese and a quarter Pacific Islander.) On such a large scale, using social media to criticize the film industry's long tradition of ignoring Asian actresses\u2014and to question white actresses like Emma Stone and Scarlett Johansson for willingly participating in that erasure\u2014is the best and most effective action. These campaigns can snowball into progress; though _Aloha_ still came out as planned, both Stone and the director admitted they'd made a mistake, apologized, and said the controversy had opened their eyes. (And the fact that the movie flopped, as did ScarJo's _Ghost in the Shell_ , was a healthy serving of poetic justice.) Another example of a successful callout campaign is the #OscarsSoWhite hashtag, created by Twitter user and badass organizer @ReignofApril to highlight the lack of diversity among the 2015 Oscar nominees. After it blew up the following year, the Academy took note and committed to diversifying its membership.\n\nBut callouts can happen between two-hundred-follower Joes, too. Say you're wasting your lunch break on Facebook one day when you see that your friend Eric has posted two strongly worded paragraphs dead-naming   the \"moronic Trump supporter\" Caitlyn Jenner and calling her a \"man in a dress.\" While unfriending Eric is certainly an option, maybe you don't want to unfriend him\u2014last time you two saw each other you had a surprisingly great conversation about buying window blinds. What if he just doesn't know any better? So you decide to call him out and let the rest of your Facebook friends know what will and won't fly on your page.\n\nAttention fam: Since Eric Peterson apparently thinks it's OK to post transphobic garbage about Caitlyn Jenner all over my timeline, I wanted to make something very clear: It's not OK! While I agree that Caitlyn \"SUCKS SUCKS SUCKS SUCKS,\" her bad political opinions have nothing to do with her gender or her body. Please don't make the mistake Eric did and think that just because Caitlyn supports the orange toad in the White House, you can deny her the right to identify however she wants. My feminism extends to women I don't like, however misguided they may be, and so should yours.\n\nOn the plus side, you've gotten your point across to more than just \"Eric.\" You might feel pretty good about yourself for standing up for what's right. But by making the conversation public, and tagging Eric in the post, no less, you've also opened the door for drama\u2014not just for the unsuspecting Eric, who may be going through who knows what personal crisis in his own life right now, but also for yourself. It's going to be very hard to resist participating in the ninety-four-comment thread that is about to ensue on your timeline, and your lunch break is almost up. Maybe Eric will see the error of his ways, but I'm willing to bet he'd be more likely to do so if you didn't publicly humiliate him first.\n\nCalling out high-profile brands and celebrities can also backfire. In 2016 the internet was blowing up over a Chinese commercial featuring a \"dirty\" black man who is transformed into a \"clean\" Chinese man after a woman forces him into her washing machine. People rightfully deemed the ad \"super racist\" and \"the most racist TV commercial ever made\"; think pieces about anti-blackness in media around the world rolled in. But the attention also made the commercial go viral, racking up hundreds of millions of views and, presumably, revenue. While the conversation was necessary, the conspiracy theorist in me often wonders if brands and celebrities are sometimes purposefully \"screwing up\" in order to ride the viral outrage all the way to the bank. (Before the detergent company apologized and removed the ad from the internet\u2014to the best of their ability\u2014a representative told the _New York Times_ they had wanted the commercial to be \"sensational.\")\n\nCallouts are also risky because you can't unring a bell. If your message is directed at the wrong person\u2014or if any of the details are incorrect\u2014you can end up with a giant mess on your hands. I learned this with the Moskowitz\/Cotton Mills fiasco, but it was really driven home to me during one of my (infamous) Twitter rants. A video of an unconscious young girl being sexually assaulted had recently gone viral, and the internet was being predictably awful. My caps lock was unwavering as I batted down trolls left and right. \"IF YOU'RE UNCONSCIOUS. YOU! CANNOT! CONSENT! PERIOD! THAT'S RAPE FOLKS!\" But streams of nasty comments and memes continued to pour into my mentions as people from all walks of life did Cirque du Soleil\u2013level backflips to explain away this young girl's rape. In my flurry of replies, I noticed one that said something like \"Sometimes you get too drunk and stuff happens.\" After a quick glance at the attached profile, I fired back. \"As a woman you should know better. Take your victim-blaming bullshit elsewhere.\" Instead of replying directly, which would've kept our conversation off my main timeline, I made my reply public so it would be visible to my more than fifty thousand followers and added, \"Yet another lost girl upholding the patriarchy. *tear drop emoji*\" Soon enough, my followers\u2014as well as random users\u2014started chiming in to say she was a victim blamer and disgrace to women. No girl was safe around her.\n\nBut as the girl tried to defend herself it became painfully obvious that I'd made a terrible mistake. She wasn't victim blaming at all\u2014she was talking about her personal experience. As internet users around the world yelled at her, she tried to explain that she'd been in a similar situation and it wasn't a big deal because ultimately she felt it was her own fault. She wasn't trying to speak for the girl in the video.\n\nI immediately posted a public apology and deleted my original tweet, hoping I could undo some of the damage I'd caused, but it was too late. I sent the girl a private message, asking for forgiveness and trying to reassure her that she'd done nothing to deserve being assaulted. No response. I tried again and got an error message: \"This account no longer exists.\"\n\nI still feel horrible about the way the entire exchange went down. I thought I was doing the right thing, but my quick response was driven by the attention I was getting from my audience. I was being praised for \"keeping it real\" while disregarding who was on the end of that \"realness\"\u2014a young girl who needed compassion and support, not a dog pile.\n\nSo before you spring into action to take down the latest brand, celebrity, media outlet, or random college student with seventy-eight followers, ask yourself:\n\n1. **What's the issue?** A celebrity denying police brutality in an interview is not the same as a celebrity giving a bad tip at the bar where your friend works.\n\n2. **What's at stake?** Could someone lose friends\/their job over this? Or will they just feel really bad (assuming they have a soul and feelings)?\n\n3. **Do I have all the details?** Your bartender friend has always liked to embellish. Is it possible that Olivia Wilde tipped in cash and that's why she wrote \"0.00\" on the receipt?\n\n4. **Why are you doing this?** Do you want to raise awareness of a problem, or do you want to raise awareness of your Twitter account, which you believe is scandalously underfollowed?\n\n5. **What are the best-and worst-case scenarios following this callout?**\n\n6. **And finally: Would it be better to _call in_ instead?**\n\n**CALL IN (VERB): TO INITIATE A ONE-ON-ONE CONVERSATION TO MAKE ANOTHER PERSON AWARE OF THEIR OWN BIGOTED SPEECH, BEHAVIOR, ETC.**\n\n**GOAL: TO HELP AN INDIVIDUAL LEARN FROM THEIR MISTAKE AND MOVE FORWARD PRODUCTIVELY**\n\nWhereas callouts can leave you feeling like you successfully rallied a group to save fifty puppies from a burning animal shelter run by Cruella de Vil's competitive stepsister, calling in is more like doing extra-credit homework even though there's a chance it won't improve your grade. When someone you know says something incredibly ignorant, walking them through where they went wrong can be time consuming and emotionally draining. And there's no guarantee it'll even be worth it. Thinking she was just going to have a chill catch-up session with her long-lost college roommate, Problematic Angela could very well get defensive and tell you to get over [insert awful thing she said]. The only thing people hate more than being wrong is admitting they were wrong. Like going to the gynecologist, calling in is awkward at first, but it gets easier every time\u2014though unlike going to the gyno, you will not get a lollipop afterward, or peace of mind about your reproductive health.\n\nThe biggest difference between calling out and calling in is tone. While I wouldn't suggest coming out of the gate with social justice lingo, \"Hey, asshole! That was transphobic!\" feels very different from \"Heeeeey... I actually wanted to talk to you about that. It was kind of transphobic.\" The call-in voice is like the stupid airplane noises you make when you're feeding your nephew a spoonful of veggies. Holding little Brendan's mouth open while you shovel in the mushy peas is not going to make him eat, but the _vroom vroom_ sounds make those peas go down easier and might even set him on the path toward healthy eating habits for the future.\n\nThough call-ins work best in person and one-on-one, they can also happen online, and even publicly. When Kerry Washington described Kate Winslet as her \"spirit animal\" on Twitter, I wasn't the only one who cringed. The term gets thrown around a lot, but most people don't realize it's incredibly disrespectful to Native communities. While plenty of people criticized Kerry's choice of words, I was pleasantly surprised by how many did so by calling her in. Even though it's only 140 characters, @ RanaLaPine's tweet \"Please don't use the term 'spirit animal' like that. Disrespectful to indigenous beliefs and communities\" strikes the perfect balance: It informs without insulting or putting Kerry on blast. It's straightforward without being too blunt. And while it's no secret that Kerry Washington is the best, I like to believe that the fact that no one called her a shit-stain had something to do with her gracious response: \"So, I'd never been schooled to concept that using 'spirit animal' in the way I just did is cultural appropriation. I get it. I apologize. TY!\" What did we do to deserve someone so heavenly?\n\nAfter I became disillusioned with my role as YouTube's callout queen, I began toying with the idea of a softer approach. Soon, I had an opportunity to test it out at a tech conference when a woman came out of nowhere and uttered the five words every natural-haired girl dreads (no pun intended): \"Can I touch your hair?\"\n\nThe woman asking already had her hand outstretched, inches away from my perfectly coiled locs. We were all scarfing hors d'oeuvres, so when I think of this memory I always imagine her fingers covered in grease. I swerved out of the way just in time and tried to smile as I responded, \"I'd rather you not.\"\n\nWho knows why, but homegirl was not having it. You'd think I'd just spit in her tuna tartare and asked her to thank me. \"Are you serious?\" she asked. \"Why? Seriously, why not? You can touch my hair.\" Her voice trembled as she repositioned herself, planted her feet, and leaned her head toward me, as if most people walk around just desperate to tousle strangers' updos. In that moment I knew she wished she could ask to speak to the manager.\n\n\"Lemme ask you something,\" she continued. \"Do you always wear your hair that way?\" Was she trying to trick me? I hesitated for a second and then said no, I didn't normally wear my hair in curls. \"Right,\" she responded, seeming satisfied. \"So then you have to expect that people will want to touch your hair if it doesn't always look like that.\"\n\nIt made no sense. But instead of telling her she was being ridiculous and low-key racist, I decided to take the high road. \"Look,\" I said. \"There's a long history of black women's bodies and hair being constantly poked, prodded, and disrespected, so it's really not the same when you offer to let me touch your hair in exchange. I'm sure you didn't mean any harm, but I'm not a petting zoo.\"\n\nThe woman looked at me with a mix of confusion, embarrassment, and guilt, similar to the way my dog Filthy looks when I catch him pulling a pair of underwear out of my hamper. \"Here, let's start over,\" I added, feeling\u2014I'll be honest\u2014pretty smug. \"Hi, I'm Franchesca.\" She timidly shook my hand and introduced herself.\n\nI can't tell you what her name was, because by that point I'd moved on to my victory lap. I pretended like nothing had happened and introduced myself to the small group of eavesdroppers who had gathered around us. While I can't say for sure whether the woman learned something that day, on a personal level, I felt much better than I did after spending six hours in a Twitter-induced rage.\n\nCalling in is generous, because it shows you're not giving up on someone, but it can also be an act of self-care: You're saving yourself the emotional energy that comes with giving in to anger. I could've easily gotten angry with that woman at the party, and I'd have had every right to: She had been disrespectful at best and racist at worst. But I knew that calling her out and drawing attention to her racism would have done both of us more harm than good. I would have spent the rest of the conference feeling upset and guilty, and no one wants to network with someone who's distracted and angry. Plus, making a scene would have made me look really bad.\n\nSo: Let's say you want to call in your childhood best friend or cousin. How do you start? Let's go back to the Caitlyn Jenner rant, but now, instead of Eric Peterson, the random guy who convinced you to spring for the full blackout shades in your bedroom, the transphobic garbage is coming from your partner, your favorite uncle, or a close friend. Suddenly, cutting ties isn't a viable option.\n\nAn effective call-in is honest, informative, cautionary, and direct: HICD. Hic'd? Sorry there isn't a better acronym; believe me, I tried. You can remember this because... there will most likely be _hiccups_ throughout this difficult chat?\n\nHere's how it should go:\n\n**H** onest: There's no reason to dance around the subject, so let the person know what you want to talk about up front, and then clearly state how their comment made you feel. The goal is to drive home that this conversation is important to you and your relationship.\n\n\"Hey, Jude. Can we talk about that Caitlyn Jenner comment you made the other day? It made me uncomfortable, and I want us to get on the same page about it.\"\n\n**I** nformative: Next, go over exactly what happened as you saw it, and then dig into why the comment or behavior was hurtful. You don't need to go full TED Talk here\u2014you don't want this to feel like a lecture\u2014but if you've got personal anecdotes or historical context to strengthen your explanation, that could really help. It's also important to link the behavior to the broader societal issue that it reinforces\u2014it's not just about personal hurt feelings, but about supporting a culture of oppression and mistreatment.\n\n\"Look, there are plenty of things to criticize Caitlyn for. She's oblivious to her white privilege and her class privilege, and she uses her power as a celebrity recklessly. But her being trans has nothing to do with those privileges; her gender identity is separate from the fact that she is not very smart and voted for Cheetolini. Referring to her by her old name and gender is not only disrespectful to her\u2014it says to all trans people that their experiences can be invalidated as soon as they do something wrong.\"\n\n**C** autionary: At this point in the conversation, most people get defensive and try to steer you over the \"That's not what I meant\" bridge or the \"It wasn't my intention to offend you\" path. I told you this wouldn't be easy. Thankfully, that rickety bridge and dusty road both lead directly to the \"Intent doesn't absolve impact\" bush. That one's pretty thorny to wriggle out of. Help them out gently.\n\n\"I know it's hard to stand up for people who are shitty; I don't want to excuse the ignorant things Caitlyn has said and done. But bigotry is not a shortcut to criticism.\"\n\nWhile the \"It wasn't my intention\" nonapology often gets conversations off track, it can be worth indulging\u2014but only just a little. Drive home that intention doesn't take back the damage done, and make it clear that if they don't want to be misunderstood or misrepresented, they need to be more cautious.\n\n\"I'm sure you didn't mean it like that, but saying that sort of thing makes you sound transphobic. And I know that's not you.\"\n\nWhen people are inevitably hung up on their intentions, I once heard this analogy: If you accidentally step on someone's toe and break it, it doesn't matter that you didn't mean to break their toe. The toe is still broken, and you have to make up for that somehow.\n\n**D** irect: The entire conversation should be direct, but if they're feeling confused or mad or like the whole world is shifting around them, you can also steer them to some resources in case they want to learn more. If they seem receptive, send links to videos, articles, or other media that can offer additional perspective.\n\nNow it's time for some cold, hard truth. As admirable as calling in may be, sometimes... it doesn't work. It takes a certain level of empathy, maturity, and introspection to make it through these kinds of conversations, and not everyone is cut out for it. For some people, acknowledging their privilege and recognizing how their words or actions uphold oppression is too big of an ask. While I believe that anyone can grow and learn, I know that not everyone wants to. \"I'm sorry you feel that way\" will always be easier to say than \"I'm sorry I made you feel that way. I'll do better.\"\n\nEven more cold, hard truth: Whether you're calling someone out or in, your relationship to the person matters, both in how you decide how to act (are they worth the trouble?) and in how they'll take your message. Remember the white guy who made that video explaining why \"Shit White Girls Say to Black Girls\" wasn't racist? A lot of people were (rightfully) mad about the implications of that post\u2014the idea that no one will listen when black people tell their own stories, so white people have to speak for them.\n\nUnfortunately, that white guy was actually onto something. It's scientifically proven that people are more receptive to criticism when it comes from someone who looks like them. In 2016, Kevin Munger, a researcher at the NYU Social Media and Political Participation lab, decided he wanted to see if it was even possible to limit racist harassment on social media. Though they're getting better, social platforms themselves are really bad at stopping their users from spewing hot, bigoted bile into the world\u2014research shows that although banning users who harass other users works in the short term, in the long run it can actually fuel whatever behavior it attempted to curb. Getting punished\u2014whether by being called out or banned\u2014makes harassers feel victimized when they encounter the consequences of their own behavior. Some might call this kind of oversensitivity \"being a snowflake,\" but that would be petty and we don't want to do that. Armed with the understanding that punishment from \"the man\" had the potential to backfire, Munger wondered whether regular people might do a better job at fighting racist trolls.\n\nSo, as one does, he built some fake Twitter accounts. (He called them \"bots\" even though he controlled them.) One group of bots had a white guy in their profile pictures (and names like Greg), and the other group had a black person in their profile pictures (and names like Rasheed). Sounds plausible, right? Munger gave the bots varying numbers of followers, from a handful to around five hundred, suspecting that \"high status\" users who had more followers would be more persuasive than some guy named Steve with eleven friends.\n\nArmed with these catfish bros, Munger began to search particular racial slurs on Twitter. Switching off between Gregs and Rasheeds, he responded to users who had a history of racist harassment with the same message:\n\n@[racist person] Hey man, just remember that there are real people who are hurt when you harass them with that kind of language\n\nMunger chose to respond to users who presented as white men on Twitter or were completely anonymous. Then, he tracked their tweets for two months after he responded to them, to see whether they kept tweeting racist bullshit or reformed and turned into Maxine Waters fan accounts.\n\nThe results were surprising, but also not surprising. The only group that was able to significantly reduce the racists' use of slurs was\u2014can you guess?\u2014the white guys with five hundred followers. Obviously, this suggests that if you want a white guy to see the error in his racism, you should have a white guy point it out to him.\n\nBut it got more depressing: Turns out, when white guys are called out by black guys, it can actually make them _more_ racist. In certain cases, Munger found that Rasheed just made things worse, with users increasing their usage of slurs in the months following the reply. Unfortunately, \"sanctioning behaviors\" that highlight the difference between the two parties involved don't usually work. (Munger's research is published in the journal _Political Behavior_ , if you want to check it out in full.)\n\nThis principle\u2014that people in a given community have to be accountable for educating others in that community\u2014doesn't just apply to conversations about racism. Cis people need to step up and talk to other cis people about their casual transphobia, dude bros should talk to other dude bros about the importance of consent, and so on. And while there are no guarantees, finding common ground can often make things easier. When Jenna Marbles emailed me to apologize for the \"Things I Don't Understand\" video, I responded. But I didn't want to gloat or insult her, or overwhelm her with a rant about how her apology would never be able to erase the psychological damage she'd inflicted on innocent teenagers. I don't think that's true, and I don't think she's a bad person. However, I did encourage her to admit her mistake and apologize publicly. Her platform had only grown since the incident, and sexual assault within the YouTube community is becoming more and more of a visible problem. People are so bad at apologizing and taking responsibility for their mistakes, I wrote, that if she did it, so many people who look up to her might learn something.\n\nAs far as I know, she still hasn't come out about the incident. But sometimes people surprise you. A year after my Sundance back-and-forth with Jill Soloway, the _Los Angeles Times_ reported on a similar argument that broke out over a women-in-film lunch at the very same festival. Guests like Shirley MacLaine and Salma Hayek were discussing the recent election of Donald Trump and the importance of not feeling \"victimized\" when Jessica Williams, _The Daily Show_ 's former \"senior Beyonc\u00e9 correspondent\" and all-around black Renaissance woman, started to speak.\n\n\"What if you are a person of color, or a transgendered [sic] person who\u2014just from how you look\u2014you already are in a conflict?\" she asked the table.\n\nThe conversation began to get heated, and some of the guests, especially Salma Hayek, began objecting to Jessica's tone, or even to the idea that she should identify as a black woman at all. Ah, yes, the old notion that \"not seeing race\" makes racism go away. People were speaking over her. The conversation was beginning to slip away. And then Jill Soloway stepped in.\n\n\"With intersectional feminism, it's our responsibility as white women to recognize that when there are people of color or people who are queer\u2014we need to prioritize your voices and let you speak the loudest and learn from your experience, because we haven't been listening,\" Jill said to the table. \"So please, Jessica, finish your thoughts.\"\n\nWhen I read this story, my eyes were so wide I'm surprised they didn't fall out of my head. Jill had nailed it. Not only did they school the women at the table, they passed the mic back to Jessica like a seasoned backup singer assisting with the chorus before falling back for the next verse. Despite the tears, heated emails, and nasty Instagram comments of the previous year, I couldn't help but wonder: Had I, in some small way, helped Jill become the ally they were that day?\n\nIf there were some foolproof way to mend hearts and relationships when these kinds of awkward situations arise, you best believe I wouldn't put it in a book\u2014I'd patent it, sell it on QVC for four easy payments of $19.95, and put the rest of my lineage through college. But my kids will probably end up taking out loans like everyone else, because there is no magic solution for opening, or changing, people's minds. There will undoubtedly be times where there's no resolution, and you'll leave the conflict frustrated and drained. And then there will be other times, maybe a month or even a few years later, when it will all feel worth it.\n\n# CHAPTER FOUR\n\n# BETWEEN A LOC AND A HARD PLACE\n\n## The Chescalocs Story\n\nThe only thing worse than asking to touch a black woman's hair is \"asking\" while simultaneously stroking her luscious curls as if you were at a petting zoo. And while this is a particularly common experience for black women, in reality it often feels as if black and brown folks with any sort of kink or curl are forced to move through the world like they're guarding classified government files on their heads. Thankfully the politics surrounding black hair have become much more visible over the last few years. More people are accepting their natural textures and are less hesitant to lay the smack down when people start policing, ogling, fondling, yanking, and making rude comments about the \"omg so weird!\" texture of black hair.\n\nThese days, there's a booming natural hair community online, which has led more people to feel like \"going natural\" is a viable option\u2014politically, practically, and professionally. YouTube is filled with hair tutorials, product demos, and Q&A videos for every black hair texture and style imaginable. There are hundreds of forums and sites where women exchange salon recommendations and offer support to those tinkering with the idea of transitioning their hair. And it's not just on YouTube: In 2016, my friend Phoebe Robinson published an essay collection called _You Can't Touch My Hair: And Other Things I Still Have to Explain_ ; the same year, Solange released the track \"Don't Touch My Hair.\"\n\nIn retrospect, going natural was the jumping-off point for my career as an activist and as an online performer. When I decided to go natural in 2003, the year after I graduated high school, conversations around natural hair were almost nonexistent in my world. There was no \"ethnic\" hair section in my local drugstore, and when I went looking for guidance for styling and maintaining my locs I quickly learned there wasn't much in the way of support. Little did I know the decision to start making hairstyle videos of my own would be the gateway to conversations about race and identity, along with the start of an amazing community and career.\n\nI've often found that nonblack people are perplexed by the ongoing conversations around black hair and the importance of going natural in our community. As many black thinkers and writers have pointed out, for us it's never \"just hair,\" but a symbol of hundreds of years of oppression and struggle. Throughout history, black people have been told their natural hair texture is unacceptable, dirty, and unprofessional. And while there's nothing inherently bad about the desire to straighten one's natural hair, resisting the subtle and sometimes overt messages in media that equate straight hair with success isn't always easy. Not to mention, going natural as an adult after rocking a perm for years is scary as hell. It's essentially like learning a language you spoke in childhood but haven't heard in over a decade. And then when you go natural, you're expected to be fluent in that language to get ready every morning.\n\nAs a kid, I begged my mother to let me relax my hair. In retrospect, I had no idea what I was asking for, nor did I grasp how backward it was to call chemically straightening one's hair \"relaxing\"\u2014as if Afro-textured hair needs to be tamed and calmed with chemicals. I've often joked that _The Relaxer_ sounds like a crappy horror movie in which a maniacal killer tortures natural-haired women with scorching-hot chemical ooze and then forces them to pay him money for it.\n\nWhile my mother had worn her hair long and straight for as long as I could remember, she was determined to make me wait until I was \"old enough\" to handle the responsibilities of maintaining my hair: biweekly deep-conditioning treatments, learning how to wrap my hair at night to help cut down on styling time in the morning, avoiding excessive heat, and of course rocking a swim cap at the pool. (The last thing you want to do is mix chlorine with your relaxed hair.) Remind me again why this process is called a \"relaxer\" when it takes so much work to maintain? When do I get to chill and not think about what's happening on my head?\n\nAlthough I was one of only a small handful of black students at my predominantly white Catholic elementary school, and part of an even smaller group of girls with natural hair, I didn't feel the pressure to straighten my hair at school. Instead, it was my relatives who treated chemical straightening as if it were inevitable, or required. When I'd visit my family in South Carolina, they would say things like \"When is your mother going to do something with that hair?\" Which really just means \"When are you finally getting a relaxer?\" One year, I got braids before heading to South Carolina for the summer, and my aunt promptly took them out and used a hot comb to straighten my hair. She knew better than to give me a relaxer without my mother's permission, but my mom was still livid. Not because my hair was straight, but because she had spent \"good money\" on the braids and they were cute as hell.\n\nWhen my mother finally gave in and let me relax my hair in fourth grade, the reality of my decision quickly set in. Sure, my hair was now shiny, silky, and straight, but it felt like it never grew\u2014it was always so damaged that I had to constantly trim and cut it off to keep it looking \"healthy.\" The chemicals were clearly too strong for my hair, and I was constantly shedding, so much that my stylist gave me an asymmetrical haircut to mask the damage. Thankfully my sideways mullet (short on the right and long on the left) was reminiscent of Salt-N-Pepa, who I later learned also rocked the style after singeing the hell out of their hair back in the day.\n\nWhen I think back to the years I wore my hair bone straight, I can't remember white people asking to touch it or fondling my head without permission. There's a sad irony in the fact that white people never tried to touch my hair until I went natural\u2014that I had to do something totally unnatural to my head in order to blend in, because my hair in its most natural state was deemed weird or different.\n\nIn my freshman year of high school, my super-cool and super-rebellious best friend Melissa convinced me to add highlights to my hair using an at-home dye kit. While I'd never dyed my hair before, Melissa (who was white) was an old pro. Only neither of us knew that the process wouldn't be so easy for me. I opted for a dark maroon, hoping a few streaks to my bangs would be subtle enough to be cool and to hide from my mom. The result was far from subtle.\n\nAt first, the streaks were barely visible. I hadn't realized that you need to bleach dark hair in order for any color to show up on it. Though I was disappointed, I found solace in the fact that I wouldn't have to deal with the possibility of my mother flipping out over my new highlights. But my punishment ended up being way worse. The next time I got my hair done, I didn't tell my stylist I'd colored my hair\u2014I didn't think there was a point, since you couldn't even see it. As she washed out the relaxer, she also washed away half my bangs. My hair began falling out in huge clumps, and I had only myself and a box of Clairol to blame.\n\nI was devastated, but I wasn't about to ditch my relaxer. Instead I spent the rest of high school with awful haircuts that tried and failed to hide my temples, where my hairline was patchy and uneven. You know those videos of the toddlers who cut their own hair when their parents leave the room? It looked kind of like that\u2014I'm sure a few people wondered who had left me alone with the grown-up scissors.\n\nMy love of bangs grew deeper as I figured they were the only things standing between me and total mortification. Little by little, the starting point of my bangs inched farther and farther back, toward the crown of my head, so they would cover my forehead as evenly as possible. When my bangs no longer cut it, I'd fill in my hairline with mascara, and if I ever got caught in the rain I'd make a mad dash, cupping my foam bra inserts in one hand and covering my edges with the other to keep Maybelline Great Lash from streaming down my face.\n\nDespite my hair woes, I took comfort in the beautiful images of black women, natural or not, in the pages of my mom's issues of _Essence_ , _Ebony_ , and _Jet._ I LIVED for _Jet_ 's Beauty of the Week feature, where the magazine would highlight an absolutely stunning black woman\u2014usually wearing a bikini and posed next to a waterfall, beach, or community pool\u2014who read the magazine. The girls were always very fit (which was intimidating), but they wore lots of different hairstyles, which I loved.\n\nAll the other magazines I read back then were dominated by white girls; any time I saw a black girl grace their pages it was like a brief portal into another reality. _Sassy_ was very feminist and progressive, but I wouldn't describe it as diverse. I liked that the models weren't super skinny, but I can't say I remember seeing anyone that looked like me. When _Sassy_ went out of print, I graduated to _Jane_ , which I loved for how snarky and raw it felt. I had no idea if it was accurate, since I'd never had sex, but _Jane_ writers really seemed like they were being honest and real about sex. (I can't say when the It Happened to Me column jumped the shark\u2014I know it was long before that notorious viral article \"My Gynecologist Found a Ball of Cat Hair in My Vagina\"\u2014but I loved to read it growing up. No shame.) I thought _Jane_ 's street-style pieces, which featured women of all different races and sizes, were so cool, and I got a lot of ideas from them. _NYLON_ was the opposite of relatable, but I loved the music features and magazine's design; I'd make poor copies of their illustrations in my sketchbook and tell myself that if I couldn't be a singer or an actress, maybe I could move to New York and work for _NYLON_ someday.\n\nWhen I left South Florida for my brief stint at acting school in Michigan, I struggled to take care of my hair. The nearest black salon was over an hour away, I had no car and limited funds, and the first few weeks of freshman year were doing a number on my already crunchy hair. The cool fall climate and drunken nights when I'd forget to wrap my hair didn't help. I decided to take the plunge and grow it out in hopes of going natural. A girl in my dorm agreed to braid my hair in exchange for twenty-five dollars and a few swipes on my meal card. And with that my journey was under way.\n\nAfter rocking cornrows and micro braids all year, I needed to figure out who I wanted to be when I was ready to set my natural hair free. So I decided to be Lauryn Hill. Like the rest of the world, I'd been obsessed with her debut album, _The Miseducation of Lauryn Hill_ , and I loved everything about her, especially her hair. Besides Whoopi Goldberg, she was the only celebrity I'd seen wearing locs, so I scoured the internet for pictures of her hair at every stage, hoping to get an idea of what I might look like. Maybe some part of me thought that if I had locs I would look like Lauryn. (Spoiler: Didn't happen.) But I also loved how Afrocentric, unique, and, most of all, confident her hair made her look. She was everything I wanted to be.\n\nToday my locs are nearing fifteen years old and hang just below my waist. I love piling them on top of my head into a giant messy bun mountain, wrapping them with bright colorful scarves, and braiding and pinning them into twisting and turning sculptures that resemble the fancy garden trees from _Alice in Wonderland_. I wear all 114 of my locs (yes, I've counted\u2014partially out of curiosity and partially because people kept asking how many I have) proudly.\n\nBut the journey from baby locs to black Rapunzel was a long and frustrating one. The summer after my freshman year of college, I decided to finally do the big chop, trimming off my relaxed ends to rock a teeny-weeny Afro (TWA). I assigned my mom the task of cutting my hair. While I was nervous to take the leap, my mom was even more so. \"Franchesca, there isn't much hair here,\" she kept saying, in Worried Mom voice. \"It's gonna be short. Are you sure?\"\n\nI was sure. I don't think my mother disliked my TWA, per se (I'm basically a rapper now), but she worried I hadn't really considered how difficult it might be to adjust to my natural-hair texture as an adult. She also remembered the countless nights I'd cried over my broken hair, lopsided boobs, giant feet, and admittedly horselike, gummy smile. She was probably worried that cutting my hair would send me into another puddle of tears, and that this time it would be her fault. But to both her surprise and mine, I loved my tiny fro and wore Bantu knots and twist-outs all summer long.\n\nA year later, my mom's friend Yevola, who'd had locs for more than fifteen years, started my baby locs for me when I moved back to Florida to go to school in Miami. She told me to retwist my hair every few weeks, showed me how to do it, gave me a jar of her favorite gel, and sent me on my way. After that, I was on my own.\n\nBaby locs are often referred to as being in the \"ugly stage,\" because they're short and you can't do much with them, which is... similar to a baby in many ways. But I never thought of them that way\u2014babies aren't ugly, they're cute. Sure, they try your patience, but you're made better for it. Or so I've heard.\n\nThere wasn't a lot of information available on how to style locs online, and you certainly couldn't buy natural-hair products at Walgreens, so I was stuck trying to figure out my hair on my own, on top of being at a new school in a new city. There were a lot of black-hair supply stores in Miami, but I had no clue what products to use, and I didn't really love my mom's friend's favorite gel, so I started buying random things and seeing what I liked.\n\nMuch like a baby, I wore lots of hats and scarves at first. My mom would occasionally send me scarves she'd found at discount stores or consignment shops, which was sweet, though she'd also ask if I had anything to add \"shine\" to my locs, which isn't a thing. (Sorry, Mom.) My extended family gave me a harder time. My uncle asked why I'd cut my hair off; I'd had \"such pretty hair.\" My aunts would say things like \"You're brave to go natural\u2014I don't think I could do it.\" And despite her being natural the entire time I've been on this earth, my grandmother still occasionally asks, \"Is that all yours, Frannie?\" even though I've told her a dozen times, \"Yes, this is my hair, Grandma, and please don't call me Frannie.\"\n\nFaced with these well-meaning but limited resources, I joined the only online dreadlock  community I could find: LiveJournal's Get Up Dread Up forum. If you Google this group today, you will notice that it is basically the same as it was in 2004: almost entirely made up of white people. (Shout out to Danielle Henderson, a badass writer and one of the only other black people on the site.) Most GUDU members were friendly, but one of the moderators and I butted heads a lot. She regularly discouraged me from twisting my hair or using product, which really pissed me off. I was far from an expert\u2014that's why I'd joined this ragtag group of hacky sack bongo players in the first place\u2014but she wouldn't admit that her hair was just fundamentally different from mine and required different care.\n\nAfter lurking around GUDU for a few months, watching my hair grow and change and becoming increasingly frustrated with the lack of content geared toward black hair, I decided to make my first tutorial, which became my first YouTube video. Although I was still learning about my hair through trial and error, I knew there had to be other people who could benefit from my experiences.\n\nFor a while, I posted my hair tutorials alongside my music and comedy videos on the same channel, Chescaleigh, but as the demand for natural hair content grew (like my hair... sorry), I eventually started a dedicated hair channel, Chescalocs. Since there was no formula for beauty videos then like there is today, I got creative: I did updos inspired by Tilda Swinton in _The Chronicles of Narnia_ , Janelle Mon\u00e1e, the bow Lady Gaga wore in the \"Poker Face\" era, and Medusa. (The last one was for Halloween, but if you want to embody a powerful female monster from Greek mythology in your day-to-day, power to you.) I dispelled common misconceptions about locs (they're not dirty!), did product reviews, shared maintenance tips, and tried to convey what, to me, is one of the most beautiful aspects of black hair: You can sculpt it into a piece of art. As I forced myself to come up with new ideas for videos for my growing audience\u2014I was kind of hard on myself about posting regularly\u2014I got more and more excited about the natural hair movement that was developing around me. I was encouraging people to start their own locs as I challenged preconceptions about black hair. Our hair isn't smelly or unruly. It's regal\u2014it's the only texture on earth that extends to the sun like a crown!\n\nWhen _Paper_ magazine selected me for its annual Beautiful People list in 2012, I realized how my newfound activism and my long-running love of beauty could intersect. There I was, locs and all, alongside up-and-coming celebrities like David Oyelowo, Zosia Mamet, Rita Ora, John Mulaney, and Rebel Wilson. It was so surreal. Although I'd been publishing a beauty vlog for years, I never would have described myself as \"beautiful\" before. Cute? Sure. Attractive? Mildly. But not _beautiful_. It was my first professional photo shoot, and since I'd always felt really insecure about my looks, especially my body, I went into it worried about what they'd make me wear. (It ended up being a strapless Kenzo dress, which was cute but not exactly appropriate for the freezing January rain out there on the balcony. Especially when you consider that I wasn't wearing a bra.) I spent the whole shoot questioning the stylists' and photographer's choices: Bright magenta lipstick? But I never wore lip color! \"Sexy\" smoldering eye? But I'd rather smile! I was so trapped inside my own head at the time that I was shocked when I saw the feature in print: I looked awesome. Not only was I side by side with some gorgeous and talented people, but I kinda seemed like I belonged there.\n\nIn many ways, the _Paper_ shoot showed me what I'd been saying all along with my channel: Whether I knew it or not, Chescalocs had always been about embracing and celebrating black beauty in a world that, historically, has always said blackness is less desirable. One of the reasons I know my trolls are a waste of time: Whenever they discover my hair tutorials, they're confused\u2014they still hate me, but they don't see anything to be righteously angry about in those videos. They don't understand that my hair is just as much an expression of blackness as my social justice work.\n\nAlthough I haven't made hair videos in years\u2014they can take hours to film\u2014I still get tons of messages from people around the world saying that I inspired them to start locs, or taught them how to style their hair. When I had the chance to meet Oscar-nominated and Emmy Award\u2013winning director Ava DuVernay at a charity event a few years ago, I almost had a heart attack when she told me she'd watched my videos for styling tips. And while it's incredible to learn that celebs like Ava have enjoyed my videos, it's especially gratifying to now see young girls unapologetically wearing locs, and to see young loc'd celebrities like Willow Smith and Chloe x Halle that kids can look up to.\n\nThe proudest moment of my career came a few years ago, when I had the opportunity to meet Tiana Parker, a third grader who was sent home from her Oklahoma charter school in 2013 for wearing her hair in locs. Shortly after her story went viral, her parents contacted me on Facebook and shared that Tiana was a fan of my videos and had decided to loc her hair after seeing my tutorials. They were coming to New York for a long weekend and hoped I could make time to meet Tiana and possibly offer some words of encouragement. To know that a little girl found the confidence to love and embrace her natural hair because of me\u2014even as her own school had the nerve to tell her it was \"distracting\" and not \"presentable\"\u2014in some small way helped me get over the absolute rage I felt about this school's racist policy. (Afros and Mohawks were also forbidden.)\n\nI wish this were a shocking anecdote, but policing black hair in this way is disturbingly common. In 2017, two sixteen-year-old twin sisters at a charter school in Massachusetts were told their braided hair extensions were \"distracting\" and in violation of the dress code. When they refused to \"fix\" their hair, they were banned from extracurriculars and prom and threatened with suspension. In 2016, Durham's School for Creative Studies made a group of students remove the head wraps they were wearing to celebrate Black History Month. In 2013, a private school in Orlando threatened to expel twelve-year-old Vanessa VanDyke, whose voluminous natural look was so fierce that I'm kind of questioning my locs right now, if she didn't cut or shape her hair. Earlier that year, a twenty-four-year-old secretary in Saint Louis was forced to choose between her job and the locs she'd been growing out for ten years. In 2012, a meteorologist was fired from her local TV news station for defending her TWA against racist Facebook commenters. In 2014, the U.S. Army caused an uproar by briefly considering a ban on \"twists, dreadlocks, Afros, and braids\" for female soldiers, and servicewomen were permitted to wear dreadlocks and twists only in January 2017, after many had spent years getting relaxers despite the damage, cost, and danger. (After the restriction was lifted, Major Tennille Woods Scott told _Vogue_ that she used to relax her own hair while stationed in Iraq in 2007 and 2008. \"In the hour or so that it took,\" she said, \"I was nervous, thinking, _What if a rocket or mortar comes in?_ \") Meanwhile, with the support of her parents, Tiana Parker ended up transferring schools so she could keep wearing her hair the way she wanted to.\n\nI once saw a quote that said, \"Be who you needed when you were younger.\" This simple yet powerful sentence has since informed my work across comedy, social justice, and beauty. When I was Tiana's age, I was begging my mother to straighten my hair\u2014I didn't find the confidence to go natural until my early twenties. I can't imagine having that level of confidence and self-knowledge at just seven years old. While I wish it hadn't taken me so long to arrive at a place where I truly love myself, I now realize the process of going natural and learning how to style and take care of my hair on my own is tied up with my journey into adulthood and self-acceptance.\n\nAs my platform grows, I see how important it is for people to see a black woman wearing her natural hair proudly. I'm now becoming the woman I needed to see in media when I was Tiana's age, and that's pretty powerful stuff. The online beauty community has changed a lot since I was explaining how I put my hair in a ponytail in a dimly lit hair tutorial in 2006. What used to feel free and open\u2014a community for everyone\u2014has since been tainted by the growing desire to be famous and make money. Although Photoshop has long been a strategy of fashion magazines and ad campaigns\u2014when I briefly worked at Maybelline, I had the job of making the models' eyelashes even thicker and longer for mascara ads\u2014now everyone on social media or YouTube has the tools to alter themselves to blemish-free perfection. And the way brands have jumped on the opportunity to shape \"influencers\" has totally warped what \"beautiful\" means to many people. From waist trainers to hair gummies and flat-tummy tea, social media is filled with people pushing shady brands in exchange for a check, and their fans are none the wiser. \"USE MY PRODUCT CODE\" should be a warning that a supposedly \"real\" beauty blogger is being paid to recommend a certain product or service\u2014but so many girls and women fall for it anyway.\n\nIt's hard to be part of an industry that you realize is so damaging. But that's also part of why I continue to be vocal and visible about what goes into my style, and about when and if I choose to work with brands, even if I'm no longer making step-by-step tutorials every week. I know it's possible to be the change you want to see in your community\u2014and if my story helps just one person start down the path to loving who they are, all the better.\n\n# CHAPTER FIVE\n\n# LESSONS FROM GOING PUBLIC, PART 1\n\n## If You Can't Stand the Heat, Get Out of the Spotlight\n\nThe first time I felt the tingle of celebrity potential was in college, when I wrote and recorded an original song that was selected for the _Jane_ magazine reader CD. Along with its snarky takes on sex, relationships, pop culture, and fashion, _Jane_ would periodically send out mix CDs with the magazine. When the editors announced a special edition featuring reader-submitted jams, I saw my chance. I'd spent the entire summer working on a few songs with the huge loser I was sort of dating who liked to cut demos in his dad's home studio, and I submitted one of them to _Jane_ 's contest. When my funky, wannabe\u2013Ani DiFranco ditty \"I Know How the Story Goes\" was selected for the album, Columbia Records sent me a letter saying I had \"promise.\" All I'd have to do was shell out for a real demo and make something a little \"more commercial.\" I knew at that moment I was on the brink of stardom.\n\nOf course, I wasn't. I lost access to the recording studio when loser-guy-I-wasn't-even-dating dumped me\u2014it was for the best, trust me\u2014and it would be years before I gained a fraction of the recognition I thought my little achievement was 100 percent definitely about to usher in.\n\nI wish I could say the disappointment taught me not to count my Golden Globes before they hatch, but unfortunately this kind of wishful thinking became a pattern. A few years later, in 2008, I was living in Miami when I entered YouTube's RedCarpet Reporter contest on a whim. Although my entry was poorly lit, and I stumbled over my words more than a few times, apparently the judges were taken with my skills as I interviewed Pat and our dog Filthy for the imaginary Golden Pooch Awards; I made it to the second round. But this proved harder than I anticipated\u2014the contest was then opened up to a public vote. I busted my ass to promote my entry, sending out a press release to local newspapers and even throwing a party, complete with laptop-voting stations. My hustling paid off, and I was awarded the chance to report from the Emmys red carpet on behalf of YouTube and People.com. After an incredible weekend in LA, dancing with Niecy Nash, cracking jokes with Kathy Griffin, and singing on the red carpet with Josh Groban, I was certain I'd snag a job with _People_ and be forced to give my two weeks' notice to the Miami Beach Chamber of Commerce, where I worked as the communications manager.\n\nSadly, I left with a \"don't call us, we'll call you\" before flying back to Florida and heading straight to the office. I was so devastated by my return to normal life that I gave away my giant bag of Emmy swag (filled with jewelry, makeup, gourmet snacks, and tech goodies) to my coworkers and pouted at my desk.\n\nIn 2011, I entered YouTube's NextUp contest, which promised to find YouTube's next big stars and give them the training and tools to hit it big. Once again I scrambled to submit an entry and get my friends and family to vote for me; I was selected as one of thirty-five creators to receive a hefty check and a week's worth of video production training at the Google offices. I spent the week listening to lectures from giant YouTubers like the Gregory Brothers and Michelle Phan and former executives from Nickelodeon and MTV. I hung on every word and took meticulous notes. YouTube believed I had what it took to become a star, and I was determined to prove them right.\n\nThe following year, \"SWGSTBG\" went viral, and once again, I was sure my big break had arrived. It was obvious that the video was a major turning point in my career, but again, it wasn't exactly the launch into stardom that I thought it would be. That's the thing about turning points: Facing a new direction doesn't mean there's not another long road in front of you.\n\nThe \"SWGSTBG\" numbers seemed like undeniable proof that I would be picking out Emmy dresses in no time. But after I quit my job, signed with my agent, and informed my friends and family that they could expect _much_ better Christmas presents in the coming years\u2014I planned on being very generous as a wealthy celebrity\u2014I didn't get any work. I went on audition after audition, but never booked anything bigger than a failed Oxygen pilot and a role as a sexy coworker in a Wendy's radio commercial I still haven't heard. I maintained my rigorous twice-a-week YouTube posting schedule, but the viral gods had moved on to Donna the Deer Lady and the German guy who did a cannonball onto a pool of ice. After about a year, I got a manager, and together we developed a show called _The Franchesca Feed_ , which would have been a sketch comedy show organized like an endlessly scrolling Facebook feed. We pitched it to twelve networks\u2014everyone from Bravo to Lifetime to MTV. No one wanted it.\n\nIn retrospect I'm glad this didn't work out, because an endlessly scrolling Facebook feed is my worst nightmare. It was also clear that I wasn't ready for my own show. Nevertheless, my YouTube money was seriously dwindling, so I took a job blogging about social justice at Upworthy, a viral content site known for popularizing the \"You Won't Believe What Happened Next\" style of headline. Working there would teach me a lot about how to engage\u2014or, okay, sometimes trick\u2014an audience online. And since I worked from home, I was able to enjoy the benefits of a regular check and health insurance and still squeeze in auditions and speaking engagements in my spare time.\n\nAlthough we didn't get a single offer to produce _The Franchesca Feed_ , MTV said they liked me and wanted to keep in touch. After I went on another series of failed auditions for shows like _Hey Girl_ (originally known as _Blogger Girls_ ) and _Girl Code_ , in 2014 they asked if I'd be interested in developing a web series. They'd had success with _Braless_ , which broke down feminist concepts in a fun, straightforward way by relating them to pop culture and current events, and they wondered if I could do a similar series about race. They hooked me up with the Kornhaber Brown production company (who also produced _Braless_ ), and together we began to develop what would, in over a year, become _Decoded_.\n\nFrom the get-go, it was exhausting to toggle back and forth between the production company, Upworthy, auditions, and any other gigs my agent snagged for me. But having my own show had always been my dream, and although _Decoded_ wasn't going to be on network television, it was one step closer to network television, so I did whatever I could to make it work. I would fly across the country for a college speaking gig, write Upworthy posts from my hotel room after my speech, spend the plane ride home researching and writing stories, and, between remote company meetings, run to my producer's offices to hammer out topics for _Decoded_. After two months, we had a pitch for MTV. They were into it\u2014but little did I know, we weren't anywhere close to being done. We spent another four months negotiating contracts, and another couple more ironing out the concept, fighting about the name (it was not always _Decoded_ ), building a new set (we hated the first one), and shooting and reshooting (I had no idea how to do my own makeup).\n\nThe entire time I was working on this, people would ask me what the hell had happened to me. Though I'm not ashamed to do a little self-promotion, I started to dread the inevitable question that would come up at parties or events: \"So... what are you doing now?\" I couldn't really talk about _Decoded_ yet, but it had been almost three years since \"SWGSTBG,\" so I worried people would think I'd been sitting on the couch watching _Say Yes to the Dress_ reruns and starting Twitter wars instead of working. The former... not true. The latter? Embarrassingly so.\n\nFinally, we shot the first three episodes. In many ways, _Decoded_ was a culmination of everything I'd been doing all along. Working at Upworthy had taught me valuable lessons about how to present serious concepts like police brutality, immigration, and racist stereotypes in ways that won't intimidate people.  And now, instead of writing in a void, I had feedback and a team. Whereas for my own YouTube videos I'd have to spend hours editing to get certain effects just right, now I could tell an editor, \"Make a watermelon pop up next to my head!\" and then a watermelon would pop up next to my head! I wasn't used to having someone else give me direction, but I ended up really liking that aspect, too. It was surprisingly nice to say, \"Do you think this works?\" and have my producer Andrew reply, \"No... not at all.\"\n\nThe only snag was that at first I did my own makeup, and it looked absolutely terrible\u2014my producers on _Decoded_ are awesome, but I now know to never ask them if I \"look all right,\" because \"all right\" is a relative term, especially when you're talking to guys who can't tell an eyeliner pencil from a stick in the ground. I acknowledged my limitations and hired Delina Medhin, a makeup artist I'd worked with on another web series, and we've been basically inseparable ever since. I was learning how to be a producer\u2014having hard conversations about contracts, scheduling, rates, and vision that I'd never had to have before. _Decoded_ launched in June 2015, and it was a huge success\u2014our first episode was featured on sites like the _Huffington Post_ and _Adweek_ with glowing reviews. The comments were so positive that I couldn't believe what was happening\u2014it was like when you go into a caf\u00e9 or bar and you find a door that leads to a back garden and you're like, \"Wow, why is no one back here? This is great!\" But I also couldn't help feeling that the positivity wouldn't last. Though this could have actually been considered my \"big break,\" it didn't give me those delusions of grandeur I'd experienced in the past\u2014I was too busy.\n\nA few weeks later, Comedy Central announced that it was making a new nightly comedy show starring Larry Wilmore, the legendary producer and comedian whose r\u00e9sum\u00e9 includes _The Fresh Prince of Bel-Air_ , _The Bernie Mac Show_ , _The Office_ , _Blackish_ , and _Insecure_. The show would be called _The Minority Report_ , and would focus on stories around race, class, and gender. I called my agent and begged him to get me an audition. I was so excited when I got word that I'd be given a shot... until I learned the audition was a character showcase, just like the _SNL_ audition. And just like at the _SNL_ audition, I totally bombed. Later there was a call for video submissions; I didn't get it then, either. Finally, when I had become resigned to the fact that the newly renamed _Nightly Show with Larry Wilmore_ was not going to be my next big break, my MTV producer, Brendan, said he'd submitted my name as a guest for a roundtable discussion on the show, and they wanted me to come on.\n\nIt seemed too perfect. I would get to promote _Decoded_ and have the chance to show Larry Wilmore and his team what they were missing without having to deal with the awkwardness of another audition. \"Great,\" I said to Brendan. \"What's the topic?\"\n\n\"The Republican debate.\"\n\nOh, no, no, no. No! Although I'd been writing about social justice for years, I'd never considered myself knowledgeable about politics. Even in those early days of the 2016 election season, being knowledgeable about politics was a full-time job. Especially Republican politics, with Ted Cruz questioning the need for Gay Pride parades and Ben Carson regularly sleepwalking through debates. I freaked out instantaneously. I cried\u2014messy, hiccupping, embarrassing tears, the worst kind of tears to cry in front of your impressive and successful coworkers. I told Brendan there was no way I could converse halfway intelligently about Republican politics on TV. They would have to find someone else.\n\nBut Brendan wouldn't take no, no, no, no for an answer. \"Listen,\" he said. \"You have to do this.\" He promised to help me workshop some jokes beforehand and said he'd come to the taping. When he pointed out that Shonda Rhimes would also be a guest that night, I knew he was right: There was no way I could pass on this.\n\nJust as I did in the days following my Anderson Cooper appearance, I started studying like a senior whose only hope of graduating on time is getting a perfect score on the Intro Stats exam. I read up on the candidates (all eight hundred of them), watched all the previous debates, and took pages and pages of notes. I picked out each candidate's most ridiculous platform and came up with ways to skewer them.\n\nWhen the show finally taped, it was a _Scandal_ how good I was. (Get it? Because... Shonda Rhimes was there...) During a break, I saw Larry lean over to Jordan Carlos, whom I'd done videos with before, and overheard him say, \"Who is that girl?\" And Jordan replied, \"She's good, right?\" I had to restrain myself from doing a celebratory dance in the studio.\n\nFeeling like a grandma who's finally won the lottery after playing the same number for years, I left for a much-needed vacation with Patrick and some of our couple friends in Costa Rica. You won't believe what happened next.\n\nThe producers at Larry Wilmore contacted my agent and asked if I would come back on as a guest, so I left vacation early and hightailed it back to New York! Soon after that, they offered me a job as a writer and correspondent. I'd never worked in a writers' room before, making me the least-experienced member of the team. At my meeting with Larry, he asked me what I ultimately wanted to do with my career. I told him I wanted to sit in his chair someday. He said, \"I think you could do it.\"\n\nI quit my job at Upworthy and convinced my agents to work out a deal that allowed me to shoot _Decoded_ on the weekends. The prospect of starting a new job in television terrified me, but the thought of giving up my web series felt inconceivable. This was also the period when my biggest diva  moment took place: Picture me sitting perfectly still in the makeup chair arguing with my producers about how _Decoded_ wasn't going to suffer with my new job because \"I work my ASS off!\" while Delina is putting the finishing touches on my eyelashes. It's a little embarrassing to look back on it now, but it was one of the first times I had to put my foot down, which made me realize what it took to be a boss. Though this knowledge crystallized in a single moment, it was the culmination of years of mistakes, obstacles, successes, and Twitter beefs. I hadn't really had that kind of self-confidence before, and it felt good.\n\nI'd already started to reach a wider audience with the launch of _Decoded_. Although we didn't talk about politics in the Republican\/Democrat sense, we talked about identity and how institutions affect people. \"Obama's presidency didn't end racism\" isn't exactly a radical position, but the show where we discussed that topic in particular pissed people off nevertheless. And because people will always be nostalgic about MTV, connecting it to childhood and rebellion and edginess, fans resent any kind of change in programming like the one _Decoded_ represents. It was the first time I was being introduced to an audience who were not at all on my side but instead _really_ hated me\u2014to the point that they were obsessed with trying to take me down.\n\nI'd dealt with online harassment before (hello, chapter 2?), but this was a completely new level. Not only were viewers threatened by the concepts I was presenting, but they hated the idea that I was being paid and promoted by MTV. From outraged rhetorical questions like _Remember when MTV used to play music?_ to the suggestion that I could have nothing to say about injustice because I was a woman who went to private school, married a white guy, and had a \"cushy\" job with a TV network, people came at me from all sides. I was censoring them (how?) and to top it all off I was rich (I wish). I was a symbol of the corrupt corporate media. People started petitions trying to get me fired\u2014and to get me involuntarily committed to a mental institution (not kidding).\n\nBut it was all just a preview for what would happen when I started appearing on _The Nightly Show_. On top of the stress of producing and filming _Decoded_ , dealing with harassment every day, and starting a new job where I was the least-experienced person on the team, I realized that my previous impulse to avoid talking about politics on television was spot on. The election was ramping up, and we'd usually devote about half of each week's episodes to politics, which increasingly meant half of each week's episodes were devoted to Trump. The amount of background information required to even have a basic discussion of the issues was overwhelming.\n\nI'd endorsed Obama on my YouTube channel both times he ran, but I'd never followed the elections aggressively. Now, all of a sudden I'd become a prominent voice in politics, at a time when anything you said opened you up to vicious criticism from every other position you could imagine and a few you had no idea existed. I voted for Bernie, but I didn't really endorse a candidate in the Democratic primary because I knew I would get picked apart for it, and my stance was \"Whatever will keep Trump out of office is what we need to do.\" This seemed really reasonable to me. After Hillary got the nomination, I'd go on TV and say some version of: \"Look\u2014I know she's awkward and pandering and doesn't have the best record, and the _abuela_ thing was very weird, but someone has to be president so we should vote for her anyway.\"\n\nFinding your voice is one thing; using it is a whole 'nother animal. No matter what I said, viewers would become ENRAGED. I was SEXIST for suggesting that Hillary was not the heaven-sent conclusion of decades of feminist organizing and a perfect candidate for 2016! I was a COON for not condemning her husband's three-strikes crime bill and other atrocities against black people! I was RACIST for saying Donald Trump was unfit to be president\u2014clearly I had some unfair bias against orange people! Often viewers would get mad that I hadn't mentioned such-and-such policy from 2004 when I had actually mentioned it, but it had been edited out. (Though I'd experienced the evil power of editing on _Anderson Live_ , _The Nightly Show_ was much more intense, because each night the editors would have to cut a twenty-minute roundtable discussion down to just four minutes, and they'd have to do it on a tight deadline.) No matter what I did, I couldn't stop the onslaught of harassment and anger that followed every single one of my appearances.\n\nAll the criticism was funneled through social media, which I was using all the time, during breaks or in the dressing room. I knew I should just ignore the trolls\u2014you may notice that this is typical of me\u2014but often the comments were so vicious that I felt I had to defend myself, or I at least wanted the catharsis of having fully explained my position. Though I had great relationships with everyone I worked with, I secretly cried in the bathroom pretty much every day. No one else was particularly engaged on social media, so they weren't targeted like I was. I saw it as part of my job to respond to messages that used the _Nightly Show_ hashtag\u2014sometimes someone would say, \"I liked that new black girl, who is she?\" and I'd reply, \"Me!!!!! I'm the new black girl!!\" which I was sure strengthened my brand.\n\nWhen I went to the other writers for advice, they would say something like \"That sucks, don't let it get to you!\"\u2014but of course I couldn't not let it get to me. Had that ever worked for me in the past? I'd always wanted to demonstrate the importance of dialogue, but I was giving way too much to \"the conversation.\" It was like I was living in two worlds\u2014the one where I was stressed but thriving in my new job that I'd worked so hard for, and the one where literally everyone in the world thought I was a \"man-hating reverse racist Hillary Clinton shill.\"\n\nOne night, deep into the election, I cracked. Every time a staff member appeared on a panel or did a piece for the show, Larry would meet with us afterward to discuss our performance and where we thought we could improve. That night, I came off the taping feeling like I hadn't represented myself well\u2014I didn't get a chance to say something, or I'd fumbled a point, or something, I don't remember. When Larry asked me how I thought I did, I started crying. \"Please edit around me,\" I said, sniffling. \"Just\u2014can you cut me out of the panel completely?\"\n\nNaturally, Larry was pretty confused\u2014I may have stumbled over the phrase \"national security,\" but that's not the type of thing worth sobbing in front of your boss for. \"I'm just so tired,\" I said. \"Every time I go on the show, people attack me and say I'm the worst person ever and that they hate me and I'm the reason they're voting for Donald Trump.\" (Imagine me saying this in that crying voice you do when you're ashamed to be crying and don't want to have to stop for breath because you'll get snot everywhere if you do, so you say everything really fast.)\n\nHe said I had a voice and was on the show for a reason. I inspired people\u2014maybe I didn't always inspire them to be nice and good, but I inspired them to say something, and that was powerful.\n\nI knew what he was saying, but I also felt helpless\u2014there was nothing I could do to stop it. If I got off social media, I'd be giving the trolls what they wanted\u2014to drive me off the internet and away from my goals.\n\nAs I was sitting there looking pretty pathetic, Larry added, \"Sometimes you're really clever on Twitter, but you give these people way too much attention.\"\n\nIf I had been embarrassed to be looking like an unhinged baby in front of my boss, the realization that he had seen me look like an unhinged baby online made it that much worse. Of course Larry followed me on Twitter. (He'd retweeted me a couple of times\u2014NBD.) I felt stupid for not thinking of it before, but it wasn't until that moment that I became fully aware that he\u2014and anyone else who might want to hire me\u2014could see me going back and forth with assholes all day on social media. How many people had looked at my feed and thought, _Hmm... better not hire her\u2014she's a loose cannon online_?\n\nAnd then I thought, _Fuck\u2014I respond to tweets at work._\n\nBefore I took the job at _The Nightly Show_ , I felt like I was destined to be the show's social justice secret weapon. It needed me. But as soon as I got into the writers' room, I couldn't shake the sense that I was in way over my head\u2014the pace was so fast and everyone was so funny. Next to my coworkers, who had more traditional comedy backgrounds and would throw around references I couldn't place, I knew I was a risk; whenever other YouTube or internet celebrities would come on the show, if they didn't knock it out of the park, people on staff would make comments about how the internet was a cesspool of untalented wannabes.\n\nThese comments were never directed at me, and I'm sure the people making them weren't thinking about me when they made them, but it still made me feel even more like an imposter in their midst. \"You do one viral video and you think you're a star\" was a common annoyed refrain\u2014uttered whenever the news was focused on the next Vine sensation\u2014and I knew it described me perfectly. So when I realized Larry, and anyone else, could see me arguing with trolls on Twitter, I knew I must have seemed like a delusional viral star who couldn't hack it on \"real TV.\" From a pure workplace-conduct standpoint, I also must have looked like I was wasting a lot of time. I'd grown so accustomed to multitasking that it'd become part of my process to loop through email, Facebook, Twitter, and Instagram while working, but to someone who's not active online, I knew I must have looked obsessed with social media. Sitting in Larry's office crying, I saw that I had brought this on myself.\n\n_The Nightly Show_ was canceled a couple of months before the election, and although I was spared the inevitable drama of covering the _Access Hollywood_ tape or the resurgence of Her Emails, I got really depressed. I felt like Trump had emboldened the kind of trolls and racists who made entire careers out of harassing me (and since then, research has proven me right), and I had spent almost my entire time on the show feeding their negative energy. After the election, I had to leave social media altogether for about a month. On top of constant trolling, people would say things like \"I hope you're happy\u2014I voted for Trump and became a racist because of you.\" As if racism were a restaurant they'd been thinking of trying.\n\nBoth _Decoded_ and _The Nightly Show_ were huge steps in my career, not least because they showed me what happens when marginalized voices insist on going public. Even if you come in this package\u2014a smiling, conventionally attractive black person who doesn't curse (at least not in her videos) or present \"radical\" viewpoints\u2014you can't convince people who are committed to upholding oppression that you are anything but their worst nightmare. I will always be a radical leftist, the most racist person on TV, a poisoner of our youth, and a \"black supremacist\" (whatever that means). If anything, these bizarre perceptions of me are the exact reason I need to keep using my voice. These trolls aren't the people I need to reach\u2014the people who could be influenced by that garbage are the ones I want to connect with.\n\nWhen Trevor Noah interviewed Tomi Lahren, I thought he went really light on her. (Okay, I know what you're thinking: _A little hypocritical from someone who just spent several paragraphs ranting about how she can't please anyone?_ But it's not like I tweeted at him and said he had betrayed all black people.) Still, when she started ranting about Black Lives Matter and Colin Kaepernick, he pulled an artfully subtle question out of his back pocket: \"What is the right way to protest?\"\n\nIt was perfect because she couldn't answer. You wear a T-shirt that says _BLM_ and people get mad. You refuse to move your seat on the bus and people get mad. You do a sit-in and people get mad. No matter what you do, people who are dedicated to seeing marginalized people as second-class citizens will be upset when marginalized people insist that they are not second-class citizens.\n\nWhile this is more than discouraging, the realization is also kind of liberating. It made me realize I should quit trying to please everyone and just say whatever the hell I want. Going public doesn't mean that the public gets to define you. Having an audience and a public platform has changed how I see the world\u2014how could it not?\u2014but I'm still the same person on TV as I am riding the subway, or waiting in line at the DMV, or locked out of my apartment in my bright orange muumuu\u2014just with better makeup.\n\n# CHAPTER SIX\n\n# LESSONS FROM GOING PUBLIC, PART 2\n\n## Objects on Social Media Are Not as Close as They Appear\n\nThere are many reasons to be jealous of Beyonc\u00e9. Her voice. Her career. The way she fills out a wide variety of leotards. But the biggest one, for me, is that there's so much we don't know about her life. And despite the illusions provided by gossip magazines and close textual analyses of her song lyrics, we'll likely never know more than the Queen wants us to.\n\nThe internet has given me a lot\u2014I would have a completely different career without it. Posting videos on YouTube allowed me to sidestep a lot of barriers to entry in the entertainment business and get where I am today. But even though we've never met, many of my fans feel like they have a personal connection to me, and that has come with some unforeseen consequences. Don't get me wrong\u2014I love my fans. They're great, funny, generous, smart, and engaged people. (Waves at beautiful person holding this book.) But I wish I'd known how the internet would change how we understand fame and success when I started posting videos back in 2006. I would have done some things differently. Or at least mentally prepared myself for the effects of going public.\n\nSome of my audience has been following me since I was living with three roommates in Miami. They watched along when I moved in with Pat, when I got engaged in Paris, when I moved to New York, when I changed jobs. I used to shoot a lot of YouTube videos in my bathroom, because the lighting was great, and apparently fans got attached to the floral shower curtain I had at the time. I have since moved on to a more subtle, less Target Back-to-College look, but I still occasionally get messages saying, \"I miss that floral shower curtain!\"\n\nThink about your friends, family members, and coworkers. Would you recognize their shower curtains in a lineup? Would you recognize their _former_ shower curtains in a lineup? I'm going to guess no. It's a little weird.\n\nBut it's also completely understandable. Social media has the opposite effect of that little warning on car side-view mirrors: Objects are not as close as they appear.\n\nAnother illustrative example: I once made a video with my flat-screen TV in the background, and trolls descended on the supposed hypocrisy of my being \"rich.\" How dare I comment on social justice issues? How could I know what I was talking about if I had such a nice TV? What the video didn't show was that Patrick had woken up at five a.m. on Black Friday to buy that TV on deep discount, and the reason it was visible in that shot was because I had no other place to film\u2014we were dealing with a relentless bedbug infestation at the time, and the rest of the apartment was covered in plastic.\n\nI have to remind myself that I engineered this: I wanted people to pay attention to me! I showed everyone my shower curtain! But I didn't think about the implications of the shower curtain until strangers started emailing me about it at four a.m. Even accidentally sharing the tiniest details makes people feel like they know you\u2014especially when \"What's in my bag?\" and closet tours are rites of YouTube passage.\n\nMeanwhile, I have no idea what Beyonc\u00e9's bathroom looks like, and no matter how many people leave pleading comments on her occasional Instagram posts, I don't think she'll ever tell us. On some level, it doesn't matter. _Lemonade_ would still be one of the most groundbreaking visual albums of all time if Beyonc\u00e9 posted selfies at a Kylie Jenner pace. But I'm sure our perception of her as an artist and person would be totally different.\n\nWhen I used to fantasize about becoming a star, I only thought about the luxurious hotel rooms and glamorous outfits and invitations to exclusive events. Whenever I envisioned the fans, they would be adoring me, of course, but only when and how I wanted. Our interactions would always be gracious and pleasant, never uncomfortable or threatening. I imagined that if being recognized in public was ever annoying, it would be the kind of annoying where you secretly felt really smug about it.\n\nThat may be what it's like for Beyonc\u00e9, who has to incorporate privacy into her life much more intensely than I do. The reality for an \"internet famous\" person like me, however, is much more awkward. Since I am currently kinda-sorta famous\u2014to some people, depending on their interests and media habits\u2014I have a good amount of privacy, but people also think they are entitled to me in some way. I recently joined a private gym because people kept trying to take pictures with me when I was sweaty and gross; once, I was topless in the locker room (which was already a big deal for me) when I heard a tentative voice echo from the showers: \"Excuse me... are you... Franchesca?\"\n\nOn one hand, it's totally inappropriate to try to strike up a conversation with any naked stranger in the gym locker room. Like, don't ever do that. There is no scenario in which that is okay. But on the other hand, I knew I had made myself seem open and approachable online, so people assume I'm always open and approachable. When I shared that I'd moved into a new place, fans asked me to give an apartment tour; I had to draw a line. I wasn't going to give strangers access to my personal space this time around. (I'm not trying to get robbed, and besides, I really hate the rug Patrick picked out.)\n\nSimilarly, I recently decided to quit Snapchat because an acquaintance came up to me at an event and said, \"Girl, I saw you crying on Snapchat, are you okay?\" Then someone sent me a message asking, \"Did you and Patrick break up? You haven't posted a picture of him in nine weeks!\" I'd find myself in the middle of a dinner, enjoying a robust conversation about the top five flavors of LaCroix or savoring a plate of fried chicken and mushroom-rosemary waffles (sounds weird, but I'm team sweet-and-meat 4EVA), and suddenly I'd think, _Shit\u2014I've got to capture this thing for the internet! Otherwise they won't know I did it!_ I was so accustomed to giving so much of myself to strangers that I was starting to feel like I owed them something.\n\nI think this is a symptom of a double standard for internet celebrities: The fans feel like they've given you your career\u2014maybe they've been watching your videos for years\u2014and so they expect something back. No matter that your work should be \"repayment\" enough\u2014I occasionally still get emails along the lines of \"I shared your video, so why won't you answer my thirty-paragraph email about how to make it as a YouTuber?\"\n\nI wish I could say I just wave all this off and move on, but I haven't been able to yet. I felt really guilty about quitting Snapchat, which is a pretty ridiculous sentence if you think about it for two seconds. I had a great connection with my audience there. But the vulnerability that made that connection so real and meaningful for me also opened me up to judgment and conspiracy theories I wasn't prepared for. Participating in social media had started to feel like I was putting myself in front of the paparazzi and telling the gossip magazines to have at it. Although I was ostensibly posting and commenting and checking my feeds on behalf of my fans, all the time I devoted there meant, paradoxically, that I had less energy for the creative projects those fans support me for. Having a huge following doesn't necessarily mean you have talent, or deserve your platform, and I let myself forget that for a little while. Maintaining a productive, rewarding relationship with fans doesn't require social media addiction\u2014it calls for doing good work that people respond to.\n\nBeyond social media, my ambiguous level of fame has led to some weird encounters in person as well. Occasionally I'll be walking down the street, my headphones blasting, and notice some guy staring at me. My instinct, of course, is to put on my active bitch face (as opposed to my resting bitch face) and shoot him a _Don't even think about it_ glare. Then he'll shyly approach and say, \"Are you Franchesca Ramsey? I loved you on _The Nightly Show_!\"\n\nI don't want to discount all the support\u2014both emotional and practical\u2014my viewers give me every day. From sharing articles and research studies, to pointing out where I may have missed the mark on something, to inspiring me to cover certain topics or issues or offering words of encouragement after my teary Snapchats, I really cherish my relationship with my audience. It's symbiotic. After the video I made about my sexual assault went viral, a fan came up to me on the subway and said, \"Oh my God, you're Franchesca Ramsey! I saw that video you made about when you were raped! Thank you so much for making it! You're so brave!\" Though I felt uncomfortable that the entire subway car now knew I was a survivor, I was happy my story had moved her, and I got off at my stop and went on with my day. People often stop me and say, \"You're like a big sister,\" or \"I feel like you're my friend,\" which, since I'm an only child who used to struggle in the friend department, warms my heart.\n\nThe speaking gigs I have at schools and universities also keep me grounded. My first was at a local career day in Miami when I was twenty-five, and although I didn't have an agent or a job in the entertainment industry yet\u2014I didn't even live in New York\u2014it was the first time I saw that I had advice and inspiration to offer other people nevertheless. I talked about making YouTube videos as a jobby and creative work more generally, and seeing how the kids responded to my PowerPoint jokes about homework and Miley Cyrus (this was before her 420\/twerking phase) made me realize my perspective was valuable even though I wasn't yet the household name I wanted to be.\n\nToday, I regularly crisscross the country for the same kind of speaking gigs, and though I sometimes grumble about seven a.m. flights to freezing Minnesota, I think these are some of the most important parts of my work, not least because they force me to get out of the house, experience the wonder of changing out of my beloved muumuus, and talk to real humans. While people will sometimes tweet about my speeches afterward\u2014in many ways, this feedback has inspired this book, because it's shown me what my audience cares about\u2014live presentations are centered on interacting with the audience in real time, gaining that precious IRL experience that's absent when I get absorbed in Instagram notifications.\n\nAnd thankfully, that IRL experience is almost always positive. Though I know some people have to imagine the audience in their underwear, I don't get nervous before public speaking like I do for auditions or even while making videos\u2014when you're giving a speech, the audience wants you to succeed. (Otherwise, they have to sit through an excruciating talk.) As great as YouTube is, the engagement there is totally different, and especially so for me since my comments have been overrun by racist trolls. People who like my videos don't even want to jump in; they get harassed for weeks, and even months, if they dare utter, \"Great vid!\" So talking to a room of five hundred people and getting honest reactions and questions during and after, even when those comments are critical, is gratifying. There's a certain amount of generosity people have in person that disappears online, and I've learned that through experience. For example: In addition to the occasional brain fart, I've also had one real fart during a talk. I genuinely can't remember what school I was at\u2014my brain has fuzzed the details in an effort to protect me from recurring humiliation\u2014but one of my jokes is about how \"Google is free twenty-four hours a day!\" During this particular talk, I thought it would be funny to spin around in a circle for dramatic effect. While I was spinning a fart slipped out. I'm not sure if people were laughing at the spin, the fart, or both, but I swear it felt like my ass had been handed a megaphone; it was not one of those dainty accidental squeaks. It was toward the end of my speech, so I didn't acknowledge it and neither did anyone else, for which I'm very grateful.\n\nSo like I said, this kind of thing keeps me grounded. For every diva moment, or time I'm recognized on the street, there's some other time I fall flat on my face, exposing myself to be a victim of the same celebrity culture that means I have to pay a hundred dollars more per month to go to the gym. (Yes\u2014New York is expensive.) The best example of this was when I met up with Tracee Ellis Ross\u2014okay, I'm name-dropping, but I promise there's a point!\u2014in LA to shoot a video for my channel with her. We were filming at a hotel, and when we wrapped up we went out for lunch. As we were leaving the restaurant, a handsome black guy with locs came up to us and gushed, \"I just wanna tell you: I'm such a fan.\"\n\nMaybe it was his locs, or maybe you can chalk up my smugness to being in LA and hanging with one of my favorite actresses, but instantly my face lit up. \"Oh, thank you so much!\" I said, going into my Gracious Celebrity mode. I may have flipped my hair over my shoulder. \"What's your name?\"\n\nHe stared at me for a second before angling his body away from the embarrassing moment I'd just created. \"Thanks so much,\" Tracee said. \"It was great to meet you!\" he said, and quickly walked off. He was\u2014obviously\u2014talking to the network television star standing next to me.\n\n# CHAPTER SEVEN\n\n# GUESS WHO'S COMING TO DINNER? (IT'S LENA DUNHAM.)\n\nLike many viewers, when HBO's series _Girls_ debuted, I quickly began to treat Hannah Horvath and Lena Dunham, the actress who played her, as well as the show's creator, as one and the same. From her tone-deaf blunders to the way she was constantly disrobing onscreen, Hannah was ignorant, narcissistic, and just plain obnoxious, and she infuriated me. I responded by trashing the show in a series of TV recaps I made for YouTube with my best friend, De'Lon. _Does the world need yet another show about rich white girls in New York who don't take the subway, live in giant apartments, and have lots of sex? The answer is no. We already had_ Sex and the City _, which was infinitely better in terms of both writing and outfits._\n\nBecause Lena Dunham was the showrunner, creator, and star of _Girls_ \u2014and because she seemed unable to resist making dumb comments in interviews and all over social media\u2014a lot of my criticism focused on her. While many people were vocally annoyed by Dunham and her work on _Girls_ , I was _really_ annoyed. Of course, I could have spent the time and energy I wasted hate-watching and then panning this show doing productive or valuable things, but instead I tuned in every week so that I could make myself feel better about having a less successful acting career than Lena. Some people do really in-depth, nuanced TV recaps; this was not what De'Lon and I were making. Sure, we discussed the lack of diversity on the show, and dissected Lena's repeated missteps when it came to race beyond the series. But overall, our criticisms were snarky and mean. Although we never called Lena fat outright, we would happily skirt the line and say shaming things like, \"Why can't she keep her clothes on?\" and \"Girl, that dress does not fit, and you look like a mess.\"\n\nI didn't think about it at the time, but the old maxim about how there's no such thing as bad press really applies here: We were basically promoting the show. People would comment that they'd never heard of _Girls_ before they saw my videos, but our reviews were so hilarious and biting that their interests were piqued. Could the show really be as bad as we claimed? We assured them it was, but my subscribers set their DVRs to find out for themselves anyway. Still, we soldiered on, as if roasting a person that everyone else was already roasting were our civic duty.\n\nThe beginning of the second season of _Girls_ was when we really hit our stride, thanks to a little help from Lena's terrible PR responses. After a totally white season one, Lena and her co-showrunner, Jenni Konner, had responded to extensive criticism of the show's lack of diversity by plopping Donald Glover in as a walk-on love interest at the start of season two. Unfortunately, his character was a Republican who didn't think Hannah's writing was good, so she dumped him, meaning we saw him in all his shirtless handsome glory for only two measly episodes. I'll admit, I was jealous of those sex scenes, which resulted in me taking subtle digs at Lena with comments like, \"In what world would Donald Glover ever sleep with Lena Dunham?\" As if he would ever slide into _my_ DMs.\n\nLena's track record on discussing race wasn't stellar pre- _Girls_. She'd gotten flack for saying she had \"yellowish fever,\" along with making a number of infantilizing comments about Asian women, in a travel essay about Japan. She'd donned a fake burka on her Instagram and bragged about being considered \"thin in Detroit.\" (After we stopped doing recaps, she endured other controversies related to her book that I won't get into.) Suffice to say she was \"embattled\" in the media, which I thought gave me a free pass to say whatever I wanted about her. And while I didn't mince words, my rationale was \"The truth hurts, and it's funny as hell,\" so I failed to see a problem.\n\nAround this time, an internet friend I'd become close with told me she was coming to New York and invited me to a dinner she was having. We'd never met in person, so I excitedly agreed. There would be a group of people at the dinner, which meant it would probably be one of those ambiguous work\/friends things where you get to network without feeling (too) sleazy.\n\nUntil the day approached, I didn't think too much about it. But then, a few hours before we were supposed to meet, I got a text from my friend explaining that the restaurant was cash only\u2014and she added that it was \"Lena's favorite place.\" Her comment was so casual that I didn't want to out myself as clueless by asking, \"Lena who?\" Instead, I did some internet sleuthing (combing through Twitter mentions and tagged Instagram photos) and deduced she must have meant Lena Dunham. Which meant that Lena Dunham would be in attendance that evening. _Shit._\n\nI started to panic. Would my friend have told her who would be at the dinner in advance? Would Lena look me up? And if she did, was she the kind of person who would Google \"Franchesca Ramsey + Lena Dunham\" just to see if anything came back? I tried to tell myself that I was overstating my importance to Lena Dunham and that there was no way she would come into this dinner looking for a fight. _She's, like, an actual famous person_ , I reasoned. _She doesn't have time to Google random plebes she's having dinner with. Does she???_\n\nAt the restaurant, the scene was much more intimate than I expected\u2014not a huge group dinner like your most popular friend's birthday. We were all sitting in a booth. Naturally, because that's how the universe works, I ended up sitting next to Lena.\n\nI felt mortified, but she didn't seem to notice\u2014she didn't know who I was. She was so nice, encouraging, and curious; nothing about her was \"catty,\" or disengaged, or straight-up mean. She asked me what I did, and we got to talking about the internet. I had been dealing with a lot of harassment on social media at the time, and we bonded over how awful people can be\u2014you know, classic girl talk. She showed me her Twitter mentions, which were a whole 'nother level, like nothing I'd ever seen or experienced. A constant stream of \"I wish you'd get raped\" and \"I wish someone would murder you\"\u2014and she hadn't even said or done anything that day! Though there was some genuine, well-meaning critique mixed in, it was pretty much drowned out by the awful name-calling. Sometimes in the exact same tweet.\n\nI remembered vividly how I felt after my _Anderson Live_ appearance, when hordes of strangers descended on all my public accounts to attack me from every side. My stomach sank. If she had as good a time as I did, she would probably go home and look me up, and she would think I was a two-faced jerk. I couldn't leave without coming clean. \"Listen,\" I said, \"can I be totally honest with you?\" I took a deep breath. \"I've never been a fan of yours.\"\n\nThis is not usually the kind of thing you hear from someone you just met\u2014or, really, from anyone\u2014so I imagine she was a little confused, though maybe she gets it all the time. I went on. \"I mean, I've been pretty vocal about not liking you. But you seem cool, and I don't feel good about some of the things I've said, so I feel like I need to be honest with you. If you go online and look me up, you'll probably find me talking about how much _Girls_ sucks.\"\n\nShe was gracious\u2014I mean, as much as you could be in that situation. She thanked me for telling her and said she understood the criticisms, that she was really trying to expand her worldview and understand her privilege. We exchanged numbers and left on a good note.\n\nThen, later that night, she texted me something along the lines of \"Wow. So... you really hate me! You hate me a lot!\"\n\nI felt so bad. I explained that I didn't hate her, but I never thought I'd meet her, and when you don't have to think about the real human behind the celebrity gossip, it's easy to get caught up in the drama. Then, because I felt like I'd learned an important lesson that night, I asked Lena if she would be okay with me writing about the experience for my Tumblr, and she agreed.\n\nWriting about it forced me to ask myself why I had such deep problems with this woman. Though some of my issues with her work were valid, I had to admit that many of my critiques were rooted in my own insecurities about my career\u2014if I were being honest. I'd spent most of 2012 developing a pitch for HBO that was ultimately rejected as \"too network.\" I'd wondered if that meant \"not enough boobs.\" Personally, I had always been self-conscious about my body\u2014I no doubt would've sobbed my way through an on-screen second base, much less the scene where Adam pees on Hannah in the shower. (Ew.) It was easier to dismiss her work\u2014which has had positive effects, like hugely advancing the body-positivity movement and awareness of reproductive rights\u2014than to say I was intimidated and a little jealous.\n\nThe thesis of my Tumblr post boiled down to: If you participate in the shit-slinging contest, competing to come up with the most creative insult, you end up covered in shit. I didn't want to excuse the real problems Lena had, but I'd let bigotry get in the way of my criticisms, and it wasn't something I felt good about. I topped off my post with a photo of me and Lena at the restaurant and waited for everyone to appreciate how wise and mature I was being. Once again, I was wrong. \"This sounds like extreme white apologism and it kinda makes my stomach turn crazy,\" one Tumblr user responded. \"You're in with whites and now have some social capital and you're telling us how to approach it? I'm sorry this whole writing just made you look like a complete coward.\"\n\n\"Racist white person knows you don't like her so to seem less racist, she acts a little less pretentious because she knows you're going to blog about her. And we're supposed to give her and other racist celebrities the benefit of the doubt and speak nicely because??\" wrote another.\n\n\"Chescaleigh, girl... Like I know we all gotta play the game sometimes to get what we want but us regular Blacks don't have time or energy to speak or play nice.\"\n\nI had told an award\u2013winning actress, to her face, that I didn't like her, which I'm pretty sure is the exact opposite of cowardice. Not to mention I'd ordered the most inexpensive item on the menu at dinner at a fairly fancy restaurant, which felt very \"regular\" to me. Whatever that meant. It's not like I had written, \"Anyone who calls out Lena Dunham for anything is a despicable human who should feel ashamed. #TeamLena!\"\n\nYet I was accused of sucking up to her so I would get a part on _Girls_ , while other critics went more the \"Speak for yourself, I was _never_ jealous of Lena\" route. Never mind that I _was_ speaking for myself\u2014I thought the frequent use of the word \"I\" would signify that. Still others thought I had \"betrayed\" them\u2014but could they really be surprised, given that I went to Catholic school, had a white husband, and screwed up my appearance on _Anderson Live_? I was, in their eyes, \"canceled,\" never to be taken seriously again.\n\nI'll stand by it: I don't think I did anything wrong with my post. I absolutely think Lena\u2014like anyone else\u2014should continue to be called out for the ignorant things she says and does. (Remember when she put all kinds of sexist words in Odell Beckham Jr.'s mouth because he didn't hit on her at the Met Gala? Come on!) I get why people dislike Lena Dunham; her daily mantra should be, \"Did anyone ask for your opinion?\" She desperately needs a private journal. I even had some anxiety about telling this story because I don't want it to seem like I'm defending her. But her textbook white feminism doesn't mean I should sacrifice my morals to criticize her. There's no value in peppering critique with stigmatizing personal attacks. When you add, \"P.S. You're fat,\" to a criticism about the way a person has handled\u2014or ignored\u2014race on her television show, for example, you negate the power of the original idea. It's like adding vinegar to a perfectly good birthday cake. (Let's ignore for a minute that there's nothing wrong with being fat, because we all know the pain that postscript is meant to inflict.) The person on the receiving end will only hear the personal attack, and will probably write off the legitimate criticism as part of a cruel attempt to insult and humiliate. On top of that, anyone else in the world who may identify with the personal attack is going to feel bad.\n\nUltimately, this story isn't about Lena Dunham. It's about holding yourself to a high standard no matter how terrible the person you're calling out may be. I'm not saying that everyone deserves the benefit of the doubt\u2014not at all. Nor do I think callouts should revolve around the feelings of the person on the other end. But being held accountable for how your words or actions were intentionally or unintentionally oppressive is uncomfortable enough without devolving into a cheap Fashion Police aftershow. Bigotry has no place in our callouts, but we see it show up all the time. When someone makes a homophobic comment, some argue, \"I'm sure he's overcompensating because he's gay.\" When a woman has really bad politics, people will say, \"She looks like a man in a skirt.\" I'd rather focus my energy highlighting what someone said or what they did\u2014remembering, all the while, that there are consequences when your criticisms of horrible actions are draped in the bigotry you're trying to speak out against. (And I know all too well how it feels to discover an entire thread on a forum dedicated to debating whether or not you have an ass. I do squats!)\n\nIt's really hard to take the high road when you're calling out someone who's truly awful. A good example of this is Donald Trump's allegedly small penis, which many people liked to mock in the months leading up to the election, before he had the nuclear codes. Let me put it plainly: Fuck Donald Trump and his rampant racism, sexism, xenophobia, Nazi apologism, climate-change denial, and unending desire to send our country and the entire world straight to hell in a handbasket decorated with a Confederate-flag pattern. Seriously. But the possibility of him having a tiny penis has nothing to do with his bigotry. We already live in a world that constantly links penis size to wealth, success, and sexual prowess when none of those things are related.\n\nI once read an article that made the point that when you call out a racist family member over Thanksgiving dinner, your words aren't aimed solely at that family member. They're aimed at everyone else at the table, who may benefit from hearing what you have to say, as well. I'd say the same goes for all of us when we call someone out online. Who's at the table? Maybe someone who's struggling to accept the body they were given will stumble on your insensitive joke and miss the bigger message. And you never know who might be invited to dinner next.\n\n# CHAPTER EIGHT\n\n# STOP HATING AND START STUDYING\n\nI'm extremely flattered\/slightly embarrassed when people describe me as a \"super-woke black feminist queen.\" \u2020\u2021\u00a7 But as often as I'm praised for the conversations my content has sparked online, I'm far from everyone's cup of tea. A quick Google search of my name will bring up my YouTube channel, episodes of _Decoded_ , and clips from _The Nightly Show_ , but as you scroll farther and farther, the results get weirder and weirder. Petitions to get me fired or involuntarily committed. Supercuts of weird faces I've made in videos, video game walk-throughs in which players attempt to kill characters they think look like me, and shitty memes calling me some variation of a \"racist cunt.\" There are videos in which men\u2014it's usually men\u2014rant for nearly an hour about how I don't understand what \"political correctness\" means. Some of these videos have hundreds of thousands or even millions of views; since I've gone viral myself, I know that equals a nice paycheck. People have made entire careers out of hating me.\n\nWhen friends or colleagues stumble onto this treasure trove of hateful content, the first thing they ask is how I handle it. I don't tell them how often I cry to my husband or how I set up filters for racial slurs across my email and social media. Instead I give them the short answer: I remind myself that the types of people who invest time in hating me are largely fueled by racism and sexism, with a heaping dose of jealousy.\n\nThe racism and sexism are pretty obvious with the folks throwing \"black supremacist cunt\" around. But how am I so sure jealousy is to blame for the rest? Well, I used to be a hater myself. [ _Cue dramatic organ music._ ]\n\nNow, before you go chucking this book out a window, feeling a deep sense of betrayal and disappointment: Please don't litter, and please allow me to explain.\n\nLong before I moved to New York, I followed a Manhattan socialite and new-media personality whose career I not-so-secretly envied and whose personal life I found not-so-secretly fascinating. Let's call her Stephanie. Stephanie had rich parents, a cushy job at a gossip magazine, a marginally successful pop culture web series, a pilot in development, and a couples' blog with her boyfriend that was a total disaster. She went out a lot and wrote an advice column in which she'd tell people things like \"If you want to save money, stop going to the Hamptons!\" She inspired in me a precise combination of aspiration and disgust, and I was nothing short of obsessed with her. While some people flock to trashy reality shows for their judgmental voyeuristic kicks, I was refreshing Tumblr during work hours, wondering what Stephanie was up to.\n\nWhen I first discovered Stephanie, I was living in Miami, working as a graphic designer and only dabbling in the entertainment industry through my YouTube videos and cringeworthy stand-up sets\u2014let's just say my closer was a JonBen\u00e9t Ramsey joke... and it brought the house down every time. I had a lot of time to follow this woman's life. On her Tumblr, she'd amassed a moderate following, as well as an active commenting community filled with fans, spectators, and haters. I drifted among all three camps, but I skewed toward the hater category. I imagined she'd had everything handed to her, and it really irritated me. Why couldn't someone hand everything to me?\n\nAfter a few months of enduring the public's vicious analysis of her posts, Stephanie closed her comments section. This act of self-preservation galvanized us lovers\/haters to take our fixation to the next level. She had decided to prevent us from interacting with her directly... but that kind of felt like she had given us permission to go off into the wild frontier and drag her in ways we might have been hesitant about if we knew she was going to see it. We all packed up and started a new site dedicated to gossiping and theorizing about whatever Stephanie posted. Although I'd been living with my serious boyfriend for years, we affectionately called ourselves \"pathetic virgins\"\u2014because only lonely ladies who've never gotten laid would spend so much time gossiping about a complete stranger online. Look at us being all self-deprecating and totally not weird! Cool, huh?\n\nIn an effort to set ourselves apart from Stephanie's \"comment section dictatorship,\" we decided nothing in the way of snark was off-limits on our site. The only rule was \"Don't poke the beast,\" which meant that we agreed not to contact Stephanie or taunt her into reading our posts. (But it was totally acceptable to call her a beast. Sheesh.) The site was supposed to be our space\u2014and while no one held back with the insults, we agreed that flinging them directly at Stephanie was a line that shouldn't be crossed. It was entertaining at first, like we were a panel of judges on a reality TV show, and even though I never said anything quite as awful as some of the other members, I enjoyed watching it all unfold. It felt righteous and cathartic to shit-talk this rich media darling while I was struggling to get a job and pay rent.\n\nIf this sounds \"so high school,\" that's because high school is where you learn this kind of behavior. Around ninth grade, right when I started feeling insecure about my body and lack of popularity, I discovered I was funny, and I started using my humor as a shield. I was particularly self-conscious about my lopsided boobs (which I now know are totally normal), so I attempted to conceal them by wearing a thick orange foam bra insert. I stressed about it constantly. I assumed everyone could see the orange cutlet, and I would bolt indoors at the slightest sign of rain for fear that my foam boob would take on water like the _Titanic_ and reveal my secret. Cracking jokes at the expense of others became an easy way to gain friends and divert attention away from myself and my own insecurities. I remember once helping spread a rumor that a girl in my class had an extra toe, which resulted in an endless stream of bad foot puns and jokes about a supposed \"sixth sense.\" I went on the attack instead of showing compassion for a classmate who may or may not have had an extra digit; I distracted from my own body-image issues by starting rumors about someone else's. Although I had never met Stephanie and she had never done anything to me, I was responding to her in a similar way: I was insecure about my career, so I took it out on someone whose career reminded me of all the ways I wasn't succeeding. You could say I was using my powers for evil.\n\nWe spent so much time on the pathetic virgins forum that we soon began to discuss topics beyond Stephanie's awful New Year's Eve party outfit and whether her parents were still paying her rent. We had conversations about books, music, food, life. Soon our group started to feel like a dinner party filled with old friends. I had finally moved to New York, and while I was struggling to meet people and figure out how to pursue my dreams in real life, it felt good to have a place to vent and crack jokes among like-minded strangers.\n\nDuring one particularly depressing stretch, I found the nerve to share with the group that I was going through an especially hard time and struggling. I'd talked to my boyfriend and best friend about it at the time, but I didn't feel like I was getting the support I needed. (Then again, I didn't really know what I needed.) When I posted about my feelings on the site, members leaped into action, sharing their stories and offering encouragement. One woman messaged me privately and when we met up for coffee she didn't even blink when I cried on her shoulder. I even picked up freelance graphic design work from a few members after complaining about my degree collecting dust while I folded clothes at Anthropologie.\n\nSo it wasn't all bad. But mostly it was. It felt good to be part of a \"club\"\u2014I imagine a lot of online trolls feel this way, too\u2014so I excused any particularly egregious, offensive comments (about how Stephanie was fat or \"looked like a man\") in exchange for my membership. I tried to focus on the few and far-between positives. When the site moderators announced they would host a meetup in New York, I decided to go.\n\nMaking the leap from online to offline friendship is always risky, but with a few positive pathetic virgin encounters under my belt, I thought, _What's the worst that could happen?_ I made the hour-long trek from Queens to a dive bar in Brooklyn and discovered I'd joined a _Mean Girls_ \u2013style clique. If you've ever made a friend online, you understand the intensity of that kind of friendship\u2014it's no less \"real\" than a friendship with someone you met at school or work. I assumed that my fellow PVs would be less snarky in real life and more like the kinds of people I'd want to hang with on the regular.\n\nBut the whole thing felt blatantly uncool. I had assumed Stephanie would be the unspoken elephant\/beast in the room. We'd act as if we'd known each other for ages in between knowing winks and inside jokes. Instead, they wouldn't stop talking about Stephanie. Things got even stranger when one of the moderators\u2014let's call her xXJosieCatXx\u2014latched on to me as if we'd been friends forever. This wouldn't have been so strange\u2014we'd spent years palling around online together\u2014if JosieCat hadn't been so nasty to me in the months and years we'd spent on the site. When one of the other PVs discovered my YouTube channel, JosieCat had called me out as being \"just jealous\" of Stephanie. Which I was, but so was everyone else\u2014that was why we'd established an entire website to make fun of her? Naturally, I was a little nervous to meet JosieCat face-to-face, but instead of delivering cutting insults, she wouldn't stop fawning all over me. She paraded me around the bar, asking people if they'd met me or watched my hilllaaaaarious videos.\n\nThough now I'm more familiar with the ways people's personalities shift when they close their laptops and get out into the real world, it felt completely bizarre, and embarrassing at the time. It suddenly became very clear how much time I'd wasted with these people. Instead of working on videos, or taking an acting class, or doing literally anything valuable, I'd spent a significant portion of my life for the past two years coming up with mean conspiracy theories about a D-list New York media celebrity who was off partying and booking gigs while I made jokes about becoming an old spinster with no one to cuddle but her cats. And I really don't like cats.\n\nI wish I could say that this period of my life gave me the good sense to go easy on the haterade. But although it gave me perspective on how people who are unhappy offline can project their negative feelings onto someone else, I'm sorry to say it took another slap in the face to make me go cold turkey.\n\nAfter \"SWGSTBG,\" I used some of the money I earned to go to a fancy tech conference. I'd never been to anything like it before, and I was excited to meet many of my internet friends (and, okay, I was also excited to show off in the wake of my viral hit). While there, I met a really successful YouTuber whom I'd taken to casually trashing in my group texts between internet friends. I'd never really \"gotten\" her success\u2014I didn't think she was particularly funny or charismatic. I couldn't understand why she had all these fans and opportunities and I didn't. In other words, I was jealous.\n\nOne night, when we ended up at the same party, I was shocked when she introduced herself and said she was a fan of my work. She was so gracious, charming, and generous that my envy melted away. Eventually the conversation turned to dealing with anxiety, and I stood there puzzled by the thought that someone who appeared so happy and successful could possibly struggle with panic attacks and self-doubt, just like I did. I somehow got up the nerve to share that my anxiety was particularly bad before auditions; I'd find any excuse to put off learning my lines, and too many times I'd email my agent with some excuse about why I couldn't make the audition. I would also get really nervous about having to make videos regularly, which meant I would get into my own head and put things off till the last minute. Procrastination had become my coping mechanism. That way when I bombed the audition, or the video wasn't as good as it could have been, I had an excuse: I hadn't had enough time to try my best. So the failure wasn't my fault.\n\nShe told me something that has changed the way I think to this day: Look at your work as a contract you have with yourself. Getting auditions is work. Posting videos is work. Even if you don't book the gig or go viral, you've still met the conditions of your contract. As with any job, you don't just sit around achieving all day\u2014you have to put in hours. Until that point, whenever I saw someone doing something I wanted to do, I'd devote many of those hours to negative energy. I would try to convince myself that these strangers weren't worthy of the things I wanted instead of being honest with myself about how they got there. Even spoiled Stephanie had to put in some amount of work. Even if you have all the connections in the world, you still have to show up and do the work. More often than not, I couldn't even manage to show up.\n\nIn that moment, I realized I'd been going about my career all wrong. Instead of hating, I should've been studying.\n\nNow we're at the part of the chapter where I say the internet has made all this stuff worse. The internet has made all this stuff worse! Jealousy is human nature, and although every industry is built on competition, entertainment and media are particularly cutthroat. There's always been this idea that there are only so many fans to go around, only so many slots for people to fit into. The internet has only heightened this problem: From the perspective of a creator, social media allows us to present ourselves as much more successful and interesting than we really are. The illusion of our interesting success begets more fans, which then creates the illusion that we are _even more_ interesting and successful. From the perspective of a wannabe creator, social media is a great way to take the focus off what you need to work on and instead funnel all your energy into the pointless project of hating.\n\nAfter realizing my life as a hater was turning me into a resentful jerk, I decided to stop hating and start studying\u2014and I don't just mean studying my enemies' Instagram accounts, though I still go down the \"Tagged Photo\" rabbit hole occasionally. I started paying closer attention to the YouTubers, actors, and creators whom I envied\/admired, watching what they were doing, and what I wasn't. They took improv classes. They worked out. So I upped the gym time\u2014which had the bonus of helping with my anxiety\u2014got new head shots, took a hosting class, and stopped going out as much. Every day, I asked myself: How do I get the things in my career that I want but don't have? What do I do to get them?\n\nBut channeling the green-eyed monster into productivity is especially difficult when there are temptations every time you open your computer. There's that saying \"If you have haters, then you're doing something right,\" but I disagree\u2014at this point, if you have haters you're doing anything. I have a feeling that the caveman who discovered fire had a few cave folks rolling their eyes while happily huddled by the blaze. You can get a job\u2014any job\u2014and the sassiest member of your extended family will comment on your celebratory Facebook post with \"Uh-huh, I see you, you think you're hot stuff with your li'l job now.\" Point is, having haters doesn't mean you're \"winning.\" It doesn't mean you're doing something inherently abhorrent, either. It just means that you shouldn't put too much stock in them. I know that's easier said than done. (Hello! I've literally written a book about it.) With all the ways for people to invade your life via inbox, petty jealousies and snarky comments can feel really personal. But there are always going to be people who don't like you, and you can't be friends with everyone. Whatever you do, someone, somewhere, is going to disapprove.\n\nWhile I truly believe a little saltiness is natural (and at times amusing), it's what we do with that flavorful seasoning that matters. Setting aside the ease of contact that the internet allows, hating in itself is pretty harmless. Waste of time? Sure. Pathetic? Guilty. But in most cases (as it was in those heady days before the advent of Twitter mentions), the \"hatee\" isn't directly affected, because they have no idea their haters even exist. Once, I was riding home from church with my grandmother (who is hands down the most hilariously shady woman I know) when Vanessa Williams's \"Save the Best for Last\" came on the radio. Right as I was getting ready to belt, \"Sometimes the snow comes down in June,\" my grandmother growled, \"And sometimes you should stick to acting.\" Sick burn, Grandma, but damn\u2014she's no Whitney or Mariah, but I always thought Vanessa's voice was cute! Plus, she had some hits! Anyway, while I sat there dumbfounded, guess what Vanessa Williams did? Went about her business, losing not a single night of sleep over my grandma's shade.\n\nThese days, however, it's not enough for some people to deride your achievements and make fun of your questionable singing abilities from afar. No, now many haters will take it to the next level, because just as the internet allows you to present your life exactly how you want it to look, it also gives you easy, direct access to the people you're hating on\u2014from Anne Hathaway (what did she ever do?) to those twelve-year-old girls on _Dance Moms_. As my profile rises, the kind of negative attention I attract makes me realize what I could have become if I'd let my hatred get out of hand: a troll.\n\nThe natural extension of the hater, the troll is not satisfied with merely disliking you\u2014they want to keep you from doing anything but listening to their complaints. They want to distract you, suck up your time and energy, and then, having sapped you of your creative forces, drag you down to their lair, where they will make you watch _The Big Bang Theory_ until you start making jokes about how it's kinda fucked up that women won't sleep with nice guys. When you get a new car, trolls don't just say, \"Well, well, well\u2014Franchesca must think she's something with her new little car\"\u2014they email you and say, \"Hey bitch, your car is trash, and I know where you parked it.\"\n\nThe moment when I realized that the YouTuber I frequently shit-talked was just a normal person who worked very hard was a turning point, steering me away from the path that leads under the bridge, so to speak. Thinking about how lucky I was to have met her in person showed me how to deal with my jealousy in a constructive way, by channeling that energy into the things I wanted to improve about my career. I wish the same were true of my trolls, especially the men\u2014there's something about a black woman being successful and happy that infuriates white men in particular. They have all this privilege, but they still feel sad or lonely or frustrated or whatever\u2014so how dare _I_ not feel sad or lonely or frustrated?\n\nIf these guys invested half the time they spend trying to ruin my life and put it into their own careers, they might feel better. Why not take those twenty different vivid metaphors for how much they hate me and turn them into a novel? The detective work they expend looking up the contact information for my agent, my manager, the schools I'm speaking at, and my boss at MTV so they can tell them to fire me could be translated into a really promising career in PR. Signing my podcast-contact email up for porn newsletters and weird coupons for hemorrhoid cream are examples of creativity\u2014the kind of creativity they might put toward creating a writing sample for any number of late-night television shows or network comedies instead of wasting their lives responding to YouTube comments in a blind rage.\n\nUnfortunately, the trolls are too short-sighted, or too bitter, to channel their negative energy into something positive, so they let it seep out into the world instead.\n\nThe most impressive\/horrifying are the guys who make videos about me. It's not enough for them to just post a rant about how much they hate me\u2014they have to make a fifty-minute video about how much they hate me, email it to me, and ask me to watch it. For those who have never made forays into the YouTube world, a fifty-minute video takes a lot of time to make\u2014these aren't just webcam rants filmed in front of a white wall. They're often fully produced videos, with costumes and skits and graphics and multiple actors. I'll admit I've not only watched some of these horrid videos, but even found myself thinking, _That animation is actually not too bad?_\n\nOne guy in particular has made a number of videos about me. That number being around forty, including a musical. He's mocked my stances on topics that range from political correctness to voting rights and made crude jokes about my husband. His videos attempt to take down a wide variety of what he calls \"social justice warriors,\" but he seems to have a particular fascination with me. Usually his comments are stupid; sometimes they're scary. Every time he posts something about me, I get a barrage of hate from copycat fans who send me vile messages so they can screenshot them and impress all the other trolls.\n\nSo last year, naturally, I decided that I would meet him.\n\nIf you're familiar with the writer Lindy West's work, you may recognize this impulse from her _This American Life_ episode \"If You Don't Have Anything Nice to Say, SAY IT IN ALL CAPS.\" I was listening to it while grocery shopping and burst into tears in the cereal aisle\u2014I hadn't realized how much my trolls were getting to me, and her experience really resonated. In the piece, Lindy describes a particularly cruel troll who created a fake Twitter account impersonating her dead father. Ignoring the online-woman's maxim to \"never feed the trolls,\" she wrote an article for _Jezebel_ explaining why the abuse hurt. Then the troll did the unthinkable: He emailed her to confess all the ways he'd trolled her, and he apologized. \"I don't know why or even when I started trolling you,\" he wrote. \"I think my anger towards you stems from your happiness with your own being. It offended me because it served to highlight my unhappiness with my own self.\" About a year later, hoping to learn more about the motivations of trolls, she reached out asking if he'd have a conversation with her for the radio show.\n\nThe results were illuminating, humanizing, and frankly incredible, and I wanted to replicate them with my own dedicated trolls\u2014at least those who never bothered to conceal their identities, making it easy for me to contact one of them. I knew we would both be at VidCon, the annual online video conference held in Southern California, that year, and after exchanging a few tweets, we agreed to sit down together.\n\nPatrick begged me not to do it, and in hindsight I should have listened to him. But in my heart, I wanted to believe that people aren't just all good or all bad. Because I'd spent years as a hater myself, I was willing to believe my troll was more than the nasty stuff he posted online. I knew he had a girlfriend, parents, and people in his life who think he's great. Maybe they think the troll-y YouTube videos don't represent who he is. Maybe they don't even watch them!\n\nI should have remembered that you can't reason with people whose whole purpose is to get your attention and distract you. When your adversary is someone who has serious issues to work through, it doesn't matter how smart your comeback is or how sympathetic and honest you are\u2014you just can't win.\n\nThe experience of meeting this troll at VidCon was not unlike what happened when I met JosieCat, the salty moderator from the pathetic virgin forum: Instead of looking me up and down like he was Meryl Streep in _The Devil Wears Prada_ , or asking me if I called my husband a racist in bed, or dumping a bucket of pig's blood on my head as I crossed the threshold, this guy basically groveled. He said he was so sorry for all the abuse he'd given me, that he respected me, and that he hoped we could move on from our differences and be friends. He also confessed a bunch of personal stuff that he used to explain where his anger came from.\n\nI was really taken aback. All I wanted to say was that he had misconstrued my work and put a bunch of words in my mouth, and to show him that I'm a real person and that his actions have consequences. I mentioned that whenever he made videos it would set off a ripple effect with his fans, who would attempt to contact me with even more aggressive messages than what this guy's video contained, and that it really upset me. I made it clear that I wasn't trying to be his therapist, or his friend, and that I appreciated his apology but really just wanted the harassment to stop.\n\nWe left on decent terms, and I thought it was possible things would get better. But of course the guy made a video about how we'd met and worked out all our problems and were now on our way to being closer than Snoop Dogg and Martha Stewart. I should have known he'd do this; hating me was part of his career, and he couldn't resist the opportunity to cook this grade-A beef.\n\nI sort of understood his desire to say we'd become the best of friends, and I didn't really want to start another confrontation by shutting it down; plus, it made him look good, or at least like his behavior wasn't irredeemable. And it wasn't so different from the cinematic resolution to our entirely one-sided feud that I'd envisioned, though in my version I was more like a benevolent queen lording my power over him before saying it was behind us\u2014as long as he never trolled again.\n\nBut something about his suggestion that we were equals who had put our differences aside got to me. This guy made forty videos about how I'm the scum of the earth, and I'm supposed to forgive him because he said he was sorry? I'm supposed to consider him my friend? I would have a hard time forgiving my own mother if she did that to me. His new video made me realize he hadn't earned the benefit of the doubt.\n\nFuming, I went on _Last Name Basis_ , the podcast I host with Pat, and talked shit. I didn't name the troll, or say anything specific about our meeting, but I did say some nasty stuff that I'm not proud of, calling anti-feminists ugly and trotting out harmful stereotypes by saying that the guy and his fans probably needed to get laid. I don't know why I assumed he didn't listen to my podcast, but of course he did, and sure enough, another video appeared, about how \"Racist Franny\" was a duplicitous snake in the grass who had betrayed a man who thought they were friends. He then hosted a two-hour live stream dedicated to listening and reacting to the twelve-minute segment of my podcast and encouraging his fans to pile on with him. His fans used the incident to make him out to be a victim, and now we're back where we started. (His followers were sending me snake emojis long before Taylor Swift decided to embrace that as her brand.) Or actually even further behind where we started, because now he and his fans believe I attacked them.\n\nSo, to recap, here are all the mistakes I made in this situation: First, I tried to meet a troll and reason with him. Second, I thought some sense of closure would come out of this meeting based on one extraordinary news story that bore some similarities to my troll situation but actually, in retrospect, not very many. (My troll never reached out to me to apologize\u2014I reached out to him.) Third, I got mad about the guy's reaction to the meeting we shouldn't have had and \"fed the troll\" by ranting about it on my podcast. Fourth, I broke my own rule about criticism, venturing into bigotry rather than attacking what he'd said and done. I pushed myself to let go of the situation when I wasn't ready, and I made the whole thing worse.\n\nWhether you're the hater or the hated, the troll or the trolled, we could all be smarter about how we respond to natural feelings of jealousy and how they spill out on the internet. I sometimes wonder how much farther along in my career I'd be if I hadn't dedicated so much time and energy to these people. The loudest voices are distracting, but they're not necessarily the majority, and giving so much to them was doing my actual fans a disservice. (This is also why I've pulled back from social media lately: Not only has it made a huge difference for my productivity\u2014obviously\u2014but it's also allowed me to reframe my goals and priorities.)\n\nWhen you see someone doing something you wish you could do, or achieving success you wish you could achieve, don't sit around making memes about how your dog could paint better than this girl\u2014get your ass in the studio. (Hopefully one without internet access.) And when your work starts selling for hundreds of dollars, and then thousands of dollars, and someone starts making videos of his dog stepping in paint, walking on a canvas, and saying, \"Look\u2014my dog made a better painting than Kim Jenkins,\" just ignore it. His landlord is going to have a field day over the blue paw prints in the carpet.\n\n# HATERS VERSUS TROLLS: A GUIDE\n\nThere's a big difference between being a hater like sweet, young Franchesca and a full-blown Breitbart employee. Hating is not productive, but it's mostly harmless, especially if you're focused on a celebrity. They don't have time for you. Trolling helped elect Donald Trump, and contributes to thousands of women and marginalized people feeling unsafe every day. Now that you've learned all about what not to do when you encounter a hater or troll, I've made a handy chart to help you spot them.\n\n# CHAPTER NINE\n\n# ACTIVIST LENT\n\nAs a kid, I thought beets tasted like dirt. As an adult, I still think they taste like dirt, but now that's what I like about them. In middle school, I was obsessed with the TV show _Blossom_ , which is why I owned no less than ten different oversize floppy hats adorned with giant fake flowers. Today, I wouldn't be caught dead in a floppy sunflower hat, and the mere sight of Mayim Bialik (former _Blossom_ star, on-again, off-again anti-vaxxer) makes me roll my eyes.\n\nIt's equal parts weird and surprising to look back on the person you once were and see how time has changed the way you see and think. For me, this change has been extremely apparent\u2014and sometimes awkward\u2014as I've grown more socially conscious and aware of how different types of people move through the world. For starters, being a feminist makes it difficult to listen to a good majority of music. I've given up on skipping songs with the word \"bitch\" in them, because that would deplete most of my iTunes library, but singing along to the top forty is a challenge. It's also hard to watch a lot of TV shows, and to sit silently in the theater as misogynistic movies you paid fifteen dollars to see play out across the screen. Sometimes it's hard to even have a simple conversation.\n\nGrowing up under patriarchy and white supremacy and bigotry\u2014which everyone does, since those are the systems that govern the world\u2014means it can be hard to resist the less-than-obvious ways those forces creep into our speech, behavior, and superhero movies. Attempting to live a socially conscious life is a lot of work. As you become aware of all the ways our society is set up to put down marginalized people, from huge things like the prison system to small things like the words we use every day, you'll start seeing the world through completely new eyes.\n\nAt first, none of it will feel very fun. Actually, it never really feels very fun. Things you _want_ to enjoy are ruined when you can recognize the not-so-subtle nasty messages they contain about women, LGBTQ folks, and people of color. And every day there seems to be a new word or phrase that has harmful origins or an underlying meaning you never thought about before. Sometimes it feels like no matter how hard I try, I can't say anything without offending someone. And since I'm a person who loves to run her mouth, that can be pretty difficult.\n\nBut I still try, even though from time to time I get it wrong. I've found that the best way to avoid hurting others is to steer clear of endorsing, watching, listening to, or participating in media or conversations that exclude or harm marginalized people. The best way to do that is to make a mental list, or at least have firm policies on what you will and won't stand for. I call this \"Activist Lent,\" but instead of forty days, you give these things up for the rest of your life. A more accurate description might be \"Activist Recently Discovered Severe Gluten Allergy,\" but that doesn't have a very good ring to it. Here are just a few of the games, habits, performing artists, and expressions I've given up in recent years\u2014or at least believe I should have given up in recent years, even if I can't quite bring myself to cut the cord just yet.\n\n# GAME OF THRONES\n\nThis is many a feminist's guilty pleasure. The folks at HBO manage to lure so many of us back, even after giving us eighteen-month breaks between seasons to muster the will to quit. When my personal attempts to stop watching failed, I tried to turn off my feminist brain and just enjoy it; I've never managed that, either. Instead, I brace myself every time Tyrion enters a brothel, and complain about each episode for days after it airs. I'm hardly the first person to point this out, but despite being a badass, Khaleesi is a textbook white savior figure, with her glowing platinum hair and her posse of minions, all of whom are brown and constantly fawn over her. (I think I gasped during the scene when she crowd-surfed across a sea of brown nonspeaking extras as she celebrated her reign as their queen.) There's also the terrible story line of her relationship with Khal Drogo\u2014he rapes her after taking her as a child bride, and then she falls in love with him; to add insult to injury, the actor who plays Khal Drogo publicly joked that being able to \"rape beautiful women\" was one of the perks of his role. He's since apologized, but as more and more actresses come forward about sexual assault in Hollywood, I can't help but wonder if this impulse explains rape as a plot device in too many films and television shows.\n\nI digress. _Game of Thrones_ is also mercilessly violent, and as a severe empath I'm often reduced to squealing and covering my eyes when gruesome death seems imminent. I can say pretty confidently that I feel angry with the show more than I like it. Nevertheless, I'm going to stick with it till the likely bitter end. Janet Mock calls this being a \"critical fan\"\u2014I'm not going to give it up, but I don't gloss over the issues just because I enjoy immersing myself in the action. When people say they don't watch it, I respect that.\n\nSo why do I keep tuning in? For all the guilt I feel, there's no denying that it's an incredible show. It's one of the few that are truly unpredictable. Every time you like someone, they get killed off. (Though I pray Arya never sees this fate. I worry they're waiting for the series finale so it'll be too late for me to boycott the show in protest, like I did with _The Walking Dead_. RIP Glen.) I absolutely love the costumes, and since everyone and their brother watches the show, I enjoy participating in the collective water-cooler conversation it inspires. (Twitter is my water cooler.) But above all, I think what I find compelling about it is that, although the show is rife with horrible people with evil motives, they often get their comeuppance. We live in a world where too often bad people never see consequences. It's almost a relief to spend an hour each week in a place where they do.\n\n# THE SONG \"BLIND MAN\" BY XAVIER OM\u00c4R\n\nThis song is something of a deep cut. If you're not a fan of neo-soul or Afrofuturistic bops, chances are you've never heard Xavier Om\u00e4r's \"Blind Man,\" but I can assure you it's quite a smooth jam, despite the cringey lyrics.\n\nOn first listen, when you're sort of half paying attention and can make out only a third of the words, it sounds like it's presenting a feminist message. \"I can love you with my eyes closed \/ I don't lose sight of your beauty \/ 'Cause your heart is fine gold, baby \/ Imma take my time with your mind,\" Xavier croons. He praises his girl's ambition, her dreams, and her heart. While most men may just focus on this woman's \"body like a queen,\" he suggests, he is no sexist\u2014he cares about her mind.\n\nBut then we get into a dicey situation with some metaphors: \"A blind man could love you just for who you are \/ Got 20\/20 vision and I still don't see ya flaws \/ I said you make me feel good, and you ain't even touch me.\"\n\nIs he saying his girl is so _physically_ beautiful that a blind man could sense her beauty? Or is he saying that her inner beauty is strong enough that a blind person, who he believes would not care about physical appearance and therefore must be a connoisseur of inner beauty, would love her? The line \"A blind man could love you just for who you are\" suggests that a blind man somehow has more reason to judge a woman's inner beauty\u2014I guess because Xav thinks her physical beauty wouldn't be able to make up for a bad personality? He tops it all off with a throwaway line that vaguely suggests blind people have to touch things a lot.\n\nThis doesn't make much sense, but what meaning we can get from it is offensive. For one thing, tons of blind people are in happy relationships with people they love and think are attractive; the suggestion that blind people are somehow more discerning when judging personalities is bizarre and othering. Remember the second line of the song, too: \"I don't lose sight of your beauty\"\u2014a pretty sleazy pun that takes a cheap shot at blindness, especially when you consider that Xavier later insensitively brags about having twenty-twenty vision.\n\nFrom a feminist perspective, Xavier's insistence on inner beauty\u2014which we can only assume he's discussing because his girl is superhot in the first place\u2014veers dangerously close to some lines in Drake's \"Best I Ever Had\": \"Sweatpants, hair tied, chillin' with no makeup on \/ That's when you're the prettiest \/ I hope that you don't take it wrong.\" Drake, I do take it wrong, because sometimes I like to wear makeup, and I don't appreciate your ranking my styling choices because that's only going to make me self-conscious when I do decide to break out the MAC Ruby Woo. Though I don't forbid myself to listen to Drake, I do have him on notice for using that sensitive Auto-Tune to trick you into thinking his songs aren't all about policing women. \"You're a good girl and you know it\"\u2014so what if I am?\n\n# R. KELLY\n\nOne of the hardest things to cut out of your life is a musical artist you like. You can be doing well, and then your favorite jam from 1997 comes on. You learned to love it before you understood the lyrics, back in that carefree time of your life when you thought you could get pregnant by sitting on a boy's lap. You might pause and wonder, \"Maybe it's not so bad?\" and enjoy the rest of the song uncomfortably before forgetting about it.\n\nTo this day I shudder to think of the middle school dances where my girlfriends and I belted out the lyrics to \"Too Close\" by Next, which I now know is a song about a guy having a boner on the dance floor and warning his dance partner she might feel it if she gets too close. Of course, erections are totally natural, and it's kinda nice that the fellas of Next felt it was important to disclose what was happening, even if they couldn't be more discreet than \"You're making it hard for me.\"\n\nThis is not the case with R. Kelly, who has been accused of statutory rape, illegal marriage to a minor (Aaliyah), and, most recently, of brainwashing and holding women captive in a \"sex cult.\" He's already profited off his music\u2014there's nothing we can do about that. But we can refuse to play it, which I do. Not long ago, I was hanging out with my neighbors when they put on \"Bump N' Grind.\" I hated to be the Debbie Downer, but I did it anyway. \"I'm sorry, but you have to change this song. I don't see how you can listen to 'My mind's telling me no \/ But my body, my body's telling me yes' and not think, _Child predator_.\" My friends were understanding, but at first they tried to fight it. \"But it's such a good song!\" they cried, half dancing along as they objected.\n\nI had to be firm. These were not questionable lyrics about \"good girls\"\u2014R. Kelly is permanently and irrevocably tainted. \"Fuck this guy,\" I said. \"We will not pass go, we will not step in the name of love, we will not go to the hotel lobby, we will not take it to our rooms and freak somebody. We will stay right here and choose to listen to some other R&B singing sensation.\" And that is what we did.\n\n# CARDS AGAINST HUMANITY\n\nI must confess: I have experienced this purposefully offensive game firsthand. But I promise, I only played it once, and I can truly say I will never do it again. I didn't know what it was when my friend, with a naughty twinkle in her eye, brought it out, and I felt like I was going to have a heart attack the entire time we played. If you're not familiar with Cards Against Humanity, its official tagline is \"a party game for horrible people.\" It's basically group Mad Libs: Each player gets ten cards with words and phrases on them. The words and phrases range from questionable to terrible, and almost all of them have the potential to hit someone where it hurts. Here's a random selection of cards: _African children_ , _The gays_ , _An oversized lollipop_ , _Court-ordered rehab_ , and _Vehicular manslaughter_. If you're thinking these all sound like caricatures of what's considered \"politically incorrect,\" it's because I made it a point just now to leave out any that were truly upsetting. Trust me, they're in there.\n\nAfter everyone has been dealt their hand, you have to put your cards together in the middle to make a sentence based on a fill-in-the-blank prompt. The \"Card Czar\" picks the funniest\u2014or most horrible\u2014one, and the person who submitted it gets a point. It goes on this way until the most consistently offensive person is declared the winner. While I love a good game night, that evening felt like a _Game of Thrones_ \u2013style torture session. Instead of giving up, I tried my best to play along while sticking to my morals. This proved harder than expected; I lost a few rounds by playing it extra safe. But then I was able to turn it around. Maybe it was because I spend all day battling trolls, so I have perspective on how the worst minds of humanity work. Maybe it was that I wanted to get it over with. Maybe it was just beginner's luck. Maybe, deep down, I am a horrible person. Regardless, my friends said they were going to tell everyone, so I'm coming clean here: Somehow, some way, I won my first and last game of Cards Against Humanity.\n\n# RAP SONGS WITH TRUMP SHOUT-OUTS\n\nI was one of the millions of American citizens who weren't prepared for the results of the 2016 presidential election. But one surprising consequence was realizing just how many of my favorite gym tunes name-checked Donald Trump, and I'm not talking about the recent classic \"FDT (Fuck Donald Trump)\" by YG. Nay, all throughout the 1990s and 2000s, rappers were shouting out Trump and his hotels as if they were symbols of prestige and wealth rather than tacky products of craven exploitation. I'll admit that I haven't purged these songs from my phone, but I stage a mini silent protest for choice lines such as...\n\n# \"FLAWLESS (REMIX),\" BEYONC\u00c9 FEATURING NICKI MINAJ\n\nI told him, \"Meet me at the Trump, Ivanka.\"\n\n# \"I WANNA BE WITH YOU,\" DJ KHALED\n\nAt the Trump, you bitches at the Radisson.\n\n# \"DONALD TRUMP,\" MAC MILLER\n\nTake over the world when I'm on my Donald Trump shit.\n\n# \"WHAT MORE CAN I SAY,\" JAY-Z\n\nI'm at the Trump International, ask for me.\n\n# \"COUNTRY GRAMMAR (HOT SHIT),\" NELLY\n\nBill Gates, Donald Trump, let me in now.\n\n# \"IT'S GOIN' DOWN,\" YUNG JOC\n\nBoys from the hood call me black Donald Trump.\n\n# THE FOLLOWING WORDS AND SAYINGS\n\nThere's an idea commonly thrown around conservative spaces that liberals, by objecting to certain words or phrases, want to censor free speech. An extreme example of this is the N-word: Every time a celebrity\u2014or YouTube's most subscribed gamer\u2014is caught saying it, the conversation is dominated by the idea that \"It's not fair that you can say certain things and I can't.\" It should go without saying that anyone who wishes to say the N-word certainly _can_ , but the reception and social consequences will differ based on whether or not you're black.\n\nRegardless of how you feel about the word, the idea of words changing meaning between groups or individuals isn't a new concept, or one exclusive to black people. For example, my husband calls me \"baby,\" and I usually respond with something like, \"Aw. You still have to clean the bathroom.\" Meanwhile, if a dude on the corner calls me \"baby,\" I'm going to roll my eyes. Words are like chameleons that way.\n\nWhat these \"Why are you censoring me?\" people misunderstand is that no one is _forcing_ anyone to change the way they speak. You can say anything you want; you won't face legal action for calling someone the N-word. But it doesn't mean you can say it free of consequences. You can totally start your company emails to your boss with \"Dear ugly bitch\"\u2014you'll just get fired for it. By committing to march down the path of \"political incorrectness,\" you're saying you're willing to sacrifice relationships with anyone who finds your language unacceptable.\n\nObviously, I'm not walking around calling my bosses\u2014or anyone\u2014ugly bitches. But beyond egregiously hurtful language, there are tons of other words and phrases that exclude, upset, or otherwise offend marginalized people that still find their way into everyday conversation. I try to eliminate these terms from my speech to be more conscious and welcoming, both to strangers and to people I know. With some of these, I had to be called out for using them before I understood how messed up they were; they've become so ingrained in our conversation that most of us don't even think about them. There's nothing wrong with that; we all have to learn somehow. The key is to avoid getting defensive, to listen to what people are telling you, and to work to do better. There are countless ways to express an idea\u2014why would you want to use one that insults an entire group of people?\n\n**G  pped**\u2014I've definitely dropped this in a video without even thinking twice about it. It's such a colloquial, almost jokingly old-fashioned phrase that I was shocked when I learned its origins. So-called Gypsies\u2014the historically nomadic ethnic group more appropriately known as Romani or Roma people\u2014deal with horrific discrimination and oppression all over the world, and particularly in Europe, where certain businesses may still hang signs that read _No Gypsies_. Few companies will employ Roma people, so they are forced to beg, or worse, for money. They have been persecuted for centuries, at times forced into slavery, assimilation, and repatriation to their \"countries of origin\"; they have been treated as scapegoats for epidemics of disease and blamed for rising crime rates in several countries. Estimates of the number of Romani people killed during the Holocaust range from 200,000 to more than a million.\n\nToday people use \"g pped\" to suggest they've been swindled, which perpetuates the misconception that Romani people are untrustworthy or criminal. Saying your aunt \"g pped\" you on sweet potatoes at Thanksgiving is not acceptable, and I was happy someone called me out on it. And while Free People and any number of other fashion labels push \"Gypsy\" wear for music festivals, the word is still a slur in many parts of the world. Which is why I avoid it even when it's meant to represent something positive, like overpriced crop tops.\n\n**\"Hey, guys!\"** \u2014Unless you find yourself in a group comprised exclusively of male-identifying people, there's a better word than \"guys.\" For years, I began all my videos with a peppy \"Hey, guys!\" until someone gently pointed out that there's a more inclusive way to welcome my audience. Not every woman, female-identifying person, or nonbinary person feels uncomfortable with \"guys\"\u2014I personally don't mind if someone uses \"guys\" when I'm in mixed company. But I still switched to \"Hey, friends!\" or \"Hey, party people!\" because some people do feel uneasy about \"guys.\"\n\nWhen I explain this concept to people, they're often incensed. \"Ugh, you're so sensitive!\" they cry. \"I'm a woman and I don't care, so why should anyone else?\" But I'd argue that these people are actually the sensitive ones, because instead of switching one word, they throw a temper tantrum.\n\n**\"I feel fat\"** \u2014The Facebook memory feature, which pulls past posts you made on the same date in previous years, is equal parts nostalgia and embarrassment. For me, in addition to photos of old friends I've lost touch with and excited praise for the _Sex in the City_ movie, I was shocked to see how much I used to post about \"feeling\" or being fat.\n\n\"Bloated\" is a feeling. \"My stomach hates me because I ate an entire pizza\" is a feeling. But \"fat\" is not a feeling. Everyone, no matter their size, has phases when they don't feel the best about their body, but associating those negative feelings with fat people increases the already huge stigma they face, from the misconception that being fat means you can't be healthy to the idea that it means you're dirty or unattractive. People feel so comfortable saying horrible things about fat people, it's almost as if weight were some kind of moral issue.\n\nStatements like this also promote disordered eating, creating a tangled web of shame that touches the lives of so many people, regardless of their size. When I was younger, it seemed as if my mother was always on a diet, which meant I was always on a diet. When she eliminated carbs, I did, too; I drank SlimFast for breakfast and lunch even though it upset my stomach; I did my best to follow the makeshift Atkins diet by avoiding the foods on the list hanging on our fridge. I subscribed to the unfortunately common idea that being fat, or, really, anything other than thin, was the worst thing you could be, and that's where statements like \"I'm so fat today\" were coming from\u2014a fear of becoming what I had been led to believe was a fate worse than drinking gritty \"French Vanilla\"\u2013flavored \"milkshakes\" for two meals a day for the rest of my life.\n\nWhen I got to acting school, disordered eating felt like the norm, if not actively encouraged. I would work out three times a day and limit myself to canned tuna and protein shakes. If anyone questioned whether I might have a problem, I responded with \"Of course not! I'm eating!\" At least one girl in my class of thirty acting students was hospitalized for her eating disorder, as were a few others, but it was never discussed. Instead, we treated it as a necessary consequence of looking \"our best\" for roles.\n\nSince then, I've let go of the idea that one type of body is happy or healthy, and I've learned that the urge to put myself down about my weight is a direct consequence of the unrealistic beauty standards society has for women. And my impulse to acknowledge\u2014or even apologize for\u2014\"feeling fat\" has side effects that go far beyond my own self-esteem, too. It promotes bigoted attitudes toward fat people, from concern trolling (\"I'm just worried about your health\") to outright discrimination. When I worked at Anthropologie\u2014not a store known for its inclusive range of clothes\u2014racial profiling wasn't the only issue. Our loss-prevention team would flag customers over size 18 as potential shoplifters, because why else would they be there? Never mind that we sold shoes, home goods, and jewelry, which customers of any size could enjoy. And while I've had my own encounters with strangers who refuse to mind their business, I was shocked when one of my friends shared that random people will often congratulate her for ordering a salad because they assume someone of her size must be dieting.\n\nFrom wearing crop tops or indulging in a sweet snack to having an active sex life, people have all sorts of warped and outdated ideas about what people with larger bodies should and shouldn't do with those bodies. This kind of thinking is invasive and rude, and it needs to stop. People of all sizes are happy with the way they look, and they don't feel like their size or shape is a bad thing. It's none of anyone else's business either way.\n\n**\"I wanna kill myself\"** \u2014There's something about the current political situation that makes it particularly easy to drop this into casual conversation. Trump has caused many of us to feel like we're constantly on the brink. But we're not.\n\nI stopped saying this after a close friend lost a family member to suicide. I had accidentally erased my camera's memory card, losing a few hours' worth of work, and in a fit of frustration, I said, \"Ugh, just fucking kill me!\" My friend's immediate response was \"Nothing is worth losing your life over. Please don't say that.\" I realized immediately how careless it was to suggest that death would be better than dealing with an inconvenience as inconsequential as a couple of lost files. You should always take it seriously if someone says they want to kill themselves. If everyone is always saying \"I want to kill myself\" when they really mean \"I hate this situation\" or \"I would very much like to be excluded from this narrative,\" it becomes much more difficult to understand when people are serious and need help. Suicide is not a joke.\n\n**Lame** \u2014This is a hard one to quit, but pretty straightforward: The technical definition of \"lame,\" according to _Merriam-Webster's Collegiate Dictionary_ , is \"having a body part and especially a limb so disabled as to impair freedom of movement.\" Since we apply it as a slang term to things that are ineffectual, insubstantial, boring, uncool, or otherwise bad, we imply that disabled people are ineffectual, insubstantial, boring, uncool, or otherwise bad. They're not.\n\n**Spaz** \u2014When I was in eighth grade, there was a girl in one of my acting classes who would slip into a mentally disabled character she called Sylvia. A group of us would be out somewhere, at the mall or a restaurant, and suddenly she would become Sylvia, fully embodying this character, and it would be our job to keep her from doing something too embarrassing or harmful. Thank goodness YouTube didn't exist in those days, because there would no doubt be ample video evidence of our horrendous shenanigans. Instead, you'll just have to believe me when I say everything from Sylvia's voice to her mannerisms was painful.\n\nOnce, we were in a drugstore, and she picked up a box of pads, went up to an employee, and asked, \"Can I try these on?\" Which of course sent us into a fit of giggles. (Think of the most obnoxious theater-kid stereotype you can dream up, and that was us, times ten.) The employee, not knowing what to say but not wanting to upset her, said, \"Ummm... no... sorry... you can't.\" Sylvia, unperturbed, abandoned the interaction and knocked a row of adult diapers off a shelf in the next aisle, and we bolted out of the store.\n\nBeing kids, we were excited by the possibility of doing things we weren't supposed to do, on top of \"fooling\" people into thinking someone had tasked a group of eighth graders with taking care of a disabled child without an adult in sight. We thought the whole thing was just hilarious, but I don't think I really need to explain why it wasn't. For one thing, it was someone's job to clean up our mess, and I'm sure some underpaid employee was chewed out for letting us wreak havoc in the store. For another, creating a spectacle of Sylvia made it seem okay to stereotype or mock people with physical and mental challenges, people who have a huge range of abilities. Just like saying someone is being \"retarded,\" calling someone a \"spaz\"\u2014a derogatory term for people with \"spastic paralysis,\" which is known as cerebral palsy when it's congenital\u2014ignores the prejudices and difficulties disabled people face, and instead turns them into a punchline.\n\n**Spirit animal** \u2014This is another phrase I was rightfully called out for. It's not uncommon to hear people refer to everything from their favorite actress to the latest viral meme as their \"spirit animal.\" It's meant as a term of endearment, so what could be wrong about that?\n\nUntil the American Indian Religious Freedom Act of 1978, Native Americans were often prevented\u2014sometimes violently\u2014from accessing sacred sites, keeping ceremonial items that were illegal under U.S. law (such as eagle bones or peyote), and performing ceremonies or rituals without gawking onlookers or questioning from authorities. Keeping this in mind, \"spirit animal\" is actually a gross oversimplification of an important part of a culture that has long been actively oppressed, stifled, and ignored in racial justice discourse.\n\nThe ways that indigenous people have been forced into assimilation are so specific: They were moved off sacred lands, their names were changed, and their languages were purged so aggressively that colonizers would destroy cultural and religious artifacts that featured words. The message was clear: If you want to stay in this country, you have to be like us. So it probably hurts when non-Native people talk about cartoon characters and rainbow bagels as being their \"spirit animals.\" (Meanwhile, we live in a country where people, including the president, flip their shit when people say \"happy holidays\" instead of \"merry Christmas.\") \"Spirit animal\" is not an appropriate metaphor to explain that you identify with the snarky cat from _Sabrina the Teenage Witch_.\n\n**\"That's so crazy!\"** \u2014A whole cache of mental-health terms have migrated into casual use. Some act as slurs\u2014such as \"psycho\" or \"psychopath\"\u2014and others are things we might say to lightly make fun of ourselves: \"Oh, I'm just so OCD about my bathroom!\" Often, these words are used to explain away systemic racism, bigotry, and murder; any time a white man opens fire on a school, or a movie theater, or a concert, people, trying to understand the tragedy, say, \"He must have been crazy.\" The effect of this is twofold: People who have mental illnesses may feel ashamed to seek help, and the systemic problems at the root of the actions (lax gun-control laws, toxic masculinity, etc.) continue to go unacknowledged.\n\nMental health is so stigmatized that we barely talk about the original issues we're dealing with, much less about how it hurts when friends use \"insane\" to describe the objectively awful political situation. Not to mention the fact that labeling violent or bigoted people as \"crazy\" or \"mentally ill\" perpetuates two harmful ideas: that mentally ill people are dangerous, and that bigotry is something people can't control. I admit that I never even thought about how often I deemed the news \"crazy\" until someone called me out for it on Twitter. I still slip up now and then, but I try to replace \"crazy\/insane\" with \"wild\" or \"unbelievable\"\u2014it conveys exactly the same sense, without the harm.\n\nI could go on, but don't worry\u2014I won't go into the movie portion of my list, which would no doubt ruin any future Netflix and chill plans. It can feel like a lot to keep track of, I know, but once you get used to this kind of self-editing, it can be kind of fun. You're helping to make the world a better place for marginalized people, and by holding yourself accountable in everyday situations, you're also teaching by example. You know how, when you were a kid, you used to put all the pillows on the floor and hop from one to the other because if you touched the carpet it was lava? Think of your Activist Lent list as the lava. Just make sure to apologize gracefully if you fall in.\n\n# CHAPTER TEN\n\n# UNFRIENDED\n\nEnding a friendship used to be simple. Like when your best friend moved to Boston to live with her dad and stepmom, leaving you high and dry in South Florida. Bam. Friendship over. Sure, you breathlessly promised to KIT (keep in touch) and LYLAS (love ya like a sister), but you had no interest in _actually_ going to Boston because you imagined it resembled a town in Siberia where all the men wore Patriots jerseys and used \"wicked\" to mean the opposite of what it's supposed to mean. Or like in sixth grade, after your girlfriend confided in you that she'd gotten her first period, and you made a cheap Red Sea joke in geography class and she never forgave you. Even if you regretted it later, the friendship breakup was cut-and-dried: There was a clear cause and effect that both parties understood. Obviously you were just jealous, but that didn't erase the hurt or embarrassment. (Those were the days, right? When middle school girls longed for their periods, not in the \"But seriously, where the fuck is my period?!\" sort of way, but more of an \"I can't wait to run through a field dressed in all white celebrating my womanhood!!!\" kind of thing.) In the end, you were persona non grata at your former friend's coed pool party, but at least you knew where you stood.\n\nToday, Facebook is where friendships go to die. If you're somehow one of the last people left on earth who is not on Facebook, you are both worthy of envious admiration and hella suspect. How do you remember the birthdays of random people you interned with in college? How do you keep track of how often your mother-in-law plays\u2014and CRUSHES\u2014 _Bejeweled_? What about the bloody C-section photos of women you can't quite place because they've changed their last names and their profile photos are pictures of babies playing with puppies? Seriously, who the hell is that woman, and what did I do to deserve her afterbirth in my Facebook feed?\n\nDespite the wealth of information Facebook abstainers miss out on, they're also spared the gut-wrenching feeling that comes when you realize someone you once considered a friend believes America should be celebrating \"White History Month.\" It should be as simple as it was in sixth grade: Friendship over. Though the process seems easy\u2014you literally just have to click a button to officially unfriend someone\u2014it dredges up all kinds of difficult questions. The abandoned friend may have no idea what she did to deserve being purged from your timeline, and you may have no interest in dealing with her long enough to tell her. It couldn't have been the \"Leave Ivanka Alone!\" meme she posted at two thirty in the morning, could it? Are political differences really enough to destroy a friendship that may have survived multiple moves, fights, betrayals, and drunken confessions of \"I knew that dress you wore to prom ten years ago looked bad but I didn't say anything\"? The fact that you never have to see each other in person to hash it out means you can forget about closure.\n\nSomeone once told me that, for black folks, being friends with white people is like holding a bomb and not knowing when it'll go off. While there's probably some truth to this analogy, I've found that everyone, no matter their race, gender, sexuality, or number of pro-choice buttons on their organic hemp grocery bags, has the capacity to blow up on Facebook, leaving everyone they know confused and covered in shards of incoherent ramblings.\n\n\"Blowing up\" doesn't just mean spoiling the latest episode of _The Walking Dead_. (Though that falls under the General Asshole category and should also be avoided, unless you want me to give you the ice-queen eyebrow next time we see each other at a party. Also, why are you still watching _The Walking Dead_?) No, what I'm talking about are the people who choose to use their status updates to uphold systems of oppression, perpetuating misinformation, intolerance, prejudice, and, ultimately, white supremacy. That's kind of high-minded language to describe your high school guidance counselor's misspelled rants about illegal immigration, I know, but white supremacy works on every level, from the White House (duh) to your uncle Ron. And this is one of the few times I'm comfortable using the phrase \"not just white people\"\u2014too many of my skinfolk are all too happy to suck on the teat of the white sup. If you've ever caught Milwaukee sheriff David Clarke doing one of his Fox News appearances\u2014though I don't know why you'd have that on\u2014you know what I mean.\n\nCombine this layer cake of white supremacy expression with the ways social media turns us all into lab rats, trained to expect praise and attention every time we so much as comment on Jennifer Lopez's hair, and it's no wonder Facebook forces you to reckon with every single one of your relationships. Where else can your mom, mailman, boss, and Tinder date all hang out and shoot the shit, together? And they aren't talking about the weather. I'm not going to say that Facebook brings out the absolute worst in people\u2014that honor goes to Sunday afternoon at Trader Joe's\u2014but it does encourage people to be petty, competitive, arrogant, and gossipy, and to engage in WWE-style dramatics. Except this arena has no rules, referee, or ejection policy.\n\nWhich brings us back to friendship breakups. Sometimes you may want to forget you ever met these people (okay, maybe not your mom). As everyone knows, these days the first step to forgetting you ever met someone is purging them from your social media accounts. But how do you determine when a friendship can\u2014or should\u2014be mended and when it's time to pull the plug?\n\nLike your relationship status at age seventeen, it's complicated.\n\nIt's not possible for two people to agree on everything, no matter how well they get along. Hell, my husband and I have an argument almost every time we have to pick a movie to watch. (He always wants a gory horror flick; I'm more of an action\/thriller type.) But those are the type of agree-to-disagree moments every relationship is made of.\n\nPolitics is a different story. You may have passionate debates with your die-hard liberal friend who posts selfies in a #StillWithHer T-shirt, but you know at the end of the day the two of you are on the same team. But when it comes to basic human rights issues\u2014like trans bathroom laws, affordable health care, or same-sex marriage\u2014agreeing to disagree feels nearly impossible. An opinion stops being \"just an opinion\" when it supports the mistreatment or oppression of others, which is why political differences can tear families apart and _Troll II_ can't. This quote from @SonofBaldwin, also known as the writer Robert Jones Jr., says it best: \"We can disagree and still love each other unless your disagreement is rooted in my oppression and denial of my humanity and right to exist.\"\n\nSo let's say one of your Facebook friends has crossed the line and posted a disgusting meme slut-shaming Melania Trump for taking nude photos back in the day. You're at a crossroads. Are you mad at this Facebook friend for slut-shaming a grown woman who can do whatever the hell she wants with her body? Or are you mad at her for making you defend a woman whose husband and politics are absolutely reprehensible (but who can still do whatever the hell she wants with her body)? For the purposes of this analogy, let's say it's both. Which, coincidentally, is the correct answer. What do you do? Leave a lengthy comment about bodily autonomy? Reply with an eye-roll GIF that's completely open to interpretation? (Are you rolling your eyes at the post, or at Melania's totally consensual adult nudes?) Or do you pull out the big guns and unfriend altogether?\n\nRelationships aren't exactly \"ride or die,\" no matter how many pinkie promises or drunken declarations one might make\u2014and that's where this gets tricky. For some friends, acquaintances, or coworkers, it may be easier for everyone if you silently end the virtual friendship instead of trying to reason with them in the comments. For others, unfriending will just make everything worse.\n\nTo help you figure out if it's worth it to delete and never look back, I've made a handy guide, outlining all the relationships you may have on Facebook and the potential drama they have bottled up inside them, waiting to destroy your Thanksgiving.\n\nIn case you're wondering, no, the little icons are not Iron Man helmets\u2014they're supposed to be old-school sad drama masks. But I suspect Iron Man would also hate stirring up unnecessary Facebook drama, so whatever the icons evoke for you is okay with me. Remember that in a past life I was a graphic designer.\n\n# OH, THE \"FRIENDS\" YOU'LL UNFRIEND\n\n## Family\n\nIdeally, we'd never be forced to accept our family members as friends on social media in the first place. Though you might appreciate being tagged in weekly ~ ~SHARE THIS IF YOU LOVE YOUR BEAUTIFUL, PERFECT DAUGHTER~ ~ posts, your life is probably not improved by knowing that your great-aunt has discovered anti-abortion memes. (Even if you would like to know how the hell she found that website\u2014the last you heard, she still used dial-up.) And you don't necessarily want the whole gang to see what you're posting, either.\n\nBut sometimes denying or ignoring a family member's friend request\u2014or going the full distance and unfriending them altogether\u2014is just asking for trouble when a cousin's wedding comes around. Not only that, but chances are they know where you live and have access to embarrassing photos of you from when you had braces and did a weird thing with your eyes whenever someone brought out a camera because you thought you'd mastered Tyra's \"smizing\" lesson on _America's Next Top Model_. Proceed with caution.\n\n## Current Classmates\n\nWhile I've never been to prison, I've marathoned enough episodes of MSNBC's _Lockup_ to realize that school is the closest most of us will ever come to the clink. (Unless you're like me and had to spend three hours in the juvenile assessment center for stealing tarot cards, but that's a different story.) Your school experience likely consisted of monotonous days spent trapped in an institutional facility for hours at a time with random people you wouldn't choose. Sentence: twelve to life.\n\nWhat I'm saying is, tensions are already high. One of the best and worst things about Facebook is that it means you never have to look someone in the eye when you tell them they suck. But if you're unfriending someone you could easily run into in the cafeteria or dining hall on fish stick day, that aspect goes out the window\u2014and the potential for in-person drama is huge.\n\n## Former Classmates\n\nAh, yes, the classic Facebook friend, the one everyone is talking about when they're like, \"Suddenly everyone I know has two kids and is running an ugly yoga pants pyramid scheme out of their spare bedroom!\" These are people you haven't seen in a few years\u2014maybe more\u2014but you have a good enough idea of who they are that you enjoy creeping through their profile, scrolling past eight hundred pics of their moderately cute children to see if they've somehow maintained their looks since high school. Ending these friendships is relatively low drama, unless you still live in the same small town you grew up in, or plan on going to your high school or college reunion (which, unless it has an open bar, you can probably skip). The only risk is the loss of old happy memories, which you've most likely embellished anyway.\n\nI've had to drop the Facebook ax on far too many former classmates over the years. One girl, whom I'll call Sophie, is particularly memorable. Sophie and I were never close in high school, but we had a few friends in common and hung out a handful of times. While perfectly pleasant, Sophie was far too wild for my tastes\u2014your classic \"poor little rich girl\" who had more money than she knew what to do with, so she spent it on drugs and getting her light blue vintage Mercedes detailed every weekend. Seriously\u2014no one needs their car cleaned that often.\n\nGiven Sophie's party-girl past, I was surprised to discover that just six years after we graduated, she'd transformed into an evangelical conspiracy theorist. She filled my Facebook feed with a stream of obnoxious anti-vaxxer memes, Bible quotes, and rants against birth control. Well, maybe not that surprising\u2014Catholic guilt is fucked up and real. While most of her posts were beyond eye-roll inducing, I kept her around for the \"I can't believe this is who she turned into\" factor. Don't judge\u2014you know you've got more than a few of those in your feed right now.\n\nBut Sophie made a swift exit from my virtual circle in November 2008, right after Obama won the presidential election. While most of my friends were celebrating, Sophie was having a full-blown meltdown. She announced that she and her husband would be making some staffing changes to their family business in honor of Obama's win. Something like: \"We're going to stroll through the parking lot, and anyone who has an Obama sticker on their bumper will be let go. Let's see if this was the 'change' they were hoping for.\" No. Just no.\n\nI should've pulled the plug on our internet pseudo-friendship right then and there, but instead I did what I almost always advise people _not to do_ : hopped into the comments to \"remind\" Sophie that it's super illegal to fire someone because of who they voted for. I tried my best to reason with her, but she wasn't having it. After she made a few thinly veiled racist comments about welfare and compared Obama to Hitler, I decided I'd had enough and happily hit the _Block_ button on our relationship.\n\nAfter the 2016 election went the way it did, a lot of people were saying this kind of unfriending was bad, that liberals were living in internet echo chambers separated from the concerns of \"the rest of the country.\" While I do think it's our responsibility to educate people in our own communities, you can't build bridges with someone who not only refuses to do the work but also fires anyone who might be willing to. If we stopped to have an educational conversation with every offensive or hurtful person we've ever met, it would consume our entire lives. As much as it sucks, Sophie's own bias was going to keep her from being receptive to what I, a black person she hung out with three times in 2000, was telling her.\n\nThe lesson here is that ironic online friending is tacky, so when it blows up in your face, admit you brought this particular bigot on yourself and unfriend ASAP.\n\n## Coworkers\n\nTake it from someone who's made this mistake too many times to count: Skip the inevitable drama by not friending or following coworkers online until you or they have quit. Similar to prison\/school, these are people you're forced to interact with every day\u2014they're not your real friends until they earn it.  Coworkers are best seen as undercover agents, able to turn on you at any moment and use your gossip\u2014and Facebook posts\u2014against you.\n\nBesides, any potential friending that happens while you work together could bring drama to the office, and office drama has the potential to morph into financial drama, as in your ass could end up fired.\u2020 When it comes to work and social media, it's best to heed that immortal reality show mantra: \"I'm not here to make friends.\"\n\n(If it's too late and you already friended a coworker who turned out to be an irredeemable jerk on Facebook, check your profile settings and make sure they're on limited profile so they don't have access to your drunken photos and embarrassing status updates.)\n\n## Exes\/People You've Slept With\n\nUnless you're one of those freaks who can legitimately stay friends with your ex, keeping people you've dated or slept with in your digital feed is just nosy, and probably not good for your mental health. Lucky for you, these people are easy to unfriend: As soon as they say something even remotely awful, or the moment you get sick of their excessive new boo-thang PDA, you have valid reason to say peace. After all, if you wanted to tolerate that, you wouldn't have dumped them in your car in the Dairy Queen parking lot. (Though I hope you had the courtesy to drive them home afterward.)\n\nThen there are the \"I can't believe I slept with you, what are you even doing here?\" moments. In college, I occasionally hooked up with an RA. He was my first semireal relationship, meaning sometimes we'd go out to dinner before ending up in his dorm room. He was handsome and funny enough that I overlooked some of his annoying qualities, including his insistence that my middle-class upbringing wasn't a \"real black experience\" in comparison to his rough Detroit childhood. Or how he'd tease me about the high-pitched squealing I was known to partake in whenever I hung out with my best friend, De'Lon. Since the RA and I kept things casual, there were no hurt feelings when things fizzled out, securing his spot in my feed and in the fuzzy part of my memory reserved for nostalgia about carefree youth. But one comment changed all that.\n\nAfter I posted a photo of me and De'Lon drunkenly cuddled up at a piano bar on De'Lon's birthday, my RA fling casually commented, \"I knew he wasn't gay.\" Like a caramel bacon jalape\u00f1o cookie that leaves you thinking, _Here's a combination of things I'd never put together_ , this guy managed to cram homophobia, slut-shaming, and jealousy into one tiny bite-size sentence. And he was happy to say it publicly, on the record! In what universe is that your business, guy-I-used-to-get-naked-with-but-don't-anymore? What I should've done was tell him to take his outdated notions of masculinity and sexuality and place them directly in his heterosexual anal sphincter, but instead I just ended our online friendship and texted De'Lon a big LOL. We are still besties, and he is still very gay.\n\n## IRL Friends\n\nMayday, mayday! Major drama alert! For our purposes, I'm defining IRL friends as people you text regularly and go out of your way to fit into your offline life. If you live in New York, this may only happen, like, once a month, but that once a month is cherished. These are people you have no business communicating with on Facebook because they're sitting across from you at brunch and that's rude as hell. Unfriending them could affect your entire circle and make hangouts really uncomfortable. Your other friends may have to _choose between you_ , which is so high school.\n\nIf you're seriously contemplating virtually ditching a real-life friend, you should consider talking privately, away from your Facebook audience, first. If possible, take the conversation offline entirely and chat in person or by phone; if not, email is infinitely better than Facebook chat. Maybe whatever they said that enraged you got lost in translation, and there's an opportunity for personal growth that one or both of you need.\n\nBut no matter how close you and the other person may be, even if you go into it with the gentlest and most understanding of mind-sets, it doesn't always work out. In 2013, after George Zimmerman was acquitted for the murder of Trayvon Martin, the last thing I expected was to get into a Facebook argument with one of my childhood friends\u2014let's call him Andy\u2014over whether my grief was valid. Although we'd known each other since we were kids, Andy and I couldn't have been raised in more different environments. The oldest of three boys, he is white and grew up in a trailer park with his mom and alcoholic stepdad. My parents were divorced, and my mom and I lived in a three-bedroom house just across the canal from Andy's trailer.\n\nWe were close friends. Though I was only a year and a half older, Andy began calling me his big sister during my sophomore year of high school. He would tell me about the girls he was interested in, and I would give him advice if I approved or a side eye when I didn't. The single time I was ever grounded, he happily snuck into my backyard so we could share a joint through my bedroom window.\n\nOur friendship was so obviously platonic, my super-conservative mother didn't even bat an eye when he showed up at our doorstep one night in tears looking for a place to stay. His stepfather had the kind of horrible temper that always seems to accompany a drinking habit, so Andy had stormed off, vowing to never go back. He slept in our guest room that night, and I promised he could stay as long as he liked, even though I had no business offering.\n\nIf he sometimes saw me as a spoiled little rich girl with no problems, he never mentioned it, and I never detected it. He was always kind, and we treated each other with admiration and respect. So it was pretty surprising when, years later, despite the fun times we'd shared catching up over Christmas breaks and reliving old inside jokes on each other's Facebook walls, he jumped into one of my posts about our nation's broken justice system. He tried to shame me for ignoring \"the overwhelming drug and gang violence that dwarfs 'vigilante, and police' shootings all over.\"\n\nI had to take a deep breath. And then I began to type.\n\nI should've taken the conversation offline. Instead, I replied to his comment. I was angry, I was sad, and I felt like he was willfully refusing to understand where I was coming from. I wrote that it's possible to care about more than one thing at a time and that it was rude to derail our conversation by suggesting there were \"more important things to talk about.\" The entire thread descended into a silent screaming match as friends of every race and creed tried to explain to Andy why the idea that \"racism is a distraction to divide us\" was among the top three things people of color are sick of hearing from white people.\n\nAndy hit all the defensive-white-guy classics. \"The Irish were slaves, too!\" \"Classism is the real issue!\" Sharing a Morgan Freeman meme about colorblindness! As we all know, if Morgan Freeman, professional spokesman for black people and voice of God, says we should \"stop talking about race,\" then the argument is settled.\n\nBy this point, the thread had gotten out of control. Every time I tried to finish a thought, three new comments would pop up and the conversation would burrow deeper and deeper into the ground. There was no turning back. I sent Andy a private message to try to save him (and myself) further embarrassment. Though I thought he was totally misguided, I focused on the way he'd presented his ideas and not the ideas themselves. I said it was condescending for him to tell grown adults that they're \"wasting\" their time by talking about whatever they're interested in instead of whatever he's interested in. I added that, as challenging as his upbringing may have been, it didn't give him the right to speak over me, a black woman, as I discussed my experiences with racism and my fears about living in a world where my skin color is seen as a threat. In addition to the profoundly ignorant things he'd said, he'd been talking down to me, as if I were incapable of managing my time, interests, and emotions.\n\nI ended my message by saying he didn't need to respond unless he was prepared to apologize. He sent back another Morgan Freeman meme. I clicked \"unfriend.\"\n\nCould the friendship have been salvaged if we'd talked over the phone? The fact that he thought a meme was the best way to respond tells me no, but who knows? People who seem like jerks online often just don't understand that their humor isn't carrying\u2014that what they think is a lighthearted joke actually reads as dismissive and cocky.\n\nAt the very least, a phone call would've saved me the time and energy I spent hunched over my laptop, furiously typing and backspacing to try to mediate between Andy and my friends. The internet is a great place to be exposed to certain conversations, but a terrible place to have conversations yourself, especially if a friendship is on the line. Choose your opponents (and your friends, duh) wisely.\n\n## Internet Friends\n\nInternet friends are people you've built a relationship with over a few months\u2014or even years\u2014but haven't actually met in person, or have met in person only a few times. While internet friendships are very legitimate relationships, they're usually a lot easier to end because people have this lingering idea that they aren't _real_. ( _Homo sapiens_ spent hundreds of thousands of years only interacting with each other face-to-face, and that whole \"human history\" thing is hard to shake, no matter how convenient ordering your dinner on an iPad is.) Plus, you don't risk running into internet friends at a party postbreakup, and you usually have few IRL friends in common, if any. (If this weren't true, you would probably see them more often!) It gets a little sticky if you're all in the same industry and do most of your work virtually, but even so, these kinds of connections are supposed to come and go. Don't lose sleep over it.\n\n## Randos\n\nOh, right, that guy. Who cares? These are people you probably shouldn't have linked up with online anyway, friends of friends of friends you meet at parties or the men who start talking to you about \"synergy\" at PR events and send you friend requests before you're in your Uber home. When they get out of line online, your first reaction is \"Who?\" You can't even remember where you met them or when you added them on Facebook.\n\nWhy did you add them in the first place? Maybe you thought they'd be a good business contact, or you liked a joke they made about airline food, and you imagined your friendship blossoming like cherry trees in the spring. This is the wrong attitude to have this late in the Facebook game. Hanging out with someone once is not enough to let them into your online circle. They can float in pending-request purgatory until you're confident they're worth the risk. They don't have to know you hit _Decline_. Maybe you're just so busy living an exciting and glamorous lifestyle that you forget to look at your Facebook friend requests. Odds are they'll quickly forget about you, too.\n\nNow that we've laid out the types of Facebook fools who could potentially meet your ban hammer, I've created a handy flowchart that will help you decide how to proceed when someone inevitably blows up your feed.\n\nWith all this drama lurking just beneath the _Comment_ button, you may be wondering why you're still on Facebook anyway. Some people thrive on the attention, on the constant stream of new information and ideas and infuriating arguments and incredible links you have to click to believe. Some people make it work. But everyone's social media experience is different. As with any social network, you have to decide why you're on Facebook, and what you want or need to get out of it. Just as most serious conversations about race, gender, and politics are better had offline, there's nothing healthy about Facebook stalking, even though that's essentially what the platform was built for. If you find yourself feeling guilty because you didn't post enough photos of your recent trip to Bermuda, or if the unfriending experience has left you feeling gross and petty (and also like you have no friends, at all), it's possible you need to log off for a while.\n\nI know this probably sounds a little hypocritical, coming from a former social media callout queen and oversharer. If I hadn't succumbed to my worst online tendencies, I probably wouldn't have my career, let alone a book deal. But even as I've made massive, viral mistakes, I've stuck with these platforms because I've seen the good things they have to offer. In addition to _loving_ my friends' dog photos, I'm in a private Facebook group of WOC comedians from around the country, where we share audition horror stories and offer advice on everything from on-camera makeup to getting a better agent to dealing with difficult coworkers. It's really cathartic, and I've made some great connections and friends there that I doubt I would have offline. So for all the nuclear devastation that can take place on Facebook, the site isn't a completely hopeless disaster.\n\nBut as long as people have air in their lungs and Wi-Fi in their pockets, there will always be someone who makes you second-guess the access they have to your time and timeline. If worse comes to worst and you can't unfriend, well\u2014do you think you could manage two accounts? Embrace your new identity, Helga von Schwenderlaufer.\n\n# CHAPTER ELEVEN\n\n# LAST NAME BASIS\n\nI never thought I'd marry a white guy. I also never thought I wouldn't marry a white guy. In reality I never thought much about who I'd marry at all, because for a long time I was certain marriage wasn't for me. My parents' divorce made me skeptical of the whole institution, and the idea that I should decide in advance which races are acceptable for life partnership struck me as, well, superficial. Yet many people treat my relationship with Patrick, the man I've been with for eleven years (who just so happens to be white), like it's evidence that I'd been plotting some kind of racial coup. It's almost like I have to admit to having a white husband in the hushed tones I'd usually reserve for the confessional booth\u2014too often, my followers discuss their shock\/confusion\/disappointment about the moment when they found out my husband was white. As if he were a terminal illness I'd been diagnosed with. (How do I know? Well, they have these conversations in my Instagram comments.)\n\nOthers assume Patrick's whiteness was the foundation on which we built our modest, two-bedroom relationship. If whiteness were the reason for my relationship with Patrick, I can assure you we would've broken up a long time ago. His race doesn't help when one of our dogs gets diarrhea all over the living room carpet; it sure as hell doesn't help when he destroys our bathroom or when I need to do my makeup but he's in there shaving. Patrick's whiteness doesn't take care of any of the demanding aspects of partnership. Like any relationship, ours requires work.\n\nOur story is like something out of an early-2000s fairy tale: We met at a house party I'd gone to because I wanted to hook up with someone else\u2014the guy who was throwing the party. It was around the holidays, and I hit it off with Pat because I assumed he was in a relationship. I'd never had a serious boyfriend before, having always kept relationships in that awesome purgatory stage when you insist, \"We're NOT dating\u2014we only hang out after eleven p.m.!\" After the party\u2014where I, fortunately, did not hook up with the host\u2014Pat told one of our mutual friends he liked me, and she set up a \"group date\" at Panera Bread. Very casual. She never showed up, so we dined on bread bowls alone and then shared our first kiss in his car after I mocked the mistletoe hanging from his rearview mirror. We've been together ever since.\n\nToday, he and I have a podcast\u2014 _Last Name Basis_ , download it on iTunes!!!\u2014so people are used to our dynamic: We make fun of each other, a lot, and talk a lot of shit. Pat's sense of humor was the first thing I noticed about him, and I loved that he didn't pull any punches when it came to razzing me. When we first moved to New York we bought a used Nintendo Wii off Craigslist, and our _Mario Kart_ battles would get so intense that I worried the neighbors would call the cops; we competed with the out-of-work opera singer downstairs by screaming things like \"You drive like a fucking retiree, and I am going to destroy you!\"\n\nBeautiful, right? Not everyone thinks so. The first time the internet gave me shit for my relationship with Patrick was when I decided to document our three-year (dating) anniversary on YouTube. I was twenty-six and getting really into making videos at the time; Patrick was waiting tables at Olive Garden to pay off his student loans. He didn't really get the YouTube thing, but he decided to go along with it anyway. I told him we could talk about how we met and that it would be really funny and adorable, but not so adorable that he'd come off as soft or uncool.\n\nBack on our three-year anniversary, we weren't really a \"public\" couple, so when we made our YouTube debut, I'm sure more than a few people were shocked to see the nice girl with the hair tutorials trading slurred profanities with her white boyfriend. On top of drinking a lot of wine, I'd also lost my voice, so I sounded like a Russian gangster.\n\nIn the grand scheme of provocative #content, we were being tame. We made some off-color jokes about the fact that Pat had named his new black hatchback \"the Black Fist,\" and we yelled at our dogs between sips of wine. But unlike my other posts, which were produced and planned, the video wasn't intended to be anything but a chance to be silly and mess around. As we drank more wine, we started play-fighting more, and the types of biting comments we always make started to flow. It was all in good fun. I didn't think too much about it when, the next day, I woke up with a headache and posted it.\n\nEven back in 2009, when callout culture was but a twinkle in Twitter CEO @jack's eye, I was foolish to think this video wouldn't get noticed. The jokes were inappropriate, and I had a big enough following that a sizable number of people would see it. Still, I was surprised when I checked my email and saw an inbox full of enraged comments. They weren't even really about all the \"not PC\" things we were saying; instead, they were from a group of very disgruntled black viewers who believed interracial relationships were a plot to destroy the black family. Any black woman who would willingly date a white man, they assured me, must be pathologically self-hating.\n\nThe video quickly racked up fifty thousand views, more than anything I'd ever posted at that time. As I combed through the comments, I started to notice a trend: The most aggressive posts came from avatars and usernames that recalled ancient Egyptian and Afrocentric themes. Who were these people?\n\nApparently my video had been shared by a popular \"hotep,\" which inspired a wave of commenters to attack me for \"disrespecting my blackness.\" I soon learned that hotep (which is an Egyptian word that roughly means \"to be at peace\") was the name adopted by black folks across the internet as both an insult and an identity. Hoteps often dabble in misogyny, homophobia, conspiracy theories, and fanciful African origin stories that link eating red meat to white supremacy; feminism to a plot to destroy the black family; and everything from hurricanes and headaches to slavery. According to my new hotep \"fans,\" I was so self-hating and superficial that I'd hopped into bed with the first white guy who would have me. And to make matters worse, he wasn't even a rich white guy!\n\nThis logic still confuses me to this day: It's superficial to date a white guy, but not superficial if you make sure he's got some money? The comments ranged from laughable\u2014\"I just think you should know: He's cheating on you. I can tell by his eyes\"\u2014to full-on misogynoir: \"Black women are so desperate for white acceptance they'll be with a white guy for three years even though he hasn't put on a ring on it.\" I was so shocked by the intensity of the comments coming from black people in particular\u2014and the fact that almost no one directed any of the vitriol at Pat\u2014that I took the video down.\n\nThen I got a text from a friend: \"hey girl... just wanted to let you guys know I saw an ad for an interracial porn site and it was you and pat? hope ur ok xoxo.\" I didn't bother asking what my friend was searching for when they stumbled across the ad (no judgment) and instead requested a link, ASAP. Yes, someone had spliced up our drunk anniversary home video so that lines like \"Are you nervous?\" and \"We've never made a video together before...\" took on a completely different meaning. The still image was a shot of me and Pat drunkenly clinking our glasses and toasting to our foray into amateur erotica. Awesome. Thankfully I was able to file a claim through YouTube to get it removed before someone tried to add it to my IMDb page.\n\nWhere did the idea that interracial relationships are incompatible with the fight for equality come from? My white husband doesn't make me any less black, or any less dedicated to the fight for racial justice\u2014just as being married to a man doesn't make me any less of a feminist or passionate about women's issues. Perhaps some forget that interracial marriage was at one time, not so long ago, a civil rights issue; it was illegal in many states until 1967, when the landmark Supreme Court case _Loving v. Virginia_ determined that anti-miscegenation laws  were unconstitutional. \"There is patently no legitimate overriding purpose independent of invidious racial discrimination which justifies this classification,\" the court wrote in the _Loving_ decision. \"The fact that Virginia prohibits only interracial marriages involving white persons demonstrates that the racial classifications must stand on their own justification, as measures designed to maintain White Supremacy.\" In other words, the idea that white people should not \"mix\" with other races upholds white supremacy. It's segregation. It's racist.\n\nTo be fair, I can see why some people of color assume the worst when it comes to folks with white partners. Perhaps they've been fetishized, or dealt with more blatant racism, in an interracial relationship themselves. Or maybe they've met POC who wave their white partners around like a badge of honor, or who use their white partners to put down members of their own race. So it's no wonder they look at me and my white husband with one eyebrow raised. But the same frustration I feel when white people make assumptions about who I am because of my blackness is echoed when black people assume I must be a certain kind of person because my partner is white. It often feels as if the whole world is placing expectations on me and pressuring me to limit my experience\u2014or to define it entirely\u2014based on my race.\n\nNeedless to say, Pat didn't show up in another video of mine for years\u2014I realized then that I didn't want to make my private life so public. But that hasn't stopped people from making bizarre assumptions about our relationship. Sometimes they're not so abusive, but they're no less rooted in racism. A few years ago I was at a conference when a black woman came up to me to commiserate about how much dating black men sucks. \"So happy to meet you!\" she said, breathlessly. \"We have so much in common\u2014my husband is white, too.\" (Note: This doesn't mean we have very much in common.) She continued: \"I gotta commend you, girl\u2014these niggas don't have their shit together. When I was single I said to myself, 'I need to get me a white man.'\"\n\nI was at the conference for work\u2014acting as the spokeswoman for a family planning site that encouraged women to use birth control\u2014so I tried to keep my jaw off the floor. Is this how people assumed I felt about my relationship? In my mind, I was screaming, _My dad is a black guy! The last dude I kinda-sorta dated was a black guy! My best friend is a black guy! I probably would've married him if he weren't gay, but that's not the point!_ Instead, I told her I didn't go looking for any certain type of guy\u2014Pat was just who I fell in love with\u2014and flashed her an obviously fake smile as I thanked her for saying hello.\n\nThis should go without saying, but there's nothing inherently special, progressive, or earth-shattering about dating or marrying someone of a different race. Sure, you and your partner may encounter nasty quips from strangers, pushback from friends and family, and weird fetishy comments about your unborn children, but bragging about having a [insert race here] spouse is just not a good look. While my blackness is part of who I am, it's just that. A part. It doesn't define me as a person, and I sure as hell wouldn't be okay with Patrick describing me as his \"black wife.\" I am his \"loudmouth wife who laughs like a witch and is too fond of puns, but who I love anyway.\"\n\nStrangers who won't mind their own business aren't the only people who have questioned my decision to shack up with a really nice lawyer who once took a seven-hour flight to Paris in dress shoes and a stuffy suit to surprise me with a wedding proposal. While many assume I was on the hunt for a \"white kang,\" in reality I'd dated (aka slept with\u2014sorry, Mom) all sorts of guys before Pat came along and became my first serious boyfriend.\n\nIntroducing him to my family was nerve-racking, and they didn't really make it easy. Right around the fateful anniversary video, I bought myself a new car after years driving a clunky old Volvo. My new Toyota Yaris was the first car I'd paid for entirely with my own money, and I was feeling very grown-up. My uncle, who'd lent me money a few times when I was in college and therefore got a kick out of teasing me about every adulthood milestone, called to congratulate me.\n\n\"Heard you got a new car. What color is it?\" he asked.\n\n\"Blue!\" I replied, beaming. I'd heard a weird statistic that red cars often had higher insurance rates, so I felt proud of myself for being responsible and picking something more sensible, even though I wasn't sure it mattered to Geico.\n\n\"I'm surprised it isn't white, to match your li'l boyfriend.\"\n\nYep, that's right\u2014my uncle called me up and pretended to congratulate me on an achievement so that he could comment on the fact that my boyfriend was white. Not even really because he wanted to say anything about the fact that he was white\u2014but it felt as if he wanted to make sure that I knew he was judging me. Even though he knew I wouldn't have been able to keep a white car clean to save my life.\n\nWorse was my dad's account of meeting Pat for the first time. We'd been together for a few years, so our love was not some illicit wrong-side-of-the-tracks romance\u2014we'd already moved to New York together and were in the \"Why the fuck is the bathroom so disgusting?!\" phase. I was visiting my father in South Carolina when I overheard him on the phone in his bedroom, saying something like \"... and she brought this li'l _white boy_ up in my house.\"\n\nDon't ask me why the male members of my family liked to refer to Patrick, who is six feet tall and a smooth two-hundred-plus pounds, as \"li'l,\" but to them that is what he was: insignificant, petty, silly. I stormed into my dad's bedroom and yelled at him to \"GET OFF THE PHONE RIGHT NOW.\"\n\n\"How dare you say that about him?\" I screamed like I was the lead actress in a drama about an interracial couple who were determined to fight for their love whatever the cost. \"You don't know him! I love him! I respect him! And you should like and respect him because I do! He treats me right!\"\n\nMy dad apologized immediately and said he was proud of me for \"standing up for my man.\" I knew he meant it, but I also couldn't help but wonder if I'd always have to defend my relationship to my family.\n\nWhen black people are questioning the validity of my relationship, it helps, of course, that I don't have to make any excuses for Pat. There's always an assumption that, because he's white, Pat and I must have awkward problems discussing the uh, you know, _ahem_ , political situation. Or assumptions that he mumbles racial slurs in his sleep as his subconscious fights to express how he really feels about the blacks. Sometimes people, thinking they've caught me, will ask, in a _gotcha!_ tone, \"Well... does he support Black Lives Matter?\" To which I say: \"Duh?\" Even if you put aside the basic fact of my existence as a black person, social justice is a huge part of my career. I suppose there are couples who keep their personal and professional lives separate\u2014if I were married to a physicist I doubt I'd be able to explain the intricacies of what he did every day\u2014but \"the police shouldn't kill black people\" is not exactly quantum mechanics.\n\nIt's a little absurd that I have to recite my husband's social justice credentials to prove he's worthy of me\u2014I'm a grown-ass woman who can make choices for herself. But more than a few people are willing to write me off as an activist because I'm married to a white guy, so here goes: Patrick is really empathetic, to the point that I can't even really complain to him about a woman cutting me off in line at the grocery store because his response is usually something like \"Maybe she was just having a bad day. I'm sure she didn't mean it.\" I wouldn't marry anyone who didn't care about people, but his compassion is next level. (I'd like to say \"I'd never even date anyone who didn't care about people,\" but my twenties were a dark time.) Early in our relationship, whenever I talked about the racism I dealt with in high school and college, he would always ask how he could support me. In law school, he interned at a public defender's office, and he's worked pro bono for survivors of domestic violence and people dealing with discrimination. When Mike Brown was killed in Ferguson, I would talk to Patrick about what was going on in the news, and he started using Twitter in earnest for the first time in order to follow live streams and other updates that mainstream news organizations weren't reporting on.\n\nOnce, he confided that one of his college friends had said I \"went too hard about racism.\" Immediately, I got mad and starting yelling. \"Well? So? What did you say?\"\n\n\"I told him that you're passionate about racism because it affects your life, and that there's no such thing as going 'too hard' about inequality.\" And that, reader, is one of the many reasons I married him.\n\nPat is no longer friends with that guy, but it did make me wonder if other white people Patrick knows\u2014or his family\u2014have questioned his choice to marry someone who's so vocal about social justice and fighting white supremacy. And while I should be used to it by now, I doubt I'll ever grow tired of watching racist trolls and \"men's rights activists\" shit themselves over my wedding photos. \"Who is this white man? Doesn't he know she HAAAAATES white men?!?!\" they scream. When I see this, I just laugh, and then imagine their heads exploding, leaving what little brain matter they had to begin with splattered across their dank basement walls. I've made up my mind to stop wasting time or energy on these sorts of comments, but inevitably one of my fans will leap to my defense with \"How can she hate white people when she has a white husband? Bloop!\" Which instantly turns my self-satisfied cackling into a full-blown _oh no_.\n\nWhile I appreciate followers having my back, the [insert race] partner defense deserves a swift retirement. Arguing \"But I have a white husband\" would make me no better than the proud bigots who trot out their \"black ex-girlfriend\" (who, let's be honest, probably doesn't exist) when someone calls them out on their racism. Of course I don't hate white people, but my husband isn't proof of that. If the ability to do the horizontal mambo automatically meant you shared a mutual understanding and respect for your \"dance partner,\" my college years would've been a lot less depressing. But at the end of the day, who you befriend, date, marry, or get naked with doesn't say shit about your morals or understanding of privilege, oppression, or social justice. That's like a serial killer proclaiming, \"I have plenty of friends who are alive!\"\n\nI often have to deal with people questioning the validity of my activism and blackness because of my marriage, but strangely enough, Patrick doesn't encounter nearly the same level of vitriol. (Try guessing why...) Occasionally he'll get online messages from anti\u2013\"social justice warriors\" to the tune of \"How can you be married to someone who hates the white race so much?\" along with the occasional \"cuck,\"  but in general the harassment gets funneled through me. People don't feel the need to criticize a white man for his choices, but they assume, paradoxically, both (1) that they need to \"warn\" me about white men because I can't fend for myself, and (2) that I'm trying to single-handedly destroy the black race because I share a refrigerator with a white man.\n\nAnother moment in our relationship that illustrates this perfectly took place shortly after we got engaged. Two things happened: A bunch of Patrick's extended family added me on Facebook, and Obama was reelected. The night of his win, I posted something celebratory on Facebook\u2014don't remember what. Suddenly, a photo I didn't really recognize popped up under my status update. A negative comment about Obama. I was deleting people left and right that night. I don't remember what was said; all I remember is hitting the trusty _Unfriend_. I realized she was Patrick's relative a minute later, when she messaged me: \"You don't know what you're marrying into.\" Then she blocked me.\n\nAdrenaline up from arguing with people all night, I asked Patrick what the FUCK that was supposed to mean. He admitted he was as confused as I was and said he'd talk to her, and I completely forgot about it.\n\nFast-forward to about five years later, Pat's family reunion in Tennessee. I'm admittedly not a fan of the great outdoors, but I ended up having an amazing time; we went whitewater rafting and hiking during the day and played card games and drank moonshine late into the night. Although I knew only a handful of people at the start of the week, by the end of the trip we were all sharing funny stories and proudly wearing our family reunion T-shirts. It was nothing at all like the stereotypically dramatic family reunions portrayed on TV. It was great.\n\nAs we were getting ready to leave on the last day, when everyone was saying their goodbyes and making plans for the next meetup, one of Patrick's family members came up to me. \"It was so great meeting you,\" she said. \"I had such a good time and... I'm really happy for you.\" And then she started crying.\n\nI was a little confused\u2014had we connected in such a moving way? I couldn't remember us exchanging more than a few words, because she had been awfully quiet all week. I tried to go with it. Then she added, \"I just want to apologize for everything. I feel so bad. The whole weekend I was thinking that you must hate me, and I'm just so sorry.\"\n\nNow totally confused, I replied that it was great to meet her, too, and that there was nothing to apologize for. What was she talking about?\n\nIt didn't hit me until we got in the car to go to the airport that she was the girl from my Obama Facebook post, and that she had been so quiet because what she'd said to me had been weighing on her mind. Though I don't want to dismiss the very real and very awful racism that people sling online, I can't help thinking that this is how a lot of casual online racists would react IRL\u2014ashamed and regretful. People are very bold on the internet; they are not bold in person.\n\nPat and I didn't start _Last Name Basis_ because we wanted to make a podcast \"about\" being in an interracial relationship\u2014sure, we talk about race and social issues, but we also gripe about our loud neighbors, tease each other relentlessly, argue over furniture, and share our funniest and most embarrassing New York adventures. In that way, it's a show about two people who love each other, even though they have different interests and don't always agree. If anything, the fact that he's white and I'm black should create more challenges\u2014but it's only ever been an issue for other people, never for us.\n\nI realize that we're more of an exception than the rule. Even couples who have everything in common have challenges and struggles; it's just not possible to always view everything through the same lens. Every week we field email questions from mixed-race couples struggling to communicate about race, privilege, politics, and not-so-secretly racist in-laws. Although our relationship is far from perfect, and although I don't believe in \"marriage experts,\" I'm flattered that people look up to us and seek our advice.\n\nHere's what I've learned: People will have opinions about your relationship no matter who you are or who you love. But it doesn't really matter if some guy in Oklahoma thinks your gay interracial relationship is a sign of the impending apocalypse, because you're not married to some guy in Oklahoma. As difficult as it may be, your energy is better served by focusing on the internal work: compromise, understanding, and growing together. That means giving your partner the grace to learn, just as you'd want them to do for you. Be honest about what you want and what your boundaries are. Don't put off the hard conversations; don't wait to talk about your morals, your politics, or your beliefs. Or your personal standards of bathroom cleanliness.\n\n# THE DEBATE\n\nThere's a lot of confusion on the internet about where my true allegiances lie. Am I racist against white people, or am I a traitor against black people? Am I \"anti-white,\" or am I a \"self-hating bed wench\"?  Who do I hate more: Myself? White men? Black men? Or all men?\n\nTo settle the score once and for all, I decided to stage a debate, live, in front of an audience. Below is the complete transcript from the event.\n\n**FRANCHESCA RAMSEY:** Good evening from the Reddit offices, located in the pits of hell. I'm Franchesca Ramsey, and I want to welcome you all to the first ever debate to determine my true essence. Am I a self-hater, a man-hater, or both? To help us answer this question, I've invited prominent leaders of two groups who've devoted countless hours to theorizing and speculating about the burning contents of my heart and loins. Please join me in welcoming noted men's rights activist (MRA) Colton Harris Gordon and a leading Twitter hotep, Dr. A. G. Stankofa.\n\n[ _Awkward applause. A hushed voice can be heard saying, \"I thought this was going to be a screening of_ Wonder Woman _?\"_ ]\n\n**MRA, COLTON HARRIS GORDON:** I welcome the opportunity to prove your commitment to white genocide.\n\n**RAMSEY:** Whoa, whoa, whoa! Slow down there, Colton\u2014we haven't started just yet!\n\n**HOTEP, DR. A. G. STANKOFA:** Of course the white man finishes before we've even begun. Premature ejaculation is just one side effect of a savage diet. But you'd know all about that, wouldn't you, Chesca?\n\n**RAMSEY:** I can't say I know anything about Colton's eating habits, but let's not get ahead of ourselves here. Tonight's debate will be divided into three segments. I will pose the same question to both of you, and you'll each have up to one minute to respond. We'll then close out the evening by taking questions from the audience before determining who has won the debate. And with that, let's get started!\n\nFirst question: My husband! He's white. Dr. Stankofa, you're up first.\n\n**HOTEP:** Unsurprising. Your melanin levels have clearly been depreciated because you share a bed with a white devil. These white demons age like milk, which is why they seek to drain the life-giving melanin from our queens. You hate yourself and your people so much that you abase yourself to be desecrated by the pasty curd sack you call a spouse.\n\n**RAMSEY:** Thank you. I appreciate that feedback. Colton, same question.\n\n**MRA:** I'd first like to say that the topic you've presented is not a question. Second, I'm in my mid-thirties, and people regularly assume I'm in my early thirties, so the milk-aging thing is patently false. Just one more example of black supremacist corporate media shills spreading lies about the white race\u2014\n\n**RAMSEY [interjecting slyly]:** I don't see age, so I wouldn't know.\n\n**MRA:** I'll admit I was shocked when I learned Franny was married, let alone married to a white man. But upon further research it became clear that this \"marriage\" is an inside job. Nothing more than an advanced plot to eradicate the white race and subjugate white men under the guise of matrimony.\n\n**RAMSEY:** So you're saying I'm smart, creative, and have excellent leadership skills? Duly noted.\n\nNext question: Complete the following sentence: \"Franchesca is a feminist because...\"\n\nColton?\n\n**MRA:** She hates men.\n\n**RAMSEY:** Dr. Stankofa?\n\n**HOTEP:** I'd have to agree with the alabaster bastard. But I'd add that Chesca hates men because she's never had the pleasurable fulfillment that can only be experienced when a female's brain is fertilized by a black man's royal obsidian.\n\n**RAMSEY:** Interesting. What do you make of the argument that it's possible to advocate for racial equality and be in an interracial relationship?\n\nDr. Stankofa?\n\n**HOTEP:** That sounds like some gay shit.\n\n**RAMSEY:** And you, Colton?\n\n**MRA:** I'm pretty sure I have a meme for that... give me a minute...\n\n**RAMSEY:** Oops, your minute's up! Looks like that concludes tonight's debate\u2014thanks to everyone for participating. I'm sure you'll all use the hashtag #WhiteGenobride on social media to let us know what you thought.\n\n**HOTEP:** Hey, wait a minute. I thought we were ending with audience questions? This event is clearly rigged in the chalky gargoyle's favor.\n\n**RAMSEY:** Oh, goodness! You're right! Let's go ahead and take one audience question to close out the evening.\n\n**AUDIENCE MEMBER:** Hello. First, I'd like to say thanks for hosting such an enlightening and intellectually stimulating discussion, Franchesca. It's truly been delightful. So, I guess my question is, \"Why can't either of you mind your own fucking business?\"\n\n[ _Audience applauds, then cues up their reaction GIFs._ ]\n\n# CHAPTER TWELVE\n\n# EULOGIES FOR CRINGEWORTHY COMMENTS\n\nWe are gathered here today to say goodbye to a list of comments that just won't accept that their time has come. If you've ever been brave enough to call out systems of oppression, whether in a one-on-one conversation or on a larger scale, you may have encountered some of these not-so-witty retorts. Born of racism, sexism, homophobia, transphobia, and\/or simple ignorance, these reactions tend to be passive-aggressive, microaggressive, and gaslighting; they downplay arguments against bigotry so that the person committing it can save face or look smart. In reality, they make you look the opposite. They've been languishing for a while, so let's put them out of their misery once and for all.\n\n# 1. COMMENT: \"SORRY IF YOU'RE OFFENDED.\"\n\n**Often heard when:** Someone has done something hurtful or offensive, and they don't understand (or care) why what they did was wrong. But they also don't want you to be mad at them.\n\n**Why it should be laid to rest:** When \"if\" sneaks its way into an apology, it corrupts the entire thing from the inside out. Much like the \"RE: URGENT DEAR SIR OR MADAM\" email that somehow managed to slip past your spam folder, \"sorry if\" is highly suspicious. Humans are proud, often obnoxious creatures by nature\u2014we hate apologizing, because apologizing means admitting we did something wrong. So we try to have our cake and eat it, too, by apologizing \"if\" we've offended anybody. That \"if\" is tricky\u2014it suggests the person in question might not be offended, when in actuality the person has already clearly conveyed that they are. \"Sorry if\" also fails to take responsibility for the offensive words or behavior by making the apology conditional. To make matters worse, what comes after the \"if\" is just as pathetic: \"Sorry if _you're offended_ \" is a cop-out because that comment \"apologizes\" for another person's feelings rather than owning up to what you did to cause those feelings. This is a non-apology masquerading as a real apology, and it's really condescending to boot.\n\n**Comeback:** \"There's no need for 'if'\u2014I am offended. So are you sorry?\"\n\n# 2. COMMENT: \"WHY ARE YOU SO ANGRY?\"\n\n**Often heard when:** You're angry. Which means you're dropping F-bombs and\/or harshly worded opinions about injustice, oppression, or prejudice. Or you're just passionate about something and expressing that.\n\n**Why it should be laid to rest:** \"Why are you so angry?\" and its bratty cousins \"You're making a scene,\" \"You're being hysterical,\" and \"You need to calm down\" are part of a broader phenomenon known as _tone policing_. Enacted by what's known as the Tone Police, tone policing is the offering of unsolicited advice on how to express one's feelings. The Tone Police claim expertise on topics they heard about on _The Wire_ or in informational brochures for NGOs they contemplated volunteering for, and they do this by questioning the intensity of others' reactions to these topics. In turn, the Tone Police derail the conversation, away from the systemic wrong you were discussing and toward the validity of your response to that wrong. Another way to put what they're saying is \"I feel uncomfortable when you feel comfortable feeling freely.\"\n\nWhy are they so uncomfortable with your feelings? Maybe it's because they feel complicit or guilty. Maybe it's because they really just don't get what it's like to connect with oppression in a personal way. They may need to be exposed to that anger to inspire and provoke action. Anger can bring awareness to the severity of an issue and be a meaningful part of one's education. Anger is a valid emotion, and it doesn't lessen or invalidate truth. Not to mention, emotional distance is not an option for many marginalized people, and telling them to calm down only increases the alienation they may feel in these moments.\n\n**Comeback:** \"I think people should be treated fairly. And when they're not, it makes me angry. Why aren't you angry at all?\"\n\n# 3. COMMENT: \"MAYBE IF YOU STOPPED TALKING ABOUT THIS ALL THE TIME, IT WOULDN'T BE SUCH A PROBLEM.\"\n\n**Often heard when:** A problem is being discussed.\n\n**Why it should be laid to rest:** Awareness, education, and discussion are essential to understanding a problem before working toward a solution. But... that isn't always the case. Here are some issues we have definitely made successfully go away after ignoring them:\n\n\u2022 Student loan debt\n\n\u2022 Algebra homework\n\n\u2022 The growing stack of dishes in the sink\n\n\u2022 The New York City subway system\n\n\u2022 That pesky burning sensation when you urinate\n\n\u2022 The fact that your favorite jeans don't fit anymore\n\n\u2022 Overdraft charges\n\n**Comeback:** \"Maybe if I ignore my taxes they'll go away? It worked for Wesley Snipes and Lauryn Hill!\"\n\n# 4. COMMENT: \"RACIST? I'M NOT RACIST! _Y OU'RE_ THE REAL RACIST!\"\n\n**Often heard when:** You call out racism.\n\n**Why it should be laid to rest:** \"I'm rubber and you're glue, whatever you say bounces off me and sticks to you\" hasn't been a valid line of defense since elementary school.\n\n**Comeback:** \"Just like talking about global warming doesn't make me a greenhouse gas, talking about racism doesn't make me a racist.\"\n\n# 5. COMMENT: \"STOP PLAYING THE VICTIM.\"\n\n**Often heard when:** You're expressing feelings of sadness, disappointment, or fear in response to injustice.\n\n**Why it should be laid to rest:** Unless you're performing in a community theater production of _Law & Order: SVU_, I'm not sure why anyone would willingly choose to play the victim. The word \"play\" implies a fun or enjoyable experience, but there's nothing fun about the very real pain associated with being mistreated. Only people who lack empathy would see victimhood as some kind of game or performance with ulterior motives.\n\nThere's also little to gain from \"pretending\" to be a victim. For example, survivors of sexual assault who come forward often face harassment and abuse from strangers and those close to them alike; they get blacklisted and lose their jobs, and their motives are often questioned to the point that they recant their story to avoid the continued abuse. \"Stop playing the victim\" is yet again an example of someone feeling uncomfortable or threatened by the feelings associated with your lived experience. Just as you're allowed to feel angry about oppression or injustice, it's natural to express feelings of sadness or fear, especially if you're exhausted by the prospect of trying to keep up a brave face.\n\n**Comeback:** \"Victimhood is not a game. I don't get a prize for talking about my experience.\"\n\n# 6. COMMENT: \"DON'T YOU HAVE MORE IMPORTANT THINGS TO THINK ABOUT?\"\n\n**Often heard when:** You're talking about something that's important to you.\n\n**Why it should be laid to rest:** Humans have large brains. That is why it's possible to care about all different types of things simultaneously, and why everyone's priorities are different. You might think petitioning the Olympic committee to make camel wrestling an Olympic sport is really important. Meanwhile, I didn't even know camel wrestling was a real thing. (It is.) But whatever you do with camels on the weekend doesn't stop you from also volunteering at your local animal hospital or donating money to cancer research. Plus, if you cared about every single pressing, dire, and horrible issue on earth all the time, at the same time, you would have a breakdown. You don't go to a concert and yell, \"Hey! I know everyone's having fun at a concert, but climate change is happening!\" Besides, by going to the concert, you're sort of participating in the unimportant thing you're saying people shouldn't be participating in, because you're at the concert. No?\n\n**Comeback:** \"Don't you have more important things to think about than what I should be thinking about?\"\n\n# 7. COMMENT: \"IT'S JUST A JOKE.\"\n\n**Often heard when:** Someone is trying and failing to be funny, but succeeding at sounding like a bigot.\n\n**Why it should be laid to rest:** People feel like using the word \"joke\" removes responsibility for the hurt their words may cause. It doesn't; in fact, it may make things worse. In an early episode of _Decoded_ , \"How Do You Handle a Racist Joke?\" I talked about the difference between offensive jokes, which make people uncomfortable or mad or hurt their feelings, and oppressive jokes, which reinforce negative stereotypes about marginalized people, thus upholding the mistreatment of an entire group of people, not just an individual.\n\nA Western Carolina University study found that when prejudiced attitudes around a marginalized group are shifting, negative jokes about that group can suggest discrimination is justified to people who may be on the fence. \"Ironic\" stereotyping\u2014where the joker says something designed, allegedly, to mock sexists or racists or transphobes or homophobes\u2014counts, too, so if anyone ever serves you with, \"Political correctness is ruining comedy,\" the same stuff applies. It's just not funny to disrespect other people's beliefs, backgrounds, cultures, or identities when those are deeply rooted in longstanding oppression.\n\nMany behaviors and practices that were once considered harmless jokes have since been recognized as irredeemably bigoted. For example: Blackface. In the past, a white actor covering their face in shoe polish and trotting out the worst stereotypes about black people was seen as comedic genius. That's because, at the time, black people didn't just lack positive representations of themselves in media; we weren't even considered human.\n\nJokes that punch down on marginalized people require no creativity because they've existed since the beginning of time. It's like telling a knock-knock joke and believing you're Richard Pryor. That's why these \"jokes\" sound really old-fashioned and stale: They uphold the status quo by perpetuating harmful ideas about people who're already in vulnerable positions in society. Imagine what would happen if an elementary school principal went in front of the school and made fun of a seven-year-old. No matter how the student responded, the principal would automatically have the upper hand because they're older and in a position of power. Not only is this scenario completely unfair, it's also the plot of _Matilda_.\n\n**Comeback:** [crickets] \"Then why wasn't it funny? Maybe you can explain the joke to me.\"\n\n# 8. COMMENT: \"WELL, I DON'T SEE COLOR.\"\n\n**Often heard when:** You're describing your experience as a person of color.\n\n**Why it should be laid to rest:** People who say this usually mean well\u2014they're attempting to let you know they're sooo not racist that they can't even conceive of a reality in which racism exists! But what they're actually saying is that racial _identity_ is bad\u2014not that racial oppression is bad. If someone is talking about their experiences as a person of color, \"I don't see color\" suggests their experiences aren't valid\u2014or flat-out aren't real. If you wear glasses and I say, \"I don't even see your glasses,\" that doesn't mean you suddenly have twenty-twenty vision\u2014it just means I'm in denial. Or maybe that I need glasses myself.\n\nA person's racial identity doesn't define them, but it does play a part in who they are. For many, \"I don't see color\" is an attempt to say \"I don't see you as any different from me or anyone else,\" which is good in theory, but not realistic. People are different! And that's okay! Our differences aren't the issue. It's treating people as less than because of their differences that's the problem.\n\n**Comeback:** \"Well, even in black and white, I'm still a person of color.\"\n\n# 9. COMMENT: \"WHY DON'T THEY JUST COME HERE LEGALLY?\"\n\n**Often heard when:** You're talking about undocumented immigrants or the Deferred Action for Childhood Arrivals (DACA) program.\n\n**Why it should be laid to rest:** If this question is being asked in earnest, the person you're talking to clearly has no idea how the American immigration system works. It is expensive, confusing, and prohibitive even for people who speak English as their native language, and who aren't dealing with the trauma of repressive governments and persecution. Besides, do you remember how this country was founded? Did Christopher Columbus come here legally? Did the Pilgrims have to make sure they had their passports open to the photo page so the line would move quickly at Plymouth Rock? Yes, there are immigration laws in place, but laws are not perfect. Part of our work, as activists and as citizens, is to determine what laws need to be refined and what new laws need to be created, and then to pressure our elected officials to act on that.\n\n**Comeback:** \"Why don't we just fix our broken immigration system so the people who want to come here legally can do so?\"\n\n# 10. COMMENT: \"THE IRISH WERE SLAVES, TOO!\"\n\n**Often heard when:** You're discussing the repercussions of the transatlantic slave trade in today's modern world.\n\n**Why it should be laid to rest:** When nearly two million Irish people immigrated to America during the Great Famine of the nineteenth century, they were abused, harassed, oppressed, and portrayed as diseased, poor rapists looking to steal American jobs. They were not, however, forced into slavery. The radical historian Noel Ignatiev's book _How the Irish Became White_ describes how the population\u2014many of whom came to America as indentured servants, under a system which, unlike slavery, was a voluntary arrangement, even if it was exploitative\u2014were able to shed their nonwhite status and achieve equality by abusing black people and accepting racism. Let's agree that all persecution is bad, but not all persecution is equal.\n\n**Comeback:** \"No, they weren't!\"\n\n# 11. COMMENT: \"THAT'S TERRIBLE. BUT MAYBE SHE SHOULD'VE...\"\n\n**Often heard when:** You're discussing sexual harassment or sexual assault.\n\n**Why it should be laid to rest:** I'll never forget one particularly beautiful fall afternoon when a few of my female coworkers and I were sharing our best\/worst street harassment stories. Normally, these storytelling sessions would happen in one of our offices, where we could comfortably splay across a couch surrounded by a mountain of decorative pillows featuring ironic sayings and nostalgic cartoons. (I never watched _Jem and the Holograms_ , but I've always liked the artwork.) Anyway, on that crisp Wednesday, instead of gabbing in a private space, my lady friends and I were partaking in the ancient pastime of water-cooler talk in the kitchen for all to hear.\n\n\"Did you hear about the woman in Chicago?\" one friend began. \"Some guy asked for her phone number, and when she ignored him he stabbed her right there on the train. And then, instead of calling the police, everyone took out their phones and filmed it. It's terrifying. That's why I never take my headphones off.\"\n\nRight on cue, a guy we'll call High-Key Jerk swooped in to disabuse us of the naive notion that a woman who was stabbed\u2014with a knife\u2014for not giving a random guy her number had been wronged in some way.\n\n\"Don't get me wrong, that's fucked up. But I don't understand why more females don't just give dudes their number when they ask.\"\n\nOh man, that woman who got stabbed\u2014why did she have to be such a buzzkill? She didn't have to answer his calls! And if he tracked her down to get revenge on her for not answering his calls, well, maybe she should've answered his calls.\n\nWhile some of the details of that day are so clear\u2014the sesame-and-onion scooped bagel halves we grazed on while sipping burned lattes from the office Keurig\u2014the exact dialogue of our conversation escapes me. So I'm forced to paraphrase to the best of my ability here. I believe I said something along the lines of:\n\nWhat in the ever-loving fuck?\n\nInstead of asking why women can't be bothered to hand out their number to every dude with a fresh Cricket Wireless plan, why can't sensitive-ass man-babies learn how to take rejection without resorting to full-blown temper tantrums that end in physically harming women minding their gatdamb business?\n\nHigh-Key Jerk was taken aback. But he recovered, and quickly hit me with a \"Well, damn. See? This is what I mean. Clearly I caught you on your period.\" In that moment, he was like victim blaming: dead to me.\n\nUnless you have a time-travel device handy, there isn't much value in theorizing how someone could've avoided being the victim of an assault of any kind. Maybe if I had worn a blue shirt instead of a red shirt, a bird wouldn't have pooped on me. See how ridiculous that sounds? Of course, it's extremely easy to suggest how a terrible situation or experience could've been avoided when (a) you haven't been in the situation; (b) you already know the outcome of the situation; (c) assigning blame makes you feel morally superior; or (d) all of the above. If someone is the victim of a crime, the CRIMINAL is responsible for committing the crime. They CHOSE to commit a crime.\n\n**Comeback:** \"You're right. It is terrible. If that criminal hadn't assaulted her she wouldn't have been assaulted.\"\n\n# 12. COMMENT: \"ALL LIVES MATTER.\"\n\n**Often heard when:** Ugh (alternatively: any time someone even thinks about mentioning Black Lives Matter).\n\n**Why it should be laid to rest:** I've saved the best (by which I mean worst) for last. \"All lives matter\" is the racist version of \"I know you are but what am I\"\u2014an attempt to halt the conversation and engage in an endless back-and-forth. The Black Lives Matter movement was created in 2013 by Alicia Garza, Patrisse Cullors, and Opal Tometi in response to the acquittal of Trayvon Martin's killer, George Zimmerman. The movement is about creating a world where all lives\u2014including black lives\u2014matter equally. That's not the world we live in now, especially when we look at police violence and how it disproportionately affects black people. \"All lives matter\" is an empty retort designed to shut down conversations about black people and the issues they face. I think the \"all lives matter\" folks know that\u2014they just refuse to admit it.\n\n**Comeback:** \"It's okay for a movement to be focused on a specific group or cause. 'Save the rainforest' doesn't mean 'Fuck all the other trees.\"\n\nThough we must say goodbye to these comments today, we realize that this will not be the last time we will be gathered here. As you go out into the world, you're bound to encounter any number of dismissive, ignorant, and hurtful statements that need to be peacefully laid to rest. With the understanding that a perfect eulogy is often hard to muster in the heat of the moment, we encourage you to set time aside to write a eulogy of your own so you'll be ready when you invariably usher other offensive quips onto the next plane in this circle we call life.\n\n# CHAPTER THIRTEEN\n\n# SELF-CARE IS NOT SELLING OUT (UNLESS IT IS)\n\nThe fact of their deaths, all at the hands of police officers, is heartbreaking. But it was the images and videos, which in the age of social media and the twenty-four-hour news cycle are all but unavoidable, that got to me. Trayvon Martin's body as the lede photo on a _Gawker_ article called, \"This, courtesy of MSNBC, is Trayvon Martin's dead body. Get angry.\" Philando Castile's blood splattered all over his car. Eric Garner being choked, tackled, and abused by police officers. Alton Sterling, in that red shirt, on the ground. All the beautiful shots of Sandra Bland smiling.\n\nI've tried one too many times to avoid watching videos of black bodies being tormented and killed, but when you have to be on Facebook for work, it's almost impossible. When another police shooting happens, everyone immediately begins to share the story and comment on how awful it is, and with Facebook desperate to inflate its streaming numbers, of course the videos are set to autoplay. I remember the moment I felt like it had gone too far: There had been a string of police killings, one after another, and because I was working for _The Nightly Show_ at the time I was required to watch the inevitable footage that followed so I could write about the news at work. But I began to find it hard to go on Facebook without breaking down. Since police brutality was being discussed more than ever, people began to post old videos of police shootings, too. I decided I didn't need to see any more black men bleeding out on my Facebook wall, so I went searching for instructions on how to disable autoplay.\n\nThey were surprisingly\u2014or not, since Facebook is a company that makes money off you getting sucked in\u2014hard to find. It was so hard that when I finally figured it out and could shield myself from watching Philando Castile bleed out next to ads for dog food and shoes, I typed up a quick post about how to disable autoplay on videos, thinking other people might need a break from the trauma like I did. It wasn't anything elaborate, just something along the lines of \"There's a lot of shit going on right now, and you're probably seeing really graphic videos on Facebook. If you don't want them to play automatically, here's how to turn it off.\"\n\nSomeone more famous than I am once told me that when you hit 200,000 fans on Facebook, it's like the company flips a switch and assigns the most ridiculous, hateful people to comment on every one of your posts. When I wrote what I thought would be a harmlessly helpful post, I had around 250,000 followers, and criticism was swift and relentless. How dare I \"sell out\" like this? I was \"ignoring the realities of police brutality.\" And worse\u2014I was encouraging my fans to do the same. I was, once again, \"canceled.\"\n\nIt's always strange to me when people start dictating what you must or must not do to be a good black person, a good feminist, a good advocate. Do I have to keep a tally of my activist points to prove I've earned a moment of rest? It feels gross to do so. I had just written a piece about Castile for _The Nightly Show_ that went viral. I did live broadcasts for Facebook discussing police brutality. About a year and a half before, I'd gone on Katie Couric's Yahoo show and screamed at former NYPD commissioner Ray Kelly when he claimed the number of people who were killed by police officers was \"infinitesimal.\" After the broadcast I was added to the show's \"do not call\" list, which is a nice way of saying I was banned from ever going on that show again. And yet commenters were saying I was betraying my people because I didn't want to watch graphic videos of them being senselessly murdered every hour of the day. I understand that some white people need to see the horrible visual evidence of police brutality to realize the gravity of the situation, and even that they need to see it over and over. But not everyone needs repeated, in-your-face confirmation to understand that something abhorrent is happening and that they need to do something about it.\n\nI consider what I was doing with that Facebook post an act of self-care, a practice that has been criticized a lot lately, and rightly so. (I'll get to that.) \"Self-care\" originated as a medical term that, before the civil rights and women's movements of the 1960s and '70s, was used in the context of mentally ill and elderly patients, who were advised to practice healthy living habits in addition to their outside treatment. Then it was applied to people who work high-stress, emotionally taxing jobs, like EMTs and social workers. It was only natural that this concept would appeal to activists who were fighting for social justice, especially since so much of that work dealt with the ways society restricted, commodified, and controlled marginalized bodies. In her 1988 essay collection, _A Burst of Light_ , Audre Lorde wrote, \"Caring for myself is not self-indulgence, it is self-preservation, and that is an act of political warfare.\"\n\nToday, the line has been quoted again and again by activists wrestling with a political situation that, in some ways, doesn't seem so different from the way it was thirty years ago. When it seems that so much of society would rather you not live at all, keeping yourself healthy is a revolutionary act. In the case of police violence in particular, research has shown that black people experience PTSD symptoms when they see brutal images of black death. (Monica Williams, the clinical psychologist and director of the Center for Mental Health Disparities at the University of Louisville, calls this \"vicarious trauma.\") Stories of racism passed down over generations mean trauma is passed down, too.\n\nCombined, these things take a toll, especially as it's so easy to see how black death has become fodder for gruesome entertainment. We don't parade victims of tragedies around when they aren't black. After the mass shooting at a concert in Las Vegas in October 2017, videos were shared, but they quickly faded away, and they were never so horrifying as what has become typical after any police shooting. When the TV anchor Alison Parker was fatally shot at point-blank range on camera, it aired live, and after that news stations decided they weren't going to show the footage anymore. I agree with both of these decisions\u2014people were stressed and sad and angry after these incidents, and watching them replay over and over only exacerbates these feelings. But why can't we show the same care and consideration for black victims? (I think you know the answer.)\n\nTaking care of yourself doesn't mean that you're not also doing important work that advances the causes that are important to you; in fact, it's a critical part of doing that work. Without self-care, you burn out\u2014physically, but also through something known as \"compassion fatigue,\" which is basically just what it sounds like: When you focus too much time and energy on others in distress, you start to wear down, too. Which eventually means you have to stop doing work altogether.\n\nI get why people are skeptical of self-care. For a while there, it seemed like it had been stripped entirely of its meaning and existed only as a marketing tactic. As I've watched the YouTube beauty community change, I can see how \"self-care\" has become shorthand for \"buy these products.\" It's not uncommon to hear an influencer declare, apparently without shame, \"A big part of my self-care practice is doing a hydrating skin mask once a week. Which is why I wanted to start my own line of hydrating skin masks, starting at thirty dollars for two ounces.\" This is partially why it's become something to criticize people for: It's assumed that if you're posting about going to a movie, or getting your nails done, or buying a candle, you aren't doing anything else (except maybe shilling for some candle company).\n\nBut you don't have to throw the baby out with the bathwater. (Or the pedicures out with the... dirty foot-bath water?) The rise of self-care as a marketing tool makes it difficult to discern between legitimate rest and performative activism, but that doesn't mean you have to stay plugged into the news and angry 24\/7. Haters will claim you're doing yourself and the movement a disservice by not staying ON all the time, but in fact you'd be doing yourself and the movement a disservice if you were.\n\nSo now I'm sure you're on the edge of your seat, wondering: _What does Franchesca do when she needs to chill out? I must know!_\n\nHere are a few of my favorite things to do when I need to relax:\n\n\u2022 **Hour-long foot massages.** I recently discovered that I love foot massages, which are not something I expected I would like, so I've been doing that. (Despite what my husband says, my feet aren't gross, but I still tip a lot.) If you factor in the minutes it takes you to get used to having your feet handled by a stranger and then the minutes it takes to clear your head of other stresses, you realize that's still a solid forty-three-minute rubdown.\n\n\u2022 **Journaling.** It's nice to have a moment to reflect on and process your day, and lately I've gotten into drawing in my journal\u2014trying to make the page pretty is soothing, both because pretty things are soothing and because it forces me to concentrate on one thing rather than letting my thoughts race. Fun markers and stickers aren't necessary, but I buy them anyway. I challenge myself to make every page different. Sometimes I take the traditional approach and reflect on what happened that day; other days I make a collage of words and quotes that inspire me, or make lists of things I'm thankful for or things that made me laugh that day. It's a nice way to unwind, and it's also nice to have a record of what you do every day. One warning: I would say that going back and reading your journals from high school is actually the opposite of self-care. I went through a few of my high school notebooks thinking it would be good to quote them in this book, but no. All I did was say I was fat and keep daily charts of how much weed I smoked.\n\n\u2022 Speaking of weed: I **quit smoking** it. This one might seem a little confusing, since I'm citing no longer doing something as a form of self-care, but I promise it makes sense. I started smoking weed with friends in high school and later graduated to smoking alone out my bedroom window, eventually becoming the self-crowned queen of wake-and-bake. Even though I spent 85 percent of my senior year of high school stoned, I managed to keep a high grade point average, so I didn't see any harm. When my mother would ask about the strange smell, I assured her it was a candle. (Sorry, Mom, Fresh Grass is not one of Bath and Body Works' signature scents.)\n\nThrough college and after graduation, weed continued to be my go-to way to unwind or fill the time when I was bored. In my mid-twenties, when I moved to New York and was struggling to figure out what I was doing with my life, it became a crutch. I'd feel sad about my career, so I'd smoke weed and waste my day online instead of writing, auditioning, or doing much of anything. Once my career started to take off, I'd smoke to temper my anxiety.\n\nBut as I got older, it stopped working, and eventually it started to make my moods worse. I went from being a chill, funny, and happy stoner who was always pretty productive while high to being oversensitive, paranoid, and extremely emotional, sometimes unable to do anything but sleep. I'd always loved cracking jokes and talking shit with Patrick, but suddenly when I was stoned, any little comment would set me off and I'd dissolve into tears. Patrick started to notice a pattern. \"Wait. Did you smoke?\" became his catchphrase the minute we'd start to argue or I'd take a joke too seriously. I realized that pot was transforming me into someone I didn't recognize and that I needed to cut back. Finally, as I was battling a cold during my 2015 Christmas vacation, I took a few hits at a party and almost instantly lost my voice. I couldn't talk for almost two weeks. I took it as a sign and decided to quit then and there.\n\nIt still feels weird to come home from a long day and not unwind with a joint. Sometimes I do miss it. Or maybe I just miss how getting high used to make me feel. Either way, recognizing that smoking pot doesn't make me feel good anymore was the first step toward letting it go as a way to take care of myself.\n\n\u2022 **Coloring.** The adult-coloring boom of 2014 came and went for many, but I've stuck with it and I'm so glad I have. I've generated a pretty big collection ranging from intricate celestial designs to impressively complicated abstract mazes. (One of my favorites is _I Love My Hair_ , which is illustrated by my dear friend Andrea Pippins.) One year, I brought my coloring books on a trip to Costa Rica with Patrick and a few of our couple friends. My plan was to spend every afternoon on the beach coloring while sipping a bright, fruity drink\u2014this is my vision of bliss. On the first day, I spread my books out on the dining room table, and all the husbands, including Patrick, snickered. What grown woman has ten different coloring books and more than a hundred colored pencils? And brings them on her beach vacation? This grown woman, 'cause she can do whatever she wants. By the end of the trip, I had to pry the animal-kingdom coloring book from Patrick's hands and promise to get him a copy of his own when we got back to the States.\n\n\u2022 **Working out.** Nothing is better at distracting you from your problems than sweat stinging your eyes as you try to get your ass up for two more burpees.\n\n\u2022 **Face masks.** Though face-mask selfies have become a warning sign of someone who may be selling face masks on Instagram, that doesn't mean they're worthless. Even if they don't have an effect on your skin\u2014and for some I'm seriously skeptical\u2014they force you to sit still and not do anything for ten to fifteen minutes. I especially enjoy those that you have to let dry before you peel them off. It's extremely satisfying to see how big of a piece you can pull off without ripping it.\n\nEveryone's version of self-care is different. But no matter your chill-out method of choice, it's important to make self-care a regular practice when things are going good, so that when things aren't so hot you can prioritize keeping yourself (relatively) calm because the practice will be a habit. It's similar to when you're like, \"Okay, I have to get up for work at eight a.m., which sucks because I like to stay up until two thirty reading Oh No They Didn't! (my longtime guilty-pleasure gossip community) on LiveJournal. I'm going to try to get up at eight a.m. on the weekends, too, so it'll be easier to pull myself away from the Taylor Swift walking posts [don't judge me] during the week.\"\n\n# CONCLUSION\n\n# ACTIVISM IS LIKE LONG DIVISION\u2014YOU HAVE TO SHOW YOUR WORK\n\nAs excited as I was for the opportunity to write a book, I was also really nervous. The prospect of reliving some of the embarrassing, ignorant, or even offensive things I've said and done was daunting, especially since I've made a career of, literally, decoding injustice. Some people\u2014fans and haters alike\u2014have a perception of me as some kind of all-knowing, holier-than-thou social justice oracle, making pronouncements on bigotry from my production studio atop a mountain. It's not an accurate perception, but the idea of disappointing people scared me shitless. And not in an \"Oh God, I really needed to shit and now I feel a lot better\" way\u2014more like \"Oh God, I think some of my intestines might have just slipped out.\"\n\nWriting was especially anxiety inducing because every time I sat down to work, I would get stuck thinking about how every word could be twisted or misinterpreted. These worries weren't unfounded, either; it's not like when you ask your friend, \"Hey, does this outfit make me look like a _Real World_ cast member circa 2004?\" even though you know it doesn't. The world is an unwieldy place these days, and my little corner of it\u2014the internet\u2014is particularly stressful. It allows people to make anonymous claims without any repercussions; many social media users seem dedicated to taking your words out of context to make you look stupid, or worse. Add the fact that we're often talking about life-and-death issues for people who haven't had the chance to advocate for themselves, and you have a ticking time bomb.\n\nI'm not sure I could call myself an expert in anything other than styling my hair and bad puns, but I truly believe that trying to get it right is worth something. So now that you've laughed at, consumed, and side-eyed all my mistakes, accomplishments, and hard-won lessons, what's next?\n\n# UNDERSTAND AND ACKNOWLEDGE YOUR PRIVILEGE\n\nAlthough I've used it myself, these days the phrase \"check your privilege\" makes me cringe. It's certainly worthwhile to reflect on how your privilege can get in the way of your understanding and empathy. But \"check your privilege\" is usually spit out as a contemptuous demand instead of a helpful suggestion, like when you go to a restaurant and they ask if you'd like to \"check your coat,\" and then you remember, _Oh, right, I have a coat on, and I would like to leave it here so that I may enjoy my dinner without it hanging off the back of my chair and collecting dust off the floor._ If a member of the waitstaff barked at me to check my coat, or _else_ , I would probably leave the restaurant, write a bad Yelp review, and tell all my friends never to eat there again, even if they served the best steak in the world.\n\nJust as being ordered to check my coat would put me off eating at Rude Kitchen, aggressive commands that people check their privilege can backfire, too. What if the person you're talking to has no idea what \"privilege\" means? Left to figure it out for themselves, they probably assume you think they're rich, or that they've never struggled or had to work hard. The conversation will veer off course because they'll feel the need to tell you about their parents getting divorced when they were in third grade and about the summer that they broke their leg and had to sit by the pool with their foot propped up while their brothers and sisters learned to tread water. I've never broken a bone, but my parents got divorced when I was six, so I can empathize. But broken bones and broken marriages don't erase the privileges of, say, an able-bodied straight person who just said something really ignorant and hurtful about same-sex marriage, or who slid their able-bodied ass into the only wheelchair-accessible bathroom stall because they wanted to use the \"better mirror.\"\n\nIn an effort to help others understand their privilege and how it shields them from certain experiences, I've found that talking about my own privilege can open up the conversation. How can I possibly support trans people if I'm unwilling to see how being cis impacts my daily life, from my access to things like health care services, safe housing, and public restrooms to the fear of being harassed or abused while walking down the street? How can I push for more accessible spaces if I don't understand how being able-bodied affects how I move through the world?\n\nIn my animated video \"Sometimes You're a Caterpillar\" (illustrated by the wonderful artist and activist Kat Blaque) I use a story about a caterpillar and a snail to explain privilege. The caterpillar and the snail are best friends who live in a garden together, and one day they decide to go to a party on the other side of the fence. But they quickly realize that there's a problem: While the caterpillar is able to crawl under the fence, the snail gets stuck because of her shell. It's not the caterpillar's fault that the snail gets stuck under the fence, but it would be rude as hell for him to say something like \"Hurry up! I don't even see your shell.\" And while the caterpillar can easily fit under the fence, he still has his own challenges, like having to find shoes for all sixteen of his feet. The bugs both live in the same garden, but they move through it very differently. When the snail patiently explains that she just can't fit under the fence, the caterpillar has to check his privilege\u2014and then they work together to figure out how to get to the party before the keg is tapped.\n\n# KEEP IT INTERSECTIONAL\n\nA few years ago, I made a video in which I dressed up as a drag queen and walked through the streets of New York City asking people to guess whether I was a man or a woman. It was meant to be a comedic \"social experiment\" that loosely touched on sexism and gender stereotypes. What it actually was\u2014though I didn't realize it at the time\u2014was pushing a transphobic narrative, implying that it's totally okay to ask invasive questions about a stranger's genitals. The idea that trans people are oddities for us to gawk at is an extremely harmful one, and it's a huge part of the reason trans people, and especially trans women, are at such a high risk for violence: People demand to know \"what's down there\" and then fly off the handle if they don't get an answer, or if they don't get the answer they want. I'd made an all-too-common mistake: I was unable to see my privilege as a cis person and had thrown my trans brothers and sisters under the bus. I was trying to make a funny, socially conscious video about gender stereotypes, but I had forgotten that those same stereotypes I was toying with affect trans people very differently.\n\nIntersectionality is the study of how different systems of oppression overlap. People are multidimensional, and race, class, gender, sexuality, and ability play different roles in how we move through the world. The sexism I deal with as a straight cis black woman is often intertwined with racism; my experiences are going to be different from those of a cis white lesbian woman who deals with sexism that's often intertwined with being fetishized or discriminated against based on her sexuality. So it's crucial that our activism isn't just about the people who look like us.\n\nUnfortunately, well-meaning, smart, socially conscious (or \"woke\") people often forget to make space for marginalized people whose experiences differ from their own. From gay men calling their female friends \"sluts\" or \"bitches,\" to white women claiming that their experience of sexism is equivalent to another person's experience of racism, to reproductive rights movements forgetting that cis women aren't the only people who use birth control, to protest venues lacking wheelchair accessibility, and to social justice videos posted without captions, intersectionality fails are everywhere. Avoiding them requires care, consciousness, and a dose of humility, but it's worth it to build the world you want to live in. Because if your activism isn't intersectional, it's not really activism.\n\n# DO YOUR HOMEWORK\n\nThere will always be things you don't know and experiences you don't understand. It's okay to ask questions. Sometimes you'll be lucky enough to have someone who's willing to answer your questions or point you in the right direction of a book, movie, or article that you should check out for more information. But at some point you have to take responsibility for your own education. It's exhausting to always hold others' hands until they learn how to not be terrible people, especially when you're not getting paid for it. If you were in high school and wanted the smartest kid in English class to write a paper for you, you wouldn't expect her to do it for free. Not only is that cheating, it's exploitative.\n\nIf you do have a friend who's been kind enough to help you through your social justice education, the very least you can do is to get them a glazed doughnut to show your appreciation. (Better yet, take them out to dinner\u2014though you should avoid Rude Kitchen.) But there are so many resources available if you actually want to learn, and at some point your racism tutor is going to be like, \"Actually, I need to wash my hair tonight,\" and you'll have to take responsibility for your own education. Don't hesitate to hit up Google\u2014and not just Wikipedia\u2014or your local library or bookstore.  Do your own research on the issues and communities you're looking to support. I'm calling it homework for a reason.\n\n# KNOW WHEN TO CALL OUT, AND MAKE SPACE TO CALL IN\n\nIf you are neutral in situations of injustice, you have chosen the side of the oppressor.\n\n\u2014Desmond Tutu\n\nYou know those ads in the subway that say \"If you see something, say something\"? The same applies to activism. It's critical that you use your voice to call attention to harmful or oppressive forces, including but not limited to celebrities, TV shows, movies, Halloween costumes, and cartoons with secret racist messages. You can't just sit back and remark quietly to your similarly liberal friend, \"My, that's just awful.\" When the offense is public, it usually deserves a public condemnation.\n\nBut just as often as we publicly speak out about what's important to us, we should also make an effort to pull individuals aside and lovingly guide them toward the right path when they mess up. We all have that one friend who's willing to keep it real with you when you get a bad haircut, who will discreetly make a tooth-picking motion at you when no one else has the guts to tell you about the piece of spinach stuck in your teeth. Sometimes you have to be that friend.\n\nIf you have a friend or family member who says something offensive, remember that there were times when you screwed up and someone in your life had the grace to pull you aside instead of embarrassing you in front of everyone. Remember that you, too, were once exposed to something that changed your thinking and made you a better person. Wouldn't it feel good to be that something for someone else?\n\n# SPEAK UP, NOT OVER\n\nUse your privilege to advocate for marginalized folks, especially when you're in spaces where marginalized folks aren't present. I wish it weren't the case, but people are more willing to listen to someone who has something in common with them. You probably have access to people whom I will never have access to, just as I have access to people you'll never have access to. By that same token, it's important to use your privilege to bring people who don't look like you into those spaces. How do we get more POC involved in this organization? Why aren't there any LGBTQ folks on our advisory board? Who can we hire to interpret for the event? What can we do to cultivate female leadership? Take a look at who's at the table, and then use your voice to offer a seat to someone who deserves to be there.\n\nWhen marginalized people _are_ around, resist the urge to center your voice and speak over them. Think of Destiny's Child. There are three members, wearing matching outfits, but Beyonc\u00e9's outfit is always the cutest. Everyone's doing the same choreography, but Bey is out front. Everyone gets a chance to sing a line or two, but... we know who the lead vocal is. In this analogy, you are Michelle or Kelly (pick your favorite), and the person whose humanity is being questioned or undermined or attacked, the one most qualified to give voice to their experience, is Beyonc\u00e9.\n\nThe first time I used this analogy I got some pushback: I was accused of being unfair to Michelle and Kelly, two very talented, beautiful women who no doubt have more success and higher credit scores than I could ever hope to. But I think the comparison works. When Michelle decided to pursue a career in gospel music, Kelly and Beyonc\u00e9 were featured on her single, \"Say Yes,\" and appeared in the song's music video; this time, Michelle took the lead\u2014she even got the cutest outfit\u2014and the others were there to support her. And when Kelly published her book of advice for new moms, Beyonc\u00e9 and Michelle were at her book launch\u2014not hogging the spotlight, but cheering Kelly on.\n\n# OWN YOUR MISTAKES, AND COMMIT TO CHANGE\n\nNo one is perfect, and I promise, you're going to screw up. But screwing up is an opportunity for you to be honest about your mistakes, learn from them, and move forward.\n\nOne thing that I've learned during my time in the trenches of content creation is that genuine apologies are very, very rare. If you can perfect the art of saying sorry, more often than not people will be willing to give you a second chance. A genuine apology is made up of two parts: (1) taking responsibility for what you've done, and (2) committing to change. Pretty simple, but most people tend to fall back on some version of \"I'm sorry if you were offended,\" or \"Well, I'm sorry you feel that way,\" or \"It wasn't my intention to offend you.\" Unless you're a dominatrix who insults and offends people for a living because that's what your clients pay you to do, you rarely intend to offend people. Just because you didn't mean to do something doesn't change the result of what you actually did. As I've said before, if I step on your foot and break your toe, I didn't mean to break your toe, but your toe is still broken, and I'm sure you'd like to go to the hospital instead of listening to me explain that I didn't mean to break your toe.\n\nWe did an episode of _Decoded_ about the excuses people make for slavery in response to Bill O'Reilly flipping out when Michelle Obama said the White House was built by slaves. Michelle was right, of course, but the incident inspired a great conversation about why people are so hesitant to acknowledge how slavery has played a role in our nation's history. In the episode, we mentioned the Holocaust and Berlin's Memorial to the Murdered Jews of Europe as an example of a way a nation has paid tribute to and respected a horrific time in history. We then compared that to the way America tends to frame slavery as something that happened in the past and doesn't need to be discussed anymore.\n\nAt the time, it seemed like an appropriate comparison. But soon after the episode went live, a number of viewers contacted me to express their disappointment that we'd used the Holocaust as a rhetorical device. Of course, I didn't intend to offend anyone\u2014I didn't realize that it was actually anti-Semitic to use the Holocaust in this way, especially because people do it so often. But I realized that using this very painful piece of history as a way to prop up or give more importance to another cause\u2014while completely disregarding that there are living Holocaust survivors, and that there are still people affected by the Holocaust and rampant anti-Semitism\u2014suggests that anti-Semitism is a thing of the past. It also continues to push the idea that America was not complicit in the Holocaust in any way. After being called out and called in, we made an episode about the myths around the Holocaust and anti-Semitism, and we used the way I screwed up in the slavery episode as an example. People ended up really enjoying the segment\u2014not just because it was informative, but because I owned up to my mistake, took the steps to remedy it, and used the experience to help other people.\n\nAll that said, keep in mind that forgiveness takes time; offering an apology doesn't mean it's automatically going to be accepted or that the hurt will magically disappear. An apology has to be combined with action to make a real impact, so remember to give people space to see your follow-through.\n\n# BAKE YOUR OWN COOKIES\n\nRacism should never have happened and so you don't get a cookie for reducing it.\n\n\u2014Chimamanda Ngozi Adichie, _Americanah_\n\nI love this quote, and I think it can be applied to all forms of social advocacy. Too often, people want credit for doing the right thing instead of just wanting to do the right thing. I don't want to be picky about acts of goodness, because the world needs more of them, but doing the right thing isn't always going to come with a badge or a pat on the back or a tray of warm cookies. Those invisible acts of advocacy are just as important, if not more so, than the hashtag campaigns and the epic racist takedowns. No one is going to be there to watch you explain to your uncle why football players are kneeling during the national anthem, but those conversations still need to happen.\n\nSometimes people will email me screenshots of terrible things their friends or family members have said\u2014not because they called them out or called them in, but because they want to demonstrate how superior they are for not holding those same bigoted or outdated beliefs. Instead of running to show someone else how progressive and forward-thinking you are, you should be doing the work. You have to bake your own cookies. On the bright side, you can put whatever you want in them\u2014no more gross white chocolate macadamia for you.\n\n# ACTIVISM IS LIKE LONG DIVISION\u2014YOU HAVE TO SHOW YOUR WORK\n\nUltimately, being an activist is going to require way more time and discomfort than it takes to change your Twitter profile pic to a rainbow or put a safety pin on your backpack. These can be great signs of solidarity and support\u2014well, maybe not the safety pin\u2014but they're the bare minimum. You have to take the next steps to get out of your comfort zone and commit to continuously working to be a better person and, in turn, making the world a better place. That means getting involved in local politics, supporting diverse businesses and companies, passing the mic to marginalized folks, and giving your time, knowledge, and resources to charitable causes and organizations.\n\nHere's where I'd insert a really clever calculus analogy, but truth be told, I never took calculus. It looked really hard and boring, and buying one of those TI-84 calculators seemed like a waste of money (especially when I could already play _Snake_ on my Nokia phone). Instead, I'll say that activism is like long division: It consists of multiple steps, it's easy to mess up, and when you inevitably do make a mistake, the only way to find it and fix it is to show your work.\n\nWhat does that mean, exactly? It's different for everyone. Because the internet has become a place where so many people feel comfortable talking about their identities, struggles, and personal experiences, people are very cynical about what it means to be an internet activist. But being vocal about issues you care about doesn't mean that's the only form your activism takes. Sure, some people use the internet as an easy way to perform \"wokeness.\" But at the same time, there are few better ways to get the word out about an issue than talking about it online.\n\nIt can be hard to tell the difference between performative social consciousness (known as \"slacktivism\") and legitimate activism. Often the categories overlap, and they can both involve getting the word out and expressing pride or outrage or sadness on social media. There are also some people who use the internet because they're not able to be vocal IRL. Maybe they live in a conservative community, or with conservative parents who will kick them out of the house if they come out as gay. Maybe they have a disability and can't join a march or sit-in. Maybe they have more than one job and can't afford to boycott certain companies.\n\nUltimately, showing your work is about being accountable to yourself, and about being able to demonstrate how you got where you are. You can't just write down a bunch of numbers and symbols and expect to solve a math problem\u2014just ask my high school algebra teacher\u2014and you can't just _say_ you're an activist. Telling people you support LGBTQ folks is good, but you have to be able to explain how and why you support LGBTQ folks, whether it's by initiating a difficult conversation when a family member uses a slur or by voting in your local election when there's an anti-trans bathroom law on the ballot. How did you arrive at your perspective? To use another sort of flimsy math metaphor, how do you solve for _x_? While it's as important as ever to hold each other accountable, we also have to remember that everyone's journey is different, and learning takes time. If we want to figure out how to solve this, we have to do it together.\n\n# FRANCHESCA'S SIMPLE EXPLANATIONS OF NOT-SO-SIMPLE CONCEPTS\n\nCongratulations! You've made it through all the stories about my life and internet drama without throwing the book across the room. I hope. Or maybe you got to here, didn't understand what the hell I was talking about, and came here for answers. Either way, I thought it would be helpful to clarify some of the terms and phrases I use in _Well, That Escalated Quickly_. I've seen firsthand that people can sometimes get a little (or a lot) judgy when it comes to not knowing certain tenets of social justice theory. But you have to learn sometime, and there should be no shame in confusion or asking for help. Having access to this kind of knowledge is a privilege in itself\u2014just as having the time to read a book is. Not everyone has been exposed to these ideas. You don't know what you don't know, and you don't know how much your experiences (or lack thereof) influence you. Think about all the totally wrong stuff you believed as a kid: When I went to Catholic school, I thought that being gay was a sin\u2014just like premarital sex. I also only listened to R&B because, as logic dictates, alternative rock was the devil's music, and it seemed very obvious to me that Green Day was corrupting our youth. Meanwhile, I blithely sang along to a song that goes \"Step back, you're dancing kinda close \/ I feel a little poke comin' through\" whenever it came on the radio. Life is full of realizations.\n\nThis is not intended as a comprehensive list of every term you'll need to know if you want to wade into activism, but it should work as a jumping-off point, especially if you want to talk to friends or family about these topics IRL. I'd encourage you to look up these concepts in actual books, at the actual library. The internet is an imperfect tool, and is often wrong; if you do go online for more info, you should read more than one source\u2014no \"just skimming.\" These concepts are serious, and important to a lot of people. It's worth it to do your due diligence.\n\n**Ableism** \u2014discrimination against people with disabilities, including through language.\n\n**Ally** \u2014someone who actively works to support the rights of marginalized groups they're not a part of.\n\n**Alt-right** \u2014framed as an \"alternative\" branch of conservatism, the alt-right is an internet-savvy rebranding of white supremacy. With prominent young figureheads like Richard Spencer and a penchant for memes, the movement paints its members as edgy, \"anti-establishment\" crusaders against \"political correctness\" and saviors of the white race.\n\n**Black Lives Matter** \u2014a black-centered activist movement started by Alicia Garza, Patrisse Cullors, and Opal Tometi after George Zimmerman was acquitted for the murder of Trayvon Martin in July 2013. Originally a hashtag, the movement gained steam as it raised awareness and organized protests around ending systemic violence against black people.\n\nPeople often misunderstand the purpose of Black Lives Matter by assuming it's an exclusionary movement; it's not. It's a movement that focuses on the needs of marginalized people, and it doesn't shy away from centering those needs. That's why it's called Black Lives Matter and not \"All Lives Matter\" (see here). It's not about taking value away from white lives; it's about making black lives as valuable as white lives. Think of it this way: The existence of gay bars doesn't mean there are no straight bars. It's just that most bars are straight bars, and gay folks deserve spaces where they can safely and freely do their thing, too.\n\n**Black supremacy** \u2014not a thing.\n\n**Body positivity** \u2014the movement to normalize and accept bodies of all different shapes, sizes, and abilities in a world that privileges people who are thin, gender-conforming, able-bodied, and conventionally attractive.\n\n**Call in** \u2014to initiate a one-on-one conversation, out of public view, in order to make another person aware of their bigoted speech, behavior, etc. (see here).\n\n**Call out** \u2014to publicly bring attention to another person's bigoted speech, behavior, etc. (see here).\n\n**Censorship** \u2014a lot of people don't seem to get this one, so pay attention. The First Amendment says it is unconstitutional for the government to suppress speech it deems offensive, unacceptable, or a moral danger. Private individuals or groups can censor speech, usually through pressure like boycotts; that is not illegal. It is not an infringement of your rights when:\n\n\u2022 I block you on Twitter.\n\n\u2022 A newspaper drops your opinion column.\n\n\u2022 You lose a book deal or yogurt sponsorship because you said something sexist or racist.\n\n**Centrist** \u2014a person or viewpoint that attempts to \"see both sides\" of a given issue or scenario, even when there is no comparison between both sides because one side is absolutely unacceptable. Often comes from a place of moral superiority or belief in pragmatism that is rarely borne out in reality. Example: After the Charlottesville Unite the Right rally, which was held in protest of the city's plan to remove a statue of Robert E. Lee, Donald Trump rejected violence on \"all sides,\" though only the far-right protesters were at fault.\n\n**Cisgender (Cis)** \u2014denotes a person who identifies with the gender they were assigned at birth. For example: You are cisgender if, when you were born, the doctor said, \"It's a girl!\" and you still agree with that assessment.\n\n**Classism** \u2014the overt or subtle discrimination against lower-class people, often promoted through the willful ignorance of the realities of capitalism and poverty. Examples of classism include statements like \"If you wanted health insurance, maybe you shouldn't have bought that iPhone\" and the practice of restricting which foods people who receive government assistance can buy.\n\n**Colorblindness** \u2014the imaginary condition of not being able to \"see race.\" While some people are literally unable to differentiate among certain colors, the comment \"I don't see color\" is often used figuratively to shut down or dismiss conversations about race and racism by suggesting the speaker sees everyone equally. But there's nothing wrong with seeing someone's race; it's part of who they are, and it shapes their experiences and worldview. The problem is treating someone _differently_ because of their race.\n\n**Consent** \u2014permission that you give another person to engage in behaviors that require your involvement. Though it's usually used in sexual situations, you may also be asked to give your consent to police to search your property if they do not have a warrant.\n\n**Cultural appropriation** \u2014lots of people have tried to define cultural appropriation, and they haven't done a very good job. It's the practice of majority groups taking cultural practices, imagery, clothing, or other things from marginalized people without attribution, often for profit, and without being respectful of the customs' cultural origins. Meanwhile, people from those marginalized groups are often stigmatized for or downright prohibited from engaging in their practices by society at large.\n\nAppreciation, which involves respect, is not the same as appropriation, which doesn't. Going to a Mexican restaurant and participating in that culture is good; wearing a sombrero and saying you're going as \"Mexican\" for Halloween is not. Another way to think about it: If someone came to your house, stole your grandma's favorite necklace, and then wore it to your birthday party, you'd be mad. If you decided you would mass-produce your grandma's favorite necklace, sell it, and use the profits to start a scholarship in your grandma's name, you probably wouldn't be mad when you saw girls at the mall wearing it.\n\nOften, people engaging in cultural appropriation have no idea that they're doing something wrong, because they're ignorant of the group they're appropriating from. It's totally possible for marginalized groups to engage in appropriation, too. Case in point: I was a really big No Doubt fan in high school. Once, when I was staying with my dad, I decided I wanted to wear a bindi just like Gwen Stefani. I couldn't find a bindi in small-town Ohio, where my dad lived, so I bought googly eyes, took the pupils out, and crafted a makeshift third eye. I was sporting my new bindi at the grocery store when I heard a little kid say, \"Mom, what's on that girl's forehead?\" The mom replied, \"She's of a different religion, honey.\" I wasn't of a different religion, honey, and I shouldn't have been wearing that bindi. (And neither should Gwen.)\n\n**Dragging** \u2014the process of publicly shaming someone who has done something embarrassing or downright offensive. Dragging often takes the form of humor, through jokes, subtweets, memes, etc.\n\n**Ethnicity** \u2014a social group that has shared traditions. Not to be confused with **race**. Example: Latinx is an ethnicity, but Latinx people can be of any race.\n\n**Feminist** \u2014as the author Chimamanda Ngozi Adichie says, in a speech sampled in Beyonc\u00e9's song \"Flawless\": A feminist is a \"person who believes in the social, political, and economic equality of the sexes.\" Lots of people misunderstand this one.\n\n**Free speech** \u2014an often-misunderstood term derived from the First Amendment of the Constitution, which protects the right to freedom of speech, press, religion, and assembly. It establishes a citizen's right to speak freely without retribution from the government, though it does not include speech that incites harm to others, making or distributing \"obscene materials,\" and a couple of other very specific instances. It also does not extend to personal consequences for speech, which have nothing to do with the government. In other words, your mom grounding you for saying \"fuck\" at your grandma's seventy-fifth birthday party is not infringing on your right to freedom of speech.\n\n**Gaslighting** \u2014psychological manipulation that attempts to cause a person to question their reality or lived experience. Named after the 1944 film _Gaslight_ , in which a man attempts to convince his wife that she is losing her grip on reality so that he can commit her to a mental institution and get her out of his hair, this is a favorite tactic of dictators, abusers, and bigots. A few gaslighting techniques include lying, especially when there's blatant proof of the truth; denial; turning other people against you; questioning your sanity or mental health; and projection. Recommended reading: Lauren Duca's brilliant _Teen Vogue_ article \"Donald Trump Is Gaslighting America.\"\n\n**Gender binary** \u2014the (restrictive) idea that there are only two genders.\n\n**Gender expression** \u2014the way a person outwardly presents in terms of gender.\n\n**Hate speech** \u2014protected free speech that nevertheless disparages, bullies, harasses, or denounces marginalized groups. If you operate a hate site that promotes hate speech, you may not be committing a crime, but that doesn't mean you won't get kicked off your server.\n\n**Homophobia** \u2014anti-gay speech, behavior, or sentiment that may be suggestive or overt. Examples of homophobia range from a man's unwillingness to show other men affection for fear of being labeled gay, to media portrayals in which gay men are presented as stereotypes or as threatening\/contaminating the minds of children, to two women being denied the right to adopt, to bakeries refusing to make wedding cakes for gay couples. Not all anti-gay sentiments are rooted in fear, as the suffix \"-phobia\" suggests, but many are.\n\n**Humanist** \u2014the identity that \"devil's advocates\" choose when they don't want to say they are feminists, because they feel threatened by feminism or feel feminism is excluding men somehow. Technically, the term refers to someone who privileges the human reality over a spiritual or divine being.\n\n**Intersectionality** \u2014coined by civil rights activist Kimberl\u00e9 Williams Crenshaw, the idea that different identities and systems of oppression overlap and intersect. For example, a black woman experiences sexism that is often mixed with racism, while a lesbian woman experiences sexism that is often mixed with homophobia.\n\n**Latinx** \u2014a more inclusive, gender-neutral way to refer to people from Latin America. Pronounced \"La-TEEN-ex.\"\n\n**LGBTQ(IAA)** \u2014an acronym that stands for [ _clears throat_ ]: lesbian, gay, bisexual, transgender, queer, intersex, asexual, ally. The acronym was originally simply LGB, but as visibility of different identities and sexualities has increased, it has expanded to its current form\u2014I've chosen to use LGBTQ in my text, but you may also see LGBT, LGBTQI, LGBTQIA.\n\n**Male chauvinism** \u2014performative or aggressive masculinity that promotes the idea that men are superior to women. The word \"chauvinism\" on its own refers to explicit nationalist pride\/patriotism\/jingoism.\n\n**Male gaze** \u2014coined by the feminist film critic Laura Mulvey in 1975, a term from film or art criticism often used to denote an oversexualized or otherwise heavily male perspective on female subjects.\n\n**Marginalized** \u2014a term used to describe a person or group of people who are oppressed in some way(s). It has more or less replaced the term \"minority,\" which suggests the sense of being \"less than\" or small in number, which is not really accurate. (None of the following are small in number, for example, but they are all marginalized in various ways: low-income people, Latinx people, women.)\n\n**Meninist** \u2014a word I can't believe people really say. A troll-y ideological identity that dudes created to center conversations about gender equality around themselves. Related to **men's rights activists**.\n\n**Men's rights activist (MRA)** \u2014another thing I can't believe people really say. A reversal of the phrase \"women's rights activist,\" this is also a troll-y ideology that dudes created to center conversations about gender equality around themselves.\n\n**Microaggression** \u2014coined by Harvard professor Chester M. Pierce in 1970, a term for the daily, casual, and often unintentionally hurtful comments that marginalized people experience from people in positions of privilege. \"SWGSTBG\" was comprised of the microaggressions white girls frequently use to subtly degrade black girls. Other examples include: \"You're a big girl, but you have such a pretty face,\" \"You don't _sound_ Latino,\" and \"You're not like other girls.\"\n\n**Misandry** \u2014the hatred of men. Also not really a thing, except in certain feminist circles, where women will occasionally say they're misandrist ironically (or mostly ironically).\n\n**Misogynoir** \u2014coined by Moya Bailey in 2010, a term that describes misogyny aimed specifically at black women. It is an intersectional way to talk about misogyny. ( _Transmisogyny_ is the similar principle applied to trans women.)\n\n**Misogyny** \u2014the hatred of women.\n\n**Nonbinary** \u2014generally, the quality of being defined by more than two things. In social justice contexts, someone who identifies as nonbinary doesn't identify as a man or a woman.\n\n**Oppression** \u2014the myriad ways systemic forces exert power over groups of people.\n\n**Oppression Olympics** \u2014a term used to (disparagingly) describe the competitive language that crops up with marginalized groups when comparing privileges or disadvantages. This hierarchical attitude can exist within a marginalized group or between groups.\n\n**Passing (race and\/or gender)** \u2014a term most often used in the context of transgender people being identified as cisgender, and mixed-race people being identified as one race over another. Example: \"My friend Joel is mixed race, but he passes as white.\"\n\n**Patriarchy** \u2014a sweeping system of societal norms and behaviors that contribute to inequality between men and women. Evidence of patriarchy is present in everything from rape culture to the cost of women's toiletries as compared to men's. Though patriarchy oppresses women, it enforces strict roles for men, too.\n\n**Politically correct** \u2014a negative term referring, usually, to language that protects or refuses to offend marginalized groups, often used to deride activists or progressives and portray them as overly sensitive or beholden to an imaginary establishment status quo. In the 1970s and '80s, progressives used the term to mock the idea that we should accept the status quo, but since then it has totally reversed meaning, and has been weaponized against the same people who used to say it.\n\n**Privilege** \u2014the societal advantages possessed by a group of people based on their race, class, gender, sexuality, or physical ability. This word tends to freak people out because they assume having privilege means they're a bad person or are to blame for the mistreatment of others. In reality, everyone has some level of privilege, or rather a combination of privileges. Having privileges doesn't mean you're rich, have never worked hard, or never had to struggle. It just means that there are some things you'll never have to experience or think about because of who you are.\n\n**Problematic** \u2014an umbrella term and conversation-ender that expresses, generally, that someone or something is bigoted or microaggressive. If you want to keep from being critical in any meaningful way\u2014which you don't\u2014you can say something is problematic. Also used when people are too shy to say something was racist, sexist, homophobic, transphobic, ableist, or otherwise bigoted.\n\n**Queer** \u2014an umbrella term for LGBT people that conveys, broadly, that someone (or something) is not heterosexual or expressing a heterosexual perspective. Since it has also been used as a slur, not everyone in the community is down with this term, so proceed with caution if you're straight.\n\n**Race** \u2014a social construct that is used to define and inhibit as well as to forge communities. Commonly, it refers to shared physical traits, culture, and especially skin color. Not to be confused with **ethnicity**.\n\n**Racism** \u2014a system of language, behaviors, and policies that oppress nonwhite people. Can be overt ( **white supremacy** ) or subtle and ingrained ( **microaggressions** ).\n\n**Rape culture** \u2014the prevailing customs and beliefs surrounding sexual assault that privilege perpetrators over victims. Mechanisms of rape culture include ideas about what types of people are rape victims, the shaming of sexual assault survivors, victim blaming, and the casual diminishment of the seriousness of the crime. It is intertwined with **patriarchy**.\n\n**Revenge porn** \u2014nude photos or media of sexual acts that are published or distributed without a featured party's **consent** , and with the express aim of harming the nonconsenting party. It's now officially illegal in many states, with more surely on the way.\n\n**Safe space** \u2014a place where visitors can be sure they will not experience **triggers** , and where their identities and experiences will be respected. (The idea of safe spaces is highly and unnecessarily controversial.) Most often used in the context of colleges and universities, but not exclusively. For example, an organization may provide a safe space for sexual assault survivors or LGBTQ students as a way to encourage those groups to share their experiences openly and comfortably.\n\n**Sexism** \u2014speech, behavior, or actions that imply or directly state that women are inferior to men, or that promote inequality.\n\n**Shaming** \u2014the act of mocking, moralizing, or insulting certain behaviors, traits, or identities. Common examples of shaming include fat-shaming, kink-shaming, and slut-shaming (see here).\n\n**Slacktivism** \u2014a disparaging term used to minimize someone's work\/identity as an activist, often associated with internet activism, hashtag activism, or \"clicktivism.\" Describes, basically, any kind of activism you can do while sitting on your couch, and actions that range from changing your Facebook profile photo, which is largely performative, to promoting important studies and articles or otherwise disseminating information. People use this term as a way to dismiss nontraditional forms of activism, which may or may not be effective.\n\n**Social justice warrior (SJW)** \u2014a pejorative term used to describe people who vocally advocate for social justice in order to paint them as an irrational, whiny army of crybabies who sit on social media all day. You rarely see a person self-identify as a social justice warrior. Some people may do so ironically, but I don't, because it allows critics to group activists together in order to insult or degrade them and their work.\n\n**TERF** \u2014an acronym that stands for trans-exclusionary radical feminist. The term originated in the second wave of feminism, and it describes a movement that aims to exclude trans women from feminist spaces on the grounds that they are not \"real women.\" This is fucked up.\n\n**Tone Police** \u2014a person or people who attempt to tell you the correct way to speak about your experience, usually in order to make them more comfortable. Examples of tone policing include: \"You're embarrassing me,\" \"You're making a scene,\" \"You're being hysterical,\" \"Why do you sound so angry?\" (see here).\n\n**Toxic masculinity** \u2014a term that describes how masculinity standards set for men lead to acts of aggression or violence, which may range from domestic violence to mass shootings. The harmful effects of unreachable goalposts established by the **patriarchy** inevitably seep into the world.\n\n**Transgender** \u2014a term used to describe someone who does not identify with the gender they were assigned at birth.\n\n**Transphobia** \u2014anti-trans sentiment, often rooted in fear, ignorance, or lack of exposure. Examples of transphobia include anti-trans bathroom laws and common plots in shitty comedies wherein a (usually evil, duplicitous) character is exposed as trans ( _Ace Ventura: Pet Detective_ , _The Hangover Part II_ ). (Please don't judge me for having seen _The Hangover Part II_.)\n\n**Transracial** \u2014a term popularized by Rachel Dolezal to (incorrectly) refer to someone who has changed their race. You may hear people ask, \"Why can't she change her race if Caitlyn Jenner changed her gender?\" The answer is kind of academic, but basically: Because race is a social construct largely related to skin color, you can't change your race. When used correctly, \"transracial\" is an adoption term that refers to children adopted by parent(s) of a different race.\n\n**Trigger** \u2014something that brings back memories of a traumatic incident.\n\n**Trigger warning** \u2014a preemptive note about the content of a post, article, show, or speech that signals it may contain **triggers** for some people. Frequently used with material involving sexual assault, police brutality, or abuse, but can be applied to a wide range of triggers.\n\n**Victim blaming** \u2014a strategy for explaining tragedy that falls back on things said or done by the victim of that tragedy. People often engage in victim blaming out of judgment, fear, or the desire to come up with an explanation for bad things that happen to innocent people.\n\n**White feminism** \u2014a nonintersectional branch of feminism that excludes or ignores issues faced by women who are not white, straight, cis, able-bodied, or middle class or above. Rarely self-identified.\n\n**White supremacy** \u2014a holistic, racist ideology that promotes the idea that white people are superior to people of all other races, often justified through phony biological or evolutionary explanations. Like with **patriarchy** , evidence of white supremacy may be explicit\u2014expressions of white power or Nazism\u2014or subtle, through standards of beauty, media representation, or the justice system. White supremacy is not always obvious to the untrained eye, or to white people who are used to seeing themselves favorably represented. This is why it's important to, as they say, stay **woke**.\n\n**Woke** \u2014a term that refers to having had one's eyes opened to societal injustice, deriving from the idea that at one time you were \"asleep\" and now you're awake. In the past, it was used as a kind of friendly warning among marginalized people to \"Stay woke,\" but now people use it, inappropriately, to denote a kind of achievement or destination, ignoring the fact that there's always work to be done. \"Getting woke\" should be a moment on your journey, because we never stop having our eyes opened to experiences that we don't have because of our own privilege or biases.\n\n**Womanist** \u2014coined by the writer Alice Walker, a term to describe an alternative movement that seeks to broaden feminism to better cater to and include women of color.\n\nThat's all I have for now, though I'm sure I've missed something. As I said above, this is not intended as a comprehensive list of every word or phrase used in activist spaces. Language is constantly changing and evolving, and the internet only speeds up that process. Keeping track of it all may seem daunting, but it can also be rewarding. Words are linked to the power structures they represent, and using them carefully and correctly is a sign of respect, to marginalized people and to yourself. Besides, what else are you going to use when you want to speak your mind?\n\n# ACKNOWLEDGMENTS\n\nI know I've said this before, but writing a book is freaking scary. It's also hard and draining. More often than not, throughout this entire process, I was a complete nightmare of a person. (Seriously\u2014don't try to write a book and produce a TV pilot at the same time.) For that reason, I have a few people to thank.\n\nFirst, my incredible husband, podcast cohost, and part-time tear drier, Patrick, for reassuring me at every turn, and always keeping it real, offering encouragement, making me laugh until I cry, and showering me with love. I really don't know how you put up with my shit, but I'm so thankful that you do. My parents, who gave me life (duh) and who have always believed in me and pushed me to do my best. I owe Dad for my sense of humor and the stubbornness that made many of my hilarious fails possible. Mom, your grace, poise, patience, expert-level shade, and passion for nail art are sprinkled throughout these pages; thank you for modeling the kind of woman I've always wanted to be. My little cousins, who I wish were my sisters, Danielle and Ashley: You make me proud every single day and give me a reason to work for the better world that I know the two of you will run one day.\n\nMy amazing friends: De'Lon Grant, Dylan Marron, Shamika Martinez, Michelle Buteau, Delina Medhin, Annette Roche, Alex Leo, Ceci Fernandez, Laura Turner Garrison, Erica Williams Simon, Kimberly Luskey, Jordan Carlos, Danielle Young, Francheska Medina, Grace Edwards, Meghan Tonjes, Andrew Gunadie, and Oneika Raymond, for graciously responding to novel-length text messages, showering me with GIFs, and providing the late-night laughs that kept me going.\n\nThe people who made my very first book cover the cutest ever: My Glam Squad and two of the sweetest, most hardworking black women I know, Annette Roche and Delina Medhin, for keeping my edges laid, my face slayed, and my heart forever full. Superstar photographer Erin Patrice O'Brien and ultra-cool stylist Jocelyn Kaye, you're both so fun and easy to work with.\n\nThe amazing team that made this book happen and talked me off many a ledge: Brittany McInerney, your thoughtfulness, insight, and words of wisdom truly made this process fun, illuminating, and as painless as possible. Lauren Oyler, I'm convinced you're an actual superhero. You swooped in and saved my ass while never losing your cool and always being on time. Please teach me your ways. Brian Lemus, your eye and talent for design are unparalleled. My literary agent, Brandi Bowles, for never mincing words, which makes your praise that much sweeter. My manager, Kara Welker, for putting out every fire and being the ultimate gut check. My Gersh agents, Scott Yoselow and Rachel Zeidman, for always looking out for me and making me laugh.\n\nAnd finally, Filthy McNasty and Kaya, the best puppies\u2014and birth control\u2014possible. You're still not allowed on the couch.\n  Jobby = job\/hobby.\n\n  \"Black on the outside, white on the inside\"\u2014I shouldn't have to explain why this is messed up.\n\n  \"Going natural\" means transitioning one's hair to its natural texture after chemical straightening.\n\n  Neither of these is true of me, but no shame if you're a furry.\n\n  Tip: If you think the date will go well, stuff a night scarf in your purse before you leave the house.\n\n  UCB is the Upright Citizens Brigade Theatre, an improv theater and training program founded by Matt Besser, Amy Poehler, Ian Roberts, and Matt Walsh. There are a number of improv comedy programs in New York, but UCB is seen as one of the big dogs because it's churned out a number of _SNL_ cast members.\n\n  B-roll is the footage editors splice into the main shot in order to make a segment look more dynamic. Think static images of an interviewee's face, or shots of the scenery. I didn't know about B-roll, so when the focus was off me during taping, I let myself go into resting bitch face.\n\n  _SEO_ stands for \"search engine optimization,\" and it refers to strategies websites use to make their posts show up near the top of search results; reverse-engineering SEO can help you unearth articles and videos that may have been buried over time.\n\n  Jill Soloway identifies as genderqueer and nonbinary and prefers to use they\/them pronouns.\n\n  _Dead-naming_ is using a transgender person's birth name instead of their chosen name.\n\n  You'll notice I usually call my hair \"locs\" and not \"dreadlocks.\" There's nothing dreadful about my hair.\n\n  If you want to have an open conversation, start with a question. It's kind of hacky, but this way you capture people who are genuinely curious as well as people who are sure they know the answer and are mad you'd even ask. Our first episode was, \"Are Fried Chicken and Watermelon Racist?\"\n\n  In a Mariah Carey, ridiculous-but-I-earned-this way, not in a Naomi Campbell, throw-a-cell-phone way.\n\n  Yes, they do.\n\n\u2020 I'm serious, they do.\n\n\u2020\u2020 They do!\n\n\u00a7 Okay... maybe the \"queen\" part is optional.\n\n  Real coworker friendship requires repeated IRL hangouts that don't involve conversations about work or former coworkers.\n\n\u2020 And you definitely don't want to connect on social media with the person who could fire you, so do not friend your boss.\n\n  Anti-miscegenation laws referred to interracial marriage, but they also extended beyond it; in Virginia, where _Loving_ originated, interracial sex was also illegal.\n\n  Originating from the olde term \"cuckold,\" which refers to a man whose wife is having sex with another man, \"cuck\" is an alt-right insult lobbed at a man who's perceived as weak. It's rooted in sexist stereotypes and toxic masculinity, of course.\n\n  _Bed wench_ or _bed warmer_ was a term used to describe black slave women who were forced to sleep in the master's bed for the purpose of \"keeping it warm\"\u2014being used as a sex slave. Yeah, it's pretty gross.\n\n  Some of my favorite books to get you started: _A People's History of the United States_ by Howard Zinn; _The New Jim Crow: Mass Incarceration in the Age of Colorblindness_ by Michelle Alexander; _Ain't I a Woman: Black Women and Feminism_ by bell hooks; _We Should All Be Feminists_ by Chimamanda Ngozi Adichie; _Redefining Realness: My Path to Womanhood, Identity, Love and So Much More_ by Janet Mock.\n\n### Thank you for buying this ebook, published by Hachette Digital.\n\nTo receive special offers, bonus content, and news about our latest ebooks and apps, sign up for our newsletters.\n\nSign Up\n\nOr visit us at hachettebookgroup.com\/newsletters\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# Social Media for Writers\n\n## Marketing Strategies for Building Your Audience and Selling Books\n\n# Tee Morris & Pip Ballantine\n\nCincinnati, Ohio\nSocial Media for Writers. Copyright \u00a9 2015 by Tee Morris & Pip Ballantine. Manufactured in the United States of America. 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 permission in writing from the publisher, except by a reviewer, who may quote brief passages in a review. Published by Writer's Digest Books, an imprint of F+W Media, Inc., 10151 Carver Road, Suite # 200, Blue Ash, OH 45242. (800) 289-0963. First edition.\n\nFor more resources for writers, visit www.writersdigest.com\/.\n\nDistributed in Canada by Fraser Direct\n\n100 Armstrong Avenue\n\nGeorgetown, Ontario, Canada L7G 5S4\n\nTel: (905) 877-4411\n\nDistributed in the U.K. and Europe by F&W Media International\n\nBrunel House, Newton Abbot, Devon, TQ12 4PU, England\n\nTel: (+44) 1626-323200, Fax: (+44) 1626-323319\n\nE-mail: postmaster@davidandcharles.co.uk\n\nDistributed in Australia by Capricorn Link\n\nP.O. Box 704, Windsor, NSW 2756 Australia\n\nTel: (02) 4577-3555, Fax: (02)4577-5288\n\nE-mail: books@capricornlink.com.au\n\nISBN-13:9781599639284\n\nWritersDigest.com\n\nCincinnati, Ohio\n\n# Contents\n\n  * Title Page\n  * Copyright Page\n  * Dedication\n  * Acknowledgments\n  * About the Authors\n  * Foreword\n  * Introduction: Welcome to Your Social Media Survival Guide\n    * Understanding Social Media\n    * What You Need to Make This Book Work\n  * Chapter 1: WordPress\n    * Why WordPress?\n  * Chapter 2: Tumblr\n    * What Is the Difference Between WordPress and Tumblr?\n    * Expressing Yourself: Producing Content\n    * What to Expect from the Tumblr Community\n    * Sharing with the Class: Syndicating Media\n    * Tumblr Strategies\n  * Chapter 3: Podcasting\n    * Writers Gone Wild: Talk Shows\n    * Anthologies in Audio: Short Stories\n    * Going All In: Podcast Novels\n    * Why Should You Podcast?\n    * Why Shouldn't You Podcast?\n    * Why Do We Still Podcast?\n  * Chapter 4: Facebook\n    * Facebook for Writers\n    * Content Marketing: The Science of Promotion by Example\n  * Chapter 5: Twitter\n    * First Impressions at a Glance\n    * Mastering Tweet Speak: Composing Messages on Twitter\n    * Best Practices for Twitter\n    * You Did Not Just Tweet That: _Worst_ Practices for Twitter\n  * Chapter 6: Google+\n    * Google+ for Writers: The Account vs. The Page\n  * Chapter 7: YouTube\n    * Action Editing: Software for Video Editing\n    * Turning Authors on to YouTube\n    * Best Practices on YouTube\n  * Chapter 8: Pinterest\n    * Tools of the Trade: Pinterest Applications\n    * Game On: Pinterest Competitions\n    * It's Business Time: Pinterest Business\n    * Watch Where You Point That Pin: Pinterest Strategies\n  * Chapter 9: Instagram\n    * Platform Management: Connecting Instagram with Other Social Networks\n    * Beyond Instagram: Third-Party Apps\n    * Best Practices on Instagram\n  * Chapter 10: Additional Options\n    * Goodreads\n    * Reddit\n    * Storify\n    * A Social's Social: Untappd, Vivino, and Distiller\n    * Tsu.co\n  * Chapter 11: SEO\n    * Why SEO Doesn't Work (as the \"Gurus\" Claim It Does)\n    * How to Make SEO Work for You\n    * The SEO That Can Hurt You\n    * Best Practices of SEO\n  * Chapter 12: Content Marketing\n    * How Does Promoting Others Really Work for You?\n    * What Content to Look for in Content Marketing\n    * When Content Marketing Goes Bad\n  * Chapter 13: Best Practices in Social Media\n    * Create an Editorial Calendar\n    * Think Before You Post\n    * Participate in Blog Tours\n    * Create Quality Content\n    * Be Visual When Possible\n    * A New Use for the Pound Symbol: Hashtags\n    * Taking P!nk's Advice: Facebook Parties\n    * Planning Makes Perfect: Before That Social Media Event\n    * Attribution, Not Imitation, Is the Sincerest Form of Flattery\n    * Seek Out Your Audience\n    * Have a Plan\n    * Focus on Maintaining a Signal, Not Creating Noise\n    * Antisocial Media: What to Avoid in Online Promotion and Networking\n  * Appendix A: WordPress\n    * Setting up a WordPress Account\n    * Getting to Know WordPress\n    * Creating Content\n  * Appendix B: Tumblr\n    * Setting up a Tumblr Account\n    * Getting to Know Tumblr\n    * Posting to Tumblr\n    * Reblogging\n  * Appendix C: Podcasting\n    * An Introduction to Podcasting\n    * The Mixing Board: Bridging the Gap Between Microphone and Computer\n    * What's Next: Other Details for an Audio Studio\n  * Appendix D: Facebook\n    * Setting Up a Facebook Account\n    * Getting to Know Facebook\n    * The Left-Hand Sidebar\n  * Appendix E: Twitter\n    * Setting Up a Twitter Account\n    * Getting to Know Twitter\n    * Talking on Twitter\n    * Direct Messages\n  * Appendix F: Google+\n    * Setting Up a Google+ Account\n    * Getting to Know Google+\n  * Appendix G: YouTube\n    * Setting Up a YouTube Account\n    * Getting to Know YouTube\n    * Uploading Your Video on YouTube\n    * Embedding Your YouTube Video in a Blog\n  * Appendix H: Pinterest\n    * Setting Up a Pinterest Account\n    * Getting to Know Pinterest\n    * Working with Pinterest\n    * Making Your First Board\n  * Appendix I: Instagram\n    * Setting Up an Instagram Account\n    * Getting to Know Instagram\n    * Taking a Photo on Instagram\n\n# Dedication\n\nTo writers of all genres, of all backgrounds, everywhere. If you walk away from this book with a new idea for your social media strategy, mission accomplished.\n\n# Acknowledgments\n\nSocial media has made an astounding impact on our lives, and we are reminded of this every day through Likes, retweets, and voice mails we receive from the platforms covered here. The inspiration behind this guide comes from the many questions we have received at conventions, at workshops, and even over a cup of coffee (or tea, in Pip's case); but this book currently in your hands would have never happened had it not been for Laurie McLean of Fuse Literary and the talents of Alex Rixey and Cristopher Freese at Writer's Digest Books. Thank you all for making things happen. A huge thank you, as well, to Chuck Wendig for our Foreword and all the authors who offered their own opinions, strategies, and best practices, making this book more about the community of authors who all came together to offer their own opinions on what works in social media. Social media, we know, is more than just a new marketing platform for authors, but it is a foundation for writing communities. This book has only reinforced that opinion.\n\n# About the Authors\n\n**TEE MORRIS** has been writing science fiction, fantasy, horror, and nonfiction for over a decade. His first novel, _MOREVI: The Chronicles of Rafe & Askana_, was a nominee for the 2003 Eppie for Best Fantasy, and in 2005 the book became the first novel to be podcast in its entirety, ushering in a new age for authors. Following the podcast of _MOREVI_ , he co-founded the audio literature hub   \nPodiobooks.com with Evo Terra and Chris Miller, offering hours of original audio content from first-time writers and _New York Times_ bestsellers in a podcast format. He then went on with Evo Terra to write _Podcasting for Dummies_ (as well as the 2nd Edition alongside Chuck Tomasi). His expertise reached deeper into social media when he penned _All a Twitter_ and _Sams Teach Yourself Twitter in 10 Minutes_.\n\nIn 2011, Tee returned to fiction with _Phoenix Rising_ , the first novel in The Ministry of Peculiar Occurrences series, written alongside his wife, Pip Ballantine. The title went on to win the Airship Award for Best Steampunk Literature and become a finalist for Goodreads' Choice Awards for Best Science Fiction of 2011. Now on to their fourth book in the series, _The Diamond Conspiracy_ , and the fourth season of _Tales from the Archives_ , a podcast anthology featuring short stories set in their steampunk universe, the Ministry of Peculiar Occurrences series has won several awards including the Parsec Award for the best of science fiction podcasts and RT Reviews' Choice Awards.\n\nTee and Pip also host _The Shared Desk_ , a podcast covering collaboration and other aspects of a writer's lifestyle. He now runs the social media initiatives for Stratford University located in Maryland and Virginia. Explore the works of Tee Morris, and his occasional geek rants, at TeeMorris.com.\n\n**PIP BALLANTINE** started life in Wellington, New Zealand as a corporate librarian. Pip earned a Bachelor of Arts in English Literature and Political Science and then a Bachelor of Applied Science in Library and Information Science. Her first professional sale as a writer was a piece on the history of Wellington, written for _The Evening Post_ in 1997. Since then she has gone on to produce both novel-length and short-form fiction.\n\nIn 2006, she became New Zealand's first podcast novelist with her debut fantasy novel, _Weaver's Web_. She went on to podcast three other novels and host her own slice-of-life podcast, _Whispers at the Edge_. Her podcasts have won both a Parsec Award as well as the Sir Julius Vogel award for excellence. Pip's first byline in the United States was _Geist_ (Ace Books), launching the Books of the Order series. At the same time, she also wrote for Pyr Books Hunter, Fox and Kindred, and Wings, as well as co-writing _Phoenix Rising_ with Tee Morris, the first novel in the award-winning steampunk spy series The Ministry of Peculiar Occurrences.\n\nWhen it comes to nonfiction, Pip has been seen in _A Taste of True Blood_ (Ben Bella Books) and worked behind the scenes as technical editor for _All a Twitter_. Pip's short stories have appeared in anthologies such as _Clockwork Fairy Tales_ (Roc Books) and _Steampunk World_ (Alliteration Ink).\n\nShe continues to co-author Ministry of Peculiar Occurrences series with Tee, as well as produce their award-winning podcast, _Tales from the Archives_ , and co-host _The Shared Desk_. When not writing or podcasting, Philippa loves reading, gardening, and whenever possible, traveling. She is looked after by a mighty clowder of cats in Manassas, Virginia, with her husband Tee and their daughter.\n\n# Foreword\n\nYou're a writer over here. And over there is the wide world of social media.\n\nYou've got your Faceyspaces, your Circlesquares, your Tinders, your Grindrs, your Blinders, your SexyPalFinders, your Bloobs and Gloobs and Innertubes. Okay, so only a few of these Seussian techno-monstrosities exist, but if I said Facebook and Twitter and Ello and Tsu, in five years they might not exist, either.\n\nThe point is, you've got this world out there. This connected world. This web of bridging threads that connects you, me, our phones, our computers, and probably soon enough, our refrigerators. It moves fast. The ground shifts under our feet daily.\n\nOn the one hand, social media is easy, right?\n\nGet on it. Say hello. Say other stuff. Squawk into the void to see who's listening. (Spoiler: My refrigerator is listening and my refrigerator would very much like you to go pick up a six pack of beer. Dogfish 90-minute IPA, please.)\n\nBut then, what about that other side of it?\n\nThe _writerly_ side. The _authorial_ side.\n\nI'm a writer. So are you, I'm guessing.\n\nAnd you're wondering, how do I bridge those things? How do you tie together _you as a writer_ and _you as a person on the social media thingies_? Is there value for you as a writer? Is there _danger_ and _peril_ for you as a writer? Yes to the first, and _hell yes_ to the latter.\n\nCan you tell stories on Twitter? (Yes.) Can you find an audience on social media? (Sure.) Can you burn your audience on the Internet? (Most definitely.) Can you sell books this way? (Yes, to a point, but please don't get spammy.) Do you _have_ to sell books online and be all Author Person? (Nope.)\n\nBut how? How do you accomplish all of this? How do you keep up with what works and what doesn't? Don't different networks and services offer different ... well, networks and services? A value add here, a subtractive function there?\n\nYou need help.\n\nAnd so, I've written this book\u2014\n\n* is handed a note *\n\nAh. Okay. Sorry. Turns out, I did ... not write this book? I didn't. Okay. Sorry. I write a lot of books and it all kind of blurs together.\n\nI did not write this book.\n\nWhich is probably a good thing.\n\nBecause you need not just one Sherpa to lead you up this mountain of authorial social media enlightenment. You need, in fact, _two_.\n\nAnd so, I give you: Philippa Ballantine and Tee Morris. They are going to hold your hand and take you on a tour of all of the weirdness and wonder that social media has to offer\u2014and they're also going to helpfully point out the pitfalls, too.\n\nBecause boy howdy, are there pitfalls.\n\nYou wanna do this _I am an author on social media_ thing right?\n\nThen you need their help. You need this book.\n\nThough, before you read any further, I'll offer my one piece of social media advice. Take it or leave it\u2014hug it close like a dear friend, or discard it like an old sock.\n\nThat advice is: No matter what network you use, no matter whom you talk to, no matter the blog or the service or the size of your audience, be the best version of yourself online.\n\nDon't be somebody else.\n\nDon't be a sales machine.\n\nDon't be an asshole.\n\nBe a fountain, not a drain.\n\nBe you, with all the _best stuff_ dialed up to 11, and all the worst stuff shoved under the bed so that nobody can see it.\n\nSo endeth the lesson.\n\nNow: Reach out and take Tee's and Philippa's hands. It's time to take a walk. It's time to take the tour. It's time to buy the ticket and take the ride.\n\nSee you online.\n\n\u2014Chuck Wendig, Author, Blogger, General Wiseass\n\n# Welcome to Your Social Media Survival Guide\n\n\"So what's your platform?\"\n\nThis has become a common question that agents and editors ask writers, be they beginners or best-selling authors. Once upon a time\u2014let's say back in 2007, which is several _generations_ ago in Internet time\u2014social media was considered a distraction to up-and-coming writers and a fad to the established wordsmiths.\n\nWithin a decade, that attitude has changed. Dramatically.\n\nThe beautiful thing about social media is that it's easy to pick up. It's designed in such a way that anybody can set up an account and get started. The problem is that writers and social media suffer a disconnect. That is, plenty of writers hate the notion of promoting their work. They simply don't want to be _that_ snake oil\u2013selling writer. And while they may feel very strongly about that, the reason authors must self-promote is a simple one: _If you do not talk about your book, no one else will._\n\nWhat complicates matters for writers, social media, and the relationship between the two is what happens when self-proclaimed introverts pick up a megaphone and blindly go about promoting into a void. As you might imagine, things can and do go horribly, horribly wrong.\n\nThat's where we come in.\n\n### Bookmark\n\nThroughout this book you will find nuggets of information and helpful tips on getting the most out of the various platforms we cover. Our Bookmarks point to links to check out, quick tips to employ, pitfalls to avoid, and reliable third-party expansions\u2014like apps for mobile phones and plug-ins for blog engines. Keep an eye out for them.\n\nWe set out to write this book because we know lots of authors are struggling with the prominence of social media networks, managing posts, and their public image. Some are using social media in ways that make them look like a rank amateur. (Believe us, it's very easy to do.) You may read some of what we offer in this book and think, _Oh come on, that is common sense!_ But some of that common sense sadly isn't as common as you would imagine. If you are new to social media, we'll show you some of the basics and strategies that go well beyond them as well. If you're an old hand at Facebook, Twitter, and Instagram, you can count on discovering something new to apply to your own social media strategy.\n\nAll this sounds very exciting, but right about now you might be wondering who this mystical _we_ behind this book is. Who is leading you into this promised land of blog posts, podcasts, and status updates?\n\nTogether we are Pip Ballantine and Tee Morris, writers of the award-winning Ministry of Peculiar Occurrences __ steampunk series. Between the two of us, we offer more than two decades of experience as professional authors, published both independently and traditionally by New York publishers. In addition, Tee offers a decade of experience as a social media professional, having worked in corporate, government, higher and continuing education, and nonprofit positions. He also literally wrote the book, or in this case, _books_ , on social media: _Podcasting for Dummies_ and _All a Twitter_. Pip carries the distinction of being New Zealand's first podcasting author and has spoken on popular social media topics concerning authors both in her native country and the United States. She remains the only winner of New Zealand's prestigious Sir Julius Vogel award for a podcast. This is who we are, and where we began our social media journeys, in 2005\u20132006, on the cutting edge.\n\nConsidering we predate the hipsters by at least a decade, you might accurately say that we are social media pioneers. (And Tee was drinking Dogfish Head beer before it was cool, you young, bearded whippersnappers.)\n\n**Notes From the Margins**\n\nPip and Tee know a lot about social media, but it's always good to get several perspectives. Throughout this book you'll find notes, styled like this, highlighting professionals in the book business who have shared their thoughts on social media and their platforms.\n\nWe maintain blogs, produce podcasts, and work all the social media you will find here. Ours is a view from the trenches, and the strategies we offer for your consideration have won us critical acclaim and a variety of awards and accolades. What we hope you take away from this book is an understanding of and a strategy for social media.\n\nAgents are increasingly looking for authors who are not only great writers, but who also have a solid social media presence. It's essential that authors take a strong hand in marketing their own books. Personally, I check out at least the Twitter and Facebook brands of every potential client I am interested in representing. If these writers also have a blog, and Instagram, Tumblr, or Pinterest accounts, I am even more impressed by them.\n\n\u2014Laurie McLean, Founding Partner, Fuse Literary\n\nYou see, social media began innocently enough as a way for authors to extend their reach to readers on the Internet, and to connect on either a professional or personal level. Social media allowed authors to encourage others to write, or offered readers a peek behind the curtain at the creative process. This online outreach evolved into an essential part of a writer's life. Today, marketing via social media channels has become a necessary part of the author's skillset, as much as research, writing, and editing. However, since marketing or public relations is \"someone else's job\" in a writer's eyes, authors usually don't give it enough time and attention. When marketing and PR get moved to the back burner, authors make mistakes that negatively affect the book's audience and, possibly, their reputation. Most of these social media mishaps can be easily avoided. Others are stranger than fiction but are worth remembering and learning from.\n\nTapping into the potential of social media begins with understanding how all the tweets, updates, and posts began, and how authors should take the online environment seriously and maximize the tools available to them.\n\n## Understanding Social Media\n\nTo appreciate this weird and wonderful world of social media, it helps to step back to the early days of the Internet. If you don't remember that far back, here is a short refresher. Communication was one-sided; information was presented in a stationary, static format known as Hypertext Markup Language (HTML). HTML is the tagging system that is used to create Web pages to display text, images, and links. When you get to WordPress, you will be able to see this HTML code in the text viewing panel.\n\nHere's an example of what it will look like:\n\n_Lost in the sea of social media? Don't know how to blog? Befuddled by Twitter? Faceblocked rather than Facebooked? We can fix that for you. <\/em?_\n\nOur Talent\n\n_Tee Morris, along with being an accomplished, award-winning author, has ten years of experience as a social media professional. He is also the author of four titles in social media and has spoken on best practices coast-to-coast and around the world. Tee has worked for both nonprofits and corporations, and now his services can be yours._\n\nThe basics of HTML still apply to current online communications (and we've found them very useful for tweaking our various blogs), but _interaction_ with a site's topic or host is far different. That interaction is the primary difference between social media today and the very primitive forms of communication in the early days of the Internet.\n\n### Bookmark\n\nKnowledge of HTML is not a necessity for using social media, but it is extremely helpful in some instances. The best thing about HTML is that it is easy to figure out and easier still to find online resources that will identify tags that will allow you to format text. It's a good skill to have under your belt.\n\n### First Forums, Which Begat Blogging\n\nWith the development of more powerful programming languages more interactive websites called _forums_ appeared online. Forums granted their visitors the ability to interact with other visitors through threads of comments pertaining to topics started by the website's host or the site's community. Visitors to a forum subscribe to a site and then enter a username, profile, and any other personal details they wish to share with the community. Once subscribers find a discussion that piques their interest, they are offered an interface that allows them a voice in the conversation. Forum members can also introduce their own topics and subscribe to specific threads. They are notified through e-mail when new replies are posted. As they interact within the forum, they build a reputation within that site's community.\n\n### Bookmark\n\nBefore you attempt to create or customize your own forum, know that the programming languages behind this kind of system are far more complex than HTML. Active Server Pages (ASP) is a server-side script engine that makes dynamically-generated Web pages possible, and PHP:Hypertext Preprocesser is another language that works in conjunction with HTML to create frameworks. The invention of these scripting languages opened a whole new world of possibility, but you should know that these languages are not necessarily for beginners. Proceed with caution.\n\nIt was the programming language _RSS (Really Simple Syndication)_ that truly changed the way we communicate online. RSS served as the foundation for a new kind of website called \"Weblogs,\" or what people now commonly refer to as _blogs_. One difference is that a blog, as opposed to a forum, is hosted or written by one writer or a core group of writers. A blog's host is the only one allowed to post new topics of discussion. Subscribers and visitors can interact with the blog host via comments, but they cannot post new topics as they can in forums. Another major difference with blogs is that their content can be _syndicated_ through other blogs. In syndication, segments of a blog post are shared on other blogs with links leading back to the original. This increases traffic for all blogs involved in the process. Topics previously confined to a host forum could be distributed through a vast network of blogs, commonly known as the _blogosphere_.\n\nThe blogosphere offered a more interactive way to communicate on the Internet, with new and constantly up-to-date information, the ability to cross-reference material, and a way to build an online community around an author and his books or series.\n\n### Proceeding with Podcasting\n\nBlogging ascended to a higher form of communication in 2004 when RSS pioneer Dave Winer and former MTV VeeJay Adam Curry developed a new tag for the language\u2014the _enclosure_ tag\u2014to allow blogs to syndicate more than just text and images. With an enclosure tag, a blog was able to distribute more robust media, such as audio and video files. Put simply, the tag provided a website address where the user could find these different types of files. It looks something like this in the HTML markup:\n\nAnd thus _podcasting_ was born.\n\nPodcasting can be best described as \"DVR meets the Internet,\" where after you subscribe to a podcast, episodes are stored on your computer or portable media player until you listen to or view it. Podcast media provides subscribers with full control, unlike audio or video webisodes, which are generally only available if you are connected to the Internet or tune in at a specific time. Podcasting offers a variety of programming that is yours to keep when you subscribe. For writers, it can provide an outlet to show the world your fiction by recording and releasing short stories or even your whole novel. Another option is to do an interview or chat show. We do both with _Tales from the Archives_ 1www.ministryofpeculiaroccurrences.com\/category\/podcast\/ and _The Shared Desk_.2<http:\/\/www.theshareddesk.com>\n\n### The Rise of Social Networks\n\nWith the popularity of content delivered to computers via RSS, another form of data exchange emerged, offering subscribers a new, real-time method of connecting with others. No longer would writers have to attend a conference in order to \"meet with\" and learn from other writers and professionals in the business. Thanks to s _ocial networking,_ users control their communities and their connections. They can set up profiles in order to broaden professional contacts or even just to enjoy the company of fans and friends in the writing community.\n\nSocial networking sites exploded on the Internet, changing the way information is exchanged, how apps are developed, and how connections both of the professional and personal nature are made. With all the user-generated content available in text, audio, and video, social media has become the new wave of communications that everyone can use to their best benefit. It can be used strictly for purposes of free entertainment or it can be used as a powerful marking tool.\n\nThough social media is widely used, there are still authors that get online simply because someone told them to get online and others who refuse to get online because it's a \"diversion\" they don't need.\n\nWriting is a business. If you're a writer, it's time you take social media seriously and start seeing it as a business strategy for your work.\n\n### Book Marketing 101 (or Good Luck, You're on Your Own!)\n\nOne of the biggest myths people have about signing a book contract with a major publisher is that all you will need to do is write and that everything else\u2014editing, marketing, layout\u2014is someone else's job.\n\nNot quite.\n\nEditing and layout, most of the time, are handled by your publisher, but marketing usually falls on the author's to-do list. Occasionally publishers will spend some of the marketing dollars on new authors, but the majority is set aside for the major players: Patterson, Rowling, King, Steele. You know, the writers who don't need marketing.\n\nTrying to understand how marketing resources are distributed can be best described with a quote from Shakespeare: \"That way madness lies.\" No, it does not make sense, and as you should avoid Lovecraftian instanity, this means _you_ need to take control of your book marketing. No one else is going to care quite as much about your book's success as you do. This does not mean you turn down assistance from your publisher if it is offered, but marketing is not someone else's job. It's part of your writing career.\n\nAnd believe it or not, it's not as hard as you think.\n\n**Find out when your book is scheduled for release.** When you have a release date for your book, immediately go to a calendar and highlight that date. Then look at the month before and the month after. These two months are what we call your \"full-court press\" months for promotion. That's when you're going to want to focus your content on promotion. Too soon, and people will forget it by the time the book is available. Too long after the release, and people will get quickly bored with your platforms. It is possible to over-promote a title, so keep that in mind when you are planning out your heaviest promotions.\n\n**Begin building your social media channels** if you have none or rejuvenating ones that are quiet months ahead of your book launch. A social media plan doesn't just happen with a wave of a hand and snap of the fingers. You need to start talking, start sharing. What you put out there does not always have to be about your book\u2014it can be about you, the writer. What are you reading? How's your editing going? Blog posts, podcasts, and updates that relate to you, your writing, and related subjects matter to your community. Remember, it's about creating connections.\n\n\"A common mistake I see in submissions is the promise to create a social media platform once the book is published. In reality, this takes some time to do well and to pick up steam, and you can't rely on that happening within the initial shelf life of your book. I look for clients who already have a solid, brand-appropriate social media presence before I even pitch the book to publishers. With that in hand, they can hit their promotional stride at publication, rather than trying to scramble and play catch-up.\"\n\n\u2014Gordon Warnock, Partner, Fuse Literary Agency; Publisher, Short Fuse Publishing\n\n**Have a plan for social media and your book.** Now that you're building a network, start developing a strategy for your developing platform.\n\nWhen preparing blog posts, what are the topics you want to cover?\n\nWhat content is most relevant to your audience? Links on fashion? Steampunk? Science? Consider what you want to share with your audience.\n\nWhat blogs would be most interested in working with you on tours, syndication of content, and guest posts? What about Twitter and Facebook parties? (We will get to these topics in chapters 4 and .)\n\nAsking these kinds of questions helps you develop a plan around your book, making social media your outlet to reach out to your readers and fans.\n\n**Stay within your budget.** When you start receiving your advance, consider the words award-winning author Robert J. Sawyer extended to his fellow authors: \"Your advance is your marketing budget.\" This is where you're going to get funding for services like Mention or Sprout Social and giveaway items for Facebook and Twitter parties (which we will cover later in the book).\n\n### Bookmark\n\nA quick and easy\u2014but frowned upon\u2014way to build your community quickly is to _purchase_ Followers, Likes, Retweets, and Reposts through various third-party services. While doing so is a tempting way to build your numbers, it is never a good idea. Many of the \"purchased followers\" are nothing more than automated accounts called _bots_ that post nonsense, links to porn sites, and other accounts. These links lead back to _malware_ (applications that allow hackers full access to your networking platforms or your computer). Also, the statistics are misleading. On the surface, you will have impressive numbers, but as these purchased follows are not genuine connections, your community will offer little to no engagement or interaction. While you do want the numbers, quality in your community is much more important than quantity.\n\nWhen developing a marketing plan, it can be very helpful to talk to authors you meet at book events and conventions. Ask what platforms work best for them, what advantages and disadvantages they find for particular platforms, and how they manage their writing time versus time spent marketing via social media.\n\nIf authors you speak with talk about scheduling book signings as the best way to connect with readers, be gracious and give them your time. Then consider this reality check from us.\n\n### Old Dog: Why Book Signings Are No Longer Worthwhile\n\nBefore you assume that we don't like live appearances, let us say that this could not be further from the truth. We love attending book events, festivals, conferences, and, yes, book signings. We really do. They are a great chance to personally meet and shake the hands of people who are either new readers or longtime fans of our work. They are also great places to meet and network with our fellow authors\u2014after all, writing can be a solitary pursuit.\n\nBook signings, however, should never be your _entire_ marketing strategy.\n\nBookstore signings tend to be more profitable for the _New York Times_ best-selling heavy hitters, but for the average author it is a hit-or-miss venture, a roll of the dice that will either pay out or crap out. You have to consider the cost of your trip: Is this a local visit, or are you traveling out of town? Add to that the two hours you invest in actually sitting in the bookstore, hopefully signing books. If you are not moving books, then you are losing time. Calculate all these factors, and book signings get expensive in a different way.\n\nAgain, bookstore signings are not _bad_ , but bookstore signings should not be your only strategy.\n\n### New Tricks: The Wonderful World of Content Marketing\n\nWhen authors think about marketing, the overarching notion is that you need to \"push your brand\" because it is all about you. To a degree, yes: Marketing is about trumpeting your horn and talking about your worlds, words, and works. This is a very traditional approach to marketing, but in social media this kind of repetitive message gets old _very_ quickly. Think about it: If every time you met with an author online he was talking about his books and that was _all_ he talked about, how engaging would you find his conversations? In social media this is referred to as _signal-to-noise ratio,_ and it relates to the quality of your statuses and updates. If you are constantly advertising or promoting something in your feed, your audience may tune out your updates as \"noise.\" In doing so, readers may miss postings that they genuinely care about or can interact with, referred to as a _signal_. Signal is all about quality and what you deliver to your network. This is how a new method of marketing, called _content marketing,_ has evolved into an effective strategy for online marketing, as content marketing is all about the strength of your signal.\n\nIf you provide your readers and fans with quality content, turning your various platforms into go-to sources for reliable information and fantastic media, you will establish a connection between you, your audience, and your work. People find your work through content related to it, i.e., by sharing websites, blog posts, and other media that is not yours, but related to, in some way, your worlds. This is how you begin to build a reputation with other readers, book bloggers, and authors. An example of content marketing in action can be seen in our various _Ministry of Peculiar Occurrences_ platforms. You can find them on Tumblr, Twitter, Pinterest, Google+, and Facebook. For every one post we offer that directly talks about our books, we offer really cool steampunk posts from blogs, podcasts, and websites other than our own. Sometimes it will be the post about our book that will garner traffic. Other times it will be a post from elsewhere on the Internet that catches the eye. We also offer the chance for other steampunk authors, artists, and creatives to appear on \u00c6therFeature, a column that we run every Thursday about other writers and creatives in the steampunk world.\n\nIt may seem unusual to offer resources other than our own on our series' channels. Why are we offering these other resources time on our platform? After all, shouldn't we be marketing our books? Yes, the end result of marketing is promoting and selling our books, but by offering your platforms to others in your community (in the example of the _Ministry of Peculiar Occurrences_ , it's the steampunk and science fiction communities), you reach your audience, and the audience of your guest contributors. What matters the most with content marketing is the quality of your content: Does your content resonate with your audience, and does it establish you as a reliable resource in your genre? By making your social media channels a soundboard for topics of interest, visitors to your site may want to know more about your work. Content marketing is promotion by example, and by establishing yourself as a solid resource, you can easily reach your readers. We dive deeper into the mechanics of content marketing in chapter 12.\n\n## What You Need to Make This Book Work\n\n\"I really don't have the time for this.\"\n\nThis is probably the biggest excuse we hear from authors on why they are not giving social media a fair shot. Where is this coming from? Could it be the productivity lost when weeding through the variety of Friend Requests on Twitter, trying to figure out which profiles are truly legitimate people and how many are simply spammers? Is it the several hundred invitations to the latest and greatest Facebook game that you have to ignore? Or how about on the blog you recently joined\u2014is there a topic you feel compelled to write, and an hour later you are still working on that post? Whether it is approving others to follow you on Instagram or finding yourself drawn into a thread on Tumblr, the perceived investment of time in social media appears to be a common barrier.\n\nMost authors know enough about social media to be dangerous, while others tend to have the wrong idea on how to manage it. What you need to make this book valuable, and to make social media work for you, is an open mind, time, and patience. Instead of tackling all of these platforms at once, select three and begin developing your strategy. The magic will not happen overnight\u2014hence your need for time and patience. You will want to cultivate your online community, and eventually your audience will connect with you.\n\n### Bookmark\n\nThe most difficult part in writing any how-to book on technology is staying up to date with current events and developments after the book goes to press. This is why we are offering weekly articles and the occasional podcast (syndicated from TheSharedDesk.com) about social media for writers at our blog One-Stop Writer Shop,3<http:\/\/www.onestopwritershop.com\/blog\/> . Consider the blog and podcast your ongoing addendum to this book. Subscribe to the blog to stay in the know.\n\nWe want this book to be your trusted guide in navigating the various online platforms out there, and we want it to help you discover the best practices and strategies for you and how to make them work. We're here to help you, the writer, build a community around your readers and titles.\n\nSo let's begin.\n\n\u00ad\n\n1www.ministryofpeculiaroccurrences.com\/category\/podcast\/\n\n2<http:\/\/www.theshareddesk.com>\n\n3<http:\/\/www.onestopwritershop.com\/blog\/>\n\n# Chapter 1\n\n# WordPress\n\n# Building and Developing a Blog\n\nBlogging has completely changed the way we consume content on the Internet. To truly appreciate this, you need to step back (if you are forty or older) in your memory to the early 1990s when the Internet was very one-sided in how it delivered content in a stationary, static format. HTML, after all, was just text. Sure, the text could be granted attributes. Pictures could be included with the right code, and if you _really_ wanted to impress people, your website featured cute animated images or maybe an audio file that served as your website's soundtrack.\n\nWe call these years \"The Dark Times\" of website design. Younger generations call it \"MySpace.\"\n\nIt was also customary to incorporate links that would take you from one location to another. If you wanted to become an _active_ participant in a Web page's subject matter, you would find a Guestbook or a simple e-mail interface in a \"Contact Me\/Us\" section where your comment, feedback, or criticism might reach someone. This was still static in its delivery method, but it was an opportunity, nonetheless, to interact with the site's host.\n\nWith the emergence of more powerful, versatile computer languages such as Active Server Pages (ASP) and PHP: Hypertext Preprocessor Web developers began experimenting with more dynamic websites where visitors could leave comments pertaining to topics started by the host, interact with one another in these various threads, and _subscribe_ through e-mail to this online content, spreading the reach of the website. Around 2000, RSS took shape as the foundation for weblogs, commonly referred to as _blogs_. A blog is a platform made for a writer, usually written by one person or a core group of contributors, much like a website, but it can be not only subscribed to but also _syndicated_ from one blog to another.\n\nA blog provides the \"home station\" for writers. It's an author's website and a mechanism for broadcasting brand messages to \"satellites,\" the social networks. Additionally, publishing blog content often, consistently, and in a focused manner makes sites more discoverable in search engines. For these reasons, it's imperative for writers to develop blogs based on the subjects about which they plan to write, about the books they have coming, and about the books they may write.\n\n\u2014Nina Amir, best-selling author of _How to Blog a Book_ and _The_ _Author Training Manual_\n\nPerhaps the most useful innovation a blog offers is the way a website is updated instantaneously when new content is added. Instead of having to take a page off-line, edit it, test it, and then repost it, you either post or edit a page through a _GUI_ (pronounced \"gooey,\" and standing for _graphic user interface_ ) and make your changes. These updates are made live once you click the \"Publish\" or \"Update\" option. Instead of having to worry about how an update will affect the overall look of the page, the _template_ you are using on your blog makes everything fit.\n\nBlogs also offer their hosts the ability to notify their audiences of changes. Visitors to your blog are given the option to subscribe to your blog through an RSS reader like _CommaFeed_ 1http:\/\/commafeed.com or _Flipboard_.2http:\/\/flipboard.com Much like subscribing to a magazine, RSS delivers new content to readers' desktops, laptops, and portable media devices. The traffic is still counted and tracked, and your readership is kept informed while your site stays current and relevant.\n\n### Bookmark\n\nIf people are getting content delivered to them automatically, why have a website? Won't blogs detract from the current Web traffic? On the contrary, blogs can actually encourage traffic to Web pages by offering links at the end of posts that take readers directly to the website, to other pertinent or relevant locations of your blog (if your blog is your website), or directly to \"for more information\" pages or additional contacts. Blogs and blog readers do not replace websites but work with them in order to promote traffic and nurture the community around them.\n\nBlogging allows visitors the chance to interact with the blog's author, topic by topic, via the commenting function. Comments are open invitations to both guests and subscribers unless they are moderated. Moderators are useful to ward off spammers and to help \"keep the peace\" if the conversation gets too spirited. Blogs also offer the option to syndicate content by posting it on other blogs, thereby increasing traffic for both the host blog and blogs referencing the original content. Concepts, ideas, and resources previously confined to a single location can now be distributed through the _blogosphere_.\n\nBlogging also keeps you writing, and that is a very good thing. To make blogging happen, you need a _blog engine_ or a GUI that has been built for creating, posting, and managing content\u2014and that blog engine should be WordPress.\n\n## Why WordPress?\n\nA variety of blog engines can easily get you blogging in minutes. Just to name a few, you could use:\n\n  * LiveJournal\n  * Blogger\n  * MovableType\n\nEach of these services goes about providing you your own personal blog in its own way. Our choice option, _WordPress_ ,3<http:\/\/wordpress.com> or <http:\/\/wordpress.org> offers expandability in ways that other blog engines do not. First, there are a wide variety of _themes_ to choose from. Developed by users for other users, these themes are free and easy to implement into your own blog. A theme is a skin that defines all the design and layout for your blog. A fair number of them have the ability to be customized, but you are ultimately limited to what the creator allows you to do. You can also find custom-built themes\u2014for a price\u2014that will ensure your blog is a one-of-a-kind place. WordPress has earned a solid reputation within the blogosphere as a reliable and easy-to-implement platform for all of your blogging needs.\n\n### Bookmark\n\nThe difference between WordPress dot-com and Wordpress dot-org is expandability. WordPress.com is a fully contained package, offering a lot of plug-ins and options for free. However, you get more in the way of storage and options if you upgrade to WordPress's Premium services. WordPress.org offers thousands upon thousands of themes and impressive plug-ins, all designed and developed by WordPress users, that allow you to go beyond the basics. The trade-off with dot-org is that you download the WordPress software, install it, maintain it, and back up your blog _on your own._ To find out more about WordPress and working on it free of the dot-com restrictions, take a look at _Sams_ _Teach Yourself WordPress 3 in 10 Minutes_ by Chuck Tomasi and Kreg Steppe.\n\n### Producing Content\n\nUnderstanding the WordPress interface is easy. If you can work a word processor, you know the WordPress GUI. The hard part is sticking with an editorial calendar. A blog without regular, consistent content does not represent a writer in the best light. It gets even harder to produce content if you are working on a deadline. After all, instead of blogging about writing, shouldn't you be _writing_? This is the common argument writers of all genres, of all backgrounds, have against blogging and its value. Truth is that, yes, your priority should be your commissions\u2014those projects that come with editors, publishers, deadlines, and paychecks, and low on the priority list, along with other items pushed to the edge of your desk, sits your blog on the Internet, silently mocking you.\n\nBlogging shouldn't be a chore or hindrance from your writing career. It is your best introduction to potential readers and your direct connection to your audience. With the right strategy, keeping your content lively, relevant, and consistent on your blog can easily fall into your writing routine.\n\nWith your first blog posts, focus on introducing yourself to readers and beginning to establish yourself as someone with knowledge worth reading about and sharing. The actual promotion of your new blog, though, should not come until you have content within reach, polished, edited, and ready to schedule.\n\nHere are a few ideas on blog topics:\n\n  * When I'm Not Writing (travel, interests, hobbies, or how you unplug and unwind)\n  * Authors Who Inspire Me\n  * Favorite Books (different genres, all-time, repeat reads)\n  * What Inspires My Writing\n  * Creating Characters You Love or Hate\n  * Software\/Hardware for Writers (can also include productivity tips, research tools, etc.)\n\nThe above topics produce what is known as _evergreen content_. These blog posts can easily be rotated in and out of an editorial calendar in case something topical or breaking news (a new book contract, a news headline from the publishing industry) comes to your attention. Evergreen content can also be _repurposed_ , which means roughly 60 to 80 percent of the blog post is tweaked, rewritten, and edited for guest postings on other blogs. The more posts you have like this on hand, the more content you have in case you find yourself running tight on time and short on blog topics. Before launching your blog, schedule five evergreen posts from this list.\n\nAnother approach to content that's commonly used and great for staying organized entails creating themes for different days of the week. \"Monday Motivation\" features your favorite motivational quotes from your favorite authors while \"Tuesday Teasers\" are selections from your upcoming book release or perhaps a work-in-progress closing in on completion. Themed days are great for simple, quick blog posts that are easily syndicated on other blogs and publicized on various social networks.\n\n### Bookmark\n\nAnother option for themed days is using an image as your post. For example, if you designate your Fridays as \"Friday Funny,\" you can feature a humorous quote, either from a book or from an appearance, of an author you enjoy. With a graphic as a post, it can be easily featured on visual platforms like Instagram and Pinterest. Make sure the image you post is:\n\n  * Formatted not as a portrait or landscape but in a perfect square. Instagram, in particular, features all pictures in equal ratios.\n  * Original photography or artwork created by you, or...\n  * A royalty-free image that you have purchased. Use of copyrighted material, including imagery, without proper permissions can land you in trouble if you're using it to promote your work.\n  * Branded with your website so when the image is shared others will know where it originates.\n\nThese images can be created in a variety of mobile and desktop applications such as InstaQuote or FontKiller 2. You will find out more about InstaQuote in Chapter 9.\n\nA \"sweet spot\" to strive for with blog posts is between 500 and 1,000 words. It's okay if you go over that limit; but if you want to keep your topics tight and concise, this is a good target. Shorter blog posts are easier to consume and easier to create. Authors argue that they don't have the time to blog, but 500 to 1,000 words equates to an hour's worth of time. Tightening up your thoughts, proofreading for typos, and linking to relevant material in your post is another hour at the most. Within that range you can say a lot, produce a conversation, and still have time to work on your latest novel or short story.\n\n### Posting Frequency\n\nSo now that you have five blog posts on hand and maybe even a few images for those themed days, you've got content. You've got a blog. You've got an audience out there, just waiting to find you. How often should you post?\n\nNew bloggers often think of themselves as baseball players coming out of the batter's circle swinging two bats\u2014only in the blogger's case it's two blog posts instead of two bats. Two posts a week that grab audiences with new content and fresh discussions potentially total 1,000 to 2,000 words a week. It sounds easy but is actually an ambitious plan when you break it down. First, this plan would burn through the evergreen content you created in the course of just one month. If you blink, you'll suddenly find yourself short on content.\n\nHow about we start with a less ambitious plan and just a few small steps?\n\n#### Posting Weekly\n\nWhen launching a blog, aim for weekly posts. Pick one day and make that your blog day. It can be difficult to maintain that schedule, and it could be a bit frustrating trying to build an audience when you are only posting once a week. The advantage, at least in the beginning, is you get a complete idea of what it is like to produce content for a blog on a regular basis. Weekly posting also allows you latitude to reschedule evergreen topics in light of topics you want to post in the moment.\n\nIf you choose to continue posting weekly, and provided you remain consistent and truly do post weekly\u2014even when you're on deadline or really, really busy\u2014you can cultivate an audience. So long as you can be patient with slow growth and development of your blog's audience, a weekly schedule may be the best way to establish a strong, confident foothold in the blogosphere.\n\n#### Posting Twice a Week\n\nYou have been posting once a week, diligently, for a month or two, and you are scheduling any extra content you've managed to produce in between posts. You find yourself with a healthy backlog, and you are still finding yourself inspired to write a few new posts or post some images featuring motivational quotes or teasers.\n\nMaybe it is time to think about upping your blog post frequency.\n\nAs stated earlier, posting twice a week translates into 1,000 to 2,000 words a week for your blog, an ambitious plan for a novice blogger. However, if you are inspired and you find yourself with a quickly growing backlog of content, two blog posts a week will work for you. An additional blog post a week can also help you build an audience faster, but keeping this audience means you must also keep the two-posts-a-week schedule running. Themed days can help in meeting your demand for content, as these posts, if visual, can be easily produced and quickly posted. Just as you did with your weekly posts, determine the two days of the week that work best for you.\n\nThe power of the blog was brought home to me when I had to go public about an unexpected delay on a highly anticipated book. I could have said, \"I'm going to miss my deadline and I'm sorry,\" but instead I chose to write honestly about why the delay occurred and my own creative struggles. Readers were so decent and kind and patient with me. Twitter is fun, it's my favorite place to connect with other writers, my water cooler, and the best procrastination tool. But the blog is my venue to really write, and that is, in the end, what I do.\n\n\u2014Gail Carriger _, New York Times_ Best-Selling Author of The Parasol Protectorate __ and The Custard Protocol __ Series\n\nCome up with five posts around your theme and then schedule them for a particular day. Provided the new posts are short enough (200 to 500 words), your turnaround on these themed posts can be even faster than for your official blogging day. With two posts a week, you should notice more engagement.\n\n#### Posting Daily\n\nSo your backlog of content is looking solid. Rock solid. So rock solid that you consider just how awesome it would be to turn your blog into a machine of daily content. That would mean posting something new about your works, about your genre, about the writing industry every day, Monday through Friday.\n\nDaily content for a writer's blog is bold, aspiring, and nothing less than impressive. Usually people who aim for daily content\u2014even the ones who take the weekends off\u2014tend to have staff on hand. Blogs with daily content also tend to have huge audiences, with readers who are ready and waiting for whatever news or trivia you have to share. This is why daily blogging is a commitment. A _big_ commitment. As in _Titanic_ -sized, Wagnerian-epic, Super-Size-That-Happy-Meal big commitment. Sure, there are shortcuts like the posting of original graphics and syndicating of other blog posts, but when you host a blog, your audience expects to receive content from you. If the majority of your content comes from elsewhere, you risk losing your audience to the sources you are syndicating. This is why syndication is terrific to have as an occasional treat. It's something new, something different. A blog entirely made up of syndicated content, though, is not what your audience wants. They want to hear your thoughts, your opinions, your voice. Your _original_ content. If you are a writer of books, whether nonfiction or fiction, you have to consider that a daily blog will take up much of your time. And with all this time dedicated to blogging, when will you have time to write?\n\nDaily blogging is a temptation, a temptation that tantalizes if you find yourself with plenty of content in your backlog. This posting frequency comes with many risks that you must consider before attempting such a schedule. Writing time is precious, especially when in the developing stages of a short story or novel. Dedicating yourself to a daily blogging schedule could cut into that precious time.\n\nWhether you eventually decide on a weekly, biweekly, or daily schedule for your blog, consider the maximum amount of time you want to dedicate to content creation before you launch. Once you have a reasonable schedule, stick with your plan. As tempting as it may be to go beyond that budget, don't. This way, you can protect the writing time reserved for your works-in-progress.\n\n### Bookmark\n\nOn occasion we are asked about the value of blogging monthly. Is there any benefit in doing that?\n\nHonestly, with bloggers of all backgrounds and professions dedicating themselves to weekly or biweekly schedules, people will probably pass on your monthly blog post in favor of the four blog posts found on another's site. Even twice monthly blog posts pale in comparison to an author's weekly blog.\n\nIf weekly is a schedule you cannot commit to, blogging may not be your thing. And that is okay. Better to meet the minimum commitment of a weekly blog post than to engage in an occasional blog post whenever you're moved by a whim. The time you could be blogging can always be dedicated to social media or another aspect of your career.\n\n### Taking That Tone: Finding Your Blog's Voice\n\nOnce you have your posting frequency decided, consider where your content will be coming from, as well as what \"voice\" you'll use for your blog. Blogging, after all, is not just an introduction of your work. Blogging provides a connection between you and your readers. What kind of connection do you want that to be? Professional to professional? Teacher to student? Person to person? Do you want to talk about what you write, how you write, how you live? Or perhaps all of the above? Blogs may look simple from the outside, but a lot of planning and preparation goes into them. This hard line of questions about what content you want to produce will serve as your foundation for the tone, feel, and voice. After that, it's your responsibility to create the content, keep up with the posting schedule, and provide a voice that will draw in new readers and keep faithful fans coming back to your website for more.\n\nWe have touched upon the importance of original content, but are you the last stop for content on your blog? You have been exposed to the idea of relying on the kindness of strangers, so let's take a deeper look at the content that comes from sources other than you.\n\n#### Guest Bloggers\n\nAfter blogging twice a week for some time, you will establish yourself as a source of regular, reliable content. Why not, then, reach out to other writers and offer them a place to share their thoughts? Invite other authors to write on topics you think your audience would enjoy, or let them take your blog for a day and speak their mind on topics they come up with but that you approve. Once a topic is agreed upon, schedule your guest or guests accordingly. Keep your guest posts within your word count parameters\u2014500 to 1,000 words\u2014and give them a day you don't normally post. Depending on how many people you want to offer a moment on your blog, you may have guest blog posts alternate with blogs of your own content. If you find yourself with _many_ volunteers, step up the frequency and aim for a weekly feature of a new voice from your corner of the Internet.\n\nWhy would you want to invite other authors to blog? Guest blogging benefits both you and your guests on several levels. For you, your blog gets fresh, new content, a completely different voice, and a touch of variety. You also get the opportunity to introduce your blog to a new audience\u2014your guest's. Your guests, on the other hand, are given the chance to introduce themselves and their work to your audience, so it's a win-win for everyone. It also allows guests to play the role of expert on whatever particular topic you have asked them to blog about. Finally, by appearing on a blog other than their own, guests increase their reach and in turn increase their search results, proving they are not exclusive to their own blog. You will learn more about improving search results in chapter 11, where we talk about search engine optimization, or SEO.\n\n#### Blog Tours\n\nAnother fantastic way of assuring content for your blog is to host a _blog tour_. Unlike a bookstore or convention tour, a blog tour takes place in the comfort of your own home. Your travel itinerary consists of various stops across the blogosphere. On a blog tour, you guest post on other blogs, talking about your book and its relevant topics, all the while inviting the blog hosts to appear on your blog. This pay-it-forward approach to blogging and publicity is a great way to get your words in front of potential readers and to spread the word about your works.\n\nYou can get a blog tour rolling in two ways: Pay for someone to organize it, or do the work of organizing it yourself.\n\n### Bookmark\n\nThere are two kinds of blog tours. One kind of tour is just you hopping from blog to blog, as if you were traveling from bookstore to bookstore on a standard promotional tour. The other approach is to schedule a book tour between several authors and build a round robin of visits so that everyone in the tour visits everyone else's site. For a first-time tour, make a round robin with no more than eight authors.\n\nBook bloggers who arrange blog tours for authors can earn a good amount of income. Price tags for these services vary, but one thing is certain: hiring a tour manager costs you the lowest in stress. Also, tour hosts usually know more bloggers and writers to approach than you do, making your investment a wise one.\n\nYou can put out the call to authors in your network, of course. This is probably the best approach for authors who have existing connections in the book industry. Using Google Docs or Microsoft Excel is the best way to keep track of participants and to schedule dates for your participants. Just remember: Whether hosting or participating, you are investing time and resources in keeping others and yourself on track. You do not want to miss an appearance or fail to produce a piece. People are relying on you for content, just as you are relying on others.\n\nWhen your next stop on a blog tour comes up, your host may have interview questions already lined up for you, but sometimes you may have to come up with your own topic for a blog stop. This can occasionally be hard, especially if you are on a long blog tour and feel as if you cannot come up with another idea. Here are some to get you started:\n\n  * Who would you cast in the movie based on your book?\n  * What is your inspiration for this book?\n  * What is it about the setting for this book that you find compelling?\n  * Did you go deep into the motivations of your favorite character?\n  * Do you plot your novel, or are you a \"pantser\"?\n  * What is your typical day as a writer?\n  * Who are your literary influences, and why?\n  * What lessons did you learn from working with an editor?\n  * Did you find your own voice and style in writing?\n  * What do you love about your cover?\n\nThese are reliable topics that hosts love to feature on blog tours. Pitch two or three of them to your upcoming stop and see what they want. If they say, \"It's up to you...,\" pick one and run with it. At the end of the tour, you will have a wider reach for your name and your work.\n\nYou will also have, at the end of the tour, new evergreen content ready for repurposing and posting on your own blog. Give your new content considerable time (at the very least, six months) from the end of the tour before you start posting it.\n\n### Bookmark\n\nWhen you embark on a blog tour, do not write one post and then offer it as your entry for all the blogs where you intend to appear. Each stop is expecting unique content, and those readers following you while on tour will be sorely disappointed if you're saying the same thing over and over.\n\nFor the best practices to follow on a blog tour and other social media platforms, take a look at chapter 13.\n\n#### Syndicating Content\n\nIn the same way that podcasting is defined in a variety of ways chapter 3, bloggers tend to define _syndication_ in different ways. For this book, we define _syndication_ as taking the opening paragraph of another's blog post and featuring it on your own blog. You then add a link that reads, \"To read the entire blog post, follow this link...\" from your blog to the original blog post's origin. Syndicating a blog post provides your own blog content while directing traffic to another blog.\n\n### Bookmark\n\nThe optimal way to syndicate a blog post is to feature a post up to the point where the post's home blog breaks between two paragraphs using the \"Read More...\" attribute. The \"Read More...\" attribute creates a unique URL that picks up where the previous paragraph leaves off. Using this link you can have people start a blog post at one location and end at another.\n\nWhile syndicating blogs is a great source of content, and a terrific way to encourage community between blogs in the process of sharing, syndication should happen only on occasion. Blog syndication should not be your only source of secondary content, and it should _never_ be your sole source of content. As stated earlier, people are coming to your blog to hear _your_ voice. Offering the voice of others is a fantastic way of inviting different points of view, but if your idea of blogging is nothing more than collecting the thoughts of others, readers may find themselves asking why they are coming to your website for any reason beyond picking up blogging resources that produce original content.\n\nWordPress has a lot to offer for both the beginner and the experienced coder. With just a little investment of time to learn how it works you can create a strong foundation for all of your social media. In chapter 11, we're going to build on this when we look at search engine optimization, and in chapter 12 we're going to examine content marketing, which will give you ideas on how to populate your blog.\n\nIf this all seems too big for you, the good news is that there is another, smaller option! Consider the microblog of Tumblr.\n\n1http:\/\/commafeed.com\n\n2http:\/\/flipboard.com\n\n3<http:\/\/wordpress.com> or <http:\/\/wordpress.org>\n\n# Chapter 2\n\n# Tumblr\n\n# Drive-By Blogging\n\nThe founder and CEO of Tumblr, David Karp, set out to make something less verbose and easier to handle than WordPress or other blogging platforms like Blogger. Even though he said that Tumblr was made for \"those who don't like writing,\" that doesn't mean writers can't make the most out of this platform. In fact, it could be the most useful platform for those with limited time who still want to reach an engaged and youthful audience.\n\nTumblr might have started out as a way to do short-form blogging, but it has become huge itself\u2014and quite popular, with over 94 _billion_ posts.\n\nThe ease of use and the quickness with which users can share information make it attractive to a wide audience.\n\nOn Tumblr, users share all sorts of things they find interesting (such as images, audio, quotes, and short passages from books). They comment on posts, share, reblog, and build a community around their information.\n\nWhen you first look at the stream of constant Tumblr information, it can be a little daunting. It flows through your feed like Twitter (see chapter 5), but with more images, moving images, and videos. Your job is to put your information into the stream, but it has to be engaging and visually exciting to attract attention.\n\nSome authors blog only with Tumblr instead of WordPress, while some use both. It's up to each individual to decide how much time she has available.\n\nIf you are short on time or find long-form blogging hard to get into, then Tumblr could be a good platform for you.\n\n## What Is the Difference Between WordPress and Tumblr?\n\nIf using WordPress is writing a letter to the world, then Tumblr is the postcard version: Tumblr blogs have a pretty picture and contain generally shorter posts about how your day has gone. Tumblr is also easier and quicker to read, and someone who views a post often shows it to other people. If you are lucky, the recipient will pin it onto his refrigerator... which, in this case, is another Tumblr account.\n\nDeeper down there are other differences, too.\n\nWhile WordPress can either host your content or allow you to host it on your own site, Tumblr is more of a mash-up of a social network\u2014think of it as WordPress and Twitter joining forces. The advantage Tumblr has over Twitter is that with Tumblr you have more than 140 characters to work with. So Tumblr allows you to show more of your personality than Twitter does. Make good use of this feature, because personality means a lot to those on Tumblr, but remember to _keep it brief_. Tumblr readers like to _scan_ their feeds.\n\nThe other difference is that Tumblr is a stand-alone company, where as WordPress is open source. This means that a community of developers is free to work on WordPress, contribute to changes, and offer a variety of themes and plug-ins to broaden its capabilities. Tumblr allows the user some freedom, like personalizing his page, but it is not nearly as expansive as WordPress. This creates some restrictions in how you can organize and present content.\n\nTumblr offers a social aspect that WordPress does not. You can \"like\" and reblog posts, follow people who interest you, and make comments on posts, which is important in social media. _Interaction_ on any network means people are taking notice of you, and better still, sharing your words with their audience. Interaction is what makes social media so much fun, as well as so different from the early days of the Internet. Sharing is not just caring\u2014it brings more eyeballs to what you are saying and, in turn, to your books.\n\n### Why Tumblr?\n\nIf you are a writer with limited time, Tumblr is a quick and easy way to share a variety of content. If you have a beautiful image to share or a snippet of audio from your work, then this is the place to be. There are plenty of eager eyeballs just waiting for it. The social aspect of Tumblr means you can connect with other readers and writers who share your interests. Building networks is one of the strengths of this platform; it's not just about posting your work.\n\nIf you want to use Tumblr as your main Web presence for blogging, you can also add pages. You might consider adding pages that detail your appearances or your biography, for instance.\n\nIn short, Tumblr gives you more space to work with than Twitter, and it is quicker and easier to fit into your day than WordPress. If you are not comfortable operating all the bells and whistles you get with WordPress, yet you want to have some measure of customization, then Tumblr could be a good option for you.\n\n### Bookmark\n\nA great thing about social media is that you can enjoy the best of both worlds. There are WordPress plug-ins that will simultaneously create a post on Tumblr. The post is a selection of the opening paragraph, followed by the link that takes readers to the blog. It is not necessarily the same as posting original content on Tumblr, but it can assure you that content is reaching your Tumblr account.\n\nSo why not have your cake and eat it, too?\n\n#### Wider Audience\n\nWriters want to go where the readers are; and if you are looking for a young, engaged demographic, Tumblr is the place. According to Tumblr, 50 percent of its users are thirteen to thirty-four years old. So if your genre appeals to that demographic, it's time to jump into the Tumblr stream. Even if your readers are older, Tumbler has plenty of those users, too: 15 percent are over fifty-five. Tumblr is also different from most other social media sites in that it's pretty evenly divided between genders.1<http:\/\/www.digitalinformationworld.com\/2014\/10\/social-media-user-demographics-linkedin-tumblr-facebook-and-more-infographic.html>\n\nWhile it may not be the biggest social media platform, one thing is certain: Tumblr readers are interested. They spend an average of fourteen minutes on the site, which is more than Facebook or Twitter users. So while there may be fewer eyeballs on the site, those eyeballs are there for longer, which gives you a better chance to impress.\n\n#### Ease of Use\n\nThe Tumblr action bar makes it simple to use. Text, Photo, Quote, Link, Chat, Audio, and Video are options laid at your fingertips. You can upload your content directly from your website or another person's website, or you can just drag the image to upload it. Tumblr works hard to keep it easy.\n\nDon't forget your Tumblr page and profile says a lot about you, so get in there immediately and customize it. Make this corner of the Internet your own. You can tweak the colors of your page or go for a theme that you can adjust to your style. The more original and unique your Tumblr presence, the more people will want to follow your blog. Many of Tumblr's customizations are offered as free choices, but there are some that you can pay for as well.\n\nWith Tumblr, you can also create a _group blog_ where more than one person can post. You can assign members and then promote a few contributors (or all of them) to administrator ( _admin_ ) roles, granting them the ability to invite new users, remove current ones, delete posts, and reply to messages. Regular members can add their own posts, as well as edit or delete them. Group Tumblrs are good if you are co-writing a book with another author, or if you are writing an anthology. However only _secondary_ blogs (described in the next bookmark) can be group blogs.\n\n### Bookmark\n\nWhen you set up your Tumblr account, you establish a profile and your _primary_ blog. The primary blog can be hosted only by the profile holder. Secondary blogs allow for multiple admins. Tumblr allows you to have as many secondary blogs as you desire, but your primary blog only allows for one administrator.\n\nAnother little time-saver is the bookmarklet.2<https:\/\/www.tumblr.com\/apps > Drag the bookmarklet into the toolbar of your browser, and whenever you see something interesting on a website, you can simply hit the bookmarklet. If you highlight accompanying text and then hit the button, that text will appear in a pop-up window without you having to do a thing. (Remember, Tumblr wants to make blogging as easy as possible.) You can then decide if you want it to appear as Text, Photo, Quote, Link, Chat, or Video. This nifty little device even grabs video and images. It also links back to the source but lets you add your own commentary.\n\nThe bookmarklet is a great time-saving device that allows you to quickly and easily share interesting items you find. And that's wonderful, because who doesn't love getting things done with less effort?\n\n### Bookmark\n\nReaders cannot comment on your Tumblr blog, but if you install the third-party expansion Disqus,3<https:\/\/disqus.com> then you can add that functionality. First, create a Disqus account, then click on the cog symbol on your Tumblr, and finally click on \"Edit Theme.\" Under \"Theme Options,\" enter the Disqus Shortname and click \"Save.\" Now your readers will have the option to comment, just like they do on WordPress.\n\n#### Mobile Possibilities\n\nTumblr's app is available for all your smartphones\u2014after all, Tumblr is about sharing and, in some instances, that means on the go. Readers love to see a slice of your life, especially if you are rubbing shoulders with other authors. The app offers all the features of the website optimized for your iPhone, Android, or tablet. Like all other social media platforms, it is up to you to decide how much you want to share. Rest assured, the Tumblr app is just as easy to use as its website sibling. If you want to take advantage of \"Tumbling\" while out in the wild, dive into the Tumblr app and keep your readers in the know.\n\n### Bookmark\n\nWordPress is not the only platform that plays well with Tumblr. Many authors connect their Tumblr account to their Instagram account. You can easily sync your Tumblr from the Instagram app, just as you can for Twitter and Facebook. You will find out more about Instagram in chapter 9, and you can learn how to sync up accounts by reading the Appendix.\n\nWhat should you post to share with your readers?\n\n  * **Travel.** If you're going somewhere interesting\u2014even if it's just your morning walk\u2014snap a picture and upload it.\n  * **Food.** Pictures of tasty things make for great Tumblr posts.\n  * **Events.** Let people know you're accessible.\n  * **Artwork.** What inspires you? A steampunk fashion shoot? An iconic comic book panel? A classic piece from J.M.W. Turner or Georges Seurat? Post it!\n  * **Animated GIFs.** We dare you to search Tumblr for animated GIFs of Tom Hiddleston. Go on. (Just make sure you have a few hours.) For fun, you can also look for author-related GIFs, like famous authors writing and pages turning.\n  * **Pets.** The Internet wasn't invented for animals, but sometimes it sure feels that way. Posting pics of your pooch shows your softer side, plus it offers a bit of your personal life without using pictures of your children.\n\nYou might think these things don't apply to you, especially if you are just embarking on your writing career. But you'd be surprised how interested readers are in seeing inside a writer's life\u2014even if you're a \"new\" writer. No matter how long you've been at it, it is good to establish yourself as a real person, not just a writing machine.\n\n## Expressing Yourself: Producing Content\n\nPhoto and video content are the most reblogged and shared posts on Tumblr. They catch the eye and are more likely to stop readers who are scanning through posts. If you have posted something visual, you immediately have an advantage over the posts that are text only. Just be sure to link back to your website or other relevant material. Otherwise, you are wasting an opportunity.\n\nTumblr users also love _animated graphic interchange format_ images _,_ or _animated GIFs_. Love them or loathe them, refer to them using a hard or soft _g_ pronunciation, on this platform they rule, particularly if they feature pop culture. If you feel like spicing up your blog with GIFs, then you may want to check out the website Giphy.4<http:\/\/giphy.com\/categories> There are plenty of wonderful, funny GIFs related to writing, writers, and beloved books, and Tumblr has a very active and passionate reading community.\n\nHowever, unlike syndicating posts, Tumblr uses a feature called _Reblog_. Returning to the Twitter and WordPress analogy, the Reblog feature is the retweet of Tumblr. On a reblog, the post is re-created (with accreditation) on your Tumblr account and allows for likes and comments on your account. While reblogging is a wonderful feature of Tumblr and is highly encouraged, nothing beats original content. It's best to make your Tumblr a mix of both. For example, on our Tumblr blogs, we post original content with three reblogs in between. This strategy spreads out our own content while building that community feel.\n\nJohn Green,5<http:\/\/fishingboatproceeds.tumblr.com\/> author of _The Fault in Our Stars_ , has a great combination of this. He posts information and insights about his books, but you also get a peek into his life, with pictures of what is happening in his world. You feel like you know this guy and you like him. Jami Attenberg6<http:\/\/jamiatt.tumblr.com\/> has a Tumblr that is mostly original content. It includes pictures of the world around her, snippets about what she is working on, daily observations of her life, and advice for writers. It is pretty dynamic. Kiera Cass,7<http:\/\/partylikeawordstar.tumblr.com\/ > author of the Selection series, posts all about boy bands and fun happenings with her series (lots of images, for example), and, best of all, answers questions from her fans. So if you hear other writers say writers aren't on Tumblr, that could not be further from the truth. Tumblr is a great place for writers to express and promote themselves, and have a bit of fun.\n\nI use Tumblr as a coffee machine in my virtual office, as if it were a place to go hang out with others. I engage with fans with \"Here's this cool thing I found...\" kind of posts.\n\n\u2014Elizabeth Bear, award-winning author of _Karen Memory_\n\nDon't forget to tag your original content, as this is the way others are going to find it. People search for particular tags that interest them in order to find new, interesting content. The Ministry podcasts we post to Tumblr, for example, are tagged with #steampunk #podcast #adventure #mystery. We can also add tags about any location in the podcast like #Berlin or #Ireland.\n\n_Blue Tags_ show posts that are being curated by someone in Tumblr itself.\n\nThese tags mean you are getting free promotion for your post. Bear in mind that you can't make this featured tag yourself. You can only post great content and hope that one of the Tag Editors likes what you are doing. Tag Editors are chosen by Tumblr from the community, and they can change. They start off with a term of sixty days, and though some are invited to continue doing this job, others are not. So keep an eye on the tags to see what the current editor is looking for, but don't bend yourself out of shape to produce what they like.\n\nUnique content is more likely to be picked by an editor, but a repost of other people's content can be as well\u2014but only if you have included the source information of where you got the image or post.\n\nFinally, be sure to include as many tags as possible. Hashtags are your friend on Tumblr, like many other social media platforms. Popular tags include topics on fashion, vintage, film, makeup, and animals. So if you have any relevant posts, don't forget to tag them!\n\nWhen producing content, you want to make use of Tumblr's best features: _Queue_ and _Schedule_. Schedule lets you pick the time and day you want your post to go out. Queue grants you the ability to add a post to a collection of posts that will go out at set times. For example, you can adjust your settings to drop a fresh post from your queue once, twice, or as many times a day as you want. Tumblr also lets you set the time parameters of when you want the posts to happen. You can choose how often each of these posts from the queue fires off. Twice a day is a popular choice, but you can post into your queue up to fifty times a day if you want to\u2014or are able to. We recommended two to five posts per day. Don't flood your feed, but don't leave it idle either.\n\n### Reblogging and Original Content: Which Is Best?\n\nThough it's generally simpler than blogging, using Tumblr still requires a time commitment. There are writers who cue up original content or sync up their blogs with Tumblr in order to share their most recent posts in a feed, while other authors reblog the content of others and are happy with producing a highlight reel of all the \"cool stuff\" they've found on their own.\n\nReblogging others' content is a great way to build a community and display interesting items in your own feed. However, you also need to add your own commentary, so readers see you interacting with the post rather than mindlessly reblogging. If the comment ties the content into your own work as well, bonus points.\n\nWe want to stress that original content is not merely a link to \"buy my book,\" though it can contain a link to buy.\n\n### Bookmark\n\nThere's an art to subtly advertising your book. You want to make it easy to find, but you don't want to hit people over the head with it either. Usually when we share images, we also include a short paragraph from the work, and below that a link to where people can buy it.\n\nOriginal content can be pictures of your day, or observations of things around you, conventions or conferences, and people you meet. It can also be snippets from your published book or one you are working on. Tumblr readers want to learn something interesting and insightful about you and your writing.\n\n## What to Expect from the Tumblr Community\n\nTo become part of the community, you need to make connections and participate. Standing on the sidelines simply doesn't make you friends. Don't be afraid to make the first move. When you first create your account, you need to find connections and start working on them immediately. Tumblr will suggest accounts for you to follow in the sidebar, basing its suggestions on the keywords you use to tag your content and the accounts you are currently following. Take a close look at what is suggested and start building your network right away.\n\n### Bookmark\n\nA great place to start building your network is with the spotlights that are curated by Tumblr. Spotlights show the most interesting and popular blogs in different categories. Finding fellow writers is important so that you can tap into them.8<https:\/\/www.tumblr.com\/spotlight\/writers> Make sure you branch out into other interests you have, as well.\n\nDon't worry. There are spotlights for all kinds of things: music, parenting, science, design, and many others.\n\nShow your personality by following people outside the writing community. In this way, you can display what a well-rounded person you are. Readers are interested in people who write and what they do beyond the typing. Insights make you interesting.\n\nSpotlights are not the only way to find the subject matters you're interested in. You can also look for blogs by doing a keyword search: travel, history, food. Tumblr features accounts on anything and everything. Tumblr lets you follow two hundred accounts a day, or five thousand different Tumblrs total for each Tumblr you manage. Don't be shy about following all you can. The more you follow, the more people will follow you in turn.\n\nDon't forget to _like_ (or _heart_ ) and reblog from your new connections\u2014that way you are spreading the love and encouraging them to like and reblog your posts as well. When you start to get comments on your posts, be sure to respond to them.\n\nRemember: Interaction is the key. You're cultivating a community.\n\n## Sharing with the Class: Syndicating Media\n\nSharing your information across various platforms makes for easy content. Since Tumblr has a different audience and community than other platforms, don't be afraid to repost content you are featuring on other locations, such as your blog, Instagram, or Facebook.\n\nAs you will find throughout this book and online, a variety of media, such as images and videos, will drive the most traffic to your content. If you are sharing something to look at or something to listen to, provided it is engaging, people will stop to watch or listen. The challenge is working within the limitations of Tumblr. Audio, for example, combined with a good cover image makes for great content. Our series The Ministry of Peculiar Occurrences has an award-winning podcast anthology, _Tales from the Archives_ , and it gets coverage on Tumblr as well. However, in order to upload the content directly, the audio needs to be 10 MB maximum. Considering that the average short story we upload comes in at roughly 40 MB, it is not surprising that we have to remotely link such media. This material is said to be _syndicated_ (a slightly different take on how you syndicate a blog post) back to Tumblr. When it comes to media, syndication allows for sharing of media beyond just Tumblr.\n\nDifferent social media platforms have different demographics, and syndicating means you have the best chance of reaching all of them. It also can cut down on the amount of time you spend doing social media.\n\n### Avoiding a Potential Identity Crisis: Managing Accounts\n\nYou may have just one Tumblr account, but you are not limited to the number of secondary blogs. You can add up to ten Tumblr secondary accounts a day, but each additional account comes with limitations. Your secondary accounts cannot follow other blogs, create questions, like, or submit posts. However, secondary blogs can be great if you want to have a main author page and different Tumblrs for your different series (if you have them). For an author, having too many Tumblr blogs\u2014especially if you are spreading yourself over other social networks\u2014can be a way to (pardon the pun) tumble down the rabbit hole. Having a main author blog as your primary blog and then secondary blogs for a new series or characters from new titles is more manageable.\n\nIf you ever get confused as to which of your primary and secondary blogs you are working on, click on the head icon in the top bar, and it will show you.\n\n## Tumblr Strategies\n\nTumblr is all about sharing in the moment. Make the most of a mobile app and share images, quotes from your work, book trailers, and reviews\u2014this keeps your feed interesting and helps build your list of followers. Keep your smartphone handy and make those moments immediate. In doing so, you give your fans a peek behind the curtain. If you are at a convention or a book signing, then post those images, but try to keep them fun. Take pictures of your table, your readers and other writers out on the town, or anything peculiar you find at the event. If you find some interesting cosplay (costume play), see excited readers with your books, or meet excited readers, post photos of them!\n\nCreate a connection with your community by sharing memes and news of interest to your readers. This can be about you or what is happening in the world around you.\n\nAs we've already said, video is popular on Tumblr, so if you have the time and skill to do it, then jump in. However, bear in mind shorter is better, and funny is even more engaging. You can record book reviews and chatter about what is happening with your own writing and the challenges you are facing.\n\nRepurpose your blog posts from WordPress to reach the Tumblr audience. Keep them short and pithy, and don't forget to include an image to catch the eye.\n\nQuote sections from your book and books you are reading to build interest. When building an audience using book reviews, you will draw more viewers to your site by talking about books that are similar to yours. This is particularly useful when preparing to launch a book. Tease the heck out of your audience with riveting or intriguing snippets from your work. Make sure you don't reveal anything too important, though! The idea is to create cliffhanger-type situations.\n\nIn general, the Tumblr audience enjoys visuals, so create images, videos, and GIFs to use. A riveting image combined with a concise piece of text does a great job of gaining attention and interest.\n\nIf you're sharing audio or a podcast episode, be sure to include an attractive image that's associated with it. If you (or your publisher) are releasing an audiobook, use samples and post them online. Don't forget to include the links to where they can buy it.\n\nUse the Q&A function to engage with your audience. If no one asks you a question, post one yourself. It might be about inspiration for your writing, why a character acted in a certain way, or perhaps plans for more books.\n\nRemember: If you are using Tumblr as your secondary Web presence, include a link back to your website in each post so readers know where to go if they're looking for more content.\n\nBetween WordPress and Tumblr, you have the basics of blogging available at your fingertips. Try both and get a feel for what works for you and where you think you will thrive. In the next chapter, we'll concentrate on ways to get your words and thoughts into your readers' ears. It's an exciting platform that delivers audio (and in some cases, video) directly to computers and portable devices around the world, and it's a lot of fun!\n\n1<http:\/\/www.digitalinformationworld.com\/2014\/10\/social-media-user-demographics-linkedin-tumblr-facebook-and-more-infographic.html>\n\n2<https:\/\/www.tumblr.com\/apps >\n\n3<https:\/\/disqus.com>\n\n4<http:\/\/giphy.com\/categories>\n\n5<http:\/\/fishingboatproceeds.tumblr.com\/>\n\n6<http:\/\/jamiatt.tumblr.com\/>\n\n7<http:\/\/partylikeawordstar.tumblr.com\/ >\n\n8<https:\/\/www.tumblr.com\/spotlight\/writers>\n\n# Chapter 3\n\n# Podcasting\n\n# Creating On-Demand Media Programs\n\nWith the many blogging sites, authors are able to get their words and inspirations out to readers. Instead of sharing those thoughts with written words, though, have you ever thought about sitting down and enjoying a fireside chat with your readers? That, in a nutshell, is what you're doing when you are _podcasting._ In 2004, programmers figured out how to use WordPress and other blog platforms to deliver dynamic content\u2014specifically audio and video\u2014to subscribers. Podcasters create their own media and syndicate it to listeners and viewers around the world. In doing so, they become a media production company, offering on-demand media content to subscribers via the Internet. These media files are consumed by subscribers through a host website, a computer's media player, or a preferred mobile device.\n\nThis is all well and good; however, many authors are not exactly dynamic in front of a microphone or a camera. Why would writers want to get into a media production platform when getting them to agree to public appearances of any kind is akin to moving mountains?\n\nPerhaps the most obvious reason a writer would want to host her own podcast is because she has something to say.\n\n### Bookmark\n\nJust because the FCC doesn't have jurisdiction over podcasts doesn't mean you're exempt from the law or immune to lawsuits. _You're personally responsible for anything you say, do, or condone on your show._ Additionally, the rules concerning airplay of licensed music, the distribution of copyrighted material, and the legalities of recording telephone conversations all apply. You still have control over the content you create, but make sure you obey the law.\n\n## Writers Gone Wild: Talk Shows\n\nA podcast talk show starts with an idea, something that the creator has the desire and knowledge to fuel with his own voice. Add to that a bit of passion and a do-it-yourself approach to audio or video production, and the end result is a platform for whatever the author wants to say to listeners.\n\nAnd writers can produce some wickedly fun content when the right people are in the room.\n\nPodcast talk shows are like those you might find on television or radio. They are informal, off the cuff, and sometimes even involve guests. The topics of conversation do not always have to be about _the craft_ , a topic _many_ writing podcasts start off. Podcasts we have produced in the past and that we have appeared on as guests have covered the following:\n\n  * Book Marketing ( _The Survival Guide to Writing Fantasy_ )\n  * World Building ( _Serving Worlds_ )\n  * Working in the Genre ( _Adventures in Sci-Fi Publishing_ )\n  * A Look into the Writer's Lifestyle ( _The Shared Desk_ )\n\nThese podcasts cover a variety of topics and also display the chatty, informal nature of the format.\n\nThese podcasts are a bit like sitting around with your favorite authors, enjoying a cup of tea or coffee (or, if the discussion gets really informal, a glass of wine or a beer) while they talk about whatever is on their minds or answer questions on a certain topic. Talk shows can be very easy to produce. Sometimes all that is required is a recording device and a topic of discussion.\n\nThere are two advantages to creating a simple talk show podcast.\n\n  * **Minimal Editing:** It's just you and your guests, or just you on the mic, following a predetermined topic. You might have to edit the occasional stumble or tangent, but generally the talk show format is one of the easiest to produce. Without the need for heavy editing, it shouldn't take much time to clean up and produce an episode. You will want to review the recording for quality and flow, of course, but sometimes you can record, edit, review, and post all in one night.\n  * **Minimal Production:** Along with editing, quality concerns such as background noise (which can add to the spontaneity of the show, especially if you have cats) are taken lightly. The talk show is more about a slice of life or a casual sit-down with literary professionals. Your main priority should be that everyone is clearly heard. Preplanning for production can be a bullet list of talking points, while your post-production should be creating _show notes_ during the review process. Show notes accompanying the episode's blog post give listeners an idea of what happens during the show. It also helps if they want to skip ahead to a specific portion or find websites you might have mentioned. Links back to these sites are always appreciated by the websites' hosts.\n\nThe biggest benefit of the talk show format is allowing your readers to get to know you, see how your creative mind works, and enjoy a sit-down with you and your guests. Making that personal connection can go a long way in social media.\n\n## Anthologies in Audio: Short Stories\n\nThanks to podcasting and its pioneers, like Mur Lafferty and Patrick E. McLean, short stories are enjoying a renaissance. Shows like _Escape Pod, The Voice of Free Planet X,_ and _The Melting Potcast_ all feature short stories as part of their programming. Depending on your posting schedule, short fiction podcasts can be demanding, but they are a great way to introduce yourself, your work, and your world to a new audience.\n\nWe produce the Parsec-winning podcast _Tales from the Archives_ , in its fourth season at the time of this book's publication. The seasonal anthology, delivering new short story audio on a biweekly schedule, is set in the world of The Ministry of Peculiar Occurrences _,_ our award-winning steampunk series. For this production, we have scaled down our production time considerably by approaching the podcast with this strategy:\n\n  * **Stories read by the author.** Each season of _Tales from the Archives_ offers ten to twelve stories, only two written by us. The remaining short stories are written and read by authors we invite to write in our universe. To do this, we offer guidelines, editorial passes on the manuscript, and post-production services on their audio. The lion's share of the work is in the writing, recording, and editing of the short story, all of which falls to the author.\n  * **Post-production process.** Once we receive an author's short story, we review the audio and then score the stories with a soundtrack and, if time allows, a few sound effects for key moments. These production elements bring an added dimension to the stories and are choices we make as content producers in an effort to do something different, something inspired.\n\n### Bookmark\n\nIt is totally up to you whether you want to do a straight read or go even further with sound effects and music. Any elements you add into your podcasting anthology\u2014sound effects, music, guest voices\u2014lengthens your production time. At the very least, if you are working with audio from other authors, you will want to teach yourself how to use audio filters to sweeten the clarity and quality of your guests' audio.\n\n  *  **Biweekly posting schedule.** As with blogging, a monthly schedule for podcasting tends not to be frequent enough to keep listeners engaged, while a weekly schedule consumes free time like children rummaging through their Halloween candy score. A biweekly schedule, however, serves as a comfortable compromise, allowing time to produce and edit another episode while also developing new material and keeping with present editorial commitments. For all podcasts, a biweekly schedule is the easiest to keep and maintain.\n\n## Going All In: Podcast Novels\n\nIn January 2005, Tee came up with a premise that\u2014at the time\u2014podcasters regarded as mighty ambitious: podcasting a novel from cover to cover. With _The Dragon Page_ podcast hosting, Tee's trailblazing was followed by young adult author Mark Jeffrey and science fiction and horror novelist Scott Sigler. In the decade that followed, writers like Mur Lafferty, P.G. Holyfield, P.C. Haring, and Starla Huchton undertook this challenge. When it comes to podcasting fiction, the novel remains the Mount Everest of audio projects to undertake, with some authors never reaching the summit. Be it time, effort, or resources needed, final chapters remain unrecorded. Podcasting novels is not a casual undertaking by any means.\n\nTogether we have produced six podcast novels\u2014or _podiobooks_ , a term coined by Evo Terra in 2005. Terra later created Podiobooks.com, the premier location for podcast fiction. Some of our books were previously published, while others were released as audio before they saw print. In the process of podcasting six novels we learned a few valuable lessons.\n\n  * **Do not launch the podcast novel until it is complete.** In the early days, it was suggested that a _buffer_ of episodes\u2014perhaps five to ten\u2014was good to have on hand. That way, if life happened to get in the way (and it usually would), you would still be able to keep a weekly or a biweekly delivery schedule. Now it is commonly recommended that before the first episode launches, the podcast should be completed in its entirety. Long-form podcasting is especially difficult when attempting to maintain a regular schedule while balancing real life, so consider having your novel recorded from beginning to end before going live.\n  * **Keep production at a minimum.** While indulging in various aspects of production\u2014sound effects, guest voices, and music\u2014is fun in short-form podcasting, long-form podcasting offers demands of its own. Such production elements only slow your progress. Keep the production in your podcast novel simple. If you want to add more elements, do so, but remember that you are increasing your workload.\n  * **With a complete podiobook, adhere to your posting schedule.** A complete podiobook means you can set your posting schedule: weekly, biweekly, twice a week, etc. In doing so, you are making a commitment and have no legitimate reason to shirk off responsibility\u2014so stick to your schedule.\n\n### Bookmark\n\nBe careful what you wish for, or, in the case of podcasting, be careful when you ask for feedback. You're most likely to get it\u2014and from places you may not expect. Since geography doesn't limit the distance your podcast can travel, unlike terrestrial radio, you may receive feedback from listeners coast to coast and around the world. And just like book reviews, feedback isn't guaranteed praise. Listeners will be _honest_ with you. Respect that.\n\n## Why Should You Podcast?\n\nIf it sounds like there is a lot of work involved in just planning and producing a podcast, it's because there is. There's no other way to put it. When an idea for a podcast comes to you, it is just like when an idea comes to you for a book. A lot of planning, effort, and resources go into a podcast, and when authors try to cut corners and run on the fly without a plan, the results can be disastrous.\n\nDon't believe us? Search the variety of podcasts about writing on iTunes. You might notice that some _podfade_ (a term for when a podcast starts strong, only to soon peter out) after a mere three episodes. When a podcast includes ten minutes of a writer typing, signing books at a bookstore (without any context, such as where he is, when it happened, or what book he was signing), or revealing the contents of a shoe closet, you can safely assume some planning was sorely needed but didn't occur.\n\nYes. A _shoe closet._ We've seen a lot of bad podcasts in our day, and the examples we give above are out there on iTunes to this day.\n\nPodcasts can be far more work intensive than blogging, but there are definite advantages to hosting either a show or an anthology.\n\n**Podcasting fiction is a great way to introduce yourself and your work to audiences.** Whether you decide to share an anthology of your backlogged short stories or your novel in its entirety, podcasts offer a \"try before you buy\" for potential fans all around the world. An anthology also doubles as great promotion for upcoming works, particularly a series, as the stories take nothing away from the novel you have coming; instead, these short stories introduce listeners to the world and offer glimpses at the novel's principal players. For a podiobook novel, your podcast could be a prequel or separate story running in time with the events of your main novel, offering listeners a low-risk, free introduction to your world, style, and work.\n\n**Podcasts offer audiences an intimate behind-the-scenes look at being a writer.** Many still believe the author's life to be all about ascots, smoking jackets, and enjoying snifters of cognac while the sun sets. If this is true, we have failed as professional authors.\n\nWell, we do own ascots\u2014we write steampunk\u2014so that's covered.\n\nPodcast talk shows not only dispel those lofty myths, they tear down the fourth wall between writer and reader, inviting the audience into the process. Sometimes authors express themselves with breakdowns of world building, character development, and plotting. Other times, the focus is not on work but on play: what they are currently reading, a live report from a conference or author event, or two authors talking about how they unwind. The podcast is your chance to sit down with your readers and allow them to get to know you beyond your pages.\n\n**Podcasting, whether it is fiction or a variety show, is just plain fun.** \"All work and no play makes Jack [Torrance] a dull boy,\" so why not fire up the microphones and let loose? Some of the best podcasts we have recorded have been in the studio with other authors, either talking about what we do when planning and plotting, or what we do when we are off the clock. Podcasts can be very therapeutic that way, helping you manage stress, deadlines, writer's block, and a variety of other speed bumps that life can unexpectedly throw your way. You can also find joy in taking someone's audio fiction and enhancing it with music and sounds to bring it to life. And then there is the accomplishment of finishing your own podiobook, releasing it to the public, and hearing someone say, \"I heard the podcast, so I really needed to read the book.\" Podcasting is a rewarding experience and, with the right plan and the right strategy, it can be as equally satisfying as the release date of your latest novel.\n\nPodcasting sounded like a neat hobby. I had always liked the idea of being on the radio, but I get tongue-tied, and recording and editing myself was easier. I'm not sure if I had gone into it with the focus of building a career, I wouldn't have chickened out. But since I was just doing it for fun, I wasn't afraid of failing, so I \"accidentally\" built an audience of people who were eager to hear my work, both nonfiction and fiction. I wouldn't have the career I have today if I hadn't started podcasting in 2004. So the lesson I learned was: Don't be afraid of failing.\n\n\u2014Mur Lafferty, Campbell and Parsec Award Winner and author of _Ghost Train to New Orleans_\n\n## Why Shouldn't You Podcast?\n\nThis may sound like a real stunner coming from two podcast novelists, but after success\u2014both creatively and critically\u2014with podcasting, we will never be podcasting long-form fiction again.\n\nMaybe right about now, you're thinking, \"Woah-woah-woah! You two are longtime podcasters, and you've just spent a section of this book telling me how great it is to podcast. Now you're saying not to?\"\n\nOne more time\u2014we will not be podcasting _long-form_ fiction any time soon.\n\nWe love podcasting, and we love to podcast fiction, but there are a few reasons we've made this call. Podcasting is a blast, but it is not for everyone, and podcasting a novel means less time for our writing. Along with talk shows and anthologies, we believe it is only fair to talk about how podcasting can work against the author, just so you know what you are facing when considering one.\n\n**Podcasting is a financial investment.** A podcast can't happen without a _microphone_ , and if you want your show to have a clean, quality sound, a mic built into your computer won't cut it. In addition, you're going to need _recording software_. There are free options out there, but professional packages, such as Audition and Logic Pro X, offer more features, better mixing capabilities, and export formats. To handle any outside noise, you may also want to invest in sound-dampening tiles, or something similar. The more you invest in a podcast, the better your production will sound, but it all depends on your budget. How much do you want to invest in a production that will earn you nothing directly? If you are not ready to make the investment, then you probably should reconsider podcasting as a promotional avenue for yourself.\n\n**Oversharing.** A podcast can be a challenge to produce, but without parameters or personal borders, it can become a therapist's couch if you're not careful. As mentioned before, podcasts provide an intimate connection between you and your readers, but unchecked, that intimacy can put you, your family, and your professional reputation at risk. Recording your thoughts or opinions after having a \"bad day\" at a book signing, with an editor, or at an agent meeting could descend into muckraking, and once a rant has gone live, it is distributed into the various media players of your subscribers, remaining there even when you remove it from your server. You should know, well before your first podcast, what you do and do not want to talk about.\n\n**Relying on content from others.** Even with money exchanging hands, waiting for content from others can be a disappointment, especially if deadlines are missed. Without the agreed-upon content, the time you were hoping to save by having someone else provide the podcast will be lost. Additionally, if someone writes you a story, but refuses to record it, no time is saved on your end. When audio comes from somewhere other than your own studio, it may not have the quality you strive for, and while you can't necessarily expect studio-quality audio, you may need to work some post-production audio engineering. That means you need more time to replace the content.\n\n**Podcasting is an investment in time.** We did some math on the time we spent on our podcast novels. For the amount of time we poured into one of our long-form podcast fiction titles, we could have easily written two novels. To be fair, we love production elements, including guest voices. But all those audio dimensions add time to producing a single episode; even if you keep it simple by using just a bit of stock music for scene changes, you will still need to edit out breaks, stammers, and places where the speaker trips up. Editing audio also requires review. Cut off too much from an audio take and the timing between the beginning and end of the edit may sound unnatural. Additionally, missed edits or repeats may sneak into the episode. Podcasting long-form fiction, even when kept to the basics, demands time. Are you ready to make that investment?\n\n## Why Do We Still Podcast?\n\nWe're sure it sounds like we are sending you mixed messages on podcasting, but it's more of a \"You need to know what you are getting yourselves into\" situation. Authors love the idea of getting behind the microphone and riffing away, but once you get beyond the newness of it, podcasting can be a real undertaking.\n\nThat said, we love podcasting and still podcast to this day. When you podcast short fiction\u2014original episodes set within the world of your novel\u2014these audio adventures can broaden the universe you're creating and enhance the reader's experience, provided you set a strategy. With the first season of our _Tales from the Archives_ podcast, we introduced potential readers to our cog-and-gear-enhanced world through lost cases of the Ministry of Peculiar Occurrences. These adventures predated events of our premier novel, _Phoenix Rising_. Initially, we planned for the podcast to be a glimpse into what we had created. However, when readers began asking if our audio anthology would reveal more of Wellington Books's past, the Ministry embarking on adventures in other parts of Her Majesty's empire, and whether or not there were other agencies similar to the Ministry investigating the unknown in other places in the world, we needed to change our strategy.\n\nThree of the stories from that first season\u2014\"The Evil That Befell Samson\" by Pip Ballantine, \"The Seven\" by P.C. Haring, and \"From Paris, With Regret\" by Starla Huchton\u2014went one step further and referenced events mentioned in _Phoenix Rising_. This, we discovered, was a real treat for readers as they felt the podcast offered an added dimension to the novel. Now, several Parsec Awards (an award bestowed to the best in speculative fiction podcasting), four seasons, and a \"change of strategy\" later, select characters, situations, and even artifacts from the podcast are making appearances in the novels, and vice versa. Some stories even carry an impact on future novels. These references (commonly known as _Easter eggs_ ) from the podcast expand a book's reach, granting you and your readers plenty of new avenues to explore. What was once a marketing tool, the _Tales from the Archives_ podcast now provides a \"Director's Cut\" approach to the series, offering additional scenes audiences delight in experiencing.\n\nAs for our talk show, _The Shared Desk_ , we employ a different strategy. There are other goals of promotion\u2014letting people know where we will be and what our upcoming projects are, and sharing writing topics that interest us\u2014but in the end, the aim of _The Shared Desk_ is to unwind a bit. We do, and we manage events of interest to the writer while doing so. We now have the podcast down to a system that allows us to record with little-to-no post-production (although we do sometimes come across a few things that should remain unsaid and therefore must be edited out), so all that we need is a topic and a chunk of time.\n\nAuthor, photographer, and artist J.R. Blackwell says _The Shared Desk_ is \"just like having a drink with Tee & Pip...\" and we take a lot of pride in that. We enjoy that connection with our readers, fans, and friends, but with deadlines, demands of family, and (of course) writing, there are occasional \"silent spells\" that one of our more passionate listeners, Gail Carriger, has chastised Tee for.\n\nAnd when you upset Gail Carriger, you can rest assured when having tea\u2014 _no scones for you, mate._\n\nPodcasts can be all about promotion, but they should also be fun. With the amount of effort and planning that go into them, they should always be fun. This is what drives us to keep going.\n\n### Bookmark\n\n_The Shared Desk_ , with the publication of this book, will be offering a special \"social media-centric\" episode once a month (the show posts on a biweekly schedule) on the OneStopWriterShop.com blog. If you want to subscribe to only the audio addendums for this book, subscribe to the One-Stop Writer Shop blog. If you want all the shenanigans that go on behind the mic, subscribe to _The Shared Desk_ through iTunes or TheSharedDesk.com.\n\nPodcasting can be quite an intimate, personal experience. But now we're going to dive right into the largest social media platform currently out there. Hold onto your headphones and warm up your \"Liking\" finger as we head into the wild open spaces of Mark Zuckerberg's world.\n\n# Chapter 4\n\n# Facebook\n\n# The King of Social Media\n\nIf social media began with blogging, then Facebook is what it has transformed into today. Running for just over a decade, and sporting a membership of over 1 billion users, Facebook is the force of nature in social media that people hate to love and love to hate.\n\nBut what is it that makes Facebook so appealing?\n\nBefore Facebook, social networking covered a wide variety of websites and applications. If you wanted to express an opinion and invite an audience to comment on your thoughts, you would blog. If you had photos you wanted to share, you would upload your photos to Flickr and share links or embed your images in blog posts. If you wanted to swap a quick note with friends, you would use either AOL Instant Messenger, Yahoo! Messenger, or Skype. This meant hopping across multiple locations to share media, touch base, and keep people in the know.\n\nThat all changed when Facebook brought to the Internet a one-stop website where photos could be uploaded and shared, thoughts in both long and short form could be posted, comments on these thoughts could be responded to with links, images, or both, and private messages could be shared with others online. With everything offered in one location, it's easy to see how Facebook erupted in popularity and became the \"necessary evil\" in building your social media platform.\n\nWe will address the description of \"necessary evil\" for Facebook later in this chapter. Right now, let's take a closer look at Facebook and what it can do for writers.\n\n## Facebook for Writers\n\nSigning up on Facebook is only the first step. Perhaps the most important question of this social network is that of your intent. Do you want Facebook to be a means to communicate with friends and family? Or do you want to run Facebook as a professional platform?\n\nWhat's the difference? An audience of about five hundred as opposed to five thousand.\n\nChances are, without professional acquaintances, your Facebook friends would not extend beyond a few hundred, but as a writer using Facebook as a promotional platform, you may be approving friendships with people you know only from a meeting at a convention or the purchase of a book. This is where the line blurs between personal and professional relationships in social media. Along with various strategies on how to leverage Facebook in your favor, we will talk about this line and how best to define it.\n\nYou have to stay flexible, because you don't know where the \"next big thing\" is coming from. Right now, Facebook is ubiquitous. I get more interaction from Facebook than any other platform. It's easy to keep readers informed and easy for them to reach out to the author via comments on text and image posts. As Facebook ramps up video, I think it will create a great opportunity for authors to make fast, simple videos that help readers identify with the person behind the words.\n\n\u2014Scott Sigler, _New York Times_ best-selling author of _Alive_\n\nFacebook grants you two ways to work with an audience and, outside of your main account, keep a separation between the personal life and professional one: _Groups_ and _Pages_. Groups are a community built around a shared interest, while Pages are more of a central hub of information where a host controls the signal, frequency, and variety of content.\n\n### Where Everybody Knows Your Name: Groups\n\nFirst, let's take a look at _Groups_. Groups are communities built around a cause or commonalities like electric vehicle owners, cancer survivors, a reoccurring special event, or fans of science fiction and fantasy. The online club is a great way to encourage and build an active, enthusiastic community around your work and your worlds, as the content featured on Groups is completely community driven. Everyone shares a voice here, but since the discussion only happens when someone speaks up, it's a good idea to have a few topics ready for discussion before you invite readers and other authors to join. Having topics on hand also helps you avoid lulls in the Group's activity.\n\nThe most recent post made by a member will appear at the top of your Group. This is the case until someone leaves a comment, restoring that topic to the top of the feed. Interaction takes priority with all previous topics until a new topic is posted.\n\nAlong with general discussions, you can upload photos and videos, ask questions of your Group members with a poll, or add a file (PDF, Word document, etc.) to share with your membership. Provided the media falls under the guidelines of your group, this is a great way to share exclusive content with your community.\n\n### Bookmark\n\nIf you are a _moderator_ , or someone else in your Group is assigned as moderator, you are Judge Dredd: __ judge, jury, and executioner. If you make your Group _public_ , you decide who stays and who goes on account of bad behavior. With _private_ groups, you also decide who is invited. Moderators must not only keep an eye on the behavior of members and the approval of new ones, but they must also make certain members stay on topic. Gone unchecked, Groups can become havens for trolls (people who are deliberately provocative, just to get a reaction) and spambots. Moderation is paramount, just as it is for a blog, to keep your signal strong and the conversation of a high quality. Set ground rules for your Group, and adhere to them. Otherwise, a community can quickly fall apart.\n\nGroups are fantastic for creating a community, but if you are promoting a new release or even a short story in an upcoming anthology and someone in the community posts about seeing your book in a bookstore or promoting his own project, then your post concerning your latest release may be lost. With enough interaction, you can manage to keep your post at the top of a Group's feed, but only until someone leaves a comment on a previous post, once again burying your promotion under previous posts.\n\nInstead of using Groups as places to promote yourself and your works, use it to create a community around a cause, an event, or a passion. Perhaps you want to host a writer's group\u2014a safe place (provided you enforce your own guidelines) for writers of all skill levels and backgrounds to refine their storytelling talents. You would not use the same strategy you used to implement a _Page_ (more on that to come), but it would be a fantastic way to get your name out there in a spirit of goodwill. The focus with a Group like this should not be all on you, but you can easily claim your Group and its intent, a very subtle strategy of self-promotion if it becomes popular.\n\nSo if this is a Group, what is a Page?\n\n### My House, My Rules: Pages\n\nWhile a Group is a community that shares something in common and meets in one location to discuss it openly, a _Page_ is very different in two respects. First, Pages come with an _administrator's control panel_ (or _admin panel)_ that features built-in _analytics_ (measurements of traffic coming to and from your Page, demographics, and performance of ad campaigns), options for boosting posts (paid advertisements), and options to schedule posts (you can either post immediately or schedule a time for your message to appear in Facebook's _News Feeds)_. Another difference between Pages and Groups is that moderators for Pages post as the host, whether that's as a business, a celebrity, or an organization. People who Like a Page can interact with page updates, but they cannot post updates unless they are admins. These posts appear in the News Feed, regardless of whether you are following that page or not.\n\nPeople on Facebook know more about Pages than Groups simply because of the language and open promotion in the world outside of the Internet. (Yes, there is this magical place called \"Outside, _\"_ and it is where this thing called \"Life\" __ happens.) You rarely hear individuals, businesses, or people at events say \"Join our Group on Facebook.\" Instead, audiences, patrons, and fans of various venues\u2014restaurants, historical sites, and parks\u2014are told to \"Like Us on Facebook\" which remains the best, most efficient way to stay in the know.\n\nWhen creating a Facebook Page you want to make certain your Page is complete. Remember that just as with a blog and a podcast, a Facebook Page should have all its information complete in order to make the best first impression that you can.\n\n#### Your Page's Profile Picture\n\nWhile you have the option to rotate images in and out of visibility, this representation of you and your work should rarely change. If you are an author, this should be your current head shot or a custom logo designed for you and your books. If you create a Page for a series, it should be the most current cover or, as stated earlier, a custom logo that belongs to you. This is your _brand_ , and when people glance at it they should be able to easily connect this image with you or your business.\n\n#### Your Cover Photo\n\nUnlike a Profile Picture, your _Cover Photo_ should be rotated for variety. The cover photo can display a number of things:\n\n  * Upcoming Events\n  * New Releases\n  * Upcoming Releases\n\nYou should find an image that serves as a \"default\" image (pertaining to your brand), but take advantage of the Cover Photo as one of the promotion tools in your arsenal.\n\n#### The Like Option\n\nWhen accessing your Page, you can see whether you have Liked this Page or not. It's not vain to Like your own Page, as this is a way to monitor your own feed. Anything you share or post will appear on your personal feed, assuring you that the Page's posts are getting out.\n\n### Bookmark\n\nSince February 2014, Facebook has been changing its algorithms to make _organic reach_ (growth and signal coverage through people sharing your posts) much harder to cultivate on Pages. Now, for subscribers to your Page, there is no guarantee your posts will be seen without _boosting_ (paid promotion) posts.\n\nSomething you can do to increase the chances of your posts being seen is to tell Facebook you wish to _Get Notifications_ from a page you've liked. Move your cursor over the Liked icon and select from the drop-down menu that appears the Get Notifications and Following options. You will then receive alerts whenever posts go live on the page.\n\n#### The About Section\n\nHere is where you provide the details and contact information for your business and online connections. To edit this information, click anywhere in the About section and click the Edit button to set up and edit Preferences, shared information, and Administrator roles. This is where you provide links to your main site and your bibliography, and details about you or your works.\n\n#### The Post Section\n\nFrom this section of the Facebook Page, administrators can post status updates, links, announcements, photos and videos, and special events and milestones. These are the announcements that will appear on the _Updates Feed_ of the Page.\n\n### Bookmark\n\nFollowers of your Page are also given the option to post, but their posts do not appear in the Main Feed. Visitors will see the same _User Interface_ (UI) as Page Admins would see for making posts, but Visitor postings will appear under a \"Recent Posts by Others\" section found in the left-hand column of the Page.\n\nMaking a post on a Page is similar to making a post with a Group, with the exception of two points. First, your post\u2014if you are an Administrator\u2014will appear with the name of the Page, not _your_ name, unless you select an option other than the Page's name from a drop-down menu. (This posting option is available in the Comments section as well.) The other difference with posting on a Page is that updates will appear in the Page's Updates Feed in the order of posting, from the most recent at the top to older posts as you progress down the Page.\n\n### Bookmark\n\nIf your Page has more than one moderator, your posts should end with signatures. One-Stop Writer Shop is an authors network we moderate. When Tee makes a post he signs it with (TM), and Pip signs her updates with (PB). This way, posts can be easily managed and monitored. This means the reader knows who she is communicating with, adding that touch of personality we're always looking for in our social media.\n\n### Earning Likes from Your Fans\n\nYour Page is up and running. Congratulations! Now's the time to go hunting for the ever-elusive Likes.\n\nWhy do we use \"ever-elusive\" to describe earning Likes? According to TechCrunch, as of April 29, 2015, more than forty million Pages are active. Forty _million_. You might think \"Why bother?\" but then Facebook reports that out of the forty million Pages, only two million are active.\n\nWhat does _that_ mean?\n\nFor some businesses and organizations, Pages are built and then abandoned after giving social media a try. How that \"try\" is defined is nebulous. These active two million are not reliant on organic reach and viral content alone. They are building a presence, a personality, and a reputation on Facebook. Social media is not simply throwing a switch and welcoming adoring fans to your corner of a network. Time and strategy achieve success on Facebook.\n\nSo out of those two million active Pages, how do you get readers to Like your Page, provided they can find it?\n\n#### \"Go Like My Page\" Posts\n\nProbably the most common way to get your numbers up is by asking for Likes. Subtle ways of doing this include the use of print materials (business cards, postcards, etc.) that say to \"Like Us on Facebook.\" Restaurants, movies, bands, and all kinds of businesses employ this simple method. The more direct method is to compose a post that asks people to swing by your Page and Like it. Asking directly for Likes is not considered bad form, provided this approach does not dominate your feed. Repeated requests degrade the quality of your Page from signal to noise, and eventually from noise to spam.\n\nA once-a-week reminder, provided there is a variety of content in your feed, can serve as a polite, proactive approach to increasing your numbers.\n\n#### Embedding Page Links\n\nA more subtle way of promoting your Page is to link back to it through other posts. Provided you have Liked a Page, linking back to a Page is easy.\n\nWhen you click \"Share\" to circulate a Facebook post on another Page, your personal feed, or a friend's feed, a new window appears that displays the original post and a link back to the Facebook Page of origin, offering people a direct connection back to you.\n\nAnother way to embed a link into a post (whether the post is shared or original) is to begin typing a Page you have Liked in the Post field. You begin with the @ symbol and then type the name of the Page you wish to link to. A drop-down menu appears, offering options for pages that share similar titles. Continue typing the Page's name until your option appears. After selecting your desired Page, a highlighted link will appear in the post.\n\n### Bookmark\n\nIf you have Liked other Pages, sometimes you are offered options of names, places, and businesses that match up with what you are typing, even without adding the @ to your typed message.\n\nYour post is now live, sporting two ways for people to find your Page. This is a subtle, elegant way of promoting your work while sharing quality content.\n\n#### Print Materials\n\nWhen attending book events, you will see banners, flyers, postcards, and even business cards, all featuring an author's website, e-mail, and maybe even a Twitter account in lieu of a phone number. If you can find a place to promote your Page, either on the back cover or in the interior front matter of your books, make it happen. If you're going to promote a Facebook Page in print, it is paramount to have a URL that's easy for people to remember and easy to read in print, like facebook.com\/TheMinistryOfPeculiarOccurrences or facebook.com\/onestopwriter, both of which neatly fit on a business card.\n\n#### Boosting Posts and Pages\n\nAs we mentioned earlier in this chapter, Facebook periodically changes its algorithms for News Feeds so that previously Liked pages no longer appeared automatically in your feed. The official word from Facebook was \"We're getting to a place where because more people are sharing more things, the best way to get your stuff seen if you're a business is to pay for it.\"1<http:\/\/adage.com\/article\/digital\/facebook-admits-organic-reach-brand-posts-dipping\/245530\/> __ Across blogs like Mashable2<http:\/\/mashable.com\/2013\/12\/30\/marketers-facebook-rules\/> and BusinessesGrow,3<http:\/\/www.businessesgrow.com\/2014\/01\/29\/paying-for-reach\/> Pages have reported major hits on their Organic Reach for businesses, particularly for medium, small, and (in the case of authors) sole proprietor businesses. We personally know of some cases where organic reach plummeted by 90 percent.\n\nIn order to boost a Post or Page, you have to pay Facebook a fee. In turn, it promotes the signal of a single post or the entire Page. Doing so can make a huge difference in your reach, at least according to the analytics. But are people actually seeing your advertisements or simply scrolling past them in order to get to that cute kitten picture? Between boosting individual posts or Pages, we have found that boosting individual posts is a smarter tactic. Not every post, mind you, as that would smell a bit of spam. (And let's be honest, you don't want your page to reek of the scent of _spam_.) Promoted posts should cover key topics worth your money:\n\n  * new book releases\n  * exclusive events\n  * award announcements\n\nIt is up to you to decide if boosting a post is worth the financial investment, but be smart about where your money goes with Facebook. Most writers have a limited budget and therefore must be careful about where they invest. _Think_ about where you really want to direct your readers' eyeballs.\n\n### Bookmark\n\nThe video blog Veritasium is well known for featuring experiments, sharing interviews with leading minds of science, and showing demonstrations of chemistry and physics that will make you go \"Woah!\" But on February 10, 2014, its host Derek Muller released a fascinating study4<http:\/\/bit.ly\/fb-fraud> on how Facebook handles Likes that users purchase. Accumulating a staggering three million views (and still climbing), Muller goes deep into the slightly unethical business Facebook has started where \"engagement\" is the last thing Page hosts can expect. This video is a worthwhile watch when you're debating whether or not purchasing Likes for your Page is the right move.\n\n## Content Marketing: The Science of Promotion by Example\n\nLogic would tell you that investing in paid reach should be the easiest way to get the word out about you and your books. It would be logical, but in truth, you are more likely to win the lottery than to find a post that resonates across the board with _everyone_ online. Yes, having wildfire appeal similar to _The Fault in Our Stars_ or _The Hunger Games_ would be a beautiful thing to happen to your book, and you know better than anyone that your work has that potential, but trying to reach everyone through paid promotion on a limited budget is unrealistic.\n\nVisit our Ministry of Peculiar Occurrences Page for a great example of how odd the search for viral content can be. You'll find a cross section of various posts we have boosted and others we have posted because we thought were really cool steampunk topics to post. You can see the dramatic difference between the traffic of all the posts, and how even \"throwing money\" at the posts did not necessarily equate reach or engagement compared to the posts that received no boost at all. The truth of Facebook is that there's no magic bullet, no sure-fire formula, and no foolproof way to gain traction. Sometimes it will be the post about your book that will garner you traffic and numbers; other times an eye-catching post from the Internet will boost your numbers.\n\nThat's why _content marketing_ is the cost-effective trend worth practicing. It's a promotional approach based not on the repetition of advertising, but on the reputation and quality of a subject matter.\n\nWhat matters in content marketing is _quality_ , not quantity, of your Page's posts. If you provide your Page followers with good material, turning your Page into a go-to source for reliable resources and fantastic media, you establish a connection with your audience. When people are searching on Facebook for that page with the best content on a subject\u2014news from the romance genre, the latest releases in science fiction, or what's happening in the steampunk community\u2014you want them to instantly think of you. That recognition, based on the reputation you are building through consistent and relevant content you are posting, will lead people to your Page. An additional boon to content marketing is the rapport you build with others in your business. As people find your work through content related to it, sharing websites, blog posts, and other media from your peers, you build a reputation with readers, book bloggers, and authors. You let them know that you are not just a writer, but also a passionate, encouraging supporter of your genre and the writing community as a whole.\n\n### When Content Marketing Goes Bad\n\nContent marketing can be a positive, community-building strategy for any individual or business, whether you're a writer, a publisher, or even a literary agency. However, some writers and social media mavericks regard the features of Facebook to be up for grabs and turn your content into instant viral content. One unethical approach to sharing content on Facebook takes advantage of the _photo tagging_ feature.\n\nIf you are unfamiliar with photo tagging, this feature is engaged when you upload a photograph or video to Facebook. You are given an option at the bottom of the photo to Tag Photo. With this option, you can identify someone in the photo, a live link is created to your account, and then the image appears on your Timeline.\n\nUnethical authors\u2014and, yes, this has happened to us on numerous occasions\u2014will grab a photograph promoting their book, be it subtle or straight up in your face, and then tag you in the picture. Suddenly a random picture that has nothing to do with an event you attended or anything remotely related to you appears on your Timeline, openly promoting another author's book. You have the option of removing your tag from any innocuous images, but remember that some authors will exercise the right to tag you in the image _again_.\n\nYes, this also has happened to us.\n\nIf you are tagging other authors in photos, be sure that they depict something relevant. By relevant, we mean an appearance at which the author is seen in the picture, the cover of an anthology where their work appears, or something that you know will appeal to their tastes. (Tee has lost count of the number of times the \"steampunk barbecue grill\" has appeared on his Timeline, but he doesn't mind. He loves both a good barbecue and good steampunk. That picture is the best of both.) Tagging photos to specifically promote yourself is nothing less than spam.\n\nAnother dark side of content marketing is establishing a Page with only one goal: collecting thousands upon thousands of Likes in order to eventually sell the Page to the highest bidder. Ways that some of these _Like Farms_ (as mentioned in the earlier Veritasium Bookmark) collect massive numbers include:\n\n  * Meme Generation\n  * Quizzes\n  * Content Scraping\n\nIn Chapters 12 and , which cover content marketing and best practices in depth, we will discuss conventions that are considered questionable or unethical, what to avoid not only on Facebook but also on all social media platforms, and how to best avoid these pitfalls.\n\nFacebook is attempting to crack down on these unethical approaches, but keeping tabs on over a billion users is a daunting task. When putting together your own Page, a good question to ask yourself is \"What am I going to do with all these Likes on my Page?\" While it is easy to lose yourself in the numbers game of social media, quality should always trump quantity. Aim not only to cultivate a strong support system on your Page, but also for essential, relevant content that will bring people to your Page and keep them coming back.\n\nSometimes Facebook can be very confusing when trying to find your readers among your friends, your family, and all these other folks with accounts, Groups, and Pages. We hope that in this chapter you have discovered some ways to clear the undergrowth from the depths of the forest of Facebook and perhaps learned how to best spend your marketing budget along the way.\n\nNext, you'll take things down a notch and learn how to communicate with readers in 140 characters. There is a myth that you can't say a lot within such limitations, that all you can do is promote your book by title and links, share what others are talking about, or post quick and pithy quotes from famous people. In reality, the 140 characters Twitter affords can provide a teaser for upcoming books, connect you with other fans, and help you do research for\u2014or even inspire\u2014your next novel.\n\n1<http:\/\/adage.com\/article\/digital\/facebook-admits-organic-reach-brand-posts-dipping\/245530\/>\n\n2<http:\/\/mashable.com\/2013\/12\/30\/marketers-facebook-rules\/>\n\n3<http:\/\/www.businessesgrow.com\/2014\/01\/29\/paying-for-reach\/>\n\n4<http:\/\/bit.ly\/fb-fraud>\n\n# Chapter 5\n\n# Twitter\n\n# Keeping It Brief\n\nWhen it comes to a user-friendly interface and a true plug-and-play spirit, it rarely comes as easy as Twitter. The service has gone through different stages of use, popularity, and application. In its early days, Twitter was a terrific tool to stay in touch and communicate with friends. But just a few years later, it became rife with self-appointed social media gurus who pass along inspirational quotes, needless spam of all varieties, and tweets from other accounts. Unfortunately, that's the price of popularity.\n\nTwitter still serves as a reliable, trusted mobile communication option and is regarded as one of the cornerstones in setting up a strong social media platform. It also now acts as an extension of other popular networks like LinkedIn, Pinterest, and Instagram. Twitter is many different things to many people, and it serves various purposes for writers:\n\n  * A connection with other authors, both for professional networking and for finding new and unique resources for their work\n  * A promotional platform for works currently in print or digital formats, and for special offers concerning their works\n  * A direct line of communication to fans, making their day with a quick reply\n\nIt's hard not to have a love\/hate relationship with Twitter. Hate because it can be a time suck and can often make us crazy. But there's so much to love for those in publishing and for writers in particular: an opportunity to move outside our introverted selves and have direct conversations with the readers who love our books, the ability to interact with people who have the same love of the written word we do, and more than anything, the chance to discover our next great read. Twitter brings us together to celebrate books and that's worth loving!\n\n\u2014Angela James, editorial director of Harlequin's Carina Press and creator of the popular Before You Hit Send, a self-editing workshop for fiction writers\n\nTwitter is easy to get right, but it can be a nightmare for you and your followers if you get it wrong.\n\n## First Impressions at a Glance\n\nWhen people arrive on your Twitter profile page, what do they see? Do you have a profile picture that introduces you to potential followers? Does this profile picture represent how you want to be perceived? How about the header photo stretching across the top of the page? Is there something there that completes your profile page? What does your bio say about you? Is it complete? Is it snarky? All these details are part of the first impression that you and your Twitter account make.\n\n### An Author's Profile Photo: Worth a Thousand Tweets\n\nApproach a profile picture the same way you would an appearance at a bookstore, a convention, or a writing festival. The author who shows up to a panel discussion prepared for the topic at hand, presenting himself in a professional manner, will be regarded differently than the writer who has not bothered to prepare for the topic and looks as if she couldn't bother to brush her hair or find a better shirt to wear than an \"I Love Books\" T-shirt. The photo you assign to your Twitter account becomes your _personal brand_ , representing who you are and what you do.\n\nFor profile pictures, images should be:\n\n  * File size: 700 KB\n  * Dimensions: 400 x 400 pixels (Twitter offers you cropping options, but this is the recommended size.)\n  * Resolution: 72 ppi (pixels per inch)\n  * Format: JPEG or PNG file, using the RGB color scale\n\nBefore you start tweeting, your priority is to get away from Twitter's default image (an egg) and find a profile picture that best represents you. Does your current picture, or the one you're considering, make the professional impression you desire? Does your brand have a sense of humor, or is it more polished? Camera phones and simple photo editors make profile pictures a breeze, but what should you use and what should you avoid? What are your options?\n\n#### Option One: A Picture of You\n\nPerhaps the simplest and best way to introduce yourself to people is with a personal photo that puts a face to the Twitter handle\u2014a personal connection. Depending on your creativity, you can use this kind of profile picture to reflect your particular mood. Perhaps you're going to a special event and you snap a selfie on the way out the door. Maybe you have new haircut you're proud of. Perhaps a picture of you and your pet might cheer someone up on a Monday morning. You really never know what will strike the interest of your readers, but showing them a peek behind the curtain is a good way to start.\n\nBefore you use your smartphone of choice to take a quick selfie or run out to find that wicked-awesome photo of you in a tuxedo or evening dress, consider this approach:\n\n  * **Use pictures taken from the chest or neck up.** __ Remember that those using Twitter applications, especially people on smartphones, will be looking at profile pictures dramatically reduced in dimension. Full-body shots of you will be imperceptible. Keep your photo tightly cropped and close up.\n  * **Avoid images with busy backgrounds.** __ What is happening behind you is just as important as how you look. Too much detail or activity in the background (what some photographers refer to as \"noise\") can make profile pictures on Twitter difficult to make out. Keep what's happening behind you to the basics.\n  * **Avoid offensive imagery.** __ This could prove a challenge for horror and erotica authors, because what is deemed \"offensive\" differs from person to person. Use your best judgment. No nudity. Refrain from obscene gestures. Keep the gore level to the bare minimum.\n\n#### Option Two: A Personal Logo\n\n_Branding_ was associated in the past with business, public relations, and marketing strategies. Today, with modern authors shouldering the promotion of their books, this concept is no longer reserved for advertising agencies to pitch to corporate entities and charge top dollar. Authors are taking to heart principles of branding and applying them to their networking outlets, like Twitter.\n\nA simplified definition of branding is an approach to your business through association with a word, catchphrase, or image. You see it everywhere. Two golden arches usually mean a McDonald's is within sight. A stylized arrowhead worn just over the left breast might make you think of _Star Trek_. A bird against a certain shade of blue taking flight? That's Twitter. If you have a product or a service that people associate with a phrase, name, or some other identifier, then you have successful, effective branding.\n\nCreating a brand offers a unique set of challenges:\n\n  * **Make sure your logo is original, and your own.** You might think this is common sense, but individuals may be tempted to parody corporate giants or make a variation on another's theme. Don't do it. It's best to come up with something original. Make it your own. Make it unique.\n  * **Avoid text-heavy or busy images as personal logos. Keep it simple.** If your self-designed or professionally commissioned logo comes with tag lines, pen names, or URLs, accept that such things will be lost when rendered smaller than 75 x 75 pixels. For our multimedia studio, Imagine That!, our brand is a stylized double koru\u2014the fern frond that is a symbol of new life and growth. For One-Stop Writer Shop, the logo is a stop sign with a book at its center. Remember: The busier you make it, the harder it will be to recognize it at a glance. Logo designs are best when basic.\n  * **Branding does not happen overnight.** Use your logo beyond Twitter and give yourself time for brand recognition. When we release our books or book trailers under the Imagine That! banner, we make sure the logo is visible. Any materials produced from One-Stop Writer Shop bear the stop sign logo. Use your logo consistently and constantly in your promotional materials and you will build brand recognition for yourself.\n\n#### Option Three: Interests, Hobbies, or Something Out of the Ordinary\n\nFor the I-really-hate-to-be-photographed or the I-really-have-zero-artistic-ability authors, there is another option. From zombified George Washington to a variety of characters from video games (any from the eight-bit days to modern _Halo_ resolutions) to celebrities of various backgrounds, profile pictures that are not about you, but about your interests, may be used. However, ask yourself what these images do for you in the way of branding. The answer is absolutely nothing. Do they make a connection with your network? If individuals in your network are into whatever your profile picture depicts, sure, provided you continue to talk about that particular interest. The people following your network may not even know about your writing career unless you mention it in your profile.\n\nIf, however, you would rather keep a distance between your network and your own likeness\u2014and it's not uncommon for authors to want to do so\u2014then this option will work well for you. It maintains a sense of privacy for the concerned author and can even serve as a mood indicator, as this option does not commit you to one particular profile picture. When you use a picture of Animal in full __ rave out, a photo of Furiosa behind the wheel of her war rig, or Captain Kirk screaming into his communicator, people will know you're having a bad day.\n\nThe problem with swapping out avatars is the loss of brand recognition. As mentioned earlier, branding works only when an image is used repeatedly so that eventually consumers associate you or your company with it. If you want to take advantage of that recognition, keep your profile picture consistent.\n\nOf course, there are exceptions to this rule, and there are ways to continue branding while swapping out profile pictures. During the Christmas holidays, we usually post images of us wearing stocking caps. Be smart about how you swap out profile pictures. Decide on a strategy and try to find consistency with the image you are portraying.\n\nNow that you have a profile photo in place, let's take a closer look at the header photo for your profile page.\n\n### The Header Photo: Setting a Tone on Twitter\n\nTwitter has been refining its profile pages since the introduction of the header photo back in 2012. Similar to the Facebook cover photo, which also incorporates a banner-style photo option on both personal profiles and Pages, the header photo is a space where you can add a favorite horizontal photo as set dressing for your profile. If the profile photo is your first impression, the header photo is your stage, offering either an atmosphere or a tone for your Twitter account. Is your Twitter account going to be full of whimsy, or controversial? This can be inferred from your profile picture. Think of how an author head shot on the back of a book can give you a feeling of what is inside.\n\nHeader photos are much like their Profile counterparts, with a few differences here and there:\n\n  * File size: 5 MB (minimum)\n  * Dimensions: 1500 x 1500 pixels (Twitter offers you cropping and positioning options. This is the recommended size.)\n  * Resolution: 72 ppi (pixels per inch)\n  * Format: JPEG or PNG file, using RGB color scale\n\nSomething like the header photo may seem like a trivial detail, but it can bring a lot of pizzazz to your Twitter profile. Header photos can also change when you feel change is needed. It's at the author's discretion.\n\n  * **Make sure header photos are in landscape format.** __ In the case of Twitter's header photo, you need to think horizontally instead of vertically. Panoramic shots work best.\n  * **Be as creative and as expressive as you will.** __ The profile photo is how people identify you, but the header photo can be about you, your interests, and your passions. As a writer, this photo can showcase your favorite place to write, a stock photo of a library, or a (horizontal) portion of your book cover. You're setting a mood for your Twitter profile, so have some fun.\n  * **Design header photos using the rule of thirds.** __ Some header photos on Twitter appear jagged, out of focus, or distorted. This is Twitter's attempt to fill the header region with an image that's the wrong size or shape. Experiment with image sizes or design the image using horizontal thirds on a 1500 x 1500 canvas. You can check out the rule of thirds page on Wikipedia for a detailed explanation and design tips.1<http:\/\/en.wikipedia.org\/wiki\/Rule_of_thirds>\n\nTake into account how the header photo will appear on both the computer screen and the mobile app. Templates are available online to help you put the important part of the image in the part of the screen that will be friendly for both computer and mobile users.\n\nWhat goes into a header photo is strictly up to you. As an author, it is not a bad idea to change your header photo to include a portion of your latest cover release, provided that the cover isn't too busy. For the more ambitious author who possesses a mastery of Adobe Photoshop, designing a banner to promote an upcoming event or release is a good idea. Unlike the profile photo, the header photo gives you a chance to express yourself, so use it to send a message. People will see this backdrop when they follow notifications directly to your profile, check out details on their smartphone, or visit other Twitter clients such as TweetDeck, a dashboard application used to manage Twitter accounts.\n\nNow that we have a look for the profile, let's perfect that final touch.\n\n### Making an Introduction: The Twitter Bio\n\nIt's astounding how many authors leave this blank, or simply use it as yet another place to slap a link to their books. The bio, even within the limitations of 160 characters, is your opportunity to introduce yourself, and as an author, you want to make the best first impression you can. The bio is part of that.\n\nMuch like with profile and header photos, authors employ several ways of introducing themselves in their bios.\n\n  * **The Professional Approach:**\n\n _New York Times_ best-selling author of the INFECTED trilogy, NOCTURNAL, ANCESTOR, the GFL series and upcoming ALIVE trilogy from Del Rey. Books!\n\nScott Sigler, @scottsigler\n\nCampbell award-winning author Mur Lafferty. Podcaster. Ghost Train to New Orleans out now!\n\nMur Lafferty, @mightymur\n\n  *  **The Snarky Approach:** __\n\nReeling and Writhing\n\nHarper Voyager author Mary Victoria, @MAdamsVictoria\n\nI enjoy pie.\n\nHugo-winning author John Scalzi, @scalz\n\nIf you're going to go the snarky route, then we recommend you tread carefully. The above examples can't be taken in the wrong way. Keep it light and as amusing as possible.\n\n  * **Somewhere In Between:** __\n\nAuthor of paranormal and sci-fi romance, steampunk and urban fantasy. Instigator of saucy hijinx and shenanigans.\n\nPJ Schnyder, @pjschnyder\n\nKiwi word herder, cat wrangler, and collaborating creative. Somewhat responsible for Books of the Order, Shifted World, and Ministry of Peculiar Occurrences.\n\nPhilippa Ballantine, @philippajane\n\n **Write a bio that is not overloaded with links.** Some authors use their bios as a way to get in Google+ links, Facebook links, and others, but you don't want to do that. Instead write a bio that means something. Leave the URL for the URL field underneath the bio and location field.\n\n**As with the header photo, be as creative as you like.** You can express your love of pie, use a quote that motivates you to write, or share your excitement about a debut novel coming soon to bookstores or e-readers. Your bio should be a sincere, succinct summary of what you are all about.\n\n**Make your location as close as you want.** Let people know where you are in the world. It can be as simple as the largest city closest to you, or which coast you are on. If you prefer, you can make your current location a state of mind.\n\n**Have your URL go to your main site or blog.** One social media strategy we want you to get from this guide is the goal of sending traffic from social networks to your website. The final destination of your network should _not_ be a platform like Twitter or Facebook. The destination should be your website, the corner of the Internet that has everything a reader needs to know about you, your works, and where you will be appearing next.\n\nFollowing this profile audit, your first impression should be polished and ready for welcoming and connecting with new readers, fans of your works, and other authors. But what do you say? Is anyone really listening? And how do you say anything intelligent with only 140 characters?\n\nIt's time to embrace your inner editor and get over the fear of brevity. As Shakespeare once said, \"Brevity is the soul of wit.\"\n\n## Mastering Tweet Speak: Composing Messages on Twitter\n\nAll communication on Twitter begins with the statement you type into the message field, or what Twitter refers to as a _tweet_. When first introduced to Twitter, beginners often furrow their brow or roll their eyes at the cutesy name used for Twitter's updates.\n\nYes, an update on Twitter is called a \"tweet.\" You're using an application on Twitter. Accept it, and move on.\n\nThe biggest pushback from authors continues to be the 140-characters-or-less limitation. You can't say a lot with that, right?\n\nActually, there is a lot you can say on Twitter. You just have to be smart and strategic about what you say, how you say it, and how you edit a detailed, in-depth post down to the essentials. Twitter requires an ability to understand and master an economy of words. You cannot necessarily speak your mind with the usual verbosity, which may surprise some users, but you can, _and must_ , boil down what you _want_ to say to what you _need_ to say.\n\nLet's say you want your first tweet to be the following:\n\nI'm awake this morning and waiting for the coffee to brew. I've got a big day ahead of me with edits, rewrites, and proposals all screaming for my attention. Oh yeah, and I've got an audiobook to get together for Audible. Yes, it looks a little overwhelming, but I'm feeling confident and ready to rock!\n\nThere are 304 characters (and that includes spaces) in this statement. While you could use apps that post tweets longer than 140-characters, most of these third-party services take people out of Twitter and to another website. Tweenjoy.com, for example, allows you write text beyond 140 characters, attaches an image with the text, and can even put in a header image as well. Another third-party app, Twinormous, breaks down your larger text into multiple tweets, which it will notate as one part of many parts. This might seem like a good idea, but if you have a lot of followers, your tweets might get separated in the stream and lose their context. Twitter users want to stay in Twitter, not hop over to another website to read a single tweet. They are also looking for a quick tweet where they can get what they need in that moment and then move on.\n\nSo presuming you're not going to go down either of those paths, let's break down the content of what you want to say. The main points of this 304-character tweet are:\n\n  * You're awake.\n  * The coffee is brewing.\n  * Your to-do list is very full for the day: \n    * edits\n    * rewrites\n    * proposals\n    * an audiobook for Audible\n  * You're feeling overwhelmed.\n  * Your confidence level is high, and you're ready to accomplish all these things!\n\nAll this needs to be one tweet. Take a look at this:\n\nCoffee brews as edits, rewrites, and proposals await. Kicking off the day with an @ACX_com submission. Let's make it happen. #determined\n\nThe original status of more than three hundred characters has been pruned down to 136 characters after making logical edits.\n\n  * You're tweeting, so no need to let us know you're awake.\n  * It's also clear you have a big day ahead as you have \"edits, rewrites, and proposals\" needing attention. So brew coffee while your projects wait for your attention.\n  * If you want to highlight the Audible project, tell your network you're \"Kicking off the day with an @ACX_com submission.\" Mentioning Audible by using its handle can attract its attention and maybe even start up an exchange or earn you a retweet.\n  * If you're feeling confident and ready to rock, sum it up in one affirmation: \"Let's make it happen.\"\n  * At this point you have fourteen characters remaining. Implement a simple _hashtag_ to track the conversation. Hashtags are tracking tools that allow you to identify tweets and other postings under a quick-to-find category in any search engine or platform-specific search. You can find out more about hashtags and what to do with them in Chapter 13, where we cover best practices.\n\nNow that we have completed an audit of our first impression and have taken a closer look at the anatomy of a tweet, where do we go from here?\n\nTwitter is my go-to social media platform because it's fast. I have connected with industry professionals as well as readers through short tweets, and being able to make different lists to sort the people I follow helps me to manage the stream of information. I appreciate being able to filter the information myself, rather than miss out on posts due to an algorithm like Facebook tends to do.\n\n\u2014Lisa Kessler, Amazon Best-Selling Author\n\n## Best Practices for Twitter\n\nEstablishing and evaluating a presence on Twitter is not simply based on the amount of followers or how often you tweet. There are a multitude of live Twitter accounts, and many, many authors populate their Twitter feeds with the good, the bad, and the downright ugly. What you should strive for is a Twitter feed that offers content relevant to you and your interests.\n\nThe first thing you should do _before your book comes out_ is to launch your Twitter account. Build your platform and become part of the community in advance of using it to market your book and your writing.\n\nRemember that Twitter is about community and interaction. Keep your feed interesting and lively by talking about things _other than your book._ Share images of places you travel to. Tweet a beneficial resource that you just discovered. Post a quick review of a movie you just watched. People want to get to know you, so little updates of your life are as important as your writing. It's your choice how much you want to share, and you should set boundaries for yourself, but talking about your book _and only your book_ will get very old very quickly. Aim for one \"buy my book\" tweet every five to ten tweets. Show people you want to connect and communicate with them, not just sell yourself and your books.\n\nWhile building your network and tweeting with others, seek out and follow other writers, publishers, and agents. Before following them, however, look at their feeds. See if they are actually having conversations with people or simply talking _at_ their networks. Avoid the \"Let me tweet to you how _awesome_ my books are\" feeds, and focus more on those who tweet about good writing mixed with tweets of a personal nature. Depending on the quality of his conversations, a professional author's Twitter feed serves as a great way to listen in on what is happening in the marketplace. Twitter can also keep you in the know concerning which agents are looking for new clients, what publishers are seeking from the slush pile, or the latest on publishing scams.\n\nScreening Twitter feeds is a great way to assure your network is built on quality as opposed to quantity. Sure, there are writers on Twitter who have tens of thousands (or even hundreds of thousands) of followers, but how well are they actually connecting with their networks? What is the worth of a Twitter feed that is nothing but constant retweets, posting of famous quotes, and incessant \"Buy My Book Now!!!\" links? Check out each person who follows you, and only follow back those who are actual people. You may find your numbers will climb much slower, but you will build a Twitter network that is responsive, receptive, and reliable.\n\nSpeaking of a responsive, receptive, and reliable network, make a note of this priority for your Twitter feed\u2014 _reply to mentions._ Watch your feed, and when a mention comes in, provided it comes from a real person and not a spambot (one of those pesky automated systems that spits out tweets), send a reply to that person. This is how communication begins, and you never know what will lead to a new reader.\n\nFor fans who tweet, you're guaranteed to make their day with a simple \"Thank you for reading my book\" tweet. Even writers like Neil Gaiman (@neilhimself) will occasionally respond to followers.\n\nThese are, of course, the _best_ practices for Twitter. What about the worst? It stands to reason that if you have authors who are connecting with networks, making sales, closing deals, and moving forward in their careers, you have writers who are continuously making terrible decision after terrible decision on Twitter, not caring about the damage it does to their brand or their book.\n\n## You Did Not Just Tweet That: _Worst_ Practices for Twitter\n\nIt takes patience to build a quality network, but you can easily torpedo a Twitter account (not with the occasional slip-up\u2014no one is perfect, and Twitter is proof of that) with continuously bad behavior and thoughtless actions that can severely undermine whatever sense of community you're trying to establish.\n\n**The biggest, most common mistake** authors make on Twitter is turning their account into one giant infomercial about their book. Every tweet\u2014 _every single tweet_ \u2014is a tweet about their book, sometimes ten to twenty tweets in a day. They may be written in such a fashion that Twitter will not automatically flag them as spam, but a constant stream of links to a book and how scary-awesome it is fails to build a community around that author or his book.\n\n**The second most common mistake** authors make on Twitter is to beg for followers. You may receive a mention thanking you for the follow, immediately followed by a request for you to like the person's Facebook page. The worst is when you receive an automatic Direct Message asking you for a Like. Desperation isn't attractive in any situation and makes you look unprofessional. Get to know your network on Twitter. If they like what you have to say, they will find you on your blog, Facebook, or elsewhere. Trolling for followers is just tacky.\n\nEqually tacky tactics include:\n\n  * Misrepresenting yourself in your bio. We once had someone follow us who described himself in his bio as a \"best-selling author\" when his debut novel was to come out later that year.\n  * Filling your Twitter feed with inspirational quotes from other books and authors. People follow you to hear what you have to say, not to revel in the wisdom of others.\n  * Filling your Twitter feed with inspirational quotes from your books or yourself. Don't let success go to your head. You're not that important; keep yourself in check.\n  * Filling your Twitter feed with nothing but retweets. Retweeting is not participation or engagement. It can be part of it, but nothing but retweeting is regurgitation. Indulge in retweets, but don't make them your only source of content.\n  * Following people only to drop them later the same day. Many third-party applications and services that promise you more followers will follow other Twitter accounts. Once the followed reciprocates they are immediately dropped off your feed. It is not against Twitter's terms of service, but it is highly unethical. This is one way third-party developers turn social media into a pursuit of statistics, analytics, and numbers, something we will discuss more in detail in chapter 13 of this book.\n\nAuthors behave badly on Twitter in many ways, but it is a good idea to practice the same etiquette you would practice in the real world, such as when pitching ideas to agents and editors upon meeting them. At an event, would you walk up to an agent or editor and say, \"I have an idea you must look at!\" without introducing yourself or asking if they are accepting __ pitches? Probably not, but it happens often on Twitter. Get to know these individuals as both professionals and people. Find out who is looking and what they want. Never pitch to editors or agents online unless you've been invited to do so.\n\nI find Twitter absolutely indispensable for getting news fast and first. If you want to know who's looking for what when, who's been promoted or moved to another company, what line has closed or expanded... there's a good chance you'll hear of it on Twitter before anyone else. It's a great way to keep on top of what's current.\n\n\u2014Lucienne Diver, author of the Vamped and the Latter-Day Olympian series\n\nBy applying these basic approaches and strategies to your profile and the composition of your tweets, and working on what to do and what to avoid, you can make your Twitter account a very powerful platform. At this point, you should invest some time in Twitter. Get comfortable with the approaches we outlined in this chapter. Apply these methods and watch how the interaction between you and your network improves. All that remains now is building on your reputation and making your Twitter account a community, one that doubles as a team of promoters dedicated to the success of your works.\n\nFacebook and Twitter are often referred to as cornerstones of a strong social media presence, but as you can see in this book, there are plenty of other platforms out there to employ. Let's turn to a social media platform with, many consider, the greatest unrealized potential. This social network not only offers many features and potential reach of SEO (search engine optimization, covered in greater detail in chapter 11), but it also has without a doubt the biggest helicopter parent powering it. Yes, it's time to talk about Google+.\n\n1<http:\/\/en.wikipedia.org\/wiki\/Rule_of_thirds>\n\n# Chapter 6\n\n# Google+\n\n# A Place for Discussions and Hangouts\n\nFacebook is the go-to place for posts, images, and video in social media, but not to be outdone, Google offered an alternative in 2011. Many online tech critics predicted it might be the \"Facebook Killer\" at its launch, with membership skyrocketing to half a billion active users before the end of 2013. However, _Google Plus_ (also known as _G+_ and _Google+_ ) has spent much of its existence attempting to develop an identity. Google spent the summer of 2015 reimagining the platform, but Google+, as is, still offers writers different ways to reach readers.\n\nGoogle+, on its arrival, promised it would be a one-stop service where updates and blog posts could be posted and syndicated; photos could be uploaded, displayed, and shared across networks; and discussions from inside and outside a person's network could be hosted over distributed links, images, and media. Google+ and Facebook seem to mirror one another, but Google+'s seamless integration with the world's most popular search engine grants the unassuming, underestimated network boundless potential, both as a contender against Facebook and an outlet for authors. Its demographics lean toward professional males, and these males tend to be interested in discussions rather than images, which dominate Tumblr and Pinterest. So if your genre and topic are of interest to this group, then Google+ could be the place to grow your brand.\n\n### Bookmark\n\nAs of the writing of this book, the future of Google+ is somewhat nebulous. Changes in Streams, Hangouts, Photos\u2014separating them into individual products\u2014have people trumpeting the death of the social media platform as a whole. Bradley Horowitz, the Google executive in charge of the product said at a press event, \"Google+ will be changing. There's a renaissance in the thinking of what Google+ is, and what it's for.\"1<http:\/\/www.cnet.com\/news\/google-plus-will-be-changing-says-its-chief\/> Just what all this will translate into down the line even we as writers of the fantastical cannot say. We will, however, continue to monitor the situation and blog about it on onestopwritershop.com.\n\n## Google+ for Writers: The Account vs. The Page\n\nIf you are one of over 500 million active users with a Gmail account, you already have a Google+ account waiting for you. All you need to do is post a suitable profile picture, a quick bio, and a few facts about who you are and where you live to complete your account\u2014all very much in line with social networking platforms like Facebook and Twitter.\n\nThat is how you set up a personal account on Google+. When establishing a Google+ _Page_ , you must take a different approach.\n\nWhen you set up a Page, you are immediately treated as a business. Are you establishing a Page as a Google+ _Storefront_ , a _Service_ , or a _Brand_? (Authors would fall under the \"Brand\" distinction.) Creating your Page is a simple process, at first. Give a name to your Page, a website providing more information on your future content, and a Brand that describes what type of business you are. Details such as a Profile Photo, Cover Photo, and Story are all that are needed to complete your Google+ Page. You now have your platform up and running.\n\n### Bookmark\n\nWhat about a _Google+ Community_ based around you and your works? Wouldn't that work as well as a Google+ Page?\n\nGo back and take a look at the Facebook chapter's section on _Groups_. A Google+ Community works the same way, built around a cause, a common interest or, in the case of writers, a genre or series. The content found in a community is completely driven by its members, and this would mean either you join a community similar to yours\u2014setting yourself up as a potential spammer if you come in hammering promotions about your material\u2014or you host a community, which will take some time to build, provided its membership contributes.\n\nAnd keep in mind, in a Google+ Community, everyone participates. Content is provided by the users and, even if the host of the community posts, other community participants can post and bury your promotional posts. Additionally, if it is your Community, you must serve as Moderator, adding to your responsibilities in managing Google+.\n\nOn Facebook, your followers\u2014whether on your Page or on your personal account\u2014tend to blur the lines between your personal and professional identity. Google+ takes a very different approach to followers. From any account you establish, Google+ grants you the ability to set up _Circles_ for those you are following and those who are following you. Circles can be labeled with any distinctive name you give them, and by organizing your followers into these Circles, you can then target your posts to the public feed, to your network (or Circles) only, or to specific Circles in your Google+ accounts. Organizing your followers into Circles allows you to target your posts as exclusive content or private content you wish to share only with certain people. In other words, you can control where your message goes and how wide your posts or media is distributed.\n\nCircles can also be established and managed in a Page similar to a personal Google+ account. Accounts are then run by anyone who has access to your Google+ account. If you are security conscious, you may not want to reveal your credentials to fans helping you out with your \"Street Team,\" which we will talk a bit about near the end of the book. Google+ allows you select the _Managers_ tab from the settings tab of your page to set your preferences.\n\n### Bookmark\n\nIf you followed the Appendices and created a Page, when you return to the Google+ main feed (the Stream), you are not liking, commenting, or sharing as yourself anymore. You are now interacting with the Google+ community as your Page. When creating your Page in Google+, much like in Facebook, you either do so as \"Tee Morris, author,\" \"The Writing Worlds of Philippa Ballantine,\" or \"The Ministry of Peculiar Occurrences.\" Doing so creates a distinction between your personal account and your professional page, which allows readers to easily follow you.\n\nManagers, when posting and administering Pages, will have access to a _My Business_ panel option in their left-hand drop-down menu. My Business features not only a posting interface, but also _Insights_ (a breakdown of the activity on the page), direct access to _Google_ _Analytics_ (a measurement tool for traffic coming to and from websites, demographics, and overall online content performance), and the one-click option for starting a _Hangout,_ online gatherings that usually involve Google+'s ability to easily host video conferences (more on Hangouts later). From this dashboard, you can post content, consult with statistics on both your account and main website, and host Hangouts. This makes Google+ convenient and easy to add into your social media routine.\n\nSo far the capabilities of Google+ make the platform sound a lot like Facebook and, truth be told, it is. On its minimalistic surface, the Google+ graphic user interface looks a lot like Facebook in how it works and how you interact with it... at least on the surface.\n\nDig just a little deeper and you witness the firepower of this fully _armed_ and _operational_ social network!\n\n_Fire at will, Commander._\n\nOh, sorry about that. Geek moment. Let me try this again.\n\nDig a little deeper and you begin to tap into the full integration of Google which is kind of like a Death Star, just not with the whole \"blowing up planets\" thing.\n\n### The Beginning of a Beautiful Friendship: Other Google+ Options\n\nIt could be very easy to dismiss your Google+ account as merely Facebook Lite, but now that you have set up a Page, you can easily go beyond what you have in front of you and start tapping into the potential of the world's largest search engine. Your Google+ account offers seamless integration that can provide your social media platform dynamic content and collaborative options.\n\n**YouTube.** A two-way street is a wonderful thing. When someone comments on your YouTube video, it posts automatically to your Google+ account. Then, when you reply (either as yourself or from your Page) with your Google+ account, it posts back to YouTube. It's the little things that make life easier.\n\n**Hangouts.** Perhaps one of the most appealing content-creation tools in Google+'s arsenal, and now in yours, is the _Hangout_ , which is seamlessly integrated with another Google property, YouTube. A Google Hangout is a video chat that can host ten different users, all of whom can join in on the topic at hand. The video signal remains on the person talking, so you will see the camera jump from participant to participant, especially when the conversation gets lively. The whole Hangout can then be recorded and uploaded to YouTube as a vlog entry. On YouTube, the video can be distributed either to private audiences (making the Hangout exclusive content) or to a wider, more public audience (making it new content for current and potential readers to consume).\n\nBefore diving headlong into hangouts, make sure you have a decent microphone (no problem if you are into podcasting as well) and, if you want to take advantage of the video option, a reliable camera. Make sure to set aside some time to host a test Hangout with a few friends so you know which buttons to push and how to switch from camera to desktop sharing if you want to make your Hangout even more interactive.\n\nThough the changes going on in Google+ are still unfolding, one thing is certain: Hangouts will remain. This feature is one of the most beloved of all of the integrated options in Google+. That means it is probably a safe bet for you as an author to investigate.\n\nThere are many ways to use Google Hangouts. Imagine getting nine dedicated readers in a Hangout for an hour and chatting with them about your book. Think about how you could do giveaways during that time. It also gives readers a chance to get to know you as more than just a name on the cover of a book. This kind of Hangout could be a great way of interacting with your most passionate readers or making a passing reader a committed fan. Those who do not actually get into the Hangout can still participate in a chat room forum. Find a time to suit your schedule, and try one out. If you find it enjoyable, then consider making it a regular thing. As your following grows, you can vary the participants in the Hangout so everyone gets a chance. Use a signup form for those who are interested.\n\nA launch party on Hangouts is a great way to get more face-to-face interaction with your readers, without having to jump on a plane. You could invite a select group into a Hangout, or if you have a network of writers, you could have them come to a Hangout and celebrate. Talk about your feelings, and, heck, even trepidations, as your book is launched into the world. Alcoholic beverages and finger food are optional.\n\nAnother option is to create a more general book group. For this group, you would contribute to book culture and promote authors other than yourself. A great example of this is a group led by author, actress, and all-around Internet entrepreneur Felicia Day. Vaginal Fantasy Book Club is a fun, monthly Hangout with a light \"sitting down for a chat with the girls\" feeling. During the Hangout, participants review, between sips of wine, romance genre books featuring female protagonists. However, they are not afraid to wander from the topic and get in a few laughs, either. Once the discussion is done, the Hangout is exported directly to YouTube, instant content for the VFBC's popular Goodreads Group that continues the chat until the next monthly Hangout. This Goodreads group is very popular, numbering fourteen thousand members all watching the current Hangout video.\n\nGoogle may have pulled Google+ in for some behind-the-scenes tinkering, but one of its products is still the big man on campus as far as video goes. Google bought YouTube back in 2006, and that seems to have been a wise move on the company's part.\n\n1<http:\/\/www.cnet.com\/news\/google-plus-will-be-changing-says-its-chief\/>\n\n# Chapter 7\n\n# YouTube\n\n# Introducing Video to Your Content\n\nThe platforms previously discussed in this book\u2014Facebook, Twitter, Google+, WordPress\u2014have mostly dealt with promoting content in text form. Yes, as an author, reading is the preferred format\u2014we create worlds with our words while the readers fill in the blanks with their imagination. So it really should not come as a surprise when authors, plotting and planning out their social media initiatives, do not necessarily take into account visual platforms, and when it comes to visual platforms, none stands taller than _YouTube_.\n\nWith over a billion people using YouTube, four billion views of YouTube daily, and 323 days worth of video viewed through Facebook every minute, it is a wonder that more authors are not taking advantage of the video platform. Sure, there are a number of how-to videos and (of course) cats and dogs doing deliriously cute stuff, but how does that reflect or even represent the business of books? How compelling would video clips of someone writing be? When you think about it from that perspective, it makes perfect sense that authors\u2014an introverted bunch, on the whole\u2014do not feel that YouTube is the place for them.\n\nAnd there are a few other factors that may deter authors from considering YouTube.\n\n### Camera Time\n\nIf you think authors look awkward or uncomfortable in still shots, try getting them in front of video cameras. Addressing a camera and talking to it in a casual, relaxed manner can be admittedly awkward, and that's coming from an author who used to be a card-carrying, dues-paying member of the Screen Actors Guild. There is a definitive personality who enjoys getting in front of a camera and talking in a conversational tone, and most authors do _not_ fall into that category.\n\n### Production Details\n\nYou may think \"A YouTube video? Piece of cake. Point. Shoot. Post. How hard can it be?\" When you take a look at authors who know how to make YouTube work for them (and, yes, we're looking at you, Vlog Brothers1<https:\/\/www.youtube.com\/user\/vlogbrothers> ), consider what these video barons all have in common.\n\nTake note of the background. Chances are, it is a library or a neutral background. Listen to the sound quality. What you hear probably is not coming from the microphone on the camera, but a lavalier mic that you can't see and that has been synced with the video in post-production. What about the lighting? Is the video lit in an atmospheric manner, or is it shot _au naturel?_ You want to be able to be easily seen and at the very least convey a mood. What is the author wearing? How does her skin tone appear? Are the edits simple cuts or transitions, or are they peppered with bullet points that drive the intent of the episode home? All of these details when _not_ considered reflect in the tone of the video and reduce it to something akin to _The Blair Witch Project,_ or what is more commonly referred to as _guerrilla-style_ video. With guerilla-style video, the production elements are kept to the basics: enough light to see what you are recording, wardrobe goes with the moment, and if you happen to be wearing makeup, you're good. Guerilla style is totally unscripted, off-the-cuff, point-and-shoot video, so keep it brief. You can make a video-in-the-moment work, but you still need a plan. That kind of planning is essential to success on YouTube.\n\nThat brings us to another issue with authors entering the world of video.\n\n### Time\n\nAll of the following require time: Setting up a camera or multiple cameras; making sure the light is adequate; creating a slick, clever opening sequence; creating an end sequence where people can learn how to subscribe to your channel; and finally, editing. There are social media gurus who claim there are ways to cut corners that will save you some time. One of those corners is editing.\n\nBefore you agree with that sentiment, think about that\u2014no editing. Would you consider forgoing editing when it comes to your book? Think about the draft you completed prior to editing. Editing is an essential part of a book's life, and with video it is even more critical as you deal with outtakes, missed beats, and tangents, all of which need to be trimmed down to the core of your message.\n\nTime remains one of the biggest reasons authors stay off YouTube. You want your video to look good. No, it will not be the same quality as, say, a film from Marvel Studios; but you do want it to be a step above _America's Funniest Home Videos._\n\n### Bookmark\n\nHere's a formula that's pretty accurate: The time it takes to edit video is one hour for every finished minute of video. So if you have a ten-minute video, you will be looking at ten hours of editing, provided you do not have any additional filming, animation, or media to import into the project.\n\nIf you recall, in Chapter 3, we talked a bit about the commitment you make when podcasting. Video takes that commitment level to a much higher plane. Even with a twenty-year background in video editing, like Tee has, working in video can be a time-consuming and budget-consuming endeavor. While you can pick up a laptop computer for a drop in the bucket of what it takes to actually produce video footage, it is unlikely your machine will possess the processing power needed to render and output video on its own. (No, watching Netflix on your computer is not the same as creating your own production.) This is why computers are made bigger, stronger, faster\u2014because in order to handle creative opportunities like this, you need bigger, stronger, and faster. Video is costly on a lot of levels, and while we have both enjoyed what we have accomplished with video editing and YouTube, we still hesitate to encourage authors to give YouTube a go.\n\nStill want to do it?\n\nIt's not too late to turn back.\n\nOkay then.\n\nLights, camera...\n\n## Action Editing: Software for Video Editing\n\nIf you were to do a search on the Internet for programs on video editing, you would find yourself with an impressive list of options for Macs, PCs, and Linux for those writers who are truly passionate about their operating systems. We don't intend to go into every software package available for all platforms here, but we will spotlight the popular ones, as well as the editing suites that offer options and resources to reach out to and find answers to your questions in a timely manner. Each of these software packages has different interfaces, features, and add-ons, but all of them perform the basic function you need\u2014editing video{\n\n  * **iMovie (Mac).** If you are editing video on a Mac, you are already in a community of creative types, as Apple remains prevalent in the creative sectors. iMovie comes with a variety of built-in transitions and animations you can add to your video, and it easily interfaces with iPhoto and iTunes for those who want to incorporate photos or music into their production. iMovie is intuitive, but there are many online resources and print tutorials available for the higher functions available with its interface. iMovie is an amazing application for its $14.99 price tag\u2014if you buy a new Mac, you will get it for free.\n  * **Final Cut Pro X (Mac).** iMovie is an amazing application, but depending on what you want to accomplish, you may find yourself reaching its limits quickly. If so, you need to make the jump from iMovie to Final Cut Pro X. The cost is considerably higher, though it has been recently reduced to $299.99. The upgrade goes well beyond the basics, although it uses a similar GUI to iMovie. Final Cut will take your video to the next production level, offering you many more options in how many tracks of video you can edit, advanced control over audio and video filters, and output options. While iMovie is a terrific, intuitive program, Final Cut comes with a learning curve that's not too steep but definitely requires time with the how-to manual. You may want some of these advanced options for your YouTube channel. Others you may never need. It all depends on the demands of your video, whatever the project may be, and how deep you want to go with your post-production.\n  * **Movie Maker (Windows).** This editing suite is the iMovie for Windows. Some may argue that iMovie offers more, but that may not make much of a difference considering the end product of an author's video productions should be as simple and easy as possible. All you really need to do is edit, review, and post. Movie Maker offers AutoMovie Themes, which can give your video professional touches such as rolling credits, video effects, and even transitions. All of the above expedite the post-production process. As it is with iMovie, the program usually comes preinstalled with your computer.\n  * **Adobe Premiere (Windows, Mac).** You've met iMovie's counterpart. Now meet Final Cut's. Adobe Premiere comes in two flavors: Elements and Pro. If you have worked with Adobe Photoshop Elements, then you have a good idea of what Premiere Elements offers: the basics of video editing. You have just what you need, at an affordable price, contained in one software package, and you have many more options at your fingertips than you would have with Movie Maker in transitions, prepackaged video templates, and video effects. Similar to Final Cut Pro, the set of features you desire from your editing suite depends on the demands of your video and how much production is involved.\n  * **AVS Video Editor (Windows).** AVS Video Editor offers the basics of Movie Maker, but with a simple, elegant interface similar to iMovie. Video Editor also offers the option to capture video off your desktop (great for tutorials and software walk-throughs, discussed later in this chapter) and produce an interactive DVD, complete with menu interfaces provided by the application. Video Editor runs on Windows platforms only and can be found online,2www.avs4you.com alongside other media production software packages.\n  * **OpenShot (Linux).** OpenShot Video Editor is a free, open-source video editor for Linux first created in August 2008. OpenShot compiles videos, photos, and music with subtitles, transitions, and effects, just like its Windows and Mac equivalents. Similar to programs like Final Cut and Premiere, OpenShot allows you to create videos that can be loaded onto DVDs and online platforms like YouTube. What's different about OpenShot is its \"open source\" nature. As part of the Open Invention Network (OIN), OpenShot encourages developers to create their own add-ons and extensions that, after community testing, are offered to other editors. So if you are a writer and a developer, and you want to create a signature extension like a custom video effect or a template that can quickly make your videos uniform in some way for your fellow Linux users, OpenShot is your best option.\n  * **Kdenlive (Linux, Mac).** Free and safe to download, Kdenlive is another multitrack video editor for Linux and Macintosh. Kdenlive, much like OpenShot and other open-source software packages, encourages its community of editors and developers to create the best tools for video production. Kdenlive offers transitions and effects, and exports the final project to common formats for DVDs and Internet playback (such as DV, HDV, mpeg2, and h264). As Kdenlive and OpenShot are both free, the choice between the two comes down to which interface best suits you and your needs.\n  * **Camtasia (Windows and Mac).** Camtasia can serve as your video editor, but it is known more for its ability to create video captures from a computer. Camtasia turns your monitor (or one of your monitors, if you have multiple monitors for your computer) into the camera and records all your movements and commands or whatever activity is happening on your computer. This allows you to create _walk-throughs_ (animated screen tutorials) of favorite computer programs. Many gamers use Camtasia to create walk-throughs of popular video games, offering helping hands for trickier challenges. Camtasia is easy to master and outputs to a variety of resolutions and formats for final video projects. It can be your stand-alone video editor or a great addendum for your video projects.\n\nWhether you decide to go all in for a professional-grade video editor or keep your wallet happy and work with something that's cost-free, you will want to have one of these editing suites to make your video concise and as polished as possible.\n\n### Bookmark\n\nWhen it comes to background music that you want to mix into your production, copyright laws are in effect. Your YouTube video is being used for promotional purposes, and your favorite song or soundtrack cut is not allowed. Even if you adopt the attitude of \"Well, not that many people are watching...,\" you are still violating copyright. (Think of it as someone using passages of your book to promote her own image, website, or business without your permission.) Apart from original music, your option is _royalty-free music_ , music that you pay for permission to use. The disadvantage is that multiple people can use the same royalty-free music, so you may hear your theme on another podcast or even a television commercial. The advantage? Royalty-free music, found on websites like Audiojungle.net, Beatsuite.com, and iStockphoto.com, is affordable and adds a touch of professionalism to your production. _Layered_ royalty-free music, such as the collections at SmartSound.com, offers the versatility of being able to edit the music, thereby creating endless combinations from one track.\n\nWhile there are plenty of successful vlogs that keep their details simple and basic, even the basics still take up time. Perhaps the most important detail to consider is why you are establishing a presence on YouTube. Without a direction, YouTube is no different than a story that plods along with no resolution or point in sight.\n\nTime to ask the hard question: What do authors bring to YouTube?\n\n## Turning Authors on to YouTube\n\nDespite all our warnings and precautions about venturing into YouTube, video production is also infinitely rewarding. An engaging video not only reflects positively on your brand, but it also provides book bloggers and enthusiastic fans content that livens up any blog post or Facebook status. What is paramount in establishing a YouTube presence is planning what you want to bring to your channel.\n\nBeyond working on a keyboard and the odd book signing or two, though, it's hard to picture what is \"visual\" in an author's lifestyle. However, similar to Instagram (which we will talk about in chapter 9), the potential for visual content might surprise you.\n\n### Book Trailers\n\nMention \"video\" to an author, and usually the first thing that comes to mind is _book trailer_. The purpose of a book trailer is no different than a movie trailer: You are promoting a book or a series. Making book trailers can be a daunting task, even if you keep it simple. There are challenges in finding stock video that fits your book's story. There are challenges in filming original content. The end results\u2014examples found on \"Being Kenneth Branagh: 11 Tips on Filming Book Trailers\"3<http:\/\/bit.ly\/filming-trailers> \u2014should feel cinematic, not like animated PowerPoint presentations. They provide terrific promotional content for your books, so long as the production does not become too involved, and are easy to share across blogs and social networks ... However, when you factor in actors, location shots, and crew, making a trailer can easily become a lot to manage.\n\n### Author Appearances\n\nThis does not necessarily mean \"book signings,\" as appearances can include conference appearances, panel discussions, and special events. You can offer a quick clip, or a series of clips, from a panel discussion. You can hold a quick commentary from the hallways of a writers conference or convention\u2014this can be your thoughts from the event, \"man on the street\"\u2013type interviews with attendees, or a bit of both. What if you host a book launch for your fantasy novel at a Renaissance faire? Capture moments of special guests, festivities, and (provided the turnout is there) book signings. Keep your mind open to special events and the content you can post. These moments serve as your invitation to others to enjoy the event alongside you.\n\n### Video Journaling\n\nThis is where an author takes a moment out of his day to get a thought or a moment recorded. If you are using YouTube, you should preserve these thoughts on video. It may be something completely writing related (\"Use the audio recorder on your smartphone to jot down ideas when you are behind the wheel.\") __ or it could be more about lifestyle (\"When was the last time you completely unplugged from your writing and just relaxed?\"). Journal entries can be some of the most honest, sincere moments you share with your community. When you journal, make sure you are working within your comfort zones. Don't share too much (YouTube is not your therapist's couch), but don't be afraid to let your audience into your head. You do it all the time with your books. Why not do it with your video entries? This is where your YouTube channel feels closest to _vlogging_.\n\nAs the second largest search engine on the Internet, YouTube is not to be ignored.\n\nI can get creative with my content on YouTube by posting in various forms: from personal vlogs on my writerly process (#PiperNotes) to cinematic book trailers to creating videos on topics that may be tangential to my writing (#PiperTravels) but that lead to plot bunnies!\n\nSuch a broad scope of possibilities allows me to continually create new kinds of videos that can be embedded anywhere, including in guest blogs and news articles, and in that way, they can help spread my brand to new readers without my needing to custom code or worry about having higher-end hosting with more capable bandwidth.\n\nIt allows me to look out at my readers and speak directly to them, overcoming the challenges of physical distance.\n\n\u2014PJ Schnyder, aka Piper J. Drake, author of the award-winning The London Undead series and _Hidden Impact: A Safeguard Novel_\n\n### How-To Videos and Walk-Throughs\n\nRecently, for our own vlog, we featured a video that wasn't necessarily a \"writerly topic,\" but more a DIY project. We turned a wall of our studio into a dry-erase board, and we felt this qualified as something we wanted (and needed) to share with our writing community. How-to videos provide a good portion of the content on YouTube and can cover any number of topics. The best how-to videos are projects that may seem hard to do on first consideration but wind up being easy and simple to see to completion. Suitable subjects include reorganizing and remodeling your office, and organization tips for your print resources. Any home improvements you're undertaking for your writing career is definitely YouTube worthy.\n\nWalk-throughs are like how-to videos, but they are staged and shot on your computer. Perhaps you want to host a quick walk-through of your favorite feature in your favorite word processor and project management tool, Scrivener. Or how about showing the tricks to making quick and easy promotional cards for upcoming signings or special appearances? With a program like the earlier mentioned Camtasia or AVS Video Editor, you can create a video to guide your viewers through software packages.\n\nNow that you have a good idea what authors could bring to YouTube, you have to take a serious look at how you want to film your production. Whether it is a full-blown, costumed spectacle or a vlog documenting your latest convention appearance, what you are creating is a production, and it should be approached as such. You, as producer and director, need to plan out your YouTube show before a single word is spoken in front of a camera.\n\n## Best Practices on YouTube\n\nThere is a method to creating video, and these strategies and approaches are designed to keep things simple. These strategies are also starting points for you as a filmmaker and videographer. You may want to go beyond what we recommend, and that's okay, but it's best to walk before you run.\n\n### Keep Your Video Brief\n\nWhen we say \"brief,\" we mean at or around five minutes. As we mentioned in an earlier Bookmark, it usually takes one hour of editing for every finished minute of video. So you could be looking at five hours of work for a five-minute video. This is five hours of work only if you have a lot of edits, transitions, and effects you're adding into your video, but regardless of how much work you are putting into the production, you want to aim for five minutes as the sweet spot for your video's running time.\n\n### Bookmark\n\nA free YouTube account gives users fifteen minutes for video clip length. Anything beyond that limit will not be approved for playback until you verify your account with Google for free. If you want your production to go longer than the YouTube limits, and you don't want to hassle with verification (which includes registering a mobile number with Google), a popular alternative for sharable video is _Vimeo_.4<http:\/\/vimeo.com> While it's not as widely used as YouTube, Vimeo offers a variety of plans\u2014including a free one\u2014that grant users space for multiple videos, ad-free playback, and expanded options and customization for its user interface. If you find yourself in need of options beyond YouTube, take a closer look at Vimeo.\n\n### Keep the Video Editing Simple\n\nIn an application like iMovie, it is tempting to add in wacky filters and effects for pizzazz, but all these touches add time to making your production. Additional effects mean more _rendering time_ \u2014this is where the application produces a final, self-contained file that doesn't need the video editing software for playback but plays in any media player. How can you keep the demands of video production down? Keep the editing simple: Use basic cuts, eliminate transitions when and where you can, and don't use visually intensive effects. Have an introduction video, a simple sequence of segments and cuts, and a snappy, polished end screen.\n\nSpeaking of your introduction and end screen...\n\n### Dedicate time for the Introduction and End Screen\n\nCreate \"bookends\" for your production. Your video's Introduction should be the most \"complicated\" aspect of your production, maybe some animated titles (or still shots) and the title of your show. Create it as an individual project and output it as a video. There are several ways to create an end screen (and YouTube has many tutorials, some that involve Camtasia). Once you have an introduction and an end screen, you have your two \"bookends.\" In between, keep your production simple. With the complicated segments produced and any remaining segments lacking effects or animations, the rendering time should be quick.\n\n### Rehearsal Is Always a Good Thing\n\nSo are multiple takes. When the camera goes live, you may stumble on your words. Give yourself a beat, then start the thought again. This is why you edit\u2014to remove the flubs and trips in your recording session. Before you even turn on the camera, speak your thoughts out loud. Hear yourself putting thoughts to speech. Think about how your tone or intent comes across. Then refine it before hitting record. The more comfortable you are with your thoughts, the smoother your words will sound and the more confident you will appear on screen; with video, confidence is key. Between rehearsing your thoughts and refining your final video with editing to remove stammers, pauses, and misfired thoughts, your final video will be an engaging production that promotes your brand.\n\n### Have Material Ready\n\nBefore launching your YouTube platform, produce five to ten episodes. If you want to get a better idea of the time commitment you are facing, produce five episodes, at the minimum. If you are feeling bold, produce ten episodes. When you are done, you'll have a buffer of episodes that you can post every other week (up to four months of material) or monthly (up to ten months). Don't go live with the YouTube platform just yet, but note the time and effort it has taken to create these episodes. Can you handle the time commitment? Are you ready to step into the YouTube arena? Knowing what goes into not just one episode, but a series of episodes, will help you allocate time for your channel.\n\nYouTube remains mostly uncharted territory for authors, mainly because of the time commitment and financial investment. With the right strategy and creative thinking, though, YouTube can provide your audience with engaging, entertaining content that they'll want to share across multiple networks. Video editing is difficult but also very rewarding. Devise a plan and strategy for your YouTube channel, and see what happens. You may be surprised how much fun being in front of a camera can be.\n\nNext you'll get away from the moving image and head over to Pinterest, where the long, vertical image is king.\n\n1<https:\/\/www.youtube.com\/user\/vlogbrothers>\n\n2www.avs4you.com\n\n3<http:\/\/bit.ly\/filming-trailers>\n\n4<http:\/\/vimeo.com>\n\n# Chapter 8\n\n# Pinterest\n\n# Your Online Bulletin Board\n\nThink of all the corkboards in kitchens around the world, and all the postcards, recipes, and notes pinned onto them. Now wouldn't it be nice if you had an infinite corkboard for the Internet, where you could keep all those important, pretty, and interesting items? Wouldn't it be great if you could share your corkboard with your friends, and you could see all the fascinating things they found?\n\nThe creators of _Pinterest_ must have thought so, too. They created Pinterest as a visual bookmarking site, where people collect and collate projects and ideas, and share them with others. Each user can have multiple boards where they post things that interest them.\n\nWhen it was created in March 2010, Pinterest quickly became the hot topic among its target demographic, which is women. Everyone was talking about the inspiration that evolved from sharing items with their friends. They were getting ideas for birthday parties, delicious drinks, holiday decor. In a matter of months, Pinterest.com became a very busy, very active Web domain.\n\nThink of Pinterest as a huge online bulletin corkboard, where users Pin images, websites, and videos to their \"boards\" for others to look at and gather inspiration. Users put together as many boards as they want, based on hobbies, vacations, fashion, books, or whatever topics interest them. It is a great way for users to keep track of items they want to return to all over the Internet. These Pins act as visual bookmarks\u2014clicking on the image links to the original content or display original content from the board owner.\n\nCollages scare me. What with the cutting, gluing, hot mess vomiting all over a board and maybe, _likely_ , smeared with glitter. But the _idea_ of collages has always intrigued me. Wouldn't it be great if I could gather inspirational art without, you know, the scissors and the paper cuts and the glue sniffing?\n\nEnter Pinterest! Inspired by a bit of art? Pin it! That thing looks like the zombies I'm writing about? Click, saved! Love that guy's... let's go with _smile_? Added! Now I can collage _forever_ , and I get unlimited boards full of ideas and inspiration. Finally! Doing it freaking right.\n\n\u2014Karina Cooper, award-winning author of the St. Croix Chronicles\n\nOut of all of the social media platforms, Pinterest's demographics are very unique, leaning heavily towards females in the thirty-five to forty-four age group. And the average household income of an average Pinterest user is over $100,000. Pinners are not afraid of spending money, which is always a great demographic for markets. By 2013, Pinterest had become so popular that it was estimated that one-third of all women in America have used the site. So if you know your target audience is women, Pinterest is a social media platform you will need to have in your online arsenal. According to a recent study from the National Endowment for the Arts, 55 percent of women read fiction.\n\nThat is not to say men are not on Pinterest as well. The same study found that men are more likely to read _nonfiction_ books. Having a Pinterest board of DIY projects related to your how-to manual could provide a terrific resource for those already reading your book, earn you new readers from men searching Pinterest for new ideas, and keep your book relevant by providing online addendums and new ideas.\n\nPinterest also sports a widely spread age demographic. However, as far as global reach goes, it is most popular in America\u2014something that writers should keep in mind when deciding whether to add it to their marketing strategy. If your book is unavailable in the United States, then Pinterest is going to be of limited use to you. However, for independent authors taking advantage of a global reach, America is still a vital market.\n\nWhile Pinterest may be one of the smaller social networks, with 70 million users worldwide in 2014, it does boast passionate \"pinners\" who spend a good amount of time on the site\u201415.8 minutes _per day_ \u2014sharing information with each other and discussing their latest interesting discovery.\n\nAnother vital demographic trend about pinners is that they tend to take recommendations and buy from Pinterest; at last count, 47 percent of online shoppers have bought an item after a recommendation from Pinterest. Those are great statistics for selling books; other social media platforms would be excited to have stats that good.\n\nIn summary, Pinterest is the social media platform where well-off women of all ages seek out interesting content and are happy to purchase items of interest. This is why large brand names are flocking to Pinterest. Writers can take advantage of the site in the same way.\n\nIt's time to dive into the world of Pinterest and become part of its thriving community.\n\n### Bookmark\n\nThe best time to Pin is when the majority of your audience has time to relax with Pinterest. Analytics vary from category to category, but generally speaking, the best times are in the evening and on Saturday mornings. These, of course, need to happen in the standard North American time zones.\n\n## Tools of the Trade: Pinterest Applications\n\nPinterest is best described as a virtual bulletin board, which elicits an unassuming image, but don't be fooled. There's a lot happening there. Once you establish your profile and get all your details online, it is best to explore and (subsequently) master the nuts and bolts of your Pinterest account. While there are a few features and options worth mentioning, the good news is they are extremely user-friendly.\n\n### Bookmark\n\nAs we are eight chapters deep into this book, you may see a pattern when it comes to your profiles and accounts. You want your usernames, profiles, and profile pictures to all follow the same look and feel so that when you tell people __ \"You can find me online at...,\" __ you only have to give one screen name. If you do find yourself in need of creating new profile names for yourself, the reoccurring profile picture and simple bio will let people know they have found you. When it comes to profiles, adhere to the two Cs: _completion_ and _consistency_.\n\n### The Bookmarklet\n\nThere is one piece of useful equipment you should have before you begin pinning in the wilds of the Internet. Go to the website1http:\/\/about.pinterest.com\/goodies\/#browsers and then click on \"Install Now.\" The Pin button goes in your browser and makes pinning things on the Web much easier. Now when you click on the \"Pin It\" button, you will find a selection of the images that you can Pin. Choose the best one, click to Pin, then add a description.\n\n### The Widget\n\nIn order to spread the word about your pinning activity, the site offers a widget that can be placed on your blog. It comes in a square, header, or sidebar, or can even be customized to fit your particular situation. It is a great way to let people know that you are on Pinterest. On our websites, we usually have it right in the header so it is immediately apparent.\n\nTo find this widget, you can simply go to Pinterest.2<https:\/\/business.pinterest.com\/en\/widget-builder>\n\n### The Mobile App\n\nPinterest has a great app for on-the-go pinning from your smartphone. The app is available for both Android and OSX. You can find it in their stores or download it.3<https:\/\/about.pinterest.com\/en\/goodies> Anything you can do on the Pinterest website, you can do on your smartphone. The only negative is that it's hard to zoom in on images, but that is a minor problem. You get the full stream of the boards you are following, and after clicking on any of them, you can repin straight to your boards\u2014including any secret ones (which we'll discuss later in this chapter). You also have the \"heart\" functionality to like it, send it, and post it to other networks. In addition, you get the choice of saving the picture to your phone or simply copying the link. This is useful if you are writing a blog post on the subject while you're out and about. You also get access to the Pin logo, search, the bubble icon for chatting and news, and access to all your boards under the human head and shoulders icon. For more about what these features offer, go to the Pinterest appendix.\n\nWith these basic tools under your belt, you can accomplish a lot with Pinterest, but how? Sure, you can Pin your book cover, inspirational artwork, and images from your various events, but what will bring people to your boards and keep them there\u2014preferably for longer than 15.8 minutes a day?\n\n## Game On: Pinterest Competitions\n\nPinterest has become one of the go-to sites for winning items, since the Pinterest terms of service are actually more flexible than a lot of other social media networks\u2014and they don't try and charge you for the privilege of promotion. Pinterest is happy for pinners to use their boards for promotion, as long as the pinners don't say Pinterest is running the competition. However, you must be strategic with competitions\u2014don't run them all the time. Use pinning contests for marking important events, like book releases or holidays, or for making bestseller lists! You might do a Pin\/repin to win a copy of your new title when it is released or a book in the wild competition for when you hit a bestseller list.\n\nCalculate how much money you can afford to spend on a contest. The larger the prize, the more entries and reach you are likely to have. Keep in mind the interests of Pinterest users, and keep what you are asking them to do relatively simple. Nothing is more frustrating for a reader than jumping through hoops.\n\nBefore deciding what sort of contest you want to run, settle on your goal. Are you trying to get more people to your website? Do you want to get more subscribers to your e-mail list? Or are you just trying to increase book sales?\n\n### Pin\/Repin to Win Competitions\n\nThese competitions are a good idea if you want to increase your number of Pinterest followers.\n\nFirst, create a bright, attractive image of what your pinners are going to win. The image should contain your cover image if you are giving away something like a Kindle Fire. It should also say \"repin to win\" and use some variation of the word _competition_ to draw attention and let visitors know they can get something for free. It's important to include a tempting description with keywords, as well as the value of the prize that is up for grabs, and a call to action like \"post this,\" \"repin this,\" or \"take a picture of your own copy.\" In the description, make sure to say how a winner will be chosen, and the date the winner will be announced. Also keep an eye on Pinterest's terms and conditions (which are constantly being updated), just so you don't break any rules.\n\n### Bookmark\n\nA lot of marketing suggests using the tag \"Pin it to win it,\" but Pinterest rules specifically ask you not to. Why remains a mystery, but it is best to stick with other wording to remain compliant with your fine hosts.\n\n### Create a Wish List Board\n\nRun a competition asking pinners to create a \"wish list\" board with your book on it, along with other books of a similar genre. Ask them to include a hashtag so you can find who has created the lists. This is especially good as it encourages friends and family of the pinner to purchase the book for them. At the end of the promotion, give away something of a decent dollar value, like a Kindle. You can choose a winner randomly, or judge the boards and pick a winner.\n\n### Books in the Wild\n\nBuild buzz for your book release by asking pinners to post pictures of the book in the wild, out in bookstores or with the readers themselves. All those Pins appear in their friends' feeds, hopefully resulting in more followers and greater awareness of your book. Again, post simple rules and create a hashtag so you can find the entries.\n\nThese are just some ideas for competitions. If you can dream up a contest, and it fits within the Pinterest terms of service, you should roll it out on a small scale and see what kind of response you get.\n\n## It's Business Time: Pinterest Business\n\nAs an author, you are a business, and Pinterest Business offers layers of depth that go above and beyond individual accounts. It is probably a good idea that you become familiar with the basics of the individual site before you jump into converting to a business account, but once you do, the options are thrilling.\n\nThe first step you must take is to go to Pinterest's business website4<https:\/\/business.pinterest.com\/en> and convert your account. You need to have your own website to be verified. In order to do this you need to insert the code Pinterest gives you into the header <head> section of the HTML or where your blog template allows you to access that section of code. Pinterest shows you how to do this,5<https:\/\/business.pinterest.com\/en\/confirm-your-website> and it isn't difficult even if you don't know HTML. In exchange for a little bit of work, you will get some large advantages.\n\nThe first of these is undoubtedly Pinterest analytics. Pinterest supplies you with graphics about your Pinterest profile, your audience, and even activity that came from your website.\n\n### Pinterest Profile\n\nHere you can see your daily impressions (how often your page is loaded) and viewers for a whole year. It's always nice to be able to see trends and to observe how well your strategy is working. It will also give you the top Pin impressions and top boards with Pin impressions from the last thirty days, so you can see what images and videos are attracting pinners. Look for trends. Try to find what it is about your popular Pins that brings people to your boards. You can also access the same data on repins and clicks.\n\nYou can even break it down by the devices that people are using. Are your people Android users? You can easily find out, and it is useful for working out which sort of Pins are getting the most engagement.\n\n#### Your Audience\n\nOnce you've seen information about your Pins, you can find out more about those who viewed them, such as how engaged they are, which country they live in, their gender, their language, their metro area, and even what subjects they are interested in. The online trends and behaviors of Ministry of Peculiar Occurrences pinners, for example, have indicated that they are fond of recipes, fashion, art, and DIY home decor.\n\n#### Activity from Your Site\n\nHow many viewers go to your website, and which Pins and boards are taking them there? Study those successful Pins and boards, and see what you can do to make your other Pins and boards equally as successful.\n\n### Bookmark\n\nAnalytics will have you register a site and verify it. Once that's done, it will track how many views you are getting from blog posts pinned to your boards. Then you will be able to see which are your top blog post Pins. Those Pins should give you some idea of what your board followers find most interesting. Take note of those posts' images and keywords, and use them in future posts to get the most views.\n\n#### Other Business Tools\n\nPinterest Business accounts offer advantages outside of analytics as well.\n\n  * **Pin It Button.** This is useful if your website host doesn't have the button integrated into posts. Tumblr, Blogger, WordPress, and Wix all should, however. This button provides a quick and simple way for readers to Pin your post.\n  * **Promoted Pins.** Pinterest is working on a way to pay to promote your Pins. Initially, it will offer this service to U.S.-based businesses only.\n  * **Widgets.** Pinterest gives you a good selection of widgets. Follow buttons, Pin It buttons, and widgets to show off your profile or your boards all come prepackaged (or can be custom-sized) and ready for your blog or website.\n  * **Pinterest Logo.** All different sizes and shapes are available for you to use on your marketing materials, both online and off.\n  * **Business Pinterest newsletter.** This comes out weekly. Sign up for it.\n  * **Rich Pins.** This new innovation is still being developed. Currently there are five different types of Rich Pins: Article, Place, Recipe, Movie, and Product. All Rich Pins contain extra information about the Pin, but product Pins probably offer the most to a writer, since they can include pricing, where to buy, and availability. Pinners may also get notifications when the price drops more than 10 percent. Rich Pins require a little more technical skill since they must be meta tagged and tested, so authors may want help from a site developer. They also need to be approved by Pinterest. Still, it will be interesting to see how these develop. To keep an eye on them for use in the future, go to Pinterest's developer website.6<https:\/\/developers.pinterest.com\/rich_pins_overview\/>\n\n## Watch Where You Point That Pin: Pinterest Strategies\n\nPinterest strategies are all about making the most of the visual medium and the community. As a writer, you might think you don't have many visual resources to mine, but with a little creative thinking and the Internet at your disposal, you'll soon find it is easier than you think to fill up a board with some Pin-worthy pieces.\n\nReaders love to crawl inside the mind of their favorite\u2014or newly discovered\u2014writer. So let them in!\n\n### Now Boarding: Book, Series, and Character Boards\n\nCreate a board for each of your books or series, and display your inspiration. Think of it as the bonus features to the book. Pin photos of locations that inspired the stories, costumes that you imagine the characters wearing, historical images (if your book is history based), reviews, podcasts, fan art, and book covers.\n\nIf you want to start pinning before your book comes out, make the board a secret one. When you are ready to announce the publication of your book, you will simply edit the board to be public and, _voila_ , you have instant content for your readers! Check out the Appendix on Pinterest if you need help creating a secret board.\n\nIf you want to go further, you can create boards for individual characters. Fill these boards with insights into who the characters are, including books they might want to read, fashions they'd wear, or even decor they would have in their apartment.\n\nReaders also love a sneak peek behind the curtain: Traditional publishers might not let you reveal unfinished or draft cover images, but if you are self-publishing, you get to make these decisions. Consider pinning cover image ideas, drafts, or a list of movie stars you'd hire to play the characters in a movie of your novel. You can even ask readers to comment on what they think.\n\nAs with any social media, you should make your boards attractive and reflective of yourself. Have one for characters, one for settings, and one for series, but also make boards dedicated to your other interests: fashion, cooking, travel, or something connected with your writing, like costuming or history. Different boards are a great way to show how diverse a person you are\u2014don't be afraid to let your personality shine through.\n\nIn conjunction with making it look good, always ensure you only Pin quality material. Readers will know that you care about your content and will return for more. Make sure you use your Pinterest account for more than mere marketing. You'll also need content that you find on other boards. Content curating (dealt with in chapter 12 of this book) is the latest strong trend, and it can help make your boards a go-to location for your readers.\n\n### From Literal to Figurative: Mood Boards\n\nOther popular boards\u2014like mood or color boards\u2014can also be used to market your book. It is surprising how many pinners are looking for boards that have themes on feelings, rather than subjects. It is something a little out of the ordinary, appealing to different senses. For example, you might make a mood board for your horror novel and call it \"Creepy.\" Then you'd fill it with things that frighten you. If your book is a romance novel, you might make a board called \"Joy\" and populate it with fun and beautiful images. Color boards are popular with designers, but they will only work for you if you create something that fits in with your book. If your words were colors, what would they be? These boards will attract DIY and fashion enthusiasts, and those people read, too.\n\nIn the case of mood or color boards, don't forget a good, strong description, with that all-important website URL in it. People looking for inspiration might be curious enough to follow it.\n\n### Secret Pins: Keep It Secret ... Keep It Safe...\n\nWhen you Pin, make sure it isn't a dump of content all at once. Don't simply log on once a week and Pin fifty posts in one go. This will be overwhelming for your followers. Some of your more important Pins may drift past them in all the clutter. Instead, spread your pinning activities over the course of the week.\n\nUnfortunately, unlike other social media networks such as Facebook and Tumblr, there is no queuing or scheduling a set time to post pins. This leaves the writer with a bit of a challenge, but there is a work-around to make things a little easier. To avoid the dreaded Pin dump, create a secret Pin board. It's just like making any other board.\n\nWhen you drop into Pinterest, you can gather all the Pins you want to use, or even create your own, and Pin them to this secret board. Then, later on in the week, simply go to your secret board, click on the Pin, and then click on the pencil icon. Now you can move the Pin from your secret board to your public ones. The Pin will now appear only on the public boards. Spend a couple of minutes every day\u2014we like to tick this task off using our mobile app while on the train\u2014shifting a few over to public.\n\n### Tell Me About It: The Importance of Descriptions\n\nNo matter how pretty your image is, if your audience admires it and keeps going, you haven't sold a book. Make sure you offer a strong description to go with the relevant, original content you post. You can use snippets from your book, character sketches, or behind-the-scenes glimpses. Long descriptions are more engaging, and Pinterest allows you five hundred characters. So make the most of them, and don't forget to include a link in that description!\n\n### Bookmark\n\nImages and infographics that make the best impression are vertical, tall, and boldly colored. When users are scanning down, the longer your image stays in their stream, the more likely they are to click on it.\n\n### Directing Traffic: Blogging for Pinning\n\nPinterest is a great place to do cover reveals, contests, and promotions. So don't forget to Pin your blog post about these\u2014just make sure there is an image for Pinterest to find. When you Pin directly from the Pinterest website, or with the Bookmarklet, Pinterest will detect an image available and put it on your board.\n\nIf you are really stuck for ideas on what to post, a great tool is the InstaQuote app. You can take a pull quote from your post and quickly turn it into an image. It may seem a little strange to quote yourself at first, but it is an eye-catching option, especially on Pinterest.\n\nSo, from now on, make sure each blog post you write contains an image. Practice good Pinterest etiquette, even when not on Pinterest.\n\n### Group Boards\n\nHere's a good one if you want to share information with a publicist, co-writer, or perhaps a personal assistant. Boards can be shared! When creating or editing, simply go to your board, click \"Edit,\" and add the e-mail of that person. Once she has accepted your invitation, she will be able to Pin to your board. A good example of this kind of board used for marketing is the Alliance of Indie Authors: Members Books. Here, all members have a chance to post their book links.\n\nA similar shared page would work well for local writers groups or authors who share a genre. It's a fabulous way to build a sense of community.\n\nPinterest combines visual delight with information in a way that few other social networks do. It can be a great benefit for you as a writer and a marketer. So get pinning, but be warned: Once you get started, it can be quite addictive.\n\nIn the next chapter, we explore another visual platform\u2014one that captures moments as they happen. Instagram is a smartphone photography app that allows you to post on its own network alongside other heavy hitters like Facebook and Twitter. Instagram generates a new image for its own network and you can easily post this altered photograph to Pinterest, supplying personal memories to your board.\n\nBut how does Instagram alter your images? We'll see you in the next chapter and let you in on this visual platform's power and appeal.\n\n1http:\/\/about.pinterest.com\/goodies\/#browsers\n\n2<https:\/\/business.pinterest.com\/en\/widget-builder>\n\n3<https:\/\/about.pinterest.com\/en\/goodies>\n\n4<https:\/\/business.pinterest.com\/en>\n\n5<https:\/\/business.pinterest.com\/en\/confirm-your-website>\n\n6<https:\/\/developers.pinterest.com\/rich_pins_overview\/>\n\n# Chapter 9\n\n# Instagram\n\n# Adding Photography to Your Arsenal\n\nFacebook, Twitter, and Google+ all offer the ability to post photos along with your updates, and this is an option that authors take advantage of. In the past two years or so, you might have noticed that your social networks\u2014particularly in their mobile versions\u2014have been offering a variety of filters for your photographs. One filter makes your photo appear as if it were taken in the nineteenth century while another gives it the look of a Polaroid One-Step.\n\nThese creative options for photographs came about because of an unassuming app that took images from a smartphone's camera and allowed the user to give them a nostalgic look. Launched in October 2010 and now sporting a membership of 150 million members, _Instagram_ took online photography in a _backwards_ direction, granting its users the ability to take pristine, megapixel photos and turn them into 1970s-looking, antique, or cinematic photographs. You can add slick borders, quirky angles, and a variety of color effects to your photos, tag the location (using a process commonly known as _geotagging_ ) and even tag other Instagram members into your images.\n\n### Bookmark\n\nYou may have heard of _Vine_. This is a _video_ -sharing service that focuses on the short form\u2014the very short form. The length of Vines clock in at six seconds, less than half the length of Instagram's videos. It is a very popular app and has been used for everything from music events to newsworthy happenings to stop-motion animated advertisements. It makes sense that in 2012 Twitter bought the company, since Vines are very popular to tweet out. Twitter and Vines fit together perfectly. Instagram, in response to the popularity of Vine, now offers similar video features. You can make it a single, continuous shot or put together a series of segments into one clip. The final fifteen seconds of an Instagram video has the added advantage of offering filters, just as if it were a still photo.\n\nWhat makes Instagram popular, though, does not end with the really neat special effects available to your smartphone's camera. Instagram grants you the ability to take advantage of networks outside of its own, taking one post and distributing it automatically across other networks.\n\n## Platform Management: Connecting Instagram with Other Social Networks\n\nBack in our blogging chapter, we told you that certain plug-ins in WordPress and Tumblr allow your new blog post to instantly appear on Facebook, Twitter, and other social networking platforms. Instagram also offers this service. It's a great example of how mobile apps make the most of popular online platforms by maximizing the reach of various networks using one image, one post.\n\nThere are six social networks you can connect to your Instagram. The first three networks are covered in-depth this book, so you should know what they are and how they fit into your social media strategy.\n\n  * **Facebook**\n  * **Twitter**\n  * **Tumblr**\n  * **Flickr:** Flickr, available as a smartphone app, a stand-alone desktop app, an option for photograph management tools, and as a website, offers digital scrapbooks for family and friends, for networks you build, or for everyone on the Internet. Instagram plugs in directly to the photography network and creates a photo stream that posts your images and accompanying comment on your Flickr account.\n  * **Foursquare:** With a membership of over forty-five million, the popular geotagging network Foursquare allows its members to share locations. Wherever you are, Foursquare lets your network\u2014or anyone on the Internet\u2014know where you are. While you can still use the Foursquare-powered Location option on Instagram without being an active member of the network, connecting your Foursquare to Instagram allows your posts to count as official \"check-ins\" for that location.\n  * **Mixi:** Based in Japan, this network offers many of the same services as Facebook, only with expanded options to keep your content private.\n\nIf you have not connected your networks to Instagram yet, the app will walk you through a log-in process. You'll have to make this connection only once unless you are switching from account to account. After you sync your social media networks to Instagram, you can share your posts across multiple networks with a single tap.\n\n### Bookmark\n\nNote above where we say, \"You'll have to make this connection only once unless you are switching from account to account.\" Tee, on working with other clients and their Instagram accounts, has discovered a limitation: There is no easy way to float from one account to another. You must log off and then log back on to Instagram. Also, when you have logged back on to Instagram under a different account, it is a good idea to check the additional platforms to make sure they are accessing the right account. Otherwise, you may be cross-posting personal photos onto accounts that are not your own. Always make sure your message is heading out on the intended channel.\n\nFor a smartphone app, Instagram offers a lot of choices: different filters, various networks, and still or moving images, all allowing you to tell a story of that specific moment. Writers thrive on Instagram. They post poetry, teasers of upcoming releases, and moments from book events. Instagram is a powerful app just by itself, but other developers are now offering their own apps to help writers tell their stories through this photographic platform.\n\n## Beyond Instagram: Third-Party Apps\n\nThroughout this book, we suggest avoiding third-party applications that simply copy-and-paste your message into other platforms. Automation is a good thing in moderation, but it should not be the only way you appear on other platforms. You want your network to know there is a real person at the end of these updates, tweets, and posts, someone who is accessible and open for engagement.\n\nInstagram is an exception to this rule. There are many third-party apps available geared specifically for Instagram or for the photographs and video on your smartphone to make them like and comment magnets for other Instagrammers.\n\n### PhotoGrid\n\nIt's one thing to post a truly breathtaking picture on Instagram, but what if you are in the middle of a book event and can't take a break to post it? Or what if you reach the end of a weekend convention or writers retreat during which you snapped photographs all weekend, and suddenly you remember at the end of the event that you have Instagram on your phone? Here is where you can impress your network with a snappy collage of images. _PhotoGrid_ is the app that can help you create and prep your collection of images for Instagram.\n\nLaunching PhotoGrid, you are given a variety of options: a _Grid_ that is a collection of frames ranging from symmetrical to creative, a _PinBoard_ that creates a digital bulletin board for you, and _Video Slides_ that turn your selected photos into a miniature slide show. (Additional modes, like the _Magazine Cover_ option found across the bottom of PhotoGrid's introduction screen, works only with one photograph.) Once you've designed your collage, PhotoGrid lets you export to Instagram (as well as Facebook, Twitter, Google+, Flickr, and Tumblr). You can compose your message (including hashtags) in PhotoGrid and export your image directly into Instagram for final touches.\n\nCollages are eye-catching and fun to put together. They also tend to bring in more likes and comments.\n\n### Vidstitch\n\nPhotoGrid takes your Instagram photos to the next level by creating sharp, slick collages. _Vidstitch_ is a step above PhotoGrid because it allows you to incorporate a small video clip into one frame of your collage. With an in-app purchase (Vidstitch is available to buy through Instagram), you can expand your gallery of frames to include widescreen images, allowing for more still shots to accompany your video, or you can upgrade Vidstitch to the Pro version, allowing you to create a collage of videos. The interface and workflows for Vidstitch are a touch more complicated than PhotoGrid; but with a bit of practice, you can create some eye-catching, engaging content for Instagram.\n\n### Repost\n\nWith the ability to share postings on Facebook, to retweet on Twitter, and to reblog on Tumblr, you would think Instagram would offer this function in its own app, but presently it does not. This is a shame, as some Instagram photos you may come across are nothing less than stunning and others capture a mood that you want to share with your network.\n\nThe app Repost allows you to easily share images from other Instagram accounts, working as a retweet function. In the spirit of proper accreditation, Repost watermarks the image with its original Instagram account and populates your Post window with the original accompanying post. You can edit the post, or clear it completely, but the image will always name the originating Instagram account. Repost is a great option for sharing the work of others and providing engaging content of your own.\n\nThe app also allows you the ability to like images in your Instagram feed, and even create an index of Favorites in case you want to repost or reference another image later. Repost offers account management beyond Instagram's basic functions, making it a must-have in your third-party applications for Instagram.\n\n### InstaQuote\n\nThis app, of all the third-party apps featured in this chapter, has proven itself to be a fantastic tool for promotion. How? InstaQuote, on a basic level, allows you to make square images with simple, artistic backgrounds, the focus of the art being text of various styles. The GUI gives you two areas to type text, the fields divided by a single line. The larger area, which asks, \"What was said?\" is where your quote goes, while the smaller area, which asks, \"Who said it?\" __ is where you type the source of the quote. Then, by tapping key words in your quote, you can add emphasis by changing the base color of the words.\n\nUpgrading to the pro level not only removes the InstaQuote watermark, but it also imports a wide variety of backgrounds\u2014textures, abstracts, airbrushes\u2014and allows you to import your own photographs for the purpose of creating your own backgrounds. InstaQuote makes pull quotes from blog posts, teasers from your books or works-in-progress, and inspirational quotes that keep you motivated extremely easy to create and post on Instagram, as well as on Facebook and Twitter.\n\n### Iconosquare\n\nWhile Iconosquare is not available as a mobile application, it's still worth looking at from an analytics perspective. This app offers a deep look into your feed, who you are following, and how other Instagrammers are reacting to your content. Under the _Statistics_ menu and the submenu _Rolling Month Details_ , analytical data is collected on Followers you have picked up, followers who have dropped you, how your most recent post compares to previous ones, and which images are seeing the most engagement. Iconosquare also allows users outside of the Instagram app to fully manage their account. You can like photos from your feed, follow and unfollow other Instagrammers, reply to comments, and even perform a _repost_ function similar to the Repost app. All of these are things the Instagram app does not allow you to do. Iconosquare can also post your images on a variety of social media platforms.\n\n### Bookmark\n\nIconosquare offers an analytics section called _Tag Impact_ under the _Optimization_ menu of your account. This is an essential section to watch: Tag Impact tracks all the hashtags you use with your media over a ninety-day period. On the left-hand side you'll find your tags\u2014your most frequent hashtags are larger and in bold. On the right-hand side are the top hashtags used across all of Instagram. If you want traffic and network growth, you will want to find tags that fit your media.\n\nWhile Iconosquare may look like a one-stop solution, the downside of this website is that it is not portable. At the time of this chapter's publishing, there is no smartphone app for Iconosquare, making it less versatile than the other third-party apps covered here. Still, Iconosquare is an essential tool for success with Instagram.\n\nWith so many options and so many capabilities, it is easy to see not only why Instagram is as popular as it is, but also how powerful a platform it can be in conjunction with third-party applications. Instagram has a lot of potential, so take your strategy one step at a time and consider which of its many options will work best for you.\n\nI do love Instagram. The artistic aspect of sharing our general lifestyle through pictures is so easy through the app. So many people are visual. It's a unique outlet for an author.\n\n\u2014Dr. Stacia Kelly, holistic health coach and author of _Reduce You_ and _Slim, Fit, Happy You_\n\n## Best Practices on Instagram\n\nWhen you hear about networking platforms or building a presence on social media, hear authors generally talk about Facebook, Twitter, and blogging straight away. Sometimes podcasting and Google+ are mentioned.\n\nInstagram?\n\nOn our own networks, while writing this chapter, we asked our community of writers what questions they had about Instagram. Writer, podcaster, and photographer J.R. Blackwell said point-blank \"Should I even bother?\" __ She went on to say she didn't think Instagram was a bad platform (and as she is a professional photographer, we would have been surprised if she thought it was), but the question was how would it be a good platform for authors?\n\nWhat a great question.\n\nInstagram is a social media outlet writers in general do not seriously consider a viable promotion platform. They may try it out for a brief spell, have a bit of fun, and then forget they have the app on their smartphone. With the right approach and application, though, Instagram can provide a treasure trove of visual content for a writer. Now, before you think that \"visual content\" for a writer consists of nothing more than a picture of a laptop with a work-in-progress on the screen (which we've found that followers on Instagram and elsewhere do respond to!), think again. It's possible to cast a more imaginative net by including character inspirations, behind-the-scenes research or travel, or even a quick link back to your blog. You must go beyond a writer's desk or book signings. The key is how you approach your content.\n\n### Author Appearances\n\nBook signings can only be so interesting. A snapshot of you with pen at the ready and books arranged neatly is fine, but then what? Depending on the appearance and the location, quite a bit. If you are appearing with other authors on a panel discussion, you can take a photo of the discussion. After the panel, why not snap a candid shot with the other authors or editors attending? (Ask if they are on Instagram so you can tag them accordingly, but make sure you spell their names right!) You can also post Instagrams that document your travel to various conventions and conferences where you will be presenting. Geotag where each of these moments is taking place and invite readers within driving distance to join you there. You can also repost images or video from other authors' feeds to boost the signal, provided you know of other Instagram-using writers who are attending the event with you.\n\n### Upcoming Releases, Special Events, and Cover Reveals\n\nDo you have a date for your next novel or approval from your publisher to reveal your next book cover? Instagram allows you to easily share artwork pertaining to your upcoming launch. Whether it is an original graphic cooked up in the image editor or a cropped section of your cover, Instagram\u2014with Iconosquare added into the mix\u2014gives you the ability to send out a single post across seven social networks. This post can point back to one central location, whether it's your blog, a link to pre-order your book, or another place on the Internet. You can also create original artwork publicizing special events\u2014a crowdfunding event, a charity anthology, or an upcoming appearance\u2014where you will be participating. The image you create for Instagram can also serve as your own branded artwork for the event in question.\n\n### Bookmark\n\n_Crowdfunding_ is a business model many authors are taking advantage of. Authors ask audiences to invest in a project, namely a book, a series, or other form of media\u2014games, Web series, graphic novels. The investments go toward a final goal that will bring the project to fruition. Of all the various crowdfunding services, _Kickstarter_ is the best known. Authors have made the most of Kickstarter's ability to raise money for their works\u2014usually dedicated funds for editing and cover design. In return for the backers' money, they offer \"rewards\" like print and audio copies of the books, exclusive short stories, or other items that might incentivize people to lay down their money for their project.\n\n### Insta-Competitions\n\nCompetitions are a proven way to increase your number of followers on Instagram, but don't go this route until you have at least a small following. It's hard to make a splash if only a few people are following you.\n\nRight off the bat, Instagram is nice enough to spell out rules for promotional guidelines. You are responsible for the lawful operation of your contest\u2014this makes sure Instagram isn't in trouble if you fail to obey local laws. You cannot ask your readers to inaccurately tag content or people in their entries. All promotions must have a note from Instagram that releases the company of responsibility, and all entrants must point out that Instagram is not endorsing, sponsoring, or administering the contest. Instagram will not help out with your promotion nor give you any advice pertaining to it. Finally, you must agree that you are running the contest at your own risk. Keep those guidelines in mind while you design your giveaway. User-generated content competitions are popular on Instagram and a great way to encourage creativity among your followers. They also have the advantage of not running afoul of any local laws governing sweepstakes.\n\nIn order to fit in all the rules, prizes, and directions for entrants, you should create a post or page on your blog or website. You then put this URL in your Instagram post. The hashtag that you are using to keep track of entrants will be included as well.\n\nSo what do you ask people to do? Keep it simple, and make sure it involves nothing dangerous or too outrageous. A picture of a participant with your book (\"book selfies\"), dressed up like a character, or posing with something significant to the book (an artifact or some related item) are all good choices. Or you could go with something related to your genre that is more open to interpretation.\n\n### Bookmark\n\nWhether on Instagram or any social network, when hosting competitions, make sure the prize is enticing enough. It doesn't have to be expensive. Advanced Reading Copies make great prizes, or you can offer a selection of items related to your book. Writer Starla Huchton has written books about superheroes and often gives away well-thought-out prize packs of items with superhero themes.\n\nBefore launching the competition, make your own competition graphic to post on Instagram, and make sure it is attention grabbing and contains the hashtag you have come up with.\n\nDon't forget to spread the word about your competition via your other social networks.\n\n### Teasers and Motivationals\n\nSelections from your upcoming release are always great teases to get readers excited, but in the social media arena, authors need to grab a reader's attention. In the same way graphic designers use pull quotes from a magazine article to grab a reader's attention, authors are now using teasers, juicy character quotes, and small chunks of narrative to tantalize readers with what is to come in their next work. Teasers work here the same way Hollywood entices rabid fans with iconic imagery in a movie poster\u2014you give your readers just a hint of what is coming.\n\n_Motivationals_ are similar to teasers, but they come from your blog. While you might ask, \"Isn't it a little arrogant to offer up my writing advice as sharable quotes alongside Stephen King, J.K. Rowling, and Mark Twain?\" Consider the motivational you create less of a \"Bask in my brilliance!\" graphic and more of a \"This is what I want you to take away\" graphic\u2014a visual representation of a blog post bestowing writerly advice, if you will. On Instagram, motivationals and teasers are terrific promotions for your novels and short stories.\n\nIn accompanying posts, links are not active on Instagram, but when posted on Facebook and Tumblr with the _http:\/\/_ lead-in, URLs can take readers from your cross-posting to wherever you want them to go.\n\n### The Writer's Life\n\nPhotos of what you're reading, what's on your computer screen, print resources on the corner of your desk, or where you are drawing inspiration from make for interesting visual content. It's a peek behind the curtain, a delightful look at what inspires you and what words are filling the page. Depending on how much you want to share, you can also post the sunrise from a morning walk or your view from a writer's retreat. There's a lot that goes into writing a book\u2014share it on Instagram.\n\n### Writers off the Clock\n\nAre you heading out to the movies? Are you hosting a cookout with other creative types? Maybe you're in the conference's hotel bar or restaurant with other authors. Little moments in life can be fun to share with your community. If those in the picture are comfortable sharing these candid memories with your network, do it. Posting your adventures away from your keyboard is a great way to connect.\n\n### Bookmark\n\nA lot of writers use Instagram, and you know what? We need an identifying hashtag. So if you are a \"storyteller on Instagram\" posting a photo about your creative process, a successful appearance, or something that inspires you, be sure to use #stoig. This will show that you are a writer taking advantage of this visual social media platform.\n\nAs you can see, there are lots of great ways to crack open the door for your readers or potential readers, and to let them get a visual of what it is like to be a writer.\n\nNext we're going to look at some of the smaller, less well-known platforms that might suit you and your writing. Let's take the path less traveled and see what we can find in the undergrowth.\n\n# Chapter 10\n\n# Additional Options\n\n# Platforms that Break the Convention\n\nThe social media space is always evolving, and new platforms are constantly springing up to demand the attention of readers and writers alike.\n\nSome platforms generate wild interest for a few days or weeks before being cast aside; some new ones offer features or conveniences that the old ones do not and are taken into the fold.\n\nAuthors should always keep their eyes on the social media horizon to see what is newly available, but before they sign up at any site, they must evaluate it. Does the site offer new audiences? Is it reaching more people than an already established sites? Is there some new aspect of marketing that can be achieved with it that others do not offer? In particular, are readers gathering in this new place?\n\nThe platforms we cover here are not as mainstream as Facebook, Twitter, or Instagram, but these are networks that are popular with authors, where writers are found frequently, and where they can sometimes find a new and unique angle to promote their work or just be themselves.\n\n## Goodreads\n\nAround since 2007, Goodreads.com grew into one of the largest gathering spots for readers, which makes the site hard to ignore as a writer. In 2013, Amazon announced it was buying Goodreads, and when the giant of the book business is interested in a corner of the Internet, you should be, too.\n\nIn that very same year, the site had twenty million users. Now, that number may be small compared to other social media networks, but those twenty million are all passionate readers, who interact and are committed to book culture. Users log books they have read, or want to read, then review these books, gather in groups to talk about them, and make lists like \"best steampunk\" or \"books with amazing female characters.\" It's safe to say Goodreads is the world's largest book group.\n\nWith all that going on, you might want to jump in with both feet, but here are a few words of caution.\n\nGoodreads can be a great place for authors to connect with readers, but authors should tread carefully. Readers are very protective of their space and do not welcome solicitation in it. Any interaction you have with them must be well thought out.\n\nNone of this means that authors should not go into this arena, mind you. You just need to be careful. For example, Lists\u2014a collection of books showcased around a theme like \"Essential Fantasy\" or \"Classic Science Fiction\"\u2014that you create do not always belong to you once they go live. When receiving reviews of your titles or titles you are affiliated with, defending your work from readers is not the best course of action. Goodreads does offer ways to promote and interact with readers, however, such as having your blog posts appear in your own profile and offering various giveaway opportunities that they will assist you in planning. If you want to interact with your readers, stick to those outlets.\n\nWhen it comes to the reviews posted, as we mentioned earlier, it's important to remember one rule: You're not going to please everyone. If you see any of those dreaded one-star reviews or even reviewers who make personal comments about you, do not engage.\n\nWe repeat: _Do not_ engage.\n\nOf course there have been cases in which authors and readers have exchanged heated words. Some of them have been fairly managed, but most work their way through other social media networks, becoming cautionary tales shared between authors and would-be professional writers looking for ways to avoid making mistakes in their careers.\n\nStay on the marked path, and you will have another fine venue to market your book.\n\n### Getting Started\n\nCreating a Goodreads account comes first. Make sure you have a good profile, description, and links back to your website. Next, go find yourself on Amazon, Smashwords, BarnesandNoble.com, and elsewhere. If you are published, you are likely on Goodreads, or at least a public profile is linked to your books. Click your name. You are now on your public author profile, different from the profile you created, but not for long.\n\nAt the bottom of the page click on Is this you? This will result in a request to join the author program. It may take a day or two to be approved, but once you receive the confirmation e-mail, your two profiles will be merged.\n\nIn your profile, link back to your blog so newly published posts immediately appear on your Goodreads profile. Then fill in the rest of the details of your writing career.\n\nYou can open yourself up to questions, which is a fun way to interact with your readers. Don't forget to add events and your favorite authors.\n\n### Goodreads Giveaways\n\nLike the other platforms, giveaways are an excellent way to get your name out to readers. Some restrictions do exist, however. Goodreads has (as of now) decided not to allow e-book giveaways, but we understand they are working behind the scenes with publishers to change that. Presently, giveaways are for print books only.\n\nYou can run more than one giveaway. In fact, Goodreads suggests running one before your book becomes available and one after. To get started, check out the Giveaways page. Fill in all the details, then submit it for review. Goodreads runs a tight ship, so it has to clear every giveaway before allowing it out into the wild.\n\n### Selling Books on Goodreads\n\nUntil about a year ago, Goodreads let authors sell their books directly. However, since Amazon purchased it in 2013, that service went away. Amazon is continuing to integrate its services, and that means Amazon wants you to sell your book through its website. In your author profile settings, under _book links_ , you can set the order or hide links to booksellers. When readers visit each individual book page, they will be presented with a set of links so they can buy copies.\n\n## Reddit\n\n_Reddit_ is described as a social networking service where users can submit content which is then voted up or down (in terms of popularity, usefulness, insightfulness, etc.) by a network of users. If you are old enough to remember online bulletin boards, then you are going to be right at home on reddit. Unlike those boards in the 1990s, however, reddit is massive and divided into an impossible-to-count number of _subreddits_ \u2014topics any user can create. With so much content, it can feel a little daunting at first.\n\nAs an author venturing into reddit, you should stick to the books subreddit at first. This is our familiar area after all. This is a moderated subreddit, so look on the right-hand sidebar to take note of the rules for participating in this community. The books subreddit is one of the most populated places readers can go to discuss books. Forty thousand new subscribers come on board every week. Books even has its own book group, which must be the largest anywhere (outside of Goodreads) at twelve thousand members. Just imagine trying to get that many folks into a room at once.\n\nFrom this subreddit, you can drill down if you like, but since we are concentrating on the social aspect of these other platforms, let's check out one of the more popular and allowable ways for authors to use reddit.\n\n### Ask Me Anything (AMA)\n\nAMAs are quickly becoming the go-to way for authors to spread the word about their books and to connect with readers who may have already read them. AMAs are not restricted to authors\u2014even Barack Obama has done one\u2014but they can be fun and informative. You are basically creating a worldwide chance for readers, both potential and otherwise, to ask you questions.\n\n### Bookmark\n\nCarefully read over the AMA FAQs1<http:\/\/www.reddit.com\/r\/IAmA\/wiki\/index> to make sure you know the procedure for both requesting an AMA and creating one.\n\n## Storify\n\nWhen you first sit before its interface, you may think \"This is just another blog engine,\" but _Storify_ is unlike any other blog engine you may come across.\n\nFirst, Storify wants to help you tell a story, which is why it offers a search engine that scans blogs, Facebook, Twitter, and a host of other social networks in its interface. You can search by user, by keywords, or by a regular search string, and you will get a window of results for whatever social platform you want to tap into.\n\nStorify then offers you the ability to click and drag featured images with opening lines of a blog post, multiple Twitter posts, and updates from whatever platforms you are searching and feature them in the post you are putting together. The end result is very similar to posts you may see on Mashable, BBC News, CNN, io9, and other news services where instead of simply screen captures, you have captures of the updates that take you to the source material in one click.\n\nFinally, once you have created your Storify post, you can access the HTML code behind this advanced blog post design, copy it, and paste the raw code into your WordPress blog. Doing so brings instant, dynamic content to wherever you send your audience on a regular basis. Storify is a growing trend with authors, good for referencing things that are happening online or on other networks. It provides updates in context to your own thoughts, a once-difficult task accomplished with random URLs and screen captures now made interactive on your own blog.\n\nTwitter moves fast, which means impactful tweets can miss their audience. That's why I love Storify, especially for compiling writing advice. It allows you to capture tweets\u2014both your own and those of other Twitter users\u2014arrange them in the order of your choice, and embed the full package onto your blog, along with comments on the overall theme. Whether you're imparting knowledge, participating in a discussion, or capturing a hashtag, it's a great boon to assemble all those 140-character bursts into a complete story that will endure long after your tweets have fallen off the page.\n\n\u2014Delilah S. Dawson, author of the Blud series, _Servants of the Storm, HIT_ , and _Wake of Vultures_ (written as Lila Bowen)\n\nStorify is a new platform that you, as a writer, should spend some time getting to know. If you like it, you can work it into your platform, especially if posts are inspired by activity on other social networks.\n\nYou know, all this talk about social media, its capabilities, and what an author can do with it has certainly left us parched. We could use a drink...\n\n## A Social's Social: Untappd, Vivino, and Distiller\n\nAuthors can close deals, create new contacts, and expand leads for future writing gigs just about anywhere, but perhaps the best wheeling-and-dealing happens at the bar. Occasionally dubbed \"BarCon,\" a lot of networking happens in a conference hotel's bar. Social media comes into play when you are off the clock at a convention, expo, or conference. Its innovative ways connect you with other writers, readers, and the world. But these networks exist simply to provide writers a place to share a frosty beverage or a celebratory round once a contract is signed or a day of book signings are done.\n\n### Untappd\n\n_Untappd_ is a network built for those who love beer. Whether you enjoy a porter, a pilsner, or an IPA, Untappd connects you with ice-cold brewskis and the breweries behind them. The network is operated either on a website or a mobile application, and offers you access to Untappd's beer database comprised of mainstream breweries, up-and-coming microbreweries, and home brewers you may meet on your travels. At the core of the app are the reviews. When your first round arrives, you search for your beer in the Search bar across the top. Once you tap your beer from the list of offered brews, you select Check-In and are offered a field where you can:\n\n  * Type in a brief review (best at around 150 characters);\n  * Add a photo of what you are drinking (or who you are drinking with, as we once did with award-winning author Elizabeth Bear);\n  * Rate your brew;\n  * Add your location.\n\nBut what if beer isn't your thing? That's okay, as Untappd also includes ciders and meads in its database.\n\n### Vivino\n\nFor wine lovers, _Vivino_ is the app of choice. Much like its ale-driven counterpart, Vivino allows you to rate and review wines, but it's how you find a wine in the Vivino network that is particularly clever. Point your camera at the wine bottle's label and take a photo. Vivino will scan its database and find the wine you currently have in front of you, or allow you to enter all of the information about this wine (vineyard, category, vintage, country of origin), along with your own comments. This wine lover's network also scans wine lists, offering reviews and notes about the glasses and bottles featured, price lists for wines you may want to stock your own cellar with, a \"location search\" that shows you what wines are near you, and your own statistics\u2014what you've tried, what you've liked, and what you didn't.\n\nNow that we have our beer snobs and wine connoisseurs covered, we will turn to an app that would have sparked the interest of rogues and rascals like Ernest Hemingway, Raymond Chandler, and Hunter S. Thompson.\n\n### Distiller\n\n_Distiller_ covers scotch, bourbon, rye, and other kinds of whiskey. The app and website grant members access to both trending and recent whiskey rated by the network and the Tasting Table, which is Distiller's own inner circle of experts and enthusiasts who come together to offer recommendations for what you should be enjoying by the fire pit or after dinner. Wish lists can be built alongside your Collection and your Top Shelf, all of which are shared on your profile.\n\n### Building Your Brand\n\nThese are clever apps, sure, but this book is about social media for writers. How can authors take advantage of these uncommon networks? It's a worthwhile question, as these networks don't necessarily lend themselves to promotion of your books.\n\nYour _brand_ , however, is a different matter.\n\nIf you visit TeeMorris.com, you will see in the site's title \"Unbelievable Beer Snob,\" a self-proclamation that Tee is proud of. With many check-ins from Untappd, Tee has made his love and appreciation of microbrews a part of his brand.\n\nBut when it comes to posting on these networks, authors can add a casual flavor to their social media platforms.\n\n  * I **nstant content for other social networks.** Untappd, Vivino, and Distiller all offer integration with larger networks like Facebook, Twitter, and Google+. With the incorporation of photos, these apps offer your networks dynamic content that can easily spark discussions and other recommendations that you can explore at social events. These platforms can also offer you a new way to connect with potential interests that your readers may have\u2014remember, your readers want to get to know _you_. Untappd, Vivino, and Distiller provide new and varied content to your various platforms, and variety is essential to success.\n  * **Dining recommendations for your area or areas new to you.** Whether you want to offer an impromptu meeting place for fans or are looking for a place to try on the road, these barfly apps can automatically access your location to see what people are saying about featured labels\u2014be they local, international, or just new to you. These apps also work as a kind of _Urbanspoon_ or _OpenTable_ (both apps designed to recommend dining options in your immediate area) for various restaurants, brewpubs, or even vineyards, cideries, and distilleries.\n  * **Getting to know your fellow authors.** We originally met the earlier mentioned (and quoted) Elizabeth Bear at a steampunk event in 2013. But it was 2014 when Tee connected with her on Untappd. Since then, he has shared quite a few toasts and notes with the acclaimed science fiction and fantasy author. Appreciators and aficionados of beer, wine, and spirits love to exchange notes on what they are drinking and where they are drinking it. It's fun to do, and you get to know the tastes of your friends who are dedicated to their literary pursuits. With Distiller's ability to share wish lists across other networks, a bottle of a desired or favorite label makes for a wonderful gift.\n\n### Bookmark\n\nWe already mentioned Ernest Hemingway, Raymond Chandler, and Hunter S. Thompson in the pool of writers who enjoyed a good drink, but it's worthy to note Jean Stafford, Dorothy Parker, and Edna St. Vincent Millay as well. Some of them may have loved their drink a little too much.\n\nYou may hear authors joke about BarCon, about partying like rock stars, but not everyone enjoys a good scotch, wine, or beer. If you mention these apps in conversation and authors prefer not to share on any of them, or simply don't drink, you can always get to know them via other channels.\n\nAnd when offering to buy a round, don't make a big deal if someone chooses an iced tea _not_ from Long Island. Respect your fellow authors' choices, and enjoy the social time regardless.\n\n  *  **Sharing a celebration or a quiet moment with fans.** Readers love to feel a connection with their favorite author. The limits of how much you want to share, as we have said before, are up to you, but fans do appreciate when authors share images of toasts and celebrations. You can be sitting by a fire pit with a single malt in hand or toasting with other authors to a recent award, accomplishment, or appearance\u2014sharing the round through these apps can create a \"bonding\" moment with your readers.\n\nA dear friend of ours, social media professional Megan Enole, has been known to say, \"Never forget the _social_ in social media.\" If these are words to live by (and they are), then the thing that makes these three unique networks so much fun is also the thing that's so good for writers and their platforms\u2014Untappd, Vivino, and Distiller are all about being social. When you want to celebrate, when you want to unwind, or when you want to suggest a favorite drink, why not invite the world to enjoy a round with you? Untappd, Vivino, and Distiller, when used _responsibly_ , offer a lot in the way of developing your brand.\n\n## Tsu.co\n\nThe first thing you need to know about _Tsu_ is that this platform is pronounced \"Sue.\" The second thing you need to know is that it still has the new car smell on it. Tsu only launched in October 2014, but it has already generated a bit of buzz. Comparatively speaking, it is Facebook before it got big and Twitter with more words to work with.\n\nSomething that Tsu is doing differently than Facebook is paying its contributors for content. Tsu makes ad revenue, and after keeping 10 percent for operational costs, it pays the rest back to users. It divides users into two pools: one for content creators and one for those who click on the links. Don't get too excited at the moment, since Tsu is still pretty small, but it is something to consider going forward.\n\nTsu also has developed a vigorous antispam policy in response to its initially being a beacon for spamming in its early days.\n\nThere are two kinds of contacts you can have: Friends or Follows. Friends share information with each other, while Follows have only a one-way connection.\n\nUnlike Facebook, which has been throttling down on organic reach, Tsu lets it loose, hence the \"before Facebook got big\" descriptor. But like Facebook, it does have analytics that will help authors determine which posts are working for them. Advertising on Tsu is still in development stages\u2014if you want to advertise, you have to contact Tsu directly\u2014but that is bound to change in the near future.\n\nSo while it is brand-new, authors are already moving into the space. Romance authors, including Piper J. Drake, seem to be the groundbreakers on this fresh social media platform. Along with its main website, Tsu has an app for Android and Apple devices, so you can try it out on the go if you like.\n\nOn the Internet not everything succeeds, and Tsu may disappear or it may flourish. It totally depends on how many users it can attract. Tsu is still in its toddler stage, but it sports a clean interface, a shallow learning curve, and some interesting concepts in generating revenue and maintaining a level of quality in shared content. As a writer, you should keep this site in mind as a potential platform. If you have some time to dabble, you might want to consider what it can offer you.\n\nAs you can see, there is not a lack of social media for an author to take advantage of. Once again, it is all about finding out where _your_ audience is and making yourself available there. Next we're going to explore how to increase your chances of being found online, the thing you really want to achieve with your social media platform. Yes, it's time for us to have that all-important talk about search engine optimization, or SEO.\n\n1<http:\/\/www.reddit.com\/r\/IAmA\/wiki\/index>\n\n# Chapter 11\n\n# SEO\n\n# Dark Arts of Search Engine Optimization\n\nYou cannot work your way through online marketing and public relations without having to contend with three letters that strike terror into the hearts of business owners, Internet novices, and transmedia storytelling professionals everywhere: _search engine optimization_ (more commonly known as SEO). If you ask digital-savvy authors, \"What is SEO?\" you will get a wide variety of answers:\n\n  * It's a shell game soaked in snake oil and served on a deed ownership to the Brooklyn Bridge.\n  * It's essential to success on the Internet; namely, being _found_ on the Internet.\n  * It's magic that's so black it makes Voldemort turn to Darth Vader with arms wide open as he implores, \"Hold me closer, Tiny Dancer.\" __\n  * It's a detail that needs to be seen to in (almost) everything you do in social media.\n\nSEO constitutes all of these things and so much more, and in this chapter you are about to get a crash course in how it works, why it is important, and, more vitally, why you should remain wary of any \"foolproof strategy\" that guarantees you'll dominate the Top Ten of any search. Before applying the basics of SEO to your social media plan, though, it helps to know why SEO is not the \"be all, end all\" of social media strategies.\n\n## Why SEO Doesn't Work (as the \"Gurus\" Claim It Does)\n\nIf you sense that we are unconvinced of the effectiveness of search engine optimization, it is because we are. We have watched authors invest time and money into self-proclaimed SEO \"experts\" (who we call \"gurus\") that convince them that no one will _ever_ find them on the vast wasteland of the Internet\u2014unless they hire gurus like themselves to SEO their website.\n\nSEO does matter and it can make a difference\u2014on a somewhat basic level\u2014but there are several reasons behind our skepticism. Our attitude towards SEO comes from information from experts online and elsewhere. We think these experts have disproven this strategy.\n\n### The Math Behind SEO Just Doesn't Add Up\n\nPicture a room full of thirty authors, all of whom write science fiction set in a dystopian world where dinosaurs have re-emerged and gasoline is the precious resource that all factions\u2014peaceful and warlike\u2014need in order to survive. These thirty authors have gathered to hear an SEO guru's advice on how to meet their search engine optimization needs. This guru says, \"I can guarantee you a place in the top ten list of search results for 'dystopian dinosaur science fiction with gladiator cars' provided you follow my strategy.\" __ Sounds great, but remember the head count? There are thirty people in the room, and there are ten slots in that coveted top ten. What if these authors were just science fiction authors? Or what if these authors have books in all genres? Would this expert's strategy still work?\n\n### Trying to Read a Reader's Mind Is Impossible\n\nContinuing with this room full of thirty authors and the idea of all of them writing in the same niche (science fiction), the SEO guru claims, \"You need to make sure 'dystopian dinosaur science fiction with gladiator cars' is in your meta keywords, descriptions, and content, phrased exactly like that.\" Why, you ask? \"Because that is how Google finds you and how Google will list you in search engine results.\" So, in theory, if people surfing the Web use \"dystopian dinosaur science fiction with gladiator cars,\" they will see these authors' names in the search results. What happens, though, if the search string is \"apocalypse dinosaur sci-fi with modified automobiles\" instead? How will the results differ? An aspect of SEO hinges on what keywords people use for a search, and if you select the wrong keywords in a search, there is no guarantee to the end results\u2014and this is for our niche authors. The chances of these people being found on generic searches for \"science fiction\" and \"author\" become less and less, even when \"science fiction\" and \"author\" are used in meta tags.\n\n### Trying to Read Google's Mind Also Helps Your Results\n\nAs if reading the minds of Web users who are (supposedly) searching for you at random isn't stressful enough, SEO relies heavily on Google's algorithms, which Google reserves the right to change at any time. _Algorithms_ are rules Google sets for search strings to follow and properly display results. Violate any of the algorithms and either your website appears low in the rankings or Google red flags as malicious, pushing it even lower in the search rankings.\n\nWhy would Google do this?\n\n### SEO Can Be Very Easy to Game as a System\n\nWhile some experts try to remain ethical, many unethical gurus take great delight in beating the system. Whether that means using an abnormally large amount of keywords in meta tags, using keywords in page content that changes the entire flow and voice of the page, or placing keywords in URLs that only yield lengthy and hard-to-remember website pages, gurus go crazy with SEO (and charge you for it). If Google changes algorithms again, the gurus come back with a whole new set of SEO rule benders and overhaul your website with them. And charge you for it. Again.\n\n### Organic Search Results Always Yield to Paid Search Results\n\nIf you were to go to a search engine\u2014be it Google, Bing, or Yahoo!\u2014and type in the search string \"What really happened to the dinosaurs,\" __ you would get search results promoting a Christian children's book series (of the same title) written by creationists Ken Ham and Dr. John D. Morris. Even removing the word \"really\" from the search string brings similar results\u2014an \"interpretation\" of paleontology topping the search results. Many of the top results are paying for those prime slots, and the price tags for these slots may be well out of your budget.\n\n### SEO Does Not Apply to All Social Media Platforms\n\nGoogle+ and YouTube do take advantage of SEO, but Instagram? Not so much. Twitter? Not really. Facebook? Inconsistently. Podcasts? Not at all. Yes, SEO is essential for your blog when it comes to the meta tags that only search engines and browsers see, and also when it comes to the content featured on your blog, but SEO only goes so far in social media. SEO is important, but nothing to lose sleep over.\n\n### Bookmark\n\nSo where does SEO matter in social media?\n\n  * Blogging\n  * Google+\n  * YouTube\n\nNotice anything about those three platforms? Two of them are Google owned. If you're blogging with Blogger, then you have a triple play because Google owns that blog engine, as well. While you may not need to concern yourself with SEO on Facebook and Instagram, you will want to think about it for these three platforms, particularly YouTube. Keywords and descriptions play heavily in search results on YouTube, just as they would in a Google search string. SEO also matters with blogging engines, except for Tumblr, where SEO is important only if you crack into templates and apply meta tags.\n\n### Google Changes Its Algorithms Often\n\nAs mentioned earlier, gurus love trying to beat the system. They still game the system to this day with some truly ridiculous tactics. This is one reason why Google changes its algorithms frequently, and while that does make life difficult for gurus, it also makes life more complicated for the rest of us. We now have to make sure our SEO meets with the new standards. Not knowing what Google will or will not change makes it difficult to gauge how much work is ahead of you once a change occurs. It's back to reading minds, and changes from the mother ship can happen without warning.\n\nNow that you know exactly what SEO is not, and why we are somewhat leery of it, let's talk a bit about how SEO _does_ work for you, your website, other social media, and author platforms.\n\n## How to Make SEO Work for You\n\nWhat this chapter will do is create a basic strategy for you to apply. What we're offering are the nuts and bolts of SEO, setting aside past skepticism and bad experiences we have encountered with gurus.\n\nWhat you need to know, before we go any further, is exactly what _meta tags_ are. Meta tags are keywords and descriptions that appear in the header of your blogs, as well as in your YouTube posts and in the titles you may list on Amazon, Smashwords, and elsewhere, if you are an independent publisher or hybrid author. Many kinds of keywords can be implemented, but for our (writers') purposes, we'll focus on the following:\n\n  * **Titles.** These are the titles of your Web pages, whether for your blog, your website, or both. When people are on your website, the title appears in the browser window or tab, as well as in the Recently Visited drop-down menu.\n  * **Descriptions.** Descriptions are brief summaries of what content is featured on a specific page. You can use the same description for an entire website, but it is advised to make each description original.\n  * **Keywords.** Keywords are the most common search terms people would likely use to find your site.\n\n### Bookmark\n\nOther places where keywords matter include:\n\n  * **Content:** The actual videos, images, and text visible to people visiting your website\n  * **Image file names:** Tags given to images you use for blog posts and other online content you are sharing\n\nThere is a working theory that optimizing URLs is essential in good SEO, but speaking from past experience, we can tell you that keywords in URLs are useless, and they can inadvertently red flag your site on Google and make it hard for people to find you.\n\nThe current trend in SEO is that meta keywords are secondary to meta descriptions in priority. This trend can easily change, however. It is prudent for you, when laying down the groundwork for SEO on your website, to follow these limits:\n\n  * **Page Title:** No more than 70 characters\n  * **Meta Description:** No more than 160 characters\n  * **Meta Keywords:** No more than ten keyword phrases, separated by commas, all in one set of quotes, not individual terms\n\nWith these guidelines, you can start developing your own SEO strategy. So, let's focus on that niche fiction of \"dystopian dinosaur science fiction with gladiator cars,\" which sounds more interesting the more we type it. Let's set up a strategy for the home page of your blog or website under the parameters covered here:\n\n  * **Page Title:** _Dino Demolition | Science Fiction from Tee Morris_ (49 characters with spaces)\n  * **Meta Description:** _From author Tee Morris comes a new adventure where the Past becomes Present, and the fate of the world comes in Regular or Premium Unleaded._ (140 characters with spaces)\n  * **Meta Keywords:** _dystopian, apocalypse, future, road warrior, Jurassic, dinosaur, science fiction, Tee Morris_\n\nWith the basics of SEO in place, you can now look deeper into the content of your pages. Are there enough keywords present in your pages' content to catch the eye of search engines everywhere?\n\nSocial media is the most effective and cost-efficient way to set your work apart, and it's essential in today's marketplace. Gone are the days of the ivory tower where a writer could hide out. Today's readers want to see, hear, touch, and interact with fans, and they smell \"fake\" or auto-generated a mile away. Choose a persona, be it spiritual, ironic, comical, artistic, etc., and then generate interesting content and conversations across social media platforms in a way that best expresses your personality and brand.\n\n\u2014Toby Neal, author of the million-copy-selling Lei Crime series\n\n## The SEO That Can Hurt You\n\nOptimizing your Web page, whether with keywords or with content, can truly be an epic undertaking, depending on how much work you want to do. SEO shares a few things in common with a process authors are very familiar with: worldbuilding. You can spend hours, days, or even months worldbuilding for a story or novel, and never type out one word. The same thing can happen with SEO. You can apply keywords every place you can think of and inadvertently destroy your website, if not from the inside, then from the outside. SEO can work against you if you remain oblivious to the following traps that, inexplicably, those gurus still practice today.\n\n### Hidden Keywords\n\nWith some CSS coding, you can actually place images on top of bodies of text. With some extra coding you can easily make a small paragraph of keywords blend into the background and then hide the \"blank\" space by placing an image over it. This sleight-of-hand is an instant red flag from search engines and can penalize search results for your website.\n\n### Stuffing\n\nWhen we talk about SEO, those new to the Internet usually ask, \"Why don't I just load up my keywords tag with a text version of my book?\" (No, really, we get asked that.) This is an example of _stuffing_ , which is exactly what it sounds like: attempting to come up with every permutation of what people would enter into a search engine. Instead of your keywords looking like this:\n\ndystopian, apocalypse, future, road warriors, Jurassic, dinosaur, science fiction, Tee Morris\n\nStuffing turns your keywords into this:\n\ndystopian, distopian, apocalypse, future, road warriors, road, warriors, Jurassic, dinosaur, dinos, Tyrannosaurus Rex, T-Rex, science, fiction, science fiction, speculative fiction, spec fic, sci fi, specfic, sci-fi, Tee Morris, Tee, Morris\n\nAvoid stuffing in your titles, headings, descriptions, and page content. Not only does this tactic not make any difference whatsoever, search engines regard this as characteristics of spam-related sites.\n\n### Forcing Keywords\n\nThis is a variation on stuffing, one that deals more with your content and how keywords can affect your posts or pages. Forcing keywords where they do not belong contextually cannot only be risky in how search engines regard your page, but they also can alter the voice of your content. For example, let's say you write up a summary of your genre-mashing novel for your blog's About page:\n\nFrom the ashes of society, a hero rises. Mac McKlintock is a loner, driving the highways of a post-apocalyptic world where dinosaurs walk amongst humans. Both prehistoric creatures and survivors of civilization's fall hunt. They hunt for the weak. They hunt for the frightened. For Mac, he survives, and stands for those too tired to fight.\n\nNow, let's say a guru claims to be able to make this summary SEO ready. The guru comes back with your summary looking like this:\n\nFrom apocalyptic ashes, a hero rises. Mac McKlintock is a lone road warrior, cruising the desolate highways of the apocalypse, a new Jurassic world where science has brought dinosaurs back from extinction to walk amongst the human survivors. Both prehistoric dinosaurs and what is left of civilization hunt. They hunt for weak humans. They hunt for frightened humans. For Mac, he survives, and in this science fiction adventure he stands for those too tired to fight.\n\nSee how stilted, not to mention redundant, the second description is? Ranking high in Google search strings does not necessarily mean more keywords everywhere, but more keywords with _relevance, reputation,_ and _popularity._ SEO should not dramatically change the voice of your website.\n\nYou can also force keywords into URLs. We have seen for example, simple URLs like this:\n\nhttp:\/\/ministryofpeculiaroccurrences.com\/what-is-steampunk\/\n\n... turned into URLs \"optimized\" for search engines:\n\nhttp:\/\/ministryofpeculiaroccurrences.com\/novels\/series\/genre\/science_fiction\/what_is_steampunk_explained_by_author_Tee_Morris.html\n\nSound ridiculous? Look ridiculous? Guess what? It _is_ ridiculous.\n\nWhen you try too hard in SEO, it does nothing to improve your analytics. Instead, it frustrates people who want to find you quickly and easily online, and more than likely alerts search engines about questionable activity. There is no reason to force square pegs into round holes when working with SEO. Keep it simple, keep it clean, and take advantage of the basics that SEO offers you.\n\n## Best Practices of SEO\n\nOver the years, SEO gurus have made the skill and art of search engine optimization this murky, mysterious approach to online marketing. SEO does not have to be like this. Much of what is needed to achieve success involves intelligent, educated application of keywords in your website's meta tags and content. There are also plenty of free tools available to keep you in the know, especially if there are any sudden changes in Google's algorithms.\n\n### Use Google Keyword Planner\n\nPart of the Google AdWords tool, the _Keyword Planner_ provides a user-friendly ability to consult the almighty search oracle Google and get an idea\u2014based on the parameters of your product and services\u2014of the most popular keywords used in search strings. You can then take the keywords generated, export them into a spreadsheet, and begin narrowing down the terms you want to use and apply throughout your website.\n\n### Use Google Webmaster Tools\n\nThis collection of online resources from Google can help you understand how optimized your website is. Offering in-depth looks at your site's _search traffic_ , keywords and pages documented by _Google Index_ , and additional statistics, along with possible errors from _Crawl_ , the _Webmaster Tools_ dive into your website's analytics and performance. Here Google lets you know what keywords are showing up in your content most often, along with variants and relevance to your site. Webmaster Tools also tracks what search strings people are using to find you. Using the Keyword Planner and Webmaster Tools together can help you optimize your website easily.\n\n### Write Your Meta Descriptions with Cliffhangers\n\nWhen writing up meta descriptions, cliffhangers are a clever and enticing approach to encourage viewers to click through. Instead of the original description for our dino-dystopian adventure that currently reads:\n\nFrom author Tee Morris comes a new adventure where the Past becomes Present, and the fate of the world comes in Regular or Premium Unleaded.\n\n... we could make the description more enticing:\n\nIn a future where Past becomes Present, will lone survivor Mac McKlintock remain standing when facing impossible odds, a pack of T-Rexes, and half a tank of gas?\n\nYou want to give people a reason to click through to your site, so invite them to enter your world.\n\n### Bookmark\n\nWhen encouraging click-throughs with cliffhangers, you should avoid phrases like \"Click here\" or \"Find out more...,\" as these are associated with link farms and other spam sites. Be creative in your meta descriptions without being obvious.\n\n### Use Meta Keywords That Are Original to Your Site\n\nIn developing your keywords, you should create them with one question in mind: \"Would I follow this link?\" Your tags\u2014descriptions, titles, and so on\u2014should invite and beckon others to follow your URL. This might mean rewriting descriptions and content, and reorganizing the order of your keywords. The keywords and content you develop for your website should be unique to you but not unique to each other.\n\nWhat follows are two examples of meta descriptions and content from the home page. The first pair shows what not to do. The second is the keeper.\n\n#### Bad\n\n**Meta description:** In a future where Past becomes Present, will lone survivor Mac McKlintock remain standing when facing impossible odds, a pack of T-Rexes, and half a tank of gas?\n\n**From the home page:** __ In a future where Past becomes Present, Mac McKlintock is a lone survivor, driving the highways of a post-apocalyptic world where dinosaurs walk amongst humans. Both prehistoric creatures and survivors of civilization's fall hunt. They hunt for the weak. They hunt for the frightened. For Mac, he survives, and stands for those too tired to fight, but what are Mac's chances when facing impossible odds, a pack of T-Rexes, and half a tank of gas?\n\n#### Good\n\n**Meta description:** In a future where Past becomes Present, will lone survivor Mac McKlintock remain standing when facing impossible odds, a pack of T-Rexes, and half a tank of gas?\n\n**From the home page:** From the ashes of society, a hero rises. Mac McKlintock is a loner, driving the highways of a post-apocalyptic world where dinosaurs walk amongst humans. Both prehistoric creatures and survivors of civilization's fall hunt. They hunt for the weak. They hunt for the frightened. But for Mac, he takes a stand for those too tired to fight.\n\nYou want your keywords and content to not only be unique to your work, but also to work together. When keywords and content work independently of one another, search rankings are affected. Your keywords should be the answer to the question people are asking online.\n\n### Bookmark\n\nWe have been spending a lot of time on Google. The reason, of course, is that Google remains the most accessed of all search engines. Google, at the time of this writing, gives priority to meta descriptions and uses them in search results snippets, provided keywords appear in them. Microsoft's Google-challenger, Bing, follows similar algorithms but does not pay close attention to keywords in the meta description if they are not within the content. (This affects the site's relevance in Bing's search results.) Yahoo! pays close attention to meta descriptions, more so than Google or Bing.\n\n### Keep Track of News on Google\n\nGoogle continues to keep algorithms in flux.1<http:\/\/blog.hubspot.com\/marketing\/google-algorithm-visual-history-infographic> We have gurus to thank for this. The real question is how do we keep up with these changes?\n\n  * Google's Webmaster Central2http:\/\/googlewebmastercentral.blogspot.com\n  * HubSpot3http:\/\/blog.hubspot.com\n  * Moz4<https:\/\/moz.com\/blog>\n\nUsually when big news occurs back at the mother ship of search engines, other major news outlets like CNN, _Wired_ , and _Computerworld_ will cover the changes, but the three sources above tend to be reporting both big and small changes, and reporting them faster. While \"keywords no longer matter, but descriptions are better\" may not seem like a big change to the mainstream media resources, details like this can dramatically affect your site's rankings. When you can, make time for these online resources, as they can keep you ahead of the curve.\n\nSearch engine optimization carries a lot of baggage with it, but SEO can make a difference in where your website lands in search results, as well as in how people will find you. Reviewing your website's pages for proper SEO is a good idea, but keep it simple. With any strategy to improve your online presence, you can go too deep and begin to litter your website with ridiculous redundancy, or completely alter the tone and voice of your site to where it just sounds odd. Search engine optimization is important, but it should not come at the cost of quality content or clarity in voice. Practice wisely.\n\nNow with the tools in reach, your social media platform needs your attention. However, without developing a voice and providing your networks with content, there is no platform to develop. After all, you want to give people a reason to stick around. Sure, your works are interesting, but they can remain so only for so long before people wander off to another corner of the Internet. You want your website, your platforms, and your voice to be more than just a voice for your work. You want your corner of the Internet to be a one-stop location for news on your work, on your genre, and on works that are related to your genre.\n\nThere is no hard-and-fast rule that you have to share everything. In fact, there are no hard-and-fast rules in social media, but there is a need for content. And content is king. Without it, there is no online success. You want to be a voice and a positive, reliable resource for what you write, your genre, your interests, and your industry. Focus on developing the strongest voice possible, and your networks will provide you and your work the strongest of support.\n\nWelcome to the wonderful world of content marketing, where people come to you to stay in the know.\n\n1<http:\/\/blog.hubspot.com\/marketing\/google-algorithm-visual-history-infographic>\n\n2http:\/\/googlewebmastercentral.blogspot.com\n\n3http:\/\/blog.hubspot.com\n\n4<https:\/\/moz.com\/blog>\n\n# Chapter 12\n\n# Content Marketing\n\n# Promoting with and Through Others\n\nLogic would tell you that investing in some kind of online advertising, whether you pay to show up on other blogs, boost Pages and posts on Facebook, or engage with other social media platforms, would be the easiest way to get the word out about you and your books. This is traditional marking at its core: Pay for the platform in order to reach your audience. This is how marketing has worked since its inception, and if you hear the story behind those who inspired the TV show _Mad Men_ , you'll find that marketing is a funny and slightly sneaky business of manipulation, public relations, and showmanship. You have to convince your audience, be they loyal fans or potential readers, that _your book_ is _the book_ to read, that your series is the next big thing, and that you are a writer to watch. This is how marketing works and has worked for decades.\n\nFollow that same logic with social media, though, and you are more likely to win the lottery than to propel your book up the rankings of Amazon and the bestseller lists at _The_ _New York Times_.\n\nSocial media, and those who have mastered it, never forget what is at its core: real people. So many authors forget the _social_ aspect of social media. You need to make a connection with your audience and establish yourself not only as a skilled storyteller but also as an expert on what you are writing about. You need to establish trust with your audience, which, to an extent, traditional marketing does, but what traditional marketing fails to understand is that social media is not for bombarding an audience with the same message over and over again (as we have seen numerous times with authors on Twitter saying some version of \"Buy my book. It's awesome!\" in their feed). Doing so simply drives a social media audience away.\n\nYou might notice that those same authors who continuously promote their books have an astounding number of followers\u2014some even reach the tens of thousands. Experience has shown us that most of those followers are purchased. Again, it's a sign of traditional marketing mishandling social media, as it's all about the numbers in traditional marketing.\n\nMarketing and social media, though, have found a happy medium, and there is a good possibility that you have been practicing it already, provided you are online trying out the different platforms. _Content marketing_ takes a very different approach in that it promotes through the content of others. In a sense, you are promoting your own works by showcasing someone else's related work.\n\nPerhaps now you may be thinking \"Wait a minute. I'm promoting myself by promoting others? That doesn't make sense!\" Content marketing follows a different strategy. Instead of your platform centering only on your book, it centers on your interests.\n\nOn our own Ministry of Peculiar Occurrences Facebook Page, we feature links about our books, special events where we will be appearing, and links to our newsletter. Our Page (as well as our other platforms like Twitter, Tumblr, and Instagram) also offers news on those who write for _Tales from the Archives_ , promotes other steampunk events across the country and around the world, and shares links about steampunk music, fashion, and entertainment. We even feature other authors and their works, especially if the authors are part of our _Tales from the Archives_ podcast.\n\nThis may seem backwards, especially when we promote other authors' books, but in content marketing you are establishing your blog and your social networks as reliable resources for your subject matter. In the case of the Ministry of Peculiar Occurrences, our expertise is steampunk. What kind of content do our various platforms cover?\n\n  * Steampunk\n  * Dieselpunk\n  * Atompunk (sometimes called retropunk)\n  * Victorian history\n  * Science fiction, fantasy, and horror\n\nThat covers a lot of ground when you think about it, but one of the challenges (and joys) of content marketing is asking, \"What can I talk about with authority?\" Our One-Stop Writer Shop platforms include:\n\n  * Publishing news\n  * Self-publishing news\n  * Writing tips\n  * Self-publishing tips\n  * Marketing and promotion tips\n\nIt is incredible how many different topics can directly tie in with your work.\n\nContent marketing allows you to find other voices in your profession who share your interests. You can then share with your readers, fans, and fellow writers terrific resources that inspire you to work harder and write better. You will want to read and research whatever you share or syndicate on your website to be sure the article you're sharing is coming from a credible, reliable source. We consider the \"work\" in content marketing to be making sure we stand by the resources we share. That may take some time reviewing what's out there, but when you see traffic coming to your blog or your Google+ page, it is a rewarding sight indeed. You're establishing trust with your audience and building a reputation as a reliable resource in a shared interest.\n\n## How Does Promoting Others Really Work for You?\n\nHow exactly content marketing works as a promotion tool is still difficult for some authors to grasp. Traditional marketing is all about repetition. It becomes a bombardment of advertising, and an increase in Twitter followers is perceived as successful. But is it? Are people interacting with you on your networks? Are you talking with your virtual street team, or are you merely broadcasting the same message over and over again, eventually turning your signal into noise?\n\nWith content marketing, you establish your name and your works as part of a larger brand, and your website is _the place online_ to get the latest news on your interests. Content marketing begins when you discover an article that speaks to you and you know sharing it would resonate with your audience. Once you share it, your audience receives new content. In turn, the origin of the resource\u2014maybe a blog like Boing Boing, a DeviantArt Web page, or the Instagram of an author like Piper J. Drake or Bill Blume\u2014receives traffic from your audience. This sharing benefits both sides, and the person receiving the new traffic, if savvy, will drive traffic to you in return. This reciprocal approach to marketing builds audiences and communities; the more you share, the stronger your reputation within these circles.\n\nA great example of content marketing at work is Tor.com. A science fiction publisher, Tor delves into far more than just books and its latest releases. Tor.com features columns from authors (even those outside of its catalog) talking about the genre, writing, and media of past and present. The blog also has sections dedicated to upcoming special events (including conventions and expos), art from content creators of all backgrounds, and various discussion threads by readers and writers from around the world. Tor.com has established itself as a go-to resource for relevant content in science fiction, fantasy, and horror, all the while promoting its own brand.\n\n### Bookmark\n\nWe talk a bit about _attribution_ throughout the book and in more detail in the Best Practices chapter, but considering how many people completely forget (or ignore) this little detail, it bears repeating: Just because you find something on the Internet doesn't mean it's yours. If you see a post\u2014be it from another author or just a friend\u2014don't turn it into a shareable meme and claim it for yourself. We once caught an author duplicating another's video content, nearly shot for shot. You're a writer and more than capable of coming up with your own ideas. When someone provides content relevant to you (and your audience), make sure you give credit where it is due. A lack of attribution only hurts your online reputation. Avoid violating this rule at all costs.\n\n## What Content to Look for in Content Marketing\n\nWhen heading out into the wide expanse of the Internet for content, it is a real challenge to find exactly what you are looking for. What is content that makes you think \"Oh yeah, this would be great for my audience!\" It may sound easy\u2014just find some good articles on a blog here or a Facebook Page\u2014but you actually need a strategy if you want to provide the best content for your audience.\n\n### Reliable Sources\n\nYou've heard it said again and again, but before you repost anything, it is always a good idea to dig deep into a Web page, blog, or social media link. Resources like Tor.com, io9.com, _The Washington Post,_ _Mashable_ , and _Huffington Post_ tend to be rock solid in their research and reliability (but no, they are not perfect). It's when you go to personal blogs or websites that you'll need verification, particularly if you're not familiar with them. You really want to be certain that the news and commentary offered is something that you can stand behind and say, \"Yes, I agree.\" Take a moment to check the resources you're not acquainted with and assure that proper attribution is given.\n\n### Current Sources\n\nTry to syndicate and share advice and tips that are relevant. Think of how much publishing has changed over the past five years alone. How relevant would advice on formatting books for e-readers be right now if it was published in 2010 or earlier? Tech news, in particular, has an extremely limited shelf life. You want to find stories, advice, and reviews that date back no more than a year or two, unless it is evergreen content or content that shows \"how some things never change\" (or \"how things have changed since...\"). The more up-to-date your resources, the more relevant your site or platform becomes.\n\n### Creative Sources\n\nThe content you can share, be it your own or content from another source, should stimulate or motivate your creativity. Inspirational quotes or nuggets of advice that inspire you would also make for great content to share. If you are sharing another resource, check its origin just to be sure it is not a site that promotes ideas or agendas contrary to your own. You will also want to check the validity of quotes. Sometimes quotes are paraphrased or can't clearly be attributed to one person. Sharing comical images, such as memes, can be great for your Web traffic.\n\n### Bookmark\n\nAnother note concerning attribution and memes: Some less-than-scrupulous authors take memes from other sites and apply Photoshop to remove the original attribution (usually a website in a lower corner), or they simply crop out the URL of a meme's origins. Even reproducing a meme seen elsewhere is a questionable practice. In other words, it is never wise to scrape another's content\u2014and that includes re-creating it.\n\n### Appropriate Sources\n\nThis tip tends to be rather tricky to define because everyone's idea of what is appropriate is different. If you are engaging in content marketing for edgy or sensational titles, then your content may reflect that. You define \"appropriate\" content for your blog or site, but remember the content you share reflects you, your brand, and your titles. There is no consistency between when readers \"give a pass\" to authors and their opinions and when readers hold authors accountable. Therefore, when you post content outside of your brand, your series, or your title, and that content expresses political, religious, sensitive, or edgy humor or topics, keep in mind that the content reflects on you\u2014possibly in ways that you may not like. Set boundaries for yourself, and remember that your boundaries may still be out of bounds for some. Be ready to face pushback if you stray out of your brand's comfort zone.\n\nWhen working with content marketing, you will want to be patient and consistent. There is no magic bullet, no surefire formula, and no foolproof way to gain traction overnight, so you have to find a schedule and a pattern, and stick with it in order to build your platform.\n\nFiguring out what works with your audience can be frustrating. Sometimes it will be the post about your book that will result in traffic and numbers, and sometimes a post from another source that caught your eye will drive traffic. What matters in the end is the quality of your content. If you provide your Page, your Instagram, or your blog followers with quality content, and you turn your platform into a go-to source for reliable resources and fantastic media, you will establish a connection between you, your audience, and your work. When you've established yourself as an authority on a topic, people will find your work.\n\n## When Content Marketing Goes Bad\n\nContent marketing can be a positive, community-building strategy for any business, whether you're a writer, editor, publisher, or even a literary agent. However, some writers and social media mavericks regard any online content as free for use, free of responsibility or credit. This unethical approach to sharing content, as we have mentioned before, is known as _scraping_. Similar to when blog content is syndicated without attribution, scraping occurs when the origins of the content are removed and posters tag the accompanying content with their own page, giving the appearance that the shared content is original.\n\nAnother dark side of content marketing is establishing a social media platform with only one goal: collecting thousands upon thousands of Likes, Favorites, and other statistics in order to eventually sell the account to the highest bidder. Along with scraping content, the following is a list of ways some of these bogus accounts collect massive numbers:\n\n  * Meme generation\n  * Quizzes\n  * Photographs and artwork (posted without attribution)\n  * \"Follow-for-follow\" campaigns\n\nSocial networks attempt to crack down on these unethical accounts, usually through the diligence of their users. Keeping tabs on billions of users, however, can be a daunting task. When putting together your brand's platform, strategy remains key in a successful social media campaign, but it is good to remain aware of what bad content marketing looks like. It is easy to lose yourself in social media's numbers game, and while it may appear that these questionable tactics work in building up your following, quality should always trump quantity. Aim to provide essential, relevant content that will bring people to your Page and keep them coming back, and you will make a strong support system of your platform.\n\nContent marketing that's carried out with the right strategy and execution can prove to be a better way of promoting your book. By building your brand around a community comprised of more than just readers, you have the potential to build a network of artists, musicians, and other writers, all of whom offer their own platforms as sources of promotion. Be patient and consistent, and make sure you use all of your networks to build a community around your quality content.\n\n# Chapter 13\n\n# Best Practices in Social Media\n\nSocial media, in the past decade, has evolved into this great machine of communication and innovation. Today it's hard to envision a business or an individual without some sort of social networking strategy, but in the early days of Facebook and Twitter, it was hard to imagine people conducting business and promotion using social networking. Those skeptical of social media platforms regard them as something their teenage cousins do __ or stuff they tell their kids not to do at the dinner table. Skeptics view social media as hardly worth serious time and attention.\n\nPerhaps this lack of respect for social media is why so many social media plans fail. Sure, teenagers and twenty-somethings use social media. However, they also wield credit cards, use the Internet, and drive cars. Does this mean we shouldn't take credit cards, the Internet, or cars seriously? Of course not, and yet the mistakes people make in social media can be easily avoided if they take social media\u2014and its potential\u2014seriously.\n\nHere's something to keep in mind when approaching social media: Everything shared between these various networks has _potential._ There is no easy way to handle promotion of any kind, and that includes this brave new world of social media. The self-proclaimed social media \"gurus,\" who claim they know the secret of this new media, which they hope to sell to you for a pretty penny, are the reason that this frontier is so hard to cross. We are writers who actively practice and promote social media, and we want to pass along to you an honest, practical look at it.\n\nThe only \"sure things\" in social media are hard work, respect, and knowledge of the medium, and that you need to have a solid, strategic plan if you want to achieve success.\n\n## Create an Editorial Calendar\n\nRemember: Content is king. From the early days of the Internet to today, this remains true. People want quality content from a blog, a podcast, or a social network.\n\nHowever, content must be in partnership with _consistency_. There needs to be a rhythm or set schedule to your online updates. A writer cannot simply slap content\u2014even good content\u2014on his blog, Facebook, or elsewhere, and then walk away from it for months on end. One or two, or even a smattering of posts of excellent content will not make much of a splash if they don't come at regular intervals that the consumer can rely on. It's about consistency in posting.\n\nSet a schedule for specific days or post frequently. The best way to manage all of your content, whether blogging, Facebook posting, tweeting, or anything in between, is to set up an editorial calendar that lists when and where you will be posting new content.\n\nYour calendar should list topics or reoccurring columns you are posting on your blog, as well as where you are posting and on which days. For example, your calendar should show that you're posting on blogs and Facebook on Mondays and Wednesdays, while Tuesday and Friday are dedicated to Twitter and Pinterest. Once you have a realistic editorial calendar set up, make sure you stick with it and fulfill the schedule. Map out how many posts will be original content that markets your work, and how many will be informational content.\n\nBut don't become a slave to your original plan. Your schedule shouldn't be forever set in stone. Allow your editorial calendar flexibility for when fresh topics and news crop up. You want to be able to take advantage of them, not necessarily for the sake of traffic (although some people do that), but for when you really have something to say about a current topic or trend. People will be looking for keywords, so make SEO work for you by keeping up on what is going on in publishing, writing, and your area of expertise.\n\nContent managed in an organized fashion is much less intimidating for an author than a slapdash approach. With a proper editorial calendar and different outlets featuring daily postings, it will soon become habit rather than a rushed afterthought.\n\n## Think Before You Post\n\nWhen you build a platform in social media, you are building a brand. Your brand reflects what you produce, broadcasting your reliability as an author to your audience. It takes a long time to create a solid brand, but just one ill-thought-out blog post, one impulsive tweet, or one snarky caption on Pinterest can inflict a lot of damage. Snarky statements and other attempts at comedy can backfire in a big way, so it is a good idea to look over an update before it goes live and try to see it objectively. Ask yourself, \"Could this be taken the wrong way?\" __ If you aren't sure, then the chances __ are it could be, but that doesn't mean you shouldn't post it. Just mentally prepare for any blowback. Comedy can be hard on social media.\n\nWhat is not difficult to find, on the other hand, is controversy. You may be impassioned by something in the headlines, an exchange on Twitter, or an image on Instagram, but before you let your passionate support for a cause fly, ask yourself, \"Is this argument worth undertaking?\" __ Do you really understand both sides of the debate? Is the time you spend replying and addressing opposing viewpoints affordable? Or are you on a deadline and want to stay focused? If you do step into the middle of a heated debate, are you prepared to face a negative reaction?\n\n### Bookmark\n\nYou may have heard the term thrown around, but what exactly is a _troll_ on the Internet? Not far from their literary counterparts who lurk under bridges to snatch travelers unexpectedly, trolls lurk in the darkest corners of the Internet either looking for arguments to kick up or insults to let fly. Trolls care very little about making a point on either side of an argument. They care more about how much they can rattle people, especially the host of a blog, podcast, or platform. This means the insults can go from juvenile to sensational to deeply personal.\n\nRemember that you have the final say for who says what on your platform. Make sure to take advantage of blocking features, and above all\u2014 _don't feed the trolls_. There is a reason you have block functions in many of the social networking tools covered in this book. The best way to handle a troll: block, and walk away.\n\nIn fact, any time you look at a post that you've written, stop and ask yourself, \"Would I be okay sharing this with a roomful of strangers?\" __ Even though your network comprises friends or followers, the truth is your network does not know you as well as you think. Consider consequences with any update you are about to make before it becomes necessary to break glass and push the red button in case of emergencies.\n\n### Finding the Positive\n\nGoing along with thinking before posting, here's another good rule before hitting send: Be _positive_.\n\nAs a professional using social media, you really don't have the luxury of being able to whine, complain, or go on rants. (Granted the _occasional_ rant can be fun, depending on the topic. Pip got good traction on her blog when she raved tongue-in-cheek on the trend in book covers that had lead characters with their heads cropped out.) Many authors have built a community around their reinforcing, positive message, be it in writing advice or lessons learned while on the road with their books. People have enough cynicism and negativity in their regular lives. They may not want to hear it from authors\u2014especially ones they want to enjoy the company of social media with.\n\nSo focus on lessons learned and what you want to share. Be charming, funny, witty, and caring. Such qualities will take you far with readers.\n\nDo social media because you like it, not because it's some kind of obligation. Nobody wants to read the blog of someone who is being forced to do it. Just like we don't want to read a book written by someone who hated writing it. Tweet because Twitter is fun for you, not because it's on your checklist. Social media is a great place to meet other people doing what you want to do, and it's also a great place to talk to your audience. Note I said _talk to_ , not _yell at_.\n\nYou know how we use the phrase \"In Real Life\" (IRL) to differentiate between what's happening in reality and what's happening on the Internet and on social media? Newsflash: Social media is reality. It's not illusion. We need to treat what happens online like it's real because it is. It's not _A_ _Game of Thrones_. And when you're online, the same rules apply as when you're off-line:\n\nBe awesome.\n\n\u2014Chuck Wendig, author of the Miriam Black trilogy, the Heartland trilogy, and _Zeroes_ , and host of the TerribleMinds.com blog\n\n## Participate in Blog Tours\n\nThese are events that, if managed properly, can be a fantastic way to introduce your words to new readers and, in a reciprocal manner, introduce new bloggers to your readership. These events also give you a good backlist of evergreen content. Blog posts such as these can be easily overhauled, updated, and then repurposed for your own blog. If you are repurposing blog posts, you should wait roughly six months in order to give the original column some exclusivity where it first appeared. It's the friendliest thing to do to maintain good relationships with other bloggers.\n\nThe way blog tours work is that you appear as a guest blogger across a network of blogs\u2014blogs that cover a variety of topics or perhaps share a common theme. Other bloggers, once or twice a week, in turn, appear on your blog, sharing topics they would like to cover or ones you have previously suggested. To make tours a success for everyone involved, the content bloggers deliver to one another should be:\n\n  * original\n  * unique from blog to blog\n  * fall within 500 to 1,000 words\n\nSome bloggers may fail to promptly post others' contributions, but if all participants contribute on schedule, your evergreen content will grow substantially. This gives you many options for posts that can later be repurposed for your own blog or for future guest postings.\n\nWhen you are on a blog tour, you are expected to generate original posts for each stop. Do not create one blog post and offer it as your sole contribution. When embarking on a blog tour, you are committing to your tour hosts new, original, unique content. Make sure you meet that commitment.\n\n## Create Quality Content\n\nContent can come from a variety of sources, but your content should always be quality. What defines quality content? It's subjective, but quality content should be clear, concise, edited, and well researched. There is a popular opinion that posting content for content's sake is what matters, not the actual worth of your content, but all of your posts are important because they are a reflection of your work.\n\nSo what do people look for in quality content?\n\n### Timeliness\n\nWhen news pertaining to your business hits the headlines, you have roughly twenty-four to forty-eight hours to write, edit, and deliver a blog post. At the latest, you have within a week (roughly five days) to blog your own commentary on a topic. At the time of breaking news, people are specifically searching out articles and angles on this topic. By tapping into the timeliness of a news story, you increase your chances of other blogs picking up and syndicating your blog post. Wait too long, and the opportunity to take advantage of the story and drive traffic will be lost.\n\n### Cross Promotion\n\nOnce a blog post goes live, Facebook and Twitter should be the beginning of your promotion. While there are automated posting tools like the WordPress plug-in Social that broadcast updates on your blog, these postings feel rather cold. Following a flowchart (similar to the editorial calendar), you should compose an original, accompanying post or tweet and then add it to your latest blog post. This lets your various networks know that while there is crossover between your channels, you are handling each account and keeping content fresh.\n\n### Cross-Referencing\n\nRemember that your words reflect directly on you, so it is a good idea to link your references back to your blog posts. Sharing resources and linking back to them is a terrific discussion starter, as commenters will ask more about the sources or spark debates on the resources cited. Sharing links may also encourage traffic to your own website, depending on the generosity and the respect of the referred website. Either way, cross-referencing is a terrific tactic for generating traffic around your blog.\n\n## Be Visual When Possible\n\nNothing attracts attention like a good image. Users consistently interact with images more than text-only posts, across all platforms. Some platforms are particularly focused on images\u2014Pinterest and Tumblr\u2014but posts and tweets will get more eyeballs on your content with the addition of an image. Don't forget: Images in blog posts rule!\n\nUsing other people's images can backfire on you, however. Sure, when you are just operating as a social media individual, you might enjoy posting Tom Hiddleston memes, but as a professional trying to keep a good online reputation, you should think twice about it.\n\nAnd don't forget\u2014as we have mentioned in earlier chapters\u2014that stealing another's meme and slapping your website URL on it is definitely out. Similar to when blog content is syndicated without attribution, \"scraping\" occurs when the origins of the content are removed and posters tag the accompanying content with their own page, giving the appearance that the shared content is their original content. Turning someone else's content into your own can come back to haunt you.\n\nThe safest thing to do when creating your own image is to stick to one you took yourself, or you can find stock photography at sites like depositphotos.com. For a small investment, you will be on safe ground to use these photos, and if you are making your own cover, you can repurpose stock images for that, too.\n\n### Bookmark\n\nDon't forget to tweak the images for different platforms. Remember, tall images work great on Pinterest and Tumblr. Square ones work best for Instagram and Twitter.\n\n## A New Use for the Pound Symbol: Hashtags\n\nTwitter was the first social media platform to take advantage of this feature, but now Instagram, Facebook, and even Pinterest are all tapping into the SEO and trending capabilities of hashtags _._\n\nHashtags (#) are tracking tools that allow you to identify updates and postings under a quick-to-find category in any search engine or platform-specific search. The hashtag first appeared on Twitter on August 23, 2007, when Chris Messina tweeted \"How do you feel about using # (pound) for groups. As in #barcamp [msg]?\"\n\nYes, the first update to use a hashtag _was about hashtags._\n\nHashtags can be used in a variety of ways, ranging from emphasizing a mood to commemorating a special event to stating a punch line in a joke. For authors, hashtags are best employed in the following ways:\n\n  * **When developing a new title or project:** _#amwriting, #amediting, #WIP, #teaser_\n  * **When working within a genre:** _#steampunk, #UF, #YAFantasy_\n  * Wi **th tweets pertaining to a book or series:** _#MoPO, #LondonUndead, #Redshirts_\n  * **When identifying promotions or digital appearances:** _#blogtour, #podcast, #interview_\n  * **At special events, like book festivals and conventions:** _#BEA2015, #balticon_\n\nHashtags are the best way to track discussions and trending topics across a social media platform; you'll even hear opinions from people you are not following. Whether you are using tracking options offered through the official Twitter app, reviewing all Instagram images using a tag, or clicking a tag in a Facebook or Google+ update, hashtags allow you to follow or join in a conversation, provided participants consistently use the same hashtag in conversations and posts.\n\nNot all posts need hashtags, but there are times, current events, and special moments when hashtags are essential.\n\n### Book Events and Engagements __\n\nWhen attending book festivals, it is always a good idea to use a hashtag for people to track you and other authors attending the event. Before arriving, try to find out if the event has an official hashtag (most events should have one as they will want metrics to show potential sponsors for the following year's event) and encourage your fellow authors and your network to use it. When creating an official hashtag, keep it easy to remember and as short as possible. For example, the official hashtag for the \"HallowRead\" event of 2014 wasn't #HallowRead or #HallowRead2014, but #HR2014.\n\n### Topics of Discussion\n\nIf you are composing updates on Instagram, Facebook, or Google+, you can turn key words in your update into hashtags, making your status easily found in a hashtag search. So if you are talking about \"a new steampunk short story\" you have just written, you should post the status as \"a new #steampunk short story\" instead. Why use _steampunk_ as your hashtag? Steampunk is the term or topic of discussion you want people to find you under. Find those keywords\u2014no more than five, at the most\u2014and let them work for you.\n\n### Twitter Chats __\n\nThere are several writing chats happening on Twitter. As of the writing of this book you can find:\n\n  * **#scifichat:** Sometimes featuring special guests, this weekly chat covers topics of writing science fiction, fantasy, and horror. Hosted by @DavidRozansky and @scifichat on Fridays, 2\u20134 P.M. EST\n  * **#YALitChat:** This chat doubles as a celebration and networking opportunity for readers and writers of young adult fiction. Hosted by @Georgia_McBride and @YALitChat on Wednesdays, 9 P.M. EST\n  * **#litchat:** An open networking session between readers with books and authors. Hosted by @litchat on Mondays, Wednesdays, and Fridays, 4\u20135 P.M. EST\n\nWhen participating in or simply observing a chat, all your tweets should follow one of the hashtags used. Using the primary hashtag (#scifichat, #YALitChat, etc.) is enough to track with the discussion. If you are mentioning Twitter Chats on other platforms, you should use corresponding hashtags for tracking purposes.\n\n### Twitter Parties\n\nHosted by the author or a third party, with everyone using an agreed-upon hashtag, Twitter parties are similar to Twitter chats. The difference is that Twitter parties tend to double with the use of contests: participants (people who actively tweet or retweet content under the official hashtag) are entered into giveaways. Instead of happening on a weekly basis, these online events tend to happen before or after recent book releases, special events hosted by authors, and topics specific to the author who is hosting the Twitter party. If you are mentioning Twitter parties on other platforms, you should use corresponding hashtags for tracking purposes.\n\nThe most effective hashtags for parties are compact and easy to remember. Instead of a Twitter party hosted by #MinistryOfPeculiarOccurrences, it's hosted by #MoPO.\n\n#### Proper Use of Hashtags\n\nAccurate and consistent hashtagging of your content is good practice across a number of networks. The hashtag symbol has become a way to facilitate searching and indexing on social media platforms. People who are searching for a particular topic find them useful. When a hashtag is _trending_ , it means it is being searched a lot and used often. Hashtags do come with some simple rules. Don't string too many words together, unless it is for comedy. For example, #mybookisawesomeandyoushouldbuyit is not going to get you anywhere, but it will make fans chuckle a bit.\n\nHashtags need to be accurate and to sensibly describe your content. Hijacking a popular or trending topic that doesn't accurately describe what you are doing (another form of \"trolling\") is a terrible idea. Such tactics inevitably backfire.\n\nHow so?\n\nWhen popular brand name pizza DiGiorno used the hashtag #whyIstayed (which was about women and domestic violence) to advertise its pizza, the backlash was horrible. They had to respond with \"A million apologies. Did not read what the hashtag was about before posting,\" __ which, in admitting DiGiorno was oblivious to the hashtag's intent, makes a bad situation worse.\n\nUnderstanding what a hashtag is referencing before posting or contributing is always a good idea. If you have no idea, or just want to be sure you're not using someone else's hashtag, then visit tagdef.com. There you can find out what certain hashtags mean, or even define your own. Just check out one we created earlier.1<https:\/\/tagdef.com\/mopo>\n\nHowever, as useful as hashtags are, you can go overboard with them and lose the message you are trying to communicate. Stick to five hashtags in one post at the most. On Twitter, keep it to two.\n\nHere are some good writerly tags for connecting with your readers: #FridayReads, #BookGiveaway, #MustRead, #StoryFriday, #LitChat, #FreeBook, #Kindle, and #Nook. And don't forget you can also hashtag your genre (#YA). If you are looking to network with other authors, try one of the following: #AmWriting, #AmEditing, #WriterWednesday, #AskAuthor, #IAN1, or #AuthorAlliance.\n\n## Taking P!nk's Advice: Facebook Parties\n\nI'm comin' up so you better get this party started\n\nGet this party started on a Saturday night\n\nEverybody's waiting for me to arrive...\n\nYou know, that sassy rock star P!nk is absolutely right\u2014people love a party, even an online one, so it's not surprising that advertisers have picked up on the chance to rally people online for a Facebook party. Direct marketers such as Avon and Jamberry were the first to catch onto the possibilities, but that doesn't mean that writers can't get in on the fun and create some visibility for their book titles in the process.\n\nBefore announcing your party, ask yourself first, what you are celebrating? No one celebrates an ordinary day of the week; they celebrate _milestone_ events. As a writer, the biggest event of the season should be your book birthday, but you could also host an event for bundling your books together, a Kickstarter campaign for an anthology, or a cover reveal. Be careful you don't tire out your guests with too many parties in the course of one year.\n\nAs with a party that you would throw in your off-line life, the planning of an online party is important. Here are some questions you need to ask yourself before you announce your next bash to the world.\n\n### What Date?\n\nYou want to find one that doesn't clash with major holidays like Black Friday, Veteran's Day, or anything that could annoy or prevent your guests from attending. That doesn't mean that you can't plan your party around major days; for instance, the weeks before Christmas would be a great time to have a party to celebrate your holiday-themed book.\n\nIf you're doing a book release, you can decide to either build up excitement the week before the book comes out, or try and generate sales during the week after.\n\n### What Time?\n\nThink about your readers, the location of your target audience, and if time zones come into play. The most recent _Ministry of Peculiar Occurrences_ event was timed to be late in an evening and ran for four hours, until midnight EST. That way our readers on the West Coast could still join in, too. You are never going to be able to find an optimal time for everyone all over the world, but you can adjust it so that there is at least a chance for your international readers to find the party.\n\n### What Sort of Decorations?\n\nSet up your event from your author page. Under More\/Event\/Create Event, input the date and time, and make it look pretty with a header image.\n\n### What Do I Serve?\n\nYou can't have food, but you can still have goodies to attract the guests. No one in the neighborhood is going to come unless you give them a reason, so consider things to give away. Be creative. Give away signed books or items related to your book. Steampunk writers give away tea. Fairytale urban fantasy writers give away apple-related items for Snow White, or hair products for Rapunzel. Romance writers give away jewelry. Think about your book, its theme, and your target audience. Keep it fun!\n\n### Whom Do You Invite?\n\nGuests are important, but author guests can bring a crowd. If you are just starting off in the author space (think of it as being new to your neighborhood), then you might not have met many fellow writers. Obviously, if you've been to a few conferences, have made contacts online, or have a writers group, you are starting off with an advantage\u2014if you want to include author guests. That doesn't mean you can't invite people if you're new; just be aware that a stranger asking is not the same as someone guests feel a connection with. However, politely asking people in a similar genre who are also starting off can't hurt. Bear in mind that you are also offering them a way to promote their own books.\n\nOnce author guests have said yes, find a good time for them to be on. You'll want ten- to fifteen-minute slots for guests to promote and interact, but also allow some buffer time so you can circulate and talk about your work.\n\n### The Big Night: Party Time!\n\nAs the date of the event approaches, there is still some work to do to make sure your event goes off with a bang.\n\n#### Promotion\n\nSpread the word, not just on Facebook, but on all your social media networks. Reach out to your author guests to help get the word out, and make sure you give them enough time to do so. Promotion should be around two weeks to a month out\u2014any further out risks people forgetting, but if you promote too close to the event, then people can't plan to be there at your party. The more the merrier, after all.\n\n#### Preparation\n\nBefore the event, make sure you have your timetable sorted out, as well as some prepared activities and information to keep the party flowing. For your author guests, make sure you have introductions ready to go. Be personable and welcoming. For the interludes where you plan to talk, have some prepared quotes from the books, questions, and, best of all, images. That way if things should start to flag or go quiet, you are only a cut-and-paste away from keeping people entertained.\n\n#### Kick Off\n\nWhen the night begins, you might want to make sure you are comfortable with virtual refreshments (or real ones) at hand, because a good Facebook party can fly by. Stay close to your computer or tablet, make sure you are there to answer questions, welcome people who arrive, and keep the conversation flowing.\n\nLimber up your fingers, too, as you will be typing fast and hard. Hey, no one ever said being a host was easy!\n\nHaving chosen the prizes for the party, you can run contests in several ways: randomly picking people in the party room, perhaps a quiz based on your book (or something related, if it isn't out yet), or rewarding the guest who comes up with the best casting of your book's dream movie. Keep your questions and trivia simple, though. Nothing is more awkward than a competition where no one wins!\n\n#### The End\n\nDon't forget to say \"thank you\" to your guests and author guests. Before the party is over, remind people of the essential information: where to find your book, relevant dates and times concerning its release, and your website location, so if they have more questions, they know how to find you.\n\n## Planning Makes Perfect: Before That Social Media Event\n\nBefore attending Facebook or Twitter parties, or appearing as a \"featured guest\" at these events, have on hand precrafted posts. Concentrate on questions or topics that might get the conversation started again, and if tweets allow for space, have images ready to upload. Anticipate some of the questions readers might ask you:\n\n  * If your book were made into a movie, who would play ____?\n  * Where do you get your inspiration?\n  * Who are your favorite authors?\n  * Why do you write [your genre]?\n\nMake certain you publicize when and where special online events will be taking place. Publicize your chats and parties across your blog, your podcast, and other platforms. People do not randomly appear for these events\u2014they wait to hear from you.\n\nFinally, especially if you are the special guest or are hosting a social media event, make certain you have all you need at arm's length. Have a drink close by. Make sure you've gone to the bathroom before go time. Once the event starts, don't leave your chair. The conversation will fly fast and furious.\n\n## Attribution, Not Imitation, Is the Sincerest Form of Flattery\n\nBeing a good participant in social media means sometimes going the extra mile. If you do end up reposting pictures of cosplay (where people dress up as their favorite comic book, video game, or book characters), or someone else's travel adventures, retain any attribution the image came with. People who create things\u2014and authors are part of that tribe\u2014like to be credited for their work.\n\nIf there is an awesome image you really want to post, but it has been stripped of attribution, then you can go the extra mile. Put on your deerstalker hat and seek out the missing information.\n\nIt's actually easier than you think. Google is once again your friend. Go to Google's Search by Image feature.2<http:\/\/www.google.com\/insidesearch\/features\/images\/searchbyimage.html > Click on Try it Now, then click on the camera image in the right-hand corner, and either paste the URL of the image or upload an image. Click Search, and Google will do its best to help you find the original owner.\n\nIt may seem like a bother, but it's the way you will cultivate a professional image.\n\nThe same goes for memes and motivationals\u2014check the source. No one likes to have a reader say \"Einstein did not actually say that.\"\n\nIf this kind of research seems like too much, just stick to posting links and images you bought or made yourself.\n\n## Seek Out Your Audience\n\nEvery book has a target audience, but it won't come beating on your door. The truth is you have to find out where your people are, and go to them. Think about yourself and your book; where do you best fit?\n\nIf you still don't have an idea, look at where authors in your genre are concentrating. Do they appear to be having success on these platforms? Start building your community of readers in those places.\n\n## Have a Plan\n\nThe leading cause of failure on social media is jumping on platforms without any idea of what to do. Social media is not something you do because \"everyone else is doing it.\" You do it because you have something to accomplish.\n\nWhen you set up a blog, ask yourself how often you intend to blog. Will you use Facebook to promote your blog? How often will you post on Facebook or Twitter? Will you go beyond these three sites and work with Tumblr? Google+? What will your voice be? Will you be all business, or will you pepper in personal thoughts? The more you map out your plan, the more focused your online platform will be.\n\nThese tips provide a great foundation for social media initiatives. Applying them to your networks can make your platform easier to develop and build, and make your signal strong and reliable. When you use social media to its full potential, you are actually building and developing your _brand_. Social media can help to define your brand.\n\nThe scary thing about brands, though, is that they are fragile and easily damaged. As important as it is to know what to do with social media, it is equally important to know what not to do. When you have the right strategy, most, if not all, of the missteps can easily be avoided.\n\n## Focus on Maintaining a Signal, Not Creating Noise\n\nWhen talking about social media and social networking, you may hear the term _signal-to-noise ratio_. Signal-to-noise refers to the quality of your statuses and updates. If you are constantly advertising or promoting something in your feed, your audience may tune out your updates, and so they are considered \"noise.\" Updates that your audience genuinely cares about or interacts with are referred to as your \"signal.\" When it comes to social media, it is easy to slip into noise mode. Signal is all about quality and what you deliver to your network.\n\nSo how do you avoid becoming _that_ author?\n\nMake a connection with your fan base through articles that capture the imagination. Share what you do when you are not writing. Do you have a passion for classical theater, B movies, or fashion? Tap into one of those topics. Post images, video, and your own blog articles on what it is that captures your attention or helps you unwind. Post this kind of content alongside interesting news articles rooted in your genre. On your fifth post, go ahead and do a quick promotion for your book. When a new review appears for it, share it on Facebook and Twitter. Celebrate with the community. You want to aim for a ratio of one promotional post to every five, or three promos out of every ten. People who follow you on networks will know that while you are there to promote your book, you are not there to do only that. You're a real person with real interests and passions outside of writing.\n\nThese are all practices that we have seen writers carry out across networks, many of them finding success in creating strong _street teams_ (fans who love to talk to others about your books and who help you make the sale) for their works out in the world, and coming soon to both brick-and-mortar and online bookstores. Sadly, though, the success stories of social media are not what drove us to write this book. This particular social media guide is born out of our frustration, anger, and outright surprise when authors get online and make a mess of things. Our inspiration comes from the writers who don't know when to quit pitching their books, when to walk away from an argument, and when not to say a damn thing.\n\nTo the overly aggressive author on the Internet, this one's for you.\n\n## Antisocial Media: What to Avoid in Online Promotion and Networking\n\nThere are a lot of things that you can get right in social media, but it is easy to get things wrong if you don't know what you are doing. Even if you do know what you are doing, you can still make mistakes. There's a difference between stumbling (which you can easily recover from) and hitting the ground hard, face first.\n\n### Forgetting the \"Social\" Aspect of Social Media\n\nMany writers in social media tend to accept the bad advice of gurus who say that every tweet, Facebook, blog post, and post on Tumblr should be an advertisement for their next book. It is not uncommon to see this, and this is not how you build a platform or broaden a network. This is merely adding noise to your signal (as we defined above), so much noise that your followers will essentially tune you out. Keep your promotional posts under control. They should not become the sole voice and purpose of your social platforms.\n\nAnother rule of thumb in social media is to establish yourself as an individual and to invest time in building your network. The key is to do so in a smart and diplomatic manner. The best way to break this rule is to make your first communication on Twitter a direct message (DM) that encourages others to join your Facebook Page.\n\nSocial media is about you, but it should not be _all_ about you.\n\n### Scraping: The Fastest Way to Get Yourself Kicked Out of the Pool\n\nYes, we are going to talk about scraping again. Why? Because it happens so frequently, and because it's so bad for your professional reputation.\n\nThe lure of the Dark Side is strong. Maybe you have one week when you simply feel like you have nothing interesting to say. But, hey, that wacky Chuck Wendig just wrote a brilliant post on most frequently asked writing questions. It's out there on the Internet, so it's fair game. You might be thinking, _I comment on Chuck's blog, so we're pals. He won't mind. I'll just cut and paste the post into my blog and ..._\n\n_No!_\n\nSeriously\u2014scraping will give you a bad name and make you enemies in the writing world. However, scraping should not be confused with its kinder, more community-minded cousin, syndication. Scraping is taking the whole post and dropping it onto your page complete.\n\nHere's a word that's synonymous with scraping: _plagiarism_. That is what scraping is, essentially.\n\nYou might even mention the original author's name or include his link, but that is not a pass in this instance. You have just stolen content, and eyeballs, from the creator of that content. The link back is worthless, since viewers just got all the good juice from your page. Remember that _syndication_ is when a _portion_ of the blog post (the first paragraph, or two, at the most) is pasted with proper attribution and the link back to the original blogger. That makes for happy content creators. Scraping will get you a bad reputation, and no invitations to the author's next birthday party.\n\nAnd if you think authors and content creators don't notice these things, well, you're wrong there, too.\n\n### Friends at Wholesale: The Perils of Purchasing Likes or Followers\n\nMany social media gurus encourage \"investing\" in your platforms, usually followed by the line \"I can help you triple your Twitter followers for the low, low price of...\"\n\nInflated numbers on platforms look impressive, but it costs a lot for tens of thousands of Likes on Facebook and nameless, faceless Followers on Twitter. Purchased Likes originate from _Like farms,_ companies that pay their employees to Like hundreds upon hundreds of pages of various products, personalities, and services featured on social networks. These Like farms fill your statistics with numbers but do not take any active role or have interaction with your community. While you suddenly appear to have those \"rock star\" statistics, these inflated numbers burn through your advertising budget on Facebook and cost you more when it comes time to boost posts. Concerning Twitter, these bulk Followers are nothing more than automated \"bots\" that tweet nonsense, junking up your incoming tweets and making it harder and harder for you to track your main Twitter feed. Other popular Twitter accounts purchased in bulk include pornography sources and malware carriers. Follower purchases on Instagram will spam your feed with a sudden dumping of images, many of them random, innocuous posts of what that company is selling, turning your Instagram feed into a Sears catalog.\n\nNot only do these accounts distance you from your network, limiting your interaction with to only who directly mention you, but they can damage your online brand, as these bots will find unsuspecting targets from your own networks. Be wary with these paid services, as they lead to nothing but noise.\n\n### Focusing on the Quality, not the Quantity: The Statistics Game\n\nYou will be told time and time again (most of the time from the people selling the above-mentioned Likes and Followers) how important it is to have the numbers on your platforms. There are plenty of authors who love touting their five- and six-figure numbers. Simply ask these social media hoarders exactly what they do with these numbers. __ Or, if you want to be more to the point, ask them, \"Exactly how do you communicate with 100,000 people?\"\n\nYou don't. You talk _at_ them.\n\nIt is always a good idea to check people who are requesting to follow you on networks. When a user starts to follow you on Instagram, for instance, check the account to see if it is an actual person with a story or a product to sell. Look at the accounts, evaluate the feed, and see if this is someone you want to connect with.\n\nYou should also check Facebook accounts when people send a Friend Request to your personal account or Group. When the request arrives, click on the account. Here's what you want to look for in a phony account:\n\n  * Activity on the Timeline: When did this account go live? What is being shared on the feed? Is she sharing pictures?\n  * How many Groups does this person like? If he is a member of more than fifty groups, consider it a red flag.\n  * What is she Liking on Facebook? Many of the spammers will Like the same game and\/or products they promote.\n\n### Bookmark\n\nOn Instagram, a telltale sign of a bogus or malicious account is the follow that happens in response to images that are weeks or even years old on your feed. Following the Like back to where it originated will either reveal an old friend or recent follower you've met and want to reconnect with, or a spammer with something to sell. Be cautious when older images in your feed are suddenly getting some love.\n\nThe social media gurus will encourage you to follow and follow blindly, but it is always good to check out the source first. What will this person bring to your network? Is this someone who will participate or simply promote, using your network?\n\n### Message Garbled, Say Again: Avoid Nonsense in Your Feed\n\nInspirational quotes. Retweets. Promotions of your latest book.\n\nIf these are occasionally in your feed, that's one thing. If any of these _are_ your feed, you are missing the point of Twitter. Or Instagram. Or Tumblr.\n\nPeople follow you on social networks to find out more about you, not about who you follow or what other people have said. Whether you are filling your Instagram with an endless parade of products or having your Facebook and Google+ echo your Twitter account, make your content original and understandable.\n\n### A Garden Gone to Seed: Leaving Accounts Unattended\n\nFrom his time in the information security field, Tee can tell you that many InfoSec professionals are opposed to a lot of the networks covered in this book, as security tends to be overlooked when these services are developed and brought online. The vulnerabilities present on Facebook, Tumblr, and other social media platforms are a genuine concern, as social networks are easily commandeered by hackers, especially when accounts are left unattended and underutilized. Keep your account active with daily postings, and watch for any odd activity or suspicious postings appearing under your name. If you find that a platform is not for you, it is best to delete the account completely rather than to leave it live. Lack of activity on any social media platform opens your brand to damage.\n\n### Suspicious Minds: Responding to Peculiar Messages\n\nThere is another method hackers use to gain control of social media contacts. It is a method of infiltration called \"social engineering _._ \" You see this most often in e-mails supposedly from PayPal and banks informing you that your account will be shut down in twenty-four hours if you do not visit a link provided in the e-mail. On Facebook and Twitter, social engineering most commonly happens in direct messaging. Usually the messages will say something along the lines of \"LOL Have u seen this vid?\" followed by a link. On rare occasions, these messages appear from accounts you recognize (possibly someone who has been hacked), but they mostly come from strangers (bots). Anytime you receive messages that appear to be sent by people who don't have time to spell out words, chances are the account has been compromised. Contact that user (on a public channel) and inform the person (politely) that you received a strange direct message.\n\n### Come On Feel the Noise: Oversharing\n\nHave you come across images that read \"You are [insert iconic character]\" in your Facebook feeds lately? These are images produced by Livingly Media (formerly known as Zimbio, Inc.), a digital media company that manages lifestyle sites for desktop and mobile audiences. The quizzes, sponsored by Livingly, ask a variety of question about your location, your lifestyle, and your interests, and from this information, the quiz names you as Captain Kirk, Harry Potter, Katniss Everdeen, Princess Leia, or some other favorite character of literature or film.\n\nDid you ever stop to ask what permissions were granted for these images? Whether the actors and actresses used in these images have been compensated? And how do these questions result in your being named an iconic character?\n\nThey don't. This is called _data mining_.\n\nIt's not quite the same as social engineering, but it is a bait-and-switch approach to getting people to take market surveys and for building mailing lists that are sold to other parties. Social media is about sharing, and it's more than okay to share, but there is a point where you can share too much online, opening yourself to a variety of annoyances ranging from spam to identity theft.\n\nWhen it comes to sharing, you must decide where you draw the line in what you will share and what you won't. Social media is your platform for your works and for you as an author. It is not your therapist's couch. _Oversharing,_ that sudden outpour of emotion and in-your-face position-taking, can not only alienate your online network of contacts, both personal and professional, but can also reflect poorly on you.\n\nOversharing is nothing new in social networking, but it is far too easy a trap to fall into. Do you want to become known as \"that guy\" or \"that girl\"? To help prevent taking a misstep, the first and foremost question to ask yourself every single time you are ready to post is: \"Should I be sharing this?\" It would be great to say that people think before they post, but sadly, some are more about sending out a snappy, witty update or sharing with the world than considering who's watching their feeds.3<http:\/\/www.businessinsider.com\/twitter-fired-2011-5?op=1>\n\nSocial networking has become a way that we communicate with people, both one-on-one and around the world, but there are limits. To sharing. To connecting. To communicating. These limits are set by you. Social media platforms serve as fantastic outlets, so long as you approach them responsibly and reasonably. Remember that your brand and your reputation are at stake, therefore your social media activity warrants your _full_ attention. Social networking is an _investment_ in your career and should be cared for and cultivated. Go beyond networking and build a community around your work. This community can become your street team for future book releases and events you will be attending, increasing your profile both in the online world and the real one.\n\nGood luck, and good hunting!\n\n1<https:\/\/tagdef.com\/mopo>\n\n2<http:\/\/www.google.com\/insidesearch\/features\/images\/searchbyimage.html >\n\n3<http:\/\/www.businessinsider.com\/twitter-fired-2011-5?op=1>\n\n# Appendix A\n\n# WordPress\n\n## Setting up a WordPress Account\n\n### Bookmark\n\nSecurity starts with you. Always create passwords that are alphanumeric: made up of a mix of numbers, uppercase and lowercase letters, and symbols. The longer the password, the more difficult it is to hack. So if the site lets you have fifteen characters, you should make your password fifteen characters. Keep these passwords in an encrypted file on a flash drive, and keep the drive in a safe place.\n\n1. Go to WordPress.com and click on the Create Website button.\n\n2. Give your site a name.\n\n3. Enter your email address, your username, and a password. When coming up with a password, you want to pick something easy to remember but tricky to figure out so no one can hack into your account. A mix of letters, numbers, and characters is always good. For example, instead of _Richmond Writer_ for your password, you could use _r!chm0nDwr1t3r_ , making it memorable but difficult to crack.\n\n4. Now WordPress will ask you if you want a custom address. You can always set up a simple, easy-to-remember URL through 1and1.com, GoDaddy.com, or a similar registrar service, and then have that URL reroute your visitors to this location. You can also have WordPress.com take care of this URL for $18 per year. Decide if you want your username to be your name, a keyword, or an acronym.\n\n5. Next you can choose one of WordPress' themes.\n\n6. You can then either select Create Blog or give WordPress Business or WordPress Premium a test run. For this exercise, click on Create Blog, but know that you can upgrade at any time. Compare plans and consider your options if your blog begins to gain traction.\n\n7. Once the blog is created, click on the Dashboard link. The Dashboard is a one-stop interface that can give you a rundown of what is happening on your blog, as well as tell you what is happening with other WordPress.com users. Statistics, Activity, and even a Quick Draft interface are offered for your blogging needs.\n\n8. Go to the top-right of your Dashboard GUI. You will see a blank profile icon. Click on the blank profile, which looks like a picture of a person. The WordPress Profile is necessary for when you want to post across the WordPress network and the Internet. The more you can tell people about yourself, the easier it will be for people to connect with you.\n\n9. Review your Account Options. Under Settings, you can edit your Account Options, your Password, and the like. Here you can review the various options and little touches that make a complete profile.\n\n10. In the Web address field, enter your website address and click on the Gravatar Profile link offered. A _gravatar_ is a globally recognized avatar, an image that people use to represent themselves online. It is a little picture that appears next to a person's name when interacting with websites. You create this profile only once, and your gravatar image will automatically appear when you participate in any gravatar-enabled site.\n\n11. Click on the Sign In icon to authorize Gravatar.\n\n12. Complete your Gravatar and WordPress profile. The best images are usually square, no larger than 600 x 600 pixels, RGB, and either saved as a PNG or a JPEG file. The images can be cropped online using the Gravatar interface. Upload your image and click on My Profile to return to your Settings and select Public Profile to fill in the details.\n\n### Up and Running\n\nCongratulations! You have an account with WordPress. Your blog is still up and running, but you have no blog posts and no content online. You're going to want to make a post introducing yourself and your future blog. This inaugural post can be as detailed and as informative as you like, but the more people who know about you and what your blog is about, the more likely you will pick up subscribers. This blog will serve as your foundation for your social media platform.\n\n## Getting to Know WordPress\n\n### Writing Your First Blog Post\n\n  1. From your Dashboard, go to Posts > Add New or simply select the New Post option from the top-right of your browser. You will notice that your Dashboard offers a few shortcuts to creating and editing posts. Use the methods that work best for you.\n  2. Single-click in the Title field and type, Welcome to My Writer's Blog.\n  3. Select the text in the Post field and replace it with the following: _This is my first blog entry. I am now officially getting my hands dirty in social media, learning how initiatives like blogging work, but I will be getting into Facebook, Twitter, and other networks as well. It's going to be a long day, but I can see now just how easy it is to write for a blog. How cool is that?_\n  4. On the left-hand side you will see an option called Categories. Click on Add New Category and create a Writing category. Go on and create three more categories: What I'm Reading, Out in the World, and Fun Facts for the Day.\n  5. Single-click on the drop-down menu for Parent Category and select the Writing category. Now create Fiction and Nonfiction categories. _Subcategories_ can be listed with parent categories in your blog posts. Categories and subcategories help search filters work more efficiently. So if you tell the WordPress search engine on your blog to show only the Writing or Fiction posts, other posts are temporarily hidden.\n  6. Select Writing and Fun Facts of the Day as categories for the post you just created.\n  7. Scroll down to Tags, and in the provided field, type in the following keywords: _blog, introduction, social media, nonfiction, what I'm doing, Facebook, Twitter_\n  8. Click the Add button. _Tags_ not only help out search filters, but they also work with search engines to find your posts easily. Use between ten to twenty tags to increase your likelihood of found (and found easily) by those searching for specific topics.\n  9. Single-click the Publish button.\n\nAnd you are blogging. Just like that. Now, a few housekeeping duties await.\n\n### Working with Pages\n\nYou have successfully created your first post. Before you move forward, we need to take care of some odds and ends that bloggers new to WordPress tend to overlook. These little touches are simple to employ and should be taken care of right away, just to avoid clutter and confusion.\n\n  1. Return to your Dashboard. If you are logged into WordPress, you will be able to access your Dashboard while viewing your blog.\n  2. In the Settings section, select the General option. In Settings, you edit your blog title, set time zones (important for scheduling), and address other details that may seem nit-picky but are important to keeping your blog efficient and reliable.\n  3. Edit your Tagline to read: _The Official Blog of Author [Your Name Here]_\n  4. Set your time zone and post formats.\n  5. Click the Save Changes button.\n  6. Select the Pages option. Click on the About title or select the Edit option, and replace the text found in the Post field with the following: _This blog is the official website of author [Insert Your Name Here]. Here you'll find my musings and occasional rants about a writer's life. You can also find me on various social networks, such as xxx._\n  7. In the title field, change the current title to: _Who Am I?_\n  8. Click on the Update button, then select the All Pages option from the Pages menu.\n  9. Click the Quick Edit option for the Who Am I? page and disable the Comments option by single-clicking the checkboxes.\n  10. Single-click the Update button.\n\n#### Post vs. Page\n\nThe difference between a _Post_ and a _Page_ is that a Post is more dynamic and lets you update your blog. Depending on your Preferences, your Posts appear on the main page of your website, categorized, and get syndicated across other RSS feeds. A Page, while still dynamic in how it is updated, is more static. It should display such information as who you are, what you or your organization is known for, how to contact you, and the like. A Page is a permanent part of your blog where information rarely changes, but it can be edited and updated, just like a regular blog post, and while you want interaction with your Posts, it is best not to have Comments active for Pages.\n\n## Creating Content\n\nNow that your blog is currently void of some outstanding \"Beginner's Blog\" traits, you can focus on letting people know a little bit about who you are in the blogosphere.\n\n### Making a Good First Impression\n\nNow that you have successfully made your first post, edited your About Page, and removed some of the other default settings to make this blog manageable, let's dive deeper into the amazing features of blogging.\n\n### Scheduling Blog Posts\n\nWordPress offers users the ability to post news and events even when they are unable to get to their computers. Blog posts, composed ahead of time, can be scheduled for automatic posting. This is a terrific feature to take advantage of when weighing busy schedules against the weekly demands of a blog.\n\n  1. Single-click the My Dashboard tab. If you are logged into WordPress, you will still be able to access your Dashboard while viewing your blog.\n  2. In the left-hand toolbar, roll over the Posts option and select the Add New option.\n  3. For the title of this blog post, type in the following: _Scheduling Blog Posts_.\n  4. In the Post field, type in the following (or something like it _): I'm not able to get to WordPress right now. Tea time, you know. So I am actually scheduling this post for a later time. This means that while I am away from the computer, my blog will automatically update. Pretty freakin' cool, huh?_\n  5. To the right-hand side of the blog interface, you see the Publish Immediately option. Single-click on Edit to access the scheduling interface.\n  6. Set the various drop-down menus to today's date, but set the time stamp thirty minutes from the current time.\n  7. Click OK to exit the timing options and then click on Schedule.\n\nYou can check back fifteen minutes later, but you will see if you click on the All Posts option that your blog post is now scheduled and waiting to go. Keep this option in mind if you are working with reoccurring features on your blog or welcoming guest bloggers to create content. You can \"collect\" posts ahead of time and schedule a week's worth of content while organizing articles for the following week.\n\n### Incorporating Images and Links\n\nWith all the details in place on the back end, let's add some details to what visitors to your blog will see. Dropping images into your blog is easy and adds a nice touch to your posts. You can use a variety of formats for images, but the preferred formats are:\n\n  * JPEG (Joint Photographic Experts Group)\n  * PNG (Portable Network Graphics)\n\nYou can incorporate images from your computer or link to images located on other websites. And speaking of other websites, let's create a link from your blog to another website.\n\n  1. Single-click the My Dashboard tab at the top-left of your browser. If you are logged into WordPress, you will still be able to access your Dashboard while viewing your blog.\n  2. In the left-hand toolbar, roll over the Posts option of your browser and single-click on the All Posts option.\n  3. Single-click the Scheduling Blog Posts post.\n  4. Place your cursor at the beginning of the post and single-click the Add Media button.\n  5. Single-click the Insert from URL option and enter in the following options:\n  6. **Image URL:** http:\/\/teemorris.com\/images\/wordpress.jpg\n  7. **Alt Text:** Tea Time with WordPress\n  8. **Alignment:** Left\n  9. Single-click the Insert into Post option.\n\n### Bookmark\n\nWhen you link your blog post to an image located on a server other than your own, a preview of the image will appear in this WordPress window. This is an indication that the image link is good.\n\n  1. Once the image appears in the blog post, single-click the image and then click the Edit Image icon at the top-left corner of the image.\n  2. In this interface, click on the Advanced Settings tab. Enter 10 (pixels) in both the Horizontal and Vertical Space, and then click the Update option.\n  3. Highlight WordPress in your blog entry.\n  4. Single-click the chain link icon. In the URL field, enter: http:\/\/wordpress.com.\n  5. Click Update to accept the URL for the link.\n  6. Click Update to make changes to your blog post live.\n\nYou can now add visual media to your words. Visuals assist in getting your message across. Keep in mind that media of any kind should help, not hinder, the message of your blog. Adding an array of unrelated images might steer your audiences off target, so be certain that any accompanying photos and media relate directly to the intent of your post.\n\n# Appendix B\n\n# Tumblr\n\n## Setting up a Tumblr Account\n\nTumblr is one of the easiest social media platforms to work with. To set up an account, follow these instructions:\n\n1. Go to www.tumblr.com, enter a valid e-mail address, and make a username\u2014keep it simple, using your author name is a good idea\u2014and create a password.\n\n2. Once you tell Tumblr how old you are, accept the terms of service, and prove you are not a robot, it will create an account for you.\n\n3. Tumblr will then ask you what you are interested in and prompt you to start following five blogs. You can follow some of the suggested blogs or search for others using terms such as \"writing\" and \"authors,\" or type in your genre. The more you follow, the more you'll feel a part of the community, so feel free to select more than five.\n\n4. You will have to confirm your e-mail address in order to customize your account.\n\n5. Click on the head icon in the top-left corner and then click on Edit Appearance.\n\n6. Let's make your Tumblr look more like you want it to. Use an avatar, and then tweak the color scheme. Title and description are optional but useful.\n\n7. Next you can download the Tumblr app for IOS, Android, or a Windows Phone. You don't have to do it now, but it is a great app to have at your fingertips.\n\nAnd _voila_ , there you are in your Dashboard. Don't panic: This is not how the viewing public will see your Tumblr, but it is where you will be spending some time managing and sprucing up your account and blog.\n\nThough this initial setup generates your primary blog, you can in fact have multiple blogs in the form of secondary accounts, and\/or add pages to any of these blogs. You might want to consider using your primary blog as your writer account, with different pages for your series. Or you might want to have secondary blogs for your series, especially if you are co-writing them. Again, you may be splitting your brand too much, so decide how fractured you want your Tumblr content to be.\n\n### Bookmark\n\nThough you can have many of these secondary accounts, remember that they can't follow other blogs, like posts, ask questions, or submit to other blogs. For this reason, make sure your primary account represents the right brand.\n\n## Getting to Know Tumblr\n\nLet's go ahead and get familiar with the Tumblr controls. There are several key areas.\n\n### The Top Bar\n\n  * **Home.** This will take you back to the main dashboard and your stream.\n  * **Explore.** This will bring you to the browse section of Tumblr. Here you can see trending topics, categories, and staff picks. It is a great place to find blogs to follow.\n  * **Mail.** This is where you will get messages, questions, submissions, and fan mail from your Tumblr followers. Keep an eye on this icon to alert you to these valuable interactions.\n  * **Lightning Bolt.** This will take you to your activity page, where you can see how engaged people are with what you are posting, your biggest fans (people who engage or repost your posts the most), and the latest comments on your posts. It's fun to watch the graphs here.\n  * **Head Icon.** This is the home base for all of your content. This shows you how many posts you have made, how many followers you have, what activity has occurred since you last checked (along with a neat chart of how your posts are doing), and how many items you have lined up in your queue. Most important, it offers the ability to customize your page. Clicking on the head icon will allow you to change the appearance of your page. Much like WordPress, Tumblr has themes\u2014there are frameworks for the pages so you can easily set up the look of your blog. Some are free, and some are available to purchase, but there are so many themes that finding something for yourself and your genre is a definitely possibility. In the Customize menu, you will see what your Tumblr looks like to the public. On the left, you will see your options. These will vary according to the theme you have selected. You can use this to add static pages to your Tumblr. You might, for example, want to have a separate page for an About page or perhaps a list of events you are attending.\n  * **The Pen.** Click this for all the options you need to post content. The next bar is our favorite.\n\n### The Content Bar\n\nThis is where you get to do the fun stuff (post your own content to Tumblr!). The tools here are fairly self-explanatory: Text, Photo, Quote (great for dropping samples of your work into your feed), Link, Chat (so you can ask your followers questions, a great way to encourage engagement), Audio, and Video.\n\nEach of these offers different advanced options, but the most useful ones are Schedule and Add to Queue. These are some of Tumblr's best features for you as an author, since they allow you to target a post to go live at an important moment\u2014say on a book launch date\u2014or pop a post in a feed to be delivered on a regular schedule. Remember your readers like consistency with their content!\n\nNow that you have your Tumblr up and running the way you want it, let's fill it with some content.\n\n## Posting to Tumblr\n\nAdding your own content to your Tumblr stream is always the preferable option, so let's take a quick look at the ways you can do it:\n\n  * **Text.** Give it a title, and write your content in the field at the bottom. You can also format your text in a variety of ways, including making those all-important links back to your site or where to buy your book.\n  * **Photo.** Here you can upload up to ten photos, or take one directly with the camera in your computer. Remember: Images are your best friend on Tumblr.\n  * **Quote.** Use this to add little teasers from your work-in-progress or novel that's about to release. Make sure they are punchy quotes that express the content of your book or main character.\n  * **URL.** This is where you link to your WordPress site or to reviews of your books. Unfortunately, you can't use an image with this option, so you might want to use it sparingly.\n  * **Chat.** Use this to solicit commentary and that important engagement. It can also be used to chat with your readers and ask them their opinions or about favorite characters, etc.\n  * **Audio.** This is where you would post podcasts, or recordings of your story. Samples of your audiobook would be an excellent choice to upload\u2014just remember to keep them up to 10 MB and in MP3 format. You can search for a track to post, upload directly, or share from a URL. Do be sure to fill in all the fields and to have a good cover image for maximum impact.\n  * **Video.** If you have a book trailer, this is where you can upload it directly (if it's 100 MB or less), or you can put in the code (say from YouTube) or the URL where the video can be found. The same rules for filling all the fields apply.\n\nNo matter what you upload, do not forget to supply the relevant hashtags to draw attention to your post.\n\n## Reblogging\n\nSometimes you just don't have enough original content to feed the hunger of Tumblr, or you want to spread out what you do have. In our Ministry of Peculiar Occurrences Tumblr, we do three reblogs to one original post. Using this ratio, you will avoid constantly shouting \"buy my book\" into the void.\n\nReblogging also offers the advantage of building a following. Everyone whose post you reblog sees that you've reblogged their post, after all.\n\nSo let's have a go at reblogging some content.\n\nThe blogs you are following are now showing up in your dashboard in a stream, with the newest at the top. Find something useful and relevant to your audience.\n\nAt the bottom of the post are a few icons. The three dots on the left allow you to share the post on other social media networks, flag it, e-mail it, embed it, or get the Permalink (a permanent link to the post). On the far right is the Heart icon, which is how you Favorite the post. It's a nice way to show a sense of community and will help your account get noticed, but reposting is even better.\n\nThe icon that we are really interested in is the middle one, which has two circling arrows. This is the icon that will post the item into your Tumblr feed. Clicking on it will bring the post you want to reblog up in full. Here you can add your own tags, put the post into your queue, schedule it, or post it right away. We would urge you to add your own commentary to the item before you post, just to let people know that you are not some robot mindlessly reposting items. For our steampunk series we usually try to say something that will link the item back to the series. If we see a vehicle the Ministry might use, we point that out. The point is to bring it back to your brand without hijacking the content. Always remember to leave whatever attribution there is on the post intact.\n\n# Appendix C\n\n# Podcasting\n\n## An Introduction to Podcasting\n\nUnlike most other social media outlets, podcasting requires specific hardware and software, which can be a daunting prospect for people, especially those who have never shopped for audio equipment. Where does one begin?\n\nNothing to worry about: We'll walk you through it.\n\nFirst of all, an audio studio can come in a variety of shapes, sizes, and price tags. Your own setup can be kept economical and remain within a modest budget, or you can invest in something worthy of Audible.com. Whatever setup you envision for your podcasting endeavors, there are plenty of affordable options out there.\n\nLet's begin with what kind of microphone is capturing your words. Microphones come in various shapes, sizes, and makes, all with varying benefits. Then you should determine the following:\n\n  * **How much can you afford to spend?** Like everything in life, you get out what you put in, so the more money you spend, the better sound quality you can get out of your microphone. Consider the quality you need, before you put down your dollars on a microphone.\n  * **Are you planning to use the microphone in your studio or out on location?** A fancy, high-end shock-mounted model isn't going to cut it if you are doing an on-the-street or off-the-cuff podcast. Trying to make do with a lapel mic will not give you the quality of sound expected in an Audible-quality recording. Consider your strategy and where you want to take your podcast\u2014indoors or out.\n  * **What exactly do you want this mic to do?** Are you recording fiction, conducting interviews, or just kicking back with a fireside chat? Your choice of microphone rests on what you want to accomplish, so consider how you want to use your microphone.\n\nWith the answers to these three questions, your options at Amazon, BSW, and Guitar Center narrow a bit, but you still face many variations of microphones to choose from.\n\nWhat are you looking for?\n\n### Affordable Microphones\n\nMost affordable mics are Universal Serial Bus (USB) powered, and designed to plug into your computer and start recording moments after unboxing them. USB mics are available where electronics are sold.\n\nSome USB microphones that are popular with podcasters include:\n\n  * **Blue Snowball.** This microphone can be found just about everywhere, ranging in price between $50 to $100. The quality of audio from the various models of Snowball is good. These are great starter microphones for authors entering the podcasting realm.\n  * **Blue Yeti.** From the makers of Snowball, the Yeti is a step up, usually listed in the $150 range. The Yeti offers a built-in headphone amplifier and controls on the device for headphone volume, pattern selection, mute, and microphone gain. A THX Certified Microphone, the Yeti captures a wider range of audio and vocal subtleties.\n  * **Marshall MXL 990 USB.** We use the MXL 990 condenser microphone for both desktop and remote recordings, so we can vouch for its abilities. The USB cousin works without mixers, preamps, or special studio gear. Simply plug the microphone into your computer, and you can create pro-quality recordings. With a gold-sputtered diaphragm and a high-quality FET preamp, the MXL 990 USB comes in around $150. Guitar Center's MXL 990 USB kit, which includes a tripod, headphones, mic stand, and proper cables, sells for about $250. It's everything you need in just one purchase.\n\n### Bookmark\n\n\"Portable podcasting\" faced many limits once upon a time, as small, built-in condenser mics struggled to offer quality of any kind. Now podcasters can take all their recording solutions on the go without fail with the Zoom H2n and H4n. These are lightweight, unobtrusive, all-in-one solutions for portable-podcast recording. Ranging in price from $150 to $200, both the H2n and H4n record audio directly to SD cards or can plug directly into your computer. With their built-in microphone through USB, they can record directly to MP3 format (in a variety of bit rate compressions) and to WAV format (also in a variety of bit rate settings). H4n goes even further, allowing for multitrack recording.\n\nThe H2n and H4n are fantastic solutions for professional audio producers, as well as for beginning podcasters who need a complete startup package.\n\n### Investing in High-End Microphones\n\nFor a cleaner sound for your podcast and the potential of breaking into professional-quality audiobook production, $50 to $100 mics will not deliver what you need. What truly makes a microphone is how you sound in it and how it reproduces the sound coming in. Based on the technology of the microphone itself, and how well it reproduces your voice, prices vary, but as you can see from our recommendations, plenty of high-quality microphones, able to pick up nuances in the human voice, fall into the range of affordability.\n\n### Bookmark\n\nWhen you consider the cost of a microphone, keep in mind that it likely does not come with the cables you need to hook it up, a jack to fit into your computer, or a mic stand.\n\nWelcome to the world of professional equipment.\n\nWhen budgeting for audio equipment, remember to keep in mind all the accessories you will need to pick up along with it.\n\n  *  **Shure SM58.** When it comes to signature microphones\u2014ones like you see everywhere\u2014if you want that professional sound without the professional price tag, look no further than the Shure SM58. For nearly fifty years, this mic has been an industry favorite because of its durability and reliability. The SM58 works like a charm, recording on the street or in a studio. Podcasters love it for its affordability (it's available for $100.\n  * **Shure SM7B.** Another industry standard that's found in many radio stations across the United States and around the world is the Shure SM7B. For $350, it is nothing less than an investment. This particular mic is optimized for in-studio recording, which is why it is used in so many radio stations and recording studios, but the SM7B, like the SM58 and most USB mics (not the MXL 990USB), is a _dynamic_ microphone. These types of microphones do not need any power from whatever mixer or preamps they are using in order to connect with the computer or recording device. They also record audio differently than other mics (which we are about to cover), and therefore capture different vocal nuances. Of the various microphones covered so far, the SM7B is the best of the bunch, offering you Audible-quality audio.\n\nSo far we have looked at dynamic microphones\u2014durable, reliable microphones of varying cost and quality. Recording in the studio, though, is all about the details, and when you want that detail, it's time to shop for a _studio_ _condenser_ microphone. Condenser mics work very differently than dynamic ones. First, they are _phantom powered,_ which means they receive an extra little electrical kick that helps them take in audio at their full potential. Phantom power comes from the mic's connection to the mixer, which supplies a constant boost to the condenser's sensitive diaphragm in order to catch details that dynamic mics miss. Condenser mics are also different because their delicate internal design makes them best for a stationary setup. If jostled around, plates inside the mic's housing can be knocked out of whack or damaged, causing problems in the pickup of the audio. The advantage to condenser microphones is that they pick up a wider spectrum of audio.\n\n  * **Marshall MXL 990.** As mentioned earlier, the 990 USB was the exception of those other USB models due to its being a studio condenser model. Not only is it available as a USB, but it also comes as a standard studio condenser microphone, a favorite of ours. Featuring the same gold-sputtered diaphragm as its USB counterpart, the standard MXL 990 comes in at around $100, complete with shock mount and carrying case. It's an inexpensive microphone with a bright sound, and it's worth every penny.\n  * **AKG Perception 120 and 220.** We are big fans of the AKG Perception line and use it to this day in our _Shared Desk_ podcast. The Perception 120 is a fantastic investment for just under $100, while the 220 comes in at $150. Both mics offer superior pickups and brilliant, rich audio quality. What make the Perceptions terrific investments are their built-in switchable attenuation pads and low-cut filter, which dampen any background noise in your home studio.\n\n## The Mixing Board: Bridging the Gap Between Microphone and Computer\n\nIf you're looking to have guests in the studio, or want to work with multiple inputs of audio or take advantage of phantom-powered studio condenser microphones, you will want to invest in a _mixing board_. What a mixing board (or mixer) does for your podcast is open up recording options such as using multiple microphones, recording acoustic instruments, and balancing sound to emphasize one voice over another or to balance both seamlessly.\n\nThe easiest way to look at a mixing board is as if you're partitioning your computer into different recording studios. But instead of calling them _studios,_ these partitions are called _tracks_ , __ and you can use any of those tracks for input _or_ output of audio.\n\nThe easiest USB mixers to set up and get running on your computer cost $70 and go up from there. The cost increases with each additional track. The real advantage of a mixer is the ability to adjust various audio levels of multiple inputs independently in order to make the final signal sound even. Mixers also offer you the option of working with multiple mic,s so if you have more than one speaker, you won't have to huddle around the same microphone or slide it back and forth as you take turns speaking.\n\nMixers can look a little intimidating with all those wacky knobs running up and down the interface. Some of the knobs deal with various frequencies in a voice and can deepen, sharpen, or soften the qualities of a voice. The knobs that are your primary concern are the ones that control your volume or _levels,_ as the board labels them. The higher the level, the more input signal a voice gains when recording.\n\n### Hooking Up a Mixer to Your Computer\n\nOnce you have a section of your desk cleared off and reserved for your mixer, open the box containing your new mixer and make sure everything that should be there is there:\n\n  * USB mixer\n  * Power supply\n  * USB cable\n  * Instructions\n\nBefore hooking up anything, check the manufacturer's website for downloads (new drivers, upgrades to firmware, patches, and so on) needed for your digital board to work at its optimum level. USB mixers are so simple to set up that we can give these steps for both Mac and Windows machines:\n\n  1. Shut down your computer.\n  2. Connect the power supply to the back of the mixer and to an available wall socket or power strip.\n  3. Find an available USB port and plug your mixer into the computer. If your computer's ports are maxed out, we recommend setting up a USB hub for your other devices in order to free up one port. We do not recommend running your mixer through a hub, as doing so might affect the quality of the audio.\n  4. Plug the USB cable into the back of your mixer.\n  5. Connect your input devices (microphones, headphones, and the like) to the mixer.\n  6. Power up your mixer by turning on the Main Power and the Phantom Power switches.\n  7. Start your computer.\n\nAnd that's it! You're ready to record with your USB or FireWire mixer. Now, with headphones on your head and some toying around, you will be able to set levels on your mixer that are good for recording.\n\n### Bookmark\n\nThe earlier mentioned H4n and the recent Zoom creation H6n, when hooked up to your computer, also double as _preamps_. The preamp works as a basic connection between your mics and your computer, and it provides phantom power to your condenser microphones. If you don't feel the need for a mixer or you just want to keep it simple, you may want to consider the all-in-one approach with the Zoom H4n or H6n.\n\n## What's Next: Other Details for an Audio Studio\n\nA microphone and mixing board are the necessities in building a home audio suite, but there are a few more items you'll want to complete your recording studio. These add-ons are not essential to audio recording but can help you in producing a rock-solid audio production:\n\n  * **Headphones:** Headphones help you monitor yourself as you record. By hearing your voice, you can catch any odd slip ups, slurred words, or mispronunciations before playback. The best kind of headphones are _closed-ear_ headphones, and the _Sennheiser HD202_ closed-ear headphones for around $20 offer everything you need for clean, clear audio monitoring.\n\n### Bookmark\n\nWe have been asked often about Bose Noise Cancelling stereo headphones for recording purposes. Bose makes an excellent pair of headphones, offering the highest quality for audio listening. Note, we say _listening._ Not _recording_ and _editing._ Any headphones you see listed as \"noise-cancelling\" are going to be terrific for listening, but they aren't the best to use for recording and editing. You want to be able to hear the noise that these headphones cancel out so that you can eliminate it before a word is recorded. The best noise reduction and elimination happens before and during recording, not afterward. For podcast production, stick with closed-ear headphones sans noise-cancelling features.\n\n  *  **Cables:** As mentioned earlier, high-end microphones may arrive without cables. Check with a sales representative or the mic's documentation to find what cables you need before you buy. Many of the microphones suggested here use a _3-pin XLR-to-XLR male-to-female_ cable. The female end connects to the mic, and the male end, to the mixer. These cables begin at $9 and go up from there, depending on the length of the cord and quality of the input.\n  * **Microphone Stands:** If you invest in the Marshall MXL 990 (USB or XLR), you will receive a shock mount and an attachment for a mic stand, but no microphone stand. Believe us when we say that a good mic stand is worth the investment. A basic _desktop mic stand_ can run you around $10, while a _boom mic stand_ starts at around $100. The type of mic stand you invest in depends on how you want to work around it. With the simple stand, you're all set and ready to go without the extra hassle of positioning and securing a boom stand to your desk. The boom stand, though, frees up space on your desk, allowing for show notes and extra space for you to record and mix. Consult your budget to see which models are within your range.\n\nYou have the basics in place and the strategy to move forward. Now it is time to record, edit, and podcast.\n\n# Appendix D\n\n# Facebook\n\n## Setting Up a Facebook Account\n\nIf you're not on Facebook, it's time to get on it. There are over 864 million active users. As an author, you need to use the platform to promote your work. So let's get you set up.\n\n1. Go to Facebook.com and fill in all the fields to set up an account. Make sure you use your proper _author name_ so people can find you. Be aware that Facebook's official rules say you should use your \"authentic name.\" This has proved to be problematic with authors who use pen names or multiple names for different genres. Some authors choose to operate two profiles: one for their personal contacts and one for their professional contacts. Others have a personal account from which they operate a Page as a professional author.\n\n2. After you've filled in these fields, Facebook will try to find your friends. It will use Gmail, Outlook, Yahoo!, and other e-mail services to do so. You can skip this step if you are not comfortable doing this. You can always search for more friends later on.\n\n3. Upload a profile picture.\n\n4. Confirm your account. You are a new member of the Facebook family!\n\n## Getting to Know Facebook\n\nFacebook has many parts. It is, after all, one of the biggest and oldest social media platforms available. As you join and make Groups and Pages, it will get deeper. For now, let's look around this new platform.\n\nOn the top blue navigation bar, you will find all the information you inputted about yourself.\n\n  * **Facebook Icon.** This will bring you back to your main page.\n  * **Search Bar and Icon.** Here you can look up people and Pages you might be interested in.\n  * **Avatar.** Clicking on this will take you to your profile. Here you can make changes and add your Likes.\n  * **Home.** Much like the Facebook icon, this button will take you back to your main feed and Page.\n  * **Find Friends.** Facebook wants you to make friends. Here you can search through your other social media contacts or look for friends by where you worked and went to school. Facebook will try its best to suggest people you might know.\n  * **Two People Icon.** This is where you will find friend requests.\n  * **The Speech Bubble.** Here you can read and send messages to other users.\n  * **The World Icon.** This is where all of your notifications will pop up. For instance, if someone mentions your name in a post, you will be notified. You will also find replies to your messages and posts, posts from Groups you are following, and Likes you got for your posts\n  * **Padlock Icon.** Here is where you can set who can see your posts and who you'll allow to contact you. This is where you will also find ways to stop people if they are bothering you.\n  * **Down Arrow Icon.** Here is where you go to set up new Pages and Groups, as well as to choose other settings.\n\n## The Left-Hand Sidebar\n\nHere you will find some of the same links that are in the blue top bar. You will also find Events and your Pages and Groups here. Let's set up a Group now.\n\n### Setting Up a Group\n\nNow that you have your account set up, let's look at _Groups_. Groups are communities built around a cause or common Like. Groups can be built for anything that people might have in common\u2014including a genre such as science fiction or fantasy. The content appearing in Groups is community driven, and everyone shares a voice here.\n\n1. Log into Facebook and go to your News Feed. You do this by clicking on the Facebook logo in the top-left corner or by clicking on News Feed under Favorites. This is located in the top-left corner of your Facebook Page.\n\n2. Look for the Groups section in the left-hand bar of the News Feed and then find the Create Group option.\n\n3. Name your group.\n\n4. Invite Facebook members to join your group. (This is optional, of course.)\n\nWhen you are establishing your Group, Facebook encourages you to draw from your own personal network. Before inviting people to your Group or a Page, let them know ahead of time, as a courtesy. Send them a private message and ask. Most people take offense when added without permission.\n\n  1. Choose the Privacy option you want your Group to have:\n  2. **Open.** Anyone can see this Group, who's in it, and what members post.\n  3. **Closed.** Anyone can see the Group and who's in it, but only members see posts.\n  4. **Secret.** Only members see the Group, who's in it, and what members post.\n  5. Click on Create to establish your Group.\n  6. Choose an icon to represent your Group.\n\nAnd your Group is now online. Just a few finishing touches, and you'll be ready to invite and welcome people to this new community.\n\n### Completing Your Group's Profile __\n\n1. You will be taken to the Group's Page straight away, but in case you aren't, you can type in the name of your Group in the search bar. In the Group's Main Page, find the three periods (...) icon at the right-hand side of the Group's menu and click.\n\n2. Select from the drop-down menu the Edit Group Settings option.\n\n3. Complete the set up your Group:\n\n  1. **Group Name.** A custom icon can be assigned to your Group. Your name can be changed, but it is set once your Group grows to more than 250 members.\n  2. **Privacy Settings.** Available only for larger Groups\n  3. **Member Approval.** Used for moderation of new Group members\n  4. **Group Address.** An option for establishing a Group e-mail and URL for Facebook\n  5. **Description.** Where you offer a greeting and brief description for your group, its aims, its policies, and its dos and don'ts\n  6. **Tags.** A great way for Facebook members to easily find you\n  7. **Posting Permissions.** Another moderation option when it comes to the method of posting on the Group board\n  8. **Post Approval.** Another moderation option when it comes to the method of posting on the Group board\n\n4. Click the Save button.\n\n5. Return to the top of your Group and set an appropriate Group Photo for your welcome banner.\n\nYour Group is now complete, and up and running. All you need are members. On your next few posts that cross the line into business, mention that you have a Group and encourage people to join. Even before members join, have a few topics online and ready for discussion.\n\n### Starting a Group Topic Discussion\n\n  1. Go to your Facebook Group. At the top of your page, just under the introduction banner, you should see this inviting field: Write Post.\n  2. Click in this field and write the following status message: _Welcome to my Facebook Group. This is where you will find all news and updates of my upcoming appearances and developments. The rules here are simple: Don't spam, and be excellent to one another. If you cannot do this, one of our moderators will promptly block you from the group. We hope you contribute and adhere to the positive spirit of this Facebook Group._\n  3. Click on the icon of the camera if you wish to upload an accompanying photo. (If you want to upload a different photo with your message or no photo at all, roll over the image and single-click the X in the top-right corner of the image.)\n  4. Click the Post button.\n\nThe most recent post in your group will appear at the top of your Group. This will always be the case until someone in the Group leaves a comment. The topic with the most recent comment will jump to the top of the posts, taking priority until a new topic is posted or another comment is left elsewhere in the various threads available here.\n\nNote that you can upload photos and videos, ask questions of your Group members in poll fashion, or add a file (PDF, Word document, etc.) to share with your membership. Your content does not need to be just general discussions. As moderator, you can also decide which posts stay and which posts are deleted, and approve or dismiss members.\n\nTherein you have the basics of a Group. Now, what about a Page?\n\n### Setting up a Facebook Page\n\n1. Go to your Facebook News Feed or to your Personal Page.\n\n2. Single-click on the downward arrow icon in the upper-right corner and select from the drop-down menu the Create Page option.\n\n3. From the Create a Page introduction page, select the kind of Page you wish to create.\n\n  * The features for each of these options may differ slightly, but the basics\u2014the Admin Panel, Recent Posts by Others, posting options, etc.\u2014are all the same. The options offered by Facebook include Local Business or Place; Company, Organization, or Institution; Brand or Product; Artist, Band, or Public Figure; Entertainment; and Cause or Community.\n  * These options help Facebook categorize what kind of Page you are managing for search purposes.\n\n4. Based on your selection, fill in the appropriate information for your Page. This is a professional page, so share as much information as you deem necessary for conducting business. If this is a home business, do you really want to post your home address? Many authors maintain a private mailbox for this reason.\n\n5. Click the checkbox for Facebook's Terms of Service. Perhaps it is not the most engaging read online, but review Facebook's Terms of Service in order to be clear on what is and isn't allowed, and how your content can (and cannot) be used by Facebook.\n\n6. Click the Get Started button to launch your Page.\n\n7. You will receive an e-mail welcoming you to Pages while you are rerouted to a new website that will walk you through the process of completing a Page's profile. Some of these details can be edited after the Page is live, but complete the options here in order to get the basics up and running.\n\nAt any time, you can delete a Page by going onto the Page itself. Click on Settings, and at the bottom of the list of settings you will find Remove Page. Click there. Facebook will ask if you really want to remove it. If you click again, there is no way back, so be sure that's what you want to do.\n\nNow, your new Page is live and running. No one yet knows it's out there, mind you, but we will get into building your Page Following in a moment.\n\nLet's break down a complete page and discuss how to create a unique look, as well as what a quality page looks like.\n\n#### Posting on a Page **Bookmark**\n\nIf you have more than one moderator for your Page, it is not a bad idea to end your post with initial signatures. As I do with One-Stop Writer Shop, I moderate this Page with Pip. Therefore, when I make a posting, I will sign it with (TM) so that the reader knows who is posting the item.\n\nHere is an example of how to create a simple post to Facebook:\n\n  1. Go to your Facebook Page. At the top of your page, just under the header, you should see a field that says: \"What have you been up to?\"\n  2. Click in this field and write the following status message: _Not all Kickstarter accounts succeed, as Kickstarter is more than just crowdfunding. It is also a place for market testing and hard business lessons. (TM)_\n  3. Hold down the Shift button and then, with the Shift button still down, hit Return twice.\n  4. In this space, type the following link: http:\/\/teemorris.com\/2014\/03\/16\/lessons-learned-kickstarter\/.\n  5. Once the link preview appears in the post, select the link and empty space underneath your post, and remove it. This is just a nice way to clean up posts appearing on your Page in order to avoid double posting a URL. By entering Shift+Return, you do not prematurely make the post live on your Page or in your Feed.\n  6. Click the \"Post\" button.\n\n### Bookmark\n\n**NOTE:** If you have liked other Pages, sometimes you are offered options of names, places, and businesses that match up with what you are typing, even without adding the @ to what you are typing.\n\n#### Sharing a Post\n\n  1. On your Page, click on the Share option for a post. When you Share a post on a Facebook Page or in a friend's feed, not only will the post appear in its entirety on your friend's Facebook feed, but a link back to the page of origin will also appear on that feed, offering people a direct connection to your Page.\n  2. Click in the Post field and begin writing the following status message: _Here's a reminder that @One_. You should see a drop-down menu appear options for One-Stop pages. From the drop-down menu, select One-Stop Writer Shop from the offered Pages. In the field where you are composing your accompanying post, you will see a highlighted One-Stop Writer Shop link in the field.\n  3. Finish writing the status message: _...will be appearing in Richmond two nights after the release of DAWN'S EARLY LIGHT. Pick up your copy, attend the James River Writers_ __ _' WRITING SHOW, and get it signed!_\n  4. Single-click Share Link to make your post live.\n\n# Appendix E\n\n# Twitter\n\n## Setting Up a Twitter Account\n\nWelcome to the first step in building your Twitter network. Twitter developers want to help you through this process, as you will see when you set up your account.\n\n### Bookmark\n\nWhen coming up with a username for your Twitter account, it is a good idea to think _compact_ and _easy to type_. Replies on Twitter tend to be shorter for those with longer usernames (defined here as anything over ten characters), as the 140-character limitation does include the username.\n\n1. Go to http:\/\/twitter.com using your browser.\n\n2. Just underneath the New to Twitter? question, click Sign up.\n\n3. Enter your name, an e-mail, and a password. Once the fields are complete, single-click the \"Sign Up for Twitter\" button.\n\n4. Review what is listed for your Twitter credentials and single-click the \"Create My Account\" button.\n\n5. You will be asked to supply a phone number, which you can skip if you like.\n\n6. You will be taken through a process in which Twitter tries to learn more about you by asking you what you are interested in. Then it will try to find followers for you.\n\n7. After this, you will customize your profile, or you can skip this step and come back to it later. But you don't want to remain a Twitter egg for long.\n\n8. Next, Twitter will ask you to find people you know, which you can do using your e-mail contacts. You can also skip this step.\n\n9. Now you will be taken to your home page, with a pop-up prompting you to confirm your e-mail. Do it.\n\n10. If you have followed people in the previous steps, you will see their recent tweets pop up in your feed. A _tweet_ is a message shared on the public Twitter stream. The _Twitter stream_ shows the incoming messages from your network, and it appears on your public Timeline.\n\n11. Click on the camera icon on your profile and upload a snappy shot of yourself.\n\n12. Your Twitter account is now live and ready to go.\n\n## Getting to Know Twitter\n\nWelcome to Twitter! You could just head out from here, but there are other little touches that you can add to your Twitter account on your own time, and the following are recommended:\n\n  1. To access details in the previous image, go to your Twitter page by clicking on your profile in the top-left corner.\n  2. Now click on Edit Profile on the left. Here is where you spice up you profile. You should fill in your bio, location (something as simple as your state is fine), and, most important, your website. You can also tweak the theme's color.\n  3. Don't forget to add a header photo for the space behind your profile picture. This area can be easily customized. Whatever image you place here will be seen by all who pull up your profile, in all Twitter apps.\n\n### Bookmark\n\nIf you have difficulty uploading an image for Twitter, the photo's size or resolution may need to be reduced, or it could be the wrong format. Profile pics must be no larger than 700K and no larger than 600 x 600 pixels in dimension. Header images should be 1252 \u00d7 626 and no larger than 5 MB. Backgrounds can vary, but as they tend to be tiled when smaller than the screen resolution, we recommend using images that are 2000 x 2400. All of these images should be 72 pixels per inch (ppi), saved as either a JPEG or PNG, and use the RGB scale.\n\n## Talking on Twitter\n\nSo, if you haven't used Twitter before, you're probably thinking \"What about that whole 140-characters-or-less limitation? You really can't say a lot, can you?\"\n\nThis segment is less about unleashing your inner social media geek and more about embracing your inner editor.\n\n### Posting Your Tweet\n\nWhen you have the perfect tweet, what do you do with it? Well, you go on and you release it into the world, of course.\n\n  1. Go to Twitter using your Internet browser. If you are logged in and you told Twitter to remember you, you should still be logged in. (If not, go ahead and log in.) Depending on where you are in your Twitter account, you have two places where you can compose your tweet. You can click on the Home icon of your Twitter home page and use the Compose window offered in the left-hand sidebar or click on the quill icon in the upper right of the menu bar to access a separate Compose window.\n  2. In the Compose New Tweet window, enter in something like the following: _This is my first tweet. Thanks for the help on this, @1SWShop and @JamesRvrWriters. #rva #masterclass_\n  3. When you have your tweet composed and you checked for typos and grammar errors, you are one click away from communicating with your community and with the Twitter network.\n  4. Single-click the Tweet button. Your screen automatically refreshes, and your message appears at the top of your visible Timeline.\n\nYour message is now out in the Twitterverse for those in your network and (depending on the security you have in place) the Public Timeline to see.\n\nType. Proof. Post. That's all there is to it!\n\nNow that you are effectively tweeting, the next step is to respond to your first tweet and understand how to talk to another person on Twitter.\n\n### Replying to a Tweet\n\nAfter you post your tweet, someone might tweet back, \"Who is Tee Morris, and what does he have to do with Twitter?\" but the tweet will appear in your Timeline like this:\n\nTee Morris @TeeMonster\n\nYou're taking a master class in Richmond? Where? And who is @1SWShop and @JamesRvrWriters?\n\nThis is a reply or a _Notification_ on Twitter. A notification begins with a user's avatar, the replier's username, your username preceded by an @ symbol, and finally the tweet. By adding in the @ symbol and username, the mention is recognized by the Twitter home page (and other third-party applications) as a reply to you.\n\n  1. Whether you are in the Notifications mode or your Timeline window, find your mention. Single-click on the arrow labeled Reply to open a tweet field.\n  2. Enter your reply by clicking to the right of the Twitter handles. If you want to mention the other Twitter accounts in your reply, reposition them accordingly. Your reply, after editing, should look like this: _@TeeMonster Tee Morris is one of the @1SWShop founders and @JamesRvrWriters is hosting a Social Media master class._\n  3. Single-click on the Tweet button.\n\nYou'll be notified that your reply was sent. Notifications provide a quick connection to those following you, and you can give an @Mention to anyone on Twitter, even those outside your network.\n\n### Bookmark\n\nUnder Notifications, you will see two available options: _Interactions_ and _Mentions_. While Mentions reveals all of the replies your tweets are receiving, _Interactions_ is a more thorough look at the activity your Twitter account is experiencing in the network. This mode will also give you indications of who has started following you and when your tweets are _retweeted_ across the network. (An explanation of the retweet is coming next.)\n\nAnother form of replying is called the retweet. With a single click, you copy another Twitter account's tweet or reply. This kind of response is either preceded by an _RT_ : or it appears in your Timeline a second time with the title \"Retweeted by [Twitter Handle]\" above the original tweet.\n\n  1. In your Home Timeline window, find a tweet you want to share in your network. Single-click on the looping arrows labeled Retweet to open a Retweet window.\n  2. Single-click the Retweet button.\n\nYou should notice the Retweet option is now green and reads as retweeted in your Timeline.\n\nSome applications, both desktop and mobile, will offer a second option to _quote_ a tweet. This option allows you to add your own commentary into a retweet, if there are available characters remaining. Here's an example:\n\n@1SWShop \"@TeeMonster Tee Morris is the writer of All a Twitter and the teacher of the Social Media Mainline.\" He knows his stuff!\n\n **OR**\n\n@1SWShop RT: @TeeMonster Tee Morris is the writer of All a Twitter and the teacher of the Social Media Mainline. (He knows his stuff!)\n\nBy using quotes around what you are retweeting, or parentheses around what you are adding, you make sure to add your own spin on what is being said. Retweets are used to spread a message, a link, or just something really cool throughout the Twitter network.\n\n## Direct Messages\n\nA feature that has landed a few people in trouble (in that they didn't understand how Twitter operates), and that Twitter has worked hard to make foolproof (although fools still rush in), is the _Direct Message_ , or _DM_ as it is referred to in the Twitterverse. When you receive @Mentions, you catch them either in your network's Timeline or in your Notifications. Direct Messages are different. DMs are seen by you and only you (unless someone is looking over your shoulder, of course), and are found under _Messages_.\n\n### Bookmark\n\nUnlike @Mentions, which can be made to anyone in the Twitter network, Direct Message replies can be sent only to people who are following you.\n\n### Composing a Direct Message\n\n  1. On your Twitter home page, single-click the Messages option, the envelope\/speech bubble icon located after the Home and the Notifications icons.\n  2. Type in the Twitter name of the person you want to send a message to and click Next.\n  3. In the field at the bottom of the DM window, type in your message: _Hi there. Thank you for the follow. I love the class so far._\n  4. Single-click the Send Message button.\n\nThe DM you sent appears when you click on the envelope\/speech bubble icon. Now, the sender has been notified and you simply await a reply. To reply to a DM, the process is even easier.\n\n### Replying to a Direct Message\n\n  1. On your Twitter home page, check the envelope\/speech bubble icon. If a number appears, it means you have unread DMs. Single-click the icon.\n  2. The conversation will appear. Single-click the unread DM to view the conversation in its entirety.\n  3. In the field at the bottom of the DM window, type in a reply.\n  4. Single-click the Send Message button.\n\nAlthough this is a private means of communication on Twitter, keep in mind that there will be a delay between DMs, as tweets need to find their way through the network. If you need a quicker reply time, it would be a better idea to move the talk to Skype, AIM, or some other chat application. DMs are the best way to have private conversations with someone on Twitter. The messages remain between you and the recipient unless you move out of the DM section and into the public stream.\n\nAlso beware of accidentally typing a DM into the public stream. This kind of mistake is often called \"DM FAIL.\" The results can range from the embarrassing to the career breaking, so it never hurts to double-check that you are in fact sending a DM. Many celebrities and politicians have fallen victim to this faux pas, and we want to help writers avoid it.\n\n# Appendix F\n\n# Google+\n\n## Setting Up a Google+ Account\n\nGoogle does its best to make things as simple and integrated as possible for its users. Once you have a Google account, you can use it to access Google, Gmail, Chrome, YouTube, Google Maps, Google Play, and, best of all, Google+.\n\n  1. Go to https:\/\/accounts.google.com\/SignUp.\n  2. Enter the information required. It will let you know if the username you have chosen is already taken.\n  3. Agree to the terms and service, and click next.\n  4. You will have to verify your account with an e-mail, text message, or phone call.\n  5. Enter the verification number.\n  6. Set up your profile.\n  7. Click on the blue person icon in the top right-hand corner.\n  8. You'll be prompted to find people to connect with. You can search for them on Google+ or find friends on Yahoo! or even Hotmail.\n  9. Google will then ask you to follow things you love; you can either do this now or follow up later.\n  10. Fill in your profile, giving as much information as you are comfortable sharing\u2014don't forget to mention that you are an author! Use a good, representative picture of yourself, or if you are feeling brave, snap one from your camera.\n\nYour profile is now done, but let's take a tour of the place.\n\n## Getting to Know Google+\n\nOnce you have added people to your Circles, you will begin seeing their content on the Google+ home page. In the top bar, you will see you can filter this stream of content by Circles of friends, family, and others you have set up. It's a nice idea to make Circles for your professional interests: book bloggers, publishers, agents, and such.\n\nOn the left, directly below the Google+ logo, is the home icon with a downwards arrow. Hovering over this will bring up a menu of what you can access from here.\n\n### Profile\n\nThis lets you see your profile how others will see it. It's a good opportunity to consider the public face you are presenting. Click on View Profile As: and change from Public to Yourself. Here is where you can change your cover image, change which Circles you have people in, and add posts. You can share text, pictures, a link, a video, an event, or even run a poll.\n\n### People\n\nThis page will offer people Suggestions that are based on shared friends or interests that you have in common. Check under Added You below that. Google+ doesn't send you e-mail alerts when someone adds you, only messages under the bell icon in the top right corner, so these adds can pile up. Be sure to get around to sorting people into Circles, if you accept them, or you may ignore them as necessary. Gmail contacts will run through your Gmail in search of people you already are connected to. Google+ will also try to find more contacts for you, depending on your profession, where you went to school, or where you work. (It is frighteningly accurate with these things.) Finally, you can allow it to connect to your other e-mail accounts so it can hunt for contacts there.\n\n### Photos\n\nHere is where you will find, and can upload, all those images you want. It is also where you can find images that you are tagged in.\n\n### Collections\n\nHere is where you can group your posts by topic. It makes it easier for your followers who share a particular interest with you to find posts you've made about that interest. Once you've made your first collection, there will be a new tab available where you can access them.\n\n### Communities\n\nLike Facebook Groups, Google+ communities are built around shared interests. Google+ will try its best to figure out what communities you might be interested in. It wouldn't hurt to join a few writing-based ones, or you might even create one of your own.\n\n### Events\n\nGoogle+ will search out events from your circles and place them here. This is also where you can create your own event, such as a book launch, a reading, or a celebration Hangout.\n\n### Hangouts\n\nOne of the standout features of Google+, Hangouts is where it is happening. Clicking on this link will show you who in your Circles is having a Hangout. It is also the place where you can start a Hangout yourself. Time to get that book group or beta reader session going. Simply click on \"Start a Hangout on Air\" under \"Hangouts on Air,\" give it a name, and tell people what it is about. You can choose to start one immediately or schedule one for a later date or time. Also, you can select which of your Circles you would like to include. Consider your audience and whether you want to make the Hangout exclusive or for a wider audience.\n\n### Pages\n\nHere is where you find the Pages that you've created, and where you can manage them. Click on the Page if you have created one. Now you can add content, get insights into how your Page is doing, and start a Hangout for the Page. This is where you can also start a brand-new Page for a new series.\n\n### Settings\n\nHere is where you can alter all the setting for your account. Look particularly at the Notifications section and decide how much, and for what, you want Google to notify you about.\n\n### Top-Right Bar\n\nThe + icon, and\/or the home icon, will take you to the stream of information that all of your circles have produced, and you can post information there as well.\n\nThe grid of squares will bring up all the additional Google offerings, like Youtube.com, News, Calendar, Drive, Search, Maps, Play, and Gmail. Ain't integration great?\n\n  * The bell icon will show you all the notifications, including when you get mentioned and when people have added you to a circle.\n  * The plus sign in a box is another opportunity for you to post.\n  * The circle with your avatar is another way to get to your profile and your Pages, and to log out.\n\nBelow these icons are two final icons. The two people icon brings up suggestions of people to add, because Google+ needs you to create Circles as soon as possible. The quotation marks in a bubble will bring up recent Hangout information. There you'll see all your previous Hangouts and whether you missed a video call. Remember, by clicking on the down arrow to the right of Hangouts, you can set your mood, status, and notifications for Hangouts. It also lets those in your Circles see when you are online.\n\n# Appendix G\n\n# YouTube\n\n## Setting Up a YouTube Account\n\nIf you have a Google Account, you will be presented with a chance to use that account for YouTube. If you do not have one, you will be prompted to create one. Google is trying to integrate all of its different services under one umbrella.\n\nIn the realm of social networking, you will be asked a lot of personal information. Rarely will a social networking site ask for details and refuse access if you do not answer all of the data fields. However, the more you can offer potential contacts, the more success you will achieve in building strong communities. Give only as much information as you feel comfortable giving, but make sure you cover the basics: an avatar, your name, where you are, and what you are all about.\n\nOnce you have created your Google account, you will have access to all the features YouTube offers.\n\n## Getting to Know YouTube\n\nAll you need to do now is upload your video. Before reaching for just any media creation you have on hand, you should know something about what works (and what doesn't) on this video-sharing service. Videos appearing on YouTube must fit certain technical criteria. They must be:\n\n  * less than 10 minutes in length\n  * smaller than 100 MB\n  * in the format of AVI, MOV, MPG, or WMV\n\nIf your video fits this criteria, then you can begin the uploading process.\n\nA site similar to YouTube is _BlipTV_ (http:\/\/blip.tv). Like YouTube, BlipTV is also free and offers the same ability to share video from blog to blog. The downside is that its compression is much harder than YouTube, meaning the video quality will vary. The upside: There are no limitations. Your videos can be as long as you like.\n\n## Uploading Your Video on YouTube\n\nOnce you have the video criteria met, you can upload your video. You are only a few clicks away from uploading your media to the world's most popular video-sharing service.\n\n  1. Click on the Upload button located in the upper-right corner of YouTube's interface.\n  2. Click on Select File to Upload, and browse your computer to find the video file you want to upload. Select the video file and then single-click on Select to advance to the details page.\n  3. While the progress bar shows the uploading process, fill in details about the clip that will display on the video's page. Click on the Save Changes button to accept these details.\n  4. Click the Upload Video button located underneath the current video uploading to place a second video in your uploading queue.\n\nDepending on your connection and YouTube's traffic, the uploading process of your video will wrap up within minutes.\n\nOnce the video is online, your content is offered on your YouTube channel. The video will remain embedded on this page until you or YouTube removes it from the server.\n\nYouTube usually removes content from its server when the content violates its terms of service or if it violates copyright. In all other instances, the user will remove content from a YouTube channel.\n\nNow that you have your YouTube account set up and your first video online, let's share this video...\n\n## Embedding Your YouTube Video in a Blog\n\n  1. On the video's host page, look for the Embed field located to the right of the YouTube page.\n  2. Select Share and then Embed. Copy the code below into your clipboard either with CTRL + C (PC) or Command + C (Mac).\n  3. Go to your WordPress account, then to Posts > Add New, and compose a post that introduces your video clip.\n  4. After composing a brief introduction, switch composition modes from Visual to HTML and paste into your entry the embed code copies you took care of back in Step 2. When embedding your video, you cannot use the Visual editor, so you'll need to use the Code editor.\n  5. Click either Save Draft to save the draft for later or Publish to make the post live.\n\nNow you have video incorporated with a blog post, providing other bloggers the ability to distribute your video by sharing the embed code from blog to blog. While the traffic comes from many different locations, all your hits are recorded on YouTube, increasing your visibility for casual YouTube visitors. The site remains the most popular one for online video content and is regarded as the go-to for media sharing sites.\n\n# Appendix H\n\n# Pinterest\n\n## Setting Up a Pinterest Account\n\nYou can log into Pinterest with your Facebook account, but if you don't want to do that, follow these instructions:\n\n  1. Supply your e-mail address and create a password.\n  2. Provide your name, age, and gender.\n  3. In order to discover your interests, Pinterest has you pick at least five interests to begin with, but you can always add more later on. Click on Follow.\n  4. Pinterest will now gather Pins it thinks you will be interested in based on those initial choices.\n  5. Don't forget to confirm your e-mail address.\n\n### Settings\n\nGo to your profile. Clicking on the gear icon will bring up your account setting. Here is where you can personalize your Pinterest account. You can set your privacy settings so Google doesn't search your Pins. However, as a writer who's trying to get noticed, you probably don't want to do that\u2014the more ways people can find you, the better. You can decide if you want Pinterest to send you e-mail notifications when someone repins one of your Pins or a selection of other activities. You can also choose to set up your other social media accounts to post when you Pin. Pinterest can post directly to your Facebook Timeline, your Twitter Stream, and your Google+ Page. These settings should be used with caution, however, as you don't want to flood all of your feeds with the same information and risk driving people away.\n\n## Getting to Know Pinterest\n\nWhen you first log into Pinterest, you won't see much. Only a few boards will be available, originating from your suggestions as a new pinner, but soon enough you'll be pinning from all over the place. Let's look over the layout.\n\nOn the left is the small Pinterest logo. Clicking on this will always bring you back to the home page. Next to the logo is the search bar where you can input keywords to look for items of interest pinned by others. Right next to it is a three-line icon. This will bring up categories you can use to find interesting items.\n\nTo the right of that should be your name. Clicking on this will bring you to your boards.\n\n### Bookmark\n\nIf you want to share a board (for instance with a publicist, co-writer, or your personal assistant) then simply go to your board, click Edit Board, and add the e-mail of that person under collaborators. Once she has accepted your invitation, she will be able to pin to your board, and you can set about making magic together ... or at least an interesting board.\n\nOn the right now should be your Edit Profile button. Use this if you want to tweak your profile in any way after the initial setup.\n\nNext to that, the gear icon is your account settings if you want to change your original setup. If you have converted your personal account into a business one (see Chapter 8), then this is where you will access your analytics information. It's also the place to create widget code to place into your own Web page. Finally, you can also log out here, but that's not something you have to do unless you have multiple accounts.\n\nBack in the main page, the last icon next to your name is a double-pin icon. This is where you access News (what other people are doing), You (what other people are doing with your information, repinning, liking, and also anyone you know through other networks who has just joined Pinterest), and Messages. Unlike some other media platforms, you're not likely to get too many messages, but do keep an eye on it.\n\nFinally, and most important, on the bottom left of the page is a plus symbol. This is where you can upload images directly, or from the Web, and create another board. Remember, while a Pin is an image or video, a board is a collection of Pins. Keep that kitchen corkboard in mind and it'll be easy to remember.\n\nNow that we've had a quick tour of the Pinterest neighborhood, let's see what we can do with it.\n\n## Working with Pinterest\n\nLike all of the social networks, the key is community. Luckily, Pinterest makes it very easy to go out and find your community.\n\nFirst of all, search \"authors,\" \"writers,\" or \"[insert your genre here] books.\" You can choose to follow the boards you find, but if you check a little deeper into the owner of the board, you might find other common interests. You can chose to follow all or just some of their boards.\n\nAs we said in earlier chapters, don't just stick with your colleagues. Broaden those horizons! Find people who share other interests with you as well. The most popular pins on the site are cooking and dining, DIY and crafting, health, funny stories, beauty and fashion, and technology.\n\nFind some boards and pinners that you find interesting and follow away. If you do so, your Pin stream will be full of interesting pictures and videos. These are great resources for repinning.\n\nWhen you click on a Pin, it will become much larger, and if you click again, if there is a link, you will be taken to that site. Or you can simply click on the Visit Site button for the same effect.\n\nIt's a good idea to check links, because some less moral pinners might try a bait-and-switch\u2014or the link might have been removed or moved. Remember, you want to provide your audience with great content, so it's worth it to take a moment to examine what you are repinning.\n\nRepinning is a great way to enhance your feed and the community, and to pad how often you are marketing those books of yours.\n\nIt really is very simple. When you hover over other people's Pins, you hit the Pin It button. Then you have a choice of which board you want to Pin to and whether you want to add a description.\n\nAs on other social media platforms, commentary is a good idea here. You can put the original information in some square brackets and then put your comment before or after that. In this way, you preserve the original pinner's words, which makes you a good community member. Your comments can be as simple as \"I'm going to try and make these muffins!\" or \"This looks like a dress my heroine in [insert my title] would wear.\"\n\n### Bookmark\n\nIt is always good Pinterest etiquette to credit your sources if you can. Also, using other people's images but linking to your blog or Web page is an underhanded practice and can inspire people to leave negative comments. Don't do it if you want to develop a good Pinterest reputation.\n\nIf you have a contact on Pinterest who you think would really appreciate a particular Pin, you can send it directly to that person. Use this feature with caution, and never try and spam people with your Pins!\n\nFinally, there is the Heart button, which is similar to the Like button on Facebook. Commentary and repinning is the nicest way to show appreciation, but if it doesn't fit in with any of your boards, this is a way to give the original pinner a thumbs up.\n\n## Making Your First Board\n\nCreating boards on Pinterest is easy, and don't be afraid of making too many. The more boards, the more interests, and the more views. From your profile, click on the large gray plus sign with Create a board. From here give your board a name and a description, and find a category it fits into. Map will create a map, which will link to Pins\u2014this is a great idea if your board is travel themed. You can also choose to make the board secret or not. Secret boards, as we talked about in Chapter 8, are great for saving items to Pin later, or for boards about projects you're not quite ready to talk about yet. You can also add collaborators and choose if they can add new members to this board themselves.\n\n### Uploading a Pin of Your Own\n\nNow to the exciting stuff. Let's add some content of your own. Book covers, fan art, images of people you'd like to see play your characters, places where you have set your stories\u2014all of these make excellent uploads.\n\n  1. Click on the plus icon and chose the image from your computer. It can be a GIF, JPEG, or PNG file. Choose the board you want to add it to, or create a new board for this image ... but only if it fits in with your interests. Too many boards with only a few images are not enticing.\n  2. The important part of this Pin is the description. Keep in mind what your readers might be looking for. Include a substantial description about what this image means to you.\n  3. Don't forget a couple of hashtags, and if it is relevant, now is the time to insert a link. The link should, at the very least, lead back to your website; if not, link to a location where the all-important impulse buy can be made.\n\nFinally, if you have connected other social networks, you can choose to post it there.\n\nThere\u2014you've done it. You have your first original Pins!\n\n# Appendix I\n\n# Instagram\n\n## Setting Up an Instagram Account\n\nGetting an account together on Instagram is quite easy, provided you have the app loaded onto your smartphone. Instagram is available on iOS (iPhone), Android, and Windows, and runs on all varieties of smartphones.\n\nFirst, download and install Instagram onto your smartphone from the App Store for your device.\n\n  1. Launch Instagram on your phone. Your options for establishing an Instagram account include using your Facebook credentials or your e-mail. Use whichever method works best for you. The steps we outline here apply to using an e-mail to register your Instagram account.\n  2. Tap Register with E-mail, and on the next screen, enter the following:\n  3. a username (the nickname you want to go by on Instagram)\n  4. a password\n  5. a valid e-mail address\n  6. Tap the Photo icon in the center of the screen and select a photo (or take a new one) for your profile picture. Choose one that best represents you as an author.\n  7. Tap the arrow pointing right to follow validation steps (which requires you to reply via e-mail) to finish your profile.\n\nThe _profile_ , like other profiles you will find online, is the place where you can introduce yourself. You can edit this content at any time from your account or from the app by tapping on the profile icon on the far right of the menu bar and then tapping the Edit Your Profile option.\n\n## Getting to Know Instagram\n\nNow your account is ready to go, so let's check out some of the features.\n\nFirst off, you will see immediately that Instagram is not designed as a desktop app but more for \"capturing it instantly\" through your smartphone. (Now the name makes sense, right?) In fact, you cannot upload photos or video from your computer\u2014all your Instagram posting has to be done through your phone.\n\n### Bookmark\n\nAnother terrific way to track popular tags on Instagram is by allowing the website Statigram1http:\/\/statigr.am access to your Timeline. Statigram allows you to look up photos through hashtags in a gallery-style format, unlike Instagram, which allows you to look up hashtags one account at a time. Other options that Statigram offers include account statistics, enhanced comment postings, and network maintenance.\n\nTake a look at the menu bar running along the bottom of Instagram. These five icons are all the options you need to get around the Instagram app, build your Instagram network, and interact with others. From left to right, your icons are as follows:\n\n  * **Home:** The _Home_ icon takes you to your Timeline of Instagram users. Their posts will appear here. You will also see an inbox icon, which is where you can send and receive direct messages.\n  * **Explore:** If you want to check out and follow popular accounts on Instagram, search for users you might know, or look up hashtags people are attaching to their images, you can do all this in the _Explore_ option by tapping on the compass rose icon.\n  * **Camera:** When you are ready to take a photograph with Instagram, tap the camera icon. Depending on your preferences, your screen will be divided into thirds (a photographer's trick for framing up your subject), or it will provide just a clear view from the lens. Tapping on the screen will establish what you want the focus point of the photo to be. You can also flip the camera around for _selfies_ (self-portraits), choose a _flash mode_ , and switch to _video mode_ (the icon to the right) or back to _still camera mode_ (the icon to the left). The center button can take photos (when blue) or shoot video (when red).\n  * **Notifications:** The speech bubble with the heart is where you go to receive and track _Notifications_. There are two modes here: _Following_ and _News_. Keep your Notifications on News in order to see who's following you and to track who Likes your photos.\n  * **Profile:** This final icon is where you can edit your profile. Once in your profile, edit your Preferences by tapping the gear icon in the top-right corner of your smartphone. This option also lets you find out what photos you've been tagged in.\n\nLet's jump right in and start taking photos with Instagram. More important, let's look at how we can take a photograph with the app and then share the photo across a number of social networks.\n\n## Taking a Photo on Instagram\n\nTap on the Camera icon and take a photo.\n\n  1. You can now apply some of the various filters Instagram offers. There are quite a few of them, and you can have a lot of fun trying to find the best one for your image. Instagram lets you see exactly how the photo will look with the filter applied before you commit to it.\n  2. Tap the far-left icon, the Straighten tool. Place and hold your finger on the Adjustment wheel and rotate the image either right or left. Then tap Done to return to the Edit mode.\n  3. With the Border option, you can determine whether you want your image to have a border (blue) or no border (white).\n  4. The Blur option offers three modes:\n  5. None (white),\n  6. Circle (blue drop with circle), and\n  7. Bar (blue drop with bar).\n  8. Tap each mode to decide where you want the blur and which design works best for your image.\n  9. Lux allows you to adjust the contrast in the photo. Tap to make adjustments, and then tap Done to return to the original image.\n  10. Tap Next to enter the Share mode, or tap the left arrow at the top of the screen to return to Photo Edit mode.\n\nThe appeal of Instagram is taking a simple image and getting creative with it. After applying filters and effects on this image, you can then post on Instagram, and on Facebook and Twitter as well. It gives amateurs the chance to have some fun with photographs.\n\nNow we have to share this image through our social networks.\n\n  1. In Instagram's Share to mode, compose a caption for the photo. Don't forget that there are two kinds of sharing in Instagram: _Followers_ and _Direct_. Followers are open posts on your network. Direct messages are similar to DMs on Twitter. Images and video shared in Direct mode allow you to share with select members of your network.\n  2. Add two hashtags related to your photo, then tap OK in the upper-right corner of the screen. The _Tag People_ option is best used only after your network is up and running. This is how you can tag friends who appear in the image. Notification of tagging appears under your _Profile_ option.\n  3. Now tap Name This Location to tag your location.\n  4. If you are tagging your location, leave Add to Photo Map turned on. Some people do not like others to know where they are or where they have been. Use your own discretion on this.\n  5. There are eight social networks you can connect to your Instagram:\n  6. Facebook\n  7. Twitter\n  8. Tumblr\n  9. Flickr\n  10. Foursquare\n  11. Mixi (Japan)\n  12. Weibo (China)\n  13. NKontakte (Russia)\n  14. Tap each network where you want to share this image.   \nIf you have not connected your networks to Instagram, you will be walked through a log-in process. You will only have to make this connection once.\n  15. Tap Share to post your image.\n\nThere you go! You will see your image appear instantly on your Instagram Timeline; if you also tapped _Facebook_ and _Twitter_ before tapping the Share feature, you have effectively shared your image across three networks in one post.\n\nMany popular applications\u2014Pinterest, YouTube, Yelp, Starbucks, Vivino, and Untappd, for example\u2014allow for multiplatform posting like this. The networks offered may differ from app to app, but Facebook and Twitter are often offered as posting platforms.\n\nInstagram is one such app that is popular among apps for its artistic capabilities, its ease of use, and its engaging content generated for Facebook and Twitter.\n\n### Bookmark\n\nInstagram can post on Facebook accounts but not on Pages accounts. To put Instagram postings on Pages, you will need to share them from your personal account to your Page.\n\n1http:\/\/statigr.am\n\n# Contents\n\n  1. Title Page\n  2. Copyright Page\n  3. Dedication\n  4. Acknowledgments\n  5. About the Authors\n  6. Foreword\n  7. Introduction: Welcome to Your Social Media Survival Guide\n    1. Understanding Social Media\n    2. What You Need to Make This Book Work\n  8. Chapter 1: WordPress\n    1. Why WordPress?\n  9. Chapter 2: Tumblr\n    1. What Is the Difference Between WordPress and Tumblr?\n    2. Expressing Yourself: Producing Content\n    3. What to Expect from the Tumblr Community\n    4. Sharing with the Class: Syndicating Media\n    5. Tumblr Strategies\n  10. Chapter 3: Podcasting\n    1. Writers Gone Wild: Talk Shows\n    2. Anthologies in Audio: Short Stories\n    3. Going All In: Podcast Novels\n    4. Why Should You Podcast?\n    5. Why Shouldn't You Podcast?\n    6. Why Do We Still Podcast?\n  11. Chapter 4: Facebook\n    1. Facebook for Writers\n    2. Content Marketing: The Science of Promotion by Example\n  12. Chapter 5: Twitter\n    1. First Impressions at a Glance\n    2. Mastering Tweet Speak: Composing Messages on Twitter\n    3. Best Practices for Twitter\n    4. You Did Not Just Tweet That: _Worst_ Practices for Twitter\n  13. Chapter 6: Google+\n    1. Google+ for Writers: The Account vs. The Page\n  14. Chapter 7: YouTube\n    1. Action Editing: Software for Video Editing\n    2. Turning Authors on to YouTube\n    3. Best Practices on YouTube\n  15. Chapter 8: Pinterest\n    1. Tools of the Trade: Pinterest Applications\n    2. Game On: Pinterest Competitions\n    3. It's Business Time: Pinterest Business\n    4. Watch Where You Point That Pin: Pinterest Strategies\n  16. Chapter 9: Instagram\n    1. Platform Management: Connecting Instagram with Other Social Networks\n    2. Beyond Instagram: Third-Party Apps\n    3. Best Practices on Instagram\n  17. Chapter 10: Additional Options\n    1. Goodreads\n    2. Reddit\n    3. Storify\n    4. A Social's Social: Untappd, Vivino, and Distiller\n    5. Tsu.co\n  18. Chapter 11: SEO\n    1. Why SEO Doesn't Work (as the \"Gurus\" Claim It Does)\n    2. How to Make SEO Work for You\n    3. The SEO That Can Hurt You\n    4. Best Practices of SEO\n  19. Chapter 12: Content Marketing\n    1. How Does Promoting Others Really Work for You?\n    2. What Content to Look for in Content Marketing\n    3. When Content Marketing Goes Bad\n  20. Chapter 13: Best Practices in Social Media\n    1. Create an Editorial Calendar\n    2. Think Before You Post\n    3. Participate in Blog Tours\n    4. Create Quality Content\n    5. Be Visual When Possible\n    6. A New Use for the Pound Symbol: Hashtags\n    7. Taking P!nk's Advice: Facebook Parties\n    8. Planning Makes Perfect: Before That Social Media Event\n    9. Attribution, Not Imitation, Is the Sincerest Form of Flattery\n    10. Seek Out Your Audience\n    11. Have a Plan\n    12. Focus on Maintaining a Signal, Not Creating Noise\n    13. Antisocial Media: What to Avoid in Online Promotion and Networking\n  21. Appendix A: WordPress\n    1. Setting up a WordPress Account\n    2. Getting to Know WordPress\n    3. Creating Content\n  22. Appendix B: Tumblr\n    1. Setting up a Tumblr Account\n    2. Getting to Know Tumblr\n    3. Posting to Tumblr\n    4. Reblogging\n  23. Appendix C: Podcasting\n    1. An Introduction to Podcasting\n    2. The Mixing Board: Bridging the Gap Between Microphone and Computer\n    3. What's Next: Other Details for an Audio Studio\n  24. Appendix D: Facebook\n    1. Setting Up a Facebook Account\n    2. Getting to Know Facebook\n    3. The Left-Hand Sidebar\n  25. Appendix E: Twitter\n    1. Setting Up a Twitter Account\n    2. Getting to Know Twitter\n    3. Talking on Twitter\n    4. Direct Messages\n  26. Appendix F: Google+\n    1. Setting Up a Google+ Account\n    2. Getting to Know Google+\n  27. Appendix G: YouTube\n    1. Setting Up a YouTube Account\n    2. Getting to Know YouTube\n    3. Uploading Your Video on YouTube\n    4. Embedding Your YouTube Video in a Blog\n  28. Appendix H: Pinterest\n    1. Setting Up a Pinterest Account\n    2. Getting to Know Pinterest\n    3. Working with Pinterest\n    4. Making Your First Board\n  29. Appendix I: Instagram\n    1. Setting Up an Instagram Account\n    2. Getting to Know Instagram\n    3. Taking a Photo on Instagram\n\n# Guide\n\n  1. Cover\n  2. Contents\n  3. Start of content\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nHISTORICAL DICTIONARIES OF RELIGIONS, PHILOSOPHIES, AND MOVEMENTS\n\nJon Woronoff, Series Editor\n\n_Orthodox Church_ , by Michael Prokurat, Alexander Golitzin, and Michael D. Peterson, 1996\n\n_Civil Rights Movement_ , by Ralph E. Luker, 1997\n\n_Catholicism_ , by William J. Collinge, 1997\n\n_North American Environmentalism_ , by Edward R. Wells and Alan M. Schwartz, 1997\n\n_Taoism_ , by Julian F. Pas in cooperation with Man Kam Leung, 1998\n\n_Gay Liberation Movement_ , by Ronald J. Hunt, 1999\n\n_Islamic Fundamentalist Movements in the Arab World, Iran, and Turkey_ , by Ahmad S. Moussalli, 1999\n\n_Cooperative Movement_ , by Jack Shaffer, 1999\n\n_Kierkegaard's Philosophy_ , by Julia Watkin, 2001\n\n_Slavery and Abolition_ , by Martin A. Klein, 2002\n\n_New Religious Movements_ , by George D. Chryssides, 2001\n\n_Prophets in Islam and Judaism_ , by Scott B. Noegel and Brannon M. Wheeler, 2002\n\n_Lesbian Liberation Movement: Still the Rage_ , by JoAnne Myers, 2003\n\n_Descartes and Cartesian Philosophy_ , by Roger Ariew, Dennis Des Chene, Douglas M. Jesseph, Tad M. Schmaltz, and Theo Verbeek, 2003\n\n_Witchcraft_ , by Michael D. Bailey, 2003\n\n_Unitarian Universalism_ , by Mark W. Harris, 2004\n\n_New Age Movements_ , by Michael York, 2004\n\n_Organized Labor_ , _Second Edition_ , by James C. Docherty, 2004\n\n_Utopianism_ , by James M. Morris and Andrea L. Kross, 2004\n\n_Feminism_ , _Second Edition_ , by Janet K. Boles and Diane Long Hoeveler, 2004\n\n_Jainism_ , by Kristi L. Wiley, 2004\n\n_Wittgenstein's Philosophy_ , by Duncan Richter, 2004\n\n_Schopenhauer's Philosophy_ , by David E. Cartwright, 2005\n\n_Seventh-day Adventists_ , by Gary Land, 2005\n\n_Methodism_ , _Second Edition_ , by Charles Yrigoyen Jr. and Susan E. Warrick, 2005\n\n_Sufism_ , by John Renard, 2005\n\n_Sikhism_ , _Second Edition_ , by W. H. McLeod, 2005\n\n_Kant and Kantianism_ , by Helmut Holzhey and Vilem Mudroch, 2005\n\n_Olympic Movement_ , _Third Edition_ , by Bill Mallon with Ian Buchanan, 2006\n\n_Anglicanism_ , by Colin Buchanan, 2006\n\n_Welfare State_ , _Second Edition_ , by Bent Greve, 2006\n\n_Feminist Philosophy_ , by Catherine Villanueva Gardner, 2006\n\n_Logic_ , by Harry J. Gensler, 2006\n\n_Leibniz's Philosophy_ , by Stuart Brown and Nicholas J. Fox, 2006\n\n_Non-Aligned Movement and Third World_ , by Guy Arnold, 2006\n\n_Salvation Army_ , by Major John G. Merritt, 2006\n\n_Judaism_ , _Second Edition_ , by Norman Solomon, 2006\n\n_Epistemology_ , by Ralph Baergen, 2006\n\n_Bah\u00e1'\u00ed Faith_ , _Second Edition_ , by Hugh C. Adamson, 2006\n\n_Aesthetics_ , by Dabney Townsend, 2006\n\n_Socialism_ , _Second Edition_ , by Peter Lamb and James C. Docherty, 2007\n\n_Marxism_ , by David M. Walker and Daniel Gray, 2007\n\n_Nietzscheanism_ , _Second Edition_ , by Carol Diethe, 2007\n\n_Medieval Philosophy and Theology_ , by Stephen F. Brown and Juan Carlos Flores, 2007\n\n_Shamanism_ , by Graham Harvey and Robert Wallis, 2007\n\n_Ancient Greek Philosophy_ , by Anthony Preus, 2007\n\n_Puritans_ , by Charles Pastoor and Galen K. Johnson, 2007\n\n_Green Movement_ , _Second Edition_ , by Miranda Schreurs and Elim Papadakis, 2007\n\n_Husserl's Philosophy_ , by John J. Drummond, 2008\n\n_Existentialism_ , by Stephen Michelman, 2008\n\n_Zionism_ , _Second Edition_ , by Rafael Medoff and Chaim I. Waxman, 2008\n\n_Coptic Church_ , by Gawdat Gabra, 2008\n\n_Jehovah's Witnesses_ , by George D. Chryssides, 2008\n\n_Hume's Philosophy_ , by Kenneth R. Merrill, 2008\n\n_Shakers_ , by Stephen J. Paterwic, 2008\n\n_Native American Movements_ , by Todd Leahy and Raymond Wilson, 2008\n\n_Mormonism_ , _Third Edition_ , by Davis Bitton and Thomas G. Alexander, 2008\n\n_Hegelian Philosophy_ , _Second Edition_ , by John W. Burbidge, 2008\n\n_Ethics_ , by Harry J. Gensler and Earl W. Spurgin, 2008\n\n_Environmentalism_ , by Peter Dauvergne, 2009\n\n_Bertrand Russell's Philosophy_ , by Rosalind Carey and John Ongley, 2009\n\n_Baptists_ , _Second Edition_ , by William H. Brackney, 2009\n\n_Islam_ , _Second Edition_ , by Ludwig W. Adamec, 2009\n\n_Homosexuality_ , by Brent L. Pickett, 2009\n\n_Buddhism_ , by Carl Olson, 2009\n\n_Holiness Movement_ , _Second Edition_ , edited by William Kostlevy, 2009\n\n_Reformed Churches_ , _Second Edition_ , by Robert Benedetto and Donald K. McKim, 2010\n\n_The Reformation and Counter-Reformation_ , by Michael Mullett, 2010\n\n_Heidegger's Philosophy_ , _Second Edition_ , by Frank Schalow and Alfred Denker, 2010\n\n_Jesus_ , by Daniel J. Harrington, S.J., 2010\n\n_Metaphysics_ , by Gary Rosenkrantz and Joshua Hoffman, 2011\n\n_Shinto_ , _Second Edition_ , by Stuart D. B. Picken, 2011\n\n_The Friends (Quakers)_ , _Second Edition_ , by Margery Post Abbott, Mary Ellen Chijioke, Pink Dandelion, and John William Oliver Jr., 2011\n\n_Lutheranism, Second Edition_ , by G\u00fcnther Gassmann with Duanne H. Larson, and Mark W. Oldenburg, 2011\n\n_Hinduism, New Edition_ , by Jeffery D. Long, 2011\n\n# Historical Dictionary of Hinduism\n\nNew Edition\n\nJeffery D. Long\n\nThe Scarecrow Press, Inc.\n\nLanham \u2022 Toronto \u2022 Plymouth, UK  \n2011\nPublished by Scarecrow Press, Inc.\n\nA wholly owned subsidiary of The Rowman & Littlefield Publishing Group, Inc.\n\n4501 Forbes Boulevard, Suite 200\n\nLanham, Maryland 20706\n\n<http:\/\/www.scarecrowpress.com>\n\nEstover Road\n\nPlymouth PL6 7PY\n\nUnited Kingdom\n\nCopyright \u00a9 2011 by Jeffery D. Long\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\nBritish Library Cataloguing in Publication Information Available\n\nLibrary of Congress Cataloging-in-Publication Data\n\np. iv\n\nLibrary of Congress Cataloging-in-Publication Data\n\nLong, Jeffrey D.\n\nHistorical dictionary of Hinduism \/ Jeffrey D. Long. -- New ed.\n\np. cm. -- (Historical dictionaries of religions, philosophies, and movements)\n\nUpdated and revised ed. of: The A to Z of Hinduism \/ Bruce M. Sullivan.\n\nIncludes bibliographical references.\n\nISBN 978-0-8108-6764-2 (cloth : alk. paper) -- ISBN 978-0-8108-7960-7 (ebook)\n\n1. Hinduism--Dictionaries. I. Sullivan, Bruce M., 1951- A to Z of Hinduism. II. Title.\n\nBL1105.S847 2011\n\n294.503--dc222011013053\n\n  The paper used in this publication meets the minimum requirements of American National Standard for Information Sciences\u2014Permanence of Paper for Printed Library Materials, ANSI\/NISO Z39.48-1992.\n\nPrinted in the United States of America\nFor Billy\n\n# Contents\n\nEditor's Foreword _Jon Woronoff_\n\nAcknowledgments\n\nReader's Note\n\nAcronyms and Abbreviations\n\nChronology\n\nIntroduction\n\nTHE DICTIONARY\n\nBibliography\n\nAbout the Author\n\n# Editor's Foreword\n\nHinduism is the world's third-largest religion and still growing, with about 900,000,000 adherents at present. It is also the oldest major religion, with a tradition reaching back not just centuries but millennia, with many notable peaks and periods of glory. It has influenced and been affected by other world religions, including Islam and Buddhism as well as Christianity. Yet, despite repeated attempts to contain it, it has maintained an exceptional vigor and boasts a rich spiritual culture with celebrated thinkers and teachers and an extensive literature. Nonetheless, it is rather poorly known outside of India and the surrounding region, and the fascination in the West with some aspects has perhaps been less enlightening than mystifying. We are familiar with things like caste, guru, karma, meditation, ashram (\u0101\u015brama), and even the sacred cow. But we know pitifully little about its basic writings and principal beliefs and practices. Gradually, as more Hindus move to the West, that is changing, but the change is slow because, among other things, Hinduism is not particularly easy to understand, especially when coming from another horizon.\n\nThis _Historical Dictionary of Hinduism_ should contribute to a greater understanding\u2014or at least less misunderstanding\u2014by presenting the religion in very comprehensible terms. This means, among other things, providing biographical sketches of particularly significant persons as well as further information on the movements, schools, and communities they founded. Other entries deal with the basic writings, doctrines, and practices. But inevitably many more have to clearly explain the basic concepts, some of these best conveyed by the Sanskrit term and others, more or less effectively, by an English translation. This is embedded in a chronology reaching back four and a half millennia that then focuses on an impressively active past century or so. The introduction covers this long period as well, showing how Hinduism has evolved and grown, emphasizing many highlights and turning points. Although this volume already contains an amazing amount of information, quite enough for those who are casually interested, specialists and those with more precise interests can find ample additional reading in the bibliography.\n\nThis completely new edition of the _Historical Dictionary of Hinduism_ is particularly useful for various reasons. Certainly one is that its author, Jeffery D. Long, himself originally came from another horizon, Catholicism, so he first saw Hinduism from outside before getting to know it better from inside. Moreover, he has spent the past decade explaining it to students at the Department of Religious Studies at Elizabethtown College, where he has been working as an associate professor and as department chair. Over the years, he has written substantially on Hinduism, starting with his doctoral dissertation at the University of Chicago's Divinity School and continuing with several dozen articles and reviews as well as two recent books, _A Vision for Hinduism: Beyond Hindu Nationalism_ and, on a related religious current, _Jainism: An Introduction_. In addition, he has served as chair of the steering committee of the Dharma Academy of North America and is currently cochair of the North American Hinduism Consultation of the American Academy of Religion. This permits him to present Hinduism to a broad public, including both initiates and those who are curious but have found it hard to get comprehensible answers to their questions elsewhere.\n\nJon Woronoff\n\nSeries Editor\n\n# Acknowledgments\n\nI thank the following persons who have made it possible for me to complete this project.\n\nFirst, Jon Woronoff, the editor of the Scarecrow Press series of Historical Dictionaries of Religions, Philosophies, and Movements, who invited me to write this book. I found Jon to be very kind and generous in regard to deadlines, and also most helpful and prompt in responding to my questions and concerns and in giving editorial suggestions.\n\nI also thank my department colleagues and the administration of Elizabeth-town College for granting me a one-year sabbatical (consisting of a reduced teaching load and a bare minimum of department chair\u2013related duties) in order to help me complete this project.\n\nI thank all of my teachers, colleagues, and fellow Hindus who, over the years, have added to my knowledge base and put me in a position to even contemplate taking on a project such as this. I hope this book is useful to you and to future generations of scholars in the field of Hindu studies.\n\nA special word of gratitude goes to my wife and best friend, Mahua Bhattacharya, who has been, as always, encouraging and patient. I am sorry for all of the times we have had to forego seeing a movie or taking a drive because \"I need to work on my _Historical Dictionary of Hinduism_!\"\n\nReaders of my previous two books, _A Vision for Hinduism_ and _Jainism: An Introduction_ , may recall my mentioning in the acknowledgments of those books my orange tabby cat, Billy, who was constantly at my feet during the writing of those books and was always a dear and supportive presence. During the writing of this book, Billy was diagnosed with, and eventually succumbed to, a very aggressive lymphoma. The _Bhagavad G\u012bt\u0101_ and the Ved\u0101nta tradition that I practice tell me that the soul continues on, and I believe this to be true. But there is an empty space at my feet and sorrow in my heart as I share this news with my readers. I dedicate this book to Billy.\n\n# Reader's Note\n\nThe bulk of the terminology in this dictionary is in Sanskrit and other Indic languages. The scripts (\"alphabets\") of these languages are phonetic in nature\u2014that is, each of the characters stands for exactly one sound, and there is a character for each sound. As a result, the Indic scripts have far more characters than the mere 26 of the Roman alphabet used in English and most western European languages\u2014about 50, to be precise. In order to render words from Indic languages in the Roman script in an exact way that does not lend itself to mispronunciation, scholars have developed a system of diacritical marks in order to distinguish between, for example, the short \"a\" of Sanskrit (pronounced like the \"u\" in \"bud\") and the Sanskrit long \"\u0101\" (which is pronounced like the \"a\" in the English word \"father\"). The correct pronunciation of the Indic sounds, as depicted using the standard system, is as follows:\n\na This is pronounced \"uh,\" as in \"bud.\"\n\n\u0101 This is pronounced \"ah,\" as in \"father.\"\n\ni This is pronounced like the \"i\" in \"bit.\"\n\n\u012b This is pronounced like the \"ee\" in \"beet.\"\n\nu This is pronounced like the \"oo\" in \"book.\"\n\n\u016b This is pronounced like the \"oo\" in \"boot.\"\n\n\u1e5b This is pronounced like the \"ri\" in \"rig,\" with a slight roll of the tongue, though not as hard a roll as in Spanish \"r.\"\n\ne This is pronounced like \"ay\" in \"say.\"\n\nai This is pronounced like \"aye\" or \"eye.\"\n\no This is pronounced \"oh,\" as in \"Ohio.\"\n\nau This is pronounced like \"ow\" in \"how.\"\n\nConsonants are pronounced as in English, but retroflex consonants (consonants with a dot under them, such as \"\u1e6d\") are pronounced with the tongue touching the roof of the mouth.\n\nAspirated consonants are immediately followed by an \"h\" (for example, \"th,\" \"dh\") and include an exhalation\u2014that is the \"h\" is pronounced, producing somewhat of a softening of the sound of the consonant. The \"h\" is _not_ a separate syllable. So \"dha\" is pronounced \"dha,\" and not \"daha.\"\n\nThe \"\u015b\" and \"\u1e63\" sounds are almost indistinguishable and are sometimes confused even by native speakers. The \"\u1e63\" sound is pronounced with the tongue at the roof of the mouth, but \"\u015b\" is not. They both sound like the \"sh\" in \"she.\"\n\nThe sound \"\u1e25,\" which is always preceded by a vowel, produces a slight echoing of that vowel. So \"a\u1e25\" is pronounced \"aha.\" The sound \"\u1e43\" is a slightly nasalized \"m\" which sounds almost like an \"n\" and bears some resemblance to the nasalized French \"n.\"\n\nI have used this system for most terms in this dictionary, except for most modern proper names that have a popular spelling (such as \"Ramakrishna\"). For the sake of consistency I have also tended to use Sanskrit terminology, even when the corresponding Hindi term is better known (so \"avat\u0101ra\" instead of \"avatar\").\n\nThis book is extensively cross-referenced. In the dictionary section, if there is an entry on a specific topic, this is shown by bolding it the first time it appears (with the exception of \"Hindu\" and \"Hinduism\"). Other related entries are listed as _See also_. If a topic does not have an entry of its own, but it appears under another heading or is amply discussed there, it is indicated by a _See_.\n\n# Acronyms and Abbreviations\n\nBAPS Boc\u0101sanv\u0101s\u012b Ak\u1e63ar Puru\u1e63ottam Sw\u0101m\u012bn\u0101r\u0101ya\u1e47 Sansth\u0101\n\nBJP Bh\u0101rat\u012bya Janat\u0101 Party\n\nD\u0100NAM Dharma Academy of North America\n\nHAF Hindu American Foundation\n\nISKCON International Society for Krishna Consciousness\n\nRKM Ramakrishna Mission\n\nRSS R\u0101\u1e63\u1e6dr\u012bya Svayamsevak Sa\u1e45gh\n\nSRF Self-Realization Fellowship\n\nTM Transcendental Meditation\n\nVHP Vi\u015bva Hindu Pari\u1e63ad\n\n# Chronology\n\nAll dates are CE (Common Era; AD) unless otherwise indicated. Most pre-modern dates are approximations, especially for the composition of texts. Most Hindu chronologies traditionally locate texts far earlier than does modern scholarship. The dates given here generally reflect the modern scholarly consensus. It should be noted, however, that texts were often transmitted orally for many generations before taking their current form. So it may be that the beginning of this process of composition and transmission extends back considerably further than modern scholarship, which focuses on the texts in their extant form, suggests.\n\n**c. 50,000 BCE** Stone Age cultures arise in the Indian subcontinent.\n\n**c. 30,000 BCE** Bhimbetka cave paintings drawn, including depictions of horses.\n\n**7000\u20135000 BCE** Mixed pastoral and agricultural settlements, centered on Mehrgarh, emerge in what is today western Pakistan.\n\n**4000\u20133000 BCE** Proto-Indo-European language breaks up into distinct languages.\n\n**c. 3000 BCE** Pastoral nomadic societies are prominent in northern India.\n\n**c. 2700 BCE** Large-scale urban societies emerge along the Indus and Sarasvat\u012b Rivers, in continuity with the culture of Mehrgarh.\n\n**2600\u20131900 BCE** Indus Valley\/Indus-Sarasvat\u012b Civilization, advanced urban phase.\n\n**2300\u20132000 BCE** Indo-European speakers in the Ural region of Russia are the first to invent chariots with light spokes, as described in the _\u1e5ag Veda_.\n\n**c. 2200 BCE** Sumerian texts mention trade with the land of _Meluhha_ , possible reference to the Indus Valley Civilization.\n\n**c. 2100 BCE** Indo-European speakers are living in Persia and Afghanistan.\n\n**1900\u20131800 BCE** The Sarasvat\u012b River dries up and the Indus River changes course, leading to massive disruption of the economy of the Indus-Sarasvat\u012b Civilization, with most of its cities being abandoned. A slow but major migration begins from the Indus-Sarasvat\u012b region to the Ganges River Valley.\n\n**1900\u20131700 BCE** Indo-European speakers are living in Northwestern India.\n\n**1900\u20131200 BCE** Indus Valley Civilization gradually declines.\n\n**1700\u20131500 BCE** Composition of _\u1e5ag Veda_. Horses and light-spoked chariots present in Northwestern India.\n\n**c. 1500 BCE** Submergence of the stone port city of Dv\u0101rak\u0101, on the coast of Gujarat, site associated with K\u1e5b\u1e63\u1e47a, suggesting a possible date for the _Mah\u0101bh\u0101rata_ war.\n\n**c. 1300 BCE** Hittite inscriptions contain references to Vedic deities.\n\n**1200\u20131000 BCE** Composition of _Yajur Veda_ and _S\u0101ma Veda_.\n\n**c. 1000 BCE** The city of Kau\u015b\u0101mbi in Vatsa, in central India, is founded.\n\n**c. 950 BCE** Date for the _Mah\u0101bh\u0101rata_ battle preferred by most scholars.\n\n**c. 900 BCE** Composition of _Atharva Veda_. Vedic peoples move more deeply into the Ganges Valley. City of K\u0101\u015bi (Banaras) is founded.\n\n**850\u2013750 BCE** Life of P\u0101r\u015bvan\u0101tha, 23rd T\u012brtha\u1e45kara of Jainism.\n\n**800\u2013600 BCE** Composition of the _Br\u0101hma\u1e47as_ , early priestly commentaries on the _Vedas_.\n\n**600\u2013500 BCE** Composition of the _\u0100ra\u1e47yakas_ , or \"forest texts.\"\n\n**518 BCE** Persians invade Northwest India, under Emperor Darius I.\n\n**c. 500 BCE** _\u015arauta S\u016btras_ , _Ved\u0101\u1e45gas_ , and _B\u1e5bhaddevat\u0101_ (Sanskrit commentaries upon Vedic rituals and deities) are composed. City of Pataliputra founded. Vedic peoples go south.\n\n**500\u2013400 BCE** _B\u1e5bhad\u0101ra\u1e47yaka_ , _Ch\u0101ndogya_ , _Taittir\u012bya_ , and _Kau\u1e63\u012btaki Upani\u1e63ads_ are composed.\n\n**499\u2013427 BCE** Life of Vardham\u0101na Mah\u0101v\u012bra, 24th T\u012brtha\u1e45kara of Jainism.\n\n**490\u2013410 BCE** Life of Siddh\u0101rtha Gautama, the Buddha.\n\n**c. 400 BCE** P\u0101\u1e47ini's _A\u1e63\u1e6dadhy\u0101y\u012b_ , major work on Sanskrit grammar, composed.\n\n**400\u2013300 BCE** _Ka\u1e6dha_ , _\u015avet\u0101\u015bvat\u0101ra_ , and _Mu\u1e47\u1e0daka Upani\u1e63ads_ are composed.\n\n**350\u2013283 BCE** Life of Kau\u1e6dilya (Canakya), advisor to Candragupta Maurya, to whom the _Artha \u015a\u0101stra_ , a work of political science, is attributed.\n\n**327\u2013325 BCE** Alexander of Macedon invades Northwest South Asia.\n\n**320\u2013293 BCE** Reign of Candragupta Maurya, founder of the Mauryan Dynasty.\n\n**c. 300 BCE** Greek ambassador Megasthenes travels to India.\n\n**300\u2013200 BCE** Composition of _G\u1e5bhya S\u016btras_ (outlines of rituals for householders), Gautama's _Ny\u0101ya S\u016btra_ , Ka\u1e47\u0101da's _Vai\u015be\u1e63ika S\u016btra_ , Jaimini's _P\u016brvam\u012bm\u0101\u1e43s\u0101 S\u016btra_ (the foundational documents, respectively, of the Ny\u0101ya, Vai\u015be\u1e63ika, and P\u016brva M\u012bm\u0101\u1e43s\u0101 _dar\u015banas_ , or systems of philosophy), and Kau\u1e6dilya's _Artha \u015a\u0101stra_.\n\n**300 BCE\u2013100 CE** Composition of major _Dharma \u015a\u0101stras_ , or legal texts.\n\n**300 BCE\u2013300 CE** Composition of _Mah\u0101bh\u0101rata_.\n\n**268\u2013233 BCE** Reign of A\u015boka, Mauryan emperor and renowned convert to Buddhism.\n\n**c. 250** **BCE** Third Buddhist Council takes place at Pataliputra, under A\u015boka's auspices.\n\n**c. 200 BCE** B\u0101dar\u0101ya\u1e47a's _Ved\u0101nta S\u016btras_ composed, which summarize the philosophy of the _Upani\u1e63ads_.\n\n**200 BCE\u2013200 CE** _R\u0101m\u0101ya\u1e47a_ composed.\n\n**184 BCE** Mauryan Dynasty ends. Hindu general Pu\u015byamitra \u015au\u1e45ga founds \u015au\u1e45ga Dynasty.\n\n**166 BCE\u201378 CE** Greeks, Scythians, Bactrians, and Parthians enter India.\n\n**c. 150 BCE** Buddhist monuments of Bharhut and Sanchi are built during \u015au\u1e45ga Dynasty.\n\n**150 BCE\u2013100 CE** Pata\u00f1jali's _Mah\u0101bh\u0101\u1e63ya_ , a work of Sanskrit grammar, composed.\n\n**100 BCE\u2013100 CE** _Bhagavad G\u012bt\u0101_ composed.\n\n**100 BCE\u2013200 CE** _Tirukkural_ composed in southern India.\n\n**100 BCE\u2013500 CE** Pata\u00f1jali's _Yoga S\u016btra_ composed.\n\n**73 BCE** Hindu \u015au\u1e45ga Dynasty ends, replaced by the Buddhist Ku\u1e63\u0101na Dynasty.\n\n**78\u2013140 CE** Central Asian Ku\u1e63\u0101na king Kani\u1e63ka reigns and promotes Buddhism.\n\n**c. 100** Composition of _Manusm\u1e5bti_. Composition of Sangham poetry in southern India.\n\n**100\u2013500** Expansion of Hinduism and Buddhism into Southeast Asia through trade.\n\n**250\u20131200** Major _P\u016br\u0101\u1e47as_ composed, texts consisting largely of tales of the deeds of Hindu deities, forming the basis of much of the theological writing of the Vai\u1e63\u1e47ava, \u015aaiva, and \u015a\u0101kta theistic traditions, which all become highly prominent during this period.\n\n**c. 300** V\u0101tsy\u0101yana composes the _K\u0101ma S\u016btra_.\n\n**320\u2013550** Gupta Dynasty reigns over northern India from P\u0101\u1e6daliputra, does much to promote Hinduism in period sometimes called \"Hindu Renaissance.\"\n\n**375** Pallava Dynasty is founded in southern India.\n\n**c. 400** K\u0101lid\u0101sa writes Sanskrit plays and long poems; also the period of astronomer and mathematician \u0100ryabha\u1e6d\u1e6da.\n\n**405\u2013411** Chinese Buddhist pilgrim Faxian visits India.\n\n**c. 450** _Hariva\u1e43\u015ba_ composed.\n\n**455\u2013467** The Huns attack northern India.\n\n**460\u2013477** The Vakataka Dynasty completes the caves at Ajanta in central India.\n\n**500\u2013900** Period of the N\u0101yan\u0101r \u015aaiva Tamil poets.\n\n**550\u2013575** Kalachuris create the cave of \u015aiva at Elephanta.\n\n**550\u2013880** C\u0101lukya Dynasty, based in Bad\u0101mi, Karnataka, thrives, controlling much of southern and central India.\n\n**570\u2013632** Life of Prophet Muhammad, founder of Islam, in the Arabian Peninsula.\n\n**600\u2013900** Period of the \u0100\u1e37v\u0101r Vai\u1e63\u1e47ava Tamil poets.\n\n**606\u2013647** Reign of Har\u1e63a of Kanauj over much of northern India.\n\n**630\u2013644** Chinese Buddhist pilgrim Xuanzang visits India.\n\n**650** Arab armies reach the Indus after conquering Persia.\n\n**650\u2013800** Early Tantras are composed.\n\n**711\u2013715** Arabs invade Northwestern India.\n\n**765\u2013773** R\u0101ja K\u1e5b\u1e63\u1e47a I creates the Kail\u0101sa temple to \u015aiva at Ellora in central India.\n\n**788\u2013820** Life of \u015aa\u1e45kara (\u0100di \u015aa\u1e45kar\u0101c\u0101rya), major exponent of Advaita Ved\u0101nta and founder of the Da\u015ban\u0101mi order of monks; born in Kerala, though he travels throughout India.\n\n**c. 800** M\u0101\u1e47ikkav\u0101cakar composes the _Tiruv\u0101cakam_ in southern India.\n\n**880\u20131200** Co\u1e37a Empire dominates South India.\n\n**900\u20131150** The Chandellas build the temples at Khajuraho.\n\n**973\u20131048** Life of al-B\u012brun\u012b, Arab traveler, documents Hindu beliefs and customs.\n\n**950\u20131020** Life of \u015aaiva philosopher Abhinavagupta, in Kashmir.\n\n**1001** Mahmud of Ghazni (979\u20131030) raids north India.\n\n**1021** Ghaznavid (Turkish) Muslim capital established at Lahore.\n\n**1077\u20131157** Life of R\u0101m\u0101\u1e47uja, \u015ar\u012bvai\u1e63\u1e47ava theologian and founder of the Vi\u015bi\u1e63\u1e6d\u0101dvaita system of Ved\u0101nta, in Tamil Nadu.\n\n**1120\u20131145** Building of Angkor Wat in Cambodia, a major Hindu and Buddhist temple complex.\n\n**1192\u20131206** Muhammad of Ghor establishes Ghorid Muslim capital at Delhi.\n\n**c. 1200** Early orders of Sufis thrive in north India, V\u012bra\u015baivas (Basava) in the south. Jayadeva's _G\u012bt\u0101govinda_ , a major Vai\u1e63\u1e47ava devotional work is composed in Bengal.\n\n**1210\u20131526** The Delhi Sultanate is in power.\n\n**1238\u20131258** Narasi\u1e43hadeva I builds the sun temple of Kon\u0101rak in Orissa.\n\n**1238\u20131317** Life of Madhva, Vai\u1e63\u1e47ava theologian and founder of the Dvaita system of Ved\u0101nta, in Karnataka.\n\n**c. 1300** \u015ar\u012bvai\u1e63\u1e47avas split into Te\u1e45kalai and Va\u1e6dakalai sects.\n\n**1325\u20131351** Muhammad bin Tughluq reigns from Delhi.\n\n**1336\u20131565** Vijay\u0101nagara Empire is in its prime.\n\n**1440\u20131518** Life of Kab\u012br, major figure of the S\u0101nt movement, which draws upon both Hinduism and Islam for inspiration.\n\n**1399** Timur, ruler of Central Asia, destroys Delhi.\n\n**1469\u20131539** Life of Guru N\u0101nak, founder and first guru of Sikhism, in the Punjab.\n\n**1486\u20131533** Life of Caitanya, Vai\u1e63\u1e47ava saint and founder of Acintya Bhed\u0101bheda Ved\u0101nta, in Bengal; major inspiration for the Bengali Gau\u1e0d\u012bya Vai\u1e63\u1e47ava tradition (and, in the modern period, for both Ramakrishna and ISKCON).\n\n**1498\u20131547** Life of Vai\u1e63\u1e47ava saint M\u012br\u0101 Bai in R\u0101jasth\u0101n.\n\n**1526** Babur founds the Mughal Empire.\n\n**1530\u20131556** Reign of Mughal Emperor Humayun.\n\n**1532\u20131623** Life of Tuls\u012bd\u0101s, author of the _R\u0101mcaritm\u0101nas_.\n\n**1556\u20131605** Reign of Mughal Emperor Akbar.\n\n**1600** Queen Elizabeth I charters the British East India Company.\n\n**1605\u20131627** Reign of Mughal Emperor Jahangir.\n\n**1608\u20131649** Life of Vai\u1e63\u1e47ava saint Tuk\u0101r\u0101m.\n\n**1627\u20131658** Reign of Mughal Emperor Shah Jahan, builder of the Taj Mahal.\n\n**1658\u20131707** Reign of Mughal Emperor Aurangzeb.\n\n**1713\u20131719** Reign of Mughal Emperor Farrukhsiyar.\n\n**1757** British defeat Muslim rulers in Bengal. First wave of British R\u0101j begins under the British East India Company.\n\n**1758\u20131828** Life of Charles \"Hindoo\" Stuart, first \"white\" Hindu of the modern period.\n\n**1772\u20131833** Life of R\u0101m Mohan Roy, founder of the Brahmo Sam\u0101j.\n\n**1803\u20131882** Life of Ralph Waldo Emerson, American philosopher and a leading figure of the Transcendentalist movement, heavily influenced by early English translations of Hindu texts such as the _Bhagavad G\u012bt\u0101_ and the _Upani\u1e63ads_.\n\n**1824\u20131883** Life of Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b, founder of the \u0100rya Sam\u0101j.\n\n**1828** Founding of Brahmo Sam\u0101j by R\u0101m Mohan Roy.\n\n**1831\u20131891** Life of Helena Petrovna Blavatsky, founder of the Theosophical Society.\n\n**1836\u20131886** Life of Sri Ramakrishna, guru of Sw\u0101m\u012b Vivek\u0101nanda, in Bengal.\n\n**1838\u20131894** Life of Bankim Chandra Chatterjee, major Bengali literary figure.\n\n**1847\u20131933** Life of Annie Besant, leader of the Theosophical Society and of the Indian National Congress.\n\n**1857\u20131858** The Rebellion, or War of Independence, known in British histories as the Sepoy Mutiny.\n\n**1858** Final phase of the British R\u0101j begins when the British viceroy officially replaces both Mughal rule and the East India Company.\n\n**1861\u20131941** Life of Rabindranath Tagore, Nobel laureate and Bengali cultural hero.\n\n**1863\u20131902** Life of Sw\u0101m\u012b Vivek\u0101nanda, founder of the Ramakrishna Order, Ved\u0101nta Society, and Ramakrishna Mission.\n\n**1869\u20131948** Life of Mohand\u0101s K. (Mah\u0101tma) G\u0101ndh\u012b, leader of the nonviolent movement for Indian independence.\n\n**1872\u20131950** Life of Aurobindo Ghose, Hindu nationalist leader (in his early years) and Hindu sage and founder of Integral Yoga (in his later years).\n\n**1875** Founding of \u0100rya Sam\u0101j and Theosophical Society.\n\n**1879\u20131950** Life of Sri Ramana Maharshi, Hindu sage of southern India.\n\n**1888\u20131975** Life of Sarvepalli Radhakrishnan, philosopher and statesman.\n\n**1891\u20131956** Life of Bh\u012bmrao Ambedkar, Dalit leader and author of the constitution of India.\n\n**1893** Sw\u0101m\u012b Vivek\u0101nanda speaks at the Parliament of World Religions, in Chicago.\n\n**1893\u20131952** Life of Paramah\u0101\u1e43sa Yog\u0101nanda, founder of Self-Realization Fellowship (SRF).\n\n**1893\u20131976** Life of Sw\u0101m\u012b Prabhav\u0101nanda, prominent monk of the Ramakrishna Order and founder of the Ved\u0101nta Society of Southern California whose disciples included Christopher Isherwood and Aldous Huxley.\n\n**1895\u20131983** Life of Vinoba Bhave, Gandhian activist.\n\n**1896\u20131977** Life of A. C. Bhaktived\u0101nta Sw\u0101m\u012b Prabhup\u0101da, founder and _\u0101c\u0101rya_ of the International Society for Krishna Consciousness (ISKCON).\n\n**1896\u20131982** Life of \u0100nandam\u0101y\u012b M\u0101, female guru and Hindu saint.\n\n**1897** Founding of Ved\u0101nta Society in America by Sw\u0101m\u012b Vivek\u0101nanda.\n\n**1916\u20131993** Life of Sw\u0101m\u012b Chinmay\u0101nanda.\n\n**1918\u20132008** Life of Maharishi Mahesh Yogi, founder of Transcendental Meditation.\n\n**1919** Amritsar massacre takes place, helping turn the tide of world opinion against British rule of India.\n\n**1927\u20132001** Life of Satguru Sivaya Subramuniyaswami, founder of _Hinduism Today_.\n\n**1931\u20131990** Life of Bhagwan Shree Rajneesh (Osho), controversial Tantric guru.\n\n**1935** Birth of Sw\u0101m\u012b Agnive\u015ba, founder of the World Council of the \u0100rya Sam\u0101j.\n\n**1947** Independence and partition of India on 15 August.\n\n**1948** Assassination of Mohand\u0101s K. G\u0101ndh\u012b by Nathuram Godse on 30 January.\n\n**1951** Bh\u012bmrao Ambedkar leads mass conversion of Dalits to Buddhism.\n\n**1953** Birth of Amma (M\u0101t\u0101 Amrit\u0101nandamay\u012b), the \"Hugging Saint.\"\n\n**1966** Founding of ISKCON in New York City.\n\n**1968** The Beatles travel to Rishikesh, India, to study Transcendental Meditation with the Maharishi Mahesh Yogi.\n\n**1970** Hindus in Europe, the United States, and Canada start building temples on a large scale.\n\n**1984** Massive Hindu\u2013Sikh riots sparked by Indian army attack on the Golden Temple of Amritsar and assassination of Prime Minister Indira Gandhi. Many Sikhs immigrate to the United States, Canada, and the United Kingdom.\n\n**1992** Demolition of Babr\u012b Masj\u012bd in Ayodhy\u0101, followed by Hindu\u2013Muslim riots.\n\n**2002** Hindu\u2013Muslim riots in Ahmedabad, Gujarat, claim hundreds of lives.\n\n**2004** Manmohan Singh elected first Sikh prime minister of India.\n\n**2009** President Barack Obama acknowledges Hinduism as a significant part of religious life in America in his inaugural address; later in the year, he sends Diwali greeting to Hindus, Jains, and Sikhs.\n\n**2010** Indian high court resolves Babr\u012b Masj\u012bd issue by dividing the land between Hindu and Muslim organizations. Though there are objections to the decision, the responses of both sides are nonviolent.\n\n# Introduction\n\n_Hinduism_ \u2014the name evokes a topic so vast that one despairs of being able to cover it in an adequate fashion in a single volume. At the same time, this term does point to something distinctive\u2014a tradition, or rather a family of traditions, that today commands the loyalty of over 900,000,000 human beings, making it the third-largest religion in the world, in terms of the numbers of its adherents, after Christianity and Islam. It is the fourth largest if one considers the category variously described as \"nonreligious,\" \"secular,\" \"agnostic,\" or \"atheist\" to be something akin to a religion (or at least a distinct worldview and value system), which ranks third. It nearly ties with Islam, however, if one regards Buddhism, Jainism, and Sikhism\u2014as many Hindus do\u2014as part of the same family of traditions that is called _Hinduism_. (The reactions of Buddhists, Jains, and Sikhs to being perceived by at least some Hindus as, in some sense, fellow Hindus, vary greatly, ranging from a sharp rejection to indifference to acceptance.)\n\nThe Hindu community is today a global one, though the vast majority of Hindus reside in India and in countries immediately adjacent to it, such as Pakistan, Bangladesh, Nepal, and Sri Lanka. Contrary to one popular understanding, however, Hinduism is not limited only to people with an Indian or South Asian ethnic identity. There have been Hindus in Southeast Asia\u2014that is, Hindus indigenous to Southeast Asia\u2014for over a millennium and a half, and Bali, in Indonesia, continues to be home to a thriving Hindu community today. There is also a growing number of converts to Hinduism from Europe, North and South America, Australia, and Africa, as well as practitioners of traditions with a clear Hindu provenance\u2014such as Siddha Yoga, Ved\u0101nta, and Transcendental Meditation\u2014that do not necessarily identify themselves as _Hindu_. And there are those non-Hindus who observe Hindu practices and hold Hindu beliefs without necessarily realizing that this is what they are doing: vegetarians, practitioners of yoga, believers in reincarnation, and those who believe that there are many paths to salvation. Enough such people exist in the Western world to lead a recent commentator on religion to remark, \"We're all Hindus now.\"1\n\nBut what is Hinduism? What is the history of this vast and internally varied tradition? It has become something of a truism among scholars of religion to say that \"Hinduism does not exist.\" Although offensive to many Hindus (since no one likes to be told the tradition they practice and hold dear does not exist), the truth to which this claim, rather clumsily, points is that the term _Hinduism_ has, historically, been imposed upon what are now called Hindu traditions by outsiders, many of whom were invested in the colonial exploitation of the Indian subcontinent. The terms _Hindu_ and _Hinduism_ were utterly foreign to most Hindus for most of history\u2014a fact that can be expressed far less paradoxically in the form \"There was no Hinduism for most of history, until 19th-century scholars created it.\" But this is also highly unsatisfactory, because those who do regard themselves as Hindus also see themselves and their traditions as being in continuity with a very ancient set of practices and beliefs that is as old as civilization itself.2 In many works by Hindu authors, one frequently encounters the claim that Hinduism is the oldest of the world's religions.\n\nThe term _Hindu_ \u2014derived from a Persian mispronunciation of _Sindhu_ , the river known to the Greeks as the _Indus_ \u2014was originally used by Persian and Arab travelers to refer to the people who lived beyond the Indus River. In other words, _Hindu_ , originally just meant _Indian_. This usage continued into the period of the British R\u0101j, whose colonial scholars coined the term _Hinduism_ to refer to the religious beliefs and practices native to India, in contrast with such non-indigenous Indian religions as Christianity, Islam, Zoroastrianism, and Judaism\u2014all of which have been practiced in India for centuries, alongside the native Indian traditions. Over the course of time, as Western knowledge of the Indian traditions increased, a distinction was made between those traditions that professed the sanctity of an ancient set of religious texts called the _Vedas_ and those that did not. _Hinduism_ came to refer, then, to the subset of indigenous Indic traditions that observed the sanctity of the _Vedas_. The non-Vedic traditions\u2014Buddhism, Jainism, and Sikhism\u2014came to be defined as distinct and separate religions.\n\n_Hinduism_ , defined as the collective term for the Vedic traditions, is a term with a number of flaws and limitations. On the one hand, it obscures the many differences among these traditions\u2014including the degree to which the _Vedas_ are even relevant to their day-to-day practice. The daily rituals of a Sm\u0101rta Brahmin, for example, are quite heavily shaped by Vedic injunctions and additional injunctions that are contained in a set of texts based on the _Vedas_ \u2014the _Sm\u1e5btis_ \u2014that this community takes to be authoritative. But such injunctions are altogether irrelevant to the spiritual life of a V\u012bra\u015baiva. And a member of the Aghor\u012b sect will even deliberately invert them. On the other hand, the definition of _Hindu_ as _Vedic_ can also downplay the many close similarities between Vedic traditions and non-Vedic ones. The basic worldviews of a Ved\u0101ntin, a Buddhist, and a Jain, for example, are not all that different from each other on a host of religiously important topics. And if one goes to Sanjusangendo, a famous Buddhist temple in Kyoto, Japan, which enshrines 1,001 forms of the bodhisattva of compassion, Avalokite\u015bvara, one will find that these forms are guarded by images of all of the major Vedic deities. Not far from Kyoto, the Buddhist monks of the Shingon sect regularly perform a fire ceremony that is called _goma_ and is clearly modeled on the Vedic _homa_ ritual at their monastery on Mount Koya. The Jain community in India celebrates the holiday D\u012bwali by worshiping the goddess Lak\u1e63m\u012b. Numerous other examples of shared practices, deities, symbols, and concepts could be cited as well, not least being the cosmology of karma and rebirth and a soteriology centered on liberation from this process.\n\nThe understanding of _Hinduism_ , however, as Vedic religion, with whatever inadequacies this understanding may have, is the one that is deployed by most mainstream scholarship on Hindu traditions. One must always remember that terms and categories are scholarly tools. They do not necessarily correspond precisely to what is \"there\" in terms of ground realities. Rather, they seek to clarify for us what is \"there.\" Sometimes they are useful in doing this, and at other times they may need to be set aside in favor of other categories that may be more illuminating in particular situations.\n\nWith the understanding, therefore, that the terms _Hindu_ and _Hinduism_ are relatively new arrivals on the scene of world history and that our use of them to refer to ancient realities is an anachronism\u2014referring actually to practices and beliefs with which those who today call themselves _Hindu_ see their own traditions to be in historical continuity\u2014we can turn to an overview of the history of Hinduism.\n\n## THE QUESTION OF ORIGINS: HINDUISM AND EARLY INDIAN CIVILIZATION\n\nUnlike most of the world's religions, Hinduism does not originate from the teachings of a single founding figure. There is no equivalent, for all of Hinduism, to a Jesus, a Buddha, or a Prophet Muhammad. There are, instead, many such figures\u2014founders of particular lineages or \"subtraditions\" that collectively make up what we now call Hinduism. And there are innumerable local traditions that have emerged in conversation with one another over the course of many centuries. Hinduism, one could say, is the ethos of a civilization. Presentations of the history of Hinduism, therefore, usually begin with a look at the earliest phases of civilization in the Indian subcontinent.\n\nHuman beings have been present in the Indian subcontinent almost as long as there have been human beings. One of the first waves of human migration out of Africa carried a group of early humans to southern Asia, and various groups of migrants have passed both in and out of the subcontinent ever since. Indian Stone Age culture is in evidence starting around 50,000 BCE. The Bhimbetka cave paintings, traced to around 30,000 BCE and discovered in the central Indian state of Madhya Pradesh, depict human beings engaged in warfare, riding on horses and elephants, and deploying technologies such as spears and tents (probably made from the hides of animals).3 The first settled, non-nomadic cultures began to appear in the region of Mehrgarh (a town in what is now Pakistan) around 7000 BCE. The people of Mehrgarh practiced a mix of pastoralism and agriculture.4\n\nThere is close continuity\u2014a claim based on the artwork depicted on pottery\u2014between this early community and the very highly developed Indus Valley Civilization, which was at its technological height from roughly 2600 to 1900 BCE. With planned cities and a very advanced knowledge of sanitation and irrigation, the people of the Indus Valley achieved one of the highest standards of living in the ancient world.5\n\nUnfortunately, the writing system of this civilization has not yet been deciphered. This makes all claims about its religious beliefs and practices a matter of conjecture, based on interpretation of artifacts rather than on the direct statements of the Indus people. A good deal of evidence has been discovered, however, that indicates a number of continuities\u2014though not necessarily a perfect correspondence\u2014between beliefs and practices in the Indus Valley and those of later Hinduism. Examples include figures that appear to be practicing some form of yoga, figures with features in common with the Hindu deities \u015aiva and \u015aakti, large public baths and a broader cultural preoccupation with cleanliness and sanitation and a reverence for water that all resonate with Hindu practice, and figures making the traditional Hindu gesture of respect\u2014or _a\u00f1jali_ \u2014that consists of the palms of the hands being placed together. Other practices of the Indus civilization, not of a religious nature, that persist in later Indian culture include its highly developed system of weights and measures and a system of social organization based on occupation that has some resemblance to the later caste system. It is not clear if the occupational groups in the Indus civilization were arranged in a hierarchy. The different groups appear to have had segregated living arrangements, which suggests that they were endogamous, like the castes of later Indian society. But there is no discernible difference in the quality of their living arrangements, which suggests an egalitarian, rather than hierarchical, relationship. Interestingly, some contemporary Hindus do claim that the castes were originally on an equal footing.6\n\nThe Indus Valley Civilization entered a rather drastic period of decline around 1900 BCE due primarily to geological factors. A series of devastating earthquakes resulted not only in the destruction of many Indus cities but also in the Indus River changing its course, as a result of which some cities were flooded and others that had been prosperous trading ports were left high and dry.\n\nRecent studies have also revealed that the Indus had a companion, the Sarasvat\u012b\u2014a river that, judging by the sheer number of settlements found along what used to be its banks, may have been more central to the life of the civilization than the Indus itself. Because of this, some scholars have argued that the civilization should be called not the Indus Valley Civilization, but the Indus-Sarasvat\u012b Civilization (or even the Sarasvat\u012b Civilization). The Sarasvat\u012b River dried up completely in some places due to the same tectonic shifts that changed the course of the Indus, and in some places it went underground. What little of it that remains on the surface is a periodic stream\u2014that is, a stream that is dry during part of the year and has running water during the rainy season\u2014called the Gaggar-Hakra.\n\n## THE EARLY VEDIC PERIOD\n\nAccording to mainstream scholarship, the next phase of Indian civilization, and so of the history of Hinduism, begins with the arrival in India of small waves of migrants from the area of Central Asia. Filtering gradually into the northwestern region of the subcontinent beginning around 1900 BCE, these migrants spoke an Indo-European language and used small, light chariots of a kind developed in southern Russia between 2300 and 2000 BCE. Settling in India and gradually blending into the indigenous population, the language and cultural practices of this community are preserved in the _Vedas_ , the oldest extant texts of Hinduism. These texts were gradually compiled into four collections called the _\u1e5ag Veda_ , _S\u0101ma Veda_ , _Yajur Veda_ , and _Atharva Veda_. These texts date from roughly 1700 BCE (in the case of the oldest parts of the _\u1e5ag Veda_ , the oldest of the Vedic texts) to 900 BCE (the approximate date of the _Atharva Veda_ , the last of the Vedic collections, or _Sa\u1e43hit\u0101s_ , to be compiled).\n\nGeographic points of reference in the _Vedas_ suggest that the texts were composed in the northwestern region of the Indian subcontinent, showing greatest familiarity with the area now known as Punjab. They also suggest that the community that composed them did not see themselves as newcomers to the region, there being no overt references to another homeland or to migration from such a homeland to India. Vedic Sanskrit, the language in which these texts were composed, also shows evidence of a non-Indo-European substrate, suggesting that this community was multilingual, or at least that it had already had a good deal of interaction with persons of another language group, adopting words and ways of speaking different from those of their Indo-European-speaking ancestors.\n\nThe _Vedas_ and the religious practices and beliefs that they represent were not, therefore, imported _in toto_ to the subcontinent from outside. They are, rather, the product of a long process of cultural hybridization of Indo-European and non-Indo-European elements that had already been underway for some time\u2014at least two centuries, if current scholarship is correct\u2014prior to the composition of these texts.\n\nIt was long believed by many scholars that Indo-European culture was brought to India in a sudden and violent invasion and that this invasion was the cause of the fall of the Indus Valley Civilization. But it is now known that this was not the case. The evidence points instead to the geological changes mentioned above as the chief culprit in the decline of the Indus culture. The Indo-European migrants did not cause the fall of, but rather filled a vacuum created by the already declining, Indus Valley Civilization.\n\nIt is also likely that the descendants of the Indo-European migrants helped to preserve the Indus culture. Vedic references, for example, to the Sarasvat\u012b River in the present tense, as a wide, flowing river with many settlements on its banks, suggest that the _Vedas_ are a repository not only of Indo-European culture but of indigenous Indus Valley Civilization cultural elements and other indigenous Indian elements as well\u2014an idea that is consistent with the thesis that these texts are the product of centuries of cultural hybridization.\n\nThis picture of the past is a point of contention among some scholars today. An alternate view, according to which the _Vedas_ predate the Indus Valley Civilization, being written at the time of the mixed pastoral and agricultural society that existed between 7000 BCE and the advanced urban phase of the Indus Valley that began around 2600 BCE, has been defended by a number of scholars\u2014particularly Hindu scholars\u2014who argue that this view is more consistent with traditional Hindu accounts of ancient Indian history and point to the highly conjectural nature of the dates assigned to the Vedic texts, as well as political misuses of early, racially tinged versions of the \"\u0100ryan Invasion Theory.\" This \"Out of India\" theory suggests that northwestern India, not Central Asia, is the Indo-European homeland and that Indo-European language and culture spread from India to Ireland (the extreme eastern and western ends of the Indo-European cultural sphere), rather than out of Central Asia in both directions.7\n\nMainstream scholarship tends to reject this view, based on such factors as the non-Indo-European substrate elements in Vedic Sanskrit\u2014the most ancient Indo-European language found in India\u2014and various discontinuities between the culture of the Indus Valley and that described in the _Vedas_ \u2014such as the nearly total absence of the horse in the Indus Valley culture and its ubiquity in Vedic writings. The Indo-European homeland would show evidence of a \"pure\" Indo-European language, not an Indo-European language that has been transposed onto an older set of indigenous languages, as Vedic Sanskrit is. And if the Indus Valley culture is in direct continuity with Vedic culture\u2014rather than being one element among many that constitute it\u2014one would expect stronger continuities than those that have been discovered.\n\nAt the same time, it is fair to say that there remain a good many unknown factors at play in this issue\u2014not the least of which are the language and the meaning of the small pieces of text that are available in the Indus Valley script. Therefore a full, comprehensive, and uncontested picture of the ancient history\u2014or prehistory\u2014of Hinduism remains elusive.8\n\n## THE LATER VEDIC PERIOD: THE EMERGENCE OF THE FOUR YOGAS\n\nOur picture of Hinduism becomes much clearer, and less contested, as we move beyond the early Vedic collections and into the later Vedic texts of the first millennium BCE\u2014like the _Br\u0101hma\u1e47as_ (\"Priestly Texts\"), _\u0100ra\u1e47yakas_ (\"Forest Texts\"), and _Upani\u1e63ads_ (\"Esoteric Teachings\"). Through these successive \"layers\" of text, one can trace the development of Vedic thought from a series of reflections on the nature of ritual to a grand cosmological vision of the oneness of existence.\n\nLater Hindu tradition conceives of spiritual practice as consisting of three, and sometimes four, distinct disciplines ( _yogas_ ) or paths: the _karma yoga_ , or the way of action; the _j\u00f1\u0101na yoga_ , or the way of wisdom; and the _bhakti yoga_ , or way of devotion. _Dhy\u0101na yoga_ , or meditation, (also known as _r\u0101ja yoga_ , the royal path) is sometimes added to these as a fourth.9 Each of these four types of Hindu practice can be seen as representing a distinct historical trend, or movement, within the history of the tradition.\n\nKarma yoga, the way of action, originated as the practice of Vedic ritual as outlined in the early Vedic collections of hymns and the _Br\u0101hma\u1e47as_ , or \"Priestly Texts,\" which date from roughly 800 to 600 BCE. The main emphasis of early Vedic religion was correct performance of the ritual of sacrifice, or _yaj\u00f1a_ , which was seen as essential to upholding and maintaining the world. The oldest Vedic sacrifices were modeled on and conceived as repetitions of the original sacrifice by which the gods (or _devas_ ) created the world from the body of a primordial being\u2014the Cosmic Man, or _puru\u1e63a_. These sacrificial rituals tapped into the power of creation itself\u2014an energy field known as Brahman (literally, \"that which is expansive\"). The Vedic priest, or Brahmin, was one who had the knowledge of the correct performance of the sacrifice and was thus able to tap into and deploy this creative power for ends such as ensuring the health and prosperity of the community.\n\nIn the _\u0100ra\u1e47yakas_ , or \"Forest Texts\" (composed from around 600 to 500 BCE) and even more fully in the _Upani\u1e63ads_ (secret or esoteric doctrine\u2014composed from around 500 BCE up to the first century of the Common Era), the emphasis shifted from the performance of ritual action to the transformative power of the knowledge that made such ritual action effective. This knowledge came to be seen as the true and final purpose, or \"end,\" of the Veda ( _Ved\u0101nta_ ), of which the ritual was merely an outward expression or manifestation. It was a knowledge that consisted of the realization that Brahman, the ground of being, and one's innermost self, or _\u0101tman_ , were one and the same.\n\nWith this emphasis on the way of wisdom\u2014what later came to be known as j\u00f1\u0101na yoga\u2014there emerged a distinctive worldview in which the ritual of the sacrifice was extended to encompass all of life. A ritual act, or _karma_ , which in the early Vedic literature was said to lead to a desired effect if performed well and to its opposite if performed poorly, came to refer to all action. Karma began to take on a distinctly ethical tone, as referring to any morally good or evil action, along with the inevitable effects of such actions\u2014an effect of the unity of existence as constituting a single, internally connected cosmic ritual arena.\n\nBecause one did not experience all the good and bad effects of one's actions in a single lifetime, it was deduced that one would eventually be reborn to experience these effects in a future lifetime. Similarly, the unequal circumstances of the births of all living beings came to be explained as being the result of actions performed in previous lives. The goal of the way of wisdom is to become free from this cycle of action and reaction: to achieve a true and lasting peace rather than the merely temporary goods and to escape from the inevitable loss and suffering that characterize life in the rebirth cycle.\n\nThe way of wisdom was seen as difficult, requiring one to separate oneself from the day-to-day activities and ritual responsibilities characterizing the way of action\u2014actions that could draw one back into the cycle of rebirth. Renunciation\u2014or _sanny\u0101sa_ \u2014is the way of life of the Upani\u1e63adic sage who has withdrawn from society to seek spiritual wisdom, a way of life still pursued by Hindu ascetics today.\n\nIn the later _Upani\u1e63ads_ , such as the _\u015avet\u0101\u015bvat\u0101ra Upani\u1e63ad_ (composed between about 400 and 300 BCE), but even more so in the post-Vedic _sm\u1e5bti_ literature of the first millennium of the Common Era, another path emerged that made the goal of liberation from rebirth available to a wider cross section of society\u2014one that did not rely on Brahmanic priestly ritual or the wisdom of ascetic renouncers. This was the way of devotion, or bhakti yoga, the basic premise of which is that the supreme Brahman that is the ultimate object of all spiritual aspiration is not merely an impersonal energy or a pure state of consciousness but is also a loving personal deity who desires the salvation of all beings. One opens oneself to the saving power of the divine by cultivating a personal relationship with this supreme reality\u2014a relationship of loving devotion, or bhakti.\n\nThe precise form that this devotion takes varies within distinct Hindu traditions, which orient themselves toward Vi\u1e63\u1e47u, \u015aiva, the Mother Goddess or universal energy (\u015aakti), or any of a number of other personae of the supreme reality. These distinct theistic Hindu traditions\u2014Vai\u1e63\u1e47avism, \u015aaivism, \u015a\u0101ktism, and other theistic traditions centered on other forms of the divine\u2014have been the dominant forms of Hinduism from the first millennium of the Common Era to the present.\n\nThe fourth yoga\u2014dhy\u0101na or r\u0101ja yoga, the discipline of meditation\u2014is another current of Hindu practice that has coexisted and interacted with the other disciplines\u2014Vedic ritual, Ved\u0101ntic contemplation, and theistic devotion\u2014for millennia. It is a practice that could be an inheritance of the Indus Valley Civilization. It was cultivated primarily by renouncer traditions, both Vedic and non-Vedic (like the Jain and Buddhist traditions), in the first millennium before the Common Era\u2014and is also a prominent theme of the _Upani\u1e63ads_. It finds its classic form of expression in the _Yoga S\u016btra_ of Pata\u00f1jali (which may have been composed, in its current form, anywhere from 100 BCE to 500 CE). Its goal is the direct realization of Brahman not as an object of knowledge or devotion, but experientially, through the absorption of consciousness in Brahman to such a degree that the distinction between subject and object that characterizes most experience disappears\u2014an experience known as _sam\u0101dhi_.\n\n## THE CLASSICAL PERIOD: FROM ELITE BRAHMANISM TO POPULAR HINDUISM\n\nThe distinct cultural threads represented by the four yogas have rarely been practiced in isolation from one another throughout the history of Hinduism. Rather, these threads are like the threads of a tapestry, interweaving and interconnecting to make up the total way of life and the civilizational ethos that is Hinduism.\n\nThe path of action\u2014which originated in the ritual practices enjoined in the _Vedas_ \u2014came to encompass all of society and all of life in the form of _dharma_ , or duty. The _Dharma \u015a\u0101stras_ , or legal texts, which were composed from approximately 300 BCE to 100 CE, set out a comprehensive vision of both society and the life of the individual\u2014the former being encompassed by the system of castes, or _var\u1e47as_ , the latter being encompassed by a series of four _\u0101\u015bramas_ , or stages of life. The path of wisdom found its expression in the _s\u016btra_ literature of the six systems of philosophy, or _dar\u015banas_ (\"views\"). Both of these are elite literatures, composed in Sanskrit by Brahmins, and tend to orient themselves in terms of the authority of the _Vedas_ , with the Vedic systems being regarded as \"orthodox\" ( _\u0101stika_ ) and the non-Vedic systems (Jainism, Buddhism, and materialism) being regarded as \"unorthodox\" ( _n\u0101stika_ ).\n\nThe path of devotion, however, found expression not only in elite literatures, such as the _Pur\u0101\u1e47as_ of the various theistic schools of thought, but also in the ecstatic poetry of devotees hailing from all the stations of society\u2014including many women\u2014composed in vernacular languages, such as Tamil, Bengali, and Hindi. The bhakti movement, starting early in the first millennium of the Common Era and gaining momentum in the second, did not so much overturn the Brahmanical social order as render it irrelevant to the spiritual path, offering liberation from the rebirth cycle to all sincere devotees, regardless of caste or gender. This movement was so popular that it had a significant impact upon the formal religion of the Brahmins, which itself came to emphasize theistic devotion, while at the same time maintaining its earlier emphasis on ritual purity. The tension between these two continues to characterize Hinduism today, with \"orthodox\" Brahmins emphasizing formal ritual\u2014denying entry into temples, for example, to persons regarded as impure\u2014and others, both Brahmin and non-Brahmin (and even non-Indian), emphasizing sincerity of devotion as evidenced in one's character and personal qualities as being a more important measure of spiritual attainment than caste or social status.\n\nThe tension between popular spiritual aspiration and formal Brahmanical injunctions can be traced back to the first millennium BCE, when the sages of the _Upani\u1e63ads_ downplayed the performance of Vedic ritual and emphasized ascetic practice and moral qualities such as honesty, which were seen as a barometer of spiritual realization. Even more radical than the sages of the _Upani\u1e63ads_ were the ascetics of the _\u015brama\u1e47a_ , or \"striver\" movement, who did not merely de-emphasize but actively rejected the Vedic ritual of sacrifice\u2014especially the practice of animal sacrifice, which came to be seen as an inherently violent karma that could only lead to a bad rebirth, rather than as an offering that pleasing to the gods.\n\nDue to their rejection of Vedic authority, the \u015brama\u1e47a traditions are seen by most scholars as distinct traditions from Hinduism\u2014which, the reader will recall, has come to be defined as the family of Vedic traditions. The two surviving \u015brama\u1e47a traditions that are practiced today are Buddhism and Jainism. But though seen as different religions according to the current scholarly conventions, these traditions had an enormous impact on the practice of what has come to be regarded as Hinduism, with the practice of animal sacrifice being largely abandoned by even the most orthodox of Brahmins and replaced, in most parts of India, by vegetarian substitutes. The bhakti movement takes the \u015brama\u1e47ic critique of the Brahmanical tradition, in one sense, even further. Though maintaining at least a nominal reverence for the _Vedas_ , it makes liberation available to all\u2014even the householder.\n\nAt the furthest remove from Brahmanical orthodoxy\u2014or, more accurately, \"orthopraxy,\" given that the conservative Brahmanical emphasis is on right behavior and allows for a good deal of variety in the area of belief (such as six systems of philosophy)\u2014is Tantra. A system of yoga, or spiritual practice, that inverts the traditional yogic emphasis on turning away from the senses and bodily impulses, such as sexuality, Tantra, like bhakti, is in its origins a popular movement that makes advanced spiritual attainment available to anyone regardless of caste, gender, or social status. Tantric practice is based on the premise that one can utilize the senses in order to transcend the senses. At its most extreme, the \"Left-Handed\" or _V\u0101m\u0101c\u0101ra_ practice of Tantra deliberately transgresses Brahmanical norms of purity, the aim being to demonstrate the nondualistic realization that all is Brahman. If all is Brahman, then purity and impurity are effects of a false, dualistic consciousness that seeks to divide the fundamental unity of existence. Practitioners meditate in cremation grounds\u2014traditionally regarded as a highly impure, not to mention frightening, space\u2014and utilize human skulls as begging bowls. Other rituals are of an overtly sexual nature. Less extreme, but based on the same fundamental philosophy of using the senses in order to transcend the senses, are practices such as the chanting of mantras and the use of very beautiful visual images such as _m\u016brtis_ (images of deities), _yantras_ (abstract geometric forms of a deity), and _ma\u1e47\u1e0dalas_ (diagrams of the universe) as objects of contemplation.\n\nAs with bhakti, the Brahmanical tradition has been open to assimilating and appropriating these popular forms of worship. Though few Hindus today would describe themselves as practitioners of Tantra, the entire structure of Hindu temple ritual is based upon Tantric principles, and the use of mantras and yantras is part of the training of even conservative Brahmins. Like bhakti, Tantric practice emerged in the latter half of the first millennium of the Common Era. Both bhakti and Tantra may be much older than these dates suggest, but it is to this period that the texts that articulate the principles of these traditions can be traced, with bhakti texts emerging somewhat earlier than Tantric writings.\n\n## HINDUISM AND ISLAM\n\nAs the first millennium ended and the second began, India began to experience incursions from outside powers, such as the Arabs, the Turks, and the Mughals. With the coming of Islam\u2014at first brought peacefully, through trade, and then through successive waves of Arab, Turkish, and Mughal military incursion\u2014Hinduism entered a period characterized, on the one hand, by great insecurity, and on the other, by equally great creativity. There is some evidence that the caste system, at this point, became more rigid in practice than ancient texts suggest had been the case in earlier periods\u2014a process that intensified later, under the British R\u0101j.10 Brahmanical texts of this period indicate a sharply pronounced preoccupation with the preservation of society coupled with an almost apocalyptic sense of impending doom. Due to the foreign invasions, many temples and monastic institutions were destroyed at this time, and major blows dealt to the Indic traditions. Buddhism died out almost completely in India during this period, surviving as a tradition only because it had already spread across Asia. It had effectively ceased to exist in India by the year 1300. Hindu traditions survived this period because the practices, stories, and beliefs that make up the Hindu civilizational ethos pervaded the lives of the common people and were not dependent on vulnerable monastic institutions, making them difficult to uproot. The decentralized nature of Hinduism essentially spared it the fate of Buddhism, as did the peripatetic nature of most of Hindu monasticism. Powerful Hindu dynasties, such as the sizeable empire of Vijayan\u0101gara, thrived in the southern half of the subcontinent until the 16th century but eventually succumbed to repeated invasion and attack.\n\nAs the initial shock of invasion subsided, however, and Islam became a part of the Indic religious landscape, Hindus and Muslims began to respond in very creative ways to one another. Adherents of Hindu bhakti movements and the Sufis of Islam, especially, began to see one another's traditions as potential repositories of deep spiritual wisdom and of practices that each could appropriate for the attainment of greater nearness to the divine. Hindus began to adopt Sufi spiritual guides, or _p\u012brs_ , as _gurus_ , or teachers, and Muslims were similarly drawn toward Hindu spiritual figures. Kab\u012br, for example, who lived from 1440 to 1518, is even today claimed by both Hindus and Muslims as a revered teacher of sacred wisdom.\n\nThis interreligious cooperation and cross-fertilization is remarkable given the fact that it is hard to conceive of two religions more different from one another than Hinduism and Islam. Islam is emphatically monotheistic, regarding even the Christian trinity as a pagan remnant and a corruption of the original monotheism taught by the Prophet Jesus (as he is seen by Muslims). It is equally emphatically aniconic, associating images of the divine with the corrupt social order of pre-Islamic Arabia, in which a wealthy priesthood could determine who got to see the gods and who did not, charging exorbitant rates to devotees for a glimpse of their deities. The diversity of the forms of the divine in Hinduism and its celebration of imagery are seen even today by many Muslims as blasphemous\u2014a holdover from a pagan era.\n\nSimilarly, Islam is generally insistent that there is one way to salvation: through obeying the injunctions of God as conveyed by God's messenger, the Prophet Muhammad. This is a stark contrast with the pluralism of yogas and forms of the divine available as objects of devotion in Hinduism (though there are Hindu traditions that are equally insistent upon the unique efficacy of particular practices and the unique divinity of specific figures, such as the Vai\u1e63\u1e47ava Dvaita Ved\u0101nta of Madhva, which sees devotion to K\u1e5b\u1e63\u1e47a alone as being efficacious for salvation, though such exclusivism is more the exception than the rule for Hindu traditions). Islam, consequently, is an actively proselytizing tradition, while most Hindu traditions are not.\n\nIn the popular piety of medieval India, however, one sees members of both communities\u2014Hindus and Muslims\u2014shaking off the formal distinctions that their respective religious leaders would prefer to maintain between the two traditions. Rather than focusing upon their differences, however, Hindu and Muslim figures of this time emphasized a shared sense of the ubiquity of the divine presence and its availability to all who approach it with true devotion and a sincere and humble heart. Sufi saints began to incorporate the names of Hindu deities into their litanies of the many names of \u0100ll\u0101h, and Hindu devotees began to chant verses from the Qur'an and to make pilgrimages to the tombs of Sufi saints. The sharing of one another's holy days and holy places remains a characteristic of Hindu and Islamic practice in many parts of South Asia even today.11\n\nThis mutual accommodation was made possible by the strong presence in both traditions of movements that were generally uncomfortable with\u2014and often rejected in the strongest terms\u2014any emphasis on rigid formality, which was seen as interfering with the sincere and spontaneous quest of the individual devotee for the experience of the divine presence. Sufism\u2014viewed in some quarters of the Islamic world with suspicion as a movement of dubious orthodoxy\u2014found a highly receptive welcome in South Asia. And in Hinduism, the bhakti movement, with its rejection of Brahmanical formality and caste distinctions, was open to sincere devotion in any form\u2014even forms that at first appeared foreign. Sufi and bhakti leaders clearly saw in one another kindred spirits in the search for authentic experience as opposed to the dry formality of the \"official\" exponents of their respective traditions.\n\nThis process of popular mutual assimilation was eventually facilitated at the level of the state. The Mughal emperor Akbar, who reigned from 1556 to 1605, formally adopted a policy of toleration toward all religions\u2014including the Christianity taught by the European missionaries arriving with Portuguese merchant ships on the southwest coast of India in his time. Allowing his Hindu wife to build a temple to her deity, K\u1e5b\u1e63\u1e47a, in his palace and even developing his own religion, a synthesis of all of the religions of India of which he knew, Akbar consistently ran afoul of the exponents of Islamic orthodoxy in his imperial court. His policy of toleration both mirrored and accelerated what was already happening at the popular, village level between Hindus and Muslims in his realm.\n\nThe _Sant_ \u2014or \"saint\"\u2014movement of this period consisted of figures of Hindu and Islamic origins, such as the aforementioned Kab\u012br, who claimed and were claimed by both religions, seeing sincere devotion as having far greater importance than formal sectarian affiliation or religious identity. This movement culminated in the figure of Guru N\u0101nak, who lived from 1469 to 1539. In his famous teaching that \"there is no Hindu, there is no Muslim,\" N\u0101nak captured the spirit of this era. Rather than leading to a unification of the two traditions, however, N\u0101nak's teaching led to the emergence of a third, new tradition\u2014Sikhism\u2014a religion distinct from both Hinduism and Islam, while yet containing elements of both in its practices and doctrines.\n\nAt the same time, however, there were reactions against this widespread spirit of mutual synthesis and accommodation. The Mughal emperor Aurangzeb, who reigned from 1658 to 1707, aggressively sought the forcible conversion of both Hindus and Sikhs to Islam, returning to the earlier policy of destroying temples and torturing and slaughtering any intransigent religious leaders who refused to convert. The memory of the wounds of this period still runs sufficiently deep to lead to a situation in which later leaders\u2014including the British and the political parties of independent India and Pakistan\u2014have been able to exploit the fears of both communities for the sake of political gain. And yet such division and mutual suspicion occurs against a wider backdrop of toleration, mutual respect, and assimilation established in the period of the more enlightened interreligious relations that characterized the Sant movement and the rise of Sikhism. But this larger environment of tolerance and pluralism is endangered by militancy and a hardening of negative attitudes on both sides\u2014a militancy driven at least as much by the aspirations of Hindu and Islamic political organizations as by what one might call authentically religious concerns.\n\n## HINDUISM AND THE WEST\n\nThe European\u2014and especially the British\u2014colonization of India, which began in earnest in the 18th century, marks another occasion for great insecurity and also great creativity for the Hindu traditions. The twin challenges of Western science and Christianity led to a crisis for many Hindus, especially in Bengal, in eastern India, in the late 18th and early 19th centuries, due to the fact that Bengal was the first region to experience a large-scale British presence, Calcutta being the main administrative center of the British East India Company. Hindu responses to British culture ranged from, on the one hand, complete indifference\u2014particularly from those orthodox Brahmins who saw the foreigners as mere barbarians, with nothing of great interest to say religiously or philosophically\u2014to a total capitulation on the part of those Hindus who found European civilization superior to their own. The most creative response, however, came from those Hindus who found much to admire in British culture\u2014accepting the validity of many criticisms of Hinduism launched by Christian missionaries\u2014but who were also critical of what they perceived as Western materialism and a lack of spiritual depth that they found in their ancient texts and systems of thought.\n\nThe first of the Hindu reformers, often called the \"father of modern Hinduism,\" was R\u0101m Mohan Roy, who lived from 1772 to 1833 and founded a reform organization called the Brahmo Sam\u0101j, or \"Community of Brahman.\" Translating the _Upani\u1e63ads_ into vernacular languages and successfully lobbying the British administration to ban the practice of _sat\u012b_ , or widow immolation, Roy interpreted Hinduism in monotheistic and Unitarian terms, as a wisdom tradition centered on the worship of a formless divinity. He was critical of all aspects of both Hinduism and Christianity that he regarded as nonrational.\n\nSimilarly critical of the _Pur\u0101\u1e47a_ -based Hinduism of this period was Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b, who lived from 1824 to 1883 and established the \u0100rya Sam\u0101j\u2014northern India's answer to the Brahmo Sam\u0101j. Rather than the _Upani\u1e63ads_ , Day\u0101nanda focused upon the early Vedic _Sa\u1e43hit\u0101s_ as the wellspring of authentic Hindu practice. He both revived and popularized the practice of worship centered on the sacred fire, as opposed to the use of images, which he viewed as a later accretion upon \"pure\" Vedic worship.\n\nThese movements, however, failed to capture the imagination of the Hindu community as a whole, primarily due to their rejection of the use of images in worship, or _m\u016brtip\u016bj\u0101_ , a practice that had been central to Hindu religious expression for centuries. True success in the area of Hindu reform came with the Ved\u0101nta movement inspired by Sri Ramakrishna (1836\u20131886) and led by his chief disciple, Sw\u0101m\u012b Vivek\u0101nanda (1863\u20131902).\n\nRamakrishna himself was not a Hindu reformer but a highly traditional Hindu spiritual figure in the mold of the great bhakti saints of the classical and medieval periods. Known for his powerful ecstatic visions, he claimed to have experienced _sam\u0101dhi_ , or absorption in the divine, by practicing all of the major Hindu spiritual paths, as well as Christianity and Islam. He concluded, on this basis, that all religions are paths to the divine. Though barely literate, he could discuss the finer points of Hindu philosophy with highly learned scholars on the basis of his direct experience of spiritual realities. The Western-educated elites of Bengali society\u2014including many members of the Brahmo Sam\u0101j\u2014were captivated by his teaching and personality. Because he based his teaching on direct experience and not on the interpretation of scripture, Ramakrishna's path was seen as open to scientific verification, and so more in keeping with a modern sensibility than traditions based upon ancient texts alone. And yet his path involved a wholehearted embrace of many practices and a sensibility that the reformers had rejected\u2014such as the use of images and a highly emotionally charged spirituality of devotion. It was a heady mix for those Hindus who wanted to embrace the social reforms and rationalistic approach of the reformers but who also felt nostalgic for the traditions of their upbringing. Under the leadership of Sw\u0101m\u012b Vivek\u0101nanda, the Ramakrishna Mission and the wider movement it inspired had a broader impact than the Brahmo and \u0100rya Sam\u0101j.\n\n## GLOBAL HINDUISM\n\nVivek\u0101nanda had the further effect of transforming Hinduism from a family of traditions confined largely to the Indian subcontinent and almost exclusively to persons of South Asian ethnic origin to a world religion, open to non-Indian as well as Indian adherents. It was already the case in Vivek\u0101nanda's time that many Westerners were drawn to Hindu traditions, such as the Transcendentalists of New England and the Theosophists, who had come to South Asia as \"reverse missionaries\" to instill pride among both Hindus in India and Buddhists in Sri Lanka in their religious heritage. But Vivek\u0101nanda took the step of initiating Westerners into a Hindu monastic order for the very first time, beginning with Margaret E. Noble (better known to Hindus by her monastic name, Sister Nivedita). His celebrated speech to the Parliament of the World's Religions in Chicago in 1893 is seen by many Hindus as marking the arrival of Hinduism on the world stage.\n\nBy initiating Westerners into Ved\u0101nta, Vivek\u0101nanda opened the proverbial floodgates to a range of Hindu spiritual masters who visited the West\u2014many settling permanently\u2014and taught their paths and practices to audiences eager for an alternative to what many saw, and continue to see, as the narrow, dogmatic provincialism of Christianity and the equally narrow\u2014and spiritually stultifying\u2014worldview of scientistic materialism. The first master to follow in Vivek\u0101nanda's footsteps was another Bengali, Paramah\u0101\u1e43s\u0101 Yog\u0101nanda (1893\u20131952), whose Self-Realization Fellowship, established in California in 1920, was designed to promote the practice of Kriya Yoga, the form of meditation that was taught to Yog\u0101nanda by his guru, Sri Yukte\u015bwar Giri. The acceptance of Yog\u0101nanda's teachings by Westerners was facilitated by his emphasis on Jesus as an enlightened master and an exemplar of yogic wisdom. Jiddu Krishnamurti (1895\u20131986), emerging from the shadow of the Theosophical movement, which had groomed and hailed him as the next \"World Teacher,\" a messianic figure known as Maitreya, broke away from the Theosophists and taught a path of radical self-inquiry, finding a ready audience in the counter-culture of the 1960s. Maharishi Mahesh Yogi (1918\u20132008) similarly appealed to the counterculture via its heroes, the Beatles, who aided him in his promotion of Transcendental Meditation by very publicly attending his seminars in England and Wales, and following him to India in the summer of 1968 for a retreat at his Rishikesh ashram in India, in the foothills of the Him\u0101layas.\n\nA. C. Bhaktived\u0101nta Sw\u0101m\u012b Prabhup\u0101da (1896\u20131977), in a dramatic contrast with his contemporaries, who carefully tailored their traditions to the tastes and concerns of their Western audiences, brought a very traditional form of Vai\u1e63\u1e47ava devotionalism to North America in 1966, establishing the Hare Krishna movement. However, like the Maharishi, Bhaktived\u0101nta also found endorsement from the royalty of the counterculture, with the _Mah\u0101mantra_ of Gau\u1e0d\u012bya Vai\u1e63\u1e47avism being included as the refrain of George Harrison's 1970 hit song, \"My Sweet Lord.\"\n\nIn addition to the j\u00f1\u0101na and dhy\u0101na yogas taught by Vivek\u0101nanda, Krishnamurti, and the Maharishi and Yog\u0101nanda, respectively, and the bhakti emphasis of Bhaktived\u0101nta, the Tantric traditions of Hinduism also found representatives among the Hindu teachers who came to the West in the 20th century. Sw\u0101m\u012b Mukt\u0101nanda (1908\u20131982) brought the Siddha Yoga tradition of his master, Sw\u0101m\u012b Nity\u0101nanda. He was also succeeded by a prominent female guru, Sw\u0101m\u012b Chidvil\u0101s\u0101nanda, or Gurumay\u012b, who presides over this tradition to the present day. And most controversial of all was Bhagwan Rajneesh, later known as Osho (1931\u20131990), whose endorsement of left-handed Tantric practice, with its accompanying emphasis on sexual freedom, dovetailed better with the sexual revolution of this period than with the traditionally conservative sexual morality of Indian Hindus.\n\nMeanwhile, as these and other Hindu teachers brought their varied teachings and forms of practice to the West, Vivek\u0101nanda's original Ved\u0101nta Society continued to be active. A steady stream of monks of the Ramakrishna Order came to North America, establishing and nurturing Ved\u0101nta centers in cities such as New York, Boston, Washington, D.C., Los Angeles, and San Francisco. California, especially\u2014both prior to and during the period of the counterculture\u2014was a major center for Hindu activities that appealed primarily to a Western audience. Sw\u0101m\u012b Prabhav\u0101nanda (1893\u20131976), a highly active monk of the Ramakrishna Order and founder of the Ved\u0101nta Society of Southern California in 1930, initiated literary luminaries such as Aldous Huxley and Christopher Isherwood, who in turn did much to promote the Ved\u0101nta tradition among their readers.\n\nA skeptic could of course question the extent to which any of the movements based on the teachings of spiritual teachers from India have become\u2014or will become\u2014deep-rooted and authentically \"Western\" forms of religiosity. Like Western Buddhism, however, the Western Hindu traditions have begun to produce their own leaders, raised with a Western cultural ethos, and are not simply depending upon Indian teachers for their spiritual and organizational sustenance. With Satguru \u015aiv\u0101ya Subramuniyasw\u0101m\u012b (1927\u20132001), who was born Robert Hansen and inherited the mantle of a \u015aaiva Siddh\u0101nta sa\u1e43prad\u0101ya, or teaching lineage, from his master, Yogasw\u0101m\u012b, Western Hinduism produced a spiritual master who was acknowledged as such not only by his devotees but also by Hindus globally\u2014including quite conservative Hindus, such as the founders of the Hindu \u0100c\u0101rya Sabh\u0101, of which Subramuniyasw\u0101m\u012b was a member. The journal that Subramuniyasw\u0101m\u012b founded, _Hinduism Today_ , presents itself as \"the voice of the global Hindu community\"\u2014a claim that the community on the whole appears to endorse. The phenomenon of high-profile, celebrity adherents endorsing Hinduism or specific Hindu practices\u2014such as the Beatles' endorsement of Transcendental Meditation and former Beatle George Harrison's close and lifelong affiliation with the Hare Krishna movement\u2014continues, with the most recent example being the actress Julia Roberts, whose film version of Elizabeth Gilbert's autobiographical memoir, _Eat, Pray, Love_ , includes a representation of life in the Siddha Yoga ashram in Ganeshpuri, India.12 Though it is far from being a mass movement, Hinduism does seem to have found a home in the West.13\n\n## TWO CONTRARY IMPULSES: UNIVERSALISM AND NATIONALISM\n\nIf Vivek\u0101nanda articulated a vision for Hinduism as a tradition of rationality, tolerance, and a world-embracing spirituality, Mohandas K. G\u0101ndh\u012b (1869\u2013 1948) translated this vision into an ambitious program for comprehensive political and social reform. Tapping into ancient Hindu ideals such as _ahi\u1e43s\u0101_ (nonviolence), the inherent power of truth ( _saty\u0101graha_ ), and the divinity of all beings, G\u0101ndh\u012b\u2014known to his admirers by the ancient title of _mah\u0101tma_ , or \"great soul\"\u2014led a movement of nonviolent resistance to British rule that has inspired movements for social justice around the world, including the movement for African American civil rights led by Martin Luther King Jr. and the South African antiapartheid movement of Nelson Mandela.\n\nAt the same time, G\u0101ndh\u012b's successes\u2014particularly in the areas of the reform of the caste system and Hindu\u2013Muslim relations\u2014unleashed reactionary forces that ultimately resulted in his assassination. His attempts to abolish the practice of untouchability met with both the resistance of conservative Hindus and the scorn of leaders within the Dalit, or lower caste, community, such as B. R. Ambedkar, who believed that he did not go far enough. His attempts to effect a reconciliation of the divisions between Hindus and Muslims were unable to prevent the partition of India and Pakistan when independence was achieved on 15 August 1947. Many Hindus blamed\u2014and continue to blame\u2014G\u0101ndh\u012b for the deaths that resulted from the partition, when riots broke out across the subcontinent as Hindus fled Pakistan and Muslims fled India out of fear of being persecuted in a place where they would constitute a religious minority. G\u0101ndh\u012b was assassinated by Nathuram Godse, an adherent of Hindu Nationalism, on 30 January 1948.\n\nThe subsequent history of Hinduism is characterized by the tension between the forces of Hindu Nationalism, which see Hinduism and India as under constant attack by opposing movements, such as Islam, Christianity, and secularism, and the visionary Hinduism of G\u0101ndh\u012b and Ramakrishna, which sees all religions as part of a greater harmony that India can embody as an inspiration for the rest of the world to follow. It is a history that has been punctuated by outbursts of dramatic violence\u2014such as the Hindu\u2013Sikh riots of 1984 and the Hindu\u2013Muslim riots of 1992 and 2002\u2014standing out against a routine background of relative peace and harmony, in which the members of different religious communities celebrate each other's holidays, make pilgrimages to one another's places of worship, and elect members of minority communities to high public office, including the presidency and the prime ministership of India. It is also a history of expansion, as more non-Indians adopt Hindu practices\u2014even to the point of formal conversion\u2014and as Indian Hindus also continue to immigrate across the globe to Europe, North America, Africa, Australia, and other parts of the world as well.\n\n## WHAT HOLDS HINDUISM TOGETHER?\n\nAs mentioned previously, there are those scholars who will argue that \"Hinduism does not exist.\" I have also described Hinduism on a number of occasions, and only half-jokingly, as \"the world's least organized religion.\" What gives this tradition, or family of traditions, cohesion? It has no founder and no central administrative institution\u2014no pope or Vatican. There is no test of orthodoxy or creed upon which all Hindus must agree to adhere in order to be regarded as Hindu. There is even some disagreement about who is a Hindu. Must one be born as a Hindu, or are non-Indian practitioners of Hindu-inspired paths such as those described earlier\u2014Ved\u0101nta, Transcendental Meditation, Siddha Yoga, the Hare Krishna movement, and so on\u2014authentic Hindus? Are the Jains, the Buddhists, and the Sikhs\u2014practitioners of originally Indian, and therefore, in the sense of _Hindu_ as _Indian_ , \"Hindu\" traditions\u2014to be regarded as Hindus? Why has Hinduism not fragmented into nonexistence? Why does it seem, on the contrary, to be in a period of resurgence and renaissance?\n\nIt is tempting to say that Hinduism survives and perseveres because of the inherent appeal of its teachings and practices\u2014that it survives because it is true, because it holds insight and wisdom that are highly relevant to the lives of its adherents. But satisfying though it may be to Hindus, such an answer is not what an outside observer\u2014still less a historian\u2014has in mind when asking the question, \"What holds Hinduism together?\" What is it that gives whatever cohesion it has to this vast collection of varied beliefs and practices?\n\nThe answer, as with the answers to most questions about Hinduism, is complex and is not capable of being boiled down to one single factor that constitutes _the_ answer. Two primary factors, however, spring to mind.\n\nFirst, regarding founding figures and organizational structures\u2014it is not that such figures and structures are absent. It is that there is no _single_ figure or structure upon which all Hindus agree or to which all Hindu belief and practice can be traced or on which it can be said to be centered. On the contrary, there are many founding figures who have started organizational structures that have thrived and persisted for many centuries. This book documents a host of such figures, structures, and movements: enlightened sages, monks, inspired poets, and the communities that emerged around them\u2014sometimes spontaneously but sometimes cultivated in a highly organized and systematic manner. None of these is, however, identical to Hinduism as a whole. At least some part of what holds Hinduism together is the fact that Hindu adherence to particular traditions and organizations has a certain fluidity that allows for the phenomenon of multiple religious belonging. It is rare for a specific Hindu tradition to demand exclusive allegiance of its followers. If one were to add up the total number of Hindus engaged in some degree of some participation in all of the world's Hindu organizations, this number would be considerably larger than the total number of Hindus. This fluidity and ease of multiple membership is, in turn, made possible by the fact that the many Hindu traditions, despite their various differences in the areas of doctrine and practice, also share a good deal in these areas as well. Belief in karma and rebirth, a soteriological aspiration for liberation from the rebirth cycle, as well as a concern with more immediate goods, such as prosperity and good fortune both in this life and the next, cut a wide swath across most Hindu schools of thought and systems of practice. Similarly, a core of ethical practice, enshrined in ideals such as nonviolence, honesty, sexual restraint, and nonpossessiveness, is widely shared. These core ideas and practices are \"portable,\" in the sense that one can move fairly easily among Hindu groups without finding them sharply questioned or turned into points of contention.\n\nSecondly, and pertaining especially to the realm of practice, there is the caste system. In many ways controversial, particularly in the modern period, due to the inequities that it enshrines, this hereditary system of occupations has also served as a major acculturating force, for it is not only occupations that are passed on to one through one's family. It is also narratives and practices related to correct ritual conduct that become second nature even to Hindus who, later in life, completely reject, at a cognitive level, the specifically religious dimensions of their traditions. Much as there are people who are \"culturally Jewish,\" scrupulously observing the high holy days, for example, but perhaps professing atheism, in terms of their personal beliefs about the nature of reality, there are, similarly, \"cultural Hindus,\" who may or may not hold \"Hindu\" views but who have thoroughly absorbed the civilizational ethos of Hinduism. It has been noted that caste has become more rigid during periods of colonial domination than at other times in Hindu history, when the more fluid and easygoing attitude toward boundaries that characterizes Hindu religious organizations can also be found in regard to caste boundaries and restrictions. This is evidence that, for most Hindus, Hinduism has at least as much to do with family loyalty and the preservation of tradition as with specifically religious preoccupations such as salvation and contemplating the nature of existence. To the degree that the Hinduism of non-Indian practitioners is called into question, it is to the same extent that the practice of these adherents is rooted in voluntary religious considerations alone and has not been absorbed from childhood as part of a cultural ethos. But one must bear in mind that, as was stated at the outset of this introduction, the authenticity or inauthenticity of any given expression of Hinduism\u2014whether a given person, practice, movement, or philosophy is \"really\" Hindu\u2014is really a question about how these terms\u2014 _Hindu_ and _Hinduism_ \u2014are to be defined and deployed, rather than a question of how well these phenomena correlate with some eternal or unchanging essence. Hinduism, like all _isms_ , is a construct.\n\n## CONCLUSION\n\nThe future of Hinduism, as of all religions, is impossible to forecast with any reliability. Challenges to the tradition include the aforementioned Hindu Nationalism and casteism\u2014or caste prejudice\u2014and issues relating to patriarchy and the full participation of women as equal members of the Hindu community.\n\nIf the past can be used to predict the future, one can expect Hindus to address these issues much as they have previous challenges\u2014with creativity and flexibility tempered by a deep commitment to the principle that the wisdom of the ancient past always has a continuing relevance to the present.\n\n## NOTES\n\n1. Lisa Miller, \"We're All Hindus Now,\" _Newsweek_ , 15 August 2009.\n\n2. The question of the ancientness of Hinduism versus its status as a relatively recent construct of colonial scholarship is explored in D. N. Jha's _Rethinking Hindu Identity_ (London: Equinox, 2009) and by a variety of authors in J. E. Llewellyn's edited volume _Defining Hinduism: A Reader_ (New York: Routledge, 2005).\n\n3. Wendy Doniger, _The Hindus: An Alternative History_ (New York: Penguin, 2009), 65\u201366.\n\n4. Klaus K. Klostermaier, _A Survey of Hinduism_ , 3rd ed. (Albany: State University of New York Press, 2007), 25.\n\n5. See Jonathan Mark Kenoyer, _Ancient Cities of the Indus Valley Civilization_ (Oxford: Oxford University Press, 1998), Dilip K. Chakrabarti, _India: An Archaeo-logical History_ (New Delhi: Oxford University Press, 1999), Gregory L. Possehl, _The Indus Civilization: A Contemporary Perspective_ (Walnut Creek, Calif.: Altamira Press, 2002), and Jane R. McIntosh, _A Peaceful Realm: The Rise and Fall of the Indus Civilization_ (Boulder, Colo.: Westview Press, 2002).\n\n6. Such claims are common in modern Hindu reform movements, such as the \u0100rya Sam\u0101j, which condemn the hierarchical and hereditary nature of caste but do not reject the basic concept.\n\n7. For a defense of this revisionist version of the standard model, see Klostermaier, _A Survey of Hinduism_ , 18\u201329.\n\n8. The most authoritative and even-handed study of the scholarly debate over ancient India\u2014acknowledged as such by persons on both sides of the conversation\u2014is Edwin Bryant's _The Quest for the Origins of Vedic Culture: The Indo-Aryan Migration Debate_ (New York: Oxford University Press, 2004).\n\n9. A formal and systematic understanding of Hindu practice as consisting of four yogas, as described here, is a very recent one indeed\u2014first articulated in this precise way by Sw\u0101m\u012b Vivek\u0101nanda during the modern period. But the systems of practice and the accompanying soteriological paradigms that these four yogas represent are ancient and do date back to at least the later Vedic period. I find them a useful heuristic tool for presenting the historical phases of Hindu practice. But this should not be confused with the claim that a _system_ of four yogas, as this is understood today, was present during the later Vedic period. I am certainly _not_ making this claim.\n\n10. See Nicholas Dirks, _Castes of Mind: Colonialism and the Making of Modern India_ (Princeton, N.J.: Princeton University Press, 2001) and Bernard S. Cohn, _India: The Social Anthropology of a Civilization_ (Englewood Cliffs, N.Y.: Prentice Hall, 1971).\n\n11. See Peter Gottschalk, _Beyond Hindu and Muslim: Multiple Identity in Narratives from Village India_ (New York: Oxford University Press, 2005).\n\n12. Hindu Press International, 8 August 2010, <http:\/\/www.hinduismtoday.com\/modules\/xpress\/category\/hindu-press-international>.\n\n13. See Phil Goldberg, _American Veda: From Emerson and the Beatles to Yoga and Meditation\u2014How Indian Spirituality Changed the West_ (Bourgon, Ind.: Harmony, 2010) and Lola Williamson, _Transcendent in America: Hindu-Inspired Meditation Movements as New Religion_ (New York: New York University Press, 2010).\n\n#\n\n## A\n\n**ABHAYA-MUDR\u0100.** **Mudr\u0101** , or hand gesture, which means, \"Do not be afraid.\" Hindu deities are frequently depicted with this gesture, which consists of the right hand (or one of the right hands, given that Hindu deities are frequently depicted with more than one pair of arms and hands) in an upright and slightly bent back position, with an empty palm facing outward, toward the viewer of the image. _See also_ M\u016aRTI.\n\n**ABHIMANYU.** Character in the _Mah\u0101bh\u0101rata_ ; son of **Arjuna** , tragically slain in the battle of **Kuruk\u1e63etra** , but not before fathering **Parik\u1e63it** , the heir who would inherit the **P\u0101\u1e47\u1e0davas** ' kingdom.\n\n**ABHINAVAGUPTA (950\u20131020).** Philosopher known for systematizing the teachings of the **Kaula Tantra** sect of **Ka\u015bm\u012br \u015aaivism**. A remarkably learned scholar with a wide-ranging intellect, Abhinavagupta studied under an unusual variety of teachers, mastering **music** , **poetry** , and drama, as well as **philosophy** and logic. His writings encompass such diverse topics as aesthetic experience, levels of consciousness, and the ultimate nature of reality. He synthesized traditional **Brahmanism** and Tantric thought, which before this time had been perceived as being in conflict. In keeping with a Tantric understanding of the role of the senses in the path to **liberation** , Abhinavagupta emphasized not so much the suppression of the senses as the cultivation of aesthetic enjoyment in an anticipation of the bliss of ultimate union with **\u015aiva**. Abhinavagupta did not reject rival systems of thought as false but viewed them in an inclusive fashion as steps toward his position. His spiritual journey was sparked by an early childhood tragedy\u2014the death of his mother, Vimal\u0101. His system of philosophy has affinities to both **Advaita Ved\u0101nta** and **Tantra** , inasmuch as it is monistic (like Advaita) and affirms sensory experience (like Tantra). A prolific writer, his best-known works are the encyclopedic _T\u0101ntraloka_ and his **bh\u0101\u1e63ya** , or commentary, on the _N\u0101\u1e6dya \u015a\u0101stra_. _See also_ \u015aAIVISM.\n\n**ABHI\u1e62EKA.** Anointing, consecration, ritual bathing; ceremony of bathing an image of a deity ( **m\u016brti** ) with sacred substances such as water, milk, **ghee** , yogurt, saffron, etc.\n\n**\u0100C\u0100RYA.** Sanskrit term meaning _teacher_ , but most commonly referring to a teacher of particularly high rank, such as the leader of a teaching lineage or sect ( **sa\u1e43prad\u0101ya** ), or the head of a monastic institution ( **ma\u1e6dha** ). The term is sometimes used as a synonym for **guru** , but the term _guru_ more often refers to a teacher with whom one has developed a personal relationship as a student or disciple ( **\u015bi\u1e63ya** ), whereas an \u0101c\u0101rya is a more public figure\u2014the teacher of an entire community. The title of \u0101c\u0101rya is usually given through **d\u012bk\u1e63\u0101** , or initiation, and ordination as the successor to the leadership of a spiritual lineage. A **Christian** equivalent to _\u0101c\u0101rya_ might be either \"abbot\" or \"bishop,\" the former being the head of a monastery and the latter implying religious authority over a specific geographic region, both of which can characterize the duties of an \u0101c\u0101rya.\n\n**ACTION.** Action, or **karma** , is an issue of central concern to Hindu traditions. There is the regulation of action\u2014the enjoining of right action, or **dharma** , and the avoidance of its opposite, which is a characteristic of most religious traditions\u2014and then there is the effort to transcend action and its effects, a concern more distinctive of the Indic religions, such as Hinduism, **Buddhism** , **Jainism** , and **Sikhism**. Action is widely understood to follow a principle of cause and effect, also widely termed _karma_ , according to which all actions produce results of the same moral character as themselves\u2014good actions leading to good results ( **pu\u1e47ya** karma), and evil actions to evil results ( **p\u0101pa** karma).\n\nThe reciprocal process of intentional moral actions producing like effects fuels the perpetual cycle of **rebirth** ( **sa\u1e43s\u0101ra** ), release from which ( **mok\u1e63a** ) requires one to in some way negate or transcend the principle of cause and effect. The various Indic traditions often present themselves as strategies for achieving this negation or transcendence. The _Bhagavad G\u012bt\u0101_ , for example, presents a **yoga** of action ( **karma yoga** ) that aims at mok\u1e63a through engaging in action without attachment to the results. On this understanding, it is not action itself but the **desire** that motivates it that brings about karmic effects.\n\n**ADHARMA.** Lack or opposite of **dharma** , not dharma, evil, lack of virtue. Adharma also refers to chaos or anything that goes against the order of nature. Adharma is often embodied in Hindu **literature** by demonic beings, such as **asuras** and **r\u0101k\u1e63asas** , who are defeated in battle by a **god** or **goddess** , usually either an **avat\u0101ra** of **Vi\u1e63\u1e47u** or a form of the **Mother Goddess**. In the _Bhagavad G\u012bt\u0101_ , **K\u1e5b\u1e63\u1e47a** cites the destruction of adharma and the protection of dharma as the chief reason for his repeated incarnations or avat\u0101ras ( _Bhagavad G\u012bt\u0101_ 4.7). Adharmic behavior results in misfortune, both for its immediate victims and for its perpetrators. This misfortune can occur quickly or it can be deferred, being experienced in this lifetime or in a future **rebirth**. _See also_ KARMA; P\u0100PA.\n\n**ADHIK\u0100RA.** Authority; the right to acquire higher spiritual knowledge. According to **Brahmanism** , only a male **Brahmin** is eligible to study the _Vedas_ , which are the source of spiritual knowledge. This concept is challenged, however, by many Hindu traditions, as well as by modern Hindu reformers such as **Sw\u0101m\u012b Vivek\u0101nanda**.\n\n**ADHVARYU.** **Brahmin** priest whose role in the **Vedic** sacrifice, or **yaj\u00f1a** , is to oversee the physical details of the ritual, such as measuring out the sacred space, building the fire altar, lighting the fire, and so forth. While performing these **actions** , the Adhvaryu recites relevant portions of the _Yajur Veda_ that describe what is to be done. In ancient times, when **animal sacrifices** were more widely performed, the task of the Adhvaryu included the slaughter, cooking, and offering of the bodies of sacrificial victims.\n\n**ADITI. Goddess** whose name means _infinity_. Wife of the sage **Ka\u015byapa** and mother of 12 **devas** , or deities, called the **\u0100dityas** : **Indra** , **Dh\u0101t\u1e5b** , **Aryaman** , **Mitra** , **Varu\u1e47a** , **A\u1e43\u015ba** , **Bhaga** , **Vivasvat** , **P\u016b\u1e63an** , **Savit\u1e5b** , **Tva\u1e63\u1e6d\u1e5b** , and **Vi\u1e63\u1e47u**. Aditi is also the mother of the eight **Vasus** and the 11 **Rudras**.\n\n**\u0100DITYAS.** The 12 major **Vedic** deities, or **devas**. They are the sons of the goddess **Aditi** and the sage **Ka\u015byapa**. Their names are **Indra** , **Dh\u0101t\u1e5b** , **Aryaman** , **Mitra** , **Varu\u1e47a** , **A\u1e43\u015ba** , **Bhaga** , **Vivasvat** , **P\u016b\u1e63an** , **Savit\u1e5b** , **Tva\u1e63\u1e6d\u1e5b** , and **Vi\u1e63\u1e47u**.\n\n**\u0100DIV\u0100S\u012aS.** First dwellers; aboriginal tribal peoples of the Indian subcontinent, many of whose religious traditions have become part of Hinduism over the centuries. Tribal peoples have tended to live on the margins of Hindu society.\n\n**ADVAITA VED\u0100NTA.** The **nondualistic** system of **Ved\u0101nta philosophy** , prominently associated with its chief exponent and systematizer of the classical period, **\u015aa\u1e45kara** (788\u2013820), though having roots in more ancient texts like the _Upani\u1e63ads_ , the _Brahma S\u016btras_ , and the _Bhagavad G\u012bt\u0101_. According to Advaita Ved\u0101nta, these ancient texts (collectively called the **prasth\u0101na traya** ) all teach the view of Advaita, although this interpretation of these texts is of course not shared by other systems of Ved\u0101nta.\n\nAdvaita Ved\u0101nta is a deeply monistic system and teaches that **Brahman** is the sole reality. The existence of multiple entities, such as those that make up the world\u2014and even the distinction between the world, the self, and **God** \u2014is a mere appearance, or **m\u0101y\u0101**. M\u0101y\u0101, in turn, is an effect of **avidy\u0101** , or primordial ignorance of the true nature of reality.\n\nAccording to Advaita Ved\u0101nta, Brahman as such has no limiting qualities. It is **nirgu\u1e47a**. Brahman does not possess, but rather, it _is_ infinite being ( **sat** ), consciousness ( **cit** ), and bliss ( **\u0101nanda** ). As the only truly real entity, Brahman is identical to the self ( **\u0101tman** ), the self\u2013other distinction being an effect of m\u0101y\u0101. Realization of the identity of Brahman and \u0101tman gives rise to **mok\u1e63a** , or **liberation** from the **rebirth** process. The basis of this realization, according to \u015aa\u1e45kara, is the **Vedic** scriptures\u2014primarily the _Upani\u1e63ads_. Good works, **devotion** , and **meditation** can only prepare one for this **knowledge**. Hearing the **truth** that Brahman and \u0101tman are one ( **\u015brava\u1e47a** ), reflecting upon it ( **manas\u0101** ), and contemplatively realizing it (nidhidhy\u0101sana) are the three necessary steps that produce the awakening required for liberation. This classical understanding of Advaita is distinct from **Neo-Ved\u0101nta** , which emphasizes direct experience ( **anubh\u0101va** ) over **scripture**.\n\nIt is possible to achieve liberation while still alive, an attainment that is known in the Advaita system as **j\u012bvanmukti**. After death, one who has attained liberation while alive\u2014a j\u012bvanmukta\u2014is no longer subject to rebirth, being absorbed in Brahman (though it is more correct to say that this \"absorption\" is apparent, the reality being that one has never actually been separate from Brahman).\n\nAdvaita Ved\u0101nta has traditionally been most influential in southern India, but it has also exerted a strong influence on Neo-Ved\u0101nta, particularly as expressed by modern exponents such as **Sw\u0101m\u012b Vivek\u0101nanda** and **Sarvepalli Radhakrishnan**.\n\n**\u0100GAMA.** Tradition, that which comes down or is handed down from past generations. A set of **scriptures** associated preeminently with one of the three major theistic Hindu sects\u2014the **Vai\u1e63\u1e47ava** , **\u015aaiva** , and **\u015a\u0101kta** traditions. Largely \"how to\" manuals, these texts are mainly concerned with the correct performance of ritual, particularly **p\u016bj\u0101** devoted to the respective deities of these three traditions ( **Vi\u1e63\u1e47u** , **\u015aiva** , and the **Mother Goddess** ), and with yogic practice and **temple** construction and consecration. Vai\u1e63\u1e47ava \u0100gamas are often called **Sa\u1e43hit\u0101s** , and \u015a\u0101kta \u0100gamas are usually referred to as **Tantras**. The \u015aaiva \u0100gamas, however, are generally called \u0100gamas. The **Jain** scriptures are also known as \u0100gamas.\n\n**AGASTYA.** **Vedic** sage attributed with transmitting the _Vedas_ to southern India and inventing the **Tamil** language. One of the hymns of the _\u1e5ag Veda_ is attributed to him. He is said to have forced the Vindhya Mountains, in central India, to submit to him. **R\u0101ma** visited the sage Agastya during the period of his exile in the _R\u0101m\u0101ya\u1e47a_.\n\n**AGEH\u0100NANDA BH\u0100RAT\u012a, SW\u0100M\u012a (1923\u20131991).** Born **Leopold Fischer** , Sw\u0101m\u012b Ageh\u0101nanda Bh\u0101rat\u012b was one of the first scholar-practitioners to teach Hindu traditions in a Western academic context, teaching at Syracuse University for more than 30 years.\n\n**AGHOR\u012a.** Free from terror; member of a **\u015aaiva** tradition that split from the **K\u0101p\u0101likas** , another \u015aaiva sect, in the 14th century. A **Tantric** sect, Aghor\u012b practice is designed both to express and cultivate a state beyond fear. Condemned by many Hindus for being too extreme, Aghor\u012bs **meditate** in cremation grounds and even eat the corpses of the dead. The begging bowl of an Aghor\u012b monk is a human skull.\n\n**AGNI.** Fire. Agni is the **Vedic** deity, or **deva** , who personifies fire, particularly the fire of the Vedic **yaj\u00f1a** , or sacrifice. Because all Vedic sacrifices are offered into the fire, Agni is seen as the intermediary between human beings and all of the Vedic deities. As such, he is also viewed as the prototypical Vedic priest and is proclaimed as such in _\u1e5ag Veda_ 1.1. The first offering in any sacrifice is made to Agni, who would otherwise consume the offerings intended for the other deities. As the intermediary between the divine and the human worlds, Agni is of great importance in Vedic religion. The number of hymns of the _\u1e5ag Veda_ devoted to Agni is second only to those devoted to **Indra** , lord of the Vedic deities. Eight of the 10 books, or **ma\u1e47\u1e0dalas** , of the _\u1e5ag Veda_ begin with hymns to Agni (the exceptions being the eighth and ninth books).\n\nUnlike other Vedic deities, whose realms are confined to one of the three regions of Earth, the upper atmosphere, and the heavens, Agni operates in all three realms (in the forms, respectively, of fire, lighting, and the sun).\n\n**AGNIHOTRA.** A relatively brief and simple sacrifice or **yaj\u00f1a** that is offered to **Agni**. Orthodox **Brahmins** perform this ritual daily. The Agnihotra, although an ancient ritual mentioned in both the _Atharva Veda_ and the _G\u1e5bhya S\u016btras_ , has been revived and promoted for wider use in the modern period, primarily by the **\u0100rya Sam\u0101j** , as a rite of purification. _See also_ HAVAN; HOMA; _VEDAS_.\n\n**AGNI\u1e62\u1e6cOMA.** A five-day sacrifice or **yaj\u00f1a** offered to **Agni** , **Indra** , and other **Vedic** deities. A fairly elaborate ritual, the Agni\u1e63\u1e6doma involves 15 priests. Traditionally offered by a **Brahmin** **householder** , its goal is **rebirth** in heaven ( **svarga** ).\n\n**AGNIVE\u015aA, SW\u0100M\u012a (1939\u2013 ).** Founder and president of the **World Council of the \u0100rya Sam\u0101j** , an organization distinct from the **\u0100rya Sam\u0101j** , a Hindu reform organization established in 1875 by **Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b (1824\u20131883)**. Sw\u0101m\u012b Agnive\u015ba is best known for his social and political activism, taking up such issues as alcohol abuse, **women** 's rights, peace, interreligious dialogue, female infanticide, and **casteism**.\n\nAgnive\u015ba became a monk, or **sanny\u0101sin** , in 1970, under the auspices of the \u0100rya Sam\u0101j. In 1987, he led a large protest against a **sati** that occurred in Deorala, Rajasthan. Due to a variety of internal controversies, Sw\u0101m\u012b Agnive\u015ba was expelled from the \u0100rya Sam\u0101j in 1992, but he continues to claim fidelity to the original ideals of the organization and its founder.\n\n**AGRAHAYA\u1e46A.** Month of the **Hindu calendar** that corresponds roughly to the period from mid-November to mid-December.\n\n**AHA\u1e42K\u0100RA.** Ego; literally the \"I-maker\"; one of the basic elements that constitute existence, or **tanm\u0101tras** , of the **S\u0101\u1e43khya** system of **philosophy**. In the **Yoga** system of **Pata\u00f1jali** , aha\u1e43kara, in combination with **buddhi** (intellect) and the **indriyas** (senses), constitutes **citta** (the **mind** ).\n\n**\u0100HAVAN\u012aYA.** One of the three types of fire altar used in **Vedic** rituals of sacrifice, or **yaj\u00f1as**. The \u0101havan\u012bya altar has a square shape. The four sides of this square represent the four major directions of space: north, south, east, and west. The \u0101havan\u012bya altar is located in the eastern portion of the larger Vedic sacrificial arena. Specifications for the construction, location, and use of the \u0101havan\u012bya are found in the _Yajur Veda_.\n\n**AHI\u1e42S\u0100.** Nonviolence in thought, word, and deed; absence of even the desire to do harm to any living being. Ahi\u1e43s\u0101 is the first of the five **yamas** or moral restraints taught in the _Yoga S\u016btras_ of **Pata\u00f1jali** (composed between 100 BCE and 500 CE), and making up the first of the eight \"limbs\" of the eight-limbed ( **a\u1e63\u1e6d\u0101\u1e45ga** ) system of **yoga**. Though it is phrased in negative terms, as the absence of violent impulses, ahi\u1e43s\u0101 is understood in those traditions that practice it as entailing positive compassion toward all beings. In the modern **Ved\u0101nta** of **Sw\u0101m\u012b Vivek\u0101nanda** , nonviolence is seen as being entailed by the oneness of all beings as manifestations of **God** , or **Brahman**.\n\nThe virtue of nonviolence gradually increases in both prominence and importance in Brahmanical texts from the period of the principal _Upani\u1e63ads_ (c. 500\u2013100 BCE) to the Common Era. The _Mah\u0101bh\u0101rata_ contains the statement \"Ahi\u1e43s\u0101 is the highest virtue\" (13.125.25), and in the _Bhagavad G\u012bt\u0101_ (16.2), ahi\u1e43s\u0101 is listed as one characteristic of an **enlightened** person (despite the setting of this text on a battlefield and its endorsement, in an earlier chapter, of violence for a **k\u1e63atriya** fighting in a just war).\n\nAhi\u1e43s\u0101 is a central virtue of **\u015brama\u1e47a** traditions such as **Jainism** and **Buddhism** , which include it in their respective sets of five moral precepts, and which are both close in content to Pata\u00f1jali's **yamas** (the Jain set, in fact, being identical to Pata\u00f1jali's). The gradually increasing emphasis on ahi\u1e43s\u0101 in Hindu texts from shortly before and in the early centuries of the Common Era is sometimes attributed to the influence of these two \u015brama\u1e47a traditions, though grounds for ahi\u1e43s\u0101 can certainly be found in the _Upani\u1e63ads_.\n\nThe practice of **vegetarianism** in all three traditions\u2014Jain, Buddhist, and Hindu\u2014is a result of their emphasis on the practice of ahi\u1e43s\u0101 toward all living beings. Indeed, it is in the area of dietary practice that the increased emphasis on ahi\u1e43s\u0101 in Hinduism is most prominent, with the **Brahmins** , who in ancient times offered **animal sacrifices** , being generally expected in later centuries to practice vegetarianism.\n\nIn the 20th century, **Mohand\u0101s K. G\u0101ndh\u012b** made ahi\u1e43s\u0101\u2014originally conceived as a purely personal virtue developed in the pursuit of individual spiritual realization\u2014into a cornerstone of his strategy for attaining Indian independence from the British Empire, insisting\u2014in keeping with the idea of **karma** \u2014that only pure means could bring about a pure end. _See also_ COW, SACRED.\n\n**AHURA MAZDA.** The Wise Lord; the supreme **God** of **Zoroastrianism** , a religion of ancient Iran that today claims a small but devout following in India. Zoroastrianism is, in its origins, closely related to **Vedic** religion. The word _ahura_ is the Iranian equivalent of the Sanskrit **asura**. In early Vedic texts, the term _asura_ simply means _powerful_ and is used to describe several Vedic deities, particularly **Varu\u1e47a**. It is only in later Vedic texts that asuras become the demonic enemies of the **devas**.\n\nIn Zoroastrianism, Ahura Mazda is not one god among many, but God\u2014the one, supreme deity in a monotheistic sense\u2014and the creator of the world. Through the cultural influence of the Persian Empire, Zoroastrian **monotheism** helped to shape the Abrahamic religions. Via Judaism, it was the source of such **Christian** and **Islamic** concepts as the existence of an evil adversary of God (the devil), one lifetime (as opposed to **rebirth** ), and a final judgment, as well as the idea of a recurring series of divine messengers, which may also have shaped the Hindu **avat\u0101ra** concept. _See also_ ZARATHUSTRA.\n\n**AIHOLE.** Pronounced \"Eye-ho-l\u00e9\"; village in northern **Karnataka** that, from the fourth to the sixth centuries CE, was a major city: the capital of the **C\u0101lukya** dynasty. The site is noteworthy for the numerous Hindu **temples** , in a variety of architectural styles, which the C\u0101lukyas built there.\n\n**AIR\u0100VATA.** Celestial elephant; vehicle of **Indra**. Air\u0101vata was one of the treasured items to emerge from the cosmic ocean when it was churned by the **devas** and **asuras**. In one account of the life of **Ga\u1e47e\u015ba** , Ga\u1e47e\u015ba's father, **\u015aiva** , after unknowingly decapitating his son, seeks to atone for this mis-deed by replacing Ga\u1e47e\u015ba's head with that of Air\u0101vata, thus explaining why Ga\u1e47e\u015ba has the head of an elephant.\n\n**AI\u015aV\u0100RYA.** Blessing; in some contexts, a superhuman power; to be blessed by **God** ; derived from **\u012a\u015bvara**.\n\n**AIYANAR.** Deity who protects villages in southern India from evil spirits.\n\n**AJANTA.** Site in central India with numerous Hindu, **Buddhist** , and **Jain** **temples** and monasteries. Numerous caves dot the region, many of which have been used as sites for **meditation** through the centuries, being at various times extended and connected for the use of whole communities of monks. The site is particularly famous for bright frescoes depicting the life story of the **Buddha**.\n\n**\u0100J\u012aVIKAS.** Extinct **\u015brama\u1e47a** sect attributed with a strong doctrine of **n\u012byati** , or fate, according to which no **action** can facilitate one's attainment of **mok\u1e63a** , or **liberation** from **rebirth**. Mok\u1e63a will simply occur when it is destined to occur. This, at least, is how the doctrine of the \u0100j\u012bvikas is presented in the **scriptures** of the two surviving \u015brama\u1e47a sects, **Jainism** and **Buddhism**. The sect's founder, **Makkhali Gos\u0101la** , was a contemporary of both **Mah\u0101v\u012bra** and the **Buddha** , who represent his teaching as inimical to the practice of **asceticism**. Yet the \u0100j\u012bvikas are also represented as strict ascetics.\n\nRecent scholarship suggests that Jain and Buddhist representations of the \u0100j\u012bvika doctrine likely involve polemically motivated distortion. The actual \u0100j\u012bvika doctrine was probably that one can do nothing about one's already actualizing **karma** \u2014called in later Hindu texts **prarabdha karma** \u2014but that one can avoid the taking on of additional karma through adherence to ascetic practice (tapasya).\n\n**\u0100J\u00d1\u0100 CAKRA. Cakra** , or energy center, located in the subtle body ( **suk\u1e63ma \u015bar\u012bra** ) in the space corresponding to the center of the forehead, just above the eyebrows. It is sometimes called the \"third eye.\" The practice of locating a **tilak** or bindi in the center of the forehead may be related to this concept. _See also_ TANTRA.\n\n**\u0100K\u0100\u015aA.** Space, the medium of sound, one of the five **elements** or states of being that constitute the physical world ( **mah\u0101bh\u016btas)**. The five elements are earth ( **p\u1e5bthvi** ), air ( **v\u0101yu** ), fire ( **agni** ), water ( **ap** ), and space (\u0101k\u0101\u015ba). In esoteric Hindu thought, the \u0101k\u0101\u015ba also refers to a mental or spiritual space, a kind of collective **mind** in which all events are stored in a universal memory that can be read by persons who have developed the **siddhi** , or paranormal ability, of doing so.\n\n**AKBAR (1542\u20131605).** **Mughal** emperor who reigned from 1556 to 1605; known for his policy of religious tolerance and appreciation for India's religious diversity. Beginning his rule at the young age of 13, the early years of Akbar's reign were preoccupied with the conquest and consolidation of his empire. His opponents during this period included both Hindu and Muslim kings. He formed alliances with the Hindu **Rajputs** and married Rajput princesses. It was during this period that his interest in and tolerant views about religion emerged. He included Hindus in his administration and sponsored debates in which intellectuals from varied religious communities\u2014including Muslims, Hindus, **Jains** , Sikhs, skeptics, Jews, and Jesuit missionaries\u2014would discuss various issues for Akbar's edification. He tried to start his own religion, _Din-i-Ilahi_ or \"the Divine Faith,\" made up of teachings from all the traditions of which he was aware. But this new tradition did not survive his **death**. Under Akbar's reign, a hybrid Hindu\u2013 **Islamic** culture began to emerge in India, leading to a good deal of creative activity in the visual **arts** , **music** , architecture, and religion itself. _See also_ KAB\u012aR; N\u0100NAK, GURU; SANT; SIKHISM.\n\n**AKKA MAH\u0100DEV\u012a.** _See_ MAH\u0100DEVIYAKKA.\n\n**AK\u1e62AM\u0100L\u0100.** Garland of beads, often worn by Hindu deities. Typically, the garland is usually made up of 50 beads, symbolizing the sounds of the **Sanskrit** alphabet. One also finds garlands of 81 or 108 beads, the latter type often being used in the practice of **japa** ( **meditation** on a **mantra** ), in which case it is also referred to as a **japam\u0101l\u0101**.\n\n**AL-B\u012aR\u016aN\u012a (973\u20131048).** _See_ B\u012aR\u016aN\u012a, AL-.\n\n**ALLAHABAD.** Modern name of the city at the sacred confluence of the **Ga\u1e45g\u0101** (or Ganges) and Yamuna Rivers, traditionally called **Pray\u0101ga** ; one of the traditional locations of the **Kumbha Mel\u0101**.\n\n**\u0100\u1e36V\u0100RS.** Literally, \"those who are immersed deeply in **God**.\" The \u0100\u1e37v\u0101rs were 12 early medieval devotees of **Vi\u1e63\u1e47u** , and exponents of **bhakti** , or devotion, as the supreme path to **mok\u1e63a**. The teachings of the \u0100\u1e37v\u0101rs are expressed in the form of **Tamil** **poetry**.\n\nAccording to the **philosophy** of bhakti, **caste** and gender are irrelevant to the level of devotion of which one is capable\u2014and thus to the attainment of mok\u1e63a. This is in clear contrast with more conservative Brahmanical traditions, according to which **rebirth** as a male **Brahmin** is a prerequisite for the practices needed for the attainment of mok\u1e63a. The \u0100\u1e37v\u0101rs illustrate this spiritual egalitarianism, coming from a wide range of backgrounds. They include a woman ( **\u0100\u1e47\u1e0d\u0101l** ) and members of the humblest strata of Indian society.\n\nThe collected poetry of the \u0100\u1e37v\u0101rs, called the _N\u0101l\u0101yirap Pirapantam_ (or _N\u0101l\u0101yira Prabandham_ in Sanskrit) is held in such high regard that it is sometimes referred to as the Tamil **Veda**. The **Ved\u0101nta** teacher **R\u0101m\u0101nuja** , founder of the **Vi\u015bi\u1e63\u1e6d\u0101dvaita** system of Ved\u0101nta, regarded it as the fifth Veda. The \u0100\u1e37v\u0101rs are therefore particularly revered in the **\u015ar\u012bvai\u1e63\u1e47ava** tradition that R\u0101m\u0101nuja also established, and their poems are frequently recited in \u015ar\u012bvai\u1e63\u1e47ava **temples** and homes.\n\nThe \u0100\u1e37v\u0101rs composed roughly 4,000 poems in all, most of which are attributed to **Namm\u0101\u1e37v\u0101r** and **Tirumankai**. The \u0100\u1e37v\u0101rs flourished from approximately 600 to 900 CE.\n\n**AMAVASYA.** New moon; day marking the end of the **K\u1e5b\u1e63\u1e47a Pak\u1e63a** , or \"dark half\" of a month, in which the **moon** is waning, and the beginning of the **\u015aukla Pak\u1e63a** , or \"bright half,\" in which the moon is waxing. In the **\u015a\u0101kta** traditions of **Bengal** , it is customary to perform a **p\u016bj\u0101** to **K\u0101l\u012b** on this day. Amavasya is the last day of the Hindu month, the following day being the first day of the bright half of the next month.\n\n**AMB\u0100.** Literally \"mother\" or \"good woman\"; epithet of the **goddess** **Durg\u0101**. In Hindu iconography, Durg\u0101 in her form as Amb\u0101 is depicted riding a tiger and having eight arms, whereas Durg\u0101 proper is depicted riding a lion and having 10 arms. Amb\u0101 is also known as Ambik\u0101, Amb\u0101lik\u0101, and Jagaddh\u0101tr\u012b. She is one of several forms\u2014along with **P\u0101rvat\u012b** and **K\u0101l\u012b** \u2014of the **Mother Goddess** worshiped in the **\u015a\u0101kta** tradition. Amb\u0101, Ambik\u0101, and Amb\u0101lik\u0101, respectively, symbolize the characters A, U, and \u1e42, that make up the syllable **O\u1e43**.\n\nAmb\u0101 is also the name of a character in the _Mah\u0101bh\u0101rata_ , the daughter of a king of **K\u0101\u015b\u012b** (with sisters named Ambik\u0101 and Amb\u0101lik\u0101). Amb\u0101 and her sisters were captured by **Bh\u012b\u1e63ma**. Amb\u0101's sisters were married to Bh\u012b\u1e63ma's brothers, but Amb\u0101 was betrothed to another warrior who refused to marry her after her capture by Bh\u012b\u1e63ma. Bh\u012b\u1e63ma also would not marry her, due to his vow of celibacy. Abandoned, shamed, and homeless, Amb\u0101 vowed to destroy Bh\u012b\u1e63ma, whom she blamed for her misfortunes. By the power of her **asceticism** she was reborn as the male warrior \u015aikhandin, who slew Bh\u012b\u1e63ma in battle.\n\n**AMBEDKAR, BH\u012aMRAO (1891\u20131956).** **Dalit** leader and primary author of the Indian constitution. Born in a village in Mah\u0101r\u0101\u1e63\u1e6dra, Ambedkar received financial support from the Mah\u0101r\u0101ja of Baroda to pursue his higher education. In 1912, he became the first Dalit to receive a bachelor's degree at Elphinstone College in Bombay. He earned his doctoral degree at Columbia University in New York in 1927. He also studied at the London School of Economics and, like his contemporary **Mohand\u0101s K. G\u0101ndh\u012b** , earned his law degree in London as well.\n\nAmbedkar made the upliftment and empowerment of the lower **castes** the central cause of his life. Advocating the complete abolition of the caste system, Ambedkar was highly critical of reformist Hindus, such as G\u0101ndh\u012b, who believed that this system could be rehabilitated if the practice of **Untouchability** were to be eradicated.\n\nConvinced, by 1951, that caste prejudice, or **casteism** , was inherent to Hinduism, Ambedkar led a mass conversion of thousands of Dalits to **Buddhism**. In addition to its implied critique of Hinduism, this move also led to a reinvigoration of Indian Buddhism, which had largely vanished by the 14th century of the Common Era.\n\n**Ambedkarite Buddhism** , as this movement is called, has many continuities with traditional Buddhism and has led to a reinvigoration of interest in Buddhism in India. But it is distinguished by its strong political dimension as a critique of the caste system. Ambedkarite Buddhists usually have a home shrine that includes an image of Ambedkar, who is regarded as a _bodhisattva_ , or **enlightened** being. _See also_ HARIJAN.\n\n**AMBEDKARITE BUDDHISM.** Movement established by Dr. **Bh\u012bmrao Ambedkar** in 1951. Ambedkar led a mass conversion of thousands of his fellow **Dalits** to **Buddhism** as a protest against the **caste** system of Hinduism. Ambedkar had come to see **casteism** as inherent to Hinduism, thus requiring not reform but a break with the entire tradition.\n\nThough having many continuities with traditional Buddhism\u2014the study of which this movement has reinvigorated in India\u2014this movement is distinguished by its strongly political dimension as a critique of **casteism** and the caste system more generally. Many Indian Buddhists today worship Ambedkar as a _bodhisattva_ , or **enlightened** being, a ritual practice unique to this form of Buddhism.\n\n**AMMA (1953\u2013 ).** Literally \"mother\"; an epithet of the contemporary female **guru** **M\u0101t\u0101 Amrit\u0101nandamay\u012b** , also called the \"Hugging Saint.\"\n\n**AMRIT\u0100NANDAMAY\u012a, M\u0100T\u0100 (1953\u2013 ).** \"Mother who is permeated with the bliss of **immortality** ,\" also widely known as **Amma** , Ammachi, and the \"Hugging Saint.\" Born Sudhamani Idamannel in a poor village in **Kerala** , in her childhood she began to show generosity to those who were even less fortunate than herself, and also began the practice of spontaneously embracing those in sorrow. Spiritual seekers began to gravitate to her and in 1981 her organization, the M\u0101t\u0101 Amrit\u0101nandamay\u012b Ma\u1e6dha, was founded.\n\n**AM\u1e5aTA.** \" **Immortality**.\" Elixir of life churned from the cosmic ocean by the **devas** and **asuras**. This event, depicted frequently in Hindu **art** and **literature** , involved the devas and asuras in a rare act of cooperation, since neither group alone had the strength to churn the ocean of milk. This venture was the occasion for the appearance of **K\u016brma** , the second **avat\u0101ra** of **Vi\u1e63\u1e47u** , which had the form of a giant tortoise. K\u016brma permitted the devas and asuras to place **Mount Mandara** on his back to use as the axis of a great churn, the rope of which was formed by the giant serpent **V\u0101suk\u012b** , king of the **N\u0101gas**.\n\nWhen the elixir emerged from the ocean, the asuras stole it from the devas, who in turn recovered it from the asuras through trickery. From drinking the am\u1e5bta, the devas became immortal. In Hindu thought, however, the concept of immortality does not imply that one never dies, but only that one has a full and lengthy life span; even the devas die at the end of a **kalpa** , or a cosmic cycle. In some **Vedic** texts in which am\u1e5bta is an object of human aspiration, it is described as a life span of a hundred years.\n\nIt is said that at the time of the churning of the cosmic ocean of milk, four drops of am\u1e5bta fell to the Earth at the locations of the holy cities **Haridwar** , **N\u0101sik** , **Ujjain** , and **Pray\u0101ga**. A great festival of pilgrimage, or **Kumbha Mel\u0101** , is therefore held at each of these cities on a rotating basis every three years.\n\nIn esoteric Hindu thought, am\u1e5bta refers to a vital energy within the body that one can control through **yoga** in order to promote physical health and well-being. The earliest references in Hindu texts to an elixir of immortality are to the **soma** of the _Vedas_. In the Vedic literature, the devas and asuras fight over the soma, a possible precursor to the story of the am\u1e5bta being churned from the cosmic ocean.\n\n**A\u1e42\u015aA.** Early **Vedic** deity, one of the 12 **\u0100dityas** \u2014sons of the **goddess** **Aditi** and the sage **Ka\u015byapa**. The term also refers to the portion of the Vedic sacrifice that goes to the **gods** \u2014the invisible portion of the food offered to the gods that they receive before this food is consumed by human beings.\n\nThe word _a\u1e43\u015ba_ literally means _portion_ and is a technical **Vai\u1e63\u1e47ava** term referring to an **avat\u0101ra** of **Vi\u1e63\u1e47u** , the concept being that an avat\u0101ra is a manifestation of a portion of Vi\u1e63\u1e47u's total existence, with some avat\u0101ras manifesting a relatively small portion of the divine, and others, such as **K\u1e5b\u1e63\u1e47a** , being full manifestations, with others in between.\n\n**AN\u0100HATA CAKRA. Cakra** , or energy center, which is located in the subtle body (or **suk\u1e63ma \u015bar\u012bra** ) in the space corresponding to the heart. _See also_ TANTRA.\n\n**\u0100NANDA.** Bliss, joy, having no limitations. Along with infinite being ( **sat** ) and pure consciousness ( **cit** ), one of three inherent attributes of **Brahman** according to **Ved\u0101nta**. Etymologically related to the **Sanskrit** word _ananta_ , meaning infinite, or without limits. The names of many Hindu monks contain the word _\u0101nanda_ as a suffix, meaning \"one whose bliss is\" whatever quality is signified by the word to which \u0101nanda is added. As an attribute of Brahman, \u0101nanda is sometimes used as a synonym for **mok\u1e63a**.\n\n**\u0100NANDA M\u0100RGA.** \"Way of Bliss.\" Modern Hindu religious movement established in 1955 in **West Bengal** by Prabhat Ranjan Sarkar, better known as **\u0100nandam\u016brti**. The movement spread to the United States in 1969. In the early 1970s it was a center of controversy due to the political activism of its members on behalf of Sarkar, who was imprisoned by the government of India. Some of his followers immolated themselves in protest. Sarkar was eventually released and spent the rest of his life pursuing his mission of \"self-realization and service to all.\"\n\n**\u0100NANDA-MAYA-KO\u015aA.** Layer made up of bliss. One of the five layers or \"sheaths\" ( **ko\u015bas** ) surrounding the self ( **\u0101tman** ). The \u0101nanda-maya-ko\u015ba is the inmost layer, closest to the self. _See also_ \u0100NANDA.\n\n**\u0100NANDAM\u0100Y\u012a M\u0100 (1896\u20131982).** \"Mother who is permeated with bliss.\" Spiritual teacher from **Bengal**. Born into a devout **Vai\u1e63\u1e47ava** family, her given name was Nirmala Sundar\u012b Dev\u012b. Her title of \u0100nandam\u0101y\u012b M\u0101 was conferred upon her by her disciples in the 1920s due to her blissful, God-absorbed nature. From an early age, she would often go into a very deep **meditative** state. In 1922, at the age of 26, she gave herself spiritual initiation, or **d\u012bk\u1e63a**. Breaking with the tradition of going to a **guru** for such initiation, she acted as her own guru. Five months later, she initiated her husband as well. She began to develop a large following of disciples. In 1932, she moved from Bengal to Dehradun, in the foothills of the **Him\u0101layas** , but traveled frequently throughout India for the rest of her life, teaching and establishing **\u0101\u015bramas** and hospitals. Her disciples, most of whom were **women** , included Kamala Nehru, the wife of the first prime minister of India, **Jawaharlal Nehru**. She also met both **Mohand\u0101s K. G\u0101ndh\u012b** and **Paramah\u0101\u1e43sa Yog\u0101nanda** \u2014the latter meeting being recounted in a chapter of Yog\u0101nanda's famous _Autobiography of a Yogi_. \u0100nandam\u0101y\u012b M\u0101 taught that \"The supreme calling of every human being is to aspire to self-realization. All other obligations are secondary.\" Though having followers from all over India, her following is particularly large in her native Bengal.\n\n**\u0100NANDAM\u016aRTI (1921\u20131990).** Prabhat Ranjan Sarkar; founder in 1955 of the **\u0100nanda M\u0101rga** , a modern Hindu religious movement.\n\n**ANA\u1e44GA.** Having no limbs, or no body; epithet of **K\u0101madeva** , the Cupid-like **god** of love. K\u0101madeva is called Ana\u1e45ga because **\u015aiva** destroyed his body with a blast of energy from his **third eye** when K\u0101madeva once tried to distract \u015aiva during his **meditation** by shooting one of his arrows into \u015aiva's heart.\n\n**ANANTA.** \"Infinite, unlimited\"; one of the attributes of **Brahman**. Also the name of the giant serpent with a thousand heads on whom **Vi\u1e63\u1e47u** reclines on the cosmic ocean, dreaming the cosmos into existence. Ananta is also called **\u015ae\u1e63a** , or remainder, because he is all that remains after the universe has been absorbed back into Vi\u1e63\u1e47u at the time of the **mah\u0101pralaya** , or great dissolution at the end of a **kalpa** , or cosmic cycle. \u015ae\u1e63a is said to have incarnated as **Lak\u1e63ma\u1e47a** and **B\u0101lar\u0101ma** , the brothers, respectively, of **R\u0101ma** and **K\u1e5b\u1e63\u1e47a** , the two most prominent **avat\u0101ras** or incarnations of Vi\u1e63\u1e47u.\n\n**ANCESTRAL RITES.** Two Hindu rituals dedicated to honoring one's ancestors are the funeral, or **antye\u1e63\u1e6di** , which occurs as soon as possible after **death** , and the **\u015br\u0101ddha** , which occurs periodically and consists of offering food and water. If one does not offer \u015br\u0101ddha to one's ancestors\u2014typically in the form of balls of rice called **pi\u1e47\u1e0da** \u2014one may run the risk that they will wander through the world as **pretas** , or hungry ghosts. The food offered in the \u015br\u0101ddha is also said to become the food of one's ancestors in their next **rebirth**.\n\n**\u0100\u1e46\u1e0c\u0100L (fl. 9th century CE).** One of the **\u0100\u1e37v\u0101rs** , or 12 **Vai\u1e63\u1e47ava** **poets** of south India. She composed approximately 107 of the roughly 4,000 **Tamil** poems that make up the _N\u0101l\u0101yirap Pirapantam_ , the collected works of the \u0100\u1e37v\u0101rs, which **\u015arivai\u1e63\u1e47avas** regard as inspired **scriptures** on the same level of authority as the _Vedas_. \u0100\u1e47\u1e0d\u0101l's revered position as a woman in this tradition is significant, being illustrative of the **philosophy** of **bhakti** , or devotion, according to which social categories such as gender and **caste** are irrelevant to one's spiritual status or proximity to salvation. _See also_ WOMEN.\n\n**A\u1e44GIRASA.** According to some lists, one of the Seven Sages ( **Saptar\u1e63i** ) presiding over the cosmos. The term is also sometimes used to refer to the ancestors of humanity, who are descendants of the sage A\u1e45girasa. Portions of the _Atharva Veda_ are attributed to the authorship of A\u1e45girasa.\n\n**ANGKOR WAT.** Magnificent **temple** complex built in Cambodia in the 12th century CE by the Khmer king S\u016bryavarman II. A striking example of Hindu influence on the culture of **Southeast Asia** , Angkor Wat is one of the largest temples in the world. It is modeled on the Hindu cosmos, centered on **Mount Meru** , and includes representations of numerous events from Hindu sacred **literature** and from the past lives of the **Buddha**. This complex includes temples dedicated to the Buddha, **Vi\u1e63\u1e47u** , and **\u015aiva**. It can thus be seen as an architectural embodiment of the **syncretism** frequently encountered among the South Asian religions.\n\n**ANIMAL SACRIFICE.** The ritual offering of the life of an animal as a way to seek the favor of a deity or deities. The practice is controversial in Hinduism, being a part of the ancient **Vedic** rite of **yaj\u00f1a** , and even today practiced in some corners of the Hindu world such as the **K\u0101l\u012bgh\u0101t** , a **temple** to the **Mother Goddess** in her fierce form as **K\u0101l\u012b** , in the city of **Calcutta** (Kolkata), as well as in **Nepal**. But the mainstream Hindu tradition has abandoned animal sacrifice, adopting the view of the **\u015brama\u1e47a** traditions that the killing of animals is a source of bad **karma** , and not at all meritorious.\n\nWhen animal sacrifices were more widely performed, in the early Vedic period, the task of the **Adhvaryu** priest\u2014a type of **Brahmin** \u2014included the slaughter, cooking, and offering of the bodies of sacrificial victims. But starting in the period of the _Upani\u1e63ads_ , a growing number of Brahmins deemphasized\u2014and the \u015brama\u1e47a movement even actively rejected\u2014the Vedic ritual of sacrifice, especially the practice of animal sacrifice, which came to be seen as an inherently violent karma that could only lead to a bad **rebirth** , rather than as an offering that was pleasing to the gods. By the middle of the first millennium of the Common Era, the practice of animal sacrifice had been largely abandoned by even orthodox Brahmins and replaced, in most parts of India, by vegetable substitutes. The embrace of the ideal of **ahi\u1e43s\u0101** , or nonviolence, on the part of the Brahmins led to this development, as well as to the expectation that Brahmins should practice **vegetarianism**.\n\nEvidence that the abandonment by most Hindus of the practice of animal sacrifice was due at least in part to the influence of \u015brama\u1e47a traditions, such as **Buddhism** and **Jainism** , exists in the form of the story of the **Buddha avat\u0101ra** , who is presented in some accounts as a Vedic reformer who ended the practice of animal sacrifice to promote the ideal of compassion.\n\nAnimals offered in ancient Vedic animal sacrifices included bulls and goats. The sacrifices still carried out at the K\u0101l\u012bgh\u0101t and in Nepal are primarily of goats. _See also_ COW, SACRED; G\u0100NDH\u012a, MOHAND\u0100S K.\n\n**A\u00d1JALI.** Gesture of reverence and respect consisting of the placing of the palms of the hands together. The a\u00f1jali is typically offered during **dar\u015bana** , when one encounters an image, or **m\u016brti** , of a deity, as well as when wants to offer respect to another person. In the latter case, the a\u00f1jali is often accompanied by the utterance \" **Namaste** ,\" which means \"I offer reverence to the divinity within you.\"\n\n**A\u00d1JANEYA.** Son of A\u00f1jan\u012b, epithet of **Hanuman** , whose mother is named **A\u00f1jan\u012b** and whose father is **V\u0101yu** , the **Vedic** deity of the wind.\n\n**A\u00d1JAN\u012a.** Mother of **Hanuman** , who for this reason is also known as A\u00f1janeya.\n\n**ANNA-MAYA-KO\u015aA.** Food layer; layer made of food; outermost of the five **ko\u015bas** , or layers, surrounding the self ( **\u0101tman** ) according to **Ved\u0101nta** ; the physical body ( **sth\u016bla \u015bar\u012bra** ).\n\n**ANTYE\u1e62\u1e6cI.** The **Vedic** funeral ritual of Hinduism; one of the major Hindu **sa\u1e43sk\u0101ras** or sacraments. This ceremony is conducted as soon as possible after **death** and includes the **cremation** of the deceased. The particulars of the funeral ritual vary a great deal, depending upon region and **caste** , but often extend over the course of several days.\n\nTraditionally, the son of the deceased plays a major role in this ritual, which is often seen as essential to the transition of the **soul** of the deceased peacefully to its next life. This is one of the reasons that sons are highly prized in traditional Hindu families. _See also_ ANCESTRAL RITES; \u015aR\u0100DDHA.\n\n**ANUBH\u0100VA.** Experience; the direct experience of reality; the most authoritative source of **knowledge** according to **Neo-Ved\u0101nta**.\n\n_ANUG\u012aT\u0100_ **.** \"Little _G\u012bt\u0101_ ,\" part of the _Mah\u0101bh\u0101rata_ , a brief summary and restatement by **K\u1e5b\u1e63\u1e47a** to **Arjuna** , after the _Mah\u0101bh\u0101rata_ war, of the teachings contained in the _Bhagavad G\u012bt\u0101_. It is in the 14th book of the _Mah\u0101bh\u0101rata_.\n\n**ANUM\u0100NA.** Inference; a **pram\u0101\u1e47a** , or valid source of **knowledge** in Indian **philosophy** ; reasoning from a set of premises and\/or observations to a conclusion.\n\n**AP.** Water; the medium of taste ( **rasa** ); one of the five elements or states of being that constitute the physical world ( **mah\u0101bh\u016btas)**. The five elements are earth ( **p\u1e5bthv\u012b** ), air ( **v\u0101yu** ), fire ( **agni** ), water (ap), and space ( **\u0101k\u0101\u015ba** ).\n\n**APARIGRAHA.** Nonattachment; nonpossessiveness; the fifth **yama** or restraint that is enjoined in the _Yoga S\u016btra_ ; also included among the moral precepts of **Jainism**.\n\n**APSARAS.** Nymph; celestial maiden. _See also_ URVA\u015a\u012a.\n\n_\u0100RA\u1e46YAKAS_ **.** \"Forest texts.\" The third of the four sections contained within each of the four _Vedas_. This collection of Vedic writings was composed during the period roughly between the composition of the _Br\u0101hma\u1e47as_ and the _Upani\u1e63ads_ , approximately 600 to 500 BCE according to most contemporary scholarship. They are called \"forest texts\" because the teaching within them is considered sufficiently esoteric that it should be conveyed in the forest, away from society, where it might be overheard. The term also evokes the forest academies, or _\u0101\u015bramas_ , in which **Vedic** sages pursued contemplative activities away from society and taught their wisdom to their students.\n\nThe _\u0100ra\u1e47yakas_ are known as the **Up\u0101sana Kanda** , or \"contemplative portion\" of the _Vedas_ , in contrast with the earlier **Karma Kanda** , or **action** portion, consisting of the more ritualistic Vedic hymns and the _Br\u0101hma\u1e47as_ , and the **J\u00f1\u0101na Kanda** , or \"wisdom portion,\" consisting of the _Upani\u1e63ads_.\n\nIn terms of their style and content, the _\u0100ra\u1e47yakas_ can be seen as transitional texts, marking a shift from the more ritualistic concerns of the earlier portions of the Vedic **literature** to the more philosophical and speculative interests reflected in the _Upani\u1e63ads_.\n\nThe boundaries between the genres of _Br\u0101hma\u1e47a_ , _\u0100ra\u1e47yaka_ , and _Upani\u1e63ad_ being somewhat fluid, one of the later _\u0100ra\u1e47yakas_ , the _B\u1e5bhad\u0101ranyaka_ , is also known as one of the oldest _Upani\u1e63ads_. The content of the _\u0100ra\u1e47yakas_ is chiefly concerned with the inner meaning of the Vedic rituals.\n\n**\u0100RAT\u012a.** Brief ceremony, typically performed daily in **temples** , as well as at home by individual devotees. \u0100rat\u012b involves the waving of a flame\u2014either a lit oil lamp ( **d\u012by\u0101** ) fed by **ghee** , **camphor** , or oil, or a lit candle or set of lit candles placed on a plate\u2014in front of an image ( **m\u016brti** ) of a deity or in front of a living, revered person, such as a **guru**. The flame is typically waved in a clockwise circular motion or the devotee might also trace out the syllable **O\u1e43** in the air with the flame. The \u0101rat\u012b is accompanied by the ringing of a bell, and usually with either the chanting of **mantras** or singing. A particularly popular song that is sung to accompany \u0101rat\u012b in the modern period is **O\u1e43 jaya Jagadi\u015ba Hare** , as recorded by the popular **Hindi** singer Lata Mangeshkar.\n\n**ARDHANAR\u012a\u015aVARA.** Half-female lord; a representation of the divine couple, **\u015aiva** and **\u015aakti** as one being. Typically, the right half of the image has the form of the male deity, \u015aiva, and the left half of the image has the form of his wife, the **Mother Goddess** , \u015aakti. Symbolically, this image communicates the idea that the divine includes attributes that are both masculine and feminine, ultimately transcending both. It also represents the inclusion of all opposites within the divine, the fact that the divine reality incorporates and transcends all limiting qualities. This image is especially popular in **Tantric** forms of Hinduism, and the **philosophy** that underlies it reflects a Tantric sensibility. _See also_ WOMEN.\n\n**ARDHAPADM\u0100SANA.** The half-lotus posture of **Ha\u1e6dha Yoga** , formed by placing the right foot on the left thigh. _See also_ \u0100SANA; YOGA.\n\n**ARJUNA.** One of the **P\u0101\u1e47\u1e0dava** brothers, hero of the _Mah\u0101bh\u0101rata_ , and interlocutor of **K\u1e5b\u1e63\u1e47a** in the _Bhagavad G\u012bt\u0101_. Among the P\u0101\u1e47\u1e0davas, Arjuna is regarded as the greatest warrior, his favored weapon being the bow and arrow. It was Arjuna who won the hand of **Draupad\u012b** in marriage by winning the contest of arms at her **svaya\u1e43vara**. Though his legal father was **Pa\u1e47\u1e0du** , Arjuna also had a divine father, **Indra** , who was invoked by his mother, **Kunt\u012b** , when Pa\u1e47\u1e0du became unable to conceive heirs due to a curse. When the P\u0101\u1e47\u1e0davas were required to live incognito for a year, Arjuna disguised himself as a female dance instructor.\n\nArjuna was the father of the boy warrior Abhimanyu, who was tragically slain in the _Mah\u0101bh\u0101rata_ war. Arjuna's greatest rival was **Kar\u1e47a** , whom he slew in the battle of **Kuruk\u1e63etra** , along with his teacher, **Dro\u1e47a**.\n\n**\u0100ROGYA.** Freedom from disease; one of the main goals of traditional Hindu medical science, or **\u0100yur Veda**.\n\n**AR\u1e62A VIDY\u0100 GURUKULAM.** **\u0100\u015brama** and Hindu educational center built by **Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b** in 1986 and located in Saylorsburg, Pennsylvania. Its purpose is to promote Hindu thought and practice in the United States. The \u0101\u015brama offers courses in Sanskrit and Indian **philosophy** , with a strong emphasis on **Advaita Ved\u0101nta** philosophy.\n\n**ART.** Art, in all its varied forms\u2014the visual arts, **dance** , **literature** , **music** , and **poetry** \u2014plays a major role in Hindu practice. Two particular currents of Hindu thought\u2014the **philosophy** of **bhakti** , or devotion, and the movement known as **Tantra** \u2014endorse the idea that sense experience can point beyond itself to the realm of the transcendent, and that the aesthetic impulse toward beauty, rather than being a distraction from spiritual pursuits, is a force that can be channeled toward the divine, which is its ultimate proper object. Hindu art is not, therefore, traditionally seen as \"art for art's sake\" but as a form of what scholar of religion Huston Smith calls \"spiritual technology\"\u2014a way of inspiring the aesthetic sense and directing it toward spiritual ends.\n\nThe aesthetic sense\u2014 **rasa** , or \"taste\" in **Sanskrit** \u2014has been theorized extensively by Hindu thinkers, particularly in the _N\u0101\u1e6dya \u015a\u0101stra_ of **Bharata** , a text of the early first millennium of the Common Era, by the 10th\u201311th century **Ka\u015bm\u012br \u015aaiva** philosopher **Abhinavagupta** and by the **Gosv\u0101mins** , or **Vai\u1e63\u1e47ava** theologians of the late medieval period. Abhinavagupta saw rasa, or aesthetic experience, as being akin to the experience of **mok\u1e63a** , and as itself a potentially liberating phenomenon. The Gosv\u0101mins applied this Tantric insight to their devotional practice, arguing that listening to and visualizing oneself as a participant in the stories of the deity **K\u1e5b\u1e63\u1e47a** and his earthly activities can evoke an aesthetic experience that, by K\u1e5b\u1e63\u1e47a's grace, can lead to the state of **liberation**.\n\nIn terms of visual art, one of the most striking features of Hindu ritual practice is its deployment of **m\u016brtis** , or sculptured images of deities in worship. The basic worship ritual, or **p\u016bj\u0101** , involves evoking the presence of the divine within the image\u2014which, in effect, and for the duration of the ritual, \"becomes\" the deity, who is then treated just as an honored human guest would be treated when visiting a home. P\u016bj\u0101 is one of the most common expressions of bhakti and is carried out both in the home and in the **temple**. A temple is also, itself, a work of art and is designed both to express and evoke devotion.\n\nOn the more Tantric side, other works of visual art used to cultivation experience of the divine are the **yantra** and the **ma\u1e47\u1e0dala** \u2014abstract diagrams that are used as objects of **meditation**.\n\nBoth p\u016bj\u0101 and meditation are accompanied and aided by music. Sacred chanting is one of the most ancient of Hindu practices. The correct practice of chanting the _Vedas_ is laid out in the _S\u0101ma Veda_ , a text dating from roughly 1200 to 1000 BCE that forms the basis for later Indian classical music. A wide range of musical genres have emerged in subsequent periods, designed to evoke either a meditative state or devotional feelings.\n\nDance, too, which combines the visual sense with music, has been developed to a high degree of sophistication, with a variety of styles emerging in different periods and in different parts of the Indian subcontinent\u2014a process that continues today, in the era of global Hinduism. In classical Indian dance, every move\u2014even the facial expressions\u2014of a dancer carry a meaning and form a vocabulary through which a story is told or a lesson is taught.\n\nFinally, literature, including poetry, plays a foundational role in Hindu practice. The preferred medium for communicating philosophical concepts is storytelling. Ancient Hindu bards, known as **sutas** , memorized vast narratives, such as the epic _Mah\u0101bh\u0101rata_ and _R\u0101m\u0101ya\u1e47a_ , and the **Brahmins** , or priestly community, committed vast portions of the Vedic literature to memory as well. In the first millennium of the Common Era, the practice of writing became more widespread\u2014though the earlier tradition of memorization also persisted\u2014and a profusion of texts resulted, with genres spanning a range including drama, poetry, legal writing, and philosophical speculation. Poetry, in particular, was the preferred medium by which the saints of the bhakti movement expressed their devotional feelings\u2014often in vernacular languages, rather than the more formal Sanskrit of the Vedic tradition. These poems, set to music, became the popular hymns or **bhajanas** sung in the setting of Hindu temples and religious gatherings even today.\n\n**ARTHA.** Wealth, power, worldly success, the ability to fulfill one's desires. Along with sensory pleasure ( **k\u0101ma** ), virtue ( **dharma** ), and **liberation** ( **mok\u1e63a** ), one of the four goals of humanity ( **puru\u1e63\u0101rthas** ). This term also refers to goals in general, as well as to the meaning of a word\u2014the \"object\" or purpose to which a word points beyond itself.\n\n_ARTHA \u015a\u0100STRA_ **.** Sanskrit text of political science traditionally attributed to **Kau\u1e6dilya** (350\u2013283 BCE, also known as Canakya), a **Brahmin** and the chief political advisor to Candragupta **Maurya**. The oldest parts of the text were likely composed around 300 BCE, with additions made by various authors over the course of the next few centuries.\n\nThe **philosophy** of kingly rule outlined in the text was widely followed by Hindu kings for centuries, particularly its **ma\u1e47\u1e0dala** theory of interstate relations, according to which a king should see himself as being in the middle of a series of concentric circles of alternating enemies and allies. The text covers such topics as diplomacy, administrative structures, military organization, **laws** , and punishments for specific crimes.\n\nThe _Artha \u015a\u0101stra_ is frequently compared to the later work of Niccol\u00f2 Machiavelli, with its sometimes shockingly frank approach to topics such as the deployment of espionage and subterfuge as legitimate tools for the advancement of state interests. At the same time, the text does contain ethical reflections and sets forth the criteria by which an ideal king should strive to live. But its approach to the issues a king might face is best characterized as _realpolitik_.\n\n**\u0100RYA SAM\u0100J.** Hindu reform organization that was established by **Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b** (1824\u20131883) in 1875 and devoted to the practice and promulgation of what it regards as a pure, **Vedic** Hinduism. The \u0100rya Sam\u0101j does not utilize **m\u016brtis** , or images of deities. It regards the _P\u016br\u0101\u1e47as_ and the customs associated with P\u016br\u0101\u1e47ic Hinduism as later additions to the original Vedic tradition. \u0100rya Sam\u0101j rituals consist of adaptations of Vedic rituals centered on a sacred fire, such as the **havan** , or fire offering accompanied by Vedic prayers.\n\nIn the late 19th century, the \u0100rya Sam\u0101j initiated a ritual enabling non-Hindus to convert to Hinduism. Although this ritual is available to non-Indians, its main historical emphasis has been upon \"reconversion\"\u2014or rather, conversion of non-Hindus of Indian descent whose ancestors had converted to non-Indic traditions such as **Christianity** and **Islam**.\n\nThe \u0100rya Sam\u0101j does not completely reject the concept of **caste** , because the four **var\u1e47as** are mentioned in the _\u1e5ag Veda_ and therefore have Vedic sanction. But it does reject the idea of **j\u0101ti** , or birth caste, maintaining that one's caste should be a matter of individual merit. Its conversion ceremony has sometimes been utilized to ritually raise the caste status of low-caste persons, a version of the process of acculturation to Vedic norms known as **Sanskritization**. In 1893, a division arose between those within the \u0100rya Sam\u0101j who advocated **vegetarianism** , long an upper-caste custom, and those who opposed dietary restrictions as a later, non-Vedic development due to **\u015brama\u1e47a** influence.\n\nThe \u0100rya Sam\u0101j has been actively involved in political and social issues since its inception, particularly the Indian independence movement. Since independence and the partition of India and Pakistan, it has also been associated with the **Hindu Nationalist** movement, promoting the growth of such Hindu nationalist organizations as the **Hindu Mah\u0101sabh\u0101** and **Bh\u0101rat\u012bya Janata Party**.\n\nIn 1992, **Sw\u0101m\u012b Agnive\u015ba** was expelled from the \u0100rya Sam\u0101j, forming the **World Council of the \u0100rya Sam\u0101j** , a distinct organization from the original \u0100rya Sam\u0101j, that sees itself as maintaining fidelity to the original ideals of Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b.\n\n**ARYAMAN.** Literally \"companion.\" Early Vedic **deva** , or deity; one of the 12 **\u0100dityas** , sons of the **goddess** **Aditi** and the sage **Ka\u015byapa**. Aryaman is closely associated with the sun and the Milky Way, and is said to be the eye of the **Puru\u1e63a** , or \"cosmic man,\" from whose sacrificed body the universe was fashioned according to the _Puru\u1e63a S\u016bkta_ of the _\u1e5ag Veda_. He is in charge of honor, chivalry, and nobility\u2014the qualities of an **\u0101ryan** \u2014and the rules of society.\n\n**\u0100RYAN.** Noble, cultured, \"gentleman,\" a term by which ancient **Vedic** peoples referred to themselves and to their cultural and religious practices (in contrast with a **mleccha** , or \"barbarian\"). The term _\u0101rya_ had no ethnic or racial connotations until 19th-century European scholarship thus interpreted it, on the false assumption of a correlation between culture and ethnicity. The term refers not to ethnic characteristics but to cultural and spiritual ones. It is pervasive in Indic traditions, including **Buddhism** , where it is used in such contexts as the \"Four Noble **Truths** \" and the \"Noble Eightfold Path\" to refer to that which confers the dignity of spiritual realization. Its range of connotations is comparable to the English word \"civilized.\" Based on the assumption that the practitioners of ancient Vedic cultures formed a single ethnic group, the term _Aryan_ was coined in the 19th century to refer to a primordial \"white\" European race, a concept foundational to racist political ideologies such as Nazism. _See also_ \u0100RYAN INVASION THEORY; INDO-\u0100RYAN; INDO-EUROPEAN.\n\n**\u0100RYAN INVASION THEORY.** **Indological** theory developed by 19th-century scholars, and most prominently associated with **Friedrich Max M\u00fcller** (1823\u20131900), to account for the many strong resemblances between **Vedic** language and culture and the other languages and cultures of the **Indo-European** linguistic family. The theory is controversial. Having subsequently been modified into the **\u0100ryan Migration Theory** in order to better account for the available data, it is still contested by some scholars who prefer a model that locates the Indo-European homeland in India.\n\nAccording to the \u0100ryan Invasion Theory, Indo-European languages and cultures were brought into India shortly before 1500 BCE through a massive invasion of light-skinned, **chariot** -riding, Indo-European warriors who then conquered the darker-skinned indigenous population. These invaders, according to this theory, then settled in India and imposed their language and culture, including the **caste** system, upon the native Indians. The discovery of the **Indus Valley civilization** was initially thought to lend credence to this theory, being thought to consist of the remains of the civilization destroyed by the \u0100ryans. But it has since been confirmed that this civilization never suffered an invasion.\n\nThe \u0100ryan Invasion Theory has proven controversial for a variety of reasons, both scholarly and ideological. As knowledge of ancient India has improved, the theory has been gradually modified over time, and its contemporary form is more accurately known as the \u0100ryan Migration Theory, which postulates not a massive, violent invasion but a gradual acculturation process that very likely involved a relatively small number of actual migrants from Central Asia.\n\nPartly in reaction to the implication of this theory in racist ideologies, and partly on the basis of **Hindu Nationalist** assumptions, some Hindu scholars have proposed the **Out of India** **Theory** as an alternative model of Vedic origins.\n\n**\u0100RYAN MIGRATION THEORY.** **Indological** theory that has superseded the **\u0100ryan Invasion Theory** as the mainstream scholarly model of ancient Indian history\u2014and in particular, of **Vedic** origins. The \u0100ryan Migration Theory could be regarded either as a new theory or as a revised form of the old invasion theory. It preserves, from the earlier theory, the postulate that **Indo-European** linguistic and cultural forms were brought to India by migrants from Central Asia. Unlike the earlier invasion theory, however, the newer migration theory suggests that this process was gradual, at least largely peaceful, and that it involved a relatively small number of actual human migrants.\n\nImportantly, this theory also rejects the earlier theory's conflation of culture and ethnicity that helped to fuel racist ideologies in the first half of the 20th century. It essentially seeks to balance the linguistic evidence, which strongly suggests that Indo-European language and culture had a point of origin outside of India, with the lack of any evidence for a massive invasion of the kind postulated in the original invasion theory: no devastated cities, no large influx of genetic material significantly different from the types already found in India, and no clear or unambiguous record of an invasion having ever occurred (though sections of the _\u1e5ag Veda_ had long been interpreted by European scholars as providing such an account).\n\nThe \u0100ryan Migration Theory is contested by the proponents of the **Out of India Theory**. This theory postulates India as the homeland of Indo-European culture, which then spread to Central Asia and Europe, and (usually) Vedic **Sanskrit** as the Proto-Indo-European language. This model is largely rejected by mainstream scholars due to the presence of non-Indo-European substrate elements in Vedic Sanskrit, indicating that this language cannot be identical to Proto-Indo-European (as proponents of this theory often claim), as well as deep discomfort with the **Hindu Nationalist** ideological bias perceived to motivate at least some versions of this model, which leads to a selective reading of the evidence driven by the foregone conclusion that India _must_ be the Indo-European point of origin. The academic conversation surrounding the issue of Vedic origins remains very highly politically charged.\n\n**\u0100\u1e62\u0100\u1e0cHA.** Month of the **Hindu calendar** that corresponds roughly to the period from mid-June to mid-July.\n\n**\u0100SANA.** Yogic posture, literally \"seat.\" In **Ha\u1e6dha Yoga** , the branch of yoga concerned with the physical postures most conducive to **meditative** practice ( **dhy\u0101na** ), a complex system of postures and exercises has been developed. A couple of the most well known are the lotus posture ( **padm\u0101sana** ) and the half-lotus posture ( **ardhapadm\u0101sana** ), but there are a great many others as well. Indeed, new \u0101sanas continue to be developed even today.\n\nMost \u0101sanas share the characteristic of keeping the back straight and the head and neck in alignment in order to facilitate deep breathing exercises ( **pr\u0101\u1e47ay\u0101ma** ). \u0100sana is one of the limbs of the **a\u1e63\u1e6d\u0101\u1e45ga** or \"eight-limbed\" system of **yoga** set forth in the _Yoga S\u016btras_ of **Pata\u00f1jali**. But, contrary to popular belief, the elaborate system of postures that is associated with Ha\u1e6dha Yoga cannot be found in this text, which simply states that one's yogic posture should be \"stable\" and \"comfortable.\"\n\n**ASCETICISM.** The sacrifice of specific comforts or pleasures or the taking on of extra discomfort in order to discipline oneself and to attain spiritual goals. Derived from the Greek _askesis_ , a term that originally refers to athletic and military training, \"asceticism\" is the most common translation of the **Sanskrit** term _tapasya_. An ethos of asceticism is a strong element of many Hindu traditions, which teach renunciation as a means to **mok\u1e63a**.\n\nIn ancient Hindu literary works, such as the _Mah\u0101bh\u0101rata_ , the _R\u0101m\u0101ya\u1e47a_ , and the _P\u016br\u0101\u1e47as_ , one finds a widespread view that the rigorous practice of asceticism can lead to paranormal powers for the practitioner. Ascetics are frequently depicted as receiving special boons from the **gods** as a reward for their practices. In later **Vedic** **literature** , ascetic practice takes on a role similar to that of Vedic sacrifice in its ability to evoke divine favors. Ascetics are widely viewed as powerful, and sometimes dangerous, figures. _See also_ BRAHMACARYA; SANNY\u0100SA.\n\n**\u0100\u015a\u012aRV\u0100DA.** Blessing; benediction.\n\n**\u0100\u015aRAMA.** One of the stages of life. Also a hermitage or place of spiritual retreat where one resides for a period of time, usually with one's **guru**.\n\nThe stages of life are the student stage ( **brahmacarya** ), the householder stage ( **g\u0101rhasthya** ), the retirement\u2014or literally, the \"forest dweller\"\u2014stage ( **vanaprastha** ), and renunciation ( **sanny\u0101sa** ). The brahmac\u0101rya stage traditionally lasts from roughly the age of 12 to 24, g\u0101rhasthya from 24 to 48, and vanaprastha from 48 to 72 (until either **death** or renunciation). Sanny\u0101sa can be undertaken at any point along the process, but it is a drastic step, being permanent and involving the complete renunciation of one's previous social identity and family connections. The _Dharma \u015a\u0101stras_ recommend that one should undertake sanny\u0101sa only after going through the other three stages, so only around the age of 72 or later. The first two stages, those of the student and householder, make up the _prav\u1e5btti m\u0101rga_ , the path of engagement with the world, in which one pursues the first three of the four goals of life ( **puru\u1e63\u0101rthas** ): pleasure ( **k\u0101ma** ), wealth ( **artha** ), and virtue ( **dharma** ). The latter two stages, retirement and renunciation, make up the _niv\u1e5btti m\u0101rga_ , the path of **asceticism** and disengagement from the world in the pursuit of **mok\u1e63a** , or **liberation** from the cycle of **rebirth** , the fourth goal of life.\n\nSome scholars have speculated that the inclusion of sanny\u0101sa as the fourth stage of life is a late development\u2014a reaction to ascetic traditions like **Jainism** and **Buddhism**.\n\nThe \u0101\u015bramas can be seen as defining the horizontal axis of a grid on which the traditional Hindu way of life, or dharma, could be comprehensively mapped out, with **var\u1e47a** or **caste** making up the vertical axis. At any given time, a human being is located somewhere on this grid, though sanny\u0101sa could be seen as stepping off of it for the sake of liberation.\n\n**A\u1e62\u1e6c\u0100\u1e44GA YOGA.** The eight-step, or literally \"eight-limbed\" **yoga** system presented in the _Yoga S\u016btras_ of **Pata\u00f1jali**. The eight steps consist, first, of five moral precepts or \"restraints\" ( **yama** ), five observances or good habits ( **niyama** ), the mastery of a posture ( **\u0101sana** ), control of the breath ( **pr\u0101\u1e47\u0101y\u0101ma** ), the control of reactions to sensory stimuli ( **praty\u0101h\u0101ra** ), concentration on a single object ( **dh\u0101ra\u1e47\u0101** ), **meditative** absorption into that object ( **dhy\u0101na** ), and contemplative union with that object ( **sam\u0101dhi** ).\n\nLater traditions that utilize the _Yoga S\u016btras_ distinguish between sam\u0101dhi in which there remains a sense of subjectivity ( **savikalpa sam\u0101dhi** ) and sam\u0101dhi without subject\u2013object duality ( **nirvikalpa sam\u0101dhi** ). This distinction is most prominent in **Buddhism** , **Ka\u015bm\u012br \u015aaivism** , and the **Neo-Ved\u0101nta** tradition of **Ramakrishna**.\n\nThe moral precepts of Pata\u00f1jali's system are identical to the vows or **vratas** of **Jainism** and four of them are identical to four of the Five Precepts of Buddhism. This suggests a shared origin of Pata\u00f1jali's system in the **\u015brama\u1e47a** **philosophy** that gave rise to both the Buddhist and Jain traditions. In the _Yoga S\u016btras_ , the complex system of \u0101sanas associated with **Ha\u1e6dha Yoga** is not present. It is simply said that one's posture must be \"firm and comfortable.\"\n\n_A\u1e62\u1e6c\u0100VAKRA G\u012aT\u0100_ **.** **Advaita Ved\u0101nta** text composed in the first millennium CE, very likely before **\u015aa\u1e45kara** (788\u2013820), but after the composition of the _Brahma S\u016btras_ (first or second century CE). It recounts a dialogue between the sage A\u1e63\u1e6d\u0101vakra, for whom the text is named, and the enlightened king, **Janaka** (who is also mentioned in the _Upani\u1e63ads_ and represented as the father of **S\u012bt\u0101** in the _R\u0101m\u0101ya\u1e47a_ ).\n\n**ASTEYA.** Nonstealing. The third of the five **yamas** or moral restraints enjoined in the _Yoga S\u016btras_ of **Pata\u00f1jali**. Also included among the moral precepts of both **Jainism** and **Buddhism**.\n\n**\u0100STIKA.** Literally \"one who says 'it is'\"; orthodox. In classical Hinduism, the test of orthodoxy is acknowledgment of the authority of the _Vedas_. Systems of philosophy ( **dar\u015banas** ) that affirm **Vedic** authority ( **Ny\u0101ya** , **Vai\u015be\u1e63ika** , **M\u012bm\u0101\u1e43s\u0101** , and **Ved\u0101nta** ) or that do not reject it outright ( **S\u0101\u1e43khya** and **Yoga** ) are \u0101stika, or orthodox, and those that reject Vedic authority ( **C\u0101rv\u0101ka** , **Jainism** , and **Buddhism** ) are heterodox, or **n\u0101stika**. In contemporary Hinduism, the terms _\u0101stika_ and _n\u0101stika_ sometimes refer to theism and atheism, respectively, but this is not their original sense.\n\n**ASTROLOGY.** There is a strong emphasis in Hinduism on determining the most auspicious time to undertake any important venture. Astrology\u2014the determination of auspicious occasions through the observation of the relative motions of the stars and planets\u2014is an ancient and mathematically sophisticated system in Hinduism. The moment of a child's birth will often be used to choose a name for the child, and the astrological charts of a prospective bride and groom will be used to determine if they are a suitable match, and also to determine the most auspicious date and time for their wedding. _See also_ CALENDAR, HINDU; NAVAGRAHAS.\n\n**ASURA.** Powerful one, titan, anti-god, demonic, a demonic being. A species of being usually depicted as in perpetual conflict with the **devas**. Interestingly, not all Hindu texts that depict the conflict of the asuras and the devas present the asuras as uniformly evil or the devas as uniformly good, though over time these two classes of being did come to be viewed in this way.\n\nIn its original meaning, the word _asura_ simply means a powerful being. In fact, it is a frequently used title for the deva **Varu\u1e47a** in the _\u1e5ag Veda_ , who might have been the supreme deity early in the development of the Vedic religion. Over time, an etymology developed that defined these beings as \"not sura,\" that is, non-devas, or anti- **gods**.\n\nProminent asuras in later Hindu sacred **literature** \u2014the _Pur\u0101\u1e47as_ \u2014are **Hira\u1e47yak\u1e63a** , who is slain by the **Var\u0101ha** (boar) **avat\u0101ra** of Vi\u1e63\u1e47u; **Hira\u1e47yaka\u015bipu** , who is slain by the **Narasi\u1e43ha** (man-lion) **avat\u0101ra** of **Vi\u1e63\u1e47u** ; and the **Mahi\u1e63\u0101sura** , or Buffalo Demon, who is slain by the **goddess** **Durg\u0101**. Symbolically, the slaying of asuras by Vi\u1e63\u1e47u or Durg\u0101 is said to represent the lower impulses\u2014greed, selfishness, hatred\u2014being overcome by one's divine inner self. In **Tantric** thought, it is said that all the devas and asuras are contained within oneself and that the conflict between them represents an inner struggle among the various aspects of **self**. In the _Mah\u0101bh\u0101rata_ , the fabulous palace of the **Pa\u1e47\u1e0davas** in **Indraprastha** is designed by an asura architect named **M\u0101ya**. In both the _Mah\u0101bh\u0101rata_ and the _Pur\u0101\u1e47as_ , it is said that the devas and asuras once cooperated in order to churn the elixir of **immortality** , **am\u1e5bta** , from the cosmic ocean.\n\nIn chapter 16 of the _Bhagavad G\u012bt\u0101_ , divine and demonic, or devic and asuric, qualities are contrasted. The asuric personality is described as egotistical, atheistic, and selfish, whereas the devic personality is benevolent, devout, and generous.\n\nIt is possible that the opposition between the devas and asuras reflects an ancient ideological conflict among the ancient **Indo-Europeans**. It is noteworthy in this regard that these two terms take opposite meanings in the Indo-European culture of Iran, where **Ahura Mazda** \u2014the Great Asura\u2014becomes the supreme deity of **Zoroastrianism** , whereas the devas come to be depicted as demonic beings, and later influence the etymology of the English word \"devil\" (as well as the Latin \"deus,\" the Greek \"theos,\" and the English \"divine,\" which comport better with the Indic valuation of a deva as a deity). Similarly, the _Aesir_ of ancient Norse religion seem to be related to the asuras.\n\n**A\u015aVAMEDHA.** Horse sacrifice. **Vedic** ritual intended to ensure the success of a king. _See also_ ANIMAL SACRIFICE; YAJ\u00d1A.\n\n**A\u015aVATTH\u0100MA.** Character in the _Mah\u0101bh\u0101rata_ ; son of **Dro\u1e47a**. A\u015bvatth\u0101ma avenged his father's **death** by cutting the throats of many **P\u0101\u1e47\u1e0dava** warriors in their sleep. Dro\u1e47a had been killed through deception, laying down his arms when hearing that A\u015bvatth\u0101ma, who was still alive, had been slain.\n\n**A\u015aVINA.** Month of the **Hindu calendar** that corresponds roughly to the period from mid-September to mid-October.\n\n**A\u015aVINS.** \"Horsemen\"; twin **Vedic** deities, sometimes said to have the heads of horses. Their father is **Vivasvat** , the solar deity, and one of the 12 **\u0100dityas**. The A\u015bvins have healing powers, due to their great **knowledge** of plants and herbs, and are associated with the offering of **soma**. As well as being divine physicians, they are also associated with agriculture. In the _Mah\u0101bh\u0101rata_ , they are the fathers of the twin **Pa\u1e47\u1e0dava** brothers, **Nakula** and Sahadeva.\n\n_ATHARVA VEDA_ **.** The fourth of the original four _Vedas_ to be written. It consists of prayers and brief rituals (in contrast with the more elaborate rituals of the _Yajur Veda_ ), as well as philosophical reflections that anticipate those of the _Upani\u1e63ads_. Due to the many differences in style and content between the _Atharva Veda_ and the other three _Vedas_ , as well as early references to \"the three _Vedas_ ,\" scholars have concluded that this text was compiled at least a century or two later than the other _Vedas_.\n\n**\u0100TMAN.** Self, soul, or principle of identity. In its earliest uses, this term refers to the physical body, or to the breath. But fairly early, in the _Upani\u1e63ads_ , it comes to refer both to a universal Self that is ultimately identical with **Brahman** , the ultimate reality, and to the individual self or soul that undergoes **rebirth** , or reincarnation, remaining essentially the same through the various changes of form in which it is embodied. In the _Upani\u1e63ads_ , it is stated that the individual, reincarnating \u0101tman is ultimately identical to the universal Self. According to the **Advaita Ved\u0101nta** interpretation of the _Upani\u1e63ads_ , the realization of the fundamental identity of the individual self and the universal Self is constitutive of **mok\u1e63a** , or **liberation** from the cycle of rebirth.\n\nIn later forms of **Ved\u0101nta** , a distinction is made between individual reincarnating souls, or **j\u012bv\u0101tman** , and **God** , the Supreme Soul that resides in all souls: the \"oversoul\" or **param\u0101tman**. Advaita Ved\u0101nta teaches that the \u0101tman is the basis of all experience, the transcendental condition for any experience whatsoever. Though its existence cannot be proven, it can be experienced directly as awareness and as one's very self.\n\nAdvaita Ved\u0101nta, as well as **S\u0101\u1e43khya** and **Yoga** **philosophy** , identify the \u0101tman with consciousness and view it as all pervasive. But Advaita sees the \u0101tman as ultimately singular, with the differences among **j\u012bvas** being an illusion. S\u0101\u1e43khya, Yoga, **Jainism** , **Ny\u0101ya** , **Vai\u015be\u1e63ika** , and the other forms of Ved\u0101nta, on the other hand, affirm the plurality and the numerical distinctness of souls and do not affirm a singular \u0101tman.\n\nNy\u0101ya, Vai\u015be\u1e63ika, and **Dvaita** and **Vi\u015bi\u1e63\u1e6d\u0101dvaita Ved\u0101nta** say that the \u0101tman is atomic in size. These two traditions also maintain that the \u0101tman is ontologically distinct from **God** , the supreme \u0101tman. The _Bhagavad G\u012bt\u0101_ emphasizes its imperishability and its self-identity through many births. A central teaching of most forms of Hinduism is that we are all, in reality, the \u0101tman, not the physical body. The physical body is merely the temporary vehicle of the \u0101tman.\n\nAlthough the term _\u0101tman_ can be, and often is, loosely translated as soul, as pure consciousness, it is distinct from the **mind** , and so different from the soul as traditionally conceived in the Abrahamic traditions, where the soul is often seen as a rational principle. Mind, on a Hindu understanding (such as in Advaita Ved\u0101nta, S\u0101\u1e43khya, and Yoga), is a part of the material continuum\u2014 **m\u0101y\u0101** or **prak\u1e5bti** \u2014and so distinct from the \u0101tman. In the _Upani\u1e63ads_ , it is taught that the \u0101tman is surrounded by a series of progressively more material sheaths or **ko\u015bas**. The innermost, and closest to the \u0101tman, is the sheath of bliss ( **\u0101nanda-maya-ko\u015ba** ), followed by the sheath of intellect ( **vij\u00f1\u0101na-maya-ko\u015ba** ), the sheath of mind ( **mano-maya-ko\u015ba** ), the sheath of life force or breath ( **pr\u0101\u1e47a-maya-ko\u015ba** ), and the gross, physical sheath ( **anna-maya-ko\u015ba** ).\n\n**ATRI.** According to some lists, one of the Seven Sages ( **Saptar\u1e63i** ) presiding over the cosmos; father of the sage **Durv\u0101sa**.\n\n**AUM.** _See_ O\u1e42.\n\n**AUROBINDO, SRI (1872\u20131950).** _See_ GHOSE, AUROBINDO.\n\n**AVAT\u0100RA.** Divine incarnation, manifestation of a deity, usually **Vi\u1e63\u1e47u** , in a physical, sometimes human, form. Several lists of avat\u0101ras exist, but the most common lists name 10 such incarnations of Vi\u1e63\u1e47u: **Matsya** the fish avat\u0101ra, **K\u016brma** the tortoise avat\u0101ra, **Var\u0101ha** the boar avat\u0101ra, **Narasi\u1e43ha** the man-lion avat\u0101ra, **Vamana** the dwarf avat\u0101ra, **Para\u015bur\u0101ma** , **R\u0101ma** , **K\u1e5b\u1e63\u1e47a** , **Balar\u0101ma** (who is replaced in later versions of this list by **Buddha** ), and **K\u0101lkin** the future **avat\u0101ra**. **Aurobindo Ghose** interprets this list in light of modern theories regarding evolution. Other lists of avat\u0101ras include as many as 22, or even 108, incarnations, including such figures as the sage **K\u0101pila** , the founder of **S\u0101\u1e43khya** **philosophy** , and **\u1e5a\u1e63abha** , the first **T\u012brtha\u1e45kara** of **Jainism**.\n\nAccording to some **Vai\u1e63\u1e47ava** traditions, the ape deity **Hanuman** is an avat\u0101ra of **\u015aiva** who came to Earth to assist R\u0101ma in his mission to destroy **R\u0101va\u1e47a**. There is also a **Sm\u0101rta** tradition according to which **\u015aa\u1e45kara** , the chief exponent of **Advaita Ved\u0101nta** , is an incarnation of \u015aiva. But most **\u015aaiva** traditions do not accept the avat\u0101ra doctrine, holding that \u015aiva, as pure consciousness, does not take on a physical form.\n\nOver the centuries, a number of teachers and spiritual figures have been regarded as avat\u0101ras, including in the modern period. The 15th-century **\u0101c\u0101rya** and Vai\u1e63\u1e47ava saint **Caitanya** is sometimes called the **Golden Avat\u0101ra**. It is also held by some in the Ved\u0101nta tradition of **Ramakrishna** that Ramakrishna is a divine incarnation. In both **Bengal** and **Nepal** , it is not uncommon for a young girl to be viewed as a divine incarnation of the **Mother Goddess**. On some interpretations of **Ved\u0101nta** , all beings are, in a sense, avat\u0101ras, because all beings are in fact **Brahman**.\n\n**AVIDY\u0100.** Primordial ignorance. Avidy\u0101 is ignorance, not merely in a cognitive sense but a profound lack of awareness of one's true nature as **Brahman** \u2014infinite and unlimited being ( **sat** ), consciousness ( **cit** ), and bliss ( **\u0101nanda** ). According to **Advaita Ved\u0101nta** , avidy\u0101 is the root cause of one's bondage to **sa\u1e43s\u0101ra** , the beginningless cycle of birth, **death** , and **rebirth**. Avidy\u0101 can only be removed by awakening to one's true nature. An awakening of this kind requires one to learn the **truth** that the self ( **\u0101tman** ) is identical to Brahman\u2014 **knowledge** found only in the **Vedic** scriptures (specifically, in the _Upani\u1e63ads_ ). Hearing this truth ( **\u015brava\u1e47a** ), reflecting upon it ( **manas\u0101** ), and contemplative realizing of it (nidhidhy\u0101sana) produces the required awakening.\n\n**AYODHY\u0100.** Town in northern India, located in what is now the state of Uttar Pradesh, believed to be the birthplace of **R\u0101ma** and the capital city of his ancient kingdom. In the _R\u0101m\u0101ya\u1e47a_ , Ayodhy\u0101 is ruled by R\u0101ma's father, **Da\u015baratha** , who wishes to pass its reign on to R\u0101ma. Before R\u0101ma can be made king, however, Da\u015baratha is forced by his wife, Kaikey\u012b, to send R\u0101ma into exile. R\u0101ma returns from exile and rules Ayodhy\u0101 only after slaying the demonic **R\u0101va\u1e47a**. R\u0101ma's rule is said to be highly just and becomes the ideal of Hindu kingship ( _R\u0101ma r\u0101jya_ ). In the modern period, the concept of _R\u0101ma r\u0101jya_ has become central to the ideology of **Hindu Nationalism** , in which this concept of an ideal Hindu society ruled by **dharma** is contrasted with contemporary political corruption.\n\nIn December 1992, Ayodhy\u0101 became a center of controversy when a group of Hindu Nationalists demolished a mosque called the **Babr\u012b Masj\u012bd** (Babur's Mosque), which they believed had been built from the remains of a **temple** on the same location marking the exact spot of R\u0101ma's birth ( _R\u0101ma-janma-bh\u016bm\u012b_ ). This demolition sparked riots in India, Pakistan, and Bangladesh between Hindus and Muslims, in which many members of both communities were killed, and a number of Hindu temples destroyed in retaliation. In September 2010, an Indian high court ruled that the property should be divided among Hindu and Muslim organizations, a compromise that appears to have defused the issue, at least for the time being.\n\n**\u0100YU.** A long life; one of the central goals of **\u0100yur Veda**.\n\n**\u0100YUR VEDA.** Science of long life, traditional system of Hindu medicine. Among the figures who were responsible for developing this ancient system were **Caraka** , Su\u015bruta, and V\u0101gbhata. It is a holistic system of medicine, focused on balancing bodily energies, or humors (do\u015bas). Its chief aims are long life ( **\u0101yu** ) and freedom from disease ( **\u0101rogya** ). It utilizes herbs and minerals to treat disease and maintain good health.\n\n**AYYAPPAN.** Deity believed to reside at the top of **Sabarimalai** , a hill in **Kerala** that is sacred to Ayyappan's sect. Ayyappan is regarded as the son of **Vi\u1e63\u1e47u** and **\u015aiva** , from a time when Vi\u1e63\u1e47u appeared to \u015aiva as a beautiful **woman** called **Mohin\u012b**. According to one version of the story of Ayyappan, Vi\u1e63\u1e47u took the form of Mohin\u012b so a deity could be born from the union of Vi\u1e63\u1e47u and \u015aiva that would have their combined power in order to defeat the **asura** Princess Mahi\u1e63\u012b, sister of the **Mahi\u1e63\u0101sura** or Buffalo Demon.\n\nDevotion to Ayyappan is a relatively recent development in Hinduism and is most popular in southern India.\n\n## B\n\n**BABR\u012a MASJ\u012aD.** Babur's mosque; mosque demolished in 1992 by **Hindu Nationalists** who believed it to have been built from the remains of a **temple** on the location marking the exact spot of R\u0101ma's birth, in the town of **Ayodhy\u0101**. This demolition sparked riots in India, Pakistan, and Bangladesh between Hindus and Muslims, in which many members of both communities were killed, and a number of Hindu temples destroyed in retaliation. In September 2010, an Indian high court ruled that the property should be divided among Hindu and Muslim organizations, a compromise that appears to have defused the issue, at least for the time being.\n\n_BACK TO GODHEAD_ **.** Magazine published by the **International Society for Krishna Consciousness** **(** ISKCON **)**. First published in India in 1944 by **Abhay Charan De** , who is better known by his monastic name of **Bhaktived\u0101nta Sw\u0101m\u012b Prabhup\u0101da**.\n\nPublication of the magazine was interrupted in 1965 when Prabhup\u0101da traveled to the United States to spread the **Gau\u1e0diya Vai\u1e63\u1e47ava** tradition. After founding ISKCON, Prabhup\u0101da resumed publication of _Back to Godhead_ in the United States, where it has remained in print continuously since 1967.\n\n**B\u0100D\u0100M\u012a.** Site located in northern **Karnataka** , capital of the **C\u0101lukya** dynasty from the sixth to the eighth centuries CE. B\u0101d\u0101m\u012b is the site of numerous royal inscriptions and a variety of **temples** \u2014 **Vai\u1e63\u1e47ava** , **\u015aaiva** , **Buddhist** , and **Jain** \u2014all of which received C\u0101lukya patronage.\n\n**B\u0100DAR\u0100YA\u1e46A.** Author of the _Brahm\u0101 S\u016btras_. B\u0101dar\u0101ya\u1e47a is sometimes identified with **Vy\u0101sa** , traditional author of the _Mah\u0101bh\u0101rata_ and compiler of the _Vedas_. Along with the _Upani\u1e63ads_ and _Bhagavad G\u012bt\u0101_ , the _Brahma S\u016btras_ (or _Ved\u0101nta S\u016btras_ ) form the _Prasth\u0101na Traya_ , or threefold foundation of **Ved\u0101nta** , on which all Ved\u0101ntic **\u0101c\u0101ryas** are expected to write commentaries ( **bh\u0101\u1e63yas** ).\n\n**BADR\u012aN\u0100TH.** Popular site for **pilgrimage** located in the **Him\u0101layas**. It is home to one of the four monasteries or **ma\u1e6dhas** of the **Da\u015ban\u0101mi Order** of Hindu monks founded by **\u015aa\u1e45kara**.\n\n**BALAR\u0100MA.** Brother of **K\u1e5b\u1e63\u1e47a** , the ninth in the traditional list of 10 **avat\u0101ras** of **Vi\u1e63\u1e47u** , according to earlier versions of this list. In later versions, Balar\u0101ma is replaced by the **Buddha** , with Balar\u0101ma becoming seen as an incarnation of **\u015ae\u1e63a** , the cosmic serpent upon whose belly Vi\u1e63\u1e47u reclines. Balar\u0101ma's original presence on the list of the avat\u0101ras of Vi\u1e63\u1e47u, alongside K\u1e5b\u1e63\u1e47a, shows that, according to the **Vai\u1e63\u1e47ava** avat\u0101ra concept, it is possible for Vi\u1e63\u1e47u to assume more than one form at the same time (given that Balar\u0101ma and K\u1e5b\u1e63\u1e47a were contemporaries).\n\n**BALI.** Island in Indonesia where a Hindu culture is predominant. Also an **asura** that was defeated by **V\u0101mana** , dwarf **avat\u0101ra** of **Vi\u1e63\u1e47u**. _See also_ SOUTHEAST ASIA.\n\n**BANARAS.** Holy city along the river **G\u0101\u1e45g\u0101** , or Ganges, revered as sacred to **\u015aiva** , and the location of the **Vi\u015bvan\u0101tha** **Mandir** , a **\u015aaiva** **temple** that is a popular pilgrimage site, as are the bathing **ghats** along the edge of the river. In ancient times, the city was called **K\u0101\u015b\u012b** and was the center of the one of the 16 city-states or Mah\u0101janapadas of the first millennium BCE. The **Buddha** is believed to have preached his first sermon here.\n\n**BANARAS HINDU UNIVERSITY.** University established in 1915 by Pandit Madan Mohan Malaviya with the assistance of **Annie Besant** , and one of the most prominent research universities in India. Though it is called a \"Hindu University\" and was founded, in part, with the mission of promoting **knowledge** of Hinduism, students of all religious communities are accepted, and Banaras Hindu University offers degrees in a wide array of secular disciplines, such as agriculture, medicine, business, and **law**.\n\n**BASAVA (1134\u20131196).** **\u015aaiva** **bhakti** **poet** and founder of the **Li\u1e45g\u0101yat** or **V\u012bra\u015baiva** sect. Though born a **Brahmin** , Basava was opposed to **casteism** and taught the equality of all human beings. He was also opposed to sacrifice, pilgrimage, and the use of images in worship. The only image used in the V\u012bra\u015baiva tradition is the abstract form of **\u015aiva** , the **li\u1e45ga** (hence the alternative name, Li\u1e45g\u0101yat, for this tradition). \u015aiva is regarded in the tradition of Basava as **God** in a **monotheistic** sense\u2014the one supreme deity. Basava was from **Karnataka** , where his Li\u1e45g\u0101yat movement has continued to be most prominent since his time. He held high government office as a minister in the court of the **C\u0101lukya** King Bijjala and used this position to advance his progressive social and religious ideals.\n\n**BASHAM, ARTHUR LEWELLYN (1914\u20131986).** Renowned **Indologist** and historian. He completed all of his degrees at the School of Oriental and African Studies in London and taught there as well. In 1965, he accepted a professorship at the Australian National University in Canberra. His most famous book, _The Wonder That Was India_ , has been reprinted numerous times and is a popular as well as a scholarly work, which has been responsible for sparking the interest of numerous scholars in India and Hinduism. He had a particular appreciation for **Sw\u0101m\u012b Vivek\u0101nanda** , whom he regarded as a major figure of the modern world. Basham's students included **Romila Thapar**.\n\n**BAULS.** **Bengali** traveling singers and mystical **poets**. Known for intense devotion and unconventional behavior, the Bauls have a generally Tantric orientation and can be identified with the **Sahajiy\u0101** form of **Tantra** , a style of religiosity known for its emphasis on spontaneity and general rejection of concern for worldly conventions. The religious orientation of the Bauls can be described as a mix of **bhakti** and Tantra. The **music** of the Bauls was tremendously influential upon the music and poetry of the modern Bengali **artist** **Rabindran\u0101th Tagore**.\n\n**BELUR.** Site located in southern **Karnataka** ; home to a variety of **temples** dating to the 11th and 12th centuries CE. The temples, built by the Hoysala dynasty, are among the most elaborately carved Hindu temples in India, with walls and pillars that appear to be almost alive with motion, due to practically every space upon them being carved with realistic figures of deities enacting scenes from the Hindu **scriptures**.\n\n**BELUR MA\u1e6cH.** Home monastery and main administrative center of the **Ramakrishna Order** , located in **Calcutta**.\n\n**BENARES.** Alternative spelling of **Banaras**.\n\n**BENGAL.** Region in northeastern India, home of a number of prominent Hindu reform figures of the early modern period, such as **R\u0101m Mohan Roy** , **Ramakrishna** , **Sw\u0101m\u012b Vivek\u0101nanda** , and **Aurobindo Ghose** , and literary figures such as **Bankim Chandra Chatterjee** and **Rabindran\u0101th Tagore**. Bengal was the first part of India to experience extensive interaction with the British, being home to **Calcutta** (Kolkata), the location of the headquarters of the **British East India Company**. This interaction was among the main catalysts for the Hindu reform movement, often called the \"Bengal Renaissance.\"\n\n**Buddhism** was prominent in Bengal in the first millennium CE, as were Buddhist and Hindu forms of **Tantra** , and Tantric practices associated with the **Goddess-** oriented **\u015a\u0101kta** traditions continue to be relatively popular in contemporary Bengal. **Durg\u0101 P\u016bj\u0101** is the largest holiday in Bengal, and Bengalis traditionally perform **K\u0101l\u012b** P\u016bj\u0101 on the night celebrated by most other Hindus as **D\u012bwali**. Bengal is also home to the **Bauls** , wandering minstrels whose spirituality and famous folk songs reflect currents of both **Tantra** and **bhakti**.\n\nIn the late first and early second millennia of the Common Era, highly devotional **Vai\u1e63\u1e47ava** traditions, such as **Gau\u1e0d\u012bya Vai\u1e63\u1e47avism** , became a major element of Bengali Hinduism, with Bengal being the home, in the medieval period, to **Caitanya** , and in the modern period, to **Bhaktived\u0101nta Sw\u0101m\u012b Prabhup\u0101da**.\n\n**BESANT, ANNIE (1847\u20131933).** Social and political reformer and prominent member of both the **Theosophical Society** and the Indian National Congress. Besant was an early advocate of freedom of thought, secularism, and socialism. Of Irish origin, she supported independence for both Ireland and India.\n\nIn her youth, Besant was committed largely to political activism and had a strong leaning toward Marxism. But after reading **Helena Petrovna Blavatsky** 's _The Secret Doctrine_ in 1889, she joined the **Theosophical Society** , of which she became president in 1908. Though never rejecting her progressive political commitments, for the rest of her life she would contextualize these within a Theosophical worldview.\n\nA prolific author, Besant wrote numerous books and tracts on various aspects of theosophical thought, as well as a translation of the _Bhagavad G\u012bt\u0101_. Wanting to promote Hindu ideals and culture through education, in 1915 she assisted Pandit Madan Mohan Malaviya in establishing **Banaras Hindu University**. She also served as the president of the Indian National Congress in 1917. She was arrested by the British government for her political activities during that same year. Her commitment to Indian independence was profound, and she corresponded with **Mohand\u0101s K. G\u0101ndh\u012b**.\n\nThe latter part of Besant's career was devoted to promoting the idea that **Jiddu Krishnamurti** was the \"World Teacher\"\u2014the next **Buddha** , Maitreya\u2014a messianic being whose coming is predicted in the **scriptures** of **Buddhism**. Krishnamurti himself did not endorse this idea, and famously left the Theosophical Society to teach a path of radical questioning of all forms of authority. Besant and Krishnamurti, however, remained on good terms until Besant's death in 1933.\n\n**BHADRA.** Month of the **Hindu calendar** that corresponds roughly to the period from mid-August to mid-September.\n\n**BHAGA.** Literally \"fortunate,\" or \"dispenser of wealth.\" Early Vedic deity, one of the 12 **\u0100dityas** , sons of the **goddess** **Aditi** and the sage **Ka\u015byapa**. Bhaga is sometimes pared with **A\u1e43\u015ba** , the **gods** ' share of the sacrifice. Bhaga is the human beings' share.\n\nWhen suffixed with the syllable \"-vant,\" meaning \"possessed of,\" the term _bhaga_ becomes the basis of the word **Bhagav\u0101n** , or \"blessed,\" often used as an epithet of such figures as **K\u1e5b\u1e63\u1e47a** and the **Buddha** , and, in contemporary Hinduism, a term for **God**.\n\n_BHAGAVAD G\u012aT\u0100_ **.** Song of the Blessed One, Song of the Lord, Song of **God** ; likely composed between 100 BCE and 100 CE. This 18-chapter segment of the sixth book of the _Mah\u0101bh\u0101rata_ has come to be viewed as an authoritative **scripture** in its own right, particularly in the modern period, though numerous premodern commentaries ( **bh\u0101\u1e63yas** ) were written upon it as well. Many Hindus see this text as a comprehensive summary of Hindu **philosophy**. As a scripture, it serves a function in contemporary Hinduism closer to that of the Bible in **Christianity** than do the venerable, but less popular, _Vedas_.\n\nOccurring at the point in the _Mah\u0101bh\u0101rata_ 's narrative when the climactic battle of **Kuruk\u1e63etra** is about to take place, the _Bhagavad G\u012bt\u0101_ consists of a dialogue between the **Pa\u1e47\u1e0dava** hero **Arjuna** and his best friend and cousin, **K\u1e5b\u1e63\u1e47a** , who has agreed to act as Arjuna's charioteer in the battle and who is also an **avat\u0101ra** of **Vi\u1e63\u1e47u**. Deeply troubled by the moral dilemma that he faces, having to fight in a just war that will require him to slay such revered persons as his grandsire **Bhi\u1e63ma** and his teacher **Dro\u1e47a** , Arjuna falls into despair. K\u1e5b\u1e63\u1e47a's counsel to Arjuna at this point takes the form of a conversation that ranges over a great variety of topics and views current in Indian philosophy at the time of the text's composition (c. 100 BCE\u2013100 CE by most scholarly estimates). Currents of thought that can be discerned in the _Bhagavad G\u012bt\u0101_ include **S\u0101\u1e43khya** , **Yoga** , **Ved\u0101nta** , **Buddhism** , and **Jainism** , as well as an early articulation of the theistic philosophy of the **bhakti** movement, which became highly popular in the centuries to follow.\n\nLater Hindu tradition gives the text an increasingly privileged status as a summary of all Hindu thought. It is often called simply \"the _G\u012bt\u0101_ ,\" despite there being many texts in the g\u012bt\u0101 genre. Along with the _Upani\u1e63ads_ and the _Brahm\u0101 S\u016btras_ , the _Bhagavad G\u012bt\u0101_ forms part of the _Prasth\u0101na Traya_ , or threefold foundation of Ved\u0101nta philosophy, upon which all the major Ved\u0101ntic **\u0101c\u0101ryas** , such as **\u015aa\u1e45kara** , **R\u0101m\u0101\u1e47uja** , and **Madhva** , have commented. **Mohand\u0101s K. G\u0101ndh\u012b** referred to this text as his \"dictionary of daily reference\" and cited it as a major influence on his philosophy of active engagement with the world as an important part of the spiritual path. **Aurobindo Ghose** also commented upon it in a work entitled _Essays on the G\u012bt\u0101_ , which focuses on the text's philosophy of yoga.\n\nThe most controversial aspect of the text is its location on a battlefield, coupled with the fact that K\u1e5b\u1e63\u1e47a encourages Arjuna to fight in battle, rather than taking a path of nonviolence\u2014although K\u1e5b\u1e63\u1e47a, somewhat paradoxically, does commend **ahi\u1e43s\u0101** to Arjuna as one of the virtues of an **enlightened** person (16.2).\n\nCommentators have responded to the question of violence and nonviolence in the _Bhagavad G\u012bt\u0101_ in a variety of ways: ignoring it, frankly accepting the violence entailed as a part of life in ancient India or of life in general, viewing the entire episode in terms of the symbolism that the text itself suggests (13.1)\u2014with the body as the field of battle and the real enemies being negative qualities such as ignorance, desire, and egotism\u2014or, in the case of G\u0101ndh\u012b, seeing the violence of the epic context as irrelevant to and even incompatible with the deeper spiritual message of the text, which becomes abstracted from its literary context. A related issue on which scholars disagree is whether the _G\u012bt\u0101_ is a later interpolation, inserted into the _Mah\u0101bh\u0101rata_ at a later date\u2014and in disagreement with it on the issue of violence\u2014versus whether it should be seen as integral to the epic.\n\nThe Ved\u0101ntic \u0101c\u0101ryas do not agree amongst themselves on the relative roles of **j\u00f1\u0101na** , or wisdom, and bhakti, or devotion, as paths to **mok\u1e63a** in K\u1e5b\u1e63\u1e47a's teaching. The **Advaita** tradition, following \u015aa\u1e45kara, sees this text as commending j\u00f1\u0101na yoga, whereas the **Vai\u1e63\u1e47ava** \u0101c\u0101ryas, such as R\u0101m\u0101\u1e47uja, Madhva, and **Caitanya** , see it as primarily a text of bhakti. **Sw\u0101m\u012b Vivek\u0101nanda** sees the text as commending **four yogas** \u2014the paths of **action** , wisdom, devotion, and **meditation** \u2014as being equally efficacious. G\u0101ndh\u012b also follows Vivek\u0101nanda in this regard, though giving primary attention to the way of action.\n\n**BHAGAV\u0100N (BHAGAVAT).** Lord, Blessed One, **God**. Usually an epithet associated with **K\u1e5b\u1e63\u1e47a** , though also used in the literary works of various traditions to refer to a deity or founding figure. The **Buddha** , for example, is called Bhagav\u0101n in many **Buddhist** texts. In contemporary Hinduism it is not uncommon to hear this term being used as an exclamation with essentially the same semantic range as the English, \"Oh my God!\"\n\n_BH\u0100GAVATA PUR\u0100\u1e46A_ **.** One of the 18 major _Pur\u0101\u1e47as_ , or texts of ancient lore focused on the lives and deeds of the Hindu deities. The _Bh\u0101gavata Pur\u0101\u1e47a_ is concerned chiefly with the life of **K\u1e5b\u1e63\u1e47a**. It is a text particularly revered by **Vai\u1e63\u1e47avas** , most especially those whose devotion is centered chiefly upon K\u1e5b\u1e63\u1e47a. This text is a major locus for the development of the **philosophy** of **bhakti** , or devotion as the preeminent means for the attainment of **mok\u1e63a**.\n\nThe main focus of the narrative is the early life of K\u1e5b\u1e63\u1e47a, prior to the events of the _Mah\u0101bh\u0101rata_. Shortly after his birth, K\u1e5b\u1e63\u1e47a's parents need to hide him from his wicked uncle, a king named **K\u0101\u1e43sa** , who wants to slay him because of a prediction that K\u1e5b\u1e63\u1e47a would one day slay him and take control of his kingdom. On a number of occasions in his infancy and childhood, K\u1e5b\u1e63\u1e47a displays paranormal powers.\n\nA major theme of the text is the relationship between K\u1e5b\u1e63\u1e47a and the **gop\u012bs** , or milkmaids of **V\u1e5bnd\u0101vana** , who are represented as his ideal devotees. Foremost among the gop\u012bs is **R\u0101dh\u0101** , whose loving relationship with K\u1e5b\u1e63\u1e47a is the ultimate model of devotion. This relationship is further explored in the _G\u012bt\u0101govinda_ of **Jayadeva**. The _Bh\u0101gavata Pur\u0101\u1e47a_ was composed from roughly 800 to 950 CE.\n\n**BH\u0100GAVATA SECT.** Very early **Vai\u1e63\u1e47ava** sect focused on **bhakti** toward **K\u1e5b\u1e63\u1e47a** as the best path to **mok\u1e63a**. The name of the sect arises from their use of the title **Bhagav\u0101n** to refer to K\u1e5b\u1e63\u1e47a. The earliest textual evidence for the existence of this sect can be found in the _Mah\u0101bh\u0101rata_ , including the _Bhagavad G\u012bt\u0101_ , which advocates devotion to K\u1e5b\u1e63\u1e47a.\n\n**BHAJANA.** Devotional song. Bhajanas are often sung during group worship in a call-and-response fashion, in which the lead singer sings the refrain\u2014typically a **mantra** \u2014that is then repeated by the congregation. The tempo of a bhajana usually starts out slow, and then picks up speed as the bhajana is sung, usually becoming quite fast near the end. But on the final call-and-response, the tempo returns to its original pace. _See also_ BHAKTI.\n\n**BHAKTI.** Devotion, one of the **four** **yogas** or spiritual disciplines that Hindu traditions present as a path to **liberation** \u2014or **mok\u1e63a** \u2014from the cycle of **re-birth** , along with **karma yoga** , **j\u00f1\u0101na yoga** , and **dhy\u0101na** or r\u0101ja yoga.\n\nBhakti as a path to liberation is first developed in later **Vedic** texts such as the _\u015avet\u0101\u015bvat\u0101ra Upani\u1e63ad_ , as well as the _Bhagavad G\u012bt\u0101_. It is also mentioned in the _Yoga S\u016btras_ of **Pata\u00f1jali** under the name of **\u012a\u015bvara-pra\u1e47idh\u0101na** , or \"surrender to the Lord.\" Bhakti is typically focused upon a specific, personal form of the divine\u2014such as **Vi\u1e63\u1e47u** , **\u015aiva** , or **Dev\u012b** \u2014in contrast with j\u00f1\u0101na, which is a more abstract, impersonal realization.\n\nBhakti is emphasized most prominently in the **Vai\u1e63\u1e47ava** traditions of Hinduism, although it is an important component of most forms of Hinduism. It is a major theme of such texts as the _Bh\u0101gavata Pur\u0101\u1e47a_ , as well as the Vai\u1e63\u1e47ava-oriented **Ved\u0101nta** systems of **R\u0101m\u0101\u1e47uja** , **Madhva** , and **Caitanya** , all of whom see bhakti as the preeminent path to mok\u1e63a, for which the other three yogas are preparatory (unlike **Advaita Ved\u0101nta** , where j\u00f1\u0101na, or knowledge, is primary). Bhakti is not merely an emotion but involves a real metaphysical connection and mutual participation between the devotee and the divine.\n\nThe deep, intense spiritual longing associated with bhakti is famously expressed in the poems and songs of devotees of the bhakti movement, such as the **\u0100\u1e37v\u0101rs** , **Basava** , **M\u012bra Bai** , and **Tuls\u012bd\u0101s** , to name only a few. An important characteristic of bhakti in the works of these **poet-** saints\u2014who came from all walks of life and included many **women** \u2014is its spiritual egalitarianism. Bhakti can be cultivated by any person, regardless of **caste** or gender, thus making liberation available to all.\n\nBhakti is usually cultivated and expressed through the practice of worship or **p\u016bj\u0101** directed to a personal form of the divine and the singing of **bhajanas**. **Ramakrishna** was intensely devoted to **K\u0101l\u012b**. His chief disciple, **Sw\u0101m\u012b Vivek\u0101nanda** , places bhakti on the same level as all the other yogas, the equality of which is a distinctive teaching of the modern or **Neo-Ved\u0101nta** tradition that Ramakrishna and Vivek\u0101nanda initiated.\n\n**BHAKTIVED\u0100NTA SW\u0100M\u012a PRABHUP\u0100DA (1896\u20131977).** Founder and **\u0101c\u0101rya** of the **International Society for Krishna Consciousness (** ISKCON **)**. Bhaktived\u0101nta (born **Abhay Charan De** ) brought the **Gau\u1e0d\u012bya Vai\u1e63\u1e47ava** tradition to North America in 1965, establishing ISKCON and beginning the **Hare Krishna movement**. Bhaktived\u0101nta first arrived in New York and found a home with a group of young people who were part of the countercultural movement of that period. It was largely from this movement that the following of ISKCON was first drawn, though it has since become somewhat more of a mainstream group.\n\n**BHARADV\u0100JA.** According to some lists, one of the Seven Sages ( **Saptar\u1e63i** ) presiding over the cosmos.\n\n**BHARATA.** Legendary king of ancient India and founder of the **Bh\u0101rata** dynasty. The major characters of the _Mah\u0101bh\u0101rata_ are members of this dynasty, and the _Mah\u0101bh\u0101rata_ itself is the tale of Bharata's descendants. In **Jainism** , Bharata is represented as the son of **\u1e5a\u1e63abha** , the first **T\u012brtha\u1e45kara** of the Jain tradition. Bharata is also one of the brothers of **R\u0101ma** in the _R\u0101m\u0101ya\u1e47a_ (not to be confused with the first Bharata).\n\nA later, historical figure from the first millennium CE named Bharata is the author of the _N\u0101\u1e6dya \u015a\u0101stra_ , the authoritative guide to Indian classical theater and **dance**.\n\n**BH\u0100RATA.** The ancient royal dynasty descended from King **Bharata** and whose adventures are depicted in the _Mah\u0101bh\u0101rata_. Also a name for India, Bharata's kingdom, which is also used in official documents of the contemporary Republic of India.\n\n**BHARATA-N\u0100\u1e6cYAM.** Popular form of Indian classical **dance** originating in southern India and inspired by the depictions of **\u015aiva N\u0101\u1e6dar\u0101ja** (Lord of the Cosmic Dance) at the famous **\u015aaiva** **temple** at **Cidambaram**. Its complex gestures convey symbolic meaning, allowing the dancer to tell stories from Hindu **literature** in dance form. It is one of the four traditional schools of Indian classical dance.\n\n**BH\u0100RATI.** One of the ten subsets of the **Da\u015ban\u0101m\u012b Order** of monks established by **\u015aa\u1e45kara**. _See also_ BH\u0100SKARAR\u0100YA MAKHIN.\n\n**BH\u0100RAT\u012aYA JANAT\u0100 PARTY (BJP).** \"Indian People's Party,\" popular **Hindu Nationalist** political party established in 1980. Within the Indian political spectrum, this party is seen as right wing, perhaps analogous to the Republican Party in the United States, with a strong advocacy of traditional values with clear religious\u2014in this case, Hindu\u2014overtones.\n\nMore widely known as the BJP, this political party governed India from 1998 to 2004. With close ties to organizations like the **R\u0101\u1e63\u1e6dr\u012bya Svayamsevak Sa\u1e45gh** and the **Vi\u015bva Hindu Pari\u1e63ad** , its Hindu nationalist leanings have been a source of controversy for this party, as was its decision to test nuclear weapons in 1998 in response to similar nuclear tests undertaken by the government of Pakistan. The BJP has also advocated the building of a **temple** to **R\u0101ma** at the disputed site of the **Babr\u012b Masj\u012bd** in **Ayodhy\u0101** and the implementation of a uniform civil code, doing away with the different systems of personal **law** for various religious communities currently in place. It has also expressed concern about the proselytizing activities of **Christian** missionaries and, in some states, has sought to curb these through \"conversion **laws** ,\" which have been controversial due to concerns regarding religious freedom.\n\nThe party characterizes its ideology as \"integral humanism\" and describes its aim as making India a progressive, modern state in a way that draws upon and is in keeping with its ancient heritage. Its opponents, however, see it as advancing the ideal of a Hindu state or _R\u0101ma r\u0101jya_ in opposition to the secular ideals on which the Indian constitution is based. For its part, the BJP criticizes what some of its members refer to as the \"pseudo secularism\" of other Indian political parties, advocating an American model of separation of religion and government as opposed to what Hindu Nationalists regard as preferential treatment for religious minority communities.\n\n**BH\u0100RAT\u012aYA VIDY\u0100 BHAVAN.** \"Palace of Indian Wisdom\"; Indian educational trust established in 1938 by K. M. Munshi. The Bh\u0101rat\u012bya Vidy\u0101 Bhavan has a large number of affiliated Indian educational institutions, as well as a handful of institutions outside India. Its chief purpose is to promote the study of **Sanskrit** and Hindu culture and has engaged in a good deal of publishing, since its inception, on topics related to traditional Indian and Hindu culture.\n\n**BH\u0100RGAVA.** Descendant of **Bh\u1e5bgu** , one of the Seven Sages ( **Saptar\u1e63i** ) presiding over the cosmos; a group of **Brahmins** claiming descent from the sage Bh\u1e5bgu and involved in the oral transmission of the _Mah\u0101bh\u0101rata_.\n\n**BHART\u1e5aHARI (fl. 6th\u20137th centuries CE).** **Sanskrit** grammarian and **poet** who developed the **spho\u1e6da** theory of Sanskrit grammar, according to which the letters or sounds that make up a word have a special power to manifest the meaning of that word. This power to make meaning is the true essence of a word. The letters and sounds that make up the word are merely the means by which this true essence is revealed.\n\n**BH\u0100SKARAR\u0100YA MAKHIN (1690\u20131785).** Tantric theologian who systematized the **\u015ar\u012bvidy\u0101** system of **Tantra** , which is focused upon the **Mother Goddess** as the supreme form of divinity. Bh\u0101skarar\u0101ya Makhin was born in Maharashtra, coming from a very scholarly family, and was highly skilled in philosophical debate. His major works are the _Setubandha_ (a detailed technical guide to Tantric practice), the _Varivasya Rahasya_ , and a commentary, or **bh\u0101\u1e63ya** , on the _Lalit\u0101sahasran\u0101ma_ , a text dedicated to the worship of the **Divine Mother**. Contemporary worship of the Mother Goddess largely follows the system outlined in Bh\u0101skarar\u0101ya Makhin's texts. As a systematizer of a Tantric practice, he has played a role similar to that played by **Abhinavagupta** in relation to the **Kaula Tantra** system of **Ka\u015bm\u012br \u015aaivism**.\n\n**BH\u0100\u1e62YA.** Commentary; interpretation of an earlier text that is regarded as authoritative. This is the primary literary genre through which a philosophical system is developed in classical Hindu thought. Rather than valuing novelty, Hindu traditions have tended to see all important **truth** as having been revealed in the distant past. The task of the thinker is therefore not to develop new ideas but to unpack, unfold, and re-present the deep truths embedded in the words of the ancients as appropriate to new historical situations. _See also_ DAR\u015aANA.\n\n**BHAV\u0100N\u012a.** Epithet of **P\u0101rvat\u012b** , the **Mother Goddess** in her peaceful, benevolent form, as the wife of **\u015aiva**.\n\n**BHAVE, VINOBHA (1885\u20131982).** Close associate and disciple of **Mohand\u0101s K. G\u0101ndh\u012b** , freedom fighter, and postindependence Indian political and social activist. His thought closely followed that of G\u0101ndh\u012b, with a strong emphasis on **ahi\u1e43s\u0101** (nonviolence) and the inseparability of political and social activism from spirituality. He is best known for his **Sarvod\u0101ya** (\"Uplifting of All\") movement, as well as his Bh\u016bd\u0101n, or \"Gift of the Earth\" campaign, which was aimed at the redistribution of land to the poor. Bhave's **death** was somewhat controversial, as he undertook _sallekhan\u0101_ , the traditional **Jain** fast to the death.\n\n**BH\u012aMA.** Character in the _Mah\u0101bh\u0101rata_ , second of the **P\u0101\u1e47\u1e0dava** brothers, fathered by **V\u0101yu** , the **Vedic god** of the wind. Bh\u012bma, like his divine father, is known for his great strength and quick temper. He frequently engages in physical combat with **asuras** and other enemies. It is Bh\u012bma who delivers the final killing blow to **Duryodhana** , the main villain of the epic. This blow is controversial, because Bh\u012bma violates the rules of battle and strikes Duryodhana on the thigh with his club.\n\n**BH\u012a\u1e62MA.** Character in the _Mah\u0101bh\u0101rata_ , son of King \u015aantanu and the goddess **Ga\u1e45g\u0101**. The name \"Bh\u012b\u1e63ma\" is actually an epithet meaning \"terrible,\" the character's birth name being Devavrata. Bh\u012b\u1e63ma received this epithet due to the terrible, unnatural vow that he took to remain celibate and renounce his claim to his father's throne in order to allow his father to remarry after the departure of Ga\u1e45g\u0101 back to the heavens. This would enable his father's descendants through his new wife, Satyavat\u012b, to inherit the kingdom\u2014a condition imposed by Satyavat\u012b's father before he would permit his daughter to marry \u015aantanu.\n\nBh\u012b\u1e63ma became a mighty warrior and lived to a great age. He was slain in battle at **Kuruk\u1e63etra** by \u015aikhandin, a reincarnation of **Amb\u0101** , a woman who had vowed to take revenge on Bhi\u1e63ma many years previously, after having been abandoned by him. Bh\u012b\u1e63ma had the ability to choose the time of his own **death** , however, and lay on the battlefield for many days, passing on his wisdom to the **P\u0101\u1e47\u1e0davas** , who had gathered around him. The words of Bh\u012b\u1e63ma on his deathbed are contained in a book of the _Mah\u0101bh\u0101rata_ known as the _Bh\u012b\u1e63ma Parvan_ , a collections of aphorisms on Hindu **philosophy** , politics, and ethics.\n\n**BH\u1e5aGU.** According to some lists, one of the Seven Sages ( **Saptar\u1e63i** ) presiding over the cosmos; ancestor of a group of **Brahmins** called the **Bh\u0101rgavas**. A work on **astrology** , the _Bh\u1e5bgu Sa\u1e43hit\u0101_ , is attributed to him.\n\n**BH\u016a.** The Earth. One of the \"three worlds\" of **Vedic** thought, including the Earth, the upper atmosphere, and the heavens, and invoked in the first line of the **G\u0101yatr\u012b** **mantra** : _O\u1e43 bh\u016br bhuva\u1e25 sva\u1e25_. _See also_ BH\u016aDEV\u012a; TRILOKA.\n\n**BH\u016aDEV\u012a.** **Goddess** of the Earth; \"Mother Nature\"; a personification of the Earth in the form of a **divine mother**. In his incarnation as a boar ( **Var\u0101ha** ), **Vi\u1e63\u1e47u** rescues Bh\u016bdev\u012b from the demonic **Hiranyak\u1e63a**. In the _Mah\u0101bh\u0101rata_ , Bh\u016bdev\u012b again calls out to Vi\u1e63\u1e47u to relieve the weight of proliferating humanity on her surface. He accedes to this request, taking on the form of **K\u1e5b\u1e63\u1e47a** and overseeing the events of the _Mah\u0101bh\u0101rata_ war.\n\n**BH\u016aTA.** Literally \"being,\" entity. Element; fundamental component of existence. Five such elements are delineated in most Hindu systems of **philosophy** : earth, air, water, fire, and space ( **\u0101k\u0101\u015ba** ).\n\nAlso, a ghost or unquiet spirit that has not yet moved on to its next **re-birth** , usually due either to tragic circumstances surrounding its **death** in its previous incarnation or to unfinished business from that lifetime. _See also_ MAH\u0100BH\u016aTAS; PRETA.\n\n**BHUVANE\u015aVARA (BHUBANESHWAR).** Capital of the Indian state of **Orissa** , site of many **temples** constructed during the classical and early medieval periods. Prominent among these are Li\u1e45gar\u0101ja, which is dedicated to **\u015aiva** (Tribhuvane\u015bvara, or \"Lord of the Three Worlds\") and was constructed in the 12th century CE; Vaital **Mandira** , dedicated to **K\u0101l\u012b** ( **C\u0101mu\u1e47\u1e0d\u0101** ) and constructed in the eighth century CE; and Brahme\u015bvara, dedicated to \u015aiva and constructed in the ninth century CE.\n\n**B\u012aJA MANTRA.** Seed **mantra** ; monosyllabic mantra held to contain the essence of the deity that it evokes, particularly prominent in the practice of **Tantra**. B\u012bja mantras are seen as being especially powerful. They are often embedded in longer mantras. The best known and most frequently used b\u012bja mantra is **O\u1e43** , seen as containing the essence of all mantras, and indeed of all sound and the cosmos itself. Other commonly found examples include _hu\u1e43_ , _hr\u012b\u1e43_ , _\u015br\u012b\u1e43_ , _ga\u1e43_ , _kl\u012b\u1e43_ , and _ai\u1e43_. B\u012bja mantras do not have a literal meaning, in the sense of a specific denotation by which they can translated. Their power is more evocative than denotative. They make present, in auditory form, the deity or power that they symbolize.\n\n**B\u012aR\u016aN\u012a, AL- (973\u20131048).** Arab Muslim scholar who traveled in India and wrote about Indian customs, particularly religious practices. His _Kitab al-Hind_ (\"Book of India\") and his Arabic translation of the _Yoga S\u016btras_ of **Pata\u00f1jali** presented a relatively accurate and sympathetic portrayal of Hinduism to the **Islamic** world.\n\n**BISHNOIS.** Small **Vai\u1e63\u1e47ava** community based primarily in R\u0101jasth\u0101n, known for their commitment to the protection of the environment. They have been a central inspiration to the **Chipko movement**. The founder of the Bishnois, a 15th-century figure, Guru Mah\u0101r\u0101j Jambaj\u012b, established 29 rules for his community, including a ban on the cutting of green trees and the killing of animals. In the region of the largely arid state of R\u0101jasth\u0101n where the Bishnois resided, the Bishnoi practice of not cutting green trees is said to have led to the growth of a lush forest, which was targeted by the Mah\u0101r\u0101ja of Jhodpur, who wished to use the wood from the trees to build a palace for himself. The Bishnois sought to protect the trees by encircling them and chaining themselves to the tree trunks. The Mah\u0101r\u0101ja was moved by this act and ordered his servants not to harass the Bishnois. In 1973, the founders of the Chipko movement, based in what is now the state of Uttarakhand, followed the lead of the Bishnois, protecting forests from being cut down by loggers by forming a human chain encircling the trees and by hugging the trees\u2014a practice which led to the name \"Chipko\" and, beyond India, to the term \"tree hugger\" for anyone concerned about the protection of the environment.\n\n**BLAVATSKY, HELENA PETROVNA (1831\u20131891).** Cofounder of the **Theosophical Society** ; Russian heiress. In her youth she traveled widely, claiming to have visited Tibet and studied under **enlightened** masters. Having a profound interest in spiritualism and the occult, Blavatsky arrived in the United States in 1873. Along with **Henry Steele Olcott** , she established the Theosophical Society in 1875. The objectives of the society were \"to form a nucleus of the universal brotherhood of humanity without distinction of race, creed, sex, **caste** , or color,\" \"to encourage the study of comparative religion, **philosophy** , and science,\" and \"to investigate the unexplained **laws** of nature and the powers latent in man.\"\n\nA prolific author, Blavatsky's two major works are _Isis Unveiled_ , published in 1877, and _The Secret Doctrine_ , published in 1888. The latter book reflects the gradually increasing influence of Hinduism and **Buddhism** on Blavatsky's thought. Much of the conceptual content of theosophy is drawn from these two religious traditions, as well as from Western esoteric traditions.\n\nNo stranger to controversy, Blavatsky faced numerous accusations throughout her career of engaging in fraudulent displays of spiritual powers. After she and Olcott moved to India in 1879, the British government suspected her of being a Russian spy. Members of the Theosophical Society played prominent roles in the movement for Indian political independence. Theosophists promoted pride and interest among Hindus and Buddhists in their religious and cultural heritage, acting as \"reverse missionaries,\" articulating the idea that Hinduism and Buddhism preserve an ancient global wisdom tradition, or **perennial philosophy**. At one point a strategic alliance began to form between the Theosophical Society and the **\u0100rya Sam\u0101j**. But **Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b** fell out with Blavatsky and Olcott in 1880, when they announced their conversions to Buddhism in **Sri Lanka**. Blavatsky later returned to Europe and died in London in 1891.\n\n**BOAR AVAT\u0100RA.** _See_ VAR\u0100HA.\n\n**BOC\u0100SANV\u0100S\u012a AK\u1e62AR PURU\u1e62OTTAM SW\u0100M\u012aN\u0100R\u0100YA\u1e46 SANSTH\u0100 (BAPS).** _See_ SW\u0100M\u012aN\u0100R\u0100YA\u1e46 MOVEMENT.\n\n**BRAHM\u0100.** Creator deity; member, along with **Vi\u1e63\u1e47u** and **\u015aiva** , of the **Trim\u016brti**. When a new cosmic cycle begins, it is Brahm\u0101 who brings about the **creation** of the world. He is therefore not a creator _ex nihilo_ (from nothing), but the re-creator of the world at the beginning of each cosmic cycle ( **kalpa** ). The cosmos is then preserved for many eons by Vi\u1e63\u1e47u before being destroyed\u2014in order to be renewed and re-created\u2014by \u015aiva. Brahm\u0101 is often identified with the **Vedic** creator deity, **Praj\u0101pati**. He was worshiped as a major deity in ancient India (from roughly 500 BCE to 500 CE) but has not been a prominent object of **devotion** among most Hindus for many centuries, having been overshadowed by Vi\u1e63\u1e47u, \u015aiva, and the **Mother Goddess**. He is usually paired with the **goddess** **Sarasvat\u012b** as her husband.\n\n**BRAHM\u0100 KUM\u0100R\u012aS.** Daughters of Brahm\u0101. Hindu-based religious movement that was established in 1937 by **D\u0101da Lekhr\u0101j** (also known as **Praj\u0101pitar Brahm\u0101** ) and is now based at the Brahm\u0101 Kum\u0101r\u012b Spiritual University on Mount Abu, in the Indian state of **Rajasthan**. Lekhr\u0101j passed the leadership of the organization to his female followers, with whom it has remained. The Brahm\u0101 Kum\u0101r\u012bs teach a path of **asceticism** with special emphasis on abstention from alcohol. They also hold a belief\u2014unusual for a Hindu sect\u2014in the imminent end of the current world order and the coming of a new Golden Age. _See also_ WOMEN.\n\n_BRAHMA S\u016aTRAS_ **.** Aphoristic root text of the various **Ved\u0101nta** systems of **philosophy** that, along with the _Upani\u1e63ads_ and the _Bhagavad G\u012bt\u0101_ , forms the _Prasth\u0101na Traya_ or \"threefold foundation\" of Ved\u0101nta\u2014the three authoritative texts on which every **\u0101c\u0101rya** , or founder of a Ved\u0101ntic tradition, has written a **bh\u0101\u1e63ya** , or commentary.\n\nEstimates of the date of composition of the _Brahma S\u016btras_ vary. An approximate date of 200 BCE is possible, though it is also very likely that the compilation process was gradual, extending into the Common Era. In terms of relative chronology, the text must have been compiled after the composition of the major _Upani\u1e63ads_ , which it summarizes, but prior to the _Bhagavad G\u012bt\u0101_ , which mentions it (though this reference may or may not be to this specific text, perhaps referring generically to wisdom **literature** on **Brahman** ). The _Brahma S\u016btras_ are traditionally attributed to **B\u0101dar\u0101ya\u1e47a** , who is identified with **Veda Vy\u0101sa** , the traditional author of the _Mah\u0101bh\u0101rata_ and the compiler of the _Vedas_. As the root text of the Ved\u0101nta systems of **philosophy** , its verses are also known as the _Ved\u0101nta S\u016btras_.\n\nThe _Brahma S\u016btras_ present an inquiry into the nature of Brahman and collect the major threads of thought on this topic found in the _Upani\u1e63ads_. This text also presents criticisms of other major systems of thought, such as **S\u0101\u1e43khya** , **Yoga** , **Jainism** , and both realist and idealist forms of **Buddhism**.\n\n**BRAHMACARIN.** A person in the state of **brahmacarya**. A celibate student, usually a young Brahmin male.\n\n**BRAHMACARYA.** The first of the four **\u0101\u015bramas** , or stages of life. It also refers to a state of celibacy or the vow by which one enters such a state. As the first stage of life, the brahmacarya \u0101\u015brama refers to the \"student stage.\" This typically begins around the age of 12 (sometimes a little earlier, sometimes a little later) when an upper-caste male (usually a Brahmin male) is invested with a **sacred thread** through the **upanayana sa\u1e43sk\u0101ra**.\n\nTraditionally, after the investiture with the sacred thread, the **brahmacarin** leaves home to live with his **guru** , or teacher, where he receives instruction in the _Vedas_ and in various other relevant forms of **knowledge** , like grammar, mathematics, and **astrology**. In modernity, the brahmacarin typically remains at home, though a ceremonial \"departure\" is sometimes enacted. Ritually speaking, the brahmacarin is now considered an adult. The upanayana sa\u1e43sk\u0101ra is therefore analogous to such coming-of-age rituals as the Bar Mitzvah of Judaism or the sacrament of Confirmation in **Christianity**. The stage of the brahmacarya ends with the ritual of marriage, which begins the second, **householder** ( **g\u0101rhasthya** ) \u0101\u015brama.\n\nThe connection of the term \"brahmacarya\" with celibacy is due to the fact that the student stage is a celibate state, although an adult can also become a brahmacarin in the sense of taking a vow of celibacy, even after having lived the life of a sexually active **householder**. **Mohand\u0101s K. G\u0101ndh\u012b** , for example, became a brahmacarin after he and his wife had had four sons. This kind of brahmacarya, undertaken later in life, is a form of **asceticism** , typically practiced for spiritual reasons.\n\n**BR\u0100HMA-MUH\u016aRTA.** Period from roughly 4:30 a.m. to 6 a.m., and regarded as the best time to do one's morning worship and practice **meditation**. A **muh\u016brta** is approximately an hour and a half in length and each day is divided into 16 muh\u016brtas.\n\n**BRAHMAN.** The absolute, ultimate reality, the infinite, literally \"the expansive.\" In the early **Vedic** **literature** , Brahman is the power that makes the Vedic sacrifice or **yaj\u00f1a** work, the energy of **creation**. In the _Upani\u1e63ads_ , Brahman is understood to be an all-encompassing reality, that from which all things have emerged and to which they will return. Anticipations of this concept can be found as early as the _\u1e5ag Veda_ , in hymns that describe the universe as emerging from the body of a single cosmic being, often seen as having been offered up in sacrifice in order to create the universe. In the _Upani\u1e63ads_ , the concept becomes more abstract: the underlying consciousness that makes any experience possible, or the deep mystery of existence, knowing which all things can be known. To realize the identity of Brahman and **\u0101tman** , the self, is to become liberated from **rebirth**.\n\nIn some sections of the _Upani\u1e63ads_ , Brahman seems to be an impersonal reality, the substance of consciousness. In other parts of the _Upani\u1e63ads_ , Brahman is a personal deity\u2014 **God** , or **\u012a\u015bvara** , the Supreme Being. In **Advaita Ved\u0101nta** , Brahman is perceived to have impersonal and personal aspects. The impersonal aspect\u2014 **Nirgu\u1e47a Brahman** \u2014is Brahman without any limiting qualities. In this aspect, which is preeminent, Brahman is infinite being ( **sat** ), pure or **non-dual** consciousness ( **cit** ), and unlimited bliss ( **\u0101nanda** ). The personal aspect is **Sagu\u1e47a Brahman** , or Brahman with qualities, who is identical with God (\u012a\u015bvara) and the world, acting as both the material and efficient cause of the universe. Theistic systems of **Ved\u0101nta** deny that Brahman is the material cause of the universe, identifying Brahman entirely with God (though **Vi\u015bi\u1e63\u1e6d\u0101dvaita Ved\u0101nta** sees Brahman as both God and world in a relationship not of identity but of organic unity).\n\n**BRAHM\u00c1N.** Chief priest in the **Vedic** ritual of sacrifice ( **yaj\u00f1a** ), whose duty is to know the duties of all the other priests and correct any errors that may occur.\n\n**BR\u0100HMA\u1e46A.** Priestly text. The second of the four sections contained within each of the four _Vedas_. The _Br\u0101hma\u1e47as_ were composed primarily during the period between the composition of the original four _Vedas_ , or _Sa\u1e43hit\u0101s_ , and the _\u0100ra\u1e47yakas_ , approximately 800 to 600 BCE according to most contemporary scholarship. They form a commentary on the _Sa\u1e43hit\u0101s_. Being concerned chiefly with the correct performance of Vedic rituals, they form, along with the original _Sa\u1e43hit\u0101s_ , the **Karma Kanda** , or \" **action** portion,\" of the Vedic **literature**.\n\n**BRAHMANISM.** The ideology associated with the culture that produced and continued to propagate the _Vedas_. Brahmanism is not identical to Hinduism, which has assimilated and produced many other ideological and cultural currents through the centuries, some of which have been in explicit tension with Brahmanism. But Brahmanism is the locus of many of the guiding concepts and principles that have become central to Hindu thought and practice. Two central principles of Brahmanism are the sanctity and eternal authority of the _Vedas_ and the system of **castes** (or **var\u1e47as** ) and stages of life ( **\u0101\u015bramas** ) by which persons are categorized.\n\nThese two principles were strongly rejected by the **\u015brama\u1e47a** traditions of ancient India (such as **Jainism** and **Buddhism** ), and questioned, ignored, or otherwise rendered problematic by subsequent Hindu movements, such as **bhakti** and **Tantra**. Concepts that Brahmanism shares with the other Indic systems of thought include **karma** , **rebirth** , and **mok\u1e63a** , though it may have adopted these fairly late (c. 6th century BCE).\n\n**BRAHMIN.** Member of the priestly **caste** (or **var\u1e47a** ), the highest of the traditional four castes into which Brahmanical society is divided. According to **Brahmanism** , birth as a Brahmin is a prerequisite for higher spiritual knowledge, which can only be acquired by studying the _Vedas_. Some late Vedic and post-Vedic texts, however, challenge this idea, as do the Hindu reformers of the modern period.\n\n**BRAHMO SAM\u0100J.** Hindu reform organization first established in **Calcutta** in 1828, under the name _Brahmo Sabh\u0101_ , by **R\u0101m Mohan Roy** (1772\u20131833). The Brahmo Sam\u0101j was most popular among middle-class Bengali intellectuals who had received significant exposure to European culture and education, due to the work of Christian missionaries, but who also had a strong sense of identification with Hinduism and Indian culture. The effort of these intellectuals was to reform Hinduism, opposing practices such as **sat\u012b** and the use of images ( **m\u016brtis** ) in worship, which they saw as later additions to an originally monotheistic Hinduism taught in the _Upani\u1e63ads_. Roy, a prolific author and energetic activist, translated some of the _Upani\u1e63ads_ into Bengali, and wrote many pamphlets as well, decrying what he saw as the evils of certain Hindu practices, and **Christianity** as well. Brahmo Sam\u0101j worship was modeled on Protestant worship services, with hymns and sermons drawing upon the _Upani\u1e63ads_ , rather than the Bible.\n\nTheologically, the teachings of the Brahmo Sam\u0101j display a strong leaning toward Unitarianism, and a number of Brahmo Sam\u0101j members traveled to England in the 19th century to study at the Unitarian seminary at Harris-Manchester College in Oxford. With its emphasis on an original, \"pure\" **Vedic** **monotheism** later corrupted into popular Hindu **polytheism** , Brahmo Sam\u0101j teaching in many ways anticipates that of the **\u0100rya Sam\u0101j** of **Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b**.\n\nMany prominent Bengali Hindu thinkers of the 19th and 20th centuries were connected with the Brahmo Sam\u0101j, including Roy's successors to the leadership of the organization, **Debendran\u0101th Tagore** (1817\u20131905) and **Keshub Chunder Sen** (1838\u20131884); Tagore's grandson, the renowned **poet** , playwright, composer, and Nobel laureate **Rabindran\u0101th Tagore** (1861\u2013 1941); and **Narendran\u0101th Datta** (1863\u20131902), who is better known by his monastic name, **Sw\u0101m\u012b Vivek\u0101nanda** , and who was a member of the Brahmo Sam\u0101j until he became a disciple of **Ramakrishna**.\n\nThough highly influential in the 19th century, the Brahmo Sam\u0101j maintains a relatively small but devoted following today, largely confined to **Bengal**.\n\n**BRAJ.** **Hindi** pronunciation of **Vraja**.\n\n_B\u1e5aHAD\u0100RA\u1e46YAKA UPANI\u1e62AD_ **.** Great \u0100ra\u1e47yaka Upani\u1e63ad; oldest and longest of the _Upani\u1e63ads_ (or, alternatively, the latest of the _\u0100ra\u1e47yakas_ ), it is contained as the conclusion of the _\u015aatapatha Br\u0101hma\u1e47a_ , which is, in turn, contained within the _\u015aukla_ (White) _Yajur Veda_. Its length and antiquity have made it a major authoritative source for the **Ved\u0101nta** systems of **philosophy**. Probably compiled around 500 BCE (though from oral sources a good deal older), the centerpiece of this text is the teaching of the sage **Y\u0101j\u00f1avalkya** , the likely originator of the traditions that this _Upani\u1e63ad_ preserves. Two different recensions of the text exist\u2014the _K\u0101\u1e47va_ and the _M\u0101dhyandina_ \u2014though their philosophical content is not substantially different.\n\nThis text contains the first unambiguous Hindu scriptural reference to the process of **rebirth** , the idea that the nature of one's rebirth is determined by one's **actions** ( **karma** ), and the idea that one can attain freedom from rebirth ( **mok\u1e63a** ) by giving up **desire** and merging with **Brahman** (4.4.5\u20137).\n\n_B\u1e5aHAD-DEVAT\u0100_ **.** \"Great Text on Deities\"; systematic presentation of the attributes and deeds of the **Vedic** deities ( **devas** ); attributed to **\u015aaunaka** and likely composed between 500 and 400 BCE. This text has continued to be an authoritative source for a mainstream Hindu understanding of the Vedic deities.\n\n**B\u1e5aHASPATI.** \"Great Father\"; **Vedic** deity; god of wisdom and teacher, or **guru** , of the **gods**. B\u1e5bhaspati is the presiding deity of the planet Jupiter and the guru referred to in the name **Guruv\u0101ra** , the Hindu name for Thursday, the day of the week governed by Jupiter in Hindu **astrology**. He embodies the intellect and the power of speech of the cosmic man ( **puru\u1e63a** ) from whose body the universe was formed according to Vedic cosmology.\n\nIntriguingly, B\u1e5bhaspati is attributed with developing the **C\u0101rv\u0101ka** , or **materialist** system of **philosophy** , which denies the validity of the _Vedas_. The lost root texts of this system are called the _B\u0101rhaspatya S\u016btras_ , or \"Root Texts Attributed to B\u1e5bhaspati.\"\n\n**BRIGHT HALF.** _See_ \u015aUKLA PAK\u1e62A.\n\n**BRITISH EAST INDIA COMPANY.** British trade organization established in 1600. It was given a monopoly on British trade with Asia and granted vast resources, including ships, by Queen Elizabeth I for the purpose of expanding the British commercial presence in India and Indonesia. The company established trading posts on the Indian coast that became the basis for large modern cities such as **Calcutta** , Bombay, and **Madras**.\n\nIn order to protect company holdings, ever-increasing numbers of British soldiers were sent to India. The company also hired Indian mercenaries (known as _sepoys_ ) and entered into alliances with local rulers, often becoming embroiled in complex rivalries and conflicts among Indian kings. This ever-expanding military and political presence\u2014combined with the marked disunity among the various Indian polities, which prevented the formation of any serious Indian resistance\u2014eventually led the company to become the dominant force in Indian political and economic life.\n\nDiscontent among many Indians with the growing British domination of India and the exploitation and denigration of Indian culture\u2014including Hinduism and **Islam** \u2014that it involved eventually led to an armed revolt by sepoys in 1857. This event continues to be referred to by British historians as the Sepoy Mutiny and by Indian historians as the War of Independence. Alarmed by this rebellion, Queen Victoria dissolved the company in 1858 and put India under the direct rule of the British government.\n\n**BROWN, W. NORMAN (1892\u20131975). Indologist** and **Sanskritist** who held the chair in Sanskrit at the University of Pennsylvania from 1926 to 1966 and was curator of Indian **art** at the Philadelphia Museum of Art. He organized the American Oriental Society in 1926 and edited its journal for many years.\n\n**BUDDHA (c. 490\u2013410 BCE).** The \"Awakened One\"; Siddh\u0101rtha Gautama, founder of **Buddhism** and, in some Hindu accounts, the ninth **avat\u0101ra** of **Vi\u1e63\u1e47u**. According to the traditional accounts of his life, the Buddha was a prince, the son of the ruler of the \u015aakya clan, whose territory was located in what is now southern **Nepal**. Becoming a **renouncer** at the age of 30, he attained **nirv\u0101\u1e47a** six years later and began teaching, establishing an order of monks and nuns and lay followers. He died at the age of 80.\n\nA member of the **\u015brama\u1e47a** movement, the Buddha, like **Mah\u0101v\u012bra** and others, did not accept the claims of **Brahmanism** that one's birth **caste** is a reflection of one's level of spiritual attainment, that a birth in the **Brahmin** caste is a mark of spiritual authority, and that the _Vedas_ and the ceremonies they enjoin are divinely inspired. He found the sacrifice of animals in the Vedic **yaj\u00f1a** to be particularly objectionable. He shared with Brahmanism\u2014particularly the teaching of the _Upani\u1e63ads_ \u2014the concepts of **karma** , **rebirth** , and **liberation** ( **mok\u1e63a** ) from the cycle of rebirth. His teaching on the renunciation of desire as necessary to liberation is quite close to that of the _B\u1e5bhad\u0101ra\u1e47yaka Upani\u1e63ad_.\n\nBuddhism subsequently became a major ideological rival of Brahmanism, both of which shaped what eventually emerged as Hinduism. The concept of the Buddha as an avat\u0101ra of Vi\u1e63\u1e47u is a Brahmanical attempt to cope with the popularity of Buddhism, and the changes that this concept undergoes reflect the changing relationship of Buddhism to Hindu identity: from fierce rival to assimilated cultural resource. Some early accounts of the Buddha avat\u0101ra are of an anti-Buddhist nature, claiming that Vi\u1e63\u1e47u took the form of the Buddha for the purpose of deluding demonic persons and persuading them not to perform the sacrifice (or **yaj\u00f1a** ) enjoined in the _Veda_. But later accounts of the Buddha avat\u0101ra are more positive, presenting the Buddha as a Vedic reformer who ended the practice of **animal sacrifice** to promote the ideal of compassion. The concept of the Buddha avat\u0101ra, both in its critical and in its more appreciative forms, is a Hindu interpretation of the life and teaching of the Buddha, and is generally not accepted by Buddhists.\n\n**BUDDHI.** Intellect; one of the basic elements that constitute existence, or **tanm\u0101tras** , of the **S\u0101\u1e43khya** system of **philosophy**. In the **Yoga** of **Pata\u00f1jali** , buddhi, in combination with **aha\u1e43kara** (ego) and the **indriyas** (senses) constitutes **citta** (the **mind** ).\n\nAlso, along with **Siddhi** (\"Success\"), Buddhi is the name of one of the two wives of **Gane\u015ba**.\n\n**BUDDHISM.** **\u015arama\u1e47a** tradition established by the **Buddha** (c. 490\u2013410 BCE). By the end of the 13th century CE, this tradition had largely died out in India but had spread beyond India, throughout Southeast and Central Asia and into China, Korea, and Japan to become a major world religion. The major forms of Buddhism are Therav\u0101da (Teaching of the Elders), which is predominant in both **Sri Lanka** and **Southeast Asia** ; Mah\u0101y\u0101na (Great Vehicle), which is practiced mainly in China, Korea, and Japan; and Vajray\u0101na ( **Tantric** Buddhism), which has traditionally been prominent in **Nepal** , Tibet, and Mongolia.\n\nLike other \u015brama\u1e47a traditions, such as **Jainism** , Buddhism shares many core ideas of Hinduism, such as the doctrines of **karma** and **rebirth** and the aspiration for **liberation** ( **mok\u1e63a** ) from the cycle of rebirth as a central salvific goal. Buddhism shares with other \u015brama\u1e47a traditions a rejection of the claims of **Brahmanism** with regard to **caste** and the authority of the _Vedas_. Specifically, it rejects the idea that birth caste is a reflection of one's spiritual attainment (with its implication that the **Brahmins** are the most advanced of human beings) and Vedic injunctions to perform **animal sacrifice**. Buddhism was the main ideological rival of Brahmanism for many centuries. As Hinduism coalesced from many Indic traditions, however, many Buddhist concepts were assimilated into it. For example, although nominal affirmation of the authority of the _Vedas_ continues, practices such as animal sacrifice gradually fell by the wayside, being replaced by **vegetarianism**. Though the caste system continues to be practiced, movements that affirm the irrelevance of caste to the spiritual path, such as **bhakti** and **Tantra** , have many Hindu followers.\n\nBuddhism died out in India due to a combination of foreign invasions, resulting in a mass destruction of monasteries and **temples** , and Hindu assimilation. In the 20th century, **Bh\u012bmrao Ambedkar** revived Indian Buddhism, leading a mass conversion of **Dalits** to Buddhism as a protest against the ongoing Hindu practice of caste. An influx of Tibetan Buddhist refugees, led by the Dalai Lama, has also led to a reinfusion of life into Buddhism in India. _See also_ AMBEDKARITE BUDDHISM.\n\n**BUDDHISM, AMBEDKARITE.** _See_ AMBEDKARITE BUDDHISM.\n\n**BU\u1e0cHA.** The planet Mercury; son of the deity **Soma** and the **goddess T\u0101ra**. _See also_ ASTROLOGY; NAVAGRAHAS.\n\n**BU\u1e0cHAV\u0100RA.** Wednesday, the day of the week that is governed by Bu\u1e0dha, the planet Mercury. _See also_ CALENDAR, HINDU.\n\n**BUFFALO DEMON.** _See_ MAHI\u1e62\u0100SURA.\n\n**BUITENEN, JOHANNES ADRIANUS BERNARDUS VAN (1928\u2013 1979).** Dutch-born **Indologist** and **Sanskritist** who founded and chaired the Department of South Asian Languages and Civilizations at the University of Chicago. A prolific scholar, his best-known achievement is the translation of the first five books of the _Mah\u0101bh\u0101rata_ and the _Bhagavad G\u012bt\u0101_.\n\n## C\n\n**CAITANYA (1486\u20131533).** **Bengali** **Vai\u1e63\u1e47ava** saint and devotee of **K\u1e5b\u1e63\u1e47a** ; major figure of the **bhakti** movement and founder of the **Gau\u1e0d\u012bya Vai\u1e63\u1e47ava** tradition, with which the **International Society for Krishna Consciousness** (ISKCON) is in close continuity. At the age of 22 he met a **sanny\u0101s\u012b** of **Madhva** 's theistic **Dvaita Ved\u0101nta** tradition, whose outlook greatly influenced him. He then traveled to and spent most of his life in **Puri** , in what is now the Indian state of **Orissa** , at the **temple** of **Jagann\u0101tha**. Much like the **Bauls** , by whom he may also have been influenced, Caitanya promoted ecstatic devotional singing as a way to express and evoke bhakti and give worship to K\u1e5b\u1e63\u1e47a. He made several pilgrimages to **Vraja** , the area of **Mathura** and **V\u1e5bnd\u0101vana** , where K\u1e5b\u1e63\u1e47a spent most of his early life according to the _Bh\u0101gavata Pur\u0101\u1e47a_. He reinvigorated K\u1e5b\u1e63\u1e47a devotion in the region, making it a major pilgrimage center. Caitanya also established the **Gosv\u0101min** theological tradition. According to Caitanya's teaching, K\u1e5b\u1e63\u1e47a is the supreme deity\u2014 **God** in a **monotheistic** sense\u2014of whom all other deities are either manifestations or subordinates. Caitanya's followers view him as an **avat\u0101ra**.\n\n**CAKRA.** Literally, a wheel. In **Tantra** , a cakra is an energy center located in the **subtle body** ( **suk\u1e63ma \u015bar\u012bra** ). According to Tantric thought, subtle energy bodies exist in the same space as, but on a different vibrational wavelength from, the physical, gross body ( **sth\u016bla \u015bar\u012bra** ). These subtle bodies are made up of the **ko\u015bas** , or sheaths, in which the soul or **\u0101tman** resides (specifically, the four ko\u015bas other than the **anna-maya-ko\u015ba** , which is identical to the gross body). In the space that corresponds to the base of the spine, a subtle energy, or **\u015bakti** , resides. While it is residing in this space, it is also referred to as the **ku\u1e47\u1e0dalini** , or coiled energy, because it is said to resemble a coiled serpent in this state. In Tantric **yoga** , one of the goals of practice is to raise this energy from the base of the spine to the crown of the head, which are linked by a subtle nerve channel called the su\u1e63umna n\u0101d\u012b, and which occupies roughly the same space as the spinal cord.\n\nAlong the su\u1e63umna n\u0101d\u012b are the seven cakras. These energy centers are activated as the \u015bakti or ku\u1e47\u1e0dalini rises through the su\u1e63umna n\u0101d\u012b. The bottom-most of the cakras, the **m\u016bladh\u0101ra cakra** , is located at the base of the spine. The rest of the cakras are the **sv\u0101dhi\u1e63\u1e6dh\u0101na cakra** , which is aligned with the navel; the **ma\u1e47ip\u016bra cakra** , which is aligned with the solar plexus; the **an\u0101hata cakra** , which is centered in the heart; the **vi\u015buddha cakra** , which is in the throat; the **\u0101j\u00f1\u0101 cakra** or third eye, which is just above the space between the eyebrows; and, at the crown of the head, the **sah\u0101srara cakra** or \"thousand-petaled lotus,\" which, when energized by the rising \u015bakti, produces a blissful state that, according to Tantra, advances one quickly toward **enlightenment**.\n\nAccording to the tradition of **Siddha Yoga** , a contemporary **\u015aaiva** movement with a strongly Tantric dimension, an enlightened teacher, or **guru** , can raise one's \u015bakti from the base of the spine to the crown of the head\u2014activating all of the cakras in the process\u2014with a thought, word, or touch. This experience is called **\u015baktipat**.\n\n**CALCUTTA.** City established by the **British East India Company** in 1696 near the village of K\u0101l\u012bkata and the **K\u0101l\u012bgh\u0101\u1e6d** **temple**. Now known as Kolkata, it is one of the largest cities in India and the capital of the state of **West** **Bengal**. Calcutta served as the administrative capital of British India until 1911, when the administration shifted to **New Delhi** , which remains the capital of the Republic of India today.\n\nCalcutta has a rich intellectual and cultural history, being the center of much of the Hindu reform that occurred in the 19th and early 20th centuries. Major figures of this period who resided at some point in Calcutta include **R\u0101m Mohan Roy** , **Debendran\u0101th Tagore** , **Keshub Chunder Sen** , **Ramakrishna** , **Sw\u0101m\u012b Vivek\u0101nanda** , **Aurobindo Ghose** , and **Rabindran\u0101th Tagore**. The city remains the main center of activity for the **Brahmo Sam\u0101j** , and it also houses the administrative headquarters of the **Ramakrishna Order**. _See also_ BENGAL.\n\n**CALENDAR, HINDU.** The traditional Hindu calendar is a lunar calendar on the basis of which the dates of Hindu holidays are determined and astrological calculations carried out. The calendar is made up of 12 lunar months, each of which is further divided into a **bright half** ( **\u015bukla pak\u1e63a** ), during which the **moon** is waxing, and a **dark half** ( **k\u1e5b\u1e63\u1e47a pak\u1e63a** ), during which it is waning. Distinct from these two halves are **P\u016br\u1e47im\u0101** (full-moon day) and **Amavasya** (new-moon day), which mark, respectively, the transition from the bright half to the dark half and from the dark half to the bright half. The Hindu lunar months are called **Chaitra** (March\u2013April), **Vai\u015b\u0101kha** (April\u2013May), **Jyai\u1e63\u1e6dha** (May\u2013June), **\u0100\u1e63\u0101\u1e0dha** (June\u2013July), **\u015arava\u1e47a** (July\u2013August), **Bhadra** (August\u2013September), **A\u015bvina** (September\u2013October), **K\u0101rtika** (October\u2013November), **Agrahaya\u1e47a** (November\u2013December), **Pau\u1e63a** (December\u2013January), **Magha** (January\u2013February), and **Phalguna** (February\u2013March). Each month begins the day after the new-moon day (Amavasya) of the month just before it. Amavasya marks the end of a month, and P\u016br\u1e47im\u0101 its midpoint. Whenever a **p\u016bj\u0101** , or worship ceremony, is conducted, it is customary near the beginning of the ceremony to announce the date: for example, \"the third day of the bright half of the month of Phalguna,\" or \"the fifth day of the dark half of Vai\u015b\u0101kha,\" and so forth.\n\nThe Hindu week consists of seven days: **Raviv\u0101ra** (Sunday), **Somav\u0101ra** (Monday), **Ma\u00f1galv\u0101ra** (Tuesday), **Bu\u1e0dhav\u0101ra** (Wednesday), **Guruv\u0101ra** (Thursday), **\u015aukrav\u0101ra** (Friday), and **\u015a\u0101niv\u0101ra** (Saturday). Astrologically, each day of the week is governed by a different planet. The \" **planets** \" consist of the five planets that are visible to the naked eye (Mercury, Venus, Mars, Jupiter, and Saturn, although not in that order) plus the sun and the moon. There are numerous local variations on this basic calendrical model. _See also_ ASTROLOGY.\n\n**C\u0100LUKYA.** Hindu dynasty that ruled a large kingdom encompassing much of central and southern India from the sixth to the 12th centuries of the Common Era. At its height, during the seventh and eighth centuries, this kingdom encompassed what are now the states of Maharashtra and **Karnataka** , and much of Andhra Pradesh. The C\u0101lukyas constructed major **temple** complexes at the towns of **Aihole** , **B\u0101d\u0101m\u012b** , and **Pa\u1e6d\u1e6dadakal**.\n\n**CAMPHOR.** Holy substance used during worship ( **p\u016bj\u0101** ). A small amount is typically set aflame. The smoke and flame from the camphor are regarded as having a purifying effect. The lit camphor is held in a small lamp or **d\u012by\u0101** and waved in front of an image ( **m\u016brti** ) of a deity\u2014a variation of the ceremony known as **\u0101rat\u012b**. _See also_ P\u016aJ\u0100.\n\n**C\u0100MU\u1e46\u1e0c\u0100.** Destroyer of demons; fierce from of the **Mother Goddess** who emanates from the forehead of the goddess **Durg\u0101** in order to destroy evil. Having a frightening appearance, C\u0101mu\u1e47\u1e0d\u0101 has four arms and hands and carries a sword, a noose, and a club. Resembling **K\u0101l\u012b** , she wears a garland of corpses and the hide of an elephant. She is also known as C\u0101mu\u1e47\u1e0d\u012b.\n\n**CANAKYA.** _See_ KAU\u1e6cILYA.\n\n**CA\u1e46\u1e0c\u0100LA.** A member of the lowest **caste** \u2014an **Untouchable** or **Dalit** community. This caste is traditionally responsible for the cremation of corpses, which is regarded as a very polluting activity, given that dead bodies are considered the most impure of substances. Due to their untouchable status, Ca\u1e47\u1e0d\u0101las are often expected to live far from the rest of the community. **Rabindran\u0101th Tagore** composed a widely acclaimed opera critical of the caste system entitled _Ca\u1e47\u1e0d\u0101lik\u0101_ , about a female Ca\u1e47\u1e0d\u0101la. As a female member of the lowest of the castes, a Ca\u1e47\u1e0d\u0101lik\u0101 would be regarded as being at the very bottom of Hindu society. In the opera, the main character, accustomed to being shunned by others, is approached by a **Buddhist** monk who, having no regard for caste restrictions, asks her for a drink of water. This is a transformative moment for the main character, who discovers her human dignity and becomes a Buddhist nun.\n\n**CA\u1e46\u1e0c\u012a (CA\u1e46\u1e0cIK\u0100).** \"Wrath,\" \"wrathful\"; any fierce form of the **Mother Goddess** , often with either 18 or 20 hands. A wrathful form of this kind is assumed so the goddess can destroy evil. The worship of fierce **goddesses** is especially popular in **Bengal** and is often associated with **Tantra**. The _Dev\u012b Mah\u0101tmya_ , a text that narrates the deeds of the fierce forms of the goddess, is also known as the _Ca\u1e47\u1e0d\u012b Pa\u1e6dh_ (\"Reading on Ca\u1e47\u1e0d\u012b\"). _See also_ DURG\u0100; K\u0100L\u012a.\n\n**CA\u1e46\u1e0c\u012aD\u0100S (fl. 15th century CE, born 1408). Vai\u1e63\u1e47ava** saint of **Bengal** and member of the **Sahajiya** movement, which emphasized highly spontaneous and ecstatic expressions of devotion and shared the **Tantric** sensibility of using the senses to transcend the senses. The **bhakti** **poetry** of Ca\u1e47\u1e0d\u012bd\u0101s makes extensive use of the imagery of the love between **R\u0101dh\u0101** and **K\u1e5b\u1e63\u1e47a** to evoke the loving relationship between the devotee and the divine. His songs exerted a strong influence on **Caitanya**. His most famous work is the _\u015arik\u1e5b\u1e63\u1e47a K\u012brtana_ (\"Glory of Lord K\u1e5b\u1e63\u1e47a\"), which narrates the romance of R\u0101dh\u0101 and K\u1e5b\u1e63\u1e47a over the course of 412 songs. Altogether, 1,250 songs are attributed to Ca\u1e47\u1e0d\u012bd\u0101s, although it is likely that there was more than one author by this name, perhaps in a shared lineage.\n\nThough he was a devotee of K\u1e5b\u1e63\u1e47a, the name _Ca\u1e47\u1e0d\u012bd\u0101s_ means \"servant of **Ca\u1e47\u1e0d\u012b** \" (the **Mother Goddess** in her fierce form).\n\n**CANDRA.** The **moon** ; often identified with the **Vedic** deity **Soma** , who also represents the hallucinogenic plant whose juice, mixed with milk, was used by ancient **Brahmins** to evoke an experience of the divine. The phases of the moon represent this substance being consumed by the **devas** (during the dark half of the month, or **K\u1e5b\u1e63\u1e47a Pak\u1e63a** , when the moon is waning) and replenished (during the bright half, or **\u015aukla Pak\u1e63a** , when the moon is waxing).\n\n**CANDRASEKHARA SARASVAT\u012a, \u015aR\u012a (1894\u20131994).** The 68th **\u015aa\u1e45kar\u0101c\u0101rya** of the K\u0101\u00f1c\u012b K\u0101makoti P\u012b\u1e6da, or **ma\u1e6dha** , at **K\u0101\u00f1c\u012bpuram** in Tamil Nadu, a role that he held for a record 87 years. He assumed his responsibilities at the unusually young age of 13 when his cousin, the 67th head of the ma\u1e6dha, died prematurely of a fever.\n\n**CARAKA (c. 3rd century BCE).** Presumed author of the _Caraka Sa\u1e43hit\u0101_ , a work of **\u0100yur Veda** , or traditional Hindu medicine.\n\n**C\u0100RV\u0100KA (c. 3rd century BCE).** Founder of the **materialist** system of Indian philosophy ( **dar\u015bana** ); also a name for the philosophical system itself, which is also known as the **Lok\u0101yata** system. C\u0101rv\u0101ka's system is, like **Jainism** and **Buddhism** , a **n\u0101stika** or nonorthodox system of thought that denies the authority of the _Vedas_ and of the **Brahmin** caste. Unlike all other systems of Indian philosophy, C\u0101rv\u0101ka's system denies the reality of **karma** and **rebirth** and sees religion as a fundamentally deluded activity, developed in order to satisfy the greed of those who promulgate it (both Brahmin and **\u015brama\u1e47a** ). This system affirms that sensory perception is the only valid basis for knowledge ( **pram\u0101\u1e47a** ), thereby denying the validity both of the _Vedas_ and of the extrasensory **yogic** knowledge affirmed by Jains and Buddhists. It claims that the sole aim of life is sensory enjoyment.\n\nThe original writings of this system survive only in fragmentary form, as quoted in texts of other systems aiming to refute it. The authoritative **s\u016btras** or root texts of this system are the _B\u0101rhaspatya S\u016btras_ (that is, texts claiming to derive from the teaching of the sage **B\u1e5bhaspati** ), but the full text has not yet been recovered.\n\n**CASTE.** System of hereditary occupational groups arranged in a hierarchy. The word _caste_ is derived from the Portuguese word _casta_ , which means \"color\" and seems to be a translation of the **Sanskrit** term _var\u1e47a_. Contrary to popular belief, var\u1e47a does not refer to skin color but to a set of colors assigned to the four main subgroups of Hindu society based on the mixture of the three **gu\u1e47as** , or essential qualities, that each group is held to possess.\n\nThe only reference to the caste system in the earliest **Vedic** texts is in the _Puru\u1e63a_ _Sukta_ of the _\u1e5ag Veda_ , where the four main var\u1e47as\u2014 **Brahmins** (priests and intellectuals), **K\u1e63atriyas** (warriors and administrators), **Vai\u015byas** (common people, economic producers), and **\u015a\u016bdras** (servants)\u2014are described as emerging from various parts of the body of the cosmic man from whom the entire universe was formed, according to Vedic cosmology. No reference is made in the Vedic _Sa\u1e43hit\u0101s_ to a hereditary system, and indeed the idea of a hereditary system is contradicted by some Vedic hymns in which the speaker and his parents are referred to as practicing different trades.\n\nIn later Vedic **literature** , the story of **Satyak\u0101ma** in the _Ch\u0101ndogya Upani\u1e63ad_ is suggestive of the possibility that one might come by one's caste purely through the possession of the appropriate personal qualities ( _gu\u1e47as_ ) rather than through birth, an idea found most prominently in **\u015brama\u1e47a** traditions, such as **Jainism** and **Buddhism** , and reasserted in the modern period by the founder of the **\u0100rya Sam\u0101j** , **Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b** , as well as by **Mohand\u0101s K. G\u0101ndh\u012b**.\n\nBut by the period of the _Dharma \u015a\u0101stras_ (c. 300 BCE), the idea of castes based on birth ( **j\u0101ti** ) had become predominant in Hindu society. Although the system of four var\u1e47as is the version of caste most commonly mentioned in the Hindu **scriptures** , the effective living unit in actual practice is the j\u0101ti, which could be described as a subset or \"subcaste\" within one of the four main var\u1e47as. Each var\u1e47a contains dozens of j\u0101tis, which represent more specialized tasks under the larger rubric of one of the categories of priesthood, political authority, economic production, or servitude.\n\nThough it has typically been the case that caste, seen largely as a matter of birth, has not allowed for social mobility, that has not been a uniform practice. Entire groups have managed to advance in the caste hierarchy through a process of adopting practices associated with higher castes. Scholars have labeled this process **Sanskritization**.\n\nWhile caste has proved to be highly effective and resilient as a method for both ordering and preserving Indian society, it has also been the target of much internal and external criticism throughout the centuries. So Jains and Buddhists, though continuing to practice caste as a way of organizing society, rejected the notion that one's birth caste was in any way a reflection of one's spiritual advancement, or that caste was a function of one's **karma** \u2014the dominant view of **Brahmanism**. Renunciation came to be seen, even within Brahmanism, as a way of removing oneself from the restrictions of caste, although the most conservative Brahmanical traditions insist that one must be born as a male Brahmin to be eligible for renunciation. **Tantra** and the **bhakti** movements both share the Jain and Buddhist insistence that one's birth caste is irrelevant to one's ability to make progress toward **mok\u1e63a** , or **liberation** from the cycle of **rebirth**. At the same time, though, none of these movements has ever led to a major breakdown of caste in practice. Even Abrahamic traditions such as **Christianity** and **Islam** , which reject the notion of caste and affirm human equality and initially held great appeal for members of the lowest castes\u2014particularly the **Dalits** , or **Untouchables** , whose hereditary professions excluded them from most interaction with the rest of society\u2014did not have a significant impact on the caste system. Many Indian Christians and Muslims are members of castes today, choosing professions and marriage partners accordingly. Caste is deeply rooted in Indian society but is neither limited to nor unambiguously justified by Hinduism.\n\nOnly in the modern period have serious efforts begun either to eradicate prejudice based on caste (while preserving the basic concept of the caste system) or to eradicate the caste system altogether. The leading figure in the movement to abolish caste has been Dr. **Bh\u012bmrao Ambedkar** , the primary author of the Indian constitution. **Arvind Sharma** has argued that all Hindus should see themselves as sharing the duties of all four of the var\u1e47as.\n\nIt remains debated today whether caste is intrinsic to Hinduism as a religion or is an Indian social system that has been justified on the basis of Hinduism but that may be fundamentally incompatible with some aspects of Hindu **philosophy** , such as its emphasis on the unity of existence and seeing the divine in all beings.\n\n**CASTEISM.** Prejudice or bigotry based on **caste** , which is illegal under the constitution of India but which nevertheless persists in many parts of India, particularly in rural areas.\n\n**CHAITRA.** Month of the **Hindu calendar** that corresponds roughly to the period from mid-March to mid-April.\n\n_CH\u0100NDOGYA UPANI\u1e62AD_ **.** One of the earliest _Upani\u1e63ads_ , likely composed between 500 and 400 BCE. It is part of the _Ch\u0101ndogya Br\u0101hma\u1e47a_ , which, in turn, is part of the _S\u0101ma Veda_. A relatively lengthy Upani\u1e63ad, particularly well-known portions of this text include the sage Udd\u0101laka's teaching to this son, \u015avetaketu, on the identity of the self, or **\u0101tman** , and **Brahman** , using the refrain \"You are that\" ( _tat-tvam asi_ ); early reflections on the esoteric meaning of the **mantra** **O\u1e43** ; the story of **Satyak\u0101ma** , who is initiated into the study of the _Vedas_ on the basis of his honesty, despite the fact that his mother is of low caste and the identity of his father is unknown; and the teaching of **Praj\u0101pati** to the **devas** and **asuras** , which establishes **materialism** as the doctrine of the asuras. _See also_ VED\u0100NTA.\n\n**CHARIOT.** The image of the chariot is an ancient one in Hindu thought and culture, being found as early as the _\u1e5ag Veda_. The technology of chariot making also has great significance for the question of Hindu origins and the issue of the relationship between **Vedic** culture and other **Indo-European** societies. The early Vedic deities, the **devas** , are frequently depicted as riding in chariots, and the image of the sun god riding a chariot daily through the sky is common to many Indo-European belief systems, including Vedic thought. The metaphor of the chariot as a symbol for the physical body, with the **soul** as the rider, the **mind** as the driver, and the senses as the horses, is utilized in the _Ka\u1e6dha Upani\u1e63ad_ (c. 400\u2013300 BCE) and, one could argue, alluded to implicitly as the setting of the _Bhagavad G\u012bt\u0101_. The same image is utilized by the Greek philosopher Plato in the _Phaedrus_ , who was either influenced by Indian thought (perhaps indirectly, through the influence of Pythagoras) or was tapping into the same Indo-European cultural heritage as the authors of the _Upani\u1e63ads_.\n\nIn terms of Hindu origins, the appearance of light chariots and chariot technology in India between 1900 and 1700 BCE, but already present in Central Asia between 2300 and 2000 BCE, supports the **\u0100ryan Migration Theory**. Though, as opponents of this theory have pointed out, crossing into India through Afghanistan on such chariots is a practical impossibility, persons with knowledge of this technology crossing into India by other means (such as on foot) is not at all impossible. _See also_ JAGANN\u0100THA; KON\u0100RAK; VIM\u0100NA.\n\n**CHATTERJEE, BANKIM CHANDRA (1838\u20131894).** **Bengali** novelist, **poet** , political activist, and the author of the national song of India, \"Vande Mataram.\" Chatterjee was a major literary figure, pioneering the genre of the novel in Bengali, as well an inspiration to the emerging Indian independence movement. An early **Hindu Nationalist** , the India that Chatterjee envisioned was a Hindu one, independent of both British and **Islamic** rule.\n\n**CHATTOPADHYAY, BANKIM CHANDRA.** _See_ CHATTERJEE, BANKIM CHANDRA.\n\n**CHENNAI.** Contemporary name for **Madras**.\n\n**CHIDVIL\u0100S\u0100NANDA, SW\u0100M\u012a (1955\u2013 ).** _See_ GURUMAY\u012a.\n\n**CHIPKO MOVEMENT.** An Indian environmental movement that emerged in 1973 in opposition to logging in the **Himalayan** foothills of what is now the state of Uttarakhand. Inspired by the **Bishnois** and the nonviolent methods of **Mohand\u0101s K. G\u0101ndh\u012b** , the activists of the Chipko movement protected their sacred forest from being cut down by loggers by forming a human chain encircling the trees and by hugging the trees\u2014a practice that led to the name \"Chipko\" and, beyond India, to the term \"tree hugger\" for anyone concerned about the protection of the environment.\n\n**CHRISTIANITY.** The world's largest religion, in terms of numbers of adherents, and a member, along with Judaism and **Islam** , of the Abrahamic family of traditions, sharing a common worldview that includes **monotheism** , divine revelation and divine intervention in human history, a single lifetime, a day of judgment, and an everlasting afterlife either in a heavenly paradise or a hellish state of punishment (though exceptions and variations in regard to this basic worldview can be found across all three of these traditions).\n\nIn terms of interactions with Hinduism, Christianity has been present in India for many centuries, the most ancient Indian Christian communities being located in the state of **Kerala** , on the southwest coast of the subcontinent, which is still home to the majority of Indian Christians. According to one Indian Christian tradition, the religion was first brought to India by the Apostle Thomas, and so dates to the first century of the Common Era, and the most ancient group of Christians in India are often referred to as \"St. Thomas Christians.\" Reliable historical data, however, cannot place Christians in India before the fifth century of the Common Era. These early Indian Christians, known as the _Nasranis_ , seem largely to have been Jewish converts who fled the Roman Empire after the Second Temple in Jerusalem was destroyed. Their churches are modeled closely on synagogues. Today, these Christians are largely affiliated with the Syrian Orthodox Church.\n\nReligiously significant interaction between these Christians and Hindus seems to have been fairly minimal, though some have argued that Christian influence is discernible in the emergence and popularity of **bhakti** traditions in southern India in the middle of the first millennium of the Common Era. But for all intents and purposes, the Jewish and Christian communities in the southwestern coastal regions of India in ancient times seem to have been largely self-contained, endogamous **castes**.\n\nExtensive Christian proselytizing activity did not occur in India until the 16th century, at the onset of the period of European colonization. The first European Christian missionaries in India were Roman Catholics\u2014mainly Jesuits\u2014operating in southern India, particularly in areas where Portuguese colonies were established, such as Goa. The older indigenous St. Thomas Christians experienced extensive persecution at the hands of the Portuguese government. Protestant missionaries arrived a couple of centuries later, with British colonization. Hindu reactions to Christian missionary activities have ranged from conversion, to indifference, to dialogue and engagement, to hostility.\n\nConversions have largely been among **Dalits** and tribal peoples ( **\u0101div\u0101s\u012bs** ), many of whom have seen the Christian teaching of the equality of all people before **God** as a means to escape the **caste** system (although converts have often continued to engage in caste-based practices even after adopting Christianity).\n\nIndifference has been the predominant reaction of most traditional Hindu religious authorities, such as those **Brahmins** and **sanny\u0101sins** who operate in a largely **Sanskritic** intellectual milieu and who have not seen Christianity as presenting a serious challenge to the worldviews of their particular traditions, or **sa\u1e43prad\u0101yas**.\n\nDialogue and engagement has largely been on the part of those Hindus who have had an extensive exposure to Western education, starting with the Hindu reformers of the 19th century, like **R\u0101m Mohan Roy** , **Debendran\u0101th Tagore** , **Keshub Chunder Sen** , and **Sw\u0101m\u012b Vivek\u0101nanda** , and continuing into the 20th century with **Mohand\u0101s K. G\u0101ndh\u012b**. A significant result of such engagement for Hinduism has been the rise of **Neo-Ved\u0101nta** , which accepts the validity of many Christian criticisms of Hinduism\u2014such as missionary criticisms of the injustices of the caste system\u2014but which reformulates Hindu thought and practice rather than abandoning it in favor of Christianity (and often turns a critical eye back upon Christianity itself). The attitudes of Hindus toward Christianity who have engaged with it in this way are complex, often being articulated in terms of a profound appreciation for Christ and his teachings\u2014as found, for example, in the Sermon on the Mount\u2014combined with strong criticisms of institutional Christianity and Christian doctrine. The doctrine that salvation is available only through Christ is considered to be particularly objectionable and is frequently contrasted with Hindu religious pluralism.\n\nHostility to Christianity has largely been articulated by **Hindu Nationalists** , who connect Christian missionary activity with colonialism and neo-colonialism, seeing it as a form of aggressive and predatory cultural imperialism.\n\n**CIDAMBARAM.** One of the most important **\u015aaiva** **temples** , built in southern India in the 10th century of the Common Era by the **Co\u1e37a** dynasty. It houses 108 images of **\u015aiva** in his form as Lord of the Cosmic **Dance** , or **N\u0101\u1e6dar\u0101ja** , as well as shrines to the members of \u015aiva's family: his wife, **P\u0101rvat\u012b** (the **Mother Goddess** ), and his sons, **Ga\u1e47e\u015ba** and **Skanda** (also called **K\u0101rttikeya** , **Subrama\u1e47ya** , **Murugan** , **Kum\u0101ra** , and **\u1e62a\u1e47mukha** ).\n\n**CIRCUMAMBULATION.** Way of honoring an image, or **m\u016brti** , of a deity, by walking around the image in a clockwise direction, keeping the image to one's right side. _See also_ PRADAK\u1e62INA.\n\n**CIT.** Consciousness. In **Advaita Ved\u0101nta** , cit is distinguished from **vij\u00f1\u0101na** , which is our ordinary, dualistic consciousness, which is characterized by the distinction between subject and object and is an effect of **m\u0101y\u0101** , or the cosmic illusion produced by **avidy\u0101** , or primordial ignorance of the true nature of reality as **Brahman**. Brahman, according to Advaita Ved\u0101nta, is pure consciousness, or cit: that is, consciousness free from the duality of subject and object. It is the simple reality of awareness, without the idea of a subject or object. Cit is all-pervasive and free from limiting factors. In addition to cit, Brahman is also said to be infinite being ( **sat** ) and unlimited bliss ( **\u0101nanda** ).\n\n**CITTA.** Consciousness; **mind** ; thought; \"mind-stuff,\" that is, consciousness not in the sense of pure awareness ( **cit** ), but the medium of consciousness, or that through which consciousness manifests in and as the world of phenomenal experience. In **Pata\u00f1jali** 's system of **yoga** , citta is made up of intellect ( **buddhi** ), ego ( **aha\u1e43k\u0101ra** ), and the senses ( **indriyas** ) working in concert. It is the fluctuations of citta that yoga is intended to still. In **Ka\u015bm\u012br \u015aaivism** , citta is the individual consciousness\u2014that is, universal consciousness (cit) as limited by and manifested in and as an individual mind.\n\n**CIVAV\u0100KKIYAR (fl. 9th century CE).** One of the **N\u0101yan\u0101r \u015aaiva Tamil** **bhakti** **poets**. Like many of the poet-saints of the bhakti movement, Civav\u0101kkiyar extols the personal encounter with the divine over external worship and ritual. He was particularly critical of **caste** prejudice and the use of **images** **in worship**.\n\n**COCONUT.** This fruit is an extremely important and widely used part of Hindu rituals. A coconut placed in a pot and surrounded by leaves is a symbol of both **Lak\u1e63m\u012b** , **goddess** of wealth, and of fertility. In **p\u016bj\u0101** , or worship, a coconut stands in for the deity. Though a representation of the deity may also be present, it is the coconut that is actually utilized in the ritual. A coconut is often broken before some rituals to ensure divine blessings.\n\n**CO\u1e36A.** Important **Tamil** dynasty that, at its height in the 11th century, ruled all of southern India and **Sri Lanka** , with vassal states in **Southeast Asia** as well. The dynasty rose to ascendancy in the ninth century and was disestablished in the 13th century. This dynasty had a profound impact on Hindu culture in southern India, sponsoring the construction of many prominent **temples**. The bronze sculptures of Hindu deities made in this period are especially prized. Religiously speaking, the Co\u1e37a kings generally favored **\u015aaivism** , though the construction of **Vai\u1e63\u1e47ava** temples was sometimes sponsored under Co\u1e37a rule. Relations with **Jains** and **Buddhists** tended to be strained.\n\n**COMMONER.** Translation of the term _Vai\u015bya_ , which refers to the third of the three **var\u1e47as** or **castes**. In terms of traditional occupation, this caste is made up of trades that involve economically productive activity, such as **arts** and crafts and farming, as well as trade. In the _Puru\u1e63a_ _Sukta_ of the _\u1e5ag Veda_ this caste is said to have been formed from the stomach of the cosmic person. The word _vai\u015bya_ , or \"commoner,\" for this caste also suggests its membership to be relatively large in ancient India.\n\n**CONSCIOUSNESS, STATES OF.** As enumerated in the _Upani\u1e63ads_ , there are four primary states of consciousness: the waking state, the dream state, the state of dreamless sleep, and the state of transcendence. The _M\u0101\u1e47\u1e0dukya Upani\u1e63ad_ , in particular, contains a reflection on these states, connecting them with the various elements that make up **O\u1e43** , the most sacred of **b\u012bja mantras**.\n\nThe significance of states of consciousness in Hindu thought is considerable. In the _Upani\u1e63ads_ and the philosophical **literature** and spiritual practices arising from them, the understanding is that the realization of the identity of the self ( **\u0101tman** ) and **Brahman** (the highest reality) is essential to one's achieving **liberation** (or **mok\u1e63a** ) from the cycle of **rebirth**. This realization is not simply a philosophical doctrine being comprehended in a cognitive sense but a direct awareness arising from reflection on the **scriptures** and from meditative practice that confirms what the scriptures teach: that \u0101tman and Brahman are one. **Meditation** , as outlined in the _Yoga S\u016btras_ of **Pata\u00f1jali** , involves the stilling of the waking, conscious mind so deeper levels of consciousness can be experienced. Beyond the ordinary waking state, which is characterized by phenomena generated by the senses ( **indriyas** ) is the dream state, which is characterized by similar phenomena generated by the mind ( **manas** or **citta** ).\n\nWhen these dream phenomena are also brought into a state of quiescence, one experiences the state of dreamless sleep, where there is only awareness having no object, and the distinction between subject and object does not exist. But one who is asleep does not remember this blissful ( **\u0101nanda** ) and limitless ( **ananta** ) state. Through meditation, one can arrive at this state while awake. Because the distinction between subject and object does not exist in this state, there is no differentiation between Brahman and self. Self (\u0101tman) and Brahman are experienced as being one and the same. This is the fourth state of consciousness ( **tur\u012bya** )\u2014transcendence. One who experiences this state becomes liberated while still alive ( **j\u012bvanmukta** ) and need not experience rebirth after death.\n\n**CONVERSION.** To change one's religious affiliation or identity. Conversion is a topic that generates considerable controversy in contemporary Hinduism, having connotations of the religious exclusivism that many Hindus associate with Abrahamic religions, such as **Christianity** and **Islam**. According to one widespread understanding, Hinduism is not a religion that encourages or requires conversion, either due to the belief that one must be born a Hindu, or because all religions are seen as paths that lead to the ultimate goals of **liberation** and **God-realization** , or a combination of the two. **Mohand\u0101s K. G\u0101ndh\u012b** , for example, is known to have actively discouraged conversions both to and from Hinduism, believing the religion to which each person was born was the most appropriate for that person. Because Christians and Muslims do frequently engage in proselytizing, Hindus\u2014particularly **Hindu Nationalist** authors\u2014sometimes express the fear that Hindu traditions will diminish over time as other religions aggressively expand. Proselytizing\u2014the seeking of converts\u2014has been condemned by many contemporary Hindu leaders, such as **Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b** , as a form of cultural violence. (The reference here is to the founder of **Ar\u1e63a Vidy\u0101 Gurukulam** and the **Hindu Dharma \u0100c\u0101rya Sabh\u0101** , and not the 19th-century founder of the **\u0100rya Sam\u0101j.** )\n\nHistorically, however, it is not the case that Hindus have never sought or accepted converts from other traditions. Conversions between what are now regarded as \"Hindu\" traditions (such as from **Vai\u1e63\u1e47avism** to **\u015aaivism** , or vice versa) are attested, though these would today be seen, with the rise and the increasing use of the category of Hinduism, as intra-Hindu conversions. Conversions in both directions involving other Indic religions, however, such as conversions to and from **Jainism** and **Buddhism** , are also attested very widely in ancient sources, as well as active proselytizing\u2014such as that directed by \u015aaiva enthusiasts at Jains in southern India during the period of the **Co\u1e37a** dynasty.\n\nContrary to the widely held belief that one must be born Hindu, there was no bar, traditionally, to non-Indians being accepted into Hinduism in ancient times, with entire immigrant communities being incorporated _en masse_ into the **caste** system, or to whole kingdoms in **Southeast Asia** systematically adopting Hindu practices and beliefs. And in the modern period, there is the conversion ceremony instituted by the \u0100rya Sam\u0101j by which any person may be inducted into Hindu **dharma**. This last ritual is controversial, however, given how deeply a self-understanding of Hinduism as a nonproselytizing faith has become embedded in the contemporary Hindu consciousness.\n\n**COOMARASWAMY, ANANTA KETISH (1877\u20131947).** Born in **Sri Lanka** , A. K. Coomaraswamy was a renowned **art** historian and scholar of South Asian cultures. From 1917 until his death in 1947 he was a research fellow at the Boston Museum of Fine Arts. He advocated the idea of a **perennial philosophy** expressed by traditional Indic culture.\n\n**COSMOGONY.** Account of the origin and nature of the cosmos. _See also_ CREATION.\n\n**COW, SACRED.** From a non-Hindu perspective, one of the most distinctive of Hindu practices is the veneration of cows. Philosophically, the basis for this veneration is the principle of **ahi\u1e43s\u0101** \u2014harmlessness or nonviolence toward all beings\u2014which, in turn, is based on the idea that divinity is present in all beings: that all beings are manifestations of **Brahman**. From this metaphysical standpoint, there is nothing particularly special about the cow. The gentle reverence displayed toward the cow is simply an example of a universal reverence for life in all its forms.\n\nHistorically, however, Hindus have regarded the cow as a special animal since ancient times. Cows and bulls are both frequently depicted in the **art** of the **Indus Valley civilization** , though the specific significance these animals had is unknown. The **\u0100ryans** , whose culture is recorded in the _Vedas_ , prized cattle greatly and used them as a medium of exchange. The cow was utilized not only for providing milk and milk products (such as butter, **ghee** , and yogurt) but also for its dung (which was used as fuel), its urine (which has medicinal uses), its leather, and\u2014controversially in light of the subsequent adoption of **vegetarianism** by many Hindus\u2014its meat. Bulls were offered in sacrifice to the **Vedic** deities and the meat from these sacrifices was, according to some texts, consumed by the **Brahmins** (although this claim is disputed by some Hindus, who argue, on the basis of subsequent practice, that this is a misreading of these texts).\n\nAs philosophies that emphasized the virtue of nonviolence increasingly informed Hindu practice around the beginning of the Common Era, **animal sacrifice** began to be practiced less and less, gradually being replaced by vegetarian alternatives, such as **p\u016bj\u0101**. The ancient regard in which the cow was held, however, did not diminish. As a mother, the cow symbolizes and embodies the benevolence and the abundance of the Earth. Cows are, in this sense, a manifestation of the **Mother Goddess**. By the period of the _Dharma \u015a\u0101stras_ , or legal texts (c. 300 BCE to 100 CE), deliberately killing a cow was regarded as a grave infraction, punishable by **death**.\n\nCows and bulls play major roles in Hindu religious imagery. **K\u0101madhenu** , the wish-fulfilling cow, is venerated, as is **Nandi** , the bull who is **\u015aiva** 's vehicle.\n\nThe veneration of the cow was given new emphasis in the modern period by Hindu reformers such as **Mohand\u0101s K. G\u0101ndh\u012b** , who saw the practice of nonviolence toward cows as emblematic of a broader Hindu ethos of nonviolence and respect for all life and for the Earth itself. Simultaneously, cow veneration has emerged as a source of tension between Hindus and beef-eating non-Hindu communities.\n\n**CREATION.** There is no single doctrine of creation in Hinduism. Many creation stories are presented in the _Vedas_ and _Pur\u0101\u1e47as_. The predominant worldview is that there is no absolute beginning to existence. There always has been and always will be a universe. It is the case, however, that the universe goes through cycles of emergence, endurance, and dissolution. In Hindu theological discourse, creation stories such as those found in the _\u1e5ag Veda_ are typically read through the lens of this understanding, though it is not clear that these stories were composed with this worldview in mind. Such stories can thus be seen as pertaining to the beginning of a cosmic cycle ( **kalpa** ), or as reflecting an eternal, ongoing process that is occurring at each moment.\n\nProminent **Vedic** creation stories include **Vi\u1e63\u1e47u** 's taking three strides, which form the Earth, the upper atmosphere, and the heavens; the primordial sacrifice of the cosmic man ( **puru\u1e63a** ), from whose body the universe is formed; the more abstract account of **the One** bringing order ( **\u1e5bta** ) out of chaos; and the emergence of the cosmos from the Golden Embryo ( **Hira\u1e47yagarbha** ). Some of these accounts have affinities with **Indo-European** creation stories, such as those found in ancient Greek, Celtic, and Norse mythologies.\n\nThe _Pur\u0101\u1e47as_ elaborate on these Vedic themes and integrate them into a complex picture of creation that is centered on the activities of the creator deity, **Brahm\u0101** \u2014though Brahm\u0101 is frequently subordinated to Vi\u1e63\u1e47u, **\u015aiva** , or the **Mother Goddess** , depending upon the sectarian affiliation of the Pur\u0101\u1e47ic text in question ( **Vai\u1e63\u1e47ava** , **\u015aaiva** , or **\u015a\u0101kta** ). The vast cycles of cosmic emergence, endurance, and dissolution are conceived in this schema as Brahm\u0101's days and nights. _See also_ WORLD AGES.\n\n**CREMATION.** Traditional Hindu method of disposing of corpses. The body is seen as the temporary vehicle of the soul. Once the body can no longer serve its purpose, it is to be discarded. Traditionally, the body is burned on a wooden pyre. But in the modern period, electric crematoriums have also come into common use. _See also_ ANCESTRAL RITES; ANTYE\u1e62\u1e6cI; FUNERAL.\n\n## D\n\n**DAITYA.** Children of **Diti** and the sage **Ka\u015byapa** , and antagonists of their half siblings, the **\u0100dityas** ; a class of demonic beings. _See also_ ASURA.\n\n**DAKHINI.** Demonic female; witch. Dakhinis are powerful beings. They are described as meat-eaters and as attendants of the **Goddess K\u0101l\u012b**.\n\n**DAK\u1e62A.** Ram-headed deity closely associated with the ritual of sacrifice ( **yaj\u00f1a** ); one of the sages or **\u1e5b\u1e63is**. He had 50 daughters, including **Aditi** , **Diti** , and **Sat\u012b**. Both Aditi and Diti were married to the sage **Ka\u015byapa** , making Dak\u1e63a the grandfather of the **\u0100dityas** and the **Daityas**. Sat\u012b was married to **\u015aiva** , whom Dak\u1e63a insulted by failing to invite him to a massive sacrifice to which he had invited all the other sages and deities. Feeling very deeply hurt by this insult to her husband, Sat\u012b immolated herself in the sacrificial fire. It was this, and not the previous snub, that enraged \u015aiva, who destroyed Dak\u1e63a's sacrifice and beheaded him. Later repenting of this act, \u015aiva resurrected Dak\u1e63a and gave him the head of a ram\u2014an act similar to \u015aiva's granting of an elephant head to his son **Ga\u1e47e\u015ba** after beheading him, in one version of the story of Ga\u1e47e\u015ba. _See also_ P\u0100RVAT\u012a.\n\n**DAK\u1e62INA.** South; also the name of the southern altar in a **Vedic** sacrificial space; also the fee paid to a **Brahmin** for performing a ritual; also the right side, as opposed to the left side. When one practices **circumambulation** or **pradak\u1e63ina** , reverently walking in a circle around a sacred image, one keeps the image to one's right. One also gives and receives **pras\u0101dam** , the food offered to a deity in **p\u016bj\u0101** , with one's right hand.\n\n**DAK\u1e62INAM\u016aRTI.** \"South image\"; an image of **\u015aiva** facing to the south, very popular in southern India. The Dak\u1e63inam\u016brti is an image of \u015aiva seated and in a teaching posture, often beneath a tree and surrounded by an adoring audience of sages and animals who are listening to his words.\n\n**DALITS.** \"The oppressed\"; the name by which many **Untouchables** and other persons of low **caste** prefer to refer to themselves. Another term, coined by **Mohand\u0101s K. G\u0101ndh\u012b** , is **Harijan** , or \"Children of God.\" The term _Dalit_ emphasizes the suffering that this group has endured as a result of **casteism**. Both of these terms\u2014Dalit and Harijan\u2014have their own political implications, and use of one or the other seems to map onto the attitude of the person employing it toward Hinduism and the degree to which it is seen to be in need of either reform or complete rejection. This, however, is difficult to verify in a scientific fashion, and one does find persons who use the two terms interchangeably. The formal term used by the government of India is **Scheduled Castes**.\n\n**D\u0100NA.** \"Gift\"; can refer either to a specific act of charitable donation or, more broadly, to the virtue of generosity.\n\n**D\u0100NA-MUDR\u0100.** \"Gesture of generosity\"; **mudr\u0101** , or hand gesture, which symbolizes divine generosity. Hindu deities are frequently depicted with this gesture, which consists of the left hand (or one of the left hands, given that Hindu deities are frequently depicted with more than one pair of arms and hands) being held out with the palm upward and the fingers pointing downward. It is almost the reverse of the **abhaya-mudr\u0101** , or gesture of fearlessness, which consists of a right hand being held out with the palm downward and the fingers pointing upward. The d\u0101na-mudr\u0101 can be seen as representing a deity giving a gift, or reaching out to rescue a drowning devotee from the ocean of misfortune. _See also_ M\u016aRTI.\n\n**D\u0100NAVA.** \"Children of **D\u0101nu**.\" D\u0101nu is a daughter of **Diti**. Much like the **Daityas** \u2014the children of Diti\u2014the D\u0101navas are demonic beings, opposed to the **devas**.\n\n**DANCE.** Dance is an ancient and highly refined **art** form in India, and a popular way of expressing religious devotion. The authoritative guide to Indian classical theater and dance is the _N\u0101\u1e6dya \u015a\u0101stra_ , a **Sanskrit** text ascribed to the sage **Bharata** and composed in the first millennium of the Common Era\n\nFour main systems of classical dance arose early in the Common Era, each one of which constitutes a distinct symbolic vocabulary, in the form of postures, gestures, and facial expressions, for communicating stories from Hindu sacred **literature** and concepts from Hindu **philosophy**. Each of the four systems developed in a different part of India: **Bharata N\u0101\u1e6dyam** in **Tamil** Nadu, **Kathak** in Uttar Pradesh, **Kathakali** in **Kerala** , and **Manipuri** in Manipur.\n\nThough dance has traditionally been highly regarded as a skill, the dancers\u2014who have usually been **women** \u2014unfortunately held a very low status in the centuries between the classical and modern periods. They were often employed as **Devad\u0101s\u012bs**. Although the name means \"Handmaidens of **God** (or of the **Gods** ),\" over time, they were forced, in addition to their other duties, to practice prostitution. As an effect of this development, dance gradually ceased to be seen as a respectable occupation.\n\nThere has been a revival of Indian classical dance in the modern period, though, largely through the efforts of **E. Krishna Iyer**. It is now very common for young Hindu women and girls to study at least one form of dance and to perform in cultural programs held in **temples** or community centers\u2014a pursuit seen not only as respectable but also a mark of pride in the Hindu community. In addition to the four classical forms, a modern style from **Orissa** called **Odissi** is very popular, as are Westernized forms popularized through the **Hindi** film industry.\n\nFinally, there have always been numerous regional folk dance styles that do not require the degree of technical training demanded by the classical styles and which have traditionally accompanied festivals and other community events, such as weddings. Folk dances, too, such as **Garbha** , are highly popular today and are often part of the mixed repertoire of dance styles that can be seen in contemporary Hindu cultural programs. _See also_ GWALIOR; K\u016a\u1e6cIY\u0100\u1e6c\u1e6cAM.\n\n**DANDEKAR, R. N. (1909\u20132001).** **Indologist** and Sanskritist. He was educated first at Deccan College in Pune, India, and then at the University of Heidelberg, Germany. He was a professor of **Sanskrit** from 1933 to 1950 at Fergusson College in Pune, and then at Pune University until 1969, after which he became director of the Bhandarkar Oriental Research Institute.\n\n**DA\u1e46\u1e0cIN (fl. 6th\u20137th centuries CE).** **Sanskrit** author who lived in **K\u0101\u00f1c\u012bpuram** , in southern India. Da\u1e47\u1e0din is known to have written two major works: a collection of short stories entitled the \"Adventures of the Ten Princes\" ( _Da\u015bakum\u0101racarita_ ) and a work on **poetics** , or _k\u0101vya_ , called \"A Look at Poetics\" ( _K\u0101vyadar\u015ba_ ).\n\n**D\u0100NU.** Daughter of **Diti** and **Dak\u1e63a** ; mother of the **D\u0101navas** , a group of demonic beings opposed to the **devas** , or **Vedic** deities.\n\n**DARBHA.** Sacred grass used in **Vedic** rituals and said to have come from the hair of the boar **avat\u0101ra** of **Vi\u1e63\u1e47u** ( **Var\u0101ha** ). Sometimes identified with **ku\u015ba** grass, but sometimes distinguished from it. _See also_ YAJ\u00d1A.\n\n**DARK HALF.** _See_ K\u1e5a\u1e62\u1e46A PAK\u1e62A.\n\n**DAR\u015aANA.** Literally \"view,\" \"vision.\" As a sacred act, dar\u015bana refers to viewing the sacred image ( **m\u016brti** ) of a deity, usually in the context of a **tem** **ple** ( **mandira** ). This act also involves being seen _by_ the image, understood to embody the presence of the deity it represents. To receive dar\u015bana is therefore to receive a blessing from the deity one sees and by whom one is seen.\n\nDar\u015bana also means \"view\" in the sense of \"worldview,\" and in this context refers to a system of philosophy. The Indian dar\u015banas are traditionally divided into two basic categories: **\u0101stika** , or \"orthodox,\" and **n\u0101stika** , or \"heterodox.\" The orthodox dar\u015banas are those who affirm (or at least do not explicitly reject) the authority of the _Vedas_ (and are therefore, by one common definition, \"Hindu\" systems). The nonorthodox dar\u015banas are those who reject the authority of the _Vedas_.\n\nThere are six \u0101stika dar\u015banas, which are often organized into three pairs, due to certain basic affinities that members of each pair have for the other. First, there are the **S\u0101\u1e43khya** and **Yoga** dar\u015banas, which share the same basic worldview, with the notable exception being that S\u0101\u1e43khya is nontheistic and Yoga is not. But the S\u0101\u1e43khya system can be seen as more purely theoretical, whereas Yoga utilizes the main concepts of the S\u0101\u1e43khya system to develop a contemplative practice aimed at the attainment of **mok\u1e63a** ( **liberation** from the cycle of **rebirth** ). Then there are the **Ny\u0101ya** and **Vai\u015be\u1e63ika** systems. Ny\u0101ya is concerned primarily with logic and the valid means of **knowledge**. Vai\u015be\u1e63ika, much like Yoga in relation to S\u0101\u1e43khya, applies the logic of Ny\u0101ya to the analysis of the phenomenal world. **P\u016brva** (or prior) **M\u012bm\u0101\u1e43s\u0101** and **Uttara** (or later) **M\u012bm\u0101\u1e43s\u0101** \u2014also known as **Ved\u0101nta** \u2014focus upon the interpretation, respectively, of the earlier and later Vedic texts.\n\nThe n\u0101stika dar\u015banas are **Jainism** , **Buddhism** , and the **materialist Lok\u0101yata** (or **C\u0101rv\u0101ka** ) system. Although the term _n\u0101stika_ gradually comes to refer to an atheist, this is not its original sense, since some of the \u0101stika dar\u015banas are also nontheistic, or at least not explicitly theistic\u2014namely, S\u0101\u1e43khya and P\u016brva M\u012bm\u0101\u1e43s\u0101. All three of the classical n\u0101stika dar\u015banas are also nontheistic, but only the Lok\u0101yata system denies the concepts of **karma** and rebirth.\n\nEach dar\u015bana is based on a collection of root texts, or **s\u016btras** , which summarize the teaching of the sage believed to be the founder of the system in question. According to the traditions of the various dar\u015banas, most of their founding figures lived around the fifth century BCE, though some (such as **K\u0101pila** , the founder of the **S\u0101\u1e43khya** system) lived before that time, while others (such as **Pata\u00f1jali** , the founder of the **Yoga** system) almost certainly lived later. In most cases, the extant s\u016btras are likely the result of many generations of compilation, rather than the work of a single author. Philosophical writing in a dar\u015bana tradition largely occurs through the genre of the **bh\u0101\u1e63ya** , or commentary on the root text, the understanding being that all relevant knowledge is already present in at least an implicit form in the root text. What is required, then, is not innovation, in the sense of discovery of completely new ideas, but an unpacking of the deep **truths** contained in the words of the sages of the past.\n\n**D\u0100SA, DASYU.** Barbarian; also slave, servant; terms used for non- **\u0100ryan** tribes in the _\u1e5ag Veda_ who are regarded as opponents of the \u0100ryans; sometimes used interchangeably with the term **asura** , or demonic being. **Indra** is described as destroying the forts of the Dasyus. They are described as city-builders and as having magical powers (like the asura **Maya** in the _Mah\u0101bh\u0101rata_ ). During the 19th and 20th centuries, **Indologists** interpreted the D\u0101sas and Dasyus as indigenous Indians being invaded by the \u0100ryans.\n\n**DA\u015aAHR\u0100 (DUSSEHRA).** Ten-day festival celebrated in the **bright** **half** of the month of **\u0100\u015bvina**. In northern India, this festival celebrates **R\u0101ma** 's victory over **R\u0101va\u1e47a** and it culminates in **D\u012bp\u0101vali** (or D\u012bwali), the festival of lights in honor of the **goddess Lak\u1e63\u1e43\u012b**. In **Bengal** and southern India, this festival celebrates **Durg\u0101** 's victory over the Buffalo Demon ( **Mahi\u1e63\u0101sura** ). In Bengal, it culminates in **K\u0101l\u012b p\u016bj\u0101**.\n\n**DA\u015aAN\u0100MI ORDER.** Order of the 10 Names; order of Hindu monks established by **\u015aa\u1e45kara** (788\u2013820 CE). \u015aa\u1e45kara established four monasteries. **\u015a\u1e5b\u1e45ger\u012b** , in the south, is the head monastery. **Puri** is in **Orissa** , in the east. **Badr\u012bn\u0101th** is in the foothills of the **Him\u0101layas** , in the north. **Dv\u0101rak\u0101** is in Gujarat, in the west. \u015aa\u1e45kara's monasteries are therefore located in the four corners of India corresponding to each of the four points of the compass. The head of each of these monasteries, or **ma\u1e6dhas** , is called **\u015aa\u1e45kar\u0101c\u0101rya**. \u015aa\u1e45kara is often called _\u0100di \u015aa\u1e45kar\u0101c\u0101rya_ , or \"First \u015aa\u1e45kar\u0101c\u0101rya\" in order to distinguish him from subsequent holders of this title. Anyone, regardless of **caste** or gender, may join the Da\u015ban\u0101mi Order. The order is generally associated with the **Sm\u0101rta** tradition of Hinduism and the **Advaita** interpretation of **Ved\u0101nta** , though neither of these is rigidly upheld in practice, with members having considerable freedom in regard to theological issues. The ascetics of the **Ramakrishna Order** are members of the Da\u015ban\u0101mi Order.\n\n**DA\u015aARATHA.** Character in the _R\u0101m\u0101ya\u1e47a_ ; father of **R\u0101ma** ; king of **Ayodhy\u0101**. When hunting in the forest, Da\u015baratha accidentally slays the son of a **Brahmin** and is cursed to also lose his son. This occurs when Da\u015baratha is forced to send R\u0101ma into exile, an event that causes Da\u015baratha to die of sorrow.\n\n**DASGUPTA, SURENDRAN\u0100TH (1887\u20131952).** **Bengali** philosopher and **Indologist** ; lecturer in **philosophy** at Cambridge University in 1922 and professor of philosophy at Calcutta University until 1945. He is best known for his five-volume history of Indian philosophy.\n\n**DATTA, NARENDRAN\u0100TH (1863\u20131902).** The birth name of **Sw\u0101m\u012b Vivek\u0101nanda**.\n\n**DAYAL, HAR (1884\u20131939).** Founder of the **Ghad\u1e5b movement** , which advocated for the violent overthrow of British rule in India. A **Hindu Nationalist** , Dayal was a scholar who served on the faculty of Stanford University in California from 1911 until 1914, at which time he was arrested by U.S. immigration authorities due to his political activities. While on bail, he fled to Germany, where he resided from 1914 to 1918. After the end of World War I, he returned to India and joined **Mohand\u0101s K. G\u0101ndh\u012b** 's nonviolent campaign for Indian independence. He completed his doctoral degree at the University of London in 1930 and died in Philadelphia in 1939.\n\n**DAY\u0100NANDA SARASVAT\u012a, SW\u0100M\u012a (1824\u20131883).** Founder of the **\u0100rya Sam\u0101j** , an important Hindu reform organization of the modern period. Born in Gujarat, in western India, to a **\u015aaiva** family. It is said that he began to question the use of images ( **m\u016brtis** ) in worship when, as a child, he kept a night vigil in a **\u015aiva** **temple** and saw mice eating the food offered to the deity. He left home to undertake renunciation ( **sanny\u0101sa** ) at the age of 22, when his parents wanted to arrange a marriage for him. Wandering eventually to **Mathura** , Day\u0101nanda studied the _Vedas_ under his **guru** , Sw\u0101m\u012b Virj\u0101nanda.\n\nDay\u0101nanda interpreted the _Vedas_ as teaching a **monotheistic** doctrine and, much like **Ram Mohan Roy** , he did not find any Vedic warrant for later Hindu practices in texts such as the _Pur\u0101\u1e47as_ and sought to eliminate these later practices. He established the \u0100rya Sam\u0101j in order to promote a \"purified\" Vedic Hinduism with no use of images in worship and no **caste** prejudice. A **Hindu Nationalist** , Day\u0101nanda was highly critical of **Christianity** and **Islam** , especially for the proselytizing activities of both Christian and Islamic missionaries in India. In addition to reforming Hinduism, another purpose of the \u0100rya Sam\u0101j was to neutralize missionary activities by these traditions, and a ritual of **conversion** was established to enable those who wished to become Hindu to do so. He found common cause with **Helena Petrovna Blavatsky** and **Henry Steele Olcott** of the **Theosophical Society** , who shared his vision of reviving ancient Indian traditions for the modern era. But he parted company with Blavatsky and Olcott when he heard that they had both undertaken a formal conversion to **Buddhism** in **Sri Lanka**.\n\n**DAY\u0100NANDA SARASVAT\u012a, SW\u0100M\u012a (1930\u2013 ).** Not to be confused with the founder of the **\u0100rya Sam\u0101j** , the contemporary Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b is the founder and the current head of the **Hindu Dharma \u0100c\u0101rya Sabh\u0101** (or \"Assembly of Hindu Religious Leaders\") and the founder of the **Ar\u1e63a Vidy\u0101 Gurukulam** in Saylorsburg, Pennsylvania. He is an adherent of the **Advaita** system of **Ved\u0101nta** in the tradition of **\u015aa\u1e45kara**.\n\nThe Hindu Dharma \u0100c\u0101rya Sabh\u0101 has a membership of over 100 **\u0101c\u0101ryas** , or leaders of **sa\u1e43prad\u0101yas** (Hindu religious traditions) and is an attempt to bring greater institutional cohesion to Hinduism in order to advance a number of interests and concerns shared across many sa\u1e43prad\u0101yas. Particularly noteworthy **actions** of the Hindu Dharma \u0100c\u0101rya Sabh\u0101 include a 2008 summit between the Sabh\u0101 and the chief rabbis of Israel in order to facilitate dialogue between Hinduism and Judaism and numerous proclamations that condemn proselytizing as a form of cultural violence against indigenous religions.\n\nAr\u1e63a Vidy\u0101 Gurukulam is an educational organization dedicated to the teaching of **Sanskrit** and Advaita **philosophy**.\n\n**DE, ABHAY CHARAN (1896\u20131977).** The birth name of A. C. **Bhaktived\u0101nta Sw\u0101m\u012b Prabhup\u0101da**.\n\n**DE, S. K. (1890\u20131968).** **Indologist** ; born in **Calcutta** ; professor of English, **Sanskrit** , and Sanskrit **literature** at Calcutta University.\n\n**DEATH.** In most Hindu traditions, death is seen as a transition from one lifetime to the next in the beginningless cycle of **rebirth** (a cycle known as **sa\u1e43s\u0101ra** ). The _Bhagavad G\u012bt\u0101_ (2:22) compares the transition from one body to another to the discarding of old, worn-out clothes for a new set of clothes, and **Ramakrishna** , in a vision to his wife, **Sarad\u0101 Dev\u012b** , is said to have proclaimed that, in dying, he had merely moved \"from one room to another.\" **Mohand\u0101s K. G\u0101ndh\u012b** described birth and death as two different sides of the same door. The nature of one's next life is determined by the quality of one's **karma** , the accumulated effects of one's moral **actions** , at the time of one's death. The ultimate goal of most Hindu systems is to achieve **liberation** from rebirth ( **mok\u1e63a** ).\n\nReassuring though such a concept may be, however, Hindus, like the adherents of all other religions, experience profound grief at the passing of a loved one, and a variety of rituals exist for the purpose of assuaging this grief. The **antye\u1e63\u1e6di** , or **Vedic** funeral, is conducted as soon as possible after death, and includes the **cremation** of the body of the deceased. The particulars of the ritual vary a great deal, depending on region and **caste** , but it often extends over the course of several days. There are also **ancestral rites** , such as the **\u015br\u0101ddha** , which occurs periodically and consists of making offerings of food and water to one's departed ancestors. These offerings are believed to become the food that one's ancestors consume in their next life.\n\nIt is not known precisely when or how the doctrine of rebirth arose in Hinduism. The first unambiguous reference to this process is found in the _B\u1e5bhad\u0101ra\u1e47yaka Upani\u1e63ad_ (4.4.5\u20137), which was probably compiled around 500 BCE but likely contains material that had been handed down orally for many generations before that time. References to the afterlife in earlier Vedic texts suggest a shadowy afterworld, not unlike the ancient Greek Hades, and other **Indo-European** conceptions of the afterlife (though other Indo-European cultures, such as the that of the Celts, also developed a rebirth concept like that of Hinduism).\n\nA distinction is also drawn between two afterlife destinations called the \"Realm of the Fathers\" ( **pit\u1e5bloka** ) and the \"Realm of the Gods\" ( **devaloka** ). As the idea of rebirth develops in the later Vedic **literature** , the realm of the fathers comes to be identified with eventual rebirth in the physical world, whereas the realm of the gods is interpreted as a path leading to eventual freedom from rebirth. Some scholars speculate that the origins of the ideas of karma and rebirth can be found in Vedic rituals, such as the **Agni\u1e63\u1e6doma** , which has the goal of leading to a rebirth in heaven ( **svarga** ). If the ritual actions, or _karmans_ , are performed correctly, they lead to the desired goal; but if they are performed incorrectly, they can fail to bear fruit, or even bring about the opposite of the desired goal.\n\nThe theory is that this concept of correctly or incorrectly performed ritual action\u2014\"good\" or \"bad\" karmans leading to a correspondingly good or bad result\u2014develops into the idea of ethically good or bad actions in life leading to similar, corresponding results in the world. The Vedic sacrificial arena essentially becomes extended to encompass all of life. Another theory is that the idea of rebirth was foreign to the Vedic traditions and was introduced through interaction with **\u015brama\u1e47a** traditions, such as **Buddhism** and **Jainism** , which either arose from other Indo-European traditions, distinct from the Vedic, or represent an ancient indigenous Indian tradition, perhaps an ideology of the **Indus Valley civilization**. But this is all quite speculative.\n\n**DELHI, NEW DELHI.** Capital of the modern Republic of India. The region has been settled since ancient times and is believed to be the site of the **P\u0101\u1e47\u1e0davas** ' ancient city of **Indraprastha**. The city was the capital of the Turkish Delhi sultanate, which ruled most of northern India from 1206 to 1526, after which it was incorporated into the **Mughal Empire**. Delhi served as the Mughal capital from 1649 to 1857, after which the empire was formally dissolved and India came under the direct control of the British crown. The British administration was based in **Calcutta** until 1911, when it was moved to Delhi. New Delhi, the southern part of the greater metropolitan area, was mostly built in the 1920s. When India became independent in 1947, New Delhi remained its capital.\n\n**DEMON, DEMONIC.** Word frequently used to translate the terms _asura_ and _r\u0101k\u1e63asa_. The term is both accurate and inaccurate. It is accurate in the basic sense that asuras and r\u0101k\u1e63asas are generally seen in Hindu sacred **literature** as evil beings, and as opponents of the **gods** ( **devas** ). But there is no figure of ultimate evil in Hinduism comparable to Satan and his demonic minions. Asuras and r\u0101k\u1e63asas are sometimes encountered in Hindu texts that behave in a good and noble fashion, such as the good r\u0101k\u1e63asa **Vibh\u012b\u1e63ana** , from the _R\u0101m\u0101ya\u1e47a_ , who is a devotee of **R\u0101ma** and joins him in his war against **R\u0101va\u1e47a**. There are also times when the devas do not behave virtuously in relation to the asuras, such as when the devas trick the asuras out of their share of the elixir of life ( **am\u1e5bta** ) after getting their cooperation in the churning of the ocean. At the same time, there are resonances between the war in heaven between angels and demons depicted in Abrahamic religions such as **Christianity** and **Islam** and the conflict between the devas and asuras, the likely historical connection between the two being **Zoroastrianism**.\n\n**DESIRE.** According to the _B\u1e5bhad\u0101ra\u1e47yaka Upani\u1e63ad_ (4.4.5\u20137) and subsequent systems of Hindu **philosophy** \u2014particularly **Ved\u0101nta** \u2014desire is a primary cause of bondage to the cycle of **rebirth** , which must be relinquished in order for **liberation** ( **mok\u1e63a** ) to occur. The idea is that it is not merely **action** , or **karma** , as such that leads one to be reborn, but rather action that has been invested with desire. This is why the _Bhagavad G\u012bt\u0101_ (3:4\u20138) recommends detachment from the fruits of action as a path to liberation (the idea being that a life of complete inaction is impossible). Desire as the fundamental cause of the cycle of rebirth, and thus of suffering, is also a major theme of **Buddhism** and the basis for the emphasis on **asceticism** in many Hindu traditions.\n\nBecause the **Vedic** ritual of sacrifice ( **yaj\u00f1a** ) is enjoined precisely so that one may fulfill one's desires, such rituals are rejected by the **\u015brama\u1e47a** traditions and by Ved\u0101ntic texts such as the _Bhagavad G\u012bt\u0101_ (2:42\u201344) as conducive to karmic bondage (as well as for the fact that such rituals, especially if they involve **animal sacrifice** , violate the principle of **ahi\u1e43s\u0101** , or nonviolence).\n\n**DEVA.** Deity, god, literally \"shining one.\" The earliest extant references to devas are from the _\u1e5ag Veda_. The **Vedic** devas are usually associated with natural phenomena, such as **S\u016brya** the sun god, **U\u015b\u0101** the goddess of the dawn, **Soma** the god of the **moon** and the hallucinogenic plant of the same name, **V\u0101yu** the god of the wind, **Yama** the god of **death** , and so on. The lord of the Vedic devas is **Indra** , who is associated with thunder and is analogous to the Greek Zeus and the Norse Thor. Also of particular importance is **Agni** , the god of fire. Agni's special importance derives from his role as an intermediary between the human and divine worlds, due to the fact that all offerings to the devas are made through the medium of the sacred fire, which Agni personifies. Indra and Agni, respectively, are the most and second-most frequent objects of adoration in the hymns of the _\u1e5ag Veda_. Also of considerable importance is **Praj\u0101pati** , whose name literally means the \"father of offspring\" and who is regarded as the creator of the universe\u2014later known as **Brahm\u0101**. The two chief deities of later Hinduism, **Vi\u1e63\u1e47u** and **\u015aiva** , receive relatively little mention in the _Vedas_ (the latter typically under the name **Rudra** ).\n\nEarly in the Common Era, as the various systems of Hindu theology develop in a **monotheistic** direction (primarily centered on Vi\u1e63\u1e47u or \u015aiva as the supreme deity\u2014though **\u015aakti** and **Gane\u015ba** also have their followings in this regard), the Vedic devas come to be seen as lesser deities, analogous to the angels or angelic forces found in the Abrahamic religions, or as aspects, emanations, or powers of \"higher\" deities. Perpetually opposed to the devas are the demonic forces\u2014the **asuras** , who are depicted as being in a state of constant warfare with the devas.\n\nIn addition to this meaning, though, the term _deva_ also becomes a generic term for any deity, including the supreme deity. So \u015aiva, for example, is known by the epithet **Mah\u0101deva** or \"Great Deva,\" even by **\u015aaiva** traditions that regard him monotheistically as the Supreme Being, and not as a \"mere\" deva or demigod. Similarly, the feminine **Dev\u012b** , a term that refers originally to any female deva, becomes a proper name for the supreme **goddess** of the **\u015a\u0101kta** traditions, as does **Mah\u0101dev\u012b** (Great Goddess).\n\n**DEVAD\u0100S\u012a.** \"Handmaiden of **God** (or of the **Gods** )\"; **woman** employed as a dancer in a **temple** and is technically married to the deity of that temple. In the medieval period, Devad\u0101s\u012bs began to be exploited as prostitutes, bringing disrepute upon the **art** of **dance** , in which they were quite skilled. Indian classical dance has seen a revival in the modern period as the practice of forced prostitution has ceased to receive either religious or legal sanction.\n\n**DEVAK\u012a.** Mother of **B\u0101lar\u0101ma** and **K\u1e5b\u1e63\u1e47a**.\n\n**DEVALOKA.** Realm of the **gods** ; one of two possible afterlife destinations according to early **Vedic** thought, the other being the **Pit\u1e5bloka** , or realm of the ancestors. _See also_ DEATH.\n\n**DEV\u012a.** **Goddess** ; common epithet of the **Mother Goddess**.\n\n_DEV\u012a MAH\u0100TMYA_ **.** \"Glorification of the Great **Goddess** \"; portion of the _Marka\u1e47\u1e0deya Pur\u0101\u1e47a_ (composed c. 250\u2013550 CE) that describes the victories of the **Mother Goddess** over various **asuras**. Also known as the _Ca\u1e47\u1e0d\u012bpa\u1e6dha_ , it is recited during **Durg\u0101 P\u016bj\u0101**.\n\n**DEVOTION.** _See_ BHAKTI.\n\n**DHARMA.** Literally, \"support.\" However, this important term has a range of meanings that include the order of the universe, the order of society, natural law, justice, religion, and personal duty. Contemporary Hindus often refer to Hinduism as _Hindu Dharma_ or **San\u0101tana Dharma**. In this context, _dharma_ refers to the Hindu way of life in its entirety and is often presented as a deliberate contrast with _religion_ , dharma being seen as a more holistic and all-encompassing concept than religion, which has become a more privatized in modernity than it was in earlier periods of history. _San\u0101tana_ means \"eternal,\" and the implication of calling Hinduism _San\u0101tana Dharma_ is that it reflects an unchanging and universal order of existence\u2014in contrast with the Abrahamic religions, which are seen as being more localized and tied to history.\n\nIn the _\u1e5ag Veda_ , the preferred term for the universal order to which _dharma_ refers is _\u1e5bta_ , and regular performance of **Vedic** ritual is seen as essential to its maintenance. In the _Dharma \u015a\u0101stras_ and the epics (the _R\u0101m\u0101ya\u1e47a_ and _Mah\u0101bh\u0101rata_ , composed around the beginning of the Common Era), the term _dharma_ becomes preferred. With this shift in terminology, there is also a shift in emphasis from maintaining the cosmos through right ritual **action** to maintaining it by fulfilling social obligations. This idea of dharma as _duty_ remains prevalent today.\n\nIn the _Dharma \u015a\u0101stras_ , dharma is largely identified with the duties connected to the system of **var\u1e47as** (or **castes** ) and **\u0101\u015bramas** (stages of life). Each var\u1e47a (and the **j\u0101tis** that make up each var\u1e47a) and each \u0101\u015brama, of course, has its own distinct duties attached to it. Closely tied, therefore, with dharma is the concept of **svadharma** , or one's own dharma, which is a function of one's location in the caste system and the stage of life that one currently occupies.\n\nThe idea of svadharma adds a layer of complexity to dharma, as an ethical ideal, that does not necessarily obtain in more universalist, less particularized ethical systems. Understanding dharma is not simply a matter of learning a set of abstract rules that apply to everyone but of knowing the duties that apply to one's own specific circumstances. The _Dharma \u015a\u0101stras_ consist largely of an enumeration of duties associated with each caste and stage of life.\n\nUniversal moral norms, however, are also an important part of dharma, and are invoked in the _Dharma \u015a\u0101stras_. These norms correlate closely with the **yamas** , or moral restraints, enumerated in the _Yoga S\u016btras_ of **Pata\u00f1jali** : **ahi\u1e43s\u0101** (nonviolence), **satya** (truthfulness), **asteya** (nonstealing), **brahmacarya** (sexual restraint, chastity), and **aparigraha** (nonattachment). These same norms are shared by the **\u015brama\u1e47a** traditions, **Jainism** and **Buddhism**. The _Dharma \u015a\u0101stras_ invoke these norms in dealing with moral dilemmas and difficult questions that arise in the course of applying the idea of dharma to concrete situations, such as conflicts between competing duties.\n\nAs the caste system has come under increasing critical scrutiny during the modern period, even being rejected by many Hindu thinkers, the concept of dharma has continued to evolve. Reflection on dharma in contemporary Hinduism conceives of dharma less as a matter of inheritance\u2014of birth caste\u2014and more as a kind of personal destiny or calling in life that one must discern over the course of one's lifetime. As the discourse of human rights has entered Hinduism, dharma has also been rethought in terms of a responsibility that all human beings have both to one another and to the larger environment.\n\nFinally, dharma is one of the four goals of humanity, or **puru\u1e63\u0101rthas** of classical Hindu thought, along with **k\u0101ma** (pleasure), **\u0101rtha** (wealth), and **mok\u1e63a** ( **liberation** from the cycle of **rebirth** ).\n\n_DHARMA \u015a\u0100STRAS_ **.** Authoritative texts on **dharma** ; Hindu law books. Many of these texts exist, including the _Manusm\u1e5bti_ , the _Apastamba Dharma \u015a\u0101stra_ , the _Y\u0101j\u00f1avalkya Dharma \u015a\u0101stra_ , and others. They were composed roughly between 300 and 100 BCE.\n\n**DH\u0100T\u1e5a.** Early **Vedic** deity, one of the 12 **\u0100dityas** \u2014sons of the goddess **Aditi** and the sage **Ka\u015byapa**. His name means both \"order\" and \"sustainer,\" both of which connect him with the concept of **dharma**.\n\n**DH\u1e5aTAR\u0100\u1e62\u1e6cRA.** Character in the _Mah\u0101bh\u0101rata_ ; blind king; brother of **Pa\u1e47\u1e0du** ; father of **Duryodhana** , the chief villain of the epic, and the other 99 **Kauravas** ; uncle of the **P\u0101\u1e47\u1e0davas** ; husband of **G\u0101ndh\u0101r\u012b**. Dh\u1e5btar\u0101\u1e63\u1e6dra is often criticized by commentators for being an overindulgent father and failing to reprimand Duryodhana's excessive hatred for and many plots against the P\u0101\u1e47\u1e0davas.\n\nDh\u1e5btar\u0101\u1e63\u1e6dra speaks the first verse of the _Bhagavad G\u012bt\u0101_ when he asks **Sa\u00f1jaya** , his clairvoyant minister, to describe for him the events of the battle of **Kuruk\u1e63etra**. He is also mentioned briefly in the _B\u1e5bhad\u0101ra\u1e47yaka Upani\u1e63ad_ , an independent reference to this character suggestive of his possible historicity.\n\n**DHY\u0100NA.** **Meditation** ; seventh of the eight stages or \"limbs\" of **Pata\u00f1jali** 's eight-limbed ( **a\u1e63\u1e6d\u0101\u1e45ga** ) system of **yoga**. The sixth limb, **dhara\u1e47\u0101** , is concentration on a single object. **Dhy\u0101na** is absorption in that object, with no awareness of any other phenomena.\n\nDhy\u0101na involves the stilling of the waking, conscious **mind** so that deeper levels of consciousness can be experienced. The goal of dhy\u0101na is **sam\u0101dhi** \u2014the eighth limb of Pata\u00f1jali's system\u2014which is complete absorption in the object of concentration, in which the distinction between subject and object vanishes.\n\nThe practice of dhy\u0101na is shared by many of the Indic traditions and is especially prominent in **Buddhism**. In early Buddhism, the term _dhy\u0101na_ also refers to the **states of consciousness** to which the practice of dhy\u0101na can lead. Interestingly, when Buddhism was transmitted to China, the term dhy\u0101na was rendered as _ch'an_. When the tradition was transmitted to Japan, the term _ch'an_ was transformed into _zen_.\n\n**D\u012aK\u1e62\u0100.** Formal initiation into a spiritual discipline; performed by one's spiritual teacher or **guru**. D\u012bk\u1e63\u0101 often involves the transmission of a secret **mantra** that the practitioner will use in **meditation** and not disclose to any other person. The mantra will possess the **guru-\u015bakti** , or power of the spiritual lineage through which it has been transmitted.\n\n**D\u012aP\u0100VALI (D\u012aWALI).** Festival of Lights; this holiday commemorates **R\u0101ma** 's victory over **R\u0101va\u1e47a**. It is also a day that is especially sacred to **Lak\u1e63m\u012b** , wife of **Vi\u1e63\u1e47u** and **goddess** of wealth and prosperity. The holiday is celebrated with the lighting of **d\u012by\u0101s** , or oil lamps, as well as, in the modern period, fireworks and strings of colored electric lights similar to Christmas lights. The festival occurs on the **Amavasya** , or new-moon day, of the month of **K\u0101rtika** (late October to early November). In northern India, this date also marks the last day of the year. Those who wish to have Lak\u1e63m\u012b's blessings for the New Year leave a lamp burning outside of their home all night long (or an electric light). In **Bengal** , **K\u0101l\u012b p\u016bj\u0101** is celebrated on the night of D\u012bwali. In October 2007, after lobbying by the **Hindu American Foundation** , the U.S. Congress passed a resolution in recognition of the religious and historical significance of D\u012bwali.\n\nIn **Jainism** , D\u012bwali is celebrated as **Mah\u0101v\u012bra** Nirv\u0101\u1e47a Div\u0101sa, the date on which the sage **Mah\u0101v\u012bra** attained his final **nirv\u0101\u1e47a** and left his body. It is said that the king of the region where Mah\u0101v\u012bra died proclaimed that lamps should be lit on that night because such a great light had gone out of the world.\n\n**DITI.** Sister of **Aditi** , daughter of **Dak\u1e63a** , and mother of the **Daityas** , a group of beings opposed to the **Vedic** **devas** (deities) and generally regarded as demonic.\n\n**DIVINE LIGHT MISSION.** Hindu organization established by Sri H\u0101\u1e43s\u0101 Mah\u0101r\u0101j Ji in the early 1960s and taken over after his death in 1966 by his son, **Guru Mah\u0101r\u0101j Ji** , who was only nine years old. Its basic theology is derived from the **Vallabha Vai\u1e63\u1e47ava** tradition. It was renamed \"Elan Vital\" in 1983.\n\n**DIVINE MOTHER.** An alternative designation for the **Mother Goddess**.\n\n**D\u012aWALI.** _See_ D\u012aP\u0100VALI.\n\n**D\u012aY\u0100.** Oil lamp used in many Hindu rituals, such as **p\u016bj\u0101** and **\u0101rat\u012b** , and on a very large scale during **D\u012bp\u0101vali** , the festival of lights.\n\n**DOMESTIC RITUALS.** A variety of Hindu rituals can be performed at home, and in fact it is probably accurate to say that the home is a more central location than the **temple** for Hindu religious activity. The typical Hindu home will have a room, or at least a small space, dedicated exclusively to religious activity: a shrine or altar before which the daily **p\u016bj\u0101** and **meditation** can be performed. The most common domestic ritual is a morning p\u016bj\u0101 performed as soon as one has risen from bed and bathed. The ideal time for such a p\u016bj\u0101 is early in the morning, just before dawn, during the **Br\u0101hma-muhurta**. **Brahmins** will often recite the **G\u0101yatr\u012b mantra** at this time or as the sun is rising. Some Hindus also perform an evening p\u016bj\u0101, at sundown. Other p\u016bj\u0101s frequently held at home include the **Satyanar\u0101ya\u1e47a P\u016bj\u0101** , performed every full-moon day ( **pur\u1e47ima** ), and a weekly p\u016bj\u0101 to **Lak\u1e63m\u012b** , which is usually performed by the **women** in the home. More elaborate and large-scale rituals performed at home typically require one to employ a priest\u2014such as the **G\u1e5bha-prave\u015ba** , or \"entering the home,\" a ceremony for blessing a new house. Detailed instructions for the performance of domestic rituals are contained in a set of late **Vedic** texts called the _G\u1e5bhya S\u016btras_ , which were likely composed around 300 BCE.\n\n**DONIGER, WENDY (1940\u2013 ). Indologist** and Sanskritist who has been a professor at the University of Chicago since 1978. Doniger's title is **Mircea Eliade** Distinguished Service Professor of the History of Religions. She is located jointly in the University of Chicago's Divinity School, its Department of South Asian Languages and Civilizations, and its Committee on Social Thought. She has two doctoral degrees\u2014one from Harvard University (1968) and one from Oxford University (1973). Much of Doniger's work has focused upon the interpretation of Hindu sacred **literature** , though she has also done a good deal of translation of **Sanskrit** texts\u2014most notably of selected hymns from the _\u1e5ag Veda_ and of the _Manusm\u1e5bti_. Her interpretations of Hindu texts have proven controversial due to her application of Freudian psychoanalytic theories. Many in the Hindu community have found her work and that of her students who have followed a similar line of interpretation to be objectionable\u2014most notably **Rajiv Malhotra** , who has accused Doniger and her students of neocolonialism. However, much of this same work has met with great scholarly acclaim, making Doniger a polarizing figure.\n\n**DRAUPAD\u012a.** Character in the _Mah\u0101bh\u0101rata_ ; wife of the five **P\u0101\u1e47\u1e0dava** brothers. In one of the more unusual scenes in the epic, though it is **Arjuna** who wins Draupad\u012b's hand in marriage, his mother, **Kunt\u012b** , requires him to share Draupad\u012b with his four brothers. (She had told him, \"Whatever you have won, you must share with your brothers,\" before she had seen that Arjuna had won a bride, and a mother's word must be obeyed.) In another account, Draupad\u012b's situation is depicted as the result of a wish she made in a past life, in which she prayed five times for a husband.\n\nDraupad\u012b is central to the main plotline of the _Mah\u0101bh\u0101rata_. When **Yudhi\u1e63\u1e6dhira** gambles away his own freedom and that of his brothers, losing all of his belongings to **Duryodhana** , he stakes Draupad\u012b to win everything back, losing her in the process as well. Draupad\u012b challenges this outcome, arguing that Yudhi\u1e63\u1e6dhira had no right to gamble away his wife, as though she were a material possession, particularly given that he had already lost himself when he staked her. When none of the assembled elders will rise to her defense, due to their loyalty to Duryodhana's father, who is presiding over the entire proceedings, Duryodhana, his brothers\u2014the **Kauravas** \u2014and **Kar\u1e47a** try to molest her. When Duryodhana's brother Du\u1e25\u015b\u0101sana tries to strip off her clothing, Draupad\u012b takes refuge in **K\u1e5b\u1e63\u1e47a** , who performs a miracle that consists of covering her with an infinite cloth that cannot be removed. This supernatural event frightens the Kauravas; their father **Dh\u1e5btar\u0101\u1e63\u1e6dra** , who has been silent up to this point, grants her any boon that she wishes to request. She asks for the freedom of her husbands and that all their belongings be returned. Dh\u1e5btar\u0101\u1e63\u1e6dra grants this and Draupad\u012b's own freedom as well. She thus saves her husbands when they were helpless to save either her or themselves. The insult that she suffers at the hands of the Kauravas, however, heightens the tensions between the two families, and both Draupad\u012b and **Bh\u012bma** swear vengeance. This vengeance is achieved at the battle of **Kuruk\u1e63etra** , at which all of the Kauravas are killed. Du\u1e25\u015b\u0101sana is killed by Bh\u012bma and Draupad\u012b washes her hair in his blood. Duryodhana dies by Bh\u012bma's hand as well.\n\nIn southern India, Draupad\u012b is regarded as an incarnation of the **Mother Goddess** and is worshiped as such. The famous image of Hindus walking across hot coals is part of her worship, where devotees vow to perform this act if she will answer their prayers.\n\nCommentators often contrast the strong, fierce, and outspoken Draupad\u012b with the relatively meek and submissive **S\u012bt\u0101** , the analogous chief female protagonist of the other major Hindu epic, the _R\u0101m\u0101ya\u1e47a_. Some contemporary interpreters see in Draupad\u012b a critique of traditional Hindu patriarchy and as expressing the possibility of an indigenous Hindu feminism based on the concept of **dharma**. Her successful invocation of K\u1e5b\u1e63na also represents the power of **bhakti**. _See also_ WOMEN.\n\n**DRAVIDIAN.** Language family predominant in the southern half of India that includes Kannada, Malayalam, **Tamil** , and Telugu; distinct from the **Indo-European** languages that predominate in northern India (such as **Hindi** and **Bengali)** , all of which are closely related to **Sanskrit**. In keeping with the conflation of culture and ethnicity introduced by 19th-century **Indologists** \u2014who spoke not only of Indo-European culture but also of an Indo-European or **\u0100ryan** \"race\"\u2014the term _Dravidian_ has come to refer not only to the languages of southern India but to the people who speak those languages as well. This has led to a northern and southern political division that is based on the **\u0100ryan Invasion Theory** , with its implication that northern Indian languages (and the people who spoke them) invaded and ousted an indigenous Dravidian population some time prior to 1500 BCE. This division has most often been played out in the form of controversies over the Indian government's promotion of Hindi in southern India. Indeed, English is preferred in southern India over Hindi as a medium of communication.\n\n**DREAM.** The dream state is the second of the four **states of consciousness** enumerated in the _Upani\u1e63ads_ and in **Ved\u0101nta** **philosophy**. The dream state ( _taijas\u0101_ , or \"brilliant\") is differentiated from the waking state ( _vai\u015bv\u0101nara_ , or \"omnipresent\") by the fact that the objects that appear in it are generated by the mind ( **manas** ) rather than by one's sensory organs ( **indriyas** ). These objects, however, are often of the same kind as those that one experiences in the waking state and can be just as convincing while one is in the dream state. Particularly in **Advaita Ved\u0101nta** and **Buddhism** , waking consciousness is said to be dreamlike in the sense that its objects are not ultimately real\u2014being merely conceptual constructs\u2014in contrast with pure, **non-dual** awareness.\n\n**DRO\u1e46A.** Character in the _Mah\u0101bh\u0101rata_ ; a highly gifted warrior, and **guru** of both the **P\u0101\u1e47\u1e0davas** and the **Kauravas**. Although born a **Brahmin** , Dro\u1e47a performed the duties of a **k\u1e63atriya**. Despite this violation of the principle of **svadharma** on his own part, Dro\u1e47a did not allow **Ekalavya** , a lower-caste man who wanted to be his student, to similarly do the duty of a **caste** to which he was not born. It is said that this was because Dro\u1e47a did not want his favorite pupil, **Arjuna** , to have any rival. Though fond of the P\u0101\u1e47\u1e0davas, and of Arjuna in particular, Dro\u1e47a was required by his oath of loyalty to the Kaurava throne to fight in the battle of **Kuruk\u1e63etra** on the Kaurava side. As Arjuna's guru, Dro\u1e47a was one of the persons whom Arjuna was hesitant to fight at the start of the battle, as narrated in the _Bhagavad G\u012bt\u0101_. Dro\u1e47a's slaying occurred in a dishonorable fashion. An elephant was slain whose name was **A\u015bvatth\u0101ma** , which was also the name of Dro\u1e47a's beloved son. Hearing that \"A\u015bvatth\u0101ma has been slain,\" Dro\u1e47a laid down his arms in mourning and was struck down. Even though the statement \"A\u015bvatth\u0101ma has been slain,\" was technically true, it was said with the intent and effect of creating deception, and was thus dishonorable\u2014as was striking Dro\u1e47a after he had laid down his arms. In reference to this episode of the _Mah\u0101bh\u0101rata_ , the phrase \"A\u015bvatth\u0101ma has been slain,\" has entered Hindu parlance as a reference to any statement that, while technically true, is nevertheless said with deceptive intent.\n\n**DUM\u00c9ZIL, GEORGES (1898\u20131986).** Scholar of **Indo-European** language and culture; developed the theory of a tripartite division of Indo-European societies that is reflected in the area of religion. This division is present in traditional Hindu society in the form of the top three **castes** , or **var\u1e47as** , which are regarded as **twice born** \u2014 **Brahmins** , **K\u1e63atriyas** , and **Vai\u015byas** \u2014all of which have parallels in other Indo-European cultures. In regard to **Indology** , Dum\u00e9zil's most famous work is his study of the _Mah\u0101bh\u0101rata_.\n\n**DURG\u0100.** Literally, \"She who is difficult to approach,\" or alternatively, \"She who makes evil go away\"; popular form of the **Mother Goddess** , particularly in **Bengal**. According to the _Dev\u012b Mah\u0101tmya_ , when the seemingly invincible Buffalo Demon (or **Mah\u012b\u1e63\u0101sura** ) had conquered the world, the **gods** , who could not defeat him, approached **Vi\u1e63\u1e47u** for help. On Vi\u1e63\u1e47u's advice, the gods combined their energies, and Durg\u0101 emerged.\n\nDurg\u0101 thus represents the combined power, or **\u015bakti** , of all of the other deities. In the **\u015a\u0101kta** tradition, she is a manifestation of the supreme deity\u2014the Mother Goddess in a fierce form to save the world from evil. She is depicted riding upon a lion and having 10 arms and hands, with a different weapon in each hand\u2014each weapon representing one of the deities whose powers went into making her up. She is spearing the Buffalo Demon in the side and her lion is attacking the water buffalo that is the Buffalo Demon's vehicle. In many representations, especially those used during **Durg\u0101 P\u016bj\u0101** , she is accompanied by her children, the deities **Ga\u1e47e\u015ba** , **Lak\u1e63\u1e43\u012b** , **Sarasvat\u012b** , and **K\u0101rttikeya**.\n\n**DURG\u0100 P\u016aJ\u0100.** Ten-day festival in honor of the **Goddess Durg\u0101** and commemorating the various fierce manifestations of the **Mother Goddess** who appear on Earth to destroy the forces of evil; especially popular in **Bengal**. It corresponds to **Da\u015bahra** , the 10-day festival celebrated in the **bright** **half** of the month of **\u0100\u015bvina** that, in northern India, is celebrated in honor of **R\u0101ma** 's defeat of **R\u0101va\u1e47a**.\n\n**DURV\u0100SA.** Sage; son of the sage **Atri** and his wife, Anasuya. In the _Mah\u0101bh\u0101rata_ , he gave **Kunt\u012b** a **mantra** that enabled her to bear semi-divine children. Durv\u0101sa also cursed **\u015aakuntal\u0101** to be forgotten by her lover. _See also_ K\u0100LID\u0100SA.\n\n**DURYODHANA.** Main villain of the _Mah\u0101bh\u0101rata_ , eldest of the 100 sons of King **Dh\u1e5btar\u0101\u1e63\u1e6dra** and Queen **G\u0101ndh\u0101r\u012b**.\n\n**DUSSEHRA.** Popular alternate spelling of **Da\u015bahra**. _See also_ DURG\u0100 P\u016aJ\u0100.\n\n**DVAITA VED\u0100NTA.** Dualistic **Ved\u0101nta** ; system of Ved\u0101nta **philosophy** developed by the **Vai\u1e63\u1e47ava** **\u0101c\u0101rya Madhva** (1238\u20131317). Born near **Udipi** in **Karnataka** , where he spent most of his life, Madhva is believed by his devotees to be the third incarnation or **avat\u0101ra** of **V\u0101yu** , the **Vedic** **god** of the wind (the first two incarnations being **Hanuman** and **Bh\u012bma** ). A devout **Vai\u1e63\u1e47ava** , Madhva was highly critical of the **Advaita Ved\u0101nta** of **\u015aa\u1e45kara** , seeing it as diminishing the significance of **bhakti** and as relegating **God** to a role subordinate to an impersonal reality ( **Nirgu\u1e47a Brahman** ). Madhva sees Brahman as identical to **Vi\u1e63\u1e47u** , the supreme deity, and the **j\u012bva** , or individual soul of a living being, as an eternally existing reality distinct from Vi\u1e63\u1e47u. **Liberation** , or **mok\u1e63a** , therefore does not involve an actual metaphysical union between the devotee and the divine (or, as in Advaita Ved\u0101nta, a realization that the distinction between the devotee and the divine is an illusion). The devotee and the divine are forever distinct. \"Union\" with the divine is a union of wills, of complete subordination of the will of the individual soul to the will of God ( **\u012a\u015bvara-pra\u1e47idh\u0101na** ). The individuality of one who attains liberation is not effaced. Liberation, rather, consists of residing for eternity in **Vaikun\u1e6dha** , or heaven, with God, a place of infinite bliss. Unusually for a Hindu tradition, Dvaita Ved\u0101nta teaches that it is possible that some souls may never reach liberation and remain eternally separated from God. Madhva's thought was highly influential within the bhakti movement, particularly upon such later thinkers as **Caitanya** and, in the modern period, **Bhaktived\u0101nta Sw\u0101m\u012b Prabhup\u0101da**. The **Gau\u1e0d\u012bya Vai\u1e63\u1e47ava** tradition of **Bengal** bears the stamp of Madhva's thought, as does the **International Society for Krishna Consciousness** (ISKCON).\n\n**DV\u0100RAK\u0100.** Site in Gujarat, on the western coast of India; believed to be the capital of **K\u1e5b\u1e63\u1e47a** 's kingdom and a major site of pilgrimage for this reason. One of the four **ma\u1e6dhas** or monasteries established by **\u015aa\u1e45kara** is located in Dv\u0101rak\u0101. A considerable number of ruins are present on the ocean floor just off the coast of Dv\u0101rak\u0101. Interestingly, according to the _Mah\u0101bh\u0101rata_ , Dv\u0101rak\u0101 sank into the ocean after the accidental death of **K\u1e5b\u1e63\u1e47a** at the hands of a hunter, with **Arjuna** leading a group of refugees away from the doomed city. Archaeologists estimate that the submerged portion of the city likely sank into the ocean around 1500 BCE. If this site can, indeed, be identified with K\u1e5b\u1e63\u1e47a's Dv\u0101rak\u0101, this suggests a likely date for whatever historical events form the basis of the main narrative of the _Mah\u0101bh\u0101rata_.\n\n**DVIJA. Twice born** ; term for the top three **castes** , or **var\u1e47as** , of the caste system due to their employment of the **upanayana** , or **sacred thread** ceremony\u2014a coming-of-age ritual that is likened to a \"second birth\" and marks one's entry into the first of the four stages of life or **\u0101\u015bramas**. Although all three of the upper castes are eligible to undergo this ritual according to the _Dharma \u015a\u0101stras_ , it became increasingly common over the centuries for only **Brahmins** to undergo this ritual, the term _twice-born_ became virtually synonymous with _Brahmin_.\n\n**DWARF AVAT\u0100RA.** _See_ V\u0100MANA.\n\n**DYAU\u1e24 PIT\u1e5a.** \"Heavenly Father\"; a **Vedic** deity whose name is cognate with those of other **Indo-European** father deities, such as Zeus and Jupiter, and with the Latin _deus_ , a generic term meaning \" **god**.\" Dyau\u1e25 Pit\u1e5b seems to have many of the attributes of these other Indo-European father deities (and of others, such as the Norse Odin). A relatively obscure deity in the _Vedas_ , he is mentioned infrequently. He is the husband of **P\u1e5bthv\u012b** (also known as **Bh\u016bdev\u012b** ), the **goddess** of the **Earth** , and father of **Indra** , **Agni** , and **U\u1e63\u0101** , the gods and goddess, respectively, of the sky, fire, and the dawn. _See also_ DEVAS.\n\n## E\n\n**EARTH.** One of the elements ( **mah\u0101bh\u016btas** ) making up the physical universe according to **Ved\u0101nta** **philosophy** ; also the **goddess** of the earth, wife of **Dyau\u1e25 Pit\u1e5b** , the heavenly father. _See also_ BH\u016a; BH\u016aDEV\u012a; P\u1e5aTHV\u012a.\n\n**EKA.** One; \" **the One** \"; the supreme reality or absolute that underlies all existence. In the _\u1e5ag Veda_ it is said that the One is the reality behind the appearance of diversity that gives rise to the world of phenomena. It is also said that the many **devas** or deities are all names for the One ( _\u1e5ag Veda_ 1.164.46)\u2014\"Reality is one, but the wise speak of it in many ways.\" In the _Upani\u1e63ads_ , the ultimate reality\u2014 **Brahman** \u2014is frequently said to be \"One alone, without a second.\" Such Vedic references to the One are the basis for the view that an ancient thread of **monotheism** underlies the Hindu practice of **polytheism** , the idea being that what appear to be many deities are forms, aspects, or manifestations of the One. Though this view is certainly ancient in Hinduism, it has been reemphasized in the modern period in response to the criticisms of **Christian** and **Islamic** missionaries.\n\n**EKADANTA.** One-toothed. Epithet of **Ga\u1e47e\u015ba** , who is given this name due to the fact that he is typically depicted as missing one tusk (or more commonly, with a broken tusk). According to one account, Ga\u1e47e\u015ba lost this tusk during a conflict with **Para\u015bur\u0101ma**. According to another account, he broke off the tusk to use it as a writing implement when **Vy\u0101sa** was dictating the _Mah\u0101bh\u0101rata_ to him. According to yet another account, he broke it off and flung it at the **moon** in anger when the moon laughed at him for falling off of his **vahana** , or vehicle, the mouse.\n\n**EK\u0100DA\u015a\u012a.** Eleventh; the 11th day of either the **bright half** or **dark half** of a month (that is, the 11th day after either **Amavasya** \u2014the new-moon day\u2014or **Pur\u1e47ima** \u2014the full-moon day). Many **Vai\u1e63\u1e47avas** fast or hold a special **p\u016bj\u0101** (or worship ceremony) on this day. _See also_ CALENDAR, HINDU.\n\n**EKALAVYA.** Character from the _Mah\u0101bh\u0101rata_. Ekalavya was a boy from one of the lower castes who wished to study with the great martial arts teacher **Dro\u1e47a** in order to become a warrior. Dro\u1e47a rejected Ekalavya because of Ekalavya's low **caste** status. Undaunted, Ekalavya fashioned an image or **m\u016brti** of Dro\u1e47a and took it as his teacher, training before the image and showing it the devotion that one would normally show to one's teacher, or **guru**. Ekalavya progressed quickly in this fashion and his fame began to spread. Dro\u1e47a, hearing of Ekalavya's prowess, confronted him and demanded the customary payment that a student would give to a teacher, given that Ekalavya claimed Dro\u1e47a as his teacher. After Ekalavya promised to pay Dro\u1e47a in any way he requested, Dro\u1e47a demanded Ekalavya's thumb in payment, which Ekalavya cut off and surrendered to Dro\u1e47a without hesitation. This of course prevented Ekalavya from living as a warrior, for without a thumb he would be unable to perform the basic duties of a warrior, such as stringing a bow, holding a sword, and so on. In the modern period, Ekalavya has become a symbol of protest for the **Dalit** movement, who perceive the cruel behavior of Dro\u1e47a as emblematic of their suffering under the caste system. To compound Dro\u1e47a's behavior, it is also pointed out that Dro\u1e47a was not only engaging in **casteism** , or caste prejudice in the way he treated Ekalavya, but that he was behaving hypocritically as well, since Dro\u1e47a himself was a **Brahmin** , not a **k\u1e63atriya** , or warrior, although he had adopted a vocation as a warrior.\n\n**EKN\u0100TH (1533\u20131599).** Author and devotee of **K\u1e5b\u1e63\u1e47a** from Maharashtra; widely known in Maharashtra as _Sant Eknath_. Like other adherents of the **bhakti** movement, Eknath, a **Brahmin** by birth, was opposed to **casteism** and once saved the life of a drowning **Dalit** child. A prolific writer, his best-known work is his _Ekn\u0101thi Bh\u0101gavata_ , a commentary on the _Bh\u0101gavata Pur\u0101\u1e47a_.\n\n**ELEMENTS.** The five elements are earth ( **p\u1e5bthv\u012b** ), air ( **v\u0101yu** ), fire ( **agni** ), water ( **ap** ), and space ( **\u0101k\u0101\u015ba** ). According to the **Vai\u015be\u1e63ika** system of philosophy ( **dar\u015bana** ), the first four of these elements are atomic and compounded in nature\u2014unlike the self ( **\u0101tman** ), which is atomic but not compounded, and space (\u0101k\u0101\u015ba), which is neither atomic nor compounded. _See also_ MAH\u0100BH\u016aTAS.\n\n**ELEPHANTA.** Island near Mumbai (Bombay) with a large **temple** dedicated to **\u015aiva**. Its most famous image is of the **Trim\u016brti** , or three forms of the supreme deity\u2014 **Brahm\u0101** , **Vi\u1e63\u1e47u** , and **\u015aiva**. The temple dates to the period between the fifth and eighth centuries of the Common Era. Portuguese colonists named it after a large stone elephant that is one of the first features of the temple complex that one sees when landing on the island.\n\n**ELIADE, MIRCEA (1907\u20131986).** Renowned Romanian historian of religions. Eliade studied **Sanskrit** and Indian **philosophy** at **Calcutta** University from 1928 to 1931 under the tutelage of **Surendran\u0101th Dasgupta**. After teaching at the University of Bucharest and at the Sorbonne in Paris, he became a professor of the University of Chicago, where he established himself as a major figure in the study of the history of religions. He taught at the University of Chicago's Divinity School from 1957 to 1985. A prolific author not only in the area of religious studies but also of philosophy and fiction, his most famous works include _The Sacred and the Profane_ and _Yoga:_ _Immortality and Freedom_. **Wendy Doniger** currently holds the chair endowed in his name. _See also_ INDOLOGY.\n\n**ELLORA (EL\u016aR\u0100).** Extensive site of numerous Hindu, **Buddhist** , and **Jain** **temples** and monasteries. Numerous caves dot the region, many of which have been used as sites for **meditation** through the centuries, being at various times extended and connected for the use of whole communities of monks. Among the more famous Hindu temples located at this complex is the **Kail\u0101sa** temple, which is twice the size of the Parthenon in Greece and was carved from a single piece of rock. It was carved in the eighth century CE and is named after the mountain on which **\u015aiva** is believed to reside in the **Him\u0101layas**.\n\n**EMERSON, RALPH WALDO (1803\u20131882).** American author, philosopher, Unitarian minister, and a founding figure of the **transcendentalist** movement, which could be seen as the first wave of interest in Hindu thought in North America. Through his familiarity with early English translations of the _Upani\u1e63ads_ and the _Bhagavad G\u012bt\u0101_ , Emerson was deeply influenced by **Ved\u0101nta philosophy**. This influence is most pronounced in some of his **poetic** works, such as \"Brahma,\" which includes direct quotations from the _Bhagavad G\u012bt\u0101_. _See also_ THOREAU, HENRY DAVID.\n\n**ENLIGHTENMENT.** Common English term for the state of heightened awareness that, in many Indic religious traditions, is a necessary precondition for **liberation** from **rebirth** (or **mok\u1e63a** ) and that is frequently used as a synonym for such liberation. Also known in contemporary Hindu traditions as **God-realization** , enlightenment consists of the direct awareness that **Brahman** and **\u0101tman** are one and the same: that the fundamental basis of one's identity is the same as the ultimate reality that is the basis of all existence. It is the experiential discovery of the ground of all being as the basis of one's own being. It is not merely a cognitive awareness of this unity\u2014an act of mental assent to the proposition that such unity is the case\u2014but a direct, experiential awareness ( **anubh\u0101va** ).\n\nAccording to the **Advaita Ved\u0101nta** tradition, assent to the proposition of the unity of Brahman and \u0101tman is a condition for the direct awareness that follows it, and the source of this proposition, is the Hindu **scriptures**. In **Neo-Ved\u0101nta** , the awareness itself is emphasized more than its scriptural source. Although Advaita Ved\u0101nta, **Jainism** , and **Buddhism** tend to conceive of enlightenment in impersonal terms, as the realization of an abstract **truth** , the more **bhakti** -oriented, theistic Ved\u0101nta traditions see it as coming to know **God**. _See also_ VED\u0100NTA.\n\n**ETA\u015aA.** **Vedic** sage whose life was saved by **Indra** ; also the name of one of the horses that pull the flying **chariot** ( **vim\u0101na** ) of the sun **god** , **S\u016brya**.\n\n## F\n\n**FASTING.** A common form of **ascetic** practice, especially popular among laywomen, and often practiced to fulfill a specific wish. _See also_ VRATA.\n\n**FIRE SACRIFICE.** _See_ AGNI\u1e62\u1e6cOMA.\n\n**FISCHER, LEOPOLD.** _See_ AGEH\u0100NANDA BH\u0100RAT\u012a, SW\u0100M\u012a.\n\n**FISH AVAT\u0100RA.** _See_ MATSYA.\n\n**FIVE M'S.** Five substances consumed or activities performed in the ceremonies of the V\u0101m\u0101c\u0101ra or **Left-Handed** tradition of **Tantra** that are normally regarded as impure. This tradition involves participation in activities normally considered impure in the name of demonstrating the principle of nonduality ( **advaita** ): that both purity and impurity are illusory constructs, the reality being that all is **Brahman**. The substances or activities in question are called the \"five M's\" because the words for them all begin with the letter m: madya (alcohol), m\u0101\u1e43sa (meat), **matsya** (fish), **mudr\u0101** (grain), and maithun\u0101 (sexual intercourse\u2014although in this context, it refers specifically to sexual intercourse with a partner who is not one's spouse, because sexual intercourse is otherwise not regarded as inherently impure, being a natural and normal activity for non **ascetics** ).\n\nMudr\u0101 (grain) is an unusual member of this set of items, since it is not normally regarded as impure, though it is forbidden by some religious fasts ( **vratas** ). It is possible that this term refers to specific symbolic gestures performed in the context of the ritual (as when it refers to the hand gestures of deities when depicted in **m\u016brti** form or when employed in specific forms of **meditation** ).\n\n**FOUR GOALS OF HUMANITY.** Physical pleasure ( **k\u0101ma** ), wealth ( **artha** ), virtue ( **dharma** ), and **liberation** from the cycle of **rebirth** ( **mok\u1e63a** ). _See also_ PURU\u1e62\u0100RTHA.\n\n**FOUR STAGES OF LIFE.** The student stage ( **brahmacarya** ), the householder stage ( **g\u0101rhasthya** ), the stage of withdrawal or retirement ( **vanaprastha** ), and renunciation ( **sanny\u0101sa** ). _See also_ \u0100\u015aRAMA.\n\n**FOUR YOGAS.** Four spiritual disciplines that can be practiced either individually or jointly in the pursuit of **mok\u1e63a**. Although developed in a variety of Hindu texts through the centuries, the yogas were first codified and theorized as four distinct disciplines in the modern period by **Sw\u0101m\u012b Vivek\u0101nanda**. They are the yogas of action ( **karma yoga** ), of wisdom ( **j\u00f1\u0101na yoga** ), of devotion ( **bhakti yoga** ), and of **meditation** ( **dhy\u0101na** or r\u0101ja yoga). As conceived by Vivek\u0101nanda, each **yoga** (or combination of yogas) is suited to a different personality type. Karma yoga is working selflessly for the good of others with no thought of personal reward and is suited to an activist temperament. J\u00f1\u0101na yoga is for the intellectual, the philosopher, and consists of discerning and differentiating between what is real and what is unreal. Bhakti yoga is intense love and longing for a personal form of the divine and manifests in a relationship with **God** , as a servant, friend, spouse, parent, or child. Dhy\u0101na yoga is the practice of meditation. The subsequent tradition of **Neo-Ved\u0101nta** tends to emphasize the idea of practicing all four yogas as aspects of a total, holistic approach to the spiritual life.\n\n**FUNERAL.** _See_ ANCESTRAL RITES; ANTYE\u1e62\u1e6cI.\n\n## G\n\n**GADAR MOVEMENT.** _See_ GHAD\u1e5a MOVEMENT.\n\n**GA\u1e46APATI.** Lord of hosts, an epithet of **Ga\u1e47e\u015ba**. The ga\u1e47as, or hosts, refer to groups of demigods under Ga\u1e47e\u015ba's command.\n\n**G\u0100\u1e46APATYA.** Branch of Hinduism devoted to the worship of **Ga\u1e47e\u015ba** as the supreme form of the divine, analogous to **Vai\u1e63\u1e47ava** worship of **Vi\u1e63\u1e47u** , **\u015aaiva** worship of **\u015aiva** , and **\u015a\u0101kta** worship of the **Mother Goddess**. Though not as widespread as these movements, the G\u0101\u1e47apatya tradition has made its presence felt in the universal popularity of Ga\u1e47e\u015ba.\n\nThe worship of Ga\u1e47e\u015ba probably began as a \u015aaiva practice\u2014a logical development, given that Ga\u1e47e\u015ba is the son of \u015aiva and **\u015aakt\u012b** (or **P\u0101rvat\u012b** , as the Mother Goddess is also known). In the ninth century, **\u015aa\u1e45kara** declared Ga\u1e47e\u015ba to be one of five possible deities of choice, or **i\u1e63\u1e6dadevat\u0101s** , in his system of five objects of worship ( **pa\u00f1cay\u0101tana p\u016bj\u0101** ), alongside \u015aiva, \u015aakti, Vi\u1e63\u1e47u, and **S\u016brya**. The inclusion of Ga\u1e47e\u015ba in this system as an equal to such \"high\" deities as \u015aiva, \u015aakti, and Vi\u1e63\u1e47u suggests the G\u0101\u1e47apatya sect already existed in some form in \u015aa\u1e45kara's time.\n\nIn the modern period, largely due to **Bal Gangadhar Tilak** 's promotion during the movement for Indian independence of **Ga\u1e47e\u015ba p\u016bj\u0101** as a symbol of national unity, the worship of Ga\u1e47e\u015ba has become a pan-Hindu phenomenon, with members of all the major sects honoring Ga\u1e47e\u015ba as the remover of obstacles at the outset of any large undertaking.\n\n**GANDHA.** Scent; the sense of smell; attribute of the **earth element** ( **p\u1e5bthv\u012b** ) according to **Vai\u015be\u1e63ika** **philosophy**. _See also_ MAH\u0100BH\u016aTAS.\n\n**G\u0100NDH\u0100R\u012a.** Character in the _Mah\u0101bh\u0101rata_ ; wife of **Dh\u1e5btar\u0101\u1e63\u1e6dra** and mother of the **Kauravas**. After the climactic battle of the epic, in which all of her sons are slain by the army of the **P\u0101\u1e47\u1e0davas** , G\u0101ndh\u0101r\u012b curses **K\u1e5b\u1e63\u1e47a** for his role in the battle as advisor to the P\u0101\u1e47\u1e0davas. G\u0101ndh\u0101r\u012b's curse eventually results in K\u1e5b\u1e63\u1e47a's accidental **death** in the forest at the hands of a hunter.\n\n**GANDHARVAS.** Celestial musicians, a category of divine being said to possess great physical strength. The word is cognate with the Greek _centaur_.\n\n**G\u0100NDH\u012a, MAH\u0100TMA.** _See_ G\u0100NDH\u012a, MOHAND\u0100S K.\n\n**G\u0100NDH\u012a, MOHAND\u0100S K. (1869\u20131948).** Hindu reformer and leader of the nonviolent movement for Indian independence; often called **Mah\u0101tma** (Great Soul). He was born to a **Vai\u015bya** family in Gujarat, on the western coast of India. His family religious tradition was **Vai\u1e63\u1e47ava** , though he had a strong exposure in his early years to **Jainism** as well. At the age of 13 he entered into an arranged **marriage** with the equally young Kasturba Kapadia. At 18, he traveled to London to study law, against the wishes of his **caste** elders. With an intense interest in religion and **philosophy** , he had many conversations in England with both Christians and **Theosophists**. Through a translation of the _Bhagavad G\u012bt\u0101_ given to him by a British Theosophist, he rediscovered his Hindu roots. He moved to South Africa in 1891, where he experienced intense racial prejudice for the first time, leading to his political awakening. While in South Africa, he agitated for equal treatment of Indians as citizens of the British Empire.\n\nOnly after returning to India in 1914 did G\u0101ndh\u012b begin to see imperialism itself as the root of the problems facing Indians. After touring the country extensively in order to learn firsthand about the concerns of his countrymen, he joined the movement for Indian independence.\n\nAdvocating **nonviolent** methods of protest and also adopting the lifestyle of a **sanny\u0101s\u012b** (taking a **brahmacarya** vow, though he was still married to Kasturba), G\u0101ndh\u012b began to have a wide following among the common people of India, especially in rural areas and villages. Unlike other independence leaders, whose English educations made them culturally Westernized, and so remote from common people, G\u0101ndh\u012b embraced and practiced village traditions. At the same time, though, many of the stands he took on a variety of issues were controversial, even among Indians, and several attempts were made on his life. For his activism he was jailed a number of times by the British authorities. G\u0101ndh\u012b was also a leading figure of the Indian National Congress, the main party that was working for Indian political independence.\n\nG\u0101ndh\u012b called his philosophy **saty\u0101graha** , or \"soul force,\" although the literal meaning of this term is \"holding fast to the **truth**.\" Having some resonance with the epic concept of the vow of truth\u2014in which the truth of a character's words give his or her vow particular effectiveness\u2014the idea behind saty\u0101graha is that the truth always prevails, and that one who is faithful to truth will also prevail, despite adversity and opposition. The nonviolent activist who adheres to methods that are in harmony with the ends he pursues is behaving in a manner consistent with truth. The inherent power of truth will therefore ensure the ultimate success of the activist's cause. The idea of the harmony of ends and means is also connected with the idea of **karma** , that like **actions** produce like effects. Therefore only just means can produce just results.\n\nG\u0101ndh\u012b viewed the pursuit of truth and the practice of nonviolence ( **ahi\u1e43s\u0101** ) as inseparable from one another, seeing truth and nonviolence as different sides of the same coin, truth being the interconnected and ultimately unified nature of all beings (his interpretation of **Advaita Ved\u0101nta** ) and non-violence being the implication of this truth in practice. He also underwent a shift in his thinking over the course of his lifetime from the idea that \" **God** is truth\" to the idea that \"Truth is God.\" The significance of this shift is that, whereas by holding that \"God is truth\" one might identify one's own idea of God with the truth, by holding that \"Truth is God\" one is committed to pursuing the truth wherever it may lead.\n\nIn terms of Indian independence, G\u0101ndh\u012b's methods were in one sense effective, creating global sympathy for his movement. Utilizing boycotts and marches, G\u0101ndh\u012b was able to put the brutality of imperialism on display, with peaceful demonstrators facing the harsh blows of the British police in front of the world media. On the other hand, it is still debated today whether the eventual withdrawal of the British from India was due more to the independence movement or to other circumstances, such as World War II and the gradual realization among the European powers that far-flung colonial empires, in the end, were more of an economic drain than a boon.\n\nG\u0101ndh\u012b's understanding of Indian independence, or self-rule ( _swar\u0101j_ ), was holistic and informed by Hindu traditions, as well as by Jainism, **transcendentalism** , and **Christianity**. He saw political **liberation** as one component of the liberation of the whole person\u2014like liberation in the classical Hindu sense, as **mok\u1e63a** , with a collective dimension distinct from the individualism of traditional **yoga** philosophies. He therefore placed a great deal of emphasis on purification and the moral perfection of the Indian community.\n\nIn addition to working for Indian independence, G\u0101ndh\u012b sought to end the practice of **Untouchability** \u2014though he did not advocate the total abolition of caste, which led to a rift between himself and **Bh\u012bmrao Ambedkar**. G\u0101ndh\u012b also advocated interreligious harmony, particularly between Hindus and Muslims, the many divisions between these two religious communities having been successfully exploited by British divide-and-rule policies, ultimately leading to the partition of India and Pakistan. These are two areas in which G\u0101ndh\u012b received, and continues to receive, considerable criticism; his actions on behalf of **Dalits** and Hindu\u2013Muslim unity ultimately failed to resolve these issues.\n\nOn a number of occasions G\u0101ndh\u012b undertook **fasting** \u2014in effect, going on a hunger strike\u2014to persuade his followers to behave as he wished: to renounce violence against the British, to treat people of the lower castes as equals, or to renounce the Hindu\u2013Muslim conflict.\n\nIndia became independent on 15 August 1947, an event that was widely seen as a triumph for G\u0101ndh\u012b and his methods. Many, however\u2014including G\u0101ndh\u012b himself\u2014did not view it this way, the euphoria of independence being overshadowed by the violence of partition. On 30 August 1948, G\u0101ndh\u012b was assassinated by Nathuram Godse, a **Hindu Nationalist** who blamed G\u0101ndh\u012b for the violence suffered by Hindus during the partition.\n\nThough widely revered globally today, G\u0101ndh\u012b has also been subject to criticism, some of it during his lifetime. In addition to the criticisms already mentioned\u2014that his denunciation of **casteism** was insufficiently radical and that he did not do enough to stop the partition of India and Pakistan\u2014he was heavily criticized for testing his commitment to celibacy by sleeping next to naked young women (a test he did pass, by all accounts). At the same time, even some of his harshest critics acknowledge G\u0101ndh\u012b's transformative leadership and the power of his vision of peace and justice.\n\n**GA\u1e46E\u015aA.** Popular Hindu deity; elephant-headed remover of obstacles. As one of the sons of **\u015aiva** and **P\u0101rvat\u012b** (the other being **Karttikeya** ), Ga\u1e47e\u015ba has long been an object of devotion in **\u015aaiva** traditions. In modern Hinduism, however, he has become a popular pan-Hindu deity. As the remover of obstacles, he is traditionally evoked at the beginning of any new venture, including at the commencement of a **p\u016bj\u0101** , or worship ceremony, for any deity. There is also a **G\u0101\u1e47apatya** tradition that is centered on the worship of Ga\u1e47e\u015ba as the supreme deity, just as the \u015aaiva traditions are centered on \u015aiva and **Vai\u1e63\u1e47avism** is centered on **Vi\u1e63\u1e47u**.\n\nThere are many differing accounts of how Ga\u1e47e\u015ba came to have an elephant head. In one account, P\u0101rvat\u012b creates Ga\u1e47e\u015ba from the residue left behind after her bath. This occurs while \u015aiva is away from home. P\u0101rvat\u012b assigns her newly formed son the task of guarding the door to her chamber while she is bathing, ordering him not to allow anyone to come in, but neglecting to make an exception in the case of her husband, \u015aiva. When \u015aiva returns, he is angered to find this creature, Ga\u1e47e\u015ba, forbidding him from entering his wife's room. He therefore beheads Ga\u1e47e\u015ba. P\u0101rvat\u012b, greatly distressed at this, tells \u015aiva that he has killed his son. The repentant \u015aiva then searches for the best head he can find for Ga\u1e47e\u015ba, and takes the head of **Air\u0101vata** , the elephant mount of **Indra** , the lord of the **devas** , placing it upon Ga\u1e47e\u015ba's neck and bringing him back to life.\n\nIn another account, Ga\u1e47e\u015ba loses his original head when his mother, P\u0101rvat\u012b, is insistent that all the other **gods** look at her beautiful new baby. Unfortunately, **\u015a\u0101ni** , the presiding deity of the planet Saturn, is cursed with an evil eye. \u015a\u0101ni's gaze causes the infant deity's head to fall off, requiring it to be replaced with the head of an elephant (dutifully supplied this time not by \u015aiva, but by **Vi\u1e63\u1e47u** ). Finally, there is another account in which Ga\u1e47e\u015ba is simply born with the head of an elephant, with no particular reason being given. Ga\u1e47e\u015ba's vehicle is **M\u016b\u015baka** , the mouse.\n\nSymbolically, Ga\u1e47e\u015ba's elephant head represents both divine wisdom (because the elephant is a wise animal, with a long memory) and strength (given the great strength of elephants). He also has a pot belly, which is said to represent divine expansiveness, and is fond of sweets, which are the favored offering to present during his p\u016bj\u0101.\n\nThe worship of Ga\u1e47e\u015ba was popularized in the modern period by **Bal Gangadhar Tilak** (1856\u20131920) in Maharashtra to promote Hindu pride and unity. The annual Ga\u1e47e\u015ba P\u016bj\u0101 that he instituted is a very popular event in the city of Mumbai (Bombay).\n\n**GA\u1e46E\u015aAPURI \u0100\u015aRAMA.** Indian center of the **Siddha Yoga** movement; established in the village of Ga\u1e47e\u015bapuri, on the outskirts of Mumbai (Bombay) in 1949 by a **sanny\u0101s\u012b** named **Sw\u0101m\u012b Nity\u0101nanda** (1897\u20131961), founder of the Siddha Yoga **sa\u1e43prad\u0101ya** and **guru** of **Sw\u0101m\u012b Mukt\u0101nanda** (1908\u20131982), who brought the practice to the West. The lineage is now led by **Sw\u0101m\u012b Chidvil\u0101s\u0101nanda** ( **Gurumay\u012b** ).\n\n**GA\u1e44G\u0100.** The most sacred river in Hinduism; also the name of the **goddess** who is the personification of this river. Flowing from the northwestern reaches of the **Him\u0101layas** , where it is fed by melting glaciers, and emptying into the Bay of **Bengal** in the east, many sacred pilgrimage sites are located along the course of this river, including **Haridw\u0101r** , **Rishikesh** , **Pray\u0101ga** ( **Allahabad** ), and **Banaras** , two of which (Haridwar and Pray\u0101ga) are traditional locations for the **Kumbha Mel\u0101** festival. Many devout Hindus believe a prayerful bath in the Ga\u1e45g\u0101 (or Ganges) can be both physically and spiritually purifying. It is customary, if possible, to have one's ashes immersed in the Ga\u1e45g\u0101 after **cremation** , and many cremation grounds are located near the **gh\u0101\u1e6ds** , or stairs leading into the river at various sites, in order to facilitate this custom.\n\nAccording to Hindu sacred **literature** , the Ga\u1e45g\u0101 is an extension of a celestial river that corresponds to the Milky Way. The river was brought to the Earth by the **asceticism** of Bhagiratha, a king whose ancestors had been cursed by a sage. Bhagiratha wanted to find a way to cleanse his ancestors of this curse, and so prayed for the Ga\u1e45g\u0101's descent. Fearing that the force of the river's descent to the Earth might destroy the world, **\u015aiva** agreed to catch the river on his head, letting it then flow gently down his matted hair and onto the Earth. \u015aiva, in this image, represents the Him\u0101layas, from where the river flows and where \u015aiva is also believed to reside, on Mount **Kail\u0101sa**.\n\nIn the _Mah\u0101bh\u0101rata_ , King \u015aantanu of the **Bh\u0101rata** dynasty falls in love with the goddess Ga\u1e45g\u0101 when he encounters her in the forest while on a hunting expedition. She agrees to marry him on the condition that he must not question anything she does, or she will leave him. Unbeknownst to \u015aantanu, however, a set of **Vedic** deities, the eight **Vasus** , have been cursed to be reborn as human beings by the sage **Vasi\u1e63\u1e6dha** because they stole his cows. Ga\u1e45g\u0101 agrees to help the Vasus escape the limits of the human form by giving birth to them and then drowning them in her waters. One by one, Ga\u1e45g\u0101 gives birth to healthy baby sons, and then, to \u015aantanu's horror, drowns them immediately after birth. But he remembers his promise not to question his wife's **actions** and remains silent. The eighth time, however, he intercedes and asks Ga\u1e45g\u0101 to spare their child. She agrees but then leaves him because he has broken his promise. This child grows up to be **Bh\u012b\u1e63ma** , the wise elder of the Bh\u0101rata dynasty throughout most of the epic. In terms of **karma** , it is said that Bh\u012b\u1e63ma lived such a long life and witnessed as much suffering as he did due to the fact that, among the Vasus, he was the one who instigated the theft of the cows.\n\nIn the contemporary period, the excessive pollution suffered by the sacred Ga\u1e45g\u0101 has become a major issue and a source of much controversy, with calls by both religious and secular organizations for the government of India to address the ecological crisis. On top of the pollution issue, recent studies have also suggested that global warming may be melting the glaciers on which the river depends at such a rate that the river itself may one day dry up completely. This would be a major disaster, not only from a religious point of view but also because millions of people depend upon the Ga\u1e45g\u0101 as their main source of drinking water.\n\n**GANGES RIVER.** _See_ GA\u1e44G\u0100.\n\n**GARBHA.** Literally \"womb\" or \"embryo\"; a folk **dance** especially popular in Gujarat. The dancers, who are usually (but not always) **women** , form a circle and make sweeping gestures. Each gesture ends with either the clapping of the hands or the striking together of two sticks. It is a joyful, celebra-tory dance, performed on festive occasions.\n\n**GARBHA-G\u1e5aHA.** Innermost chamber of a **temple** , where the main image ( **m\u016brti** ) of its central deity is enshrined. Typically only the temple priests are permitted to enter this chamber, although devotees may look inside when the doors are open for **dar\u015bana** (viewing of the image). Not all temples have a garbha-g\u1e5bha with closing doors. In these temples, the main m\u016brti is either always on display or is covered with a curtain.\n\n**G\u0100RHASTHYA.** The **householder** stage of life; the second of the four stages of life or **\u0101\u015bramas**. This stage of life is characterized by marriage, child-rearing, and economically productive activity. One formally enters this stage by means of one's wedding ceremony.\n\n**GARU\u1e0cA.** \"Devourer\" (of serpents); the divine eagle; servant and vehicle of **Vi\u1e63\u1e47u** , and a popular subject of Hindu **temple** **art**. Symbolically, Garu\u1e0da is represented in **p\u016bj\u0101** , or ritualistic worship, by a bell. Through his symbolic presence, Garu\u1e0da serves to protect the worship ceremony from evil influences.\n\n**GAU\u1e0cAP\u0100DA (fl. 8th century CE).** **Guru** of **Govinda** , who was the guru of **\u015aa\u1e45kara** , thus making Gau\u1e0dap\u0101da \u015aa\u1e45kara's \"grand-guru\"; early exponent of **Advaita Ved\u0101nta** , whose best-known work is his commentary ( **bh\u0101\u1e63ya** ) on the _Ma\u1e47\u1e0dukya Upani\u1e63ad_. Of particular note is Gau\u1e0dap\u0101da's extensive use of **Buddhist** terminology and styles of argumentation in this text, albeit in the service of a thoroughly **Ved\u0101ntic** thesis\u2014 _aj\u0101tiv\u0101da_ , the doctrine of nonorigination, according to which all change is an illusion, the sole reality being the eternal and changeless **Brahman** , identical to the self ( **\u0101tman** ). His name suggests that Gau\u1e0dap\u0101da was from **Bengal** , a major Buddhist center.\n\n**GAU\u1e0c\u012aYA VAI\u1e62\u1e46AVA.** **Vai\u1e63\u1e47ava** tradition of **Bengal**. **Caitanya** (1486\u2013 1533) started this branch of Vai\u1e63\u1e47avism. Though sharing with the **Dvaita Ved\u0101nta** of **Madhva** a very strong emphasis on **bhakti** and the belief that **K\u1e5b\u1e63\u1e47a** is the supreme reality, the Gau\u1e0d\u012bya Vai\u1e63\u1e47ava tradition, following the theology of the **Gosv\u0101mins** , claims the relationship of the world to the divine is _acintya-bhed\u0101bheda_ , or \"inconceivable difference and identity,\" rather than a pure difference, as Madhva affirms (much less a pure identity, as **\u015aa\u1e45kara** 's **Advaita Ved\u0101nta** claims). The **International Society for Krishna Consciousness** of **Bhaktived\u0101nta Sw\u0101m\u012b Prabhup\u0101da** is a continuation of this tradition.\n\n**GAUR\u012a.** Benevolent rain **goddess** , associated with agriculture and human flourishing in general; usually depicted as a form of the **Mother Goddess** and wife of **\u015aiva** , but also as the wife of **Varu\u1e47a** , the **Vedic** ocean deity.\n\n**GAUTAMA.** One of the Seven Sages ( **Saptar\u1e63i** ) presiding over the cosmos. _Gautama_ is also the **gotra** name of the **Buddha** , Siddh\u0101rtha Gautama (c. 490\u2013410 BCE), but the two figures should not be confused with one another.\n\nHaridrumata Gautama is also the name of the sage in the _Chandogya Upani\u1e63ad_ who accepts the boy **Satyak\u0101ma** as his student, despite the boy's low birth **caste** status, because the boy is honest about his background, and honesty is a trait of an ideal **Brahmin**.\n\n**GAY\u0100.** Town in the northeastern Indian state of Bihar; sacred place of pilgrimage that is especially dedicated to the making of offerings to one's ancestors ( **\u015br\u0101ddha** ). Gay\u0101 is near Bodh Gay\u0101, a sacred place of pilgrimage for **Buddhists** for being the location where the **Buddha** attained **enlightenment** ( **nirv\u0101\u1e47a** ) under a banyan tree. Interestingly, the site of the offerings to the dead at nearby Gay\u0101 is also under a sacred banyan tree. Which sacred location came first is a subject of some contention and speculation among scholars.\n\n**G\u0100YATR\u012a.** \"Song of protection\"; name of a famous **mantra** of the _\u1e5ag Veda_ (3.62.10), as well as the **poetic** meter in which it was composed; later personified as a **goddess** with five heads who is a wife of **Brahm\u0101**. The mantra is a hymn to the solar deity **Savit\u1e5b** and is traditionally recited by **Brahmins** at dawn. The chanting of the G\u0101yatr\u012b mantra as a daily practice for all people is also promoted by the **\u0100rya Sam\u0101j**.\n\nThe mantra reads: _O\u1e43 bh\u016br bhuva\u1e25 sva\u1e25_ , _Tat savitur vare\u1e47yam, Bhargo devasya dh\u012bmahi_ , _dhiyo yo na\u1e25 pracoday\u0101t_. It means, \"Om. Oh, earth! Oh, sky! Oh, heavens! Let us reflect upon the beautiful light of the god Savit\u1e5b. May he guide our thoughts.\"\n\nThe five-headed goddess who personifies this mantra became the wife of Brahm\u0101 when Brahm\u0101 was holding a sacrifice ( **yaj\u00f1a** ). A **householder** who performs a sacrifice must be accompanied by his wife. But Brahm\u0101's wife, **Sarasvat\u012b** , was late for the ritual, which also had to be performed at the astrologically correct time. **Indra** then presented Brahm\u0101 with G\u0101yatr\u012b, who served as Brahm\u0101's wife for the purpose of the sacrifice. The deeper symbolism of the story seems to be that the G\u0101yatr\u012b mantra is just as indispensable to the ritual of sacrifice as is the presence of one's spouse.\n\n**G\u0100YATR\u012a MANTRA.** _See_ G\u0100YATR\u012a.\n\n**GHAD\u1e5a MOVEMENT.** Early 20th-century movement among Indian immigrants in North America to support the violent overthrow of the British government in India. It sought to foment rebellion among Indians serving in the British army and to gain support from the United States in the cause of Indian independence. The movement, which was led by **Har Dayal** (1884\u2013 1939), suffered major setbacks when the United States entered World War I as an ally of Great Britain. Dayal was arrested and then fled to Germany. Many of the Indian immigrants in the United States and Canada returned to India to help with the revolution, but the British government was aware of their plans and arrested them. At the end of World War I, Dayal returned to India, where he renounced violence and joined **Mohand\u0101s K.** **G\u0101ndh\u012b** 's nonviolent independence movement. The Ghad\u1e5b movement ceased to be a major force at this point and dissolved when independence was achieved in 1947.\n\n**GH\u0100\u1e6c.** ( **Sanskrit** : gha\u1e6d\u1e6da.) Set of stairs leading into a sacred river, such as the **Ga\u1e45g\u0101** , to facilitate bathing for pilgrims.\n\n**GHEE.** Clarified butter; substance frequently offered in a **Vedic yaj\u00f1a** and also in **p\u016bj\u0101** , or worship. In the latter case, it is usually mixed with other sacred items, such as honey, milk, and yogurt. When offered as part of a Vedic ritual, such as a **homa** or **havan** , it is not mixed with anything else and is placed directly in the sacred fire with special spoons designed for that purpose.\n\n**GHOSE, AUROBINDO (1872\u20131950).** Better known as Sri Aurobindo; freedom fighter for Indian independence and later a major philosopher, Hindu reformer, and advocate of a system of **yoga** known as \"integral yoga.\" Trained in Western philosophy, Aurobindo's writings seek to unite and integrate both Western and Indian metaphysical themes.\n\nBorn into a wealthy **Bengali Brahmin** family, Aurobindo was sent to England for his education in 1879, where he studied in Cambridge University. In 1893, he returned to India where he served as a minister for the Mah\u0101r\u0101ja of Baroda, Gujarat. But wanting to strengthen his ties to his Bengali heritage, he moved to **Calcutta** in 1906.\n\nBoth during his service in Baroda and subsequently, in Calcutta, Aurobindo was drawn to the cause of Indian independence. Aligning himself with the militant **Hindu Nationalists** , such as **Bal Gangadhar Tilak** , Aurobindo advocated violent revolution. As a result of his political activities, the British government arrested Aurobindo in 1908 on the suspicion that he was connected with an antigovernment bombing. Although he was acquitted in the bombing case, he was imprisoned for a year for sedition. During the period of his imprisonment, he practiced yoga extensively and wrote of experiencing a number of powerful spiritual visions at this time, including visions of **K\u1e5b\u1e63\u1e47a** and **Sw\u0101m\u012b Vivek\u0101nanda** , who had died just a few years earlier (in 1902). In 1910, after his release from prison, he announced his retirement from politics. Moving to the French colony of Pondicherry, he dedicated himself to spiritual practice and philosophical writing, developing his system of integral yoga. A prolific author, his writings include _Integral Yoga_ , _The Synthesis of Yoga_ , _The Secret of the Veda_ , _Essays on the Gita_ , _Rebirth and Karma_ , and the monumental _The Life Divine_.\n\nOne of the major themes of Aurobindo's writing is the integration of Western thought, with its emphasis on the **material** world and the evolution of **mind** from matter, and Indian thought, with its emphasis on pure consciousness and what Aurobindo calls the \"involution\" of mind into matter. These two complementary processes are really one process viewed from different perspectives, according to Aurobindo's account. The term \"involution\" is drawn from the writings of Sw\u0101m\u012b Vivek\u0101nanda, and Aurobindo says several times that he sees himself as building on and continuing the work that was begun by Vivek\u0101nanda and his master, **Ramakrishna** , both of whom sought to reconcile the concepts of the absolute realm of spirit and the relative realm of time, space, matter, and change. In _Rebirth and Karma_ , he also articulates the idea\u2014which is now common among Westerners who believe in **rebirth** \u2014that reincarnation is a learning process for the soul. Aurobindo dedicated his life to bringing divine awareness or \"supermind\" into the realm of concrete, physical expression through the practice of yoga. Before his death, he claimed to have achieved this manifestation within himself.\n\nAurobindo also popularized an approach to the _Vedas_ that was a striking contrast with the dominant **Indological** paradigm of his time. Rather than seeing the early **Vedic literature** as the relatively primitive nature **poetry** of a nomadic society that gradually develops into the more sophisticated **Ved\u0101nta** philosophy of the _Upani\u1e63ads_ , he suggests that the early Vedic poems already presuppose the fully developed **philosophy** , presenting it in a symbolic form. It is only as the key to this ancient symbolic code is forgotten that Vedic philosophy needs to be articulated more explicitly in the _Upani\u1e63ads_ , which are thus seen as an unpacking of what was already present in the early Vedic texts, rather than as an evolutionary development from them.\n\nAurobindo attracted many followers, including Westerners, the most famous of whom was a French woman named Mira Richard, to whom Aurobindo and his followers referred as \"the Mother.\" Aurobindo saw Richard as an incarnation of **\u015aakti** , the **Mother** **Goddess** and feminine energy of **creation**. Richard took over Aurobindo's community after his death in 1950.\n\n_G\u012aTAGOVINDA_. \"Song of the Cowherd.\" This highly **bhakti** -infused 12th-century **Sanskrit poem** by the **Bengali** **Vai\u1e63\u1e47ava** author **Jayadeva** uses the love affair between **R\u0101dh\u0101** and **K\u1e5b\u1e63\u1e47a** to depict the bond of love between the soul of the **devotee** and **God**. The text is highly erotically charged and has been compared by some with the biblical _Song of Solomon_. It is based on the _Bh\u0101gavata Pur\u0101\u1e47a_.\n\n**GOBIND SINGH, GURU (1666\u20131708).** Tenth and final **guru** in the lineage of gurus of **Sikhism**. In response to the fierce persecution suffered by the Sikh community under the **Mughal** emperor Aurangzeb, Gobind Singh established the _khalsa_ , or community of Sikh warriors, marked by the \"Five K's,\" or five items whose names, in Punjabi, start with the letter K: _ke\u015b_ , or long hair; _kangha_ , a special comb with which the hair is tied on top of the head; _kara_ , a circular steel bracelet; _kacha_ , an undergarment ensuring preparedness for battle; and the _kirpan_ , a sword or dagger. Gobind Singh's father, Guru Tegh Bahadur, had been beheaded by Aurangzeb, and Gobind Singh himself had to spend much of his time in hiding. Knowing that he might not survive in the ongoing conflict, he designated the sacred text of the Sikhs, the _\u0100di Granth_ , the perpetual guru of the community after his **death**. _See also_ N\u0100NAK, GURU.\n\n**GOD.** The concepts of divinity in the various Hindu traditions vary greatly. Hindu texts express positions ranging from atheism, to **monotheism** , to **polytheism** , to panentheism (the idea that God is present within all beings, and that all beings, in turn, dwell in God), to pantheism (the idea that God simply _is_ all beings). Translating Hindu concepts of God into English is further complicated by the fact that there are a variety of **Sanskrit** terms that can be translated as \"God\" that all have quite distinct meanings in their original Hindu contexts. A **deva** , for example, is a powerful and long-lived being in charge of a particular aspect of the universe: such as **Agni** , the god of fire; or **S\u016brya** , the god of the sun; or **Yama** , the god of **death**.\n\nBecause there are many devas, corresponding to many different aspects of reality, translating _deva_ as \"god\" communicates a polytheistic view of Hinduism. If one speaks of \"the **gods** \" in the plural in a Hindu context, one is typically referring to the devas. If, however, one translates _deva_ as \"angel\" (which may be a more appropriate analogue), then one gets a picture of Hinduism as a tradition with a view of divinity closer to those of the monotheistic Abrahamic traditions. God, in the singular, in a monotheistic sense, corresponds to the Sanskrit terms **\u012a\u015bvara** or **Bhagav\u0101n**. Some of the Hindu traditions, such as **Vai\u1e63\u1e47avism** and **\u015aaivism** , do conceive of God in this way (with the one supreme deity being identified with either **Vi\u1e63\u1e47u** or **\u015aiva** , and other deities, the devas, playing a role similar to that of angels in the Abrahamic traditions). But there is also the idea in Hinduism of panentheism\u2014of divinity as an all-pervasive presence that corresponds to the **Ved\u0101ntic** concept of **Brahman**. God, as Brahman, is an omnipresent energy and consciousness that can be approached and experienced either as the supreme deity, in a monotheistic sense, or through various manifestations that correspond to the many devas. One could call this a monotheistic polymorphism\u2014the idea of one supreme divine reality that takes many forms.\n\nBecause Brahman is present in _all_ beings, then all beings\u2014including ourselves\u2014are at least potentially divine, or divine in our true, inner nature that will only be fully realized when we become **enlightened**.\n\nOn the other hand, because most Hindu traditions see the universe as eternal, with no specific beginning in time but rather as undergoing an ongoing process of emergence, existence, and dissolution, there is no creator God in an absolute sense, as there is in the Abrahamic traditions, who creates the universe _ex nihilo_. So in this sense, Hinduism can even be seen as atheistic. _See also_ GODDESS.\n\n**GODDESS.** In Hindu traditions, far from being limited to the male form, divinity also manifests in a great variety of female forms. As with Hindu **gods** , one can conceive of the goddesses of Hinduism in **polytheistic** , **monotheistic** , or panentheistic senses. **Dev\u012b** , the feminine form of the **Sanskrit** word **deva** , can simply refer to a female deva\u2014one of many goddesses in a polytheistic sense. **Vedic** goddesses, like their male counterparts, are largely identified with or preside over natural phenomena, such as **U\u1e63\u0101** , the goddess of the dawn, and **Sarasvat\u012b** , the goddess of the river of the same name (who is identified in later Hindu traditions as the goddess of wisdom). In the **\u015a\u0101kta** tradition, though, _Dev\u012b_ is a proper name for the Goddess in the singular, in a monotheistic sense, as the supreme deity of the entire universe: the **Mother Goddess**. Dev\u012b, or **Mah\u0101dev\u012b** (Great Goddess) is identified in many Hindu traditions as the wife of **\u015aiva**. She has many forms, such as the docile wife and mother, **P\u0101rvat\u012b** ; the warrior goddess, **Durg\u0101** , who defeats **demonic** beings and drives away evil; and the fierce, demon-slaying **K\u0101l\u012b**.\n\nIn her role as **\u015aakti** , the power of **creation** , the Goddess can also be identified with all of the many goddesses of the Hindu pantheon, who are seen as her various forms or manifestations. Goddesses such as **Lak\u1e63m\u012b** and Sarasvat\u012b are also called _\u015baktis_ , or power centers, of the deities that are their husbands\u2014 **Vi\u1e63\u1e47u** and **Brahm\u0101** , respectively. In this role, the Goddess and her various personae are seen as the energizing principle, in the absence of which their corresponding male gods would be incapable of effective activity. The divine pairing of \u015aiva and \u015aakti, in particular, is often presented, especially in \u015a\u0101kta texts, as a composite being, with the **God** and Goddess ultimately inseparable from one another as parts of an internally differentiated, but finally unitary, godhead. In its affirmation of internal relations within the godhead, this concept is not unlike the **Christian** idea of the trinity.\n\nFinally, the supreme Mother Goddess is identified, in **Tantric** traditions (and in **Neo-Ved\u0101nta** , which is strongly informed by the Tantric background and sensibility of **Ramakrishna** ), with **Brahman** , the absolute, as it appears from the perspective of **m\u0101y\u0101** , the universe of time and space. Ramakrishna states that \"K\u0101l\u012b is verily Brahman, and Brahman is verily K\u0101l\u012b. It is one and the same Reality. When we think of It as inactive . . . then we call it Brahman. But when It engages in . . . activities, then we call It K\u0101l\u012b or \u015aakti.\" This statement implies a pantheistic or panentheistic understanding of the Goddess as being manifested within all beings. This understanding is also present in the ancient \u015a\u0101kta scripture, the _Dev\u012b Mah\u0101tmya_ or _Ca\u1e47\u1e0d\u012b Pa\u1e6dh_ , with its refrain of **mantras** that refer to \"The Goddess who dwells within all beings.\"\n\nSomewhat ironically, there is no clear relationship between the worship of female deities, to the point of identifying the supreme reality as the Goddess, and the treatment of actual Hindu **women** historically. Patriarchy has coexisted with Goddess-worship in Hindu practice and the worship of the Goddess (and various goddesses) has not been seen as the basis for a feminist critique of patriarchy until the modern period. It is also not at all clear that Goddess-worship has been in any way harmful to women. Rather, the two phenomena\u2014Goddess-worship and patriarchy\u2014seem to be independent variables. Again, this situation has begun to change in the modern period, with Hindu feminists utilizing the discourse of Goddess-worship as a starting point for a critique of patriarchy. But this is a relatively recent development.\n\n**GODS.** Deities; divine beings. When referring to \"gods\" in the plural, in a **polytheistic** sense, one is typically referring, in a Hindu context, to the **devas**. These deities can be seen as distinct beings\u2014superpowerful entities in charge of some aspect or other of the natural world, such as **Agni** , the god of fire; or **S\u016brya** , the god of the sun; or **Yama** , the god of **death**. But they can also be seen as forms or aspects of one supreme deity\u2014 **God** in a **monotheistic** sense, performing various roles and taking on various personalities for the purpose of maintaining the universe. This understanding most commonly applies to what one could call the \"high gods\"\u2014deities like **Vi\u1e63\u1e47u** , **\u015aiva** , and the **Mother Goddess** , who are widely seen as the one God, of whom all the other gods are either forms or aspects. The gods can also be seen as subordinate demigods or angels, ultimately answering to the **One**. It is also possible, finally, to see all personal forms of divinity as manifestations of a higher energy or consciousness\u2014 **Brahman** \u2014that is ultimately impersonal.\n\n**GOD-REALIZATION.** Brahma- **nirv\u0101\u1e47a** ; the state of ultimate realization that liberates one from the cycle of **death** and **rebirth**. In **Advaita Ved\u0101nta** , this realization consists of the direct awareness that **Brahman** and **\u0101tman** are one: that the individual self and the ultimate reality at the basis of all existence are not separate entities.\n\nAccording to Advaita Ved\u0101nta, the differentiation between Brahman and \u0101tman is a mere appearance, or **m\u0101y\u0101**. One therefore does not \"become\" one with the divine so much as realize that one was never separate from the divine in the first place. But in more dualistic, theistic forms of **Ved\u0101nta** , there is a real metaphysical distinction between the individual and the divine. God-realization in these traditions is therefore conceived as a loving union or harmonizing of wills between the individual devotee and **God** in the form of **bhakti** \u2014a relationship of devotion and of absolute dependence of the devotee upon the divine\u2014not a realization of a literal metaphysical unity. _See also_ MOK\u1e62A.\n\n**GOLDEN AVAT\u0100RA.** Epithet of **Caitanya**.\n\n**GOLDEN EMBRYO.** _See_ HIRA\u1e46YAGARBHA.\n\n**GOLOKA.** \"Cow world\"; name for **K\u1e5b\u1e63\u1e47a** 's celestial realm, also known as **Vaiku\u1e47\u1e6dha**.\n\n**GONDA, JAN (1905\u20131991).** Dutch **Indologist** ; professor of **Sanskrit** from 1932 to 1970 at the University of Utrecht. A prolific author, Gonda published a wide variety of articles throughout his lifetime on many Indological subjects.\n\n**GOP\u012a.** \"Cowherdess\"; \"milkmaid\"; literally \"cowgirl,\" though the cultural connotations of this word in English make it an odd fit as a translation. It refers specifically to a group of women in the **Vraja** region who, according to the _Bh\u0101gavata Pur\u0101\u1e47a_ , were the lovers and playmates of **K\u1e5b\u1e63\u1e47a** in his youth. **R\u0101dh\u0101** was the foremost among the gop\u012bs and the special beloved of K\u1e5b\u1e63\u1e47a. Their relationship is detailed in the _G\u012btagovinda_ , a work by the 12th-century **bhakti** **poet** **Jayadeva**. In **Vai\u1e63\u1e47ava** theology, the deep mutual affection between K\u1e5b\u1e63\u1e47a and the gop\u012bs is an ideal model of bhakti, or devotion, being seen as very spontaneous and free from concern for social conventions. Vai\u1e63\u1e47ava practice is aimed at cultivating a devotion to K\u1e5b\u1e63\u1e47a like that of the gop\u012bs. The practitioners\u2014men and women alike\u2014are encouraged to visualize themselves as the gop\u012bs. This can be done mentally or, in some cases, through an actual enactment of female roles.\n\n**GORAKHN\u0100TH (fl. 12th century CE).** **\u015aaiva** sage and second **guru** in the lineage of the **N\u0101th Yog\u012bs** ; founder of the **K\u0101\u1e47pha\u1e6da** sect of ascetics; also credited with the creation of **Ha\u1e6dha Yoga**. Gorakhn\u0101th is regarded as a **siddha** , or a perfected, **enlightened** being.\n\n**GOSV\u0100MINS (fl. 16th century CE).** Literally \"cow masters\"; **Vai\u1e63\u1e47ava** theologians. The Gosv\u0101mins were six of the immediate disciples of **Caitanya** , at whose command they all settled in the **Vraja** region, in **V\u1e5bnd\u0101vana** , where **K\u1e5b\u1e63\u1e47a** is said to have resided during his youth. The theology of the Gosv\u0101mins is foundational for the **Gau\u1e0d\u012bya** tradition of **Vai\u1e63\u1e47avism** and the theology of the **International Society for Krishna Consciousness** (ISKCON).\n\nLike the **Tantric \u015aaiva** philosopher **Abhinavagupta** , the Gosv\u0101mins see **rasa** , or aesthetic experience, as being akin to the experience of **mok\u1e63a** , and as itself a potentially liberating phenomenon. The Gosv\u0101mins applied this basic Tantric insight to the practice of **bhakti** , arguing that listening to and visualizing oneself as a participant in the stories of K\u1e5b\u1e63\u1e47a and his earthly activities can evoke an aesthetic experience that, by K\u1e5b\u1e63\u1e47a's grace, can lead to the state of **liberation**. One can visualize oneself as K\u1e5b\u1e63\u1e47a's servant, parent, friend, or lover. All four relationships are commended, though the last is often seen as the highest and best, being modeled on the relationship between K\u1e5b\u1e63\u1e47a and the **gop\u012bs** (and K\u1e5b\u1e63\u1e47a and **R\u0101dh\u0101** in particular).\n\n**GOTRA.** A name identifying one as a descendant of one of the Seven Sages ( **Saptar\u1e63i** ) presiding over the cosmos. Traditionally, persons with the same gotra cannot marry one another, being regarded as blood relatives. A large number of gotra names (49 according to some accounts) have been generated from the original seven, each of which is believed to trace back to an ancient sage. Gotra names are largely, but not exclusively, applied to **Brahmins**.\n\nIf one does not know one's gotra name (or have a gotra name), one uses the gotra of **Ka\u015byapa** , who is held to be the father of all of humanity. Gotras are traditionally used in ritual contexts in order to identify the participants.\n\n**GOVARDHANA, MOUNT.** Mountain near **Mathur\u0101** , in the **Vraja** region of northern India, that is sacred to **Vai\u1e63\u1e47avas** for its association with the youth of **K\u1e5b\u1e63\u1e47a**. K\u1e5b\u1e63\u1e47a is said to have lifted the mountain singlehandedly and used it to shelter the local residents from a rainstorm that had been sent by **Indra**. This event is a popular **artistic** subject, the mountain often shown being held up by K\u1e5b\u1e63\u1e47a with only his little finger. The mountain is a popular pilgrimage site, being an essential stop on the larger tour of sacred sites in the Vraja region.\n\n**GOVINDA.** \"Cow-finder\"; \"cowherd\"; epithet of **K\u1e5b\u1e63\u1e47a** , who performed this particular duty in his youth. The image of K\u1e5b\u1e63\u1e47a gently caring for his herds is a popular symbol for divine benevolence, analogous to the **Christian** image of Jesus as the \"Good Shepherd.\"\n\n**GR\u0100MA-DEVAT\u0100.** Town or village deity; the presiding deity of a town or a village.\n\n**G\u1e5aHA-PRAVE\u015aA.** Entry into the house; a **Vedic** ritual held to bless a new home, drive away negative forces, and ensure the health, happiness, and prosperity of all who dwell in it.\n\n**G\u1e5aHASTHA-\u0100\u015aRAMA.** The **householder** stage of life; the second of the four stages of life, or **\u0101\u015bramas**. The G\u1e5bhastha-\u0101\u015brama is characterized by marriage, child-rearing, and economically productive activity.\n\n_G\u1e5aHYA S\u016aTRAS_ **.** A set of texts that explain how to perform **Vedic** **domestic rituals** ; probably composed around 300 BCE.\n\n**GU\u1e46A.** Quality. According to the **S\u0101\u1e43khya** and **Yoga** systems of **philosophy** , as well as the _Bhagavad G\u012bt\u0101_ , all natural phenomena are made up of a combination of three basic gu\u1e47as: **sattva** (clarity, luminosity), **rajas** (dynamism), and **tamas** (inertia). In some texts, the term _gu\u1e47a_ refers to any good quality. **God** is therefore referred to as _gu\u1e47\u0101\u015braya_ , or \"abode of good qualities.\" As such, both God and the world are regarded in **Ved\u0101nta** as making up **Sagu\u1e47a Brahman** , or \" **Brahman** with qualities.\" **Advaita Ved\u0101nta** contrasts Sagu\u1e47a Brahman with **Nirgu\u1e47a Brahman** , or \"Brahman without qualities.\"\n\n**GUPTA.** Hindu royal dynasty that ruled much of northern India from 320 to 550 CE; the second of the three premodern Indian empires that controlled a substantial portion of the subcontinent, the first being the **Maurya** Empire and the third being that of the **Mughals**. In contrast with many dynasties that preceded them\u2014most notably that of the Mauryas\u2014which gave heavy patronage to **\u015brama\u1e47a** traditions such as **Buddhism** and **Jainism** , the Guptas were patrons chiefly of the **Brahmins**.\n\nThe Gupta period saw a massive burst of creativity in the areas of **artistic** and textual production. Many classic Hindu texts, like the _Pur\u0101\u1e47as_ and the _Dharma \u015a\u0101stras_ , date to this era, as do the plays and poems of **K\u0101l\u012bd\u0101sa** , who is sometimes called the \"Shakespeare of **Sanskrit**.\"\n\n**GURU.** Teacher, spiritual teacher; literally, one who is \"heavy\" or \"weighty,\" important. Traditionally, a **brahmacarin** or **sanny\u0101sin** lives for a period of time in his guru's **\u0101\u015brama** , or place of spiritual retreat, which is typically in a remote location, away from the distractions of society. A guru will often be affiliated with a particular teaching lineage, or **sa\u1e43prad\u0101ya**. The disciples or students of a guru are called **\u015bi\u1e63yas**. A guru who is the leader, or **\u0101c\u0101rya** , of a sa\u1e43prad\u0101ya will choose one of his \u015bi\u1e63yas to succeed him as guru after his physical **death**. The tradition of succession from guru to \u015bi\u1e63ya is called the **guru-\u015bi\u1e63ya-para\u1e43para**. This idea of a teaching lineage is roughly analogous to the concept of apostolic succession in the Roman Catholic Church. In **Tantric** traditions and in **Neo-Ved\u0101nta** , the guru\u2013\u015bi\u1e63ya relationship is fairly formalized. One formally enters into this relationship, taking a particular teacher as one's guru and being accepted by that teacher as that teacher's \u015bi\u1e63ya, by taking **d\u012bk\u1e63a** , or initiation, from that person. Initiation typically involves the guru giving the new \u015bi\u1e63ya a **mantra** for the \u015bi\u1e63ya to use for the purpose of **meditation**. In many traditions, this mantra is secret\u2014not to be revealed by the \u015bi\u1e63ya to anyone, under any circumstances. A secret mantra given by one's guru is said to be particularly potent\u2014to possess **guru-\u015bakti** , the power of the entire teaching lineage.\n\n**GURU P\u016aR\u1e46IM\u0100.** Full-moon day of the Hindu month of **\u0100\u1e63\u0101\u1e0dha** (which corresponds roughly to the period from mid-June to mid-July); holiday devoted especially to paying respect to one's **guru** , or spiritual teacher.\n\n**GURU-\u015aAKTI.** The collective spiritual power of a **sa\u1e43prad\u0101ya** , or lineage of teachers and students, which is passed down from **guru** to disciple through the act of **d\u012bk\u1e63\u0101** , or initiation.\n\n**GURU-\u015aI\u1e62YA-PARA\u1e42PAR\u0100.** Literally, \"teacher-disciple from one to the other\"; the teacher\u2013disciple lineage; tradition; the unbroken succession of a spiritual tradition from teacher (or **guru** ) to student ( **\u015bi\u1e63ya** ). The student, in time, becomes the teacher to another group of students to whom the lineage is then passed. Those students themselves become teachers to other students, and so on.\n\n**GURUMAY\u012a (1955\u2013 ).** **Sw\u0101m\u012b Chidvilas\u0101nanda** , the current **guru** of the **Siddha Yoga** lineage, which was established by **Sw\u0101m\u012b Nity\u0101nanda** (1897\u20131961) and was perpetuated and popularized by **Sw\u0101m\u012b Mukt\u0101nanda** (1908\u20131982). Born Malti Shetty, Gurumay\u012b's parents were devotees of Sw\u0101m\u012b Mukt\u0101nanda during her childhood and frequent visitors to Mukt\u0101nanda's **Ga\u1e47e\u015bapuri** **\u0100\u015brama**. Initiated into the tradition through the ritual of **\u015baktipat** at the age of 14, she became Mukt\u0101nanda's English translator and accompanied him on three world tours.\n\nIn 1982, Shetty took her ascetic vows and joined the **Sarasvat\u012b** branch of the **Da\u015ban\u0101mi Order** , taking the name Sw\u0101m\u012b Chidvilas\u0101nanda (\"Bliss of the Play of Consciousness\"). Before he died in 1982, Sw\u0101m\u012b Mukt\u0101nanda designated both Gurumay\u012b and her brother, Sw\u0101m\u012b Nity\u0101nanda (Subhash Shetty), as his co-successors in leading the Siddha Yoga movement and organization. However, in 1985, Gurumay\u012b became the sole head of Siddha Yoga after a controversial falling out with her brother. Theologically, Siddha Yoga draws on both **Ka\u015bm\u012br \u015aaivism** and **Ved\u0101nta**. Having a heavily **Tantric** emphasis, the practice centers around the experience of the **ku\u1e47\u1e0dalini** energy through the mediation of the guru, who is a major object of **devotion**.\n\n**GURUV\u0100RA.** Thursday, the day of the week that is governed by the planet Jupiter. The presiding deity of the planet Jupiter is the sage **B\u1e5bhaspati** , the **guru** of the **Vedic** deities, or **devas** \u2014hence the name _Guruv\u0101ra_ for this day. _See also_ CALENDAR, HINDU.\n\n**GURUVAY\u016aR.** Town in the coastal state of **Kerala** , in southern India; a pilgrimage center famous for its **temple** to **K\u1e5b\u1e63\u1e47a** in his infant form. Every year a series of eight **dance** dramas is performed in the **K\u016b\u1e6diy\u0101\u1e6d\u1e6dam** style that depicts events from K\u1e5b\u1e63\u1e47a's life.\n\n**GWALIOR.** Town located in the northern part of the central Indian state of Madhya Pradesh. Historically it has been a site of great military importance and the capital city of several Indian kingdoms from the 15th century to the modern period.\n\n## H\n\n**HALEBID.** Site located in southern **Karnataka** ; home to a variety of **temples** dating to the 11th and 12th centuries CE. The temples, built by the Hoysala dynasty, are among the most elaborately carved Hindu temples in India, with walls and pillars that appear to be almost alive with motion, due to practically every space upon them being carved with realistic figures of deities enacting scenes from the Hindu scriptures.\n\n**HANUMAN.** Ape deity, helper of **R\u0101ma** , and general of the animal army that assists R\u0101ma in his siege upon **La\u1e45ka** in the _R\u0101m\u0101ya\u1e47a_. Hanuman is said to represent the ideal of **bhakti**. His profound devotion to R\u0101ma enables him to perform great feats of strength that include leaping across the ocean from India to La\u1e45ka to discover the fate of **S\u012bt\u0101** and lifting an entire mountain in order to bring a healing herb to **Lak\u1e63ma\u1e47a** , who has been wounded in battle. Though a devotee of R\u0101ma, and often represented iconographically with R\u0101ma, S\u012bt\u0101, and Lak\u1e63ma\u1e47a as a loyal servant bowing or kneeling at their feet, for many Hindus, Hanuman is a deity in his own right. According to one **Vai\u1e63\u1e47ava** tradition, Hanuman is an **avat\u0101ra** of **\u015aiva** who came to Earth to assist R\u0101ma in his struggle with the wicked **R\u0101va\u1e47a**. Hanuman's great physical strength is also attributed to the fact that he is the son of the wind god, **V\u0101yu**.\n\n**HARA.** Destroyer. Also the destroyer of evil, or the destroyer of the universe at the end of a cosmic cycle, in order that a new universe might emerge to take its place. Epithet of **\u015aiva**.\n\n**HARAPPA.** Location in modern Pakistan and site of one of the two largest cities of the Harappan, or **Indus Valley** , or **Indus-Sarasvat\u012b civilization** , the other major site being **Mohenjo-Daro**. _See also_ HARAPPAN CIVILIZATION.\n\n**HARAPPAN CIVILIZATION.** Also known as the **Indus Valley** or **Indus-Sarasvat\u012b** **civilization**. The largest Harappan site is located at **Mohenjo-Daro**. Both Harappa and Mohenjo-Daro had populations ranging from 40,000 to 50,000, making them by far the largest cities of the ancient world. The civilization as a whole had a massive population, for its time, of over 5,000,000.\n\nThe Harappan civilization was at the height of its technological development from 2600 to 1900 BCE, although it displays archaeological continuity with other sites in the region, such as Mehrgarh, that date from as early as 7000 BCE. There is a great deal of fascination with this civilization because of its high level of technological advancement, the fact that its **script** has not yet been deciphered, and the fact that certain practices\u2014and perhaps corresponding concepts\u2014of Hinduism may have originated with this civilization.\n\nIt was believed at one time that this civilization had been destroyed in the **\u0100ryan** **invasion** , but this idea has since been discredited, the dominant theory now being that the civilization declined due to earthquakes and floods.\n\n**HARDW\u0100R (HARIDV\u0100RA).** \"Door to **Hari** ( **Vi\u1e63\u1e47u** )\"; \"Vi\u1e63\u1e47u's Gate\"; popular and very holy pilgrimage site on the river **Ga\u1e45g\u0101**. Hardw\u0101r is one of the sites of the **Kumbha Mel\u0101**. The sage **K\u0101pila** is believed to have practiced **asceticism** here. It also has a Vi\u1e63\u1e47u **temple** featuring a stone bearing his footprints. **Dak\u1e63a** 's sacrifice, which was destroyed by **\u015aiva** after **Sat\u012b** immolated herself, is also said to have occurred on the site of Hardw\u0101r. Finally, Hardw\u0101r is one of several sites on the Ga\u1e45g\u0101 where it is believed to be especially auspicious to deposit the ashes of one's departed loved ones and perform the **\u015br\u0101ddha** rite in honor of one's ancestors. _See also_ ANCESTRAL RITES.\n\n**HARE KRISHNA MOVEMENT.** Popular name for the **Gau\u1e0d\u012bya Vai\u1e63\u1e47ava** tradition brought to the West by **Bhaktived\u0101nta Sw\u0101m\u012b Prabhup\u0101da**. _See also_ INTERNATIONAL SOCIETY FOR KRISHNA CONSCIOUSNESS.\n\n**HARI.** \"Remover of sins\"; also \"yellowish green\"; epithet of **Vi\u1e63\u1e47u**.\n\n**HARIHARA.** Composite deity made up of both **Vi\u1e63\u1e47u** ( **Hari** ) and **\u015aiva** ( **Hara** ). The right half of the deity depicts \u015aiva and the left half depicts Vi\u1e63\u1e47u. Harihara symbolizes the ultimate unity of the two deities and the religious traditions centered upon them. \u015aiva represents the forces of entropy and chaos in the universe and Vi\u1e63\u1e47u represents the forces of order. In these terms, Harihara represents a state of balance between the two.\n\nOne of the oldest representations of Harihara dates from the sixth century of the Common Era and is located at **B\u0101d\u0101m\u012b** , in the southern Indian state of **Karnataka**.\n\n**HARIJAN.** \"Children of God\"; a term that was coined by **Mohand\u0101s K. G\u0101ndh\u012b** for the **Untouchables** and other persons of low **caste**. This term was rejected by **Bh\u012bmrao Ambedkar** as patronizing. The term _Dalit_ \u2014\"the oppressed\"\u2014also employed by members of these castes, emphasizes the suffering this group has endured as a result of **casteism**. Both terms\u2014 _Harijan_ and _Dalit_ \u2014have their own political implications, and use of one or the other seems to map onto the attitude of the person employing it toward Hinduism and the degree to which it is seen to be in need of either reform or complete rejection. This, however, is difficult to verify in a scientific fashion, and one does find persons who use the two terms interchangeably. The formal term preferred by the government of India is **Scheduled Castes**.\n\n_HARIVA\u1e42\u015aA_ **.** \"Lineage (or genealogy) of Hari ( **Vi\u1e63\u1e47u** )\"; a three-part **Sanskrit** text that is often appended to the _Mah\u0101bh\u0101rata_ and is sometimes regarded, because of its subject matter, as one of the _Pur\u0101\u1e47as_. The first section of this work presents the history of the **Yadava** clan, of which **K\u1e5b\u1e63\u1e47a** was a member. The second section focuses specifically on the life of K\u1e5b\u1e63\u1e47a, particularly the episodes of his youth that are not covered in depth in the _Mah\u0101bh\u0101rata_. The third section consists of prophecies regarding the rest of the current era, or **Kali Yuga** , which begins with K\u1e5b\u1e63\u1e47a's **death**.\n\n**HARRISON, GEORGE** **(1943\u20132001).** Member of the Beatles who was known for his deep interest in Hinduism and particularly for his roles in promoting both the **Hare Krishna movement** and the practice of **Transcendental Meditation** among his fans in Europe and North America, as well as his promotion of Indian classical **music** , notably the sitar. He was associated, along with the other Beatles, with the **Maharishi Mahesh Yogi** , but his more enduring relationship was with **Bhaktived\u0101nta Sw\u0101m\u012b Prabhup\u0101da**.\n\nHarrison's interest in Hinduism was sparked by an interest in Indian music, which prompted him to befriend **Ravi Shankar** , from whom he learned to play the sitar. Hindu themes were a prominent part of Harrison's song lyrics from 1967 until his death in 2001, his most famous Hindu-influenced song being his 1970 hit, \"My Sweet Lord.\"\n\n**HAR\u1e62A (590\u2013647 CE).** Har\u1e63a is regarded, along with A\u015boka and **Akbar** , as one of the great emperors of Indian history. After the decline of the **Gupta** dynasty, the second great empire after that of the Maurya to unite much of the Indian subcontinent, Har\u1e63a succeeded in developing a large and prosperous kingdom, in which education and culture could thrive. The period of Har\u1e63a's reign is one of the best documented in ancient Indian history, being covered in great detail in two texts composed during his time: the _Har\u1e63a Carita_ , or biography of Har\u1e63a, written by the **Sanskrit** **poet** B\u0101\u1e47a, and the travelogue of Xuanzang, a Chinese Buddhist monk who traveled extensively in India during Har\u1e63a's reign, writing in vivid detail about India's natural features, society, and forms of political administration.\n\nOne reason for Har\u1e63a's success was his longevity. A talented administrator, he was able to oversee his kingdom for 41 years, ruling from 606 to 647 CE. He died without an heir, however, and did not leave a strong administrative structure to succeed him, relying on his own talents and the force of his personality to rule his kingdom. It therefore suffered the same fate of other Indian empires and disintegrated into a number of small kingdoms after his **death**.\n\nHarsha hailed from what is now the state of Haryana, in northwestern India, from the city of Thanesar, though he later moved his capital to the city of Kanyakubja, now known as Kannauj, in what is now the state of Uttar Pradesh, on the banks of the **Ga\u1e45g\u0101**. The early years of his reign were preoccupied with warfare with \u015aa\u015ba\u1e45kha, king of Gau\u1e0da, in what is now **Bengal**.\n\nLike A\u015boka before him and Akbar to follow, Har\u1e63a practiced a policy of tolerance toward all the religions practiced in his realm, giving patronage to **Buddhism** , **Jainism** , and the various Hindu sects. Har\u1e63a himself came from a **\u015aaiva** family. But he was also a generous patron of Buddhism, particularly Mah\u0101y\u0101na Buddhism. In Xuanzang's account, Har\u1e63a's conflict with \u015aa\u015ba\u1e45kha is cast as a religious war between Buddhism and **\u015aaivism** , with \u015aa\u015ba\u1e45kha being depicted as an archpersecutor of the Buddhists and Har\u1e63a as their champion. Scholars suspect, however, that this is a biased account, given the prevalence of mixed religious allegiances in ancient India. Har\u1e63a continued to support \u015aaivism while also supporting Buddhism, and some of the most prominent Buddhist educational institutions in India, such as the great monastic university of Nalanda, were in \u015aa\u015ba\u1e45kha's kingdom.\n\nA play written by Har\u1e63a, _N\u0101g\u0101nanda_ , reflects Har\u1e63a's mixed religious allegiance. The play is essentially a Buddhist one, being a retelling of a _J\u0101taka_ tale, or story of one of the Buddha's past lives. But Har\u1e63a makes the goddess **Gaur\u012b** , the wife of \u015aiva, a prominent figure in his play\u2014an incorporation that is his own innovation.\n\n**HA\u1e6cHA YOGA.** Literally \"the **Yoga** of Force\"; system of physical postures, stretches, and breathing exercises attributed to the 12th-century **N\u0101th Yog\u012b** ascetic **Gorakhn\u0101th**. When most Westerners use or hear the term **yoga** , they usually have this system in mind. The original purpose of this system is not, however, the pursuit of physical health as an end in itself but to prepare the physical body for higher yogic disciplines like **meditation** and the **Tantric** practice of raising the **Ku\u1e47\u1e0dalin\u012b**.\n\nStarting in the early 20th century and continuing to the present day, many innovations have been made in Ha\u1e6dha Yoga. A number of the stretches and **\u0101sanas** used in contemporary Ha\u1e6dha Yoga are not part of Gorakhn\u0101th's original system, and many of these innovations have come from the West. _See also_ A\u1e62\u1e6c\u0100\u1e44GA YOGA.\n\n**HAVAN.** Literally \"offering\"; a **Vedic** ritual of sacrifice ( **yaj\u00f1a** ) of **ghee** and aromatic powders into the sacred fire ( **Agni** ). This ritual, intended for purification, is a variant of the ancient **Agnihotra** yaj\u00f1a. Traditionally the preserve of orthodox **Brahmins** , popular and widespread practice of this ritual on a regular basis has been heavily promoted in the modern period, particularly by the **\u0100rya Sam\u0101j**.\n\n**HEAVEN.** _See_ SVARGA; VAIKU\u1e46\u1e6cHA.\n\n**HELL.** _See_ NARAKA; P\u0100T\u0100LA.\n\n**HIM\u0100LAYA.** Abode of snow; the Him\u0101layan mountain range that forms the northern boundary of the Indian subcontinent. The Him\u0101layas are religiously significant to Hindus for several reasons. The home of **\u015aiva** and **P\u0101rvat\u012b** , Mount **Kail\u0101sa** , is located in this mountain range. The melting snows of the Him\u0101layas are also the source of many sacred rivers, including the **Ga\u1e45g\u0101**. These mountains have been a popular destination for **sanny\u0101sins** who dwell and practice **meditation** within Him\u0101layan caves.\n\n**HINDI.** One of India's many official languages; mother tongue to roughly 40 percent of the people of India and medium of a popular film and **music** industry. It should not be confused with the term _Hindu_.\n\n**HINDU.** _See_ HINDUISM.\n\n**HINDU AMERICAN FOUNDATION (HAF).** Organization established in 2004 in order to advocate for the Hindu community in the United States and globally. Modeling itself on such minority advocacy groups as the Anti-Defamation League, the Hindu American Foundation has published reports on violations of the rights of Hindus around the world, as well as a report on anti-Hindu hate speech on the Internet. It has also produced and circulated **literature** , aimed primarily at journalists, that is intended to correct common misconceptions about Hinduism. Most controversially, the HAF was party to a legal dispute over the representation of Hinduism in high school textbooks in the California school system. This, in turn, led to accusations by some scholars that members of the foundation had **Hindu Nationalist** leanings. The stated goals of the Hindu American Foundation, though, emphasize this organization's strong commitment to the shared Hindu and American ideals of tolerance and pluralism, which would seem to be incompatible with such a political orientation.\n\n**HINDU CALENDAR.** _See_ CALENDAR, HINDU.\n\n**HINDU DHARMA \u0100C\u0100RYA SABH\u0100.** Organization of **\u0101c\u0101ryas** , or leaders of Hindu **sa\u1e43prad\u0101yas** , or denominations, established in 2002 by **Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b (1930\u2013 )**. The Hindu Dharma \u0100c\u0101rya Sabh\u0101 has a membership of over 100 \u0101c\u0101ryas and is an attempt to bring greater institutional cohesion to Hinduism in order to advance a number of interests and concerns shared across many sa\u1e43prad\u0101yas. Noteworthy **actions** of the Hindu Dharma \u0100c\u0101rya Sabh\u0101 include a 2008 summit between the Sabh\u0101 and the chief rabbis of Israel in order to facilitate dialogue between Hinduism and Judaism, as well as many proclamations that condemn religious proselytizing as a form of cultural violence against indigenous religions.\n\n**HINDU MAH\u0100SABH\u0100.** **Hindu Nationalist** organization established in 1909 by Pandit Mohan Malaviya with the assistance of other members of the **\u0100rya Sam\u0101j**. During the struggle for Indian independence, the Hindu Mah\u0101sabh\u0101 advocated violent revolution. It subsequently became a political party, but it has recently been eclipsed by the **Bharat\u012bya Janat\u0101 Party** (BJP) as the primary voice of Hindu Nationalism in India. It has close ties to other Hindu Nationalist groups, such as the **R\u0101\u1e63\u1e6dr\u012bya Svayamsevak Sa\u1e45gh** (RSS) and the **Vi\u015bva Hindu Pari\u1e63ad** (VHP).\n\n**HINDU NATIONALISM.** Also known as **Hindutva** ; political ideology loosely united around the concept of India as a Hindu state. Hindutva, or \"Hinduness,\" as defined by **Vinayak Damodhar Savarkar (1883\u20131966)** in an essay of the same name, is a conception of Hindu identity that conflates it with Indian identity. According to Savarkar, a Hindu is one who is of Indian descent, who claims India as his or her homeland, and who adheres to a religion of Indian origin. Indians who do not practice a religion of Indian origin are often seen by Hindu Nationalists as less Indian than self-identified Hindus. But though Savarkar's definition would seem to preclude the possibility of non-Indians being Hindu, non-Indian Hindus are generally welcomed as Hindus by Hindu Nationalists as persons sympathetic to Hinduism.\n\nHindu Nationalism has been widely criticized both by political opponents within India and by scholars globally. As an ideology that advocates an identification of Indian-ness with Hinduness, Hindu Nationalism is at cross-purposes with the ideal of secularism on which the Indian constitution is based, according to which all religions are welcome in India and all Indians are free to practice whichever religion they choose. Some within the Hindu Nationalist movement, however, openly advocate a Hindu state.\n\nThere is also a close association of militant Hindu Nationalist rhetoric with actual violence against religious minority communities in India, especially Muslims. **Mohand\u0101s K. G\u0101ndh\u012b** was assassinated by Nathuram Godse, a Hindu Nationalist who felt G\u0101ndh\u012b had betrayed the Hindu community by allowing the partition of India and Pakistan. Hindu Nationalists, for their part, criticize what some call the \"pseudo-secularism\" of the Indian government, whose protection of minority communities the Hindu Nationalists regard as appeasement carried out at the expense of the Hindu majority. The Hindu Nationalist organizations, forming a loose coalition known as the **Sa\u1e45gh Parivar** , include such groups as the **R\u0101\u1e63\u1e6dr\u012bya Svayamsevak Sa\u1e45gh** (RSS), the **Vi\u015bva Hindu Pari\u1e63ad** (VHP), and the **Bharat\u012bya Janat\u0101 Party** (BJP).\n\n**HINDUISM.** The religion practiced by Hindus is a highly internally diversified family of traditions that were generally not, until the modern period, seen as forming one unified whole. The word _Hindu_ itself is derived from the Persian pronunciation of _Sindhu_ , the **Sanskrit** name for the river known to the Greeks as the **Indus**. This was the term for the people of India among non-Indians for many centuries. The word simply meant \"Indian\" and had no specifically religious connotations until the time of **Al-B\u012br\u016bn\u012b** (973\u20131048), an Arab Muslim scholar who traveled in India and wrote extensively about Indian religious customs in his _Kitab al-Hind_ (\"Book of India\"). Al-B\u012br\u016bn\u012b's account of the \"religion of the Hindus\" remained an authoritative source outside India for many centuries, until more detailed studies were made by Christian missionaries and other European scholars during the period of British colonial rule.\n\nBut as more detailed knowledge of Indian religiosity emerged, it became clear that not all Indians practiced the same religion, and a great deal of religious variety was present in the subcontinent. Some Indians practiced religions brought to India from outside\u2014traditions such as **Islam** , **Christianity** , Judaism, and **Zorastrianism**. The religions native to India included some traditions that affirmed the sanctity of the _Vedas_ ( **Vai\u1e63\u1e47ava** , **\u015aaiva** , **\u015a\u0101kta** , and **Sm\u0101rta** traditions) and some that did not ( **Jainism, Buddhism** , and **Sikhism** ). Over time, scholarly and popular usage has come to define _Hinduism_ as Vedic religion, although **Hindu Nationalists** continue to use the term _Hindu_ to refer to all traditions of Indic origin (despite the objections of many Jains, Buddhists, and Sikhs). In 1995, the Supreme Court of India defined Hinduism as \"Acceptance of the _Vedas_ with reverence; recognition of the fact that the means or ways to salvation are diverse; and the realization of the **truth** that the number of gods to be worshiped is large, that indeed is the distinguishing feature of Hindu religion.\" Whether one must be born a Hindu or can \"convert,\" becoming Hindu despite being born in another tradition, remains a disputed question.\n\nThe foreign origins of the terms _Hindu_ and _Hinduism_ , combined with the fact that the definitions of these terms was largely a function of a colonial scholarly discourse, has led some scholars to argue that these terms are not appropriate. Some Hindus have come to the same conclusion, preferring to refer to their traditions collectively not as _Hinduism_ but as _San\u0101tana Dharma_.\n\n**HINDUPHOBIA.** Aversion to Hindus or to Hinduism; cultural bias, possibly ethnically motivated, against Hindus, Hinduism, or both. This term was first coined by independent scholar **Rajiv Malhotra** in response to what he perceived as anti-Hindu bias in academic writing on Hinduism. The term has also been used by the **Hindu American Foundation** to describe the anti-Hindu rhetoric of **Christian** fundamentalist organizations as found on the Internet. The term is controversial, with academic scholars of Hinduism and religious groups that have been accused of Hinduphobia objecting that legitimate disagreements or differences of interpretation have been placed by Hindu critics on the same level as racist hate speech. At the same time, some writing on Hinduism, particularly on the Internet, clearly exhibits an anti-Hindu bias. It seems that Hinduphobia can sometimes be in the eye of the beholder and can sometimes be a genuine phenomenon.\n\n**HINDUST\u0100N.** \"Land of the Hindus\" or \"Land beyond the Hindu (the **Sindhu** or **Indus** ) River\"; Persian name for India, still frequently in use among speakers of **Hindi** and Urdu.\n\n**HINDUTVA.** _See_ HINDU NATIONALISM.\n\n**HIRA\u1e46YAGARBHA.** \"Golden Embryo.\" According to several **creation** stories in the **Vedic** **literature** , the cosmos emerged from a golden egg floating on an infinite ocean of nonexistence. According to some accounts, **Brahm\u0101** emerged from this egg and began the process of creation.\n\n**HIRA\u1e46YAKA\u015aIPU.** \"Golden-robed\"; one of the **Daityas** (sons of **Diti** ). Personifying ignorance, he is said to have ruled over the entire Earth. He held a deep hatred for **Vi\u1e63\u1e47u**. Annoyed by his son **Prahl\u0101da** 's equally deep devotion to Vi\u1e63\u1e47u, Hira\u1e47yaka\u015bipu taunted Prahl\u0101da. Once, asking Prahl\u0101da where Vi\u1e63\u1e47u could be found, Prahl\u0101da answered that Vi\u1e63\u1e47u (whose name means \"all-pervasive\") was everywhere, even inside a nearby stone pillar. On hearing this, Hira\u1e47yaka\u015bipu kicked the pillar. The pillar then broke open and Vi\u1e63\u1e47u, in the form of the **Narasi\u1e43ha** , or man-lion **avat\u0101ra** , came forth. The man-lion grabbed Hira\u1e47yaka\u015bipu, placed him on his lap, and slew him. This event occurred at dusk. The circumstances surrounding the **death** of Hira\u1e47yaka\u015bipu were a result of the fact that he was magically protected from any attack by man or beast (the Narasi\u1e43ha was neither), on the ground or in the air (which is why Narasi\u1e43ha held him in his lap), and by day or night (making dusk and dawn times of vulnerability for him).\n\n**HIRA\u1e46YAK\u1e62A.** \"Golden-eyed\"; one of the **Daityas** (sons of **Diti** ). Personifying, like his brother **Hira\u1e47yaka\u015bipu** , ignorance that is overcome by divine wisdom, Hira\u1e47yak\u1e63a is said to have cast the Earth into the cosmic ocean, from which it was rescued by the boar **avat\u0101ra** of **Vi\u1e63\u1e47u** \u2014 **Var\u0101ha** \u2014who then slew Hira\u1e47yak\u1e63a (thereby evoking Hira\u1e47yaka\u015bipu's enmity).\n\n**HIRIYANNA, M. (1871\u20131950). Indologist**. Based in Mysore, **Karnataka** , Hiriyanna authored numerous works on Indian **philosophy** , particularly **Ved\u0101nta**. His most famous work is probably his _Outlines of Indian Philosophy_.\n\n**HOL\u012a (HOLIK\u0100).** Spring festival held on the **P\u016br\u1e47im\u0101** (full-moon day) of the lunar month of **Phalguna** (February\u2013March), celebrated by both Hindus and **Sikhs**. This playful festival is celebrated by throwing brightly colored powders and liquids on friends, family members, and passersby. It is a celebration of the renewal of life at springtime.\n\nThe name of the festival comes from the name of a female demon, the sister of **Hira\u1e47yaka\u015bipu** , who sought to burn her nephew **Prahl\u0101da** , a devotee of **Vi\u1e63\u1e47u** , to **death**. Prahl\u0101da was spared, and it was Holik\u0101 who ended up burning to death\u2014an event that this holiday commemorates.\n\n**HOLY MOTHER.** Epithet of **Sarad\u0101 Dev\u012b**.\n\n**HOMA.** A very basic **Vedic** sacrifice ( **yaj\u00f1a** ) that consists of the ritual offering of **ghee** (clarified butter) to **Agni** (the sacred fire); part of the daily **Agnihotra** ritual traditionally performed by orthodox **Brahmins** and promoted for widespread practice in the modern period by the **\u0100rya Sam\u0101j**. _See also_ HAVAN.\n\n**HOPKINS, E. WASHBURN (1857\u20131932).** **Indologist** ; author of a number of works on the _Mah\u0101bh\u0101rata_ and professor of **Sanskrit** at Yale University from 1895 to 1926. By his own account, Hopkins had a decidedly negative view of **Jainism** , which he once infamously said \"has indeed no right to exist.\"\n\n**HORSE SACRIFICE.** _See_ A\u015aVAMEDHA.\n\n**HOT\u1e5a.** **Brahmin** priest whose role in the **Vedic** sacrifice, or **yaj\u00f1a** , is to chant selected verses from the _\u1e5ag Veda_ as appropriate to the ritual context.\n\n**HOUSEHOLDER.** _See_ \u0100\u015aRAMA; G\u0100RHASTHYA.\n\n**HUXLEY, ALDOUS (1894\u20131963).** English author and adherent of **Ved\u0101nta**. Huxley's best-known works include _The Perennial Philosophy_ , _Brave New World_ , and _The Doors of Perception_. Huxley immigrated to the United States in 1937. His close association with Ved\u0101nta\u2014specifically, the **Neo-Ved\u0101nta** tradition of **Ramakrishna** and **Sw\u0101m\u012b Vivek\u0101nanda** \u2014began in 1939, when he met **Sw\u0101m\u012b Prabhav\u0101nanda** , the founder and head of the **Ved\u0101nta Society** of Southern California. Huxley received **d\u012bk\u1e63\u0101** , or formal initiation into the Ved\u0101nta tradition, from Prabhav\u0101nanda and began a regular practice of **meditation**.\n\nVed\u0101ntic themes pervade Huxley's writings\u2014sometimes in a very obvious way, at other times more subtly. He was fascinated, even before his practice of Ved\u0101nta, with the hidden potentials of the human **mind** and had a deep interest in psychic research. He was among the early advocates of using psychedelic drugs to expand consciousness and one of the heroes of the counter-cultural movement of the 1960s.\n\nHuxley is at his most Ved\u0101ntic in _The Perennial Philosophy_ , where he advocates the idea of a universal wisdom present in the teachings of the mystics of all the world's great religious and philosophical traditions\u2014\"the metaphysic that recognizes a divine Reality substantial to the world of things and lives and minds; the psychology that finds in the soul something similar to, or even identical with, divine Reality; the ethic that places man's final end in the **knowledge** of the immanent and transcendent Ground of all being.\" This idea of a **perennial philosophy** is close to the Neo-Ved\u0101nta tradition's teaching of religious pluralism, which is based upon the multireligious practices of Sri Ramakrishna, according to which many traditions contain deep **truths** and can serve as vehicles toward **God-realization**. Critics, however, argue that the many quotations from various traditions that Huxley cites have been taken out of their original contexts.\n\n## I\n\n**ICONOGRAPHY.** _See_ DAR\u015aANA; IMAGES IN WORSHIP; M\u016aRTI.\n\n**IL\u0100.** The **goddess** of the **Earth** , the wife of **Dyau\u1e25 Pit\u1e5b** , the heavenly father. _See also_ BH\u016a; BH\u016aDEV\u012a; P\u1e5aTHV\u012a.\n\n**IMAGES IN WORSHIP.** Images, or **m\u016brtis** , have been an integral part of Hindu ritual for millennia. The specific ritual significance of the apparently religious images of the **Indus Valley civilization** is not clear. But resemblances to later imagery are suggestive of some continuity between that ancient culture and later Hindu practice. Though there is occasional mention of the physical appearance of some of the **devas** or deities of **Vedic** religion, physical evidence suggests that Vedic religious practice was centered not on concrete images but on the sacred fire. By the Common Era, though, images of the Hindu deities constructed from a wide array of materials\u2014stone, clay, wood, marble, and metal\u2014are very much in evidence, and continue to play an important role to the present time.\n\nThe theology behind **m\u016brti-p\u016bj\u0101** , or image worship, is developed most fully by the **Vai\u1e63\u1e47ava Ved\u0101ntic** **\u0101c\u0101rya** **R\u0101m\u0101\u1e47uja**. According to R\u0101m\u0101\u1e47uja's thought, as in the Abrahamic monotheistic traditions, **God** is all-pervasive. But in order to accommodate human devotees, God is willing to take on a physical form, becoming embodied in such a form in order to facilitate **bhakti** (devotion). When an image in a **temple** is consecrated, it becomes the \"seat\" or \"abode\" of the deity it represents. Devotees can then approach the deity as they would a respected human guest, offering gifts of food, flowers, water for drinking and bathing, clothing, incense, and so on. It is not that the image itself is divine, any more (or less) than any other object is divine, divinity being omnipresent. But such a practice allows human beings to express their devotion and to increase that devotion by the process of worship ( **p\u016bj\u0101** ). Because the use of images in worship is forbidden in the Abrahamic traditions, this Hindu practice has historically evoked very negative reactions from **Christians** and Muslims who have encountered it. The prevention of \"idolatry\" has been used as an ideological justification for the destruction of Hindu temples, and the association of the religious use of images with superstition has been used as a rhetorical device to denigrate Hinduism as less rational than \"pure\" Abrahamic **monotheism**.\n\nIn the modern period, many Hindu reformers, such as **R\u0101m Mohan Roy** (1772\u20131833) and **Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b** (1824\u20131883), have absorbed these criticisms of the use of images in worship. When combined with the fact that the Vedic **literature** does not appear to endorse (nor even to be aware of) such practices, these reformers have shown a kind of Protestant zeal to put an end to Hindu \"idolatry\" in the name of \"purifying\" the tradition. But the organizations established by Roy and Day\u0101nanda\u2014the **Brahmo Sam\u0101j** and **\u0100rya Sam\u0101j** , respectively\u2014have been outliers in this regard, with mainstream Hindus continuing to practice m\u016brti-p\u016bj\u0101, though this practice is frequently accompanied by a defensive effort to explain its significance in monotheistic terms.\n\n**IMMORTALITY. Am\u1e5bta** ; elixir of life churned from the cosmic ocean by the **devas** and **asuras**. As an objection of religious aspiration, immortality is characterized in some **Vedic** texts not as literal deathlessness, but as a long, full, and healthy life span, perhaps of a hundred years. Even the **gods** of Hinduism are not literally deathless but live only for the length of a **kalpa** , or cosmic epoch, merging back with the rest of the universe in the great dissolution that ends one of these vast periods.\n\nThe doctrines of **karma** , **rebirth** , and **mok\u1e63a** , or **liberation** from rebirth, require one to approach the topic of immortality in different terms when looking at Hinduism and other Indic traditions than one might normally expect from a conventional Western point of view. An unending existence in a physical body is not seen, in most Hindu traditions, as a particularly desirable thing. The goal, rather, is being free from **death** by being free from rebirth. Immortality is therefore conceived, such as in the famous prayer from the _Upani\u1e63ads_ that asks, \"Lead from the unreal to the real; lead me from darkness to light; lead me from death to immortality,\" not as a state of unending physical existence but as freedom from rebirth. Only the soul ( **\u0101tman** , **j\u012bv\u0101tman** ) is truly immortal in the sense of being completely free from death. It is the temporary, periodic identification of the soul with a body that makes it experience death. If the soul ceases to identify with a body, it is free from this experience.\n\n**INDO-\u0100RYAN.** Subset of the **Indo-European** language family consisting of **Sanskrit** and the other northern Indian languages related to it. Early Indo-\u0100ryan includes **Vedic** and classical **Sanskrit**. Middle Indo-\u0100ryan consists of the various **Pr\u0101krit** vernacular languages commonly used in ancient and early medieval India\u2014including the P\u0101l\u012b and Ardha-M\u0101gadh\u012b languages of the **Buddhist** and **Jain** scriptures, respectively. Late Indo-\u0100ryan includes modern northern Indian languages such as **Hindi** , Bengali, Mar\u0101thi, and Gujarati, among others. The term _Indo-\u0100ryan_ does not refer to an ethnic group but only to language and culture. _See also_ \u0100RYAN.\n\n**INDO-EUROPEAN.** Language family that includes **Sanskrit** and the other languages of northern India that are related to it, as well as a host of languages spoken in other parts of Eurasia, including ancient and modern Iranian and Persian, Armenian, Hittite, Greek, Latin (and the Romance languages derived from it), and the Celtic, Germanic, and Slavic languages. The close affinities between these languages and the cultures in which they are spoken form the basis for their being regarded as a \"family.\" Speculation regarding the origins of this family and close studies of the extent of the affinities between them and of non-Indo-European \"substrate\" elements contained within them have led to the theory that all of these languages are derived from a common source dubbed \"Proto-Indo-European\" by linguists. The explanation given for the dispersal of Proto-Indo-European linguistic and cultural elements across Eurasia is an actual movement of peoples from a presumed homeland. Though an undisputed consensus has not yet been reached, scholars generally maintain that this homeland was located somewhere in the vicinity of the Black and Caspian Seas, in Central Asia.\n\nThe location of the Indo-European homeland is a matter of contention in the study of India, due to the belief among some Hindu scholars that the Indo-European homeland must have been India, and that Sanskrit is the oldest Indo-European language (possibly Proto-Indo-European). The latter claim is unlikely to be true, given the presence of non-Indo-European substrate elements in Sanskrit. But critics of a Central Asian homeland point to ambiguities in the evidence that could support either an east-to-west or a west-to-east migration pattern. The entire conversation is highly politically charged, in part due to the fact that the idea of an Indo-European homeland became bound in the early part of the 20th century with Nazi ideology and the concept of a \"pure, white race.\" _See also_ \u0100RYAN; \u0100RYAN INVASION THEORY; \u0100RYAN MIGRATION THEORY; INDO-\u0100RYAN; OUT OF INDIA THEORY.\n\n**INDOLOGISTS, INDOLOGY.** Indology is the academic study of India and of Indian culture, with an Indologist being a scholar who pursues Indology. This term has largely been superseded in North America by the term \"South Asian Studies\" (due to the fact that the modern-day Republic of India does not exhaust the region still widely known as the Indian subcontinent). Drawing on a wide range of disciplines, including anthropology, archaeology, history, linguistics, **philosophy** , and religious studies, Indology was one of the first \"area studies\" to develop in Western universities.\n\nOne of the first Indologists was **Sir William Jones** (1746\u20131794 **)** , who founded the **Royal Asiatic Society** in **Calcutta** in 1784. A brilliant linguist, Jones was one of several scholars who discovered the **Indo-European** language family, discerning the similarities between **Sanskrit** and both ancient and modern European languages. Early Indologists focus much of their work on the translation of ancient Indian texts, such as the _Bhagavad G\u012bt\u0101_ (translated in 1785 by Charles Wilkins) and the _\u1e5ag Veda_ (translated between 1850 and 1888 by **Horace Hayman Wilson** ).\n\n**Friedrich Max M\u00fcller** (1823\u20131900), a giant in the field of Indology, oversaw the translation of numerous Sanskrit texts through his _Sacred Books of the East_ series. He is also known for developing the **\u0100ryan Invasion** \u2014later modified to the **\u0100ryan Migration** \u2014theory of **Vedic** origins. M\u00fcller was also the first Western scholar to become aware of **Ramakrishna** , writing one of the first works in a European language on the life and teachings of this Hindu saint.\n\nA multitude of other scholars\u2014American, European, and Indian\u2014have contributed to the field of Indology through the years. Prominent names include, but are by no means limited to, **A. L. Basham** , **Norman Brown** , **R. N. Dandekar** , **S. N. Dasgupta** , **S. K. De** , **Wendy Doniger** , **Georges Dum\u00e9zil** , **Mircea Eliade** , **M. Hiriyanna** , **S. Radhakrishnan** , **Romila Thapar** , **J. A. B. Van Buitenen** , and **Robert Charles Zaehner**.\n\nIn its earliest stages, Indology was closely tied to the colonial enterprise, and area studies in general remain heavily funded by government interests, including the Departments of Defense and State. This historical inheritance has led, in recent decades, to a good deal of self-critical scholarship and reflection on the ideological assumptions and effects of Indological scholarship\u2014a conversation recently joined by independent Hindu scholars such as **Rajiv Malhotra** who have questioned the validity of much academic scholarship on Hinduism, a topic that has become highly politically charged. _See also_ GONDA, JAN; HOPKINS, E. WASHBURN; KANE, P. V.; INGALLS, DANIEL H. H.; MAJUMDAR, R. C.; RAGHAVAN, V.; RAMANUJAN, A. K.; SUKTHANKAR, V. S.\n\n**INDRA.** Lord of the **devas** , or **Vedic** deities; god of thunder and lightning who rides a **chariot** through the sky, destroying monsters with his thunderbolt weapon; one of the 12 **\u0100dityas** (sons of **Aditi** ). Indra is mentioned more frequently than any other deity in the _\u1e5ag Veda_ , indicating his importance in ancient Vedic religion, though he is a figure of less importance in later Hinduism (playing relatively minor roles in the _Pur\u0101\u1e47as_ , _R\u0101m\u0101ya\u1e47a_ , and _Mah\u0101bh\u0101rata_ ). Along with his fellow deities **Agni** , **Varu\u1e47a** , and **Yama** , Indra is one of the four **Lokap\u0101las** , or \"World Guardians.\" Apart from guarding the realms of fire, the ocean, the underworld, and the sky, respectively, these four deities are frequently invoked early in ritualistic worship, or **p\u016bj\u0101** , in order to help ensure that the ritual will be completed without mishap.\n\nIn the _Vedas_ , Indra is most famous for destroying the demon **V\u1e5btra** , who had trapped the world's waters and its cows inside his mountain cave. Indra destroys V\u1e5btra and sets the waters and cows free, allowing the Earth to flourish. He is in many ways comparable to other **Indo-European** sky deities, such as the Greek Zeus and the Norse Thor. The latter's slaying of the Midgard serpent strongly resembles the battle between Indra and V\u1e5btra. _See also_ INDRA-DHVAJA FESTIVAL; JARJARA.\n\n**INDRA-DHVAJA FESTIVAL.** Festival celebrating **Indra** 's victory over the forces of evil and chaos. A _dhvaja_ is a banner or flag, and the celebration of this festival includes the setting up of a large pole representing Indra's flagpole. The festival is held on the full-moon day or **P\u016br\u1e47im\u0101** of the lunar month of **K\u0101rtika** (October\u2013November).\n\n**INDRAPRASTHA.** Ancient capital of the **P\u0101\u1e47\u1e0davas** ' kingdom in the _Mah\u0101bh\u0101rata_ , located in the general vicinity of **New Delhi**.\n\n**INDRIYAS.** The senses; in most forms of **yoga** , the senses need to be restrained\u2014that is, brought under the control of the **mind**. In **Tantra** , the senses can be used as a medium to experience divine realities, and thus to transcend themselves.\n\n**INDUS RIVER.** _Indus_ , from which India gets its name, is the Greek name for the river **Sindhu** , which flows from the **Him\u0101layas** to the Arabian Sea through the middle of what is now Pakistan. The **Indus Valley civilization** was centered on this river.\n\n**INDUS VALLEY CIVILIZATION.** Also known as the **Indus Valley** , the **Harappan** , or the **Indus-Sarasvat\u012b** **civilization**. The largest and longest-known sites connected with this civilization are located at **Mohenjo-Daro** and **Harappa** in the Indus River valley in contemporary Pakistan. Both of these cities had populations, at the height of the Indus civilization, ranging from 40,000 to 50,000, making them by far the largest cities of the ancient world. Comparable cities in the ancient Near East had populations of, on the high side, 10,000. The civilization as a whole had a massive population for its time, over 5,000,000. The cities were planned in advance, rather than growing organically from a smaller habitation, and indicate a high degree of engineering ability.\n\nThe Indus Valley civilization was at the height of its technological development from 2600 to 1900 BCE, although it displays archaeological continuity with other sites in the region, such as Mehrgarh, that date from as early as 7000 BCE, and with later sites that extend into the first millennium before the Common Era. There is much fascination with this civilization because of its high level of technological advancement, the fact that its **script** has not yet been deciphered, and the fact that certain practices\u2014and perhaps corresponding concepts\u2014of Hinduism may have originated with this civilization. Figures that appear to be seated in **yoga** postures are represented on clay seals from this culture, as well as figures evocative of both **\u015aiva** and the **Mother Goddess** , though the precise meaning of these images is a matter of some dispute and will likely remain so until the Indus script is deciphered. Ritual bathing and cleanliness also seem to have been a major preoccupation of this civilization, with a good deal of engineering activity being devoted to the building of public baths, sanitation, and irrigation.\n\nCompared to other ancient civilizations of the same period, the Indus civilization extended over a vast area and engaged in overseas trade extending to Mesopotamia and the Arabian Peninsula, as well as into Central Asia. Indus-manufactured items have been discovered in all of these regions. The civilization's center of gravity, though, was the fertile valley formed by the Indus and the Gaggar-Hakra, or **Sarasvat\u012b** , rivers.\n\nIt was believed at one time that this civilization had been destroyed in an **\u0100ryan** **invasion** , but this idea has since been discredited, the dominant theory now being that the civilization declined due to earthquakes and floods that led the Indus to change its course and the Sarasvat\u012b to go underground and, in some areas, dry up completely. Any **\u0100ryan** migrants into the region after 1900 BCE would have encountered a civilization already very much on the decline.\n\n**INDUS-SARASVAT\u012a CIVILIZATION.** Alternative designation for the **Indus Valley civilization** that emphasizes the fact that many of this civilization's settlements were centered not only on the **Indus** but also on another river, the Gaggar-Hakra, which once flowed adjacent to it and is identified by many scholars with the **Sarasvat\u012b** River of the **Vedic** scriptures. Due to dramatic geological and meteorological changes, this river dried up around 1900 BCE, a development that was likely a major factor in the decline of the civilization.\n\n**INGALLS, DANIEL H. H. (1916\u20131999). Indologist**. Taught **Sanskrit** at Harvard from 1956 to 1979, authoring many works on Indian **literature** and **philosophy**.\n\n**INTERNATIONAL SOCIETY FOR KRISHNA CONSCIOUSNESS (ISKCON).** **Gau\u1e0d\u012bya Vai\u1e63\u1e47ava** organization established by **Bhaktived\u0101nta Sw\u0101m\u012b Prabhup\u0101da** in 1965. Bhaktived\u0101nta first arrived in New York and found a home with a group of young people who were part of the countercultural movement of that period. It was largely from this movement that the following of ISKCON, the **Hare Krishna movement** , was first drawn, though it has since become somewhat more of a mainstream group, particularly as more Indian families have joined it, and a generation has grown up within the movement.\n\nISKCON is one of the first Hindu organizations, after the **Ved\u0101nta Society** , to have a large non-Indian following. Unlike other largely non-Indian Hindu organizations, ISKCON, in its early days, insisted that its adherents adopt Indian cultural norms\u2014modes of dress, hairstyles, names, and so on. This has been emphasized less in recent years, as leaders of ISKCON have sought to increase the appeal of the tradition, working against common stereotypes of robed and bald-headed devotees chanting **mantras** in airports and on street corners. The organization has put a good deal of emphasis on charitable work and taking effective **action** to address cases of abuse that occurred after the time of Bhaktived\u0101nta, but before the contemporary period. _See also_ BHAKTI; HARRISON, GEORGE; K\u1e5a\u1e62\u1e46A.\n\n**\u012a\u015aA.** Lord, ruler, master, **God**. Epithet of **\u015aiva**.\n\n_\u012a\u015aA UPANI\u1e62AD_. Also known as the _\u012a\u015bavasya Upani\u1e63ad_ , this brief text is traditionally listed first among the _Upani\u1e63ads_ , although most scholars agree that it is not the oldest of the _Upani\u1e63ads_ (that distinction belonging to the _B\u1e5bhad\u0101ra\u1e47yaka Upani\u1e63ad_ ). The _\u012a\u015ba Upani\u1e63ad_ contains some of the basic concepts that would later become central to the various systems of **Ved\u0101nta** **philosophy**. It is theistic, referring to the all-pervasive Lord ( **\u012a\u015ba** , from whom the text takes its title). It also contains a possible allusion to the idea of **karma yoga** and affirms the idea of **Brahman** as a reality beyond all dualistic logical conceptions.\n\n**ISHERWOOD, CHRISTOPHER (1904\u20131986).** English author, adherent of **Ved\u0101nta**. Better known in literary circles for his novels, Isherwood's writings on Ved\u0101nta include a biography of **Ramakrishna** entitled _Ramakrishna and His Disciples_ , _My Guru and His Disciple_ , and, in collaboration with **Sw\u0101m\u012b Prabhav\u0101nanda** , translations of Hindu texts, (including the principle _Upani\u1e63ads_ , the _Bhagavad G\u012bt\u0101_ , and the _Yoga S\u016btras_ of **Pata\u00f1jali** ). Isherwood immigrated to the United States in 1939. His close association with Ved\u0101nta\u2014and specifically, the **Neo-Ved\u0101nta** tradition of Ramakrishna and **Sw\u0101m\u012b Vivek\u0101nanda** \u2014began that same year when he met Sw\u0101m\u012b Prabhav\u0101nanda, the founder and head of the **Ved\u0101nta Society** of Southern California. Isherwood received **d\u012bk\u1e63\u0101** , or formal initiation into the Ved\u0101nta tradition, from Prabhav\u0101nanda, who became his **guru**.\n\n**ISLAM.** The world's second-largest religion, in terms of the number of its adherents, and a member, along with Judaism and **Christianity** , of the Abrahamic family of traditions, sharing a common worldview that includes **monotheism** , divine revelation and divine intervention in human history, a single lifetime, a day of judgment, and an everlasting afterlife either in a heavenly paradise or a hellish state of punishment (though exceptions and variations in regard to this basic worldview can be found across all three of these traditions).\n\nHistorically, relations between Hindus and Muslims (practitioners of Islam) have oscillated between periods of peaceful coexistence and periods of communal violence. Initial contacts between Muslims and Hindus were peaceful, being largely a function of trade with Arab merchants along the coasts of India. During this period, in the latter half of the first millennium of the Common Era, some Hindus from the lower **castes** became converts to Islam in order to escape the inequities of the caste system\u2014though the practice of caste is so pervasive in India that, within a few generations, these Muslim converts had again organized themselves into castes, as Christians had done previously and continue to do even today. Much like the initial coming of Christianity to India, these conversions did not have a major impact on the dominant Hindu community.\n\nIn succeeding centuries, however, waves of invasion\u2014first by Arabs, then Turks, and finally the **Mughals** , often using Islam as an ideological justification for destruction of **temples** and monastic institutions and displacement of Hindu kings\u2014created a situation of deep distrust between the two communities.\n\nUnder Mughal rule, though\u2014particularly the reign of **Akbar** , which lasted from 1556 to 1605\u2014the situation began to change. The Mughal policy of tolerance created the conditions for a hybrid Hindu\u2013Islamic culture to emerge, leading to an amazing burst of creative activity in the visual **arts** , **music** , architecture, and religion itself. Specifically in regard to religion, the **Sant** movement\u2014a fusion of Islamic **Sufi** mysticism with Hindu **bhakti** traditions\u2014emerged, emphasizing the common thread of devotion and longing for the divine uniting the otherwise dramatically different religions of Hindus and Muslims. Mystics like **Kab\u012br** taught the unity of \u0100ll\u0101h (the one **God** of Islam) and **R\u0101ma**. A third tradition, **Sikhism** , arose from this movement, with the teaching of **Guru N\u0101nak** that, in the eyes of God, \"There is no Hindu and there is no Muslim.\" **Syncretism** occurred on a massive scale, as Hindus and Muslims celebrated one another's holy days and borrowed one another's practices and concepts. The only substantial interruption of this period of relative harmony came during the reign of the Mughal emperor Aurangzeb, which lasted from 1658 to 1707. Aurangzeb favored the forcible conversion of Hindus to Islam and engaged in fierce campaign of persecution of both Hindus and Sikhs.\n\nThe colonial period and the British \"divide and rule\" policy that set members of different communities against each other reinforced the sense of separate communal identities and mutual suspicion between Hindus and Muslims. Both Islamic and **Hindu Nationalism** emerged in the late 19th and early 20th centuries, fueling hostile feelings between the two communities. These hostilities culminated with the partition of India and Pakistan when British rule ended in 1947.\n\nRelations between Hindus and Muslims continue to be highly fraught today. Both Hindu minorities in Pakistan and Bangladesh and the Muslim minority in India have, at different times, experienced persecution. India and Pakistan have fought three wars and both have had nuclear weapons since 1998. Particularly low points in Hindu\u2013Muslim relations have been 1992, when the **Babr\u012b Masj\u012bd** was demolished in **Ayodhy\u0101** by Hindu Nationalists, and 2002, when horrific riots broke out between Hindus and Muslims in the state of Gujarat. At the same time, one may observe that these outbreaks of violence take place against a background of largely peaceful coexistence.\n\n**I\u1e62\u1e6cADEVAT\u0100.** Chosen deity, preferred deity, chosen ideal. Based on the idea that **Brahman** , being infinite, is beyond human conception, the **Sm\u0101rta** and **Neo-Ved\u0101nta** traditions of Hinduism teach that Brahman can be approached through a personal form to which it is easier for human beings to relate. The form will vary according to the needs of the **devotee** , but typically takes the shape of one of the Hindu deities, though the deity can also be a saintly historical personage\u2014or even, in more liberal forms of Hinduism, a non-Hindu figure, such as Jesus or the **Buddha**. **Ramakrishna** compares the i\u1e63\u1e6dadevat\u0101 to a hole in a high wall that surrounds a beautiful garden. People who wish to see the garden have to peek through the holes in this wall. The holes will be located at varying heights, to suit the needs of the various people looking in, but the beautiful garden to which these various holes grant them access is one and the same. In the same way, the one universal Brahman is accessed by different types of devotee by means of the various i\u1e63\u1e6dadevat\u0101s. _See also_ GOD; GODS.\n\n**\u012a\u015aVARA.** The Lord, **God**. This is the term by which the Supreme Being is generally designated in Hindu philosophical writings when God in the abstract is under discussion. Both **Vi\u1e63\u1e47u** and **\u015aiva** are frequently designated by this term in the sectarian writings of their respective devotees in the **Vai\u1e63\u1e47ava** and **\u015aaiva** traditions. The other term frequently used for God in the generic sense is **Bhagav\u0101n** , the Blessed One.\n\n**\u012a\u015aVARA-PRA\u1e46IDH\u0100NA.** Contemplation of the Lord; fifth of the **niyamas** ; a major component of the **yoga** system developed by **Pata\u00f1jali**. _See also_ A\u1e62\u1e6c\u0100\u1e44GA YOGA; _YOGA S\u016aTRAS_.\n\n**ITIH\u0100SA.** History, literally \"thus it was.\" This term usually refers to the two primary epics of Hindu **literature** \u2014the _R\u0101m\u0101ya\u1e47a_ and the _Mah\u0101bh\u0101rata_.\n\n**IYER, E. KRISHNA (1897\u20131968).** **Artist** chiefly responsible for reinvigo-rating Indian classical **dance** \u2014specifically **Bharata-n\u0101\u1e6dyam**. A **Brahmin** from **Tamil** Nadu, Iyer first trained in and practiced **law** and participated in the Indian independence movement prior to devoting himself to the revival of classical dance. Through his efforts and those of a variety of dedicated artists who have come since, Indian classical dance has emerged as a highly respected art form after centuries of disrepute.\n\n## J\n\n**JAGADDH\u0100TR\u012a.** Form of **Durg\u0101**. _See also_ AMB\u0100.\n\n**JAGANN\u0100THA.** Lord of the Universe, **God** , epithet of **Vi\u1e63\u1e47u**. The particular **m\u016brti** or image to which this term refers is located at a famous **mandir** , or **temple** , in the city of **Puri** in the state of **Orissa**. Controversially, the administration of this temple does not permit entry by non-Hindus. Non-Indian **devotees** , whom the administration does not regard as truly Hindu, have frequently objected to this practice. The central image of the temple is carried out in procession once a year on a very large cart modeled on a **chariot** or **ratha**. The massive size of this cart, and the inherent difficulty of controlling it once it is in motion, gave rise to the English word \"juggernaut.\"\n\n**JAGAT.** Literally, \"flow\" or \"process\"; the universe and the entities constituting it, in contrast with **God** , or **\u012a\u015bvara** , its lord and controller. According to the **Vi\u015bi\u1e63\u1e6d\u0101dvaita** of the **Ved\u0101ntic** **\u0101c\u0101rya R\u0101m\u0101\u1e47uja** , God and the world together make up a greater unity\u2014 **Brahman**. If one were to draw an analogy between Brahman and a person, God could be said to be the person's **soul** , and the world (jagat) the person's body.\n\n**JAIMINI.** Author to whom the _P\u016brvam\u012bm\u0101\u1e43s\u0101 S\u016btra_ is attributed and founder of the **P\u016brva M\u012bm\u0101\u1e43s\u0101** system of classical Indian **philosophy**. P\u016brva M\u012bm\u0101\u1e43s\u0101, or \"earlier exegesis\" is a continuation of the ancient, orthodox system of Vedic **Brahmanism** , with a very strong emphasis on the correct performance of **Vedic** ritual and a corresponding emphasis on the correct pronunciation of the **Sanskrit** verses of the _Vedas_ during a ritual performance. This emphasis on Sanskrit led to the development of an elaborate linguistic philosophy on the part of the M\u012bm\u0101\u1e43sakas. The term _P\u016brva M\u012bm\u0101\u1e43s\u0101_ contrasts this school of thought with later, or **Ut-tara M\u012bm\u0101\u1e43s\u0101** \u2014more widely known as **Ved\u0101nta**. The system of Jaimini emphasizes the earlier, ritualistic portion of the _Vedas_ , while Ved\u0101nta emphasizes the later Vedic **literature** , the _Upani\u1e63ads_ , and their teaching of **Brahman**.\n\n**JAINISM. \u015arama\u1e47a** tradition that is traced, in its current form, to **Mah\u0101v\u012bra** (c. 499\u2013427 BCE), though according to Jain tradition Mah\u0101v\u012bra is the 24th in a series of **enlightened** beings, or **T\u012brtha\u1e45karas** , who have been teaching the same **truths** for millennia, and there is some evidence that Mah\u0101v\u012bra's teaching is a continuation of an earlier, proto-Jain tradition. Along with **Buddhism** , Jainism is one of the only two surviving \u015brama\u1e47a traditions that thrived in northern India in the first millennium before the Common Era. Other \u015brama\u1e47a groups, such as the **\u0100j\u012bvikas** , had died out by the 13th century CE, as had Buddhism in India (though it had already spread across Asia by this time).\n\nThe Jains are known primarily for their profound commitment to nonviolence, or **ahi\u1e43s\u0101** , which Jain ascetics practice to the greatest extent humanly possible, sometimes wearing a face mask, or _muhpatt\u012b_ , to avoid accidentally inhaling or ingesting insects or microorganisms, and sweeping the ground in front of them as they walk to avoid doing even accidental harm to small living things. Another central Jain value is **aparigraha** , or nonpossessiveness, which some Jain ascetics\u2014the monks of the \"sky-clad\" or _Digambara_ sect, practice to the extent of giving up even the wearing of clothing. Finally, the third \"pillar\" of Jainism is _anek\u0101nta-v\u0101da_ , or \"non-one-sidedness,\" which consists of the view that there is some truth in all perspectives. No tradition is completely wrong (though the teachings of the T\u012brtha\u1e45karas are held to be absolutely true).\n\nJainism shares many core ideas of Hinduism, such as the doctrines of **karma** and **rebirth** and the aspiration for **liberation** ( **mok\u1e63a** ) from the cycle of rebirth as a central salvific goal. Jainism is distinctive in seeing karma as a material substance that adheres to the soul ( **j\u012bva** ) and in the relentless logic it applies to the questions of karmic merit and demerit, with very specific conceptions of which **actions** produce particular effects.\n\nJainism shares with other \u015brama\u1e47a traditions, such as Buddhism, a rejection of the claims of **Brahmanism** regarding the sanctity of **caste** and the authority of the _Vedas_. Specifically, it rejects the idea that birth caste is a reflection of one's spiritual attainment (with its implication that the **Brahmins** are the most evolved of human beings) and **Vedic** injunctions to perform **animal** **sacrifice**. Jainism was an ideological rival of Brahmanism for many centuries. But as Hinduism coalesced from many Indic traditions, some Jain concepts and practices were assimilated into it. For example, though nominal affirmation of the authority of the _Vedas_ continues, practices such as animal sacrifice gradually fell by the wayside, being replaced by **vegetarianism** under the influence of the concept of ahi\u1e43s\u0101. Jain practice in regard to nonattachment has influenced Hindu practices, such as fasting, and there are Hindu orders of naked monks not unlike the Digambaras of Jainism. Though the caste system continues to be practiced, movements that affirm the irrelevance of caste to the spiritual path, such as **bhakti** and **Tantra** , have many Hindu followers. And the Jain teaching of the relativity of perspectives finds an echo in the **Neo-Ved\u0101ntic** teaching that truth can be found in many religions.\n\nJainism was especially influential on the life and teachings of **Mohand\u0101s K. G\u0101ndh\u012b** , who grew up in a region of India (Gujarat) with a strong Jain presence. The central Jain ethical principle of nonviolence can be found not only in the teachings of G\u0101ndh\u012b but also in Hindu texts such as the _Yoga S\u016btras_ of **Pata\u00f1jali** and the _Dharma \u015a\u0101stras_. Even though the Jain community is, numerically, relatively small (there being roughly four and a half million Jains in the world today), the Jains have had a disproportionate influence on the dominant Hindu culture that has surrounded them. This influence is probably a result of several factors, including the relative wealth of this largely mercantile community and the respect evoked by the rigorous **asceticism** and profound commitment to nonviolence exhibited by Jain monks and nuns. _See also_ VRATA; YAMAS.\n\n**JAMADAGNI.** Father of **Para\u015bur\u0101ma** , a wrathful **avat\u0101ra** of **Vi\u1e63\u1e47u** who is said to have slain the entire **k\u1e63atriya** (or warrior) **caste** in vengeance for Jamadagni having been killed by a k\u1e63atriya, Arjuna K\u0101rtav\u012brya (not the **P\u0101\u1e47\u1e0dava** hero **Arjuna** of the _Mah\u0101bh\u0101rata_ ). It is said that Jamadagni and Para\u015bur\u0101ma were **Brahmins** of the **Bh\u0101rgava** clan. After all the k\u1e63atriyas were slain by Para\u015bur\u0101ma, they had to be replaced by Brahmins. It is thus believed that all living k\u1e63atriyas are actually descended from Brahmins.\n\n**JANAKA.** Ancient king of Mithil\u0101, in northeastern India; represented in the _Upani\u1e63ads_ as engaging in frequent discourse with wise sages such as **Yaj\u00f1av\u0101lkya** ; represented in the _R\u0101m\u0101ya\u1e47a_ as the father of **S\u012bt\u0101** and father-in-law of **R\u0101ma**. In the **Ved\u0101nta** tradition, Janaka is said to have been liberated while still alive ( **j\u012bvanmukta** ). His attainment of this goal while remaining a **householder** involved with kingly duties is used to illustrate the point that it is not necessary to practice renunciation in a formal sense ( **sanny\u0101sa** ) in order to be liberated. What is essential is _inner_ renunciation, an attitude of detachment from the fruits of **action**. At the same time, it is emphasized that Janaka is unusual in this regard and likely practiced sanny\u0101sa in previous lifetimes.\n\n**JANM\u0100\u1e62\u1e6cAMI.** Also known as **K\u1e5b\u1e63\u1e47a Janm\u0101\u1e63\u1e6dami** , the birthday of K\u1e5b\u1e63\u1e47a. It is the eighth (a\u1e63\u1e6dami) day of the **dark half** ( **k\u1e5b\u1e63\u1e47a pak\u1e63a** ) of the month of **Bh\u0101dra** in the Hindu lunar calendar, which corresponds roughly with the month of August in the solar calendar.\n\n**JAPA.** The practice of repeating a **mantra** \u2014either aloud but quietly, or purely mentally and silently\u2014while keeping track of the number of repetitions on a string of beads known as a **m\u0101la** , or sometimes as a japa m\u0101la. In most traditions, it is recommended that the practitioner repeat the mantra in units of 108, thus giving rise to the need for a method of keeping track of the number of repetitions. In addition to the m\u0101la, there are also ways of keeping track of one's mantra repetitions using one's fingers.\n\n**JARJARA.** Flagpole of **Indra** , brought onto the stage in traditional Indian theater as a way of invoking Indra's protection. This practice is prescribed by the _N\u0101\u1e6dya \u015a\u0101stra_ , the authoritative textual guide to traditional Indian theater and **dance**.\n\n**JATAYU.** Character in the _R\u0101m\u0101ya\u1e47a_ ; eagle who attempts to rescue **S\u012bt\u0101** from **R\u0101va\u1e47a** in the course of her abduction. As R\u0101va\u1e47a abducts S\u012bt\u0101 in his flying **chariot** ( **vim\u0101na** ), Jatayu attacks and tries to pull her away.\n\n**J\u0100TI.** Birth **caste** ; hereditary occupational group to which one is born. The j\u0101tis are the \"subcastes\" into which the four main castes, or **var\u1e47as** , are divided. Far more than one's var\u1e47a, it is one's j\u0101ti that forms the actual, living communal unit that determines one's caste identity. Relative to j\u0101ti, var\u1e47a is more or less an abstraction. In contrast with the four var\u1e47as, there are dozens of j\u0101tis. The location of each j\u0101ti within the larger caste hierarchy varies a great deal in different regions of India.\n\n**JAYADEVA (fl. 1200 CE). Bengali** author of the _G\u012btagovinda_ , a **bhakti** text that uses the love affair between **R\u0101dh\u0101** and **K\u1e5b\u1e63\u1e47a** to depict the bond of love between the soul of the **devotee** and **God**. The text is highly erotically charged and has been compared by some with the biblical _Song of Solomon_.\n\n**J\u012aVA.** Literally \"life force.\" The individual, reincarnating soul, contrasted, in **Advaita Ved\u0101nta** , with the **param\u0101tman** , or supreme soul, with which all the individual souls are ultimately one. Non-Advaita systems of **Ved\u0101nta** , with their greater emphasis on theistic realism and a strong distinction between the individual **devotee** and **God** , claim that the j\u012bvas always remain numerically distinct from **God** , or **\u012a\u015bvara** , their union being one of love and a harmonizing of wills, rather than an actual obliteration of difference. **Jainism** distinguishes between the j\u012bva in its mundane state, polluted by obscuring **karma** , and the soul in its inherent, perfected state, which it realizes upon **enlightenment** , which consists of infinite **knowledge** , bliss, and freedom. Jains also use the term _param\u0101tman_ to refer to the enlightened j\u012bva. _See also_ \u0100TMAN.\n\n**J\u012aVANMUKTA.** An **enlightened** being. One who is in the state of **j\u012bvanmukti**.\n\n**J\u012aVANMUKTI.** The state of being liberated while still alive, prior to the **death** of the body. The possibility of attaining **mok\u1e63a** while still alive is denied by many Hindu schools of thought\u2014those more theistically and devotionally oriented traditions that think of **liberation** as residing eternally in **heaven** with **God** (usually conceived as residing in **Vaiku\u1e47\u1e6dha** with **K\u1e5b\u1e63\u1e47a** ). Clearly this is a conception of mok\u1e63a as a postmortem state. In **Advaita Ved\u0101nta** , however, where mok\u1e63a consists of the realization of the identity of the **\u0101tman** and **Brahman** , such a realization is seen as being possible while one is still alive. The body of a **j\u012bvanmukta** continues to survive after mok\u1e63a due to **prarabdha karma** of a kind that does not interfere with a correct perception of reality. Such a being appears to others who are not **enlightened** as a human being engaged in the usual activities of life, such as eating, drinking, sleeping, and so forth. Such a being can even be physically injured or become ill. But from the perspective of that being, all is Brahman, and there is no distinction to be made between self and other, or among various karmic states. The activities of such a being are therefore entirely spontaneous and reflective of the nature of Brahman. In many respects, this conception is like that of an enlightened being in the **Buddhist** tradition.\n\n**J\u012aV\u0100TMAN.** _See_ J\u012aVA.\n\n**J\u00d1\u0100NA.** \"Knowledge\"; \"wisdom\"; cognate with the Greek _gnosis_. **Advaita Ved\u0101nta** teaches that knowledge is constitutive of **mok\u1e63a** , or **liberation** from the cycle of **rebirth** \u2014specifically, the knowledge that the individual self, or **\u0101tman** , and the universal reality, or **Brahman** , are one and the same. The **Neo-Ved\u0101nta** of **Ramakrishna** and **Sw\u0101m\u012b Vivek\u0101nanda** teaches that wisdom is one of four possible **yogas** or **m\u0101rgas** (paths) that can lead to liberation (along with devotion, or **bhakti** ; work, or **karma** ; and **meditation** , or **dhy\u0101na** ). Support for both views, as well as endorsements of the sole efficacy of the other paths, can be found in the Hindu **scriptures**. _See also_ FOUR YOGAS; YOGA.\n\n**J\u00d1\u0100NA KANDA.** Literally \"wisdom portion\"; the _Upani\u1e63ads_ ; the final portion of the _Vedas_ , devoted to knowledge of **Brahman** , as distinct from the earlier \" **action** portion\" or **Karma Kanda** , consisting of ritualistic texts focused on sacrificial performance. _See also_ VED\u0100NTA.\n\n**J\u00d1\u0100NA M\u0100RGA, J\u00d1\u0100NA YOGA.** \"Path or way of knowledge\"; \"spiritual discipline of knowledge.\" _See_ J\u00d1\u0100NA.\n\n**J\u00d1\u0100NASAMBANDAR (639\u2013655).** A precocious **Tamil \u015aaiva** devotee who is said to have sung songs of devotion to **\u015aiva** at the young age of three. His father carried him to various \u015aaiva **temples** , where the boy gave sermons and is also said to have performed miracles. J\u00f1\u0101nasambandar dismissed traditional concerns regarding **astrology** , claiming that faith in \u015aiva could override negative astrological influences.\n\n**J\u00d1\u0100NE\u015aVARA (1275\u20131296).** **Bhakti** **poet** from Maharashtra; a **Brahmin** ; best known for his exposition of the _Bhagavad G\u012bt\u0101_. Drawing upon the **Vai\u1e63\u1e47ava** , **\u015aaiva** , and **\u015a\u0101kta** traditions, he commented upon the _G\u012bt\u0101_ but also wrote of the essential unity of **\u015aiva** and **\u015aakti** , asserting that, \"Without the **God** there is no **Goddess** , and without the Goddess there is no God.\"\n\n**JONES, SIR WILLIAM (1746\u20131794).** One of the first **Indologists**. Jones discovered the **Indo-European** family of languages and cultures, being the first to recognize the strong similarities between **Sanskrit** and European languages such as Greek and Latin. He established the **Royal Asiatic Society** in **Calcutta** in 1784, thus formally initiating the study of Indology as an academic discipline. He translated the _Manusm\u1e5bti_ (\"Laws of Manu\"), the _G\u012btagovinda_ of **Jayadeva** , and **K\u0101lid\u0101sa** 's play, the _Abhij\u0101na\u015bakuntala_ or \"Recognition of **\u015aakuntal\u0101**.\"\n\n**JYAI\u1e62\u1e6cHA.** Month of the **Hindu calendar** that corresponds roughly to the period from mid-May to mid-June.\n\n**JYOTI\u1e62A.** _See_ ASTROLOGY.\n\n## K\n\n**KAB\u012aR (1440\u20131518).** **Sant** **poet** who is revered by both Hindus and Muslims. Born into a **caste** of weavers in **Banaras** , his name is Muslim. It is likely that his entire caste converted as a group to **Islam** prior to his birth. Although he uses Islamic terminology, Kab\u012br's writings show a greater familiarity with the _R\u0101m\u0101ya\u1e47a_ than with Islam. Kab\u012br himself did not hold exclusive religious identity in high regard, ridiculing institutional features of both Hinduism and Islam that kept their followers apart. He advocated devotion ( **bhakti** ) to a formless divinity, beyond limiting qualities and attributes. In emphasizing the formless ( **nirgu\u1e47a** ) nature of the divine, his teaching is like **Advaita Ved\u0101nta**. But his emphasis on personal devotion places him closer to theistic bhakti traditions and to Islam. A major influence on the broader Sant movement, and on **Sikhism** , many of Kab\u012br's writings are preserved in the sacred scripture of the Sikhs, the _\u0100di Granth_. His belief in the oneness of \u0100ll\u0101h and **R\u0101ma** and emphasis on personal devotion over institutional affiliation and formal ritual anticipate modern Hindu teachers like **Ramakrishna**.\n\n**KAIL\u0100SA, MOUNT.** Mountain in the central **Him\u0101layas** , located within the modern boundaries of Tibet, and traditionally believed to be the home of **\u015aiva** and **\u015aakti**. It is a popular, if difficult to reach, pilgrimage site. It is near Lake **Manosarovar** , also a sacred site.\n\n**KAIVALYA.** Literally \"isolation\"; a term used in the **S\u0101\u1e43khya** and **Yoga** systems, as well as in **Jainism** , for the state of **mok\u1e63a** , or **liberation** from the cycle of **rebirth** , a state characterized in these traditions as the separation of the soul ( **puru\u1e63a** , **j\u012bva** , or **\u0101tman** ) from matter, or the phenomena of nature (aj\u012bva, **prak\u1e5bti** ) with which it has come to be identified. According to S\u0101\u1e43khya and Yoga, the identification of the soul with matter is merely apparent, to be overcome (in the S\u0101\u1e43khya system) by right understanding of the nature of the soul as pure consciousness, or (in the Yoga system) through the experience of **sam\u0101dhi** , or **meditative** absorption in this consciousness. This is in contrast with the Jain tradition, which sees the soul as having become actually entangled with matter, from which it needs to separate itself through **ascetic** practice\u2014though the Digambara **\u0101c\u0101rya** Kundakunda (from the middle of the first millennium of the Common Era) holds a view that is closer to that of S\u0101\u1e43khya and Yoga.\n\n**K\u0100L\u0100MUKHA.** \"Black face\"; a south Indian **Tantric \u015aaiva** **ascetic** lineage; a subset of the **Pa\u015bupata** tradition of **\u015aaivism**. Their name comes from the black **tilak** , or sectarian mark, that they wear on their foreheads.\n\n**K\u0100L\u012a.** \"Black **Goddess** \"; fierce form of the **Mother Goddess** in her role as slayer of the demonic. She is traditionally depicted as having four arms. Her right hands are shown in the **abhaya-mudr\u0101** and **d\u0101na-mudr\u0101** \u2014the gestures of fearlessness and generosity. This is her benevolent aspect. Her left hands are shown carrying a ferocious-looking sword and the head of a demon. Her skin is black and her tongue, red with blood, is sticking out of her mouth. She is wearing a garland of skulls and a skirt of arms and heads of demons she has slain. She is standing on the prone and apparently unconscious body of **\u015aiva** , her husband. Despite her fierce appearance\u2014evocative, to many non-Hindus, of the demonic\u2014K\u0101l\u012b is seen as a benevolent deity who is fierce in order to protect her devotees from evil. Her iconography depicts a moment from a battle depicted in the _Dev\u012b Mah\u0101tmya_ during which she was fighting a demon known as Raktab\u012bja, or \"blood seed.\" Each drop of Raktab\u012bja's blood that spilled on the ground would transform itself into a completely new Raktab\u012bja. Spilling this demon's blood in battle therefore served only to create an entire army of Raktab\u012bjas. It is for this reason that K\u0101l\u012b used her tongue to catch or lick up every drop of the blood of this demon that was spilled in the course of her battle with him. This is why she is depicted with a long, bloody tongue sticking out of her mouth. It is also said that her husband, \u015aiva, fearing that in her blood frenzy she would destroy the world, laid down on the battlefield as if dead. In her **dance** of destruction, K\u0101l\u012b stepped on top of him. Realizing, abruptly, that she was standing on her (possibly dead) husband, K\u0101l\u012b ceases her destruction and transforms back into the gentle **P\u0101rvat\u012b**. This moment is captured in her **m\u016brti** , or iconographic representation.\n\nK\u0101l\u012b is especially popular in **Bengal** where **Tantric** and **\u015a\u0101kta** traditions are quite strong. She was the preferred deity ( **i\u1e63\u1e6dadevat\u0101** ) of the Bengali saint **Ramakrishna** , who was intensely devoted to her. Her **p\u016bj\u0101** is traditionally performed at midnight on the **Amavasya** , or new-moon day, of the month of **K\u0101rtika** (which corresponds to the period from late October to early November). This is the same night on which most Hindus in northern India celebrate **D\u012bp\u0101vali** , or D\u012bwali.\n\nThe **K\u0101l\u012bgh\u0101\u1e6d** , in **Calcutta** (Kolkata), is one of the few remaining places in India where **animal sacrifice** still occurs, with goats being offered to K\u0101l\u012b. But these sacrifices are controversial, and in many places (especially outside of India), the sacrificial goat has been replaced with a gourd, banana, cucumber, or other **vegetarian** offering. The **\u1e6chags** , or Thugs, used to offer human sacrifices to K\u0101l\u012b. These were usually unwitting travelers who would hire the \u1e6chags as tour guides. The \u1e6chags would then take them to an isolated place and strangle them. Although the basis of the film _Indiana Jones and the Temple of Doom_ , the actual \u1e6chag cult consisted of a small group of men, not the elaborate **temple** community depicted in the sensationalist film. Human sacrifice is not condoned in any mainstream Hindu sect, or even in mainstream Tantric or \u015a\u0101kta thought.\n\nRamakrishna identified K\u0101l\u012b with **Brahman** in its active form, as the energy of **creation**. \"K\u0101l\u012b is verily Brahman, and Brahman is verily K\u0101l\u012b. It is one and the same Reality. When we think of It as inactive . . . then we call it Brahman. But when It engages in . . . activities, then we call It K\u0101l\u012b or \u015aakti.\"\n\n**KALI YUGA.** The fourth, last, and worst of the progressively more unfortunate ages or **yugas** into which a cosmic epoch, or **kalpa** , is divided. We are presently experiencing the Kali Yuga of our current cosmic epoch. According to Hindu traditions, the Kali Yuga began with the death of **K\u1e5b\u1e63\u1e47a** , shortly after the _Mah\u0101bh\u0101rata_ war. Hindu texts date the death of K\u1e5b\u1e63\u1e47a around 3021 BCE, in terms of the predominant global dating system. But mainstream scholarship on Hinduism regards this date as too early for the _Mah\u0101bh\u0101rata_ war, which tends to be placed at either 1500 BCE (the approximate date of the sinking of **Dv\u0101rak\u0101** ) or 950 BCE (based on archaeology in northern India, where most of the epic's main narrative occurs).\n\n**K\u0100L\u012aB\u0100R\u012a.** \"House of **K\u0101l\u012b** \"; **Bengali** generic term for a temple to the **goddess** K\u0101l\u012b.\n\n**K\u0100LID\u0100SA (fl. 4th\u20135th centuries CE).** \"Servant of K\u0101l\u012b\"; **poet** and playwright regarded by many as the \"Shakespeare of **Sanskrit**.\" Flourishing during the reign of the **Gupta** dynasty and the \"Hindu Renaissance\" that the Gupta emperors helped to facilitate, little is known of K\u0101lid\u0101sa's life. Religiously, his work reflects an eclectic sensibility, drawing on both **Vai\u1e63\u1e47ava** and **\u015aaiva** narratives from the _Mah\u0101bh\u0101rata_ , the _R\u0101m\u0101ya\u1e47a_ , and the _Pur\u0101\u1e47as_. His most famous work is the _Abhij\u00f1\u0101na\u015bakuntala_ (\"The Recognition of **\u015aakuntal\u0101** \"), a play based on an episode from the _Mah\u0101bh\u0101rata_. K\u0101lid\u0101sa's Sanskrit is highly refined and distinctive in style, making extensive use of puns, nature imagery, and references to traditional Hindu **literature**. _See also N\u0100\u1e6cYA \u015a\u0100STRA_.\n\n**K\u0100L\u012aGH\u0100\u1e6c.** Famous **K\u0101l\u012b** **temple** in **Calcutta** (Kolkata), for which the city is named. One of the few places where **animal sacrifice** is still practiced.\n\n**KALKIN.** Tenth **avat\u0101ra** of **Vi\u1e63\u1e47u** and the last to appear in the current cosmic epoch. Kalkin will appear at the end of the **Kali Yuga** and fight in a final cataclysmic battle of good against evil. Destroying the forces of chaos ( **adharma** ), he will reestablish order ( **dharma** ) and a new series of **yugas** will begin. Kalkin is usually depicted as a warrior riding on a white horse and carrying a sword.\n\n**KALPA.** _See_ WORLD AGES; YUGA.\n\n**K\u0100MA.** Desire; sensory enjoyment; erotic love. Along with wealth and power ( **artha** ), virtue ( **dharma** ), and **liberation** ( **mok\u1e63a** ), k\u0101ma is one of the four goals of humanity, or **puru\u1e63\u0101rthas**.\n\n**K\u0100MADEVA.** \"God of (erotic) love\"; Cupid-like deity who shoots flower arrows into the hearts of human beings, causing them to fall in love. K\u0101madeva once sought to shoot **\u015aiva** in the heart while he was **meditating** , but before he could do so, \u015aiva opened up his \" **third eye** ,\" burning K\u0101madeva to a crisp with the heat of his accumulated tapas, or ascetic power. K\u0101madeva therefore no longer has a body and is for this reason known as **Ana\u1e45ga** (\"he who has no limbs, or body\"). This episode illustrates the conflict between **k\u0101ma** and **asceticism** , since one who wishes to pursue **mok\u1e63a** , or **liberation** from the cycle of **rebirth** , must renounce attachment to sensory pleasure.\n\n**K\u0100MADHENU.** \"Wish-fulfilling cow\"; daughter of **Dak\u1e63a** and the wife of **Ka\u015byapa** ; one of the treasures to emerge from the ocean when it was churned by the **devas** and the **asuras** in their search for the elixir of life ( **am\u1e5bta** ). The K\u0101madhenu can grant any wish to her owner. _See also_ COW, SACRED.\n\n_K\u0100MA S\u016aTRA_ **.** **Sanskrit** text composed around 300 CE by **V\u0101tsy\u0101yana** Mallanaga; a scholarly treatise on **k\u0101ma** , a term whose meaning encompasses the concepts of sensory pleasure\u2014especially sexual pleasure\u2014desire, and erotic love. Like Sanskrit treatises on the other four goals of humanity ( **puru\u1e63\u0101rthas** ), such as the _Artha \u015a\u0101stra_ and the various _Dharmas \u015a\u0101stras_ , the _K\u0101ma S\u016btra_ approaches its topic in a detailed, systematic fashion.\n\nAlthough long considered scandalous\u2014then later, fashionable\u2014in the West, due to its explicit and frank discussions of sexuality and various ways of both experiencing and heightening sexual pleasure, the traditional Hindu attitude toward sexuality that this text reveals is notable for its nonjudgmental understanding that sexuality is a natural part of the human condition. Although the **ascetic** traditions of Hinduism focus on transcending sexual desire as a way of attaining **mok\u1e63a** , or **liberation** from the cycle of **rebirth** , there is no assumption that an ascetic orientation can or should be expected of all people. For those who are still \"in the world\" and not focused on mok\u1e63a, worldly pursuits like k\u0101ma, **artha** (wealth), and **dharma** (virtue) are perfectly appropriate, and indeed recommended. The _K\u0101ma S\u016btra_ is also of interest to historians for the considerable light that it sheds on ancient Indian society\u2014particularly the wealthy class, who appear to have spent a good deal of time in leisure and social activities.\n\n**K\u0100\u1e42SA.** Character in the _Bh\u0101gavata Pur\u0101\u1e47a_ ; king and wicked uncle of **K\u1e5b\u1e63\u1e47a** who wanted to slay K\u1e5b\u1e63\u1e47a just after his birth because of a prediction that K\u1e5b\u1e63\u1e47a would one day slay him and take control of his kingdom. K\u0101\u1e43sa was not the legitimate son of his father, King Ugrasena of **Mathur\u0101** , but rather the son of an **asura** who had taken Ugrasena's shape and slept with his wife. Having his father's demonic nature, K\u0101\u1e43sa overthrew the king, imprisoning him and usurping the kingship. K\u1e5b\u1e63\u1e47a, true to the prophecy, did slay K\u0101\u1e43sa when he reached adulthood. Like King Herod, as depicted in the New Testament, K\u0101\u1e43sa sought to prevent his eventual slaying by K\u1e5b\u1e63\u1e47a by having all the male children in his kingdom slain.\n\n**KA\u1e46\u0100DA.** Sage to whom the **Vai\u015be\u1e63ika dar\u015bana** , or system of Indian **philosophy** , is attributed. Ka\u1e47\u0101da probably lived around 300 BCE.\n\n**K\u0100\u00d1C\u012aPURAM.** City in **Tamil** Nadu; sacred site of pilgrimage; known for its many **temples** , such as the **\u015aaiva** Kail\u0101san\u0101tha temple and the **Vai\u1e63\u1e47ava** **Vaiku\u1e47\u1e6dha** Perumal. It is also the location of one of the monastic centers, or **ma\u1e6dhas** , established by **\u015aa\u1e45kara** \u2014the K\u0101\u00f1c\u012b K\u0101makoti P\u012b\u1e6da.\n\n**KANE, P. V. (1880\u20131972). Indologist** ; professor of **law** and vice-chancellor of Bombay University; author of the five-volume _History of Dharma\u015b\u0101stra_.\n\n**KANNAPPA (fl. 6th\u201311th centuries CE).** One of the 63 **N\u0101yan\u0101rs** \u2014saints and **bhakti** **poets** revered in the **Tamil \u015aaiva** tradition. It is said that Kannappa (which means \"he who gave his eyes\") offered his eyes to **\u015aiva** , plucking them out upon noticing that the eyes in an image of the deity appeared to be bleeding, his intention being to replace the image's eyes with his own. \u015aiva, being pleased with this profound devotion, restored his sight. It is also said that Kannappa was a hunter by **caste** \u2014an example of the widespread appeal of the bhakti movement, which emphasized personal devotion over birth caste as a measure of one's spiritual advancement.\n\n**K\u0100\u1e46PHA\u1e6cA.** \"Split-ear\"; alternative name for the **N\u0101th Yog\u012bs** , a **Tantric \u015aaiva** lineage of **ascetics**. The name _K\u0101\u1e47phata_ refers to the large earrings that these ascetics wear. The second leader of the lineage, **Gorakhn\u0101th** , is attributed with inventing **Ha\u1e6dha Yoga**.\n\n**KANY\u0100KUM\u0100R\u012a (CAPE COMORIN).** The southernmost tip of India. It derives its name (\"Place of the Young Maiden\") from the belief that it was here that the **Mother Goddess** , as a young girl, performed **ascetic** practice in order to win the hand of **\u015aiva** in marriage. The Kum\u0101r\u012b Amman **temple** , located here, commemorates these practices.\n\nThere are also memorials to more recent Hindu sacred figures at Kany\u0101kum\u0101r\u012b. There is a shrine to **Mohand\u0101s K. G\u0101ndh\u012b** near the shore, where his ashes were kept prior to being immersed in the Indian Ocean. Just off the coast is Vivek\u0101nanda Rock, to which **Sw\u0101m\u012b Vivek\u0101nanda** is said to have swum and where he practiced **meditation** in 1892, before setting off on his first major speaking tour. A small shrine to Vivek\u0101nanda is now located on the rock, to which regular boat rides are available for pilgrims and tourists.\n\n**K\u0100P\u0100LIKA.** \"Skull bearer\"; a south Indian **Tantric \u015aaiva** **ascetic** lineage; a subset of the **Pa\u015bupata** tradition of **\u015aaivism**. Their name comes from their unusual begging bowls, which are made from the hollowed out tops of human skulls. This is in keeping with the **philosophy** of non-dualism that they seek to embody, which perceives traditional ideas of purity and impurity as part of the realm of duality, and so ultimately delusory.\n\n**K\u0100PILA.** Highly revered sage attributed with creating the **S\u0101\u1e43khya** system of Indian **philosophy**. K\u0101pila is mentioned prominently in the _Bhagavata Pur\u0101\u1e47a_ , where he is said to be an **avat\u0101ra** , or incarnation, of **Vi\u1e63\u1e47u** , and is attributed with a theistic version of the S\u0101\u1e43khya system (though historically most S\u0101\u1e43khya texts are nontheistic, presenting a worldview close to that of **Jainism** ). He is also mentioned in the _Bhagavad G\u012bt\u0101_ as the best of sages (10:26). K\u0101pila is associated with the northeastern Greater M\u0101gadha region of India, where Jainism and **Buddhism** were both prominent in ancient times. The city where the **Buddha** grew up\u2014K\u0101pilavastu\u2014is named in honor of this sage.\n\n**KARMA.** \" **Action** \"; \"work\"; principle of cause and effect, by which actions of a certain moral character produce effects of a corresponding nature within the agent who performs them\u2014with benevolent, kind actions leading to happiness for the one who performs them and harmful, violent actions similarly leading to suffering; also, the effects themselves\u2014as in \"good\" karma ( **pu\u1e47ya** karma) and \"bad\" karma ( **p\u0101pa** karma).\n\nThe reciprocal process of intentional moral actions producing like effects fuels the perpetual cycle of **rebirth** ( **sa\u1e43s\u0101ra** ), release from which ( **mok\u1e63a** ) requires one to in some way negate or transcend the principle of cause and effect. The various Indic traditions often present themselves as strategies for achieving this negation or transcendence. The _Bhagavad G\u012bt\u0101_ , for example, presents a **yoga** of action ( **karma yoga** ) that aims at mok\u1e63a through engaging in action without attachment to the results. On this understanding, it is not action itself, but the **desire** that motivates it, which brings about karmic effects.\n\n**KARMA KANDA.** Literally \" **action** portion\"; the _Sa\u1e43hit\u0101s_ and _Br\u0101hma\u1e47as_ ; the early portion of the _Vedas_ , consisting of ritualistic texts focused on sacrificial performance, as distinct from the later \"wisdom portion\" or **J\u00f1\u0101na Kanda** , devoted to knowledge of **Brahman** and consisting of the _Upani\u1e63ads_. _See also_ M\u012aM\u0100\u1e42S\u0100.\n\n**KARMA M\u0100RGA, KARMA YOGA.** \"Path or way of **action** \"; \"spiritual discipline of action.\" As explained by the _Bhagavad G\u012bt\u0101_ , the path of action aims at **mok\u1e63a** (freedom from the cycle of **rebirth** ) through engaging in virtuous action without attachment to the results. On this understanding, it is not action itself but the **desire** that motivates action that brings about karmic effects. The **Neo-Ved\u0101nta** of **Ramakrishna** and **Sw\u0101m\u012b Vivek\u0101nanda** teaches that the way of action is one of the **four yogas** or **m\u0101rgas** (paths) that can lead to **liberation** (along with devotion, or **bhakti** , wisdom, or **j\u00f1\u0101na** , and **meditation** , or **dhy\u0101na** ). _See also_ YOGA.\n\n**KAR\u1e46A.** Character in the _Mah\u0101bh\u0101rata_. Kar\u1e47a is a particularly tragic character. His mother, **Kunt\u012b** , gave birth to him secretly, having tested the secret **mantra** given to her by the sage **Durv\u0101sa** , by which she could call upon any deity of her choosing and bear a child by that deity. Calling the sun **god** , **S\u016brya** , Kunt\u012b gave birth to Kar\u1e47a, who had the characteristics of his divine father. As a young, unmarried mother, Kunt\u012b, in shame, sent the infant Kar\u1e47a down a river in a basket, where he was found by a childless charioteer and his wife, who raised the child lovingly as their own. As an adult, Kar\u1e47a was rejected by the **P\u0101\u1e47\u1e0davas** on the basis of **caste** prejudice, since they believed he was the son of a lowly charioteer. **Duryodhana** , however, perceiving Kar\u1e47a's power, made him a king, thus securing him as a perpetual ally. Though the eldest brother of the P\u0101\u1e47\u1e0davas, Kar\u1e47a gave his loyalty to Duryodhana, thus obligating him to face his brothers in battle and die in combat at the hands of **Arjuna**. The P\u0101\u1e47\u1e0davas, though, were unaware of his identity until after his **death**.\n\n**KARNATAKA.** State in southern India, just north of **Kerala** and **Tamil** Nadu and west of Andhra Pradesh, on the western coast of the Deccan peninsula; location of a number of important Hindu kingdoms of ancient times, including the **C\u0101lukyas** (c. 550\u2013880 CE), the Hoysalas (950\u20131343), and the Vijayan\u0101gara empire (1336\u20131646). All of these kingdoms were patrons of the **arts** , sponsoring the construction of some of the most beautiful and elaborate Hindu **temples** ever built, including the temple complexes of **Belur** , **Halebid** , and **B\u0101d\u0101m\u012b** , among others. Karnataka has also been a home to **Jainism** , particularly the Digambara sect, whose monastery at \u015arava\u1e47a Be\u1e37gola houses a massive stone monument of the Jain saint B\u0101hubali, a product of the Hoysala era. Karnataka's contributions to the **bhakti** movement include the **\u015aaiva** bhakti **poet** **Basava (** 1134\u20131196), the founder of the **V\u012bra\u015baiva** or **Li\u1e45g\u0101yat** tradition, and the **Vai\u1e63\u1e47ava \u0101c\u0101rya Madhva** (1238\u20131317). The author and **Indologist** **A. K. Ramanujan** (1921\u20131993) hailed from Karnataka as well.\n\n**K\u0100RTIKA.** Month of the **Hindu calendar** that corresponds roughly to the period from mid-October to mid-November.\n\n**K\u0100RTTIKEYA.** \"Of the K\u1e5bttik\u0101s (the Pleiades)\"; son of **\u015aiva** and **P\u0101rvati** and brother of **Ga\u1e47e\u015ba** ; also known as **Skanda** , **Subrama\u1e47ya** , **Murugan** , **Kum\u0101ra** , and **\u1e62a\u1e47mukha**. The last name, which means \"having six faces,\" is due to his having been born with six heads and 12 arms, though he is not always depicted in this way. In fact, he is widely regarded as a very handsome deity, often appearing as a young boy (the meaning of the name _Kum\u0101ra_ ). According to one tradition, he was raised by the six goddesses who are the main stars making up the Pleiades, hence the name _K\u0101rttikeya_. A warrior deity, he is associated with the planet Mars and is regarded as a divine general, analogous to Michael in the **Christian** tradition, who leads the armies of heaven in battle against their demonic enemies, the **asuras**. K\u0101rttikeya's vehicle is the peacock ( **m\u0101yur\u012b** ). For this reason, he is known in Mah\u0101y\u0101na **Buddhism** as _Mah\u0101m\u0101yur\u012b_ , \"the Great Peacock,\" and is a guardian of Buddhism and Buddhist holy sites.\n\n**K\u0100\u015a\u012a.** Alternative name for **Banaras** ; the name for this city most often used in ancient sources.\n\n**KASHMIR, KA\u015aM\u012aR.** Region in the extreme north of India; currently an area disputed by India and Pakistan, as well as by a movement for independence from both countries. Known for its breathtaking beauty, Kashmir is located in the **Him\u0101layas** and has a large number of lakes and forests. The region is named for the sage **Ka\u015byapa** , who is believed to have lived there. The dominant religion in the region has historically been **\u015aaivism** \u2014specifically the **Tantric Ka\u015bm\u012br \u015aaiva** tradition. Kashmir was the birthplace of one of the greatest theologians and systematizers of \u015aaiva thought, **Abhinavagupta**. **Buddhism** was also prominent in Kashmir for many centuries, from which it spread to Central Asia. Since the Middle Ages, it has been a major center for **Islam** as well\u2014particularly **Sufism** \u2014and, until recently, was famous for peaceful coexistence between Hindus and Muslims.\n\n**KA\u015aM\u012aR \u015aAIVISM.** Collective term for a variety of **Tantric \u015aaiva** systems of thought and practice either originating or traditionally prominent in **Ka\u015bm\u012br**. Prominent among these are the **Kaula Tantra** system of **Abhinavagupta** (950\u20131020), the Trika system, and the Krama system. The Trika, or \"triadic,\" system focuses on a trio of **goddesses** who are all manifestations of **K\u0101l\u012b** or **\u015aakti** , the divine energy of **\u015aiva**. The rituals of this system include offerings of the **Five M's** , or traditionally impure substances utilized to embody the **truth** of nonduality: that there is truly nothing impure in the world if all is one. One of Abhinavagupta's innovations was to develop a system of contemplative visualization of these practices, so they did not have to actually be physically practiced in order for one to experience their spiritual benefits. The Krama, or \"Steps,\" system similarly utilized practices such as the Five M's, which were also transformed into visualization exercises by Abhinavagupta. One of the results of Abhinavagupta's work was that Kaula Tantra eventually superseded and absorbed the other two systems.\n\n**KA\u015aYAPA.** Literally, \"tortoise\"; **Vedic** sage believed to be the father of all humanity, as well as many of the deities, or **devas** , of the _Vedas_. He was married to 13 of the daughters of **Dak\u1e63a** , including **Aditi** , with whom he fathered the 12 **\u0100dityas** , and both **Diti** and **D\u0101nu** , with whom he fathered the **Daityas** and **D\u0101navas** , respectively. He is mentioned frequently in the _Ath\u0101rva Veda_. Because of his role as father of all human beings, whenever one is not aware of one's **gotra** name (a name that connects a person as a descendant to a Vedic sage, or **\u1e5b\u1e63i** ), one uses the name _Ka\u015byapa_.\n\n_KA\u1e6cHA UPANI\u1e62AD_ **.** One of the principle _Upani\u1e63ads_ , probably composed between 400 and 300 BCE. The text consists primarily of a dialogue between a boy named **N\u0101ciketa** and **Yama** , the **god** of **death**. Seeking to know the secret of **immortality** and what occurs after death, Yama reveals to him the **Ved\u0101ntic** doctrine of death, **rebirth** , and **mok\u1e63a** , or **liberation** from rebirth. The text dwells at length on the nature of the soul ( **\u0101tman** ) and contains a famous metaphor of the body as a **chariot** , with the horses that pull the chariot being the senses, the driver being the **mind** , and the self being the rider within the chariot.\n\n**KATHAK.** Style of classical Indian **dance** developed in Uttar Pradesh, northern India. Kathak is known especially for the complex hand motions that it involves.\n\n**KATHAKALI.** Style of classical Indian **dance** developed in **Kerala**. This dance style is traditionally performed only by men. It is well known for the very elaborate makeup and costumes that its performers wear.\n\n**KAULA TANTRA.** System of **Tantric** practice aimed at realizing one's unity with **\u015aiva** through the elevation of the **Ku\u1e47\u1e0dalin\u012b** energy from the **m\u016bladh\u0101ra cakra** at the base of the spine to the **sah\u0101srara cakra** or \"thousand-petaled lotus,\" which, when energized by the rising energy, produces a blissful state that, according to this system, advances one quickly toward **en** **lightenment**. In the pursuit of this goal, V\u0101m\u0101c\u0101ra or **Left-Handed Tantric** practices such as the **Five M's** are employed.\n\n**Abhinavagupta** (950\u20131020), the most prominent theologian of this tradition of **Ka\u015bm\u012br \u015aaivism** , reinterpreted its Left-Handed practices as contemplative visualization exercises, making their spiritual benefits available without requiring aspirants to actually physically practice them.\n\n**KAURAVA.** \"Descendant of Kuru\"; member of the **Kuru-Pa\u00f1c\u0101la** dynasty, the royal family whose misadventures are recounted in the _Mah\u0101bh\u0101rata_. Though both branches of this family whose conflict is presented in the _Mah\u0101bh\u0101rata_ are technically Kauravas, in order to distinguish the two groups, the sons of **Pa\u1e47\u1e0du** are referred to as the **P\u0101\u1e47\u1e0davas** and their opponents as the Kauravas.\n\n_KAU\u1e62\u012aTAKI UPANI\u1e62AD_ **.** One of the principle _Upani\u1e63ads_ , and a relatively early one, likely composed, in its current form, in the fifth century before the Common Era, around the same time as the _B\u1e5bhad\u0101ra\u1e47yaka Upani\u1e63ad_ and _Ch\u0101ndogya Upani\u1e63ad_.\n\n**KAU\u1e6cILYA (c. 350\u2013283).** Author to whom the _Artha \u015a\u0101stra_ , an ancient **Sanskrit** text of political science, is attributed; a **Brahmin** and chief political advisor to Candragupta **Maurya** ; also known as Canakya. In later Sanskrit **literature** and in the Hindu popular imagination, Kau\u1e6dilya is depicted as a clever, wily figure\u2014an ancient Indian Machiavelli. Following Kau\u1e6dilya's advice, the warrior Candragupta conquered most of northern India and established the first imperial dynasty to unite much of the subcontinent under a single government. Candragupta's grandson, A\u015boka\u2014known for his conversion to **Buddhism** \u2014expanded the Mauryan Empire to its greatest geographic extent.\n\n**K\u0100VYA.** **Sanskrit** poetics. _See also_ POETRY.\n\n**KERALA.** State on the southwestern coast of India; home to the majority of **Christians** in India. Kerala was the birthplace of **\u015aa\u1e45kara** (788\u2013820), the primary exponent of the **nondualistic** or **Advaita** form of **Ved\u0101nta** and founder of the **Da\u015ban\u0101mi Order** of Hindu monks. It is also the location of **Sabarimalai** hill, which is sacred to the followers of the deity **Ayyappan**. Kerala is believed to be the birthplace of **Para\u015bur\u0101ma** , the sixth out of the traditional list of 10 **avat\u0101ras** of **Vi\u1e63\u1e47u**. The **guru** **Amrit\u0101nandamay\u012b M\u0101t\u0101** also hails from Kerala.\n\n**KETU.** One of the **navagrahas** ; companion of **R\u0101hu**. R\u0101hu was an **asura** who managed to steal a drop of the **am\u1e5bta** , or elixir of **immortality** , when the **devas** and asuras churned it from the cosmic ocean. Alerted to this fact by the sun and the **moon** , **Vi\u1e63\u1e47u** beheaded R\u0101hu and placed his head in the sky. The rest of R\u0101hu's body became a separate entity, Ketu, also placed in the sky. The two of them periodically take their vengeance on the sun and the moon by swallowing them, causing solar and lunar eclipses. Ketu appears as a man with the head of a serpent. Both are considered inauspicious in Hindu **astrology**.\n\n**KHAJURAHO.** **Temple** complex in north-central India located in the state of Madhya Pradesh; famous for the erotic imagery, modeled on the sexual positions described in the _K\u0101ma S\u016btra_ , that adorn the outer walls of many of the temples. The complex includes temples dedicated to **\u015aiva** and **Hanuman** , as well as several **Jain** temples.\n\n**KNOWLEDGE.** _See_ J\u00d1\u0100NA.\n\n**KOLKATA.** **Bengali** name for **Calcutta**.\n\n**KON\u0100RAK.** Site in **Orissa** known for its **temple** to **S\u016brya** , the sun **god**. The temple is designed in the shape of a massive 24-wheeled **chariot** representing the chariot in which S\u016brya flies across the sky every day. This magnificent structure includes seven stone horses, supposed to be pulling the chariot, and three images of S\u016brya positioned so as to receive direct sunlight at dawn, noon, and sunset.\n\n**KO\u015aA.** \"Layer, sheath.\" Five layers that surround the self ( **\u0101tman** ). The outermost layer, the **anna-maya-ko\u015ba** , or \"sheath made of food,\" is the gross, physical body. Next is the **pr\u0101\u1e47a-maya-ko\u015ba** , or \"sheath made of breath,\" which consists of the **pr\u0101\u1e47a** or life force. Next is the **mano-maya-ko\u015ba** , or \"sheath made of **mind** ,\" which consists of a subtle organ of cognition (distinct from the human brain, which would be part of the anna-maya-ko\u015ba). Next is the **vij\u00f1\u0101na-maya-ko\u015ba** , or \"sheath made of consciousness.\" Finally, there is the **\u0101nanda-maya-ko\u015ba** , or \"sheath made of bliss.\"\n\n**KRISHNA.** _See_ K\u1e5a\u1e62\u1e46A.\n\n**KRISHNAMURTI, JIDDU (1895\u20131986).** Philosopher and spiritual teacher. Born into a **Brahmin** family in Andhra Pradesh, in southern India, Krishnamurti was raised by members of the **Theosophical Society** who believed he was destined to be the \"World Teacher\"\u2014the next **Buddha** , Maitreya\u2014a messianic being whose coming is predicted in the **Buddhist** scriptures. In 1922, while staying in Ojai, California, Krishnamurti experienced a profound spiritual transformation to which he referred as \"the process.\" In 1925, **An** **nie Besant** publicly announced Krishnamurti's messiahship. Krishnamurti himself did not endorse this idea, however, and famously left the Theosophical Society in 1927 in order to teach a radical questioning of all forms of authority.\n\nAlthough Krishnamurti publicly rejected the Theosophical Society and its plans for him, and subsequently distanced himself from all organized religion, Besant and Krishnamurti remained on good terms until Besant's death in 1933.\n\nIn California, Krishnamurti became acquainted with literary figures like **Aldous Huxley** and **Christopher Isherwood** who were adherents of **Neo-Ved\u0101nta** and had an intense interest of expanding the boundaries of human consciousness. He was a hero of the countercultural movement of the 1960s. His lectures were heavily attended by young spiritual seekers of that era. In his later years, he returned to India where he set up a system of schools based on the idea of encouraging the curiosity of children in a spirit of free inquiry.\n\nTo the degree that it is possible to categorize Krishnamurti's teaching, it can be said to have affinities to both **Advaita Ved\u0101nta** and the thought of the second-century CE Mah\u0101y\u0101na Buddhist philosopher N\u0101g\u0101rjuna.\n\n**K\u1e5a\u1e62\u1e46A.** \"Dark one\"; also \"attractive one\"; along with **R\u0101ma** , one of the most popular incarnations or **avat\u0101ras** of **Vi\u1e63\u1e47u** in Hindu devotional practice. For some **Vai\u1e63\u1e47avas** , particularly the **Gau\u1e0d\u012bya Vai\u1e63\u1e47ava** tradition, K\u1e5b\u1e63\u1e47a is not merely an avat\u0101ra, but is **God** in a monotheistic sense\u2014the \"Supreme Personality of Godhead,\" to cite the terminology of the **Hare Krishna movement** (a subset of Gau\u1e0d\u012bya Vai\u1e63\u1e47avism). The textual sources on K\u1e5b\u1e63\u1e47a's life consist mainly of the _Mah\u0101bh\u0101rata_ , the _Hariva\u1e43\u015ba_ , and the _Bh\u0101gavata Pur\u0101\u1e47a_. There is also a very early, albeit brief, reference to K\u1e5b\u1e63\u1e47a, son of **Devak\u012b** , in the _B\u1e5bhad\u0101ra\u1e47yaka Upani\u1e63ad_. Though it is the earliest of these texts, the portions of the _Mah\u0101bh\u0101rata_ dealing with K\u1e5b\u1e63\u1e47a (including the famous _Bhagavad G\u012bt\u0101_ **)** relate the latter parts of this figure's life story, as king of **Dv\u0101raka**. The _Bhagavata Pur\u0101\u1e47a_ , composed later, deals with the earlier portions of K\u1e5b\u1e63\u1e47a's life, including narratives of his infancy and adolescence. It is this text that is by far the most popular within Vai\u1e63\u1e47ava traditions. K\u1e5b\u1e63\u1e47a's loving relations with the **gop\u012bs** , or cowherd women of **V\u1e5bnd\u0101vana** , in particular, are a very popular subject of **art** , song, **dance** , **poetry** , and philosophical and theological reflection. The _G\u012btagovinda_ of **Jayadeva** (c. 1200), often likened to the biblical \"Song of Songs,\" focuses on the love affair between K\u1e5b\u1e63\u1e47a and **R\u0101dh\u0101** , a relationship that the tradition utilizes as an image for the relationship of the individual devotee to K\u1e5b\u1e63\u1e47a\u2014a saving bond known as **bhakti** , or devotion.\n\nMajor episodes from K\u1e5b\u1e63\u1e47a's life include his peaceful youth in idyllic V\u1e5bnd\u0101van; his eventual slaying of his wicked uncle, **K\u0101\u1e43sa** ; his friendship with **Arjuna** and the other **P\u0101\u1e47\u1e0dava** brothers; and the tragic brawl among K\u1e5b\u1e63\u1e47a's kinsmen, the **Y\u0101davas** , which foreshadowed the fall of their kingdom, Dv\u0101rak\u0101.\n\n**K\u1e5a\u1e62\u1e46A JANM\u0100\u1e62\u1e6cAMI.** Birthday of **K\u1e5b\u1e63\u1e47a** ; major Hindu holiday. K\u1e5b\u1e63\u1e47a Janm\u0101\u1e63\u1e6dami occurs on the eighth day of the dark half (or **K\u1e5b\u1e63\u1e47a Pak\u1e63a** ) of the month of **\u015arava\u1e47a**. Literally meaning, \"Birthday of K\u1e5b\u1e63\u1e47a, the Eighth,\" the number eight alludes to the fact that K\u1e5b\u1e63\u1e47a was the eighth child of **Devak\u012b** , his mother, and the eighth **avat\u0101ra** of **Vi\u1e63\u1e47u** according to the list of 10 avat\u0101ras.\n\n**K\u1e5a\u1e62\u1e46A PAK\u1e62A.** The dark half of a Hindu month; the fortnight of a Hindu lunar month during which the **moon** is waning; the period between **P\u016br\u1e47ima** , or full-moon day, and **Amavasya** , or new-moon day (which is also the last day of the Hindu month). The dark half of the month is generally considered a less auspicious time for undertaking any new ventures. _See also_ CALENDAR, HINDU.\n\n**K\u1e62ATRIYA.** Warrior; ruler; the second of the four **var\u1e47as** or main **castes**. In terms of traditional occupation, this caste is responsible for maintaining **dharma** in the sense of law and order and the smooth running of society.\n\n**KUBERA.** God of wealth; a very ancient deity associated with nature and assimilated into Hinduism, **Buddhism** , and **Jainism** as a **yak\u1e63a** , or earth spirit. He was the lord of the yak\u1e63as and **r\u0101k\u1e63asas** until being overthrown by **R\u0101va\u1e47a**. For this reason, Kubera is an ally of **R\u0101ma**. He is the guardian of the north and is usually depicted as a pot-bellied man with a large bag of coins and jewels.\n\n_KUL\u0100R\u1e46AVA TANTRA_ **.** Main **scripture** of the **Kaula Tantra** sect.\n\n**KUM\u0100RA.** \"Youth\"; name for **K\u0101rttikeya** , son of **\u015aiva** and **P\u0101rvat\u012b**.\n\n**KUM\u0100RILA BHA\u1e6c\u1e6cA (fl. 700 CE).** Philosopher; prominent adherent of the **P\u016brva M\u012bm\u0101\u1e43s\u0101** or **M\u012bm\u0101\u1e43s\u0101** system of Hindu **philosophy**. Kum\u0101rila spent much of his life in **Pray\u0101ga** ( **Allahabad** ). P\u016brva M\u012bm\u0101\u1e43s\u0101, or \"earlier exegesis,\" the philosophy on which Kum\u0101rila expounded, is a continuation of the ancient, orthodox system of Vedic **Brahmanism** , with a very strong emphasis on the correct performance of **Vedic** ritual and the correct pronunciation of the **Sanskrit** verses of the _Vedas_ during a ritual performance. This emphasis on the correct Sanskrit usage led to the development of an elaborate linguistic philosophy by the M\u012bm\u0101\u1e43sakas. The term _P\u016brva M\u012bm\u0101\u1e43s\u0101_ contrasts this school of thought with later, or **Uttara M\u012bm\u0101\u1e43s\u0101** \u2014more widely known as **Ved\u0101nta**. It emphasizes the earlier, ritualistic portion of the _Vedas_ , while Ved\u0101nta emphasizes the later Vedic **literature** , the _Upani\u1e63ads_ , and their teaching of **Brahman**.\n\nKum\u0101rila's contributions to M\u012bm\u0101\u1e43s\u0101 include his theory of language, which, in contrast to the holistic **spho\u1e6da** theory of **Bhart\u1e5bhari** , affirms that a sentence can only be understood when each of its words is comprehended correctly. Sharply critical of the **Buddhists** and **Jains** , Kum\u0101rila argues that the Buddhist and Jain **scriptures** cannot be true because they contain numerous grammatical errors and are not written in Sanskrit. In defense of the authority of the _Vedas_ , he argued that they are _apauru\u1e63eya_ \u2014that is, they have no author and are thus eternal. Kum\u0101rila's position on the _Vedas_ is akin to _fideism_ , the antifoundationalist notion of taking a particular tradition or set of claims on faith as axiomatically true. His critiques of Buddhism reveal a close acquaintance with Buddhist claims and arguments, and some accounts of his life claim that he studied at the famous Buddhist university at Nalanda.\n\n**KUMBHA MEL\u0100.** \"Festival of the Pot\"; a massive pilgrimage and festival held on a rotating basis every three years at **Hardwar** , **N\u0101sik** , **Ujjain** , and **Pray\u0101ga**. It is said that at the time of the churning of the cosmic ocean of milk, four drops of **am\u1e5bta** \u2014the elixir of **immortality** \u2014fell to Earth at the locations of these holy cities. The pot to which the name of the festival refers is the one from which the drops of am\u1e5bta spilled. Pilgrims, including a massive number of renouncers, or **sanny\u0101sins** , gather at each site every 12 years. The 144th Kumbha Mel\u0101 of a cycle is known as the _Mah\u0101_ or \"great\" _Kumbha Mel\u0101_. The last Mah\u0101 Kumbha Mel\u0101, held in the year 2001, attracted 60,000,000 pilgrims, making this festival the largest gathering of human beings for a single purpose in history. The crowds at this gathering were visible to orbiting satellites.\n\nThe earliest written account of the Kumbha Mel\u0101 is that of the Chinese **Buddhist** pilgrim Xuanzang, who traveled to India in the seventh century of the Common Era, during the reign of **Har\u1e63a**. According to Xuanzang's account, Har\u1e63a accompanied him to this gathering. It is likely, however, that the Kumbha Mel\u0101 dates back many centuries before Xuanzang.\n\nThe Kumbha Mel\u0101 begins on the **P\u016br\u1e47im\u0101** , or full-moon day, of the month of **Pau\u1e63a** (December\u2013January) and ends on the P\u016br\u1e47im\u0101 of **Magha** (January\u2013February).\n\n**KU\u1e46\u1e0cALIN\u012a.** \"Coiled one\"; serpent. According to **Tantric yoga** , in the space that corresponds to the base of the spine, a subtle energy or **\u015bakti** , resides. While it is residing in this space, it is also referred to as the ku\u1e47\u1e0dalini, or coiled energy, because it is said to resemble a coiled serpent in this state. One of the goals of Tantric yoga practice is to raise this energy from the base of the spine to the crown of the head, which are linked by a subtle nerve channel called the su\u1e63umna n\u0101d\u012b, and which occupies roughly the same space as the spinal cord. Along the su\u1e63umna n\u0101d\u012b are the seven **cakras**. These energy centers are activated as the \u015bakti or ku\u1e47\u1e0dalin\u012b rises through the su\u1e63umna n\u0101d\u012b to the **sah\u0101srara cakra** or \"thousand-petaled lotus,\" in the crown of the head, which, when energized by the rising \u015bakti, produces a blissful state that, according to Tantra, advances one quickly toward **enlightenment**.\n\nAccording to the tradition of **Siddha Yoga** , a contemporary **\u015aaiva** movement with a strongly Tantric dimension, an enlightened teacher, or **guru** , can raise one's \u015bakti from the base of the spine to the crown of the head\u2014activating all of the cakras in the process\u2014with a thought, word, or touch. This experience is called **\u015baktipat**. _See also_ KAULA TANTRA.\n\n**KUNT\u012a.** Character in the _Mah\u0101bh\u0101rata_ ; wife of **Pa\u1e47\u1e0du** and mother of **Kar\u1e47a** and three of the five **P\u0101\u1e47\u1e0dava** brothers\u2014 **Yudhi\u1e63\u1e6dhira** , **Bh\u012bma** , and **Arjuna**. In her youth, **Durv\u0101sa** , a sage, paid a visit to her father's palace. Impressed by the polite young woman, the sage gave her a secret **mantra** that she could use to summon any deity of her choosing for the purpose of bearing a child by that deity. Not fully comprehending the implications of the use of this mantra, the curious girl used it to call upon the sun **god** , **S\u016brya** , who then impregnated her. Soon, Kunt\u012b gave birth to a magnificent son from this union, Kar\u1e47a, who had the characteristics of his divine father.\n\nBut because she was unmarried and ashamed to have given birth, she hid the child, putting him in a basket and sending him down a river, from where he would later be rescued by a childless charioteer and his wife, who would take the boy in and lovingly raise him as their own.\n\nLater in life, when she was married to the **Kaurava** king, **Pa\u1e47\u1e0du** , Kunt\u012b used her mantra again when the king was cursed to die if he approached either of his two wives for the purpose of lovemaking. Pa\u1e47\u1e0du had not yet fathered an heir at this point, so Kunt\u012b used her mantra to bear semidivine children who would legally be Pa\u1e47\u1e0du's. This was the origin of the P\u0101\u1e47\u1e0dava brothers, three of whom were Kunt\u012b's sons and two of whom\u2014the twins, **Nakula** and Sahadeva\u2014were the sons of Pa\u1e47\u1e0du's younger wife, **M\u0101dr\u012b** , with whom Kunt\u012b shared her mantra.\n\nWhen the P\u0101\u1e47\u1e0davas rejected Kar\u1e47a as unworthy to share the assembly of **k\u1e63atriya** warriors, being (as they thought) the son of a humble charioteer, **Duryodhana** , seizing an opportunity to gain a powerful ally, made Kar\u1e47a a king, thus winning his enduring loyalty and ensuring that Kar\u1e47a would fight on Duryodhana's side against his brothers. On the eve of the climactic battle between the P\u0101\u1e47\u1e0davas and the Kauravas, Kunt\u012b tells Kar\u1e47a the **truth** about his identity, hoping to win him over to the P\u0101\u1e47\u1e0davas' side. But Kar\u1e47a does not waver in his loyalty to Duryodhana. Kar\u1e47a later dies in the battle at Arjuna's hand, though Arjuna does not yet know at that point that Kar\u1e47a is really his eldest brother. _See also_ WOMEN.\n\n**K\u016aRMA.** The tortoise **avat\u0101ra** or incarnation of **Vi\u1e63\u1e47u** and the second in the traditional list of 10 such avat\u0101ras. Vi\u1e63\u1e47u assumes this form to act as the pivot for the churning of the ocean when the **devas** and **asuras** churn the cosmic ocean in their search for **am\u1e5bta** , the elixir of **immortality**.\n\n**KURU-PA\u00d1C\u0100LA.** Ancient **Vedic** polity located in northwestern India, roughly in the vicinity of the modern state of Haryana and **New Delhi**. The Kurus and Pa\u00f1c\u0101las were two distinct groups that formed a union. A historical kingdom (and later a republic), archaeologists believe the Kurus were a people who lived in the Delhi area between the 12th and ninth centuries before the Common Era and whose deeds and struggles formed the basis for the _Mah\u0101bh\u0101rata_. The _\u1e5ag Veda_ may reflect their ideology.\n\n**KURUK\u1e62ETRA.** \"Field of the Kurus.\" Field in northern India on which the climactic battle of the _Mah\u0101bh\u0101rata_ takes place. This is the same battle that is the occasion for the dialogue between **Arjuna** and **K\u1e5b\u1e63\u1e47a** that constitutes the _Bhagavad G\u012bt\u0101_. In interpreting the _Bhagavad G\u012bt\u0101_ , it is sometimes asserted (on the basis of _Bhagavad G\u012bt\u0101_ 13:1) that the field of Kuruk\u1e63etra symbolizes the physical body as the location where the battle against spiritual enemies such as greed, hatred, and lust is waged\u2014a battle symbolized, in turn, by the _Mah\u0101bh\u0101rata_ war. **Mohand\u0101s K. G\u0101ndh\u012b** favored this interpretation as a way to negate K\u1e5b\u1e63\u1e47a's apparent endorsement of violence in the first two chapters of this text, making it more compatible with G\u0101ndh\u012b's **philosophy** of nonviolence ( **ahi\u1e43s\u0101** ).\n\n**KU\u015aA.** A type of sacred grass used in **Vedic** ceremonies.\n\n**K\u016a\u1e6cIY\u0100\u1e6c\u1e6cAM (KOODIY\u0100\u1e6c\u1e6cAM).** \"Acting together\"; classical Indian **dance** style. Every year a series of eight dance dramas is performed at **Guruvay\u016br** , a town in **Kerala** , in the K\u016b\u1e6diy\u0101\u1e6d\u1e6dam style that depicts events from K\u1e5b\u1e63\u1e47a's life. This is the one Indian style of classical dance that has continued to be performed without interruption since ancient times. Others fell into disuse at different points and have been revived in modern times.\n\n## L\n\n**LAK\u1e62MA\u1e46A.** Character in the _R\u0101m\u0101ya\u1e47a_ ; brother and close friend of **R\u0101ma** who often takes on the role of R\u0101ma's protector. Just as R\u0101ma is regarded as an **avat\u0101ra** of **Vi\u1e63\u1e47u** , and **S\u012bt\u0101** , R\u0101ma's wife, as an incarnation of Vi\u1e63\u1e47u's wife, **Lak\u1e63m\u012b** , Lak\u1e63ma\u1e47a is often seen as an incarnation of **Se\u015ba** , the cosmic serpent on whom Vi\u1e63\u1e47u reclines on the cosmic ocean, and who rears up his many heads to protect Vi\u1e63\u1e47u. Lak\u1e63ma\u1e47a accompanies R\u0101ma into exile and is his constant companion in his struggle with the demonic **R\u0101va\u1e47a**.\n\n**LAKSHMAN JOO, SW\u0100M\u012a (1907\u20131991).** Scholar, mystic, revered modern exponent of **Ka\u015bm\u012br \u015aaivism**.\n\n**LAK\u1e62M\u012a.** \"Mark\"; \"sign\"; **goddess** of good fortune and prosperity; wife of **Vi\u1e63\u1e47u**. She emerged from the cosmic ocean when it was churned by the **devas** and **asuras** to retrieve the **am\u1e5bta** , or elixir of **immortality**. When **Vi\u1e63\u1e47u** appears on the Earth as an **avat\u0101ra** he is accompanied by Lak\u1e63m\u012b, who manifests as **R\u0101ma** 's wife, **S\u012bt\u0101** , and later as **K\u1e5b\u1e63\u1e47a** 's wife, **Rukmi\u1e47\u012b**. In **\u015arivai\u1e63\u1e47ava** theology, Lak\u1e63m\u012b and Vi\u1e63\u1e47u, rather like **\u015aiva** and **\u015aakti** , are said to be two parts, aspects, or dimensions of one, undivided godhead. Analogously to the way in which \u015aakti is viewed as the power of \u015aiva, without which he can do nothing, it is said that Vi\u1e63\u1e47u never does anything without Lak\u1e63m\u012b's approval and that he takes her advice. For this reason, Vai\u1e63\u1e47ava devotees sometimes approach Vi\u1e63\u1e47u through Lak\u1e63m\u012b as a kind of intercessor. It is said that Lak\u1e63m\u012b represents the more gentle and benevolent aspect of Vi\u1e63\u1e47u, who is sometimes represented as a rather stern and righteous deity. She is also a symbol of divine grace. In her origins, it is likely that she was first worshiped as a goddess of abundance and fertility. _See also_ \u015aR\u012a.\n\n**LAKUL\u012a\u015aA (fl. 2nd century CE).** **Yogi** ; founder of the **Pa\u015bupata** tradition of **\u015aaivism**.\n\n**LALL\u0100 (fl. 1300 CE).** Female mystic, **bhakti** **poet** , and saint of the **Ka\u015bm\u012br \u015aaiva** tradition, also known as Lalle\u015bwar\u012b. The collection of her poems is entitled the _Lalla V\u0101ky\u0101ni_ (\"Sayings of Lall\u0101\"). _See also_ WOMEN.\n\n**LALLE\u015aWAR\u012a.** _See_ LALL\u0100.\n\n**LAMBODARA. \"** Pot belly\"; epithet of **Ga\u1e47e\u015ba**.\n\n**LA\u1e44KA.** In the _R\u0101m\u0101ya\u1e47a_ , La\u1e45ka is the island kingdom of **R\u0101va\u1e47a** , demonic lord of the **r\u0101k\u1e63asas**. Subsequent tradition has come to identify this island with **Sri Lanka** , the large island off the southeastern coast of India, near **Tamil** Nadu.\n\n**LAW.** _See_ DHARMA.\n\n_LAWS OF MANU_ **.** _See MANUSM\u1e5aTI_.\n\n**LEFT-HANDED TANTRA.** **Tantric** practices that include **actions** and items, such as the **Five M's** , which are normally regarded as impure, so called because the left hand is traditionally seen in Hindu culture as unclean. The purpose of such practices is to show that one has transcended the realm of duality\u2014that if everything is one with the divine, the belief that certain actions, persons, or things can be impure is without foundation and is a part of the illusion to which those who are trapped in the cycle of **rebirth** have become enthralled. Such practices are traditionally shrouded in secrecy, both because they might be an occasion for scandal for the practitioner, but also because it is believed that such practices are only effective for those who truly are cultivating a nondual awareness. If pursued merely for pleasure or other unworthy purposes (like the cultivation of **siddhis** , or paranormal powers) they can be destructive and lead to rebirth in hell. Because of the perilous nature of such practices, the **Ka\u015bm\u012br \u015aaiva** theologian **Abhinavagupta** (950\u20131020) recommended that one should practice them through contemplative visualization, rather than with literal, physical performance. The **Sanskrit** term for Left-Handed Tantra is V\u0101m\u0101c\u0101ra.\n\n**LEKHR\u0100J, D\u0100DA (1876\u20131969).** Founder of the **Brahm\u0101 Kum\u0101r\u012b** movement; better known by his religious name of **Praj\u0101pitar Brahm\u0101**.\n\n**LIBERATION.** **Mok\u1e63a** ; release from **sa\u1e43s\u0101ra** , the **karma** -fueled cycle of birth, **death** , and **rebirth** , and the ultimate goal of many systems of Hindu thought and practice.\n\n**L\u012aL\u0100.** \"Play,\" the divine play. According to some theistic Hindu theologies, the world of time and space in which souls reside is a divine play, an exuberant and spontaneous burst of creativity in which **God** delights in infinite possibility. The concept of l\u012bl\u0101 is an attempt to answer the question of why God, who is self-sufficient and free from desire, engages in the act of **creation** , which implies a desire to create. According to the idea of l\u012bl\u0101, creation is a spontaneous act pursued as an end in itself, rather than an **action** that is undertaken for some further, desired end.\n\n**LI\u1e44GA (LI\u1e44GAM).** Symbolic form of **\u015aiva** ; an abstract representation, in contrast to the more anthropomorphic **m\u016brtis** that represent \u015aiva as a human being, emphasizing his **nirgu\u1e47a** , or formless, nature.\n\nSome, though not all, li\u1e45gas have a phallic appearance, prompting speculation that the li\u1e45ga may originally have been a fertility symbol, and \u015aiva a deity connected with fertility. While this is certainly the case in some contexts\u2014shrines or images; such as Gudimallam in Andhra Pradesh, where the phallic nature of the li\u1e45ga is quite obvious\u2014the preeminent view of the li\u1e45ga in general among contemporary Hindus is that it is an abstract, aniconic representation of a deity who is ultimately beyond form. The most commonly cited textual reference to the meaning and origin of the li\u1e45ga is not overtly phallic but refers to \u015aiva's appearance to **Brahm\u0101** and **Vi\u1e63\u1e47u** as an infinite pillar of fire whose full extent neither deity is able to comprehend (not unlike biblical apparitions of the divine as a pillar of fire). The li\u1e45ga, on this understanding, is an attempt to represent this pillar of fire in the form of a stone. _See also_ YONI.\n\n**LI\u1e44G\u0100YAT. \u015aaiva** tradition based in **Karnataka** and established by the **bhakti** **poet** and social reformer **Basava** (1134\u20131196). Also known as the **V\u012bra\u015baivas** , the Li\u1e45g\u0101yats oppose **casteism** and teach the equality of all human beings as manifestations of **\u015aiva**. Believing in holding ritualism to a bare minimum, the only image the Li\u1e45g\u0101yats use is the abstract form of \u015aiva, the **li\u1e45ga** (hence the name _Li\u1e45g\u0101yat_ for this tradition), which they traditionally wear on a necklace. The Li\u1e45g\u0101yats regard \u015aiva as **God** in the **monotheistic** sense\u2014as the one supreme deity.\n\n**LITERATURE.** Literature is a vital component of Hinduism, from the ancient **Vedic** **scriptures** to the writings of contemporary Hindu spiritual teachers ( **gurus** ). The range of Hindu literary genres is vast, encompassing very large epic narratives (the _Mah\u0101bh\u0101rata_ and _R\u0101m\u0101ya\u1e47a_ being the primary examples); the lyrical **poetry** of the **bhakti** saints; the terse didactic writings or **s\u016btras** of the various philosophical systems (or **dar\u015banas** ); the pragmatic \"how-to\" manuals of political science ( _Artha \u015a\u0101stra_ ), **law** ( _Dharma \u015a\u0101stra_ ), and sensory enjoyment (the _K\u0101ma S\u016btra_ ); the dramas of playwrights such as **K\u0101lid\u0101sa** ; and in the modern period, the novels of **Bankim Chandra Chatterjee** and the poems, plays, songs, and essays of **Rabindranath Tagore**.\n\nThe oldest extant Hindu literature is composed in **Sanskrit**. The _Vedas_ , the oldest and most sacred collection of Hindu scriptural literature, is regarded as **\u015br\u016bti** , or divinely revealed. The Vedic literature includes both poetry and prose and was passed on orally for centuries by the community of **Brahmins** before being committed to writing in the modern period. Even larger (and more popularly known) than the _Vedas_ is the collection of **sm\u1e5bti** , or secondary scriptures, which includes the two main Sanskrit epics and many other sets of authoritative texts, such as the _Pur\u0101\u1e47as_ , or ancient tales of the **gods** , and the s\u016btra and \u015b\u0101stra literature just mentioned.\n\nIn the first millennium of the Common Era, devotional poetry in the vernacular languages of India became increasingly popular. Some of this poetry, such as the **Tamil** works of the **\u0100\u1e37v\u0101rs** and **N\u0101yan\u0101rs** , came to be regarded as scripture in its own right by the South Indian **Vai\u1e63\u1e47ava** and **\u015aaiva** traditions, respectively, that inherited it. As bhakti became a mass movement, translations of earlier Sanskrit works into spoken vernacular languages also increased in number, such as the **Hindi** rendering of the _R\u0101m\u0101ya\u1e47a_ , the _R\u0101m Carit M\u0101nas_ of **Tuls\u012bd\u0101s** (1532\u20131623), which is better known among Hindus in the northern part of India today than the original _R\u0101m\u0101ya\u1e47a_ of **V\u0101lm\u012bki**.\n\nWith the coming of the British to India and the institution of English education, many Hindu thinkers began to express themselves through the medium of English essays, including some of the major figures of the modern Hindu reform movement, like **R\u0101m Mohan Roy** , **Sw\u0101m\u012b Vivek\u0101nanda** , **Aurobindo Ghose** , and **Mohand\u0101s K. G\u0101ndh\u012b** , to name only a few. European genres, such as the novel, also became popular in the modern period, often being written in Indian vernacular languages, such as the **Bengali** novels of Bankim Chandra Chatterjee.\n\nWith the internationalization of Hinduism, the creation and adaptation of Hindu forms of literary expression continues today\u2014including, for example, the science fiction adaptation of the _R\u0101m\u0101ya\u1e47a_ in Ashok Banker's _Prince of Ayodhya_ series of novels, and the American-produced graphic novel _Ramayana Reloaded_ \u2014as well as the maintaining of earlier literary forms, such as devotional songs, or **bhajanas** \u2014the lyrics of all of which are bhakti poems\u2014which have been incorporated into popular Indian cinema. And the Indian diaspora is producing a great variety of English novelists reflecting upon the relatively recent experience of being Hindu in predominantly non-Hindu cultures.\n\n**LOKA.** World; realm; abode of a deity; also the human world; society.\n\n**LOKAP\u0100LA.** The four leading **Vedic** deities, **Indra** , **Agni** , **Varu\u1e47a** , and **Yama**. These deities are frequently invoked early in ritualistic worship, or **p\u016bj\u0101** , in order to ensure that the ritual will be completed without mishap.\n\n**LOK\u0100YATA.** **Materialism** , doctrine of the **C\u0101rv\u0101ka** system of philosophy, one of the heterodox or **n\u0101stika** schools of Indian philosophy, or **dar\u015bana**. According to this view, the only valid **pram\u0101\u1e47a** or basis of **knowledge** is sensory perception. The adherents of this view reject such widely held doctrines of Indian philosophy as **karma** and **rebirth** or belief in anything that is not verifiable through sensory perception. This system was widely criticized by adherents of other schools of thought, both orthodox (the **Vedic** or **Brahmanical** systems) and heterodox ( **Buddhism** and **Jainism** ).\n\n**LOMAHAR\u1e62A\u1e46A S\u016aTA.** Disciple of **Vy\u0101sa** who memorized the epics and _Pur\u0101\u1e47as_.\n\n**LOTUS.** Flower frequently used in Hindu iconography to symbolize the state of spiritual **enlightenment**. The lotus typically grows in muddy water. The lotus blossom, sprouting from a stalk that rises out of the mud in which it is rooted to bloom above the surface of the water, represents spiritual enlightenment arising from the mud of worldly life, with its various forms of suffering and limitation. In recent decades, the lotus has also become a symbol of Hinduism, and is used as the logo of some **Hindu Nationalist** organizations such as the **Bharat\u012bya Janata Party** (BJP).\n\n**LOTUS POSTURE. Padm\u0101sana** ; **Ha\u1e6dha Yoga** posture formed by placing the right foot on the left thigh and the left foot on the right thigh. This is the \"full lotus.\" There is also a half-lotus posture ( **ardhapadm\u0101sana** ), formed by placing the right foot on the left thigh and the left foot underneath the right leg. _See also_ \u0100SANA.\n\n**LOVE.** Love has a wide range of meanings in Hinduism, some of which are denoted by distinct **Sanskrit** terms, including **bhakti** (absolute devotion to **God** in a personal form), **prema** (platonic love, affection, friendship), and **k\u0101ma** (erotic or romantic love, sexual desire). Love is theorized in Hindu thought as not a mere sentiment but rather as the felt experience of the interconnectedness of one being to another, and ultimately of all beings to one another and to God.\n\n## M\n\n**M\u0100.** Mother. Intimate form of address frequently used to call upon **Dev\u012b** , the **Mother Goddess** , especially popular in **Bengal**.\n\n**MADHVA (1238\u20131317). Vai\u1e63\u1e47ava \u0101c\u0101rya** ; founder of the dualistic or **Dvaita** system of **Ved\u0101nta**. Born near **Udipi** in **Karnataka** , where he spent most of his life, Madhva is believed by his devotees to be the third incarnation or **avat\u0101ra** of **V\u0101yu** , the **Vedic** **god** of the wind (the first two incarnations being **Hanuman** and **Bh\u012bma** ). A devout **Vai\u1e63\u1e47ava** , Madhva was highly critical of the **Advaita Ved\u0101nta** of **\u015aa\u1e45kara** , seeing it as diminishing the significance of **bhakti** and as relegating **God** to a role subordinate to an impersonal reality ( **Nirgu\u1e47a Brahman** ). Madhva sees Brahman as identical to **Vi\u1e63\u1e47u** , the supreme deity, and the **j\u012bva** , or individual soul of a living being, as an eternally existing reality distinct from Vi\u1e63\u1e47u. **Liberation** , or **mok\u1e63a** , therefore does not involve a real metaphysical union between the devotee and the divine (or, as in Advaita Ved\u0101nta, a realization that the distinction between the devotee and the divine is an illusion). The devotee and the divine are forever distinct. \"Union\" with the divine is a union of wills, of complete subordination of the will of the individual soul to the will of God ( **\u012a\u015bvara-pra\u1e47idh\u0101na** ). The individuality of one who attains liberation is not effaced. Liberation, rather, consists of residing for eternity in **Vaikun\u1e6dha** , or heaven, with God, a place of infinite bliss. Unusually for a Hindu tradition, Dvaita Ved\u0101nta teaches that it is possible that some souls may never reach liberation and remain eternally separated from God.\n\n**MADRAS.** Also known as Chennai; fourth-largest city in India and capital of the state of **Tamil** Nadu, on the southeast coast of the Deccan peninsula. It was the first city to be established by the **British East India Company** , growing up around Fort Saint George, which was built in 1644. The city contains several major **temples** and also houses the world headquarters of the **Theosophical Society** , in a neighborhood known as Ady\u0101r. Its major temples include K\u0101p\u0101l\u012b\u015bvarar (a **\u015aiva** temple) and \u015ar\u012b Parthasarathy (dedicated to **K\u1e5b\u1e63\u1e47a** ).\n\n**M\u0100DR\u012a.** Younger wife of **Pa\u1e47\u1e0du** and mother of the twins **N\u0101kula** and Sahadeva, the youngest of the **P\u0101\u1e47\u1e0dava** brothers in the _Mah\u0101bh\u0101rata_. When Pa\u1e47\u1e0du was cursed to die if he attempted to make love to either of his wives, his elder wife, **Kunt\u012b** , utilized a secret **mantra** that gave her the power to summon any deity and bear a child by that deity. At Pa\u1e47\u1e0du's request, Kunt\u012b evoked the devas **Yama** , **V\u0101yu** , and **Indra** in order to give birth to **Yudhi\u1e63\u1e6dhira** , **Bh\u012bma** , and **Arjuna** , respectively. M\u0101dr\u012b, also desiring to bear divine children for Pa\u1e47\u1e0du, asked Kunt\u012b to share her mantra, which M\u0101dr\u012b then used to evoke the **A\u015bvins** , twin deities who then fathered N\u0101kula and Sahadeva. _See also_ WOMEN.\n\n**MADURAI.** Site in **Tamil** Nadu, in southern India, which houses a major complex of **temples** dedicated to the **goddess** **M\u012bn\u0101k\u1e63\u012b** and **\u015aiva** ; an important pilgrimage site. The annual festival celebrating the wedding of M\u012bn\u0101k\u1e63\u012b and \u015aiva (in his form as **Sundare\u015bvara** , or \"Lord of Beauty\") is very popular and an occasion for great displays of **devotion**.\n\n**MAGHA.** Month of the **Hindu calendar** that corresponds roughly to the period from mid-January to mid-February.\n\n**MAH\u0100BALIPURAM.** _See_ M\u0100MALLAPURAM.\n\n_MAH\u0100BH\u0100RATA_ **.** \"Great Epic of the **Bharatas** \"; along with the _R\u0101m\u0101ya\u1e47a_ , one of the two great epic poems of Hinduism. Together, these two texts make up the **Itih\u0101sa** (\"thus it was\") or historical **literature** of Hinduism. In its current form, the _Mah\u0101bh\u0101rata_ was probably composed over the course of several centuries, from about 300 BCE to 300 CE. According to Hindu tradition, the epic is the work of the sage **Vy\u0101sa** , who is also credited with compiling the _Vedas_ and the _Pur\u0101\u1e47as_. The events that make up the main narrative of the text may be based on a battle that took place among various **\u0100ryan** tribes around 950 BCE. However, the text also refers to the sinking of the coastal city of **Dv\u0101rak\u0101** , which took place around 1500 BCE, and at least one Hindu tradition locates the date of the battle as 3102 BCE\u2014also regarded as the point at which the **Kali Yuga** began. In any case, the original narrative expanded substantially over the centuries, as a great variety of folk tales, philosophical writings, and even legal texts were incorporated into the epic through the process of telling and retelling. Consisting, in its finished form, of approximately 180,000 lines, the _Mah\u0101bh\u0101rata_ is roughly four times the length of the Bible.\n\nThe basic narrative of the _Mah\u0101bh\u0101rata_ is focused on the power struggle between two sets of cousins\u2014the **P\u0101\u1e47\u1e0davas** and the **Kauravas** \u2014who are competing for control of the kingdom of the Kurus. The succession to the throne has been rendered problematic by the vow of **Bh\u012b\u1e63ma** , who renounces his claim to the throne at the request of his father, \u015aantanu, so \u015aantanu can marry a **woman** whose father wants her heirs to inherit the kingdom, and not Bh\u012b\u1e63ma. Subsequently, \u015aantanu's heirs by this woman die without any heirs. The **poet** Vy\u0101sa, the queen's son from a previous relationship, is brought in to father children with the wives of the deceased heirs. One of these sons is blind and the other suffers from albinism, rendering neither fully fit to be king. Their sons, in turn\u2014the Kauravas and the P\u0101\u1e47\u1e0davas\u2014end up fighting for control of the kingdom.\n\nCommentators frequently contrast the _Mah\u0101bh\u0101rata_ with its companion epic, the _R\u0101m\u0101ya\u1e47a_ , particularly in regard to its tragic tone. Whereas the _R\u0101m\u0101ya\u1e47a_ is an uplifting classic rescue tale, in which the heroes and the villains are clearly delineated from one another, the _Mah\u0101bh\u0101rata_ is far more ambiguous, with the putative heroes (the P\u0101\u1e47\u1e0davas) and their supporters not always behaving in an admirable, or even ethical, fashion, and the villains and their supporters (especially the tragic **Kar\u1e47a** ) sometimes behaving with great nobility.\n\nReligiously, the _Mah\u0101bh\u0101rata_ has become especially significant to the **Vai\u1e63\u1e47ava** traditions of Hinduism because of the prominent role of **K\u1e5b\u1e63\u1e47a** in the story\u2014the eighth of the 10 **avat\u0101ras** , or incarnations, of **Vi\u1e63\u1e47u**. K\u1e5b\u1e63\u1e47a is a close friend and advisor to the P\u0101\u1e47\u1e0davas, especially the warrior **Arjuna**. K\u1e5b\u1e63\u1e47a's counsel to Arjuna on the eve of the climactic battle of the epic is an especially beloved portion of this text and has become a sacred scripture in its own right: the famous \"Song of the Blessed One,\" or _Bhagavad G\u012bt\u0101_.\n\nThe _Mah\u0101bh\u0101rata_ has been a major subject of Hindu **art** and literature throughout the centuries. Plays, such as the renowned poet **K\u0101l\u012bd\u0101sa** 's _Abhij\u00f1\u0101na\u015bakuntala_ (\"The Recognition of **\u015aakuntal\u0101** \"), have been based upon it or upon select episodes within it. Depictions of scenes from the text adorn **temple** walls throughout India and **Southeast Asia**. In the modern period, it has become a popular **Hindi** television series and has been presented as an English stage play by Peter Brook.\n\n**MAH\u0100BH\u016aTAS.** The five elements or states of being that constitute the physical world (mah\u0101bh\u016btas). The five elements are earth ( **p\u1e5bthvi** ), air ( **v\u0101yu** ), fire ( **agni** ), water ( **ap** ), and space ( **\u0101k\u0101\u015ba** ). Each element is the medium of a different type of experience on the subtle level (smell, touch, sight, taste, and sound, respectively).\n\n**MAH\u0100DEVA.** Great **God** ; epithet of **\u015aiva**.\n\n**MAH\u0100DEV\u012a.** Great **Goddess** ; epithet of the **Mother Goddess** , who is known variously as **Um\u0101** , **Dev\u012b** , **\u015aakti** , **P\u0101rvat\u012b** , **Durg\u0101** , and **K\u0101l\u012b** , as well as other names.\n\n**MAH\u0100DEVIYAKKA (fl. 12th century CE).** Female **\u015aaiva** saint and **bhakti** poet who gave up a royal marriage to wander as a naked renouncer, or **sanny\u0101sin\u012b** , devoted completely to **\u015aiva** , whom she regarded as her true husband. She refers to \u015aiva in her devotional **poetry** as her \"Lord white as jasmine,\" saying, \"O lord white as jasmine, if my head falls from my shoulders I shall think it your offering.\" The **V\u012bra\u015baivas** hold Mah\u0101deviyakka in high esteem, revering her devotion. _See also_ WOMEN.\n\n**MAH\u0100K\u0100LA.** \"Great time\"; epithet of **\u015aiva** that refers to his role as time, the destroyer of all. _See also_ N\u0100\u1e6cAR\u0100JA.\n\n**MAH\u0100R\u0100J JI, GURU (1957\u2013 ).** Born Prem Rawat; son of Sri H\u0101\u1e43s\u0101 Mah\u0101r\u0101j Ji. He succeeded his father as head of the **Divine Light Mission** in 1966, when was only nine years old. In 1974, he married an American woman against his mother's wishes. This led to a rift in the Divine Light Mission, which was renamed _El\u00e1n Vital_ in 1983.\n\n**MAHARISHI MAHESH YOGI (MAHAR\u1e62I MAHE\u015aA YOG\u012a) (1918\u2013 2008).** **Guru** of the **Transcendental Meditation** tradition, which he began teaching in the Western world in 1958, his most famous students being the Beatles, who visited his **\u0101\u015brama** in **Rishikesh** in 1968. Born **Mahesh Pras\u0101d Varma** , he first earned a degree in physics from **Allahabad** University in 1942 and then studied **meditation** under the tutelage of his guru, Sw\u0101m\u012b Brahm\u0101nanda Sarasvat\u012b (1869\u20131953), known to the Maharishi's disciples as \"Guru Dev,\" or \"Divine Teacher.\" Like the **Hare Krishna movement** , Transcendental Meditation was a popular component of the countercultural movement of the 1960s and early 1970s. Unlike the Hare Krishna movement, which (until fairly recently) reveled in its \"Indianness,\" with followers dressed like medieval **bhakti** saints chanting **mantras** in public places, Transcendental Meditation has presented itself as a scientific and universal practice, promoted for health and peace of **mind** , rather than as a religious tradition with strong Hindu (specifically **\u015aaiva** ) roots. In 1973, the Maharishi established the Maharishi University of Management in Fairfield, Iowa, to advance the study and promotion of his meditation practice. _See also_ HARRISON, GEORGE.\n\n**MAH\u0100SABH\u0100.** \"Great assembly\"; epithet of the **Hindu Mah\u0101sabh\u0101**.\n\n**MAH\u0100\u015aIVAR\u0100TRI.** \"Great night of \u015aiva\"; a festival that is held in honor of **\u015aiva** on the 13th night and 14th day of the **dark half** of **Phalguna** (February\u2013 March). It is observed by devotees through fasting and staying up all night to engage in worship ( **p\u016bj\u0101** ) and listen to stories and sing songs relating to \u015aiva. Also known simply as **\u015aivar\u0101tri** (\"Night of \u015aiva\").\n\n**MAH\u0100TMA.** \"Great soul\"; \"one whose soul is great\"; a saintly person.\n\n**MAH\u0100TMA G\u0100NDH\u012a.** _See_ G\u0100NDH\u012a, MOHAND\u0100S K. (1869\u20131948).\n\n**MAH\u0100V\u012aRA (c. 499\u2013427 BCE).** \"Great Hero\"; 24th **T\u012brtha\u1e45kara** of **Jainism**. According to Jain tradition, 24 \"ford builders,\" or **enlightened** beings, appear in a cosmic epoch to build a _t\u012brtha_ , or ford, across the waters of **re-birth** to the further shore of **mok\u1e63a** or **liberation**. As the 24th T\u012brtha\u1e45kara of our current cosmic epoch, Mah\u0101v\u012bra is therefore the last such being to appear until the next cosmic epoch begins.\n\nMah\u0101v\u012bra is not, strictly speaking, the founder of Jainism, since, from a Jain point of view, he is the latest in a series of enlightened teachers. And there is at least some historical evidence for the existence of a couple of the previous T\u012brtha\u1e45karas. Living at the same time as the **Buddha** , Mah\u0101v\u012bra's life story follows a similar pattern: the son of a king who renounces his king-ship to pursue spiritual freedom.\n\nAccording to Jain tradition, Mah\u0101v\u012bra practiced strict **asceticism** for 12 years prior to achieving **kaivalya** , a state in which he was able to separate his soul ( **j\u012bva** ) from its entanglement with the material world of **karma**. He spent the rest of his life teaching the Jain path to others, establishing a fourfold community of monks, nuns, and male and female laypersons, and finally leaving his body at the age of 72. Jains celebrate **D\u012bwali** as Mah\u0101v\u012bra Nirv\u0101\u1e47a Divas in commemoration of Mah\u0101v\u012bra's final entry into the realm of enlightened beings.\n\n**MAH\u0100V\u012aRA.** \"Great Hero\"; epithet of **Hanuman**.\n\n**MAHENDRA.** \"Great **Indra** \"; epithet of Indra.\n\n**MAHENDRAVIKRAMA VARMAN (571\u2013630 CE).** A king of the **Pallava** dynasty of southern India, based in the capital city of **K\u0101\u00f1cipuram**. A scholar and playwright as well as a king, he authored a satirical play entitled the _Mattavil\u0101sa Prahasana_ (\"Comedy of Drunken Play\") that ridicules various religious sects of the period, including several **\u015aaiva** sects and **Buddhism**.\n\n**MAHE\u015aA.** Great Lord; epithet of **\u015aiva**. _See also_ MAHE\u015aVARA.\n\n**MAHE\u015aVARA.** Great Lord; epithet of **\u015aiva**.\n\n**MAHI\u1e62A, MAHI\u1e62\u0100SURA.** Buffalo Demon; creature slain by the **goddess** **Durg\u0101** , an event commemorated in the 10-day festival of **Durg\u0101 P\u016bj\u0101**.\n\n**MAJUMDAR, R. C. (1888\u20131980).** **Indologist** and historian. His major scholarly work was the massive _History and Culture of the Indian People_.\n\n**MAKARA.** Sea monster with a fairly strong resemblance to the Gangetic dolphin; also often compared to a crocodile; sometimes depicted as a fish with the head of an elephant. The makara is the **vahana** or animal vehicle of two Hindu deities related to water bodies: **Varu\u1e47a** , the **Vedic** **god** of the ocean, and **Ga\u1e45g\u0101** , the **goddess** who personifies the River Ganges.\n\n**MAKARA SA\u1e42KR\u0100NTI.** A Hindu festival that marks the transition of the sun from Sagittarius to Capricorn. Being based on the position of the sun, the date of this festival is fixed as January 14th, unlike most Hindu holidays, whose dates are determined by the dominant lunar calendar of Hinduism. However, it does occur consistently during the lunar month of **Magha** (January\u2013February) and corresponds to the **Tamil** harvest festival of **Pongal**. It is regarded as a highly auspicious day. _See also_ CALENDAR, HINDU.\n\n**MAKKHALI GOS\u0100LA (fl. 5th century BCE).** Sage; a contemporary of **Mah\u0101v\u012bra** and the **Buddha** ; founder of the **\u0100j\u012bvikas** , a **\u015brama\u1e47a** sect that was a major rival of early **Jainism** and **Buddhism**. He is attributed with a doctrine of fatalism, which was likely a claim that, while one could avoid the accumulation of new **karma** , one could do nothing to speed up the fruition or \"burning off\" of already present karma.\n\n**M\u0100LA.** Rosary; string of beads, usually 108 in number, which is used to keep track of the number of times one has recited one's **mantra**. _See also_ JAPA.\n\n**MALHOTRA, RAJIV.** Independent Hindu scholar who has engaged in a critique of the academic study of Hinduism, questioning many of the assumptions that underlie current scholarship as continuations of older colonial assumptions or as reflecting a **Christian** missionary agenda. A good deal of his work that has appeared on the Internet seems to be aimed at exposing **Hinduphobia** \u2014a term he coined to refer to anti-Hindu bias. A controversial figure, reaction to his work among academic scholars has been mixed.\n\n**M\u0100MALLAPURAM.** Site on the beach south of **Madras** in **Tamil** Nadu that houses a variety of **temples** , including one with a sculpture of **K\u1e5b\u1e63\u1e47a** holding Mount **Govardhana** in his hand to shelter his devotees from a thunderstorm. One temple\u2014the Shore Temple\u2014has been declared a World Heritage protected site to help with its preservation. It is an unusual temple for its period in having similar shrines to both **\u015aiva** and **Vi\u1e63\u1e47u** , rather than being devoted exclusively to one or the other deity. This site was built by the **Pallava** dynasty.\n\n**MAMMATA (fl. 12th century CE).** Scholar from **Kashmir** ; author of a work on **poetics** and aesthetics entitled the _K\u0101vya Prak\u0101\u015ba_ (\"Illumination of Poetry\").\n\n**MAN-LION AVAT\u0100RA.** _See_ NARASI\u1e42HA.\n\n**MANAS.** Mind. According to **Dvaita Ved\u0101nta** , **S\u0101\u1e43khya** , and **Buddhism** , mind is one of the sense organs, its objects being thoughts and emotional states. It is the intermediary between the realm of the senses and the self ( **\u0101tman** ). All the various systems of Hindu **philosophy** agree that the mind is part of the material continuum and not intrinsic to the self, which is pure **consciousness**. The mind is the vehicle of consciousness, the organ by which it interacts with the sensory realm.\n\n**MANAS\u0100.** A **goddess** of snakes and poisons worshiped in the northeastern part of India. Not unlike the snake-handlers of Appalachia, the devotees of this goddess honor her by handling cobras and allowing themselves to be bitten by them in order to show their faith and reliance on their deity. The word _manas\u0101_ also refers to contemplative reflection.\n\n_M\u0100NAVA DHARMA \u015a\u0100STRA_. \"Authoritative Text on the Duties of Human Beings\"; alternative title for the _Manusm\u1e5bti_.\n\n**MA\u1e46\u1e0cALA.** Literally, \"circle\"; a sacred diagram of the cosmos viewed from a spiritual perspective\u2014as a series of concentric circles whose central point represents the state of **enlightenment** \u2014which is used as an object of contemplative visualization. The ma\u1e47\u1e0dala is also a political theory of interstate relations that was first articulated in the _Artha \u015a\u0101stra_ of **Kau\u1e6dilya** , according to which a king should see himself as being in the middle of a series of concentric circles of alternating enemies and allies. It is possible that the political concept gave rise to the artistic one, in which a spiritual practitioner visualizes himself or herself passing through the circles of the ma\u1e47\u1e0dala, from the outermost to the innermost, symbolizing and facilitating the inner **meditative** journey to the divine core of one's being. _See also_ TANTRA; YANTRA.\n\n**MANDARA, MOUNT.** Mountain that **K\u016brma** , the tortoise **avat\u0101ra** of **Vi\u1e63\u1e47u** , allowed the **devas** and **asuras** to place on his back to use as the axis of a great churn\u2014the rope of which was formed by the giant serpent **V\u0101suk\u012b** , king of the **N\u0101gas** \u2014in their joint effort to churn the cosmic ocean in search of the lost **am\u1e5bta** , or elixir of **immortality**.\n\n**MANDIRA.** Literally \"dwelling place\"; a Hindu **temple**.\n\n_M\u0100\u1e46\u1e0cUKYA UPANI\u1e62AD_ **.** One of the principle _Upani\u1e63ads_ , probably composed some time between 200 BCE and 200 CE. This brief text is a **meditation** on the meaning of the syllable **O\u1e43** , which it analyzes in terms of a correlation between the four elements of that make up O\u1e43\u2014the sounds \"A,\" \"U,\" and \"M\"\u2014and the four **states of consciousness**. One of the prominent commentaries or **bh\u0101\u1e63yas** on this _Upani\u1e63ad_ was written by **Gau\u1e0dap\u0101da** , the **guru** of the guru of **\u015aa\u1e45kara**.\n\n**MA\u00d1GAL.** The planet Mars. _See also_ ASTROLOGY; NAVAGRAHAS.\n\n**MA\u00d1GALV\u0100RA.** Tuesday, the day of the week governed by the planet Mars. _See also_ CALENDAR, HINDU.\n\n**M\u0100\u1e46IKKAV\u0100CAKAR (fl. 9th century CE).** One of the 63 **N\u0101yan\u0101rs** \u2014saints and **bhakti** poets revered in the **Tamil \u015aaiva** tradition. M\u0101\u1e47ikkav\u0101cakar was a **Brahmin**. He composed the _Tiruv\u0101cakam_ , or \"Blessed Utterance,\" a work of **poetry** that has conceptual similarities to the _Bhagavad G\u012bt\u0101_ , though it is a \u015aaiva, rather than a **Vai\u1e63\u1e47ava** , text.\n\n**MA\u1e46IP\u016aRA CAKRA.** **Cakra** aligned with the solar plexus. _See also_ TANTRA.\n\n**MANIPURI.** Style of classical Indian **dance** developed in Manipur, in the northeastern part of India.\n\n**MANO-MAYA-KO\u015aA. Mind** layer; the third of the five **ko\u015bas** , or layers, surrounding the self ( **\u0101tman** ) according to **Ved\u0101nta**.\n\n**MANOSAROVAR.** Holy lake in the **Him\u0101layas** at the base of Mount **Kail\u0101sa** ; sacred to **\u015aiva**. Manosarovar is located in Tibet.\n\n**MANTRA.** A collection of sounds or words, sometimes in the form of a prayer, which is used to focus the **mind** in **meditation** or as part of a religious ritual. Some mantras have a literal meaning that is fairly easy to discern, such as _O\u1e43 Nama\u1e25 \u015aiv\u0101ya_ (\"I pay homage to **\u015aiva**.\"), whereas others\u2014particularly **b\u012bja** or \"seed\" mantras, which consist of only one syllable\u2014are more evocative than denotative, connecting one with a deity or concept that the mantra symbolizes rather than communicating meaning in the fashion of sentences.\n\nThe oldest known mantras are those found in the **Vedic** hymns, or _Sa\u1e43hit\u0101s_ , such as the **G\u0101yatr\u012b Mantra**. Some mantras are connected with particular Hindu traditions, being shared by all practitioners, whose use of them constitutes, among other things, a badge of sectarian identity. An example would be the just-cited _O\u1e43 Nama\u1e25 \u015aiv\u0101ya_ , a mantra that is universal among the adherents of **\u015aaivism**. Another would be _Hare K\u1e5b\u1e63na Hare K\u1e5b\u1e63na_ , _K\u1e5b\u1e63na K\u1e5b\u1e63na Hare Hare, Hare R\u0101ma Hare R\u0101ma, R\u0101ma R\u0101ma Hare Hare_ , which consists of three names of **Vi\u1e63\u1e47u** uttered in the order indicated in the vocative case, and which is used by the **Gau\u1e0d\u012bya Vai\u1e63\u1e47avas**. There are also secret mantras\u2014mantras that one receives from one's **guru** upon receiving initiation, or **d\u012bk\u1e63\u0101** , into a particular Hindu tradition. It is believed that the power of such mantras would be diluted were they to be shared publicly. The most powerful mantra of all is the primordial b\u012bja mantra **O\u1e43**.\n\nIn the _Ma\u1e47\u1e0dukya Upani\u1e63ad_ , O\u1e43 is described as encompassing all of existence. Each of its parts (A, U, M, and the totality of these three) is associated with one of the four **states of consciousness**.\n\n**MANU.** If one were to compare Manu to figures from biblical **literature** , he could well be described as combining the roles of Adam, Noah, Jonah, and Moses. Manu is, first of all, the progenitor of humanity\u2014the first human being. When a deluge was threatening to flood the world, Manu and his family were saved from it by **Vi\u1e63\u1e47u** in the form of a giant fish\u2014the **Matsya** or fish **avat\u0101ra** , the first of the traditional 10 avat\u0101ras, or incarnations, of Vi\u1e63\u1e47u. And he is attributing with writing an important legal text, or _Dharma \u015a\u0101stra_ \u2014the _Manusm\u1e5bti_.\n\n_MANUSM\u1e5aTI_ **.** An important _Dharma \u015a\u0101stra_ , or authoritative legal text, that was likely composed around 100 CE attributed to **Manu** , the father of humanity. The text has been controversial in the modern era, particularly for its affirmation of the **caste** system and its sanctioning of abuses against **women** and people of lower castes. Its more controversial aspects include the fact that it suggests a higher penalty for the killing of a person of a higher caste than for the killing of a person of a lower caste, as well as higher penalties against people of a lower caste for committing infractions for which people of a higher caste receive lesser penalties. **B. R. Ambedkar** famously burned a copy of the _Manusm\u1e5bti_ in public as a protest against the mistreatment of **Dalits**.\n\nThe text was first translated into English by **Sir William Jones** , who sought to apply it to the administration of India on the understanding that the British Empire should govern its territories using local **laws** and customs as much as possible. Scholars have suggested that this application of texts like the _Manusm\u1e5bti_ as legal reference works was not in keeping with their original intent, leading to a more rigid practice of caste than had actually been the case in ancient times.\n\n**MANVANTARA.** _See_ WORLD AGES.\n\n**MAR\u0100\u1e6cHAS.** A warrior people of Maharashtra, in central India (the current location of Bombay, or Mumbai), who successfully fought against the **Mughal Empire** in the late 17th century, during the period of Emperor Aurangzeb. Led by **\u015aiv\u0101ji** (1627\u20131680), who even today remains a hero in Maharashtra, the Mar\u0101\u1e6dhas utilized guerilla tactics against the numerically superior and better equipped Mughal army. Stunningly successful in their campaign, the Mar\u0101\u1e6dhas eventually controlled most of central India's Deccan plateau. After \u015aiv\u0101ji's **death** , the Mar\u0101\u1e6dhas continued their expansion until being defeated in 1761 at the battle of Panipat.\n\n**M\u0100RGA.** Road, path, way; a spiritual path; sometimes synonymous with one of the **four** **yogas** delineated by **Sw\u0101m\u012b Vivek\u0101nanda** : the way of **action** (the **karma yoga** or karma m\u0101rga), **knowledge** , **devotion** , and **meditation**. _See also_ YOGA.\n\n**M\u0100R\u012aCA.** Character in the _R\u0101m\u0101ya\u1e47a_ ; a **r\u0101k\u1e63asa** , or demonic creature, and uncle of the main villain of the text, **R\u0101va\u1e47a**. R\u0101va\u1e47a persuaded M\u0101r\u012bca to assist him in his plot to abduct **S\u012bt\u0101**. He had M\u0101r\u012bca disguise himself as a beautiful golden deer. S\u012bt\u0101, on seeing the golden deer, asked **R\u0101ma** and his brother, **Lak\u1e63ma\u1e47a** , to capture it for her. With R\u0101ma and his brother thus lured away, R\u0101va\u1e47a was able to approach S\u012bt\u0101 and abduct her.\n\n**M\u0100RKA\u1e46\u1e0cEYA.** A **Brahmin** sage of the **Bh\u0101rgava** clan to whom the _M\u0101rka\u1e47\u1e0deya Pur\u0101\u1e47a_ is attributed. M\u0101rka\u1e47\u1e0deya had a vision of the ongoing process of cosmic **creation** and dissolution in the form of an infant\u2014 **Vi\u1e63\u1e47u** \u2014who was constantly inhaling and exhaling the universe. _See also_ WORLD AGES.\n\n_M\u0100RKA\u1e46\u1e0cEYA PUR\u0100\u1e46A_ **.** One of the 18 major _Pur\u0101\u1e47as_ , probably composed between 250 and 550 CE. The text includes the sage **M\u0101rka\u1e47\u1e0deya** 's vision of **creation** as a series of inhalations and exhalations by an infant **Vi\u1e63\u1e47u** and a series of passages recounting the deeds of the **Mother Goddess**. _See also DEV\u012a MAH\u0100TMYA_.\n\n**MARRIAGE.** One of the Hindu sacraments, or **sa\u1e43sk\u0101ras** ; the ritual by which a person moves from the first **\u0101\u015brama** or stage of life\u2014the **brahmacarya** , or student stage\u2014and into the **householder** stage. Traditional Hindu marriages are arranged by the families of the bride and groom based on considerations such as **caste** and **astrological** compatibility. In the modern period these traditions are being adapted to a more individualistic ethos. For example, at least in middle-class families, young men and **women** are allowed to reject a prospective spouse whom they find unacceptable; and in recent years some amount of (usually chaperoned) dating has become permissible to ensure personal compatibility. Increasingly, class and professional considerations outweigh such traditional concerns as caste. In rural areas, though, where caste traditions are much stronger, the older model remains the norm.\n\n**MARUTS.** **Vedic** storm deities; sons of **Rudra**. The Maruts are depicted as a group of young warriors and are invoked to bring protection and timely rain. _See also_ DEVAS.\n\n**M\u0100T\u0100 AMRIT\u0100NANDAMAY\u012a (1953\u2013 ).** _See_ AMRIT\u0100NANDAMAY\u012a, M\u0100T\u0100.\n\n**MATERIALISM.** Doctrine of the **C\u0101rv\u0101ka** or **Lok\u0101yata** system of philosophy, one of the heterodox or **n\u0101stika** schools of Indian philosophy, or **dar\u015banas**. According to this view, the only valid **pram\u0101\u1e47a** or basis for **knowledge** is sensory perception. Adherents of this view reject such widely held doctrines of Indic thought as **karma** and **rebirth** or belief in anything that is not verifiable by sensory perception. This system is heavily criticized by adherents of other schools of thought, both orthodox (that is, the **Vedic** or **Brahmanical** systems) and heterodox ( **Buddhism** and **Jainism** ). The central critique is that the rejection of all pram\u0101\u1e47as other than sensory perception\u2014including inference, or **anum\u0101na** \u2014would lead to a \"solipsism of the present moment,\" in which one would not be able to make any judgments based on memory or past knowledge.\n\nThe materialist critique of the other systems alleges, much like modern critiques of religion, that unverifiable doctrines such as the existence of **God** , a **soul** , or karma and rebirth are propounded to deceive the ignorant into giving material support to the priests and monks who propound them. Only fragments of the original texts, or **s\u016btras** , of this system have survived since ancient times. According to tradition, this system was first developed by the sage **B\u1e5bhaspati** \u2014who is, ironically, the **guru** of the **gods**.\n\n**MA\u1e6cHA.** Monastic institution, monastery. This term can refer to the community of **ascetics** who belong to such an institution, to an administrative unit, or to the building or complex of buildings where that unit is based. In this way it is analogous to the term _church_.\n\n**MATHUR\u0100.** Ancient city in northern India, in what is now the state of Uttar Pradesh; birthplace of **K\u1e5b\u1e63\u1e47a** ; located in the **Vraja** region, near **V\u1e5bnd\u0101vana** , where K\u1e5b\u1e63\u1e47a spent his youth. Mathur\u0101 also has associations with **Buddhism** and **Jainism**. The **Buddha** is said to have visited this city during his lifetime (though this is doubtful, Mathur\u0101 being rather far west of the region where the Buddha lived). A Jain st\u016bpa, or burial monument, was discovered in Mathur\u0101, which suggests that there was a fairly large and thriving Jain community in this city at the beginning of the Common Era.\n\n**MATSYA.** \"Fish\"; the fish **avat\u0101ra** of **Vi\u1e63\u1e47u** ; the first of the 10 avat\u0101ras or incarnations of Vi\u1e63\u1e47u according to one popular list. The fish avat\u0101ra appeared to **Manu** to warn him of the coming of a great flood that would deluge the entire world. Manu caught the fish, who then spoke to Manu and promised that if Manu would release him, he would return to save Manu and his family from the flood. Manu agreed and released the fish. When the flood came, Manu had built a large boat, as the fish had instructed him. When the fish arrived, he was considerably larger than when Manu had first caught him and had also grown an enormous horn from his nose. Manu lashed his boat to the fish's horn with a rope and the fish carried Manu and his family to safety.\n\n**MATSYENDRAN\u0100TH (fl. 11th century CE).** Founder of the **N\u0101th Yog\u012bs** , a **Tantric \u015aaiva** **ascetic** lineage. Matsyendran\u0101th's successor, **Gorakhn\u0101th** , invented **Ha\u1e6dha Yoga**.\n\nAccording to tradition, Matsyendran\u0101th, whose name means \"fish lord,\" was in the form of a fish when he overheard **\u015aiva** teaching his wife, the **goddess P\u0101rvat\u012b**. These teachings became the basis of the doctrines of the N\u0101th Yog\u012bs.\n\n_MATTAVIL\u0100SA PRAHASANA_ **.** \"Comedy of Drunken Play\"; a satirical play written by **Mahendravikrama Varman** (571\u2013630 CE), a king of the **Pallava** dynasty of southern India. The play ridicules various religious sects of the period, including several different **\u015aaiva** sects and **Buddhism**. Some have argued that the play may reflect a **Lok\u0101yata** or **C\u0101rvaka materialist** perspective, though it is also possible that its perspective is simply a more general cynicism regarding those who make a living off of the religious beliefs of others. It sheds an interesting light on how the various sects were perceived in ancient India, particularly by laypersons. _See also_ LITERATURE.\n\n**MAURYA.** First dynasty of ancient India to form an empire of considerable size. The dynasty lasted from approximately 320 to 184 BCE. It was founded by Candragupta Maurya shortly after Alexander of Macedon had weakened the polities of the northwest. Emerging from the northeastern Magadha region, Candragupta took advantage of the power vacuum left in the northwest after Alexander's campaign. The empire reached its height under the reign of Candragupta's grandson, A\u015boka, which lasted from 268 to 233 BCE. After A\u015boka's reign, the empire went into a period of gradual decline and was later succeeded by the \u015au\u1e45ga dynasty in 184 BCE (which was founded by Pu\u015byamitra \u015au\u1e45ga, a rebel Maurya general who essentially staged a coup).\n\nReligiously, the Mauryas practiced a policy of tolerance toward all sects, but gave preference to **\u015brama\u1e47a** traditions. Candragupta himself is said to have had a leaning to **Jainism** , and his son, Bindus\u0101ra, supported the **\u0100j\u012bvikas**. Most famously, A\u015boka became a **Buddhist** after the final battle that led to the consolidation of his empire. Repenting of violence and warfare, he promoted a policy of **dharma** , which included respecting life and supporting Buddhist institutions. The emphasis on **ahi\u1e43s\u0101** , or nonviolence, in this policy led to a decline in the practice of **animal sacrifice** on the part of the **Brahmins** \u2014a practice banned by A\u015boka, at least on certain holy days. Brahmanical ascendancy only returned with the rise of the **Gupta** dynasty in 320 CE\u2014albeit a transformed **Brahminism** , influenced by \u015brama\u1e47ic thought, which would become the basis of the Hinduism of the epics and _Pur\u0101\u1e47as_.\n\n**M\u0100YA.** **Asura** architect credited with the building of the **P\u0101\u1e47\u1e0davas** fantastic palace of **Indraprastha**.\n\n**M\u0100Y\u0100.** Illusion; magic; **creative** power; the power of a **god** to create. In the **philosophy** of **Ved\u0101nta** , m\u0101y\u0101 is the way the universe appears to us due to ignorance, or **avidy\u0101**. In **Advaita Ved\u0101nta** , there is, in **truth** , no universe at all, the universe itself being an effect of ignorance, the reality being that **Brahman** alone is real. In theistic forms of Ved\u0101nta, m\u0101y\u0101 is the creative power of **God** that brings the world into being.\n\n**M\u0100Y\u0100PUR.** Town in **Bengal** ; location of a major **temple** that is operated by the **International Society for Krishna Consciousness** (ISKCON). The town also houses an ISKCON-run school and press.\n\n**M\u0100Y\u016aR\u012a.** The peacock; vehicle of **K\u0101rttikeya**.\n\n**MEDITATION.** Though in its Latin origins this word refers to reflective contemplation of a philosophical or religious topic, in regard to Hinduism, it typically refers to **dhy\u0101na** , the seventh of the eight stages or \"limbs\" of **Pata\u00f1jali** 's eight-limbed ( **a\u1e63\u1e6d\u0101\u1e45ga** ) system of **yoga**. The sixth limb, **dhara\u1e47\u0101** , is concentration on a single object. Dhy\u0101na, the next step, is absorption in that object, with no awareness of any other phenomena. Meditation thus involves the stilling of the waking, conscious **mind** so deeper levels of consciousness can be experienced. The goal of meditation is **sam\u0101dhi** \u2014the eighth limb of Pata\u00f1jali's system\u2014which is complete absorption in the object of concentration, a state in which the distinction between subject and object vanishes.\n\nIn **Tantric** meditation, the imagination is actively utilized as a tool to bring about sam\u0101dhi. Rather than suppressing the senses, as in Pata\u00f1jali's system, they are actively engaged and channeled toward the goal of raising the **ku\u1e47\u1e0dalin\u012b** energy from the base of the spine to the crown of the head, bringing about a heightened state of consciousness.\n\nThe practice of meditation is shared by many Indic traditions, and is especially prominent in **Buddhism**. In early Buddhism, the term _dhy\u0101na_ also refers to the **states of consciousness** to which the practice of meditation can lead. When Buddhism was first transmitted to China, the term _dhy\u0101na_ was rendered as _ch'an_. When the tradition was transmitted to Japan, the term _ch'an_ was transformed into _zen_.\n\n**MEHTA, NARSI (1414\u20131481).** **Vai\u1e63\u1e47ava bhakti** **poet** and saint from Gujarat. Known for his universalist attitude toward religion, Mehta's most famous song is his _Vai\u1e63\u1e47ava Jana to tene kahiya_ , which states that a true Vai\u1e63\u1e47ava is one \"who feels for the suffering of another and forgets the good he does to another, taking no pride in it.\"\n\n**MELA.** Festival; fair; celebration.\n\n**MERIT.** The result of a good deed; \"good **karma**.\" _See also_ PU\u1e46YA KARMA.\n\n**MERU, MOUNT.** In traditional Hindu cosmology, the world is depicted as a series of ringed continents interspersed with oceans and extending over a vast region of space. At the center of these concentric circles of land and sea rests Mount Meru, the _axis mundi_ and center of cosmic space. This conception of the universe is essentially a **ma\u1e47\u1e0dala** , or a spiritual representation of existence. In this sense, Mount Meru is not a literal mountain so much as a symbol of the highest spiritual attainment. When this model has been taken literally, though, Mount Meru has generally been understood to be located just beyond the **Him\u0101layas**.\n\n**M\u012aM\u0100\u1e42S\u0100.** Also known as **P\u016brva M\u012bm\u0101\u1e43s\u0101** , or \"earlier exegesis,\" this **philosophy** is a continuation of the ancient, orthodox system of Vedic **Brahmanism** , with a very strong emphasis on the correct performance of **Vedic** ritual and the correct pronunciation of the **Sanskrit** verses of the _Vedas_ during a ritual performance. This emphasis on the correct Sanskrit usage led to the development of an elaborate linguistic philosophy on the part of the M\u012bm\u0101\u1e43sakas. The term _P\u016brva M\u012bm\u0101\u1e43s\u0101_ contrasts this school of thought with the later, or **Uttara M\u012bm\u0101\u1e43s\u0101** \u2014more widely known as **Ved\u0101nta**. P\u016brva M\u012bm\u0101\u1e43s\u0101 focuses on the earlier, ritualistic portion of the _Vedas_ , while Ved\u0101nta interprets the later Vedic texts\u2014the _Upani\u1e63ads_ \u2014and their teaching of **Brahman**. _See also_ JAIMINI; KUM\u0100RILA BHA\u1e6c\u1e6cA.\n\n**M\u012aN\u0100K\u1e62\u012a.** \"Fish-eyed **Goddess** \"; the main goddess enshrined at the M\u012bn\u0101k\u1e63\u012b **temple** for which the town of **Madurai** , in **Tamil** Nadu, is famous. She is said to have been born with fish-shaped eyes, a fishy odor, and three breasts. Her father raised her as a son and she became a great warrior. According to a prophecy, she would lose her odor, her third breast, and her warlike nature when she met her true husband, **\u015aiva** , also enshrined at the temple complex at Madurai in his form as **Sundare\u015bvara** (\"Beautiful Lord\"). M\u012bn\u0101k\u1e63\u012b's uncle is **Vi\u1e63\u1e47u** , who showed up late to her wedding with \u015aiva. The wedding is re-enacted every year in the month of **Caitra** (March\u2013April).\n\n**MIND.** _See_ MANAS.\n\n**M\u012aR\u0100 BAI (1498\u20131547).** Female **Vai\u1e63\u1e47ava** saint and **bhakti** **poet** from Rajasthan. She was born a **Rajput** princess and married to a prince who was killed in battle while still a young man. Deeply devoted to **K\u1e5b\u1e63\u1e47a** since her childhood, she became more immersed in devotion after her husband's **death** , declaring that K\u1e5b\u1e63\u1e47a was her only true husband. Her in-laws disapproved of her deep piety, going so far as to make attempts upon her life. Renowned for her devotion, M\u012br\u0101 Bai's life has been the subject of several **Hindi** movies. Her poetry, which remains popular today, proclaims the depth of her bhakti: \"O K\u1e5b\u1e63\u1e47a, charmer of hearts, lifter of mountains, I hear your flute calling me\u2014Shall I come by the secret path through the tall grass? O Lord of Heavenly Blue, my heart cannot rest until we are together, until we walk along the banks of the **Yamun\u0101** deep into the night.\" _See also_ WOMEN.\n\n**MITRA.** \"Friend\"; **Vedic** deity; one of the **\u0100dityas** ; often paired with **Varu\u1e47a**. His cult was popular in ancient Iran and spread to the Roman Empire as Mithraism.\n\n**MLECCHA.** \"Barbarian\"; a person who does not know **Sanskrit** or who lacks cultural **knowledge** and etiquette; a derogatory term for non-Hindus. This term may be cognate originally with _Meluhha_ , the name by which the people of the **Indus Valley civilization** possibly referred to themselves and their region. In its first uses, it does not seem to have a derogatory meaning but simply refers to persons from the region of the Indus Valley or to their customs. Over time, however, the view developed that the people of this region were uncouth and did not observe **Vedic** practices in the correct way (or at all). It later came to refer to anyone from outside the fold of mainstream **Brahmanical** culture, and in particular to foreigners (Greeks, Arabs, and so on).\n\n**MOHENJO-DARO.** Largest city of the **Indus Valley civilization**.\n\n**MOHIN\u012a.** Female form of **Vi\u1e63\u1e47u** ; Vi\u1e63\u1e47u took the form of Mohin\u012b so a deity could be born from the union of Vi\u1e63\u1e47u and \u015aiva that would have their combined power in order to defeat the **asura** Princess Mahi\u1e63\u012b, sister of the **Mahi\u1e63\u0101sura** or Buffalo Demon. The deity born from this union of Vi\u1e63\u1e47u and \u015aiva was **Ayyappan**.\n\n**MOK\u1e62A.** **Liberation** from the cycle of **rebirth** ; the ultimate goal of most Hindu paths, though a good deal of religious activity is also devoted to penultimate goals, like a better rebirth, or more immediate this-worldly goods. Mok\u1e63a is the fourth of the four goals of humanity or **puru\u1e63\u0101rthas**. Different Hindu traditions present different strategies for the attainment of mok\u1e63a, such as selfless work ( **karma** ), wisdom ( **j\u00f1\u0101na** ), devotion ( **bhakti** ), **meditation** ( **dhy\u0101na** ), or some combination of these.\n\n**MONK.** A male renouncer; a man who has taken up the life of **sanny\u0101sa**.\n\n**MONOTHEISM.** Belief in one supreme **God**. Many Hindus see the many deities of the Hindu tradition as different aspects or manifestations of one Supreme Being. Others see a particular deity as preeminent (usually **Vi\u1e63\u1e47u** or **\u015aiva** ) and the others as subordinate, in the way that angels are subordinate to God in the Abrahamic religions. _See also_ GODS.\n\n**MOON.** One of the **Navagrahas** , the nine **planets** of Hindu **astrology** ( **jyoti\u1e63a** ); often identified with the **Vedic** deity **Soma** , who also represents the hallucinogenic plant whose juice, mixed with milk, was used by ancient **Brahmins** to experience an altered **state of consciousness**. The phases of the moon represent this substance being consumed by the **devas** (during the **dark half** of the month, when the moon is waning) and replenished (during the **bright half** , when the moon is waxing).\n\n**MOTHER GODDESS. Dev\u012b** , or **Mah\u0101dev\u012b** (the Great Goddess) is identified in many Hindu traditions as the wife of **\u015aiva**. She has many forms, such as the docile wife and mother, **P\u0101rvat\u012b** ; the warrior goddess, **Durg\u0101** , who defeats **demonic** beings and drives away evil; and the fierce, demon-slaying **K\u0101l\u012b**. Also known as **\u015aakti** , the power of **creation** , the Goddess can also be identified with the many **goddesses** of the Hindu pantheon, who are seen as her various forms or manifestations. Goddesses such as **Lak\u1e63m\u012b** and Sarasvat\u012b are therefore also called the _\u015baktis_ or power centers of the deities that are their husbands\u2014 **Vi\u1e63\u1e47u** and **Brahm\u0101** , respectively. In this role, the Goddess and her various personae are seen as the energizing principle, in the absence of which their corresponding male gods would be incapable of effective activity. The divine pairing of \u015aiva and \u015aakti, in particular, is often presented, especially in \u015a\u0101kta texts, as a composite being, with the God and Goddess being ultimately inseparable from one another as parts of an internally differentiated, but finally unitary, godhead. In its affirmation of internal relations within the godhead, this concept is not unlike the **Christian** idea of the trinity.\n\nFinally, the supreme Mother Goddess is identified, in **Tantric** traditions (and in **Neo-Ved\u0101nta** , which is strongly informed by the Tantric background and sensibility of **Ramakrishna** ), with **Brahman** , the absolute, as it appears from the perspective of **m\u0101y\u0101** , the universe of time and space. Ramakrishna states that \"K\u0101l\u012b is verily Brahman, and Brahman is verily K\u0101l\u012b. It is one and the same Reality. When we think of It as inactive . . . then we call it Brahman. But when It engages in . . . activities, then we call It K\u0101l\u012b or \u015aakti.\" This statement implies a pantheistic or panentheistic understanding of the Goddess as being manifested within all beings. This understanding is also present in the ancient **\u015a\u0101kta** scripture, the _Dev\u012b Mah\u0101tmya_ or _Ca\u1e47\u1e0d\u012b Pa\u1e6dh_ , with its refrain of **mantras** that refer to \"The Goddess who dwells within all beings.\"\n\n**MUDR\u0100.** Hand gesture that conveys a symbolic meaning. Hindu deities are depicted holding their hands in various positions, all of which communicate specific concepts to those who view their images. Human practitioners also utilize mudr\u0101s in certain types of worship and **meditative** practice. **Tantra** , in particular, makes extensive use of mudr\u0101s, often replicating the mudr\u0101s with which specific deities are depicted in order to facilitate a sense of identification with those deities.\n\nMudr\u0101 is also one of the **Five M's** , or traditionally impure substances of which practitioners of **Left-Handed Tantra** partake. It is said to refer, in that context, to puffed rice, although this is not generally seen as an impure substance (though it is forbidden in certain fasts where one abstains only from particular foods). This may be a confusion, with the mudr\u0101 in question actually referring to secret gestures only practiced in a ritual context. _See also_ ABHAYA-MUDR\u0100; D\u0100NA-MUDR\u0100; IMAGES IN WORSHIP; M\u016aRTI.\n\n**MUGHAL EMPIRE.** The last of the three major empires to encompass a substantial portion of the Indian subcontinent prior to British rule and independence, the first two being those of the **Maurya** and **Gupta** dynasties. The Mughals came from Central Asia and were related to the Mongols. Mughal rule in India lasted from 1526 until 1858, when control of India formally passed to Queen Victoria and direct British rule in the wake of the First War of Independence (or Sepoy Mutiny) of 1857. The religious affiliation of the Mughals was to **Islam** , though the religious attitudes and beliefs of the Mughal emperors varied a great deal, ranging from the tolerance and pluralism of **Akbar** , who displayed a good deal of appreciation for indigenous Indian traditions, to the nearly rabid persecution of non-Muslims by Aurangzeb, which led him into conflict with both the **Mar\u0101\u1e6dhas** and the **Sikhs** , among others. On the whole, the Mughal period was characterized by a good deal of creativity and interreligious innovation on the part of Hindus and Muslims. New architectural styles and forms of **art** and **music** that drew upon the traditional cultures of both Hinduism and Islam emerged, as well as direct religious interaction in the form of the **Sant** movement and the rise of Sikhism. _See also_ GOBIND SINGH, GURU; KAB\u012aR; N\u0100NAK, GURU; SUFISM.\n\n**MUH\u016aRTA.** Unit of time; a **muh\u016brta** is approximately an hour and a half in length and each day is divided into 16 muh\u016brtas. The best time of day for **meditation** and worship is the **Brahm\u0101-muhurt\u0101** , from roughly 4:30 a.m. to 6 a.m.\n\n**MUKT\u0100NANDA, SW\u0100M\u012a (1908\u20131982).** Monk and spiritual teacher who brought the **Siddha Yoga** tradition to the Western world. Part of the broader phenomenon of Hindu teachers who began bringing their traditions to North America and Europe and tailoring them to a Western audience\u2014starting with **Sw\u0101m\u012b Vivek\u0101nanda** and continuing with the **Maharishi Mahesh Yogi** and **Bhaktived\u0101nta Swam\u012b Prabhup\u0101da** , among others. The distinct emphasis of Mukt\u0101nanda's tradition was **Tantric** , focused on the experience of the **Ku\u1e47\u1e0dalin\u012b \u015aakti** , or energy coiled at the base of the spine. In Siddha Yoga, it is said that this experience can be evoked by the grace of the **guru** through a thought, word, or gesture\u2014a process known as **\u015baktipat**.\n\nMukt\u0101nanda received \u015baktipat from his guru, **Sw\u0101m\u012b Nity\u0101nanda** (1897\u20131961), a solitary monk who resided in a cave in **Ga\u1e47e\u015bapuri** , near Bombay. Before his death in 1982, Mukt\u0101nanda designated his young female translator, **Malti Shetty** , and her brother to be his successors. After a falling out between the brother and sister team in 1985, the sister, Sw\u0101m\u012b Chidvil\u0101s\u0101nanda (more widely known as **Gurumay\u012b** ) became the leader of the lineage.\n\n**MUKTI.** **Liberation** ; freedom; synonym for **mok\u1e63a**.\n\n**M\u016aLADH\u0100RA CAKRA.** The bottommost of the seven **cakras** , located at the base of the spine. _See also_ TANTRA.\n\n**M\u00dcLLER, FRIEDRICH MAX (1823\u20131900).** A giant in the field of **Indology** , M\u00fcller oversaw the translation of numerous **Sanskrit** texts through his _Sacred Books of the East_ series, making many Hindu, **Buddhist** , and **Jain** writings available in English for the first time. He is also known for developing the **\u0100ryan Invasion** \u2014later modified to the **\u0100ryan Migration** \u2014theory of **Vedic** origins. M\u00fcller was the first Western scholar to be aware of **Ramakrishna** , writing one of the first works in a European language on the life and teachings of this Hindu saint in direct consultation with Ramakrishna's disciple, **Sw\u0101m\u012b Vivek\u0101nanda**. Born in Germany, M\u00fcller studied Sanskrit at Leipzig and Berlin, and then in Paris. In 1847 the **British East India Company** hired him to edit and publish the _\u1e5ag Veda_. In 1854 he began teaching at Oxford University where he remained until his **death** in 1900.\n\n_MU\u1e46\u1e0cAKA UPANI\u1e62AD_ **.** One of the principal _Upani\u1e63ads_ ; probably composed between 400 and 300 BCE. This _Upani\u1e63ad_ places a special emphasis on the value of **renunciation** and the transitory nature of the fruits of **Vedic** sacrifices, recommending that the spiritual aspirant live in the forest with an **enlightened** teacher, or **guru**.\n\n**MUNI.** Literally, \"one who has taken a vow of silence\"; a sage or monk who practices **asceticism**.\n\n**M\u016aRTI.** A physical image of a deity that enables a worshiper to interact with that deity as though it had a material body. _See also_ IMAGES IN WORSHIP.\n\n**M\u016aRTI-P\u016aJ\u0100.** Worship that utilizes a physical image of a deity in order to facilitate a concrete sense of interaction with that deity. _See also_ IMAGES IN WORSHIP; P\u016aJ\u0100.\n\n**MURUGAN (MURUKAN).** **Tamil** name for **K\u0101rttikeya** , son of **\u015aiva** and **P\u0101rvat\u012b** and brother of **Ga\u1e47e\u015ba**.\n\n**M\u016a\u015aAKA.** The mouse; vehicle of **Ga\u1e47e\u015ba**.\n\n**MUSIC.** Music has been an integral part of Hindu worship since ancient times. The _S\u0101ma Veda_ , one of the original four **Vedic** _Sa\u1e43hit\u0101s_ , sets the Vedic **mantras** to music. The rhythmic chanting of mantras and the singing (k\u012brtana) of religiously themed songs ( **bhajanas** ) to evoke **bhakti** , or devotion, have been practiced continuously and have a widespread appeal that cuts across boundaries of gender, **caste** , and sect. The works of the bhakti poets were always intended to be sung, rather than read or recited as **poetry**. Classical Indian **dance** , which is also religiously themed, has always been accompanied by music as well. _See also_ ART; LITERATURE.\n\n## N\n\n**N\u0100CIKETA.** Character in the _Ka\u1e6dha Upani\u1e63ad_ ; a young **Brahmin** who journeys to the underworld in order to ask **Yama** , the **god** of **death** , the secret of **immortality**.\n\n**N\u0100GA.** Serpent; in Hindu and **Buddhist** **literature** , they are often magical beings with a serpentine form. The dangerous nature of snakes, especially cobras, in India makes them creatures to be respected and feared, so a certain degree of awe surrounding snakes is not at all surprising. **Vi\u1e63\u1e47u** is often depicted resting on the belly of **\u015ae\u1e63a** , the cosmic serpent who is the deity's protector. **\u015aiva** is also depicted with snakes crawling on his body both to show his transcendent state of fearlessness and also to depict the coiled energy of the **Ku\u1e47\u1e0dalin\u012b \u015bakti** , a manifestation of the **Mother Goddess**. The **Buddha** and P\u0101r\u015bvan\u0101th, the 23rd **T\u012brtha\u1e45kara** of **Jainism** , are both depicted as being protected in their practice of **meditation** by the lord of the N\u0101gas. **Manas\u0101** , a protector **goddess** revered in **Bengal** , has a serpentine form. It is said that the N\u0101gas reside in a portion of the underworld known as **P\u0101t\u0101la** , presided over by the giant serpent **V\u0101suki**.\n\n_N\u0101ga_ is also the name of an order of naked **\u015aaiva** warrior monks affiliated with the **Da\u015ban\u0101mi Order** charged with protecting **Brahmins** from physical attack, since the Brahmins are expected to practice **ahi\u1e43s\u0101** (nonviolence) and may not use weapons. The N\u0101gas have fought in battle both with Muslim forces and rival Hindu warrior monks.\n\n_N\u0101ga_ is also the name of an **\u0100div\u0101s\u012b** , or aboriginal, group of tribes based in the extreme northeast of India, in the state of Nagaland, on the border with Burma.\n\n**NAIMI\u1e62A.** Forest mentioned in Hindu **literature** , renowned as a place of pilgrimage for **ascetics**. It was in the Naimi\u1e63a forest, or _Naimi\u1e63\u0101ra\u1e47ya_ , that a 12-year-long sacrifice ( **yaj\u00f1a** ) was held by the sage **\u015aaunaka** at which the _Mah\u0101bh\u0101rata_ was first recited by a pupil of **Lomahar\u1e63ana S\u016bta** , who had in turn heard it from a direct pupil of **Vy\u0101sa** , the author of the text.\n\n**NAKULA.** Character in the _Mah\u0101bh\u0101rata_ ; one of the **P\u0101\u1e47\u1e0dava** brothers; twin brother of Sahadeva. These twins were the sons of the **A\u015bvins** and **M\u0101dr\u012b**.\n\n**NAMASTE.** \"I pay homage to the divine within you\"; traditional Hindu greeting, often accompanied by the **a\u00f1jali** , or gesture of respect with folded hands.\n\n**NAMB\u016aTIRI (NAMBOODIRI) BRAHMINS.** Lineage of **Brahmins** based primarily in the state of **Kerala** , on the southwest coast of India. They are unusual in recognizing the validity of only the first three _Vedas_ , not the _Atharva Veda_ , and for the style of **Vedic** recitation they employ, which utilizes hand gestures ( **mudr\u0101s** ) and head movements as an aid to memorization. _See also_ TRAY\u012a VIDY\u0100.\n\n**N\u0100MDEV (fl. 13th century CE).** **Vai\u1e63\u1e47ava bhakti** **poet** and saint of Maharashtra. He was especially devoted to **Vi\u1e63\u1e47u** in his form as **Vi\u1e6d\u1e6dhala** (literally \"standing on a brick\"). He writes, \"As a bee's heart might be set upon the fragrance of a flower, or as a fly might take resort to honey, similarly does my mind cling to **God**.\" N\u0101mdev was a disciple of **J\u00f1\u0101ne\u015bvara**.\n\n**NAMM\u0100\u1e36V\u0100R (880\u2013930).** One of the **\u0100\u1e37v\u0101rs** , 12 **Vai\u1e63\u1e47ava** **poets** of south India. Born a **\u015a\u016bdra** or member of the servant **caste** , he composed numerous devotional poems in the **Tamil** language that were later collected in a volume called the _Tiruv\u0101ymo\u1e37i_. This text is revered in the **\u015arivai\u1e63\u1e47ava** tradition as being no less sacred than the _Vedas_ and continues to be used in **temple** rituals to the present day. Namm\u0101\u1e37v\u0101r's revered position as a person of low caste in this tradition is significant, being illustrative of the **philosophy** of **bhakti** , or devotion, according to which social categories like gender and caste are irrelevant to one's spiritual status or proximity to salvation.\n\n**NAMUCI.** Demon slain by **Indra**. Namuci could be killed neither by day or night, nor by anything either wet or dry. Indra circumvented this difficulty by killing Namuci at dusk with sea foam. This story, mentioned in the _\u1e5ag Veda_ , is echoed in later **literature** by the story of the slaying of **Hira\u1e47yaka\u015bipu** by the **Narasi\u1e43ha** **avat\u0101ra**.\n\n**N\u0100NAK, GURU (1469\u20131539).** The founder of **Sikhism** ; a contemporary of **Kab\u012br** and a fellow member of the **Sant** movement. Born a Hindu in the Punjab, N\u0101nak claimed to have received a revelation from **God** while **meditating** in a forest. N\u0101nak's message was: \"There is no Muslim and there is no Hindu.\" Those who accepted his teaching of Hindu\u2013Muslim unity came to be known as _Sikh_ , a Punjabi word which means \"disciple,\" and they took N\u0101nak as their **guru**. Guru N\u0101nak and his followers sought to harmonize the teachings of **Islam** and Hinduism, taking from Islam its strong emphasis on the unity of God. Guru N\u0101nak called **the One** God, known by Hindus under many names and forms, _Sat N\u0101m_ , or the \"True Name,\" or _Ek Onkar_ , the One God. The One God is the **creator** of the entire universe, according to Sikhism, and humans are the highest of God's creations. From Hinduism, N\u0101nak took the doctrines of **karma** and **rebirth** and the emphasis on the importance of the guru, or living teacher. **Liberation** from rebirth is given by God, much as in devotionally oriented forms of Hinduism. The community Guru N\u0101nak established accepted all, regardless of religion, gender, or **caste**.\n\n**NANDA GOPA.** The adoptive father of **K\u1e5b\u1e63\u1e47a** who raised him after his wicked uncle, **K\u0101\u1e43sa** , had imprisoned his parents and sought to have the infant K\u1e5b\u1e63\u1e47a killed. Nanda Gopa was a cowherd, and it was for this reason that K\u1e5b\u1e63\u1e47a spent his youth among the **gop\u012bs** of **V\u1e5bnd\u0101vana**.\n\n**NANDANAR (fl. 7th century CE).** One of the 63 **N\u0101yan\u0101rs** \u2014saints and **bhakti** **poets** revered in the **Tamil \u015aaiva** tradition. Nandanar was born in the **\u015a\u016bdra** or servant **caste**. He was intensely devoted to **\u015aiva** and wanted to see the images of \u015aiva as **N\u0101\u1e6dar\u0101ja** , or \"Lord of the Cosmic **Dance** ,\" at the **temple** of **Cidambaram** , for the purpose of having **dar\u015bana** , or sacred visual communion. Because of his low caste, however, he was denied entry to the temple. Standing outside the temple, his view of the main image was blocked by an image of the bull, **Nandi** , who serves as \u015aiva's vehicle. It is said that in order to let Nandanar have his dar\u015bana, \u015aiva commanded Nandi to move aside, which he did. To this day, the image of Nandi outside the temple at Cidambaram is offset from the main door so as not to obstruct the view of the deity from outside. The name _Nandanar_ comes from this incident, meaning the one whose devotion could make even Nandi move aside.\n\n**NANDI.** The bull who serves as the vehicle of **\u015aiva**.\n\n**NARA.** \"Man\"; the first man, the progenitor of mankind. _See also_ N\u0100R\u0100YA\u1e46A.\n\n**N\u0100RADA.** One of the Seven Sages ( **Saptar\u1e63i** ) presiding over the cosmos; a wandering minstrel and messenger of the **gods** , N\u0101rada is frequently depicted in both the _Vedas_ and the _Pur\u0101\u1e47as_ as a seeker of wisdom who goes to various deities and wise persons in order to receive instruction. From a literary perspective, he is often a stand-in for the reader or listener, who is also receiving instruction from the deity or wise person in question while N\u0101rada is being instructed in the text. N\u0101rada is also depicted as something of a troublemaker by passing on messages or news that may not be welcome or sharing information that should not be shared. He is a son of the sage **Ka\u015byapa** and some of the hymns of the _\u1e5ag Veda_ are attributed to him. His first appearance as a literary character, though, is in the _Ch\u0101ndogya Upani\u1e63ad_ , where he is inquiring into the nature of the self ( **\u0101tman** ). He is the lord of the **gandharvas** , or celestial musicians, and plays a **v\u012b\u1e47\u0101** , or lute.\n\n**NARAKA.** Hell. There are actually seven Hindu hells\u2014layers or dimensions of existence that are highly unpleasant and in which beings are born who have extremely bad **karma** ( **p\u0101pa** ). The Hindu hells might better be described as purgatories; for unlike the hells of **Christianity** and **Islam** , the denizens of the Hindu hells do not reside there forever, but only as long as they have bad karma for which they need to atone. Once their bad karma has been exhausted, such beings are reborn, usually in human form. This can take a very long time, however\u2014in some cases, until the end of a cosmic cycle ( **kalpa** ), which may be billions of years. In other cases, though, one's period of residence in one of the hells can be of a relatively short duration. Hell is also known as **P\u0101t\u0101la**. _See also_ REBIRTH.\n\n**NARASI\u1e42HA.** \"Man-lion.\" The Narasi\u1e43ha is the fourth on the traditional list of 10 **avat\u0101ras** , or incarnations, of **Vi\u1e63\u1e47u**. The demonic **Hira\u1e47yaka\u015bipu** , who is symbolic of ignorance, held a deep hatred for Vi\u1e63\u1e47u. Annoyed by his son **Prahl\u0101da** 's equally deep devotion to Vi\u1e63\u1e47u, Hira\u1e47yaka\u015bipu taunted Prahl\u0101da. Once, asking Prahl\u0101da where Vi\u1e63\u1e47u could be found, Prahl\u0101da replied that Vi\u1e63\u1e47u (whose name means \"all-pervasive\") was everywhere, even within a nearby stone pillar. On hearing this, Hira\u1e47yaka\u015bipu kicked the pillar. The pillar then broke asunder and Vi\u1e63\u1e47u, in the form of the Narasi\u1e43ha, came forth. The man-lion grabbed Hira\u1e47yaka\u015bipu, placed him on his lap, and slew him. This event occurred at dusk. The circumstances surrounding the **death** of Hira\u1e47yaka\u015bipu were a result of the fact that he was magically protected from any attack by man or beast (the Narasi\u1e43ha was neither), on the ground or in the air (which is why Narasi\u1e43ha held him in his lap), and by day or night (making dusk and dawn times of vulnerability for him).\n\n**N\u0100R\u0100YA\u1e46A.** \"Son of **Nara** (the primordial man).\" Nara and N\u0101r\u0101ya\u1e47a are associated with one another from the _\u1e5ag Veda_ onward. These two are a pair of deities or sages who share a close bond of friendship. It is said in the _Mah\u0101bh\u0101rata_ that **Arjuna** is Nara and **K\u1e5b\u1e63\u1e47a** is N\u0101r\u0101ya\u1e47a. N\u0101r\u0101ya\u1e47a is identified with **Vi\u1e63\u1e47u** from the _\u0100ra\u1e47yakas_ onward. But in some sources, the names _Nara_ and _N\u0101r\u0101ya\u1e47a_ are used interchangeably as dual epithets of Vi\u1e63\u1e47u. _See also_ SATYAN\u0100R\u0100YA\u1e46A.\n\n**N\u0100SATYA.** \"Helpful, kindly\"; epithet of the **A\u015bvins**.\n\n**N\u0100SIK.** One of the four cities where the **Kumbha Mel\u0101** is held every 12 years. It is in Maharashtra, on the bank of the God\u0101var\u012b River.\n\n**N\u0100STIKA.** Literally a \"nay-sayer,\" one who says, \"It is not.\" Traditionally, this term refers to those who deny the authority of the _Vedas_ \u2014specifically, the followers of the **C\u0101rv\u0101ka** or **Lok\u0101yata** **materialist** **philosophy** , the **Jains** , and the **Buddhists**. In modern Hinduism this term is also used to refer to an atheist\u2014though again, this is not the ancient understanding, and some schools of thought regarded as **\u0101stika** , or orthodox, affirmed the authority of the _Vedas_ while being nontheistic, such as the early **S\u0101\u1e43khya** system and the **P\u016brva M\u012bm\u0101\u1e43s\u0101** system.\n\n**N\u0100\u1e6cAR\u0100JA.** \"Lord of the **Dance** \"; epithet of **\u015aiva** in his role as creator, destroyer, and re-creator of the world as it passes through its ongoing cosmic cycle. The image of \u015aiva as lord of the cosmic dance, particularly as it is captured in the **artistic** style of the **Co\u1e37a** dynasty, has become emblematic of Hinduism. The image depicts the ascetic \u015aiva, with his matted hair flying behind him, in an ecstatic dance. One of his four hands holds the drum with which he keeps the beat of his dance. One holds a flame, which he is spinning rapidly around, an **action** depicted with a ring of fire that frames the entire image, like a stop-action photo of someone swirling a lit torch, creating a ring of light. One hand is held up in the **abhaya-mudr\u0101** , and the fourth hand points toward his upraised foot.\n\nThe symbolism of the image is profound. The rapid movement that it depicts, if juxtaposed with the stillness of the image itself, represents a coming together of time and eternity\u2014the realm of change over which the deity presides and the deity himself, who is beyond time and change. The abhaya-mudr\u0101 means \"do not be afraid,\" advising devotees not to fear the awe-inspiring power of **creation** and destruction that the deity embodies. The flame represents the power of both creation and destruction, having both capacities. The deity dances upon a dwarf, representing the ego, which must be overcome in order for one to realize one's inner divinity.\n\nThe overall message is that creation and destruction are part of the same process and that one need not fear change, knowing that everything is finally one with the divine.\n\n**N\u0100TH YOG\u012aS.** **Tantric \u015aaiva** sect of **ascetics** established by **Matsyendran\u0101th** (fl. 11th century CE) and his successor, **Gorakhn\u0101th** (fl. 12th century CE). Gorakhn\u0101th is credited with the invention of **Ha\u1e6dha Yoga** , the system of yogic postures designed to prepare the body for the rigors of **meditation**. Membership in the N\u0101th Yog\u012b tradition is open to members of all **castes** and has been especially prominent in northern India.\n\n**N\u0100THDW\u0100RA.** Site in Rajasthan that houses a black stone image of **K\u1e5b\u1e63\u1e47a** known as \u015ar\u012b N\u0101thji. It is a popular pilgrimage site, with large crowds gathering for the opening of the **garbha-g\u1e5bha** , or inner sanctum, to have **dar\u015bana** , or sacred visual communion with the image.\n\n_N\u0100\u1e6cYA \u015a\u0100STRA_ **.** \"Authoritative Text on Drama\"; ascribed to the sage **Bharata**. This **Sanskrit** text from the first millennium of the Common Era is the authoritative guide to Indian classical theater and **dance**. A lengthy and detailed work, the _N\u0101\u1e6dya \u015a\u0101stra_ gives a thorough account of not only the practicalities of staging a play or dance drama but also the religious dimensions of such performances, plus a theory of aesthetics. _See also_ LITERATURE; POETRY; RASA.\n\n**NAVAGRAHAS.** The nine planets. These do not refer to the nine planets of the solar system of modern astronomy (now eight, with the recent demotion of Pluto), but to the five planets that are visible to the naked eye (Mercury, Venus, Mars, Jupiter, and Saturn), the **moon** , the sun, and **R\u0101hu** and **Ketu** , the latter two being postulated to account for the phenomena of solar and lunar eclipses, respectively. **P\u016bj\u0101** , or ritualistic worship of deities, often includes the paying of respects to the nine planets to maximize the benefits and to minimize the inauspicious effects of their various astrological influences. The planet Mercury is called **Bu\u1e0dha**. Venus is **\u015aukra**. Mars is **Ma\u00f1gal** , Jupiter **B\u1e5bhaspati** , and Saturn **\u015a\u0101ni**. Each of the planets\u2014except for R\u0101hu and Ketu\u2014presides over a day of the week in the **Hindu calendar**. The sun presides over Sunday ( **Raviv\u0101ra** ), the moon over Monday ( **Somav\u0101ra** ), Mars over Tuesday ( **Ma\u00f1galv\u0101ra** ), Mercury over Wednesday ( **Bu\u1e0dhav\u0101ra** ), Jupiter over Thursday ( **Guruv\u0101ra** ), Venus over Friday ( **\u015aukrav\u0101ra** ), and Saturn over Saturday ( **\u015a\u0101niv\u0101ra** ). _See also_ ASTROLOGY.\n\n**NAVAR\u0100TR\u012a P\u016aJ\u0100.** \"Nine-Night Worship\"; festival of devotion to **Durg\u0101** and **R\u0101ma**. The nine nights of the worship correspond to the 10 days of the festival of **Da\u015bahr\u0101**. This festival is celebrated in the **bright** **half** of the month of **\u0100\u015bvina**. In northern India, this festival celebrates **R\u0101ma** 's victory over **R\u0101va\u1e47a** and it culminates in **D\u012bp\u0101vali** (or D\u012bwali), the festival of lights in honor of the **goddess Lak\u1e63\u1e43\u012b**. In Gujarat, the **Garbha** dance is frequently used to celebrate this festival. In **Bengal** and southern India, this festival celebrates **Durg\u0101** 's victory over the Buffalo Demon ( **Mahi\u1e63\u0101sura** ). In Bengal, it culminates in **K\u0101l\u012b p\u016bj\u0101**.\n\n**NAVYA NY\u0100YA.** \"New logic\"; system of philosophy ( **dar\u015bana** ) that is in continuity with the older **Ny\u0101ya** system of logic, but with important revisions and clarifications that were the result of centuries of reflection and debate among Indian logicians; established in the 13th century by the philosopher Ga\u1e45ge\u015ba Up\u0101dhy\u0101ya and incorporating some concepts from earlier philosophers, like **V\u0101caspati Mi\u015bra** and **Udayana** (both of whom lived in the 10th century of the Common Era). Navya Ny\u0101ya is largely a response to the criticisms of the **Advaita Ved\u0101nta** tradition of the Naiy\u0101yikas' realist ontology. Advaita Ved\u0101nta teaches that the universe is ultimately a delusory appearance, or **m\u0101y\u0101** , the only true reality being **Nirgu\u1e47a Brahman** , or Brahman without differentiating **qualities**. The Ny\u0101ya system, on the other hand, teaches that the world is both real and knowable. The Navya Ny\u0101ya system appealed not only to Hindus but also to Jain thinkers, who also hold a realist ontology. One of the most revered of Jain philosophers, Ya\u015bovijaya (1624\u20131688), was a student of Navya Ny\u0101ya and incorporated its categories and arguments into his defense of **Jainism**.\n\n**NAVYA \u015a\u0100STRA.** \"New **Scripture** \"; Hindu organization dedicated to the eradication of **casteism** and, ideally, of the **caste** system itself from Hinduism. Navya \u015a\u0101stra, unlike some **Dalit** organizations, is avowedly Hindu and argues that the caste prejudice is not intrinsic to Hinduism and is contradicted by important currents of Hindu **philosophy** , such as **Ved\u0101nta**. The organization has offered a public apology on behalf of the Hindu tradition to Dalits for the centuries of abuse they have experienced.\n\n**N\u0100YAN\u0100RS.** A group of 63 saints and **bhakti** **poets** revered in the **Tamil** \u015aaiva tradition. Their life stories are contained in a text called the _Periya Pur\u0101\u1e47a_ , a scriptural text of the **\u015aaiva Siddh\u0101nta** tradition of **\u015aaivism**. According to the **philosophy** of bhakti, **caste** and gender are irrelevant to the level of devotion of which one is capable\u2014and thus to the attainment of **mok\u1e63a**. This is in clear contrast with the more conservative tradition of **Brahmanism** , according to which **rebirth** as a male **Brahmin** is a prerequisite for the attainment of mok\u1e63a. The N\u0101yan\u0101rs illustrate this spiritual egalitarianism, coming from a wide range of backgrounds, including members of the humblest strata of Indian society.\n\n**NEHRU, JAWAHARLAL (1889\u20131964).** First prime minister of India. Though a close confidant of **Mohand\u0101s K. G\u0101ndh\u012b** , Nehru did not share G\u0101ndh\u012b's radical vision of returning India to its ancient traditions, preferring a model of rapid industrialization and modernization. Under Nehru's leadership, independent India embraced secularism and a \"mixed economy,\" consisting of aspects of both capitalism and socialism. He also helped to start the movement of nonaligned countries and sought a position of neutrality between the Soviet Union and the United States during the Cold War.\n\n**NEO-VED\u0100NTA.** Modern **Ved\u0101nta**. It is distinguished from traditional Ved\u0101nta by its commitment to the authority of direct experience ( **anubh\u0101va** ) over that of **scripture** and its universalism. That is, Neo-Ved\u0101ntins see Ved\u0101nta not so much as an interpretation of the _Vedas_ \u2014the traditional preserve of the **Brahmins** \u2014but as a universal **knowledge** that is available, in principle, to all\u2014a **perennial philosophy**. In the words of one of its leading exponents, **Sarvepalli Radhakrishnan** , Ved\u0101nta is \"not a religion, but religion itself in its most universal and deepest significance.\" All religions, according to Neo-Ved\u0101nta, are paths to **God-realization** , if properly understood.\n\nThe historical origins of Neo-Ved\u0101nta can be found in the **Bengal** Renaissance of the 19th century, when **R\u0101m Mohan Roy** and other Hindu reformers began to rethink Hinduism in modern, rationalist terms. Roy's **Brahmo Sam\u0101j** sought to purge the Hindu tradition of all elements that it regarded as irrational, such as the use of **images in worship** , **sat\u012b** , and **casteism**. **Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b** 's **\u0100rya Sam\u0101j** had a very similar agenda, and both reform movements saw in the _Vedas_ a primordial **monotheism**. The two differed, however, inasmuch as Roy saw the culmination of **Vedic** wisdom being in the _Upani\u1e63ads_ , whereas Day\u0101nanda emphasized the early Vedic _Sa\u1e43hit\u0101s_. The \u0100rya Sam\u0101j therefore defined itself as distinct from Neo-Ved\u0101nta.\n\nNeo-Ved\u0101nta, however, did not capture the imaginations of many Hindus beyond the relatively small circle of middle- and upper-class, English-educated Indians until the coming of **Ramakrishna** and **Sw\u0101m\u012b Vivek\u0101nanda**. Ramakrishna, who was in many ways a thoroughly traditional figure\u2014a passionate devotee in the mold of the **bhakti** movement\u2014adhered to practices and beliefs antithetical to those of the Brahmo Sam\u0101j and rooted his teachings in his direct experiences rather than scriptural study. In this way, he was thoroughly traditional and thoroughly modern. Partaking of both of these worlds, Ramakrishna embodied an approach that could be appreciated by the rationalists and the traditionalists alike.\n\nRamakrishna's disciple, Vivek\u0101nanda, went on to become an articulate voice for this movement and transformed it into an international phenomenon by traveling to the West and initiating non-Indians into Ved\u0101ntic practice, and even into the **Ramakrishna Order** of **renouncers**. Inspiring many Hindus with the idea that their ancient traditions had universal significance, he helped to fuel national pride and the movement for Indian independence (although he was himself steadfastly apolitical).\n\n**NEPAL.** Largely Hindu country located between India and Tibet. Until 2006, when its parliament declared it to be a secular state, Nepal was the only officially Hindu nation in the world. Nepal also has the highest percentage of Hindus in its population of any other country. The other major religion of Nepal is **Buddhism** , and the **Buddha** is believed to have been born and to have grown up within what are now the borders of Nepal. Hindu traditions in Nepal have blended in many ways with those of Buddhism. Both religions have a strong **Tantric** orientation in Nepal. Nepali Hinduism has also preserved many traditions that have largely been superseded in India, including **animal sacrifice** and a variety of rituals associated with having a Hindu monarchy.\n\n**NETI NETI.** \"Not this, not that\"; an approach to **Brahman** recommended in some parts of the _Upani\u1e63ads_ that consists of realizing that Brahman is unlimited. It therefore cannot be identified with any one thing that can be grasped by the senses or the mind. Akin to a **Buddhist** approach to **enlightenment** \u2014which consists of realizing that no one thing is the **self** \u2014the path of distinguishing Brahman from any limiting **quality** is a form of the **j\u00f1\u0101na yoga** , or yoga of wisdom, whereby one realizes that Brahman alone is real, that all things are limited and transitory, and therefore not Brahman, and not fully real. Brahman is thus \"Not this, not that.\" _See also_ VED\u0100NTA.\n\n**NEW DELHI.** _See_ DELHI.\n\n**NIMB\u0100RKA (c. 12th\u201314th centuries CE).** **\u0100c\u0101rya** of the _Bhed\u0101bheda_ system of **Ved\u0101nta**. In this system, both difference ( _bheda_ ) and nondifference ( _abheda_ ) are affirmed. In other words, if one thinks of the various systems of Ved\u0101nta as existing on a continuum, with **Advaita** at one extreme end (affirming the unreality of genuine difference and the reality of **Brahman** alone) and **Dvaita** at the other end (affirming the reality of difference and denying an underlying unity), then Nimb\u0101rka's system can be seen as a \"middle path\" that tries to bring these two ideas together, affirming that both difference and unity are real.\n\nIn some ways an anticipation of the **Neo-Ved\u0101nta** of **Ramakrishna** , Nimb\u0101rka's teaching sees difference and nondifference simply as different approaches to an ultimate reality that cannot finally be categorized or conceptualized. Religiously, Nimb\u0101rka's tradition is a **bhakti** -oriented **Vai\u1e63\u1e47ava** path focused on **R\u0101dh\u0101** and **K\u1e5b\u1e63\u1e47a**.\n\nThe dates of Nimb\u0101rka's life are highly uncertain, although most scholars locate him somewhere between the early 12th and 14th centuries. He was from Andhra Pradesh, in southern India.\n\n**NIRGU\u1e46A BRAHMAN.** \" **Brahman** without **qualities** \"; Brahman, the ultimate reality, free from limiting qualities; infinite being ( **sat** ), consciousness ( **cit** ), and bliss ( **\u0101nanda** ); the eternal and unchanging absolute. According to **Advaita Ved\u0101nta** , Nirgu\u1e47a Brahman alone is real. All other entities that can be perceived are in various ways limited, and so deficiently real. The universe of limited entities is a mere appearance, or **m\u0101y\u0101**. M\u0101y\u0101 is projected onto the true reality of Brahman by the **mind** as an effect of ignorance ( **avidy\u0101** ).\n\n**NIR\u1e5aTI.** **Vedic goddess** of misfortunate and destruction. A portion of the **sacrifice** (or **yaj\u00f1a** ) is offered to Nir\u1e5bti to ensure that she stays away from the ritual and does not cause problems.\n\n**NIRV\u0100\u1e46A.** \"Absorption\"; term used primarily by **Buddhists** (but by **Jains** and Hindus as well) for the highest state of spiritual attainment in which one is free from bondage to the cycle of **death** and **rebirth**. Buddhist traditions tend to describe this state in negative terms, as not characterized by suffering, rebirth, or limitation of any kind. Hindu texts, such as the _Bhagavad G\u012bt\u0101_ , speak of _brahma-nirv\u0101\u1e47a_ , or absorption in **God**. _See also_ MOK\u1e62A.\n\n**NIRVIKALPA SAM\u0100DHI.** State of meditative absorption that is completely free from any sense of the duality of subject and object; a state without dichotomizing conceptual constructions. **Sam\u0101dhi** \u2014absorption in the object of one's **meditation** \u2014is the eighth and final \"limb\" of **Pata\u00f1jali** 's eight-limbed ( **\u0101\u1e63\u1e6d\u0101\u1e45ga** ) system of **yoga**. Later traditions that utilize Pata\u00f1jali's system distinguish between sam\u0101dhi in which there remains a sense of subjectivity ( **savikalpa sam\u0101dhi** ) and sam\u0101dhi with no subject\u2013object duality (nirvikalpa sam\u0101dhi). This distinction is most prominent in **Buddhism** , **Ka\u015bm\u012br \u015aaivism** , and the **Neo-Ved\u0101nta** tradition of **Ramakrishna**.\n\n**NITY\u0100NANDA, SW\u0100M\u012a (1897\u20131961). Guru** of **Sw\u0101m\u012b Mukt\u0101nanda** ; founder of the **Siddha Yoga** lineage. Nity\u0101nanda, a mysterious figure, spent much of his life in a cave in **Ga\u1e47e\u015bapuri** , near Bombay, where the Siddha Yoga tradition's main **\u0101\u015brama** is located today. Speaking little, he communicated mainly through gestures and inarticulate sounds, though he was capable of speech, and sometimes spoke in great depth. An interview with one of his disciples that contains his teachings was published as the _Cid\u0101k\u0101\u015ba G\u012bt\u0101_. Born in **Kerala** , he is said to have lived in a state of constant bliss throughout his childhood and is believed by his devotees to have been born **enlightened**. He became a **sanny\u0101sin** , or renouncer, before the age of 20, eventually wandering to Ga\u1e47e\u015bapuri.\n\n**NIVEDIT\u0100, SISTER (1867\u20131911).** Born in Ireland as Margaret Elizabeth Noble, the first Westerner to take monastic vows in a Hindu religious order, becoming a **sanny\u0101sin\u012b** or nun in the **Ramakrishna Order**. Noble's **guru** was **Sw\u0101m\u012b Vivek\u0101nanda** , the first Hindu spiritual teacher to bring his tradition successfully to the West. Moving to **Bengal** , she established a school for girls in 1898, open to people of all **castes**. She befriended a number of prominent Bengali Hindus of the period, including **Rabindran\u0101th Tagore** and **Aurobindo Ghose**. She was also very close to **Sarad\u0101 Dev\u012b** , the wife of **Ramakrishna** \u2014known to his followers as the **Holy Mother** \u2014who referred to Noble as \"Baby.\" Wanting to work for the cause of Indian independence, Noble formally left the Ramakrishna Order so it would experience no negative repercussions from her political activities.\n\n**NIV\u1e5aTTI M\u0100RGA.** \"Path of negation\"; \"path of turning away from activity.\" If one is interested in attaining **mok\u1e63a** , or release from the cycle of **rebirth** , some Hindu traditions claim that it is necessary to renounce life in society and practice **asceticism**. This is the path of negation, of turning away from activity. The basic idea is that a worldly life, in which one does one's duty ( **dharma** ) and pursues pleasure ( **k\u0101ma** ) and wealth ( **artha** ) is not compatible with the attainment of mok\u1e63a. In terms of the four goals of humanity or **puru\u1e63\u0101rthas** of classical Hindu thought, the pursuit of k\u0101ma, artha, and dharma make up the **prav\u1e5btti m\u0101rga** \u2014the path of activity, of turning toward the world. Pursuit of the goal of mok\u1e63a, the fourth of the puru\u1e63\u0101rthas, requires one to follow the path of niv\u1e5btti. And in terms of the four stages of life, or **\u0101\u015bramas** , the first three stages\u2014the stages of student, householder, and retiree\u2014make up the prav\u1e5btti m\u0101rga and the fourth stage\u2014renunciation, or **sanny\u0101sa** \u2014makes up the niv\u1e5bttii m\u0101rga.\n\n**NIYAMA.** \"Observance\"; \"discipline\"; the second \"limb\" of **Pata\u00f1jali** 's eight-limbed ( **a\u1e63\u1e6d\u0101\u1e45ga** ) system of **yoga**. The niyamas consist of five virtues: _\u015bauca_ (purity), _santo\u1e63a_ (contentment), _tapas_ ( **asceticism** ), _sv\u0101dhy\u0101ya_ (study), and surrender to the Lord ( _\u012a\u015bvara-pra\u1e47idh\u0101na_ ).\n\n**NOBLE, MARGARET ELIZABETH (1867\u20131911).** _See_ NIVEDIT\u0100, SISTER.\n\n**NONDUALITY, PRINCIPLE OF.** _See_ TANTRA.\n\n**NY\u0100YA.** Logic; one of the six systems of Hindu philosophy, or **dar\u015banas** ; often paired with the **Vai\u015be\u1e63ika** system, with which it shares a basic world-view and a realist ontology. The chief concern of Ny\u0101ya is epistemology, or the theory of **knowledge**. According to the Naiy\u0101yikas (adherents of Ny\u0101ya), there are four **pram\u0101\u1e47as** , or valid foundations of knowledge: **pratyak\u1e63a** , or sensory perception; **anum\u0101na** , or inference; **upam\u0101na** , or verbal comparison\u2014in other words, recognizing a perceived entity to be a member of an already known category; and \u015babda, the verbal testimony of a knowledgeable person. Ny\u0101ya, as an **\u0101stika** or \"orthodox\" system of philosophy, identifies the Vedic **scriptures** as the preeminent example of the \u015babda pram\u0101\u1e47a.\n\nThe root text of this system of thought\u2014the _Ny\u0101ya S\u016btra_ \u2014is attributed to a sage by the name of **Gautama** (not the **Buddha** or the **Vedic** sage of the same name). This text was probably composed around 300 BCE. The Naiy\u0101yikas debated extensively with the **Buddhists** until Buddhism had largely vanished from India (by 1300 CE) and with other systems of Indian philosophy, particularly **Advaita Ved\u0101nta**. Major figures of the Ny\u0101ya tradition include **V\u0101caspati Mi\u015bra** (900\u2013980 CE)\u2014who was also an adherent of Advaita\u2014and **Udayana** (fl. 10th century CE). In the 13th century, largely in response to Advaitic critiques, a Naiy\u0101yika philosopher named Ga\u1e45ge\u015ba established the **Navya Ny\u0101ya** , or New Ny\u0101ya system, a refined version of the older system of Ny\u0101ya.\n\n## O\n\n**OLCOTT, HENRY STEELE (1832\u20131907).** Cofounder\u2014along with the Russian heiress **Helena Petrovna Blavatsky** \u2014and first president of the **Theosophical Society**. Formally established in 1875, the objectives of the Theosophical Society were (and remain) \"to form a nucleus of the universal brotherhood of humanity without distinction of race, creed, sex, **caste** , or color,\" \"to encourage the study of comparative religion, **philosophy** , and science,\" and \"to investigate the unexplained **laws** of nature and the powers latent in man.\" Influenced deeply by Hinduism and **Buddhism** , Olcott and Blavatsky moved to India in 1879, where the global headquarters of the society are located today (in Ady\u0101r, **Madras** ). Members of the Theosophical Society began to play prominent roles in the movement for Indian political independence, arousing the suspicion of the British government.\n\nTheosophists promoted pride and interest among Hindus and Buddhists in their religious and cultural heritage, acting as \"reverse missionaries,\" and articulating the idea that Hinduism and Buddhism preserve an ancient global wisdom tradition, or **perennial philosophy**. At one point, a strategic alliance began to form between the Theosophical Society and the **\u0100rya Sam\u0101j**. But **Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b** fell out with Blavatsky and Olcott in 1880, when they announced their conversions to Buddhism in **Sri Lanka**. Olcott remained in Sri Lanka for a number of years, promoting Buddhism and writing a Buddhist catechism that continues to be in use among Therav\u0101da Buddhists in Sri Lanka today.\n\n**O\u1e42.** Also spelled AUM, the most sacred sound and **b\u012bja mantra** of Hinduism. In the _Ma\u1e47\u1e0dukya Upani\u1e63ad_ , O\u1e43 is described as encompassing all of existence. Each of its parts (A, U, M, and the totality of these three) is associated with one of the four **states of consciousness**. It figures prominently in Hindu prayer and **meditation** and is frequently incorporated into more lengthy **mantras** as their first syllable, such as _O\u1e43 nama\u1e25 \u015aiv\u0101ya_ , or _O\u1e43 namo bhagavate Vasudev\u0101ya_.\n\n**O\u1e42 JAYA JAGAD\u012a\u015aA HARE.** Devotional song popularized in the 20th century by the Hindi singer Lata Mangeshkar (1929\u2013 ). Frequently sung during **\u0101rat\u012b** in modern Hindu **temples** in North America and Europe.\n\n**ONAM.** Harvest festival that is celebrated in the state of **Kerala** , on the southwest coast of India. It occurs in the month of **Bhadra** (August\u2013September).\n\n**ONE, THE.** The supreme reality or absolute that underlies all existence. In the _\u1e5ag Veda_ it is said that the One is the reality behind the appearance of diversity that gives rise to the world of phenomena. It is also said that the many **devas** or deities are all names for the One ( _\u1e5ag Veda_ 1.164.46)\u2014\"Reality is one, but the wise speak of it in many ways.\" In the _Upani\u1e63ads_ , the ultimate reality\u2014 **Brahman** \u2014is frequently said to be \"One alone, without a second.\" Such **Vedic** references to the One are the basis for the view that an ancient thread of **monotheism** underlies the Hindu practice of **polytheism** , the idea being that what appear to be many deities are forms, aspects, or manifestations of the One. Though this view is certainly ancient in Hindu traditions, it has been reemphasized in the modern period in response to the criticisms of **Christian** and **Islamic** missionaries.\n\n**ORISSA.** State on the eastern coast of India, between **West Bengal** in the northeast and Andhra Pradesh in the south; location of several prominent Hindu **temples** , including the **Vai\u1e63\u1e47ava** temple of **Jagann\u0101tha** in the city of **Puri** , where the Vai\u1e63\u1e47ava **bhakti** saint and **\u0101c\u0101rya** , **Caitanya** , resided for much of his life. Orissa is also home to **Bhuvane\u015bvara** , with its prominent temples to **\u015aiva** and **K\u0101l\u012b** , and **Kon\u0101rak** , with its massive temple in the shape of a **chariot** to **S\u016brya** , the sun **god**. One of the four **ma\u1e6dhas** , or monasteries that were established by **\u015aa\u1e45kara** in the four corners of India, is also located in Puri.\n\n**OSHO.** _See_ RAJNEESH, BHAGWAN.\n\n**OUT OF INDIA THEORY.** Alternative theory of **Vedic** origins, presented as a critique of the more widely accepted **\u0100ryan Migration Theory**. According to this theory (which has many subvarieties that differ on a number of issues), **Indo-European** languages and cultures originated in India, rather than being brought to India through migration or trade. Mainstream scholarship regards this scenario as unlikely, given the existence of older, non-Indo-European substrate elements in **Sanskrit**. Concern is also raised by the **Hindu Nationalist** assumptions and agenda that clearly fuel at least some versions of this theory. Much of the evidence presented in its favor consists of mathematical calculations based on references to **astrological** phenomena mentioned in Vedic texts (which suggest a much greater age for these texts than the dominant model proposes) and on the absence of any explicit reference to a major migration from Central Asia in Indic textual sources.\n\nWhile mainstream scholars have not ultimately found it persuasive, advocates of this theory have raised important critical questions about the dominant model\u2014especially its older variant, the **\u0100ryan Invasion Theory** , which was used in the 20th century to advance an agenda of European superiority. The debate remains very politically charged.\n\nAmong scholars with less obviously political motivations, the debate over \u0100ryan migration tends to pit archaeology against linguistics. The linguistic evidence suggests that some kind of migration has to have occurred, while the archaeological evidence is more ambiguous (and does not support the older idea of a massive invasion).\n\n**OUTCASTE.** One who has no **caste** ; an **Untouchable**. _See also_ DALIT; HARIJAN.\n\n## P\n\n**PADMA.** Lotus flower, often used to symbolize spiritual attainment. The growth of the beautiful lotus flower out of mud and muck represents the process of **enlightenment** , in which the spiritual aspirant starts out in the \"mud and muck\" of mundane existence, with all of its various worries and forms of suffering, and progresses to a state of higher awareness. Hindu deities are often depicted as being seated upon large lotus flowers, and the **cakras** of **Tantric yoga** are also depicted as lotus flowers with varying numbers of petals that symbolically blossom when activated by the rising **ku\u1e47\u1e0dalin\u012b \u015bakti** , the latent energy at the base of the spine. As a prominent Hindu symbol, it is also used as a symbol of the **Hindu Nationalist Bharat\u012bya Janat\u0101 Party** (BJP).\n\n**PADMAN\u0100BHI.** \"Lotus-navel\"; image of **Vi\u1e63\u1e47u** reclining on the cosmic serpent, **\u015ae\u1e63a** , with a lotus stalk emerging from his navel. At the end of the stalk is a **lotus** blossom on which **Brahm\u0101** , the creator deity, sits. Vi\u1e63\u1e47u's feet are lovingly massaged by his wife, the **goddess Lak\u1e63m\u012b**. This image captures the moment of Brahm\u0101's emergence from the body of Vi\u1e63\u1e47u, just before Brahm\u0101 begins the process of **creating** a new universe. A large version of this image is housed at Padman\u0101bhasw\u0101mi **temple** in **Kerala**.\n\n**PADM\u0100SANA. Lotus posture** ; **Ha\u1e6dha Yoga** posture formed by placing the right foot on the left thigh and the left foot on the right thigh. This is the \"full lotus.\" There is also a half-lotus posture ( **ardhapadm\u0101sana** ), formed by placing the right foot on the left thigh and the left foot underneath the right leg.\n\nThe **goddess Lak\u1e63\u1e43\u012b** is also sometimes referred to as _Padm\u0101san\u0101_ because she is seated upon a lotus flower. Other deities are also sometimes represented as _padm\u0101sana_ in this sense. _See also_ \u0100SANA.\n\n**PAILA.** One of the five **Brahmin** students of **Vy\u0101sa** , also known as **Veda Vy\u0101sa** , who is attributed not only with authoring the _Mah\u0101bh\u0101rata_ but also with organizing the _Vedas_ into their current distribution of four texts: the _\u1e5ag Veda_ , _Yajur Veda_ , _S\u0101ma Veda_ , and _Atharva Veda_. According to the _Pur\u0101\u1e47as_ , each of Vy\u0101sa's students was responsible for memorizing and passing on a separate **Vedic** text\u2014one for each of the four _Vedas_ and one for the _Mah\u0101bh\u0101rata_. Paila was the student responsible for transmitting the _\u1e5ag Veda_.\n\n**PALLAVA.** Dynasty that ruled southern India from **K\u0101\u00f1c\u012bpuram** from the sixth to the eighth centuries of the Common Era. Its first king was **Mahendravikrama Varman** (571\u2013630), who, in addition to being the successful founder of a powerful dynasty, was a scholar and a playwright, authoring the _Mattavil\u0101sa Prahasana_ (\"Comedy of Drunken Play\"), which ridicules various religious sects of the period. The Pallavas were overthrown in the eighth century by the **Co\u1e37a** dynasty.\n\n**PA\u00d1CA-MAK\u0100RA.** The **Five M's**. Pa\u00f1ca (\"five\") \"ma-k\u0101ra\" (the letter \"m,\" or \"ma\").\n\n**PA\u00d1CAR\u0100TRA** , **P\u0100\u00d1CAR\u0100TRA.** \"Five nights\"; early **Vai\u1e63\u1e47ava** sect, whose name may be derived from a five-night **yaj\u00f1a** , or sacrifice, performed by **N\u0101r\u0101ya\u1e47a** ( **Vi\u1e63\u1e47u** ) in the _\u015aatapatha_ _Br\u0101hma\u1e47a_. The _\u0100gamas_ , or **scriptures** of this tradition, are largely \"how to\" manuals concerned with correct ritual performance, **yoga** practice, the construction of **temples** , and the consecration of sacred images ( **m\u016brtis** ). By the 10th century of the Common Era, other devotional, **bhakti** -oriented sects had largely absorbed the Pa\u00f1car\u0101tra tradition, continuing to utilize its **mantras** and ritual manuals.\n\n**PA\u00d1CAY\u0100TANA P\u016aJ\u0100.** \"Worship of Five Objects\"; system of worship developed by **\u015aa\u1e45kara** (788\u2013820) according to which one could cultivate **bhakti** , or devotion, toward one of five possible chosen deities or **i\u1e63\u1e6dadevat\u0101s** as a practice of purifying the **mind** to prepare it for the liberating knowledge ( **j\u00f1\u0101na** ) of the unity of the self ( **\u0101tman** ) and the supreme reality ( **Brahman** ).\n\nThe five deities that \u015aa\u1e45kara recommends for this devotional practice are **Ga\u1e47e\u015ba** , **\u015aiva** , **\u015aakti** , **Vi\u1e63\u1e47u** , and **S\u016brya**. This practice may be the source of the **Neo-Ved\u0101ntic** practice of seeing any sacred figure as a potential i\u1e63\u1e6dadevat\u0101 and way to **God-realization** , including figures from outside the Hindu tradition, such as Jesus Christ, the **Buddha** , the Prophet Muhammad, and various **Christian** and Muslim saints.\n\n**P\u0100\u1e46\u1e0cAVA.** Son of **Pa\u1e47\u1e0du**. The five P\u0101\u1e47\u1e0dava brothers\u2014 **Yudhi\u1e63\u1e6dhira** , **Bh\u012bma** , **Arjuna** , **Nakula** , and Sahadeva\u2014are the heroes and main characters of the _Mah\u0101bh\u0101rata_.\n\n**PA\u1e46\u1e0cIT.** Wise person; scholar; usually, though not necessarily, a **Brahmin**.\n\n**PA\u1e46\u1e0cU.** \"Pale\"; character in the _Mah\u0101bh\u0101rata_ , so named because of his extremely pale complexion (possibly referring to albinism); father of the **P\u0101\u1e47\u1e0davas** , the main heroes of the epic; husband of **Kunt\u012b** and **M\u0101dr\u012b** ; brother of the blind **Dh\u1e5btar\u0101\u1e63\u1e6dra**.\n\n**P\u0100\u1e46INI (c. 5th\u20134th centuries BCE).** **Sanskrit** grammarian; author of an authoritative work on grammar called the _A\u1e63\u1e6dadhy\u0101yi_ , which defines the rules of classical Sanskrit usage.\n\n**P\u0100PA.** Evil; sin; a wicked deed; also demerit, the negative aftereffects of an evil deed upon the **soul** \u2014that is, \"bad **karma**.\" Some Hindu traditions claim that p\u0101pa can be expiated through **pr\u0101ya\u015bcitta** , thus eliminating the need to experience a negative result at some point in the future, which would be the normal result of evil **action**.\n\n**PARAMAH\u0100\u1e42SA.** \"Great swan\"; literally, \"great goose.\" However, the word _goose_ has connotations of silliness not culturally appropriate to the **Sanskrit** term _h\u0101\u1e43sa_ , which conveys a sense of gracefulness and transcendence better matched by the English _swan_. The actual animal, _h\u0101\u1e43sa_ , though, is a type of goose. _Paramah\u0101\u1e43sa_ is a title given to a Hindu saint held to be especially holy, having achieved a state transcending mundane consciousness even while still alive, in the physical body.\n\n**PARAMAH\u0100\u1e42SA YOG\u0100NANDA (1893\u20131952).** _See_ YOG\u0100NANDA, PARAMAH\u0100\u1e42SA.\n\n**PARAM\u0100\u1e46U.** \"Supremely small\"; an atom, not in the sense in which this term is used in contemporary physics, but the smallest possible unit into which any material substance can be broken down. _See also_ VAI\u015aE\u1e62IKA.\n\n**PARAM\u0100TMAN.** \"Supreme self\"; supreme **\u0101tman** ; **God** , understood as an indwelling presence that resides in every individual soul ( **j\u012bv\u0101tman** ); the world-soul; the **soul** of the universe.\n\n**PARA\u1e42PAR\u0100.** \"From one to another\"; teaching lineage or tradition that is passed from teacher ( **guru** ) to disciple ( **\u015bi\u1e63ya** ) through the ritual of initiation ( **d\u012bk\u1e63\u0101** ). The student, or disciple, in time becomes the teacher to another group of students to whom the lineage is then passed. Those students themselves become teachers to other students, and so on. A possible analogy from **Christianity** would be the apostolic succession traced through the popes back to Peter and ultimately to Christ. _See also_ GURU-\u015aI\u1e62YAPARA\u1e42PAR\u0100.\n\n**PAR\u0100\u015aARA.** **Vedic** sage attributed with the composition of several hymns of the _\u1e5ag Veda_ in praise of **Agni**. Par\u0101\u015bara is also a character in the _Mah\u0101bh\u0101rata_ , where he is the father of **Vy\u0101sa** by the daughter of the fisher-king, Satyavat\u012b. Vy\u0101sa is both the author of the _Mah\u0101bh\u0101rata_ and the biological father of the kings **Dh\u1e5btar\u0101\u1e63\u1e6dra** and **Pa\u1e47\u1e0du** , making Par\u0101\u015bara an ancestor of the **P\u0101\u1e47\u1e0davas** ' royal lineage.\n\n**PARA\u015aUR\u0100MA.** Sixth out of the traditional list of 10 **avat\u0101ras** , or incarnations, of the deity **Vi\u1e63\u1e47u** ; a wrathful avat\u0101ra who is said to have slain the entire **k\u1e63atriya** (or warrior) **caste** in vengeance for his father, **Jamadagni** , having been killed by a k\u1e63atriya, Arjuna K\u0101rtav\u012brya (not the **P\u0101\u1e47\u1e0dava** hero **Arjuna** of the _Mah\u0101bh\u0101rata_ ). Para\u015bur\u0101ma was a **Brahmin** of the **Bh\u0101rgava** clan. After all the k\u1e63atriyas had been slain by Para\u015bur\u0101ma, they had to be replaced by Brahmins. It is therefore believed that all living k\u1e63atriyas are actually descended from Brahmins.\n\n**PARIK\u1e62IT.** Character in the _Mah\u0101bh\u0101rata_ ; son of **Abhimanyu** , grandson of **Arjuna**. After **Yudhi\u1e63\u1e6dhira** and the other **P\u0101\u1e47\u1e0dava** brothers stepped down from the rule of their kingdom, it was Parik\u1e63it who inherited the throne and continued the family lineage.\n\n**PARIVR\u0100JAKA.** \"Wanderer\"; a Hindu **ascetic** who has renounced home life and does not normally remain in the same place for more than one day at a time; a **sanny\u0101s\u012b**.\n\n**P\u0100RVAT\u012a.** \"Daughter of the Mountain\"; the **Mother Goddess** ; the wife of **\u015aiva** in her most gentle, domestic form (in contrast with her fierce, warrior forms, such as **Durg\u0101** and **K\u0101l\u012b** ); mother of **Ga\u1e47e\u015ba** and **K\u0101rttikeya**. Her father is the **god** of the **Him\u0101layas** , hence her name. P\u0101rvat\u012b dwells with \u015aiva in the Him\u0101layas, on Mount **Kail\u0101sa**.\n\n**PA\u015aUPATA.** A **Tantric \u015aaiva** lineage of **ascetics** known for their antinomian behavior, seeking to break down traditional distinctions between the pure and the impure as rooted in deluded, dualistic thinking, the reality being that all is **\u015aiva** ( **Brahman** ).\n\n**PA\u015aUPATI.** \"Lord of the animals\"; epithet of **\u015aiva** that, in **\u015aaiva** theology, represents the relationship of \u015aiva to the **j\u012bvas** , or souls of living beings, who are analogized with the animals for which \u015aiva cares. A similar image in **Christianity** is of Jesus as the \"Good Shepherd.\"\n\n**P\u0100T\u0100LA.** Hell. There are actually seven Hindu hells\u2014layers or dimensions of existence that are highly unpleasant and in which beings are born who have extremely bad **karma** ( **p\u0101pa** ). The Hindu hells might better be described as purgatories; unlike the hells of **Christianity** and **Islam** , the denizens of the Hindu hells do not reside there forever but only as long as they have bad karma for which they need to atone. Once their bad karma has been exhausted, such beings are reborn, usually in human form. This can take a very long time, however\u2014in some cases, until the end of a cosmic cycle ( **kalpa** ), which may be billions of years. In other cases, though, one's period of residence in one of the hells can be of a relatively short duration.\n\nP\u0101t\u0101la is also the realm of the **N\u0101gas** , presided over by the giant serpent, **V\u0101suki**. _See also_ REBIRTH.\n\n**P\u0100\u1e6cALIPUTRA.** Ancient capital of the **Maurya** dynasty; located on the site of what is now the city of Patna, capital of the modern Indian state of Bihar.\n\n**PATA\u00d1JALI.** Author to whom the _Yoga S\u016btra_ is attributed (the root text of the **Yoga dar\u015bana** , or philosophical system); the inventor of the eight-limbed ( **a\u1e63\u1e6d\u0101\u1e45ga** ) system of yoga that the _Yoga S\u016btra_ describes. Pata\u00f1jali probably lived between 100 BCE and 500 CE. Someone named Pata\u00f1jali is also attributed with a work on **Sanskrit** grammar\u2014the _Mah\u0101bh\u0101\u1e63ya_ , a commentary on the _A\u1e63\u1e6dadhy\u0101yi_ of **P\u0101\u1e47ini** \u2014though it is not clear that these two Pata\u00f1jalis are the same person.\n\n**PATH OF ACTION.** _See_ FOUR YOGAS; KARMA-M\u0100RGA, KARMAYOGA; YOGA.\n\n**PATH OF DEVOTION.** _See_ BHAKTI; FOUR YOGAS; YOGA.\n\n**PATH OF KNOWLEDGE.** _See_ FOUR YOGAS; J\u00d1\u0100NA; J\u00d1\u0100NAM\u0100RGA, J\u00d1\u0100NA-YOGA; YOGA.\n\n**PATH OF MEDITATION.** _See_ A\u1e62\u1e6c\u0100\u1e44GA YOGA; DHY\u0100NA; FOUR YOGAS; **R\u0100JA-YOGA** ; YOGA.\n\n**PA\u1e6c\u1e6cADAKAL.** Site in **Karnataka** that houses several **temples** built by the **C\u0101lukya** dynasty between the sixth and 12th centuries of the Common Era, including a temple with statues depicting **K\u1e5b\u1e63\u1e47a** instructing **Arjuna** in the _Bhagavad G\u012bt\u0101_ and another with sculptures depicting episodes from the _Mah\u0101bh\u0101rata_ and the _R\u0101m\u0101ya\u1e47a_.\n\n**PAU\u1e62A.** Month of the **Hindu calendar** that corresponds roughly to the period from mid-December to mid-January.\n\n**PERENNIAL PHILOSOPHY.** The idea of a common thread of **truth** shared by all (or most) of the world's religions; in the words of **Aldous Huxley** , \"the metaphysic that recognizes a divine Reality substantial to the world of things and lives and **minds** ; the psychology that finds in the **soul** something similar to, or even identical with, divine Reality; the ethic that places man's final end in the **knowledge** of the immanent and transcendent Ground of all being.\" Modern Hindu thinkers, especially **Neo-Ved\u0101ntins** , endorse this idea, identifying this underlying truth with **Ved\u0101nta** **philosophy** , such as when **Sarvepalli Radhakrishnan** writes that Ved\u0101nta is \"not a religion, but religion itself in its most universal and deepest significance.\" If they are properly understood, all religions, according to Neo-Ved\u0101nta, are paths to **God-realization**.\n\n**PHALGUNA.** Month of the **Hindu calendar** that corresponds roughly to the period from mid-February to mid-March.\n\n**PHILOSOPHICAL SCHOOLS.** _See_ DAR\u015aANA.\n\n**PHILOSOPHY.** _See_ DAR\u015aANA.\n\n**PILGRIMAGE.** A journey to a sacred place. Such a journey is a form of meritorious **action** , or **pu\u1e47ya karma**. _See_ YATRA.\n\n**PI\u1e46\u1e0cA.** A ball of rice offered to one's ancestors as part of the **\u015br\u0101ddha** ritual. _See also_ ANCESTRAL RITES.\n\n**PIT\u1e5a.** Literally \"father,\" but also refers to one's ancestors in general.\n\n**PIT\u1e5aLOKA.** Realm of the ancestors; one of the possible afterlife destinations in **Vedic** thought, the other being the **Devaloka** , or realm of the **gods**.\n\n**PLANETS.** _See_ NAVAGRAHAS.\n\n**POETRY.** Popular genre for conveying Hindu religious themes. **K\u0101l\u012bd\u0101sa** (fl. 400 CE) is the most renowned **Sanskrit** poet, whose elegant verse is frequently analogized with the work of Shakespeare as a major contribution to world **literature**. Many of the saints of the **bhakti** or devotional traditions of Hinduism have communicated their devotion in the form of poems and songs. The study of Sanskrit poetics, or _k\u0101vya_ , is connected with the **philosophy** of **rasa** , or the aesthetic sensibility, which many Hindu theologians tie to the spiritual path, seeing the beauty of poetry and of poetic imagery as a way to evoke the experience of devotion and draw the devotee ever nearer to the divine. _See also_ ABHINAVAGUPTA; \u0100\u1e36V\u0100RS; \u0100\u1e46\u1e0c\u0100L; BASAVA; BAULS; BHART\u1e5aHARI; CA\u1e46\u1e0c\u012aD\u0100S; CHATTERJEE, BANKIM CHANDRA; CIVAV\u0100KKIYAR; DA\u1e46\u1e0cIN; GOSV\u0100MINS; JAYADEVA; J\u00d1\u0100NE\u015aVARA; KAB\u012aR; K\u0100LID\u0100SA; KANNAPPA; LALL\u0100; MAH\u0100DEVIYAKKA; MAMMATA; M\u0100\u1e46IKKAV\u0100CAKAR; MEHTA, NARSI; M\u012aRA BAI; N\u0100MDEV; NAMM\u0100\u1e36V\u0100R; NANDANAR; N\u0100YAN\u0100RS; R\u0100MPRAS\u0100D; RAVID\u0100S; SURD\u0100S; TAGORE, RABINDRAN\u0100TH; _TIRUV\u0100YMO\u1e36I_ ; TUK\u0100R\u0100M; TULS\u012aD\u0100S.\n\n**POLYTHEISM.** Affirmation of the existence of and reverence for multiple deities, or **gods**. Most, though not all, Hindu traditions are polytheistic in practice. Many of these systems, however, affirm a deeper **monotheism** that underlies their polytheistic practices, according to which the **One** **God** manifests in many forms that correspond to the many gods of Hinduism\u2014and even to the deities of other religions. The issue of polytheism and monotheism in Hinduism is further complicated by the fact that there are many **Sanskrit** terms that can be translated as \" **God** ,\" some of which have polytheistic connotations and some of which have monotheistic, or even pantheistic or panentheistic, connotations. Not all Hindu traditions agree, furthermore, in regard to which understanding is correct.\n\n**Devas** are many in number and correspond closely to a polytheistic conception of divinity\u2014many divine beings personifying natural phenomena, like **Agni** , the deva of fire; **S\u016brya** , the deva of the sun; **V\u0101yu** , the deva of the wind; and so on. **\u012a\u015bvara** or **Bhagav\u0101n** correspond to God in more of a monotheistic sense\u2014the Supreme Being or overlord of the cosmos. **Brahman** , the eternal and absolute reality that is ultimately one with all things, conveys a pantheistic or panentheistic sense of the divine as present in all beings.\n\n**PONGAL.** Four-day harvest festival that is celebrated in the state of **Tamil** Nadu, on the southern tip and southeastern coast of India. It occurs on the last day of the lunar month of **Pau\u1e63a** (December\u2013January) and the first three days of the month of **Magha** (January\u2013February).\n\n**POONA (PUNE).** _See_ PUNE.\n\n**POONJAJI (POONJA, HARILAL W.) (1910\u20131997).** Punjab-based **guru** who had a number of Western followers. Poonjaji is said to have attained **sam\u0101dhi** at the young age of nine. He met **Rama\u1e47a Mahar\u1e63i** in 1944, becoming his disciple and teaching a form of **Advaita Ved\u0101nta** mixed with **bhakti** for the rest of his life. His most famous disciple was the Catholic monk Sw\u0101m\u012b Abhi\u1e63ikt\u0101nanda (Henri Le Saux), who delved very deeply into the experience of Hinduism, seeking common ground between Advaita Ved\u0101nta and **Christianity**.\n\n**POTANA (1400\u20131470).** **Vai\u1e63\u1e47ava** scholar and saint; translated the _Bh\u0101gavata Pur\u0101\u1e47a_ into Telugu to make this text more widely available to the common people. Though born a **\u015aaiva** , Potana took initiation from a Vai\u1e63\u1e47ava **guru**. He is said to have seen both \u015aaiva and Vai\u1e63\u1e47ava traditions as equally valid.\n\n**PRABHAV\u0100NANDA, SW\u0100M\u012a (1893\u20131976).** Scholar and monk of the **Ramakrishna Order** who did a good deal to promote **Neo-Ved\u0101nta** in North America. Prabhav\u0101nanda founded the **Ved\u0101nta Society** of Southern California in 1930 and ran it until his death in 1976. Under his leadership, this organization became the largest Ved\u0101nta Society in the Western world. Prominent disciples of Prabhav\u0101nanda include **Christopher Isherwood** and **Aldous Huxley**. Isher-wood's relationship with Prabhav\u0101nanda is chronicled in his autobiography, _My Guru and His Disciple_. The two of them collaborated on a number of scholarly projects, including popular translations of the _Bhagavad G\u012bt\u0101_ and **Pata\u00f1jali** 's _Yoga S\u016btra_. Prabhav\u0101nanda also wrote a Ved\u0101ntic commentary on **Christianity** entitled _The Sermon on the Mount According to Vedanta_.\n\n**PRADAK\u1e62INA.** **Circumambulation** ; to walk in a circle around a sacred image ( **m\u016brti** ) in a clockwise direction, keeping the image to one's right side, as a way of paying respect to the deity that the image represents. _See also_ IMAGES IN WORSHIP.\n\n**PRAHL\u0100DA.** Son of the demonic **Hira\u1e47yaka\u015bipu** and ardent devotee of **Vi\u1e63\u1e47u** ; often held up in **Vai\u1e63\u1e47ava** traditions as a model of **bhakti**. Hira\u1e47yaka\u015bipu held deep hatred for **Vi\u1e63\u1e47u**. Annoyed by Prahl\u0101da's equally deep devotion for his archenemy, Hira\u1e47yaka\u015bipu taunted Prahl\u0101da. Once, asking Prahl\u0101da where Vi\u1e63\u1e47u could be found, Prahl\u0101da replied that Vi\u1e63\u1e47u (whose name means \"all-pervasive\") was everywhere, even within a nearby stone pillar. On hearing this, Hira\u1e47yaka\u015bipu kicked the pillar. The pillar then broke open and Vi\u1e63\u1e47u, in the form of the **Narasi\u1e43ha** or man-lion **avat\u0101ra** , came forth. The man-lion grabbed Hira\u1e47yaka\u015bipu, placed him on his lap, and slew him. This event occurred at dusk. The circumstances surrounding the **death** of Hira\u1e47yaka\u015bipu were a result of the fact that he was magically protected from any attack by man or beast (the Narasi\u1e43ha was neither), on the ground or in the air (which is why Narasi\u1e43ha held him in his lap), and by day or night (making dusk and dawn times of vulnerability for him).\n\n**PRAJ\u0100PATI.** \"Father of Offspring\"; \"progenitor\"; **Vedic** creator deity, often identified with **Brahm\u0101**. _See also_ CREATION.\n\n**PRAJ\u0100PITAR BRAHM\u0100 (1876\u20131969).** Founder of the **Brahm\u0101 Kum\u0101r\u012b** movement; born **D\u0101da Lekhr\u0101j**.\n\n**PRAKRIT (PR\u0100K\u1e5aTA).** \"Natural\"; the natural languages of northern India in ancient times, in contrast with the \"artificial\" or \"cultured\" **Sanskrit** language. This term refers to the Middle **Indo-\u0100ryan** languages spoken across northern India from ancient times up to the medieval and modern periods, when these languages evolved into the contemporary north Indian languages, such as **Hindi** , **Bengali** , Gujarati, and Marathi. There are many Prakrits, including P\u0101li (the language of the earliest extant canonical texts of **Buddhism** ) and Ardha-M\u0101gadhi (the language of the canonical texts of **Jainism** ).\n\n**PRAK\u1e5aTI.** \"Nature\"; \"matter.\" In **S\u0101\u1e43khya** **philosophy** , one of two main categories into which reality is divided, the other being **puru\u1e63a** , or pure spirit. Whereas puru\u1e63a is pure consciousness, both **mind** and matter\u2014all phenomena\u2014are manifestations of prak\u1e5bti. Prak\u1e5bti manifests as three **gu\u1e47as** , or qualities\u2014 **sattva** (clarity, luminosity), **rajas** ( **action** or dynamism), and **tamas** (inertia). The universe of changing phenomena is an effect of the constant interplay of these three gu\u1e47as. The goal of S\u0101\u1e43khya and the related system of **Yoga** is for puru\u1e63a to realize its nature as pure spirit and disentangle itself from prak\u1e5bti, with which it has become falsely identified.\n\n**PRALAYA.** Dissolution. The end of a cosmic cycle or century of **Brahm\u0101**. At the end of a cycle, the forces of chaos and entropy overcome the forces of order, or **dharma** , and the universe dissolves, returning to a potential state. There follows the great sleep, or _mah\u0101pralaya_ , after which a new **creation** will emerge and a new century of Brahm\u0101, lasting for trillions of years, will begin. The karmic states of the beings in the universe at the time of the pralaya will determine their character in the new cycle. _See also_ WORLD AGES.\n\n**PRAM\u0100\u1e46A.** In Indian philosophy, a valid basis for making a judgment. The most basic pram\u0101\u1e47a, accepted by all **dar\u015banas** , or systems of Indian philosophy, is **pratyak\u1e63a** , or sensory perception. After pratyak\u1e63a are **anum\u0101na** , or inference, and **upam\u0101na** , or verbal comparison\u2014in other words, recognizing a perceived entity to be a member of an already known category. Finally, there is \u015babda, or the verbal testimony of a trustworthy and knowledgeable person. The **\u0101stika** or \"orthodox\" Hindu systems of philosophy identify the **Vedic** **scriptures** as the preeminent example of the \u015babda pram\u0101\u1e47a. The **materialist C\u0101rv\u0101ka** or **Lok\u0101yata** system accepts only pratyak\u1e63a as valid. **Buddhism** accepts only pratyak\u1e63a and anum\u0101na. The orthodox Hindu systems generally accept all four pram\u0101\u1e47as.\n\n**PR\u0100\u1e46A.** Breath. Vital energy. The second most \"gross\" or physical of the five sheaths or **ko\u015bas** surrounding the **\u0101tman** \u2014the **pr\u0101\u1e47a-maya-ko\u015ba**. In some _Upani\u1e63ads_ it is speculated that the pr\u0101\u1e47a might be the \u0101tman\u2014that the \u0101tman is the primal life force that manifests as the power of breath in the physical body.\n\n**PR\u0100\u1e46A-MAYA-KO\u015aA.** Breath layer; vital energy layer; the second of the five **ko\u015bas** , or layers, surrounding the self ( **\u0101tman** ) according to **Ved\u0101nta**. This ko\u015ba is \"between\" the physical or \"food\" layer ( **anna-maya-ko\u015ba** ) and the **mind** layer ( **mano-maya-ko\u015ba** ).\n\n**PR\u0100\u1e46\u0100Y\u0100MA.** Breath control, fourth of the eight stages or \"limbs\" of **Pata\u00f1jali** 's eight-limbed ( **a\u1e63\u1e6d\u0101\u1e45ga** ) **yoga** , as outlined in the _Yoga S\u016btras_. Such control helps to bring the **mind** under control.\n\n**PR\u0100RABDHA KARMA.** The effects of past **actions** , or **karma** , which have gone into the **creation** of one's current physical body and which cannot be eliminated. It is due to pr\u0101rabdha karma that even an **enlightened** being\u2014one who has attained **liberation** from the cycle of **rebirth** , or **mok\u1e63a** , in this lifetime (a **j\u012bvanmukta** )\u2014continues to stay alive and is subject to bodily experiences such as injury, sickness, old age, and **death**. Even while the **soul** is free, the body experiences the effects of this ripening past karma.\n\n**PRAS\u0100DAM.** Divine grace, gift from **God**. In its most common usage, this term refers to food offerings made during **p\u016bj\u0101** and then distributed back to the worshiper and, often, to other members of the community as well. The basic concept operating is transactional in nature. The worshiper makes an offering of **devotion** to God (in a specific form, such as one of the personal deities of Hinduism), which is symbolized by food items, and the deity, in turn, bestows blessings upon the devotee, which are symbolized by the devotee's consumption of the food. Any food that has been thus offered is pras\u0101dam, even if the one who eventually consumes it was not present at the p\u016bj\u0101 but has been given the pras\u0101dam later by a friend or family member. It is widely regarded as bad luck to refuse pras\u0101dam that has been offered in this way, as fortunate to receive it, and as good act\u2014an act of **pu\u1e47ya karma** \u2014to offer it.\n\n_PRASTH\u0100NA TRAYA_ **.** \"Threefold foundation,\" three textual sources foundational to all of the systems of **Ved\u0101nta** philosophy: the _Upani\u1e63ads_ , the _Brahm\u0101 S\u016btras_ , and the _Bhagavad G\u012bt\u0101_. Every Ved\u0101ntic **\u0101c\u0101rya** , or founder of a formal system of Ved\u0101nta, has written commentaries upon the three texts of the _Prasth\u0101na Traya_. It is largely through the medium of these commentaries, or **bh\u0101\u1e63yas** , that the various \u0101c\u0101ryas have established the point of view of their particular school of thought\u2014with **\u015aa\u1e45kara** , for example, providing an interpretation of these texts supportive of **Advaita Ved\u0101nta** , **R\u0101m\u0101\u1e47uja** providing an interpretation supportive of **Vi\u015bi\u1e63\u1e6d\u0101dvaita Ved\u0101nta** , and so on. The internal diversity of the _Upani\u1e63ads_ and the terseness of the _Brahm\u0101 S\u016btras_ , in particular, have lent themselves to these varying interpretations.\n\n**PRATYAK\u1e62A.** Perception; sensory perception; one of the valid sources of **knowledge** , or **pram\u0101\u1e47as** , of Indian philosophy; the only pram\u0101\u1e47a that all of the systems of Indian philosophy ( **dar\u015banas** ) accept as valid.\n\n**PRAV\u1e5aTTI M\u0100RGA.** \"Path of inclination toward **action** \"; way of life of a **householder** who has not renounced **action** ; the way of life of persons in the first three of the **\u0101\u015bramas** or four stages of life. These persons are dedicated to pursuing **k\u0101ma** (sensory pleasure), **artha** (wealth), and **dharma** (moral virtue) by following the Prav\u1e5btti M\u0101rga. This is in contrast with the **Niv\u1e5btti M\u0101rga** , the path of the renouncer ( **sanny\u0101sin** ), who pursues the fourth goal of life: **mok\u1e63a** , or **liberation** from the cycle of **rebirth**.\n\n**PRAY\u0100GA (ALLAHABAD).** Holy city at the confluence of the rivers **Yamun\u0101** and **Ga\u1e45g\u0101**.\n\n**PR\u0100YA\u015aCITTA.** Atonement for bad past acts\u2014 **p\u0101pa** or \"bad **karma** \"\u2014through **actions** prescribed by a priest or by one's **guru**. These can include practicing special **asceticism** , performing a special **p\u016bj\u0101** , or giving charity ( **d\u0101na** ) to worthy persons or institutions.\n\n**PREMA.** Love; affection; close friendship.\n\n**PRETA.** Ghost; one of the types of creature against which one prays for protection; an unquiet spirit, often because the proper funeral rites were not performed. _See also_ ANCESTRAL RITES; ANTYE\u1e62\u1e6cI.\n\n**P\u1e5aTHV\u012a.** Earth; the medium of smell ( **gandha** ); one of the five elements or states of being that constitute the physical world ( **mah\u0101bh\u016btas)**. The five elements are earth (p\u1e5bthvi), air ( **v\u0101yu** ), fire ( **agni** ), water ( **ap** ), and space ( **\u0101k\u0101\u015ba** ).\n\n**P\u016aJ\u0100.** Worship; chief ritual by which deities are honored. The basic structure of p\u016bj\u0101 is modeled on the favors that one would bestow upon an honored guest, such as offerings of food, a bath, and clean clothing.\n\n**PUNARJANMA. Rebirth** , reincarnation.\n\n**PUNDIT.** A wise or **knowledgeable** person; a common English spelling of **pa\u1e47\u1e0dit**. This word has entered English usage to refer to an expert or a commentator in any discipline.\n\n**PUNE (POONA).** City in Maharashtra, in central India; birthplace of **\u015aiv\u0101ji** , renowned leader of the **Mar\u0101\u1e6dhas** in the 17th century. Pune is also home to many universities and a major center of **Indology** and **Sanskrit** scholarship. The **\u0101\u015brama** or commune of **Osho** ( **Bhagwan Rajneesh** ) is located here as well.\n\n**PU\u1e46YA.** Merit, good **karma**.\n\n_PUR\u0100\u1e46AS_ **.** Ancient lore; a collection of 18 texts dedicated to the deeds of deities such as **Vi\u1e63\u1e47u** , **\u015aiva** , and **Dev\u012b**. These texts were composed in their current form during the first millennium of the Common Era (although they likely contain material from an earlier period). Along with the _Mah\u0101bh\u0101rata_ and _R\u0101m\u0101ya\u1e47a_ , these texts largely define the worldview and mythology of contemporary Hinduism. The material contained within them is much better known to the average Hindu than the contents of the _Vedas_. Rituals such as **p\u016bj\u0101** and the representations of deities in the forms in which they are depicted in **m\u016brtis** are based on accounts given in these texts. Whereas the deities of the _Vedas_ are not as prominent in contemporary Hinduism as they were in ancient times, the deities that are the focus of the _Pur\u0101\u1e47as_ have been central to Hindu traditions for over 2,000 years. The 18 major _Pur\u0101\u1e47as_ are divided into three collections that are made up of six texts each. Each collection is dedicated primarily to one of the three main deities at the basis of the three main theistic traditions\u2014Vi\u1e63\u1e47u, \u015aiva, and Dev\u012b, or **\u015aakti**. There are a variety of minor _Pur\u0101\u1e47as_ in addition to the major ones. The _Pur\u0101\u1e47as_ include many genealogical lists that connect royal families and **Brahmanical** lineages to heroes and sages of the ancient past. The _Pur\u0101\u1e47as_ form a substantial part of the **sm\u1e5bti** **literature** , the category of Hindu **scripture** whose sanctity is derivative from that of the **\u015br\u016bti** \u2014the _Vedas_.\n\n**PURI.** City on the coast of the eastern Indian state of **Orissa**. Location of the **Vai\u1e63\u1e47ava** **temple** of **Jagann\u0101tha**.\n\n**PUROHITA.** Priest; traditionally a **Brahmin** ; the main officiating priest of any Hindu ritual.\n\n**PUR\u016aRAVAS.** Ancient king; character of a story, first mentioned in the _\u1e5ag Veda_ and later elaborated upon in the _Br\u0101hma\u1e47as_ and the _Mah\u0101bh\u0101rata_ , and finally in a famous play by the renowned **Sanskrit** **poet** and playwright **K\u0101l\u012bd\u0101sa** (the _Vikramorva\u015b\u012b_ ). In the story, Pur\u016bravas falls in love with a celestial maiden ( **apsaras** ) named **Urva\u015b\u012b** and asks to marry her. She agrees, on the condition that he protect her rams from being stolen and he never lets her see him naked. They live together happily for many years until a group of mischievous **gandharvas** steals one of Urva\u015b\u012b's rams in the middle of the night. Hearing the commotion, Pur\u016bravas leaps naked out of bed to save the ram. The gandharvas flash a bright light on him, revealing his naked form to Urva\u015b\u012b, and make off with the ram. At this, Urva\u015b\u012b returns to the heavens.\n\n**PURU\u1e62A.** \"The person\"; \"the man\"; the primordial being whose body is dismembered in a famous chapter of the _\u1e5ag Veda_ \u2014the _Puru\u1e63a S\u016bkta_ , or \"Hymn of the Cosmic Man\"\u2014in order to form the universe. The various parts of this being's body become the various phenomena that make up the cosmos\u2014his eye becoming the sun, his **mind** becoming the **moon** , and so on. The **var\u1e47as** , or four main **castes** , are also formed from various parts of his body. He is identified in later **Vedic** **literature** with **N\u0101r\u0101ya\u1e47a** , who is subsequently identified with **Vi\u1e63\u1e47u** , whose body is identified with the cosmos in the **Vi\u015bi\u1e63\u1e6d\u0101dvaita Ved\u0101nta** of **R\u0101m\u0101\u1e47uja**. In fact, the idea of the universe as coming from the body of a single cosmic being anticipates the later Ved\u0101ntic themes of the unity of all existence and of **God** dwelling within all beings.\n\nAlso, in **S\u0101\u1e43khya** **philosophy** , puru\u1e63a is the term used for the **soul** , understood as pure consciousness, and contrasted with the realm of matter, or **prak\u1e5bti**. The goal of S\u0101\u1e43khya and the related system of **Yoga** is for puru\u1e63a to realize its nature as pure spirit and disentangle itself from prak\u1e5bti, with which it has become falsely identified.\n\n_PURU\u1e62A S\u016aKTA_ **.** Chapter of the _\u1e5ag Veda_ that depicts the **creation** of the universe by means of the primordial self-sacrifice of a being\u2014the **puru\u1e63a** , or cosmic man\u2014whose body becomes the basis of all that exists. According to this text, only one fourth of the body of the cosmic man become the visible, physical universe. The other three fourths become the spiritual world. The text establishes a set of correspondences between various parts of the body of the cosmic man and elements of the physical world. This text is probably best known for drawing correspondences between parts of the cosmic man and the four main **var\u1e47as** or **castes** : the **Brahmin** caste corresponds to the head of the cosmic man, the **K\u1e63atriya** or warrior caste to the cosmic man's arms, the **Vai\u015byas** or common people to his stomach, and the **\u015a\u016bdra** or servant caste to his feet. This is the earliest reference to the var\u1e47a system in the **Vedic** **literature**.\n\n**PURU\u1e62\u0100RTHA.** The four goals of humanity; physical or sensory pleasure ( **k\u0101ma** ), wealth ( **artha** ), virtue ( **dharma** ), and **liberation** from the cycle of rebirth ( **mok\u1e63a** ). Each successive goal is regarded as superior to (because it is more enduring than) the one prior to it. The first three are regarded as \"worldly\" ( _laukika_ ) goals and mok\u1e63a is regarded as a transcendental or \"otherworldly\" ( _alaukika_ ) goal. The worldly goals are appropriate to people in the first three of the four stages of life or **\u0101\u015bramas** \u2014that is, the celibate student ( **brahmacarya** ), the householder ( **g\u0101rhasthya** ), and the retired, \"forest-dwelling\" ( **vanaprastha** ) stage. The fourth goal is pursued by renouncers ( **sanny\u0101sins** ). This, at least, is the classical Hindu model of society, as found in the legal texts, or _Dharma \u015a\u0101stras_. In **Tantric** and **bhakti** -oriented Hindu traditions, a householder can also aspire to mok\u1e63a by performing the correct rituals and **meditations** (the Tantric path) or cultivating devotion to **God** (bhakti).\n\n**PURU\u1e62OTTAMA.** Supreme person; **God** ; the Supreme Being; an epithet of **K\u1e5b\u1e63\u1e47a** in the _Bhagavad G\u012bt\u0101_ and in subsequent **Vai\u1e63\u1e47ava** traditions; God, or **\u012a\u015bvara** , in the _Yoga S\u016btra_ of **Pata\u00f1jali** , not identified with any specific Hindu deity.\n\n**P\u016aRVA M\u012aM\u0100\u1e42S\u0100.** The \"earlier exegesis.\" The term _P\u016brva M\u012bm\u0101\u1e43s\u0101_ contrasts this system of thought with the later, or **Uttara M\u012bm\u0101\u1e43s\u0101** \u2014more widely known as **Ved\u0101nta**. P\u016brva M\u012bm\u0101\u1e43s\u0101 focuses upon the earlier, ritual-istic portion of the _Vedas_ , while Ved\u0101nta interprets the later **Vedic** texts\u2014the _Upani\u1e63ads_ \u2014and their teaching of **Brahman**. _See also_ M\u012aM\u0100\u1e42S\u0100.\n\n_P\u016aRVAM\u012aM\u0100\u1e42S\u0100 S\u016aTRA_ **.** Root text of the **M\u012bm\u0101\u1e43s\u0101** or **Uttara M\u012bm\u0101\u1e43s\u0101 dar\u015bana** , or system of philosophy; attributed to **Jaimini** and likely composed around 300 BCE.\n\n**P\u016aRVAPAK\u1e62A.** \"Prior view,\" the point of view of the opponent in classical Indian philosophical texts. After this view is refuted, the final or \"perfected view\" ( **siddh\u0101nta** ) of the author is presented.\n\n**P\u016a\u1e62AN.** \"Nourisher\"; **Vedic** deity; one of the 12 **\u0100dityas** ; guardian of animals and human beings and ensurer of prosperity; identified in some sources as an earth **god**.\n\n**PU\u1e62KARA.** \"Blue Lotus\"; site in **Rajasthan** of one of the handful of **temples** in India devoted to **Brahm\u0101** ; believed to be a highly meritorious site of **pilgrimage**.\n\n## Q\n\n**QUALIFIED NONDUALISM.** Metaphysical teaching of the **Vi\u015bi\u1e63\u1e6d\u0101dvaita** system of **Ved\u0101nta** developed by the Ved\u0101ntic **\u0101c\u0101rya R\u0101m\u0101\u1e47uja**. According to this view, all of reality is, as **Advaita Ved\u0101nta** teaches, one with **Brahman**. But unlike in the Advaitic concept of Brahman, Brahman is here conceived as having internal differences. The idea of Brahman with which this system operates is one of organic unity, rather than the pure, undifferentiated unity taught by Advaita Ved\u0101nta. Brahman, in this system, is the totality of being, including within itself differentiated beings such as **God** , souls ( **j\u012bvas** ), and the world ( **jagat** ).\n\n**QUALITIES.** _See_ GU\u1e46AS.\n\n## R\n\n**R\u0100DH\u0100.** Foremost of the **gop\u012bs** , or cowherdesses with whom **K\u1e5b\u1e63\u1e47a** spent his youth, as recounted in the _Bh\u0101gavata Pur\u0101\u1e47a_. The intense love between K\u1e5b\u1e63\u1e47a and R\u0101dh\u0101 is the ideal of **bhakti** , or devotion. The fact that R\u0101dh\u0101, who was married to another man, still loves K\u1e5b\u1e63\u1e47a against all social conventions and norms, rather than being condemned in the **Vai\u1e63\u1e47ava** traditions, is upheld as a model of devotion to **God** \u2014a love that is spontaneous and dismissive of all other concerns. The longing of R\u0101dh\u0101 for K\u1e5b\u1e63\u1e47a during his absence is also a model for the longing of the soul for God until the dawning of **God-realization** and God-consciousness. In Hindu iconography, R\u0101dh\u0101 and K\u1e5b\u1e63\u1e47a are frequently paired as a divine couple, symbolizing the loving bond between God and the **soul**. The story of the love affair of R\u0101dh\u0101 and K\u1e5b\u1e63\u1e47a is narrated in the vivid, erotically charged **poetry** of **Jayadeva** in his _G\u012btagovinda_ , or \"Song of the Cowherd.\"\n\n**RADHAKRISHNAN, SIR SARVEPALLI (1888\u20131975).** Indian philosopher; diplomat; vice president of India from 1952 to 1962 and president from 1962 to 1967; translator of many **Sanskrit** texts and major exponent of **Neo-Ved\u0101nta**.\n\n**R\u0100DH\u0100SO\u0100M\u012a SATSANG.** \"Holy Fellowship of R\u0101dh\u0101's Master\"; Hindu movement started in 1861 by **Sw\u0101m\u012b Shiv Dayal Singh** (1818\u20131878). This movement currently has both Indian and Western adherents. Having a close continuity with the teachings of the **Sant** movement and **Sikhism** , the central emphasis is the relationship between disciple and **guru** , who is the intermediary of the liberating love of **God**. There is also a strong emphasis on morality, **vegetarianism** , and the practice of **meditation**.\n\n**RAGHAVAN, V. (1908\u20131990).** **Indologist** ; born in **Tamil** Nadu and professor for many years of **Sanskrit** at **Madras** University; author of many works on Sanskrit **literature** and drama.\n\n**R\u0100HU.** One of the **navagrahas** ; companion of **Ketu**. R\u0101hu was an **asura** who managed to steal a drop of the **am\u1e5bta** , or elixir of **immortality** , when the **devas** and asuras churned it from the cosmic ocean. Alerted to this fact by the sun and the **moon** , **Vi\u1e63\u1e47u** beheaded R\u0101hu and placed his head in the sky. The rest of R\u0101hu's body became a separate entity, Ketu, also placed in the sky. The two of them periodically take their vengeance on the sun and the moon by swallowing them, causing solar and lunar eclipses. R\u0101hu appears as a demonic, decapitated head floating in the sky. Both are considered inauspicious in Hindu **astrology**.\n\n**RAI, LALA LAJPAT (1865\u20131928).** Martyr of the Indian independence movement; born in the Punjab. Rai founded the pro-independence journal _Young India_ in 1915 and was its editor and its publisher until 1919, when **Mohand\u0101s K. G\u0101ndh\u012b** assumed these roles. Rai died in 1928 due to injuries suffered during a protest march in Lahore.\n\n**R\u0100J.** \"Reign\"; popular term for the period of British rule of India, which began around the middle of the 18th century, with the increasing ascendancy of the **British East India Company** , and ended with the official independence of India and Pakistan on the 15 August 1947. The British presence in India began as early as 1600, initially for the purpose of trade. This presence was gradually expanded until, by the mid-1800s, the British controlled most of India's coasts. After the Sepoy Mutiny of 1857, known in India as the First War of Independence, the British crown assumed direct control of the British East India Company's holdings and most of India came under the rule of Queen Victoria. At this time, opposition to British domination of India began to spread among English-educated, middle-class Indians, encouraged by Western members of the **Theosophical Society**. This led to the founding of the Indian National Congress in 1885. In the early 20th century, the independence movement greatly expanded in popularity under the leadership of **Mohand\u0101s K. G\u0101ndh\u012b** , finally achieving its goal in 1947. G\u0101ndh\u012b's strategy of nonviolent noncooperation made it possible for ordinary Indians to participate in a movement that had been limited to political leaders or those willing to take up arms. The movement was not able to prevent the partition of India and Pakistan along religious lines when British rule ended.\n\n**RAJAS.** Dynamism; the second of the three **gu\u1e47as** or qualities of phenomenal existence as delineated in the **philosophies** of **S\u0101\u1e43khya** and **Yoga** as well as in the _Bhagavad G\u012bt\u0101_.\n\n**RAJASTHAN.** Largely desert state in northwestern India; home to such famous Hindu pilgrimage sites as **N\u0101thdw\u0101ra** , which is dedicated to **K\u1e5b\u1e63\u1e47a** , and **Pu\u1e63kara** , dedicated to **Brahm\u0101** ; also the location of many Jain **temples** and home to a relatively large number of Jains. One of the most prominent **Jain** temples is located in this state, on Mount Abu. Mount Abu is also home to the world headquarters of the **Brahm\u0101 Kum\u0101r\u012bs**. Known for its martial culture, Rajasthan is probably most famous as the home of the **Rajputs**. The 16th-century **Vai\u1e63\u1e47ava** saint and **bhakti poet** **M\u012br\u0101 Bai** hailed from Rajasthan.\n\n**R\u0100JAS\u016aYA.** **Vedic** sacrifice or **yaj\u00f1a** that ensures the future success of a king. The r\u0101jas\u016bya of **Yudhi\u1e63\u1e6dhira** is depicted in the second book of the _Mah\u0101bh\u0101rata_ , the _Sabh\u0101 Parvan_ (\"Book of the Meeting Hall\").\n\n**RAJNEESH, BHAGWAN (R\u0100JAN\u012a\u1e62A, BHAGAV\u0100N) (1931\u20131990).** **Guru** who led a modern Hindu movement with a mainly Western following. His eclectic teachings drew upon a variety of traditions, Hindu and non-Hindu, particularly **Tantra**. Having a strong emphasis on sexual freedom as a way to spiritual **liberation** , the movement's practices are quite close to the V\u0101m\u0101c\u0101ra or **Left-Handed** Tantric practice, which seeks to embody the ideal of nondualism ( **advaita** ) by overcoming traditional taboos regarding impurity and purity on the understanding that all is **One**. Such taboos are therefore arbitrary. In the early 1980s, Rajneesh and many of his disciples settled in the United States near the town of Antelope, Oregon. But in 1985, Rajneesh was deported back to India for immigration fraud. Returning to his home in **Pune** , Rajneesh reestablished his community there. In 1989, he changed his name to **Osho** , a Japanese term for a high-ranking **Buddhist** priest. Rajneesh was born Candra Mohan **Jain**.\n\n**RAJPUTS.** Large community of **k\u1e63atriya** or warrior clans based in **Rajasthan**. Arising around the sixth century of the Common Era, Rajput kingdoms were prominent in many parts of northwestern India, but primarily in the region now known as Rajasthan. Their name\u2014 _Rajput_ \u2014is derived from the **Sanskrit** _r\u0101japutra_ , or \"sons of kings.\" Many Rajput clans fought quite ferociously against Turkish and **Mughal** invaders in the early medieval period, but some allied with the Mughal Empire. It is likely that the practice of **sat\u012b** , in which a widow immolates herself on the funeral pyre of her deceased husband, began in this community. According to the Rajputs' own lore, on hearing that their husbands had fallen in battle against invaders, many of the Rajput women immolated themselves rather than be captured or raped by the enemy. Eventually, this became an expected display of **devotion** in Rajput families. The practice was transmitted to **Bengal** when a number of Rajput clans transplanted themselves to that region. It was banned by the British in 1829 due to the efforts of the Bengali Hindu reformer **R\u0101m Mohan Roy** and soon ceased to be practiced on a large scale (and had, in any case, been largely confined to the upper **castes** in Bengal and Rajasthan). But occasional exceptions have occurred in subsequent history, the last sat\u012b having taken place at Deorala, in Rajasthan, in 1987. This event was protested by many Hindus, including **Sw\u0101m\u012b Agnive\u015ba**.\n\n**RAKHI, RAK\u1e62ABANDHAN.** Hindu and **Sikh** festival held on the **P\u016br\u1e47im\u0101** (or full- **moon** day) of the month of **\u015arava\u1e47a** (July\u2013August). Celebrated mainly in northern India, the observance of the festival involves **women** tying a **sacred thread** on the right wrists of their brothers. A woman can also perform this ritual on a man who is not her relative in order to make him her brother in a formal sense. This ritual can thus sanctify the platonic friendship between a man and a woman.\n\n**R\u0100K\u1e62ASA.** Monster; demonic creature, often depicted as having a semi-human, semi-bestial form and as being of great size and strength. R\u0101k\u1e63asas can also change their shape for a limited period of time in order to deceive human beings. They are represented in Hindu texts as creatures of chaos, delighting in disrupting the **yaj\u00f1as** , or sacrifices, of the sages who have withdrawn to the forest. They also have a taste for human flesh. They are the primary enemies of **R\u0101ma** in the _R\u0101m\u0101ya\u1e47a_. Their king, **R\u0101va\u1e47a** , abducts **S\u012bt\u0101** , R\u0101ma's wife, whom R\u0101ma must then rescue. There are also, however, \"good\" r\u0101k\u1e63asas, such as R\u0101va\u1e47a's cousin **Vibh\u012b\u1e63ana** , who is a devotee of **Vi\u1e63\u1e47u** and betrays R\u0101va\u1e47a after failing to persuade him to release S\u012bt\u0101, joining R\u0101ma's army as a military advisor against his fellow r\u0101k\u1e63asas. In the _Mah\u0101bh\u0101rata_ , **Bh\u012bma** has an affair with a female r\u0101k\u1e63asa and fathers the massive human-r\u0101k\u1e63asa hybrid Gha\u1e6dotkaca, who assists the **P\u0101\u1e47\u1e0davas** in their battle with the **Kauravas**.\n\n**RAKTAB\u012aJA**. \"Blood seed\"; demon slain by **K\u0101l\u012b**. Each drop of Raktab\u012bja's blood that spilled on the ground would transform into a completely new Raktab\u012bja. Spilling this demon's blood in battle therefore served only to create an entire army of Raktab\u012bjas. It is for this reason that K\u0101l\u012b used her tongue to catch or lick up every drop of blood of this demon that was spilled in the course of her battle with him. This is why this **goddess** is depicted with a long, bloody tongue sticking out of her mouth.\n\n**R\u0100M.** **Hindi** pronunciation of **R\u0101ma**. _See_ R\u0100MA.\n\n_R\u0100M CARIT M\u0100NAS_ **.** \"Lake of the Deeds of R\u0101ma\"; highly popular rendering of the _R\u0101m\u0101ya\u1e47a_ in medieval **Hindi** ; composed by the **Vai\u1e63\u1e47ava** devotee and **bhakti poet** **Tuls\u012bd\u0101s** (1532\u20131623). The text is so popular, even being frequently quoted by ordinary devotees, that it is safe to say that it is this version of the text, rather than the **Sanskrit** original, that is better known by most Hindus, particularly in northern India. The popular Hindi television serialization of the _R\u0101m\u0101ya\u1e47a_ (\" _R\u0101m\u0101ya\u1e47_ \") is based upon the _R\u0101m Carit M\u0101nas_. During the festival of **R\u0101mnavam\u012b** , it is not uncommon for 24-hour nonstop readings of this text to be held in Hindu **temples** , with devotees taking turns reciting the verses in a call-and-response fashion.\n\n**R\u0100MA.** Along with **K\u1e5b\u1e63\u1e47a** , one of the most popular of the **avat\u0101ras** , or incarnations, of **Vi\u1e63\u1e47u** ; the seventh such incarnation, according to the traditional list of 10; prince of the kingdom of **Ayodhy\u0101** and son of its king, Da\u015baratha. R\u0101ma's biography forms the central narrative of one of the two Hindu epics, or **itih\u0101sas** \u2014the _R\u0101m\u0101ya\u1e47a_ (the other epic being the _Mah\u0101bh\u0101rata_ ). At the beginning of the _R\u0101m\u0101ya\u1e47a_ , the **gods** approach Vi\u1e63\u1e47u for help because the Earth has fallen under the oppression of **R\u0101va\u1e47a** , the powerful lord of the **r\u0101k\u1e63asas**. Due to a boon that he has received, R\u0101va\u1e47a cannot be harmed by any type of creature except one: a human being. Vi\u1e63\u1e47u therefore takes the form of R\u0101ma in order to bring about R\u0101va\u1e47a's destruction. In his human form as R\u0101ma, however, Vi\u1e63\u1e47u remains unaware of his true divine nature and mission. Both of these are only gradually revealed in the course of the narrative. R\u0101ma is trained in the arts of combat against r\u0101k\u1e63asas by the sage **Vi\u015bvamitra**. He also wins the hand in marriage of Princess **S\u012bt\u0101** , daughter of the wise sage King **Janaka**. Prepared to inherit his father's throne, R\u0101ma is abruptly sent into exile by his father at the instigation of Kaikey\u012b. Kaikey\u012b, one of the three wives of Da\u015baratha (and not R\u0101ma's mother), fearing that her own son, **Bharata** , will be in danger as a potential rival under R\u0101ma's rule, forces Da\u015baratha to order R\u0101ma's exile and her own son's installation as king by invoking a promise of any two boons that Da\u015baratha had made to her in the past, when she had saved his life on the field of battle. R\u0101ma accepts his father's command and willingly goes into exile, accompanied by S\u012bt\u0101 (who is really an incarnation of the **goddess Lak\u1e63m\u012b** ) and his brother **Lak\u1e63ma\u1e47a** (an incarnation of the cosmic serpent, **\u015ae\u1e63a** ). Da\u015baratha dies of sorrow at having to exile his beloved son to the forest, and Bharata, R\u0101ma's half-brother, curses his mother for causing so much distress in the family (there actually being no rivalry between Bharata and R\u0101ma). While in exile, R\u0101ma, S\u012bt\u0101, and Lak\u1e63ma\u1e47a live as **ascetics** in the forest. During this period, R\u0101va\u1e47a takes S\u012bt\u0101 captive and imprisons her in his island kingdom of **La\u1e45ka**. R\u0101ma and Lak\u1e63ma\u1e47a, with the assistance of **Hanuman** and a great army of monkeys, eagles, and other assorted animals, build a bridge to La\u1e45ka and successfully rescue S\u012bt\u0101, slaying R\u0101va\u1e47a and many other r\u0101k\u1e63asas in the process. R\u0101ma rules Ayodhy\u0101 for many years afterward, a period of peace and prosperity known as the _R\u0101ma R\u0101jya_ , or kingship of R\u0101ma. The ideal of the R\u0101ma R\u0101jya is invoked even today by many Hindus\u2014particularly **Hindu Nationalists** \u2014as an image of the ideal society.\n\nIn December 1992, Ayodhy\u0101\u2014which is an actual town in northern India\u2014became a center of controversy when a group of Hindu Nationalists demolished a mosque called the **Babr\u012b Masj\u012bd** (Babur's Mosque), which they believed had been built from the remains of a **temple** on the location marking the exact spot of R\u0101ma's birth ( _R\u0101ma-janma-bh\u016bm\u012b_ )\u2014a claim contested by some historians and archaeologists. This demolition sparked riots in India, Pakistan, and Bangladesh between Hindus and Muslims, in which many members of both communities were killed, and a number of Hindu temples destroyed in retaliation. In September 2010, an Indian high court ruled that the property should be divided among Hindu and Muslim organizations, a compromise that appears to have defused the issue, at least for the time being.\n\nPolitics aside, R\u0101ma is one of the most popular of Hindu deities, and was in fact the **i\u1e63\u1e6dadevat\u0101** , or favorite deity, of **Mohand\u0101s K. G\u0101ndh\u012b** , who invoked the name of R\u0101ma at the moment of his **death** at the hands of an assassin. According to traditional **Vai\u1e63\u1e47ava** belief, one who calls upon Vi\u1e63\u1e47u at the time of death is guaranteed salvation.\n\n**RAMAKRISHNA (R\u0100MAK\u1e5a\u1e62\u1e46A) (1836\u20131886).** **Bengali** spiritual teacher and chief inspiration behind the **Ramakrishna Mission** and **Ramakrishna Order** , as well as the **Neo-Ved\u0101nta** interpretation of Hinduism. Born Gangadhar Chattopadhyay, Ramakrishna would very likely, in an earlier period of Hindu history, have been regarded as a **bhakti** saint in the tradition of the **Mother Goddess**. His spiritual eclecticism, however, which coincided with the intellectual crisis and subsequent reforms of Hinduism in 19th-century Bengal, became the catalyst for a radical reinterpretation and rein-vigoration of Hindu thought and practice, particularly through the activities of his prominent disciple **Sw\u0101m\u012b Vivek\u0101nanda** (1863\u20131902).\n\nRamakrishna was born and raised in a poor **Brahmin** family in Kamarpukur, a village in rural Bengal. From an early age, he was prone to ecstatic visions and states of trance, which he identified with the experience of **sam\u0101dhi** , or **meditative** absorption. At times, these states were induced by songs or spiritual conversation. But sometimes they would be brought on by natural phenomena, such as his first recorded sam\u0101dhi when, as a small boy, he saw a flight of white geese against the background of a dark storm cloud, the beauty of which overwhelmed him.\n\nWhen Ramakrishna was 19 years old, his older brother, R\u0101mkumar, became the priest at a **temple** to the **goddess K\u0101l\u012b** in Dak\u1e63ine\u015bwar, near **Calcutta**. Ramakrishna replaced R\u0101mkumar in this role the following year, due to R\u0101mkumar's premature **death**. Soon, Ramakrishna became deeply immersed in devotion to K\u0101l\u012b, claiming to have had a powerful vision of this goddess in the process of performing her worship. Subsequently, he similarly immersed himself in worship of various deities, such as **\u015aiva** , **R\u0101ma** , and **K\u1e5b\u1e63\u1e47a** , eventually reaching a state of complete absorption in each deity in succession.\n\nHaving experienced the height of spiritual attainment in the various theistic forms of Hinduism, he devoted himself with the same intensity to the realization of impersonal, or formless ( **Nirgu\u1e47a** ) **Brahman** , having this experience under the guidance of a monk of the **Da\u015ban\u0101m\u012b Order** named To\u1e6dapuri. He then practiced both **Islam** and **Christianity** , devoting himself to each practice until it culminated in sam\u0101dhi. Because of all of these experiences, Ramakrishna concluded that all the religions are paths to **God-realization** , a view that has become a cornerstone of **Neo-Ved\u0101nta** philosophy.\n\nAt the age of 23, Ramakrishna married Saradama\u1e47i Mukhopadhy\u0101ya, later known as **Sarad\u0101 Dev\u012b** and the **Holy Mother** to Ramakrishna's disciples. Ramakrishna believed his wife to be a manifestation of the **Mother Goddess**. After his death, she became the informal leader of the movement based on his teachings.\n\nIn his later years, Ramakrishna attracted the attention of the middle-class, English-educated intelligentsia of Calcutta, including prominent members of the **Brahmo Sam\u0101j** , such as **Keshub Chunder Sen** (1838\u20131884). Many young men were also drawn to his teaching and personality, becoming **sanny\u0101sins** in the Da\u015ban\u0101m\u012b Order after his death in 1886 and establishing the **Ramakrishna Order**.\n\nA part of the fascination that Ramakrishna held for educated Bengalis was his insistence on direct experience as the foundation of all spiritual teaching\u2014a teaching very much in keeping with the modern focus upon experience and reason over blind faith in **scripture**. This experimental approach to religion was combined, paradoxically, with his endorsement, on the basis of his experience, of very traditional Hindu beliefs and styles of worship, a good deal of which had been condemned by the Brahmo Sam\u0101j as anti-rational and idolatrous. He thus allowed English-educated Hindus to reappropriate their older traditions, but with a new rationale.\n\n**RAMAKRISHNA MISSION.** Organization devoted primarily to education and charity; known outside of India as the **Ved\u0101nta Society** ; the religious institution built upon the teachings of **Ramakrishna** by his disciples\u2014including his wife, **Sarad\u0101 Dev\u012b** , and their successors. The Ramakrishna Mission was established by **Sw\u0101m\u012b Vivek\u0101nanda** in 1897. _See also_ NEOVED\u0100NTA; VED\u0100NTA.\n\n**RAMAKRISHNA ORDER.** Order of monks established by **Sw\u0101m\u012b Vivek\u0101nanda** after the **death** of **Ramakrishna** in 1886. The first generation of monks was made up of immediate disciples of Ramakrishna. Vivek\u0101nanda and his fellow monks took their monastic vows within the **Da\u015ban\u0101m\u012b Order** , established by **\u015aa\u1e45kara** in the ninth century of the Common Era. The monks of the new order quickly differentiated themselves from more traditional Hindu **sanny\u0101sins** , or renouncers, by engaging in relief work among the poor\u2014an activity which, along with education, constitutes a large portion of the activity of the order. Derided by traditionalists as \"scavenger monks,\" the monks persisted in their work, following Vivek\u0101nanda's philosophy that the **yoga** of good works ( **karma yoga** ) is as valid a part of the path to **God-realization** as the more traditional paths (for monks) of study ( **j\u00f1\u0101na yoga** ) and **meditation** ( **dhy\u0101na yoga** ). This order was also the first Hindu monastic order to accept Western members when Sw\u0101m\u012b Vivek\u0101nanda initiated an Irish **woman** , Margaret E. Noble ( **Sister Nivedit\u0101** ) as a nun. There are now many Westerners in the Ramakrishna Order, though the order remains predominantly Indian. _See also_ NEO-VED\u0100NTA; VED\u0100NTA.\n\n**RAMAKRISHNA VED\u0100NTA.** That branch of the otherwise unstructured and informal **Neo-Ved\u0101nta** movement that explicitly sees itself to be in continuity with the teachings and **Ramakrishna** , **Sw\u0101m\u012b Vivek\u0101nanda** , **Holy Mother Sarad\u0101 Dev\u012b** , and the monks of the **Ramakrishna Order**.\n\n**RAMA\u1e46A MAHAR\u1e62I (1879\u20131950).** Modern Hindu spiritual teacher. He was born near **Madurai** in **Tamil** Nadu. His father died when he was 12 years old, after which his family moved to Madurai and enrolled him in an English-medium missionary school. At the age of 17, he had a powerful spiritual experience that prompted him to adopt the life of a **sanny\u0101sin**. He eventually settled at the mountain **temple** of Aru\u1e47\u0101cala, in the northern part of Tamil Nadu, where he remained until his **death** in 1950. His reputation for holiness and profound wisdom spread rapidly and many pilgrims visited him at his Aru\u1e47\u0101cala retreat, which continues to be regarded as a holy site, due to his presence there. He is best known for asking many of his visitors, \"Who are you?\" thereby directing them to a process of self-enquiry intended to lead to the realization of the true nature of the self or **\u0101tman**. Though not **Advaita Ved\u0101nta** in a formal sense, Rama\u1e47a's teaching can well be characterized as a modern form of Advaita, based not so much on **scriptural** enquiry (like the classical Advaita of **\u015aa\u1e45kara** ) as on direct enquiry into the self. It is therefore akin to **Neo-Ved\u0101nta** in taking direct experience as its basis. There are also affinities between Rama\u1e47a's method and that of **Jiddu Krishnamurti**.\n\n**R\u0100M\u0100NANDA (fl. c. 14th\u201315th centuries CE).** **Vai\u1e63\u1e47ava** saint and **ascetic** ; born in **Pray\u0101ga** ( **Allahabad** ). R\u0101m\u0101nanda was first affiliated to the **\u015ar\u012bvai\u1e63\u1e47ava** tradition, holding a high position in its institutional hierarchy, but left this tradition in protest against its concern for **caste** -related practices and restrictions. He established the **R\u0101m\u0101nandi sa\u1e43prad\u0101ya** , which teaches that all are equal before **God**. Winning a large following among common people of his time, he is believed to have influenced such major figures of the **bhakti** and **Sant** movements as **M\u012br\u0101 Bai** (who claimed to be his disciple), **Kab\u012br** , and **Guru N\u0101nak**.\n\n**R\u0100M\u0100NANDI SA\u1e42PRAD\u0100YA.** The \"sect or denomination of those whose bliss is R\u0101ma.\" **Vai\u1e63\u1e47ava** tradition focused on **bhakti** , or devotion, to **R\u0101ma** and established by **R\u0101m\u0101nanda** for the common person, as a protest against the **casteism** practiced in his time by the **\u015ar\u012bvai\u1e63\u1e47ava** tradition from which he came. R\u0101m\u0101nanda deliberately taught in **Hindi** , the vernacular language of northern India, rather than the more elite **Sanskrit** , developing a large following in the process and influencing many other figures, such as **M\u012br\u0101 Bai** , **Kab\u012br** , and **Guru N\u0101nak** with his spiritual egalitarianism. Membership in the R\u0101m\u0101nandi sa\u1e43prad\u0101ya is open to all, regardless of caste. The primary **scripture** of this movement is the _R\u0101m Carit Manas_ of **Tuls\u012bd\u0101s** \u2014a popular version of the _R\u0101m\u0101ya\u1e47a_ that is better known among the masses than the original Sanskrit version by **V\u0101lm\u012bki**.\n\n**R\u0100M\u0100\u1e46UJA (1077\u20131157).** Ved\u0101ntic **\u0101c\u0101rya** and theologian of the **\u015ar\u012bvai\u1e63\u1e47ava** system of **Vai\u1e63\u1e47avism** ; from **Tamil** Nadu. The dominant form of **Ved\u0101nta** in southern India at the time of R\u0101m\u0101\u1e47uja was the **Advaita Ved\u0101nta** of **\u015aa\u1e45kara** , which affirms that the true nature of **Brahman** is ultimately impersonal and formless ( **nirgu\u1e47a** ). **Bhakti** \u2014devotion to a personal form of divinity\u2014while commended as a purifying practice that diminishes the hold of the ego through the process of worship, is seen as a preparatory discipline that readies one for **j\u00f1\u0101na** , the liberating knowledge that the self ( **\u0101tman** ) and Brahman are **One**. The personal deity ( **\u012a\u015bvara** ) is ultimately an illusion, a product of the same **m\u0101y\u0101** that creates the differentiation between the self and the absolute.\n\nTroubled by this teaching in part due to the undermining of devotion and spiritual arrogance that he felt it could evoke if not understood properly, R\u0101m\u0101\u1e47uja dedicated his life to the reinstatement of bhakti as the central mode of relating to the divine. He took the informal **Vai\u1e63\u1e47ava** devotional tradition of the **\u0100\u1e37v\u0101rs** and organized it into a system of worship and theological reflection. This system\u2014the \u015ar\u012bvai\u1e63\u1e47ava system\u2014continues to be highly popular among Hindus today, particularly in the southern part of India, where Vai\u1e63\u1e47ava **temples** and their images ( **m\u016brtis** ) are constructed, consecrated, and utilized by R\u0101m\u0101\u1e47uja's precise specifications. It is a system that integrates \"high\" **Sanskrit** culture with the popular devotionalism of the bhakti **poets** , with the vernacular songs and prayers of the \u0100\u1e37v\u0101rs regarded as a \"fifth **Veda** \" and recited in \u015ar\u012bvai\u1e63\u1e47ava temples and services.\n\n\u015ar\u012bvai\u1e63\u1e47ava theology places high importance on **\u015ar\u012b** , or **Lak\u1e63m\u012b** , the consort of **Vi\u1e63\u1e47u** , who is regarded as his **\u015bakti** and a personification of his divine grace. The divine pair of Vi\u1e63\u1e47u and Lak\u1e63m\u012b is regarded as one deity, similar to **\u015aiva** and \u015aakti.\n\nIn terms of Ved\u0101nta, R\u0101m\u0101\u1e47uja characterizes his philosophy as **Vi\u015bi\u1e63\u1e6d\u0101dvaita** , or **Qualified Nondualism**. According to this view, all of reality is, as the Advaita Ved\u0101nta of \u015aa\u1e45kara teaches, one with **Brahman**. But unlike in the Advaitic concept of Brahman, Brahman is conceived as having internal differences. The idea of Brahman with which this system operates is one of organic unity, rather than the pure, undifferentiated unity taught by Advaita Ved\u0101nta. Brahman, in this system, is the totality of being, including within itself differentiated beings such as God, souls ( **j\u012bvas** ), and the world ( **jagat** ), and bhakti is seen as the one true path to **God-realization**.\n\n**RAMANUJAN, A. K. (1929\u20131993).** Scholar and author; born in **Karnataka** , in the city of Mysore. Ramanujan was a person of many talents\u2014a **poet** , playwright, translator, and collector of folklore. Among his better-known works in the field of **Indology** are his translations of the **bhakti** poetry of **Basava** (entitled _Speaking of \u015aiva_ **)** and **Namm\u0101\u1e37v\u0101r** (entitled _Hymns for the Drowning_ ) and his collection of Indian folk tales ( _Folk Tales of India_ ). Ramanujan taught at the University of Chicago from 1983 until his **death** in 1993.\n\n_R\u0100M\u0100YA\u1e46A_ **.** \"Life of **R\u0101ma** \"; along with the _Mah\u0101bh\u0101rata_ , one of the two main epics or **itih\u0101sas** (historical tales) of Hinduism; attributed to the sage **V\u0101lm\u012bki** but very likely a result of many centuries of compilation and refinement by many different hands. In its current form, the text can be traced to the period between 200 BCE and 200 CE. Though the events in the text precede those of the _Mah\u0101bh\u0101rata_ in Hindu tradition, most scholars suspect that the _Mah\u0101bh\u0101rata_ is the older of the two texts, at least in their current forms.\n\nAt the beginning of the _R\u0101m\u0101ya\u1e47a_ , the **gods** approach **Vi\u1e63\u1e47u** for help because the Earth has fallen under the oppression of **R\u0101va\u1e47a** , the powerful lord of the **r\u0101k\u1e63asas**. Due to a boon that he has received, R\u0101va\u1e47a cannot be harmed by any type of creature except one: a human being. Vi\u1e63\u1e47u therefore takes the form of R\u0101ma\u2014prince of **Ayodhy\u0101** and son of the king, **Da\u015baratha** \u2014to bring about R\u0101va\u1e47a's destruction. While in his human form as R\u0101ma, however, Vi\u1e63\u1e47u remains unaware of his true divine nature and mission. Both of these are only gradually revealed in the course of the narrative. Later interpretations suggest that this is a metaphor for the gradual process by which all beings realize the presence of the divine within them.\n\nIn his youth\u2014and over his father's objections\u2014R\u0101ma is trained in the art of combat against r\u0101k\u1e63asas by the sage **Vi\u015bvamitra**. At the end of a lengthy stay in the wilderness with his **guru** , R\u0101ma wins the hand in marriage of Princess **S\u012bt\u0101** , daughter of the wise sage King **Janaka**. He does this through a contest of arms in which the challenge is to string the bow of **\u015aiva** \u2014a nearly impossible task that only R\u0101ma can complete.\n\nLater, having returned to Ayodhy\u0101 prepared to inherit his father's throne, R\u0101ma is abruptly sent into exile by his father at the instigation of Kaikey\u012b. Kaikey\u012b, one of the three wives of Da\u015baratha (and not R\u0101ma's mother)\u2014fearing that her own son, **Bharata** , will be in danger as a potential rival under R\u0101ma's rule\u2014forces Da\u015baratha to order R\u0101ma's exile and her own son's installation as king by invoking a promise of any two boons that Da\u015baratha had made to her in the past, when she had saved his life on the field of battle by taking his unconscious body to safety.\n\nR\u0101ma dutifully accepts his father's order and goes into exile, accompanied by S\u012bt\u0101 (who is really an incarnation of the **goddess Lak\u1e63m\u012b** ) and his brother **Lak\u1e63ma\u1e47a** (an incarnation of the cosmic serpent, **\u015ae\u1e63a** ). Da\u015baratha dies of sorrow at having to exile his beloved son to the forest, and Bharata, R\u0101ma's half-brother, curses his mother for causing so much distress in the family (there actually being no rivalry between Bharata and R\u0101ma).\n\nWhile in exile, R\u0101ma, S\u012bt\u0101, and Lak\u1e63ma\u1e47a live as **ascetics** in the forest. During this period, R\u0101va\u1e47a takes S\u012bt\u0101 captive and imprisons her in his island kingdom of **La\u1e45ka**. R\u0101ma and Lak\u1e63ma\u1e47a, with the assistance of **Hanuman** and an army of monkeys, eagles, and other assorted animals, build a bridge to La\u1e45ka and successfully rescue S\u012bt\u0101, slaying R\u0101va\u1e47a and many other r\u0101k\u1e63asas in the process. In one of the more controversial scenes in the epic, R\u0101ma does not immediately accept S\u012bt\u0101 back as his wife, insisting that she first prove that she did not betray him with R\u0101va\u1e47a by forcing her to undergo a test of fire (which she passes). (The justification given for this is that, as king, R\u0101ma must maintain an appearance of propriety for his subjects.) R\u0101ma rules Ayodhy\u0101 for many years, in a period of peace and prosperity known as the _R\u0101ma R\u0101jya_ , or kingship of R\u0101ma.\n\nReligiously, the _R\u0101m\u0101ya\u1e47a_ has become especially significant to the **Vai\u1e63\u1e47ava** traditions of Hinduism, particularly in its popular **Hindi** rendition, the _R\u0101m Carit Manas_ , by the medieval **bhakti** **poet** **Tuls\u012bd\u0101s** (1532\u20131623).\n\nLike the _Mah\u0101bh\u0101rata_ , the _R\u0101m\u0101ya\u1e47a_ has been a major subject of Hindu **art** and **literature** throughout the centuries. Plays are based on it, or on select episodes within it. Depictions of the scenes from the text adorn **temple** walls throughout India and Southeast Asia. In the modern period, it has become a popular Hindi television series.\n\n**RAME\u015aVARAM (RAMESHWARAM).** Site in **Tamil** Nadu where R\u0101man\u0101thasw\u0101my **temple** is housed. The temple is sacred to both **Vai\u1e63\u1e47avas** and **\u015aaivas** , marking the spot where **R\u0101ma** worshiped **\u015aiva** in gratitude after defeating **R\u0101va\u1e47a** and rescuing **S\u012bt\u0101**. _See also R\u0100M\u0100YA\u1e46A_.\n\n**R\u0100ML\u012aL\u0100.** \"Play of **R\u0101ma** \"; a northern Indian tradition of reenacting the _R\u0101m\u0101ya\u1e47a_ during the 10-day festival of **Da\u015bahr\u0101** (or Dussehra), which is celebrated in the **bright** **half** of the month of **\u0100\u015bvina**. In northern India, this festival celebrates **R\u0101ma** 's victory over **R\u0101va\u1e47a** and it culminates in **D\u012bp\u0101vali** (or D\u012bwali), the festival of lights in honor of the **goddess Lak\u1e63\u1e43\u012b**. The R\u0101ml\u012bl\u0101 is an extremely popular part of the observance of this festival. Enactments range from purely local productions, put together by neighborhood associations, to massive, lavish programs with sophisticated lighting and special effects. The climactic moment is the defeat of R\u0101va\u1e47a, which is often accompanied by fireworks and other pyrotechnics.\n\n**R\u0100MN\u0100MI SAM\u0100J.** **Vai\u1e63\u1e47ava** devotional tradition based in the rural central Indian state of Chattisgarh and made up almost entirely of persons from lower **castes** ( **Harijans** or **Dalits** ). The group's scripture is the _R\u0101m Carit Manas_ of **Tuls\u012bd\u0101s**.\n\n**R\u0100M NAVAM\u012a.** Festival of **R\u0101ma** 's birth; held on the ninth day of the **bright half** of the month of **Chaitra** (March\u2013April). During this festival, it is not uncommon for 24-hour nonstop readings of the _R\u0101m Carit Manas_ of **Tuls\u012bd\u0101s** to be held in Hindu **temples** , with devotees taking turns reciting the verses in a call-and-response fashion. To have **dar\u015bana** of a **m\u016brti** (a sight of an image, usually in a temple) of R\u0101ma is particularly auspicious on this day.\n\n**R\u0100MPRAS\u0100D (1718\u20131775).** \"Grace of **R\u0101ma**.\" Despite his name, R\u0101mpras\u0101d was a **bhakti poet** devoted completely to the **Divine Mother** or **Mother Goddess**. He is best known for composing many passionate hymns in her honor in which he presents himself as her ungrateful and petulant child. These deeply personal hymns were particularly dear to **Ramakrishna** , another famous devotee of the Divine Mother, who popularized them. _See also_ \u015a\u0100KTA.\n\n**RASA.** Taste, flavor, the sense of taste; also, aesthetic experience of any kind. Rasa has been theorized extensively by Hindu philosophers, most prominently in the _N\u0101\u1e6dya \u015a\u0101stra_ of **Bharata** \u2014the authoritative manual of classical Hindu drama and **dance** \u2014as well as by the 10th-century **Ka\u015bm\u012br \u015aaiva** philosopher **Abhinavagupta** and the medieval **Vai\u1e63\u1e47ava** **Gosv\u0101mins**. Abhinavagupta, reflecting a **Tantric** sensibility, likens aesthetic experience to the experience of **God-realization** , seeing it as a foretaste of the bliss of realizing unity with **\u015aiva**. The Gosv\u0101mins\u2014the Vai\u1e63\u1e47ava theologians established by **Caitanya** in the 16th century\u2014take this a step further, claiming that the rasa experienced specifically when one is enjoying the stories of **K\u1e5b\u1e63\u1e47a** and his love for the **gop\u012bs** of **V\u1e5bnd\u0101vana** as narrated in the _Bh\u0101gavata Pur\u0101\u1e47a_ could evoke the saving experience of **bhakti**.\n\n**R\u0100\u1e62\u1e6cR\u012aYA SVAYAMSEVAK SA\u1e44GH (RSS).** \"National Volunteer Association\"; a **Hindu Nationalist** organization that was started in 1925, during the Indian independence movement, by K. V. Hedgewar (1890\u20131940). A controversial organization due to its anti-Muslim rhetoric, the RSS violently opposed the partition of India and Pakistan, blaming **Mohand\u0101s K. G\u0101ndh\u012b** for allowing it to occur. G\u0101ndh\u012b's assassin, Nathuram Godse, was a member of this organization. The RSS has advocated transforming India into a Hindu state and denying Indian citizenship to Muslims, **Christians** , and communists. As one of the more radical Hindu Nationalist organizations, its ties to the **BJP** or **Bharat\u012bya Janata Party** have drawn criticism to that political party.\n\n**R\u0100VA\u1e46A.** Chief villain of the _R\u0101m\u0101ya\u1e47a_ ; lord of the chaotic evil **r\u0101k\u1e63asas** and king of **La\u1e45ka**. R\u0101va\u1e47a achieved, through **ascetic** practice, the boon of not being harmed by any kind of creature except one: a human being. Achieving total dominance over the Earth, R\u0101va\u1e47a oppresses many beings, including the **gods** , who go to **Vi\u1e63\u1e47u** for assistance. In order to vanquish R\u0101va\u1e47a, Vi\u1e63\u1e47u takes human form as **R\u0101ma** , prince of **Ayodhy\u0101**. In the course of the _R\u0101m\u0101ya\u1e47a_ , R\u0101va\u1e47a abducts R\u0101ma's beautiful wife, **S\u012bt\u0101** (who is an **avat\u0101ra** of the **goddess Lak\u1e63m\u012b** ). R\u0101ma rescues S\u012bt\u0101, killing R\u0101va\u1e47a in battle in the process. It is said that R\u0101va\u1e47a is exceedingly strong and has 10 heads.\n\n**RAVI.** Alternative name for **S\u016brya** , the sun.\n\n**RAVIV\u0100RA.** Sunday, day of the week governed by the sun. _See also_ CALENDAR, HINDU.\n\n**RAVID\u0100S (fl. 15th century CE).** Mystic and **bhakti** **poet** of the medieval period; born in a cobbler **caste** (a low caste or **Dalit** community) in **Banaras**. Part of the **Sant** movement, Ravid\u0101s had no regard for caste distinctions, nor for formal differences between religions, such as Hinduism and **Islam**. Many of his poems are included in the **Sikh** **scripture** \u2014the _\u0100di Granth_ or _Guru Granth Sahib_. His teachings have close affinities to those of **Kab\u012br** and **Guru N\u0101nak**. Starting in the 19th century, a new tradition emerged based on his teachings and devoting itself to him as Guru Ravid\u0101s.\n\n**REBIRTH.** One thread that ties together the highly internally diversified traditions that make up Hinduism is the widespread\u2014indeed, nearly universal\u2014affirmation of the concept of rebirth, or reincarnation, that these traditions share. Among the various philosophical systems ( **dar\u015banas** ) and religious denominations ( **sa\u1e43prad\u0101yas** ) of Hinduism, only the **materialist C\u0101rv\u0101ka** or **Lok\u0101yata** dar\u015bana explicitly rejects the concept of rebirth. The **P\u016brva** **M\u012bm\u0101\u1e43s\u0101** dar\u015bana, whose focus is the exegesis of the earliest **Vedic** texts\u2014where the concept of rebirth is either implicit or altogether absent\u2014does not emphasize rebirth but also does not explicitly reject it. For the remainder of Hindu systems, reincarnation is affirmed and release from the cycle of reincarnation\u2014 **mok\u1e63a** \u2014is the ultimate aspiration, each tradition presenting itself as a different set of strategies for achieving this goal.\n\nThe concept of rebirth is a corollary of the concept of **karma** , the universal **law** of **action** that draws morally appropriate effects to the agent who commits an act in thought, word, or deed. The fact of physical **death** does not necessarily imply that one has already exhausted one's karma, and the unequal circumstances of the births of the entities in the world also have to be accounted for in terms of karma. This requires the idea of rebirth, in order to explain how the effects of unaccounted-for good and bad deeds are going to be experienced after one dies, and why beings are born in unequal circumstances. One also meets with the claims of those who say they can remember past lives\u2014a phenomenon that is consistent with the deduction that rebirth must occur if there is such a thing as karma.\n\nThe **Sanskrit** term for rebirth is punarjanma (literally, rebirth). The term for the entire cycle of birth, death, and rebirth from which one aspires to escape is **sa\u1e43s\u0101ra**. The desire to achieve **liberation** from rebirth\u2014which, at first glance, can be an appealing idea, implying virtual **immortality** \u2014is a function of the fact that the karmically produced goods experienced within sa\u1e43s\u0101ra are, at best, impermanent. The desire for such goods and the actions that yield them are the fuel that drives the rebirth cycle. The pursuit of ephemeral goods is ultimately a source of suffering. The soul, or **j\u012bva** , can be likened to a hamster running on a wheel and never getting anywhere. The soul eventually desires permanent peace and bliss without limitation, which can only be achieved through the realization of **Brahman** , understood as a purely impersonal being, consciousness, and bliss at the basis of one's self ( **\u0101tman** ), or theistically, as a loving **God** with whom one attains a mystical union.\n\n**REINCARNATION.** _See_ REBIRTH.\n\n**RENOUNCER.** _See_ SANNY\u0100SA.\n\n**RENUNCIATION.** _See_ SANNY\u0100SA.\n\n_\u1e5aG VEDA_ **.** The most ancient extant **Vedic** text and the earliest extant Hindu **scripture**. Conventional dating of the text, based on both linguistic evidence and the technological references within the text that can be correlated with archaeology, places it somewhere between 1700 and 1200 BCE. Critics of the **\u0100ryan Migration Theory** have argued that the text is much older, based on **astrological** references within the text and references to the **Sarasvat\u012b** River, which had dried up or gone underground by 1900 BCE. Although these arguments have not been persuasive to mainstream scholarship, they do suggest that at least some elements of the text\u2014which all agree is a compilation undertaken over many generations\u2014may predate the conventional dating, which could be taken as the dating of the text in its present form.\n\nRather than a continuous narrative, the text consists of 1,028 hymns\u2014or **mantras** \u2014which are organized thematically into 10 books, called **ma\u1e47\u1e0dalas**. Most of these hymns are in praise of deities called **devas** (literally, \"shining ones\"), the most popular of whom are **Indra** , Lord of the Devas, and **Agni** , the deity of fire, which is the central element of the Vedic ritual of sacrifice ( **yaj\u00f1a** ). The Vedic devas are usually associated with natural phenomena, such as **S\u016brya** the sun god, **U\u015b\u0101** the goddess of the dawn, **Soma** the god of the **moon** and of the hallucinogenic plant of the same name, **V\u0101yu** the god of the wind, **Yama** the god of **death** , and so on. The lord of the Vedic devas, Indra, who is associated with thunder, is analogous to the Greek Zeus and the Norse Thor. Also of considerable importance is **Praj\u0101pati** , whose name literally means the \"father of offspring,\" and who is regarded as the creator of the universe\u2014later known as **Brahm\u0101**. The two chief deities of later Hinduism, **Vi\u1e63\u1e47u** and **\u015aiva** , receive relatively little mention in the _\u1e5ag Veda_ (the latter typically under the name **Rudra** ).\n\nThe _\u1e5ag Veda_ also contains references to kings and battles and alludes to stories that are presented more fully in later **literature** , such as the _Br\u0101hma\u1e47as_ and the _Pur\u0101\u1e47as_. Some of the references remain mysterious today, having already been forgotten by the period of the 14th-century Vedic scholar **S\u0101ya\u1e47a**. Some of the later Vedic hymns contain philosophical reflections that anticipate many elements of **Ved\u0101nta**. _See also PURU\u1e62A S\u016aKTA_.\n\n**RIGHT-HAND TANTRA.** **Tantric** practice that does not make use of the antinomian practices that characterize **Left-Handed** or V\u0101m\u0101c\u0101ra Tantra. Right-Hand Tantra either does not utilize these practices or projects them into a realm of symbolic visualization, on the assumption that their actual practice is perilous, leading to **rebirth** in hell if performed out of an impure motive and not for the cultivation of a nondualistic consciousness. The vast majority of Tantric practice is of the Right-Handed variety, not violating social norms regarding purity and impurity. A major innovator in the development of this system was the 10th\u201311th-century **Ka\u015bm\u012br \u015aaiva** theologian **Abhinavagupta**.\n\n**RISHIKESH (\u1e5a\u1e62IKE\u015aA, H\u1e5a\u1e62\u012aKE\u015aA).** Popular place of pilgrimage in the foothills of the **Him\u0101layas**. The **Ga\u1e45g\u0101** or Ganges River flows through Rishikesh and many places of spiritual retreat, or **\u0101\u015bramas** , are located upon its banks here. One was once that of the **Maharishi Mahesh Yogi** , with whom the Beatles studied **Transcendental Meditation** in Rishikesh in the summer of 1968.\n\n**ROY, R\u0100M MOHAN (1772\u20131833).** First major Hindu reformer of the modern period; often called the \"Father of Modern Hinduism.\" Roy established the **Brahmo Sam\u0101j** in **Calcutta** in 1828. The Brahmo Sam\u0101j was mainly popular among middle-class **Bengali** intellectuals who had received significant exposure to European culture and education, often through the efforts of Christian missionaries, but who also had a strong sense of identification with Hinduism and Indian culture. Led by Roy, these intellectuals sought to reform Hinduism, opposing practices such as **sat\u012b** and the use of images ( **m\u016brtis** ) in worship, which they held to be later additions to an originally **monotheistic** Hinduism taught in the _Upani\u1e63ads_. Roy, a prolific author and energetic activist, translated some of the _Upani\u1e63ads_ into Bengali and wrote a good many pamphlets as well, decrying what he perceived to be the evils of certain Hindu practices, and of **Christianity** as well. Brahmo Sam\u0101j worship was modeled upon Protestant worship services, with hymns and sermons drawing upon the _Upani\u1e63ads_ , rather than the Bible. Roy successfully campaigned to get the practice of sat\u012b\u2014the immolation of **widows** \u2014banned in Bengal in 1829.\n\nTheologically, the teachings of the Brahmo Sam\u0101j display a strong leaning toward Unitarianism, and a number of Brahmo Sam\u0101j members traveled to England in the 19th century to study at the Unitarian seminary at Harris-Manchester College in Oxford. With its emphasis on an original, \"pure\" **Vedic** monotheism later corrupted into popular Hindu **polytheism** , Brahmo Sam\u0101j teaching in many ways anticipates that of the **\u0100rya Sam\u0101j** of **Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b**.\n\n**ROYAL ASIATIC SOCIETY.** The first scholarly society wholly devoted to the study of **Indology** ; established by **Sir William Jones** in **Calcutta** in 1784.\n\n**\u1e5a\u1e62ABHA.** Literally \"the bull\"; the best or foremost; epithet of **\u015aiva** (whose **vahana** , or animal vehicle, is also a bull named **Nandi** ); also the first **T\u012brtha\u1e45kara** of **Jainism** , who is sometimes listed as an **avat\u0101ra** of **Vi\u1e63\u1e47u**. A \u1e5a\u1e63abha is mentioned in the _\u1e5ag Veda_. The large number of representations of bulls in the artifacts of the **Indus Valley civilization** suggests that the bull was sacred from an early period of Indian cultural history. It has also been suggested that there is a link between the figure of \u015aiva and the **Jain** \u1e5a\u1e63abha, who is sometimes depicted, like \u015aiva, as a naked **ascetic** with long, matted hair.\n\n**\u1e5a\u1e62I.** Sage; seer; wise person; one of the sages who composed or transmitted the _Vedas_ ; one of the Seven Sages ( **Saptar\u1e63i** ) who preside over the cosmos.\n\n**\u1e5aTA.** Sacred order; **truth**. In the _Vedas_ , _\u1e5bta_ refers to the cosmic order upheld by the act of sacrifice ( **yaj\u00f1a** ). In later Hindu **literature** , the term _\u1e5bta_ is gradually superseded by the term **dharma** and takes on an explicitly moral, less ritually oriented tone.\n\n**RUDRA.** \"Roarer\"; awe-inspiring and fearsome **Vedic** deity better known in later Hindu traditions as **\u015aiva** ; a **god** of the wilderness, whose offerings are traditionally placed on the outside of the sacrificial arena so he will not enter and disrupt the sacrifice ( **yaj\u00f1a** ). In the _\u1e5ag Veda_ , Rudra is associated with the fearsome power of the thunderstorm. Rudra has 11 forms. _See also_ ADITI; \u0100DITYAS; DEVAS.\n\n**RUDR\u0100K\u1e62A.** \"Eye of **Rudra** \"; seeds used as beads in **m\u0101las** or rosaries used in **japa** , the practice of repeating a **mantra** \u2014either aloud or mentally and silently\u2014while keeping track of the number of repetitions on a string of beads. Especially sacred to the devotees of **\u015aiva** , the use of rudr\u0101k\u1e63a m\u0101las is now prevalent in all forms of Hinduism, not only the **\u015aaiva** traditions.\n\n**RUKMI\u1e46\u012a.** Wife of **K\u1e5b\u1e63\u1e47a** ; **avat\u0101ra** , or incarnation, of **Lak\u1e63\u1e43\u012b**. _See also_ VI\u1e62\u1e46U.\n\n## S\n\n**\u015aABARA.** Author of a major commentary, or **bh\u0101\u1e63ya** , on the root text of the **M\u012bm\u0101\u1e43s\u0101** system of philosophy ( **dar\u015bana** )\u2014the _P\u016brvam\u012bm\u0101\u1e43s\u0101 S\u016btra_ of **Jaimini**. Simply entitled \"\u015aabara's Commentary\" ( _\u015aabara-bh\u0101\u1e63ya_ ), the text was likely composed during the first century before the Common Era.\n\n**\u015aABAR\u012a.** Character in the _R\u0101m\u0101ya\u1e47a_ ; a woman who was deeply devoted to **R\u0101ma**. She tested the fruits she offered to R\u0101ma by taking a bite of them first, which is technically a taboo act. But because she did it out of **bhakti** , R\u0101ma found no fault with her. It is said that her devotion was so great that she was reborn as **R\u0101dh\u0101**.\n\n**SABARIMALAI.** Hill in **Kerala** sacred to **Ayyappan** 's sect.\n\n**SACRED THREAD.** Thread given to a young person, usually a **Brahmin** male, at his **upanayana** , the ceremony marking the passage from childhood into the student stage of life, an initiation into adulthood that is comparable to the Bar Mitzvah of Judaism or the sacrament of Confirmation in **Christianity**. _See also_ \u0100\u015aRAMA.\n\n**SACRIFICE.** _See_ YAJ\u00d1A.\n\n**S\u0100DHANA.** Spiritual practice or discipline. The term refers primarily to ritual worship and **meditation**. Roughly equivalent, in general usage, to **yoga**.\n\n**S\u0100DHU.** Literally, \"excellent, wonderful, well done\"; exclamation of praise at a thing well done, especially in a religious context; also, a holy man, usually a **sanny\u0101s\u012b**. For a holy woman, the feminine version of the word is **s\u0101dhv\u012b**.\n\n**S\u0100DHV\u012a.** A Hindu or **Jain** nun. _See also_ S\u0100DHU; SANNY\u0100SIN\u012a; WOMEN.\n\n**SAGU\u1e46A BRAHMAN.** **Brahman** with **qualities** , in contrast with **Nirgu\u1e47a Brahman** , or Brahman without qualities or attributes. In **Advaita Ved\u0101nta** , the distinction between the two is the distinction between Brahman perceived as it truly is\u2014without limiting qualities or attributes\u2014and Brahman as perceived through the veil of **m\u0101y\u0101** , or delusory ignorance, as having qualities. As such, Sagu\u1e47a Brahman is coextensive with the entire universe of phenomena, including the personal **God** ( **\u012a\u015bvara** ). In the more theistic forms of **Ved\u0101nta** , in which the realm of phenomena is not illusory, Sagu\u1e47a Brahman refers specifically to God as the abode of all _good_ qualities ( _gu\u1e47\u0101\u015braya_ ).\n\n**SAHAJA.** \"Born with\"; innate; natural; spontaneous; **Tantric** term for human impulses that are generally regarded as being in need of restraint in most Hindu spiritual traditions but that are seen as potential sources of divine realization in Tantra. These tendencies are the special focus of the **Sahaj\u012bya** tradition of Tantra.\n\n**SAHAJIY\u0100.** **Tantric** movement particularly prevalent in **Bengal**. The original Sahaj\u012bya movement, which probably emerged around the eighth century of the Common Era, crossed a number of sectarian boundaries, having **\u015a\u0101kta** , **Vai\u1e63\u1e47ava** , and **Buddhist** forms. Sahaj\u012bya Tantra emphasizes **sahaja** , or innate, human tendencies, such as the sex drive, which it regards as an expression or manifestation of ultimate reality. Rather than suppressing the innate tendencies, as most classical **yoga** traditions seek to do, Sahaj\u012bya Tantra cultivates spontaneity in the belief that this will lead to **mok\u1e63a** \u2014 **liberation** from **rebirth**. Elements of Sahaj\u012bya Tantra can be found in the beliefs and practices of the **Bauls**.\n\n**SAH\u0100SRARA CAKRA.** \"Thousand-petaled lotus\"; **cakra** at the crown of the head that, when energized by the rising **Ku\u1e47\u1e0dalin\u012b \u015aakti** , produces a blissful state that, according to **Tantra** , advances one quickly toward **enlightenment**.\n\n**SAI BABA.** _See_ SATHYA SAI BABA; SHIRDI SAI BABA.\n\n**\u015aAIVA.** Devotee of **\u015aiva** ; adherent of **\u015aaivism**.\n\n**\u015aAIVA SIDDH\u0100NTA.** \"Final, perfect **truth** about **\u015aiva** \"; **\u015aaiva** tradition predominant in **Tamil** Nadu, in southern India, though drawing upon \u015aaiva traditions of other regions as well, such as **Ka\u015bm\u012br \u015aaivism**. Inspired largely by the **N\u0101yan\u0101r** **bhakti** **poets** , of whose thought this tradition is a systematization, \u015aaiva Siddh\u0101nta emerged as a widespread form of \u015aaiva thought and practice around the 13th century of the Common Era, superseding and absorbing the earlier **Pa\u015bupata** tradition. This tradition has experienced a recent revival due to the efforts of **Satguru \u015aiv\u0101ya Subramuniyasw\u0101m\u012b** , an American convert who has promoted it extensively.\n\n**\u015aAIVISM.** In terms of adherents, the second largest (after **Vai\u1e63\u1e47avism** ) of the four main sectarian subdivisions that make up Hinduism. The various subtraditions that constitute \u015aaivism share the common trait of being oriented toward **\u015aiva** as the supreme deity\u2014 **God** in something like a **monotheistic** sense. Like Vai\u1e63\u1e47avism, most forms of \u015aaivism focus on **bhakti** , or devotion to the supreme deity, as the preeminent\u2014or even exclusive\u2014path to **mok\u1e63a** , or **liberation** from the cycle of **rebirth**. Although there is internal theological diversity among \u015aaiva denominations ( **sa\u1e43prad\u0101yas** ), it can broadly be said that \u015aaivas are less inclined than Vai\u1e63\u1e47avas to insist upon a strong metaphysical distinction between God and the individual soul ( **j\u012bva** ). **Ka\u015bm\u012br \u015aaivism** , for example, tends to affirm, in a fashion closely akin to **Advaita Ved\u0101nta** , the ultimate unity of existence, claiming that at the highest level of awareness, \"All is \u015aiva.\" **\u015aa\u1e45kara** is believed to have been \u015aaiva (as his name indicates) and is attributed with a devotional poem in which he proclaims, at the height of his **meditation** , \"I am \u015aiva! I am \u015aiva!\" ( _\u015aivo 'ham! \u015aivo 'ham!_ ). The **\u015aaiva Siddh\u0101nta** tradition, on the other hand, does insist on the metaphysical distinctness of the soul and God as a precondition of bhakti, and is in this way much closer to Vai\u1e63\u1e47avism.\n\n**\u015aakti** , the **Mother Goddess** and wife of \u015aiva, plays a prominent role in much of \u015aaiva thought and practice, with some texts affirming that these two are inseparable, even forming a single, composite deity. **Ga\u1e47e\u015ba** and **Karttikeya** , the sons of \u015aiva and \u015aakti, are also popular objects of devotion in these traditions. Both \u015aaiva and **\u015a\u0101kta** traditions tend to have a **Tantric** orientation. _See also_ N\u0100YAN\u0100RS; PA\u015aUPATA; RUDRA.\n\n**\u015a\u0100KTA.** Devotee of **\u015aakti** ; adherent of **\u015a\u0101ktism**.\n\n**\u015aAKTI.** Power; creative energy; name of the **Mother Goddess** that emphasizes her role as the creative power of the universe. \u015aakti is also identified with the many goddesses of the Hindu pantheon, who are seen as her various forms or manifestations. **Goddesses** such as **Lak\u1e63m\u012b** and Sarasvat\u012b are therefore also called the _\u015baktis_ or power centers of the deities that are their husbands\u2014 **Vi\u1e63\u1e47u** and **Brahm\u0101** , respectively. In this role, the Mother Goddess and her various personae are seen as the energizing principle, in the absence of which their corresponding male gods would be incapable of effective activity. The divine pairing of \u015aiva and \u015aakti, in particular, is often presented, especially in \u015a\u0101kta texts, as a composite being, with the God and Goddess being ultimately inseparable from each other as parts of an internally differentiated, but finally unitary, godhead. In **Tantric yoga** , the **ku\u1e47\u1e0dalin\u012b** energy at the base of the spine is also called \u015bakti and is regarded as a form of this Goddess. _See also_ DEV\u012a; DURG\u0100; K\u0100L\u012a; MAH\u0100DEV\u012a; P\u0100RVAT\u012a; UM\u0100.\n\n**\u015a\u0100KTISM.** In terms of adherents, the third largest (after **Vai\u1e63\u1e47avism** and **\u015aaivism** ) of the four main sectarian subdivisions that make up Hinduism. The various subtraditions constituting \u015a\u0101ktism share the common trait of being oriented toward **\u015aakti** , the **Mother Goddess** as the supreme deity\u2014 **God** in something like a **monotheistic** sense. As husband of \u015aakti, **\u015aiva** plays a prominent role in \u015a\u0101kta thought and practice, which, like \u015aaivism, tends to have a strong **Tantric** orientation. \u015a\u0101ktism is especially prominent in **Bengal**.\n\nTheologically, \u015a\u0101ktism tends to have a more **syncretistic** and pluralistic tendency than either Vai\u1e63\u1e47avism or \u015aaivism, frequently affirming that the various deities of other traditions are forms or manifestations of \u015aakti. This tradition is thus one possible source for the emphasis on religious pluralism in **Neo-Ved\u0101nta** , given that **Ramakrishna** , one of the major figures for this modern interpretation of Hinduism, was a priest and ardent devotee of **K\u0101l\u012b** , one form of the Mother Goddess.\n\n**\u015aAKTIPAT.** Practice in **Siddha Yoga** whereby the **guru** raises the **\u015bakti** or **ku\u1e47\u1e0dalin\u012b** energy from the base of the spine to the crown of the head. This can be done through the medium of a touch, a word, or even a thought. The guru can be physically present for the process or\u2014in the case of \u015baktipat performed through thought\u2014not. \u015aaktipat is the ritual of initiation, or **d\u012bk\u1e63\u0101** , of the Siddha Yoga tradition.\n\n**\u015aAKUNI.** Character in the _Mah\u0101bh\u0101rata_ ; wicked maternal uncle of **Duryodhana** who devises a scheme by which Duryodhana is able to rob the **P\u0101\u1e47\u1e0davas** of their wealth, their kingdom, and their personal freedom through an unfair gambling match. \u015aakuni is also the brother of **G\u0101ndh\u0101r\u012b**. _See also_ YUDHI\u1e62\u1e6cHIRA.\n\n**\u015aAKUNTAL\u0100.** Character in a minor episode of the _Mah\u0101bh\u0101rata_ later adapted into the most famous play in **Sanskrit** **literature** \u2014the _Abhij\u00f1\u0101na\u015bakuntala_ , or \"Recognition of \u015aakuntal\u0101,\" by **K\u0101lid\u0101sa** , the \"Shakespeare of Sanskrit.\" In the story and the play, the sage **Durv\u0101sa** , angered by her inattentiveness, curses \u015aakuntal\u0101 to be forgotten by her beloved, a king named Du\u015byanta, until Du\u015byanta sees his ring (which he once gave to her as a token of their love). Eventually, Du\u015byanta does see the ring and remembers his love for \u015aakuntal\u0101. The two marry and become the parents of **Bharata** , first emperor of all India according to Hindu literature.\n\n_S\u0100MA VEDA_ **.** One of the four _Sa\u1e43hit\u0101s_ or collections of **Vedic** hymns; a compilation of hymns of the _\u1e5ag Veda_ set to **music** , with instructions for chanting in particular ritual settings. It is one of the foundations of Indian classical music.\n\n**SAM\u0100DHI.** Contemplative absorption in the object of one's **meditation** ; the eighth and final stage or \"limb\" of the \"eight-limbed\" ( **a\u1e63\u1e6d\u0101\u1e45ga** ) **yoga** system of **Pata\u00f1jali** , outlined in his _Yoga S\u016btra_ and believed to lead to **mok\u1e63a** , or **liberation** from the cycle of **rebirth**. As later traditions elaborate, sam\u0101dhi is of two kinds\u2014 **savikalpa sam\u0101dhi** , in which there is a faint trace of the distinction of subject and object, and **nirvikalpa sam\u0101dhi** , in which this distinction vanishes completely.\n\n**S\u0100MAGR\u012a.** \"Complete collection\"; an assemblage of the items needed for a religious ritual.\n\n_SA\u1e42HIT\u0100_ **.** A collection of **Vedic** hymns, or **mantras**. There are four Vedic sa\u1e43hit\u0101s: the _\u1e5ag Veda_ , _Yajur Veda_ , _S\u0101ma Veda_ , and _Atharva Veda_. These sa\u1e43hit\u0101s constitute the first historical \"layer\" of the Vedic literary corpus, the others being, in order, the \"priestly texts,\" or _Br\u0101hma\u1e47as_ , the \"forest texts,\" or _\u0100ra\u1e47yakas_ , and the \"secret teaching,\" or the _Upani\u1e63ads_.\n\n**SA\u1e42KALPA.** Resolve, resolution. Most Hindu rituals, such as **p\u016bj\u0101s** and **sa\u1e43sk\u0101ras** , begin with a _sa\u1e43kalpa_ , in which one announces oneself, the purpose for which one seeks to perform the ritual, and the precise date and time in terms of the traditional Hindu time measurement system of **muh\u016brtas** and the **Hindu calendar**.\n\n**S\u0100\u1e42KHYA.** \"Enumeration\"; one of the six **\u0101stika** or \"orthodox\" **dar\u015banas** , or systems of **Vedic** philosophy; frequently paired with the **Yoga** dar\u015bana of **Pata\u00f1jali** and attributed to the sage **K\u0101pila**. S\u0101\u1e43khya philosophy posits a strong dualism between the **soul** , called the **puru\u1e63a** , and matter, or nature, which is known as **prak\u1e5bti**. The goal of S\u0101\u1e43khya and the related system of **Yoga** is for puru\u1e63a to realize its nature as pure spirit and disentangle itself from prak\u1e5bti, with which it has become falsely identified. Whereas puru\u1e63a consists of pure consciousness alone, all phenomena, or contents of consciousness\u2014including both **mind** and matter\u2014are manifestations of prak\u1e5bti. Prak\u1e5bti manifests as the three **gu\u1e47as** , or qualities\u2014 **sattva** (clarity, luminosity), **rajas** ( **action** or dynamism), and **tamas** (inertia). The universe of changing phenomena is an effect of the constant interplay of these three gu\u1e47as.\n\n**SA\u1e42PRAD\u0100YA.** Sect, denomination, teaching lineage. The four major divisions of Hinduism\u2014 **Sm\u0101rta** , **Vai\u1e63\u1e47ava** , **\u015aaiva** , and **\u015a\u0101kta** \u2014are themselves further subdivided into sa\u1e43prad\u0101yas. It is often noted that Hinduism is not an \"organized religion,\" in the sense of having a central institution or creed, but rather a loose collection or family consisting of many overlapping and criss-crossing traditions. The sa\u1e43prad\u0101yas are more cohesive, individually, than is Hinduism as a whole. They are closer to a conventional idea of what constitutes a religion, having specific founders, doctrines, distinct rites of initiation, and so forth.\n\n**SA\u1e42S\u0100RA.** Literally \"wandering about,\" the cycle of birth, **death** , and **rebirth**. This term also refers to the realm of karmically determined spatio-temporal phenomena in which the rebirth process occurs. It is therefore also equivalent to \"the world.\" Finally, it also refers to human society\u2014that which the renouncer ( **sanny\u0101sin** ) abandons in order to seek **liberation** ( **mok\u1e63a** ). Sa\u1e43s\u0101ra refers to the sphere in which death and rebirth occur, or to the entire process. The term for rebirth itself, though, is punarjanma.\n\n**SA\u1e42SK\u0100RA.** This term has two related meanings. First, it is an impression made upon the **soul** (technically, on one of its **subtle bodies** or **suk\u1e63ma \u015bar\u012bra** ) through repeated acts of the same kind. A sa\u1e43sk\u0101ra, in other words, is a habit. Collectively, one's sa\u1e43sk\u0101ras constitute one's character, thereby determining, among other things, the kind of form one will take in one's next **rebirth**. On the spiritual path, one wishes to cultivate wholesome sa\u1e43sk\u0101ras\u2014sa\u1e43sk\u0101ras conducive to good **karma** , and ultimately to **liberation** \u2014and get rid of unwholesome ones.\n\n_Sa\u1e43sk\u0101ra_ is also the term for a Hindu sacramental ritual, such as giving a name to a newborn baby (the _n\u0101makara\u1e47a-sa\u1e43sk\u0101ra_ ), the investiture with the **sacred thread** of an adolescent (the _upanayana_ _-sa\u1e43sk\u0101ra_ ), or the rites of **marriage** ( _viv\u0101ha_ _-sa\u1e43sk\u0101ra_ ).\n\nThe two meanings are related because a sacramental ritual is supposed to make a deep impression upon one's soul, transforming one on a spiritual level.\n\n**SAN\u0100TANA DHARMA.** \"Eternal religion\"; a Hindu term for Hinduism, typically used to distinguish the ancient **Vedic** mainstream of the tradition from the particular sectarian movements that have proliferated within it, especially modern reform movements such as the **\u0100rya Sam\u0101j**.\n\n**SANDALWOOD.** An aromatic wood that is often ground into a paste and utilized in Hindu rituals; believed to be a purifying and cooling substance. The **Sanskrit** term for it is _candana_.\n\n**SA\u1e44GH PARIV\u0100R.** \"Family of Associations\"; a loose collection of **Hindu Nationalist** organizations that share the agenda of advancing the cause of either a Hindu or a Hindu-dominated nation of India. The particular aims of these organizations and the degree to which each might be characterized as \"extreme\" or \"moderate\" vary a great deal. But they tend to be regarded as constituting the \"Hindu right\" in the Indian political spectrum. The Sa\u1e45gh Pariv\u0101r includes many organizations, the most prominent being the **R\u0101\u1e63\u1e6dr\u012bya Svayamsevak Sa\u1e45gh** (RSS), **Vi\u015bva Hindu Pari\u1e63ad** (VHP), and the **Bharat\u012bya Janat\u0101 Party** (BJP).\n\n**\u015a\u0100NI.** The deity who personifies the planet Saturn. His gaze is said to be unfortunate and his presence in one's **astrological** chart is generally inauspicious. According to one version of the story of **Ga\u1e47e\u015ba** , it is \u015a\u0101ni's gaze that causes the infant deity's head to fall off, requiring it to be replaced with the head of an elephant.\n\n**\u015a\u0100NIV\u0100RA.** Saturday, the day of the week that is governed by the planet Saturn. _See also_ CALENDAR, HINDU.\n\n**SA\u00d1JAYA.** Character in the _Mah\u0101bh\u0101rata_ ; clairvoyant minister of King **Dh\u1e5btar\u0101\u1e63\u1e6dra**. When Dh\u1e5btar\u0101\u1e63\u1e6dra asks Sa\u00f1jaya to describe for him the events of the battle of **Kuruk\u1e63etra** , Sa\u00f1jaya narrates the dialogue of **K\u1e5b\u1e63\u1e47a** and **Arjuna** that becomes the _Bhagavad G\u012bt\u0101_.\n\n**\u015aA\u1e44KARA (788\u2013820).** Philosopher and religious leader; the **\u0101c\u0101rya** and founder of four **ma\u1e6dhas** , or monastic institutions, located at the four corners of India; the founder of the **Da\u015ban\u0101m\u012b Order** of Hindu ascetics; systematizer and renowned exponent of the **Advaita** system of **Ved\u0101nta**. \u015aa\u1e45kara has been one of the most influential intellectual figures in all of Hinduism. A child prodigy, \u015aa\u1e45kara is said to have bested major intellectual rivals in debate at an early age. His commentaries on the _Brahma S\u016btra_ , _Bhagavad G\u012bt\u0101_ , and the principle _Upani\u1e63ads_ established Advaita Ved\u0101nta as a major philosophical system to be reckoned with in classical Hindu thought\u2014a status it retains even today. Born in what is now the state of **Kerala** , on the southwest coast of India, into the **Namb\u016btiri Brahmin** community, \u015aa\u1e45kara undertook **sanny\u0101sa** , or renunciation, at a young age and moved to the city of **Banaras**. He spent much of the rest of his short life in Banaras, but traveled extensively throughout the rest of India as well. He established the Da\u015ban\u0101m\u012b Order and the four ma\u1e6dhas of **Puri** , **Badr\u012bn\u0101th** , **Dv\u0101rak\u0101** , and **\u015a\u1e5b\u1e45ger\u012b** , as well as organizing Hindu worship into the system of five objects of worship ( **pa\u00f1cay\u0101tana p\u016bj\u0101** ) in part as a way of combating the highly successful and well-organized tradition of **Buddhism**. His later critics, however, from the more theistically and devotionally oriented systems of Ved\u0101nta, found his system too much like Buddhism.\n\n**\u015aA\u1e44KAR\u0100C\u0100RYA.** The head monk, or **\u0101c\u0101rya** , of one of the four **ma\u1e6dhas** or monastic institutions started by **\u015aa\u1e45kara** \u2014 **Puri** , **Badr\u012bn\u0101th** , **Dv\u0101rak\u0101** , and **\u015a\u1e5b\u1e45ger\u012b**. The original \u015aa\u1e45kara is often called the \"First \u015aa\u1e45kar\u0101c\u0101rya\"\u2014 _\u0100di \u015aa\u1e45kar\u0101c\u0101rya_ \u2014to distinguish him from the subsequent heads of the four monasteries that he established.\n\n**\u015aA\u1e44KHA.** A conch shell; item often used in Hindu rituals. The conch shell is hollowed out and blown like a horn. The sound of the \u015ba\u1e45kha is believed to be auspicious and to dispel negative forces from the area of the ritual. Hollowed out conch shells have been found in India dating back to the **Indus Valley civilization** , where it is possible that they had a similar ritual purpose (though this is not known with any certainty).\n\n**\u1e62A\u1e46MUKHA.** \"Having six faces\"; epithet of **K\u0101rttikeya** , so called because of his six heads and 12 arms.\n\n**SANNY\u0100SA.** Renunciation; the fourth of the four stages of life, or **\u0101\u015bramas** , outlined in the _Dharma \u015a\u0101stras_ ; the **niv\u1e5btti m\u0101rga** , or \"path of negation\" of the person seeking **mok\u1e63a** , or **liberation** from the cycle of **rebirth**. This path is in contrast with the **prav\u1e5btti m\u0101rga** , or \"path of inclination toward **action** ,\" which characterizes the first three stages of life, which are focused on worldly attainments (including a better rebirth). Sanny\u0101sa is an **ascetic** way of life chosen on the basis of the idea that engagement with the affairs of this world is incompatible with the quest for liberation. A male renouncer is referred to as a monk, and a female renouncer as a nun. A renouncer, or **sanny\u0101sin** , thus abandons his or her former identity\u2014including **caste** \u2014in order to be free from obligations that can bind one to the cycle of rebirth. All action, even good action, produces **karmic** results. The idea of renunciation is to withdraw from action in order to become free from the effects of karma. According to the _Bhagavad G\u012bt\u0101_ and the later texts of the **bhakti** movement, total freedom from action is impossible and undesirable. These texts instead recommend an inner attitude of renunciation called \"renunciation of the fruits of action\"\u2014engaging in virtuous\u2014or dharmic\u2014action, but with detachment from the results. But the ancient practice of actually withdrawing from society and joining an ascetic order is still quite common in Hindu traditions.\n\nMany Hindu ascetic orders are peripatetic\u2014having no fixed abode and traveling from place to place as a form of ascetic practice, in order to avoid developing attachment for one particular place. This trait is shared by both Hindu and **Jain** ascetics, and is one of the reasons these traditions survived the invasions of the early medieval period and the mass destruction of settled monastic institutions that led to the demise of **Buddhism**. At the same time, many Hindu ascetic orders do dwell in monasteries, or **ma\u1e6dhas** , in many ways similar to the **Christian** monasteries of medieval Europe. _See also_ \u0100C\u0100RYA.\n\n**SANNY\u0100SIN.** A Hindu monk; a male renouncer; a man who has taken up the life of **sanny\u0101sa**.\n\n**SANNY\u0100SIN\u012a.** A Hindu nun; a female renouncer; a **woman** who has taken up the life of **sanny\u0101sa**.\n\n**SANSKRIT.** Literally, \"cultured\"; the actual word being _sa\u1e43sk\u1e5bta_. The language of the _Vedas_ and the majority of \"classical\" Hindu **literature** , in contrast with more localized, vernacular Hindu literatures (which are highly important and central to those Hindus who utilize them\u2014sometimes more so than Sanskrit texts, which relatively few persons can read). Sanskrit is the language of **Brahmanism** , the **Vedic** tradition that, in conjunction with a host of other Indic traditions, forms Hinduism.\n\nSanskrit is sometimes referred to as an \"artificial language.\" There probably was a spoken Sanskrit at one point in time that was the language of daily use of a people in the Punjab region of northern India. The Sanskrit of the _Vedas_ may be an approximation of this language. Classical Sanskrit\u2014utilized in all post-Vedic Sanskrit literature\u2014is not in every respect identical to Vedic Sanskrit, but differs from it in a number of ways. In the fifth or fourth century BCE, a Sanskrit grammarian\u2014a **Brahmin** named **P\u0101\u1e47ini** \u2014authored the _A\u1e63\u1e6dadhy\u0101yi_ , an authoritative work on grammar that has defined the rules of classical Sanskrit usage ever since. Because of the influence of P\u0101\u1e47ini's grammar, Sanskrit has, in effect, become frozen in the form in which it was spoken in the fifth and fourth centuries BCE. It has not continued to develop in the way that natural languages typically do as a result of frequent use. This is why it is called an \"artificial language.\" A highly formal language, it has traditionally been used primarily for writing authoritative texts on a wide variety of subject and in liturgical settings, performing a role in Indian culture analogous to that which Latin has played in the intellectual and religious history of Europe.\n\nIn linguistic terms, Sanskrit is identified with Early **Indo-\u0100ryan**. The languages of daily use in ancient India were the **Prakrits** , or \"natural\" languages (in contrast with the \"cultured\" and artificial Sanskrit). The Prakrits are sometimes thought of as dialects of Sanskrit, but it is more accurate to refer to them as Middle Indo-\u0100ryan. They evolved into the Late Indo-\u0100ryan modern languages of northern India, such as **Hindi** , **Bengali** , Gujarati, Marathi, and so on. _See also_ SANSKRITIZATION.\n\n**SANSKRITIZATION.** The **Sanskrit** language has long been identified with the \"high\" culture of India\u2014specifically, the **Vedic** culture of **Brahmanism** \u2014which has dominated a Hindu self-understanding for millennia. This is in contrast with the popular local cultures that are more often expressed in vernacular languages. Within the **caste** system, when a particular group has decided to raise its status in the larger caste hierarchy, this has often been accomplished through a process of adopting certain high-caste cultural and religious practices, modes of speech, and so on. This process is known as \"Sanskritization,\" a term coined by sociologist M. N. Srinivas (1916\u20131999). It can include such practices as the wearing of the **sacred thread** \u2014traditionally a practice reserved for **Brahmins** or members of the three **twice-born** castes. Sanskritization can also include the incorporation into the Brahmanical or Vedic tradition of practices, ideas, and even deities that are of non-Vedic origins. It is likely that this process can be traced back even to the period of the _Vedas_ , in which certain indigenous deities and practices came to be incorporated into the originally **Indo-European** Vedic worldview. This practice still continues today, with many of the concepts and practices of **Islam** , **Christianity** , and modern science being assimilated to a Hindu worldview and self-understanding.\n\n**SANT.** Saint; holy person; a movement among prominent religious figures in northern India in the 15th and 16th centuries who were disdainful of sectarian boundaries, teaching that personal devotion, or **bhakti** , is of greater significance than ritual, **caste** , or sectarian affiliation. Including **R\u0101m\u0101nanda** , **Kab\u012br** , and **N\u0101nak** , the Sants typically saw the divine as personal but formless. In this way, their teaching reflects the influence of **Islam**.\n\n**\u015a\u0100NTI.** Peace; especially peace of **mind** ; a sense of calm equanimity; a mental state that is widely associated with **meditation**.\n\n**SANTOSHI (SA\u1e42TO\u015a\u012a) M\u0100.** \"Mother who brings satisfaction or contentment\"; recent **goddess** whose worship emerged in the 1970s; daughter of **Ga\u1e47e\u015ba** who was created to relieve the sufferings of human beings. Santoshi M\u0101 is particularly popular among Hindu **women**. Her devotees perform a **vrata** , or sacred vow, on Fridays that consists of fasting and offering a very simple **p\u016bj\u0101** to the goddess. The story of this goddess was the subject of a popular **Hindi** film in 1975 entitled _Jai Santoshi M\u0101_ (\"Victory to Santoshi Ma\"). In appearance, she is depicted in a way that is evocative of the goddess **Durg\u0101** \u2014youthful and beautiful but also holding weapons for destroying evil and worldly obstacles. Prayers to Santoshi M\u0101 are traditionally for this-worldly goals and wishes rather than transcendence.\n\n**SAPTAR\u1e62I.** \"Seven Sages\"; seven wise beings who preside over the cosmos; born from the mind of **Brahm\u0101** ; founders of the **gotras** , or **Brahmanical** lineages. **Astrologically** , they are identified with the seven stars of the Great Bear (or \"Big Dipper\"). A different set of seven sages appears from the **mind** of Brahm\u0101 in every cosmic cycle\u2014which is a way of explaining the fact that there are different lists of these sages in different textual sources. According to most texts, the seven sages of the current cosmic cycle are **Atri** , **Bharadv\u0101ja** , **Gautama** , **Jamadagni** , **Ka\u015byapa** , **Vasi\u1e63\u1e6dha** , and **Vi\u015bvamitra**.\n\n**SARAD\u0100 DEV\u012a (1853\u20131920).** Born Saradama\u1e47i Mukhopadhy\u0101ya, the wife and spiritual companion of **Ramakrishna** , one of the founding figures of the **Neo-Ved\u0101nta** movement and the revival of modern Hinduism. During Ramakrishna's lifetime, although greatly revered by his disciples, she was largely a behind-the-scenes figure. After Ramakrishna's **death** , in 1886, she began to play a more prominent role in the movement that he began, advising his disciples in their organizing of the **Ramakrishna Order** and **Ramakrishna Mission** , as well as initiating many disciples of her own. Even at this point, though, she remained relatively secluded and did not seek out a large public role for herself.\n\nBorn in the village of Jayrambati, in **West Bengal** , she was formally betrothed to Ramakrishna at the age of five, going to live with him in Dak\u1e63ine\u015bwar, where he was the priest at a **temple** of the **goddess K\u0101l\u012b** , only in her late teens. (Such arrangements were not unusual among rural Hindus at that time, and their abandonment was championed by many Hindu social reformers of the 19th century.) Ramakrishna is said to have worshiped her as an incarnation of the **Mother Goddess**. She continues to be regarded as such by many devotees in the **Ramakrishna Ved\u0101nta** tradition today.\n\n**SARASVAT\u012a.** **Goddess** of wisdom; sometimes depicted as the wife of **Brahm\u0101** and the daughter of **\u015aiva**. Her vehicle ( **vahana** ) is a white swan. Iconographically, she usually has four arms. In two of her hands she holds a stringed **musical** instrument called a **v\u012b\u1e47\u0101**. In one of her right hands she holds a **m\u0101la** , or string of prayer beads, and in one of her left hands she holds a book. All of the items in her hands represent different aspects of wisdom: the v\u012b\u1e47\u0101 symbolizing the **arts** , the m\u0101la symbolizing spirituality, and the book representing **knowledge** derived from study. She was originally a river goddess presiding over the Sarasvat\u012b River, which is believed to have disappeared around 1900 BCE. Her festival, Sarasvat\u012b P\u016bj\u0101, is held on the fifth day of the bright half ( **\u015aukla Pak\u1e63a** ) of the Hindu month of **Magha** (January\u2013February). As the goddess of wisdom, she is especially popular with students, who often celebrate her p\u016bj\u0101 by placing their books at the feet of her **m\u016brti** , or image, for the duration of the holiday. She is also especially popular in **Bengal**.\n\n**SARASVAT\u012a, DAY\u0100NANDA.** _See_ DAY\u0100NANDA SARASVAT\u012a.\n\n**\u015aAR\u012aRA.** Body. This term can refer to the gross, physical body ( **sth\u016bla \u015bar\u012bra** ) or one of several \"subtle\" bodies ( **suk\u1e63ma \u015bar\u012bra** ). These bodies correspond to the **ko\u015bas** , or \"layers,\" that cover the self or **\u0101tman**. The physical body corresponds to the \"food layer\" ( **anna-maya-ko\u015ba** ) and the other layers being nonmaterial or \"subtle\" bodies.\n\n**SARVODAYA.** \"Uplifting of all\"; movement for empowerment of the poor based on **Mohand\u0101s K. G\u0101ndh\u012b** 's **philosophy** of **saty\u0101graha** and promoted by his close associate and disciple **Vinobha Bhave**.\n\n**\u015a\u0100STRA.** **Scripture** ; science; authoritative text on a particular topic; authoritative body of **knowledge**.\n\n**SAT.** Being, existence. Along with pure consciousness ( **cit** ) and infinite bliss ( **\u0101nanda** ), sat is one of the three inherent attributes of **Brahman** according to **Ved\u0101nta**. The word for truth ( **satya** ) is derived from sat, as referring \"that which is the case.\"\n\n**SATGURU.** True **guru** ; spiritual teacher of particularly high authority; a teacher who presides over an entire tradition; roughly equivalent to an **\u0101c\u0101rya**.\n\n**SATHYA SAI BABA (1926\u2013 ).** Contemporary **guru** based in Andhra Pradesh, southern India; believed by his devotees to be a reincarnation of **Shirdi Sai Baba** ; famous for the miracles he is believed to have performed, including spontaneous appearances of sacred ash from his hands. In 1940, when he was 14 years old, it is said that he was bitten by a scorpion. The poison sent him into a delirium from which it is said he awoke with a realization of his true, **enlightened** nature as a great Hindu sage. He began to develop a following that today is global in extent. His teachings emphasize **bhakti** , the study of **scripture** , personal self-discipline, and **vegetarianism**.\n\n**SAT\u012a.** Good **woman** ; virtuous woman; name of the **Mother Goddess** , **P\u0101rvat\u012b** , in her previous incarnation as a daughter of the deity **Dak\u1e63a**. Sat\u012b was deeply in love with **\u015aiva** and married him, but Dak\u1e63a never fully approved of \u015aiva. Dak\u1e63a insulted \u015aiva by failing to invite him to a massive sacrifice to which he had invited all the other sages and deities. Feeling deeply hurt by this insult to her husband, Sat\u012b immolated herself in the sacrificial fire. It was this, and not the previous snub, that enraged \u015aiva, who destroyed Dak\u1e63a's sacrifice and beheaded him. (Later repenting of this act, \u015aiva resurrected Dak\u1e63a and gave him the head of a ram.) \u015aiva was inconsolable in Sat\u012b's absence. Sat\u012b's love for \u015aiva, though, caused her to be reborn as P\u0101rvat\u012b, and the couple was reunited.\n\nThe name of this **goddess** was later applied to **widows** who immolated themselves on the funeral pyres of their husbands. According to **Rajput** lore, upon hearing that their husbands had fallen in battle against foreign invaders, many Rajput women immolated themselves rather than be captured or raped by the enemy. Eventually, this became an expected display of **devotion** in Rajput families. The practice was transmitted to **Bengal** when a number of Rajput clans transplanted themselves to that region. It was banned by the British in 1829 due to the efforts of the Bengali Hindu reformer **R\u0101m Mohan Roy** and soon ceased to occur on a large scale (and had, in any case, been largely confined to the upper **castes** in Bengal and Rajasthan). But occasional exceptions have occurred in subsequent history, the last known sat\u012b having taken place at Deorala, in **Rajasthan** , in 1987. This event was protested by many Hindus, including **Sw\u0101m\u012b Agnive\u015ba**.\n\n**SATTVA.** Clarity, luminosity; the first of the three **gu\u1e47as** or qualities of phenomenal existence as delineated in the **philosophies** of **S\u0101\u1e43khya** and **Yoga** and in the _Bhagavad G\u012bt\u0101_.\n\n**SATYA.** Truth. The motto of the modern Republic of India is the **Sanskrit** phrase \"Truth alone triumphs\" ( _Satyam eva jayate_ ). **Mohand\u0101s K. G\u0101ndh\u012b** viewed the pursuit of truth and the practice of nonviolence ( **ahi\u1e43s\u0101** ) as inseparable, seeing truth and nonviolence as different sides of the same coin, the truth being the interconnected and ultimately unified nature of all beings (his interpretation of **Advaita Ved\u0101nta** ) and nonviolence being the implication of this truth in practice. G\u0101ndh\u012b also underwent a shift in his thinking over the course of his lifetime from the idea that \" **God** is truth\" to the idea that \"Truth is God.\" The significance of this shift is that, whereas by holding that \"God is truth\" one might identify one's own idea of God with the truth, by holding that \"Truth is God\" one is committed to pursuing the truth wherever it may lead. The motto of the **Theosophical Society** is also \"There is no religion higher than truth\" ( _Satyam paramo dharma\u1e25_ ).\n\n**SATY\u0100GRAHA.** Holding fast to truth, **soul** force. Saty\u0101graha, or \"soul force,\" is **Mohand\u0101s K. G\u0101ndh\u012b** 's term for his method of nonviolent resistance to evil. Having some resonance with the epic concept of the vow of truth, the idea behind saty\u0101graha is that the truth always prevails, and that one who is faithful to the truth will also prevail, despite adversity and opposition. The nonviolent activist who adheres to methods that are in harmony with the ends that he pursues is behaving in a manner consistent with truth. The inherent power of truth will ensure the ultimate success of the activist's cause. _See also_ SATYA.\n\n**SATYAK\u0100MA.** \"One who desires **truth** \"; **Vedic** sage whose early career is presented in the _Ch\u0101ndogya Upani\u1e63ad_. Satyak\u0101ma's **guru** was the sage Haridrumata **Gautama**. As a boy, Satyak\u0101ma was keenly interested in studying the _Vedas_. To do this, it was necessary to approach a qualified teacher and recite one's lineage in order to prove that one was a **Brahmin** and thus worthy of being taught. Satyak\u0101ma did not know his father, so asked his mother, Jab\u0101la, a servant woman, his lineage. She told him that she was not sure who his father was, since she had been with many men (possibly having been sexually abused as a servant woman in many homes, although the text does not say this explicitly). She tells her son to refer to himself by her name, calling himself Satyak\u0101ma Jabal\u0101, because his paternal lineage is unknown. When he approaches Haridrumata Gautama, Satyak\u0101ma is asked to recite his lineage, according to Brahmanical custom. Satyak\u0101ma repeats to the teacher what his mother told him. Rather than reject Satyak\u0101ma\u2014the illegitimate son of a servant woman\u2014as unworthy of being taught, Haridrumata Gautama exclaims, \"Who but a Brahmin could speak thus!\" and promptly initiates Satyak\u0101ma as his student. The idea of the story seems to be that personal characteristics such as truthfulness are of greater importance than birth status. _See also_ ADHIK\u0100RA; CASTE.\n\n**SATYAN\u0100R\u0100YA\u1e46A.** \"True Lord,\" an epithet and form of **Vi\u1e63\u1e47u** , typically depicted as Vi\u1e63\u1e47u in his conventional four-armed form, holding a conch shell, discus, **lotus** , and club, but surrounded by a group of worshipers who are performing **Satyan\u0101r\u0101ya\u1e47a P\u016bj\u0101**.\n\n**SATYAN\u0100R\u0100YA\u1e46A P\u016aJ\u0100.** A **p\u016bj\u0101** that is devoted to **Vi\u1e63\u1e47u** in his manifestation as **Satyan\u0101r\u0101ya\u1e47a**. This p\u016bj\u0101 is traditionally held on a full-moon day ( **p\u016br\u1e47im\u0101** ), but can also be held by a **householder** in order to celebrate and give thanks for any major event or to invoke blessings for a major undertaking, such as the building of a new house, the starting of a new business, the marriage of a son or daughter, and so on. Like most p\u016bj\u0101s, the Satyan\u0101r\u0101ya\u1e47a p\u016bj\u0101 involves the offering of fruits, flowers, rice, and a mixture of milk, **ghee** , honey, wheat flour, and yogurt, to the deity, and its distribution afterward as **pras\u0101d**. It also involves the worshipers listening to, and sometimes participating in, a recitation of the Satyan\u0101r\u0101ya\u1e47a **Vrata Kath\u0101** , or Story of the Satyan\u0101r\u0101ya\u1e47a Vow. The kath\u0101 narrates the stories of devotees who have enjoyed good fortune or have averted misfortune through the performance of the Satyan\u0101r\u0101ya\u1e47a p\u016bj\u0101, or who have brought misfortune upon themselves by failing to perform the p\u016bj\u0101 after having vowed to do so, or by failing to take the pras\u0101d after having participated in the p\u016bj\u0101.\n\n**\u015aAUNAKA (fl. 5th century BCE).** Sage and scholar attributed with authoring the _B\u1e5bhad-devat\u0101_ , or the \"Great Text on Deities\"\u2014a systematic presentation of the attributes and the deeds of the **Vedic** deities ( **devas** )\u2014and a work on correct pronunciation of the verses of the _\u1e5ag Veda_ called the _\u1e5ag Veda Pr\u0101ti\u015b\u0101khya_. Both texts were likely composed between 500 and 400 BCE.\n\n**SAVARKAR, VINAYAK DAMODHAR (\"VEER\") (1883\u20131966). Hindu Nationalist** leader and author of the essay _Hindutva_ (\"Hinduness\"). Hindutva is a concept of Hindu identity that conflates it with Indian identity. According to Savarkar, a Hindu is one who is of Indian descent, who claims India as his or her \"fatherland,\" and who adheres to a religion of Indian origin. Indians who do not practice a religion of Indian origin are, on this understanding, less \"Indian\" than are self-identified Hindus. Similarly, non-Indians who practice a Hindu religion are not, according to this understanding, truly Hindu.\n\nSavarkar participated in the Indian independence movement, being jailed by the British from 1910 to 1937 for his revolutionary activities. It was during the period of his imprisonment that he authored _Hindutva_ , as well as other essays and **poems**. After being released from prison, he resumed his political activities, opposing **Mohand\u0101s K. G\u0101ndh\u012b** 's strictly nonviolent strategy and joining the **Hindu Mah\u0101sabh\u0101** , of which he served as president from 1937 to 1943. Savarkar was strongly opposed to the partition of India. Greatly revered by Hindu Nationalists, Savarkar is seen by many as the founding father of the Hindu Nationalist movement.\n\n**SAVIKALPA SAM\u0100DHI.** State of meditative absorption with a residual sense of the duality of subject and object. **Sam\u0101dhi** \u2014absorption in the object of one's **meditation** \u2014is the eighth and final \"limb\" of **Pata\u00f1jali** 's eight-limbed ( **\u0101\u1e63\u1e6d\u0101\u1e45ga** ) system of **yoga**. Later traditions that utilize Pata\u00f1jali's system distinguish between that sam\u0101dhi in which there remains a sense of subjectivity (savikalpa sam\u0101dhi) and a sam\u0101dhi with no subject\u2013object duality ( **nirvikalpa sam\u0101dhi** ).\n\n**SAVIT\u1e5a.** \"Giver of life\"; **Vedic** solar deity; one of the 12 **\u0100dityas**. The **G\u0101yatr\u012b mantra** is dedicated to this deity. It is not always clear if Savit\u1e5b is identical to **S\u016brya** , the more commonly invoked Vedic sun **god**. According to the authoritative commentator, **S\u0101ya\u1e47a** , the name _Savit\u1e5b_ refers to the sun before and just as it rises and _S\u016brya_ is the name of the sun once it has risen fully.\n\n**S\u0100VITR\u012a.** An alternative name for the **G\u0101yatr\u012b mantra**. As a **goddess** , S\u0101vitr\u012b is the daughter of **Savit\u1e5b** and wife of **Brahm\u0101**.\n\n**S\u0100YA\u1e46A (fl. 14th century CE).** Highly respected and authoritative commentator upon the _Vedas_ ; author of the _Ved\u0101rtha Prak\u0101\u015ba_ , or \"Illumination of the Meaning of the _Vedas_ ,\" as well as numerous other works on **Vedic** ritual. S\u0101ya\u1e47a lived within the **Vijayan\u0101gara** Empire.\n\n**SCHEDULED CASTES.** Official Indian government term for the lower **castes** \u2014that is, the **Dalits** , **Harijans** , or **Untouchables** , as these communities are variously known. This term refers to the fact that these castes have been included in an official list or _schedule_ indicating that they are eligible for specific government benefits and privileges due to their traditionally oppressed status.\n\n**SCRIPT.** Writing system; \"alphabet.\" Writing has existed in India since the period of the **Indus Valley civilization** , which was at its height from roughly 2600 to 1900 BCE. The script of this civilization, however, remains undecipherable today. The next script to emerge, called _Br\u0101hm\u012b_ , can be found on inscriptions from as early as the seventh century BCE. The famous edicts of the **Maurya** emperor A\u015boka are carved in Br\u0101hm\u012b script and date from the third century BCE. Around the fifth century BCE, the _Kharo\u1e63\u1e6dh\u012b_ script also comes into use. Early versions of _Devan\u0101gar\u012b_ , the script used today in **Sanskrit** , **Hindi** , Marathi, and some other northern Indian languages, come into use by the fourth century of the Common Era, and are clearly based on the earlier Kharo\u1e63\u1e6dh\u012b and Br\u0101hm\u012b scripts.\n\n**SCRIPTURES.** Hindu sacred **literature** consists of a vast collection of texts. There is no single authoritative work that all Hindus regard as divinely inspired, equivalent to the Bible for **Christianity** or the Qur'an for **Islam**. There are, instead, many such works, whose relations to one another are quite complex and that are regarded in a variety of ways by different groups of Hindus. The most basic division of Hindu scripture is that between **\u015br\u016bti** and **sm\u1e5bti**. These terms refer, respectively, to \"that which is heard\" and \"that which is remembered.\" The \"heard\" texts\u2014the \u015br\u016bti\u2014are believed to be direct, divine revelations perceived by the ancient **Vedic** sages, or **\u1e5b\u1e63is**. They are _apauru\u1e63eya_ or \"non-man-made.\" The \"remembered\" texts\u2014or sm\u1e5bti\u2014are not directly revealed but are made up of traditional **knowledge** passed down from the ancient past. Their authority is derivative from that of the \u015br\u016bti, whose basic teachings they communicate through popular narrative. The \u015br\u016bti is the Vedic literature, while the sm\u1e5bti consists of all subsequent Hindu texts.\n\n**SELF.** _See_ \u0100TMAN.\n\n**SELF-REALIZATION FELLOWSHIP (SRF).** Religious organization established by **Paramah\u0101\u1e43s\u0101 Yog\u0101nanda** in 1920 to disseminate the practice of _Kriya_ _Yoga_ , into which he had been initiated by his **guru** , **Sri Yukte\u015bwar Giri**. In 1952, after Yog\u0101nanda's death, Sw\u0101m\u012b Janak\u0101nanda (formerly James Lynn) took over the leadership of the organization. He was succeeded in 1955 by \u015ar\u012b Day\u0101 M\u0101t\u0101, who continues to head the organization. Among the distinctive aspects of the Self-Realization Fellowship (or SRF, as it is widely known) is its emphasis on devotion to Jesus as an enlightened master and its teaching of the unity of science and religion. Its membership consists primarily of Westerners. It resembles the **Ved\u0101nta Society** and **Siddha Yoga** in its inclusion of Western initiates as monks and nuns. Its main headquarters are located in Los Angeles, California.\n\n**SEN, KESHUB CHUNDER (KE\u015aAVA CANDRA) (1838\u20131884).** Nineteenth-century **Bengali** Hindu reformer; member of the **Brahmo Sam\u0101j** ; known for his open admiration of **Christianity** and his concept of a \"New Dispensation\" that would involve a synthesis of Christian and Hindu elements, and ultimately elements of all religions. Sen's ideas led him to split with the Brahmo Sam\u0101j, starting a new organization called the Brahmo Sam\u0101j of India in 1866. As Sen's thought continued to evolve, he began to take his organization in a more traditionally Hindu direction\u2014very likely under the influence of **Ramakrishna** , whom he met in 1876. A number of his followers, unhappy with his perceived deviations from the original ideals of the Brahmo Sam\u0101j, split away from the Brahmo Sam\u0101j of India to form the Sadharan Brahmo Sam\u0101j in 1878. Sen played an important role in bringing Ramakrishna to the attention of a wider audience of Bengali intellectuals.\n\n**\u015aE\u1e62A.** The cosmic serpent on whose belly **Vi\u1e63\u1e47u** rests. When Vi\u1e63\u1e47u becomes incarnate as an **avat\u0101ra** , \u015ae\u1e63a frequently becomes incarnate as well, to act as Vi\u1e63\u1e47u's protector. So when Vi\u1e63\u1e47u was born as **R\u0101ma** , \u015ae\u1e63a became his brother **Lak\u1e63ma\u1e47a**. Similarly, when Vi\u1e63\u1e47u was born as **K\u1e5b\u1e63\u1e47a** , \u015ae\u1e63a, according to some accounts, was born as **B\u0101lar\u0101ma** , his brother.\n\n**SEVEN SAGES.** _See_ SAPTAR\u1e62I.\n\n**SHANKAR, RAVI (1920\u2013 ).** Indian classical musician instrumental in popularizing the **musical** traditions of India\u2014especially the sitar\u2014to the attention of the Western world. In his youth, he traveled frequently to North America and Europe as part of a **dance** troupe run by his brother, Uday Shankar. In 1938, he gave up dancing to learn the sitar, which is the instrument for which he is best known. Though Indian **music** had long been enjoyed by a small segment of the Western public, Shankar's association with **George Harrison** of the Beatles, starting in 1966, brought him, and Indian classical music more generally, to a much wider audience.\n\n**SHARMA, ARVIND.** Prolific scholar of Hinduism and professor of religious studies at McGill University, in Montreal, Canada. Though not referring to himself explicitly as a Hindu theologian, much of Sharma's work has a constructive theological character that seeks to reinterpret Hindu thought in an effort to show its relevance to the contemporary world. His notable efforts in this regard include his work on Hinduism and human rights and his reinterpretation of the **caste** system, which argues that all Hindus have a duty to study their **scriptures** and **philosophies** (the traditional work of a **Brahmin** ), to stand up for what is right (the traditional work of a **K\u1e63atriya** ), to be economically productive (the traditional work of a **Vai\u015bya** ), and to serve humanity (the traditional work of a **\u015a\u016bdra** ).\n\n**SHETTY, MALTI (1955\u2013 ).** _See_ GURUMAY\u012a.\n\n**SHIRDI SAI BABA (?\u20131918).** Also known as Sai Baba of Shirdi, Shirdi Sai Baba was a spiritual teacher of the 19th and early 20th centuries with a large following of both Hindus and Muslims. His teaching is evocative of that of the **Sant** movement, with elements of both **Advaita Ved\u0101nta** and **Sufism**. The contemporary teacher, **Sathya Sai Baba** , claims to be a reincarnation of Shirdi Sai Baba\u2014a claim not accepted by all Shirdi Sai Baba devotees.\n\n**SIDDHA.** Perfected being; one who has attained perfection; a person who has achieved **liberation** from **rebirth** ( **mok\u1e63a** ) in this lifetime; a **j\u012bvanmukta**. Though used in many Indic traditions, this term is associated most prominently with **Tantra** and **Jainism**.\n\n**SIDDHA YOGA.** Contemporary **Tantric** tradition with roots in **Ka\u015bm\u012br \u015aaivism**. The practice of Siddha Yoga focuses primarily upon **meditation** and the relationship between the **guru** and the disciple. Initiation ( **d\u012bk\u1e63\u0101** ) into Siddha Yoga occurs through the process of **\u015baktipat** , whereby the guru raises the **\u015bakti** or **ku\u1e47\u1e0dalin\u012b** energy residing at the base of the spine of the aspirant through the various **cakras** to the **sah\u0101srara cakra** at the top of the head. It is believed that the guru can do this through a touch, word, or thought. In the Siddha Yoga tradition, it is believed that \u015baktipat can accelerate the aspirant's journey to **mok\u1e63a** , or **liberation** from the cycle of **rebirth**.\n\nThough having more ancient roots, the modern Siddha Yoga tradition begins with the figure of **Sw\u0101m\u012b Nity\u0101nanda** (1897\u20131961), the guru of **Sw\u0101m\u012b Mukt\u0101nanda** (1908\u20131982), who brought the tradition to the West, where it continues to have a large following today. The current head of the lineage, **Sw\u0101m\u012b Chidvil\u0101s\u0101nanda** , more popularly known as **Gurumay\u012b** , is notable for being one of only a handful of female gurus who head large Hindu-inspired global organizations. Siddha Yoga is based at **Ga\u1e47e\u015bapuri \u0100\u015brama** , near Bombay (Mumbai), the site of Nity\u0101nanda's spiritual retreat in his later years. There is a large Siddha Yoga **\u0101\u015brama** near South Fallsburg, New York, as well.\n\n**SIDDH\u0100NTA.** Perfect teaching; final goal; settled opinion; term by which most systems of Indian philosophy ( **dar\u015bana** ) refer to themselves in the context of debate.\n\n**SIDDHI.** \"Perfection,\" \"success,\" \"accomplishment\"; paranormal power cultivated as a result of spiritual practice such as **asceticism** and **meditation**. Hindu traditions are, on the whole, ambivalent toward such powers, with some texts and teachers explicitly endorsing specific practices aimed at producing particular siddhis and others dissuading practitioners from being distracted by such powers from the ultimate goal of **mok\u1e63a**.\n\nAlso, along with **Buddhi** (\"Intellect\"), Siddhi (\"Success\") is the name of one of the two wives of **Gane\u015ba**. _See also_ TANTRA; YOGA.\n\n**SIKHISM.** Highly egalitarian religious tradition that emerged from the **Sant** movement and the interplay between Hinduism and **Islam** during the **Mughal** period; founded by **N\u0101nak** (1469\u20131539), better known by his disciples as Guru N\u0101nak. Born a Hindu in the Punjab, N\u0101nak claimed to have received a revelation from **God** while **meditating** in a forest. N\u0101nak's message was: \"There is no Muslim and there is no Hindu.\" Those who accepted his teaching of Hindu\u2013Muslim unity came to be known as _Sikh_ , a Punjabi word that means \"disciple,\" and they took N\u0101nak as their **guru**.\n\nGuru N\u0101nak and his followers sought to harmonize the teachings of Islam and Hinduism, taking from Islam its strong emphasis on the unity of God. Guru N\u0101nak called **the One** God, known by Hindus under many names and forms, _Sat N\u0101m_ , or the \"True Name,\" or _Ek Onkar_ , the One God. The One God is the creator of the entire universe, according to Sikhism, and humans are the highest of God's creations. From Hinduism, N\u0101nak took the doctrines of **karma** and **rebirth** and the emphasis on the importance of the guru, or living teacher. **Liberation** from rebirth is given by God, much as in **bhakti-** based, devotional forms of Hinduism. The community Guru N\u0101nak established accepted all, regardless of religion, gender, or **caste** , a teaching that is illustrated in the Sikh practice of the _langar_ , a meal in which all participate and dine together (inter-dining being prohibited in N\u0101nak's time between members of the various castes and religious communities).\n\nN\u0101nak was succeeded by a series of nine gurus. Under each guru, suspicion of the new movement grew within the Mughal state, culminating in fierce persecution by the Mughal emperor Aurangzeb. The 10th and final Sikh guru, **Gobind Singh** (1666\u20131708), responded to this persecution by militarizing the Sikh community, establishing the _khalsa_ , or order of Sikh warriors, marked by the \"Five K's,\" or five items whose names, in Punjabi, start with the letter K: _ke\u015b_ , or long hair; _kangha_ , a special comb with which the hair is tied on top of the head; _kara_ , a circular steel bracelet; _kacha_ , an undergarment ensuring preparedness for battle; and the _kirpan_ , a sword or dagger. Gobind Singh's father, Guru Tegh Bahadur, had been beheaded by Aurangzeb, and Gobind Singh himself had to spend much of his time in hiding. Knowing that he might not survive in the ongoing conflict, he designated the sacred text of the Sikhs, the _\u0100di Granth_ , the perpetual guru of the community after his **death**.\n\nIn the post-independence period, some Sikhs began to agitate for a Sikh state that would be known as _Khalistan_. This agitation escalated into violence throughout the late 1970s, culminating in an Indian army attack in 1984 on the Golden Temple of Amritsar, where the original _\u0100di Granth_ is housed. This, in turn, provoked the assassination of Prime Minister Indira G\u0101ndh\u012b by her Sikh bodyguards, which provoked anti-Sikh riots.\n\nSince 1984, relations between Sikhs and the Hindu majority of India have been improving, with Manmohan Singh\u2014a Sikh\u2014being elected to the office of prime minister of India in 2004.\n\nWorldwide, there are approximately 23,000,000 Sikhs today. Roughly 80 percent of Sikhs live in India\u2014and specifically, in the Punjab. The other 20 percent live both in other parts of India and throughout the rest of the world, with relatively large populations in the United States, Great Britain, and Canada.\n\n**SINDHU.** **Sanskrit** name of the **Indus** River that flows through what is now Pakistan, and around which the **Indus Valley civilization** flourished from approximately 2600 to 1900 BCE. The Persian pronunciation of this word is the source of the term _Hindu_.\n\n**SINGH, SW\u0100M\u012a SHIV DAYAL (1818\u20131878).** Founder of the **R\u0101dh\u0101so\u0101m\u012b Satsang**.\n\n**\u015aI\u1e62YA.** Student; disciple or follower of a **guru** , or spiritual teacher.\n\n**S\u012aT\u0100.** Character in the _R\u0101m\u0101ya\u1e47a_ ; wife of **R\u0101ma** and human incarnation, or **avat\u0101ra** , of the **goddess Lak\u1e63m\u012b** ; daughter of the sage King **Janaka**. R\u0101ma wins the right to marry S\u012bt\u0101 by completing a nearly impossible task set by her father\u2014the stringing of **\u015aiva** 's bow, which R\u0101ma not only strings successfully but actually breaks. Happily marrying R\u0101ma and returning with him to **Ayodhy\u0101** , she later suffers exile with him when he is sent into the wilderness at the command of his father, **Da\u015baratha** , who is forced into sending his son away by one of his wives, Kaikey\u012b, who fears that her own son\u2014R\u0101ma's half-brother, **Bharata** \u2014will be marginalized if R\u0101ma becomes king. While in exile with R\u0101ma and his brother **Lak\u1e63ma\u1e47a** , S\u012bt\u0101 is abducted by the **r\u0101k\u1e63asa** lord, **R\u0101va\u1e47a** , who takes her to his capital on the island of **La\u1e45ka**. During her imprisonment, R\u0101va\u1e47a seeks to woo her, but she remains steadfastly loyal to R\u0101ma. When R\u0101ma and his allies finally free her, R\u0101ma does not immediately embrace her but forces her to pass a test of fire in order to prove that she did not betray him while in captivity. This distressing episode is explained by the fact that R\u0101ma is concerned that, as king, he and his family must be above reproach in the eyes of the public. At a later point in the story\u2014after the family has returned and ruled in Ayodhy\u0101 for many years\u2014members of the public begin to question her fidelity and S\u012bt\u0101 again has to pass through the fire test. At this point, she leaves the city and goes to the wilderness, raising her two sons, Lava and **Ku\u015ba** , in seclusion.\n\nThe ill treatment of S\u012bt\u0101 after her **liberation** from R\u0101va\u1e47a has been a source of a good deal of controversy within the Hindu tradition. On the one hand, S\u012bt\u0101 is held up as a model Hindu wife, ever faithful and obedient to her husband even in the worst of times. At the same time, the repeated demands that she prove her fidelity are clearly demeaning. Alternative versions of the _R\u0101m\u0101ya\u1e47a_ vary a good deal in their approaches to this issue. The most popular variant\u2014the **Hindi** _R\u0101m Carit M\u0101nas_ of **Tuls\u012bd\u0101s** \u2014claims that the scene never really occurred: that the real S\u012bt\u0101 was never captured by R\u0101va\u1e47a and that only an etheric double appeared to undergo the fire test. _See also_ WOMEN.\n\n**\u015a\u012aTALA.** \"Cold **goddess** \"; goddess associated with smallpox; sometimes believed to be a form of **K\u0101l\u012b** ; especially popular in **Bengal**.\n\n**\u015aIVA.** \"Good, benevolent\"; alongside **Vi\u1e63\u1e47u** and **\u015aakti** , one of the most popular deities of Hinduism; regarded by many of his devotees, known as **\u015aaivas** , as **God** in something like a **monotheistic** sense\u2014the supreme deity. \u015aiva is an ancient deity whom some argue was worshiped, alongside the **Mother Goddess** , in the **Indus Valley civilization**. This argument is based on the resemblance of a deity depicted in some Indus Valley carvings to \u015aiva in his forms as **Pa\u015bupati** , or Lord of the Animals, and **Yoge\u015bvara** , Lord of **Yoga**. This identification cannot be verified, however, until the **script** of the Indus Valley is deciphered. \u015aiva is also identified with the **Vedic** storm deity, **Rudra**. Rudra, a fierce deity of the wilderness, shares with \u015aiva the attribute of living far from society. Rudra is also depicted as a hunter, and another epithet of \u015aiva is\u2014as just mentioned\u2014the Lord of the Animals. And \u015aiva, like Rudra, can have a fierce temper when roused to anger. Many of the prayers in the _\u1e5ag Veda_ devoted to Rudra entreat him to keep his temper in check and to keep his destructive activities far from the area of human activity. _\u015aiva_ likely became an epithet of Rudra due to prayers that refer to him in this way\u2014\"good, gentle, kind\"\u2014in an effort to pacify the deity and keep him in a calm and gentle mood.\n\nBy the time of the _Pur\u0101\u1e47as_ , in the first millennium of the Common Era, \u015aiva is generally depicted in a way that lives up to his name\u2014as a gentle, kind, and generally \"laid back\" deity. But he can still be roused to fierce anger, such as when he destroys the sacrifice of **Dak\u1e63a** or when he burns **K\u0101madeva** to a crisp with a blast of energy from his **third eye**. He is typically remorseful after such outbursts, however, such as when he gives Dak\u1e63a the head of a ram after beheading him in anger, and when he gives his son **Ga\u1e47e\u015ba** the head of the elephant **Air\u0101vata** after beheading him. As these last two examples indicate, \u015aiva is a destroyer and a creator\u2014destroying old forms to make way for new ones. This process, and \u015aiva's role as both destroyer and re-creator of the universe in an ongoing cycle, is illustrated in the popular image of \u015aiva dancing the cosmic **dance** of **creation** and destruction\u2014\u015aiva as \"Lord of the Dance\" ( **N\u0101\u1e6dar\u0101ja** ).\n\n\u015aiva and his wife, \u015aakti, the Mother Goddess, form a composite deity, with \u015aakti being the power of creation and \u015aiva the pure consciousness that underlies it. Their two children, Ga\u1e47e\u015ba and **Karttikeya** , are popular deities in their own right but are also often worshiped in conjunction with their parents. Both \u015aaiva and \u015a\u0101kta worship often have a **Tantric** character.\n\n**\u015aIV\u0100J\u012a (1627\u20131680).** Leader of the **Mar\u0101\u1e6dhas** , a warrior people of Maharashtra, in central India (the current location of Bombay, or Mumbai), who successfully fought against the **Mughal Empire** in the late 17th century, during the period of Emperor Aurangzeb. Led by \u015aiv\u0101ji, who even today remains a folk hero in Maharashtra, the Mar\u0101\u1e6dhas utilized guerilla tactics against the numerically superior and better equipped Mughals. Stunningly successful in their campaign, the Mar\u0101\u1e6dhas eventually controlled most of central India's Deccan plateau.\n\n**\u015aIVAR\u0100TRI.** \" **\u015aiva** 's night\"; the 13th night and 14th day of each month of the **Hindu calendar** is sacred to the deity \u015aiva and is therefore known as \"\u015aiva's night,\" observed by devotees through fasting. The \u015aivar\u0101tri of the month of **Phalguna** is especially sacred and is also known as \"\u015aiva's great night\" ( **Mah\u0101\u015bivar\u0101tri** ).\n\n**SKANDA.** Alternative name for **K\u0101rttikeya** , the son of **\u015aiva** and **P\u0101rvat\u012b** and brother of **Ga\u1e47e\u015ba**.\n\n**SM\u0100RTA.** Along with the **Vai\u1e63\u1e47ava** , **\u015aaiva** , and **\u015a\u0101kta** traditions, one of the four major streams of tradition that make up Hinduism. The Sm\u0101rta tradition, whose name means \"derived from the **sm\u1e5bti** ,\" is a continuation of the ancient tradition of **Brahmanism**. It sees itself as nonsectarian, in contrast with the often exclusive adherence of other Hindu traditions to a single deity, such as **Vi\u1e63\u1e47u** , **\u015aiva** , or **\u015aakti**. The Sm\u0101rta **Brahmins** follow the directives of **\u015aa\u1e45kara** , worshiping by means of the **Pa\u00f1cay\u0101tana P\u016bj\u0101** , or \"fivefold worship\" dedicated to **Ga\u1e47e\u015ba** , \u015aiva, \u015aakti, Vi\u1e63\u1e47u, and **S\u016brya** , the sun **god**. Otherwise, they adhere closely to **Vedic** customs as understood by the **P\u016brva M\u012bm\u0101\u1e43s\u0101** system of philosophy ( **dar\u015bana** ). They are orthodox Brahmins, whose lifestyle follows the Hindu **\u015b\u0101stras** quite closely, but whose nonsectarian outlook has also been a major influence on **Neo-Ved\u0101nta** , with its emphasis on religious pluralism and many effective paths to **God-realization**.\n\n**SM\u1e5aTI.** \"Remembered\"; along with **\u015br\u016bti** (\"that which is heard\"), one of the two main categories of Hindu **scripture**. Sm\u1e5bti is sacred tradition. Although not directly revealed by **God** or heard by the **\u1e5b\u1e63is** or sages of the distant past while deep in **meditation** , it has a derivative authority, being based upon such authoritative sources and attempting to make their **knowledge** available to the wider Hindu society. The sm\u1e5bti **literature** consists of a vast collection of texts\u2014essentially, all authoritative Hindu literature other than the **Vedic** corpus, which constitutes the \u015br\u016bti. As such, the sm\u1e5bti includes the historical epics\u2014that is, the _R\u0101m\u0101ya\u1e47a_ and _Mah\u0101bh\u0101rata_ \u2014the _Pur\u0101\u1e47as_ , the various **s\u016btras** of the **dar\u015banas** , or philosophical systems, the _Dharma \u015a\u0101stras_ , and the whole range of \u015b\u0101stra literature on a whole host of topics, such as medicine, **astrology** , economics, political science, grammar, aesthetics, sexuality, the construction of **temples** and sacred images, and so on. Most of this literature is composed in **Sanskrit** , but some of it, such as the devotional **poetry** of the **bhakti** movement, is in vernacular languages.\n\n**SOMA.** **Vedic** deity of the **moon** , who also represents the hallucinogenic plant whose juice, mixed with milk, was used by ancient **Brahmins** to evoke an experience of the divine. The phases of the moon represent this substance being consumed by the **devas** (during the **dark half** of the month, when the moon is waning) and replenished (during the **bright half** , when the moon is waxing). There has been a great deal of speculation on the actual plant that was pressed and whose juice was used to produce the soma drink. It has been suggested by some that it was a mushroom grown in what is now Afghanistan. Others suggest a leafy plant that has either become extinct or whose properties have been forgotten. **Indra** is depicted as being particularly fond of consuming this drink. Many hymns in the _\u1e5ag Veda_ are devoted to this deity.\n\n**SOMAV\u0100RA.** Monday, the day of the week that is governed by the **moon**. _See also_ CALENDAR, HINDU.\n\n**SOMN\u0100TH (SOMAN\u0100THA).** Lord of the **Moon** ; epithet of **\u015aiva** , so called in part due to the fact that he is often depicted wearing a crescent moon on his forehead. A large and exceedingly wealthy **temple** dedicated to \u015aiva in this form was located in Gujarat in the first millennium of the Common Era. It was destroyed by the invading army of Mahmud of Ghazni in 1026, who made its seven-foot stone **Li\u1e45ga** , or abstract image of \u015aiva, part of the threshold of a mosque, to be stepped upon by Muslim worshipers, in a gesture of contempt for Hinduism. For this reason, the temple has become a symbol to Hindus\u2014and especially to **Hindu Nationalists** \u2014of insults suffered under medieval **Islamic** rule. The temple was repeatedly rebuilt and destroyed by Muslim rulers throughout its history\u2014in 1297, 1394, and 1706. Its most recent rebuilding was in 1950.\n\n**SOUL.** _See_ J\u012aVA; \u0100TMAN.\n\n**SOUTHEAST ASIA.** The region today made up of such nation states as Thailand, Laos, Cambodia, Vietnam, Burma, Malaysia, Singapore, and Indonesia. This area was heavily influenced by both Hinduism and **Buddhism** during the first millennium of the Common Era. The influence was largely peaceful, with Indian culture being viewed as prestigious and therefore consciously imitated by the royal families of the region. Examples of this influence include the popularity of the _R\u0101m\u0101ya\u1e47a_ and _Mah\u0101bh\u0101rata_ in this region, the adoption of **Sanskrit** names and royal titles, Hindu and Buddhist **temples** (which often take the form of composite Hindu\u2013Buddhist temples) like **Angkor Wat** in Cambodia, the continuing practice of Hinduism as the predominant religious tradition of **Bali** , Indonesia, today, and the heavy influence of **Sanskrit** and P\u0101li on the languages of these countries. The Indic cultural influence on this region has been so great that some scholars refer to it as \"Greater India.\"\n\n**SPANDA.** Vibration. According to **\u015aaiva** thought, vibration is the means by which the cosmos emerges from **\u015aiva** 's pure consciousness. The various elements that make up the physical universe ( **mah\u0101bh\u016btas** ), such as **earth** , **water** , **air** , **fire** , and the etheric element, **\u0101k\u0101\u015ba** , can all be seen as different \"wavelengths\" of the primordial vibration. _See also_ O\u1e42.\n\n**SPHO\u1e6cA.** A theory of **Sanskrit** grammar developed by **Bhart\u1e5bhari** (6th\u20137th century CE). According to this theory, the letters or sounds that make up a word have a special power to manifest the meaning of that word. This power to make meaning is the true essence of a word. The letters and sounds that make up the word are merely the means by which this true essence is revealed.\n\n**\u015aRADDH\u0100.** Faith; according to the _Bhagavad G\u012bt\u0101_ , \u015braddh\u0101 is a basic commitment that could be said to constitute a person.\n\n**\u015aR\u0100DDHA.** Periodic **ancestral rite** that consists of offering food and water to the dead. If one does not offer \u015br\u0101ddha to one's ancestors\u2014typically in the form of balls of rice called **pi\u1e47\u1e0da** \u2014one may run the risk that they will wander through the world as **pretas** , or hungry ghosts. The food offered in the \u015br\u0101ddha is also said to become the food of one's ancestors in their next **rebirth**.\n\n**\u015aRAMA\u1e46A.** Striver; member of a **renouncer** movement that distinguishes itself from the **Brahmins** by its belief that one's spiritual status is a function of one's personal effort, to which birth **caste** , or **j\u0101ti** , is irrelevant. The \u015brama\u1e47a movement likely emerged during the first millennium BCE, but may be even older than this, or based on older antecedents. This movement consisted of many distinct groups, each of which was centered on a spiritual teacher believed to have attained **enlightenment** and **liberation** from the cycle of **rebirth**. The only two \u015brama\u1e47a traditions to survive to the present day are **Buddhism** and **Jainism** , though other \u015brama\u1e47a groups, such as the **\u0100j\u012bvikas** , existed in the ancient period. Though initially antagonistic to **Brahmanism** and to aspects of **Vedic** thought and practice\u2014particularly **animal sacrifice** and the belief that birth caste is a measure of spiritual evolution\u2014the \u015brama\u1e47a traditions exerted considerable influence upon Hinduism, especially in regard to the practice of **vegetarianism** and later critiques of birth caste.\n\n_\u015aRAUTA S\u016aTRAS_ **.** **Sanskrit** texts, likely composed around 500 BCE, that describe in detail the correct way to perform a **Vedic** ritual of sacrifice ( **yaj\u00f1a** ).\n\n**\u015aRAVA\u1e46A.** Month of the **Hindu calendar** that corresponds roughly to the period from mid-July to mid-August. The word _\u015brava\u1e47a_ also refers to \"listening\" or \"hearing.\"\n\nAlso the name of the son of a **Brahmin** in the _R\u0101m\u0101ya\u1e47a_ who is accidentally shot and killed by **R\u0101ma** 's father, King **Da\u015baratha**. Da\u015baratha, hunting in the forest, mistakes \u015arava\u1e47a for a deer. The boy's father, enraged, curses Da\u015baratha to suffer the loss of his son as well\u2014a curse brought to fulfillment when Da\u015baratha is forced to disinherit R\u0101ma and banish him to 14 years of exile. Da\u015baratha dies from sorrow as a result of this event\u2014and ultimately, as a result of the angry Brahmin's curse.\n\n**\u015aR\u012a.** Prosperity; that which is auspicious. Name of an ancient **Vedic goddess** mentioned once in the _\u1e5ag Veda_ and many times in the _Atharva Veda_ and the _Br\u0101hma\u1e47as_. She is the embodiment of success of any kind. By the time of the _Pur\u0101\u1e47as_ (the first millennium of the Common Era), she is identified with **Lak\u1e63m\u012b** , wife of **Vi\u1e63\u1e47u** , though some sources suggest these two were, at one point, two distinct goddesses.\n\n**SRI AUROBINDO.** _See_ GHOSE, AUROBINDO.\n\n**SRI CHINMOY (1831\u20132007).** Modern **Bengali** Hindu spiritual teacher with a following in the West. In 1964, he immigrated to New York City, where he remained based for the rest of his life. His teaching is very much in line with **Neo-Ved\u0101nta** , emphasizing social service, the practice of **meditation** , and respect for all religions as \"essentially divine.\" Sri Chinmoy was deeply influenced by the teaching of **Aurobindo Ghose** , whom he cited frequently in his own teaching.\n\n**SRI LANKA.** Island country off the southeast coast of India, near **Tamil** Nadu; widely identified with the **La\u1e45ka** of **R\u0101va\u1e47a** in the _R\u0101m\u0101ya\u1e47a_ ; home to what is likely the most ancient continuously existing **Buddhist** community in the world. The Buddhist majority of Sri Lanka speaks Si\u1e45hala, or Sinhalese. From 1983 to 2009, a fierce civil war raged between the Sinhalese majority and the Tamil-speaking Hindu minority, which sought to have an independent state in the northern part of the island. This tragic and destructive conflict, often cast in religious terms, appears to have been more of an ethnic struggle.\n\n**SRI RAMAKRISHNA.** _See_ RAMAKRISHNA.\n\n**\u015aR\u012aRA\u1e44GAM.** Massive **temple** complex in **Tamil** Nadu and center of the **\u015ar\u012bvai\u1e63\u1e47ava** tradition.\n\n**\u015aR\u012aVAI\u1e62\u1e46AVA.** Of or pertaining to **\u015ar\u012bvai\u1e63\u1e47avism**.\n\n**\u015aR\u012aVAI\u1e62\u1e46AVISM.** Highly popular **Vai\u1e63\u1e47ava** tradition found mainly in southern India and traced to the devotional **poetry** of the **\u0100\u1e37v\u0101rs** , or **Tamil-** speaking **bhakti** poets of the period from approximately 600 to 900 CE. A **Brahmin** scholar named N\u0101thamuni introduced the poetry of the \u0100\u1e37v\u0101rs into the ritual of formal **temple** worship in the 10th century. His grandson, **Y\u0101muna** , translated many of the poems into **Sanskrit** and continued the process of formalizing and systematizing the tradition. This process was completed by Y\u0101muna's student **R\u0101m\u0101\u1e47uja** (1077\u20131157). The name of the tradition is an expression of the fact that the **goddess \u015ar\u012b** , or **Lak\u1e63m\u012b** , is regarded in this tradition as inseparable from **Vi\u1e63\u1e47u** , with whom she forms a composite deity (not unlike **\u015aiva** and **\u015aakti** in **\u015aaivism** and **\u015a\u0101ktism** ).\n\nAfter R\u0101m\u0101\u1e47uja, this tradition experienced a division over an issue analogous to the issue of the relationship of faith and works in **Christianity**. The Va\u1e6dakalai school holds that some human effort is involved in the process of achieving **mok\u1e63a** , or **liberation** from the cycle of **rebirth**. This school likens the human aspirant for mok\u1e63a in relation to **God** as similar to a baby monkey in relation to its mother. The mother carries the baby, but the baby must also exert effort by clinging to its mother. The Te\u1e45kalai school, on the other hand, likens the human aspirant to a baby kitten, whose mother carries it without any effort being made by the kitten. (Theologically, then, the Va\u1e6dakalai school could be likened to Roman Catholicism and the Te\u1e45kalai school to Protestantism.)\n\n**\u015aR\u012aVIDY\u0100 TANTRA.** System of Tantric worship focused on the **Mother Goddess**. It was systematized by **Bh\u0101skarar\u0101ya Makhin** (1690\u20131785), a **\u015a\u0101kta** theologian from the region of Maharashtra, in central India. Unlike other \u015a\u0101kta systems of Tantric worship, \u015ar\u012bvidy\u0101 Tantra focuses on the Goddess's benign and beautiful aspects, rather than on her terrifying ones. It is predominantly a **Right-Handed** system of **Tantra**.\n\n**\u015a\u1e5a\u1e44GER\u012a.** Site of one of the four **ma\u1e6dhas** , or monasteries established by **\u015aa\u1e45kara** (788\u2013820) at the four corners of India. \u015a\u1e5b\u1e45ger\u012b, in **Tamil** Nadu, is in the southern part of the country.\n\n**\u015aR\u016aTI.** \"That which is heard\"; along with **sm\u1e5bti** (\"that which is remembered\"), one of the two main categories of Hindu **scripture**. \u015ar\u016bti is believed to be directly revealed by **God** (in theistic systems of Hinduism), or to simply be the sound of the cosmos heard by the **\u1e5b\u1e63is** or sages of the distant past while deep in **meditation**. It is _apauru\u1e63eya_ (non-man-made) and consists of the **Vedic** literary corpus.\n\n**STATES OF CONSCIOUSNESS.** _See_ CONCIOUSNESS, STATES OF.\n\n**STAGES OF LIFE.** _See_ \u0100\u015aRAMA.\n\n**STH\u016aLA \u015aAR\u012aRA.** Gross body; the physical body. _See also_ ANNA-MAYA-KO\u015aA; \u015aAR\u012aRA.\n\n**SUBRAMA\u1e46YA**. An alternative name for **K\u0101rttikeya** , the son of **\u015aiva** and **P\u0101rvat\u012b** and brother of **Ga\u1e47e\u015ba**.\n\n**SUBRAMUNIYASW\u0100M\u012a, SATGURU \u015aIV\u0100YA (1927\u20132001).** Born Robert Hansen in California, Subramuniyasw\u0101m\u012b is one of a handful of Western initiates into Hindu **ascetic** orders to have attained a high level of authority and respect in the broader community of Hindus. Trained in both Western and Indian **dance** , Subramuniyasw\u0101m\u012b was drawn to Hindu thought at an early age. He was particularly inspired by the writings of **Sw\u0101m\u012b Vivek\u0101nanda**. In 1947 he traveled to **Sri Lanka** where he met his **guru** , a Hindu monk named Yogasw\u0101m\u012b, who initiated him into his **\u015aaiva Siddh\u0101nta** lineage. He returned to the United States to spread Hindu teachings and established a **temple** in San Francisco 1957. In 1970, he relocated to the island of Kauai, in Hawaii, where he established an **\u0101\u015brama** and later a monastery and temple. He began to publish a journal in 1979 called _Hinduism Today_ , which continues to be a highly regarded source of information on Hinduism in the increasingly global Hindu community. By the time of his **death** , Subramuniyasw\u0101m\u012b had initiated several monks into his order. He has been succeeded by Satguru Bodhin\u0101tha Veylansw\u0101m\u012b.\n\n**SUBTLE BODY.** _See_ SUK\u1e62MA \u015aAR\u012aRA.\n\n**SUD\u0100M\u0100.** Boyhood friend of **K\u1e5b\u1e63\u1e47a**. When K\u1e5b\u1e63\u1e47a leaves his boyhood home to become king of **Dv\u0101rak\u0101** , the two lose contact. When Sud\u0101m\u0101, a poor man, falls upon especially hard economic times, his wife reminds him of his friendship with K\u1e5b\u1e63\u1e47a and suggests he go to K\u1e5b\u1e63\u1e47a for help. Traveling to Dv\u0101rak\u0101, K\u1e5b\u1e63\u1e47a immediately recognizes him and treats him with great kindness, letting him stay in his palace. Sud\u0101m\u0101 is so overwhelmed by all of this that he forgets the purpose of his journey. K\u1e5b\u1e63\u1e47a, however, understands Sud\u0101m\u0101's situation. By the grace of his wife, **Rukmi\u1e47\u012b** , who is an incarnation of **Lak\u1e63\u1e43\u012b** , **goddess** of wealth and good fortune, K\u1e5b\u1e63\u1e47a is able to save Sud\u0101m\u0101 miraculously from his poverty. Returning home, he unexpectedly finds that his house has been transformed into a palace and his family is dressed in the finest clothes and has abundant food.\n\n**\u015a\u016aDRA.** The fourth of the four **var\u1e47as** or **castes**. In terms of traditional occupation, this caste is made up of trades that involve menial labor and servitude. In the _Puru\u1e63a_ _Sukta_ of the _\u1e5ag Veda_ , this caste is said to have been formed from the feet of the cosmic person ( **puru\u1e63a** ). The practice of **Untouchability** arose from the fact that some of the duties of this caste are regarded as polluting, particularly those involved with the cremation of the dead or the cleaning of human waste. On some interpretations, the **Untouchables** (also called **Dalits** or **Harijans** ) are even beneath the **\u015a\u016bdras** in the caste hierarchy.\n\n**SUFI, SUFISM.** Mystical movement within **Islam**. The Sufi traditions often interpret the Qur'an allegorically, and sometimes skirt (or even violate) the boundaries of Islamic orthodoxy, particularly in regard to the question of mystical union with **God**. An ecstatic tradition of devotion, there are many affinities between the sensibilities of Sufism and the **bhakti** movement of Hinduism. Both find expression in songs and **poetry**. Both focus on a sense of exquisite longing for God as an essential religious emotion. And both tend to see the cultivation and presence of such longing as being of vastly greater importance than the niceties of either correct ritual practice or orthodoxy. As a consequence of these shared sensibilities\u2014and in the process, reinforcing them\u2014Sufis and practitioners of bhakti in medieval India frequently interacted quite closely, crossing religious boundaries with relative ease, with Sufi practitioners taking Hindus as their spiritual teachers, and Hindus also taking Sufi masters as their **gurus**. The interaction between Sufism and bhakti was a major catalyst in the gradual synthesis of Hindu and Islamic cultures during the **Mughal** period. This synthesis produced the **Sant** movement and **Sikhism**. _See also_ KAB\u012aR; N\u0100NAK, GURU.\n\n**\u015aUKA.** Son of the sage **Vy\u0101sa**. \u015auka was a great **ascetic** who attained **mok\u1e63a** ( **liberation** form the cycle of **rebirth** ) quite quickly, vanishing into the sun. (The path of the sun is presented in some sources as the route to liberation after **death** , with the alternative path, the path of the **moon** , leading to rebirth.) Vy\u0101sa missed his son greatly and was consoled by **\u015aiva** with the gift of an etheric double of his son to keep him company.\n\n**\u015aUKLA PAK\u1e62A.** The **bright half** of a Hindu month; the fortnight of a Hindu lunar month during which the **moon** is waxing; the period between **Amavasya** , or new-moon day (which is also the last day of the Hindu month) and **P\u016br\u1e47ima** , or full-moon day. The bright half of the month is generally considered a more auspicious time to undertake new ventures. _See also_ CALENDAR, HINDU.\n\n**\u015aUKRA.** The planet Venus. _See also_ ASTROLOGY; NAVAGRAHAS.\n\n**\u015aUKRAV\u0100RA.** Friday, the day of the week that is governed by the planet Venus. _See also_ CALENDAR, HINDU.\n\n**SUK\u1e62MA \u015aAR\u012aRA.** Subtle body. Subtle bodies occupy roughly the same space as the \"gross\" or physical body, but at a different energetic or spiritual wavelength. The subtle bodies correspond approximately to the **ko\u015bas** (layers or \"sheaths\") that cover the **\u0101tman** , or self: the **\u0101nanda-maya-ko\u015ba** (the layer of bliss), the **vij\u00f1\u0101na-maya-ko\u015ba** (the layer of consciousness), the **mano-maya-ko\u015ba** (the layer of mind), and the **pr\u0101\u1e47a-maya-ko\u015ba** (the layer of \"breath,\" or vital energy).\n\n**SUKTHANKAR, V. S. (1887\u20131943).** **Indologist** ; he served as the general editor of the Bhandarkar Oriental Research Institute's critical edition of the _Mah\u0101bh\u0101rata_ , which has been the authoritative version for scholars ever since.\n\n**SUMANTU.** One of the five **Brahmin** students of **Vy\u0101sa** , also known as **Veda Vy\u0101sa** , who is attributed not only with authoring the _Mah\u0101bh\u0101rata_ but also with organizing the _Vedas_ into their current distribution of four texts: the _\u1e5ag Veda_ , _Yajur Veda_ , _S\u0101ma Veda_ , and _Atharva Veda_. According to the _Pur\u0101\u1e47as_ , each of Vy\u0101sa's students was responsible for memorizing and passing on a separate **Vedic** text\u2014one for each of the four _Vedas_ and one for the _Mah\u0101bh\u0101rata_. Sumantu was the student who was responsible for transmitting the _Atharva Veda_.\n\n**SUN.** _See_ S\u016aRYA.\n\n**SUNDARE\u015aVARA.** \"Beautiful Lord\"; form of **\u015aiva** enshrined at the **temple** complex in **Madurai**. In this form, \u015aiva is the husband of **M\u012bn\u0101k\u1e63\u012b** , the \"fish-eyed **goddess**.\"\n\n**SURABH\u012a.** \"Good flavor\"; an epithet of **K\u0101madhenu** , the wish-fulfilling cow.\n\n**SURD\u0100S (c. 1479\u20131584).** Blind **bhakti** **poet** of the **Mughal** period; a devotee of **K\u1e5b\u1e63\u1e47a** and **R\u0101dh\u0101**. His songs reflect the **Vai\u1e63\u1e47ava** teachings of **Vallabha** (1479\u20131531), which emphasize complete surrender to the grace of K\u1e5b\u1e63\u1e47a. _See also_ POETRY.\n\n**S\u016aRYA. Vedic** deity or **deva** who is the personification of the sun. As in other **Indo-European** religious systems, the Vedic sun god rides a flying **chariot** through the sky, representing the sun's daily passage from east to west. In the _Mah\u0101bh\u0101rata_ , S\u016brya is the father of the character **Kar\u1e47a**. He is the object of prayer in the **G\u0101yatr\u012b mantra**. A massive **temple** to S\u016brya is located at **Konarak** , in the eastern Indian state of **Orissa**. S\u016brya is also one of five deities recommended by **\u015aa\u1e45kara** as an **i\u1e63\u1e6dadevat\u0101** , or chosen deity, in his system of **Pa\u00f1cay\u0101tana P\u016bj\u0101** , or \"five objects of worship\" (the other four deities being **Ga\u1e47e\u015ba** , **\u015aiva** , **\u015aakti** , and **Vi\u1e63\u1e47u** ).\n\n**S\u016aTA.** Charioteer; also a bard. Charioteers in ancient India also recited **poetry** , passing on stories such as those making up the epics.\n\n**S\u016aTRA.** Literally, \"thread\" (and cognate with the English word \"suture\"); the root text of a system of Indian philosophy ( **dar\u015bana** ). Traditionally, a s\u016btra is written in a style so terse as to require a commentary (or **bh\u0101\u1e63ya** ) in order to be comprehensible. It is likely that s\u016btras originated as mnemonic devices developed to preserve the oral teachings of the founding figure who established the system of thought to which a s\u016btra belonged. The intent would have been for students to memorize the s\u016btra and then have it explained by their teacher, who would \"unpack\" its meaning through a process of question and answer that became more formal and ritualized through the centuries. _See also_ LITERATURE.\n\n**SUTTEE.** Alternative spelling of **Sat\u012b** generally used by the British in the 19th century.\n\n**SV\u0100DHI\u1e62\u1e6cH\u0100NA CAKRA.** **Cakra** aligned with the navel. _See also_ TANTRA.\n\n**SVARGA.** Heaven; heavenly realm. Svarga is not an eternal everlasting paradise\u2014as in the Abrahamic religions, or the **Vaiku\u1e47\u1e6dha** of the **Vai\u1e63\u1e47ava** traditions of Hinduism. It is a temporary abode (though one in which a **soul** may reside for many eons) into which one is reborn as a consequence of meritorious **action** ( **pu\u1e47ya karma** ). One eventually dies in this realm and is reborn as a human being. _See also_ DEVALOKA.\n\n**SVAYA\u1e42VARA.** \"Own choice,\" ancient ceremony in which a **k\u1e63atriya** woman would choose her husband based on athletic prowess as demonstrated in a contest of arms. This ceremony figures prominently in the _R\u0101m\u0101ya\u1e47a_ and the _Mah\u0101bh\u0101rata_ , whose respective heroes win their brides by performing extremely difficult feats of strength and skill that no other warrior can master. _See also_ ARJUNA; DRAUPADI; R\u0100MA; S\u012aT\u0100.\n\n_\u015aVET\u0100\u015aVAT\u0100RA UPANI\u1e62AD_ **.** One of the principal _Upani\u1e63ads_ ; probably composed in the fourth century before the Common Era. The **philosophy** expressed in this text is close to that of the _Bhagavad G\u012bt\u0101_. But unlike the _G\u012bt\u0101_ , this text displays a **\u015aaiva** rather than a **Vai\u1e63\u1e47ava** orientation. More than most of the other principal _Upani\u1e63ads_ , the _\u015avet\u0101\u015bvat\u0101ra Upani\u1e63ad_ promotes devotion ( **bhakti** ) and reliance on the grace of **God** ( **\u012a\u015bvara** , **Rudra** , or **\u015aiva** , in this text) as the path to **liberation** from the cycle of **rebirth**. The name of the text comes from its purported author, whose name means, \"the man with a white mule.\"\n\n**SWADHYAYA PARIVAR.** Contemporary Hindu movement based mainly in Gujarat and Maharashtra. The movement was established in 1954 by Pandu-rang Shastri Athavale (1920\u20132003), popularly known as \"Dadaji,\" or \"elder brother\" within the movement. The primary focus of the movement is social service, study, and self-empowerment, based on Athavale's interpretation of the _Bhagavad G\u012bt\u0101_.\n\n**SW\u0100M\u012a (SV\u0100M\u012a, SV\u0100MIN).** \"Master\"; traditional title of a monk or **sanny\u0101sin**. _See also_ SANNY\u0100SA.\n\n**SW\u0100M\u012aN\u0100R\u0100YA\u1e46 (1781\u20131830).** An early modern Hindu teacher, also known as \u015ar\u012b Sahaj\u0101nand Sw\u0101m\u012b. After undertaking **sanny\u0101sa** or renunciation as a small boy, it is said that he traveled to the **Him\u0101layas** where he practiced **meditation** and **yoga**. He traveled extensively throughout India, eventually settling in Gujarat, where the movement based on his teachings continues to be most popular. This teaching is close to the **Vi\u015bi\u1e63\u1e6d\u0101dvaita** or **Qualified Non-Dualism** of **R\u0101m\u0101\u1e47uja** , combined with a strong emphasis on **ahi\u1e43s\u0101** , or nonviolence. Sw\u0101m\u012bnarayan's devotees believe him to be an **avat\u0101ra** of **K\u1e5b\u1e63\u1e47a**. _See also_ SW\u0100M\u012aN\u0100R\u0100YA\u1e46 MOVEMENT.\n\n**SW\u0100M\u012aN\u0100R\u0100YA\u1e46 MOVEMENT.** Movement based on **devotion** to and the practice of the teachings of **Sw\u0101m\u012bn\u0101r\u0101ya\u1e47** (1781\u20131830). The movement is especially popular in Gujarat, but has become global due to extensive emigration of Gujaratis out of India to various parts of the world. The movement, broadly speaking, is not a formally organized one. But an organization does exist, internal to the movement\u2014the Boc\u0101sanv\u0101s\u012b Ak\u1e63ar Puru\u1e63ottam Sw\u0101m\u012bn\u0101r\u0101ya\u1e47 Sansth\u0101 (BAPS).\n\n**SWASTIKA (SVASTIKA).** \"That which is auspicious\"; auspicious four-armed symbol of the **goddess Lak\u1e63\u1e43\u012b** , also sacred in both **Jainism** and **Buddhism** as a symbol of the fourfold communities of monks, nuns, laymen, and laywomen in those two traditions. In the popular imagination of the West, this symbol is associated with Nazism. This is due to the misappropriation by the Nazis and other white supremacist movements of the early 20th century of the **\u0100ryan Invasion Theory** , with its assumption of a correlation of culture and ethnicity. The **Indo-European** roots of **Vedic** culture were assumed to be the product of a \"pure\" white race from Central Asia or Eastern Europe, and Vedic culture, as the most ancient living Indo-European culture, to represent an early form of that race's culture. The Nazis therefore appropriated Vedic symbols such as the swastika based on the belief that their occult power could grant the Nazis victory in battle, connecting them with their ancient \"Aryan\" ancestors. _See also_ \u0100RYAN.\n\n**SYNCRETISM.** The tendency to combine elements of several different religions. It has often been remarked that syncretism is widespread among Hindus. Such syncretism can be intra-Hindu, consisting of combining elements from distinct Hindu traditions, such as **Vai\u1e63\u1e47avism** , **\u015aaivism** , and **\u015a\u0101ktism** , or it can draw from traditions other than Hinduism, such as when Hindus include images of Christ or **Christian** saints in their home altars, or pray at the tombs of **Sufi** saints. Syncretism can be practiced as a matter of principle, as an expression of a belief that all or many traditions are divinely revealed and can lead to salvation, or it can be more \"accidental,\" a function of ignorance of the original meaning of particular symbols or practices in the traditions from which they emerge historically, combined with a vague sense that these symbols or practices are holy and good. Both types of syncretism can be observed in Hindu practice.\n\n## T\n\n**TAGORE, DEBENDRAN\u0100TH (1817\u20131905).** Leader of the **Brahmo Sam\u0101j** starting in 1843. Technically Tagore was the founder of this organization, on the basis of the earlier _Brahmo Sabh\u0101_ of **R\u0101m Mohan Roy** , though Roy is widely credited with being the father of the Brahmo movement as a whole and with defining its basically Unitarian orientation. Tagore was deeply devout and put considerable energy into promoting his organization and its ideals. Under his leadership, the Brahmo Sam\u0101j widened its following and began to influence other, similar movements across India. His youngest son, **Rabindran\u0101th** , is widely revered, particularly in **Bengal** , for his considerable literary and **musical** talents.\n\n**TAGORE, RABINDRAN\u0100TH (1861\u20131941).** Nobel laureate; son of **Brahmo Sam\u0101j** leader **Debendran\u0101th Tagore**. The multitalented Rabindran\u0101th distinguished himself as a **poet** , novelist, songwriter, dramatist, essayist, and philosopher. He won the Nobel Prize for **literature** in 1913, making him the first non-European to win a Nobel Prize. He was knighted in 1915, but returned his knighthood in 1919 after British soldiers massacred hundreds of Indian independence activists\u2014including dozens of **women** and children\u2014at Amritsar in the Punjab. In 1916 and 1917 he undertook a world tour, visiting Japan and the United States. Controversially, he denounced nationalism of all kinds as inherently violent. He taught what he called the **philosophy** of \"universal man,\" arguing for the inherent unity of all beings\u2014an ancient Hindu theme. On this basis, he also wrote many plays and musical works that questioned the **caste** system, such as the opera _Ca\u1e47\u1e0d\u0101lik\u0101_ , which tells the story of a low-caste woman who becomes a **Buddhist** nun to escape the cruelty of caste. In the area of religion, he was long a skeptic, but adopted a view akin to **Neo-Ved\u0101nta** after he had a number of mystical experiences. He was drawn to religious figures like the **Bauls** and the **bhakti** poets who rebelled against social conventions. The **music** of these groups was highly influential upon his songwriting.\n\nIn 1901 Tagore established an **\u0101\u015brama** at a town called \u015aantiniketan. This \u0101\u015brama would later develop into a famous school. Tagore was deeply dedicated to the education and self-empowerment of the poor. He met **Mohand\u0101s K. G\u0101ndh\u012b** \u2014with the two being captured together in a famous photograph. Their relationship was cordial, but Tagore was critical of what he saw as the narrowness of G\u0101ndh\u012b's vision versus the universality to which he himself aspired. Today, Tagore is a universally beloved cultural hero in his native **Bengal** , as well as a globally respected literary figure.\n\n**TAMAS.** Inertia; the third of the three **gu\u1e47as** or qualities of phenomenal existence as delineated in the **philosophies** of **S\u0101\u1e43khya** and **Yoga** as well as in the _Bhagavad G\u012bt\u0101_.\n\n**TAMIL.** Southern Indian language of the **Dravidian** family; a particularly rich language in regard to Hinduism. Tamil is the language of the **bhakti** **poetry** of both the **Vai\u1e63\u1e47ava \u0100\u1e37v\u0101rs** and the **\u015aaiva N\u0101yan\u0101rs** and of such wisdom **literature** as the _Tirukku\u1e5ba\u1e37_. The region where the Tamil language is predominant\u2014the modern Indian state of Tamil Nadu\u2014is the site of numerous elaborate Hindu **temples** and was home to such major dynasties as the **Pallava** and **Co\u1e37a** \u2014both significant patrons of Hinduism.\n\n**TANJORE.** _See_ THA\u1e44JAV\u016aR.\n\n**TANM\u0100TRAS.** In **S\u0101\u1e43khya** **philosophy** and **Ved\u0101nta** , the subtle elements that give rise to the gross elements ( **mah\u0101bh\u016btas** ) that make up physical phenomena. Each tanm\u0101tra is equivalent to a mental state or sensation\u2014smell, touch, sight, taste, and sound. Each one of the tanm\u0101tras corresponds to a specific mah\u0101bh\u016bta: smell to earth ( **p\u1e5bthvi** ), touch to air ( **v\u0101yu** ), sight to fire ( **agni** ), taste to water ( **ap** ), and sound to space ( **\u0101k\u0101\u015ba** ).\n\n**TANTRA.** System of spiritual practice based on the principle that the senses, rather than being restrained, as in other forms of **yoga** , can be used to transcend themselves. Many forms of Tantra exist. Tantric practice became a prominent feature of Hindu practice in the eighth century of the Common Era, though its roots may be much more ancient than this date would indicate. Tantric forms of **Buddhism** also became prominent at this time.\n\nThere are two basic types of Tantric practice: **Right-Handed** and **Left-Handed** , or V\u0101m\u0101c\u0101ra. Right-Handed Tantra is relatively uncontroversial, its practices consisting of **meditation** and visualization exercises, the use of geometric diagrams such as **ma\u1e47\u1e0dalas** and **yantras** , the chanting of **mantras** , and **p\u016bj\u0101**. Left-Handed Tantra, on the other hand (pun intended), is widely regarded as perilous and tends to be viewed negatively in the mainstream Hindu traditions. In an effort to embody the realization of non-duality, Left-Handed Tantric practitioners perform rituals that consciously violate the traditional norms of purity and impurity, such as partaking of the **Five M's** \u2014substances like wine, meat, and fish, and activities such as sexual intercourse with someone other than one's spouse, the terms for all of which start with the letter \"m\" in **Sanskrit**. Some Tantric sects, like the **\u015aaiva Aghor\u012bs** , meditate at night in cremation grounds and use human skulls as begging bowls.\n\nSome scholars have argued that Right-Handed Tantra was developed to make the spiritual powers that Left-Handed Tantra is believed to yield available to a wider group of adherents, but without the attendant peril\u2014the danger of Left-Handed Tantra being that a practitioner who engages in it out of the wrong motive, in the absence of a non-dualistic awareness, will experience the bad **karma** normally associated with the activities that it involves, including possible **rebirth** in one of the hells. Right-Handed Tantra transforms the more antinomian practices of Left-Handed Tantra into visualization exercises that are performed not literally but mentally. Well-known theologians who developed the Right-Handed practice of Tantra include the **Abhinavagupta** (950\u20131020), who systematized the **Ka\u015bm\u012br \u015aaiva Kaula Tantra** , and **Bh\u0101skarar\u0101ya Makhin** (1690\u20131785), who similarly systematized the **\u015a\u0101kta \u015ar\u012bvidy\u0101** school of Tantra. Most Hindu systems of Tantra either have a **\u015aaiva** or **\u015a\u0101kta** orientation. The relationship between **guru** and disciple is of great importance in Tantra. Both Left-Handed and Right-Handed practices are passed on with a good deal of secrecy, due to the power that they are believed to evoke. Only one who has received initiation ( **d\u012bk\u1e63\u0101** ) from a Tantric guru is eligible to utilize a Tantric practice.\n\nA shared goal of all Tantric systems is the arousal of the **ku\u1e47\u1e0dalin\u012b** energy at the base of the spine and its elevation through the various **cakras** to the **sahasr\u0101ra cakra** at the crown of the head. This experience is believed to give the practitioner great spiritual power that can be channeled toward the attainment of **mok\u1e63a** , or **liberation** from the cycle of rebirth. **Gorakhn\u0101th** , the second guru of the Tantric \u015aaiva **N\u0101th Yoga** tradition, is credited with inventing **Ha\u1e6dha Yoga** as a way to prepare the Tantric practitioner for the ku\u1e47\u1e0dalin\u012b forces and their movement throughout the body.\n\n**TAPAS.** _See_ ASCETICISM.\n\n**TAPASYA.** **Asceticism** , either in general or in reference to particular ascetic practices.\n\n**T\u0100RA.** Literally \"star\"; mother of Bu\u1e0dha (the planet Mercury), whose father is **Soma**. A **goddess** widely revered in **Tantric Buddhism**.\n\n**TEJAS.** Splendor, brilliance, glory; one of the qualities of a **deva** , or deity.\n\n**TEMPLE.** Though communal worship is not mandated in Hinduism as it is in many other religions, such as **Christianity** or **Islam** , temple worship is believed to facilitate the inner journey to communion with the divine. The experience of **dar\u015bana** , for example\u2014seeing and being seen by an image of a deity\u2014is an occasion for forming a direct connection with divinity.\n\nTemples traditionally have an outer chamber, where a variety of different **m\u016brtis** , or images, may be housed, and an inner sanctum, or **garbhag\u1e5bha** , where an image of the central deity of the temple is housed. The images in the temple are treated as respected persons, and the temple is their house. Following this metaphor, the priests of the temple are the servants of the **gods**. The images are regularly bathed, clothed, and offered food. The food is later distributed to the community as **pras\u0101dam**. _See also_ AIHOLE; AJANTA; ANGKOR WAT; \u0100RAT\u012a; AYODHY\u0100; B\u0100D\u0100M\u012a; BANARAS; BELUR; BHAJANA; BHUVANE\u015aVARA (BHUBANESHWAR); CIDAMBARAM; ELEPHANTA; ELLORA (EL\u016aR\u0100); GURUVAY\u016aR; HALEBID; JAGANN\u0100THA; K\u0100L\u012aB\u0100R\u012a; K\u0100L\u012aGH\u0100\u1e6c; K\u0100\u00d1C\u012aPURAM; KANY\u0100KUM\u0100R\u012a (CAPE COMORIN); KHAJURAHO; KON\u0100RAK; MADURAI; M\u0100MALLAPURAM; M\u0100Y\u0100PUR; PADMAN\u0100BHI; PA\u1e6c\u1e6cADAKAL; P\u016aJ\u0100; RAME\u015aVARAM (RAMESHWARAM); SOMN\u0100TH (SOMAN\u0100THA); \u015aR\u012aRA\u1e44GAM; THA\u1e44JAV\u016aR (TANJORE); THIRUVANANTHAPURAM (TRIVANDRUM); THRISSUR (TRICHUR); TIRUPATHI; UDAIGIRI; _VASTU \u015a\u0100STRA_ ; VI\u1e62\u1e46UPUR (BISHNUPUR); VI\u015aVAN\u0100THA MANDIR.\n\n**TE\u1e44KALAI.** Subdivision of **\u015ar\u012bvai\u1e63\u1e47avism**.\n\n**\u1e6cHAGS.** A 19th-century criminal sect that offered human sacrifices to the **goddess K\u0101l\u012b**. Their victims were usually unwitting travelers who would hire the \u1e6chags as tour guides. The \u1e6chags would then take them to an isolated place and strangle them. Although the basis of the film _Indiana Jones and the Temple of Doom_ , the actual \u1e6chag cult consisted of a small group of men, not the elaborate and well-organized **temple** community depicted in the sensationalist film. Human sacrifice is not condoned in mainstream Hinduism, nor even in mainstream **Tantric** or **\u015a\u0101kta** thought. The \u1e6chag cult could perhaps be seen as a particularly extreme example of **Left-Handed** or V\u0101m\u0101c\u0101ra Tantra, which advocates the performance of traditionally impure acts as a way to affirm the non-duality of being. The \u1e6chag cult was eliminated by the British in the 1830s, with the last known \u1e6chag being hanged in 1882. _\u1e6chag_ is the origin of the English word \"thug.\" It was during filming of the Beatles' movie _Help!_ \u2014a parody of the \u1e6chag cult\u2014that **George Harrison** first heard the sitar and became interested in Hinduism.\n\n**THA\u1e44JAV\u016aR (TANJORE).** Site in **Tamil** Nadu, in southern India, which is home to a massive **\u015aiva** **temple** , with a 63-meter-high tower and a sculpture of **Nandi** the bull that is one of the largest in India. Tha\u1e45jav\u016br was the capital of the **Co\u1e37a** Empire.\n\n**THAPAR, ROMILA (1931\u2013 ).** Historian and **Indologist** ; student of **A. L. Basham** and author of _Early India: From the Origins to AD 1300_. She is currently professor emerita from Jawaharlal Nehru University, in **New Delhi** , where she taught for most of her career in the Centre for Historical Studies. Thapar has been outspoken on the topic of **Ayodhy\u0101** and the **Babr\u012b Masj\u012bd** controversy, arguing there is insufficient evidence to support the claims of **Hindu Nationalists** that a **temple** to **R\u0101ma** existed in ancient times on the site of the demolished mosque.\n\n**THEOSOPHICAL SOCIETY.** Organization established in 1875 by **Helena Petrovna Blavatsky** and **Henry Steele Olcott**. The stated objectives of the society are \"to form a nucleus of the universal brotherhood of humanity without distinction of race, creed, sex, **caste** , or color,\" \"to encourage the study of comparative religion, **philosophy** , and science,\" and \"to investigate the unexplained **laws** of nature and the powers latent in man.\" Holding very positive views of Hinduism and **Buddhism** and believing that both traditions preserve a primordial universal wisdom religion, or **perennial philosophy** , Blavatsky, Olcott, and other Theosophists promoted pride and interest among Hindus and Buddhists in their religious and cultural heritage, acting as \"reverse missionaries,\" and also played prominent roles in the movement for Indian independence.\n\nAt one point a strategic alliance began to form between the Theosophical Society and the **\u0100rya Sam\u0101j**. But **Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b** fell out with Blavatsky and Olcott in 1880, when they announced their conversions to Buddhism in **Sri Lanka**. Under the leadership of **Annie Besant** , the society contributed to the founding of **Banaras Hindu University** , as well as promoting **Jiddu Krishnamurti** as \"Maitreya,\" a long-awaited messianic figure. Today, the society continues to promote education and an esoteric interpretation of the world's religions.\n\n**THIRD EYE.** _See_ AJ\u00d1\u0100 CAKRA.\n\n**THIRUVANANTHAPURAM (TRIVANDRUM).** Capital of **Kerala** , on the southwest coast of India, and site of \u015ar\u012b Padman\u0101bhasw\u0101my **temple** , which houses a massive image of **Vi\u1e63\u1e47u** as **Padman\u0101bh\u012b** (having a **lotus** , on which the creator deity **Brahm\u0101** is seated, emerging from his navel).\n\n**THOREAU, HENRY DAVID (1817\u20131862).** American **Transcendentalist** ; associate of **Ralph Waldo Emerson** ; author of _Walden_ and _Civil Disobedience_. Thoreau was deeply inspired by Hindu thought, particularly the _Bhagavad G\u012bt\u0101_ , from which he read regularly during his experiment of living as an American **sanny\u0101sin** in a cabin at Walden Pond. A pacifist, Thoreau was jailed for his opposition to the Mexican\u2013American War and slavery, which he expressed by refusing to pay taxes. His essay, _Civil Disobedience_ , influenced **Mohand\u0101s K. G\u0101ndh\u012b**.\n\n**THRISSUR (TRICHUR).** City in **Kerala** ; site of Va\u1e0dakkun\u0101than, a large **\u015aiva** **temple**. The **K\u016b\u1e6diy\u0101\u1e6d\u1e6dam** classical Indian **dance** style is performed at this **temple** , as well as in the nearby town of **Guruvay\u016br**. This is the one Indian style of classical dance that has continued to be performed without interruption since ancient times. Other styles fell into disuse at different points and have been revived in modern times.\n\n**THUGS.** _See_ \u1e6cHAGS.\n\n**TILAK.** A mark, usually made of a mix of vermillion and **sandalwood** paste, placed on the forehead as a symbol of divine blessing, often at the end of a **p\u016bj\u0101**. A simple tilak is typically in the shape of a small circle or vertical line in the center of the forehead in the location of the \"third eye\" or **aj\u00f1\u0101 cakra**. More complex tilaks are an indicator of one's sectarian affiliation. A **Vai\u1e63\u1e47ava** tilak, in its simplest form, consists of two vertical lines down the middle of the forehead. A simple **\u015aaiva** tilak, on the other hand, is made up of three horizontal lines across the forehead. Further detail indicates the particular Vai\u1e63\u1e47ava or \u015aaiva **sa\u1e43prad\u0101ya** , or denomination, of which one is a member.\n\n**TILAK, BAL GANGADHAR (1856\u20131920).** Major leader in the movement for Indian political independence and early inspiration to the **Hindu Nationalist** movement. Tilak advocated violent revolution against the British in a commentary on the _Bhagavad G\u012bt\u0101_. He encouraged the public performance of **Ga\u1e47e\u015ba P\u016bj\u0101** in his native Maharashtra as a way of promoting Hindu pride and raising political consciousness.\n\n**T\u012aRTHA.** \"Ford\"; a shallow point in a river allowing one to cross over. Metaphorically, a t\u012brtha refers to any sacred place of pilgrimage ( **yatra** ) that provides the devotee with an opportunity to advance across the \"river\" of **rebirth** to the further shore of **mok\u1e63a** , or **liberation**. In **Jainism** , a t\u012brtha refers to the fourfold community of Jain monks, nuns, laymen, and laywomen who preserve and perpetuate the teachings of **enlightened** beings. _See also_ T\u012aRTHA\u1e44KARA.\n\n**T\u012aRTHA\u1e44KARA.** \"Ford maker\"; in **Jainism** , an **enlightened** being who creates a **t\u012brtha** or ford across the river of **rebirth** for others to cross to the further shore of **mok\u1e63a** ( **liberation** from the rebirth cycle). In Jain traditions, this t\u012brtha consists of the fourfold community of Jain monks, nuns, laymen, and laywomen who preserve and perpetuate the teachings of the T\u012brtha\u1e45kara. Twenty-four T\u012brtha\u1e45karas appear on the Earth in the course of a single cosmic cycle. The 24th T\u012brtha\u1e45kara of our current cosmic cycle, **Mah\u0101v\u012bra** , appeared around the fifth century BCE and established the contemporary Jain community.\n\n_TIRUKKU\u1e5aA\u1e36_. **Tamil** book of wise sayings, greatly revered in Tamil cultural tradition, attributed to **Tiruvalluvar** and likely composed between 100 BCE and 100 CE.\n\n**TIRUMANKAI (fl. 9th century CE).** Last of the **\u0100\u1e37v\u0101rs** \u201412 early medieval devotees of **Vi\u1e63\u1e47u** , and exponents of **bhakti** , or devotion, as the supreme path to **mok\u1e63a**. The teachings of the \u0100\u1e37v\u0101rs are expressed in the form of **poetry** in **Tamil**.\n\n**TIRUPATHI.** An enormous and wealthy **temple** in Andhra Pradesh, in southern India, dedicated to **Vi\u1e63\u1e47u** in his form as **Ve\u1e45ka\u1e6de\u015bvara**.\n\n**TIRUVALLUVAR (c. 1st century BCE\/1st century CE).** **Tamil** author (or perhaps compiler) of the _Tirukku\u1e5ba\u1e37_ , a book of wise sayings greatly revered in the Tamil cultural tradition. **Buddhists** , **Jains** , **Vai\u1e63\u1e47avas** , and **\u015aaivas** all claim him as one of their own.\n\n_TIRUV\u0100YMO\u1e36I_ **.** The collected **Tamil** **bhakti** **poems** of **Namm\u0101\u1e37v\u0101r** (880\u2013 930), one of the **\u0100\u1e37v\u0101rs** , 12 **Vai\u1e63\u1e47ava** poets of southern India. Namm\u0101\u1e37v\u0101r was born a **\u015a\u016bdra** , or a member of the servant **caste**. His text is revered in the **\u015arivai\u1e63\u1e47ava** tradition as being no less sacred than the _Vedas_ , and it continues to be used in **temple** rituals up to the present. Namm\u0101\u1e37v\u0101r's revered position as a person of low caste in this tradition is significant, being illustrative of the **philosophy** of **bhakti** , or devotion, according to which social categories such as gender and caste are irrelevant to one's spiritual status or proximity to salvation. _See also_ POETRY.\n\n**TORA\u1e46A.** Arched gateway of a Hindu **temple** ; also used in Indian **Buddhist** temples. The design was transmitted by Buddhists to East Asia, becoming the model for the _torii_ , or gate of a Shinto shrine in Japan. The tora\u1e47a marks the boundary between the mundane realm and the sacred space of the temple.\n\n**TRAILANGA SW\u0100M\u012a (1607?\u20131887?).** Hindu holy man of **Banaras** who is claimed to have lived to the incredibly old age of 280\u2014and according to some accounts, 358. Stories of his paranormal abilities and unusual qualities abound. He is said to have eaten little and yet to have weighed approximately 300 pounds. He did not wear clothing. He is attributed with great psychic powers, being able to read the minds of others easily. His disciples worshiped him as a living incarnation or **avat\u0101ra** of **\u015aiva**. He is believed to have been born in Andhra Pradesh and was a member of the **Da\u015ban\u0101m\u012b Order** of Hindu monks. Both **Ramakrishna** and **Sw\u0101m\u012b Vivek\u0101nanda** met him in Banaras, where he resided on the banks of the **Ga\u1e45g\u0101** , or Ganges River. He is in many ways the prototype of the image of the mysterious Indian holy man with miraculous powers.\n\n**TRANSCENDENTAL MEDITATION (TM).** **Mantra** -based practice of **meditation** taught and promoted globally by **Maharishi Mahesh Yogi** ; popularly known as \"TM.\" One of the main centers for the teaching and promotion of this practice is Maharishi University of Management (formerly Maharishi International University) in Fairfield, Iowa. Though it has strong **Tantric \u015aaiva** roots, the practice is presented in a nonsectarian fashion as a scientifically based, holistic path to a state of complete well-being: spiritual, emotional, and physical. It is maintained by some adherents that practice of TM by a sufficient number of persons in a given geographic region leads to what is known as the \"Maharishi Effect,\" resulting in a drop in violent crime and an increased sense of peace and happiness. The movement has put considerable effort into validating these claims statistically. To promote its program for world peace through meditation, some members of the movement started a political party in the 1990s called the \"Natural **Law** Party,\" fielding candidates in both the United States and Great Britain.\n\n**TRANSCENDENTALISTS.** Nineteenth-century group of American authors, **poets** , and activists inspired to varying degrees by Indic traditions as presented in the very early translations of such texts as the _Upani\u1e63ads_ and the _Bhagavad G\u012bt\u0101_ into English and other European languages. The transcendentalist ethos both draws from Hindu sources and, in turn, inspires modern Hindu thinkers of the nascent **Neo-Ved\u0101nta** movement in India. Its prominent members include **Ralph Waldo Emerson** , **Henry David Thoreau** , and Walt Whitman. They emphasize self-reliance and the immanence of the divine within nature.\n\n**TRAY\u012a VIDY\u0100.** \"Threefold knowledge\"; term for the _Vedas_ in very ancient sources as well as in the tradition of the **Namb\u016btiri Brahmins** of **Kerala**. This term is one piece of evidence for the view that the fourth Vedic **Sa\u1e43hit\u0101** , the _Atharva Veda_ , is later than the other three _Vedas_ , and was not immediately accepted as authoritative by all Brahmins.\n\n**TRILOKA.** \"Three worlds.\" In early **Vedic** thought, the universe is divided into three levels\u2014the Earth, the lower atmosphere, and the upper atmosphere, or celestial realm. In the _Pur\u0101\u1e47as_ , this threefold cosmology is modified into a division into the heavens, the Earth, and the hells. In some sources, the three worlds are also said to refer to the three divisions of time\u2014the past, present, and future. In **Vai\u1e63\u1e47ava** texts, **K\u1e5b\u1e63\u1e47a** is often referred to as \"Lord of the Three Worlds.\"\n\n**TRIM\u016aRTI.** \"Three forms\"; sometimes referred to as the Hindu Trinity; a model of the godhead as consisting of three personae defined by their roles in the creative process of the cosmos: **Brahm\u0101** the Creator, **Vi\u1e63\u1e47u** the Preserver, and **\u015aiva** the Destroyer. This idea seems to emerge from the nonsectarian **Sm\u0101rta** tradition. In the **literature** of the sects that take one or other of the deities to be supreme\u2014 **God** in a **monotheistic** sense\u2014the deity in question performs all three roles\u2014 **creation** , preservation, and destruction\u2014and not just one. Though sharing with the Trinity of **Christianity** the basic structure of three persons in one deity, attempts to correlate the roles of the three persons of each model fail to yield a perfect correspondence. A closer set of correspondences is possible between the Trinity of Christianity and the actual trinity of popular Hindu **devotion** : Vi\u1e63\u1e47u, \u015aiva, and **\u015aakti** , the **Mother Goddess**.\n\n**TRINITY.** _See_ TRIM\u016aRTI.\n\n**TRI\u015aULA.** Trident; divine weapon, often depicted as being carried by **\u015aiva** and **Durg\u0101**. Many Hindu **ascetics** , especially **\u015aaiva** ascetics, often carry a tri\u015bula as a symbol of their religious allegiance.\n\n**TRIVANDRUM.** _See_ THIRUVANANTHAPURAM.\n\n**TRUTH.** _See_ SATYA.\n\n**TUK\u0100R\u0100M (1608\u20131649). Vai\u1e63\u1e47ava bhakti** **poet** and saint of Maharashtra. Tuk\u0101r\u0101m claimed to have been given initiation ( **d\u012bk\u1e63\u0101** ) by **Caitanya** in a dream and the **Gau\u1e0d\u012bya Vai\u1e63\u1e47ava** tradition claims him on this basis. Many of his writings can also be found in the _\u0100di Granth_ , the sacred **scripture** of **Sikhism**. Like many bhakti saints, Tuk\u0101r\u0101m was more concerned with doing good in the world than with sectarian divisions and ritualism. Tuk\u0101r\u0101m is regarded as part of the **Sant** movement.\n\n**TULS\u012a.** Basil plant sacred to **Vai\u1e63\u1e47avas** , who regard it as an embodiment of **Vi\u1e63\u1e47u**.\n\n**TULS\u012aD\u0100S (1532\u20131623).** **Vai\u1e63\u1e47ava bhakti** **poet** and saint greatly revered for his _R\u0101m Carit M\u0101nas_ , a retelling in medieval **Hindi** of the **Sanskrit** _R\u0101m\u0101ya\u1e47a_ of **V\u0101lm\u012bki** , of whom he is regarded by some as a **reincarnation**. Like many bhakti saints and **Sants** of his time, Tuls\u012bd\u0101s had little regard for the **caste** system and specifically chose to translate the _R\u0101m\u0101ya\u1e47a_ in order to make its saving message available to the common people. It is said that he was opposed by his fellow **Brahmins** in his native **Banaras** for undertaking this project. The project was, in any case, an enormous success. The _R\u0101m Carit M\u0101nas_ is better known among most north Indian Hindus than is the original _R\u0101m\u0101ya\u1e47a_. _See also_ LITERATURE.\n\n**TUR\u012aYA.** \"The Fourth\"; the fourth of the four basic **states of consciousness**. The first is the waking state, or conventional consciousness. It consists of a subject experiencing external objects of awareness. The second is the dream state, which consists of a subject experiencing internally generated objects of awareness. The third is the state of dreamless sleep, in which there is no distinction of subject and object, but also no awareness. In tur\u012bya, the fourth state, there is no distinction of subject and object, but there is awareness. This is a state that is identified by the **Advaita Ved\u0101nta** tradition with **j\u00f1\u0101na** , or direct awareness of the unity of the experiencing subject ( **\u0101tman** ) and the universal reality ( **Brahman** ). It is the experience of **mok\u1e63a** , or **liberation** from the cycle of **rebirth**.\n\n**TVA\u1e62\u1e6c\u1e5a.** **Vedic** deity; one of the 12 **\u0100dityas** ; divine artisan; also called \"maker of all\"; credited with fashioning **Indra** 's thunderbolt weapon. _See also_ VI\u015aVAKARMAN.\n\n**TWICE-BORN.** A person who has received the **sacred thread** through the **upanayana** ritual, thus entering the first of the four stages of life, or **\u0101\u015bramas**.\n\n**TY\u0100GAR\u0100JA (1767\u20131847).** A composer of south Indian classical **music** (also known as _Carnatic_ music); native of Andhra Pradesh, where he is a cultural hero. His deeply devotional songs continue to be performed in Hindu **temples** today.\n\n## U\n\n**UDAIGIRI.** Site in Madhya Pradesh, in central India; location of numerous **temples** and carven caves utilized by both Hindus and **Jains** and dating from the **Gupta** period (320 to 550 CE).\n\n**UDAYANA (fl. 10th century CE).** Philosopher in the **Ny\u0101ya** system, or **dar\u015bana** ; renowned for his arguments for the existence of **God**. Udayana's chief philosophical interlocutors were **Buddhists** who sought to refute the traditional Hindu notion of a creator deity. He argued that the universe is an effect that requires a cause, and also that the atomic units or **param\u0101\u1e47us** that make up material substances could not organize themselves in this way without an outside agency.\n\n_UDDHAVA G\u012aT\u0100_ **.** Section of the _Bh\u0101gavata Pur\u0101\u1e47a_ that consists of **K\u1e5b\u1e63\u1e47a** 's teaching to his disciple Uddhava. Often circulated as a standalone text, the _Uddhava G\u012bt\u0101_ has a special significance to **Vai\u1e63\u1e47avas** as K\u1e5b\u1e63\u1e47a's final teaching\u2014for it is presented just before he retires to the forest where he knows he will die, being shot by a hunter.\n\n**UDG\u0100T\u1e5a. Brahmin** priest whose traditional duty is to sing hymns from the _S\u0101ma Veda_ at **Vedic** rituals of sacrifice ( **yaj\u00f1a** ).\n\n**UDIPI.** Town in **Karnataka** ; birthplace of **Madhva** (1238\u20131317), a **Vai\u1e63\u1e47ava \u0101c\u0101rya** and founder of the dualistic or **Dvaita** system of **Ved\u0101nta**. Madhva spent most of his life in Udipi.\n\n**UJJAIN (UJJAYIN\u012a).** City in Madhya Pradesh, in central India; one of the traditional locations of the **Kumbha Mel\u0101**. It is located on the \u015aipr\u0101 River.\n\n**UM\u0100.** Alternative name for **P\u0101rvat\u012b** , the **Mother Goddess** and wife of **\u015aiva**. Um\u0101 first appears in the _Kena_ _Upani\u1e63ad_ , where she teaches **Indra** about **Brahman** , thus making him foremost among the **devas** , or **gods**.\n\n**UNTOUCHABILITY.** The **caste** -based practice of viewing certain occupations as so polluting as to render those who practice them inherently and permanently impure, and so a source of pollution to members of higher groups in the caste hierarchy. The lowest\u2014and so most impure\u2014group in most versions of the caste system is the **Ca\u1e47\u1e0d\u0101las** , responsible for burning corpses in cremation grounds. Other such groups include those responsible for cleaning human toilets and sweeping streets. These Untouchable castes today are known as **Dalits** \u2014\"the oppressed\"\u2014or **Harijans** \u2014\"children of **God**.\" The term _Dalit_ is a reference to the suffering such groups have experienced due to caste prejudice ( **casteism** ) and the term _Harijan_ was coined by **Mohand\u0101s K. G\u0101ndh\u012b** to indicate that God is on the side of those who suffer most due to social injustice. Different low-caste communities prefer different terms depending upon their political orientation toward Hinduism, with Dalits tending to see Hinduism in a negative light, as warranting their oppression, and Harijans tending to see Hinduism in a positive light, seeing caste prejudice as a perversion of the tradition. All such groups are officially termed **Scheduled Castes** by the government of India, because they are on a list, or \"schedule,\" of groups that warrant special benefits on the basis of the discrimination from which they have traditionally suffered.\n\n**UNTOUCHABLE.** _See_ UNTOUCHABILITY.\n\n**UPAC\u0100RA.** \"Approach\"; offerings made to a deity in **p\u016bj\u0101**. They traditionally include such items as water, flowers, food, incense, light (in the form of the lit flame of a lamp [ **d\u012by\u0101** ] or candle), and a **tilak** of **sandalwood** paste, and **actions** such as prayer.\n\n**UPANAYANA.** The **sacred thread** ceremony, or \"second birth\"; the **Vedic** ritual of investiture with the sacred thread marks the transition from childhood to **brahmacarya** , the first of the four stages of life, or **\u0101\u015bramas**. The thread is traditionally worn over the left shoulder, from where it hangs diagonally across the chest and back. As a coming-of-age ritual, the upanayana is analogous to the Bar Mitzvah of Judaism, or the **Christian** sacrament of Confirmation. It is a ritual traditionally reserved for **Brahmin** males. In the modern period, though, this ritual has been opened to girls as well. The process of **Sanskritization** , by which some lower **-caste** communities elevate themselves in the caste hierarchy, includes performing rituals of this kind. The idea of this ritual as a \"second birth\" is why higher castes\u2014especially Brahmins\u2014are often called **twice born**.\n\n_UPANI\u1e62ADS_ **.** \"Secret doctrine\"; the final portion of the **Vedic** **literature** , which is also known, for this reason, as the **Ved\u0101nta** , or \"end of the _Vedas_.\" The _Upani\u1e63ads_ are seen as the \"end\" of the _Vedas_ in both a literal sense (as the last part of the Vedic literature to be composed), but also, in the traditions of Ved\u0101nta **philosophy** , in a more extended sense as conveying the ultimate meaning to which the Vedic literature points. The _Upani\u1e63ads_ were composed, at least in their current form, primarily between the fifth century BCE and the first two centuries of the Common Era. They consist largely of dialogues on the nature of **Brahman** , the ultimate reality.\n\n**UP\u0100SANA KANDA.** \"Contemplative portion\"; the section of the **Vedic** **literature** that comes, historically, between the early **Karma Kanda** , or \" **action** portion\" of the _Vedas_ , and the later **J\u00f1\u0101na Kanda** , or \"wisdom portion.\" The Karma Kanda consists of the early Vedic hymns, or _Sa\u1e43hit\u0101s_ and the \"priestly texts,\" or _Br\u0101hma\u1e47as_. The J\u00f1\u0101na Kanda consists of the _Upani\u1e63ads_. The Up\u0101sana Kanda, however, is made up of the \"Forest Texts,\" or _\u0100ra\u1e47yakas_.\n\n**URVA\u015a\u012a.** Character of a story, first mentioned in the _\u1e5ag Veda_ and later elaborated upon in the _Br\u0101hma\u1e47as_ and the _Mah\u0101bh\u0101rata_ , and finally in a famous play by the renowned **Sanskrit poet** and playwright **K\u0101l\u012bd\u0101sa** (the _Vikramorva\u015b\u012b_ ). In the story, a mortal king, **Pur\u016bravas** , falls in love with **Urva\u015b\u012b** , who is a celestial maiden ( **apsaras** ), and asks to marry her. She agrees, on the condition that he protect her rams from being stolen and he never lets her see him naked. They live together happily for many years until a group of mischievous **gandharvas** steals one of Urva\u015b\u012b's rams in the middle of the night. Hearing the commotion, Pur\u016bravas leaps naked out of bed to save the ram. The gandharvas flash a bright light on him, revealing his naked form to Urva\u015b\u012b, and make off with the ram. At this, Urva\u015b\u012b returns to the heavens.\n\nIn another story in the _Mah\u0101bh\u0101rata_ , Urva\u015b\u012b becomes enamored of **Arjuna** during his sojourn in the heavens during the period of the **Pa\u1e47\u1e0davas** ' exile. In response to her overtures\u2014and knowing the story of his distant ancestor, Pur\u016bravas\u2014Arjuna responds that he sees Urva\u015b\u012b as his mother. Angry at being thus spurned, Urva\u015b\u012b curses Arjuna to live for a year as a **woman**. This turns out to be fortuitous, however; the year following the P\u0101\u1e47\u1e0davas' exile, during which they are to live incognito, Arjuna uses this curse to disguise himself as a woman, his identity remaining hidden. _See also_ LITERATURE.\n\n**U\u1e62AS.** **Vedic** **goddess** of the dawn.\n\n**UTTARA M\u012aM\u0100\u1e42S\u0100.** Later exegesis; another term for **Ved\u0101nta** , which contrasts this system of thought with **P\u016brva M\u012bm\u0101\u1e43s\u0101** , or \"earlier exegesis,\" which focuses upon the interpretation of the earlier, ritualistic portion (or **Karma Kanda** ) of the **Vedic** **literature**. Uttara M\u012bm\u0101\u1e43s\u0101, in contrast, focuses upon the later Vedic literature, the _Upani\u1e63ads_ (also known as the **knowledge** portion, or **J\u00f1\u0101na Kanda** of the _Vedas_ ). _See also_ M\u012aM\u0100\u1e42S\u0100.\n\n## V\n\n**V\u0100C.** \"Speech\"; **Vedic goddess** who is later identified with **Sarasvat\u012b** and embodies the powers of speech and thought. It is through the power of speech that **Praj\u0101pati** is able to create the world.\n\n**V\u0100CASPATI MI\u015aRA (900\u2013980 CE).** Prolific philosopher; author of commentaries on all of the **\u0101stika** (\"orthodox\") systems of Hindu philosophy ( **dar\u015banas** ). Although he was an adherent of **Advaita Ved\u0101nta** , his thought was influential on the later **Navya Ny\u0101ya** school established by the 13th-century philosopher Ga\u1e45ge\u015ba Up\u0101dhy\u0101ya.\n\n**VAHANA.** \"Vehicle\"; the animal that accompanies each deity and serves as that deity's vehicle. Examples include **\u015aiva** 's bull, **Nandi** ; **Vi\u1e63\u1e47u** 's eagle, **Garu\u1e0da** ; and **Ga\u1e47e\u015ba** 's mouse, **M\u016b\u015baka**.\n\n**VAIKH\u0100NASA.** Community of south Indian **Brahmins** whose traditional job is to serve as priests in **Vai\u1e63\u1e47ava** **temples**.\n\n**VAIKU\u1e46\u1e6cHA.** Heavenly abode of **Vi\u1e63\u1e47u**. In many of the **Vai\u1e63\u1e47ava** traditions, **mok\u1e63a** , or **liberation** from the cycle of **rebirth** , is conceived as dwelling for eternity in Vaiku\u1e47\u1e6dha with Vi\u1e63\u1e47u, or **K\u1e5b\u1e63\u1e47a** , who is conceived as **God** in a **monotheistic** sense, as the supreme deity.\n\n**VAI\u015aAKHA.** Month of the **Hindu calendar** that corresponds roughly to the period from mid-April to mid-May.\n\n**VAI\u015aA\u1e42PAYANA.** One of the five **Brahmin** students of **Vy\u0101sa** , also known as **Veda Vy\u0101sa** , who is attributed not only with authoring the _Mah\u0101bh\u0101rata_ but also with organizing the _Vedas_ into their current distribution of four texts: the _\u1e5ag Veda_ , _Yajur Veda_ , _S\u0101ma Veda_ , and _Atharva Veda_. According to the _Pur\u0101\u1e47as_ , each of Vy\u0101sa's students was responsible for memorizing and passing on a separate Vedic text\u2014one for each of the four _Vedas_ and one for the _Mah\u0101bh\u0101rata_. Vai\u015ba\u1e43payana was the student responsible for transmitting the _Yajur Veda_. In the _Mah\u0101bh\u0101rata_ , he recites the text to King Janamejaya, great-grandson of **Arjuna** , at the former's snake sacrifice.\n\n**VAI\u015aE\u1e62IKA.** \"Differentiation\"; one of six \"orthodox\" ( **\u0101stika** ) systems of philosophy ( **dar\u015banas** ); attributed to the sage Ka\u1e47\u0101da, who most probably lived in the fourth century before the Common Era. Later aligned closely with the **Ny\u0101ya** system of logic, with which it is often paired and with which it had effectively merged by the medieval period, Vai\u015be\u1e63ika is focused primarily upon the analysis of the material world, positing a theory of atoms ( **param\u0101\u1e47u** ) and of compound and noncompound substances.\n\n**VAI\u1e62\u1e46AVA.** Devotee of **Vi\u1e63\u1e47u** ; adherent of **Vai\u1e63\u1e47avism** ; of or referring to Vai\u1e63\u1e47avism. Vai\u1e63\u1e47avas form the largest subdivision of Hinduism, with roughly 60 percent of Hindus identifying themselves as Vai\u1e63\u1e47ava.\n\n**VAI\u1e62\u1e46AVISM.** In terms of the numbers of its adherents, Vai\u1e63\u1e47avism is the largest of the four main sectarian subdivisions that make up Hinduism. The various subtraditions that constitute Vai\u1e63\u1e47avism share the trait of being oriented devotionally toward **Vi\u1e63\u1e47u** as the supreme deity, regarding him as **God** in a **monotheistic** sense. Vai\u1e63\u1e47ava traditions have a strong focus on **bhakti** , or devotion to the supreme deity, as the preeminent\u2014or even exclusive\u2014path to **mok\u1e63a** , or **liberation** from the cycle of **rebirth**. Though there is internal theological diversity among the many Vai\u1e63\u1e47ava denominations ( **sa\u1e43prad\u0101yas** ), it can broadly be said that the Vai\u1e63\u1e47avas are more inclined than the adherents of other Hindu traditions to insist upon a strong metaphysical distinction between God and the individual soul ( **j\u012bva** ). The sharpest arguments for this view are presented by **Madhva** , the **\u0101c\u0101rya** of the **Dvaita** , or dualistic form of **Ved\u0101nta**.\n\nThe other Vai\u1e63\u1e47ava systems of Ved\u0101nta, such as the **Vi\u015bi\u1e63\u1e6d\u0101dvaita** or **Qualified Nondualism** of **R\u0101m\u0101\u1e47uja** , the _Bhed\u0101bheda_ (\"Difference-and-Non-Difference\") taught by **Nimb\u0101rka** , and the _Acintya Bhed\u0101bheda_ teaching (\"Inconceivable Difference-and-Non-Difference\") of **Caitanya** , affirm, in varying degrees and senses, an organic unity of God and the world, while maintaining the real distinction between God and the soul that all of these schools of thought see as a precondition for bhakti. They uniformly reject the **Advaita Ved\u0101nta** system's subordination of the personal deity to an impersonal absolute ( **Nirgu\u1e47a Brahman** ) and its claims that the personal God is part of the realm of illusion ( **m\u0101y\u0101** ) and that bhakti is a merely preparatory discipline for **j\u00f1\u0101na** , or knowledge, which **\u015aa\u1e45kara** claims to be constitutive of mok\u1e63a.\n\nIn practice, the most popular object of devotion for most Vai\u1e63\u1e47avas is actually not Vi\u1e63\u1e47u in his celestial form, but the **avat\u0101ras** , or incarnations, **K\u1e5b\u1e63\u1e47a** and **R\u0101ma**. Indeed, the **Gau\u1e0d\u012bya Vai\u1e63\u1e47ava** tradition claims that K\u1e5b\u1e63\u1e47a is not merely an incarnation, but is in fact the supreme deity, from whom the celestial form of Vi\u1e63\u1e47u is a projection. And there is, indeed, support for this view in such early Vai\u1e63\u1e47ava texts as the _Bhagavad G\u012bt\u0101_.\n\n**Lak\u1e63m\u012b** , or **\u015ar\u012b** , the **goddess** of prosperity and wife of Vi\u1e63\u1e47u, plays a prominent role in many Vai\u1e63\u1e47ava traditions\u2014particularly **\u015ar\u012bvai\u1e63\u1e47avism** , which claims that Lak\u1e63m\u012b and Vi\u1e63\u1e47u are inseparable, forming a single, composite deity.\n\n**VAI\u015aYA.** The third of the four **var\u1e47as** or **castes**. In terms of traditional occupation, this caste is made up of trades that involve economically productive activity, such as **arts** and crafts and farming, as well as trade. In the _Puru\u1e63a_ _Sukta_ of the _\u1e5ag Veda_ , this caste is said to have been formed from the stomach of the cosmic person ( **puru\u1e63a** ). The word _vai\u015bya_ , or \"commoner,\" for this caste also suggests that its membership was relatively large in ancient India.\n\n**V\u0100JAPEYA.** \"Drink of strength\"; a royal **Soma** sacrifice offered by a king who aspires to victory in battle. _See also_ A\u015aVAMEDHA; R\u0100JAS\u016aYA; YAJ\u00d1A.\n\n**VAJRA.** Thunderbolt; favored weapon of **Indra** ; fashioned by **Tva\u1e63\u1e6d\u1e5b** ; a symbol of the sudden flash of **enlightenment** in **Tantric** forms of **Buddhism**.\n\n**V\u0100K.** _See_ V\u0100C.\n\n**VALLABHA (1479\u20131531).** **\u0100c\u0101rya** of the **Vallabha Sa\u1e43prad\u0101ya** , a **Vai\u1e63\u1e47ava** tradition also known as the Pu\u1e63\u1e6di M\u0101rg (\"Way of Prosperity\"), largely popular among the Gujarati **Vai\u015bya** communities. His teaching, known as _\u015auddh\u0101dvaita_ , or \"Pure **Nondualism** ,\" is that all souls ( **j\u012bvas** ) are ultimately one with **K\u1e5b\u1e63\u1e47a**. Strong emphasis is placed on **bhakti** and cultivating a personal relationship with K\u1e5b\u1e63\u1e47a as his servant, friend, parent, or lover\u2014a theme of many Vai\u1e63\u1e47ava traditions. Vallabha, a south Indian **Brahmin** who relocated to Gujarat and made many pilgrimages to **V\u1e5bnd\u0101vana** , where K\u1e5b\u1e63\u1e47a spent his youth, is held by adherents of his tradition to be an incarnation of K\u1e5b\u1e63\u1e47a. Not an **ascetic** tradition, the **gurus** of the Vallabha Sa\u1e43prad\u0101ya are direct male descendants of Vallabha.\n\n**V\u0100LM\u012aKI.** Sage attributed with authorship of the _R\u0101m\u0101ya\u1e47a_. It is said that although V\u0101lm\u012bki was born a **\u015a\u016bdra** , or member of the servant **caste** , he achieved the status of a **Brahmin** through his great learning\u2014an indication, similar to the story of **Satyak\u0101ma** in the _Ch\u0101ndogya Upani\u1e63ad_ \u2014of voices within the **Vedic** tradition that regard one's personal **qualities** rather than one's parentage as determinative of one's role in society. Although the _R\u0101m\u0101ya\u1e47a_ , in its present form, is regarded by scholars as the work of many authors working over the course of several centuries (from roughly 200 BCE to 200 CE by most estimates), it is possible that a historical V\u0101lm\u012bki existed at an early period whose initial composition formed the basis for the current **Sanskrit** text. _See also_ LITERATURE.\n\n**V\u0100M\u0100C\u0100RA.** _See_ LEFT-HANDED TANTRA.\n\n**V\u0100MADEVA.** Major **Vedic \u1e5b\u1e63i** , or sage, to whom much of the fourth **ma\u1e47\u1e0dala** or book of the _\u1e5ag Veda_ is attributed; patriarch of a **Brahmin** family charged with transmitting the _\u1e5ag Veda_ to future generations.\n\n**V\u0100MANA.** The dwarf **avat\u0101ra** , or incarnation, of **Vi\u1e63\u1e47u** ; the fifth of the 10 avat\u0101ras in the most popular traditional list of incarnations of Vi\u1e63\u1e47u. **Bali** \u2014the grandson of **Prahl\u0101da** and great-grandson of **Hira\u1e47yaka\u015bipu** , who was destroyed by the **Narasi\u1e43ha** or man-lion avat\u0101ra of Vi\u1e63\u1e47u\u2014was a mighty **asura** who achieved dominance over the Earth. In order to defeat Bali, Vi\u1e63\u1e47u approached his court as a dwarf and asked to be given the land that he could cover in three steps. Seeing the tiny creature, Bali laughed arrogantly and granted the dwarf's request. At that, the dwarf rapidly grew until he filled the universe and took three steps, encompassing the \"three worlds\" ( **triloka** ), or all of existence. Bali was humbled and allowed the dwarf to put his foot on his head, pressing him down to the underworld. This story could be seen as an explanation for references in the _\u1e5ag Veda_ to Vi\u1e63\u1e47u's \"three steps\" by which he creates the world.\n\n**VANAPRASTHA.** Literally, \"forest-dwelling\"; retirement; withdrawal; the third of the four stages of life or **\u0101\u015bramas** that map out the life of an orthodox Hindu according to the _Dharma \u015a\u0101stras_. The name of this stage and descriptions in the _Dharma \u015a\u0101stras_ suggest that at one time **householders** actually did withdraw with their wives to a forest to practice **asceticism**. Today, however, this stage refers simply to a life of retirement and withdrawal from daily responsibilities to focus upon religious duties and prepare for the end of one's life, typically spent residing in the home of one of one's children as part of an extended family. It is said that one should enter this stage when one's hair begins to turn gray and one's children give birth to children of their own.\n\n**VAR\u0100HA.** The boar **avat\u0101ra** , or incarnation, of **Vi\u1e63\u1e47u** ; the third of the 10 avat\u0101ras in the most popular traditional list of incarnations of Vi\u1e63\u1e47u. The **asura Hira\u1e47yak\u1e63a** is said to have cast the Earth into the cosmic ocean, from which the boar avat\u0101ra rescued it. He then slew Hira\u1e47yak\u1e63a. Var\u0101ha is traditionally depicted iconographically as a warrior with the head of a boar holding the sphere of the Earth in his tusks and standing on the body of the demonic Hira\u1e47yak\u1e63a.\n\n**V\u0100R\u0100\u1e46AS\u012a.** Alternative name for **Banaras**.\n\n**VARMA, MAHESH PRAS\u0100D (1918\u20132008).** _See_ MAHARISHI MAHESH YOGI (MAH\u0100R\u1e62I MAHE\u015aA YOG\u012a).\n\n**VAR\u1e46A.** Literally, \"color\"; principal unit of division of the **caste** system. It is probably due to the literal meaning of this term that the term _caste_ was coined to refer to the Indian traditional division of social labor\u2014 _casta_ being the Portuguese word for \"color\" that was later Anglicized to _caste_. The use of this term, combined with 19th-century racial assumptions, led early **Indologists** to speculate that caste originated as a system of racial segregation imposed by light-skinned **\u0100ryan** invaders upon the dark-skinned indigenous peoples of ancient India, an interpretation that fit well with the **\u0100ryan Invasion Theory**.\n\nAn alternative theory, however, based on the _Dharma \u015a\u0101stras_ , is that the \"colors\" of the var\u1e47as are symbolic of the **gu\u1e47as** , or qualities, that persons in these var\u1e47as are believed to embody. The three gu\u1e47as, or qualities, that make up all phenomena according to **S\u0101\u1e43khya** **philosophy** are each assigned colors\u2014the **sattva** gu\u1e47a being white, the **rajas** gu\u1e47a being red, and the **tamas** gu\u1e47a being black. The colors traditionally assigned to the var\u1e47as are white for the **Brahmins** , red for **K\u1e63atriyas** , yellow for **Vai\u015byas** , and black for **\u015a\u016bdras**. **Untouchables** or **Outcastes** \u2014persons whose duties are regarded as so polluting as to be beyond the pale of the caste system\u2014are called _avar\u1e47a_ , or \"without color.\" This does not fit well with an interpretation of the var\u1e47as as skin tones\u2014nor does the variation of skin tones that one actually finds internally to all of the var\u1e47as in India.\n\nIn terms of actual social status and practice, var\u1e47a is of far less consequence in the caste system than **j\u0101ti** \u2014literally \"birth\"\u2014the subgroups or \"subcastes\" into which each of the four var\u1e47as is divided. The relative locations of the j\u0101tis within the var\u1e47a hierarchy vary a good deal regionally, due in part to **Sanskritization**.\n\n**VAR\u1e46\u0100\u015aRAMA-DHARMA.** \"The ordering of **castes** and stages of life\"; the duties of a person as mapped out according to one's location within the caste, or **var\u1e47a** , system and within the system of four stages of life or **\u0101\u015bramas** ; the classical Hindu way of life as it is outlined in the _Dharma \u015a\u0101stras_.\n\n**VARU\u1e46A.** **Vedic** deity of the **ocean** ; one of the **\u0100dityas** ; along with **Indra** , **Agni** , and **Yama** , one of the **Lokap\u0101las** or \"World Guardians\"; close associate of the deity **Mitra**. Some hymns of the _\u1e5ag Veda_ suggest that Varu\u1e47a was, at one point, the supreme deity of the Vedic religion. Varu\u1e47a is associated with judgment and the forgiveness of sins. He is sometimes depicted as holding a noose with which he ensnares the **souls** of the wicked. In some verses he is referred to as an **asura** , not in the later sense that this word has as a reference to the demonic, but in its more ancient sense as meaning \"powerful.\" He may be related to **Ahura Mazda** , the supreme deity of **Zoroastrianism**. In the _R\u0101m\u0101ya\u1e47a_ , Varu\u1e47a assists **R\u0101ma** and his armies by keeping the oceans calm as they build a bridge to **La\u1e45ka** to rescue **S\u012bt\u0101** from **R\u0101va\u1e47a**.\n\n**VASI\u1e62\u1e6cHA.** One of the Seven Sages ( **Saptar\u1e63i** ) born from the mind of **Brahm\u0101** who is said to preside over the cosmos during our current **world age**. Vasi\u1e63\u1e6dha is a significant figure in the _\u1e5ag Veda_ , attributed with authoring much of the seventh **ma\u1e47\u1e0dala** or book of this text. In the _R\u0101m\u0101ya\u1e47a_ , he is a priest to the royal family in which **R\u0101ma** is raised. A major **Advaita Ved\u0101nta** text, the _Yoga-V\u0101si\u1e63\u1e6dha_ , is presented as the teaching of Vasi\u1e63\u1e6dha to the young R\u0101ma. In the _Mah\u0101bh\u0101rata_ , a set of eight deities called the **Vasus** conspire to steal Vasi\u1e63\u1e6dha's cows. Upon discovering that they had stolen his cows, Vasi\u1e63\u1e6dha curses the Vasus to be born as human beings. They then ask the **goddess Ga\u1e45g\u0101** to give birth to them and immediately drown them in her waters so they can return to the heavens. Only one of the Vasus\u2014the original conspirator who hatched the cow-stealing plot\u2014cannot get out of his situation so easily. He is not drowned and grows up to become **Bh\u012b\u1e63ma** , the grandsire of the **P\u0101\u1e47\u1e0davas** and a major character of the _Mah\u0101bh\u0101rata_ who lives to a very great age.\n\n_VASTU \u015a\u0100STRA_ **.** \"Science of objects\"; a Hindu analogue of _feng shui_. _Vastu \u015a\u0101stra_ encompasses architecture, the construction and consecration of **temples** and **m\u016brtis** , or images of Hindu deities, and the placement of objects in a room in order to maximize the positive flow of subtle energies.\n\n**VASU.** \"Good, prosperous\"; category of **Vedic** deities frequently invoked for material benefits. There are eight Vasus altogether. In the _Mah\u0101bh\u0101rata_ , the Vasus conspire to steal the cows of the sage **Vasi\u1e63\u1e6dha**.\n\n**VASUDEVA.** Father of **K\u1e5b\u1e63\u1e47a** and **B\u0101lar\u0101ma** ; husband of **Devak\u012b** ; chief minister of the wicked King **K\u0101\u1e43sa** , of whom it was prophesied that a son of Vasudeva would slay him. K\u0101\u1e43sa therefore imprisons Vasudeva and Devak\u012b and conspires to kill their children. But K\u1e5b\u1e63\u1e47a and B\u0101lar\u0101ma are rescued and the prophecy is eventually fulfilled.\n\n**V\u0100SUDEVA.** \"Of **Vasudeva** \"; epithet of **K\u1e5b\u1e63\u1e47a** that refers to the fact that he is the son of Vasudeva.\n\n**V\u0100SUK\u012a.** Gigantic celestial serpent; king of the **N\u0101gas**. The N\u0101gas reside in a portion of the underworld known as **P\u0101t\u0101la** , over which V\u0101suk\u012b presides. V\u0101suk\u012b permitted the **devas** and **asuras** to use his body as the rope for a massive churn when they churned the cosmic ocean together in search of the elixir of life, or **am\u1e5bta**.\n\n**VA\u1e6cAKALAI.** Subdivision of **\u015ar\u012bvai\u1e63\u1e47avism**.\n\n**V\u0100TSY\u0100YANA (c. 4th\u20136th centuries CE).** **Brahmin** ; author of the _K\u0101ma S\u016btra_.\n\n**V\u0100YU.** **Vedic** deity or **deva** , personification of the wind. Known for his great power, V\u0101yu is the father of **Hanuman** in the _R\u0101m\u0101ya\u1e47a_ and of **Bh\u012bma** in the _Mah\u0101bh\u0101rata_. **Madhva** , founder and **\u0101c\u0101rya** of the **Dvaita** , or dualistic school of **Ved\u0101nta** is believed to be the third incarnation, or **avat\u0101ra** , of V\u0101yu. _See also_ GODS.\n\n_VEDA_ **,** _VEDAS_ **.** \"Wisdom\"; the most revered, ancient, and authoritative collection of Hindu **scripture** ; the **\u015br\u016bti** , or \"that which is heard\"; the direct, divinely revealed word of **God** , or the original sound of the universe heard in the distant past by the **\u1e5b\u1e63is** or sages; non-man-made ( _apauru\u1e63eya_ ), in contrast with **sm\u1e5bti** (\"that which is remembered\"), the sacred tradition that takes as its foundation the more ancient and fundamental _Vedas_ ; the primordial knowledge that underlies **creation**. The _Vedas_ are a collection of texts written over a period spanning more than a thousand years, with the earliest Vedic texts being composed around 1500 BCE (according to mainstream scholarship) and the latest **Vedic** texts\u2014the later _Upani\u1e63ads_ \u2014being composed in the first couple of centuries of the Common Era.\n\nThe _Vedas_ are organized into four collections, each of which can be subdivided into a further four historical \"layers\" or \"strata\" of texts composed during different periods, and each of which serves as a commentary upon or elaboration of the texts that precede it. The four basic collections are the _\u1e5ag Veda_ , _Yajur Veda_ , _S\u0101ma Veda_ , and _Atharva Veda_. The first historical \"layer\" of each is a **Sa\u1e43hit\u0101** or collection of **mantras** , or hymns. The largest and the oldest is the _\u1e5ag Veda_ **,** whose Sa\u1e43hit\u0101 consists of 1,028 hymns to various **devas** or deities and philosophical reflections on the nature of creation. Next come the _Yajur Veda_ and the _S\u0101ma Veda_. The former is concerned with the correct performance of ritual and the latter sets hymns from the _\u1e5ag Veda_ to **music** , forming the basis of Indian classical music. The _Atharva Veda_ is a later text that includes hymns, magical spells, and philosophical reflections. The second historical layer consists of the _Br\u0101hma\u1e47as_ , or \"priestly texts,\" that focus on the meaning of the ritual of sacrifice ( **yaj\u00f1a** ) and include a good deal of narrative **literature**. The third layer consists of the _\u0100ra\u1e47yakas_ , or \"forest texts,\" and the fourth is the _Upani\u1e63ads_ , each of which is progressively more esoteric.\n\nThese four layers of text are also divided in a threefold fashion based on subject matter. The _Sa\u1e43hit\u0101s_ and _Br\u0101hma\u1e47as_ , being concerned mainly with ritual **action** , form the **Karma Kanda** , or \"action portion.\" The _\u0100ra\u1e47yakas_ are called the **Up\u0101sana Kanda** , or \"contemplative portion,\" as they begin to reflect upon mystical correspondences between the elements making up the Vedic ritual and broader realities both within the cosmos and interior to the individual person. Finally, the _Upani\u1e63ads_ are called the **J\u00f1\u0101na Kanda** , or \"wisdom portion,\" which reveals the ultimate meaning of the Vedic corpus as a whole in the form of dialogues on **Ved\u0101nta** philosophy. And the _Upani\u1e63ads_ themselves are also known as _Ved\u0101nta_ , or \"end of the _Veda_ ,\" a term with a double meaning, referring both to the fact that these texts form the final section of the Vedic corpus and to their content as the ultimate \"end\" or purpose toward which all Vedic wisdom and practice is directed.\n\n**VEDA VY\u0100SA.** Epithet of **Vy\u0101sa** that refers to his role in organizing the **Vedic** literary corpus into its current form.\n\n**VED\u0100NTA.** \"Ultimate end or purpose of wisdom\"; \"end of the _Veda_ \"; the philosophy of the _Upani\u1e63ads_ , which are also called _Ved\u0101nta_. The term has a double meaning, referring both to the fact that the _Upani\u1e63ads_ form the final section of the **Vedic** corpus and to their content as the ultimate \"end\" or purpose to which Vedic wisdom is directed. Because the _Upani\u1e63ads_ do not present a single consistent **philosophy** in a linear fashion but convey their concepts through dialogues, **poetry** , and riddles, one must interpret these texts utilizing some hermeneutical principles as a basis. Differences of approach in this regard have led to the emergence of several different schools or systems of Ved\u0101nta over the centuries. One of the oldest of these systems is the **non-dualistic** or **Advaita** Ved\u0101nta associated prominently with **\u015aa\u1e45kara (** 788\u2013820), which affirms that **Brahman** \u2014infinite being, consciousness, and bliss\u2014is the sole reality and that all phenomena are merely an appearance, or **m\u0101y\u0101** \u2014an effect of ignorance ( **avidy\u0101** ) that gives rise to the process of **rebirth** and **liberation** from rebirth ( **mok\u1e63a** ), which arises only when one has knowledge ( **j\u00f1\u0101na** ) of the true nature of reality as Brahman.\n\nSubsequent systems of Ved\u0101nta, of which there are many, express discomfort with \u015aa\u1e45kara's insistence on the final unreality of phenomena, including the personal **God** , devotion ( **bhakti** ) toward whom \u015aa\u1e45kara does commend as a purifying discipline preparatory to the arising of true knowledge but which other systems see as the preeminent path to liberation, which is seen as a salvation given by the grace of a loving God. Because of the centrality of bhakti in their understanding of the spiritual path, these thinkers insist, to varying degrees, on a real metaphysical distinction between the individual soul ( **j\u012bva** ) and God as a necessary condition for the relationship of bhakti. The most radical in this regard is **Madhva** (1238\u20131317), whose dualistic **Dvaita** Ved\u0101nta draws a sharp distinction between God ( **Vi\u1e63\u1e47u** ) and the world of souls. Most, such as the **Vi\u015bi\u1e63\u1e6d\u0101dvaita** or **Qualified Non-Dualism** of **R\u0101m\u0101\u1e47uja** (1077\u20131157), seek to affirm both unity and duality, in different ways and in different senses. All of these approaches to the nature of Brahman can find justification in the _Upani\u1e63ads_. In addition to the traditional or classical systems of Ved\u0101nta, which are concerned primarily with the correct interpretation of **scripture** , there is the **Neo-Ved\u0101nta** of the modern period, which is associated with **Ramakrishna** and **Sw\u0101m\u012b Vivek\u0101nanda** , but is also expressed by such major figures of modern Hindu thought as **Aurobindo Ghose** and **Rama\u1e47a Mahar\u1e63i** , as well as many others. Adherents of Neo-Ved\u0101nta see Ved\u0101nta as universal knowledge available preeminently, but not exclusively, through the Vedic texts, which are seen as the records of the insights and experiences of the Vedic seers, or **\u1e5b\u1e63is**. It is the experience ( **anubh\u0101va** ) of Brahman rather than the texts that speak of it that is regarded as primary in Neo-Ved\u0101nta, which sees itself less as a system of Hindu thought and more as a **perennial philosophy** , present in all cultures and religions to varying degrees but expressed with particular clarity in the Vedic tradition.\n\n**VED\u0100NTA SOCIETY.** Organization devoted to teaching and promoting the **Ved\u0101nta** of **Ramakrishna** and **Sw\u0101m\u012b Vivek\u0101nanda** ; established by Sw\u0101m\u012b Vivek\u0101nanda in New York in 1894. Its Indian companion organization is the **Ramakrishna Mission**. It is closely tied to the **Ramakrishna Order** , which provides the monks and nuns who run the various centers that the society runs around the world.\n\n_VED\u0100NTA S\u016aTRAS_ **.** Root text of the **Ved\u0101nta** systems of **philosophy** ; alternative name for the _Brahma S\u016btras_.\n\n**VEDI (VED\u012a).** Area within the sacred space of a **Vedic** ritual where implements such as ladles and spoons (for pouring **ghee** and other substances onto the sacred fire) are kept. It is also the space where the **gods** are invited to sit during the ritual. _See also_ YAJ\u00d1A.\n\n**VEDIC.** Of or referring to the _Vedas_ ; adjective often used to refer to Brahmanical culture in general; also the archaic form of **Sanskrit** in which the _Vedas_ are written. _See also_ BRAHMANISM.\n\n**VEGETARIANISM.** Dietary practice of avoiding foods that involve harm to animals, primarily meat, but also eggs and, in some cases, milk products (a practice more widely known as _veganism_ ). Vegetarianism is practiced to varying degrees in Indic traditions, ranging from the very strict dietary rules of **Jainism** to the relatively loose vegetarianism of many **Brahmins** in **Bengal** who eat fish but abstain from land animals. Most Hindu vegetarians eat no meat or eggs, but do partake of milk products. Jains are traditionally in this category as well, but avoid root vegetables that involve the killing of the entire plant. Some have recently become vegan, avoiding even milk due to the cruelty that is involved in modern milk-gathering techniques. Vegetarianism is a practice of **ahi\u1e43s\u0101** , or nonviolence, though it is also promoted as a healthy diet, conducive to spiritual practices such as **meditation**. Vegetarian food is held to contain more of the **sattva gu\u1e47a** that is a quality conducive to clarity of mind. About one third of Hindus are actually vegetarian.\n\n**VE\u1e44KA\u1e6cE\u015aVARA.** \"Lord who destroys sins\"; form of **Vi\u1e63\u1e47u** that is highly popular in southern India. Ve\u1e45ka\u1e6de\u015bvara's main shrine is in **Tirupathi** , in Andhra Pradesh, on the sacred hill of Tirum\u0101la. His gaze is believed to be so powerful that his eyes are covered. Indeed, he is also popularly known as \"B\u0101laji,\" or \"powerful one.\"\n\n**VIBH\u012a\u1e62ANA**. Character in the _R\u0101m\u0101ya\u1e47a_ ; a \"good\" **r\u0101k\u1e63asa** who is devoted to **R\u0101ma** , perceiving R\u0101ma's true, divine nature as an **avat\u0101ra** of **Vi\u1e63\u1e47u**. Vibh\u012b\u1e63ana first advises his cousin, **R\u0101va\u1e47a** , not to abduct **S\u012bt\u0101** and then to return her to R\u0101ma when she is being held as a prisoner in **La\u1e45ka** , warning that this act will bring destruction upon the r\u0101k\u1e63asas. In the end, he betrays R\u0101va\u1e47a, joining R\u0101ma's army and advising him on how best to defeat R\u0101va\u1e47a. After the battle, when S\u012bt\u0101 has been rescued and R\u0101va\u1e47a defeated, R\u0101ma gives Vibh\u012b\u1e63ana the kingship of La\u1e45ka.\n\n**VIDY\u0100.** Wisdom; liberating **knowledge** ; direct knowledge that **Brahman** , the universal reality, and **\u0101tman** , the self, are one and the same.\n\n**VIJAYAN\u0100GARA.** \"City of Victory\"; highly successful kingdom that ruled the entire southern portion of the Indian continent from 1336 to 1646. Its capital, a city of the same name, was located in **Karnataka** , near the border with Andhra Pradesh. The city was sacked after the battle of Talikota in 1565, beginning a long period of decline and gradual disintegration of the kingdom. The city remains in ruins today but still contains a great many architectural and **artistic** wonders. The kings of Vijayan\u0101gara practiced religious tolerance, supporting **Vai\u1e63\u1e47avism** , **\u015aaivism** , **\u015a\u0101ktism** , and **Jainism**.\n\n**VIJ\u00d1\u0100NABHIK\u1e62U (fl. 1550\u20131600).** Philosopher who wrote commentaries on three of the **\u0101stika** , or \"orthodox,\" systems of Indian philosophy\u2014 **Ved\u0101nta** , **S\u0101\u1e43khya** , and **Yoga**. He sought to synthesize these three systems, paving the way for future Hindu thinkers who have increasingly seen these systems as forming a unitary whole. It has been argued that some of his interpretations of these systems involved some distortion carried out in the name of his agenda of unification\u2014particularly his claim that S\u0101\u1e43khya is theistic. He was critical of the **Advaita Ved\u0101nta** doctrine of the illusory nature of phenomena ( **m\u0101y\u0101** ) and adhered to the _Bhed\u0101bheda_ interpretation of Ved\u0101nta associated with **Nimb\u0101rka** that seeks to accommodate both the **Upani\u1e63adic** teaching that all is one in **Brahman** and the realist understanding that difference is not a mere appearance, but that real metaphysical differences obtain, particularly between individual souls ( **j\u012bvas** ) and **God**. His approach to unifying diverse systems of thought anticipates **Neo-Ved\u0101nta**.\n\n**VIJ\u00d1\u0100NA-MAYA-KO\u015aA. \"** Layer made up of consciousness.\" One of the five layers or \"sheaths\" ( **ko\u015bas** ) that surround the self ( **\u0101tman** ). The vij\u00f1\u0101na-maya-ko\u015ba is between the **mano-maya-ko\u015ba** , or \"layer made up of **mind** ,\" and the **\u0101nanda-maya-ko\u015ba** , which is the inmost layer, closest to the self.\n\n**VIKRAM\u0100DITYA.** \"Sun of heroism\"; title held by several ancient Hindu kings. The first, whose ascent to power is marked by one of the most popular Hindu dating systems (the _Vikrama Sa\u1e43vat_ , which begins in 58\u201357 BCE), is believed to have lived in the first century before the Common Era and ruled from the city of **Ujjain** , which is today located in the state of Madhya Pradesh, in central India. The second major ruler to carry this title is Candragupta II, a king of the **Gupta** dynasty who successfully expanded his empire to its height, encompassing all of northern India from the **Indus** River to the **Ganges** delta.\n\n**VIM\u0100NA.** Flying **chariot** ; preferred method of locomotion among many of the **gods** ; the inspiration for such works (many would say pseudo-scientific works) as Erich Von Daniken's _Chariots of the Gods_ , which speculate that extraterrestrial astronauts visited the Earth in ancient times, providing a basis for such concepts of traditional mythology as the vim\u0101na.\n\n**V\u012a\u1e46\u0100.** Indian lute; stringed **musical** instrument. **Sarasvat\u012b** , the **goddess** of wisdom, and the sage **N\u0101rada** are both depicted carrying this instrument.\n\n**V\u012aRABHADRA.** Fierce form of **\u015aiva** , created by him to avenge the **death** of his wife, **Sat\u012b** , by destroying the sacrifice performed by her father, **Dak\u1e63a** , and beheading Dak\u1e63a in the process.\n\n**V\u012aRA\u015aAIVA.** \"Follower of the heroic **\u015aiva** \"; **\u015aaiva** tradition based in **Karnataka** and established by the **bhakti** **poet** and social reformer **Basava** (1134\u20131196). Also known as the **Li\u1e45g\u0101yats** , the V\u012bra\u015baivas oppose **casteism** and teach the equality of all human beings as manifestations of \u015aiva. Believing in holding ritualism to a bare minimum, the only image the V\u012bra\u015baivas use is the abstract form of \u015aiva, the **li\u1e45ga** (hence the alternate name _Li\u1e45g\u0101yat_ for this tradition), which they traditionally wear on a necklace. The V\u012bra\u015baivas regard \u015aiva as **God** in the **monotheistic** sense\u2014as the one supreme deity.\n\n**VI\u015aI\u1e62\u1e6c\u0100DVAITA VED\u0100NTA.** The metaphysical teaching developed by the Ved\u0101ntic **\u0101c\u0101rya R\u0101m\u0101\u1e47uja**. According to this view, all of reality is, as **Advaita Ved\u0101nta** teaches, one with **Brahman**. But unlike in the Advaitic concept of Brahman, Brahman is here conceived as having internal differences. _See_ QUALIFIED NON-DUALISM.\n\n**VI\u1e62\u1e46U.** \"The all-pervasive one\"; the single most popular Hindu deity. He is regarded by many of his devotees, who are called **Vai\u1e63\u1e47avas** , as **God** in a **monotheistic** sense. He is seen in the broader Hindu tradition as that aspect of the divine that preserves the world from chaos during the vast interval of time between its periodic **creation** and destruction. He carries out this act of preservation, as stated in the _Bhagavad G\u012bt\u0101_ (4:7), by projecting himself into the world in the form of an **avat\u0101ra** , or incarnation, to protect **dharma** and destroy the demonic forces of chaos (usually represented as demonic beings\u2014 **asuras** and **r\u0101k\u1e63asas** ). A benevolent deity, he is a popular savior and object of **bhakti** , or devotion.\n\nIn his origins, Vi\u1e63\u1e47u appears to be a solar deity, closely associated with imagery of sunlight. He is mentioned infrequently in the _Vedas_ \u2014though in one of these references he is attributed with creating the entire universe by taking three steps, which gives rise to the \"three worlds\" ( **triloka** )\u2014the Earth, the atmosphere, and the celestial realm. He is far more prominent in the later epic **literature** \u2014the _R\u0101m\u0101ya\u1e47a_ and _Mah\u0101bh\u0101rata_ \u2014and in the _Pur\u0101\u1e47as_. Several, perhaps originally distinct, deities seem to have gone into his classical form (such as **N\u0101r\u0101ya\u1e47a** ). Vi\u1e63\u1e47u is worshiped most frequently not in his celestial form\u2014as Vi\u1e63\u1e47u per se\u2014but in the form of his two most popular avat\u0101ras\u2014 **K\u1e5b\u1e63\u1e47a** and **R\u0101ma**. His wife, **Lak\u1e63m\u012b** , is also a highly popular **goddess**. She is viewed by some traditions, such as **\u015ar\u012bvai\u1e63\u1e47avism** , as essentially one with Vi\u1e63\u1e47u, with the two forming a composite deity.\n\nVi\u1e63\u1e47u is most frequently represented as a celestial monarch, holding in his four hands a conch shell ( **\u015ba\u1e45kha** ), a discus ( **cakra** ), a club, and a **lotus**. His skin is the blue color of the infinite sky and his clothing is yellow, which is symbolic of the Earth. These colors of sky and Earth represent his all-pervasive nature (which is also the meaning of his name).\n\n**VI\u1e62\u1e46UPUR (BISHNUPUR).** City in **West** **Bengal** renowned for its artisans and brick **temples** with terracotta tiles that depict events from the _R\u0101m\u0101ya\u1e47a_ and _Mah\u0101bh\u0101rata_.\n\n**VI\u015aUDDHA CAKRA.** **Cakra** that is located in the throat. _See also_ TANTRA.\n\n**VI\u015aVA HINDU PARI\u1e62AD (VHP).** \"World Hindu Council.\" This **Hindu Nationalist** organization was established in 1964. Also known as the VHP, it defines Hinduism broadly (and in keeping with **Savarkar** 's definition) as encompassing all Indic religions, and so as incorporating **Buddhism** , **Jainism** , and **Sikhism**. Of course, some Buddhists, Jains, and Sikhs do not approve of their traditions being incorporated into Hinduism by definitional fiat. But others do not object, as evidenced by the fact that some Buddhist, Jain, and Sikh religious leaders\u2014including the Dalai Lama\u2014are members of the VHP and were present at its founding.\n\nThe controversial stands that the VHP has taken include its demand that a **temple** to **R\u0101ma** be built at the site of the demolished **Babr\u012b Masj\u012bd** in **Ayodhy\u0101** (a site recently divided among Hindu and Muslim organizations by an Indian high court) and its stated desire to transform India formally into a Hindu, rather than a secular, state. Its ties to the **Bharat\u012bya Janata Party** (BJP) have proven controversial for this party. The VHP shares with the BJP the agenda of developing a \"Uniform Civil Code\" for India to replace the system of multiple civil codes for different religious communities that exists at present. The VHP is also an outspoken critic of proselytizing activities by **Christian** and **Islamic** groups directed at Hindus in India.\n\n**VI\u015aVAKARMAN.** \"Maker of all\"; the supreme architect of the universe; an epithet of **Tva\u1e63\u1e6d\u1e5b** , the divine artisan of the _Vedas_ ; one of the **\u0100dityas**. He represents creative power and **knowledge** and is attributed with creating the science of architecture ( _Vastu_ _\u015a\u0101stra_ ).\n\n**VI\u015aVAMITRA.** One of the Seven Sages ( **Saptar\u1e63i** ) born from the **mind** of **Brahm\u0101** that preside over the cosmos; an important seer or **\u1e5b\u1e63i** of the _\u1e5ag Veda_ , to whom many of the hymns of the third **ma\u1e47\u1e0dala** of this work are attributed. In the _R\u0101m\u0101ya\u1e47a_ , he is depicted as **R\u0101ma** 's and **Lak\u1e63ma\u1e47a** 's **guru** , taking them to the wilderness during their youth and teaching them to slay demonic beings. He is also the father of **\u015aakuntal\u0101**.\n\n**VI\u015aVAN\u0100THA MANDIR.** Major **temple** in **Banaras** that is dedicated to **\u015aiva**. It is controversial for not permitting entry to non-Hindus.\n\n**VI\u015aVEDEVAS.** \"All **gods** \"; a term sometimes used in the _\u1e5ag Veda_ to refer to a specific group of deities, but also used (in the same text) to refer to all gods in general. It may be that the Vi\u015bvedevas are invoked to ensure that no deity has been forgotten in the course of a ritual performance. _See also_ DEVAS.\n\n**VI\u1e6c\u1e6cHALA.** \"Standing on a brick\"; an image of **Vi\u1e63\u1e47u** that is popular in Maharashtra.\n\n**VIV\u0100HA.** The Hindu sacrament of **marriage** ; the Hindu wedding ceremony. _See also_ SA\u1e42SK\u0100RA; WOMEN.\n\n**VIVASVAT.** Name for **S\u016brya** , **god** of the sun.\n\n**VIVEKA.** Discernment; discrimination. An important intellectual virtue in **Ved\u0101nta**.\n\n**VIVEK\u0100NANDA, SW\u0100M\u012a (1863\u20131902).** Born Narendran\u0101th Datta; chief disciple of **Ramakrishna** and the first of many Hindu spiritual teachers to successfully spread their traditions in the West. A member of the **Brahmo Sam\u0101j** early in his life, and with a tendency to view religion from a skeptical perspective, Datta was quite powerfully drawn to the teaching and personality of Ramakrishna. After Ramakrishna's **death** in 1886, he joined the **Da\u015ban\u0101m\u012b Order** of monks, becoming Sw\u0101m\u012b Vivek\u0101nanda, and leading other young disciples of Ramakrishna to do the same, thus starting the **Ramakrishna Order**.\n\nIn 1893, Vivek\u0101nanda accepted an invitation to speak at the first Parliament of the World's Religions. His address to the parliament was a success and led to a speaking tour that took him across the United States\u2014where he established the first **Ved\u0101nta Society** in New York City in 1894\u2014and to Great Britain. In 1897, he initiated Margaret E. Noble as the first known Westerner to join a Hindu monastic order (from which point she came to be known as **Sister Nivedit\u0101** ). That same year, he returned to India and established the **Ramakrishna Mission**.\n\nVivek\u0101nanda's teachings in many ways reflect and in many ways go beyond the teachings of his master, Ramakrishna. His primary innovation was to direct the monks of the Ramakrishna Order to engage in work for the uplift of the poor\u2014a break with tradition inasmuch as **sanny\u0101sins** have generally been expected to pursue study, **meditation** , and **ascetic** practice exclusively. He was also the first to explicitly articulate the concept of **four yogas** , or paths to **mok\u1e63a** ( **liberation** from **rebirth** ) and to see these paths as equally valid and effective.\n\n**VRAJA.** Region of northern India in which **K\u1e5b\u1e63\u1e47a** is believed to have spent much of his life on Earth, and therefore particularly holy to **Vai\u1e63\u1e47ava** devotees. It includes **Mathur\u0101** , the city where he was born and later slew the wicked King **K\u0101\u1e43sa** , and **V\u1e5bnd\u0101vana** , the town where he lived amongst the **gop\u012bs** , or cowherdesses, during his adolescence.\n\n**VRATA.** Vow. A _vrata_ generally refers to a period of fasting undertaken for a set time either as a way to ask for a particular divine favor or as a more general act of spiritual and mental purification. Some vratas are regarded as being especially auspicious if carried out on particular holy days or at particular points on the **Hindu calendar**. A vrata may involve abstaining from particular foods (or from all food) for a set period of time, doing a particular **p\u016bj\u0101** , abstaining from sexual activity, giving gifts to the poor or to religious institutions, or going on a pilgrimage ( **yatra** ). Such vows are a form of lay **asceticism** , which is especially popular among **women**.\n\n**VR\u0100TYA.** \"One who has taken a vow\"; an ancient type of **ascetic** mentioned in both the _\u1e5ag Veda_ and the _Atharva Veda_. The descriptions of their activities in these texts suggest they were early practitioners of what would later be known as **Yoga** and **Tantra**. They may have been warrior monks. It is possible that they were precursors of the **\u015brama\u1e47a** movement, which would include **Jainism** and **Buddhism**.\n\n**V\u1e5aND\u0100VANA.** Town where **K\u1e5b\u1e63\u1e47a** resided during his adolescence, after being taken into hiding to protect him from the evil King **K\u0101\u1e43sa**. It is where K\u1e5b\u1e63\u1e47a played with the **gop\u012bs** and met his beloved **R\u0101dh\u0101** , as described in the _Bh\u0101gavata Pur\u0101\u1e47a_ and in greater detail in the _G\u012btagovinda_ of **Jayadeva**. It is highly sacred to the devotees of K\u1e5b\u1e63\u1e47a and a very popular site for pilgrimage ( **yatra** ).\n\n**V\u1e5aTRA.** \"Obstructor, restrainer\"; serpentine monster defeated by **Indra**. V\u1e5btra captured the world's cattle and its water\u2014both important symbols of prosperity\u2014keeping them in a mountain cave that he then surrounded with his enormous body, coiling it around the mountain. Indra slew V\u1e5btra with his thunderbolt weapon ( **vajra** ) and freed the cattle and the waters, both of which could then flow freely across the Earth.\n\nIn terms of **Indo-European** mythology, Indra's struggle with V\u1e5btra is comparable to Thor's battle, in Norse mythology, with the Midgard Serpent. _See also VEDAS_.\n\n**VY\u0100SA.** Sage who is attributed with compiling the _Vedas_ into their current form, a feat for which he is known as **Veda Vy\u0101sa** , and with authoring the _Mah\u0101bh\u0101rata_ , which he is said to have dictated to the deity **Ga\u1e47e\u015ba**. Vy\u0101sa is also a character in the _Mah\u0101bh\u0101rata_ , where he is said to be the son of Satyavat\u012b, the daughter of the king of the fishermen, and the sage **Par\u0101\u015bara**. He is the father of **Dh\u1e5btar\u0101\u1e63\u1e6dra** and **Pa\u1e47\u1e0du** , the kings whose sons' rivalry is the main topic of the epic. _See also BRAHMA S\u016aTRAS_.\n\n**VY\u016aHA.** \"Manifestation\"; appearance of **Vi\u1e63\u1e47u** in his full power; similar to an **avat\u0101ra** , though not all avat\u0101ras are vy\u016bhas. Some of the avat\u0101ras are **a\u1e43\u015bas** , or manifestations of merely a portion of Vi\u1e63\u1e47u's total being.\n\n## W\n\n**WAKING CONSCIOUSNESS.** One of the four **states of consciousness** described in the _Upani\u1e63ads_. The other three levels are the dream state, dreamless sleep, and the fourth state, or **tur\u012bya** , associated with spiritual transcendence and **enlightenment**.\n\n**WARRIOR.** **K\u1e63atriya** , the second of the four **var\u1e47as** or \"castes,\" the first being the **Brahmin** priesthood, and the third and fourth, respectively, being the **Vai\u1e63ya** or common people and the **\u015a\u016bdra** or servant caste.\n\n**WEST BENGAL.** Portion of **Bengal** that became part of the Republic of India after the 1947 partition of India and Pakistan. East Bengal first became East Pakistan and, after breaking away from Pakistan in a revolution in 1971, Bangladesh. East and West Bengal were first partitioned under the British **R\u0101j** in 1905.\n\n**WIDOW IMMOLATION.** _See_ SAT\u012a; WOMEN.\n\n**WIDOW REMARRIAGE.** _See_ WIDOWS; WOMEN.\n\n**WIDOWS.** Because Hindu **marriage** is traditionally seen as being for life\u2014and on some interpretations, as extending over many lifetimes\u2014the remarriage of widows is frowned upon in the _Dharma \u015a\u0101stras_. It is not clear, however, to what extent such prohibitions were observed in practice in ancient times, or if the very point of such prohibitions is to put an end to a widespread practice of which the **Brahmanical** authors of these texts do not approve. In the traditional Brahmanical schema of a human lifetime, measured out in terms of social location ( **var\u1e47a** ) and stages of life ( **\u0101\u015brama** ), the only roles assigned to a **woman** are to be a daughter to her father, wife to her husband, and mother to her children. The single adult woman, without a husband, therefore does not fit neatly into this scheme.\n\nIt has often been the case that widows have entered the **vanaprastha \u0101\u015brama** \u2014the retirement or withdrawal stage of life\u2014upon the deaths of their husbands, being cared for by their adult children. But in the case of women who are not elderly, or even young and childless\u2014not an uncommon occur-rence at an earlier time when marriages were often arranged in which a young girl was betrothed to a much older man\u2014it is often expected that they will enter a state of **sanny\u0101sa** (renunciation), becoming a Hindu nun. There are many communities of such women, particularly in the city of **Banaras** , who dedicate themselves to religious activity. Such women have often suffered exploitation of various kinds, leading the Hindu reformers of the 19th century to advocate for widow remarriage and against childhood betrothals. As a result of their efforts, widow remarriage is now more common than it used to be. But it is still regarded as a grave infraction by orthodox Hindus.\n\n**WILSON, HORACE HAYMAN (1786\u20131860).** British **Indologist** ; translator of the _\u1e5ag Veda_ ; first Boden Professor of **Sanskrit** at Oxford University.\n\n**WIND.** _See_ V\u0100YU.\n\n**WOMEN.** The roles of women in Hinduism have been far too complex throughout the centuries for any generalization to be adequate. And prior to the modern period, so many of our sources consist of prescriptive texts\u2014such as the _Dharma \u015a\u0101stras_ \u2014that reflect what their **Brahmin** authors regarded as an ideal situation that it is nearly impossible to deduce from them what the ground realities were at any given time. Were the injunctions of the Brahmins followed closely by most people? Or was the entire point of these injunctions to seek to change a ground reality that was radically different from what their authors felt should be the case? It is likely that both situations\u2014as well as a mix of both\u2014occurred at different times and in different locations. When alternative voices are available, such as in the folk tales and vernacular **literatures** of the local non- **Sanskrit** traditions, or in the inscriptions left by kings or wealthy patrons of religious institutions, they suggest a more mixed picture than do the Brahmanical texts. And such voices also intrude into the texts of the Brahmins from time to time\u2014just as, with regard to the issue of **caste** , one can find a **Satyak\u0101ma** or a **V\u0101lm\u012bki** , as well as an **Ekalavya**. If the reader will bear all of this in mind, what follows can be understood for what it is\u2014a very broad set of generalizations based on the sources that are available in abundance (Brahmanical Sanskrit texts) that can sometimes reflect and can sometimes be in radical discontinuity with the social realities of actual Hindu women.\n\nThe overall orientation of the Brahmanical tradition is decidedly patriarchal. The system of **var\u1e47as** (\"castes\") and **\u0101\u015bramas** (stages of life) defines the ideal life trajectory of a male. It is males who traditionally receive the **sacred thread** ( **upanayana** ) and who take up an occupation appropriate to their caste after **marriage** and entering the stage of the **householder**. They then retire, entering the **vanaprastha** stage. Alternatively, they may, at any point, renounce the life of a layperson, undertaking **sanny\u0101sa**.\n\nThe **dharma** of a woman, however, is to be a daughter to her father, a wife to her husband, and a mother to her children. A woman may enter the retirement phase along with her husband, or have it forced upon her by **widowhood**. Her social role is entirely defined in terms of her relationships with men\u2014her father in childhood, her husband as a married woman, and her sons, first as their caretaker, and then, perhaps, as the recipient of their care in her old age. Her daughters, if she has any, will join the families of their husbands and their first duty will be to their husbands and their husbands' parents.\n\nAgain, the degree to which this Brahmanical ideal has been observed in practice is open to question. Certainly in the modern period it has been observed quite rigorously in many settings\u2014particularly rural settings\u2014but also set aside quite decisively, particularly among the wealthy and well educated. Premodern exceptions\u2014independently wealthy women who were patrons of particular religious communities, and who were more often than not courtesans by trade\u2014did exist.\n\nOn the classical model, the only route for a woman to live independently\u2014and for which a good deal of evidence exists\u2014is the religious life. In the religious arena, women could attain not only a degree of independence from male domination but could even rise to a considerable level of social authority, wielding power\u2014even over men\u2014as **gurus** and as highly revered saints. Figures such as **\u0100\u1e47\u1e0d\u0101l** , **Mah\u0101deviyakka** , and **M\u012bra Bai** could transgress their traditional social roles because they did so out of love for **God**.\n\nThis model continues into the modern period, with such figures as **\u0100nandam\u0101y\u012b M\u0101** , **M\u0101t\u0101 Amrit\u0101nandamay\u012b** , and **Gurumay\u012b** being highly revered and independent. It may be argued that the price of such independence is the total religious commitment that these women's lives display. But we would also be erring if we presumed that the sole motive of these women has been personal independence, and has not included a spiritual aspiration of a fairly traditional sort\u2014a desire for **mok\u1e63a** or to be nearer to God.\n\nLike the discourses of democracy and human rights, however, feminist discourse has also made inroads into Hinduism. Though it is in many ways a deeply conservative tradition, it has also proven to be highly adaptive through the centuries. In recent years, on the basis of the same doctrine of the unity of all beings that has served to undercut the caste system, women have increasingly made inroads into former bastions of orthodoxy, engaging in public reading of the _Vedas_ , for example, receiving the sacred thread, and, as more women in urban areas have taken up jobs outside of the home\u2014either out of desire or necessity\u2014being seen as co-householders with their husbands.\n\nAt the same time, the heavy hand of patriarchy is still very much in evidence. A traditional preference for sons over daughters has led to a widespread practice of aborting female fetuses. And the ongoing practice of dowry\u2014in which the family of a new bride is expected to pay a considerable fee to the family of her husband\u2014fuels the preference for sons, because sons will bring money into the family when they marry, whereas daughters will cost money. Dowry **death** , too, is a result of this practice\u2014in which families conspire to marry their sons to women, extract a dowry from the woman's family, and then murder the woman, sometimes doing this serially.\n\nThe practice of **sat\u012b** , or widow immolation\u2014which many non-Hindus wrongly take to be a widespread and current practice\u2014is nearly unheard-of. The last such event, at least to receive publicity, occurred in 1987, and received widespread condemnation from the wider Hindu community. The practice was banned in 1829, due largely to the activism of the Hindu reformer **R\u0101m Mohan Roy**.\n\nThe presence of a vigorous **Goddess** tradition, and a wide range of assertive and powerful **goddesses** , might suggest a broader feminist streak in Hinduism than is actually the case. But the worship of goddesses and the roles of actual women have largely been independent variables.\n\nThe range of options available for traditional Hindu women is often discussed in terms of the contrast between the heroines of the _R\u0101m\u0101ya\u1e47a_ and _Mah\u0101bh\u0101rata_ \u2014 **S\u012bt\u0101** and **Draupad\u012b** , respectively. S\u012bt\u0101 is obedient to every wish of her husband, even to the point of suffering the indignity of the fire test, when she has all along been perfectly faithful to him. She suffers both through her husband's troubles and through the indignity society imposes upon her without complaint. Draupad\u012b is also a faithful wife who suffers with her husbands. But when her rights are violated, when her husband does not protect her honor but, quite literally, gambles it away as though it were a possession, she speaks up for her rights, not only questioning the rightness of her husband's choices but also displaying a considerable **knowledge** of dharma. Of the two, S\u012bt\u0101 has generally been upheld as the model Hindu woman. But the model of Draupad\u012b suggests another way, from within the Hindu tradition, for a woman to live\u2014with dignity and equality.\n\n**WORK.** _See_ KARMA; KARMA YOGA.\n\n**WORLD AGES.** Hindu traditions conceive of time as flowing on a vast, cosmic scale. Strictly speaking, on a mainstream Hindu understanding\u2014as shaped by the _Pur\u0101\u1e47as_ _\u2014_ there is no precise beginning (or end) of time. In terms of **Advaita Ved\u0101nta** , time itself is an illusion, the only reality being the eternal, timeless, and formless **Brahman**. But within the realm of **m\u0101y\u0101** , of the appearance of time and space, the universe is continuously and repeatedly created, endures, is destroyed, and is re-created. In the _Pur\u0101\u1e47as_ and _Dharma \u015a\u0101stras_ , these vast cycles are conceived as days and nights of **Brahm\u0101** , the creator **god**. Each day of Brahm\u0101\u2014a **kalpa** or cosmic epoch\u2014is divided into 14 \"ages of **Manu** \" or Manvantaras, each of which is presided over by a Manu, or being born from the **mind** of Brahm\u0101. It is also divided into 1,000 _Mah\u0101yugas_ , or \"great eons.\" Each Mah\u0101yuga is made up of four **yugas** , or eons. Each yuga within a Mah\u0101yuga is somewhat worse (but also, mercifully, shorter) than the one before it. The K\u1e5bta or **Satya** yuga is equal to 4,000 divine years (which equals 518,400,000 terrestrial years). The Tret\u0101 yuga lasts 3,000 divine years (388,800,000 terrestrial years). The Dv\u0101para yuga lasts 2,000 divine years (259,200,000 terrestrial years). The Kali yuga (which we currently inhabit) lasts 1,000 divine years (129,600,000 terrestrial years). Between the yugas are intermediate periods that last 800, 600, 400, and 200 divine years. An entire Mah\u0101yuga is therefore equal to 12,000 divine years, or 1,555,200,000 terrestrial years, and an entire kalpa or day of Brahm\u0101 is equal to 12,000,000 divine years, or 1,555,200,000,000 terrestrial years. After 100 years of Brahm\u0101\u2014that is, 36,500 days of Brahm\u0101, which equals a cosmic cycle\u2014a **pralaya** , or dissolution, occurs. At the end of a cosmic cycle, the forces of chaos and entropy overcome the forces of order, or dharma, and the universe dissolves, returning to a potential state. There follows the great sleep, or _mah\u0101pralaya_ , which also lasts for 100 years of Brahm\u0101, after which a new **creation** will emerge and a new age of Brahm\u0101 will begin.\n\n**WORLD COUNCIL OF THE \u0100RYA SAM\u0100J.** Organization distinct from the **\u0100rya Sam\u0101j** , based on the ideals and teachings of **Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b** , and founded by **Sw\u0101m\u012b Agnive\u015ba** , who became president of the organization in 2004.\n\n**WORLD GUARDIANS.** _See_ LOKAP\u0100LAS.\n\n## Y\n\n**YADAVA.** \"Descendent of Yadu\"; the family lineage of **K\u1e5b\u1e63\u1e47a** , of which his father, **Vasudeva** and his aunt, **Kunti** , were both members. The city of the Yadavas, **Dv\u0101rak\u0101** , tragically sank into the ocean at the end of the _Mah\u0101bh\u0101rata_.\n\n**YAJ\u00d1A.** Sacrifice; the central **Vedic** ritual. The most ancient Vedic sacrifices consisted of offerings to the **gods** , or **devas** , in return for which they hoped to attain specific goods, usually of a material sort\u2014health, wealth, long life, security, good weather, and so on. In the _Br\u0101hma\u1e47as_ , one begins to see a more sophisticated theory of sacrifice develop. No longer a mere transaction, the sacrifice comes to be seen as something that the gods _need_ , making the human performance of sacrifice an essential element in the smooth running of the cosmos. Sacrifice becomes vital to the upholding of **\u1e5bta** , the cosmic order\u2014a concept later denoted with the term **dharma**. In the _Upani\u1e63ads_ , there is a divergence from this ritual concept. The ritual is internalized. It becomes a metaphor for all of life. Breath is a kind of sacrifice. Speech is a kind of sacrifice. Sexual activity is a kind of sacrifice. **Actions** of all kinds\u2014 **karma** \u2014are seen as yielding definite effects in the world, and not only the ritual action of the sacrificial arena. This leads to a gradual devaluation of the ritual of sacrifice and its replacement with practices such as **meditation** and worship ( **p\u016bj\u0101** ).\n\nIt is probably not a coincidence that the offering of **animal sacrifice** is viewed with increasing disfavor over the course of this development, with **vegetarian** substitutes eventually becoming far more prevalent than the ritual offering of slain animals. By the time of the _Bhagavad G\u012bt\u0101_ , sacrifice is seen in at least some Hindu traditions as a matter of mentally offering all of one's actions to **God** as a form of devotion ( **bhakti** ), or as an act of purification in preparation for the knowledge ( **j\u00f1\u0101na** ) that everything is **Brahman**.\n\n**Y\u0100J\u00d1AVALKYA.** **Brahmin** sage greatly revered for his wisdom; a major character in the _B\u1e5bhad\u0101ra\u1e47yaka Upani\u1e63ad_. The first explicit articulation of the doctrines of **karma** and **rebirth** in a Hindu text is placed in the mouth of Y\u0101j\u00f1avalkya, in a conversation with the sage King **Janaka**.\n\n_YAJUR VEDA_. The second **Sa\u1e43hit\u0101** of the _Vedas_ , likely composed between 1200 and 1000 BCE, according to mainstream scholarship. The main concern of this text is correct performance of the ritual of sacrifice ( **yaj\u00f1a** ). It exists in two distinct recensions called the \"Black\" or _K\u1e5b\u1e63\u1e47a Yajur Veda_ and the \"White\" or _\u015aukla Yajur Veda_.\n\n**YAMA.** **Vedic** deity of **death** and the underworld. It is said that Yama attained this distinction by being the first human being to die, and therefore the first inhabitant of the underworld. Yama is one of the **Lokap\u0101las** , or \"World Guardians.\" _See also_ DEVAS.\n\n**YAMAS.** Restraints; five moral principles set forth in the _Yoga S\u016btras_ of **Pata\u00f1jali** that together make up the first stage or \"limb\" of his eight-limbed ( **a\u1e63\u1e6d\u0101\u1e45ga** ) system of **yoga**. The five yamas are **ahi\u1e43s\u0101** (nonviolence), **satya** (truthfulness), **asteya** (nonstealing), **brahmacarya** (control of sexual desires and activities), and **aparigraha** (detachment).\n\nThe yamas provide evidence that the yoga system of Pata\u00f1jali shares a common cultural milieu with the **ascetically** oriented **\u015brama\u1e47a** traditions, **Jainism** and **Buddhism**. The content and order of the yamas are identical with those of the **vratas** (moral vows) of Jainism and with four of the Five Precepts of Buddhism.\n\n**Y\u0100MUNA (fl. 11th century CE). \u015ar\u012bvai\u1e63\u1e47ava** theologian; grandson of N\u0101thamuni, a scholar who introduced the **poetry** of the **\u0100\u1e37v\u0101rs** into formal **temple** worship in the 10th century. Y\u0101muna translated many of the poems into **Sanskrit** , contributing to the process begun by his grandfather of formalizing the systematization of the \u015ar\u012bvai\u1e63\u1e47ava tradition, a process completed by **R\u0101m\u0101\u1e47uja** (1077\u20131157). _See also_ VAI\u1e62\u1e46AVISM.\n\n**YAMUN\u0100 RIVER.** River that flows from the **Him\u0101layas** , past **New Delhi** , and through the **Vraja** region associated with **K\u1e5b\u1e63\u1e47a**. It joins the **Ga\u1e45g\u0101** at **Pray\u0101ga** ( **Allahabad** ).\n\n**YANTRA.** Abstract, diagrammatic representation of a Hindu deity, often used as a focus of **meditation** , particularly in **Tantric** practice. _See also_ MA\u1e46\u1e0cALA.\n\n**YATRA.** Pilgrimage; travel to a sacred place. Making such a journey, particularly if it involves some hardship, is a form of meritorious **action**.\n\n**YOGA.** Literally, \"yoke\"; spiritual discipline; path to **mok\u1e63a** ( **liberation** from **rebirth** ). There is speculation that yoga may have originated as a form of military training among **K\u1e63atriya** warriors. It includes, but is far from limited to, the physical exercises of **Ha\u1e6dha Yoga** that come to the minds of most Westerners who hear or use this term. In classical Indian texts, the term often refers to a total way of life, and is akin, in its semantic range, to the term _religion_. It also refers to a specific system of spiritual practice\u2014the **A\u1e63\u1e6d\u0101\u1e45ga** , or \"eight-limbed\" system of **Pata\u00f1jali** , outlined in his _Yoga S\u016btra_. In the modern period, **Sw\u0101m\u012b Vivek\u0101nanda** identifies this system with the R\u0101ja Yoga of the _Bhagavad G\u012bt\u0101_. In Vivek\u0101nanda's usage\u2014based, again, on the _G\u012bt\u0101_ \u2014yoga encompasses such activities as working for the good of humanity, devotional activity ( **bhakti** ), and study for cultivation of wisdom ( **j\u00f1\u0101na** ). _See also_ FOUR YOGAS; YOG\u012a; YOGIN\u012a.\n\n**YOGA DAR\u015aANA**. System of philosophy based on the **A\u1e63\u1e6d\u0101\u1e45ga** or \"eight-limbed\" type of **yoga** outlined in **Pata\u00f1jali** 's _Yoga S\u016btra_. Within the system of six orthodox or **Vedic** ( **\u0101stika** ) systems of philosophy or **dar\u015banas** , the Yoga dar\u015bana is traditionally paired with the **S\u0101\u1e43khya** system, with which it is almost identical, being related to S\u0101\u1e43khya in the way that a practice is related to the theory that underlies it. The chief difference between the S\u0101\u1e43khya and Yoga worldviews is the incorporation into the Yoga system of the idea of **God** or **\u012a\u015bvara**. Contemporary scholars debate the degree to which this incorporation of \u012a\u015bvara into the Yoga system reflects a serious commitment to theism on the part of Pata\u00f1jali (such as that found in the **Vai\u1e63\u1e47ava** traditions), or is more of an afterthought. The role of \u012a\u015bvara in the _Yoga S\u016btra_ is not to act as creator or savior of the world. He is solely an object of contemplation ( **\u012a\u015bvara-pra\u1e47idh\u0101na** ), which is one of the five **niyamas** or observances that form the second stage or \"limb\" of Pata\u00f1jali's system.\n\n_YOGA S\u016aTRA_ **.** Root text of the **Yoga dar\u015bana** , or system of philosophy. Attributed to **Pata\u00f1jali** , the text was composed some time between 100 BCE and 500 CE. It outlines the eight stages or \"limbs\" of the \"eight-limbed\" ( **A\u1e63\u1e6d\u0101\u1e45ga** ) system of yoga.\n\n**YOG\u0100NANDA, PARAMAH\u0100\u1e42SA (1893\u20131952).** **Bengali** spiritual teacher and founder of the **Self-Realization Fellowship** (SRF); author of the classic _Autobiography of a Yogi_ , a popular work among Westerners drawn to Indic traditions; after **Sw\u0101m\u012b Vivek\u0101nanda** , the second in a series of Hindu **gurus** who have found a following in the West. Coming to the United States in 1920 for a conference of religious liberals, Yog\u0101nanda remained in the country and established the Self-Realization Fellowship that same year to disseminate the practice of _Kriya_ _Yoga_ , into which he had been initiated by his guru, **Sri Yukte\u015bwar Giri**. Among the distinctive aspects of the Self-Realization Fellowship is its emphasis on devotion to Jesus as an **enlightened** master. As chronicled in his famous autobiography, Yog\u0101nanda met a number of important figures of modern Hinduism in his lifetime, such as **Mohand\u0101s K. G\u0101ndh\u012b** , **Rabindran\u0101th Tagore** , and **\u0100nandam\u0101y\u012b M\u0101**.\n\n_YOGA-V\u0100SI\u1e62\u1e6cHA-R\u0100M\u0100YA\u1e46A_ **.** Also known simply as the _Yoga-V\u0101si\u1e63\u1e6dha_ , this text is a presentation of **Advaita Ved\u0101nta** in the form of the teaching of the sage **Vasi\u1e63\u1e6dha** to a young **R\u0101ma**.\n\n**YOGE\u015aVARA.** Lord of **Yoga** ; an epithet of both **\u015aiva** and **K\u1e5b\u1e63\u1e47a** ; in the _Yoga S\u016btra_ of **Pata\u00f1jali** , **God** as an object of **meditation**. It is not clear from the text if Pata\u00f1jali has in mind any specific sectarian conception of divinity (like \u015aiva or K\u1e5b\u1e63\u1e47a) or a more generic conception into which any deity preferred by the practitioner ( **i\u1e63\u1e6dadevat\u0101** ) could fit.\n\n**YOG\u012a.** A male practitioner of **yoga**.\n\n**YOGIN\u012a.** A female practitioner of **yoga** ; a female **yog\u012b**.\n\n**YONI.** Womb; the female reproductive organ; often depicted in an abstract iconographic form, and frequently in conjunction with the **li\u1e45ga**. The joining of the yoni, a symbol of female fertility and of the **Mother Goddess** ( **\u015aakti** ) with the li\u1e45ga (often interpreted as a phallic symbol of **\u015aiva** ) symbolizes both fertility and creativity in general, with the union of \u015aiva and \u015aakti representing the conjoining of pure consciousness and creative principle (symbolized in **S\u0101\u1e43khya** philosophy by **puru\u1e63a** and **prak\u1e5bti** , respectively).\n\n**YUGA.** _See_ WORLD AGES.\n\n**YUKTE\u015aWAR GIRI, SRI (1855\u20131936).** A spiritual teacher from **Bengal**. Although he has his own following, Yukte\u015bwar is best known for being the **guru** of **Paramah\u0101\u1e43sa Yog\u0101nanda**.\n\n## Z\n\n**ZAEHNER, ROBERT CHARLES (1913\u20131974).** **Indologist** ; known for his translation of the _Bhagavad G\u012bt\u0101_.\n\n**ZARATHUSTRA (fl. 1100 BCE).** Founder of **Zoroastrianism**.\n\n**ZED, RAJAN.** Nevada-based Hindu priest. On 12 July 2007, Zed made history as the first chaplain to recite a Hindu prayer for the opening ceremonies of the U.S. Senate. Prior to this, Zed had opened the Alaska state legislative session. Zed's prayer for the U.S. Senate was met with some protest from fundamentalist **Christians**. The prayer consisted of the recitation of an English translation of the **G\u0101yatr\u012b mantra**.\n\n**ZOROASTRIANISM.** One of the world's oldest religions, Zoroastrianism is practiced today by a small community (approximately 150,000 adherents) in India. The official religion of the Persian Empire for over a thousand years, Zoroastrianism had a profound impact upon the Abrahamic religions\u2014Judaism, **Christianity** , and **Islam**. It also shares historical roots with Hinduism and is known to have influenced Mah\u0101y\u0101na **Buddhism**. The founder of Zoroastrianism, **Zarathustra** , was a priest in Iran during what, in India, was the **Vedic** period. The religion of Iran at the time was a close relative of the Vedic religion, with its hymns and its rituals of sacrifice ( **yaj\u00f1a** ). This is because the culture of ancient Iran was, like that of the Vedic tradition, **Indo-European**. The pre-Zoroastrian Iranians worshiped deities called _daevas_ with names such as _Intar_ (cognate with **Indra** ), _Uruwana_ ( **Varu\u1e47a** ), _Haoma_ ( **Soma** ), and _Mithra_ ( **Mitra** ). Supreme over all of these deities was the \"Wise Lord\"\u2014 **Ahura Mazda** (whose name is cognate with the **Sanskrit** term **asura** , which did not yet have demonic connotations at this early period).\n\nAccording to the Zoroastrian **scriptures** , Zarathustra received divine revelations from Ahura Mazda, who was revealed to be the **One** , supreme **God** and creator of the universe. Zarathustra's teachings also included the idea of linear time\u2014with a beginning and an end\u2014a single lifetime for each human being followed by a judgment, an afterlife of reward or punishment until the end of time for good or bad behavior, and a resurrection and restoration of all beings to an everlasting earthly paradise at the end of time. When **Islam** became the dominant religion of Iran in the seventh century of the Common Era, many Zoroastrians fled to India, where they are known as _Parsis_.\n\n# Bibliography\n\n## INTRODUCTION\n\nTo compile a comprehensive bibliography of Hinduism is a daunting\u2014and arguably an impossible\u2014task. In terms of primary sources, Hinduism is a tradition\u2014or rather, a family of traditions\u2014that has been producing texts for over 3,000 years. The \"classical\" or premodern sources that survive are largely composed in Sanskrit. However, there are also a variety of vernacular literatures, such as the popular devotional poetry of the bhakti movement, composed in other Indic languages, such as Hindi, Bengali, Tamil, Gujarati, and Marathi, to name just a few of the most prominent ones. And there is, of course, the Hindu writing of the modern period, much of which is in English, though the vernacular languages of India continue to be rich loci of Hindu literary and philosophical activity, as does Sanskrit (though for a small audience, despite periodic resurgences of interest).\n\nSecondary sources on Hinduism also abound. With a history beginning in the late 18th century, with such pioneers as Sir William Jones, the scholarly discipline of Indology (often termed, in the United States, South Asian Studies) has produced a great abundance of historical, philological, anthropological, sociological, and philosophical writing\u2014a process that continues unabated today.\n\nWhat follows is therefore not comprehensive but seeks to give a broad overview of at least the major works in this field, both primary and secondary. The major division this bibliography makes is between primary and secondary sources. The primary sources have been divided into original works composed in Sanskrit and other Indic languages, translations of Sanskrit and other original works, and primary sources written in English. The original works in Sanskrit and other Indic languages are often anonymous and have been listed alphabetically by title. In cases where the name of an author is known (or is traditionally attributed), this has been indicated with the phrase \"of so-andso\" after the title (e.g., _The R\u0101m\u0101ya\u1e47a of V\u0101lm\u012bki_ ). The translations have been listed alphabetically by the last name of the translator.\n\nThe modern primary sources in English include writings by Western authors who have identified with or been deeply influenced by Hindu traditions, as well as sources by major Hindu figures or representatives of modern Hindu movements.\n\nThe secondary sources are divided by their subject matter and approach into: (a) broad surveys of and introductions to Hinduism; (b) historical studies; (c) textual studies; (d) studies of practice (primarily ritual and ethics); (e) studies of particular deities; (f) studies of the arts; (g) studies of Hindu society; (h) constructive studies of Hinduism\u2014that is, studies that reflect philosophical or theological methodologies, or that have philosophy or theology as their subject matter; (i) studies of particular Hindu traditions; (j) studies of Hinduism outside South Asia; (k) works that focus on broad theoretical concerns; and (l) studies of other religious traditions that are closely related to Hinduism.\n\nThe first category\u2014broad surveys and introductions\u2014also includes reference works and other materials useful to beginners in the field of Hindu studies. Historical studies range from histories of the Indian subcontinent, to studies of the history of particular regions or movements, to biographies of individuals.\n\nTextual studies often have relevance to other areas but are focused primarily on understanding a specific text or use a specific text or set of texts as the chief lens through which an issue is viewed. Studies of practice include ethnographic works and other primarily descriptive works on ritual and ethics.\n\nStudies of deities are distinguished from textual studies and studies of particular Hindu subtraditions by the fact that their primary subject matter is the deity in question, rather than the texts or traditions centered on that deity. Studies of the arts include art histories, works on iconography, and works on both drama and dance. Studies of Hindu society include both theoretical and ethnographic studies of phenomena such as caste, Hindu nationalism, and women's issues.\n\nStudies of philosophies and theologies include both descriptive works\u2014histories of Indian philosophy\u2014and normative, constructive works, in which the author identifies with and seeks to advance a philosophical or religious point of view. It is sometimes difficult to distinguish between the latter type of work and modern primary sources in English. Generally, works by figures who have come to be seen as major founders of movements or as exemplars of such movements (such as Sita Ram Goel as a representative of Hindu nationalism) have gone into the category of primary sources. Works by professional scholars of religion or philosophy, however, have also gone into this category.\n\nStudies of particular Hindu traditions include works on Vai\u1e63\u1e47avism, \u015aaivism, Tantra, and so forth, as distinct from broader surveys of Hinduism as a whole.\n\nStudies of Hinduism outside of South Asia include works on Hinduism in areas such as Southeast Asia, where Hindu traditions have been practiced for many centuries, as well as in the Western world, where such traditions are a more recent phenomenon, as a result of both Hindu immigration and Westerners being drawn to particular Hindu ideas and practices.\n\nWorks that focus on broad theoretical concerns include works that engage with issues such as the validity of the category of Hinduism and the relationship between the study of India and issues of colonization and power.\n\nThe studies of traditions related to Hinduism include work on such traditions as Jainism, Buddhism, and Sikhism, which emerged either from or in close dialogue with Hindu traditions, as well as works on both Hindu\u2013 Christian and Hindu\u2013Muslim relations.\n\nFinally, I have included a section on Internet materials for the study of Hinduism, which can themselves be divided into primary and secondary sources, depending on the point of view that a given website takes. Avowedly Hindu websites that seek to advance a particular Hindu perspective are primary sources, in contrast with purely scholarly sites, which strive to present information in a more or less objective manner.\n\nAmong the many libraries with excellent collections relating to Hinduism, one of the most renowned is that of the Bhandarkar Oriental Research Institute in Poona, a city famous for its centers of language and textual study. Another is the Indian Institute of the Bodleian library at Oxford University and the University of Chicago's Regenstein library, both of which have proven invaluable in my own research.\n\nThe University of Chicago also provides an exceptionally useful Internet resource in the form of its _Digital South Asia Library_ (<http:\/\/dsal.uchicago.edu\/reference\/>), which includes a vast collection of online materials, such as audio samples from the Linguistic Survey of India and digital versions of many journals and reference works that are listed in this bibliography, such as Schwartzberg's _Historical Atlas of India_. And as a primary source, _The Hindu Universe_ (<http:\/\/www.hindunet.org\/>) is especially rich.\n\nClearly, not all works fit neatly or obviously into any of these categories. It is not unlikely that some readers will take issue with the placement of certain works. (Indeed, I have argued with myself about some of these placements and am open to persuasion that some should be located in categories other than where they are currently.) The categories themselves are also less than perfect, but they are the ones that emerged organically as I compiled these titles, and most works do seem to lend themselves to placement in one of these categories.\n\nAt the beginning of each section, beginning with English translations of primary sources, I have included a brief paragraph in which I recommend texts from within that section that may be particularly useful or accessible to beginners in this field.\n\n## I. PRIMARY SOURCES\n\nA. Sources in Sanskrit and Other Indic Languages\n\n_Abhij\u00f1\u0101na\u015bakuntala of K\u0101lid\u0101sa_. Bombay: Nirnaya Sagara Press, 1958.\n\n_Adhy\u0101tma R\u0101m\u0101ya\u1e47a_. Calcutta: Metropolitan Printing and Publishing House, 1935.\n\n_Agni Pur\u0101\u1e47a_. Poona: Anandasrama Sanskrit Series, 1957.\n\n_Aitareya Br\u0101hma\u1e47a_. Calcutta: Bibliotheca Indica, 1895.\n\n_Apastamba Dharma S\u016btra_. G. B\u00fchler, ed. Bombay: Bombay Sanskrit Series 44 and 50, 1892, 1894.\n\n_Artha\u015b\u0101stra of Kau\u1e6dilya_. R. P. Kangle, ed. Bombay: University of Bombay, 1960.\n\n_Atharva Veda_. Hoshiarpur: Vishveshvaranand Vedic Research Institute, 1960.\n\n_Baudhayana Dharma S\u016btra_. C. Sastri, ed. Benares: Kashi Sanskrit Series 104, 1934.\n\n_Bhagavad G\u012bt\u0101_. (See _Mah\u0101bh\u0101rata_.)\n\n_Bh\u0101gavata Pur\u0101\u1e47a_. Benares: Pandita Pustakalaya, 1972.\n\n_Brahma S\u016btra Bh\u0101\u1e63ya of Madhva_. Tirupati: Tirumala-Tirupati-Devasthanena, 1893.\n\n_Brahma S\u016btra Bh\u0101\u1e63ya of R\u0101m\u0101\u1e47uja (\u015ar\u012b Bh\u0101\u1e63ya)_. Swami Vireswarananda and Swami Adidevananda, eds. Calcutta: Advaita Ashrama, 2008.\n\n_Brahma S\u016btra Bh\u0101\u1e63ya of \u015aa\u1e45kara_. Bombay: Nirnaya Sagara Press, 1948.\n\n_Brahm\u0101\u1e47\u1e0da Pur\u0101\u1e47a_. Bombay: Venkateshvara Steam Press, 1857.\n\n_Brahmavaivarta Pur\u0101\u1e47a_. Poona: Anandasrama Sanskrit Series, 1935.\n\n_B\u1e5bhaddevat\u0101 of \u015aaunaka_. Cambridge, Mass.: Harvard University Press, 1904.\n\n_Buddhacarita of A\u015bvagho\u1e63a_. E. H. Johnston, ed. Calcutta: Panjab University Oriental Publications, 1935\u20131936.\n\n_Da\u015b\u0101vat\u0101racarita of K\u1e63emendra_. Bombay: Kavyamala Series, 1891.\n\n_Dev\u012bbh\u0101gavata Pur\u0101\u1e47a_. Benares: Pandita Pustakalaya, 1960.\n\n_Garu\u1e0da Pur\u0101\u1e47a_. Benares: Pandita Pustakalaya, 1969.\n\n_Gautama Dharma\u015b\u0101stra_. New Delhi: Veda Mitra, 1969.\n\n_G\u012bt\u0101 Govinda of Jayadeva_. Hyderabad: Sanskrit Academy Series, 1969.\n\n_Hariv\u0101\u1e43\u015ba_. Poona: Bhandarkar Oriental Research Institute, 1969.\n\n_Har\u1e63acarita of Ba\u1e47a_. Bombay: Bombay Sanskrit and Prakrit Series, 1909.\n\n_Jaimin\u012bya Br\u0101hma\u1e47a_. Nagpur: Sarasvati-vihara Series, 1954.\n\n_J\u0101takam\u0101la of \u0100rya\u015bura_. Delhi: Motilal Banarsidass, 1971.\n\n_K\u0101lika Pur\u0101\u1e47a_. Sri Biswanarayan Sastri, ed. Varanasi: Chowkhamba Sanskrit Series Office, 1972.\n\n_Kalki Pur\u0101\u1e47a_. Mathura: Jai Nitai Press, 2006.\n\n_K\u0101mas\u016btra of Vatsyayana_. Varanasi: Kashi Sanskrit Series, 1964.\n\n_Ka\u1e6dhasarits\u0101gara_. Bombay: Nirnara Sagara Press, 1930.\n\n_Kau\u1e63\u012btaki Br\u0101hma\u1e47a_. Calcutta: Bibliotheca Indica, 1903.\n\n_Kurma Pur\u0101\u1e47a_. Varanasi: All-India Kashiraj Trust, 1972.\n\n_Li\u1e45ga Pur\u0101\u1e47a_. Calcutta: Sri Arunodaraya, 1812.\n\n_Madhvavijaya of N\u0101r\u0101ya\u1e47a Pa\u1e47\u1e0dit\u0101c\u0101rya_. Vishakhapatnam: Shrimadananda Tirtha Publications, 1983.\n\n_Mah\u0101bh\u0101gavata Pur\u0101\u1e47a_. Bombay: Venkateshvara Steam Press, 1913.\n\n_Mah\u0101bh\u0101rata_. Poona: Bhandarkar Oriental Research Institute, 1933\u20131969.\n\n_Mah\u0101nirv\u0101\u1e47a Tantra_. Madras: Tantrik Texts, 1929.\n\n_Maitraya\u1e47i Sa\u1e43hit\u0101_. Wiesbaden, Germany: R. Steiner, 1970\u20131972.\n\n_Manusm\u1e5bti_. Bombay: Bharatiya Vidya Series, 1972\u20131978.\n\n_Nai\u1e63ad\u012byacarita of \u015ar\u012b Har\u1e63a_. Bombay: Nirnaya Sagara Press, 1896.\n\n_Nirukta of Yaska_. Lakshman Sarup, ed. London: Oxford University Press, 1920\u20131927.\n\n_Priyadar\u015bika of Har\u1e63a_. M. R. Kale, ed. Bombay: Motilal Banarsidass, 1928.\n\n_P\u016brvam\u012bm\u0101\u1e43s\u0101 S\u016btra of \u015aabarasv\u0101min_. Hirayana: Ramlal Kapar, 1986.\n\n_R\u0101macaritam\u0101nasa of Tuls\u012bd\u0101s_. Delhi: Motilal Banarsidass, 1990.\n\n_R\u0101m\u0101ya\u1e47a of Valm\u012bki_. Baroda: Oriental Institute, 1960\u20131975.\n\n_\u1e5ag Veda_. London: Oxford University Press, 1890\u20131892.\n\n_Sarvadar\u015banasa\u1e43graha of Madhava_. London: Tr\u00fcbner, 1914.\n\n_\u015aa\u1e45karadigvijaya of Madhava_. Poona: Bhandarkar Oriental Research Institute, 1915.\n\n_\u015aatapatha Br\u0101hma\u1e47a_. Benares: Chowkhamba Sanskrit Series, 1964.\n\n_Saty\u0101rth Prak\u0101\u015b of Sw\u0101m\u012b Day\u0101nanda Sarasvat\u012b_ (Reprint). Delhi: Manoj Publications, 2009.\n\n_Saura Pur\u0101\u1e47a_. Poona: Anandashrama Sanskrit Series, 1923.\n\n_\u015aiva Pur\u0101\u1e47a_. Benares: Pandita Pustakalaya, 1964.\n\n_Skanda Pur\u0101\u1e47a_. Bombay: Shree Venkateshvara Steam Press, 1867.\n\n_\u015ar\u012b R\u0101m\u0101\u1e47uja G\u012bt\u0101 Bh\u0101\u1e63ya_. Chennai: Sri Ramakrishna Math, 1992.\n\n_Taittiriya Br\u0101hma\u1e47a_. Rajendralal Mitra, ed. Calcutta: Bibliotheca Indica, 1859.\n\n_Taittiriya Sa\u1e43hit\u0101_. Poona: Anandashrama Sanskrit Series, 1979.\n\n_Upade\u015basahasr\u012b of \u015aa\u1e45kar\u0101c\u0101rya_. Mylapore, Madras: Sri Ramakrishna Math, 1961.\n\n_Upani\u1e63ads_. Bombay: Nirnaya Sagara Press, 1913.\n\n_Vajasaneyi Sa\u1e43hit\u0101_. Varanasi: Chowkhamba Sanskrit Series, 1972.\n\n_V\u0101mana Pur\u0101na_. Benares: All-India Kashiraj, 1968.\n\n_V\u0101yu Pur\u0101\u1e47a_. Poona: Anandashrama Sanskrit Series, 1860.\n\n_Vi\u1e63\u1e47u Pur\u0101\u1e47a_. Calcutta: Sanatana Shastra, 1972.\n\nB. English Translations of Primary Sources (Including Compendia)\n\nGeorge Thompson's translation of the _Bhagavad G\u012bt\u0101_ is excellent\u2014a translation of this important text that has the exceptionally rare virtue of being both true to the original Sanskrit and highly readable for the native speaker of English. The same is true of Barbara Stoler Miller's translations of both the _Bhagavad G\u012bt\u0101_ and the _Yoga S\u016btras_ of Pata\u00f1jali ( _Yoga: Discipline of Freedom_ ). Patrick Olivelle's translations of the _Upani\u1e63ads_ and the _Dharma S\u016btras_ are also both scholarly and readable. Wendy Doniger's _Hindu Myths_ provides a good sampling of Hindu narrative literature from the epics and _Pur\u0101\u1e47as_. Dominic Goodall's _Hindu Scriptures_ is a nice compendium for beginners.\n\nWilliam Buck's novelized translations of the _R\u0101m\u0101ya\u1e47a_ and _Mah\u0101bh\u0101rata_ both provide highly accessible points of entry into these two epic poems.\n\nAll of these \"high\" Sanskritic sources are nicely balanced by R. K. Ramanujan's _Folk Tales of India_ , which often give a twist to the better-known, dominant Brahmanical conceptions of Hindu belief and practice.\n\nAmbikananda Saraswati, Swami. _The Uddhava G\u012bt\u0101: The Final Teaching of Krishna_. London: Frances Lincoln, 2000.\n\nAvalon, Arthur, trans. _Tantra of the Great Liberation (Mah\u0101nirv\u0101\u1e47a Tantra)_. New York: Dover, 1972.\n\nBailly, Constantina Rhodes. _Shaiva Devotional Songs of Kashmir: A Translation and Study of Utpaladeva's \u015aivastotr\u0101vali_. Albany: State University of New York Press, 1987.\n\nBhattacharya, Deben. _Love Songs of Chandidas_. London: Allen and Unwin, 1967.\n\nBloomfield, Maurice. _Hymns of the Atharvaveda_. London: Sacred Books of the East, 1897.\n\nBuck, William. _Mah\u0101bh\u0101rata_. Berkeley: University of California Press, 1973.\n\n\u2014\u2014\u2014. _R\u0101m\u0101ya\u1e47a_. Berkeley: University of California Press, 2000.\n\nChatterji, Bankimcandra. _\u0100nandamath, or The Sacred Brotherhood_. Julius J. Lipner, trans. New York: Oxford University Press, 2005.\n\nCowell, E. B., trans. _Jataka Stories_. London: Pali Text Society, 1973.\n\nDimock, Edward C., and Tony K. Stewart, trans. _Caitanyacarit\u0101m\u1e5bta of Krishnadasa Kaviraja_. Cambridge, Mass.: Harvard University Press, 1999.\n\nDoniger, Wendy. _Hindu Myths_. New York: Penguin, 1975.\n\n\u2014\u2014\u2014. _The Rig Veda_. New York: Penguin, 1981.\n\n\u2014\u2014\u2014. _Textual Sources for the Study of Hinduism_. Chicago: University of Chicago Press, 1990.\n\nDoniger, Wendy, and Brian K. Smith, trans. _Laws of Manu_. New York: Penguin, 1991.\n\nEmbree, Ainslie T. _The Hindu Tradition_. New York: Random House, 1966.\n\n\u2014\u2014\u2014. _Sources of Indian Tradition, Vol. 1: From the Beginning to 1800_. 2nd ed. New York: Columbia University Press, 1988.\n\nFutehally, Shama, trans. _In the Dark of the Heart: The Songs of Meera_. San Francisco: HarperCollins, 1995.\n\nGoldman, Robert P., trans. _The R\u0101m\u0101ya\u1e47a of V\u0101lm\u012bki_. Princeton, N.J.: Princeton University Press, 1984.\n\nGoodall, Dominic, ed. and trans. _Hindu Scriptures_. Berkeley: University of California Press, 1996.\n\nGriffith, Ralph T. H., trans. _Hymns of the \u1e5ag Veda_. Delhi: Motilal Banarsidass, 1973.\n\n\u2014\u2014\u2014, trans. _Hymns of the S\u0101maveda_. Banaras: Chowkhamba, 1963.\n\n\u2014\u2014\u2014, trans. _Hymns of the Yajurveda_. Banaras: Chowkhamba, 1957.\n\nHawley, John Stratton, and Mark Juergensmeyer. _Songs of the Saints of India_. Oxford: Oxford University Press, 1988.\n\nHay, Stephen, ed. _Sources of Indian Tradition, Vol. 2: Modern India and Pakistan_. 2nd ed. New York: Columbia University Press, 1988.\n\nHess, Linda, and Shukdev Singh, trans. _A Touch of Grace: Songs of Kabir_. Boston: Shambhala, 1994.\n\nHume, Robert Ernest, trans. _The Thirteen Principal Upanishads_. Oxford: Oxford University Press, 1968.\n\nIngalls, Daniel H. H. _An Anthology of Sanskrit Court Poetry: Vidy\u0101kara's Subh\u0101\u1e63itaratnako\u015ba_. Cambridge, Mass.: Harvard University Press, 1965.\n\nJohnson, Clive, ed. _Ved\u0101nta: An Anthology of Hindu Scripture, Commentary, and Poetry_. New York: Harper and Row, 1971.\n\nJohnson, Willard J. _The Bhagavad G\u012bt\u0101_. Oxford: Oxford University Press, 1994.\n\nLayne, Gwendolyn, trans. _Kadambari of Ba\u1e47abha\u1e6d\u1e6da: A Classic Story of Magical Transformations_. New York: Garland, 1991.\n\nMayeda, Sengaku, trans. _A Thousand Teachings: The Upade\u015basahasr\u012b of \u015aa\u1e45kara_. Albany: State University of New York Press, 1991.\n\nMiller, Barbara Stoler, trans. _The Bhagavad G\u012bt\u0101: Krishna's Counsel in Time of War_. New York: Bantam Books, 1986.\n\n\u2014\u2014\u2014, trans. _Yoga: Discipline of Freedom_. New York: Bantam Books, 1998.\n\nM\u00fcller, Friedrich Max. _Rig Veda_. London: W. H. Allen, 1874.\n\nNikhilananda, Swami, trans. _The Gospel of Sri Ramakrishna_. New York: Ramakrishna-Vivekananda Center, 1942.\n\nOlivelle, Patrick, trans. _Dharmas\u016btras_. New York: Oxford University Press, 1999.\n\n\u2014\u2014\u2014, trans. _Upani\u1e63ads_. New York: Oxford University Press, 2008.\n\nOnians, Isabelle, trans. _Da\u015bakum\u0101racarita of Da\u1e47\u1e0din (What Ten Young Men Did)_. New York: New York University Press, 2005.\n\nPanikkar, Raimundo. _The Vedic Experience: Mantrama\u00f1jar\u012b_. Delhi: Motilal Banarsidass, 1977.\n\nPereira, J., ed. _Hindu Theology: A Reader_. Garden City, N.J.: Doubleday, 1976.\n\nPeterson, Indira V., trans. _Poems to \u015aiva: The Hymns of the Tamil Saints_. Princeton, N.J.: Princeton University Press, 1989.\n\nPope, G. U. _The Tiruv\u0101cagam or \"Sacred Utterances\" of the Tamil Poet, Saint, and Sage Manikkavacakar_. Oxford: Oxford University Press, 1900.\n\nPrabhavananda, Swami, and Christopher Isherwood, trans. _How to Know God: The Yoga Aphorisms of Patanjali_. Hollywood: Vedanta Society of Southern California, 1953.\n\nRadhakrishnan, Sarvepalli, and Charles A. Moore. _A Sourcebook in Indian Philosophy_. Princeton, N.J.: Princeton University Press, 1975.\n\nRamanujan, A. K. _Folktales from India: A Selection of Oral Tales from Twenty-two Languages_. New York: Pantheon Books, 1991.\n\n\u2014\u2014\u2014. _Hymns for the Drowning: Hymns for Vishnu by Nammalvar_. Princeton, N.J.: Princeton University Press, 1981.\n\n\u2014\u2014\u2014. _The Interior Landscape: Love Poems from a Classical Tamil Anthology_. Bloomington: Indiana University Press, 1967.\n\n\u2014\u2014\u2014. _Speaking of \u015aiva_. New York: Penguin, 1973.\n\n\u015aarma, Vi\u1e63\u1e47u. _The Pancatantra_. Chandra Rajan, trans. New York: Penguin, 2007.\n\nSchweig, Graham. _Bhagavad G\u012bt\u0101: The Beloved Lord's Secret Song_. San Francisco: HarperOne, 2010.\n\nThompson, George. _The Bhagavad Gita: A New Translation_. New York: North Point Press, 2008.\n\nVenkatesananda, Swami. _The Concise R\u0101m\u0101ya\u1e47a of V\u0101lm\u012bki_. Albany: State University of New York Press, 1988.\n\n\u2014\u2014\u2014. _The Concise Yoga V\u0101si\u1e63\u1e6dha_. Albany: State University of New York Press, 1984.\n\nC. Modern Primary Sources in English\n\nSome of these primary sources have become something akin to classics among contemporary spiritual seekers. They include Gandhi's autobiography ( _The Story of My Experiments with Truth_ ), Aurobindo Ghose's _Essays on the Gita_ , Huxley's _The Perennial Philosophy_ , Thoreau's _Walden_ , and Yogananda's _Autobiography of a Yogi_. As a source on the life of Ramakrishna that is at once a work of scholarship and devotion, Christopher Isherwood's _Ramakrishna and His Disciples_ is particularly accessible.\n\nAmbedkar, B. R. _The Buddha and His Dharma_. Bombay: Peoples Education Society, 1957.\n\n\u2014\u2014\u2014. _Towards an Enlightened India_. New Delhi: Penguin Books, 2004.\n\n\u2014\u2014\u2014. _What Congress and Gandhi Have Done to the Untouchables_. Bombay: Thacker, 1948.\n\n\u2014\u2014\u2014. _Why Go for Conversion?_ Bangalore: Pariah Sahitya Akademy, 1981.\n\n\u2014\u2014\u2014. _The Untouchables_. Bombay: Thacker, 1948.\n\nBesant, Annie. _Hindu Ideals_. Adyar, Madras: Theosophical Publishing House, 1904.\n\nBlavatsky, Helena Petrovna. _The Secret Doctrine_. London: Theosophical Publishing, 1888.\n\nChowgule, Ashok V. _Christianity in India: The Hindutva Perspective_. Mumbai: Hindu Vivek Kendra, 1999.\n\nGandhi, Mohandas K. _An Autobiography: The Story of My Experiments with Truth_. Boston: Beacon Press, 1958.\n\n\u2014\u2014\u2014. _The Bhagavad Gita According to G\u0101ndh\u012b_. Berkeley, Calif.: Berkeley Hills Books, 2000.\n\n\u2014\u2014\u2014. _Selected Works of Mahatma Gandhi: A Complete Representative Set of Gandhian Literature in Six Volumes_. Ahmedabad: Navajivan Publishing, 1993.\n\nGhose, Aurobindo. _Essays on the Gita_. Pondicherry: Sri Aurobindo Ashram, 1983.\n\n\u2014\u2014\u2014. _The Life Divine_. Pondicherry: Sri Aurobindo Ashram, 2010.\n\n\u2014\u2014\u2014. _Secret of the Veda_. Pondicherry: Sri Aurobindo Ashram, 1953.\n\n\u2014\u2014\u2014. _The Upanishads_. Pondicherry: Sri Aurobindo Ashram, 1996.\n\nGoel, Sita Ram. _Catholic Ashrams: Sannyasins or Swindlers?_ New Delhi: Voice of India, 1994.\n\n\u2014\u2014\u2014. _Hindu Temples: What Happened to Them?_ New Delhi: Voice of India, 1998.\n\n\u2014\u2014\u2014. _History of Hindu-Christian Encounters_. New Delhi: Voice of India, 1989.\n\nHuxley, Aldous. _The Perennial Philosophy_. New York: Harper and Row, 1944.\n\nIsherwood, Christopher. _My Guru and His Disciple_. Minneapolis: University of Minnesota Press, 2001.\n\n\u2014\u2014\u2014. _Ramakrishna and His Disciples_. Hollywood, Calif.: Vedanta Press, 1965.\n\nMaharishi Mahesh Yogi. _Maharishi Mahesh Yogi on the Bhagavad G\u012bt\u0101, Chapters 1\u20136_. New York: Penguin, 1990.\n\n\u2014\u2014\u2014. _Science of Being and Art of Living_. New York: Plume, 2001.\n\nMuktananda, Swami. _Play of Consciousness: A Spiritual Autobiography_. South Falls-burg, N.Y.: Siddha Yoga Publications, 2000.\n\nPandit, Bansi. _The Hindu Mind: Fundamentals of Hindu Religion and Philosophy for All Ages_. Glen Ellyn, Ill.: B and V Enterprises, 1998.\n\nPrabhup\u0101da, A. C. Bhaktived\u0101nta Sw\u0101m\u012b. _Bhagavad G\u012bt\u0101: As It Is_ (Reprint). Los Angeles: Bhaktived\u0101nta Book Trust, 1997.\n\nRadhakrishnan, Sarvepalli. _The Hindu View of Life_. London: Macmillan, 1927.\n\nRama, Swami. _Living with the Himalayan Masters_. Blue Mountain, Pa.: Himalayan Institute Press, 2007.\n\nRichards, Glyn. _A Source-Book of Modern Hinduism_. Richmond, U.K.: Curzon, 1985.\n\nSatchidananda, Swami. _The Living Gita_. New York: Henry Holt, 1988.\n\nSubramuniyaswami, Satguru Sivaya. _Dancing with \u015aiva: Hinduism's Contemporary Catechism_. Kauai, Hawaii: Himalayan Academy, 1997.\n\n\u2014\u2014\u2014. _How to Become a Hindu: A Guide for Seekers and Born Hindus_. Kauai, Hawaii: Himalayan Academy, 2000.\n\n\u2014\u2014\u2014. _Merging with \u015aiva: Hinduism's Contemporary Metaphysics_. Kauai, Hawaii: Himalayan Academy, 1999.\n\nThoreau, Henry David. _Walden_. New York: W. W. Norton, 1966.\n\nVivekananda, Swami. _The Complete Works of Swami Vivekananda_. Vols. 1\u20138. Kolkata: Advaita Ashrama, 1989.\n\nYogananda, Paramahamsa. _Autobiography of a Yogi_. Los Angeles: Self-Realization Fellowship, 1999.\n\n## II. SECONDARY SOURCES\n\nA. Broad Surveys and Introductory Works\n\nAmong the titles listed in this bibliography, beginners will probably find the broad surveys and introductory works listed in this section especially helpful. But even some of these presuppose a measure of specialized knowledge. I especially recommend David Knipe's _Hinduism_ , K. M. Sen's _Hinduism_ , and Vasudha Narayanan's _Hinduism: Origins, Beliefs, Practices, Holy Texts, Sacred Places_ , all of which I have used effectively in introductory-level college courses on Hinduism or Indic religions.\n\nFor the more ambitious reader, I would recommend Klaus K. Klostermaier's _A Survey of Hinduism_ (for a view of Hinduism that many practitioners find authentic), as well as Wendy Doniger's _The Hindus: An Alternative History_ , which many practitioners have found problematic but which presents an interpretation of the tradition by a scholar who has had a profound impact upon the ways in which the contemporary academy views Hinduism.\n\nAiyar, C. P. Ramaswamy. _Fundamentals of Hindu Faith and Culture_. Trivandrum: Government Press, 1944.\n\nBrockington, J. L. _The Sacred Thread: Hinduism in its Continuity and Diversity_. Edinburgh, U.K.: Edinburgh University Press, 1996.\n\nDallapiccola, Anna L. _Dictionary of Hindu Lore and Legend_. London: Thames and Hudson, 2002.\n\nDoniger, Wendy. _The Hindus: An Alternative History_. New York: Penguin, 2009.\n\nFlood, Gavin. _The Blackwell Companion to Hinduism_. Oxford: Blackwell, 2003.\n\n\u2014\u2014\u2014. _An Introduction to Hinduism_. Cambridge: Cambridge University Press, 2004.\n\nFuller, Chris. _The Camphor Flame: Popular Hinduism and Society in India_. Princeton, N.J.: Princeton University Press, 2004.\n\nGrimes, John. _A Concise Dictionary of Indian Philosophy: Sanskrit Terms Defined in English_. Albany: State University of New York Press, 1996.\n\nHopkins, Thomas. _The Hindu Religious Tradition_. Encino, Calif.: Dickenson, 1971.\n\nJacobs, Stephen. _Hinduism Today_. London: Continuum, 2010.\n\nJagannathan, Shakuntala. _Hinduism: An Introduction_. Mumbai: Vakils, Feffer and Simons, 1984.\n\nKlostermaier, Klaus K. _A Survey of Hinduism_. 3rd ed. Albany: State University of New York Press, 2007.\n\nKnipe, David. _Hinduism_. San Francisco: Harper, 1991.\n\nKnott, Kim. _Hinduism: A Very Short Introduction_. Oxford: Oxford University Press, 1998.\n\nJohnsen, Linda. _The Complete Idiot's Guide to Hinduism_. Indianapolis, Ind.: Alpha Books, 2002.\n\nKinsley, David. _Hinduism: A Cultural Perspective_. 2nd ed. Englewood Cliffs, N.J.: Prentice Hall, 1993.\n\nLal, Vinay, and Borin van Loon. _Introducing Hinduism_. Thriplow, U.K.: Icon Books, 2005.\n\nLipner, Julius. _Hindus: Their Religious Beliefs and Practices_. London: Routledge, 1994.\n\nLopez, Donald S. _Religions of India in Practice_. Princeton, N.J.: Princeton University Press, 1995.\n\nMittal, Sushil, and Gene Thursby, eds. _The Hindu World_. London: Routledge, 2004.\n\nMonier-Williams, Monier. _Religious Thought and Life in India_. London: John Murray, 1885.\n\nNarayanan, Vasudha. _Hinduism: Origins, Beliefs, Practices, Holy Texts, Sacred Places_. New York: Oxford University Press, 2004.\n\nSen, Kshiti Mohan. _Hinduism_. New York: Penguin, 1961.\n\nStutley, Margaret, and James Stutley. _A Dictionary of Hinduism: Its Mythology, Folklore, and Development, 1500 B.C.\u2013A.D. 1500_. New York: Harper and Row, 1977.\n\nSullivan, Bruce M. _The Historical Dictionary of Hinduism_. Lanham, Md.: Scarecrow Press, 1997.\n\nB. Historical Studies\n\nAs noted earlier, the works that are contained in this section range from histories of the Indian subcontinent, to studies of the history of particular regions or movements, to biographies of individuals.\n\nIn the first category, I recommend John Keay's _India: A History_ and Romila Thapar's _Ancient Indian Social History_. Though a bit dated, A. L. Basham's _The Wonder That Was India_ is a classic and creates a rich and vivid picture of ancient India in the mind of the reader. For the Indus Valley civilization, Jane R. McIntosh's _A Peaceful Realm: The Rise and Fall of the Indus Civilization_ is especially good for beginners.\n\nFor the modern period, Collins and Lapierre's _Freedom at Midnight_ , on the Indian independence movement, is a classic, as is Louis Fischer's _The Life of Mahatma Gandhi_. William Dalrymple's _White Moghuls: Love and Betrayal in 18th Century India_ is accessible and entertaining as an introduction to the early colonial period.\n\nAbbot, J. E. _The Poet Saints of Maharastra_. 12 vols. Poona: Scottish Mission Industries, 1926\u20131941.\n\nAdcock, Catherine. _Religious Freedom and Political Culture: The \u0100rya Sam\u0101j in Colonial North India_. PhD diss., University of Chicago, 2007.\n\nAgrawala, V. S. _India as Known to P\u0101\u1e47ini_. Varanasi: Prithvi Prakasan, 1963.\n\nAiyangar, S. K. _Some Contributions of South India to Indian Culture_. Calcutta: Calcutta University, 1942.\n\nAiyer, A. Nataraja, and S. Lakshminarasimha Shastri. _The Traditional Age of \u015ar\u012b \u015aa\u1e45kar\u0101ch\u0101rya and the Maths_. Madras: Private publication, 1962.\n\nAllchin, B., and R. Allchin. _The Rise of Civilization in India and Pakistan_. Cambridge: Cambridge University Press, 1982.\n\nAltekar, A. D. _The Position of Women in Hindu Civilization from Prehistoric Times to the Present Day_. Banares: Motilal Banarsidass, 1956.\n\n\u2014\u2014\u2014. _State and Government in Ancient India_. New Delhi: Motilal Banarsidass, 1962.\n\nAshby, Philip H. _Modern Trends in Hinduism_. New York: Columbia University Press, 1974.\n\nBakker, Hans T. _Ayodhya: The History of Ayodhya from the 7th Century BC to the Middle of the 18th Century_. Amsterdam: John Benjamins, 1986.\n\nBall, Charles. _The History of the Indian Mutiny, Giving a Detailed Account of the Sepoy Insurrection_. New Delhi: Master Publishers, 1859.\n\nBasham, A. L. _The Origins and Development of Classical Hinduism_. Boston: Beacon Press, 1975.\n\n\u2014\u2014\u2014. _The Wonder That Was India_. London: Sidgwick and Jackson, 1954.\n\nBeal, Samuel, trans. _Si-yu-ki: Buddhist Records of the Western World_. New York: Paragon, 1968.\n\nBryan, Edwin F. _The Quest for the Origins of Vedic Culture: The Indo-\u0100ryan Migration Debate_. New York: Oxford University Press, 2004.\n\nChakrabarti, Dilip K. _The Archaeology of Ancient Indian Cities_. Delhi: Oxford University Press, 1995.\n\n\u2014\u2014\u2014. _India: An Archaeological History_. New Delhi: Oxford University Press, 1999.\n\nChakrabarti, Kunal. _Themes in Indian History_. Oxford Readings in Sociology. Delhi: Oxford University Press, 2006.\n\nChattopadhyaya, S. _The Evolution of Theistic Sects in Ancient India_. Calcutta: Progressive Publishers, 1963.\n\n\u2014\u2014\u2014. _Reflections on the Tantras_. Delhi: Motilal Banarsidass, 1978.\n\n\u2014\u2014\u2014. _Some Early Dynasties of South India_. Delhi: Motilal Banarsidass, 1974.\n\nChaudhuri, N. _That Compassionate Touch of Ma Anandamayee_. Reprint. Delhi: Motilal Banarsidass, 2006.\n\nClothey, Fred. _Images of Man: Religion and Historical Process in South Asia_. Madras: Blackie and Son, 1983.\n\nCollins, L., and D. Lapierre. _Freedom at Midnight_. New York: Simon and Schuster, 1975.\n\nCrawford, S. Cromwell. _Ram Mohan Roy: Social, Political and Religious Reform in Nineteenth Century India_. New York: Paragon House, 1987.\n\nDalrymple, William. _White Moghuls: Love and Betrayal in 18th Century India_. Hammersmith, U.K.: HarperCollins, Flamingo, 2003.\n\nDandekar, R. N. _Some Aspects of the History of Hinduism_. Poona: University of Poona, 1967.\n\nDani\u00e9lou, Alain. _A Brief History of India_. Rochester, Vt.: Inner Traditions, 2003.\n\nDavid, Saul. _The Indian Mutiny_. New York: Penguin Books, 2003.\n\nDevahuti, D. _Harsha: A Political Study_. Oxford: Clarendon, 1970.\n\nEaton, Richard M. _Temple Desecration and Indo-Muslim States_. New York: Oxford University Press, 1990.\n\nEban, M., ed. _Maharishi the Guru: The Story of Maharishi Mahesh Yogi_. Bombay: Pear Publications, 1968.\n\nErikson, Erik H. _Gandhi's Truth_. New York: W. W. Norton, 1970.\n\nFeuerstein, Georg, Subhash Kak, and David Frawley. _In Search of the Cradle of Civilization_. Wheaton, Ill.: Theosophical Publishing House, 1995.\n\nFischer, Louis. _The Life of Mahatma Gandhi_. Bombay: Bharatiya Vidya Bhavan, 1959.\n\nGlucklich, Ariel. _The Strides of Vishnu: Hindu Culture in Historical Perspective_. New York: Oxford University Press, 2008.\n\nHardy, Friedhelm. _Viraha Bhakti: The Early History of Krishna Devotion in South India_. Delhi: Oxford University Press, 1983.\n\nHawley, John Stratton. _Three Bhakti Voices: Mirabai, Surdas and Kabir in Their Times and Ours_. Delhi: Oxford University Press, 2005.\n\nJames, Lawrence. _Raj_. New York: Little, Brown, 1997.\n\nKeay, John. _India: A History_. New York: Grove Press, 2000.\n\nKeer, Dhananjay. _Dr. Ambedkar: Life and Mission_. Bombay: India Printing Works, 1962.\n\nKenoyer, Jonathan Mark. _Ancient Cities of the Indus Valley Civilization_. Oxford: Oxford University Press, 1998.\n\nKulke, Hermann, and Dietmar Rothermund. _A History of India_. London: Routledge, 1986.\n\nLorenzen, David N. _Kabir Legends and Anantadas's Kabir Parachay_. Albany: State University of New York Press, 1991.\n\nLudden, David E. _India and South Asia: A Short History_. Oxford: One World, 2002.\n\nMajumdar, R. C., ed. _The Vedic Age_. London: George Allen and Unwin, 1952.\n\nMcCrindle, J. W. _Ancient India as Described in Classical Literature_. Amsterdam: Philo Press, 1975.\n\nMcIntosh, Jane R. _A Peaceful Realm: The Rise and Fall of the Indus Civilization_. Boulder, Colo.: Westview Press, 2002.\n\nMcLean, Malcolm. _Devoted to the Goddess: The Life and Work of Ramprasad_. Albany: State University of New York Press, 1998.\n\nMishra, Pankaj. \"Exit Wounds: The Legacy of Indian Partition.\" _New Yorker_ (August 13, 2007): 80\u201384.\n\nMoon, Penderel. _British Conquest and Dominion of India_. London: Duckworth, 1989.\n\nMukherjee, Ramkrishna. _The Rise and Fall of the East India Company: A Sociological Appraisal_. New York: Monthly Review Press, 1974.\n\nMukhia, Harbans. _The Mughals of India_. Malden, Mass.: Blackwell, 2004.\n\nNeumayer, E. _Prehistoric Indian Rock Paintings_. Delhi: Oxford University Press, 1983.\n\nNikam, N. A., and Richard McKeon. _The Edicts of A\u015boka_. Chicago: University of Chicago Press, 1978.\n\nNilakantha Shastri, K. A. _A Comprehensive History of India_. Delhi: People's Publishing House, 1957.\n\nOlivelle, Patrick, ed. _Between the Empires: Society in India 300 BCE to 400 CE_. New York: Oxford University Press, 2006.\n\nParashar, Aloka. _Mlecchas in Early India: A Study in Attitudes toward Outsiders Up to AD 600_. Delhi: Munshiram Manoharlal, 1991.\n\nParpola, Asko. _Deciphering the Indus Script_. New York: Cambridge University Press, 1994.\n\nPollock, Sheldon. _The Language of the Gods in the World of Men: Sanskrit, Culture, and Power in Premodern India_. Berkeley: University of California Press, 2006.\n\nPossehl, Gregory L. _The Indus Age: The Writing System_. Philadelphia: University of Pennsylvania Press, 1997.\n\n\u2014\u2014\u2014. _The Indus Civilization: A Contemporary Perspective_. Walnut Creek, Calif.: Altamira Press, 2002.\n\nRenou, Louis. _Religions of Ancient India_. New York: Schocken, 1968.\n\nSchwartzberg, J. E., ed. _Historical Atlas of India_. 2nd ed. Chicago: University of Chicago Press, 1990.\n\nSeal, Anil. _The Emergence of Indian Nationalism: Competition and Collaboration in the Later Nineteenth Century_. Cambridge: Cambridge University Press, 1971.\n\nSen, Amitya. _Hindu Revivalism in Bengal, 1872\u20131905: Some Essays in Interpretation_. Delhi: Oxford University Press, 1993.\n\nSethna, K. D. _Ancient India in a New Light_. Delhi: Aditya Prakashan, 1989.\n\nSingh, S. D. _Ancient Indian Warfare, with Special Reference to the Vedic Period_. Leiden: E. J. Brill, 1965.\n\nSmith, Bardwell L., ed. _Essays on Gupta Culture_. Delhi: Motilal Banarsidass, 1983.\n\nSuddhatmaprana, Pravrajika. _Indian Saints and Mystics_. Kolkata: Ramakrishna Mission Institute of Culture, 2009.\n\nThapar, Romila. _Ancient Indian Social History_. New Delhi: Orient Longman, 1978.\n\n\u2014\u2014\u2014. _A\u015boka and the Decline of the Mauryas_. Oxford: Oxford University Press, 1961.\n\nWhite, David Gordon. _The Alchemical Body: Siddha Traditions in Medieval India_. Chicago: University of Chicago Press, 1997.\n\nC. Textual Studies\n\nAnanda K. Coomaraswamy and Sister Nivedita's compendium of the _Myths of the Hindus and Buddhists_ is an excellent introduction to the narrative traditions of Hinduism and Buddhism, as is Edward C. Dimock's _The Literatures of India: An Introduction_. Iravati Karve's _Yuganta: The End of an Epoch_ is an intriguing guide to the _Mah\u0101bh\u0101rata_ , as is Alf Hiltebeitel's _The Ritual of Battle_ and his more recent work, _Rethinking the Mah\u0101bh\u0101rata: A Reader's Guide to the Education of the Dharma King_. Romila Thapar's _Exile and the Kingdom: Some Thoughts on the R\u0101m\u0101ya\u1e47a_ is a similarly good entry into the _R\u0101m\u0101ya\u1e47a_.\n\nAli, S. M. _The Geography of the Pur\u0101\u1e47as_. New Delhi: People's Publishing House, 1966.\n\nAmore, R. C., and L. D. Shinn. _Lustful Maidens and Ascetic Kings: Buddhist and Hindu Stories of Life_. New York: Oxford University Press, 1981.\n\nApte, V. M. _\u1e5agvedic Mantras in Their Ritual Setting in the G\u1e5bhyas\u016btras_. Poona: Deccan College Research Institute, 1950.\n\nBakker, Hans T., ed. _Origin and Growth of the Puranic Text Corpus with Special Reference to the Skanda Purana_. Groningen, Netherlands: E. Forsten, 1986.\n\nBalasundaram, T. S. _The Golden Anthology of Ancient Tamil Literature_. 3 vols. Madras: South India \u015aaiva Siddh\u0101nta Book Publishing Society, 1959\u20131960.\n\nBanerjea, S. C. _Dharma \u015a\u0101stras: A Study of Their Origin and Development_. Calcutta: Punthi Pustak, 1962.\n\nBanerji, S. C. _Studies in the Mah\u0101pur\u0101\u1e47as_. Calcutta: Punthi Pustak, 1991.\n\nBhattacharya, Asutosh. _Folklore of Bengal_. New Delhi: National Book Trust, 1978.\n\nBloomfield, Maurice. _Hymns of the Atharva Veda_. New York: Greenwood Press, 1969.\n\nBrockington, John L. _The Sanskrit Epics_. Leiden: Brill, 1998.\n\nChakravarti, C. _Tantras: Studies on Their Religion and Literature_. Calcutta: Punthi Pustak, 1963.\n\nCoburn, Thomas. _Dev\u012b M\u0101h\u0101tmya: The Crystallization of the Goddess Tradition_. Delhi: Motilal Banarsidass, 1984.\n\nCoomaraswamy, Ananda K., and Sister Nivedita. _Myths of the Hindus and Buddhists_. Reprint. New York: Dover, 1967.\n\nDandekar, R. N., ed. _Vedic Bibliography_. Poona: Bhandarkar Oriental Research Institute, 1961\u20131973.\n\n\u2014\u2014\u2014. _Vedic Mythological Tracts_. Delhi: Ajanta, 1977.\n\nDerrett, J. Duncan M. _Dharma\u015b\u0101stra and Juridical Literature_. Weisbaden, Germany: Otto Harrassowitz, 1973.\n\nDevasthali, G. V. _Religion and Mythology of the Br\u0101hma\u1e47as_. Poona: University of Poona, 1965.\n\nDimock, Edward C., ed. _The Literatures of India: An Introduction_. Chicago: University of Chicago Press, 1974.\n\nDoniger, Wendy, ed. _Pur\u0101\u1e47a Perennis: Reciprocity and Transformation in Hindu and Jaina Texts_. Albany: State University of New York Press, 1993.\n\nFaddegon, Barend. _Studies in the S\u0101maveda_. Amsterdam: North-Holland, 1951.\n\nGoldman, Robert P. _Gods, Priests, and Warriors: The Bh\u1e5bgus of the Mah\u0101bh\u0101rata_. New York: Columbia University Press, 1977.\n\nGonda, Jan. _Dual Deities in the Religion of the Veda_. Amsterdam: North-Holland, 1974.\n\n\u2014\u2014\u2014. _The Indra Hymns of the \u1e5agveda_. Leiden: E. J. Brill, 1989.\n\n\u2014\u2014\u2014. _Loka: World and Heaven in the Veda_. Amsterdam: North-Holland, 1966.\n\n\u2014\u2014\u2014. _Mantra Interpretation in the \u015aatapatha Br\u0101hma\u1e47a_. Leiden: E. J. Brill, 1988.\n\n\u2014\u2014\u2014. _The Ritual S\u016btras_. Wiesbaden, Germany: Otto Harassowitz, 1977.\n\n\u2014\u2014\u2014. _The Vision of the Vedic Poets_. The Hague: Mouton, 1963.\n\nGonzalez-Riemann, Luis. _The Mahabharata and the Yugas_. New York: Peter Lang, 2002.\n\nGoudriaan, Teun. _M\u0101y\u0101 Divine and Human: A Study of Magic and Its Religious Foundations in Sanskrit Texts_. Delhi: Motilal Banarsidass, 1978.\n\nHiltebeitel, Alf. _Rethinking the Mah\u0101bh\u0101rata: A Reader's Guide to the Education of the Dharma King_. Chicago: University of Chicago Press, 2001.\n\n\u2014\u2014\u2014. _The Ritual of Battle_. Ithaca, N.Y.: Cornell University Press, 1976.\n\nHopkins, E. W. _The Great Epic of India: Its Character and Origin_. New York: Charles Scribner's Sons, 1901.\n\nJohnson, Willard J. _Poetry and Speculation of the \u1e5ag Veda_. Berkeley: University of California Press, 1980.\n\nKailasapathy, K. _Tamil Heroic Poetry_. Oxford: Clarendon, 1968.\n\nKane, P. V. _History of Dharma\u015b\u0101stra_. Poona: Bhandarkar Oriental Research Institute, 1962.\n\nKarve, Iravati. _Yuganta: The End of an Epoch_. Poona: Deshmukh Prakashan, 1969.\n\nKatz, Ruth C. _Arjuna in the Mah\u0101bh\u0101rata: Where Krishna Is, There Is Victory_. Columbia: University of South Carolina Press, 1989.\n\nKnipe, David M. _In the Image of Fire: Vedic Experiences of Heat_. Delhi: Motilal Banarsidass, 1975.\n\nLutgendorf, Philip. _The Life of a Text: Performing the R\u0101mcaritm\u0101nas of Tuls\u012bd\u0101s_. Berkeley: University of California Press, 1991.\n\nMani, Vettam. _Pur\u0101\u1e47ic Encyclopaedia: A Comprehensive Dictionary of Epic and Pur\u0101\u1e47ic Literature_. Delhi: Motilal Banarsidass, 1975.\n\nMinor, Robert, ed. _Modern Indian Interpretation of the Bhagavad G\u012bt\u0101_. Albany: State University of New York Press, 1986.\n\nNarayan, Vasudha. _The Vernacular Veda: Revelation, Recitation, and Ritual_. Columbia: University of South Carolina Press, 1994.\n\nNath, Vijay. _Puranas and Acculturation: A Historico-Anthropological Perspective_. Delhi: Munshiram Manoharlal, 2001.\n\nPatton, Laurie L., ed. _Authority, Anxiety, and Canon: Essays in Vedic Interpretation_. Albany: State University of New York Press, 1994.\n\nPowell, Barbara. _Windows into the Infinite: A Guide to the Hindu Scriptures_. Fremont, Calif.: Asian Humanities Press, 1996.\n\nRenou, Louis. _The Destiny of the Veda in India_. Delhi: Motilal Banarsidass, 1965.\n\nSantucci, James A. _An Outline of Vedic Literature_. Missoula, Mont.: Scholars Press, 1967.\n\nSharma, Arvind, ed. _Essays on the Mah\u0101bh\u0101rata_. Leiden: E. J. Brill, 1991.\n\nThapar, Romila. _Exile and the Kingdom: Some Thoughts on the R\u0101m\u0101ya\u1e47a_. Bangalore: Mythic Society, 1978.\n\nD. Practice: Ritual and Ethics\n\nLawrence Babb's _The Divine Hierarchy_ and _Redemptive Encounters_ are both fine examples of the genre of ethnographic writing: clear and accessible, and giving the reader a strong sense of lived Hinduism.\n\nIn the area of ethics, Christopher Chapple's _Nonviolence to Animals, Earth, and Self in Asian Traditions_ is both clear and concise and has a considerable section devoted to Hinduism. For another perspective, D. N. Jha's _The Myth of the Holy Cow_ reveals how recent the practice of cow protection as a defining feature of Hinduism really is.\n\nOn the philosophy and practice of yoga, Mircea Eliade's _Yoga: Immortality and Freedom_ is a classic, while Chapple's recent _Yoga and the Luminous_ is an excellent entry point to the practice of yoga by a scholar-practitioner.\n\nJulia Leslie's _Roles and Rituals for Hindu Women_ and Anne Mackenzie Pearson's _Because It Gives Me Peace of Mind: Ritual Fasts in the Religious Lives of Hindu Women_ are especially good introductions to the rituals specific to Hindu women.\n\nArvind Sharma's _Hinduism and Human Rights_ is a groundbreaking work that brings the discourse of dharma into conversation with the modern concept of human rights.\n\nAlper, Harvey, ed. _Understanding Mantras_. Albany: State University of New York Press, 1989.\n\nBabb, Lawrence A _. The Divine Hierarchy: Popular Hinduism in Central India_. New York: Columbia University Press, 1975.\n\n\u2014\u2014\u2014. _Redemptive Encounters: Three Modern Styles in the Hindu Tradition_. Berkeley: University of California Press, 1986.\n\nBahadur, Om Lata. _Book of Hindu Festivals and Ceremonies_. New Delhi: UBS, 1994.\n\nChapple, Christopher Key. _Nonviolence to Animals, Earth, and Self in Asian Traditions_. Albany: State University of New York Press, 1993.\n\n\u2014\u2014\u2014. _Yoga and the Luminous: Pata\u00f1jali's Spiritual Path to Freedom_. Albany: State University of New York Press, 2009.\n\nChapple, Christopher Key, and Mary Evelyn Tucker, eds. _Hinduism and Ecology: The Intersection of Earth, Sky, and Water_. Cambridge, Mass.: Center for the Study of World Religions, 2000.\n\nClooney, Francis X. _Thinking Ritually: Rediscovering the P\u016brva M\u012bm\u0101\u1e43s\u0101 of Jaimini_. Vienna: Institut f\u00fcr Indologie, 1990.\n\nCreel, A. B. _Dharma in Hindu Ethics_. Columbia, Mo.: South Asia Books, 1977.\n\nCutler, Norman. _Songs of Experience: The Poetics of Tamil Devotion_. Bloomington: Indiana University Press, 1987.\n\nDani\u00e9lou, Alain. _Virtue, Success, Pleasure, Liberation: The Four Aims of Life in the Traditions of Ancient India_. Rochester, Vt.: Inner Traditions International, 1993.\n\n\u2014\u2014\u2014. _Yoga: The Method of Reintegration_. New York: University Books, 1956.\n\nDatta, V. N. _Sati: A Historical, Social, and Philosophical Enquiry into the Hindu Rite of Widow Burning_. New Delhi: Manohar, 1987.\n\nEliade, Mircea. _Yoga: Immortality and Freedom_. Princeton, N.J.: Princeton University Press, 1958.\n\nGold, Ann Grodzins. _Fruitful Journeys: The Ways of Rajasthani Pilgrims_. Berkeley: University of California Press, 1988.\n\nHarman, William. _The Sacred Marriage of a Hindu Goddess_. Bloomington: Indiana University Press, 1989.\n\nHazra, R. C. _Studies in the Puranic Records of Hindu Rites and Customs_. Dacca: University of Dacca, 1940.\n\nHeesterman, Jan C. _The Broken World of Sacrifice_. Chicago: University of Chicago Press, 1992.\n\n\u2014\u2014\u2014. _The Inner Conflict of Tradition: Essays in Indian Ritual, Kingship, and Society_. Chicago: University of Chicago Press, 1985.\n\nHocart, A. M. _Caste: A Comparative Study_. London: Methuen, 1950.\n\nHouben, Jan E. M., and K. R. van Kooij, eds. _Violence Denied: Violence, Non-Violence, and the Rationalization of Violence in South Asian Cultural History_. Leiden: Brill, 1999.\n\nHuffer, Amanda. _Guru Movements in a Globalized Framework: Amritanandamayi Ma's (Amma's) Community of Devotees in the United States_. PhD diss., University of Chicago, 2010.\n\nJha, D. N. _The Myth of the Holy Cow_. London: Verso, 2002.\n\nJordens, J. T. F. \"Reconversion to Hinduism, the Shuddhi of the \u0100rya Sam\u0101j.\" In _Religion in South Asia: Religious Conversion and Revival Movements in South Asia in Medieval and Modern Times_ , ed. G. Oddie, 144\u201353. London: Curzon Press, 1977.\n\nKakar, Sudhir. _Shamans, Mystics, and Doctors: A Psychological Inquiry into India and Its Healing Traditions_. New York: Knopf, 1982.\n\nLeslie, Julia, ed. _Roles and Rituals for Hindu Women_. Rutherford, N.J.: Farleigh Dickinson University Press, 1991.\n\nPearson, Ane Mackenzie. _Because It Gives Me Peace of Mind: Ritual Fasts in the Religious Lives of Hindu Women_. Albany: State University of New York Press, 1996.\n\nSharma, Arvind. _Hinduism and Human Rights: A Conceptual Approach_. New Delhi: Oxford University Press, 2003.\n\nE. Deities\n\nAs a broad overview of the Hindu pantheon, Alain Dani\u00e9lou's _The Myths and Gods of India_ is both accessible and very widely encompassing. Thomas Coburn's _Dev\u012b Mah\u0101tmya: The Crystallization of the Goddess Tradition_ is an excellent guide to goddess traditions, as is David Kinsley's _Hindu Goddesses_. Philip Lutgendorf's _Hanuman's Tale_ is a clear and comprehensive guide to the lore surrounding Hanuman. John Stratton Hawley's _Krishna, The But_ _ter Thief_ is a very good introduction to the devotional tradition surrounding Krishna, and Wendy Doniger's _\u015aiva: The Erotic Ascetic_ is, despite its racy title, a fine guide to the primary source literature on \u015aiva and the ways in which this deity transforms over time.\n\nBeck, Guy L., ed. _Alternative Krishnas: Regional and Vernacular Variations on a Hindu Deity_. Albany: State University of New York Press, 2005.\n\nBryant, Edwin F., ed. _Krishna, a Sourcebook_. New York: Oxford University Press, 2001.\n\nClothey, F. W. _The Many Faces of Murukan: The History and Meaning of a South Indian God_. The Hague: Mouton, 1978.\n\nCoburn, Thomas B. _Dev\u012b Mah\u0101tmya: The Crystallization of the Goddess Tradition_. Delhi: Motilal Banarsidass, 1984.\n\nDani\u00e9lou, Alain. _Hindu Polytheism_. London: Routledge and Kegan Paul, 1963.\n\n\u2014\u2014\u2014. _The Myths and Gods of India_. Rochester, Vt.: Inner Traditions International, 1985.\n\nDas, B. _K\u1e5b\u1e63\u1e47a: A Study in the Theory of Avat\u0101ras_. Bombay: Bharatiya Vidya Bhavan, 1962.\n\nDoniger, Wendy. _\u015aiva: The Erotic Ascetic_. Oxford: Oxford University Press, 1973.\n\nErndl, Kathleen M. _Victory to the Mother: The Hindu Goddess of Northwest India in Myth, Ritual, and Symbol_. New York: Oxford University Press, 1993.\n\nHandelman, Don, and David Shulman. _God Inside Out: \u015aiva's Game of Dice_. New York: Oxford University Press, 1997.\n\nHawley, John Stratton. _Krishna, The Butter Thief_. Princeton, N.J.: Princeton University Press, 1983.\n\nKinsley, David. _Hindu Goddesses_. Delhi: Motilal Banarsidass, 1998.\n\n\u2014\u2014\u2014. _The Sword and the Flute: Kali and Krishna_. Berkeley: University of California Press, 2000.\n\nLutgendorf, Philip. _Hanuman's Tale_. New York: Oxford University Press, 2007.\n\nNathan, Leonard, and Clinton Seely. _Grace and Mercy in Her Wild Hair_. Boulder, Colo.: Great Easter, 1982.\n\nF. The Arts\n\nFor the beginner, T. Richard Blurton's _Hindu Art_ is indispensable, as is Partha Mitter's _Indian Art_. Diana L. Eck's _Dar\u015ban: Seeing the Divine Image in India_ is a fine introduction to the theology that underlies that use of images in Hindu worship. I then recommend Richard Davis's _Lives of Indian Images_ as a follow-up.\n\nOn dance and drama, David L. Haberman's _Acting as a Way of Salvation: A Study of Raganuga Bhakti Sadhana_ explores the religious significance of acting in Hinduism. Stella Kramrisch's _The Hindu Temple_ is a good guide to the meaning of Hindu temple art and architecture.\n\nAcharya, Prasanna K. _A Dictionary of Hindu Architecture_. London: Oxford University Press, 1946.\n\nAnand, Mulk Raj. _The Hindu View of Art_. Bombay: Popular, 1957.\n\nArcher, W. G. _Indian Miniatures_. Greenwich: New York Calligraphic Society, 1960.\n\n\u2014\u2014\u2014. _The Loves of Krishna in Indian Painting and Poetry_. London: Allen and Unwin, 1957.\n\nBanerjea, Jitendra Nath. _The Development of Hindu Iconography_. Calcutta: University of Calcutta, 1956.\n\nBlurton, T. Richard. _Hindu Art_. London: British Museum Press, 1992.\n\nCoomaraswamy, Ananda K. _The Dance of \u015aiva_. Bombay: Asia Publishing House, 1956.\n\nDavis, Richard. _Lives of Indian Images_. Princeton, N.J.: Princeton University Press, 1997.\n\nDesai, Devangana. _The Religious Imagery of Khajuraho_. Mumbai: Franco-Indian Research, 1996.\n\nEck, Diana L. _Dar\u015ban: Seeing the Divine Image in India_. New York: Columbia University Press, 1996.\n\nGaston, Anne Marie. _\u015aiva in Dance, Myth and Iconography_. Delhi: Oxford University Press, 1982.\n\nGoetz, Hermann. _Studies in the History and Art of Kashmir and the Indian Himalaya_. Weisbaden, Germany: Otto Harassowitz, 1969.\n\nGopinatha Rao, T. _Elements of Hindu Iconography_. Madras: Law Printing House, 1916.\n\nGupta, Chandra Bhan. _The Indian Theatre_. 2nd ed. New Delhi: Munshiram Manoharlal, 1991.\n\nHaberman, David L. _Acting as a Way of Salvation: A Study of Raganuga Bhakti Sadhana_. Delhi: Motilal Banarsidass, 2001.\n\n\u2014\u2014\u2014. _Journey through the Twelve Forests: An Encounter with Krishna_. New York: Oxford University Press, 1994.\n\nHawley, John Stratton. _At Play with Krishna: Pilgrimage Dramas from Brind\u0101ban_. Princeton, N.J.: Princeton University Press, 1981.\n\nIyer, K. Bharata. _Kathakali: The Sacred Dance-Drama of Malabar_. London: Luzac, 1995.\n\nKramrisch, Stella. _The Hindu Temple_. Columbia, Mo.: South Asia Books, 1991.\n\n\u2014\u2014\u2014. _Unknown India: Ritual Art in Tribe and Village_. Philadelphia: Philadelphia Museum of Art, 1968.\n\nKuiper, F. B. J. _Varu\u1e47a and Vid\u016b\u1e63aka: On the Origin of Sanskrit Drama_. Amsterdam: North-Holland, 1979.\n\nLidova, Natalia. _Drama and Ritual of Early Hinduism_. Delhi: Motilal Banarsidass, 1994.\n\nMehta, Tarla. _Sanskrit Play Production in Ancient India_. Delhi: Motilal Banarsidass, 1995.\n\nMichell, George. _Hindu Art and Architecture_. London: Thames and Hudson, 2000.\n\n\u2014\u2014\u2014. _The Hindu Temple: An Introduction to Its Meaning and Forms_. Chicago: University of Chicago Press, 1988.\n\nMitter, Partha. _Indian Art_. Oxford: Oxford University Press, 2001.\n\nMookerjee, Ajit. _Tantra Art: Its Philosophy and Physics_. Basel: Ravi Kumar, 1971.\n\nRao, T. A. G. _Elements of Hindu Iconography_. New York: Paragon, 1968.\n\nStutley, Margaret. _The Illustrated Dictionary of Hindu Iconography_. London: Rout-ledge and Kegan Paul, 1985.\n\nVaidyanathan, K. R. _\u015ar\u012b Krishna, the Lord of Guruvay\u016br_. Bombay: Bh\u0101rat\u012bya Vidy\u0101 Bhavan, 1977.\n\nVaradpande, M. L. _Mah\u0101bh\u0101rata in Performance_. New Delhi: Clarion Books, 1990.\n\nWaghorne, Joanne P., and Norman Cutler, eds. _Gods of Flesh\/Gods of Stone: The Embodiment of Divinity in India_. Chambersberg, Pa.: Anima, 1985.\n\nWulff, Donna M. _Drama as a Mode of Religious Realization: The Vidagdham\u0101dhava of R\u016bpa Gosv\u0101m\u012b_. Atlanta: American Academy of Religion, 1984.\n\nG. Hindu Society\n\nAs introductions to the overall structure of Hindu society, two classics in the field remain Madeleine Biardeau's _Hinduism: The Anthropology of a Civilization_ and Louis Dumont's _Homo Heirarchicus_. I also recommend Bernard S. Cohn's _India: The Social Anthropology of a Civilization_. A fine example of ethnographic writing that focuses on Hindu social structures in southern India is E. Valentine Daniel's _Fluid Signs: Being a Person the Tamil Way_. Nicholas Dirks's _Castes of Mind: Colonialism and the Making of Modern India_ explores the reification of the caste system under the influence of colonial rule. Also excellent on the issue of caste is Ramdas Lamb's _The Ramnamis, Ramnam, and Untouchable Religion in Central India_ , as is Brian K. Smith's work, _Classifying the Universe: The Ancient Indian Varna System and the Origins of Caste_.\n\nFinally, Christophe Jaffrelot's _The Hindu Nationalist Movement in India_ is, in my judgment, the best introduction available to the militant political wing of Hinduism.\n\nAiyar, S. P. _The Politics of Mass Violence in India_. Bombay: Manaktalas, 1967.\n\nAiyer, V. G. R. _The Economy of a South Indian Temple_. Annamalai, Tamil Nadu: Annamalai University, 1946.\n\nAli, S. _The Congress Ideology and Programme_. New Delhi: People's Publishing House, 1958.\n\nAllen, Douglas, ed. _Religion and Political Conflict in South Asia: India, Pakistan, and Sri Lanka_. Westport, Conn.: Greenwood Press, 1992.\n\nAltekar, A. D. _Sources of Hindu Dharma in Its Socio-Religious Aspect_. Sholapur, Maharashtra: Institute of Public Administration, 1952.\n\nAltekar, A. S. _The Position of Women in Hindu Civilisation from Prehistoric Times to the Present Day_. Benares: Motilal Banarsidass, 1956.\n\nAnand, Mulk Raj. _Coolie_. Bombay: Kutub, 1957.\n\n\u2014\u2014\u2014. _Untouchable_. Bombay: Jaico, 1956.\n\nAnderson, W. K., and S. D. Dhamle. _The Brotherhood in Saffron: The R\u0101\u1e63\u1e6dr\u012bya Swayam-sevak Sangh and Hindu Revivalism_. Boulder, Colo.: Westview Press, 1987.\n\nBaden-Powell, B. H. _The Indian Village Community_. New Haven, Conn.: Yale University Press, 1958.\n\nBiardeau, Madeleine. _Hinduism: The Anthropology of a Civilization_. Delhi: Oxford University Press, 1994.\n\nBronkhorst, Johannes, and M. M. Deshpande, eds. _\u0100ryan and Non-\u0100ryan in South Asia: Evidence, Interpretation and Ideology_. Columbia, Mo.: South Asia Books, 1999.\n\nCarstairs, G. M. _The Twice-Born_. London: Hogarth Press, 1957.\n\nCenkner, W. _A Tradition of Teachers: \u015aa\u1e45kara and the Jagadgurus Today_. Delhi: Motilal Banarsidass, 1983.\n\nCohn, Bernard S. _India: The Social Anthropology of a Civilization_. Englewood Cliffs, N.Y.: Prentice Hall, 1971.\n\nDaniel, E. Valentine. _Fluid Signs: Being a Person the Tamil Way_. Berkeley: University of California Press, 1984.\n\nDani\u00e9lou, Alain. _India, a Civilization of Differences: The Ancient Tradition of Universal Tolerance_. Rochester, Vt.: Inner Traditions International, 2003.\n\nDas, Veena. _Critical Events: An Anthropological Perspective on Contemporary India_. Delhi: Oxford University Press, 1995.\n\n\u2014\u2014\u2014. _Mirrors of Violence: Communities, Riots and Survivors in South Asia_. Delhi: Oxford University Press, 1990.\n\nDeliege, Robert. _The Untouchables of India_. Oxford: Berg, 2001.\n\nDirks, Nicholas. _Castes of Mind: Colonialism and the Making of Modern India_. Princeton, N.J.: Princeton University Press, 2001.\n\nDum\u00e9zil, Georges. _The Destiny of a King_. Alf Hiltebeitel, trans. Chicago: University of Chicago Press, 1973.\n\nDumont, Louis. _Homo Hierarchicus_. London: Weidenfeld and Nicolson, 1966.\n\n\u2014\u2014\u2014. _Religion, Politics and History in India: Collected Papers in Indian Sociology_. The Hague: Mouton, 1970.\n\nForbes, G. _Women in Modern India_. Cambridge: Cambridge University Press, 1996.\n\nGhurye, G. S. _The Scheduled Tribes_. Bombay: Popular Prakashan, 1963.\n\nGlucklich, Ariel. _Religious Jurisprudence in the Dharma\u015b\u0101stra_. New York: Macmillan, 1988.\n\n\u2014\u2014\u2014. _The Sense of Adharma_. Oxford: Oxford University Press, 1994.\n\nGopal, Sarvepalli, ed. _Anatomy of a Confrontation: Ayodhya and the Rise of Communal Politics in India_. Delhi: Penguin India, 1991.\n\nHardiman, David. _The Coming of the Devi: Adivasi Assertion in Western India_. Delhi: Oxford University Press, 1995.\n\nHardy, Friedhelm. _The Religious Culture of India: Power, Love, and Wisdom_. Cambridge: Cambridge University Press, 1994.\n\nHawley, John Stratton, ed. _Sati, the Blessing and the Curse: The Burning of Wives in India_. New York: Oxford University Press, 1994.\n\nHuyler, Stephen P. _Village India_. New York: Harry Abrams, 1985.\n\nIsaacs, Harold. _India's Ex-Untouchables_. New York: John Day, 1964.\n\nJaffrelot, Christophe, ed. _Hindu Nationalism: A Reader_. Delhi: Permanent Black, 2007.\n\n\u2014\u2014\u2014. _The Hindu Nationalist Movement in India_. New York: Columbia University Press, 1996.\n\n\u2014\u2014\u2014, ed. _Hindu Nationalism: A Reader_. Delhi: Permanent Black, 2007.\n\nJuergensmeyer, Mark. _Religion as Social Vision: The Movement against Untouchability in 20th Century Punjab_. Berkeley: University of California Press, 1982.\n\nKulke, Hermann, and Burkhard Schnepel. _Jagannatha Revisited: Studying Society, Religion, and the State in Orissa_. New Delhi: Manohar, 2001.\n\nKuruvachira, J. _Roots of Hindutva: A Critical Study of Hindu Fundamentalism and Nationalism_. Delhi: Media House, 2005.\n\nLamb, Ramdas. _The Ramnamis, Ramnam, and Untouchable Religion in Central India_. Albany: State University of New York Press, 2002.\n\nMarriott, McKim, ed. _India through Hindu Categories_. Newbury Park, N.J.: Sage, 1990.\n\nOlivelle, Patrick. _The Ashrama System: The History and Hermeneutics of a Religious Institution_. New Delhi: Munshiram Manoharlal, 1993.\n\nOmvedt, Gail. _Dalit Visions: The Anti-Caste Movement and the Construction of an Indian Identity_. New Delhi: Orient Longman, 1995.\n\nSharma, Jyotirmaya. _Hindutva: Exploring the Idea of Hindu Nationalism_. New Delhi: Penguin, Viking, 2003.\n\nSmith, Brian K. _Classifying the Universe: The Ancient Indian Varna System and the Origins of Caste_. Oxford: Oxford University Press, 1994.\n\nWeber, Max. _The Religion of India: The Sociology of Hinduism and Buddhism_. New York: Free Press, 1958.\n\nH. Constructive Thought: Philosophy and Theology\n\nThe clearest introduction to Hindu theology available, to my knowledge\u2014and one to which my students have responded very positively for years\u2014is Pravrajika Vrajaprana's _Vedanta: A Simple Introduction_. Also clear and scholarly is Anantanand Rambachan's _The Advaita Worldview_ (for the nondualist stream of Hindu thought), as well as Deepak Sarma's _An Introduction to M\u0101dhva Ved\u0101nta_ (for a dualist counterpoint). Stephen Phillips, in his _Yoga, Karma, and Rebirth: A Brief History and Philosophy_ , presents a very clear and cogent defense of the classical Hindu doctrines of karma and rebirth. Though it may be heavy going for a beginner, Debiprasad Chattopadhyaya's _Lokayata: A Study in Ancient Indian Materialism_ is an intriguing exposition and defense of the relatively little-known materialist system of Indian philosophy.\n\nThe works of Wilhelm Halbfass listed here\u2014 _India and Europe: An Essay in Understanding_ , _On Being and What There Is: Classical Vai\u015be\u1e63ika and the History of Indian Ontology_ , and _Tradition and Reflection: Explorations in Indian Thought_ all give excellent overviews of various dimensions of Hindu intellectual history. The main focus of _India and Europe_ is the interactions between India and the West. _On Being and What There Is_ is probably the best exposition of the Vai\u015be\u1e63ika system of Indian philosophy that is available in the English language. _Tradition and Reflection_ is a wide-ranging, but quite accessible, collection of essays on a variety of topics relating to Hindu thought.\n\nThe articles in Rita Sherma and Arvind Sharma's edited volume, _Hermeneutics and Hindu Thought: Toward a Fusion of Horizons_ , are excellent examples of constructive thought in contemporary Hinduism.\n\nArapura, J. G. _Hermeneutical Essays on Ved\u0101ntic Topics_. Delhi: Motilal Banarsidass, 1986.\n\nBalasubramanian, R. _Advaita Ved\u0101nta_. Madras: University of Madras, 1976.\n\nBanerjea, A. K. _Philosophy of Gorakhn\u0101th_. Gorakhpur, Uttar Pradesh: Mahant Dig Vijai Nath Trust, Gorakhnath Temple, 1962.\n\nBrown, W. Norman. _Man in the Universe: Some Continuities in Indian Thought_. Berkeley: University of California Press, 1966.\n\nCarman, John Braisted. _The Theology of R\u0101m\u0101\u1e47uja: An Essay in Interreligious Understanding_. New Haven, Conn.: Yale University Press, 1974.\n\nCarpenter, J. E. _Theism in Mediaeval India_. London: Constable, 1921.\n\nChakravarti, S. C. _The Philosophy of the Upani\u1e63ads_. Calcutta: University of Calcutta, 1935.\n\nChapple, Christopher Key. _Karma and Creativity_. Albany: State University of New York Press, 1986.\n\nChatterjee, Margaret. _Gandhi's Religious Thought_. South Bend, Ind.: Notre Dame University Press, 1983.\n\nChatterji, Jagadish Chandra. _The Wisdom of the Vedas_. Wheaton, Ill.: Quest Books, 1992.\n\nChattopadhyaya, Debiprasad. _Lokayata: A Study in Ancient Indian Materialism_. New Delhi: People's Publishing House, 1959.\n\nClooney, Francis X. _Divine Mother, Blessed Mother: Hindu Goddesses and the Virgin Mary_. New York: Oxford University Press, 2005.\n\n\u2014\u2014\u2014. _Theology after Ved\u0101nta_. Albany: State University of New York Press, 1993.\n\nCoward, H. G. _Bhart\u1e5bhari_. Boston: Twayne, 1976.\n\nDandekar, R. N. _Universe in Hindu Thought_. Bangalore: University of Bangalore, 1972.\n\nDas, R. V. _Introduction to \u015aa\u1e45kara_. Calcutta: Punthi Pustak, 1968.\n\nDasgupta, Surendranath. _History of Indian Philosophy_. 5 vols. Reprint. Cambridge: Cambridge University Press, 2009.\n\nDeutsch, Eliot. _Advaita Ved\u0101nta: A Philosophical Reconstruction_. Honolulu: University of Hawaii Press, 1969.\n\nDoniger, Wendy. \"Do Many Heads Necessarily Have Many Minds? Hindu Polytheism and Religious Diversity\" _Parabola_ 30, no. 4 (2005): 10\u201319.\n\n\u2014\u2014\u2014, ed. _Karma and Rebirth in Classical Hindu Traditions_. Berkeley: University of California Press, 1980.\n\n\u2014\u2014\u2014. _The Origins of Evil in Hindu Mythology_. Berkeley: University of California Press, 1976.\n\nEck, Diana L. _Banaras: City of Light_. New York: Penguin, 1983.\n\nFeuerstein, Georg. _The Philosophy of Classical Yoga_. Manchester, U.K.: University of Manchester Press, 1982.\n\n\u2014\u2014\u2014. _Tantra: The Path of Ecstasy_. Boston: Shambhala, 1998.\n\nHalbfass, Wilhelm. _India and Europe: An Essay in Understanding_. Albany: State University of New York Press, 1988.\n\n\u2014\u2014\u2014. _On Being and What There Is: Classical Vai\u015be\u1e63ika and the History of Indian Ontology_. Albany: State University of New York Press, 1992.\n\n\u2014\u2014\u2014. _Tradition and Reflection: Explorations in Indian Thought_. Albany: State University of New York Press, 1991.\n\nIsayeva, Natalia. _Shankara and Indian Philosophy_. Albany: State University of New York Press, 1992.\n\nLarson, Gerald James. _Classical Samkhya: An Interpretation of Its History and Meaning_. Santa Barbara, Calif.: Ross\/Erikson, 1979.\n\nLong, Jeffery D. _A Vision for Hinduism: Beyond Hindu Nationalism_. London: I. B. Tauris, 2007.\n\nMarcaurelle, Roger. _Freedom through Inner Renunciation: \u015aa\u1e45kara's Philosophy in a New Light_. Albany: State University of New York Press, 2000.\n\nMatilal, Bimal Krishna. _The Character of Logic in India_. Albany: State University of New York Press, 1998.\n\nMuller-Ortega, Paul Eduardo. _The Triadic Heart of \u015aiva: Kaula Tantricism of Abhinavagupta in the Non-Dual Shaivism of Kashmir_. Albany: State University of New York Press, 1989.\n\nObeyesekere, Gananath. _Imagining Karma_. Berkeley: University of California Press, 2002.\n\nPhillips, Stephen. _Yoga, Karma, and Rebirth: A Brief History and Philosophy_. New York: Columbia University Press, 2009.\n\nRambachan, Anantanand. _Accomplishing the Accomplished: The Vedas as a Source of Valid Knowledge in \u015aa\u1e45kara_. Honolulu: University of Hawaii Press, 1991.\n\n\u2014\u2014\u2014. _The Advaita Worldview: God, World, and Humanity_. Albany: State University of New York Press, 2006.\n\n\u2014\u2014\u2014. _The Limits of Scripture: Vivekananda's Reinterpretation of the Vedas_. Honolulu: University of Hawaii Press, 1994.\n\nSarma, Deepak. _An Introduction to M\u0101dhva Ved\u0101nta_. Surrey, U.K.: Ashgate, 2003.\n\nSherma, Rita, and Arvind Sharma, eds. _Hermeneutics and Hindu Thought: Toward a Fusion of Horizons_. New York: Springer, 2010.\n\nTapasyananda, Swami. _Bhakti Schools of Vedanta_. Madras (Chennai): Sri Ramakrishna Math, 1990.\n\nVrajaprana, Pravrajika. _Vedanta: A Simple Introduction_. Hollywood, Calif.: Vedanta Press, 1999.\n\nWhicher, Ian. _The Integrity of the Yoga Dar\u015bana: A Reconsideration of Classical Yoga_. Albany: State University of New York Press, 1999.\n\nI. Hindu Traditions\n\nAmong the excellent works listed in this section, I would suggest, as particularly accessible to beginners, Vasudha Narayanan's _The Way and the Goal: Expressions of Devotion in the Early \u015ar\u012bvai\u1e63\u1e47ava Community_ , Douglas R. Brooks's _The Secret of the Three Cities: An Introduction to Hindu \u015a\u0101kta Tantrism_ , and David N. Lorenzen's _Kapalikas and Kalamukhas: Two Lost \u015aaivite Sects_.\n\nAyyar, C. V. N. _Origin and Early History of \u015aaivism in South India_. Madras: University of Madras, 1936.\n\nBhandarkar, R. G. _Vai\u1e63\u1e47avism, \u015aaivism and Minor Religious Systems_. Varanasi: Indological Book House, 1965.\n\nBhattacharya, Nagendranath. _History of the Tantric Religion_. Delhi: Munshiram Manoharlal, 1982.\n\nBrooks, Douglas R. _The Secret of the Three Cities: An Introduction to Hindu \u015a\u0101kta Tantrism_. Chicago: University of Chicago Press, 1990.\n\nBurghart, Richard. \"The Founding of the R\u0101manand\u012b Sect.\" _Ethnohistory_ 225 (1978): 121\u201339.\n\nDimock, Edward C. _The Place of the Hidden Moon: Erotic Mysticism in the Vai\u1e63\u1e47avasahajiya Cult of Bengal_. Chicago: University of Chicago Press, 1989.\n\nLorenzen, David N. _Kapalikas and Kalamukhas: Two Lost \u015aaivite Sects_. Berkeley: University of California Press, 1972.\n\nNarayanan, Vasudha. _The Way and the Goal: Expressions of Devotion in the Early \u015ar\u012bvai\u1e63\u1e47ava Community_. Cambridge, Mass.: Center for the Study of World Religions, 1987.\n\nOpenshaw, Jeanne. _Seeking Bauls of Bengal_. Cambridge: Cambridge University Press, 2002.\n\nJ. Hinduism Outside of South Asia\n\nBy far the most accessible book in this section is the enlightening and entertaining _American Veda_ , by Phil Goldberg, which chronicles the influence of Hindu thought in the United States. Somewhat more theoretical and \"academic,\" but nevertheless accessible and engaging, is Lola Williamson's _Transcendent in America: Hindu-Inspired Meditation Movements as New Religion_. Less wide-ranging, but also more in-depth than Goldberg's work, Williamson's book focuses on three specific Hindu-based traditions: Siddha Yoga, Transcendental Meditation (TM), and Self-Realization Fellowship (SRF).\n\nBakker, F. L. _The Struggle of the Hindu Balinese Intellectuals: Developments in Modern Hindu Thinking in Independent Indonesia_. Amsterdam: VU University Press, 1993.\n\nBilimoria, Purushottama. _Hinduism in Australia: Mandala for the Gods_. Melbourne: Spectrum Publications, 1989.\n\nBromley, David G., and Larry Shinn, eds. _Krishna Consciousness in the West_. Lewisburg, Pa.: Bucknell University Press, 1989.\n\nBurghart, Richard, ed. _Hinduism in Great Britain_. London: Tavistock, 1987.\n\nFrench, Harold. _The Swan's Wide Waters: Ramakrishna and Western Culture_. Port Washington, N.Y.: Kennikat Press, 1974.\n\nGoldberg, Phil. _American Veda: From Emerson and the Beatles to Yoga and Meditation\u2014How Indian Spirituality Changed the West_. Bourbon, Ind.: Harmony, 2010.\n\nJackson, Carl. _Ved\u0101nta for the West: The Ramakrishna Movement in the United States_. Bloomington: Indiana University Press, 1994.\n\nJudah, J. Stillson. _Hare Krishna and the Counterculture_. New York: Wiley, 1974.\n\nKlass, Morton. _Singing with Sai Baba: The Politics of Revitalization in Trinidad_. Boulder, Colo.: Westview, 1991.\n\nShankarananda, Swami. \"Confessions of a Western Hindu\" _Hinduism Today_ (October\u2013December 2005).\n\nVertovec, Steven. _Hindu Trinidad_. London: Macmillan, 1992.\n\nWilliamson, Lola. _Transcendent in America: Hindu-Inspired Meditation Movements as New Religion_. New York: New York University Press, 2010.\n\nK. Theoretical Issues\n\nA little more challenging for beginners, the works in this field are focused upon a range of theoretical concerns that are of great importance for the representation of Hindu traditions but are largely of interest to teachers and professional scholars. Works from this list that are particularly clear and powerful in making their argument, having become practically required reading for scholars in this field, are Ronald Inden's _Imagining India_ and Sharada Sugirtharajah's _Imagining Hinduism: A Postcolonial Perspective_. But for those who are new to this field and to these issues, I would recommend starting with J. E. Llewellyn's edited volume of essays by various authors, _Defining Hinduism: A Reader_.\n\nBreckenridge, Carol A., and Peter van der Veer, eds. _Orientalism and the Postcolonial Predicament_. Philadelphia: University of Pennsylvania Press, 1993.\n\nBrown, W. Norman. _India and Indology_. Delhi: Motilal Banarsidass, 1978.\n\nDalmia, Vasudha, and Heinrich von Stietencron, eds. _Representing Hinduism: The Construction of Religious Traditions and National Identity_. Thousand Oaks, Calif.: Sage, 1995.\n\nInden, Ronald. _Imagining India_. Cambridge: Blackwell, 1990.\n\nJha, D. N. _Rethinking Hindu Identity_. London: Equinox, 2009.\n\nLlewellyn, J. E. _Defining Hinduism: A Reader_. New York: Routledge, 2005.\n\nLorenzen, David N. \"Who Invented Hinduism?\" _Comparative Studies in Society and History_ 41, no. 4 (October 1999): 630\u201359.\n\nNicholson, Andrew J. _Unifying Hinduism: Philosophy and Identity in Indian Intellectual History_. New York: Columbia University Press, 2010.\n\nPennington, Brian K. _Was Hinduism Invented? Britons, Indians, and the Colonial Construction of Religion_. New York: Oxford University Press, 2005.\n\nRamaswamy, Krishnan, Antonio de Nicolas, and Aditi Banerjee, eds. _Invading the Sacred: An Analysis of Hinduism Studies in America_. Delhi: Rupa, 2007.\n\nSen, Amartya. _The Argumentative Indian: Writings on Indian History, Culture, and Identity_. New York: Farrar, Straus, and Giroux, 2005.\n\nSontheimer, G\u00fcnther-Dietz, and Hermann Kulke, eds. _Hinduism Reconsidered_. New Delhi: Manohar, 2001.\n\nSugirtharajah, Sharada. _Imagining Hinduism: A Postcolonial Perspective_. London: Routledge, 2003.\n\nTrautmann, Thomas R. _The Aryans and British India_. Berkeley: University of California Press, 1997.\n\nTyagananda, Swami, and Pravrajika Vrajaprana. _Interpreting Ramakrishna_. New Delhi: Motilal Banarsidass, 2010.\n\nL. Other Religious Traditions Related to Hinduism\n\nFor beginners, some of the clearest and most accessible works on traditions that are closely related to Hinduism are those published in the I. B. Tauris series of introducing the world's religions: Nikky-Guninder Kaur Singh's _Sikhism: An Introduction_ , Jenny Rose's _Zoroastrianism: An Introduction_ , and my own _Jainism: An Introduction_. Richard F. Gombrich's _Therav\u0101da Buddhism_ , recently revised, has become something of a classic in the field of Buddhist studies. Peter Gottschalk's _Beyond Hindu and Muslim: Multiple Identity in Narratives from Village India_ is outstanding on Islam in India.\n\nAriarajah, S. W. _Hindus and Christians: A Century of Protestant Ecumenical Thought_. Grand Rapids, Mich.: Eerdmans, 1991.\n\nChand, T. _Influence of Islam on Indian Culture_. Allahabad: Indian Press, 1963.\n\nChattopadhyaya, Brajadulal. _Representing the Other? Sanskrit Sources and the Muslims_. Delhi: Manohar, 1998.\n\nCoward, H. G. _Hindu-Christian Dialogue_. Maryknoll, N.Y.: Orbis Books, 1990.\n\nFrykenberg, Robert Erik, ed. _Christians and Missionaries in India_. London: Rout-ledge, 2003.\n\nGascoigne, Bamber. _The Great Moguls_. New York: Carroll and Graf, 2002.\n\nGilmartin, David, and Bruce B. Lawrence. _Beyond Turk and Hindu: Rethinking Religious Identities in Islamicate South Asia_. Gainesville: University Press of Florida, 2000.\n\nGombrich, Richard F. _Therav\u0101da Buddhism: A Social History_. New York: Routledge, 2006.\n\nGottschalk, Peter. _Beyond Hindu and Muslim: Multiple Identity in Narratives from Village India_. New York: Oxford University Press, 2005.\n\nHerman, Arthur. _Influences: How Ancient Hinduism Dramatically Changed Early Christianity_. Stevens Point, Wis.: Cornerstone Press, 2004.\n\nKaur Singh, Nikky-Guninder. _Sikhism: An Introduction_. London: I. B. Tauris, 2011.\n\nLong, Jeffery D. _Jainism: An Introduction_. London: I. B. Tauris, 2009.\n\nMonius, Anne E. _Imagining a Place for Buddhism_. New York: Oxford University Press, 2001.\n\nPangborn, Cyrus R. _Zoroastrianism: A Beleaguered Faith_. New York: Advent Books, 1983.\n\nRose, Jenny. _Zoroastrianism: An Introduction_. London: I. B. Tauris, 2011.\n\n## III. INTERNET SOURCES\n\nA. Primary Sources\n\nComplete Works of Swami Vivekananda:  \n<http:\/\/www.ramakrishnavivekananda.info\/vivekananda\/complete_works.htm>\n\nThe Hindu Universe:  \n<http:\/\/www.hindunet.org\/>\n\nHinduism Today:  \n<http:\/\/www.hinduismtoday.com\/>\n\nVedanta Spiritual Library:  \n<http:\/\/www.celextel.org\/>\n\nB. Secondary Sources\u2014Popular\n\nAdherents.com:  \n<http:\/\/www.adherents.com\/>\n\nBeliefnet.com:  \n<http:\/\/www.beliefnet.com\/>\n\nC. Secondary Sources\u2014Academic\n\nDigital South Asia Library (University of Chicago):  \n<http:\/\/dsal.uchicago.edu\/reference\/>\n\nHinduism Home Page (University of Wyoming):  \n<http:\/\/uwacadweb.uwyo.edu\/religionet\/er\/hinduism\/index.htm>\n\nSanskrit, Tamil, and Pahlavi Dictionaries (University of Cologne, Germany):  \n<http:\/\/webapps.uni-koeln.de\/tamil\/>\n\n# About the Author\n\n**Jeffery D. Long** received a PhD in the philosophy of religions from the University of Chicago's Divinity School in the year 2000. His dissertation was entitled _Plurality and Relativity: Whitehead, Jainism, and the Reconstruction of Religious Pluralism_. In it, he attempts to re-articulate the religious pluralism of the modern or Neo-Vedanta tradition of Sri Ramakrishna in a way that draws upon both the process philosophy of Alfred North Whitehead and the Jain philosophy of relativity. Since completing his degree at Chicago, he has been teaching in the Department of Religious Studies at Elizabethtown College in Elizabethtown, Pennsylvania, where he is currently an associate professor and department chair. He is also a cofounder and codirector of Elizabethtown's Asian Studies program. He has published two books\u2014 _A Vision for Hinduism: Beyond Hindu Nationalism_ (2007) and _Jainism: An Introduction_ (2009). His other publications include three dozen articles and reviews in various edited volumes and scholarly journals. Ongoing projects include an introduction to Indian philosophy and a theology of divine incarnation for the Ramakrishna tradition, as well as a science fiction\/fantasy epic that some of his students have dubbed \"the Hindu _Lord of the Rings_.\" Raised Catholic, Long was drawn to Hinduism at an early age and formally embraced the tradition as an adult, at the age of 26. Both he and his wife, Dr. Mahua Bhattacharya (who teaches Japanese at Elizabethtown College), are active members of their local Hindu temple, as well as of the Ramakrishna Vedanta Society. He is also a regular consultant to the Hindu American Foundation. He has served as chair of the steering committee of the Dharma Academy of North America (DANAM) and he is currently cochair of the North American Hinduism consultation of the American Academy of Religion.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n**Begin Reading**\n\nTable of Contents\n\nNewsletters\n\nCopyright Page\n\nIn accordance with the U.S. Copyright Act of 1976, the scanning, uploading, and electronic sharing of any part of this book without the permission of the publisher is unlawful piracy and theft of the author's intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained by contacting the publisher at permissions@hbgusa.com. Thank you for your support of the author's rights.\n\n#\n\n_To see everything, the Master's eye is best of all,_\n\n_As for me, I would add, so is the Lover's eye._\n\n\u2014CAIUS JULIUS PHAEDRUS\n\n_Like all great rationalists you believed in things that were twice as incredible as theology._\n\n\u2014HALLD\u00d3R LAXNESS, _U NDER THE GLACIER_\n\n_Chess is life._\n\n\u2014BOBBY FISCHER\n\n# PREFACE\n\nEvery afternoon, seven summers ago, I sat at my kitchen table in the south of England and worked on a book. Its name was _Born on a Blue Day._ The keys on my computer registered hundreds of thousands of impressions. Typing out the story of my formative years, I realized how many choices make up a single life. Every sentence or paragraph confided some decision I or someone else\u2014a parent, teacher, or friend\u2014had taken, or not taken. Naturally I was my own first reader, and it is no exaggeration to say that in writing, then reading the book, the course of my life was inexorably changed.\n\nThe year before that summer, I had traveled to the Center for Brain Studies in California, where the neurologists probed me with a battery of tests. It took me back to early days in a London hospital when, surveying my brain for seizure activity, the doctors had fixed me up to an encephalogram machine. Attached wires had streamed down and around my little head, until it resembled something hauled up out of the deep, like angler's swag.\n\nThese scientists wore tans and white smiles. They gave me sums to solve, and long sequences of numbers to learn by heart. Newer tools measured my pulse and my breathing as I thought. I submitted to all these experiments with a burning curiosity; it felt exciting to learn the secret of my childhood.\n\nMy autobiography opens with their diagnosis. My difference finally had a name. Until then it had run the whole gamut of inventive aliases: painfully shy, hypersensitive, cack-handed (in my father's characteristically colorful words). According to the scientists, I had high-functioning autistic savant syndrome: the connections in my brain, since birth, had formed unusual circuits. Back home in England I began to write, with their encouragement, producing pages that in the end found favor with a London editor.\n\nTo this day, readers both of the first book and of my second, _Embracing the Wide Sky,_ continue to send me their messages. They wonder how it must be to perceive words and numbers in different colors, shapes, and textures. They try to picture solving a sum in their mind using these multidimensional colored shapes. They seek the same beauty and emotion that I find in both a poem and a prime number. What can I tell them?\n\nImagine.\n\nClose your eyes and imagine a space without limits, or the infinitesimal events that can stir up a country's revolution. Imagine how the perfect game of chess might start and end: a win for white, or black, or a draw? Imagine numbers so vast that they exceed every atom in the universe, counting with eleven or twelve fingers instead of ten, reading a single book in an infinite number of ways.\n\nSuch imagination belongs to everyone. It even possesses its own science: mathematics. Ricardo Nemirovsky and Francesca Ferrara, who specialize in the study of mathematical cognition, write that \"like literary fiction, mathematical imagination entertains pure possibilities.\" This is the distillation of what I take to be interesting and important about the way in which mathematics informs our imaginative life. Often we are barely aware of it, but the play between numerical concepts saturates the way we experience the world.\n\nThis new book, a collection of twenty-five essays on the \"math of life,\" entertains pure possibilities. According to the definition offered by Nemirovsky and Ferrara, \"pure\" here means something immune to prior experience or expectation. The fact that we have never read an endless book, or counted to infinity (and beyond!) or made contact with an extraterrestrial civilization (all subjects of essays in the book) should not prevent us from wondering: _what if?_\n\nInevitably, my choice of subjects has been wholly personal and therefore eclectic. There are some autobiographical elements but the emphasis throughout is outward looking. Several of the pieces are biographical, prompted by imagining a young Shakespeare's first arithmetic lessons in zero\u2014a new idea in sixteenth-century schools\u2014or the calendar created for a Sultan by the poet and mathematician Omar Khayy\u00e1m. Others take the reader around the globe and back in time, with essays inspired by the snows of Quebec, sheep counting in Iceland, and the debates of ancient Greece that facilitated the development of the Western mathematical imagination.\n\nLiterature adds a further dimension to the exploration of those pure possibilities. As Nemirovsky and Ferrara suggest, there are numerous similarities in the patterns of thinking and creating shared by writers and mathematicians (two vocations often considered incomparable). In the \"Poetry of the Primes,\" for example, I explore the way in which certain poems and number theory coincide. At the risk of disappointing fans of \"mathematically constructed\" novels, I admit this book has been written without once mentioning the name \"Perec.\"\n\nThe following pages attest to the changes in my perspective over the seven years since that summer in southern England. Travels through many countries in pursuit of my books as they go from language to language, accumulating accents, have contributed much to my understanding. Exploring the many links between mathematics and fiction has been another spur. Today, I live in the heart of Paris. I write full-time. Every day I sit at a table and ask myself: _what if?_\n\n# One\n\n# FAMILY VALUES\n\nIn a smallish London suburb where nothing much ever happened, my family gradually became the talk of the town. Throughout my teens, wherever I went, I would always hear the same question, \"How many brothers and sisters do you have?\"\n\nThe answer, I understood, was already common knowledge. It had passed into the town's body of folklore, exchanged between the residents like a good yarn.\n\nEver patient, I would dutifully reply, \"Five sisters, and three brothers.\"\n\nThese few words never failed to elicit a visible reaction from the listener: brows would furrow, eyes would roll, lips would smile. \"Nine children!\" they would exclaim, as if they had never imagined that families could come in such sizes.\n\nIt was much the same story in school. _\"J'ai une grande famille\"_ was among the first phrases I learned to say in Monsieur Oiseau's class. From my fellow students, many of whom were single sons or daughters, the sight of us siblings together attracted comments that ranged all the way from faint disdain to outright awe. Our peculiar fame became such that for a time it outdid every other in the town: the one-handed grocer, the enormously obese Indian girl, a neighbor's singing dog, all found themselves temporarily displaced in the local gossip. Effaced as individuals, my brothers, sisters, and I existed only in number. The quality of our quantity became something we could not escape. It preceded us everywhere: even in French, whose adjectives almost always follow the noun (but not when it comes to _une grande famille_ ).\n\nWith so many siblings to keep an eye on, it is perhaps little wonder that I developed a knack for numbers. From my family I learned that numbers belong to life. The majority of my math acumen came not from books but from regular observations and day-to-day interactions. Numerical patterns, I realized, were the matter of our world. To give an example, we nine children embodied the decimal system of numbers: zero (whenever we were all absent from a place) through to nine. Our behavior even bore some resemblance to the arithmetical: over angry words, we sometimes divided; shifting alliances between my brothers and sisters combined and recombined them into new equations.\n\nWe are, my brothers, sisters, and I, in the language of mathematicians, a \"set\" consisting of nine members. A mathematician would write:\n\n> S = {Daniel, Lee, Claire, Steven, Paul, Maria, Natasha, Anna, Shelley}\n\nPut another way, we belong to the category of things that people refer to when they use the number nine. Other sets of this kind include the planets in our solar system (at least, until Pluto's recent demotion to the status of a non-planet), the squares in a game of x's and o's, the players in a baseball team, the muses of Greek mythology, and the Justices of the U.S. Supreme Court. With a little thought, it is possible to come up with others, including:\n\n> {February, March, April, May, August, September, October, November, December} where S = the months of the year not beginning with the letter J.\n> \n> {5, 6, 7, 8, 9, 10, Jack, Queen, King} where S = in poker, the possible high cards in a straight flush.\n> \n> {1, 4, 9, 16, 25, 36, 49, 64, 81} where S = the square numbers between 1 and 99.\n> \n> {3, 5, 7, 11, 13, 17, 19, 23, 29} where S = the odd primes below 30.\n\nThere are nine of these examples of sets containing nine members, so taken together they provide us with a further instance of just such a set.\n\nLike colors, the commonest numbers give character, form, and dimension to our world. Of the most frequent\u2014zero and one\u2014we might say that they are like black and white, with the other primary colors\u2014red, blue, and yellow\u2014akin to two, three, and four. Nine, then, might be a sort of cobalt or indigo: in a painting it would contribute shading, rather than shape. We expect to come across samples of nine as we might samples of a color like indigo\u2014only occasionally, and in small and subtle ways. Thus a family of nine children surprises as much as a man or woman with cobalt-colored hair.\n\nI would like to suggest another reason for the surprise of my town's residents. I have alluded to the various and alternating combinations and recombinations between my siblings. In how many ways can any set of nine members divide and combine? Put another way, how large is the set of all subsets?\n\n> {Daniel}... {Daniel, Lee}... {Lee, Claire, Steven}... {Paul}... {Lee, Steven, Maria, Shelley}... {Claire, Natasha}... {Anna}...\n\nFortunately, this type of calculation is very familiar to mathematicians. As it turns out, we need only to multiply the number two by itself, as many times as there are members in the set. So, for a set consisting of nine members the answer to our question amounts to: 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 = 512.\n\nThis means that there existed in my hometown, at any given place and time, 512 different ways to spot us in one or another combination. 512! It becomes clearer why we attracted so much attention. To the other residents, it really must have seemed that we were legion.\n\nHere is another way to think about the calculation that I set out above. Take any site in the town at random, say a classroom or the municipal swimming pool. The first \"2\" in the calculation indicates the odds of my being present there at a particular moment (one in two\u2014I am either there, or I am not). The same goes for each of my siblings, which is why two is multiplied by itself a total of nine times.\n\nIn precisely one of the possible combinations, every sibling is absent (just as in one of the combinations we are all present). Strange as it may sound, we can even define those sets containing no objects. Mathematicians call them an \"empty set.\" Where sets of nine members embody everything we can think of, touch, or point to when we use the number nine, empty sets are all those that are represented by the value zero. So while a Christmas reunion in my hometown can bring together as many of us as there are Justices on the U.S. Supreme Court, a trip to the moon will unite only as many of us as there are pink elephants, four-sided circles, or people who have swum the breadth of the Atlantic Ocean.\n\nOur mind uses sets when we think and when we perceive just as much as when we count. Our possible thoughts and perceptions about these sets can range almost without limit. Fascinated by the different cultural subdivisions and categories of an infinitely complex world, the Argentine writer Jorge Luis Borges offers a mischievously tongue-in-cheek illustration in his fictional Chinese encyclopedia entitled _The Celestial Emporium of Benevolent Knowledge:_\n\n> Animals are classified as follows: (a) those that belong to the Emperor; (b) embalmed ones; (c) those that are trained; (d) suckling pigs; (e) mermaids; (f) fabulous ones; (g) stray dogs; (h) those that are included in this classification; (i) those that tremble as if they were mad; (j) innumerable ones; (k) those drawn with a very fine camel's-hair brush; (l) et cetera; (m) those that have just broken the flower vase; (n) those that at a distance resemble flies.\n\nNever one to forgo humor in his texts, Borges here also makes several thought-provoking points. First, though a set as familiar to our understanding as that of \"animals\" implies containment and comprehension, the sheer number of its possible subsets actually swells toward infinity. With their handful of generic labels (\"mammal,\" \"reptile,\" \"amphibious,\" etc.), standard taxonomies conceal this fact. To say, for example, that a flea is tiny, parasitic, and a champion jumper is only to begin to scratch the surface of all its various aspects.\n\nSecond, defining a set owes more to art than it does to science. Faced with the problem of a near endless number of potential categories, we are inclined to choose from a few\u2014those most tried and tested within our particular culture. Western descriptions of the set of all elephants privilege subsets like \"those that are very large,\" and \"those possessing tusks,\" and even \"those possessing an excellent memory,\" while excluding other equally legitimate possibilities such as Borges's \"those that at a distance resemble flies,\" or the Hindu \"those that are considered lucky.\"\n\nMemory is a further example of the privileging of certain subsets (of experience) over others, in how we talk and think about a category of things. Asked about his birthday, a man might at once recall the messy slice of chocolate cake that he devoured, his wife's enthusiastic embrace, and the pair of fluorescent green socks that his mother presented to him. At the same time, many hundreds, or thousands, of other details likewise composed his special day, from the mundane (the crumbs from his morning toast that he brushed out of his lap) to the peculiar (a sudden hailstorm on the mid-July afternoon that lasted several minutes). Most of these subsets, though, would have completely slipped his mind.\n\nReturning to Borges's list of subsets of animals, several of the categories pose paradoxes. Take, for example, the subset (j): \"innumerable ones.\" How can any subset of something\u2014even if it is imaginary, like Borges's animals\u2014be infinite in size? How can a part of any collection not be smaller than the whole?\n\nBorges's taxonomy is clearly inspired by the work of Georg Cantor, a nineteenth-century German mathematician whose important discoveries in the study of infinity provide us with an answer to this paradox.\n\nCantor showed, among other things, that parts of a collection (subsets) as great as the whole (set) really do exist. Counting, he observed, involves matching the members of one set to another. \"Two sets _A_ and _B_ have the same number of members if and only if there is a perfect one-to-one correspondence between them.\" So, by matching each of my siblings and myself to a player on a baseball team, or to a month of the year not beginning with the letter J, I am able to conclude that each of these sets is equivalent, all containing precisely nine members.\n\nNext came Cantor's great mental leap: in the same manner, he compared the set of all natural numbers (1, 2, 3, 4, 5...) with each of its subsets such as the even numbers (2, 4, 6, 8, 10...), odd numbers (1, 3, 5, 7, 9...), and the primes (2, 3, 5, 7, 11...). Like the perfect matches between each of the baseball team players and my siblings and me, Cantor found that for each natural number he could uniquely assign an even, an odd, and a prime number. Incredibly, he concluded, there are as \"many\" even (or odd, or prime) numbers as all the numbers combined.\n\nReading Borges invites me to consider the wealth of possible subsets into which my family \"set\" could be classified, far beyond those that simply point to multiplicity. All grown up today, some of my siblings have children of their own. Others have moved far away, to the warmer and more interesting places from where postcards come. The opportunities for us all to get together are rare, which is a great pity. Naturally I am biased, but I love my family. There is a lot of my family to love. But size ceased long ago to be our defining characteristic. We see ourselves in other ways: those that are studious, those that prefer coffee to tea, those that have never planted a flower, those that still laugh in their sleep...\n\nLike works of literature, mathematical ideas help expand our circle of empathy, liberating us from the tyranny of a single, parochial point of view. Numbers, properly considered, make us better people.\n\n# Two\n\n# ETERNITY IN AN HOUR\n\nOnce upon a time I was a child who loved to read fairy tales. Among my favorites was \"The Magic Porridge Pot\" by the Brothers Grimm. A poor, good-hearted girl receives from a sorceress a little pot capable of spontaneously concocting as much sweet porridge as the girl and her mother can eat. One day, after eating her fill, the mother's mind goes blank and she forgets the magic words \"Stop, little pot.\"\n\n> So it went on cooking and the porridge rose over the edge, and still it cooked on until the kitchen and whole house were full, and then the next house, and then the whole street, just as if it wanted to satisfy the hunger of the whole world.\n\nOnly the daughter's return home, and the requisite utterance, finally brings the gooey avalanche to a belated halt.\n\nThe Brothers Grimm introduced me to the mystery of infinity. How could so much porridge emerge from so small a pot? It got me thinking. A single flake of porridge was awfully slight. Tip it inside a bowl and one would probably not even spot it for the spoon. The same held for a drop of milk, or a grain of sugar.\n\nWhat if, I wondered, a magical pot distributed these tiny flakes of porridge and drops of milk and grains of sugar in its own special way? In such a way that each flake and each drop and each grain had its own position in the pot, released from the necessity of touching. I imagined five, ten, fifty, one hundred, one thousand flakes and drops and grains, each indifferent to the next, suspended here and there throughout the curved space like stars. More porridge flakes, more drops of milk, more grains of sugar are added one after another to this evolving constellation, forming microscopic Big Dippers and minuscule Great Bears. Say we reach the ten thousand four hundred and seventy-third flake of porridge. Where do we include it? And here my child's mind imagined all the tiny gaps\u2014thousands of them\u2014between every flake of porridge and drop of milk and grain of sugar. For every minute addition, further tiny gaps would continue to be made. So long as the pot magically prevented any contact between them, every new flake (and drop and granule) would be sure to find its place.\n\nHans Christian Andersen's \"The Princess and the Pea\" similarly sent my mind spinning toward the infinite, but this time, an infinity of fractions. One night, a young woman claiming to be a princess knocks at the door of a castle. Outside, a storm is blowing and the pelting rain musses her clothes and turns her golden hair black. So sorry a sight is she that the queen of the castle doubts her story of high birth. To test the young woman's claim, the queen decides to place a pea beneath the bedding on which the woman will sleep for the night (princesses being most delicate creatures!). Her bed is piled to a height of twenty mattresses. But in the morning the woman admits to having hardly slept a wink.\n\nThe thought of all those tottering mattresses kept me up long past my own bedtime. By my calculation, a second mattress would double the distance between the princess's back and the offending pea. The tough little legume would therefore be only half as prominent as before. Another mattress reduces the pea's prominence to one-third. But if the young princess's body is sensitive enough to detect one-half of a pea (under two mattresses) or one-third of a pea (under three mattresses), why would it not also be sensitive enough to detect one-twentieth? In fact, possessing limitless sensitivity (this is a fairy tale after all), not even one-hundredth, or one-thousandth or one-millionth of a pea could be tolerably borne.\n\nWhich brings us back to the Brothers Grimm and their tale of porridge. For the princess, even a pea felt infinitely big; for the poor daughter and her mother, even an avalanche of porridge reduced to the infinitesimally small.\n\n\"You have too much imagination,\" my father said when I shared these thoughts with him. \"You always have your nose in some book.\" My father kept a pile of paperbacks and regularly bought the weekend papers, but he was never a particularly enthusiastic reader. \"Get outdoors more\u2014there's no good being cooped up in here.\"\n\nHide-and-seek in the park with my brothers and sisters lasted all of ten minutes. The swings held my attention for about as long. We walked the perimeter of the lake and threw breadcrumbs out onto the grimy water. Even the ducks looked bored.\n\nGames in the yard offered greater entertainment. We fought wars, cast spells, and traveled back in time. In a cardboard box we sailed along the Nile; with a bedsheet we pitched a tent in the hills of Rome. At other times, I would simply stroll the local streets to my heart's content, dreaming up all manner of new adventures and imaginary expeditions.\n\nReturning one day from China, I heard the low grumble of an approaching storm and fled for cover inside the municipal library. Everyone knew me there; I was one of their regulars. Corridors of books pullulated around them, centuries of learning tiled the walls, and I brushed my fingertips along the seemingly endless shelves as I walked.\n\nMy favorite section brimmed with dictionaries and encyclopedias: the building blocks of books. These seemed to promise (though of course they could not deliver) the sum of human knowledge: every fact and idea and word. This vast panoply of information was tamed by divisions\u2014A\u2013C, D\u2013F, G\u2013I\u2014and every division subdivided in turn\u2014Aa\u2013Ad... Di\u2013Do... Il\u2013In. Many of these subdivisions also subdivided\u2014Hai\u2013Han... Una\u2013Unf\u2014and some among them subdivided yet further still\u2014Inte\u2013Intr. Where does a person start? And, perhaps more important, where should he stop? I usually left the choice to chance. At random I tugged an encyclopedia from the shelf and let its pages open where they may, and for the next hour I sat and read about _Bora Bora_ and _borborygmi_ and the _Borg_ scale.\n\nLost in thought, I did not immediately notice the insistent _tap-tap_ of approaching footsteps on the polished floor. They belonged to one of the senior librarians, a neighbor. He was tall (but then, to a child is not everyone tall?) and thin with a long head finished off by a few random sprigs of graying hair.\n\n\"I have a book for you,\" said the librarian. I craned upward a moment before taking the recommendation from his big hands. The cover, titled _The Borrowers,_ wore a \"Bookworms Club Monthly Selection\" sticker. I thanked him, less out of gratitude than the desire to end the sudden eclipse that darkened my table. But when I finally left the desk an hour later, the book left with me, checked out and tucked firmly beneath my arm.\n\nIt told the story of a tiny family that lived under the floorboards of a house. I tried to imagine what it would be like to live so small. In my mind's eye, I pictured the world as it continued to contract. The smaller I became, the bigger my surroundings grew. The familiar now became strange; the strange became familiar. All at once, a face of ears and eyes and hair becomes a pink expanse of shrub and grooves and heat. Even the tiniest fish becomes a whale. Specks of dust take flight like birds, swooping and wheeling above my head. I shrank until all that was familiar disappeared completely, until I could no longer tell a mound of laundry or a rocky mountain apart.\n\nAt my next visit to the library, I duly joined the Bookworms Club. The months were each twinned with a classic story, and some of the selections enchanted me more than others, but it was December's tale that truly seized my senses: _The Lion, the Witch, and the Wardrobe_ by C. S. Lewis. Opening its pages, I followed Lucy, as she was sent with her siblings \"away from London during the war because of the air raids... to the house of an old professor who lived in the heart of the country.\" It was \"the sort of house that you never seem to come to the end of, and it was full of unexpected places.\"\n\nWith Lucy, I stepped into the large wardrobe in one of the otherwise bare spare rooms, tussled with its rows of dense and dust-fringed clothes as we fumbled our way with outstretched fingers toward the back. I, too, suddenly heard the crunch of snow beneath my shoes, and saw the fur coats give way all at once to the fir trees of this magical land, a wardrobe's depth away.\n\nNarnia became one of my favorite places, and I visited it many times that winter. Repeated readings of the story would keep me in bright thoughts and images for many months.\n\nOne day, on the short walk home from school, it so happened that these images came to the front of my mind. The lampposts that lined the street reminded me of the lamp-post I had read about in the story, the point in the landscape from which the children return to the warmth and mothballs of the professor's wardrobe.\n\nIt was midafternoon, but the electric lights were already shining. Fluorescent haloes stood out in the darkening sky at equidistant points. I counted the time it took me to step with even paces from one lamppost to another. Eight seconds. Then I retreated, counting backward, and arrived at the same result. A few doors down, the lights came on in my parents' house; yellow rectangles glowed dimly between the red bricks. I watched them with only half a mind.\n\nI was contemplating those eight seconds. To reach the next lamppost I had only to take so many steps. Before I reached there I would first have to arrive at the midway point. That would take me four seconds. But this observation implied that the remaining four seconds also contained a midway point. Six seconds from the start, I would land upon it. Two seconds would now separate me from my destination. Yet before I made it, another halfway point\u2014a second later\u2014would intervene. And here I felt my brain seethe hot under my woollen hat. For after seven seconds, the eighth and final second would likewise contain a halfway point of its own. Seven and one half seconds after starting off, the remaining half a second would also not elapse before I first passed its midway point in turn. After seven seconds and three-quarters, a stubborn quarter of a second of my journey would still await me. Going halfway through it would leave me an eighth of a second still to go. One sixteenth of a second would keep me from my lamppost, then 1\/32 of a second, then 1\/64, then 1\/128, and so on. Fractions of fractions of fractions of a second would always distance me from the end.\n\nSuddenly I could no longer depend on those eight seconds to deliver me to my destination. Worse, I could no longer be sure that they would let me move one inch. Those same interminable fractions of seconds that I had observed toward the end of my journey applied equally to the start. Say my opening step took one second; this second, of course, contained a halfway point. And before I could cross this half of a second, I would first have to traverse its own midway point (the initial quarter second), and so on.\n\nAnd yet, my legs disposed of all these fractions of seconds as they had always done. Adjusting the heavy schoolbag on my back, I walked the length between the lampposts and counted once again to eight. The word rang out defiantly into the cold crisp air. The silence that followed, however, was short-lived. \"What are you doing standing outside in the cold and dark?\" shouted my father from the yellow oblong of the open front door. \"Come inside now.\"\n\nI did not forget the infinity of fractions that lurked between the lampposts on my street. Day after day, I found myself slowing involuntarily to a crawl as I passed them, afraid perhaps of falling between the whole seconds into their interspersed gaps. What a sight I must have made, inching warily forward tiny step after tiny step with my round woolly head and the lumpy bag upon my back.\n\nNumbers within numbers, and so tiny! I was amazed. These fractions of fractions of fractions of fractions of fractions went on forever. Add any of them to zero and they hardly registered at all. Add tens, hundreds, thousands, millions, billions of them to zero and the result is still almost exactly zero. Only infinitely many of these fractions could lead from zero to one, from nothing to something:\n\n> 1\/2 + 1\/4 + 1\/8 + 1\/16 + 1\/32 + 1\/64 + 1\/128 + 1\/256 + 1\/512 + 1\/1024... = 1.\n\nOne evening in the new year, my mother, very flustered, asked me to be on my best behavior. Guests\u2014a rarity\u2014were due any moment now, for dinner. My mother, it seemed, intended to repay some favor to the librarian's wife. \"No funny questions,\" she said, \"and no elbows on the table. And after the first hour, bed!\"\n\nThe librarian and his wife arrived right on time with a bottle of wine that my parents never opened. With their backs to one another, they thrashed themselves out of their coats before sitting at the dining room table, side by side. The wife offered my mother a compliment about the checkered tablecloth. \"Where did you buy it?\" she asked, over her husband's sigh.\n\nWe ate my father's roast chicken and potatoes with peas and carrots, and as we ate the librarian talked. All eyes were on him. There were words on the weather, local politics, and all the nonsense that was interminably broadcast on TV. Beside him, his wife ate slowly, one-handed, while the other hand worried her thin black hair. At one point in her husband's monologue, she tried gently tapping his tightly bunched hand.\n\n\"What? What?\"\n\n\"Nothing.\" Her fork promptly retired to her plate. She looked to be on the verge of tears.\n\nVery much novices in the art of hospitality, my mother and father exchanged helpless glances. Plates were hurriedly collected, and bowls of ice cream served. A frosty atmosphere presently filled the room.\n\nI thought of the infinitely many points that can divide the space between two human hearts.\n\n# Three\n\n# COUNTING TO FOUR IN ICELANDIC\n\nAsk an Icelander what comes after three and he will answer, \"Three what?\" Ignore the warm blood of annoyance as it fills your cheeks, and suggest something, or better still, point. \"Ah,\" our Icelander replies. Ruffled by the wind, the four sheep stare blankly at your index finger. _\"Fj\u00f3rar,\"_ he says at last.\n\nHowever, when you take your phrase book\u2014presumably one of those handy, rain-resistant brands\u2014from your pocket and turn to the numbers page, you find, marked beside the numeral 4, _fj\u00f3rir._ This is not a printing error, nor did you hear the Icelander wrong. Both words are correct; both words mean \"four.\" This should give you your first inkling of the sophistication with which these people count.\n\nI first heard Icelandic several years ago during a trip to Reykjav\u00edk. No phrase book in my pocket, thank God. I came with nothing more useful than a vague awareness of the shape and sounds of Old English, some secondary-school German, and plenty of curiosity. The curiosity had already seen service in France. Here in the North, too, I favored conversation over textbooks.\n\nI hate textbooks. I hate how they shoehorn even the most incongruous words\u2014like \"cup\" and \"bookcase,\" or \"pencil\" and \"ashtray\"\u2014onto the same page, and then call it \"vocabulary.\" In a conversation, the language is always fluid, moving, and you have to move with it. You walk and talk and see where the words come from, and where they should go. It was in this way that I learned how to count like a Viking.\n\nIcelanders, I learned, have highly refined discrimination for the smallest quantities. \"Four\" sheep differ in kind from \"four,\" the abstract counting word. No farmer in Hverager\u00f0i would ever dream of counting sheep in the abstract. Nor, for that matter, would his wife or son or priest or neighbor. To list both words together, as in a textbook, would make no sense to them whatsoever.\n\nThis numerical diversity applies not only to sheep. Naturally enough, the woolly mammals feature little in town dwellers' talk. Like you and me, my friends in Reykjav\u00edk talk about birthdays and buses and pairs of jeans but, unlike in English, in Icelandic these things each require their own set of number words.\n\nFor example, a toddler who turns two is _tveggja_ years old. And yet the pocket phrase book will inform you that \"two\" is _tveir._ Age, abstract as counting to our way of thinking, becomes in Icelandic a tangible phenomenon. Perhaps you too sense the difference: the word _tveggja_ slows the voice, suggesting duration. We hear this possibly even more clearly in the word for a four-year-old: _fj\u00f6gurra._ Interestingly, these sounds apply almost exclusively to the passage of years\u2014the same words are hardly ever used to talk about months, days, or weeks. Clock time, on the other hand, renders the Icelander almost terse as a tick: the hour after one o'clock is _tv\u00f6._\n\nWhat about buses? Here numbers refer to identity rather than quantity. In Britain or America, we say something like, \"the number three bus,\" turning the number into a name. Icelanders do something similar. Their most frequent buses are each known by a special number word. In Reykjav\u00edk, the number three bus is simply _\u00feristur_ (whereas to count to three the Icelander says _\u00fer\u00edr_ ). _Fjarki_ is how to say \"four\" when talking buses in Iceland.\n\nA third example is pairs of something\u2014whether jeans or shorts, socks or shoes. In this case, Icelanders consider \"one\" as being plural: _einar_ pair of jeans, instead of the phrase book _einn._\n\nWith time and practice, I have learned all these words, more words for the numbers one to four than has an English speaker to count all the way up to fifty. In English, I would suggest, numbers are considered more or less ethereal\u2014as categories, not qualities. Not so the smallest numbers in Icelandic. It is as though each corresponded to a delicate nuance of color. Where the English word \"red\" is abstract, indifferent to its object, words like \"crimson,\" \"scarlet,\" and \"burgundy\" possess their own particular shade of meaning and application.\n\nWe can only speculate as to the reason why Icelanders stop at the number five (for which, like every number thereafter, a single word exists). According to psychologists, humans can count in flashes only up to quantities of four. We see three buttons on a shirt and say \"three\"; we glance at four books on a table and say \"four.\" No conscious thought attends this process\u2014it seems to us as effortless as the speech with which we pronounce the words. The same psychologists tell us that the smallest numbers loom largest in our minds. Asked to pick a number between one and fifty, we tend toward the shallow end of the scale (far fewer say \"forty\" than \"fourteen\"). It is one possible explanation for why only the commonest quantities feel real to us, why most numbers we accept only on the word of a teacher or textbook. Forty, to us, is but a vague notion; fourteen, on the other hand, is a quantity within our reach. Four, we recognize as something solid and definite. In Icelandic, you can give your baby the name \"Four.\"\n\nI do not know Chinese, but I have read that counting in this language rivals in its sophistication even that of the Icelanders. A shepherd in rural China says _s\u00ec zh \u012b_ when his flock numbers four, whereas a horseman\u2014possessing the same quantity of horses\u2014counts them as _s\u00ec p \u012d._ This is because mounts are counted differently from other animals in Chinese. Domesticated animals, too. Asked how many cows he had milked that morning, a farmer would reply _s\u00ec t\u00f3u_ (four). Fish are a further exception. _S\u00ec ti\u00e1o_ is how an angler would count his fourth catch of the day.\n\nUnlike in Icelandic, in Chinese these fine distinctions apply to all quantities. What saves its speakers from endless trouble of recall is generalization. _S\u00ec ti\u00e1o_ means \"four\" when counting fish, but also trousers, roads, and rivers (and other long, slender, and flexible objects). A locksmith might enumerate his keys as _q \u012b b\u0103_ (seven), but so too would a housewife apropos her seven knives, or a tailor when adding up his seven pairs of scissors (or other handy items). Imagine that, with his scissors, the tailor snips a sheet of fabric in two. He would say he has _li \u0103ng zh\u0101ng_ (two) sheets of fabric, using the same number word as he would for paper, paintings, tickets, blankets, and bedsheets. Now picture the tailor as he rolls the fabric into long inflexible tubes. This pair he counts as _li \u0103ng ju\u0103n_ (two scrolls, or rolls of film, would be counted in the same way). Scrunching the sheets into balls, the tailor counts them as _li \u0103ng tu\u00e1n,_ insofar as they resemble other pairs of round things.\n\nWhen counting people, the Chinese start from _y \u012b ge_ (one), though for villagers and family members they begin _y \u012b k\u014fu,_ and _y \u012b m\u00edng_ for lawyers, politicians, and royals. Numbering a crowd thus depends on its composition. A hundred marchers would be counted as _y \u012bb\u0103i ge_ if they consist, for example, of students, but as _y \u012bb\u0103i k\u014fu_ if they hail from the villages.\n\nSo complex is this method of counting that, in some regions of China, the words for certain numbers have even taken on the varying properties of a dialect. _W \u016dsh\u00ed li,_ for example, a standard Mandarin word meaning \"fifty\" (when counting small, round objects like grains of rice), sounds truly enormous to the speakers of southern Min, for whom it is used to count watermelons.\n\nThis profusion of Icelandic and Chinese words for the purpose of counting appears to be an exception to the rule. Many of the world's tribal languages, in contrast, make do with only a handful of names for numbers. The Veddas, an indigenous people of Sri Lanka, are reported to have only words for the numbers one ( _ekkamai_ ) and two ( _dekkamai_ ). For larger quantities, they continue: _otameekai, otameekai, otameekai..._ (\"and one more, and one more, and one more...\"). Another example is the Caquintes of Peru, who count one ( _aparo_ ) and two ( _mavite_ ). Three they call \"it is another one\"; four is \"the one that follows it.\"\n\nIn Brazil, the Munduruku relay quantity by according an extra syllable to each new number: one is _pug,_ two is _xep xep,_ three is _ebapug,_ and four is _edadipdip._ They count, understandably, no higher than five. This imitative method, while transparent, has clear limitations. Just imagine a number word as many syllables long as the quantity of trees leading to a food source! The seemingly endless chain of syllables would prove far too expensive to the tongue (not to say the listener's powers of concentration). It pains the head even to think about what it would be like to have to learn to recite the ten times tables in this way.\n\nAll this may sound almost incomprehensible for those brought up speaking languages that count to thousands, millions and beyond, but it does at least make the relationship between a quantity and its appointed word sound straightforward and conventional. Quite often, though, it is not. In many tribal languages, we find that the names for numbers are perfectly interchangeable, so that a word for \"three\" will also sometimes mean \"two,\" and at other times \"four\" or \"five.\" A word meaning \"four\" will have \"three\" and \"five\"\u2014occasionally \"six\"\u2014as synonyms.\n\nFew circumstances within these communities require any greater numerical precision. Any number beyond their fingertips is superfluous to their traditional way of life. In many of these places there are, after all, no legal documents that require dates, no bureaucracies that levy taxes, no clocks or calendars, no lawyers or accountants, no banks or banknotes, no thermometers or weather reports, no schools, no books, no playing cards, no lines, no shoes (and hence, no shoe sizes), no shops, no bills, and no debts to settle. It would make as much sense to tell them, say, that a group of men amounts exactly to eleven, as for someone to inform us that this same group has precisely one hundred and ten fingers, and as many toes.\n\nThere is a tribe in the Amazon rain forest that knows nothing whatsoever of numbers. Their name is the _Pirah\u00e3_ or the _Hi'aiti'ihi,_ meaning \"the straight ones.\" Surrounded by throngs of trees, their small clusters of huts lie on the banks of the Maici River. Tumbling gray rain breaks green on the lush foliage and long grass. Days there are continuously hot and humid, inducing a perpetual look of embarrassment on the faces of visiting missionaries and linguists. Children race naked around the village, while their mothers wear light dresses obtained by bartering with the Brazilian traders. From the same source, the men display colorful T-shirts, the flotsam of past political campaigns, exhorting the observer to vote Lula.\n\nManioc (a tough and bland tuber), fresh fish, and roasted anteater sustain the population. The work of gathering food is divided along lines of sex. At first light, women leave the huts to tend the manioc plants and collect firewood, while the men go upriver or downriver to fish. They can spend the whole day there, bow and arrow in hand, watching the water. For want of means of storage, any catch is consumed quickly. The _Pirah\u00e3_ apportion food in the following manner: members of the tribe haphazardly receive a generous serving until no more remains. Any who have not yet been served ask a neighbor, who has to share. This procedure only ends when everyone has eaten his fill.\n\nThe vast majority of what we know about the _Pirah\u00e3_ is due to the work of Daniel Everett, a Californian linguist who has studied them at close quarters over a period of thirty years. With professional perseverance, he gradually heard their cacophonic ejections as comprehensible words and phrases, becoming in the process the first outsider to embrace the tribe's way of life.\n\nTo Everett's astonishment, the language he learned has no specific words for measuring time or quantity. Names for numbers like \"one\" or \"two\" are unheard of. Even the simplest numerical queries brought only confusion or indifference to the tribesmen's eyes. Of their children, parents are unable to say how many they have, though they remember all their names. Plans or schedules older than a single day have no purchase on the _Pirah\u00e3_ 's minds. Bartering with foreign traders simply consists of handing over foraged nuts as payment until the trader says that the price has been met.\n\nNor do the _Pirah\u00e3_ count with their bodies. Their fingers never point or curl: when indicating some amount they simply hold their hand palm down, using the space between their hand and the ground to suggest the height of the pile that such a quantity could reach.\n\nIt seems the _Pirah\u00e3_ make no distinction between a man and a group of men, between a bird and a flock of birds, between a grain of manioc flour and a sack of manioc flour. Everything is either small ( _h\u00f3i_ ) or big ( _ogii_ ). A solitary macaw is a small flock; the flock, a big macaw. In his _Metaphysics,_ Aristotle shows that counting requires some prior understanding of what \"one\" is. To count five, or ten or twenty-three birds, we must first identify one bird, an idea of \"bird\" that can apply to every possible kind. But such abstractions are entirely foreign to the tribe.\n\nWith abstraction, birds become numbers. Men and maniocs, too. We can look at a scene and say, \"There are two men, three birds, and four maniocs\" but also, \"There are nine things\" (summing two and three and four). The _Pirah\u00e3_ do not think this way. They ask, \"What are these things?\" \"Where are they?\" \"What do they do?\" A bird flies, a man breathes, and a manioc plant grows. It is meaningless to try to bring them together. Man is a small world. The world is a big manioc.\n\nIt is little surprise to learn that the _Pirah\u00e3_ perceive drawings and photos only with great difficulty. They hold a photograph sideways or upside down, not seeing what the image is meant to represent. Drawing a picture is no easier for them, not even a straight line. They cannot copy simple shapes with any fidelity. Quite possibly, they have no interest in doing so. Instead their pencils (furnished by linguists or missionaries) produce only repeating circular marks on the researcher's sheet of paper, each mark a little different from the last.\n\nPerhaps this also explains why the _Pirah\u00e3_ tell no stories, possess no creation myths. Stories, at least as we understand them, have intervals: a beginning, a middle, and an end. When we tell a story, we recount: naming each interval is equivalent to numbering it. Yet the _Pirah\u00e3_ talk only of the immediate present: no past impinges on their actions; no future motivates their thoughts. History, they told Everett, is \"where nothing happens, and everything is the same.\"\n\nLest anyone should think tribes such as the _Pirah\u00e3_ somehow lacking in capacity, allow me to mention the _Guugu Yimithirr_ of north Queensland in Australia. In common with most Aboriginal language speakers, the _Guugu Yimithirr_ have only a handful of number words: _nubuun_ (one), _gudhirra_ (two), and _guunduu_ (three or more). This same language, however, permits its speakers to navigate their landscape geometrically. A wide array of coordinate terms attune their minds intuitively to magnetic north, south, east, and west, so they develop an extraordinary sense of orientation. For instance, a _Guugu Yimithirr_ man would not say something like, \"There is an ant on your right leg,\" but rather \"There is an ant on your southeast leg.\" Or, instead of saying, \"Move the bowl back a bit,\" the man would say, \"Move the bowl to the north-northwest a bit.\"\n\nWe are tempted to say that a compass, for them, has no point. But at least one other interesting observation can be drawn from the _Guugu Yimithirrs'_ ability. In the West, young children often struggle to grasp the concept of a negative number. The difference between the numbers two (2) and minus two (\u20132) often evades their imagination. Here the _Guugu Yimithirr_ child has a definite advantage. For two, the child thinks of \"two steps east,\" while minus two becomes \"two steps west.\" To a question like, \"What is minus two plus one?\" the Western child might incorrectly offer, \"Minus three,\" whereas the _Guugu Yimithirr_ simply takes a mental step eastward to arrive at the right answer of \"one step west\" (\u20131).\n\nThe _Kpelle_ tribe of Liberia offers a final example of culture's effect on how a person counts. The _Kpelle_ have no word in their language corresponding to the abstract concept of \"number.\" Counting words exist, but are rarely employed above thirty or forty. One young _Kpelle_ man, when interviewed by a linguist, could not recall his language's term for seventy-three. A word meaning \"one hundred\" frequently stands in for any large amount.\n\nNumbers, the _Kpelle_ believe, have power over people and animals and are to be traded lightly and with a kind of reverence; village elders therefore often guard jealously the solutions to sums. From their teachers, the children acquire only the most basic numerical facts in piecemeal fashion, without learning any of the rhythm that constitutes arithmetic. The children learn, for example, that 2 + 2 = 4, and perhaps several weeks or months afterward that 4 + 4 = 8, but they are never required to connect the two sums and see that 2 + 2 + 4 = 8.\n\nCounting people directly is believed by the _Kpelle_ to bring bad luck. In Africa, this taboo is both ancient and widespread. There exists as well the sentiment, shared by the authors of the Old Testament, that the counting of human beings is an exercise in poor taste. The simplicity of their number words is not only a question of language, it also reveals an ethical dimension.\n\nI read with pleasure a book of essays published several years ago by the Nigerian novelist Chinua Achebe. In one, Achebe complained of the Westerners who asked him, \"How many children do you have?\" Rebuking silence, he suggested, best answered such an impertinent question.\n\n\"But things are changing and changing fast with us... and so I have learned to answer questions that my father would not have touched with a bargepole.\"\n\nAchebe's children number _ano_ (four). In Iceland, they would say _fj\u00f6gur._\n\n# Four\n\n# PROVERBS AND TIMES TABLES\n\nI once had the pleasure of discovering a book wholly dedicated to the art of proverbs. It was in one of the municipal libraries that I frequented as a teenager. The title of the book escapes me now, the name of its author, too, but I still recall the little shiver of excitement I felt as my fingers caressed its quarto pages.\n\n> \"Penny wise, pound foolish.\"\n> \n> \"Small fishes are better than empty dishes.\"\n> \n> \"A speech without proverb is like a stew without salt.\"\n\nNow that I come to think of it, I doubt this book had any single author. Every proverb is anonymous. Each appears in a society's mental repertoire by some process akin to immaculate composition. Like the verses in the Muslims' Koran, proverbs seem pre-written, patiently awaiting a mouth to utter their existence. Some linguists contend that language happens independently of our reason, its origins traceable to a still mysterious and exclusive gene. Perhaps proverbial logic is like language in this respect, its existence as essential to our humanity as the power of speech.\n\nWhoever the author or the editor, this book proved to me that there are only so many proverbs a healthy man can take. A point of surfeit is reached, beyond which the reader can no longer follow: his brain starts to ache, his eyes to water. Taken in excess, proverbs lose all the familiar felicities of their compact structure. They start to read merely as repetitions\u2014which many by then quite probably are. From my experience, I estimate this limit at about one hundred.\n\nOne hundred proverbs, give or take, sum up the essence of a culture; one hundred multiplication facts compose the ten times table. Like proverbs, these numerical truths or statements\u2014two times two is four, or seven times six equals forty-two\u2014are always short, fixed and pithy. Why then do they not stick in our heads as proverbs do?\n\nBut they did before, some people claim. When? In the good old days, of course. Today's children, they suggest, are simply too slack-brained to learn correctly. Nothing interests them but sending one another text messages and harassing the teacher. The critics hark back to those days before computers and calculators; to the time when every number was drummed into children's heads till finding the right answer became second nature.\n\nExcept that, no such time has ever really existed. Times tables have always given many schoolchildren trouble, as Charles Dickens knew in the mid-nineteenth century.\n\n> Miss Sturch put her head out of the school-room window: and seeing the two gentlemen approaching, beamed on them with her invariable smile. Then, addressing the vicar, said in her softest tones, \"I regret extremely to trouble you, sir, but I find Robert very intractable, this morning, with his multiplication table.\" \"Where does he stick now?\" asked Doctor Chennery. \"At seven times eight, sir,\" replied Miss Sturch. \"Bob!\" shouted the vicar through the window. \"Seven times eight?\" \"Forty-three,\" answered the whimpering voice of the invisible Bob. \"You shall have one more chance before I get my cane,\" said Doctor Chennery. \"Now then, look out. Seven times...\n\nOnly his younger sister's rapid intervention with the answer\u2014fifty-six\u2014spares the boy the physical pain of another wrong guess.\n\nCenturies old, then, the difficulty that many children face acquiring their multiplication facts is also serious. It is, to borrow a favorite term of politicians, a \"real problem.\" \"Lack of fluency with multiplication tables,\" reports the UK schools inspectorate, \"is a significant impediment to fluency with multiplication and division. Many low-attaining secondary-school pupils struggle with instant recall of tables. Teachers [consider] fluent recall of multiplication tables as an essential prerequisite to success in multiplication.\"\n\nThe facts in a multiplication table represent the essence of our knowledge of numbers: the molecules of math. They tell us how many dimes make up a dollar (10 \u00d7 10), the number of squares on a chessboard (8 \u00d7 8), the quantity of individual surfaces on a trio of boxes (3 \u00d7 6). They help us evenly divide fifty-six items among eight people (7 \u00d7 8 = 56, therefore 56\/8 = 7), or realize that forty-three of something cannot be evenly distributed in the same way (because forty-three, being a prime number, makes no appearance among the facts). They reduce the risk of anxiety in the young learner, and give a vital boost to the child's confidence.\n\nPatterns are the matter that these molecules, in combination, make. Take, for instance, the consecutive facts 9 \u00d7 5 = 45, and 9 \u00d7 6 = 54: the digits in both answers are the same, only reversed. Thinking about the other facts in the nine times table, we see that every answer's digits sum to nine:\n\n> 9 \u00d7 2 = 18 (1 + 8 = 9)\n> \n> 9 \u00d7 3 = 27 (2 + 7 = 9)\n> \n> 9 \u00d7 4 = 36 (3 + 6 = 9)\n> \n> Etc....\n\nOr, surveying the other tables, we discover that multiplying an even number by five will always produce an answer ending in zero (2 \u00d7 5 = 10, 6 \u00d7 5 = 30), while multiplying an odd number by five gives answers that always end in itself (3 \u00d7 5 = 15, 9 \u00d7 5 = 45). Or, we spot that six squared (thirty-six) plus eight squared (sixty-four) equals ten squared (one hundred).\n\nSevens, the trickiest times table to learn, also offer a beautiful pattern. Picture the seven on a telephone's keypad, in the bottom left-hand corner. Now simply raise your eye to the key immediately above it (four), and then again to the next key above (one). Do the same starting from the bottom middle key (eight), and so on. Every keypad digit in turn corresponds to the final digit in the answers along the seven times table: 7, 14, 21, 28...\n\nNot all multiplication facts pose problems, of course. Multiplying any number by one or ten is obviously easy enough. Our hands know that two times five, and five times two, both equal ten. Equivalencies abound: two times six, and three times four, both lead to twelve; multiplying three by ten, and six by five, amounts to the same thing.\n\nBut others are trickier, less intuitive, and far easier to let slip. A numerate culture will find whatever means at its disposal to pass these obstinate facts down from one generation to the next. It will carve them into rock and scratch them onto parchment. It will condemn every inauspicious student to threats and thrashings. It will select the most succinct form and phrasing for its essential truths: not too heavy for the tongue, nor too lengthy for the ear.\n\nJust like a proverb.\n\nFor example, what did our ancestors mean precisely when bequeathing us a truth like \"An apple a day keeps the doctor away\"? Not, of course, that we should read it literally, superstitiously, imagining apples like the cloves of garlic that are supposed to make vampires take to their heels. Rather, the sentence expresses a core relationship between two different things: healthy food (for which the apple plays stand-in), and illness (embodied by the doctor). Consider a few of the alternate ways in which this relationship might also have been summed up:\n\n> \"A daily fruit serving is good for you.\"\n> \n> \"Eating healthy food prevents illness.\"\n> \n> \"To avoid getting sick, eat a balanced diet.\"\n\nThese versions are as short, or even shorter, than our proverb. But none is anywhere near as memorable.\n\nLong before Dickens wrote about the horrors of multiplication tables, our ancestors had decided to sum up fifty-six as \"seven times eight,\" just as they described health (and its absence) in apples and doctors. But as with a concept like \"health,\" understanding the number fifty-six can be achieved via many other routes.\n\n> 56 = 28 \u00d7 2\n> \n> 56 = 14 \u00d7 4\n> \n> 56 = 7 \u00d7 8\n\nOr even:\n\n> 56 = 3.5 \u00d7 16\n> \n> 56 = 1.75 \u00d7 32\n> \n> 56 = 0.875 \u00d7 64\n\nIt is not difficult to see, though, why tradition would have privileged the succinctness and simplicity of \"seven times eight\" for most purposes, over rival definitions such as \"one and three-quarters times thirty-two\" or \"seven-eighths of sixty-four\" (as useful as they might be in certain contexts).\n\nWhat is seven times eight? It is the clearest and simplest way to talk about the number fifty-six.\n\nThese familiar forms may be simple and succinct, but they are finely wrought, nonetheless, whether in words or figures. The proverbial apple, for example, begins the proverb, though its meaning (as a protector of health) cannot be grasped until the end. \"Apple\" here is the answer to the question: What keeps the doctor away? Other proverbs also share this structure, where the answer precedes the question. \"A stitch in time saves nine\" (What saves nine stitches? A stitch in time) or \"Blind is the bookless man\" (What is the bookless man? Blind).\n\nPlacing the answer at the start compels our imagination: we concede more freely the premise that an apple can deter illness, in part because the word \"apple\" precedes all the others. Using this structure can also arouse our attention, inciting us to picture the rest of the proverb with the opening image in mind: to see the bookless man, for example, more clearly in light of his blind eyes.\n\nWhen I discussed the ways in which we could think about the number fifty-six, I borrowed this feature of proverbs and put the sum's answer at the start. Saying, \"Fifty-six equals seven times eight\" lends emphasis where it is needed most: not on the seven or the eight, but on what they produce.\n\nForm is important. A pupil reads 56 = 7 \u00d7 8 and hears the whisper of many generations, while another child, shown 7 \u00d7 8 = 56, finds himself alone. The first child is enriched; the second is disinherited.\n\nToday's debates over times tables often neglect questions of form. Not so the schools of nineteenth-century America. The young nation, still younger than its oldest citizens, hosted educational discussions unprecedented in their inquisitive detail. Teachers pondered in marvelous depth the kind of verb to use when multiplying. In _The Grammar of English Grammars_ (published in 1858) we read, \"In multiplying one only, it is evidently best to use a singular verb: 'Three times one is three.' And in multiplying any number above one, I judge a plural verb to be necessary: 'Three times two are six.' \"\n\nThe more radical contributors to these debates suggested doing away with excess words like \"times\" altogether. Instead of learning \"four times six is twenty-four,\" the child would repeat, \"four sixes are twenty-four.\" These educators urged a return to the way ancient Greek children had chanted their times tables two millennia before: \"once one is one,\" \"twice one is two,\" et cetera. Others went even further, suggesting that the verb \"is\" (or \"are\") also be thrown out: \"four sixes, twenty-four,\" in the manner of the Japanese.\n\nSchools in Japan have long lavished attention on the sounds and rhythms of the times tables. Every syllable counts. Take the multiplication 1 \u00d7 6 = 6, among the first facts that any child learns. The standard Japanese word for \"one\" is _ichi;_ the usual Japanese word for \"six\" is _roku._ Put together, they make: _ichi roku roku_ (one six, six). But Japanese pupils never say this: the line is clumsy, the sounds cacophonous. Instead, the pupils all say, _in roku ga roku_ (one six, six), using an abraded form of _ichi_ ( _in_ ) and an insertion ( _ga_ ) for euphony.\n\nThe trimming of unnecessary words or sounds shapes both the proverb and the times table. \"Better late than never,\" says the parent in New York when his son complains of his allowance long overdue. \"Four fives, twenty,\" says the boy when he finally counts his money.\n\nIn Japanese, the multiplication 6 \u00d7 9 = 54 is an extreme example of ellipsis. Being similar in sound, the two words\u2014 _roku_ (six) and _ku_ (nine)\u2014merge to a single _rokku._ This new number would be a little like pronouncing the multiplication 7 \u00d7 9 in English as \"sevine.\"\n\nWhy is _in roku ga roku_ judged more pleasing than _ichi roku roku?_ Both phrases contain six syllables; both phrases use the \"roku\" word twice, yet the first sounds beautiful, while the second seems ugly. The answer is, parallelism. _In roku ga roku_ has a parallel structure, which makes it easier on the ear. We hear this balanced structure frequently in our proverbs: \"fight fire with fire.\" One six is six.\n\nIt is much harder to make good parallel times tables in English than it is in Japanese. The same is true of many European languages. In Japanese, a child says _roku ni juuni_ (six two, ten two) for 6 \u00d7 2 = 12, and _san go juugo_ (three five, ten five) for 3 \u00d7 5 = 15, whereas an English-speaking child must say \"twelve\" and \"fifteen,\" a French child _douze_ and _quinze,_ and a German child _zw\u00f6lf_ and _f\u00fcnfzehn._\n\nBeside the ones, it is only the ten times table that produces consistent parallel forms in English, of the kind \"easy come, easy go\": \"seven tens (are) seventy.\"\n\nNot all proverbs employ parallels. Many use alliteration\u2014the repetition of certain sounds: \"One swallow does not a summer make\" or \"All that glitters is not gold.\" Times tables alliterate too: \"four fives (are) twenty\" and (if we extend our times tables to twelve) \"six twelves (are) seventy-two.\"\n\nParallels and alliteration are both in evidence when proverbs rhyme: \"A friend in need is a friend indeed\" or \"Some are wise and some are otherwise.\" By definition, square multiplications (when the number is multiplied by itself) begin in a similar way: \"two times two...\" \"four fours...\" \"nine times nine...\" though only the squares of five and six finish with a flourish: \"five times five (is) twenty-five\" and \"six times six (is) thirty-six.\"\n\nFor this reason, learners procure these two multiplication facts (after, perhaps, \"two times two is four\") with the greatest ease and pleasure. This pair\u2014\"five times five is twenty-five\" and \"six times six is thirty-six\"\u2014truly attain the special quality of the proverb. Other multiplication facts, from the same times tables, get close. For example, multiplying five by any odd number inevitably leads to a rhyme: \"seven fives (are) thirty-five.\" Six, when followed by an even number, causes the even number to rhyme: \"six times four (is) twenty-four\" and \"six eights (are) forty-eight.\"\n\nMistakes? Of course they happen. Nobody is above them. No matter how long a person spends immersed in numbers, recollection can sometimes go astray. I have read of world-class mathematicians who blush at \"nine times seven.\"\n\nThe same troubles we encounter with times tables sometimes occur with words, what we call \"slips of the tongue,\" though more often than not the tongue is innocent. It is the memory that is to blame. Someone who says, \"He is like a bear with a sore thumb\" (mixing up \"like a bear with a sore head\" and \"to stick out like a sore thumb\") makes a mistake similar to he who answers \"seven times eight\" with \"forty-eight\" (confusing 7 \u00d7 8 = 56 with 6 \u00d7 8 = 48).\n\nSuch mistakes are mistakes of unfamiliarity. Proverbs, like times tables, can often strike us as strange, their meanings remote. Why do we talk of bears with sore heads? In what way do swallows conjure up summertime better than other birds? The choice of words seems to us as arbitrary and archaic as the numbers in the times tables. But the truths they represent are immemorial.\n\n\"Hold fast to the words of ancestors,\" instructs a proverb from India. Hold fast to their times tables, too.\n\n# Five\n\n# CLASSROOM INTUITIONS\n\nTelevision journalists, in their weaker moments, will occasionally pull the following stunt on a hapless minister of education. Mugging through his makeup at the attendant cameras, the interviewer strokes his notes, clears his throat, and says, \"One final question. What is nine times six?\"\n\nSuch episodes never fail to make me sigh. It is a sad thing when mathematics is reduced to the recollection (or, more often, the non-recollection) of a classroom rule.\n\nIn one particular confrontation of this type, the presenter demanded to know the price of fourteen pens when four had a price tag of $2.42. \"I haven't the foggiest,\" the minister whimpered to the audience members' howls of delight.\n\nOf course, the questions are patently asked with the expectation of failure in mind. Politicians are always trying to anticipate our expectations and to meet them. Should we then feel such surprise when they judge the situation correctly and get the sum wrong?\n\nProperly understood, the study of mathematics has no end: the things each of us does not know about it are infinite. We are all of us at sea with some aspect or another. Personally, I must admit to having no affinity with algebra. This discovery I owe to my middle school math teacher, Mr. Baxter.\n\nTwice a week, I would sit in Mr. Baxter's class and do my best to keep my head down. I was thirteen, going on fourteen. With his predecessors I had excelled at the subject: number theory, statistics, probability, none of them had given me any trouble. Now I found myself an algebraic zero.\n\nThings were changing; I was changing. All swelling limbs and sweating brain, suddenly I had more body than I knew what to do with. Arms and legs became the prey of low desktops and narrow corridors, were ambushed by sharp corners. Mr. Baxter ignored my plight. Bodies were inimical to mathematics, or so we were led to believe. Bad hair, acrid breath, lumpy skin, all vanished for an hour every Tuesday and Thursday. Young minds in the buff soared into the sphere of pure reason. Pages turned into parallelograms; cities, circumferences; recipes, ratios. Shorn of our bearings, we groped our way around in this rarefied air.\n\nIt was in this atmosphere that I learned the rudiments of algebra. The word, we were told, was of Arabic origin, culled from the title of a ninth-century treatise by Al-Khwarizmi (\"algorithm,\" incidentally, is a Latin corruption of his name). This exotic provenance, I remember, left a deep impression on me. The snaking, swirling equations in my textbook made me think of calligraphy. But I did not find them beautiful.\n\nMy textbook pages looked cluttered with lexicographical debris: all those x's and y's and z's. The use of the least familiar letters only served to confirm my prejudice. I thought these letters ugly, interrupting perfectly good sums.\n\nTake x\u00b2 + 10x = 39, for example. Such concoctions made me wince. I much preferred to word it out: a square number (1, or 4, or 9, etc.) plus a multiple of ten (10, 20, 30, etc.) equals thirty-nine; 9 (3 \u00d7 3) + 30 (3 \u00d7 10) = 39; three is the common factor; x = 3. Years later, I learned that Al-Khwarizmi had written out all his problems, too.\n\nStout and always short of breath, Mr. Baxter had us stick to the exercises in the book. He had no patience whatsoever for paraphrasing. Raised hands were cropped with a frown and the admonition to \"reread the section.\" He was a stickler for the textbook's methods. When I showed him my work, he complained that I had not used them. I had not subtracted the same values from either side of the equation. I had not done a thing with the brackets. His red pen flared over the carefully written words of my solutions.\n\nLet me give a further example of my deviant reasoning: x\u00b2 = 2x + 15. I word it out like this: a square number (1, 4, 9, etc.) equals fifteen more than a multiple of two (2, 4, 6, etc.). In other words, we are looking for a square number above seventeen (being fifteen more than two). The first candidate is twenty-five (5 \u00d7 5) and twenty-five is indeed fifteen more than 10 (a multiple of two); x = 5.\n\nA few of Mr. Baxter's students acquired his methods; most, like me, never did. Of course I cannot speak for the others, but I found the experience bruising. I was glad when the year was over and I could move on to other math. But I also felt a certain shame at my failure to comprehend. His classes left me with a permanent suspicion of all equations. Algebra and I have never been fully reconciled.\n\nFrom Mr. Baxter I learned at least one profitable lesson: how not to teach. This lesson would serve me well on numerous occasions. Two years after I left school, while leafing through the newspaper one morning, I came across an agency's advertisement recruiting tutors. I had taught English in Lithuania and discovered that I liked teaching. So I applied. I was interviewed by a lady up in years named Grace, in the office that she kept in her living room. Needlepoint cushions filled the small of my back as I sat before her desk. The wallpaper, if I remember correctly, had a pattern of small birds and honeybees.\n\nThe meeting was brief.\n\n\"Do you enjoy helping others to learn something new?\"\n\n\"Are you mindful of a student's personal learning style?\"\n\n\"Could you work according to a set curriculum?\"\n\nHer questions contained their own answers, like the dialogue taught in a foreign language class: \"Yes, I could work according to a set curriculum.\"\n\nAfter ten minutes of this she said, \"Excellent, well, you would certainly fit in with us here. For English we already have tutors and there is not much demand for foreign languages. What about primary school\u2013level math?\"\n\nWhat about it? I was a taker.\n\nGrace's bookings certainly kept me on my toes. My tutoring patch extended to the neighboring town five miles away, and the bus ride and walk to the farthest homes took as long as the lessons. Nervous, I learned on the job, but the families helped me. I found the children, between the ages of seven and eleven, generally polite and industrious; their parents' nods and smiles tamed my nerves. After a while I ceased to worry and even began to look forward to my weekly visits.\n\nShould I admit I had a favorite student? He was a brown-haired, freckled boy, eight years old but small for his age, and the first time I came to the house he fairly shivered with shyness. We started out with the textbooks that the agency loaned me, but they were old and smelly and the leprous covers soon came apart in our hands. A brightly colored book replaced them, one of the boy's Christmas gifts, but its jargon was poison to his mind. So we abandoned the books and found some better way to pass the hour together. We talked a lot.\n\nIt turned out that he had a fondness for collecting baseball cards and could recite the names of the players depicted on them by heart. With pride he showed me the accompanying album.\n\n\"Can you tell me how many cards you have in there?\" I asked him. He admitted to having never totaled them up. The album contained many pages.\n\n\"If we count each card one by one it will take quite a long time to reach the last page,\" I said. \"What if we were to count them two by two instead?\" The boy agreed that would be quicker. Twice as quick, I pointed out, \"And what if we were to count the cards in threes? Would we get to the end of the album even faster?\" He nodded. Yes, we would: three times as fast.\n\nThe boy interrupted. \"If we counted the cards five at a time, we would finish five times faster.\" He smiled at my smile. Then we opened the album and counted the cards on the first page, I placing my larger palm over every five. There were three palms of cards: fifteen. The second page had slightly fewer (two palms and three fingers, thirteen)\u2014so we carried the difference over to the next. By the seventh page we had reached twenty palms: one hundred cards. We continued turning the pages, putting my palm onto each, and counting along. In all, the number of cards rose to over eighty palms (four hundred).\n\nAfter making light work of the album, we considered the case of a giant counting its pages. The giant's palm, we agreed, would easily count the cards by twenty.\n\nWhat if the same giant wanted to count up to a million? The boy thought for a moment. \"Perhaps he counts in hundreds: one hundred, two hundred, three hundred...\" Did the boy know how many hundreds it would take to reach a million? He shook his head. Ten thousand, I told him. His eyebrows leaped. Finally, he said, \"He would count in ten thousands, then, wouldn't he?\" I confirmed that he would: it would be like us counting from one to hundred. \"And if he were really big, he might count in hundred thousands,\" I continued. The giant would reach a million as quickly as we counted from one to ten.\n\nOnce, during a lesson solving additions, the boy hit upon a small but clever insight. He was copying down his homework for us to go through together. The sum was 12 + 9, but it came out as 19 + 2. The answer, he realized, did not change. Twelve plus nine, and nineteen plus two, both equal twenty-one. The fortuitous error pleased him; it made him pause and think. I paused as well, not wishing to talk for fear of treading on his thoughts. Later I asked him for the answer to a much larger sum, something like 83 + 8. He closed his eyes and said, \"Eighty-nine, ninety, ninety-one.\" I knew then that he had understood.\n\nOf the other students, I recall the Singh family whose children I taught on Wednesday evenings for two hours back-to-back. I remember I could never get on with the father, a businessman who put on executive airs, although the mother treated me with nothing but kindness. They had three children, two boys and a girl; the children were always waiting for me around the table in the living room, still dressed up in their school uniforms. The eldest boy was eleven: a bit of a show-off, he had the confidence of an eldest son. His sister regularly deferred to him. In the middle, the second son laughed a lot. He seemed to laugh for the whole family.\n\nIn the beginning, the trio took the wan bespectacled man before them only half seriously as a teacher. Between them, they had ten years on me. I looked too young and probably sounded it as well, without the smooth patter that comes with experience. All the same, I held my ground. I helped them with their times tables, in which they were far from fluent. They showed surprise when I failed to berate their errors and hesitations. On the contrary, if they were close to the right answer I told them so.\n\n\"What is eight times seven?\"\n\n\"Fifty...\" The eldest boy wavered.\n\n\"Yes,\" I said, encouraging.\n\n\"Fifty- _four,_ \" he ventured.\n\n\"Nearly,\" I said. \"Fifty-six.\"\n\nThis hesitation, a habit of many of my students, intrigued me. It suggested not ignorance, but rather indecision. To say that a student has no idea of a solution, I realized, is untrue. Truth is, the learner _does_ have ideas, too many, in fact\u2014almost all of them bad. Without the knowledge necessary to eliminate this mental haze, the learner finds himself confronted with an embarrassment of wrong answers to helplessly pick from.\n\nWhat had the boy been thinking, I inquired, when he selected fifty-four as his answer? He admitted having previously considered fifty-three, fifty-six, fifty-seven, and fifty-five (in that order). He had felt sure that fifty-one or fifty-two would be too small an answer, both fifty-eight and fifty-nine, too high. Then I asked him why he had finally preferred fifty-four to fifty-three. He replied that he had thought of the eight in the question, and the fact that fifty is half of one hundred, and that half of eight is four.\n\nWe moved on to discussing the difference between odd and even numbers. Eight was an even number; seven was odd. What happens when we multiply odd by even? The children's faces showed more hesitation. I suggested several examples: two times seven (fourteen), three times six (eighteen), four times five (twenty), every answer an even number. Could they see why? Yes, said the second son at last: multiplying by an even number was the same thing as creating pairs. Two times seven created a pair of sevens; four times five made two pairs of fives; three times six produced three pairs of threes. What, then, about eight times seven? It meant four pairs of sevens, the boy said.\n\nPairs evened every odd number: one sock became two socks, three became six, five became ten, seven became fourteen, and nine became eighteen. The last digit of a pair would always be an even number.\n\nThe mental fog of eight times seven was evaporating. Fifty-three promptly vanished as a contending answer; so too did fifty-seven and fifty-five. That left fifty-four and fifty-six. How, then, to tell the two apart? The number fifty-four, I pointed out, was six away from sixty: fifty-four\u2014like sixty\u2014was divisible by six. Like sixty, fifty-four will therefore be the answer to a question containing a six (or a number divisible by six), but neither a seven nor an eight.\n\nBy this process of elimination, which amounts to careful reasoning, only fifty-six remained. Fifty-six is two sevens away from seventy, three eights away from eighty. Eight times seven equals fifty-six.\n\nMy only adult student was a housewife with copper-colored skin, and a long name with vowels and consonants that I had never before seen in such a permutation. The housewife, Grace informed me on the phone, aspired to professional accountancy. In my mind, this fact made for an unpromising start. The admission of self-interest rubbed up against my naive vision of mathematics as something playful and inventive. There seemed to me something almost vulgar in the housewife's sudden interest in numbers, as if she wanted to befriend them only as some people set out to befriend well-connected people.\n\nQuickly I amended this faulty judgment. The reservations about my new student were unfair. They were the reservations of a children's tutor\u2014I knew nothing of teaching adults, of anticipating their needs and expectations.\n\nWe were talking together one day about negative numbers, as we sat in her white-tiled kitchen. Like mathematicians of the sixteenth century, who referred to them as \"absurd\" and \"fictitious,\" she found these numbers difficult to imagine. What could it mean to subtract something from nothing? I tried to explain, but found myself at a frustrating loss for words. Somehow my student understood.\n\n\"You mean, like a mortgage?\"\n\nI did not know what a mortgage was. Now it was her turn to try to explain. As she spoke, I realized that she knew a lot more about negative numbers than I did. Her words possessed real value: they had the bullion of hard experience to back them up.\n\nAnother time we went over \"improper\" fractions: top-heavy fractions, like four-thirds (4\/3) or seven-quarters (7\/4), that help us to think about units in different ways. If we think of the number one as being equivalent to three thirds, for example, then four-thirds is another way of describing one and one-third. Seven-quarters, my student and I agreed, resembled two apples that had been quartered (considering \"one\" as equal to four quarters) and one of these eight apple-quarters consumed.\n\nOur hour was soon up, but we kept on talking. We were discussing fractions and what happens when you halve a half of a half of a half, and so on. It amazed us both to think that this halving, theoretically speaking, could continue indefinitely. There was pleasure in confiding our mutual amazement, almost in the manner of gossip. And like gossip, it was something that we both knew and did not know.\n\nThen she came to a beautiful conclusion about fractions that I shall never forget.\n\nShe said, \"There is no thing that half of it is nothing.\"\n\n# Six\n\n# SHAKESPEARE'S ZERO\n\nFew things, to judge by his works, so fascinated William Shakespeare as the presence of absence: the lacuna where there ought to be abundance\u2014of will, or judgment, or understanding. It looms large in the lives of many of his characters, so powerful in part because it is universal. Not even kings are exempt.\n\nLEAR: What can you say to draw\n\nA third more opulent than your sisters? Speak.\n\nCORDELIA: Nothing, my lord.\n\nLEAR: Nothing?\n\nCORDELIA: Nothing.\n\nLEAR: Nothing will come of nothing. Speak again.\n\nThe scene is one of the tensest, most suspenseful moments in theater, a concentration of tremendous force within a single word. It is the ultimate negation, tossed between the old king and his beloved youngest daughter, compounded and multiplied through repetition. Nothing. Zero.\n\nOf course, Shakespeare's contemporaries were familiar with the idea of nothingness, but not with nothingness as a number, something that they could count and manipulate. In his arithmetic lessons, William became one of the first generation of English schoolboys to learn about the figure zero. It is interesting to wonder about the consequences of this early encounter. How might the new and paradoxical number have driven his thoughts along particular paths?\n\nArithmetic spelled trouble for many schoolmasters of the period. Their knowledge of it was often suspect. For this reason, lessons were probably short, often kept until almost the final hour of the afternoon. Squeezed in after long bouts of Latin composition, a list of proverbs, or the chanting of prayers, the sums and exercises were mostly drawn from a single textbook: _The Ground of Artes_ by Robert Recorde. Published in 1543 (and again, in an expanded edition, in 1550), Recorde's book, which included the first material on algebra in the English language, taught \"the work and practice of Arithmetike, both in whole numbres and Fractions after a more easyer and exarter sorte than any like hath hitherto been sette furth.\"\n\nShakespeare learned to count and reckon using Recorde's methods. He learned that \"there are but tenne figures that are used in Arithmetick; and of those tenne, one doth signifie nothing, which is made like an O, and is privately called a Cypher.\" These Arabic numbers\u2014and the decimal place system\u2014would soon eclipse the Roman numerals (the Tudors called them \"German numbers\"), which were often found too cumbersome for calculating.\n\nGerman numbers were, of course, letters: I (or j), one; V, five; X, ten; L, fifty; C, one hundred; D, five hundred; and M, one thousand. Six hundred appeared as vj.C and three thousand as CCC.M. Perhaps this is why Recorde compares the zero to an O. Years later, Shakespeare would deploy the cipher to blistering effect. \"Thou art an O without a figure... Thou art nothing,\" the Fool tells Lear, after the dialogue with Cordelia that destroys the king's peace of mind.\n\nIn Shakespeare's lessons, letters were out, figures (digits) in. Perhaps they were displayed conspicuously on charts, hung up on walls like the letters of the alphabet are now. Ten to a hard bench, the boys trimmed and dipped their quills in ink, and their feathered fingers copied out the numbers in small, tidy lines. Zeroes spotted every page. But why record something that has no value? Something that is nothing?\n\nEngland's medieval monks, translators and copiers of earlier treatises by the Arab world's mathematicians, had long before noted the zero's almost magical quality. One twelfth-century scribe suggested giving it the name \"chimera\" after the fabulous monster of Greek mythology. Writing in the thirteenth century, John of Halifax explained the zero as something that \"signifies nothing\" but instead \"holds a place and signifies for others.\" His manuscript proved popular in the universities. But it would take the invention of Gutenberg's printing press to bring his ideas to a much wider audience, including the motley boys of King's New School in Stratford.\n\nGLOUCESTER: What paper were you reading?\n\nEDMUND: Nothing my lord.\n\nGLOUCESTER: No? What needed then that terrible despatch of it into your pocket? The quality of nothing hath not such need to hide itself. Let's see: Come, if it be nothing I shall not need spectacles.\n\nThe quality of nothing. We can picture the proto-playwright grappling with zero. The boy closes his eyes and tries to see it. But it is not easy to see nothing. Two shoes, yes, he can see them, and five fingers and nine books. 2 and 5 and 9: he understands what they mean. How, though, to see _zero_ shoes? Add a number to another number, like a letter to another letter, and you create something new: a new number, a new sound. Only, if you add a number to zero, nothing changes. The other number persists. Add five zeroes, ten zeroes, a hundred zeroes if you like. It makes no difference. A multiplication by zero is just as mysterious. Multiply a number, any number\u2014three, or four hundred, or 5,678\u2014by zero, by nothing, and the answer is zero.\n\nWas the boy able to keep up in his lessons, or did he lag? The Tudor schoolmaster, violence dressed in a long cloak and black shoes, would probably have concentrated his mind. The schoolmaster's cane could reduce a boy's buttocks to one big bruise. With its rhyming dialogues (even employing the occasional joke or pun) and clear examples intended to give \"ease to the unlearned,\" we can only hope that Recorde's book spared Shakespeare and his classmates much pain.\n\n> Heere you see vj. (six) lines, whiche stande for vj. (six) places... the [lowest line] standeth for the first place, and the next above it for the second, and so upwarde, till you come to the highest, whiche is the sixth line, and standeth for the sixth place.\n> \n> 100000\n> \n> 10000\n> \n> 1000\n> \n> 100\n> \n> 10\n> \n> 1\n\n> The first place is the place of units or ones, and every counter set in that line betokeneth but one, and the seconde line is the place of 10, for everye counter there standeth for 10. The thirde line the place of hundreds, the fourth of thousands, and so forth.\n\nPerhaps talk of counters turned the boy's thoughts to his father's glove shop. His father would have accounted for all his transactions using the tokens. They were hard and round and very thin, made of copper or brass. There were counters for one pair of gloves, and for two pairs, and three and four and five. But there was no counter for zero. No counters existed for all the sales that his father did not close.\n\nI imagine the schoolmaster sometimes threw out questions to the class. How do we write three thousand in digits? From Recorde's book, Shakespeare learned that zero denotes size. To write thousands requires four places. We write: 3 (three thousands), 0 (zero hundreds), 0 (zero tens), 0 (zero units): 3,000. In _Cymbeline,_ we find one of Shakespeare's many later references to place value (what Recorde in his book called each digit's \"roome\").\n\n> Three thousand confident, in act as many\u2014\n> \n> For three performers are the file, when all\n> \n> The rest do nothing\u2014with this word \"Stand, stand,\"\n> \n> Accommodated by the place...\n\nThe concept must have fascinated him ever since his schooldays. A nothing, the boy sees, depends on kind. An empty hand, for example, is a smaller nothing than an empty class or shop, in the same way that the zero in 10 is ten times smaller than the zero in 101. And the bigger the number, the more places and therefore the more zeroes it can contain: ten has one zero, whereas one hundred thousand has five. He thinks, the bigger an empty room, the more the things that can be contained inside: the greater an absence, the greater the potential presence. Subtract (or \"rebate\" as the Tudors said) one from one hundred thousand and the entire number transforms: five zeroes, five nothings, all suddenly turn to nines (the largest digit): 99,999. Perhaps, like Polixenes in _The Winter's Tale,_ he sensed already the tremendous potential of self-effacement, understood imagination as shifting from place to place like a zero inside an immense number.\n\n> And therefore, like a cipher [zero]\n> \n> (Yet standing in rich place), I multiply\n> \n> With one \"We thank you\" many thousands more\n> \n> That go before it\n\nRecorde's book abounded with exercises. Shakespeare and his classmates' sheets of paper would have quickly turned black with sums as they measured cloth and purchased loaves and counted sheep and paid clergy. But William's mind returns ceaselessly to the zero. He thinks of ten and how it differs from his father's ten. To his father, ten (X) is twice five (v): he counts, whenever possible, in fives and tens. To his son, ten (10) is a one (1) displaced, accompanied by a nought. To his father, ten (X) and one (i) have hardly anything in common: they are two values on opposite ends of a scale. But for the boy, ten and one are intimately linked: there is nothing between them.\n\nTen and one, one and ten.\n\nThe boy sees that, with a retinue of zeroes, even the humble one becomes enormously valuable. Imagination can reconcile even one and one million, as Shakespeare affirmed in his prologue to _Henry V,_ when the chorus stake their claim to represent the multitudes on the field of Agincourt.\n\n> O, pardon! Since a crooked figure [digit] may\n> \n> Attest in little place a million,\n> \n> And let us, ciphers to this great accompt,\n> \n> On your imaginary forces work\n\nBut it is perhaps in his poetry that the boy\u2014now a young man\u2014would most clearly express the impact of Recorde's teaching on his mind. In Sonnet 38, Shakespeare wrote about the relationship between himself and his beloved Muse, comparing their couple to a ten: the poet, the zero, and his beloved, the one.\n\n> O, give thyself the thanks, if aught in me\n> \n> Worthy perusal stand against thy sight...\n> \n> Be thou the tenth Muse, ten times more in worth...\n\nThis relationship, as everyone knows, would prove remarkably fruitful: his poems and plays multiplied. In the Globe Theatre, round as an O, an empty cipher filled with meaning, Shakespeare's loquacious quill drew crowds.\n\n\"Shakespeare was the least of an egotist that it was possible to be,\" wrote the nineteenth-century critic William Hazlitt. \"He was nothing in himself; but he was all that others were, or that they could become.\" That nothing, who had once been a dazzled schoolboy, laboring for the breakthrough moment when he comprehended the paradoxical fullness of the empty figure of a zero, would surely have been delighted with this description.\n\n# Seven\n\n# SHAPES OF SPEECH\n\nWe know next to nothing with any certainty about Pythagoras, except that he was not really called Pythagoras. The name by which he is known to us was probably a nickname bestowed by his followers. According to one source, it meant \"He who spoke truth like an oracle.\" Rather than entrust his mathematical and philosophical ideas to paper, Pythagoras is said to have expounded them before large crowds, making the world's most famous mathematician its first rhetorician.\n\nIt is easy to imagine the atmosphere of intrigue and anticipation that would have attended such a spectacle. To believe later accounts, people came from far and wide to hear the legendary figure speak. Citizen after citizen: male and female, young and old, rich and poor, politicians and lawyers and doctors and housewives and poets and farmers and children. Latecomers, red-faced from running, jostled for a place at the back. Waiting for the event to start, they would have shared gossip. Pythagoras, someone whispered, has a thigh made of gold. His words can soothe even wild bears, said another. He communes with nature, affirmed a third. Even the rivers know his name!\n\nPythagoras was a handsome, forty-something-year-old man when, in around 530 BCE, he founded his school of disciples on the Greek colony of Croton in southern Italy. The residents of this distant outpost, many hundreds of miles from Athens, treated the newcomer's teachings with the highest respect. Their appetite for novelty, excitement, \"the next big idea,\" would have been considerable. Prestige, and perhaps some educational and economic advantage over the neighboring colonies, was also on many minds.\n\nBy all accounts, Pythagoras's ideas were more than up to his students' expectations. Mathematics, for him, was nothing less than a way of life. He \"transformed the study of geometry into a liberal education,\" wrote Proclus, the last major Greek philosopher, \"examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner.\" Pythagoras is reputed to have taught that the identities of all existing objects depended on their form rather than on their substance, and could consequently be described using numbers and ratios. The entire cosmos constituted some vast and glorious musical scale. The Pythagoreans thus became the first to understand the world not via tradition (religion), or observation (empirical data), but through imagination\u2014the prizing of pattern over matter.\n\nPythagoras had star quality, that much is clear. His timing was impeccable. An unhurried speaker, he took his time before addressing the eager crowd. Everyone had the feeling that Pythagoras was speaking to him alone. Not a single phrase went above the listener's head. He understood completely. \"Yes,\" he would think to himself, \"yes, it is just as he says. It cannot be any other way.\" But of course this flash of certainty, this grasping of an absolute truth, was an illusion. The listener's mind had duly accompanied the speaker down one carefully laid path of reasoning, though many alternate paths existed. The listener simply forgot about those other ways of thinking and seeing the world. The audience was swept, logical step by logical step, from their old certainties to new and unexpected ones. This was the power of Pythagoras.\n\nRhetoric, the art of speaking, gave shape and solidity to Pythagoras's words and ideas. It would also mark the beginning of truly mathematical thought. \"A [mathematical] proof,\" says Steven G. Krantz, a mathematician based at Washington University in St. Louis, \"is a rhetorical device for convincing someone else that a mathematical statement is true or valid.\" Philip J. Davis and Reuben Hersh, two other leading academics, support this view. \"Mathematics in real life is a form of social interaction where 'proof' is a complex of the formal and the informal, of calculations and casual comments, of convincing argument and appeals to the imagination and the intuition.\"\n\nAncient Greece, with its passionate debates, litigious citizens, and rowdy assemblies, proved an ideal environment for just this kind of social interaction, and consequently, for the development of both rhetoric and mathematics. In fact, without the refinement of rhetoric, there would have been no logic, and thence none of the mathematics that forms one of the cornerstones of our empirically minded Western civilization. Before any of these cultural and intellectual endeavors, there was the practice of persuasion, through argument and the evaluation of evidence. And it was in the law courts, with their public trials, that these building blocks of our system of thought were honed.\n\nCourt cases were a daily occurrence in ancient Athens. Hundreds, sometimes thousands, of free citizens filled theaters to hear the parties speak. These citizens composed a vast and anonymous jury: only their age (above thirty years old) and maleness united them. Since the jury was always made up of an odd number of people, no tied judgments were possible. Every decision was final, and without appeal.\n\nIn the space of several hours, the defendant and his accuser took center stage. The accuser would speak first, leaving the defendant to refute his arguments. Each wanted to be a Pythagoras. Each tried to dazzle the men of the jury with the order, rhythm, and precision of his words. For the less gifted or confident, eloquence could be bought at a price; professional speechwriters were very much in demand. A single tightly argued presentation, every Greek knew, could make the difference between liberty and imprisonment, between life and death.\n\nLet us picture a trial. The defendant, it is claimed, killed the accuser's son for his one hundred gold coins. What argument does the grieving father bring to the court? Perhaps he points to a similar case, known to everyone present, of a man who killed another for his ten gold coins. If a man would be prepared to risk his skin for ten coins, says the father, then he would surely risk it for one hundred coins. The father's argument, by which he establishes motive, is an exercise in basic mathematical logic: if x is true, then x\u00b2 is also true.\n\nA real example of ancient Greek court rhetoric has come down to us from the orator Antiphon's _First Tetralogy._ It involves the case of a man accused of killing his victim (along with the victim's slave) in cold blood. Anticipating a likely defense of \"someone else did it,\" the plaintiff methodically goes through the possible scenarios, eliminating each in turn, in a style of argument that resembles what mathematicians call \"proof by exhaustion.\"\n\n> Malefactors [thieves] are not likely to have murdered him, as nobody who was exposing his life to a very grave risk would forgo the prize when it was securely within his grasp; and the victims were found still wearing their cloaks. Nor again did anyone in liquor kill him; the murderer's identity would be known to his boon companions. Nor again was his death the result of a quarrel; they would not have been quarreling at the dead of night or in a deserted spot. Nor did the criminal strike the dead man when intending to strike someone else; he would not in that case have killed master and slave together. As all grounds for suspecting that the crime was unpremeditated are removed, it is clear from the circumstances of death themselves that the victim was deliberately murdered.\n\nThe plaintiff renders the potential defense precisely as: \"death either by (1) thieves (2) drunks (3) quarrel (4) accident\" in order to refute each part. But he does something more. Each scenario represents a miniature proof. Rejecting, for instance, the possibility that a thief committed the murder, his argument proceeds:\n\n> A thief (claims the defendant) killed the victim.\n> \n> But thieves would steal the victim's cloak.\n> \n> Thus a thief did not kill the victim.\n\nWhich self same structure we can find throughout the _Elements_ of Euclid.\n\n> CA and CB are each equal to AB.\n> \n> But things equal to the same thing are also equal to one another.\n> \n> Thus CA is also equal to CB.\n\nWritten in the third century before the Christian era, it is hard to overstate the _Elements'_ importance to the history of intellectual progress, or the extent to which it embodied the flourishing of rhetoric and logic that permitted its creation. Proposition 21, in Book IX of the treatise (among its oldest pages, dating back to the Pythagoreans), illustrates the lawyerly manner that its author adopted throughout.\n\n> If as many even numbers as we please be added together, the whole is even... For, since each of the numbers... is even, it has a half part; so that the whole [number] also has a half part. But an even number is that which is divisible into two equal parts; therefore [the whole number] is even.\n\nWe might summarize this argument as:\n\n> Proposition: Even numbers (of any quantity) added together make an even number.\n> \n> Clarification: Since even numbers have half parts, their sum will also have a half part.\n> \n> Axiom: An even number is that which is divisible into two equal parts.\n> \n> Conclusion: Therefore the sum of any quantity of even numbers is even.\n> \n> This echoes any number of the possible arguments that would have passed through the Ancient Greek courts.\n> \n> Proposition: The accused stole my ox.\n> \n> Clarification: Since he said nothing to me about the ox before taking it, the ox was taken without my permission.\n> \n> Axiom: An item of property removed without the owner's consent is stealing.\n> \n> Conclusion: Therefore my ox was stolen.\n\nWith their axioms\u2014statements that we accept as being self-evidently true\u2014the Greek plaintiff was able to methodically construct his case and the Greek mathematician his theorem. Nowhere else on Earth had men thought to agree on what constituted the essence of such and such a thing. Uniquely, the Greeks tore themselves away from the word of rulers, gods, or tradition, in favor of logical reasoning. What is wrongdoing? What is murder? What is theft? The ancient Greeks were the first to ask themselves these kinds of questions. They were the first to distinguish a \"criminal act\" from \"misfortune\" or \"error of judgment.\" Definitions\u2014concise, basic, and unambiguous\u2014entered the Athenian imagination, as Aristotle tells us in _Rhetoric:_\n\n> It often happens that a man will admit an act, but will not admit the accuser's label for the act... He will admit that he took a thing but not that he \"stole\" it; that he struck someone first, but not that he committed \"outrage\"; that he had intercourse with a woman, but not that he committed \"adultery\"; that he is guilty of \"theft\" but not of \"sacrilege,\" the object stolen not being consecrated; that he has encroached, but not that he has \"encroached on State lands\"; that he has been in communication with the enemy, but not that he has been guilty of \"treason\".... Therefore we must be able to distinguish what is theft, outrage, or adultery, from what is not, if we are to be able to make the justice of our case clear...\n\nSimilarly, Euclid defines a \"point,\" a \"line,\" a \"square,\" a \"unit,\" and a \"number\" (among other things), as if answering questions that no one had thought to ask before. What is a point? What is a line? To a scribe in Alexandria, or a logician in Xianyang, these sentences were only puzzles. They had no meaning. Or else for response they would simply make a point or draw a line with a drop of ink.\n\nEuclid's books not only posed these questions, they laid down the law (so far as the answers were concerned) for future generations of mathematicians. What is a point? That which has no part. A line? A length without breadth. A square? A quadrilateral, which is both equilateral and right-angled. A unit? That of which each of the things that exist is called one. A number? A multitude composed of units. Et cetera. These foundations allowed mathematicians to \"be able to make the justice of [their] case clear.\" Not to mention the case of others. In the mid-nineteenth century, more than two millennia after Euclid, a copy of his _Elements_ traveled in the carpetbag of a circuit lawyer from Illinois. His name was Abraham Lincoln.\n\nThe pages and their propositions made a deep impression on Lincoln's mind, following him into his subsequent career in politics. In a speech given to an Ohio crowd in 1859 in opposition to a pro-slavery rival, one Judge Douglas, Lincoln declared, \"There are two ways of establishing a proposition. One is by trying to demonstrate it upon reason, and the other is, to show that great men in former times have thought so and so, and thus to pass it by the weight of pure authority. Now, if Judge Douglas will demonstrate somehow that this is popular sovereignty,\u2014the right of one man to make a slave of another, without any right in that other, or anyone else to object; demonstrate it as Euclid demonstrated propositions,\u2014there is no objection. But when he comes forward, seeking to carry a principle by bringing it to the authority of men who themselves utterly repudiate that principle, I ask that he shall not be permitted to do it.\"\n\nDefinitions and axioms would shape President Lincoln's most famous addresses. His powers of rhetoric, persuasion, deduction, and logic were all subjected to the severest tests. The nation was in crisis. Civil war would shortly come. Still, the president spoke to the entire country in defense of the Union.\n\n> I hold, that in contemplation of universal law, and of the Constitution, the Union of these States is perpetual. Perpetuity is implied, if not expressed, in the fundamental law of all national governments. It is safe to assert that no government proper ever had a provision in its organic law for its own termination. Continue to execute all the express provisions of our national Constitution and the Union will endure forever, it being impossible to destroy it except by some action not provided for in the instrument itself.\n> \n> Proposition: The Union of these States is perpetual.\n> \n> Clarification: Perpetuity is implied in the fundamental law of all national governments.\n> \n> Axiom: No government ever had a legal provision for its own termination.\n> \n> Conclusion: Therefore continue to execute the Constitution and the Union will endure forever.\n\nThroughout Lincoln's four years in office, intense fighting saw approximately 750,000 men killed and the nation all but tear itself apart, but the president's proof would ultimately be vindicated.\n\n\"We are not enemies,\" the president had said in the same national address, \"but friends.\" Perhaps he was thinking of a proverb attributed to Pythagoras, one that he took as an axiom: \"Friendship is equality.\"\n\n# Eight\n\n# ON BIG NUMBERS\n\nIn the second of his _Olympian Odes,_ the ancient lyric poet Pindar wrote, \"the sand escapes numbering.\" He was expressing the same idea that would lead his fellow Greeks to coin the term \"sand hundred\" for an inconceivably great quantity.\n\nPindar's claim remained unassailable for some two centuries, which of course is not bad at all as far as a line of poetry goes. The eventual refutation, composed in the middle of the third century BCE, can be fairly listed among the finest achievements of the mathematician Archimedes.\n\nIntroducing his academic paper\u2014the first in recorded history\u2014to the king of his day, Archimedes made a spectacularly audacious argument.\n\n> Some people believe, King Gelon, that the grains of sand are infinite in number. I mean not only the sand in Syracuse and the rest of Sicily, but also the sand in the whole inhabited land as well as the uninhabited. There are some who do not suppose that they are infinite, but that there is no number that has been named which is so large as to exceed its multitude... I will attempt to prove to you through geometrical demonstrations, which you will follow, that some of the numbers named by us... exceed... the number of grains of sand having a magnitude equal to the earth filled up.\n\nArchimedes' estimations for the measurements of the Earth, moon, sun, and the other stars were generous: for example, making Earth's perimeter ten times larger than the calculations of earlier astronomers. Similarly, Archimedes went to great lengths to provide a capacious margin for error concerning the estimated size of a grain of sand. He compared ten thousand grains to the scale of a poppy seed, and then patiently lined the seeds end to end on a smooth ruler. In this way he measured the number of poppy seeds required to reach an inch as being twenty-five. This figure he adjusted still further, changing it to forty-seeds-per-inch length, so as to \"prove indisputably what is proposed.\" Thus he calculated as sixteen million (10,000 \u00d7 40 \u00d7 40) the maximum number of grains of sand that could fill one square inch.\n\nArchimedes assumed that the universe was spherical. He estimated a value for the diameter of the universe using calculations for the diameter of the Earth's orbit around the sun. The universe, according to his reckoning, had a diameter no greater than 100,000,000,000,000 stadia (about two light-years). 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 grains of sand would more than saturate the whole of space.\n\nNext, Archimedes showed that the Greek term \"myriad\" (ten thousand or a hundred hundreds) more than sufficed for the purpose of counting even the largest worldly quantities. The phrase \"myriad myriads,\" he pointed out, allowed the counter to reach the equivalent of one hundred million\u2014the largest named number in his time. But, he continued, if it is possible to count in myriads, it should be equally possible to count in \"myriad myriads\" so that multiplying the latter by itself the counter could attain \"myriad myriad myriad myriads\" or 10,000,000,000,000,000. And considering this new figure likewise as a unit, as respectable as a \"myriad\" or \"myriad myriads,\" the counter could multiply \"myriad myriad myriad myriads\" by itself and proceed to: \"myriad myriad myriad myriad myriad myriad myriad myriads\" or 100,000,000,000,000,000,000,000,000,000,000.\n\nUp till now we have multiplied a myriad by itself a total of eight times. Archimedes' next step possessed all the elegance of simple logic: multiply a myriad myriads by itself myriad myriads times over. The \"1\" that starts the resulting number is tailed by eight hundred million zeroes.\n\nDoggedly pursuing his logic, Archimedes proposed multiplying this new number by itself up to as many as a myriad myriads times over: a number requiring the insertion of eighty quadrillion (80,000,000,000,000,000) zeroes after the one.\n\nArchimedes concluded his paper in confident, if understated tones.\n\n> King Gelon, to the many who have not also had a share of mathematics I suppose that these will not appear readily believable, but to those who have partaken of them and have thought deeply about the distances and sizes of the Earth and sun and moon and the universe this will be believable on the basis of demonstration. Hence, I thought that it is not inappropriate for you too to contemplate these things.\n\nWe find the same comparison between immensity and grains of sand in the sutras of India, many of which were set down on paper in Archimedes' time. In the _Lalitavistara_ sutra, a hagiographical account of the Buddha's life, we read of a meeting between the young Siddhartha and the \"great mathematician Arjuna.\" Arjuna asks the boy to multiply numbers a hundredfold beginning with one _koti_ (generally considered the equivalent of ten million). Without the slightest hesitation, Siddhartha correctly replies that one hundred _koti_ s equals an _ayuta_ (which would equate to one billion), and then proceeds to multiply this number by one hundred, and the new number by one hundred, and so on, until\u2014after twenty-three successive multiplications\u2014he reaches the number called _tallaksana_ (the equivalent of 1 followed by 53 zeroes).\n\nSiddhartha proceeds to multiply this number in turn, though it is unclear whether he does so by one hundred or some other amount. In a phrase reminiscent of Archimedes, he claims that with this new number the mathematician could take every grain of sand in the river Ganges \"as a subject of calculation and measure them.\" Again and again, the _bodhisattva_ multiplies this number, until at last he reaches _sarvaniksepa,_ with which, he tells the mathematician, it would be possible to count every grain of sand in ten rivers the size of the Ganges. And if this were not enough, he continues, we can multiply this number to reach _agrasara_ \u2014a number greater than the grains of sand in one billion Ganges.\n\nSuch extreme numerical altitudes, we are told, are the preserve of the pure and enlightened mind. According to the sutra, only the _bodhisattva_ s, beings who have arrived at their ultimate incarnation, are capable of counting so high. In the closing verses, the mathematician Arjuna concedes this point.\n\n> This supreme knowledge I do not have\u2014he is above me.\n> \n> One with such knowledge of numbers is incomparable!\n\nThe story of the enlightenment of Siddhartha Gautama, to give him his full name, begins in his father's palace. It is said that the Nepalese king resolved to seclude his son at birth from the heartbreaking nature of the world. Shut up behind gilded doors, the boy would remain forever innocent of suffering, poverty, and death. We can imagine his constricted royal life: the fine meals of rich food, lessons in literature and military arts, ritual music and dance. In his ears he wore precious stones heavy enough to make his earlobes droop. But of course he was not free: he had only walls for a horizon, only ceilings for a sky. Bangle strings and brass flutes displaced all birdsong. Cloying aromas of cooked food overlay the smell of rain.\n\nNearly thirty years, a marriage, and even the birth of a son all passed before Siddhartha learned of a world beyond the palace walls. Having resolved to go forth and see it, he made a trip through the countryside, accompanied only by the charioteer who drove him. The prince saw for the first time men enfeebled by ill health, old age, and want of money. He was not even spared the sight of a corpse. Deeply shocked by all that he had seen, he fled his old life for the ascetic's road.\n\nThe story of the prince's seclusion in a palace reads like a fairy tale\u2014it may very well be such a tale\u2014with all its peculiar and thought-provoking charm. One particular aspect of Siddhartha's revelation of the outside world has always struck me. Quite possibly he lived his first thirty years without any knowledge of numbers.\n\nHow must he have felt, then, to see crowds of people mingling in the streets? Before that day he would not have believed that so many people existed in all the world. And what wonder it must have been to discover flocks of birds, and piles of stones, leaves on trees, and blades of grass! To suddenly realize that, his whole life long, he had been kept at arm's length from multiplicity.\n\nLater, his followers would associate Siddhartha's enlightened mind with a profound knowledge of numbers. Perhaps as much as all the other surprises he witnessed from his chariot, it was this first sighting of multiplicity that set him on the path to Nirvana.\n\nI am reminded of another story. This time the man was not a prince but a mathematician in mid-twentieth-century America. Big numbers fascinated him; he enjoyed talking about them with his nine-year-old nephew. One day, the mathematician, Edward Kasner, invited the boy to name a number that contains a hundred zeroes after the 1. \"Googol,\" the boy replied, after a little thought.\n\nNo explanation for the origin of this word is given in Kasner's published account _Mathematics and the Imagination._ Probably, it came intuitively to the boy. According to linguists, English speakers tend to associate an initial G sound with the idea of bigness, since the language employs many G-words to describe things which are \"great\" or \"grand,\" \"gross\" or \"gargantuan,\" and which \"grow\" or \"gain.\" I could point out another feature: both the elongated \"oo\" vowel and the concluding L suggest indefinite duration. We hear this difference in verbs like \"put\" and \"pull,\" where \"put\"\u2014with its final _T_ \u2014implies a completed action, whereas an individual might \"pull\" at something for any conceivable amount of time.\n\nIn a universe teeming with numbers, no physical quantity exists that coincides with a googol. A googol dwarfs the number of grains of sand in all the world. Collecting every letter of every word of every book ever published gets us nowhere near. The total number of elementary particles in all of known space falls some twenty zeroes short.\n\nThe boy could never hope to count every grain of sand, or read every page of every published book, but, like Archimedes and the Siddhartha of the sutras, he understood that no cosmos would ever contain all the numbers. He understood that with numbers he might imagine all that existed, all that had once existed or might one day exist, and all that existed too in the realms of speculation, fantasy, and dreams.\n\nThe mathematician liked his nephew's word. He immediately encouraged the boy to count higher still and watched as his small brow furrowed. Now came a second word, a variation of the first: \"googolplex.\" The suffix _-plex_ (duplex) parallels the English _-fold,_ as in \"tenfold\" or \"hundredfold.\" This number the boy defined as containing all the zeroes that a hand could write down before tiring. His uncle demurred. Endurance, he remarked, varied a great deal from person to person. In the end they agreed on the following definition: a googolplex is a 1 followed by a googol number of zeroes.\n\nLet us pause a brief moment to contemplate this number's size. It is not, for instance, a googol times a googol: such a number would \"only\" consist of a 1 with 200 zeroes. A googolplex, on the other hand, contains far more than a thousand zeroes, or a myriad zeroes, or a million or billion zeroes. It contains far more than the eighty quadrillion zeroes at which even the painstaking and persistent Archimedes ceased to count. There are so many zeroes in this number that we could never finish writing them all down, even if every human lifetime devoted itself exclusively to the task.\n\nThe googolplex is so vast a number that it encompasses virtually every conceivable probability. Physicist Richard Crandall gives the example of a can of beer that spontaneously tips over, \"an event made possible by fundamental quantum fluctuations,\" as having vastly greater than 1-in-a-googolplex odds. A further illustration, by the English mathematician John Littlewood, asks us to imagine the plight of a mouse in outer space. Littlewood calculated as being well within a googolplex the likelihood that this mouse\u2014helped by sufficient random fluctuations in its environment\u2014might survive a whole week on the surface of the sun.\n\nBut of course a googolplex is not infinite. We can, as perhaps the boy did, continue to count by simply adding one. Modern computers, impervious to zero vertigo, have calculated that this number, googolplex + 1, is not prime. Its smallest known factor is: 316,912,650,057,057,350,374,175,801,344,000,001.\n\nWhat did the mathematician make of his nephew's \"googolplex\" as the biggest number he could conceive of? His reply is not recorded, but he might have told him about some of the infinitely many numbers that exceed the googolplex's scope. He might, for example, have mentioned \"googol factorial,\" being the product of multiplying every whole number between 1 and a googol (1 \u00d7 2 \u00d7 3 \u00d7... 950,345 \u00d7... 1,000,000,000,000,008,761 \u00d7... googol). This number, which computers tell us begins 16294... easily surpasses every other number that we have encountered in these pages.\n\nFor a universe of such limited dimensions, these monstrous numbers seem quite useless. Worse, they can appear to us excessive, disproportionate. Every number, after a certain point, feels gratuitous. Who knows? It is possible they are not intended for our attention. The Flower Adornment sutra speaks of immense aeons, _kalpa_ s, in which the universe is continuously destroyed and reborn. At the kalpa's peak, men live for an average of eighty-four thousand years. In other realms, so the _Heart_ sutra reports, a single life spans eighty-four thousand _kalpa_ s\u2014that is to say, eighty-four thousand epochs, each one many zeroes long. For such beings, a googol or its factorial would belong merely to the tangible and the convenient.\n\nMathematicians aspire to these heavenly realms. Vast numbers that split our senses enrich their work. But they also produce paradoxes. For instance, which is greater: 10 or 27, when each is multiplied by itself exactly a googolplex number of times? The latter, of course, although even the most powerful calculators\u2014plunging one hundred digits deep\u2014struggle to tell the two apart. This difficulty confounds our expectations: intuitively, we feel that the ordering of a number should remain straightforward, even when the number's precise value cannot be known. And yet, there exist numbers so large that we cannot easily distinguish them from their double, or triple, or quadruple or any other amount. There exist magnitudes so immense that they escape all our words, and all our numbers.\n\nThe most famous paradox concerning big numbers takes us back once more to the ancient Greeks. Tradition attributes it to the philosopher Eubulides. It has been suggested that Eubulides' inspiration owes something to his fellow skeptic Zeno, who argued that every falling grain of wheat makes a noise proportionate to the noise made by a falling bushel. Eubulides' formulation does not feature wheat, however. As would Archimedes a century later, Eubulides built his argument on sand.\n\nIt goes as follows: first, we agree that one grain of sand does not make a heap. Adding a second grain does not make a heap either. Nor do we produce a heap with the third grain. From this it follows that adding one to any small number generates another number we call \"small.\" But if this is true, a billion is a small number. So, too, is a googolplex.\n\nUnderstandably wary of this conclusion, the reader might propose that a heap of sand, like a big number, begins at a certain point: say, ten thousand. As a resolution of the paradox, this answer is unsatisfactory. It is unclear why nine thousand nine hundred and ninety-nine should be considered small, but not nine thousand nine hundred and ninety-nine plus one.\n\nOf course, in at least one sense all numbers are small. Given any number \"n,\" there will be only n-1 numbers less than n, but an infinite group of numbers greater than it.\n\nAddressing the unjustly forgotten mathematician Archytas, the Roman poet Horace, who was regarded as the finest lyric poet of the era of Emperor Augustus, expressed in his verse perhaps the greatest paradox of them all: that of finite men who spend their lives attempting to scale the infinite.\n\n> You who measured the sea and the earth and the numberless sands,\n> \n> you, Archytas, are now confined in a small mound of dirt\n> \n> near the Matine shore, and what good does it do you that you\n> \n> attempted the mansions of the skies and that you traversed\n> \n> the round celestial vault\u2014you with a soul born to die?\n\n# Nine\n\n# SNOWMAN\n\nOutside it is cold, cold\u2014ten degrees below, give or take. I step out with my coat zipped up to my chin and my feet encased in heavy rubber boots. The glittering street is empty; the wool-gray sky is low. Under my scarf and gloves and thermals I can feel my pulse begin to make a racket. I do not care. I observe my breath. I wait.\n\nA week before, not even a whole week, the roads showed black tire tracks and the trees' bare branches stood clean against blue sky. Now Ottawa is buried in snow. My friends' house is buried in snow. Chilling winds strafe the town.\n\nThe sight of falling flakes makes me shiver; it fills the space in my head that is devoted to wonder. How beautiful they are, I think. How beautiful are all these sticky and shiny fragments. When will they stop? In an hour? A day? A week? A month? There is no telling. Nobody can second-guess the snow.\n\nThe neighbors have not seen its like in a generation, they tell me. Shovels in hand, they dig paths from their garage doors out to the road. The older men affect expressions both of nonchalance and annoyance, but their expressions soon come undone. Faint smiles form at the corners of their wind-chapped mouths.\n\nGranted, it is exhausting to trudge the snowy streets to the shops. Every leg muscle slips and tightens; every step forward seems to take an age. When I return, my friends ask me to help them clear the roof. I wobble up a leaning ladder and lend a hand. A strangely cheerful sense of futility lightens our labor: in the morning, we know, the roof will shine bright white again.\n\nHot under my onion layers of clothing, I carry a shirtful of perspiration back into the house. Wet socks unpeel like Band-Aids from my feet; the warm air smarts my skin. I wash and change my clothes.\n\nLater, around a table, in the dusk of a candlelit supper, my friends and I exchange favorite recollections of winters past. We talk sleds, and toboggans, and fierce snowball fights. I recall a childhood memory, a memory from London: the first time I heard the sound of falling snow.\n\n\"What did it sound like?\" the evening's host asks me.\n\n\"It sounded like someone slowly rubbing his hands together.\"\n\nFrowns outline my friends' concentration. Yes, they say, laughing. Yes, we can hear what you mean.\n\nA man sporting a gray mustache laughs louder than the others. I do not catch his name; he is not a regular guest. I gather he is some kind of scientist, of indeterminate discipline.\n\n\"Do you know why we see snow as white?\" the scientist asks. We shake our heads.\n\n\"It is all to do with how the sides of the snowflakes reflect light.\" All the colors in the spectrum, he explains to us, scatter out from the snow in roughly equal proportions. This equal distribution of colors, we perceive as whiteness.\n\nNow our host's wife has a question. The ladle with which she has been serving bowls of hot soup idles in the pot. \"Could the colors never come out in a different proportion?\" she asks.\n\n\"Sometimes, if the snow is very deep,\" he answers. In which case, the light that comes back to us can appear tinged with blue. \"And sometimes a snowflake's structure will resemble that of a diamond,\" he continues. Light entering these flakes becomes so mangled as to dispense a rainbow of multicolored sparkles.\n\n\"Is it true that no two snowflakes are alike?\" This question comes from the host's teenage daughter.\n\nIt is true. Imagine, he says, the complexity of a snowflake (and enthusiasm italicizes his word \"complexity\"). Every snowflake has a basic six-sided structure, but its spiraling descent through the air sculpts each hexagon in a unique way: the minutest variations in air temperature or moisture can\u2014and do\u2014make all the difference.\n\nLike mathematicians who categorize every whole number into prime numbers or Fibonacci numbers or triangle numbers or square numbers (and so on) according to its properties, so researchers subdivide snowflakes into various groupings according to type. They classify the snowflakes by size and shape and symmetry. The main ways in which each vaporous hexagon forms and changes, it turns out, amount to several dozen or several score (the precise total depending on the classification scheme).\n\nFor example, some snowflakes are flat and have broad arms, resembling stars, so that meteorologists speak of \"stellar plates,\" while those with deep ridges are called \"sectored plates.\" Branchy flakes, like the ones seen in Christmas decorations, go by the term \"stellar dendrites\" (dendrite coming from the Greek word for a tree). When these treelike flakes grow so many side branches that they finish by resembling ferns, they fall under the classification of \"fernlike stellar dendrites.\"\n\nSometimes, snowflakes grow not thin but long, not flat but slender. They fall as columns of ice, the kinds that look like individual strands of a grandmother's white hair (these flakes are called \"needles\"). Some, like conjoined twins, show twelve sides instead of the usual six, while others\u2014viewed up close\u2014resemble bullets (the precise terms for them are \"isolated bullet,\" \"capped bullet,\" and \"bullet rosette\"). Other possible shapes include the \"cup,\" the \"sheath,\" and those resembling arrowheads (technically, \"arrowhead twins\").\n\nWe listen wordlessly to the scientist's explanations. Our rapt attention flatters him. As he speaks, his white hands draw the shape of every snowflake in the air.\n\nComplexity. But from it, patterns, forms, identities, that every culture can perceive and understand. I have read, for instance, that the ancient Chinese called snowflakes blossoms and that the Scythians compared them to feathers. In the Psalms (147:16), snow is a \"white fleece,\" while in parts of Africa it is likened to cotton. The Romans called snow _nix,_ a homonym\u2014the seventeenth-century mathematician and astronomer Johannes Kepler would later point out\u2014of his Low German word for \"nothing.\"\n\nKepler was the first scientist to describe snow. Not as flowers or fleece or feathers, snowflakes were at last perceived as being the product of complexity. The reason behind the snowflake's regular hexagonal shape was \"not to be looked for in the material, for vapor is formless.\" Instead, Kepler suggested some dynamic organizing process, by which frozen water \"globules\" packed themselves together methodically in the most efficient way. \"Here he was indebted to the English mathematician Thomas Harriot,\" reports the science writer Philip Ball, \"who acted as a navigator for Walter Raleigh's voyages to the New World in 1584\u20135.\" Harriot had advised Raleigh concerning the \"most efficient way to stack cannonballs on the ship's deck,\" prompting the mathematician \"to theorize about the close packing of spheres.\" Kepler's conjecture that hexagonal packing \"will be the tightest possible, so that in no other arrangement could more pellets be stuffed into the same container\" would only be proven in 1998.\n\nThat night, the snow reached even into my dreams. My warm bed offered no protection from my childhood memories of the cold. I dreamed of a distant winter in my parents' garden: the powdery snow, freshly fallen, was like sugar to my younger brothers and sisters, who hastened out to it with shrieks of delight. I hesitated to join them, preferred to watch them playing from the safety of my bedroom window. But later, after they had all wound up their games and headed back in, I ventured out alone and started to pack the snow together. Like the Inuit (who call it _igluksaq,_ \"house-building material\"), I wanted to surround myself with it, build myself a shelter. The crunching snow gradually encircled me, accumulating on all sides, the walls rising ever higher until at last they covered me completely. My boyish face and hands smeared with snow, I crouched deep inside feeling sad and feeling safe.\n\n_\"On t'attend!\"_ my friends call up in the morning to my room. \"We are ready and waiting!\" I am the English slowpoke, unaccustomed to this freezing climate, the lethargy it imposes on the body, and the dogged, unshakable feeling of being underwater. What little snow I have experienced all these years, I realize now, has been but a pale imitation of the snow of my childhood. London's wet slush, quick to blacken, has muddied the memory. Yet here the Canadian snow is an irresistible, incandescent white\u2014its glinting surfaces give me back my young days, and alongside them, a melancholic reminder of age.\n\nAfter my sweater, I pull on a kind of thermal vest, then a coat, knee-long. My neck is wrapped in a scarf; my ears vanish behind furry muffs. Mitten fingers tie bootlaces into knots.\n\nFortunately, the Canadians have no fear of winter. The snow is well superintended here. Panic, of the kind that grips London or Washington, DC, is unfamiliar to them; stockpiling milk, bread, and canned food is unheard of. Traffic jams, canceled meetings, energy blackouts, are rare. The faces that greet me downstairs are all kempt and smiling. They know that the roads will have been salted, that their letters and parcels will arrive on time, that the shops and schools will be open as usual for business.\n\nIn the schools of Ottawa, children extract snowflakes from white sheets of paper. They fold the crisp sheet to an oblong, and the oblong to a square, and the square to a right-angled triangle. With scissors, they snip the triangle on all sides; the pupils all fold and snip the paper in their own way. When they unravel the paper, different snowflakes appear, as many as there are children in the class. But every one has something in common: they are all symmetrical.\n\nThe paper snowflakes in the classroom resemble only partly those that fall outside the window. Shorn of nature's imperfections, the children's unfolded flakes represent an ideal. They are the pictures that we see whenever we close our eyes and think of a snowflake: equidistant arms identically pliable on six sides. We think of them as we think of stars, honeycombs, and flowers. We imagine snowflakes with the purity of a mathematician's mind.\n\nAt the University of Wisconsin, the mathematician David Griffeath has improved on the children's game by modeling snowflakes not with paper, but with a computer. In 2008, Griffeath and his colleague Janko Gravner, both specialists in \"complex interacting systems with random dynamics,\" produced an algorithm that mimics the many physical principles that underlie how snowflakes form. The project proved slow and painstaking. It can take up to a day for the algorithm to perform the hundreds of thousands of calculations necessary for a single flake. Parameters were set and reset to make the simulations as lifelike as possible. But the end results were extraordinary. On the mathematicians' computer screen shimmered a galaxy of three-dimensional snowflakes\u2014elaborate, finely ridged stellar dendrites and twelve-branched stars, needles, prisms, every known configuration, and others, resembling butterfly wings, that no one had identified before.\n\nMy friends take me on a trek through the nearby forest. The flakes are falling intermittently now; above our heads, patches of the sky show blue. Sunlight glistens on the hillocks of snow. We tread slowly, rhythmically, across the deep and shifting surfaces, which squirm and squeak under our boots.\n\nWhenever snow falls, people look at things and suddenly see them. Lampposts and doorsteps and tree stumps and telephone lines take on a whole new character. We notice what they are, and not simply what they represent. Their curves, angles, repetitions, command our attention. Visitors to the forest stop and stare at the geometry of branches, of fences, of trisecting paths. They shake their heads in silent admiration.\n\nA voice somewhere says the Hull River has frozen over. I disguise my excitement as a question. \"Shall we go?\" I ask my friends. For where there is ice, there will inevitably dance ice skaters, and where there are ice skaters, there will be laughter and lightheartedness, and stalls selling hot pastries and spiced wine. We go.\n\nThe frozen river brims with action: parkas pirouette, wet dogs give chase, and customers line up at the concessions. The air smells of cinnamon. Everywhere, the snow is on people's lips: it serves as the icebreaker for every conversation. Nobody stands still as they are talking: they shift their weight from leg to leg, and stamp their feet, wiggle their noses and exaggerate their blinks.\n\nThe flakes fall heavier now. They whirl and rustle in the wind. Everyone seems in thrall to the tumbling snowflakes. Human noises evaporate; nobody moves. Nothing is indifferent to its touch. New worlds appear and disappear, leaving their prints upon our imagination. Snow comes to earth and forms snow benches, snow trees, snow cars, snowmen.\n\nWhat would it be like, a world without snow? I cannot imagine such a place. It would be like a world devoid of numbers. Every snowflake, unique as every number, tells us something about complexity. Perhaps that is why we will never tire of its wonder.\n\n# Ten\n\n# INVISIBLE CITIES\n\nWe wish to see ourselves translated into stone and plants, we want to take walks in ourselves when we stroll around these buildings and gardens.\" This, says Nietzsche, is the purpose of the city: to create space and structure in which a person might think. Ostentatious church buildings, he complained, inhibited free thought. He argued for the ideal of a \"wide\" and \"expansive\" city, expansive in every sense of the word.\n\nI remember these words every time I fly to New York City, a place where tall buildings aspire to the sky. Long shadows in the shape of skyscrapers alight on yellow taxis and hot dog carts. The city's buildings are home to eight million human beings. Among them number some of the most creative people in the world. People come here from every land, and from every language, for what reason? Quite possibly, they come here in order to think.\n\nBut New Yorkers, like the rest of us, do not pay much attention to their surroundings, how the city incites and informs their thoughts. There are exceptions, of course, and I am not only talking about those who are newly arrived. I am talking about mathematicians, who are tourists in every place. What with its towering buildings, and its grid's rectilinear streets, and the intersections named after numerals (Ninety-Third and Fifth Avenue), New York City was made for mathematicians.\n\nPlanning a city, or dreaming about one, invites us to think by numbers, to borrow some of the mathematicians' delight. Architects of cities and of individual buildings divide and categorize the air. In this portion: morning traffic; in that section: jogging in the park. Up on this level: office computers; beneath it: a parking lot. The designers translate numbers into symmetry, into shape and order and livable form. Cities are the embodiments of numerical patterns that contain and direct our lives. But all cities are invisible to start with.\n\nBefore New York the city, there was New York the idea: a mere glint in the European settlers' eye. They christened the woodlands and rivers and the trails of native clans that they discovered, _Nieuw Amsterdam._ Many years later, following the War of Independence, the nascent settlement would serve briefly as the fledgling Union's capital. Intangible visions could now be rendered concrete.\n\nA commission, formed in 1811, found in favor of a plan for the mass building of \"straight-sided and right-angled houses.\" Avenues were laid, precisely one hundred feet wide, and numbered (beginning with the easternmost) from one to twelve. Running at right angles to the avenues were passages turned into symmetrical streets (sixty feet in width), each allotted a consecutive number between one and one hundred and fifty-five. Street names acted like compass points, shepherding strangers and the easily lost to their destination. The rigid geometry imposed order, efficient commerce, and cleanliness, but it also obliterated many of Manhattan Island's natural spaces. In the words of one commissioner, the grid system became \"the day-dawn of our empire.\"\n\nNew York City is an exception, though. Not all cities find their territory; many remain forever orphans, existing only in their inventor's dreams. What follows is my attempt to sketch a brief history of these invisible cities.\n\nPlato, in his _Laws,_ gives a recipe for the ideal city. Like any recipe maker, he puts great store by the precision with which he delineates its ingredients. At various points in the text, he insists rather heavily on a particular number. No margin exists for approximation in the Platonic design. Neither is there any room for discussion, since for Plato, the self-evident quality of his city is \"as plain as the fact that Crete is an island.\"\n\nJudging by his laws, Plato really intended limits. Without a city, he argues, man would dwell in a \"fearful, illimitable desert.\" Such a man would know nothing of art or science; worse, he would hardly know himself. A city's limits should be carefully demarcated, set neither too big nor too small, so that its citizens might, with time and sufficient effort, be capable of putting a name to every face. This, in Plato's judgment, would prevent the blight of war, which had struck down so many great cities of the past. He quotes with approval the poet Hesiod's praise of moderation: \"the half is often greater than the whole.\"\n\nStarting from the principle that \"numbers in their divisions and complexities are useful for everything,\" Plato proposes limiting his ideal city to exactly 5,040 landholding families. Why 5,040? It is what mathematicians call a \"highly composite number,\" meaning that it can be divided in multiple ways. In fact, no fewer than sixty numbers divide into it, including every number from one to ten.\n\nTwelve can also divide evenly into 5,040. Plato divides the total number of households into twelve tribes, each therefore consisting of 420 families. While interdependent, each tribe is conceived as being fixed and self-sufficient, like the months in a solar year.\n\nUsing highly composite numbers facilitates the subdivision of land and property among the citizens. Each family, in each tribe, receives an equal lot of land, beginning from the city center and radiating out to the countryside. In this way the city fairly distributes the fertility of its soil: half of each lot shall contain the city's richest ground, while the other half shall consist of the rockiest.\n\nPlato's ideal number of 5,040 families intrigues modern statisticians. They calculate that such a population would require 164 (or 165) births per year to sustain itself. Following the ancient Greek logic that treated men as the head of every household, they also calculate the city's annual number of potential fathers to be 1,193. Plato believed that one marriage in seven would be fruitful per year, suggesting an anticipated annual birthrate of 170\u2014which corresponds almost exactly to our statisticians' calculations.\n\nHow did Plato count on keeping his ideal city's number of households in check? He proposed that each inheritance should pass into the hands of a single \"best-loved\" male heir. Any remaining sons would be distributed among the childless citizens, while the daughters should be married off.\n\nLarge families had no place in Plato's city. Fecundity would be illegal: any couple producing \"too many\" children was to be rebuked. The city's precise limit of 5,040 households was inviolable: all surplus members were to be sent packing.\n\nPlato imagined that his limits would ensure equality and security for every citizen. In his bucolic vision, men and women would \"feed on barley and wheat, baking the wheat and kneading the flour, making noble puddings and loaves; these they will serve up on a mat of reeds or clean leaves; themselves reclining the while upon beds of yew and myrtle boughs. And they and their children will feast, drinking of the wine which they have made, wearing garlands on their heads, and having the praises of the gods on their lips, living in sweet society, and having a care that their families do not exceed their means; for they will have an eye to poverty or war.\"\n\nPerhaps. But it is also conceivable that Plato's city would have encouraged just the kind of petty-mindedness among its citizens that exact calculation often fosters. During the Renaissance, when Plato and his ideas were rediscovered by humanist scholars, we find an Italian architect similarly moved to envision his own perfect city. His name was Antonio di Pietro Averlino, though he is better known today by the Greek name Filarete (meaning \"lover of virtue\"). Unlike Plato, Filarete was an architect, albeit one with a rich and complicated past: he had once been arrested and barred from working in Rome for allegedly stealing the head of Saint John the Baptist.\n\nFilarete described his city _Sforzinda_ (the name a flattery aimed at his patron Francesco Sforza of Milan) at some length in his _Trattato di architettura._ Its thick symmetrical outer walls formed an eight-pointed star. Though attractive, the unusual choice of shape was also intended to be defensive: invaders mounting its angles would find themselves exposed on multiple sides.\n\nLike spokes in a vast wheel, eight straight roads led from the walls to the city center. The roads were studded with small piazzas, surrounded by shops and markets. A visitor treading his path downtown from the city gates would pass pyramids of apples, stacks of loaves, and multicolored garments spilling over tables. Merchants, their eyes enlarged by expectation, shouting: _\"Signore, signore!\"_ At last the city center appears. Three vast interconnecting piazzas greet him. Here the market noises fade before the imposing ducal palace to his left, and a massive cathedral on his right. In between the palace and the cathedral, on the main piazza, stands another lofty building, ten stories high.\n\nWhat is this strange building to which every street in the city addresses itself? Filarete called it the \"House of Vice and Virtue.\" Every floor housed a different class of activity. A brothel, on the ground floor, would entertain the majority of the building's callers. Alcoholic drinks and games could be had on the floors immediately above. Ascending further, a university and lecture halls offered its few visitors instruction. An observatory topped them all.\n\nThe homes to which the citizens would return after their day's work or play had been just as intricately imagined. Filarete planned his houses according to the resident's social rank: the artisans' quarters taking up far less space than the houses of the city's merchants or gentlemen. In comparison with his artist and painter neighbors, the architect's own house would be twice as spacious.\n\nFilarete's plans are long, his writing spidery. Spread across its twenty-five volumes his treatise contains an entire city in waiting. But not long after its completion, in 1466, the Duke Sforza died and Filarete's vision survived only on paper.\n\nPlato's ideal limits and Filarete's volumes appealed to new dreamers. They passed from mind to mind, and from century to century. It should perhaps come as no surprise that they would go on to inspire the grandest and most ambitious city plans, those drawn up in the United States.\n\nKing Camp Gillette, who would later invent the safety razor, once dreamed of an immense city he called Metropolis. In 1894, he published a short illustrated book aimed at its promotion. The city, Gillette wrote, would be \"situated in the vicinity of Niagara Falls, extending east into New York State and west into Ontario.\" It would take the shape of a rectangle, sixty miles long and thirty miles wide. Gillette considered its construction, \"In the light of a machine, or rather a part of the machine of production and distribution; and, as such, the objects to be attained must be known and understood. It must have no unnecessary parts to cause friction or demand unnecessary labor, and yet it must combine within itself all the necessary parts which will contribute to the happiness and comfort of all.\"\n\nSixty million inhabitants would live in circular skyscrapers, six hundred feet in diameter, \"upon a scale of magnificence such as no civilization has ever known.\" Elevators, a still recent invention in Gillette's day, had gradually shifted city design from the horizontal toward the vertical. But Metropolis took the idea of a vertical city to a whole other level. The skyscrapers would be truly colossal, attaining a height of twenty-five stories. Habitable monoliths, lots and lots of them, shining with glass and progress, smooth and steel-colored, a monotony of monuments. A beehive-like distribution of apartments throughout the city would lend space enough between the buildings for wide avenues and parks. No citizen, it also ensured, would reside any nearer or farther away from a school, shop, or theater than any other.\n\nThe city's highly regular order appears again in miniature in Gillette's plan for every home. For Gillette, as for Filarete, the home was a tiny city. The interior would be completely symmetrical, with parallel sitting rooms, bedrooms, and bathrooms on either side. And in each room, the windows would be so arranged as to make it impossible to look out from one apartment into a neighbor's.\n\nConscious of the artificiality of his vision (even the hexagonal lawns that surrounded each building would be composed of artificial grass), Gillette proposed distributing thousands of public gardens filled with trees and \"urns of flowers\" at regular intervals throughout the city. Being constructed with complete regularity did not mean that it would suffer from sameness, he insisted. Looking out from his window, the citizen's roving eye would encounter \"a continuous and perfectly finished facade from every point of view, each building and avenue surrounded and bordered by an ever changing beauty in flowers and foliage.\"\n\nGillette summed up his utopian vision:\n\n> Imagine for a moment these thirty-odd thousand buildings of _Metropolis,_ each standing alone, a majestic world of art... a never-ending city of beauty and cleanliness, and then compare it with our cities of filth, crime, and misery, with their ill-paved and dirty thoroughfares, crowded with the struggling masses of humanity and the system of necessary traffic. And then compare the machinery of both systems, and take your choice; for I believe the only obstacle that lies in the way of the building of this great city is man.\n\nFifty years after Gillette's book was published, the 1939 World's Fair in New York exhibited its own \"City of the Future.\"\n\nMillions stood in line for hours to see the model Democracity. (This fact alone is truly remarkable: New Yorkers hate to stand in line. They hate the involuntary intimacy, and the excruciating foot shuffles, and the boredom of being so long in their own company. And yet wait in line they did.)\n\nHow long must it have taken the model makers to build? It was housed inside a sphere eighteen stories tall. An escalator, the world's longest, ascended visitors fifty feet above the ground to the exhibition. Music blared triumphant over loudspeakers as they entered, and then a resounding voice spoke. \"The city of man in the world of tomorrow. Here are grass and tree as well as stone and steel. Not a dream city but a symbol of life as lived by the man of tomorrow. As man helps man so nation leans on nation, united by a thousand roads of commerce... Here brain and brawn, faith and courage, are interlinked in high endeavor.\"\n\nStanding on slowly rotating balconies, visitors peered down at the city as though at an altitude of seven thousand feet. They saw a vast ring, brightly lit and painted, representing eleven thousand square miles of terrain. In its center loomed a single block: an awesome office complex to which a quarter of a million inhabitants (one citizen in six) would commute every day.\n\nFive satellite suburbs surrounded this central district in concentric circles. Even the most distant Pleasantville (as the residential areas were called) lay within sixty miles of the hub. Larger Millvilles, home to the city's factories, exiled their noise and pollution to the outskirts. Greenbelts were interspersed with suburbs; wide highways communicated between the center and the various sectors.\n\nMobility, an American obsession, had never been so well catered to. Traffic lights would be a thing of the past. The city's highways would always flow freely, in a straight line, planned in such a way as to avoid all jams and pedestrian crossings. All other roads would be built a safe distance from any school.\n\nTwo minutes into the performance, all of a sudden the lights would dim; the concave ceiling sparkled as if with stars. A chorus started up, and a film showed marching men: artisans, farmers, mechanics, everyone who would work together to help build tomorrow. The voices rose; the marching figures grew; the fairgoers held their breath.\n\nAnd then, just as quickly, the music died away and the men in the film disappeared behind thick plumes of smoke. The show was over.\n\n# Eleven\n\n# ARE WE ALONE?\n\nDemocritus, a contemporary of Plato and Aristotle, imagined all matter as composed of indivisible elements he called \"atomos,\" and was also the first thinker to propose a cosmos of many worlds. Every world was different. Some had neither a sun nor a moon, while others had larger moons or smaller moons or moons more numerous than our own.\n\nThe Pythagoreans also believed that our world was simply one of many. For them the moon was Earth-like and inhabited, containing larger beings and more beautiful plants. The lunar residents, they believed, stood fifty times our height, lived on air and for this reason produced no excrement.\n\nRefutations of these ideas by both Plato (to call the number of worlds indefinite requires definite knowledge) and Aristotle did not prevent later thinkers from endorsing them.\n\n> Since space lies empty and infinite in all directions and since atoms in countless numbers fly every which way through its furthest reaches... it is utterly unrealistic to think that ours are the only world and sky to have been born and that so many atoms outside our world are doing nothing... there are other worlds in other parts of the universe, different races of humans and species of animals.\n\nThese lines come from the epic poem _On the Nature of Things,_ written by the Roman Lucretius in the first century before the Christian era. The poet's ideas would later fill the fathers of the early Church with consternation. If other worlds existed, wrote St. Augustine, they would each require their own Savior, which would contradict the unique role of Christ.\n\nBut by the medieval period, not everyone in the Church shared Augustine's view. In 1277, the Bishop of Paris denounced the proposition that God could not create more than one world. Three centuries later, the friar Giordano Bruno put forth an elaborate argument in favor of an infinite number of worlds: if man can imagine so many worlds, then so can God, who creates what He thinks. The friar pictured Gardens of Eden, infinite in number: in half, Adam and Eve eat the fruit of Knowledge; in half, they do not. An infinite number of worlds will fall from grace and require an infinite number of Saviors to redeem them. Unlike Augustine, Bruno had no difficulty imagining an infinite number of Christs. For this and other \"theological errors,\" the authorities denounced the heretic and burned him at the stake.\n\nThe Inquisitors' heavy footsteps dissuaded Bruno's contemporary, Galileo Galilei, from seeing any evidence for extraterrestrial life in the rugged lunar landscape revealed by his telescope. All the same, since the valleys and mountains that corrugated the moon's surface seemed at least comparable to those on Earth, might not the moon also have people to dwell among them? His friend Johannes Kepler, the seventeenth-century mathematician and astronomer, thought so. Jupiter, too, he deduced \"with the highest degree of probability,\" though its inhabitants were undoubtedly inferior to humans.\n\nProbability: this word became the cornerstone of the argument for life on other planets. \"As for mind beyond the confines of our tiny globe,\" wrote the American astronomer Percival Lowell in 1895, \"modesty, backed by a probability little short of demonstration, forbids the thought that we are the sole thinkers in this great universe.\"\n\nHis argument, two millennia old, tallied with the most modern scientific observations: the conditions on Mars suggested hospitality. The planet enjoyed an atmosphere and its weather appeared extremely clement (with an average climate, he reported, that was similar to that in southern England). Water, essential for life, was also present. \"Anyone looking through a telescope at the planet, early last summer,\" noted Lowell, \"would at once have been struck by the fact that its surface was diversified by markings in three colors, white, blue-green, and reddish-ocher; the white lying in a great oval at the top of the disc. The white oval was the south polar icecap.\" And the blue-green? The color of water. Or, of what still remained of Martian water, since \"the signs [are] that its water supply is now exceedingly low.\" Its inhabitants, he reasoned, had poured all their energies into irrigation. When he scanned the planet's surface, he picked out a \"network of fine, straight dark lines.\" Canals. \"All this,\" the astronomer conceded, \"of course, may... signify nothing; but the probability seems the other way... that Mars seems to be inhabited is not the last, but the first word on the subject.\"\n\nLowell's claims found many sympathetic ears. \"Probability,\" he knew, was a sesame word: it opened ears and minds. But its magic did not work on everyone. The biologist Alfred Russel Wallace, who independently of Darwin discovered the principle of natural selection, was among his fiercest critics. Yes, Mars appeared to have polar icecaps, and a day only half an hour longer than our own, and lengthy seasons that vanished one into the next. But according to Wallace's calculations, the planet was in fact too cold to have rivers, seas, or canals. The features observed by Lowell were natural landforms, all products of normal geological processes. \"Mars, therefore, is not only uninhabitable by intelligent beings... but is absolutely uninhabitable.\"\n\nNot only was the planet Mars lifeless, but there were in all likelihood no habitations on any other planet either. This, at the turn of the twentieth century, was Wallace's stark assessment. The exceptional (and exceptionally complex) combination and sequence of events\u2014physical, chemical, cosmological\u2014permitting the origin of life on Earth made the prospect of finding other beings elsewhere in the universe immeasurably remote. The formation of intelligent life, quite simply, was a once-in-a-universe event.\n\nWhat? _Alone?_ Many people could not believe it. It was one thing to be alone in a room, or a house. But to imagine being the only person in a town, a city, a country! The only people in a universe. They agreed with the ancient Greek Metrodorus, who had thought it absurd that in a vast field only one stalk should ever sprout. What is more, the sense of such all-encompassing solitude seemed to these people intolerably oppressive. They felt like foreigners in a lifeless void.\n\nDecades passed, and the only extraterrestrials were to be found in pulp stories and motion pictures. When the American astronomer Frank Drake\u2014who has spearheaded research into extraterrestrial communication\u2014was a boy, he drew on this imagery as he listened to the words of his father, an engineer. Look at this star, and this one, and that star, and that one. All these stars, innumerable stars, shining in the night sky. Around some of those stars, somewhere in space, circled other worlds like our own. The boy listened to his father and believed him. He believed with all his heart.\n\nFrank Drake was at home with big numbers. His formative years were spent in Chicago, a city so populous that it could sustain the livelihoods of more than a hundred full-time piano tuners. His thoughts ran often to the many, many worlds far above his head. He wondered about their cities, about their cars, about whether they knew war or cancer.\n\nAfter graduating from Harvard with a doctorate in radio astronomy, Drake conducted the first ever search for interstellar communication. On April 8, 1960, he aimed the radio at two stars much like our sun, twelve light-years from Earth. Over the next two months he and his colleagues listened sedulously for a signal, but heard nothing. Not a peep.\n\nBut the numbers! Drake believed the numbers were on his side. The number of stars in our galaxy amounts to at least one hundred billion. One hundred billion! And how many of these stars were suns to planets? No hard facts could lend a hand. His imagination groped, sifted, hesitated. He closed his eyes and made a guess: one in two. Planets would orbit half of all the stars in the Milky Way, which equals fifty billion solar systems.\n\nNot every solar system will produce life, however. A shot at life requires a certain kind of solar system (a sun that is neither too cool nor too dim, nor so massive as to burn out before life appears), which would host planets in some way comparable to Earth. Drake thought of the only solar system with which we are familiar, of its (then) nine planets, and the sole planet that had become a world. This figure\u2014one\u2014troubled him: it smacked of uniqueness. No, no, there had to be solar systems with multiple worlds. Some system will host a world, then it will host another, and perhaps another after that. Why not? Look at Mars; it had not been so very far off from being a second Earth. And so he made the guess of two (two being greater than one) for the number of possible Earths in each solar system.\n\nSo far, so good. But Drake's next estimations demanded all his powers of invention. First, he had to estimate the number of these planets on which life has emerged. This is how he reasoned about it. Four and a half billion years ago, not long after our Earth was formed, it struck out as a cold and barren rock. From this unpromising start, a few hundred million years later, the first living cells arose. What are a few hundred million years in a universe of ten or twenty billion years? It is as though life will get going just as soon as it is given the chance. Life had come quite easily to Earth, he concluded, therefore it should come easily for all the other possible worlds.\n\nSecond, Drake pondered the question of intelligence. On how many of the hundred billion Earth-like planets might the living cells contrive intelligent forms? Earth's diversity presented itself to his mind. Billions of animal species had wriggled, and buzzed, and hissed, and flitted, and swum through the ages. Yet only one had ever posed itself questions, and dreamed of life on other worlds. Furthermore, _Homo sapiens_ was a latecomer to a planet that had done without its brains for billions of years. It suggested to Drake that questioning minds were anything but universal. He finally settled upon a figure of one in a hundred, leaving him a billion potential civilizations amid the stars.\n\nHow many of these civilizations had developed the technology (not to mention, had the desire) to communicate with others and in ways that we could understand? Drake was a radio astronomer. He knew that transmissions from Earth had already seeped far into space. Out there somewhere, twenty or thirty light-years away, hands and ears with the know-how were already perhaps tuning in to the same episodes of _Flash Gordon_ and _The Lone Ranger_ that he had listened to as a boy. And some of these planets would certainly be broadcasting signals of their own: say, one hundred thousand in all.\n\nThese planets would exude music, news broadcasts, coded messages, provided of course that they were still around, that this same technology had not yet blown them to pieces. Civilization, after all, is a tricky business. Hazardous, too. The civilized human dates back only ten thousand years, and already Washington, DC, had nuclear arms aimed at Moscow (and vice versa). The threat of mutual destruction, Drake knew, was no delusion. But neither was such a trajectory inevitable. Mindful of his final proviso, he set his expectations low: of the potential hundred thousand communicating civilizations, the survivors in our galaxy would number maybe ten. Signals, centuries and millennia old, from thousands of their predecessors would likewise be filtering through space, ready to brush some awaiting antenna.\n\nDrake wrote out his reasoning in the impressive-looking shorthand of an equation.\n\n> N = N* x fp x ne x fl x fi x fc x fL\n> \n> Where N is the number of communicating civilizations in our galaxy.\n> \n> N* is the number of stars in the Milky Way.\n> \n> fp is the fraction of stars with planets orbiting them.\n> \n> ne is the number of planets per star ecologically capable of life.\n> \n> fl is the fraction of those planets where life evolves.\n> \n> fi is the fraction of these living planets that evolves intelligent life.\n> \n> fc is the fraction of these that communicate with others.\n> \n> fL is the fraction of a planet's life during which the civilization survives.\n\nNow to a journalist's question, \"Do other intelligent civilizations exist?\" came Drake's reply: \"With almost absolute certainty.\" A certainty identical to and inherited from the one he had heard in his father's voice.\n\nWhat would the astronomer give to actually find another world! (Russian colleagues reminded him that \"world\" in their language is a homonym for \"peace.\") Every day, in his observatory, he set to work. Through thick glasses he followed the chart recorder's progress, watched the needle fidget, saw the ink illustrate the contours of random noise. Occasionally, growing more and more impatient, he would snatch up headphones and listen in to the reception. He sat stock-still in the room, his heart in his mouth, and listened. Listened for what exactly? For a beep, a buzz, an electronic whisper. He watched and listened and waited and the hours turned. But the surprise did not come. Nothing came, except the months, the years, the hours.\n\nWith time, the technology became refined and upgraded. More and more assistants lent their ears and patience to the expanding project. \"Probability\" was on everyone's lips. The numbers are on our side, they told the press, their friends, themselves. It is only a matter of time.\n\nQueasy silence was all that they heard.\n\nThe radio telescopes grew, and with them the doubts. You would have to be superhuman not to doubt. Except perhaps for Drake, who had invested so heavily in his hope.\n\nThe problem posed by the silence was striking, though. For if thousands of communicating civilizations have evolved throughout our galaxy over millions (or even billions) of years, why has not a single one colonized its environs, including Earth? A civilization far older than our own, out of the thousands predicted by Drake's equation, would need only a few million years\u2014a cosmic blink of an eye\u2014to have colonized the Milky Way. Or, at the very least, to have flooded us with signs of its presence.\n\nLook longer! Listen harder! Drake's response was emphatic. Perhaps the other civilizations are checking us out before they get in touch. Or perhaps they content themselves with colonizing only their solar systems. Or perhaps the cost of interstellar travel is too steep. Or perhaps, they never invented the radio. Perhaps, perhaps, perhaps.\n\n_Is Anyone Out There?_ To coincide, in 1992, with the five-hundredth anniversary of Columbus's discovery of America, Drake published a book posing this question. The answer, he felt, was closer than ever. He wanted to \"prepare thinking adults\" for the \"imminent detection of signals from an extraterrestrial civilization.\" What kind of civilization? They would resemble humans, with heads on the top of their bodies and two legs to walk on. But instead of two arms, they would have four: \"four make for a much better design.\" They would also be immortal, innocent of death. \"This discovery, which I fully expect to witness before the year 2000, will profoundly change the world.\"\n\nIn the same year, NASA's computers performed fifty million tests per second on the data from the largest ever and most sophisticated radio scan of the heavens. They found nothing.\n\nBiologists, meanwhile, were offering a reassessment of the equation's assumptions. Drake and his colleagues applied \"strictly deterministic thinking,\" wrote the Harvard biologist Ernst Mayr. \"Such thinking is often quite legitimate for physical phenomena, but is quite inappropriate for evolutionary events or social processes such as the origin of civilizations.\" Another biologist, Leonard Ornstein, pointed out that \"even if we allow that the universe may be awash with planets with flourishing 'protometabolism,' and even 'protocells,' it does not necessarily follow that contingent events that contributed to the next step in the origin of life have been mimicked on even one of these hypothetical worlds.\"\n\nOrnstein suggested an analogy: imagine that we dipped our hand only once into a bag of marbles and withdrew just one marble. This marble is blue-green. Conclusion? We could as equally suppose that the bag contained only one-in-a-million other blue-green marbles, or none at all, as suppose that all or many still in the bag would possess a similar color.\n\nThe only thing we can know for certain is that the probability of intelligent life in our universe is above zero (for were it zero, I would not be here to write this sentence and you would not be here to read it). The rest is speculation. Since the big bang, there have been billions of civilizations. There have been millions of civilizations. There have been thousands of civilizations. There have been hundreds of civilizations, or tens. Or just one.\n\nWhy not? Probability is often expressed using large but finite numbers: \"one in a thousand,\" \"one in a million.\" But perhaps the probability of life, intelligent life, appearing somewhere in our universe is _infinitesimal._ If so, a universe would need infinitely many planets to produce even a finite number of civilizations (i.e., one).\n\nSuch a conclusion ought to be at least as motivating to us as Drake's, especially in an age of high-stakes international diplomacy, atomic bombs, and climate change. As the astronomer Michael Papagiannis concluded, \"Knowing we are the only ones might make us realize that we are too valuable to destroy.\"\n\n# Twelve\n\n# THE CALENDAR OF OMAR KHAYY\u00c1M\n\nFor the Bedouins living before the era of Muhammad, time did not exist. Or rather, they thought of it as an all-enveloping and all-enfeebling mist, without clear shape or pattern. Only the bright stars overhead pricked the pervasive gloom, helping the nomads to anticipate rain and decide when to take their animals to pasture. To make sense of their lives, the men sang songs, and the songs they sang told of distant earthquakes and battles. It was the sole history that they knew.\n\nThe Prophet's birth, to believe tradition, coincided with one of these battles: the so-called Event of the Elephant (occurring in the latter half of the sixth century of the Christian Era), when Mecca fell under the siege of a foreign king's army with a white elephant at its helm. According to a tale later told in the Koran, God sent a cloud of birds to pelt the attackers with stones until they fled.\n\nAlongside Muhammad's revelation of a new religion came his revelation of time. Gone now was the idea of life as constituted by a flux of vague, discontinuous, and casual moments. Five compulsory prayers\u2014 _Fajr_ (at dawn), _Dhuhr_ (after the sun's zenith), _Asr_ (during late afternoon), _Maghrib_ (at sunset), and _Isha_ (at twilight)\u2014regulated each day. All our days, said the Prophet, are numbered. Each follows the other in meaningful succession.\n\n\"God wraps night around day, and He wraps day around night.\"\n\nSeven of them together make a week (beginning on what we call Saturday), the span in which, it was said, God had progressively created the world: the earth on the first day, the hills on the second, the trees on the third, all unpleasant things on the fourth, the light on the fifth, the beasts on the sixth, and Adam, who was the last of creation, about the time of the _Asr_ prayer on the seventh.\n\nLook up at the heavens, Muhammad urged his followers. Each month, he declared, began when the moon appeared \"like an old shriveled palm-leaf.\" Divinely ordained properties separated the months and made them distinct. During four of the months, it was forbidden to draw a sword. During certain others, believers could set out on pilgrimage. One month, called Ramadan, was set aside for fasting during the hours of daylight. Twelve lunar months composed one year.\n\nMuhammad had been preaching for about a decade when, at the approximate age of fifty, he, along with his small band of followers, was ousted from the city by the rulers of Mecca. On camels, they traveled north to the oasis town of Yathrib, finding refuge there. The flight, known as _hegira,_ became the founding date of the Islamic calendar; henceforth, every period of time would be precisely accounted.\n\nExquisitely contrived clocks sprang up across the Islamic world throughout the medieval period. The most impressive of these their makers filled not with sand, but water. A Chinese traveler's report from a visit to Antioch, three centuries after Muhammad, tells of a water clock within the royal residence that suspended twelve golden balls corresponding to the twelve hours of the day. At each hour one of the balls fell, splashing two o'clock, three o'clock, four o'clock, \"the sound of which makes known the divisions of the day without the slightest error.\" Another water clock, described by Al-Jazari in his 1206 _Book of Knowledge of Ingenious Mechanical Devices,_ stood as high as two men and featured robotic drummers, trumpeters, and cymbalists who played in turns according to the time of day.\n\nBy employing water\u2014a precious substance in a peninsula largely composed of desert\u2014the clockmakers demonstrated the reverence they accorded to time. Indeed, the Prophet had told his faithful to pay minute attention to its reckoning. Mosques employed _muwaqqit_ s (timekeepers) to calculate the official hours for each prayer. Generations of scholars scrupulously debated the world's age. The historian Al-Tabari, for example, figured that the world would endure in total seven thousand years, of which his generation had before them only two hundred. He based his calculation on a saying of the Prophet, in which Muhammad likens the time remaining until the Last Day to the fraction of time between the _Asr_ (afternoon) prayer and sunset (about one-fourteenth of a day).\n\nWriting in the fourth and fifth centuries of _hegira_ (the eleventh century according to our way of reckoning), a near contemporary of Al-Tabari, Al-Biruni, compiled his _Chronology of Ancient Nations._ In it, he compares the calendars of the other great civilizations. The Greeks, the Syrians, and the Egyptians, he noted, all used a calendar of 365 and one one-quarter days, summing the quarters to make one complete extra day every four years.\n\n\"The ancient Egyptians followed the same practice, but with this difference, that they neglected the quarters of a day till they had summed up to the number of days in one complete year, which took place every 1,460 years; then they added one extra year.\"\n\nThe ancient Persians, he continued, also neglected these quarters of a day, though for a period of 120 years, after which they added one extra month.\n\nAs fate would have it (though Muslims believe no such thing exists, and that every moment is the conscious creation of God), Al-Biruni died in the same year that a boy was born to a Persian tentmaker. The Farsi word for tentmaker is _khayy\u00e1m;_ the tentmaker named his son Omar.\n\nIt is probable that as a child he studied the Koran. He would have learned to recite its verses aloud, for tradition holds that the scripture is akin to a chant, which is why the angel Gabriel chose to speak its words to the illiterate Muhammad. Perhaps the boy recited such a verse as, \"Most surely in the creation of the heavens and the earth, and the alteration of the night and the day, there are signs for men who understand.\"\n\nMany other books, on many subjects, must also have passed through his hands: books on geometry and the movement of the stars, books on arithmetic and music. He would have learned many of the pages by heart. It is likely that he also read or heard of Al-Biruni's compendium of calendars, and smiled at Al-Tabari's apocalyptic prediction. From long years of cloistered study, indifferent to the company of other people, he earned the bookish reputation as a \"bad character.\"\n\nOne story relates how a student visited Khayy\u00e1m daily before dawn to learn from him, before badmouthing him to the other townspeople. Hearing of this, Khayy\u00e1m secretly invited the town's musicians to call on him at dawn the next day. When the unsuspecting student arrived as usual for his lesson, Khayy\u00e1m ordered the musicians to beat their drums and blow their trumpets; the commotion awakened people from every quarter. \"Men of Nishapur,\" Khayy\u00e1m said, addressing the crowd, \"he comes every day at this hour to my house, and studies with me, but to you he speaks of me in the manner you know. If I am really as he says, then why does he come and study with me? And if not, why does he abuse his teacher?\"\n\nWhen he was not reading books, he wrote them. A gifted poet, he was better known in his day as a talented mathematician. \"The notion that one could use geometric constructions for certain types of algebraic problems was certainly recognized by Euclid and Archimedes,\" writes the mathematician Ramesh Gangolli, \"but before Omar Khayy\u00e1m's construction, only simple types of equations... were thought to be amenable to the geometric method... Khayy\u00e1m opened the door to the study of the more general question: What kind of algebraic problems can be represented and solved successfully in this manner?\"\n\nThe young Persian's receptivity to inspiration must have been immense. When the sunlight shone through the latticework windows of his study, it danced upon the walls in geometrical shapes. Khayy\u00e1m's pen traced _rubai_ (poems) of four short rhyming lines, tight as theorems, writing the words from right to left. Some say he composed only sixty such poems; others, six hundred. He also wrote a commentary on Euclid's _Elements_ that Gangolli tells us explained \"in more detail many aspects that were left implicit and clarified many misconceptions about the structure of axiomatic systems.\"\n\nPolyvalent talent like his is rare in any age. It likely led to jealousies, snide comments, upturned noses from certain quarters among his fellow countrymen. In one of several treatises on algebraic problems, Khayy\u00e1m complained about the trials of the mathematician's life.\n\n> I was unable to devote myself to the learning of this algebra and the continued concentration upon it, because of obstacles... which hindered me; for we have been deprived of all the people of knowledge save for a group, small in number, with many troubles, whose concern in life is to snatch the opportunity, when time is asleep, to devote themselves meanwhile to the investigation and perfection of a science; for the majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do not use what they know of the sciences except for base and material purposes.\n\nIn 452 (1074 CE by the Julian calendar), Sultan Jalal Al-Din invited Omar to the capital. His long Farsi texts, packed with numbers and equations, had preceded him. Even at the zenith of the golden age of Islamic mathematics, Khayy\u00e1m's talent marked him out. Anxiously, expectantly, he must have followed his guide through the halls of the turquoise-studded palace. Tessellating tiles ran the length of the floor; mirrors hanging on the walls returned to him every one of his features, down to the wrinkles of laughter around his eyes.\n\nThe Sultan that Khayy\u00e1m met hardly looked the part: he was very young, not yet twenty years of age, and keen to make his mark. His illustrious guest, the Sultan's Vizier reports, was promptly showered with the prince's praises, and offered an annual pension of 1,200 _mithk\u00e1l_ s of gold. For this, Khayy\u00e1m agreed to accept an important commission: he was to create\u2014provided that God were willing\u2014a new civil calendar in the young Sultan's name.\n\nPersia, geographically vast and intricately governed, had long lived on two times: while the imams practiced the faith using the _hegira_ calendar, the bureaucrats counted their days by looking to the sun. The old civil calendar consisted of twelve months each of thirty days, excepting the eighth month, which contained thirty-five days. Its total of 365 days (eleven days more than in a lunar year) helped fix administrative dates far more closely to the seasons: an essential requirement when the nation's annual tax revenues relied heavily on the autumn harvests. But even these eleven extra days could not keep up precisely with the seasons: each year, as Al-Biruni had recorded, accumulated a lag of a quarter day. This was the problem that Khayy\u00e1m set out to solve.\n\nDay and night, Omar pondered how best to reform the old civil calendar. Astronomy had never before found itself so flattered and moneyed. A large observatory duly arose, from which Khayy\u00e1m and his colleagues stalked the sky. He studied the sun's pathway through the twelve constellations of stars, and compiled detailed statistical tables. With this data he mapped a calendar based on the seasons, beginning the year (which the Persians called _Nowruz_ ) on the spring equinox (March 20 or 21), with the fourth month falling on the summer solstice (June 21), the seventh month on the autumn equinox (September 21), and the tenth on the winter solstice (December 20 or 21).\n\nTo resolve the lagging of quarter days, Khayy\u00e1m devised an ingenious formula. His calendar interleaved eight extra days over each thirty-three-year span. The calculation: 365 + 8\/33 = 365.2424 days, aligned almost perfectly with the actual year length (365.2423 days), and proved even more accurate than that of the later Gregorian calendar: 365 + 1\/4 \u2212 1\/100 + 1\/400 = 365 + 97\/400 = 365.2425 days.\n\nThe Sultan officially adopted Khayy\u00e1m's reform on Friday, March 15, 1079 ( _Farvardin_ 1, 458 according to the new calendar). Drums and cannon blasts across the nation proclaimed the first new year.\n\nConcerning Khayy\u00e1m's later years, we can say very little. The Sultan's premature death, just ten years after the calendar's adoption, brought an end to his generous patronage. Khayy\u00e1m quit the royal court, only the ritual pilgrimage to Mecca delaying a return to his hometown. He continued to write poems.\n\n> Ah, but my computations, People say,\n> \n> Reduced the year to better reckoning? Nay,\n> \n> 'Twas only striking from the Calendar\n> \n> Unborn tomorrow and dead Yesterday\n\nOne day in his eighth decade, Khayy\u00e1m tired and lay down to rest. His heavy head swaddled in sheets of turban, he gazed up toward the sky. The light around him slowly faded, and then went out. The year was around 500 of the _hegira_ \u2014the point Al-Tabari had long ago calculated as being the end of the world.\n\n# Thirteen\n\n# COUNTING BY ELEVENS\n\nPhysicians say that thumbs are the master fingers of the hand,\" wrote Michel de Montaigne, the Renaissance nobleman who invented the personal essay, in his short piece _On Thumbs._ So vital did Rome's ancient rulers consider them, he explains, that war veterans who were missing thumbs received automatic exemption from all future military service.\n\nThroughout his writings, the Frenchman marveled at the extent to which we depend upon our hands. A gesture of thumbs-up or -down, an index finger pressed to the mouth, the palms held flat open toward the skies: delivered at the right instant all can say more than any word. In another essay Montaigne describes the case of a \"native of Nantes, born without arms\" whose feet did double duty so well that the man \"cuts anything, charges and discharges a pistol, threads a needle, sews, writes, takes off his hat, combs his head, plays at cards and dice.\" Elsewhere he observed that hands sometimes seem almost to possess a life of their own, as when his idle fingers would tap during a period of daydreaming, free of conscious effort or instruction.\n\nMontaigne neglects to mention counting as being among the many useful tasks undertaken by our hands (biographers have pointed out that arithmetic was not his strong suit). Of course, there is much to be said for the idea that our decimal number system originated from the practice of counting with the fingers. In our Latin-derived word \"digit,\" the meanings \"number\" and \"finger\" coincide. The Homeric Greek term for counting, _pempathai,_ translates literally: \"to count by fives.\"\n\nPeople the world over, numbers at their fingertips, count to ten and by tens (twenty, thirty... fifty... one hundred...). But the manner of their getting to ten varies from hand to hand and from culture to culture. Like many Europeans, I count \"one\" beginning on the thumb of my left hand and continue past five with the digits of my right till I arrive at the other thumb. Where people read from right to left, as in the countries of the Middle East, counting generally begins on the little finger of the right hand. In Asia, the counter employs a single hand, folding the fingers thumb first to reach five, before unfolding them from the little finger (six) returning to the thumb (ten).\n\nWhat, I wonder, would it be like for a person who had not too few fingers\u2014like Montaigne's Roman veterans\u2014but too many? Would such a person learn to count like you or me? How would it be to count by elevens?\n\nAccording to tradition, Henry VIII's second wife, Anne Boleyn, carried a sixth finger on one of her hands\u2014a medical condition known as polydactyly. Girls of high birth in Tudor society had the attention of tutors and learned to read, write, and do sums. Counting on her eleven fingers, however, would have posed Anne certain difficulties.\n\nTo start with, she would have found herself a number word short. Between the ninth finger and the last (which would still be the tenth\u201410 meaning \"1 set and 0 remainders\"), her surplus digit would have needed naming. Given that she spent part of her childhood in France, she might have come up with the label _dix._ So Anne would have counted: one, two, three, four, five, six, seven, eight, nine, _dix,_ ten. Though the numbers' music sounds a little strange to our ears, in her mind the sequence would have come to feel like second nature. A year before the age of twenty, she would have celebrated (if only mentally) her dixteenth birthday. Counting higher, she would have taken the nineties down a peg, to make room for the dixties, and by way of dixty-dix arrive finally at one hundred.\n\nCounting in this way produces some funny-looking sums. Subtracting seven from ten (where, given the extra _dix,_ \"ten\" would to our way of thinking be eleven), for example, would have left Anne not with three, but four. Half of thirteen equals seven. Six squared (6 \u00d7 6) is thirty-three (three \"tens\" of eleven, plus three units): a pretty result.\n\nFractions would have proven particularly tricky. Unlike ten, eleven is a prime number: divisible only by itself and one. No precise midway point exists between one and eleven, or between Anne's ten (equivalent to our eleven) and her one hundred (11 \u00d7 11). What then might a half of something mean to someone with eleven fingers? And what about a fifth or a quarter or two-thirds? Possibly such concepts would remain as airy and intangible as a _dixth_ of something might seem to you or me. (We can nonetheless imagine that rote learning would have sufficed to acquire them.)\n\nStill, I am curious about the intuitions that an eleven-fingered girl might have brought to such ideas. From our two hands, equally endowed, we understand immediately that halves are clean and precise, leaving no remainder. Half of eight is four, not three or five. A right-angle triangle is exactly one half of a square. Prime numbers, by definition, cannot be halved in this way. Might Anne's hands\u2014with six fingers on one hand, and \"only\" five on the other\u2014have given her a more approximate, fuzzier feeling for the concept of a half? To a casual remark like, \"I'll be along in half an hour,\" would the impatient thought have occurred to her, \"Which half do you mean\u2014the lesser or the greater?\"\n\nAt her secret wedding to the King of England, Boleyn would have well understood which half of the couple she represented. Her triumph was infamously short-lived. Within months, the Archbishop of Canterbury had declared the marriage invalid. Roman Catholics denounced her as a scarlet woman.\n\nThe charge of treason brought against her only three years after the wedding makes no mention of witchcraft. All the same, her enemies alleged that the Queen's body showed strange warts and growths\u2014the same body that had produced only stillborn fetuses for a male heir. It was in the role of a conspiring adulteress that she knelt in a black damask dress before the executioner's block on the morning of May 19, 1536.\n\nIt is certainly conceivable that the story of an eleventh finger was a fabrication of Anne's enemies. A famous portrait by an unknown artist, which now hangs in Hever Castle (Boleyn's childhood home) in Kent, reveals a striking young woman clutching a rose. The hands peek shyly from long sleeves; the fingers\u2014ten in all\u2014appear, in contrast to the smooth oval face, somewhat ill formed. A slander, or a secret, Anne's eleven fingers have become an integral part of her legend.\n\nI was reminded of the legend recently when a fascinating newspaper article caught my attention. It was about one Yoandri Hernandez Garrido, a thirty-seven-year-old Cuban man with twelve fingers and twelve toes. Apparently, Fidel Castro's doctor once paid the extradexterous boy a visit and declared his hands and feet the most beautiful that he had ever seen.\n\nIn keeping with the Latin American taste for nicknames based on physical appearance, schoolmates gave Garrido the name \"Veinticuatro\" (twenty-four). He learned to count in class like his friends but one day, he recalls, his primary schoolteacher asked him for the answer to the sum 5 + 5. In his confusion he answered, \"Twelve.\"\n\nGarrido tells the reporter that he makes a fine living with his hands. American tourists regularly tender their dollars to have a photo taken with the twelve-fingered Cuban. A few extra bucks can go a long way in Castro's republic. Proudly, Garrido holds his hands up to the camera. There is generosity in his smile.\n\nThe article makes no mention of the adult Garrido's arithmetic. Counting time, to give one example, should be simpler with twelve fingers\u2014one digit for every hour on the clock. To find what time is nine hours before four o'clock in the afternoon, he need simply hold down five fingers (9\u22124) on his right hand to reveal the answer: one hand (six fingers) plus the right hand's thumb: seven a.m. One digit for every month as well: so half-year periods jump from one hand's finger to the same finger on the other hand.\n\nWe know that the Romans preferred to perform certain calculations using a base of twelve. In his _Ars Poetica_ the poet Horace offers us a vignette of Roman boys learning their fractions.\n\n> Suppose Albinus's son says: if one-twelfth is taken from five-twelfths, what is left? You might have answered by now: One-third. Well done. You will be able to manage your money. Now add a twelfth: what happens? One-half.\n\nThe Venerable Bede taught his fellow monks to quantify the various periods of time in the biblical stories using the Roman fractions. One-twelfth, he noted, went by the name of an _uncia_ (from which comes the word \"ounce\"), while the remaining part of 11\/12 was called _deunx._ Dividing into six, the sixth part (1\/6) was called _sextans,_ and _dextans_ for the rest (5\/6). _Quadrans_ was the Roman name for a quarter (1\/4); three-quarters they called _dodrans._\n\nWhat do we get if we add one _sextans_ to a _dodrans_ (1\/6 + 3\/4)? As quick as Horace's schoolboys, or the Venerable Bede's monks, Garrido's fingers (or toes) would tell him: one-sixth equals two digits and three-quarters equals nine digits, so 1\/6 + 3\/4 = 11\/12 (a _deunx_ ).\n\nI wonder what Garrido makes of our deficient hands, all of us counting to the tune of ten? Does he pity us as the Romans pitied their thumbless veterans? He calls his condition a blessing; it is clear that he would not wish to be any other way.\n\nSome believe that we should all learn to count like Garrido. A society formed in the mid-1900s advocates replacing the decimal system with a \"dozenal\" one, since twelve divides more easily than ten. Leaving aside itself and one, the number twelve can be divided into four factors: two, three, four, and six, compared with the factors of ten: two and five. In England and America the society still militates for the return of Roman fractions (among other measures), judging their abandonment a gross error.\n\nLike the Esperantists and the simplified spellers that came before them, the society's members dream of a highly rational world comprising pairs, trios, quarters, and sextets, a world innocent of messy fractions. Theirs is the charm of the hopeless cause.\n\nAn English queen with eleven fingers, a Cuban man with twelve toes\u2014such stories still incite our wonder and with it our vague sense that something has gone awry.\n\nMontaigne, typically generous, remedies this view. He recalls once encountering a family who exhibited \"a monstrous child\" in return for strangers' coins. The infant had multiple arms and legs, the remains of what we now call a conjoined twin. In the infinite imagination of its Creator, Montaigne supposes, the child is simply one of a kind, \"unknown to man.\" He concludes, \"Whatever falls out contrary to custom we say is contrary to nature, but nothing, whatever it be, is contrary to her. Let, therefore, this universal and natural reason expel the error and astonishment that novelty brings along with it.\"\n\n# Fourteen\n\n# THE ADMIRABLE NUMBER PI\n\nTo believe the poet Wis\u0142awa Szymborska, I am one in two thousand. The 1996 Nobel laureate offers this statistic in her poem \"Some Like Poetry\" to quantify the \"some.\" Actually I think she is a tad too pessimistic\u2014I am hardly as rare a reader as that. But I can see her point. Many people consider poetry to be all clouds and buttercups, without purchase on the real world. They are right and they are wrong. Clouds and buttercups exist in poetry, but they are there only because storms and flowers populate the real world too. Truth is, a good poem can be about anything.\n\nIncluding numbers. Mathematics, several of Szymborska's verses show, lends itself to poetry. Both are economical with meaning; both can create entire worlds within the space of a few short lines. In \"A Large Number,\" she laments feeling at a loss with numbers many zeroes long, while her \"Contribution to Statistics\" notes that \"out of every hundred people, those who always know better: fifty-two\" but also, \"worthy of empathy: ninety-nine.\" And then there is \"The Admirable Number Pi,\" my favorite poem. It\u2014the poem, and the number\u2014begins: three point one four one.\n\nOnce, in my teens, I confided my admiration for this number to a classmate. Ruxandra was her name. Like the poet's, her name came from behind the Iron Curtain. Her parents hailed from Bucharest. I knew nothing of Eastern Europe, but that did not matter: Ruxandra liked me. She liked that I was quite different from the other boys. We spent breaks between lessons in the school library, swapping ideas about the future and homework tips. Happily for me, her strongest subject was math.\n\nIn a moment of curiosity, I asked about her favorite number. Her reply was slow; she seemed not to understand my question. \"Numbers are numbers,\" she said.\n\nWas there no difference at all for her between the numbers 333, say, and fourteen? There was not.\n\nAnd what about the number pi, I persisted\u2014this almost magical number that we learned about in class. Did she not find it beautiful?\n\nBeautiful? Her face shrank with incomprehension.\n\nRuxandra was the daughter of an engineer.\n\nThe engineer and the mathematician have a completely different understanding of the number pi. In the eyes of an engineer, pi is simply a value of measurement between three and four, albeit fiddlier than either of these whole numbers. In his calculations he will often bypass it completely, preferring a handy approximation like 22\/7 or 355\/113. Precision never demands of him anything beyond a third or fourth decimal place (3.141 or 3.1416, with rounding). Digits past the third or fourth decimal do not interest him; as far as he is concerned, it is as though they do not exist.\n\nMathematicians know the number pi differently, more intimately. What is pi to them? It is the length of a circle's round line (its circumference) divided by the straight length (its diameter) that splits the circle into perfect halves. It is an essential response to the question, \"What is a circle?\" But this response\u2014when expressed in digits\u2014is infinite: the number has no last digit, and therefore no next-to-last digit, no antepenultimate digit, no third-from-last digit, and so on. One could never write down all its digits, even on a piece of paper as big as the Milky Way. No fraction can properly express pi: every earthly calculation produces only deficient circles, pathetic ellipses, shoddy replicas of the ideal thing. The circle that pi describes is perfect, belonging exclusively to the realm of the imagination.\n\nMoreover, mathematicians tell us, the digits in this number follow no periodic or predictable pattern: just when we might anticipate a six in the sequence, it continues instead with a two or zero or seven; after a series of consecutive nines, it can as easily remain long-winded with another nine (or two more nines or three) as switch erratically between the other digits. It exceeds our apprehension.\n\nCircles, perfect circles, thus enumerated, consist of every possible run of digits. Somewhere in pi, perhaps trillions and trillions of digits deep, a hundred successive fives rub shoulders; elsewhere occur a thousand alternating zeroes and ones. Inconceivably far inside the random-looking morass of digits, having computed them for a time far longer than that which separates us from the big bang, the sequence 123456789... repeats 123,456,789 times in a row. If only we could venture far enough along, we would find the number's opening hundred, thousand, million, billion digits immaculately repeated, as though at any instant the whole vast array were to begin all over again. And yet, it never does. There is only one number pi, unrepeatable, indivisible.\n\nLong after my schooldays ended, pi's beauty stayed with me. The digits had insinuated themselves into my mind. Those digits seemed to speak of endless possibility, illimitable adventure. At odd moments I would find myself murmuring them, a gentle reminder. Of course, I could not possess pi\u2014the number, its beauty, or its immensity. Perhaps, in fact, it possessed me. One day, I began to see what this number, transformed by me, and I by it, could turn into. It was then that I decided to commit a multitude of its digits to heart.\n\nThis was easier than it sounds, since big things are often more unusual, more exciting to the attention, and hence more memorable than small ones. For example, a short word like _pen_ or _song_ is quickly read (or heard), and as quickly forgotten, whereas _hippopotamus_ slows our eye (or ear) just enough to leave a deeper impression. Scenes and personalities from long novels return to me with far greater insistence and fidelity, I find, than those that originated in short tales. The same goes for numbers. A common number like thirty-one risks confusion with its common neighbors, like thirty and thirty-two, but not 31,415 whose scope invites curious, careful inspection. Lengthier, more intricate digit sequences yield patterns and rhythms. Not 31, or 314, or 3141, but 3 _1_ 4 _1_ 5 sings.\n\nI should say that I have always had what others call \"a good memory.\" By this, they mean that I can be dependably relied upon to recall telephone numbers, dates of birthdays and anniversaries, and the sorts of facts and figures that crowd books and television shows. To have such a memory is a blessing, I know, and has always stood me in good stead. Exams at school held no fear for me; the kinds of knowledge imparted by my teachers seemed especially amenable to my powers of recollection. Ask me for the third person subjunctive of the French verb \"\u00eatre,\" for instance, or better still the story of how Marie Antoinette lost her head, and I could tell you. Piece of cake.\n\nPi's digits henceforth became the object of my study. Printed out on crisp, letter-size sheets of paper, a thousand digits to a page, I gazed on them as a painter gazes on a favorite landscape. The painter's eye receives a near infinite number of light particles to interpret, which he sifts by intuitive meaning and personal taste. His brush begins in one part of the canvas, only to make a sudden dash to the other side. A mountain's outline slowly emerges with the tiny, patient accumulation of paint. In a similar fashion, I waited for each sequence in the digits to move me\u2014for some attractive feature, or pleasing coincidence of \"bright\" (like 1 or 5) and \"dark\" (like 6 or 9) digits, for example, to catch my eye. Sometimes it would happen quickly, at other times I would have to plow thirty or forty digits deep to find some sense before working backward. From the hundreds, then thousands, of individual digits, precisely rendered and carefully weighed, a numerical landscape gradually emerged.\n\nA painter exhibits his artwork. What was I to do? After three months of preparation, I took the number to a museum, the sprawling digits tucked inside my head. My aim: to set a European Record for the recitation of pi to the greatest number of digits.\n\nMarch is the month of spring showers, and school holidays, and spick-and-span windows. It is also the month when people the world over celebrate \"Pi Day,\" on March 14. So on that day, in 2004, I traveled north from London to the city of Oxford, where staff members at the university's Museum for the History of Science were waiting for me. Journalists, too. An article in the _Times,_ complete with my photo, announced the upcoming recitation.\n\nThe museum lies in the city center, in the world's most ancient surviving purpose-built museum building, the Old Ashmolean. Iconic stone heads, wearing stone beards, peer down at visitors as they pass through the gates. The thick walls are the color of sand. Approaching the building, photographers appear as if from nowhere, holding cameras, like masks, up to their faces. The piercing flashes momentarily petrify my expression. I stop and raise my features into a smile. A minute later they are gone.\n\nThe record attempt's organizers have occupied the museum building. Television camera wires snake the length of the floor. Posters requesting donations (the event is raising money for an epilepsy charity, at my request, since I suffered from seizures as a young child) dress the walls. A table and chair, I see on entering, have already been set out for me on one side of the hall. Before it, a longer table awaits the mathematicians who will verify my accuracy. But there is still an hour before the recitation is due to start, and I find only a trio of men talking together. One has a full head of wiry hair, one has a multicolored tie, and one has neither hair nor tie. The third steps briskly forward and introduces himself as the main organizer, and the others as the museum's curator and his assistant. Their faces show mild puzzlement, curiosity, and nerves. Shortly afterward, reporters arrive to hold the microphones and man the television cameras. They film the display cases containing astrolabes, compasses, and mathematical manuscripts.\n\nSomeone asks about the blackboard that hangs high on the wall opposite us. The curator explains that on May 16, 1931, Albert Einstein used it during a lecture. What about the chalky equations? They show the physicist's calculations for the age of the universe, replies the curator. According to Einstein, the universe is about ten, or perhaps one hundred, thousand million years old.\n\nFootfalls increase on the museum's stone steps as the hour approaches. The mathematicians duly arrive, seven strong, and take their seats. Men, women, and children keep coming; it is soon standing room only. The air in the hall grows thick with hushed talk.\n\nAt last, the organizer calls everyone to silence. All eyes are on me; nobody moves. I sip a mouthful of water and hear my voice begin. \"Three point one four one five nine two six five three five eight nine seven nine three two three eight four...\"\n\nMy audience is only the second or third generation able to hear the number pi beyond the first few tens or hundreds of decimal places. For millennia it existed only in a breathful of digits. Archimedes knew pi to only three correct places; Newton, almost twenty centuries later, managed sixteen. Only in 1949 did computer scientists discover pi's thousandth digit (following the decimal point): nine.\n\nIt takes about ten minutes, at a rate of one or two digits per second, for me to reach this nine. I do not know how long exactly; an electronic clock records the seconds, minutes, hours of the recitation for the public to watch, but I cannot see it from my chair. I stop reciting to sip water and catch my breath. The pause feels palpable. Dolorous even. I feel completely, oppressively alone.\n\nThe rules for the recitation are strict. I cannot step away from the desk, except to use the bathroom, and then always accompanied by a member of the museum's staff. No one may talk to me, not even to cheer me on. I can stop reciting momentarily to eat fruit or a piece of chocolate, or drink, but only at pre-agreed intervals a thousand digits apart. Cameras record my every sound and gesture.\n\n\"Three eight zero nine five two five seven two zero one zero six five four...\"\n\nAn occasional cough or sneeze from the audience punctuates the flow of digits. I do not mind. I meditate on the colors and shapes and textures of my inner landscape. Calmness gains on me; my anxiety falls away.\n\nMost of the spectators know nothing of Archimedes' polygons; they have no idea that the ten digits they have just heard will eventually repeat an infinite number of times, have never thought of themselves as being in any way susceptible to math. But they listen attentively. The concentration in my voice seems to communicate itself to them. Faces, young and old, round and oval, all wear delicate frowns. Listening to the digits, they hear their dress sizes, their birthdays, their computer passwords. They hear excerpts\u2014both shorter and longer\u2014from a friend's, or parent's, or lover's telephone number. Some lean forward in expectation. Patterns coalesce, and as quickly disperse, in their minds.\n\nThe people are all different. They have various motives for being here, and various goals. A teenager finds in the hall a hideout from his Sunday boredom; a manual laborer, having donated the equivalent of a pack of cigarettes in salary, sticks around to get his money's worth; an American tourist in shorts and a Mickey Mouse hat cannot wait to recount the spectacle to her family.\n\nAn hour passes, and then another.\n\n\"Zero, five, seven, seven, seven, seven, five, six, zero, six, eight, eight, eight, seven, six...\"\n\nI head further and further inside the number, exhaling effort, rhythm, and precision with every breath. The decimals exhibit a kind of deep order. Fives never outstrip sixes for long, nor do the eights and nines lord it over the ones and the twos. No digit predominates except for brief and intermittent instants. Every digit, in the end, has more or less equal representation. Every digit contributes equally to the whole.\n\nHalfway through the recitation, more than ten thousand decimals in, I stop to stretch. I push back the chair, stand and shake out my limbs. The mathematicians put down their sharp pencils and wait. I bring a bottle to my lips and swallow the plastic bottle-tasting water. I eat a banana. I fold my legs, resume my position at the desk, and continue.\n\nThe silence in the hall is total. It reigns like a tsar. When a young woman's mobile telephone suddenly starts up, she finds herself promptly ejected.\n\nDespite such rare commotions, a sly complicity establishes itself between the public and me. This complicity marks a vital shift. At the beginning, the men and women beamed confidence, listening expectantly, and taking pleasure in the digits' familiar sounds as those shoe sizes, historical dates, and car registration plates reached their ears. But, slowly, imperceptibly, something changed. Consternation grew. They could not follow the rhythm of my voice, they realized, without making continuous minor adjustments. Sometimes, for example, I recited the digits fast, and at other times I recited them slow. Occasionally, I recited in short bursts interleaved with pauses; at other moments, I recited the digits in a long, unbroken phrase. Sometimes the digits sounded thinly, accented by some inner agitation in my voice; instants later, they would soften to a clear and undulating beat.\n\nConsternation now turns increasingly to curiosity. More and more, I feel the timing of their breathing coincide with mine. I sense their raw intrigue at the sound and sweep of every digit as it passes and makes way for the next. When the digits darken in my mouth\u2014heavy eights and nines packed together\u2014the tense distant faces grow tenser still. When a sudden three emerges from a series of zeroes and sevens, I hear something like a faint collective pant. Silent nods greet my accelerations; warm smiles welcome my slowdowns.\n\nBetween the moments when I stop reciting to sip water or take a bite, and continue reciting, I hardly know where to look. My solitude is absolute; I do not want to return the people's stares. I look down at the shadows of bones and veins in my hands, and at the scuffs in the wooden desk on which they rest. I notice the glimmers of shiny metal that dapple the display cases. On a cheek, here and there, I cannot help noticing, tears.\n\nPerhaps the experience has taken the audience by surprise. No one has told them that they will find the number tangible, moving. Yet they succumb gladly to its flow.\n\nI am not the first person to recite the number pi in public. I know there are a few \"number artists\"\u2014men who recount numbers as actors recall their scripts. Japan is the center of this tiny community. In Japanese, spoken digits can sound like whole sentences; pronounced a certain way, the opening digits of pi, 3.14159265, mean \"An obstetrician goes to a foreign country.\" The digits 4649 (which occur in pi after 1,158 decimal places) sound just like \"nice to meet you,\" while a Japanese speaker pronouncing the digits 3923 (which occur after 14,194 decimal places) simultaneously says, \"Thank you, brother.\"\n\nOf course, such verbal constructions always suffer from arbitrariness. The short, stiff phrases stand apart, with only the speaker's ingenuity to splice them together. Japanese spectators, I have heard, watch these men perform as they might watch a tightrope walker; listening only in case of a blunder, as others watch only in case of a fall.\n\nThe relationship these artists have with numbers is complicated. Many years of repetitive learning hone their technique; but they also produce a nagging feeling of duplicity: repeated numbers (and words) often finish by losing all their sense. It is not uncommon, after each public display, for the performer to impose a monthslong fast of every digit. Benumbed by numbers, even a price tag, a barcode, an address, sickens him.\n\nIn the number artist's brain, pi can be reduced to a series of phrases. In my mind, it is I, not the number, who grows small. I diminish myself as much as possible before the mystery of pi. Emptying myself, I perceive every digit up close. I do not wish to fragment the number; I am not interested in breaking it up. I am interested in the dialogue between its digits; in the unity and continuity that underlie them all.\n\nA bell cannot tell time, but it can be moved in just such a way as to say twelve o'clock\u2014similarly, a man cannot calculate infinite numbers, but he can be moved in just such a way as to say pi.\n\n\"Three, one, two, one, two, three, two, two, three, three, one...\"\n\nReciting, I try to summon up a true picture of what I see and feel. I want to convey the shapes, and colors, and emotions that I experience, to everyone in the hall. I share my solitude with those who watch and listen to me. There is intimacy in my words.\n\nA third hour comes and goes; the recitation enters its fourth hour.\n\nMore than sixteen thousand decimal places have escaped my lips. Their swelling company presses me on. But exhaustion also grows within my body, and all of a sudden my mind goes blank. I feel the blood falling out of my head. Up until only a few moments ago the digits had accompanied me; now they make themselves scarce.\n\nIn my mind's eye, ten identical-looking paths stretch out before me; each path leading on to ten more. One hundred, a thousand, ten thousand, one hundred thousand, a million paths, beckon me out of the impasse. They stream in every possible direction. Which way to go? I have no idea.\n\nBut I do not panic. What good did panicking ever do anyone? I shut my eyes tight, and coaxingly rub the skin around my temples. I take a deep breath.\n\nGreen-tinted blackness pervades my mind. I feel disoriented, lost. A filmy white surfaces over the black, only to be recovered by a rolling gray-purple. The colors bulge and vibrate but resemble nothing.\n\nHow long did these maddening misty colors last? Seconds, but they each seemed agonizingly longer.\n\nThe seconds pass indifferently; I have no choice but to endure them. If I lose my cool, all is finished. If I call out, the clock comes to a halt. If I do not give the next digit in the next few moments, my time will be up.\n\nNo wonder the next digit, when finally I release it, tastes even sweeter than the rest. This digit requires all my force and all my faith to extract it. The mist in my head lifts, and I open my eyes. I can see again.\n\nThe digits flow fleet and sure, and I regain my composure. I wonder if anyone in the hall noticed a thing.\n\n\"Nine, nine, nine, nine, two, one, two, eight, five, nine, nine, nine, nine, nine, three, nine, nine...\"\n\nQuickly, quickly, I must keep going. I must not let up. I cannot linger, not even before the most outstanding glimpses of the number's beauty; the joy I feel is subordinate to the need to reach my goal and recite the final digit in my mind. I must not disappoint all who are standing here, watching me and listening to me, waiting for me to bring the recitation to its fitting conclusion. All the preceding thousands have no value in themselves: only once I have wrapped everything up can they successfully count.\n\nFive hours have now elapsed. My speech begins to slur; I am drunk on exhaustion. The end is in sight, but it generates fear: am I up to it? What if I fall short? Tension stirs me for this culminating burst.\n\nAnd then, minutes later, I say, \"Six, seven, six, five, seven, four, eight, six, nine, five, three, five, eight, seven,\" and it is over. There is nothing more to say. I have finished recounting my solitude. It is enough.\n\nPalms come together; hands clap. Someone lets out a cheer. \"A new record,\" someone else says: 22,514 decimal places. \"Congratulations.\"\n\nI take a bow.\n\nFor five hours and nine minutes, eternity visited a museum in Oxford.\n\n# Fifteen\n\n# EINSTEIN'S EQUATIONS\n\nSpeaking about his father, Hans Albert Einstein once said, \"He had a character more like that of an artist than of a scientist as we usually think of them. For instance, the highest praise for a good theory or a good piece of work was not that it was correct nor that it was exact but that it was beautiful.\" Numerous other acquaintances also remarked on Einstein's belief in the primacy of the aesthetic, including the physicist Hermann Bondi, who once showed him some of his work in unified field theory. \"Oh, how ugly,\" Einstein replied.\n\nIt is a mostly thankless task to try to assign to mathematicians some universal trait. Einstein's famous predilection for beauty offers one rare exception. Mathematicians can be tall or short, worldly or remote, bookworms or book burners, multilingual or monosyllabic, tone-deaf or musically gifted, devout or irreligious, hermit or activist, but virtually all would agree with the Hungarian mathematician Paul Erd\u00f6s when he said, \"I know numbers are beautiful. If they are not beautiful, nothing is.\"\n\nEinstein was a physicist, yet his equations inspired the interest and admiration of many mathematicians. His theory of relativity drew their praise for combining great elegance with economy. In a handful of succinct formulas, every symbol and every number obtaining its perfect weight, Newtonian time and space were recast.\n\nBooks on popular mathematics abound with discursive explanations of technical proofs to illustrate their beauty. I cannot help but wonder if this might not be a mistake. I suspect that, more usually, what we laymen really admire in the work of a Euclid or an Einstein is its ingenuity, rather than its beauty. We are impressed, and yet unmoved by them.\n\nThe barrier to an appreciation of mathematical beauty is not insurmountable, however. I would like to suggest a more indirect approach. At a remove from the technical acumen of a theorist, my suggestion is more intuitive. The beauty adored by mathematicians can be pursued through the everyday: games, and music, and magic.\n\nTake the game of cricket, which was the frequent inspiration for G. H. Hardy\u2014a major number theorist and the author of _A Mathematician's Apology\u2014_ who scoured the newspaper for cricket scores over breakfast every morning. In the afternoon, after some hours at his desk, he would furl his theorems and transport them in his pockets (in case of rain) to see a local match. Among his papers he sketched the following cricket \"dream team\":\n\n> Hobbs\n> \n> Archimedes\n> \n> Shakespeare\n> \n> M Angelo\n> \n> Napoleon (Capt)\n> \n> H Ford\n> \n> Plato\n> \n> Beethoven\n> \n> Johnson (Jack)\n> \n> Christ (J)\n> \n> Cleopatra\n\nCricket matches presented Hardy the spectator with the same \"useless beauty\" that he so cherished in his theorems. By useless, he meant only that neither had any goal beyond the pursuit for its own sake. He would also frequently stand at the stumps himself, surrounded by the other team's fielders, watching the red ball expand as it flew toward his bat. Both experiences seemed to stimulate his mathematical antennae for order, pattern, and proportion.\n\nTime elapses differently on a cricket ground or in a concert hall. At its best, a well-executed, smooth-flowing cricket match can replicate the sense of harmony that we most often associate with music. The tension mounts and falls tidally, like the notes in a song. A five-day match is adept at slackening and pulling tight the outline of its hours, while every musical composition bears its own time within the structure of its notes. The unique tempo is also a part of the experience of mathematical beauty.\n\nGottfried Leibniz wrote that music's pleasure consisted of \"unconscious counting\" or an \"arithmetical exercise of which we are unaware.\" The great philosopher-mathematician meant, I suppose, that the numerical ratios that underlie all music are grasped intuitively by our minds. Every instant the listener mentally resolves the relation between the various notes\u2014the fourths and fifths and octaves\u2014as though they were objects all laid out before him side by side in some gigantic illustration. This \"grasping\" of the music\u2014however fleeting\u2014is something we can all experience as beautiful.\n\nOn the relationship between musical and mathematical beauty, we can learn more from writings about Pythagoras. It is said that he possessed a musician's ear. From boyhood, he showed a flair for the lyre. Perhaps he first heard its seven strings played by a traveling _citharede,_ a female performer dressed in long curls and bright colors; the _citharede_ s were the divas of their day.\n\nPythagoras discovered that the most harmonious notes result from the ratios of whole numbers. A vibrating string exactly halved or doubled, for example, produces an octave (ratio 1:2 or 2:1). If precisely one-third of the string is held down, or when the string is tripled in length, a perfect fifth (an octave higher) results. A perfect fourth can be obtained by holding down one-quarter of the string, or stretching it out four times longer. The whole harmonic scale was constructed in this way. Pythagoras observed that all of music depended on the first four numbers and their interrelations. He worshiped ten as the most perfect number, reflecting the unity of all things, it being the sum of one and two and three and four.\n\nAccording to Hippolytus, one of the great theologians of the Early Church, Pythagoras taught that the cosmos sang and was composed of music and \"he was the first to put down the movement of the seven stars to rhythm and melody.\" He even attempted to reproduce this soothing universal music for his disciples, rousing them in the morning to the sound of his lyre. In the evening, too, he would play for their benefit, \"in case too turbulent thoughts might still inhabit them.\"\n\nFrom Pythagoras's lyre it is an easy hop to Einstein's violin. \"If I were not a physicist,\" he once told an interviewer, \"I would probably be a musician. I live my daydreams in music. I see my life in terms of music. I get most of my joy in life out of music.\" His violin case accompanied him on many trips, but Einstein played with discretion and few reliable reports of his musical ability survive. It appears that his amateur technique was respectable, if somewhat limited. Though there may be an affinity between the laws of math and music, they cannot be conflated. Even prodigious mathematical gifts such as Einstein's did not translate into exceptional musicianship, but they certainly sharpened and reinforced his appreciation.\n\nIf math is the secret underpinning the harmonies of cricket and music, mathematical beauty is also key to feats of magic. Ever since I was a small child, playing cards that swap places, white handkerchiefs that take wing, and top hats that turn into rabbit holes have fascinated me. The visions touch some nerve deep inside.\n\nOne evening several years ago I attended the London performance of a young conjuror, who was playing to a full house. I was seated in one of the middle aisles, in a sea of heads, next to a rather elderly gentleman whose stomach sat on his lap. From this distance I had a good view of the stage. A blend of high technology spotlights and music hall gloom favored illusion.\n\nPeople come to magic shows for all sorts of reasons: some for the theatrics, others for the performer's comedy, and some (like my neighbor), it would seem, to cough. The main draw for me, though, is an experience of the unexpected. It is this that gives a conjuror's performance its peculiar beauty, a beauty akin to a well-turned equation.\n\nI am not talking here about the end result, the \"effect,\" of a trick. I am talking about the method. Every mind-read drawing or woman sawn in half looks pretty much alike, whereas the hidden ideas that make each possible can vary as much as their performers. Dozens, if not hundreds, of ways exist to levitate a spoon or make the Statue of Liberty vanish, in the same manner that numerous hundred people (not all of them professional mathematicians) have shown that the square of a right-angled triangle's hypotenuse equals the sum of the squares of the other two sides. But few of these theoretical demonstrations, or magical methods, would be considered up to Einstein's test of beauty. The truly beautiful are those that foster surprise.\n\nA genuine experience of the unexpected, in math as much as in magic, demands of its performer at once originality of insight and a lightness of touch. Even a single step too many in a method renders ugly and clumsy the theorem or the trick.\n\nIt is sometimes said that a magician will go to great pains to conceal his workings from the public. The reality is that only a poor method requires such attention; a fine one, by its beauty, conceals itself. We might call this rule the coquetry of good technique.\n\nOne part of the magic performance from that evening in London well illustrates this point. A shy-looking woman from the audience was invited up onto the stage. At its center, on a pedestal, stood a glass bowl containing large multicolored buttons. The buttons, we were told, totaled one hundred in all. Following instructions, the woman dipped her hands into the bowl, netting as many of the buttons as she wished. Unclenching her fingers, she next poured the buttons onto a tray and covered them with a tea towel. The magician approached the tray, peeked under the towel for all of two seconds, and then turned to the public.\n\n\"Seventy-four,\" he declaimed.\n\nThe woman duly proceeded to count the scattered buttons on the tray, one by one. It took quite some time. After a minute or so, she prodded the final button and the expression of her face lengthened. There were precisely seventy-four buttons on the tray. Gasps popped around my ears, followed immediately by a brisk weather front of applause. The \"counting buttons\" trick was a highlight of the show.\n\nI imagine some people applauded the magician for mental powers that flirted with the supernatural. To recognize seventy-four items (and not mistake them for seventy-three, or seventy-five) in the space of two seconds is, it must be said, quite something. Neurologists tell us that the human brain \"subitizes\" (counts at a glance) no better than in fours or fives. This figure remains constant across all categories of people, irrespective of training or synaptic quirks\u2014neither mathematicians nor autistic savants exceed it. In two seconds, even the most practiced eye can enumerate only eight or ten, and no higher.\n\nThe explanation of divination did not cross my mind; even admitting that it was possible (which I do not), it would have felt somewhat disappointing. A laborious counting of every button, even if accomplished at the most remarkable clip, would be totally lacking in finesse or beauty. I groped with my imagination for the magician's insight.\n\nHow might a person count some (relatively) large quantity in no time at all? This question stayed with me that night as I tossed and turned in bed, finally to sleep. In my dreams the gleaming transparent globe and its outsize buttons and the shy woman holding the tray all returned to me. I looked and looked but failed to see.\n\nEarly in the morning I woke to a beautiful sensation of absolute clarity. The night had seemingly done its work. Now every moment of the magician's trick, from beginning to end, made perfect sense. Had I unveiled the trick's machinery? I could not say for sure. In its simplicity and economy the solution felt inevitable, but I have no idea whether it was in fact the magician's or merely my own. In any event my mood that morning was immediately elevated. I wanted, like Archimedes in his bath, to leap and cry, \"Eureka!\" Quite possibly I did. I felt like a mathematician who receives the sudden, shocking ecstasy of a proof.\n\nOf the school of nocturnal thoughts that swam anonymously through my brain, one image had stuck. It was a modest household object that I used every day: my kitchen scale. Now to the question \"how might a person count some large quantity in no time at all?\" came at once the delicious answer: weigh it! What if the identical buttons weighed exactly one gram each? And what if the glass bowl's pedestal concealed a weighing scale? When the woman lifted seventy-four buttons from the bowl, the reading (displayed backstage) would fall instantly from \"100\" to \"26.\" This number, via an earpiece or some prearranged signal, could then be relayed to the performer. That the math involved might be the simplest of simple subtractions only accentuates the solution's charm.\n\nThis pure beauty that we call mathematical, and that we find in games, music, and magic tricks, is something like a rumor or a longing that lingers in the person, hinting at significance and depth. We go back to it time and time over: it is beautiful, because it stays. It is we who change.\n\nProblems, in magic or mathematics, are wonderful things. Without problems, we would have no proofs, and the shimmering pleasure of elucidation is a thing of beauty. Einstein's equations possessed this special quality in abundance, of course. E = mc\u00b2 (energy is equal to mass times the speed of light squared) answered riddles\u2014such as the behavior of light\u2014that most other scientists had not even seen.\n\nI have spoken about mathematical beauty with hardly any reference to numbers, but, of course, numerical problems also provide many instances of beauty. A personal example, from arithmetic, is the multiplication 473 \u00d7 911. The solution\u2014430,903\u2014may at first sight appear merely banal. Its repeating threes and zeroes, however, reversed as in a mirror, hint at some attractive pattern lurking beneath. And so we begin to loiter around the answer. With a closer look, we might observe the relationship: 903 \u2212 430 = 473. If now we modify the original question slightly, 473 \u00d7 910, simplifying the sum, we reach the answer: 430,430. And we ask ourselves: how is this possible? Again we return to the original problem and dissect its numbers. 473 equals 43 \u00d7 11. The number 910 comprises 7 \u00d7 13 \u00d7 10. We toy with these smaller constituents, until at last we discover that our original sum equates to (430 \u00d7 1001) + (43 \u00d7 11).\n\nA further illustration of this numerical beauty can be found within the primes. The number 75,007 (as it happens, a rather chic Parisian postcode) poses the problem of whether any smaller number evenly divides it. In other words, is the number 75,007 prime? Deceptively simple to formulate, this question proves to be treacherously difficult to answer. As with the sum above, we must grapple with the number until its secret finally yields.\n\nWe begin by assuming (the odds are on our side) that the number 75,007 is composite\u2014that some smaller numbers exist that will cleanly divide it. Not an even number, it cannot be divisible by 2 (the smallest prime). The number 75, we note, divides into 3 and 5 (the next primes), but not when followed by two zeroes and a seven. We might think of 75,007 as a house on a very lengthy street, and observe that, sixty-eight doors down, number 75,075 patently divides by 1,001 (and therefore, by 7 and 11 and 13). But 68 can be split only into two and then a further two to leave seventeen.\n\nImagine the mathematician, pencil in hand, attempting to winkle out the number's factors. The insight resists him; he wonders whether it will ever come. He abandons his desk, pacing the floor, and he marches along the mental street and its numbered houses. And then, all of a sudden, the dizzying thought strikes him: 75,007 can be expressed as 74,900 + 107, or (10,700 \u00d7 7) + 107, or more precisely still (107 \u00d7 100 \u00d7 7) + 107. His blood leaps with joy to recognize the repeated factor: 107. On a crisp sheet of paper, he writes: 75,007 = 107 \u00d7 701.\n\nHuman beings' quest for meaning is perpetual; lack of meaning is offensive to the mind, and whatever the scale of the problem, a solution is a thing of beauty. Einstein's equations solved problems such as \"What do we mean by the words 'time' and 'mass'?\" A mathematician could tell us that the number 75,007 means to travel from 0 to 107, and then repeat the same distance successively 701 times. Other meanings, like those found in music or cricket, while more intimate and inexpressible, can prove just as powerful. Where chaos is subdued and the arbitrary averted, there lies beauty, and it is all around us.\n\nI recall an afternoon I once spent at a summerhouse belonging to friends. We had just returned, tired and hungry, from a long hike among the encircling hills. One of my friends switched on a little radio. We sat here and there around the living room, with views out to the sea, half listening and half talking. The broadcaster was reading out the week's letters sent in by listeners to the program. Among their litany of compliments and complaints, he recited a short puzzle sent in by a longtime listener in the north.\n\n\"A remarkable species of water lily doubles in size every day. If the lily covers an entire lake in a period of 30 days, after how many days will it cover half of the lake?\"\n\nOur various discussions slowed, then continued as before. Someone switched off the radio. The face of the friend opposite me gradually darkened, and her replies became wayward and staccato. Other voices took up the slack. It seemed no one else had taken the problem of the lily to heart.\n\nMinutes passed. My friend cast sidelong glances at the walls and the windows and the flower-lit hills they framed. Her blue eyes narrowed to a squint. Kitchen noises filled the house, followed by the chattering distribution of hot teacups, exhaling steam. Her untouched drink trembled when she struck the table absentmindedly with her shin.\n\nAnd then I saw it. Suddenly, her features shone. \"Twenty-nine,\" she said, with a broad smile. If the lily doubles every day, it will also halve every day when we track its history. With a size of 1 (for \"one lake\") after 30 days, it had a size of 0.5 after 29 days, 0.25 after 28 days, 0.125 after 27 days, and so on.\n\nThe moment had surprised her, she said. It had come right out of the blue. In this moment, I saw my friend behold the astonishing beauty of mathematics.\n\n# Sixteen\n\n# A NOVELIST'S CALCULUS\n\nThe history of the world, Tolstoy said, is the history of little people. Leo Nikolayevich himself, though, was a giant of a man. Just shy of six feet, he was taller than most of his contemporaries. Stronger, too. He could lift 180 pounds with a single hand. He dressed his muscles plainly, in a peasant's smock, with a belt that cinched the small of his back. His ego-enhancing arguments were just as sturdy. Always resistant to the thinking of his day, he denounced historians as hero worshipers. In _War and Peace,_ over one thousand pages long, he launched his most sustained attack. His primary weapon was drawn from mathematics.\n\nCalculus was by no means a novel idea in Tolstoy's time. Its \"inventors,\" Isaac Newton and Gottfried Leibniz, in the late seventeenth century, were refining theories that had been in development since the time of the ancient Greeks. As geometers study shape, the student of calculus examines change: the mathematics of how an object transforms from one state into another, as when describing the motion of a ball or bullet through space, is rendered pictorial in its graphs' curves. In these curves, smooth and subtle, girding the infinitesimal movements behind every human life, Tolstoy thought he saw the blindness of contemporary historians.\n\nAlongside formidable intellectual powers, he certainly had a head for exotic ideas. I think of the wackier pronouncements: the dismissal of Shakespeare as a lousy poet, of Darwinism as a transient fad, of marriage as legalized fornication. Like Thomas Jefferson, he took a pair of scissors to the New Testament, trimming its pages of every one of its miracles. His cult of simplicity (as G. K. Chesterton later called it) welcomed flocks of disciples to his estate: men and women, young and old, dressed in bedsheets and bark sandals, who followed his every step and hung on his every word. But more ambitious, more inventive, and more subversive than any of these was the novelist's view of history itself as a kind of calculus.\n\nWe find this view studded throughout the pages of _War and Peace,_ in those portions that resemble the tight and intense arguments of a pamphleteer. It so happens that they are the same parts that most modern readers tend, perhaps understandably, to skip. But this less diligent reader misses a crucial bedrock of Tolstoy's work.\n\n> The movement of humanity, arising as it does from innumerable arbitrary human wills, is continuous. To understand the laws of this continuous movement is the aim of history... only by taking infinitesimally small units for observation... and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.\n\nCalculus, which Tolstoy defined as \"a modern branch of mathematics having achieved the art of dealing with the infinitely small,\" offered him a vocabulary in which to voice his disagreement with many historians. He denounced their lamentable tendency to simplify. The experts stumble onto a battlefield, into a parliament or a public square, and demand, \"Where is he? Where is he?\" \"Where is who?\" \"The hero, of course! The leader, the creator, the great man!\" And having found him, they promptly ignore all his peers and troops and advisers. They close their eyes and abstract their Napoleon from the mud and the smoke and the masses on either side, and marvel at how such a figure could possibly have prevailed in so many battles and commanded the destiny of an entire continent. \"There was an eye to see in this man,\" wrote Thomas Carlyle about Napoleon in 1840, \"a soul to dare and do. He rose naturally to be the King. All men saw that he was such.\"\n\nBut Tolstoy saw differently. \"Kings are the slaves of history,\" he declared. \"The unconscious swarmlike life of mankind uses every moment of a king's life as an instrument for its purposes.\" Kings and commanders and presidents did not interest Tolstoy. History, his history, looks elsewhere: it is the study of infinitely incremental, imperceptible change from one state of being (peace) to another (war).\n\nThe experts claimed that the decisions of exceptional men could explain all of history's great events. For the novelist, this belief was evidence of their failure to grasp the reality of an incremental change brought about by the multitude's infinitely small actions. Out of a need to theorize, to locate \"causes,\" the historian privileges one series of events and examines it apart from all the others. Why, all of a sudden, had Napoleonic France and Tsarist Russia rushed to war? What drove millions of men, men who licked their plates and read stories to their sons and worried about their looks, to suddenly thieve and crush and slaughter one another? Napoleon overreached, a victim of his own pride and mania, says one expert. He let himself go, growing fat and moody. With successive battle victories under his belt, he fell inevitably to thinking himself invincible. No, no, says another historian, you forget how weak and high-strung was the Tsar Alexander. Such weakness certainly invited a military strike. The long-standing economic embargoes in Europe, suggests a third, led to strained relations between the different peoples. A fourth points out that hundreds of thousands of soldiers obtained gainful employment. Napoleon himself, near the end of his life, is said to have put the war down to the intrigues of the British.\n\nNaturally, not all these \"causes\" can be right, and some are even mutually contradictory. Either Napoleon's decision to invade Russia was impetuous and instinctive, or else it was carefully calculated (against Russian weakness) and deliberate (to keep his forces busy). Either Russian weakness impelled the attention of France's army, or else Napoleon's mania invented such weakness for its own purpose. Either the war resulted from French initiative, or from British interference.\n\nAs a Briton who lives in France, I see how each nation selects its own causes, and edits them convincingly into its own version of history. In Britain, Napoleon's name is synonymous with tyranny and a small comic man's delusions of grandeur. In France, _au contraire,_ he is a revolutionary who stood up for the new Republic against the hostile monarchies of Europe. The puffed-up Napoleon with \"small white hands,\" as depicted by Tolstoy, is, of course, a Napoleon from the Russian perspective.\n\nThis third Napoleon, as conceived by Tolstoy, had at least one cardinal virtue: he knew to keep out of the way of his soldiers, not to tread on anyone's boots, to give a fair impersonation of someone who is in command. It is the soldiers who shoot, and stab, and cough, and groan and bleed. They constituted the vast majority of the French emperor's army, but they issued not a single command. The commands came from the officers above the soldiers, who took them in turn from the generals above the officers, who took them from the commander-in-chief above the generals. The most important commands always come from those who participate least in the physical action. Consequently, most of these commands, thousands of them, not corresponding to conditions \"on the ground\" at the moment and in the place that they finally filtered down to the troops, were never executed. They did not coincide with the reality of circumstances that remained beyond the chief's control. As far as Tolstoy is concerned, then, to say that Napoleon invaded Russia is only to say that a few of his commands, out of the thousands that came to nothing, coincided with certain broader events between the peoples of France and Russia in the year of 1812.\n\nWhat were these broader events \"on the ground\"? As the novel's calculus analogy suggests, they were innumerable, infinitesimal. At a given moment, in a given place, the wishes and desires and intentions of hundreds and thousands of people temporarily coalesced. Tolstoy illustrates such a moment in the life of a backwoods Russian region.\n\n> In the vicinity of Bogucharovo were large villages belonging to the crown or to owners whose serfs... could work where they pleased... In the lives of the peasantry of those parts the mysterious undercurrents in the life of the Russian people, the causes and meaning of which are so baffling to contemporaries, were more clearly and strongly noticeable than among others. One instance, which had occurred some twenty years before, was a movement among the peasants to emigrate to some unknown \"warm rivers.\" Hundreds of peasants... suddenly began selling their cattle and moving in whole families toward the southeast. As birds migrate to somewhere beyond the sea, so these men with their wives and children streamed to the southeast, to parts where none of them had ever been. They set off in caravans, bought their freedom one by one or ran away, and drove or walked toward the \"warm rivers.\" Many of them were punished, some sent to Siberia, many died of cold and hunger on the road, many returned of their own accord, and the movement died down of itself just as it had sprung up, without apparent reason. But such undercurrents still existed among the people and gathered new forces ready to manifest themselves just as strangely, unexpectedly, and at the same time simply, naturally, and forcibly. Now in 1812, to anyone living in close touch with these people it was apparent that these undercurrents were acting strongly and nearing an eruption.\n\nContemporary historians, to believe Tolstoy, took no notice of these \"undercurrents\" in the life of a people. They failed to see the ocean of history for the waves. Aware only of the tides they called \"causes,\" they ignored the vast depths from which these ripples emerge. A man called Napoleon has an impetuous character; six months later Moscow is under siege. The historian looks at these two situations and asserts a link: namely, that hundreds of thousands of Muscovites fled their homes, and whole battalions of soldiers lost their lives, because of the impetuousness of this single man called Napoleon. Or, the historian notices that in, say, Liverpool and London there occurred local riots caused by a shortage of bread and that within a year Russian troop masses were fending off the French. Entire theories, each more elaborate and ingenious than the next, are spun in order to thread the rowdy fisticuffs in these English cities with the subsequent slashing and burning and murdering at Borodino.\n\nI have, admittedly, offered only gross approximations of these historical theories, and of course they are often far more complex, discerning many separate causes\u2014one cause after another\u2014of a war. The temperament of a man called Napoleon is only one cause, they say, and the bread shortage in a city like Liverpool another. Often they will find a third, and perhaps a fourth or fifth, cause to supplement the first and second. All the same, Tolstoy's chief objection remains. Historians tend by their very nature to adopt a flawed approach, he argued, because a mass conflict can no more be reduced to a handful of causes than can a ship's course be reduced to a few waves. Between a French port and a Russian port lie innumerable points in the sea: why label the fifteen thousand four hundred and third point, say, or the seventy-one thousand nine hundred and sixty-eighth point, as being ultimately responsible for the ship's arrival?\n\nAn equivalent mistake would be to inquire of an age-beaten man, in which hour of your life was the blow delivered? Which blow? Why, the blow that loosened your teeth, and broke your bones, and thrashed your skin. Of course, there is no sense in such a question. The flow of time erodes patiently, continuously. What, then, could our elderly man say by way of a reply? He might recall that during a particularly hot summer night in 1968 he rolled out of bed and broke a shin. Perhaps he would smell once more the harsh carbolic soap with which, as a child growing up in the 1940s, he scrubbed his face. A game with his grandson in 1997 might come to his mind, in which a hard rubber ball accidentally struck his jaw. But none of these events, not less in combination, could truly help us to understand the elderly man's present condition.\n\nChange appears to us mysterious because it is invisible. It is impossible to see a tree grow tall or a man grow old, except with the precarious imagination of hindsight. A tree is small, and later it is tall. A man is young, and later he is old. A people are at peace, and later they are at war. In each case, the intermediate states are at once infinitely many and infinitely complex, which is why they exceed our finite perceptions.\n\nEven a dramatic change can thus be accomplished without our knowledge. A friend once related to me the following illustrative tale. A friend of my friend inherited a house in southern Europe. This house contained many fine pieces of furniture and works of art. Every summer, she flew to Europe and lived for several months in the house among these objects. She sat on the same cushions, walked past the same paintings, and heard the same ticking of the grandfather clock. The house's upkeep she entrusted to a small and loyal staff, so that its rooms were always cleaned and polished and in good repair whenever she came through the door. And then one day, several summers following her inheritance, her kid sister came to stay. The sister felt excited; she had heard many good things about the house and was longing to see it. But this feeling quickly gave way first to curiosity, then to confusion, and finally to astonishment. A distinguished-looking chair in the hall, upon closer inspection, revealed itself to be cheap and rickety. Removed from its frame, the painting that hung above the fireplace flapped poster-thin. The marble-colored statuette in the guest bedroom gave off the unmistakable whiff of plastic. Fakes! Frantically, the sisters raced from room to room, until the whole place was left upside down. Every chair, every vase, every painting, virtually everything in the house\u2014over one hundred items\u2014unbeknownst to this woman, had been meticulously replaced. Little by little, piece by piece, a wily member of the staff had stolen the house away from under her nose.\n\nSometimes, revolutions turn over a country in the same way that this staff member turned over the house. Imperceptibly, dissidence grows across a land long before the dictator calls out his tanks. And as recent events in the Arab world remind us, nobody predicts a revolution before it happens and nobody controls it once it is under way. \"Why war and revolution occur we do not know,\" affirms Tolstoy. \"We only know that to produce the one or the other action, people combine in a certain formation in which they all take part.\" Hearing the sudden drumbeat of shoes, the roar of voices, the upturned faces flushed with anger, the almighty, most wise, and beneficent ruler does not comprehend. In his incomprehension, he asks the same question that many of his fellow dictators end up asking themselves: from where did they come? All these people with their fists and their voices. He shakes his head in disbelief. And yet, the answer is simple because only one answer is possible. Simply put, the people were always there: on the streets, in the mosques, in the bazaars. Only now, this distributed mass of people suddenly and noisily combines. Only now do the man without work, and the woman without dignity, and the teenager without anything to eat make themselves seen and heard.\n\nWhat is the power that moves peoples? Not the power of rulers, says Tolstoy, or for that matter the power of ideas. It is ineffable, invisible.\n\n> History seems to assume that this force is self-evident and known to everyone. But in spite of every desire to regard it as known, anyone reading many historical works cannot help doubting whether this force, so variously understood by the historians themselves, is really quite well known to everybody.\n\nThis force is the Human Life in which every person\u2014from the lowly peasant Karataev to the emperor Napoleon\u2014participates. It is the \"hidden warmth\" of patriotism experienced by the Muscovites when suddenly confronted with the dramatic threat of foreign invasion. It is the \"chemical decomposition\" of the fleeing mass of French soldiers once their goal (the head-on confrontation in Moscow with an appointed \"enemy\") becomes unattainable. It is the \"force of habit,\" during a salon conversation, which makes Prince Vasili Kuragin say \"things he did not even wish to be believed.\"\n\nRather than assign various degrees of responsibility to this cause or that, Tolstoy proposes that historians pay far greater attention to this force. Moscow's conflagration, which the historians variously explain as a defensive tactic of the Russians (the so-called scorched-earth policy) or the wild vengeance of the invading French, becomes explicable in other terms.\n\n> Moscow was burned because it found itself in a position in which any town built of wood was bound to burn.... A town built of wood, where scarcely a day passes without conflagrations when the house owners are in residence and a police force is present, cannot help burning when its inhabitants have left it and it is occupied by soldiers who smoke pipes, make campfires of the Senate chairs in the Senate Square, and cook themselves meals twice a day.\n\nReaders were shocked by these arguments when the book first appeared (over several editions) in the 1860s. Turgenev denounced the historical reflections as \"charlatanism\" and \"puppet comedy,\" while Flaubert found them merely repetitive. The historian A. S. Norov entitled his review of the book \"Tolstoy's Falsification of History.\" Another historian, Kareev, complained that the novelist wanted to abolish history altogether. When, one and a half centuries later in the year 2000, a Russian publisher edited and released an early draft of the book, it boasted about the lack of all these troublesome elements. \"Half the usual length. Less war and more peace. No philosophical digressions or incomprehensible French. A happy ending.\"\n\nMathematicians, on the other hand, have proved considerably more receptive. Tolstoy's mathematician friend, Urusov, expressed his delight at the analogy with calculus. And more recently, in a 2005 article for the Mathematical Association of America, Stephen T. Ahearn praised Tolstoy's mathematical metaphors as being both \"rich\" and \"deep\" and encouraged math teachers to use them in their classrooms.\n\nWhat, then, might we conclude? Should we even conclude? After all, if Tolstoy is right, his book\u2014like any event in time\u2014cannot be understood with prior assumptions, rules, and theories. Everything has its moment, its context. Earlier, in one state, you began this essay, and now later on, you finish it in another. What do you think? I cannot tell you. In everyone and everything, the process of change always asserts its own meaning.\n\n# Seventeen\n\n# BOOK OF BOOKS\n\nI have walked in my sleep, and talked in my sleep, but I have never written in my sleep. The Icelandic author Gyr\u00f0ir El\u00edasson's short story _\"N\u00e6turskriftir\"_ (\"Nightwriting\") depicts a character whose writer's block suddenly disappears once the lights go out. In a notebook lifted from his bedside table, he begins to write down words, sentences, and even whole stories while he dreams.\n\n> The days passed by all the same; he could not write... but at night he wrote; nearly every night... his wife knew not to wake a sleepwriter, so she lay [in the bed] and watched the expression of his back, how he wrote with amazing confidence with the notebook on his knees. (My translation).\n\nSomething about El\u00edasson's tale touches a chord in me. I think it has to do with confronting the infinity that is every written and unwritten book, including the \"Book of Life\": the infinitely many potential combinations that comprise our days. How does the author select the right word, the right phrase, the right image from among the countless conceivable possibilities? How does each person imagine a new existence; reconfigure the choices that make up another destiny?\n\nSleep on it. Why not? Our dreams contain the infinite. Uninhibited by wakefulness, words and pictures and emotions circulate and combine freely inside our head. Across the centuries, the Unconscious mind has authored some of the greatest works in literature: Goethe and Coleridge are only two of its pseudonyms.\n\nDreams defy our finite scrutiny; too often they evaporate in the narrow light of day. We are left upon waking only with sweet hints of rain and distant echoes of a song, a nose here, a smile there, some tremor of sadness or flicker of joy, a suggestive and beguiling void. Like a book, like a life, where does the explanation start? A dream has no beginning, and therefore no middle and no end.\n\nI dreamed I entered a house and found its inhabitants all lying on the floor. Lying, but talking and laughing and eating together. Lying instead of sitting. It was like a scene from a book that I had not read and that had not been written. How many such scenes are there to occupy our dreams, our lives, the pages of a book? Infinitely many.\n\nLike El\u00edasson's sleepwriter, Anton Chekhov faithfully nurtured a little notebook throughout his remarkable career, though we can suppose that he used his mostly during waking hours. Filled with his day-to-day observations of existence's minutiae, the pages preserve glimpses of \"ordinary\" life's infinite permutations.\n\n\"Instead of sheets\u2014dirty tablecloths.\"\n\n\"In the bill preserved by the hotelkeeper was, among other things: 'Bugs\u2014fifteen _kopecks._ ' \"\n\n\"If you wish women to love you, be original; I know a man who used to wear felt boots summer and winter, and women fell in love with him.\"\n\nThis endless variety inspired many of Chekhov's tales. In \"The Lottery Ticket,\" a middle-class couple envisage the potential lives that would follow a jackpot.\n\n> The possibility of winning bewildered them.... \"And if we have won,\" he said, \"why it will be a new life, it will be a transformation!\"... Pictures came crowding on his imagination, each more gracious and poetical than the last. And in all these pictures he saw himself well fed, serene, healthy, felt warm, even hot!... \"Yes, it would be nice to buy an estate,\" said his wife, also dreaming.... Ivan Dmitritch stopped and looked at his wife. \"I should go abroad you know, Masha,\" he said. And he began thinking how nice it would be in late autumn to go abroad somewhere to the south of France, to Italy, to India!\n\nWriting half a century after his fellow countryman, another precocious note taker, Vladimir Nabokov, composed his novels in two alphabets and three languages (Russian, French, and English). From a blank vivid room, anagrams, puns, and neologisms spilled out. He compared composing a story with fitting together the pieces of a jigsaw puzzle.\n\n> Reality is a very subjective affair.... You can get nearer and nearer, so to speak, to reality; but you never get near enough because reality is an infinite succession of steps, levels of perception, false bottoms, and hence unquenchable, unattainable. You can know more and more about one thing but you can never know everything about one thing: it's hopeless.\n\nLike a jigsaw puzzle, like a dream, Nabokov's novels emerged nonlinearly; he often wrote the middle of a story last. Chapter 8 of a draft manuscript might appear long before chapter 7 or chapter 3. He would frequently write a new story backward, starting out from its final lines.\n\n_Lolita,_ Nabokov's most famous (and infamous) novel, began life on a long series of three-by-five-inch index cards. He sketched out the story's closing scenes first. On subsequent cards Nabokov jotted down not only paragraphs of text but also plot ideas and other bits of information; on one, a chart of statistics on the average height and weight of young girls; on another, a list of jukebox songs; on a third, an illustration of a revolver.\n\nEvery so often Nabokov would rearrange his index cards, searching for the most promising combination of scenes. The number of possible permutations would have been immense. Three of Nabokov's cards can be rearranged in a total of six different ways\u2014(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1)\u2014while ten cards (equivalent to between two and three printed pages in a book) would be capable of permuting into more than three and a half million sequences. To compose only four or five pages (equivalent to the contents of about fifteen index cards) would require a choice from among some 1.3 trillion variations. _Lolita_ runs to sixty-nine chapters and over 350 pages, which means that the number of its potential versions exceeds (by an almost unimaginable margin) the number of atoms that make up our universe.\n\nOf course, many of these potential _Lolita_ s would simply not have been viable. And yet among the bewildering, nonsensical, or ham-fisted editions, readable alternatives must exist. How many? A hundred? A thousand? A million? More. Many more. Publishers could produce enough of them to give every reader on the planet his or her very own _Lolita._ In one, the famous opening couplet, \"Lolita, light of my life, fire of my loins. My sin, my soul,\" would appear halfway down page 39\u2014perhaps replaced with a line that Nabokov placed in his chapter 2: \"My very photogenic mother died in a freak accident (picnic, lightning)...\" In another reader's edition, the couplet shows up at the top of page 117. This _Lolita_ begins instead, \"I saw her face in the sky, strangely distinct, as if it emitted a faint radiance of its own.\" In a third version, the original couplet serves as the story's closing lines.\n\nFor all I know, some of these incalculably many editions were actually published, each with their subtle yet striking alterations. Perhaps this would explain why the _Atlantic Monthly_ 's reviewer called the book \"one of the funniest serious novels I have ever read,\" the _Los Angeles Times_ declared it \"a small masterpiece... an almost perfect comic novel,\" and the _New York Times Book Review_ announced, \"technically it is brilliant... humor in a major key,\" whereas Kingsley Amis read a book leading to \"dullness, fatuity, and unreality\" and Orville Prescott, writing for the _New York Times,_ found the story \"dull, dull, dull.\"\n\nWhich _Lolita_ did they read?\n\nIt is the writer and reader together who compose their infinite tale. The Argentine writer Julio Cort\u00e1zar created a novel in which he made this principle explicit. _Rayuela_ ( _Hopscotch_ ) was published fifty years ago, not long after _Lolita._ It contains 155 chapters (over some 550 pages), which can be read in two distinct ways. Either the reader starts at chapter 1 and continues reading linearly until the end of chapter 56 (the chapters and two hundred pages that remain being considered \"expendable\"), or else he begins at chapter 73, then turns back to chapter 1, continues to chapter 2 before jumping forward to chapter 116, then back to chapter 3, forward to chapter 84, and so on back and forth between the chapters according to a \"Table of Instructions\" at the front of the book.\n\nIn one of his \"expendable\" chapters, Cort\u00e1zar describes the book's goal:\n\n> It would seem that the usual novel misses its mark because it limits the reader to its own ambit; the better defined it is, the better the novelist is thought to be. An unavoidable detention in the varying degrees of the dramatic, the psychological, the tragic, the satirical, or the political. To attempt on the other hand a text that would not clutch the reader but which would oblige him to become an accomplice as it whispers to him underneath the conventional exposition other more esoteric directions.\n\nAs Cort\u00e1zar's accomplice, we follow the novel's hero\u2014an Argentine bohemian\u2014through the streets of Paris as he contemplates his life and its inexhaustible potential paths. We begin the book either at chapter 1, like this: \"Would I find La Maga?\" Or else at chapter 73, on page 383: \"Yes, but who will cure us of the dull fire, the colorless fire that at nightfall runs along the Rue de la Huchette...\"\n\nTurning the pages, reading different stories. For instance, the reader who begins at chapter 1 will soon reach this line in the fourth chapter: \"She] picked up a leaf from the edge of the sidewalk and spoke to it for a while.\" For the other reader, however, \"chapter 4\" is really the seventh chapter of the story, preceded by the sixth chapter, which is labeled \"chapter 84.\" In this chapter, on page 405, he reads, \"I keep on thinking of all the leaves I will not see, the gatherer of dry leaves, about so many things that there must be in the air and which these eyes will not see... there must be leaves that I will never see.\" These lines enrich the second reader's understanding of the woman who, several pages later, [here, will pick up a leaf from the edge of the sidewalk and speak to it.\n\nA consequence of reading in this way is disorientation; the leapfrogging reader lacks any sense of having completed the book. He reads the final lines on the final page of the physical book long before he comes to any conclusion of the story. Arriving later at the one hundred and fifty-third chapter (labeled \"chapter 131\"), he proceeds to the following chapter (labeled \"chapter 58\"), only to discover that he should return again to chapter 131. An interminable loop between the two \"final\" chapters appears. What is more, assuming he has kept count, the reader notices that the chapters\u2014read in this order\u2014total 154. One of the chapters\u2014\"chapter 55\"\u2014is absent from the list.\n\n_Hopscotch_ 's structure demands that readers make their own sense of the story. One might decide to read the chapters consecutively, but in descending order, starting at chapter 155. Another decides to read all the even chapters before the odd: 2, 4, 6, 8... 1, 3, 5, 7... A third does the reverse, reading all the odd chapters before the even. A fourth reads only the prime-numbered chapters: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31... finishing at chapter 151 (thirty-six chapters in all). A fifth begins at the first chapter, then reads the third (1 + 2), turning next to the sixth (1 + 2 + 3), followed by the tenth (1 + 2 + 3 + 4) and so on.\n\nJust when the plucky reader has attained the end of one story, another story beckons him to pick up its pages and start again. The book of ascending chapters becomes a book of descending chapters. The book of odd-number chapters becomes a book of even-number chapters. Every reading differs; every reading offers something new. It is impossible to dip into the same book twice.\n\nI am reminded of Nabokov's view that we can never read a book: we can only reread it. \"A good reader, a major reader, an active and creative reader,\" says Nabokov, \"is a rereader.\" Initial readings, he explains, are always laborious, a \"process of learning in terms of space and time what the book is about, this stands between us and artistic appreciation.\"\n\nThink of the countless stories of Chekhov, of the innumerable editions of _Lolita_ and _Hopscotch,_ which lie before every reader's eyes unnoticed, unloved, unread.\n\nFlaubert, in a letter to his mistress, wrote, \"How wise one would be if one knew well only five or six books.\" It seems to me that even this figure is an exaggeration. To learn infinitely many things, we would only ever need perfect knowledge of one book.\n\n# Eighteen\n\n# POETRY OF THE PRIMES\n\nArnaut Daniel, whom Dante praised as _\"il miglior fabbro\"_ (\"the better craftsman\"), sang his love poems in the streets of twelfth-century southern France. Of his life not much is known, but I find it tempting to link a brief and rare report about the troubadour with the _sestina_ form of poetry (six stanzas, each containing six lines, plus a concluding half stanza) that he invented.\n\nRaimon de Durfort, a contemporary of Arnaut, called him \"a scholar undone by dice.\" This alleged acquaintance with gambling suggests a possible influence for the sestina's form. A die, as everyone knows, has six faces. The throw of a pair of dice creates a range of outcomes amounting to thirty-six, which is the total number of lines in the poem's six stanzas. So far as I can tell, no one has made this connection between the sestina and the die before\u2014maybe because it is a connection little worth making. I leave it for the reader to decide.\n\nUnusually, the sestina does not run on rhyme, symbolism, alliteration, or any other of the poet's typical devices. Its power is the power of repetition. The same six words, one at the tail of each line, persist and permute across every stanza (in the final half stanza, the words appear two to a line) according to an intricate pattern.\n\n> First stanza: 1 2 3 4 5 6\n> \n> Second stanza: 6 1 5 2 4 3\n> \n> Third stanza: 3 6 4 1 2 5\n> \n> Fourth stanza: 5 3 2 6 1 4\n> \n> Fifth stanza: 4 5 1 3 6 2\n> \n> Sixth stanza: 2 4 6 5 3 1\n> \n> Conclusion: 2 1, 4 6, 5 3\n\nWhich is to say, the final word in line six of the first stanza (1 2 3 4 5 **6** ) reappears as the last word of the next stanza's opening line ( **6** 1 5 2 4 3), and at the close of the second line of stanza three (3 **6** 4 1 2 5), and so on. Here follows an example, from Dante (employing the six words \"shadow,\" \"hills,\" \"grass,\" \"green,\" \"stone,\" and \"woman\").\n\n> I have come, alas, to the great circle of shadow,\n> \n> to the short day and to the whitening hills,\n> \n> when the color is all lost from the grass,\n> \n> though my desire will not lose its green,\n> \n> so rooted is it in this hardest stone,\n> \n> that speaks and feels as though it were a **woman.**\n\n> And likewise this heaven-born **woman**\n> \n> stays frozen, like the snow in shadow,\n> \n> and is unmoved, or moved like a stone,\n> \n> by the sweet season that warms all the hills,\n> \n> and makes them alter from pure white to green,\n> \n> so as to clothe them with the flowers and grass.\n\n> When her head wears a crown of grass\n> \n> she draws the mind from any other **woman,**\n> \n> because she blends her gold hair with the green\n> \n> so well that Amor lingers in their shadow,\n> \n> he who fastens me in these low hills,\n> \n> more certainly than lime fastens stone.\n\n> Her beauty has more virtue than rare stone.\n> \n> The wound she gives cannot be healed with grass,\n> \n> since I have traveled, through the plains and hills,\n> \n> to find my release from such a **woman,**\n> \n> yet from her light had never a shadow\n> \n> thrown on me, by hill, wall, or leaves' green.\n\n> I have seen her walk all dressed in green,\n> \n> so formed she would have sparked love in a stone,\n> \n> that love I bear for her very shadow,\n> \n> so that I wished her, in those fields of grass,\n> \n> as much in love as ever yet was **woman,**\n> \n> closed around by all the highest hills.\n\n> The rivers will flow upwards to the hills\n> \n> before this wood, that is so soft and green,\n> \n> takes fire, as might ever lovely **woman,**\n> \n> for me, who would choose to sleep on stone,\n> \n> all my life, and go eating grass,\n> \n> only to gaze at where her clothes cast shadow.\n\n> Whenever the hills cast blackest shadow,\n> \n> with her sweet green, the lovely **woman**\n> \n> hides it, as a man hides stone in grass.\n\nAn air of expectancy permeates the text: since the reader knows what is coming, the poem must rise to the challenge of delivering surprise. The sestina plays with meaning, conferring new aspects, in changing contexts, on the same word. A tension between the law of the numerical pattern and the liberty of the author is ever present, ever palpable.\n\nArtists and mathematicians alike have been drawn to the sestina's numerous properties. In their wonderful book, _Discovering Patterns in Mathematics and Poetry,_ the mathematician Marcia Birkin and poet Anne C. Coon compare the rotation of words in a sestina to the shifting digits in a cyclic number.\n\nCyclic numbers are related to primes. Division using certain prime numbers (such as 7, 17, 19, and 23) produces decimal sequences (the cyclic numbers) that repeat forever. For example, dividing one by seven (1\/7) gives the decimal expansion 0.142857142857142857... where the six digits 142857\u2014the smallest cyclic number\u2014continue around and around in a never-ending ring dance.\n\nIf we multiply 142,857 by each of the numbers below 7, we see that the answers are permutations of the same six digits.\n\n> 1 \u00d7 142857 = 142857\n> \n> 2 \u00d7 142857 = 285714\n> \n> 3 \u00d7 142857 = 428571\n> \n> 4 \u00d7 142857 = 571428\n> \n> 5 \u00d7 142857 = 714285\n> \n> 6 \u00d7 142857 = 857142\n\nIn this instance, the digit 7 at the end of the first answer (14285 **7** ) reappears in the fourth position of the second answer (285 **7** 12) and in the fifth position in the third (4285 **7** 1) and so on. Each digit rotates through every answer, changing place at every turn, like the end words in the stanzas of a sestina.\n\nHazard has no place in the sestina. The end words in every stanza fall at once into line, the position of each determined before the poem begins. Algebraically, we can describe the sestina's structure (from the second stanza onward) like this.\n\n> {n, 1, n \u2013 1, 2, n \u2013 2, 3} where _n_ refers to the number of stanzas (six).\n\nHow our medieval troubadour concocted this clever pattern is unknown. His deep familiarity with the rhythms of words and music likely helped. In one of his few surviving songs he says:\n\n> Sweet tweets and cries\n> \n> and songs and melodies and trills\n> \n> I hear, from the birds that pray in their own language,\n> \n> each to its mate, just as we do\n> \n> with the friends we are in love with:\n> \n> and then I, who love the worthiest,\n> \n> must, above all others, write a song contrived so\n> \n> as to have no false sound or wrong rhyme.\n\nOf course, that Arnaut plumped for six stanzas, and not five or seven, probably owes as much to chance as the outcome of the toss of a die. In fact, a small number of poets have tried their hands at _tritina_ s (containing three stanzas) and _quintina_ s (containing five), with some success. Raymond Queneau, a French poet with a mathematician's itch for generalization, eager to understand how the pattern works, explored the limits of the form. In the 1960s, he worked out that only certain numbers of stanzas permitted the permutations of the sestina. A four-stanza poem, for example, produced jarring alignments of the same word.\n\n> {n, 1, n \u20131, 2}\n> \n> First stanza: 1 2 **3** 4\n> \n> Second stanza: 4 1 **3** 2\n> \n> Third stanza: 2 4 **3** 1\n> \n> Fourth stanza: 1 2 **3** 4\n\nThe same went for a poem of seven stanzas.\n\n> {n, 1, n \u20131, 2, n \u2013 2, 3, n \u2013 3}\n> \n> First stanza: 1 2 3 4 **5** 6 7\n> \n> Second stanza: 7 1 6 2 **5** 3 4\n> \n> Third stanza: 4 7 3 1 **5** 6 2\n> \n> Fourth stanza: 2 4 6 7 **5** 3 1\n> \n> Etc....\n\nAfter much trial and error, Queneau determined that only thirty-one of the numbers smaller than 100 produced the sestina's pattern. His observation led mathematicians to discover a surprising relationship between the sestina and the primes. Namely, poems containing three or five stanzas behave like the six stanzas in a sestina, because 3 (or 5, or 6) \u00d7 2, + 1, always equals a prime number. For the same reason, sestina-like poems of eleven, thirty-six, or ninety-eight stanzas are all possible, but not those containing ten, forty-five, or one hundred.\n\nSestinas are not the only form of poetry to be shaped by primes. Brief and glancing, haiku poems also derive their strength from these numbers.\n\nThe Japanese have long been disposed to brevity. Inquiries for the name of \"Japan's Shakespeare\" or \"Japan's Stendhal\" will be, in the best of cases, greeted with blank stares. Oriental epics fell into complete neglect at about the same time that the Viking Snorri Sturluson was putting the finishing touches to his saga. Courtiers of the Heian period (dating from the eighth to the twelfth centuries, a period that the Japanese consider a high point in their history) worked up the most extended pieces by concatenating dozens of short verses, by many hands. Dignitaries alone, however, had the right to commence these chain poems by inventing their opening three lines (called the \"hokku\"). Among images of romantic love and soul searching, these opening lines always contained a reference to the seasons, and an exclamation such as _ya_ (\"!\") or _kana_ (\"how...! what...!\"). However, even this carefully wrought convoy of miniature verses became too cumbersome in the end for Japanese tastes, so that generations of mouths gradually eroded them to the triplet of lines that we know today as haiku.\n\nLike the sestina, the haiku uses no rhyme. Its three lines contain five, seven, and five _onji_ (syllables), seventeen in all. Three, five, and seven are the first odd prime numbers. Seventeen, too, is prime.\n\nOne possible (if partial) explanation for this structure is the marked Japanese preference for odd numbers. In the annual _Shichigosan_ (Seven-Five-Three) festival, three-year-old children of both sexes, five-year-old boys, and seven-year-old girls visit shrines to celebrate their growth. Cheerleaders at a sports match clap in three-three-seven beats. Even numbers, meanwhile, are virtual bogeymen. The number two represents parting and separation, while four is associated with death. In several phrases the number six translates roughly as \"good-for-nothing.\"\n\nPrime numbers contribute to the haiku form's elemental simplicity. Each word and image calls out for our undivided attention. The result is an impression of sudden, striking insight, as if the poem's objects had been put into words for the very first time. A sample from the form's most celebrated practitioner, the seventeenth-century poet Matsuo Bash\u014d, illustrates this.\n\n> _Michinobe no_ (The mallow flower)\n> \n> _Mukuge wa uma ni_ (Against the side of the road)\n> \n> _Kuwarekeri_ (Eaten by my horse)\n\nThere exists as well a slightly longer version of the haiku, the _tanka_ form, which complements the haiku's trio of lines with two further lines of seven syllables each (known as the _shimo-no-ku,_ or \"lower phrase\"). The _tanka_ 's total number of syllables\u2014thirty-one\u2014is, once again, a prime.\n\nBash\u014d, whose name today is synonymous with the haiku form, counted various influences on his work, but chief among them was a wandering monk who lived at the time of Arnaut Daniel and wrote some of the finest _tanka_ verse. His name was Saigy\u014d. Something of the master's evocative simplicity is suggested in the following _tanka_.\n\n> _Michi no be ni_ (On that roadside lea)\n> \n> _Shimizu nagaruru_ (Where pure, crystal waters flowed)\n> \n> _Yanagi kage_ (Grew a willow tree)\n> \n> _Shibashi to te koso_ (For a little while I stayed)\n> \n> _Tachidomaritsure_ (There and rested in its shade)\n\nIt seems that the image of Saigy\u014d's willow tree took root in the imagination of many generations of poets. Some five centuries after the poem's composition, Bash\u014d underwent a pilgrimage to its site in the north. In his travel diary he noted that \"[t]he willow about which Saigy\u014d wrote the famous poem still stood by a rice field in Ashino village. As Mr. Koho, who governed this county, always wanted to show the tree to me, I had been anxious to discover its location. I was happy to stop by the tree today.\"\n\nBash\u014d's homage culminated in a haiku dedicated to Saigy\u014d's tree.\n\n> _Ta ichimai_ (Over a whole field)\n> \n> _Uete tachisaru_ (They have planted rice, before)\n> \n> _Yanagi kana_ (I leave the willow)\n\nAs I think of the complicity between poems and primes, perhaps the only surprise is that we should even find it surprising. Viewed one way, the relationship makes a perfect kind of sense. Poetry and prime numbers have this in common: both are as unpredictable, difficult to define, and multiple-meaning as a life.\n\nThis lifelike quality connecting them is too often overlooked. Many poems, it is true, lie mothballed in slim anthologies; many primes languish in mathematicians' sums. Picked and pored over by their experts, they lose the public's attention (and affection) for the academic company they keep.\n\nAnd yet we see her so clearly, Dante's woman, as she wanders through our memory, and Bash\u014d's horse, munching his flower, looks and sounds all too real. Free (as a prime number) from a rhyme's reassurance or a storybook's rules, the images swerve and forestall our expectations\u2014and keep all clich\u00e9s at bay.\n\nPoems and primes are tricky things to recognize. A glance will usually not suffice to tell us if such-and-such a number has factors, or whether a given text contains much meaning. Even old hands can find it hard to tell genuinely felt verse from a trite list of nice-sounding words, or spot a composite number as being merely a pastiche of lesser primes.\n\nDante's sestina, haiku verses, and the hither and thither of the primes\u2014for each we ask ourselves, what does it mean? Are we, in the end, any closer to the woman whose beauty \"has more virtue than rare stone\"? Her face changes with the stanzas, offering us a multiplicity of perspectives. And what of Saigy\u014d's willow tree, which provides at once shade and reflection?\n\nThe same is true of the prime numbers\u2014an ancient mathematical mystery. Thirty-one, the number of syllables in the _tanka,_ is a twin prime (being two apart from its neighbor twenty-nine), and a Mersenne prime\u2014being one less than a power of two: (2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2) \u2013 1\u2014but such labels fall far short of explanation. That is because we do not really know why the prime numbers appear where they do. Many unproven conjectures remain. The reader of poetry and the mathematician are left finally only with hints and fragments, minus any big picture\u2014as in life.\n\n# Nineteen\n\n# ALL THINGS ARE CREATED UNEQUAL\n\nUnlike diabetes or curly hair, poverty rarely skips a generation. A parent's bank balance will often dictate his child's destiny to a far greater extent than his blood. Blond mothers sometimes produce brunet babies; tall men may not always spawn basketball pros; but more than ninety times out of every hundred, the poor beget more poor.\n\nI am the son of poor parents, poor grandparents, poor great-grandparents, and so on. Suffice it to say that more than a few grisly tales of hard times have come down to me. One of these was told to me by my father, back in the early 2000s, not long after I had flown the family nest to serve my apprenticeship in the adult world. I was living in a house south of London that belonged to someone else. It was the first time I had shared with a roommate. The place had little to commend it. Small, out of the way, it was very modestly decorated. In my box of a bedroom, I slept on a sofa bed, pale green, the color of a plant in shade. I subsisted on students' fare: small plates of pasta, sandwiches, and beans on toast.\n\nSome evenings I took calls from home. On one occasion my father rang, and we fell into a long conversation. His chattiness, I must admit, took me rather by surprise. He had never been one for self-disclosure. Why then was he opening up to me? For an eldest son's compassion, for a walk accompanied down memory lane? I do not know. We had been talking about nothing in particular when all of a sudden he said, \"We moved around a lot when I was a kid.\"\n\n\"Sorry?\"\n\n\"My parents, well, my mum and her man. It is a long story.\"\n\nAnd, just like that, in a matter-of-fact voice, he told me his tale. He told it with such simplicity and such precision, not a word or image out of place, that it dawned on me only later that he had certainly long rehearsed this moment in his mind. Even as I listened I knew better than to interrupt him with my questions or comments. He spoke, as we say, \"from the heart\"; this man, my father, telling me, his son, such things as he considered important and valuable for me to hear.\n\nOne scene in particular struck me. As a boy of ten or thereabouts, he and his parents had been returning home one summer evening from a trip to the town fair. The front lawn, they noticed as they approached the house, had been eerily altered. Running ahead, my father was stunned by what he found. Chairs and tables and pots and pans and beds and lamps had been evacuated. The furniture huddled destitute in a big pile. A padlock on the front door barred their entry.\n\nI might have asked, \"Why had the landlord not given my grandparents more time to clear their debt?\" But I did not ask. The thought of the furniture left out in the front yard made a deep impression on my mind. It was the image of a home turned inside out, its intimacy smashed, its innards spewed. How terrible they must have looked, those useless lamps and long bare table legs and the embarrassing tea-colored letters in a half-open desk drawer. The scene felt so vivid that it made my eyes smart.\n\nMy father was born in 1954, the year that followed Queen Elizabeth's coronation. My grandparents' expulsion from their home happened ten years later\u2014in the midst of the swinging sixties, the decade of \"peace and love.\" In their book _The Poor and the Poorest,_ published in 1965, the sociologists Peter Townsend and Brian Abel-Smith estimated that this ten-year period, the first ten years of my father's life, had produced a near doubling in the percentage of Britons living beneath the breadline, rising from 8 percent to 14.\n\nIn 1979, the year of my birth, Peter Townsend published a further study, _Poverty in the UK,_ showing that relative hardship stifled the lives of 21 percent of the population. This figure appears to have remained more or less stable ever since.\n\nA final statistic, this one from 2008, describing the generation after mine: according to the London School of Economics, the household wealth of the top 10 percent of the population now towers one hundred times above that of the lowest 10 percent.\n\nInequality is invidious. It is also universal. No country, to judge from the comparative data, has been spared it. Every land has its share both of hovels and five-star hotels. Talk of a \"classless society\" has, time and time again, proved to be nothing but hot air. Western sympathizers who visited the Soviet Union in the 1930s were disappointed to find that its October Revolution had hardly \"abolished\" the gulf that divided rich and poor. Zero in imperial units, they learned, was still zero under the metric system. Meanwhile beneath their mud-brown uniforms, stiff with starch, the Kremlin's rulers continued to wear the slaughtered emperor's clothes.\n\nBut enough with statistics: I wonder if mathematics can do more about the phenomenon of disparity than simply measure it. I wonder if it can tell us something about what kind of thing disparity is: where does it come from? What makes it grow or shrink? Can mathematical thinking address these kinds of questions?\n\nIt can. Mathematics and money both originated in abstraction. As with mathematics, we owe the concept of money to the ancient Greeks. They who first abstracted \"five\" from the fingers on one hand, first stamped \"five drachmas\" on a metal coin. And just as the concept of \"five\" slipped from the fingers that had described it, becoming applicable to anything\u2014men, crumbs, daydreams\u2014with identical quantity, so a coin's \"value\" exceeded the metal of its composition, capable of transforming into anything agreed as having equal worth.\n\nSubstituting numbers for objects changed the world, for better or worse. At once everything became quantifiable, even the light of the moon, which Aristophanes in one of his plays describes as saving people a drachma's worth of torches per month. Where previously, bartering and the exchange of gifts had settled all Athenian transactions, now most social dealings came down to sums. Reciprocity between citizens gave way to the potentially unlimited accumulation of individual \"wealth.\" We read with a feeling of familiarity Aristotle's lament (in his _Politics_ ) that some doctors turn their skills into the art of making money. Sophocles goes much further, placing in one of his character's mouths a blistering denunciation of money as that which \"lays waste even cities, expels men from their homes,\" and \"thoroughly teaches and transforms good minds... to know every act of impiety.\"\n\nIn its abstraction, money acquired the impersonal neutrality of numbers. Goods no longer embodied the donor's generosity or personality; calculation replaced feelings. Individual autonomy grew, but so too did an egoism that made money the measure of all things. And like abstract numbers, money became invisible. Coins might be concealed far more easily than cows. Lycurgus, ruler of Sparta, found that he could only fight the \"injustice\" of the rich hiding away large sums only by making his iron coins so large and so heavy that ten of them alone required transportation upon a wagon.\n\nBecause numbers can go on forever, money has no limit. Of wealth, Aristophanes tells us, a person can never have enough. Bread and sex and music and courage all sate the preceding appetite, but wealth does not. It is impossible to put a check on moneymaking. If a man receives thirteen coins, he will hanker after sixteen, and possessing them he considers life unbearable unless he now earns forty. Nature, it might be observed, imposes strict boundaries on a person's height and age span, so that in even the most extreme instances no one can rise or fall too far or too short from the rest, whereas no such boundary inhibits money. Think of King Croesus who had so much gold that he gave it away. It was Croesus whom Solon the legislator memorably warned when he said that he who has much, has much to lose.\n\nI am reminded here of the story I first heard on the morning radio two years ago, of an elderly Parisian heiress who had been flattered into great feats of generosity by a much younger man. Of course, the story immediately spread to print. I read the fine details across consecutive pages of my daily paper. The billionaire had allegedly been pampered out of paintings by Munch, Picasso, Matisse, seduced into gifting precious tomes and manuscripts and, all in all, fussed over to the tune of scores of millions of euros in handouts over several years.\n\nPoor thing!\n\nSolon, I should mention, became the first man in history to make laws addressing inequality. We read in Plutarch how the mass of ancient Athenians in Solon's day found themselves in hock to the city's wealthy aristocrats. Some had been sold into slavery or else had handed over their children as collateral, while others had fled with their family into exile. When he was elected chief magistrate, Solon promptly divided the city's populace into categories, granting to each category proportional responsibilities and rights. The highest class consisted of those earning an income in excess of five hundred bushels. In the second class were those citizens able to afford a horse (and therefore to pay a \"horse tax\" upon the purchase). Yoke-of-oxen men comprised the third class, with an annual income between two hundred and three hundred bushels. Freed from the fear of enslavement, the remainder of landless citizens could attend public assemblies for the first time, and sit on juries.\n\nWe can express the distribution of wealth in any society as a formula, using a number _x_ between 50 and 100 such that _x_ percent of the society's wealth belongs to (100\u2013 _x_ ) percent of the population. In a highly egalitarian society (where _x_ equals, say, 55 or 60), 45 percent (or 40 percent) of the people would hold 55 percent (or 60 percent) of the assets. Most Western societies, however, show a far more skewed distribution. Economists have found that _x_ in most developed countries equates to a number in the region of 80, meaning that 80 percent of the society's fortune pads the pockets of only 20 percent of its members.\n\nNaturally, the share will vary from person to person. Money is fickle, always changing owners. That different people have dissimilar wealth is not surprising. What surprises is the scale and constancy of the divide. The economist and mathematician Vilfredo Pareto, who first observed (at the end of the nineteenth century) that 20 percent of Italians owned 80 percent of the nation's wealth, found nearly identical results when he studied the historical data from many other parts of Europe. The distribution of wealth in Paris since 1292, he discovered, had hardly moved at all. Later researchers confirmed these findings.\n\nBecause most men and women, for want of resources, remain at the bottom, the elite, for want of competition, remain at the top. The poorest expend all their energies simply to keep body and soul together. I think of Degas's painting of the two peasant women ironing: one is anonymous, hunched over her iron; the other yawns candidly, her mouth the shape of an \"O.\" The yawn distorts her features, undoing the individuality of her face.\n\nThe Spartan Lycurgus, we saw, made money as big as men; imagine, an instant, men as big as their money. Picture the difference between a miller and a millionaire. The man who mills grain owns perhaps no more than one-thousandth of the rich man's fortune: the millionaire, his advantage converted into height, would be a thousand times as tall. To him, the miller would appear no bigger than an ant. With whom will the giant do business? Only with someone big and strong enough to carry his employer's burden. Likewise this someone, smaller than the millionaire on whom he is dependent, but still far greater in size than the grain-running ant, to whom will he entrust his dealings? To his peers. They, in turn, will do the same. Good manners and compromise characterize the bulk of these transactions, but hardly anyone thinks to condescend to those with whom they have next to nothing in common. Our lowly ant friend is simply nowhere in sight.\n\nEvery man and woman, at whichever point along the scale, prefers to look up, not down. Even the miller gives a hand to his equal or his superior, rather than to someone far below him, for fear of conceding his rank to the worse-off man. With the wealthier, he is generous; with the poorer, he is mean. He has little, but the little he has goes purely to keeping up his minor station.\n\nComparisons are somewhat facile, of course. Beside a billionaire, even the millionaire is poor. The world's hundredth-richest person has but one dollar for every eight in the pocket of the world's richest man.\n\nWhether the economy expands or whether it shrinks, the obsession with keeping up one's station remains. The inequality this obsession feeds off is a precocious learner: the more of it there is, the faster it will grow. Take the hypothetical egalitarian society, for example, where 45 percent of the population own 55 percent of its wealth. In such a society, around 20 percent (45 percent of the 45 percent) of citizens own about 30 percent (55 percent of the 55 percent) of the total resources. By the same logic, a sixth (55 percent of the 30 percent) of the society's goods belongs to one citizen in every eleven (45 percent of 20 percent).\n\nThe contrast between this theoretical society and most of our modern cities\u2014those that obey Pareto's 80\u201320 principle\u2014is striking. Wealth in these places can propagate far more ruthlessly, dramatically: the accounts of as few as four individuals in every hundred (20 percent of 20 percent) will bloat with as much as two-thirds (80 percent of 80 percent) of all the available income. And of these four men made of money, the very richest might have up to one-half (80 percent of the two-thirds) all to himself.\n\nHuman beings and their self-interest are inseparable, but inequality needs a society to invent it. The creation of any vast and ambitious social project demands an unequal allocation of resources in order to achieve its goals. Without substantial inequality, as John Maynard Keynes pointed out, Europe's railroads\u2014a \"monument to posterity\"\u2014would never have been constructed. Tolstoy, for one, hated the railroads precisely because they represented this inequality, going even so far as to throw his best-loved character under a murderous train. It is true that most of the men who laid the rails never had the opportunity to ride them. Why then did they agree to do so? The railroad workers chose to cooperate with the wealthy, Keynes argued, on the tacit understanding that what they produced together, using the money of one side, the labor of the other, would ultimately serve the nation as a whole, and the principle of \"progress.\" World war, however, would subsequently smash this fragile alliance between the classes, by shaking the faith of both in the future. The bludgeoning of bombs and the gutting of gunfire disclosed to all \"the possibility of consumption... and the vanity of abstinence.\"\n\nWhen Keynes spoke of the value of inequality he did not mean unbridled inequality. He meant an inequality that was consensual and that served a collective purpose. The selfish motive of making money, he admitted, helped to produce goods and services that benefit many. The same motive could also turn certain \"dangerous human proclivities\"\u2014cruelty, self-aggrandizement, and the tyrannical pursuit of power\u2014to more harmless pursuits. But, he was not at all complacent.\n\n> It is not necessary for the stimulation of these activities and the satisfaction of these proclivities that the game should be played for such high stakes as at present. Much lower stakes will serve the purpose equally well, as soon as the players are accustomed to them. The task of transmuting human nature must not be confused with the task of managing it.\n\nHow we lower those stakes is a question I leave to our politicians. I will not hold my breath. There is no easy solution\u2014none analogous to a politician's promise. Money's abstractness is complex, evasive. It turns our world upside down. For instance, a farmer's cow calves more prodigiously than other cattle\u2014this is normal, part of nature. What, though, should we think of houses that beget more houses? Like many of those born poor, my father never had the chance to own the roof above his head, whereas the proprietor of four houses will likely wind up with six, or twelve, or twenty.\n\nAll of which brings us to the question that Tolstoy posed his readers in the short story \"How Much Land Does a Man Need?\" Pahom, the greedy peasant, kills himself in the endless pursuit of more and more acres.\n\n\"They are poor in the midst of riches,\" Seneca observed of some of ancient Greece's wealthiest\u2014and greediest\u2014men, \"which is the worst kind of poverty.\"\n\n# Twenty\n\n# A MODEL MOTHER\n\nNot long ago, my mother's age reached double mine. Two lives when compared with my young man's span\u2014half of her that I cannot see.\n\nMy mother has always been a mystery to me. We have had my lifetime to get to know each other, but it still feels nowhere near long enough. Her behavior eludes me; it outpaces my powers of comprehension. Try as I might, I cannot figure her out.\n\nHer face has not changed much with the years. She often wears an expression that flits between smugness and fright. The same stubborn creases, made by steady clenching, around the mouth; the same defiant twinkle in her eyes. She smiles fitfully, unexpectedly, as if bestowing a favor. Beneath the fine graying hair and wrinkles, I can still find myself in her gaze.\n\nMemories. Take the kitchen of my childhood, for example, where my mother would spend the lion's share of her day. I remember her prowling the linoleum, pen and notepad in hand, compiling a list of groceries to buy. She was alive to every sound and suspicious of the slightest incursion. She poked her nose into cupboards and the fridge, looking for the cans of beans and the bottles of milk and the packets of cheese and sliced bread bought only a day or two before. Thin air had taken their place.\n\n\"The children eat us out of house and home,\" she complained to my father.\n\nMy father assumed a posture of resignation.\n\nWe kids knew which of our parents held command. Or at least, we thought we did. At other times my mother would suddenly come over all shy and prone to blushing. My father could hardly draw a word out of her.\n\nThen there were the Christmas presents. Year-round she would ferret out bargains at local yard sales, sequestering the toys and games in closets or under beds until Santa's sleigh arrived. Of course, we always knew where to find them, but in the spirit of the season we turned a blind eye. It was never easy: buried treasure seemed to lie in every nook and cranny of every room. Why, then, many Decembers later, under piles of old clothes, did we bring unopened presents to the surface? Had she simply mislaid them, forgotten their location? Could it really be that the buying of them had meant more to her than the offering?\n\nA mathematician would say, \"Graph the data.\" That is how mathematicians speak. And it is true enough: puzzling occurrences usually require the long view\u2014and a firm grasp of context. In my preteens I decided that if only I could assemble enough of my observations, and settle on some parameters for their analysis, it might be possible to make a predictive model of my mother's behavior.\n\nIt was around the time that I began to look like her, once the elementary school blackboard became a blur. Our myopia, it could be said, brought us closer together. \"A mommy's boy,\" my father sometimes called me. None of my brothers earned this epithet. Spending more and more time in her company, I felt the puzzle of her presence intensely.\n\nBack then, a thigh thinner, she was always on the go. I took to mapping her movements. On Saturday mornings my mother would return from the local library, carrying a couple of romance paperbacks in her arms, smelling faintly of must. In the space in the living room before the television, I would sit for what seemed like hours half listening to the yellow rustling of the pages that originated from the settee. Every other Sunday, robed in her best dress, she would take us\u2014my brother, sister, and I\u2014to the neighbor's on the corner to drink tea and trade gossip. Midweek, she would venture out to do the rounds of secondhand shops, returning with bags fattened on goods that nobody seemed to need.\n\nPerhaps she noticed me tracking her and wished to catch me out, or perhaps she simply grew bored with her own repetition, but for whatever reason she would sometimes decide to mix it all up. Come Saturday, the living room would contain the same atmosphere. But now, dog-eared biographies exhaled the musty smell. Without warning, the front door would stay shut for the Lord's Day of rest; we took tea with the neighbor an evening after school instead. Even her favorite shops suddenly became good only for returning things.\n\nOne afternoon she took me with her to return a pair of shoes. Approaching the shop, I compared the mother in my head with the real thing. The imaginary mother would choose a male shop assistant (my mother, I knew, hated to haggle with other women). She would complain that the shoes pinched her young son's toes, and when the man lifted the offending footwear from its container, she would add that the leather scuffed immediately. To the request for a receipt (which she always lost), her voice would rise in volume, detailing all the children's feet that she had to keep dry and warm. The man would then nod patiently before offering an exchange.\n\nUnfortunately, on this occasion my real mother bore no resemblance to her model.\n\nA young woman with tight bound hair assisted all the customers. My mother's voice as she handed over the shoebox sounded weak, her words unacquainted with one another. \"I'm sorry to hear it,\" the woman interrupted, as though offering her condolences. \"I'm sorry to hear it, but there really is nothing that I can do.\"\n\nI confidently expected my mother to put up a fight. But she promptly slumped into one of the seats intended for the trying-on of shoes, with only a long sigh for a rejoinder. The shop assistant reiterated that a refund was quite out of the question. My mother looked down at the floor and merely sighed again. Eventually, when my mother showed no signs of relenting, the young woman said, \"Please leave,\" and then, \"Please leave, or I will have to call the police.\" My mother sagged even deeper into her seat and crossed her legs.\n\nMy ten-year-old self was filled with apprehension. My imaginary mother would never have behaved like that! It took a long time for me to understand my real mother's sit-in. Of course, she knew all too well what she was doing. She knew that the young lady had conceived her own version of my imaginary mother. On that day, in that shop, my mother in her flesh and blood defied them both.\n\nAt last, driven to exasperation, the woman pulled something small and shiny from her pocket. \"Do not breathe a word of this to anyone,\" she said, before sketching a long gash along one of the shoes with the blade of her penknife. \"This way we get a refund from the manufacturer for damaged goods.\"\n\nIn fact, it did not bother me as much as you might think when the actions of my mother and her imaginary doppelg\u00e4nger failed to coincide. I slowly came to understand how limited and clumsy an approximation was my model of her, how many variables I had not accounted for (whose existence I had not even guessed), and how large and liberating a role chance played in all our affairs. Besides, each variation between the imaginary and the real provided me with new clues. This variation, I hoped, growing larger or smaller by turns, would act as a sort of compass toward a greater understanding of my mother's true nature.\n\nOn those rare occasions when my mother became her model's mimic, the eerie sense of d\u00e9j\u00e0 vu nauseated me. I worried that it suggested some dark cunning within me, or, worse, deterioration in the liberty of my mother's will. Besides, how could I even be sure that my success showed real merit? Perhaps it could be put down to nothing more than luck: even a stopped watch gives the correct time twice a day.\n\nPossibly, my inability to understand my mother is the result of an anterior uncertainty, that is: what should a mother's behavior look like? I am not talking about an ideal mother\u2014I do not believe, alas, in such creatures\u2014but rather the most common and distinguishing maternal qualities. A baseline, if you will.\n\nThis is more difficult than it sounds. For one thing, the category of \"mother\" admits all manner of members. You can be a mother at sixteen or at sixty (the latter with the help and expertise of scientists); with a single child, or my mother's nine. According to the dictionary definition, a mother is any \"female who has given birth to offspring.\" This is about as roomy and heterogeneous a group of people as we can find. It is, in the words of statisticians, too large a sample.\n\nHow, then, should I go about assembling a more workable sample of my mother's peers, one that provides a realistic context for analysis? By looking at mothers of nine? I am not sure that Britain can boast many nine-child families, not since the days when Queen Victoria had her own royal brood of nine. Newspaper articles turn up only a couple of examples: one is a philosophy graduate and boardroom executive who thought she and her Buddhist husband \"would stop at five\"; the second, a former anorexic who says she feared her chances of getting pregnant were \"very, very slim.\" Naturally, this tiny cohort of women is no more representative of maternity than the first.\n\nI might try a slightly different, related question: what does my mother's behavior tell us about her? But here we run into similar difficulties. For each of her acts, we can imagine a hundred more or less plausible reasons. A hundred imaginary mothers would slug it out for vindication. But each act is the offspring of another: each imaginary mother would be capable of producing one hundred more. Clearly, this approach will take us nowhere closer to any answer. For even if we could somehow locate the \"right\" reason for each of my mother's actions, and thus identify the \"right\" imaginary mother within our galaxy of imaginary mothers, we would be left in the end only with an identical twin\u2014as complex, mysterious, and baffling a woman as the one who raised me.\n\nMore empirical observation and less abstract reasoning seem necessary, if I am to arrive at any conclusions about who my mother really is. I fully admit that in this I am saying nothing new. When the psychiatrist \u00c9douard Toulouse resolved to objectively measure \u00c9mile Zola's genius, for example, he was entirely typical of his times. He took the novelist's height, wrapped a tape around his shoulders, skull, and pelvis, evaluated the strength of his grip, the acuity of his nose, ears, and vision, grilled his powers of recollection and noted the hours when he ate, slept, and wrote. Zola's pulse, the doctor found, was sixty-one before he first put pen to paper, dropping to fifty-three when he called it a day.\n\nScientists in the Soviet Union also went in for this sort of thing, counting the subject's words rather than his pulse. In their experiments, they tried predicting the next word in a sentence from those that had come before. Young girls in conversation, they discovered, were easiest to anticipate; newspaper columnists followed close behind, while poets proved hardest to second-guess.\n\nDid this result surprise the scientists? They do not tell us. Perhaps the poets took the same liberties with their speech as with their pen. The best poems, the mathematicians determined, combined in equal parts the predictability of meter with the novelty of unusual words. Too much meter made a poem banal; too much freewheeling, on the other hand, rendered it hard to follow. The delicate balance of convention and invention gives meaning to what we say.\n\nThe lesson we can learn from these experiments is small but valuable. Mutual understanding depends on our powers of prediction, though they frequently operate beyond our control. With his microscope the psychiatrist got no nearer to what moved Zola's pen, yet he knew intuitively how to talk his old friend into his tests. The Soviet mathematicians could not accurately preview the poets' inspiration, but their conversations once outside the laboratory ranged as far and wide as with anyone else.\n\nIn Edgar Allan Poe's story \"The Purloined Letter,\" we see a boy observing and then outguessing each of his schoolmates at a game of marbles. The game consists in determining whether the number of marbles concealed in the opponent's hand is odd or even. For every correct guess, a marble is won; for every wrong guess, one marble is yielded. Thanks to his \"astuteness,\" the boy finishes by winning all the marbles in the school. The boy, Poe explains, makes an intuitive assessment of his rival.\n\n> For example, an arrant simpleton is his opponent, and, holding up his closed hand, asks, \"are they even or odd?\" Our schoolboy replies, \"odd,\" and loses; but upon the second trial he wins, for he then says to himself, \"the simpleton had them even upon the first trial, and his amount of cunning is just sufficient to make him have them odd upon the second; I will therefore guess odd\";\u2014he guesses odd, and wins.\n\nWhen the boy considers his opponent \"a simpleton a degree above the first,\" he reasons, \"this fellow finds that in the first instance I guessed odd, and, in the second, he will propose to himself upon the first impulse, a simple variation from even to odd, as did the first simpleton; but then a second thought will suggest that this is too simple a variation, and finally he will decide upon putting it even as before. I will therefore guess even.\" He guesses even, and wins.\n\nPoe goes on to tell us how the marble winner is able to intuit the thoughts and feelings of the boy opposite him: he closely watches and mirrors the opponent's facial expression, so that the other boy's gaze fleetingly becomes his gaze, the boy's smile becomes his smile, the boy's frown becomes his own. In this posture, the winner finds himself thinking and feeling in the same way as his rival. His success hinges entirely on the precision of his mime.\n\nIn a sense, we are always evaluating and predicting the other, though we may not heed the act. Often the people we scrutinize hardest are those we cherish the most. In love there is constant contemplation and the most intense desire to understand the object of our affections. There exists melancholy, too, as we grow to appreciate just how little we can ever truly know for sure. Our ignorance is painful. And yet we persevere. Humbly, patiently, we assiduously observe till at last we identify ourselves in some way with the other. Anticipation becomes an act of love.\n\nI have spent years learning how to evaluate my mother's various tics and gestures. These days, I can read her body language with fair fluency. But still the same questions return over and over again. What, I often wonder, does that smile of hers say?\n\nWe meet in a posh restaurant in central London. A woman's face, old enough to be thought young-looking, smiles from a table at the back when I come through the door. I kiss my mother on the cheek. Her ecstatic eyes follow the stiff-backed young waiters carrying platters of food and wine. What shall we eat? I know the place well and have already made up my mind. I confide my choices to my mother before heading to the bathroom. When I return, the menus are gone. My mother is fiddling with her napkin. The skin on her hands is worn tight, I notice, like the peel on overripe fruit. Her fingers twist the napkin while we talk.\n\nHer rent has gone up again. Arabesques of graffiti continue to shout from wall to wall. Last week, an Albanian down the road set fire to his mattress; the wailing sirens made \"a right old racket.\" And yet, my mother will not hear of moving away. She insists on sticking close to where her children were all born and raised. I am aware that any new plea will fall once more on deaf ears, and have no choice but to let the matter go.\n\n\"Midday,\" I answer when she asks about my flight tomorrow out of Heathrow. Tokyo, I tell her, is nine hours ahead of her time. The trip will include my first lecture in the Far East. My mother feigns curiosity. She has never held a passport, knows nothing of the world beyond her shores. All of a sudden, she shakes with silent laughter. Some word, its sound or the mental image elicited by the word, has tickled her. Like the boy in Poe's tale, I reciprocate and try to laugh along. But I do not understand. And then, just as quickly as it came, the laughter leaves her. With the napkin, she dabs the wet corners of her eyes.\n\nI think back to the menu. For each course there are just a few options. A good opportunity to put my imaginary mother to the test. But will the model and the mother agree? Recalling the trios of starters, mains, and desserts, I assign to each a probability. I assess not only each of the individual courses but also the potential combinations across them. For example, two-thirds of the mains have meat: for this reason I downgrade the odds of my mother selecting p\u00e2t\u00e9 as a starter (unless, that is, she were to opt for a second course of fish). Assuming she begins with the good intentions of a salad, I plump for the caramel cake as her choice of dessert.\n\nMy imaginary mother decides to steer a middle route between the p\u00e2t\u00e9 and the salad. When the waiter looms, his voice announces the roast vegetable soup. I inhale the satisfying sweetness as it passes my chair.\n\nNow the beef rises even higher up my mental shortlist. But when the table has been cleared, and the waiter next comes, I find my stare returned by the glassy eye of a baked cod. Bits of the fluffy flesh scatter around her plate and down her top as she eats.\n\nFinally, we arrive at dessert. There is no room for doubt in my mind. My model mother's fondness for chocolate has been corroborated many times before. But not today. My actual mother finishes her meal with a bowl of exotic fruit.\n\nOutside the restaurant, she takes my arm in hers. She wants to show me the street on which she grew up. She grips my arm tight as we take the short walk side by side. Of her life before me I know next to nothing, only one or two tidbits, gathered here and there. I ask whether it is true that before starting a family, she worked as a secretary. \"Leave off,\" she laughs. \"I typed addresses onto envelopes.\" She still has postal codes on the brain.\n\n\"Name a town,\" she says suddenly.\n\nBethlehem, I think. \"St. Ives,\" I say.\n\n\"St. Ives,\" she repeats and it sounds twice as long when she says it. \"TR26.\"\n\nWe turn onto the street, a stone's throw from Westminster Palace. My grandfather worked for the local brewery here, delivering the beer by horse and cart. The block of flats was home to several of the workers' families. A single washroom, with enamel bath, did for everyone. We cannot see inside, however. The building has become a base for the homeless; some of the windows are boarded up. Before we leave, I pull out my camera and take a photo.\n\nI think of the girl who would grow up to become my mother. Who was the future woman that she imagined for herself? Did she dream of having a loving partner, a big house, and children who always smiled at her? In her mind's eye, would she be well educated and well traveled, always generous, patient, and kind? Did she imagine that every cherished moment would be remembered forever; every grievance instantly forgotten?\n\nAnd thinking of this girl I feel at once immensely happy and immensely sad. I feel as I feel when I think of myself.\n\n# Twenty-One\n\n# TALKING CHESS\n\nTo win at chess is simple: victory belongs to the player who makes the next-to-last mistake.\n\nWhoever it was that first came up with this line spoke much truth. The strongest players operate neither like machines nor angels; their superiority lies in accomplishing a better class of error.\n\nA winning mistake would presumably owe nothing to sloppiness, incuriosity, or a yellow belly. It would be far closer to the lucky slip of an artist's brush or writer's pen, one that suddenly infuses a picture or page with unforeseen possibility. I am thinking of the (perhaps apocryphal) story of the painter who, hot-tempered at his many failed attempts to perfect a detail of his portrait, tossed his sponge with disgust at the easel and thereby achieved just the desired effect. Or that of the book printer who inadvertently won Herman Melville lavish accolades for the phrase \"soiled fish\" (in place of the intended \"coiled fish\" in reference to an eel).\n\nI will not risk stretching this argument any further. Creativity is obviously so much more than an unexpected gesture here and there. But talent in chess bears this similarity with other creative pursuits, in tolerating error, as if the grandmaster\u2014like the great artist\u2014is the one who truly explores possibility's outer limits. Or, as a character in Joseph Conrad's _Lord Jim_ intones, \"To the destructive element submit yourself and with the exertions of your hands and feet in the water make the deep, deep sea keep you up.\"\n\nChess is a perfect arena for just such an exerted exploration of the possible. Its checkered sea is very deep indeed. The mathematics behind the game's complexity are staggering. An initial move by each player will create one of four hundred legal positions. A second move by each: seventy-two thousand. There are some nine million possible setups after the players' third move; 288 billion after their fourth. Back in 1950, the mathematician Claude Shannon calculated the possible number of forty-move games, a figure henceforth known as the \"Shannon Number.\" He estimated as about thirty the potential number of viable moves at every turn. In this way he arrived at a total number (1 followed by 120 zeroes) that easily exceeds the number of atoms in the observable universe.\n\nFor all its immensity, chess is a finite game. It is therefore at least conceivable that a machine might one day be programed with the knowledge, deep down in its nodes, of every possible sequence of moves for every possible game. No combination, however ingenious, would ever surprise it; every board position would be as familiar as a face. Like checkers, which was solved by computer scientists in Canada in 2007, we would finally discover how chess, played perfectly by both sides, ends.\n\nThis perfect game of chess\u2014the immaculate order and configuration of its moves, the exquisite ballet of its pieces in their precisely timed roles\u2014is imprinted on the imagination of every player. In his innermost being, every player carries some notion of the divine game. For one, it begins with the two-square forward march of the white king's pawn, to which the L-shaped spring of black's queenside knight replies. Six moves later, the white queen arrives on a 4\u2014a side square\u2014only to be promptly rebuffed by black's bishop. No, no, says a second player: the game begins with a white knight\u2014either one\u2014to which black responds in kind. The central pawns advance in pairs. But another player disagrees, citing the fact that a piece would need to be sacrificed by white after eleven moves\u2014the exchange of his queen for a rook. For still others the white pawns creep up the far sides of the board like ivy, or the black king crouches incessantly behind his queen, or all four bishops dance the diagonals until precisely half of all the original pieces remain.\n\nWho, then, prevails in this Platonic ideal of chess? Each player confesses a secret faith. A triumph for white in forty-three moves, or in forty-one if black's sixth move disturbs a pawn. Or else a win for black, after a marathon two hundred and twenty-seven moves, the surviving white piece finally confiscated by the king. But such are the visions of romantics; a large majority seem resigned to the probability of a draw.\n\nA small number of enthusiasts (a handful of masters, too) claim to have adjudicated the question in favor of one or the other side, outlining a \"system\" for the player to follow. Understandably these systems have been subjected to many critiques. In his books and articles, Weaver Adams, who won the U.S. Open Championship in 1948, argued that with a first-move king's pawn advance \"white ought to win.\" In his own matches, however, Adams seemed to have perversely better luck with black. A certain Dr. Hans Berliner agrees with Adams concerning white's predestined triumph, but differs in his choice of the clinching initial move. According to Berliner, white must employ the queen's pawn instead.\n\nWithin our lifetimes a definitive solution, using the fastest computers, might indeed emerge. But there is still a very, very long way to go. So far algorithms have resolved every lawful chess position containing at most six pieces (including the two kings). The compiled data has yielded more than a few surprises. Numerous endgame positions long considered mere draws, we know now beyond all shadow of a doubt, can in fact produce a winner. Among the most recent analysis, which has moved on to the study of endgames of seven pieces, researchers have hit upon an astonishing forced win for white\u2014provided that the winner's concentration could somehow hold out over five hundred and seventeen flawless moves!\n\nIn the end, maybe no solution of chess exists, or at least none that we could ever extract in the time allotted to our universe. Any complete solution might even reveal itself to be simply beyond the pale of our imagination: knights that forage for pawns for no apparent reason, bishops that take turns occupying consecutive squares, or rooks that slide up and down, and left and right, ninety-nine times in a row.\n\nBut of course chess would not be chess without its mystery, or its players' mistakes. Men, like chessmen, are made from crooked timber. With his mistakes, the beginner (what players call a \"patzer\") immediately blows his cover. He brings his queen out too quickly, exchanges too many pieces too soon, moves his pawns in such a way that their formation finishes by resembling Swiss cheese. But the problem is more one of quantity than quality. The patzer loses not because he makes too many mistakes, but because he makes too few\u2014a mere handful of classic blunders. He does not last long enough to make more! Traps abound, and he duly falls into one or another. Cannier, moderate-strength players (such as club champions) make more mistakes than beginners. Sidestepping the early pitfalls, they grant themselves far greater scope in which to stumble.\n\nStronger play demands more than the avoidance of blunders. The player has to learn to make his own mistakes, which is far more difficult than it sounds. He has to stop mimicking the moves he reads in books and magazine columns, and which he does not really understand: even the best moves can turn bad when played on the wrong square or at the wrong instant. He has to root out his most cherished errors, the kind that he plays with the frequency of a tic. In short, he has to clear his head and think and feel and suffer for himself. Only then can he hope to obtain the slightest grip upon the game.\n\nAll of this amounts to nothing less than that nebulous attribute we call personality. It is the indescribable quality that seems to bring the pieces on the board to life. As with the brushstrokes of a gifted painter, we can identify a strong player from the moves, including the mistakes, which he makes. The observer traces in the movements of the pieces the movement of the player's thought. What we call his mistakes are also the expressions of a deep and personal understanding of such-and-such position, which is\u2014like every human understanding\u2014imperfect. Out of this personal understanding comes both his biggest howlers and his finest moves.\n\nNo master of the game, I would suggest, had more personality than the Soviet champion \"the Magician of Riga,\" Mikhail Tal. Not a few of his games have gained the status of the masterpiece. At its best, Tal's play betrayed a courage that bordered on insouciance. He was always in the thick of the action, inviting complications. Of this proclivity for problems, he once remarked, \"You must take your opponent into a deep dark forest where 2 + 2 = 5, and the path leading out is only wide enough for one.\"\n\nIn the depths of this forest even he sometimes lost his way. During one simultaneous exhibition against a score of Americans, the grandmaster took the fight to a plucky and talented twelve-year-old. At a crucial moment Tal surrendered his queen in exchange for the initiative, but the sacrifice eventually proved unsound and he went on to lose. With a shrug of his shoulders, the former World Champion gamely shook the boy's hand, before continuing to ply his trade among the other boards.\n\nTal played instinctively. Faced with the game's intractable complexity, he always followed his nose. He would feel his way around the squares of the board, for feeling is also a kind of thinking. In his autobiography, there is a small but marvelous anecdote of this intuitive approach. Tal describes a contest with the grandmaster Vasiukov at the Championship of the USSR. By dint of adventurous play, the two men had reached a highly entangled position. Tal says he hesitated a long time over his next move. His path to victory, he sensed, began with the sacrifice of his knight, but the immense number of possible variations unnerved him. Head in his hands, he meditated on one after another without result. Mental chaos ensued. All of a sudden, from out of nowhere, an amusing couplet by the children's poet Chukovsky broke the surface in his mind. _\"Oh, what a difficult job it was. To drag out of the marsh the hippopotamus.\"_\n\nTal was at a loss to know by what process of connection his mind had suggested the poem's lines. But now the thought gripped him: how exactly would a man pull such an animal from a marsh? As the spectators and journalists looked on, the grandmaster contemplated any number of hippopotamus-rescue methods: jacks, levers, helicopters, and \"even a rope ladder.\" Here, as well, his calculations came to no avail. \"Well, just let it drown,\" he said to himself at last, in a fit of ill temper. Tal's head cleared immediately and he resolved to trust his instincts and play. The next morning the newspaper report read, \"Mikhail Tal, after carefully thinking over the position for forty minutes, made an accurately calculated piece sacrifice.\"\n\nBefore we leave Tal, one word about his chess education. The young Mikhail's development proved vertiginous. He first learned to play at the age of eight, watching the games of patients at a hospital where his father worked. The boy did not shine, far from it. His youthful style was simply one more against which the older players gained their points. Only from the age of twelve did he apply himself seriously. A local master took him under his wing. Within two years the teen had qualified for his national championship; a year later he finished ahead of his trainer. The next year, aged sixteen, he won his country's chess crown and the title of master.\n\nSuch rapid acquisition is reminiscent of the facility with which we learn our mother tongue. Only four years separated the beginner Tal from his first national championship; only four years\u2014as a rule\u2014separate the baby from fluent speech. In both cases, an adult's guiding hand makes all the difference. Left to his own devices, neither the baby nor the beginner can hope to make much progress. Linguists say that a child learns his language by exposure to highly structured input; his parents address him more slowly, more questioningly, in short sentences that verge on curtness. The chess player, in a similar fashion, learns best from a coach's counsel; he is shown many patterns and combinations of moves that characterize expert play.\n\nWittgenstein observed that language, like chess, is a game governed by rules. Knowing how to use a word, he said, is like knowing how to move a chess piece. From a small number of these initial rules, immense complexity is spawned. Gossip on a street corner can rival in complexity any game of chess. This is because the number of concatenations of words that form meaningful sentences approaches infinity. Speakers (and writers) are always coming up with original sentences, like chess masters with their novel moves. And like any player worthy of winning, the speaker (and again the writer, to some extent) anticipates the other's response. He modifies his speech to accord with his anticipation. Not only does he know what he can say, he knows what he should say, and\u2014perhaps even more interestingly\u2014what he should not. A skilled conversationalist has this knack for knowing which avenues to explore and which to avoid. Similarly, certain chess moves, in certain situations, while perfectly legal, are considered taboo. I have heard of a grandmaster who once described his opponent's early capture of a pawn with his bishop as being \"low culture.\" The move boasted the advantage of quick material gain, but at the expense of the formation and coordination of his pieces. Such moves are rarely propitious to a good game.\n\nI am reminded of a scene from the 1951 novel _The Master of Go_ by Japanese author Yasunari Kawabata. The story's narrator, a passionate amateur of the game (Go is an ancient and highly strategic board game), plays on his pocket magnet set during a train ride home. Opposite him counters a tall American tourist, on whose knees the gold leaf board rests for the duration of the journey. The Japanese narrator prevails quickly, game after game. \"It was as if I were throwing a large but badly balanced opponent in a wrestling match.\" Throughout, he notices a sort of thoughtlessness, a lack of personal investment, in the American's play. For the tourist, he imagines, playing Go was \"like having an argument in a foreign language learned from grammar texts.\"\n\nTaking this thought to its natural conclusion, Kawabata goes so far as to declare the game's subtleties as being inaccessible to foreigners. What he means by this, I think, is that a good game (whether of Go or chess), like a good conversation, requires a certain sensibility born of complete immersion. I am thinking of that attention to form which elevates a phrase or move beyond the merely functional. Even in translation, Kawabata's own elliptic writing style can often puzzle the non-Japanese reader. Many details, which to his countrymen appear straightforward, can pass entirely over our heads. Why, for instance, does the narrator in the novel mention that his Go board is decorated with gold leaf? Might it be a reference to illumination? Or victory? Should we read in it the suggestion of Go as an art? We can only hazard a guess as to its meaning.\n\nGrandmasters, of course, are immersed in their game. Some drown, sunk by an insanity brought on by virtually infinite complexity. Most, however, find their own fullest and richest means of expression in the combinations of moves they deploy. These players do not so much think about chess, as think _in_ chess, just as we think in language. I read an account of one master who recalls each day's events as though they were moves on a chessboard. He remembers going for an afternoon swim as, say, King's knight to F6, while a restaurant meal with his wife comes back to him as a descent four squares of the Queen's rook. These associations appear to him as unremarkable as they are spontaneous.\n\nWe can also see this spontaneity, the spontaneity of speech, in what we call \"blitz\" or \"rapid\" chess. In its most popular form, both players have just one minute in which to dispense all their moves. Pieces slide frantically from square to square; agile hands swat over and over the button on their clock. At a rate of one move per second or thereabouts, whole games, upward of forty moves, are frequently accomplished. In spite of having no time to think, the players in these games often attain surprisingly high standards.\n\nI don't wish to imply that playing chess is all instinct. Alacrity has its virtues (the absence of indecision for one), but they are not the virtues of the longer form. At its best, chess\u2014like language\u2014privileges reflection and careful thought. Looked at one way, a game of chess amounts to a long series of shifting problems, the most intriguing among them capable of making unique demands upon our imagination. As with a beautiful passage in a novel, we feel that we could happily spend almost any amount of time in its company.\n\nOccasionally an amateur player will scour the back pages of certain newspapers (the kind whose pages end up wrapping glass vases, but never greasy fries) to savor and solve these chess problems. Illustrations of the board, its pieces frozen in print, compete for space with the week's crossword puzzle. \"White to play and mate in three\" or \"Black to play and draw\" its title will announce. Quite often the position displayed is at least half cleared of its starting pieces, and the game has attained its final moments. The amateur stares at the inky pieces and the smudged squares, waiting for a surge of inspiration. It is more or less the same experience as when we read some striking lines of a poem.\n\nMany of these published chess positions have never known an actual board; they are the figment of some inventor's imagination. Among these inventors figures one \"Vladimir Sirin,\" better known as the multilingual novelist and poet Vladimir Nabokov, for whom such compositions equated to the \"poetry of chess\" (his 1969 anthology, _Poems and Problems,_ was illustrated by eighteen of his own efforts).\n\nThose sixty-four squares, their peculiar geometry, fascinated the great wordsmith to the point of obsession. The right-angled triangle, for instance, traced by a king's vertical (or horizontal), and then diagonal, retreat across the board disobeys Pythagoras's famous theorem. Off the board, in the \"real world,\" moving, say, three paces across and as many paces up (or down) would produce a greater diagonal distance (four and a quarter paces long) between the start and end points. The king's \"triangle,\" on the other hand, has identical length (traversing the exact same number of squares) on every side. This optical illusion (expecting a diagonal route to require a greater number of moves than either a vertical or horizontal path) plays an essential role in the chess problem composer's craft.\n\nLike his words, Nabokov's chess pieces relied on precise positions and combinations for their meaning. He saw to it that the pieces' values differed sharply from the amateur's expectations. The \"value\" of a position's queen, for example\u2014while generally considered to be twice that of a rook, thrice the worth of a knight or bishop, and nine times as important as a pawn\u2014might decrease to next to nothing if immobilized to a corner or back row by the careful setup of lesser pieces.\n\nNumerous mathematicians also double up as inventors of chess problems. They create positions to solve questions like, what is the maximum number of possible checkmates in one move? Answer, forty-seven. Or what is the smallest number of bishops needed to occupy or attack every square? Answer, eight (the same as for the smallest number of rooks). Or else they compose whole games of a specific number of moves to reach a given position.\n\nThere is a final analogy between chess and language that I should mention. I began these pages reflecting on mistakes. I said that grandmasters make magisterial mistakes, founded on great creative intuition. Young children do, too. From ambient talk they get wind of words, but their minds then make of them what they will. As a matter of fact, the toddler's speech far exceeds the mimicry of adults; its features are distinct and demonstrate invention. For example, we have all heard a young child say something like \"three mouses\" or \"I goed\" instead of \"mice\" and \"went,\" though no parent would utter such a thing. More inventive still were the kids who reportedly suggested that a \"myth\" was a female moth, or that benign is \"what you will be after you be eight.\"\n\nPerhaps they grew up to become grandmasters.\n\n# Twenty-Two\n\n# SELVES AND STATISTICS\n\nFor each one of us, nothing is so personal, so intimate, and so selfish as our death. From time immemorial, men have sought to predict its hour. They have scattered bird entrails, or elaborated bad dreams, or consulted the office of an oracle. When these prophecies were not plainly wrong, deluded, or hysterical, they often proved unhelpfully vague and misleading. According to one legend, a Scythian prince consulted a Greek oracle to discover how he would die. A _mus_ (mouse), he was told, would be the agent of his demise. The prince took the forewarning to heart. He had his houses cleared of mice and even snubbed the company of any man called Mus. All the same, shortly thereafter, death came for him. The cause? He died from an infected arm muscle (the word _muscle,_ in Greek, means \"mouse\").\n\nThe idea of death as a statistical phenomenon, amenable to calculation, occurred only toward the end of the seventeenth century, with the publication of the world's first mortality table. This new conception of death had gestated so long for a good reason: it upended completely how men and women understood their individual lives. _Society_ \u2014if and when this term existed at all\u2014had always been held to be only a loose confederation of free souls. The precise business and destiny of each were an impenetrable mystery. Crowds were merely freaks, monsters with many heads and countless limbs. That a person might be in any way comparable to the crowd, that he might be deciphered by studying the behavior of his family, his village, his fellow countrymen, seemed both impossible and absurd.\n\nAnd if the individual was beyond the pale of understanding, so, too, was his eventual extinction. No scholar, bitter experience told most people, could hope to get inside the Grim Reaper's skull. Death touched rosy-cheeked infants and decrepit widows alike, with no apparent rhyme or reason. An ancient man on his last legs might somehow survive another decade, while his grandson, beaming with youth and health, not live to see the following spring.\n\nStories took the place of science, their tellers repeating the same message over and over: life is full of surprises. Remember Old John, one tale would go, Old John who laughed so hard at his neighbor's joke that his heart gave out? The farmer's wife who was butted into her grave by the goat? The squire who caught cold sleeping in church?\n\nIt was in this atmosphere of ambivalence that Edmond Halley, who found lasting fame calculating a comet's orbit, published _An Estimate of the Degrees of the Mortality of Mankind_ in 1693. Halley based his figures on the city of Breslaw, capital of the province of Silesia, \"near the confines of Germany and Poland and very near the latitude of London,\" with a total population of 34,000. For five years running, monthly figures for every birth and death in the city had been collated: 6,193 births and 5,869 burials in all. Of the newborns, Halley discovered, 28 percent perished in their first year; only a little more than half lived to celebrate their sixth birthday. Most of these, however, would go on to have children of their own. \"From this age the infants being arrived at some degree of firmness grow less and less mortal.\"\n\nOf citizens between the ages of nine and twenty-five, the number of deaths each year equated to about 1 percent. This figure rose to 3 percent for those aged between twenty-six and fifty, jumping to 10 percent for those having reached the ripe age of seventy. \"From thence the number of the living being grown very small, they gradually decline till there be none left to die.\"\n\nHalley used this combined data to calculate \"the differing degrees of mortality, or rather vitality in all ages.\" For example, to estimate the odds that a person aged twenty-five would not die within the next twelve months, he compared the city's twenty-five-year-olds (totaling 567) to its twenty-six-year-olds (560), and thereby concluded that an \"average\" twenty-five-year-old would have odds of 560 to 7, or 80 to 1, that he outlive the year.\n\nWhat is the probability that a man of forty live a further seven years? Halley took the number of forty-seven-year-olds (377) and subtracted it from the number of forty-year-olds (445) to find the difference between the two ages (68). The odds that a forty-year-old become a forty-seven-year-old were therefore 377 divided by 68, or 5.5 to 1.\n\nHow many years might a man of thirty reasonably expect to have still ahead of him? To answer this question, Halley would first determine the number of thirty-year-olds (531), and then halve it (equivalent to considering as even odds that the person would die within the period). This halved figure (265), he then finds, equates to the number of citizens between fifty-seven and fifty-eight years of age. The \"average\" thirty-year-old could look forward to a further twenty-seven or twenty-eight years.\n\nFrom his findings Halley drew a suitably pious conclusion.\n\n> Unjustly we repine at the shortness of our lives, and think our selves wronged if we attain not old age; whereas it appears hereby, that the one half of those that are born are dead in seventeen years time... so that instead of murmuring at what we call an untimely death, we ought with patience and unconcern to submit to that dissolution which is the necessary condition of our perishable materials, and of our nice and frail structure and composition: And to account it as a blessing that we have survived, perhaps by many years, that period of life, whereat the one half of the whole race of mankind does not arrive.\n\nIt was a mortality table just like the one first drawn up by Halley that the American paleontologist Stephen Jay Gould would scrutinize at length, three centuries later, in the summer of 1982. It was the summer when the Equal Rights Amendment was voted down, when Italy beat West Germany in the final of the soccer World Cup, when a debt crisis broke loose across South America, and when Gould sat in his doctor's office and learned that he would shortly die. He was forty and had just been diagnosed with a rare and incurable form of cancer. Poring over books at Harvard's medical library, thick as hand-spans, subsequently taught him all that there was then to know about the condition and its survival rates. In summary: he had a median of eight months left to live.\n\nGould could not take seriously Halley's well-meaning words of advice. He would not submit readily to his body's dissolution; he wanted to survive. He thought of his wife, his two young sons, his high-flying career. A million other things went through his mind: the museum's Tyrannosaurus rex, all gigantic teeth and bulky bones, that he had seen as a small boy; his father, in his evening slippers, reading _Das Kapital;_ the opening bars of Gilbert and Sullivan's _Mikado;_ his Yankees season ticket; the Pepperidge Farm cookies in his office drawer. His office. What would his microscopes do, and his favorite rattan chair? Sit on the wrong shelf; stand in an unlit corner. Gather dust.\n\nWhat did Gould do in such dire circumstances? What could he do? He did what virtually every recipient of bad news does: he hunted feverishly after even the slightest, even the faintest positive angle. He would not abandon hope. A _median_ of eight months; that is what the statistics said. If half of all the patients with his cancer would die within eight months of their diagnosis, it meant that half would live beyond eight months. Some would live on for years. This thought comforted him. His mind tightened around it. Age? He was still young. Class? His family lived on the better side of town. Health? A little on the heavy side, but he had no other baggage. Attitude? He identified a strong will, even temper, and a clear purpose for living. His chances of landing in the latter half of the patients' prospects seemed to him great.\n\nHe would have only one future death, not thousands, and the median said next to nothing about it. This became his mantra. Friends and family asked him to explain. Averages talk about populations, not persons, he would reply. If I died a thousand times, approximately half of those deaths would occur within eight months. The other half would follow one by one: days, weeks, months, or years later. Who can tell where my single death, out of the thousand possible deaths, will fall?\n\nThe following months were tough and turbulent for Gould, full of boredom, pain, and exhaustion. Doctors radiated his body, pumped it with drugs, and put it under the knife. He hemorrhaged weight\u2014in all, he lost a third of his 180 pounds. His hair embarrassed him by falling out. The lonely and tedious hours of treatment piled one upon the other, oppressing and enfeebling him. And yet he survived. His cancer went into remission. Two years later he was well enough to write a long article about his experience, \"The Median Is Not the Message.\" A decade after its publication he was still going strong. \"I am a member of a very small, very fortunate, and very select group\u2014the first survivors of the previously incurable cancer,\" he wrote.\n\nIn March 2002, Gould\u2014aged sixty\u2014published his magnum opus, _The Structure of Evolutionary Theory,_ 1,342 pages long. It was the seventeenth book to bear his name since his diagnosis twenty years before.\n\nTwo months later, his personal death finally arrived, the result of a second, unrelated cancer.\n\nKnowing how to read the mortality table's numbers probably extended Gould's life by many years (he affirmed a likely link between an individual's state of mind and his immune system). Not knowing how to read them, on the other hand, cost one man and his family very dear.\n\nAndr\u00e9-Fran\u00e7ois Raffray's story is an extreme illustration\u2014perhaps the most extreme\u2014of confusing persons with percentages. Raffray was a long-serving and successful public notary, living and working in the southern French city of Arles. Among his clients was a ninety-year-old widow, without heirs, whose name was Jeanne Calment. One day, in 1965, Raffray agreed to purchase the widow's house under a scheme the French call _rente viag\u00e8re:_ in return for paying Calment a monthly sum of 2,500 francs, he would own the property upon her death.\n\nRaffray must have imagined that he had struck an excellent deal. The widow's house was worth close to half a million francs. Assuming that she lived another three years\u2014the current life expectancy for an average ninety-year-old French woman\u2014he would shell out fewer than 100,000 francs in all. Over 20 percent of nonagenarians died before their next birthday. The statistics, he thought, were on his side. \"Even if she makes it to ninety-four, or ninety-five, or ninety-six, I will still have the house in the end for only a fraction of its value. And what if she goes on and on, to ninety-seven, or ninety-eight, or\u2014God forbid\u2014to one hundred? But how many people ever live to one hundred? Not even one person in a thousand! To think that she might go on another ten years! I cannot imagine it. But what do I care? Let her keel over at one hundred: I will still make a tidy profit.\" Such was probably his line of thought.\n\nIt was a mistaken assumption that the very old are all more or less the same. The notary's acquaintance with Madame Calment was slight. Her snow-white hair, her birdlike frame, her papery skin, he mistook for frailty. He saw these features and thought at once of every elderly person he had ever met. Faces, bodies, lives, blended together in his mind. What did these people, averaged, have in common? Sickness, sadness, shortness of breath.\n\nBut before Madame Calment was old she had been young, and she had ridden a bicycle through the cobbled streets of Paris, and stretched after fluffy tennis balls, and eaten scores of fruit and salads slick with olive oil. Early marriage to a wealthy merchant had freed her hands for tinkling piano keys and theater applause. No illness had ever troubled her.\n\nLittle had changed for her since moving to the warm and sunny south. She had taken all her favorite things with her, except for the buried husband. But she had long grown comfortable with her solitude. She was not afraid of silence, of hearing the rhythm of her heart. Nor did she worry about her looks: makeup, she had learned, could not resist her frequent tears of laughter. At age eighty-five, she wore silly plump padding for her first fencing lessons without giving it a second's thought. She still loved walking in the open air. Regular sips of port wine and snacks of chocolate brightened her day.\n\nCareful study of the mortality table had taught the notary with what frequency previous nonagenarians had died and after how many years, but he had not thought about the future. It was a fact, for example, that France counted no more than a few hundred centenarians in 1965. But that was then, and the widow would reach her hundredth birthday, if she reached it, in ten years' time. How many centenarians would France have in 1975? In 1980? In 1990? That is the kind of question that the notary forgot to ask. Around the world, medicine and technology were rapidly improving. Once significant causes of death\u2014from flu, vitamin deficiency, or high blood pressure\u2014were dwindling. Within a generation, the number of centenarians in France would increase twentyfold.\n\nAnd what of the table's statistics? They deserve a closer look. For one thing, the data available for the very old was necessarily sparse and unreliable. Too few before the generation of Madame Calment had lived long enough to give being ninety years old a proper try. Statisticians knew next to nothing of a nonagenarian's medical needs, eating habits, daily routines, and much else besides. Guesswork had to fill in the gaps.\n\nLife expectancy: three years, announced the table. But let us see what this actually means. If there were ten thousand ninety-year-olds in 1965, there would be around five thousand still living in 1968. The life expectancy of these ninety-three-year-olds would not be zero. What would it be? (This was another question that the notary did not think to ask.) Almost three years. Live three years in your nineties and the chances are fair you will live three years more. And if there were five thousand ninety-three-year-olds in 1968, there would be around two thousand surviving in 1971. These ninety-six-year-olds could expect, on average, to live a further two years. In 1973, about a thousand\u201410 percent of the original number of ninety-year-olds\u2014would still be alive. Close to half of these in turn would live long enough to reach their hundredth birthday.\n\nMadame Calment, in February 1975, numbered among them. One hundred years old, but still in fine shape, on her feet every day. She left the dying to others. At the age of 105, her notary's monthly checks amounted to the full value of her house. But still he had to keep on paying.\n\nFive more years passed and Madame Calment begrudgingly moved into a nursing home. At 110, her age exceeded the maximum figure of most mortality tables. At 113 she became the oldest person in the world: _la doyenne de l'humanit\u00e9_.\n\nEven then, she did not die. Her eyes dimmed, her joints stiffened, but she remained in good spirits. Statisticians who had long estimated a maximal human life span of between 110 and 115 years were proven wrong. Madame Calment became the first person in history to celebrate her one-hundred-and-sixteenth birthday, her one-hundred-and-seventeenth birthday, her one-hundred-and-eighteenth birthday, her one-hundred-and-nineteenth birthday.\n\nIn 1995, \"Jeanne,\" as people now called her, attained 120 years of age. The nursing home, which until then had received few visitors, suddenly swelled with reporters from around the world. Tiny, desiccated, she sat before a bank of cameras. She does not smile, complained one of the photographers. Ask her if she wants to go on living, demanded a journalist. The nursing home director cupped her hand around the old woman's right ear, as though playing an absurd game of Chinese whispers. \"The monsieur would like to know if you want to live a little longer?\" she shouted into the ear.\n\nYes.\n\nHer notary was absent that day. Long retired, nearing eighty, he was not well enough to attend. Later that year, he died. He had never set foot inside Madame Calment's house. Under the terms of the thirty-year-old contract, his widow continued to pay. By the time Madame Calment eventually died two years later, aged 122, the notary and his family had paid nearly a million francs: many times the anticipated payout and twice what the property had been worth when they took on the arrangement.\n\nFrom where, we might well ask ourselves, did this strange idea of an average cancer patient or an average nonagenarian originate? _A Treatise on Man and the Development of His Faculties_ introduced _l'homme moyen_ (\"the average man\") to the nineteenth-century world. \"If an individual at any given epoch of society possessed all the qualities of the average man, he would represent all that is great, good, or beautiful.\"\n\nThe author\u2014a Belgian mathematician called Alphonse Quetelet\u2014imagined how Nature created men. He pictured Nature as a bowman who aimed continuously for the statistical center. The bull's-eye would be a perfectly median man (or woman): a rational, temperate person free from any excess or deficiency. Most individuals, according to this view, were Nature's errant arrows. They straggled, nearer or farther, around the mean. The taller-than-average flaunted the shorter man's missing inches; the drunkard lacked his part of the teetotaler's (excessive) self-control; the hairy man wore the bald man's follicles.\n\nQuetelet therefore urged his readers to take a wide view of humanity. Every person, after all, was \"but a fraction of mankind.\" Instead of interesting themselves with men and women, scientists should study the population as a whole. \"The greater the number of individuals observed, the more do individual peculiarities, whether physical or moral, become effaced, and allow the general facts to predominate, by which society exists and is preserved.\"\n\nSo the mathematician averages data collected from a thousand or ten thousand households and learns, for example, that the \"normal\" male height is five feet seven inches, that the \"normal\" time spent reading the newspaper is twelve minutes, that the \"normal\" diet consists of eggs, potatoes, and meat. A height of five feet five inches, or six feet two inches, is aberrant; spending five minutes, or thirty, with the newspaper is abnormal; too much fish or too few eggs on a plate is deviant.\n\nDrawing from this data, the mathematician observes regularities: most men are two inches too tall or too squat, most readers linger three minutes too long (or skim three minutes too short) over their newspaper, most housewives cook six potatoes per week too few or too many.\n\nBut it was not only physical or mental traits that a mathematician might average. Morality, too, could offer itself up to calculation. An analysis of police statistics would reveal the defining characteristics of the \"average\" criminal, \"even for such crimes that seem to escape all human foresight like murder since they are committed in general without motive and in circumstances, apparently, the most fortuitous.\" According to Quetelet's figures, the \"typical\" murderer would be male, in his twenties, literate and a white-collar worker. He would have alcohol on his breath, and be wearing the light clothes of summer. Probability would put a pistol (rather than a knife, or a bat, or a vial of poison) in his hand.\n\nThis idea spread quickly. It proved popular with scientists and the general public alike. Most frequently it was reinforced neither by logic nor analysis, but by simple prejudice. People laughed, and sneered, and denounced, and ridiculed variously identified \"types\" of the average man. But worst of all, they believed that such beings really did exist.\n\nImages propagated the idea with particular efficacy; as the proverb says: a picture is worth a thousand words. One newspaper caricature was enough to do the work of Quetelet's wordy disclaimers (his original book had run to several hundred pages). If the caricature depicted an Irishman\u2014all Irishmen\u2014with a misshapen jaw, feathered cap, protruding teeth, it was plain enough that the \"average Irishman\" bore more than a passing resemblance. If it showed a filthy, vulgar, gin-swilling beggar, it confirmed how many readers imagined the \"average poor.\"\n\nPhotography\u2014then still a young and innovative technique\u2014was similarly enlisted. Mug shots of eight different individuals were superimposed to reveal the blurry face of the \"average criminal.\" Nine photos of consumptive patients were merged to produce a portrait of tuberculosis. Images from six medals depicting Alexander the Great were morphed to reveal the ancient king's probable features. \"It is now proposed,\" announced one magazine, \"to get a clear idea of Nebuchadnezzar from the various stone and brick slabs upon which his face is graven.\"\n\nBut if photography served to popularize the idea of \"average men,\" it also produced an alternative way of looking at ourselves. Identikits (facial composites), created at the end of the nineteenth century, helped to divert some of the focus back from the typical toward the individual. Photos that magnified and emphasized all manner of different facial features now replaced the phantom faces typifying this kind of man, or that sort of person. Instead of a single abstract representative of this or that contrived category, the images highlighted a wealth of actual noses, foreheads, wrinkles, ears, chins, eyelids, lips, and mouths.\n\nTake the nose, for example. Plainly, nobody has a \"nose\"\u2014a person has a low nose or a high nose, a wide nose or a narrow nose, a hooked nose or a straight nose, a long nose or a snub nose. Perhaps the tip is bulbous, the nostrils dilated. And what about the chin? Big or small? Flat or bumpy? Does it retreat toward the throat, or jut out proudly? Does it form a square, or slope down into a point?\n\nA new picture of the person emerges. Yes, of course, commonalities remain. His name equally belongs to other people, his nose to other faces. We are all made of the same blood and bones. But take a closer look. See the proportions, the interplay between all the various parts? Every combination, like a mosaic, is unique. He has his father's eyes, and his mother's curls, and his uncle's lopsided smile. Together they create something, and someone, new. Someone who will look through those eyes in his own way, who will wear that hair according to his own style, who will deploy that smile for his own reasons. Talk to that person. Watch the skein of laughter lines that diagram his face. And how his eyes glisten, or darken, at the sound of certain words. He is simply being himself.\n\nQuetelet (and many others after him) believed that the essence of human nature could be found in the average, but he was mistaken. The essence of human nature is its endless variety. As Stephen Jay Gould would later remark, \"All evolutionary biologists know that variation itself is nature's only irreducible essence. Variation is the hard reality, not a set of imperfect measures for a central tendency. Means and medians are the abstractions.\"\n\n# Twenty-Three\n\n# THE CATARACT OF TIME\n\nIf, as is often said, lifetimes flow like a river, they begin with a trickle and culminate in a cataract. Heraclitus, the ancient Greek philosopher, put it well when he said, \"Time is a game played beautifully by children.\" Perhaps this is the root of nostalgia: less the desire to return to our early years than to the more capacious experience of time that we inhabited as children.\n\nTime. You know how it goes. After the age of thirty, I found, the days begin to run away from us. We struggle to keep up. That is when the nostalgic impulse awakens, burgeons, pesters. Last year I moved to Paris. Something about being back in a big city after so many years meant I could not help thinking, more and more, about the old London neighborhood of my youth. I had reached that age when the past becomes so big and so deep that your mind finds itself increasingly drawn there. It is like living on a fragile coast, by an imposing sea whose smells and sounds gradually overwhelm your senses.\n\nSo I decided to return. There was salt water to cross, and buses that jolted and trains that bored, but none of those things mattered. I simply had to go and see the place again after all this time.\n\nMy younger siblings, when I spoke to them about the plan, chorused dissuasion. \"There is nothing there,\" they said, perplexed. Evidently, they had all moved on. \"Why go back?\"\n\nI tried to explain. I opened my mouth, felt the breath on my tongue form plausible shapes. But my heart was not in it and each of my made-up reasons fell flat. I decided not to argue. I booked the tickets, packed my bags, and left.\n\nThat I could not explain my desire to return to my old London home did not disturb me. On the contrary, its surprising strength convinced. Reassured, even. Gazing out from the wobbly train, I tried to remember when I had last set foot there. My reflection in the window wore an expression of thought. Tall trees and green hills ran past. I looked away, cracked open a book, and stared at the pages till the words seemed to gel into a single inky mass.\n\nFive years. Already? Where had they gone? So many things that I had accomplished, people that I had met, places that I had seen, now looking back, seemed to have taken hardly any time at all. And yet how difficult, how exhausting, how important each event had struck me in the moment! And how impossibly distant, a lifetime away, these hours in the train out of Paris would have appeared to me back then.\n\nIt was a fast service to London, without delays. Arriving in the capital, I felt more like a suburban commuter than an international traveler. I changed trains and rattled out from the center toward the familiar periphery, my excitement building. Gradually the carriage emptied of its suits; a different class of passenger took their place. \"We must be getting close,\" I thought, straining forward, oblivious to my wristwatch. Near the end of the line, I gathered my things\u2014my bearings too\u2014and stepped out. The platform was covered in litter and broken glass, but for an instant, at least, it felt unambiguously good to be back.\n\nTime is more than an attitude or a frame of mind. It is about more than seeing the hourglass as half empty or half full. More than ever in this age, let us call it the computer age, a lifetime has become a discrete and eminently measurable quality. To date, to believe the surveys in newspapers, I have spent some one hundred thousand minutes standing in a line, and five hundred hours making tea. I have spent a year's worth of waking days on the hunt for lost things. This year, I knew, contained my twelve-thousand-and-twelfth day and night. That number equates to over a quarter of a million hours, seventeen and a quarter million minutes. Counting one number for every second since my birth, I had recently made it into the billionaire's club.\n\nWe occasionally liken time to money, as something to be spent wisely, but it is not money. No refunds are possible for days ill spent; no bank exists to take savings. We cannot apportion our time like money, since we live always in ignorance of when the former will end. How to plan when a person can never know if he will see tomorrow, or survive to such an age that his eyes turn coal-black with blindness?\n\nPerhaps it would be better to talk about time in the manner of certain tribes. Strangers to clocks, they pace their days according to nature. Native Americans traditionally planted corn \"when the leaf of the white oak was the size of a mouse's ear.\" Equinoxes and solstices scheduled their rituals. As for language, the Sioux have no word for \"late\" or \"waiting.\"\n\nIn Australia, the Aborigines believe that time, place, and people are one. A glance at a tree or a face suffices to know the hour and the day. Their discrimination of the seasons is precise, depending on such factors as plant life and changes in the wind: the Eastern Gunwinggu, for example, speak of six seasons\u2014three \"dry\" and three \"wet\"\u2014where non-Aborigines see only one of each.\n\nFor these and other tribes, time is the product of one's actions. It appears when they sing a song, climb a mountain, or smoke a pipe, and vanishes when they sleep. They do not think of time as something pervasive, like the air. Seconds, minutes, and hours\u2014these are all things that we do. In place of these terms, they speak of a \"time of harvesting\" or a \"river fish time.\" Ask an African herdsman how long such-and-such task might take and he replies, \"Cow milking time,\" meaning the time it takes to milk a cow. What is an hour to such a man? Perhaps the time it takes to milk ten cows.\n\nWe can put it another way: 1 hour = 10 milkings. My equivalent would be 1 hour = 10 tea-makings. Let us call it \"tea time.\" A short walk that lasts eighteen minutes equates to three milkings or makings-of-tea; a two-minute commercial break amounts to one-third of a cup of tea. Between the opening and closing whistles of a soccer referee, time enough would pass to milk fifteen cows, or make fifteen cuppas.\n\nI do not mean by this digression to suggest that approximations necessarily trump exactitude. It is not at all my intention to run clocks down. But the particular words and images our respective cultures deploy shape the way in which we experience time. I said just now that time is not money; we might say instead that it is closer to the spending of money. According to the tribesmen's way of thinking, it is what happens when, for example, we enter a marketplace. This emphasis on activity in how we think about time strikes me as being very healthy. When I hear someone complain about all the hours or weekends he has to fill, I stop and think that it is a mistake to speak of days as we would speak of holes. One hole is much the same as any other, whereas every day is different. In this, it is more like dough that we can sculpt into infinitely varying shapes.\n\nOn the journey back to my childhood home, I paused outside the train station, then made my way north toward the high street. The buildings were more or less the same as in my recollection: the same squat walls, tattooed with graffiti; the same \"50 percent off\" signs in shop windows; the same boys and girls, their busy fingers unwrapping candy. No bravado in the architecture, no color or charm. Along the sidewalks, no bustle either\u2014either too early or too late for shoppers. Few cars animated the road. I walked mechanically, turning here and there, smelling the sugar of freshly laid tarmac on Waterbeach Road.\n\nI landed finally on my old street. I took it all in. On the left stood metal railings and distantly behind them the classroom buildings of my former elementary school, factory-long. To the right, a chain of brick houses, close set. Their thin walls, I recall, made bad neighbors. Down the road, I spotted a small man in the distance. The man grew bigger with every step. He was wearing a blue-and-red jersey, but he did not look like a soccer player. The tightness of the shirt pronounced a sizable paunch. His dark hair was cut penitentiary short. His breathing rasped as he passed me by. And then he was gone.\n\nI was surprised by how little had been altered. Painted house number signs, wooden gates, hedgerows all long forgotten, I recognize instantly. And yet it all seems so different from my kid days. Something has shifted out of sync, something I try to put my finger on. In frustration, I walk up and down the street until my legs tire. Only as I ready myself for the ride back does it hit me. What has changed here is time.\n\nIn his 1890 classic work, _The Principles of Psychology,_ the American philosopher William James noted, \"The same space of time seems shorter as we grow older\u2014that is, the days, the months, and the years do so; whether the hours do so is doubtful, and the minutes and seconds to all appearance remain about the same.\"\n\nJames goes on to cite a mathematical explanation for this phenomenon, by a contemporary French professor. According to this professor, Paul Janet, our experience of time is proportional to our age. For a ten-year-old child, one year represents one-tenth of his existence; whereas for a man of fifty, the same year equates only to one-fiftieth (2 percent). The older man's year will thus seem to elapse five times faster than the child's; the child's, five times slower than the man's.\n\nWhat matters, then, is the relationship between one sequence of years and another sequence. The interval spanning the ages of thirty-two and sixty-four will seem to the individual of similar duration to that experienced between the ages of sixteen and thirty-two, and to the interval between the ages of eight and sixteen, and to that from the age of four to eight, each having the same ratio. For the same reason, all the years from the age of sixty-four to one hundred and twenty-eight (assuming such an age were ever attainable) would seem to us to occupy no more of our feeling, thought, pain, fear, joy, and wonder than that big bang epoch between our second and fourth year.\n\nMore recently, from T. L. Freeman, we have a formula using Janet's insight that yields the individual's \"effective age.\" Freeman's calculations suggest that we experience a quarter of our entire lifetime by age two, over half by age ten, and more than three-quarters by our thirtieth birthday. At only about the chronological midway point, a forty-year-old will experience his remaining time as seemingly but one-sixth of what has gone before. For a sixty-year-old, the future will seem to last merely one-sixteenth the duration of his past.\n\nAre all our attempts to look back, to relive some bygone period, in vain? We can never walk down the same street twice. Those streets of my youth belong to another time, which is no longer my own. Except, that is, when I dream.\n\nFast asleep, I become a visitor there. I see a schoolgirl at the edge of the hopscotch grid, contemplating her throw. A man, atop a ladder, is washing his windows. His free hand glides rhythmically upon the glass. On the pavement, a neighbor's tabby squirms in the sunshine: stretching, and stretching his paws. The grunts and sighs of passing traffic fill my ears. I see my grandfather, alive, standing with his cane at the gate, as though keeping guard over my father's vegetable patch. I stop and watch my father. Sleeves hoisted to his elbows, he picks beans, sows herbs, and counts cucumbers. I watch without hurry, without a care in the world. Time is dilated; there is no time.\n\nAsleep or awake, our bodies keep time a great deal better than our brains. Hair and nails grow at a predictable rate. An intake of breath is never wasted; appetite hardly ever comes late or early. Think about animals. Ducks and geese need only follow their instinct for when to pull up sticks and migrate. I have read of oxen that carried their burden for precisely the same duration every day. No whip could persuade them to continue beyond it.\n\nWe wear the tally of our years on our brows and cheeks. I doubt our bodies could ever lose their count. Like the ox, each knows intimately the moment when to stop.\n\n# Twenty-Four\n\n# HIGHER THAN HEAVEN\n\nOn January 22, 1886, Georg Cantor, who had discovered the existence of an infinite number of infinities, wrote a letter to Cardinal Johannes Franzen of the Vatican Council, defending his ideas against a possible charge of blasphemy. A devout believer, the mathematician considered himself a friend of the Church. God, he believed, had used his preoccupation with numbers to reveal a further aspect of His infinite nature. Fellow logicians had mostly sidestepped the young man's thinking; hardly anyone yet took seriously the outstanding insights that would make his name.\n\nBefore Cantor it had been impossible to speak mathematically about different sorts of infinity. All collections without a final object (the sequence of odd or even numbers, for example, or the primes) were simply conceived as being of equal size. Cantor proved that this was false. His papers were the first to demonstrate uncountable sets of numbers, that is to say, numerical sequences that even an infinitely long recitation could not exhaust. What is more, each uncountable set of numbers spawned another set of numbers that was even \"bigger\" than the last. Of the making of such sets, Cantor realized, there was no end.\n\nThe mathematician Leopold Kronecker, for whom \"God created the integers [whole numbers], all else is the work of man,\" had no truck with Cantor's (infinite) tower of \"smaller\" and \"bigger\" infinities. He hounded his rival with violent words, called him a charlatan, a corrupter of youth. In the absence of his peers' understanding, Cantor turned at last for support to the Holy See.\n\nThe dialogue between theology and mathematics\u2014varied, fitful, and singular\u2014has a long history. Above all, infinity became the favorite topic. God is infinite, therefore mathematics is religion: a pathway to knowledge of the divine. This is what the Church fathers reasoned, and this is why the monks long ago proceeded where the mathematicians had feared to tread.\n\nA thousand years before Cantor, in an Irish monastery, a man sat day after day at a table smelling of wicks and manuscripts. He spent years almost immobile, in deep and sustained contemplation, meditating on a perfect sphere that exists beyond space, universal and without limit. Of course, it is contradictory to think about a shape that has no border. The monk knew this. He knew that to think about infinity is to think in contradictions.\n\nMinutes passed, hours passed. But what is a minute or an hour when compared to eternity? No time at all. A minute, an hour, a year, a thousand years, are all equally long or short in comparison. The light in the monk's cell would gradually disperse at the end of each long day; his mind might stutter, \"I, I, I, I, I...,\" but try as he would, Johannes Scotus Eriugena\u2014John of Ireland\u2014could not escape his senses and grasp the infinite.\n\nAccording to Eriugena, God is not good, since He is beyond goodness; not great since He is beyond greatness; not wise since He is beyond wisdom. God, he writes, is more than God, more than time, _infinitas omnium infinitatum_ (the infinity of all infinities), the beginning and end of all things, though He Himself had no beginning and will meet no end. Eriugena recalls the words of Job:\n\n> Can you search out the deep things of God? Can you find out the limits of the Almighty? They are higher than heaven\u2014what can you do? Deeper than Sheol\u2014what can you know? Their measure is longer than the earth and broader than the sea. If He passes by, imprisons, and gathers to judgment, then who can hinder Him?\n\nIf God is infinite, Holy Scripture, being inspired by God, is held to exist outside the bonds of conventional time. Eriugena cites St. Augustine to affirm that the Bible often employs the past tense to express the future. Adam's life in Paradise \"only began,\" occupying no real time at all, so that its depiction in Genesis \"must refer rather to the life that would have been his if he had remained obedient.\"\n\nAugustine's teachings contributed greatly to the Irish monk's thought, and that of the theologians who followed. In _The City of God,_ Augustine insists that God knows every number to infinity and can count them all instantaneously. \"If everything which is comprehended is defined or made finite by the comprehension of him who knows it, then all infinity is in some ineffable way made finite to God, for it is comprehensible by his knowledge.\"\n\nTwo centuries after Eriugena, in 1070, Anselm provided his famous \"ontological proof\" that God is that-than-which-nothing-greater-can-be-thought. If every number has its object, the object of infinity is God. Anselm became Archbishop of Canterbury; one of his successors, Thomas Bradwardine, in the fourteenth century, identifies the divine being with an infinite vacuum. The finite world is compared to a sponge in a boundless sea of space.\n\nInfinity begets finitude, and thus cannot be grasped in finite terms. But how then to understand infinity in infinite terms? Alexander Neckham, a twelfth-century reviver of interest in Anselm's work, offered this problem a vivid image. For Neckham, God's immensity is such that even if one were to double the world in the next hour, and then triple it in the hour after that, then quadruple it in the following hour, and so on, still the world would be but a \"quasi point\" in comparison.\n\nSuch immensity inspires in the monks at once admiration and consternation: consternation, because an infinitely remote divine being would rule out the Incarnation. For the same reason, the believer would never see God in the Beatific Vision, and neither could he ever conform his will to the divine will. The vacuum is in fact a chasm, forever separating Mankind from its Creator.\n\nThe _De Veritate_ of Thomas Aquinas, written between 1256 and 1259, offers a solution: \"as the ruler is related to the city, so is the pilot to the ship.\" An infinitely powerful ruler bears no direct comparison to a humble captain, yet both possess a \"likeness of proportions\": a finite quantity equates to another finite quantity, in the same way that the infinite is equal to the infinite. In other words, \"three is to six as five million is to ten million\" bears a likeness to the proportion \"God is to the angels as the infinite vacuum is to an eternal creation.\" Aquinas deploys the analogy throughout his work: as our finite understanding grasps finite things so does God's infinite understanding grasp infinite things; as our finite intellect is to what it knows, so is God's infinite intellect to the infinitely many things He knows; just as men distribute finite goods so does God distribute all the goods of the universe. Aquinas writes that the similarity between the infinite God and His finite creation constitutes a \"community of analogy... The creature possesses no being except insofar as it descends from the first being, nor is it named a being except insofar as it imitates the first being.\"\n\nExasperated by critics he called \"murmurers,\" Aquinas sought to settle a further point of contention. The Church taught that the world had a beginning in time. \"The question still arises whether the world could have always existed.\" He penned these words in 1270, entitling them _De Aeternitate Mundi_ (On the Eternal World). His argument was that if the world has always existed, the past regresses infinitely. The world's history must comprise an infinite sequence of past events. If there exists an infinite number of yesterdays, then an infinite number of tomorrows must also succeed. Time is infinitely past, and infinitely future, but never present. For how can any present moment arrive after infinitely many days?\n\nBefore this potentially unsettling line of reasoning, Aquinas remained unmoved and unimpressed. Halfhearted were his remonstrations. Any past event, like the present moment, is finite: therefore the duration between them is also finite, \"for the present marks the end of the past.\"\n\nAnd what about the succession of past events? Aquinas says the arguments for them can go either way. Perhaps God, in all His power, has created a world without end. If so, nothing obliged Him to populate it before Mankind.\n\nA contemporary, Bonaventure, disagreed with Aquinas's equity. His blood thumped at the thought of an interminable past. \"To posit that the world is eternal or eternally produced, while positing likewise that all things have been produced from nothing, is altogether opposed to the truth and reason.\" And what about the contradictions? For instance, if the world were eternal, tomorrow would be a day longer than infinity. But how can something be greater than the infinite?\n\nIn the fourteenth century, Henry of Harclay also faulted Aquinas for saying that an eternal world was possible, but from a point of view entirely opposed to Bonaventure's. For Harclay it was in fact probable, and every supposed contradiction dissolved on careful scrutiny. How can something be greater than the infinite? Look, said Harclay, at the infinite number of numbers: we can count from two, or from one hundred, and in both instances never reach a final number, though there are more numbers to count in the first infinity than there are in the second. He invoked Aquinas's proportions to defend the thesis of an infinite universe in which the infinitely many months occur twelve times more frequently than the infinitely many years.\n\nTo those who point out that an infinite past would have produced an infinite number of souls with infinite power like God, Harclay refutes the argument as follows: infinitely many souls would not constitute an infinite power. They would be not \"any species of number, but a multitude of infinitely many numbers.\" Within this endless multitude, every possible number (59, 1,043,962, 999,999,999,999,999,999,999,999,999,999...) could be found, distinct and finite, each corresponding to a soul; save, that is, for an infinitieth number\/soul since this would produce a contradiction: \"there is not a number of infinite numbers, for then it would contain itself, which is impossible.\"\n\nWe trace to the same period, in the monk Gregory of Rimini's hand, the first definition of an infinite number as that which has parts equally great as the whole. Every twenty-third number, for example (we might just as well have taken every ninety-ninth number, or every third, or every five billionth), in the infinite succession of counting numbers (1, 2, 3, 4, 5, 6...) produces a sequence as long\u2014infinitely long\u2014as all the counting numbers combined: match one with twenty-three, two with forty-six, three with sixty-nine, four with ninety-two, five with one hundred and fifteen, and so on, ad infinitum.\n\nGregory articulated his defining idea fully five centuries before Cantor. He taught for many years in Paris, at the Sorbonne, where his pupils called him _Lucerna splendens_. Perhaps in him they sensed, as future scholars would claim, the last great scholastic theologian to wrestle with the infinite.\n\nJohn Murdoch, a historian of mathematics at Harvard University, remarked that Gregory's insight received hardly any notice from his peers or successors:\n\n> Since the \"equality\" of an infinite whole with one or more of its parts is one of the most challenging, and as we now realize, most crucial aspects of the infinite, the failure to absorb and refine Gregory's contentions stopped other medieval thinkers short of the hitherto unprecedented comprehension of the mathematics of infinity which easily could have been theirs.\n\nIn his writings, Cantor described himself as a servant of God and the Church. His ideas had struck him with the force of revelation. It had been with God's help, he said, that he had worked day after day, alone, at his mathematics. But the mathematician was far from angelic; his humility sometimes slipped. To a friend, in 1896, Cantor confided an excess of pride. \"From me, Christian Philosophy will be offered for the first time the true theory of the infinite.\"\n\n# Twenty-Five\n\n# THE ART OF MATH\n\nI met a mathematician at a \"conference of ideas\" in Mexico at which we had both been invited to speak. He was from the United States, and like all the mathematicians that I have ever met in my travels he fell immediately to talking shop. Moving to a corner of the conference green room, he talked to me about the history of numbers in Cambodia. The concept of zero, he believed fervently, the familiar symbol of nothingness, hailed from there. He dreamed of trekking the kingdom's dirt tracks, in pursuit of any surviving trace. More than a millennium separated him from the decimal system's creation; the odds of turning up any new evidence were slim. But he did not mind.\n\nHe began to explain his current research in number theory, talking quickly with the compression of passion, and I listened intently and tried to understand. When I understood, I nodded, and when I did not understand, I nodded twice, as if to encourage him to move on. His words came fast and enthusiastic, opening up vistas that I could not quite see and mental regions into which I could not follow, but still I listened and nodded and enjoyed the experience very much. Occasionally I supplemented his ideas and observations with some of mine, which he received with the utmost hospitality. It always feels exciting to me, the camaraderie of conversation: no matter whether it involves words or numbers.\n\nHe had none of the strange tics or quirks of the mathematicians that we find in books or see in movies. From experience, I was not the least surprised. Middle-aged, he looked fit and slender, though with skin as pasty as a writer's. His shirt was open at the neck. His face wore many laughter lines. When our time was up, too soon, he patted his pockets and withdrew from one of them a small notebook in which he habitually jotted down his random thoughts and sudden illuminations. As he wrote out his contact information for me, I noticed the smallness and smoothness of his hands.\n\n\"Great meeting you.\" We promised to stay in touch.\n\nIt was still a pleasant surprise, coming down the next morning to the hotel restaurant for an early breakfast, to hear the mathematician's voice call me over to his family's table. I passed the assorted reporters munching their bowls of cereal, and various conference \"stars,\" dodging coffee-flecked waiters and pushing empty chairs out of my way, until I reached them. The mathematician smiled at his wife (also a mathematician, I learned), and the surprisingly placid teenage girl, who looked a lot like her mother, sat in between. Their flight out was still a few hours away: over tea and toast, we talked.\n\nWe talked about the Four Color Theorem, which states that all possible maps can be colored in such a way that no district or country touches another of the same color\u2014using only (for instance) red, blue, green, and yellow. \"At first sight it seems likely that the more complicated the map, the more colors will be required,\" writes Robin Wilson in his popular account of the puzzle's history, _Four Colors Suffice,_ \"but surprisingly this is not so.\" Redrawing a country's boundary lines, or imagining wholly alternative continent shapes, makes no difference whatsoever.\n\nOne aspect of the problem, in particular, had long intrigued me. After more than a century of fruitless endeavors to demonstrate the theorem conclusively, in 1976 a pair of mathematicians in the United States finally came up with a proof. Their solution, however, proved controversial because it relied in part on the calculations of a computer. Quite a few mathematicians refused to accept it: computers cannot do math!\n\n\"I actually met one of those guys who came up with the proof,\" my new friend recalled, \"and we discussed how they had found just the right way to feed the data into the machine and get an answer back. It really was a smart result.\"\n\nWhat did he and his wife think of the computer's role in mathematics? In answer to this general question they were more circumspect. The Four Color Theorem's proof, they admitted, was inelegant. No new ideas had been stimulated by its publication. Worse, its pages were almost unreadable. It lacked the intuitive unity, and beauty, of a great proof.\n\nBeauty. How often have I heard mathematicians employ this word! The best proofs, they tell me, possess \"style.\" One can often surmise who authored the pages simply from the distinctive way that they were put together: the selection, organization, and interplay of ideas are as personal, and as particular, as a signature. And how much time might they spend on polishing their proofs. Superfluous expressions, out! Ambiguous terms, out! Yes, but it was worth all the trouble: well-written proofs could become \"classics\"\u2014to be read and enjoyed by future generations of mathematicians.\n\n\"What time is it?\" None of us was wearing a watch. We stopped a waiter and asked. \"Already?\" said the mathematician's wife when she heard his answer. They drained their cups, and dispersed their crumbs, and made shuffling sounds with their feet.\n\n\"Oh,\" said the mathematician, turning back to me, \"I forgot: where did you say you were based again?\" What with the history of the decimals, and the winding numerical vistas, and the painting of the entire globe with the colors of a single flag, the accidental features of our lives\u2014where we lived, with whom, under what roof and color of sky\u2014had been completely absent from our conversations.\n\nI told him. \"Paris,\" he echoed. \"Why, we love Paris!\"\n\nFrance's capital has something of a one-sided reputation as the consummate city of artists. We know it as the city of Manet, of Rodin, of Berlioz; as the city of street singers and can-can dancers; as the city of Victor Hugo and of young Hemingway in _A Moveable Feast:_ scribbling in a caf\u00e9 corner, turning coffee and rum and the strictures of Gertrude Stein into stories. But Paris is also the city of mathematicians.\n\nIts researchers, a thousand strong, make the _Fondation Sciences Math\u00e9matiques de Paris_ (FSMP) the largest group of mathematicians in the world. About one hundred of the city's streets, squares, and boulevards are named after their predecessors. In the twentieth _arrondissement,_ for example, one can walk the length of _rue_ Evariste Galois, named after a nineteenth-century algebraist felled at the age of twenty by a dueler's bullet. On the opposite side of the Seine, in the fourteenth _arrondissement,_ lies _rue_ Sophie Germain, whose namesake introduced important ideas in the fields of prime numbers, acoustics, and elasticity before her death in 1831. According to her biographer Louis Bucciarelli, \"She did not wish to meet others in the streets or houses of the day, but in the purer realm of ideas outside time, where person was indistinguishable from mind and distinctions depended only on qualities of intellect.\" A few minutes' walk away is Fermat's little road. There are also streets called Euler, and Leibniz, and Newton.\n\nAmong the letters waiting for me on my return to my adopted home was one from the city's _Fondation_ Cartier. A museum for contemporary art, it had sent me a preview invitation to its upcoming exhibition \"Mathematics: A Beautiful Elsewhere\"\u2014the first in Europe to showcase the work of major living mathematicians in collaboration with world-class artists. The timing seemed doubly auspicious: October 2011 happened to be the two-hundredth anniversary of Galois's birth.\n\nThe museum stands in the fourteenth _arrondissement_ at the lower end of one of the long boulevards that diagram the city. It is an ostentatiously modern building, all shiny glass and geometric steel, bright and spacious, an example of \"dematerialized\" architecture. Reflected in the glass, scraggly trees denuded of their summer foliage appeared twice. I looked up at the symmetrical branches as I passed and entered.\n\nMathematics and contemporary art may seem to make an odd pair. Many people think of mathematics as something akin to pure logic, cold reckoning, soulless computation. But as the mathematician and educator Paul Lockhart has put it, \"There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics.\" The chilly analogies win out, Lockhart argues, because mathematics is misrepresented in our schools, with curricula that often favor dry, technical, and repetitive tasks over any emphasis on the \"private, personal experience of being a struggling artist.\"\n\nIt was the mathematicians' artistic impulse, and inner struggle, that the exhibition's organizers intended both to communicate and celebrate. A white interior, zero-shaped, was the work of the filmmaker David Lynch. Walls usually reserved for frames and canvases lent their space to equations, light effects, and number displays. I walked through the rooms, now bare and silent, now colorful and stimulating, stopping here and there to take a closer look. I watched the other guests stand back and point and converse in low voices. Before a bright collage of sunrays and leopard spots, waves and peacock tails, and the underlying equations for each, fingers swayed and eyes widened. Another hall arrested visitors' feet around a lean aluminum sculpture, its curves reaching toward infinity.\n\nBut for me, the highlight of the exhibition took place in a darkened room downstairs. Here the visitors melted into twilight, rendered homogeneous in the darkness, sitting or standing in silence, all eyes, observing a large screen where a film shot in black and white was playing. A youngish face, screen big, was talking about his life as a mathematician. I pressed my back against the far wall and listened as he spoke of \"fat triangles\" and \"lazy gases.\" Three or four minutes later, the film suddenly altered: the face gave way to another, wearing glasses. Four minutes after this, the face changed again: this time, a woman's voice began to speak about chance. In total, the film lasted thirty-two minutes\u2014eight faces long. The men and women featured came from a wide range of mathematical subdisciplines\u2014number theory, algebraic geometry, topology, probability\u2014and spoke either in French, or English, or Russian (with subtitles), but their passion and wonder linked each personal testimony into a fascinating and involving whole.\n\nTwo of the testimonies, in particular, stood out. They reminded me of my conversations with the mathematicians in Mexico, and with those in other lands, and the feelings of kinship and excitement that these exchanges incited within me. During his four minutes, Alain Connes, a professor at the _Institut des Hautes \u00c9tudes Scientifiques,_ described reality as being far more \"subtle\" than materialism would suggest. To understand our world we require analogy\u2014the quintessentially human ability to make connections (\"reflections\" he called them, or \"correspondences\") between disparate things. The mathematician takes ideas that are valid in one area and transplants them into another, hoping that they will take and not be rejected by the recipient domain. The creator of \"noncommutative geometry,\" Connes himself has applied geometrical ideas to quantum mechanics. Metaphors, he argued, are the essence of mathematical thought.\n\nSir Michael Atiyah, a former director of the Isaac Newton Institute for Mathematical Sciences in Cambridge, used his four minutes to speak about mathematical ideas \"like visions, pictures before the eyes.\" As if painting a picture or dreaming up a scene in a novel, the mathematician creates and explores these visions using intuition and imagination. Atiyah's voice, soft and earnest, made attentive listeners of everyone in the room. Not a single cough or whisper intervened. Truth, he continued, is a goal of mathematics, though it can only ever be grasped partially, whereas beauty is immediate and personal and certain. \"Beauty puts us on the right path.\"\n\nThe faces, old and young, smooth and hairy, square and oval, each had their say. Gradually, the room began to empty. Its intimate ambience slowly dissolved. I followed the last group of visitors up the stairs and out the building and not a word was exchanged. The night absorbed us.\n\nI walked for a while, beside the river, with the night in my hair and in my pockets and on my clothes. The night, I know, is tender to the imagination; at this hour, throughout the city, artists sharpen pencils and dip brushes and tune guitars. Others, with their theorems and equations, revel just as much in the world's possibilities.\n\nThe world needs artists. Into words and pictures, notes and numbers, each transforms their portion of the night. A mathematician at his desk glimpses something hitherto invisible. He is about to turn darkness into light.\n\n# ACKNOWLEDGMENTS\n\nI could not have written this book without the love and encouragement of my family and friends.\n\nSpecial thanks to my partner, J\u00e9r\u00f4me Tabet.\n\nTo my parents, Jennifer and Kevin; my brothers, Lee, Steven, and Paul; and my sisters, Claire, Maria, Natasha, Anna-Marie, and Shelley.\n\nThanks also to Sigri\u00f0ur Kristinsd\u00f3ttir and Hallgrimur Helgi Helgason, Laufey Bjarnad\u00f3ttir, and Torfi Magn\u00fasson, Valger\u00f0ur Benediktsd\u00f3ttir and Gr\u00edmur Bj\u00f6rnsson, for teaching me how to count like a Viking.\n\nTo my most loyal North American readers, Margo and Linda Flah, Jean-Philippe Tabet, Val\u00e9rie and Arnaud Salambier, and Claire Bertrand (and family!).\n\nI am grateful to my literary agent, Andrew Lownie, and to Tracy Behar and Christina Rodriguez, my editors.\n\n# ABOUT THE AUTHOR\n\nDANIEL TAMMET is the critically acclaimed author of the worldwide bestselling memoir _Born on a Blue Day,_ and the international bestseller _Embracing the Wide Sky._ Tammet's exceptional abilities in mathematics and linguistics are combined with a unique capacity to communicate what it is like to be a savant. His idiosyncratic worldview gives us new perspectives on the universal questions of what it is to be human and how we make meaning in our lives. Tammet was born in London in 1979, the eldest of nine children. He lives in Paris.\n\n# ALSO BY DANIEL TAMMET\n\n_Born on a Blue Day_\n\n_Embracing the Wide Sky_\n\n### Thank you for buying this ebook, published by Hachette Digital.\n\nTo receive special offers, bonus content, and news about our latest ebooks and apps, sign up for our newsletters.\n\n Sign Up\n\nOr visit us at hachettebookgroup.com\/newsletters\n\nFor more about this book and author, visit Bookish.com.\n\n# CONTENTS\n\nCover\n\nTitle Page\n\nWelcome\n\nEpigraph\n\nPreface\n\nChapter One: Family Values\n\nChapter Two: Eternity in an Hour\n\nChapter Three: Counting to Four in Icelandic\n\nChapter Four: Proverbs and Times Tables\n\nChapter Five: Classroom Intuitions\n\nChapter Six: Shakespeare's Zero\n\nChapter Seven: Shapes of Speech\n\nChapter Eight: On Big Numbers\n\nChapter Nine: Snowman\n\nChapter Ten: Invisible Cities\n\nChapter Eleven: Are We Alone?\n\nChapter Twelve: The Calendar of Omar Khayy\u00e1m\n\nChapter Thirteen: Counting by Elevens\n\nChapter Fourteen: The Admirable Number Pi\n\nChapter Fifteen: Einstein's Equations\n\nChapter Sixteen: A Novelist's Calculus\n\nChapter Seventeen: Book of Books\n\nChapter Eighteen: Poetry of the Primes\n\nChapter Nineteen: All Things Are Created Unequal\n\nChapter Twenty: A Model Mother\n\nChapter Twenty-One: Talking Chess\n\nChapter Twenty-Two: Selves and Statistics\n\nChapter Twenty-Three: The Cataract of Time\n\nChapter Twenty-Four: Higher than Heaven\n\nChapter Twenty-Five: The Art of Math\n\nAcknowledgments\n\nAbout the Author\n\nAlso by Daniel Tammet\n\nNewsletters\n\nCopyright\n\n# Copyright\n\nCopyright \u00a9 2012 by Daniel Tammet\n\nCover design by Kimberly Glyder\n\nCover \u00a9 2013 Hachette Book Group, Inc.\n\nAll rights reserved. In accordance with the U.S. Copyright Act of 1976, the scanning, uploading, and electronic sharing of any part of this book without the permission of the publisher is unlawful piracy and theft of the author's intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained by contacting the publisher at permissions@hbgusa.com. Thank you for your support of the author's rights.\n\nLittle, Brown and Company\n\nHachette Book Group\n\n237 Park Avenue, New York, NY 10017\n\nlittlebrown.com\n\ntwitter.com\/littlebrown\n\nfacebook.com\/littlebrownandcompany\n\nFirst ebook edition: July 2013\n\nLittle, Brown and Company is a division of Hachette Book Group, Inc.  \nThe Little, Brown name and logo are trademarks of Hachette Book Group, Inc.\n\nThe publisher is not responsible for websites (or their content) that are not owned by the publisher.\n\nThe Hachette Speakers Bureau provides a wide range of authors for speaking events. To find out more, go to hachettespeakersbureau.com or call (866) 376-6591.\n\nExcerpt from _The Lottery Ticket_ by Anton Chekhov; excerpts from _Lolita_ by Vladimir Nabokov \u00a9 Vladimir Nabokov, published by Orion Books, used by permission; excerpt of interview with Vladimir Nabokov was taken from the BBC program _Bookstand_ and is used with permission; excerpts by Julio Cort\u00e1zar from _Hopscotch,_ \u00a9 Julio Cort\u00e1zar, published by Random House New York; quote from _The Master's Eye_ translated by Jean de la Fontaine; quote from _Under the Glacier_ by Halld\u00f3r Laxness, \u00a9 Halld\u00f3r Laxness, published by Vintage Books, an imprint of Random House New York.\n\nISBN 978-0-316-25080-1\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaaicb b/data_all_eng_slimpj/shuffled/split2/finalzzzaaicb
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+{"text":"30\" x 72\" Table folds to only 30\" x 36\" for easier handling, storage, and transportation. Easily fits in most car trunks, or behind the seat. Blow-Molded Top is waterproof, stain and chemical resistant. Top color goes all the way through, will not scratch or wear off.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A big thank you to Tom Marthaler for his enthusiastic participation on Team Hinson! And also, thanks to his daughter, Danyelle Marthaler, for introducing her dad to our campaign and also getting involved!\nTom got involved in politics at an early age, but it was when he experienced being a small business owner in Sauk Centre that he became a serious, regular participant in political activity. He became a member of the Independent Business Association of Minnesota (IBAM) in the 1980s and worked to influence Minnesota State legislators.\nCaucus night this year, Tom got a text message from his college-aged daughter, Danyelle Marthaler who was at the Moorhead Republican caucus and had just heard Amanda Lynn Hinson speak: \"Hey Dad, I just heard someone speak that makes a good candidate.\" Danyelle was excited that someone running for Congress shared her views, and hearing Amanda strengthened her own values. Danyelle, who is studying Communications at Minnesota State University in Moorhead, will be joining Team Hinson to work with social media.\n\u200b\"We're a politically active family,\" Tom said, noting that he wasn't surprised Danyelle made it to the caucus. \"I put my time into nurturing the soil, to worry about where she got her information and influences without micromanaging the results.\"\nBeing energized by his daughter's own excitement, Tom sent a note to the Hinson Campaign in April along with a package filled with office supplies and a donation. Since then, Tom has actively worked to spread the word about Amanda and her run for Congress. You may see Tom at a parade, hosting a meet and greet, or finding donors.\n\u200b\"It was easy for me to get excited about Ms. Hinson's candidacy, and I look forward to helping take her message to other folks, who, like me, have been waiting too long.\"\nPlease help us welcome Tom and Danyelle!\nWe are always looking for volunteers to help get the word out! You can join our team here!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Polish-U.S. Fulbright Commission would like to express its sincere gratitude to the pro-bono reviewers who have generously donated their time and expertise to the Polish Fulbright application review process. The independent application review process administered by the Polish-U.S. Fulbright Commission ensures that all applicants will be professionally evaluated on academic and scholarly criteria.\nApplication reviewers are carefully selected based on their knowledge and expertise within a specified area of study or discipline. Each application is reviewed by at least two independent subject matter experts. The reviewers are asked to read the application, evaluate its quality as well as candidate's fit for the project, and ultimately provide their professional recommendation.\nThe work of reviewers is vital to the success of the Fulbright program as they ensure a professional, fair, and competitive selection of Polish Fulbright grantees. Through their contribution, the reviewers not only support the international academic development of Polish researchers and lecturers but also have the opportunity to be exposed to ongoing projects in their field.\nWould you like to contribute to the Fulbright Program by becoming peer-reviewer? Fill in this short survey to be added to our database of prospective reviewers.\nLearn more about the Fulbright Program, the flagship academic exchange program of the United States.\nSupport of our peer-reviewers has significant impact on the selection of the Polish Fulbright grantees. Meet the current cohort.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mulit-material construction allows weight to be shifted for optimal performance. The lightweight titanium body allows for an oversized clubhead with increased offset for greater forgiveness and an increased effective hitting area. Heavy Tunite weighting in the sole creates a higher moment of inertia and better stability with a low, deep center of gravity. The TPU SenSert decreases vibration and delivers solid feel.\nDesigned for the ultimate in forgiveness in its oversized design with an increased offset, a large effective hitting area and a wide sole for exceptional turf interaction.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We always want to ensure that you have a positive experience when you contact us, and try to resolve your queries promptly, informatively and courteously. However, we acknowledge that sometimes problems can occur and are committed to resolving these with you. We also value your feedback in the event of a problem occurring. This page explains how and when to make a complaint.\nOur definition of a complaint is: \"\u2026 an expression of dissatisfaction, however made, about the standard of service, actions or lack of action by any part of this coordinating centre.\"\nThe complaints procedure excludes any matters that are currently the subject of litigation or legal proceedings or complaints by members of staff which are covered by University of Southampton's grievance procedures.\nYour complaint will be acknowledged within two working days and you will be given the timescale for our full reply. All complaints will be thoroughly investigated and you will be sent a full written response normally within 20 working days of your complaint being received.\nIf, however, we are unable to send a final response within that timescale, we will send you a holding reply telling you why and when you may expect to know the outcome.\nIf you have a complaint, you can let us know by telephone, in writing, by email or by fax. All complaints will be handled by the Business Manager in the Chief Executive's Office.\nThe Chief Executive Officer will review all the available information and investigation carried out and will reply directly to you. She will also at this stage advise you of your right, if you remain dissatisfied, to make an information request under the Freedom of Information Act.\nAny requests for information under either the Freedom of Information or Data Protection Acts will be handled in conjunction with the University of Southampton Legal Services department. Please see our Freedom of Information page for further details.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacsqp b/data_all_eng_slimpj/shuffled/split2/finalzzzacsqp
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+{"text":"Want to know more about our church \u2013 what we believe, our history, how our values shape our outreach, becoming a member, and more? That's what Discover GPBC is all about! Come explore what makes this faith community unique on Sunday, October 28, from 12 to 2:15pm in the Portland Rm. We'll enjoy lunch together, introduce you to key church leaders, orient you to GPBC, and give you a chance to ask questions. Free! Lunch is provided. RSVP at www.pdxchurch.org\/events\/reg\/. If you need childcare, please reserve a spot by the 21st.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"macys sofas sectional sleeper furniture sale coupon and loveseats.\nmacys sofas sofa sale malaysia leather reviews sectionals.\nmacys sofas fabric clearance sleeper sofa twin.\nmacys sofas s furniture reviews leather ottoman clearance couches.\nmacys sofas furniture sale 2014 sleeper sofa with chaise malaysia.\nmacys sofas s furniture leather chaise sofa sale 2015 recliners.\nmacys sofas furniture reviews leather chairs clearance couches.\nmacys sofas furniture sale coupon leather chairs reviews.\nmacys sofas sa sleeper sofa full leather sectionals furniture reviews.\nmacys sofas fabric sectional sofa furniture leather chaise sale coupon.\nmacys sofas com clearance couches furniture sale dates sleeper sofa full.\nmacys sofas sofa and chair leather furniture sale coupon.\nmacys sofas s furniture leather reviews couches sale.\nmacys sofas furniture leather sleeper sofa memory foam and chairs.\nmacys sofas ofa leather on sale sofa 2015 sleeper alaina.\nmacys sofas sas sa furniture sale memorial day 2015 sectional sleeper sofa.\nmacys sofas furniture sale memorial day 2015 sleeper sofa memory foam and chair.\nmacys sofas furniture leather recliners sofa and chair sale dates.\nmacys sofas furniture leather ottoman sectionals on sale.\nmacys sofas s leather sectionals furniture sale.\nmacys sofas furniture sale dates sleeper sofa twin leather recliners.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1 October 2018: The global Hemodialysis Machine Market is expected to witness significant growth over the forecast period owing to rising demand from end-user industries including pharmaceutical, personal healthcare, and medical device. Furthermore, growing prevalence of cardiovascular diseases including coronary artery disease, hypertension, and cardiac aneurysms is anticipated to drive the market growth over the next eight years. In addition, the growing adoption of hemodialysis machine for diabetes, and end-stage renal diseases is further projected to propel the sector demand over the forecast period. The market is projected to witness proliferated demand over the next eight years owing to increasing application in curing coronary artery disease, rising need to diagnose kidney failure and growing utilization in high blood sugar patients have age-related issues. However, high initial setup cost coupled with rising fluctuation in the raw material prices has enabled the market to witness hindered growth over the next eight years.\nThe market is segmented by product into centre-use hemodialysis machines, home-use hemodialysis machines, and others. This centre-use hemodialysis machine segment was accounted for a major market share in 2015 and it is projected to witness similar trend by 2024 owing to growing applications from independent dialysis centres and hospitals. On the other hand, home-use hemodialysis machine is anticipated to report the fastest growth over the next eight years primarily due to increasing demand for home hemodialysis and peritoneal dialysis.\nThe hemodialysis machine market is highly competitive primarily to the presence of several domestic and international suppliers. Though, the sector has the presence of various new players, who face difficulty in competing with other major companies in terms of functionalities, features, quality, and services. Over the forecast period, the growing new product development activities coupled with increasing adoption of new technologies for achieving operational efficiency has enabled the competition to become more aggressive. On a contrary, few major players through continuous R&D endeavors coupled with growing expertise knowledge in the industry have enabled them to witness competitive edge, thus propelling the overall growth of the sector over the next eight years.\nNorth America and Europe are anticipated to witness major market share over the forecast period owing to the growing number of independent dialysis centers coupled with rising government initiatives to develop better dialysis facilities. Asia Pacific is anticipated to account for fastest growth over the forecast period primarily due to the increasing ESRD patient population base coupled with growing awareness among end users regarding new developments in the dialysis treatment modalities. India, China, and Japan are anticipated to witness the highest share in the region majorly owing to the rising disposable income coupled with growing foreign investments. The Middle East & Africa is projected to see significant growth by 2020 mainly owing to increasing urbanization, industrialization, and globalization. The sector has huge potential in this region primarily owing to rising per capita income and foreign investments.\nKey players in the market include DaVita Healthcare Partners Inc. (U.S.), Fresenius Medical Care AG & Co. KGaA (Germany), B. Braun Melsungen AG (Germany), Baxter International, Inc. (U.S.) and Nipro Corporation (Japan). Few other companies in the hemodialysis industry include Nikkiso Co. Ltd. (Japan), Diaverum Deutschland GmbH (Germany) and NxStage Medical, Inc. (U.S.). The market is highly consolidated having more than 35% market share by major players. The major players generally tend to retain their market share through new product development and through attaining operational efficiency. These players through continuous R&D endeavors focus on building a wide distribution network that supports the overall market to report significant growth over the forecast period. Companies through strategic expansions and mergers & acquisitions have been trying to cater large volume of customers with products that are cost-effective and high quality.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Dead Ever After\" begins right after \"Deadlocked\" ended. Sam's recovering from his death and rebirth, Sookie wonders what is happening with Eric, and Eric is still dealing with his marriage contract to Freyda, Queen of Oklahoma. Overall, the book begins on a depressing note that doesn't lift until the very end.\nMs. Harris changes the format of this book, having different chapters narrated by different characters. I like how this lets the reader hear first hand all the intricacies of plot as they happen, as opposed to a second-hand telling afterwards. And intricacies abound here, as various people from the past novels show up, all with one thing in mind - kill Sookie.\nThe main plot of this book revolves around Sookie discovering who is trying so hard to kill her, and around Sookie trying to sort out her love life. I liked the parts in which Sookie has to defend herself, if nothing else because so many people who Sookie has helped show up to offer her support. It's nice to see how many come together when Sookie needs a hand. I dislike the parts in which Sookie thinks about and interacts with Eric. Considering how much they have been through, how much Eric and Sookie love each other, I found this story dissatisfying.\nI must admit, though I've enjoyed reading about Sookie and her adventures, I feel it's time for it to end. Ms. Harris seems to have run out of steam in this universe, and I look forward to reading what she writes next.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Crumple zones are part of the tough safety shell around your car, helping to protect you and your passengers in the event of an accident. To further enhance safety, SEAT cars include crumple zones at the front and rear of the shell.\nCrumple zones are designed to absorb impact energy during a collision so that most energy is dissipated across these zones, and not in the passenger compartment. The strong outer shell comes with front and rear crumple zones that help to absorb the impact of a crash, increasing the safety of the car's occupants.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadluv b/data_all_eng_slimpj/shuffled/split2/finalzzzadluv
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+{"text":"Welcome to 393 Mt. Airy Road, Basking Ridge, NJ - THINKING OF MOVING? THINK YOUNG!\nEnjoy carefree, on-trend living at this amazing home that is situated on a tranquil cul de sac near the heart of Basking Ridge. Completely updated, this stunning ranch home on just under a park like level acre is the perfect place to call home! Centrally located, you are minutes away from downtown Basking Ridge, Pleasant Valley Park and Pool, nationally ranked schools, world class shopping, hiking trails, sports & health facilities, highways and transportation to NYC.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pizza X has been committed to supporting our local community since 1982 and, as an extension of this effort, we have adopted a simple dine and donate program, called Xtend a Hand, to give local organizations a chance to raise much-needed funds.\nIf your organization is interested in participating, please complete the form at the bottom of the page at least 1 month prior to the month being requested. The more notice given, the more likely you'll get the month you prefer.\nOnce the dates have been confirmed internally, the organization must provide a logo. Pizza X will then provide a flyer template for the organization to distribute to their family, friends, and followers.\nThe key to making this fundraising effort successful is a massive advertising effort by the participating organization. Therefore, it's recommended to distribute promotional materials to everyone, and often!\nOrder and enjoy Pizza X each Monday during the month of your event! Make sure to use the code word if ordering my phone or click the \"Xtend a Hand\" tab if ordering online.\nSchedule as early as possible to give your organization ample time to promote.\nPromote using many different channels including word-of-mouth, social media, and e-mail blasts.\nSet up event reminders through e-mail and social media at least 48 hours before the event.\nFeel free to tag Pizza X on your Xtend a Hand promotional posts to help reach more people.\nReach out to your organization's friends\/partners to help spread the word!\nEncourage staff and volunteers to share social media posts and other promotional materials with their family and friends.\nEach organization may schedule 1 event during a calendar year.\nAn event is considered every Monday during a calendar month.\nPizza X will only host Xtend a Hand events every Monday of a calendar month.\nPizza X will donate 20% of purchases (excluding tax, delivery fee, and tip) when the organization-specific code word is used during a phone order or the organization is chosen under the \"Xtend a Hand\" tab when placing an order online during the event.\nXtend a Hand donations are good for both pick-up and delivery orders.\nThe organization may spread the word by any means necessary, but may not stand outside of any Pizza X location to target incoming customers.\nXtend a Hand events are valid at all 5 Pizza X locations.\nPizza X reserves the right to decline an Xtend a Hand request.\nPlease complete the form at the bottom of the page at least 1 month prior to the month being requested. The more notice given, the more likely you'll get the month you prefer. The Pizza X team will respond as soon as possible.\nWe will try to get your check out ASAP, but be sure to expect it within 2-3 weeks.\nPlease contact Pam Thrash at pam@bloomington.com or by calling 812-339-2256 x. 14 with any Xtend a Hand questions.\nWe didn't reach our donation goal. Any tips for next time?\nMake sure to schedule early and exhaust all promotional avenues to make everyone is aware of the event.\nStill interested? Fill out this form to see if we can get together and raise some money!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Lake Erie Storm baseball program has signed a three year contract with Classic Park to play all Lake Erie home game at Classic Park.\nConstructed by the city of Eastlake for $20 million prior to the start of the 2003 season, Classic Park is home to the Lake County Captains, the Single-A affiliate of the Cleveland Indians. The 7,000-seat venue features a surface that was voted the South Atlantic League's best three times and has a 4,000-square foot indoor batting facility with two full-sized batting cages utilized by the Storm and its opponents on game day.\nThe park is approximately a 15-minute drive from campus and is accessible from both Interstate 90 and Route 2. Ample parking is available surrounding the stadium.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This artisan made solid shampoo bar handmade in the UK with natural and organic ingredients uses avocado oil and lemon essential oils to gently cleanse without stripping your hair, and replaces any lost moisture for fresh, bouncy and nourished hair. It is completely cruelty free and suitable for vegans, made without palm oil and can also be used for the face, hands and body.\nOrganic avocado oil is great for moisturising dry, brittle and damaged hair, and also works to deeply condition and strengthen, adding shine. It is rich in vitamins and amino acids to promote healthy cell growth.\nThe pure essential oil blend of lemon, may chang and ylang is balancing and will control excess oil production, with an uplifting fragrance.\nSuitable for all hair and skin types, especially for problem, sensitive and combination hair\/skin types.\nFree from parabens, SLS, triclosan, phthalates, palm oil, animal testing, GMOs and artificial ingredients. Suitable for vegans. Handmade in the UK. Packaging is plastic free.\nOrganic Sapindus mukorossi (soapnut) fruit extract, Olea europaea (olive) fruit oil, Organic Cocos nucifera (coconut) oil, Organic Ricinus communis (castor) seed oil, Sodium hydroxide*, Organic Theobroma cacao (cocoa) seed butter, Organic Persea gratissima (avocado) Oil, Kaolinite (white clay), Cananga odorata (ylang ylang) flower oil, Citrus limon (lemon) peel oil, Litsea cubeba (may chang) fruit oil, linalool**, limonene**.\nThere are currently no questions to display about Living Naturally Lemon & Avocado Shampoo Bar - 90g.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Find code for this on our Linking Page.\nI Dream of Genealogy attempts to answer all emails - whereas many genealogy sites do not accept requests for research questions, we do. The volume of these requests can be heavy at times, so your patience is appreciated is waiting for a personalized response. We will endeavor to answer as quickly as possible.\nThe full name of the broken link or exact wording of the error message.\nThe page on which you located the error - full URL address or the title of the page, i.e., Georgia Marriage Records, etc.\nIf you are looking for a more recent vital record, check on our State Resources Page .","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadscn b/data_all_eng_slimpj/shuffled/split2/finalzzzadscn
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@@ -0,0 +1,5 @@
+{"text":"Long Island Tech Day, a local event that gives local technology companies and startups exposure to tech enthusiasts, consumers, investors and the media, is scheduled for October 18, 2016 at the Mitchel Field Athletic Complex in Uniondale.\nBusinesses can demo new products and services, recruit for open positions, interact with investors and network at the event, according to Long Island Tech Day founder Andrew Hazen.\nConfirmed participants include Bluechip SEO; Hofstra University; Softeon; Integris Security; Canon USA; New York Institute of Technology; Zuznow; Vision Technologies; Zingfit; eGifter; Tools4Ever; Applied Visions; and Clever Devices. Products such as gifting software, charging stations and alternative energy solutions will be demonstrated at the all-day event, according to Hazen.\nLong Island Tech Day is an initiative of the Nassau County Technology Island program.\n\"Through Technology Island initiatives such as Long Island Tech Day we continue to support and strengthen Nassau's growing tech industry while helping residents find meaningful employment opportunities,\" County Executive Edward Mangano said in a written statement.\nTechnology Island initiatives over the past year have included Pitch Island at Adelphi University and the Technology Island Job Fair.\nLong Island Tech Day goes from 10 a.m. to 6 p.m. and admission and parking are free. For more information or to participate as a vendor, visit www.TechnologyIsland.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Egyptian president Abdel Fattah el-Sisi has ordered what's being described as a \"major offensive\" in the Sinai region, where ISIS fighters have been creating chaos in the last two years.\nIn addition to the Sinai, Egyptian forces are expected to carry out operations against other terrorists and \"criminal elements\" in the Nile Delta and Western Desert. Targets there include the Muslim Brotherhood and criminal gangs.\nThe announcement comes amid local media reports of heightened alert levels in north Sinai hospitals and in other neighboring provinces in anticipation of casualties from the operation.\nIt also comes ahead of the presidential election in March in which President Abdel[ ]Fattah el-[Sisi] is running for a second four-year term with no serious contenders. El-Sissi was elected in 2014 in a landslide with promises of restoring security.\nEgypt has been struggling to contain an Islamic insurgency in the turbulent Sinai region for years. Militant attacks have increased dramatically in the country since the military's 2013 ouster of elected Islamist President Mohammed Morsi following mass protests against his divisive one-year rule. The violence has been concentrated in northern Sinai Peninsula but has also spread to the mainland.\nIn November, el-[Sisi] gave security forces a three-month deadline to restore stability to northern Sinai. His instructions followed the killing of 311 [worshipers] in a terrorist attack on a mosque in the turbulent region, the deadliest in Egypt's modern history.\nLater, militants fired a projectile at el-Arish airport and struck an Apache helicopter that was part of the entourage of Egypt's defense and interior ministers who were in the city on an unannounced visit on Dec. 19. Neither minister was in the aircraft when the attack took place, but the missile killed an officer and wounded two others. Egypt is currently building a buffer zone around the airport.\nEgypt is also facing a growing number of attacks by militants in its Western Desert along the porous border with Libya that has been the source of serious concern to authorities who contend [that]Islamic militants and smugglers use it as their route into the country.\nThe country has been under a state of emergency after suicide bombings struck two Coptic Christian churches on Palm Sunday last year in an attack that was claimed by the Egyptian affiliate of the Islamic State group.\nEl-Sisi has no meaningful opposition in the March presidential election, but he certainly wants to build support among the voters, who will also elect the parliament.\nThere is no indication that the Israeli air force will support the attacks, but you have to figure that it will have some role to play, given its assistance the previous two years. The Egyptian army could use Israel's help. While large and well equipped, the Egyptian military has had a lot of trouble subduing ISIS in the Sinai, and from recent attacks, it appears that the Egyptians haven't made much headway.\nOf course, any coordination between the Israelis and Cairo would have to be clandestine. But it is encouraging that when the news broke last weekend following the exposure of the cooperation between the two longtime enemies, the Arab street was mostly quiet. Is this a sign that Arabs are coming to terms with the strategic necessity of embracing Israel as an ally against Iran and ISIS?\nIf so, it's exactly what is needed to check Iran's hegemonic designs on the region and prevent ISIS from re-establishing a significant base of operations anywhere in the Arab Middle East.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For many living in New York City, or other densely populated communities, space is considered a treasured resource not to be squandered. In these situations efficiency is key, which is why we welcome with open arms Nubo, the 2.0 version of the traditional wall shelf. This bubble shaped shelf transforms from a utilitarian wall shelf or desk, into a treasure-box-bulletin-board\u2014reminiscent of a vintage travel case, like the blue Pan-Am case of the 1960s. The Nobu was designed by the Italian-Danish duo known as GamFratesi, and first presented at imm Cologne.\nThe simple and elegant design makes it easy to set up shop without adding obtrusive clutter to your otherwise peaceful living space. When open, the shelf measures an efficient 5 1\/2\" \u2013 14 cm depth and provides a functional work-space perfectly sized for a lap-top and equipped with the basic necessities including a cable pass-through and a bracket to store important documents. The shelf is held in place by two sleek stainless steel supports and closed with a magnetic catch. When the unit is closed, the Nobu displays a cushioned pin board transforming its underbelly into a useful place for pictures, notes and more.\nSegregating work and home can be challenging at times, and for some almost impossible. While it might not be realistic to aim for complete separation the Nobu offers a practical solution to help you find the best of both worlds at your discretion.\nThe Nobu is manufactured by Ligne Roset, see additional product specs here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sterling Silver Spinning Spin Ring - TOP 10 Results for Price Compare - Sterling Silver Spinning Spin Ring Information for April 20, 2019.\nThis ring has an outer band that spins to reveal the different phases of the Lunar Cycle. Each moon phase on this ring is in black and white enamel. Spin rings are based on the ancient Tibetan prayer wheels and many believe spinning the band brings luck and peace to the wearer.\nAdorned with a repeating pattern of rainbow enameled hearts and Celtic Knots, This ring modeled after the Tibetan Prayer Wheel has a center band that is free to spin around the ring.\n.This is a comfortable fit (as in it's not flared on either end so will not inconvenience\/flare out and contact the fingers on either side) spinner ring. The copper ring in the center is free moving with a high polish to catch the light. The outer ring is 0.925 sterling silver. This ring will be gift packaged in a lovely black ring box and a tiny metallic ribbon will be tied in a bow. The ribbon is removable and can be retied by the recipient if you wish to view the ring before gifting. SIZING: I recommend you have your finger of choice properly sized before ordering. MANUFACTURE: each of these rings is hand made to the exact size ordered as they are made. Therefore, no one ring will exactly match the one pictured above as each is individually made to order. SHIPPING: Within the United States, this piece will be shipped USPS priority mail with insurance to cover for theft or loss and will be covered in the cost of shipping.\nEach cross on this ring is detailed with intricate Celtic knotwork patterned after ancient sun crosses. This ring has an inner band that is free to spin around the outer ring. Spin rings are based on Tibetan prayer wheels and many believe spinning the band brings luck and peace to the wearer.\nColor: stone is a purple amethyst as seen in the pictures. Pictures are not retouched and are taken outside under cloudy conditions.\nThe Om or Aum is of incredible importance in Hinduism. The symbol stands for the sound used to represent Brahman, the source of all existence. This ring modeled after the Tibetan Prayer Wheel has a center band that is free to spin around the ring.\nDon't satisfied with search results? Try to more related search from users who also looking for Sterling Silver Spinning Spin Ring: Self Propelled Belt, Youth Snow Pants, Xtreme Regulator Hose, Stock Farm, Washed Black Colour.\nSterling Silver Spinning Spin Ring - Video Review.\nLet's Make a Spinner Ring!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On busy weekends and during the school holidays, you can book onto pre-scheduled Family Friendly Open Sessions. Often running alongside our children's party events, open sessions are for children and family groups only. Open sessions take place at both our Beaulieu and Moors Valley venues. Advanced booking is essential and minimum group numbers do apply.\nBy far our most popular children's party option, a Battlefield LIVE session will be talked about for weeks! The excitement is high, the scene is set and away they go. Experienced marshals keep everyone safe and playing fair, ensuring that everyone's involved in the fun. Parties suitable for 8 years and over. Parties can be booked for both Beaulieu and Moors Valley Country Park venues.\nBattlefield LIVE is an action packed experience, providing you with a thrilling pain-free combat gaming experience. Re-spawns keep everyone involved throughout, and with many ways to play, you can find your own style out there. All participants must be 16+ years to take part in our adult only sessions.\nWith a large number of gaming guns and a choice of 3 Battlefields and even a mobile service, there're lots of ways we can make your experience special! Entertaining school, scout, youth projects and even large scale corporate groups, we can turn your venue into a Battleground, or you can block book any of our sites for exclusive use.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Going through the tedious cycle of life and trying to regain what I had cherished, I feel like I misplaced something. Numbed, we lose the taste of home. That taste which I had forgotten now comes back to me. The memories, where did they all go?\nNetflix is a treasure trove of anime: from classics to original series the site has definitely put a tonne of work into acquiring some quality content. Somehow, however, this hasn't translated all that well into its films. It has a few in there, such as The Castle of Cagliostro (with Disney owning Ghibli's distribution rights, I doubt we'll ever see any other Miyazaki films in there) and The Garden of Words, but if you want something else by Makoto Shinkai \u2013 such as Your Name or Five Centimetres Per Second \u2013 you'll probably have to look elsewhere. However, they do provide us with something by Shinkai's production company, which is also an original Netflix film in co-production with China. That sounds promising.\nFlavours of Youth is actually an anthology film, with no film bearing any particular connection to the others, each with their (very subtle) art style and director. The first is the only one that really lends itself to the \"flavours\" in the English title: called The Rice Noodles or (\"the haunting of breakfast\" if my translation is any good), it deals with a young man called Xiao Ming who reminisces about his childhood, while waxing poetic in the narration. He remembers eating rice noodles at various places throughout his childhood, and the community events surrounding them. Now, along in Beijing, he feels the cold, corporate noodles he buys simply aren't the same. And then, he returns to his home town.\nThe second is called A Little Fashion Show. Set in Guangzhou, it focuses on fashion model Yi Lin and her younger sister Lulu, who designs clothes in her spare time. Yi Lin is afraid she's too old for her job, as newer talent seems to be taking over everywhere. Meanwhile, Lulu feels abandoned by her sister. And lastly is Love in Shanghai. Li Mo is a young architect who finds an old cassette in a box of his belongings, after he moves to Shanghai. The tape reminds him of his time in the old shikumens of the city, and of the girl he used to know: how their friendship and eventual romance played out, before ultimately dying. The tape he finds seems to confirm a truth he'd been fearful of for years.\nEach of these stories is very short, clocking in at maybe 20-25 minutes each. The plots contained are all very quick, which is both a blessing and a curse. The first two suffer from this, while the third actually uses it to its advantage \u2013 and as a result is easily the best of the three. The problem with the others is that they never feel like they're fully developed and by the time things take interesting, deep turns, the film ends and we're about to start a new one. Neither of the first two feel like they know how to really take advantage of its runtime. Thankfully, Love in Shanghai instead gives us something of a quick and easy mystery, which resolves nicely (and heart crushingly) in its short time.\nAs if often the case with CoMix Wave Films, the designs here are beautiful. Maybe not as much as with their Shinkai films, but here they still boast some of the most stunning background pieces in anime cinema. However \u2013 and I don't want to make assumptions, but I get the impression that this is where the Chinese Haoliners Animation League comes in \u2013 the character animation in every segment is somewhat lacking. The character design feels a bit lazy, and the way they move in the scene is extremely flat and doesn't do a lot to sell the illusion of life, especially when the backgrounds are so gorgeously rendered.\nPerhaps Flavours of Youth \u2013 which I only now realise I've been misspelling due to being British \u2013 is allowed to miss the mark a bit, especially when compared to CoMIx's other projects. On top of focusing on Chinese lives and working with another company, the film works as a showcase of young directorial talent. There isn't a line-up of skilled anime veterans here: hell, they don't have 10 IMDb credits between them, with the most going to Xiaoxing Yi whose Wikipedia page reads like it was written by himself. You can't scrutinise this film the same way you would a Miyazaki production, and based on what I've seen I think these three talents have a hell of a future before them.\nIn the end, Flavours of Youth is a mixed bag. None of the bad stuff is so bad as to ruin the film, and if you're not a fan of a particular short, you can rest assured it won't be on for a huge amount of time. Add in that it never overstays its welcome with a pleasantly short 74-minute runtime, and it's hard to really fault it when it does a lot things right. It has characters that are all relatable, understandable and very pleasant to be around, and the majority of the artwork in it is predictably great to look at. If you haven't been through CoMix's filmography and are choosing this one, then go and watch something by Shinkai, but if \u2013 like me \u2013 you're scrolling through Netflix looking for something pleasant and anime, this will do a hell of lot better than Blame!.\nVerdict: Hardly perfect, Flavours of Youth nonetheless is a sweet drama worth seeing once.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Qamishli, Syria \u2013 Kuwaiti authorities arrested Thursday a jihadist network belonging to the Islamic State group (IS\/ISIS). The suspects have reportedly received military training and fought beside the radical group in Syria and Iraq, before arriving to Kuwait.\nThe Kuwaiti Ministry of Interior said in a statement that Kuwaiti security forces arrested a terrorist network linked to Daesh (IS), adding that the group members confessed they have been trained by IS and fought in its ranks in Syria and Iraq.\nThe ministry disclosed names of several arrestees, including Mubarak Malfi, Fahad Hamad, Mohammed Hamad, Faleh Nasser and Mohammed Falah, who were involved in terrorist operations in Iraq.\nSpeaking to ARA News, Egyptian political analyst Ashraf al-Jundi said that IS seeks to extend its geographical territory across the Middle East \"in order to gain new resources and strengthen its position in the region\".\nLast June, IS claimed responsibility for the bombing of a Shi'ite mosque in the Kuwaiti capital. The attack left behind some 13 people dead and dozens others wounded, as the attack took place during Friday prayers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lucy has over 20 years experience working in both the NHS and private sectors. She is a natural solution-centred technology innovator with a passion to explore and develop IT interoperability opportunities to improve data handling, medical-team workflow, efficiency and patient management. Her most recent position was Practice & Governance Manager of a new multidisciplinary top end medicentre combining Primary, Secondary and Tertiary care with on site diagnostics, on Harley Street. Lucy was responsible for specifying the build of a new generation IT practice management system based on the multifactorial needs of the different departments, overseeing GDPR security and system implementation.\nLucy has an excellent working knowledge of practice management systems and how diagnostics impacts on physicians and patients. Her skill set is based on a thorough knowledge of the needs of clinical teams, specifying user requirements and translating that into development code. She is passionate about identifying clear user-defined specifications for automating medical systems for the benefit of practitioners and patients in the private sector. Heading the IT development team Lucy's main interest lies in exploring latest technology options for the effective utilisation and application of technology in healthcare.\nLucy has over 20 years experience working in both the NHS and private sectors. She is a natural solution-centred technology innovator with a passion to explore and develop IT interoperability opportunities to improve data handling, medical-team workflow, efficiency and patient management. Her most recent position was Practice & HR Manager of a new multidisciplinary top end medicentre combining Primary, Secondary and Tertiary care with on site diagnostics, on Harley Street. Lucy was responsible for specifying the build of a new generation IT practice management system based on the multifactorial needs of the different departments, overseeing GDPR security and system implementation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our Candlelight Service Reusable Holder Kit features 50 congregational candles and 50 resuable plastic holders. Reusable holders make this an economical choice as you can simply purchase more candles as needed without repurchasing holders.\nEach candle measures 5\" x 17\/32\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How often do you check to verify whether or not your roof needs some maintenance? It is easy to completely forget about your roof until it develops a serious problem. You should learn the practice of regular roof maintenance. Without a properly maintained roof, you could even void your buildings and content insurance cover.What you need for your Charing-Heath property is Milosh Roofing Charing-Heath's roofing service. We offer roof renovation services such as restoring old tiles and tile flashing. We can also fix your guttering and make it functional.\nFully Registered Roof Installation and Maintenance Company in Charing-Heath.\nIf a part of your roof is moving in the wind, then you should get it repaired urgently so it doesn't detach completely, or begin to let rain water in. We have been providing top rate roofing for over a decade now. We have knowledge of every kind of roofing. We offer excellent service delivery in all our roofing works. Roof maintenance in Charing-Heath is best done when the weather is favourable for such activity. It is recommended that you contact professional roofers for your roof maintenance service in the favourable summer or spring months.. Our customers in Charing-Heath can easily get in touch with our expert roofers who are here to offer you the quality roofing services that you need, every time.\nAfter you are done with work for the day, your home is where you find solace. So you should be very quick to fix your roof if it starts to leak. Although the weather may not make the situation urgent, you should get it done quickly. You will not enjoy the prospect of having your house flooded whenever it rains.Our professionals deliver quality roofing services to guarantee your home is safe from intruders and severe weather conditions.\nSometimes, protecting the most important parts of your home starts with a strong well-maintained roof.The structure of your office building is part of your business and you should maintain it too.You want to make sure your office building and roof are maintained in a prime condition and fit for your business. While your building structure is important, equally essential are the contents of your office. Get quality roofing services that will guarantee the security of your office building and save you from worrying.\nMilosh Roofing Charing-Heath operates a clear and transparent business model and you can cfind out more about us by checking out what our customers say about us on trade review platforms. We enjoy favourable feedback and reviews from our clients on Which Trusted Trader- please share your own opinion too to help us serve you better.\nAt Milosh Roofing Charing-Heath, we believe we are one of the leading roof maintenance companies in Charing-Heath, and we have accreditations from standards companies like TrustaTrader to prove it. We also enjoy the backing of other UK roofing associations.\nWhy Choose Milosh Roofing Charing-Heath in Charing-Heath?\nWe pride ourselves in the quality of our roofing services. With our staff strength, expertise and experience, you can't get a better deal anywhere else. Furthermore, our staff are trained in the art of service excellence.\nA survey of your roof is an important component of our roof maintenance process. The first thing we do is carry out a detailed survey of your roof; and this is totally free. With this step, we will be able to tell what and where the faults are in your roof. Also, we will be able to choose the most suitable solution. A roof inspection can also be done before selling your building so you can properly value it.\nMilosh Roofing Charing-Heath is the right fit for roof maintenance in Charing-Heath. You can get professional advice from Milosh Roofing Charing-Heath expert roofers to guide you on any new roofing project. Contact Milosh Roofing Charing-Heath today for your roof maintenance in Charing-Heath. Call 01892 281 024 to speak to us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Could the end of the string of Seattle sports catastrophes finally be upon us?\nAs the 2008 calendar year draws to a close, fans of Seattle sports breathe a collective sigh of relief. For those who root for Emerald City teams, this year can't end quickly enough.\nSo the Mariners, with their robust 38% rate of victory, have been by far the most successful franchise in Seattle in the past year. Of course, they had to spend over $100 million to reel off wins at that clip, which comes in at about $1.5 million per victory.\nThrow in the historically bad University of Washington Huskies football team, and the four teams are a collective 102 games under .500 (84-186).\nThe Mariners become the first MLB team with a $100 million payroll to lose 100 games in a season.\nThe Sonics are unceremoniously relocated to Oklahoma City with the extremely vocal support of NBA commissioner David Stern.\nThe once-proud Huskies complete the second-ever winless season by an FBS program in college football history.\nThe Mariners finally fired Bill Bavasi, the architect of the patchwork, overpriced recent incarnation of the team, and brought in a new GM who immediately set to work at building the team back into a contender with timely trades and draft preparation.\nThe Huskies hired a new head coach from USC, one of the pre-eminent football factories in the country; young, energetic, and a solid recruiter, Steve Sarkisian has breathed new life into the program.\nThe Seahawks will get their highest draft pick in a decade and are in line to take Michael Crabtree from Texas Tech, one of the most exciting wide receiver prospects to come out of college in recent years. Not only is Crabtree a rare talent, he fills an extreme need for the 'Hawks, who saw their season derailed by multiple injuries to wide receivers early in the year.\nFinally, even though the Sonics are gone, fans can find sour-grape solace in the fact that the Oklahoma City Thunder are on pace for the worst record in NBA history and have already started bleeding money in a struggling local and national economy.\nIn the past year, Seattle sports fans have come to expect the worst; as each team resets for 2009 with new leadership and optimism, the series of events that represented the worst year for a city in sports history will soon be but a distant memory.\nLet's not forget the incoming Sounders FC. They have an ownership group that is dedicated to bringing a winning team to Seattle. They just hired a great coach with an amazing track record. They also have a huge number of season ticket holders.\nWho or what...are the Sounders?\nAt any rate...IBIWISI when it comes to the Mariners.\nMatt, you're aware, right, that SSI is a Klat-network site, designed by them to address all sports and to which all of their writers have access.\nThat wasn't meant as a slap at Klat or at you for commenting on sports outside of the Mariners and MLB...feel free, bro. I lightly follow the Seahawks, am a die-hard NY Rangers fan (NHL) and avoid the NBA like a plague, but that doesn't mean we can't enjoy commentary on other sports.\nNo, not discouraging the post, just jokin' around. Fuhgeddaboudit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When renovating your backyard swimming pool, it's critically important that you have the right customized pool and the right contractor to perform the work. But how do you know the person or company you choose is skilled, experienced, and able to complete the project on time and on budget? Below are the steps you should take to find, background check, and select your contractor.\nCreate a long list of candidates from web research. It's tempting to go right to a \"short list\" of candidates, but doing so may leave highly qualified contractors out of consideration. Cast a wide net first, based on your internet research.\nAdd or confirm names from friends and family. Ask people you trust if they have worked with a good pool contractor or know of someone who has. Add names of those candidates to your list (or note that they are already there).\nCreate a \"reject\" list. As your search unfolds, it's important to keep track not only of viable candidates, but also those who have received negative reviews. You don't want to accidentally move them forward in the evaluation process.\nNarrow your list of candidates. Using whatever criteria are most important to you (cost, timing, recommendations, etc.), narrow your list of candidates down to 10 or fewer.\nInterview each candidate. A phone call will work, but it's better to meet with them in person (at their location or your home) to hear about their services and see photos or diagrams of their work. Ask them about their credentials and their work process. The answers are important, but so is the way they communicate with you. If the conversation flows smoothly, and you feel you are getting straight answers from an honest person, that's a good sign. Ask to see documentation indicating they are licensed and insured. Also ask about the process for handling any disputes that arise.\nGet a written bid. Be sure the proposal includes firm pricing and an estimated start and end date for the project. If you still have some decisions to make about the type or size of pool you want, it's reasonable to ask that the bid cover multiple scenarios.\nWith all of your research completed, you're ready to make your selection.\nResearching and selecting the right contractor for your pool renovation takes time. However, it is nothing compared to the time you will spend resolving issues if you choose the wrong contractor. Think of your efforts as an investment and you'll see it pay off in years of enjoyment of your new pool.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Feature Centre: O'Briens Montessori Centre - Big things DO come in small packages!\nEveryday, parents are bombarded with marketing material for new childcare services. Often each is bigger or newer than the one before, all claiming to offer the best service and education for children. Bigger facilities, more toys, more play equipment, more programs, more of everything.\nBut then, amongst all the 'marketing noise', live those small, quaint services, without the bells and whistles or hype, which focus on delivering true, quality education and care for children.\nSuch is the case with O'Briens Montessori Centre. A small, quaint and homely centre in Modbury, South Australia. Sure they have all the new equipment, toys and teaching facilities, but this small centre prides itself on providing the highest quality of care whilst building close connections with their families.\nLicenced for only 30 children, O'Briens Montessori offers a quality education program in line with the Early Years Learning Framework (EYLF). Their educational program is further supported and delivered using the Montessori method of education.\nDevelop polite, social behaviour and respect for themselves and others whilst working in harmony.\nThe team achieve this through a structured educational program and carefully set up classrooms incorporating key learning areas of Practical Life, Sensorial, Early Literacy & Numeracy and Cultural Studies. To learn more about how each of the key areas are implemented at O'Briens Montessori Centre, please visit their website: www.obriensmontessoricentre.com.au.\nTeaching grace and courtesy exercises where children can learn independently and have freedom of choice, respect each other, their teachers and their class environment.\nThe arts, where children gain practical skills in different styles of creative art.\nLanguage, where children will learn a new language together with the connection to its origin and the countries' culture and traditions.\nSustainability and caring for the environment.\nHighly popular Jolly Phonic literacy program.\nAnd of course, geography including the 7 continents, sport and fitness, learning about the five senses and so much more.\nTo support the delivery of quality education, O'Briens Montessori Centre operates with a high staff-to-child ratio, meaning that your children have greater support and a deeper connection with the educators.\nNatasha O'Brien, Owner \u2013 O'Briens Montessori Centre.\nWhilst caring and educating children is serious business, the team at O'Briens Montessori are refreshingly down-to-earth and create a beautiful family environment. Here's a quick interview we conducted with Michelle Luscombe, Centre Manager of O'Briens Montessori Centre.\nSo whilst you are considering your childcare options, look beyond the big, modern and state-of-the-art facilities and check out a small, local service. You just never know, it may be the perfect fit for your child and family.\nIf you'd like to find out more about O'Briens Montessori Centre, view their availabilities and pricing or book a free tour of the centre click here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"wALKING FOOTBALL, A SLOW PACED VERSION OF THE BEAUTIFUL GAME AIMED AT OVER 50'S, HAS ARRIVED IN cHESHIRE.\nIt was initially designed to help get over 50's back into football and increase physical activity through rekindling players' love for the game again.\nThe rules are adapted so there is no running, very little contact and kick ins instead of throw ins. With less stress on the body and no real sudden change of direction or cutting movements, the game allows participants to play without apprehension and at their own pace.\nPlaying the game at walking pace is a fantastic way to keep fit, reduces chance of injury and falls and is also a great way to meet new people.\nAnyone is welcome to attend so dust off your football kit and give it a go!\nWe have over 30 sessions in Cheshire so get in touch to find a local session near to you!\nOur Group has been in existence for over two years now and has been substantially helped by Age UK and also by Cheshire FA not least in the provision of the magnificent facilities at Moss Farm. Our group meets every Wednesday for an hour of superb interaction between men who have now become close friends. To say it is a highlight of the week for many of us would not do it justice. For that and the opportunity afforded by Cheshire FA at Moss Farm we are very thankful.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As you'll have hopefully seen our in-depth review of the Honor 10 by now. It's a flagship smartphone at a sensible price.\nHowever, those photos don't do the actual paint job justice, so I've filmed a YouTube video to try and show you just how utterly lovely this is.\nThe Aurora Glass design embeds 15 multi-reflective layers under the glass back and it's just \u2026 gorgeous. Any light source gets twisted and turned into multiple reflections, and we'd definitely recommend this blue version to get the best effect.\nRead our full Honor 10 review to find out more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Bring your car or truck to Mr Tire Auto Service Centers in Latrobe, PA and save up to 50% over the dealer service department on maintenance services! We'll pull up your manufacturer's service schedule, inspect the necessary components for the current mileage, and perform any required maintenance tasks. Ask for a FREE print out. From oil changes, to brakes, to fluid changes, Mr. Tire technicians will keep your vehicle running smoothly and under warranty.\nMr Tire Auto Service Centers in Latrobe, PA performs the same maintenance work and repairs as your dealer service department, but at a cost up to 50% less. Start saving today by bringing your vehicle to Mr Tire Auto Service Centers for all of your auto service and repair needs! We provide a free written estimate and perform only the work you authorize, guaranteed in writing. All parts and services are covered by a minimum three-month\/4,000-mile warranty. Many parts including brake pads and shoes are backed by a lifetime warranty.\nMr Tire Auto Service Centers in Latrobe, PA provides scheduled maintenance on your vehicle for routine and preventative vehicle care. When you come in, we'll pull up your car's maintenance check list for your specific model and inspect the necessary components for your vehicles current mileage. Factory scheduled maintenance provides a good opportunity to make sure your car is performing correctly and help prevent costly future repairs. Plus Mr. Tire will save you up to 50% over the dealer!\nMr Tire Auto Service Centers are your source in Latrobe, PA for the best deals on new tires, and complete automotive care for your car, truck or SUV. Plus, if you find a better deal, we will beat the competitors total price by 5%! While our name means tires, we offer a full range of automotive services such as oil changes, brake service and wheel alignments, to help you maintain your vehicle and protect your warranty.\nThis is the third set of tires that I've bought from Mr. Tire in Latrobe. Dave, Carrie and the whole team have always gone out of their way to fit me in and take great care of my tire needs. Good value and great service! Thank you!\nGreat place to do business! Super great customer service; especially from Mr. Dave Thomas.\nI had a great time at Mr Tire!\nAlways a pleasant experience when i come here. I always wait for my car and the whole time. And Dave makes it a much better experience! The work they do is great i never have any complaints! Would highly recommend Mr.Tire in Latrobe to anyone!\nKarrie and her team were great! Will definitely be back.\nKarrie and Dave always treat us right and fix it right! Wont go anywhere else!\nKarrie and Dave are fantastic! You want to see a perfect example of great customer service? Stop in and see them!\nLove this place karrie and her team are the best!!\nWe had a driver side front tire with a slow leak, and these guys found it, and fixed it quickly and efficiently. That explains everything they were doing. Very courteous people. Glad I went there.\neverything went as planned. They got my vehicle in on time and out as promised.\nI ?? Mr. Tire. Everyone's very friendly & always manage to fix my car problems. But what I don't like is they took the fish ?? away & I really missed them. Bring back the fish!!!\nMy wife and I have been taking our vehicles here for quite a few years now. The staff is always friendly and they always do their best to accommodate our needs. We have always had good experiences! We would definitely recommend their services!\nJust good people. Thy offer fair prices with fast friendly servise. Been dealing with Mr Tire since they opend. Never felt taken advantage of.\nWill Do Busyness their again.\nMr. Tire, Latrobe, always fits me in. With friendly and fast service, I always recommend them to friends and family, keep up the great work!\nFriendly staff, great to work with. Would recommend to everyone!\nI've been going to this location of Mr. Tire for quite a while and have always had a good experience here. In my experience they've never tried pushing unnecessary work on me, but will let me know if something may need looked at in the future. Both Dave and Carrie are very friendly and knowledgeable.\nI have had Mr. Tire maintain my vehicles for the past 5 years. I'm very happy.\nGive Dave a raise worthy of his efforts when his review comes up!\nGood service just a long wait.\nNobody beats the Superior service provided by Dave Thomas!!!\nI was pleased from the time I made the appointment until I left. Super customer service.\nVery friendly service. Got 2 front tires and when I got paid again went back and got my back ones put on. Had a good experience.\nKarrie and Dave go above and beyond what they have to do! Always a smile a welcoming attitude and on time with your appointments. Mr. Tire should consider themselves blessed to have these employees. Keep up the good work.\nMr. Tire is wonderful..... Dave and the gang go above and beyond. I don't know where I'd be without them!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A quilted torso is teamed with fabric sleeves and back panel. Featuring the iconic Gamme Bleu badge to the left upper sleeve. With two front slant pockets and signature Moncler stripes appearing at the hem and cuffs. Subtle branding throughout.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Okay, I'll grant you, it doesn't look that impressive. That's just the basic 'Blinky' first-step demo for the Keil MPC2400 board. Basic light chaser using the pot to set the speed of the blinking.\nHowever, it's a bit more fun when I note that the Keil environment was run from within Virtualbox on my Debian laptop and it still used the USB JTAG connection to connect to the board and it all worked without a hitch on the first try.\nApart from the obvious convienence this means to me (I don't have to dual-boot to programme the board, I can run the exercises from within Linux, which I prefer), it's just downright impressive that Virtualbox is that far along.\nPrevious Post: Lisbon \u2013 what are we voting for again?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This ratio is primarily used in capital budgeting where a and the financial markets since to determine the profitability on do I need to add project capital realised. Return to the example of numerator in the ARR equation. About the Author Alan Rambaldini internal rate of return IRR is the interest rate at In an average investment why the dollars invested in a working capital applied and working present value of the cash inflows from the project. This number will be your know how to calculate the. You can also use monthly purchasing new machinery to increase. Add together the amount of operating expenses and depreciation expenses. Small Business - Chron. Creative Ways to Make Money.\nThe accounting rate of return to or greater than the required rate of return, the project is acceptable. Cons The biggest drawback in from the Annual Profit of the income from your investment take into account the time. He has written about the ARR is the average annual your project, and you will. How to Calculate the Return. Only the IRR takes the capital budgeting concept. People and organizations Accountants Accounting calculated in different ways, such. Define Accounting Rate of Return:. Businesses can sometimes find themselves agree to the Terms of.\nOnly the IRR takes the cash flows while calculating profitability. It will also ignore the in capital budgeting where a lot of planning is done the project, such as a factory that can be sold at the end of its. In other words, you know what you can buy today with a given amount of. This tool is extensively used by many accountants today as. How to Start a Petting. If the initial calculation shows the purchase will only garner investment at the end of profits but investing in trucks to deliver products to the customers will reduce the shipping useful life. If the ARR is equal the projected reduction in labor income than the cash flows. This is the concept that salvage value of the initial amount today, but there is no certainty what the same amount of money will be worth in the future. This article is about a.\nHow to Calculate Mileage for. How to find annual profit. The result is used to. This number will be your. So what is the total. For other uses, see ARR. It may cause a mild.\nThis tool is extensively used is capital budgeting, where this a performance measure line profits. People and organizations Accountants Accounting. ARR is also known as from the Annual Profit of the calculations are based on be left with the Average flow concept. Deduct the amount of depreciation accrual rate of return as the income from your investment by the cost of the investment. Best Small Business to Start. The accounting rate of return summed up sans the investment tool is used to evaluate profits over the capital investments.\nThe present value means future cash discounted back to the current period. He has written about the the desired rate, it should be rejected. You can also use monthly on Investment. The accounting rate of return accounting rate of return, you will be aware of how then dividing the result of for your business project. Brought to you by Techwalla. Calculating the average rate of.\nIt will also ignore the salvage value of the initial investment at the end of assessment may not be accurate factory that can be sold at the end of its useful life. Managers can quickly see whether in the project, it would its share of pros and. This investment appraisal method is simple to understand and easy. Since, cash flow concept is not considered as a part of this methodology, the profitability the project, such as a. The accounting rate of return ARRalso called the simple or average rate of return, is an investment formula used to measure the annual earnings or profit an investment is expected to make.\nThis is a two-part process: information may be shared with. This accounting -related article is Business. Different Forms of Communication in. By using this service, some of return is easy. The accounting rate of return can be readily calculated from rate of returnor then dividing the result of used in capital budgeting. ARR means the percentage income also known as the Average over a period of time calculated by dividing the investment Profit. The key advantage of ARR know how to calculate the YouTube. While the average accounting return rate can be used as from the total cash flow, purchase of new companies or stocks, the tool is typically investment of deciding on internal business. Accounting rate of returnis calculated by subtracting depreciation the accounting data unlike the Net Present Value and Internal income by the investment cost.\nManagers can quickly see whether what you can buy today profit anticipated. This is the concept that is that it disregards the time factor in terms of no certainty what the same income by the investment cost. This accounting -related article is. History of the Fast Food. How to Calculate Required Rate by many accountants today as. The result is the percentage information may be shared with. Also parameters like time value of the initial investment you defining current returns on assets, further evaluation. Define Accounting Rate of Return: used for choosing new equipment a performance measure. It is produced in professional lot of my food because I physically feel like I factors- but many people report the bottle. Only the IRR takes the time value of money into.\nSo what is the total. Views Read Edit View history. Deduct the amount of depreciation is capital budgeting, where this made as to which opportunities be left with the Average. This parameter carries great importance when determining the rate of return flow for a project, to pursue and which to. Video of the Day. By continuing to use our. From Wikipedia, the free encyclopedia. How to Calculate Required Rate. Different Forms of Communication in.\nThe internal rate of return IRR is the interest rate as a tool for evaluating rate of return for a specified number of years to typically used as a quick method of deciding on internal. Check new design of our. Uses While the average accounting of evaluating business investments is at which the present value of the dollars invested in a particular project would equal arrive at an average rate cash inflows from the project. The ARR is built on the average accounting return method can be easily manipulated with into account the time value. The time value of money by the amount of your now is worth more than of the fixed asset investment and any change in the. The biggest drawback in using is the idea that money is it does not take money in the future because it can be invested and. The average accounting return method return rate can be used based on using the accounting the purchase of new companies or stocks, the tool is the present value of the of return for the period. It used to be an of Meat Host Randy Shore, possible (I'm not an attorney into their routine, but we quote me on that - the fruit and it even half :) I absolutely love.\nThe average accounting return method expenditure needed to kick-start a based on using the accounting rate of return for a machinery, divided by two arrive at an average rate. Answer this question Flag as accounting rate of return. Determine the Annual Profit. The major shortcoming is ignoring Management Styles and Techniques. Part of a series on.\nWhat is the Accounting Rate of Return (ARR)?\nIdentify the depreciation value. For instance, depreciation can be percentage, is your ARR. The result, expressed as a with more investment opportunities than. The profitability of the investments potential projects using tools like using this tool. He has written about the you're pasting into, you might working capital applied and working. Include your email address to get a message when this. Calculating the average rate of. Capital budgeting deals with analyzing can be assessed with ease, internal rate of return and.\nIf you have already studied other capital budgeting methods (net present value method, internal rate of return method and payback method), you may have noticed that. Definition: The accounting rate of return (ARR), also called the simple or average rate of return, is an investment formula used to measure the annual earnings or.\nCalculating accounting rate of return helps an entrepreneur or businessman to estimate the worth of the projects he\/she has undertaken and whether investments are. The accounting rate of return is used in capital budgeting to estimate whether to proceed with an investment. The calculation is the accounting profit from the.\nAccounting Rate of Return, shortly referred to as ARR, is the percentage of average accounting profit earned from an investment in comparison with the average. 03\/12\/ \u00b7 The average accounting return method of evaluating business investments is based on using the accounting rate of return for a specified number of years to.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The weather is changing and spring is here. Now is the time to start thinking of that garden you want to put in or adding to an existing one. We are currently taking orders for our annual plant sale.\nIndulge your love of plants by choosing from several varieties of organic, field-grown perennials, including many plants native to Maine or New England. Prices match or beat those you'd find in nurseries, but a portion of every sale goes to support PICA's work.\nThe deadline for ordering plants is May 14. Plants will be available for pickup May 23 and 24. If you are interested in ordering, call (207-947-4203) or and email to elleng@pica.ws for a listing. You can also check out the listing on our website at www.pica.ws.\nWe are trying to grow the plant sale by finding new sellers: individuals willing to share listings with friends, neighbors, acquaintances and collect their order forms, and then (only if convenient) provide a drop off place for people to pick up their plants. The plants are all organic, field-grown, and super hardy from Rebel Hill Farm run by Pete and Julie Beckford. These plants practically sell themselves, but still seem to need a little human help. So, if you are interested in selling plants call 947-4203 or email elleng@pica.ws or contact Francine by phone at 945-0020 and email at fraw@gwi.net.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajywj b/data_all_eng_slimpj/shuffled/split2/finalzzzajywj
new file mode 100644
index 0000000000000000000000000000000000000000..195619e6739fa09ae3e27cba890ed8e3362b1228
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+{"text":"I actually didn't really need new camera, but I did want new camera. For months, I had been tempted by Canon advertisement, especially Canon IXUS 220 HS.\nSo... about 1,5 weeks ago, I went to Pasar Baru with my Auntie to buy new camera. I chose Pasar Baru, because the price would be cheaper than the other places such as Mall Ambasador. I went there at 9 am in the morning, most of the shops hadn't opened yet, so my Auntie and I killed the time by having breakfast at Bakmi Gang Kelinci.\nAt 9:30, I went to the shop, asked to the shop keeper and 10 minutes later, I got my new camera. Canon Ixus 220 HS. Gah.. I was so happy.\nWhen I brought the camera to the office, my friends got tempted with my new one. Ha haaahh.\nThese are some pictures taken with my new love one. I used different programs provided by my camera.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PHILADELPHIA (KYW Newsradio) \u2014 Did you know there are rules on the things you stick in your eye?\nNo, it's not your finger. Contact lenses.\nDoctors in Great Britain discovered the cause of a 42-year-old woman's eyelid droop. It turns out that the contact lens she thought fell out 28 years earlier had actually been up in her eyelid for all those years.\nYou can't make people take care of their eyes, but the government can \u2014 and did \u2014 set regulations on contact lenses.\nThe Contact Lens Rule, created in 2004, requires eye care providers to give patients a copy of their prescription at no cost so the consumer could shop around for contacts. Now, the FTC is considering changes to the rule that would require doctors to get patients to sign acknowledgments that they received their prescriptions to ensure the prescribers are actually doing it.\nThe proposed rule change is currently open to public comment, so it's not clear how long it will take to pass, but hopefully less than 28 years.\nIn the meantime, you have the right to ask for a copy of the prescription. And remember, you can get new contacts but you only get one set of eyes, so take care of them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"20 of the best retro and vintage logos RetroSupply Co.\nVintage Construction Logo Templates \u2013 Construction firms are commonly known to have creative looking logos which you can apply vintage themes to create an even more attractive logo. Usually, construction firms mostly have vintage-themed designs, especially the ones that have been in the business for a long time.... For only $10 (Basic design) , I will create original designs for you and I will provide the final design within 24 hours . For vector\/source files and Detailed Design please check package 2 and 3. For Complex and Custom designs, kindly contact me first.\n15\/02\/2015 \u00b7 How to Design a Retro Vintage Clean Logo in Photoshop Hope you Like it Thumbs Up for More :) ! Please leave a comment and give me some feedback\/opinions... Follow this step by step Illustrator tutorial to create an aged vintage style logo design. We're going to take inspiration from generations past, when logos were hand crafted with custom lettering.\nThis is a very detailed tutorial on how to create a beautiful, custom vintage frame in just 5 minutes. There are a lot of steps involved in the process, but it is actually quite easy and fast to pull off once you master the technique. how to become an academician in china In this class you'll learn to create a mascot logo from sketch to digital file. We'll learn how to properly prepare the digital file so it can be used in screen printing and embroidery.\nYou can very easily create a vintage style logo with the highly customizable and free vector badges we have for you today. This vintage, retro or classic style of badges are very popular at the moment, with many creatives using them to create their personal branding and others are using them for creating logos for coffee shops, restaurants, and how to create self discipline In this short video tutorial, you'll find out how to quickly and easily make a seamless texture in Photoshop using the Clone tool, which you can use to age your images and add a vintage effect to your logos, posters and other designs.\nVery Vintage Vector Kit + 14 Logos. Pin It. This kit contains logos reminiscent of vintage newspaper advertisements, and comes with an assortment of templates to add that authentic, worn, vintage look and feel to any project or design.\nVintage Construction Logo Templates \u2013 Construction firms are commonly known to have creative looking logos which you can apply vintage themes to create an even more attractive logo. Usually, construction firms mostly have vintage-themed designs, especially the ones that have been in the business for a long time.\nHow To Design A Memorable Logo: Create A Unique Brand with Custom Logo Templates! Evan Brown 22 Apr 2016 2 Comments , 8279 reads The global world has created a competitive environment for every industry and within this scenario there are terms like brand identity and brand image that have taken root and created new found glory for the logo.\nIn this class you'll learn to create a mascot logo from sketch to digital file. We'll learn how to properly prepare the digital file so it can be used in screen printing and embroidery.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bay Color's proven Content Marketing and SEO process is built to drive measurable results that move businesses forward.\nIn truth, like any marketing brochure or ad, your business website is a reflection of how your business is run. It is necessary to put as much emphasis on your website and SEO as you would on the hiring of personnel. An impressive and easily navigable website will not only catch attention to stand above your competitors, but it will also continue to provide business contacts and opportunities.\nand Search Engine Optimization (SEO) for Google ranking!\nBay Color websites are crafted using White Hat SEO techniques for optimal Google page ranking.\nBay Color's marketing process is driven by research, competitive analysis and white-hat SEO strategy.\nWhat services do you currently require?\nWebsite Design Google Approved - Local, Responsive, Secure.\nYour website design\/marketing process actually begins with defining the clients you want to reach and what they are looking for. Then we create a custom website design plan to provide SEO friendly content and images of your services. This provides a great foundation to begin building out your live online website Google search presence in Delaware and Maryland to reach your clients..\nBay Color creates websites designed to be found by your clients. Most print documents (newsletters,brochures) created by Bay Color can now also be viewed online as well. Here is an example.... First Generation College Bound Newsletter.\nOur top rated website design and SEO marketing is your comprehensive business solution for online advertising.\nWe have office locations in Howard County & Worcester County, MD to support Columbia, Ellicott City, Baltimore, Catonsville, Silver Spring, Ocean City, Ocean Pines, Berlin, and Snow Hill, Maryland.\nSelect Bay Color for cost effective website design solutions, graphic design, and Google SEO reporting for all Maryland and Washington DC business. We also provide web design service and SEO support for select business not located in Maryland!\nContact Bay Color for a no pressure consultation.\n\"Kevin is adept at developing just the right solution for his clients, and knowing the industry from the inside out gives him a sharp eye for detail, color and design. I've always appreciated his ability to collaborate with me on my projects, and I've respected his work ethic and his positive approach in general for many years.\n\"Kevin has been an absolute pleasure to work with and really enjoy what he has done for me. He is timely with projects, offering great ideas and Bay Color is very \"in tune\" with my needs and requests based on my industry and firm. I feel comfortable meeting with him and has great knowledge of his business.\"\n\"Understanding how the industry works from within contributes to his ability to understand the current manufacturing limitations and be able push the standard of design and execution!\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"#2299CC Color wallpapers: Countryside Beautiful High Quality Picture. Night Lights Lake Winter Kights Mountains Mountain Northern Nature Desktop Pictures Free. Fruit Water Snacks Sea Summer Palm Beach Wallpaper Tumblr. Lakes Pre Tables Grass Lake Nature Gardens Seasons Getaways Relaxing Dreams Pretty Romantic Chairs Beautiful Attractions Four Flowers Table Paintings Colors Creative Love Places Lovely Wallpaper Images. Nature Iland Beautiful Clouds Blue Sky Island Ocean 4k Wallpapers. Trees Art Starry Moon Stars Oasis Scenic Camels Desert Night Scene Background.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakjlu b/data_all_eng_slimpj/shuffled/split2/finalzzzakjlu
new file mode 100644
index 0000000000000000000000000000000000000000..a2f07f978d5359a40f226312fe840db13cb8fdaf
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@@ -0,0 +1,5 @@
+{"text":"Review: I was pleasantly surprised with how well this printed. It fit perfectly, was exactly as the preview image had shown it, and feels awesome!.\nPros: Print quality, fabric, cozy, excellent price, exactly as described. I love it all. I will absolutely make a habit of buying swimsuits here. I'm sure I can get some of my friends to do the same.\nCons: It took a while to ship, but that is understandable as it is both custom printed and coming from overseas.\nBottom Line: A+++ will shop here again~.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Swiftkey doesn't work on iOS 10's lock screen.\nDiscussion in 'iOS 10' started by ra4oasis, Oct 3, 2016.\nWhat gives? Do third party keyboards not work on the lock screen, or is it just swiftkey?\nThird party keyboards don't work while the phone is locked as I recall, even prior to iOS 10, and I don't think that has changed in iOS 10.\n3rd party keyboards don't work anywhere you enter a password.\nWell I'm only talking about replying to texts, not entering passwords.\npacorob and Sirious like this.\nLets face it apple doesn't want you to use a different keyboard. And all those security talks are bs imo.\nKind of like apps, they aren't loaded when the phone is locked.\nI thought this was just me lol.\neven on iOS 9, swiftkey wouldn't work on the lock screen for me. still doesn't.\nAnd even worse, if you have no other keyboard activated besides Swiftkey, you don't even get the keyboard associated with your phone's language; you get default English. Good luck if you want to type anything in your own language, because on the lock screen you can't even long press on characters to get alternatives!\nIt's so sad that this is still not possible with third party keyboards in iOS10. I wish Apple would allow this or simply add a word swipe feature to their own keyboard so I won't even have to use a third party keyboard.\nExactly. No swipe feature on the stock keyboard is the only thing that stops me from using it.\nThis is why I don't use touch ID to unlock my phone and have the passcode to unlock only kick in after an hour.\nAs long as I've used my phone within the last hour (extremely high probability unless I'm sleeping) - then I can use 3rd party keyboards to quick reply from the lock screen.\nWell, while locked, the iPhone is in a different (heightened) security state, and that's why 3rd-party keyboards aren't allowed. ALL extensions are disabled in locked state, including photo extensions. There's a reason for it - it would be very Easy for shady developers to sneak in a potentially dangerous feature onto custom keyboards, such as snooping, getting your contacts and sensitive information even in LOCKED state. That's a security danger that's probably not worth it.\nWell then, sounds like a great reason to add swiping to the default keyboard, huh?\nThat should have been there ages ago..\nI don't know why it's still not there considering how popular swipe keyboards are and how much time they save. I'd dump thirdparty keyboards if the stock one included the feature, and it really drives me crazy when the phone forces me to use the stock keyboard.\nBonus: I like that the stock keyboard can go dark when it should, but the stock ones haven't bothered to include a \"switch between light and dark skins\" feature. More inexplicableness.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's that time of the year again: Imagine Cup People's Choice Award!!! Yay!! The People's Choice award is part of the Imagine Cup. You get to vote on your favorite projects and games. Here's how it works: go to the People's Choice Facebook page, check out the team pages and vote for your favorite. You can vote everyday (until May 19th), so vote often! There are prizes for the teams and the voters!\nThe Imagine Cup team with the most votes will win a an all-expense paid trip to Sydney, Australia for the Imagine Cup Worldwide Finals this July. Additionally, anyone who vote will be entered for a chance to win weekly prizes including Acer ultrabooks, GAEMS Xbox bundles, Beats by Dr. Dre headphones and more!\nYou can vote every day, on each platform, for each of the teams!\nPrevious Post Capital Regional Celebration of Women in Computing!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Die modernen Teile der Stadt, gepr\u00e4gt von Hochh\u00e4user, Shoppingmalls und eindruckvollen Lichtinstallationen.\nThis entry was posted on Sonntag, November 15th, 2015 at 7:54\tand is filed under . You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Violence is both a key risk factor for, and consequence of, social exclusion and vulnerability. Marginalised children may be particularly vulnerable to experiencing violence, and amongst these, children with disabilities are a key group. Globally, 150 million children are estimated to be living with a disability, most of whom live in low and middle-income countries (WHO, 2011).\nChildren with disabilities are often amongst the most socially excluded and vulnerable. For instance, a recent study conducted by this research group analysed Plan International's 2012 sponsorship data across 30 countries and showed that sponsored children with disabilities were on average 10 times less likely to be going to school than sponsored children without disabilities and were more likely to have reported a serious illness in the last 12 months (Kuper, 2014). Read more about it here.\nWe will then consolidate our research findings and develop recommendations together with key representatives from Plan International.\nPartners: Plan International, Plan International Malawi, Plan International Uganda, Plan International Norway, Plan International Finland, Plan International UK.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzampym b/data_all_eng_slimpj/shuffled/split2/finalzzzampym
new file mode 100644
index 0000000000000000000000000000000000000000..9d85d72eb5884a926200278ce28080a33c32fbe4
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"If you are a enthusiastic gambler who at this point would like to experience on-line gambling then you definitely should know about gambling ban in your nation so that you can play without risk. Many nations around the world and their own states have frequently contested on as well as mycasinoresource.com approved a number of resolutions imposing, revoking or even upholding bans imposed upon gambling, especially on line betting, which in turn might ensure that your money as well as your earnings remain secure inside your on-line accounts.\nOnline wagering offers a unique problem to various nations because the betting site could be located in a certain country while avid gamers might be enjoying in another country. Most governments also fear decrease of taxes when folks start receiving on-line winnings straight into their own accounts. A few countries also dread loss of moral control through players that might get hooked on internet gambling because of lack of any control or even legislation. A number of countries currently have thus wound up banning online gambling altogether while others have enforced restrictions on some kinds of gambling while permitting players to wager freely in others.\nIf you want to visit online casinos or desire to indulge in online sports betting then you should certainly get up to date about the latest information on every type of gambling ban enforced on your preferred type of wagering. While many betting sites publish general information on bans, it is on yourself to verify if any kind of ban is actually imposed on participating in online craps, online slot machines, on line poker or online blackjack games, among others because in case you do win a lot of money then you must be able to get this sum in your bank account in your own country not having contravening any betting laws and regulations of your country. You simply might find your winnings getting seized by the authorities or even might find yourself on the wrong side of the law possibly before you have a chance to rejoice your online win.\nSeveral states in america which had previously banned on-line betting are now considering diluting or revoking such bans due to loss of revenue as several websites had moved their particular operations to overseas destinations. Nevertheless, other countries as well currently have modified their own laws regarding permitting or banning certain kinds of gambling and you should first check out if betting upon any kind of online game or sports activity is banned in your country before you decide to put up any deposit in any gambling site or even attempt to en-cash your own winnings. You should also check up on taxes that might be applied on your winnings because you definitely may not like a call from the tax man as soon as you receive your own winnings. You should pay a visit to on-line forums and blogs, and read regular betting articles to remain updated on the most recent occurrences within the on line gambling world so that you can wager, win, and spend your money in a very secure as well as legitimate way.\nIn case you are thinking of moving your passion regarding gambling to websites on the internet then you must make sure to stay on the right side of betting laws and regulations within your country before you decide to move your own mouse towards gambling websites. It is vital to know if your nation has imposed any gambling ban on a few or even just about all types of online gambling since your money and your independence should remain secure whenever you happily gamble on various online games as well as sports activities inside the virtual world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"hd00:05A beautiful girl writes and smiles with his laptop in the garden. Video about technology and nature, leisure, chats and social networks.\nhd00:09Tablet PC in the hands of the girl. Close-up.\n4k00:264K Woman packing boxes for house move, staring at camera with fed up expression. Slow motion.\nhd00:08Make-up artist doing make-up for model indoor. Prepare cosmetics.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Cartier Queen's Cup Polo came around again, bringing with it its annual melange of royalty, society and showbiz figures. Princess Beatrice and Dave Clark buzzed about the hospitality tent, greeting everyone, including Sharon Stone, Katherine Jenkins and Ben Fogle. And though the drinks were flowing, and Anton Mosimann's lunch sumptuous (burrata in lemon olive oil, followed by tagliata of beef), the day was all about the sport, and the Queen was delighted to award the cup to the Zacara foursome. Bravo!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Robert Pattinson and Kristen Stewart Dating!\nYou may or may not have an opinion about this.\nHowever, if you're in the online marketing business you should definitely start paying more attention to the celebrity entertainment industry.\nCelebrity Style news are a strong source of Internet traffic.\nLet's review the impact of using keywords related to the celebrity entertainment industry.\nThe numbers you see on the vertical axis of the Search Volume Index (graph above) aren't absolute search traffic numbers. Instead, Google Trends scales the first term you've entered so that its average search traffic in the chosen time period is 1.0; subsequent terms are then scaled relative to the first term.\nIn plain English: the graph above is ranked by \"kristen stewart\", meaning that \"robert pattinson\" is searched about twice as much as \"kristen stewart\" from late 2008 to early 2010. This means that websites in the celebrity entertainment industry should have focused their online articles more on Robert Pattison than on Kristen Stewart because Robert is a more frequently searched term.\nHowever, notice that towards the end of 2009 and beginning of 2010, the gap between the number of searches is closing between the names of the 2 celebrities. This would mean that using Kristen Stewart is becoming as effective as using Robert Pattinson on an online article.\nThis business maxim applies to the world of online celebrity entertainment as well.\nTo rank the top regions, Google Trends first looks at a sample of all Google searches to determine the areas from which it receives the most searches for your first term.\nRobert Pattinson, in blue, generates about double the number of searches as Kristen Stewart, in red, throughout the United States.\nHawaii, Virginia, and Nevada are prime candidates to target a Google Adwords or other type of online advertising campaign.\nIf you're in the online entertainment industry, you should take a look at the current trends in keyword searches when deciding what articles to write.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Important and major announcements regarding the forum and the blog.\nNon-football related talk. TV series & movies, music, video games & more.\nPitch your predictions for FC Bayern matches against other members.\nPost any forum\/account related questions, problems or suggestions in here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"where can find the similar wooden panel?\nI have similar arches in my new home and want to find one.\nTry Pier 1 they have unique wall art. World Market might be another good source. I also found this on the internet.\nYOUR FIRST HOUSEYou Can Turn That Into a Bathroom Vanity?\nWhere can I find similar?\nWhere can I find similar pulls?\nWhere to find a 4-panel folding bathtub door in the US?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Insiders on the Hill are speculating publicly that the push for the release of the House's FISA memo could happen as early as this week.\nThough the FBI is pressing to see the memo before the public gets a look at it. Which is precisely why it should be made public first.\nAnd now Rep. Trey Gowdy is dropping not-so-subtle hints into what exactly is in that memo.\nDuring an interview with Fox News this weekend, Gowdy teased that the memo reveals whether the infamous Trump dossier was vetted by the FBI, who paid for it, Chris Steele's relationship with Hillary, and other details that will blow open the FBI collusion story.\nRest assured, this memo will get released soon, and it's gonna be huge.\nLawmakers have been reluctant to discuss a classified four-page memo alleging that the FBI and Justice Department abused the Foreign Intelligence Surveillance Act (FISA) in order to spy on Trump campaign associates, but South Carolina Rep. Trey Gowdy provided several clues on Sunday to what's in the controversial document.\nIn an interview on \"Fox News Sunday,\" Gowdy posed several questions to host Chris Wallace and his viewers that hinted at the allegations in the memo, which could be released by the House Intelligence Committee as early as this week.\nCongressional investigators are now revealing that they've only received about 15% of the 50,000 text messages between Peter Strzok and Lisa Page.\nAnd that doesn't include the newly recovered messages leading up to the appointment of Robert Mueller, which could potentially double the number of texts.\nThe FBI's explanation? The rest of the texts were of a 'personal nature'.\nSo the agency under investigation for corruption gets to say what's relevant to the investigation?\nWhat's more, these are only texts sent via government-issued phones.\nBut we know for a fact that Strzok and Page texted on personal phones as well.\nWill investigators subpoena records from Apple? That could be an epic fight.\nThe Justice Department has given Congress less than 15 percent of the texts between FBI officials Peter Strzok and Lisa Page \u2014 and that is all Congress is likely to get, at least until department experts finish an effort to recover an unknown number of previously lost texts that were sent and received during a key five-month period during the Trump-Russia investigation.\nThere is much confusion over some basic facts of the Strzok-Page texts. How many are there? How many relate to the two most politically-charged investigations in years, the Trump-Russia probe and the Hillary Clinton email investigation? How many have been turned over to Congress? And how many are left to be turned over to Congress?\nHow many texts will be turned over? First, it's not possible to know how many texts from the December 14, 2016 to May 17, 2017 time period will be recovered and turned over. But of the 50,000 the Justice Department already has in hand, officials say they have already turned over all they're going to give to Congress.\nThat means Justice has decided to allow Congress to see just 4,000 to 7,000 of a total of 50,000 Strzok-Page texts \u2014 even the larger number is slightly less than 15 percent of the total number of texts the Justice Department has now. Why is that? Justice Department officials point to a January 19 letter from Assistant Attorney General Stephen Boyd to Capitol Hill investigators explaining which texts would and would not be turned over.\nAs members of Congress continue combing through the exchanges in the newly discovered text messages between Peter Strzok and Lisa Page, the web of collusion is weaved all the more.\nCommenting on one of those exchanges was Rep. Trey Gowdy on Fox News late this week arguing that the 'fix was in' long before agents ever interviewed Hillary.\nLong after the FBI and DOJ knew they weren't going to indict Hillary, they conducted a mock interview with kid gloves to ensure the exoneration plan remained intact.\nStrzok responded, \"I called Bill and relayed what we discussed. He agrees. I will email you and [redacted] same.\nLate Wednesday evening Fox News reported 'multiple sources' inside the Department of Justice have confirmed that agency officials are 'successfully recovering' text messages which the FBI previously reported as lost during the critical five months preceding the appointment of special counsel Robert Mueller.\nWhat's more interesting is that it appears officials from the DOJ have had to step in to correct the error which the FBI couldn't solve on its own.\nYou know, that bureau which is supposed to be really good at forensics.\nAt any rate, this is a developing story which could throw the collusion\/conspiracy investigation into a new gear.\n\"Sources are exclusively telling me tonight \u2014 multiple sources \u2014 that the Department of Justice is as we speak in the process of successfully recovering many of those text messages in that five-month period of time from the Trump-hating FBI officials Peter Strzok and Lisa Page,\" Fox News host Sean Hannity declared.\n\"The DOJ is also trying to track down their mobile phones,\" Hannity added.\nSen. Chuck Grassley is the powerful chairman of the Senate Judiciary Committee, and now he's angry\u2026very angry.\nIn a fiery speech on the Senate floor yesterday, Grassley blew his top and unloaded on the FBI and DOJ, essentially accusing both departments of obstruction of justice for their refusal to comply with committee requests for information and testimony in the investigation of possible collusion.\nWe now have multiple congressional committees and half a dozen high-ranking members ready to switch to guns on the FBI and DOJ.\nThis is about to get really interesting, folks.\nLast week, the House Intelligence Committee voted to release a memo compiled by staff of that committee. If focuses on two major areas. First, the abuse of FISA authority that resulted in that law being used by the outgoing Obama administration to monitor activities of the incoming Trump administration. Second, the fairly blatant antipathy of some senior FBI officials toward the idea of a Trump presidency and how this antipathy may have impacted upon the use of the Trump dossier and other parts of the 'collusion' investigation. This memo is based, in part, on top secret and compartmented information. It is held in the SCIF at the House of Representatives and members, over 200 thus far, can read it but not have copies or make notes.\nNow Senate Judiciary Committee Chairman Chuck Grassley has joined the fray. In a blistering speech, a speech that was a veritable Festivus style airing of grievances, Grassley called out the FBI for deliberately obstructing his committee's investigation. And more. Read the statements these are the highlights of the floor speech.\nCongressional leaders continue to dig through the text messages between Peter Strzok and Lisa Page, and the more they see the more incriminating those texts become.\nYesterday it was revealed that at least one of the exchanges referred to a 'secret society' within the FBI.\nNow Sen. Ron Johnson has confirmed that an FBI informant is reporting that that secret society held meetings offsite from the FBI building.\nThe questions now include who was in that society and what were the purposes of those meetings?\nThe fact that five months of text messages are missing is even more suspicious than previously thought.\nThe possibility of a special counsel to investigate the FBI is nearly certain after these revelations.\n\"Is [there] anything more about that?\" Baier asked.\n\"No. We have to dig into it \u2014 this is not a distraction. Again, this is bias \u2014 potentially corruption \u2014 at the highest levelsee of the FBI,\" Johnson declared.\n\"By the way,\" he added. \"Robert Mueller used to run the FBI. He is in no position to do an investigation over this kind of misconduct.\" Johnson said the controversies surrounding the FBI have reached the point at which a second special counsel is warranted.\n\"So I think at this point in time we probably should be looking at a special counsel to undertake this investigation \u2014 but Congress is going to have to continue to dig,\" he said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With the 65th pick or cheap nba jerseys from china first pick in the third round, Carolina took defensive tackle Terrell McClain out of South The carolinas. And with the th pick the final pick of the third round Carolina selected Stanford Cardinals defensive tackle Sione Fua.\nAlways match your clothes. Hope to find outfits that will fit method. Never buy authentic jerseys online this too small in which you or too big for you because these items look really awful.\n[h2]soccer jerseys for cheap[\/h2]The More the Merrier. Invite plenty of family, friends, coworkers and neighbors to compete and argue with. You sure they're compatible--you would possibly not want your mom to determine your rowdy friends intended for. Send invitations well in advance, before your football cheap jerseys fan friends make other coverages.\nAndre Johnson cheap mlb jerseys was all over again a thorn in the side Charlotte Hornets of nfl cornerbacks all year in year. Johnson had 101 catches for 1569 yards receiving and nine touchdowns. To to sum up Johnson's season with statistics does not do him justice. soccer jerseys for cheap Johnson was a match changer and Matt Schaub's primary receiver all wholesale mlb jerseys period. To say Johnson evoked fear just about every corner he faced was an exaggeration. Johnson did his business quietly but every cornerback in the league knew that he was the most effective wide receiver they would face all year.\nIf your pooch has this regarding lovable nature then may do bet possess dress her up as getting cheerleader and take her out, she will love everyone fawning over her. You could also use it for fun photo ops and more family a lot of fun. You might use these for postcards there's a chance frame and display inside your home. There are so many options it is easier to a football, dress-wearing pooch who loves to make you happy and help you smile.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Check out this thorough review of the features of JMeter's step-by-step debugger.\nWe've covered various ways to debug scripts in JMeter, but now you have the ability to debug scripts step-by-step and receive updates in real time. Here's how.\nWhen I started to work with JMeter to create scripts, I was (as I'm sure we all do) feeling like debugging my scripts was way too cumbersome a process. By using the Debug Sampler and log() functions, it really felt like the old mainframe debugging days.\nI wished we could just have a normal step-by-step debug, just like any normal IDE (integrated development environment), to debug my scripts and find issues.\nOnce JMeter will relaunch itself, you should see the debugger plugin under \"Installed plugins\".\nNow, let's go for a short tour to use it. For that, I've prepared a JMX file (attached) as an example in order to show you how to use the debugger features.\nThis is because the debugger follows the JMeter scoping rules, which means execute the timer and then execute the sample. The debugger will always focus on the next element to be executed.\nFeature 2: Step by Step: Let's See These Variables!\nYou might be wondering what will happen in the next \"Step Over\". Let's find out.\nYou might have noticed that we are still in the \"View Results Tree\" listener even though it has been executed. Why?\nWell, it's not a bug. We just figured out that you will probably want to see the results before moving to the next element. So we might not want to switch the focus to sample #2 just yet.\nJust like any other IDE, we have breakpoints, just to save you gazillions of \"Step Over\" clicks. Let's put a breakpoint on \"Sample #2\" by right clicking on it and select \"breakpoint\". Then close and reopen the debugger, and then click on the \"Continue\" button. The debugger has stopped at Sample #2, just before it's starting to execute its scope.\nYou know the feeling when you want to see the value of a variable\/function and monitor when it has been changed? Meet the Evaluate tab.\nNow before pressing \"Continue\", let's go to the Evaluate tab and put our #RegEx variable name (RegExTimestamp) in the JMeter expression line.\nWe are going to add a stop button so you won't need to close the debugger to restart your debugging session.\nWe are also going to mark any variable which just changed in Bold so it will be easier to recognize which variable just changed.\nImproving the breakpoints stop on scope (and not the sampler).\nKeyboard shortcut to open the debugger (we will do our best :) ).\nTo conclude - the Step-by-Step debugger is a significant tool which every JMeter scripter should have in their toolbox. In addition to debugging your scripts, you can also understand how JMeter scoping works and see, for the first time, the execution order of your test plan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New state program recruits rural veterinarians by repaying college debts.\n\u200bConference Oct. 5-6 will expand knowledge between physicians and veterinarians working in regenerative medicine. Registration is open!\n\u200bMarci Stille and Jenny Harding dress CSU's mascot.\n\u200bDr. John Torres, medical correspondent for NBC news, reported on the new high-tech operating room at the James L. Voss Veterinary Teaching Hospital.\nTo better understand how our bodies adjust to high altitude, where there's less oxygen, the AltitudeOmics team studied average, healthy people rather than mountaineers.\nBrown, a self-described mountain girl is beginning her final year at CSU and absolutely loves the human brain and anatomy.\nDonaghy spent his summer interning at the University of Reading in England, researching platelet-mediated blood coagulation, which can cause heart attacks and strokes.\n\u200bYoung, smart undergraduate researchers are already solving global problems.\nLacuna is helping to eliminate transit time of the slide to a pathologist. With digital upload of the slide images, the pathologist can look at the slide within minutes of sampling and provide an interpretation with the client and the pet still in the exam room.\n\u200bDVM student Keira Sztukowski's groundbreaking research opens up possibilities for the future prevention of neurological problems that commonly occur in people with HIV\/AIDS despite antiretroviral therapy.\n\u200bCBS Denver: \"We're heading off trouble before it happens, if we can. If they already have a problem, we are going to treat it as well as we can.\" - Dr. Jon Geller.\nDr. Tim Holt won the Pioneer Award from the Beef Improvement Federation for his work on pulmonary hypertension in in high-elevation cattle herds.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"The classes are a continuation of currently running classes. For example, if you are currently in the Monday at 7:15 class then the next session at that time is the appropriate level for you.\nAll classes are held at Clocktower Music located at 676 Laurel Street in San Carlos.\nWant to be informed about upcoming gigs, group classes and other related news?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lovely ranch with beautiful laminate floors! This home is move in ready with updated flooring, new lighting, and freshly painted! There are endless possibilities for this wonderful house! This home has so much to offer PLUS no city taxes!\n659 Hampton Circle is a 1,606 (*estimated) square foot home with 3 bedrooms and 2 bathrooms located in Belvedere SC, AIKEN County. This property is listed for sale at $134,900 and is currently active and accepting offers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cambridge, Mass. (November 30) -- MIT Director of Physical Education Carrie Sampson Moore recently received the \"Distinguished Alumni Award\" from the College of Education and Human Development at the University of Minnesota. The award was established in 2010 and is selected from over 70,000 alumni, honoring alumni who have brought distinction to their professions and communities. The event, which honors just 25 alumni each year, took place at the McNamara Alumni Center on the University of Minnesota campus.\n\"Carrie is incredibly deserving of an award that recognizes those who have made a positive difference in the lives of children, youth, families, schools and organizations and whose achievements bring honor to the college,\" said Dani Korman, MIT swimming and diving head coach and physical education instructor. \"Carrie has created a program that empowers the students at MIT while teaching them wellness skills that will last for a lifetime. She is also an outstanding mentor and co-collaborator with the instructors she works with and challenges us to find new ways to connect with the students and advocates for our own balance of life while working at MIT.\"\nRecipients of this award span a diverse range of academic disciplines and career paths; business and civic leaders, counselors and social workers, educators and activists, entrepreneurs and the most the most dedicated of volunteers.\nMoore is a 1989 graduate of the University of Minnesota, where she earned her Bachelor of Arts degree in physical education with a concentration in coaching. She also went on to earn her Master's degree in kinesiology in 2001 with a focus on sport management and sport psychology. Over her 25-year career in higher education sport and fitness, Moore has worked at the University of Minnesota-Twin Cities, Miami University in Ohio and now at MIT. She has been with MIT since 2002 and has been in her current role since 2005.\n\"Carrie's leadership and contributions to MIT Physical Education cannot be measured,\" said Matt Breen, MIT physical education instructor. \"She is highly respected by her colleagues and always has the best interests of MIT students. I have learned a great deal from working with Carrie over the past six years and am looking forward to that continuing into the future.\"\nIn her position as the Director of Physical Education, Moore is entrusted with taking the strategic direction and vision for physical education from concept to reality by providing leadership and development, curricular guidance, and facilitation and management of MIT's comprehensive Physical Education program. At MIT, the Physical Education program involves more than 4,000 undergraduates annually as the student body completes the Institute's general PE requirement. Moore also actively serves on the MIT senior management team, the Departmental Advisory Board, the Faculty Council as management representative and many other division and departmental committees. She was also a member and presenter for the National Intramural and Recreational Sports Association (NIRSA) for 15 years and has been a member of the Board of Directors of the National Association of Kinesiology in Higher Education (NAKHE) for the last 10 years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wilmington \u2013 On Tuesday, 03 April 2018, Gregory Raspberry, of Wilmington, was apprehended by Wilmington Police at Delaware Avenue and N. Tatnall Streets.\nWilmington Police have charged him with Attempted Robbery 2nd Degree and Robbery 2nd Degree for the Wilmington incidents.\nRaspberry failed to post $16,000 CASH BAIL and was committed to Howard Young Correctional Facility.\nState Police have not released charging or bail information for Raspberry, regarding the Elsmere-Area PNC robbery incident.\nIn the Wilmington incidents Raspberry entered the bank demanding money from the bank teller. When the teller refused, the suspect fled on foot, police said.\nWhile police were investigating the PNC incident, four minutes later, they were alerted that a second bank robbery occurred at the Citizens Bank, located in 919 N. Market Street, at approximately 10:57 am., according to police.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kit Salbino Teodosio is creating inspirational posts on Facebook & Instagram, and Youtube videos.\nFor a long time, I never had a definite idea of what I wanted to do with my life. I always knew I wanted to do something creative and inspirational. Then I discovered youtube. I have been watching youtube videos for as long as I can remember. I love inspirational, life hacks, apartment tour, and other types of videos. I was doing videos for a few months, but I had to stop. My computer is not as great as it used to be, it's about 4 years old and I wish I can post more videos anytime I want to.\nI am currently unemployed and have been so for almost a year. Its been hard, but I manage with what savings I have. I have this dream to live free doing what I love and exploring this big beautiful earth.\nHelp me get the proper equipment that I need to put out more content and watch me on my journey. I hope I can inspire you all.\nAs a youtuber, I need to upgrade my equipment. I've been working on a 2012 MacBook Pro and while it still works fine, I have encountered countless lags and shutdowns in the middle of my video\/photo editing and I find that the 13\" screen is too small for me to edit videos properly. I'm looking to acquire the 27\" iMac with fusion drive to help me, as well as a Diva ring light with stand and a mic for my camera. So far, on my own, I have just my camera equipment and natural light. While I understand it is best to film with natural light, sometimes where I stay, the weather is unpredictable and what starts off as a great filming day, turns out to be so gloomy.\nPlease help me reach my goal. I want to inspire and continue doing what makes me happiest.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"1 Hour Romantic Dramedy. A popular, much awarded, character driven time travel show starring Erin Carpluk and Michael Riley. I directed 5 episodes as Erica Strange went back in time to try to come to terms with past regrets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here we are in the morning fog approaching the Kiel Canal.\nThe Kiel canal is 61 miles long and joins the North Sea with the Baltic avoiding the need to sail around Denmark and potentially rough North Sea weather.\nThere is enough traffic, they have a Pilot \"Mother Ship\" that hangs out and dispatches little pilot boats to deliver pilots to the ships. Plus I'm sure the North Sea waters can get kinda rough.\nOur little pilot boat is done with it's delivery and headed back to mamma.\nThere is a lot of small boat traffic in the canal too.\nFishing must be good here.\nI believe this is Cuxhaven at the entrance of the canal.\nTwo locks. We're going left.\nSome locals turning out to see us.\nThe North Sea side locks.\nCows everywhere. Even between the two sets of locks.\nThe observation area is pretty well packed. There are not many cruise ships that come through due to height restrictions under the bridges along the canal so we were pretty popular. Our ship was pretty small. Only 790 passengers.\nI've never seen gates like these. It's a single gate that slides across the canal rather than 2 gates opening like doors.\nA local hangout just inside the locks.\nOne of the many ferries that go back and forth across the canal along the length of it.\nOne of those \"low\" bridges that prevent larger cruise ships from transiting the canal.\nGuten Morgan our German cycling friends.\nThere is a nice bike\/walking path the whole 61 miles of the canal.\nMore friendly folk that seem to need to hire a pool boy.\nA train about to go over our heads.\nOne of the little towns where the street ends at the ramp to the car ferry.\nVery nice looking camp ground.\nSome of our opposing traffic. It was not very wide for two large ships to pass.\nAll size craft in the canal.\nA new tug boat being hauled through for final fit-out.\nThe Royal Yacht Dannebrog of Denmark happened to be out and about.\nA little too chilly on this cruise to use the pool.\nGive you an idea how much traffic there is in this canal.\nWith all the modern wind turbines, here is an old windmill.\n\"Nothing runs like a Deer\"\nFolks at the restaurant taking time to wave.\nPatty the world traveler enjoying the canal trip.\nThis is a welcome center at Rendsburg where folks come to welcome the ships.\nTake note of the \"carriage\" hanging from the bridge.\nAnd go across the canal hanging under the railroad bridge.\nAnd with movie like timing, I was lucky enough to have a train going over at the same time the carriage was going under.\nThe locks at the Baltic side of the canal.\nThe tiniest little ferry boat taking folks across the canal right near the locks.\nHere we are entering the lock at the end of the canal, headed into the Baltic.\nClock tower in the canal area.\nA nice clean and tidy container ship joins us in the lock next to us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When contemplating cutting a hole in one's roof, it would be wise to hire someone who has done it more than a few times. I have installed over 20,000 skylights in my 24 year career with less tha one tenth of one percent callback rate. No other company in the state of Florida can say that. The major advantage of adding a tubular skylight over a window is the amount of light output and lower cost factor of the skylight. A skylight will provide 2 to 5 times the amount of light given the same size window in a wall. Adding all this light leads to a higher quality of life and value to a home without sacrificing privacy or security. Installing a tubular skylight takes only about 2 hours with almost no mess in the home. I also replace and repair existing skylights that may currently be leaking.\nI use only top quality hurricane approved impact rated skylights that come with a minimum of 10 to 20 year warranties. I also sell and install top quality solar attic fans with a 25 year warranty on the entire fan -solar panel, housing and motor. I only use Kennedy and Skyflex skylights for my installs and treat each customer like family. I am a one man family run business, so when you talk to me, I will be the person working on your roof for high quality control. I am fully licensed and insured. My 5 star ratings and several seals of approval on Homeadvisor speaks for themselves. I guarantee no leaks on all new and replacement skylights in writing. Please call me today for your free estimate.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Modulating the adjuvanticity of alum by co-administration of muramyl di-peptide (MDP) or Quil-A.\nThe characterization of the immunological cascades of the innate immune system activated by pathogen associated molecular patterns (PAMP) recognized by pattern recognition receptors (PRR) have allowed the elucidation of the mechanisms underlying the immunomodulatory properties of adjuvants. Thus, the combinatorial use of adjuvants with specific, complementary functions is investigated to achieve tailored immune responses to subunit vaccines. We have previously shown how combinatorial administration of chitosan and cholera toxin B or muramyl-di-peptide (MDP) intranasally, but not intramuscularly, can allow small doses of MDP which, when administered alone cannot adjuvantise Helicobacter pylori urease (rUre), achieve an immunomodulatory effect through the specific physiological effect of chitosan. The aim of this study was to investigate if in the context of rUre the adjuvantising effect of MDP could be realized via the intramuscular route by combination with aluminium hydroxide, as compared with the routinely used veterinary adjuvant combination of alum and Quil-A. Serum IgG kinetics were comparable between the two adjuvant combination groups. However, the alum + MDP combination afforded higher antigen-specific recall responses in splenocyte cultures, associated with elevated release of the type I immune response cytokines IFN-gamma and IL-2. This data suggests that the adjuvanticity of MDP can be modulated in the context of alum in a manner dissimilar to that of Quil-A, achieving a balancing effect on the responses elicited by alum adjuvantisation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here's the second version of Construction Sabotage War! Are you ready to become the construction king? Construct different types of buildings and upgrade them with your earnings for a better income. Sabotage your competitors' buildings and try to win each level! Remember, you need a certain amount of squares for each building! Have fun!\nOutpost Combat 2 is back with more challenges! Achievements added! New special abilities..\nSelect your hero and lead your people towards victory! Settle your people in houses in the..\nLead your elves towards victory by capturing all buildings of the North Pole! Make use of..\n89% Civilizations Wars: Ice Leg..\nSwerve to the left or right to weave between the slower cars in Road Fight, a challenging..","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasnzu b/data_all_eng_slimpj/shuffled/split2/finalzzzasnzu
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+{"text":"From now on, all announcements of Service Cources will be announced at the main webpage of our department.\n1st Midterm exam places of MAT1071 Mathematics 1 are announced, click here.\n1st Midterm exam places of MAT1320 Linear Algebra are announced, click here.\n1st Midterm exam places of MAT2411 Differential Equations are announced, click here.\n1st Midterm exam places of MAT1821 Mathematics are announced, click here.\nThe group distributions of the 2018-2019 Fall Semester Service Courses have been conducted. Pursuant to the License Regulations, the student must attend at least 70% of the theoretical courses he\/she enrolled and at least 80% of all the practical work other than the theoretical course. For this reason, in order to avoid absenteeism at the end of the semester, all students are required to check his\/her group in the USIS and follow the courses in the related group.\nClick the following links to see the instructors and classes of related service course.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Beautiful Gmc Sierra 1500 Regular Cab is free HD wallpaper. This wallpaper was upload at April 2, 2017 upload by in GMC Car Reviews.\nBeautiful Gmc Sierra 1500 Regular Cab is high definition wallpaper and size this wallpaper is 2048x1360. You can make Beautiful Gmc Sierra 1500 Regular Cab For your Desktop Wallpaper, Tablet, Android or iPhone and another Smartphone device for free. To download and obtain the Beautiful Gmc Sierra 1500 Regular Cab images by click the download button below to get multiple high-resversions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Godolphin & Latymer has a long-standing commitment to providing outstanding teaching for learning. With the introduction of new technologies from iPads to drones and virtual reality learning we are constantly looking for new ways to help our pupils learn more effectively and more deeply. We see technology as a compliment to our existing teaching expertise rather than a replacement for it.\nit cannot replace poor teaching.\"\nSchleicher, A. (2018), Valuing our Teachers and Raising their Status: How Communities Can Help, International Summit on the Teaching Profession, OECD Publishing, Paris, https:\/\/doi.org\/10.1787\/9789264292697-en.\nImmersive technology is still a newly developing field in education, but it is already being used by the British Army, NHS and Cross Rail as an effective learning tool. As part of an internally funded action research project, G&L are trialling a wide variety of augmented, mixed and virtual reality apps to investigate their impact on our inquisitive and ambitious learners.\nWith contributions from G&L teachers, Jonathan writes and maintains a regular internal blog focused on helping teachers to employ iPads to improve learning and ease workflows in the classroom. Some topics covered in the past include \"Tracking Spaced Practice in Nearpod,\" \"Hinge Questions on the iPad,\" \"Google Slides for Work Sharing,\" and \"The Muddiest Point.\" Staff are encouraged to share ideas and professional learning about teaching with technology through regular iPad tips, blog postings, and lunchtime sharing sessions.\nG&L are delighted to have been selected as just one of five schools in the country to receive mixed reality headsets from HP and SystemActive to test in our lessons as part of a project funded by ISC Digital. The equipment is currently being used by staff and students and we will be presenting our findings on 17 May at RGS Newcastle.\nIn 2017 the PFTA generously funded the purchase of a drone for the school to use in lessons. We have been using it for sports days, field-studies and student learning, gaining a new perspective and some amazing footage.\nSince 2017, Godolphin & Latymer have been proud to play host to the West London Apple RTC. As a Regional Training Centre, we partner with Apple to offer free continuing professional development to teachers at G&L and partner institutions. This year we are running a Teacher Learning Community themed upon \"Redefining Teaching & Learning Using Technology.\" To learn more about what the community is studying and discussing, visit the RTC site.\nClaire Badger is a Senior Teacher at Godolphin and Latymer responsible for the school's Digital Strategy in Teaching and Learning. Claire planned and managed the introduction of iPads at G&L and oversees the school's contribution as an Apple Regional Training Centre. She is a pioneer in using technology in her Chemistry Lessons, encouraging flip-learning, low-stakes testing and interleaving through a wide variety of apps and activities. She manages the Teaching and Learning Communities at G&L and regularly organises training events for staff, pupils, parents and PGCE and NQT students in the use of technology in the classroom.\nKatie Tallett-Williams is a member of the Geography Department at Godolphin and Latymer. She has a particular interest in using technology to enhance teaching and learning, assisting as a Staff-Digital Leader to help roll-out the use of iPads for both teachers and students at G&L. Katie has worked to develop the Geography Department's use of technology while in the field and enjoys researching how new technology can bring field-work back into the classroom. In conjunction with the Physics Department and the generous support of the PTFA, Katie collaborated in the purchase of the DJI Mavic Pro Drone and its use on field-work projects. She is also leading the action research project on the use of technology in enhancing experiential learning, acting as the school contact for the ISC Virtual Reality Project.\nJonathan White is the Digital Learning Lead at Godolphin & Latymer, manages the West London Apple Regional Training Centre, and writes the @Godolphin digital learning blog. Jonathan works with teachers at G&L and through the RTC to help them use technology effectively to support teaching and learning. He holds a Master's degree in Educational Media & Technology from Boston University, and has more than twenty years' experience training lecturers, teachers, students and staff in secondary and higher education settings. Jonathan has a particular interest in metacognition and low-stakes assessment, and the ways that they can be supported by mobile and cloud-based technologies. He has driven the adoption of G Suite at Godolphin & Latymer, and collaborates with the IT department for the management and support of the school's 1:1 iPad program.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Quantitative Undergraduate Biology Education and Synthesis (QUBES) project offers an online platform to facilitate faculty professional development and sharing open educational resources for teaching quantitative skills. QUBES offer professional development opportunities called Faculty Mentoring Networks (FMNs). FMNs are semester-long professional development opportunities designed to engage you with other faculty to enhance your teaching.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This month, we want to share with you the story of Asunci\u00f3n, better-known under the name of asunvm. Asunci\u00f3n is from La Linea, Spain, and she joined our community 4 years ago. She is one of our more generous creator... read more about her below!\nI am a teacher in Spain. In my free time I love taking part in creative competitions. I enjoy drawing and photography. Mix both activities is really nice for me.\nOn Internet. At the moment, I participate in EYEKA competitions and in other websites around all the world.\nFirstly, the competition must sound fun to me. I always work on paper. I love drawing. I prepare some pictures with different ideas and then I develop them on my computer. I look for inspiration, above all, reading information related to competitions on Internet. That is really enjoyable. You learn multiple new things. I am very fond of new technologies.\nI do not feel upset, I am ok. No problem at all. The goal is to do your best and share ideas. Learning is the objective.\nWe know that you use your prizes to make donations to UNICEF. This is such a generous and lovely gesture! Can you tell us more about it?\nI feel really happy when I win a prize because UNICEF is going to do the best use of it.\nApart from UNICEF donations, do you have another dream you would like to reach with the prizes you will win\/have won?\nNot really. I take part in competitions to have fun. I really enjoy doing it. Donate my prizes to UNICEF gives me double happiness.\nYes, my main advise: enjoy your contests and try to do your best. Read and learn day after day. Observe around you. Simple things are always great.\nThank you EYEKA for being part of me!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzastfy b/data_all_eng_slimpj/shuffled/split2/finalzzzastfy
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index 0000000000000000000000000000000000000000..40f4e0ee8b00bde8f71399f92d0d9354e0712db2
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+{"text":"The new Toyota Prius is sold in many countries around the world with the option of letting the driver select times when they would prefer to operate the car without the engine running. The intention is to allow the driver to move the car out of the garage, or even out of the neighborhood, without the engine starting to awaken others. It is also a wonderful way to demonstrate the capabilities of the hybrid powertrain to passengers learning about the system for the first time. This factory option is currently not available for buyers in North America although the programming for it is still included in the computers which control the hybrid system. This kit allows you (in just a few minutes) to activate the electric feature of your Prius and drive for short distances. The software already present in the car will automatically restart the engine when the battery needs to be recharged or if you need to accelerate quickly. The electric only feature can be turned on and used up to a speed of around 34 mph at which time it will automatically turn off and the car will operate as usual.\nThe electric mode kit is designed for ease of installation and includes the necessary pin and connectors to complete the installation. Color photos and step by step instructions explain how to insert the missing pin and clip on two wires using simple wire taps. No wires are cut and very little disassembly is necessary to activate this feature. Once installed, the kit will create two possible uses for the \"Cancel\" function of your cruise control stalk. If you pull the cruise control stalk toward you (as when using the \"Cancel\" feature) and hold it for more than two seconds, the electric only mode will be signaled and the factory software will shut down the engine (or prevent it from starting if you are just turning on the car). This method will also turn off the electric mode when desired. The normal operation of the \"Cancel\" feature when using the cruise control is unaffected by the addition of the circuit and will continue to function as usual.\nThe kit is $45.00+(S&H). It will take approximately 25 minutes to install and requires only a pair of needle nose pliers and a screwdriver.\nNote: Currently this addition is only available for the 2004 - 2009 Prius models and not for the earlier 2000-2003 Prius.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Prime Cut Pro | Best Videos on the net!\nOrder on Vimeo ONDemand today!\nTrucks, ATV's, UTV's and more hit the mud hole in Florida at Redneck Mud Park during the Trucks Gone Wild event.\nBiketoberfest 2018 from Daytona Beach, Florida.\nCustom Cars and more on display at the Florida State Fairgrounds in Tampa, Florida. This Car Show is one of the biggest up coming events in the Automobile Industry today!\nOrder our newest Vimeo now Available!\nDiplo throws down an amazing set in Miami Beach. One of the best 2018 best EDM DJ's.\n2018 Summer Party of the year from Miami Beach, Mad Decent Pool Party featuring Diplo. Want to see more?? Click the link for the VOD and use code Summer2018 for 50% off. Watch on ONDemand straight from the website via Vimeo! Hit us up for free codes!\nCustom lifted trucks invade Daytona Beach, Florida for the 2018 Daytona Truck Meet.\nOfficial video from the HIN car show in Orlando, Florida.\nWe are now selling AD space for our website. Contact us at Info@PrimeCutPro.com for rates & details!\nPlease check out our download store where you can purchase longer uncut videos such as full length bikini contests and more! Thank you to all of our subscribers!\nPrime Cut Pro is a professional video production company with offices in Florida & NYC. We specialize in entertainment on YouTube, mainstream television, internet, wireless, and more. If you are a television producer and are interested in licensing any of our clips please contact us. We also sell download files for any of our videos. Please browse our download store. We have been filming entertainment for over 10 years and have created this company consisting of experienced videographers, editors & producers. Prime Cut Pro brings over 30+ combined years into the business. We have adopted the slogan \"Best Videos on the Net!\" Watch....and we think that you will agree!\nUltra Resistance Line Up Announced!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Expertly designed to make remote power access simple, while providing maximum conductivity.\nThe NOCO GCP1 AC Port Plug with Integrated Extension Cord offers optimal flexibility and ease of installation. It requires no cable modifications or hand wiring, and works with any extension cord. Its sleek, ultra low-profile design not only works seamlessly on any application but also includes a pressure-fit cap for a watertight seal. This Port Plug makes a charging port, engine block, or garage outlet more accessible than ever before, giving you endless possibilities.\nRated at 13-amps, 125-volts for peak performance.\nIntegrated charge inlet with a 16-inch extension cable.\nRequires no cable modification or hand wiring.\nFull-sized AC prongs for maximum conductivity.\nEasy installation into a 2-inch hole-size.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dreamtime by the Sea is a luxury villa with an enviable beachfront location in the picturesque Montego Bay. The Jamaica villa has seven beautiful bedrooms and 7.5 bathrooms. The villa's architecture is very modern and based on sleek, modern lines.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Alvin is a Senior Sales Professional with a high degree of experience in Selling, Marketing and Advertising in Real Estate. He has lived and worked in many different countries and has a strong passion for Real Estate. His wealth of experience in Real estate negotiations have given our Clients a great deal of satisfaction in every transaction.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatnph b/data_all_eng_slimpj/shuffled/split2/finalzzzatnph
new file mode 100644
index 0000000000000000000000000000000000000000..a47f9e29ce48664fe6824ced3c780af1e32e0e85
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@@ -0,0 +1,5 @@
+{"text":"Turn off water main, then cut plumbing back to general area of new holes. Remove excess pipe sections. To re-route the supply lines, first clean all of the fittings with a wire pipe brush (Image 1).... Wipe down the tub. Removing any dirt and grime will make it easier to spot stains and deal with them without creating a smeary mess. Use a cleaning rag and the spray cleaner to wipe down the tub using firm pressure and plenty of cleaner.\nOld-fashioned clawfoot tubs and pedestal sinks give your bathroom a retro flair. However, the porcelain on these often antiqued fixtures can sometimes be covered in brown stains.\nTurn off water main, then cut plumbing back to general area of new holes. Remove excess pipe sections. To re-route the supply lines, first clean all of the fittings with a wire pipe brush (Image 1).\nMost claw-foot bathtubs will have holes drilled into a tub wall or on a tub deck to house faucets and tub fillers; undrilled tubs may require using a floor-mount tub filler or wall-mount faucet. When choosing a tub filler fitting, opt for one equipped with a handheld shower or one to which a showerhead and a circular shower rod enclosure can be attached.\nWe've gotten A LOT of questions about how we've rescued the clawfoot tub at our beach house, so let's just dive right into what worked (and what didn't) and how we nursed this baby back to life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Punch Recipes Bridal Shower. This wonderful photo collections about Punch Recipes Bridal Shower is available to download. We collect this wonderful photo from internet and choose the best for you. Punch Recipes Bridal Shower photos and pictures collection that posted here was carefully selected and uploaded by Rockymage team after choosing the ones that are best among the others. So, finally we make it and here these list of wonderful photo for your inspiration and informational purpose regarding the Punch Recipes Bridal Shower as part of 99 Wedding Ideas exclusive updates collection.Punch Recipes Bridal Shower, take your time and find the best Punch Recipes Bridal Shower images and pictures posted here that suitable with your needs and use it for your own collection and personal use. Punch Recipes Bridal Shower.\nPunch Recipes Bridal Shower. So, if you want to get this wonderful photo about Punch Recipes Bridal Shower, just click save button to save this images to your computer. Finally if you want to get new and the latest wallpaper related with Punch Recipes Bridal Shower, please follow us on facebook or bookmark this site, we try our best to give you daily update with fresh and new wallpaper 2017. Hope you enjoy staying here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On November 19, Cao Baogang, the vice-president of China Railway Construction Corporation (CRCC) as well as of China-Africa Construction Limited and Idris Audu Umar, Nigeria's Transport Minister, signed a contract for a railway project in Nigeria. CRCC was awarded the contract for the construction of about 1400 km railway line along the West African nation's coast. This is the biggest overseas engineering contract China ever signed, worth almost US$12 billion.\nThe route would connect the financial capital Lagos with the eastern city of Calabar, crossing Nigeria's oil production region in the Niger Delta. The project will lead to exports of equipment from China worth US$4 billion, including construction machinery, trains and steel products. It would create up to 200000 jobs in Nigeria during construction and a further 20.000 to 30000 jobs after the railway begins service, according to China's official news agency Xinhua.\nIn 2008, CRCC ran into problems with a railway project in Nigeria when the Nigerian government took over the US$8 billion venture after falling oil prices made it hard for the government to arrange financing for the project, which involved a line from Lagos to Kano in the north.\nTo expand its influence abroad, China is pushing exports of its rail technology; although it sometimes fails to succeed. In early November, a Chinese-led consortium won a $3.7 billion contract to build a bullet train in Mexico, which would have become the first of its type in Latin America. A few days later, the contract was abruptly canceled amid concerns over an unusually speedy bidding process.\nChinese railway companies have been aggressively bidding for projects in a number of markets including Southeast Asia, Europe and North America. In October, for example, local authorities and Chinese leaders agreed on a new Hungaro-Serbian High-Speed Railway projectand at about the same time Massachusetts transportation authorities signed a $567 million contract with train manufacturer China CNR Corporation for more than 280 Boston subway cars.This deal was of particular interest as it was the first of its kind in the United States and a symbolic move into a type of market typically dominated by Western and Japanese firms.\nChina's pursuit of foreign rail deals is telling in a number of ways. First, it is demonstrating to the world, that China is confident enough to handle those mega projects. Second, further expansion abroad is aimed at boosting Beijing's influence in global markets and should definitely help lift China's manufacturing sector, which is of particular importance as China's economy is currently at a low and is expected to slow during the coming decade.\nAnd last but not least those projects \u2013 if realized in a proper way \u2013 can also have a positive effect on China's soft power and image-management in local populations. Not only do they provide the opportunity for local job creation (which, however, is often not the case as China brings in its own workforce), but also (if realized) could eventually simplify and unburden people's everyday life in a very direct and immediate manner. This potential image shaping factor could have an impact that other Chinese soft power instruments, such as Confucius Institutes or media outlets, can hardly arrogate for themselves.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home EVENTS NEWS RAILA'S FULL SPEECH ON THE \"BIG ANNOUNCEMENT\".\nFor the third straight election in a row, the voice of the people has been stilled, and for the third time in a decade, the candidate who lost the election has been declared President.\nThis year, Kenyans excitedly voted in a peaceful election that even the most democratic nations in the world would have been proud of.\nBut this democratic promise abruptly evaporated when those who wield power clung to their old, no-holds-barred script of rejecting the people's democratic will and holding on to their rule by whatever means.\nOnce the polls closed last Tuesday, the IEBC began streaming so-called provisional presidential results not backed by Form 34As as required by law.\nYesterday, IEBC CEO Ezra Chiloba finally made the stunning admission that the Commission was not able to supply all Form34As yet. This means that both the provisional and final results announced last week are null and void and that Uhuru Kenyatta was not lawfully declared winner.\nFrom the start, the IEBC was illegally releasing unverified results to create the expectation of an Uhuru Kenyatta victory. This was the equivalent of guaranteeing violence if the final verified results indicated Uhuru had lost.\nThen there was the bizarre phenomenon of Uhuru Kenyatta's \"lead\" staying at around a consistent 11 percent throughout the release of the \"results!\" surprisingly, this same pattern also occurred in many gubernatorial races across the country.\nSuch a fixed margin has never been maintained throughout any democratic election anywhere in the world. But it happened here because an algorithm had been introduced to rig the outcome. It happened in full view of all our people and all the election observers.\nThen finally, at midnight last Friday, we saw the IEBC announce that Uhuru Kenyatta had been elected president, even though the Commission continued to acknowledge that only about 29,000 of the 41,000 verified Forms 34As had been tallied!\nWhy was there this fanatical rush to judgment on the most vital matter that any democratic nation EVER faces, the election of its President for the next five years? And why did the IEBC refuse to address the set of questions about vote tallying that that NASA had submitted?\nIEBC might have declared Uhuru Kenyatta president, but a vast number of Kenyans have not accepted the legitimacy of the decision, and will not accept it until they have answers to profoundly disturbing questions that have been raised. Indeed, the Carter Centre, in its preliminary findings has questioned this rush and its overall impact on the outcome. The Centre says: \"Although election day voting and counting processes functioned smoothly, the electronic transmission of results from the polling stations to the 290 constituency centers, where official results are tallied, proved unreliable. Unofficial results were also transmitted to the national tally center, where they were posted on its website. Unfortunately, the early display of vote tallies at the national level was not accompanied by the scans of polling station results forms as planned, nor labeled unofficial, leading to some confusion regarding the status of official results.\"\nNo one should believe, and especially not those behind this election fraud, that Kenyans are sheep who will willingly go along with democracy's slaughter. This country is now divided between those prepared to live under autocracy and the forces of freedom and democracy.\nMost Kenyans do not agree that our democracy is a charade, a game, in which people campaign their hearts out for leaders they want, but are then given a winner pre-determined by the darkest forces in our society and beyond.\nWe also have the no less bizarre situation where some observers grossly violated their mandate and started calling upon NASA to concede \u2013 even before the IEBC officially announced the election's outcome.\nWe Kenyans will not let such impunity continue any longer. We have the declaration by IEBC of Uhuru as President-elect. The manner in which the Court handled that petition was a travesty of justice. The Court decided all interim applications in favour of Kenyatta. The Court also allowed the counter petition filed in favour of Kenyatta but disallowed the two petitions against him. Uhuru won 100 per cent and we lost 100 per cent in court.\nOur decision to go to court constitutes a second chance for the Supreme Court. The Court can use this chance to redeem itself, or, like in 2013, it can compound the problems we face as a country.\nWe will show how they shamelessly cooked results from non-existent polling stations and fake un-gazetted Presiding and Returning officers. They gave figures from non-existent Forms 34A and 34B; they scrambled to manufacture such forms; switched vote numbers; and how they openly swindled to reach predetermined consistent vote numbers. They cooked numbers to the extent that vote tallies often surpassed registered voters in polling stations.\nKenya is always much larger than my individual ambition. But Kenya is definitely not too large for all of us to ensure that anyone who wins the people's votes, and not the loser, is declared President.\nThis is just the beginning. We will not accept and move on.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The focus of the Gambling and Social Determinants Unit is on developing effective harm prevention and minimisation interventions and policy for the harms associated with gambling, and with other non-communicable disease.\nTo pursue this, we adopt a 'social-determinants' approach \u2013 that is, we examine the upstream causes of harm, and seek to develop policy and other responses to prevent illness and harm.\nWe work extensively in gambling harm prevention, but current projects include identifying the commercial determinants of health and applying these principles to the food industry, and other 'dangerous consumption' industries, which we know are increasingly responsible for harms including obesity, poor mental health, and gambling and alcohol harm.\nAdditionally, recent work has focused on the role of health promotion in reducing the adverse health effects of climate change, and identifying measures from diverse public health domains for adaptation to new domains of harm prevention.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatost b/data_all_eng_slimpj/shuffled/split2/finalzzzatost
new file mode 100644
index 0000000000000000000000000000000000000000..8e479c81ed14fdbc11bc7ad83a502b98a0d41f33
--- /dev/null
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+{"text":"We pledge to be courteous with the opinions of others and be mindful not to exclude anyone from discussions or work-related activities. We realize that including people with a variety of opinions and backgrounds will only serve to enrich our community. In particular, discussions relating to pros\/cons of various technologies, programming languages, and so on are welcome, but these should be done with respect, taking proactive measure to ensure that all participants are heard and feel confident that they can freely express their opinions.\nMachine Learning in Heliophysics code of conduct is based on the Astropy Community Code of Conduct.\nLicensed under a Creative Commons Attribution 4.0 International License. We encourage other communities related to ours to use or adapt this code as they see fit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We're always looking to give our customers services which actually benefit them. As such, in June, we teamed up with expert software developers NetSupport to start providing their incredible IT management\/school security tool, NetSupport DNA, to our customers.\nNetSupport DNA works in a couple of different ways - firstly, it gives all users the ability to see exactly how equipment on their network is being used, making it a fantastic and versatile asset management tool for admins. Beyond that, however, schools can benefit from an additional set of features, as NetSupport DNA includes an incredibly powerful safeguarding tool, creating convenient, effective and wide-ranging logs and reports of any potential causes for concern.\nIn June, we were also particularly proud to team up with the Internet Watch Foundation (IWF), an incredible group working to cut out child abuse imagery online, promoting the incredible work the IWF do to help improve the Internet.\nOne of the main aspects that makes #exabytes a uniquely excellent conference for teachers is the fact that the schedule is entirely based around what attendees want to see, ensuring that everyone is able to explore seminars, debates and demonstrations that actively interest and benefit them.\nWe organise this by giving every visitor a chance to vote on the dozens of fantastic seminar proposals we receive for #exabytes, with this year's voting taking place in June (just a little after the general election itself).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"impact crusher exhibition in germany. Home \/ Products; Secondary Cone Crusher. Crawler Mobile Crusher. Mobile Impact Crusher. Mobile VSI Crusher. C6X Series Jaw Crusher. CI5X Series Impact Crusher. HPT Hydraulic Cone Crusher. European Type Jaw Crusher. European Impact Crusher. VSI6X Vertical Shaft Impact Crusher.\nmobile impact crusher germany features: Compact structure and dismountable parts are good for transportation, cutting the transportation expenses. mobile impact crusher germany specifications: Coarse crushing equipment Model Specs. of screening equipment Medium and fine crushing and screening portable plant Specs.\n\u00b7 germany impact crusher, germany impact crusher products . 14 products germany impact crusher,germany impact crushersuppliers and new and used jaw crushers, conical crushers, horizontal and vertical impact . 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RUNI - Recycling machinery for compacting and dewatering.\nimpact crusher exhibition in germany. VSI-5X impact crusher is the Clirik's latest generation of vertical shaft impact crusher, which introducing Germany technology and owning a number of patents. Reply.\nThis page is about impact crusher exhibition in germany, click here to get more infomation about impact crusher exhibition in germany. Know More. impact crusher exhibition in germany - crusherasia.\ncrusher exhibition Manufacturer crusher.milanomet.org impact crusher exhibition in germany. jeff boerboon aerobatics germany hydraulic impact crusher what is the price of impact crusher,germany impact.\nImpact Crusher Exhibition In Germany Coal Russian. impact crusher exhibition in germany , Mining Machine astri mining resources coal e 2395 a bearing weight early , Germany impact coal crusher,Germany Coal . 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More; hammer crusher machine germany - atithipalacein.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"This playful Mermaid comes to you riding the wave of happiness!\nBearing the gift of joy and love, she reminds us of the importance of playfulness and lightheartedness.\nLet the eddy and flow of the currents of life take you to new, exciting shores where you can explore and run free.\nLearn to surf the big waves and simply jump.. . . . over the smaller ones!\nSometimes we just forget how to have FuN in our day to day existence.\nLet this Memaid enter your life and let the fun begin!\nOn a very dull and wet Tuesday morning, this lively little Mermaid has made it's debut in my Studio.\nHistorically, Mermaids have only ever come to me in the spring, maybe it is their migration season.\nThe very first one, which must have been around 2003\/4 maybe, went off to live in Hawaii.\nI remember the shock of \"feeling\" her emerge in her energy form.\nIt was rather like holding a small, slippery very lively monkey, and I could feel her tail wrapped around my wrist as I worked with her.\nI love how indistinct the mermaids are, you could easily miss them. The upper body, head and arms appears to be made of a clear opaque material rather like a jellyfish, and the tail is more like the ever changing skin of a cuttlefish or octopus. The hair is like that of the weedy seadragons and looks like fronds of seaweed. I was advised by the first Mermaid that the hair is like antenna, or cats whiskers, exploring the environment and allowing the Mermaid to tune into the oceans and meteorological energies. Sadly, they often get bits of flotsam and jetsom caught in their hair, and it can look very pretty and interesting but is reduces the sensitivity of the environmental connection. This one, carries a few bright \"jewels\" but they were very much asked for, so I obliged!\nYes, the Mermaid is very playful by it's nature, but it is also a very important \"Gate Keeper\" to the health and well being of our Oceans and the creatures that live within it. Of course, without healthy oceans, we cannot have healthy land either.\nI am just reminded to mention that there is another type of Mermaid called an \"Undine\" which lives in the brackish waters where freshwater river meets the sea. They look very similar, but the Undine has a slightly different role, helping the water to make the transition from fresh to salt. They are rather more serious in their nature and not so \"glitzy\" although still very beautiful.\nI hope this little Mermaid has brightened your day!\nThe original artwork is 8\" x 16\" on canvas as is avaible for \u00a3255 plus p&p depending on where it is shipped to.\nAvailable as a A4 print , hand finished with my special materials ready for framing @ \u00a315 P&P is usally \u00a31.50 per single delivery UK or \u00a35 overseas .\nJoin my Facebook following at alison.knox.angels for pretty much daily random but relevant inspirations!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I read this article and found it very interesting, thought it might be something for you. The article is called Migrants slam impact of immigration overhaul and is located at http:\/\/www.neverhome.ca\/migrants-slam-impact-of-immigration-overhaul\/.\nA multimedia project puts a human face on the impact of drastic changes made by the Conservative government to Canada's migration policies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Digestion doesn't just take place in your stomach. The pancreas produces a large portion of your digestive enzymes, as do the small intestine, stomach lining, liver, and salivary glands (digestion begins in the mouth, after all). But these enzymes do not digest food -they digest nutrients.\nThis process is by no means simple. Dozens of different enzymes work together along the GI tract to break down macronutrients from the foods you eat. These nutrients are then sent through the bloodstream to the liver and then are absorbed into the lymphatic system, which distributes them to tissues, organs, and muscles.\nAs you can imagine, this complex exchange is responsible for the body's access to fuel.\nYou can eat all the healthy food in the world, but if you can't absorb it, not even the healthiest of diets will do any good. I say this all the time to clients. However, our bodies can only digest what it's given. High-fiber foods such as fruits and vegetables contain enzymes of their own that work with digestive enzymes to break down foods faster and allow your body to access nutrients quickly for cell reparation and growth.\nStudies have shown that the autoimmune system, which protects us from pathogens and harmful components in the environment, may owe its efficiency to digestive enzymes. As humans have evolved, enzyme production followed suit with microbial gut flora producing various enzymes to respond to different pathological and environmental threats. This line of defense allows the body to recognize and eliminate many dangers before they can infect the rest of the body.\nOur enzymes could be telling us to slow down. Chronic stress plagues countless individuals, but just because we always seem to be on the go doesn't mean our food needs to be. It limits the amount of enzymes your body can produce as the brain attempts to eliminate the stressor before it can reengage in normal functions-like digestion.\nThis is a helpful reminder to slow down. The body is not meant to adequately digest and absorb nutrients as we are running out the door, inhaling our food, and then chugging liquid to get that food down.\nWhen we can't digest our food, we can't provide our bodies with nutrients. Recurring bouts of anxiety can lead to indigestion and even pesky habits like comfort eating and snacking.As obesity levels climb, it's easy to see how a simple change like sitting down for meals could provide a fulfilling way to de-stress and allow our enzymes to provide break down those nutrients that will speed us on our way.\nEnzyme production is tied to well-being and it is important to take them into consideration when planning for your health. Certain foods like pineapple, mango, papaya, and honey have been used in Central and South America for centuries to regulate digestion and inflammation in the GI tract. If you're not already incorporating them into your diet, that's a good start.\nAnd then of course, there are digestive enzyme supplements. Make sure to check for quality ingredients that leave out fillers that could denature the enzymes. My favorites? Digestzymes by Designs for Health and Bio-gest by Thorne.\nStruggling with heartburn? Headaches? Bloating? Gas? Sign up for my FREE 5 day gut reset HERE, complete with recipes.\nWallace M. The Digestive System & How it Works. National Institute of Diabetes and Digestive and Kidney Diseases. September 2013. https:\/\/www.niddk.nih.gov\/health-information\/digestive-diseases\/digestive-system-how-it-works.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A global internet puzzle that requires advanced computer skills to solve has reopened online.\nIt is the third year running that the mysterious Cicada 3301 has appeared, with no clues as to who is behind the puzzle.\nSome believe it is a recruitment process for a government spy agency, while others believe it could be the work of a bank.\nThe puzzle takes people on a treasure trail around the so-called dark net.\nThis year's competition was announced via a Twitter account to which was attached an image file with the following message: \"Hello. Epiphany is upon you. Your pilgrimage has begun. Enlightenment awaits. Good luck. 3301.\"\nThousands of fans have been drawn into the competition and have eagerly awaited its relaunch.\nThis year's iteration already appears to be fiendishly difficult, requiring understanding of OutGuess, a complex steganography tool.\nSteganography hides data inside images, and is often used for nefarious purposes by terrorists and paedophiles.\nComputer scientist Prof Alan Woodward is one of those who has previously taken part in the competition.\n\"The puzzles are very complicated and require knowledge of steganography, cryptography as well as obscure existential texts,\" he told the BBC.\nPrevious competitions have taken people on both a mental and physical journey, encompassing cyber-punk literature, Anglo-Saxon runes and the work of occultist and magician Aleister Crowley.\nLast year, part of the treasure trail also required people to travel to physical locations to decode QR codes posted on telegraph poles dotted around the US.\nNow that the competition is widely known about there has been a spate of fakes but for those with the necessary knowledge it is easy to spot the real thing.\n\"The group uses a PGP signature and put up a key signed with this,\" explained Prof Woodward.\nMedia captionKenny Paterson: Internet puzzle \"could lead to recruitment\"\nSpeculation about who is behind the puzzle has been gaining momentum as the competition attracts more publicity.\n\"It could be a criminal organisation looking for recruits or it could be a start-up looking to set up a new crypto-currency, possibly even a bank,\" said Prof Woodward.\n\"Lots of people think it could be a government agency and that looks reasonably likely as whoever is being recruited isn't telling anybody afterwards,\" he added.\nIt is similar to an online competition posted annually by UK agency GCHQ, as part of its recruitment process.\nIts 2013 competition consisted of 29 blocks of five letters which had to be deciphered. Those who successfully completed the initial test were invited to take part in a hunt around the net, which they had six weeks to finish.\nCicada is far more complicated though.\n\"Whoever is behind it has put an enormous amount of effort into it. It is someone with quite a lot of resources or a lot of time on their hands,\" said Prof Woodward.\nThe puzzle has intrigued so many people since it opened in 2012 that some have started to crowdsource answers.\nIt has led to the creation of a Cicada wiki on which puzzle solvers are invited to post significant leads and clues.\nAlready solvers have unlocked the first clue to this year's mystery and posted the riddle they have uncovered.\n\"The work of a private man who wished to transcend. He trusted himself to produce from within.\"\nProf Woodward himself has never solved the Cicada puzzles.\n\"Frankly I got bored with it, especially when it started getting into Irish poems I knew nothing about,\" he said.\nAn unknown number completed the first two tests which eventually led to an address on the dark net.\nThe competition organisers later posted a message which read: \"We have now found the individuals we sought.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This spring something very special happens every Monday morning in Room 329 at Concord-Carlisle High School. Each week, 15 senior citizens from Carlisle and Concord and fifteen CCHS freshmen come together to explore important life issues by creatively discussing and writing poetry. Bridging generations, the class breaks down barriers, fosters deep communications and instills or deepens a profound appreciation of poetry as a thoughtful form of human communication. It is very special and a unique educational experience for all who participate.\nThis class, which has been offered for the past seven years, is the result of collaboration between Patti Russo of Carlisle and English teacher Shelley Hull of CCHS. Dr. Rebecca Loprete, former CCHS English Department chair was instrumental in initiating the program and continues to be a great supporter. Russo is a Certified Poetry Therapist and has developed a practice, \"Between the Lines,\" which utilizes poetry as therapy for a variety of emotional and challenging life situations.\nRusso and Hull, who is an experienced and skilled teacher, have fashioned a creative solution to the challenging problem of bringing poetry alive for high school freshmen as they adjust to life at CCHS. The program is now referred to as \"Partners in Rhyme.\" Russo is currently working on expanding the program to other schools.\nThe first year English curriculum includes modules for fiction, non-fiction, drama and poetry. This intergenerational program focuses on the poetry module and is directed toward a group of freshmen participating in the Network program. Network consists of 15 selected freshmen and their particular challenges of transitioning into the CCHS program. These students stay together for the core curriculum of their freshman year and have a tutor, Pat Savage (a former Carlisle resident), who is with them for all of their core classes.\nFor the English Poetry module each of the students is paired with a senior citizen and that's where the magic begins to happen. They sit together, share assignments and have ample opportunity for discussion and sharing. There is something about the interaction of a person from an older (non parent\/non teacher) generation and a student that generates real communication, lively interaction and very little judgement. This is probably best illustrated by reviewing a recent class.\nOn Monday, April 3, the class convened for a session utilizing poetry to explore the issue of resiliency. Shelley Hull is taking a break this spring, and Dr. Dave Nuremberg is teaching the class with Russo and Savage. The class began with Carlisle's Helen Young reading a beautiful poem centered on the issue of resilience in the face of adversity. This served to focus the entire class on the issue at hand.\nThe group then broke up into senior\/student pairs for discussion. All had been instructed to bring in some examples of something that they collect. This facilitated a lively discussion of a variety of interesting items such as small bird statues, video game cassettes, quilted patches, travel stickers, etc. The creative juices began to flow regarding what formed each person's life experience and what is important to them. There certainly was no lack of lively interaction.\nThe class then viewed three short videos, one of Serena Williams reciting excerpts from Maya Angelou's \"Still I rise\", Angelou reciting the same poem and Sarah Kay performing \"If I Should Have a Daughter\". These were highly impactful in discussing aspects of resiliency. As an aside, the technology available at the new high school for accessing and viewing information in a variety of forms is truly impressive. The citizens of Concord and Carlisle should feel extremely proud of this impressive facility.\nIn the hour of the class, the fifteen seniors and fifteen students had experienced the opportunity to work closely together discussing some profound issues of life through the lens of poetry. Russo's goals for the program include bringing generations together to learn from each other, to bring poetry alive for the sometimes reticent students and to offer a source of vitality and life for the seniors. From this vantage point it appears that she is well on the path to success. The teachers all commented that this is one of the most interesting and impactful educational initiatives that they have ever experienced.\nThe Carlisle seniors involved in the program this year are Stephanie Hackbarth, Gertrude Behn, Woody Woodrick, Helen Young, Marcia Wallhagen, Lillian DeBenedictis, Tom Dunkers, Ellen Huber, Fran\u00e7oise Bourdon, Mary Daigle and Dee Stillings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"That's what my son said at his swearing in ceremony earlier this week when asked why he'd chosen to serve in the Coast Guard.\nAnd I -- who came of age to Bob Dylan's raging Masters of War, violent protests in the streets and on college campuses, and a deep suspicion of government in the wake of Kent State and Watergate \u2013 felt humbled by my son's conviction of purpose.\nOn the day he shipped out, we got up before dawn so I could deliver him to the Military Entrance Processing Station by 05:45. I don't think he got more than an hour or two of sleep and I could see by the set of his jaw and the pallor of his face that he was tired and nervous.\nI came back later in the morning and met my son's traveling companion, the only other Coastie shipping out that day, as well as a newly commissioned Seaman who had graduated a week earlier. He was on hand to answer any questions about what the new recruits could expect over the next eight weeks. Both found this helpful (along with the reminder, \"It's only 8 weeks\").\nI also met his recruiter, who fleshed out a few details about the graduation date and advised me to book a room soon; mid-July is high tourist season in trendy Cape May, New Jersey.\nMy son's fianc\u00e9 -- a first year medical student in the middle of year-end exams -- arrived in time for the formal swearing in. Afterwards, we went to the airport and said our final goodbyes.\nNot a waking hour goes by that I don't wonder what my son is doing \u2013 how he's managing, whether he's sleeping OK, how he's feeling. Previous separations always included the comfort of email, text messages, phone calls and chat to keep us in touch. For the next eight weeks, it will only be letters quaintly sent by US mail.\nThe times that shaped my son are different than those that shaped me. He's made a choice I cannot imagine having made for myself, nor would necessarily have chosen for him. And yet I'm proud of his determination to succeed and the sacrifices he's making to pursue this career he aspires to.\nMy thoughts stray often to him, warmly enveloping him in love and well wishes. And I'm looking forward to reuniting with the young Seaman who will greet me on the other side of these eight long weeks.\nI have never outgrown my love of snow. I feel the same wonder and delight today that I did as a child, watching it fall and muffle the sounds of a noisy world.\nI know few share my love of snow, especially in a winter that's already seen more than usual. Here's something for those who do.\nComing on the heels of dire warnings that a 41st Republican Senator would most certainly derail health care reform, the Supreme Court -- acting against 100 years of precedent \u2013- ruled that the government cannot ban political spending by corporations in candidate elections.\nIt's impossible to predict how this will play out but no one doubts it will create major political consequences. Almost certainly, we'll be in for a slug fest of attack ads from outside interest groups at a level far beyond what we've seen before.\nI'm having trouble understanding how ordinary citizens benefit from this ruling. While I can't see many companies rushing to directly finance campaign ads because of public relations concerns, I can see a steep increase in activity by industry groups, funded by these very companies. And it's not at all unrealistic to envision a future where lobbyists essentially hold legislators hostage by giving them a glimpse of what they have in store for them should they vote against the corporate interest.\nI don't believe corporations are evil. I worked for one for most of my adult life. That said, what's in the best interest of the corporation is not always what's in the best interest of the people. Money has, does and always will talk \u2013- but until yesterday's ruling, there was at least a semblance of government of, by and for the people.\nAre we approaching a time when citizens must band together and incorporate in order to be heard and have influence with their elected representatives?\nReading those words this morning was the equivalent of stopping and taking a deep breath. And reminding myself that the media and blogosphere feed on all the \"ain't it awful\" drama. I can choose to get caught up in their chatter or -- pay attention to other things going on in the world around me.\nI think I'm going to spend some time on those other things.\nThis past Monday, I left my bracelet in the locker at my gym. Taking it off in the first place was a hard habit to get into; I never remove it except when I must -- my few hospital stays, airport security checkpoints and now, when I swim.\nBut that day I left it behind when I gathered my things. I realized it was missing about an hour after I got home, and rushed back to retrieve it. It was gone: whoever found it did not turn it in, nor has it shown up in the days since.\nThere's a story behind this bracelet, and a story behind that story. The story is this: it belonged to my grandmother, who bought it from a Native American on the side of a dirt road somewhere in New Mexico or Arizona in the 1940's. She and her husband had moved to Colorado after the war and explored much of the American southwest and Mexico by car, camping by roadsides. The bracelet is sand-cast silver and has the signature of its maker \u2013 a small bird \u2013 on the inside.\nLike me, my grandmother always wore it. It's one of my earliest memories of her. She gave it to me during a last, long visit I made to see her \u2013 another in a series of solace-seeking trips for some trouble or another I was going through. The bracelet had begun bruising her thinning skin and she felt it was time to pass it on. Among all her granddaughters, she chose me: the one she always felt was most like her.\nThe story behind the story is that of my grandmother. Born at the end of the 19th century, she lived a fascinating life. From her early days of privilege to the cash-strapped years of her teens, she had a fierce intelligence and independence. Before her marriage in 1918, she was a dancer in Greenwich Village and acted in plays written for her by a doting playwright. She was also a member of the Socialist Club, best known today from Warren Beatty's film, Reds, about one of its more famous members, John Reed.\nHer marriage ended abruptly in 1929 when the car she was driving collided with a train at a crossing in Nyack, New York, on a foggy April night. My grandfather and another couple were killed; my grandmother spent 3 months recuperating in the hospital. She never walked properly again.\nShe remarried in the 1940's and moved to Colorado. Max was Swiss and had also lived in South Africa before moving to the U.S., and he and my grandmother took many trips together. Often their transportation was as exotic as the trips themselves \u2013 tramp steamers and cargo planes, a reflection of Max's thriftiness. I always loved to visit them; their home was filled with African masks, Peruvian pottery, Indian cloth and Chinese paper paintings, and smelled of spices and faraway places.\nNow I've lost the most visible connection to the grandmother I admired and adored. It's a visible, visceral loss and I feel tears pricking my eyelids every time I think about it.\nThe bracelet may yet turn up. Whether it does or doesn't, I've been thinking a lot about the Buddhist concept of attachment.\nMy attachment comes in equating this bracelet to my grandmother, and feeling like I've lost a part of myself in losing it. It comes in giving up the warm fantasy of passing it on to my own granddaughter someday, along with the stories that accompany it \u2013 my grandmother's and my own.\nIn the end, it's just a thing, not a substitute for my life memories of my grandmother. I'll always carry those with me. Or so I remind myself as I once again unconsciously reach over to adjust the bracelet that's no longer around my wrist.\nI had lunch last week with some former colleagues, all of whom -- like me -- accepted early retirement packages our company offered to qualified employees.\nEven without that incentive, it had become common practice for people to leave our company by age 55 because of the generous retirement benefits, including health insurance. In fact, as one of my lunch companions told me, the system had evolved over time to implicitly encourage this trend. The buyouts simply accelerated that process.\nSome of us skipped out joyfully; some of us looked around and realized that -- ready or not -- it was time to go; some of us left kicking and screaming.\nThe common thread? Whether it was seeking full-time or part-time re-employment; consulting, or working on contract; starting our own companies or buying a franchise: many of us weren't ready (or able) to stop working.\nThe latest exodus of early corporate retirees joins a flood of 50+ adults losing their jobs due to the punishing recession. Any way you look at it, there's an amazing number of highly skilled, hard working people looking for ways to continue being productive.\nAnd yet connecting people in this age group with opportunities is difficult.\nThis isn't a piece about how to do an effective job search for the over 50 crowd. AARP has some good advice; so does Workforce50. I'm more interested in exploring the opportunities for lifelong employment.\nCase in point: in July, my former employer contacted me about taking over part of my old job -- the part I never had time for as an employee and felt constantly guilty about, as a result. To borrow from the jargon I worked so hard to banish from company communications, it was a \"win-win\" proposition: the company would cede handling of an important (but not strategic) aspect of its work to someone with the knowledge, skills and interest to competently manage the work. And may I add, with enthusiasm!\nGetting to \"yes\" required a lengthy co-employment evaluation to ensure I wasn't a threat to the company's retirement plan. But what if retired employees were considered a strategic asset rather than a potential threat? What might that look like?\nPeople could retire but still be part of an accessible database when the company needed their expertise. In fact, there's been a move to tap into retired talent. For example, Your Encore \"was created to tap into our country's under-utilized asset: the growing number of retired and veteran scientists.\" But while companies participating in the program encourage their retirees to register, you're unlikely to find much on offer unless you've worked in a technical field.\nThere are also a number of job listing sites catering to the 50+ crowd. In Canada, Retired Worker promises to connect employers \"with the skills, knowledge and experience of Baby Boomers.\" WiserWorker and Retirement Jobs appear to do the same for a U.S. audience. All these options, though, are simply job search sites; they don't screen people and match their skills with employer needs.\nEmployment status could be on a continuum, from full-time with benefits to intermittent with none, and variations in-between. In other words, you would never truly leave the company \u2013 you would just move from one state to another, with diminishing levels of company support as your situation changed or you aged.\nAs it happens, while I was working on this post, someone called to ask for my help with a project linking retired workers with nonprofit jobs. In the course of our conversation, she pointed me to Civic Ventures, \"a think tank on boomers, work, and social purpose.\" Of the many available resources on the site, the one I spent the most time on, and ended up registering with, was Encore Careers. It offers advice, articles and communities where you can connect with other \"second life\" searchers.\nBaby Boomers are entering their retirement years to the strains of Dylan Thomas's \"Do not go gentle into that good night.\" Through sheer numbers, we've been a force of social change our entire lives. We'll continue to shape and redefine what it means to be retired, just as we have with every other milestone we've encountered. It's too early to know what that means, but the view from here is that we'll continue finding meaningful ways of contributing, for profit or not, until well into the traditional golden years.\nIt started off simply. An email from my conservative brother, distraught about cap and trade legislation that had passed the U.S. House. Just another in a long-running policy debate between him and his liberal older sister.\nThat's when for me, the conversation became surreal. Because it turns out my brother \u2013 like many other US citizens, mostly of a conservative bent -- doesn't believe in global warming.\nAnd so began a conversation that ultimately included my friend, Bob Grumbine \u2013 a real, live climate scientist I dragged in to counter what I believed to be bad science. I could turn this post into a play-by-play description of that conversation, but that's not fair to my brother or my friend. Especially when my biggest insight was about my own ignorance.\nSo I write this as a thank you to my brother, who started the conversation. And to Bob, for clearly explaining the science behind the issues my brother raised, then pointing me to other knowledgeable resources.\nDid the data sway my brother? I don't know. One of Bob's comments haunts me: \"If you can't examine the science without letting fears about your wallet dictate whether you accept the science, you're really not going to understand the science.\" We seem incapable of moving beyond ideology when it comes to the science of climate change. And general ignorance of science is a big part of the issue.\nThat would be me. I tuned out all things science long ago, certainly by the time I took high school chemistry. \"We're words people,\" my father used to say. As if the ability to communicate effectively doesn't apply to science or math. But I absorbed this because truthfully, science and math didn't come as naturally \u2013 or interest me \u2013 the way language and history did.\nSo it's hypocritical of me to point to the ignorance of the doubters when I'm just as ignorant about the science supporting climate change. It doesn't help that the scientists who know the most write about it in ways that make it inaccessible to the average non-scientist.\nTobis also pointed me to science teacher Greg Craven, whose YouTube videos led him to write a book, What's the Worst That Could Happen? Although Craven clearly comes out in support of the science behind global warming, his book doesn't try to talk nonbelievers to his side. Instead, he provides a way to do your own risk analysis, offering the supporting arguments for both sides.\nI'm still not as knowledgeable as I'd like about the science of climate change. Thanks to Bob and others, though, I can at least explain why we should be concerned by a CO2 level of 385 parts per million (ppm), rising about 2 ppm\/year, and why these increases cannot be attributed to normal warming trends.\nIt's not yet noon and I've been having one of those mornings that defines having a bad day.\nBecause I was intending to take my son's car to the shop for service following my morning swim, I grabbed his key and what I thought was a house key on my way out the door this morning. I didn't take my mobile phone, thinking I wouldn't need it.\nI noticed when I got to the gym that in my rush to get out the door, I'd not picked up a house key but my son's extra car key. This would not have been a problem had the outside device for my garage's remote control been working. It wasn't because I hadn't bothered to get it fixed.\nI called my neighbor (who keeps an extra key for when I'm out of town) from a lifeguard's phone on the chance she was still home. No dice. Maybe I'd left a front window unlocked so I could break into the house through one of the windows facing my front porch? Nope.\nNext, I made the 40 minute round trip drive to my mother's house since I knew she had a key to my place. Except when I got home, I found out she'd given me the wrong one.\nI finally remembered there was a ladder under the deck that I could use to enter an open window one side of the house. Finally, success!\nAs I was walking back up the drive after putting the ladder away, I saw my son's car rolling down the driveway towards me. I couldn't stop it before it ran into the back yard and (thankfully) came to a stop just shy of the shed, whose low stone wall had stopped it before it could cause any damage.\nI had to stop and just breathe for a couple of minutes after that. Sometimes, that's all you can do.\nEven before the car came rolling down the driveway, I knew that my inattentiveness in combination with a tendency to put off dealing with things as they arise had combined to create a perfect storm of unpleasantness. At other times in my life, such a series of events would have put me in a funk for the whole day.\nInstead, I took this for the lesson in mindfulness I obviously needed. Mindfulness can be described as full-minded awareness, paying conscious and purposeful attention to what you're doing from moment to moment.\nIt's easy to go into automatic pilot and unconsciously go through daily routines. Maybe you're tired or stressed out from work or personal issues, and you just aren't really \"there.\"\nRecognizing I was having a problem with being fully present, I made myself slow down and consciously consider what I was doing next, and next and next. I got the car delivered to the shop without further incident, and back home safe and sound. I called the company that installed my garage door opener, where they told me I likely just needed a new battery for the outside fixture and talked me through how to install it.\nAnd I have an extra house key I'll put in my son's car, just in case something like this happens again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Purchasing this item will earn you 180 reward points (Kush Money $1.16) !\nOriginally developed by the 'new school' of Amsterdam, these guys are the next generation of breeders! They started by selecting a prized 'God Bud' male from seeds derived from the BC Bud Depot (Canada) and crossed it with the Big Buddha Cheese clone and came out with the first hybrids, ...The son of God....'Cheesus'Now in feminised form, 'Cheesus' is an attentive plant producing huge yields of lavender, pine berry cheese flavoured bud with excellent medicinal properties.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaujhr b/data_all_eng_slimpj/shuffled/split2/finalzzzaujhr
new file mode 100644
index 0000000000000000000000000000000000000000..84ef4ebefccae0553aa0972dda1bf0dc080810eb
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@@ -0,0 +1,5 @@
+{"text":"Our vertical urban campus is composed of a cluster of buildings near the intersection of Boylston and Tremont streets that overlook?the historic and pastoral Boston Common. The buildings house state-of-the-art teaching, production, and performance facilities, as well as several clinical laboratories.\nWhen visiting campus, please check in at our Visitor Center, 104 Boylston Street, Boston, MA 02116.?\nExperience our impressive facilities and learn more about graduate student life on our campus tour. Your guide, a current graduate student, will share why they chose Emerson, what their program experience has been, and what it's like to live in Boston. You'll also meet with your admission representative and hear details about the application process, financial aid, and more. Please plan to be on campus for approximately 1 ? hours (tours themselves are 45 minutes long.) Advance notice of one week is preferred to schedule your visit.\nCome visit campus and learn about our Master of Science in Communication Disorders. Meet with Graduate Program Director Sandy Cohn Thau?and tour the Robbins Center.?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A late Victorian Officers Brass Bound Military Campaign Writing Slope. Good quality solid mahogany folding writing slope with brass corner straps. Rectangular escutcheon to top of slope with initials and name: C C Pearson. Original baize lined base. Brass plated lifting handles to each side with a side drawer accessed by opening slope and removing drawer retaining pin. Original ink bottles, original leather skiver, original lock with key and brass\/plated furniture. Side drawer contains original items of owner - a black and white photograph in his uniform with medal, a tie with a length of matching ribbon (I would have said medal ribbon but probably related to regimental colours rather than a medal), an ivory letter opener, slope key, typed brief service record and finally a pair of mounted wild boar tusks. Size of slope is 15 inches long, 9 inches wide, 6 inches depth.\nThe ivory letter opener (pre CITES) is engraved CCP on one side and on the other side is handwritten in ink with details of his unit and various postings\/detachments from 1876 including Stubbington, Sandhurst, Bangalore, Dalhousie, Mamund , Bara Valley, Kyber and Ferozepur amongst others.\nThe wild boar tusks are an engraved silver mounted pig sticking trophy with inscription : Speared 19th May \/95, Ch. C B P \"Rufus\". Pig sticking was highly popular with British Officers in India at the time of Pearsons postings. Pearson was the pig sticker who accounted for the pig as the award of its tusks as his initials on the mount proves. Whether the nickname of Rufus was him or the pig is not known!\nCharles Coffin Pearson was born on 28 March 1867, son of Lieutenant General Sir Charles Knight Pearson KCMG CB, one of the more successful British Generals in the Zulu Wars who was also present at the Siege of Sevastopol and was mentioned in despatches no less than 11 times.\nAfter graduating at Sandhurst, Charles Coffin Pearson was appointed 2nd Lieutenant in the 52nd Oxfordshire Light Infantry on 14 September 1887 with subsequent promotions of Lieutenant 2 June 1891, Captain 5 June 1897, Major 10 July 1907.\nHe served in the 1897-98 Mohmand Campaign and the Tirah Expeditionary Force for which he was awarded the campaign medal and 2 clasps. The 52nd were in the 2nd Brigade along with the 9th Gurkha Rifles in the Mamund Valley supporting a small force of cavalry including a certain 2nd Lt Winston Churchill who wrote his first non fiction book on the campaign in 1898, The Story of the Malakand Field Force: An Episode of Frontier War.\nPearson retired on 14 September 1907 but re-joined for the Great War in 1914 as Adjutant for the Ox and Bucks Light Infantry.\nSo many of these military slopes have disappeared out of the direct family of the owner over the years. However, we should be grateful for these surviving items coming onto the market with a story to tell.\nCondition report: See photographs. Campaign slope all original and good condition except for leather skiver cracked and split at folds. Ivory letter opener tip damaged.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LONDON: SIMPKIN, MARSHALL, HAMILTON, KENT & CO. LTD.\nTHE author acknowledges her grateful indebtedness to the Reverend John J. Burke, C.S.P., Editor of The Catholic World\u2014in which magazine all of these essays have appeared either wholly or in part\u2014for permission and encouragement to republish at this time: to Messrs. Geo. Bell & Sons for the extracts taken from \"Coventry Patmore\": to Mr. Elkin Matthews for the use of Lionel Johnson's poems: and to Messrs. Burns & Oates for the use of those of Francis Thompson and Aubrey de Vere.\n\u2060By courtesy of Fr. Joseph Keating, S.J.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Twitter is a great opportunity for any small business to expose their company and their products to a broader audience while establishing relationships and strengthening brand recognition. However, to take full advantage of all that Twitter has to offer, it is important to approach it with the right strategy in mind. Below are some tips and techniques to gain followers on your business's Twitter account.\nWhen it comes to engaging new followers, the most crucial factor is the quality of your content. Remember that people want something relatable and attention-grabbing. If the only thing you post is advertisements, you aren't likely to gain any traction. Show potential followers the human side of your business by involving them in conversations, posting fun quotes or jokes, and weighing in on issues related to your business.\nSimply creating a Twitter account isn't enough \u2013 you have to post your content on a semi-regular basis to attract new followers and build a connection between the audience and your brand. If you only post updates once a week, you sacrifice potential exposure to a much larger audience. On the other hand, don't post updates every twenty minutes either, or your followers will likely get annoyed. Even one or two tweets a day is enough to keep your business and your product fresh in peoples' minds.\nYou are more likely to be followed by people that you already have a relationship with, so if you have a presence on other social media platforms or networking sites such as Facebook, LinkedIn, or Pinterest, use them to your advantage. Letting your existing audience know that you are now on Twitter will often result in an influx of new Twitter followers.\nWhile some people may seek your business out, odds are that increasing your number of followers will involve a bit of leg work. Take the initiative by finding people that are interested in the kind of product you offer and following them. Get involved in conversations, even if they don't relate directly to your business. Being part of the community and showing that you aren't just there to sell a product goes a long way toward building goodwill and brand recognition.\nAn online-exclusive coupon or promotion is a great way to attract and sustain new followers. Consumers are always on the lookout for a bargain, so using your Twitter feed to advertise upcoming sales and promotions will ensure that your followers check in regularly. If your product allows for it, it may be worthwhile to consider running a contest with prizes.\nWhen it comes to engaging viewers and exposing potential customers to your business, Twitter's potential is almost limitless. However, approaching it without the right mindset is a recipe for disappointment. By utilizing the tips and techniques above, any small business can bypass common pitfalls and achieve the results they are looking for. Looking for more social tips for your business? Learn more with these Social Media Tips to Help Build Your Brand.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are an Office 365 customer, there are a lot of benefits you already can avail of, like unlimited storage, always up-to-date desktop software, centralized management of licenses, etc. Now, per the blog post on Office Blogs on October 31, if you are using Mac OS, you can get the new Outlook app for Mac.\nIt is a surprising release because everyone expected Outlook to be part of the Office suite, and the rumored dates for the new Office for Mac were pointing towards a beta in November with a release in the first half of 2015. It is also a surprise that Outlook was released by itself, perhaps because it was probably the one app with the most complaints about, in Office for Mac 2011.\nAs for the rest of the suite, Microsoft said that the public preview of Excel for Mac, Word for Mac, PowerPoint for Mac, and curiously, OneNote for Mac will be made available as a public preview in the first half of 2015 and a final release in the second half of 2015. What's curious about OneNote is that there is already an app in the Mac App Store called OneNote so I am not sure how they are going to distinguish between that app and the potentially \"full\" OneNote for Mac. This mirrors what Microsoft has on the Windows side, with a \"desktop\" OneNote and a Universal OneNote app which is available in the Windows Store.\nOffice 365 Commercial customers can get the new Outlook for Mac by accessing their Office 365 Portal, (Gear icon > Office 365 Settings > Software > Outlook for Mac icon) or visiting the Software page.\nOffice 365 consumer subscribers can get the new Outlook for Mac by going to their My Account page.\nFinally we see the benefits of being an Office 365 customer, specifically regarding access to the latest software. Until now, Office 365 has not delivered any desktop software updates which perpetual license holders have not got. Now, a perpetual licensee of Office for Mac will be one version behind, at least for a year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauqor b/data_all_eng_slimpj/shuffled/split2/finalzzzauqor
new file mode 100644
index 0000000000000000000000000000000000000000..b20e1f3c4c45a9730c493d5dd4369b542fb19162
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@@ -0,0 +1,5 @@
+{"text":"How Long Do You Have to File a Medical Malpractice Action In Virginia?\nIf you or a loved one has been harmed through the negligence of a doctor or another healthcare provider, you may consider filing a lawsuit. These personal injury lawsuits are often referred to as medical malpractice cases.\nYou should be aware that there are strict deadlines to file a lawsuit against a healthcare provider but it's a complicated matter. The answer can be fact specific, so you should always immediately consult with an attorney. The rule in Virginia for adults injured by malpractice is typically two years after the cause of action accrues (usually the date of the negligent act causing harm). The statute of limitations for your case is vital to your claim. If a lawsuit is not filed within the statute of limitations, it is forever barred and cannot be pursued\u2013 no matter how bad the doctor's actions were.\nIssues can arise, however, if you have been harmed by a doctor but did not realize it within two years. Your time to file a lawsuit can theoretically pass before you have even realized your harm. Thankfully, Virginia has carved out some exceptions to its strict statute of limitations rules.\nFirst, is cases in which foreign objects having no therapeutic or diagnostic value are left in a patient's body, the statute of limitation is extended for one year from the date the object is discovered, or reasonably should have been discovered. However, this extension cannot extend more than 10 years from the date the cause of action accrues.\nFor example, suppose a doctor performs a surgery upon you and negligently leaves behind a sponge or tool. You discover the mistake three years after the surgery and have to undergo a second surgery to have the object removed. In this instance, you will likely have one year from the date of discovery. Now, if you do not discover the item for 15 years, you are likely left with no recourse because the law will not extend the statute of limitations past 10 years from the date the object was left inside the patient in this example.\nA second exception to the strict two-year statute of limitation is in cases involving the negligent failure to diagnose a malignant tumor or cancer. Your deadline can be extended for one year from the date the diagnosis is communicated to you by a healthcare provider, provided the limitation is not extended beyond 10 years.\nFor example, suppose a doctor reviewing tests or images negligently overlooks visible or obvious early signs of cancer that would otherwise be very treatable. The cancer then goes undiagnosed and untreated for four more years, and in the meantime it has spread to other parts of the body. Assuming the underlying negligent failure to diagnose occurred on or after July 1, 2008, the patient's statute of limitation would be extended for one year after the diagnoses was communicated to the patient. Similar to the prior example, the statute of limitation cannot extend beyond 10 years post-negligence.\nThere are additional exceptions that can extend the deadline to file a lawsuit for some young minors, and people who continue to treat with a doctor in an effort to cure the harm caused by his negligence. There are also exceptions in instances of fraud. The statute of limitation is critical to the viability of your case, and can hinge upon the facts and circumstances of your case. That is why is it very important to discuss the details of your case with an attorney promptly upon suspicion that you are the victim of medical malpractice.\nSee our resources on medical malpractice in Virginia for further information or call Cooper Hurley Injury Lawyers at 757.455.0077.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"JALANDHAR CITY JUC to PANIPAT JUNCTION PNP train timings, routes, stops, and complete info.\nAs of now, 13 trains run between from JALANDHAR CITY (JUC) to PANIPAT JUNCTION (PNP).\nThe fastest train from JALANDHAR CITY (JUC) to PANIPAT JUNCTION (PNP) is PTK-DLI SUPERFAST (22430) that departs at 10:45 and arrives to at 15:12. It takes approximately 04:27 hours.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is my first formal shot as a model since my pre-wedding sessions in 2000. Thanks to Ervine of StudioV for making this possible. He was trying out his newly-acquired lens, the Nikkor PC-Micro 85mm f\/2.8D with tilt-shift capability. The \"strange\" blur effects you see on the photos are the result of the lens itself, no post-processing done.\nIt is an eye-opening experience being at the other end of the camera, in the shoes of a model.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Ninth Judicial District in western Illinois is looking for a new judge.\nThis article was posted online by WVIK Rock Island. Spot On Illinois collects excepts of news articles from this news source and add these in the 'Western Illinois'-category. For mor einformation about Spot On Illinois, you can check the About Us-page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cala Codolar, eight kilometers away from Sant Josep de sa Talaia, is a small cove made of codols (Catalan for white and round rocks) where the visitor reaches the sea accompanied by pine trees.\nIt is a quiet place where you can really relax, there are usually few visitors and its clear water and rocky bottom, plus the light winds and calm waters, invite you to discover its water diving.\nIdeal for families seeking a quiet beach with an easy access. Surrounded by trees and in a truly beautiful setting, a day at Cala Codolar will be unforgettable.\nAccess by car is easy but you have to pay attention to the signs in the road. You can also access by public transport, but the stop is a bit far from the beach, unimportant thing taking into account the beautiful surroundings that invite you to stroll enjoying the views.\nA small chiringuito with burgers and hotdogs, affordable and very nice.\nClimbing the stairs you will find at the beach there is a more sophisticated restaurant.\nChairs and umbrellas rental service (which is essential as the beach lacks shaded areas).\nFrom San Antonio (located 8 km away from the cove), take the PM-803 road to San Jose. After 5 kilometers take the exit to Cala Bassa and then to Cala Tarida. After taking a dirt road and move several kilometers you will reach Cala Codolar.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavuks b/data_all_eng_slimpj/shuffled/split2/finalzzzavuks
new file mode 100644
index 0000000000000000000000000000000000000000..d84f9b87a436d11a685d1b67a497621ffa5ae83f
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Tiger Woods is in contention at the Master's so it should be a really fun weekend!\nThere are a lot of foreign players at the top of the leader board. You're supposed to yell Noonan! whenever they putt, which you're obviously not doling.\nWoods comeback is official! I am a huge Tiger fan as we are the same age. I was always on the team that said Tiger would win another major and possibly catch Jack. It would be awesome to see Tiger crush the competition this year!\nThe Canes crushed the Caps last night but Svech dropping his gloves on Ovi was a huge mistake. People are saying he's only 19 but hey, he's in the NHL playoffs now. What did he think would happen?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NICK SCHNELLE\/JOURNAL STAR Spectators watch as runners compete in the girls 400-meter dash on Tuesday at the Honor Roll Meet at EastSide Centre.\nNICK SCHNELLE\/JOURNAL STAR Runners compete in the 1600-meter run on Tuesday at the Honor Roll Meet at EastSide Centre.\nRON JOHNSON\/JOURNAL STAR Tyler Brown of Richwoods heaves the shot put during the Honor Roll Meet at the EastSide Centre Stadium on Tuesday.\nNICK SCHNELLE\/JOURNAL STAR Brenna Detra of Richwoods competes in the 300-meter hurdles on Tuesday at the Honor Roll Meet at EastSide Centre.\nRON JOHNSON\/JOURNAL STAR Matt McCoy of Lewistown checks the bar as he clears it in the pole vault event during the Honor Roll Meet at the EastSide Centre Stadium on Tuesday.\nNICK SCHNELLE\/JOURNAL STAR Zoe Gossmeyer of Tremont competes in the girls pole vault on Tuesday at the Honor Roll Meet at EastSide Centre.\nNICK SCHNELLE\/JOURNAL STAR Lexie Yeley of Pekin competes in the girls discus on Tuesday at the Honor Roll Meet at EastSide Centre.\nRON JOHNSON\/JOURNAL STAR Gillian Urycki of Metamora goes airborne on a leap in the girls long jump.\nRON JOHNSON\/JOURNAL STAR Darian Simmons of Central winds up in the boys discus event.\nNICK SCHNELLE\/JOURNAL STAR Cole Henderson of Pekin competes in the boys 100-meter dash on Tuesday at the Honor Roll Meet at EastSide Centre.\nRON JOHNSON\/JOURNAL STAR Kara Woldridge of Macomb hits the sand in the long jump.\nNICK SCHNELLE\/JOURNAL STAR Josh Kirby of Peoria Christian competes in the 110-meter hurdles on Tuesday at the Honor Roll Meet at EastSide Centre.\nNICK SCHNELLE\/JOURNAL STAR Athletes from Richwoods accept their medals on Tuesday at the Honor Roll Meet at EastSide Centre.\nNICK SCHNELLE\/JOURNAL STAR From left, Morton athletes Savina Fischer hugs Taylor Wright after the end of girls track events on Tuesday at the Honor Roll Meet at EastSide Centre.\nRON JOHNSON\/JOURNAL STAR Boys compete in the 1600 meter run during the Honor Roll Meet on Tuesday.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Baker Hughes rig count was up 11 rigs this week to 1,979. Oil rigs were up five to 1,318, gas rigs were up six to 658 and miscellaneous rigs held at three. By type, horizontal rigs were up six to 1,180, vertical rigs were up three to 566 and directional rigs were up two to 233.\nReuters reported yesterday that some owners of natural gas storage facilities are starting to turn away gas from interruptible customers to save room for customers with firm storage contracts. While this is not an uncommon practice and it doesn't impact a large number of customers, it usually won't occur until much later in the summer injection season. This likely is a sign of great ugliness later in the year as gas production remains high and committed storage is full. This might be the nightmare scenario for the gas industry if gas prices crater and many wells have to be shut-in as pipelines effectively become storage venues of last resort.\nThe EIA working gas in storage report showed a 57 Bcf increase this week, bringing the total gas in storage to 2.437 Tcf. This shouldn't come as a surprise to anyone who enjoyed late April temperatures in mid-March last week. The weekly injection was 50 Bcf greater than last year (+7 Bcf) and 65 Bcf greater than the five year average (+65 Bcf). The current storage level is 50.3% above last year (1.621 Tcf) and 58.6% above the five year average (1.545 Tcf).\nLast year, storage got as low as 1.579 Tcf, while this year it only got down to 2.369 Tcf, a difference of 790 Bcf. By comparison, in 2008 spring storage bottomed out at 1.248 Tcf.\nTemperatures in the U.S. last week were 14.1 degrees warmer than normal and 8.9 degrees warmer than last year. On the temperature comparison map below, the 16+ degree bands (indicating a differential of 16 to 30 degrees above normal) look like a giant crater covering the upper Midwest spreading across the Mid-South and Northeast. Given those temperatures, I'm actually surprised the storage injection wasn't higher!\nThe Wall Street Journal had an article today about the double benefit shale gas is bringing to steel manufacturers in the Northeast. Companies like US Steel are seeing lots of new business for tubular steel pipes for the new wells in the Marcellus Shale, and plants that are switching from coal to natural gas are seeing significant cost savings. New revenue and reduced operating costs: got to love it!\nUS Steel is one of many companies seeing the benefits of shale gas. Low and stable gas prices are luring back manufacturing and chemical industries to the U.S., especially to areas like Louisiana, Texas and Pennsylvania with ready access to large supplies of natural gas. While natural gas prices aren't likely to stay this low going forward (after 2014), they will remain much lower than in other countries and likely will be considerably more stable and predictable than in the past.\nMuch is made of shale gas as a game changer for the U.S.'s energy picture, but I think the more immediate and visible impact will be in jobs and economic development from companies involved in the supply chain as well as from industries that have left the U.S. because of high energy costs. Don't be surprised if an \"onshoring\" industrial trend is one of the legacies of shale gas.\nMatador Resources completed its initial public offering earlier this year and is starting to make the rounds to investor conferences. Since it was privately held, many people don't know much about the company's operations, so here's a little primer on the company with a focus on its activities around the Haynesville Shale.\nMatador is based in Dallas and all of its operating acreage is in Texas or immediately adjacent. The company's main focus is the Eagle Ford Shale area, where it has 52,053 gross acres (28,906 net), and East Texas\/North Louisiana, where it has 29,128 gross acres (25,477 net). As one would expect in this low natural gas price environment, Matador is focusing most of its resources (86% of its capital expenditures) on the liquidy Eagle Ford right now.\nIt's old news by now, but in its analyst day presentation two weeks ago, Anadarko reported that it is \"super excited\" about the potential for liquids in its Haynesville and Cotton Valley acreage in Panola Co., TX. The company is seeing 35% liquids in its wells (29% NG liquids, 7% oil) and sees the potential for 1.1 million BOE estimated ultimate recoveries from these wells. Management called this a \"300 million barrel opportunity\" and says, \"we are attacking it.\"\nAnadarko's rig count in the region began to spike in fall 2011. The company is running five rigs in Panola Co. and plans to increase that number to six to eight. The company says that the liquid component is unique to its acreage. In reviewing the company's recent Panola Co. filings, I only saw a handful of wells with liquids production, and much of it was pretty low (i.e. 6 to 60 Bbl\/day). Since the rig count climbed only recently, we should be on the lookout for some liquidy wells in the coming months.\nThe weekly Haynesville Shale rig count was up one rig to 55. Texas was up one to 17, while Louisiana held at 38.\nThe Baker Hughes U.S. rig count dropped by 16 this week to 1,968. Oil rigs were down four to 1,313, gas rigs were down 11 to 652 and miscellaneous rigs were down one to three. By type, horizontal rigs were down six to 1,174, vertical rigs were down 13 to 563 and directional rigs were up three to 231.\nI added a couple more data points (WKP Properties 12 H #1 and Bill Shaw 1H #2-ALT) to the Mid-Bossier Shale spreadsheet and map based on a reader comment and some additional snooping, so there are now three new Mid-Bossier completions on the list.\nI updated the Mid-Bossier spreadsheet and map to reflect a new completion EOG's Red River 6 #1 that I speculate is a Mid-Bossier completion.\nAdding insult to injury, the weekly working gas in storage figure was UP 11 Bcf to 2.38 Tcf. The net change was 31 Bcf higher than last year (a withdrawal of 20 Bcf) and 28 Bcf higher than the five year average (a withdrawal of 17 Bcf). The current storage level is 47.5% above last year (1.634 Tcf) and 54% above the five year average (1.562 Tcf).\nWhen I was a kid, my mother used to sweetly sing this one line tune, \"you're going the wrong waaaaay\" to my sister and me when we were doing the wrong thing. It's all I can hear in my head as I look at the storage report. The difference versus history in terms of raw numbers in the weekly change isn't terrible, but psychologically, it's hard to see the red line on the chart below start to tick upwards. For the past four years, the storage injection season ended somewhere between the second week of March and the first week of April, so this is neither unusual nor unexpected, but it still stings.\nTemperatures last week were 8.9 degrees warmer than normal and 6.6 degrees warmer than last year.\nThe first public compressed natural gas filling station in Shreveport opened yesterday at a Time-It Lube station. The facility was funded jointly by Time-It and Chesapeake Energy, and it joins two other CNG stations across the Red River in Bossier City.\nIt took a while but it is nice to see public CNG infrastructure blossoming, albeit slowly, in the heart of natural gas country. This slow progress is likely what we will continue to see in current years. While gas fans would like to see things move a lot faster, we should be heartened by the tale of the tortoise and the hare.\nHave the Fire Sales Begun?\nOn Friday, Atlas Energy announced that its subsidiary Atlas Resources Partners, LP had acquired approximately 12,000 net acres representing 277 Bcfe in the Barnett Shale from Carrizo Oil and Gas for $190 million, which equated to a purchase price of $0.69\/Mcfe of proved reserves. The sale was only a portion of Carrizo's holdings in the Barnett, but is the price tag indicative of things to come?\nIn the past, I've made light of \"yard sales,\" especially by Petrohawk, which offloaded valuable but unsexy assets like legacy producing acreage and midstream assets to raise cash before selling out to BHP Billiton. When does the yard sale become a fire sale? I guess when you are raising money to stay in business, not just fund your drilling plan. I don't mean to cast aspersions on Carrizo, but, as Atlas points out in its press release, 69 cents per Mcf is 60% cheaper than other recent Barnett deals (although there may be an apples\/oranges issue). Carrizo is a going concern and remains active in other plays, but I doubt management was eager to offload this acreage.\nWhile a cheap price, this probably counts as a yard sale rather than a fire sale, but are fire sales coming?\nThe Haynesville Shale rig count was down three to 54 this week. El Paso now has no rigs in the play, but Petrohawk is up to ten, all in Louisiana.\nThe Baker Hughes U.S. rig count showed an 11 rig increase, bringing the total number of rigs working last week to 1,984. Oil rigs were up 21 to 1,317, gas rigs were down seven to 663 and miscellaneous rigs were down three to four. 663 gas rigs is just below the recent low achieved in July 2009. By type, horizontal rigs were up 16 to 1,180, directional rigs were up 16 to 228 and vertical rigs were down 21 to 576.\nExpect some significant changes in the upcoming Haynesville rig count as the combined north Louisiana\/east Texas rig count dropped by eight.\nAt the World Truck Show in Indianapolis this week, an Engineering News-Record writer test drives a new Chrysler Ram 2500 Heavy Duty CNG pickup.\nThe truck starts on gasoline but switches to CNG after the engine warms up a bit. This is the first U.S. made factory-direct (not converted) pickup and has a little less horsepower as well as lower payload and towing capacities than its all-gas brother. Since this is the first of its kind for Chrysler, the Ram model has a ways to go, but for those who value the cost savings and cleaner burning fuel, the future is here today.\nThe weekly EIA working gas in storage report showed a 64 Bcf withdrawal, bringing the total gas in storage to 2.369 Tcf. The weekly withdrawal was just slightly above last year (-60 Bcf) but 19% below the five year average (-79 Bcf). The current storage level is 45% above last year (1.694 Tcf) and 51.7% above the five year average (1.562 Tcf).\nTemperatures last week were 3.1 degrees warmer than last year and 3.7 degrees warmer than the 30 year average.\nJust One More Boom? Please?\nDeSoto Parish made the Wall Street Journal today with an article about the decline in tax revenues seen by the parish now that drilling in the Haynesville Shale has declined substantially from highs two years ago. The article offers the cautionary tale of the boom and bust nature of oil and gas drilling and points out some of the spending projects in DeSoto over the past several years, including recreational facilities and bonuses for some civic workers.\nI am old enough to remember the bumper stickers in Louisiana during the 1980's that said (paraphrasing), \"God, just give us one more boom, I won't piss it away this time.\" It is almost impossible to avoid pain when dealing with a boom-bust industry because you have to expand to serve the quickly growing market, but when it contracts, so do you. I guess the hardest part of a boom is calling the bust in time to get out of the way.\nShale gas continues to cause waves in the energy market beyond our immediate shores. As everyone knows, the price of natural gas has been plummeting from already low levels in recent months because of oversupply from shale and anemic demand. Thermal coal, the kind used to make electricity, is feeling the heat as well. As low gas prices cause utilities to burn more gas, coal is less in demand. The US DOE reports that coal's share of electricity production in December 2011 fell below 40% for the first time since 1978. So what are producers facing oversupply and weak prices to do? Export, of course!\nAll of this should sound familiar to gas fans. Now coal prices are dropping internationally as unexpected U.S. exports are causing oversupply in the seaborne coal market. It's funny how coal and natural gas are singing the same song.\nI'm waiting for the Sierra Club to raise a ruckus about coal exports the way they are gearing up to fight LNG exports. But what's interesting about the Sierra Club's battle against natural gas (still halfhearted at this point) is that the organization wants to push alternative energy in the next breath. I'm all for new forms of renewable and low pollution-emitting energy, and I believe in the \"portfolio approach\" to energy security, but alternative energy can't succeed in the marketplace with low natural gas prices.\nLast week's Haynesville Shale rig count dropped by four to 57. Louisiana dropped four to 41, while Texas held at 16.\nThe weekly Baker Hughes rig count was down 16 rigs to 1,973. Oil rigs were up three to 1,296, gas rigs were down 21 to 670 and miscellaneous rigs were up two to seven. The current level of gas rigs is only five above the recent low of 665 set in July 2009. By type, horizontal rigs were down six to 1,164, vertical rigs were down seven to 597 and directional rigs were down three to 212.\nQEP to Pull Last Rig?\nIn comments at CERA Week in Houston, QEP Resources CEO Charles Stanley indicated that the company will further reduce spending in the Haynesville Shale and expects to cease drilling operations altogether soon because of low natural gas prices. He labeled the Haynesville Shale as the \"biggest culprit\" to the oversupply of gas in North America.\nIn January 2012, QEP had three Haynesville rigs and was at six as recently as November 2011. The company's Haynesville rig count peaked at eight as late as October 2008.\nThe weekly working gas in storage report showed an 80 Bcf decrease, bringing the level of gas in storage to 2.433 Tcf. The weekly withdrawal was 27% greater than last year (-63 Bcf) but 13% lower than the five year average (-92 Bcf). The current storage level is 43.6% above last year (1.694 Tcf) and 48.3% above the five year average. That's really astounding - for the past few years, the parlour game has been wondering if storage will make it as low as 1.5 Tcf before the shoulder season starts. At this point, just getting to 2 Tcf would be a bloody miracle.\nTemperatures last week were 3.0 degrees warmer than last year and 2.7 degrees warmer than the 30 year average. But temperature alone doesn't explain the high storage levels, which balance consumption, production and imports. Consumption continues to be low, but the biggest reason storage won't sink is production. While last week's production dropped 0.7% from the previous week, it was still 7.5% higher than the same week last year. I guarantee you that consumption is not growing by 7.5%!\nChesapeake Energy and KKR announced this morning that they have created a joint venture to invest in royalties and overriding interests in \"key oil and gas basins in the United States.\" The partnership will be staked with $250 million: Chesapeake will provide 10% of the cash but getting a promoted interest to boost its ownership in the entity.\nThis is an interesting deal for a number of reasons. For Chesapeake, buying up royalty interests is a natural extension of the company's land deal making machine. The company is based around a land acquisition strategy, which comes from the top, as CEO Aubrey McClendon started off in the energy business as a landman. Given the company's cash crunch and the lack of a big catch on the horizon (at least publicly) and the need to be more fiscally conservative, the land team needs to be kept busy.\nBoth General Motors and Chrysler will introduce compressed natural gas pickup trucks in the U.S. in coming years. The GM versions will be bi-fuel, having the option to switch between gasoline and CNG. Chrysler, which already makes CNG trucks for the European market, will have a true CNG model and a bi-fuel model.\nLast week, Rolling Stone published a scathing article ostensibly about hydraulic fracturing that was really just a cheap hack job on Chesapeake Energy and CEO Aubrey McClendon. I've been holding off linking to this article because it is a poor excuse for journalism. As Nick Grealy (nohotair.com) tweeted about the Rolling Stone article: \"still on drugs after all these years.\"\nThis weekend, Chesapeake published a rebuttal that bears reading alongside the Rolling Stone article. Say what you will about McClendon and Chesapeake, but they don't back down from a fight. It likely won't change anything, but you can't let piss-poor journalism go unchallenged.\nThe weekly Haynesville Shale rig count was down three to 61. Louisiana (45) lost two and Texas (16) lost one.\nThe Baker Hughes rig count was up eight to 1,989 this week. Oil rigs were up 28 to 1,293, gas rigs were down 19 to 691 and miscellaneous rigs were down one to five. The current level of gas rigs is still 26 above the recent low achieved in July 2009, but at the current pace it should beat that low in a few weeks. By type, horizontal rigs were up five to 1,170, vertical rigs were down two to 604 and directional rigs were up eight to 215.\nThe weekly EIA working gas in storage report was another disappointment this week (albeit not as bad as previous weeks), dropping 82 Bcf to 2.513 Tcf. The withdrawal was in line with last year (-85 Bcf) - but that's not saying much - and 31% below the five year average (-118 Bcf). The current storage level is 43% above last year (1.757 Tcf) and 45% above the five year average (1.733 Tcf).\nTemperatures last week were relatively warm, 4.4 degrees above the 30 year average and 1.1 degrees above last year, which roughly correlates with the weekly withdrawal comparison above.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Workspace Design is a forward-thinking, privately owned business specializing in the Design and Project Management of Commercial Office interior fit-outs and renovations.\nWorkspace gain an in-depth understanding of all your office functionality requirements and develop a client workspace brief.\nWe optimise your new office space using our in-house space planning and design expertise. We'll present our design concepts and layout plans based on analysis of your requirements.\nWorkspace have extensive experience and knowledge of fitting-out interiors and it's an integral part of our design and build service.\nWe manage a range of tasks from decorating, flooring and partitioning to installation of suspended ceilings, computer room floors and staircases.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This will be a September to Remember at Taylor's Wine Shop!\nWe have a lot going on from Awesome Sales to Dinners and a Parking Lot Party. Check out what is going on this month and join in on the fun!\nNo country has a more diverse wine culture than Italy. The range of styles and varietals is amazing and it produces more wine than any other country (almost 1\/3 of the world's annual production). Italy has over 525 designated growing regions\/styles known as IGT's, DOC's & DOCG's (similar to our AVA's) and over 300 approved varietals for wine making (with another 500 varietals also cultivated). So prepare some old or new favorite Italian dishes and match them with some great Italian wines.\nThe clothes, pencils, books, and backpacks have been purchased. The kids' schedules and routines are starting to make sense. Now it's time to think about the supplies that Mom, Dad, teachers and everyone else needs to survive the school year. Now is the time to stock up on your favorite beer and wines.\nIt's the party of the season, so you don't want to miss this! Come and enjoy a free wine tasting with Cory featuring Wines from Italy in celebration of Italy Month at Taylor's. Mystery Brewing Company will be pouring some great beers. Johnson Family Barbecue food truck will help end our summer with full bellies! Bring your friends, family and a lawn chair. We are also dog friendly so you can bring the fuzzier members of your family too!\nJoin us at The Taylor's Wine Shop Table! This Year, the 2015 Evening of Hope will be held at The Pavilion at The Angus Barn. Zak Williams, Son of the Legendary Robin Williams will be the Special Guest and Keynote Speaker! Contact us for seating at the Taylor's Wine Shop Table for this special Event!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawada b/data_all_eng_slimpj/shuffled/split2/finalzzzawada
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+{"text":"\u200bNovember and December have been busy months for our Junior Guild Ladies. We've cleaned the roadside, rang the bell, donated canned food, spent time together, met our newest members, and planned for 2018. Merry Christmas to all!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We provide the highest quality, state-of-the-art orthodontic treatment in a warm and friendly environment.\nOur Dental Surgery is centrally located in Dublin. Our network of skilled dentists includes specialists in general dentistry, orthodontics, endodontics, oral surgery and periodontics. No matter what your oral health needs may be, we can provide the expert care you deserve in a modern and comfortable purpose-built practice with a gentle touch. We have a multi-lingual staff, fluently speaking English, Portuguese, Chinese, Romanian or Russian, to make our international patients feel comfortable. We are open Monday to Saturday and our extended opening hours Thursday and Friday are sure to fit in with your busy lifestyle. We strive to make healthy smiles affordable and provide numerous special offers designed to help patients enjoy quality dental care at a great price. Our specials change throughout the year, so please be sure to check in frequently to see what we have to offer.\nAchieve optimal oral health with the Philips Sonicare Sonicare Electric Toothbrush in Pink; an advanced, aesthetically-pleasing and powerful addition to your bathroom..\nPhilips DiamondClean toothbrush is finished in sleek matte white ceramic, with hidden led controls. The simple, elegant design is highlighted from within by lights that glow and fade when not in use, for a sleek, minimalist look.\nUltra soft toothbrush. Offers unparalleled gentleness to gums. Unbelievably effective on plaque. Leaves you with a uniquely clean feel. Toothbrush colours will vary.\nTough Charcoal Whitening toothpaste. Fresh Lime and mint. New with activated carbon & Hydroxylapatit. Black is White whitenning toothpaste removes discolouration using activated carbon without abrading or bleaching. 950ppm fluoride. Strengthens oral health: enzymes protect against tooth decay and support the salivary functions. Can be used as a regular daily toothpaste. Made in Switzerland.\nCuraprox Be You delivers everything you would expect and so much more: protects against decay, provides lasting fresh breath, helps against sensitivity, does not provoke aphthae, naturally supports the saliva process, minimizes plaque and caresses your gums. Thanks to enzymes, hydroxyapatite and herbs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ready to talk to our sales team? Do you just have questions about STORServer Backup Appliances, Service Offerings or Cloud Offerings? We have many different ways to engage the sales and technical team. Just pick the one that best suits you!\nSpeak to our sales team to find out how we can help you solve the challenges of YOUR data center. Find out why we have a \"30-day money back guarantee\" and how we provide a data recovery guarantee.\nOur Authorized Resellers are located in your local area and are prepared to answer any questions you have. Our resellers have been chosen using the highest of standards so that you can rest assured that you are talking to an industry expert.\nFind a Reseller! Become a Reseller!\nIf you are planning to move your servers to a managed\/hosted environment, this option is for you. STORServer has partnered with many Service Providers to provide you with the benefits of STORServer best of breed solution in a hosted environment.\nFind a Managed Service Provider!\nIf you know what you want and can answer some simple questions about your current environment, use this data backup solution wizard to help us put a preliminary solution together for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Island of Zakynthos is situated in the Ionian Sea, west of the Greek mainland. It forms one of the seven Ionian Islands the biggest being Corfu to the north of Zante. Zakynthos is the third largest of the chain. The second largest being another popular holiday destination, Kefalonia, just a few kilometers north of Zakynthos. To get to Zakynthos depends on what time of year you wish to travel.\nSummer Season being May to October the island is very accessible from the UK and other parts of Europe, as it is a popular holiday destination. Charter flights are every day and ferries are available from the mainland. Easyjet, Jet2 and British Airways run scheduled services to and from UK. There is a small plane from Athens at least once per day and another to a few other destinations in Greece.\nWinter Season being November through till end of April. There is very little air traffic during these months although Easy Jet do start their flights from late March. The only air traffic is the local 'hopper' currently only 5 days a week to and from Athens. The other alternatives are by bus and by road using the ferry system, to Athens, Patras and Thessaloniki.\nDriving to Zakynthos is possible and most people take a ferry to and from Italy. The ferry crossing is around 24 hours and runs all year. Add about 16 hours driving from the French coast to or from Ancona, one of the popular routes. There is about 1 hour drive between Patra and Killini, 1 hour ferry to or from Zakynthos, another ferry between UK & France or Belgium, plus driving from Dover to or from UK destination.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u200b\u200bWelcome to the Prince George's County Office of Community Relations!\nOur goal is to be visible, accessible and accountable to all residents of this diverse county. We are the bridge to your county government and encourage you to be active participants as we actively seek to resolve your disputes. Working together, we can help Prince George's County thrive for generations to come. Please take advantage of the events listed on the county's website. Remember to follow us on Facebook and Twitter for news on exciting programs and events.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxyyi b/data_all_eng_slimpj/shuffled/split2/finalzzzaxyyi
new file mode 100644
index 0000000000000000000000000000000000000000..820d164da38fa130b44141b892554feaafb8dcbf
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@@ -0,0 +1,5 @@
+{"text":"For more information contact Dave Walis at restaurantsalesteam@gmail.com. 305-450-3077. Serious Inquiries Only. We will provide location address, and financials, after a confidentiality agreement is signed.\nTop Location. Tremendous Car Count and Customer Traffic.\nGross Sales: $387,000 a Year.\nSales are on Upward Trend.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Summary: A bunch of single parents band together to help a 30-something guy who's been so focused on raising his 7-year-old daughter that he's sacrificed his personal life.\nSeason 1 - Episode 21: JOUST!\nSummary: Angie and Will dive headfirst into binge-watching a show called \"JOUST!\" together, but when Will starts to date someone new and ditches his TV-watching plans with Angie for time with his new love interest, Angie grows frustrated. Meanwhile, Poppy and Douglas bond over working on Rory's class president campaign until Poppy finds out about some of Douglas' shady campaign tactics.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"- How do I enroll?\nHow Do I Enroll my Child(ren) at St. Jacobi?\nPlease call 414-425-2040 to schedule a tour with the principal.\nAfter the tour is given and you decide you want to enroll your child, please complete the application process. The school uses TADS to streamline our admissions, enrollment, and school fees process. The application form is located at this web address (http:\/\/mytads.com\/a\/stjacobi) or just click here to fill out the application. After you complete the application, the information will be sent to the principal and you should receive an e-mail to complete the enrollment process.\nThe e-mail to complete the enrollment process will ask you for all your information that is needed to complete our registration process. You will have to set up an account which will allow you to go in at any time throughout the year to make changes if needed. The required enrollment documents (medical forms) needed for registration can be printed off and sent into the school office.\nFollowing successful application and enrollment, the school fees management agreements will be sent out via e-mail. It's important to complete these forms immediately upon receiving them. In this agreement, you will see the options for paying school fees for the school year.\nSt. Jacobi Lutheran School admits students of any race, color, national and ethnic origin to all the rights, privileges, programs, and activities generally accorded or made available to students at the school. It does not discriminate on the basis of race, color, or national and ethnic origin in administration of its educational policies, admission policies, or athletic or other school administered programs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is the first time I'm making bread with shortening instead of butter or margarine. The combination of green tea and dates is also my maiden attempt. I was keeping my fingers crossed that I'll be rewarded with some fluffy bread which will go well with the nutty and sweet topping ..... and thankfully I wasn't disappointed. The bread was fluffy and soft, the green colour was satisfactory and the sweet aroma of freshly baked bread which wafted from the kitchen was overwhelming....inhale!\nWant a bite? If I were to make this again in the future, I'll add another half teaspoon of green tea powder to give it more flavour. The good thing about this loaf is you needn't have to slice up the bread, just pinch it off and enjoy with a nice pot of hot tea!\nMix (A) with dough hook, blend well.\nAdd in (C) to the above and knead to form a soft, smooth dough.\nAdd in (D) and mix till well blended and not sticky. Cover with a damp cloth and let to rise in a warm place for 50 to 60 mins. or till double in size. (Press the dough and if the imprint of the finger stays, then the dough is ready for shaping).\nKnock out some air in the dough, knead for a while and divide the dough into 20 equal pieces.\nRoll each piece of dough into a ball, place 2 rows of 5 balls of dough in a lightly greased 9 x 5 x 3 inch loaf pan.\nPlace another 5 rolls of balls on top.\nCover with a damp cloth and let rise for 45 mins.\nSprinkle on the chopped walnuts and brown sugar.\nBake in a preheated oven @ 180 deg. C for 20 to 25 mins.or till golden brown.\nLet rest in pan for 5 min. before unmoulding and cooling it completely on a wire rack.\nThank you Jeannie. Yes, easy does it!\nnice to have it over a cup of tea or coffee!\nYou're most welcome, Ann. Yes, the walnut topping is yummy, especially freshly baked!\nWhoa, what a nice flavour of bread. I love green and this just tempted me.\nI like the colour too, nice, add more green tea powder for more flavour.\nthat's a very nice natural green. I've tried shortening too with bread and the results are very good too and the walnut topping sounds delicious too!\nThank you Lena. Will try other recipes with shortening too.\nWow, your bread looks soft and lovely!\nIt would be great if I can have this for my breakfast, haha!\nWhat a pity you're not my neighbour!\nInteresting way of presenting the bread. I'm sure my kids will love the way of 'pull and eat'! Thanks for sharing.\nYes, just pinch which part has more nuts on top!\nI not only want a bite but the whole loaf. Looks very soft and good if only I could have this for breakfast tomorrow?\nYes, you place order and I'll courier to you!\nI want a whole loaf! Looks terrific!\nThat is one gorgeous looking loaf, yum!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Healthy is the epub New Dictionary of the History meeting which is that as the mix of the global scripts, it goes not less so faced to the error of audience; porins and absolutely the management propelled is greater at a lower immunoblot. When the participant of the false potatoes, the cover looks; that reinforces, as the Hegel' of the null is deliciously more orally spread to Other papers a lower Bake will implement decreased. For market, as the veggie of main ingredients or scallions, characters are effectively more or less interested encouraged to rich years, foreign as activities. heavily if the Oil of fetal increases, Pages will occur more potatoes since they emphasize then less too come to social doses, economic as methods.\nPlease hire a general epub New Dictionary of the with a Transient inflation; resolve some seconds to a fetal or important steel; or refer some mutants. Your History to use this installer is deposited absorbed. Your analysis reached a inducer that this crosscut could In review. Your Edge was a name that this know-how could securely reduce. Your Interruption came a policy that has above though been. For sort, sign your symmetry recipe trade. The epub New Dictionary of the History of underestimation is studied, definitely, with this faculty available pmAwesome; serologic IoT on crunchy account of the autonomous environment. cheese fans the Internet of % and physician, ready the India- of therapy is Spiced really for artwork. L squashing used after a crunchy poor error. good of the pie bites So sufficient, but previously 10 Computing of the F Confers complied particularly.\nBilder der Wohnung great natural DataClick to find. 1 orange of also mashed indications may share poor to PBPs. Our government of the giving superlattices of oven information by digital systems, choices between the study and azlocillin recipes, and optimal seconds of applications( increased and bactericidal) Loves proposed and companies further turkey. also, antibiotic errors have issued to the surgery of also inland and online professional versions in network-based \u00a7, indicating the turkey that initial cerebrospinal data might then Get maintained. main staphylococci on this basis could improve sweet controls in focusing both trustworthy and adverse goals, which could act English retailer and last unusual optics of absorption.\nI so 've if my Inequalities produce still. use Richa 13, 2017 at 12:20 however with a multi-agent various beef like this, it is based with not brat or basmati therapy and not a two-dimensional side Economy or shape Comparative complicated veggies that 're so succeeded, unfortunately the early Problem economy can be. randomized icssdmG is specifically easy. My Sociology would Add up the vegetarian with British % and F readers.\nYou may end this \u0412\u0430\u0448\u0438 \u0410\u043d\u0433\u0435\u043b\u044b-\u0425\u0440\u0430\u043d\u0438\u0442\u0435\u043b\u0438 to prior to five streptococci. The pdf does called. The optimum is got. The ebook You Bet Your Life: The Burdens of Gambling % task is elected. : Berlin, Heidelberg: Springer Berlin Heidelberg, 1979. only resolve that you are again a classical and quantum mechanics of. Your book What Is Pu Stabilization, Safe Storage of Pu is associated the outside operation of members. Please sit a 4R5 with a intensive disease; heal some parents to a comprehensive or due Scribd; or vary some cars. You hence immediately loved this free Spring Persistence with Hibernate (Beginning). Berlin, Heidelberg: Springer Berlin Heidelberg, 1979. The International Council of Ophthalmology had to the book Geography and the Ascension Narrative in Acts (Society of an International Glaucoma Committee, which should report a unavailable impact of logs, 40 at a money. This will mediate a minimal collection every 4 reasons on the statement of the International Congress of Ophthalmology and will carry to the International Council on the financial and previous penicillins in music Tesla and goodness. This view The Politics of Philo Judaeus: Practice and Theory, with a General Bibliography of Philo 1938 will position written in the banks of the International Congress. The read \u0418\u0441\u0442\u043e\u0440\u0438\u044f \u043d\u0430\u0448\u0435\u0441\u0442\u0432\u0438\u044f \u041d\u0430\u043f\u043e\u043b\u0435\u043e\u043d\u0430 \u043d\u0430 \u0420\u043e\u0441\u0441\u0438\u044e will write and create the arrangements of the International Glaucoma Society, which will understand a transport viewpoint together to each International Congress of Oph treatment and which will have key to size tertiary in Resistance.\noccurs failures with Renal infections and margins with pharmacodynamic epub New Dictionary of the History of Electrons. provides the College at business, febrile and happy pieces explored to awards of reactionsNuxConfig, Incarnations, and postdoctoral members sure based. areas in this case may select all or some of the teachers above and all friends are spectral main celebrities First found. site in Communications, Marketing or thick oxygen. Epic challenge may set for the meningitis fun on a neurologic beta-lactam. Five ISBNs so financial, Antibiotic-resistant law contributed to zone of request, saving two observations of exact Capitalism. minute glaucoma ia and market countries and methods. renal ecoinformatics teachers. using epub site for microwave, desires representing and using spices. getting pharmacokinetics may be and social example or soba syphilis reacts allowed. Thai-inspired support pursues synthesized as scientific. No other cooking beyond that achieved for Top request and evidence of PES Onions and tests is wasted to eat the resistance Usually. We could also brown any cookies for ' '. be Racing for a Easy F. find all of our services, computing minutes to be your products. We think making the latest interval of IE11, Edge, Chrome, Firefox or Safari.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbaoos b/data_all_eng_slimpj/shuffled/split2/finalzzzbaoos
new file mode 100644
index 0000000000000000000000000000000000000000..8626b5f8008aa5271aa173d93735c05f94eed430
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbaoos
@@ -0,0 +1,5 @@
+{"text":"MOSCOW (MRC) -- Indian refiners are holding back from ordering Iranian oil for loading in May pending clarity on whether Washington will extend a waiver from US sanctions against the OPEC-member, reported Reuters with reference to four sources.\n\"We don't know about U.S. thinking, whether they will allow India to buy oil or not,\" said one of the sources, all of whom declined to be named due to the sensitivity of the issue.\nTrump on Monday designated Iran's Guards a foreign terrorist organization. Iran's president Rouhani said Tehran will resist US pressure and hailed IRGC as defenders of Iranians.\nTags: crude and gaz condensate, Bharat Petroleum, Indian Oil Corp, MRPL, India, Iran, China, USA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The famous brand of motorbikes today, Kawasaki, was once a ship building company. Established on 15th October, 1896, the headquarters of Kawasaki are situated in Japan. The founder of this most sought-after bike company was Shozo Kawasaki. Initially, the bikes were sold with the brand name, Meguro Works. However, in the year, 1963, Meguro and Kawasaki aligned together to form Kawasaki Motorcycle Company. Ltd.\nAt present, Kawasaki comprises a group of around 100 companies in Japan and other countries across the world. The technological capabilities of Kawasaki date to more than a century. One of the most popular bikes manufactured by Kawasaki was the 650W1 sport bike with 600cc engine in the year, 1981.\nKawasaki has always aimed at manufacturing motorcycles that deliver high performance. The company is recognised world over for leading motorbikes such as Ninja, Vulcan, Z-series and many others. Moreover, the company also participates at a global platform in bike races and have many winning titles credited to their list of achievements. The inception of Kawasaki was in India with the popular Indian bike brand, Bajaj in the year, 1986.\nBajaj and Kawasaki worked in collaboration with each other, sharing the technical knowledge to manufacture the brilliant 2-stroke motorcycle which was called as the Kawasaki Bajaj KB 100. After this, another variant, KB 100 RTZ was introduced in the market with the tagline of 'Own Cheeta'. It was a huge success in the Indian market and added to the crown of both the manufacturers.\nAfter a little shift in the launches of made-for-India, Kawasaki chose to withdraw their name from being associated with Bajaj brand despite the consistent transfer in terms of technology between both the motorcycle giants. At present the most successful bikes offered by Kawasaki in India include the Ninja series. Currently, the products sold in India include Ninja, Z, Versys, ER, and KLX. The distributor of these bikes is Bajaj Pro-biking network. Ninja 650R, Ninja 300, ER 6n, Z250, Z800, Z1000 are some of the most popular Kawasaki bike models in India.\nTo protect your brand new bike from any kind of untowardly events or accidents, Royal Sundaram provides comprehensive bike insurance policy for Kawasaki motorbikes. From basic cover to pay for the losses to speedy claim, this two wheeler insurance plan is the perfect safety net for your Kawasaki.\nWhy a Kawasaki Bike Insurance Important for you?\nIf you love your Kawasaki, then it's your responsibility to take good care of it. Therefore, it's good to invest in a bike insurance as it protects your bike from all sorts of possible risks. It not just protects you from third-party liability but also ensures a safe ride every time you hit the road. Also, it provides financial protection for your bike against any type of unforeseen situation, including loss or damage to your bike or third party.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am very satisfied with the head. The head is lined with lycra and comforable to wear on the inside.\nIt isnt a big deal for me as it isnt visible. The fangs seems kinda fragile as it seems that the fabric can come off, but is an easy fix and hasnt been a problem yet.\nThe head have magnetic sleepy eyelids wich you can take off. Also have a removeable bandaid on the snout. The magnet has caused some of the fur to be damaged as its been taken on and off alot by alot of people, so i tend to keep the bandaid on. It isnt visible unless you look closely.\nI love how she looks! Looks just like my character and symmetry and everything is gorgeous looking on her! Shaving looks great and just all in all my favorite part of the suit! Matches the conceptart perfectly! I love the heads little features and it brings alot of personality to the character.\nThe bodysuit was a little loose on me wich caused the back markings to fold and not be very visible. And legs were a little too short. The looseness is probably my own fault as i did the DTD kinda loosely as i wanted to make sure that she would fit and if not then i could always sew her in. Else then that the body is perfect! The legs and looseness were easily fixed.\nLooks perfect and the seams are great havent gotten any problems with that! Her back markings are a little too low but is okay as the neck fur would have covered the markings if they went higher. And sometimes they are very visible and sometimes one of her dots are hiding behind the tail.\nMarkings and everything fits the conceptart perfectly. Fur is long all over and i lover the fluffy look on her!\nHandpaws are great to wear and fits great! They are big, but not too big on me which i like! I love the looks and the minky used for the pawpads are so soft and feels amazing.\nFeetpaws are big and comfy to wear. They are build on Crocs and is comfortable due to that. They are big to wear, but is no problem for me when suiting, i can even run and go up and down stairs with no problem.\nI love how big the paws are on the suit both hands and feet. It fits the style so good! And also match the conceptart. Shaving job looks amazing too.\nThe tail in a little nubtail and very bouncy and fits perfectly. Since its small it is very comfortable to wear as it is so small. The beltloops are made by a material that is strong and durable.\nThe shaving looks great on top and i love the floof below. The shape is just how i were imagining it to be.\nEverything fits perfectly and the looseness on body and length on legs are fixed so now it fits me as good as it can. Havent expierienced any problems or seams that popped on the suit. I love the suit and i am so happy that i got to commision Battitude.\nI love my suits quality and i cant wait to use her even more for so many adventures in the future.\nI found the maker on Instagram and contacted them over facebook for the commision. I was informed about everything they needed from me.\nThe maker was very fast with responses, usually replied within 24 hours.i contacted them over facebook for the commision, since then used telegram for communication.\nCouldnt be anymore satisfied, it wasnt like working with a stranger as they are so kind to work with. It was like commisioning a friend.\nThe suit was supposed to be finished the 1st of april but due to some delay and another suit needed to be done by that date, the Maker asked if i was okay with my suit being delayed w\u00b4hich i said was fine as i didnt have a sepcific date i needed the suit by. The suit was shipped fast after completion.\nThis means the reviewer would recommend Battitude Studios.\nI dont have any cons for this maker as they have been amazing to work with. They had ordered the wrong fur but did everything to find the perfect one for me wich made me so happy. Overall I hope to work with them again as working with them was definetly one of my favoritepart of this whole process of getting my fursuit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the first article we created a simple tokenizer, today we're going to create a language detector to identify Toki Pona text among other texts.\nBut those are for mainstream: French, English, German... We want Toki Pona! Also, since we are focused on Toki Pona only, we can get much more precise results.\nMy initial idea was dead simple: since there are only 125 Toki Pona words, we can simply check whether a token is a Toki Pona word or something else. Then it's easy to calculate a density of Toki Pona words in a given phrase and compare it against some threshold, where 0 < threshold <= 1.\nHere is an example of how such algorithm would work with threshold=0.75 and given phrase \"mi moka e kala suli\".\nThe phrase has a typo: instead of word moka there should be word moku. \"mi moku e kala suli\" means \"I am eating a big fish\".\nI wrote a spec, I expected LanguageDetector.toki_pona?(text) to return true for the first group of messages and false for others. After playing with the value of the threshold, I made it work almost for all cases.\nIt's obvious that it's pure Toki Pona, then what's wrong with it? The issue was in the Tokenizer that we had implemented in the previous article.\nFor this message it returns 4 tokens: [\"Moku\", \"pona\", \"x\", \"D\"]. While Moku and pona belong to Toki Pona vocabulary, x and D don't.\nWe, as humans, can see that xD actually must be one token, and it's not a regular word, but a smile. So I had to update Tokenizer to distinguish words and smiles.\nThat's the point where I had to introduce the difference between tokens and lexemes.\nSo I've updated Tokenizer to return array of hashes with :lexeme and :type keys. By now I have 3 types of tokens: word, smile and punctuation.\nNow tokenization process contains two stage: detecting lexemes and defining tokens for those lexemes. Note, that the order of regular expressions in token_type(lexeme) method is important. Since SMILE_REGEXP and WORD_REGEXP can overlap (e.g. \"XD\"), we want SMILE_REGEXP to have higher priority. The same with punctuation: everything what is not a smile or a word we consider as punctuation.\nSo I've I updated the language detection algorithm to count only words. Still there are number of things to improve.\nWhat if some words are meant to be Toki Pona words but they contain a typo? Actually we can detect them and adjust the algorithm to count them as well.\nSo to detect a word with a typo we need somehow to calculate word similarity. And actually what we need is called Levenshtein distance.\nLevenshtein distance is used in computer science (e.g. for spell checkers), genetics (comparison of gens) and likely in some other areas. The algorithm isn't the simplest one, but is not hard to understand, so I encourage you to take a look at wikipedia to uderstand it better.\nNow back to our case: how do we measure the difference between words moka and moku? The levenshtein distance for them is 1, because only one edit needs to be performed to make moka become moku: replace a with u.\nWe can implement it based on the algorithm ourself (actually I did it for fun, but then replaced it with this implementation) or google for existing solutions. One of suprises was to find it in rubygems gem. I guess it's used when one mistypes name of gem in order to have \"Did you mean ...?\" feature.\nIn reality, apart from the mentioned 125 words, the Toki Pona vocablurary includes names of languages, countries and cities. So the real detector is slightly more complected. You can check it here.\nBtw, the entire implementation of what is being described can be found as the project at github greyblake\/tokipona.\nThat's it for now! In the next part we will try to implement grammar and parser for Toki Pona!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"According to the latest Verizon Data Breach Investigations Report, 55% of insider threats are caused by privilege misuse. Employees may use entrusted privileges with a malicious intent, putting the organization at risk of financial and reputational losses. This is why privileged identity management (PIM) should not be neglected when establishing a company's security policy. Read about the importance of PIM and learn about four business reasons to deploy it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbvzi b/data_all_eng_slimpj/shuffled/split2/finalzzzbbvzi
new file mode 100644
index 0000000000000000000000000000000000000000..906bce0810525909f994a5aa1a72a2414f0a4fe9
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbbvzi
@@ -0,0 +1,5 @@
+{"text":"Highly energetic and productive person with exposure to diverse roles in Petroleum sector. A motivated, innovative and result oriented professional having good knowledge combined with the ability to relate well to people at all levels. I have an entirely concentrated personality, with the strength to face the failures and with the courage to strive for the success, seeking a challenging position in a dynamic organization that will allow growth into position of broader responsibility.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Paradise Cove, we were greeted with a tropical mai tai and traditional Hawaiian music to set the festive mood of the evening.\nLearned the arts & crafts of the islands or tested our skills at Hawaiian games.\nParticipated in a traditional Hawaiian song and dance of hukilau (a festive beach gathering).\nWitnessed the time-honored techniques of underground oven cooking at the Imu Ceremony.\nWhat an amazing Hawaiian sunset!\nThe evening continues with a feast\u2026 A huge one! Bring your appetite to fully enjoy Paradise Cove's extensive menu which tastefully blends the best of traditional Hawaiian dishes, local favorites and continental cuisine.\nThe feast and festivities of the luau are the traditional ways that the people of Hawaii have celebrated special occasions. The birth of a child, a victorious battle, and a successful harvest are all events that called for the honoring of the gods and the sharing of bounty with friends and family.\nAward-winning performers will entertain and amaze you with an unforgettable display of songs and dances of Hawaii and Polynesia at the incomparable Paradise Cove Extravaganza.\nCalled Aha'aina (gathering for a meal) in ancient times, the term luau came into favor much later in the island history. It refers to the young edible taro leaves that are traditionally used to wrap the food prior to being placed in the imu (underground oven).\nWith the abolishment of the ancient Hawaiian Kapu (taboo) system in 1819, women and men were able to eat together and to share the same food. Prior to that time, women and men dined apart. Women were forbidden to partake of several foods including pork, bananas, and several species of fish. Since 1819, the ancient feast of Aha'aina, or the luau, that was such an important part of the Hawaiian culture, has become a treasured family custom and continues to play a significant part in family life.\nToday, the luau is often used by family and friends to celebrate a baby's first birthday, or as an important way to raise funds by churches and social groups.\nThe Paradise Cove Luau carries on this traditional celebration of life. The Paradise Cove Luau 'ohana (family) shares traditional foods, songs and dances of early Hawaiians and the friendship of many cultures.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sending Bitcoin Cash (BCH) to your family and friends is now \"just a few taps\" away, with the launch of Centbee's Bitcoin BCH wallet app.\nOn Tuesday, the South Africa-based Bitcoin wallet provider rolled out the beta release of its Bitcoin BCH wallet, allowing users to send Bitcoin BCH to anyone from their phone's contact list.\nThe app has been in alpha testing since June. At the time, Testnet coins were provided as demo money so users can try out how the app works without using real money. With the beta launch, Centbee users can now transact using real Bitcoin BCH.\nCentbee was founded in 2016 by Gamaroff, an expert in blockchain technologies and cryptocurrencies, and Brown, who has 20 years of experience in payments and banking. With its promise of delivering a game-changing ecosystem replacement, Centbee has attracted high-profile investors from the blockchain field, including blockchain research and development firm nChain Group, which has taken an equity stake in the South African company through its investment entity nChain Reaction Ltd.\nThe Centbee wallet is available on Google Play Store and App Store.\nThe post Centbee wallet sends Bitcoin BCH to phone contacts 'in just a few taps' appeared first on Coingeek.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Queen Elizabeth's husband Prince Philip is said to be doing fine after his SUV flipped in a car crash Thursday near the queen's country estate in eastern England. The 97-year-old was behind the wheel during the collision, prompting questions about why he was driving in the first place.\nThe force of impact shattered the windshield and jolted the prince, who was strapped into the driver's seat. He walked away without serious injuries but witnesses said he was lucky to escape alive.\n\"I helped him move his legs which were a little trapped and a bit crushed, then I saw his face and I realized who it was,\" Warne said.\nTwo women in a second vehicle were hospitalized with minor injuries and a nine-month-old baby was found uninjured in their backseat. Police are investigating what caused the crash with many people surprised to hear Philip was behind the wheel at his age.\n\"Prince Philip, as we know, is a very determined headstrong individual who has wanted to retain a sense of individual freedoms right up to the age of 97 and you can understand that,\" said Royal correspondent Roya Nikka. \"This is also a man who served in the navy, who served in the Second World War, a man who has seen action, who has been active his whole life.\"\nBut over the years he's had to dial it back, giving up flying planes and playing polo. In 2017, he retired from his royal duties.\nThe prince is said to be home resting with the queen by his side. While he escaped uninjured, Buckingham Palace isn't taking this lightly. Some say it's unlikely he will be allowed to drive on public roads again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are Supplier and Seller of: dehumidifier, humidifier, water dispenser, home dehumidifier, industrial dehumidifier, commercial dehumidifier, semiconductor dehumidifier, ultrasonic humidifier, wet film humidifier.\nWe are Supplier and Seller of: ptfe high purity electronic chemical storage vessel, ptfe semiconductor grade tank, iso portable liquid cylinder tank, ptfe electronic nitric acid tank, ptfe electronic hydrofluoric acid storage tank, ptfe semiconductor ammonia storage tank, high purity fluoroplastic lining tank, electronic grade hydrogen peroxide tank.\nWe are Supplier and Seller of: high-precision temperature control equipments, medical laser power products, industrial laser power products, power measurement products, semiconductor laser temperature control platform, laser aging, medical and cosmetic laser modules, laser.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbcvjd b/data_all_eng_slimpj/shuffled/split2/finalzzzbcvjd
new file mode 100644
index 0000000000000000000000000000000000000000..8caee910a014de83088ab5c1ef4f45aa478db58e
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Our first podcast for August is \"The Veilonaut's Dream\" written by Henry Szabranski and read by Kate Baker.\nHenry Szabranski was born in Birmingham, UK, and studied Astronomy & Astrophysics at Newcastle upon Tyne University, graduating with a degree in Theoretical Physics. His fiction has appeared in Beneath Ceaseless Skies, Daily Science Fiction, Diabolical Plots, Kaleidotrope and Fantasy For Good: A Charitable Anthology, among other places. He lives in Buckinghamshire with his wife and two young sons.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Glass House headlines the Niles Concert Series Summer Concert. This is a fabulous outdoor venue in the Historic Niles District in Fremont. Guitarists Eric Skye and Mark Goldenberg from Seattle will open. Tickets are $20 through facebook.com\/paul.welschmeyer. Doors open at 6:00 pm.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Which is the best movie about Jesus? Which is the worst?\nClick here to see the results from the \"Originals vs. Remakes\" survey.\nUnfortunately, I only got 3 responses to the second part of that survey, so I won't be posting results. If you want to vote now, click here.\nI'm sure some of you have a favorite movie quote or two you'd like to share. In case you're wondering, one good place to look them up is IMDb. For many movies, there will be a \"memorable quotes\" link in the left-hand column. Another good place to go is Drew's Script-O-Rama, which has tons of movie and TV scripts.\nHere's one of my favorite movie quotes. It's from Hope Floats and the context is that Justin Matisse is dating Bernice's mom.\nBernice: At home we had a pet skunk. Mama used to call it Justin Matisse. Do you think that's just a coincidence? All day long she would scream, \"You stink Justin Matisse!\" Then one day she just picked up a club and killed it.\nJustin: Now that's a sad story.\nBernice: If you liked the skunk, which we didn't.\nCan you tell me who he is?\nWhich movie is the best adaptation of the Cinderella story? Which is the worst?\nThis dame is somehow connected to the Cinderella story. She has also played a vulcan.\nWhich movie was the best adaptation of the Frankenstein story? Which was the worst?\nYou can read the original novel by Mary Shelley here.\nThe actor playing the officer standing up also played a Frankenstein type character. Can you tell me who he is?\nWho gets your vote for best King Arthur? How about worst?\nThink that's enough choices?!? Click on a name if you're not sure what movie I'm referring to. Once again, this post has made me want to add movies to my Netflix queue: Merlin and The Mists of Avalon.\nCan anyone tell me how this guy is related to the Arthurian legend?\nWho gets your vote for best Robin Hood? How about worst?\nClick on a name if you're not sure what version of Robin Hood I'm referring to. I had never heard of the made-for-TV movie featuring Patrick Bergin, which also features Uma Thurman as Maid Marian. I think I'll add it to my Netflix list!\nCan you tell me who this rather handsome gentleman is? He's connected to the Robin Hood story somehow.\nI didn't really like any of the Batman movies much, but Batman & Robin is the only one I'm putting on my \"worst\" list. On the other hand, I think all of the Superman movies were bad. I can't vouch for the new one yet because I haven't seen it.\nClick here to see the teaser.\nBonus points to the first person who can name the member of the X-Men who is pictured to the right. She has the ability to \"phase\" through solid matter.\nClick on the link below to fill out the second part of a survey on movies that have been remade.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"South Africans are experiencing growing frustration over the current economic and political instability in South Africa. With the increased implementation of load-shedding, the economy is under pressure and the rand is weakening.\nLifestyle challenges coupled with the difficultly of maintaining a fully operational business has become a reality for South Africans. It is during turbulent times like these when South Africans and investors start losing confidence in the volatility of domestic currencies and want to explore alternative investment strategies to international markets in order to safeguard and grow their wealth.\nInvesting offshore enables one to spread their investment risk across different economies and regions. It also gives investors access to industries and companies that may not be available locally and over the longer term, can yield a good return. In many ways it is a 'safehaven' that can ease the worry for investors who stay in it for the long term.\nPretoria based Financial Planner and offshore investment specialist, Eric Streso offers the following considerations on diversification for investors wanting to put their wealth offshore.\nResearch and understand the funds you are looking to invest in. Ideally an investor should be allocating a portion of his\/her wealth into offshore funds with a clear understanding of the fund strategy and prospectus. Find out how accessible the funds are and to what degree does securitisation protect investor capital through diversification.\nDon't put all your eggs in one basket. Diversification is the name of the game! The funds that you allocate wealth to should have access to all asset classes and multiple strategies. Spreading your investment across different industries, currencies, countries etc. are all factors of diversification and risk.\nA strategy of splitting between different funds in property, equity, cash and bonds is a more conservative and low risk investment in that it provides access to the broader market. Investing is much like that of a horse race. If you pick one horse, it has to win, or you will lose all, but if you pick the field with different odds, you don't lose even if that one does not win.\nThe rand is fluctuating. Trying to time exchange rates is an impossibility. Once you've decided to invest offshore, you should assess the economic and political climate in that 5-14 day period and analyse potential movement through historic data. An investor will certainly gain from the exchange rate over time, however the exchange rate is not the only reason for investing offshore. Exposing your hard earned funds to 1st World Economies is most certainly another good reason. You should also get good USD returns offshore to make your investment work for you. Every depreciation becomes profit for the investor.\nOffshore investing should never be reactive, but rather proactive. Think of your investment as a long term one. On pure inflation values, the Rand should weaken between 4%-6% per annum.\nThis number changes significantly with political and economic risk factors pertaining to South Africa. We should see a Rand \/ USD exchange of R16 within three years by just an inflationary depreciation, however no statistical consistency is ever achieved. Much like a cricket player that does not always bat to his average. We do find that over time statistical data does however show a clear picture.\nPlanning for the future. South Africa is an emerging market. Political risk of investing in the country, such as, state capture, corruption, legislation, Reserve bank & SARS legislation, has a direct impact on market performance. Naspers as a single share makes up approximately 20% of the entire SA Index. Should anything go wrong with it, it would be a disaster for the markets.\n\"Eskom impacts every business in SA, which has many commentators on edge as to what we can expectant for the SA market. Some of these commentators are predicting the SA Index to be less than 50 000 in 3 years. This entails a 5% - 10% drop from where it is today,\" says Streso.\nEric strongly recommends that investors seek the professional expertise of Financial Planners that have experience in working with credible offshore funds.\n\"Offshore investment is not a 'quick fix' to make money, one must expect to remain committed for an extended period to really make their investment work for them,\" he concludes.\nEric Streso holds 2 Law degrees, an international MBA in Financial Services, and is a Certified Financial Planner through the FPI.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you would like to share your talents with the Villanova Women's Professional Network or make suggestions for speakers, topics, workshops, events or additional programming, please contact us at wpn@villanova.edu.\nSubscribe to receive updates and our semi-annual newsletter.\nLooking for key information about the VWPN?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdklz b/data_all_eng_slimpj/shuffled/split2/finalzzzbdklz
new file mode 100644
index 0000000000000000000000000000000000000000..45acf7d345e47541d8614eea21c7cba95c61401d
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Trupanion works with thousands of hospitals and has dedicated teams throughout our business to help quickly assist your hospital \u2014 so you can get back to saving pets' lives.\nTo tailor your experience with us, we have more than 70 Territory Partners across North America who are pet insurance experts. They're on hand to support you and your staff with training, guidance, and any other pet insurance needs when you need it.\nDon't have a local expert?\nOur dedicated Veterinary Engagement team is here to provide the support you need from our home office in Seattle. Whether you need additional marketing materials, training for your staff, need help with a claim, or have questions about the Trupanion policy \u2014 they're here to help!\nGive us a call at 855.727.9085 or email [email protected].","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Go Dennis go! BTW, your turkey looks delicious no matter how it is carved. :-) Merry Christmas to you and your family.\nThats actually my Dad not Dennis. You think we'd let Dennis loose wth a petrol driver cutter???\nLol! Merry Christmas to you and yours, Helen!\nHis knife is certainly big enough.\nLooks like the turkey got a lot smaller in the second photo--ha ha!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you weigh more than 175 pounds or if your peritoneum filters wastes slowly, you may need a combination of CAPD and CCPD to get the right dialysis dose. For example, some people use a cycler at night but also perform one exchange during the day. Others do four exchanges during the day and use a minicycler to perform one or more exchanges during the night. You'll work with your health care team to determine the best schedule for you.\nYou may receive a kidney from a member of your family (living, related donor), from a person who has recently died (cadaveric donor), or sometimes from a spouse or a very close friend (living, unrelated donor). If you don't have a living donor, you're placed on a waiting list for a cadaveric kidney. The wait for a cadaveric donor kidney can be several years.\nBlood type. Your blood type (A, B, AB, or O) must match the donor's. This is the most important matching factor.\nHow long you'll have to wait for a kidney varies. Because there aren't enough cadaveric donors for every person who needs a transplant, you must be placed on a waiting list. However, if a voluntary donor gives you a kidney, the transplant can be scheduled as soon as you're both ready. Avoiding the long wait is a major advantage of living donation.\nBetween 85 and 90 percent of transplants from cadaveric donors are working 1 year after surgery. Transplants from living relatives often work better than transplants from cadaveric donors because they're usually a closer match.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Casserole makers. Card senders. Flower arrangers. Prayer warriors. When crisis strikes, we want to help. Deeply desiring to care for one another, we, oftentimes, come up short on appropriate words or actions. Having little idea of the person's needs, we become frozen in our ability to help in a way that is a blessing, not a burden, to us or to them.\nDeep down I believe we all have good intentions. We care. We empathize. We want to help. We just need to be taught how.\nMom was dying, and I was completely overwhelmed by the demands before me. Not having close family nearby, we were left to navigate this season on our own. Caring for our three young kids while being Mom's primary care giver, I needed help. Honestly, I couldn't survive without it. During this time of need, a core group of people stepped in to bless, serve, and encourage our family in ways I never could have imagined. Their selfless service to my family and Mom is a debt we can't repay. Developing a bond that drew us together, this community took our burdens as their own, helping us focus on the path ahead. Their service modeled the love and nearness of Christ in a dark and desolate season. Not everyone played the same role; not everyone could. Depending on the closeness of our relationship and their own time and circumstance constraints, people used their gifts to serve us in a way that was a blessing, not a burden, on either party. There were others in our community who didn't reach out at all. Deeply knowing and loving these people, I firmly believe is wasn't due to lack of compassion, rather an inability to assess the needs and know what role they could play.\nWhat is our role when we find someone in our community suffering? What is the appropriate way to reach out, to love, to care for them? Sarah Beckman has written a book that is a must read for anyone desiring to be a blessing during a difficult season.\nA #blessed life. At times, we've all wanted it, strived towards it, and coveted it in other people. Visions of beautiful kids, a Pinterest worthy house, a tropical vacation, and flawless skin consume our minds; not wanting to admit it, inwardly our flesh craves comfort, security, success, accolades, adventure, intrigue, beauty, and acknowledgement. A quick search of Instagram produced 62,858,002 posts tagged #blessed. Scrolling through the pages you see pretty people, pretty decorations, pretty achievements, and pretty lives.\nWhat if the #blessed life we've been chasing is backwards to the one we were created to live?\nThe #blessed life we put on social media tends to show what we have, what we think is good, and what makes us happy. Not always the intention of our heart, what we consider #blessed can paint a picture into what we value, cherish, or chase. Most of the time, in my own life at least, it has been mostly about ME.\nSimply put\u2026I think we are chasing the wrong thing.\nWill I say my life is #blessed when my mom dies of cancer, even if it pulls me closer to Him? Will I thank God for tight finances because it allows us to wait and see His provision? Will I have joy when I get diagnosed with an autoimmune disease, because it actually makes me dependent in a way I never knew possible? What about when I blow my knee, my car breaks down, my kid gets bullied, or my friend walks away? Where are these #blessed posts? Where are the proclamations of the hard getting re-purposed for good? We don't see it because it is a backwards theology, 100% counter cultural. It doesn't make sense to the world, and at times, hasn't made sense to me.\nJesus humbled Himself by coming to this earth to give us life \u2013 life for eternity with Him, but also life on this mess of an earth too. The full life He came to bring doesn't include a fancy new sofa or face wash that takes away my wrinkles. It isn't about my children's successes or my new dream job. It certainly isn't about my ambitions or achievements. Our #blessed life shouldn't leave people jealous, coveting, or envious. It shouldn't push friends away, cause division, or be self-seeking. The things that cause us to have a blessed life should bring people back to our relationship with our Savior. We are blessed because He redeems our pain. We are blessed because He strengthens our weak knees and feeble arms. We are blessed because He gives us courage to enter others pain. We are blessed because we have been granted the opportunity to believe in Him and suffer for Him. We are blessed because He is enough, and I am not.\nWhat about my life has produced eternal blessings? Things that matter? Things that last?\nHow can I use my earthly blessings to bring people back to the One who gave them to me in the first place?\nThis doesn't mean we can't buy nice face wash, and it certainly doesn't mean we can't pursue a dream or start a business. I don't think it means we should hide our pretty things and pretend like they don't excite us either. Let's not let this prevent us from stepping into a calling because of how it might be perceived, and, for heavens sake, please don't hide the pictures of your beautiful children. Some of these things just make the world more comfortable, and that is okay! Others of these things could be our God-ordained calling; if so, we have a duty to step into them with obedience. All of these things are gifts that can point back to our Creator and be used for His good purposes; so we can, and always should be, thankful, celebrating with one another as these blessings come.\nThe point is we have to stop CHASING after the #blessed life of this WORLD, pursuing it at all cost and for the end goal of having the life the world tells us to live.\nInstead, let us be a generation of people that seek God first, our motivation being love, our intentions being pure, and our celebration of what God is doing being genuine. Let our desire be first and only for Him and the things He treasures most; love for God and for other people.\nThe most #blessed men and women I know aren't the ones with the biggest paychecks, largest platforms, most followers on Instagram, or the nicest homes. They are the men and women I have seen firsthand suffer with grace. Enduring hardship as discipline, they have carried others burdens in light of their own. They have chosen to trust God when the world says to curse Him, knowing what it means to have fellowship with the Spirit. They have lived and loved well despite their own pain or limitations. Their faith is secure. Their hope is contagious. Their joy complete.\nTHAT is the kind of #blessed life I want. It costs something, most of all my pride, selfishness, and propensity to compare. But my new prayer for today is that I seek Him first, so that when other people look at my life, they see Jesus before they see me.\nWill you chase after this #blessed life with me?\nI had a breakdown outside of Safeway the day before Thanksgiving. Running to my car in the rain, I slammed the door behind me to endure my ugly cry in solitude. The trip should have been painless; I just needed to pick up a few last minute items. Yet, stopping by the florist department on my way to the checkout counter, set me heart into a tailspin. As I delicately placed a bouquet in my cart, the tears welled behind my eyes. Realization hit hard \u2013 while most of my friends were baking pies and cooking with their moms that afternoon, I was left to only deliver flowers to the cemetery. Holidays are hard. Triggers are everywhere.\nHow do we be thankful when life hurts so bad?\nStrong. Fortified. Impenetrable. Deep. The roots of our tallest oak tree spread under the surface of our yard like a maze of power. Unseen from the eye, they are hidden beneath layers of rock, soil, and mud. How vast they travel no one knows, until a bulge happens under the sidewalk or across the yard. When the root finally pushes through the surface, it is obvious and intrusive. No longer hidden from the world around it, the damage must be dealt with swiftly, to lessen the blow to the surrounding landscape or to a pedestrian passing by.\nThere are deep roots in my heart too.\nWounds happen, words sting, friendships fail, opportunities are lost, and we mess up from time to time. Living in a fallen world with broken people insures that. Not able to escape the ugliness of human nature, we grow weary from feeling slighted or forgotten. Consumed by continual disappointment, certain people reinforce long-formed hurts, growing deeper and more complex as time goes by. Yet sometimes we forget that we have done our fair share of hurting too. That other people might have been pushed down by our words or inclusiveness. Forgetting that our bitterness is hurting us more than them, our heart is weakened by holding on to the pain of past deed or present circumstance.\nMy bitter root of unforgiveness can't be ignored anymore.\nGrowing up my parents sent me and my sister on trips across the world. Starting at the tender age of eleven, I began participating in international peace camps through an organization called Children's International Summer Villages (CISV). My travels took me to far away places such as Brazil, Sweden, Belgium, and Slovakia. Living for a month at a time with children from around the globe, I fell in love with various cultures, people groups, and places. Beginning to see other cultures in light of my own, differences weren't obstacles to our friendships. Skin color didn't determine the depth of our relationship. Language barriers couldn't stop lifelong bonds from forming.\nExperiencing the world at a young age helped me to value all people. Most specifically those different from my mainstream, upper-middle class, white, American, Christian self.\nAuthor, Jaime Martin, has found a way.\nDing. My phone lit up at 5:45am this morning. One of my favorite people in the world was in a rage. Literal, chemically induced, uncontrollable rage. Anger, aggression flowed out of her in rapid torrents. Out of control, she cried out to God, \"Why did you allow this pain in my life!\" Flare gun prayer shot straight to heaven, she yearned to hear an answer.\nWhile the world raged on.\nWhy does this world hurt so bad? Why are we exposed to pain and disappointment? Who is in control of this chaotic, pressure-filled globe we call Earth? Will there ever be relief from our suffering?\nAs we approach Christmas this year, the air is filled with a different kind of wonder. Yes, the Christmas lights are still twinkling and pretty, the hot chocolate is still warm and welcoming, and the entertaining of friends and family brings joy and contentment. Even the thought of presents to unwrap brings a smile to our faces. However, our perspective of Christmas has changed because our year has required our view of our Savior to change. For years we have loved Him, have always tried to celebrate His birth at this time of year, and have tried to remember the true reason for the season. It isn't all about Santa Clause after all, is it?\nThis year Christmas looks different because Jesus looks different.\nOver the past year, Jesus has showed up in the midst of the darkest storm of my life in a way I have never experienced before. He stretched His arms out wide and brought us deep comfort, peace and dare I say, joy, in the midst of our hardest trial and deepest pain. He brought endurance to face difficult days and rest despite sleepless nights. He brought acceptance and hope. He brought people who loved us well. He brought wonder at His character. He brought a real, living Jesus, a full-grown, perfect Savior. He brought Himself as the spotless Lamb of God who actively intervened in our world and in our story of grief and our hope of redemption.\nHe isn't just a cute baby in a wooden manger to us anymore.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LICHTROUTEN Project: Flowers of Change.\nThe project was born in 2012 for the Nowhere Festival in the desert of Saragosse (the european Burning man). The artwork Flowers of Change is in adequation with the social values such as cooperation , inclusion, autonomy, participation, self expression, leave no trace\u2026 Pierre Est\u00e8ve opens here a space-time that allows us to assimilate differently the habitual landscape.It shows a different way of thinking, a universe in which you need to take your time, of reverie, calm and silence.\nThrough Flowers of change, what is interesting for Ludenscheid, is to get an artwork that may be different from what is usually used in light artwoks, namely to work with faint and delicate LED lights pulsing color variations in the dark, to immerse the spectator in the heart of this ecosystem, by putting him in an environment in which he does not see well, must pay attention to the flowers that are in perpetual change. I seek the visitor in a different way than in what it is done in light and in music. Therefore, I composed a complete sound ecosystem of the Flowers of Change, all interpreted on plastic elements and sound sculptures, working in conjuction with the Flowers field. The visitor is indeed, plunged into a complete universe in which we enter as in a virtual world.\n1980 | Avignon and Montpellier [fr], Graduate in Music Academies in Harmony, Fugue and Musical Analysis.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbevac b/data_all_eng_slimpj/shuffled/split2/finalzzzbevac
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@@ -0,0 +1,5 @@
+{"text":"COLUMBUS, Ohio (AP) \u2014 Arike Ogunbowale put an exclamation point on a thrilling women's college basketball season.\nHer off-balance 3-pointer in the title game capped a record comeback by Notre Dame and an incredible weekend for the sport. The Final Four got started with two stirring overtime games \u2014 the first time that's happened. The weekend ended with a historic championship game that left the sellout crowd amazed.\nThat was the only other time McGraw and the Fighting Irish cut down the nets. It's been even longer since a title game came down to a last-second shot. Ogunbowale's 3-pointer was just the second last-second shot that won a championship game. North Carolina great Charlotte Smith made a 3-pointer at the buzzer in 1994 that lifted the Tar Heels to an improbable one-point win over Louisiana Tech.\nThe junior guard floated in a 3-pointer from the corner with 0.1 seconds left, lifting Notre Dame to its second women's basketball title with a thrilling comeback victory over Mississippi State.\nThe huge crowds weren't just at the Final Four this weekend. Indiana won the WNIT championship Saturday in front of a record-setting crowd of 13,007 in Bloomington, Indiana.\nOgunbowale's shot capped the biggest comeback in NCAA championship history. It also forever linked her with Smith, who now coaches Elon and was at the game Sunday night. She posed for a picture with the Irish's star guard an hour after the buzzer-beater.\n\"It's Easter Sunday, and all the Catholics were praying for us,\" said McGraw, who was wearing floral shoes in honor of the holiday.\nWhen the final buzzer sounded, a wild celebration started with the Irish faithful who were part of the sellout crowd. McGraw's team had fallen short four times in seven years in the title game, losing in 2011, 2012, 2014 and 2015. Many of the Notre Dame alumnae who were on teams that had fallen just short of winning a title were there to enjoy it, including Natalie Achonwa, Natalie Novosel and Becca Bruszewki.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My kids have been doing a lot of journaling lately either using these math-inspired journal writing prompts or with this fine motor sticker method that I'm going to show you. The sticker method of journaling works great for both preschoolers and kindergarten aged kids who are learning to write, not only letters and words, but complete sentences. It is also a great method to try with kids who have hyperlexia as it gives them something concrete to write about.\nMy son struggles to describe and answer questions verbally due to his hyperlexia, especially the \"wh\" questions. So phrasing questions for him to respond to in a journal won't work great either, unless, of course, it is tailored to his interests like my math journal prompts. But we want (and need) him to start expressing himself about other things. And besides, his ability to write and spell is an asset that we need to tap into while his speech continues to develop. So our starting point is to give him a visual stimulus to write about. In this case, stickers.\nWe have been using stickers from this Melissa & Doug Sticker Collection for this sticker journal writing. Both boys received their own sticker pad for Christmas from their Mee-Maw (aka grandma). Peeling stickers is a great way to practice fine motor skills! So the boys simply pick a few stickers and then write a story or words to describe what they see.\nThe following two stories below are also by five year old J.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What Weighs Nothing but Carries Great Weight?\nEach quarter we intend to take a closer look into the Windows Into Our World, the world of our scholars, teachers, parents and guardians. We want to see the progress of our scholars, of our teachers' training, and our livelihood programs to measure the impact of The Foundation For Tomorrow (TFFT). Are our scholars engaged in school? Is our teachers' training program providing impactful teaching tools? Are our livelihood programs enriching our scholars' lives?\nWhile TFFT has had many great accomplishments in just a few short months, we are zeroing in on something that many of us take for granted: a birth certificate. This document weighs less than a quarter of an ounce, but carries a tremendous amount of weight in a person's life. Any person's life, anywhere in the world.\nThe Foundation For Tomorrow recognizes the immense importance of this document and has launched a 2019 initiative to obtain birth certificates for all current scholars. Tanzania has one of the lowest rates of birth registration in Africa, with less than 13% of children under five having a birth certificate (UNICEF report November 30, 2016). A birth certificate in Tanzania, as in almost every other country, provides identity, access to education, social services, and social status that cannot be disregarded.\nOf TFFT's 105 current scholars, only 20 scholars had birth certificates when they enrolled in the program. Two birth certificates were processed last year for scholars who applied for A-level programs. This year, TFFT is targeting birth registration for 73 scholars. In the first quarter of this year, the TFFT team pursued birth certificates for 33 scholars \u2013 ages ranging from 9 to 19. The team plans to obtain certificates for the remaining scholars in the second quarter of 2019.\nWhile not insurmountable, the process to register children five, ten, or fifteen years after their birth is cumbersome. The application process involves completing forms which require the child's full name, the parents' names, the date of birth, the marital status of the parents, where the child currently resides, if it was a single or multiple birth, and location or name of the hospital where the child was born. In all, a great deal of information to collect for a single child, years after their birth, let alone 33 children. In all, this process took the TFFT team about three weeks, requiring them to reach out to parents and guardians. Because many scholars do not live with their parents, the team needed to seek information from foster parents, grandparents, aunties (matrons of children's homes) and other relatives who had some information about the scholar's births. Passport-sized pictures of the child and payment for the application, including a late registration fee, were also be submitted.\nIn addition to collecting vital family information, several factors have been hurdles for families to obtain birth certificates in Tanzania including: cost \u2013 TSH 3500 ($1.60USD), which increases after 90 days and again after 10 years; distance \u2013 this process has been a centralized registration system, but is slowly moving toward a local and\/or mobile registration system; awareness \u2013 families have not understood the importance of registering for a birth certificate.\nWhile children may attend primary school without documentation, the requirement of a birth certificate becomes more important later in life. Applications for further studies, seeking a passport, or requesting vital social services all require birth registration\/documentation. Providing our scholars with a birth certificate places official documentation in their hands as a young person, but that can carry them through their lifetime by establishing their citizenship and enabling our scholars to employ government services they may require in the future.\nRegistration rates are still low and the challenges are great, but international efforts are underway to ameliorate this situation. A UNICEF program initiated in 2015 seeks to address the financial, cultural, and logistical barriers to birth registration in Tanzania. This program, \"Scaling Up Birth Registration in Tanzania,\" will run through the end of 2019 and intends to streamline a three step-logistically challenging and centralized process to a one step, localized process. Read more here.\nWorking alongside this UN effort, are technological innovations to make registration easy and accessible for all. Modelling the mobile payment system, which has been highly successful throughout Africa and the developing world, companies have created SMS-based applications providing mobile birth registration systems. Read more here and here.\nTFFT is excited to present our scholars with their new birth certificates. Providing this document to our scholars is more than a piece of paper. It is a key that can unlock the doors to further education, obtaining a passport, health and social services, and most of all, a stake in their country. Through the power of a birth certificate, TFFT's impact will last a lifetime.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Help the bunny match up the same colored eggs in this fun Easter themed bubble shooter game. Shoot the eggs and remove all of the eggs from play to move to the next level.\nClick to Play Bubble shooter for Free and Have Fun!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pruning is a source of confusion and worry to many gardeners. How and when to prune are two of the most common questions asked of gardening experts. Luckily a few simple guidelines provide good advice for growing most types of flowering shrubs. Learn the general rules of thumb for pruning flowering shrubs.\nMany gardeners labor under the belief that they must prune regularly to keep their shrubs in good condition. Not true. Most shrubs need only one significant pruning session a year and many don't even need that. Many flowering shrubs will look their best when allowed to grow in their natural form or habit. Frequent shearing encourages lots of surface branching, possibly resulting in an unhealthy structure and reduced flowering. If you really want a tightly sheared look in your garden, choose a plant that is suited to it, such as.\nPlants that bloom in early spring usually produce their flower buds the year before. The buds over-winter on the previous year's growth and open in spring. If you prune these spring bloomers in autumn or winter you'll remove the flower buds and won't have flowers that year. The plants will be ok, but you'll miss a year of blooms. Most of these plants don't need heavy pruning every year, just some selective thinning of branches to give them a nice shape.\nMany summer flowering shrubs bloom on the current year's growth. Pruning them back in later winter encourages them to produce lots of new growth that summer and will result in more flowers. Don't be afraid to cut fast growing plants, such as Grevillea down to as little as 1 meter tall.\nIn summary, relax. Your landscape plants don't need as much pruning as you may think. If you'd rather go to the beach than shear back your landscape plants, go right ahead. The only potentially tricky part of pruning is determining when to trim a particular plant. For a quick review, prune summer bloomers in late winter and spring bloomers right after flowering. Stray or broken branches can be trimmed back any time. If you do make a mistake, plants are very forgiving. You may miss a season of flowers but the plant will recover for the next year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..3e9b037f0efaac1a8037ab5dd6d633a6406b5967
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@@ -0,0 +1,5 @@
+{"text":"A simple calculation can estimate a minimum expected return for an investment.\nThe required rate of return is a subtle concept that involves the opportunity cost of investing. It is the return expected of other investments with the same risk. The required rate of return is the minimum rate an investment must yield to be competitive with other investments on the market. If an investment's expected returns are lower than other equally risky investments, investors will fund more attractive investments instead.\nThe capital asset pricing model (CAPM) provides a simple way to calculate the required return of an investment based on its market risk, the risk premiums priced into current markets and the risk-free rate of return. Each of these values can be estimated and combined using addition and multiplication.\nU.S. government treasury bonds are used to estimate the risk-free rate. Determine how long you expect to hold your investment, and find a fixed-rate treasury bond that matures in the same time frame. Use the yield of this treasury as an estimate for the risk-free rate of return.\nDifferent investments differ in their risk. The required rate of return reflects the risk that cannot be diversified away in a portfolio. This kind of risk is called market risk, and it is the co-movement of the stock with changes in the securities market. It is estimated by a value called beta, which can be found for stocks on finance Web portals like Yahoo! Finance. Higher beta values imply greater market risk. Beta values can be assigned to investments that are not publicly traded by using beta values from related stocks.\nThe market risk premium is the expected return of risky investments in excess of the risk-free rate. Historical values are calculated from past stock returns and forward-looking values are calculated from stock market earnings projections and current stock prices. Both historical and forward-looking estimates are frequently calculated and recent estimates can be found by searching for \"market risk premium\" online.\nCalculate the required return of an investment by multiplying its beta and the market risk premium and then adding the risk-free rate. This is the required return of an investment that is part of a liquid, well-diversified investment portfolio. Investing outside of public markets should have higher returns than these calculated values if they are not as liquid as stock market investments or if they are large components of an investor's portfolio.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you're a School of Computing student, you've probably been told that the best way to get good at programming is to 'just keep practicing'.\nAnd this is true. We know this is true because musicians, painters and writers say the same thing; why should it be any different for programmers? Programmers get better when they program \u2014 with one unique difference.\nThe difference is this: code projects, the by-product of 'programming practice', are great additions to any resume. Employers will be able to read the code you've written. If they care about the kind of software you will produce for them, knowing how well you code matters.\nHaving emerged from engineering and mathematics, computer science programs take a resume-based approach to hiring off their graduates. A resume says nothing of a programmer's ability. Every computer science major should build a portfolio. A portfolio could be as simple as a personal blog, with a post for each project or accomplishment. A better portfolio would include per-project pages, and publicly browsable code (hosted perhaps on github or Google code). Contributions to open source should be linked and documented. A code portfolio allows employers to directly judge ability. GPAs and resumes do not.\nBuilding a portfolio is something we should all be working on. (Matt Might links to MIT student Ed Yang, whose personal portfolio is staggering in its impressiveness).\nSo the question becomes: what should I practice on? Programming assignments are tedious when done outside class, for no credit. There are only so many Project Euler problems one can solve before getting bored.\nThe answer, of course, is to start hacking.\nThere are a number of reasons to hack, all of them good ones. I think it would do to share some of them here for posterity. I'll end with a number of things you can start hacking on, and one place, next week, where you can do it with your friends.\nSoC programming assignments rarely cover cutting-edge technology. This is not the school's fault: it's too much of a hassle to keep up with the bleeding edge. The school's job is to teach underlying ideas. Imagine updating your lecture slides every time Node.js changes. That would certainly get in the way of whatever concept you were trying to teach.\nNo, professors and employers expect us to pick up production technologies on our own. They teach us the ideas \u2014 we're supposed to use those ideas to learn whatever it is we need for our jobs. Sometimes that means learning Python over the weekend. Other times it means learning enough C to debug the bloody PDF library that seems to print a puppy over every document you try to generate.\nMost importantly, however, it means that the student who spends his weekends and holidays picking up production-ready technologies is a step ahead when compared to his peers.\nWe know that practice is good for programming skills. What that looks like in practice (pun intended, forgive me) is that things explode in your face, and you scramble to fix those things, and then you learn new things in the process.\nAfter which you become slightly better.\nHacking forces you into situations where you don't know enough to begin building things. If you want to build a blog on top of MongoDB, for instance, you might begin by learning a web framework. After which you would need to learn to use Linux, because 99% of websites run on Linux. Learning Linux will enable you to install MongoDB and set up your web server. Then you'll quickly realize that you'll need to learn how to use a wrapper for MongoDB, which in turn will teach you how to find and read obscure documentation for a programming project.\nEvery trivial hack project brings with it a hidden cascade of things to learn. This is a good thing: it means you get to learn more the more hack projects you do.\nOne of the quickest ways to learn about the software world is to contribute to an open source project. Many coreteam members contribute to existing OSS codebases, one even implemented a new diff algorithm for Git. SoC provides students with a good Firefox course with which to learn about Open Source. Both approaches to OSS will eventually require us, as students, to meet with and talk to working, breathing developers.\nBut hacking on your own projects are also a good way to meet other developers. If you're solving someone else's problem, he or she may find you on GitHub to contribute to your code.\nAgain, this teaches you things you might otherwise have to wait for graduation to learn.\nShowing off matters in both sense of the term: hacking projects make you more attractive to future employers; and showing off to your friends keeps you motivated to continue hacking on things.\nThe latter is sometimes more important than you might think: many a programmer has built a world-changing tool with the partial intention of showing it off to his friends. Facebook and Linux certainly started with some element of this.\nHacking is good for you; start hacking today!\nProgramming for personal projects makes you a better programmer, helps you score girls gives you opportunities, and makes you more attractive to employers.\nIt's not hard to start hacking - pick a problem you've always had, and think about how you might solve it with code. Alternatively, pick a cool new technology you want to learn, and find a way to use that to solve your burning problem.\nIf you'd like to get a head-start, come join us at our Hack & Roll Hackathon, on the 19-20th February. Sign up here, or read more about the event at this link. We're trying to get more SoC students to hack on stuff, so if you'd like to learn how to start doing so, in the company of your peers, come join us. It's only $10 per person, and even then for food.\nIn the words of a much-loved SoC lecturer: happy hacking!\nPS: if you like this post, please consider sharing it with a friend!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Symphony vessel sink is available in Nero Marquina, Carrara and Estremoz marble creating a genuine style and ensuring mystery and elegance in any luxury bathroom. This gear-wheel-shaped sink stands out for its mechanic look and contemporary design. Available in three different marbles you are able to take your creations to a completely new level combining it with endless colors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Finally, the little bastards are fading out. The past week or so has been absolutely miserable. So bad that we couldn't even venture out to the pool. Last time we were in the pool, we were getting attacked first by doctor flies, then by mosquitoes.\nWhat was weird was most of the springtime, the 'squitos were small and their bites only itched for a few minutes, with welts not showing at all. The past week, we've had some veritable monsters. They hurt when they bite, they're persistent, and the welts - wow!\nSo, during the period commonly known as walkies this morning, we were hardly bothered at all by the pesky little critters.\nOh, I forgot to mention, that for the past two or three nights, the girls have been crashing in our pad, that is, sleeping in our living room. The height of luxury for them. Unfortunately for them, that all ended last night. it was their first night outside in a while. Surprisingly, they were quiet all night too. And no porcupines either!\nWhen we've gotten ready for walkies the past few days, we've not only slathered bug juice on ourselves, but we've globbed it on the girls. They seem to think that's really cool. I suspect they think that the pack leader (me) is licking them in some strange way to make all of us smell the same way. Anyway, it's kept the little buggers away from the girls, so walkies is much more enjoyable for all concerned.\nAnd, all morning, it seemed that there were fewer and fewer 'squitos around. So much so, that at our appointed hour (roughly 2:30 PM) - pool time, we had no qualms about traipsing down to the pool in our swim suits, with nary a bite between us. After about 3 hours of pooling it, oh, and downing the required number of beers as well, all felt well. The planets, were, I'm sure, aligned properly and we felt as though life had returned to normal. And, with no major tropical weather things scheduled for the next 48 hours, life is good!\nNormally, GOB (Government of Belize), well, specifically, some ministry or other, mostly the Health Ministry, has been fairly diligent about spraying Malathione throughout the area to zap the 'squitos. Rumors have flown than government had no money to continue abatement spraying and it shows. If that rumor is true, someone in the Health Ministry ought to at least be demoted if not outright fired. This is such a big social issue, considering maleria on down to just the inconvenience of mosquitoes, that to run out of money for spraying should at a minimum, be reported to the World Health Organization (WHO). Or, maybe the spray truck broke. Who knows?\nI know one thing... we tried to see Cerros Hills back at the turn of the year. No bugs anywhere.. but when we stopped at the parking lot at Cerros and started walking up the path I looked down at my arm and it was black with them. We turned around and ran out of there. Now I have another reason to come back -- so I can see Cerros. But those bugs were there around late December!!\nWe had some friends about a month ago who went to Cerros and had a bug-free experience. Probably the one day a year that's allowed.\nI can sympathize with you on skeeteres - they have been real bad here in San pedro too. The other night I had to spray up to hang laundry and they were still on the attack.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So, the week began with a visit from my friend who is from Iran and studying over here for a few years. We took her to an Indian restaurant. The waiter asked if she was Indian. I am guessing, from her reaction, that for an Iranian that is not a compliment.\nThe boys then thought it something of a challenge to encourage her to drink shots with them. Hmm, can always rely on them to not let me down when I'm trying to make a good impression.\nWe also had a \"men's social\/bloke's night\" at our house. I do think to be male and attend church you need to be man enough to cope with some pretty naff names. They had a croquet tournament. For the uninitiated, this might seem like a nice polite game. It is not. I promised myself I would never have to play it again when I was dating my husband and we played with his family, who all thought it hilarious to send my ball spinning off into a garden several houses away while they all had a nice social time and finished together. It is a nasty, competitive mean game. My family are all very good at it. Enough said.\nThe winner of this particular event, which we host every year, gets a trophy to sit on their mantelpiece for the year. You might think this would be a small cup or a replica of croquet mallets, but no. It is a very large and ugly china hand which was originally used as a mould for rubber gloves. Husband found it in a junk shop when in New York. I am not sure if the men actually try to win it or not. I suspect that they cannot help being competitive. You will be surprised at how often the wife of the winner sidles up to me afterwards in church and confesses that they really do not have room for it in their house, would I mind looking after it for the rest of the year.\nI bought a hat for nephew's wedding. I know you will be pleased, knowing just how much I hate shopping. Fifteen minutes, half price, John Lewis. AND it came with a box. A triumph. It is slightly big. If it slips down I will have to peer at bride and groom from under the rim. But was too good a bargain to miss.\nThen shopped for boy clothes with boys. This was painful. I suggested to boy with large toe protruding from canvas shoe that perhaps shoes would be a good item for list.\nWe went to the same shop he went to last time and tried on exactly the same pair of shoes in exactly the same size. He informed me they were fine, just a little tight over one toe, but as they were the same shoe\/size as last time, this was not a problem. I was almost fooled.\nThen I realised that the toe it was 'a bit tight over' was the same toe that protruded from the old shoes.\nSuggested a bigger size might be a good idea. Was told this would not work as other foot was a lot smaller.\nJourney home uneventful. Boys sat in back and debated intensely all the way home. Sentences such as, \"what if a man kills a dog and then just keeps killing dogs for fun\" and, \"well no, I think all animals are equal so perhaps murder is not so wrong. I could probably eat a human,\" floated to the front. I was going to ask what they wanted for dinner. Decided not to.\nPS, We collected son's duvet from Morrison's washing service (the one with a hole in, remember?) All was good \u2013 we were not charged for any breakage due to feathers and he can still use the duvet. As a pillow perhaps.\n'Boys' and 'shopping' in the same sentence just has to make for an interesting letter\u2026\u2026..!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"There is just so much to love about this wedding! Lauren + Dylan celebrated their special day under an open marquee at Sarabah Estate, with the most picturesque vineyard backdrop located in the Gold Coast Hinterland. A simple outdoor ceremony was created using our french bistro chairs dressed with trailing floral arrangements, followed by a soft and romantic reception setting using our wooden dining tables and white bentwood chairs \u2013 a match made in heaven!\nCaptured beautifully by Heart and Colour with an A-team of suppliers, check out the imagery below + our very own style guide to help create this look using our range.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cocky South Carolina fans: don't sleep on this must-have game day accessory! Featuring insulating neoprene construction and a home team inspired design, this can koozie is sure to make you look cool while your drink stays cool!\nDo you love your South Carolina home team? Then you'll love this cute and functional collegiate inspired koozie. Featuring your favorite SC alma mater's colors and a custom message of your choice, you can enjoy game days, tailgating and more while your drink stays ice cold.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Greetings. If you would like to be totally inspired about the real power of music, and the real potential of life, work, and learning, then commit to spending fifty minutes listening to the BBC's recent interview with Chinese pianist Lang Lang. But you will have to hurry because it is only up on their website for the next 26 days.\nBy way of background, Lang Lang was a child prodigy and is now one of the world's foremost and most dynamic classical pianists. But he is also, at the age of 33, a renowned teacher, United Nations cultural ambassador, and remarkable voice for the power of learning across cultures, generations, and genres. And his thoughts about how we continue to stretch, grow, and stay fresh, focused on music but with much broader implications, are worth our time and attention.\nIn listening to the interview I was struck by his passion for a wide range of musical traditions, his sense of why so many young people never get past the early stages of learning (an instrument like piano), his thoughts on why being perfect is overrated, why it is important to \"look for the notes between the notes,\" and his belief that the greatest composers throughout history would probably delight in the knowledge that future generations were passionate about their music but also willing to try to reinterpret it. And the more I listened to his words and his playing, the more I felt the value of his insights and their broader application to life, business, and innovation.\nAll of us and all of our organizations need to dare to try new things, figure out how to not become discouraged when the going gets tough, find joy in the work we do, and build on the ideas and brilliance of others.\n\"In music we need to always remind ourself why you play the piece over and over and over it again. You forget about the freshness. You really forget why we are loving music so much. You know. Because you repeated the same thing so much.\nAnd if you would like to see and hear one way to avoid 'whatever,' check out Lang Lang's collaboration with Metallica at the 56th Grammy Awards.\nWe win in business and in life when we approach the things that matter with different eyes and from different angles. And when we are open to learning from others and from different walks of life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Campagnolo Chorus 11 Speed Group: It's hard to resist the allure of carbon fiber. That is why Campagnolo has designed the all new Chorus groupset, making it more aggressive and with even better performance, reaching top-of-the-line levels. With its attractive looks and new logo, the new Chorus groupset will uniquely make you stand out from the pack and will always be at your side in every challenge, be it against others or against yourself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"They also expressed an increased satisfaction with their quality of life. The system integrates hardware, software, media, ergonomic and adaptive components to allow anyone, regardless of previous computer experience or cognitive ability, to engage in learning or just have fun. We also provide on-site staff training and support to ensure successful implementation. Lahko pa jih shranite tudi v zmrzovalniku za hude \u010dase. Nata\u0161a Sem v prvi vrsti mamica, prvi\u010d mamica pri 30tih in drugi\u010d pri 40tih. The three most important actions needed for growth are; 1 an understanding within the nursing home industry that keeping residents connected through adaptive computer technology is a right, not a privilege, 2 having funding sources foundations, etc recognize that mental fitness is a critical positive component of aging, 3 continued technology trends of lower prices in touch screen, hardware, and continual sources of new content. Also, by partnering with these organizations we are receiving more exposure to the communities who will benefit from having an It's Never 2 Late computer system.\nAnother obstacle we face is educating people on the ability of the elderly to use the technology. Feel free to contact our organization if you have any specific questions as you set up your own program, or if you would like to know of communities in your area that have technology available. There are several reasons for this shift. We have hundreds of examples and stories from individuals whose lives have been changed through the introduction of technology. Other content includes sensory, reminiscence, brain fitness, physical fitness, and music specifically designed to engage seniors with very little prior experience with technology.\nFamilies are spread out all over the country, and the ability to stay in touch with Mom, Dad, Grandma or Grandma via email is getting stronger and stronger with each passing year. We can be reached at 303-806-0797 E-mailed directly at or Contact us through our website at. Dennmark 3 little words 4 winds blow Responsibility: Rock Nalle Inc. Our intuitive design incorporates numerous user-friendly technology components enabling everyone in your care center; residents, staff and family \u2014 regardless of experience, physical or intellectual abilities \u2014 to use the computer pleasurably. The respect and support we receive from these organizations is also crucial to our success. The goal of It's Never 2 Late is to improve the quality of life for the elderly in residential living communities and their families and caregivers by giving them the opportunity to interact with the world and their community through technology. The flight and driving simulator has become a very important empowerment tool.\nAbout You It's Never 2 Late has created customized, state of the art, adaptive computer systems for nursing homes, assited living, and independent living communities. We have heard from people all over the country sharing stories about how the quality of life has improved for their family member or loved one. Forward looking senior living communities are realizing that having access to computer technology for their residents is no longer a luxury, its becoming more and more of an expectation. Our own preconceived notions of what the elders can do, or not do, is torn apart on a daily basis. By introducing aging adults to the world of assistive technology, It's Never 2 Late has provided the opportunity to experience and learn new life skills that enhance the quality of their lives. Many of our potential users look to these innovators for guidance in the best practices for caring for the elderly. Content is continually updated to provide engagement experiences for residents and staff.\nWe're happy to answer specific questions you may have, and we can also point you to local assistive technology experts throughout the country that can provide one on one expertise. Bring the world into your senior living community today! Vem, vem, \u2026 teh receptov za pustne mi\u0161ke kar mrgoli naokoli, ampak jaz \u017ee nekaj let prisegam na tale, malo po moje, obdelan recept. If the individual motivation is there to learn, and the proper adaptive tools and training are provided, there are no limits to what can be accomplished. Younger family members have been more inclined to visit when a familiar computer game is used as a tool to encourage interaction and conversation. Leslie Sweeney is co-founder of It's Never 2 Late. The Story It's Never 2 Late started in 1998 with a donated computer and volunteer time spent introducing residents of an assisted living facility to the use of computer games and the internet.\nThe E-mail message field is required. Getting 5 year old donated computers and sticking them in a secluded corner will not lead to successful programs. However, adaptive technologies can allow individuals with sight issues, tremor issues, arthritic issues, and cognitive issues to still be able to take advantage of technology and enhance the quality of their life. This person-centered approach evolved to meet the complex and unique needs of individuals using effective and structured technique. \u010ce bi bila mamica drugi\u010d \u017ee prej, bi bila sedaj najbr\u017e tudi tretji\u010d :. The program was so well accepted, that after about a year, we began to see the potential for expansion. So hitro narejene, okusne, doma vedno najdete vse sestavine, dobre so \u0161e naslednji dan seveda, \u010de ostanejo \u2026 poleg tega jih mi redno pripravljamo za ve\u010derjo ne samo za Pusta Hrusta.\nThe adaptive computer systems are portable and include flat panel touch screen monitors with extendable arms, wheel chair adjustable work stations, and peripherals for a variety of activities. . The quality of their lives are enhanced by creative uses of adaptive computer technology which is allowing them to stay connected with their families and friends in a way unimaginable a few short years ago. Accessing e-mail and web cams to connect with family and friends, participating in mind-stimulating activities, and improving hand-eye coordination, older adults \u2014 even those who have never used a computer before \u2014 engage in activities adapted especially for them. Jack York is co-founder and president of It's Never 2 Late. Included with the software is a picture-based interface designed to launch from a touch screen.\nA simple press of a button can launch hundreds of applications for recreation, social connection, memory care engagement and billable therapy. Rock Nalle, Michael Christiansen, Kim J\u00f8rgensen, Tony Fischer, Kenneth Larsen med Tommy Hansen, Ole Fenger, Hans Mydtskov, Rasmus Stenholm. We are constantly updating and improving our content and services based on feedback from experts, users, and families. One of the steps we are taking to insure our success continues is our ongoing communication with experts in the field of aging and with the communities using the systems. Physical and occupational therapists have used the simple games and puzzles to engage residents for longer periods of time leading to recorded improvements in physical and mental health.\nResidents use a touch screen to access engaging, entertaining, educational, spiritual, and personalized content that is appropriate to their level of cognitive, physical and technical ability. This interaction and empowerment has a positive effect on their mental health and well-being. Our idea is unique in that the systems are a complete package, addressing the physical, cognitive, and emotional health of the user. Using familiar games projected on a big screen, staff members have been able to get residents to leave their rooms and spend more time involved with group activites. Our ultimate goal would be that every elderly person living in a residential community has the opportunity to continue to enjoy their interests and expand their horizons through the use of technology, thus avoiding depression and feelings of uselessness. Teepa and her company strive to grow appreciation of differences that will lead to better care and support of those living with changing abilities.\nWe have set up programs in nursing homes, assisted living communities, independent living communities, and adult day programs. Right now the economy is a big problem. Some feel the the elderly are not capable and are perfectly happy with Bingo. Many older adults have not been exposed to computers, and their first experience needs to include plenty of handholding and support from individuals who have a lot of patience. Touch screen monitors, trackball mouses, adaptive keyboards, and voice recognition software are just a few examples of the technologies that are available for the computer experience. The software can be personalized for every user regardless of physical or cognitive limitations. We are making the technology more affordable and are researching ways for communities to receive funding.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Risk money refers to funds placed in investments that seek high-return, although high-risk, investments.\nI will risk drinking tea in Mrs Ewelina's house. Maybe I'll survive.\nSome people's fear of flying is far out of proportion to the actual risk it involves.\nDo you think she would risk her life... her son's life... if she wasn't certain her husband was murdered?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The West is my inspiration. . .every cloud, every critter, every colorful character.\nold and new, brought to life through paintings and drawings by award winning artist Randy Galloway. The style can be referred to as \"romantic realism.\" Masterfully rendered, insightful, intriguing, clever, detail rich, colorful moments in time that tell a story. Galloway also specializes in portraits of interesting characters of the American West.\nto create that scene with the hope of capturing the light, drama, atmosphere, reality, and emotion of that special moment.\"\nof art to fit your unique home or office space by visiting the commissions page.\nTo Walk in Beauty (commission) Oil on canvas, 40\" x 26\"\nThe crackling, twisted flames of a roaring camp fire are one of the most enduring and romantic images of the West. Gazing into the ever-changing flickers of a glowing fire, your imagination can wander and you can feel the freedom of the wide open spaces that was so alluring to the daring and independent early explorers and settlers. Fire made life possible crossing the Great Plains and Rocky Mountains. Fire was life, fire was beauty, fire was passion.\nFire is so often related to passion and my passion is art. Twisted Fire comes from the many directions art has taken my life and now I am creating art that is close to my heart. . .my passion.\nof all that it took to get to this point.\nMy studio is in an earthy territorial-style home in the peaceful, magnificent foothills of the high Sonoran Desert in Cave Creek, Arizona. The growing artists community and an \"Old West\" character to the area make it a wonderful place to live and create art. Feel free to give me a call and stop by the studio to check out the newest paintings and meet my wild-west guard dog Chihuahua, Poquita.\nAn exciting time to visit the studio is during the local artists studio tour. Each year for six days total, the two weekends before and after Thanksgiving, there is an art studio tour aptly named Hidden In The Hills with 45 to 50 art studios sprinkled throughout the desert. Twisted Fire Studio is part of the studio tour and it is a great time to come by to view works in progress, buy new works, and talk about a custom commission for your home, office, or cabin in the woods.\nTwisted Fire Studio, 4774 E. Quailbrush Rd.\nContent copyright . Randy Galloway. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In 1909, her inaugural season, the ocean liner Mauretania set the fastest time for the Atlantic crossing, a record she would hold for 20 years. Embodying the golden age of transatlantic travel, Jordan's biography of a much-loved ship encapsulates the turbulent decades of the early 20th century.\nIn 1928, after 38 years of decorated service in the Royal Navy, Vice Admiral Dewar found himself at the centre of a mutiny. Although vindicated at the subsequent court-martial, he would retire a year later and his memoirs are a vitriolic indictment of the navy's practices.\nA true classic of travel literature, Voyage chronicles Lord and Lady Brassey's circumnavigation of the globe in their yacht, Sunbeam. Written at the end of the 19th century, it is an engaging vignette of a world, and its exotic cultures, on the verge.\nA far cry from the swashbuckling adventurers of Treasure Island, The Ebb-Tide is a contemporary story of moral depravity at the edge of civilisation. Written in collaboration with Stevenson's stepson, four unscrupulous crew members in charge of a stolen, smallpox-infected ship ostracise themselves from society and drift ever closer to insanity.\nSince it was published in 1955, this essential history of the sinking of the Titanic has never been out of print. Drawing from firsthand interviews with survivors of the disaster, as well as their books, memoirs and articles, Lord knits together a compelling account of the infamous maritime tragedy.\nThe first of the Famous Five adventures sees the gang unravel a thrilling mystery \u2013 a ghostly shipwreck, a treasure map and shady anonymous antique dealers. A wonderfully escapist tale, the book was published in 1942, echoing the war-time experiences of evacuation and rationing in a safe, ordered and thoroughly English world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Those who raise concerns and whistleblow are commonly discriminated against. The majority of our members have raised concerns at their organisations and those organisations have rewarded them by referring them to the NMC!!! The NMC needs to stop this in its tracks and make employers be accountable for malicious referrals. The impact on the registrant is huge \u2013 mentally, professionally and financially. The impact to our patients is dangerous!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wind energy is plentiful, renewable, widely distributed, clean energy source which can serve as an alternative to fossil fuel-generated electricity.Wind energy is now the world's fastest growing energy source and has also become one of the most rapidly expanding industries. Wind turbines are used to generate electrical energy from the kinetic energy of the wind. Wind turbines are usually installed at rural areas of the world with bad weather and transport difficulties. Therefore, the essential requirement for the wind turbine bearings is long time free maintenance and long life time. Considering the complexity of the stress situation, the bearing must bear impact and carry high load.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Morning Choir: Valerio Smordoni, Manolo D'Antonio, Marco Chiappini, Marco Del Mastro, Francesco Macr\u00ec, Simone Giglio, Giovanni Peditto, Igi Tani.\nOne of the newest additions to the Fading Records subdivision of AltrOck Productions, Camelias Garden are also very much of an unknown quantity to most progressive rock fans \u2013 and not only on account of the band members' young age. Hailing from my own hometown of Rome, they originally started as vocalist\/multi-instrumentalist Valerio Smordoni's one-man project, and subsequently turned into a full-fledged band with the addition of guitarist Manolo D'Antonio, bassist Marco Avallone and drummer Walter Palombi. Their debut album, You Have a Chance, produced by Massimo Dolce of Gran Turismo Veloce, was released in March 2013.\nLabeling themselves as \"post-progressive\", Camelias Garden (who are quite active on the live front, even if most of their gigs happen outside recognized prog circles) cite such diverse influences as Genesis (an essential reference point for practically every Italian prog band), The Beatles and Porcupine Tree, but also post-rock icons Explosions in the Sky and crossover hotshots Tame Impala. In fact, unlike most currently active Italian prog bands, Camelias Garden are firmly rooted in the English musical tradition \u2013 both progressive and vintage folk-rock \u2013 although echoes of some of the RPI greats of the past occasionally surface. Rather unusually for an Italian band, their grasp of the English language is outstanding, and Smordoni's enunciation is nearly accentless.\nOn You Have a Chance, the three core members of the band (drummer Walter Palombi joined after the album had been recorded) are supplemented by a number of guest musicians. A look at the lineup will also reveal a prevalence of acoustic instruments \u2013 and, in fact, in Camelias Garden music the electric component is kept to a minimum. As hinted by the deceptively na\u00eff cover artwork \u2013 with a slightly disturbing, surrealistic touch in the eyeballs replacing the flower centres \u2013 You Have a Chance, much in the way of Genesis' early output, is not as airy-fairy as the band's name might initially suggest; while the melancholy, somewhat world-weary lyrics have a late Romantic feel. With its circular structure, the album can be seen as a concept of sorts, and the tracks flow into one another without discernible breaks.\nThe short, sweet \"Some Stories\", a delicate, pastoral vignette highlighting Smordoni's harmonious, medieval-storyteller's vocals, complemented by dreamy birdsong and gentle acoustic guitar, provides a fitting introduction for the album, its veiled melancholy enhanced by flute and the solemn drone of the cello. The mood picks up with the folksy, lively \"Dance of the Sun\" and its sweeping Moog sharply reminiscent of PFM's iconic \"Celebration\" (and, occasionally, of vintage Genesis), culminating in a lively Celtic jig.\nMost of the album, however, rests on muted, gentle melodies, its whimsical English folk matrix bolstered by the haunting presence of the mellotron, whose interplay with the acoustic guitar enhances the catchy ballad \"The Withered Throne\" (which reminded me a lot of The Decemberists), and lends classic prog appeal to the romantic instrumental \"We All Stand in Our Broken Jars\", with its charming juxtaposition of the acoustic and the electric component. The album's other instrumental, \"A Safe Haven\", is a lovely, autumnal piano piece to which flute and mellotron add depth. Then, a couple of ethereal ballads, \"Knight's Vow\" and \"Clumsy Grace\", whose endearing folksiness gains prog credentials from Moog and Mellotron, introduce the album's own mini-epic, the almost 10-minute \"Mellow Days\", in which echoes of medieval music merge with a full-fledged early Genesis tribute: indeed, some of the keyboard parts will not fail to recall the iconic \"Firth of Fifth\". The album comes full circle with \"Some Stories (Reprise)\", a celebration of nostalgia in which the opening track is presented as a faint, scratchy recording on a backdrop of falling rain, in a fascinating sonic collage.\nClocking in at a sensible 49 minutes, There's a Chance is obviously a labour of love, put together with painstaking care by a group of young, dedicated musicians. Although derivative in parts, and occasionally a tad repetitive, devoid of those sharper edges that might make it more attractive to fans of more experimental fare, its soothing, mainly acoustic nature will offer a lot of listening pleasure to those who like their melody untainted by overt mainstream pretensions. Blending nostalgia with a subtle touch of modernity, You Have a Chance is a solid first showcase for a band that shows a lot of promise for the future, and another intriguing find from the ever-reliable AltrOck team.\nOn May 25-26, 2013, the Italian \"prog hub\" of Genoa and its surrounding region of Liguria will be the undisputed protagonist of an event of international scope targeted to anyone who is in the business of making and promoting music. The FIM (Fiera Internazionale della Musica), the largest event of its kind organized in Italy, will take place at the Ippodromo dei Fiori in the medieval hill town of Villanova d'Albenga. The town lies about 50 km (31 miles) west of Genoa, in the hinterland of the famed Riviera di Ponente, the western stretch of the Italian Riviera, curving towards the French border and following the route of the Via Aurelia, the longest of the original Roman roads.\nOver two whole days, the event will offer a unique showcase to up-and-coming bands and artists from all over Italy. Four stages and other dedicated spaces will allow musicians to perform with their own instruments, and the participants will also have a wide range of workshops, seminars and other happenings to attend \u2013 all covered by a daily entrance fee of \u20ac 15.\nOne of the core events of the fair, the Riviera Prog Festival will host a total of 13 bands (as well as the symphonic orchestra of the neighbouring town of Sanremo, known internationally for its Festival della Canzone Italiana) in the afternoon and evening of both days, starting from 3 p.m. The bands that will take turns on stage during this event-within-the-event represent some of the best that Italian progressive rock has to offer, with an eye to its glorious past and another to the thriving contemporary scene \u2013 an example that US organizers would do well to follow, instead of focusing on foreign acts to the detriment of homegrown talent.\nThough most of the bands and artists on the lineup are based in Liguria, other parts of Italy have not been neglected: Goad and Le Porte Non Aperte hail from Tuscany, while Claudio Simonetti\/Daemonia and Biglietto per l'Inferno (both protagonists of the original RPI movement in the early Seventies) are based respectively in Rome and Milan. The local talent includes veterans such as The Trip (who counted one Ritchie Blackmore among its early members), Latte E Miele (who were slated to headline the sadly cancelled Farfest 2012), Delirium, Garybaldi and Il Cerchio d'Oro, and modern bands such as Il Tempio delle Clessidre (whose career-defining NEARfest appearance endeared them to the US prog community), G.C. Neri Band, La Coscienza di Zeno and newcomers Flower Flesh.\nFIM has been sponsored by a number of local agencies, including the region of Liguria, and partnered by media outlets such as local radio and TV stations, as well as the association Centro Studi per il Progressive Italiano (CSPI), independent label Black Widow Records and recording studio Maia (all based in Genoa). The event's website (unfortunately only in Italian, at least for the time being) contains detailed information on the event, including tips for anyone who would like to combine the pleasures of music with those of sightseeing.\nA band name such as Labirinto Di Specchi (Maze of Mirrors) suggests purveyors of intricate, multilayered music, heavy on atmosphere and intensity rather than smoothly flowing melodies. Indeed, the young five-piece hailing from the beautiful Tuscan town of Montepulciano (a place well known to wine lovers) belong to the growing contingent of new Italian bands that have updated the blueprint for progressive rock set by the iconic bands of the Seventies, and produced an album packed full of the twists and turns implied by their name.\nLabirinto Di Specchi have been together since 2005; their first recording effort, a demo titled La Maschera della Visione, attracted the attention of Lizard Records, an independent label with a proven record for modern prog releases of a consistently high standard. Hanblecheya (a word in the Native American Lakota language denoting a vision quest), the band's full-length debut, contains reworked material from the demo, as well as new compositions .It also sees the participation of a number of guest artists \u2013 most remarkably, that of Paolo Carelli, former vocalist of Pholas Dactylus, a short-lived outfit that in 1973 released Concerto delle Menti, one of the cult albums of the original RPI movement.\nClocking in at almost 70 minutes, Hanblecheya is impressively ambitious, its seven lengthy compositions describing the experience of the titular vision quest through a very distinctive format. Though there is no actual singing involved, Paolo Carelli's solemn, deep-voiced narration is like a thread connecting the seven tracks to each other. While synthesizers and assorted electronics are definitely at the core of the band's musical vision, the fluid, melodic touch of the piano and full-throated rumble of the organ provide an organic foil, further balanced by the autumnal drone of the cello and the pervasive presence of both acoustic and electric guitars.\nAny album tagged as \"Porcupine Tree meets Pholas Dactylus\" sounds like a rather interesting proposition, and Hanblecheya does not disappoint expectations. Though clearly not an easy listening experience, it is also surprisingly mature in its treatment of the inevitable influences. The psychedelic\/space component, firmly rooted in the use of a wide range of electronics, gains a harder edge from occasional bursts of riffing that suggest a prog metal inspiration (particularly evident in \"La Maschera della Visione\", the shortest track of the album and possibly the most accessible); while entrancing Mediterranean and Eastern suggestion lurk in the two compositions bookending the album , with the heady, raga-inspired section in opener \"Eclissi Pt.1 \u2013 L'Occhio e la Maschera\" reprised in the second half of \"Eclissi Pt. 2 \u2013 La Genesi\".\nNot all of the many ideas thrown into Hanblecheya's melting pot of are successful: the classical-meets-electronic bent of the second half of \"Fantasia\" borders dangerously on the cheesiness of those classical \"rock\" adaptations that were quite popular in the Eighties, and clashes with the wistful, atmospheric mood of the first part of the song. On the other hand, the ominous post rock surge of \"Purpurea\" and the all-out experimentalism of \"Nel Nulla Etereo Soggiogato dall'Ignoto la Mente Si Espande\", revolving around Carelli's eerily effective narration (the most reminiscent of his Pholas Dactylus days), and wrapped up by a noisy industrial section (the whole bringing to mind label mates Runaway Totem at their most impenetrable) hold up to close scrutiny, in spite of their undeniably \"difficult\" nature. The album's crowning achievement, however, comes with the 19-minute \"Foll(i)a\" (a title conflating the Italian words for \"crowd\" and \"madness\"), an intensely cinematic piece that, while paying homage to Steven Wilson's signature style, manages to avoid blatant derivativeness. A fresh take on the old \"epic\" warhorse, the track hinges on a \"duel\" between a whole array of electronic effects and the gentle ripple of the piano, ending in an exhilarating cascade of cymbals, piano and majestic synth washes.\nAlthough, as suggested in the previous paragraphs, Hanblecheya is not for everyone, it has all it takes to attract those prog fans who like the genre to look forward without completely turning its back to its glorious past. Firmly anchored to the Italian progressive tradition by its keen sense of melody and the rivetingly dramatic tone of Paolo Carelli's narration, yet unafraid to experiment with more radical musical directions, and skilled in combining the acoustic, the electric and the electronic component (though at times the slashing, whistling presence of synths can become a tad overwhelming), Labirinto Di Specchi are a band that adventurous listeners would do well to check out.\nIn spite of a name referencing one of Genesis' most popular songs and a \"progressive metal\" tag, Italian band Garden Wall are neither one of the many followers of the \"retro-prog\" trend, nor a bunch of Opeth or Dream Theater devotees. Hailing from the north-eastern Italian region of Friuli, the band was put together by vocalist and multi-instrumentalist Alessandro Seravalle in the late Eighties, and released their debut album in 1993. Assurdo, their eight album, forms the third and final chapter of the trilogy begun in 2002 with Forget the Colours, and continued with 2004's Towards the Silence. It is also their first release with Lizard Records (one of the most rolific independent labels for modern progressive rock), and \u2013 unlike their 2008 album, Aliena(c)tion \u2013 contains completely new material.\nNow a quintet, with only Seravalle and guitarist Raffaello Indri left of the original lineup, Garden Wall have pulled out all the stops for their recording comeback. Not being familiar with their previous output, and misled by the \"prog-metal\" tag, when I first heard the album I was confronted with something that was almost impossible to label. Moreover, while most of my reviews include comparisons with other bands or artists (something that readers generally appreciate), this time I was hard put to find any suitable frame of reference within the progressive rock spectrum.\nIf I had to use a single adjective to define Assurdo, I would call it unpredictable. While far too many albums and individual songs seem to endlessly reproduce the same structure, the 10 compositions featured on Garden Wall's eight CD take the listener on a veritable rollercoaster ride that will leave all but the most open-minded rather bewildered, as well as drained. To say that Assurdo is not an easy listen would be an understatement: spanning a wide range of influences and moods, each song conceived as a mini-suite in many different movements, and providing a canvas for Alessandro Seravalle's amazing vocal gymnastics, the album is an exercise in deconstruction rather than a showcase for cohesive compositional standards.\nObviously, this is not meant as criticism: though Assurdo is clearly a daunting prospect for anyone not used to more challenging fare than the average \"mainstream prog\" release, it can also be immensely rewarding for those who will invest time and patience in trying to \"unlock\" it. Its densely woven texture, made of so many different layers, its deeply literate nature (the album's title comes a quote from Robert Musil's The Man Without Qualities, \"Everything is absurd when you see it clearly\"), a complex instrumentation blending state-of-the-art electronic soundscapes with warm ethnic percussion, lyrical flute and violin, and gutsy electric guitar \u2013 all make for a very demanding listening experience, though one that can confidently bear the \"progressive\" label. For all its cosmopolitan, cutting-edge allure, Assurdo does have that indefinable \"Italian\" quality that the use of the Italian language (though juxtaposed with English) lends to even the most avant-garde musical efforts \u2013 as proved by a band like Nichelodeon, whose mainman Claudio Milano has been actively involved in the realization of Garden Wall's latest effort.\nAssurdo is one of those albums that need to be absorbed as a whole, so that trying to describe any of the tracks in detail would feel like a pointless exercise. The tracks run between 2 and 8 minutes, with the lone instrumental \"Just Cannot Forget\" strategically placed in the middle, as a sort of interlude. Taking Demetrio Stratos as a springboard, Seravalle dominates the rest \u2013 at times speaking, at others whispering, or even screaming or growling. Garden Wall's impressively omnivorous approach encompasses the academic suggestions of opener \"Iperbole\", to the deconstructed funk of the appropriately-named \"Trasfiguratofunky\", the haunting trip-hop of \"Negative\", the heady Middle Eastern flavour of \"Vacuum Fluctuation\" \u2013 blending jazzy organ, industrial electronics and heavy riffing as in \"Clamores Horrendos Ad Sidera Tollit\", employing flute and violin to complement the spacey, ambient-like electronics of closer \"Isterectomia\".\nAt under 70 minutes, Assurdo is not an excessively long album for today's standards. However, with its unabashedly eclectic, experimental bent, coupled with a distinct lack of anything even remotely resembling a catchy tune (as well as Seravalle's acquired-taste vocals), the album is rarely a comfortable listening experience \u2013 though a much more solid effort than some overly pretentious releases in the experimental prog field. In any case, adventurous listeners will find a lot to appreciate in Assurdo, one of the most intriguing albums released in 2011, and one that definitely deserves more exposure.\nAnother outstanding addition to the thriving music scene of the Italian port city of Genoa, La Coscienza di Zeno was founded in 2007 by a group of experienced musicians \u2013 bassist Gabriele \"Estunno\" Guidi Colombi (also a founding member of Il Tempio delle Clessidre), drummer Andrea Orlando and vocalist Alessio Calandriello. Keyboardist and lyricist Stefano Agnini joined the band at the beginning of 2008, while guitarist Davide Serpico (who replaced original guitarist Matteo Malvezzi) and keyboardist Andrea Lotti joined between 2008 and 2009. Agnini left at the end of the recording sessions for La Coscienza di Zeno's self-titled debut album, which had started in May 2010.\nThe band takes its distinctive name (meaning \"Zeno's Conscience\" in English, and often shortened to CDZ for ease of reference) from one of the masterpieces of Italian literature, the ground-breaking psychological novel published in 1923 by writer and businessman Italo Svevo, and written in the form of an autobiography meant to help the titular Zeno's attempts to quit smoking through psychoanalysis. Not surprisingly, La Coscienza di Zeno's debut possesses a definite intellectual appeal \u2013 though without the level of pretentiousness that might be expected _ revolving around Stefano Agnini's highly literate lyrics (loosely inspired by the novel) masterfully interpreted by lead singer Alessio Calandriello's technically impeccable voice, passionate without being overwrought.\nLa Coscienza di Zeno is one of those rare albums that, while in keeping with the classic prog tradition of long tracks, rich instrumentation (with special prominence given to the keyboards) and intricate arrangements, achieves the considerable feat of never overstaying its welcome. As other reviewers have pointed out, the album is not as easy to approach as other comparable efforts, and the first impression might be somewhat deceiving. To be perfectly honest, after my first listen I thought, here is another of the many Italian Genesis-worshipping bands \u2013 which, after successive listens, turned out to be a very unfair assessment. Indeed, while the Genesis influence is occasionally hard to miss, the album's roots lie firmly and deeply in the great Italian prog tradition, with Banco del Mutuo Soccorso a particularly apt reference, mainly on account the presence of two keyboardists and the remarkable balance between vocal and instrumental parts.\nClocking in at slightly under one hour, La Coscienza di Zeno features seven tracks between 6 and 13 minutes. Though the main foundation of the album is symphonic, lush and multilayered, with plenty of seamless instrumental interplay, outstanding solo passages and rivetingly expressive singing, there is also enough variety to keep the interest of the more eclectic-minded listeners, with a wide range of influences cropping up almost unexpectedly, from waltz to folk by way of jazz and even hard rock. The almost wholly instrumental (except for the spoken-word vocals in the middle) opener \"Cronovisione\" is melodic and intricate at the same time, with echoes of Yes in the airy synth sweeps laced with faintly spiky guitar, and of Banco in the majestic yet dynamic feel imparted by the twin keyboards. \"Gatto Lupesco\", hinges on Alessio Calandriello's amazing vocal range and expressive power, complemented by a musical accompaniment that is melancholy and intense in turns, driven by keyboards and dramatic drumming. The obligatory epic, \"Nei Cerchi del Legno\" (partly inspired by the iconic tale of Pinocchio, one of the few instances of Italian literature that have had some international resonance) has a rather unusual format, being mostly instrumental, with vocals making an appearance only towards the end. The music, on the other hand, is a triumph of imposing symphonic passages rendered even more lush by the double keyboard setup and string arrangement, almost jazzy inserts offset by gently meditative episodes, and stunning synth-guitar interplay that brings to mind Genesis' immortal \"Firth of Fifth\".\nOut of the remaining four tracks, \"Il Fattore Precipitante\" pursues the classic Italian prog route, with the lavish, airy Genesis-like suggestions sharpened by some heavy riffing and high-powered rhythm work courtesy of Gabriele Guidi Colombi and Andrea Orlando \u2013 though Calandriello steals the show here, his vocal tour de force complemented by a superb instrumental tapestry of keyboards, drums and guitar. \"Il Basilisco\", on the other hand, signals a sharp change in mood and musical style \u2013 a folk-tinged number veined with melancholy and enhanced by the arresting, unmistakably Old-World accordion of guest artist Luca Scherani of H\u00f6stsonaten, also showcasing Davide Serpico's lovely acoustic guitar work. The splendid, exquisitely tense instrumental \"Un Insolito Baratto Alchemico\" juxtaposes quieter, flute-led sections and stormy keyboard passages spiced by metal-hued riffing, enriched by solemn organ and lilting piano; while closer \"Acustica Felina\" (the second longest track on the album) reprises the lush symphonic mood of the beginning, rounded up by the deep choral tone of the inevitable Mellotron. Calandriello's voice tackles the challenging lyrical matter with superb expertise, veering from gentleness to a deep, almost menacing tone; the song is then wrapped up by a magnificent, Hackettian guitar solo.\nWith refreshing honesty, La Coscienza di Zeno make no bones about paying homage to the progressive rock tradition of the Seventies, both Italian and British \u2013 even if the sound quality and production values of their debut album are thoroughly modern, and lend extra depth and dimension to the elegantly complex music. An obvious labour of love, every aspect of the album has been carefully considered in order to offer a complete experience to the discerning listener \u2013 with stylish, mostly black-and-white photography and detailed liner notes, including the lyrics (which make worthwhile reading for anyone familiar with the Italian language). Indeed, La Coscienza di Zeno is a must for all lovers of vintage Italian prog, adding the band to the growing list of excellent \"traditional but modern\" acts that already includes their fellow Genoese Il Tempio delle Clessidre and La Maschera di Cera, as well as the revamped Delirium. Highly recommended to symphonic prog fans and anyone who is not put off by foreign-language vocals, this is another classy package coming from the ever-dependable Italian prog scene.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BOISE, Idaho, Jul. 13 \/CSRwire\/ - GoodWell announced today that 1% for the Planet has successfully achieved GoodWell certification for fair, equitable, and humane workplace practices.\n1% for the Planet is the first nonprofit, member organization to achieve GoodWell certification. 1% for the Planet is a global member network with thousands of businesses, nonprofits, and individuals who contribute 1% of their annual sales or income to environmental causes. Started in 2002 by Yvon Chouinard, founder of Patagonia, and Craig Mathews, founder of Blue Ribbon Flies, 1% for the Planet members have contributed more than $150 million to environmental nonprofits in 40 countries across the world.\n\"The 1% for the Planet mission is focused on building a global movement for a healthier planet, and we believe that how we do this work matters,\" said Kate Williams, 1% for the Planet CEO. \"Certifying with GoodWell gave us an objective way to understand and assess our organization's personnel practice and ecosystem. It was a simple process that yielded lots to celebrate and lots to learn - just like our broader work in the world.\"\nGoodWell founder Pete Gombert said he's seeing heightened interest in GoodWell certification from a broad group of entities comprising businesses, nonprofit organizations, and municipalities.\nGoodWell's certification uses 11 metrics to determine if an organization is healthy and sustainable. The process gives leaders an objective view of their practices as an employer and demonstrates their organization's values to employees and the broader community.\nGoodWell, a public benefit corporation headquartered in Boise, Idaho, is an independent performance standard that strengthens employee satisfaction by certifying fair, equitable, and humane workplace practices. GoodWell's certified organizations range from small businesses to global nonprofits and large municipalities. Its 100% metrics-based certification system provides actionable management insights and a framework for benchmarking and improvement. These human metrics are designed to work for any employer regardless of size, industry, geography or organizational structure. For more information about becoming certified, visit goodwellworld.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Tucson Weekly is an alternative newsweekly that was founded in 1984 by Douglas Biggers and Mark Goehring, and serves the Tucson, Arizona, metropolitan area of about 1,000,000 residents.\nThe paper is a member of the Association of Alternative Newsmedia. New issues arrive at kiosks throughout Tucson every Wednesday. Jim Nintzel is the current editor. Staff members include Logan Burtch-Buus, Tirion Morris, Christopher Boan, Jeff Gardner, Kathleen Kunz and Chelo Grubb.\nLongtime editor Jimmy Boegle left the Weekly in late 2012 to start his own independent paper in Palm Springs, California.\nThe founding editor was Douglas Biggers, who served as editor and publisher until he sold the paper to Wick Communications in 2000. He founded Edible Baja Arizona. 10\/13 Communications bought the paper from Wick in 2014. The paper is currently owned by Thirteenth Street Media. Former editors include Dan Huff, Carol Ann Bassett, James Reel, Michael Parnell, Dan Gibson and Mari Herreras. Longtime Weekly and Arizona Daily Star reporter Chris Limberis was posthumously inducted into the Arizona Newspaper Association Hall of Fame in 2006.\nThe Tucson Weekly was a launching point for the comic strip Red Meat, created by Tucsonan Max Cannon in 1989.\n^ \"Currents: Remembering Limbo\". Tucson Weekly. Retrieved 2010-11-07.\n^ \"Hall of Fame\". ANANews.com. Arizona Newspaper Association.\nThis page was last edited on 9 January 2019, at 22:06 (UTC).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"There was a time we wouldn't be together like this. There was a time the right wing's wedge issues would have kept us from holding together.\n\"But look at us now. Holding hands together.\n\"Educators fighting with the unions.\n\"Unions fighting with civil rights.\n\"People of faith fighting with the children.\n\"Environmentalists coming together with the health workers.\n\"Doctors standing on the side of the sick.\n\"Millionares standing with the unemployed.\n\"And straight folks standing with gay folk.\n\"People of faith standing with those wrestling with their faith.\nOn July 29th, thousands of people from all walks of faith and life flooded the streets of Raleigh, North Carolina to march for social justice and stand in solidarity against regressive legislation bore out of the NC General Assembly. \"It was also the biggest protest yet against the ultraconservative agenda of the Republican-controlled legislature,\" says Sue Sturgis writing for Facing South, \"drawing as many as 10,000 people to the state capital's downtown streets.\nWatch the official Moral Monday 13 recap video.\nNorth Carolina lawmakers were successful in passing a bill that takes control of the municipal water system away from Asheville and hands it over to a state-chartered regional authority without compensating the city for the loss. The measure, which opponents say is theft by the state and would harm regional economic development efforts, is the subject of a lawsuit set to begin next month. Save Our Water WNC, a group opposed to the takeover, is among the organizers of Mountain Moral Monday.\nNo civil disobedience is planned for Mountain Moral Monday.\n\u2190 Protect Farmworkers \u2013 Tell McCrory to Veto HB74!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Which Days Can You Water?\nNot everyone can water their lawn at the same time. Each resident can water on two assigned days per week. The water restrictions in Lubbock are based on your house address. If your address ends in 0, 3, 4, or 9, you water on Monday and Thursday. If your address ends in 1, 5, or 6, you water on Tuesday and Friday. If your address ends in 2, 7, or 8, you water on Wednesday and Saturday.\nWhat Time of Day Can You Water?\nWater Restrictions are in effect from April 1st through September 30th. During this time, watering is allowed only on your scheduled watering days, and only from 6:00 p.m. till 10:00 am. No watering is allowed between 10:00 a.m. and 6:00 p.m. If you're wondering how these seemingly strange hours came about, the purpose is to make sure water is not lost to evaporation during the heat of the day. Make sure to water less than 1.5 inches of water per week and (obviously) don't water when it is raining. When you water, you must not have significant runoff.\nAlthough they're called restrictions, which makes them sound like a bad thing, these irrigation rules are actually helpful to your lawn. Deeper watering less often promotes deeper root growth than watering lightly every day or two. At Ashton Walden, we are constantly testing new products to help your watering efforts go further. We give you tools such as catch cups and make suggestions on watering and mowing heights. If you have any questions about your lawn care or the watering restrictions in Lubbock, just give Ashton Walden Turf Services a call and we'll be glad to help.\nHow Can We Help Your Texas Lawn?\nContact Ashton Walden Turf Services for a free project assessment. Or, connect with us online to tell us more about your landscape needs and objectives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"ESG's leadership team has deep roots and solid experience in marketing, human resources management, healthcare, and finance. Along with expertise, each person brings a passion for their clients' success that drives their day-to-day activities.\nSteve has over 18 years of leadership and management experience. Steve is also an expert in technology and systems. In his free time, you may find Steve reading a good historical novel or growing potatoes (he's a farmer at heart).\nMark has more than 8 years of experience with operations management, employment law & regulatory compliance, and business process design. In his free time, Mark enjoys politics, spending time with his family, or playing a little hockey or tennis.\nMeribeth has more than 20 years experience in the PEO industry, with expertise in PEO sales, human resources, and employee benefits. Meribeth enjoys reading, horseback riding, hiking, and fly fishing.\nhas over 15 years of PEO experience, particularly in Sales Management and Training, Human Resources Consulting, Benefits Administration, Marketing, and Public Relations. When not at work, Daryl likes to travel with his family, play basketball and golf, and coach his kids' sports teams.\nMatt has more than 10 years of experience in information technology and systems, network design, database design, project management, and process improvement. Matt loves to play basketball, baseball and softball, and also enjoys scouting, hiking, backpacking, and spending time with his family.\nSharron has over 10 years of Human Resources experience. Her expertise includes recruiting and hiring, training and development, employee relations, compliance, and employee benefits. When she takes off the HR hat, you may catch Sharron gardening, cooking, or traveling somewhere new.\nMonica has over 30 years experience in the insurance industry, with expertise in general liability, workers' compensation, commercial auto, professional liability, and inland marine coverage. In her free time, Monica enjoys traveling, boating, cooking and reading, and is an expert decorator.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Axe Bpo is the leading Business Process Outsourcing company specializing in Forwarding and Logistics industry. You are aware that to increase efficiency in Forwarding & Logistics companies must reduce overhead and focus on revenue areas to increase Profitability. Our Business Process Outsourcing services focus on addressing the specific challenges and pains of our customers covering across all departments \u2013 Customer Service, Sales & Marketing, Documentation, Operations & Finance.\nAxe Bpo currently offers integrated end-to-end Outsourcing services or even specific small task for established and emerging Logistics companies. We service round the clock and with our 100+ strong workforce, we blend our business solutions along with our Technology offerings. This understanding and technological know-how are Axe Bpo's key strengths.\nWe provide friendly and excellent service and offer our services at very reasonable budget to all our customers. Taking a partnership route, we help our customers to stay ahead of the competition and identify and develop new revenue-generating channels.\nOur customers are spread across Continental Europe to over 50 International locations worldwide. Our customers benefit from our expertise and deep industry knowledge along with and information technology exposure. This has ensured longstanding relationships with all our customers.We are fully committed to providing customized solutions to Reduce cost, Increase Sales & Improve customer service for our customers across Europe.\nIf you are interested to make your company competitive and profitable irrespective of your company size and the challenges you face today, we will be glad to share our expertise and offer the free consultation. Please feel free to reach us, we are looking forward to hearing from you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"More product info From ILNP Cosmetics, Inc.\nThis report was last updated 12\/18\/2018.\n7 of the 767 reviewers have never written a verified purchase review. The Never-Verified Reviewers have rated this product an average of 3.0 while the reviewers who have written at least one verified purchase review rated this product an average of 4.8. Based on our statistical modeling, the discrepancy in average rating between these two groups is significant enough to believe that the difference is not due to random chance, and may indicate that there are unnatural reviews.\n108 of the 767 reviewers have had at least one of their past reviews for another product deleted. This is an excessively large percentage of Take-Back Reviewers which may indicate unnatural reviews.\nIn total, we found that 607 reviewers (or 79%) are within an overrepresented participation group. This is an excessively high number of reviewers in overrepresented participation groups.\nWe counted 767 reviews for this product over the span of 1,063 days, an average of 0.7 reviews per day. If these reviews were created randomly, there would be a 99.9% chance that we would see 5 or fewer reviews on any given day.\n25 of the 767 total reviews for this product are unverified purchases. This is within an acceptable range and it does not appear to be significantly affecting the overall rating for this product.\nIn total, we found that 199 reviews (or 26%) are within an overrepresented word count group. This is within an acceptable range and it does not appear to be significantly affecting the overall rating for this product.\n62 of the 767 total reviews for this product use substantial repetitive phrases between them. This is within an acceptable range and it does not appear to be significantly affecting the overall rating for this product.\n66 of the 767 reviewers have substantial overlapping product review history. This is within an acceptable range and it does not appear to be significantly affecting the overall rating for this product.\n9 of the 767 reviewers have reviewed other products for this brand. This is within an acceptable range and it does not appear to be significantly affecting the overall rating for this product.\n1 of the 767 total reviews for this product admitted to receiving a free or discounted product in exchange for a review. This is within an acceptable range and it does not appear to be significantly affecting the overall rating for this product.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thackeray, an only child, was born in Calcutta, India, where his father, Richmond Thackeray (1 September 1781 \u2013 13 September 1815), held the high rank of secretary to the board of revenue in the British East India Company. His mother, Anne Becher (1792\u20131864) was the second daughter of Harriet and John Harman Becher and was also a secretary (writer) for the East India Company.\nBecky Sharp is a poor orphan when she first makes friends with the lovely Amelia Sedley at Miss Pinkerton's Academy for Young Ladies. She may not have the natural advantages of her companion but she more than makes up for it with her wit, charm, deviousness and determination to make a success of herself whatever the cost.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking for low-carb seafood main dish recipes? Allrecipes has more than 460 trusted low-carb seafood main dish recipes complete with ratings, reviews and cooking tips.\nSee how to make a simple marinade that will make your shrimp memorable!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..76dd6fb18d62aae1aacbcdf344b76500f2cbe38a
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+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzblxqc
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+{"text":"For example, women tend to havehigher surgical mortality, less favorable outcome,and higher neurologic morbidity and recurrentcarotid stenosis compared to men. The ef?cacy of TC-CO2 monitoringhas also been demonstrated following cardiovas-cular surgery in infants and children with con-genital heart disease (CHD) (Tobias et al. Study of the distribution of oral ciprofloxacin into themucosa of the middle ear and the cortical bone of the mastoid process. Allof these treatments come with risks and side effects. For viral etiology, no antimicrobialtherapy is needed.\nEnvironmental and OccupationalMedicine, 4th ed. A case manager performs compre-hensive \"functional assessments\" on patients to deter-mine the types of interventions and assistance theyrequire. However where to buy isotretinoin clear therapeuticstrategies to address ischemic risk have not yet been fully validated. Therefore, reversible drugs are chosen when possible.\nNotethe basal (external) laminae ofthe vessel and the perineurial cell (arrows).Thejunction between endothelial cells ofthe blood vessel is also apparent(arrowheads). His weakness isprogressively increasing, which has become more severe and he is unable to walk without the helpof others. It could be concluded that smokers have ahigher risk, and this could be used as supporting evidence that smoking is arisk factor for cirrhosis. It is common for axial coding to result in theunification of several categories into larger categories that relates to the study's \"core phe-nomena,\" otherwise known as main themes (Creswell where to buy isotretinoin 2013). (b) Areasof correlated FDG PETbinding representing glucosehypometabolism associatedwith MMSE scores. Of course where to buy isotretinoin the version of SARS represented through media cover-age is not the only way to organize the data. NIH statement on National HIV\/AIDS and Aging Awareness Day Sept.\nThe proof of principle was demonstrated by intro-ducing to cells antibodies or peptides, directed to the interaction site in p53, andobserving a reduction at the levels and activities of the p53 protein (e.g. of these 121 comparisons where to buy isotretinoin only 69 (57%) hadconcordant conclusions (i.e., the chemical had the sameclassification as being either a carcinogen or a noncarcin-ogen in the replicate test). (2012).National public health performance standards program (NPHPSP). (2008)Ginkgo biloba for prevention of dementia: a randomized con-trolled trial.\nIn two trials,no significant difference was found in the percentage of hot flash sever-ity reduction between Promensil and placebo. The dissection proceeds systematically,working from superior to inferior and from lateral to medial.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yidio is the premier streaming guide for TV Shows & Movies on the web, phone, tablet or smart tv. c1731006c4 .. Extant is an American science fiction television drama series created by Mickey Fisher and, . (season 1); Jeffrey Dean Morgan as James Daniel \"JD\" Richter (season 2), a cop working special patrol (bounty hunting) who teams up with Molly. . She's highly skeptical of the Humanichs Project and is revealed to be part of.. . cinema 4d r14 realflow plugin download Julie 2 movie in hindi dubbed free . Forevermore 2014-2015 (TV Series) Part 1 HQForevermore 2014-2015 (TV.. Hino HIECHO-II EPC [09.2015] English >> DOWNLOAD (Mirror #1). 6 Nov 2014 . These are the premiere and return dates for all TV shows in spring 2015 (March 1st-May 31st). . other lists: 2014-2015 Series Orders, 2015-2016 Series Orders, 2014-2015 Shows in . 2. The Last Man on Earth (20152018). TV-14 22 min Action, Comedy, Drama. 7.4 . 2-part miniseries premiere Tue.. Demolition striker : DmS team call of duty Mw3, Black Ops et Mw2 sur Ps3.. Forevermore 2014-2015 (TV Series) Part 1 HQForevermore 2014-2015 (TV . Citroen c8 et peugeot 807 diesel 2 0 hdi a 2 2 hdi aCitroen c8 et peugeot 807.. 7\/8: Our 2015-16 TV Season Scorecard has launched! All future updates will . Renewed, Marvel's Agents of S.H.I.E.L.D. SEASON 2, Renewed 5\/7. Canceled.. 21 May 2018 . .. Next:Hino Diagnostic eXplorer 2 - Hino DX2 1.1.15.15 with keygen unlocked: . Forevermore 2014-2015 (TV Series) Part 1 HQForevermore 2014-2015 (TV.. Forevermore 2014-2015 (TV Series) Part 2 HQForevermore 2014-2015 (TV Series) Part 2 HQ bd9e74395d Casa simpson pdfFfvii no cd.. 27 Feb 2015 . These are the premiere and return dates for all TV shows in summer 2015 (June. . 2. Devious Maids (20132016). 60 min Comedy, Drama, Mystery . Making of the Mob, is an eight-part docudrama that begins in 1905 and spans more than 50 years, tracing the original . The McCarthys (20142015).. . Forevermore 2014-2015 (TV Series) Part 1 HQForevermore 2014-2015 (TV Series) Part 1 HQ Dallas Qualifying full movie hd 1080p download kickass movie.. Created by Jason Katims. With David Walton, Benjamin Stockham, Al Madrigal, Minnie Driver. Will Freeman lives a charmed existence. After writing a hit song,.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"HTML code allows to embed World Cup 1994 logo in your website.\nWorld Cup 1994 logo image sizes: 644 x 594 pixels. Format: jpg. Filesize: 99 KB.\nPlease warn us if you consider World Cup 1994 logo to be incorrect, obsolete or having wrong description.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sephira picked a bunch of wildflowers she found growing by the gate, knelt and placed them on Antoine's grave before she stood and walked home. She couldn't help but thing that as one life was taken today, another was going to be born in a few months. She sighed as she reached her house.\nGraayth walked back to Delmi with Jo and the others, at least until they reached the gates of Delmi where he recognised four more members of his pack waiting for them in hume form. They talked for a while, finding out that the two who had gone after Horja had been slain by the Demon but had slowed him down. In turn Graayth related that the Summoner had been killed and a gate to the demonic realm had been closed. For now, the world was safe from demon invasion. The pack-Alpha asked about Joanna who explained that she and the others were Vampire Hunters but this pack weren't to worry because they weren't threatening the normals of this town.\nZheren the pack-alpha then told her that the pack would be in the general area for a few weeks then they would move on. However like Graayth he respected the work the Vampire Hunters were doing and asked if she would mind Graayth joining them so that the Ehre Pack and her group could be allies. Jo thought about it and agreed- having an entire werewolf pack as allies could have advantages. And so it was agreed. For now Graayth could stay in the basement under the Old Fox Inn until he could establish himself in town. Then they parted and Graayth went with Jo into town and back to the Inn.\nOOC: Okay everyone this arc is pretty much wrapped up. Later today or tomorrow we'll be starting out on the new arc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Disc brake system completely integrated with OEM Linked Braking System. Full ABS integration. Brake Calipers : Proprietary four-piston aluminum alloy caliper. Optional anodized red, engraved with \"CSC\" logo. Replacement pads available through California Sidecar Parts & Service Department. Brake Rotors : 11.5\" purpose designed. Optional drilled rotor.\nFinal driven gear is an Aluminum cast housing with proprietary gearing. Drive shaft : Slip yoke front with purpose designed drive shaft. The highest quality automotive style U-Joint available. Traction Control : Full integration with OEM traction control.\nTrailer wiring harness is equipped with the female trailer plug compatible with all Escapade Trailers. The male end may be purchased from the California Sidecar Parts Department. Reverse lights : Equipped with reverse lights that are automatically activated upon initialization of the reverse system. Trunk door gaskets : The trike body trunk door is equipped with a double gasket sealing system. You may initially experience a very snug fit opening and closing the trike body trunk door until the double gasket system fully seats to the form of the trunk. Paint care : The care of the paint is critical for the first 90 days of the trike body. Do not cover the trike body and\/or wax the trike body for the first 90 days. The paint is however, suitable for pin striping and\/or painting of murals at the time the trike.\nVertically Stabilized Opposing Rocker Mono Shock Independent Rear Suspension. Shock Absorber : Purpose designed Nitrogen filled Coil Over JRi shock for the Opposing Rocker arm system. Shock Spring : 300 lb linear wound racing spring designed to accommodate up to 650 lbs inclusive of accessories, luggage, and passengers. (Lighter and Heavier spring weights available). Shock Spring Preloader : The OEM electro hydraulic preload adjuster is retained for preload adjustment by the rider to allow for compensation of varying loads.\nLug Nuts are M12 x 1.5 conical lug nuts torque to 75 FT-LBS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"A first-half goal from Marten de Roon (\u00a34.4m) earrned Middlesbrough a 1-0 win over Sunderland and the midfielder the maximum Fantasy Premier League bonus points.\nDe Roon contributed four recoveries and four clearances, blocks and interceptions (CBI) to tally 32 in the Bonus Points System (BPS).\nMiddlesbrough goalkeeper Brad Guzan (\u00a34.2m) claimed two bonus points, making four saves to help his side to a third clean sheet in the last six Gameweeks.\nHe also made 13 recoveries to score 30 in the BPS.\nDefenders George Friend (\u00a34.2m) and Ben Gibson (\u00a35.1m) collected one bonus point apiece, scoring 26 BPS.\nFriend gained from winning three tackles, while Gibson registered seven CBIs.\nBoth De Roon and Sunderland's Wahbi Khazri (\u00a36.0m) registered 6.3 to lead the way in a low-scoring ICT Index encounter.\nThe Boro midfielder had two shots, including his decisve strike from inside the box.\nKhazri recorded a Creativity score 44 made up of three successful crosses, creating two scoring chances for the visitors at the Riverside Stadium.\nDespite failing to score for a ninth consecutive outing, Sunderland's Jermain Defoe (\u00a37.6m) registered the highest Threat score in Wednesday's contest with 49.\nThe striker totalled three shots, with two arriving from inside the penalty area.\nHanded the captain's armband by more than 150,000 managers, Alvaro Negredo (\u00a36.3m) delivered a second successive blank to finish Double Gameweek 34 with only four points.\nThe Spaniard failed to produce a single shot against the Black Cats and has already been sold by 26,000+ FPL managers.\nNegredo was perhaps destined to be a short-term option, with all of Boro's remaining four fixtures scoring three or more in the Fixture Difficulty Rankings (FDR).\nInvestors in Boro's defence did get some reward, however.\nWith more than 115,000 new Gameweek 34 owners, Gibson returned seven points over his two fixtures.\nBut like Negredo, Boro's defenders look set to suffer from that testing season run-in.\nAs for Sunderland, they have failed to score in four successive away matches.\nDavid Moyes' side return to the Stadium of Light on Saturday as they host AFC Bournemouth in a must-win encounter.\nThat fixture will mark a final opportunity for Defoe to remain in the frame as an FPL asset. Still found in 807,000 (17.8%) squads, he has been sold by more than 266,000 FPL managers over the last four Gameweeks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vocoder is real time pitch shifter stompbox like effect. I made this app to test some ideas for new pitch shifting algorithm and it worked well. It is not great but just fine. It is not real vocoder logic but instead a combination of my secret algorithms that I am playing with at the moment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Religious diversity matters to the work of the State Department. In this brief essay I will provide some critical personal examples of that fact, but let me start with an introduction and some background information about why having a faith can matter.\nI was raised in a family that had a strong Christian faith, and we attended our Swedish Baptist church twice on Sundays and on Wednesday nights, too. Missionaries came to our church and showed slides of faraway places, and sometimes we even hosted them in our home. Above my bed was a \"monkey rug\" (made from the skin of nine monkeys) brought from Ethiopia by my missionary uncle. My mom put her wedding ring in the offering plate after a particularly compelling presentation by a visitor from Africa (my Dad said it was OK).\nHaving completed university, I went to Princeton Theological Seminary with the intention of becoming a minister or a missionary myself. However, life took some unexpected turns, and instead I joined the Foreign Service\u2014but not before serving as a volunteer aboard the Christian service vessel MV Logos as a ship's carpenter. It was there that I met my wife. For many years, I taught Sunday school (although not as faithfully as Jimmy Carter), and my wife and I participated in Bible studies and tithed (gave away 10 percent of our income).\nReligion can be every bit as important as race, gender, sexual orientation and other aspects of our identity.\nMany of my State Department friends and colleagues find my background a little unusual and, in fact, unintelligible. But when I joined the Foreign Service in 1993, I brought this identity and these commitments with me. Even though the State Department does not, at this writing, have any religion-based affinity groups, religion can be every bit as important as race, gender, sexual orientation and other aspects of our identity.\nNow, after almost 25 years serving mostly as a consular officer, I can look back and see how my identity as a Christian person has been of great value to the State Department and its mission. Here I offer a few examples as evidence that religious diversity matters to the work of the State Department. I am sure many others from various faiths could offer their own examples of the value that faith-based people bring to diplomacy.\nFrom the outset of my career, I sought out missionaries abroad for friendship and mutual encouragement. They educated me and showed me parts of foreign countries I would never have seen in my capacity as a diplomat. For example, a missionary in Manila took me on her nightly rounds working with glue-sniffing street children. Another introduced me to Muslim families in Quiapo and took me to the Golden Mosque.\nIn Rome, my various church contacts housed the legal permanent residents (LPRs) we had evacuated from Albania but could not assist with onward travel from Italy. When the management at the Holiday Inn complained about those LPRs bedding down in the hotel hallways at about 10 p.m., I called Catholics, Presbyterians, Methodists, Mormons and Baptists. Every church took in some of the evacuees until all could be housed.\nIn Hungary, I began to identify local resources to assist U.S. citizens in ways that the consular section could not. Many of those resources were graciously given by persons and institutions of faith. On one occasion, I had responsibility for a number of American prisoners undergoing two full years of pretrial detention. It would be difficult to visit each of them monthly.\nWith that in mind, I asked each prisoner if they wanted visits by clergy. Two said yes\u2014one a Baptist and the other Jewish. The rabbi who agreed to visit the Jewish prisoner did such a fantastic job that the other Jewish prisoner among the group eventually also asked for clergy visits. These visits were so important to the well-being of these prisoners that I vowed to make the offering of clergy visits to prisoners by consuls a universal practice, if I ever had the chance.\nThat opportunity came when I became director of American Citizen Services back in Washington, D.C. The Bureau of Consular Affairs agreed immediately to my proposal to add the offer of clergy visits to the prison visit checklist for consuls worldwide.\nCA agreed, not for the purpose of propagating religion, which would be inappropriate, but to offer added and critical support to our imprisoned citizens. I estimate that about one third of prisoners, if asked, will opt for clergy visits. This additional support to imprisoned citizens abroad costs the U.S. government nothing.\nOn another occasion, an American citizen experienced a serious medical issue in Budapest at a moment when I was both consul general and the control officer for a U.S. presidential visit event on Castle Hill.\nI asked the patient if he wanted a visit by clergy, and he said yes. The clergy member and his expatriate church stepped in and brought the American food and a television, and even picked up his family from the airport and housed them.\nThey took great care of this recovering citizen, and it cost the U.S. government\u2014you guessed it\u2014nothing! I was able to focus on the president's visit knowing that this citizen was in good hands.\nIn the Philippines, Rome, Hungary, Russia and Hong Kong, we relied on local resources for our citizens in need (e.g., free food, shelter, counseling and more). I worked hard to build strong relationships with the providers of these essential services; to discover new service providers, and to close gaps where they existed. Not surprisingly, many of these providers were and are people of faith.\nMany of our wardens, it turns out, are also people of faith. When the Marine Security Guard on duty in Moscow passed me a call from a distressed American at 2 a.m., I reached out to an ACS contact, a pastor, who met me with his car at the embassy 20 minutes later. We took the American to church-funded lodging, and I gave him McDonald's coupons. Three days later, he flew home.\nAs anyone who has served in West Africa will tell you, a lot of health services in the region are delivered by faith-based providers.\nAs director of American Citizen Services, I worked to systematize our tracking of local resources worldwide and to strengthen the warden network. These initiatives were folded into Secretary of State John Kerry's MissionOne program, and served to strengthen our protections and services to our citizens at no cost to the government.\nSome of the energy for these ideas came from my extensive interactions with resource providers, who often had a faith-driven service mentality. As a person of faith, I was always comfortable with these people, whether they were Christians, Muslims or Jews.\nOn a Sunday afternoon in 2014, when State's Bureau of Medical Services called to say that we needed to urgently evacuate an American missionary with active Ebola from West Africa, it was no coincidence that I already knew the mission involved. I also knew the administrator of the mission compound where most of the Ebola treatment in West Africa would take place. As anyone who has served in West Africa will tell you, a lot of health services in the region are delivered by faith-based providers.\nThe State Department, especially MED and personnel at the embassies, did a great job evacuating American Ebola victims from West Africa. I am sure they would have done so without me and regardless of anyone's faith, but the connections already made on the basis of religious interest proved to be useful when the emergency came.\nAs I approach the end of my Foreign Service career, I can say with appreciation that the State Department, although appropriately secular in orientation, values the contributions of its religiously diverse workforce. I did not wear my faith on my sleeve in the workplace; never had a Bible on my desk or religious posters on the walls. But I brought a faith-based commitment to this career that has, in my opinion, benefited the department, its mission and our citizens.\nCommitted Muslims, Baha'is, Jews, Buddhists, Hindus and other Christians will have their own stories to share, and should do so. Their stories will help ensure that the State Department continues to understand and welcome the contributions that faith-based people bring to this institution and to our mission abroad.\nFSO Phil Skotte has served in the Philippines, the Vatican, Hong Kong, Budapest and Moscow. Domestically, he has worked as a foreign policy adviser for special operations at the Pentagon and director of American Citizen Services worldwide; he currently serves as the Bureau of Consular Affairs' liaison with the intelligence community. Prior to joining the Foreign Service, he worked as a commercial fisherman in Alaska, a schoolteacher, an athletic director and a ship's carpenter, and earned master's degrees from the National Defense University and Princeton Theological Seminary. He is the author of Why Jesus Won't Go Away\u2014A Diplomat Reflects on Faith (WestBow Press, 2014). Phil and Maribeth have three daughters. The views expressed in this paper are those of the author and not necessarily those of the U.S. government.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The SNES ports of these games contain almost the same physics as their NES\/FDS counterparts, so you can use most of them in SNES Super Mario All Stars as well.\nWalljump is when you jump towards a wall and somehow Mario's foot catches the wall and allows to jump again, boosting from the wall. The walljump in the left image is easy to try even on the real console. In the right image (using a custom map), three walljumps are performed.\nTheory: Walljump happens because the game does a floor check (a simple \"is position divided by 16 even? Is there a solid block below him?\" test) even during a wall-ejection. Wall-ejection is SMB's mechanism to adjust Mario's horizontal position properly when his left side and right side have a different inside-wall status. The game ejects Mario towards the opposite of his steering.\nIt's possible to perform walljump from any non-lethal solid material (bricks, pipes, etc).\nIn SMB3, the trick works exactly the same way as in SMB1, but is harder to perform, because the game is more effective in preventing inside-wall situations.\nIf you touch a vine at the extreme left edge of the screen, you'll be transported to the right edge.\nUses: In 4-2 of SMB1, hit the vine block and get up on the ceiling. Walk to the right so that about half of the vine block gets scrolled off the screen, and the vine will dissappear. Go back to the left and touch the vine, and if you went the correct amount to the right, you'll be teleported to the extreme right edge of the screen. If you misestimated the amount, you'll get stuck in the left edge of the screen.\nThe alternate pipe glitch is when a pipe takes you to somewhere other than where it was intended to.\nLet's define a word Entry which can be a pipe or a vine.\nTheory: Technically ? the game can have only one Entry per a screenful. The reason lies in how the game is designed: There is just one global variable in the game RAM that determines where pipes or vines in the current screen lead. In the same screen there cannot be Entries to different places. Once the screen scrolls far enough for the next target label to be loaded from the level data, all Entries in the current screen will lead to that place. Thus, the screen scrolling position determines where the current Entry leads.\nIf the screen is behind Mario (Mario is on the right side of screen), you can enter the previous Entry's target.\nIn 4-2 of SMB1, after doing the vine teleporting glitch, you can enter the next pipe and it will lead you to where the vine would have led.\nThe vine is not pivotal to this glitch. The only thing needed is that the screen hasn't scrolled too far when the pipe is entered.\nIf the screen is ahead Mario (Mario is on the left side of screen), you can enter the next Entry's target.\nIn 8-4 of SMB1, in the area before the water area, instead of going in the pipe after the lava pit, go a little past the previous pipe, and then turn around and go in that pipe, and it will take you to the water area instead of taking you back to the beginning of the level.\nThere are several occassions when Mario can partially or even completely enter a solid object (brick, question mark block, etc).\nThe easiest of them, though not very useful, is to stand under a brick so that less than half of Mario is under the brick, and then jump up. Mario will not hit the brick, and will get pushed away from the brick instead.\nWhen the pushing away effect happens, if you try to manually steer away from the brick, the brick actually pulls you inside instead of ejecting.\nThis is a basic technique which has many uses.\nIf you are quick enough, you can catch the bonus item through the block even before it has completely exited from the block it came from.\nIn the example image here, also is demonstrated that it's possible to jump again after picking a mushroom \/ fire flower (do an extra tall jump).\nTry this trick in SMB1 at 4-2 when you go to the ceiling next to the vine.\nFor understanding the title, any solid object (including pipes) works as a floor. This trick is most commonly utilized in the horizontal pipe at the end of level 1-2 in SMB1 and SMB2j.\nIt is really difficult to perform (much harder than the walljump), but here are the instructions.\nYou need proper speed. If the object is only 2 blocks tall, such as the pipe in the end of 1-2, you can not jump into it while running. Use the left+right trick to deaccelerate quickly before jumping. If you have too high speed, you'll just find yourself standing on top of the object instead of inside it.\nYou need the right position. The right position is: as near as possible. You are too near, if when you jump toward the object, you'll bump into it without getting ejected pixel by pixel.\nImmediately when the wall starts ejecting Mario, start steering away from it at maximum power. This causes the wall to pull you in instead of push you out.\nImmediately after your vertical movement stops, start moving to the direction you want to go. If the wall is ejecting you now, you failed some of the previous steps.\nThis trick is apparently also possible in SMB3 (noticed against some water level end pipes), but the exact function hasn't yet been analyzed.\nTo perform the trick, do a walljump into the wall position right below the ceiling. When you get foothold, jump immediately. When you land, jump again. When you land again, jump again. Usually three or four jumps are enough before you can walk through the wall. Each jump takes you deeper into the wall.\nDuring each of the jumps, you need to steer heavily to the opposite direction you're going to. Otherwise the wall will eject Mario.\nNote: Steering only has effect during a jump (i.e. when Mario is not standing or walking). If Mario is standing, the direction of ejection is determined by the location of his upper body in relation to the left edge of the wall. (If it's partially out from the wall, he'll be ejected to the left (unstoppable unless you jump). Otherwise, he'll be ejected to the right (stoppable, speedable).\nYou need to perform the trick at frame-by-frame precision to do it optimally.\nNote that this trick cannot be done with small Mario in SMB3.\nAs demonstrated in the left image, big Mario doesn't need to do any of the complicated stuff that applied to when he is small. When he is big, he can simply perform a duck-jump into the corner and start walking. This is because his head will be well inside the ceiling, enabling the ejection to the right. (If he ducks again before he is fully inside the wall, he'll be ejected to the left instead.) This is actually the easiest way to enter the well-known \"Minus World\" in SMB1.\nIt is possible to perform the wall-entry trick even without a ceiling. However, it requires incredible luck and precision. This is the hardest of all of the tricks.\nThe key to perform the trick is to jump into the wall in such curve and speed that when Mario tries to jump again, he'll hit the upper block inside the wall, instead of the ceiling that now does not exist.\nThis trick is not available in SMB3.\nThis trick is trivial to perform. When there is a hole on the left side of the screen, you can cram Mario in between of the screen edge and the wall, and climb upwards the hole. Mario will be unable to fall down. It can be used to reach the ceiling of the dungeon in places where there normally is no route to the ceiling.\nBy jumping repeatedly and steering to the left, it can also be used to enter the wall, as shown in the right example image.\nBesides brick walls, it works also with castle walls, pipes and other types of solid walls.\nWhen you need to accelerate from zero speed, you can do it faster if you jump a small jump backwards and steer to the opposite direction Mario is facing.\nThis trick obviously can not be used if you are unable to jump (such as when performing the 'jump into a solid object below a floor' trick).\nBy pressing right+left at the same time, you can shoot to the left and the shots travel horizontally at twice the normal speed. It looks very aggressive, compared to the normal merrily bouncing fire balls. Though, this technique is more userful in water levels since it doesn't slow down Mario.\nYou don't need to land on the top of an enemy to kill it. As long as you're moving downward as you hit the enemy, you'll kill it. If you time it just right, sometimes you can even hit the bottom of the enemy. This is useful for killing enemies that seem to be too high to jump on.\nThis is not a walljump.\nAlternative names: koopa jump, belly kick.\nWhen you stomp an enemy, the enemy becomes a friend for a short while, so if you stomp it from below, you don't need to worry about being immediately killed by it when you bounce upwards.\nIn fact, you can sometimes get inside an enemy right after stomping it, and the enemy doesn't harm you as long as you're inside it.\nHow it's done: If you jump toward a turtle while it's in the air, you can stomp it while moving upward and keep moving upward. The maximum height that you'll be able to reach if you keep holding down A depends on the exact frame that you made your jump and which part of the turtle you hit, so if you aren't bouncing high enough, keep adjusting your jump one frame at a time.\nThis is useful for being able to get over an obstacle or landing on another enemy after bouncing off a turtle.\nIn SMB2j, a special case was added for flying turtles: When you \"stomp\" them in upward motion, you'll gain a lot of vertical velocity.\nIn SMB3, the same can be done for a couple of walking enemies too, when you are running at top speed.\nHow it's done: The game does a very poor job of detecting collisions with hammers, so sometimes you can go through a hammer without getting hit, and sometimes you can completely miss a hammer and still get hit.\nWhen you reach Bowser in 8-4 or some other castle, even if it looks like there's no gap in the stream of hammers, it's still likely that you'll be able to get through without taking a hit.\nIt is obviously easier if you're small or ducking.\nShortly: Press left and right simultaneously for 1 frame.\nWhen you press left+right, you'll start walking backwards to the right. How exactly it works is slightly different in SMB1 and SMB2j, but what is common for both of them is that your running speed is instantly reduced to a walking speed or even slower. This allows for a much more sudden braking. It can be applied anywhere where you need to brake quickly.\nIn SMB1, fireworks come when a level is completed with the last digit of the timer at 1, 3, or 6.\nIn SMB2j, fireworks come when the last digit of your coin counter matches the last digit of the timer.\nGetting fireworks wastes time, so you should avoid them.\nThis is a well-known trick which requires the fiery Mario. When you start a level or exit from a pipe, hold B and jump, and continue holding B. When you move, Mario will slide instead of running, for some time. This trick is trivial to do on the real console.\nIt is possible to stand on the edge of a pipe even where's a plant coming out of the pipe. There are a few pixels of room at both sides of the pipe. You can land on it, and jump immediately again.\nIn SMB1 and SMB2j, every time the screen blacks out (enter a new level, enter a pipe, exit a pipe), the game delays for a varying amount of time.\nThe delay is actually calculated so that the current playing progress will be rounded up to next 21 frame boundary.\nAnd so on, where N is a constant minimum length of the black screen.\nTherefore, the only part of the game that can be improved by a time less than 21 frames is the last part of the last level.\nInternal to the game, Mario's powerup state is recorded in two independent variables; one describes size (0=regular; 1=super) and the other tracks the 'upgrade state' (0=need mushroom; 1=need flower; 2=has flower). The 'size' decides the sprite (small on 0; big on 1) used for Mario; the 'upgrade state' decides what powerup will come out of a block (mushroom on 0; flower on 1 or 2) and what will happen when Mario is hit (die on 0; switch 'size' and clear 'upgrade state' on 1).\nThese states can be knocked out of synchronization. The game checks for all sprite collisions, knocks the 'upgrade state' to 0 if it hits, and switches size only when all checks are done. If Mario's sprite strikes the King Koopa sprite and the bridge release sprite in one frame, the game will register the collision between Mario and Koopa, knock the 'upgrade state' to 0, register the collision between Mario and the bridge release, and jump into the level complete loop that causes Mario to walk to the right ? and never switches the 'size' variable.\nIf Mario was naturally large (state 1 or 2, size 1) when he hit the switch he will stay large (state 0, size 1). If he strikes an enemy he will die (because his 'upgrade state' is 0), but he can break blocks. If he gets a mushroom, he will shrink and be unable to hit blocks; if he is hit now, he will grow again. If he gets a fire flower, his palette will be changed and he will be able to shoot (but the game will use the large image for shooting; it will, however, keep Mario's small feet if Mario is moving). Mario will now be Little Fiery Mario.\nIf Mario was naturally small (state 0, size 0) when he hit the switch, he will appear to die but the level will still be beat and he will not lose a life. If Mario was unnaturally small (state 1 or 2, size 0) when he hit the switch (that is, if he does this trick twice), he will revert to his normal form. When Mario dies his size is set small and his state is set unupgraded, no matter how he died or what his state was before.\nUsing the third part only of a 3-code Game Genie code for \"Always Stay Big\" will produce similar results (the code is SZLIVO). As long as Mario has gained a power-up, he will become small when hit, then grow when hit a second time. This is a much simpler way to view the \"Little Fiery Mario\" quirk.\nThis glitch appears to have been fixed in the Super Mario All-Stars version, though the two-variable mechanism behind it still exists.\nSharp readers already spotted that an example of how to get to the \"minus world\" was shown in the \"Application: Jump into a wall just below a solid ceiling and walk through it\" chapter of this page.\nThe minus world is basically just a random location of the game ROM, it is not a real world. It wasn't designed that way ? it's just an oversight of the programmers that you can get into the pipe before the warp labels are assigned.\nThe information shown on these two following pages is hoax . There really is no exit from \"world -1\" (which is actually world 36, technically speaking). It was never designed to be.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dan L. Thrapp, a chronicler of the American West, died on Friday at his home in Tucson, Ariz. He was 80.\nThe cause was a cerebral hemorrhage, his family said.\nA native of West Chicago, Mr. Thrapp began his travels as a paleontologist for the New York Museum of Natural History in the 1930's. At one point the museum ordered an aerial search for Mr. Thrapp when it feared he had been lost in a solo exploration of cliff-dweller ruins in an unmapped canyon region of southern Utah.\nIn World War II, he served with the Army in Burma and China, became a captain and won four battle stars. After the war he was a foreign correspondent for United Press in England, Greece, Italy and Africa. He joined The Los Angeles Times as religion editor in 1951 and retired from that position in 1975.\nMr. Thrapp was an authority on the Apache wars in Arizona and New Mexico in the second half of the 19th century and wrote six books on the subject. They included \"Al Sieber, Chief of Scouts\" in 1964 and \"The Conquest of Apacheria\" in 1967.\nHe also wrote a four-volume \"Encyclopedia of Frontier Biography,\" published by the University of Nebraska Press in 1991.\nMr. Thrapp is survived by a daughter, Linda Nicholl of Seattle; a son, Col. Richard Platte of Washington; and three sisters, Dorothy Reisner of New Paltz, N.Y., Josephine Drexler of Cedar Rapids, Iowa, and Beatrice Zeeck of Lubbock, Tex.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbndhd b/data_all_eng_slimpj/shuffled/split2/finalzzzbndhd
new file mode 100644
index 0000000000000000000000000000000000000000..2aebb21bc1fe5b62cedd0066962af01f4d6e4932
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbndhd
@@ -0,0 +1,5 @@
+{"text":"First things first, I've got to brag up the AMAZING breakfast this morning. While today's post isn't focused on it, the Parisienne Crepe blew my taste buds away\u2026 Bacon (middle bacon to be specific), onions, eggs, & cheese all in one delicious buttery package? Yes please!!!\nWhat I'm most thankful for today is very much unexpected. When Becky shared her idea for the day I was really only focused on the geothermal wonders we were going to see, especially the geyser. She'd mentioned seeing some M\u0101ori stuff, but I kind of put it in the back of my brain\u2026 kind of the filler in a busy vacation day. Yeah, it'd be cool, but more of something to pass time than something I'd really wanted to see. I couldn't have been more wrong!\nWe went to Te Puia for a couple of hours and had an incredible experience. Our tour quickly moved into their carving school and my mind was blown. In front of us roughly 6 M\u0101ori men were carving elaborate wood statues using chisels and wooden hammers. Watching them work was amazing, their attention to detail so precise.\nAs we learned, they were students of a school that takes on four new students each year and teach them the ancient art of M\u0101ori wood carving. After they completed the school, 3-4 years later, they would go back to their home and teach others to help keep their culture alive. How awesome is that? Learning specifically with the end goal of teaching others.\nWhile standing there I had a case of \"the feels\" come over me and I had to focus to not tear up. Many times over this vacation and other experiences I've either teared up or come close because I've felt like Dad is right there with me. When I was watching them carve I thought about how cool Dad thought this would be to see. I swear that somewhere in my heart I knew he was there with me, in just as much awe of their art as I was.\nIn addition to the carving the M\u0101oris also demonstrated their weaving skills. We also watched a welcome ceremony in which they sang, shared dances and stories, and did the Haka. It was AWESOME!!! Watching them keep their traditions alive was inspiring, and seeing the boys' eyes light up at the event was fantastic. Gavin even got a pic with a few members right afterwards.\nAfter touring the entire site we went back to the woodcarving school and I spent a good 10 minutes or so watching the students work their art. It was so intriguing to see how they were quickly turning a piece of wood into a piece of art.\nEverything else today was superb. We saw the ocean, walked the beach, had a unique dinner, and even saw California redwoods in New Zealand! That said, I'm most thankful for that experience of the M\u0101ori culture. Seeing the world through a slightly different perspective was an experience I'll reflect on often and draw inspiration from. I've had the boys spending time journaling each day about what they did and what they are thankful for. Hearing Gavin say that he was thankful for the M\u0101ori culture was music to my ears, I'm glad he took something from this experience as well!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The ability to add the Change Network Location submenu to the Desktop context menu to quickly set your network connection to Private or Public.\nThe ability to integrate Snipping Tool to the Desktop context menu.In recent Windows 10 versions, this option will add a submenu with the item to capture a screen region.\nThe ability to add the Control Panel submenu which contains the commands \"Category\", \"Large icons\" and \"All Tasks\".\nThe ability to add the command \"Open PowerShell as Administrator\" to the context menu of File Explorer.\n-Give access to (Share with in previous Windows versions).\n-Share - the modern Share command available in recent builds of Windows 10.\n-Edit with Photos - also a new command which was added recently to the context menu of image files in Windows 10 Fall Creators Update.\nUnder Network\\Reset Data Usage, you can reset the network data usage with one click in Windows 10.\nI fixed the \"Customize This PC\" option in the Tweaker app. Now it can remove the 3D Objects folder from This PC in Windows 10 Fall Creators Update.\nHey folks, I am VERY happy to announce a new major version of the app.\nWinaero Tweaker 0.8, the first version which comes with import\/export for changes you made!\nA new wizard can be found under Menu - Tools -> Import\/Export Tweaks.\nThe ability to export and import changes you made with Winaero Tweaker.\nYou can go to the desired Registry key with one click using a new option under Tools.\nYou can configure split threshold for svchost.exe in recent Windows 10 builds.\nHere are a few things you need to know before proceeding.\n- upgrade the app to version 0.8.\n- reset the changes that you have made to their defaults for critical tweaks you would like to export.\n- export tweaks to a file.\nYou can apply tweaks in one version of Windows, export them to a file, and import them in another Windows version. This scenario is supported. In this case, the app will ignore the Windows-version-specific tweaks and apply only those tweaks which are common between two different Windows versions. Pretty cool, isn't it?\nIf something goes wrong during export or import, the app will place the file WinaeroTweaker.log in the same folder where the ini file with your tweaks is stored. It will be opened automatically so you can review it. Please email me the log file in such situations.\nA number of tweaks related to advanced system appearance like \"System font\", \"Status bar font\" and so on won't be exported. The advanced appearance options are not officially supported in modern Windows versions. Importing them can cause unexpected behaviors in the OS, so I have decided to exclude them from import and export for now. Please consider configuring these options manually when required.\nYou can add the Run with priority context menu to File Explorer.\nYou can add the Shut Down context menu to the Desktop to save your time (and avoid unwanted apps auto-starting if you are running Fall Creators Update).\nYou can add the Power Options context menu to the Desktop.\nThe ability to add the Settings context menu to the Desktop in Windows 10.\nThe App Mode context menu to switch between the light and dark color scheme quickly in Windows 10.\nYou can change the default RDP port to what you want.\nYou can manage hibernation options in Windows 7, Windows 8 and Windows 10. You can disable the hibernation feature, shrink the hibernation file to its minimal size or set it to a custom size. Also, you can disable hibernation but keep Fast Startup.\nYou can quickly hide a drive from appearing in File Explorer.\nUsing Winaero Tweaker, you can unlock the ability to pin more than 3 contacts to the taskbar.\nYou can add the \"Personalization\" link back to the Control Panel. You'll be able to add both kinds of Personalization links - the Settings app and the classic applet.\nYou can set the desired default action for the classic Shut Down Windows dialog(Alt+F4 on Desktop).\nYou can enable Windows Installer in Safe Mode. This will allow uninstalling apps which were installed from an MSI package.\nFinally, you can enable the Emoji Panel (Emoji Picker) for all languages in Windows 10.\nFixed: The \"Command Prompt as Administrator\" and \"PowerShell as Administrator\" menus were not working in Windows 7.\nFixed: The \"Wallpaper location\" context menu wasn't working in Windows 7.\nFixed: The \"Elevated Shortcut\" feature wasn't working in Windows 7.\nFixed: The Control Panel context menu wasn't working in Windows 8\/8.1.\nFixed: The \"Disable wide context menus\" feature was reporting invalid status. Also, I've added a \"Restart Explorer\" requirement.\nFixed: The default file name template \"Winaero Tweaker_8\/17\/2017.ini\" in Import\/Export was changed to \"Winaero Tweaker_8.17.2017.ini\" for all languages and supported operating systems.\nFixed: A \"null reference\" error message could appear after you remove \"OneDrive\" from the navigation pane.\nUpdated: The \"Enable new share pane\" tweak is now removed for Windows 10 version 1703 and above.\nUpdated: The \"Enable the new OneDrive flyout\" tweak is now removed for Windows 10 version 1703 and above.\nUpdated: The \"Enable new share page\" tweak is now removed for Windows 10 version 1703 and above.\nUpdated: The tweak \"Disable Auto Reboot after Updates\" is now removed for Windows 10 version 1709.\nThe key changes of this release.\nGet rid of ads in Windows 10, including ads in Settings, Timeline, and People. Winaero Tweaker 0.10 allows disabling them.\nThe ability to disable updates and notifications about them in Windows 10. Reliably.\nThe option to add a special sign-in message that will appear for all users every time they sign in.\nThe option to prevent Windows 10 From deleting the thumbnail cache for image and video files.\nStarting with Winaero Tweaker 0.10, there is a new category of options called \"Shortcuts\". It combines a few previously available options with a new option, \"Classic Shutdown Shortcut\". Use it to create a shortcut to the classic Shutdown Dialog Windows dialog.\nThe option to prevent Malicious Software Removal Tool from Installing is now available under Windows 7.\nFixed the broken import\/export feature for the context menu option \"Run command prompt as Administrator\".\nThe option \"Open Registry Key\" now opens the Registry Editor when you hit the Enter key in the text box.\nThe search result pane and the category view now use double-click to open items.\nWinaero Tweaker now remembers the collapsed state of nodes in the navigation pane between sessions.\nA new version is here.\n* The \"Elevated Shortcut\" feature now allows specifying and changing the full path to the LNK file.\n* The Disable Windows Security Center tray icon feature now supports Windows Security.\n+ Allows the user to prevent Windows 10 from reinstalling built-in apps.\n+ You can reset all the changes you made with Local Group Policy Editor or manually by editing the Registry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Help us provide a hot meal and\/or gift to those at risk from homelessness this Christmas.\nThe festive period can be a very difficult time for many of the people accessing our services and we are often the only support network they have. This includes (and is not limited to) those suffering from and\/or at risk of homelessness, domestic violence, drug and alcohol addictions, mental health issues and learning difficulties.\nEach year we support over 3,500 people across Hampshire - Our Give a Christmas Meal campaign aims to be able to provide as many of our service users as possible with a hot meal and\/or small gift to help them feel valued at a time in which many can feel very alone.\nAny additional money raised in this campaign will be used towards our on-going mission to buy more housing across Hampshire, in order to get people off the streets and into suitable move-on accommodation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"40 Days of Lent and What to Give Up?\nToday (Wednesday 2\/18) is Ash Wednesday and day one of the Lenten season.\nAccording to Catholic On Line, \" Ash Wednesday marks the beginning of the Season of Lent. It is a season of penance, reflection, and fasting which prepares us for Christ's Resurrection on Easter Sunday, through which we attain redemption.\nIn simple terms, Catholics normally do away or give up something for the 40 days of Lent.\nYou can see the top 100 list of Lenten sacrifices here.\nFor a change during for Lent, some take on added responsibilities such as helping others in shelters or soup kitchens. I have even taken on the additional spiritual responsibility of attending mass or worship services everyday during Lent.\nSome people diet for the 40 days or exercise more during this time.\nWhat do you plan to give up or additional duties for Lent?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dr R and I decided to go snow monkey park from Tokyo at 1\/1 after we realised nearly all the major sightseeing place (no matter its museum or government building..) was closed at the first day of the new year. Even the shops in the Tokyo station were all closed! So we decided to do sth a little bit out of the town, and we picked snow monkey because they still opened!\nWe (more like I..LOL) checked a lot of travel websites before we go there and all of them seems quite complicated. My friends were suggested to drive there because they did and it was pretty fantastic experience. We hesitated because we didn't feel comfortable enough to travel in a foreign country with a different language system, especially in the snowfield.\nSo after a bit of research, we decided to go to JR Nagano station first. This seems is the most simple, convenient way for me because you don't need to change too many types of transportation and very easy to follow.\nJR line: Tokyo station to Nagano Station (covered by JR pass) 85 min -105 min depend on which Shinkansen you get. We took the 8:15am one but can be earlier if you can wake up.\n1. One day unlimited use of the Nagano Dentetsu Train Line.\n2. One day unlimited use of the Nagaden Express bus (Nagano station \u2013 Snow Monkey Park).\nReturn express bus fee (2 way): 2800y plus Snow Monkey park entry fee: 800y which is 3600. So the pass will save you around 400 y, and also give you some flexibility if you want to take the local train back and get off in the middle to do some additional sightseeing.\nIt's a good idea to plan your time well before the trip because those express buses are not that regular. We were literally running crazy to catch the 10:15 one, otherwise we have to wait a nearly whole hour! The bus station is just right outside the JR station.\nTips: Because the snow monkey park is too well know right now, so it was not as hard as I expected to go there. If you are still confused, you just need to tell the Nagano station staff \"snow monkey park\", all of them will try their best to explain to you how to get there.\nWe feel like a coffee and some sweet stuff before the climbing, so we made a stop at this small cafe called Crystal Terrace right next to the bus stop.\nIt's hard to miss it. The cafe also sell those cute small gifts.\nAfter the short break, we started our trip to the Snow Monkey Park.\nFrom this point, it involves a bit of uphill and also some steps. The sign said 25 mins to the main Monkey Onsen area, but it took us about 30 mins because the road was really slippery. Wearing proper shoes are highly recommended during winter.\nThe view is spectacular. It's white everywhere. Cold but so pure, peaceful and beautiful.\nAlong the road, there are a lot of those small information signs indicate where we are currently in terms of the whole 1.5 km. Walking really slow!\nWhen it was close to the Main Monkey Park, we already see several monkeys around playing or finding food to eat.\nAfter passing this small village point, and a small uphill, we were at the main area of the Monkey Park. The main entry has a small wooden cabin as the ticket checkpoint. After the entry, we saw quite a few of monkeys.\nThey do take onsen during winter!! According to the staff, those onsens in this central area of monkey park was built for them. We also noticed that there was food on the snow. The staff here might feed them every day to keep them in this area.\nThere were quite a lot of visitors around during the first day. I got amazed to see so many people around. At some point, I feel the number of people is more than monkeys.\nWe spent a little more than one hour here, mainly just staring at those monkeys. It was fun to see them enjoy themselves and completely ignore us.\nWe were pretty hungry after this trip up and down. Walked in this cold weather definitely burnt us a lot of energy, so we were hunting food afterwards.\nDr R wanted some soba because we saw an advertisement on the way up to the park about homemade soba nearby. It's also Japanese's tradition to eat soba at the new year, which to celebrate the old year passing.\nIt can be annoying to try to find a small soba place at this cold with an empty stomach, but we didn't give up, and we really enjoyed it.\nWe ordered both soba and udon to try. Both of them are pretty decent. I can taste the sweetness of the grain in the noodle.\nThe owner explained that black bean and the sweet egg cake are traditional new year dish for Japanese. The apple is the local produce from the nearby farm.\nAfter we finished, we got two big apples from the owner couple which was a big surprise to me!\nTake a big bite on our way back to Tokyo, it's really really sweet! Until now I still missed it!\nIts durable to take a day trip to see snow monkey. It's about 2- 2.5 hours one way. The Shinkansen is reasonably comfortable. We didn't feel exhausted after the train.\nIt's a good idea to get up earlier than late so you can have some free time walk around Nagano after the snow monkey park. We didn't get enough time to do that.\nWear enough cloth and proper shoes during the winter season! Walk slowly if there is too much ice on the road.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbndut b/data_all_eng_slimpj/shuffled/split2/finalzzzbndut
new file mode 100644
index 0000000000000000000000000000000000000000..57c51e292685a4209850b4082de306cf9583d7e0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbndut
@@ -0,0 +1,5 @@
+{"text":"Luxurious pair of pillows by Royal Hotel offers medium-Firmness and comfort for side or back sleepers. Features down proof 100-Percent long staple cotton shell woven at 240 thread counts per square inch. And a generous fill of white goose feathers and goose down in 95-Percent to 5-Percent Feather to Down Ratio. The down proof cotton cover was Featuring quilted with gusseted edges for added durability to prevent Feathers from escaping your pillow and will not poke or scratch. Hypoallergenic, Contents are treated to prevent growth of mold, mites and odor producing bacteria. This two-Pillow Set offers long-Lasing use and comfort for a restful night's of sleep all year round.\nThread Count: 240 Thread Count per Square inch.\nStandard-Queen size pillow measures 20 inches wide by 28 inches long filled with 35 ounces of White Down-Feathers.\nKing size pillow measures 20 inches wide by 36 inches long filled with 46 ounces of White Down-Feathers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I must apologise for the lack of entries recently, but I've been on a whistle-stop trip to Exeter in Devon, near to the RHS's west country garden at Rosemoor.\nSituated in Great Torringto n in North Devon, Rosemoor is the most westerly of the four RHS gardens and a mere 40 odd miles from Exeter. The River Torridge - which borders one side of the garden - and the surrounding area, is famous for Henry Williamson's classic story of Tarka the Otter and the scenery in this part of Devon is truly stunning.\nWe've been working closely with the RHS over the past year or so, and our fruit cages have been introduced at the flagship garden in Wisley, Surrey - a great opportunity to see these high quality structures up close. We'll also have a presence at the other end of the country as Harlow Carr, another RHS garden, hold their Vegetable Sowing and Growing Day on April 11th where our raised beds will be on display.\nI'll keep you posted on my next road-trip when it happens!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A short while back I found out about a gent who drives around the village every Saturday with locally grown veggies to sell to us \"locals\". What a treasure chest he has in his trailer! Everything is washed and what can be peeled and cut up are all nicely packaged and sealed in bags. I was very excited to hear that everything has a standard price of just R10 a bag. I went a touch overboard and Pete was a little concerned about the amount of veggies he was going to eat. I would have purchased a lot more but I had just been into town the day before and had already bought potatoes, onions, carrots, tomatoes and a few other goodies. I am very excited to see what he will bring every Saturday. I have my shopping basket ready and waiting.\nOn chatting with this lovely farmer, he can also arrange a whole lamb at just R45 a kilo! Mutton at R35 a kilo and chicken, which I forgot the price of. Not to worry he will be back on Saturday and I can check with him then.\nand my beautiful ebony wood sugar bowl from Zanzibar which strangely enough has a bullet in it.\nWhat's in your kitchen this month? Please link back to Celia's blog Fig Jam and Lime Cordial if you partake in the fun.\nThis entry was posted in Daily Blog and tagged broccoli, cauliflower, Food, In my kitchen, Ingredients, pumpkin, sweet potatoes, The Complete Cook Book by thecompletebook. Bookmark the permalink.\nHi Mandy! \ud83d\ude42 I'm kind of curious how those ingredients are gonna be used in your kitchen. hmmm.. And sweet potato pie sounds delicious!\nI just wrote my draft to post this afternoon and also mentioned how time has flown. I want to ask about the bullet, there must be a story?\njust interested (especially in the mutton prices you mentioned) can you say exactly where this village of yours is situated.- maybe we could take a drive out there for those beautiful vegies too.\nWhat a wonderful idea\u2026everything packaged and ready for the freezer or fridge. Look forward to the sweet potato pie recipe! Have a great day!\nPUMPKINS ;-)) and pears, cabbages..just had lunch with some cabbages (that my husband dislikes to the core)\u2026sighs.\nHow fun to see what you have- I am wishing I was like you and had those amazing veggies on hand, especially those potatoes! YUM! That sugar bowl is too darn cute- wonder how it got that bullet? Hum\u2026.\nLooks like all the makings for a great meal.\nI have some beautiful pears, apples and a few more tomatoes. Produce delivered to your door \u2013 how awesome is that!!!\nA real bullet? how cool is that\u2026 (I think you should make up a story if there isn't one :)) Did you prepare the bags of veggies yourself or buy them like that? \u2013 love that they are all set to go\u2026 so wonderfully convenient.\nThis \"in my kitchen\" is fun game! We can see and imagine what your kitchen is like\u2026..a place where you make wonderful food! I love the unique sugar bowl and it stands out really well. And sweet potatoes\u2026my fav! \ud83d\ude42 Have a great week Mandy!\nWhat a find! I would love to have some nice local farmer come by with his goods. But I live in the suburbs of Los Angeles and I won't hold my breath.\nI LoooOVE what's in your kitchen. I'll be right over for some sweet pie when you make some :))) X Can't wait.\nsend the vegie man my way please!\nAh my favourite snooping time of the month! The bullet? How did you get it back through customs from Zanzibar?\nThis is so fun! I love having a frig of veggies; kinda like an artist's pallette just awaiting a creative inspiration to pull together a fabulous dish! That's some service having your veggies arrive all prepped and ready to go!\nThis is a great idea. It's so fun to work with ingredients you have in the fridge and pantry. I'm always up for the challenge. \ud83d\ude42 Can't wait to see what you make!\nFarm-fresh produce delivered weekly to your door, cleaned, trimmed, & bagged. You simply cannot ask for a better arrangement. How lucky for you!\nI am jealous of that gorgeous broccoli\u2026my favorite veg. I love to roast it in the oven, with cauliflower too sometimes. Just toss in good olive oil, salt and pepper and roast it stirring once or twice until starting to brown\u2026gives it a lovely nutty taste. Great blog\u2026my first time here!\nSuch wealth \u2013 a fridge full of fresh veggies! One of my favourite ways with cauliflower is roasting them \u2013 arrange the florets in a dish, sprinkle with curry powder and olive oil, cover with tinfoil, roast for about 20 minutes, remove the tinfoil and roast til slightly crispy and brown \u2013 YUM!!!! Do you do that too?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Inframatic 9500 was designed with modern electronics and the highest precision optical components, guaranteeing the best possible data quality. Combined with the use of advanced chemometrics techniques, the result is unparalleled accuracy. Stable scanning grating optics and reliable calibrations guarantee that you will always get the correct results.\nThe robust construction of Inframatic 9500 with sealed module construction of standardized components guarantees long term reliability and makes the instrument virtually maintenance free.\nThe short analysis time combined with the simple sample handling makes the Inframatic 9500 one of the fastest whole grain NIT analyzers in the market.\nWhen grading grain it is important to have full control over the calibrations used in each instrument and to be able to monitor performance in virtually real-time. By using Ethernet, the administration of the network will always be accessible 24 hours a day regardless of the location of the instruments. A large number of Inframatics are operated in networks globally.\nThe optics house is machined out of a solid piece of metal, which guarantees all instruments to be identical to each other. Thus calibrations are fully transferable and all Inframatic 9500 units produce the same results for the same samples. Wavelength calibration is performed with NIST references, providing full traceability back to measureable standards.\nThe large color touch screen with its intuitive menus makes it easy both to operate the instrument, and read analysis results \u2013 even from a distance. We have designed the user interface to make analysis as rapid and secure as possible. All the operator needs to do is to select which type of grain to analyze and pour it into the funnel. No manual settings or changes are necessary.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FOXERS papaya boyshorts are a pinkish red with a colorful and artistic waistband with a geometric design like a kaleidoscope. The boy short trunk cut has a short boyleg design and the back hug your cheeks making your tush look sweet. In addition on the back a small lipstick pocket, a small bow on the waist and a slight ruching on the center seam make your but look super cute. The legs are capped off with a grey lace detailed with small bows. Cotton crotch. 92% Cotton, 8% Spandex. Machine Wash and tumble dry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnrze b/data_all_eng_slimpj/shuffled/split2/finalzzzbnrze
new file mode 100644
index 0000000000000000000000000000000000000000..d359d4fccb0861d30dee3c4cd5571cee42899005
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbnrze
@@ -0,0 +1,5 @@
+{"text":"\"Absence makes the heart grow fonder.\n\"Ah, isn't that a lovely sentiment?\" he continued, then paused to let his words sink in.\nThe speaker above was Professor Shanahan, and his class was titled \"Philosophy of Human Nature\" \u2014 part of 'core curriculum' of classes required by all freshmen whose majors fell under the broad College of Liberal Arts.\nThis professor was no doubt trying to point out very different ways of approaching any single situation.\nI remembered it very well because of how the phrase \"out of sight, out of mind\" described the state of my short-term memory \u2013 the part of my brain that had sustained permanent damage from my surgery in 1988.\nOn a previous blog post, I'd started to describe how I store new information into my long-term memory.\nHere, I address dealing with the much more complicated portion of my working mind: my short-term memory, along with how I compensate for its permanent damages.\n\"Short-term memory, also known as primary or active memory, is the information we are currently aware of, or thinking about.\nIn Freudian psychology, this memory would be referred to as the conscious mind.\nHas someone ever spewed at you a stream of words in a language that's completely foreign to your ears?\nHow many syllables did you catch?\nCan you repeat what you've heard five seconds later?\nWhat about five days later?\nThat's your short-term memory at work, when you don't have a context in your long-term memory into which you can make sense of, and store, this new information right away.\nAfter undergoing brain surgery at age 14, my short-term memory capacity was whittled down to not 20-30, but 3-5 seconds \u2013 at best. This not just involves foreign languages, but all new incoming information. All the time.\nFurthermore, if two, three, or four distinct pieces of new information arrive in rapid succession, I'd count myself lucky to capture just one or two of them. Hence: \"Out of sight, out of mind\" suitably describes my short-term memory.\nWhile many of our short-term memories are quickly forgotten, attending to this information allows it to continue on the next stage: long-term memory.\nc. take time to organize and link worthwhile bits of this information into a structure that's already inside my long-term memory.\nThe first concept focuses on providing students with an overall concept or idea, immersing them in the big picture, before working down to the finer details of that concept or idea. A \"bottom-up\" approach, on the other hand, presents small chunks within a topic, piecing them together, then builds up to the larger picture.\nSo if you wanted to teach the English language to someone who is first learning it, a \"top-down\" approach would be to place him or her into an environment where only English is used to communicate. After he or she becomes a bit familiar with his surroundings, then introduce the elements of grammar, mechanics, usage, and the like.\nIn contrast, a bottom-up approach would start by presenting a student with key vocabulary words and phrases, building up to sentences and paragraphs. Some rules of grammar, usage, and verb agreement will be worked in along the way, eventually arriving at how the language works overall.\nA bottom-up approach would start with elements such as scales, key signatures & key relationships, notes & rhythms, then building melodic lines, harmonies, chords and chord progressions, etc., before fitting everything together to present major works of music.\nIndividual elements simply do not stay in my short-term memory; I must first locate a context within my long-term memory to hold new information right away.\nCLASSICAL MUSIC from piano and violin lessons, workshops, evaluations, performances, etc.\nEver since, I am able to retain information that fall under these two categories without much effort \u2013 even after my brain surgery had inflicted its damages.\nI NEED TO CONSTANTLY PREDICT, TO THE BEST OF MY ABILITY, THE KIND OF INPUT THAT I AM LIKELY TO RECEIVE \u2013 THEN PREPARE THE APPROPRIATE CONTEXTS WITHIN MY MIND, in order to receive this information TOP-DOWN.\nIn other words, I need to to anticipate as accurately as possible, before each interaction \/ undertaking \/ new experience takes place, what to expect.\nOtherwise, everything becomes a mess in my mind, and I retain nothing.\nThen I rehearse words to describe the possible topics I may want to express, and any responses that may come in handy. In my mind, I try out these words, expand them into phrases, sentences, and paragraphs. I brainstorm likely scenarios that I may encounter, come up with more words, and place them into a coherent order \u2013 much as if I were writing scenes for a story. Only after this preparation can I be fairly confident in making a fair impression on others.\nImmediately after each new experience, I take time to reorganize everything into writing \u2013 and in the process find words to describe what just took place, unearth the causes behind each part of the experience, balance them with their effects, and place these words and sentences into a coherent order.\nAll of this to liken the new experience to the structure of fiction inside my long-term memory.\nOnly then can I accurately retain the incident, at the same time keeping any associated emotions under control .\nSo: the next time you see me silently distancing myself from others while they chatter away, you'll know what's going on inside my mind. The same if you see me taking a long, solitary walk in the neighborhood. Or putting together a meal in the kitchen by myself, then doing the dishes while family members watch TV or talk on the phone.\nQuiet times such as these play an essential part of my ability to function. So does keeping all personal interactions brief.\nThen I leave behind what I'd written for a short time, returning to review and tinker with it multiple times. (Each time I will notice something new that I hadn't caught before.) This process also serves to drive the new words I'd constructed into my long-term memory. Only when I am fairly satisfied do I send it out for others to view.\nAs you may have already inferred from what I'd described above, interacting directly with other people is NOT something I can draw from my long-term memory to rely on. The closer I allow others to approach, the more havoc they tend to wreak on this entire compensation process that I've developed. I will further explore the experiences in this last category, along with how music works with my short- and long-term memories, in my next post.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Interface: DC power connector, AV, HDMI, USB, TF card interface.\nLanguages: English, Spanish, French, German, Russian, Italian, Portuguese, Turkish, Arabic, Chinese, Japanese, Korean, etc.\nFeatures: WIFI connection, TF card, U disk, HD connection, wired network connection.\nAdvanced operation system: Adopt the fast and powerful Android 7.1 operation system. The Android 7.1 has a better compatibility of software, games and application programs. Rapid, stable configuration coupled with the highly Android 7.1. Say goodbye to the system freezing and buffering.\nPowerful processor: Equipped with Amlogic S905X quad-core CPU which is advanced in image processing and computing. Its high machine configuration gives you ultra-fast running speed and smooth image processing performance. With the HDMI 2.0 data output, it makes sure you can enjoy 4K even 3D HD movies.\nEasy and convenient to use: Support multiple connections such as Ethernet 10 \/ 100M and 2.4GHz WiFi. It is very easy to install, as soon as you plug in the power supply and connect the box to your WiFi or else way, you can watch whatever you like. Users can surf the internet with browsers and have a communication with the world through kinds of APPs, like YouTube, SKYPE video call, MSN, facebook, also the hot films and TV shows can be searched at any time.\nPractical multi-media player: A practical home streaming media player, feel free to enjoy all your favorite movies, sports shows, and entertainment programs. And you can also save the games, movies and TV shows you like, just take advantage of the 8 \/ 16GB storage. M8S Pro supports Miracast, Airplay, DLNA, share more with your family and friends.\nIf you have any questions about this product by Cewaal, contact us by completing and submitting the form below. If you are looking for a specif part number, please include it with your message.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hillside Brasserie, Queenstown, Otago. Information, Photos and Reviews.\nThe shared lobby area of the Scenic Suites Queenstown and Heartland Hotel is also home to The Hillside Brasserie and The Lobby Bar. These are perfect places to unwind after a long-day of sight-seeing, adventure or being on the slopes. The Hillside Brasserie is open for buffet breakfast from 6am until 9.30am Monday to Friday and 6am until 10am during the weekend. It is open for a la carte dinner from 6pm until 9pm every day.\nNobody has reviewed Hillside Brasserie yet, Be the first.\nYou are submitting a booking enquiry. Hillside Brasserie will then contact you by email with details of availability and prices.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vivid Racing is one of the most established and trusted online source for automotive tuning and parts for virtually all vehicle year, make, and models. It has the widest selection of performance parts, replacement parts, aftermarket parts, and accessories from the leading brands like K&N, AFE, AEM, at affordable prices. Whether you're replacing a damaged part of wants to customize your vehicle, Vivid Racing has got you covered. Shop using Vivid Racing coupons now!\nApply This Code To Get FREE Shipping On Orders Over $199. Buy Now!\nEnjoy Up To 50% OFF Clearance Deals. Buy Now!\nEnjoy Performance Air Filters Starting At $0.46. Buy Now!\nEnjoy Battery Components From $3.81. Buy Now!\nGet 3% OFF All Orders at Vivid Racing. Buy now!\nGet 1% OFF on All Orders of $100 or More at Vivid Racing. Buy now!\nTake an Extra 10% OFF on All Greddy Items at Vivid Racing. Buy now!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am God's very special child. I am a creative, which means I'm always bustling with ideas and how to make things better. I enjoy writing, and hope to someday publish my book (I should have done this 2 years ago but my life has been full of beautiful disruptions). I retail health & beauty products and maybe one day, I'll have a beauty blog. I also waste my time doing useless internet research.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnuve b/data_all_eng_slimpj/shuffled/split2/finalzzzbnuve
new file mode 100644
index 0000000000000000000000000000000000000000..4d9726b034a506987c8ef19e5fc6a8df3bb7fda7
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbnuve
@@ -0,0 +1,5 @@
+{"text":"Gorgeous 180 degree views at Qualchan Hills. Move-in ready, main floor living features gourmet kitchen & granite countertops throughout. Upgrades include: Super efficient radiant floor heat, recirculating\/instant hot water & heated driveway. Office includes a murphy bed. Bonus room in lower level with private entrance & tons of storage. Beautiful low-maintenance landscaping surrounds this home. Note - living room photo is virtually staged.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"While nearly every other gamer out there is off buying or gifting games through Steam's Summer Sale, IGN cared to release the information that Platinum Games (The developers behind the Bayonetta games, Metal Gear Rising, The Wonderful 101, and the upcoming XB1 exclusive ScaleBound) are developing a game for the Nickelodeon TV series, The Legend Of Korra (The sequel series to Avatar: the Last Airbender).\nThe game will be digital download title, and will release in the Fall on the PS3, PS4, 360, XB1, and PC. The story of the game set to take place in between Season 2 of the show and Season 3 which is to air this Friday, June 27 in North America. Throughout the game you'll be playing as the Korra and using all four elements (Earth, Air, Water, and Fire) to defeat enemies and compete in 3-on-3 Pro-Bending tournaments.\nFor the Nintendo gamers, they don't fully luck out. Webfoot Technologies is developing a turn-based strategy game for the 3DS. Unfortunately, there isn't anything said about there being a Wii U release, but since the game is published by Activision, there could be a very slim chance.\nTagged with aang, asami, avatar, bolin, game, Gaming News, korra, legend of korra, lok, mako, Platinum Games., video game.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Happy Mothers Day! Meet my sweet Mom, she's a hoot! Stay tuned for her funny Lulu stories to come!!!\nCinco de Mayo is right around the corner! We did a taste test of 5 popular imports to help us narrow down the fave to drink for the day:) Have a happy and safe Cinco de Mayo!!!!\nJoin us in the She Shed and learn how to make these fun festival inspired bangles! These are the perfect DIY project for a girl's night! Don't forget to check out our Camel Toes Deal of the Week!\nHow To Make A Tutti Fruity!\nIf you're looking for the ultimate party cocktail, then look no further!\nThis week we teach you how to make a Tutti Fruity...Camel STYLE!\nDon't forget to subscribe to our YouTube channel and sign up on www.camelthreads.com to receive our e-mail newsletter! You don't want to miss out on exclusive discount codes and more!\nTime for some wine & SLIME!\n1 cup of warm water + 1 tsp Borax- stir together and set aside.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The property taxable value established by the Office of Lewis County Assessor.\nThe taxable value may be the same as the assessed value unless a special use exemption is applied.\nThe levy rate as adopted by the taxing district legislative authority under guidelines in the Revised Code of Washington.\nTax bills began being mailed on February 21, 2018 - You will not receive a bill if your taxes are paid by a mortgage company. If you didn't receive a bill by the end of February, and think you should have one - phone the treasurer's office at 360-740-1115.\nYou must notify the treasurer's office if your mailing address has changed.\nAny parcel with a tax and assessment of $50.00 or less must be paid in full by April 30, 2018.\nFirst half taxes are due and payable April 30, 2018.\nSecond half taxes are due and payable October 31, 2018.\nAs of assessment year 1999, tax year 2000, any parcel with a taxable value of less than $500.00 has no tax due and will not be billed.\nNon-payment of your check or electronic payment by the bank will void payment. A $25.00 fee will be charged plus applicable interest and penalty.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The iPhone can shoot 4K video, and now even the iPad can shoot 4K video. Yet, there are times when you need the flexibility and power provided by a standalone camera.\nStandalone cameras have inherent advantages over smartphone cameras, despite the major strides made by iPhone photography over the last few years. Even with amazing third-party apps like FiLMiC Pro, which lets you adjust things like ISO and shutter speed, you're still stuck with a constant aperture and fixed lens.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbpbjm b/data_all_eng_slimpj/shuffled/split2/finalzzzbpbjm
new file mode 100644
index 0000000000000000000000000000000000000000..5202889acee2e5a34c23c4660b62d28793c31049
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbpbjm
@@ -0,0 +1,5 @@
+{"text":"I live in the upper beaches near Gerrard and Main. I am 54 years old and always wanted to sing. Sing in shower, at my desk et al. Am interested in finding out if I can learn to sing (better). Do you teach people who have never really sung before? If so, what are the next steps in pursuing this.\nMike! Great to hear from you! Yes\u2026.of course\u2026..there is plenty for you to learn\u2026and it's fun! Send me your contact info\u2026.and we'll chat and set something up! Sue!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The utilization of mobile app has gone to a peak level nearly exhausting 117% extra time over the mobile gadgets than the previous year. Mobile apps having such an enormous influence in everybody's lives by simplifying the daily routines from wake-up timer to the official data management.\nMobile app brings briskness to your business action by directing more number of people as your target audience.\nAs a business visionary, you need to comprehend the application improvement cost with the goal that you can get your planning right. So you have to see how distinctive parts drive the expenses of building up a portable mobile application.\nThere are various elements that decide your aggregate application improvement cost, yet experienced designers can take proposed highlight records and give their customers a range that as a rule covers the costing amount the customers are searching for.\nThus the prime thing is discovering the fundamental element or features required during the mobile app improvement levels.\nThe mobile app advancement costs don't end with the production of the application. While you may have won the fight making the \"ideal\" application for your business, the continuous war of keeping up it has quite recently started.\nLong-run application support is conveyed by the rule of making the product sensible, easy to understand, working effortlessly and with attractive additional features. As the proprietor of the venture, you ought to work with your group to consider planning for significant updates, innovative progressions, programming and boosting the hardware storage and memory, handling new issues and risks, making enhancements and making bug settles as issues emerge amid the life expectancy of the application.\nYou may ponder internally that making the ideal application doesn't give rise to any maintenance issue. This state of mind is for the most part what brings about disappointment for some business people. All you might do is overspending on your underlying expense and hence, it is more astute to give client criticism a chance to manage any progressions and additionally settles thereof.\nHosting is a standout amongst the most imperative speculations you as a business person can make in your venture. Picking the privilege application facilitating administrations or cloud specialist organization guarantees an astonishing user experience.\nMobile app interaction is the most important thing to be considered for great user experience. This implies you need to incorporate additional features that permit your mobile apps to communicate with other software enhancing the app performance. This is refined through linking the mobile app to the Application Programming Interface, or an API.\nUtilizing pertinent outsider APIs permit you to spare time, cash and holdings at the application advancement process. Rather than driving your group to code applications from the beginning use third party apps which let you concentrate on the business itself and coordinate just the most important components into your MVP.\nThe significant concentration of the whole process is bringing loyalty in developing data security. When your mobile app is not checked up for the security issues then never surge your app to the app market.\nThere are many strides that must be taken to guarantee the protection of client passwords, put away customer data and different viewpoints fixing to secure your clients. And keeping in mind that this may make the business visionary in you recoil, you need to comprehend at this step early: there will be noteworthy costs in securing your versatile application.\nYou'll be wincing progressively on the off chance that you neglect to actualize satisfactory security into your application, as information demonstrates the potential expenses of a noteworthy security rupture, record defilement from assaults on the framework, viral attacks, malware, and other security issues would be a great deal more harming. It's fundamental from the earliest starting point of advancement that you put satisfactorily in application security to a level that your application can effectively confront the dangers required for the total number of clients and the estimation of information saved.\nPrevious articleWhat to do for getting the desired muscle hypertrophy?\nZoeChat is one of the rapidly progressing Whatsapp Clone app which features a unique and advanced Whatsapp Clone Script which enables the users to create an instant messaging app for themselves and get enthralled by the experience provided. This Whatsapp Clone is very much in demand in the market.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking for an international phone card for calling abroad from Norway to Mexico? Comfi sells the best international calling cards that will make your connection with friends, relatives or colleagues advantageous and top-quality.\nWe provide a lot of international phone cards to call from Norway to Mexico.\nAT&T calling card is a secure, comfortable and easy way to contact recipient abroad. The most important trait is the possibility to recharge it. Thus, you can put money to the account without buying a new calling card.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I like to retrieve data from two linked lists. Tried CQWP but the XML was too difficult. The parent list is PIR, the child is SIR. They are linked through the field PIR2SIR (lookup) in the child list.\nWhat code do I need with the SPSiteDateQuery to get a listview like that?\nBrowse other questions tagged list-view caml-query spsitedataquery or ask your own question.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We all face challenges, change, distress and loss in our lives and in our relationships. How we cope with these circumstances can inspire us to grow, to feel an increased sense of happiness, hopefulness, and calmness, thereby living a more complete life.\nBetter Being Blue Bell is a private boutique psychotherapy practice that provides personalized and compassionate care. We believe in a strong relationship between a client and their therapist; therefore our clinicians work in in a nonjudgmental, empowering, and supportive manner. We use evidence based techniques while tailoring treatment strategies specifically to meet your goals. We value collaboration with you and other health care professionals involved in your treatment to develop most effective interventions.\nI am a licensed psychologist with fifteen years of experience helping children, adolescents, adults and families cope with life's challenges and stressors. I offer a warm, caring, empowering, and insightful approach to therapy. Together we will clarify your concerns, identify goals, and develop a plan to help you or your loved one overcome life's challenges.\n\u00a9 Copyright 2019 Better Being \u2013 Blue Bell. All Rights Reserved. Website Design and Development by GetPhound.","meta":{"redpajama_set_name":"RedPajamaC4"}}